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IN MEMORIAM 
 FLOR1AN CAJOR1 
 
PLANE GEOMETRY 
 
 DEVELOPED BY THE 
 
 SYLLABUS METHOD 
 
 BY 
 
 EUGENE RANDOLPH SMITH, A.M. 
 
 /! 
 
 HEAD OF THE DEPARTMENT OF MATHEMATICS, POLYTECHNIC PREPARATORY 
 
 SCHOOL, BROOKLYN, NEW YORK 
 (FORMERLY HEAD OF THE DEPARTMENT OF MATHEMATICS, 
 
 MONTCLAIR HIGH SCHOOL) 
 
 <x>>*i< 
 
 NEW YORK : CINCINNATI : CHICAGO 
 
 AMERICAN BOOK COMPANY 
 
COPYRIGHT, 1909, BY 
 EUGENE RANDOLPH SMITH. 
 
 ENTERED AT STATIONERS' HALL, LONDON. 
 
 SMITH SYL. GEOM. 
 
 W. P. I 
 
PREFACE 
 
 THE belief that the proofs of Geometry should be, as 
 far as possible, worked out by the pupils, either in class 
 discussions or individually, is becoming more widespread 
 every year. The day of memorizing proofs will soon be 
 past, and the most efficient method for mental training 
 along logical lines will be the one generally adopted. 
 This syllabus is written with the hope of encouraging 
 teachers to undertake Geometry by the " no text " method. 
 The author believes very decidedly that this method gives 
 a maximum of mental training with 1 a minimum waste 
 of energy. 
 
 The list of theorems is based on the latest reports of 
 the Mathematical Associations, and, while much shorter 
 than that in many of the text-books, it will be found 
 sufficient to prepare the pupils for any of the colleges. 
 It contains all the theorems of the " New England List " 
 with a few additions that simplify proofs. 
 
 The order is the development of ten years' class use, 
 and will be found different from that of any text. When- 
 ever a theorem has seemed to be simplified, either in 
 content or in proof, by making a change in its place in 
 the order of theorems, that change has been tried in class, 
 and has been made permanent if it proved of advantage. 
 Any teacher using this book should feel equally free to 
 make changes in the order if he is convinced that there 
 is a decided advantage in the change. 
 
 3 
 
4 PREFACE 
 
 The chapter on Logic has been found of great assist- 
 ance in helping the pupils to think accurately, and it is 
 certain to save more time for a class than its discussion 
 requires. 
 
 The definitions and axioms, are given in quite complete 
 form, not for assignment to the class, as this part of the 
 work should be developed before the text is given to a 
 class for study; but as a guide for the teacher, both in 
 order and in subject-matter, and as a reference book for 
 the pupil. Good results can be secured by withholding 
 the book from the pupils until part, at least, of the pre- 
 liminary matter has been discussed. 
 
 The subjects of " existence " and " betweenness " have 
 not been considered to any great extent, as they do not 
 seem worth the time and effort required, except to a 
 student of the more advanced pure mathematics. " Loca- 
 tion " and " intersection," on the other hand, are of such 
 vital importance in considering the correctness of proofs 
 that they have received some attention. The aim through- 
 out has been to arrange a system of Geometry that should 
 be natural, reasonably complete, and suitable to afford as 
 much mental training as the maturity of the pupils would 
 allow. The author has not hesitated to assume any axiom 
 that would help more than its presence would complicate ; 
 on the other hand, he has left out things that seemed to 
 require more than they gave. 
 
 Geometry itself has no concern with measurement by 
 means of a unit. The applications of Geometry to such 
 measurement are, however, very frequent and very im- 
 portant, and while this book presupposes geometrical 
 proofs to as great a degree as seems possible without 
 unnecessarily complicating the subject, there has been 
 no attempt to draw a hard and fast line of demarcation 
 
PREFACE 5 
 
 between Geometry and its applications. If a teacher 
 believes in distinguishing sharply between the different 
 branches of mathematics, the study of the lengths of line 
 sects and the calculation of areas can be put under the 
 head of MENSURATION. 
 
 The exercises are in two divisions, those under the 
 theorems and those in general lists. The exercises under 
 the theorems have been chosen to illustrate the uses of 
 the various theorems, and they should therefore be of 
 great help to the teacher. The general lists give the 
 pupils practice in finding for themselves what princi- 
 ples underlie the proofs. Probably no class could finish 
 all the exercises in the book in one year, but the teacher 
 can easily choose those best suited to his purposes. There 
 are several pages of college examination questions. Some 
 of these are duplicates of exercises scattered through the 
 book, but the differences in wording, as well as the desire 
 to let students know what type of questions examiners 
 ask, has prompted leaving them in the book. 
 
 This book has been written with little reference to the 
 order and methods of other texts, for it is a compilation 
 that has grown naturally from class work. The author 
 is, of course, indebted for many of the ideas used to 
 numerous works on mathematics and its pedagogy, but 
 in many cases it is now impossible to tell from what 
 source the suggestion first arose. He wishes, however, 
 to acknowledge his special indebtedness to Dr. William H. 
 Metzler of Syracuse University, for assistance and encour- 
 agement in the writing of this book. 
 
 EUGENE R. SMITH. 
 
SUGGESTIONS TO TEACHERS 
 
 Preliminary Definitions and Axioms. Do not assign 
 these to the class to read until after they have been thor- 
 oughly discussed in class. As far as possible, let the 
 pupils frame the definitions for themselves, and lead up 
 naturally to the simple deductions from them (starred in 
 the text), so that the pupils can begin to discover these 
 truths from the very beginning. 
 
 Every word of this part need not be digested before 
 going on to the theorems ; but before any new work is 
 undertaken the teacher should make sure that the pupils 
 have a perfect grasp of the particular facts to be used in 
 developing the new matter. Certain parts can be touched 
 upon lightly on the first reading, and cleared up thor- 
 oughly just before being used. Make haste slowly. 
 
 Theorems. Develop the very difficult ones in class by 
 the question and answer method of analysis ; assign those 
 less difficult for outside preparation or to be worked out 
 in class, either at the board or at the pupils' seats. Clear 
 up the work frequently by review recitations covering 
 the theorems recently done. Use a great deal of oral 
 work, asking for every viewpoint on both new and old 
 work. Never tell the pupils what to do; ask questions so 
 framed that the pupils are made to think. Summarize 
 frequently, and by the use of exercises cultivate the pupils' 
 power to choose the correct method. 
 
 6 
 
SUGGESTIONS TO TEACHERS 7 
 
 Notebooks. The author advises that the pupils be re- 
 quired to keep notebooks in which they write out all 
 the theorems, unless the conditions under which the 
 teacher is working make it impossible. All necessary 
 corrections can be noted by the teacher and made by the 
 pupil, after which the theorems can be filed, and so be- 
 come at the end of the course a complete reference book 
 compiled by the pupil. A loose-leaf system will be found 
 convenient. 
 
 Time Required. Do not hurry the early work. What- 
 ever time is used in obtaining a thorough understanding 
 of the foundations and of the methods will be more than 
 repaid by increased speed later on. 
 
 The first quarter of the year's course should be spent 
 on the work up to the section on Locus in Book I, and 
 if this assignment is not quite finished, no great harm will 
 be done. 
 
 The second quarter should almost, if not quite, finish 
 the second book. 
 
 The third quarter should take about to the construc- 
 tions of the fourth book. 
 
 The last quarter should finish the syllabus and leave 
 at least a month for general review and additional exer- 
 cise work. 
 
 This is an estimate of a fair average rather than of an 
 excellent record, and many classes can do much better ; 
 it supposes that a good deal of original work on exer- 
 cises, and almost daily oral work on uses of theorems, 
 methods of attack, and other important topics have been 
 taken. 
 
 Logic. Make frequent use of the section on Logic. Ask 
 the pupils to state the converse and the contraposite of 
 very many of the theorems, and to discover whether these 
 
8 SUGGESTIONS TO TEACHERS 
 
 are new theorems. In this way the proofs of many theo- 
 rems and of many excellent exercises will be found. 
 
 Limits and the Incommensurable Case. In common 
 with many of the best educators, the author believes that 
 the proofs of these theorems are too difficult for imma- 
 ture minds. He therefore recommends that the Appendix 
 proofs be omitted, but that some explanation of the possi- 
 bility of the incommensurable case be given. 
 
 Methods of Attack. It is the experience of the author 
 that the methods of pure logic and the classification and 
 elimination method are the ones of principal importance 
 for proving theorems. The so-called " Indirect Method " 
 and " Reductio ad Absurdum " are unnecessary if logic 
 is understood. " Intersection of Loci," while following 
 directly from the classification method, 'has its uses, and 
 the analysis method for constructions is invaluable. The 
 analyses of the sample theorems will serve as a guide to 
 the class method of discovering the proof for a theorem. 
 
CONTENTS 
 BOOK L RECTILINEAR FIGURES 
 
 PAGES 
 
 I. LOGIC 15-21 
 
 II. POINTS, LINES, AND SURFACES .... 22-27 
 
 III. EQUALITY 28-31 
 
 IV. ANGLES 32-38 
 
 V. POLYGONS 39-41 
 
 VI. INEQUALITIES 42 
 
 VII. PROPOSITIONS . . 43-46 
 
 ORAL AND REVIEW QUESTIONS . . . . 46, 47 
 
 VIII. TRIANGLE THEOREMS . . . . . 48-64 
 
 IX. PARALLELS AND PARALLELOGRAMS . . . 65-74 
 
 X. Loci AND CONCURRENCE 75-79 
 
 SUMMARY OF PROPOSITIONS . . . . 79, 80 
 
 ORAL AND REVIEW QUESTIONS . . . . 80, 81 
 
 GENERAL EXERCISES 81-86 
 
 BOOK II. CIRCLES 
 
 I. CIRCLE DEFINITIONS 87-93 
 
 ORAL AND REVIEW QUESTIONS .... 93 
 
 II. CONSTRUCTIONS 94-100 
 
 III. CIRCLE THEOREMS 101-109 
 
 IV. CONSTRUCTIONS (Usi\G CIRCLE THEOREMS) . . 110-111 
 
 SUMMARY OF PROPOSITIONS 112, 113 
 
 ORAL AND REVIEW QUESTIONS .... 113 
 
 GENERAL EXERCISES 114-118 
 
 BOOK III. EQUIVALENCE AND AREA 
 
 I. DEFINITIONS AND FORMULAS 119-122 
 
 II. THEOREMS 123-132 
 
 III. CONSTRUCTIONS 133-134 
 
 SUMMARY OF PROPOSITIONS 134-136 
 
 ORAL AND REVIEW QUESTIONS .... 136, 137 
 
 GENERAL EXERCISES 137, 138 
 
 9 
 
10 CONTENTS 
 
 BOOK IV. SIMILAR FIGURES: PROPORTIONS 
 
 I. RATIO AND PROPORTION 
 SUMMARY 
 
 
 
 . 139-141 
 . 141, 142 
 
 II. PROPORTIONAL SECTS . 
 
 . 
 
 . 145,144 
 
 III. SIMILAR FIGURES 
 
 . 
 
 . ... . 145-151 
 
 IV. CONSTRUCTIONS . 
 
 . 
 
 . 152-154 
 
 SUMMARY OF PROPOSITIONS, 
 
 . 154,155 
 
 ORAL AND REVIEW QUESTIONS . 
 
 . 155, 156 
 
 GENERAL EXERCISES . 
 
 . 
 
 156-158 
 
 BOOK V. REGULAR POLYGONS AND CIRCLES 
 
 I. THEOREMS . . 159-163 
 
 II. CONSTRUCTIONS 164, 165 
 
 SUMMARY QF PROPOSITIONS . . . . . 165 
 
 ORAL AND REVIEW QUESTIONS ..... 166 
 
 GENERAL EXERCISES 166, 167 
 
 GENERAL 
 
 THE FORMULAS OF GEOMETRY 168-170 
 
 COLLEGE EXAMINATION QUESTIONS 171-182 
 
 APPENDIX 
 
 CONTRAPOSITE LAW 183 
 
 LAW OF CONVERSE 183 
 
 PROOFS OF THE OBVERSE AND CONVERSE OF A SINGLE 
 
 STATEMENT 184 
 
 GENERAL CONDITION . . . . . . . . . 185 
 
 AXIOMS ,. 185 
 
 SYMMETRY 185 
 
 POSITIVE AND NEGATIVE SECTS AND ANGLES .... 186 
 
 DISTANCE 186 
 
 LIMITS 187 
 
 INCOMMENSURABLE CASE 189 
 
 SIMILAR FIGURES 190 
 
 THE EVALUATION OF Pi . . 190 
 
INDEX 
 
 TOPIC REFERENCES AND ABBREVIATIONS 
 
 TERM ABBREV. ART. 
 Acute angle (d) 63, 80 
 
 Addition of polygons 226, 228 
 
 Adjacent (angles) Adj. 46 
 Alternate (angles) Alter. 115 
 Altitude Alt. 81, 245 
 
 Ambiguous Case 130 
 
 Analysis Anal. 92, 94-5, 178 
 
 Angle (Sect. IV) Z 44, 45 
 
 of polygon 47 
 
 Antecedent 258 
 
 Apothern Ap v 313 
 
 Approach (as a limit) = 318 
 Arc 161 
 
 Area 69, 160, 236, 245 
 
 Arm 44 
 
 Axiom Ax. 14, 23, 27, 38-42, 
 
 68,82,119,321,313 
 Axis 344 
 
 76, 79, 135 
 42 
 
 Bisect 
 Broken line 
 
 Center 160, 313, 344 
 
 Center line or center sect 168 
 
 Central angle 188 
 
 Centroid 157 
 
 Chord 161 
 
 Circle O 160 
 
 Circumcenter 152 
 
 Circumference Circum. 160 
 
 Circumscribed Circumsc. 165 
 
 Classification 88 
 
 TERM ABBREV. ART. 
 
 Closed line 27 
 
 Collinear 164 
 
 Commensurable Commen. 203 
 Common chord 172 
 
 Compass 176 
 
 Complements Comp. 61, 193 
 
 Composition 261 
 
 Concave polygon 73 
 
 Concentric circles 198 
 
 Conclusion Concl. 2, 85 
 
 Concurrent 151 
 
 Concyclic ' 218 
 
 Condition Cond. 2, 85 
 
 Conditional statements 2 
 
 Congruent ^ 32-3 
 
 Consequents 258 
 
 Constant 318 
 
 Construction Const. 84 
 
 Contact 173 
 
 Continued proportion 259 
 
 Contraposite Cont. 6, 8, 339 
 
 Converse Conv. 4, 9, 340-1 
 
 Convex polygon 73 
 
 Corollary Cor. 84 
 
 Corresponding angles Cor. A 115 
 Corresponding parts Cor. Pts. 93 
 Cross polygon 73 
 
 Curved line 28 
 
 Decagon 
 
 Definition 
 
 Degree 
 
 75 
 
 Def. 13 
 Deg., 134,206 
 
 11 
 
INDEX 
 
 TERM ABBREV. ART. 
 
 TERM ABBREV. 
 
 ART. 
 
 Describe 179 
 
 Intersection 68, 
 
 167, 170 
 
 Determine (line) 89, 98, 110 
 
 Inversely proportional Inv. 
 
 267 
 
 Diagonal 72 
 
 Isosceles Isos. 
 
 79 
 
 Diameter 161 
 Difference 47 
 Dimensions 16, 17, 18 
 
 Law of contraposite 
 Law of converse 
 
 8 
 10 
 
 Distance Dist. 346 
 
 Legs 79, 80, 135 
 
 Division (prop.), (sect) 261, 279, 306 
 
 Length 
 Limit 
 
 236 
 318 
 
 Elimination 90 
 
 Line 
 
 18 
 
 Equality (Sect. Ill) = 32-38 
 
 Linear 
 
 236 
 
 Equiangular Equiang. 74 
 
 Location 
 
 96, 110 
 
 Equilateral Equilat. 74 
 
 Locus 
 
 146 
 
 Equimultiples 262 
 
 Logic (Sect. I) 1-11, 
 
 339-342 
 
 Equivalence = 34, 225 
 Escribed 166 
 Evaluation of Pi 350 
 Excenter 155 
 
 Major arc 
 Mean and extreme ratio 
 Mean proportion 
 
 Means 
 
 193 
 306 
 259 
 258 
 
 Explements Exp. 56, 193 
 
 Measurement 203, 
 
 206, 236 
 
 Exterior angles Ext. A 71 
 
 Median 
 
 81 
 
 Ext. Interior angles Ext. Int. 115 
 
 Mensuration 
 
 Preface 
 
 Extremes 258 
 
 Methods 
 
 90, 147 
 
 Figure Fig. 29 
 
 Minor arc 
 
 193 
 
 Formulas 23Q, 245, 246, 338 
 
 Minute ' 
 
 134, 206 
 
 Fourth proportional 258 
 
 Negative converse 
 
 6 
 
 General and special cases 11 
 
 Negative sects and angles 
 
 345 
 
 Geometry 12 
 
 Negative statement 
 
 5 
 
 TTrtl-C >i n 
 
 Numerical 
 
 242 
 
 Halt 42 
 
 
 
 Harmonic division 279 
 
 Oblique 
 
 58 
 
 Hexagon 75 
 
 Obtuse angle (d) 
 
 63,80 
 
 Homologous 93 
 
 Obverse Obv. 
 
 5,341 
 
 Hypotenuse Hyp. 80 
 
 Octagon 
 
 75 
 
 Hypothesis 2, 85 
 
 Order of proof 
 
 87 
 
 Incenter 154 
 
 Orthocenter 
 
 156 
 
 Incommensurable Incom. 203, 348 
 
 Parallel II 
 
 118 
 
 Inequality (Sect. VI) Ineq. 83 
 
 Parallelogram O 
 
 135 
 
 Inscribed Insc. 165 
 
 Pencil of lines 
 
 275 
 
 Inscribed angle 188 
 
 Pentagon 
 
 75 
 
 Interior angles Int. A 
 
 Perigon 
 
 54 
 
 of polygon, 71 ; Us, 115 
 
 Perimeter Per. 
 
 69 
 
INDEX 
 
 13 
 
 TERM ABBREV. ART. 
 
 TERM ABBREV. ART. 
 
 Perpendicular JL 58 
 
 Similar ~ 282, 289, 349 
 
 Pi TT 324 
 
 Solid 16 
 
 Plane 30 
 
 Space 15 
 
 Plane geometry 31 
 
 Special cases 11 
 
 Point Pt. 19 
 
 Square Q 135, 229 
 
 Point cutting sect 279 
 
 Stand on 188 
 
 Point of tangency 167, 173 
 
 Straight angle St. Z 49 
 
 Polygon (Sect. V) 69 
 
 Straight edge 176 
 
 Positive sects and angles 345 
 
 Straight line 22 
 
 Postulate Post. 176 
 
 Subtend 188 
 
 Problem Prob. 84 
 
 Subtraction 227-8 
 
 Projection Proj. 243 
 
 Sum + 47, 227-8 
 
 Proof 86-7, 260 
 
 Summary 89, 158, 223, 256, 274, 
 
 Proportion (al) Prop. 202, 258 
 
 310, 336 
 
 Proposition (Sect. VII) 84 
 
 Superimpose 41 
 
 Pythagorean proposition 240 
 
 Supplements Sup. 51, 193 
 
 Quadrant of arc 192 
 Quadrilateral Quad. 75 
 
 Surface 17 
 Symmetry 344 
 
 Radius Rad.; r 161,313 
 
 Tangent Tang. 167,173 
 
 Ratio 202 
 
 Terms 202 
 
 Rectangle CH 135,229 
 
 Theorem Th. 84 
 
 Rectilinear 29 
 
 Third 42 
 
 Reflex 63 
 
 Third proportional 259 
 
 Regular Reg. 74 
 
 Touch 173 
 
 Representation 21 
 
 Transversal '115 
 
 Rhomboid 135 
 
 Transverse set Trans set 115 
 
 Rhombus 135 
 
 Trapezium 135 
 
 Right angle (d) Rt. Z 58,80 
 
 Trapezoid Trap. 135 
 
 Scalene 97 
 
 Triangle A 75, 77-81 
 
 
 Trisect 42 
 
 Secant 167 
 
 
 Second " 134, 206 
 
 Unit 236 
 
 Sect 25 
 
 
 Sector 187 
 
 Variable Var. 318 
 
 Segment 25, 187 
 
 Vertex angle Vert. Z 78 
 
 Semicircle Semi O 193 
 
 Vertical angle of triangle 78 
 
 Semicircumference Semicir. 193 
 
 Vertical angles Vert. A 48 
 
 Side S. 70 
 
 Vertices of polygons 70 
 
SUMMARY OF GEOMETRICAL SIGNS 
 
 > 
 
 plus, sign of addition 
 minus, sign of subtraction 
 times, sign of multiplication 
 , /, : , divided'by, sign of division 
 square root sign 
 is (or are) equal, or equivalent 
 
 is not equal, or equivalent, to 
 is identical to 
 is congruent to 
 approaches as a limit 
 is similar to 
 is greater than 
 
 is not reater than 
 
 < is less than 
 
 jC is not less than 
 
 II is parallel to 
 
 _L is perpendicular to 
 
 Z or 2jL angle 
 
 A triangle 
 
 O parallelogram 
 
 n rectangle 
 
 Q square 
 
 O circle 
 
 ^ arc 
 
 .. therefore 
 
 . because, since 
 
 The signs for figures become plural by the addition of s, often within 
 the sign, as U) for rectangles. 
 
 OTHER ABBREVIATIONS 
 
 eq. -f eq., for "equals plus equals," and similarly for the other axioms. 
 
 2 s. incl. Z, for "two sides and the included angle," etc. 
 
 In referring to propositions or axioms as authority, it is permissible to 
 use any easily understood abbreviation for the theorem. The ones given 
 above are good examples of abbreviations that, if used correctly in a 
 theorem, could hardly be misunderstood. 
 
PLANE GEOMETRY 
 
 BOOK I. RECTILINEAR FIGURES 
 
 SECTION I. LOGIC 
 
 1. Need of Logic. In all discussions, more or less logic 
 is used, though often unknowingly. For example, if a 
 person, noticing that a flag is hanging limp on a pole, says, 
 " There is no wind to-day," that person has reasoned as 
 follows : 
 
 " If the wind were blowing, the flag would be waving. 
 But the flag is not waving, so the wind is not blowing." 
 
 Because such reasoning is so instinctive that we seldom 
 realize the steps through which the mind arrives at the 
 conclusion, the need of a knowledge of the most important 
 laws of logic is not fully appreciated. 
 
 In Geometry, the same kind of reasoning must be 
 applied, and as instinct with regard to the figures to be 
 studied is not reliable, a brief study of the most commonly 
 applied laws of logic is necessary. 
 
 2. Conditional Statements. Practically all statements are 
 conditional, though the condition is sometimes implied 
 rather than stated. Even the statement, "It rains," de- 
 pends upon the place and time. 
 
 When a statement asserts that the truth of one thing- 
 assures the truth of another thing, the one upon which the 
 
 15 
 
16 LOGIC 
 
 other depends is the condition (or hypothesis), the other 
 the conclusion of the statement. 
 
 EXAMPLES: 
 
 CONDITION CONCLUSION 
 
 If it rains to-day, I will stay at home. 
 
 If a triangle has two equal sides, the opposite angles are 
 
 equal. 
 
 A horse has four legs. 
 
 If John goes down town, I will go with him. 
 
 NOTE. In the second example, the word "triangle" simply in- 
 dicates the figure in which the reasoning is to be, and for the purposes 
 of Geometry it does not need to be considered as part of either the 
 condition or the conclusion. The terms used in this illustration have 
 not been defined as yet, but they are common enough to be familiar 
 to most students. 
 
 3. Four Related Statements. There are four related 
 conditional statements which can be formed from two pos- 
 sible truths used in the positive and the negative. Let the 
 two possible truths be " it rains," and " the sidewalk is 
 wet"; then the following related statements might be made 
 (with no consideration at present as to whether they are 
 true statements or not) : 
 
 (1) If it rains, the sidewalk will be wet. 
 
 (2) If the sidewalk is wet, it has rained. 
 
 (8) If it does not rain, the sidewalk will not be wet. 
 (4) If the sidewalk is not wet, it has not rained. 
 
 If the two possible facts were represented by A and B, 
 where A and B stand for any two possibilities such that 
 one might depend upon the other, these statements could 
 be written as follows : 
 
 (1) If A, then B. (3) If not A, then not B. 
 
 (2) If B, then A. (4) If not J5, then not A. 
 
LOGIC 17 
 
 4. Converse. The first and second statements have the 
 condition of each the same as the conclusion of the other ; 
 or, the condition and conclusion of either interchanged will 
 the other. They are called converse to each other. 
 
 5. Negative or Obverse. The first and third have the 
 condition and conclusion of either the negative of the 
 condition and the conclusion of the other ; or, the condition 
 and conclusion of either made negative will' form the other. 
 They are called negative or obverse to each other. 
 
 6. Negative Converse or Contraposite. The first and 
 fourth have the condition of each the negative of the 
 conclusion of the other ; or, the condition and the conclusion 
 of either interchanged and made negative will form the other. 
 They are called negative converse or contraposite to each 
 other. The reason for the name " negative converse " is 
 evident from the definitions of these three relations in 
 4, 5, and 6. 
 
 7. Exercises. 
 
 1. Write three conditional statements, separate each into condition 
 and conclusion, and write the converse, negative, and negative con- 
 verse of each. 
 
 2. What relation have statements 2 and 3 ? 2 and 4? 3 and 4? 
 
 3. What relation to the original statement has 
 (a) the converse of its negative? 
 
 (Z>) the converse of its negative converse? 
 
 (c) the negative of its negative converse? 
 
 (d) the negative of the converse of its negative converse? 
 
 8. Contraposite (or Negative Converse) Law. The con- 
 traposite of any true conditional statement is also true. 
 
 NOTE. This law is discussed more fully in the Appendix, 339. 
 SMITH'S SYL. PL. GEOM. 2 
 
18 LOGIC 
 
 It is true also that the negative converse of any false 
 statement is false, although this is of less importance than 
 the former statement. 
 
 The most common way of using this method of reason- 
 ing is to make some true statement that a certain con- 
 clusion follows a certain condition ; then, on finding that 
 the conclusion is no't true, to say that the condition is not 
 true. The statement about the flag in 1 was an example 
 of this. Anoth'er example of a very self-evident kind is: 
 One is looking for a book with a green cover, and finding 
 a book the cover of which is not green, knows at once 
 that the book is not the book wanted. Definitely put, 
 the argument would be : 
 
 The book wanted has a green cover. 
 
 This book has not a green cover, so it is not the right 
 book. 
 
 Contraposite reasoning is probably the most common 
 kind of reasoning, for almost every one uses it many times 
 daily. It is, however, used more or less unconsciously 
 in daily life, while in Mathematics and Science, and in 
 many other places where the reasoning needs to be per- 
 fectly accurate, one must know definitely just what steps 
 are being taken, in order that no error can creep in. 
 
 EXERCISES. Suppose that if A is true, B is also true ; what other 
 statement regarding A and B is known to follow? Write five true 
 statements, then see if their coutraposites are also true. See if their 
 converses are true. 
 
 9. Truth of the Negative and the Converse of a True 
 Statement. The negative and the converse of a single true 
 statement are not necessarily true. They are either both 
 true, or both false, for they are negative converse to each 
 other. The discussion of when they are true will be 
 
LOGIC 19 
 
 taken up in 10. The following examples will show that 
 they are not always true : 
 
 u An apple tree has leaves." It does not follow that 
 
 (1) If it is not an apple tree, it has not leaves. 
 
 (2) If it has leaves, it is an apple tree. 
 
 "This desk is made of wood." It does not follow that 
 
 (1) If it is not this desk, it- is not made of wood. 
 
 (2) If it is made of wood, it is this desk. 
 
 " A donkey has a head." It does not follow that be- 
 cause you have a head you are a donkey. 
 
 WARNING. Never assume that the negative or the converse of ONE 
 true statement is also true. 
 
 10. Law of Converse. If conditional statements such 
 that their conditions cover all possibilities, and no two 
 conclusions can be true at once, are true, then the con- 
 verses of those statements are also true. 
 
 NOTE. This law is discussed more fully in the Appendix, 340. 
 
 This law will be understood when it is applied to defi- 
 nite cases, but the following examples make its meaning 
 somewhat clearer. 
 
 (1) If A is true, X is true. 
 
 If A is not true, X is not true. 
 
 These two conditions (true and untrue) cover all pos- 
 sibilities, and the two conclusions (true and untrue) are 
 such that they cannot both be true at the same time, so the 
 converses are also true ; i.e. 
 
 If X is true, A is true. 
 
 If X is not true, A is not true. 
 
 (2) HA>BI X>Y. 
 If A = B, x = Y. 
 
 If A<B,X< Y. 
 
 These three conditions (>,=,<) cover all possibilities, 
 
20 LOGIC 
 
 and but one of the conclusions (>,=,<) can be true at 
 once, so the converses are also true ; i.e. 
 If X> F, A >B. 
 
 If X < F, A > B. 
 
 11. General and Special Cases. 
 
 NOTE. This need not be read until the pupil is ready to begin 
 the theorems. 
 
 Anything known of mankind is known of each man 
 separately; but anything known of certain men only, would 
 not necessarily be true of all men. If it is true for each 
 man in existence, it is true for all mankind. In other 
 words, anything known of a class as a whole, or of all 
 members of the class, is known to be a characteristic of 
 that class, both as a whole and by individuals. On the 
 other hand, anything known of part of a class is not 
 known of the class as a whole, or of other members of 
 that class. 
 
 So in Geometry, proofs should be made for the general 
 case whenever that is possible, and when that does not 
 seem possible, the proof should be worked for each of the 
 different cases separately. For example, in working with 
 triangles, the triangle used should always be one about 
 which no assumption (other than the given of the theo- 
 rem) is made. The triangle should neither be isosceles, 
 nor be assumed to have any certain sized angle, such as a 
 right or acute angle. It is better to draw the triangle in 
 the figure so that it does not even appear to have any 
 special characteristic, or the one studying the figure may 
 carelessly assume that the characteristic which the figure 
 appears to have really belongs to it. 
 ' In theorems where it does not seem possible to find a 
 
LOGIC 21 
 
 proof for the general case (and these are comparatively 
 rare), it is necessary to prove enough cases to cover all 
 possibilities in order that the general case may be known; 
 for example, a proof for right, acute, and obtuse-angled 
 triangles would be true for all triangles. In the same 
 way, a proof for triangles having three equal sides, two 
 equal sides, and no equal sides, would be true for all 
 triangles. 
 
 Sometimes a proof is true only for a special case on ac- 
 count of points or line's that are added to the figure and 
 are assumed to lie in certain positions, when as a matter 
 of fact they can equally well lie in other positions. If 
 points or lines are added to a figure, the proof must hold 
 for all possible positions in which they can lie. This is 
 discussed in more detail in 110. 
 
SECTION II. POINTS, LINES, AND SURFACES 
 
 12. Geometry. This subject studies points, lines, and 
 figures formed by them. It proves facts about the figures, 
 
 *and uses as a basis for the reasoning definitions and axioms. 
 
 13. Definitions. A definition is such a description of the 
 thing defined as will distinguish it from all other things; 
 it might be said to be an agreement as to what a term 
 shall be used to indicate. Some things are of such simple 
 nature that it is difficult, if not impossible, to define them 
 in terms still simpler, and in such cases an explanation in 
 regard to them may well take the place of a definition. 
 
 14. Axioms. A truth that is taken as one of the 
 foundation facts of a subject is called an axiom. It is 
 often defined as a truth so simple that it cannot be de- 
 rived from truths still simpler ; but for Elementary 
 Geometry this is not strictly true (Appendix, 343). It 
 will be found that the axioms of Geometry are facts so 
 self-evident that there is no doubt as to their truth. 
 
 15. Space. The space in which everything exists is, as 
 far as experience shows, unlimited. At any rate, the 
 space studied in Elementary Geometry (sometimes called 
 Euclidean Space) is unlimited. Space is evidently divis- 
 ible, for all bodies occupy portions of space. 
 
 16. Solids. Any limited portion of space, such as the 
 space occupied by any body, is called irrespective of 
 the nature of the body which may occupy it a geometric 
 
 22 
 
POINTS, LINES, AND SURFACES 23 
 
 solid, or simply a solid. Solids are said to have three 
 dimensions: length, breadth, and thickness. 
 
 17. Surfaces. That which separates one portion of space 
 from an adjoining portion is called a surface. If two 
 adjoining lots are considered as extending down into the 
 ground so that any distance down there is still a boundary 
 between the lots, that boundary is a surface. It evidently 
 does not occupy space, for any particle of soil belongs to 
 one lot or to the other, yet there is a distinct boundary 
 such that all on one side of it belongs to one lot, while all 
 on the other side of it belongs to the other lot. 
 
 NOTE. The terms "side," "between," "within," "outside," will 
 be used in this syllabus in their ordinary meaning without any 
 attempt to define them geometrically. 
 
 Another example of a surface is the outside of any object, 
 as a box. It separates the space occupied by the object 
 from the space outside the object, but itself occupies no 
 space. 
 
 A surface may be limited or unlimited in extent, and 
 may have limited portions ; it is said to be two dimen- 
 sional, having length and breadth. 
 
 18. Lines. That which separates one portion of a sur- 
 face from an adjoining portion is called a line. The 
 surface boundary between two lots is a line. The line 
 occupies no space, yet definitely divides one lot from the 
 other. 
 
 A line may be indefinite in extent, but has limited 
 portions ; it is said to be one dimensional, having length 
 only. 
 
 19. Points. That which separates one portion of a 
 line from an adjoining portion is called a point. If four 
 lots come together in what is ordinarily called a corner, 
 
24 RECTILINEAR FIGURES 
 
 that corner is a point. It is a place, for it is fixed in 
 position, but it occupies no space, and no portion of the 
 surface. There is evidently no unowned spot at the 
 corner, for all four lots extend to the point, leaving no 
 unoccupied surface. 
 
 A point has no dimensions and is indivisible; *it has 
 position only. 
 
 20. These definitions might have been taken in reverse 
 order; that is, starting with the point. A moving 
 point passes through (or describes) a line ; a moving line 
 usually describes a surface; a moving surface usually 
 describes a solid. It can be seen that it might be possible 
 for a line to move along itself in such a way as not to 
 describe a surface, and for a surface to fail to describe a solid. 
 
 21. Representations. Points, lines, and surfaces are 
 represented by various things. The corner of a lot may 
 be marked by a fence post, and the line by a fence, but it 
 is evident that the post and the fence are not the point 
 and the line. So in Geometry, a pencil or chalk mark 
 may be used to represent a point or a line, but it is under- 
 stood that they are used only to represent the things, and 
 that they are not the points and lines themselves. 
 
 22. Straight Lines. The straight line is the most im- 
 portant and the most familiar kind of line. It is not 
 easy to define, but some discussion of it is necessary. 
 Different straight lines, or parts of the same straight line, 
 need have but two points in common to coincide through- 
 out, irrespective of how the lines are placed in other 
 respects, and this characteristic of straight lines is the 
 foundation of the definition. 
 
 The two following are probably the best definitions of 
 the straight line: 
 
POINTS, LINES, AND SURFACES 25 
 
 (0) A line such that any part will, however placed, lie 
 wholly on any other part, if its extremities are made to 
 fall upon that other part, is called a straight line. 
 
 (5) A line that is determined by any two of its points 
 is a straight line. 
 
 The word " determined " here means that the line is 
 distinguished from any other line of the same kind by the 
 fact that it goes through these particular two points ; or, in 
 other words, that it is the only straight line through the 
 two points. 
 
 23. However the definition is worded, the fundamental 
 fact about straight lines is the following : 
 
 Straight-line Axiom. Through two points but one 
 straight line can pass. 
 
 An illustration to show that this distinguishes the 
 straight line from all others can be made by folding a 
 sheet of paper smoothly and drawing any line through 
 two points on the edge formed, inside the fold. If the line 
 is drawn in ink, and the sheet is folded firmly together, a 
 second line exactly like the first, but in the opposite posi- 
 tion, will appear from the trace of the ink, unless the line 
 is drawn along the edge, in which case no second line will 
 appear. The edge represents the straight line through 
 the two points, and is the only one. 
 
 24. A second fact about straight lines which is based 
 directly on the straight-line axiom is 
 
 * Two different straight lines can intersect in but one 
 point. 
 
 Whenever a statement is marked with the mark *, that statement 
 requires proof, the proof in the first part of the Geometry being always 
 closely associated with the definitions and axioms. These proofs are 
 sometimes given in the form of explanations, but the pupil should in all 
 such cases be able to explain why the statement is true, 
 
26 RECTILINEAR FIGURES 
 
 This is the first example of contraposite reasoning in 
 the Geometry. It can be done as follows: 
 
 If the two lines met in a second point, there would be 
 two different straight lines through two points. 
 
 There cannot be two different straight lines through 
 two points, by the straight-line axiom; therefore, by con- 
 traposite argument (i.e. the conclusion being untrue, the 
 condition is also untrue), the two straight lines do not 
 meet in two points. 
 
 In places where no ambiguity results, "line" will be 
 used to mean straight line, since when any other kind of 
 line is meant, the kind is always stated. 
 
 25. Sects. A limited portion of a line is called a sect, 
 or a line segment. That part of a line that lies between 
 the points A and B is a sect, A R 
 
 and is called AB. Where there ' ' 
 
 is nothing to indicate otherwise, sect will mean straight- 
 line sect. 
 
 R * 
 
 26. Broken Lines. A line ^^\/^^ ^-^^ 
 composed of sects of differ- A c ^ 
 ent lines is called a broken 
 
 line^ as ABODE. A broken line is said to be closed if it 
 
 is continuous; that is, if the line, when traced from 
 
 any point through its entire 
 
 length, is found to return 
 
 to the starting point. 
 
 ABODE is not a closed line, 
 
 but BSTUV is closed. 
 
 The straight-line axiom 
 shows that a straight line is not continuous, and that it 
 does not intersect itself, 
 
POINTS, LINKS, AND SURFACES 27 
 
 27. Closed-line Intersection Axiom. // a straight line 
 of indefinite length passes through a point within the 
 surface inclosed by any closed line, it intersects the closed 
 line at least twice. 
 
 28. Curved Lines. A line, 
 no part of which is straight, 
 is called a curved line. 
 
 A curved line also is called 
 closed if it is continuous. (See 26.) 
 
 29. Geometrical Figures. Any combination of points, 
 lines, and surfaces, formed under given conditions, is a 
 geometrical figure ; as, a triangle might be defined as the 
 figure formed by three lines meeting in pairs. If the 
 figure is formed by straight lines, it is called a rectilinear 
 figure. 
 
 30. Planes. A surface in which, any two points being 
 taken, the straight line which joins them lies wholly in 
 the surface is called a plane surface, or simply a plane. 
 
 A plane might be defined as the surface determined by 
 any three of its points that are not in the same straight 
 line. 
 
 The plane occupies the same position among surfaces 
 that the straight line holds among lines. As indicated 
 .by the first definition given, it is straight through any two 
 of its points. The surface of a blackboard is a familiar 
 example of a plane. 
 
 31. Plane Geometry. Plane Geometry treats only of 
 geometric figures that lie entirely in the same plane. 
 
SECTION III. EQUALITY 
 
 32. There are three words used in Geometry to denote 
 equality : congruent, equivalent, and equal. 
 
 33. Congruence. Two figures are congruent when they 
 can be made to coincide in every point. 
 
 34. Equivalence. Closed figures (or figures bounded 
 by closed lines, see 26, 28) are said to be equivalent 
 when their boundaries inclose the same amount of surface. 
 
 35. Equality. The word equal is used somewhat in 
 both senses, but in this syllabus it will be used only in 
 those places where there can be no confusion between the 
 ideas of congruence and equivalence. For example, sects 
 will be said to be equal when they can be made to coin- 
 cide, even though this fulfills the definition of congruence, 
 for since a sect cannot inclose surface, there can be no 
 confusion with equivalence. 
 
 36. Congruence includes equivalence, whereas equiva- 
 lence does not imply congruence ; a figure inclosed by a' 
 curved line might be equivalent to a figure inclosed by a 
 broken line although it would be impossible to make them 
 coincide. 
 
 37. Geometric Equality. In the strictest geometric 
 sense, equality means that coincidence is possible, and in 
 this the test for geometric equality differs from the test 
 for arithmetic equality, for two arithmetic magnitudes are 
 equal if they contain the same unit the same number of 
 
 28 
 
EQUALITY 29 
 
 times. Evidently, then, equivalence is to some extent an 
 arithmetic property, but Geometry is applied so often 
 to calculations of magnitudes in terms of a unit, that it is 
 neither necessary nor desirable to attempt to distinguish 
 too carefully between it and other mathematical subjects. 
 In practical work, Geometry will be found to have many 
 parts that involve Arithmetic and Algebra, and while the 
 distinctions between the subjects may be kept in mind, 
 their combined use is entirely legitimate. 
 
 38. Equality Axioms. 
 
 . (1) Things equal to the same thing, or to equal things, 
 are equal to each other. 
 
 This axiom applies to both congruence and to equiva- 
 lence ; the first four following apply only to sects, angles 
 ( 44), and equivalent closed figures ; they cannot be ap- 
 plied to the congruence of closed figures. 
 
 (2) If equals are added to the same thing or to equal 
 things, the results are equal. 
 
 (3) If equals are taken from the same thing or from 
 equal things, the results are equal. 
 
 (4) If equals are multiplied by the same thing or by 
 equal things, the results are equal- 
 
 (5) If equals are divided by the same thing or by equal 
 things, the results are equal. 
 
 (6) The whole equals the sum of all its parts. 
 
 39. Substitution Axiom. A magnitude may be put 
 in place of an equal magnitude in any equation or state- 
 ment of inequality- 
 
 This axiom is not independent of the equality axioms, 
 but it is more convenient for many purposes. For example, 
 if ak~b=r, and k = l; then, substituting I for &, al b = r. 
 
30 RECTILINEAR FIGURES 
 
 40. General and Geometric Axioms. Certain of the 
 axioms apply not only to geometric magnitudes, but to all 
 magnitudes, and are therefore called general axioms. The 
 six equality axioms, the substitution axiom, and the axiom 
 of inequality ( 82), are general axioms. The straight-line 
 axiom, and all others that refer to geometric conceptions 
 only, are geometric axioms. 
 
 41. Axiom of Motion. Geometric figures can be moved 
 about in space without altering them in any way. 
 (Sometimes stated, " without altering their size or 
 shape-") This axiom is used in testing congruence, for 
 one figure is sometimes supposed to have been placed 
 upon (or superimposed upon) another figure, and what- 
 ever is known about the two figures is then used to deter- 
 mine whether or not they- coincide. 
 
 42. Axiom of Division. Any magnitude can be divided 
 into any number of equal parts. (The number must be 
 a positive integer.) 
 
 If the magnitude is divided into two equal parts, it 
 is said to be bisected; if into three equal parts, to be 
 trisected, and the parts are called halves, and thirds, 
 respectively. 
 
 * 43. A sect can be bisected by but one point. 
 
 If P and Q were both points of bisection (or midpoints) 
 of AB, then AP and AQ would be equal, since each is 
 AB-t-2 (eq. -*- eq.). Therefore AP and AQ coincide, and P 
 falls on Q, making but one bisection point. 
 
 It might be thought at first that this fact was self-evi- 
 
EQUALITY 31 
 
 dent, but many things can be bisected in different ways, 
 and it is important to know which ones have but one pos- 
 sible bisection. An apple, for instance, could be cut into 
 halves in many different places. 
 
 The method of proof used to show that a sect has but 
 one midpoint can also be used to show that a sect has 
 but one point which cuts off one third of the sect from 
 a certain end, and but one point which cuts off two 
 thirds from that end. The method can also be extended 
 to cover all cases of the division of a sect into equal parts. 
 
 NOTE. A method of proof can often be used for various purposes 
 with very little change except for minor details. It has already been 
 said that the proof in 43 can be used for other divisions besides 
 bisection. It is also true that it can be used for the bisection of an 
 angle as well as of a sect, and for any other divisions of an angle. 
 (See 64.) It is very important that a student should become famil- 
 iar with the method of work for a proof, rather than to attempt to 
 memorize its relatively unimportant details. If the method is thor- 
 oughly understood, the details of the proof are not likely to prove 
 troublesome. 
 
SECTION IV. ANGLES 
 
 44. Angles. If two lines meet in a point, they are said 
 to form an angle. The common point is called the 
 vertex, and the lines the 
 arms of the angle. The size 
 of the angle is measured 
 by the amount of rotation 
 through which a line would 
 pass in going from the posi- 
 tion of one arm to the position of the other arm. 
 
 In the accompanying figure, the lines AO and BO form 
 an angle, the size of the angle being measured by the 
 amount of rotation necessary to pass from the position 
 AO to the position BO. 
 
 Since there are two ways in which to rotate from one 
 line to the other, two angles are formed by two lines diver- 
 ging from a point; when it is necessary to distinguish 
 between the two, the letters on the arms are named in 
 the order of direction contrary to that taken by the 
 hands of a clock, the vertex letter being between the 
 other two. The angle marked by the arrow (which 
 shows the correct direction) would be named AOB, while 
 the other angle would be BOA. Ordinarily the smaller 
 of the two angles is meant, such an angle as BOA seldom 
 being considered in Geometry. 
 
 If there is little chance of confusion as to which angle 
 at a certain vertex is being used, the vertex letter alone 
 is often used to name the angle ; as, angle 0. 
 
 32 
 
ANGLES 33 
 
 It should be noticed that the size of an angle in no 
 way depends on the length of its "arms. One angle would 
 be greater than, equal to, or less than, another, according 
 as it would include, coincide with, or fall within, that 
 other, if it were placed upon it with the vertex and one 
 arm coinciding. An angle both of whose arms lie between 
 the arms of a second angle is evidently smaller. 
 
 45. Sign for "Angle." The sign commonly used for 
 "angle" is Z ; as Z AOB. If there is a chance for con- 
 fusion with the signs for "greater" and " less " (> and <), 
 aline is sometimes drawn across it, as ^. The commonest 
 signs and abbreviations will be found following the index 
 of terms. 
 
 46. Adjacent Angles. If two angles have a common 
 vertex and lie on opposite sides of a common arm, they 
 are called adjacent 
 
 angles. 
 
 /.AOB and Z BOC 
 are adjacent angles. 
 They are adjacent 
 whether or not the arms AO and OC lie in the same 
 straight line. 
 
 47. Sum and Difference of Angles. The sum of two 
 
 angles is the angle obtained by placing the angles adjacent 
 to each other, and ignoring the common arm. The sum 
 of Z AOB and Z BOC is Z AOC (in 46). 
 
 The difference of two angles is the angle obtained by 
 placing the angles so that they have a common vertex, 
 and lie on the same side of the common arm. The dif- 
 ference between Z AOC and Z BOC is Z AOB (in 46). 
 
 SMITH'S SYL. PL. GEOM. 3 
 
34 
 
 RECTILINEAR FIGURES 
 
 48. Vertical Angles. If two lines intersect, forming four 
 angles, any two angles that are not adjacent to each other 
 are called opposite, 
 
 or vertical angles. 
 
 Z A OB and Z COD 
 are vertical, as are 
 Z BOG and Z DOA. 
 
 The arms of one of. two vertical angles are the arms of 
 the other angle extended through the vertex. 
 
 49. Straight Angles. If the arms of an angle lie in the 
 same straight line, but on opposite sides of the vertex, 
 the angle is called a 
 
 straight angle. 
 
 Z A OB is a straight 
 
 angle. The sum of 
 
 all the successive angles around a point, on one side of 
 a straight line, is equal to a straight angle; for example, 
 Z AOX+ Z XOB = st. Z. 
 
 *50. All straight angles are equal. 
 
 For the arms of one straight angle can be made to 
 coincide with the arms of any other straight angle, since 
 they form, in each case, a straight line, and so need have 
 but two points in common to coincide throughout. 
 
 51. Supplements. If the sum of two angles is a straight 
 angle, the angles are called supplements of each other. 
 In 49 Z.AOX and /.XOB are supplements. 
 
 *52. Supplements of the same angle, or of equal angles, 
 are equal to each other. 
 
ANGLES 35 
 
 For if equal parts are taken from straight angles (which 
 are equal), equal parts must be left, by equality axiom 3. 
 In the figure : st. /.AOC st. ZXPZ. 
 
 If Z.AOB = Z XPY, 
 then Z.BOC = Z YPZ. (eq.-eq.). 
 
 NOTE. When one wishes to refer to some authority, as in this 
 case to the equality axiom, any clear abbreviation may be used. The 
 reference (eq. eq.) means " equals minus equals," and refers to 
 equality axiom 3. It is not best to refer to authorities by number; 
 all authorities should be quoted in such a way that one could readily 
 understand the meaning of the reference, and words or simple ab- 
 breviations are usually used. 
 
 *53. Any two vertical angles are equal. 
 
 For they are supplements of the same angle. 
 
 In the figure of 48 Z AOB and Z COD are both sup- 
 plements of Z BOG (or of ZDO^4); similarly, /.BOG and 
 Z DO A are both supplements of Z AOB (or of Z COD). 
 
 54. Perigons. If the arms of an angle lie in the same 
 straight line and on the same side of the vertex, one arm 
 having rotated through two straight angles, the angle is 
 called a perigon. 
 
 Z AOB is a perigon, since OB is supposed to have rotated 
 around O (as indicated by the arrow) to the position OB 
 on OA. 
 
 A perigon is the sum of the successive angles around 
 a point; and it equals 
 two straight angles. It 
 is in the sense of a sum 
 rather than as a single 
 angle that perigon is 
 most frequently used in Geometry. Z AOX + Z XOB a 
 perigon. 
 
36 RECTILINEAR FIGURES 
 
 *55. All perilous are equal. 
 
 For all straight angles are equal, and since a perigon is 
 twice a straight angle, perigons are equal (eq. x eq.). 
 
 56. Explements. If the sum of two angles is a perigon, 
 the angles are called explements of each other. In the 
 figure of 54 /_AOX and Z.XOB are explements of each 
 other. 
 
 *57. Explements of the same angle or of equal angles 
 are equal. 
 
 Notice the similarity 
 to 52. 
 
 58. Right Angles. If 
 
 one line meets another 
 
 line so as to make the 
 
 two adjacent angles 
 
 equal, the angles are 
 
 called right angles, and 
 
 the lines are said to be perpendicular to each other. 
 
 If /.AOB = /-BOC, then these angles are right angles. 
 
 A line not perpendicular to a second line is said to be 
 
 an oblique to that line. 
 
 *59. A right angle is one half a straight angle. 
 For by the definition of right angle, a straight angle is 
 the sum of two equal right angles. 
 
 *60. All right angles are equal. 
 
 For they are halves of equal straight angles. What 
 axiom is used? 
 
 61. Complements. If the sum of two angles is a right 
 angle, the angles are called complements of each other. In 
 58 Z AOX and /.XOB are complements of each other. 
 
ANGLES 
 
 37 
 
 *62. Complements of the same angle or of equal angles 
 are equal. 
 
 See 52 and 57. 
 
 63. Acute, Obtuse, Reflex Angles. An angle less than 
 a right angle is called acute; one greater than a right 
 angle, but less than a straight angle, is called obtuse ; one 
 more than a straight angle, but less than a perigon, is 
 called reflex. 
 
 *64. An angle can be bisected by but one line. 
 If OX and OF both 
 bisect Z AOB, then 
 
 Z AOB -^ 2, and eq. -*- 
 eq.). Therefore the 
 angles coincide, and OX 
 falls on OF; that is, 
 there is but one line 
 which bisects the angle. 
 
 *65. At a given point in a given line there can be but 
 one perpendicular in the same plane. 
 
 For the perpendicular bisects the straight angle having 
 its vertex at its foot ( 59), and there can be but one 
 bisector of an angle (64). 
 
 *66. The bisectors of two vertical angles lie in one 
 straight line. 
 
 ZA'OF is one half the perigon, or a straight angle, for 
 the Zs XOB, BOC, 
 COF, are respec- 
 tively equal to the 
 Zs AOX, DO A, YOD. 
 Therefore Z.YOF is 
 a straight angle, and XY is a straight line. 
 
38 RECTILINEAR FIGURES 
 
 *67. The bisectors of two adjacent supplements are 
 perpendicular to each other. 
 
 For the angle between them is one half the straight 
 angle. 
 
 68. Axiom of Intersection. An indefinite line drawn 
 from the vertex* of an angle that is less than a straight 
 angle, between the arms of the angle, intersects any line 
 that joins a point on one arm of the angle with a point 
 on the other arm of the angle. 
 
 For example, the 
 
 line OK meets ES, or s. 
 
 any other line that 
 joins any point on OX 
 with a point on OF. 
 
 A consequence of 
 this axiom that is 
 needed for some of the proofs will be found in the Appen- 
 dix, 343. 
 
SECTION V. POLYGONS 
 
 69. Polygons. The limited portion of a plane bounded 
 by a broken line which is closed ( 26) is called a polygon. 
 The amount of surface within the polygon is called the 
 area of the polygon, and the length of the boundary is called 
 the perimeter of the polygon. The term "perimeter" 
 is often used for the broken line bounding the polygon 
 when no idea of length is involved, but when this is done, 
 the context usually makes clear which meaning is to be 
 attached to the word. 
 
 70. Vertices and Sides. The vertices of the angles 
 formed by the sects of the broken line are called the 
 vertices of the polygon, and the sects themselves are called 
 the sides of the polygon. The figure ABODE is a polygon. 
 
 71. Angles of a 
 Polygon. The angles 
 on the left in passing 
 around a polygon in 
 the direction contrary 
 to that taken by the 
 hands of a clock are 
 called interior angles. 
 The angles of ABODE 
 
 which are marked with the arrows are interior angles. 
 
 The angle at any vertex of a polygon having as arms 
 one side of the polygon, and the continuation of another 
 side, is called an exterior angle, as Z XBC. In speaking 
 
 39 
 
 X 
 
40 RECTILINEAR FIGURES 
 
 of the exterior angles of a polygon, the angles formed by 
 producing the sides in succession, each through the vertex 
 formed with the following one, in passing around the 
 polygon, are meant. There are evidently two sets of 
 exterior angles of a polygon, formed by passing around 
 clockwise or counterclockwise. These sets are, however, 
 equal, for the two exterior angles at any vertex are ver- 
 tical. 
 
 72. Diagonals. A line joinihg two non-consecutive 
 vertices of a polygon is called a diagonal. 
 
 73. Concave, Convex, and Cross Polygons. A polygon 
 is said to be convex if no side when produced could cut the 
 surface of the polygon. Unless otherwise stated, convex 
 polygon will be meant whenever the term " polygon " is 
 used. 
 
 A polygon is concave when at least one side, if produced, 
 would cut its surface ; it is called cross when its perimeter 
 intersects itself. 
 
 74. Equilateral, Equiangular, and Regular Polygons. A 
 
 polygon that has all its sides equal is called equilateral; 
 one that has all its angles equal is called equiangular. A 
 polygon that is both equilateral and equiangular is called 
 regular. 
 
 75. Number of Sides. A polygon of three, four, five, 
 six, eight, ten sides is called, respectively, a triangle, quad- 
 rilateral, pentagon, hexagon, octagon, decagon, etc. 
 
 76. Base. The side of a polygon on which it appears 
 to stand is called its base. Any side of a polygon might 
 be considered the base. 
 
POLYGONS 41 
 
 TRIANGLES 
 
 77. Parts of a Triangle. A triangle has six parts, three 
 sides and three angles. An angle and a side are spoken 
 of as opposite to each other when the side 'is not one of 
 the arms of the angle. A side is sometimes spoken of as 
 included between two angles, and an angle as included 
 between two sides, when the order in which they lie is 
 meant. 
 
 78. Vertex Angle. The angle opposite the base of a 
 triangle is called the vertex angle, or the vertical angle. 
 
 79. Equality of Sides. If a triangle has two equal 
 sides, it is called isosceles ; if no equal sides, scalene. In 
 an isosceles triangle, the equal sides are sometimes called 
 legs, the third side the base. 
 
 80. Angles of a Triangle. If all the angles of a 
 triangle are acute, it is called an acute-angled triangle; if 
 one angle is right, it is called a right triangle ; and if one 
 angle is obtuse, it is called an obtuse- angled triangle. In a 
 right triangle the side opposite the right angle is called the 
 hypotenuse, and the other sides the legs. 
 
 81. Lines of a Triangle. There are four kinds of lines 
 of importance in work with a triangle: the bisectors of the 
 angles, the perpendicular bisectors of the sides, the altitudes, 
 and the medians. The first two explain themselves ; an 
 altitude is a perpendicular from a vertex to the opposite 
 side, and a median is a line from a vertex to the midpoint 
 of the opposite side. If the altitude of a triangle is spoken 
 of, the altitude to the base is meant. 
 
SECTION VI. INEQUALITIES 
 
 82. Axiom' of Unequals. The whole is greater than 
 any of its parts. 
 
 83. Inequalities. There are certain truths relating to 
 statements of inequality that depend very closely on the 
 general axioms. Their proofs are given under the head of 
 Inequalities in nearly all Algebras, so they will not be con- 
 sidered here. They ^ are used for those magnitudes for 
 which equality axioms 2-5 are used. 
 
 (1) If equals are added to, taken from, multiplied ~by, 
 or divided into, unequals, the results are unequal in the 
 same sense. 
 
 (That is, the greater quantity remains greater after the 
 operation is performed.) 
 
 (2) If unequals are taken from, or divided into equals, 
 the results are equal in the opposite sense. 
 
 (3) If unequals are added to, or multiplied by, un- 
 equals in the same sense, the results are unequal in the 
 same sense. 
 
 (4) If the first of several magnitudes is greater than 
 the second, the second greater than the third, the third 
 greater than the fourth, and so on, then the first is greater, 
 than the last. 
 
 NOTK. (-t) holds for "the first less than the second," etc.; also 
 any pairs might be equal without changing the result, if there is at 
 least one inequality. 
 
 These statements are all in regard to positive magni- 
 tudes ; they are not all true when negative quantities are 
 used. 
 
 WARNING. Unequals should not be taken from, or divided into, un- 
 euuals, for the results cannot, in general, be determined. 
 
 43 
 
SECTION VII. PROPOSITIONS 
 
 84. Propositions. Proposition is a general term includ- 
 ing: 
 
 (1) Theorem, which is a truth to be proved. 
 
 (2) Corollary, which is also a truth to be proved, but 
 generally one that follows quite directly, and often very 
 simply, from a known truth most often from a theorem 
 that has just been proved. 
 
 (3) Problem, or Construction Theorem, which requires 
 that a certain figure be drawn from given parts. A 
 more definite understanding of problem will be given in 
 
 I??- 
 
 85. Parts of a Theorem. A theorem is composed of the 
 condition, or hypothesis, and the conclusion. It usually 
 takes the form, If a certain condition is true, a certain 
 conclusion is also true. (See 2.) 
 
 86. Proof of a Proposition. To prove any proposition, 
 the student has certain materials from which to work, 
 namely : the foregoing definitions, axioms, and truths con- 
 cerning them ; the theorems preceding the one that is 
 being proved ; and the condition of the proposition in 
 question. From these known truths the proof must be 
 deduced, and the required conclusion must be reached. 
 
 The proof must always be a logical one ; the truth of a 
 proposition must not be judged by measurements as in 
 Concrete Geometry or from the appearance of the figure, 
 for such methods have no place in this subject. It might 
 be said, however, that a carefully drawn figure will some- 
 
 43 
 
44 RECTILINEAR FIGURES 
 
 times give the idea which suggests the correct proof, al- 
 though the appearance of the figure cannot be quoted as 
 authority for the truth of any statement. 
 
 87. Order of Proof. The following order of proof has 
 been found very convenient, and the student is advised to 
 follow it in all work. 
 
 STATEMENT OF THEOREM 
 Figure 
 
 Given. Condition, in terms of the letters of the figure. 
 
 To prove. Conclusion, in terms of the letters of the 
 figure. 
 
 Proof. I. The proof, in numbered steps, with the author- 
 ity for each step following it. 
 
 NOTES. Any special case or interesting fact concerning the 
 theorem. 
 
 COROLLARIES. Those connected with the theorem. 
 Examples of this form will be found in the proofs given 
 later. 
 
 88. Classification of Theorems. All theorems of Plane 
 Geometry may be divided into certain groups which might 
 be called classes ; for example, some theorems prove angles 
 equal, others prove lines unequal, still others prove tri- 
 angles congruent, etc. 
 
 It is evident that any theorem of a certain class must 
 depend upon something which will prove the particular re- 
 sult desired ; that is, upon something in its own class, unless 
 it can be obtained by logic from, another class. Except in 
 the cases of logic, which are readily recognized, each geo- 
 metric truth depends directly upon some preceding truth 
 of its own class, and the foundation truths of each class are 
 the definitions and axioms that concern the things used. 
 
CLASSIFICATIONS 45 
 
 89. Important Classes Already Started. 
 
 (1) Congruence. To prove figures congruent, they 
 must be shown to coincide ; the axiom of motion can be 
 used to suppose one of two figures placed on the other in 
 order to test the coincidence. 
 
 (2) Angles Equal. Right angles, straight angles, peri- 
 gons, complements of equal angles, supplements of equal 
 angles, explements of equal angles, vertical angles. 
 
 (3) Magnitudes Unequal. The inequality axiom : after 
 having obtained one inequality, the inequality statements 
 can be used. It follows from this that the only way to 
 prove two things unequal at this stage of the work is to 
 show that one is a part of the other, or that one equals a 
 part of the other. 
 
 (4) Line Straight. A line is straight if its sects are 
 the arms of a straight angle ; the bisectors of vertical 
 angles lie in a straight line. 
 
 (5) Lines Perpendicular. If the adjacent angles formed 
 by them are equal ; if they bisect adjacent supplemental 
 angles. 
 
 (6) Lines Coincide (or are determined). If they go 
 through the same two points, if they bisect the same angle, 
 if they are perpendicular to the same line at the same 
 point (in a plane). 
 
 (7) Points Coincide (or are determined). If they are 
 intersections of the same two lines, if they are midpoints 
 of the same sect, if they are correspondingly placed points 
 of equal sects which are made to coincide. 
 
 (8) Lines Equal. This would be a case of congruence ; 
 the lines would have to be shown to coincide. 
 
 These are, of course, not all the classes to be found in 
 Plane Geometry, but they serve to show that a founda- 
 
46 RECTILINEAR FIGURES 
 
 tion has already been laid, upon which certain classes of 
 geometric truths can be established. 
 
 90. Method of Attack. 
 
 (1) Determine the class of the theorem. 
 
 (2) List the known ways of proving the required con- 
 clusion (or, in other words, list the ones of the same 
 class). 
 
 (3) Examine each way found as to the possibility of its 
 being applied to the figure in question, and especially to ' 
 conditions of the theorem. 
 
 (4) Having decided on one or more ways as possibili- 
 ties (probably by eliminating those which seem impossible 
 of application), try to reason from the condition to the 
 conclusion by the method chosen. 
 
 WARNING. Remember that it is necessary to obtain the conclusion, 
 and to use the condition; more trouble is made by neglect to think of these 
 two points than by anything else in Geometry. 
 
 91. ORAL AND REVIEW QUESTIONS 
 
 Define contraposite, straight line, adjacent angles, converse, ver- 
 tical angles, axiom, perigon, right angle, complements, theorem. 
 Are two supplemental angles ever equal? always equal? What is 
 the complement of one half a right angle? of a right angle? of a 
 zero angle ? of three quarters of a right angle ? What is the sup- 
 plement of a right angle? of a right angle and a half? of two right 
 angles? What does a perpendicular do to a straight angle? Why 
 are all straight angles equal? all right angles? all perigons ? How 
 many points are needed to fix a line? How many lines are needed 
 to fix a point? How many bisecting points can a sect have? Why? 
 How many bisecting lines can an angle have? Why? How does 
 the angle case apply to perpendiculars? What method of proof 
 applies to proving that complements, supplements, and explements 
 of equal angles are equal? Apply one of these to show that vertical 
 angles are always equal. State the equality axioms ; the straight- 
 line axiom; the di\ 7 ision axiom; the substitution axiom; the inter- 
 
ORAL AND REVIEW QUESTIONS 47 
 
 section axiom. Upon what foundation must the proofs of the propo- 
 sitions be based?. If a statement is known to be true, what related 
 statement is always true? What statement is sometimes true, and 
 when? State the obverse of "This statement is true." Of what is 
 a straight angle the sum? a perigon? Of what two equal angles is a 
 straight angle the sum ? a perigon ? Of what four equal angles is a 
 perigon the sum? If an angle is twice its complement, how large is 
 the angle ? If an angle is three times its explement, how large is it? 
 Upon what does the size of an angle depend ? What ways are known 
 to prove figures congruent? angles equal? angles unequal? lines 
 equal? lines unequal ? lines perpendicular? lines not perpendicular? 
 How can a line be determined? a point be determined? What order 
 of proof can be used for a theorem? Define each of the parts of the 
 proof. Distinguish between the terms " proposition," " theorem," 
 " problem." 
 
SECTION VIII. TRIANGLE THEOREMS 
 
 92. Theorem I. // two triangles have two sides and 
 the included angle of the one respectively equal to two 
 sides and the included angle of the other, the triangles 
 are congruent. 
 
 ANALYSIS 
 
 Class. Triangles congruent. 
 
 Known Methods. Coincidence, using the axiom of 
 motion. 
 
 Method to be used. One triangle will be supposed to 
 be placed on the other, and the given facts will then be 
 used to determine whether they would coincide. 
 
 PROOF 
 
 Given. A ABC, A DEF ; BC = EF, CA = FD, Z c = ^ F. 
 
 To prove. A ABC ^ A DEF. 
 
 Proof. I. Suppose A ABC to be placed on A DEF, C on 
 
 F, and BC along EF (ax. of motion). 
 Then CA would lie along FD, vZc=^F 
 (given). 
 
 48 
 
TRIANGLE THEOREMS 49 
 
 II. B -would fall on E,-.-BC=EF (given). 
 A would fall on D, . CA = FD (given). 
 
 III. AB would coincide with DE (but one straight 
 
 line through two points). 
 
 IV. .-. &ABC ^ ADEF (def. ^). 
 
 NOTE. It often makes the conditions of the theorem more clear if 
 those conditions are indicated in the figure. The usual way of show- 
 ing equal parts is to place a like mark on any two parts that are 
 known to be equal. In the figure used in Th. I, the equal parts are 
 indicated by such marks. Where equal parts are used in the theorem, 
 although not given, the same method is sometimes used. 
 
 93. Corresponding Parts of Congruent Figures. When 
 two figures coincide, each part (side or angle) of one 
 coincides with a part of the other, and is therefore equal 
 to it. Two parts of congruent figures that would coin- 
 cide if the figures were made to coincide are called corre- 
 sponding, or homologous parts. 
 
 When two figures are known to be congruent on account 
 of their having certain equal parts, as by Th. I, the 
 other corresponding parts can be told by their position 
 relative to the known parts; as, by their being opposite 
 to known parts, or between two known parts. In Th. I, 
 AB = DE, Z ^t = Z D, Z # = Z j. 
 
 The most important use of congruence of figures is to prove 
 equality of lines and of angles. 
 
 94. Theorem II. If two triangles have two angles and 
 the included side of the one equal to two angles and the 
 included side of the other, the triangles are congruent. 
 
 ANALYSIS 
 
 This theorem is of the same class as Th. I, so must be 
 done in the same way, or else by means of Th. I. It can 
 be proved in either of these ways, and the superposition 
 SMITH'S SYL. PL. GBOM. 4 
 
50 RECTILINEAR FIGURES 
 
 method is much the easier. It is so much like Th. I that 
 the proof will be left for the student to do for himself. 
 
 EXERCISES 
 
 1. Prove Th. II by means of Th. I. 
 
 HINT. A second side is all that is needed. Suppose that the second 
 side of one is not equal to the corresponding side of the other, and cut off 
 a sect on it which is equal to the corresponding side of the other, thus farm- 
 ing a triangle which is congruent to the second triangle. Now examine the 
 angles in the Jig tire. 
 
 2. A point in the perpendicular bisector of a sect is equidistant 
 from the ends of the sect. 
 
 3. In an equilateral triangle, the bisector of any angle forms two 
 congruent triangles. 
 
 4. If the diagonals of a quadrilateral bisect each other, the quad- 
 rilateral has two pairs of equal sides. 
 
 5. If a diagonal of a quadrilateral bisects two angles, the quadri- 
 lateral has two pairs of equal sides. 
 
 WARNING. In exercises 4 an( l > be certain that the sides used are 
 corresponding parts of the triangles found. 
 
 6. In a regular pentagon, the diagonals are equal. 
 
 7. If a line is drawn from the end of a line sect to a point in its 
 perpendicular bisector, the line from the other end of the sect making 
 the same angle with the sect will meet the perpendicular bisector at 
 the same point. 
 
 8. If two triangles are congruent, medians drawn to corresponding 
 sides are equal. 
 
 9. If the bisector of an angle of a triangle is perpendicular to the 
 opposite side, the triangle is isosceles. 
 
 10. Two equal lines AC and AD are drawn on opposite sides of 
 AB, making equal angles with AB. Prove that BC and BD will also 
 make equal angles with AB. 
 
TRIANGLE THEOREMS 51 
 
 11. If two triangles are congruent, the bisectors of corresponding 
 angles are equal. 
 
 12. If one line is perpendicular to a second line, then two lines 
 drawn from a point in the first line to the second line are equal if they 
 make equal angles with the first line. 
 
 NOTE. It was said in 93 that the most important use of con- 
 gruent figures was to prove lines equal and angles equal. It is often 
 necessary to choose between different congruent theorems, and the re- 
 sult desired affects greatly the choice of the theorem. 
 
 For example, if lines are to be proved equal, one is much more likely 
 to be able to use Th. II than Th. I, for it requires less knowledge 
 about equal lines the thing that is being worked for. So, if angles 
 are to be proved equal, Th. I is better than Th. II, for one is more 
 likely to know about the lines than about the angles when the de- 
 sired end is equality of angles. 
 
 Such considerations as the above are often of use in finding the 
 proof of a theorem, and the pupil should give the question under 
 examination very careful thought before attempting to write the 
 proof. 
 
 95. Theorem III. If a side of a triangle is extended, 
 the exterior angle formed is greater than 
 
 (1) the angle opposite the side extended; 
 
 (2) either angle not adjacent. 
 [Note that (2) includes (1).] 
 
 NOTE. This theorem .is a hard one to discover at the beginning 
 of Geometry. It is, however, a very useful theorem, and a very com- 
 plete example of the classification method of finding a construction 
 and proof. The pupil need not feel discouraged by its apparent diffi- 
 culty, but should rather regard this analysis as an example of the 
 method to be applied to the following theorems, most of which are 
 much less difficult- 
 
 The proof of this theorem will be easy to understand after the 
 auxiliary lines (or extra lines which must be added to the figure in 
 order to obtain the proof) are found. Notice that the classification 
 method shows what additional lines are needed in the figure, when the 
 given figure does not itself include all the material necessary. 
 
52 RECTILINEAR FIGURES 
 
 ANALYSIS. (PART I) 
 
 Class. Angles unequal. 
 
 Preceding Methods. Inequality axiom. 
 
 Application. 
 
 It is necessary to show that Z XBC > Z C ; that is, 
 that Z C is part of Z XBC, which is evidently untrue, or 
 that Z C is equal to part of Z.XBC. Since this requires 
 angles equal, a new class must be used. 
 
 Class. Angles equal. 
 
 Preceding Methods. Those in the list in 89, (2) ; cor- 
 responding parts of congruent triangles. 
 
 Elimination. Z C is not known to be a rt. Z, st. Z, or 
 a perigon ; there are no equal angles of which Z C and 
 part of Z XBC could be complements, supplements, or ex- 
 piemen ts; it is not vertical to any angle at B. There 
 remains only the possibility that Z C and an angle at B 
 may be corresponding angles of congruent triangles. 
 
 This requires still a third classification. 
 
 Class. Triangles congruent (triangles, because no ap- 
 plicable way of proving other figures congruent is known 
 as yet). 
 
 Preceding Methods. 2 s. incl. Z ; 2 A incl. s. Prob- 
 ably 2 s. incl. Z will be used because angles are to be 
 proved equal. 
 
TRIANGLE THEOREMS 58 
 
 Application. Any line meeting BC and CA, as RS, will 
 form a triangle of which Z <7 is a part, and any line from 
 
 B to the extension of RS, as BK, will form a triangle con- 
 taining a part of Z XBC. It remains only to draw these 
 lines in such a way that the triangles formed will be con- 
 gruent. Since Z C is to be corresponding to Z KBS, the 
 opposite sides must be made equal, and this can be done by 
 cutting off SK= RS. 
 
 But Z.C8E = Z KSB (vert.), and the only other lines 
 needed are CS and SB, for these, with the parts named, 
 would make two sides and the included angle. As RS can 
 be drawn anywhere, let 8 be the midpoint of BC. 
 
 Therefore, to make the triangles congruent, but two things 
 are needed : That a line be drawn from any point on CA 
 through the midpoint of BC, and extended its own length, the 
 extremity being joined to B. 
 
 The proof has now been discovered if Z KB 8 is a part of 
 Z XBC. It is a part of it if it lies within Z XBC-, that is, 
 if the point K is between BX and BC. K does lie between 
 these arms unless line RSK meets one arm of /.XBC after 
 intersecting BC at 8. It cannot meet BC again (two st. 
 lines meet in but one point), and it cannot meet BX again 
 if it has already met it. The simplest way to have it meet 
 
54 
 
 RECTILINEAR FIGURES 
 
 BX and still be drawn from a point on CA is to draw it 
 from A. (See 96.) 
 
 PROOF 
 
 Given. A ABC-, AB extended to X. 
 To prove. (1) ZxC>Zc; (2) 
 Proof. I. Draw AK from A, through M, the midpoint 
 of BC, to -ET, so that MK = AM. Draw BK. 
 II. CM = MB (bisection) . 
 Z CM A = Z KMB (vert.). 
 AM = MK (const.). 
 
 .-. ACMA ^AKMB (2s. incl. Z). 
 
 III. .-. Z C = Z KBM (cor. pts.). 
 
 IV. But Z KBM is part of Z XBC (AMK having 
 
 met both arms). 
 .-.Z.XBOZ.KBM (ineq. ax.). 
 V. . . Z XBC > Z C (sub. ax.). 
 
 NOTE. This proves an exterior angle greater than the interior 
 angle opposite the side produced. 
 
 VI. 
 
 VII. 
 
 (2) Extend CB to F; then 
 
 since Z A is opposite the extended side 
 
 CB (part 1). 
 
 But ZCBX= /.ABY (vert.). 
 .-. /.CBX>/.A. (sub. ax.). 
 
TRIANGLE THEOREMS 55 
 
 96. Location. That part of Th. Ill which shows that 
 point K lies between the arras of /. XBC, and all other 
 parts of the Geometry where the place in which a point 
 or line must lie is considered, are said to deal with the 
 location of the thing in question. It is very important 
 that the place of each point or line in the figure be fixed 
 absolutely, as otherwise incorrect proofs will often be 
 given. (See 110.) 
 
 97. COR. 1. There can be but one perpendicular from, 
 a given point to a given line. 
 
 Assume one perpendicular, and show that any other line 
 from the same point will make an obtuse angle. 
 
 98. Determination of Lines. 97 is another way to de- 
 termine, or fix a line. Three ways have already been 
 mentioned in 89 (6) ; they are, by two points, by its bi- 
 secting an angle, by its being perpendicular to a line at a 
 point in the line. The new one is by its being perpendic- 
 ular to a line from a point outside the line. The last two 
 are usually spoken of as perpendicular to a line at a point 
 and from a point. 
 
 In making a construction line to help in obtaining a 
 proof, the line can be drawn so as to do any one of these 
 four things, but no more than one. For example, in Th. 
 Ill, the line AM is drawn, and it is determined by A 
 and 3f; then it is extended to K, so that MK = AM. This 
 extension simply represents another part of the line de- 
 termined by A and 3f, of which AM is one sect. The line 
 SB is determined by the points K and B. 
 
 It should be noticed that the determination of a line 
 usually depends upon the determination of one or more 
 points ; as in the case just discussed, the point A is given, 
 and the point M is determined, because there is but one 
 
56 RECTILINEAR FIGURES 
 
 midpoint in a sect. K is determined because the sect MK 
 equals the sect AM. 
 
 WARNING. Never attempt to make a line do two determining things 
 at the same time ; as, bisect an angle and go from the vertex to a fixed 
 point ; or join tico known points and be perpendicular to a Jixed line. 
 Always determine each line, but by one only of the determining conditions. 
 
 The lines discussed in this paragraph auxiliary lines 
 are only representations of lines that do exist. Each 
 point or line added to a figure is the representation of 
 some actual point or line that can exist in the figure, 
 therefore it is necessary to be especially careful not to 
 attempt to draw lines that are not possible. 
 
 99. COR. 2. If a triangle has one right or obtuse 
 angle, the other two angles are acute. 
 
 Use Th. III. 
 
 13. In an isosceles triangle the base angles are acute. 
 
 14. The sum of any two angles of any triangle is less than a 
 straight angle. 
 
 15. If from a point a perpendicular and other lines are drawn to a 
 given line, then the angle formed by any of the lines on its side 
 which is away from the perpendicular, is obtuse, and the greatest 
 angles in the figure are the ones formed by the lines farthest from 
 the perpendicular on its two sides. 
 
 16. If from the ends of a side of a triangle lines are drawn to a 
 point within, the angle formed is greater than the angle opposite that 
 side. 
 
 100. Theorem IV. // two sides of a triangle are 
 equal, the angles opposite those sides are equal. 
 
 An auxiliary line is necessary, but the analysis will 
 show what line is necessary. What method of proving 
 angles equal uses the given, that is, uses equal lines? 
 
 17. An equilateral triangle is equiangular. 
 
TRIANGLE THEOREMS 57 
 
 18. The bisectors of the base angles of an isosceles triangle are 
 equal. 
 
 19. The medians to the legs of an isosceles triangle are equal. 
 
 20. Two points on the base of an isosceles triangle equally dis- 
 tant from the ends of the base are equally distant from the vertex. 
 
 21. If equal distances are laid off on the sides of an equilateral 
 triangle taken in order, the points obtained are the vertices of an 
 equilateral triangle. 
 
 22. Assuming that the bisectors of the base angles of an isosceles 
 triangle meet, prove that they form an isosceles triangle. 
 
 101. COR. 1 . In an isosceles triangle, the bisector of the 
 vertex angle, the median to the base, the altitude to the 
 base, and the perpendicular bisector of the base, are all 
 one line. 
 
 The proof of Th. IV can be extended to prove this. 
 It is very largely a matter of determination of the lines. 
 
 102. COR. 2. There can be but two equal lines from a 
 given point to a given line. 
 
 Draw three lines supposed to be equal, and examine the 
 base angles. 
 
 103- Theorem V. If two sides of a triangle are une- 
 qual, the angles opposite those sides are unequal, the 
 greater angle being opposite the greater side. 
 
 The classification shows that it must be done by the 
 inequality axiom or by exterior angle. Try to form a 
 new angle equal to one of the angles or to part of it, but 
 exterior to the other. 
 
 23. In the quadrilateral A BCD, if AB is greater than EC, EC 
 than CD, and CD than DA, then angle D is greater than angle B. 
 Is angle C always greater than angle A ? 
 
 24. ABCD is a quadrilateral of which DA is the longest side, 
 BC the shortest. Prove angle B greater than D, and C than A. 
 
58 RECTILINEAR FIGURES 
 
 104. Theorem VI. (a) If two angles of a triangle are 
 equal, the opposite sides are equal. (&) // two angles of a 
 triangle are unequal, the opposite sides are unequal, the 
 greater side being opposite the greater angle. 
 
 This is evidently the converse of Th. IV and Th. V, 
 and since they cover all possibilities, the law of converse 
 can be used. The following form is recommended for 
 all theorems proved by the law of converse. 
 
 Given. AABC; /.A = Z#, ZJ. >ZJ3, or /.A</.B. 
 To prove. BC = CA, BC > CA, or BC < CA, respec- 
 tively. 
 
 Proof. I. If BC ;= CA, then Z A = Z B (opp. = sides). 
 If BO CA, then Z A > Z B (opp. = sides). 
 If BC<CA, then Z A<Z..B (opp. Asides). 
 II. These statements cover all possibilities, and 
 no two of the conclusions can be true at once. 
 III. .. If Z A = Z J5, then BC= CA. 
 If Z A >Z B, then BC> CA. 
 If Z A < Z B, then BC< CA (law of con- 
 verse). 
 
 25. An equiangular triangle is equilateral. 
 
 26. In a right-angled triangle, the hypotenuse is the longest side. 
 
 27. In an obtuse-angled triangle, the side opposite the obtuse 
 angle is the longest side. 
 
TRIANGLE THEOREMS 59 
 
 28. Assuming that the bisectors of the angles of a triangle which 
 is not isosceles meet, prove fche bisector of the smaller angle the 
 longer line. 
 
 29. Angle A of triangle ABC is bisected by a line meeting BC 
 at P. Prove that AB is longer than PB, CA than PC. 
 
 30. If a perpendicular and two other lines on the same side of 
 the perpendicular are drawn from a point to a given line, the line 
 which is farther from the perpendicular is the longer. 
 
 105. Theorem VII. If two triangles have three sides 
 of one equal respectively to the three sides of the other, 
 the triangles are congruent. 
 
 Will superposition work ? If not, what other methods 
 of proving triangles congruent are known ? Don't neglect 
 the given. Read Appendix, 344 (2). 
 
 31. In an equilateral quadrilateral, (1) the diagonals bisect the 
 angles and are the perpendicular bisectors of each other; (2) the 
 opposite angles are equal. 
 
 32. Two triangles are congruent if they have two sides and the 
 median to one of those sides respectively equal. 
 
 33. If one diagonal of a quadrilateral divides it into two. isosceles 
 triangles, the other diagonal bisects two of the angles and is the 
 perpendicular bisector of the first diagonal ; also, one pair of opposite 
 angles are equal. 
 
 106. Theorem VIII. If two right triangles have the 
 hypotenuse and a side of the one respectively equal to the 
 hypotenuse and a side of the other, the triangles are con- 
 gruent. 
 
 NOTE. Investigate also the congruence of triangles having two 
 sides and an angle not included equal, when the angle is not right. 
 
 34. If two isosceles triangles have the equal sides of one the same 
 length as the equal sides of the other, and the base of one double the 
 altitude of the other, the triangles are equivalent. 
 
 35. If a quadrilateral has a pair of opposite sides equal, and a pair 
 of opposite angles right angles, one diagonal .divides it into congruent 
 triangles. 
 
60 RECTILINEAR FIGURES 
 
 107. Theorem IX. The sum of any two sides of a tri- 
 angle is greater than the third side. 
 
 Draw the sum of the two sides in the simplest position 
 possible ; then classify. 
 
 108. COR. I. The difference of any two sides of a tri- 
 angle is less than the third side. 
 
 If AB + BC > CA, why is CA BC < AB ? 
 
 109. COR. II. A straight line between two points is 
 less than any broken line between those points. (See 
 Appendix, 346.) 
 
 36. If two sides of a triangle are 6 ft. and 10 ft. long, what can be 
 told about the length of the third side V 
 
 * 37. If two sides of a triangle are a and b, what can be told of the 
 length of the third side ? 
 
 38. The sum of the sides of a triangle is greater than two thirds 
 the sum of the medians, but is less than twice that sum. 
 
 39. The sum of the sides of a quadrilateral is greater than the 
 sum of the diagonals, but is less than twice their sum. 
 
 40. If from the ends of a side of a triangle lines are drawn to a 
 point within, the sum of the lines so drawn is less than the sum of 
 the other two sides of the triangle. 
 
 41. The sum of the lines from the vertices of a triangle to any 
 point is greater than half the perimeter of the triangle. 
 
 42. The difference of the diagonals of a quadrilateral is less than 
 the sum of either pair of opposite sides. 
 
 110. Incorrect Proofs depending on Location. Many 
 apparently correct proofs are entirely incorrect, or are 
 incomplete, because some point or line appears to be in a 
 certain position, and is therefore assumed to be in that posi- 
 tion in the proof, when as a matter of fact it is never in 
 that position at all, or is in that position for certain cases 
 only. 
 
TRIANGLE THEOREMS 61 
 
 If the assumed location is never right, the proof is en- 
 tirely incorrect ; if the location is right for some cases, 
 but not for all, the proof applies only to those cases for 
 which the location is correct, that is, it is not a correct 
 general proof. On this account great care should be 
 taken that no point or line is assumed to be in a certain 
 position unless it has been proved to lie in that position 
 always. 
 
 EXAMPLE: INCOMPLETE PROOF OF THEOREM IX 
 
 Drop a perpendicular from c to AB at F. 
 
 Then , ' BC> FB (hyp. rt. A). 
 
 CA > ^LF(hyp. rt. A). 
 
 And adding, BC + CA > AF+ FB (or AB). 
 
 The fault here is in assuming that AF+FB = AB, 
 which is true only if F falls on AB. It is not necessary 
 that F shall fall on AB, for the perpendicular can just as 
 readily meet the extension of AB. 
 
 This proof can be completed so as to be correct for all 
 cases .by adding a proof that applies to the case where the 
 point F is in the extension of AB. 
 
 111. Theorem X. If two triangles have two sides of the 
 one equal to two sides of the other, but the included angles 
 unequal, then the third sides are unequal, the greater side 
 being opposite the greater angle. 
 
62 
 
 RECTILINEAR FIGURES 
 
 ANALYSIS 
 
 Class. Lines unequal. 
 
 Preceding Methods. Ineq. ax. ; opp. uneq. A ; 2 s. > 3 d. 
 
 Elimination. The inequality axiom has been used to ob- 
 tain the two following ways, so it is more likely that it will 
 be used indirectly through them than that it alone will 
 give the proof. Neither of the other methods will apply 
 if the triangles are entirely separate, so they must be put 
 together, either by being placed side by side, or by super- 
 position. The given will have to determine which method 
 will be best. The fact that one angle is greater than the 
 other can be used by cutting off the smaller on the greater, 
 or by taking enough from the greater so that when it is 
 added to the smaller, the angles will be equal. The secpnd 
 way is nothing more than bisecting the sum of the two 
 angles. It is possible to get a proof by following any of 
 the methods mentioned, but the following way is perhaps 
 the easiest. 
 
 Given. A ABC, DEF-, BC=EF, CA = FD,Z.C>-Z.F. 
 
 To prove. AB > DE. 
 
 F A 
 
 K 
 
 Application. Let the triangles be placed with FD coin- 
 ciding with its equal CA, B and E lying on opposite sides 
 of the common line. Draw a line bisecting the angle 
 BCE; it will meet the opposite side AB of A ABC, for 
 
TRIANGLE THEOREMS 63 
 
 Z C> Z jr, and so this line will lie within Z c. Call 
 the point where it meets AB, K. 
 
 Now the condition has changed from two sides equal 
 and the included angles unequal, to two sides (#C and CK, 
 and EC and CK) and the included angles (Z BCK and 
 Z ECK) equal. This suggests congruent triangles, so the 
 line EK is drawn, forming AECE^ABCK. Then EK = 
 KB, and the proof by two sides greater than the third 
 follows. 
 
 NOTE. This proof holds even when EA and AB happen to fall 
 in one straight line. 
 
 WARNING. " opp. > Z," and "opp. > s." are often used as author- 
 ity. It must be kept in mind that this applies only in the same triangle, 
 or in triangles having two sides equal. Lines and angles must not be 
 judged unequal on account of the opposite parts in any other cases. 
 
 43. In triangle ABC, if CA is greater than AB, and P and Q are 
 taken on AB and CA, respectively, so that PB equals QC, prove BQ 
 less than CP. 
 
 44. If a quadrilateral has a pair of opposite sides equal, but the 
 angles formed by those sides with one of the other sides unequal, 
 then the diagonals are unequal. 
 
 45. If no median of a triangle is perpendicular to a side, then the 
 triangle has no equal sides. 
 
 46. If an equilateral pentagon is not equiangular, the longest diag- 
 onal is that which joins the vertices on either side of the greatest 
 angle. 
 
 112. Theorem XI. // two triangles have two sides of 
 the one equal to two sides of the other, but the third sides 
 unequal, then the angles opposite those sides are unequal, 
 the greater angle being opposite the greater side. 
 
 What relation has this to Th. X? Is Th. X all that is 
 required to prove this theorem ? 
 
 47. In the figure of exercise 44, the angles formed by the equal 
 sides with the fourth side are also unequal, the greater angle being 
 opposite the greater of the two given angles. 
 
64 RECTILINEAR FIGURES 
 
 48. If lines are drawn from the ends of the base of a triangle so as 
 to cut off equal distances (from the base) on the opposite sides, the 
 triangle is isosceles when those lines are equal; when those lines are 
 unequal, the longer line is drawn to the shorter side. 
 
 113. Theorem XII. Of all lines drawn to a given line 
 from a given external point 
 
 (a) the perpendicular is the shortest. (See Appendix, 
 346.) 
 
 (5) those making equal angles with the perpendicular, 
 or cutting off equal distances from its foot, are equal. 
 
 (c) those making unequal angles with the perpendicu- 
 lar, or cutting off unequal distances from Us foot, are un- 
 equal, the one making the greater angle, or cutting off the 
 greater distance, being the greater. 
 
 114. COR. 1. (a) Equal obliques from a point to aline 
 make equal angles with the perpendicular and cut off 
 equal distances from its foot. 
 
 (6) Unequal obliques from a point to a line make un- 
 equal angles with the perpendicular, and cut off unequal 
 distances from its foot, the longer oblique making the 
 greater angle, and cutting off the greater distance. 
 
SECTION IX. PARALLELS AND PARALLELOGRAMS 
 
 115. Angles formed by Two Lines and a Transversal. 
 A line cutting other lines is called a transversal of those 
 lines. 
 
 If two lines are cut by a transversal at two points, 
 eight angles are formed at those points, and certain sets 
 of those angles have names / 
 
 as follows : / 
 
 The two sets of angles 
 
 in the figure A 1, 3, 5, 7 
 and Zs 2, 4, 6, 8 are called 
 transverse sets of angles, 
 and any two angles in the 5 / 6 
 
 7 1 1* 
 v 
 
 The angles Z 1, Z 2, Z 7, Z 8 are called exterior angles, 
 and the angles Z 3, Z 4, Z 5, Z 6 are called interior angles. 
 
 The pairs of angles Z 1 and Z 7 ; Z 2 and Z 8 ; Z 3 and 
 Z 5 ; Z 4 and Z 6 are called alternate pairs of angles. 
 
 The pairs Z 1 and Z 5 ; Z 2 and Z 6 ; Z 3 and Z 7 ; Z 4 
 and Z 8 are called corresponding or exterior interior angles. 
 It is evident that any pair of alternate or corresponding 
 angles are also in the same transverse set. 
 
 116. Theorem XIII. If a transversal of two lines 
 makes one pair of transverse angles, which are not ver- 
 tical, equal, 
 
 (1) any two transverse angles are equal; 
 
 (2) any two angles not transverse are supplemental. 
 SMITH'S SYL. PL. OEOM. 5 65 
 
66 RECTILINEAR FIGURES 
 
 117. COR. 1. If two non-adjacent angles, which are 
 not transverse, are supplemental, 
 
 (1) any two transverse angles are equal ; 
 
 (2) any two angles not transverse are supplemental. 
 
 NOTE. These propositions are introductory to the subject of 
 parallels, and should be used wherever possible to shorten the fol- 
 lowing proofs. They show that the angles formed by two lines and 
 a transversal separate into two groups when there is any question of 
 equality of the angles, and that the angles of different groups are 
 supplemental when angles of the same group are equal. 
 
 118. Parallels. Two straight lines in the same plane 
 which never meet, however far produced, are called 
 parallel lines. 
 
 119. Parallel Axiom. Through a given point there can 
 be but one parallel to a given line. 
 
 This is sometimes stated : Two intersecting lines cannot 
 both be parallel to the same line. 
 
 * 120. Lines parallel to the same line are parallel to 
 each other. For, if two lines that are parallel to the 
 same line should meet, there would be more than one par- 
 allel through a point. 
 
 121 . Theorem XIV. If a transversal of two lines makes 
 
 (1) a pair of transverse angles, which are not vertical, 
 equal, or 
 
 (2) two non-adjacent angles, which are not transverse, 
 supplemental, 
 
 then the two lines are parallel. 
 
 Is there any difference between the two cases ? If the 
 lines meet, examine the angles ; use contraposite. 
 
PARALLELS AND PARALLELOGRAMS 67 
 
 122. COK. 1. If a transversal of two lines makes 
 
 (1) a pair of transverse angles unequal, 
 
 (2) two angles which are not transverse not supple- 
 mental, then the lines are not parallel. 
 
 This is the obverse of 121. The best way to prove the 
 obverse of a single statement is to use both " givens," and 
 show that the two conclusions could not be the same. 
 Applied to this statement, the work would be about as 
 follows : 
 
 If a pair of transverse angles were equal, the lines 
 would be parallel. 
 
 If a second line through the same point on the trans- 
 versal makes an angle not equal to the other of the pair, 
 that line cannot also be parallel, by the parallel axiom. 
 This is, of course, in very condensed form ; the proof 
 should be in terms of the letters of the figure. 
 
 Note that the lines, if not parallel, meet on the side of 
 the smaller of the two alternate interior angles. Why ? 
 
 123. COR. 2. Lines perpendicular to the same line are 
 parallel. 
 
 49. The bisector of the exterior vertex angle of an isosceles tri- 
 angle is parallel to the base. 
 
 50. If A C and BD bisect each other, prove that AB is parallel to 
 CD. 
 
 51. The bisectors of two consecutive angles of any polygon inter- 
 sect one another. 
 
 52. If all the angles of a quadrilateral are right angles, the figure 
 has two pairs of parallel sides. 
 
 124. Theorem XV. If two parallels are cut by a trans' 
 versal, 
 
 (1) any two transverse angles are equal; 
 
 (2) any two angles that are not transverse are sup- 
 plemental. 
 
68 RECTILINEAR FIGURES 
 
 What relation has this to Th. XIV? to Th. XIV, Cor. 
 1 ? After proving this by the simplest way, see Appendix, 
 341. 
 
 125. COR. 1. A line perpendicular to one of two par- 
 allels is perpendicular to the other. 
 
 It is necessary first to prove that it meets the other; 
 in other words to prove that it cannot be parallel to it. 
 If it were parallel to one of the two, while given perpen- 
 dicular to the other, what impossible figure would be 
 formed ? 
 
 WARNING. Never assume that two lines meet without proving why 
 they meet. 
 
 126. COK. 2. Two lines that are perpendicular to two 
 intersecting lines are not parallel. 
 
 127. COR. 3. If the arms of one angle are parallel to 
 the arms of a second angle, the angles are equal, or supple- 
 mental (according to their relative positions) . 
 
 128. COR. 4. If the arms of one angle are perpendicu- 
 lar to the arms of a second angle, the angles are equal, or 
 supplemental (according to their relative positions). 
 
 Through the vertex of one of the angles draw parallels 
 to the arms of the other angle, thus using angles at one 
 vertex ( 127). 
 
 53. The bisectors of a pair of alternate angles formed by parallels 
 with a transversal are parallel. 
 
 54- A line through the vertex of an isosceles triangle parallel to 
 the base bisects the exterior angle at the vertex. 
 
 55. If the bisector of an exterior angle of a triangle is parallel to 
 a side of the triangle, the triangle is isosceles. 
 
 56. If from a point in the base of an isosceles triangle, lines are 
 drawn parallel to the equal sides, those lines form, with the equal 
 sides, a quadrilateral whose perimeter equals twice one of the equal 
 sides. 
 
PARALLELS AND PARALLELOGRAMS 69 
 
 57. A line parallel to the base of an isosceles triangle makes equal 
 angles with the legs, or the legs produced. 
 
 58. A line parallel to the base of an isosceles triangle is perpen- 
 dicular to the bisector of the vertex angle. 
 
 129. Theorem XVI. In any triangle, 
 
 (1) any exterior angle equals the sum of the two non- 
 adjacent interior angles ; 
 
 ( '2) the sum of all the interior angles equals a straight 
 angle. 
 
 What is known about the exterior angle? Use it. 
 
 130. COR. 1. If two triangles have a side and any 
 two angles of the one equal to a side and the respective 
 angles of the other, the triangles are congruent. Use the 
 theorem. 
 
 This is the last of the propositions on the congruence of 
 triangles by equal parts. It should be noticed that three 
 parts are always necessary, and that any three correspond- 
 ing parts will prove the triangles congruent except 
 
 (1) three angles. (See 286, 289.) 
 
 (2) two sides and an acute angle not included (called 
 the Ambiguous Case). (See 106.) 
 
 131. COR. 2. If a triangle has one right angle, the 
 other two angles are complements ; in an equilateral tri- 
 angle, each angle is one third of a straight angle. 
 
 59. Using (1) of 129, prove (2) by drawing a line from the vertex 
 to any point in the base. 
 
 60. The bisectors of two interior angles of a triangle meet (ex- 
 amine the angles made with the included side), and form an angle 
 equal to a right angle plus one half the third angle. 
 
 61. The bisectors of two exterior angles of a triangle meet, and 
 form an angle equal to one half the sum of the interior angles supple- 
 mental to the ones bisected. 
 
70 RECTILINEAR FIGURES 
 
 68. If an angle of a triangle is bisected, and a line is drawn per- 
 pendicular to that bisector, that line makes an angle with either arm 
 of the bisected angle equal to half the sum of the other angles of the 
 triangle, and makes an angle with the* third side equal to half the 
 difference of those angles. 
 
 63. Any two altitudes of a triangle make equal angles each with 
 the side to which the other is drawn. 
 
 64- Find the sum of the angles of a quadrilateral by drawing a 
 diagonal. 
 
 132. Theorem XVII. The sum of the interior angles of 
 a polygon of n sides is (n 2} straight angles. 
 
 65. What is the sum of the interior angles of a pentagon ? a hexa- 
 gon? a decagon? an octagon? a 29-sided figure? 
 
 66. How many sides has the polygon the sum of whose interior 
 angles is 12 st. angles? 3 st. angles? 17 st. angles? 
 
 67. How large is one angle of a regular pentagon? hexagon? 
 decagon ? octagon ? 34-sided figure ? 
 
 133. Theorem XVIII. The sum of the exterior angles of 
 a polygon is two straight angles. 
 
 68. How many sides has the polygon the sum of whose interior 
 angles equals the sum of the exterior angles? 
 
 69. How many sides has a polygon if the sum of the interior 
 angles is seventeen times the sum of the exterior angles? 
 
 70. How many sides has the polygon one third the sum of whose 
 interior angles is ten times the sum of its exterior angles? 
 
 71. How many sides has the polygon each of whose exterior angles 
 is one tenth of a perigon ? 
 
 72. How many sides has the polygon each of whose interior angles 
 is six sevenths of a straight angle ? 
 
 134. Degrees, Minutes, and Seconds. For convenience 
 in numerical calculations, a perigon is diyided into 360 
 equal parts, called degrees; each degree is divided into 
 
PARALLELS AND PARALLELOGRAMS 71 
 
 60 equal parts called minutes, and each minute into 60 
 equal parts called seconds. 
 
 73. How many degrees are there in an interior angle of a regular 
 polygon of 15 sides ? of n-sides? 
 
 74. How many degrees are there in an exterior angle of a regular 
 polygon of 20 sides? of n-sides? 
 
 135. Quadrilaterals having Two or More Parallel Sides. 
 
 A quadrilateral having two pairs of parallel sides is called 
 a parallelogram ; one which has one pair of parallel sides 
 is called a trapezoid ; one which has no parallel sides is 
 called a trapezium. 
 
 The parallel sides of a trapezoid are called its bases, the 
 other two sides its legs. 
 
 Either pair of parallel sides may be called the bases of a 
 parallelogram, but the side on which it appears to stand is 
 usually called its base. 
 
 A parallelogram having all its angles right angles is 
 called a rectangle; one having all its sides equal is called 
 a rhombus ; a rectangular rhombus is called a square. A 
 parallelogram that is neither a rectangle nor a rhombus is 
 called a rhomboid. 
 
 136. Theorem XIX. Any two consecutive angles of a 
 parallelogram are supplemental, and any two opposite 
 angles are equal. 
 
 137. COR. 1. A parallelogram having a right angle is 
 a rectangle. 
 
 138. Theorem XX. In any parallelogram, 
 
 (1) either diagonal divides it into congruent triangles; 
 
 (2) the opposite sides are equal. 
 
 139. COR. 1. The diagonals of a parallelogram bisect 
 each other. 
 
72 RECTILINEAR FIGURES 
 
 75. The diagonals of a parallelogram are equal if the figure is a 
 rectangle. 
 
 76. A parallelogram having two consecutive sides equal is a 
 rhombus. 
 
 77. Two trapezoids are congruent if their sides are respectively 
 equal. 
 
 140. Theorem XXI. A quadrilateral is a parallelo- 
 gram if 
 
 (1) two opposite sides are equal and parallel; or 
 
 (2) the opposite sides are equal. 
 
 What is the definition of a parallelogram ? What ways 
 of proving this do you know? Keep in mind that the 
 given must be used. 
 
 141. Theorem XXII. A quadrilateral is a parallelo- 
 gram if the opposite angles are equal. 
 
 142. Theorem XXIII. A quadrilateral is a parallelo- 
 gram if the diagonals bisect each other. 
 
 78. Js a quadrilateral a parallelogram if one diagonal divides it 
 into congruent triangles? if each diagonal divides it into congruent 
 triangles? 
 
 79. If each side of a square is extended from each end a length 
 half as long as the diagonal, the figure formed by joining the ends of 
 the sects obtained is a regular octagon. 
 
 80. On diagonal BD of parallelogram A BCD, points K and L are 
 taken so that BK LD. Prove that AKCL is a parallelogram. 
 
 81. Perpendiculars from opposite vertices of a parallelogram to 
 the diagonal joining the other vertices are equal. 
 
 82. Lines through the vertices of a triangle parallel to the oppo- 
 site sides form a triangle having sides twice the length of the sides of 
 the given triangle. 
 
 83. Lines drawn through the vertices of a quadrilateral parallel to 
 the diagonals form a parallelogram twice as large as the quadrilateral. 
 
PARALLELS AND PARALLELOGRAMS 73 
 
 84. Any quadrilateral is equal to a triangle having as sides the 
 diagonals of the quadrilateral, and the included angle equal to the 
 angle between the diagonals. 
 
 85. The sum (or the difference) of the perpendiculars from one 
 pair of opposite vertices of a parallelogram to any line equals the sum 
 (or the difference) of the perpendiculars from the other vertices to 
 the same line. 
 
 86. The sum of the perpendiculars dropped from a point in the 
 base of an isosceles triangle to the legs is constant. 
 
 87. The difference of the perpendiculars from any point in the 
 extension of the base of an isosceles triangle to the legs is constant. 
 
 143. Theorem XXIV. If two or m,ore sects cut off on one 
 transversal by parallels are equal, the corresponding sects 
 cut off on any other transversal by the same parallels are 
 also equal; including as a special case, 
 
 A line through the midpoint of a side of a triangle, 
 parallel to a second side, bisects the third side. 
 
 144. COR. 1. A line joining the midpoints of two sides 
 of a triangle is parallel to the third side, and equals one 
 half that side. 
 
 145. COR. 2. If through the vertices of a triangle, 
 lines are drawn parallel to the opposite sides, a new tri- 
 angle is formed, such that the sides of the given triangle 
 join the midpoints of the sides of the new triangle. 
 
 88. The lines from two opposite vertices of a parallelogram to the 
 midpoints of two parallel sides trisect the diagonal joining the other 
 two vertices. 
 
 89. The lines joining the midpoints of the opposite sides of a quad- 
 rilateral bisect each other. 
 
 When do lines bisect each other? What way is there of using 
 midpoints to obtain this ? 
 
 90. The lines of 89 intersect at the midpoint of the line which 
 joins the midpoints of the diagonals. 
 
74 RECTILINEAR FIGURES 
 
 91. A line through the midpoint of one of the legs of a trapezoid, 
 parallel to the bases, bisects the other leg. 
 
 92. The line joining the midpoints of the legs of 'a trapezoid is 
 parallel to the bases. 
 
 93. What line passes through the midpoints of all the lines drawn 
 from a point to a line? 
 
 94. If, in an isosceles triangle, any number of parallels to the base 
 are drawn, the perpendicular bisector of the base goes through the 
 intersections of the diagonals of each trapezoid formed. 
 
 NOTE. The trapezoid, while not used in the theorems, has been 
 taken up quite frequently in exercises. Many of the triangle theorems 
 have corresponding facts for trapezoids, and on this account, a good 
 way to study the trapezoid is to word the triangle theorems so that 
 they apply to trapezoids, and investigate their truth. Probably the 
 most valuable auxiliary line for trapezoids is the line from the vertex 
 between the shorter base and a leg, parallel to the other leg. This 
 line divides the trapezoid into a triangle and a parallelogram, and so 
 allows the theorems for those figures to be applied. 
 
SECTION X. LOCI OF POINTS AND CONCURRENCE 
 
 146. Locus. That place, or places, in which all points 
 which fulfill a certain condition and no others lie, is called 
 the locus of points which fulfill that condition. 
 
 If all pupils of a certain class are in a certain room 
 (that is, no pupil is absent), and all in the room belong 
 to that class (that is, there are no visitors present), then 
 that room contains all pupils of that class and no others. 
 Similarly, if a line contains every point of a certain kind 
 (where " kind " means in respect to some condition), and 
 every point on it is of that kind, then it is the locus of 
 such points. Evidently a locus is nothing but the place 
 which may be said to belong to a certain kind of point. 
 
 To prove a locus, it is necessary to prove two things : 
 
 (1) all points of the kind lie in the place ; 
 
 (2) all points in the place are of the same kind. 
 These statements are converses of each other, and their 
 
 contraposites will also be true. 
 
 (3) all points not in this place, are not of this kind ; 
 
 (4) all points not of this kind do not lie in this place. 
 It is sometimes easier to prove (3) or (4) instead of its 
 
 contraposite, the most common way being 'to prove (3) 
 instead of (1). 
 
 147. Attack of a Locus Theorem. The best way to dis- 
 cover a locus theorem is to take a point of the required 
 kind and study it for characteristics which will show that 
 it must lie in some certain place. After the place is 
 
 75 
 
76 RECTILINEAR FIGURES 
 
 found, it is also necessary to prove that every point in the 
 place is of the same kind. 
 
 148. Theorem XXV. Find the locus of points equidis- 
 tant frojn two given points. 
 
 Take any point equidistant from two given points, join 
 the three points, and what kind of a triangle will be 
 formed? What will pass through the vertex? Then 
 what is the place which appears to be the locus ? Now 
 prove that all points on the line found are equidistant 
 from the given points. 
 
 149. Con. 1. A line containing two points that are 
 equidistant from the ends of a sect is the perpendicular 
 bisector of that sect. 
 
 150. COR. 2. The midpoint of the hypotenuse of a 
 right triangle is equidistant frojn tlxe three vertices. 
 
 95. Find the locus of points equidistant from three points. (Use 
 the points two at a time.) 
 
 151. Concurrence. If two or more lines meet in a 
 point, the lines are said to be concurrent at that point. 
 
 152. Theorem XXVI. Tl:e perpendicular bisectors of 
 the three sides of a triangle are concurrent. 
 
 Show that two meet, then that the third must pass 
 through the same point. 
 
 This point of concurrence is equidistant from the three 
 vertices, and is "called the circumcenter of the triangle. 
 (See 165.) 
 
 96. If the circumcenters of the triangles formed by a diagonal of 
 a parallelogram are joined to the ends of the diagonal, a parallelo- 
 gram is formed. 
 
LOCI OF POINTS AND CONCURRENCE 77 
 
 97. The circumcenters of the four triangles into which a quadri- 
 lateral is divided by its diagonals are the vertices of a parallelogram. 
 
 98. In triangle ABC, if the midpoints of BC and CA are joined, 
 the circumcenters of the original triangle ($) and of the new triangle 
 (/$*') and C are in one straight line, and #' is the midpoint of the 
 line. 
 
 153. Theorem XXVII. Find the locus of points equi- 
 distant from two intersecting lines. 
 
 NOTE. Locus does not always mean equidistance, although it 
 involves equidistance in the first few cases. 
 
 99. Find the locus of points equidistant from two parallel lines. 
 
 100. Find the locus of points at a fixed distance from a given line. 
 
 101. If two points between the arms of an angle are equidistant 
 from those arms, the line through the points is the bisector of the 
 angle. 
 
 102. Find the locus of points equidistant from three lines that 
 intersect in no point; in one point; in two points ; in three points. 
 
 103. Find the locus of points equidistant from two intersecting 
 lines, and at the same time equidistant from two points. 
 
 104. Find the locus of points equidistant from two given lines 
 and at a fixed distance from a given line. 
 
 154. Theorem XXVIII. The bisectors of the interior 
 angles of a triangle are concurrent. 
 
 The point of concurrence is called the incenter, and 
 it is equidistant from the three sides of the triangle. 
 (See 166.) 
 
 155. COR. 1. The bisectors of any two exterior angles 
 of a triangle, and the third interior angle, are con- 
 current. 
 
 There are three such points of concurrence, all equi- 
 distant from the sides of the triangle ; they are called 
 excenters. (See 166.) 
 
 Of what are the incenter and the excenters the locus ? 
 
78 RECTILINEAR FIGURES 
 
 105. If the incenter of an equilateral, triangle is joined to the 
 vertices, the triangles formed are congruent. 
 
 106. If the incenters of the triangles formed in 105 are joined to 
 each other, the incenter of the original triangle is the circumcenter 
 of the new triangle. 
 
 107. If the incenters of the triangles into which a diagonal 
 divides a parallelogram are joined to the ends of the diagonal, a 
 parallelogram is formed. 
 
 108. The incenters and the circumcenters of the triangles into 
 which a diagonal divides a parallelogram are the vertices of a 
 p aralle logr am . 
 
 156. Theorem XXIX. The altitudes of a triangle are 
 concurrent. 
 
 Use 145. The point is called the orthocenter. 
 
 109. In triangle ABC, if YX joins the midpoints of BC and CA, 
 and O and 0' are the orthocenters of triangles ABC and XCY, then 
 O, 0', and C are collinear, and CO' = 00'. 
 
 110. In the figure o 109, if S and S' are the circumcenters, / and 
 /' the incenters, then O'S 1 is parallel to OS and equals one half OS, 
 O'l' is parallel to OI and equals one half OI. 
 
 157. Theorem XXX. The medians of a triangle are 
 concurrent in a trisection point of each. 
 
 Show that two meet. Draw the Hue from the third 
 vertex through their intersection, and show that it bisects 
 the third side, and is therefore the third median. This 
 line may be extended its own length past the intersection 
 point, and shown to be one diagonal of a parallelogram. 
 
 The point of concurrence is called the centroid. 
 
 111. The sum of the three medians of a triangle is greater than 
 three fourths the sum of the sides. 
 
 112. In triangle AB C, if YX joins the midpoints of BC and CA, 
 and if T and T are the centroids of triangles ABC and XCY, T, T, 
 and C' are collinear, and CT 1 = TT'. 
 
SUMMARY OF PROPOSITIONS 79 
 
 113. In the figure of 112, if ', /, 0', and S, /, 0, are the circum- 
 center, incenter, and orthocenter, respectively, of triangles JTGTand 
 ABC, then T S' is parallel to TS and equals one half of it; TT is 
 parallel to TI and equals one half of it; TO' is parallel to TO and 
 equals one half of it. 
 
 158. SUMMARY OF THEOREMS AND COROLLARIES, BOOK I 
 
 (Numbers in parentheses refer to black- faced section numbers.) 
 
 I. CONGRUENCE OF TRIANGLES. Triangles are congruent when 
 they have three parts (of which at least one is a side) of one equal to 
 the corresponding parts of the other, except when the parts are two 
 sides and an acute angle not included between those sides. 
 
 Cases. 2 s. incl. Z (92) ; 2 A incl. s. (94) ; 3 s. (105) ; rt. Z, hyp., 
 leg (106) ; 2 A any s. (130) ; diag. par. (138). 
 
 II. LINES ARE DETERMINED, OR" COINCIDE. One _L from a point 
 (97) ; lines in an isosceles A (101) ; one I! through a point (119). 
 (See also 89.) 
 
 III. LINES EQUAL. Congruence; opp. = A in a A (104); obliques 
 (113, 114) ; opp. s. n (138) ; diag. O bisect (139) ; II s cutting off equal 
 sects on transversals (143) ; lines through midpt. of s. of a A (144, 
 145) ; midpt. hyp. rt. A equidist. (150). 
 
 IV. LINES UNEQUAL. Opp. = A in a A (104) ; but two = lines 
 from a pt. (102) ; sum of two s. of a A > 3d, diff. less (107, 108, 109) ; 
 opp. = A in A having two = s. (Ill) ; obliques (113, 114). 
 
 V. ANGLES EQUAL, AND ANGLE EQUATIONS. Opp. = s. (100) ; 
 obliques (114) ; Us (116, 117, 124) ; II arms (127) ; arms (128) ; opp. 
 A a (136, 137). (See also 89.) 
 
 Ext. Z of a A = two int. A, sum A of a A = st. Z (129) ; acute A 
 of a rt. A = rt. Z (131) ; int. A of a polygon = (n - 2) st. A (132) ; 
 ext. A = 2 st. A (133). 
 
 VI. ANGLES UNEQUAL. Ext. Z of a A (95) ; in rt. or obtuse angled 
 A other A acute (99) ; opp. = s. in a A (103) ; opp. =. s. in two A 
 having two = s. (112) ; obliques (114). 
 
 VII. ANGLES SUPPLEMENTAL. Us (116, 117, 124) ; II arms (127) ; 
 A. arms (128) ; con. sec. A of a O (136). 
 
80 RECTILINEAR FIGURES 
 
 VIII. LINES PARALLEL. A = (121); A sup. (121); JL same line 
 (123) ; II same line (120) ; opp. s. O. (See XI of this set) ; line 
 through midpts. of the s. of a A (144.) 
 
 IX. LINES NOT PARALLEL. But one || through a pt. (119) ; A = 
 (122) ; A not sup. (122) ; intersecting lines (126). 
 
 X. LINES PERPENDICULAR. Line _L one of two Us _L other (125); 
 contains two points equidist. (149). (See also 89.) 
 
 XL A QUADRILATERAL A PARALLELOGRAM. Opp. s. || (135); 
 opp. s. = (140) ; two opp. s. = and || (140) ; opp. A = (141) ; diag. 
 bisect (142). 
 
 XII. Loci. Points equidist. from two points (148); equidist. from 
 two intersecting lines (153). (See exercises 99, 100.) 
 
 XIII. CONCURRENCE. _L bisectors of sides (152) ; bisectors int. A 
 (154); bisectors 2 ext. A and int. (155); altitudes (156) ; medians 
 (157). 
 
 These classifications should be carefully studied, and made use of 
 in solving exercises depending on the respective classes. The pupil 
 should not depend on the list here, but should know the important ways 
 of doing theorems of each class. 
 
 159. ORAL AND REVIEW QUESTIONS 
 
 What are the two most common ways of proving lines equal ? the 
 three most common of proving angles equal? the ways to prove lines 
 parallel ? not parallel ? a figure a parallelogram ? Name the points of 
 concurrence of the sets of lines in a triangle. What is known about 
 the circumcenter ? the incenter? the excenters? the centroid? If 
 you wish to prove that two lines bisect each other, how ought you 
 to start? If it is necessary to prove two lines equal, and two angles 
 are given equal, how are they likely to be used ? How can correspond- 
 ing parts of congruent triangles be distinguishe'd without superim- 
 posing? What would be the sum of the interior angles of a polygon 
 of K sides ? of 79 sides ? the exterior angles of each ? If the interior 
 angles of a polygon have a sum of 29 straight angles, how many sides 
 has the polygon? a sum of (2 K + 6) right angles? If the interior 
 angles of a polygon are eleven times the exterior angles, how many 
 
ORAL AND REVIEW QUESTIONS 81 
 
 sides has the polygon ? If each exterior angle is one twenty-second 
 of a straight angle, how many sides has the polygon? If each interior 
 angle is nineteen twentieths of a straight angle, how many sides has 
 the polygon ? If two sides of a triangle are 17 and 28, how long is 
 the third side ? How many points need to be known to determine a 
 perpendicular bisector? What ways are known to determine a line? 
 In how many ways should the same line be determined? What class 
 of theorems would be used to prove that two lines meet? Outline 
 in as brief a manner as possible the methods used in proving the fol- 
 lowing : base angles of an isosceles triangle are equal ; sum of interior 
 angles of a polygon ; quadrilateral having two pairs of equal sides is 
 a parallelogram ; lines perpendicular to the same line are parallel ; 
 but one perpendicular from a point to a line ; exterior angle of a tri- 
 angle equals the sum of the non-adjacent interior angles; opposite 
 angles of a parallelogram are equal ; opposite sides of a parallelogram 
 are equal; angles having parallel arms are equal or supplemental; 
 perpendicular bisectors of the sides of a triangle are concurrent. 
 
 NOTE. Be ready to explain why each construction used in these 
 proofs is the best one to use. 
 
 If the sides of a triangle are a, b, c, give air the length relations 
 possible between the sides. How many equal lines are there from a 
 point to a line ? Which statement (converse, obverse, or contraposite) 
 always follows from a true statement? which sometimes follows? 
 when ? State three places where converse is used to prove theorems 
 or corollaries. Try to explain why these propositions are not in 
 reverse order. Give some examples of contraposite reasoning used in 
 proving theorems. What is the locus of points equidistant from three 
 points not in the same line? Can the locus of points equidistant from 
 four or more points be found? Why? From four or more lines? 
 Why? What are the two most important things to keep in mind in 
 trying to prove a theorem ? which should be thought of first, in gen- 
 eral ? Why is it necessary to be careful of the location of points and 
 lines? 
 
 GENERAL EXERCISES 
 
 114' If two lines drawn from the ends of the base of an isosceles 
 triangle to the opposite sides cut off equal distances from the vertex, 
 those lines are equal. 
 
 115. If one angle of a triangle is twice a second angle, the bisector 
 of the larger angle forms a triangle equiangular to the given triangle. 
 SMITH'S SYL. PL. GEOM. 6 
 
82 RECTILINEAR FIGURES 
 
 116. If one of the opposite sides of a quadrilateral is the longest 
 side, the other the shortest side, examine the size of the opposite 
 angles. 
 
 117. The sum of the lines from the vertices of a triangle to any 
 point is greater than one half the perimeter of the triangle. 
 
 118. The sum of the lines from the vertices of a quadrilateral to 
 any point is greater than the sum of the diagonals. 
 
 119. The difference of the diagonals of a quadrilateral is less than 
 half the sum of its sides. 
 
 120. If two points K and L are on the same side of the line AB, 
 to what point P in AB must lines be drawn from K and L, so that 
 KP + LP shall be the shortest possible length? (Use symmetry.) 
 
 121. If a pair of base angles of a trapezoid are equal, the legs are 
 equal, and conversely. 
 
 122. If the legs of a trapezoid are unequal, the base angles are 
 unequal, and conversely. 
 
 123. The diagonals of an isosceles trapezoid are equal. 
 
 124-- The lines joining the midpoints of the consecutive sides of a 
 quadrilateral form a parallelogram. Examine the special cases, such 
 as parallelogram, rhombus, rectangle, square. 
 
 125. Find out everything possible about the diagonals of a 
 rhombus. 
 
 126. The perpendicular bisector of a side of a triangle intersects 
 the greater of the other two sides, if the triangle is not isosceles. 
 
 127. In a right triangle, the altitude to the hypotenuse divides the 
 triangle into two triangles whose angles equal the angles of the 
 original triangle. 
 
 128. The line through the midpoints of the bases of an isosceles 
 trapezoid passes through the intersection of the diagonals, and the 
 intersection of the legs. 
 
 129. Two lines through a point equidistant from two parallels cut 
 off on the parallels, if they meet them, equal sects. 
 
 130. The midpoint of any transversal of two parallels is equidis- 
 tant from them. 
 
 131. Lines drawn from any point in the bisector of an angle 
 parallel to its arms form a rhombus with those arms. 
 
GENERAL EXERCISES 83 
 
 132. A line through the incenter of a triangle parallel to a side 
 equals the sum of the sects it cuts off on the other two sides. 
 
 138. If the three sides of one triangle are parallel respectively to 
 the three sides of another triangle, the angles of the first triangle are 
 respectively equal to the angles of the second triangle. 
 
 134. If the sides of one triangle are respectively perpendicular to 
 the sides of a second triangle, the angles of the first triangle are 
 respectively equal to the angles of the second triangle. 
 
 135. The angle between the bisectors of two exterior angles of a 
 triangle equals the complement of half the third interior angle. 
 
 136. The angle between the bisectors of one interior and one ex- 
 terior angle of a triangle equals half the interior angle at the third 
 vertex. 
 
 137. If the vertex of an isosceles triangle is joined to the trisec- 
 tion points of the base, the middle of the three angles formed is the 
 largest. 
 
 138. If one acute angle of a triangle is double another, the tri- 
 angle can be divided into two isosceles triangles by a line through 
 the third vertex. 
 
 139. Two exterior angles of a triangle less the third interior angle 
 equals a straight angle. 
 
 140. The angle between the bisectors of two angles of a triangle 
 equals the third angle plus half the sum of the bisected angles. 
 
 141. Two equilateral triangles are congruent if an altitude of one 
 equals an altitude of the other. 
 
 142. An angle of a triangle is acute, right, or obtuse, according as 
 the median from it is greater than, equal to, or less than one half the 
 opposite side. 
 
 143. The angle between the bisectors of the base angles of an 
 isosceles triangle equals the exterior angle at the base. 
 
 144. If equal distances are measured off on the sides of a square 
 taken in order, the points obtained will be the vertices of a second 
 square. 
 
 145. A transversal of the legs of -an isosceles triangle, one pro- 
 duced, which is perpendicular to the base, forms an isosceles triangle 
 with the legs. 
 
84 RECTILINEAR FIGURES 
 
 146. Divide the right angle of a right triangle into parts equal to 
 the acute angles, and so prove that the midpoint of the hypotenuse 
 is equidistant from the three vertices. 
 
 147. If one of the equal sides of an isosceles triangle is extended 
 through the vertex its own length, the line joining the end of the 
 extension to the other end of the base is perpendicular to the base. 
 
 148. If the angles of a quadrilateral are bisected, the angles 
 formed by the bisectors of any pair of adjacent angles is supple- 
 mental to the angle formed by the bisectors of the other angles. 
 
 149. A trapezoid whose diagonals make equal angles with a base 
 is isosceles. 
 
 150. Prove the theorem for the sum of the angles of a polygon 
 by joining one vertex to all the others. How many diagonals can be 
 drawn in any n-sided polygon ? How many triangles will be formed 
 in an n-sided polygon ? 
 
 151. How many sides has a polygon the sum of whose interior 
 angles is four times the sum of the interior angles of a quadrilateral? 
 
 152. How many sides has a polygon the sum of whose interior 
 angles is equal to the sum of the interior angles of a hexagon plus 
 the sum of the exterior angles of an octagon ? 
 
 153. The "triangles formed by producing the sides of an equi- 
 angular hexagon through both vertices of each side are equilateral. 
 
 154- The sum of the angles formed by producing the sides of a 
 hexagon both ways until they meet equals a perigon. 
 
 155. If the sides of any polygon of w-sides are produced from 
 both ends until they meet, the sum of the angles so formed is n 4 
 straight angles. 
 
 156. If each consecutive pair of angles of a quadrilateral are sup- 
 plemental, is the figure a parallelogram ? 
 
 157. A parallelogram which has supplemental opposite angles is 
 a rectangle. 
 
 158. Is a quadrilateral a parallelogram if one pair of opposite 
 sides are equal, and the other pair are parallel ? 
 
 159. If each side of a square is produced in order, the extensions 
 of the opposite sides being equal, the points obtained are the vertices 
 of a parallelogram. 
 
GENERAL EXERCISES 85 
 
 160. The sum of the perpendiculars from any point to the three 
 sides of an equilateral triangle is constant. (In certain cases some 
 of the lines become negative, and the difference would be taken.) 
 
 161. If X and Y cut off equal sects from A and C on the sides 
 AB and CD of the parallelogram A BCD, AXCY is a parallelogram. 
 
 162. Parallelograms having two consecutive sides and an angle 
 of one equal to the corresponding parts of the other are congruent. 
 
 163. Rectangles having two consecutive sides of the one equal to 
 the corresponding parts of the other are congruent. 
 
 164- I D anv parallelogram, a line through the midpoint of a side 
 parallel to a consecutive side divides it into two congruent parts. 
 
 165. The median to any side of a triangle is less than one half the 
 sum of the other two sides. 
 
 166. The midpoints of two opposite sides of any quadrilateral and 
 of its diagonals are the vertices of a parallelogram. What if the 
 given figure is a parallelogram ? 
 
 167. A line drawn from A of triangle ABC, through the mid- 
 point of the median to CA, to K on BC, cuts off one third of BC. 
 
 168. If three parallels are drawn from the ends and the midpoint 
 of a sect to meet a given line, the parallel from the midpoint is half 
 the sum of the parallels from the ends. 
 
 169. If equal sects are cut off on one leg of an isosceles triangle 
 and the other leg extended through the base, the line joining the 
 
 points obtained is bisected by the base. 
 
 170. A line from a point on a side of an equilateral triangle to a 
 point twice as far from the common vertex on another side is per- 
 pendicular to one of the sides of tlve triangle. 
 
 171. A line from the vertex of the right angle of a right triangle 
 to the middle of the hypotenuse divides the triangle into two isosceles 
 triangles. 
 
 172. The bisectors of the two interior adjacent angles formed by 
 a transversal with one of two parallels cut off equal sects from the 
 transversal on the other parallel. 
 
 173. If two consecutive sides of a parallelogram are extended, 
 not from the common vertex, their own lengths, the line joining their 
 extremities passes through the fourth vertex. 
 
86 RECTILINEAR FIGURES 
 
 174. One angle of a triangle is double another; the included side 
 is constant, but the angles change. Find the locus of the intersection 
 of the bisector of the larger angle with the opposite side. 
 
 175- If the circumcenters of the triangles at the vertices, formed 
 by joining the midpoints of the sides of a triangle, are joined to the 
 midpoints of the two sides which help form each triangle, an equi- 
 lateral hexagon (not always convex) is formed. 
 
 176. In triangle ABC, if YX joins the midpoints of the sides BC 
 and CA, the incenters 7 and /', of triangles ABC and XCY, and C 
 are collinear, and CI' = II'. 
 
 177. In the figure of 176, if S and S f are the circumcenters, ST 
 is parallel to SI, and equals one half SI. 
 
 NOTE. Compare Exercises 109, 110, 112, 113, 176, 177, and note 
 that the facts proved of each corresponding pair of points are the 
 same. Upon what one fact about the figure do all these proofs 
 depend ? 
 
BOOK II. CIECLES 
 
 SECTION I. DEFINITIONS 
 
 160. Circles. The limited portion of a plane bounded 
 by a closed line, every point of which is equidistant from 
 a point within called the center, is a circle. The closed 
 line that bounds a circle is called its circumference; the 
 term "circumference" is often used for the length of the 
 circumference, the context usually showing which meaning 
 is to be used. The amount of surface in a circle is called 
 the area of the circle. In the advanced subjects of Mathe- 
 matics, the word " circle " is used for the circumference, 
 and some books on Geometry are now using " circle " in 
 that sense. The meaning is usually clear, and the pupil 
 should not be confused by either use. Some of the exer- 
 cises will be worded in each way, although in the text the 
 definitions will be adhered to. 
 
 *Jl circumference is a curve. 
 
 For if any part were straight, it could not have all its 
 points equidistant from the center ( 102). 
 
 *A circle has but one center. 
 
 For if it had two, they would both be midpoints of the 
 sect through them with its ends on the circumference. 
 
 A circle is named by the letter at its center, unless that 
 would cause ambiguity ; as, Oo would mean the circle 
 having its center at O. 
 
 A part of a circumference is called an arc. 
 
 87 
 
88 CIRCLES 
 
 161. Radii, Diameters, Chords. A line from the center of 
 a circle to any point on the circumference is called a radius. 
 
 A line joining two points of 
 a circumference is called a 
 chord; a chord through the 
 center is called a diameter. 
 
 OB is a radius, AB is a diam- 
 eter, CD is a chord. 
 
 The parts of the circumfer- 
 ence, BKA, ACDB, are arcs, and 
 are named, in the direction 
 contrary to that taken by the 
 hands of a clock, BA, AB. 
 
 * Radii of the same circle are equal ( 160). 
 
 * Diameters of the same circle are equal. 
 
 162. A point is within a circle, on the circumference, or 
 outside the circle, according as its distance from the center 
 is less than, equal to, or greater than, a radius. 
 
 163. Equality of Circles. If two circles have equal 
 radii, they are congruent, for if superimposed they coin- 
 cide. If two circles are equivalent, they are congruent, 
 for if, when superimposed with the centers coinciding, any 
 point of the circumference of one fell within the other, 
 all points of that circumference would fall within the 
 other, and the circle would be a part of the other, and 
 therefore smaller. 
 
 On this account, equal will be used for circles that are 
 equivalent, or congruent, or have equal radii, for there is 
 no distinction between these cases. 
 
 From the foregoing it is evident that but one circum- 
 ference can be drawn with a given center and a given 
 radius, 
 
DEFINITIONS 89 
 
 A circle is sometimes supposed to be rotated about its 
 center, so that any particular point of the circle takes a 
 different position. The circle as a whole would evidently 
 be in the same position as before rotation, but chords, arcs, 
 radii, etc., would have changed position. This can be 
 used to superimpose one figure in the circle on the place 
 occupied by another figure. (In this case, one figure 
 would be supposed to rotate, while the other would 
 remain in the same position.) 
 
 *164. Through three points not in the same straight 
 line, one, and but one circumference can be drawn 
 ( 152). Points in the same straight line are called 
 collinear. 
 
 165. Inscribed and Circumscribed Figures. A circum- 
 ference that passes through the vertices of a polygon is 
 said to be circumscribed about the polygon, and the poly- 
 gon is said to be inscribed in the circle. 
 
 A circumference that lies within a polygon and touches 
 each of its sides in one point only is said to be inscribed 
 in the polygon, and the polygon is said to be circumscribed 
 about the circle. 
 
 164 might have been stated One, and but one, cir- 
 cumference can be circumscribed about a given triangle. 
 
 166. One, and but one, circumference can: be inscribed 
 in a given triangle. 
 
 See 154. Note (by 155) that three circumferences 
 can be drawn outside a triangle, touching the sides each in 
 one point. Such circumferences are said to be escribed to 
 the triangle. 
 
 167. Intersection of Lines and Circumferences. A line 
 that meets a circumference at one point only is called 
 a tangent to the circle ; one that meets the circumference 
 
90 CIRCLES 
 
 at two points is called a secant. The point where a tan- 
 gent meets a circumference is called the point of tangency, 
 or point of contact. 
 
 (1) *A line perpendicular to a radius at its intersection 
 with the circumference is a tangent. For all other points 
 on the line are at a distance from the center greater than 
 the radius. 
 
 (2) * A line which meets a circumference at a point, 
 and' is not perpendicular to the radius to that point, is a 
 secant. 
 
 For a perpendicular can be drawn to it, and a second 
 oblique can be found equal to the radius, thus showing 
 a second point of intersection with the circle ( 113). 
 
 From the closed-figure intersection axiom, and the pre- 
 ceding statement, it follows that (3) a line at less than a 
 radius distance from the center of a circle that is, a line 
 through a point within a circle is a secant. 
 
 (4) *A tangent to a circle is perpendicular to the 
 radius drawn to the point of contact. Follows from (2). 
 
 (5) *The perpendicular to a tangent at the point of con- 
 tact passes through the center of the circle. 
 
 (6) *A line at more than a radius distance from the 
 center of a circle does not meet the circumference. 
 
 168. Center Lines. The line through the centers of 
 two circles is called their center line ; that part of it be- 
 tween the centers is called their center sect. 
 
 169. Length of the center sect dependent on the position 
 of the two circles in regard to each other. 
 
 (1) *// two circumferences do not meet, their center 
 sect is greater than the sum, or less than the difference, of 
 the radii, according as the center of one is outside, or in- 
 side, the other. 
 
DEFINITIONS 91 
 
 (2) *If two circumferences meet on their center line, 
 the center sect equals the sum, or the difference, of the 
 radii, according as the center of one circle is outside, or 
 inside, tJw other. 
 
 (3) * If two circumferences meet at a point not on tlwir 
 center line, the center sect is less than the sum, and greater 
 than the difference, of the radii. 
 
 If the figures are drawn, the reasons for the statements 
 will be found to depend upon the inequality axiom, and 
 the length relations between the sum or difference of two 
 sides of a triangle and the third side. 
 
 170. Intersection of Two Circumferences. (Converses 
 
 of 169.) 
 
 *If the center sect of two circles 
 
 (1) is greater than the sum of the radii, the circumfer- 
 ences do not meet, and lie outside each other. 
 
 (2) is less than the difference of the radii, the circum- 
 ferences do not meet, and one circle is within the otJwr. 
 
 (3) equals tlie sum of the radii, the circumferences 
 meet on the center line, and are outside of each other. 
 
 (4) equals the difference of the radii, the circumfer- 
 ences meet on the center line, and one circle is inside the 
 other. 
 
 (5) is less than the sum and greater than the difference 
 of the radii, the circumferences meet at a point not on the 
 center line. 
 
 These statements are used in all cases where it is neces- 
 sary to show whether or not two circumferences meet. 
 
 tfoxE. There are five values for the center sect: less than the 
 difference of the radii, equal to that difference, between the difference 
 and the sum, equal to the sum, and greater than the sum. The pupil 
 should understand the results from each of these lengths. 
 
92 CIRCLES 
 
 171. *Two circumferences that meet at a point not on 
 their center line, intersect twice. 
 
 If the circum- 
 ferences of Os O 
 and o f meet at P, 
 a second triangle 
 OO'Q could be con- 
 structed congruent 
 to OO'P, by drawing 
 angles on the other Q 
 
 side of O0 r equal respectively to Z o'OP and Z oo f P. 
 Why does this show that Q is a second point of inter- 
 section ? 
 
 NOTE. To show that two circumferences intersect twice it is 
 necessary to show. 
 
 (1) the center sect less than the sum of the radii. 
 
 (2) the center sect greater than the difference of the radii. 
 
 172. Common Chord. If two circumferences meet twice, 
 the line joining the points of intersection of the circles is 
 called their common chord. 
 
 *If two circumferences intersect twice, their center line 
 is the perpendicular bisector of their common chord. 
 Use equidistance. 
 
 173. Tangency of Circles. Two circumferences that 
 meet at one point only are said to be tangent to each other, 
 or to touch each other. The point where they meet is 
 called the point of tangency, or the point of contact. The 
 circles also are sometimes spoken of as tangent to each 
 other. 
 
 *If two circumferences meet at a point on their center 
 line, they are tangent to each other. If they meet at a 
 
ORAL AND REVIEW QUESTIONS 93 
 
 second point on the center line, they are identical ; if at 
 a second point which is not on the center line, the center 
 sect would have two different lengths, which is impos- 
 sible [ 169 (2), (3)]. 
 
 *Two circles tangent to the same line at the same point 
 are tangent to each other. 
 
 174. The shortest sect from a given point to a circum- 
 ference is a sect of the line from that point through the 
 center. Why ? Prove it when the point is inside and 
 when it is outside. 
 
 175. ORAL AND REVIEW QUESTIONS 
 
 If the radii of two circles are 10 ft. and 6 ft. long, tell what is 
 known about the position of the circles if the center sect is (1) 12 ft. 
 long, (2) 16 ft. long, (3) 4 ft. long, (4) 2 ft. long, (5) 20 ft. long. 
 Why is a circumference a curve? Why has it but one center? How 
 many points not in a straight line determine a circumference? Can 
 a circumference always be drawn through four points? Can it always 
 be drawn tangent to three lines? When are circles tangent? When 
 is a line tangent to a circle? What is known about the center line 
 of two intersecting circumferences and their common chord? What 
 relation is there between the distance of a line from the center of a 
 circle and the intersection of the line and circumference. If it is 
 necessary to draw two circles which are tangent externally (i.e. out- 
 side of each other), how long must the center sect be made? If it 
 is necessary to have their circumferences meet twice? 
 
SECTION II. CONSTRUCTIONS 
 
 176. Postulates. In plane geometry, the use of two 
 instruments for construction is allowed, or postulated. 
 These instruments are the straight edge (or ruler, except 
 that no measurements may be taken with it) and the 
 compass. 
 
 The postulates that allow the use of these instruments 
 are: 
 
 (1) A straight line may be drawn from any one point to 
 any other point. 
 
 (2) A sect may be produced to any length in that line. 
 
 (3) A circumference may be described with any point as 
 a center and any sect as a radius. 
 
 177. Constructions. In the theorems of Book I, aux- 
 iliary lines have been added to the given figure, but only 
 as representations of existing lines about which it was 
 necessary to reason in order to establish the proof. The 
 question of the accuracy of the drawing of these lines 
 had no bearing on the truth of the theorem, for the 
 reasoning was entirely about the figures that the lines 
 were taken to represent. In the problems that are to 
 be done, the proof consists in showing that the figure 
 constructed is within the limits of accuracy of the 
 instruments used a correctly drawn figure according 
 to the requirements of the proposition. The theorem 
 proves a fact ; the construction makes a required figure, 
 then proves the correctness of the method of construction. 
 
 94 
 
CONSTRUCTIONS 95 
 
 178. Analysis of a Construction. To discover the 
 method of drawing a required construction, first draw 
 the completed figure as accurately as possible, without 
 actually constructing it. By the study of the com- 
 pleted figure, attempt to find what lines must be con- 
 structed in order to make the figure in such a way 
 that it can be shown to be the required one. This 
 is really working backwards from the completed figure 
 in an attempt to find upon what it is based. Hav- 
 ing analyzed the figure, draw the lines found neces- 
 sary, and so build up a figure that can be proved to be 
 correct. 
 
 Do not neglect the classification method, for it will 
 make the analysis method unnecessary in many cases. 
 For example, a perpendicular can be obtained by finding 
 two points equidistant from the ends of the sect to which 
 the line is to be perpendicular. 
 
 179. Construction Uses of the Circumference. The cir- 
 cumference is the locus of points at a certain fixed distance 
 from a fixed point. The definition shows this fact, and it 
 is of great importance, both in proofs, and in construction 
 work. The two following hints will show common uses 
 of the circumference. 
 
 To draw a line of a given length from a given point, 
 use the compass to draw a circumference around the given 
 point, with the given sect as a radius. This is called 
 describing a circumference with a certain center and 
 radius. 
 
 To construct a point equidistant from two given points, 
 use each of the given points as a center, and use the same 
 radius. What relation must hold between the center sect 
 and the radii? 
 
96 
 
 CIRCLES 
 
 178. Construct an equilateral triangle on a given sect as a side. 
 
 179. Construct a triangle having its sides equal to three given 
 sects. Is it always possible? 
 
 180. Construct an angle equal to two thirds of a right angle. 
 
 181. Construct an isosceles triangle on a given base, having each 
 leg twice the length of the base. 
 
 182. Construct a regular hexagon on a given sect as a side. 
 
 183. Construct three equal circles, each tangent to the other two. 
 184- Construct seven equal circles, six of them surrounding the 
 
 seventh and each tangent to three others. 
 
 185. Construct two sects, given their sum and their difference. 
 
 186. Construct a sect, given a second sect and the sum of the two 
 sects. 
 
 187. Construct a sect, given a second sect and the difference of 
 the two sects. 
 
 180. CONSTRUCTION I. To draw the perpendicular bi- 
 sector of a given sect. 
 
 ANALYSIS 
 
 Two points equidistant from the ends of the given 
 sect will determine the perpendicular bisector. 
 
 PROOF 
 
 B 
 
CONSTRUCTIONS 
 
 97 
 
 Given. 
 Required. 
 
 Construction. I. 
 
 II, 
 
 III. 
 
 To prove. 
 Proof. 
 
 II. 
 
 AB. 
 
 To draw the JL bisector of AB. 
 
 With center A, and radius AB (or any 
 radius > \ AB) describe a circum- 
 ference. With center B and the 
 same radius describe a second cir- 
 cumference (circum. post.). 
 
 These circumferences meet at two 
 points. (Center sect is less than 
 the sum, but greater than the dif- 
 ference of the radii.) Call the 
 points of intersection P and Q. 
 
 Draw PQ (line post.). 
 
 PQ _L AB at its midpoint. 
 PA = PB, QA = QB (radii of equal 
 circles). 
 
 . . P and Q are equidistant from the 
 ends of AB, and PQ _L AB at its 
 midpoint M. (A line through two 
 points equidistant from the ends 
 of a sect is the perpendicular bi- 
 sector of that sect.) 
 
 188. Find by construction the point equidistant from three given 
 non-collinear points. 
 
 189. Describe a circumference having its center in a given circle, 
 and passing through two given points. 
 
 190. Draw a line through a given point, so that it will be equi- 
 distant from two other given points. 
 
 191. Construct an isosceles triangle, given the base and the length 
 of the altitude. 
 
 SMITH'S SYL. PL. GEOM. 7 
 
98 CIRCLES 
 
 181. CONST. II. To draw a perpendicular to a given 
 line through a given point 
 
 (1) in the line ; 
 
 (2) outside the line. 
 Use equidistance. 
 
 192. Construct an isosceles triangle, given the altitude and a leg. 
 
 193. Construct a triangle, given 
 
 (1) Two sides and the altitude to the third side. 
 
 (2) The base, one of the sects cut off on the base by the altitude, 
 another side. 
 
 (3) The base, its median, and its altitude. 
 
 194- Construct a rectangle, given a side and a diagonal. 
 
 195. Given an angle and a point, draw a line through the point, 
 so as to make equal angles with the arms of the given angle. 
 
 196. Construct the point in a given line, such that lines from two 
 given points on the same side of the given line to the point found 
 may make equal angles with the given line. 
 
 182. CONST. III. To draw a perpendicular to a given 
 sect at the end, without extending the sect. 
 
 Let the sect be a leg of the required right triangle, 
 and use the fact known about the midpoint of the hypote- 
 nuse. 
 
 183. CONST. IV. To bisect a given angle. 
 
 Draw the lines so that the two parts of the angle may 
 be proved to be equal ; that is, use angles equal. 
 
 197. Construct the points that are equidistant from three lines 
 that are neither concurrent nor all parallel. 
 
 198. Construct an angle equal to one half a right angle ; to one 
 third a right angle. 
 
 NOTE. It is impossible to trisect any given angle with no instru- 
 ments except the compass and ruler. Such exercises as the second 
 
CONSTRUCTIONS 99 
 
 part of 198 must be done by using some angle that is known to be 
 one third of a right angle, or that can be made into one third of a 
 right angle by bisection. 
 
 199. Construct a point in one side of a given triangle equidistant 
 from the other sides. 
 
 184. CONST. V. At a given point in a given line, to con- 
 struct a line making a given angle with the given line. 
 
 What is the simplest way to prove two angles in differ- 
 ent places equal ? 
 
 200. Construct a triangle, given two sides and the included angle. 
 
 201. Construct a triangle, given two angles and the included side. 
 
 202. Construct a triangle, given two angles and any side. 
 
 203. Construct an isosceles triangle, given the altitude and the 
 vertex angle. 
 
 204. Construct an isosceles triangle, given the altitude and a base 
 angle. 
 
 205. Construct a right triangle, given the perimeter and either leg. 
 
 185. CONST. VI. Through a given point to draw a par- 
 allel to a given line. 
 
 206. From a point outside a given line, draw a line making a 
 given angle with that line. 
 
 207. Construct a triangle, given an altitude and two angles. 
 
 208. Construct a triangle, given an angle, one of the sides includ- 
 ing the angle, the median from the same vertex. (Use the midpoint 
 of the given side.) 
 
 209. Draw a sect terminated by the arms of a given angle, and 
 through a given point within the angle, so that the point will bisect 
 the sect. 
 
 210. Draw a line from a given point outside a given angle, 
 through the arms of the angle, so one of the arms will bisect the sect 
 from the point to the other arm. 
 
100 CIRCLES 
 
 211. Construct a rhombus, given 
 (a) A side and an altitude. 
 
 (6) The altitude and a diagonal. 
 
 (c) The sum of the altitude and a side, an angle. 
 
 186. CONST. VII. To divide a given sect into any num- 
 ber of equal parts. (Use a system of parallels.) 
 
 212. Divide a given sect so that one part of it equals twice the 
 other part; so that one part equals three fifths of the other. 
 
 218. Divide a~given sect so that the second of three parts equals 
 twice the first, the third equals three times the second. 
 
 214. Divide a given sect into four parts, so that the second equals 
 twice the first, the third equals three times the first, the fourth equals 
 four times the first. 
 
 187. Sectors and Segments. A sector of a circle is that 
 surface bounded by two radii and the arc between their 
 ends. Two radii evidently divide the circle into two 
 sectors. The order of lettering (reverse to the direction 
 taken by the hands of a clock) shows which sector is 
 meant. 
 
 'A segment of a circle is that surface bounded by a chord 
 and the arc between its ends. A chord evidently divides 
 the circle into two segments. Here again, the order of 
 lettering shows which one is meant. 
 
 188. Angles in a Circle. A central angle is the angle 
 between two radii. Any two radii form two ' central 
 angles, which are distinguished by the order of their 
 lettering. 
 
 An inscribed angle is an angle between two chords 
 that meet on the circumference. 
 
 The arc that lies between the arms of an inscribed 
 angle or a central a.ngle is said to be subtended by the 
 angle, and the angle is said to stand on the arc. An arc 
 is also said to be subtended by its chord. 
 
SECTION III. CIRCLE THEOREMS 
 
 189. Theorem I. In the same circle, or in equal circles, 
 if two central angles are equal, the arcs on which they 
 stand are equal also; and of two unequal angles, the 
 greater angle stands on the greater arc. 
 
 190. COR. 1. In the same circle, or in equal circles, if 
 two arcs are equal, the central angles standing on those 
 arcs are equal also ; and of two unequal arcs, the greater 
 has the greater central angle standing on it. 
 
 191. COR. 2. In the same circle, or in equal circles, 
 sectors having equal central angles are equal ; and of two 
 sectors having unequal central angles, that which has the 
 greater central angle is the greater. 
 
 192. COR. 3. A diameter bisects the circle and its 
 circumference. 
 
 215. An arc is one quarter of a circumference if its central angle 
 is a right angle, and conversely. 
 
 NOTE. One quarter of a circumference is called a quadrant of arc. 
 
 216. An arc is greater than, or less than, a quadrant, if its central 
 angle is greater than, or less than, a right angle. 
 
 193. Semicircle; Major and Minor Arcs; Complements, 
 Supplements, Explements. One half a circle is called a 
 semicircle, and one half a circumference is called a semicir- 
 cumference. An arc less than a semicircumference is 
 called a minor arc; one greater than a semicircumference 
 is called a major arc. 
 
 101 
 
102 CIRCLES 
 
 Two arcs whose sum is a quadrant are called comple- 
 ments of each other ; two whose sum is a semicircumfer- 
 ence are called supplements of each other ; two whose sum 
 is a circumference are called explements of each other. 
 
 Notice that the part of the surface of a circle cut off by 
 a diameter (a semicircle) is a segment, and also a sector. 
 
 217. If two arcs are equal, their complements are equal; their 
 supplements are equal ; their explements are equal. 
 
 194. Theorem II. In the same circle, or in equal cir- 
 cles, if two arcs are equal, they are subtended by equal 
 chords; and of two unequal minor arcs, the greater is sub- 
 tended by the longer chord. 
 
 195. COR. 1. In the same circle, or in equal circles, if 
 two chords are equal, they subtend equal minor and equal 
 major arcs ; and of two unequal chords, the longer sub- 
 tends the greater minor arc. 
 
 218. If two equal chords of a circle intersect, they include a pair 
 of equal arcs. 
 
 219. An inscribed equilateral polygon is regular. 
 
 220. An inscribed equilateral hexagon has each side equal to a 
 radius. 
 
 221. If a chord equals a radius, its arc is one sixth of the circum- 
 ference. 
 
 196. Theorem III. A diameter that is perpendicular 
 to a chord bisects the chord and its subtended arcs. 
 
 197. COR. 1. A diameter that bisects a chord is per- 
 pendicular to the chord. 
 
 198. COR. 2. The perpendicular bisector of a chord 
 passes through the center of the circle. 
 
 222. The perpendicular from the center of a circle to a side of an 
 inscribed equilateral triangle equals one half the radius. 
 
THEOREMS 103 
 
 223. The sects of any line intercepted between the circumferences 
 of two circles that have the same center (concentric circles') are equal. 
 
 224. Find the locus of the center of a circumference that passes 
 through two given points. 
 
 225. Find the chord through a given point in a circle, and bisected 
 by that point. 
 
 226. Find the locus of the midpoints of parallel chords. 
 
 199. Theorem IV. In the same circle, or in equal 
 circles, equal chords are equidistant from the center; and 
 of two unequal chords, the longer is nearer the center. 
 
 If the chords are placed so that they diverge from a 
 common point on the circumference, and the line joining 
 their centers is drawn, the proof can be easily found. Why 
 is this position a good one in which to place the chords ? . 
 
 200. COR. 1. In tlw same circle, or in equal circles, 
 chords that are equidistant from the center are equal; 
 and of two chords that are unequal distances from the 
 center, the one nearer the center is the longer. 
 
 201. COR. 2. ji diameter of a circle is greater than 
 any other chord. 
 
 227. If two sects from a given point to a given circumference are 
 equal, those lines are equidistant from the center of the circle ; and if 
 the lines intersect the circumference in two points each, the sects from 
 the given point to the second points on the circumference are also 
 equal. 
 
 228. If two equal chords intersect, the sects of one equal the sects 
 of the other. 
 
 229. Find the locus of the midpoints of the equal chords of a 
 circle. 
 
 230. If two equal chords meet at a point on a circumference, the 
 bisector of the angle formed by them is a diameter. 
 
 231 . If two circles are concentric, all chords of the larger which 
 are tangent to the smaller are equal. 
 
104 CIRCLES 
 
 202. Ratio, Proportion. The ratio of one magnitude to 
 another like magnitude is that number by which the sec- 
 ond magnitude must be multiplied to give the first magni- 
 tude. For example, the ratio of a line two inches long to 
 one four inches long is one half; the ratio of a line four 
 feet long to one two feet long is two. A ratio is evidently 
 a quotient. 
 
 If the ratio of one pair of magnitudes is equal to the 
 ratio of a second pair of magnitudes, the four magni- 
 tudes are said to form a proportion, and are said to be 
 proportional. The four magnitudes used in a proportion 
 are called terms. 
 
 WARNING. It needs four magnitudes to form a proportion. Do not 
 say that two things are proportional. 
 
 203. Measures: Commensurable, Incommensurable. The 
 
 measure of a magnitude is its ratio to a unit of the same 
 kind ; as, the measure of a line is its ratio to an inch, a 
 foot, or some other unit of length. The numerical measure 
 of a magnitude is its ratio to a unit, expressed numeric- 
 ally ; as, five feet. 
 
 Two magnitudes are commensurable, if some third magni- 
 tude is contained a whole number of times in each, with no 
 remainder. 
 
 Two magnitudes are incommensurable, if no third mag- 
 nitude is contained evenly in both. 
 
 The ratio of two like commensurable magnitudes is the 
 ratio of the numbers of times some third magnitude is 
 contained in them ; as, the ratio of a line six feet long to 
 one eleven feet long is six to eleven. 
 
 NOTE. Incommensurable quantities will be treated in the Appen- 
 dix, 348. 
 
THEOREMS 105 
 
 204. Theorem V. In the same circle, or in equal cir- 
 cles, central angles are proportional to the arcs on which 
 they stand. 
 
 The commensurable part of this theorem is all that is re- 
 quired at this time; that is, the pupil may assume that the 
 central angles are commensurable (have a common divisor), 
 and so show that the ratio obtained for them will be th'e 
 same as the ratio obtained for their arcs. Instead of 
 using numbers to show how many times the assumed com- 
 mon divisor will go into the angles, letters should be used, 
 so that the proof will be general. For the incommensur- 
 able discussion, see Appendix, 348. 
 
 205. COR. 1. In the same circle, or in equal circles, 
 sectors are proportional to their central angles (com. 
 case only). 
 
 206. Measurement of Central Angles. As a central 
 angle is shown by Theorem V to be the same part of a 
 perigon that its arc is of a circle, it is customary to say 
 that a central angle is measured by its arc. 
 
 The reason for this can be shown more clearly by the 
 following explanation of the common method of measuring 
 angles : 
 
 If a circumference is divided into 360 equal parts called 
 arc degrees, the radii joining the points of division to the 
 center will divide the perigon at the center into 360 equal 
 parts called degrees of angle; also, if each arc degree is 
 divided into 60 equal parts, called minutes, and each arc 
 minute into 60 equal parts, called seconds, the radii to 
 these points of division will form minutes, and seconds, of 
 angle. Then if any arc can be expressed in degrees, 
 minutes, and seconds, its central angle will evidently be 
 
106 CIRCLES 
 
 composed of the same numbers of degrees, minutes, and 
 seconds of arc; that is, the angle is measured by its arc. 
 
 Theorem V might then have been stated: A central 
 angle is measured by its arc. 
 
 207. Theorem VI. din angle inscribed in a circle is 
 measured by one half its arc. 
 
 To make this theorem complete, it must be proved for 
 an inscribed angle having 
 
 (1) one arm a diameter; 
 
 (2) one arm each side of the center; 
 
 (3) both arms the same side of the center. 
 Upon what must this theorem depend ? 
 
 208. COR. 1. In the same circle, or in equal circles, 
 inscribed angles on equal arcs are equal ; and of two in- 
 scribed angles on unequal arcs, that on the larger arc is 
 the larger angle, and conversely. 
 
 209. COR. 2. (1) An angle inscribed in a semicircle 
 is a right angle. (2) An angle inscribed on an arc 
 greater than, or less than, a semicircumference, is gre-ater 
 than, or less than, a right angle. 
 
 232. What would be the locus of the vertices of all triangles hav- 
 ing a fixed base and a constant vertex angle? 
 
 233. Prove that the bisectors of the vertex angles of all triangles 
 having the same base and equal vertex angles on the same side of 
 that base, are concurrent. 
 
 234- If a right triangle is inscribed in a circle, its hypotenuse is 
 the diameter of the circle. 
 
 235. A circumference described on any side of a triangle as a 
 diameter passes through the feet of the altitudes to the other sides. 
 In an isosceles triangle, it bisects the base if a leg is used as diameter. 
 
 236- A circumference described on the hypotenuse of a right tri- 
 angle as a diameter passes through the vertex of the right angle. 
 
THEOREMS 107 
 
 237. Circumferences described on two sides of a triangle as diam- 
 eters intersect on the third side. 
 
 238. Three consecutive sides of an inscribed quadrilateral subtend 
 arcs of 68, 97, 59, respectively. Find each angle of the quadrilat- 
 eral, and the angles between the diagonals. 
 
 239. Two angles of an inscribed triangle are 75, 95. Find the 
 arcs subtended by the sides. 
 
 210. Theorem VII. An angle formed by a tangent and 
 a chord of a circle is measured by one half the arc sub- 
 tended by the chord. Notice that there are two angles 
 between the chord and the tangent, and that each is meas- 
 ured by one half the arc on the same side of the chord as 
 the angle measured. 
 
 211. COR. 1. Tangents from the same point to a cir- 
 cumference are equal. 
 
 212. COR. 2. Theline joining the point of intersection 
 of two tangents to the center of tfo circle to which they are 
 tangent bisects the angles between the tangents, and the 
 central angle between the radii drawn to the points of 
 contact. 
 
 240. Tangents from a point a radius distant from a circle make 
 with their chord of contact an equilateral triangle. 
 
 241 The sum of two opposite sides of a circumscribed quadrilat- 
 eral equals the sum of the other two sides. 
 
 - 242. A parallelogram circumscribed about a circle is a rhombus. 
 243. If the vertices of a circumscribed quadrilateral are joined to 
 the center of the circle, the non-adjacent central angles formed are 
 supplemental. 
 
 244- The angle between two tangents to a circle is double the 
 angle between the chord of contact and the radius" to a point of 
 contact. 
 
 245. If two circles are tangent to each other, the common tangent 
 at their point of tangency bisects the other common tangents. 
 
108 CIRCLES 
 
 213. Theorem VIII. If two parallel lines meet a cir- 
 cumference, the arcs cut off between them are equal. 
 
 246. Lines through the points of intersection of two circles, parallel 
 to each other, are equal. 
 
 247- If the ends of two unequal parallel chords are all joined, the 
 points of intersection and the center of the circle are collinear. 
 
 248. If the ends of two equal arcs of a circle are all joined, two 
 of the lines are parallel, the other two cut off equal sects on each other, 
 
 214. Theorem IX. An angle formed by two chords 
 that intersect within a circle is measured by one half the 
 sum of the arcs intercepted by the arms of the angle. 
 
 215. Theorem X. An angle formed by two secants, two 
 tangents, or a secant and a tangent, that intersect outside 
 the circle, is measured by one half the difference of the 
 intercepted arcs. 
 
 NOTE. To complete the discussion of angles whose arms, or arms 
 extended, meet a circumference, the measurement of the supplements 
 of the angles mentioned in 207, 210, 214, and 215 should be 
 investigated. 
 
 249. An exterior angle of an inscribed quadrilateral equals the 
 opposite interior angle. 
 
 250. The sum of the angles inscribed in the segments of a circle 
 cut off by the sides of an inscribed quadrilateral equals three straight 
 
 251. The ends of two arcs of 120 and 60 are joined. How large 
 are the angles formed? (Two cases.) 
 
 252. The opposite sides of an inscribed quadrilateral, no two of 
 whose sides are parallel, are extended until they meet. Find the 
 angle between the bisectors of the angles formed. 
 
 253. If two chords do not intersect, and the midpoint of one minor 
 arc of one chord is joined to the ends of the other chord, the two 
 triangles formed are mutually equiangular (that is, the angles of one 
 are equal respectively to the angles of the other). 
 
THEOREMS 109 
 
 $54. If three angles have their arms pass through the ends of the 
 same chord of a circle, their vertices being on the same side of the 
 chord, one inside the circle, one on the circle, the other outside the circle, 
 the angle whose vertex is outside is the largest, the one whose vertex 
 is inside is the smallest. 
 
 216. Theorem XI. The opposite angles of an inscribed 
 quadrilateral are supplemental. 
 
 217. COR. 1. A parallelogram inscribed in a circle is 
 a rectangle. 
 
 218. COB. 2. If the opposite angles of a quadrilateral 
 are supplemental, the quadrilateral is inscriptible. 
 
 Through how many vertices can a circumference be 
 drawn ? What if the other fell within the circle ? outside 
 the circle ? 
 
 Four or more points that lie on the same circumference 
 are said to be concyclic. 
 
 255. A rectangle is inscriptible, and its diagonals intersect at the 
 center. 
 
 256. In an inscribed hexagon, the sum of three angles, no two 
 adjacent, equals the sum of the other three. What general statement 
 for polygons of an even number of sides can be made? 
 
 257. A circle described on the line joining the orthocenter of a 
 triangle to a vertex passes through the feet of two altitudes. 
 
SECTION IV. CONSTRUCTIONS DEPENDING ON CIRCLE 
 THEOREMS 
 
 219. CONST. VIII. To bisect a given arc. 
 
 258. Bisect an inscribed angle, without using its arms. 
 259- Draw a tangent to a given arc at a given pointj without 
 using the center of the circle. 
 
 220. CONST. IX. To find the center of a circle, given 
 any arc. 
 
 260. Find the center of a circle, given the position of a chord and 
 the length of the diameter. 
 
 261. Describe a circle, given the positions of two chords. 
 
 221. CONST. X. To draw a tangent from a given point 
 to a given circle. 
 
 262. Draw a chord of a given circle, through a given point, so that 
 it will have a given length. 
 
 263. Draw a line through a given point, so that its distance from 
 a given point may equal a given sect. 
 
 264- Construct a triangle, given the base, and the altitudes to the 
 other sides. 
 
 265. Construct a triangle, given the base, the altitude to the base, 
 and a second altitude. 
 
 222. CONST. XI. To construct a circumference, so that 
 a given angle will be inscribed on the arc subtended ly a 
 given chord. 
 
 What other kind of an angle is measured by the same 
 arc as an inscribed angle ? Can this angle be constructed, 
 given the chord in a fixed position ? How can the center 
 then be found ? 
 
 110 
 
CONSTRUCTIONS USING THEOREMS 111 
 
 This theorem is the foundation of many exercises on 
 loci. The most common way of using this as an example of 
 locus is : " Find the locus of the vertex of a given angle op- 
 posite a given fixed sect" 
 
 When a locus gives a circumference for its result, it is 
 because of one of two things : 
 
 (1) because the point in question is always at a fixed 
 distance from a given fixed point ; 
 
 (2) because a given angle is always opposite a given 
 fixed sect, the most common and most important case 
 being that of a right angle opposite the fixed sect, when 
 the locus is the circumference on the sect as diameter. 
 
 266. Construct a triangle, given the base, the vertex angle, the 
 median to the base. 
 
 267. Construct a triangle, given the base, the vertex angle, the 
 altitude. 
 
 268. Construct a triangle, given the base, another side, the angle 
 between the median to the base, and the third side. 
 
 269. Construct a triangle, given the base, the altitude to the base, 
 the angle between the median to the base, and a side. 
 
 270. Find the locus of the vertex of a triangle, given the base and 
 the vertex angle ; the base and the angle between the median and a 
 side. 
 
 271. Find the locus of the midpoints of all chords whose lines pass 
 through a given fixed point. 
 
 272. Find the locus of the midpoint of a sect of fixed length, 
 whose ends are on the arms of a given fixed right angle. 
 
 273. Construct a parallelogram, given a diagonal, an angle opposite, 
 the angle between the diagonals. 
 
 274. Construct a trapezoid, given the bases, a side, and the angle 
 between the diagonals. 
 
112 CIRCLES 
 
 223. SUMMARY OF THEOREMS AND COROLLARIES, BOOK II 
 
 (Numbers in parentheses refer to black-faced section numbers.) 
 I. CIRCUMFERENCES MEET TWICE, c. s. <sum radii and > ditf. 
 (170). 
 
 II. CIRCLES TANGENT. c. s. = sum radii or = diff. (170) ; 
 tangent same line (173). 
 
 III. LINE MEETS A CIRCUMFERENCE TWICE. Not radius (167). 
 
 IV. LINE TANGENT TO A CIRCLE. _L radius at end (167). 
 
 V. LINES EQUAL. Center 1 in e_L bisector common chord (172) ; 
 chords of equal arcs (194) ; diameter _L chord (106) ; = chords equi- 
 distant from center, and conversely (199, 200) ; tangents (211). 
 
 VI. LINES UNEQUAL. Chords of unequal arcs (194) ; ^ chords 
 unequal distances from the center, and conversely (199, 200); diam- 
 eter greatest chord (201). 
 
 VII. LINES PERPENDICULAR. Diameter bisecting chord (197); 
 Oinsc. in circle (217). 
 
 VIII. LINE PASSES THROUGH CENTER. _L bisector chord (198); 
 _L tangent at point of tangency (167). 
 
 IX. ANGLES EQUAL. Central angles having equal arcs (190); 
 inscribed angles on equal arcs (208) ; angles bisected by line from 
 center to intersection of tangents (212). 
 
 X. ANGLES UNEQUAL. Central angles on unequal arcs (190); 
 inscribed angles on unequal arcs (208, 209). 
 
 XI. ANGLES SUPPLEMENTARY. Opp. angles insc. quad. (216). 
 XII. ANGLES MEASURED. Central angles, by their arcs (204, 
 206). 
 
 (a) Angles with vertices outside the circle, by one half 
 
 the difference of the intercepted arcs (215). 
 (6) Angles with vertices on the circumference, in- 
 cluding inscribed and tangent and chord angles, 
 by one half the intercepted arc : an angle on a 
 semicircumference equals a right angle; one on 
 more, or less than, a semicircumference, is obtuse 
 or acute (207, 210, 209). 
 (c) Angles with vertices inside the circle, by one half 
 
 the sum of the intercepted arcs (214). 
 
 XIII. ARCS EQUAL. Equal chords (195) ; equal central angles 
 (189) ; J. bisector chord bisects arc (196) ; diam. bisects circumference 
 (192); equal insc. angles (208) ; between parallel lines (213). 
 
SUMMARY OF PROPOSITIONS 113 
 
 XIV. ARCS UNEQUAL. Unequal central angles (189) ; unequal 
 chords (195); unequal inscribed angles (208). 
 
 XV. SECTORS EQUAL. Equal central angles (191) ; diameter 
 bisects circle (192). 
 
 XVI. SECTORS UNEQUAL. Unequal central angles (191). 
 XVII. SECTORS PROPORTIONAL. To their central angles (205). 
 XVIII. QUADRILATERAL INSCRIPTIBLE. Opp. angles sup. (218); 
 
 CONSTRUCTIONS 
 
 XIX. LINE PERPENDICULAR. 
 
 (a) Bisector (180). 
 
 (b) At point in the line (181). 
 
 (c) From a point to a line (181). 
 
 (d) At the end of the sect, without extending (182) . 
 XX. BISECT ANGLE (183). 
 
 XXL ANGLE EQUAL TO A GIVEN ANGLE (184). 
 
 XXII. PARALLEL TO A GIVEN LINE (185). 
 
 XXIII. DIVIDE SECT INTO EQUAL PARTS (186). 
 
 XXIV. BISECT ARC (219). 
 
 XXV. FIND CENTER OF A GIVEN CIRCLE (220). 
 XXVI. TANGENT (221). 
 
 XXVII. CIRCUMFERENCE HAVING A GIVEN ANGLE OPPOSITE A 
 GIVEN FIXED SECT (222). 
 
 LOCI 
 
 XXVIII. POINTS EQUIDISTANT FROM A FIXED POINT (179). 
 
 XXIX. VERTEX OF A GIVEN ANGLE OPPOSITE A FIXED SECT 
 (222). (See XXVII.) 
 
 224. ORAL AND REVIEW QUESTIONS 
 
 Why is the intersection of the perpendicular bisectors of the sides 
 of a triangle called the circumcenter ? Why are the intersections of 
 the angle bisectors called the in- and ex-centers? If the center sect of 
 two circles is 10 ft. long, the radii being 8 ft. and 4 ft., what is known 
 about the meeting of circumferences ? For what length center sect, 
 with the same radii, would the circles be internally tangent ? When 
 is a line tangent to a circle? How can you construct a tangent at 
 a point on a circle? from an external point to a circle? How can 
 SMITH'S SYL. PL. GEOM. 8 
 
114 CIRCLES 
 
 you get a right angle opposite a given sect? any angle? Express this 
 as a locus. Of what is a circumference the locus? Define chord. 
 What special kind of chord is there? Which is the longest chord 
 of a circle? Why? WJien are arcs equal? central angles equal? 
 inscribed angles equal? How can you measure an angle with its 
 vertex inside the circle? its vertex on the circle? its vertex outside 
 the circle ? What kinds of angles have their vertices on the circumfer- 
 ence? outside the circle? What is known about the angles of an 
 inscribed quadrilateral? How would an exterior angle of the quadri- 
 lateral compare with the opposite interior angle? Why? What 
 principle underlies the construction of perpendiculars? Is it also 
 used for the proofs? Why does the perpendicular bisector of a chord 
 pass through the center? Give a second proof. What kind of 
 triangle is always formed by two radii and a chord? Can you tell 
 any proofs where this fact is of use? What is the key to drawing 
 lines parallel? to making equal angles? to dividing a sect into any 
 number of equal parts? How much of a circle is needed to find 
 its center? Are lines joining the ends of equal arcs always parallel? 
 What is known about all equal chords of a circle ? What is the locus 
 of the midpoints of all such chords? Upon what two characteristics 
 of a circle do the circle loci depend? How many points need to be 
 fixed in the first? in the second? Explain what is meant by a 
 central angle being measured by its arc. What is the shortest line 
 from a point to a circumference? Why? What relation has the 
 angle between two tangents to the angle between the radii to the 
 points of tangency ? What two ways are there of measuring the angle 
 between two tangents? If the arms of a central angle, an inscribed 
 angle, a tangent and chord angle, are all respectively parallel, how do 
 their arcs compare? Where does the bisector of an inscribed angle 
 meet the arc? If two tangents to a circle are parallel, what part of 
 the circumference is between them? 
 
 GENERAL EXERCISES 
 
 275. If two circles are tangent internally, and the radius of the 
 larger is the diameter of the smaller, any chord of the larger drawn 
 from the point of contact is bisected by the smaller. 
 
 276. Construct a rhombus, given the sum of the diagonals and an 
 angle. 
 
GENERAL EXERCISES 115 
 
 277. Construct a right triangle, given the hypotenuse and the sum 
 of the other sides. 
 
 278. An inscribed equiangular polygon of odd number of sides is 
 equilateral; of even number of sides, has each side equal to the 
 second from it. 
 
 279. The tangents at the vertices of an inscribed quadrilateral, of 
 which two opposite angles are right angles, form a trapezoid. 
 
 280. Construct a rectangle, given the difference between two con- 
 secutive sides and a diagonal. 
 
 281. If two circles intersect twice, the angle formed by joining a 
 point of intersection to the ends of a secant through the other point 
 of intersection is always the same. 
 
 282. If the feet of the altitudes of a triangle are joined, the angles 
 of the triangle formed (called the pedal triangle) are bisected by the 
 altitudes. 
 
 283. Construct a triangle, given the altitude, a base angle, the 
 sect, of the base between the foot of the altitude and the second 
 
 vertex. 
 
 284~ Construct a triangle, given the ba'se, its median, a base angle. 
 
 285. Construct a rectangle, given a diagonal and the angle between 
 the diagonals. 
 
 286. Construct a rectangle, given the sum of two consecutive sides, 
 and the angle between a side and a diagonal. 
 
 287. A circumscribed equiangular polygon is equilateral. 
 
 288. A circumscribed equilateral polygon of odd number of sides 
 is regular, and has its sides bisected by the points of contact. 
 
 289. A circumscribed equilateral polygon of even number of sides 
 has its alternate angles equal. 
 
 290. A circumscribed equilateral triangle has each altitude equal 
 to three radii. 
 
 291. Construct an isosceles trapezoid, given the longer base, the 
 sum of altitude and side, an angle. 
 
 292. Construct a parallelogram, given the sum of two consecutive 
 sides, an angle, and a diagonal, 
 
116 CIRCLES 
 
 293. Construct the common tangent to two given circles. (Draw 
 the tangent, the radii to the points of contact, see what parts of the 
 trapezoid formed are given, then try to construct it from those given 
 parts. Notice that four common tangents can be drawn in certain 
 cases.) 
 
 294. If two circles are internally tangent at P, and the chord AB 
 of the larger circle is tangent to the smaller circle at T, prove that PT 
 bisects the angle APB. 
 
 295. If triangle ABC is circumscribed about a circle 0, with CA 
 and A B tangent at fixed points T and T', and BC any third tangent, 
 then CA + AB BC is constant, and equals the diameter if Z.A is a 
 right angle. 
 
 296. If two parallel tangents are crossed by a third tangent, the 
 third tangent subtends a right angle at the center. 
 
 297. In 295, if the triangle is escribed, AB -f BC + CA is con- 
 stant. 
 
 298. Construct a trapezoid, given the four sides. 
 
 299. Construct a trapezoid, given the parallel bases and the diago- 
 nals. 
 
 300. Construct a square, given the sum of a diagonal and a side. 
 
 301. Construct a square, given the difference of a diagonal and a 
 side. 
 
 302. If the sum of two opposite sides of a quadrilateral equals the 
 sum of the other two sides, the quadrilateral can be circumscribed 
 about a circle. 
 
 303. If two circles intersect in two points, and diameters are drawn 
 from one of those points, the line joining the other ends of the diame- 
 ters is double the center sect, and passes through the other point of 
 intersection. 
 
 304. Through a point within a circle, draw "the chord which is 
 bisected by that point, and prove that it is the shortest chord through 
 the point. 
 
 305. Describe a circle tangent to two given lines, to one at a given 
 point. 
 
GENERAL EXERCISES 117 
 
 306. Describe a circle through a given point, touching a, given line 
 at a given point. 
 
 307. Describe a circle tangent to a given line at a given point, and 
 tangent to a given circle. (Two cases.) 
 
 308. In a circumscribed cross quadrilateral the difference of two 
 opposite sides equals the difference of the other two opposite sides. 
 
 309. Find the locus of the center of a circle which has a given 
 radius, and cuts a given circle at the ends of a movable diameter. 
 
 310. Find the locus of the extremities of tangents of a certain 
 length drawn to a given circle. 
 
 311. If perpendiculars are drawn from the ends of a moving diam- 
 eter to a fixed chord, the sum of those perpendiculars is constant. 
 
 312. Construct the bisector of an angle without using its vertex. 
 
 313. Construct a right triangle, given the two sects into which the 
 bisector of the right angle divides the hypotenuse. 
 
 314- Construct a right triangle having a given fixed hypotenuse, 
 and having the vertex of the right angle in a given fixed line. 
 
 315. Construct a right triangle, having a fixed hypotenuse, and 
 having its vertex at a given distance from a given point. 
 
 316. Construct a right triangle, having a fixed hypotenuse, and 
 having its vertex at a given distance from a given line. 
 
 317. Draw a circle with a given radius, tangent to two given circles. 
 
 318. Construct a right triangle, given the radius of the inscribed 
 circle and the altitude to the hypotenuse. 
 
 319. Draw a line at a given distance from two given points. 
 
 320. Construct a right triangle, given the radius of the inscribed 
 circle and the bisector of the right angle. 
 
 321. Construct a right triangle, given the radius of the circum- 
 scribed circle, and (a) an acute angle, (6) the difference of the acute 
 angles. 
 
 322. Construct a triangle, given the radius of the circumscribed 
 circle, and (a) two sides, (6) two angles, (c) the sects of the base made 
 by the altitude. 
 
118 CIRCLES 
 
 323. Describe a circle of given radius, tangent to two given lines. 
 
 324' Describe a circle tangent to two given lines at a given dis- 
 tance from their intersection. 
 
 325. Draw a right triangle, given the altitude and the bisector of 
 the right angle. 
 
 326. Construct a triangle, given a side, the bisector of the vertex 
 angle, the sect of the base between the foot of the altitude and the end 
 of the bisector of the vertex angle. 
 
 327. Construct a triangle, given the sum of the sides, two angles. 
 
 328. Construct a triangle, given the sum of two sides, and two 
 angles. 
 
 329. Construct a triangle, given the sum of two sides, one of the 
 opposite angles, the altitude to the third side. 
 
 330. Construct a triangle, given the altitude to the base, a base 
 angle, and the sect from that vertex to the foot of the bisector of the 
 vertex angle. 
 
 331. Draw a circle on one side of a triangle as a diameter, and so 
 show that the altitudes from the ends of that side make equal angles 
 with the other sides to which they are not perpendicular. 
 
 332. If two circles are tangent externally, and two secants are 
 drawn through their point of tangency, the chords joining the ends of 
 the secants are parallel. 
 
 NOTE. There are many other triangle constructions of the same 
 kind as those in the preceding set, and the pupil can make exercises 
 for himself, by taking any three parts which are independent (that is, 
 such that no one could be obtained from the other two) and trying to 
 form the triangle from those parts. Some of the triangle constructions 
 cannot be done until the fourth book has been studied. 
 
 Notice that when two circles are used, the center line is almost 
 always necessary to the figure, and that if the two circles intersect, or 
 are tangent, the common chord, or the common tangent at their 
 point of tangency, is very likely to be needed. 
 
BOOK III. EQUIVALENCE AND AREA 
 
 SECTION I. DEFINITIONS AND THEIR DISCUSSION; 
 FORMULAS 
 
 225. Equivalence. Two closed figures have been defined 
 as being equivalent when they contain the same amount 
 of surface ( 34). The question at once arises as to what 
 ways of proving figures equivalent are known. The dis- 
 tinction between the different kinds of equality, as shown 
 in 32-37, and the equality axioms ( 38), give the 
 foundation for this class ; the two following methods show 
 how these definitions and axioms can be used. 
 
 (1) Congruent figures are equivalent, and equivalent 
 figures added, subtracted, and multiplied or divided by 
 the same number, give equivalent results (although the 
 results need not be congruent). 
 
 (2) The whole equals the sum of all its parts, and the 
 other equality axioms can be applied to the equation 
 obtained. 
 
 The equivalence class must then depend on these two 
 methods : the equivalence of two figures usually on the 
 first, equivalence equations usually on the second. When 
 the first is used, the congruent figures are often added or 
 subtracted in different positions, thus forming new figures 
 which are equivalent, but not congruent. 
 
 226. Addition of Polygons. The addition of two poly- 
 gons is accomplished by placing the polygons entirely 
 outside of each other, as regards their surface, but with 
 some portion of the perimeter (either a point, or some one 
 
 119 
 
120 EQUIVALENCE AND AREA 
 
 or more sects) in common, and then taking the whole 
 figure thus formed. The common part of the boundary 
 line is considered as omitted, unless it is a point. It is 
 evident that this is simply the ordinary understanding of 
 a sum, the only difference being that the polygons have 
 to be placed in such a position that their sum can be 
 shown as a single figure. 
 
 227. Subtraction of Polygons. The subtraction of two 
 polygons is accomplished by placing the smaller polygon 
 entirely inside the larger, and then omitting the part 
 occupied by the smaller ; the remaining part of the larger 
 polygon is the difference. Here again the usual idea of 
 subtraction is employed, the figures being placed so that 
 the difference shows as a new figure. 
 
 228. Addition and Subtraction of Parallelograms. From 
 the definitions of addition and subtraction of polygons, 
 it can be seen that two parallelograms which have a side 
 and an angle of one equal to a side and an angle of the 
 other can be added or subtracted so as to give a new 
 parallelogram having the same side, and the same angle, 
 as the result. This applies to rectangles having one 
 equal side, and the fact can be put in formal statement as 
 follows : 
 
 Two rectangles having an equal side can be added so 
 as to form a rectangle having the equal side for one of 
 its sides, the sum of the two other length sides of the 
 given rectangles for its other side. Similarly for the 
 difference. 
 
 229. Rectangles and Squares. A rectangle having the 
 sides X and Y is spoken of as the n X, Y. This is per- 
 missible, since all rectangles having those sides are con- 
 
FORMULAS 121 
 
 gruent. Similarly, a square on the side X is spoken of 
 as the D X. 
 
 Remembering that all rectangles which have two sides 
 equal are congruent, it is easily seen that all that it is 
 necessary to do in order to multiply a rectangle, is to 
 multiply one side, for that makes a new rectangle, composed 
 of as many congruent rectangles as the number by which 
 the side was multiplied. This is most easily seen by 
 keeping in mind that multiplication is here strictly a kind 
 of addition, and that the line is multiplied by continuing 
 it until it contains the required number of equal parts. 
 In exactly the same way, a rectangle can be divided into 
 any number of equal parts by dividing one side into that 
 number of equal parts, and so forming a new rectangle, 
 having one side the required part of the given side. 
 
 NOTE. Dividing both sides of a rectangle, or multiplying both 
 sides, performs the operation on the rectangle ttcice. For example, 
 if each side of a rectangle is doubled, the rectangle is made four 
 times as large. 
 
 230. Formulas. By the use of the axiom, "the whole 
 equals the sum of all its parts," it is possible to obtain 
 many equations between rectangles and squares, the fol- 
 lowing of which are the most important: 
 
 *(1) The square on the sum of two sects is equivalent 
 to the sum of their squares plus twice their rectangle. 
 
 Take two given sects, 4raw the square on their sum, 
 and. see if it contains the required parts. 
 
 *(2) The square on twice a sect is equivalent to four 
 times the square on the sect. 
 
 *(3) The square on the difference of two sects is equiv- 
 alent to the sum of their squares, less twice their rec- 
 tangle. 
 
122 EQUIVALENCE AND AREA 
 
 Try to add the squares of the sects, and subtract the 
 two rectangles in such a manner that a square will be left. 
 
 *(4) The difference of the squares on two sects equals a 
 rectangle, having one side equal to the sum of the sects, 
 the other side equal to the difference of the sects. 
 
 Subtract the smaller square, then make one rectangle 
 out of the remainder. 
 
 These geometric formulas, which deal entirely with the 
 surfaces of the figures, correspond very closely to certain 
 algebraic formulas ; the only differences between the two 
 are that where " square on a sect" appears in the Geometry, 
 "square of a number" appears in Algebra, and where 
 "rectangle of two sects" appears in Geometry, "product 
 of two numbers " appears in Algebra. This correspond- 
 ence follows throughout ; wherever an algebraic formula 
 is entirely of the second degree, the corresponding geo- 
 metric statement concerning squares and rectangles is also 
 true. The reason for this correspondence will be evident 
 from the third theorem and its corollaries. 
 
 333. Find the formula for the square on the sum of three sects. 
 Try to show what general formula can be obtained. 
 
 334. What does the square on three times a sect equal? What 
 general formula is there? 
 
SECTION II. THEOREMS 
 
 231. Theorem I. Parallelograms (or triangles) on the 
 same base, or on equal bases, and between the same paral- 
 lels, are equivalent. 
 
 Do not add, for this method does not work for all posi- 
 tions of the parallelograms. 
 
 232. COR. 1. Parallelograms (or triangles) on the 
 same base, or on equal bases, and having equal altitudes, 
 are equivalent. 
 
 Place their bases on the same line, then show what ? 
 
 335. Triangles having two sides equal, and the included angles 
 supplemental, are equivalent. 
 
 336. Any two medians of a triangle form, with the side from 
 whose ends they are drawn, and the halves of the sides to which they 
 are drawn, two equivalent triangles. 
 
 337. Any median divides the triangle into two equivalent triangles. 
 
 838. The three medians of a triangle divide it into six equivalent 
 triangles. 
 
 NOTE. In the following theorems, it is the surface of the figure 
 which is being considered. 
 
 233. Theorem II. Two rectangles having equal alti- 
 tudes are proportional to their bases. (Commensurable 
 case only.) 
 
 234. COR. 1. Two rectangles having equal bases are 
 proportional to their altitudes. 
 
 235. Theorem III. The ratio betiveen two rectangles 
 
 equals the product of the ratio of their bases by tJie ratio 
 
 123 
 
124 EQUIVALENCE AND AREA 
 
 of their altitudes. Can their ratio be found at once, if 
 they have no equal dimension? To what could the ratio 
 of each be found ? 
 
 In this theorem, and, in fact, in most of the theorems 
 of this book, it is convenient to use a single letter to rep- 
 resent each side of the figure rather than to name it by its 
 end points. 
 
 236. Area. The area of a figure has been defined as 
 the amount of surface inclosed, and this area is usually 
 stated in' terms of some area unit. It is customary to call 
 the number of length, or linear units in a sect, its length; 
 and to call the number of area units in a surface its area. 
 The unit of length is any convenient sect, such as a sect 
 one inch long, or one yard long ; the unit of area is a 
 square having a linear unit for a side. 
 
 In dealing with area, it is necessary to use like units ; 
 that is, if the inch is used as a length unit, then the square 
 inch must be used as the area unit, and so for all units. 
 
 To find the area of a figure, then, it is necessary to take 
 the ratio of the figure to a square unit ; while to find the 
 length of a sect, it is necessary to take its ratio to a linear 
 unit. While the terms "area" and "length" will be 
 used in this way, the student should bear in mind that it 
 is the number of units which is being used, both in area 
 and in length. 
 
 237. Theorem IV. The area of a rectangle equals the 
 product of its base by its altitude. Read 236 carefully, 
 then take the ratio of the given rectangle to a unit of area. 
 
 238. COR. 1. The area of any parallelogram equals 
 the product of its base by its altitude; the area of any tri- 
 angle equals one half the product of its base by its altitude. 
 
THEOREMS 125 
 
 239. COR. 2. The area of a trapezoid equals one half 
 the sum of its bases times its altitude. 
 
 339. The area of a trapezoid equals its median (the line joining 
 the midpoints of the legs) times its altitude. 
 
 340. The area of a circumscribed polygon equals one half its 
 perimeter times the radius of the circle. 
 
 341. A line through the midpoint of the median of a trapezoid, 
 cutting the bases, divides the trapezoid into two equivalent parts. 
 
 240. Theorem V. The square on tlxe hypotenuse of a 
 right triangle is equivalent to the sum of the squares on 
 the other two sides. 
 
 This theorem is the famous Pythagorean proposition, so 
 called because it is said to have been discovered by 
 Pythagoras. It is probably the most important theorem 
 in Geometry as far as numerical applications are concerned. 
 It has been claimed that about 350 proofs for this theorem 
 have been discovered ; it is certain, at any rate, that the 
 number of proofs is very great, and as they are all applica- 
 tions of the equivalence relations that have been given 
 in this third book, the student ought not to have great 
 difficulty in finding several. The only thing to be avoided 
 is the use of area in its arithmetical sense, for this theorem 
 must be proved in a strictly geometrical sense, and so 
 the area relations (that is, the formulas for the area of 
 rectangles, triangles, etc.) must not be used. Many of 
 the proofs are obtained by adding and subtracting equals, 
 so as to form the square on the hypotenuse in one case, 
 and the sum of the squares on the other sides in the other 
 case. Probably the best of these proofs is obtained by 
 drawing the square on the hypotenuse (so as not to include 
 the triangle) and completing the rectangle through its 
 vertices, using the legs continued as two sides of the con- 
 
126 EQUIVALENCE AND AREA 
 
 structed rectangle. This new figure can be shown to have 
 two different equivalence values ; one is obtained from the 
 fact that it can be proved to be the square on the sum of 
 the legs, the other by the axiom of the whole. These two 
 values will quickly give the required relation between the 
 squares. 
 
 241. COB. 1. The square on one leg of aright triangle 
 is equivalent to the square on the hypotenuse less the 
 square on the other leg. 
 
 242. Numerical Applications. Since the area of a figure 
 is the number of units in the figure, many of the equiva- 
 lence relations can be used for numerical applications. 
 For example, the area of a triangle in square units can be 
 obtained when the lengths of base and altitude are known 
 in length units ; as, if the altitude of a triangle is 4 ft. 
 long, its base 10 ft. long, then the area of the triangle is 
 4 x 10 -v- 2 (or 20) sq. ft. This fact applied to the right 
 triangle formulas enables one to find any side of a right 
 triangle if. the other two sides are known ; as, in a right tri- 
 angle having legs 3 ft. long, and 4 ft. long, respectively, 
 the hypotenuse is 5 ft. long, for the area of a square is 
 evidently the square of the number of units in one side, 
 and 3 2 + 4 2 =5 2 . Similarly, if the hypotenuse is 13 ft. 
 long, and one leg is 12 ft. long, 13 2 - 12 2 = 169 - 144 = 25 
 = 5 2 ; showing that the other leg must be 5 ft. long. 
 
 342. Find the area of a parallelogram of base 12 ft. and altitude 
 14 ft.; of a triangle having the same base and altitude. 
 
 343. In order that a triangle should be equivalent to a parallelo- 
 gram of the same base, what must be true of their altitudes? 
 
 344. Find the area of a trapezoid of bases 10 and 18 ft., altitude 
 lift. 
 
 345. If one leg of a right triangle is 3 ft. long, the other 8 ft. long, 
 find the length of the median to the side 8. 
 
THEOREMS 
 
 127 
 
 846. If two chords of a circle are perpendicular to each other, the 
 sum of the squares on the sects formed is equivalent to the square on 
 the diameter. 
 
 347. In right triangle ABC, C being the right angle, BD is drawn 
 to D on AC. Prove that the square on BD plus the square on AC is 
 Equivalent to the square on AB plus the square on CD. 
 
 348. In acute triangle ABC, let AB = 7, BC = S, CV1=9; call 
 the altitude to 7, li, and the sect from A to the foot of the altitude, p. 
 Then two right triangles are formed, and an equation can be written 
 for each. Write these two equations, and solve them for h and p. 
 Find the area of ABC. Draw the median to AB and find its length 
 from the new right triangle formed. 
 
 NOTE. This exercise shows that the use of the theorem for the 
 squares on the sides of a right triangle will give, when the lengths of 
 the sides of a triangle are known, 
 
 (1) the length of each altitude ; 
 
 (2) the length of each median ; 
 
 (3) the lengths of the sects of the sides made by the altitudes ; 
 
 (4) the area, using the altitude found. 
 
 243. Projections. The foot of the perpendicular from a 
 point to a line is called the projection of that point on the 
 line. If both ends of a sect are projected on a line, the 
 sect between the projections of the ends of the given sect 
 is called the projection of the sect on the line. 
 
 P' 
 
 B' R 
 
 S 
 
 The projection of P is P', of AB is A f B f , of US is RS r , of 
 KL is K f L f . 
 
128 EQUIVALENCE AND AREA 
 
 244. Theorem VI. The square on a side of an oblique 
 triangle equals the sum of the squares on the other two 
 sides 
 
 ( 1 ) plus twice the rectangle of either of those sides and 
 the projection of the other side on its line, if the opposite 
 angle is obtuse ; 
 
 (2) less twice the rectangle of either of those sides and 
 the projection of the other side on its line, if the opposite 
 angle is acute. 
 
 The acute angle case will have to be proved in both the 
 acute and the obtuse angled triangle. In what kind of a 
 triangle is the square on a side known ? Then what must 
 be done ? 
 
 This theorem is also adaptable to numerical work, and 
 if the letters #, J, c, stand for the sides of a triangle, with 
 a 1 for the projection of a on 5, and b' for the projection of 
 b on , the formula obtained would be (P = a 2 + b 2 . 2a'b 
 (or 2 >'). Note that if three of these parts are known, 
 the others can be found, and that the kind of angle oppo- 
 site a side can be told by comparing its square with the 
 sum of the other squares. 
 
 849. If one side of a triangle is 5, the projection of this side on a 
 second side is 2, and the other sect of that side is 4, find the third 
 side. 
 
 350. Does Th. VI apply to right triangles also? See what the 
 projection would be in the case of a right triangle. Can you make a 
 general statement for the square on any side of any triangle, regard- 
 ing the projection as negative when it takes off from a side, positive 
 when it adds to the side ? 
 
 351. In triangle of sides a, &, c, find the lengths of the sects of c 
 made by the altitude to c. 
 
 352. Using Th. VI, find the projection of b on c, if a = 7, b = 10, 
 
THEOREMS 129 
 
 245. Area and Altitude Formulas. It is evident that 
 the altitude and the area of a triangle can be found when 
 the three sides are known, either by the right-triangle 
 method, or by the use of the formulas for the square on a 
 side in an oblique triangle. If the altitude and area are 
 worked out for a triangle of sides a, 5, <?, a general for- 
 mula can be obtained, which can be applied very rapidly. 
 If h is the altitude to <?, b f the projection of b on c, show 
 that 
 
 (l)^*^ 2 -^-" 2 ). 
 (2) h = 
 
 = V(2 bo + b 2 + <? - a 2 ) (2 be - b 2 - & + a 2 ), 
 
 or h = -V(a + b + C )(-a + b + c) (a-l + c) (a + 5-c). 
 
 (3) Calling the sum of the sides 2 s, and substituting, 
 area = Vs (s a) (s b) (s c). 
 
 APPLICATION. If the sides are 7, 8, 9, then s (or half 
 the sum) = 12. Subtracting each of the sides, the three 
 remainders, 5, 4, 3, are obtained. Then 
 
 area = Vl2 x 5 x 4 x 3 = 12V5. 
 
 To find any altitude by this method, divide the area by 
 half the side to which the altitude is drawn ; for example, 
 
 24 
 
 the altitude to 5 is -r-V57 
 o 
 
 246. The Half Equilateral Triangle ; the Isosceles Right 
 Triangle. There are two right triangles in which the 
 relative lengths of the sides can be told at once, and these 
 
 SMITH'S SYL. PL. GEOM. 9 
 
130 EQUIVALENCE AND AREA 
 
 triangles must be used when angles are given to find the 
 lengths of sides, or sides are given to find the size of 
 angles. 
 
 If an altitude is drawn in an equilateral triangle, two 
 right triangles are formed, called half equilateral trian- 
 gles. The altitude meets the opposite side at its mid- 
 point, so one leg is half the hypotenuse; the other leg is 
 found by the squares on the sides. The altitude is always 
 half the side of the equilateral triangle times the square root 
 of three; and the area of the equilateral triangle is the square 
 of half the side times the square root of three. If the side 
 is 10, the altitude is 5V3^ and the area is 25 V& This 
 triangle has angles equal to one third a straight angle, 
 and one sixth a straight angle ; or, in degrees, of 60 and 
 30. 
 
 In an isosceles right triangle, the hypotenuse is a leg 
 times the square root of two. (Why ?) The angles here 
 are one half a right angle, or 45. 
 
 853. If the sides of a triangle are 29, 31, 50, find the area and the 
 altitude to 50, by the area formula. 
 
 354. How long is the altitude of an equilateral triangle of side 14? 
 
 355. If the altitude of an equilateral triangle is 20 V3, find the 
 side. 
 
 356. If the area of an equilateral triangle is 49 V3, find the altitude. 
 
 357. If the altitude of an equilateral triangle is 10, find the side ; 
 if the area is 10. 
 
 358. If a leg of an isosceles right triangle is 5, find the hypotenuse. 
 
 359. If two thirds of one side of an equilateral triangle, and one 
 third of a second side, are cut off from the same vertex, the line join- 
 ing those points is perpendicular to the side from which one third is 
 cut off. Does this need to be one third ? What general statement 
 can be made? 
 
 360. If the vertex angle of an isosceles triangle is 30, and each 
 leg is 10, find the base ; if each leg is a. 
 
THEOREMS 131 
 
 361. In a triangle, having an angle of 120, show that the square 
 on the opposite side equals the sum of the squares on the other two 
 sides, plus the rectangle of those sides. 
 
 362. In a triangle having one angle equal to one third of a straight 
 angle, the square on the opposite side equals the sum of the squares 
 on the other two sides less the rectangle of those sides. 
 
 247. Theorem VII. The sum of the squares on any 
 two sides of a triangle is equivalent to twice the square 
 on one half the third side, plus twice the square on the 
 median to that side. 
 
 The previous theorems allow us to take but one square 
 at a time ; therefore, for many purposes this theorem is 
 better, for it takes two squares at once. It has a second 
 purpose, for from it one can find the median in terms of 
 the sides, or one side in terms of the other sides and a 
 median. 
 
 248. COR. 1. The difference of any two sides of a 
 triangle is equivalent to twice the rectangle of the third 
 side and the projection of its median on it. 
 
 363. Find the median to 5 in the triangle of sides 4, 5, 6. 
 
 364' Find the projection of the median to 5 on that line, if the 
 sides of the triangle are 4, 5, 6. 
 
 365. If the median to a side of a triangle is 7, the other sides are 
 8 and 9, find the side. 
 
 366. Show that the sum of the squares on the sides of any quadri- 
 lateral equals the sum of the squares on the diagonals, plus four times 
 the square on the line joining their midpoints. What does this 
 formula become if the quadrilateral is a parallelogram? 
 
 367. The sum of the squares on the diagonals of a trapezoid is 
 equivalent to the sum of the squares on the legs, plus twice the rec- 
 tangle of the bases. 
 
 368. In the triangle of sides a, b, c, find the lengths of the three 
 medians. 
 
132 EQUIVALENCE AND AREA 
 
 369. Prove that the sum of the squares on the medians of a tri- 
 angle equals three fourths the sum of the squares on the sides. 
 
 370. If two points in a diameter of a circle are equidistant from 
 the center, the sum of the squares of their distances from a moving 
 point on the circle is constant. 
 
 NOTE. The number of numerical exercises founded on the theo- 
 rems of this book is practically unlimited. The pupil can familiarize 
 himself with many of the different kinds by taking each one of the 
 formulas and assuming values for all but one of the sects involved, 
 and solving for that one. For example, if the sides of a triangle are 
 
 a, by c, and the median to c is m, then a 2 + 6 2 = 2 f -J + 2m 2 ; assume 
 
 values for (1) a, ft, c, (2) a, ft, w, (3) a, c, m, (4) ft, c, m; thus using 
 this formula in all possible ways. 
 
 In assuming lengths for lines in a triangle, care must be taken 
 that the figure formed is possible ; for instance, a = 3, b = 7, c = 10 is 
 not possible. Why ? 
 
 The best way to master a formula is to investigate each way it can 
 be used, so that no question about it can be entirely unexpected. 
 
SECTION III. CONSTRUCTIONS 
 
 249. CONST. I. (1) To construct a polygon equivalent 
 to a given polygon, but having one less side. 
 
 (2) To construct a triangle equivalent to a given polygon. 
 
 Note that any two consecutive sides add to the surface 
 only a triangular surface, which could equally well have 
 been contained by one of these sides and the extension of 
 one of the sides already used ; in other words, it is a waste 
 of a side to use two extra sides to inclose an additional trian- 
 gular surface. 
 
 250. CONST. II. To construct a rectangle equivalent to 
 a given triangle. 
 
 251. CONST. III. To construct a square equivalent to 
 the sum of two given squares. 
 
 252. COR. 1. To construct a square equivalent to 
 the difference of two given squares. 
 
 253. COR. 2. To construct a square equivalent to 
 the sum of any number of given squares. 
 
 254. COR. 3. To construct a square equivalent to 
 any whole number of times a given square. 
 
 371. Given the diagonals of two squares, construct the diagonal of 
 the square equal to the sum of the given squares. 
 
 372. Construct a parallelogram equivalent to a given triangle, 
 having its diagonals equal to two sides of the triangle. 
 
 373. Construct a rhombus equivalent to a given triangle, having 
 one of its sides equal to a side of the triangle. 
 
 133 
 
134 EQUIVALENCE AND AREA 
 
 874- Construct a rectangle equivalent to any given polygon. 
 
 375. Construct a parallelogram equivalent to a given triangle, 
 having one side equal to a side of the triangle, and having a given 
 angle. 
 
 376. Construct a parallelogram equivalent to a given triangle, 
 having its two sides equal to two of the sides of the triangle. 
 
 255. CONST. IV. To construct a square equivalent to a 
 given rectangle. 
 
 Any of the formulas that give equivalences between 
 squares and rectangles could be used to make this con- 
 struction, for the given rectangle could represent the rec- 
 tangle in the formula, and the squares could then be added 
 or subtracted as stated in the formula. The formula for 
 the difference of two squares [ 230, (4)] is probably the 
 best, although the difference of the squares on two sides 
 of a triangle is almost, if not quite, as good. If the first 
 way is used, what must the sides of the rectangle be 
 called ? Try to find as many constructions for this as 
 possible, for they are excellent practice in the use of the 
 equivalence formulas; there are at least seven methods 
 that can be easily seen from the work done in this book. 
 
 377. Construct a rectangle equivalent to a given square, if one 
 side of the rectangle is given. 
 
 378. Construct a rectangle equivalent to a given square, if the 
 perimeter of the rectangle is given. 
 
 379. Construct a rectangle equivalent to a given square, if the 
 diagonal of the rectangle is given. 
 
 256, SUMMARY OF THEOREMS AND COROLLARIES, BOOK III 
 
 (Numbers in parentheses refer to black-faced section numbers.) 
 I. PARALLELOGRAMS EQUIVALENT. = bases, bet. || s (231) ; = bases 
 
 and = alt. (232); prod, bases and altitudes = (238). 
 
 IT. TRIANGLES EQUIVALENT. = bases, bet. Us (231) ; bases and 
 
 = alt. (232) ; prod, bases and alt. = (238). 
 
SUMMARY OF PROPOSITIONS 135 
 
 TIL COMPARISON OF RECTANGLES. Having = bases, proportional 
 to alt. (233) ; having = alt., proportional to bases (234) ; proportional 
 to the prod, of base and alt. (235). 
 
 IV. AREAS. (1) rectangle = base x alt. (237). 
 
 (2) Parallelogram = base x alt. (238). 
 
 (3) Triangle = J base x alt. (238) ; equilateral, = the square of 
 half the side, times >/3 (246). 
 
 (4) Trapezoid \ sum of bases x alt. (239). 
 
 V. EQUIVALENCE FORMULAS INVOLVING SECTS NOT SIDES OF A 
 TRIANGLE. (1) Square on sum of two sects equivalent to the sum of 
 their squares plus twice their rectangle (230). 
 
 (2) Square on the difference of two sects equivalent to the sum of 
 their squares less twice their rectangle (230). 
 
 (3) Square on twice a sect equivalent to four squares on the 
 sect (230). 
 
 (4) Difference of the squares on two sects equivalent to the rec- 
 tangle of their sum and their difference (230). 
 
 VI. SQUARE ON A SIDE OF A TRIANGLE. General Theorem: 
 The square on any side of any triangle is equivalent to the sum of the 
 squares on the other sides plus twice the rectangle of one of those 
 sides and the projection of the other on its line, that projection being 
 considered positive or negative, according as it adds to, or takes away 
 from, the side on which it is projected. 
 
 (1) In a right triangle, square on hypotenuse equivalent to the 
 sum of the squares on the other two sides ; square on a leg equivalent 
 to the difference of the squares on the hypotenuse and the other leg 
 (240, 241). 
 
 (2) Square on a side opposite an obtuse angle equivalent to the 
 sum of the squares on the other sides plus twice the rectangle of one 
 of those sides and the projection of the other on it (244). 
 
 (3) Square on a side opposite an acute angle equivalent to the sum 
 of the squares on the other sides less twice the rectangle of one of 
 those sides and the projection of the other on it (244). 
 
 VII. SQUARES ON Two SIDES OF A TRIANGLE. (1) Sum equiva- 
 lent to twice the square on half the third side plus twice the square 
 on the median to that side (247). 
 
 (2) Difference equivalent to twice the rectangle of the third side 
 and the projection of its median on it (248). 
 
136 EQUIVALENCE AND AREA 
 
 CONSTRUCTIONS 
 
 VIII. TRIANGLE, equivalent to a polygon (249). 
 IX. RECTANGLE, equivalent to a triangle (250). 
 X. SQUARE, equivalent to 
 
 (1) sum of two squares (251). 
 
 (2) difference of two squares (252). 
 
 (3) sum of any number of squares (253). 
 
 (4) any number times a square (254). 
 
 (5) a rectangle (255). 
 
 257. ORAL AND REVIEW QUESTIONS 
 
 When is a square equivalent to the sum of two squares ? to the dif- 
 ference? State the formula for the square on a side of a triangle 
 opposite an obtuse angle ; an acute angle. Does this apply to right 
 triangles? Why? What is the method for finding the altitude of 
 an equilateral triangle from the side ? the area from the side ? If the 
 side is 24, find the altitude; the area. If the altitude is 19 \/3, what 
 is the side? If the area is 9V3, what is the side? If the side is 
 8V3, how far is it from the centroid to a vertex? How can the area 
 of a triangle be most easily found from the sides? What kind of a 
 triangle is one of sides 3, 4, 5? of sides 9, 7, 5? of sides 7, 8, 9? In 
 what way can the sides of a trapezoid be used to find the area ? Can 
 the sides alone of a parallelogram be used to find the area t Why ? 
 What is needed? What angles can be found from the lengths of 
 sides of triangles? in what kinds of triangles? What are the rela- 
 tive lengths of the sides of a triangle having angles of 90, 60, 30? of 
 90, 45, 45? If a leg of an isosceles triangle were known, and the 
 vertex angle was 45, how could one find the base ? if the vertex 
 angle were 30 ? In constructing a triangle equivalent to a parallelo- 
 gram, or vice versa, what relation between the surfaces should be kept 
 in mind? Explain in a few words the general method used in reduc- 
 ing a polygon to a triangle. If it were necessary to draw a square 
 equivalent to a given polygon, through what steps would it be neces- 
 sary to go ? a square equivalent to a triangle ? to a parallelogram ? to 
 the sum of five squares? If a rectangle were to be constructed equiv- 
 alent to a given square-, and having a given side, what would it be 
 best to call the side of the rectangle? Why? What three theorems 
 
GENERAL EXERCISES 137 
 
 give ratios between rectangles ? Are all likely to be used for farther 
 work, or have some served their purpose ? Why was an extra figure 
 constructed in one of them? What does area mean? When a 
 theorem asks for the ratio of two like magnitudes for the first time, 
 what must be done with those magnitudes? Then in terms of what 
 is the ratio expressed? (NOTE. The commensurable case only is 
 being considered.) Which formula do you regard as likely to be 
 used the most frequently of all those in this book? Why? Explain 
 two methods of finding the projection of one side of a triangle on 
 another, when the sides are known. Explain two methods of finding 
 the area of a triangle from the sides. Which is shorter? Which 
 can be most readily used if formulas have been forgotten ? What two 
 methods of finding the median from the sides do you think of? 
 Which is shorter? Which can be used if formulas are forgotten? 
 What is the construction for finding the square on a side of an oblique 
 triangle? Why? What is the formula for the area of a rectangle? 
 a square? a parallelogram? a triangle? an equilateral triangle? a 
 trapezoid ? 
 
 GENERAL EXERCISES 
 
 380. If a square field and a rectangular field have the same area, 
 which would require the longer fence? 
 
 381. Find the locus of a point such that the squares of its dis- 
 tances from two fixed points shall have a constant sum. (To prove 
 a quantity constant, it is usually best to show that it equals another 
 quantity which is constant.) 
 
 382. Find the locus of a point such that the squares of its distances 
 from two fixed points shall have a constant difference. 
 
 383. If any point within a rectangle is joined to the vertices, the 
 sum of the squares of the sects to two opposite vertices equals the 
 sum of the squares of the sects to the other vertices. 
 
 384' The sum of tne squares on the diagonals of any quadrilateral 
 is equivalent to twice the sum of the squares on the sects joining the 
 midpoints of the opposite sides. 
 
 385. Construct a rectangle equivalent to a given square, if the 
 difference of the sides of the rectangle is given. 
 
 386. Find the three altitudes of the triangle of sides a, a, b. 
 
138 EQUIVALENCE AND AREA 
 
 387. The vertex angle of an isosceles triangle is 45. " Find the 
 base, if the leg equals 10 ; if it equals a. 
 
 388. Find the area of a trapezoid of bases 12, 18, legs 4, 5. 
 
 389. Find the area of a trapezoid of bases a, c, legs b, d. 
 
 390. Find the side of an equilateral triangle of altitude 12. 
 
 391. If the altitude of an equilateral triangle is 10 in., what are 
 the radii of the inscribed and circumscribed circles? if the side is 
 14 in.? 
 
 392. Compare the area of a square and a rhombus of the same 
 sides, but with a 30 angle; a 60 angle; a 45 angle. 
 
 393. The sides of a parallelogram are 12 and 8. Find the area 
 if an angle is 30; if 60; if 45. 
 
 394. Construct a triangle having a given angle, a given side of a 
 certain triangle, and equivalent to that triangle. 
 
 395. Construct a parallelogram equivalent to a given triangle, 
 having its diagonals equal to two sides of the triangle. 
 
 396. Find the diagonals of the trapezoid in 388. 
 
BOOK IV. SIMILAR FIGURES. 
 PROPORTIONS 
 
 SECTION I. RATIO AND PROPORTION 
 
 258. Terms of a Proportion. Ratio and proportion have 
 already been defined ( 202). The numerator of the first 
 ratio and the denominator of the second ratio (the first 
 and last terms) are called extremes, the other two terms 
 are called means. The numerators of the two ratios are 
 called antecedents, the denominators are called consequents. 
 The last term of a proportion is called the fourth propor- 
 tional. 
 
 259. Mean Proportion. If the means of a proportion 
 are equal, the proportion is called a mean proportion ; the 
 mean is called the mean proportional, or simply the mean, 
 and the last term is called the third proportional. 
 
 A continued proportion is a series of equal ratios in 
 which any two successive ratios form a mean proportion. 
 
 260. Proportion Proofs. In proving the following 
 proportion theorems, the letters a, b, c, d, may be used to 
 represent like geometrical magnitudes expressed in terms 
 of a common unit of measure. Then, while the ratio may 
 be of two sects, or of two surfaces, that ratio takes a 
 numerical form of expression ; as v the ratio of two sects 
 might be f . 
 
 139 
 
140 PROPORTIONS 
 
 These proofs must, of course, depend upon the equality 
 axioms. 
 
 261. Composition ; Division. Four quantities are said 
 to be in proportion by composition when the antecedents 
 become the sums of the terms of the ratios ; as, 
 
 b d 
 
 Four quantities are said to be in proportion by division 
 when the antecedents become the differences of the terms 
 
 of the ratios; as, a-^^c^d,^ 
 
 b d 
 
 If the ratios of the sums to the differences are used, 
 the quantities are said to be in proportion by composition 
 
 a + b c -+- d 
 
 and division; as, 
 
 a b c d 
 
 262. Equimultiples. Equimultiples of two quantities 
 are the results obtained by multiplying those quantities by 
 the same number. 
 
 NOTE. In proving the theorems in Ratio and Proportion, use the 
 fractional form for the ratios, and use the proportion as an equation ;* 
 
 as, - = - In this way the equality axioms can be used more easily. 
 b d 
 
 THEOREMS 
 
 263. Theorem I. If four quantities are in proportion, 
 the product of their means equals the product of their 
 extremes. 
 
 264. COB. 1. In a mean proportion, the square of the 
 mean equals the product of the extremes. 
 
 265. COR. 2. The value of any term of a proportion 
 can be expressed in the other terms of the proportion. 
 
RATIO AND PROPORTION 141 
 
 266. Theorem II. If the product of two quantities 
 equals the product of two other quantities, either pair can 
 be made the means, and the other pair the extremes, of a 
 proportion. 
 
 267. COR. 1. If four quantities are in proportion, they 
 are in proportion in any way in which the means of the 
 given proportion are either both means, or both extremes, 
 in the new proportion. 
 
 397. Any two sides of a triangle are inversely proportional to the 
 altitudes drawn to them. (Inversely proportional means that one of 
 the ratios is inverted.) Use area formula. 
 
 268. Theorem III. Four quantities which are in pro- 
 portion are in proportion by composition. 
 
 269. Theorem IV. Four quantities which are in pro- 
 portion are in proportion by division. 
 
 270. Theorem V. Four quantities which are in pro- 
 portion are in proportion by composition and division. 
 
 271. Theorem VI. If four quantities are in proportion, 
 equimultiples of the antecedents are in proportion to equi- 
 multiples of the consequents. 
 
 272. Theorem VII. If four quantities are in propor- 
 tion, like powers of those quantities are in proportion. 
 
 273. Theorem VIII. In a series of equal ratios, the 
 ratio of the sum of the antecedents to the sum of the con- 
 sequents equals any of the given ratios. 
 
 274. SUMMARY 
 
 If a proportion is written in fractional form, 
 and the four terms are considered as forming the vertices of a rec- 
 tangle, its terms will be in proportion in any order in which the pairs 
 are taken along opposite sides of the rectangle, in the same direction ; 
 
142 PROPORTIONS 
 
 that is, both to the right, both down, etc. For example, starting from 
 
 d and going toward the left, - = -. 
 
 b a 
 
 Notice that the proportion never goes along a diagonal, as from 
 a to d\ this can be kept in mind because the diagonals form a multi- 
 plication sign, and diagonal terms can be multiplied but not divided. 
 
 The same method applies to composition and to division, the 
 same operations being applied to opposite sides to form the new 
 antecedents, and to form the new consequents. For example, starting 
 
 from b and going up, ^_^ = *L. 
 b a d c 
 
 This method can be extended to apply to equimultiples, to powers, 
 and to composition and division forms involving equimultiples aiid 
 powers, and in this way it serves as a test of the correctness of pro- 
 portion forms. 
 
 This is not a proof, but simply a test for correctness, which also 
 acts as a help to the memory by combining all the most important 
 proportion forms in one rule. 
 
SECTION II. PROPORTIONAL SECTS 
 
 275. Pencil of Lines. Lines that are concurrent are 
 spoken of as a pencil of lines. In the same way a number 
 of lines that are all parallel are spoken of as a pencil of 
 parallels. 
 
 276. Theorem I. If aline is cut by a pencil of parallels, 
 its sects are proportional to the sects of any other line cut 
 by the same pencil of parallels, including as a special case, 
 
 A line parallel to the base of a triangle cuts the sides, 
 or the sides extended, so that the sects are proportional. 
 (Com. case.) 
 
 277.' COR. 1. A line parallel to the base of a triangle 
 has the same ratio to the base as the lengths it cuts off on 
 the other sides (from their common vertex) have to the 
 whole sides. 
 
 278. COR. 2. If parallel lines are cut by a pencil of 
 lines, the sects cut off on the parallels are proportional. 
 
 398. In the quadrilateral A BCD, having angle B and angle D right 
 angles, PE and PF are drawn from P in A C perpendicular to EC and 
 DA , respectively. Prove that BE : E C = A F : FD. 
 
 399. If EC, of triangle ABC, is extended to A r , and A Y is cut off 
 on AB equal to CX, then XY is cut by CA in the ratio AB:BC. 
 
 400. The diagonals of a trapezoid cut each other proportionally, 
 and their sects are proportional to the bases. 
 
 401. If aline cuts the sides of a triangle, extended if necessary, 
 the product of three non-consecutive sects equals the product of the 
 other three sects. 
 
 143 
 
144 PROPORTIONS 
 
 279. Points. Cutting a Sect ; Harmonic Division. A point 
 on a sect, or on the sect extended, is said to cut the sect 
 in the ratio of its distances from the ends of the sect; as, 
 if P is on AB, or on AB extended, the ratio in which it 
 cuts AB is PA : PB. If the point is three fourths as far 
 from A as from JS, it cuts AB in the ratio 3 : 4, and this is 
 the same whether P is in the sect itself or not. 
 
 If two points cut the same sect in the same ratio, one 
 internally, the other externally, they are said to cut the 
 sect harmonically. The equal ratios must, of course, be 
 taken from corresponding ends of the sect ; as, if P and Q 
 cut AB harmonically, PA : PB = QA : QB. 
 
 Notice that a sect cannot be cut externally in the ratio 
 1, for if P is in the extension of AB, PA cannot equal PB. 
 
 280. Theorem II. A sect can be cut in the same ratio 
 internally by but one point, and externally by but one 
 point. 
 
 402. The interior common tangents of two circles (those between 
 the circles) meet the center line at the same point. 
 
 403. The exterior common tangents of two circles meet the center 
 line at the same point. 
 
 404. The interior and exterior common tangents to two circles cut 
 the center sect harmonically. 
 
 405. A line through the ends of two parallel radii of two circles 
 meets the center line at the same point as the common tangents. 
 
 NOTE. The points where the common tangents meet the center 
 line are called the inverse, and direct centers of similitude. 
 
 406. If two points cut a sect harmonically, they include a second 
 sect, which is cut harmonically by the ends of the first sect. 
 
 281. Theorem III. A line that cuts two sides of a 
 triangle proportionally is parallel to the third side. 
 
 407. If a sect joins the one third points of two sides of a triangle 
 (taken from their common vertex), what part of the third side is it? 
 
SECTION III. SIMILAR FIGURES 
 
 282. Similar Figures. Polygons are said to be similar 
 if their corresponding angles are equal and their corre- 
 sponding sides are proportional. See also Appendix, 349. 
 
 There are now three things which can be proved about 
 polygons : that they are congruent, equivalent, or similar. 
 Equivalent means of the same size (as regards surface), 
 similar means of the same shape, while congruent includes 
 both size and shape. Notice that the sign for congruent 
 is composed of the equivalent sign and the similar sign. 
 These facts are not definitions of the words, but serve to 
 show the distinction in meaning in a somewhat different 
 
 light. 
 
 *283. Polygons similar to the same polygon are similar 
 to each other. 
 
 *284. Perimeters of similar polygons are proportional to 
 any pair of corresponding sides. 
 
 *285. Regular polygons of the same number of sides are 
 similar. 
 
 408. If two similar polygons are placed with a pair of correspond- 
 ing sides parallel (the polygons lying on the same sides of those lines), 
 the lines through the corresponding pairs of vertices will form a pen- 
 cil, which is cut proportionally by the vertices of the polygon. 
 
 409. If two polygons lie in a pencil of lines, and their vertices cut 
 the lines proportionally, the polygons are similar. 
 
 SMITH'S SYL. PL. GEOM. 10 145 
 
146 SIMILAR FIGURES 
 
 286. Theorem IV. Two triangles are similar if two 
 angles of one are equal to two angles of the other. 
 
 410. All lines through the point of tangency of two circles are cut 
 proportionally by the circles. 
 
 411. If AB is a diameter of a circle, CD a chord perpendicular to 
 AB, then any chord A Y cutting CD at A' has the product AX x A Y 
 constant. 
 
 412. The product of two sides of a triangle equals the product of 
 the altitude to the third side by the diameter of the circumscribed 
 circle. 
 
 413. The product of two sides of a triangle equals the product of 
 the bisector of the included angle by the sect of thac bisecting line 
 from the v,ertex of the angle to the circumscribed circle. 
 
 287. Theorem V. Two triangles are similar if two sides 
 of one are proportional to two sides of the other, and the 
 included angles are equal. 
 
 414. In any triangle, the orthocenter, the centroid, and the circum- 
 center, lie in a straight line, and the distance between the first two is 
 double the distance between the second two. 
 
 288. Theorem VI. Two triangles are similar if the 
 sides of one are proportional to the sides of the other. 
 
 If one had the included angle equal to that of the other, 
 the triangles would be similar ; cut it off equal, and show 
 that the triangle obtained is the same triangle as that 
 given. 
 
 289. Similar Triangles. Similar triangles are obtained 
 much as congruent triangles were obtained, namely, by 
 three parts. The angles are given equal, but the sides 
 are given proportional instead of equal. Any three parts 
 will do, except two sides proportional and a pair of angles, 
 not included, equal. The following table will serve to 
 show the relation between congruence and similarity. 
 
SIMILAR FIGURES 
 
 147 
 
 GIVEN PARTS IF SIDES ABE EQUAL IF SIDES ARE PROPORTIONAL 
 
 3 sides : 
 
 2 sides, angle 
 
 included, 
 not included, 
 
 1 side, 2 angles 
 
 3 angles : 
 
 figures congruent. 
 
 figures congruent ; 
 figures congruent if 
 angle is right or ob- 
 tuse; or if acute, with 
 the greater side oppo- 
 site. Otherwise, other 
 angles not included 
 are supplemental, 
 figures congruent. 
 
 figures not necessa- 
 rily congruent. 
 
 figures similar. 
 
 figures similar, 
 same conditions as 
 for congruent. 
 
 figures similar ; side not 
 needed as two sects cannot 
 be proportional ; same as 3 
 angles, 
 figures similar. 
 
 Similar figures, and especially similar triangles, have 
 many practical applications, such as in finding the height 
 of trees and buildings, and the distances between objects. 
 The principle involved is that if two triangles are similar, 
 any one side, of two pairs of corresponding sides, can be 
 found if the other three are known. Some of the follow- 
 ing exercises illustrate this method. 
 
 415. How tall is the tree that casts a shadow 50 ft. long at the 
 same time that a pole 6 ft. long casts a shadow 8 ft. long ? 
 
 416. A building casts a shadow 64 ft. long. A projection on one 
 corner of the building that is found to be 8 ft. from the ground 
 casts a shadow 9 ft. long. How high is the building? 
 
 417. It is necessary to find the distance from B to an inaccessible 
 point X. A line BA perpendicular to the sighted line BX is laid off 
 146 ft. long, and at K, 60 ft. from A on A B, a perpendicular to AB 
 is erected, meeting the sighted line AX at L. If KL is found to be 
 30 ft. long, how long is BX'! 
 
148 SIMILAR FIGURES 
 
 418. Two triangles are similar if the sides of one are parallel to 
 the sides of the other. 
 
 419. Two triangles are similar if the sides of one are perpendicu- 
 lar to the sides of the other. 
 
 420. If two chords of a circle cut each other, the four sects are 
 proportional. 
 
 421' If two secants of a circle intersect, the four sects from the 
 vertex to the circle are proportional. 
 
 NOTE. There are now two principal methods by which to find 
 four sects proportional. What are they? 
 
 290. Theorem VII. Tlw areas of triangles, or of paral- 
 lelograms, having an angle of one equal to an angle of the 
 other, have the same ratio as the product of the sides in- 
 cluding that angle. 
 
 291. COR. 1. The areas of similar triangles have the 
 same ratio as the squares of their corresponding sides. 
 
 292. COR. 2. If two triangles that have an angle of 
 one equal to an angle of the other are equivalent, the prod- 
 uct of the sides including the angle in one equals the 
 product of the sides including the angle in tlxe other, 
 and conversely. 
 
 422. If BC, of triangle ABC, is 8, and CA is 6, how long must 
 CFbe, so that a line from F, on BC, to X, the two thirds point of 
 CA, will cut the triangle into equivalent parts. 
 
 423. A line from the midpoint of a side of a triangle must go to 
 what point on a second side to form an equivalent triangle? 
 
 424. In a right triangle of legs 3 and 4 ft., the hypotenuse is 
 extended 10 ft. How long must a leg be extended at the same 
 vertex so that the line joining the extremities of the extensions will 
 form a triangle double the given triangle? (Two cases.) 
 
 4%5. If similar triangles are drawn upon the sides of a right 
 triangle as corresponding sides, the triangle on the hypotenuse equals 
 the sum of the other triangles. 
 
SIMILAR FIGURES 149 
 
 293. Theorem VIII. // two polygons are similar, they 
 can be divided into the same number of triangles, simi- 
 lar each to each, and similarly placed. 
 
 294. Theorem IX. If two polygons are composed of 
 the same number of triangles, similar each to each, 
 and similarly placed, the polygons are similar. 
 
 295. Theorem X. TJw areas of similar polygons have 
 the same ratio as the squares of any pair of correspond- 
 ing sides. 
 
 426. If similar polygons are constructed on the sides of a right 
 triangle as corresponding sides, the polygon on the hypotenuse equals 
 the sum of the other two polygons. 
 
 427. If each side of one polygon is double the corresponding side 
 of a second similar polygon, what relation have the areas? 
 
 428. If the area of one polygon is 36 times the area of a similar 
 polygon, what relation have the sides? 
 
 NOTE. Area ratios in similar figures are always the squares of line 
 ratios; line ratios are the square roots of area ratios. 
 
 296. Theorem XI. In a right triangle, the altitude 
 to the hypotenuse divides the triangle into two triangles 
 similar to each other, and to the whole triangle. 
 
 297. COR. 1. (1) The altitude to the hypotenuse is the 
 mean proportional between the sects of the hypotenuse. 
 
 (2) Either leg of the triangle is the mean propor- 
 tional between the hypotenuse and its own projection on 
 the hypotenuse. 
 
 429. In a right triangle of legs 5 and 12, find the projections of 5 
 and of 12, on the hypotenuse; find the altitude to the hypotenuse. 
 
 430. In a right triangle of legs , />, hypotenuse c, find the projec- 
 tions of a and of b on c ; find the altitude to c. 
 
 NOTE. The results of exercise 430 are formulas that hold for all 
 right triangles. 
 
150 SIMILAR FIGUllKS 
 
 431. Tangents from a point to a circle of radius 6 are of length 
 8. Find the chord of contact. 
 
 432. In a right triangle, the sects of the hypotenuse made by the 
 altitude are 4 and 5. Find the other sides and the altitude. 
 
 433. The squares of two chords drawn from a point on a circle 
 have the same ratio as their projections on the diameter from that 
 point. 
 
 434. The half chord perpendicular to a diameter is the mean 
 between the sects of the diameter. 
 
 298. Theorem XII. A line that bisects an angle, 
 interior or exterior, of a triangle, divides the opposite 
 side, internally or externally, into sects proportional to 
 the other sides of the triangle. Note that the two lines 
 divide harmonically. 
 
 435' If a line drawn from the vertex of an angle of a triangle 
 divides the opposite side, internally or externally, in the ratio of the 
 other two sides, the line bisects the angle, interior or exterior, from 
 whose vertex it is drawn. 
 
 436. In a triangle of sides 6, 7, 8, find the sects of 7 made by the 
 bisector of the opposite angle ; of the exterior angle at the opposite 
 vertex. 
 
 437. In a triangle of sides a, b, c, find the sects of c made by the 
 bisector of the opposite angle ; of the exterior angle at the opposite 
 vertex. 
 
 NOTE. The results of 437 are formulas that hold for all triangles. 
 
 438. If two lines from the vertex of the right angle of a right 
 triangle make equal angles with one of the legs, they cut the hypote- 
 nuse harmonically. 
 
 299. Theorem XIII. If a pencil of lines cuts a cir- 
 cumference, the lines are cut proportionally, so that 
 the product of the two sects from the vertex to the cir- 
 cumference on one line is the same as that product on 
 any other line. 
 
SIMILAR FIGURES 151 
 
 300. COR. 1. A tangent from the vertex of a pencil 
 to a circumference is the inean proportional between 
 the two sects, from the vertex to the circumference, 
 of any other line of the pencil that ' is cut by the 
 circumference. 
 
 439. If two points are taken on each line of a pencil, so that the 
 product of the two sects from the vertex is the same for all the lines 
 of the pencil, the two points being on opposite sides of the vertex, 
 the four points on any two lines are concyclic. 
 
 440. In the figure of 439, if the two points on each line are on 
 the same side of the vertex, the four points on any two lines are 
 concyclic. 
 
 441. Find the locus of the point P on a secant to a given circle 
 which cuts the circle in changing points, A and B, so that PA x PB 
 is constant. 
 
 442- If two intersecting lines are cut, one by one point, the other 
 by two points on the same side of the vertex, so that the sect cut off 
 by the one point is the mean proportional between the sects from the 
 vertex on the other line, a circle through the three points would be 
 cangent to the line cut by the one point. 
 
 443. Construct a circle through two given points tangent to a 
 given line. 
 
 444. The product of two sides of a triangle equals the square of 
 the bisector of the included angle, plus the product of the sects into 
 which the bisector divides the opposite side. 
 
 SUGGESTION. Inscribe the triangle. 
 
 445- If three circles intersect, their common chords are concurrent. 
 
 446. If two circles intersect, tangents from any point in their 
 common chord extended, to the two circles, are equal. 
 
SECTION IV. CONSTRUCTIONS 
 
 301. CONST. I. To divide a given sect internally, and 
 externally, in a given ratio (or, harmonically). 
 
 NOTE. A given ratio is always represented by two given sects, the 
 given ratio being that of those sects. 
 
 302. COR. 1. To divide a given sect into parts propor- 
 tional to any number of given sects. 
 
 447. Given a sect, and one point of harmonic division, find the 
 other. 
 
 448- Given a sect, cut it into parts having the ratio 1:2:5. 
 
 303. CONST. II. To find the fourth proportional to three 
 given sects. 
 
 304. COR. 1. To find the third proportional to three 
 given sects. 
 
 449> Draw a line through a given point so that it will cut off sects 
 having a given ratio on the arms of a given angle. 
 
 450. Draw a line from a given point to a given line, so that it will 
 have a given ratio to the perpendicular from that point. 
 
 451. Draw a line through a given point so that it will be cut in a 
 given ratio by the arms of the angle. 
 
 452- Construct two sects, given their sum and their ratio. 
 453. Construct two sects, given their difference and their ratio. 
 
 305. CONST. III. To find the mean proportional between 
 two given sects. Find three methods. 
 
 This construction is the foundation of all square root 
 questions in the Geometry. Since the square of the mean 
 
 152 
 
CONSTRUCTIONS 153 
 
 equals the product of the extremes, it follows that the 
 mean equals the square root of the product of the extremes. 
 It is, therefore, possible to construct the square root of 
 any required number by making the extremes of such 
 length that their product equals the number of which the 
 square root is asked. This is used also in constructing 
 figures whose areas have a certain ratio, for if the figures 
 are similar, the line ratio is the square root of the given 
 area ratio, and so can be found by the mean proportional. 
 
 454. Construct a sect V2in. long, given the sect 1 in. long. 
 
 455. Explain how the sect Vn in. long could be constructed for 
 any value of n, given a sect of 1 in. 
 
 456. Find another way to construct the square root of a number. 
 
 306. Mean and Extreme Ratio. If a sect is divided so 
 that the longer part is the mean between the whole sect 
 and the shorter part, the sect is said to be divided in 
 mean and extreme ratio. If AB is divided by P so that 
 
 = -, then P divides AB in mean and extreme ratio. 
 AP PB 
 
 There is an external mean and extreme division as well as 
 an internal ; the words " mean " and " extreme " do not 
 refer to the interior and exterior cases, but to the position 
 in which the parts occur in the proportion. 
 
 It should be noticed that, as the proportion stands, there 
 are two unknown sects used ; by a proper transforming of 
 the proportion the parts can be combined (in different 
 ways for the two cases) so that the given sect AB will 
 appear more often, thus displacing the unknowns. The 
 new form, in which the unknowns occur as seldom as 
 possible, is the one with which to work in attempting 
 the construction; it will be found very easy if attacked 
 logically. 
 
154 PKOPORTIONS 
 
 307. CONST. IV. To divide a given sect in mean and 
 extreme ratio. 
 
 437. If a line 10 in. long is divided in mean and extreme ratio, 
 how long is the mean sect? 
 
 458. If the bisector of a base angle of an isosceles triangle cuts off 
 the longer sect of the opposite side equal to the shorter, of the sides 
 including the bisected angle, that side is cut in mean and extreme 
 ratio. 
 
 308. CONST. V. To construct a polygon similar to a 
 given polygon, on a given sect as a side corresponding to a 
 certain side of the given polygon. 
 
 459. Construct a polygon similar to a given polygon and having 
 twice as great an area; n times as great an area (n a whole number). 
 
 460. Construct a polygon similar to a given polygon and having 
 an area one half as great; three fifths as great; times as great. 
 
 461. Given two similar polygons, construct a polygon similar to 
 them and equal to their sum. 
 
 462. Given two similar polygons, construct a polygon similar to 
 them and equal to their difference. 
 
 309. - CONST. VI. (1) To draw a rectangle having a 
 given ratio to a given square. (2) To draw a square 
 having a given ratio to a given square. 
 
 Notice that the construction " To draw a square equiva- 
 lent to a given rectangle " uses a method of construction 
 which is the same as drawing a mean proportional. 
 Why? 
 
 463. Construct a square three fourths as large as a given square. 
 
 464. Construct a square five times as large as a given square 
 by the method of 309. 
 
 310. SUMMARY OF THEOREMS AND COROLLARIES, BOOK IV 
 
 (Numbers in parentheses refer to black-faced section numbers.) 
 I. SECTS PROPORTIONAL. If cut off on transversals by Us (270) ; 
 cut off on Us by a pencil (278) ; sides of A cut by a II to third side (276) ; 
 
SUMMARY OF PROPOSITIONS 155 
 
 line parallel to side of A has the same ratio to it as the length cut 
 off on another side has to that side (277) ; sides of ~ polygons (282) ; 
 perimeters of ~ polygons proportional to sides (284); in a right 
 triangle the altitude is the mean, either leg is a mean (297) ; line 
 bisecting an angle of a A divides the opposite side (298) ; pencil 
 cutting a circle, the product of the sects is constant (299); the 
 tangent is the mean (300). 
 
 II. SECT CUT. By but one internal point in a given ratio, by 
 but one external point in a given ratio (280). 
 
 III. LINES PARALLEL. Line cutting two sides of A propor- 
 tionally (281). 
 
 IV. FIGURES ARE SIMILAR. ~ to the same polygon (283); regu- 
 lar, same number of sides (285) ; A, if they have 2 A = (286), 2 sides 
 proportional, incl. A (287), 3 sides proportional (288) ; in rt. A alti- 
 tude forms A ~ each other and to the whole (296) ; ~ polygons can 
 be divided into ~ A (293) ; polygons composed of ~ A are ~ (294). 
 
 V. AREA RATIOS. A having an Z (290) ; [U having an 
 = Z (290) ; ~ A proportional to squares of sides (291) ; ~ polygons 
 proportional to squares of sides (295) ; equivalent when an angle , 
 product including sides the same (292) . 
 
 CONSTRUCTIONS 
 
 VI. DIVIDE A SECT. Internally and ext. in a given ratio (301) ; 
 into parts proportional to given sects (302) ; in mean and extreme 
 ratio (307). 
 
 VII. FIND A PROPORTIONAL. Fourth to three given sects (303); 
 third to two given sects (304) ; mean between two given sects (305). 
 
 VIII. CONSTRUCT A POLYGON. Similar to a given polygon (308). 
 
 IX. CONSTRUCT A SQUARE. Having a given ratio to a given 
 square (309). 
 
 311. ORAL AND REVIEW QUESTIONS 
 
 What are the two principal ways of getting sects proportional? 
 Upon what do area ratios always depend? How many parts are 
 needed to prove triangles similar? Will any three parts do? If one 
 side of a polygon is double the corresponding side of a similar poly- 
 gon, what can be told about the areas ? If the area is double, what 
 can be told about the side? If each side of a triangle including one 
 of the angles is doubled, what does it do to the area of the triangle? 
 
156 PROPORTIONS 
 
 if one side is made three halves as long, the other two thirds as long? 
 How could a triangle be made isosceles without changing the vertex 
 angle or the area? Why are polygons similar to the same polygon 
 similar to each other? In proving polygons similar, what two things 
 must be considered ? Why are regular polygons of the same number 
 of sides mutually equiangular? Why are their sides proportional? 
 What is used when the ratio of a sum of several magnitudes to a 
 second sum is needed? What changes can be made in a proportion 
 without making it untrue? If the sides of a right triangle are 
 3, 4, 5, how long is the projection of 3 on 5? of 4 on 5? How long 
 is the altitude to 5? How long are the sects of 5 made by the 
 bisector of the opposite angle ? Tell three ways to construct a mean 
 proportional. What is the foundation of the constructions which 
 make four sects proportional ? Name three such constructions. What 
 constructions in this book have special cases which have been done 
 before? What new way of finding that a line is parallel to another 
 line occurs in this book? What special case of it has been taken 
 before? What other theorem is the general case corresponding to 
 a special case done before? Define mean and extreme ratio, cutting 
 harmonically, similar polygons, fourth proportional, composition, 
 division. In what form is the mean and extreme ratio proportion 
 put before trying to find the internal point of division? the external? 
 What is meant by a point cutting a sect in a given ratio internally? 
 externally? In what ratio is it not possible to cut a sect? What is 
 the fundamental way of proving four sects proportional? What two 
 cases of it are there ? If a line were drawn through the two thirds 
 points of the legs of a trapezoid, would it be parallel to the bases? 
 Why? What new way of proving triangles equivalent has been 
 found? Give a numerical example illustrating it. How can a line 
 of length \/2 be constructed? the length Vn? Is the length exact 
 or approximate (in so far as the instruments are accurate) ? 
 
 GENERAL EXERCISES 
 
 465. If two lines through the vertex of a tria-ngle divide the op- 
 posite side harmonically, and are at right angles to each other, they 
 bisect the angles at that vertex. 
 
 466. Find the locus of a point whose distances from two given 
 points are in a given ratio. 
 
GENERAL EXERCISES 157 
 
 467. Find the locus of a point such that the angle between the 
 tangents from the point to one given circle equals the angle between 
 the tangents from the point to a second given circle. 
 
 468. Construct a square inscribed in a semicircle. 
 
 469. A line is drawn from a vertex of a parallelogram, cutting the 
 other diagonal, and the other two sides, one extended. Prove that 
 the sect from the vertex to the diagonal is the mean between the sects 
 from the diagonal to the sides. 
 
 470. If the center line of two circles that do not meet intersects 
 the exterior common tangents at P, and the circles at A, B, C, D, 
 and a secant from P meets the circles at A', Y, Z, W, then PX x PW 
 = PYx PZ = PB x PC. 
 
 471. Find the locus of the vertex of a triangle, given the base, and 
 the ratio of the other sides. 
 
 472. Find the locus of a point whose distances from two inter- 
 secting lines have a given ratio. 
 
 473. Find the locus of a point whose distances from the sides of a 
 triangle are in a given ratio. 
 
 474- Cut off equal parts from two given sects, leaving them in a 
 given ratio. 
 
 475. Inscribe a triangle similar to a given triangle in a given 
 circle. 
 
 476. Circumscribe a triangle similar to a given triangle about a 
 given circle. 
 
 477. From a given point on a side of a triangle, draw a line 
 dividing the triangle into equivalent parts. 
 
 478. Construct an equilateral triangle equivalent to a given 
 triangle. 
 
 479. Construct a sect, given the greater sect obtained by dividing 
 it in mean and extreme ratio. 
 
 480. Inscribe a square in a given triangle. 
 
 481. Inscribe a rectangle similar to a given rectangle in a given 
 semicircle. 
 
 482. Inscribe a rectangle similar to a given rectangle in a given 
 triangle. 
 
158 PROPORTIONS 
 
 483. In a triangle of sides a, b, c, find the radius of the circum- 
 scribed circle. 
 
 484' Draw a square equivalent to a given rhombus. 
 
 485. Describe a circle through a given point, tangent to two given 
 lines. 
 
 486. Construct a triangle, given the three altitudes. Note that 
 the ratio of the altitudes can be used to find the ratio of the sides. 
 
 487. Construct an isosceles triangle, given the vertex angle and 
 the sum of the base and its altitude. 
 
 488. Construct a triangle, given the base angles, and the difference 
 between the base and its altitude. 
 
 NOTE. In many triangle and other constructions, a similar figure 
 can be used to advantage; as here, a triangle similar to the required 
 triangle should be constructed first. 
 
 489. Construct a triangle, given the base, a base angle, and the 
 ratio of the other including side to the radius of the circumscribed 
 circle. 
 
 490. Construct a triangle, given the three medians. 
 
 NOTE. Many other triangle constructions can be solved by the 
 use of proportions and of similar figures. It is good practice for 
 students to try to invent new combinations of parts of triangles from 
 which the triangles can be constructed. See also note at end of 
 Bk. II. 
 
BOOK V. REGULAR POLYGONS AND 
 CIRCLES 
 
 SECTION I. THEOREMS 
 
 312. Theorem I. In any regular polygon there is a 
 point that is equidistant from the vertices and equi- 
 distant from the sides. 
 
 Prove that the bisectors of all the angles meet in a 
 point, and use locus. 
 
 313. Center, Radius, Apothem. That point in a regular 
 polygon which is equidistant from the vertices, and also 
 equidistant from the sides, is called the center of the poly- 
 gon ; the line from the center to a vertex is called the 
 radius of the polygon, and a perpendicular from the center 
 to a side is called the apothem of the polygon. 
 
 It is evident that the radius of the polygon is also the 
 radius of the circumscribed circle, and that the apothem 
 of the polygon is the radius of the inscribed circle. 
 
 314. COR. 1. Tlw area of a regular poly gon equals one 
 half tlw product of the perimeter by tlw apothem. 
 
 315. Con. 2. The perimeters of regular polygons of the 
 same number of sides are proportional to their sides, 
 apothems, or radii. 
 
 316. COR. 3. Tlw areas of regular polygons of the 
 same number of sides are proportional to the squares of 
 their sides, apothems, or radii. 
 
 159 
 
160 REGULAR POLYGONS AND CIRCLES 
 
 Note again that the area ratio is the square of the line 
 ratio. 
 
 49 1. If from a point within a polygon of n sides perpendiculars 
 are drawn to all the sides, the sum of those perpendiculars is n times 
 the apothem. 
 
 492. Find the area of a regular hexagon of side 10 ; of side a. 
 
 493. If the apothem of one regular polygon is 12, that of another 
 of the same number of sides is 15, and the area of the first is 477.16, 
 find the area of the second. 
 
 317. Theorem II. An equilateral polygon inscribed in 
 a circle is regular; an equiangular polygon circum- 
 scribed about a circle is regular. 
 
 318. Variables and Limits. A constant quantity is one 
 that keeps the same value throughout the investigation 
 in question. A quantity may be constant in 'one discus- 
 sion, but not in another. 
 
 A variable quantity is one that takes different suc- 
 cessive values during an investigation. 
 
 The limit of a variable is that constant to which the 
 variable can approach so near that the difference is less 
 than any possible fixed quantity, but which the variable 
 cannot equal. 
 
 If the sum of the numbers 1, J, J, J, T ^, -fa, etc., is 
 taken, that sum will never equal 2, no matter how large 
 a number of terms is added. However, it is not possible 
 to name a number less than 2, such that the sum cannot 
 become greater than that number ; that is, be nearer to 2 
 (its limit) than any fixed number. 
 
 It is evident from the definition that when a variable 
 approaches its limit, the difference between the limit and 
 the variable approaches the limit zero ; and, conversely, 
 that when the difference between a constant and a vari- 
 
THEOREMS 161 
 
 able approaches zero as a limit, the variable must be 
 approaching the constant as its limit. 
 
 319. Limit Theorems. (Given without proof ; see Ap- 
 pendix, 347.) 
 
 (1) If two variables approaching limits are equal for 
 all values, their limits are equal. 
 
 (2) If a variable is approaching a limit, that variable 
 multiplied by, or divided by, any constant will approach 
 its limit multiplied by, or divided by, that constant. 
 
 (3) If two variables are proportional to two constants, 
 their limits are proportional to the same constants. 
 
 320. Theorem III. If the number of sides of a regular 
 polygon inscribed in a circle is increased indefinitely, the 
 apothem of the polygon will approach the radius of the 
 circle as a limit. 
 
 Show that r a < , where r, a, s, stand for radius, 
 
 t 
 
 apothem, and side ; then show that & = 0, when the 
 number of sides is increased indefinitely. 
 
 321. Circumference Axiom. The circumference of a 
 circle is the limit which the perimeters of regular in- 
 scribed and circumscribed polygons approach when the 
 number of sides is increased indefinitely. 
 
 322. Theorem IV. The area of a circle is the limit 
 which the areas of regular inscribed and circumscribed 
 polygons approach when the number of sides is increased 
 indefinitely. 
 
 323. Theorem V. The ratio of the circumference to the 
 diameter is the same for all circles (or, circumferences 
 are proportional to their diameters'). 
 
 SMITH'S SYL. PL. GEOM. 11 
 
162 REGULAR POLYGONS AND CIRCLES 
 
 Regular inscribed polygons have perimeters propor- 
 tional to the diameters ; apply limit Th. III. 
 
 324. Value of the Ratio of Circumference to Diameter. 
 
 The ratio of circumference to diameter is represented by 
 the Greek letter TT (called pi). This letter is the initial 
 letter of the Greek word for circumference. The value of 
 TT can be found numerically by Geometry, and the method 
 employed is shown in the Appendix ( 350); the value 
 commonly used in calculations is 3.14159, or for less 
 accurate results, 3.1416, or even 3^. The last value is 
 sufficiently accurate for many of the numerical exercises 
 of the Geometry, but the student should become accus- 
 tomed to the use of the more accurate values also. 
 
 The value of TT cannot be expressed exactly in the deci- 
 mal system ; that is, it is an incommensurable number. 
 It has, however, been calculated to over 700 decimal 
 places, and there is no limit to the accuracy with which a 
 calculation can be carried out, if it is considered worth 
 while. 
 
 325. COR. I. The circumference of any circle equals 
 
 2 TT tunes its radius. 
 
 494- If the radius of one circle is double the radius of a second 
 circle, and the circumference of the second circle is 30 ft. long, how 
 long is the circumference of the first circle ? 
 
 495. What is the width of the ring between two concentric circles 
 whose circumferences are 100 and 200 ft. 
 
 496. If one third of the' circumference of one circle equals one 
 fourth the circumference of a second circle, how do the radii compare V 
 
 497. Find the circumference of a circle of radius 10 in. 
 
 498. Find the radius of a circle of circumference 22 ft. 
 
 326. Theorem VI. The area of a circle is one half the 
 product of the circumference by the radius. 
 
 Use a circumscribed regular polygon. 
 
THEOREMS 163 
 
 327. COR. 1. The area of a circle equals TT times the 
 square of the radius. 
 
 328. COR. 2. The areas of two circles are propor- 
 tional to the squares of their radii. 
 
 499. Find the area of a circle of radius 10. 
 
 500. Find the radius of a circle of area 49. 
 
 501. Find the circumference of a circle of area 3.14159. 
 
 502. If the radius of one circle is four times the radius of a second 
 circle, the area of the first is how many times the area of the second? 
 
 503. If the circumference of one circle is twice the circumference 
 of a second, how do the areas compare ? 
 
 504- What is the area of the ring between concentric circles of 
 circumferences 100 and 200 ft.? 
 
SECTION II. CONSTRUCTIONS 
 
 329. CONST. I. (1) To inscribe a circle in a given reg- 
 ular polygon. 
 
 (2) To circumscribe a circle about a given regular 
 polygon. 
 
 330. CONST. II. (1) G-iven a regular inscribed poly- 
 gon, to draw the regular circumscribed polygon of the same 
 number of sides. 
 
 (2) G-iven a regular circumscribed polygon, to draw the 
 regular inscribed polygon of the same number of sides. 
 
 331. CONST. III. (1) G-iven a regular inscribed poly- 
 gon, to draiv the regular inscribed polygon of double the 
 number of sides. 
 
 (2) G-iven a regular circumscribed polygon, to draw 
 the regular circumscribed polygon of double the number 
 of sides. 
 
 332. CONST. IV. To inscribe a square in a given circle. 
 
 333. CONST. V. To inscribe a regular hexagon in a 
 given circle. 
 
 334. CONST. VI. To inscribe a regular decagon in a 
 given circle. 
 
 505. Construct a regular pentagon inscribed in a given circle. 
 
 164 
 
SUMMARY OF PROPOSITIONS 165 
 
 506. Construct a regular 12-sided figure inscribed in a given circle. 
 
 507. Circumscribe about a given circle a polygon of 4, 5, 6, 8, 10, 
 sides. 
 
 335. Regular Polygons which can be Constructed. It is 
 clear from the constructions in this book that polygons of 
 3, 4, 5, sides, or those numbers multiplied by any power 
 of 2, are possible. There are certain other regular poly- 
 gons which can be constructed with compass and ruler, 
 such as the polygon of 17 sides, but the ones mentioned, 
 and possibly the one of 15 sides, are the only ones of 
 importance to Elementary Geometry. 
 
 508. Construct a regular polygon of 15 sides. 
 
 336. SUMMARY OF THEOREMS AND COROLLARIES. BOOK V 
 
 (Numbers in parentheses refer to black-faced section numbers.) 
 
 I. REGULAR POLYGON. Has a center (312); area = \ ap. x per. 
 (314) ; perimeters prop, to sides, apothems, radii (315) ; areas prop, 
 to squares of sides, apothems, radii (316) ; equilateral insc. polygon 
 regular, equiangular circumscribed polygon regular (317). 
 
 II. LIMIT APPLICATIONS. Apothem of regular inscribed polygon 
 approaches radius (320) ; circumference axiom (321) ; area regular 
 insc. or circum. polygon approaches the area of the circle (322). 
 
 III. CIRCLES. Ratio of circumference to diameter (TT) the same 
 for all circles (323) ; circum. = 2 irr (325) ; area = r x circum. 
 (326) ; area -rrr 2 (327) ; areas prop, to squares of radii (328). 
 
 CONSTRUCTIONS 
 
 IV. To DRAW A CIRCLE. Inscribed in a given regular polygon 
 (329) ; circumscribed about a given regular polygon (329). 
 
 V. To DRAW A REGULAR POLYGON. Circumscribed, of same 
 number of sides as a given inscribed, or inscribed, of the same number 
 of sides as a given circumscribed (330) ; of double the number of sides 
 of a given polygon, inscribed or circumscribed (331); of 4 sides in- 
 scribed (332) ; of 6 sides inscribed (333); of 10 sides inscribed (334). 
 
166 REGULAR POLYGONS AND CIRCLES 
 
 337. ORAL AND REVIEW QUESTIONS 
 
 How is the center of a regular polygon found ? Tell a second way. 
 To what are the perimeters of regular polygons proportional? the 
 areas? the circumferences of circles? the areas of circles? Give a 
 formula for the circumference of a circle, two formulas for the area 
 of a circle. To what is the side of a regular inscribed hexagon equal? 
 the side of a regular inscribed decagon ? the side of a circumscribed 
 square? of an inscribed square? of a circumscribed equilateral trian- 
 gle? of an inscribed equilateral triangle? What must be known to 
 prove an inscribed polygon regular? a circumscribed polygon? Ex- 
 plain the inscribed case. What is the construction upon which the 
 regular decagon depends? How can a regular pentagon be drawn? 
 Tell which of the following regular polygons can be constructed with 
 compass and ruler, and explain; 7, 8, 9, 12, 14, 16 sides. What values 
 of TT are most commonly used? What is the circumference of a circle 
 of radius 10? the area? How can you find the radius from the cir- 
 cumference? the radius from the area? the circumference from the 
 area? the area from the circumference? How can circumferences be 
 added? subtracted? multiplied? How can the areas of circles be 
 added ? subtracted ? multiplied ? divided ? Explain how to draw a 
 circle having its circumference double that of a given circle; its area 
 double that of the given circle. 
 
 GENERAL EXERCISES 
 
 509. The area of a regular dodecagon (12) equals three squares 
 on its radius. 
 
 510. What is the radius of a circle, if the area of the regular in- 
 scribed hexagon is 6 V3 ? 
 
 511. The area of the ring between two concentric circles equals 
 that of a circle whose diameter is that chord of the larger which is 
 tangent to the smaller. 
 
 512. If two chords of a circle are perpendicular to each other, the 
 sum of the circles on the sects as diameters equals the original circle. 
 
 613. On the sides of a square of side o, as diameters, circles are 
 drawn. Find the area of the parts into which the square is divided. 
 
 614. On the sides of an equilateral triangle of side a as diameters, 
 circles are drawn. Find the areas of the parts of the figure formed. 
 
GENERAL EXERCISES 167 
 
 51 5. With the vertices of an equilateral triangle of side a as cen- 
 ters, and a radius equal to half the side, circles are drawn. Find the 
 area of the entire figure, and of the figure inside the triangle bounded 
 by the arcs. 
 
 516. Construct a regular octagon on a given sect as side. 
 517' Construct a regular hexagon on a given side. 
 
 518. Construct a regular decagon on a given side. 
 
 51 9. Construct a regular pentagon on a given side. 
 
 520. Construct a regular hexagon, given the shorter diagonal. 
 
 521. Construct a circle equal to the sum of two given circles. 
 
 522. Construct a circle equal to the difference of two given circles. 
 
 523. Construct a circle equal to the sum of any number of given 
 circles. 
 
 524- Construct a circle equal to any number of times a given circle. 
 
 525. Construct a circumference equal to the sum of two given cir- 
 cumferences. 
 
 526. Construct a circumference equal to the difference of two given 
 circumferences. 
 
 527. Construct a circumference equal to the sum of any number of 
 given circumferences. 
 
 528. Construct a circle whose area has any given ratio to the area 
 of a given circle. 
 
 529. Construct a circle whose area is one half the area of a given 
 circle. 
 
 530. Divide a given circumference into parts having the ratio 3 : 7, 
 by a line through a given point. 
 
 531. Divide the surface of a circle into equal parts, by a concen- 
 tric circle. 
 
 532. Divide the surface of a circle into any number of equal parts 
 by concentric circles. 
 
 533. Given the radius of a circle, find the perimeters of regular 
 inscribed and circumscribed polygons of 3, 4, 5, 6, 8, 10 sides. 
 
 53^. Cut off two thirds of a circle by a line through a given point, 
 
GENERAL 
 
 THE FORMULAS OF GEOMETRY 
 
 338. I. IN AN n-siDED POLYGON. 
 
 (1) The sum of the interior angles equals (n 2) st. ^. 
 
 (2) The sum of the exterior angles equals 2 st. A. 
 
 II. ANGLES FORMED BY LINES MEETING A CIRCUMFER- 
 ENCE. 
 
 (1) Vertex on the circumference, measured by half the 
 arc ; includes inscribed angles (three cases), tangent 
 and chord angles. 
 
 (2) Vertex inside the circle, measured by half the sum 
 of the arcs. 
 
 (3) Vertex outside the circle, measured by half the dif- 
 ference of the arcs; includes angles between two 
 secants, two tangents, tangent and secant ; angle be- 
 tween two tangents supplemental to central angle 
 between radii to points of tangency. 
 
 III. AREA. 
 
 '(1) Rectangle, equals base times altitude. 
 
 (2) Parallelogram, equals base times altitude. 
 
 (3) Triangle, equals half base times altitude. 
 
 (4) Triangle, given the sides, equals the square root of 
 s(s a)(s b)(s c), where s is half the sum of 
 the sides. 
 
 (5) Equilateral triangle, equals the square of half the 
 side times the square root of three. 
 
 (6) Trapezoid, equals half the sum of the bases times 
 the altitude. 
 
 (7) Regular polygon, equals half the perimeter times 
 the apothem. 
 
 (8) Circle, equals TT times the square of the radius. 
 
 168 
 
FORMULAS OF GEOMETRY 169 
 
 IV. EQUIVALENCE FORMULAS BASED ON THE AXIOM OF THE 
 WHOLE. 
 
 (1) The square on the sura of two sects is equivalent to 
 the sum of their squares plus twice their rectangle. 
 
 (2) The square on the difference of two sects is equivalent 
 to the sum of their squares less twice their triangle. 
 
 (3) The square on twice a sect is equivalent to four 
 squares on the sect. 
 
 (4) the difference of the squares on two sects is equiva- 
 lent to the rectangle of their sum and their difference. 
 
 V. THE SQUARE ON A SIDE OF A TRIANGLE. 
 
 (1) Opposite a right angle, is equivalent to the sum of the 
 squares on the other sides. 
 
 (2) Opposite an obtuse angle, is equivalent to the sum of 
 the squares on the other sides plus twice the rectangle 
 of one side by the projection of the other on its line. 
 
 .(3) Opposite an acute angle, is equivalent to the sum of 
 the squares on the other sides, less twice the rectangle 
 of one side by the projection of the other side on its 
 line ; or, in a right triangle, to difference of squares. 
 
 VI. THE SUM OF THE SQUARES ON Two SIDES OF A TRIANGLE. 
 
 Is equivalent to twice the sum of the squares on one half 
 the third side and on the median to that side ; difference, 
 to two rectangles of the base by its median's projection. 
 
 VII. PROPORTIONS BETWEEN SECTS. 
 
 (1) Parallels cutting transversals, including triangle case. 
 
 (2) Sides of similar polygons are proportional. 
 
 (3) In a right triangle with the altitude to the hypotenuse, 
 (a) the altitude is the mean between the sects of the 
 
 hypotenuse ; 
 
 (ft) either leg is the mean between the hypotenuse and 
 the projection of that leg on the hypotenuse. 
 
 (4) If two secants cut a circle, the product of the sects from 
 the vertex is the same ; a tangent is the mean between 
 the sects of a secant from the same point. 
 
 (5) The bisector of an angle of a triangle cuts the opposite 
 side (internally or externally) into sects proportional to 
 the sides including the angle. 
 
170 FORMULAS OF GEOMETRY 
 
 (6) Perimeters of regular polygons are proportional to 
 sides, apothems, or radii. 
 
 (7) Circumferences are proportional to radii. 
 
 VIII. PROPORTIONS BETWEEN AREAS. 
 
 (1) Areas of triangles having an angle equal are propor- 
 tional to the product of the sides including the angle ; 
 if the triangles are equivalent, the product of the sides 
 in one equals the product in the other, and conversely. 
 
 (2) Areas of any similar polygons are proportional to the 
 squares of corresponding sides. 
 
 (3) Areas of circles are proportional to the squares of their 
 radii. 
 
 IX. THE CIRCUMFERENCE OF A CIRCLE equals 2 irR. 
 X. ALTITUDE. 
 
 (1) Of any triangle, is found from III, 4, or by first find- 
 ing the projection in V, (2) or (3). 
 
 (2) In an equilateral triangle, equals half the side times 
 the square root of three. 
 
 XL PROJECTION OF A SIDE OF A TRIANGLE, from V, (2), or (3). 
 XII. MEDIAN OF A TRIANGLE, from VI. 
 
COLLEGE EXAMINATION QUESTIONS 
 
 535. If ABODE is an inscribed pentagon, and arc DE is 40, 
 find the number of degrees in the sum of angles A and C. 
 
 536. In the triangle ARC, AB equals 7, BC equals 8, and CA 
 equals 5. Find the projection of BC on CA. 
 
 537. Construct a square whose area is to that of a given square as 
 2 is to 5. 
 
 538. Construct a circumference equal to the difference of two 
 given circumferences. 
 
 539. Construct a triangle, given two sides and the altitude on one 
 of those sides. 
 
 540. In a circle of center O, two perpendiculars, OD and OE, are 
 drawn to chords AB and CF, respectively, and angle EDO equals 
 angle DEO. Prove AB equal to CF. 
 
 541. Upon the four sides of a square as chords arcs of 90 are 
 constructed within the square. If the diagonal of the square equals 
 2 M in., find (a) the area bounded by the four arcs ; (b) the radius 
 of a circle equal to this area. 
 
 542. Two equivalent triangles have a common vertex y, and equal 
 bases AB and CD. If A, B, C, D have fixed positions not in the 
 same Hue, find the locus of the point Y. 
 
 543. The side of an inscribed equilateral triangle is parallel to the 
 side of a regular decagon inscribed in the same circle. Find the 
 number of degrees contained in the intercepted arcs, if the center of 
 the circle lies between the two lines. 
 
 544. If the sides of a triangle are 7 and 9 cm., respectively, and the 
 median to the third side equals 7 cm., find the third side. 
 
 545. The sides of a polygon are 4, 5, 6, 7, 8 cm. Find the sides 
 of a similar polygon whose area equals four times the area of the 
 given polygon. 
 
 171 
 
172 EXAMINATION QUESTIONS 
 
 546. Two vertices A and B of a triangle have fixed positions. 
 Find the locus of the third vertex C, if angle C is equal to a given 
 angle M, and prove your result. 
 
 547. If the base of an isosceles triangle be trisected, and from each 
 point of division a perpendicular be drawn upon the nearest side, 
 prove the perpendiculars equal. State the converse. 
 
 548' State two propositions which may be used to prove the 
 equality of arcs. If the lines which join a point in a circumference 
 to the midpoints of two radii are equal, prove that the point bisects 
 the arc subtended by the two radii. 
 
 549. Describe briefly a method for proving that the product of 
 two lines is equal to the product of two other lines. If through a 
 point A in a circumference, a tangent AB and a chord AC be drawn, 
 and from C a diameter CD and a perpendicular CE upon AB be 
 drawn, prove that the square of A C equals CE times CD. 
 
 550. Construct a circle whose area equals five times the area of a 
 given circle. 
 
 551 . Prove that the square of a line drawn from the vertex of an 
 isosceles triangle to any point in the base is equal to the square of 
 the leg, diminished by the product of the segments of the base. 
 
 552. Determine how many sides the polygon has, the sum of 
 whose interior angles equals the sum of its exterior angles. Explain 
 the method in full. 
 
 553. If a fixed arc AB of a circle equals 120, and a movable 
 arc CD on arc BA equals 60, find the locus of the intersection of 
 (a) AC and BD; (b) AD and BC. 
 
 554- A chord 24 in. long is 9 in. from the center of a circle. Find 
 the length of the tangents drawn from the extremities of the chord 
 and produced till they meet. 
 
 555. In the triangle ABC, the angle A is acute and BD is drawn 
 perpendicular to AC. AB equals 10 ft., BC equals 12 ft., and AC 
 equals 14 ft. Find A D. 
 
 556. The bases of a. trapezoid are 8 in. and 12 in., the area 30 
 sq. in. Find the length of a line drawn between the legs parallel to 
 the lower base and 2 in. from it. 
 
 557. Find the area of the sector of a circle whose radius is 15 
 in., the angle of the sector being 32. 
 
EXAMINATION QUESTIONS 173 
 
 558. Show how to construct a circle of given radius tangent to 
 two given circles. 
 
 559. Show how to draw a line parallel to the base of an isosceles 
 triangle so that in the trapezoid thus formed the legs shall be equal 
 to the upper base. 
 
 560. Prove that the perpendiculars dropped from the midpoints 
 of two sides of a triangle to the third side are equal. 
 
 561. Two equal chords produced meet outside the circle. Prove 
 that the secants thus formed are equal. 
 
 562. Find the locus of the point at which a given segment of a 
 straight line subtends a given constant angle. 
 
 563. A quadrilateral is formed by the diameter AB of a circle, 
 the tangents at .4 and B, and a third tangent which meets the other 
 tangents at C and D, respectively. Prove that the area is equal to 
 one half the product of the opposite sides AB and CD. 
 
 564. A circle of radius 4 has its center at the intersection of the 
 diagonals of a square whose side is 12. Find the length of the circum- 
 ference of a circle which touches two adjacent sides of the square and 
 also the circle. 
 
 565. How many sides has the polygon each of whose interior angles 
 is 175? 
 
 566. If M and N are two lines, construct a line equal to V2 MN. 
 
 567. If triangle ABC is equivalent to triangle DEF, A equal to 
 D, AB = 6, and DE = 9, find the ratio of CA to FD. 
 
 568. ABC is a right triangle, C the right angle ; EC = 5, CA = 12, 
 AB is extended to D so BD = 10. If CB is extended to E, and DE 
 drawn, how long must BE be in order to make triangle BED equiva- 
 lent to triangle ABC. 
 
 569. If a quadrilateral ABCD is inscribed in a circle, and the 
 diagonals A C and BD meet in E so that BE = CE, prove that arc 
 AB = arc CD. 
 
 570. Construct a triangle, having given a side and the medians to 
 the other two sides. 
 
 571. Each side of a triangle is 2n cm., and from each vertex 
 as a center with radii equal to n centimeters, circles are drawn. Find 
 the areas bounded (a) by the three arcs that lie without the triangle ; 
 (&) by the three arcs that lie within the triangle. 
 
174 EXAMINATION QUESTIONS 
 
 572. Given a square with the side 3 in. long. Find the locus 
 of a point P such that the distance from P to the nearest point on 
 the perimeter of the square is 1 in. Describe the locus accurately. 
 
 573. Semicircles are drawn with their centers at the middle points 
 of the sides of an equilateral triangle, forming a triangle less the area 
 of three semicircles. Prove that if the perimeter of this figure is one 
 fifth greater than that of the triangle, its area is about one third less 
 than that of the triangle. If the side of the triangle is 10 in., 
 what is the area of the figure, correct to 1 per cent ? 
 
 574' On the sides of an equilateral triangle ARC as bases, equal 
 isosceles triangles ABP, ACQ, BCR, are constructed ; the first two are 
 exterior to the given triangle, while R is on the same side of .BC as 
 A. Prove that A PR Q is a rhombus. 
 
 575. Find the altitude of an equilateral triangle whose area is 
 
 576. If the non-parallel sides of a trapezoid are equal, the angles 
 which they make with the base are equal. 
 
 577. Prove that the bisectors of the angles of a quadrilateral form 
 a second quadrilateral of which the opposite angles are supplementary. 
 
 578. Through the point M in the base of a triangle parallels to 
 the other two sides are drawn, forming a parallelogram. Find the 
 locus of the center of the parallelogram as M moves along the base of 
 the triangle. 
 
 579. Find to one place of decimals the area of a six-pointed star 
 formed by joining the alternate vertices of a regular hexagon inscribed 
 in a circle of radius 3 in. 
 
 580. The three angles of a triangle are 48, 82, and 50. Find the 
 three angles formed by the bisectors of the angles of the triangle. 
 Verify by using the theorem involving the sum of the angles about a 
 point in a plane. 
 
 581. A chord 1 ft. long is 4 in. from the center of a circle. 
 How far from the center is a chord 9 in. long? 
 
 582. A circle has an area of 80 sq. ft. Find the length of an 
 arc of 80. 
 
 583. Prove that if a median of a triangle is equal to half the side 
 to which it is drawn, the triangle is a right triangle. 
 
EXAMINATION QUESTIONS 175 
 
 584. Prove that if AB is a diameter of a circle, and BC a tangent, 
 and A C meets the circumference at D, the diameter is a mean pro- 
 portional between AC and AD. 
 
 585. Given three lines a, 6, c. Construct a line x so that a : b : :c :x. 
 
 586. Two parallelograms are equal if two sides and the included 
 angle of one are equal to two sides and the included angle of the other. 
 
 587. The area of a regular inscribed hexagon is a mean propor- 
 tional between the areas of the inscribed and circumscribed equilateral 
 triangles. 
 
 588. Three equal circles are described each tangent to the other 
 two. If the common radius is /?, find the area contained between 
 the circles. 
 
 589. The radius of a circle is 10 ft. ; the area of a sector of that 
 circle is 130 sq. ft. What is its arc in degrees? 
 
 590. Two sides and a diagonal of a parallelogram are 7, 9, and 8, 
 respectively. Find the length of the other diagonal. 
 
 591. One of two secants meeting without a circle is 12.5 in., and 
 its external segment is 4 in. ; the other secant is divided into two 
 equal parts by the circumference. Find the length of the second 
 secant. 
 
 592. A BCD is any parallelogram, and E any point within it. 
 Prove that the sum of the triangles EAD and EBC equals half the 
 area of the parallelogram. 
 
 593. Three consecutive angles of an inscribed quadrilateral sub- 
 tend arcs of 70, 85, and 98, respectively. Find each angle of the 
 quadrilateral and the angle between the diagonals. 
 
 594- The diameters of two concentric circles are 16 and 40 ft., 
 respectively. Find the length of a chord of the larger which is 
 tangent to the smaller. 
 
 595. The area of a rhombus is 96 sq. ft. and its side is 10 ft. 
 Find the lengths of its diagonals. 
 
 596. Show how to inscribe a circle in a given sector. 
 
 597. Show how to divide a triangle into two equivalent parts by 
 a line parallel to one of its sides. 
 
 598. Prove that the tangents to two intersecting circles from 
 any point in their common chord produced are equal. 
 
176 EXAMINATION QUESTIONS 
 
 599. Two tangents drawn from the same external point to a 
 circle form an angle 01 64. Find the number of degrees in each 
 of the arcs intercepted by these tangents. 
 
 600. Find the radius of a circle whose circumference numerically 
 equals its area. 
 
 601. From a point 12 in. from the center of a circle 16 in. in 
 diameter, two tangents are drawn to the circumference. Find the 
 length of the chord joining the points of contact. 
 
 602. Prove that if a circle is circumscribed about an isosceles 
 triangle, the tangents through the vertices form another isosceles 
 triangle. 
 
 603. Two circles are tangent externally and through the point 
 of contact two straight lines are drawn terminating in the circum- 
 ferences. Prove that the corresponding segments of the lines are 
 proportional. 
 
 604. Show how to draw a line terminating in the arms of an 
 angle, which shall be equal to one given line and parallel to another. 
 
 605. Two circles are tangent internally. If chords of the larger 
 circle are drawn through the point of contact, prove that they are 
 divided proportionally by the smaller circle. 
 
 606. Prove that if one acute angle of a triangle is double another, 
 the triangle can be divided into two isosceles triangles by a straight 
 line through the vertex of the third angle. 
 
 607. Given a regular hexagon each side of which is 6 in. With 
 three of the alternate vertices as centers, arcs of circles are drawn 
 passing through the center of the polygon. Find the area of the 
 three loops thus formed. 
 
 608. A point A is 4 ft. from the circumference of a circle ; the 
 length of a tangent from A to the circle is 10 ft. Find the diameter. 
 
 609. The bases of a trapezoid are respectively 29 ft. and 37 ft. 
 and its area is 247.5 sq. ft. Find its altitude. 
 
 610. Show how to divide a given rectangle into four equivalent 
 parts by lines drawn from one of the vertices of the rectangle. 
 
 611. Given a straight line and two points on the same side of that 
 line and at unequal distances from it. Construct a circumference 
 passing through the two points and having its center in the given 
 line. 
 
EXAMINATION QUESTIONS 177 
 
 612. Prove that the area of a square inscribed in a given circle is 
 twice the square of the radius of the circle. 
 
 613. ABC is a triangle inscribed in a circle with center 0. Take 
 D the middle point of the arc EC and draw OD and AD. Prove that 
 the angle ADO equals half the difference of the angles B and C. 
 
 614. If a line is drawn from the vertex A of triangle ABC to any 
 point D of the opposite side, and any point on AD is joined to 
 
 A ABC AD 
 ande , prove __ = _. 
 
 615. From a point at a distance of 10 in. from the center of a 
 circle of radius 5 in., two tangents are drawn. Compute the area 
 bounded by the tangents and their included arc. 
 
 616. Construct a triangle having given the midpoints of its three 
 sides. 
 
 617. The legs of a right triangle are 6 and 8, respectively. Find 
 their projections on the hypotenuse. 
 
 618. Compute the area of a square inscribed in a circle whose 
 perimeter is 63 ft. 
 
 619. The legs of a right triangle are respectively 15 ft. and 8 ft. 
 Find the length of each segment made on the hypotenuse by the 
 bisector of the right angle. 
 
 620. Find the product of the segments of any chord drawn 
 through a point 12 in. from the center of a circle whose diameter 
 is 18 in. 
 
 621. Find the area of an equilateral triangle inscribed in a circle 
 of radius 4 in. 
 
 622. Show how to construct a square which shall be one third 
 of a given square. 
 
 623. Prove that if in a right triangle one angle is 30, the hypote- 
 nuse is double the shorter leg. 
 
 624. Prove that two chords drawn perpendicular to a third chord 
 at its extremities are equal. 
 
 625. Draw a square ABCD. On the diagonal AC take the point 
 E so that AE AB, and draw through E a perpendicular to AE, 
 cutting BC in F. Prove that BF = FE = EC. 
 
 SMITH'S SYL. PL. GEOM. 12 
 
178 EXAMINATION QUESTIONS 
 
 626. Take five points on the circumference of a circle (^4, 5, C, Z), Z?), 
 in the order named. Let the middle point of the arc ABC be F, 
 and the middle point of the arc AED be G. Let the chord FG cut 
 the chord A C in H and the chord AD in K. Prove that AH is equal 
 fc>4& 
 
 627. Prove that if in a right triangle the hypotenuse is double 
 the shorter leg, one acute angle is double the other. 
 
 628. Show how to construct a square equivalent to a given paral- 
 lelogram. Prove the correctness of your method. 
 
 629. Prove that if one leg of a right triangle is the diameter of a 
 circle, the tangent at the point where the circumference cuts the 
 hypotenuse bisects the other leg. 
 
 630. The apothem of a regular hexagon is 6\/3. Find the area 
 of the hexagon and the area of the inscribed circle. 
 
 631. Prove that if a tangent be drawn from one end of a diameter 
 meeting a secant from the other end, the product of the secant and 
 the internal segment will be the same for all directions of the secant. 
 
 632. Prove that if two circles are tangent internally at A and a 
 straight line intersects the two circles in B, C, Z>, and E, then angle 
 
 = DAE. 
 
 633. If in triangle ABC the line from C to D on AB bisects 
 angle C, and AB = 2.09, C = 3.14, CA =4, compute values of DB 
 and DA, respectively. 
 
 634. To measure the height of a church spire, a rod 10 ft. long 
 is planted vertically at position A, then at position B. In each case 
 the observer takes such a position that the top of the spire, the top 
 of the rod, and his eye are all in line when he stands erect. He 
 measures the distance each time from his position to the rod. and 
 obtains the measurements 4 ft. and 8.3 ft. He also measures the 
 distance between his two positions, finding it to be 138 ft. If the 
 observer's eye is 5 ft. above the ground, what is the height of 
 the spire? Explain your solution. 
 
 635. What is the locus of the midpoint of one leg of a right 
 triangle whose hypotenuse is fixed? Prove the correctness of your 
 answer. 
 
 636. In a quadrilateral ABCD the lengths of AB and EC are 
 
EXAMINATION QUESTIONS 179 
 
 equal and angle A is greater than angle C. Which is the longer 
 side, AD or CZ>? Give the reason for your answer. 
 
 637. Prove that in a convex quadrilateral the angle between the 
 bisectors of two adjacent angles is one half the sum of the other two 
 angles. 
 
 638. An arc of a certain circle is 100 ft. long, and subtends an 
 angle of 25 at the center. Compute the radius of the circle correct 
 to three significant figures. 
 
 639. Three successive vertices of a regular octagon are A, B, C, 
 respectively. If the length AB is a, compute the length A C. 
 
 640. The areas of similar segments of circles are proportional to 
 the squares of their radii. 
 
 641. Given a circle whose radius is 16. Find the perimeter and 
 the area of the regular inscribed octagon. 
 
 61+2. Two circles intersect at points A and B. Through A a 
 variable secant is drawn, cutting the circles in C and D. Prove that 
 the angle CBD is constant for all positions of the secant. 
 
 643. Let A and B be two fixed points on the circumference of a 
 given circle and P and Q the extremities of a variable diameter of 
 the same circle. Find the locus of the point of intersection of the 
 straight lines AP and BQ. 
 
 644- Prove that in any right triangle the line drawn from the 
 right angle to the middle of the hypotenuse is equal to one half the 
 hypotenuse. 
 
 645. The area of a regular decagon is 108 sq. in. Find the 
 radius of the circumscribed circle. 
 
 646. Two secants are drawn from the same point to the same 
 circle. The external segment of the first is 5 in. and its internal 
 segment is 19 in. The internal segment of the other secant is 7 in. 
 Find the length of the second secant. 
 
 647. On the diameter AB of a circle mark a point P. Through 
 P draw the chord CPD at right angles to AB. Prove that if AP, 
 BP, CP, and DP be taken as diameters of circles, the sum of the 
 areas of the four circles is equal to the area of the original circle. 
 
 648. To construct a rectangle, having given the perimeter and the 
 diagonal. 
 
180 EXAMINATION QUESTIONS 
 
 649. Find how far from the base of a triangle of altitude a lines 
 parallel to the base must be drawn to divide the area of the triangle 
 into three equal parts. 
 
 650. Prove that two triangles are similar if the sides of one are 
 respectively parallel to the sides of the other. 
 
 651. Derive the numerical value of ?r.. 
 
 652. Of all triangles having the same base and equal areas, the 
 isosceles triangle has the minimum perimeter. 
 
 653. How high is a tree which casts a shadow 70 ft. long, when a 
 man 6 ft. high casts a shadow 8| ft. long? 
 
 654. Construct a triangle, given the base, vertical angle, and 
 median drawn to the base. 
 
 655. State six propositions concerning parallel lines and prove 
 any one of them. 
 
 656. An interior angle of an equiangular polygon is 150. Find 
 the number of sides. 
 
 657. Three cylindrical barrels, diameter of each being 20 in., are 
 placed in a pile with axes horizontal so that each just touches the 
 other two. Find the height of the pile, and the length of the 
 shortest rope to go over the pile and touch the floor on each side. 
 
 658. AB is the diameter of a circle of radius 2 in., and A C is a 
 chord such that BAG is 30. Find area and perimeter of BACB 
 correct to two decimal places. 
 
 659. If from a fixed point D, within a triangle ABC, lines are 
 drawn to all points in the perimeter of the triangle, what is the locus 
 of the middle points of those lines? 
 
 660. Show that if the radius of a circle is a, the side of the 
 regular inscribed decagon is ?(V5 1) and the side of the regu- 
 
 lar inscribed pentagon is ^ 
 
 661. The base of a triangle is 32 ft. and its height is 20 ft. What 
 is the area of the triangle formed by drawing a line parallel to the 
 base 5 ft. from the vertex ? 
 
 662. The sides of a triangle are 5, 12, 13. Find the segments into 
 which each side is divided by the bisector of the opposite angle. 
 
EXAMINATION QUESTIONS 181 
 
 663. The sides of a triangle are a, J, c, and the area is k. What 
 is the radius of the inscribed circle ? 
 
 664. A and B are fixed points, AC is drawn in any direction, and 
 BP is drawn perpendicular to A C, meeting it at P. What is the 
 locus of P ? 
 
 665. Upon a line about ] in. long construct a segment to contain 
 an angle of 60. 
 
 666. Three circles of radius a touch each other, and another circle 
 is circumscribed about them. Find its radius, circumference, and 
 area. 
 
 667. AB is a fixed line, angle A CB = 45. Construct the locus 
 of C. 
 
 668. Two sides of a triangle are a and b, the included angle 135. 
 What is the square of the third side c ? 
 
 669. The lengths of the circumferences of two concentric circles 
 differ by 6 in. Compute the width of the ring to three significant 
 figures. 
 
 670. Prove the area of the triangle formed by joining the middle 
 point of one of the nonparallel sides of a trapezoid to the extremities 
 of the opposite side equals one half the trapezoid. 
 
 671 . The three sides of a triangle are 4 ft., 13 ft., and 15 ft. long. 
 Show that the altitude upon the side of length 15 is 3.2. 
 
 672. Through the vertex A of the parallelogram A BCD draw a 
 secant. Let this line cut diagonal BD in E, and the sides BC, CD 
 (or these sides produced) in F and G, respectively. Prove that AE 
 is a mean proportional between EF and EG. 
 
 673. Let ABC be an equilateral triangle, and on the sides AB, 
 BC, CA, lay off AD, BE, CF, each equal to one third AB, and join 
 the points D, E, F, with one another. Prove that the triangle DEF 
 is equilateral, and that its sides are respectively perpendicular to 
 the sides of the given triangle. 
 
 674- On the circumference of a circle take two points subtending 
 a right angle at the center, and a third point on the arc between 
 these two. Prove that the perimeter of the triangle formed by the 
 tangents at these three points is equal to the diameter of the circle. 
 
 675. An indefinite straight line moves in such a way that it 
 always passes through at least one vertex of a given square, but 
 
182 EXAMINATION QUESTIONS 
 
 never crosses the square. What is the locus of the foot of the per- 
 pendicular dropped on the moving line from the center of the square ? 
 Describe the locus accurately and prove the correctness of your' 
 answer. 
 
 676. Prove the correctness of the following construction for 
 bisecting an angle ABC] upon AB produced beyond B take BD 
 equal to BC and draw a line through B parallel to DC. 
 
 677. Show how to construct a chord through a given point A 
 within a circle, so that the extremities of the chord shall be equi- 
 distant from another point B. 
 
 678. A rod of length a is free to more within a semicircular area 
 of radius a. Describe accurately the boundary of the region within 
 which the middle point of the rod will always be found. 
 
 679. A roadway 60 ft. wide is cut through the middle of a circu- 
 lar field 120 ft. in diameter. Compute the area of the remainder of 
 the field correct to 1 per cent of its value. 
 
 680. The radii of two circles are 1 in. and V3 in., respectively, 
 and the distance between their centers is 2 in. Compute their 
 common area to three significant figures. 
 
 681. Determine a point P without a given circle so that the sum 
 of the tangents from P to the circle shall be equal to the distance 
 from P to the farthest point of the circle. 
 
 682. The image of a point in a mirror is, apparently, as far 
 behind the mirror as the point itself is in front. If a mirror revolves 
 about a vertical axis, what will be the locus of the apparent image 
 of a fixed point one foot from the axis? 
 
 683. The hypotenuse of a right triangle is 10 in. long and one of 
 the acute angles is 30. Compute the lengths of the segments into 
 which the short side is divided by the bisector of the opposite angle. 
 
 684- A chord BC of a given circle is drawn, and a point A moves 
 on the longer arc BC. Draw triangle ABC and find the locus of the 
 center of a circle inscribed in this triangle. 
 
 685. Three equal circular plates are so placed that each touches 
 the other two, and a string is tied tightly around them. If the 
 length v of the string is 10 ft., find the radius of the circles correct 
 to three figures. 
 
APPENDIX 
 
 339. Contraposite Law. This law was stated in 8, bat 
 no explanation of why it was true was attempted. The 
 following explanation is simple, and shows very plainly 
 that the law holds for all statements. 
 
 Given. If A, then B. 
 
 To prove. If not B, then not A. 
 
 Proof. I. If not B, then either (1) A, 
 
 or (2) not A (all possibilities). 
 II. But, if A, then B (given). 
 
 .-. using (1), if not , then A, then B. 
 III. This is impossible, for it contradicts itself. 
 .-. If not B, then not A 
 (only other possibility). 
 
 (This proof depends upon the "Law of Excluded 
 Mean," but for the purpose of this proof it is not 
 necessary to discuss that Law.) 
 
 340. Law of Converse. Stated in 10. 
 Given. If -4, then JT, 
 
 If B, then F, 
 If <7, then Z, etc. 
 
 Where A, B, <7,*. . . cover all possibilities, 
 
 and no two of the conclusions X, F, Z, 
 
 . . . can be true at once. 
 
 To prove. If X, then A ; 
 If F, then B ; 
 If Z, then C, etc. 
 183 
 
184 APPENDIX 
 
 Proof. I. If JT, then not F, Z, or any other conclusion (no 
 
 two can be true at once, by the given). 
 II. .*. not B, not C, not any condition but A 
 
 (contraposite). 
 III. .*. If J$T, then A (only remaining possibility). 
 
 341. Proofs of the Obverse and the Converse of a Single 
 Statement. (1) When it is necessary to prove the obverse 
 of a single statement, take both conditions, and show that 
 they cannot give the same conclusion. 
 
 If A, then B. Take " not A" and show that it cannot 
 also give B. 
 
 See 122 ; in thiscase A represents equal angles, B rep- 
 resents parallel lines. Then " not A " represents unequal 
 angles, and the reason it cannot also give U, that is, 
 parallel lines, is that there can be but one parallel through 
 a point. 
 
 (2) To prove the converse of a single statement, when 
 that is necessary, show that the condition and the conclu- 
 sion give one and the same thing. 
 
 If A, then B. Show that B gives the same as A. 
 
 In 124, if it is to be proved as the converse of 121, 
 the proof would be as follows, the work being much 
 abbreviated. 
 
 If A (that is, the angles are equal), then B (that is, the 
 lines are parallel). 
 
 If B (that is, the lines are parallel). But there is but 
 one parallel through the point in question, therefore this fig- 
 ure is identical with the other, and so the angles are equal. 
 
 Note the application of this method to 144. 
 
 In both these methods the proof follows because of 
 some element of exclusion, that is, some statement that 
 there can be but one of a certain thing. 
 
APPENDIX 185 
 
 342. General Condition. In many of the theorems of 
 Geometry the hypothesis contains two kinds of conditions, 
 one of which might be called the general condition, for it 
 states the kind of figure about which the statement is to 
 be made, while the other might be called the special condi- 
 tion, for it is that on account of which the conclusion 
 follows. For example, in Th. I, the general condition is 
 the given triangles ; the special condition, the fact that 
 they have two sides and the included angle equal. If 
 the converse, obverse, or contraposite is to be stated, the 
 general condition should be left just as in the original state- 
 ment, the special condition and the conclusion being used. 
 
 The subject could be made very complicated by going 
 into all the different ways of using the two conditions, 
 but the statements obtained by following the direction 
 given above will be sufficient for all practical purposes. 
 
 343. Axioms. The axioms, as used in this book, are not 
 entirely independent ; for example, the substitution axiom 
 could be derived from the equality axioms, as far as the 
 equal cases are concerned. The position of the author in 
 thig respect has been defined in the preface. The follow- 
 ing axiom can be derived from the axiom of intersection, 
 but the work is not worth while, although the axiom is 
 needed for certain work in the theorems and exercises. 
 
 Diagonal Intersection Axiom. The diagonals of any 
 convex quadrilateral intersect each other at a point 
 within the quadrilateral. 
 
 344. Symmetry. 
 
 (1) With respect to a center. Two points are said to be 
 symmetrical with regard to a point (or center) when they 
 are on opposite sides of the point on the same straight 
 line, and at the same distance from that point. Two 
 
186 APPENDIX 
 
 figures are said to be symmetrical with regard to a center 
 when all their corresponding points are respectively sym- 
 metrical with regard to the center. 
 
 (2) With respect to an axis. Two points are said to be 
 symmetrical with regard to an axis when they are on 
 opposite sides of a line (or axis) on a perpendicular to 
 the axis, and equidistant from the axis. Two figures are 
 said to be symmetrical with regard to an axis when any 
 two corresponding points are symmetrical with regard to 
 that axis. 
 
 The placing of figures in symmetrical positions, as in 
 Bk. I, Th. VII ( 105), is quite common, as is the use of 
 symmetrical points. Note also the figure of 100, and 
 exercise 120. 
 
 345. Positive and Negative Sects and Angles. There 
 are many theorems in Geometry where different cases of 
 the theorem require that sects or angles be added in one 
 case, subtracted in a second, and in the third one of the 
 things added or subtracted is zero. If these are arranged 
 in one general statement, it can be shown that the sects, 
 or angles, are all positive in the addition case, that one 
 pair has become zero in the zero case, and then negative in 
 the subtraction case. Examples of this can be found in 
 the proof of Th. VII, Bk. I (105), in 207, 214, and 
 215, and in 240 and 244. 
 
 346. Distance. Where the word " distance " is used, it 
 always means the shortest possible distance. 
 
 The distance between two points means the straight-line 
 distance. That this is the shortest line has been partly 
 proved in 109, where the straight line between two points 
 is shown to be less than any broken line between the 
 points. That it is less than any curved line between the 
 
APPENDIX 187 
 
 points has not been shown, but it can be proved quite 
 easily by showing that the shortest line between two 
 points must pass through any point on the straight line 
 between those points, and therefore through every point 
 on the straight line. 
 
 The distance 'from a point to a line is the perpendicular 
 from the point to the line ( 113). 
 
 The distance from a point to a circumference is the sect 
 from the point to the circumference on the line through 
 the center ( 174). 
 
 347. Limits. There is some doubt as to the perfect 
 accuracy of any of the proofs for the first limit theorem ; 
 the following proof is probably as free from objections 
 as any. 
 
 I. If two variables approaching limits are equal for all 
 values, their limits are also equal. 
 
 Given. v = v r ; v = L, v f =L. 
 
 To prove. L = L'. 
 
 Proof. I. v = L - x. 
 
 V' = L'X', where x and x r are quantities, 
 
 either positive or negative, which can 
 
 become indefinitely small, but cannot equal 
 
 zero. 
 
 II. v = v f (given), . . L x = L' x f (eq. same), 
 
 and L L r = x x' (eq. + , eq.). 
 III. L, L f and L L f are constants; x and x' can 
 each be made less than any constant except 
 zero. If L L r is not zero, x and x' can 
 each become less than l (z, i'), and then 
 x x r would be less than L L f , even 
 though the signs were such that the values 
 of x and x 1 were added. 
 
188 APPENDIX 
 
 IV. But this is impossible, since L L 1 x x f . 
 .-. L-L'=0. 
 
 II. (1) If a variable approaches the limit zero, its quo- 
 tient by a constant, and its product by a constant (other than 
 zero), approaches zero. 
 
 Given. v = ; c. 
 
 To prove. - = ; vc = 0. 
 
 Proof. I. Let k be any constant quantity, however 
 small. Then ck is a constant quantity, 
 and v can become less than ck. 
 
 1) 
 
 II. But if v < ck, then - <k (uneq. -5- eq.) . 
 
 
 
 HI. And -=0, since v^O (def.). 
 
 
 
 IV. Since - is less than k, a constant quantity, 
 
 
 
 however small, but is not zero, . . - = (def.). 
 
 c 
 
 Similarly v<? = 0. 
 
 (2) If a variable approaches any limit, its quotient by a 
 constant, and its product by a constant {other than zero), 
 will approach the quotient of its limit by the constant, and 
 the product of its limit by the constant. 
 
 Given. v = L ; c. 
 
 V L 
 
 To prove. c = c ; VC = LC. 
 Proof. I. If v = L, then L-v = (def.). 
 II. ,.() = 
 
 III. .-. = (def.). 
 
 Similarly vc = Lc. 
 
APPENDIX 189 
 
 III. If two variables are proportional to two constants, 
 their limits are proportional to the same constants. 
 
 v v' 
 
 Given. - = ; v = L, v' = L . 
 
 c c 
 
 L L' 
 
 To prove. ~ = ~J' 
 
 c c/ 
 
 Proof. I. Since v = L, - = (II). 
 
 c c ^ 
 
 II. Since v' = L' , -7 = ^- 
 
 III. But- = T- - = 7(0- 
 
 c c' c c' 
 
 348. Incommensurable Case. The incommensurable 
 case of Th. V, Bk. II, will be taken as an example of the 
 method to be applied to all theorems having the two cases. 
 
 Let the central angles AOB and COD in circle O have 
 no common divisor. Suppose an exact divisor of Z AOB 
 to be applied to Z COD as of ten* as possible, leaving a 
 remainder Z XOD, which is therefore less than the divisor 
 used. 
 
 Then Z AOB and Z COX have a common divisor, so 
 
 Z AOB AB , ^ ,1 
 
 = - by the commensurable case. 
 Z cox ex 
 
 But, if the divisor of Z AOB is taken smaller and 
 smaller, that is, is made to approach the limit zero, the 
 remainder Z XOD, being still smaller, will also approach 
 the limit zero. Therefore Z COX will approach the limit 
 Z COD. Also its arc, CX, will approach the limit CD. 
 
 But, since the variables Z COX and CX are proportional 
 to the constants* Z AOB and AB, their limits are also pro- 
 portional to those constants ; that is, = ^=- - 
 
 Z COD CD 
 
190 APPENDIX 
 
 This method of proof will apply to Th. II, Bk. Ill, to 
 Th. I, Bk. IV, and in fact to all proofs where the method 
 of a common divisor is used. 
 
 349. Similar Figures. Similar figures have already 
 been defined ( 282j) as those which have their correspond- 
 ing angles equal, and their corresponding sides propor- 
 tional. This, of course, applies only to polygons. The 
 following definition is sometimes given, and while it is 
 not as convenient for use in Plane Geometry, it has the 
 merit of applying to all kinds of figures, including those 
 of Solid Geometry. 
 
 Two figures that may be placed in a pencil (or sheaf, 
 in Solid Geometry) of lines, so that all pairs of correspond- 
 ing points of the figures cut the respective lines of that 
 pencil in the same ratio, are similar. 
 
 These two definitions are identical in result, as far as 
 polygons are concerned, as may be shown by proving the 
 two following statements : 
 
 1. If two polygons that are mutually equiangular and 
 have their corresponding sides proportional are placed with 
 one pair of corresponding sides parallel, the polygons lying 
 on the same side of those lines, the lines joining their 
 corresponding vertices will form a pencil which is cut pro- 
 portionally by the vertices of the polygons. 
 
 2. If two polygons lie in a pencil of lines, and their 
 vertices cut the lines proportionally ', the polygons are mutually 
 equiangular, and their corresponding sides are proportional. 
 
 350. The Evaluation of Pi. The ratio of the circum- 
 ference of a circle to its diameter is obtained by the 
 use of the ratios of the perimeters of regular inscribed 
 and circumscribed polygons of a large number of sides to 
 the diameter. The circumference is longer than the 
 
APPENDIX 191 
 
 perimeter of any' inscribed polygon, and shorter than 
 the perimeter of any circumscribed polygon, so that if the 
 lengths of the perimeters of two regular polygons of the 
 same number of sides, one inscribed, the other circum- 
 scribed, can be obtained in terms of the diameter, the 
 length of the circumference will lie between these values, 
 and an approximate value can therefore be obtained. 
 
 c 1 
 The value , or TT, cannot be obtained exactly, for it 
 
 lias been proved to be an incommensurable number. It is 
 evident, however, that the value obtained will be more 
 exact as the polygons used have a larger number of sides. 
 
 The approximation is started by finding the perimeters 
 of some regular polygon, inscribed and circumscribed ; 
 squares or hexagons are the easiest to use. The perim- 
 eters of polygons of twice the number of sides are then 
 worked out, and the doubling is continued until the value 
 is found to the required degree of accuracy. The value 
 has been carried to o'ver 700 decimal places, but for ordi- 
 nary purposes the value 3. 14159+ is sufficiently accurate. 
 3.1416 is also much used, and 3^ is a fair approximation 
 for rough work. 
 
 The following theorems give the material for the 
 numerical calculation : 
 
 I. The perimeter of a regular inscribed hexagon 
 equals 3D; the perimeter of a regular circumscribed 
 hexagon equals 2 D v3. 
 
 II. If the perimeters of regular inscribed and cir- 
 cumscribed polygons of any number of sides are known, 
 the perimeter of the regular circumscribed polygon of 
 double the number of sides equals twice their product 
 divided by their sum. 
 
 III. If the perimeters of the regular inscribed poly- 
 
192 APPENDIX 
 
 
 ij>on of any number of sides, and the regular circum- 
 scribed polygon of double that number of sides, are 
 known, the perimeter of the regular inscribed polygon 
 having double the first number of sides equals the 
 square root of their product. 
 
 Applying the formula in II to the regular hexagons, 
 calling the perimeter of a regular circumscribed dodeca- 
 gon P : 
 
 p= 2(3D)(2W3) = 12W3 =12 
 
 3D + 2W3 3 + 2V3 
 or, p = (3.21539+)z>. 
 
 Applying the formula in III to this value and the perim- 
 eter of the regular inscribed hexagon, and calling the 
 perimeter of the regular inscribed dodecagon p : 
 
 Applying these methods to the dodecagons, the perime- 
 ters of circumscribed and inscribed regular polygons of 24 
 sides are, respectively, (3.15966-)Z) and (3.13263~)D. 
 
 It is already evident that the circumference of the 
 circle, since its value is between those found, must be 
 (3.1 + )D, and a continuance of the work will determine 
 additional figures. 
 
YB 17314 
 
 M306177 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY