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A SYLLABUS 
 
 MODERN PLANE GEOMETRY. 
 
 
 7i is requested that any observations or criticisms upon this Syllabus 
 
 may be addressed to the Sccretaiy, the Rev. J. J. Milne, Invermark, 
 
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 7 
 

 A SYLLABUS OF 
 MODERN PLANE GEOMETRY. 
 
 I. -INTRODUCTION. 
 
 (i) In Elementary Geometry a straight line of definite length, 
 or a SEGMENT, is considered wj^th reference to absolute length only, 
 and the question of whether it is to be added to or subtracted from 
 another line is a matter of explicit statement. 
 
 In the Modern Plane Geometry account is taken of the direction 
 of measurement along a straight line, and this is expressed with 
 the help of the signs + and — . 
 
 The fundamental rules with regard to these signs are that 
 (i) if A and B be any two points on a straight line, 
 
 AB = - BA, BA = - AB ; 
 
 an immediate consequence of which is that 
 
 -(-AB) = AB, 
 
 (ii) if A, B, P be any three points on a straight line, 
 
 AP + PB = AB. 
 
 (2) Two fixed points (A and B) being taken on a straight line, 
 the position of any other point P (on the same straight line) is 
 given when the ratio of the segments AP : PB is given. 
 
4 .-{ SYLLABUS OF 
 
 If AP : PB is positive, AB is internally divided in P. 
 If AP : PB is negative, AB is externally divided in P. 
 In either case P is nearer to A or to B, according as AP : PB 
 numerically < or > unity. 
 
 AP 
 
 If — = + I, then P bisects AB. 
 PB 
 
 \P 
 
 If : — = - I, then P is at an infinite distance. 
 PB 
 
 The point at an infinite distance beyond A is identically the 
 same point as that at an infinite distance beyond B ; for 
 when P is at A, AP : PB is zero ; 
 when P is at the middle point of AB then AP : PB is + i, 
 
 as P approaches B the ratio continually increases in value, 
 becoming equal in the limit to + =c ; 
 
 as P passes through B the sign of the ratio changes and 
 gradually decreases numerically from - x (in the limit) to 
 - I, which is the value it assumes when P is at an infinite 
 distance beyond B. 
 
 Up to this point the value of the ratio has undergone a continuous 
 change, and the motion of P has also been continuous. As P 
 moves from the point at an infinite distance beyond A to A, the 
 ratio gradually decreases numerically in value from - i till in the 
 limit it assumes the valae zero. The changes in the value of the 
 ratio have been continuous throughout. This, then, points to the 
 conclusion that the motion of P has also been continuous ; that is, 
 that the point at an infinite distance beyond B may be considered 
 as identically the same as that at an infinite distance beyond A, 
 and that the straight line may be considered as a continuous 
 locus. 
 
 Note. — When a straight line AB is mentioned, a confusion is 
 likely to arise in the mind of the student as to whether the Finite 
 straight line AB is meant, or the continuous Locus of which AB 
 
MODERN PLANE GEOMETRY. 5 
 
 is a portion. It may be safely itiferred that tJie latter is always 
 meant except where the lengtii AB is expressed or implied. 
 
 (3) Ratios may be compounded to any extent in the way of 
 either multiplication or division. It is usual to write 
 
 a h c d e , , 
 
 _._._._._= J og ahcde = p(]rst : 
 p q r s t ' 11^ 
 
 AB, AB _ I _^ I _ ^ 
 
 Ac"^Ab ~ "' ""' AC ^ AD AB' 
 
 and so forth ; but it must be distinctly understood that the second 
 
 equations are simply convenient modes of representing the first, 
 
 and are not to be taken as implying that there is any geometrical 
 
 2 
 meaning assignable to the quantities abcde, — ;.' ^^'^•' taken sepa- 
 
 AB 
 
 rately. 
 
 (4) The following are illustrations of relations between the 
 segments of a straight line. 
 
 [i] If A, B, C be three points in a straight line, 
 AB + BC ^- CA = o. 
 
 [ii] If P be any other point in the straight line, 
 
 PA . BC + PB . CA + PC . AB = o. 
 
 [iii] PA2. BC + PB2. CA + PC2. AB = - BC . CA . AB. 
 
 [iv] If P bisect BC, AP = \ (AB + AC). 
 
 [v] If A, B, C, D be four points in a straight line, and P bisect 
 AB and Q bisect CD, 
 
 PQ = 5 (AC + BD) = I (AD + BC). 
 
6 A SYLLABUS OF 
 
 II.— PROPERTIES OF A TRIANGLE. 
 Definitions. 
 (i) Any straight line drawn across a system of lines is called 
 
 a TRANSVERSAL. 
 
 (2) If three or more straight lines intersect in the same point, 
 they are said to be concurrent. 
 
 (3) If three or more points lie on the same straight line they 
 are said to be collinear. 
 
 i (4) If four or more points lie on the circumference of the same 
 circle, they are said to be concyclic. 
 
 Theorem. 
 
 1 If transversals through the angular points A, B, C of a triangle 
 are concurrent and intersect the opposite sides in D, E, F, 
 respectively, then 
 
 BD . CE . AF = DC . EA . FB (Ceva's Theorem). 
 
 2 If D, E, F, be points in the sides BC, CA, AB, respectively, 
 of a triangle ABC, such that 
 
 BD . CE . AF = DC . EA . FB, 
 then AD, BE, CF are concurrent (converse of Theorem i). 
 
 3 The three internal and three external bisectors of the angles 
 of a triangle meet three by three in four points, the centres 
 of the inscribed and escribed circles, (the in- and ex-centres). 
 
 4 The straight lines joining the angular points of a triangle to 
 the middle points of the opposite sides (the medians) meet 
 in a point (the centeoid). 
 
 5 The perpendiculars from the angular points of a triangle on 
 the opposite sides (the altitudes) meet in a point (the 
 orthocentre). 
 
 6 The lines joining the angular points of a triangle to the points 
 where the incircle touches the sides are concurrent. 
 
MODERN PL AXE GEOMETRY. 7 
 
 7 If a transversal meet the sides BC, CA, AB of a triangle 
 in D, E, F, respectively, then 
 
 BD . CE . AF= -DC . EA . FB (Menelaus' Theorem). 
 
 8 The converse of Theorem 7. 
 
 9 The three internal and three external bisectors of the angles 
 of a triangle meet the opposite sides in six points, which lie 
 three by three in four straight lines. 
 
 ID The feet of the perpendiculars drawn from any point in the 
 circumference of the circumscribing circle of a triangle on the 
 sides are collinear (the Simson or pedal line). 
 
 In the rest of this chapter the notation for the triangle ABC 
 will be as follows : 
 
 A', B', C, are the middle points of the sides BC, CA, AB, 
 
 respectively ; 
 AD, BE, CF, the perpendiculars on these sides; 
 G, H, the centroid aad orthocentre respectively ; 
 
 0, the circumcentre, N, the nine-point centre ; 
 
 1, I^, Ig, I3, the incentre and excentres respectively ; 
 X, Y, Z, the points where the incircle, and 
 
 X^, Yj, Zj^, the points where the excircle opposite A touches 
 
 the sides BC, CA, AB respectively ; 
 R, r, Tj, r.„ To, the circum -, in -, and exradii respectively ; 
 a, b, c, s, A, the lengths of the sides, semiperimeter, and area. 
 Theorem 
 
 11 AZ + BC = BX + CA = CY + AB - semiperimeter; 
 
 or AZ = YA = 5 — <7, etc. 
 
 12 AZ]^ = BX^ + AB = X^C -f CA = semiperimeter; 
 or AZi = YiA = :r; BX^ = BZ^ = s-c ; X^C = YjC = J--6. 
 Cor. XX^ = ^ - ^, XX, = b, XX3 = c 
 
 13 ^ = rs = 7\ {s-a) = r^ (s~b) = r.^ (■*--'•)■ 
 
 14 AA'=3GA'; BB' = 3GB'; CC' = 3GC'. 
 
8 .4 SYLLABUS OF 
 
 15 If the straight lines which bisect the interior and exterior 
 angles between AB and AC meet the circumcircle in P, P', 
 then PP' is the diameter of the circle which bisects EC. 
 
 16 PI = PB = PC. 
 
 17 If PM be perpendicular to AB or AC, then 
 
 2AM = AB + CA; 2MB = AB-CA. 
 
 18 4RA = abc. 
 
 19 Of the four points A, B, C, H, any one is the orthocentre 
 of the triangle formed by joining the other three. 
 
 20 Angle BHF = CHE - CAB ; 
 Angle CHD = AHF = ABC: 
 Angle AHE = BHD = BCA. 
 
 21 The triangles AEF, DBF, DEC are similar to ABC. 
 
 22 AI, BI, CI, bisect the angles OAH, OBH, OCH, respec- 
 tively. 
 
 23 A, B, C, H are the centres of the four circles which touch the 
 sides of DEF (the orthocentric or pedal triangle). 
 
 24AH = 20A'; BH = 20B'; CH = 20C'. 
 
 25 If AD be produced to meet the circumcircle in Q, then 
 HD = DQ. 
 
 26 Rectangle AH . HD = BH . HE = CH . HF. 
 
 27 A circle passes through A', B', C, D, E, F, and the middle 
 points of HA, HB, HC (the nine-point circle). 
 
 Cor. — The centre of this circle is the middle point of OHj 
 and its diameter = R. 
 
 28 G lies on OH, and OH = 3OG. 
 
 29 I02 = R2-2R;-. 
 
 30 IH2=2r2-AH . HD. 
 31. OH2-R2-2AH . HD. 
 32 NI = m-r. 
 
 Cor. The nine-point circle touches the incircle and each 
 excircle. 
 
MODERN PLANE GEOMETRY. 9 
 
 ANTIPARALLELS, SYMMEDIAN LINES AND POINT. 
 
 If ABC be a triangle and any points b, c be taken on CA, AB 
 respectively in such a manner that the angles Kbc, ABC are equal, 
 then will also the angles kcb^ ACB be equal and the triangles Kbc 
 ABC similar ; and the base be is said in this case to be antiparallel 
 to the base BC. (Compare Euclid I. 28, Syllabus of Pla7ie 
 Geometry, p. 51.) 
 
 The points B, c, b, C are concyclic. 
 
 The antiparallel be is fixed as regards direction, being always 
 parallel to the tangent to the circumcircle at A. 
 
 Let PQR be the triangle formed by the tangents to the circum- 
 circle at A, B, C so that PB = PC, QC = QA, and RA = RB. Then 
 if yP^ be drawn parallel to QR meeting CA, AB in yS, 7 
 respectively 
 
 Py = PB = PC = VjB. 
 
 Hence every antiparallel to BC is bisected by AP, Similarly 
 antiparallels to CA, AB are bisected by BQ, CR. 
 
 A line drawn through the vertex of a triangle so as to bisect 
 an antiparallel to the base is called a symmedian. 
 
 If A', B', C be the middle points of the sides of ABC, parallels 
 to those sides are bisected by the "medians" AA', BP/, CC 
 concurring at G the centroid. Analogously, antiparallels are 
 bisected by the " symmedian " lines AP, BQ, CR concurring 
 at K the symmedian point (point de Lemoine). 
 
 The lines joining K, G to any vertex of the triangle are equally 
 inclined to the bisector of the angle. Pairs of points possessing 
 this property are termed " isogonal conjugates," and the rectangles 
 under their perpendiculars on the three sides are equal. 
 
 The symmedian point K has its perpendicular distances from 
 the sides respectively proportional to those sides, and the sum 
 of the squares of those distances a minimum. 
 
 The sides of the orthocentric triangle are antiparallel to the sides 
 of the original triangle, and are respectively perpendicular to the 
 corresponding radii of the circumcircle. . 
 
 B 
 
A SYLLABUS OF 
 
 Brocard Points. 
 
 If a circle be described touching AB in A and passing through 
 C, and AR be the chord parallel to BC, then BR will meet the 
 circle again in a point O such that 
 
 the angle QBC = fiCA = fiAB. 
 
 Each of these angles is called the Brocard angle of the triangle 
 and is denoted by w. 
 
 The Brocard angle of a triangle depends only upon the shape 
 of the triangle and is therefore the same for all similar triangles. 
 It lies between o° and 30°. 
 
 The point O is called the posi'/ive Brocard point. A negative 
 Brocard point fi' exists, for which 
 
 the angle li'CB = fi'AC = fi'BA = w. 
 
 The points fi, fi' are " isogonal conjugates " and consequently 
 the rectangle under their perpendiculars upon any side is the 
 
 same. 
 
 The angle BflC = supplement of C = Crj'A 
 CfiA = „ A = AQ'B 
 
 AQB = „ B = BO'C. 
 
 In the following articles the triangle ABC is supposed to be 
 lettered in such a manner that A is the upper vertex, B the left and 
 C the right extremity of the base. 
 
 Triplicate Ratio Circle. 
 
 If through the symmedian point K parallels F'KE, D'KF, E'FD 
 be drawn to the sides, 
 
 (i) The six points D, D', E, E', F, F' all lie on a circle whose 
 centre o- is the mid-point of KO, O being the circum-centre. 
 
 (2) DD' : EE' : FF' = a^ : l)^ : r', and this circle is consequently 
 termed the Triplicate-ratio circle. 
 
 (3) The triangles FDE, ET'D' are equal to one another, and 
 are each similar to ABC. 
 
MODERN PLANE GEOMETRY. ii 
 
 (4) If fi, O' be the Brocard points of ABC, then fi, K and 
 K, O' are the Brocard points of FDE, E'F'D'. 
 
 Brocard Circle. 
 
 The circle described upon OK as diameter, and which has con- 
 sequently the same centre cr as the T. R. circle, is termed 
 the Brocard circle. 
 
 The points fi, O' lie upon the Brocard circle ; and 
 the angle OcrK = fi'o-K = 20). 
 
 Cosine Circle. 
 
 If through the symmedian point K be drawn the antiparallels 
 /K/, ^K/', ^K^' to BC, CA, AB respectively, 
 
 (i) The six points </, d', e, e',f,f' all lie on a circle termed the 
 "Cosine circle," whose centre is K. 
 
 (2) Any figure such as def'd' is a rectangle. 
 
 Taylor Circle. 
 
 If a, /?, 7 be the mid-points of the sides of the orthocentric 
 
 triangle 
 
 and /3y meet CA, AB in v\ w respectively 
 
 ya ,, AB, BC in w, u 
 
 ttyS ,, BC, CA in ii , v 
 then the six points u, ?/, t\ v, w, w all lie on the circumference of 
 a circle termed the "Taylor circle" of the triangle, whose centre 
 is the in-centre of the triangle a^y. 
 
 Tucker Circles. 
 
 If upon KA, KB, KC points A', B', C be taken such that 
 KA' : KA = KB' : KB = KC : KC = a constant ratio, 
 
 and B'C meet the sides CA, AB in Y, Z' respectively 
 CA' „ AB, BC in Z, X' „ 
 
 A'B' „ BC, CA in X, Y' 
 
 B 2 
 
■^e—-.: 
 
 12 A SYLLABUS OF 
 
 (i) Then the six points X, X', Y, Y', Z, Z', all ].ie upon the cir- 
 cumftt-ence of a circle, termed a "Tucker circle," whose centre 
 bisects the distance between the circumcentres of AEC, A'B'C 
 (coUinear with K). 
 
 (2) The triangles ZXY, Y'Z'X' are equal to each other, and are 
 each similar to ABC. 
 
 The circum-, Triplicate ratio, Cosine, and Taylor circles are 
 each particular cases of Tucker circles. 
 
 III.— HARMONIC RANGES AND PENCILS. 
 Definitions. 
 
 If A, B be two points in a straight line, and C, D two other 
 points in the line so taken that 
 
 AC AD 
 
 BC ^ " BD' 
 then the points A, C, B, D form a harmonic range ; and C and 
 D are harmonic conjugates with respect to A and B ; AC, AB, 
 AD are said to be in harmonic progression ; and AB is said 
 to be the harmonic mean between AC and AD. [These latter 
 definitions agree with those given in algebra.] 
 
 In order that the above proportion may hold, one and only one 
 of the points C, D must lie between A and B. 
 
 Theorem. 
 
 1 If C and D are Harmonic Conjugates with respect to A and 
 B, then A and B are Harmonic Conjugates with respect 
 to C and D. 
 
 2 If A, C, B, D form a harmonic range, then 
 
 112 
 
 AC "^ AD " AB" 
 
 Note. — In this and the follo^aiJig Proposition it should be 
 ttnder stood that account is taken of the direction of vieasureme7it 
 along the straight line. 
 
MODERN PLANE GEOMETRY. .. 13 
 
 3 If O be the middle point of AB, then 
 
 0A2 = 0B2 = OC. OD, 
 also DA. DB = DC. DO. 
 
 Cor. If C be the middle point of AB, then the Harmonic 
 Conjugate of C with respect to A and B is at an infinite 
 distance. 
 
 4 If S be a point without the straight line in which is the 
 harmonic range ACBD, and if through C a line be drawn 
 parallel to SD meeting SA, SB at G, H respectively, then 
 
 GC = CH. 
 
 5 If in the figure of the last proposition any other transversal be 
 drawn meeting SA, SB, SC, SD in A', B', C, D' respectively, 
 then A'C'B'D' is a harmonic range. 
 
 Definitions. 
 
 A system of four straight lines SA, SC, SB, SD met by a 
 transversal ACBD, so that A, C, B, D is a harmonic range, is 
 called a harmonic pencil ; the point S is called the vertex of 
 the pencil ; and each of the four straight lines is called a ray ; 
 SC, SD are said to be harmonic conjugates of SA, SB. 
 Theorem 
 
 6 The legs of any angle form with its internal and external 
 bisectors a harmonic pencil. 
 
 7 If in a harmonic pencil one ray bisect the angle between the 
 two rays not conjugate to it, then it is perpendicular to its 
 conjugate. 
 
 8 If in a harmonic pencil one pair of conjugate rays be at right 
 angles, then these are the internal and external bisectors of 
 the angle between the other pair. 
 
 9 If two harmonic ranges are taken, one in each of two straight 
 lines, and if three of the straight lines which connect corre- 
 sponding points meet in a point, then the fourth connector 
 passes through the same point. 
 
14 A SYLLABUS OF 
 
 Cor. If two corresponding points coincide in the inter- 
 section of the two lines, then the three connectors of the 
 three other pairs of corresponding points are concurrent. 
 
 10 If two harmonic pencils are such that the three points of 
 intersection of three pairs of corresponding rays are in a 
 straight line, then the point of intersection of the fourth pair 
 of rays is in the same straight line. 
 
 Cor. If two of the rays, one in each pencil, are coincident, 
 then the three points of intersection of the other three pairs 
 of rays are coUinear. 
 
 IV.— PROPERTIES OF A COMPLETE QUADRILATERAL 
 AND COMPLETE QUADRANGLE. 
 
 Definitions. 
 
 A system of four straight lines (no three of which are con- 
 current) in which the six points of intersection are taken into 
 account is called a complete quadrilateral. The four lines 
 are called the sides of the quadrilateral ;' and the six points the 
 vertices. Two vertices which do not lie on the same " side " 
 are called opposite vertices. The three straight lines joining 
 " opposite vertices " are called diagonal lines, and the triangle 
 formed by the " diagonal lines " is called the diagonal triangle 
 of the complete quadrilateral. 
 
 Note. — This is Cremona s 7Wtatio7i. It linll he obseri'ed that a 
 complete quadrilateral with its diagonal triangle cofisists of seven 
 straight lines and nine points of intersection. 
 
 Theore!\i 
 
 I In a complete quadrilateral each pair of opposite vertices 
 forms with two of the angular points of the diagonal triangle 
 a harmonic range. 
 
MODERN PLANE GEOMETRY. 15 
 
 Note.— ^_y this proposition tae obtain a method of finding 
 with the ruler only the Jiarmonic conjugate of a give?i stf-aight 
 line with respect to two other straight lifies with which it is 
 concurrent. 
 
 Definitions. 
 
 A system of four points (no three of which are collinear) in 
 which the six straight lines of connection are taken into account 
 is called a complete quadrangle. The four points are called 
 VERTICES of the quadrangle, and the six straight lines the sides. 
 Two sides which do not pass through the same " vertex " are 
 called OPPOSITE sides. The three points of intersection of 
 " opposite sides " are called diagonal points, and the triangle 
 formed by joining the "diagonal points " is called the diagonal 
 TRIANGLE of the complete quadrangle. 
 
 Note. — This again is Cremona's notation. It zuill be observed 
 that a complete quadrangle with its diagonal triangle consists of seven 
 points and nine connecting straight li7ies. 
 Theorem 
 
 2 In a complete quadrangle each pair of opposite sides forms 
 with two of the sides of the diagonal triangle a harmonic 
 pencil. 
 
 Note. — By this propositioji we obtain a method of finding 
 with the ruler only the harmo?iic conjugate of a given point 
 7vith respect to tiuo other points with which it is collinear. 
 
 Principle of duality. The definitions relating to a complete 
 quadrilateral and complete quadrangle are such that each can be 
 obtained from the other by the interchange of the terms " point " 
 and " line," " vertex " and " side." The reason of this is that in 
 the first case the figure is regarded as consisting of a system of 
 straight lines which intersect in points, whilst in the latter we re- 
 gard it as a system of points which are connected by straight lines, 
 just in the same way as a curve may be considered either as the 
 envelope of a series of straight lines or as a system of points. 
 
1 6 A SYLLABUS OF 
 
 Introducing the idea of harmonic section we obtain Theorem i 
 or Theorem 2 according as we take lines or points as elements in 
 the description of the figures, and each Theorem can be obtained 
 from the other by interchanging the terms 'point' and Mine,' 
 ' range ' and pencil. 
 
 It will be seen that every descriptive theorem or theorem of 
 position is such that from it can be derived another corresponding 
 or correlative theorem by means of the above and similar changes. 
 Thus corresponding to "concurrent lines" we have "collinear 
 points," to "line joining two points" we have "point of inter- 
 section of two lines," to " points forming a harmonic range " we 
 have "lines forming a harmonic pencil," &:c. This correlation of 
 " point " and " line " is termed the " principle of duality." Each 
 of two descriptive theorems so correlated is said to be the dual 
 of the other, and it will be found that if any descriptive property 
 is demonstrated, its dual also holds. 
 
 3 The middle points of the portions of the three diagonal lines 
 of a complete quadrilateral intercepted between opposite 
 vertices are collinear. 
 
 v.— PROPERTIES OF A CIRCLE. 
 
 Theorem 
 
 1 If C, D be harmonic conjugates with respect to A, B the ex- 
 tremities of a diameter of a circle, and if P be any point on the 
 circumference, then, regarding only the magnitudes of the lines, 
 
 PC _ AC ^ EC 
 PD AD BD' 
 
 Cor. — PA and PB are the bisectors of the angles between 
 PC and PD. 
 
 2 If C, D be two fixed points, and P be a point such that the 
 ratio PC : PD is constant, then the locus of P is a circle except 
 when the constant is unity, and then the locus becomes a 
 straight line. 
 
MODERN PLANE GEOMETRY. 17 
 
 Def. — If C, D be harmonic conjugates with respect to 
 A, B the extremities of any diameter of a given circle, then 
 C, D are termed inverse points with respect to the circle. 
 The term inverse comes from the fact that if O be the centre 
 of the circle, OC . OD = OA'', and consequently OC, OD 
 vary inversely. 
 
 3 If C, D be inverse points with respect to a circle whose centre 
 is O, and one of the two points C describe a circle, the inverse 
 point D will describe another circle, except when the circle 
 which C describes passes through O, in which case D will 
 describe a straight line. 
 
 Note. — // is from the principle of inve?-sion in the latter 
 portioti of tills theorem that Peaucellier Invented the system of 
 linkages by which rectilinear motion may be derived from circular 
 motion. 
 
 4 Every straight line through O cuts any curve and its inverse 
 at supplementary angles. 
 
 5 Two loci cut at the same angle as their inverses cut. 
 
 Poles and Polars. 
 
 Def. — If C, D be harmonic conjugates with respect to 
 A, B the extremities of a diameter of a circle, and if through 
 1) a line DE be drawn parallel to the tangent at A or B, then 
 DE is called the polar of C, and C the pole of DE with 
 respect to the circle. 
 
 Harmonic Property of Pole and Polar. 
 
 6 Any straight line through the pole is cut harmonically by the 
 circle, pole, and polar. 
 
 Cor. — If a straight line is drawn through a point C in the 
 plane of a circle, the locus of the harmonic conjugate of C 
 with respect to the points in which the line cuts the circle is 
 a straight line. 
 
3 A SYLLABUS OF 
 
 Reciprocal Property of Pole and Polar. 
 
 7 If any point E is taken on the polar DE of a point C, then 
 the polar of E passes through C. 
 
 8 The polar of any point with regard to a circle is the locus of 
 the intersections of tangents at the extremities of chords 
 through the point. 
 
 Cor. — The polar of a point external to the circle is the 
 chord of contact of tangents drawn from that point, and the 
 polar of a point on the circumference is the tangent at the 
 point. 
 
 Def. — A triangle of which each side is the polar of the 
 opposite vertex with regard to a circle is said to be self- 
 conjugate with respect to the circle. 
 
 9 The centre of the circle to which a given triangle is self-con- 
 jugate is at the orthocentre of the triangle, and its radius is a 
 mean proportional between HA and HD (see Notation of 
 Sect. II.). 
 
 Note. — Ifi order that there may be such a circle, the ortho- 
 centre must be exteriial to the triangle, that is the triangle must 
 be obtuse. 
 
 Pole and Polar Properties of a Quadrangle. 
 ID If a complete quadrangle be inscribed in a circle, its diagonal 
 triangle is self-conjugate with respect to the circle. 
 
 Note. — This theorem enables us with a ruler only, i° to draiv 
 the polar of a given point, 2° to find the pole of a gii'en straight 
 line with respect to a given circle, 3° to drata a tangent to the 
 circle from a gii'eii point. 
 
 11 The diagonal triangle of a complete quadrangle inscribed in a 
 circle is also the diagonal triangle of the complete quadrilateral 
 whose sides touch the circle at the vertices of the quadrangle. 
 
 12 The polars of the four points of a harmonic range form a 
 harmonic pencil ; and, conversely, the poles of the four rays 
 of a harmonic pencil form a harmonic range. 
 
MODERN PLANE GEOMETRY. 19 
 
 VI.— PROPERTIES OF TWO OR MORE CIRCLES. 
 Orthogonal Circles. 
 
 Definition. 
 
 Two circles which cut each other so that the tangents at the 
 points of section are at right angles are said to be orthogonal. 
 
 Theorem 
 
 1 If two circles are orthogonal, the tangent to either at a point 
 of section passes through the centre of the other. 
 
 2 If two points C, D are harmonic conjugates with respect to 
 two others A, B, the circles described on AB and CD as 
 diameters are orthogonal. 
 
 3 If two circles are orthogonal, every diameter of either which 
 meets the other is cut harmonically. 
 
 4 Circles are ortliogonal if the angles in the segments on 
 opposite sides of the chord of intersection are complementary. 
 
 5 The locus of the points of intersection of the straight lines 
 joining two fixed points on a circle to the extremities of a 
 variable diameter is the circle through the fixed points ortho- 
 gonal to the given circle. 
 
 Radical Axis. 
 
 6 The locus of a point from which tangents drawn to two given 
 circles are equal is a straight line at right angles to the line 
 joining the centres of the circles, and dividing this line so that 
 the /difterence of the squares on the segments is equal to the 
 difference of the squares on the radii of the circles.^ 
 
 CoR. — If the two circles intersect, this locus is the common 
 chord of the two circles. 
 
.o A SYLLABUS OF 
 
 Def. — The straight Hne which is the locus of the point from 
 which tangents drawn to two given circles are equal is called 
 the RADICAL AXIS of the two circles. 
 
 7 If circles be drawn having their centres in a fixed diameter 
 (produced) of a given circle and orthogonal to it, the radical 
 axis of any pair of the circles is the straight line through the 
 centre of the given circle at right angles to the fixed 
 diameter. 
 
 Cor. — No pair of such a system of circles intersect each 
 other. 
 
 Defs. — If a system of circles be such that every pair has 
 the same radical axis, then the circles are said to be 
 
 CO-AXAL. 
 
 With respect to a series of co-axal circles described as in 
 Theorem 7, the extremities of the fixed diameter of the given 
 circle are called the limiting points of the system, and may 
 be considered as circles of the system whose radii are inde- 
 finitely diminished. 
 
 Note. — In a system of intersecting co-axal circles there are 
 no limiting points ; or, as it is more iisual to say, the limiting 
 points ai'e imaginary. 
 
 8 The circle on a common tangent to two of a system of 
 co-axal circles as diameter meets the line of centres in the 
 limiting points, and these points are inverse to each other 
 with respect to each circle of the system. 
 
 9 The three radical axes of a system of three circles are 
 concurrent. 
 
 Def. — The point of concurrence of the three radical axes 
 of a system of three circles is called the radical centre of 
 the circles. 
 
 10 The radical axes of a fixed circle and each of a system of 
 co-axal circles are concurrent. 
 
MODERN PLANE GEOMETRY. 21 
 
 11 The polars of a fixed point with regard to a system of co-axal 
 circles are concurrent. 
 
 12 If there are two systems of co-axal circles, one of which passes 
 through two fixed points and the other has these two fixed 
 points for its limiting points, then any circle of either system 
 is orthogonal to all the circles of the other system. 
 
 13 The circumcircle of a triangle, the nine-point circle, and the 
 circle to which the triangle is self-conjugate are co-axal. 
 
 Centres of Similitude. 
 
 14 The straight lines joining the extremities of two parallel 
 diameters of two given circles intersect in two fixed points 
 which divide the line joining the centres of the circles inter- 
 nally and externally into segments which have the same ratio 
 as the radii of the circles. 
 
 Def. — These two fixed points are called respectively the 
 internal and external centres of similitude of the two 
 circles. 
 
 15 The common tangents of two circles pass through one or 
 other of the centres of similitude. 
 
 16 Any line drawn through two circles and one of their centres 
 of similitude is cut by the circles in four points, which are 
 two and two at the extremities of parallel radii ; and the 
 portion of the line between either pair of parallel radii is 
 divided by the centre of similitude into segments which have 
 the same ratio as the radii. 
 
 17 If through a centre of similitude of two circles, two straight 
 lines be drawn, meeting one circle in R, R', S, S', and the 
 other in r, r', j-, s' ; then the pairs of chords RS, rs; R'S', 
 rs' ; RS', rs' ; R'S, rs, are parallel ; and the pairs RS, r's ; 
 R'S', rs ; RS', r's ; R'S, rs', intersect on the radical axis of 
 the circles. 
 
'.2 A SYLLABUS OF 
 
 i8 The centres of similitude of three circles are the six vertices 
 of a complete quadrilateral of which the centres of the circles 
 are the angular points of the diagonal triangle. 
 
 Defs. — The four straight lines on which the six centres of 
 similitude of three circles lie are called axes of similitude. 
 The circle described on the line joining the centres of simili- 
 tude of two circles as diameter is called the circle of 
 
 SIMILITUDE. 
 
 19 The three circles of similitude of three given circles are 
 co-axal. 
 
 VIL— GEOMETRICAL ^lAXIMA AND MINIMA. 
 
 Theorem 
 
 1 The rectangle of the segments of a given straight line is a 
 maximum, and the sum of the squares on them a minimum, 
 when the segments are equal. 
 
 2 The maximum or minimum straight line from a given point 
 to a circle is the normal through the point. 
 
 3 Given a circle and a point within it, the chord which is 
 bisected by the point (i) is a minimum, (2) cuts off a min- 
 imum area. 
 
 4 Given two straight lines and a point within the angle formed 
 by them, of all straight lines drawn through the point that 
 which is bisected by it cuts off the minimum triangle. 
 
 5 Given two sides of a triangle in magnitude, the area is a 
 maximum when they contain a right angle. 
 
 6 Of all triangles on the same base the isosceles has 
 
 (i) Minimum perimeter when the area is given. 
 (2) Maximum area when the perimeter is given. 
 
 7 Of all triangles the equilateral has 
 
 (i) Minimum perimeter when the area is given. 
 (2) Maximum area when the perimeter is given. 
 
MODERN PLANE GEOMETRY. 23 
 
 8 Of all polygons of the same number of sides the regular 
 polygon has 
 
 (i) Minimum perimeter when the area is given. 
 (2) Maximum area when the perimeter is given. 
 
 9 If a tangent to a circle cut two other tangents drawn from a 
 fixed point, that tangent so drawn will be a minimum which 
 is bisected at its point of contact, and will cut off a maximum 
 or a minimum triangle according as it includes or excludes 
 the given circle. 
 
 10 Of all polygons of a given number of sides circumscribing a 
 circle, that which is regular is a minimum both in perimeter 
 and area. 
 
 11 Of all triangles inscribed in a circle on a given chord as base, 
 the isosceles triangle is a maximum both in perimeter and 
 area. 
 
 12 If two convex polygons be on the same base and the vertices 
 of one lie upon the sides or within the area of the other, this 
 other has the greater perimeter. 
 
 13 Of all polygons of a given number of sides inscribed in a 
 circle, that which is regular is a maximum both in perimeter 
 and area. 
 
 14 Of all quadrilaterals which can be formed from four straight 
 lines of given lengths, the maximum is that which can be 
 inscribed in a circle. 
 
 Cor. I. — Of all polygons which can be formed from straight 
 lines which are given both in number and length, the maxi- 
 mum is that about which a circle can be described. 
 
 Cor. 2. — The area of a circle is greater than that of any other 
 closed figure having the same perimeter. 
 
 15 If two of the lines in 14 be crossed, the difference of the 
 areas of the triangles so formed is a minimum when the 
 vertices are concyclic. 
 
24 A SYLLABUS OF 
 
 VIIL— CROSS RATIOS, INVOLUTION, AND 
 RECIPROCAL POLARS. 
 
 Definitions. 
 
 A system of collinear points is called a pange; the straight line 
 in which the points lie is called the base of the range ; each of the 
 collinear points is called a point or element of the range. A 
 system of concurrent straight lines is called a pencil ; the point 
 where the lines meet is called the vertex of the pencil ; each of 
 the concurrent lines is called a ray or element of the pencil. 
 
 AC AD 
 
 If a range consists of four points A, B, C, D. the ratio vrp;; : -57^ is 
 
 called the cross ratio or anharmonic ratio of the range, and is 
 
 written (ABCD) ; the points A and B are considered in this 
 
 definition to be conjugate points. 
 
 Note. — T/ie cross or anharmonic ratio (ABCD) is defined in 
 
 various mays by diffet'ent ti'riters^ the most usual being the ratio 
 
 AB CB AB.CD rr^^ j r ■.■ , • • , ^ 
 
 -r^fi : -z^TT^ or , ihe dennition 7ve have nven is that of 
 
 AD CD AD.CB ■' <•> y 
 
 Cremona, tvhich has the advantas^es of inaking A and B conjugate 
 points, and also of fitting in readily with our notation in the chapter 
 
 071 Harmonic ranges a?id pencils. It will be observed that if 
 
 {ABCD)= — I, the7i ACBD is a harmonic range. 
 
 Since the four letters A, B, C, D, may be arranged ifi tiueiity- 
 four different ways, there are tioenty-four cross ratios with the same 
 four points. In these tiveniyfour cases each 7-atio appears four 
 
 times, and the twe7ityfour 7-atios are thus reduced to six differc7it 
 
 1-atios, of which tJiiee are the reciprocals of the other thz-ee. 
 
 Theorem 
 
 I If the cross ratio (AbCD) = X, then the other five ratios 
 derived from the same four collinear points are 
 
 I .1 A— I X 
 
 X I -X A A— I 
 
MODERN PLANE GEOMETRY. 25 
 
 2 If S be a point without the range ABCD, and if through C a 
 line be drawn parallel to SD, meeting SA, SB, in G, H, 
 respectively, then GC : HC = (ABCD). 
 
 3 If in the figure of the last proposition any other transversal be 
 drawn meeting SA, SB, SC, SD, in A', B', C', D', respec- 
 tively, then the cross ratio (A'B'C'D') = (ABCD). 
 
 Note. — We may speak of the cross or atiharmojiic ratio of a 
 pe?icil SA, SB, SC, SD, of four rays, mea?iing thereby the cross 
 ratio in a?ty trafisversal cutting the pencil. The cross ratio of 
 the pe7icil is writte7i S {ABCD). 
 
 4 Equiangular pencils have equal cross ratios. 
 
 Def. — If two ranges or pencils have equal cross ratios, they 
 are said to be equi-cross, or equi-anharmonic. 
 
 5 The lines joining corresponding elements of two equi-cross 
 ranges which have two corresponding elements coincident 
 are concurrent. 
 
 6 The points of intersection of corresponding elements of two 
 equi-cross pencils which have two corresponding rays co- 
 incident are collinear. 
 
 7 Pencils whose rays pass through four fixed points on a conic, 
 and whose vertices lie on the conic, are equi-cross. 
 
 Cor. — The vertices of equi-cross pencils whose rays pass 
 through four fixed points lie on a conic which passes through 
 the four points. 
 
 8 Ranges whose points lie on four fixed tangents to a conic, and 
 whose bases touch the conic, ar-e equi-cross. 
 
 Cor. — The bases of equi-cross ranges whose elements lie on 
 four fixed straight lines are tangents to a conic which touches 
 the four lines. 
 
26 A SYLLABUS OF 
 
 Involution. 
 
 Defs. — If a system of pairs of collinear points A, A'; B,B'j 
 C,C' ; etc., be so situated with regard to another point O in 
 the same straight hne that OA.OA' = OB.OB' = OC.OC = etc., 
 they are said to be in involution. The point O is called the 
 cenrf:re, and A,A' ; B,B'; C,C'; etc., are called conjugate 
 POINTS of the involution. 
 
 The points E, F, situated on the range on opposite sides 
 of O such that 
 
 0E2 = 0F2 = OA.OA' = OB.OB' = etc. 
 
 are called the double points of the involution. 
 
 If lines be drawn from a point S outside the range to A, A'; 
 B,B'; etc., they form a pencil in involution, and SE, SF 
 are called the double rays of the pencil. 
 
 Note. — // should be obseived tliat the do2tble poifits and 
 double rays are real 07ily whe7i the conjugate points of the 
 involution are on the same side of the centre. 
 
 9 The two double points and any pair of conjugate points of an 
 involution form a harmonic range. 
 
 10 In a system of points or rays in involution the cross ratio of 
 any four points or rays is equal to that of their conjugates. 
 
 11 A straight line is cut in involution by any system of co-axal 
 circles, the point of intersection of the radical axis with the 
 straight line being the centre of involution' 
 
 12 If two pairs of conjugate rays of a pencil in involution be at 
 right angles, then every pair of conjugate rays are at right 
 angles. 
 
 Cor. — The rays of a series of right angles at the same vertex 
 form a system in involution. 
 
MODERN PLANE GEOMETRY. 27 
 
 Reciprocal Polars. 
 Def. — If with respect to a given fixed circle (centre S, radius k) bs 
 
 taken the •< ^ , ^ of the { . ^ > of a figure 2, we obtain a 
 ( polars J ( points j o > 
 
 second figure 2', interchangeably related to the first. Either of 
 
 the figures 5, 2' is termed the polar reciprocal of the other ; 
 
 and any geometrical property of the one has its correlative for 
 
 the other. 
 
 Thus to the line joining two points of the one corresponds the 
 
 point of intersection of two lines of the other (Chap. V., Theor. 7) ; 
 
 to a number of collinear points, a number of concurrent lines ; to 
 
 parallel lines, points collinear with S ; to the angle between two 
 
 lines, the angle (or its supplement) subtended by two points 
 
 at S ; &;c. 
 
 13 A triangle may be its own reciprocal polar (Chap. V., Theor. 9). 
 
 14 A range reciprocates into a pencil having the same cross ratio ; 
 and conversely. 
 
 15 If 2 be a curve (the locus of a moving point), then 2' is also 
 a curve (the envelope of a moving line). To a point on 2 corre- 
 sponds a tangent to 2' ; and to a tangent to 2 (passing through 
 two indefinitely near points) corresponds a point on 2' (the inter- 
 section of two indefinitely near tangents). 
 
 16 If 2 be a circle (centre O, radius r), then 2' is a conic 
 having S for its focus, SO/r for eccentricity, k^-'.r for semi-Iatus 
 rectum, and its transverse axis in the direction SO. To the 
 centre O corresponds the S-directrix ; to the tangents at the 
 extremities of the diameter through S, the two vertices ; to the 
 tangents at E, E' passing through S, the points w, w' at infinity 
 (Sw, Sod' being perpendicular to SE, SE' respectively) ; to the 
 points E, E', the asymptotes ; to the chord EE', the centre ; to 
 the line FF' bisecting SE, SE', the other focus S'. 
 
28 J SYLLABUS OF 
 
 IX.— PROJECTION. 
 Definitions. 
 
 If two figures in different planes be so related to a fixed point 
 outside both planes that the straight line joining this fixed point 
 with any point in one figure passes through a point in the other 
 figure, each figure is said to be the central or conical projec- 
 tion of the other. 
 
 The fixed point is called the centre or vertex of projection; 
 and the intersection of the planes the axis of projection. 
 
 Points in which any straight line through the centre of projec- 
 tion meets the two planes are called corresponding points. 
 
 When the centre of projection is at an infinite distance, the 
 projection becomes parallel. 
 
 Theorem 
 
 1 The projection of a straight line is a straight line which inter- 
 sects the former in the axis of projection. 
 
 Cor. — The straight line joining two points projects into the 
 straight line joining the corresponding points ; the intersection 
 of two lines projects into the intersection of the corre- 
 sponding lines ; and the tangent to a curve at any point 
 projects into the tangent to the corresponding curve at the 
 corresponding point. 
 
 2 If two figures in different planes are so related that correspond- 
 ing straight lines meet (in the axis), then the straight lines 
 joining corresponding points are concurrent. 
 
 Definitions. 
 
 If a plane be drawn through the centre of projection 
 parallel to the plane containing one of the figures, its inter- 
 section with the other plane is called the vanishing line, 
 and may be considered to be the line corresponding to the 
 line at infinity of the first plane. 
 
MODERN PLANE GEOMETRY. 29 
 
 The intersection of a line in either figure with the vanishing 
 Hne in its plane is termed its vanishing point, and corre- 
 sponds to the point at infinity on the corresponding line in the 
 other figure. 
 
 3 Any line of one figure is parallel to the line joining the 
 centre with the vanishing point of the corresponding line in 
 the other. 
 
 4 The angle contained by any two straight lines in one figure is 
 equal to the angle subtended at the centre by the vanishing 
 points of the corresponding lines in the other. 
 
 5 If through the centre and in the plane perpendicular to the 
 axis of projection a straight line be drawn making equal 
 angles with the planes of the two figures, the two correspond- 
 ing points thus determined are such that the angle contained 
 by any pair of lines through one is equal to the angle between 
 the corresponding lines. 
 
 Cor. and Def. — ^Two such points may be found in each 
 plane, and they are called limiting points. 
 
 6 Parallel lines correspond to lines intersecting on the vanishing 
 line. 
 
 7 Any range or pencil of four elements projects into an equi- 
 cross range or pencil ; and any range of points or pencil of 
 rays in involution projects into a range or pencil in involution. 
 
 Cor. — In particular, a harmonic range or pencil projects 
 into a harmonic range or pencil. 
 
 8 Any two angles of a figure can be projected into angles of 
 given magnitude, and at the same time a straight line of the 
 figure can be projected to infinity. 
 
 Cor. — -Any quadrilateral can be projected into a parallelo- 
 gram of any form and size. 
 
 9 A circle can be projected into a conic of any eccentricity and 
 magnitude. 
 
30 A SYLLABUS OF 
 
 Cor. — The conic will be a hyperbola, parabola, or ellipse 
 according as the vanishing line cuts, touches, or does not 
 meet the circle. 
 
 Note. — The definition of Pole and Polar and the properties 
 e7iunciatcd in Tlieorems d-Z, 10-12 of Chapter V. obtain for 
 any conic, being all projdtii'e. 
 
 10 When the plane of one figure is turned round the axis of 
 projection, the figures are still projections of one another, and 
 the locus of the centre of projection is a circle ; and when 
 the two planes coincide, the centre of projection coincides 
 with one or otiier of the limiting points. 
 
 Definitions. 
 
 When the two planes thus coincide, the figures are said to 
 be in plane perspective or homology, and the axis and 
 centre are called the axis and centre of perspective, or 
 homology. 
 
 11 If the lines joining corresponding vertices of two triangles 
 are concurrent, then the intersections of corresponding sides 
 arc coUinear ; and, conversely, if the intersections of corre- 
 sponding sides of two triangles are collinear, then the lines 
 joining the corresponding vertices are concurrent. 
 
 Note. — This theorem is coiyimonly stated thus: — Co-polar 
 triatigles are also co-axial : a7id, conversely, co-axial triangles 
 aj-e CO -polar. 
 
 12 The perspective of a circle whose centre is the centre of per- 
 spective is a conic whose focus is the centre of perspective 
 and whose directrix is the vanishing line of its own figure. 
 
 Note on imaginary Points and Lines. 
 
 As a straight line which cuts a circle or conic does so in two and 
 only two points, it is usual and convenient in modern geometry to 
 
MODERN PLANE GEOMET;^Y. 31 
 
 say that every straight line in the plane of a circle or conic cuts 
 it in two points real, coincident, or imajinary ; a tangent cutting it 
 in two coincident points, and a line which does not meet it being 
 said to cut it in two imaginary points. The second point where a 
 diameter of a parabola, or a line parallel to an asymptote of a 
 hyperbola cuts the curve is at infinity, and the asymptotes of a 
 hyperbola meet the curve at two points at infinity ; and we might 
 thus speak of the imaginary asymptotes of an ellipse. Many 
 theorems in this syllabus may be generalized by introducing the 
 idea of imaginary points and lines : thus (see Chap. V., Theor. 8, 
 Cor.) the polar of a point may always be regarded as the chord 
 of contact of tangents real, coincident or imaginary drawn from 
 that point. Again, (see Chap. VI., Theor. 6, Cor.) the radical 
 axis of two circles is the common chord of the circles whether they 
 meet in real, or imaginary points. Or again, when the conjugate 
 points or rays of a system in involution are on opposite sides of 
 the centre, the double points or rays become imaginar)^ Further, 
 if the conception be allowed that imaginary points and lines may 
 be projected into real points and lines, the value and power of the 
 principle of projection becomes greatly extended. 
 
 The following particulars should be noted : — 
 
 That imaginary points occur in pairs, so that to every imaginai'v 
 point corresponds another imaginary point which might be called 
 its conjugate. The straight line joining a pair of conjugate imagi- 
 nary points is real, and no other straight lines through either 
 imaginary point can be real. 
 
 Two conies in the same plane intersect in four points of which 
 all may be real, two real and two imaginary, or all imaginary : in 
 the first case the six chords of intersection are all real, and in the 
 latter two cases two only of the chords are real, the other four 
 being imaginarj'. 
 
 In the case of two intersecting circles, the second real chord of 
 intersection is the line at infinity (see Theor. 13, below). When 
 two circles in a plane do not intersect, the two real chords of con- 
 
32 A SYLLABUS OF MODERN PLANE GEOMETRY. 
 
 tact are the radical axis and the line at infinity, and these coincide 
 in the case of concentric circles (see Theorem 15 below). 
 
 Theorem 
 
 13 Every circle in a plane passes through the same two imaginary 
 points on the line at infinity. 
 
 Definition. 
 These two points in any plane are called the circular 
 POINTS at infinity, or the focoids. (The latter term is due 
 to Dr. C. Taylor. See Taylor's Ancie?it a?id Modern Geojnetry 
 of Co flics, p. 308.) 
 
 14 Every conic which passes through the focoids is a circle. 
 
 15 Concentric circles touch each other at the focoids. 
 
 16 Co-axal circles may be projected into a system of conies 
 passing through four fixed points ; and concentric circles may 
 be projected into a system of conies touching one another at 
 two fixed points. 
 
 17 Every conic may be considered as inscribed in a quadrilateral 
 which has for its six vertices, the two real foci, the two 
 imaginary foci, and the two focoids. 
 
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