P C'B' 333o. Let A, A', B, B', C, C' be six points in involution, and let O be the centre, Draw any line OPQ through 0, and take P and Q so that OP.OQ=OA, OA', and join PA, PB, PC, and PC', and also QA', QB, QC, and QC. Then, :: A' A B A, P, Q, A' are concyclic... LOPA=LOA'Q. Similarly, B, P, Q, B’ are concyclic, and LOPB=LOB’Q, etc. LAPB=LA'QB. Similarly, LBPC=_B'QC, LCPC'=LC'QC, etc. Hence the pencils P(ABCC') and Q(A'B'C'C) are equianharmonic, or ABCC'} = {A'B'C'C}. Hence also {ABB‘C} = A'B'BC'}, {AA’BC} = {A’AB'C'}. And any one of these relations expresses the condition that the six points symbolized may be in involution. 334o. As involution is only a species of homography, the relations constantly existing between homographic ranges and their corresponding pencils, hold also for ranges and pencils in involution. Hence 1. Every range in involution determines a pencil in involution at every vertex, and conversely. 2. If a range in involution be projected rectilinearly through any point on a circle it determines a system in involution on the circle, and conversely. Ex. The three pairs of opposite connectors of any four points cut any line in a six-point involution. A, B, C, D are the four points, =BCARD (302) P D R А P B T {PQRR'} = {P'Q'R'R}, and the six points are in involution. Cor. 1. The centre o of the involution is the radical centre of any three circles through PP', QQ', and RR'; and the three circles on the three segments PP", QQ', and RR' as diameters are co-axal. When the order of PQR is opposite that of P'Q'R' as in the figure, and the centre 0 lies outside the points, the co-axal circles are of the l.p.-species, and when the two triads of points have the same order, the co-axal circles are of the c.p.-species. Cor. 2. Considering ABC as a triangle and AD, BD, CD three lines through its-vertices at D, we have-- The three sides of any triangle and three concurrent lines through the vertices cut any transversal in a six-point involution. EXERCISES. 1. A circle and an inscribed quadrangle cut any line through them in involution. 2. The circles of a co-axal system cut any line through them in involution. 3. Any three concurrent chords intersect the circle in six points forming a system in involution. 4. The circles of a co-axal system cut any other circle in involution. 5. Any four circles through a common point have their six radical axes forming a pencil in involution. INDEX OF DEFINITIONS, TERMS, ETC. The Numbers refer to the Articles. IOI Angles, Vertical, 39 of the same Affection, 65 289 146 136 1752 3 230 254 294 ܕܕ 1 36 Addition Theorem for Sine 236 87 66 31 40 49 49 49 40 89 32 43 213 40 213 32 35 73 39 , IOI 49 213 22 7 Basal Angles, 43 320 320 Diameter, 95 27 109, 328 22 86, 97 Eidograph, 2013 223 27, 136 53 131 49 193 21 Centre-line, 95 129 85 92 256 288 36 97 92 97 95 114 273 150 93 4 51 97 291 8 274 213 15 Finite Line, Generating Point, 69 Harmonic Division, 208 299 160 196 88, 168 4. 2II 36 I2 21 12 102 IO Interadjacent Angles, Orthogonal Projection, 167, 229 73 179, 256 260 211) 332 70 53 80 319 167 319 Peaucellier's Cell, 148 203 274 Perigon, Perimeter, 146 Perspective, 254 Perspective, Axis of, 254 69 Centre of, 254 274 40 Physical Line, . 190 IO, 17 Plane Geometric Figure, 175 Plane Geometry, 238 13 169 30 193 109 150 Point, Double, 55 Pole, 183 266 150 268 175 266 132 Prime Vector, . 1265, Ex. 2 167 89 192 Proportional Compasses, · 2111 40 123 39 1 80, 89 88 Quadrilateral, 115 Quadrilateral, complete, . 247 II 109, 328 |