. The principles of projective geometry applied to the straight line and conic . {PBPA) = {PAPB). Therefore A A, BB, PP arethree pairs of conjugate points ofan involution. Let q, r, s, t be any four fixedtangents to a circle, let thetangents from any point S to thecircle be p, p and let the linesjx)ining 8 to qr, st, qt, rs be b, b,a, a. Then (q . prpt) = (s. prpt)by the anharmonic property oftangents to a circle. But (q .prpt) = (pbpa),and (s. prpt) = (papb). Therefore {pbpa) = {papb). Therefore aa , bb, pj) are threepairs of conjugate rays of an in-volution pencil. 11—2 1G4 Principles of Pro
. The principles of projective geometry applied to the straight line and conic . {PBPA) = {PAPB). Therefore A A, BB, PP arethree pairs of conjugate points ofan involution. Let q, r, s, t be any four fixedtangents to a circle, let thetangents from any point S to thecircle be p, p and let the linesjx)ining 8 to qr, st, qt, rs be b, b,a, a. Then (q . prpt) = (s. prpt)by the anharmonic property oftangents to a circle. But (q .prpt) = (pbpa),and (s. prpt) = (papb). Therefore {pbpa) = {papb). Therefore aa , bb, pj) are threepairs of conjugate rays of an in-volution pencil. 11—2 1G4 Principles of Projective Geometry (ii) It will be shown hereafter (Art. 101) that every circle may beregarded as passing through two imaginary fixed points at infinity andthat a circle may be considered to be a conic through these two fixedpoints at infinity. It follows therefore from the general form ofDesargucs theorem (Art. 101) that for the circle it may be stated in asecond form as follows: Every transversal cuts a system of coaxal circles in pairs of conjugatepoints of an Let the transversal s meet the circles in A A, BB, CC\ and theradical axis in 0. If 0 is external to the circles the tangents from 0 to the circles areequal, and the squares of these tangents are equal to OA . OA, OB. OB,00. 00,.... Hence OA . OA = 0B .OB= ... and the pairsof points A A, BB, CO,... are pairs of conjugate points of an involutionof which 0 is the centre. If 0 is inside the circles, the radical axis must meet the circles intwo real points L and M. Then ^OA. OA = = = ....Hence in this case also A A, BB, GO, ... are pairs of conjugate pointsof an involution of which 0 is the centre. In the former case the double points of the involution are real andin the latter they are imaginary. ABC the vertices of a triangle parallel lines are drawn cutting thecircumcircle in ABC. Show that the lines joining A, B, C to any other point onthe circumcircle cut BC, CA, AB
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