. The principles of projective geometry applied to the straight line and conic . drs to S be b, b, a and (q . prpt) = (s. prpt). But (q. prpt) = (pbpa) and (s. prpt) = (papb). Therefore (pbpa) = (papb). Therefore aa, bb, pp are pairsof conjugate rays of an involutionpencil. If any other conic be describedto touch q, r, s, t, the tangents Desargues Theorem 195 in a pair of points of the sameinvolution, namely that determinedby the two pairs of conj ugate pointsA, A and B, B. The points where SQ and RTmeet s are also a pair of conjugatepoints of the same involution.(Art. 56.) Converse : L
. The principles of projective geometry applied to the straight line and conic . drs to S be b, b, a and (q . prpt) = (s. prpt). But (q. prpt) = (pbpa) and (s. prpt) = (papb). Therefore (pbpa) = (papb). Therefore aa, bb, pp are pairsof conjugate rays of an involutionpencil. If any other conic be describedto touch q, r, s, t, the tangents Desargues Theorem 195 in a pair of points of the sameinvolution, namely that determinedby the two pairs of conj ugate pointsA, A and B, B. The points where SQ and RTmeet s are also a pair of conjugatepoints of the same involution.(Art. 56.) Converse : Let the vertices of the trianglebe Q, R, S and let the involutionbe determined by the pairs ofconjugate points A, A and B, B. from 8 will be a pair of conjugaterays of the same pencil, namelythat determined by the two pairsof conjugate rays a, a and h, h. The lines joining *Si to sq and rtare also a pair of conjugate raysof the same involution. (Art. 56.) Let the sides of the triangle beq, r, s and let the involution bedetermined by the pairs of con-jugate lines a, a and 6, Describe two conies throughthe points A, A, Q, R, S andB, B, Q, R, S respectively tointersect in some point T. Describe a third conic throughQ, R, S, T and (7, some given pointon the transversal. This conic meets the transversalin C the conjugate of G in theinvolution determined by ^, ^ andB, B. Since only one conic can be Describe two conies to touch thelines a, a, q, r, s and h, h, q, r, srespectively. Draw the fourth com-mon tangent, t, to these conies. Describe a third conic to touchq, r, s, t and c, some given linethrough S, the centre of theinvolution. This conic will touch c the conjugate to c in the involution determined by a, a and b, b. Since only one conic can be 13—2 106 Prinrij^leti of Projective Geoinetrij described through five points, thisconic may be looked upon as theconic through Q, R, S, G and this conic passes through T,the theorem is proved. described to touch five lines, t
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
