. The principles of projective geometry applied to the straight line and conic . In the figure, let BiA^ meetAB in G-i and B2A0 meet AB in Then by the anharmonicproperty of a conic{B,X\C,B,A,) = {A,.G,G,B,A,). Taking intercepts on -45 iG,C,AG,) = (C,GoG,B).Hence AB, , C^G^ are pairsof conjugate points of an involutionand therefore (Art. 58)AC, AG, AC, AG, anharmonicto a conicCCi, GC„BB, BCi jdC/2 BC, BC, .(1). In the figure, let ^^1. BB, ^ be projected from G onAB into the points C3 and Ctrespectively. Then by theproperty of tangentsthe intersections ofAA„ BBo with AA, andform tw

---

. The principles of projective geometry applied to the straight line and conic . In the figure, let BiA^ meetAB in G-i and B2A0 meet AB in Then by the anharmonicproperty of a conic{B,X\C,B,A,) = {A,.G,G,B,A,). Taking intercepts on -45 iG,C,AG,) = (C,GoG,B).Hence AB, , C^G^ are pairsof conjugate points of an involutionand therefore (Art. 58)AC, AG, AC, AG, anharmonicto a conicCCi, GC„BB, BCi jdC/2 BC, BC, .(1). In the figure, let ^^1. BB, ^ be projected from G onAB into the points C3 and Ctrespectively. Then by theproperty of tangentsthe intersections ofAA„ BBo with AA, andform two projective ranges. Projecting these ranges from Con AB {Gfi,AG,)^{Cfi,C,B).Hence AB, C-^C^, C3C, are pairsof conjugate points of an involutionand therefore (Art. 53) AG, AC,_AC3 AG,BG,BC,~BCsBG, .(1). 1S8 Pnnciphs of Projective Geometry But by Menelaus Theorem But by Cevas Theorem AG, 1 AG, - 1 BG, BA, GBi BG, BA, GB, , GA, • AB, anrl ^^^ ^ , AG, - 1 i^C, BA, GB, ■ •* iyc, ii^, Gi^, • GA,AB, GA,AB, Substituting these vahies of Substituting these val m:-^:^o

---

Size: 1623px × 1538px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No

Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective