A first course in projective geometry . Fig. 29. Let CA cut the polar of A in , since AMBN is cyclic, CM . CA = CB. CM . CA = square of radius; .*. CB. CN = square of radius;.. AN is the polar of B. §4. Salmons Theorem. The distances of two points from the centre of a circle areproportional to the distances of each from the polar of theother. 64 PROJECTIVE GEOMETRY Let S be the centre of the circle (Fig. 30), and let KX andQL be the polars of O and Fig. 30. Let OT, the parallel through O to QL, cut SPL in , PN be perpendicular to KX and SO SX . SO = squar
A first course in projective geometry . Fig. 29. Let CA cut the polar of A in , since AMBN is cyclic, CM . CA = CB. CM . CA = square of radius; .*. CB. CN = square of radius;.. AN is the polar of B. §4. Salmons Theorem. The distances of two points from the centre of a circle areproportional to the distances of each from the polar of theother. 64 PROJECTIVE GEOMETRY Let S be the centre of the circle (Fig. 30), and let KX andQL be the polars of O and Fig. 30. Let OT, the parallel through O to QL, cut SPL in , PN be perpendicular to KX and SO SX . SO = square of radius = SP. SL. Also, since OTPN is cyclic, SN . SO = SP . ST ; Let SP SXso SL and also ;ST SP SX- SN NX PK SO SL- ST LT OQ § 5. Conjugate Points and Lines. Def. a pair of points such thateach lies on the polar of the otherwith respect to a given circle arecalled conjugate points with re-spect to that circle. Def. a pair of straight linessuch that each contains the poleof the other with respect to agiven circle are called conjugatelines with respect to that circle. POLES AND POLARS 65 Prop. On any straight line aninfinite number of pairs of conju-gate points with respect to thecircle can be found. For taking any point A on theline a, tlie point A, where thepolar of A cuts a, is conjugate to A. A and A are therefore a pair ofconjugate points on the line a ;and clearly an infinite number ofsuch pairs can be found. Prop. Through any point aninfinite num
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