A first course in projective geometry . Fig. 16. OA therefore meets OA at a fixed point; the locus of Pis a straight line perpendicular to OA and passing through theinverse of A. § 6. Prop. The inverse of a circle with respect to anypoint is a circle. Let O and k be as before. Let 0 (Fig. 17) be the centre of the circle to b?^^inverted. Then if P is the inverse of P, OP. OP = k^. If OP meet the circle again in Q, OP. 0Q = 0T2. OP k^ Whence — = —i^, which is constant for all positions of the,. ^„^ OQ 0T2line OPQ. OX Divide OC at X so that —- = this constant ratio. OC INVERSION 45 Then X is
A first course in projective geometry . Fig. 16. OA therefore meets OA at a fixed point; the locus of Pis a straight line perpendicular to OA and passing through theinverse of A. § 6. Prop. The inverse of a circle with respect to anypoint is a circle. Let O and k be as before. Let 0 (Fig. 17) be the centre of the circle to b?^^inverted. Then if P is the inverse of P, OP. OP = k^. If OP meet the circle again in Q, OP. 0Q = 0T2. OP k^ Whence — = —i^, which is constant for all positions of the,. ^„^ OQ 0T2line OPQ. OX Divide OC at X so that —- = this constant ratio. OC INVERSION 45 Then X is a fixed point. Also, since PX is parallel to QC, PX — = this same ratio; .. PX is constant. .*. the locus of P is a circle whose centre is Pig. 17. It is a good exercise in the application of the principle ofcontinuity to deduce from this the result of § 5. § 7. Prop. If two curves cutone another, their inverses cut atthe same angle. Consider first a single curve andits inverse. Take two near positions of theradius vector, and let P, P, Q, Q bepairs of inverse points. (Fig. 18.) Then, since = F = , the quadrilateral PQQP is cyclic;.. angle OPQ = angle if P, Q are points on the rQgiven curve which move up to Pio. is.
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