A first course in projective geometry . ts for every circle of thesystem. COAXAL CIRCLES 57 For if C be the centre of any circle of the system, CLj. CLg= sq. on the tangent from C to the circle whose centre is X = radius of circle whose centre is C. .*. Lj and Lg are inverse points for this circle. It follows that any circle through L^ and Lg cuts all circles of the system orthogonally and has its centre on the radical axis. Ex. 1. The student should repeat the exercises in § 16, taking someor all of the given points inside the circles. Ex. 2. If each member of one set of circles cuts ea
A first course in projective geometry . ts for every circle of thesystem. COAXAL CIRCLES 57 For if C be the centre of any circle of the system, CLj. CLg= sq. on the tangent from C to the circle whose centre is X = radius of circle whose centre is C. .*. Lj and Lg are inverse points for this circle. It follows that any circle through L^ and Lg cuts all circles of the system orthogonally and has its centre on the radical axis. Ex. 1. The student should repeat the exercises in § 16, taking someor all of the given points inside the circles. Ex. 2. If each member of one set of circles cuts each member ofanother set orthogonally, the two sets are each coaxal, one having realand the other imaginary limiting points. Ex. 3. Prove that a common tangent to any two circles of thesystem subtends a right angle at either limiting point. § 20. Prop. If T be any point, TP, TP tangents from itto two circles whose centres are C, C, TP2-TP2 = , where N is the foot of the perpendicular from T on the radicalaxis of the two Fig. 27. Let X be as before, TM perpendicular to CC, O the middlepoint of CC. 58 PROJECTIVE GEOMETRY Then TP2 - TP2 - CT2 - CP^ - (CT2 - CP2) = CT--CT2-(CP2-CP2)= CT2 - CT2 - (CX2 - CX2) (§ 14) 2CC. TN. Cor. 1. If T be on a circle, centre C, coaxal with theother two, TP2=2CCMN, TP2 = TP2 CC TP2 CC constant for all positions of T on the circle. Cor. 2. The locus of a point which moves so that thetangents from it to two given circles are in a constant ratio, isa circle coaxal with the given circles. For, by the preceding,T lies on a circle, centre C, coaxal with the given circles, where CC —r^, = the square of the given c We have shown that only one such circle can be drawn. Historical Note. The method of inversion is said to havebeen known to geometers from very early times; Chasleseven states that it was used by Ptolemy (2nd century ). Its modern resuscitation is, however, due to Drs. Stubbsand Ingr
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