[alogo] Mid circle property by inversions

Consider two circles {a(A, ra), b(B, rb)} and the inversion on circle c(C) which interchanges the two circles (see InversionInterchanging.html ). Invert the whole system with respect to a fourth circle w(W,r). Then the image circles a*, b*, c* are so that c* is the circle whose inversion interchanges a* and b*. In other words the [Inversion interchanging property] of circle c(C) is preserved by inversions.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]
[3_0] [3_1] [3_2]

The proof depends on a property of the conjugation of inversions by other inversions (see ConjugateInversion.html ). Denote by Fa, Fb, etc. ... the inversions with respect to the circles a, b, ... etc. Then Fw is the inversion interchanging c and c* and we have (Fc*)(b*) = (Fw)(Fc)(Fw)(b*) = (Fw)(Fc)(b) = (Fw)(a) = a*. Thus the inversion (Fc*) interchanges circles a* and b* as stated.

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