INVERSION - Circle to Circle

Suppose now that P describes a circle C which does not pass through point O.

Theorem 12.1 The inverse of a circle not through O is a circle not through O.

Proof 

Let C be the circle described by P, M the inverse of O with respect to circle C, M' the inverse of M with respect to circle .

We show that the inverse P' of P describes a circle with centre M'.

Now (P, P'), (M, M') are inverse with respect to P, P', M', M are concyclic

OPM = OM'P' (show; opposite angles of a cyclic quadrangle are supplementary) 

OPM ~ OM'P' (equal angles) MP / P'M' = OP / OM' P'M' = (MP / OP).OM'.

Finally, since M was chosen to be the inverse of O with respect to circle C, by the converse of Apollonius’ Theorem, the ratio MP / OP is constant. Length OM' is also fixed. We deduce that P'M' is constant for all positions of P'. That is, the locus of P' is a circle with centre M'. 

This completes the proof of the theorem.
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