INVERSION - Circle to Line

Theorem 12.2 The inverse of a circle through O is a straight line not through O, and the inverse of a straight line not through O is a circle through O.

Proof 

Let C be a circle on OM as diameter, and let M' be the inverse of M with respect to the circle of inversion . 

Let line m be the polar of M with respect to ; this line meets OM at right angles at M'. 

Since M and M' are inverse, and P and P' are given inverse, we know that points P, P' M', M are concyclic points.

We show that P lies on circle C P' lies on line m.

Now P lies on C OPM = 90° OM'P = 90° P lies on line m.

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