Inversion with respect to a circle and orthogonal circles.

Theorem: If C2 is orthogonal to  C1  (with center O) and A is a point  inside C1 that is on C2 then the ray OA will intersect C2 at the point A' where A and A' are inverses with respect to the circle C1.

Proof:  Consider the following figure.

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We need to show that m(OA)m(OA') = m(OP)2.
Let m(OA)=a, m(AM)=m, m(OP)=R, m(O'M)=h, m(O'A) = S, m(OO') = T.
Then we want to show that a(a+2m) = R^2.