Inversion in the circle C : |z| = r

Inversion in C sends z to r2/z* (for non-zero z), and interchanges 0 and Ñ.

Take z non-zero, and let A be the point with complex coordinates z,
and suppose that A has inverse B (coordinates w) with respect to C.

By the definition of inverse points,
B lies on the ray OA, so w = kz, with k > 0,
and OA.OB = r2, i.e. |z|.|w| = r2.

Hence k|z|2 = r2, so k = r2/|z|2

Then w = kz = k.r2/(z.z*) = r2/z*.

The result for 0 and Ñ is obvious.

The Algebraic Inversion Theorem