Wilson Stothers' Inversive Geometry and CabriJava Pages

Inversion in the circle C : |z| = r

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

Proof
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

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