Inversion in C sends z to r^{2}/z* (for nonzero z), and interchanges 0 and Ñ.


Proof Take z nonzero, 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, Hence kz^{2} = r^{2}, so k = r^{2}/z^{2} Then w = kz = k.r^{2}/(z.z*) = r^{2}/z*. The result for 0 and Ñ is obvious. 
The Algebraic Inversion Theorem 