Involutions from Concurrency of Four Algebraic Curves

Gergonne-Steiner Involution on Miquelian Benz Planes?

Gibert-Burek-Moses(Incenter) Inversion on the Extended Complex Plane?

Given a quadrangle ABCD on the extended complex plane, we can get the four incenters A'B'C'D' of ABCD. And given a point P, four circles PAA',PBB',PCC',PDD' concur in a point P'. The transformation P↦P' is the Gibert-Burek-Moses inversion.

Isodynamic Involution on the Extended Complex Plane?