Formalizing Complex Plane Geometry

Deep connections between complex numbers and geometry had been well known and carefully studied centuries ago. Fundamental objects that are investigated are the complex plane (usually extended by a single infinite point), its objects (points, lines and circles), and groups of transformations that act on them (e.g., inversions and Möbius transformations). In this paper we treat the geometry of complex numbers formally and present a fully mechanically verified development within the theorem prover Isabelle/HOL. Apart from applications in formalizing mathematics and in education, this work serves as a ground for formally investigating various non-Euclidean geometries and their intimate connections. We discuss different approaches to formalization and discuss the major advantages of the more algebraically oriented approach.