Noname manuscript No. (will be inserted by the editor) Formalizing Complex Plane Geometry Filip Mari´c · Danijela Petrovi´c Received: date / Accepted: date Abstract Deep connections between complex numbers and geometry had been well known and carefully studied centuries ago. Fundamental objects that are in- vestigated 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 M¨ obius 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 con- nections. We discuss different approaches to formalization and discuss the major advantages of the more algebraically oriented approach. Keywords Interactive theorem proving · Complex plane geometry · M¨ obius transformations 1 Introduction Connections between complex numbers and geometry are deep and intimate. Al- though complex numbers have been recognized for more than 450 years, their geometric interpretation came only at the end of 18th century in works of Wes- sel, Argand and Gauss [26]. Their most significant applications in geometry were developed by Cauchy, Riemann, M¨ obius, Beltrami, Poincar´e and others during the 19th-century [26]. Complex numbers present a very suitable apparatus for in- vestigating properties of objects in very different geometries. Geometry has been studied analytically since Descartes, and the Cartesian plane (R 2 ) is often used as This work is partially supported by the Serbian Ministry of Education and Science grant ON174021, and Serbian-French Technology Co-Operation grant EGIDE/,,Pavle Savi´c” 680- 00-132/2012-09/12 (“Formalization and automation of geometry”). Faculty of Mathematics University of Belgrade Studentski Trg 16 1100 Belgrade, Serbia