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Formalizing complex plane geometry

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Abstract

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.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Studentski Trg 16, 1100, Belgrade, Serbia

    Filip Marić & Danijela Petrović

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  1. Filip Marić
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  2. Danijela Petrović
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Correspondence to Filip Marić.

Additional information

This work is partially supported by the Serbian Ministry of Education and Science grant ON174021, and Serbian-French Technology Co-Operation grant EGIDE/“Pavle Savić” 680-00-132/2012-09/12 (“Formalization and automation of geometry”).

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Marić, F., Petrović, D. Formalizing complex plane geometry. Ann Math Artif Intell 74, 271–308 (2015). https://doi.org/10.1007/s10472-014-9436-4

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  • DOI: https://doi.org/10.1007/s10472-014-9436-4

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