Pappus’s Hexagon Theorem in Real Projective Plane

In this article we prove, using Mizar [2], [1], the Pappus’s hexagon theorem in the real projective plane: “Given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear”2. More precisely, we prove that the structure ProjectiveSpace TOP-REAL3 [10] (where TOP-REAL3 is a metric space defined in [5]) satisfies the Pappus’s axiom defined in [11] by Wojciech Leonczuk and Krzysztof Prazmowski. Eugeniusz Kusak andWojciech Leonczuk formalized the Hessenberg theorem early in the MML [9]. With this result, the real projective plane is Desarguesian. For proving the Pappus’s theorem, two different proofs are given. First, we use the techniques developed in the section “Projective Proofs of Pappus’s Theorem” in the chapter “Pappos’s Theorem: Nine proofs and three variations” [12]. Secondly, Pascal’s theorem [4] is used. In both cases, to prove some lemmas, we use Prover93, the successor of the Otter prover and ott2miz by Josef Urban4 [13], [8], [7]. In Coq, the Pappus’s theorem is proved as the application of Grassmann-Cayley algebra [6] and more recently in Tarski’s geometry [3].This work has been supported by the “Centre autonome de formation et de recherche en mathématiques et sciences avec assistants de preuve” ASBL (non-profit organization). Enterprise number: 0777.779.751. Belgium.Rue de la Brasserie 5, 7100 La Louvière, BelgiumGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: Stateof-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Gabriel Braun and Julien Narboux. A synthetic proof of Pappus’ theorem in Tarski’s geometry. Journal of Automated Reasoning, 58(2):23, 2017. doi:10.1007/s10817-016-9374-4.Roland Coghetto. Pascal’s theorem in real projective plane. Formalized Mathematics, 25(2):107–119, 2017. doi:10.1515/forma-2017-0011.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599–603, 1991.Laurent Fuchs and Laurent Thery. A formalization of Grassmann-Cayley algebra in Coq and its application to theorem proving in projective geometry. In Automated Deduction in Geometry, pages 51–67. Springer, 2010.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi:10.1007/s10817-015-9333-5.Adam Grabowski. Solving two problems in general topology via types. In Types for Proofs and Programs, International Workshop, TYPES 2004, Jouyen-Josas, France, December 15-18, 2004, Revised Selected Papers, pages 138–153, 2004. doi:10.1007/11617990 9. http://dblp.uni-trier.de/rec/bib/conf/types/Grabowski04.Eugeniusz Kusak and Wojciech Leonczuk. Hessenberg theorem. Formalized Mathematics, 2(2):217–219, 1991.Wojciech Leonczuk and Krzysztof Prazmowski. A construction of analytical projective space. Formalized Mathematics, 1(4):761–766, 1990.Wojciech Leonczuk and Krzysztof Prazmowski. Projective spaces – part I. Formalized Mathematics, 1(4):767–776, 1990.Jürgen Richter-Gebert. Pappos’s Theorem: Nine Proofs and Three Variations, pages 3–31. Springer Berlin Heidelberg, 2011. ISBN 978-3-642-17286-1. doi:10.1007/978-3-642-17286-1_1.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.292697