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A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry

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Abstract

In this paper, we report on the formalization of a synthetic proof of Pappus’ theorem. We provide two versions of the theorem: the first one is proved in neutral geometry (without assuming the parallel postulate), the second (usual) version is proved in Euclidean geometry. The proof that we formalize is the one presented by Hilbert in The Foundations of Geometry, which has been described in detail by Schwabhäuser, Szmielew and Tarski in part I of Metamathematische Methoden in der Geometrie. We highlight the steps that are still missing in this later version. The proofs are checked formally using the Coq proof assistant. Our proofs are based on Tarski’s axiom system for geometry without any continuity axiom. This theorem is an important milestone toward obtaining the arithmetization of geometry, which will allow us to provide a connection between analytic and synthetic geometry.

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Notes

  1. http://dpt-info.u-strasbg.fr/~narboux/AreaMethod/AreaMethod.examples_4.html.

  2. http://www-sop.inria.fr/marelle/CertiGeo/pappus.html.

  3. Note that we do not use the same notation as in the book [21].

  4. This definition is called R in [21]. We call it Per because we want to keep single letter notations for points.

  5. Note, however that for arithmetization of geometry we will need to use this axiom to obtain the standard axioms of an ordered field expressed using functions instead of relations [1].

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  1. ICube, UMR 7357, University of Strasbourg - CNRS, Strasbourg, France

    Gabriel Braun & Julien Narboux

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  1. Gabriel Braun
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  2. Julien Narboux
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Correspondence to Julien Narboux.

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Braun, G., Narboux, J. A Synthetic Proof of Pappus’ Theorem in Tarski’s Geometry. J Autom Reasoning 58, 209–230 (2017). https://doi.org/10.1007/s10817-016-9374-4

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