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

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 Schwabhauser, 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.