Two New Ways to Formally Prove Dandelin-Gallucci's Theorem

Abstract

Mechanizing proofs of geometric theorems in 3D is significantly more challenging than in 2D. As a first noteworthy case study, we consider an iconic theorem of 3D geometry: Dandelin-Gallucci's theorem. We work in the very simple but powerful framework of projective incidence geometry, where only incidence relationships are considered. We study and compare two new and very different approaches to prove this theorem. First, we propose a new proof based on the well-known Wu's method. Second, we use an original method based on matroid theory to generate a proof script which is then checked by the Coq proof assistant. For each method, we point out which parts of the proof we manage to carry out automatically and which parts are more difficult to automate and require human interaction. We hope these first developments will lead to formally proving more 3D theorems automatically and that it will be used to formally verify some key properties of computational geometry algorithms in 3D.