Cancellation Patterns in Automatic Geometric Theorem Proving

Abstract

This article is about the equivalence of two seemingly different methods for proving incidence theorems in projective geometry. The first proving method is essentially an algebraic certificate for the non-existence of a counterexample—via biquadratic final polynomials [13]. For the second method the theorems of Ceva and Menelaus are elementary building blocks and are used as faces of an oriented topological 2-cycle, with their geometric structure on the edges identified appropriately. The fact that the cycle finally closes up translates into the proof of the theorem. We start by formalizing both methods. After this we present a bijective translation process that establishes the equivalence of the two methods. The proving methods and the translation process will be illustrated by a (quite well-natured) example. Using our methods one gains additional structural insight in the purely algebraic proofs (biquadratic final polynomials) by reconstructing an underlying topological structure of the proof.