Automated Theorem Proving in Incidence Geometry — A Bracket Algebra Based Elimination Method

Abstract

In this paper we propose a bracket algebra based elimination method for automated generation of readable proofs for theorems in incidence geometry. This method features three techniques, the first being heuristic automated reordering of geometric constructions for the purpose of producing shorter proofs, the second being some heuristic elimination rules which improve the performance of the area method of Zhang and others without introducing signed length ratios, the third being a simplification technique called contraction, which reduces the size of bracket polynomials. More than twenty theorems in incidence geometry have been proved, for which short proofs can be produced very fast, together with the corresponding nondegeneracy conditions. An interesting phenomenon is that a proof composed of polynomials of at most two terms can always be found for any of these theorems, similar to that by the biquadratic final polynomial method of Richter-Gebert.