Automated production of traditional proofs for theorems in Euclidean geometry I. The Hilbert intersection point theorems

Abstract

We present a method which can produce traditional proofs for a class of constructive geometry statements in Euclidean geometry. The method is a mechanization of the traditional area method used by many geometers. The key idea of our method is to eliminate dependent (constructed) points in a geometry statement using a few basic geometry propositions about the area of triangles. The method has been implemented. Our program, calledEuclid, can produce traditional proofs of many hard geometry theorems such as Pappus' theorem, Pascal's theorem, Gauss point theorem, and the Pascal conic theorem. Currently, it has produced proofs of 110 nontrivial theoremsentirely automatically. The proofs produced byEuclid are elegant, short (often shorter than the proofs given by geometers) and understandable even to high school students. This method seems to be the first that can produce traditional proofs for hard geometry theorems automatically.