Deciding a Class of Euclidean Geometry Theorems with Buchberger’s Algorithm

Abstract

Buchberger’s method of Gröbner bases can be used to decide a certain class of theorems in elementary Euclidean geometry. Moreover, the method can also be used to find subsidiary conditions that are necessary in order to transform an “almost valid” formulation of a geometry theorem into a valid one. The introduction surveys all alternative approaches to automated geometry theorem proving, giving references to the corresponding literature. After explaining how to obtain correct algebraic translations of geometry theorems, Buchberger’s method of Gröbner bases is shortly reviewed. Then the application of Buchberger’s algorithm to geometry theorem proving is explained in all details. Finally, a computing time statistics on 20 plane Euclidean geometry theorems of growing complexity is given.Key wordsautomated geometry theorem provingBuchberger’s algorithmGröbner bases