A Survey of Geometric Reasoning Using Algebraic Methods

Abstract

In the past decade highly successful algebraic methods for mechanical geometry theorem proving have been developed. The first step in these methods is to assign coordinates to key points and to translate the hypotheses and conclusion of a geometry statement into multivariate polynomial equations and inequalities. The second step is to prove the corresponding algebraic statements using various algebraic techniques. To date the most practically successful algebraic techniques have been Ritt-Wu’s characteristic set (CS) method and the Gröbner basis (GB) method. Also Collins’ method, a quantifier elimination method for real closed fields of Tarski’s type, has been practically improved to such an extent that many non-trivial geometry problems can now be solved by computer programs based on this method.This survey mainly concentrates on the applications of the characteristic set method (the CS method) and the Gröbner basis method (the GB method) to automated reasoning in elementary geometries, differential geometries, and mechanics. We will use elementary and understandable examples to show the essence of the techniques we are using, letting the reader to consult the related references for more detailed issues underlying these techniques.KeywordsTheorem provingformula derivationelementary geometrydifferential geometrycharacteristic set methodGröbner basesTarski’s decision procedureCollins’ methodalgebraic varietydecomposition of algebraic setSimson’s theoremMorley’s trisector theoremKepler-Newton problemHeron’s formulaPeaucellier’s Linkage