Geometry theorem proving in vector spaces by means of Gröbner bases

Within the last few years several approaches to automated geometry theorem proving have been developed and proposed that are based 1 ) on the formulation of a geometric statement as the implication of a polynomial equation (the “conclusion” ) from a set of polynomial equations (the “hypotheses” ), and 2) the proof of the implication by algebraic methods, namely C,robner bases and Ritt’s bases. All these approaches require the introduction of coordinates for the points involved. Many geometric theorems, however, can be formulated as relations between points directly, without needing coordinates. In this paper we develop a new method, based on Grobner bases in vector spaces, that can prove geonletric theorems that are formulated as relations between points directly. Our approach has the advantages that theorems can be formulated more naturally and fewer variables are needed for their formulations. This results in shorter and faster proofs.