Algebraic proofs for geometric statements.
It is well-known that in the first half of the 17th century Ren ́e Descartes proposed a program of problem solving based on the assumption that solving a problem in mathematics means essentially solving a problem in algebra, more precisely, solving a system of algebraic equations. It is less known that precisely the same method was in current use in China long before Descartes, as witnessed by the book Jade Mirror of Four Elements by Shi-jie Zhu, dated 1303 (cf. [3]). However, as a consequence of Godel’s work, in the first part of the 20th century it became clear that not all geometric statements can be derived in this way from a small set of axioms. The advent of fast computers has revived the interest in algebraization of geometric reasoning. The most successful methods currently available are based on either Grobner-bases theory or on Ritt-Wu characteristic sets method. Other approaches are described in various chapters of [5]. An impressive bibliography can be found on Dongming Wang’s web site [10]. The automatic methods for proving theorems can be classified according to several criteria. Thus one has to distinguish those who use coordinates from the coordinate-free approaches. The method of the former kind have a common structure: