A completion of hypotheses method for 3D-geometry. 3D-extensions of Ceva and Menelaus theorems

Abstract

A method that automates hypotheses completion in 3D-Geometry is presented. It consists of three processes: defi ning the geometric objects in the confi guration; determining the hypothesis conditions of the confi guration (through a point-on-object declaration method); and applying an algebraic automatic theorem proving method to obtain and prove the sufficiency of complementary hypothesis conditions. To avoid as much as possible the appearance of rational expressions, projective coordinates are used (although affine and Euclidean problems can also be treated). A Maple implementation of the method has been used to extend to 3D classic 2D geometric theorems like Ceva's and Menelaus'.