A Maple Package for Automatic Theorem Proving and Discovery in 3D-Geometry

Abstract

Abstract.A package for investigating problems about configuration theorems in 3D-geometry and performing mechanical theorem proving and discovery is presented. It includes the preparation of the problem, consisting of three processes: defining the geometric objects in the configuration; determining the hypothesis conditions through a point-on-object declaration method; and fixing the thesis conditions. After this preparation, methods based both on Groebner Bases and Wu's method can be applied to prove thesis conditions or to complete hypothesis conditions. Homogeneous coordinates are used in order to treat projective problems (although affine and Euclidean problems can also be treated). A Maple implementation of the method has been developed. It has been used to extend to 3D some classic 2D theorems.

A Brief Note on the Approach to the Conic Sections of a Right Circular Cone from Dynamic Geometry

“…Let us note that the constructive approach requires of effective algebraic methods that can be performed directly by the DGS, if it has algebraic capabilities [2], or externally by a CAS [8]. Let us observe that GeoGebra 5 offers a Locus tool for the 2D case, but it is unavailable in the 3D case (although computer packages on automatic 3D loci determination do exist [4,8]).…”

“…Let us observe that GeoGebra 5 offers a Locus tool for the 2D case, but it is unavailable in the 3D case (although computer packages on automatic 3D loci determination do exist [4,8]). …”

Connecting the 3D DGS Calques3D with the CAS Maple

“…As said above, paramGeo3D package [39] had already been used by the authors in automatic theorem proving and discovery [38][39][40][41][42]. In those works we had to directly type in Maple the details of the geometric configurations, instead of drawing the construction with the 3D DGS (as proposed in this article).…”

“…And even some 3D DGSs like Cabri3D and the free However, a gap between these two types of system still exists. On one hand, powerful packages devoted to Euclidean geometry have been implemented in CASs like Maple or Derive [20,37]. But they have incorporated neither mouse drawing capabilities, nor dynamic capabilities 4 .…”

“…The best known CASs are probably Mathematica [45,50], Maple [12,16,25,37,51], Derive [20,22,30] (recently discontinued) [52], Reduce [24,26,53], Axiom [17,54] Dynamic Geometry Systems (DGSs) are interactive systems that allow to draw "ruler and compass" geometric constructions using only the mouse, and to deform the construction by dragging and dropping the points that do not depend on previously defined objects. The best-known 2D DGSs are probably Cabri Geometry [59], The Geometer's Sketchpad [60], Cinderella [19,61] and GeoGebra [14,62], although many other exist.…”