Automatic Deduction in Dynamic Geometry using Sage

TL;DR

This paper introduces a Sage-based symbolic tool for automatic deduction in dynamic geometry, capable of computing loci and verifying geometric statements using algebraic methods, with novel algorithms for locus simplification.

Contribution

It presents a new Sage worksheet integrating GeoGebra and Intergeo formats, and introduces an algorithm to eliminate extraneous parts in computed loci.

Findings

Successfully computes geometric loci and verifies statements.

Integrates multiple dynamic geometry formats in Sage.

Provides an algorithm to refine locus results by removing degenerate parts.

Abstract

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.