Parametric Root Finding for Supporting Proving and Discovering Geometric Inequalities in GeoGebra

Abstract

We introduced the package/subsystem GeoGebra Discovery to GeoGebra which supports the automated proving or discovering of elementary geometry inequalities. In this case study, for inequality exploration problems related to isosceles and right angle triangle subclasses, we demonstrate how our general real quantifier elimination (RQE) approach could be replaced by a parametric root finding (PRF) algorithm. The general RQE requires the full cell decomposition of a high dimensional space, while the new method can avoid this expensive computation and can lead to practical speedups. To obtain a solution for a 1D-exploration problem, we compute a Groebner basis for the discriminant variety of the 1-dimensional parametric system and solve finitely many nonlinear real (NRA) satisfiability (SAT) problems. We illustrate the needed computations by examples. Since Groebner basis algorithms are available in Giac (the underlying free computer algebra system in GeoGebra) and freely available efficient NRA-SAT solvers (SMT-RAT, Tarski, Z3, etc.) can be linked to GeoGebra, we hope that the method could be easily added to the existing reasoning tool set for educational purposes.