Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

Tomás Recio, Rafael Losada, Zoltán Kovács and Carlos Ueno
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Tomás Recio: Departamento de Ingeniería Industrial, Escuela Politécnica Superior, Universidad Antonio de Nebrija, C/Santa Cruz de Marcenado 27, 28015 Madrid, Spain
Rafael Losada: Sociedad Asturiana de Educación Matemática “Agustín de Pedrayes”, Federación Española de Sociedades de Profesores de Matemáticas, Plaza Club Patín Gijón Solimar 1, 33213 Gijón, Spain
Zoltán Kovács: The Private University College of Education of the Diocese of Linz, Salesianumweg 3, 4020 Linz, Austria
Carlos Ueno: CEAD Profesor Félix Pérez Parrilla, C/Dr. García Castrillo, 22, 35005 Las Palmas de Gran Canaria, Spain

Mathematics, 2021, vol. 9, issue 20, 1-29

Abstract: Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p ( x , y ) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.

Keywords: automated theorem proving in geometry; automated deduction in geometry; automated reasoning in geometry; Dynamic Geometry; GeoGebra; computational algebraic geometry (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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