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Keywords = automated theorem proving in geometry

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29 pages, 5608 KiB  
Article
Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools
by Tomás Recio, Rafael Losada, Zoltán Kovács and Carlos Ueno
Mathematics 2021, 9(20), 2548; https://doi.org/10.3390/math9202548 - 11 Oct 2021
Cited by 3 | Viewed by 3886
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symbolic Computation for Mathematical Visualization)
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17 pages, 3581 KiB  
Article
Dealing with Degeneracies in Automated Theorem Proving in Geometry
by Zoltán Kovács, Tomas Recio, Luis F. Tabera and M. Pilar Vélez
Mathematics 2021, 9(16), 1964; https://doi.org/10.3390/math9161964 - 17 Aug 2021
Cited by 4 | Viewed by 2438
Abstract
We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in [...] Read more.
We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components. Full article
(This article belongs to the Special Issue Computer Algebra and Its Applications)
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