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.

Highlights

  • In recent years we are witnessing a rich interaction between two geometric instruments which were initially born with different intentions in mind, but are converging and joining forces nowadays to create a new generation of software for doing Geometry

  • This fruitful cooperation was put into practice in several Dynamic Geometry Software (DGS) programs such as Geometry Expert, GDI or GCLC, with GeoGebra joining this list in recent years

  • Since 2014, GeoGebra is making a systematic and permanent effort in order to develop new Geometric Automated Reasoning Tools (GARTs) based on algebraic computations that go beyond proving geometric statements, providing better feedback than just deciding the truth or falsehood of a statement, and introducing instruments that lead to the discovery of geometric facts

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Summary

Introduction

In recent years we are witnessing a rich interaction between two geometric instruments which were initially born with different intentions in mind, but are converging and joining forces nowadays to create a new generation of software for doing Geometry. Dynamic Geometry programs turned out to be an ideal interface to introduce geometric information, and the extension of this software with convenient algebraic tools allowed the creation of a natural environment to work on automated proving in Geometry This fruitful cooperation was put into practice in several DGS programs such as Geometry Expert, GDI or GCLC, with GeoGebra joining this list in recent years. Our research goal is to show how to approach this Proposition through the combination of a dynamic color scanning method and (real algebra) symbolic computations, exemplifying the need to automatize this combination of tools, to address and to solve some automated reasoning challenging issues that arise in the current GeoGebra Discovery version.

The GeoGebra Automated Reasoning Tools
New Tools in GeoGebra Discovery
Method
A Case Study
The Visual Approach Using the Dynamic Color Scanning Method
The Symbolic Approach Using Maple Commands
Findings
Conclusions and Further Research
Full Text
Published version (Free)

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