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Solving with GeoGebra Discovery an Austrian Mathematics Olympiad problem: Lessons Learned

Belén Ariño-Morera, Zoltán Kovács, Tomás Recio, Piedad Tolmos

TL;DR

This paper demonstrates GeoGebra Discovery's capacity to solve an Austrian Mathematics Olympiad problem using automated reasoning. It shows how the tool not only confirms the statement but also yields a high complexity score via a Gröbner-basis approach and enables automatic generalization beyond the original rhombus configuration. The authors also explore a challenging locus computation, revealing numerous degenerate cases and promoting a shift from locus equation plotting to study of intrinsic geometric features. These results illustrate the practical value and current limitations of automated geometric reasoning for teaching, exploration, and rapid verification of geometric conjectures.

Abstract

We address, through the automated reasoning tools in GeoGebra Discovery, a problem from a regional phase of the Austrian Mathematics Olympiad 2023. Trying to solve this problem gives rise to four different kind of feedback: the almost instantaneous, automated solution of the proposed problem; the measure of its complexity, according to some recent proposals; the automated discovery of a generalization of the given assertion, showing that the same statement is true over more general polygons than those mentioned in the problem; and the difficulties associated to the analysis of the surprising and involved high number of degenerate cases that appear when using the LocusEquation command in this problem. In our communication we will describe and reflect on these diverse issues, enhancing its exemplar role for showing some of the advantages, problems, and current fields of development of GeoGebra Discovery.

Solving with GeoGebra Discovery an Austrian Mathematics Olympiad problem: Lessons Learned

TL;DR

This paper demonstrates GeoGebra Discovery's capacity to solve an Austrian Mathematics Olympiad problem using automated reasoning. It shows how the tool not only confirms the statement but also yields a high complexity score via a Gröbner-basis approach and enables automatic generalization beyond the original rhombus configuration. The authors also explore a challenging locus computation, revealing numerous degenerate cases and promoting a shift from locus equation plotting to study of intrinsic geometric features. These results illustrate the practical value and current limitations of automated geometric reasoning for teaching, exploration, and rapid verification of geometric conjectures.

Abstract

We address, through the automated reasoning tools in GeoGebra Discovery, a problem from a regional phase of the Austrian Mathematics Olympiad 2023. Trying to solve this problem gives rise to four different kind of feedback: the almost instantaneous, automated solution of the proposed problem; the measure of its complexity, according to some recent proposals; the automated discovery of a generalization of the given assertion, showing that the same statement is true over more general polygons than those mentioned in the problem; and the difficulties associated to the analysis of the surprising and involved high number of degenerate cases that appear when using the LocusEquation command in this problem. In our communication we will describe and reflect on these diverse issues, enhancing its exemplar role for showing some of the advantages, problems, and current fields of development of GeoGebra Discovery.
Paper Structure (5 sections, 9 figures)

This paper contains 5 sections, 9 figures.

Table of Contents

  1. Introduction
  2. Solving the Problem
  3. Generalizing the Problem
  4. Grading Problem 2
  5. Conclusions

Figures (9)

  • Figure 1: Problem 2. Österreichische Mathematik-Olympiade Regionalwettbewerb für Fortgeschrittene 30. März 2023.
  • Figure 2: Confirming the truth of the statement in Problem 2.
  • Figure 3: Confirming the truth of the statement in Problem 2 and showing the circle through $A,S,D,E$.
  • Figure 4: Confirming the truth of the statement in Problem 2, and many others, through the Stepwise-discover command.
  • Figure 5: Locus of $C$ (in red color) for the concyclicity of $A,S,D,E$, assuming only that $B,D$ are in a circle centered at $A$.
  • ...and 4 more figures