A Computational Approach to the Perimeter-Area Inequality in a Triangle

This paper explores the application of automated reasoning tools, specifically those implemented in GeoGebra Discovery, to the perimeter-area inequality in triangles. Focusing on the computational complex and real algebraic geometry methods behind these tools, this study analyzes a geometric construction involving a triangle with arbitrary side lengths and area, investigating the automated derivation of the relationship between the area and perimeter of a triangle, and showing that only equilateral triangles satisfy the exact perimeter-area equality. The main contribution of this work is to describe the challenges, and potential ways to approach their solutions, still posed by the use of such automated, symbolic computation, methods in dynamic geometry, in particular concerning the discovery of loci of points that satisfy specific geometric conditions, suggesting relevant improvements for the future development of these symbolic AI-based educational tools in geometry.