A conjectura de Toeplitz para polígonos regulares

Abstract

This work aims to demonstrate the Toeplitz conjecture, an open mathematical problem that proposes the following sentence about plane curves: "any closed simple plane curve (Jordan Curve) contains the four vertices of a square", specifically for regular polygons using only Euclidean Euclidean plane geometry. The first part of the development of this work consists of demonstrations of the square inscribed in the equilateral triangle, the square, the regular pentagon and the regular hexagon. Based on these cases and geometric constructions using the dynamic geometry software (GeoGebra), we demonstrate the Toeplitz Conjecture for regular polygons with even numbers of sides. Furthermore, we present a way to choose the four sides of a regular polygon with an odd number of sides that contains the vertices of an inscribed square. Thus, we conclude that the results presented in this paper contribute to solving the problem of the square inscribed in a regular polygon with an even number of sides.