Aplicação da distância de ponto a reta em problemas de minimização: generalização do ponto de Lemoine

Abstract

The minimum point of the two-variable function, which represents the sum of the squares of the distances, weighted by non-negative constants, from a point P(x,y) to the sides of a polygonal chain, was obtained analytically. Although using the point-to-line distance as a function of two real variables introduces some technical difficulties, this approach allowed for the expression of the minimum point to be derived using Differential Calculus techniques in two variables. The determinant of the Hessian matrix was related to the Cauchy-Schwarz inequality to ensure the minimum point condition. The subsequent investigation included the analysis of specific cases, such as that of a closed polygonal chain and the scenario in which the weighting constants are identical. The results can be interpreted as a generalization of the classical problem of minimizing the sum of the squares of the distances to the sides of a triangle, which is a particular case where a closed polygonal chain and identical constants are considered. In all cases, both general and special, the GeoGebra software was used to implement the analytical solutions and illustrate the visualization of the polygonal chains, the minimum points, the minimizing values, as well as the two-variable functions and their components. The analytical results are presented in an interactive and dynamic manner, offering an alternative interpretation of the geometric problem associated with the Lemoine point.