Analysis of the centroid as a critical point of the harmonic mean function in triangles

Inspired by previous works, considering an arbitrary triangle ABC, where AM is the median relative to side BC and P ? AM, in this article, we propose to investigate the behavior of the harmonic mean function of P B2 and P C2 . By analyzing the critical points of this function, we show that the relative extrema coincides with the centroid of the triangle if and only if the sides of the triangle are roots of a certain homogeneous polynomial of degree six. We also present some numerical simulations to illustrate the study conducted. The study is relevant as it brings previously unknown properties of the centroid of a triangle and has the potential for further investigations by considering other functions besides the harmonic mean between P B2 and P C2 . In this work, we use differential calculus, the law of cosines, Stewart's theorem, and the GeoGebra software.