O baricentro como ponto crítico de uma função

Given the median AM of a triangle ABC, let P be a point sliding along it.In this article, we will find the point on AM that optimizes the ratio PC PB and also verify under which condition such a critical point coincides with the centroid.The conclusions we have reached are that the critical point P is such that PM = BC 2 and this point coincides with the centroid if and only if the triangle satisfies the relationship 5BC 2 = AC 2 + AB 2 .For the proof, we will employ Stewart's theorem, the Law of Cosines, e Differential Calculus.