Weighted sums of the squares of the distances of a point to the sidelines of a triangle

Abstract

We study a function, which is a weighted sum of the squares of the distances of an arbitrary point to the sidelines of a triangle. The given weights, considered as barycentric coordinates, determine a point $M$. We prove that the function reaches its minimum (maximum) at a point, which is isogonal conjugate to $M$.

arXiv comment8 pages, 2 figures