Weighted sums of the squares of the distances of a point to the   sidelines of a triangle
  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 $M$. 
Georgi Ganchev and Nikolai Nikolov
math.MG