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

TL;DR

This paper investigates a weighted sum of squared distances from a point to a triangle's sides, proving that the extremum occurs at a point isogonal conjugate to a specific point determined by barycentric weights.

Contribution

It establishes a novel relationship between weighted distance sums and isogonal conjugates within triangle geometry.

Findings

The function attains its minimum and maximum at isogonal conjugates of a point defined by barycentric weights.

The extremal points are characterized geometrically as isogonal conjugates.

The results deepen understanding of distance functions in triangle geometry.

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 $M$.