An optimization problem for triangles

TL;DR

This paper investigates an optimization problem involving the product of distances from a point to triangle vertices, analyzing specific cases for isosceles triangles and providing explicit conditions.

Contribution

It introduces a detailed analysis of the distance product optimization problem, including explicit conditions for isosceles triangles, which was not previously characterized.

Findings

Explicit conditions for when both cases occur in isosceles triangles

Analysis of the optimization problem for the product of distances

Characterization of cases in general triangles

Abstract

We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.