The Fermat-Torricelli Problem in the Projective Plane

TL;DR

This paper investigates the Fermat-Torricelli problem in the projective plane using sine distance, showing that for triangles with sides longer than sin 60°, the Fermat-Torricelli point is at the vertex opposite the longest side.

Contribution

It provides a novel characterization of the Fermat-Torricelli point in the projective plane for triangles with sufficiently long sides, including a complete analysis of the equilateral case.

Findings

Fermat-Torricelli point is at the vertex opposite the longest side when sides > sin 60°

Complete characterization of the equilateral case

Uses deformation argument in proof

Abstract

We pose and study the Fermat-Torricelli problem for a triangle in the projective plane under the sine distance. Our main finding is that if every side of the triangle has length greater than $\sin 60^\circ$, then the Fermat-Torricelli point is the vertex opposite the longest side. Our proof relies on a complete characterization of the equilateral case together with a deformation argument.