Isogonal Conjugacy and Fermat Problems

TL;DR

This paper explores isogonal conjugacy types in triangles and applies these concepts to uniquely solve the Fermat problem with positive weights, providing geometric characterizations of solutions.

Contribution

It introduces a geometric characterization of isogonal conjugacy types and applies this to uniquely determine solutions to the weighted Fermat problem.

Findings

Unique solution to the Fermat problem determined by isogonal conjugacy

Geometric equalities characterize isogonal conjugacy types

Solutions extend to cases with mixed weights

Abstract

We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the general case the given weights determine uniquely a point X and the solution to the Fermat problem is the point Y, which is isogonally conjugate of type I to the point X. We obtain a similar characterization of the solution to the Fermat problem in the case of mixed weights as well.