Two new analytical solutions and two new geometrical solutions for the weighted Fermat-Torricelli problem in the Euclidean plane

TL;DR

This paper presents two new analytical solutions and two new geometric methods for solving the weighted Fermat-Torricelli problem in the Euclidean plane, enhancing understanding of optimal point location with weighted distances.

Contribution

The paper introduces novel analytical and geometric solutions for the weighted Fermat-Torricelli problem, expanding existing methods for finding minimal weighted distance points.

Findings

Two new analytical solutions derived for the problem.

Two geometric solutions based on equilibrium and rotation principles.

Enhanced methods for solving the weighted Fermat-Torricelli problem.

Abstract

We obtain two analytic solutions for the weighted Fermat-Torricelli problem in the Euclidean Plane which states that: Given three points in the Euclidean plane and a positive real number (weight) which correspond to each point, find the point such that the sum of the weighted distances to these three points is minimized. Furthermore, we give two new geometrical solutions for the weighted Fermat-Torricelli problem (weighted Fermat-Torricelli point), by using the floating equilibrium condition of the weighted Fermat-Torricelli problem (first geometric solution) and a generalization of Hofmann's rotation proof under the condition of equality of two given weights (second geometric solution).