Finding the Fermat point via analysis

TL;DR

This paper explores finding the Fermat point in a plane by applying advanced calculus techniques such as the extreme value theorem, Fermat's theorem, and the intermediate value theorem, offering an analytical approach to a classical geometric problem.

Contribution

It introduces an analytical method for locating the Fermat point using calculus, complementing traditional geometric solutions.

Findings

Successfully identified the Fermat point using calculus methods.

Provided a rigorous proof of the Fermat point's properties.

Bridged geometric and analytical approaches to the problem.

Abstract

Let $P_1,P_2,P_3$ be three given points in $\mathbf{R}^2$, and $P$ be an arbitrary point in $\mathbf{R}^2$. The classical Fermat's problem to Torricelli asks for the location of the point $P$ such that $|PP_1|+|PP_2|+|PP_3|$ is a minimum. There exist several elegant geometrical solutions in the literature.In this note, we consider finding the Fermat point by using methods in advanced calculus. The main tools we use are the extreme value theorem,Fermat's theorem, and the intermediate value theorem.