Finding the Fermat point via analysis

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.