A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle

Abstract

The weighted Fermat–Torricelli problem with positive weights α, β, and γ asks for the point in the plane of a given triangle ABC that minimizes the function \({f(P) = \alpha \|PA\| + \beta \|PB\| +\gamma \|PC\|}\). This paper provides a complete, fully analytical, and self-contained solution to this problem. The solution starts, as is most natural, with the gradient equation ∇f = 0, and obtains all the desired results by some delicate algebraic manipulations of this equation.