The Fermat-Torricelli problem revisited

Abstract Given three points, one can ask to find a fourth point for which the sum of the distances to the given points becomes minimal. This is known as the Fermat-Torricelli problem; various generalizations and modifications of this problem are also associated—not always in a historically correct way—with the names of Weber and Steiner. The purpose of this paper is twofold. First, we present in a succinct (and, as we hope, appealing) way, suitable for use in schools, this interesting old problem and its rich history which, in our opinion, deserves as much popularity as it can get. Second, we add two new facets to the picture, one concerning the analytic solvability of the problem, the other concerning the duality between this problem and Moss’ problem to be described below.