The Fermat-Torricelli problem on surfaces

In this note a simple proof of the famous Fermat-Torricelli problem is given. For the vertices of a given triangle, Fermat asks for a fourth point such that the sum of its Euclidean distances to the three given points is minimized. Many authors present geometric approaches to the Fermat-Torricelli problem. We solve the problem by analytic and geometrical method and extend it to the sphere, we also characterize the median point P on the general regular surface.