It is well-known that Vecten and Fasbender observed the duality of two problems: the Fermat–Torricelli problem for three points A1,A2,A3 in the plane, and the determination of the largest equilateral triangle circumscribed about the triangle A1A2A3. As Kuhn pointed out, from the historical point of view this seems to be the first example of dualization in the sense of nonlinear programming. We give a geometric proof of the corresponding duality between the generalized Fermat–Torricelli problem, also called the Weber problem, for three weighted points and the determination of largest triangles of prescribed shape circumscribed about a given triangle. Preparing this result, we give an extensive historical introduction to the mathematical subjects related to the proof.
