Pólya's Mechanical Model for the Fermat Point

By a geometric theorem, the Fermat point (the ball) coincides with the isogonic point, that is, the unique point at which the sides of the triangle subtend equal angles [1], [3]. To construct it, erect on the sides of the triangle three outwardly equilateral triangles. The three lines joining the vertices of the equilateral triangles to the opposite vertices of ABC meet in the isogonic point F.

The Hungarian mathematician George Pólya [2] modeled the geometric problem by a physical system consisting of a perforated horizontal plane and weighted ropes passing through the perforations. In mechanical equilibrium, the potential energy of the system is at a minimum. Applying equal weights means that the angles between the force vectors at F are equal to 120 degrees.

[1] H. S. M. Coxeter, Introduction to Geometry, 2nd ed., New York: Wiley, 1989.

[2] George Pólya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton: Princeton Univ. Press, 1954.

[3] Gergely Szmerka, "A Taste of the Fermat–Torricelli Circle of Problems", KöMaL (Mathematical and Physical Journal for Secondary Schools), 58(4), 2008 pp. 194–201. (in Hungarian)