Pólya's Mechanical Model for the Fermat Point

The Fermat point (also called Torricelli point) is the point inside a triangle (provided no angle exceeds 120 degrees) such that the sum of its distances from the vertices is a minimum.
Three strings are attached to a ball and passed through holes in a horizontal table, with equal weights attached at the ends of the strings under the table. Then the ball is at the Fermat point.
One of the holes can be shifted. As the hole moves the ball's position changes, constantly keeping the strings aligned at 120 degrees.
This model serves as an interesting (and rare) example in which a geometric theorem is proved by means of physical principles.


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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)
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