Fagnano's problem

Fagnano's original proof used calculus methods and an intermediate result given by his father Giulio Carlo de' Toschi di Fagnano.

Later however several geometric proofs were discovered as well, amongst others by Hermann Schwarz and Lipót Fejér.

These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.

A solution from physics is found by imagining putting a rubber band that follows Hooke's Law around the three sides of a triangular frame

The tension inside the rubber band is the same everywhere in the rubber band, so in its resting position, we have, by Lami's theorem,