Fagnano's problem
Fagnano's problem asks: in a given acute triangle, which inscribed triangle has the smallest possible perimeter? The answer is the orthic triangle—the triangle formed by the feet of the altitudes of the original triangle.
Why is it minimal? A helpful way to see it uses reflections or a rubber-band analogy. If you imagine a path that goes along the three sides and back to the start, the shortest such closed path appears when you reflect the triangle across its sides and look for a straight-line route. This reasoning forces the contact points with the sides to be the feet of the altitudes, giving the orthic triangle. This reflection idea also leads to simple geometric proofs, and explains why the orthic triangle has the smallest perimeter.
Fagnano proved the result using calculus, and later geometers such as Hermann Schwarz and Lipót Fejér found purely geometric proofs. The same ideas give a physical picture: a rubber band placed around the three sides settles into a position of minimal length, which again corresponds to the orthic triangle.
The problem also relates to billiards: for an acute triangle, the corresponding path that reflects off the sides in a certain way can form a periodic path inside the triangle. Whether such a path exists for all triangles is still an open question. The fact that the orthic triangle minimizes perimeter also holds in more general geometric settings beyond the usual Euclidean space.
This page was last edited on 3 February 2026, at 08:01 (CET).