Fagnano's problem

The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of 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 $ABC$, such that it could slide around smoothly. Then the rubber band would end up in a position that minimizes its elastic energy, and therefore minimize its total length. This position gives the minimal perimeter triangle.

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, $\backslash angle\; bcA\; =\; \backslash angle\; acB,\; \backslash angle\; caB\; =\; \backslash angle\; baC,\; \backslash angle\; abC\; =\; \backslash angle\; cbA$

File:Altitudes and orthic triangle SVG.svg

Therefore, this minimal triangle is the orthic triangle.

The minimality of the perimeter of the orthic triangle can be proven in a more general setting, that of absolute geometry and even weaker settings.{{citation|first1=Victor|last1=Pambuccian|first2=Horst|last2=Struve|first3=Rolf|last3=Struve|title=A comprehensive analysis of the axiomatic frameworks in which Fagnano’s theorem holds|journal=Journal of Geometry|volume=116 |year=2025|pages=1–28|doi=10.1215/00294527-2017-0019 |url=https://link.springer.com/article/10.1007/s00022-025-00744-x}}

• Set TSP problem, a more general task of visiting each of a family of sets by the shortest tour

• {{citation

| last = Dörrie | first = Heinrich

| translator-last = Antin | translator-first = David

| contribution = Fagnano's altitude base point problem

| isbn = 978-0-486-61348-2

| pages = 359–361

| publisher = Dover

| title = 100 Great Problems of Elementary Mathematics

| year = 1965}}

• Paul J. Nahin: When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. Princeton University Press 2004, {{ISBN|0-691-07078-4}}, p. 67
• Coxeter, H. S. M.; Greitzer, S. L.:Geometry Revisited. Washington, DC: Math. Assoc. Amer. 1967, pp. 88–89.
• H.A. Schwarz: Gesammelte Mathematische Abhandlungen, vol. 2. Berlin 1890, pp. 344–345. ([https://archive.org/stream/gesammeltemathem02schwuoft#page/344/mode/2up online] at the Internet Archive, German)
• {{Citation|first1=Victor|last1=Pambuccian|first2=Horst|last2=Struve|first3=Rolf|last3=Struve|title=A comprehensive analysis of the axiomatic frameworks in which Fagnano’s theorem holds|journal=Journal of Geometry|volume=116 |year=2025|pages=1–28|doi=10.1215/00294527-2017-0019 |url=https://link.springer.com/article/10.1007/s00022-025-00744-x}}

{{reflist}}

• [https://www.cut-the-knot.org/triangle/Fagnano.shtml Fagnano's problem at cut-the-knot]
• [https://encyclopediaofmath.org/wiki/Fagnano_problem Fagnano's problem] in the Encyclopaedia of Mathematics
• {{usurped|1=[https://web.archive.org/web/20120705102919/http://www.pballew.net/orthocen.html Fagnano's problem at a website for triangle geometry]}}
• {{MathWorld|urlname=FagnanosProblem|title=Fagnano's problem}}

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Category:Triangle problems