Fagnano's problem

In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775:

For a given acute triangle determine the inscribed triangle of minimal perimeter.

The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle.

Solution

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, among others by Hermann Schwarz and Lipót Fejér. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.

Physical principles

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

A

B

C

{\displaystyle 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,

∠

b

c

A

=

∠

a

c

B

,

∠

c

a

B

=

∠

b

a

C

,

∠

a

b

C

=

∠

c

b

A

{\displaystyle \angle bcA=\angle acB,\angle caB=\angle baC,\angle abC=\angle cbA}

Therefore, this minimal triangle is the orthic triangle.