Fagnano's problem

The Fagnano problem is a named after Giovanni Fagnano optimization problem.

This is understood to mean an inscribed triangle of a triangle, a triangle whose vertices lie on the sides of the triangle, that is, and. For the Höhenfußpunktdreieck that its circumference is less than that of any other inscribed triangle and thus it is the solution of the Fagnano problem.

First showed Giovanni Giulio Carlo Fagnano Fagnanos father that you can construct on W and at any fixed point U to 2 points on V so that the scope of the triangle is minimal. Giovanni Fagnano related this result to then use the calculus of all possible U to to determine the one in which the size of the triangle is minimal. Later, several elementary geometric proofs were found, among other things, by Leopold Fejér and Hermann Amandus Schwarz. These proofs use mostly properties of reflections to determine a minimal path.