# Minimum-perimeter triangle embedded inside a given triangle

A friend gave me this problem but I have no idea how to approach it.

Suppose you are given a triangle $ABC$. Pick points $P, Q$ and $R$ on $BC, CA$ and $AB$ such that the perimeter of the triangle $PQR$ ($PQ+QR+RQ$) is minimized.

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Assuming ABC is acute angled, this is Fagnano's problem, and has an elegant solution using reflection. More proofs : Cut the Knot page on Fagnano's problem.

For acute angled triangles, it is the orthic triangle (whose vertices are the feet of the perpendicular from the vertices to the opposite sides).

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