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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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Problem 1 (Page 203) and Problem 7 (Page 207) from the section on Geometry Gems in the book "Challenges and Thrills of Pre college Mathematics" answers your question. The google books link is here.

( This was one of the beautiful constructions I remember having seen while preparing for my olympiad, years back :))

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Thanks for pointing to what seems to be a very interesting book. – Alex Smart Apr 25 '12 at 3:56

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