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Fagnano's Problem

Let ABC be an acute triangle. Let D, E, and F be the points where the altitudes from A, B, and C intersect the sides of the triangle. Then DEF is the inscribed triangle of smallest perimeter.

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"Fagnano's Problem" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/FagnanosProblem/
Contributed by: Jay Warendorff
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