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Brazilian Mathematical Olympiad

Geometry Difficulty 4.7 AIME Prove it Brazil

$ABC$ is a triangle with $\angle A = 90^\circ$ . For a point $D$ on the side $BC$ , the feet of the perpendiculars to $AB$ and $AC$ are $E$ and $F$ . For which point $D$ is $EF$ a minimum?

Solution

The foot of the perpendicular from $A$ . $AEDF$ is a rectangle, so $EF = AD$ . $AD$ is obviously a minimum for the altitude.

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