Library / /127 of 740

, 2015

Geometry Difficulty 4.7 AIME Prove it United States

Problem:
Let $ABC$ be an acute triangle with orthocenter $H$ . Let $D, E$ be the feet of the $A$ -, $B$ -altitudes respectively. Given that $AH = 20$ and $HD = 15$ and $BE = 56$ , find the length of $BH$ .

Solution

Solution:
Let $x$ be the length of $BH$ . Note that quadrilateral $ABDE$ is cyclic, so by Power of a Point, $x(56 - x) = 20 \cdot 15 = 300$ . Solving for $x$ , we get $x = 50$ or $6$ . We must have $BH > HD$ so $x = 50$ is the correct length.

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