HMMT Spring, 2021

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

In triangle $A B C$ , let $M$ be the midpoint of $B C$ , $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$ . Suppose that $O A = O N = 11$ and $O H = 7$ . Compute $B C^{2}$ .

Solution

Solution:

Let $\omega$ be the circumcircle of $\triangle A B C$ . Note that because $O N = O A$ , $N$ is on $\omega$ . Let $P$ be the reflection of $H$ over $M$ . Then, $P$ is also on $\omega$ . If $Q$ is the midpoint of $N P$ , note that because
$N H = H M = M P,$
$Q$ is also the midpoint of $H M$ . Since $O Q \perp N P$ , we know that $O Q \perp H M$ . As $Q$ is also the midpoint of $H M$ ,
$O M = O H = 7$
With this,
$B M = \sqrt{O B^{2} - B M^{2}} = 6 \sqrt{2}$
and $B C = 2 B M = 12 \sqrt{2}$ . Therefore, $B C^{2} = 288$ .

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