EGMO 2023 | Problem 2 | Solution - 17 April 2023

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Problem 2.
We are given an acute triangle $ABC$ . Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$ , respectively, and that $DK$ and $DL$ are tangent to circle $AKL$ .
Show that line $KL$ passes through the orthocenter of triangle $ABC$ .

Solution
Let $H$ be the orthocenter of $ABC$ , and let $M$ be the midpoint of $\overline{KL}$ . Notice that $\angle DBK=\angle DMK=\angle DCL=\angle DML=90^\circ,$ so $BDMK$ and $CDML$ are cyclic. We have $\angle ABM=\angle KDM=90^\circ-\angle DKM=90^\circ-\angle BAC=\angle ABH,$ and similarly, $\angle ACM=\angle ACH$ . This means $M=H$ , so $H$ lies on $\overline{KL}$ , as desired..