ELMO 2018 Problem 4

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ . Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$ . Let $X$ be the foot of the altitude from $O$ onto line $PT$ . Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$ .

*The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$ , $\overline{BH}$ , and $\overline{CH}$ .

Proposed by Zack Chroman