CN-TST 2023 Problem 19

Let $A,B$ be two fixed points on the unit circle $\omega$ , satisfying $\sqrt{2} < AB < 2$ . Let $P$ be a point that can move on the unit circle, and it can move to anywhere on the unit circle satisfying $\triangle ABP$ is acute and $AP>AB>BP$ . Let $H$ be the orthocenter of $\triangle ABP$ and $S$ be a point on the minor arc $AP$ satisfying $SH=AH$ . Let $T$ be a point on the minor arc $AB$ satisfying $TB || AP$ . Let $ST\cap BP = Q$ .

Show that (recall $P$ varies) the circle with diameter $HQ$ passes through a fixed point.