Question 163
Given two fixed points $A$ and $B$ on the unit circle $\omega$, satisfying $\sqrt{2} \lt AB \lt 2$. Let $P$ be a moving point on $\omega$ such that $\triangle ABP$ is an acute-angled triangle and $AP \gt AB \gt BP$.
For this moving point $P$, let $H$ be the orthocenter of $\triangle ABP$. Take a point $S$ on the arc $\widehat{AP}$ such that $SH = AH$, and take a point $T$ on the arc $\widehat{AB}$ such that $TB \parallel AP$. Let $Q$ be the intersection of lines $ST$ and $BP$.
Prove that there exists a fixed point in the plane such that the circle with diameter $HQ$ passes through it.
For this moving point $P$, let $H$ be the orthocenter of $\triangle ABP$. Take a point $S$ on the arc $\widehat{AP}$ such that $SH = AH$, and take a point $T$ on the arc $\widehat{AB}$ such that $TB \parallel AP$. Let $Q$ be the intersection of lines $ST$ and $BP$.
Prove that there exists a fixed point in the plane such that the circle with diameter $HQ$ passes through it.