Problem: Let $ABC$ be an acute triangle, and let $X$ be a variable interior point
on the minor arc $BC$ of its circumcircle.
Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively.
Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$.
Let $\ell$ be the line through $P$ parallel to $XR$.
Prove that as $X$ varies along minor arc $BC$,
the line $\ell$ always passes through a fixed point.

Solution: The fixed point is the orthocenter,
since $\ell$ is a Simson line.
See Lemma 4.4 of \emph{Euclidean Geometry in Math Olympiads}.