Question 45
Let $\triangle ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $\widehat{BC}$ 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 $\widehat{BC}$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $\widehat{BC}$, line $\ell$ passes through $F$.)