Question 64
Let $ABC$ be an acute-angled triangle in which $BC \lt AB$ and $BC \lt CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcentre of triangle $APQ$, $H$ the orthocentre of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$ and $S$ are collinear.