11th European Girls' Mathematical Olympiad | EGMO 2022

Problem 1. Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$ . 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 circumcenter of triangle $APQ$ , $H$ the orthocenter of triangle $ABC$ , and $S$ the point of intersection of the lines $BQ$ and $CP$ . Prove that $T$ , $H$ , and $S$ are collinear.;