26th Junior Balkan Mathematical Olympiad | JBMO 2022

Problem 2

Let $ABC$ be an acute triangle such that $AH = HD$ , where $H$ is the orthocenter of $ABC$ and $D \in BC$ is the foot of the altitude from the vertex $A$ . Let $\ell$ denote the line through $H$ which is tangent to the circumcircle of the triangle $BHC$ . Let $S$ and $T$ be the intersection points of $\ell$ with $AB$ and $AC$ , respectively. Denote the midpoints of $BH$ and $CH$ by $M$ and $N$ , respectively. Prove that the lines $SM$ and $TN$ are parallel.