EGMO 2025 Problem 3

Let $A B C$ be an acute triangle. Points $B$ , $D$ , $E$ , and $C$ lie on a line in this order and satisfy $B D =$ $D E = E C$ . Let $M$ and $N$ be the midpoints of $A D$ and $A E$ , respectively. Let $H$ be the orthocentre of triangle $A D E$ . Let $P$ and $Q$ be points on lines $B M$ and $C N$ , respectively, such that $D$ , $H$ , $M$ , and $P$ are concyclic and $E$ , $H$ , $N$ , and $Q$ are concyclic. Prove that $P$ , $Q$ , $N$ , and $M$ are concyclic.

The orthocentre of a triangle is the point of intersection of its altitudes.