Question 65
Let $ABC$ be an acute triangle. Points $B$, $D$, $E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Let $H$ be the orthocentre of triangle $ADE$. Let $P$ and $Q$ be points on lines $BM$ and $CN$, 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.
The orthocentre of a triangle is the point of intersection of its altitudes.