MathNet 2017-part-01 — 176 problems (EN–VI) Question 24

Let $ABC$ be an acute-angled triangle, $H$ its orthocentre, and $D$, $E$ and $F$ the feet of the altitudes through $A$, $B$ and $C$, respectively. Let $P$ be the intersection point of $DF$ with the altitude through $B$. The perpendicular to $BC$ through $P$ intersects the side $AB$ at $Q$. Let $N$ be the intersection point of $EQ$ with the altitude through $A$.
Prove that $N$ bisects the segment $AH$.
(Karl Czakler)