Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$