It is equiavlent to proving that $N'$ is midpoint of $BC$ which isn't too hard, since the condition tells us that $N$ is inside the circumcircle $N'$ is on. Invalid URL

The Iranian's problem $ 24 $ of its Mathematical Olympiad 2003 is almost the same: "In an acute triangle $ABC$ points $D, E, F$ are the feet of the altitudes from $A, B$ and $C$, respectively. A line through $D$ parallel to $EF$ meets $AC$ at $Q$ and $AB$ at $R$. Lines $BC$ and $EF$ intersect at $P$. Prove that the circumcircle of triangle $PQR$ passes through the midpoint of $BC$." I found solution in the book with title: "Narrative approaches to the International mathematical problems." on page 74.