VN-TST 2013 Problem 5

Let $ABC$ be a triangle with $\angle BAC= 45^o$ . Altitudes $AD, BE, CF$ meet at $H$ . $EF$ cuts $BC$ at $P$ . $I$ is the midpoint of $BC$ , $IF$ cuts $PH$ in $Q$ . a) Prove that $\angle IQH = \angle AIE$ . b) Let $(K)$ be the circumcircle of triangle $ABC$ , $(J)$ be the circumcircle of triangle $KPD$ . $CK$ cuts circle $(J)$ at $G$ , $IG$ cuts $(J)$ at $M$ , $JC$ cuts circle of diameter $BC$ at $N$ . Prove that $G, N, M, C$ lie on the same circle.