Question 64
Let $ABC$ be an acute and non-isosceles triangle with $\angle A = 45^\circ$. The altitudes $AD, BE, CF$ of triangle $ABC$ are concurrent at orthocenter $H$. The line $EF$ meets the line $BC$ at $P$. Point $I$ is the midpoint of segment $BC$ and $IF$ cuts $PH$ at $Q$.
1. Prove that $\angle IQH = \angle AIE$.
2. Let $K$ be the orthocenter of triangle $AEF$ and $(J)$ is the circumcircle of the triangle $KPD$. The line $CK$ intersects circle $(J)$ at $G$ (different to $K$) and the line $IG$ intersect circle $(J)$ at $M$ (different to $G$), the line $JC$ intersect the circle with the diameter $BC$ at $N$ (different to $C$). Prove that $G, M, N, C$ are cyclic.
1. Prove that $\angle IQH = \angle AIE$.
2. Let $K$ be the orthocenter of triangle $AEF$ and $(J)$ is the circumcircle of the triangle $KPD$. The line $CK$ intersects circle $(J)$ at $G$ (different to $K$) and the line $IG$ intersect circle $(J)$ at $M$ (different to $G$), the line $JC$ intersect the circle with the diameter $BC$ at $N$ (different to $C$). Prove that $G, M, N, C$ are cyclic.