Question 5
Let $ABC$ be an acute, non-isosceles triangle with $\angle A=45^\circ$. Its altitudes $AD,BE,CF$ meet at $H$. Let $P=EF\cap BC$, let $I$ be the midpoint of $BC$, and let $Q=IF\cap PH$.
1. Prove $\angle IQH=\angle AIE$.
2. Let $K$ be the orthocentre of $AEF$, and let $(J)$ be the circumcircle of $KPD$, with centre $J$. Let $G\ne K$ be the second intersection of $CK$ with $(J)$, $M\ne G$ the second intersection of $IG$ with $(J)$, and $N\ne C$ the second intersection of $JC$ with the circle of diameter $BC$. Prove that $G,M,N,C$ are concyclic.
The second-intersection qualifications make explicit the intended construction in the source.
1. Prove $\angle IQH=\angle AIE$.
2. Let $K$ be the orthocentre of $AEF$, and let $(J)$ be the circumcircle of $KPD$, with centre $J$. Let $G\ne K$ be the second intersection of $CK$ with $(J)$, $M\ne G$ the second intersection of $IG$ with $(J)$, and $N\ne C$ the second intersection of $JC$ with the circle of diameter $BC$. Prove that $G,M,N,C$ are concyclic.
The second-intersection qualifications make explicit the intended construction in the source.