Question 5
An acute triangle $ABC$ is inscribed in a fixed circle $(O)$, with $B,C$ fixed and $A$ moving on $(O)$. On rays $AB,AC$, respectively, choose $M,N$ such that $MA=MC$ and $NA=NB$. The circumcircles of $AMN$ and $ABC$ meet again at $P\ne A$. Line $MN$ meets line $BC$ at $Q$.
(a) Prove that $A,P,Q$ are collinear.
(b) Let $D$ be the midpoint of $BC$. The circles centered at $M,N$ and passing through $A$ meet again at $K\ne A$. The line through $A$ perpendicular to $AK$ meets $BC$ at $E$. The circumcircle of $ADE$ meets $(O)$ again at $F\ne A$. Prove that $AF$ passes through a fixed point.
(a) Prove that $A,P,Q$ are collinear.
(b) Let $D$ be the midpoint of $BC$. The circles centered at $M,N$ and passing through $A$ meet again at $K\ne A$. The line through $A$ perpendicular to $AK$ meets $BC$ at $E$. The circumcircle of $ADE$ meets $(O)$ again at $F\ne A$. Prove that $AF$ passes through a fixed point.