Question 4
Let $ABC$ be an acute non-isosceles triangle inscribed in circle $(O)$. The line through $O$ and the midpoint $I$ of $BC$ meets lines $AB,AC$ at $E,F$, respectively. Let $D,G$ be the reflections of $A$ in $O$ and in the circumcenter of $AEF$, respectively. Let $K$ be the reflection of $O$ in the circumcenter of $OBC$.
(a) Prove that $D,G,K$ are collinear.
(b) Choose $M,N$ on lines $KB,KC$, respectively, such that $IM\perp AC$ and $IN\perp AB$. The perpendicular bisector of $IK$ meets $MN$ at $H$. Let $P,Q$ be the intersections of $IH$ with $AB,AC$, respectively. Prove that the circumcircle of $APQ$ meets $(O)$ again at a point on $AI$.
(a) Prove that $D,G,K$ are collinear.
(b) Choose $M,N$ on lines $KB,KC$, respectively, such that $IM\perp AC$ and $IN\perp AB$. The perpendicular bisector of $IK$ meets $MN$ at $H$. Let $P,Q$ be the intersections of $IH$ with $AB,AC$, respectively. Prove that the circumcircle of $APQ$ meets $(O)$ again at a point on $AI$.