Question 187
Given an acute, scalene triangle $ABC$ with circumcircle $(O)$. The line passing through $O$ and the midpoint $I$ of $BC$ intersects $AB$, $AC$ at $E$, $F$. Let $D$, $G$ be the reflections of $A$ over $O$ and over the circumcenter of triangle $AEF$, respectively. Let $K$ be the reflection of $O$ over the circumcenter of triangle $OBC$.
a) Prove that $D$, $G$ and $K$ are collinear.
b) Take $M$ on $KB$ and $N$ on $KC$ such that $IM$ is perpendicular to $AC$ and $IN$ is perpendicular to $AB$. The perpendicular bisector of $IK$ intersects $MN$ at $H$. Suppose that $IH$ meets $AB$, $AC$ at $P$, $Q$ respectively. Prove that the circumcircle of triangle $APQ$ intersects $(O)$ again at a point on $AI$.
a) Prove that $D$, $G$ and $K$ are collinear.
b) Take $M$ on $KB$ and $N$ on $KC$ such that $IM$ is perpendicular to $AC$ and $IN$ is perpendicular to $AB$. The perpendicular bisector of $IK$ intersects $MN$ at $H$. Suppose that $IH$ meets $AB$, $AC$ at $P$, $Q$ respectively. Prove that the circumcircle of triangle $APQ$ intersects $(O)$ again at a point on $AI$.