Question 172
Let $ABC$ be an acute triangle inscribed in the circle $(O)$ and $I$ is the circumcenter of triangle $OBC$. Point $G$ belongs to the arc $BC$ (not contains $O$) of $(I)$. The circle $(ABG)$ intersects $AC$ at $E$ and circle $(ACG)$ intersects $AB$ at $F$ (points $E, F$ differ from $A$).
1. Denote $K$ as the intersection of $BE$ and $CF$. Prove that $AK, BC$ and $OG$ are concurrent.
2. Let $D$ be a fixed point on the arc $BC$ that contains $O$ of $(I)$ and $GB$ meets $CD$ at $M$, $GC$ meets $BD$ at $N$. Suppose that $MN$ intersects $(O)$ at $P, Q$. Prove that when $G$ moves on $(I)$, the circumcircle of triangle $GPQ$ always pass through two certain fixed points.
1. Denote $K$ as the intersection of $BE$ and $CF$. Prove that $AK, BC$ and $OG$ are concurrent.
2. Let $D$ be a fixed point on the arc $BC$ that contains $O$ of $(I)$ and $GB$ meets $CD$ at $M$, $GC$ meets $BD$ at $N$. Suppose that $MN$ intersects $(O)$ at $P, Q$. Prove that when $G$ moves on $(I)$, the circumcircle of triangle $GPQ$ always pass through two certain fixed points.