Question 7
Let an acute triangle $ABC$ be inscribed in $(O)$. Let $G$ lie on the arc $BC$ not containing $O$ of the circumcircle $(I)$ of triangle $OBC$. The circumcircle of $ABG$ meets $AC$ again at $E\ne A$; the circumcircle of $ACG$ meets $AB$ again at $F\ne A$.
a) Put $K=BE\cap CF$. Prove that $AK,BC,OG$ are concurrent.
b) Fix a point $D$ on the arc $BOC$ containing $O$ of $(I)$. Put $M=GB\cap CD$ and $N=GC\cap BD$. Suppose $MN$ meets $(O)$ at two points $P,Q$. Prove that, as $G$ varies on the arc $BC$ not containing $O$, the circumcircle of $GPQ$ always passes through two fixed points.
a) Put $K=BE\cap CF$. Prove that $AK,BC,OG$ are concurrent.
b) Fix a point $D$ on the arc $BOC$ containing $O$ of $(I)$. Put $M=GB\cap CD$ and $N=GC\cap BD$. Suppose $MN$ meets $(O)$ at two points $P,Q$. Prove that, as $G$ varies on the arc $BC$ not containing $O$, the circumcircle of $GPQ$ always passes through two fixed points.