Question 8
Let $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ be a point on a line $BA$ such that $A$ lies between $B$ and $D$. Let $O_1$ be the circumcircle of triangle $DAC$. $O_1$ meets $BC$ at point $E$. Let $F$ be the point on the line $BC$ such that $FD$ is tangent to circle $O_1$, and let $O_2$ be the circumcircle of triangle $DBF$. Two circles $O_1, O_2$ meet at point $G$ ($G \neq D$). Let $O$ be the circumcenter of triangle $BEG$. Prove that the line $FG$ is tangent to circle $O$ if and only if $DG$ is perpendicular to $FO$.