FKMO 2014 Problem 4

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