FKMO 2017 Problem 1

A acute triangle $\triangle ABC$ has circumcenter $O$ . The circumcircle of $OAB$ , called $O_1$ , and the circumcircle of $OAC$ , called $O_2$ , meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$ . Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.