SHARYGIN 2010 Problem 18

A point $B$ lies on a chord $AC$ of a circle $\omega$. Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F$, and the rays $AD$ and $CE$ do in a point $G$. Prove that the line $FG$ passes through the midpoint of the segment $AC$.