Question 19
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. Its diagonals $AC$ and $BD$ intersect at the point $O$. Let $E$ and $F$ be points on the segments $AO$ and $DO$ respectively. The line $EF$ intersects $\omega$ at the points $E_1$ and $F_1$. Circumcircles of triangles $ADE$ and $BCF$ intersect the line $EF$ at the points $E_2$, $F_2$ respectively. Prove that $E_1E_2 = F_1F_2$.