Question 184
The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$, respectively, intersect at two different points $E$ and $F$. The line $O_1E$ intersects for the second time $\omega_1$ at the point $A$, and $\omega_2$ at the point $C$. The line $O_2E$ intersects for the second time $\omega_2$ at the point $B$, and $\omega_1$ at the point $D$. Let $M$ and $N$ be the midpoints of the segments $AD$ and $BC$, respectively.
Prove that the circumcircles of the triangles $FMD$ and $FNC$ intersect at the line $DC$.
Prove that the circumcircles of the triangles $FMD$ and $FNC$ intersect at the line $DC$.