Question 180
The circles $\mathcal{K}_1$ and $\mathcal{K}_2$ with centers $O_1$ and $O_2$ intersect at the points $A$ and $B$. The distance between the centers is greater than the radii of both circles. Let $C_1$ and $C_2$ be those intersection points of the line $O_1 O_2$ with the circles $\mathcal{K}_1$ and $\mathcal{K}_2$ which do not lie on the segment $O_1 O_2$. Denote the second intersection of the line $C_2 A$ and the circle $\mathcal{K}_1$ by $D_1$, and the second intersection of the line $C_1 A$ and the circle $\mathcal{K}_2$ by $D_2$.
The lines $D_1 B$ and $D_2 A$ meet at $E$, and the lines $D_1 A$ and $D_2 B$ meet at $F$. Show: if the quadrilateral $A O_1 B O_2$ is cyclic, then the quadrilateral $A E B F$ is cyclic as well.
The lines $D_1 B$ and $D_2 A$ meet at $E$, and the lines $D_1 A$ and $D_2 B$ meet at $F$. Show: if the quadrilateral $A O_1 B O_2$ is cyclic, then the quadrilateral $A E B F$ is cyclic as well.