DE-BUNDESWETTBEWERB 2005 Problem 7

Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ . Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$ ), and the circle $k_2$ at a point $E$ (apart from $B$ ). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$ ), and the circle $k_2$ at a point $F$ (apart from $B$ ). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$ .

Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$ .

Prove that the triangles $ACD$ , $AEF$ and $AMN$ are similar to each other.