BxMO 2009 Problem 4

Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$ , let $E$ be a point on line $BC$ outside segment $BC$ , such that segment $AE$ intersects segment $CD$ . Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$ . Denote by $I$ the point of intersection of $CD$ and $EF$ , and by $J$ the point of intersection of $AB$ and $EF$ . Let $K$ be the midpoint of segment $EF$ , and assume that $K$ is different from $I$ and $J$ .

Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$ .