Školjka

%V0 Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, assume that there exist points $E$ on line $BC$ outside segment $BC$, and $F$ inside segment $AD$ such that $\angle DAE = \angle CBF$. 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 $EF$, assume it does not lie on line $AB$. Prove that $I$ belongs to the circumcircle of $ABK$ if and only if $K$ belongs to the circumcircle of $CDJ$. Proposed by Charles Leytem, Luxembourg