Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE = \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. Utari Wijayanti, Bandung