On side \( AB \) of an isosceles trapezoid \( ABCD \) (\( AD \parallel BC \)), points \( E \) and \( F \) are chosen such that a circle can be inscribed in quadrilateral \( CDEF \). Prove that the circumcircles of triangles \( ADE \) and \( BCF \) are tangent to each other. Proposed by Matthew Kurskyi