ÖMO 2001 Problem 3

A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$ . A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$ . Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$ . Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.