IR-MO 2002 Problem 5

$\omega$ is circumcirlce of triangle $ABC$ . We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$ . Assume that $M$ is midpoint of $BC$ . Let $\omega'$ be circumcircle of $UMV$ . We know that $R(ABC)=R(UMV)$ . $ME$ and $\omega'$ intersect at $T$ , and $FT$ intersects $\omega'$ at $S$ . Prove that $EF$ is tangent to circumcircle of $MCS$ .