DE-MO 2009 Problem 5

Let a triangle $ABC$ . $E,F$ in segment $AB$ so that $E$ lie between $AF$ and half of circle with diameter $EF$ is tangent with $BC,CA$ at $G,H$ . $HF$ cut $GE$ at $S$ , $HE$ cut $FG$ at $T$ . Prove that $C$ is midpoint of $ST$ .