TH-TMO 2007 Problem 4

A triangle $\vartriangle ABC$ has $AC = 16$ and $BC = 12$ . $E$ and $F$ are points on $AC$ and $BC$ , respectively, so that $CE = 3CF$ . Let $M$ be the midpoint of $AB$ , and let lines $EF$ and $CM$ intersect at $G$ . Compute the ratio $EG : GF$ .