9. In $\triangle A B C$, $A B=12, A C=16, M$ is the midpoint of $B C$, $E, F$ are on $A B, A C$ respectively, $E F$ intersects $A M$ at $G$, and $A E=2 A F$. Find the ratio $\frac{E F}{G F}$.
(IMO - 29 Shortlist)
9. Let $A F=a, A E=2 a$, then $\frac{A F}{A C}=\frac{a}{16}, \frac{A E}{A B}=\frac{2 a}{12}=\frac{a}{6}$.

Transform $\triangle A B C$ into a right triangle, establish an affine coordinate system, set $B(6,0), C(0,16), A(0,0)$, then $E(a, 0)$, $F(0, a), M(3,8)$. From the equations of line $A M: y=\frac{8}{3} x$, and line $E F: x+y=a$, solving them simultaneously, we get $G\left(\frac{3 a}{11}, \frac{8 a}{11}\right)$, hence $\frac{E G}{G F}=$ $\frac{y_{G}}{y_{F}-y_{G}}=\frac{\frac{8}{11}}{1-\frac{8}{11}}=\frac{8}{3}$ is the desired result.