Example 1 In $\triangle ABC$, $AB=12, AC=16$, $M$ is the midpoint of $BC$, $E, F$ are on $AB, AC$ respectively, $EF$ intersects $AM$ at $G, AE=2AF$. Find the ratio $\frac{EG}{GF}$.
Solution: Connect $C E$ intersecting $A M$ at $O, \triangle C B E$ and $\triangle C E F$ are intersected by the same...
line $A M$, using Menelaus Theorem,
we get (as shown in the figure):
$$
\frac{B M}{M C} \cdot \frac{C O}{O E} \cdot \frac{E A}{A B}=1, \frac{E O}{O C} \cdot \frac{C A}{A F} \cdot \frac{F G}{G E}=1 .
$$

Multiplying the two equations, noting that $\frac{B M}{M C}=1$, we have
$$
\frac{E G}{G F}=\frac{A C}{A B} \cdot \frac{A E}{A F}=\frac{16}{12} \cdot \frac{2 A F}{A F}=\frac{8}{3} .
$$