In triangle $\triangle A B C$, point $F$ is on side $A C$ and $F C=2 A F$. If $G$ is the midpoint of segment $B F$ and $E$ is the intersection point of the line passing through $A$ and $G$ with segment $B C$, calculate the ratio $E C / B E$.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-031.jpg?height=303&width=406&top_left_y=2233&top_left_x=1473)
Let's choose point $H$ on segment $B C$ such that segment $F H$ is parallel to segment $A E$, as shown in the given figure. It follows that triangles $\triangle A E C$ and $\triangle F H C$ are similar, as they have parallel sides.

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-102.jpg?height=397&width=534&top_left_y=1892&top_left_x=795)

Since $F C=2 A F$, it follows from the similarity that $H C=2 E H$. On the other hand, triangles $\triangle B H F$ and $\triangle B E G$ are also similar, as they have parallel sides. From this similarity and the fact that $G$ is the midpoint of segment $B F$, we conclude that $E$ is the midpoint of segment $B H$. Thus, $B E=E H$ and, therefore,

$$
E C=E H+H C=E H+2 E H=3 E H=3 B E
$$

Consequently, $E C / B E=3$.