The given diagram shows two triangles, $\triangle ABC$ and a smaller triangle within it, with segments parallel and proportional. Since $FE \parallel BC$, we can apply the concept of similar triangles. The triangles $\triangle AFE$ and $\triangle ABC$ are similar by the Basic Proportionality Theorem (or Thales' theorem).

We are asked to find the length of $FE$, and we are given that $BC = 24$. The similar triangles give us a proportionality relationship between the corresponding sides.

Let's assume the ratio of the smaller triangle $AFE$ to the larger triangle $ABC$ is denoted by $k$. Since the lengths of the corresponding segments $FG \parallel GD$, and so on, are equal, the diagram suggests that the ratio between the sides of the smaller and larger triangles is 1:3.

Thus, we can set up the proportion: $\frac{FE}{BC} = \frac{1}{3}$

Since $BC = 24$, we can solve for $FE$: $FE = \frac{1}{3} \times 24 = 8$

Thus, the length of $FE$ is 8.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you apply the Basic Proportionality Theorem in similar triangles?
2. What are other ways to identify if triangles are similar?
3. How can you verify the proportionality of other sides in this problem?
4. What would happen if the ratio was different from 1:3?
5. How can the length of $FG$ or $GD$ be determined from this diagram?

Tip: In similar triangles, all corresponding angles are equal, and the ratios of the corresponding sides are proportional!