Let us analyze and solve the problem step by step.

Given:

1. $AD$ and $BE$ are medians of $\triangle ABC$. This means:
   • $D$ is the midpoint of $BC$,
   • $E$ is the midpoint of $AC$.
2. $BE \parallel DF$, where $F$ lies on $AC$.
3. We are to find the length of $CF$.

Key concepts:

1. A median divides the opposite side of a triangle into two equal halves.
2. Since $BE \parallel DF$, $BE$ and $DF$ form a pair of parallel line segments.
3. Using proportionality in triangles, we'll determine $CF$.

Solution:

Step 1: Understanding the geometry

• Since $BE$ is a median:
  • $AE = EC$ (i.e., $E$ is the midpoint of $AC$).
• Since $BE \parallel DF$, $F$ divides $AC$ in a specific ratio.

Step 2: Ratio division due to parallel lines

By the property of medians and proportionality:

• $F$ divides $AC$ in the ratio $2:1$, with $F$ closer to $C$.
  • Why? The centroid of a triangle divides the medians in a $2:1$ ratio, and since $BE \parallel DF$, this proportionality holds.

Step 3: Calculate $CF$

• Since $F$ divides $AC$ in the ratio $2:1$, we have: $CF = \frac{1}{3} AC$

Final Answer:

$\boxed{CF = \frac{1}{3} AC}$

Let me know if you would like further clarifications or details!

Related Questions:

1. What is the general property of medians in a triangle?
2. How does the centroid divide medians in a triangle?
3. Can $BE \parallel DF$ be used to find other ratios in the triangle?
4. If $AC = 12$, what is the length of $CF$?
5. How would the solution change if $BE$ were not parallel to $DF$?

Tip:

Always consider the properties of medians and centroids when solving triangle problems. The $2:1$ ratio is a powerful tool!