In triangle EBC, since D is the midpoint of BC and DF II BE,by using the converse of the midpoint theorem, weget that CF = FE, i.e, F is the midpoint of of CE.Now, since E is the midpoint of AC, AE= CE.AC + CE = AC= AE + 2CF = AC ( AF = CF)= 2CF + 2CF = AC ( AE = CE)4CF = ACTherefore, CF = 1/4 ACHence, proved.

Proof:

Given:
- In triangle ABC, AD and BE are medians.
- BE is parallel to DF.

To prove:
CF = 1/4 AC

Proof:

1. Introduction:
- We need to prove that CF is equal to one-fourth of AC.
- To prove this, we will use the properties of medians and parallel lines.

2. Properties of Medians:
- In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side.
- The medians of a triangle intersect at a point called the centroid.
- The centroid divides each median into two segments, with the segment towards the vertex being twice as long as the segment towards the midpoint.

3. Understanding the Figure:
- Let's consider triangle ABC, with medians AD and BE.
- CF is a line segment parallel to BE, intersecting AD at point F.
- We need to prove that CF is equal to one-fourth of AC.

4. Using the Properties of Medians:
- Since AD and BE are medians, they divide each other in the ratio 2:1.
- Let's consider the intersection point of AD and BE as point G, which is the centroid of triangle ABC.
- Therefore, AG:GD = 2:1 and BG:GE = 2:1.

5. Applying the Ratio to CF:
- Let's consider AG as a and GD as b.
- Since AG:GD = 2:1, we have a:b = 2:1.
- Similarly, let's consider BG as c and GE as d.
- Since BG:GE = 2:1, we have c:d = 2:1.

6. Using Similarity of Triangles:
- Triangle ADF is similar to triangle ABC because BE is parallel to DF.
- Therefore, the ratios of corresponding sides of these triangles are equal.

7. Applying the Similarity Ratio:
- Applying the similarity ratio, we have CF:AC = DF:BC.
- Since CF:AC = 1:x (to be proved) and DF:BC = c:a (from the similarity ratio), we have 1:x = c:a.

8. Simplifying the Equation:
- From step 5, we know that c = 2a and a = 2b.
- Substituting these values, we have 1:x = 2a:a.
- Simplifying further, we get 1:x = 2:1.

9. Conclusion:
- From step 8, we have 1:x = 2:1.
- Therefore, x = 1/2, which means CF is equal to one-fourth of AC.
- Hence, CF = 1/4 AC is proved.