Given:
- Triangle ABC
- AD is a median on BC
- E is the midpoint of AD
- B is produced to meet AC at F

To Prove:
AF = 1/3 AC

Proof:

1. Draw a Diagram:
It is important to visualize the given information. Draw a clear diagram of triangle ABC with all the given points labeled.

2. Use the Property of Medians:
Since AD is a median of triangle ABC, it divides BC into two equal parts. Therefore, BD = CD.

3. Use the Property of Midpoints:
Since E is the midpoint of AD, it divides AD into two equal parts. Therefore, AE = ED.

4. Apply the Triangle Proportionality Theorem:
According to the Triangle Proportionality Theorem, if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

Since BF is parallel to DE and intersects AC at point F, we can apply the Triangle Proportionality Theorem to triangle ABC.

According to the theorem, we have the following proportions:

AF/FC = DE/EB

Since AE = ED and BD = CD, we can substitute these values into the proportion:

AF/FC = (AE + ED)/(BD + DE)

5. Substitute Known Values:
We know that AE = ED and BD = CD, so we can substitute these values into the proportion:

AF/FC = (AE + AE)/(BD + AE)

Simplifying further:

AF/FC = 2AE/(BD + AE)

6. Use the Property of Medians:
Since AD is a median, BD = CD, so we can substitute BD with CD:

AF/FC = 2AE/(CD + AE)

7. Use the Property of Midpoints:
Since E is the midpoint of AD, AE = ED, so we can substitute AE with ED:

AF/FC = 2ED/(CD + ED)

8. Simplify the Proportion:
Since ED = AE and BD = CD, we can simplify the proportion further:

AF/FC = 2/3

9. Conclusion:
From the proportion, we can see that AF is 2/3 of FC. Since AF + FC = AC, AF must be 1/3 of AC.

Therefore, AF = 1/3 AC, which proves the statement.