Given:
- ABC is an isosceles triangle with AB = AC.
- BD and CE are medians of triangle ABC.

To prove:
BD = CE

Proof:

1. Properties of Medians:
- A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
- In triangle ABC, BD is a median that joins vertex B to the midpoint of AC, and CE is a median that joins vertex C to the midpoint of AB.

2. Midpoint of a Line Segment:
- The midpoint of a line segment divides it into two equal halves.
- Since AB = AC, the midpoint of AC is the same as the midpoint of AB.
- Let's denote the midpoint of AC and AB as M.

3. Definition of Isosceles Triangle:
- An isosceles triangle is a triangle in which two sides (AB and AC) are equal in length.
- In triangle ABC, AB = AC, which makes it an isosceles triangle.

4. Property of Isosceles Triangle:
- In an isosceles triangle, the medians from the base to the opposite sides are equal in length.
- Since BD and CE are medians from the base (BC) to the opposite sides (AC and AB), they are equal in length.

5. Conclusion:
- From step 4, we know that BD = CE.
- Therefore, we have proved that BD = CE in triangle APC.