Question

ABC IS a isoceles triangle in which AB=AC 'IF AB and AC are produced to D and E respectively such that AB = CE.prove that BE =CD

Answer 1

You don't say, but seem to imply that AB=BD. That is, you haven't said anything about how far AB is extended.

It appears that DE is parallel to BC.

In triangle BCE and CBD,
BC = CB
angle BCE = angle CBE
CE = BD

So, by SAS, the triangles are congruent.
Thus, BE=CD