1, make AF⊥BC from A to BC, and intersect BC at point F.

Because AB=AC, for △ABC, the height on the bottom edge is the center line on the bottom edge (the height on the bottom edge of an isosceles triangle, the bisector of the center line and the vertex angle are three in one).

So f is the midpoint of BC, BF = CF

Similarly, f is also the midpoint of DE, and DF=EF.

So BD=BF-DF=CF-EF=EC.

2. Prove with congruent triangles:

Because AD=AE, ∠ADE=∠AED.

So ∠ ADB =180-∠ ade =180-∠ aed = ∠ AEC.

Because AB=AC, ∠ B = ∠ C.

So between △ABD and △AEC

∠ADB=∠AEC，∠B=∠C，AB=AC

So △ Abd△ AEC

So BD=EC.