Because BC is perpendicular to m,

So the angle BMD = 90, so the angle EMC = 90 (equal to the vertex angle).

So angle DEC+ angle BCE=90.

Because angle A = 90, angle ABC+ angle ACB =180-90 = 90.

So angle DEC= angle ABC

So in triangle ABC and triangle CED,

AB = EC

Angle BAC= angle ECD

Angle DEC= angle ABC

So triangle ABC is equal to triangle CED (ASA)

So BC=DE (the corresponding sides of congruent triangles are equal).