We have:
ab = bc (hypothetical)
so the triangle ABC is isosceles at B
we have: bd is the height of the triangle ABC, so BD is perpendicular to AB
Consider the triangle ABD and the right triangle ACD have:
AB = BC (hypothesis)
BD is the common edge
So the triangle ABD = triangle ACD (hypotenuse - acute angle)
hence the angle ABD = angle ACD
