How is bisector of one side of a right angled triangle, drawn from right angled corner equal to the half of the bisected side?

Question

In a right angled triangle ABC with right angle at B and D being the mid-point of side AC, is it possible to prove BD=AD=CD without using co-ordinate geometry or circle theorems etc? (Just by using other elementary theorems: This has bothered me since my school years)

Answer 1

A start: Complete the triangle to a rectangle in the obvious way. (Make a copy of the triangle, and rotate it into place.)