Finding a line segment within an angle based on given midpoint

Question

Given an angle ∠ABC and an arbitrary point D somewhere within the angle, how would you (using only a compass and straight edge) draw a line segment where one end lies on AB and the other on BC, with D being the midpoint of the segment?

Answer 1

1. Let $B'$ be the symmetric of $B$ with respect to $D$;
2. Let $A'\in BC$ be such that $B'A'\parallel BA$;
3. Let $C'\in BA$ be such that $B'C'\parallel BC$;
4. $BC'B'A'$ is a parallelogram, hence $D$ is the midpoint of $A'C'$.