Minimal Trigonometry Triangle ABC has angle BAC given and the lengths |AB| and |AC| satisfy 
$$|AB|\cdot |AC| = 1.$$ Let AD be the angle bisector of BAC. Express the length |AD| as a function of the length |AB|. 
I keep running into an issue where I need the length of one more side or the measure of another angle to come up with a formula. Can anyone supply a solution to this problem?

Hint:

 $\frac{AD}{\sin B}=\frac{BD}{\sin \frac A2}=\frac{ac}{(b+c)\sin \frac A2}$, and $[ABC]=\frac 12bc\sin A=\frac 12 ca\sin B\implies ca\sin B=\sin A$.

