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?

The area of the triangle is
$T
=\frac12 |AB|\ |AC| \sin(BAC)
$.
Therefore
$T
=\frac12 \sin(BAC)
$.
Since we know
$\angle BAC$,
we know
$\theta = \frac12 \angle BAC$.
The area of
$BAD$
is
$\frac12 |AB| |AD| \sin \theta$
and the area of
$DAC$
is
$\frac12 |AC| |AD| \sin \theta$.
The sum of these
is the area of the triangle,
so,
canceling the $\frac12$,
$\sin(BAC)
=|AB| |AD| \sin \theta + |AC| |AD| \sin \theta
= |AD| \sin \theta(|AB| + |AC|)
$.
Therefore
$\begin{array}\\
|AD|
&=(|AB| + |AC|)\frac{\sin(BAC)}{\sin \theta}\\
&=(|AB| + |AC|)\frac{\sin(2\theta)}{\sin \theta}\\
&=(|AB| + |AC|)\frac{2\sin(\theta)\cos(\theta)}{\sin \theta}\\
&=2(|AB| + |AC|)\cos(\theta)\\
&=2(|AB| + \frac1{|AB|})\cos(\theta)\\
\end{array}
$
