Length of Angle Bisector/Proof 1

Theorem
Let $\triangle ABC$ be a triangle.

Let $AD$ be the angle bisector of $\angle BAC$ in $\triangle ABC$.


 * LengthOfAngleBisector.png

Let $d$ be the length of $AD$.

Then $d$ is given by:


 * $d^2 = \dfrac {b c} {\left({b + c}\right)^2} \left({\left({b + c}\right)^2 - a^2}\right)$

where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.

Proof
Similarly, or by symmetry, we get:
 * $BD = \dfrac {a c} {b + c}$

From Stewart's Theorem, we have:
 * $b^2 \cdot BD + c^2 \cdot DC = d^2 \cdot a + BD \cdot DC \cdot a$

Substituting the above expressions for $BD$ and $DC$: