Length of Angle Bisector

Theorem
The length of an angle bisector is given by the following equations:

$$b_\alpha^2=\frac{cb}{(c+b)^2}[(c+b)^2-a^2]$$

$$b_\gamma^2=\frac{ab}{(a+b)^2}[(a+b)^2-c^2]$$

$$b_\beta^2=\frac{ac}{(a+c)^2}[(a+c)^2-b^2]$$

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

And $$b_\alpha$$, $$b_\beta$$, and $$b_\gamma$$ are the angle bisectors from $$A$$, $$B$$, and $$C$$ respectively.

Proof 1
We look at one of the angle bisectors, WLOG $$b_\gamma$$:

$$ $$ $$ $$ $$

$$ $$ $$ $$ $$

We now plug these results in to Stewart's Theorem, also noting that $$CP=b_\gamma$$:

$$ $$ $$ $$ $$

A similar argument can be used to show that the statement holds for the others bisectors.

Proof 2
We look at one of the angle bisectors, WLOG $$b_\gamma$$:



Let $$u$$ and $$v$$ be the segments of $$AB$$ created by the angle bisector.

$$u=\frac{bc}{a+b}$$

$$v=\frac{ac}{a+b}$$

we have

$$ $$

Then by AA similarity we have $$\triangle CAD \sim \triangle CFB$$

$$ $$

Now we use the chord theorem, which gives us $$v \cdot u=b_\gamma \cdot DF$$.

$$ $$ $$ $$ $$

A similar argument can be used to show that the statement holds for the others bisectors.