Length of Angle Bisector/Proof 2

Proof

 * LengthOfAngleBisector2.png

From Length of Angle Bisector: Proof 1, we have:


 * $BD = \dfrac {a c} {b + c}$


 * $DC = \dfrac {a b} {b + c}$

Then we have:

Then from Triangles with Two Equal Angles are Similar we have:
 * $\triangle ABD \sim \triangle AFC$

So:

Now we use the Intersecting Chord Theorem, which gives us $BD \cdot DC = d \cdot DF$.