bisectors theorem

Bisectors theorem.
Let $ABC$ a triangle, and $AP$ the angle bisector of $\mathrm{\angle }CAB$. Then

$$\frac{BP}{PC}=\frac{BA}{CA}.$$

Generalization of bisectors theorem.
Let $ABC$ a triangle, and $P$ any point on the side $BC$. Then

$$\frac{BP}{PC}=\frac{BA\sin BAP}{CA\sin PAC}.$$

The latest theorem even holds when $P$ is not on the segment $BC$ but anywhere on the line (possibly at the infnity), but in this case, the ratio $BP/PC$ must be done with directed segments and the angles considered as directed angles.