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.

Title	bisectors theorem
Canonical name	BisectorsTheorem
Date of creation	2013-03-22 14:49:23
Last modified on	2013-03-22 14:49:23
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Theorem
Classification	msc 51A05
Synonym	angle bisector theorem
Synonym	generalization of bisectors theorem
Related topic	CevasTheorem
Related topic	HarmonicDivision