bisectors theorem

Bisectors theorem.
Let $ABC$ a triangle, and $AP$ the angle bisector of $\angle CAB$. 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 BisectorsTheorem 2013-03-22 14:49:23 2013-03-22 14:49:23 drini (3) drini (3) 9 drini (3) Theorem msc 51A05 angle bisector theorem generalization of bisectors theorem CevasTheorem HarmonicDivision