User:Ghazer~enwiki/bisector

Generalization
Generalized angle bisector says that if D lies on the line BC, and the point A doesn't, the following is true:
 * $${\frac {|BD|} {|DC|}}={\frac {|AB| \sin \angle DAB}{|AC| \sin \angle DAC}} $$

Proof of generalization
If we define B1 and C1 as the bases of altitudes in the triangles ABD and ACD through, respectively, B i C, it is true that:


 * $$|BB_1|=|AB|\sin \angle BAD$$
 * $$|CC_1|=|AC|\sin \angle CAD$$

It is also true that both the angles DB1B and DC1C are right, while the angles B1DB and C1DC are congruent if D lies on the segment BC and they are identical otherwise, so the triangles DB1B and DC1C are similar (AAA), which implies:


 * $${\frac {|BD|} {|CD|}}= {\frac {|BB_1|}{|CC_1|}}=\frac {|AB|\sin \angle BAD}{|AC|\sin \angle CAD}$$

Q.E.D.