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.