User:Ghazer/bisector

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[edit] 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}}

[edit] 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.