Angle bisector theorem

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In this diagram, BD:DC = AB:AC.
In this diagram, BD:DC = AB:AC.

In geometry, the angle bisector theorem relates the length of the side opposite one angle of a triangle to the lengths of the other two sides of the triangle.

Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC.

The generalized angle bisector theorem states that if D lies on BC, then {\frac {|BD|} {|DC|}}={\frac {|AB| \sin \angle DAB}{|AC| \sin \angle DAC}}

If AD is the bisector of BAC, then \sin \angle DAB = \sin \angle DAC, reducing to the non-generalized version of theorem.

[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.

[edit] External links