Definitions

Angle subtraction theorem

Angle bisector theorem

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 ACL

{frac
{|AC}={frac {|BD{|DC}. >

The generalized angle bisector theorem states that if D lies on BC, then

{frac {|DC}={frac {|AB| sin angle DAB}{|AC| sin angle DAC}}. >

This reduces to the previous version if AD is the bisector of BAC.

Proof of generalization

Let B1 be the base of altitude in the triangle ABD through B and let C1 be the base of altitude in the triangle ACD through C. Then,

begin{align}
&= |AB|sin angle BAD, &= |AC|sin angle CAD. end{align}

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 that

{frac {|CD}= {frac {|BB_1{|CC_1>}=frac {|AB|sin angle BAD}{|AC|sin angle CAD}. Q.E.D.

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