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In geometry, the **angle bisector theorem** is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative 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

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

The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.

An angle bisector of an isosceles triangle will also bisect the opposite side, when the angle bisect bisects the vertex angle of the triangle

In the above diagram, use the law of sines on triangles *ABD* and *ACD*:

- ..... (Equation 1)

- ..... (Equation 2)

Angles *BDA* and *ADC* form a linear pair, that is, they are adjacent supplementary angles. Since supplementary angles have equal sines,

Angles *BAD* and *DAC* are equal. Therefore the Right Hand Sides of Equations 1 and 2 are equal, so their Left Hand Sides must also be equal:

which is the Angle Bisector Theorem.

If angles *BAD* and *DAC* are unequal, Equations 1 and 2 can be re-written as:

Angles *BDA* and *ADC* are still supplementary, so the Right Hand Sides of these equations are equal, so the Left Hand Sides are equal to:

which rearranges to the "generalized" version of the theorem.

An alternative proof goes as follows, using its own diagram:

Let *B*_{1} be the base of the altitude in the triangle *ABD* through *B* and let *C*_{1} be the base of the altitude in the triangle *ACD* through *C*. Then,

*DB*_{1}*B* and *DC*_{1}*C* are right, while the angles *B*_{1}*DB* and *C*_{1}*DC* are congruent if *D* lies on the segment *BC* and they are identical otherwise, so the triangles *DB*_{1}*B* and *DC*_{1}*C* are similar (AAA), which implies that

- A Property of Angle Bisectors at cut-the-knot
- Proof of angle bisector theorem at PlanetMath
- Another proof of angle bisector theorem at PlanetMath
- On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem by G.W.I.S Amarasinghe, Global Journal of Advanced Research on Classical and Modern Geometries, Vol 01(01), pp. 15 - 27, 2012