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 between B and C. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:

and conversely, if a point D on the side BC of △ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle ∠ A.

The generalized angle bisector theorem (which is not necessarily an angle bisector theorem, since the angle ∠ A is not necessarily bisected into equal parts) states that if D lies on the line BC, then

This reduces to the previous version if AD is the bisector of ∠ BAC. When D is external to the segment BC, directed line segments and directed angles must be used in the calculation.

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 immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.

There exist many different ways of proving the angle bisector theorem. A few of them are shown below.