Angle bisector theorem

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. | A B | | B D | = sin ⁡ ∠ B D A sin ⁡ ∠ B A D {displaystyle {frac {|AB|}{|BD|}}={frac {sin angle BDA}{sin angle BAD}}}     (1) | A C | | D C | = sin ⁡ ∠ A D C sin ⁡ ∠ D A C {displaystyle {frac {|AC|}{|DC|}}={frac {sin angle ADC}{sin angle DAC}}}     (2) 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 DC 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 triangle 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 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.