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. Consider a triangle △ABC. Let the angle bisector of angle ∠ A intersect side at a point D between B and C. The angle bisector theorem states that the ratio of the length of the line segment to the length of segment is equal to the ratio of the length of side to the length of side : and conversely, if a point D on the side of △ABC divides in the same ratio as the sides and , then is the angle bisector of angle ∠ A. The generalized angle bisector theorem states that if D lies on the line , then This reduces to the previous version if is the bisector of ∠ BAC. When D is external to the segment , 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. As shown in the accompanying animation, the theorem can be proved using similar triangles. In the version illustrated here, the triangle gets reflected across a line that is perpendicular to the angle bisector , resulting in the triangle with bisector . The fact that the bisection-produced angles and are equal means that and are straight lines. This allows the construction of triangle that is similar to . Because the ratios between corresponding sides of similar triangles are all equal, it follows that . However, was constructed as a reflection of the line , and so those two lines are of equal length. Therefore, , yielding the result stated by the theorem.