Warning: This name is used differently in different textbooks. In some textbooks, it refers to the theorem which states that any point on an angle bisector is equidistant from the two sides of the angle.
What most textbooks call the Angle Bisector Theorem is this:
An angle bisector in a triangle divides the opposite side into two segments which are in the same proportion as the other two sides of the triangle.
In the figure above, bisects , so .