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.
To bisect an angle means to cut it into two equal parts or angles. Say that we wanted to bisect a 50-degree angle, then we would divide it into two 25-degree angles.
Angle Bisector Theorem
In the figure above, let and let be the length of the bisector of angle .
In its simplest form, the angle bisector theorem states that
The angle bisector theorem also states that the length of the angle bisector satisfies
Proof of (1):
Applying the sine rule on and gives
Using the equalities and since is the angle bisector we get
which is what we want.
Proof of (2):
Stewart's theorem states that (remember )
Rearranging gives Then using the angle bisector theorem and we have
Another proof of (1):
Let us consider and let be the angular bisector of .
Now extend to such that be parallel to .
Then = and = which implies
Since = = by isosceles property which implies
In , . Let be a point on side such that bisects . Then what is the length of
Let , then we are now looking for
Using the angle bisector theorem,
Since or plugging this into , we have
This next example is the same as the previous, but we are instead solving for the length of the angle bisector .
In , . Let be a point on side such that bisects , then what is the length of
In our previous example, we already found and .
We are given a triangle with the following property: one of its angles is quadrisected (divided into four equal angles) by the height, the angle bisector, and the median from that vertex.
Find the measure of the quadrisected angle.
The base is partitioned into four segments in the ratio .
Suppose the length of the left-hand side of the triangle is .
Then the length of the angle bisector is also .
Applying the angle bisector theorem to the large triangle, we see that the length of the right-hand side is
But if we apply the angle bisector theorem to the left half of the triangle, we obtain for the same length. Therefore,
Now apply the angle bisector theorem a third time to the right triangle formed by the altitude and the median. The segments in the base are in the ratio , so the altitude and the median form the same ratio. As this is a right triangle, it must be a 45-45-90 triangle.
So the quadrisected angle is right.
Some additional problems are as follows:
Find the length of the internal bisector of the right angle in the right triangle with side lengths .
Let be a triangle with angle bisector with on side . If , and , what are and