Angle Bisector Theorem
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Introduction
The Angle Bisector Theorem states that given triangle and angle bisector AD, where D is on side BC, then
. Likewise, the converse of this theorem holds as well.
![Anglebisector.png](https://wiki-images.artofproblemsolving.com//3/39/Anglebisector.png)
Proof
Method 1
Because of the ratios and equal angles in the theorem, we think of similar triangles. There are not any similar triangles in the figure as it now stands, however. So, we think to draw in a carefully chosen line or two. Extending AD until it hits the line through C parallel to AB does just the trick:
Since AB and CE are parallel, we know that and
. Triangle ACE is isosceles, with AC = CE.
By AA similarity, . By the properties of similar triangles, we arrive at our desired result:
![$\frac cm = \frac bn.$](http://latex.artofproblemsolving.com/c/f/1/cf1574360a930676f4336884fe3bed4fc7d32024.png)
Method 2
Let . Now, we can express the area of triangle ABD in two ways:
![$[ABD] = \frac 12 cd\sin \angle BAD = \frac 12 md \sin \angle ADB.$](http://latex.artofproblemsolving.com/9/1/e/91ed453f1f6383386b7e7c2316f952e2d76c6f93.png)
Thus, .
Likewise, triangle ACD can be expressed in two different ways:
![$[ACD] = \frac 12 bd \sin \angle CAD = \frac 12 dn \sin \angle ADC.$](http://latex.artofproblemsolving.com/6/1/9/6194b407398cbb9835d9512e991c68bfc3aaac74.png)
Thus, .
But and
since
. Therefore, we can substitute back into our previous equation to get
.
We conclude that , which was what we wanted.
In both cases, if we reverse all the steps, we see that everything still holds and thus the converse holds.
Method 3
Since B,D, and C are collinear, . Now consider the perpendicular from
to
and
to
. Every point on the angle bisector of an angle is equidistant to the sides of the angle, so the height
to
is equal to the height to
. Thus
. Thus
, so
. We can prove the converse by the Phantom Point Method, since we can find
and
in terms of
,
, and
, and prove that the points are the same.
Examples
- Let ABC be a triangle with angle bisector AD with D on line segment BC. If
and
, find AB and AC.
Solution: By the angle bisector theorem,or
. Plugging this into
and solving for AC gives
. We can plug this back in to find
.
- In triangle ABC, let P be a point on BC and let
. Find the value of
.
Solution: First, we notice that. Thus, AP is the angle bisector of angle A, making our answer 0.
- Part (b), 1959 IMO Problems/Problem 5.