Angle Bisector Theorem
Introduction
The Angle Bisector Theorem states that given triangle and angle bisector AD, where D is on side BC, then
.
![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 as we shall see!
Since AB and CE are parallel, we know that and
. Triangle ACE is isosceles meaning that AC = CE.
By AA, . 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.