|Moor Xu on 2013 Pi Day|
|peterthedestroyer on 2013 Pi Day|
|Against the “R… on Contest Math|
|The Ram on Putnam vs USAMO|
|pi on Analysis of the 24 Game|
- 48,064 hits
Given the sides of a triangle, what is the length of the angle bisector? Consider the following image of triangle ABC, where the sides opposite points A, B, C have lengths a, b, c. We want to find length d.
First, we can apply the Angle Bisector Theorem to see that , so that and . Plugging these into the equation yields and , so that and .
We can now apply Stewart’s Theorem to see that . Plugging in our expressions for x and y, we see that
Therefore, the length of the angle bisector at vertex C is
The lengths of the other two angle bisectors can be found analogously and are given by a simple permutation of coordinates.
Here’s another way to compute the length of the angle bisector that might be a bit simpler: Let be , so that the angle bisector separates two angles with measure . The area of triangle ABC is the sum of the areas of the two smaller triangles, which can be expressed as . Simplifying and rearranging, we see that , so the length of the angle bisector is therefore
This is simpler than what we had before, though it also involves an angle.
We can use this expression for the length of the angle bisector to show that the sum of lengths of the angle bisectors is less than the perimeter. The length of the angle bisector to angle C is , and the other angle bisectors are similar. Therefore, it is sufficient to show that
This inequality actually follows as a simple application of the arithmetic mean – harmonic mean (AM-HM) inequality, which states that
Applying this three times yields
which is what we wanted to show. Therefore, we've proven that the sum of lengths of angle bisectors in a triangle is less than the perimeter.