Homework Help: Lengths of line secgment that bisect angle

1. Feb 23, 2008

ER901

1. The problem statement, all variables and given/known data
Let t_i be the lengths of the line segments that bisect angle Ai of a triangle. The segments go from A_i to the opposite side. Let a_i be the lengths of the sides opposite angle A_i. Prove the following inequalities:

$$\sum$$from i=1to 3 1/t$$_{}i$$ > $$\sum$$ from i=1 to 3 1/a$$_{}i$$

$$\sum$$from i=1 to 3 t$$_{}i$$ < $$\sum$$ from i=1 to 3 a$$_{}i$$

3. The attempt at a solution
I'm not sure where to start.. but this is what I have in mind:
a_2, a_3 > t_1 ; a_1, a_3 > t_2 and so on...

but then I don't know what to do next

Last edited: Feb 24, 2008
2. Feb 24, 2008

mjsd

this is what I would try:
use the fact that the sum of length of 2 sides of any triangle is bigger than the remaining side
also, note that the three bisecting line segments cut the original triangle into 6 smaller triangles