Lengths of line secgment that bisect angle

[ER901]

Homework Statement

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:

\sumfrom i=1to 3 1/t_{}i > \sum from i=1 to 3 1/a_{}i

\sumfrom i=1 to 3 t_{}i < \sum from i=1 to 3 a_{}i

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

 

[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