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Lengths of line secgment that bisect angle

  1. Feb 23, 2008 #1
    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:

    [tex]\sum[/tex]from i=1to 3 1/t[tex]_{}i[/tex] > [tex]\sum[/tex] from i=1 to 3 1/a[tex]_{}i[/tex]

    [tex]\sum[/tex]from i=1 to 3 t[tex]_{}i[/tex] < [tex]\sum[/tex] from i=1 to 3 a[tex]_{}i[/tex]


    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. jcsd
  3. Feb 24, 2008 #2

    mjsd

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    Homework Helper

    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
     
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