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Determine which triangle is this

  1. Dec 31, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    If in a ΔABC , [itex]\left( 1-\dfrac{r_1}{r_2} \right) \left( 1-\dfrac{r_1}{r_3} \right) = 2 [/itex]
    then which triangle is it?

    r1, r2, r3 denotes ex-radii.

    3. The attempt at a solution

    I get this condition

    [itex]a^2+2b^2+2c^2+3bc=3ac+3ab [/itex]

    Also

    [itex] \sum tan \frac{A}{2} tan \frac{B}{2} = tan^2 \frac{A}{2} [/itex]

    where sigma denotes the sum of products of tan of the half-angles taken pairwise.

    But it is difficult to figure out the type of triangle by just looking at the above two eqns.
     
    Last edited: Dec 31, 2013
  2. jcsd
  3. Dec 31, 2013 #2

    Ray Vickson

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    What do ##r_1, r_2, r_3## mean? Are they angles, sides, or what?
     
  4. Dec 31, 2013 #3

    utkarshakash

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    They are the ex-radii.
     
  5. Dec 31, 2013 #4

    SteamKing

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  6. Dec 31, 2013 #5
    Rewrite the given equation as

    $$r_1r_2+r_2r_3+r_1r_3=r_1^2$$

    The formulas mentioned in the following link are helpful to proceed further.

    http://mathworld.wolfram.com/Exradius.html
     
  7. Jan 1, 2014 #6

    utkarshakash

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    Thanks. The formulas given in Wolfram were of great help.
     
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