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Math Help - Geometric construction

  1. #1
    Super Member Showcase_22's Avatar
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    Geometric construction

    Suppose that a,b,c  \in \mathbb{R} satisfy:

    0<a<b+c \ \ \ \ \ 0<b<a+c \ \ \ \ \ 0<c<a+b

    i). Give a geometric construction to prove there exists a triangle in the Euclidean plane \mathbb{E}^2 (that is \mathbb{R}^2 with the usual metric) with sides of length equal to a,b,c.
    I'm not really sure if I got this, but I chose a 5,12,13 triangle. It satisfies the inequalities, but i'm not sure if they want a more general answer.

    ii). What special thing happens in your construction if one of the inequalities becomes equality, say, a=b+c?
    With equality (I chose b=c), we get 0<a<2c and 0<c<a+c. The second inequality seems a bit redundant since a>0.

    I don't really see the special thing that happens. My 5,12,13 triangle still works!
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  2. #2
    MHF Contributor
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    It seems to me that you have to prove that a triangle exists for arbitrary a, b, and c (subject to the constraints). Maybe you can consider a line segment of length a and two circles whose centers are the ends of the segment and whose radiii are b and c. Then show that the circles intersect when the inequalities hold.
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