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Math Help - Triangle

  1. #1
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    Triangle

    Prove that if the line through the centroid and the incenter is parallel to a side then the sides of the triangle form an arithmetic sequence.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Suppose that IG\parallel BC (I=incenter, G=centroid).
    Let  r be the inradius, h_a the height from the vertex A, S the area of triangle and p the semiperimeter.

    Then we have r=\frac{h_a}{3}=\frac{2S}{3a}

    But r=\frac{S}{p}\Rightarrow\frac{2S}{3a}=\frac{S}{p}\  Rightarrow\frac{1}{3a}=\frac{1}{a+b+c}\Rightarrow a=\frac{b+c}{2} and that means that b, a, c (or c, a, b) are in arithmetic progression.
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