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Math Help - Area of an arbitrary triangle?

  1. #1
    Super Member fardeen_gen's Avatar
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    Area of an arbitrary triangle?

    Prove that area of an arbitrary triangle is less than one fourth of the square of its perimeter.

    (The question is from my sequences and series text)
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello fardeen_gen
    Quote Originally Posted by fardeen_gen View Post
    Prove that area of an arbitrary triangle is less than one fourth of the square of its perimeter.

    (The question is from my sequences and series text)
    Do you know the formula for the area of a triangle: \Delta = \sqrt{s(s-a)(s-b)(s-c)}, where the semi-perimeter s = \tfrac12(a+b+c)?

    Well, each of (s-a), (s-b), (s-c) is less than s. So \Delta < \sqrt{s^4} = \tfrac14(a+b+c)^2.

    Grandad
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