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Math Help - Regular heptagons

  1. #1
    MHF Contributor alexmahone's Avatar
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    Regular heptagons

    Let A_1A_2...........A_7, B_1B_2.........B_7, C_1C_2............C_7 be regular heptagons with areas S_A, S_B, S_C respectively. Let A_1A_2 = B_1B_3 = C_1C_4. Prove that \frac {1}{2} < \frac {S_B + S_C}{S_A} < 2 - \sqrt2.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Using trigonometry, I've found that area of a regular heptagon=3.633 (side)^2.
    Let a, b, c be the side lengths of the respective sides.
    I was able to prove b=0.556a or a/2<b<a.
    Similarly I could express c in terms of a.
    With all this data, I could find the exact value of the fraction, let alone prove the inequality.
    But I think there is a more elegant way of doing this than brute force calculation.
    Suggestions, anyone?
    Last edited by alexmahone; November 6th 2008 at 04:03 AM.
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