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Math Help - constructible angle

  1. #1
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    constructible angle

    The measure of a given angle is
    180o

    n
    , where n is a positive integer
    not divisible by 3. Prove that this angle can be trisected by Eucliden means

    (straightedge and compass).
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by anncar View Post
    The measure of a given angle is
    180o
    n
    , where n is a positive integer
    not divisible by 3. Prove that this angle can be trisected by Eucliden means

    (straightedge and compass).
    your question is not clear, please rephrase
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  3. #3
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    It is possible to construct a regular pentagon whose angles are 72 and an equilateral triangles with angles 60. So it is possible to construct 72 - 60 = 12degrees. Bisect this angle with compass to get that 6 is constructible. Bisect again to show that 3 is constructible. However, 1 is not constructible because 3 cannot be trisected*. Also 2 cannot be constructed for that will imply that 1 (after bisection) can be constructed a contradiction. Thus, the smallest contructible angle is 3 degrees. Then any multiple of this can be constructed. Since 180/n does not contain a factor of 3 it means after this division this integer is a multiple of three. Thus, it can be constructed. Q.E.D.




    *)Because constructible numbers are algebraic over \mathbb{Q} of degree power of two. Note that 4\cos^3 x - 3\cos x - \cos 3x = 0 showing that this polynomial is of degree 3 over the rationals.
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