Tuesday, December 16, 2008

Archimedes' Book of Lemmas, Proposition #8

Angle Trisection. Neusis Construction
Exercise your brain. Archimedes wrote the "Book of Lemmas" more than 2200 years ago. Solve the proposition #8 (high school level) and lift up your geometry skills.

Archimedes' Book of Lemmas #8.
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  1. Not that difficult at all:

    As OD=r, AOD is isosceles, and therefore DOA=OAD=alpha. Then ADO=180°-2alpha and ODC=2alpha. Now OCD also is isosceles (two radii) and therefore DCO=ODC=2alpha and COD=180°-4alpha. As BOC+COD+DOA=180°, it follows, that beta=BOC=180°-COD-DOA=3alpha.

    The proof still holds, if the angle COD=180°-4alpha is negative. (by Thomas)

  2. ∠DOA = ∠OAD = α
    ∠OCD = ∠ODC = ∠DOA + ∠OAD = 2α
    β = ∠OAC + ∠OCA = α + 2α = 3α