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.

Continue reading at:

gogeometry.com/ArchBooLem08.htm

## Tuesday, December 16, 2008

### Archimedes' Book of Lemmas, Proposition #8

Labels:
angle,
Archimedes,
book of lemmas,
chord,
circle,
neusis,
radius,
secant,
trisection

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Not that difficult at all:

ReplyDeleteAs 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)

∠DOA = ∠OAD = α

ReplyDelete∠OCD = ∠ODC = ∠DOA + ∠OAD = 2α

β = ∠OAC + ∠OCA = α + 2α = 3α