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
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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
https://photos.app.goo.gl/tB92ZCnkwYgzHgCp9
ReplyDelete