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Geometry Problem 647: Archimedes' Book of Lemmas: Prop. 8, Angle trisection. Neusis construction

If CD be any chord of a circle whose center is O, and if CD be produced to A so that AD is equal to the radius; if further AO meets the circle in B, then the angle BOC will be equal to three times the angle BAC.

This is the Archimedes' neusis construction to the angle trisection problem. "Neusis" means use of a marked ruler in Greek.

Arimedes Lemma 8, Square, Angle Trisection, Neusis Construction

Problem 647, Archimedes' Book of Lemmas, Proposition 8 on Twitter:

#Geometry Problem 647: #Archimedes ' Book of Lemmas: Prop. 8, Angle #trisection. #Neusis construction at https://t.co/tQDLIidW4Y pic.twitter.com/6p8taCXrvG
— Antonio Gutierrez (@gogeometry) August 20, 2017

pic.twitter.com/Oc8aW9sP2T
— Mathmorphosis (@Mathamorphoz) August 20, 2017

pic.twitter.com/rx4WfonFUY
— José Luis da Vila (@jldavilaa01) August 20, 2017

Here's a version used by the ancient Greeks: not a true "compass & straightedge" const., but it works: https://t.co/ntq7cJn7QI @geogebra
— Tim Brzezinski (@dynamic_math) August 20, 2017

Books of Lemmas Index

Proposition 1
Proposition 2
Proposition 3
Proposition 4
Proposition 5
Proposition 6
Proposition 7
Proposition 8
Proposition 9
Proposition 10
Proposition 11
Proposition 12
Proposition 13
Proposition 14
Proposition 15