If we wanted to use circles and no additional lines, how many would it take?
(Count carefully!)
WOW! Only four circles. A birdie! (Right-click to zoom in.)
This is in fact the best possible, since three circles cannot suffice (details). Nor are two circles and two additional lines enough to trisect the segment (details), nor three circles and one additional line (details).
Here is a proof. This construction can be generalized to construct a segment of length 1/n by replacing the circle of radius 3 by one of radius n.
By the way, it is impossible to trisect an arbitrary angle with unmarked ruler
and compass!
MORE! (the Mohr-Mascheroni construction.)
Villanova Home Page
Trisection Index of
Pages 26 January 2006