What if one allows a third circle?
Hartshorne in his wonderful textbook Companion to Euclid (references) says the average good geometer can trisect a segment with a "par" of 5 (average of five steps.) Here is a slick version based on the classical construction.
We have used three circles and two additional lines, "par 5." (The proof is very similar to the 2 circle 3 line proof.)
If we wanted to use circles and no additional lines, how many would it take?
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