In high school geometry, one of the most useful ruler and compass constructions is bisecting a line segment. This beautiful construction can be done with only two circles and one line.
Euclid's very first proposition contains the essence of this construction.
But what if we wanted to trisect the line segment? Given a line with
segment AB, construct a point F on the segment so that
AF = (1/3) AB, using the classical straightedge and compass.
Scott Coble found a clever construction, reprinted in the wonderful book Proofs without Words (references). (proof).
This construction uses two circles and four additional lines. Certainly two is the fewest number of circles that is possible, since one needs two circles to construct a point not on the given line through AB.
Can we do better?
NEXT!
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Pages revised 26 June 2003