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).
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?
This elegant construction takes two circles and only three additional lines.
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."
If we wanted to use circles and no additional lines, how many would it take?
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).
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! (see references for more on the impossible angle trisections.)
Even More on Trisections (using only circles or only lines)
Villanova Home Page Math Dept Home Page 25 April 2001