In the 1800s, Poncelet and Steiner showed that all Euclidean constructions can be done with a ruler only provided one is given a circle, its center, and a couple suitable points off the circle. Milos Tatarevic (emails dated 7 June 2003) found a construction using twelve lines. Here is his construction; he notes the key point is constructing the parallel line A'C' and the midpoint E. (We cannot prove this is the shortest such construction.)
Martin (references) points out that when one wishes to construct a rational point, then one need not use the circle. Following his convention, we begin with the points (0,1), (1,0), (0,2), (2,0). Of course we easily construct the origin (0,0).
Here is a Poncelet-Steiner trisection using eight lines with Martin's starting convention; this is the least number of lines needed: details. Here is a proof that it trisects, illustrated with a slightly more general starting convention.
For more information on trisections and geometric constructions, see the annotated reference page.
Next: Trisecting Angles
Villanova Home Page
Trisection Index of
Pages 26 June 2003