Trisecting a Line Segment (With World Record Efficiency!)
Appendix: Trisecting Angles
A very famous problem is trisecting arbitrary angles. This problem occupied mathematicians for thousands of years, and although in 1837 Pierre Wantzel proved that angles cannot be trisected by Euclidean methods, people still keep trying (see references, especially Dudley's Budget of Trisections ).
If you change the rules and allow something other than a straightedge and compass, you can often trisect angles. The most famous method is Archimedes' who used a straightedge with two marked points on it; see the Geometry Forum's http://www.geom.uiuc.edu/docs/forum/angtri/. Origami paper folding is another elegant way to trisect an angle.
Here are a few of many web pages discussing angle trisection:
MacTutor History of Mathematics
The Mathematical Atlas
One of the earliest methods, the Quadratrix of Hippias
Styer, Robert, "Trisecting a Line Segment (With World Record Efficiency!)," Loci (October 2009), DOI: 10.4169/loci003342
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