Abstract 

The book Proofs without Words is a collection of many short simple proofs, including a construction to trisect a line segment.  This short construction naturally leads one to wonder what the shortest construction is (that is, the construction using the least number of circles and lines).  We find the shortest trisection of a line segment, then the shortest construction using only circles (Mohr-Mascheroni) and using only lines (Poncelet-Steiner).

• Trisecting a Line
  • Introduction
  • Trisecting with Two Circles and Four Lines
  • Trisecting with Two Circles and Three Lines
  • Trisecting with Three Circles and Two Lines
  • Trisecting with Four Circles
• Proofs of Constructions with Five or Six Steps
  • Picturing the Proof of the Coble Construction (Two Circles and Four Lines)
  • Picturing the Proof of the Two Circles and Three Lines Construction
  • Picturing the Proof of the Three Circles and Two Lines Construction
• Proofs of Constructions with Four Steps
  • Picturing the Proof of the Four Circles Construction 
  • Proof that Three Circles and One Additional Line are Not Enough
  • Proof that Two Circles and Two Lines are Not Enough
• Mohr-Mascheroni and Poncelet-Steiner Constructions
  • The Mohr-Mascheroni Construction
  • The Poncelet-Steiner Construction
  • Picturing the Proof of the Mohr-Masheroni Construction
  • Proof that Six Circles are Not Enough for Mohr-Mascheroni
  • Picturing the Proof of Martin's Poncelet-Steiner Construction
  • Proof that Seven Lines are Not Enough for Poncelet-Steiner
• Appendix: Trisecting Angles
• Instructor's Guide (also available as a Word document)
• Annotated References

This article uses Flash animations to illustrate many of the constructions.  You will need a reasonably current version of the Flash Player plug-in installed for your browser in order to view the animations.