Trisecting a Line Segment (With World Record Efficiency!)
Mohr-Mascheroni and Poncelet-Steiner Constructions
Someone might object that we began with the line through \(A\) and \(B\). What if we do not want to begin with this line? So our goal is to find a point \(F\) such that the distance from \(A\) to \(F\) is one third that of \(A\) to \(B\) without using any line.
Mascheroni (1797) showed that all Euclidean constructions could be done just using a compass without a ruler. A few decades later, a work by Georg Mohr in 1672 was discovered proving the same result. (details in )
Here is a Mohr-Mascheroni (uses no lines) trisection construction with seven circles. (Here is a proof that this construction works. Seven circles is in fact the least number possible: see the details.)
I personally love circles, but... What if we like straight lines and really do not like circles?
In the 1800s, Poncelet and Steiner showed that all Euclidean constructions can be done with a ruler only provided one is given a single 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  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 \((0,1)\), \((1,0)\), \((0,2)\), \((2,0)\). Let the point \(A\) be \((1,0)\) and the point \(B\) be \((2,0)\) so we want to construct the trisecting point \((4/3, 0)\). Of course we easily construct the origin \((0,0)\).
Here is a Poncelet-Steiner construction, using Martin's starting convention, of the trisecting point \((4/3, 0)\) using eight lines. Here is a proof that it trisects, illustrated with a slightly more general starting convention. This is the least number of lines needed: see the details.
For more information on trisections and geometric constructions, see the annotated reference page.
Styer, Robert, "Trisecting a Line Segment (With World Record Efficiency!)," Loci (October 2009), DOI: 10.4169/loci003342
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