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.
Mascheroni (1797) showed that all Euclidean constructions could be done just using a compass without a ruler. Later, a work by Georg Mohr in 1672 was discovered proving the same result. (details in Eves: references)
Here is a construction that uses no lines, just a compass to draw nine circles. (I have no idea if this is the shortest such construction.)
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. For our case of trisecting a segment, we can actually do better; following the lead of Martin (references) we begin with the points (0,1), (1,0), (0,2), (2,0). Of course we easily construct the origin (0,0). (All we really need is six generic points from two intersecting lines, three points on each line, one point being the midpoint for the segment determined by the other two points.)
Here is a Poncelet-Steiner trisection using ten lines. I have no idea if this is the shortest such construction.
Here is another version of the same Poncelet-Steiner construction, except the points A and B are placed differently.
See the annotated reference page for more information on segment trisections.
Return to Main Trisection Page
Villanova Home Page Math Dept Home Page 4 April 2001