Trisecting a Line Segment

Date: 08/13/99 at 05:40:34
From: cbr53530
Subject: Geometry Constructions

I have a question on a geometry construction that I hope you can help 
me solve. How can I measure one-third of a line of an unknown length 
using a compass and a straightedge?

Date: 08/13/99 at 11:51:58
From: Doctor Rob
Subject: Re: Geometry Constructions

Thanks for writing to Ask Dr. Math.

Start with the given line segment AB. At A, draw a new line through A 
not coinciding with AB. Using any convenient length, mark off points C 
and D on that line such that AC = CD.

Through B, draw a line parallel to AD.  On this line, using the same 
length as before, mark off points E and F such that BE = EF = AC = CD, 
and E and F are on opposite sides of line AB from points C and D.

Connect CF and DE. Those two lines will cut AB into equal thirds.

Diagram:

             /
          D o.
           /  `.
          /     `.
       C o.       `.
        /  `.       `.
       /     `.       `.
    A o--------+---------+--------o B
                 `.       `.     /
                   `.       `.  /
                     `.       `o E
                       `.     /
                         `.  /
                           `o F
                           /


- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/