Trisecting a Line

Date: 01/30/98 at 20:48:32
From: Ryan
Subject: Constructing a trisected line using compass and straightedge

How do I construct a trisected line using only a straightedge and 
compass? Keeping in mind that I can't measure the line, how do I do 
this?

Thank you,
Ryan

Date: 01/31/98 at 01:12:38
From: Doctor Guy
Subject: Re: Constructing a trisected line using compass and 
straightedge

Let's suppose that your original line segment is called AB. What you 
do is use your straightedge to construct ray AC that has the same 
endpoint, A, as your original segment AB, and that meets AB at some 
convenient angle (anywhere from 20 to 90 degrees works fine). 

Then use your compass to mark off three equal distances along ray AC, 
starting at A. Let's call the three new points D, E, and F. 

Now construct segment FB by using your straightedge to connect F to B. 
You now have triangle ABF.

I assume you know how to copy angles with your compass and a 
straightedge. Use your compass to measure angle AFB and copy that 
angle to make angles AEG and AEH, where G and H are on your original 
segment AB.

That's it. You have trisected segment AB.

The reason this works is that since all those angles are congruent, 
all the segments that look as if they are parallel to BF really are, 
because the corresponding angles are congruent (you made them that 
way). And equal distances along ray AC mean equal distances along AB.

I will attempt to make a sketch of what I mean.

                       G here     H here
       A---------------------------------------B
         \           /           /           /
           \       /           /           /
             \   /           /           /
               D           /           /                             
                 \       /           /
                   \   /           /   
                     E           /
                       \       /
                         \   /
                           F
                             \
                               \
                                 C
                                   \
                                     \



I hope this helps. It's a lot harder to TYPE a drawing than to make it 
with a pencil, a compass, and a straightedge.

-Doctor Guy,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/