Trisecting a Right Angle

Date: 12/16/96 at 15:50:32
From: forwarded by Eric Sasson
Subject: trisecting an angle

I have figured out how to trisect a right angle!  1st - draw your 
right angle and mark off 2 points on both legs  2nd - using the
same measurement on the compass, draw two arcs, one from each point. 
3rd - draw the lines connecting the intersection and the points on the 
legs (making equilateral triangles). 4th - draw the lines connecting 
the vertex of the right angle and the points on the equilateral 
triangles that you made.  The two lines made trisect the angle!  :)

Date: 12/16/96 at 16:30:54
From: Doctor Pete
Subject: Re: trisecting an angle

I'm sorry to burst your bubble, but trisecting a 90-degree angle is 
equivalent to constructing a 60-degree angle, which is easily done as 
you mention in your message.  But the real goal is to trisect an 
*arbitrary* angle with straightedge and compass; that is, to trisect 
any angle, not just one particular angle.  Your construction relies on 
the fact that your given angle is ninety degrees.  To trisect an 
arbitrary angle with straightedge and compass is impossible, as the 
ancient Greeks knew, but were unable to prove.  It took hundreds of 
years before the tools of abstract algebra and Galois Theory came 
along to show it was indeed impossible.

When I was in high school, I too tried to achieve the impossible; I 
researched the topic and discovered ways that were indeed trisections, 
but did not involve a "legal" use of straightedge and compass.  One 
method I particularly liked involved using a marked straightedge, 
which was known to Archimedes.  Can you discover it?  Another method 
involved drawing many arcs, enough to approximate a non-circular curve 
called a limacon; this curve was then used to trisect the angle, but 
this is not allowed - the construction must be exact.

Speaking of constructions and the ancient Greeks, did you know that 
Euclid knew how to inscribe regular polygons with 3, 4, 5, 8, 10, 12, 
and 15 sides, as well as those obtained from bisecting these?  Do you 
know how to construct a regular pentagon?  Did you also know that 
Euclid missed a few?  In particular, he missed the regular 17-gon!  
Karl Gauss, at the young age of 19, discovered that one can construct 
the regular 17-, 257- and 65537-gon with straightedge and compass!  To 
see how it's done, check out:

     http://www.ugcs.caltech.edu/~peterw/studies/17gon/   

for some constructions that will *really* impress your teachers.

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