Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Constructions
High School Euclidean/Plane Geometry
High School Geometry
High School History/Biography
Middle School Geometry
Middle School History/Biography
Middle School Two-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search