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

### Trisecting an Angle

```
Date: 06/15/99 at 19:00:49
From: Eric Kovach
Subject: Trisecting an angle

I've been looking for someone to talk to about the trisecting an angle
problem from ancient Greece. I understand the whole modern algebra
proof thing but it doesn't disprove the possibility of angle
trisection. At least not from the way I have read it and understood
it. I came up with a method when I was 16 and in geometry class and so
far no one can disprove it. I tried proving it geometrically for a
while but gave up. All I know is that with best drawing programs using
only circles and unmeasured lines, it works at least to the naked eye
for any angle from 5 to 175 degrees. I've done numerous examples
trying to find one where it is off by more than the error when using a
compass and so far everything I've done is a perfect trisection. Who
can I show this to? I think I have something here...
```

```
Date: 06/15/99 at 20:05:42
From: Doctor Peterson
Subject: Re: Trisecting an angle

Hi, Eric.

I presume you looked at our FAQ on this subject:

http://mathforum.org/dr.math/faq/faq.impossible.construct.html

As I'm sure you know, algebra does prove that you can't trisect a
general angle precisely using only compass and straightedge under the
traditional Greek rules. It doesn't prove that you can't trisect a
particular angle, or trisect it using modified tools, or only
approximately. In your case, it sounds as if you don't necessarily
claim a perfect trisection, and don't claim to have proved it, but you
must have an interesting construction. I'd like to see it; we might be
able to work together either to find how accurate it is, or to see
whether it is accurate but twists some rule a little, or whatever.

If you want to send it in, please make sure it's stated clearly, to
make it easy on us. If you can describe it step by step so that I can
duplicate it easily, I'll see what I can do to prove or disprove it.

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

```
Date: 06/16/99 at 18:07:02
From: Eric
Subject: Re: Trisecting an angle

Thanks for the feedback and your interest. I will give you the method
I used for my construction.

1. Take any angle and label the vertex A.

2. Draw a circle centered at A. Label the intersections with the legs
of the angle B and C. So we have an angle labeled BAC.

3. Bisect angle BAC and extend the bisection line downward so it
intersects the circle at point D.

4. Draw another circle of equal radius as the first centered at
point D.

5. Extend the angle's bisection line downward until it intersects with
the second circle. Label this point E.

6. Draw line segments BE and CE. This will form angle BEC.

7. Take angles EBA and ECA. Bisect them. Extend their bisection lines
until they intersect with line segments CE and BE respectively.
Label these points F and G. The bisection lines will be called
BF and CG.

8. Draw the lines FA and GA. These lines will trisect the original
angle (or closely approximate it).

I hope the instructions are easy to follow.

Thanks.
Eric
```

```
Date: 06/16/99 at 23:04:49
From: Doctor Peterson
Subject: Re: Trisecting an angle

Hi, Eric. Thanks for writing back.

lets me adjust the angle and measure the results, and although it's
clear your construction is not precise (so it doesn't go against what
has been proven), it is remarkably accurate. The resulting three
angles are the same to within 0.001 degree for ABC up to 36 degrees;
to within 0.01 degree up to about 60 degrees; to within 0.1 up to
100 degrees; and to within 1 degree up to about 150 degrees. Even for
ABC = 180 degrees, the thirds are 60.774, 58.389, and 60.755 - and
since I'd expect the first and last to be the same, I'm probably past
the accuracy of the software at that point.

The only trouble I had following your instructions was that I didn't
catch the meaning of "downwards" at first. You expressed it very well.

I'll look at it a little more closely as I have time, and see if I can
determine what the angles are trigonometrically to confirm my quick
measurements. One thing that intrigues me is that F and G are close to
circle A, and as they move away, the angles depart from true
trisectors. It may be that if they stayed on the circle, they would be
correct. I'll let you know what I find. I may also look around to see
if this construction is well known.

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

```
Date: 06/17/99 at 17:04:26
From: Doctor Peterson
Subject: Re: Trisecting an angle

Hi, Eric. Here's a little more.

I've been playing with your construction, and it turns out that my
hunch was correct: if, rather than making point E twice as far along
the bisector as D, we position it so that F and G are exactly on the
circle A, then the lines FA and GA do exactly trisect the angle.
That's because if we continue BA to K on the far side of the circle,
the fact that the inscribed angles KBF and FBG are equal implies that
arcs KF and FG are equal; similarly if we continue CA to L on the
circle, FG = GL, so we have trisected arc KL which is equal to BC. So
if you compare this drawing of your construction:

with this drawing, where I have moved E in closer

you will see that I have (mostly) corrected the slight error in the
trisection.

The problem, of course, is that there is no way with only compass and
straightedge to position E correctly. Your construction puts F and G
very close to the circle, so the construction comes close enough for
practical purposes (though not enough for a mathematician!). I had to
use a large angle to make the deviation in points F and G visible, and
even then the error in the trisectors is less than a fifth of a
degree. Yet to a mathematician, even if we can't measure the
difference, if we know that it isn't precise (even when drawn with
impossibly perfect tools), then it just isn't right. Useful, maybe,
but not right.

I'm curious: how did you come up with this - trial and error, or
deliberate planning? I've looked around the Web for other approximate
trisections and didn't find any quite like this, though some share a
few features. There's a book called _The Trisectors_, by Underwood
Dudley, which tells about many such attempts, but I don't have access
to it. You might want to check your library.

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

```
Date: 06/17/99 at 18:16:05
From: Eric
Subject: Re: Trisecting an angle

That is really cool that you can use that program to get exact
measurements. I will have to get a copy of that for myself. It would
have saved me lots of time when I was trying out examples on different
angles.

Thanks for your interest in this. I had this idea in class when I
advanced studies in geometry back in junior high, so when the class
came up in tenth grade, I didn't really need to spend class time
listening to lectures.

The idea was based on the fact that two angles that have vertices on
the same circle and have the same arc are equal. That, and the fact
that an angle at the center of the circle is half of an angle on the
circumference. My idea was to extend the idea with an angle 2 radii
from the center that would be three times the angle in the center.
Since that wasn't true I expanded on the idea into what I gave you.

I really hadn't given this much thought since high school until the
other day, but it is good to finally know a little more about the
quality of my trisection method. I've really been too busy with
college to give math any more thought. I am studying Chemical
Engineering, which is definitely a challenge to my math skills. I had
thought about being strictly a mathematician because that is what I am
best at, but I didn't really see a whole lot of jobs in that outside
of teaching, so I went into engineering.

I will have to look for that book just to see if there is anything
similar to what I came up with. Thank you very much.

Eric
```
Associated Topics:
High School Constructions
High School Euclidean/Plane Geometry
High School 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