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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:   

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   

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).

Try it, and let me know what you think about this construction method. 
I hope the instructions are easy to follow.


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

Hi, Eric. Thanks for writing back.

I just made your construction on Geometer's Sketchpad, software that 
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   

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 

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   

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 

Thanks for your interest in this. I had this idea in class when I 
first learned about the classical three problems. I had already done 
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.

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

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