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Angle Trisection: Construction vs. Drawing

Date: 10/17/2001 at 15:31:36
From: Joe Maddox
Subject: Trisecting an angle

Has anyone ever divided an angle into three equal parts by 
construction? I have been told it has not been accomplished.

Thank you, 
Joe Maddox, Ocala, FL

Date: 10/17/2001 at 17:14:04
From: Doctor Tom
Subject: Re: Trisecting an angle

Hi Joe,

Using the standard methods of construction with a straightedge and 
compass, it can be proven that it is impossible to trisect an 
arbitrary angle.

People who think they have solved the problem usually make one
of two mistakes:

1) They trisect a particular angle that happens to allow a
   trisection. For example, anyone can trisect a 90-degree angle.

2) They do not understand the "rules" of straightedge and compass 
construction. For example, if you are allowed to make two marks on the 
straightedge to turn it into a sort of ruler, you can trisect any 
angle. But the offical rules do not allow this.

- Doctor Tom, The Math Forum   

Date: 10/17/2001 at 21:08:42
From: Joe Maddox
Subject: Re: Trisecting an angle

Thanks for the quick reply.

I have trisected arbitrary angles up to 90 degrees. Don't laugh until 
you see it. Only a straightedge and a compass. No measuring. The 
drawing must be precise for the angle to be correct. Where could I 
submit my effort for confirmation?

Joe Maddox

Date: 10/17/2001 at 23:07:52
From: Doctor Peterson
Subject: Re: Trisecting an angle

Hi, Joe.

I notice something in what you just said that indicates where you are 
misunderstanding the trisection problem. In the Dr. Math archives, at:

   Trisecting an Angle   

Dr. Tom listed two mistakes people commonly make (trisecting only a 
specific angle, or using the wrong tools). But there is another that 
is even more common: not recognizing what we mean by an exact 

When you say that "the drawing must be precise," you show that it is 
the drawing itself that you have been focusing on. But to a 
mathematician, the drawing itself is nothing. It is only a 
representation of something that really happens in an ideal world 
where lines have no thickness, and so on. In that world, we can prove 
that a construction is ABSOLUTELY exact; either it meets the precise 
point you claim, or it is a false construction. And since this is the 
world of the mind, ONLY proofs count. It doesn't matter how good a 
drawing you make, it proves nothing.

So unless you can prove that your construction really works exactly, 
you have nothing to show anyone. And we know that you can't, because 
it has been PROVEN that such a construction can't be done.

I've seen many constructions that come remarkably close, usually just 
because they are very complex; there is nothing at all impressive 
about a close approximation. Please don't waste your time on this, as 
so many people have. And see, from the Dr. Math FAQ:

   "Impossible" Geometric Constructions   

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Constructions
High School Euclidean/Plane Geometry
High School Geometry

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