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### Angle Trisection: Construction vs. Drawing

Date: 10/17/2001 at 15:31:36
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,

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
http://mathforum.org/dr.math/

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

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?

Thanks,

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
http://mathforum.org/dr.math/problems/henson11.21.96.html

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
trisection.

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
so many people have. And see, from the Dr. Math FAQ:

"Impossible" Geometric Constructions
http://mathforum.org/dr.math/faq/faq.impossible.construct.html

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

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

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