Attempt at Trisecting an Angle

Date: 11/30/1999 at 08:48:52
From: Hollis Mellon
Subject: Geometry

We have a bisector of an angle of 30 degrees (or any degree) that 
extends into the angle 1" and extends outside the angle 2". Then we 
have bisector line 3" long. From the vertex of the angle, a circle 
with a 1" radius is drawn. Then 2" outside the angle, and on the 
bisector, a circle with a 3" radius is drawn. The two circles meet on 
the bisector of the angle 1" inside the angle. 

Question: Are the arcs of the two circles within the angle the same 
length?

Arc a-b of small circle = arc A-B of larger circle.

Both arcs are 1" within the angle.

Thanks.

Date: 11/30/1999 at 16:47:25
From: Doctor Peterson
Subject: Re: Geometry

Hi, Hollis.

I could show you the trig to work out the actual arc lengths, but it's 
pretty ugly and probably not worth the effort. I'll just tell you they 
are not the same.

Your construction is reminiscent of some attempts I've seen to trisect 
an angle. In particular, the figure looks much like Archimedes' 
method, which requires a marked straightedge, and that in itself is 
enough to tell me that the answer will be no. If the arcs were the 
same, then the angles would be in a 3:1 ratio (since they have the 
same arc length and the radii are in 3:1 ratio), and you would have 
trisected the angle; since I know that's impossible with a 
construction that can be done with compass and straightedge, I don't 
really have to do the extra work.

If that's what you're trying to do, don't feel it's foolish to try. As 
the alchemists discovered useful things in the process of trying to do 
the impossible, you may learn a lot about what does work in trying to 
do what won't. There are some fascinating ways to trisect an angle 
using special tools, as our FAQ tells you if you dig deep enough - see

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

- and the attempt to prove that any given method might work can help 
you get a better feel for geometry, and stretch your mind 
considerably. What you'll want to do, though, is to develop the skills 
to prove for yourself whether something is true, rather than make a 
conjecture and have to ask whether it is true. That's the fun part of 
geometry, and what makes it more challenging than any other part of 
elementary math.

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