Attempt at Trisecting an AngleDate: 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/ |
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