Trisecting a Right AngleDate: 12/16/96 at 15:50:32 From: forwarded by Eric Sasson Subject: trisecting an angle I have figured out how to trisect a right angle! 1st - draw your right angle and mark off 2 points on both legs 2nd - using the same measurement on the compass, draw two arcs, one from each point. 3rd - draw the lines connecting the intersection and the points on the legs (making equilateral triangles). 4th - draw the lines connecting the vertex of the right angle and the points on the equilateral triangles that you made. The two lines made trisect the angle! :) Date: 12/16/96 at 16:30:54 From: Doctor Pete Subject: Re: trisecting an angle I'm sorry to burst your bubble, but trisecting a 90-degree angle is equivalent to constructing a 60-degree angle, which is easily done as you mention in your message. But the real goal is to trisect an *arbitrary* angle with straightedge and compass; that is, to trisect any angle, not just one particular angle. Your construction relies on the fact that your given angle is ninety degrees. To trisect an arbitrary angle with straightedge and compass is impossible, as the ancient Greeks knew, but were unable to prove. It took hundreds of years before the tools of abstract algebra and Galois Theory came along to show it was indeed impossible. When I was in high school, I too tried to achieve the impossible; I researched the topic and discovered ways that were indeed trisections, but did not involve a "legal" use of straightedge and compass. One method I particularly liked involved using a marked straightedge, which was known to Archimedes. Can you discover it? Another method involved drawing many arcs, enough to approximate a non-circular curve called a limacon; this curve was then used to trisect the angle, but this is not allowed - the construction must be exact. Speaking of constructions and the ancient Greeks, did you know that Euclid knew how to inscribe regular polygons with 3, 4, 5, 8, 10, 12, and 15 sides, as well as those obtained from bisecting these? Do you know how to construct a regular pentagon? Did you also know that Euclid missed a few? In particular, he missed the regular 17-gon! Karl Gauss, at the young age of 19, discovered that one can construct the regular 17-, 257- and 65537-gon with straightedge and compass! To see how it's done, check out: http://www.ugcs.caltech.edu/~peterw/studies/17gon/ for some constructions that will *really* impress your teachers. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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