```Date: Jul 16, 2002 8:59 AM
Author: mark
Subject: Re: Trisection

Thanks for the hard work Eric. > I have algebraically checked this construction. It will effectivelyconverge> to the trisection, with a cubic convergence rate. More precisely, ifwe take A> as the origin and C as the unit point on the X axis, 3*t the angleto trisect,> if h is the difference in abscissa between the first guess and theresult > (i.e. the> difference in cosine), and h' the same difference after oneiteration, we have:> > h' = h^3 / (48*sin(t)^2) + O(h^4)I was unable to follow your suggestion for finding a longerconstruction line so I found my own (see below). In response to John's suggestion of relettering, I am reposting myconstruction based on AOB.>[It might also bea good idea to change the lettering, since, to the extent that>there's a standard, it's usually an angle  AOB  that gets trisected.]Near Exact Trisection:1. Start with an unknown angle <90 deg., label the vertex O. 2. Draw an arc with origin at O crossing both lines of the angle atpoints A and B. 3. Draw line AB making an isosceles triangle.4. Using point A as the origin, draw an arc crossing line AB and theearlier arc somewhere between ÃÂ¼ and ÃÂ½ way between points A and B.Label where this new arc crosses line AB point D.Label where this new arc crosses the first arc point E.5. Draw line DE and extend it well past O .  If  line DE passesexactly through point O (it wonÃÂt) stop, your first guess was an exacttrisection.	6. Extend line OA well past point O, step off  3 times length OA frompoint O and label the new point F.7. Swing an arc of length OF with O as the origin that crosses theextended line DE near point F.Label the intersection G.8. Draw line GO and extend it to intersect the original arc from step2.Label the intersection EÃÂ.Line OEÃÂ is a good trisection. However this is only the start.Repeating the process from step 4 using AEÃÂ as the arc radius resultsin a trisection to within 10E-11 degrees. Each subsequent iterationimproves the trisection by several orders of magnitude. It takes an extra step, but an alternate construction using a line from point B perpendicular to line OA (line AB is no longer necessary)yields exactly the same results if your step 4 circle has its originat point B, is between 1/2 and 3/4 angle AOB, and point D is on thenew perpendicular line instead of line AB. This gives a longer workingline for the construction.
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