Subject: sci.math FAQ: Trisection of an Angle
Summary: Part 22 of many, New version,
Date: Fri, 17 Nov 1995 17:15:33 GMT
Nntp-Posting-Host: neumann.uwaterloo.ca


Archive-Name: sci-math-faq/trisection
Version: 6.2




   
The Trisection of an Angle

   
   
   Theorem. It is not possible to split an arbitrary angle in three equal
   parts using a compass and an unmarked ruler.
   
   This problem, together with Doubling the Cube, Constructing the
   regular Heptagon and Squaring the Circle were posed by the Greeks in
   antiquity, and remained open until modern times.

   The solution all of them is rather inelegant from a geometric
   perspective. No geometric proof has been offered [check?], however, a
   very clever solution was found using fairly basic results from
   extension fields and modern algebra.

   It turns out that trisecting the angle is equivalent to solving a
   cubic equation. Constructions with ruler and compass may only compute
   the solution of a limited set of such equations, even when restricted
   to integer coefficients. In particular, the equation for theta = 60
   degrees cannot be solved by ruler and compass and thus the trisection
   of the angle is not possible.

   It is possible to trisect an angle using a compass and a ruler marked
   in 2 places.

   Suppose X is a point on the unit circle such that angle XOE is the
   angle we would like to ``trisect''. Draw a line AX through a point A
   on the x -axis such that |AB| = 1 (which is the same as the radius of
   the circle), where B is the intersection-point of the line AX with the
   circle.

   [IMAGE]





   Let theta be angle BAO . Then angle BOA = theta , and angle XBO =
   angle BXO = 2 theta Therefore:

   angle AOB + angle BOX + angle XOE = pi/2 = angle XBO + angle BXO +
   angle BOX + theta + angle BOX + angle XOE = 2 theta + 2 theta + angle
   BOX Therefore: angle XOE = 3 theta . QED.




     _________________________________________________________________



    alopez-o@barrow.uwaterloo.ca
    Tue Apr 04 17:26:57 EDT 1995
   
   
   
   
   
   
     _________________________________________________________________
   
   
   
    alopez-o@barrow.uwaterloo.ca
    Sun Nov 20 20:45:48 EST 1994


