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Dividing an angle into an arbitrary ratio

Back in my school days, probably in the year 1997, I first read about a famous and ancient geometrical problem, the problem of trisecting an angle using only a compass and an unmarked ruler. It took me some time to understand that a general method for trisecting all angles is impossible arbitrary angle \theta. Of course with a marked ruler, we can always trisect an arbitrary angle, and moreover, in case of some special angles such as \pi, \pi/2 or some non-constructible angles such as 3\pi/7, we can trisect even with an unmarked ruler. I found some approximate trisection methods, each of which gave a dominant term \theta/3 and an error terms that was much smaller than the dominant term. At that time, I thought is it possible to find a method that gives an approximate division of an arbitrary angle into an arbitrary ratio. In formal terms, this problem can be stated as:

Problem: Let n be positive integer and x be any positive realGiven an angle \theta and a line that divides it in the ratio 1/x find a method with finite number of steps to construct an line that approximately divides the angle in the ratio 1/(x+n) using only a compass and an unmarked ruler.

Solution: We shall use the notation [A] to denote the angle A. In the diagram let [COB]=\theta be an acute angle and let [IOB]=\theta/x be given.

Steps:

  1. With O as center and radius OC, draw an arc to cut OB at D.
  2. Join CD and draw DJ perpendicular to OB.
  3. Let OI intersect DJ at I and the arc CD at K.
  4. Produce DK to intersect OC at G.
  5. Let OI intersect CD at E.
  6. Extent GE to intersect OB at H.
  7. Join IH to intersect GD at F.
  8. Draw a line OA through F. [AOB] \approx \theta/(x+1).
  9. Repeating this method n times, to construct a line that approximately divides \theta in the ratio 1/(x+n).

As \theta increases, the lines OC and DG tend to become parallel to each other and therefore the paper size required for construction increases. To overcome this problem, we can consider an obtuse angle as the sum of a right angle and an acute angle and apply our method on each part separately and then add up the final angle resulting form each of the two parts.