An angle "trisection"

A highly accurate approximate construction by Mark Stark

Drag the point B to change the angle AOB
Drag the point E to change the initial guess
The angle AOE' is a very close to being 1/3 of angle AOB
Note how insensitive G and E' are with respect to displacements of E
Type "r" to reset the diagram to its initial state

The construction shown above, which trisects an arbitrary angle with great accuracy, was first proposed by Mark Stark in the geometry-puzzles discussion list. In a followup article, Eric Bainville noted that the iteration of this trisection algorithm "will effectively converge to the trisection with a cubic convergence rate." Here is the outline of the construction, as restated by Mark in a later article:

  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 at points 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 the earlier arc somewhere between 1/4 and 1/2 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 passes exactly through point O (it wont) stop, your first guess was an exact trisection.
  6. Extend line OA well past point O, step off 3 times length OA from point O and label the new point F.
  7. Swing an arc of length OF with O as the origin that crosses the extended line DE near point F. Label the intersection G.
  8. Draw line GO and extend it to intersect the original arc from step 2. Label the intersection E'. Line OE' is a good trisection. However this is only the start. Repeating the process from step 4 using AE' as the arc radius results in a trisection to within 10E-11 degrees.

Remark 1: It is possible to derive a precise expression for the error of this construction.

Remark 2: Also see an alternative trisection method, proposed by Mark Stark.

This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 16, 2002.