An angle "trisection"

A pretty simple approximate construction due to C. R. Lindberg and Free Jamison

Drag the point B to change the angle AOB
The angle E'OB is approximately 1/3 of angle AOB
Type "r" to reset the diagram to its initial state

The construction

The construction shown above, which trisects an arbitrary angle with a pretty good accuracy, is described in:

Free Jamison, Trisection Approximation, American Mathematical Monthly, vol. 61, no. 5, May 1954, pp. 334-336.

The construction, the main idea of which, according to Jamison, comes from an unpublished work by C. R. Lindberg, is as follows:

  1. Draw a circle centered at the angle's vertex O. Let the circle intersects the angle's sides at A and B.
  2. Extend BO to intersect the circle at a point C.
  3. Draw the bisector of the angle AOB and let it intersect the circle at D.
  4. Draw the line CD and extend it to a point E such that DE equals the circle's diameter.
  5. Draw the line OE and Let it cut the circle at the point E'. Then the angle E'OB approximately 1/3 of angle AOB
Jamison states that if the angle AOB is "a", and 0 < a < Pi, then the error e(a) in the construction is:

e(a) = a/12 - arctan[ sin(a/4) / (2 + cos(a/4)) ] < a^3/4000.

The function e(a) is monotonically increasing, therefore the worst error occurs at a=Pi. We have: e(Pi) = 0.0063 radians = 0.361 degrees. This corresponds to a relative error of approximately 0.2%.

If we confine the construction to the 0 < a < Pi/2 range, the worst error will be e(Pi/2) = 0.000757 radians = 0.0434 degrees. This corresponds to a relative error of approximately 0.05%.

Explanation

Jamison then explains why the construction works. The key lies in the observation that (i) in the triangle ODE the angles O and E are "small", and (ii) the side ED is twice as long as the side OD. Therefore from the law of sines we have sin(O)/sin(E) = ED/OD = 2 which implies that the angle O is approximately twice the angle E in the triangle ODE.

Let the measure of the angle OED be x. Then the triangle's external angle at D, that is the angle ODC, is the sum of the internal angles O and E, therefore it is approximately 3x. Therefore the angle OCD is 3x. Therefore the angle BOD is 6x. Since the angle E'OD is 2x, we conclude that the angle BOE' is 4x. Furthermore, since the angle BOD is 6x, the angle BOA is 12x. This shows that BOA is 3 times BOE', as asserted.

An extension

Although this construction is pretty good, Jamison proceeds to give an extension of Lindberg's method which requires a bit more work but is substantially more accurate.

This applet was created by Rouben Rostamian using David Joyce's Geometry Applet on July 22, 2002.