Trisecting an Angle

By the time they finish geometry most high school students have heard that it is impossible to trisect an angle with the traditional tools of a straightedge and compass (which should be referred to in the plural, but Americans seldom do). What they seldom hear, or see, is a way to trisect a general angle with a marked straightedge, or what the Greeks called a neusis (from the Greek word for "incline toward", apparently because of the way it was used). More than that, you can find a simple way to show the double angle formula for the cosine of an angle.

In the figure above, the angle to be trisected is EAB with the "3x" in there. To create the construction, use the marked segment (call this one unit) of the straightedge as a radius, and construct a circle with the radius of one with the vertex at the center of the circle.

Now put one edge of the straightedge on the point at E and slide the edge along this point until the marked segment touches the circle (F), and the endpoint is on the diameter of the circle(D). The angle ADF is 1/3 of the angle EAB, and has a measure of x.

To see that this is true, draw the radius AF... and the two perpendicular lines shown dotted. Note that both triangles ADF and FEA are isosceles. If we begin by letting FDA =t then EFA is an exterior angle for that triangle, and so it, and its congruent angle AEF both have measure of 2t. Now if FA is extended, the exterior angle F'AE would have a measure of 4t, and this exterior angle would consist of the vertical reflection of FAE with a measure of x, and the remaining original angle of 3x... so 4x =4t or t=h.

The drawing can now be used to derive the cos(2x). Notice that DG and AG are both Cos(x). Since DH and DA are the adjacent leg and hypotenuse of angle D with measure x, we know that Cos(x)*DA = DH. Now DA=1+cos(2x), and AC = cos(x)+ cos(x), so we get 1+cos(2x) = cos(x) *cos(x)+ cos(x).

This simplifies to 1 + cos(2x) = cos²(x) + cos²(x) or...

1 + cos (2x) = 2 cos²(x)....and with one more step..

cos(2x) = 2 cos²(x) -1.

You can actually extend this with a little algebra to calculate cos(3x).

A Mechanical Trisection of theAngle

In the June issue following the Maskelyne trisection article I mentioned, there was a post by J. R. Cotter, that proposed a simple, exact (within mechanical limits) and apparently original trisection device. I have created a geogebra aplet to emulate the method for those who wish to explore.
The idea, shown in the image above is as follows. A mechanical linage with four pens labeled A, B, C, and D is created so that A is fixed at the vertex of the angle, and B is allowed to slide along one ray of the angle. Pens B, C, and D are constrained to a straight line, and the distances AC, BC, AD are all equal. AS B is moved along the ray of the angle, the point D will be moved around the fixed point at A.

When D is aligned with the line along the terminal ray of the angle, the angle ABD is 1/3 of the original angle.
The aplet allows you to change the original angle, and includes construction devices to emulate the mechanical linkage described in the paper and help visualize a proof of the trisection. Enjoy...