# The angle trisection problem

You have an angle and you have a pen, paper, compass and a straight edge. You don't know how big the angle is, divide this angle into three equal part using only the material that is listed here? If not possible what other tool is needed (protractors are not allowed)?

• Welcome to math.SE. Please consider glancing over FAQ and please refrain from using imperative style when asking questions. – Sasha Aug 18 '11 at 19:24
• Have you looked at Wikipedia: Angle_trisection ? – Sasha Aug 18 '11 at 19:25
• In general this is not possible, although there are some specific angles that do permit trisection. – user12998 Aug 18 '11 at 19:27
• @Sasha: "neusis" simply means a marked straightedge is allowed. There are also a number of curves that one can use, as well as a carpenter's square, for the trisection of angles. – J. M. isn't a mathematician Aug 18 '11 at 19:41
• Also, there's the tomhawk thingy(Look it up in link), if you have a pen along with your compass and straightedge,you can build this. – kneidell Aug 18 '11 at 19:47

As was already noted in comments, the problem admits no solution using compass and straight edge. You can render the problem solvable by either allowing neusis or tomahawk.

Here I will explain the Archimedes solution using neusis. Given an angle $\alpha$, draw a circle centered at its tip point $\mathbf{O}$. Draw a chord $\mathbf{AC}$. Let $\beta = \angle \mathbf{BOC}$, and let $\gamma = \angle \mathbf{BCO}$.

It follows elementary that $\angle \mathbf{OBA} = \beta+\gamma$ and $\angle \mathbf{OAB} = \alpha - \gamma$. Since $\mathbf{OA} = \mathbf{OB}$ as radii, $\alpha - \gamma = \beta+\gamma$, giving $\gamma = \frac{\alpha - \beta}{2}$. If we further impose $\beta = \gamma$, we get $\beta=\gamma=\frac{\alpha}{3}$.

In this configuration, $\mathbf{CB} = \mathbf{OB}$ as sides opposite to equal angles, which is how the neusis comes in.

One would use the marked ruler, to make $\mathbf{CB}$ equal to the radius of the circle.

The wording suggests a riddle, it can be done if you are allowed to fold the paper, so: LINK_1 and: LINK_2