Things I know:
1. We can divide any angle into two
2. We can prove that $\pi/3$ can not be trisected. Thus trisection is not true in general.
3. A regular $n$-gon can only be constructed if and only if $\phi(n)=2^t$ for some integer $t$. Where $\phi(n)$ is the number of relatively prime numbers less than $n$ (Euler totient function)
I also went over this problem. But the thing that I don't understand is in those cases they talk about constructing a regular $n$-gon. That is we divide $360^{\circ}$ into $n$ equal parts. Which in my case will be $n=5$. But what I need is different.
I need to check whether any angle $\theta$ can be divided into $5$ equal parts.
Appreciate your help
Best Answer
No, it is not possible to pentasect any angle using straightedge and compass. If it were possible to do so, then it would be possible to construct a regular $25$-gon. However this is impossible by the Gauss-Wantzel theorem, which implies that if an $n$-gon is constructible, then the odd prime factors of $n$ are distinct.
So in general for any $n$ that is not a power of two, there is no construction that divides any angle into $n$ equal parts, otherwise you could construct a $n^2$-gon.