Proof of Fermat primes and constructible n-gon

Question

Prove that if a regular n-gon is constructible, then $n = 2^kp_1 ···p_r $ where $p_1,...,p_r$ are distinct Fermat primes using the following facts.

1. If the regular $n$-gon is constructible and $n = qr$, the regular $q$-gon is also constructible.
2. If $\xi = \cos(2\pi/p^2) + i \sin(2\pi/p^2)$ then $\xi$ is a root of $f(x)=1+x^p +x^{2p} +···+x^{(p−1)p}$. And $f(x)$ is irreducible over $\mathbb{Q}$.

I proved both of these facts, but I'm having trouble seeing how they help me prove the first statement. I think I'm supposed to use the following theorem, but I'm not sure: "If $p$ is a prime for which a regular $p$-gon is constructible, then $p$ is a Fermat prime.

Any help would be great, thank you in advance!