BG-MO 1987 Problem 2

Let there be given a polygon $P$ which is mapped onto itself by two rotations: $\rho_1$ with center $O_1$ and angle $\omega_1$ , and $\rho_2$ with center $O_2$ and angle $\omega_2~(0<\omega_i<2\pi)$ . Show that the ratio $\frac{\omega_1}{\omega_2}$ is rational.