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double rotation, rational ratio of angles

Source: Bulgaria 1987 P2

June 15, 2021
geometrygeometric transformationrotationratio

Problem Statement

Let there be given a polygon PP which is mapped onto itself by two rotations: ρ1\rho_1 with center O1O_1 and angle ω1\omega_1, and ρ2\rho_2 with center O2O_2 and angle ω2 (0<ωi<2π)\omega_2~(0<\omega_i<2\pi). Show that the ratio ω1ω2\frac{\omega_1}{\omega_2} is rational.
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