A number theoretic question arising in the geometry of plane curves and in billiard dynamics

We prove that if $\rho \mathrm{\ne }1\mathrm{/}2$ is a rational number between zero and one, then there is no integer $n>1$ such that $$n\tan (\pi \rho )=\tan (n\pi \rho )\mathrm{.}$$ This has interpretations both in the theory of bicycle curves and that of mathematical billiards.