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We prove that if $\rho\neq1/2$ is a rational number between zero and one,then there is no integer $n>1$ such that $$ n\tan(\pi\rho)=\tan(n\pi\rho). $$This has interpretations both in the theory of bicycle curves and that ofmathematical billiards.

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