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

Abstract

We prove that if $\rho \ne 1/2$ 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 of mathematical billiards.