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

Abstract

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 of mathematical billiards.