A number theoretic question arising in the geometry of plane curves and   in billiard dynamics
  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. 
Van Cyr
math.NT math.DG math.DS