  Let $\Omega$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is
left invariant under the action of the dihedral group $D_n$ of isometries of
$\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction
must intersect the boundary of $\Omega$, under suitable conditions on
$\frac{\partial}{\partial \theta}$ .
