We introduce $L$-involutions for any positive number $L$ and we give a characterization of the class $(L)$ of all $L$-involutions. Then we define so-called $\nu$-involutive pairs of points of a curve $C\in \Cal M$ where $\Cal M$ is the family of all $C^1$ plane closed curves. For arbitrary $C\in \Cal M$ of length $L$ and for arbitrary $\nu\in (L)$ there exists a $\nu$-involutive pair of $C$ such that the tangent lines at the points ot this pair are parallel. Applications of this fact are given.