  R\'edei and Megyesi proved that the number of directions determined by a $p$
element subset of $\mathbb{F}_p^2$ is either $1$ or at least $\frac{p+3}{2}$.
The same result was independently obtained by Dress, Klin and Muzychuk. We give
a new and short proof of this result using a Lemma proved by Kiss and the
author. The new proof further on a result on polynomials over finite fields.
