A new proof of Rédei's theorem on the number of directions

Rédei 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.