Another Proof of Segre's Theorem about Ovals

In 1955 B. Segre showed that any oval in a projective plane over a finite field of odd order is a conic. His proof constructs a conic which matches the oval in some points and tangents, and then shows that it actually coincides with the oval. The different proof given here parametrizes an affine piece of the oval and shows directly that the parametrization is given by a polynomial of degree $2$.