  The subject of this paper is the study of small complete arcs in
$\mathrm{PG}(2,q)$, for $q$ odd, with at least $(q+1)/2$ points on a conic. We
give a short comprehensive proof of the completeness problem left open by Segre
in his seminal work [20]. This gives an alternative to Pellegrino's long proof
which was obtained in a series of papers in the 1980s. As a corollary of our
analysis, we obtain a counterexample to a misconception in the literature [6].
