Segre's theorem on ovals in Desarguesian projective planes

Segre’s theorem on ovals in projective spaces is an ingenious result from
the mid-twentieth century which requires surprisingly little background to prove. This
note, suitable for undergraduates with experience of linear and abstract algebra, provides a complete and self-contained proof. All necessary pre-requisites, principally
evaluation of homogeneous polynomials at projective points and Desargues’ theorem
are presented in full. While following the broad outline of Segre’s proof, careful parameterisation of certain tangent lines results in shorter and simpler computations
than the original.