Variations on Pascal's Theorem

In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for $d+4$ points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in ${\text{bbP}}^3$. Finally we reprove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown, for quadrics in ${\text{bbP}}^4$ containing five general lines.