Circumscribing Conics around Convex Quadrilaterals
by Thomas F. Banchoff and Philip J. Davis

This problem proves that a necessary a sufficient condition for four points in the plane lying on an ellipse is that no one points lies in or on the triangle formed by the other three. The proof follows from elementary facts about invertible affine transformations and general conic sections. An alternate shorter proof, due to Alan Landman, is given which uses basic algebraic geometry.