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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.