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Newton's quadrilateral theorem can be phrased as follows. If H is a circlethat is tangent to the four extended sides of a non-parallelogram quadrilateralQ, the center of H lies on the Newton line of Q. We prove that the theoremremains true if H is an arbitrary hyperbola or ellipse. A quadrilateral canhave at most one circle tangent to it but infinitely many ellipses andhyperbolas. We also prove a converse of Newton's theorem, namely that everypoint on the Newton line, excepting three singular points, is the center ofsome ellipse or hyperbola tangent to the four extended sides of Q. Using thesame proof techniques we give an elementary proof of the (lesser known)Minthorn's quadrilateral theorem, which concerns quadrilaterals passing throughthe four vertices of Q. Our proofs are analytic; they rely on linear algebraand affine transformations.

A generalization of Newton's quadrilateral theorem and an elementary proof of Minthorn's quadrilateral theorem