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

TL;DR

This paper extends Newton's quadrilateral theorem to ellipses and hyperbolas, proves a converse, and provides an elementary proof of Minthorn's theorem using linear algebra and affine transformations.

Contribution

It generalizes Newton's theorem to conic sections beyond circles and offers a new elementary proof of Minthorn's quadrilateral theorem.

Findings

Newton's theorem holds for ellipses and hyperbolas tangent to quadrilaterals.

Every point on the Newton line (except three) is a center of such conics.

An elementary proof of Minthorn's theorem is provided.

Abstract

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