论文标题
牛顿四边形定理的概括和米特恩四边定理的基本证明
A generalization of Newton's quadrilateral theorem and an elementary proof of Minthorn's quadrilateral theorem
论文作者
论文摘要
牛顿的四边形定理可以用如下。如果H是一个与非平行四边形四边形Q的四个延伸侧相切的圆，则H的中心位于Q的Newton线上。我们证明，如果H是任意的双曲线或椭圆形，则该定理仍然是正确的。四边形最多可以有一个切线，但无限的椭圆和双曲线无限。 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;他们依靠线性代数和仿射变换。
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.
