A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry

A porism concerning cyclic quadrilaterals

We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as the circle.

四边形折叠、锯齿形与Arnold-Liouville可积性

Folding of quadrilaterals, zigzags, and Arnold-Liouville integrability

本文将Darboux四边形折叠porism和Bottema锯齿形porism纳入Arnold-Liouville可积性框架，揭示其可积结构。

Deformation of quadrilaterals and addition on elliptic curves

The space of quadrilaterals with fixed side lengths is an elliptic curve. Darboux used this to prove a porism on foldings. In this article, the space of oriented quadrilaterals is studied on the base of biquadratic equations between their angles. The space of non-oriented quadrilaterals is also an elliptic curve, doubly covered by the previous one, and is described by a biquadratic relation between the diagonals. The spaces of non-oriented quadrilaterals with the side lengths $(a_1, a_2, a_3, a_4)$ and $(s-a_1, s-a_2, s-a_3, s-a_4)$ turn out to be isomorphic via identification of two quadrilat…

Two generalizations of the Butterfly Theorem

We establish two direct extensions to the Butterfly Theorem on the cyclic quadrilateral along with the proofs using the projective method and analytic geometry of the Cartesian coordinate system.