Random Cyclic Quadrilaterals

TL;DR

This paper explores the properties of uniform cyclic quadrilaterals, revealing their unique angle and side correlations, contrasting with cyclic triangles, and providing insights into their geometric dependencies.

Contribution

It characterizes the correlation structures of sides and angles in uniform cyclic quadrilaterals, highlighting differences from cyclic triangles.

Findings

Sides of uniform cyclic quadrilaterals are negatively correlated.

Adjacent angles in uniform cyclic quadrilaterals are uncorrelated but dependent.

Cyclic quadrilaterals exhibit distinct correlation patterns compared to cyclic triangles.

Abstract

The circumcircle of a planar convex polygon P is a circle C that passes through all vertices of P. If such a C exists, then P is said to be cyclic. Fix C to have unit radius. While any two angles of a uniform cyclic triangle are negatively correlated, any two sides are independent. In contrast, for a uniform cyclic quadrilateral, any two sides are negatively correlated, whereas any two adjacent angles are uncorrelated yet dependent.