The Ptolemaean Inequality in the closure of complex hyperbolic plane

TL;DR

This paper proves the Ptolemaean Inequality and Ptolemaeus' Theorem within the closure of the complex hyperbolic plane using the Cygan metric, extending classical geometric inequalities to this setting.

Contribution

It establishes the Ptolemaean Inequality and Ptolemaeus' Theorem in the context of the complex hyperbolic plane with the Cygan metric, a novel extension of classical results.

Findings

Proves Ptolemaean Inequality in the complex hyperbolic plane

Establishes Ptolemaeus' Theorem in this setting

Extends classical Euclidean inequalities to complex hyperbolic geometry

Abstract

We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.