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

TL;DR

This paper explores a geometric porism involving cyclic quadrilaterals passing through four collinear points, providing two proofs—one using cross-ratios and another via hyperbolic isometries—highlighting deep connections in geometry.

Contribution

It introduces a new porism related to cyclic quadrilaterals and offers two distinct proofs, enriching the understanding of geometric configurations and hyperbolic geometry.

Findings

Existence of infinitely many cyclic quadrilaterals passing through four collinear points.

Two different proofs of the porism: one using cross-ratios, another using hyperbolic isometries.

Deepens understanding of geometric properties and hyperbolic geometry connections.

Abstract

If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism; one based on cross-ratios, the other on compositions of hyperbolic isometries.