Two generalizations of the Butterfly Theorem

TL;DR

This paper extends the classical Butterfly Theorem to more general cases within cyclic quadrilaterals, providing new proofs via projective geometry and Cartesian analytic methods.

Contribution

It introduces two novel generalizations of the Butterfly Theorem and offers multiple proof techniques for these extensions.

Findings

Two new generalizations of the Butterfly Theorem

Proofs using projective geometry and Cartesian coordinates

Enhanced understanding of cyclic quadrilaterals

Abstract

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.