Beyond Conway's concyclicity theorem: generalization and alternatives

TL;DR

This paper generalizes Conway's concyclicity theorem using a parametrized approach, explores an anti-Conway configuration, and introduces new related theorems, expanding understanding of geometric concyclicity conditions.

Contribution

It introduces a parametrization of Conway's theorem, demonstrates the existence of an infinite family of invariant triplets, and extends related theorems with new concyclicity results.

Findings

Generalization of Conway's theorem with arbitrary triplets

Existence of an infinite family of triplets preserving the theorem

Identification of a unique triplet related to triangle sides that preserves Dussau's theorem

Abstract

The famous concyclicity theorem stated by John H. Conway is here reconsidered by means of a parametrisation of the associated triangular configuration with arbitrary triplets of real numbers ($\alpha$;$\beta$;$\gamma$). This theorem, thus corresponding to the case ($\alpha$;$\beta$;$\gamma$)=(1;1;1), is generalized while demonstrating that there always exist an infinite family of such triplets which keeps unchanged the conclusion. The "anti-Conway" configuration corresponding to the case ($\alpha$;$\beta$;$\gamma$)=(-1;-1;-1) is also investigated : Xavier Dussau's theorem of concurrent lines is redemonstrated and completed by another concyclicity theorem. It is also proved that there exist in general a unique triplet ($\alpha$;$\beta$;$\gamma$)$\ne$(-1;-1;-1) which is a function of the sides of the considered triangle and which keeps unchanged the conclusion of Dussau's theorem.