Conics associated with triangles, or how Poncelet meets Morley

TL;DR

This paper explores the connection between Poncelet's theorem and Morley's theorem through a new criterion for six points on a triangle's sides to lie on a conic, uniting two classical geometric results.

Contribution

It introduces a novel criterion linking six points on a triangle's sides to conics and demonstrates a unified construction connecting Poncelet's and Morley's theorems.

Findings

Established a criterion for six points on a triangle to be conic-related.

Unified Poncelet's and Morley's theorems in a single geometric construction.

Provided new insights into classical projective and Euclidean geometry connections.

Abstract

We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem of Euclidean geometry - can meet in a one construction.