Projective Line Revisited

TL;DR

This paper offers a new perspective on projective line geometry, clarifies classical results about the projective plane, and generalizes the nine-point circle theorem to affine planes over fields with characteristic not equal to 2.

Contribution

It introduces a novel viewpoint on projective line geometry and extends the nine-point circle theorem to broader affine plane contexts.

Findings

New perspective on projective line geometry

Generalization of the nine-point circle theorem

Clarification of classical projective plane results

Abstract

This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the nine-point circle theorem valid for any affine plane over any field of characteristic different from 2.