Affine Maps and Feuerbach'sTheorem

TL;DR

This paper presents an affine proof of Feuerbach's theorem, constructing an explicit affine map that relates the nine-point circle to the incircle and is valid in any Hilbert plane satisfying the parallel postulate.

Contribution

It provides a new affine proof of Feuerbach's theorem using an explicit affine map, applicable in Hilbert planes with the parallel postulate.

Findings

Constructed an explicit affine map relating nine-point circle and incircle.

Proved the affine map fixes the Feuerbach point.

Validated the proof in any Hilbert plane with the parallel postulate.

Abstract

We give an affine proof of Feuerbach's theorem, by constructing an explicit affine map which takes the nine-point circle of any given Euclidean triangle to the incircle and fixes the Feuerbach point. The proof is shown to be valid in any Hilbert plane which satisfies the parallel postulate.