A simple proof of Feuerbach's theorem

TL;DR

This paper presents a straightforward proof of Feuerbach's theorem, which states that the nine-point circle of any non-equilateral triangle is tangent to its incircle and excircles.

Contribution

The paper provides a simplified and elegant proof of Feuerbach's theorem, making the geometric relationship more accessible.

Findings

Nine-point circle is tangent to incircle and excircles in non-equilateral triangles.

The proof simplifies understanding of Feuerbach's theorem.

Enhances geometric theory with an accessible proof.

Abstract

The theorem of Feuerbach states that the nine-point circle of a nonequilateral triangle is tangent to both its incircle and its three excircles. We give a simple proof of this theorem.