A Simple Vector Proof of Feuerbach's Theorem

TL;DR

This paper provides a straightforward vector-based proof of Feuerbach's Theorem, demonstrating the tangency relationships between the nine-point circle, incircle, and excircles of a triangle.

Contribution

It introduces a simple, vector-based proof of Feuerbach's Theorem, including all necessary preliminaries for completeness.

Findings

Proof confirms the tangency of the nine-point circle with incircle and excircles

Vector computations simplify the understanding of Feuerbach's Theorem

All necessary preliminaries are established for clarity

Abstract

The celebrated theorem of Feuerbach states that the nine-point circle of a nonequilateral triangle is tangent to both its incircle and its three excircles. In this note, we give a simple proof of Feuerbach's Theorem using straightforward vector computations. All required preliminaries are proven here for the sake of completeness.