Michael J. G. Scheer, A simple vector proof of Feuerbach’s theorem,
Forum Geometricorum, 11 (2011) 205--210.
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.
[ps file] [pdf]