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

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