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A Short Proof of McDougall’s Circle Theorem

Marc Chamberland and Doron Zeilberger
The American Mathematical Monthly
Vol. 121, No. 3 (March 2014), pp. 263-265
DOI: 10.4169/amer.math.monthly.121.03.263
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.121.03.263
Page Count: 3
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A Short Proof of McDougall’s Circle Theorem
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Abstract

Abstract This note offers a short, elementary proof of a result similar to Ptolemy’s theorem. Specifically, let di, j denote the distance between Pi and Pj. Let n be a positive integer and Pi, for 1 ≤ i ≤ 2n, be cyclically ordered points on a circle. If \documentclass{article} \pagestyle{empty}\begin{document} $$R_i := \prod_{\substack{1\leq j\leq 2n \\ j\neq i}}$$ \end{document} then \documentclass{article} \pagestyle{empty}\begin{document} $$\sum_{i=1}^n \frac{1}{R_{2i}} = \sum_{i=1}^n \frac{1}{R_{2i-1}}.$$ \end{document}

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