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A short proof of McDougall’s circle theorem. (English) Zbl 1303.51006

Summary: This note offers a short, elementary proof of a result similar to Ptolemy’s theorem. Specifically, let \(d_{i,j}\) denote the distance between \(P_i\) and \(P_j\). Let \(n\) be a positive integer and \(P_i\), for \(1\leq i\leq 2n\), be cyclically ordered points on a circle. If \[ R_i:=\prod\limits_{\underset{j\neq i}{1\leq j\leq 2n}}\,d_{i,j}, \] then \[ \sum\limits^n_{i=1}\frac{1}{R_{2i}}=\sum\limits^n_{i=1}\frac{1}{R_{2i-1}}.\eqno{(1)} \]

MSC:

51M04 Elementary problems in Euclidean geometries
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