Chamberland, Marc; Zeilberger, Doron A short proof of McDougall’s circle theorem. (English) Zbl 1303.51006 Am. Math. Mon. 121, No. 3, 263-265 (2014). 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 \[ \sum\limits^n_{i=1}\frac{1}{R_{2i}}=\sum\limits^n_{i=1}\frac{1}{R_{2i-1}}.\eqno{(1)} \] Cited in 1 Document MSC: 51M04 Elementary problems in Euclidean geometries Keywords:Lagrange interpolation formula; Ptolemy’s theorem PDFBibTeX XMLCite \textit{M. Chamberland} and \textit{D. Zeilberger}, Am. Math. Mon. 121, No. 3, 263--265 (2014; Zbl 1303.51006) Full Text: DOI