Centers of mass of Poncelet polygons, 200 years after.

*(English)*Zbl 1351.01013The starting point for this article is a letter, dated 1814 but apparently never sent, written by a certain Konstantin Shestakov to his younger brother Alexander. A copy of the letter was given to the authors by Shestakov’s great great great grandson. It appears that both Konstantin Shestakov and his brother had mathematical connections: Alexander had studied in Kazan as a contemporary of N. I. Lobachevskii, whilst Konstantin had befriended Jean-Victor Poncelet whilst the latter was a prisoner of war in Saratov. Indeed, Poncelet seems to have inspired in Konstantin an interest in geometry, and thus provided a reason for his writing this letter. Only the first paragraph of the Russian original is reproduced here, but this is followed by an English translation of the full letter. In the letter, Shestakov communicated a new result in geometry which he hoped that his brother would pass on to Lobachevskii for comment. The authors of the article under review interpret Shestakov’s theorem as the following: Let \(\gamma\subset\Gamma\) be a pair of nested ellipses that admit a \(1\)-parameter family of Poncelet \(n\)-gons \(P_t\). Then, both loci \(CM_0(P_t)\) and \(CM_2(P_t)\) (the centre of mass of the vertices of \(P_t\), and the centre of mass of \(P_t\) considered as a homogeneous lamina, respectively) are ellipses homothetic to \(\Gamma\) (or single points). A proof of this result is given, along with a statement and further proof of Weill’s theorem on polygons between nested circles, framed in similar language (i.e., centres of mass) to the preceding theorem.

Reviewer: Christopher Hollings (Oxford)

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\textit{R. Schwartz} and \textit{S. Tabachnikov}, Math. Intell. 38, No. 2, 29--34 (2016; Zbl 1351.01013)

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