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Centers of mass of Poncelet polygons, 200 years after. (English) Zbl 1351.01013
The 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.

MSC:
01A55 History of mathematics in the 19th century
51-03 History of geometry
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[1] W. Barth, Th. Bauer. Poncelet theorems. Exposition. Math. 14 (1996), 125-144.
[2] H. J. M. Bos, C. Kers, F. Oort, D. W. Raven. Poncelet’s closure theorem. Exposition. Math. 5 (1987), 289-364. · Zbl 0633.51014
[3] I. Dolgachev. Classical algebraic geometry. A modern view. Cambridge U. Press, Cambridge, 2012. · Zbl 1252.14001
[4] V. Dragović, M. Radnović. Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Birkhäuser/Springer Basel AG, Basel, 2011. · Zbl 1225.37001
[5] V. Dragović, M. Radnović. Bicentennial of the great Poncelet theorem (1813-2013): current advances. Bull. Amer. Math. Soc. 51 (2014), 373-445. · Zbl 1417.37034
[6] L. Flatto. Poncelet’s theorem. Amer. Math. Soc., Providence, RI, 2009. · Zbl 1157.51001
[7] M. Gardner. Mathematical Games. Six sensational discoveries that somehow or another have escaped public attention, Scientific American, (1975), 126-130.
[8] Ph. Griffiths, J. Harris. On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math. 24 (1978), no. 1-2, 31-40. · Zbl 0384.14009
[9] W. M’Clelland. A treatise on the geometry of the circle and some extensions to conic sections by the method of reciprocation. Macmillan and Co., London and New York, 1891.
[10] O. Romaskevich. On the incenters of triangular orbits on elliptic billiards. Enseign. Math. 60 (2014), 247-255. · Zbl 1371.37073
[11] R. Schwartz, S. Tabachnikov. Elementary surprises in projective geometry. Math. Intelligencer 32 (2010) no. 3, 31-34. · Zbl 1204.51024
[12] A. Shen. Poncelet Theorem Revisited. Math. Intelligencer 20 (1998) no. 4, 30-32.
[13] A. Skutin. On rotation of an isogonal point. J. Classical Geom. 2 (2013), 66-67.
[14] M. Weill. Sur les polygones inscrits et circonscrits à la fois à deux cercles. Journ. de Liouville 4 (1878), 265-304. · JFM 10.0359.04
[15] A. Zaslavsky, G. Chelnokov. Poncelet theorem in Euclidean and algebraic geometry (in Russian). Matemat. Obrazovanie (2001), no. 4 (19), 49-64.
[16] A. Zaslavsky, D. Kosov, M. Muzafarov. Trajectories of remarkable points of the Poncelet triangle (in Russian). Kvant (2003), no. 2, 22-25.
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