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Parallelograms inscribed in a curve having a circle as \(\frac{\pi}{2}\)-isoptic. (English) Zbl 1182.52004

Summary: Jean-Marc Richard observed in [Eur. J. Phys. 25, No. 6, 835–844 (2004; Zbl 1162.70320)] that maximal perimeter of a parallelogram inscribed in a given ellipse can be realized by a parallelogram with one vertex at any prescribed point of ellipse. Alain Connes and Don Zagier gave in [Am. Math. Mon. 114, No. 10, 909–914 (2007; Zbl 1140.51010)] probably the most elementary proof of this property of ellipse. Another proof can be found in [M. Berger, Geometry. I, II. (2009; Zbl 1153.51001)].
In this note we prove that closed, convex curves having circles as \(\pi/2\)-isoptics have the similar property.

MSC:

52A10 Convex sets in \(2\) dimensions (including convex curves)
51M04 Elementary problems in Euclidean geometries
51M16 Inequalities and extremum problems in real or complex geometry
53A04 Curves in Euclidean and related spaces
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