, Volume 131, Issue 1, pp 5395-5400

Quadrangles Inscribed in a Closed Curve and the Vertices of a Curve

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Abstract

Let ABCDE be a pentagon inscribed in a circle. It is proved that if $\mathcal{O}$ is a C4-generic smooth convex planar oval with four vertices (stationary points of curvature), then there are two similarities φ such that the quadrangle φ(ABCD) is inscribed in $\mathcal{O}$ and the point φ(E)lies inside $\mathcal{O}$ , as well as two similarities ψ such that the quadrangle ψ(ABCD) is inscribed in $\mathcal{O}$ and ψ(E)lies outside $\mathcal{O}$ . Itisalsoprovedthatif n is odd, then any smoothly embedded circle γ ↪ ℝn contains the vertices of an equilateral (n + 1)-link polygonal line lying in a hyperplane of ℝn. Bibliography: 7 titles.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 241–251.