Journal of Mathematical Sciences

, Volume 131, Issue 1, pp 5395–5400

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

Authors

  • V. V. Makeev
    • St. Petersburg State University
Article

DOI: 10.1007/s10958-005-0412-8

Cite this article as:
Makeev, V.V. J Math Sci (2005) 131: 5395. doi:10.1007/s10958-005-0412-8

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.

Copyright information

© Springer Science+Business Media, Inc. 2005