Let ABCDE be a pentagon inscribed in a circle. It is proved that if \(\mathcal{O}\) is a C^{4}-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 ↪ ℝ^{n} contains the vertices of an equilateral (n + 1)-link polygonal line lying in a hyperplane of ℝ^{n}. Bibliography: 7 titles.