Geodesic
On quadrangles inscribed in a closed curve and the vertices of the curve
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 8, Tome 299 (2003), pp. 241-251 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $ADCDE$ be a pentagon inscribed in a circle. It is proved that if $\gamma$ is a $C^4$-generic smooth convex planar oval with 4 vertices (stationary points of curvature), then there are 2 similarities $\varphi$ such that the quadrangle $\varphi(ABCD)$ is inscribed in $\gamma$ and the point $\psi(E)$ lies inside $\gamma$, as well as 2 similarities $\psi$ such that the quadrangle $\psi(ABCD)$ is inscribed in $\gamma$ and $\psi(E)$ lies outside $\gamma$. It is also proved that any circle $\gamma\hookrightarrow\mathbb R^n$ smoothly embedded in the space $\mathbb R^n$ of odd dimension contains the vertices of an equilateral $(n+1)$-link polygonal line lying in a hyperplane of $\mathbb R^n$. 
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V. V. Makeev. On quadrangles inscribed in a closed curve and the vertices of the curve. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 8, Tome 299 (2003), pp. 241-251. http://geodesic.mathdoc.fr/item/ZNSL_2003_299_a14/

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