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 
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/
@article{ZNSL_2003_299_a14,
     author = {V. V. Makeev},
     title = {On quadrangles inscribed in a~closed curve and the vertices of the curve},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {241--251},
     year = {2003},
     volume = {299},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_299_a14/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $\mathbb R^n$.

[1] S. Mukhopadhayaya, “New methods in the geometry of planar arc-I, cyclic, and sextactic points”, Bull. Calcutta Math. Soc., 1 (1909), 31–37

[2] O. R. Musin, “Ekstremumy krivizny i teorema o 4 vershinakh dlya mnogougolnikov i mnogogrannikov”, Zap. nauchn. semin. POMI, 280, 2001, 251–271 | MR | Zbl

[3] V. Makeev, “O chetyrekhugolnikakh, vpisannykh v zamknutuyu krivuyu”, Mat. zametki, 57:1 (1995), 129–132 | MR | Zbl

[4] L. G. Shnirelman, “O nekotorykh geometricheskikh svoistvakh zamknutykh krivykh”, Uspekhi mat. nauk, 1944, no. 10, 34–44 | MR | Zbl

[5] R. C. Bose, “On the number of circles of curvature perfectly enclosing or perfectly enclosed by a closed oval”, Math. Ann., 35 (1932), 16–24 | MR

[6] V. Makeev, “Vpisannye i opisannye mnogogranniki vypuklogo tela”, Mat. zametki, 55:4 (1994), 128–130 | MR | Zbl

[7] H. Griffits, “The topology of square pegs in round holes”, Proc. London Math. Soc., 363 (1991), 647–672 | DOI | MR