Geodesic
On polygons inscribed into a convex figure
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 15-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper contains a survey of results about the possibility to inscribe convex polygons of particular types into a plane convex figure. It is proved that if $K$ is a smooth convex figure, then $K$ is circumscribed either about four different reflection-symmetric convex equilateral pentagons or about a regular pentagon. Let $S$ be a family of convex hexagons whose vertices are the vertices of two negatively homothetic equilateral triangles with common center. It is proved that if $K$ is a smooth convex figure, then $K$ is circumscribed either about a hexagon in $S$ or about two pentagons with vertices at the vertices of two hexagons in $S$. In the latter case, the sixth vertex of one of the hexagons lies outside $K$, while the sixth vertex of anther one lies inside $K$. 
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V. V. Makeev. On polygons inscribed into a convex figure. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 12, Tome 415 (2013), pp. 15-20. http://geodesic.mathdoc.fr/item/ZNSL_2013_415_a1/

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