Geodesic
Cyclic polygons are critical points of area
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 238-245

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It is shown that typical critical points of the signed area function on the moduli space of a generic planar polygon are given by cyclic configurations, i.e., configurations that can be inscribed in a circle. Several related problems are briefly discussed in conclusion. Bibl. – 14 titles. 
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     author = {G. Panina and G. N. Khimshiashvili},
     title = {Cyclic polygons are critical points of area},
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     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_360_a10/}
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G. Panina; G. N. Khimshiashvili. Cyclic polygons are critical points of area. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamics systems, combinatorial methods. Part XVI, Tome 360 (2008), pp. 238-245. http://geodesic.mathdoc.fr/item/ZNSL_2008_360_a10/