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Feb 8 '15 at 13:20 history edited Alexandre Eremenko CC BY-SA 3.0
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Feb 8 '15 at 4:25 comment added Aaron Meyerowitz The conditions are sufficient for a cyclically ordered list of angles to be realized by some polygons having a polygonal path touching each side. A regular pentagon has such a path but an equiangular pentagon with sides of lengths $1,\epsilon,1.6,\epsilon,1$ seems unlikely to. For a triangle the angles do determine the sides (up to similarity) so acute triangle have paths and obtuse ones do not.
Feb 7 '15 at 20:50 comment added Alexandre Eremenko Yes, I think these considerations permit to describe explicitly all such polygonal tables.
Feb 7 '15 at 19:59 comment added Will Sawin It is not too hard to see that the interior angles of the polygon you get by doing this are the averages of the adjacent interior angles of the billiard path. From this you get an explicit system of inequalities that the angles satisfy if and only if there is a polygonal billiard path touching every edge.
Feb 7 '15 at 15:11 history edited Alexandre Eremenko CC BY-SA 3.0
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Feb 7 '15 at 14:28 history answered Alexandre Eremenko CC BY-SA 3.0