Suppose you start with a regular n-gon, where n is at least 5, and you draw in all the diagonals that don't pass through the center. this divides the region enclosed by the regular n-gon into several parts, one of which contains the center.What is the shape of the part containing the center? How do you know?What is the area of the part containing the center, as a function of n and of the area of the entire region.

For even: Only look at the diagonals 2nd in sizeFor odd: Only look at largest diags.Next you realise that smaller diagonals cannot cut this piece. By symmetry it's a small n-gon... too tired to calc the area.So for 8-gon, draw from 0 to 3, 3 to 6, 6 to 1, 1 to 4, 4 to 7, 7 to 2, 2 to 5, 5 to 0.Gives a smaller 8gon

For the area, calculate the distance from the center to the sides of the small and large n-gon, take the ratio and square.Distance to the sides of large n-gon: R*cos(pi/n).Distance to the sides of small n-gon: R*sin(pi/n), if n even,R*sin(pi/2n), if n odd.Area of small n-gon as a fraction of large one's area:(tan(pi/n))^2, n even(sin(pi/2n)/cos(pi/n))^2, n odd.

In Mathematica:ngon[p_, q_] := Polygon[Table[{Cos[2 Pi k q/p], Sin[2 Pi k q/p]}, {k, p}]];vrtngon[p_, q_] := Table[{Cos[2 Pi k q/p], Sin[2 Pi k q/p]}, {k, p}];linesngon[p_, q_] := Tuples[vrtngon[p, q], 2];nocenterlinesngon[p_, q_] := Module[{lista, iguais, opostos}, lista = linesngon[p, q]; iguais = Map[(#[[1]] == #[[2]]) &, lista]; opostos = Map[(#[[1]] == -#[[2]]) &, lista]; Pick[lista, Map[Apply[And, #] &, Transpose[{Map[Not, iguais], Map[Not, opostos]}]]]];GraphicsGrid[ Transpose[{Table[ Graphics[Map[Line, nocenterlinesngon[n, 1]], ImageSize -> Medium], {n, 5, 12}]}]]