Theorem 3.4: If k < n/2, then in an {n/k} star there are 1 + nk - n regions. They consist of

1 regular convex n-gon, and

if k > 1, there will be n triangles, and

n(k - 2) quadrilaterals.

proof: First note that the {n/1} star in the center of the figure is a regular convex n-gon. Next, note that all the other regions have a uniquely determined extreme vertex point which is a point of one of the {n/j} stars, j = 2, . . . , k, so there will be a one to one correspondence between the points which are not on the convex n-gon and the other regions. The points of the {n/2} star will be vertices of triangles each of whose base is a side in the convex n-gon, so there will be n triangles. The other points will be vertices of {n/j} stars where j > 2. The base of each of those regions will be an angle between the points of the {n/j-1}. As a result each of those regions will be quadrilaterals.

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