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This is a more general example of the midpoint porism. I don't know of a way to make the polygon have the midpoints lay on the surface, except by moving some key points until the polygon closes. The gray polygon is what determines the size and shape of the inner circle. By moving points A,B,C,D you can change the polygon and the inner circle. By moving E you can see what would happen if you started at a different starting point for the same inner circle. Currently the polygon has 7 sides. If you move A,B,C,D around you can create polygons that have fewer sides by overlapping the segments. If you have arranged the gray polygon so that the polygon closes, It seems that the blue polygon will also close - no matter where point E is placed. This is what I call the midpoint porism.

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