Given a polygon with an even number of sides, the derived polygon is obtained by joining the points which
are a fractional distance 
along each side. If 
, then the derived polygons are called midpoint
polygons and tend to a shape with opposite sides parallel and equal in length.
Furthermore, alternate polygons have approximately the same length, and the original
and all derived polygons have the same centroid.
Amazingly, if 
,
the derived polygons still approach a shape with opposite sides parallel and equal
in length, and all have the same centroid. The above illustrations show 20 derived
polygons for ratios 
,
0.5, 0.7, and 0.9. More amazingly still, if the original polygon is skew, a plane
polygonal is approached which has these same properties.
