A centroid hexagon is the hexagon obtained from an arbitrary hexagon by connecting, in order, the centroids of the six triangles formed by three consecutive vertices. Its opposite sides are equal and parallel (Wells 1991). A proof of the centroid hexagon result, as well as a generalization to octagons, decagons, and so on, is given by de Villiers (2007).
More generally, for a 
-polygon with an integer 
, let 
be the geometric centroid
of the 
consecutive vertices starting at 
, with indices interpreted cyclically. If 
and 
are the corresponding position
vectors, then
 |
(1)
|
and
 |
(2)
|
Thus opposite sides of the derived polygon are equal and parallel, and each has 
the length of the corresponding
diagonal 
of the original polygon
(Lord 2008). The case 
gives Varignon's theorem, while 
gives the centroid hexagon.
Writing 
for the geometric centroid of the original
vertices gives
 |
(3)
|
so each line segment joining opposite vertices of the derived polygon has midpoint 
. The original and derived vertices
have the same geometric centroid (Lord 2008).
The vector identities hold in any Euclidean
space, so the construction and common geometric
centroid property also apply to nonplanar polygons.
