General Theorem: The 2n centroids of the n-gons, A1A2A3...An, A2A3A4...An+1, etc. sub-dividing a 2n-gon, A1A2A3...A2n (where n ≥ 2), form a 2n-gon with opposite sides equal and parallel.
The above theorem is a generalization of Varignon's theorem (1731), which states that the midpoints of the sides of any quadrilateral form a parallelogram. The dynamic sketch below illustrates it for a hexagon.
Read my 2007 article in The Montana Mathematics Enthusiast A hexagon result and its generalization via proof.
Also see Nick Lord's note "Maths bite: averaging polygons" in The Mathematical Gazette, Vol. 92, No. 523, March 2008, p. 134 that gives the same generalization to 2n-gons, and proving it very easily with vectors.
Generalization to 3D: Perhaps surprisingly, the general theorem is also true in 3D space as illustrated in the YouTube video above. Below is a dynamic Cabri 3D sketch shown for a spatial, non-planar hexagon ABCDEF, where a 'spatial' parallelo-hexagon GHIJKL is formed1. Note that even though the hexagon GHIJKL is non-planar, it's opposite side are still equal and parallel (the length measurements are shown and the configuration can be rotated to align opposite sides and see that they are parallel).
Note: This dynamic 3D applet unfortunately no longer works on Internet Explorer or newer versions of Safari and Firefox, in which case it only gives a static image. The dynamic 3D applet also requires the downloading & installation of the free Cabri 3D Plug In, available at Windows (4 Mb) or Mac OS (13.4 Mb).
Spatial Hexagon Centroids
Basic manipulation: 1) Right click (or Ctrl + click) and drag to rotate the whole figure (glassball).
2) Click to select and hold down the left button to drag any of the vertices A, C or D of the hexagon ABCDEF.
Or click Summary of manipulation to open & resize a separate window with instructions.
1Acknowledgement: I'm indebted to Zalman Usiskin from the University of Chicago who in my ICME-12 paper in Seoul, Korea in July 2012, when I referred to this 2D generalization of Varignon's theorem, raised the question of whether it generalizes to 3D, and even more to Roger Howe from Yale University who afterwards showed me a simple vector proof, similar to that of Nick Lord that I'd seen before. But Roger Howe then pointed out that this proof immediately shows that it is also valid in 3D, since vectors are not dimension specific, something which I knew, but had somehow not thought of before! So this is another excellent example of the 'discovery' function of proof, whereby Polya's 'looking-back' strategy, done in the right way, produces an immediate generalization.
HTML export of second sketch by Cabri 3D. Download a 30 day Demo, or for more information about purchasing this software, go to Cabri 3D.
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Revised by Michael de Villiers, 23 July 2012; updated 4 January 2019; 16 June 2022.