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.
Parallelo-2ngon formed by centroids
Explore
1) Drag any of the vertices of the hexagon, and ensure that you also drag it into concave & crossed cases.
2) Click on the 'Link to octogon centroids' button to navigate to a new sketch which shows the corresponding construction for a octogon.
3) Explore the new sketch by dragging, and ensure that you also drag it into concave & crossed cases.
Note: the general result stated above holds for vertex centroids; i.e. the balancing point of a polygon if equal point masses are placed at each of its vertices.
References
De Villiers, M. (2007). A hexagon result and its generalisation via proof. The Montana Mathematics Enthusiast, Vol. 4, No. 2 (June), pp. 188-192.
Lord, N. (2008). Maths bite: averaging polygons. The Mathematical Gazette, Vol. 92, No. 523 (March), p. 134.
Generalization to 3D
Perhaps unexpectedly, 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.)
Related Links
Napoleon's Regular Hexagon (Special case of above theorem)
The Center of Gravity of a Triangle (Rethinking Proof activity)
Kite Midpoints (generalizes to Varignon's theorem, Rethinking Proof activity)
Balancing Weights in Geometry as a Method of Discovery & Explanation
Point Mass (Vertex) Centroid (centre of gravity or balancing point) of Quadrilateral
A side trisection triangle concurrency
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Van Aubel Vertex Centroid & its Generalization
Geometry Loci Doodling of the Centroids of a Cyclic Quadrilateral
External Links
Centroid (geometric centre) (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created with JavaSketchpad, 2007, by Michael de Villiers, revised 23 July 2012; updated 4 January 2019; 16 June 2022; updated to WebSketchpad, 7 Jan 2026.