Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral

The respective centroids C', D', A' and B' of triangles ABD, ABC, BCD and CDA form a quadrilateral A'B'C'D', similar to the original and scale factor -1/3 (a halfturn and reduction by 1/3), with lines AA', BB', CC' and DD' concurrent in G. Then this point of concurrency G (centre of similarity between ABCD and A'B'C'D') is defined as the centroid of the quadrilateral.

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Point Mass Centroid (centre of gravity) of Quadrilateral

Can you prove this result? If stuck, see my paper at Generalizations involving maltitudes. This concurrency (centroid) result generalizes to any polygon, and is applied in another, interesting related generalization at Octagon Centroids form Parallelo-Octagon as well as in Generalizing the Nagel line to Circumscribed Polygons by Analogy

In the physical, real world context, this centroid is the centre of gravity or balancing point of equal point masses placed at the vertices of a polygon. Click on the Show Varignon Parallelogram button to see why this is the case for a quadrilateral - for example the masses at the vertices can be replaced by point masses at the midpoints of the sides, which in turn will balance at the centre of the parallelogram (the intersection of its diagonals).

Note that the centroid of a cardboard quadrilateral (a planar quadrilateral of uniform density), unlike the case for a triangle, does NOT always coincide with the point mass centroid illustrated above. See for example, the dynamic geometry sketch at Centroid of Cardboard Quadrilateral


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Michael de Villiers, 6 April 2010.