Van Aubel Centroid & its Generalization

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Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B, BQ₁Q₂C, CR₁R₂D, DS₁S₂A with respective vertex centroids P, Q, R, S. Further let F, G and V be the respective midpoints of the segments AC, BD and GF.
It is well-known and easy to prove geometrically that V is the vertex centroid (centre of mass/gravity) of equal point masses placed at the vertices of quadrilateral ABCD.
(Analytically, the vertex centroid of any polygon with equal point masses at its vertices can also be easily determined by taking the arithmetic means of the coordinates of the vertices of the polygon).

1) How has the placement of the directly similar triangles on the sides affected the centre of mass (gravity) of the whole configuration?
2) Click on the 'Show Coordinates' button to show the coordinates of A, B, C, D, P, Q, R, S, as well as calculations with them.
3) What do you notice about the coordinates of V and the arithmetic mean of the measured coordinates in 2)?
4) Explore your conjecture in 3) above by dragging the vertices of ABCD or P₁ or P₂.
5) Can you prove your conjecture? (For a hint, click on the 'Link to Proof' button, and on that page, click on the 'Show Proof Hint' button).

Van Aubel Centroid & its Generalization

Van Aubel Centroid Theorem
Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B, BQ₁Q₂C, CR₁R₂D, DS₁S₂A with respective vertex centroids P, Q, R, S. Further let F, G and V be the respective midpoints of the segments AC, BD and GF. Then the vertex centroid of the whole configuration coincides with V, the original vertex centroid of the quadrilateral ABCD.

Further Generalization
As can be seen in the theorem & its illustration above, the result is not really about directly similar quadrilaterals, but about directly similar triangles (with point masses at their respective apex vertices) constructed on the sides of a quadrilateral. This therefore leads us to the following neat generalization:
If directly similar triangles are placed on the sides of a polygon, then the vertex centroid of the whole configuration coincides with the original vertex centroid of the polygon.
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6) Click on the 'Link to Pentagon' button to navigate to a new sketch illustrating the above generalization for a pentagon.
7) Drag any of the red vertices in the figure to explore dynamically.
Challenge
Can you explain why (prove that) the above generalization is true?

Application
Apart from obviously applying to the standard version of Van Aubel's theorem with squares on the sides of a quadrilateral, the generalization above also applies to equilateral triangles erected on the sides of a triangle (Napoleon's theorem) or squares erected on the sides of a triangle (Vecten's configuration).

Published Paper
A joint paper The Vertex Centroid of a Van Aubel Result involving Similar Quadrilaterals and its Further Generalization by Hans Humenberger (University of Vienna) & myself about the above results has been published Open Source in IJMEST (July 2024) - all rights reserved.

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Michael de Villiers, created with WebSketchpad, 13 April 2024; updated 15 July 2024.

Van Aubel Centroid & its Generalization

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Van Aubel Centroid & its Generalization