An associated result of the Van Aubel configuration and some generalizations

The following result associated with Van Aubel's quadrilateral theorem (1878) is apparently not as well-known as the theorem itself:
If squares are constructed on the sides of any quadrilateral, then the respective midpoints F and G, of the diagonals AC and BD of ABCD, and the respective midpoints I and H of the segments connecting the centres of the squares on opposite sides form a square GHFI (Tabov, 1994).
Tabov, J. (1996). Solutions to a Posed Problem, Mathematics & Informatics Quarterly, 6:4, pp. 213-214.

Here the theorem is first generalized to similar quadrilaterals, and then to similar triangles, all placed exterior or interior on the sides of a quadrilateral.

1a. Van Aubel Associated Similar Quadrilateral Generalization: Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B, BQ₁Q₂C, CR₁R₂D, DS₁S₂A with respective centroids P, Q, R, S. Further let F, G, H, I be the midpoints of the segments AC, BD, QS, PR respectively. Then GHFI is a parallelogram.

Van Aubel Associated Similar Quadrilateral Generalization

Some special cases: To view & manipulate some special cases of this result, navigate to it using the appropriate button in the ABOVE dynamic sketch.

1b. Van Aubel Associated Similar Quadrilateral Variation: Two other different placements of the similar quadrilaterals on the sides of a quadrilateral, together with some special cases, can be viewed & manipulated at Differently Placed Similar Quadrilaterals. For the one placement of similar quadrilaterals in this link GHFI is a kite, and in the other, a placement of similar parallelograms, a cyclic quadrilateral.

2. Further Generalizations to Similar Triangles: By considering two (possibly different) pairs of directly similar triangles on opposite sides of quadrilateral ABCD we obtain three further generalizations as can be viewed & manipulated at Similar Triangles. For different placements of pairs of similar triangles on the opposite sides of ABCD in this link, GHFI respectively is: 1) a quadrilateral with opposite angles equal, 2) a kite, and 3) a cyclic quadrilateral.

Published paper
De Villiers, M. (2023). An associated result of the Van Aubel configuration and its generalization. International Journal of Mathematical Education in Science and Technology, Vol 54, no 3, pp. 462-472.

Related Links
An associated result of the Van Aubel configuration and its generalization: Different Quadrilateral Arrangements
Some further generalizations of an associated result of the Van Aubel configuration using pairs of similar triangles
A Fundamental Theorem of Similarity
Van Aubel Centroid & its Generalization
Van Aubel's Theorem and some Generalizations
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Generalizations of a theorem of Sylvester (about forces acting on a point in a triangle)
Napoleon's Theorem: Generalizations & Converses
Some Variations of Vecten configurations
Parallelogram Squares (Rethinking Proof activity)

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