## 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 as follows:
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.

#### Web Sketchpad .sketch_canvas { border: medium solid lightgray; display: inline-block; } 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.

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.

Further Generalizations: 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: My 2022 paper in the Int. Journal of Math Ed in Sci & Technol. discussing these results has been published online. The first 50 copies are free to download at: An associated result of the Van Aubel configuration and its generalization. (If free download copies are no longer available, contact me directly).