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 *AP _{1}P_{2}B*,

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).

**Related Links**

Van Aubel's Theorem and some Generalizations

Van Aubel Centroid & its Generalization

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

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Created with *WebSketchpad* 18 May 2021 by Michael de Villiers: updated 19 February 2022; 15 July 2024.