**Quadrilateral Balancing theorem**: Given four points *A*, *B*, *C*, *D*, and four directly similar quadrilaterals *AP _{1}P_{2}B, CQ_{1}Q_{2}B*,

(i)

(ii)

(iii)

Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem (2007)

For a proof of the above, read my 2007/2008 article in the *Mathematical Gazette*: *A Question of Balance: An Application of Centroids*.

**Similar Triangles on Sides of Quadrilateral Theorem**: Given a quadrilateral *ABCD* with similar triangles *PBA*, *QBC*, *RDC* and *SDA* constructed on the sides so that say the first and third are exterior to the quadrilateral, while the second and the fourth are interior to the quadrilateral, then quadrilateral *PQRS* is a parallelogram.

**Important**: To view & manipulate the *dynamic version* of this result, navigate to it using the appropriate button in the **ABOVE** dynamic sketch; the picture below is static.

Directly Similar Triangles on Sides of Quadrilateral Theorem (1994)

A proof of the above result appears on pp. 145-148 of the (1994/2009) edition of my book Some Adventures in Euclidean Geometry as well as in my paper "*The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections*", a chapter in the 1997 MAA book, *Geometry Turned On!*, edited by Doris Schattschneider & Jim King.

**Lemma: A Trio of Parallelograms**: Given two parallelograms *ABCD* and *IJKL*, the midpoints *E*, *F*, *G*, *H* of the segments *AI*, *BJ*, *CK*, *DL* form another parallelogram. More-over, the centre *Y* of *EFGH* is the midpoint of *XZ* where *X* and *Z* are respective centres of *ABCD* and *IJKL*. (The diagram shows a case with the parallelograms both labelled anticlockwise, but both lemma and proof work equally well in all cases.)

**Important**: To view & manipulate the *dynamic version* of this result, navigate to it using the appropriate button in the **ABOVE** dynamic sketch; the picture below is static.

Lemma: A Trio of Parallelograms (2007)

The lemma above, together with the Directly Similar Triangles Theorem, was used to prove the Quadrilateral Balancing Theorem in my 2007/2008 article in the *Mathematical Gazette*: *A Question of Balance: An Application of Centroids*.

**Van Aubel's theorem**: Here is a link to Van Aubel's theorem and some Generalizations.

**An Associated Result of Van Aubel & its Generalization (2022)**:
For an interesting associated result of Van Aubel's theorem also involving similar quadrilaterals on the sides, go to An associated result of the Van Aubel configuration and some generalizations.

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Created 2013 by Michael de Villiers. Most recent updates, 16 March 2020; 19 February 2022.