## Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem

**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, *CR*_{1}R_{2}D, *AS*_{1}S_{2}D with respective centroids *P*, *Q*, *R*, *S*. Let *K*, *L*, *M* and *N* be the midpoints of the segments *P*_{1}Q_{2}, *Q*_{1}R_{2}, *R*_{1}S_{2} and *S*_{1}P_{2} respectively (or of *P*_{1}S_{2}, *S*_{1}R_{2}, *R*_{1}Q_{2} and *Q*_{1}P_{2}), and let *V*, *W*, *X* be the centroids of the quadrilaterals *ABCD*, *PQRS*, *KLMN* respectively.

Then: (i) *PQRS* is a parallelogram: (ii) *KLMN* is a parallelogram: and (iii) *W* is the midpoint of the segment *VX*.

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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.

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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.)

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Lemma: A Trio of Parallelograms (2007)

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

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

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By Michael de Villiers. Most recent update, 13 July 2013.