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.

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

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.

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.)

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