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

Quadrilateral Balancing theorem
Given four points A, B, C, D, and four directly similar quadrilaterals AP₁P₂B, CQ₁Q₂B, CR₁R₂D, AS₁S₂D with respective vertex centroids P, Q, R, S. Let K, L, M and N be the midpoints of the segments P₁Q₂, Q₁R₂, R₁S₂ and S₁P₂ respectively (or of P₁S₂, S₁R₂, R₁Q₂ and Q₁P₂), and let V, W, X be the vertex 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)

Published Paper
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 by clicking on the 'Link to Similar Triangles' button in the ABOVE dynamic sketch; the picture below is static.

Directly Similar Triangles on Sides of Quadrilateral Theorem (1994)

1) 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.
2) However, this result has appeared earlier in Finney (1970) who references Van Aubel (1881) for the original result.
3) More recently, Humenberger (2026) revisited the problem with a spiral similarity proof, historical background & interesting special cases.

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 by clicking on the 'Link to Lemma Parallelograms' 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.
(Note: Hungerbühler (2025) has provided an interesting further generalization of the 'Directly Similar Triangles' Theorem which also suggests a further generalization of the Quadrilateral Balancing Theorem above. But that is left to the interested reader to explore further).

References
De Villiers, M. (1994/2009). Some Adventures in Euclidean Geometry (free download), Lulu Publishers.
De Villiers, M. (1997). The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections, in D. Schattschneider & J. King (Editors), Geometry Turned On!, MAA.
De Villiers, M. (2007/2008). A Question of Balance: An Application of Centroids, Mathematical Gazette, Nov 2007, pp. 525-528; March 2008, pp. 167-169.
Finney, R. L. (1970). Dynamic Proofs of Euclidean Theorems, Math Magazine, 43, pp. 177-185.
Humenberger, H. (27 Aug 2026). A parallelogram arising when erecting similar triangles on the sides of a quadrilateral — and corresponding historic notes. Mathematical Gazette, DOI: 10.1080/00255572.2026.2714216.
Hungerbühler, N. (2025). When Varignon meets Napoleon. Journal of Geometry, 116(2), 1-14. DOI: https://doi.org/10.1007/s00022-025-00746-9
Van Aubel, M.H. (1881). Question 56, Mathesis, no. 1, p. 167.

Related Links
Van Aubel's theorem and some Generalizations
Parallelogram Squares (Rethinking Proof activity, special case of Van Aubel)
An associated result of the Van Aubel configuration and some generalizations
Van Aubel Centroid & its Generalization
Triangle Centroids of a Hexagon form a Parallelo-Hexagon (A generalization of Varignon's Theorem)
Centroid Properties of a Bisect-diagonal Quadrilateral
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
A Fundamental Theorem of Similarity
Napoleon's Theorem (Rethinking Proof activity)
Napoleon's Theorem: Generalizations & Converses
Some Variations of Vecten configurations
Weighted Airport Problem
The Center of Gravity of a Triangle (Rethinking Proof activity - concurrency of medians, Ceva's theorem)
Balancing Weights in Geometry as a Method of Discovery & Explanation
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices
Point Mass (Vertex) Centroid of Quadrilateral
Three different centroids (balancing points) of a quadrilateral
A generalization of a Parallelogram Theorem to Parallelo-hexagons, Hexagons and 2n-gons in general
The Equi-partitioning Point of a Quadrilateral
More Properties of a Bisect-diagonal Quadrilateral

External Related Links
Similar Triangles on Sides of a Quadrilateral (Cut The Knot)
Squares on Sides of a Quadrilateral (Cut The Knot)
Van Aubel's theorem (Wikipedia)
Balanced areas in quadrilaterals - on the way to Anne's Theorem (PDF)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)