The dynamic geometry activity below is from my book Rethinking Proof (free to download).
Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Parallelogram Squares.
Prerequisites: Knowledge of side-angle-side condition for congruent triangles; symmetry properties of parallelograms, rhombuses, and squares, as well as their hierarchical relationships.
Parallelogram Squares
Notes
1) Click on the 'Show EFGH' button to show the quadrilateral formed by the centres of the squares.
2) What type of quadrilateral is EFGH?
3) To measure the sides of EFGH, use the 'Distance' tool on the left.
4) Alternatively, click on the 'Show sides & angles of EFGH' button to show the sides & angle measurements.
5) As directed in the accompanying worksheet (see link at the top)), navigate to a new sketch by clicking on the 'Link to Proving' button that will provide you with some hints for a proof.
Further Exploration
6) To explore what type of quadrilateral EFGH is formed when squares are constructed of an arbitrary quadrilateral ABCD, click on the 'Link to squares' button.
7) The theorem in 6) above is called Van Aubel's theorem (click the link for more info).
8) Navigate to dynamic sketches of even further generalizations by clicking on the corresponding 'Link to' buttons.
9) For other generalizations & variations of Van Aubel's theorem use the link in 7) or those in the Related Links below.
References
De Villiers, M. (1997). The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections. Chapter in Schattschneider, D. & King, J. Geometry Turned On! Washington: MAA, pp. 15-24.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Some Related Links
Van Aubel's Theorem and some Generalizations
Twin Circles for a Van Aubel configuration involving Similar Parallelograms
An associated result of the Van Aubel configuration and some generalizations
The Vertex Centroid of a Van Aubel Result involving Similar Quadrilaterals and its Further Generalization
Dào Thanh Oai's Perpendicular Lines Van Aubel Generalization
Some Corollaries to Van Aubel Generalizations
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
A Van Aubel like property of an Equidiagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
A Fundamental Theorem of Similarity
Finsler-Hadwiger Theorem plus Gamow-Bottema's Invariant Point
Pompe's Hexagon Theorem
Sum of Two Rotations Theorem
Some Variations of Vecten configurations
A Vecten area variation (Cross's theorem) & generalizations to quadrilaterals
Napoleon's Theorem: Generalizations, Variations & Converses
Some External Links
De stelling van Van Aubel en algemenisering daarvan (in Dutch)
Van Aubel's theorem (Wikipedia)
van Aubel's Theorem (Wolfram MathWorld)
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.)
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Michael de Villiers, created with WebSketchpad, 24 August 2025.