In the dynamic geometry activity at Parallelogram Squares we discovered as a generalization of the 'parallelogram squares' theorem, the remarkable theorem (named after Van Aubel) that the quadrilateral EFGH formed by the centres of squares constructed on the sides of any quadrilateral ABCD has equal and perpendicular diagonals. Such a quadrilateral is called a mid-square quadrilateral since the midpoints of its sides form a square.
What if?
Let's suppose quadrilateral ABCD is constrained by a fixed circle, i.e. its vertices lie on a circle of fixed radius. What would be the maximum area of EFGH, and when would that occur?
Maximizing the Area: Explore
The dynamic sketch below shows the areas of both EFGH and ABCD. Drag the vertices A, B, C and D until EFGH has maximum area.
1) What do you notice? Formulate a conjecture.
Optimizing the Area of Van Aubel's Quadrilateral
Challenge
2) Can you prove your conjecture in 1) above? Can you prove it in more than one way?
Dual Result
Let's suppose ABCD circumscribes a fixed circle, i.e. its sides touch a circle of fixed radius. What would be the minimum area of EFGH, and when would that occur?
Minimizing the Area: Explore
Click on the 'Link to minimum' button to navigate to a new dynamic sketch.
The new sketch shows the areas of both EFGH and ABCD. Drag the tangent points P, Q, R and S until EFGH has minimum area.
3) What do you notice? Formulate a conjecture.
4) Can you prove your conjecture?
(Note: The proof of this result is quite challenging & requires advanced mathematical techniques.)
Related Problem
We can investigate a similar 'optimizing area' question in relation to Napoleon's Theorem. For a dynamic sketch go to: Optimizing the Area of Napoleon's Triangle.
Submitted paper
A paper "Napoleon Triangles and Van Aubel Quadrilaterals – Area Formulas and Optimization Problems" by Hans Humenberger (University of Vienna) and myself discussing this problem has been submitted for consideration for publication. All rights reserved.
Related Links
Parallelogram Squares (Rethinking Proof activity, generalizes to Van Aubel)
Van Aubel's Theorem and some Generalizations
Napoleon's Theorem (Rethinking Proof activity)
Optimizing the Area of Napoleon's Triangle
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization
Minimum Area of Miquel Circle Centres Triangle
A generalization of Neuberg's Pedal Theorem
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
Attached Regular Pentagons form Congruent Equilateral Triangles
Bride's Chair Concurrency & Generalization
Some Variations of Vecten configurations
Kite Midpoints (Rethinking Proof activity, generalizes to Varignon & orthodiagonal quad)
Isosceles Trapezoid Midpoints (Rethinking Proof activity, generalizes to Varignon & equidiagonal quad)
External Links
Van Aubel's theorem (Wikipedia)
Mid-square quadrilateral (Wikipedia)
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.)
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Created by Michael de Villiers with WebSketchpad, 18 April 2026; updated 22 April 2026.