Van Aubel's Theorem and some Generalizations

Van Aubel
Henri van Aubel (1830-1906)

Van Aubel's theorem: If squares are constructed on the sides of any quadrilateral, then the segments connecting the centres of the squares on opposite sides are perpendicular and of equal length.

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Van Aubel's Theorem

Historical Note: Van Aubel taught pre-university mathematics at the Koninklijk Atheneum in Antwerpen ( Belgium) and it seems that his theorem first appeared in Nouvelles Corresp. Mathematique 4 (1878), pp 40-44.

Read Yutaka Nishiyama's paper in the 2011 issue of the International Journal of Pure and Applied Mathematics at: The beautiful geometric theorem of Van Aubel.

A Generalization of Van Aubel's Theorem to Similar Rectangles (1994 or earlier)

If similar rectangles with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD as shown, then ∠FOG = 90o and EG/FH = XA/AB. Further, if J, K, L and M are the midpoints of the dashed segments shown, then KM = JL, and the angle of JL and KM equals the angle of the diagonals of the rectangles. (Also note that lines EG, FH, KM and JL are concurrent in O, and that the line FH bisects ∠KOL, etc.)

Important: To view & manipulate the dynamic version of this result, navigate to it using the appropriate button in the ABOVE dynamic sketch; the picture below is static.

similar rectangles

A Generalization of Van Aubel's Theorem to Similar Rectangles

A Generalization of Van Aubel's Theorem to Similar Rhombi (1994)

If similar rhombi with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD as shown, then EG = FH and ∠FWG = ∠ABY. Further, if J, K, L and M are the midpoints of the dashed segments shown, then KM/JL = YA/XB, and the angle of JL and KM equals the angle of the diagonals of the rhombi (= 90o).

Important: To view & manipulate the dynamic version of this result, navigate to it using the appropriate button in the ABOVE dynamic sketch; the picture below is static.

similar rhombi

A Generalization of Van Aubel's Theorem to Similar Rhombi (1994)

A Generalization of Van Aubel's Theorem to Similar Parallelograms (Hessel Pot, 1997)

If similar parallelograms with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD as shown, then FH/EG = XY/YB , and the angle of EG and FH equals the angle of the sides of the parallelograms. Further, if J, K, L and M are the midpoints of the dashed segments shown, then KM/JL = YA/XB, and the angle of JL and KM equals the angle of the diagonals of the parallelograms.

Important: To view & manipulate the dynamic version of this result, navigate to it using the appropriate button in the ABOVE dynamic sketch; the picture below is static.

similar rhombi

A Generalization of Van Aubel's Theorem to Similar Parallelograms (1997)

Some Corollaries of Van Aubel Generalizations

For some interesting corollaries, click on Some Corollaries of Van Aubel Generalizations to open them in a new window.

Some Articles

Read my chapter in Schattschneider, D. & King, J. (1997). Geometry Turned On! Washington: MAA, pp. 15-24. The Role of Proof in Investigative, Computer-based Geometry: Some Personal Reflections (1997).

Read my 1998 article in the "Mathematical Gazette", also about similar rectangles or similar rhombi on the sides of a quadrilateral, but different proofs to those in the chapter above Dual Generalizations of Van Aubel's Theorem.

My "Mathematics Magazine" article in 2000 Generalizing Van Aubel using Duality, which mentions, without proof, the further generalization to similar parallelograms, and even further generalizations due to Chris Fisher.

John Silvester's excellent 2006 "Mathematical Gazette" article gives different proofs and some further Extensions of a theorem of Van Aubel.

The website of Dick Klingens (in Dutch) provides proofs, and a related result for congruent rectangles on the opposite sides of a cyclic quadrilateral with interactive Cabri applets and proofs. De Stelling van Van Aubel en algemenisering daarvan.

For another 'Van Aubel-like' theorem discovered in 2007 with directly similar quadrilaterals on the sides of a quadrilateral and their centroids go to Quadrilateral Balancing Theorem.

In 2011, Dick Klingens posed an interesting problem of two intersecting circles or two adjacent isosceles triangles, which turns out to be a special case of the 'similar rectangles' generalization of Van Aubel.

In 2019, Dario Pellegrinetti found an intriguing 6-point circle in relation to the basic Van Aubel configuration of squares constructed on the sides of a quadrilateral. Read his paper at: The Six-Point Circle for the Quadrangle.

Historical note: Using dynamic geometry, I first discovered and then proved the similar rectangles and rhombi generalizations of Van Aubel in the early 1990's. Subsequently I published it in the 1st draft in 1994 of my "Some Adventures in Euclidean Geometry" book as well as in a 1997 MAA book "Geometry Turned On!" and with different proofs in a 1998 paper in the Mathematical Gazette. In subsequent communication with Hessel Pot from Woerden in the Netherlands in 1997, he mentioned that he had seen the similar rectangles generalization before, but not the similar rhombi one. He then also pointed out the neat further generalization to similar parallelograms on the sides.



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First posted 26 March 2009, by Michael de Villiers. Most recent update, 1 October 2020.