## Van Aubel's Theorem and some Generalizations

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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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.)

#### Please install Java (version 1.4 or later) to use JavaSketchpad applets.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 = ∠FWG. 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).

#### Please install Java (version 1.4 or later) to use JavaSketchpad applets.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.

#### Please install Java (version 1.4 or later) to use JavaSketchpad applets.A Generalization of Van Aubel's Theorem to Similar Parallelograms (1997)

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

Read my article in 2000 Generalizing Van Aubel using Duality which mentions further generalizations due to Chris Fisher.

Or an earlier 1998 one with similar rectangles or rhombi as special cases and proofs Dual Generalizations of Van Aubel's Theorem.

Also read John Silvester's excellent 2004 article on further extensions Extensions of a theorem of Van Aubel.

The website of Dick Klingens (in Dutch) provides 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.

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.