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The beautiful geometric theorem of van Aubel. (English)
Int. J. Pure Appl. Math. 66, No. 1, 71-80 (2011).
Van Aubel’s theorem states that if squares are erected outwardly on the sides of an arbitrary convex quadrilateral, then the line segment joining the centers of two opposite squares is perpendicular and equal in length to the line segment joining the centers of the other two. In the paper under review, the author narrates how he came across this theorem, how he was struck by its beauty, how he was intrigued by a proof using complex numbers, and how he endeavored to understand the theorem fully and to supply his own proofs. He presents proofs, not claimed to be necessarily new, that make use of complex numbers, vectors, and elementary geometry, and he tells how he felt the need of ‒ and how he found ‒ the appropriate dynamic software. He also presents a personal perspective of why geometry was removed from high school mathematics curricula in Japan and why these curricula may be reformed to accommodate more geometry. This reviewer would like to join the author in this appeal for such a reform worldwide. Beautiful discussions and proofs of van Aubel’s theorem and other related theorems can be found in Chapter 8 of the delightful book [Charming proofs. A journey into elegant mathematics. The Dolciani Mathematical Expositions 42. Washington, DC: The Mathematical Association of America (MAA) (2010; Zbl 1200.00021, ME 2010f.00600)] by {\it C. Alsina} and {\it R. B. Nelsen}.
Reviewer: Mowaffaq Hajja (Irbid)
Classification: G40 G70
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