id: 05896059
dt: j
an: 2012b.00715
au: Nishiyama, Yutaka
ti: The beautiful geometric theorem of van Aubel.
so: Int. J. Pure Appl. Math. 66, No. 1, 71-80 (2011).
py: 2011
pu: Academic Publications, Sofia
la: EN
cc: G40 G70
ut: Van Aubel’s theorem; complex numbers; rotation; quadrilateral
ci: Zbl 1200.00021; ME 2010f.00600
li:
ab: 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}.
rv: Mowaffaq Hajja (Irbid)