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Properties of bisect-diagonal quadrilaterals

Published online by Cambridge University Press:  15 June 2017

Martin Josefsson
Martin Josefsson*
Affiliation:
Västergatan 25d, 285 37 Markaryd, Sweden e-mail: martin.markaryd@hotmail.com

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The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?

We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties of convex bisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 551 , July 2017, pp. 214 - 226
Copyright
Copyright © Mathematical Association 2017 

References

1. Graumann, G., Investigating and ordering Quadrilaterals and their analogies in space — problem fields with various aspects, ZDM 37 (3) (2005) pp. 190–198, also available at: http://subs.emis.de/journals/ZDM/zdm053a8.pdf Google Scholar
2. Ramachandran, A., The four-gon family tree, At Right Angles 1 (1) (2012) pp. 53–57, also available at http://www.teachersofindia.org/sites/default/files/12_four_gon_family_tree.pdf Google Scholar
3. de Villiers, M., Some adventures in Euclidean geometry, Dynamic Mathematics Learning (2009).Google Scholar
4. Coxeter, H. S. M. and Greitzer, S. L., Geometry revisited, Math. Ass. Amer. (1967).Google Scholar
5. Pop, O. T., Minculete, N. and Bencze, M., An introduction to quadrilateral geometry, Editura Didactică şi Pedagogică, Bucharest, Romania (2013).Google Scholar
6. Alsina, C. and Nelsen, R. B., Charming proofs – a journey into elegant mathematics, Math. Ass. Amer. (2010).CrossRefGoogle Scholar
7. Josefsson, M., Characterizations of trapezoids, Forum Geom. 13 (2013) pp. 23–35.Google Scholar
8. Josefsson, M., On the classification of convex quadrilaterals, Math. Gaz. 100 (March 2016) pp. 68–85.CrossRefGoogle Scholar
9. Josefsson, M., Characterizations of orthodiagonal quadrilaterals, Forum Geom. 12 (2012) pp. 13–25.Google Scholar