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On the classification of convex quadrilaterals

Published online by Cambridge University Press:  14 March 2016

Martin Josefsson
Martin Josefsson*
Affiliation:
Västergatan 25d, 285 37 Markaryd, Sweden e-mail: [email protected]
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Extract

We live in a golden age regarding the opportunities to explore Euclidean geometry. The access to dynamic geometry computer programs for everyone has made it very easy to study complex configurations in a way that has never been possible before. This is especially apparent in the study of triangle geometry, where the flow of new problems, properties, and papers is probably the highest it has ever been in the history of mathematics. Even though it has increased a bit in recent years, the interest in quadrilateral geometry is significantly lower. Why are triangles so much more popular than quadrilaterals? In fact, we think it would be more logical if the situation were reversed, since there are so many classes of quadrilaterals to explore. This is the primary reason we think that quadrilaterals are a lot more interesting to study than triangles.

Type
Research Article
Information
The Mathematical Gazette , Volume 100 , Issue 547 , March 2016 , pp. 68 - 85
Copyright
Copyright © Mathematical Association 2016 

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References

1.Josefsson, M., Characterizations of bicentric quadrilaterals, Forum Geom. 10 (2010) pp. 165–173.Google Scholar
2.Josefsson, M., More characterizations of tangential quadrilaterals, Forum Geom. 11 (2011) pp. 65–82.Google Scholar
3.Josefsson, M., When is a tangential quadrilateral a kite?, Forum Geom. 11 (2011) pp. 165–174.Google Scholar
4.Josefsson, M., Characterizations of orthodiagonal quadrilaterals, Forum Geom. 12 (2012) pp. 13–25.Google Scholar
5.Josefsson, M., Similar metric characterizations of tangential and extangential quadrilaterals, Forum Geom. 12 (2012) pp. 63–11.Google Scholar
6.Josefsson, M., Characterizations of trapezoids, Forum Geom. 13 (2013) pp. 23–35.Google Scholar
7.Josefsson, M., Properties of equidiagonal quadrilaterals, Forum Geom. 14 (2014) pp. 129–144.Google Scholar
8.Josefsson, M., A few inequalities in quadrilaterals, International Journal of Geometry 4 no. 1 (2015), pp. 11–15.Google Scholar
9.Joyce, D. E., Euclid's Elements, Book I (1996), http://aleph0.clarku.edu/∼djoyce/java/elernents/bookl/bookl.htmlGoogle Scholar
10.Athanasopoulou, A., An inquiry approach to the study of quadrilaterals using Geometer's Sketchpad: a study with pre-service and in-service teachers, Proquest, Umi Dissertation Publishing (2011).Google Scholar
11.Puttaswamy, T. K., Mathematical achievements of pre-modern Indian mathematicians, Elsevier Science Publishing (2012).Google Scholar
12.Usiskin, Z. and Griffin, J., The classification of quadrilaterals. A study of definition, Information Age Publishing (2008).Google Scholar
13.Wheeler, R. F., Quadrilaterals, Math. Gaz. 42 (December 1958) pp. 275–276.CrossRefGoogle Scholar
14.Fritsch, R., Bemerkungen zur Viereckslehre, Schriften der Sudetendeutschen Akademie der Wissenschaften und Künste, 19 (1998) pp. 69–94, also available at http://www.mathematik.uni-muenchen.de/∼fritsch/viereck.pdfGoogle Scholar
15.Radic, M., Kaliman, Z. and Kadum, V., A condition that a tangential quadrilateral is also a chordal one, Math. Commun. 12 (2007) pp. 33–52.Google Scholar
16.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.pdfGoogle Scholar
17.Olson, A. T., Exploring skewsquares, Mathematics Teacher 69 (November 1976) pp. 570–573.Google Scholar
18.de Villiers, M., An extended classification of quadrilaterals (1996), available at http://mysite.mweb.co.za/residents/profmd/quadclassify.pdfGoogle Scholar
19.de Villiers, M., Some adventures in Euclidean geometry, Dynamic Mathematics Learning (2009).Google Scholar
20.de Villiers, M., Hierarchical quadrilateral tree (2011), available at http://frink.machighway.com/∼dynamicm/quad-tree-web.htmlGoogle Scholar
21.Rusczyk, R., Art of problem solving: classifying quadrilaterals (2012), available at https://www.youtube.com/watch?v=xoXLbOWRBMAGoogle Scholar
22.Keyton, M., Trapezoids Talk (2008), available at http://www.fresnarus.com/keyton/Mathematical%20Talks%20and%20Papers/Trapezoids%20Talk.pdfGoogle Scholar
23.Chase, Mr., Random Walks. Mr. Chase blogs about math (2011), available at http://mrchasemath.wordpress.com/2011/02/03/why-i-hate-the-definition-of-trapezoids/CrossRefGoogle Scholar
24.Craine, T. V. and Rubenstein, R. N., A quadrilateral hierarchy to facilitate learning in geometry, Mathematics Teacher 86 (January 1993) pp. 30–36.CrossRefGoogle Scholar
25.de Villiers, M., Using dynamic geometry to expand mathematics teachers’ understanding of proof, Int. J. Math. Educ. Sci. Technol. 35 5 (2004), pp. 703–724.CrossRefGoogle Scholar
26.Hardy, G. H., A mathematician's apology, Cambridge University Press (1940).Google Scholar
27.Josefsson, M., Maximal area of a bicentric quadrilateral, Forum Geom. 12 (2012) pp. 237–241.Google Scholar
28.Josefsson, M., Minimal area of a bicentric quadrilateral, Math. Gaz. 99 (July 2015) pp. 237–242.Google Scholar
29.Zirkel, W., Das Ankreisviereck, Praxis der Mathematik 5 (1963) pp. 129–132.Google Scholar
30.Roller, J., Hierarchie der Vierecke (2007), available at http://www.mathematische-basteleien.de/viereck.htmGoogle Scholar