Hostname: page-component-788cddb947-kc5xb Total loading time: 0 Render date: 2024-10-11T19:06:45.335Z Has data issue: false hasContentIssue false

Properties of Pythagorean quadrilaterals

Published online by Cambridge University Press:  14 June 2016

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

Extract

There are many named quadrilaterals. In our hierarchical classification in [1, Figure 10] we included 18, and at least 10 more have been named, but the properties of the latter have only scarcely (or not at all) been studied. However, only a few of all these quadrilaterals are defined in terms of properties of the sides alone. Two well-known classes are the rhombi and the kites, defined to be quadrilaterals with four equal sides or two pairs of adjacent equal sides respectively. The orthodiagonal quadrilaterals are defined to have perpendicular diagonals, but an equivalent defining condition is quadrilaterals where the consecutive sides a, b, c, d satisfy a2 + c2 = b2 + d2. Then it is possible to prove that the diagonals are perpendicular and that no other quadrilaterals have perpendicular diagonals (see [2, pp. 13-14]). In the same way tangential quadrilaterals can be defined to be convex quadrilaterals where a + c = b + d. Starting from this equation, it is possible to prove that these and only these quadrilaterals have an incircle (since this equation is a characterisation of tangential quadrilaterals, see [3, pp. 65-67]).

Type
Research Article
Copyright
Copyright © Mathematical Association 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Josefsson, M., On the classification of convex quadrilaterals, Math. Gaz. 100 (March 2016) pp. 6885.CrossRefGoogle Scholar
2.Josefsson, M., Characterizations of orthodiagonal quadrilaterals, Forum Geom. 12 (2012) pp. 1325.Google Scholar
3.Andreescu, T. and Enescu, B., Mathematical Olympiad Treasures, Birkhäuser, Boston (2004).CrossRefGoogle Scholar
4.Maynard, P. and Leversha, G., Pythagoras’ theorem for quadrilaterals, Math. Gaz. 88 (March 2004) pp. 128130.CrossRefGoogle Scholar
5.Ivanoff, V. F., Pinzka, C. F. and Lipman, J., Problem E1376: Bretschneider's formula, Amer. Math. Monthly 67 (March 1960) p. 291.CrossRefGoogle Scholar
6.Loeffler, D., An extension of Ptolemy's theorem, Crux Math. 27 (September 2001) pp. 326327.Google Scholar
7.Harries, J., Area of a quadrilateral, Math. Gaz. 86 (July 2002) pp. 310311.CrossRefGoogle Scholar
8.Josefsson, M., Similar metric characterizations of tangential and extangential quadrilaterals, Forum Geom. 12 (2012) pp. 6377.Google Scholar
9.de Villiers, M., Some adventures in Euclidean geometry, Dynamic Mathematics Learning (2009).Google Scholar
10.Ostermann, A. and Wanner, G., Geometry by its history, Springer (2012).CrossRefGoogle Scholar
11.Posamentier, A. S. and Salkind, C. T., Challenging problems in geometry, Dover (1996).Google Scholar