Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-27T04:43:58.871Z Has data issue: false hasContentIssue false

Heron quadrilaterals with sides in arithmetic or geometric progression

Published online by Cambridge University Press:  17 April 2009

R.H. Buchholz
Affiliation:
Department of Defence, Locked Bag 5076, Kingston ACT 2605 Australia e-mail: ralph@defcen.gov.au
J.A. MacDougall
Affiliation:
Department of Mathematics, University of Newcastle, Callaghan NSW 2308 e-mail: mmjam@cc.newcastle.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Beauregard, R.A. and Suryanarayan, E.R., ‘Arithmetic triangles’, Math. Mag. 70 (1997), 105115.CrossRefGoogle Scholar
[2]Beauregard, R.A. and Suryanarayan, E.R., ‘The Brahmagupta triangles’, College Math. J. 29 (1998), 1317.CrossRefGoogle Scholar
[3]Eves, H., An introduction to the history of mathematics, 5th ed (Saunders College Publishing, Philadephia, PA, 1983).Google Scholar
[4]Fleenor, C.R., ‘Heronian triangles with consecutive integer sides’, J. Rec. Math. 28 (19961997), 113115.Google Scholar
[5]MacDougall, J.A., ‘Heron triangles with sides in arithmetic progression’, (submitted).Google Scholar
[6]Mordell, L.J., Diophantine equations (Academic Press, London, 1969).Google Scholar
[7]Silverman, J.H. and Tate, J., Rational points on elliptic curves (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar