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Heron quadrilaterals with sides in arithmetic or geometric progression

  • R.H. Buchholz (a1) and J.A. MacDougall (a2)
Abstract

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.

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References
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[1]Beauregard R.A. and Suryanarayan E.R., ‘Arithmetic triangles’, Math. Mag. 70 (1997), 105115.
[2]Beauregard R.A. and Suryanarayan E.R., ‘The Brahmagupta triangles’, College Math. J. 29 (1998), 1317.
[3]Eves H., An introduction to the history of mathematics, 5th ed (Saunders College Publishing, Philadephia, PA, 1983).
[4]Fleenor C.R., ‘Heronian triangles with consecutive integer sides’, J. Rec. Math. 28 (19961997), 113115.
[5]MacDougall J.A., ‘Heron triangles with sides in arithmetic progression’, (submitted).
[6]Mordell L.J., Diophantine equations (Academic Press, London, 1969).
[7]Silverman J.H. and Tate J., Rational points on elliptic curves (Springer-Verlag, Berlin, Heidelberg, New York, 1992).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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