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    IZADI, FARZALI KHOSHNAM, FOAD MOODY, DUSTIN and ZARGAR, ARMAN SHAMSI  2014. ELLIPTIC CURVES ARISING FROM BRAHMAGUPTA QUADRILATERALS. Bulletin of the Australian Mathematical Society, Vol. 90, Issue. 01, p. 47.


    2013. Finite Sums of the Alcuin Numbers. Mathematics Magazine, Vol. 86, Issue. 4, p. 280.


    Buchholz, Ralph H. and MacDougall, James A. 2008. Cyclic polygons with rational sides and area. Journal of Number Theory, Vol. 128, Issue. 1, p. 17.


    Chisholm, C. and MacDougall, J.A. 2008. Rational tetrahedra with edges in geometric progression. Journal of Number Theory, Vol. 128, Issue. 2, p. 251.


    Chisholm, C. and MacDougall, J.A. 2006. Rational and Heron tetrahedra. Journal of Number Theory, Vol. 121, Issue. 1, p. 153.


    Chisholm, C. and MacDougall, J.A. 2005. Rational tetrahedra with edges in arithmetic progression. Journal of Number Theory, Vol. 111, Issue. 1, p. 57.


    ×
  • Bulletin of the Australian Mathematical Society, Volume 59, Issue 2
  • April 1999, pp. 263-269

Heron quadrilaterals with sides in arithmetic or geometric progression

  • R.H. Buchholz (a1) and J.A. MacDougall (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700032883
  • Published online: 01 April 2009
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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]R.A. Beauregard and E.R. Suryanarayan , ‘Arithmetic triangles’, Math. Mag. 70 (1997), 105115.

[2]R.A. Beauregard and E.R. Suryanarayan , ‘The Brahmagupta triangles’, College Math. J. 29 (1998), 1317.

[7]J.H. Silverman and J. Tate , 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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