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Heron quadrilaterals with sides in arithmetic or geometric progression

Published online by Cambridge University Press:  17 April 2009

R.H. Buchholz  and
J.A. MacDougall
R.H. Buchholz
Affiliation:
Department of Defence, Locked Bag 5076, Kingston ACT 2605 Australia e-mail: [email protected]
J.A. MacDougall
Affiliation:
Department of Mathematics, University of Newcastle, Callaghan NSW 2308 e-mail: [email protected]
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Abstract

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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
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 2 , April 1999 , pp. 263 - 269
Copyright
Copyright © Australian Mathematical Society 1999

References

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