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Heron triangles and elliptic curves

Published online by Cambridge University Press:  17 April 2009

Ralph H. Buchholz  and
Randall L. Rathbun
Ralph H. Buchholz
Affiliation:
Department of Defence, Defence Science and Technology Organisation, Locked Bag 5076, Kingston ACT 2604, Australia e-mail: [email protected]
Randall L. Rathbun
Affiliation:
403 Marcos St Apt C, San Marcos CA 92069–1509, United States of America e-mail: [email protected]
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Abstract

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In this paper we present a proof that there exist infinitely many rational sided triangles with two rational medians and rational area. These triangles correspond to rational points on an elliptic curve of rank one. We also display three triangles (one previously unpublished) which do not belong to any of the known infinite families.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 3 , December 1998 , pp. 411 - 421
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Buchholz, R.H., On triangles with rational altitudes, angle bisectors or medians, (Doctoral Dissertation) (Newcastle University, New South Wales, 1989).Google Scholar
[2]Buchholz, R.H. and Rathbun, R.L., ‘An infinite set of Heron triangles with two rational medians’, Amer. Math. Monthly 104 (1997), 107–115.CrossRefGoogle Scholar
[3]Cremona, J.E., Algorithms for modular elliptic curves (Cambridge University Press, Cambridge, 1992).Google Scholar