New York Journal of Mathematics
Volume 4 (1998) 1-15


David J. Rusin

Rational Triangles with Equal Area

Published: January 21, 1998
Keywords: rational triangles, Heron surfaces, elliptic curves
Subject: 11G05

We consider the set of triangles in the plane with rational sides and a given area A. We show there are infinitely many such triangles for each possible area A. We also show that infinitely many such triangles may be constructed from a given one, all sharing a side of the original triangle, unless the original is equilateral. There are three families of triangles (including the isosceles ones) for which this theorem holds only in a restricted sense; we investigate these families in detail. Our explicit construction of triangles with a given area may be viewed as a dynamical system in the plane; we consider its features as such. The proofs combine simple calculation with Mazur's characterization of torsion in rational elliptic curves. We discuss the isomorphism classes of the elliptic curves involved.


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Department of Mathematical Sciences, Northern Illinois Univ., DeKalb IL, 60115, USA