From mark@csc.albany.edu Wed Jan 21 09:10:37 1998
Date: Wed, 21 Jan 1998 01:46:08 -0500
From: Mark Steinberger
Reply-To: Abstracts from the New York Journal of Mathematics
To: NYJMTH-A@CNSIBM.ALBANY.EDU
Subject: Rational Triangles with Equal Area
New Release from the New York Journal of Mathematics
Title: Rational Triangles with Equal Area
Author: David J. Rusin
Date of Publication: January 21, 1998
Keywords: rational triangles, Heron surfaces, elliptic curves
Subject Classification: 11G05
Abstract:
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.
This paper's home page is http://nyjm.albany.edu:8000/j/1998/4-1.html.
It contains links to the various graphical formats of the paper.