 

David J. Rusin
Rational Triangles with Equal Area


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


Links
Maple input file used in the paper
The APECS package


Author information
Department of Mathematical Sciences, Northern Illinois Univ., DeKalb IL, 60115, USA
http://www.math.niu.edu/~rusin

