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