Rational Triangles with Equal Area

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. In this paper a rational triangle means a triangle in the plane whose sides are of rational length. We consider the set of such triangles with a given area A (necessarily the square root of a positive rational number). We show there are infinitely many rational triangles for each such area A (Theorem 1). This was previously known for rational A; the known proofs are, like ours, algebraic. So we ask whether there is a simple geometric manipulation of a rational triangle which yields infinitely many others with the same area. The answer is affirmative: these triangles may be selected to have a side in common with the original triangle, unless that triangle is equilateral (Theorems 2 and 4). The proof of this stronger statement combines simple calculation with Mazur’s characterization of torsion in rational elliptic curves. We describe the construction of new rational triangles from old as a dynamical system. Three families of triangles (including the isosceles ones) play a special role; they are treated in depth (Theorem 3). Finally we discuss the isomorphism classes of the elliptic curves involved (Theorem 5). Fermat (cf. [3]) showed that it is possible for infinitely many rational triangles to have the same area; all the triangles in his example are right triangles. When the triangles are all assumed to be right triangles, this subject is the “congruent number problem” (see, e.g., [5]): there are indeed infinitely many triangles with a given area if there are any at all; they may all be constructed in a simple way from just a few; and (assuming some unproven conjectures in algebraic geometry) there is a characterization by Tunnell [9] of which numbers do indeed arise as areas of rational right triangles. We seek results parallel to these in the general case. Received November 21, 1996. Mathematics Subject Classification. 11G05.