Rational triangles with equal area
The key result is that, given a rational triangle \((p,q,r)\) with area \(A\), there are infinitely many rational triangles with area \(A\) and one side equal to \(r\), unless \(p=q\) or one of two other conditions hold. This is obtained by studying the surface defined by \[ C=-(p+q+r)(p+q-r)(p-q+r)(-p+q+r) \] with \(C\) a negative real, which can be transformed into an elliptic curve whose group structure is analyzed. The author studies in detail triangles satisfying one of the exceptional conditions, and ends up with the result that infinitely many triangles can be constructed sharing a side and the area with a given non-equilateral triangle. Finally, he looks at when the elliptic curves are isomorphic. There are suggestions for further investigation.
scientific article; zbMATH DE number 1144094