Abstract
We solve a problem of Bill Sands, to find pairs of Heron triangles and rectangles, such as (5,5,6) & [2 × 6] or (13,20,21) & [6 × 21] which have a common area and a common perimeter. The original question was posed for right-angled triangles, but there are no nondegenerate such. There are infinitely many isosceles triangles and these have been exhibited by Guy. Here we solve the general problem; the triangle-rectangle pairs are parametrized by a family of elliptic curves.
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