  Starting from any given rational-sided, right triangle, for example the
$(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there
are infinitely many other rational-sided, right triangles of the same area. We
show further that the set of all such triangles of a given area is finitely
generated under our geometric construction. Such areas are known as "congruent
numbers" and have a rich history in which all the results in this article have
been proved and far more. Yet, as far as we can tell, this seems to be the
first exploration using this kind of geometric technique.
