For all such right triangle, there exist a positive rational number $k$ that the length of two legs of the triangle is equal to $k\times (m^2-n^2)$ and $k\times (2mn)$ for a pair of positive integers $(m,n)$. Note that $mn(m^2-n^2)k^2=S$ We choose the right triangle that legs length equal to $h\times (x^2-y^2)$ and $h\times (2xy)$ where $$h=\frac{8k}{|m^6-5m^4n^2-5m^2n^4+n^6|},x=\Big( \frac{m^2+n^2}{2}\Big)^2 ,y=mn(m^2-n^2).$$

Note that $$m^6-5m^4n^2-5m^2n^4+n^6=(m^2+n^2)(m^4-6m^2n^2+n^4) \neq 0$$since $m^4+n^4\neq 6m^2n^2$ for every positive integers $m,n$.