Pairs of Pythagorean triangles with given catheti ratios

In this note we investigate the problem of finding pairs of Pythagorean triangles $(a,b,c),(A,B,C)$, with given catheti ratios $A\mathrm{/}a,B\mathrm{/}b$. In particular, we prove that there are infinitely many essentially different (“non-similar”) pairs of Pythagorean triangles $(a,b,c),(A,B,C)$ satisfying given proportions, provided that $Aa\mathrm{\ne }Bb$.