We investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given cathetus ratios $A/a,\, B/b$. 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 $Aa\neq Bb$.
"Pairs of Pythagorean triangles with given ratios between catheti." Rocky Mountain J. Math. 49 (8) 2779 - 2789, 2019. https://doi.org/10.1216/RMJ-2019-49-8-2779