# Pairs of Pythagorean triangles with given catheti ratios

## TL;DR

This paper explores the existence of infinitely many non-similar Pythagorean triangle pairs with specified catheti ratios, under certain conditions, expanding understanding of Pythagorean triples and their proportional relationships.

## Contribution

It proves the existence of infinitely many non-similar Pythagorean triangle pairs with prescribed ratios, given that the product of one triangle's hypotenuse and leg differs from the other.

## Key findings

- Infinitely many non-similar Pythagorean pairs exist for given ratios.
- Such pairs are characterized by the condition $Aa
eq Bb$.
- The result generalizes the understanding of Pythagorean triangle ratios.

## Abstract

In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti 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 $Aa\neq Bb$.