Rational triangle pairs and cyclic quadrilateral pairs with areas and perimeters in certain proportions

Abstract

The problem of finding a pair of distinct rational triangles (resp. cyclic quadrilaterals) with the same area and the same perimeter has been solved. Now we generalize these results and get several parametric solutions to find infinitely many pairs of rational triangles (resp. cyclic quadrilaterals) with areas and perimeters in fixed proportions $(\alpha,\beta)$ respectively, where $\alpha$ and $\beta$ are positive rational numbers.