Beyond the unique pair: a two-parameter family of rational triangles with equal perimeter and area

We study pairs consisting of a rational isosceles triangle and a rational triangle having a prescribed angle $θ$, with equal perimeter and equal area. Writing $ρ=\cos(θ)\in\mathbb{Q}$, we obtain a two-parameter description of the solutions and, for fixed $ρ$, a family of curves $C_ρ$ which are gener