$\theta$-triangle and $\omega$-parallelogram pairs with areas and perimeters in certain proportions

Abstract

By the theory of elliptic curves, we show that given a convex angle 𝜃, there exist, except for finitely many exceptions, infinitely many pairs of rational 𝜃-triangle and ω-parallelogram with areas and perimeters in fixed proportions (α,β) respectively, satisfying that sinω is a previously fixed rational multiple of sin𝜃, where α and β are positive rational numbers.