$\theta$-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 $\omega$-parallelogram with areas and perimeters in fixed proportions $\left(\alpha ,\beta \right)$ respectively, satisfying that $sin\omega$ is a previously fixed rational multiple of $sin𝜃$, where $\alpha$ and $\beta$ are positive rational numbers.