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By the theory of elliptic curves, we show that there are infinitely many integer right triangle and integer parallelogram pairs with a common area and a common perimeter. [ABSTRACT FROM AUTHOR]

We say that a triangle $$T$$ tiles the polygon $$\mathcal A $$ if $$\mathcal A $$ can be decomposed into finitely many non-overlapping triangles similar to $$T$$ . A tiling is called regular if there are two angles of the triangles, say $$\alpha $$ and $$\beta $$ , such that at each vertex $$V$$ of the tiling the number of triangles having $$V$$ as a vertex and having angle $$\alpha $$ at $$V$$ is the same as the number of triangles having angle $$\beta $$ at $$V$$ . Otherwise the tiling is called irregular. Let $$\mathcal P (\delta )$$ be a parallelogram with acute angle $$\delta $$ . In this paper we prove that if the parallelogram $$\mathcal P (\delta )$$ is tiled with similar triangles of angles $$(\alpha , \beta , \pi /2)$$ , then $$(\alpha , \beta )=(\delta , \pi /2-\delta )$$ or $$(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$$ , and if the tiling is regular, then only the first case can occur. [ABSTRACT FROM AUTHOR]