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Using the theory of elliptic curve, we show that all right triangles, such that the sum of the area and the square of the sum of legs is a square, are given by an infinite set. Similarly, we get all right triangles such that the sum of the area and the square of the semi-perimeter is a square. Using the theory of Pell's equation, we prove that there are infinitely many non-primitive right triangles such that the sum of the area and the hypotenuse (or the smaller leg) is a square, and an infinity of primitive right triangles such that the sum of the area and the smaller leg (or the perimeter, the semi-perimeter, the larger leg) is a square. [ABSTRACT FROM AUTHOR]

We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann's Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors. [ABSTRACT FROM AUTHOR]

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]