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.

中文翻译：

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关于直角三角形面积的某些Diophantine方程

使用椭圆曲线理论，我们证明了所有直角三角形（使面积之和与腿之和的平方成正方形）是由无限集给出的。类似地，我们得到所有直角三角形，使得面积和半周长的平方之和为一个正方形。使用Pell方程的理论，我们证明了无穷多个非本构直角三角形，使得面积和斜边（或小腿）的总和是一个正方形，而本构直角三角形的无穷大，使得该和面积和较小的腿（或周长，半周长，较大的腿）的平方成正方形。