A positive integer N is called a \(\theta \)-congruent number if there is a \({\theta }\)-triangle (a, b, c) with rational sides for which the angle between a and b is equal to \(\theta \) and its area is \(N \sqrt{r^2-s^2}\), where \(\theta \in (0, \pi )\), \(\cos (\theta )=s/r\), and \(0 \le |s|<r\) are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a \({\theta }\)-congruent number if and only if the elliptic curve \(E_N^{\theta }: y^2=x (x+(r+s)N)(x-(r-s)N)\) has a point of order greater than 2 in its group of rational points. Moreover, a natural number \(N\ne 1,2,3,6\) is a \({\theta }\)-congruent number if and only if rank of \(E_N^{\theta }({{\mathbb {Q}}})\) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational \({\theta }\)-triangle for a \({\theta }\)-congruent number N from given ones by generalizing the Fermat’s algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle \({\theta }\) satisfying the above conditions. We show that this generalization is analogous to the duplication formula in \(E_N^{\theta }({{\mathbb {Q}}})\). Then, based on the addition of two distinct points in \(E_N^{\theta }({{\mathbb {Q}}})\), we provide a way to find new rational \({\theta }\)-triangles for the \({\theta }\)-congruent number N using given two distinct ones. Finally, we give an alternative proof for the Fujiwara’s Theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in \(E_N^{\theta }({{\mathbb {Q}}})\) with corresponding rational \({\theta }\)-triangles.

整数的正Ñ称为\（\ THETA \） -congruent数量，如果有一个\（{\ THETA} \）女士三角（一个，  b，  c ^）与合理的侧面为在它们之间的角度一个和b等于到\（\ theta \），其面积为\（N \ sqrt {r ^ 2-s ^ 2} \），其中\（\ theta \ in（0，\ pi）\），\（\ cos（\ theta）= s / r \）和\（0 \ le | s | <r \）是互质整数。归因于藤原（Number Theory，de Gruyter，第235–241页，1997年），N是\（{\ theta} \）当且仅当椭圆曲线\（E_N ^ {\ theta}：y ^ 2 = x（x +（r + s）N）（x-（rs）N）\）时，才有一个全等数2在其组中的合理点。此外，当且仅当\（E_N ^ {\ theta}（{{\}的阶数，自然数\（N \ ne 1,2,3,6 \）是\（{\ theta} \）一致数。 mathbb {Q}}}）\）大于零。在本文中，我们对一个问题进行了肯定的回答，该方法涉及通过从给定数N到给定\（{\ theta} \）-同数N来创建新的有理\（{\ theta} \）-三角形的方法的存在。费马算法，该算法针对给定数的给定数在任何角度生成新的有理直角三角形满足上述条件的\（{\ theta} \）。我们证明了这种概括类似于\（E_N ^ {\ theta}（{{\ mathbb {Q}}}）\）\中的复制公式。然后，基于\（E_N ^ {\ theta}（{{\ mathbb {Q}}}）\）中两个不同点的相加，我们提供了一种找到新的有理\（{\ theta} \）的方法-使用给定的两个不同的\（{\ theta} \）-全等数N的三角形。最后，我们给出了藤原定理2.2和定理2.3的另一种证明。特别是，我们提供了\（E_N ^ {\ theta}（{{\ mathbb {Q}}}）\）中所有扭转点的列表，并带有相应的有理\（{\ theta} \）-三角形。