On θ-congruent numbers on real quadratic number fields

Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < \theta< \pi $ has rational cosine, say $\cos (\theta)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive integer $n$ is called a $(\mathbb K,\theta)$-congruent number if there is a triangle, called the $(\mathbb K,\theta, n)$-triangles, with sides in $\mathbb K$ having $\theta$ as an angle and $n\alpha_\theta$ as area, where ${\alpha_\theta}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,\theta)$-congruent number elliptic curve $E_{n,\theta}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb K$. Denote the squarefree part of positive integer $t$ by ${\rm sqf}(t)$. In this work, it is proved that if $m\neq {\rm sqf}(2r(r-s))$ and $mn\neq 2, 3, 6$, then $n$ is a $(\mathbb K,\theta)$-congruent number if and only if the Mordell-Weil group $E_{n,\theta}(\mathbb K)$ has positive rank, and all of the $(\mathbb K,\theta, n)$-triangles are classified in four types.