  A positive integer $n$ is called a $\theta$-congruent number if there is a
triangle with sides $a,b$ and $c$ for which the angle between $a$ and $b$ is
equal to $\theta$ and its area is $n\sqrt{r^2 - s^2}$, where $0 < \theta <
\pi$, $\cos \theta = s/r$ and $0 \leq |s| < r$ are relatively prime integers.
The case $\theta=\pi/2$ refers to the classical congruent numbers. It is known
that the problem of classifying $\theta$-congruent numbers is related to the
existence of rational points on the elliptic curve $y^2 =
x(x+(r+s)n)(x-(r-s)n)$. In this paper, we deal with a variant of the congruent
number problem where the cosine of a fixed angle is $\pm \sqrt{2}/2$.
