Strictly Inscribed Similar Triangles

Two triangles $ABC$ and $XYZ$ are similar if their corresponding sides are proportional (equivalently, if their corresponding angles are equal). We say that $ABC$ and $XYZ$ are similar in order if $A$ corresponds to $X$, $B$ corresponds to $Y$, and $C$ corresponds to $Z$. That is,

$\frac{|AB|}{|XY|} = \frac{|BC|}{|YZ|} = \frac{|AC|}{|XZ|},$

where $|MN|$ denotes the length of the segment from $M$ to $N$.

Triangle $XYZ$ is strictly inscribed in triangle $ABC$ if each vertex of $XYZ$ lies in the interior (not at a vertex) of a different edge of $ABC$. This means no edge of $XYZ$ can be contained in an edge of $ABC$. If $XYZ$ is similar in order to $ABC$ and strictly inscribed in $ABC$, we call $XYZ$ a strictly inscribed similar triangle of $ABC$.

If the line through $X$ and $Y$ makes an angle $\theta$ with the line through $A$ and $B$, there are four possible orientations, illustrated below: $X$ and $Y$ may be at either end of the segment, and the third vertex $Z$ may be on either side of the line. In the figure, the line through $X$ and $Y$ makes an angle of $30^\circ$ with the line through $A$ and $B$.

Depending on the shape of the outer triangle $ABC$ and the angle $\theta$ between the line through $X$ and $Y$ and the line through $A$ and $B$, there may be 0, 1, 2, 3, or 4 strictly inscribed similar triangles of $ABC$ with angle $\theta$.

Given the vertices of triangle $ABC$ and an angle $\theta$, determine how many strictly inscribed similar triangles of $ABC$ exist for which the line through $X$ and $Y$ makes an angle $\theta$ with the line through $A$ and $B$.

The first line contains a positive integer $n$, the number of triangle datasets that follow. Each dataset consists of four lines: the first line has the $x$ and $y$ coordinates of vertex $A$, the second line has the $x$ and $y$ coordinates of vertex $B$, the third line has the $x$ and $y$ coordinates of vertex $C$, and the fourth line has the angle $\theta$ (in degrees) between the line through $X$ and $Y$ and the line through $A$ and $B$.

For each dataset, output a single line containing one integer: the number of strictly inscribed similar triangles of $ABC$ for which the line through $X$ and $Y$ makes an angle $\theta$ with the line through $A$ and $B$.