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We have to find out which lines intersect the positive half axis of $x$.

According to this formula we can determine if the angle between two points $(A[x_1, y_1]$ and $B[x_2, y_2]$ ) of the line ($\angle AOB$ where $O$ is the center point of the coordinate system) is less than $180^\circ$:

if $(y_1<0)$ then if $x_2y_1-x_1y_2>0$ then the $\text{angle} < 180^\circ$

if $(y_1 > 0)$ then if $x_2y_1-x_1y_2<0$ then the $\text{angle} < 180^\circ$

My question is how did we get this formula?

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The term $x_2\cdot y_1 - x_1\cdot y_2$ in your question corresponds to the determinant $\begin{vmatrix}x_2 & x_1 \\ y_2 & y_1\end{vmatrix}$ which can be interpreted as twice the oriented area of the triangle $BOA$. The triangle $AOB$ has the opposite orientation.

If that determinant is positive, then $BOA$ is oriented counter-clockwise, and $AOB$ is oriented clockwise. This implies that when you look from $A$ towards $O$, then $B$ is somewhere to your right. So if $A$ is somewhere below the $x$ axis, then viewed from $A$ the point $B$ is visible somewhere to the right of $O$, so the connecting line $AB$ has to intersect the $x$ axis in its positive half. The same argument holds if $A$ is avove te $x$ axis but $B$ off to the left of $O$.

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