Intercept theorem/Three rays/Fact

Let ${\displaystyle {}V}$ be a two-dimensional Euclidean vector space, let ${\displaystyle {}s_{1},s_{2},s_{3}\in V}$ be vectors, and suppose that ${\displaystyle {}v\in V}$ is linearly independent to each of these vectors. Let ${\displaystyle {}S_{i}=\mathbb {R} s_{i}}$, ${\displaystyle {}i=1,2,3}$, be the lines defined by the ${\displaystyle {}s_{i}}$. Let ${\displaystyle {}P}$ and ${\displaystyle {}Q}$ be points in ${\displaystyle {}V}$ with den corresponding parallel lines ${\displaystyle {}G=P+\mathbb {R} v}$ and ${\displaystyle {}H=Q+\mathbb {R} v}$. We denote the intersecting points of the lines (which exist uniquely due to the conditions) by

${\displaystyle A_{1}=S_{1}\cap G,\,A_{2}=S_{2}\cap G,\,A_{3}=S_{3}\cap G,\,B_{1}=S_{1}\cap H,\,B_{2}=S_{2}\cap H,\,B_{3}=S_{3}\cap H,}$

and suppose that ${\displaystyle {}B_{1}\neq B_{2}}$.