Intercept theorem/Two rays/Fact

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

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

and suppose that ${\displaystyle {}A_{1},B_{1}\neq 0}$.