Intercept theorem/Three rays/Fact

Intercept theorem (three rays)

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

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

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

Then

${}{\frac {d(B_{2},B_{3})}{d(A_{2},A_{3})}}={\frac {d(B_{1},B_{2})}{d(A_{1},A_{2})}}\,.$

Proof, Write another proof