Intercept theorem/Two rays/Fact/Proof

Without loss of generality, we may assume that ${\displaystyle {}s_{1}=A_{1}}$, ${\displaystyle {}s_{2}=A_{2}}$, and ${\displaystyle {}v=A_{2}-A_{1}}$, as this does not change the lines involved. We write ${\displaystyle {}B_{1}=tA_{1}}$. We have ${\displaystyle {}A_{2}=A_{1}+v}$; therefore, we obtain

${\displaystyle {}tA_{2}=tA_{1}+tv=B_{1}+tv\,.}$

This point belongs to ${\displaystyle {}S_{2}}$ and also to ${\displaystyle {}H}$. This means that this point is just ${\displaystyle {}B_{2}}$. Hence, ${\displaystyle {}B_{2}=tA_{2}}$, and

${\displaystyle {}{\frac {\Vert {B_{1}-B_{2}}\Vert }{\Vert {B_{1}}\Vert }}={\frac {\Vert {tA_{1}-tA_{2}}\Vert }{\Vert {tA_{1}}\Vert }}={\frac {\Vert {A_{1}-A_{2}}\Vert }{\Vert {A_{1}}\Vert }}\,.}$