Intercept theorem/Three rays/Fact/Proof

We apply fact twice, to the situation given by ${\displaystyle {}s_{1},s_{2}}$ and to the situation given by ${\displaystyle {}s_{2},s_{3}}$, and we obtain

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