Intercept theorem/Two rays/Fact/Proof

Proof

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

t A_{2} = t A_{1} + t v = B_{1} + t v .

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

\frac{‖ B_{1} - B_{2} ‖}{‖ B_{1} ‖} = \frac{‖ t A_{1} - t A_{2} ‖}{‖ t A_{1} ‖} = \frac{‖ A_{1} - A_{2} ‖}{‖ A_{1} ‖} .

To fact