the problem is :
if $p_1,p_2,p_3$ are collinier points in the space , and for an arbitrary line $l$ we have that $q_1,q_2,q_3$ are the image of $p_1,p_2,p_3$ on $l$ respectivly. then does the following hold? $$\frac{d(p_1,p_2)}{d(p_2,p_3)}=\frac{d(q_1,q_2)}{d(q_2,q_3)}$$
i only trid to take a plane $W$ such that $l \in W$ and then put image of $p_1,p_2,p_3$ on $W$. but i could not move forward.
Hint: Let $\hat l$ denote the unit vector along $l$ and $\theta$ be the angle between the line containing $p_i$ and $l$. Then$$d(q_1,q_2)=\left|\left[\vec{p_2}-\vec{p_1}\right]\cdot\hat l\right|=d(p_1,p_2)\left|\cos\theta\right|$$