$m$ and $n$ are given lines. $p$ and $q$ are two intersecting lines that intercept $m$ and $n$ at $C,D$ and $A,B$. I want to show that $m\parallel n$ iff $\dfrac{OC}{CD}=\dfrac{OA}{AB}=k$.
I am trying to solve the problem using vectors.
$\vec{OC}=k\cdot\vec{CD}$
$\vec{OA}=k\cdot\vec{AB}$
How to continue?
Altering the notation makes the equations slightly easier.
Let $\vec{OB}=k\cdot \vec{OA}$ and $\vec{OD}=l\cdot \vec{OC}$.
Then $\vec{BD}= \vec{BO}+\vec{OD}=-k\cdot\vec{OA}+l\cdot\vec{OC}=(l-k)\cdot\vec{OA}+l\cdot\vec{AC}$
$\vec{OA}$ and $\vec{AC}$ are not parallel and so $\vec{BD}$ and $\vec{AC}$ are parallel if and only if $l=k$.