Point at Infinity of Intersection of Parallel Lines

Theorem
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$ such that $\LL_1$ and $\LL_2$ are parallel.

By Condition for Straight Lines in Plane to be Parallel, $\LL_1$ and $\LL_2$ can be expressed as the general equations:

The point at infinity of $\LL_1$ and $\LL_2$ can thence be expressed in homogeneous Cartesian coordinates as $\tuple {-m, l, 0}$.

Proof
Let $\LL_1$ be expressed in the form:
 * $l x + m y + n = 0$

Hence let $\LL_2$ be expressed in the form:
 * $l x + m y + k n = 0$

where $k \ne 1$.

Let their point of intersection be expressed in homogeneous Cartesian coordinates as $\tuple {X, Y, Z}$

Then: