Sign of Half-Plane is Well-Defined

Theorem
Let $\LL$ be a straight line embedded in a cartesian plane $\CC$, given by the equation:
 * $l x + m y + n = 0$

Let $\HH_1$ and $\HH_2$ be the half-planes into which $\LL$ divides $\CC$.

Let the sign of a point $P = \tuple {x_1, y_1}$ in $\CC$ be defined as the sign of the expression $l x_1 + m y_1 + n$.

Then the sign of $\HH_1$ and $\HH_2$ is well-defined in the sense that:
 * all points in one half-plane $\HH \in \set {\HH_1, \HH_2}$ have the same sign
 * all points in $\HH_1$ are of the opposite sign from the points in $\HH_2$
 * all points on $\LL$ itself have sign $0$.

Proof
By definition of $\LL$, if $P$ is on $\LL$ then $l x_1 + m y_1 + n = 0$.

Similarly, if $P$ is not on $\LL$ then $l x_1 + m y_1 + n \ne 0$.

Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two points not on $\LL$ such that the line $PQ$ intersects $\LL$ at $R = \tuple {x, y}$.

Let $PR : RQ = k$.

Then from Joachimsthal's Section-Formulae:

If $u_1$ and $u_2$ have the same sign, then $k$ is negative.

By definition of the position-ratio of $R$, it then follows that $R$ is not on the ine segment $PQ$.

Hence $P$ and $Q$ are in the same one of the half-planes defined by $\LL$.

Similarly, if $u_1$ and $u_2$ have the opposite signs, then $k$ is positive.

Again by definition of the position-ratio of $R$, it then follows that $R$ is on the ine segment $PQ$.

That is, $\LL$ intersects the ine segment $PQ$.

That is, $P$ and $Q$ are on opposite sides of $\LL$.

Hence $P$ and $Q$ are in opposite half-planes.