Position of a Point Relative to a Line

Two points lie on the same side of ax + by + c = 0 if the expressions ax₁ + by₁ + c and ax₂ + by₂ + c have the same sign, and on opposite sides if signs differ.

For the line $L: ax+by+c=0$, define $f(x,y) = ax+by+c$.

On the line: $f = 0$. On one side: $f > 0$. On the other: $f < 0$.

Why signs agree for points on the same side

Take two points $P_1$ and $P_2$ on the same side of $L$. The segment $P_1P_2$ does not cross $L$, so $f$ does not change sign along it. Since $f$ is continuous and nonzero on the segment, it maintains one sign throughout — so $f(P_1)$ and $f(P_2)$ have the same sign.

Conversely, if $P_1$ and $P_2$ are on opposite sides, the segment crosses $L$ where $f=0$, so $f$ changes sign — $f(P_1)$ and $f(P_2)$ have opposite signs.

Practical test

To check if $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$ are on the same side of $ax+by+c=0$:

Compute $(ax_1+by_1+c)$ and $(ax_2+by_2+c)$.

• Same sign $\Rightarrow$ same side.
• Opposite sign $\Rightarrow$ opposite sides (the segment $P_1P_2$ crosses the line).

This is also the basis of the sign analysis in angle bisector problems.