An Interesting Property of a Straight Line

An Interesting Property of a Straight Line

Consider any straight line in a plane; it splits every point not on the line into two groups — the points on one side, and the points on the other. So altogether the plane divides into three parts: the points on the line, the points on one side, and the points on the other side.

If the line is $ax+by+c=0$, then for every point $(x_1,y_1)$ on one side, $ax_1+by_1+c>0$, and for every point $(x_2,y_2)$ on the other side, $ax_2+by_2+c<0$. (Which side gets the positive sign and which gets negative is just a matter of labelling, but the two sides always come out with opposite, consistent signs.)

Illustration. For the line $y-2x-3=0$: the points $P(-2,0)$, $Q(-2,4)$, $R\left(\tfrac12,5\right)$ all give $y-2x-3>0$ (check $P$: $0+4-3=1>0$), so they lie on one side. The points $A(0,0)$, $B\left(\tfrac12,3\right)$, $C(8,4)$ all give $y-2x-3<0$ (check $A$: $0-0-3=-3<0$), so they lie on the other side. …