# The Two Points Form of the Equation of a Line

The equation of a non-vertical line passing through two points $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$ is given by
$\frac{{x – {x_1}}}{{{x_2} – {x_1}}} = \frac{{y – {y_1}}}{{{y_2} – {y_1}}}$

To prove this equation let $P\left( {x,y} \right)$ be any point on the given line $l$. Also this line passes through $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$, as shown in the given diagram.

From $A$ and $B$ draw $AL$ and $BN$ perpendicular to the X-axis and from point $P$ draw $PM$ also perpendicular to the X-axis. Also from $A$ draw perpendicular to $AD$ on $BN$. Now from the given diagram, consider the similar triangles $ADB$ and $ACP$, and by the definition of a slope we take
$\frac{{PC}}{{BD}} = \frac{{AC}}{{AD}}\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Also from the given diagram we have
$\begin{gathered} PC = PM – CM = y – {y_1} \\ \Rightarrow BD = BN – DN = {y_2} – {y_1} \\ \Rightarrow AC = LM = OM – ON = x – {x_1} \\ \Rightarrow AD = LN = ON – OL = {x_2} – {x_1} \\ \end{gathered}$

Putting these all values in the above equation (i) we have
$\frac{{x – {x_1}}}{{{x_2} – {x_1}}} = \frac{{y – {y_1}}}{{{y_2} – {y_1}}}$

This is the equation of a line passing through two points $A\left( {{x_1},{y_1}} \right)$ and $B\left( {{x_2},{y_2}} \right)$. This equation can also have the form $\frac{{x – {x_2}}}{{{x_1} – {x_2}}} = \frac{{y – {y_2}}}{{{y_1} – {y_2}}}$

In determinant form, the given equation of a line through two points is
$\left| {\begin{array}{*{20}{c}} x&y&1 \\ {{x_1}}&{{y_1}}&1 \\ {{x_2}}&{{y_2}}&1 \end{array}} \right| = 0$

NOTE: There is an alternate way to prove the two points form of the equation of a straight line.

Consider the slope point form of the equation of a line, we have
$y – {y_1} = m\left( {x – {x_1}} \right)\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

Since the line is passing through the point $\left( {{x_1},{y_1}} \right)$ in the above equation and the slope of the line is $m = \frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}$, so equation (i) becomes
$\begin{gathered} y – {y_1} = \left( {\frac{{{y_2} – {y_1}}}{{{x_2} – {x_1}}}} \right)\left( {x – {x_1}} \right) \\ \frac{{x – {x_2}}}{{{x_1} – {x_2}}} = \frac{{y – {y_2}}}{{{y_1} – {y_2}}} \\ \end{gathered}$