Equation of Straight Line in Plane/Two-Point Form

Theorem
Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.

Let $\mathcal L$ be the straight line passing through $p_1$ and $p_2$.

Then $\mathcal L$ can be described by the equation:
 * $\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$

or:


 * $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Proof
From the gradient-intercept form of the equation of the straight line:


 * $(1): \quad y = m x + c$

which is to be satisfied by both $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.

We express $m$ and $c$ in terms of $\paren {x_1, y_1}$ and $\paren {x_2, y_2}$:

Substituting for $m$ and $c$ in $(1)$: