Equation of Straight Line in Plane/Two-Point Form/Determinant Form

Theorem
Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:
 * $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Then $\LL$ can be expressed in the form:


 * $\begin {vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end {vmatrix} = 0$

Proof
Consider the general equation for $\LL$:


 * $l x + m y + n = 0$

Since $\LL$ passes through both $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$, we have:

Eliminating $l$, $m$ and $n$ from these three equations:

we obtain:


 * $\begin {vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end {vmatrix} = 0$