Equation of Straight Line in Plane/Two-Point Form/Proof 2

Proof
Let $\tuple {x, y}$ be an arbitrary point on the straight line through $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.

The area of the triangle formed by $\tuple {x, y}$, $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ is equal to $0$.

Hence from Area of Triangle in Determinant Form:


 * $\AA = \dfrac 1 2 \size {\paren {\begin {vmatrix}

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

Hence: