Collinearity via Determinant

Equivalent determinant condition for collinearity of three points.

Three points $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ are collinear when slope $AB$ equals slope $AC$:

$$(y_2-y_1)(x_3-x_1) = (y_3-y_1)(x_2-x_1)$$

Expanding and collecting:

$$x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) = 0$$

This is exactly the cofactor expansion of:

$$\boxed{\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0}$$

The determinant form is preferred because it is symmetric in all three points — no point is singled out as a base — and it directly generalises to the area formula:

$$\Delta = \frac{1}{2}\left|\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\right|$$

Collinearity is the special case $\Delta = 0$.