If three points $A(a_1, b_1), B(a_2, b_2)$ and $C(a_3, b_3)$ are collinear and $D=\begin{vmatrix}a_1&b_1&1\\a_2&b_2&1\\a_3&b_3&1\end{vmatrix}$, then:

$D=0$ $D=±1$ $D^2 = 0$ or 1 $D= (a_1+a_2+ a_3)-(b_1+b_2+b_3)$

$D=0$

Point A, B and C are collinear. $⇒A'=(a_1, b_1,1),B'(a_2, b_2,1), C'(a_3, b_3,1)$ are collinear as well so $[A'\,\,B'\,\,C']=0$ (for collinear points) $⇒\begin{vmatrix}a_1&b_1&1\\a_2&b_2&1\\a_3&b_3&1\end{vmatrix}=0$