If three points, **THE**(x_{THE}, y_{THE}), **B**(x_{B}, y_{B}) and **Ç**(x_{Ç}, y_{Ç}), are aligned, so:

To demonstrate this theorem we can consider three cases:

a) three horizontally aligned points

In this case, the ordinates are the same:

y_{THE} = y_{B} = y_{Ç}

and the determinant is null, since the 2nd and 3rd column are proportional.

b) three vertically aligned points

In this case, the abscissae are equal:

x_{THE} = x_{B} = x_{Ç}

and the determinant is null, since the 1st and 3rd column are proportional.

c) three points on a line not parallel to the axes

From the figure, we find that the triangles ABD and BCE are similar. So:

Developing, comes:

How:

So .

Note: The reciprocal of the stated statement is valid, ie if , then the points A (x_{THE}, y_{THE}), B (x_{B}, y_{B}) and C (x_{Ç}, y_{Ç}) are aligned.