If three points, THE(xTHE, yTHE), B(xB, yB) and Ç(xÇ, yÇ), 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:
yTHE = yB = yÇ
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:
xTHE = xB = xÇ
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:
Note: The reciprocal of the stated statement is valid, ie if , then the points A (xTHE, yTHE), B (xB, yB) and C (xÇ, yÇ) are aligned.Next: General Equation