In details

Three-point alignment conditions


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:

Developing, comes:

How:

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.

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