Three-point alignment condition

When three points belong to the same straight, they are called aligned dots.

In the figure below, the points $\dpi{120} \mathrm{A}(x_1,y_1)$ , $\dpi{120} \mathrm{B}(x_2,y_2)$ and $\dpi{120} \mathrm{C}(x_3,y_3)$ they are aligned dots.

dots lined up

Three-point alignment condition

If points A, B and C are aligned, then triangles ABD and BCE are similar triangles, therefore, have proportional sides.

Alignment condition

$\dpi{120} \boldsymbol{\frac{x_2-x_1}{x_3-x_2} = \frac{y_2-y_1}{y_3-y_2}}$

So, the three-point alignment condition $\dpi{120} \mathrm{A}(x_1,y_1)$ , $\dpi{120} \mathrm{B}(x_2,y_2)$ and $\dpi{120} \mathrm{C}(x_3,y_3)$ any, is that the following equality is satisfied:

$\dpi{120} \boldsymbol{\frac{x_2-x_1}{x_3-x_2} = \frac{y_2-y_1}{y_3-y_2}}$

Examples:

Check that the dots are aligned:

a) (2, -1), (6, 1) and (8, 2)

We calculate the first side of the equality:

$\dpi{120} \frac{x_2-x_1}{x_3-x_2} = \frac{6 -2}{8-6} = \frac{4}{2}=2$

We calculate the second side of the equality:

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$\dpi{120} \frac{y_2-y_1}{y_3-y_2} = \frac{1-(-1)}{2-1} = \frac{2}{1}=2$

Since the results are equal (2 = 2), then the points are aligned.

b) (-2, 0), (4, 2) and (6, 3)

We calculate the first side of the equality:

$\dpi{120} \frac{x_2-x_1}{x_3-x_2} = \frac{4-(-2)}{6-4} = \frac{6}{2}=3$

We calculate the second side of the equality:

$\dpi{120} \frac{y_2-y_1}{y_3-y_2} = \frac{2-0}{3-2} =\frac{2}{1} =2$

Since the results are different (3 ≠ 2), then the points are not aligned.

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Observation:

It is possible to show that if: $\dpi{120} \frac{x_2-x_1}{x_3-x_2} = \frac{y_2-y_1}{y_3-y_2}$

Then the matrix determinant of coordinates of the points is zero, that is:

$\dpi{120} \mathrm{\begin{vmatrix} x_1& y_1 & 1\\ x_2& y_2 & 1\\ x_3& y_3 & 1 \end{vmatrix} = 0}$

Therefore, another way to check whether three points are aligned is by solving the determinant.

You may also be interested:

straight equation
perpendicular lines
parallel lines
How to calculate the distance between two points
Differences between function and equation

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