Collinearity of Three Points

In geometry, a point determines the location on a plane. We can mark any number of points on a plane. Suppose if you mark three points on a paper, we would be required to label them by a single capital letter like A, B, C or P, Q, R since we need to represent the points using capital letters. Also, we can draw several shapes that pass through these three points such as a line, ray, line segment when they are collinear points otherwise, we can draw a triangle or circle, etc. Here, you can learn how to find collinearity of three points in different ways using slope, distance, area, and so on.

Condition of Collinearity of Three Points

In general, lines can be parallel, perpendicular, intersection, etc. In all these cases, the slopes of lines are related to each other in some way. As we know, the slopes of two parallel lines are equal. If two lines have the same slope pass through a common point, then two lines will coincide. In other words, if A, B, and C are three points in the XY-plane, they will lie on a line, i.e., three points are collinear if and only if the slope of AB is equal to the slope of BC.

This scenario can be observed in the below figure.

Collinearity of three points

From the above, we can derive the condition for collinearity of three points A, B and C using slope formula.

Let (x1, y1), (x2, y2) and (x3, y3) be the coordinates of three points, say A, B and C, respectively.

Slope of AB = Slope of BC {that means A, B and C are collinear}

(y2 – y1)/ (x2 – x1) = (y3 – y2)/ (x3 – x2)

(x3 – x2)(y2 – y1) = (x2 – x1)(y3 – y2)

x3(y2 – y1) – x2(y2 – y1) = x2(y3 – y2) – x1(y3 – y2)

Rearranging the terms,

x2(y3 – y2) + x1(y2 – y3) + x2(y2 – y1) – x3(y2 – y1) = 0

x1(y2 – y3) + x2(y3 – y2 + y2 – y1) + x3(y1 – y2) = 0

x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2) = 0

This is known as the collinearity of three points formula.

How to Prove Collinearity of Three Points

The following conditions are used to prove the collinearity of given points.

Suppose the points A(x1, y1), B(x2, y2) and C(x3, y3) are collinear, then the Conditions for Collinearity of Three Points are:

(i) Slope of AB = Slope of BC

(ii) AB + BC = AC (or) AB + AC = BC (or) AC + BC = AB

This can be proved using the distance formula in coordinate geometry.

(iii) Area of triangle ABC = 0, i.e. (½)|x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)| = 0

(iv) If the third point satisfies the equation through any two of the given three points, then the three points A, B, and C will be collinear.

Solved Examples

Example 1:

Find the value of p for which the points (p, -1), (2, 1) and (4, 5) are collinear.

Solution:

Let the given points be:

A(p, -1) = (x1, y1)

B(2, 1) = (x2, y2)

C(4, 5) = (x3, y3)

Given that A, B, and C are collinear.

Slope of AB = Slope of BC

(y2 – y1)/(x2 – x1) = (y3 – y2)/(x3 – x2)

Substituting the values of coordinates of given points,

(1 + 1)/(2 – p) = (5 – 1)/(4 – 2)

2/(2 – p) = 4/2

2/(2 – p) = 2

⇒ 2 – p = 1

⇒ p = 2 – 1

⇒ p = 1

Hence, the value of p is 1.

Example 2:

Using the equation method, check the collinearity of the points A(7, -2), B(2, 3) and C(-1, 6).

Solution:

We know that the equation of a line passing through the points (x1, y1) and (x2, y2) is:

y – y1 = [(y2 – y1) /(x2 – x1)] (x – x1)

Let A(7, -2) = (x1, y1) and B(2, 3) = (x2, y2).

So, the equation of a line passing through the points A(7, -2) and B(2, 3) is given by:

y + 2 = [(3 + 2)/(2 – 7)] (x – 7)

y + 2 = (5/-5) (x – 7)

y + 2 = -x + 7

x + y + 2 – 7 = 0

x + y – 5 = 0

Now, substituting the point C(-1, 6) in the above equation,

-1 + 6 – 5 = 0

0 = 0

Thus, the third point satisfies the equation of the line passing through the two of given three points.

Therefore, the given points A, B and C are collinear.

Frequently Asked Questions – FAQs

How do you find Collinearity with 3 points?

We can find the collinearity with three points in different methods such as:
(i) Using slope formula
(ii) Using distance formula
(iii) Using area of triangle formula
(iv) Using equation method

What are 3 collinear points?

The points A, B and C are collinear points if they lie on the same line in a plane. In that case, the slope of AB is equal to the slope of BC.

What is the formula of collinear points?

If the points (x1, y1), (x2, y2) and (x3, y3) are collinear, then the below formula can be given:
x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2) = 0

Are three points always collinear?

No, the given three points cannot always be collinear. That means, they can be collinear or noncollinear depending on the position of these points in a plane.

How do you prove collinearity of three points using the distance formula?

Suppose three points A, B, and C are collinear, then using distance formula we can show the collinearity of these points when they satisfy either of the following conditions:
(i) AB + BC = AC
(ii) AB + AC = BC
(iii) AC + BC = AB

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