Collinear Points – Definition, formula and examples

In Mathematics, Collinear points are a set of three or more points that lie on the same straight line. According to the definition, points stand on the same straight line but do not necessarily lie in the same plane(coplanar). 

According to Latin words, ‘col’ means together, and ‘linear’ means on the same straight line; this is how the collinear word is derived. 

Collinearity is the characteristic of the points lying on the same line. Examples of collinearity are students standing in rows for prayers or a carton of eggs arranged in a row. In general, collinear means that if we consider that three objects are placed in a line, then the objects are collinear. The term collinear can be used as “in a row” or “in a line”.

According to Euler’s theory, the collinear points in a triangle exist as an Euler line in a triangle, in which three points of concurrency of the triangle lie. The three points on the Euler’s line are the orthocenter, the circumcenter, and the centroid. 

Hence, all the points of concurrencies are the collinear points in a triangle. Refer to Collinear Points in Geometry for a more detailed explanation of collinear points.

Definition of collinear points and non-collinear points

Collinear points: These are points that line on the same straight line; any two points are always collinear points by drawing a straight line. Three or more points are collinear but don’t have to be on the same line. 

In general, a straight line does not necessarily pass through any three or more distinct points. If these three or more distinct points are aligned in a way such that there is exactly one straight line passing through them, then those points are said to be collinear. The below figure shows the examples for collinear points:

Non-collinear points: The two or more points that do not stand on the same straight line are called non-collinear points. That means a single straight line cannot be drawn through these points. The below figure shows the example of non-collinear points is:

• The points A, B and C on the line m are collinear points. And 
• The points D, B and C on the line are collinear points.
• The points D, B, A and C, B, E are not in the same line, so they are non-collinear points.

Coplanar Points: This is defined as a group of points that lie in the same plane. Two or three points are always coplanar, but four or more points may or may not be coplanar.

Collinear point Formula:

The collinear point formula helps to find out whether the points lie on the same line or not. Below are the various methods to find out collinear points

Here are some common ways,

1. First method:  

For the distance formula, we need to calculate the distance between point A and point B, and then the distance between point B and point C, and then check if the sum (AB + BC) of these two distances is equal to the distance between point A and point C. 

It will only be possible when the three points ABC are collinear. We use the distance formula for those whose coordinates of two points are known.

By using the  distance formula, if ABC is a straight line then, AB + BC=AC

2. Second method :

In case of geometry,

•  if you are given 3 points, A(x,y,z) ,B(a,b,c),C(p,q,r)
•  first Find the distance between AB = =√(x-a)^2 + (y-b)^2 + (z-c)^2,
•  then find BC and  AC in a similar way.

•   If AB + BC=AC, then points are collinear.

3. Third method:

Use the concept that the area of the triangle formed by three collinear is zero. One way is by Using determinants.

4. Forth method:

The direction ratios of three vectors a,b, and c are proportional; they are collinear.

Examples for collinear points:

• The circum-centre, the in-centre and the ortho-centre of a triangle are distinct; then, they are collinear. The straight line of the circum-centre, the in-centre and the ortho-centre is the Euler line of that triangle.
• In the given diagram, a hexagon ABCDEF is inscribed in a circle. If there exists three pairs of points that are directly opposite, non-parallel sides, say AB & DE, BC & EF and CD & FA, and we produce these three pairs of opposite sides to meet at three points G, H and K, then G, H, K are collinear. This is known as Pascal’s Theorem.

 

• Solved example

 The Slope of the given line segment joining two points, say (x1, y1) and (x2, y2), is given by the formula:m = (y2 – y1)/ (x2 – x1)

Show that the three points P(4, 6), Q(6, 8) and R(8, 10) are collinear.

Solution: The definition says, if the given three points P(4, 6), Q(6, 8) and R(8, 10) are collinear, then the slopes of any two pairs of points PQ, QR & PR will be equal.

Now, we can find the slopes of the respective pairs of points by using a slope formula such that:

Slope of PQ = (8 – 6)/ (6– 4) = 2/2 = 1

Slope of QR = (10 – 8)/ (8 – 6) = 2/2 = 1

Slope of PR = (10 – 6) /(8 – 4) = 4/4 = 1

From the above example, the slopes of all the pairs of points are equal.

Hence it says, the three points P, Q and R are collinear.

Conclusion

Collinear points are two or more collinear points; then, they lie on the same line. If they are collinear, points that lie on the same line. The two parallel lines never intersect. And also not necessary to lie in the same plane. There exist no two such points that a straight line cannot pass through them. 

Therefore any two points are always collinear. There are many methods to find out whether the points are collinear or non-collinear. The basic and most frequently used formulas to find collinear points are the Distance Formula, Slope Formula or Area of Triangle Formula.