eguruchela

Calculate Collinearity of Three Points


Calculate Collinearity for given 3-points A, B, C and tells points are collinear or non-collinear.

 

Point A (x1,y1)
Point B (x2,y2)
Point C (x3,y3)

 

Three or more points are said to be collinear if they lie on a single straight line. The condition for three points A, B, and C to be collinear can also be expressed as the distance between any one point and the line determined by the other two is zero means resultant value is equal to zero, then the points are collinear otherwise points are non-collinear and if points are non-collinear then a triangle can be formed with these three points as vertices.
collinearty

Formula used

Area = 1/2{ (x1 y2 + x2 y3 + x3 y1) - ( x2 y1 + x3 y2 + x1 y3) }

Methods to Prove if Points are Collinear

There are following methods to find whether the three points are collinear or not collinear:

Slope Formula Method

If the slope of any two pairs of points is the same than three points are collinear. As shown in fig. with three points A, B and C, three pairs of points can be formed, they are: AB, BC and AC. If Slope of BC = slope of BC = slope of AC, then A, B and C are collinear points.

Area of Triangle Method

Three points are collinear if the value of the area of the triangle formed by the three points is equal to zero, means If the result for the area of the triangle is zero, then the given points are collinear.

Example of non-collinear

Lets find the collinearty for three points (3, 2), (33, 44) and (4, 3)

Area	= 1/2{ (x1 y2 + x2 y3 + x3 y1) - ( x2 y1 + x3 y2 + x1 y3) }
 	= 1/2{(132+99+8) - (66+176+9 )}
 	= 1/2(239 - 251)
 	= 1/2(-12)
 	= -6
Area != 0; The given points are non collinear	

Example of collinear

Lets find the collinearty for three points (3, 2), (5, 4) and (7, 6)

				
Area	= 1/2{ (x1 y2 + x2 y3 + x3 y1) - ( x2 y1 + x3 y2 + x1 y3) }
 	= 1/2{(12+30+14) - (10+28+18 )}
 	= 1/2(56 - 56)
 	= 1/2(0)
 	= 0
Area = 0; The given points are collinear