Collinear Points | Brilliant Math & Science Wiki

Collinearity tests are primarily focused on determining whether a given 3 points \(A, B\), and \(C\) are collinear. This is because it is easily extensible: e.g. showing 4 points \(A, B, C, D\) are collinear can be done by first showing \(A, B, C\) are collinear then showing \(B, C, D\) are collinear as well.

When the coordinates of the points are given, this problem is relatively simple, if sometimes computationally challenging. One simple test simply finds the equation of the line between \(A\) and \(B\) using standard methods, then finding the equation of the line between \(B\) and \(C\), and comparing.

Let \(A = (1, 2), B = (2, 4), C = (3, 6)\). Are \(A, B, C\) collinear?

The slope of the line between \(A\) and \(B\) is \(\frac{4 - 2}{2 - 1} = 2\), making the equation of the line between \(A\) and \(B\) \(y = 2x + b\), and setting \(x = 1, y = 2\) we determine \(b = 0\). Thus, the equation of the line becomes \(y = 2x\). Similarly, the slope between \(B\) and \(C\) is \(\frac{6 - 4}{3 - 2} = 2\), making the equation of the line between \(B\) and \(C\) also \(y = 2x\). This is the same line, so \(A, B, C\) are collinear.

This can be slightly improved:

Yes No Reveal the answer

Suppose we have points \(A, B\) and \(C\), such that the slope between \(A\) and \(B\) is equal to the slope between \(B\) and \(C\). Are \(A, B\) and \(C\) necessarily collinear points?

The correct answer is: Yes