Section formula is just finding in what ratio does a point divide two points. and when a point divides two points in some ratio, it basically means they exist in one line or there exist a line in which all the three points lies.

Think of it as- any or every point P between two points A and B  in a LINE, divides AB in some ratio. so if we show that P divides AB in some ratio and conclude that the point P must exist on the line joining A and B.

We use this simple concept to show any three points collinear.

like here,

A (7,-5)  B (9,-3)  C (13,1)

Let a point B(9,-3) divides the line joining the points A(7,-5) and C(13,1) in the ratio 1:k 


as per section formula,

x-coordinate of B =

\frac{13+7k}{1+k}=9\Rightarrow 13+7k=9+9k\Rightarrow 4=2k\Rightarrow k=2

y-coordinate of B =

\frac{1-5k}{1+k}=-3\Rightarrow 1-5k=-3-3k\Rightarrow 4=2k\Rightarrow k=2

From x-coordinate and y-coordinate of B, we get the same ratio of AB/BC. Hence all the three points A, B, and C are collinear.