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.
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 =
y-coordinate of B =
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.