We know that the slopes of two parallel lines are equal. If two lines having the same slope pass through a common point, then they will coincide. Hence, If A, B and C are three points in the XY-plane, then they will lie on a line i.e. three points are collinear if and only if slope of AB=slope of BC.

Explanation

Let `(a,0), B(0,b)and C(x,y)` be the given collinear points, then,
Slope of `Ab=` Slope of BC
`implies(b-0)/(0-a)=(y-b)/(x-0)`
`implies(-b)/(a)=(y-b)/(x)`
`-bx=ay-ab`
`impliesbx+=ay-ab`
`impliesx/a+y/b=1.`
Hence Proved.