Prove that the points (a, b),(a1,b1) and (a - a1, b - b1) are collinear if ab1 = a1b
Consider the following points A(a,b), B(a1,b1), C(a−a1,b−b1)
Since the given points are collinear, we have area(△ABC)=0
First find the area of area(△ABC) as follows:
area(△ABC)=1/2 |x1(y1−y3)+x1(y3−y1)+x3(y1−y1)|
=|a(b1−(b−b1))+a1((b−b1)−b)+(a−a1)(b−b1)|
= |a(b1−b+b1)+a1(b−b1−b)+a(b−b1)−a1(b−b1)|
=|−ab−a1b1+ab−ab1+a1b+a1b1|
=|−(ab1−a1b)|
= (ab1−a1b)
This gives, ab1−a1b=0
∴ ab1 = a1b