If x1,x2,x3  as well as y1,y2,y3  are in G.P. with the same common ratio, then the points (x1,y1),(x2,y2)  and (x3,y3)

Let x1=a,x2=ar,x3=ar2 y1=b,y2=br,y3=br2 Let A≡(x1,y1),B≡(x2,y2),C≡(x3,y3) Slope of AB=y1−y2x1−x2=b(1−r)a(1−r)=ba Slope of BC=y2−y3x2−x3=br(1−r)ar(1−r)=ba Hence, AB‖BC  and points A,B,C are collinear.