Show collinear points in 3d project to collinear points in 2d

Question

Let $A,B,C$ be collinear points in 3d. Show that their projections $a,b,c$ respectively, onto 2d image plane are also collinear. If $A=[x_1,y_1,z_1], B=[x_2,y_2,z_2], C=[x_3,y_3,z_3]$ then $a=[x_1/z_1, y_1/z_1,1], b=[x_2/z_2, y_2/z_2,1], c=[x_3/z_3, y_3/z_3,1]$. Is a correct approach here to find the projected line $l'$ and show that $l'.a=0, l'.b=0, l'.c=0$ meaning that each projected point belongs on the projected line? If so how can this be shown?