Can one prove Pappus hexagon theorem with cross ratios alone

Question

Can one prove Pappus theorem knowing that projections don't change the cross ratio and that $(A,B;C,D) = (A,B;C,E) \iff D=E$?

I was reading this in exercise 6 they say we can prove Pappus theorem with projections and cross ratio, but they don't quite do it.

Pappus theorem is this: Given $A,B,C$ colinear in $r$ and $D,E,F$ colinear in $s$, then $X = AE \cap BD, Y =BF \cap CE, Z= CD \cap AF$ then $X,Y,Z$ are colinear. They hint use to define $P = BF \cap DC$ and $Z' = XY \cap CD$.

Answer 1

This is a proof using cross ratios from S. Dobos Cross ratio in use, The Mathematical Gazette, Vol. 95, No. 534 (November 2011), pp. 444-453:

Answer 2

Refer to above picture. We shall prove that AF Π XY = CD Π XY = Z. Suppose that AC and XY intersect on the right hand side at point K (with each line’s extension). From point A projection the Cross Ratio: (X, AF Π XY; Y, K ) = (E, AF Π CE; Y, C ) (point A) = (D, CD Π XY; OP, C) (point F) = (X, CD Π XY; Y, K) (point B)

Therefore AF Π XY = CD Π XY = Z (end)

Answer 3

Refer to above (top) picture. Suppose that $AC$ and $XY$ intersect on the right hand side at point $K$ (with each line’s extension). Note:

Z’ = AF ∩ XY;

Z’’ = CD ∩ AF;

Z’’’= BZ’’ ∩ XY;

P = BF ∩ CD;

Q= AF ∩ CE.

From point A, then point F and Point B central projection the Cross Ratios evaluation as below:

(X, Z’; Y, K ) = (E, Q; Y, C ) (point A) ;

= (D, Z’’; P, C) (point F);

= (X, Z’’’; Y, K) (point B);

Therefore

Z’= Z’’’ , hence Z'= Z''= Z'''=Z .

(end)