I am trying to prove the Pappus configuration with homogeneous coordinates in projective geometry:
I have, $A=\{1,1,1\}$ $B=\{0,0,1\}$ $P=\{1,0,0\}$ $Q=\{0,1,0\}$, from here my professor sees that $A'=\{1,0,a\}$ and $B'=\{b,1,1\}$ for some $a,b\in K$. I undestand the $y$ coordinate of $A'$ must be $0$ because it is in the line $y=0$ and the $y$ and $z$ coordinates of $B'$ must be equal because it is in the line $y=z$. Finally $C=\{1,1,a\}$ and $C'=\{1-ab,1-a,1-ab\}$.
I would really appreciate if somone could explain how, step by step, can I get the coordinates of the points $A',B',C$ and $C'$.