The Construction: Observation One 

Since we know that \(\rho(x,y)=\rho(\nu x, \nu y)\) for all \(\nu \in S_{3}\) ,we have that \(\rho(\nu x, \mu y)=\rho(\mu^{-1} \nu x, y)\) for all \(\mu,\nu \in S_{3} \) .Thus by permuting triples points in Pappus' theorem we can construct six distinct lines, \(\rho(x, y)\) , \(\rho(\sigma x, y)\) , \(\rho(\sigma^{2} x, y)\) , \(\rho(\tau x, y)\) , \(\rho(\sigma \tau x, y)\) ,and \(\rho(\sigma^{2} \tau x, y)\) . 
 

Observation 1   The three lines corresponding to the even permutations, i.e.. \(\rho(x, y)\) , \(\rho(\sigma x, y)\) ,and \(\rho(\sigma^{2} x, y)\) are coincident. Similarly the three lines corresponding to the odd permutations, i.e. \(\rho(\tau x, y)\) , \(\rho(\sigma \tau x, y)\) ,and \(\rho(\sigma^{2} \tau x, y)\) are coincident. Below is an applet demonstrating this fact.

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The applet allows you to move the blue dots and the green dots and shows the resulting lines from the six Pappus' theorem constructions. The lines corresponding to even permutations are in red and the odd permutations are shown in yellow.

Next Step: Duality in the Projective Plane