Pappus's hexagon theorem

For two triples of distinct points on two projective lines, the intersections $r_{ij}=p_i{q}_j\cap p_j{q}_i$ are collinear when the configuration is defined. Normalize a projective frame so $p_1=[0,1,0]$, $p_2=[1,1,0]$, $q_1=[0,0,1]$, $q_2=[1,0,1]$, and write $p_3=[1,\alpha ,0]$, $q_3=[1,0,\beta ]$. Then representatives of the three intersections are $(1,1,1)$, $(1,\alpha ,\beta )$, and $(\alpha \beta -1,\alpha (\beta -1),\beta (\alpha -1))$. The last is $\alpha \beta$ times the first minus the second, proving collinearity by linear dependence. This gives a coordinate proof over any field when the normalizations are available.