= Pappus's hexagon theorem
{c}
{wiki}
For two triples of distinct points on two projective lines, the intersections $r_{ij}=p_iq_j\cap p_jq_i$ 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.
Back to article page
