 Complete quadrangle

A complete quadrangle consists of four projective points in a projective plane, no three collinear, and their six joining projective lines. The intersections of opposite joining projective lines are its three diagonal points. The construction turns incidence relations into a harmonic conjugate points construction.

Over a field of characteristic different from two, choose a projective frameThe diagonal points are $E=AB\cap CD=[1:1:0]$, $F=AC\cap BD=[1:0:1]$ and $G=AD\cap BC=[0:1:1]$. The projective line $FG$ meets $AB$ at $P=[1:-1:0]$. On $AB$, the affine coordinate $y/x$ assigns $A,B,E,P$ the values $0,\infty ,1,-1$. ConsequentlyThus $E$ and $P$ are harmonic conjugate points relative to the explicitly specified reference pair $A,B$. Permuting the four vertices gives the corresponding construction on every side. In characteristic two, $1=-1$ and this pair collapses, explaining the restriction.

A **complete quadrangle** is a geometric configuration consisting of four points (vertices) that are not all on the same line, along with the six lines that connect each pair of points. More specifically, these four points form a set of lines, and every pair of distinct points is connected by a line segment.