 Harmonic conjugate points

Distinct projective points $C,D$ on the projective line are harmonic conjugates relative to distinct $A,B$ when $(AB:CD)=-1$. Use the cross-ratio convention $(AB:CD)=[a,c][b,d]/([a,d][b,c])$, where the brackets are two-by-two determinants. In an affine coordinate with $A=0$, $B=\infty$, the condition is $D=-C$. A complete quadrangle constructs the fourth point using joins and intersections only. Characteristic two is excluded because then $-1=1$, and the four points could not be distinct.