= Harmonic conjugate points
{title2=$(AB:CD)=-1$}
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.
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