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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.

The four points form a projective frame: no three are collinear. By the allowed uniqueness theorem and cross-ratio invariance, useIntersecting the indicated projective lines givesA point of $Q_0{Q}_2$ has coordinates $[a:b:a+b]$. Intersecting with $P_0{P}_1$, whose equation is $z=0$, gives $Q_3=[1:-1:0]$. On that line use its first two homogeneous coordinates. The cross-ratio from (a)(ii) isThe characteristic of a field being zero ensures that $Q_3\ne Q_1$. The construction is the diagonal-point property of a complete quadrangle, producing harmonic conjugate points.