a harmonic conjugate

Among all the ways four points can sit on a line, one arrangement is so balanced and natural that it appears everywhere — in the diagonal of a square, in the optics of a lens, in the rungs of a perspective drawing. It is the harmonic arrangement, and the fourth point that completes it from the first three is called their harmonic conjugate. It is the projective plane's most basic 'special position', defined purely by incidence, with no ruler needed.

Start from the cross-ratio. Four collinear points A, B, C, D form a harmonic set, and D is the harmonic conjugate of C with respect to A and B, exactly when their cross-ratio (A, B; C, D) = -1. Geometrically that one value says C and D divide the segment AB 'internally and externally in the same ratio': if C lies between A and B cutting it in ratio AC : CB, then D lies outside cutting AD : DB in the same numerical ratio but with opposite sign. The beautiful part is that you can find D from A, B, C with straightedge alone, using a complete quadrangle: draw any four points in general position whose six sides pass appropriately through A, B, C, and the fourth diagonal point falls exactly on D. No measuring, no compass — incidence alone pins down the harmonic conjugate.

Harmonic conjugates matter because '-1' is the one cross-ratio value preserved by every projective map and recognisable without coordinates, so it gives projective geometry a notion of perfect balance it can use to build everything else, including the pole-polar relationship of a conic. Two honest notes. First, harmonic conjugacy is symmetric in the right way: if D is the harmonic conjugate of C with respect to A and B, then C is the harmonic conjugate of D, and the roles of the pair {A, B} and the pair {C, D} can be exchanged. Second, the midpoint of AB is the harmonic conjugate of the point at infinity in that direction — which is the projective way of seeing why a 'midpoint' is not a projective notion until you fix a line at infinity.