Definition

In some symmetrical cases, the six values reduce to three or even two. When two points coincide, the possible values are 0, 1 and . With certain symmetrical configurations of complex points the possible values are and . Finally, when

Hence, the harmonic cross ratio is invariant under interchange of points in each couple and also (as usual) under interchange of couples, so the couples can be considered to be unordered. Given (

The previous exercise explains why the harmonic conjugate is considered to be a projective extension of the notion of an affine mid-point.