For four harmonic points the six cross-ratios become equal two and two:
| λ = −1, 1 − λ = 2, | λ | = ½, | 1 | = −1, | 1 | = ½, | λ − 1 | = 2. |
| λ − 1 | λ | 1 − λ | λ |
Hence if we get four points whose cross-ratio is 2 or ½, then they are harmonic, but not arranged so that conjugates are paired. If this is the case the cross-ratio = −1.
§ 19. If we equate any two of the above six values of the cross-ratios, we get either λ = 1, 0, ∞, or λ = −1, 2, ½, or else λ becomes a root of the equation λ² − λ + 1 = 0, that is, an imaginary cube root of −1. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are all-important.
§ 20. From the definition of harmonic points, and by aid of § 11, the following properties are easily deduced.
If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are “separated” by C and D.
For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B.
If A and B are different points, and if C coincides with A or B, D does. But if A and B coincide, one of the points C or D, lying between them, coincides with them, and the other may be anywhere in the line. It follows that, “if of four harmonic conjugates two coincide, then a third coincides with them, and the fourth may be any point in the line.”
If C is the middle point between A and B, then D is the point at infinity; for AC : CB = +1, hence AD : DB must be equal to −1. The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.
This important property gives a first example how metric properties are connected with projective ones.