 The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator.If collinear points A, B, C, and D are such that D is the harmonic conjugate of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression.Specifically, each of the sequences
AC, AB, AD; BC, BA, BD; CA, CD, CB; and DA, DC, DB are harmonic progressions, where each of the distances is signed according to a fixed orientation of the line.