The proof of this we leave to the student.
If A, B, C, D are four harmonic points (see Fig. 6, p. *22), and a quadrilateral KLMN is constructed such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D, then, projecting A, B, C, D from L upon KM, we have (ABCD) = (KOMD), where O is the intersection of KM with LN. But, projecting again the points K, O, M, D from N back upon the line AB, we have (KOMD) = (CBAD). From this we have
(ABCD) = (CBAD),
or
whence a = 0 or a = 2. But it is easy to see that a = 0 implies that two of the four points coincide. For four harmonic points, therefore, the six values of the anharmonic ratio reduce to three, namely, 2,
, and -1. Incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value, then the four points are harmonic.
Fig. 49