Fig. 5
Let O be the intersection of KM and LN (Fig. 5). Join O to A and C. The joining lines cut out on the sides of the quadrangle four points, P, Q, R, S. Consider the quadrangle P, K, Q, O. One pair of opposite sides [pg 19] passes through A, one through C, and one remaining side through D; therefore the other remaining side must pass through B. Similarly, RS passes through B and PS and QR pass through D. The quadrangle P, Q, R, S therefore has two opposite sides through B, two through D, and the remaining pair through A and C. A and C are thus harmonic conjugates with respect to B and D. We may sum up the discussion, therefore, as follows:
30. If A and C are harmonic conjugates with respect to B and D, then B and D are harmonic conjugates with respect to A and C.
31. Importance of the notion. The importance of the notion of four harmonic points lies in the fact that it is a relation which is carried over from four points in a point-row u to the four points that correspond to them in any point-row u' perspective to u.
To prove this statement we construct a quadrangle K, L, M, N such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D. Take now any point S not in the plane of the quadrangle and construct the planes determined by S and all the seven lines of the figure. Cut across this set of planes by another plane not passing through S. This plane cuts out on the set of seven planes another [pg 20] quadrangle which determines four new harmonic points, A', B', C', D', on the lines joining S to A, B, C, D. But S may be taken as any point, since the original quadrangle may be taken in any plane through A, B, C, D; and, further, the points A', B', C', D' are the intersection of SA, SB, SC, SD by any line. We have, then, the remarkable theorem:
32. If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic.