28. In a complete quadrangle the line joining any two points is called the opposite side to the line joining the other two points.  The result of the preceding paragraph may then be stated as follows: 

Given three points, A, B, C, in a straight line, if a pair of opposite sides of a complete quadrangle pass through A, and another pair through C, and one of the remaining two sides goes through B, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed.

29.  Four harmonic points. Four points, A, B, C, D, related as in the preceding theorem are called four harmonic points.  The point D is called the fourth harmonic of _B__ with respect to __A__ and __C_.  Since B and D play exactly the same role in the above construction, B_ is also the fourth harmonic of __D__ with respect to __A__ and __C_. B and D are called harmonic conjugates with respect to _A__ and __C_.  We proceed to show that A and C are also harmonic conjugates with respect to B and D—­that is, that it is possible to find a quadrangle of which two opposite sides shall pass through B, two through D, and of the remaining pair, one through A and the other through C.

[Figure 5]

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 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.