Fig. 4
For, by the converse of the last theorem, KK', LL', and NN' all meet in a point S (Fig. 4). Also LL', MM', and NN' meet in a point, and therefore in the same point S. Thus KK', LL', and MM' meet in a point, and so, by Desargues's theorem itself, A, B, and D are on a straight line.
27. Importance of the theorem. The importance of this theorem lies in the fact that, A, B, and C being given, an indefinite number of quadrangles K', L', M', N' my be found such that K'L' and M'N' meet in A, K'N' and L'M' in C, with L'N' passing through B. Indeed, the lines AK' and AM' may be drawn arbitrarily through A, and any line through B may be used to determine L' and N'. By joining these two points to C the points K' and M' are determined. Then the line [pg 18] joining K' and M', found in this way, must pass through the point D already determined by the quadrangle K, L, M, N. The three points A, B, C, given in order, serve thus to determine a fourth point D.
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 rôle 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.
