An Elementary Course in Synthetic Projective Geometry | Page 9

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 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 r?le 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.
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 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._

*33. Four harmonic lines.* We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem
_Four harmonic lines are cut by any transversal in four harmonic points._

*34. Four harmonic planes.* We also define four harmonic planes as four planes through a line which pass one through each of four harmonic points, and we may show that
_Four harmonic planes are cut by any plane