An Elementary Course in Synthetic Projective Geometry | Page 9

a __ point __Q__; and __LN__,
__L'N'__ meet in a point __B__ on the line __AC__; then the lines
__KM__ and __K'M'__ also meet in a point __D__ on the line
__AC__._
[Figure 4]
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 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.
[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