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 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_,