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_,