separated from each other by _A_ and _C_.




*39. Harmonic conjugate of the point at infinity.* It is natural to
inquire what position of _B_ corresponds to the infinitely distant
position of _D_. We have proved (§ 27) that the particular quadrangle _K_,
_L_, _M_, _N_ employed is of no consequence. We shall therefore avail
ourselves of one that lends itself most readily to the solution of the
problem. We choose the point _L_ so that the triangle _ALC_ is isosceles
(Fig. 7). Since _D_ is supposed to be at infinity, the line _KM_ is
parallel to _AC_. Therefore the triangles _KAC_ and _MAC_ are equal, and
the triangle _ANC_ is also isosceles. The triangles _CNL_ and _ANL_ are
therefore equal, and the line _LB_ bisects the angle _ALC_. _B_ is
therefore the middle point of _AC_, and we have the theorem

_The harmonic conjugate of the middle point of __AC__ is at infinity._

[Figure 7]

FIG. 7




*40. Projective theorems and metrical theorems. Linear construction.* This
theorem is the connecting link between the general protective theorems
which we have been considering so far and the metrical theorems of
ordinary geometry. Up to this point we have said nothing about