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._