are so
situated that the lines __AA'__, __BB'__, and __CC'__ all meet in a point,
then the pairs of sides __AB__ and __A'B'__, __BC__ and __B'C'__, __CA__
and __C'A'__ all meet on a straight line, and conversely._

[Figure 3]

FIG. 3

Let the lines _AA'_, _BB'_, and _CC'_ meet in the point _M_ (Fig. 3).
Conceive of the figure as in space, so that _M_ is the vertex of a
trihedral angle of which the given triangles are plane sections. The lines
_AB_ and _A'B'_ are in the same plane and must meet when produced, their
point of intersection being clearly a point in the plane of each triangle
and therefore in the line of intersection of these two planes. Call this
point _P_. By similar reasoning the point _Q_ of intersection of the lines
_BC_ and _B'C'_ must lie on this same line as well as the point _R_ of
intersection of _CA_ and _C'A'_. Therefore the points _P_, _Q_, and _R_
all lie on the same line _m_. If now we consider the figure a plane
figure, the points _P_, _Q_, and _R_ still all lie on a straight line,
which proves the theorem. The converse is established in the same manner.

*26. Fundamental theorem concerning two complete quadrangles.* This
theorem throws into our hands the following fundamental theorem concerning
two complete quadrangles, a _complete quadrangle_ being defined as the
figure obtained by joining any four given points by straight lines in the
six possible ways.

_Given two complete quadrangles, __K__, __L__, __M__, __N__ and __K'__,
__L'__, __M'__, __N'__, so related that __KL__, __K'L'__, __MN__, __M'N'__
all meet in a point __A__; __LM__, __L'M'__, __NK__, __N'K'__ all meet in
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