20
__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.