[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 be found such that K’L’ and M’N’ meet in A, K’N’ and L’M’ in C, with L’N’ passing through B.  Indeed, the lines AK’ and AM’ may be drawn arbitrarily through A, and any line through B may be used to determine L’ and N’.  By joining these two points to C the points K’ and M’ are determined.  Then the line joining K’ and M’, found in this way, must pass through the point D already determined by the quadrangle K, L, M, N. The three points _A__, __B__, __C__, given in order, serve thus to determine a fourth point __D__._