*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.