*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__._ *28.* In a complete quadrangle the line joining any two points is called the _opposite side_ to the line joining the other two points. The result of the preceding paragraph may then be stated as follows: Given three points, _A_, _B_, _C_, in a straight line, if a pair of opposite sides of a complete quadrangle pass through _A_, and another pair through _C_, and one of the remaining two sides goes through _B_, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed.