_Given two complete quadrangles, __K__, __L__, __M__, __N__ and __K’__, __L’__, __M’__, __N’__, so related that __KL__ and __K’L’__ meet in __A__, __MN__ and __M’N’__ in __A’__, __KN__ and __K’N’__ in __B__, __LM__ and __L’M’__ in __B’__, __LN__ and __L’N’__ in __C__, and __KM__ and __K’M’__ in __C’__, then, if __A__, __A’__, __B__, __B’__, and __C__ are in a straight line, the point __C’__ also lies on that straight line._ The theorem follows from Desargues’s theorem (Fig. 32). It is seen that _KK’_, _LL’_, _MM’_, _NN’_ all meet in a point, and thus, from the same theorem, applied to the triangles _KLM_ and _K’L’M’_, the point _C’_ is on the same line with _A_ and _B’_. As in the simpler case, it is seen that there is an indefinite number of quadrangles which may be drawn, two sides of which go through _A_ and _A’_, two through _B_ and _B’_, and one through _C_. The sixth side must then go through _C’_. Therefore, *122.* _Two pairs of points, __A__, __A’__ and __B__, __B’__, being given, then the point __C’__ corresponding to any given point __C__ is uniquely determined._ The construction of this sixth point is easily accomplished. Draw through _A_ and _A’_ any two lines, and cut across them by any line through _C_ in the points _L_ and _N_. Join _N_ to _B_ and _L_ to _B’_, thus determining the points _K_ and _M_ on the two lines through _A_ and _A’_, The line _KM_ determines the desired point _C’_. Manifestly, starting from _C’_, we come in this way always to the same point _C_. The particular quadrangle employed is of no consequence. Moreover, since one pair of opposite sides in a complete quadrangle is not distinguishable in any way from any other, the same set of six points will be obtained by starting from the pairs _AA’_ and _CC’_, or from the pairs _BB’_ and _CC’_. *123. Definition of involution of points on a line.* _Three pairs of points on a line are said to be in involution if through each pair may be drawn a pair of opposite sides of a complete quadrangle. If two pairs are fixed and one of the third pair describes the line, then the other also describes the line, and the points of the line are said to be paired in the involution determined by the two fixed pairs._ [Figure 33] FIG. 33