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

124.  Double-points in an involution. The points C and C’ describe projective point-rows, as may be seen by fixing the points L and M.  The self-corresponding points, of which there are two or none, are called the double-points in the involution.  It is not difficult to see that the double-points in the involution are harmonic conjugates with respect to corresponding points in the involution.  For, fixing as before the points L and M, let the intersection of the lines CL and C’M be P (Fig. 33).  The locus of P is a conic which goes through the double-points, because the point-rows C and C’ are projective, and therefore so are the pencils LC and MC’ which generate the locus of P.  Also, when C and C’ fall together, the point P coincides with them.  Further, the tangents at L and M to this conic described by P are the lines LB and MB.  For in the pencil at L the ray LM common to the two pencils which generate the conic is the ray LB’ and corresponds to the ray MB of M, which is therefore the tangent line to the conic at M.  Similarly for the tangent LB at L. LM is therefore the polar of B with respect to this conic, and B and B’ are therefore harmonic conjugates with respect to the double-points.  The same discussion applies to any other pair of corresponding points in the involution.

[Figure 34]

FIG. 34