141. Introduction of infinite point; center of involution. We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called the center of the involution. Since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points, the center is midway between the double points when they exist. To construct the center (Fig. 39) we draw as usual through A and A' any two rays and cut them by a line parallel to AA' in the points K and M. Join these points to B and B', thus determining on AK and AN the points L and N. LN meets AA' in the center O of the involution.
142. Fundamental metrical theorem. From the figure we see that the triangles OLB' and PLM are similar, P being the intersection of KM and LN. Also the triangles KPN and BON are similar. We thus have
OB : PK = ON : PN
and
OB' : PM = OL : PL;
whence
OB · OB' : PK · PM = ON · OL : PN · PL.
In the same way, from the similar triangles OAL and PKL, and also OA'N and PMN, we obtain
OA · OA' : PK · PM = ON · OL : PN · PL,