the minus sign being given to the ratio on account of the fact that A and C are always separated from B and D, so that one or three of the segments AB, CD, AD, CB must be negative.
45. Writing the last equation in the form
CB : AB = -CD : AD,
and using the fundamental relation connecting three points on a line,
PR + RQ = PQ,
which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written
(CB - BA) : AB = -(CA - DA) : AD,
or
(AB - AC) : AB = (AC - AD) : AD;