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,
(AB — AC) : AB = (AC — AD) : AD;
so that AB, AC, and AD are three quantities in hamonic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third. Also, from this last proportion comes the familiar relation
which is convenient for the computation of the distance AD when AB and AC are given numerically.
46. Anharmonic ratio. The corresponding relations between the trigonometric functions of the angles determined by four harmonic lines are not difficult to obtain, but as we shall not need them in building up the theory of projective geometry, we will not discuss them here. Students who have a slight acquaintance with trigonometry may read in a later chapter (§ 161) a development of the theory of a more general relation, called the anharmonic ratio, or cross ratio, which connects any four points on a line.
1. Draw through a given point a line which shall pass through the inaccessible point of intersection of two given lines. The following construction may be made to depend upon Desargues’s theorem: Through the given point P draw any two rays cutting the two lines in the points AB’ and A’B, A, B, lying on one of the given lines and A’, B’, on the other. Join AA’ and BB’, and find their point of intersection S. Through S draw any other ray, cutting the given lines in CC’. Join BC’ and B’C, and obtain their point of intersection Q. PQ is the desired line. Justify this construction.
2. To draw through a given point P a line which shall meet two given lines in points A and B, equally distant from P. Justify the following construction: Join P to the point S of intersection of the two given lines. Construct the fourth harmonic of PS with respect to the two given lines. Draw through P a line parallel to this line. This is the required line.
3. Given a parallelogram in the same plane with a given segment AC, to construct linearly the middle point of AC.
4. Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made.
5. Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point.