*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;*

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

[formula]

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.