30
ction: 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.
*6.* A line is drawn cutting the sides of a triangle ABC in the points _A'_, _B'_, _C'_ the point _A'_ lying on the side BC, etc. The harmonic conjugate of _A'_ with respect to B and C is then constructed and called _A"_. Similarly, _B"_ and _C"_ are constructed. Show that _A"B"C"_ lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point.
CHAPTER III
- COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS
[Figure 9]
FIG. 9
*47. Superposed fundamental forms. Self-corresponding elements.* We have seen (§ 37) that two projective point-rows may be superposed upon the same straight line. This happens, for example, when two pencils which are projective to each other are cut across by a straight line. It is also possible for two projective pencils to have the same center. This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pencils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen,