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, indeed, that if B and D are harmonic conjugates with respect to A and C, then the point-row described by B is projective to the point-row described by D, and that A and C are self-corresponding points.  Consider more generally the case of two pencils perspective to each other with axis of perspectivity u’ (Fig. 9).  Cut across them by a line u.  We get thus two projective point-rows superposed on the same line u, and a moment’s reflection serves to show that the point N of intersection u and u’ corresponds to itself in the two point-rows.  Also, the point M, where u intersects the line joining the centers of the two pencils, is seen to correspond to itself.  It is thus possible for two projective point-rows, superposed upon the same line, to have two self-corresponding points.  Clearly M and N may fall together if the line joining the centers of the pencils happens to pass through the point of intersection of the lines u and u’.

[Figure 10]

FIG. 10

48. We may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all.  Thus we may take two lines revolving about a fixed point S and always making the same angle a with each other (Fig. 10).  They will cut out on any line u in the plane two point-rows which are easily seen to be projective.  For, given any four rays SP which are harmonic, the four corresponding rays SP’ must also be harmonic, since they make the same angles with each other.  Four harmonic points P correspond, therefore, to four harmonic points P’.  It is clear, however, that no point P can coincide with its corresponding point P’, for in that case the lines PS and P’S would coincide, which is impossible if the angle between them is to be constant.