By Derrick Norman Lehmer
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Additional resources for An Elementary Course in Synthetic Projective Geometry
If the two point-rows are perspectively related, then their common point is evidently a self-corresponding point. The converse is also true, and we have the very important theorem: 52. If in two protective point-rows, the point of intersection corresponds to itself, then the point-rows are in perspective position. Let the two point-rows be u and u' (Fig. 11). Let A and A', B and B', be corresponding points, and let also the point M of intersection of u and u' correspond to itself. Let AA' and BB' meet in the point S.
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.
Four harmonic points. Four points, A, B, C, D, related as in the preceding theorem are called four harmonic points. The 29. Four harmonic points 19 point D is called the fourth harmonic of B with respect to A and C. Since B and D play exactly the same rôle in the above construction, B is also the fourth harmonic of D with respect to A and C. B and D are called harmonic conjugates with respect to A and C. We proceed to show that A and C are also harmonic conjugates with respect to B and D—that is, that it is possible to find a quadrangle of which two opposite sides shall pass through B, two through D, and of the remaining pair, one through A and the other through C.
An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer