By Peter Sinn Nachtrieb
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The quadrangle P, Q, R, S therefore has two opposite sides through B, two through D, and the remaining pair through A and C. A and C are thus harmonic conjugates with respect to B and D. We may sum up the discussion, therefore, as follows: 30. If A and C are harmonic conjugates with respect to B and D, then B and D are harmonic conjugates with respect to A and C.  31. Importance of the notion. The importance of the notion of four harmonic points lies in the fact that it is a relation which is carried over from four points in a point-row u to the four points that correspond to them in any point-row u' perspective to u.
Also LL', MM', and NN' meet in a  18 An Elementary Course in Synthetic Projective Geometry point, and therefore in the same point S. Thus KK', LL', and MM' meet in a point, and so, by Desargues's theorem itself, A, B, and D are on a straight line.  27. Importance of the theorem. The importance of this theorem lies in the fact that, A, B, and C being given, an indefinite number of quadrangles K', L', M', N' my be found such that K'L' and M'N' meet in A, K'N' and L'M' in C, with L'N' passing through B.
Show that a line at right angles to the direction of the gun at its muzzle will pass through any point in the plane twice or not at all. ) 7. Two lines u and u' revolve about two points U and U' respectively in the same plane. They go in the same direction and at the same rate of speed, but one has an angle a the start of the other. Show that they generate a point-row of the second order. PROBLEMS 39 8. Discuss the question given in the last problem when the two lines revolve in opposite directions.