By Derrick Norman Lehmer

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For four harmonic points must always correspond to four harmonic points. In the same way the harmonic conjugate of D with respect to B and C must correspond to itself. Combining new points with old in this way, we may obtain as many self-corresponding points as we wish. We show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points. Thus, let a point P lie between A and B. Construct now D, the fourth harmonic [31] 34 [32] An Elementary Course in Synthetic Projective Geometry of C with respect to A and B.

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 32 An Elementary Course in Synthetic Projective Geometry FIG. 9 [30] 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.

By similar reasoning the point Q of intersection of the lines BC and B'C' must lie on this same line as well as the point R of intersection of CA and C'A'. Therefore the points P, Q, and R all lie on the same line m. If now we consider the figure a plane figure, the points P, Q, and R still all lie on a straight line, which proves the theorem. The converse is established in the same manner. 17 26. Fundamental theorem concerning two complete quadrangles. This theorem throws into our hands the following fundamental theorem concerning two complete quadrangles, a complete quadrangle being defined as the figure obtained by joining any four given points by straight lines in the six possible ways.

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An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer

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