Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In projective geometry, the harmonic conjugate point of a triple of points on the real projective line is defined by the following construction due to Karl von Staudt: “Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively. If AN and BM meet at K, and LK meets AB at D, then D is called the harmonic conjugate of C with respect to A, B.” (See Goodstein and Primrose (1953)). What is remarkable is that the point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N. This fact follows from Desargues theorem; it can also be defined in terms of the cross-ratio as (A, B; C, D) = –1. The four points are sometimes called a harmonic range on the real projective line. When this line is endowed with the ordinary metric interpretation via real numbers, then the projective tool of cross-ratio is in force.

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