Projective harmonic conjugates

In projective geometry, a pair of harmonic conjugate points on the real projective line is defined by the following harmonic construction:
“Given three collinear points A, B, C, let L be a point not lying on their join and let any line through C met LA, LB at M, N respectively. If AN and BM met at K, and LK meets AB at D, then D is called the harmonic conjugate of C with respect to A, B.”

So is the harmonic construction introduced by 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.

Crossratio criterion

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. Given this metric context, the harmonic range is characterized by a crossratio of minus one.

Projective conics

A conic in the projective plane is a curve which has the following property: If P is a point not on the conic, and if lines through P meet the conic at points A and B, then the harmonic conjugate of P with respect to A and B forms a line. The point P is called the pole of that line of harmonic conjugates, and this line is called the polar line of P with respect to the conic.


  • R. L. Goodstein & E. J. F. Primrose (1953) Axiomatic Projective Geometry, University College Leicester (publisher). This text follows synthetic geometry. Harmonic construction on page 11.
  • Juan Carlos Alverez (2000) Projective Geometry, see Chapter 2: The Real Projective Plane, section 3: Harmonic quadruples and von Staudt's theorm.
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