Projective harmonic conjugates

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.

References

* 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) [http://www.math.poly.edu/~alvarez/teaching/projective-geometry Projective Geometry] , see Chapter 2: The Real Projective Plane, section 3: Harmonic quadruples and von Staudt's theorm.