Projective Harmonic Conjugate
In projective geometry, the harmonic conjugate point of a triple of points on the real projective line is defined by the following construction:
- 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.
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.
Read more about Projective Harmonic Conjugate: Cross-ratio Criterion, Of Midpoint, From Complete Quadrangle, Projective Conics
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