Back Harmonische Teilung German Cuaterna armónica Spanish Division harmonique French 調和共役 (幾何学) Japanese Проективен хармоничен конјугат Macedonian Harmonische ligging Dutch Harmonisk deling NB Conjugado harmônico projetivo Portuguese Diviziune armonică Romanian Гармоническая четвёрка Russian
A, D, B, C form a harmonic range.
KLMN is a complete quadrangle generating it.
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points 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 and B.[1]
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.
In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as (A, B; C, D) = −1.
- ^ 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