A Sylvester-Gallai result for concurrent lines in the complex plane. (English) Zbl 1530.52011
The Sylvester-Gallai theorem says that if a finite set of points in the real plane is not contained in a line, then there exists a line containing exactly two points of the set. In the complex plane, the analogous theorem does not hold. The author studies conditions which ensure that the conclusion of the Sylvester-Gallai theorem also holds in the complex plane. The main result says that if a finite set \(S\) of non-collinear points in the complex plane is contained in the union of \(m\) concurrent lines, and if one of those lines contains more than \(m-2\) points of \(S\) (not including the point of concurrency), then there exists a line containing exactly two points of \(S\). The bound is optimal because it is achieved in the Fermat configurations. The new idea in the proof consists in ordering complex numbers by their real part.
Reviewer: Norbert Knarr (Stuttgart)
MSC:
| 52C30 | Planar arrangements of lines and pseudolines (aspects of discrete geometry) |
| 51A45 | Incidence structures embeddable into projective geometries |
| 30C99 | Geometric function theory |
Keywords:
Sylvester-Gallai theorem; incidence geometry; Green’s identity; Hesse configuration; Fermat configurationsReferences:
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