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Secant line
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==Sets and {{mvar|n}}-secants== The concept of a secant line can be applied in a more general setting than Euclidean space. Let {{mvar|K}} be a finite set of {{mvar|k}} points in some geometric setting. A line will be called an {{mvar|n}}-secant of {{mvar|K}} if it contains exactly {{mvar|n}} points of {{mvar|K}}.<ref>{{citation|first=J. W. P.|last=Hirschfeld|author-link=James William Peter Hirschfeld|title=Projective Geometries over Finite Fields|year=1979|publisher=Oxford University Press|page=[https://archive.org/details/projectivegeomet0000hirs/page/70 70]|isbn=0-19-853526-0|url=https://archive.org/details/projectivegeomet0000hirs/page/70}}</ref> For example, if {{mvar|K}} is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or ''bisecant'') and a line passing through only one of them would be a 1-secant (or ''unisecant''). A unisecant in this example need not be a tangent line to the circle. This terminology is often used in [[incidence geometry]] and [[discrete geometry]]. For instance, the [[Sylvester–Gallai theorem]] of incidence geometry states that if {{mvar|n}} points of Euclidean geometry are not [[collinearity|collinear]] then there must exist a 2-secant of them. And the original [[orchard-planting problem]] of discrete geometry asks for a bound on the number of 3-secants of a finite set of points. Finiteness of the set of points is not essential in this definition, as long as each line can intersect the set in only a finite number of points.
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