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five points determine a conic
(詳細はgeometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines. In projective geometry, a line is defined via three points, all families of circles are conics passing through two points of a line at infinity, and coordinates are the basic conic objects of study, and are defined as all conics passing through the line and two points at infinity.
Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the affine plane and projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.
== Proofs ==
This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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