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In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only ...
The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line. The three-point circuits are the complements of the four-point circuits, so it is possible to partition the seven points of the plane into two circuits, one of each kind. Thus, the Fano plane is also Eulerian.
Finite point sets in general position (no three collinear) do always contain a big clique, so the conjecture is true for .Additionally, finite point sets that have no five mutually-visible points (such as the intersections of the integer lattice with convex sets) do always contain many collinear points, so the conjecture is true for .
A complete quadrangle consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the Fano axiom, often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete ...
Points that are incident with the same line are said to be collinear. The set of all points incident with the same line is called a range. If P 1 = (x 1, y 1, z 1), P 2 = (x 2, y 2, z 2), and P 3 = (x 3, y 3, z 3), then these points are collinear if and only if
Three or more collinear points, where the circumcircles are of infinite radii. Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical. In this case the Voronoi diagram contains vertices of degree four or greater and its dual graph contains polygonal faces with four or more sides.
However, not all sets of four points, no three collinear, are linearly separable in two dimensions. The following example would need two straight lines and thus is not linearly separable: Notice that three points which are collinear and of the form "+ ⋅⋅⋅ — ⋅⋅⋅ +" are also not linearly separable.
The interval AB is the segment AB and its end points A and B. The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB]. The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear. An angle consists of a point O (the vertex) and two non-collinear rays out from O ...