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is of rank 2 or less, the points are collinear. 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 ...
A quadrilateral that is not a parallelogram has one and only one pedal point, called the Simson point, with respect to which the feet on the quadrilateral are collinear. [6] The Simson point of a trapezoid is the point of intersection of the two nonparallel sides. [7]: p. 186 No convex polygon with at least 5 sides has a Simson line. [8]
By extension, k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes. In Euclidean geometry , the Euclidean distance d ( a , b ) between two points a and b may be used to express the collinearity between three points by: [ 3 ] [ 4 ]
This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points ABC and abc (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines Ab, aB, Ac, aC, Bc, and bC, and their three intersection points X = Ab · aB, Y = Ac · aC, and Z = Bc · bC.
Points of the line X = Y have coordinates which may be written as (1,1,c). Three points, one from each of these lines, are collinear if and only if a = b + c. By selecting all the points on these lines where a, b and c are the field elements with absolute trace 0, the condition in the definition of a projective triad is satisfied.
A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.
A collineation, automorphism, or symmetry of the Fano plane is a permutation of the 7 points that preserves collinearity: that is, it carries collinear points (on the same line) to collinear points. By the Fundamental theorem of projective geometry , the full collineation group (or automorphism group , or symmetry group ) is the projective ...
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated ...