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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 no-three-in-line drawing of a complete graph is a special case of this result with =. [12] The no-three-in-line problem also has applications to another problem in discrete geometry, the Heilbronn triangle problem. In this problem, one must place points, anywhere in a unit square, not restricted to a grid. The goal of the placement is to ...
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. Edges of the Voronoi diagram going to infinity are not defined by this relation in case of a finite set P.
The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun. [2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three ...
In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. [1] The line through these points is the Simson line of P, named for Robert Simson. [2] The concept was first published, however, by William Wallace in 1799, [3] and is sometimes called the Wallace line. [4]
The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between ...
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 .
In the case of three non-collinear points in the plane, the triangle with these points as its vertices has a unique Steiner inellipse that is tangent to the triangle's sides at their midpoints. The major axis of this ellipse falls on the orthogonal regression line for the three vertices. [ 7 ]