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From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. [3]: p.199
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 converse is also true; if the three closest points to P on three lines are collinear, and no two of the lines are parallel, then P lies on ...
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]
The Pappus graph. The Levi graph of the Pappus configuration is known as the Pappus graph.It is a bipartite symmetric cubic graph with 18 vertices and 27 edges. [3]Adding three more parallel lines to the Pappus configuration, through each triple of points that are not already connected by lines of the configuration, produces the Hesse configuration.
Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise ...
The three apex points always define a plane in three dimensions, and all three centers of similarity must lie in the plane containing the circular bases. Hence, the three centers must lie on the intersection of the two planes, which must be a line in three dimensions. [2] Monge's theorem can also be proved by using Desargues' theorem.
There exist four points such that no three are collinear (points not on a single line). In an affine plane, two lines are called parallel if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: [2] Given a point and a line, there is a unique line which contains the point and is parallel to the line.
There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are: G1: Every line contains at least 3 points; G2: Every two distinct points, A and B, lie on a unique line, AB. G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).