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In geometry, collinearity of a set of points is the property of their lying on a single line. [1] A set of points with this property is said to be collinear (sometimes spelled as colinear [2]). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
Möbius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight. Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence. Such a mapping ...
In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear.
Menelaus's theorem, case 1: line DEF passes inside triangle ABC. In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A ...
Collinear – in the same line; Parallel – in the same direction. Transverse – intersecting at any angle, i.e. not parallel. Orthogonal (or perpendicular) – at a right angle (at the point of intersection). Elevation – along a curve from a point on the horizon to the zenith, directly overhead.
Thus, in Euclidean geometry three non-collinear points determine a circle (as the circumcircle of the triangle they define), but four points in general do not (they do so only for cyclic quadrilaterals), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by ...
Known as the cyclos, the device draws circles similarly to the compass, but does so not by defining a radius or providing a center, but by two points defining a diameter, or by three non-collinear points defining the arc. In either case, a single application of the tool is used, by definition, to draw a complete circle.
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
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