Collinearity (geometry)

In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".

Points on a line

In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry, where lines are represented in the standard model by great circles of a sphere, sets of collinear points lie on the same great circle. Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being ''in a row''.

A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property.
The linear maps (or linear functions) of vector spaces, viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations. In projective geometry these linear mappings are called '' homographies'' and are just one type of collineation.

Examples in Euclidean geometry

Triangles

In any triangle the following sets of points are collinear:

*The orthocenter, the circumcenter, the centroid, the Exeter point, the de Longchamps point, and the center of the nine-point circle are collinear, all falling on a line called the Euler line.
*The de Longchamps point also has other collinearities.
*Any vertex, the tangency of the opposite side with an excircle, and the Nagel point are collinear in a line called a splitter of the triangle.
*The midpoint of any side, the point that is equidistant from it along the triangle's boundary in either direction (so these two points bisect the perimeter), and the center of the Spieker circle are collinear in a line called a cleaver of the triangle. (The Spieker circle is the incircle of the medial triangle, and its center is the center of mass of the perimeter of the triangle.)
*Any vertex, the tangency of the opposite side with the incircle, and the Gergonne point are collinear.
*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.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).
*A triangle's incenter, the midpoint of an altitude, and the point of contact of the corresponding side with the excircle relative to that side are collinear.Altshiller-Court, Nathan. ''College Geometry'', Dover Publications, 1980.
* Menelaus' theorem states that three points $P_1, P_2, P_3$ on the sides (some extended) of a triangle opposite vertices $A_1,A_2, A_3$ respectively are collinear if and only if the following products of segment lengths are equal:

::$P_1A_2 \cdot P_2A_3 \cdot P_3A_1=P_1A_3 \cdot P_2A_1 \cdot P_3A_2.$
* The incenter, the centroid, and the Spieker circle's center are collinear.
*The circumcenter, the Brocard midpoint, and the Lemoine point of a triangle are collinear.
*Two perpendicular lines intersecting at the orthocenter of a triangle each intersect each of the triangle's extended sides. The midpoints on the three sides of these points of intersection are collinear in the Droz–Farny line.

Quadrilaterals

*In a convex quadrilateral whose opposite sides intersect at and , the