Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

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.

Such points do not lie on a "straight line" in the Euclidean sense, and are not thought of as being in a row.

): This determinant is, by Heron's formula, equal to −16 times the square of the area of a triangle with side lengths d(AB), d(BC), d(AC); so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices A, B, C has zero area (so the vertices are collinear).

Equivalently, a set of at least three distinct points are collinear if and only if, for every three of those points A, B, C with d(AC) greater than or equal to each of d(AB) and d(BC), the triangle inequality d(AC) ≤ d(AB) + d(BC) holds with equality.

Two numbers m and n are not coprime—that is, they share a common factor other than 1—if and only if for a rectangle plotted on a square lattice with vertices at (0, 0), (m, 0), (m, n), (0, n), at least one interior point is collinear with (0, 0) and (m, n).

Given a partial geometry P, where two points determine at most one line, a collinearity graph of P is a graph whose vertices are the points of P, where two vertices are adjacent if and only if they determine a line in P. In statistics, collinearity refers to a linear relationship between two explanatory variables.

More commonly, the issue of multicollinearity arises when there is a "strong linear relationship" among two or more independent variables, meaning that where the variance of