the triangle with those points as vertices has zero area. Collinearity of points whose pairwise distances are given A set of at least three distinct points is called straight, meaning all the points are collinear, if and only if, for every three of those points , the following determinant of a Cayley–Menger determinant is zero (with meaning the distance between and , etc.): This determinant is, by Heron's formula, equal to −16 times the square of the area of a triangle with side lengths ; so checking if this determinant equals zero is equivalent to checking whether the triangle with vertices 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 with greater than or equal to each of and , the triangle inequality holds with equality. Number theory Two numbers and 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 , at least one interior point is collinear with and . Concurrency (plane dual) In various plane geometries the notion of interchanging the roles of "points" and "lines" while preserving the relationship between them is called plane duality. Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity. Collinearity graph Given a partial geometry , where two points determine at most one line, a collinearity graph of is a graph whose vertices are the points of , where two vertices are adjacent if and only if they determine a line in . Usage in statistics and econometrics In statistics, collinearity refers to a linear relationship between two explanatory variables. Two variables are perfectly collinear if there is an exact linear relationship between the two, so the correlation between them is equal to 1 or −1. That is, and are perfectly collinear if there exist parameters and such that, for all observations , we have This means that if the various observations are plotted in the plane, these points are collinear in the sense defined earlier in