collinearity

Collinearity is a geometric relation among points: a set of points is collinear if they all lie on a single straight line. For two points this is always true; for three or more points, the property means that there is a line that contains every point in the set.

In the Euclidean plane, three distinct points A(x1,y1), B(x2,y2), and C(x3,y3) are collinear if and only if the area of triangle ABC is zero, equivalently the determinant |x1 y1 1; x2 y2 1; x3 y3 1| equals zero. This is also equivalent to (x2 - x1)(y3 - y1) = (x3 - x1)(y2 - y1), or that the slopes AB and AC are equal (when defined). In three-dimensional space, three points are collinear if the vectors AB and AC are parallel, i.e., their cross product AB × AC is the zero vector; equivalently AB and AC are linearly dependent.

A set of more than three points is collinear if every triple among them is collinear, or equivalently all points lie on a single line. Degenerate cases include repeated points, which do not affect the definition.

Collinearity has applications in various fields, including geometry problem solving, computer graphics, geographic information systems, and data analysis. It is used to detect alignment, simplify geometric models, perform line fitting, and test spatial relationships. The concept contrasts with non-collinearity, where no single line contains all the points.