Three or more points P_1, P_2, P_3, ..., are said to be collinear if they lie on a single straight line L. A line on which points lie, especially if it is related to a geometric figure such as a triangle, is sometimes called an axis.

Two points are trivially collinear since two points determine a line.

Three points x_i=(x_i,y_i,z_i) for i=1, 2, 3 are collinear iff the ratios of distances satisfy


A slightly more tractable condition is obtained by noting that the area of a triangle determined by three points will be zero iff they are collinear (including the degenerate cases of two or all three points being concurrent), i.e.,

 |x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|=0

or, in expanded form,


This can also be written in vector form as


where Tr(A) is the sum of components, x=(x_1,x_2,x_3), and y=(y_1,y_2,y_3).

The condition for three points x_1, x_2, and x_3 to be collinear can also be expressed as the statement that the distance between any one point and the line determined by the other two is zero. In three dimensions, this means setting d=0 in the point-line distance


giving simply


where x denotes the cross product.

Since three points are collinear if x_3=x_1+c(x_2-x_1) for some constant c, it follows that collinear points in three dimensions satisfy

det(x_1x_2x_3)=|x_1 x_2 x_1+c(x_2-x_1); y_1 y_2 y_1+c(y_2-y_1); z_1 z_2 z_1+c(z_2-z_1)|

by the rules of determinant arithmetic. While this is a necessary condition for collinearity, it is not sufficient. (If any single point is taken as the origin, the determinant will clearly be zero. Another counterexample is provided by the noncollinear points x_1=(16,20,20), x_2=(5,6,6), x_3=(15,9,9), for which det(x_1x_2x_3)=0 but d=22898!=0.)

Three points alpha_1:beta_1:gamma_1, alpha_2:beta_2:gamma_2, and alpha_3:beta_3:gamma_3 in trilinear coordinates are collinear if the determinant

 |alpha_1 beta_1 gamma_1; alpha_2 beta_2 gamma_2; alpha_3 beta_3 gamma_3|=0

(Kimberling 1998, p. 29).

Let points P_1, P_2, and P_3 lie, one each, on the sides of a triangle DeltaA_1A_2A_3 or their extensions, and reflect these points about the midpoints of the triangle sides to obtain P_1^', P_2^', and P_3^'. Then P_1^', P_2^', and P_3^' are collinear iff P_1, P_2, and P_3 are (Honsberger 1995).

See also

Axis, Concyclic, Configuration, Directed Angle, Droz-Farny Theorem, General Position, Line, N-Cluster, Point-Line Distance--3-Dimensional, Sylvester's Line Problem Explore this topic in the MathWorld classroom

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Coxeter, H. S. M. and Greitzer, S. L. "Collinearity and Concurrence." Ch. 3 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 51-79, 1967.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 153-154, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

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Cite this as:

Weisstein, Eric W. "Collinear." From MathWorld--A Wolfram Web Resource.

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