Collinear

Three or more points , , , ..., are said to be collinear if they lie on a single straight line . 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 for , 2, 3 are collinear iff the ratios of distances satisfy

(1)

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.,

(2)

or, in expanded form,

(3)

This can also be written in vector form as

(4)

where is the sum of components, , and .

The condition for three points , , and 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 in the point-line distance

(5)

giving simply

(6)

where denotes the cross product.

Since three points are collinear if for some constant , it follows that collinear points in three dimensions satisfy

			(7)
			(8)

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 , , , for which but .)

Three points , , and in trilinear coordinates are collinear if the determinant

(9)

(Kimberling 1998, p. 29).

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

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