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).
