The intersection of two lines 
and 
in two dimensions with, 
containing the points 
and 
, and 
containing the points 
and 
, is given by
 |  |  | (1) |
 |  |  | (2) |
where 
denotes a determinant. This corresponds to simultaneously solving
 |  |  | (3) |
 |  |  | (4) |
for 
and 
. Other treatments are given by Antonio (1992) and Hill (1994).
The intersections of two lines given in trilinear coordinates as
 |  |  | (5) |
 |  |  | (6) |
is
 | (7) |
Pseudocode for segment intersection is given by de Berg et al. (2000).
Three lines in trilinear coordinates
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
concur if their trilinear coordinates satisfy
 | (11) |
in which case the point is
 | (12) |
Three lines in Cartesian coordinates concur if the coefficients of the lines
 |  |  | (13) |
 |  |  | (14) |
 |  |  | (15) |
satisfy
 | (16) |
In three dimensions, the algebra becomes more complicated. The intersection of two lines containing the points 
and 
, and 
and 
, respectively, can also be found directly by simultaneously solving
 |  |  | (17) |
 |  |  | (18) |
together with the condition that the four points be coplanar (i.e., the lines are not skew),
![|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|=(x_3-x_1)·[(x_2-x_1)x(x_4-x_3)]=0](http://mathworld.wolfram.com/images/equations/Line-LineIntersection/NumberedEquation5.svg) | (19) |
for 
, eliminating 
and 
. This set of equations can be solved for 
to yield
 | (20) |
where
 |  |  | (21) |
 |  |  | (22) |
 |  |  | (23) |
(Hill 1994).
The point of intersection can then be immediately found by plugging back in for 
to obtain
 | (24) |
A slightly more symmetrical and concise form can obtained by additionally defining
 |  |  | (25) |
 |  |  | (26) |
 |  |  | (27) |
where 
denotes a unit vector, then
 | (28) |
(Goldman 1990).
