Given collinear points 
, 
, 
, and 
, 
and 
are harmonic conjugates with respect to 
and 
if
 |
(1)
|
and 
are also harmonic conjugates with respect to 
and 
.
The distances between such points are said to be in a harmonic range, and the line segment depicted above is
called a harmonic segment. In other words, harmonic points divide a line
segment internally and externally in the same ratio. If 
, then
 |  |  |
(2)
|
 |  |  |
(3)
|
Harmonic conjugates are also defined for a triangle. If 
and 
have trilinear coordinates 
and 
, then the trilinear
coordinates of the harmonic conjugates are
 |  |  |
(4)
|
 |  |  |
(5)
|
(Kimberling 1994).
