Through a point 
in the plane of a triangle 
, draw parallelians
through a point as illustrated above. Then there exist four points 
for which 
, i.e., for which the
segments of the parallels have equal length.
To restrict these four points, let the length of 
be considered negative if 
lies on the extension of 
beyond 
and 
lies on the extension of 
beyond 
, and positive otherwise. Define the lengths of the other two
parallelians to be signed in the analogous manner. With this sign convention, there
is a unique point 
for which the signed parallelians have equal length. This
point is called the equal parallelians point of 
.
It has equivalent triangle center functions
 |  |  |
(1)
|
 |  |  |
(2)
|
and is Kimberling center 
(Kimberling 1998, p. 104).
The length 
of the equal parallelians is
 |  |  |
(3)
|
 |  |  |
(4)
|
As is true for general parallelians, those for the 
lie on an ellipse. The center of this ellipse has triangle
center function
 |
(5)
|
which is not a Kimberling center.
The equal parallelians point is also the perspector of the incentral triangle 
and anticomplementary
triangle 
of 
(Kimberling 1998, p. 105).
