There exist points 
, 
, and 
on segments 
, 
, and 
of a triangle, respectively, such that
 |
(1)
|
and the lines 
,
,
concur. The point of concurrence is called the trisected perimeters point, which
is Kimberling center 
. Near the end of the 20th century, P. Yff found
trilinears for 
in terms of the unique real root 
of the cubic polynomial
 |
(2)
|
The triangle center function is then given by
![alpha=bc[r^2-(2c+a)r+(-a^2+b^2+2c^2+2bc+3ca+2ab],](/images/equations/TrisectedPerimeterPoint/NumberedEquation3.svg) |
(3)
|
as shown by Yff in a geometry conference held at Miami University of Ohio, October 2, 2004 (Kimberling).
It can be derived by noting that the trilinears for Cevians from 
and 
passing through the point 
are given by 
and 
, respectively. Computing the some of distances
to these points from the vertex 
(1:0:0) and analogously for vertices 
and 
gives the three equation
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Finding a Gröbner basis for
 |
(7)
|
where 
is the semiperimeter of the reference
triangle, simultaneously together with the condition
 |
(8)
|
for the trilinears to be exact then gives a solution for 
in terms of a sixth-degree polynomial (which is third-degree
in 
).
