Trisected Perimeter Point
There exist points A^', B^', and C^' on segments BC, CA, and AB of a triangle, respectively, such that
| A^'C+CB^'=B^'A+AC^'=C^'B+BA^' |
(1)
|
and the lines AA^', BB^', CC^' concur. The point of concurrence is called the trisected perimeters point, which is Kimberling center X_(369). Near the end of the 20th century, P. Yff found trilinears for X_(369) in terms of the unique real root r of the cubic polynomial
| 2t^3-3(a+b+c)t^2+(a^2+b^2+c^2+8bc+8ca+8ab)t-(b^2c+c^2a+a^2b+5bc^2+5ca^2+5ab^2+9abc)=0. |
(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)], |
(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 B and C passing through the point alpha:beta:gamma are given by alpha:0:gamma and alpha:beta:0, respectively. Computing the some of distances to these points from the vertex A (1:0:0) and analogously for vertices B and C gives the three equation
Finding a Gröbner basis for
| s_A=s_B=s_C=2/3s, |
(7)
|
where s is the semiperimeter of the reference triangle, simultaneously together with the condition
for the trilinears to be exact then gives a solution for alpha in terms of a sixth-degree polynomial (which is third-degree in alpha^2).
See also
Cevian, Perimeter, Semiperimeter, TrisectionExplore with Wolfram|Alpha
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References
Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Clark Kimberling's Encyclopedia of Triangle Centers--ETC." https://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X369.Referenced on Wolfram|Alpha
Trisected Perimeter PointCite this as:
Weisstein, Eric W. "Trisected Perimeter Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrisectedPerimeterPoint.html