Information from Kimberling's Encyclopedia of Triangle Centers |
Trilinears x : y : z (see below)
Barycentrics ax : by : cz
There exist points A', B', C' on segments BC, CA, AB, respectively, such that AB' + AC' = BC' + BA' = CA' + CB' = (a + b + c)/3, and the lines AA', BB', CC' concur in X(369). Near the end of the 20th century, 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 - (cb^{2} + ac^{2} + ba^{2} + 5bc^{2} + 5ca^{2} + 5ab^{2} + 9abc),
as follows: x = bc(r - c + a)(r - a + b). Here x(a,c,b) ≠ x(a,b,c), so that y and z are not obtained from x by cyclically permutating a,b,c. At the geometry conference held at Miami University of Ohio, October 2, 2004, Yff, proved that X(369) is also given by x_{1} : y_{1} : z_{1} where y_{1} : z_{1} are given by cyclic permutations of a,b,c, in x_{1}, where
x_{1} = bc[r^{2} - (2c + a)r + (- a^{2} + b^{2} + 2c^{2} + 2bc + 3ca + 2ab].
His presentation included a proof that there is only one point for which AB' + AC' = BC' + BA' = CA' + CB' .