## X(369) (1ST TRISECTED PERIMETER POINT)

 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

2t3 - 3(a + b + c)t2 + (a2 + b2 + c2 + 8bc + 8ca + 8ab)t - (cb2 + ac2 + ba2 + 5bc2 + 5ca2 + 5ab2 + 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 x1 : y1 : z1 where y1 : z1 are given by cyclic permutations of a,b,c, in x1, where

x1 = bc[r2 - (2c + a)r + (- a2 + b2 + 2c2 + 2bc + 3ca + 2ab].

His presentation included a proof that there is only one point for which AB' + AC' = BC' + BA' = CA' + CB' .

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.