Information from Kimberling's Encyclopedia of Triangle Centers |
Trilinears x : y : z (see below)
Barycentrics ax : by : czThere exist points A', B', C' on segments BC, CA, AB, respectively, such that B'C + C'B = C'A + A'C = A'B + B'A = (a + b + c)/3, and the lines AA', BB', CC' concur in X(3232). Near the beginning of the 21st century, trilinears x : y : z were found for X(3232) in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X(369), the 1st trisected perimeter point. The proof is given in Sadi Abu-Saymeh, Mawaffaq Hajja, and Hellmuth Stachel, "Another cubic associated with the triangle," forthcoming in Journal for Geometry and Graphics.