There 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 X3232. Near the beginning of the 21st century, trilinears \(x : y : z\) were found for X3232 in terms of the unique real root of a cubic polynomial related to the cubic polynomial shown at X369, 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.
X3232 (2nd TRISECTED PERIMETER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶
