X₃₆₉

1st TRISECTED PERIMETER POINT

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 X₃₆₉. Near the end of the 20th century, Yff found trilinears for X₃₆₉ in terms of the unique real root, r, of the cubic polynomial

5, 6, 7 3, 4, 5 Kepler Kimberling

X₃₆₉ (1st TRISECTED PERIMETER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(x : y : z (see below)\)

Barycentrics: \(ax : by : cz\)

Search Value (6, 9, 13): \(2.053651811125\)

Notes

\(2t^{3}\) - 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 X₃₆₉ 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' .