X₃₄₆₄

X₃₀-CEVA CONJUGATE OF X₁

X₃₄₆₄ = X₃₀-Ceva conjugate of X₁

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

X₃₄₆₄ (X₃₀-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 74}, {57, 1354}

• Related Centers: X₃₀ (EULER INFINITY POINT), X₁ (INCENTER), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₁₃₅₄ (BRISSE TRANSFORM OF X₇₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = x(- x/u + y/v + z/w)\), where \(u : v : w\) = X₃₀,\(x : y : z\) = X₁

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-1.686182544157\)

X₃₄₆₄ lies on the Bevan circle, the Neuberg cubic.

▸ Incident Central Lines7 lines

{1, 74}, {3, 3466}, {4, 1768}, {13, 1652}, {14, 1653}, {57, 1354}, {1054, 1724}

▸ Functional Relations1 relation

• X₃₄₆₄ = X₃₀-Ceva conjugate of X₁

Notes

Let A'B'C' be the triangle homothetic to ABC and passing through the excenters (i.e., A'B'C' is the tangential triangle of the excentral triangle.) X₃₄₆₄ = X₁₃₅₄-of-A'B'C'. (Randy Hutson, July 11, 2019)