X₃₁₂₈

X₄-CEVA CONJUGATE OF X₁₁₆₃

X₃₁₂₈ = X₄-Ceva conjugate of X₁₁₆₃

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

X₃₁₂₈ (X₄-CEVA CONJUGATE OF X₁₁₆₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1163, 1164}

• Related Centers: X₄ (ORTHOCENTER), X₁₁₆₃ (INNER GREBE-ORTHIC PERSPECTOR), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₁₆₄ (POINT ALTAIR)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₄ and \(u : v : w\) = X₁₁₆₃

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\), where \(f(a,b,c) = u(-u/x + v/y + w/z)\)

Search Value (6, 9, 13): \(8.501102643127\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b^{2} + c^{2} - S)/(b^{2} + c^{2} - a^{2}) (E\). Danneels)

▸ Incident Central Lines4 lines

{2, 3}, {1163, 1164}, {1588, 1899}, {1853, 3071}

▸ Functional Relations1 relation

• X₃₁₂₈ = X₄-Ceva conjugate of X₁₁₆₃

Notes

As a point on the Euler line, X₃₁₂₈ has Shinagawa coefficients \((F,E + F - S)\).