X₃₁₄₄

X₂₂₅-CEVA CONJUGATE OF X₄

X₃₁₄₄ = X₂₉₀₇-cross conjugate of X₄

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

X₃₁₄₄ (X₂₂₅-CEVA CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {34, 978}

• Related Centers: X₂₂₅ (-CEVA CONJUGATE OF X₆₅), X₄ (ORTHOCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₃₄ (-BETH CONJUGATE OF X₄), X₉₇₈ (-CEVA CONJUGATE OF X₁)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₂₂₅ and \(u : v : w\) = X₄

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): \(0.130071747120\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = [a^{3} + 2a^{2}(b + c) + abc - b^{3} - c^{3}]/(b^{2} + c^{2} - a^{2}) (E\). Danneels)

▸ Incident Central Lines8 lines

{2, 3}, {34, 978}, {158, 2752}, {225, 1247}, {242, 1838}, {1068, 1148}, {1193, 1870}, {1714, 3186}

▸ Functional Relations2 relations

• X₃₁₄₄ = X_i-Ceva conjugate of X_j for these (i,j): (225, 4), (1940, 1148), (2907, 1046)

• X₃₁₄₄ = X₂₉₀₇-cross conjugate of X₄

Notes

As a point on the Euler line, X₃₁₄₄ has Shinagawa coefficients \((2FS^{2}, -ES^{2} - \$bcS_{B}S_{C}\$)\).