X₂₄₇₈

INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X₃₇₇

X₂₄₇₈ = anticomplement of X₄₇₄

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

X₂₄₇₈ (INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X₃₇₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {8, 210}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₈ (NAGEL POINT), X₂₁₀ (-CEVA CONJUGATE OF X₃₇)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[a^{4} - 2a^{2}bc - 2abc(b + c) - (b^{2} - c^{2})^{2}]\)

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

Search Value (6, 9, 13): \(3.312286630385\)

▸ Incident Central Lines22 lines

{1, 908}, {2, 3}, {8, 210}, {10, 1479}, {11, 958}, {12, 1001}, {55, 1329}, {63, 1210}, {78, 950}, {145, 1058}, {149, 1145}, {264, 1896}, {283, 1724}, {329, 938}, {355, 392}, {388, 1319}, {496, 956}, {499, 993}, {1038, 1877}, {1125, 1478}, {1441, 1882}, {1714, 2328}

▸ Functional Relations2 relations

• X₂₄₇₈ = complement of X₄₁₉₀

• X₂₄₇₈ = anticomplement of X₄₇₄

Notes

As a point on the Euler line, X₂₄₇₈ has Shinagawa coefficients \((abc\$a\$, -2S^{2})\).