X₉₆₄

INTERCEPT OF EULER LINE AND LINE X₆X_8

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

X₉₆₄ (INTERCEPT OF EULER LINE AND LINE X₆X_8) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 8}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₈ (NAGEL POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[a^{4} + (b + c)(a^{3} + ab^{2} + ac^{2} + abc + (b + c)(a^{2} + bc))]\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(2.536291462852\)

▸ Incident Central Lines4 lines

{1, 321}, {2, 3}, {6, 8}, {10, 31}

Notes

As a point on the Euler line, X₉₆₄ has Shinagawa coefficients \((2(E + F)^{2} + 2(E + F) \$bc\$ + abc \$a\$, 2S^{2})\).