X₁₀₁₀

INTERCEPT OF EULER LINE AND LINE X₁X_75

X₁₀₁₀ = crossdifference of every pair of points on line X₆₄₇X₇₉₈

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

X₁₀₁₀ (INTERCEPT OF EULER LINE AND LINE X₁X_75) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {10, 58}

• Related Centers: X₁₂₄₅ (ISOGONAL CONJUGATE OF X₁₀₁₀), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER), X₅₈ (ISOGONAL CONJUGATE OF X₁₀)

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

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): \(2.548255626440\)

▸ Incident Central Lines8 lines

{1, 75}, {2, 3}, {8, 81}, {10, 58}, {72, 894}, {283, 1065}, {312, 975}, {759, 833}

▸ Functional Relations2 relations

• X₁₀₁₀ = isogonal conjugate of X₁₂₄₅

• X₁₀₁₀ = crossdifference of every pair of points on line X₆₄₇X₇₉₈

Notes

As a point on the Euler line, X₁₀₁₀ has Shinagawa coefficients \(((E + F)^{2} + \$bc\$(E + F) + abc\$a\$, S^{2})\).