X₁₀₁₁

INTERCEPT OF EULER LINE AND LINE X₆X_31

X₁₀₁₁ = crossdifference of every pair of points on line X₅₁₄X₆₄₇

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

X₁₀₁₁ (INTERCEPT OF EULER LINE AND LINE X₆X_31) 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, 31}

• Related Centers: X₁₂₄₆ (ISOGONAL CONJUGATE OF X₁₀₁₁), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT)

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

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): \(4.415363174892\)

▸ Incident Central Lines6 lines

{2, 3}, {6, 31}, {9, 228}, {35, 43}, {51, 573}, {184, 572}

▸ Functional Relations4 relations

• X₁₀₁₁ = isogonal conjugate of X₁₂₄₆

• X₁₀₁₁ = inverse-in-orthocentroidal circle of X₃₁₃₆

• X₁₀₁₁ = crosssum of X₈₃₄ and X₁₀₈₆

• X₁₀₁₁ = crossdifference of every pair of points on line X₅₁₄X₆₄₇

Notes

As a point on the Euler line, X₁₀₁₁ has Shinagawa coefficients \((E + \$bc\$, -\$bc\$)\).