X₁₀₀₆

INTERCEPT OF EULER LINE AND LINEX(9)X₄₈

X₁₀₀₆ = isogonal conjugate of X₁₂₄₃

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

X₁₀₀₆ (INTERCEPT OF EULER LINE AND LINEX(9)X₄₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {9, 48}

• Related Centers: X₁₂₄₃ (ISOGONAL CONJUGATE OF X₁₀₀₆), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₉ (MITTENPUNKT), X₄₈ (CROSSPOINT OF X₁ AND X₆₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{6} - a^{5}(b + c) - a^{4}(2b^{2} + bc + 2c^{2})\), + \(2a^{3}(b^{3} + c^{3})\) + \(a^{2}\)[\(b^{4}\) + \(c^{4}\) + 2\(bc(b^{2}\) + \(c^{2})\) + \(2b^{2}\)\(c^{2}\)], - a[(\(b^{5}\) + \(c^{5})\) - \(bc(b^{3}\) + \(c^{3})\)] - \(bc(b^{2}\) - \(c^{2})\)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): \(4.498758304589\)

▸ Incident Central Lines9 lines

{1, 201}, {2, 3}, {9, 48}, {35, 950}, {36, 226}, {54, 72}, {238, 1064}, {944, 958}, {954, 999}

▸ Functional Relations1 relation

• X₁₀₀₆ = isogonal conjugate of X₁₂₄₃

Notes

As a point on the Euler line, X₁₀₀₆ has Shinagawa coefficients \((\$aS_{A}\$ + abc, - \$aS_{A}\$)\).