X₁₀₁₂

INTERCEPT OF EULER LINE AND LINE X₁X_84

X₁₀₁₂ = center of circle {X₁,X₁₇₀₉,PU(4)}

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

X₁₀₁₂ (INTERCEPT OF EULER LINE AND LINE X₁X_84) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1, 84}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁ (INCENTER), X₈₄ (ISOGONAL CONJUGATE OF X₄₀)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{7} - a^{5}u + 2a^{4}bc(b + c) + a^{3}v - a(b + c)^{2}(b^{2} - c^{2})^{2} - 2bc(b + c)(b^{2} - c^{2})^{2}\), where \(u = 3b^{2} - 2bc + 3c^{2}\) and \(v = 3b^{4} + 2b^{2}c^{2} + 3c^{4}\)

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): \(-3.364047289326\)

▸ Incident Central Lines10 lines

{1, 84}, {2, 3}, {7, 104}, {40, 958}, {55, 515}, {56, 946}, {63, 517}, {268, 281}, {516, 993}, {954, 971}

▸ Functional Relations1 relation

• X₁₀₁₂ = center of circle {X₁,X₁₇₀₉,PU(4)}

Notes

As a point on the Euler line, X₁₀₁₂ has Shinagawa coefficients \((S^{2}, abc\$a\$ - S^{2})\).