X₄₀₆

EULER LINE INTERCEPT OF LINE X₁₀X_33

X₄₀₆ = anticomplement of X₃₄₁₂₀

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

X₄₀₆ (EULER LINE INTERCEPT OF LINE X₁₀X_33) 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, 33}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₀ (SPIEKER CENTER), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(a + b + c) + abc \sec A\)

Barycentrics: \(b + c + a(1 + \sec A) : c + a + b(1 + \sec B) : a + b + c(1 + \sec C)\)

Search Value (6, 9, 13): \(3.640581424626\)

▸ Incident Central Lines8 lines

{2, 3}, {8, 1061}, {10, 33}, {37, 158}, {92, 1068}, {108, 388}, {208, 226}, {261, 317}

▸ Functional Relations3 relations

• X₄₀₆ = anticomplement of X₃₄₁₂₀

• X₄₀₆ = polar conjugate of isogonal conjugate of X₃₆₇₄₄

• X₄₀₆ = inverse-in-orthocentroidal-circle of X₄₇₅

Notes

As a point on the Euler line, X₄₀₆ has Shinagawa coefficients \((\$a\$F, abc)\).