X₄₇₂

X₆₂-CROSS CONJUGATE OF X₃₀₃

X₄₇₂ = orthocentroidal-circle-inverse of X₄₇₃

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

X₄₇₂ (X₆₂-CROSS CONJUGATE OF X₃₀₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {13, 275}

• Related Centers: X₆₂ (ISOGONAL CONJUGATE OF X₁₈), X₃₀₃ (ISOTOMIC CONJUGATE OF 2nd NAPOLEON POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₃ (1st ISOGONIC CENTER), X₂₇₅ (CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT)

Trilinears: \(\cos(A + \pi /3) \csc 2A : \cos(B + \pi /3) \csc 2B : \cos(C + \pi /3) \csc 2C\)

Barycentrics: \(\cos(A + \pi /3) \sec A : \cos(B + \pi /3) \sec B : \cos(C + \pi /3) \sec 2C\)

Search Value (6, 9, 13): \(0.453874122304\)

▸ Alternate Coordinates1 form

• Barycentrics: \(\sqrt{3} \tan A - 1 : :\)

▸ Incident Central Lines7 lines

{2, 3}, {13, 275}, {53, 395}, {264, 298}, {299, 317}, {343, 621}, {394, 622}

▸ Functional Relations6 relations

• X₄₇₂ = isogonal conjugate of X₃₂₅₈₆

• X₄₇₂ = orthocentroidal-circle-inverse of X₄₇₃

• X₄₇₂ = anticomplement of X₄₆₆

• X₄₇₂ = X₆₂-cross conjugate of X₃₀₃

• X₄₇₂ = polar conjugate of X₁₈

• X₄₇₂ = {X₂,X₄}-harmonic conjugate of X₄₇₃

Notes

As a point on the Euler line, X₄₇₂ has Shinagawa coefficients \((F, -3^{1/2}S)\).