X₄₇₃

X₆₁-CROSS CONJUGATE OF X₃₀₂

X₄₇₃ = orthocentroidal-circle-inverse of X₄₇₂

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

X₄₇₃ (X₆₁-CROSS CONJUGATE OF X₃₀₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {14, 275}

• Related Centers: X₆₁ (ISOGONAL CONJUGATE OF X₁₇), X₃₀₂ (ISOTOMIC CONJUGATE OF 1st NAPOLEON POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₄ (2nd 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): \(7.198177455364\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines7 lines

{2, 3}, {14, 275}, {53, 396}, {264, 299}, {298, 317}, {343, 622}, {394, 621}

▸ Functional Relations6 relations

• X₄₇₃ = isogonal conjugate of X₃₂₅₈₅

• X₄₇₃ = orthocentroidal-circle-inverse of X₄₇₂

• X₄₇₃ = anticomplement of X₄₆₅

• X₄₇₃ = X₆₁-cross conjugate of X₃₀₂

• X₄₇₃ = polar conjugate of X₁₇

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

Notes

As a point on the Euler line, X₄₇₃ has Shinagawa coefficients \((F, 3^{1/2}S)\).