X₄₇₁

X₁₆-CROSS CONJUGATE OF X₂₉₉

X₄₇₁ = inverse-in-orthocentroidal-circle of X₄₇₀

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

X₄₇₁ (X₁₆-CROSS CONJUGATE OF X₂₉₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {17, 275}

• Related Centers: X₁₆ (2nd ISODYNAMIC POINT), X₂₉₉ (ISOTOMIC CONJUGATE OF 2nd ISOGONIC CENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₇ (1st NAPOLEON POINT), X₂₇₅ (CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT)

Trilinears: \(\sin(A - \pi /3) \csc 2A : \sin(B - \pi /3) \csc 2B : \sin(C - \pi /3) \csc 2C\)

Barycentrics: \(\sin(A - \pi /3) \sec A : \sin(B - \pi /3) \sec B : \sin(C - \pi /3) \sec 2C\)

Search Value (6, 9, 13): \(1.750793657835\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines7 lines

{2, 3}, {17, 275}, {264, 300}, {299, 340}, {303, 317}, {343, 633}, {394, 634}

▸ Functional Relations14 relations

• X₄₇₁ = isogonal conjugate of X₃₆₂₉₇

• X₄₇₁ = complement of X₁₉₇₇₃

• X₄₇₁ = cevapoint of X_i and X_j for these {i,j}: {16, 8740}, {6110, 23715}, {6138, 30468}

• X₄₇₁ = crosspoint of X₂ and X₁₉₇₇₅

• X₄₇₁ = crosssum of X₆ and X₁₁₂₄₄

• X₄₇₁ = trilinear pole of line X₆₁₁₁X_6783

• X₄₇₁ = inverse-in-orthocentroidal-circle of X₄₇₀

• X₄₇₁ = X₁₆-cross conjugate of X₂₉₉

• X₄₇₁ = X₄-Hirst inverse of X₄₇₀

• X₄₇₁ = pole wrt polar circle of trilinear polar of X₁₄ (line X₃₉₆X₅₂₃)

• X₄₇₁ = polar conjugate of X₁₄

• X₄₇₁ = polar-circle inverse of X₃₂₄₆₀

• X₄₇₁ = X₆₃-iaoconjugate of X₃₄₅₈

• X₄₇₁ = {X₂,X₄}-harmonic conjugate of X₄₇₀

Notes

As a point on the Euler line, X₄₇₁ has Shinagawa coefficients \((3^{1/2}F, -S)\).