X₄₇₀

X₁₅-CROSS CONJUGATE OF X₂₉₈

X₄₇₀ = orthocentroidal-circle-inverse of X₄₇₁

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

X₄₇₀ (X₁₅-CROSS CONJUGATE OF X₂₉₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {18, 275}

• Related Centers: X₁₅ (1st ISODYNAMIC POINT), X₂₉₈ (ISOTOMIC CONJUGATE OF 1st ISOGONIC CENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₈ (2nd 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): \(3.743678655959\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines7 lines

{2, 3}, {18, 275}, {264, 301}, {298, 340}, {302, 317}, {343, 634}, {394, 633}

▸ Functional Relations14 relations

• X₄₇₀ = isogonal conjugate of X₃₆₂₉₆

• X₄₇₀ = complement of X₁₉₇₇₂

• X₄₇₀ = cevapoint of X_i and X_j for these {i,j}: {15, 8739}, {6111, 23714}, {6137, 30465}

• X₄₇₀ = crosspoint of X₂ and X₁₉₇₇₄

• X₄₇₀ = crosssum of X₆ and X₁₁₂₄₃

• X₄₇₀ = trilinear pole of line X₆₁₁₀X_6782

• X₄₇₀ = orthocentroidal-circle-inverse of X₄₇₁

• X₄₇₀ = polar-circle-inverse of X₃₂₄₆₁

• X₄₇₀ = X₁₅-cross conjugate of X₂₉₈

• X₄₇₀ = X₄-Hirst inverse of X₄₇₁

• X₄₇₀ = X₆₃-isoconjugate of X₃₄₅₇

• X₄₇₀ = pole wrt polar circle of trilinear polar of X₁₃ (line X₃₉₅X₅₂₃)

• X₄₇₀ = polar conjugate of X₁₃

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

Notes

As a point on the Euler line, X₄₇₀ has Shinagawa coefficients \((3^{1/2}F, S)\).