X₅₄₂

DIRECTION OF VECTOR AX + BX + CX, where X = X₉₈

X₅₄₂ = crossdifference of every pair of points on line X₆X₅₂₆

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

X₅₄₂ (DIRECTION OF VECTOR AX + BX + CX, where X = X₉₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 13}, {3, 67}

• Related Centers: X₈₄₂ (ISOGONAL CONJUGATE OF X₅₄₂), X₆ (SYMMEDIAN POINT), X₁₃ (1st ISOGONIC CENTER), X₃ (CIRCUMCENTER), X₆₇ (ISOGONAL CONJUGATE OF X₂₃)

Trilinears: \((2ax - by - cz)/a : :\), where \(x = x(A,B,C) = \sec(A + \omega )\), \(y = x(B,C,A)\), \(z = x(C,A,B)\)

Barycentrics: \(2a^{6} - 2a^{4}(b^{2} + c^{2}) + a^{2}(b^{4} + c^{4}) - (b^{2} - c^{2})^{2}(b^{2} + c^{2}) : :\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(2 a \sec(A + \omega ) - b \sec(B + \omega ) - c \sec(C + \omega ) : :\)

• Barycentrics: \((3 S_{A} - 2 S_{\omega}) S^{2} + 3 S_{\omega} S_{B} S_{C} : :\)

▸ Linear Combinations1 combination

• X₅₄₂ = X₆ - X₁₃

▸ Incident Central Lines25 lines

{2, 98}, {3, 67}, {4, 576}, {5, 575}, {6, 13}, {30, 511}, {68, 1177}, {69, 74}, {141, 549}, {146, 148}, {159, 2931}, {161, 1619}, {230, 2030}, {428, 1112}, {858, 3292}, {1350, 3534}, {1365, 2606}, {1550, 1551}, {1569, 3094}, {1648, 2502}, {1843, 1986}, {1853, 3167}, {3023, 3028}, {3024, 3027}, {3043, 3044}

▸ Functional Relations18 relations

• X₅₄₂ = isogonal conjugate of X₈₄₂

• X₅₄₂ = isotomic conjugate of X₅₆₄₁

• X₅₄₂ = crossdifference of every pair of points on line X₆X₅₂₆

• X₅₄₂ = orthopoint of X₆₉₀

• X₅₄₂ = X₂-Ceva conjugate of X₂₃₉₆₇

• X₅₄₂ = X₃₀-Hirst inverse of X₅₂₃

• X₅₄₂ = X₃₀-of-1st-Brocard triangle

• X₅₄₂ = X₅₁₁-of-4th-Brocard-triangle

• X₅₄₂ = X₅₁₁-of-orthocentroidal-triangle

• X₅₄₂ = X₅₁₁-of-X₄-Brocard-triangle

• X₅₄₂ = X₅₂₄-of-McCay-triangle

• X₅₄₂ = X₅₂₄-of-anti-McCay-triangle

• X₅₄₂ = intercept of Fermat axis and the line at infinity

• X₅₄₂ = perspector of hyperbola {A, B, C,X₂,X₄₇₆}}

• X₅₄₂ = Cundy-Parry Phi transform of X₁₄₃₅₇

• X₅₄₂ = Cundy-Parry Psi transform of X₁₄₂₄₆

• X₅₄₂ = 1st-Brocard-isogonal-conjugate of X₆₇₉₅

• X₅₄₂ = homothetic center of 2nd Ehrmann triangle and Ehrmann vertex-triangle