X₄₅₀

X₃-HIRST INVERSE OF X₄

X₄₅₀ is the X₃-hirst inverse of X₄.

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

X₄₅₀ (X₃-HIRST INVERSE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {107, 511}

• Related Centers: X₁₉₄₂ (TRILINEAR POLE OF U(17)), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

Trilinears: \((\sec A)[\cos^{4} A - (\cos B \cos C)^{2}] : :\)

Barycentrics: \((\tan A)[\cos^{4} A - (\cos B \cos C)^{2}] : :\)

Search Value (6, 9, 13): \(1.524838623044\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\cos A)[\sec^{4} A - (\sec B \sec C)^{2}] : :\)

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{4} - a^{2} b^{2} + a^{2} b c - b^{3} c - a^{2} c^{2} + 2 b^{2} c^{2} - b c^{3}) \cdot (a^{4} - a^{2} b^{2} - a^{2} b c + b^{3} c - a^{2} c^{2} + 2 b^{2} c^{2} + b c^{3}) : :\)

▸ Incident Central Lines28 lines

{2, 3}, {107, 511}, {155, 1075}, {184, 15466}, {264, 5651}, {275, 5943}, {648, 3292}, {653, 17975}, {685, 17974}, {1069, 7049}, {1092, 1093}, {1148, 3157}, {1897, 17976}, {1935, 1940}, {1993, 3168}, {2052, 9306}, {2451, 2501}, {3260, 18020}, {3291, 6531}, {3462, 9820}, {5972, 14165}, {6090, 9308}, {6331, 12215}, {6335, 17977}, {6530, 11064}, {8062, 22382}, {8764, 17973}, {13346, 14249}

▸ Functional Relations8 relations

• X₄₅₀ = isogonal conjugate of X₁₉₄₂

• X₄₅₀ = crossdifference of every pair of points on line X₁₈₅X₆₄₇

• X₄₅₀ = X₃-Hirst inverse of X₄

• X₄₅₀ = crossdifference of PU(17)

• X₄₅₀ = perspector of hyperbola {A, B, C,PU(17)}

• X₄₅₀ = intersection of trilinear polars of P(17) and U(17)

• X₄₅₀ = X₂-Ceva conjugate of X₃₉₀₃₄

• X₄₅₀ = inverse-in-circumconic-centered-at-X₄ of X₂₅

Notes

As a point on the Euler line, X₄₅₀ has Shinagawa coefficients \(((E - 2F)F,(E + F)F - S^{2})\).