X₄₂₆

EULER X₁₉-4th-SUBSTITUTION POINT

X₄₂₆ = isogonal conjugate of the isotomic conjugate of X₄₄₁₄₁

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

X₄₂₆ (EULER X₁₉-4th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {157, 1853}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₅₇ (-OF-TANGENTIAL-TRIANGLE), X₁₈₅₃ (ZOSMA TRANSFORM OF X₂₀₄)

Trilinears: \((v^{2} + w^{2})\cos A : (w^{2} + u^{2})\cos B : (u^{2} + v^{2})\cos C\), where \(u : v : w\) = X₁₉; e.g., \(u = u(A,B,C) = \tan A\)

Barycentrics: \((v^{2} + w^{2})\sin 2A : (w^{2} + u^{2})\sin 2B : (u^{2} + v^{2})\sin 2C\)

Search Value (6, 9, 13): \(1.875882662069\)

▸ Alternate Coordinates2 forms

• Barycentrics: \((a + b) \cdot (a + c) \cdot (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{6} - a^{5} b - a^{4} b^{2} + a^{3} b^{3} - a^{5} c + a^{4} b c + a^{3} b^{2} c - b^{5} c - a^{4} c^{2} + a^{3} b c^{2} + 2 a^{2} b^{2} c^{2} - 2 a b^{3} c^{2} + a^{3} c^{3} - 2 a b^{2} c^{3} + 2 b^{3} c^{3} - b c^{5}) : :\)

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

▸ Incident Central Lines31 lines

{2, 3}, {6, 13409}, {97, 7998}, {125, 2351}, {157, 1853}, {182, 46832}, {184, 6509}, {216, 43650}, {305, 6394}, {394, 2972}, {577, 3917}, {1073, 6090}, {1092, 16391}, {1899, 6389}, {3162, 9475}, {3964, 4176}, {5085, 26898}, {5422, 30258}, {5651, 46831}, {9306, 44436}, {11245, 41005}, {14059, 14152}, {22057, 22421}, {22352, 26880}, {23332, 37813}, {26905, 34828}, {36752, 42441}, {39243, 41462}, {40913, 43975}, {41204, 46717}, {47409, 47412}

▸ Functional Relations12 relations

• X₄₂₆ = isogonal conjugate of the isotomic conjugate of X₄₄₁₄₁

• X₄₂₆ = isotomic conjugate of the polar conjugate of X₃₉₆₄₃

• X₄₂₆ = isogonal conjugate of the polar conjugate of X₆₃₈₉

• X₄₂₆ = X₂₅₅-complementary conjugate of X₃₇₈₆₄

• X₄₂₆ = X_i-Ceva conjugate of X_j for these (i,j): {3, 40947}, {4563, 32320}, {6389, 39643}

• X₄₂₆ = X₁₀₉₆-isoconjugate of X₃₄₄₀₅

• X₄₂₆ = X_i-Dao conjugate of X_j for these (i,j): {264, 3767}, {1093, 6389}, {6503, 34405}, {6524, 14713}

• X₄₂₆ = crosspoint of X_i and X_j for these (i,j): {3, 3964}, {6389, 44141}

• X₄₂₆ = crosssum of X₄ and X₆₅₂₄

• X₄₂₆ = barycentric product X_i*X_j for these {i,j}: {3, 6389}, {6, 44141}, {69, 39643}, {326, 2083}, {394, 1899}, {577, 41009}, {1092, 41760}, {3767, 3964}, {3926, 40947}, {4176, 42295}, {6507, 17871}, {6751, 34386}, {16391, 41770}

• X₄₂₆ = barycentric quotient X_i/X_j for these {i,j}: {394, 34405}, {1632, 15352}, {1899, 2052}, {2083, 158}, {3767, 1093}, {3964, 42407}, {6389, 264}, {6751, 53}, {17871, 6521}, {39643, 4}, {40947, 393}, {41009, 18027}, {42295, 6524}, {44141, 76}

• X₄₂₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3, 6641}, {3, 441, 3148}, {3, 6617, 25}, {3, 6638, 22}, {3, 7395, 26897}, {3, 15781, 378}, {3, 26865, 34003}, {25, 6617, 852}, {2972, 23606, 394}, {7386, 28412, 427}, {15246, 26874, 3}

Notes

As a point on the Euler line, X₄₂₆ has Shinagawa coefficients \(((2E + F)F - S^{2},-(E + F)F + S^{2})\).