X₄₁₀

EULER X₂₉-1st-SUBSTITUTION POINT

X₄₁₀ = crosspoint of PU(82)

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

X₄₁₀ (EULER X₂₉-1st-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)

• Related Centers: X₂₆₅₆ (TRILINEAR POLE OF PU(82)), X₂₆₅₉ (CROSSDIFFERENCE OF PU(83)), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(u^{2} + vw : v^{2} + wu : w^{2} + uv\), where \(u : v : w\) = X₂₉; e.g., \(u = u(A,B,C) = (\sec A)/(\cos B + \cos C)\)

Barycentrics: \(a(u^{2} + vw) : b(v^{2} + wu) : c(w^{2} + uv)\)

Search Value (6, 9, 13): \(1.700177224401\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a + b) \cdot (a + c) \cdot (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{8} b^{2} - 3 a^{6} b^{4} + 3 a^{4} b^{6} - a^{2} b^{8} + 3 a^{8} b c - a^{7} b^{2} c - 5 a^{6} b^{3} c + a^{5} b^{4} c + a^{4} b^{5} c + a^{3} b^{6} c + a^{2} b^{7} c - a b^{8} c + a^{8} c^{2} - a^{7} b c^{2} - 3 a^{6} b^{2} c^{2} + a^{5} b^{3} c^{2} + 2 a^{4} b^{4} c^{2} + a^{3} b^{5} c^{2} + a^{2} b^{6} c^{2} - a b^{7} c^{2} - b^{8} c^{2} - 5 a^{6} b c^{3} + a^{5} b^{2} c^{3} + 8 a^{4} b^{3} c^{3} - 2 a^{3} b^{4} c^{3} - a^{2} b^{5} c^{3} + a b^{6} c^{3} - 2 b^{7} c^{3} - 3 a^{6} c^{4} + a^{5} b c^{4} + 2 a^{4} b^{2} c^{4} - 2 a^{3} b^{3} c^{4} + a b^{5} c^{4} + b^{6} c^{4} + a^{4} b c^{5} + a^{3} b^{2} c^{5} - a^{2} b^{3} c^{5} + a b^{4} c^{5} + 4 b^{5} c^{5} + 3 a^{4} c^{6} + a^{3} b c^{6} + a^{2} b^{2} c^{6} + a b^{3} c^{6} + b^{4} c^{6} + a^{2} b c^{7} - a b^{2} c^{7} - 2 b^{3} c^{7} - a^{2} c^{8} - a b c^{8} - b^{2} c^{8}) : :\)

▸ Incident Central Lines3 lines

{2, 3}, {73, 2659}, {823, 2660}

▸ Incident Central Lines4 lines

{1, 1248}, {2, 3}, {73, 2659}, {823, 2654}

▸ Functional Relations5 relations

• X₄₁₀ = X₂₆₅₆-Ceva conjugate of X₂₆₅₉

• X₄₁₀ = barycentric product X₂₆₆₂*X₃₁₆₂₃

• X₄₁₀ = barycentric quotient X₂₆₆₂/X₁₂₁₄

• X₄₁₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {21, 29, 414}

• X₄₁₀ = crosspoint of PU(82)

Notes

As a point on the Euler line, X₄₁₀ has Shinagawa coefficients \(([S^{6} - (2E - F)FS^{4} + 3\$bc\$FS^{4} - 3\$bcS_{B}S_{C}\$FS^{2}]F, -[S^{4} - 4(E - F)FS^{2} + 3\$bc\$FS^{2}]FS^{2} + \$bcS_{B}S_{C}\$[S^{4} - 2(E - 2F)F S^{2} + abc\$aS_{A}\$F])\).