X₄₃₈

EULER X₂₀₄-8th-SUBSTITUTION POINT

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

X₄₃₈ (EULER X₂₀₄-8th-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: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER)

Trilinears: \((u^{2} + vw)\csc A : (v^{2} + wu)\csc B : (w^{2} + uv)\csc C\), where \(u : v : w\) = X₂₀₄; e.g., \(u(A,B,C) = (\tan A)(\tan B + \tan C - \tan A)\)

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

Search Value (6, 9, 13): \(1.578456191935\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (5 a^{12} - 7 a^{10} b^{2} - 8 a^{8} b^{4} + 18 a^{6} b^{6} - 11 a^{4} b^{8} + 5 a^{2} b^{10} - 2 b^{12} - 7 a^{10} c^{2} + 26 a^{8} b^{2} c^{2} - 18 a^{6} b^{4} c^{2} - 16 a^{4} b^{6} c^{2} + 17 a^{2} b^{8} c^{2} - 2 b^{10} c^{2} - 8 a^{8} c^{4} - 18 a^{6} b^{2} c^{4} + 54 a^{4} b^{4} c^{4} - 22 a^{2} b^{6} c^{4} - 6 b^{8} c^{4} + 18 a^{6} c^{6} - 16 a^{4} b^{2} c^{6} - 22 a^{2} b^{4} c^{6} + 20 b^{6} c^{6} - 11 a^{4} c^{8} + 17 a^{2} b^{2} c^{8} - 6 b^{4} c^{8} + 5 a^{2} c^{10} - 2 b^{2} c^{10} - 2 c^{12}) : :\)

▸ Incident Central Lines3 lines

{2, 3}, {5895, 6330}, {35711, 42287}

Notes

As a point on the Euler line, X₄₃₈ has Shinagawa coefficients \((4(E + F)^{2}F^{2} - (E + 3F)FS^{2}, -[4(E - F)F - S^{2}]S^{2})\)