X₄₃₇

EULER X₂₁₄-7th-SUBSTITUTION POINT

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

X₄₃₇ (EULER X₂₁₄-7th-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: \((u^{2} + vw)\sec A : (v^{2} + wu)\sec B : (w^{2} + uv)\sec C\), where \(u : v : w\) = X₂₁₄

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

Search Value (6, 9, 13): \(-2.773444273579\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (5 a^{6} - 6 a^{5} b - 8 a^{4} b^{2} + 11 a^{3} b^{3} - 5 a b^{5} + 3 b^{6} - 6 a^{5} c + 18 a^{4} b c - 5 a^{3} b^{2} c - 12 a^{2} b^{3} c + 12 a b^{4} c - 5 b^{5} c - 8 a^{4} c^{2} - 5 a^{3} b c^{2} + 21 a^{2} b^{2} c^{2} - 11 a b^{3} c^{2} - 2 b^{4} c^{2} + 11 a^{3} c^{3} - 12 a^{2} b c^{3} - 11 a b^{2} c^{3} + 12 b^{3} c^{3} + 12 a b c^{4} - 2 b^{2} c^{4} - 5 a c^{5} - 5 b c^{5} + 3 c^{6}) : :\)

▸ Incident Central Lines1 line

{2, 3}

Notes

As a point on the Euler line, X₄₃₇ has Shinagawa coefficients \((2(E+F)^{2}F+10FS^{2}+7\$bcS_{A}\$F-5\$ab\$(E+F)F, (5E-16F)S^{2}-\$abS_{C}^{2}\$ +41\$bcS_{B}S_{C}\$+\$ab\$[(E+F)^{2}-19S^{2}])\).