X₄₃₃

EULER X₁₅₉-6th-SUBSTITUTION POINT

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

X₄₃₃ (EULER X₁₅₉-6th-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: \((v^{2} + w^{2})\sec A : (w^{2} + u^{2})\sec B : (u^{2} + v^{2})\sec C\), where \(u : v : w\) = X₁₅₉

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

Search Value (6, 9, 13): \(8.967335720280\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{12} b^{2} + 2 a^{10} b^{4} - a^{8} b^{6} - 4 a^{6} b^{8} - a^{4} b^{10} + 2 a^{2} b^{12} + b^{14} + a^{12} c^{2} - 4 a^{10} b^{2} c^{2} + a^{8} b^{4} c^{2} + 8 a^{6} b^{6} c^{2} - 5 a^{4} b^{8} c^{2} - 4 a^{2} b^{10} c^{2} + 3 b^{12} c^{2} + 2 a^{10} c^{4} + a^{8} b^{2} c^{4} - 8 a^{6} b^{4} c^{4} + 6 a^{4} b^{6} c^{4} - 2 a^{2} b^{8} c^{4} + b^{10} c^{4} - a^{8} c^{6} + 8 a^{6} b^{2} c^{6} + 6 a^{4} b^{4} c^{6} + 8 a^{2} b^{6} c^{6} - 5 b^{8} c^{6} - 4 a^{6} c^{8} - 5 a^{4} b^{2} c^{8} - 2 a^{2} b^{4} c^{8} - 5 b^{6} c^{8} - a^{4} c^{10} - 4 a^{2} b^{2} c^{10} + b^{4} c^{10} + 2 a^{2} c^{12} + 3 b^{2} c^{12} + c^{14}) : :\)

▸ Incident Central Lines1 line

{2, 3}

Notes

As a point on the Euler line, X₄₃₃ has Shinagawa coefficients \((4(E + F)^{3}F^{2} - E^{2}FS^{2}, (E + F)[4(E + F)^{3}F - (E + 8F)ES^{2}])\).