X₄₅₆

EULER X₁₉₅-11th-SUBSTITUTION POINT

{X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {15770, 15779, 3}

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

X₄₅₆ (EULER X₁₉₅-11th-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: \(u^{2}\sec A : v^{2}\sec B : w^{2}\sec C\), where \(u : v : w\) = X₁₉₅

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

Search Value (6, 9, 13): \(3.729690948512\)

▸ Alternate Coordinates1 form

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

Notes

As a point on the Euler line, X₄₅₆ has Shinagawa coefficients \(((9E + 16F)EF^{2} - 64F^{2}S^{2}, -E^{2}F^{2} + 16(E + 4F)FS^{2})\).