X₄₅₄

EULER X₁₅₅-11th-SUBSTITUTION POINT

X₄₅₄ = X₃₅₄₂-Ceva conjugate of X₁₅₅

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

X₄₅₄ (EULER X₁₅₅-11th-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: X₃₅₄₂ (EULER LINE INTERCEPT OF LINE X), X₁₅₅ (EIGENCENTER OF ORTHIC TRIANGLE), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(u^{2}\sec A : v^{2}\sec B : w^{2}\sec C\), where \(u : v : w\) = X₁₅₅; e.g., \(u(A,B,C) = (\cos A)[\cos^{2} B + \cos^{2} C - \cos^{2} A]\)

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

Search Value (6, 9, 13): \(-0.003927668967\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines9 lines

{2, 3}, {155, 6503}, {1609, 34853}, {3964, 44405}, {9908, 16391}, {9937, 23181}, {12164, 34333}, {15316, 47195}, {20975, 38260}

▸ Functional Relations5 relations

• X₄₅₄ = X₃₅₄₂-Ceva conjugate of X₁₅₅

• X₄₅₄ = X₂₅₄-isoconjugate of X₉₂₁

• X₄₅₄ = barycentric product X_i*X_j for these {i,j}: {155, 6515}, {1609, 40697}, {3542, 6503}

• X₄₅₄ = barycentric quotient X_i/X_j for these {i,j}: {155, 6504}, {1609, 254}, {6515, 46746}

• X₄₅₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {3, 6617, 3548}

Notes

As a point on the Euler line, X₄₅₄ has Shinagawa coefficients \((E(E + 4F) - 4S^{2},-E^{2} + 4S^{2})\).