X₄₅₅

EULER X₁₅₉-11th-SUBSTITUTION POINT

X₄₅₅ = X₁₃₇₀-Ceva conjugate of X₃₁₆₂

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

X₄₅₅ (EULER X₁₅₉-11th-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: X₁₃₇₀ (ANTICOMPLEMENT OF X₂₅), X₃₁₆₂ (-CEVA CONJUGATE OF X₂₅), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(u^{2}\sec A : v^{2}\sec B : w^{2}\sec C\), where \(u : v : w\) = X₁₅₉

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

Search Value (6, 9, 13): \(-0.233349250637\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines3 lines

{2, 3}, {159, 17407}, {2386, 40144}

▸ Functional Relations6 relations

• X₄₅₅ = X₁₃₇₀-Ceva conjugate of X₃₁₆₂

• X₄₅₅ = X₃₉₇₃₃-isoconjugate of X₅₂₀₄₁

• X₄₅₅ = X₂₅-Dao conjugate of X₁₃₅₇₅

• X₄₅₅ = barycentric product X_i*X_j for these {i,j}: {159, 41361}, {1370, 3162}, {23115, 41766}

• X₄₅₅ = barycentric quotient X_i/X_j for these {i,j}: {3162, 13575}, {41361, 40009}

• X₄₅₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {}

Notes

As a point on the Euler line, X₄₅₅ has Shinagawa coefficients \((4(E + F)^{3}F^{2} - E^{2}FS^{2}, -4(E + F)^{4}F + (E + F)(E + 4F)ES^{2})\).