X₄₅₇

EULER X₃₉₉-11th-SUBSTITUTION POINT

X₄₅₇ = barycentric product X₁₂₇₂*X₅₂₁₆₆

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

X₄₅₇ (EULER X₃₉₉-11th-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 \(u : v : w\) = X₃₉₉

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

Search Value (6, 9, 13): \(-0.492700786670\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines4 lines

{2, 3}, {14993, 52166}, {16240, 51345}, {37496, 47215}

▸ Functional Relations3 relations

• X₄₅₇ = barycentric product X₁₂₇₂*X₅₂₁₆₆

• X₄₅₇ = barycentric quotient X₅₂₁₆₆/X₁₁₃₈

• X₄₅₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {15766, 15773, 3}

Notes

As a point on the Euler line, X₄₅₇ has Shinagawa coefficients \((9(E + 16F)EF - 64FS^{2},-81E^{2}F + 16(E + 4F)S^{2})\).