X₄₁₃

EULER X₂₁-3rd-SUBSTITUTION POINT

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

X₄₁₃ (EULER X₂₁-3rd-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: \(u^{3}(v^{2} + w^{2} - vw) : v^{3}(w^{2} + u^{2} - wu) : w^{3}(u^{2} + v^{2} - uv)\), where \(u : v : w\) = X₂₁; e.g., \(u = u(A,B,C) = 1/(\cos B + \cos C)\)

Barycentrics: \(au^{3}(v^{2} + w^{2} - vw) : bv^{3}(w^{2} + u^{2} - wu) : cw^{3}(u^{2} + v^{2} - uv)\)

Search Value (6, 9, 13): \(3.646041986345\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations1 relation

• X₄₁₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {21, 29, 409}, {21, 13614, 13588}

Notes

As a point on the Euler line, X₄₁₃ has Shinagawa coefficients \((\$aS_{A}\$[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] - 2abcFS^{2}, -(\$aS_{A}\$ - abc)[(\$aS_{A}\$)^{2} - \$bcS_{B}S_{C}\$] - 2[\$bcS_{B}S_{C}\$ + \$aS_{A}\$F - abcF]S^{2})\).