X₃₈₅₁

X₃₅₂₃com(EULER TRIANGLE)

X₃₈₅₁ = complement of X₃₅₂₈

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

X₃₈₅₁ (X₃₅₂₃com(EULER TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 3528}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₅₂₈ (EULER LINE INTERCEPT OF LINE X), X₃ (CIRCUMCENTER)

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{4} - 4b^{4} - 4c^{4} + 3a^{2}b^{2} + 3a^{2}c^{2} + 8b^{2}c^{2}\)

Search Value (6, 9, 13): \(-0.337055565874\)

▸ Incident Central Lines7 lines

{2, 3}, {355, 3244}, {946, 3626}, {1352, 3629}, {1479, 3614}, {1482, 3632}, {3519, 3527}

▸ Functional Relations9 relations

• X₃₈₅₁ = reflection of X₁₅₇₀₀ in X₂

• X₃₈₅₁ = complement of X₃₅₂₈

• X₃₈₅₁ = anticomplement of X₁₄₈₆₉

• X₃₈₅₁ = inverse-in-orthocentroidal-circle of X₅₅₀

• X₃₈₅₁ = homothetic center of Euler triangle and mid-triangle of medial and anticomplementary triangles

• X₃₈₅₁ = homothetic center of 2nd isogonal triangle of X₄ and mid-triangle of orthic and circumorthic triangles

• X₃₈₅₁ = homothetic center of X₁₄₀-altimedial and X₄-anti-altimedial triangles

• X₃₈₅₁ = {X₂,X₄}-harmonic conjugate of X₅₅₀

• X₃₈₅₁ = {X₂,X₅}-harmonic conjugate of X₅₀₇₉

Notes

As a point on the Euler line, X₃₈₅₁ has Shinagawa coefficients \((3, 5)\)