X₆₂₆

COMPLEMENT OF X₃₂

X₆₂₆ is the complement of X₃₂.

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

X₆₂₆ (COMPLEMENT OF X₃₂) 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, 32}

• Related Centers: X₂ (CENTROID), X₃₂ (3rd POWER POINT)

Trilinears: \((b^{4} + c^{4})/a : (c^{4} + a^{4})/b: (a^{4} + b^{4})/c\)

Barycentrics: \(b^{4} + c^{4} : c^{4} + a^{4} : a^{4} + b^{4}\)

Search Value (6, 9, 13): \(3.803696965640\)

▸ Incident Central Lines8 lines

{2, 32}, {3, 114}, {5, 141}, {10, 760}, {37, 746}, {39, 325}, {76, 115}, {316, 384}

▸ Functional Relations15 relations

• X₆₂₆ = midpoint of X_i and X_j for these (i,j): (32, 315), (639, 640)

• X₆₂₆ = isogonal conjugate of X₃₈₈₂₆

• X₆₂₆ = isotomic conjugate of isogonal conjugate of X₂₀₈₅₉

• X₆₂₆ = complement of X₃₂

• X₆₂₆ = anticomplement of X₆₆₈₀

• X₆₂₆ = complementary conjugate of X₃₉

• X₆₂₆ = X₃₀₁-Ceva conjugate of X₂

• X₆₂₆ = crosspoint of X₂ and X₁₅₀₂

• X₆₂₆ = crosssum of X₆ and X₁₅₀₁

• X₆₂₆ = crossdifference of every pair of points on line X₃₀₀₅X₃₀₅₀

• X₆₂₆ = X_i-isoconjugate of X_j for these {i,j}: {1, 38826}, {32, 38847}, {560, 38830}, {1923, 3115}, {2084, 33515}

• X₆₂₆ = trilinear product X_i*X_j for these {i,j}: {2, 4118}, {6, 20627}, {9, 7217}, {10, 18167}, {37, 16891}, {57, 4178}, {75, 20859}, {76, 2085}, {81, 16894}, {561, 8265}, {3118, 18833}

• X₆₂₆ = trilinear quotient X_i/X_j for these (i,j): (1, 38826), (76, 38847), (561, 38830), (2085, 32), (3118, 1923), (4118, 6), (4178, 9), (4593, 33515), (8265, 560), (16891, 81), (16894, 37), (18833, 3115), (20627, 2), (20859, 31)

• X₆₂₆ = barycentric product X_i*X_j for these {i,j}: {1, 20627}, {7, 4178}, {8, 7217}, {10, 16891}, {75, 4118}, {76, 20859}, {86, 16894}, {321, 18167}, {561, 2085}, {1502, 8265}

• X₆₂₆ = barycentric quotient X_i/X_j for these (i, j): (6, 38826), (75, 38847), (76, 38830), (308, 3115), (2085, 31), (3118, 3051), (4118, 1), (4178, 8), (4577, 33515), (7217, 7), (8265, 32), (16891, 86), (16894, 10), (18167, 81), (20627, 75), (20859, 6)

Notes

X₆₂₆ is the center of the inellipse that is the barycentric square of the de Longchamps line. The Brianchon point (perspector) of the inellipse is X₁₅₀₂. (Randy Hutson, October 15, 2018)