X₂₀₈₅

NK-TRANSFORM OF X₃₁

X₂₀₈₅ = crosspoint of X₃₁ and X₇₅

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

X₂₀₈₅ (NK-TRANSFORM OF X₃₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 1581}, {31, 1917}

• Related Centers: X₁ (INCENTER), X₁₅₈₁ (ISOGONAL CONJUGATE OF X₁₅₈₀), X₃₁ (2nd POWER POINT), X₁₉₁₇ (TRILINEAR 6th POWER POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a^{2}(b^{4} + c^{4})\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.283778185143\)

▸ Incident Central Lines4 lines

{1, 1581}, {31, 1917}, {75, 1928}, {1755, 1923}

▸ Functional Relations8 relations

• X₂₀₈₅ = isogonal conjugate of X₃₈₈₄₇

• X₂₀₈₅ = complement of X₃₈₈₄₄

• X₂₀₈₅ = crosspoint of X₃₁ and X₇₅

• X₂₀₈₅ = crosssum of X₃₁ and X₇₅

• X₂₀₈₅ = trilinear product X_i*X_j for these {i,j}: {2, 8265}, {4, 4173}, {6, 20859}, {25, 20819}, {31, 4118}, {32, 626}, {37, 16717}, {83, 3118}, {213, 18167}, {264, 23209}, {560, 20627}, {710, 14946}, {1397, 4178}, {1502, 8023}, {1918, 16891}, {1974, 4121}, {2175, 7217}, {2206, 16894}, {3051, 16890}, {8039, 9233}, {21110, 32739}

• X₂₀₈₅ = trilinear quotient X_i/X_j for these (i,j): (1, 38847), (2, 38830), (32, 38826), (83, 3115), (626, 76), (3118, 39), (4118, 75), (4121, 305), (4173, 3), (4178, 3596), (7217, 6063), (8023, 1501), (8265, 6), (14946, 711), (16717, 81), (16890, 308), (16891, 310), (16894, 313), (18167, 274), (20627, 561), (20819, 69), (20859, 2), (21110, 3261), (23209, 184)

• X₂₀₈₅ = barycentric product X_i*X_j for these {i,j}: {1, 20859}, {6, 4118}, {10, 16717}, {19, 20819}, {31, 626}, {32, 20627}, {41, 7217}, {42, 18167}, {75, 8265}, {92, 4173}, {213, 16891}, {604, 4178}, {692, 21110}, {1333, 16894}, {1917, 8039}, {1928, 8023}, {1964, 16890}, {1969, 23209}, {1973, 4121}, {3112, 3118}

• X₂₀₈₅ = barycentric quotient X_i/X_j for these (i,j): (1, 38830), (6, 38847), (82, 3115), (560, 38826), (626, 561), (1964, 18833), (3118, 38), (4118, 76), (4173, 63), (4178, 28659), (7217, 20567), (8023, 560), (8265, 1), (16717, 86), (16891, 6385), (16894, 27801), (18167, 310), (20627, 1502), (20819, 304), (20859, 75), (23209, 48)