X₃₀₀₁

(BROCARD AXIS)∩(DE LONGCHAMPS LINE)

X₃₀₀₁ = X₃₁-isoconjugate of X₂₃₆₇

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

X₃₀₀₁ ((BROCARD AXIS)∩(DE LONGCHAMPS LINE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {325, 523}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₃₂₅ (-HIRST INVERSE OF X₆₉), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀)

Trilinears: \(a(b^{6} + c^{6} - a^{2}b^{4} - a^{2}c^{4}) : :\)

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

Search Value (6, 9, 13): \(5.295529902196\)

▸ Linear Combinations1 combination

• X₃₀₀₁ = X₂₃₇ - \(3\) X₂₂₀₈₇, \(3\) X₂₂₀₈₇ - \(2\) X₃₄₉₉₀

▸ Incident Central Lines52 lines

{3, 6}, {67, 36214}, {86, 22150}, {141, 20819}, {157, 20806}, {160, 12220}, {206, 33801}, {231, 6036}, {237, 9019}, {311, 34845}, {316, 36187}, {323, 9142}, {325, 523}, {338, 21531}, {524, 20975}, {868, 34827}, {1503, 44716}, {1576, 22151}, {1594, 44145}, {1634, 2393}, {1990, 2967}, {2450, 44388}, {3964, 34777}, {3981, 34811}, {4383, 21493}, {4558, 11416}, {5094, 7778}, {6033, 7574}, {6593, 9407}, {6660, 18374}, {7495, 7792}, {7774, 16063}, {7819, 42442}, {8705, 9155}, {9766, 31152}, {9969, 35222}, {9971, 11328}, {13409, 26905}, {13857, 16186}, {14570, 14957}, {15270, 28710}, {16776, 37338}, {17710, 20775}, {18181, 37796}, {19151, 43722}, {21284, 44089}, {23333, 39113}, {34834, 38998}, {35296, 39231}, {37444, 41761}, {38987, 44380}, {40073, 40361}

▸ Functional Relations10 relations

• X₃₀₀₁ = midpoint of X₁₄₅₇₀ and X₁₄₉₅₇

• X₃₀₀₁ = reflection of X_i in X_j for these {i,j}: {237, 34990}, {338, 21531}, {1634, 36212}, {5201, 3003}

• X₃₀₀₁ = isotomic conjugate of X₂₃₆₇

• X₃₀₀₁ = isotomic conjugate of the isogonal conjugate of X₂₃₈₇

• X₃₀₀₁ = X₃₁-isoconjugate of X₂₃₆₇

• X₃₀₀₁ = crosssum of X₆ and X₂₁₁₇₇

• X₃₀₀₁ = crossdifference of every pair of points on line {32, 523}

• X₃₀₀₁ = barycentric product X₇₆*X₂₃₈₇

• X₃₀₀₁ = barycentric quotient X_i/X_j for these {i,j}: {2, 2367}, {2387, 6}

• X₃₀₀₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {216, 3313, 8266}, {237, 22087, 34990}, {325, 45279, 45921}, {325, 45921, 45918}, {570, 11574, 41328}, {3014, 45918, 45921}, {9737, 11511, 5063}, {20819, 23635, 141}, {22151, 37183, 1576}, {45279, 45921, 3014}