X₃₂₆₉

CROSSPOINT OF X₆ AND X₆₄₇

X₃₂₆₉ is the crosspoint of X₆ and X₆₄₇.

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

X₃₂₆₉ (CROSSPOINT OF X₆ AND X₆₄₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3331, 232}

• Related Centers: X₃₃₃₁ (INTERSECTION OF LINES X), X₂₃₂ (-CEVA CONJUGATE OF X₁₃₂)

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

Search Value (6, 9, 13): \(2.819128564738\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin A \cos^{2} A \sin^{2}(B - C) : :\)

• Trilinears: \(\cos A \sin 2A \sin^{2}(B - C) : :\)

X₃₂₆₉ lies on the Brocard inellipse, the inconic with perspector X₂₀₅₂.

▸ Incident Central Lines14 lines

{3, 248}, {4, 1987}, {6, 74}, {32, 1204}, {39, 185}, {64, 2207}, {99, 287}, {115, 125}, {184, 574}, {186, 1971}, {339, 525}, {1425, 1500}, {1636, 2972}, {1899, 2549}

▸ Functional Relations11 relations

• X₃₂₆₉ = reflection of X₃₃₃₁ in X₂₃₂

• X₃₂₆₉ = isogonal conjugate of X₂₃₅₈₂

• X₃₂₆₉ = isotomic conjugate of polar conjugate of X₂₀₉₇₅

• X₃₂₆₉ = X_i-Ceva conjugate of X_j for these (i,j): (6, 647), (64, 512), (66, 3005), (287, 684), (394, 520), (1177, 351), (2052, 523)

• X₃₂₆₉ = crosspoint of X₆ and X₆₄₇

• X₃₂₆₉ = crosssum of X_i and X_j for these (i,j): (2, 648), (3, 1625), (4, 112), (6, 1624), (99, 315), (107, 393), (110, 577), (162, 2326)

• X₃₂₆₉ = crossdifference of every pair of points on the line X₁₀₇X₁₁₀

• X₃₂₆₉ = X₁₉-isoconjugate of X₁₈₀₂₀

• X₃₂₆₉ = X₉₂-isoconjugate of X₂₅₀

• X₃₂₆₉ = crosssum of X₂₄₇₉ and X₂₄₈₀

• X₃₂₆₉ = barycentric square of X₆₅₆