X₈₀₀

CROSSPOINT OF X₂ AND X₆₄

X₈₀₀ is the crosspoint of X₂ and X₆₄.

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

X₈₀₀ (CROSSPOINT OF X₂ AND X₆₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {53, 115}

• Related Centers: X₂ (CENTROID), X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA)

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

Barycentrics: \(\sin^{2} A (\cos^{2} B + \cos^{2} C) : :\)

Search Value (6, 9, 13): \(0.670564699772\)

▸ Alternate Coordinates2 forms

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

• Barycentrics: \((\sin^{2} A) (-2 + \sin^{2} B + \sin^{2} C) : :\)

▸ Incident Central Lines15 lines

{3, 6}, {19, 1945}, {41, 2197}, {53, 115}, {232, 1196}, {393, 1093}, {1015, 3554}, {1033, 2207}, {1194, 5304}, {1195, 1409}, {1500, 3553}, {1713, 2238}, {2257, 2277}, {2331, 5336}, {3289, 3787}

▸ Functional Relations8 relations

• X₈₀₀ = isogonal conjugate of X₈₀₁

• X₈₀₀ = complement of X₁₄₆₁₅

• X₈₀₀ = crosspoint of X_i and X_j for these (i,j): (2, 64), (6, 393)

• X₈₀₀ = crosssum of X_i and X_j for these (i,j): (2, 394), (6, 20)

• X₈₀₀ = perspector of circumconic centered at X₂₈₈₃

• X₈₀₀ = center of circumconic that is locus of trilinear poles of lines passing through X₂₈₈₃

• X₈₀₀ = X₂-Ceva conjugate of X₂₈₈₃

• X₈₀₀ = {X₃,X₆}-harmonic conjugate of X₅₀₆₅