X₆₄₁

COMPLEMENT OF X₄₈₅

X₆₄₁ is the complement of X₄₈₅.

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

X₆₄₁ (COMPLEMENT OF X₄₈₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {485, 488}

• Related Centers: X₄₈₅ (VECTEN POINT), X₄₈₈ (ANTICOMPLEMENT OF X₄₈₅)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = [2S_{B}S_{C} + 3S_{A}a^{2} + 2(b^{2} + c^{2}) \operatorname{area}]/a\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = af(a,b,c)\)

Search Value (6, 9, 13): \(3.535230763926\)

▸ Incident Central Lines5 lines

{2, 372}, {3, 639}, {39, 615}, {140, 141}, {371, 492}

▸ Functional Relations10 relations

• X₆₄₁ = midpoint of X₄₈₅ and X₄₈₈

• X₆₄₁ = complement of X₄₈₅

• X₆₄₁ = complementary conjugate of X₆₃₉

• X₆₄₁ = crosspoint of X₂ and X₄₉₂

• X₆₄₁ = X₃-of-outer-Vecten-triangle

• X₆₄₁ = outer-Vecten-isogonal conjugate of X₄₈₅

• X₆₄₁ = perspector of circumconic centered at X₅₉₀

• X₆₄₁ = center of circumconic that is locus of trilinear poles of lines passing through X₅₉₀

• X₆₄₁ = X₂-Ceva conjugate of X₅₉₀

• X₆₄₁ = perspector of medial triangle and outer Vecten of outer Vecten triangle (note: the medial triangle is the inner Vecten of outer Vecten triangle.)

Notes

Erect squares outwards on the sides of triangle ABC; the circumcenter of the centers of the squares is X₆₄₁. (Peter Moses, 7/16,2003)