X₇₅₉

EVEN (1, 3) CIRCUMCIRCLE POINT

X₇₅₉ = trilinear pole of line X₆X₆₆₁

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

X₇₅₉ (EVEN (1, 3) CIRCUMCIRCLE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {10, 21}, {1, 60}

• Related Centers: X₇₅₈ (EVEN (1, 3) INFINITY POINT), X₁₀ (SPIEKER CENTER), X₂₁ (SCHIFFLER POINT), X₁ (INCENTER), X₆₀ (ISOGONAL CONJUGATE OF X₁₂)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/[a^{0}(b^{3} + c^{3}) - a^{2}(b^{1} + c^{1})]\)

Search Value (6, 9, 13): \(4.495915743960\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1/((b + c) \cot A - b \cot B - c \cot C) : :\)

• Trilinears: \(1/((b + c) (b^{2} + c^{2} - a^{2} - b c)) : :\)

X₇₅₉ lies on the circucircle.

▸ Incident Central Lines19 lines

{1, 60}, {10, 21}, {19, 112}, {28, 108}, {31, 994}, {37, 101}, {58, 65}, {75, 99}, {82, 827}, {91, 925}, {107, 158}, {214, 662}, {270, 933}, {484, 901}, {691, 897}, {741, 876}, {833, 1010}, {840, 1019}, {934, 1014}

▸ Functional Relations11 relations

• X₇₅₉ = isogonal conjugate of X₇₅₈

• X₇₅₉ = isotomic conjugate of X₃₅₅₅₀

• X₇₅₉ = reflection of X₆₀₁₁ in X₃

• X₇₅₉ = anticomplement of X₃₁₈₄₅

• X₇₅₉ = trilinear pole of line X₆X₆₆₁

• X₇₅₉ = Ψ(X₆, X₆₆₁)

• X₇₅₉ = Ψ(X₁, X₅₂₃)

• X₇₅₉ = trilinear product of circumcircle intercepts of line X₁X₅₂₃

• X₇₅₉ = trilinear pole, wrt 2nd circumperp triangle, of line X₇X₂₁

• X₇₅₉ = circumcircle-antipode of X₆₀₁₁

• X₇₅₉ = Conway-circle-inverse of X₃₈₄₈₂