X₆₁

ISOGONAL CONJUGATE OF X₁₇

X₆₁ is the isogonal conjugate of X₁₇.

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

X₆₁ (ISOGONAL CONJUGATE OF X₁₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {4, 13}

• Related Centers: X₁₇ (1st NAPOLEON POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₁₃ (1st ISOGONIC CENTER)

Trilinears: \(\sin(A + \pi /6) : \sin(B + \pi /6) : \sin(C + \pi /6)\)

Barycentrics: \(\sin A \sin(A + \pi /6) : \sin B \sin(B + \pi /6) : \sin C \sin(C + \pi /6)\)

Search Value (6, 9, 13): \(1.921708280860\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\cos(A - \pi /3) : \cos(B - \pi /3) : \cos(C - \pi /3)\)

• Trilinears: \(\cos A + \sqrt{3} \sin A : :\)

X₆₁ lies on the Napoleon cubic.

▸ Incident Central Lines11 lines

{1, 203}, {2, 18}, {3, 6}, {4, 13}, {5, 14}, {30, 397}, {56, 202}, {140, 395}, {299, 636}, {302, 629}, {618, 627}

▸ Functional Relations24 relations

• X₆₁ = reflection of X₆₂ in X₅₀₀₇

• X₆₁ = reflection of X₆₃₃ in X₆₃₅

• X₆₁ = isogonal conjugate of X₁₇

• X₆₁ = isotomic conjugate of X₃₄₃₈₉

• X₆₁ = complement of X₆₃₃

• X₆₁ = anticomplement of X₆₃₅

• X₆₁ = Brocard-circle-inverse of X₆₂

• X₆₁ = eigencenter of cevian triangle of X₁₄

• X₆₁ = eigencenter of anticevian triangle of X₁₆

• X₆₁ = X₁₄-Ceva conjugate of X₁₆

• X₆₁ = crossdifference of every pair of points on line X₅₂₃X₁₄₄₄₆

• X₆₁ = crosspoint of X₃₀₂ and X₄₇₃

• X₆₁ = point of concurrence of Brocard axes of BCX₁₅, CAX₁₅, ABX₁₅

• X₆₁ = perspector of ABC and centers of circles used in construction of X₁₃₃₇

• X₆₁ = X₆₁-of-circumsymmedial-triangle

• X₆₁ = orthocorrespondent of X₁₆

• X₆₁ = {X₁₅,X₆₂}-harmonic conjugate of X₃

• X₆₁ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₁₅

• X₆₁ = perspector of inner Napoleon triangle and orthocentroidal triangle

• X₆₁ = Cundy-Parry Phi transform of X₁₅

• X₆₁ = Cundy-Parry Psi transform of X₁₃

• X₆₁ = Kosnita(X₁₅,X₃) point

• X₆₁ = Kosnita(X₁₅,X₁₅) point

• X₆₁ = antigonal conjugate of X₃₄₂₁₉

Notes

X₆₁ is the {X₃,X₆}-harmonic conjugate of X₆₂.