X₆₂

ISOGONAL CONJUGATE OF X₁₈

X₆₂ is the isogonal conjugate of X₁₈.

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

X₆₂ (ISOGONAL CONJUGATE OF X₁₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {4, 14}

• Related Centers: X₁₈ (2nd NAPOLEON POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₄ (ORTHOCENTER), X₁₄ (2nd 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): \(-0.374701511313\)

▸ Alternate Coordinates2 forms

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

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

X₆₂ lies on the Napoleon cubic.

▸ Incident Central Lines11 lines

{1, 202}, {2, 17}, {3, 6}, {4, 14}, {5, 13}, {30, 398}, {56, 203}, {140, 396}, {298, 635}, {303, 630}, {619, 628}

▸ Functional Relations24 relations

• X₆₂ = reflection of X₆₁ in X₅₀₀₇

• X₆₂ = reflection of X₆₃₄ in X₆₃₆

• X₆₂ = isogonal conjugate of X₁₈

• X₆₂ = isotomic conjugate of X₃₄₃₉₀

• X₆₂ = complement of X₆₃₄

• X₆₂ = anticomplement of X₆₃₆

• X₆₂ = Brocard-circle-inverse of X₆₁

• X₆₂ = eigencenter of cevian triangle of X₁₃

• X₆₂ = eigencenter of anticevian triangle of X₁₅

• X₆₂ = X₁₃-Ceva conjugate of X₁₅

• X₆₂ = crosspoint of X₃₀₃ and X₄₇₂

• X₆₂ = crossdifference of every pair of points on line X₅₂₃X₁₄₄₄₇

• X₆₂ = point of concurrence of Brocard axes of BCX₁₆, CAX₁₆, ABX₁₆

• X₆₂ = perspector of ABC and centers of circles used in construction of X₁₃₃₈

• X₆₂ = X₆₂-of-circumsymmedial-triangle

• X₆₂ = orthocorrespondent of X₁₅

• X₆₂ = {X₁₆,X₆₁}-harmonic conjugate of X₃

• X₆₂ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₁₆

• X₆₂ = perspector of outer Napoleon triangle and orthocentroidal triangle

• X₆₂ = Cundy-Parry Phi transform of X₁₆

• X₆₂ = Cundy-Parry Psi transform of X₁₄

• X₆₂ = Kosnita(X₁₆,X₃) point

• X₆₂ = Kosnita(X₁₆,X₁₆) point

• X₆₂ = antigonal conjugate of X₃₄₂₂₀

Notes

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