X₁₆₃₆

TRIPOLAR CENTROID OF X₃

X₁₆₃₆ is the tripolar centroid of X₃.

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

X₁₆₃₆ (TRIPOLAR CENTROID OF X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {110, 112}, {520, 647}

• Related Centers: X₆₄₈ (TRILINEAR POLE OF EULER LINE), X₁₂₉₄ (COLLINGS TRANSFORM OF X₁₂₂), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)), X₅₂₀ (ISOGONAL CONJUGATE OF X₁₀₇), X₆₄₇ (CROSSDIFFERENCE OF X₂ AND X₃)

Trilinears: \((\cos A)(\tan B - \tan C)(\tan B + \tan C - 2 \tan A) : :\)

Barycentrics: \((\sin 2A)(\sin 2B - \sin 2C)(\sin 2B + \sin 2C - 2 \sin 2A) : :\)

Search Value (6, 9, 13): \(-7.428419383177\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cos A [\sec B \csc(A - C) + \sec C \csc(A - B)] : :\)

▸ Incident Central Lines4 lines

{110, 112}, {520, 647}, {525, 3268}, {1637, 1651}

▸ Functional Relations5 relations

• X₁₆₃₆ = crosspoint of X₆₄₈ and X₁₂₉₄

• X₁₆₃₆ = crosssum of X_i and X_j for these (i,j): (4, 1637), (523, 1990)

• X₁₆₃₆ = X₂-Ceva conjugate of X₃₈₉₉₉

• X₁₆₃₆ = perspector of hyperbola {{A, B, C,X₃,X₃₀}}

• X₁₆₃₆ = intersection of perspectrix of ABC and orthocentroidal triangle (line X₁₆₃₆X₁₆₃₇) and perspectrix of ABC and anti-orthocentroidal triangle (line X₅₂₀X₆₄₇)