X₃₂₆₅

ISOTOMIC CONJUGATE OF X₁₀₇

X₃₂₆₅ is the isotomic conjugate of X₁₀₇.

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

X₃₂₆₅ (ISOTOMIC CONJUGATE OF X₁₀₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {647, 2525}

• Related Centers: X₆₄₇ (CROSSDIFFERENCE OF X₂ AND X₃), X₂₅₂₅ (RADICAL CENTER OF {NINE-POINT, BROCARD,)

Trilinears: \((\csc^{2} A)(\cos A)(\sin 2B - \sin 2C)\)

Barycentrics: \(\cot A (\tan B - \tan C) : :\)

Search Value (6, 9, 13): \(15.089545516393\)

▸ Alternate Coordinates2 forms

• Barycentrics: \((b^{2} - c^{2}) (b^{2} + c^{2} - a^{2})^{2} : :\)

• Barycentrics: directed distance from A to van Aubel line : :

X₃₂₆₅ lies on the Kiepert Parabola.

▸ Incident Central Lines6 lines

{2, 2419}, {99, 1304}, {325, 523}, {441, 525}, {520, 4131}, {2501, 2799}

▸ Functional Relations14 relations

• X₃₂₆₅ = midpoint of X₆₄₇ and X₂₅₂₅

• X₃₂₆₅ = isogonal conjugate of X₃₂₇₁₃

• X₃₂₆₅ = isotomic conjugate of X₁₀₇

• X₃₂₆₅ = complement of X₃₃₂₉₄

• X₃₂₆₅ = X_i-Ceva conjugate of X_j for these (i,j): (99, 69), (3267, 525)

• X₃₂₆₅ = X_i-cross conjugate of X_j for these (i,j): (122, 2), (520, 525), (2972, 394)

• X₃₂₆₅ = crosspoint of X₆₉ and X₉₉

• X₃₂₆₅ = crosssum of X_i and X_j for these (i,j): (6, 2485), (25, 512), (1974, 2489)

• X₃₂₆₅ = crossdifference of every pair of points on the line X₂₅X₃₂

• X₃₂₆₅ = anticomplement of X₆₅₈₇

• X₃₂₆₅ = perspector of hyperbola {{A, B, C,X₆₉,X₇₆}} (the isotomic conjugate of the van Aubel line)

• X₃₂₆₅ = pole wrt polar circle of trilinear polar of X₆₅₂₉ (line X₂₅X₃₉₃)

• X₃₂₆₅ = polar conjugate of X₆₅₂₉

• X₃₂₆₅ = trilinear pole of line X₃₂₆₉X₁₅₅₂₆