X₂₃₉₆

GIBERT-SIMSON TRANSFORM OF X₉₉

X₂₃₉₆ = trilinear pole of line X₁₁₄X₃₂₅ (the Simson line of X₉₉)

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

X₂₃₉₆ (GIBERT-SIMSON TRANSFORM OF X₉₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 39}, {99, 110}

• Related Centers: X₂₄₂₂ (ISOGONAL CONJUGATE OF X₂₃₉₆), X₂ (CENTROID), X₃₉ (BROCARD MIDPOINT), X₉₉ (STEINER POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where f(a,b,c) is as given just before X₂₃₉₄

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-34.638689508336\)

▸ Alternate Coordinates1 form

• Barycentrics: \((b^{4} + c^{4} - a^{2} b^{2} - a^{2} c^{2})/(b^{2} - c^{2}) : :\)

X₂₃₉₆ lies on the cubic K010.

▸ Incident Central Lines6 lines

{2, 39}, {99, 110}, {325, 868}, {880, 2395}, {892, 2408}, {1316, 1975}

▸ Functional Relations7 relations

• X₂₃₉₆ = isogonal conjugate of X₂₄₂₂

• X₂₃₉₆ = isotomic conjugate of X₂₃₉₅

• X₂₃₉₆ = anticomplement of isotomic conjugate of X₃₉₂₉₂

• X₂₃₉₆ = anticomplement of anticomplement of X₁₁₀₅₂

• X₂₃₉₆ = cevapoint of X_i and X_j for these (i,j): (230, 804), (511, 2491), (523, 2023)

• X₂₃₉₆ = trilinear pole of line X₁₁₄X₃₂₅ (the Simson line of X₉₉)

• X₂₃₉₆ = X_i-cross conjugate of X_j for these (i,j): (2421, 877), (2491, 511)