X₂₈₀₁

ISOGONAL CONJUGATE OF X₂₇₁₇

X₂₈₀₁ is the isogonal conjugate of X₂₇₁₇.

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

X₂₈₀₁ (ISOGONAL CONJUGATE OF X₂₇₁₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {9, 48}, {7, 80}

• Related Centers: X₂₇₁₇ (SR(X₁₀₀, X₁₀₃)), X₉ (MITTENPUNKT), X₄₈ (CROSSPOINT OF X₁ AND X₆₃), X₇ (GERGONNE POINT), X₈₀ (REFLECTION OF INCENTER IN FEUERBACH POINT)

Trilinears: \(1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b)\), where f(a,b,c) is as at X₂₇₁₇

▸ Alternate Coordinates1 form

• Trilinears: \(a^{4}(b + c) - 2a^{3}(b^{2} + c^{2}) + a^{2}bc(b + c) + 2a(b - c)^{2}(b^{2} + bc + c^{2}) - b^{5} + b^{3}c^{2} + b^{2}c^{3} - c^{5} : :\)

▸ Incident Central Lines13 lines

{1, 651}, {7, 80}, {9, 48}, {11, 118}, {30, 511}, {63, 100}, {116, 119}, {149, 152}, {480, 2932}, {984, 991}, {990, 1814}, {1458, 1736}, {1776, 2078}

▸ Functional Relations6 relations

• X₂₈₀₁ = isogonal conjugate of X₂₇₁₇

• X₂₈₀₁ = isotomic conjugate of X₃₅₁₆₄

• X₂₈₀₁ = X₂-Ceva conjugate of X₃₅₁₁₆

• X₂₈₀₁ = crosssum of X₅₅ and X₂₂₄₆

• X₂₈₀₁ = X₅₁₁-of-Fuhrmann-triangle

• X₂₈₀₁ = endo-homothetic center of Ehrmann vertex-triangle and 2nd Ehrmann triangle; the homothetic center is X₅₄₂

Notes

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically. X₂₈₀₁ = X₅₁₆-of-I_aI_bI_c; I_aI_bI_c is the inner Garcia triangle. (Randy Hutson, February 10, 2016)