X₁₁₅₆

ISOGONAL CONJUGATE OF X₁₁₅₅

X₁₁₅₆ is the isogonal conjugate of X₁₁₅₅.

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

X₁₁₅₆ (ISOGONAL CONJUGATE OF X₁₁₅₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {144, 149}

• Related Centers: X₁₄₄ (ANTICOMPLEMENT OF X₇), X₁₄₉ (REFLECTION OF X₂₀ IN X₁₀₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = 1/[(b - c)^{2} + a(b + c - 2a)]\)

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

Search Value (6, 9, 13): \(4.542109263484\)

X₁₁₅₆ lies on the Darboux septic.

▸ Incident Central Lines14 lines

{1, 651}, {4, 653}, {7, 11}, {8, 190}, {9, 100}, {21, 662}, {44, 294}, {80, 516}, {90, 411}, {104, 971}, {144, 149}, {314, 799}, {390, 952}, {673, 885}

▸ Functional Relations11 relations

• X₁₁₅₆ = midpoint of X₁₄₄ and X₁₄₉

• X₁₁₅₆ = reflection of X_i in X_j for these (i,j): (7, 11), (100, 9)

• X₁₁₅₆ = isogonal conjugate of X₁₁₅₅

• X₁₁₅₆ = isotomic conjugate of X₃₀₈₀₆

• X₁₁₅₆ = antigonal conjugate of X₇

• X₁₁₅₆ = symgonal of X₉

• X₁₁₅₆ = trilinear pole of line X₁X₆₅₀

• X₁₁₅₆ = polar conjugate of X₃₇₈₀₅

• X₁₁₅₆ = pole wrt polar circle of trilinear polar of X₃₇₈₀₅ (line X₆₃₆₆X_12831)

• X₁₁₅₆ = intersection of the Feuerbach circumhyperbola and the circumellise centered at X₉

• X₁₁₅₆ = BSS(a^2 → a) of X₇₄

Notes

Let B_c be the reflection of B in the internal angle bisector of angle C, and let C_b be the reflection of C in the internal angle bisector of angle B. Let Γ_a be the circle that passes through A, B_c, C_b. Let A' be the pole of BC, with respect to Γ_a and define B ‘and C' cyclically. Then X₁₁₅₆ is the perspector of A'B'C' and ABC, and X₃₅₄₄₅ is the perspector of A'B'C' and the excentral triangle. (Angel Montesdeoca, January 13, 2021)