X₄₃

X₆-CEVA CONJUGATE OF X₁

X₄₃ is the {X₂,X₄₂}-harmonic conjugate of X₁. X₄₃ is the external center of similitude of the Bevan circle and Apollonius circle; the internal center is X₁₆₉₅.

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

X₄₃ (X₆-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 995}

• Related Centers: X₁ (INCENTER), X₉₉₅ (MIDPOINT OF X₁ AND X₄₃)

Trilinears: \(ab + ac - bc : bc + ba - ca : ca + cb - ab\)

Barycentrics: \(a(ab + ac - bc) : b(bc + ba - ca) : c(ca + cb - ab)\)

Search Value (6, 9, 13): \(0.257220860135\)

▸ Alternate Coordinates1 form

• Trilinears: \(\csc B + \csc C - \csc A : \csc C + \csc A - \csc B : \csc A + \csc B - \csc C\)

X₄₃ lies on the Kiepert hyperbola of the excentral triangle.

▸ Incident Central Lines18 lines

{1, 2}, {6, 87}, {9, 256}, {31, 100}, {35, 1011}, {40, 970}, {46, 851}, {55, 238}, {57, 181}, {58, 979}, {72, 986}, {75, 872}, {81, 750}, {165, 573}, {170, 218}, {210, 984}, {312, 740}, {518, 982}

▸ Functional Relations17 relations

• X₄₃ = reflection of X₁ in X₉₉₅

• X₄₃ = isogonal conjugate of X₈₇

• X₄₃ = isotomic conjugate of X₆₃₈₄

• X₄₃ = anticomplement of X₃₈₄₀

• X₄₃ = X₆-Ceva conjugate of X₁

• X₄₃ = X₁₉₂-cross conjugate of X₁

• X₄₃ = crosssum of X₂ and X₃₃₀

• X₄₃ = X₅₅-Hirst inverse of X₂₃₈

• X₄₃ = inverse-in-excircles-radical-circle of X₅₁₂₁

• X₄₃ = perspector of ABC and extraversion triangle of X₈₇

• X₄₃ = trilinear product of extraversions of X₈₇

• X₄₃ = excentral-isogonal conjugate of X₁₇₆₆

• X₄₃ = polar conjugate of isotomic conjugate of X₂₂₃₇₀

• X₄₃ = perspector of Gemini triangle 5 and cross-triangle of Gemini triangles 3 and 5

• X₄₃ = X_i-aleph conjugate of X_j for these (i,j): (1, 9), (6, 43), (55, 170), (100, 1018), (174, 169), (259, 165), (365, 1), (366, 63), (507, 362), (509, 57)

• X₄₃ = X₆₆₀-beth conjugate of X₄₃

• X₄₃ = {X₂,X₈}-harmonic conjugate of X₃₇₄₁

Notes

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B' and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle at X₄₃. (Randy Hutson, November 30, 2018)