X₁₂₀₉

ISOGONAL CONJUGATE OF X₁₁₆₆

X₁₂₀₉ is the isogonal conjugate of X₁₁₆₆.

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

X₁₂₀₉ (ISOGONAL CONJUGATE OF X₁₁₆₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {52, 973}

• Related Centers: X₅₂ (ORTHOCENTER OF ORTHIC TRIANGLE), X₉₇₃ (1st EHRMANN POINT)

Barycentrics: \((S^{2} + S_{B} S_{C}) (a^{2} (S_{A}^{2} + S^{2}) + 2 S_{A} (S^{2} + S_{B} S_{C})) : :\)

Search Value (6, 9, 13): \(7.037932714967\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines7 lines

{2, 54}, {3, 161}, {5, 51}, {6, 17}, {12, 942}, {125, 128}, {127, 129}

▸ Functional Relations9 relations

• X₁₂₀₉ = reflection of X₅₂ in X₉₇₃

• X₁₂₀₉ = isogonal conjugate of X₁₁₆₆

• X₁₂₀₉ = complement of X₅₄

• X₁₂₀₉ = complementary conjugate of X₁₄₀

• X₁₂₀₉ = crosspoint of X₂ and X₃₁₁

• X₁₂₀₉ = perspector of circumconic centered at X₅₇₀

• X₁₂₀₉ = center of circumconic that is locus of trilinear poles of lines passing through X₅₇₀

• X₁₂₀₉ = X₂-Ceva conjugate of X₅₇₀

• X₁₂₀₉ = X₅₄ of X₅-Brocard triangle

Notes

In the plane of a triangle ABC, let

• A'B'C' = medial triangle;

• B_A = reflection of B' in BC, and define C_B and A_C cyclically;

• C_A = reflection of C' in BC, and define A_B and B_C cyclically;

• O_A = circle {{B', C', B_A, C_A}}, and define O_B and O_C cyclically.

Then X₁₂₀₉ = radical center of O_A, O_B, O_C. (Dasari Naga Vijay Krishna, September 8, 2021)

X₁₂₀₉ is the {X₁₇,X₁₈}-harmonic conjugate of X₂₃₁.