X₆₆

ISOGONAL CONJUGATE OF X₂₂

X₆₆ is the isogonal conjugate of X₂₂.

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

X₆₆ (ISOGONAL CONJUGATE OF X₂₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2892, 3448}

• Related Centers: X₂₈₉₂ (ANTICOMPLEMENTARY CONJUGATE OF X₂₃), X₃₄₄₈ (ISOGONAL CONJUGATE OF X₃₄₄₇)

Trilinears: \(bc/(b^{4} + c^{4} - a^{4}) : :\)

Barycentrics: \(1/(b^{4} + c^{4} - a^{4}) : :\)

Search Value (6, 9, 13): \(9.640588374561\)

▸ Incident Central Lines9 lines

{2, 206}, {3, 141}, {6, 427}, {68, 511}, {73, 976}, {193, 895}, {248, 571}, {290, 317}, {879, 924}

▸ Functional Relations16 relations

• X₆₆ = midpoint of X₂₈₉₂ and X₃₄₄₈

• X₆₆ = reflection of X_i in X_j for these (i,j): (159, 141), (1177, 125)

• X₆₆ = isogonal conjugate of X₂₂

• X₆₆ = isotomic conjugate of X₃₁₅

• X₆₆ = complement of X₅₅₉₆

• X₆₆ = anticomplement of X₂₀₆

• X₆₆ = cyclocevian conjugate of X₂₉₉₈

• X₆₆ = cevapoint of X₁₂₅ and X₅₁₂

• X₆₆ = X₃₂-cross conjugate of X₂

• X₆₆ = crosssum of X₃ and X₁₅₉

• X₆₆ = trilinear pole of line X₆₄₇X₈₂₆ (radical axis of Brocard and polar circles)

• X₆₆ = antigonal image of X₁₁₇₇

• X₆₆ = orthocenter of X₃X₄X₂₄₃₅

• X₆₆ = X₃₁₇₄-of-orthic-triangle if ABC is acute

• X₆₆ = polar conjugate of isotomic conjugate of X₁₄₃₇₆

• X₆₆ = X₆₃-isoconjugate of X₈₇₄₃

Notes

Let P be a point on the circumcircle, and let L_P be its Steiner line. The locus of the orthopole of L_P, as P varies, is an ellipse with center X₄ and perspector X₆₆. (Randy Hutson, March 29, 2020)

In the plane of a triangle ABC, let

• A'B'C' = anticomplementary triangle;

• O_a = circle with diameter BC, and define O_b and O_c cyclically;

• A_b = A'BC'∩O_a, and define B_c and C_a cyclically;

• A_c = B'CA'∩O_a, and define B_a and C_b cyclically;

• A″= B_cB_a∩C_aC_b, and define B″ and C″ cyclically.

The triangle A″B″C″ is perspective to ABC, and the perspector is X₆₆. (Dasari Naga Vijay Krishna, April 15, 2021)