X₂₇₄

ISOGONAL CONJUGATE OF X₂₁₃

X₂₇₄ is the isogonal conjugate of X₂₁₃.

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

X₂₇₄ (ISOGONAL CONJUGATE OF X₂₁₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 39}, {1, 75}

• Related Centers: X₂₁₃ (-CEVA CONJUGATE OF X₄₂), X₂ (CENTROID), X₃₉ (BROCARD MIDPOINT), X₁ (INCENTER), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

Trilinears: \(b^{2}c^{2}/(b + c) : :\)

Barycentrics: \(bc/(b + c) : ca/(c + a) : ab/(a + b)\)

Search Value (6, 9, 13): \(3.221143406647\)

▸ Alternate Coordinates1 form

• Trilinears: \([a \csc(A - \omega )]/(b + c) : :\)

▸ Incident Central Lines19 lines

{1, 75}, {2, 39}, {7, 959}, {10, 291}, {21, 99}, {28, 242}, {57, 85}, {58, 870}, {69, 443}, {81, 239}, {88, 799}, {110, 767}, {183, 474}, {213, 894}, {264, 475}, {278, 331}, {315, 377}, {325, 442}, {961, 1014}

▸ Functional Relations14 relations

• X₂₇₄ = isogonal conjugate of X₂₁₃

• X₂₇₄ = isotomic conjugate of X₃₇

• X₂₇₄ = complement of X₁₆₅₅

• X₂₇₄ = X₃₁₀-Ceva conjugate of X₃₁₄

• X₂₇₄ = cevapoint of X_i and X_j for these (i,j): (2, 75), (85, 348), (86, 333)

• X₂₇₄ = X_i-cross conjugate of X_j for these (i,j): (2, 86), (75, 310), (81, 286), (333, 314)

• X₂₇₄ = crossdifference of every pair of points on line X₆₆₉X₇₉₈

• X₂₇₄ = trilinear pole of line X₃₂₀X₃₅₀ (anticomplement of antiorthic axis)

• X₂₇₄ = pole wrt polar circle of trilinear polar of X₁₈₂₄

• X₂₇₄ = X₄₈-isoconjugate (polar conjugate) of X₁₈₂₄

• X₂₇₄ = trilinear product of vertices of Gemini triangle 1

• X₂₇₄ = trilinear product of vertices of Gemini triangle 2

• X₂₇₄ = trilinear product of vertices of Gemini triangle 3

• X₂₇₄ = trilinear product of vertices of Gemini triangle 4

Notes

Let A_b, A_c, B_c, B_a, C_a, C_b be as in the construction of the Conway circle; see http://mathworld.wolfram.com/ConwayCircle.html. Let O_a be the circumcircle of AB_aC_a, and define O_b and O_c cyclically. Then X₂₇₄ is the radical center of O_a, O_b, O_c. see also X₈ and X₂₁. (Randy Hutson, April 9, 2016)