X₅₂₄

ISOGONAL CONJUGATE OF X₁₁₁

As the isogonal conjugate of a point on the circumcircle, X₅₂₄ lies on the line at infinity.

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

X₅₂₄ (ISOGONAL CONJUGATE OF X₁₁₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {53, 317}

• Related Centers: X₁₁₁ (PARRY POINT), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE), X₃₁₇ (ISOTOMIC CONJUGATE OF X₆₈)

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

Barycentrics: \(2 \cot A - \cot B - \cot C : :\)

▸ Linear Combinations1 combination

• X₅₂₄ = X₂ - X₆

▸ Incident Central Lines57 lines

{2, 6}, {5, 576}, {23, 2930}, {30, 511}, {53, 317}, {67, 858}, {74, 2696}, {76, 598}, {98, 2709}, {99, 843}, {100, 2721}, {101, 2729}, {102, 2735}, {103, 2740}, {104, 2746}, {105, 2753}, {106, 2759}, {107, 2763}, {108, 2767}, {109, 2768}, {110, 2770}, {140, 575}, {182, 549}, {187, 2482}, {237, 1634}, {239, 320}, {249, 1691}, {287, 1494}, {297, 340}, {316, 671}, {319, 594}, {332, 2305}, {338, 3260}, {376, 1350}, {381, 1351}, {397, 633}, {398, 634}, {428, 1843}, {441, 3284}, {468, 2192}, {487, 1152}, {488, 1151}, {551, 1386}, {620, 2030}, {637, 3070}, {638, 3071}, {694, 3228}, {1030, 1444}, {1084, 3229}, {1146, 1944}, {1238, 2965}, {1330, 1834}, {1901, 2893}, {1989, 2987}, {2094, 2097}, {3056, 3058}, {3241, 3242}

▸ Functional Relations27 relations

• X₅₂₄ = isogonal conjugate of X₁₁₁

• X₅₂₄ = isotomic conjugate of X₆₇₁

• X₅₂₄ = complementary conjugate of X₁₂₆

• X₅₂₄ = X_i-Ceva conjugate of X_j for these (i,j): (4, 126), (67, 141)

• X₅₂₄ = X₁₈₇-cross conjugate of X₄₆₈

• X₅₂₄ = crosssum of X₆ and X₁₈₇

• X₅₂₄ = crossdifference of every pair of points on line X₆X₅₁₂

• X₅₂₄ = X_i-line conjugate of X_j for these (i,j): (4, 126), (67, 141)

• X₅₂₄ = intercept of line X₂X₆ and the line at infinity (trilinear polars of X₉₉ and X₂)

• X₅₂₄ = perspector of circumconic centered at X₂₄₈₂

• X₅₂₄ = center of circumconic that is locus of trilinear poles of lines passing through X₂₄₈₂

• X₅₂₄ = X₂-Ceva conjugate of X₂₄₈₂

• X₅₂₄ = X₂₈₅₄-of-4th-Brocard-triangle

• X₅₂₄ = X₂₈₅₄-of-orthocentroidal-triangle

• X₅₂₄ = X₂₈₅₄-of-X₄-Brocard-triangle

• X₅₂₄ = crosspoint of X₆ and X₂₉₃₀ wrt both the excentral and tangential triangles

• X₅₂₄ = X₆-isoconjugate of X₈₉₇

• X₅₂₄ = polar conjugate of X₁₇₉₈₃

• X₅₂₄ = X₆₃-isoconjugate of X₈₇₅₃

• X₅₂₄ = barycentric cube root of X₈₀₃₀

• X₅₂₄ = perspector of ABC and unary cofactor triangle of 2nd Ehrmann triangle

• X₅₂₄ = X₈₇₀₅-of-circumsymmedial-triangle

• X₅₂₄ = Cundy-Parry Phi transform of X₁₃₆₀₈

• X₅₂₄ = Cundy-Parry Psi transform of X₁₄₂₆₂

• X₅₂₄ = Thomson-isogonal conjugate of X₁₂₉₆

• X₅₂₄ = Lucas-isogonal conjugate of X₁₂₉₆

• X₅₂₄ = trilinear pole of line X₃₅₁X₆₉₀ (line is perspectrix of ABC and 2nd Brocard triangle, and the tangent to the Steiner inellipse at X₂₄₈₂)

Notes

Let A'B'C' be the 2nd Brocard triangle. Let A″ be the trilinear pole of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ are parallel, and meet the line at infinity at X₅₂₄. (Randy Hutson, July 20, 2016)