X₅₁₁

ISOGONAL CONJUGATE OF X₉₈

As the isogonal conjugate of a point on the circumcircle, X₅₁₁ lies on the line at infinity.

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

X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {2, 51}

• Related Centers: X₉₈ (TARRY POINT), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(\cos(A + \omega ) : \cos(B + \omega ) : \cos(C + \omega )\)

Barycentrics: \(\sin A \cos(A + \omega ) : \sin B \cos(B + \omega ) : \sin C \cos(C + \omega )\)

▸ Alternate Coordinates8 forms

• Trilinears: \(\sin A - \sin(A + 2\omega ) : \sin B - \sin(B + 2\omega ) : \sin C - \sin(C + 2\omega )\)

• Trilinears: \(\cos A + \cos(A + 2\omega ) : \cos B + \cos(B + 2\omega ) : \cos C + \cos(C + 2\omega )\)

• Trilinears: \(a(a^{2}b^{2} + a^{2}c^{2} - b^{4} - c^{4}) : :\) (M. Iliev, 5/13/07)

• Trilinears: \(\cos A - \sin A \tan \omega : :\)

• Trilinears: \(a^{2}\cos B \cos C - bc \cos^{2} A : : (R. Hutson, 1/29,15)\)

• Trilinears: \(\cos A + \cos 2A \cos(B - C) : :\)

• Trilinears: \(a - 2R \cos A \cot \omega : :\)

• Barycentrics: \((\cot A + \cot B + \cot C) \tan A - (\tan A + \tan B + \tan C) \cot A : :\)

▸ Linear Combinations1 combination

• X₅₁₁ = X₃ - X₆

▸ Incident Central Lines94 lines

{1, 256}, {2, 51}, {3, 6}, {4, 69}, {5, 141}, {20, 185}, {22, 184}, {23, 110}, {24, 1092}, {25, 394}, {26, 206}, {30, 512}, {35, 2330}, {36, 1428}, {40, 1045}, {49, 2937}, {54, 1176}, {55, 611}, {56, 613}, {66, 68}, {67, 265}, {74, 691}, {83, 3399}, {98, 385}, {99, 2698}, {100, 2699}, {101, 2700}, {102, 2701}, {103, 2702}, {104, 2703}, {105, 2704}, {106, 2705}, {107, 450}, {108, 2707}, {109, 2708}, {111, 352}, {112, 2710}, {114, 325}, {125, 858}, {140, 143}, {154, 3167}, {155, 159}, {165, 3097}, {171, 181}, {186, 249}, {195, 2916}, {199, 1790}, {230, 2023}, {232, 2211}, {238, 3271}, {242, 1944}, {283, 3145}, {287, 401}, {291, 3510}, {295, 3509}, {298, 1080}, {299, 383}, {343, 427}, {355, 3416}, {376, 1992}, {381, 599}, {399, 2930}, {403, 1568}, {468, 1112}, {549, 597}, {550, 1353}, {631, 3567}, {694, 3229}, {843, 1296}, {852, 2972}, {982, 1401}, {1113, 2105}, {1114, 2104}, {1194, 3051}, {1196, 1613}, {1292, 2711}, {1293, 2712}, {1294, 2713}, {1295, 2714}, {1297, 2715}, {1364, 1936}, {1370, 1899}, {1385, 1386}, {1437, 2915}, {1482, 3242}, {1757, 3507}, {1818, 2183}, {1976, 2065}, {2070, 3447}, {2095, 2097}, {2323, 3220}, {2653, 2670}, {3100, 3270}, {3124, 3291}

▸ Functional Relations33 relations

• X₅₁₁ = isogonal conjugate of X₉₈

• X₅₁₁ = isotomic conjugate of X₂₉₀

• X₅₁₁ = anticomplementary conjugate of X₁₄₇

• X₅₁₁ = complementary conjugate of X₁₁₄

• X₅₁₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 11672), (4, 114), (290, 2), (297, 232)

• X₅₁₁ = cevapoint of X₃₈₅ and X₄₀₁

• X₅₁₁ = X_i-cross conjugate of X_j for these (i,j): (4, 114), (290, 2), (297, 232)

• X₅₁₁ = crosspoint of X_i and X_j for these (i,j): (2, 290), (297, 325)

• X₅₁₁ = crosssum of X_i and X_j for these (i,j): (2, 385), (6, 237), (11, 659), (523, 868)

• X₅₁₁ = crossdifference of every pair of points on line X₆X₅₂₃

• X₅₁₁ = orthopoint of X₅₁₂

• X₅₁₁ = X₃-Hirst inverse of X₆

• X₅₁₁ = X_i-line conjugate of X_j for these (i,j): (3, 6), (30, 523)

• X₅₁₁ = intercept of Brocard axis and the line at infinity (trilinear polars of X₁₁₀ and X₂)

• X₅₁₁ = trilinear pole of line X₆₈₄X₂₄₉₁

• X₅₁₁ = radical center of Lucas(-2 tan ω) circles

• X₅₁₁ = {X₃,X₆}-harmonic conjugate of X₁₈₂

• X₅₁₁ = X₅₁₁-of-2nd-Brocard-triangle

• X₅₁₁ = X₅₄₂-of-orthocentroidal-triangle

• X₅₁₁ = X₅₄₂-of-X₄-Brocard triangle

• X₅₁₁ = X_i-isoconjugate of X_j for these (i,j): (6, 1821), (63, 6531), (92, 248)

• X₅₁₁ = X₅₁₂-of-3rd-Parry-triangle

• X₅₁₁ = ideal point of PU(i) for these i: 29, 145

• X₅₁₁ = bicentric sum of PU(145)

• X₅₁₁ = X₅₄₂-of-4th-anti-Brocard-triangle

• X₅₁₁ = Thomson-isogonal conjugate of X₉₉

• X₅₁₁ = Lucas-isogonal conjugate of X₉₉

• X₅₁₁ = perspector of 2nd Neuberg triangle and cross-triangle of 1st and 2nd Neuberg triangles

• X₅₁₁ = Cundy-Parry Phi transform of X₃₂

• X₅₁₁ = Cundy-Parry Psi transform of X₇₆

• X₅₁₁ = perspector of ABC and side-triangle of pedal triangles of PU(1)

• X₅₁₁ = perspector of ABC and side-triangle of reflection triangles of PU(1)

• X₅₁₁ = polar conjugate of isotomic conjugate of X₃₆₂₁₂

Notes

Let L denote the line having trilinears of X₅₁₁ as coefficients. Then L is the line passing through X₆ perpendicular to the Euler line.

X₅₁₁ is the perspector of triangle ABC and the tangential triangle of the hyperbola {A, B, C, X₂, and X₁₁₀}}.