X₁₁₄₁

GIBERT POINT

In Jean-Pierre Ehrmann and Bernard Gibert, “Special Isocubics,” downloadable from Cubics in the Triangle Plane, the point X₁₁₄₁ is labeled E, barycentrics are given, and it is established that this point also lies on the line X₅-to-X₁₁₀ [listed below as 5,49], two other cubics, and the hyperbola that passes through the points A, B, C, X₄, X₅.

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

X₁₁₄₁ (GIBERT POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 49}, {2, 128}

• Related Centers: X₅ (NINE-POINT CENTER), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE), X₂ (CENTROID), X₁₂₈ (-OF-ORTHIC-TRIANGLE)

Trilinears: \(bc/[16D^{2} + (a^{2}+c^{2}-b^{2})(a^{2}+b^{2}-c^{2})][16D^{2} - 3(b^{2}+c^{2}-a^{2})^{2}] : :\), where \(D = \operatorname{area}(ABC) = \sec(B - C)/(1 - 4 \cos^{2} A) : \sec(C - A)/(1 - 4 \cos^{2} B) : \sec(A - B)/(1 - 4 \cos^{2} C)\) (Eric Weisstein, Nov. 17, 2005)

Search Value (6, 9, 13): \(-0.574877742883\)

▸ Linear Combinations1 combination

• X₁₁₄₁ = \(3\) X₅₄₉ - \(2\) X₆₅₉₂

X₁₁₄₁ lies on the conic of {A, B, C, X₃, X₄₉}, the conic of {A, B, C, X₆, X₅₆₇}, and the conic of {A, B, C, X₇₀, X₂₅₃, X₂₅₄}.

X₁₁₄₁ lies on the circumcircle, and cubics K060, K112, K466, K467, K491, the circumconic {{A, B, C,X₄,X₅}}.

▸ Incident Central Lines33 lines

{2, 128}, {3, 252}, {4, 137}, {5, 49}, {20, 11671}, {30, 1157}, {53, 112}, {55, 7159}, {56, 3327}, {79, 109}, {94, 96}, {95, 99}, {101, 7110}, {107, 3518}, {140, 11016}, {476, 2070}, {549, 6592}, {621, 10409}, {622, 10410}, {1303, 5890}, {1304, 5627}, {1487, 10619}, {2166, 2222}, {2413, 5966}, {3153, 10420}, {3459, 12254}, {5994, 11582}, {5995, 11581}, {6069, 11464}, {6240, 6799}, {7418, 9076}, {7731, 9512}, {8800, 12225}

▸ Functional Relations20 relations

• X₁₁₄₁ = midpoint of X₂₀ and X₁₁₆₇₁

• X₁₁₄₁ = reflection of X_i in X_j for these (i,j): (4, 137), (5, 12026), (930, 3)

• X₁₁₄₁ = isogonal conjugate of X₁₁₅₄

• X₁₁₄₁ = isotomic conjugate of X₁₂₇₃

• X₁₁₄₁ = anticomplement of X₁₂₈

• X₁₁₄₁ = X₂₃₁-cross conjugate of X₂

• X₁₁₄₁ = polar conjugate of X₁₄₉₁₈

• X₁₁₄₁ = antipode of X₄ in hyperbola {{A, B, C,X₄,X₅}}

• X₁₁₄₁ = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A, B, C,X₄,X₅}}

• X₁₁₄₁ = Collings transform of X₁₃₇

• X₁₁₄₁ = cevapoint of X_i and X_j for these {i,j}: {3, 539}, {13, 6104}, {14, 6105}, {54, 1157}, {265, 5961}

• X₁₁₄₁ = X₁₁₀-of-Lucas-triangle (defined at X₉₅)

• X₁₁₄₁ = X_i-cross conjugate of X_j for these (i,j): (4, 5627), (231, 2), (2070, 1166), (10412, 476), (11063, 288)

• X₁₁₄₁ = isoconjugate of X_j and X_j for these (i,j): {1, 1154}, {2, 2290}, {5, 6149}, {31, 1273}, {63, 11062}, {323, 1953}, {526, 2617}, {662, 2081}, {2179, 7799}

• X₁₁₄₁ = trilinear pole of line {6,2623}

• X₁₁₄₁ = inverse of X₅₄ in the circle having diameter OH

• X₁₁₄₁ = X₁₁₀-of-circumorthic-triangle

• X₁₁₄₁ = trilinear pole, wrt circumorthic triangle, of line X₃X₉₅

• X₁₁₄₁ = barycentric product X_i*X_j for these {i,j}: {54, 94}, {95, 1989}, {97, 6344}, {264, 11077}, {265, 275}, {328, 8882}, {930, 2413}, {2166, 2167}

• X₁₁₄₁ = barycentric quotient X_i/X_j for these (i,j): (i,j}: (2, 1273), (6, 1154), (25, 11062), (31, 2290), (54, 323), (94, 311), (95, 7799), (231, 128), (265, 343), (275, 340), (512, 2081), (1989, 5), (2148, 6149), (2623, 526), (6344, 324), (8737, 6116), (8738, 6117), (8882, 186), (11060, 51), (11071, 1263), (11077, 3)

Notes

X₁₁₄₁ was first noted (Hyacinthos #1498, September 25, 2000) by Bernard Gibert as a point of intersection of the circumcircle and certain cubic, denoted K_n. To define K_n, note first that the Neuberg cubic is the locus of a point M such that the reflections of M in the sidelines of triangle ABC are the vertices of a triangle perspective to ABC. The locus of the perspector is the cubic K_n, and X₁₁₄₁ is the point, other than A, B, C, in which K_n meets the circumcircle. Also, X₁₁₄₁ is the perspector when M = X₁₁₅₇.

Let Ref_a be the reflection of A in line BC, and define Ref_b and Ref_c cyclically. Let A_B be the reflection of Ref_a in AB, and define A_C, B_C, B_A, C_A, C_B cyclically. Let

A₁ = BA_B∩CA_C, and define B₁ and C₁ cyclically,

• A₂ = BA_C∩CA_B, and define B₂ and C₂ cyclically,

• A₃ = BB_A∩CC_A, and define B₃ and C₃ cyclically,

• A₄ = BB_C∩CC_B, and define B₄ and C₄ cyclically,

• A₅ = BC_A∩CB_A, and define B₅ and C₅ cyclically,

• A₆ = BC_B∩CB_C, and define B₆ and C₆ cyclically.

Then triangle A_nB_nC_n is perspective to ABC, for n = 1,2,3,4,5,6. The six perspectors are X₁₁₄₁, X₁₈₆, X₄, X₅₄, X₂₆₅, X₅, respectively. (Keith Dean, #4953, 3/12/02; coordinates by Paul Yiu, #4963; summary by Dean, #4971)

X₁₁₄₁ is the antipode of X₉₃₀ on the circumcircle, and X₁₁₄₁ lies on the line of the nine-point center, X₅, and its isogonal conjugate, X₅₄.

See X₂₀₂₁₂ for an additional comment about X₁₁₄₁; also, 24183.