X₃₅₁₉

ISOGONAL CONJUGATE OF X₃₅₁₈

X₃₅₁₉ is the isogonal conjugate of X₃₅₁₈.

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

X₃₅₁₉ (ISOGONAL CONJUGATE OF X₃₅₁₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 17}, {4, 93}

• Related Centers: X₃₅₁₈ (EULER LINE INTERCEPT OF LINE X), X₆ (SYMMEDIAN POINT), X₁₇ (1st NAPOLEON POINT), X₄ (ORTHOCENTER), X₉₃ (ISOGONAL CONJUGATE OF X₄₉)

Trilinears: \(1/(4 \cos A - 3 \sec A) : :\)

Barycentrics: \(1/(\cot A - 3 \tan A) : :\)

Search Value (6, 9, 13): \(33.233746429069\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\csc A \tan 3A + \sec 3A : :\)

• Barycentrics: \(1/((3 S^{2}-S_{A}^{2}) S_{B} S_{C}) : :\)

• Barycentrics: \((a^{2} - b^{2} - c^{2}) \cdot (a^{4} - a^{2} b^{2} + b^{4} - 2 a^{2} c^{2} - 2 b^{2} c^{2} + c^{4}) \cdot (a^{4} - 2 a^{2} b^{2} + b^{4} - a^{2} c^{2} - 2 b^{2} c^{2} + c^{4}) : :\)

• Barycentrics: \(\tan(A + \pi /3) + \tan(A - \pi /3) : :\)

X₃₅₁₉ lies on the the Jerabek hyperbola, the cubic K618.

▸ Incident Central Lines56 lines

{2, 1493}, {3, 539}, {4, 93}, {5, 1173}, {6, 17}, {20, 13452}, {49, 343}, {54, 140}, {64, 1657}, {65, 2962}, {68, 12606}, {69, 12363}, {70, 14791}, {74, 550}, {185, 14861}, {265, 5562}, {290, 7768}, {340, 8795}, {394, 15317}, {399, 14862}, {511, 13433}, {524, 5576}, {542, 13564}, {567, 12899}, {895, 11585}, {1176, 3564}, {1503, 9935}, {1899, 11577}, {2889, 3448}, {2914, 14643}, {2918, 5898}, {2979, 12291}, {3426, 5073}, {3431, 3523}, {3521, 13754}, {3522, 11270}, {3527, 3574}, {3850, 14483}, {4857, 6286}, {5056, 13565}, {5059, 11738}, {5068, 14491}, {5449, 15002}, {5504, 12359}, {6101, 12226}, {6145, 7574}, {6515, 9827}, {7517, 15069}, {10018, 11597}, {10625, 14864}, {10628, 11744}, {11138, 11600}, {11139, 11601}, {11225, 15047}, {11412, 12280}, {12936, 15232}

▸ Functional Relations13 relations

• X₃₅₁₉ = midpoint of X_i and X_j for these {i,j}: {2888, 12325}, {11412, 12280}

• X₃₅₁₉ = reflection of X_i in X_j for these {i,j}: {195, 1209}, {6243, 6152}, {6288, 2888}, {11271, 1493}, {12226, 6101}, {12316, 3574}, {13431, 12242}, {13432, 13431}

• X₃₅₁₉ = isotomic conjugate of X₃₂₀₀₂

• X₃₅₁₉ = complement of X₁₁₂₇₁

• X₃₅₁₉ = anticomplement of X₁₄₉₃

• X₃₅₁₉ = X_i-anticomplementary conjugate of X_j for these (i,j): {1487, 8}, {2962, 2889}

• X₃₅₁₉ = cevapoint of X_i and X_j for these (i,j): {633, 634}, {14813, 14814}

• X₃₅₁₉ = X₁₁₁₄₀-Ceva conjugate of X₂₉₆₃

• X₃₅₁₉ = X₁₂₁₆-cross conjugate of X₃

• X₃₅₁₉ = barycentric product X_iX_j for these {i,j}: {3, 11140}, {63, 2962}, {69, 2963}, {93, 394}, {252, 343}, {525, 930}

• X₃₅₁₉ = barycentric quotient X_i/X_j for these {i,j}: {3, 1994}, {5, 14129}, {6, 3518}, {17, 472}, {18, 473}, {48, 2964}, {51, 14577}, {69, 7769}, {93, 2052}, {184, 2965}, {216, 143}, {252, 275}, {562, 14165}, {570, 6152}, {577, 49}, {647, 1510}, {930, 648}, {2962, 92}, {2963, 4}, {8603, 10633}, {8604, 10632}, {11140, 264}, {14111, 11547}

• X₃₅₁₉ = X_i-isoconjugate of X_j for these (i,j): {1, 3518}, {4, 2964}, {19, 1994}, {49, 158}, {92, 2965}, {143, 2190}, {162, 1510}, {1973, 7769}, {2148, 14129}, {2167, 14577}, {2216, 6152}

• X₃₅₁₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 11271, 1493), (17, 18, 2963), (93, 562, 14111), (195, 1656, 12242), (195, 13432, 13431), (1209, 12242, 1656), (1209, 13431, 12242), (1656, 13432, 195), (12242, 13431, 195), (19778, 19779, 11140)

Notes

Let Ref_aRef_bRef_c be the reflection triangle. Let O_a be the circle centered at Ref_a and passing through A, and define O_b and O_c cyclically. The radical center of O_a, O_b, O_c is X₃₅₁₉. Let N_a be the reflection of X₅ in the perpendicular bisector of BC, and define N_b and N_c cyclically. The lines AN_a, BN_b, CN_c concur in X₃₅₁₉. Let A″B″C″ be the Trinh triangle. Let A* be the orthopole of line Ref_bRef_c, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₃₅₁₉. (Randy Hutson, October 13, 2015)

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 26850.