X₃₅₉₈

1st LIU POINT

Let O_A be the circle passing through B and C, internally tangent to the incircle, and likewise for O_B and O_C. Let P_A be the point where O_A meets the incircle, and likewise for P_B and P_C. Let Q_A be the point of intersection of the tangents to the incircle at P_B and P_C, and likewise for P_B and P_C. Then X₃₅₉₈ is the perspector of the triangles P_AP_BP_C and Q_AQ_BQ_C; this point is also X₇-of-Q_AQ_BQ_C. This point and X₃₅₉₉ were discovered by Kang-Ying Liu of St. Andrew’s Priory School, Honolulu, Hawaii, during 2010.

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

X₃₅₉₈ (1st LIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 7}, {56, 105}

• Related Centers: X₂ (CENTROID), X₇ (GERGONNE POINT), X₅₆ (EXSIMILICENTER), X₁₀₅ (Λ(INCENTER, SYMMEDIAN POINT))

Trilinears: \(bc(a - b + c)(a + b - c)(3a^{2} + b^{2} + c^{2} - 2bc) : :\)

Barycentrics: \((a - b + c)(a + b - c)(3a^{2} + b^{2} + c^{2} - 2bc) : :\)

Search Value (6, 9, 13): \(0.207265042975\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a + b - c) \cdot (a - b + c) \cdot (3 a^{2} + b^{2} - 2 b c + c^{2}) : :\)

X₃₅₉₈ lies on the circumconic {{A, B, C, X₉, X₁₀₅, X₂₆₉, X₃₉₀, X₃₅₉₈, X₇₂₁₈, X₉₃₀₉}}.

▸ Incident Central Lines94 lines

{1, 7218}, {2, 7}, {20, 3673}, {22, 347}, {25, 1119}, {55, 3672}, {56, 105}, {75, 7172}, {77, 7191}, {85, 3600}, {145, 3212}, {165, 3663}, {193, 33891}, {241, 26242}, {244, 7204}, {269, 479}, {273, 6995}, {320, 37668}, {348, 5265}, {354, 3056}, {388, 7198}, {612, 4328}, {910, 4000}, {942, 7390}, {1002, 1469}, {1014, 4228}, {1086, 7735}, {1111, 4293}, {1122, 7248}, {1155, 4346}, {1323, 13462}, {1418, 3290}, {1420, 3160}, {1429, 2280}, {1442, 17024}, {1467, 4350}, {1477, 9086}, {1788, 30617}, {2266, 18162}, {2348, 37681}, {3091, 4911}, {3340, 5543}, {3361, 10481}, {3616, 3674}, {3664, 10980}, {3665, 7288}, {3705, 10513}, {3785, 33940}, {3920, 7190}, {4056, 10591}, {4225, 18600}, {4294, 7264}, {4308, 9312}, {4352, 37575}, {4441, 10030}, {4644, 37665}, {4848, 32003}, {4872, 5274}, {4888, 24239}, {5228, 5276}, {5232, 25631}, {5261, 7247}, {5268, 7274}, {5272, 7240}, {5542, 9746}, {5838, 24600}, {6049, 25716}, {6180, 33854}, {6356, 7494}, {6904, 20880}, {7081, 31995}, {7146, 29624}, {7185, 17081}, {7223, 7268}, {7238, 7778}, {7249, 30712}, {7263, 8667}, {7269, 29815}, {7272, 10590}, {7282, 7378}, {7465, 37541}, {7485, 15804}, {7736, 17365}, {9058, 15728}, {10106, 31994}, {10404, 24805}, {11038, 14828}, {12632, 17158}, {14986, 17170}, {16823, 32086}, {22464, 26228}, {24796, 32636}, {26236, 26245}, {34253, 37657}

▸ Functional Relations6 relations

• X₃₅₉₈ = X₇₂₉₀-cross conjugate of X (5222)

• X₃₅₉₈ = X_i-isoconjugate of X_j for these (i,j): {220, 21446}, {663, 37223}

• X₃₅₉₈ = crosssum of X₄₅₁₃ and X (37658)

• X₃₅₉₈ = barycentric product X_i*X_j for these {i,j}: {7, 5222}, {85, 7290}, {269, 30854}, {279, 390}, {651, 30804}, {1434, 3755}, {4626, 14330}, {24002, 35280}

• X₃₅₉₈ = barycentric quotient X (i)/X_j for these {i,j}: {269, 21446}, {390, 346}, {651, 37223}, {3755, 2321}, {4989, 3686}, {5222, 8}, {5575, 10322}, {7290, 9}, {14330, 4163}, {28017, 21450}, {30804, 4391}, {30854, 341}, {35280, 644}

• X₃₅₉₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {7, 1447, 2}, {7, 5435, 9436}, {56, 7195, 279}, {57, 1423, 672}, {75, 15589, 7172}, {3705, 21296, 10513}