X₁₀₂

Λ(INCENTER, ORTHOCENTER)

X₁₀₂ = anticomplement of X₁₁₇

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

X₁₀₂ (Λ(INCENTER, ORTHOCENTER)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {40, 78}, {1, 108}

• Related Centers: X₄₀ (BEVAN POINT), X₇₈ (ISOGONAL CONJUGATE OF X₃₄), X₁ (INCENTER), X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER))

Trilinears: \(1/[\sin B (\sec A - \sec B) + \sin C (\sec A - \sec C)] : :\)

Barycentrics: \(a^{2} (a^{4} - 2 a^{2} b^{2} + b^{4} - a^{3} c + a^{2} b c + a b^{2} c - b^{3} c + a^{2} c^{2} - 2 a b c^{2} + b^{2} c^{2} + a c^{3} + b c^{3} - 2 c^{4}) \cdot (a^{4} - a^{3} b + a^{2} b^{2} + a b^{3} - 2 b^{4} + a^{2} b c - 2 a b^{2} c + b^{3} c - 2 a^{2} c^{2} + a b c^{2} + b^{2} c^{2} - b c^{3} + c^{4}) : :\)

Tripolars: \(|\sin B (\sec A - \sec B) + \sin C (\sec A - \sec C)| : :\)

Search Value (6, 9, 13): \(13.403799631213\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a/[2a^{4} - (b + c) a^{3} - (b - c)^{2} a^{2} + (b - c)^{2} (b + c) a - (b^{2} - c^{2})^{2}] : :\)

• Barycentrics: \((\sin A)/[\sin B (\sec A - \sec B) + \sin C (\sec A - \sec C)] : :\)

▸ Linear Combinations1 combination

• X₁₀₂ = X₁₀₆₉₆ - \(4\) X₁₁₇₁₃, \(3\) X₂ - \(4\) X₆₇₁₁, \(4\) X₁₁₇ - \(3\) X₁₀₇₀₉, X₁₅₁ - \(4\) X₆₇₁₁, \(2\) X₁₅₁ - \(3\) X₁₀₇₀₉, \(8\) X₆₇₁₁ - \(3\) X₁₀₇₀₉, \(3\) X₃ - X₃₈₅₇₉, \(3\) X₃ - \(2\) X₃₈₆₀₇, \(2\) X₃ + X₃₈₆₆₇, \(4\) X₃ - X₃₈₆₇₄, \(2\) X₃ - \(3\) X₃₈₆₉₁, \(4\) X₃ - \(3\) X₃₈₆₉₇, X₁₀₉ + \(2\) X₃₈₅₇₃, \(3\) X₁₀₉ - \(2\) X₃₈₅₇₉, X₁₀₉ - \(4\) X₃₈₆₀₀, \(3\) X₁₀₉ - \(4\) X₃₈₆₀₇, X₁₀₉ - \(3\) X₃₈₆₉₁, \(2\) X₁₀₉ - \(3\) X₃₈₆₉₇, \(3\) X₃₈₅₇₃ + X₃₈₅₇₉, X₃₈₅₇₃ + \(2\) X₃₈₆₀₀, \(3\) X₃₈₅₇₃ + \(2\) X₃₈₆₀₇, \(4\) X₃₈₅₇₃ + X₃₈₆₇₄, \(2\) X₃₈₅₇₃ + \(3\) X₃₈₆₉₁, \(4\) X₃₈₅₇₃ + \(3\) X₃₈₆₉₇, X₃₈₅₇₉ - \(6\) X₃₈₆₀₀, \(2\) X₃₈₅₇₉ + \(3\) X₃₈₆₆₇, \(4\) X₃₈₅₇₉ - \(3\) X₃₈₆₇₄, \(2\) X₃₈₅₇₉ - \(9\) X₃₈₆₉₁, \(4\) X₃₈₅₇₉ - \(9\) X₃₈₆₉₇, \(3\) X₃₈₆₀₀ - X₃₈₆₀₇, \(4\) X₃₈₆₀₀ + X₃₈₆₆₇, \(8\) X₃₈₆₀₀ - X₃₈₆₇₄, \(4\) X₃₈₆₀₀ - \(3\) X₃₈₆₉₁, \(8\) X₃₈₆₀₀ - \(3\) X₃₈₆₉₇, \(4\) X₃₈₆₀₇ + \(3\) X₃₈₆₆₇, \(8\) X₃₈₆₀₇ - \(3\) X₃₈₆₇₄, \(4\) X₃₈₆₀₇ - \(9\) X₃₈₆₉₁, \(8\) X₃₈₆₀₇ - \(9\) X₃₈₆₉₇, \(2\) X₃₈₆₆₇ + X₃₈₆₇₄, X₃₈₆₆₇ + \(3\) X₃₈₆₉₁, \(2\) X₃₈₆₆₇ + \(3\) X₃₈₆₉₇, X₃₈₆₇₄ - \(6\) X₃₈₆₉₁, X₃₈₆₇₄ - \(3\) X₃₈₆₉₇, \(4\) X₁₂₄ - X₁₀₇₂₆, \(2\) X₅ - \(3\) X₃₈₇₇₆, X₁₀₇₄₀ - \(3\) X₃₈₇₇₆, \(3\) X₁₀₇₁₆ - X₁₀₇₃₂, \(3\) X₁₀₇₁₆ - \(2\) X₁₀₇₄₇, \(3\) X₃₅₇₆ - \(2\) X₁₁₇₀₀, \(4\) X₁₄₀ - \(5\) X₃₈₇₈₆, \(3\) X₁₆₅ - \(2\) X₁₄₆₉₀, \(3\) X₃₇₆ - \(2\) X₃₈₇₈₅, X₃₈₂ - \(3\) X₃₈₇₇₉, \(4\) X₅₄₈ - \(3\) X₃₈₇₇₈, \(5\) X₆₃₁ - \(4\) X₆₇₁₈, \(5\) X₆₃₁ - \(6\) X₃₈₇₈₄, \(2\) X₆₇₁₈ - \(3\) X₃₈₇₈₄, \(7\) X₃₀₉₀ - \(10\) X₃₈₇₈₇, X₃₁₄₆ - \(4\) X₃₈₇₈₁, \(5\) X₃₅₂₂ - \(4\) X₃₈₇₈₃, \(5\) X₃₆₁₆ - \(4\) X₁₁₇₂₇, \(7\) X₃₈₃₂ - \(12\) X₃₈₇₈₂, \(3\) X₅₆₀₃ - \(4\) X₁₁₇₃₄

X₁₀₂ lies on the circumcircle, the conic {A, B, C,X₁, X₃}}, the cubics K269, K685.

▸ Incident Central Lines144 lines

{1, 108}, {2, 117}, {3, 109}, {4, 124}, {5, 10740}, {6, 10757}, {11, 10771}, {19, 282}, {20, 33650}, {24, 41401}, {29, 107}, {30, 2689}, {36, 1795}, {40, 78}, {55, 1361}, {56, 1364}, {57, 12016}, {63, 43347}, {64, 18237}, {73, 947}, {74, 2773}, {77, 934}, {98, 2785}, {99, 332}, {101, 198}, {103, 928}, {104, 3738}, {105, 2814}, {106, 2815}, {110, 283}, {111, 2819}, {112, 284}, {140, 38786}, {165, 14690}, {186, 40081}, {226, 1065}, {376, 38785}, {382, 38779}, {386, 34455}, {476, 7424}, {511, 2701}, {512, 2708}, {513, 2716}, {514, 2723}, {515, 1309}, {516, 929}, {517, 1807}, {518, 2730}, {519, 2731}, {520, 2732}, {521, 2733}, {522, 2734}, {523, 2695}, {524, 2735}, {548, 38778}, {550, 38777}, {572, 29044}, {580, 15440}, {631, 6718}, {901, 2077}, {927, 31637}, {930, 16113}, {933, 35196}, {944, 32704}, {949, 2301}, {958, 3042}, {959, 37530}, {1036, 32691}, {1067, 12053}, {1069, 11249}, {1183, 37732}, {1292, 2835}, {1293, 2841}, {1294, 2846}, {1295, 2849}, {1296, 2852}, {1297, 2853}, {1304, 2075}, {1305, 4329}, {1311, 2399}, {1350, 28291}, {1376, 3040}, {1385, 7100}, {1420, 30239}, {1457, 36067}, {1499, 2768}, {1503, 2769}, {1633, 33810}, {1657, 38780}, {1794, 15439}, {1897, 31866}, {2291, 2432}, {2338, 40116}, {2359, 8687}, {2715, 5060}, {2728, 14203}, {2743, 13528}, {2751, 3309}, {2757, 3667}, {2762, 6000}, {2765, 6001}, {2968, 18339}, {3090, 38787}, {3146, 38781}, {3422, 36076}, {3430, 6011}, {3478, 9088}, {3522, 38783}, {3616, 11727}, {3832, 38782}, {4262, 26716}, {5053, 32685}, {5450, 21228}, {5603, 11734}, {5759, 44876}, {5840, 10777}, {6014, 6244}, {6614, 7215}, {7015, 29055}, {7152, 8064}, {8607, 32683}, {8686, 41343}, {9057, 36007}, {9058, 35996}, {9059, 10327}, {9107, 35973}, {10680, 37489}, {10902, 40442}, {11014, 26711}, {13329, 32682}, {13397, 36986}, {14127, 14987}, {15379, 35187}, {15501, 39763}, {16132, 44063}, {18446, 26706}, {21740, 30250}, {22765, 34921}, {24466, 39444}, {26712, 30272}, {32643, 32689}, {32667, 32688}, {32674, 47411}, {36984, 46964}

▸ Functional Relations50 relations

• X₁₀₂ = midpoint of X_i and X_j for these {i,j}: {3, 38573}, {20, 33650}, {109, 38667}, {1657, 38780}

• X₁₀₂ = reflection of X_i in X_j for these {i,j}: {1, 11713}, {3, 38600}, {4, 124}, {109, 3}, {117, 6711}, {151, 117}, {1897, 31866}, {10696, 1}, {10709, 2}, {10726, 4}, {10732, 10747}, {10740, 5}, {10757, 6}, {10771, 11}, {18339, 2968}, {38579, 38607}, {38667, 38573}, {38674, 109}, {38697, 38691}, {38777, 550}

• X₁₀₂ = isogonal conjugate of X₅₁₅

• X₁₀₂ = isotomic conjugate of X₃₅₅₁₆

• X₁₀₂ = anticomplement of X₁₁₇

• X₁₀₂ = complement of X₁₅₁

• X₁₀₂ = anticomplement of the anticomplement of X₆₇₁₁

• X₁₀₂ = anticomplement of the isogonal conjugate of X₁₅₃₇₉

• X₁₀₂ = complement of the isogonal conjugate of X₃₄₁₈₀

• X₁₀₂ = isogonal conjugate of the anticomplement of X₅₁₅

• X₁₀₂ = isogonal conjugate of the complement of X₅₁₅

• X₁₀₂ = isotomic conjugate of the anticomplement of X₈₆₀₇

• X₁₀₂ = isogonal conjugate of the isotomic conjugate of X₃₄₃₉₃

• X₁₀₂ = Thomson isogonal conjugate of X₅₂₂

• X₁₀₂ = Collings transform of X_i for these i: {124, 3137, 38983}

• X₁₀₂ = X₁₅₃₇₉-anticomplementary conjugate of X₈

• X₁₀₂ = X₃₄₁₈₀-complementary conjugate of X₁₀

• X₁₀₂ = X₂₁-beth conjugate of X₁₀₈

• X₁₀₂ = circumcircle-antipode of X₁₀₉

• X₁₀₂ = Λ(X₁, X₄)

• X₁₀₂ = trilinear pole of line X₆X₆₅₂

• X₁₀₂ = Ψ(X₆, X₆₅₂)

• X₁₀₂ = Ψ(X₁, X₅₂₁)

• X₁₀₂ = Ψ(X₁₀₈, X₁)

• X₁₀₂ = Ψ(X₆₅₁, X₆₃)

• X₁₀₂ = Ψ(X₆₅₃, X₂)

• X₁₀₂ = trilinear product of circumcircle intercepts of line X₁X₅₂₁

• X₁₀₂ = trilinear pole wrt 2nd circumperp triangle of line X₉₇₁X₁₀₀₁

• X₁₀₂ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X₄,X₅₈}}

• X₁₀₂ = reflection of X₂₆₉₅ in the Euler line

• X₁₀₂ = reflection of X₂₇₀₈ in the Brocard axis

• X₁₀₂ = reflection of X₂₇₁₆ in line X₁X₃

• X₁₀₂ = X₁₃₁-of-excentral-triangle

• X₁₀₂ = X₁₃₆-of-hexyl-triangle

• X₁₀₂ = X₉₂₅-of-2nd-circumperp-triangle

• X₁₀₂ = Thomson-isogonal conjugate of X₅₂₂

• X₁₀₂ = Lucas-isogonal conjugate of X₅₂₂

• X₁₀₂ = Cundy-Parry Phi transform of X₁₀₅₇₁

• X₁₀₂ = Cundy-Parry Psi transform of X₁₀₅₇₀

• X₁₀₂ = Ψ(X₁₉, X₆₅₀)

• X₁₀₂ = polar-circle-inverse of nine-point circle antipode of X₃₈₉₇₇

• X₁₀₂ = X₃₆₁₀₀-Ceva conjugate of X₁₅₆₂₉

• X₁₀₂ = X_i-cross conjugate of X_j for these (i,j): {3, 15379}, {1457, 1}, {2342, 2316}, {8607, 2}, {8677, 109}, {46359, 34393}

• X₁₀₂ = X_i-isoconjugate of X_j for these (i,j): {1, 515}, {2, 2182}, {4, 46974}, {8, 1455}, {9, 34050}, {31, 35516}, {63, 8755}, {80, 11700}, {102, 24034}, {108, 39471}, {109, 14304}, {521, 23987}, {649, 42718}, {650, 2406}, {652, 24035}, {653, 46391}, {656, 7452}, {2425, 4391}, {6087, 13138}, {23986, 36100}, {34393, 42076}, {36037, 42755}, {36121, 38554}

• X₁₀₂ = cevapoint of X_i and X_j for these (i,j): {48, 2361}, {55, 2183}, {523, 3137}

• X₁₀₂ = crosssum of X₄₀ and X₆₃₂₆

• X₁₀₂ = crossdifference of every pair of points on line {23986, 46391}

• X₁₀₂ = barycentric product X_i*X_j for these {i,j}: {1, 36100}, {6, 34393}, {7, 15629}, {63, 36121}, {75, 32677}, {83, 46359}, {92, 36055}, {109, 2399}, {664, 2432}, {1262, 15633}, {4391, 36040}, {6081, 14837}, {6332, 36067}, {32643, 35519}, {32667, 35518}

• X₁₀₂ = barycentric quotient X_i/X_j for these {i,j}: {2, 35516}, {6, 515}, {25, 8755}, {31, 2182}, {48, 46974}, {56, 34050}, {100, 42718}, {108, 24035}, {109, 2406}, {112, 7452}, {604, 1455}, {650, 14304}, {652, 39471}, {1946, 46391}, {2182, 24034}, {2399, 35519}, {2432, 522}, {3310, 42755}, {6081, 44327}, {7113, 11700}, {8607, 117}, {15379, 2988}, {15629, 8}, {15633, 23978}, {32643, 109}, {32667, 108}, {32674, 23987}, {32677, 1}, {32683, 9056}, {32700, 26704}, {32720, 26715}, {34393, 76}, {35183, 44765}, {36040, 651}, {36055, 63}, {36067, 653}, {36100, 75}, {36121, 92}, {46359, 141}

• X₁₀₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 151, 117}, {3, 109, 38697}, {3, 38501, 38559}, {3, 38579, 38607}, {3, 38600, 38691}, {3, 38667, 38674}, {109, 38691, 3}, {117, 151, 10709}, {117, 6711, 2}, {6718, 38784, 631}, {10716, 10732, 10747}, {10740, 38776, 5}, {38573, 38600, 109}, {38573, 38691, 38674}, {38579, 38607, 109}, {38600, 38667, 38697}, {38667, 38691, 109}, {38674, 38697, 109}