X₃₀₀₃

(BROCARD AXIS)∩(ORTHIC AXIS)

lies on the cubics K277, K489, K1170, K1171, and these lines: on lines {2, 3260}, {3, 6}, {30, 6128}, {31, 8606}, {37, 7359}, {44, 13006}, {53, 235}, {54, 43753}, {74, 32681}, {111, 9060}, {112, 32710}, {114, 45921}, {115, 11799}, {157, 1974}, {160, 6467}, {186, 8749}, {206, 40947}, {230, 231}, {237, 2393}, {248, 1177}, {393, 847}, {395, 40696}, {396, 40695}, {419, 1632}, {524, 34990}, {607, 2178}, {608, 2164}, {615, 8963}, {648, 44375}, {672, 22059}, {895, 35298}, {1084, 23967}, {1100, 2649}, {1194, 5306}, {1196, 34811}, {1213, 16699}, {1249, 3147}, {1495, 9142}, {1503, 44437}, {1513, 45279}, {1545, 5321}, {1546, 5318}, {1576, 44102}, {1634, 8681}, {1885, 6748}, {1942, 1987}, {1989, 14993}, {2079, 37933}, {2174, 2197}, {2252, 8776}, {2269, 22058}, {2351, 44077}, {2386, 21177}, {2790, 44227}, {3133, 12235}, {3148, 19136}, {3163, 18334}, {3164, 41760}, {3199, 13881}, {3580, 18609}, {3815, 30739}, {4383, 21494}, {4558, 35296}, {5181, 11672}, {5191, 9407}, {5305, 16618}, {5412, 44193}, {5413, 44192}, {5486, 43718}, {5702, 46453}, {6509, 26958}, {6663, 13383}, {7113, 7117}, {7202, 43034}, {7297, 14936}, {7493, 7735}, {7669, 18374}, {7736, 15302}, {7745, 34664}, {8623, 18371}, {8962, 32787}, {9027, 9155}, {9145, 32127}, {9220, 39565}, {9300, 43957}, {9475, 38987}, {9723, 40318}, {9822, 35222}, {9969, 23635}, {10313, 37929}, {11188, 37465}, {11328, 29959}, {13567, 46832}, {14060, 37183}, {14569, 15508}, {14575, 41593}, {14581, 35372}, {14984, 44221}, {15073, 37114}, {15355, 39576}, {16237, 44138}, {16238, 45847}, {17907, 41770}, {18122, 34827}, {18362, 36430}, {18877, 21663}, {19118, 44200}, {20775, 32366}, {21639, 39231}, {23357, 44174}, {23980, 35090}, {34104, 39170}, {34394, 36296}, {34395, 36297}, {35370, 44895}, {35486, 39575}, {36982, 41373}, {37778, 41678}, {39176, 44281}, {40350, 44533}, {40423, 46230}, {41758, 45172}

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

X₃₀₀₃ ((BROCARD AXIS)∩(ORTHIC AXIS)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Trilinears: \(a[(a^{2} - b^{2})\sin 2B + (a^{2} - c^{2})\sin 2C] : :\)

Barycentrics: \(a^{2} (a^{4} b^{2} - 2 a^{2} b^{4} + b^{6} + a^{4} c^{2} + 2 a^{2} b^{2} c^{2} - b^{4} c^{2} - 2 a^{2} c^{4} - b^{2} c^{4} + c^{6}) : :\)

Search Value (6, 9, 13): \(0.463616828748\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin A - \sin 2A \cos(B - C) : :\) (Joe Goggins, 11/26/08)

• Trilinears: \((\sin A)(1 + \cos 2B + \cos 2C) : :\)

▸ Linear Combinations1 combination

• X₃₀₀₃ = X₃₀₀₁ - \(3\) X₄₆₁₂₇, X₅₂₀₁ + \(3\) X₄₆₁₂₇

▸ Functional Relations32 relations

• X₃₀₀₃ = midpoint of X_i and X_j for these {i,j}: {237, 20975}, {3001, 5201}

• X₃₀₀₃ = reflection of X₃₆₂₁₂ in X₃₄₉₉₀

• X₃₀₀₃ = isogonal conjugate of X₂₉₈₆

• X₃₀₀₃ = isotomic conjugate of X₄₀₈₃₂

• X₃₀₀₃ = complement of X₃₂₆₀

• X₃₀₀₃ = Brocard-circle-inverse of X₅₀₆₃

• X₃₀₀₃ = Moses-circle-inverse of X₁₅₅₄₄

• X₃₀₀₃ = Moses-radical-circle-inverse of X₁₆₃₁₉

• X₃₀₀₃ = Schoutte-circle-inverse of X₁₀₅₆₄

• X₃₀₀₃ = Moses-Parry-circle-inverse of X₃₀₁₈

• X₃₀₀₃ = complement of the isogonal conjugate of X₄₀₃₅₂

• X₃₀₀₃ = complement of the isotomic conjugate of X₇₄

• X₃₀₀₃ = isogonal conjugate of the isotomic conjugate of X₃₅₈₀

• X₃₀₀₃ = isotomic conjugate of the polar conjugate of X₄₄₀₈₄

• X₃₀₀₃ = isogonal conjugate of the polar conjugate of X₄₀₃

• X₃₀₀₃ = polar conjugate of the isotomic conjugate of X₁₃₇₅₄

• X₃₀₀₃ = perspector of circumconic centered at X₁₁₃

• X₃₀₀₃ = center of circumconic that is locus of trilinear poles of lines passing through X₁₁₃

• X₃₀₀₃ = Brocard-axis intercept of line through X₃₇₁-Ceva conjugate of X₃₇₂ and X₃₇₂-Ceva conjugate of X₃₇₁

• X₃₀₀₃ = center of bicevian conic of X₁₅ and X₁₆

• X₃₀₀₃ = crossdifference of every pair of points on line X₃X₅₂₃

• X₃₀₀₃ = X_i-complementary conjugate of X_j for these (i,j): {31, 113}, {74, 2887}, {560, 3163}, {810, 16177}, {1304, 21259}, {1494, 21235}, {2159, 141}, {2349, 626}, {2433, 21253}, {8749, 20305}, {9139, 21256}, {15627, 21244}, {18877, 18589}, {23995, 31945}, {32640, 4369}, {32715, 8062}, {33805, 40379}, {35200, 1368}, {36034, 512}, {36119, 21243}, {36131, 30476}, {40351, 16583}, {40352, 10}, {40354, 226}, {44769, 42327}

• X₃₀₀₃ = X_i-Ceva conjugate of X_j for these (i,j): {2, 113}, {186, 1495}, {403, 44084}, {476, 512}, {687, 924}, {1299, 25}, {2407, 526}, {3580, 13754}, {6344, 51}, {8749, 6}, {15329, 21731}, {18609, 1725}, {18879, 110}, {32640, 647}, {38534, 184}, {46106, 6000}

• X₃₀₀₃ = X₂₁₇₃₁-cross conjugate of X₁₅₃₂₉

• X₃₀₀₃ = X_i-isoconjugate of X_j for these (i,j): {1, 2986}, {2, 36053}, {31, 40832}, {63, 1300}, {75, 14910}, {92, 5504}, {162, 15421}, {525, 36114}, {656, 687}, {661, 18878}, {662, 15328}, {1109, 18879}, {1577, 10420}, {2173, 40423}, {2349, 15454}, {6149, 40427}, {10419, 14206}, {14208, 32708}, {15470, 32680}, {24006, 43755}, {36102, 39986}, {36151, 39988}

• X₃₀₀₃ = crosspoint of X_i and X_j for these (i,j): {2, 74}, {6, 1989}, {54, 43766}, {110, 18879}, {249, 1304}, {403, 3580}

• X₃₀₀₃ = crosssum of X_i and X_j for these (i,j): {2, 323}, {4, 15262}, {6, 30}, {115, 9033}, {1650, 32320}, {3936, 32933}, {5504, 14910}

• X₃₀₀₃ = trilinear pole of line {686, 21731}

• X₃₀₀₃ = crossdifference of every pair of points on line {3, 523}

• X₃₀₀₃ = barycentric product X_i*X_j for these {i,j}: {1, 1725}, {3, 403}, {4, 13754}, {6, 3580}, {30, 14264}, {37, 18609}, {67, 12824}, {69, 44084}, {74, 113}, {92, 2315}, {99, 21731}, {112, 6334}, {131, 1299}, {155, 16172}, {184, 44138}, {186, 39170}, {265, 1986}, {399, 18781}, {523, 15329}, {526, 41512}, {647, 16237}, {648, 686}, {895, 12828}, {1177, 12827}, {1300, 34333}, {1989, 34834}, {5663, 39985}, {10419, 34104}, {10693, 12826}, {11557, 33565}, {11744, 12825}, {18879, 39021}, {38534, 46085}

• X₃₀₀₃ = barycentric quotient X_i/X_j for these {i,j}: {2, 40832}, {6, 2986}, {25, 1300}, {31, 36053}, {32, 14910}, {74, 40423}, {110, 18878}, {112, 687}, {113, 3260}, {184, 5504}, {403, 264}, {512, 15328}, {647, 15421}, {686, 525}, {1495, 15454}, {1576, 10420}, {1725, 75}, {1986, 340}, {1989, 40427}, {2315, 63}, {3580, 76}, {5663, 39988}, {6334, 3267}, {12824, 316}, {12827, 1236}, {12828, 44146}, {13754, 69}, {14264, 1494}, {14270, 15470}, {14583, 39375}, {15329, 99}, {16172, 46746}, {16237, 6331}, {18609, 274}, {18781, 40705}, {21731, 523}, {23357, 18879}, {32661, 43755}, {32676, 36114}, {34397, 38936}, {34834, 7799}, {39170, 328}, {40352, 10419}, {40354, 40388}, {41512, 35139}, {44084, 4}, {44138, 18022}

• X₃₀₀₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {3, 6, 5063}, {6, 50, 3284}, {6, 216, 570}, {6, 566, 39}, {6, 570, 5421}, {6, 1609, 571}, {6, 8553, 577}, {6, 8573, 13345}, {6, 11063, 50}, {6, 13338, 5007}, {6, 18573, 566}, {15, 16, 10564}, {32, 5158, 6}, {39, 216, 566}, {39, 566, 570}, {50, 11063, 187}, {53, 9722, 1879}, {74, 36896, 40353}, {187, 3284, 50}, {187, 40135, 3284}, {216, 800, 6}, {230, 1990, 16310}, {230, 2493, 3291}, {230, 16303, 3018}, {230, 16306, 6103}, {230, 16308, 2493}, {230, 16310, 231}, {231, 3018, 16310}, {232, 3291, 2493}, {232, 6103, 14580}, {371, 372, 13352}, {566, 18573, 216}, {577, 46432, 6}, {1990, 11062, 232}, {1990, 16310, 3018}, {1990, 16328, 11062}, {2028, 2029, 15544}, {2493, 16308, 232}, {3284, 40135, 6}, {5063, 40320, 571}, {5201, 46127, 3001}, {7669, 18374, 42671}, {8105, 8106, 3018}, {8607, 8608, 8609}, {11081, 11086, 1495}, {13338, 41335, 6}, {15166, 15167, 6}, {16303, 16310, 1990}, {23635, 40981, 9969}, {34569, 46216, 50}, {35296, 37784, 4558}, {40135, 46216, 18365}, {41196, 41197, 14961}, {41678, 44893, 37778}, {46203, 46222, 50}