X₆₅₅

TRILINEAR POLE OF LINE X₁X_5

X₆₅₅ = trilinear pole of line {1, 5}

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

X₆₅₅ (TRILINEAR POLE OF LINE X₁X_5) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {59, 523}, {80, 516}

• Related Centers: X₆₅₄ (CROSSDIFFERENCE OF X₁ AND X₅), X₅₉ (ISOGONAL CONJUGATE OF X₁₁), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀), X₈₀ (REFLECTION OF INCENTER IN FEUERBACH POINT), X₅₁₆ (ISOGONAL CONJUGATE OF X₁₀₃)

Trilinears: \(1/[\cos(A - B) - \cos(C- A)] : :\)

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

Search Value (6, 9, 13): \(-0.737967855306\)

▸ Linear Combinations1 combination

• X₆₅₅ = \(3\) X₂ - \(4\) X₄₀₅₃₆

X₆₅₅ lies on the curve CC₉.

▸ Incident Central Lines44 lines

{2, 40536}, {7, 37131}, {54, 45238}, {57, 14628}, {59, 523}, {80, 516}, {88, 2006}, {100, 522}, {162, 7012}, {190, 4391}, {476, 15439}, {514, 651}, {517, 40437}, {527, 36101}, {653, 17924}, {658, 24002}, {662, 4552}, {664, 4604}, {673, 2161}, {885, 36086}, {908, 36100}, {925, 36078}, {1020, 38340}, {1025, 37143}, {1087, 3460}, {1090, 2957}, {1155, 14204}, {1411, 4318}, {1807, 3100}, {2349, 17484}, {2401, 2406}, {2595, 2597}, {3218, 18359}, {3322, 4542}, {3904, 13136}, {4581, 36098}, {6740, 37142}, {7672, 36815}, {10015, 23592}, {10030, 37130}, {11041, 34232}, {16548, 21368}, {30379, 43760}, {37214, 40704}

▸ Functional Relations12 relations

• X₆₅₅ = isogonal conjugate of X₆₅₄

• X₆₅₅ = isotomic conjugate of X₃₉₀₄

• X₆₅₅ = polar conjugate of X₄₄₄₂₈

• X₆₅₅ = isotomic conjugate of the isogonal conjugate of X₃₂₆₇₅

• X₆₅₅ = X_i-cross conjugate of X_j for these (i,j): {484, 7045}, {650, 40437}, {654, 1}, {900, 7}, {1983, 6742}, {2245, 59}, {2600, 3615}, {3465, 24032}, {10015, 2}, {16548, 7012}, {23703, 664}, {24029, 651}, {24715, 39293}, {30725, 14628}, {35466, 1016}, {40663, 4998}

• X₆₅₅ = cevapoint of X_i and X_j for these (i,j): {1, 654}, {6, 39200}, {9, 2804}, {57, 30725}, {216, 8677}, {513, 8609}, {514, 3911}, {517, 650}, {523, 2245}, {2599, 2600}, {4559, 23981}

• X₆₅₅ = crosssum of X₂₀₈₁ and X₂₆₂₄

• X₆₅₅ = trilinear pole of line {1, 5}

• X₆₅₅ = X_i-isoconjugate of X_j for these (i,j): {1, 654}, {2, 8648}, {6, 3738}, {8, 21758}, {11, 1983}, {21, 21828}, {31, 3904}, {36, 650}, {41, 4453}, {48, 44428}, {54, 2600}, {55, 3960}, {60, 2610}, {281, 22379}, {320, 3063}, {513, 2323}, {514, 2361}, {522, 7113}, {523, 4282}, {647, 17515}, {649, 4511}, {652, 1870}, {657, 1443}, {663, 3218}, {667, 32851}, {758, 7252}, {1021, 1464}, {1639, 16944}, {1835, 23090}, {1946, 17923}, {2148, 6369}, {2150, 6370}, {2185, 42666}, {2194, 4707}, {2222, 35128}, {2245, 3737}, {2432, 11700}, {2624, 3615}, {3271, 4585}, {3724, 4560}, {4242, 7117}, {4895, 40215}, {5081, 22383}, {8641, 17078}, {17455, 23838}, {18344, 22128}, {18593, 21789}, {36110, 38353}

• X₆₅₅ = barycentric product X_i*X_j for these {i,j}: {1, 35174}, {57, 36804}, {75, 2222}, {76, 32675}, {80, 664}, {100, 18815}, {109, 20566}, {190, 2006}, {311, 36078}, {476, 40999}, {651, 18359}, {658, 36910}, {668, 1411}, {1414, 15065}, {1807, 18026}, {2161, 4554}, {2594, 35139}, {3257, 14628}, {3904, 23592}, {4551, 14616}, {4552, 24624}, {4555, 14584}, {4566, 6740}, {4572, 6187}, {4585, 34535}, {4625, 34857}, {6358, 37140}, {16577, 32680}, {34388, 36069}, {35175, 37630}, {38340, 41226}

• X₆₅₅ = barycentric quotient X_i/X_j for these {i,j}: {1, 3738}, {2, 3904}, {4, 44428}, {5, 6369}, {6, 654}, {7, 4453}, {12, 6370}, {31, 8648}, {57, 3960}, {80, 522}, {100, 4511}, {101, 2323}, {108, 1870}, {109, 36}, {162, 17515}, {163, 4282}, {181, 42666}, {190, 32851}, {226, 4707}, {476, 3615}, {603, 22379}, {604, 21758}, {651, 3218}, {653, 17923}, {654, 35128}, {658, 17078}, {664, 320}, {692, 2361}, {759, 3737}, {934, 1443}, {1020, 18593}, {1168, 23838}, {1400, 21828}, {1411, 513}, {1415, 7113}, {1807, 521}, {1813, 22128}, {1897, 5081}, {1953, 2600}, {1983, 34544}, {2006, 514}, {2149, 1983}, {2161, 650}, {2171, 2610}, {2222, 1}, {2341, 1021}, {2594, 526}, {3676, 4089}, {4551, 758}, {4552, 3936}, {4554, 20924}, {4559, 2245}, {4564, 4585}, {4566, 41804}, {4572, 40075}, {5219, 23884}, {6187, 663}, {6740, 7253}, {7012, 4242}, {14584, 900}, {14616, 18155}, {14628, 3762}, {15065, 4086}, {16577, 32679}, {18359, 4391}, {18815, 693}, {20566, 35519}, {21741, 2624}, {21859, 4053}, {23703, 214}, {23706, 1845}, {23981, 34586}, {24029, 16586}, {24624, 4560}, {27818, 27836}, {32671, 2150}, {32675, 6}, {34079, 7252}, {34242, 21189}, {34857, 4041}, {35013, 45950}, {35174, 75}, {35308, 21012}, {36061, 1789}, {36069, 60}, {36078, 54}, {36804, 312}, {36815, 3716}, {36910, 3239}, {37140, 2185}, {37630, 2802}, {39200, 38984}, {40172, 4895}, {40437, 43728}, {40999, 3268}

• X₆₅₅ = {X₂₅₉₅,X₂₅₉₉}-harmonic conjugate of X₃₆₁₅