X(5159) updated

X(5159) = INVERSE-IN-NINE-POINT-CIRCLE OF X(1368)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b² + c² - a²)(3b⁴ + 3c⁴ - 2a⁴ + a²b² + a²c² - 6b²c²)
Barycentrics    (a^2-b^2-c^2)*(2*a^4-3*(b^2-c^2)^2-a^2*(b^2+c^2)) : :
Barycentrics    SA*(6*SB*SC-SB*SW-SC*SW) : :
X(5159) = 9*X(2)-X(23), 3*X(2)+X(858), 5*X(2)-X(7426), 7*X(2)+X(10989), 3*X(2)+5*X(30745), 6*X(2)-X(37897), 7*X(2)-X(37904), 11*X(2)-3*X(37907), 9*X(2)+X(46517), 5*X(2)+X(47311), 11*X(2)-X(47312), 6*X(2)+X(47315), 9*X(2)-2*X(47316), 3*X(2)+2*X(47629), X(3)+3*X(2072), X(3)-3*X(10257), 2*X(3)-3*X(16976), 3*X(3)+X(18323), 5*X(3)+3*X(18403), 7*X(3)-3*X(44246)

As a point on the Euler line, X(5159) has Shinagawa coefficients (E -5*F, -E - F)

X(5159) = (inverse-in-de-Longchamps-circle of X(22)) = (radical trace of nine-point circle and de-Longchamps circle) = (radical trace of polar circle and de-Longchamps circle) = (reflection of X(23) in de Longchamps line)    (Randy Hutson, August-September, 2013)

For a construction of X(5159), see David Nguyen and Ercole Suppa, euclid 7508.

X(5159) lies on these lines: {1, 47489}, {2, 3}, {6, 47546}, {8, 47533}, {10, 47492}, {39, 47182}, {69, 40920}, {98, 44877}, {114, 16177}, {122, 16188}, {125, 3292}, {126, 38971}, {127, 31655}, {131, 46436}, {141, 8542}, {193, 47463}, {216, 3055}, {230, 3284}, {265, 55981}, {325, 16315}, {339,3266}, {343, 8538}, {373, 37511}, {385, 47237}, {394, 8548}, {511, 6723}, {523, 4885}, {524, 6698}, {525, 22264}, {551, 47491}, {575, 23292}, {576, 13567}, {577, 3054}, {597, 47460}, {599, 47280}, {625,40544}, {647, 47256}, {850, 47248}, {892, 16103}, {895, 15128}, {1007, 2452}, {1038, 7286}, {1040, 5160}, {1092, 61544}, {1125, 51725}, {1154, 64689}, {1351, 37643}, {1352, 59767}, {1353, 26869}, {1503, 5972}, {1514, 36518}, {1531, 38727}, {1560, 52951}, {1568, 38729} ,{1698, 47321}, {1853, 59543}, {1899, 38398}, {1992, 47462}, {2697, 44060}, {2770, 58096}, {3066, 38136}, {3167, 23291}, {3231, 14965}, {3241, 47536}, {3258, 31842}, {3291, 14961}, {3576, 47469}, {3580, 15059}, {3589, 20113}, {3618, 32220}, {3619, 47279}, {3620, 47281}, {3634, 37613}, {3679, 47490}, {3763, 32113}, {3815, 5158}, {3818, 61507}, {4669, 47564}, {5093, 63081}, {5099, 31275}, {5203, 53895}, {5272, 10149}, {5297, 18447}, {5622, 41615}, {5650, 9967}, {5651, 18358}, {5886, 47471}, {6000, 65095}, {6390, 15398}, {6425, 18289}, {6426, 18290}, {6509, 15850}, {6593, 15116}, {6688, 58481}, {6697, 13562}, {6721, 62490}, {6776, 62708}, {7292, 18455}, {7736, 47184}, {7745, 15820}, {7763, 59766}, {7868, 16325}, {7925, 47155}, {7998, 18438}, {8263, 23327}, {8280, 43879}, {8281, 43880}, {8705, 11574}, {8780, 32064}, {8854, 53513}, {8855, 53516}, {9019, 35370}, {9140, 32272}, {9148, 47159}, {9165, 44401}, {9300, 15860}, {9306, 23332}, {9716, 45968}, {9745, 63633}, {9771, 16333}, {9775, 46637}, {9820, 18914}, {9970, 15131}, {10116, 15129},{10173, 11594}, {10192, 64196}, {10256, 46987}, {10415, 34897}, {10510, 47558}, {10516, 47474}, {10564, 23515}, {10634, 43103}, {10635, 43102}, {11059, 41009}, {11245, 11422}, {11427, 53092}, {11433, 11482}, {11477, 26958}, {11513, 32789}, {11514, 32790}, {11580, 22121}, {12041, 58885}, {12079, 14919}, {12358, 40685}, {12900, 14915}, {13202, 20725}, {13292, 15120}, {13445, 15029}, {13561, 31831}, {13611, 65620}, {13754, 16270}, {13857, 41586}, {13869, 30741}, {14156, 15115}, {14341, 41357}, {14389, 51732}, {14561, 47571}, {14984, 60774}, {15025, 50435}, {15027, 22115}, {15034, 25739}, {15066, 61545}, {15124, 50708}, {15125, 15311}, {15139, 15142}, {15448, 29012}, {15471, 32300}, {15491, 16324}, {15526, 22110}, {15533, 47466}, {15812, 47355}, {15819, 47568}, {15905, 62992}, {16092, 39061}, {16227, 64854}, {16303, 31489}, {16310, 44529}, {16313, 50666}, {16760, 34844}, {18583, 37648}, {18911, 61690}, {19126, 32217}, {19131, 22112}, {19862, 51693}, {19875, 47488}, {19883, 47495}, {20190, 58447}, {20208, 37690}, {20299, 59659}, {20582, 47446}, {21243, 53415}, {21356, 47551}, {21358, 47276}, {21639, 53778}, {21850, 61506}, {21968, 64059}, {21970, 51212}, {22104, 34841}, {22151, 32251}, {23878, 47247}, {25055, 47472}, {29181, 32223}, {29639, 47178}, {31174, 46983}, {31274, 47326}, {31277, 47004}, {31279, 47173}, {31804, 64181}, {31843, 46664}, {32111, 64101}, {32227, 34153}, {32269, 51360}, {32411, 63709}, {32767, 64035}, {33522, 55602}, {33924, 47164}, {34128, 51391}, {34988, 44533}, {35259, 39884}, {35707, 58437}, {35968, 42426}, {36900, 47257}, {37638,48876}, {37644, 61624}, {37647, 62698}, {37688, 41008}, {37689, 38292}, {38028, 47476}, {38047, 47506}, {38053, 47507}, {38057, 47508}, {38098, 47534}, {38110, 54012}, {38314, 47493}, {38317, 47581}, {38795, 51425}, {39220, 52058}, {39899, 64177}, {40112, 41724}, {40341, 47465}, {40349, 40350}, {40995, 63098}, {41139, 46998}, {42424, 53832}, {44381, 47239}, {44436, 47213}, {46264, 61680}, {46686, 58871}, {46982, 63440}, {46986, 47245}, {46989, 47254}, {47169, 62196}, {47175, 47259}, {47207, 65760}, {47249, 47261}, {47253, 47442}, {47271 ,54320}, {47352, 47458}, {47447, 63121}, {47451, 51128}, {47456, 52238}, {47461, 51171}, {47494, 53620}, {47499, 47761}, {47500, 47760}, {47537, 51071}, {47541, 59373}, {47544, 48310}, {47575, 59403}, {47576, 59404}, {51358, 59661}, {52520, 63632}, {52987, 61646}, {53777, 62375}, {58451, 58639}, {59399, 63084}, {60414, 60415}, {60420, 60424}, {60421, 60425}, {60422, 60426}, {60423, 60427}

X(5159) = midpoint of X(i) and X(j) for these (i, j): {2, 47097}, {3, 10297}, {5, 15122}, {23, 46517}, {69, 47277}, {125, 11064}, {325, 16315}, {403, 47090}, {427, 16387}, {441, 37987}, {468, 858}, {625, 40544}, {850, 47248}, {892, 16103}, {1529, 46620}, {2071, 10151}, {2072, 10257}, {3153, 37931}, {3679, 47593}, {5189, 37899}, {5196, 47098}, {5203, 53895}, {6390, 51258}, {7426, 47311}, {7575, 47341}, {9148, 47159}, {10295, 47339}, {10510, 47558}, {10989, 37904}, {12041, 58885}, {13202, 20725}, {13473, 16386}, {13857, 44569}, {15115, 15123}, {15118, 19510}, {16188, 47570}, {18323, 47308}, {18572, 47335}, {22110, 46980}, {23323, 34152}, {31174, 46983}, {32269, 51360}, {36170, 56370}, {37897, 47315}, {37900, 47095}, {37911, 47629}, {37938, 44452}, {37950, 47336}, {38971, 54075}, {46686, 58871}, {46982, 63440}, {47091, 47096}, {47092, 62344}, {47280, 47552}, {47310, 54995}, {47312, 47314}, {47612, 47613}
X(5159) = reflection of X(i) in X(j) for these (i, j): (4, 63821), (23, 47316), (403, 44912), (468, 37911), (858, 47629), (10257, 63860), (15471, 32300), (16976, 10257), (32217, 47454), (37897, 468), (37899, 47630), (37910, 37897), (37935, 44452), (37942, 44911), (37984, 5), (41357, 14341), (47004, 47262), (47114, 16976), (47239, 44381), (47261, 47249), (47296, 6723), (47315, 858), (47338, 11799), (47442, 47253), (47457, 3589), (47549, 47460), (51725, 1125), (65154, 6677)
X(5159) = complement of X(468)
X(5159) = inverse-in-nine-point circle of X(1368)
X(5159) = inverse-in-{circumcircle, nine-point circle}-inverter of X(20)
X(5159) = inverse-in-complement-of-polar-circle of X(2)
X(5159) = intersection, other than A, B, C, of circumconics {A, B, C, X(3), X(40349)} and {A, B, C, X(4), X(53419)}
X(5159) = barycentric product X(i)*X(j) for these (i,j): (69, 53419), (76, 21639), (264, 40349), (305, 40350), (525, 53351), (5485, 53778)
X(5159) = barycentric quotient X(i)/X(j) for these {i,j}: {40349, 3}, {40350, 25}, {53351, 648}, {53419, 4}, {53778, 1992}
X(5159) = trilinear product X(i)*X(j) for these (i,j): (63, 53419), (75, 21639), (92, 40349), (304, 40350), (656, 53351), (53778, 55923)
X(5159) = trilinear quotient X(i)/X(j) for these (i,j): (21639, 31), (40349, 48), (40350, 1973), (53351, 162), (53419, 19), (53778, 36277)
X(5159) = (X(i), X(j))-harmonic conjugate of X(k) for these (i, j, k): (2, 20, 52290), (2, 22, 52297), (2, 427, 6677), (2, 468, 37911), (23, 468, 47316), (597, 47549, 47460), (599, 47280, 47552), (3291, 47298, 43291), (3618, 32220, 47459), (5651, 45303, 18358), (10510, 62376, 47558), (24855, 47298, 3291), (26869, 37645, 1353), (30786, 37804, 62310), (37648, 61743, 18583), (37804, 62310, 6390), (51360, 61691, 32269), (59767, 61735, 1352)

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