X(54) = KOSNITA POINT

Trilinears

\(sec(B - C) : sec(C - A) : sec(A - B)\)

\(a/(b cos B + c cos C) : :\)

Barycentrics

\(sin A sec(B - C) : sin B sec(C - A) : sin C sec(A - B)\)

\(a^2/(S^2 + SB SC) : :\)

\(a^2/(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :\)

\(a^2*(a^4 - 2*a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2)*(a^4 - a^2*b^2 - 2*a^2*c^2 - b^2*c^2 + c^4) : :\)

Notes

X(54) = 3 X[2] - 4 X[6689], X[3] + 2 X[1493], 3 X[3] - X[12307], 3 X[3] + X[12316], 2 X[3] + X[15801], X[4] + 2 X[10619], X[4] - 4 X[12242], 3 X[4] - 2 X[32340], 5 X[5] - 4 X[20584], X[5] + 4 X[20585], 3 X[51] - 2 X[11808], 2 X[52] + X[12226], X[64] + 3 X[17824], X[64] - 3 X[32345], X[74] + 2 X[2914], X[74] + 4 X[32226], X[110] + 2 X[15089], 4 X[140] - X[3519], 2 X[195] + X[7691], X[195] + 2 X[10610], 3 X[195] + X[12307], 3 X[195] - X[12316], 3 X[381] - 2 X[22804], 4 X[389] - X[6242], 2 X[389] + X[21660], X[550] + 2 X[11803], 4 X[575] - X[9972], 5 X[631] + X[11271], 5 X[631] - X[12325], 5 X[631] - 4 X[32348], 5 X[632] - 3 X[21357], 4 X[973] - 5 X[3567], 4 X[973] - 3 X[7730], 4 X[973] + X[12291], 4 X[973] - X[13423], X[1141] + 2 X[27423], 3 X[1157] - 4 X[12060], 4 X[1493] + X[7691], 6 X[1493] + X[12307], 6 X[1493] - X[12316], 4 X[1493] - X[15801], 5 X[1656] - 4 X[13565], X[2888] - 4 X[6689], 3 X[2917] - 5 X[17821], 3 X[2917] - X[17846], 7 X[3090] - 8 X[32396], 7 X[3523] + 2 X[13431], 11 X[3525] - 4 X[15605], 5 X[3567] - 2 X[6152], 5 X[3567] - 3 X[7730], 5 X[3567] + 4 X[11577], 5 X[3567] + X[12291], 5 X[3567] - X[13423], 2 X[3574] + X[12254], 3 X[3574] - X[32340], X[3627] - 4 X[30531], 8 X[5462] - 3 X[41713], X[5889] + 2 X[12606], 3 X[5890] - X[32339], X[5898] - 3 X[32609], 3 X[5946] - X[13368], 3 X[5946] + X[15532], 3 X[6030] - 2 X[13564], 2 X[6102] + X[22815], 2 X[6152] - 3 X[7730], X[6152] + 2 X[11577], 2 X[6152] + X[12291], 4 X[6153] - 3 X[41713], X[6241] + 2 X[12300], X[6241] + 4 X[15739], X[6242] + 2 X[21660], X[6242] + 4 X[40632], X[6288] - 4 X[8254], 5 X[6288] - 8 X[20584], X[6288] + 8 X[20585], X[6288] + 2 X[36966], X[6759] - 3 X[10274], 2 X[6759] - 3 X[32379], X[7691] - 4 X[10610], 3 X[7691] - 2 X[12307], 3 X[7691] + 2 X[12316], 3 X[7730] + 4 X[11577], 3 X[7730] + X[12291], 3 X[7730] - X[13423], X[7979] + 2 X[9905], X[7979] - 4 X[12266], 5 X[8254] - 2 X[20584], X[8254] + 2 X[20585], 2 X[8254] + X[36966], X[8718] + 3 X[13482], X[8718] + 4 X[37472], 3 X[9730] - 2 X[11802], 7 X[9781] - 4 X[11576], 8 X[9827] - 11 X[15024], 4 X[9827] - 3 X[41578], X[9905] + 2 X[12266], X[9935] - 4 X[10282], 4 X[10115] + X[12226], 6 X[10610] - X[12307], 6 X[10610] + X[12316], 4 X[10610] + X[15801], X[10619] + 2 X[12242], 3 X[10619] + X[32340], X[11271] + 4 X[32348], 3 X[11402] + X[32333], 3 X[11402] - X[32341], X[11412] - 4 X[12363], 4 X[11577] - X[12291], 4 X[11577] + X[13423], 4 X[12007] + X[13622], 4 X[12242] + X[12254], 6 X[12242] - X[32340], 3 X[12254] + 2 X[32340], X[12280] - 7 X[15043], 2 X[12307] + 3 X[15801], 2 X[12316] - 3 X[15801], X[12325] - 4 X[32348], 4 X[13365] - 5 X[15026], X[13432] + 11 X[15720], 3 X[13482] - 4 X[37472], 2 X[14049] + X[33565], 4 X[14865] - X[16835], 11 X[15024] - 6 X[41578], 5 X[15034] - 2 X[25714], X[15800] - 4 X[22051], 5 X[17821] - X[17846], 5 X[17821] - 6 X[32391], X[17846] - 6 X[32391], X[18368] + 2 X[34564], X[20584] + 5 X[20585], 4 X[20584] + 5 X[36966], 4 X[20585] - X[36966], X[32352] + 2 X[40632], 2 X[32369] - 3 X[32395]. See John Rigby, “Brief notes on some forgotten geometrical theorems,” Mathematics and Informatics Quarterly 7 (1997) 156-158.

Let O be the circumcenter of triangle ABC, and OA the circumcenter of triangle BOC. Define OB and OC cyclically. Then the lines AOA, BOB, COC concur in X(54). For details and generalization, see

Darij Grinberg, A New Circumcenter Question.

The above construction of X(54) generalizes. Suppose that P and Q are points (as functions of a,b,c). Let A’ = Q-of-BCP, B’ = Q-of-CAP, C’ = Q-of-ABP. If the lines AA’, BB’, CC’ concur, the perspector is called the Kosnita(P,Q) point, denoted by K(P,Q). (Randy Hutson, 9/23/2011)

Let NANBNC be the reflection triangle of X(5). Let OA be the circumcenter of ANBNC, and define OB and OC cyclically. Triangle OAOBOC is perspective to ABC at X(54), homothetic to the orthic-of-orthocentroidal triangle at X(54), and orthologic to the reflection triangle at X(54). (Randy Hutson, June 7, 2019)

Let (Na) be the reflection of the nine-point circle in BC, and define (Nb) and (Nc) cyclically. X(54) is the radical center of (Na), (Nb), (Nc). The tangents at A, B, C to the Napoleon-Feuerbach cubic K005 concur in X(54). Let A’B’C’ be the reflection triangle. Let La be the trilinear polar of A’, and define Lb and Lc cyclically. Let A″ be Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(54). (Randy Hutson, July 23, 2015)

If you have GeoGebra, you can view X(54).

let Na = reflection of X(5) in the line BC, and define Nb and Nc cyclically. The medial triangle of NaNbNc is perspective to ABC, and the perspector is X(54). (Dasari Naga Vijay Krishna, June 8, 2021)

X(54) lies on the Jarabek circumhyperbola, the cubics K005, K073, K112, K316, K361, K364, K373, K388, K439, K464, K466, K467, K469, K471, K499, K500, K502, K526, K566, K569, K589, K590, K629, K633, K646, K668, K822, K919, K942, K947, K976, K1107, K1180, the curves Q023, Q029, Q089, Q110, Q141, and these lines: {1, 2599}, {2, 68}, {3, 97}, {4, 184}, {5, 49}, {6, 24}, {10, 9562}, {11, 2477}, {12, 215}, {13, 3206}, {14, 3205}, {15, 10678}, {16, 10677}, {17, 3201}, {18, 3200}, {19, 16031}, {20, 4846}, {22, 36747}, {23, 5446}, {25, 3527}, {26, 3060}, {28, 1243}, {30, 3521}, {32, 9985}, {33, 9638}, {35, 6286}, {36, 73}, {39, 248}, {51, 288}, {52, 1166}, {55, 9666}, {56, 9653}, {59, 5399}, {60, 5396}, {61, 3166}, {62, 3165}, {64, 378}, {65, 1870}, {66, 3541}, {67, 5622}, {69, 95}, {70, 1899}, {71, 572}, {72, 1006}, {74, 185}, {98, 3203}, {99, 39805}, {112, 217}, {113, 3047}, {114, 3044}, {115, 9697}, {118, 3046}, {119, 3045}, {125, 3043}, {136, 6801}, {137, 33545}, {140, 252}, {143, 2070}, {154, 10594}, {155, 7503}, {156, 381}, {186, 389}, {193, 19131}, {206, 14853}, {219, 26915}, {222, 26914}, {251, 37123}, {262, 3202}, {276, 290}, {287, 37125}, {311, 34385}, {323, 1216}, {324, 37127}, {371, 6414}, {372, 6413}, {376, 10984}, {382, 18550}, {394, 7509}, {397, 11134}, {398, 11137}, {399, 33539}, {402, 12797}, {403, 12241}, {411, 34800}, {418, 2055}, {427, 31804}, {436, 8794}, {476, 36159}, {477, 36179}, {493, 12998}, {494, 12999}, {496, 40450}, {511, 1176}, {523, 36161}, {526, 38897}, {542, 18125}, {546, 10540}, {548, 13623}, {549, 34483}, {550, 11803}, {568, 1658}, {575, 895}, {576, 7556}, {577, 26876}, {632, 21357}, {695, 14153}, {826, 879}, {930, 35720}, {953, 36078}, {970, 1798}, {1062, 9637}, {1075, 21449}, {1087, 2595}, {1113, 14374}, {1114, 14375}, {1192, 35472}, {1204, 11270}, {1246, 7554}, {1263, 6343}, {1291, 15907}, {1296, 9226}, {1329, 9702}, {1351, 9715}, {1352, 18124}, {1353, 19129}, {1437, 6905}, {1439, 1443}, {1495, 10110}, {1498, 11455}, {1503, 15321}, {1506, 9696}, {1511, 12006}, {1587, 13440}, {1588, 13429}, {1593, 3426}, {1594, 6145}, {1595, 16659}, {1598, 3531}, {1656, 9703}, {1698, 9586}, {1699, 9587}, {1853, 38433}, {1879, 9378}, {1907, 16658}, {1976, 37334}, {2051, 9563}, {2071, 40647}, {2393, 32367}, {2574, 14709}, {2575, 14710}, {2616, 3737}, {2620, 7136}, {2623, 10097}, {2781, 34437}, {2886, 9701}, {2904, 34438}, {2929, 42016}, {2937, 10263}, {2981, 14818}, {3048, 5512}, {3068, 8995}, {3069, 13986}, {3090, 9306}, {3091, 9544}, {3147, 11433}, {3167, 7395}, {3292, 7550}, {3311, 6416}, {3312, 6415}, {3336, 3468}, {3357, 13452}, {3398, 36214}, {3431, 11438}, {3432, 32409}, {3448, 10116}, {3470, 38933}, {3471, 38935}, {3515, 11432}, {3517, 9777}, {3523, 13336}, {3524, 37515}, {3525, 15605}, {3526, 11935}, {3528, 37480}, {3529, 31371}, {3530, 13339}, {3532, 10605}, {3542, 14457}, {3545, 15749}, {3548, 18911}, {3563, 32692}, {3575, 34397}, {3580, 7542}, {3613, 11816}, {3627, 30531}, {3628, 40111}, {3796, 10323}, {3815, 9603}, {5050, 6391}, {5067, 5651}, {5085, 34817}, {5092, 41435}, {5093, 16195}, {5133, 12134}, {5157, 10519}, {5198, 14530}, {5254, 9604}, {5418, 9676}, {5422, 6642}, {5447, 15246}, {5462, 6153}, {5486, 35486}, {5494, 10693}, {5498, 15061}, {5504, 9730}, {5562, 34986}, {5587, 9622}, {5597, 12480}, {5598, 12481}, {5640, 7506}, {5643, 12099}, {5663, 11559}, {5721, 38850}, {5870, 10262}, {5871, 10261}, {5891, 41597}, {5898, 13363}, {5900, 25563}, {5907, 35500}, {5946, 13368}, {5972, 19481}, {6000, 14865}, {6030, 13391}, {6151, 14819}, {6198, 11429}, {6239, 12231}, {6240, 12233}, {6243, 7502}, {6400, 12232}, {6515, 41594}, {6561, 9677}, {6636, 10625}, {6640, 18952}, {6643, 37645}, {6644, 8907}, {6696, 16623}, {6794, 7765}, {6800, 7387}, {6815, 12318}, {6853, 18123}, {6875, 13323}, {7393, 15066}, {7505, 39571}, {7507, 32402}, {7514, 11444}, {7516, 7998}, {7517, 26881}, {7525, 37484}, {7526, 12111}, {7527, 12162}, {7529, 35264}, {7544, 34116}, {7547, 7699}, {7549, 41608}, {7574, 13470}, {7575, 16881}, {7576, 34782}, {7577, 16000}, {7689, 39242}, {7728, 11805}, {7731, 19362}, {8227, 9621}, {8562, 14380}, {8743, 40823}, {8795, 41204}, {8889, 38442}, {8918, 8930}, {8919, 8929}, {9140, 13561}, {9418, 39283}, {9590, 31760}, {9625, 31757}, {9652, 10895}, {9667, 10896}, {9729, 22962}, {9786, 14528}, {9818, 11441}, {9932, 15045}, {9971, 15582}, {10018, 13567}, {10024, 12370}, {10095, 13621}, {10202, 28787}, {10205, 35729}, {10226, 15055}, {10295, 13568}, {10299, 13347}, {10575, 12086}, {10601, 11465}, {10602, 11458}, {10721, 11744}, {10950, 40437}, {11004, 37478}, {11077, 41335}, {11202, 13472}, {11263, 38535}, {11264, 34826}, {11381, 13596}, {11403, 32063}, {11416, 15074}, {11439, 31861}, {11440, 18570}, {11443, 38263}, {11460, 19350}, {11461, 19354}, {11462, 19355}, {11463, 19356}, {11466, 19363}, {11467, 19364}, {11477, 19127}, {11491, 20986}, {11591, 34864}, {12007, 13622}, {12023, 12024}, {12041, 35498}, {12084, 15072}, {12106, 15019}, {12110, 40643}, {12112, 13474}, {12229, 12509}, {12230, 12510}, {12281, 19457}, {12282, 19458}, {12283, 19459}, {12284, 19456}, {12285, 19461}, {12286, 19462}, {12287, 19463}, {12288, 19464}, {12359, 41730}, {12834, 13365}, {13011, 13035}, {13012, 13036}, {13017, 19465}, {13018, 19466}, {13351, 37813}, {13364, 18369}, {13382, 21663}, {13420, 18368}, {13432, 15720}, {13433, 34565}, {13445, 13491}, {13488, 32111}, {13598, 37925}, {13754, 14118}, {13856, 38618}, {14070, 37493}, {14071, 25150}, {14106, 14111}, {14152, 26897}, {14220, 34210}, {14371, 14379}, {14491, 34417}, {14518, 34756}, {14542, 18533}, {14587, 18114}, {14641, 37944}, {14788, 37649}, {14805, 15091}, {15053, 16867}, {15056, 15068}, {15059, 24572}, {15093, 32448}, {15121, 15124}, {15232, 32381}, {15305, 32139}, {15328, 38936}, {15340, 27371}, {15401, 15537}, {15644, 22352}, {15646, 16665}, {15712, 26861}, {15760, 18433}, {16252, 16657}, {16766, 31674}, {16837, 34449}, {16868, 18390}, {17702, 34007}, {17711, 23329}, {18324, 37490}, {18374, 22336}, {18376, 40276}, {18474, 34799}, {18559, 34785}, {18874, 21308}, {18945, 32393}, {19123, 19125}, {19124, 39874}, {19136, 38005}, {19142, 22829}, {19151, 37473}, {19186, 19408}, {19187, 19409}, {19212, 33971}, {19349, 19368}, {19358, 19414}, {19359, 19415}, {19440, 19502}, {19441, 19503}, {19460, 22535}, {20190, 32599}, {20193, 30551}, {20421, 23040}, {21394, 30504}, {21849, 37939}, {22233, 41448}, {22330, 37953}, {22533, 32375}, {22950, 22972}, {23128, 26216}, {23293, 25738}, {24385, 36837}, {26877, 26889}, {26896, 26898}, {26916, 26920}, {31376, 34837}, {32110, 38448}, {32248, 39562}, {32249, 32251}, {32284, 37784}, {32321, 41715}, {32357, 34207}, {32661, 41334}, {32713, 42873}, {32737, 38394}, {33543, 37483}, {33695, 35909}, {33992, 35728}, {34351, 41596}, {34384, 39287}, {34418, 35467}, {34664, 41615}, {35480, 40242}, {35602, 37514}, {35724, 35885}, {37489, 38444}, {39808, 39810}, {39837, 39839}

X(54) = midpoint of X(i) and X(j) for these {i,j}: {1, 9905}, {3, 195}, {4, 12254}, {5, 36966}, {125, 14049}, {389, 40632}, {973, 11577}, {1263, 6343}, {1493, 10610}, {2929, 42016}, {3574, 10619}, {5889, 32338}, {6276, 6277}, {7691, 15801}, {11271, 12325}, {11597, 15089}, {12026, 31675}, {12291, 13423}, {12307, 12316}, {13368, 15532}, {17824, 32345}, {21660, 32352}, {27196, 27423}, {32333, 32341}, {32346, 32354}

X(54) = reflection of X(i) in X(j) for these {i,j}: {1, 12266}, {3, 10610}, {4, 3574}, {5, 8254}, {52, 10115}, {110, 11597}, {195, 1493}, {265, 11804}, {1141, 27196}, {1209, 6689}, {2888, 1209}, {2914, 32226}, {2917, 32391}, {3519, 21230}, {3574, 12242}, {6145, 32351}, {6152, 973}, {6153, 5462}, {6242, 32352}, {6288, 5}, {7691, 3}, {7728, 11805}, {7979, 1}, {9972, 9977}, {9977, 575}, {11412, 41590}, {12254, 10619}, {12300, 15739}, {12785, 10}, {12797, 402}, {13121, 10066}, {13122, 10082}, {13423, 6152}, {15062, 14130}, {15800, 20424}, {15801, 195}, {20424, 22051}, {21230, 140}, {21660, 40632}, {23061, 15137}, {32196, 143}, {32338, 12606}, {32352, 389}, {32379, 10274}, {33565, 125}, {41590, 12363}

X(54) = isogonal conjugate of X(5)

X(54) = isotomic conjugate of X(311)

X(54) = inverse-in-circumcircle of X(1157)

X(54) = complement of X(2888)

X(54) = anticomplement of X(1209)

X(54) = circumcircle-inverse of X(1157)

X(54) = Brocard-circle-inverse of X(18335)

X(54) = polar conjugate of X(324)

X(54) = antigonal image of X(33565)

X(54) = symgonal image of X(11597)

X(54) = Thomson-isogonal conjugate of X(35885)

X(54) = psi-transform of X(14656)

X(54) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {1166, 8}, {2216, 2888}

X(54) = X(i)-complementary conjugate of X(j) for these (i,j): {3432, 10}, {40140, 21231}

X(54) = X(i)-Ceva conjugate of X(j) for these (i,j): {5, 2120}, {95, 97}, {97, 33629}, {275, 8882}, {288, 6}, {933, 23286}, {1141, 1157}, {1166, 25044}, {8884, 8883}, {14533, 26887}, {14587, 110}, {18315, 2623}, {18831, 15412}, {20574, 1614}, {23286, 19208}, {35196, 2169}, {39287, 95}

X(54) = X(i)-cross conjugate of X(j) for these (i,j): {3, 96}, {6, 275}, {184, 14533}, {186, 74}, {389, 4}, {523, 110}, {570, 2}, {1199, 1173}, {2594, 1}, {2623, 18315}, {3269, 39181}, {8603, 2981}, {8604, 6151}, {13366, 6}, {13367, 3}, {14533, 97}, {16030, 95}, {16035, 8884}, {19189, 1298}, {21638, 8795}, {21660, 3519}, {23286, 933}, {30451, 4558}, {32352, 6145}, {39199, 109}, {39201, 112}, {39478, 901}, {40632, 13418}, {41218, 654}

X(54) = cevapoint of X(i) and X(j) for these (i,j): {3, 1147}, {6, 184}, {15, 3200}, {16, 3201}, {32, 3202}, {39, 3203}, {55, 3204}, {58, 9563}, {61, 3205}, {62, 3206}, {215, 2245}, {523, 8901}, {572, 9562}, {654, 41218}, {3270, 9404}

X(54) = crosspoint of X(i) and X(j) for these (i,j): {1, 3461}, {3, 3463}, {4, 3459}, {5, 2121}, {95, 275}, {3467, 3469}, {3489, 3490}

X(54) = crosssum of X(i) and X(j) for these (i,j): {1, 3460}, {2, 17035}, {3, 195}, {4, 3462}, {11, 8819}, {51, 216}, {54, 2120}, {61, 8839}, {62, 8837}, {233, 3078}, {288, 38816}, {523, 8902}, {627, 628}, {1953, 7069}, {2600, 41218}, {3336, 3468}, {3470, 38933}, {3471, 38935}, {6368, 35442}, {8918, 8930}, {8919, 8929}, {12077, 41221}, {17434, 41219}, {21011, 21807}

X(54) = trilinear pole of line {50, 647}

X(54) = crossdifference of every pair of points on line {2081, 2600}

X(54) = trilinear pole of line X(50)X(647) (the polar of X(324) wrt polar circle)

X(54) = pole wrt polar circle of trilinear polar of X(324)

X(54) = X(48)-isoconjugate (polar conjugate) of X(324)

X(54) = X(92)-isoconjugate of X(216)

X(54) = intersection of tangents to hyperbola {A,B,C,:ref:X(4) <X(4)>,:ref:X(5) <X(5)>}} at X(4) and X(3459)

X(54) = {X(2595),:ref:X(2596) <X(2596)>}-harmonic conjugate of X(1087)

X(54) = trilinear product of vertices of circumnormal triangle

X(54) = intersection of tangents at X(3) and X(4) to Neuberg cubic K001

X(54) = exsimilicenter of nine-point circle and sine-triple-angle circle

X(54) = homothetic center of orthocevian triangle of X(3) and circumorthic triangle

X(54) = perspector of ABC and unary cofactor triangle of reflection triangle

X(54) = X(3)-of-reflection-triangle-of-X(5)

X(54) = perspector of ABC and cross-triangle of ABC and circumorthic triangle

X(54) = perspector of ABC and Hatzipolakis-Moses triangle

X(54) = X(191)-of-orthic-triangle if ABC is acute

X(54) = trilinear product of vertices of X(4)-altimedial triangle

X(54) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5}, {2, 1953}, {6, 14213}, {7, 7069}, {8, 1393}, {10, 18180}, {19, 343}, {25, 18695}, {31, 311}, {37, 17167}, {38, 17500}, {48, 324}, {51, 75}, {52, 91}, {53, 63}, {54, 1087}, {69, 2181}, {76, 2179}, {79, 35194}, {81, 21011}, {86, 21807}, {92, 216}, {94, 2290}, {100, 21102}, {110, 2618}, {143, 2962}, {158, 5562}, {162, 6368}, {163, 18314}, {217, 1969}, {255, 13450}, {293, 39569}, {304, 3199}, {318, 30493}, {326, 14569}, {467, 1820}, {523, 2617}, {561, 40981}, {610, 13157}, {655, 2600}, {656, 35360}, {661, 14570}, {662, 12077}, {811, 15451}, {823, 17434}, {897, 41586}, {920, 8800}, {921, 41587}, {1154, 2166}, {1209, 2216}, {1263, 1749}, {1474, 42698}, {1568, 36119}, {1577, 1625}, {1707, 27364}, {1826, 16697}, {1895, 8798}, {1956, 32428}, {1972, 2313}, {1973, 28706}, {2081, 32680}, {2153, 33529}, {2154, 33530}, {2167, 36412}, {2180, 5392}, {2184, 42459}, {2222, 6369}, {2595, 7135}, {2596, 2603}, {2599, 3615}, {2621, 18114}, {2964, 25043}, {4560, 35307}, {4575, 23290}, {8769, 41588}, {10015, 35321}, {17438, 31610}, {18070, 35319}, {18833, 27374}, {20577, 36148}, {23181, 24006}, {24000, 35442}, {24041, 41221}, {27371, 34055}, {32678, 41078}, {36035, 36831}

X(54) = barycentric product X(i)*X(j) for these {i,j}: {1, 2167}, {3, 275}, {4, 97}, {6, 95}, {25, 34386}, {32, 34384}, {39, 39287}, {48, 40440}, {63, 2190}, {69, 8882}, {75, 2148}, {83, 16030}, {92, 2169}, {96, 1993}, {99, 2623}, {110, 15412}, {140, 288}, {182, 42300}, {184, 276}, {226, 35196}, {249, 8901}, {252, 1994}, {253, 33629}, {264, 14533}, {287, 19189}, {290, 41270}, {323, 1141}, {338, 14587}, {340, 11077}, {371, 16032}, {372, 16037}, {394, 8884}, {401, 1298}, {520, 16813}, {523, 18315}, {525, 933}, {571, 34385}, {577, 8795}, {578, 37872}, {647, 18831}, {648, 23286}, {662, 2616}, {801, 16035}, {850, 14586}, {1073, 38808}, {1092, 8794}, {1105, 19180}, {1157, 13582}, {1166, 37636}, {1502, 14573}, {1577, 36134}, {1634, 39182}, {2052, 19210}, {2245, 39277}, {2888, 40140}, {2984, 11245}, {3051, 41488}, {3431, 4993}, {3904, 36078}, {4551, 39177}, {6504, 8883}, {6563, 32692}, {7763, 41271}, {11140, 25044}, {13366, 31617}, {14096, 39283}, {14528, 19188}, {14618, 15958}, {15351, 19208}, {15414, 32713}, {19166, 41890}, {19170, 40448}, {19174, 28724}, {20574, 40684}, {22052, 39286}, {35311, 39181}, {37225, 39274}, {39201, 42405}

X(54) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 14213}, {2, 311}, {3, 343}, {4, 324}, {6, 5}, {15, 33529}, {16, 33530}, {24, 467}, {25, 53}, {31, 1953}, {32, 51}, {41, 7069}, {42, 21011}, {50, 1154}, {51, 36412}, {58, 17167}, {63, 18695}, {64, 13157}, {69, 28706}, {72, 42698}, {95, 76}, {96, 5392}, {97, 69}, {110, 14570}, {112, 35360}, {154, 42459}, {160, 41480}, {163, 2617}, {184, 216}, {186, 14918}, {187, 41586}, {213, 21807}, {232, 39569}, {251, 17500}, {252, 11140}, {275, 264}, {276, 18022}, {288, 40410}, {323, 1273}, {389, 34836}, {393, 13450}, {512, 12077}, {523, 18314}, {526, 41078}, {560, 2179}, {570, 1209}, {571, 52}, {577, 5562}, {604, 1393}, {647, 6368}, {649, 21102}, {654, 6369}, {661, 2618}, {850, 15415}, {933, 648}, {1141, 94}, {1157, 37779}, {1166, 40393}, {1173, 31610}, {1298, 1972}, {1333, 18180}, {1437, 16697}, {1501, 40981}, {1510, 20577}, {1576, 1625}, {1609, 41587}, {1843, 27371}, {1953, 1087}, {1971, 32428}, {1973, 2181}, {1974, 3199}, {1993, 39113}, {2148, 1}, {2167, 75}, {2168, 91}, {2169, 63}, {2174, 35194}, {2190, 92}, {2207, 14569}, {2501, 23290}, {2616, 1577}, {2623, 523}, {2963, 25043}, {2965, 143}, {3049, 15451}, {3053, 41588}, {3124, 41221}, {3202, 40588}, {3269, 35442}, {3284, 1568}, {3518, 14129}, {5063, 5891}, {6748, 14978}, {8648, 2600}, {8739, 6117}, {8740, 6116}, {8770, 27364}, {8795, 18027}, {8882, 4}, {8883, 6515}, {8884, 2052}, {8901, 338}, {9409, 14391}, {10311, 39530}, {10312, 30506}, {11077, 265}, {13338, 13364}, {13342, 27355}, {13366, 233}, {14270, 2081}, {14533, 3}, {14573, 32}, {14575, 217}, {14579, 1263}, {14585, 418}, {14586, 110}, {14587, 249}, {14642, 8798}, {15109, 21230}, {15412, 850}, {15958, 4558}, {16029, 1591}, {16030, 141}, {16032, 34391}, {16034, 1592}, {16035, 13567}, {16037, 34392}, {16813, 6528}, {18315, 99}, {18353, 565}, {18831, 6331}, {19180, 41005}, {19189, 297}, {19208, 39352}, {19210, 394}, {19306, 1749}, {20574, 31626}, {21449, 9291}, {21461, 36300}, {21462, 36301}, {21741, 2599}, {23195, 42445}, {23286, 525}, {25044, 1994}, {26887, 3164}, {32445, 42453}, {32640, 36831}, {32661, 23181}, {32692, 925}, {33629, 20}, {33872, 14845}, {34384, 1502}, {34386, 305}, {34397, 11062}, {34756, 39114}, {35196, 333}, {36078, 655}, {36134, 662}, {37636, 1225}, {38808, 15466}, {39109, 41536}, {39177, 18155}, {39201, 17434}, {39287, 308}, {40440, 1969}, {40633, 13595}, {41213, 41222}, {41270, 511}, {41271, 2165}, {41331, 27374}, {41373, 41481}, {41488, 40016}, {42293, 34983}, {42300, 327}

X(54) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3460, 2599}, {2, 2888, 1209}, {2, 9545, 1147}, {2, 18912, 26917}, {3, 1493, 15801}, {3, 1993, 11412}, {3, 7592, 5890}, {3, 11402, 7592}, {3, 12161, 5889}, {3, 12316, 12307}, {3, 15087, 6102}, {3, 16266, 2979}, {3, 19210, 97}, {3, 25044, 1157}, {3, 32046, 5012}, {4, 184, 1614}, {4, 275, 4994}, {4, 578, 15033}, {4, 1614, 14157}, {4, 19467, 12289}, {5, 49, 110}, {5, 567, 13434}, {5, 9706, 9705}, {5, 14516, 41171}, {6, 24, 3567}, {6, 2917, 973}, {6, 14533, 8882}, {6, 14585, 10312}, {6, 15073, 8537}, {6, 19189, 9792}, {6, 19357, 24}, {6, 19468, 6152}, {24, 12291, 12380}, {24, 19357, 11464}, {25, 9707, 26882}, {26, 36749, 3060}, {36, 35197, 7356}, {49, 110, 9705}, {49, 567, 5}, {51, 3518, 38848}, {51, 10282, 3518}, {52, 18475, 7488}, {110, 9706, 49}, {110, 13434, 5}, {140, 11245, 26879}, {140, 40631, 252}, {143, 5944, 2070}, {154, 10982, 10594}, {155, 7503, 11459}, {155, 37506, 7503}, {182, 1092, 631}, {184, 275, 26887}, {184, 578, 4}, {184, 3574, 32379}, {184, 11424, 6759}, {184, 15033, 14157}, {185, 3520, 74}, {185, 11430, 3520}, {186, 1199, 389}, {195, 5012, 10203}, {195, 10610, 7691}, {195, 12307, 12316}, {217, 1970, 112}, {275, 8884, 4}, {275, 38808, 8884}, {288, 20574, 1173}, {323, 37126, 1216}, {378, 1181, 6241}, {381, 9704, 156}, {389, 13366, 1199}, {389, 13367, 186}, {389, 21660, 6242}, {394, 7509, 7999}, {394, 37476, 7509}, {427, 31804, 34224}, {569, 1147, 2}, {578, 6759, 11424}, {578, 10274, 3574}, {627, 628, 1273}, {631, 14912, 18916}, {973, 6152, 7730}, {973, 32391, 24}, {1157, 25042, 3}, {1173, 38848, 51}, {1173, 39667, 288}, {1181, 11425, 378}, {1209, 6689, 2}, {1216, 37513, 37126}, {1263, 10285, 31392}, {1493, 12363, 1993}, {1495, 10110, 34484}, {1498, 35502, 11455}, {1511, 36153, 12006}, {1593, 11456, 12290}, {1593, 19347, 11456}, {1594, 6146, 25739}, {1614, 15033, 4}, {1899, 37119, 23294}, {1994, 7488, 52}, {2070, 14627, 143}, {2595, 2596, 1087}, {2917, 13423, 12380}, {2937, 10263, 15107}, {3091, 9544, 10539}, {3518, 37505, 1173}, {3520, 15032, 185}, {3541, 6776, 11457}, {3567, 7730, 973}, {3567, 11464, 24}, {3567, 12291, 6152}, {3567, 13423, 7730}, {3567, 19468, 12380}, {3574, 21659, 32365}, {3796, 37498, 10323}, {5012, 34148, 3}, {5422, 6642, 15024}, {5622, 32245, 32234}, {5889, 11422, 12161}, {5889, 19167, 19194}, {5890, 11423, 7592}, {5946, 15532, 13368}, {6102, 32136, 15087}, {6143, 33565, 14076}, {6146, 23292, 1594}, {6150, 18016, 3}, {6152, 11577, 12291}, {6640, 18952, 26913}, {6644, 36753, 15043}, {6750, 35717, 4}, {6759, 11424, 4}, {7526, 18445, 12111}, {7542, 13292, 3580}, {7547, 18396, 18394}, {7592, 11402, 11423}, {7592, 16030, 19168}, {7592, 32333, 32339}, {7699, 18394, 7547}, {7722, 32607, 74}, {7730, 12291, 13423}, {7730, 13423, 6152}, {8254, 20585, 36966}, {8254, 36966, 6288}, {8882, 14533, 33629}, {9706, 13434, 110}, {9707, 11426, 9781}, {9730, 12038, 22467}, {9781, 26882, 25}, {9818, 11441, 15058}, {9905, 12266, 7979}, {10066, 10082, 1}, {10274, 12254, 1614}, {10282, 37505, 51}, {10605, 35477, 11468}, {10619, 12242, 4}, {10984, 13346, 376}, {11402, 16030, 19170}, {11402, 32333, 32341}, {11425, 17809, 1181}, {11427, 18925, 4}, {11427, 32354, 3574}, {11430, 15032, 74}, {11449, 15043, 6644}, {11464, 13423, 2917}, {12006, 36153, 15037}, {12038, 22467, 15035}, {12161, 12606, 15801}, {12227, 32607, 7722}, {12234, 23358, 32352}, {13121, 13122, 7979}, {13198, 15463, 74}, {13353, 22115, 140}, {13366, 13367, 389}, {13366, 21660, 12234}, {13367, 21638, 19185}, {13367, 32352, 23358}, {13621, 15038, 10095}, {13630, 43394, 3}, {14389, 14516, 5}, {16029, 16034, 6}, {16030, 16035, 3}, {16030, 19170, 19209}, {16031, 16036, 19}, {16032, 16037, 2}, {16035, 19210, 8883}, {17821, 17846, 2917}, {17928, 36752, 15045}, {18388, 21659, 4}, {18570, 34783, 11440}, {18925, 32346, 12254}, {19095, 19096, 6}, {19172, 19180, 19206}, {19459, 39588, 12283}, {21638, 21660, 19207}, {37481, 37814, 15053}