X(3) = CIRCUMCENTER¶
Trilinears
\(cos A : cos B : cos C\)
\(a(b2 + c2 - a2) : b(c2 + a2 - b2) : c(a2 + b2 - c2)\)
Barycentrics
\(sin 2A : sin 2B : sin 2C\)
\(tan B + tan C : tan C + tan A: tan A + tan B\)
\(S^2 - SB SC : :\)
\(1 - cot B cot C : :\)
Notes
As a point on the Euler line, X(3) has Shinagawa coefficients (1, -1).
X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by R = a/(2 sin A) = abc/(4*area(ABC)).
The tangents at vertices of excentral triangle to the McCay cubic K003 concur in X(3). Also, the tangents at A,B,C to the orthocubic K006 concur in X(3). (Randy Hutson, November 18, 2015)
Let A’B’C’ be the cevian triangle of X(4). Let A″ be X(4)-of-AB’C’, and define B″, C″ cyclically. The lines A″, BB″, CC″ concur in X(3). (Randy Hutson, June 27, 2018)
Let P be a point in the plane of ABC. Let P’ be the isogonal conjugate of P. Let P” be the pedal antipodal perspector of P. X(3) is the unique point P for which P’ = P”. (Randy Hutson, June 27, 2018)
Taking a reference triangle ABC and a variable point P on the plane, P=:ref:X(3) <X(3)> is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=:ref:X(3) <X(3)> gives the global maximum. (Mark Helman, July 10, 2020)
A slightly similar thing happens regarding the area of the antipedal triangle of P. P=:ref:X(4) <X(4)> has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X(4) is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=:ref:X(4) <X(4)> gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)
View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.
If you have The Geometer’s Sketchpad, you can view Circumcenter. If you have GeoGebra, you can view Circumcenter.
Let T be any one of these trianges: {Aries, X(3)-Ehrmann, X3-ABC reflections, 3rd pedal of X(3), 3rd antipedal of X(3), inner-Le Viet An, outer-Le Viet An}. Let OA be the circle centered at the A-vertex of T and passing through A; define OBand OC cyclically. X(3) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)
In the plane of a triangle ABC, let Oa = circle with diameter BC, and define Ob and Oc cyclically; A′ = reflection of A in Oa, and define B′ and C′ cyclically; Ab = BA’∩Oa, and define Bc and Ca cyclically; Ac = CA’∩Oa, and define Ba and Cb cyclically; A1 = BcBa∩CaCb, and define B1 and C1 cyclically. The triangle A1B1C1 is perspective to ABC, and the perspector is X(3). (Dasari Naga Vijay Krishna, April 19, 2021)
Let O be a point (not necessarily X(3)), and let be a AOB be a fixed-angle sector of a circle C=(O,R), rigidly rotating about center O. Let P be an arbitrary point. The locus of X(3)-of-PAB is a conic E whose major axis is OP. This conic is an ellipse (resp. hyperbola) if P is interior (resp. exterior) to C. One of its foci is O. Figure (ellipse). –> Figure (ellipse). Figure (hyperbola). (Dan Reznik, December 10, 2021) –> Figure (hyperbola). (Dan Reznik, December 10, 2021)
Let A’B’C’ be the anticevian triangle of X(3), and let Ea be the ellipse passing through A’ and having foci B’ and C’. Define Eb and Ec cyclically. The 6 major vertices of the three ellipses lie on a circle that is concentric with the circumcircle of A’B’C’. Figure. (Dan Reznik, December 19, 2021) –> Figure. (Dan Reznik, December 19, 2021)
X(3) lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, the McCay cubic, then Darboux quintic, and these lines: {1,35}, {2,4}, {6,15}, {7,943}, {8,100}, {9,84}, {10,197}, {11,499}, {12,498}, {13,17}, {14,18}, {19,1871}, {31,601}, {33,1753}, {34,1465}, {37,975}, {38,976}, {41,218}, {42,967}, {43,5247}, {47,1399}, {48,71}, {49,155}, {51,3527}, {54,97}, {60,1175}, {63,72}, {64,154}, {66,141}, {67,542}, {68,343}, {69,332}, {73,212}, {74,110}, {76,98}, {77,1410}, {80,5445}, {81,5453}, {83,262}, {85,5088}, {86,1246}, {90,1898}, {95,264}, {96,5392}, {101,103}, {102,109}, {105,277}, {106,1293}, {107,1294}, {108,1295}, {111,1296}, {112,1297}, {113,122}, {114,127}, {115,2079}, {119,123}, {125,131}, {128,1601}, {142,516}, {143,1173}, {144,5843}, {145,1483}, {147,2896}, {149,1484}, {158,243}, {161,1209}, {164,3659}, {169,910}, {172,2276}, {191,1768}, {193,1353}, {194,385}, {200,963}, {201,1807}, {207,1767}, {214,2800}, {217,3289}, {223,1035}, {225,1074}, {226,4292}, {227,1455}, {230,2549}, {232,1968}, {238,978}, {248,3269}, {252,930}, {256,987}, {269,939}, {295,2196}, {296,820}, {298,617}, {299,616}, {302,621}, {303,622}, {305,1799}, {315,325}, {323,3431}, {329,2096}, {345,1791}, {347,1119}, {348,1565}, {351,2780}, {352,353}, {356,3278}, {358,6120}, {373,3066}, {380,2257}, {388,495}, {390,1058}, {392,3420}, {393,1217}, {395,398}, {396,397}, {476,477}, {480,5223}, {485,590}, {486,615}, {489,492}, {490,491}, {496,497}, {501,5127}, {513,3657}, {518,3433}, {519,3654}, {523,5664}, {524,5486}, {525,878}, {528,3813}, {532,5859}, {533,5858}, {539,3519}, {541,5642}, {543,5569}, {551,3653}, {595,995}, {604,2269}, {607,1951}, {608,1950}, {609,5280}, {611,1469}, {612,5322}, {613,1428}, {614,5310}, {618,635}, {619,636}, {623,629}, {624,630}, {639,641}, {640,642}, {653,1148}, {659,2826}, {662,1098}, {667,1083}, {669,1499}, {690,6334}, {691,842}, {692,2807}, {695,1613}, {732,6308}, {741,6010}, {758,5884}, {759,6011}, {805,2698}, {840,2742}, {843,2709}, {846,2944}, {847,925}, {895,4558}, {901,953}, {902,1201}, {905,1946}, {915,2969}, {917,1305}, {920,1858}, {927,2724}, {929,2723}, {934,972}, {935,2697}, {938,3488}, {945,1457}, {947,5399}, {950,1210}, {951,1407}, {955,1170}, {960,997}, {962,1621}, {968,6051}, {974,5504}, {984,3497}, {1000,1476}, {1014,3945}, {1015,2241}, {1018,4513}, {1033,1249}, {1037,1066}, {1046,4650}, {1047,2636}, {1054,1283}, {1055,1334}, {1056,3600}, {1057,1450}, {1069,6238}, {1072,3011}, {1075,1941}, {1093,1105}, {1104,3752}, {1107,4386}, {1124,2066}, {1131,3316}, {1132,3317}, {1135,6121}, {1137,6122}, {1138,3471}, {1139,3370}, {1140,3397}, {1167,1413}, {1177,1576}, {1180,1627}, {1184,1194}, {1196,1611}, {1199,1994}, {1203,5313}, {1211,5810}, {1213,5816}, {1247,2640}, {1263,3459}, {1270,5874}, {1271,5875}, {1276,5240}, {1277,5239}, {1290,2687}, {1298,1303}, {1301,5897}, {1304,2693}, {1308,2717}, {1309,2734}, {1330,4417}, {1331,1797}, {1335,2067}, {1337,3489}, {1338,3490}, {1348,2040}, {1349,2039}, {1364,1795}, {1386,3941}, {1389,2320}, {1397,1682}, {1398,1870}, {1400,2268}, {1406,1464}, {1411,1772}, {1412,2213}, {1425,3561}, {1427,1448}, {1433,2188}, {1445,5728}, {1446,3188}, {1447,3673}, {1452,1905}, {1453,2999}, {1471,2293}, {1475,2280}, {1495,3426}, {1500,2242}, {1506,5475}, {1568,3521}, {1575,4426}, {1587,3068}, {1588,3069}, {1602,2550}, {1603,2551}, {1612,4000}, {1625,1987}, {1630,3197}, {1632,2790}, {1633,5698}, {1661,2883}, {1672,3238}, {1673,3237}, {1676,5403}, {1677,5404}, {1696,3731}, {1698,4413}, {1699,3624}, {1709,3683}, {1714,5721}, {1723,2264}, {1724,3216}, {1728,1864}, {1737,1837}, {1745,1935}, {1762,2939}, {1770,1836}, {1779,1780}, {1788,3486}, {1794,3173}, {1796,3690}, {1808,4173}, {1810,4587}, {1811,4571}, {1813,3270}, {1834,5292}, {1901,5747}, {1914,2275}, {1916,3406}, {1918,2274}, {1939,6181}, {1960,2821}, {1986,2904}, {2007,3235}, {2008,3236}, {2053,2108}, {2120,3463}, {2121,3482}, {2130,3343}, {2131,3350}, {2133,5670}, {2163,2334}, {2174,2911}, {2183,2267}, {2197,2286}, {2222,2716}, {2292,3724}, {2329,3501}, {2346,3296}, {2407,2452}, {2548,3815}, {2688,2690}, {2689,2695}, {2691,2752}, {2692,2758}, {2694,2766}, {2696,2770}, {2699,2703}, {2700,2702}, {2701,2708}, {2704,2711}, {2705,2712}, {2706,2713}, {2707,2714}, {2710,2715}, {2718,2743}, {2719,2744}, {2720,2745}, {2721,2746}, {2722,2747}, {2725,2736}, {2726,2737}, {2727,2738}, {2728,2739}, {2729,2740}, {2730,2751}, {2731,2757}, {2732,2762}, {2733,2765}, {2735,2768}, {2783,4436}, {2792,4655}, {2797,6130}, {2801,3678}, {2810,3939}, {2814,3960}, {2827,4491}, {2854,5505}, {2886,4999}, {2888,3448}, {2916,3456}, {2951,3646}, {2971,3563}, {3006,5300}, {3058,4309}, {3061,3496}, {3065,3467}, {3092,5413}, {3093,5412}, {3100,6198}, {3101,6197}, {3165,5669}, {3166,5668}, {3177,3732}, {3200,3205}, {3201,3206}, {3218,3418}, {3219,3876}, {3224,6234}, {3229,3360}, {3272,3334}, {3276,3280}, {3277,3282}, {3305,5927}, {3306,5439}, {3332,4648}, {3341,3347}, {3351,3354}, {3366,3391}, {3367,3392}, {3373,3387}, {3374,3388}, {3381,5402}, {3382,5401}, {3399,3407}, {3413,6178}, {3414,6177}, {3417,3869}, {3436,5552}, {3437,5224}, {3440,5674}, {3441,5675}, {3447,6328}, {3452,6259}, {3460,3465}, {3461,3483}, {3462,5667}, {3464,3466}, {3474,3485}, {3555,3870}, {3582,4330}, {3583,4324}, {3584,4325}, {3585,4316}, {3589,5480}, {3614,5326}, {3620,5921}, {3632,5288}, {3647,3652}, {3667,4057}, {3679,5258}, {3681,4420}, {3687,5814}, {3694,5227}, {3705,5015}, {3710,3977}, {3711,5531}, {3733,6003}, {3734,3934}, {3740,5302}, {3824,5715}, {3849,6232}, {3874,4973}, {3877,4881}, {3889,3957}, {3901,4880}, {3925,6253}, {4001,4101}, {4317,4995}, {4338,4870}, {4340,5323}, {4549,4846}, {4653,6176}, {4720,5372}, {4850,5262}, {4993,4994}, {5226,5714}, {5260,5818}, {5268,5345}, {5275,5277}, {5284,5550}, {5286,5305}, {5306,5319}, {5346,5355}, {5436,5437}, {5441,5442}, {5443,5444}, {5530,5725}, {5541,6264}, {5590,5594}, {5591,5595}, {5606,5951}, {5638,6141}, {5639,6142}, {5640,5643}, {5656,6225}, {5658,5811}, {5672,6191}, {5673,6192}, {5735,6173}, {5962,5963}, {5971,6031}, {6082,6093}, {6118,6250}, {6119,6251}, {6228,6230}, {6229,6231}, {6233,6323}, {6236,6325}, {6294,6295}, {6296,6298}, {6297,6299}, {6300,6302}, {6301,6303}, {6304,6306}, {6305,6307}, {6311,6313}, {6312,6314}, {6315,6317}, {6316,6318}, {6391,6461}, {6413,6458}, {6414,6457}, {6581,6582}
X(3) is the {X(2),:ref:X(4) <X(4)>}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(3), click Tables at the top of this page. If triangle ABC is acute, then X(3) is the incenter of the tangential triangle and the Bevan point, X(40), of the orthic triangle.
X(3) = midpoint of X(i) and X(j) for these (i,j): (1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)
X(3) = reflection of X(i) in X(j) for these (i,j): (1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)
X(3) = isogonal conjugate of X(4)
X(3) = isotomic conjugate of X(264)
X(3) = complement of X(4)
X(3) = anticomplement of X(5)
X(3) = complementary conjugate of X(5)
X(3) = anticomplementary conjugate of X(2888)
X(3) = nine-point-circle-inverse of X(2072)
X(3) = orthocentroidal-circle-inverse of X(5)
X(3) = 1st-Lemoine-circle-inverse of X(2456)
X(3) = 2nd-Lemoine-circle-inverse of X(1570)
X(3) = Conway-circle-inverse of X(38474)
X(3) = eigencenter of the medial triangle
X(3) = eigencenter of the tangential triangle
X(3) = exsimilicenter of 1st and 2nd Kenmotu circles
X(3) = exsimilicenter of nine-point circle and tangential circle
X(3) = X(1)-of-Trinh-triangle if ABC is acute
X(3) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)
X(3) = cevapoint of X(i) and X(j) for these (i,j): (6,154), (48,212), (55,198), (71,228), (185,417), (216,418)
X(3) = X(i)-cross conjugate of X(j) for these (i,j): (48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)
X(3) = crosspoint of X(i) and X(j) for these (i,j): (1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)
X(3) = crosssum of X(i) and X(j) for these (i,j): (1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)
X(3) = crossdifference of every pair of points on the line X(230)X(231)
X(3) = X(i)-Hirst inverse of X(j) for these (i,j): (2, 401), (4,450), (6,511), (21,416), (194, 385)
X(3) = X(2)-line conjugate of X(468)
X(3) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1046), (21,3), (188,191), (259,1045)
X(3) = X(i)-beth conjugate of X(j) for these (i,j): (3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)
X(3) = center of inverse-in-de-Longchamps-circle-of-anticomplementary-circle
X(3) = perspector of inner and outer Napoleon triangles
X(3) = trilinear product of vertices of 2nd Brocard triangle
X(3) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (1,8,5556), (1,9,885), (2,6,1640), (2,10,4049), (3,6,879), (3,66,2435), (4,6,879), (7,8,885), (67,74,879), (6,64,2435), (4,66,2435)
X(3) = intersection of tangents at X(3) and X(4) to Orthocubic K006
X(3) = homothetic center of tangential triangle and 2nd isogonal triangle of X(4); see X(36)
X(3) = trilinear pole of line X(520)X(647)
X(3) = crossdifference of PU(4)
X(3) = trilinear product of PU(16)
X(3) = barycentric product of PU(22)
X(3) = midpoint of PU(i) for these i: 37, 44
X(3) = bicentric sum of PU(i) for these i: 37, 44, 63, 125
X(3) = vertex conjugate of PU(39)
X(3) = PU(63)-harmonic conjugate of X(351)
X(3) = PU(125)-harmonic conjugate of X(650)
X(3) = intersection of tangents to orthocentroidal circle at PU(5)
X(3) = X(3398) of 5th Brocard triangle (see X(32))
X(3) = X(182) of 6th Brocard triangle (see X(384))
X(3) = homothetic center of 1st anti-Brocard triangle and 6th Brocard triangle
X(3) = similitude center of antipedal triangles of the 1st and 2nd Brocard points (PU(1))
X(3) = inverse-in-polar-circle of X(403)
X(3) = inverse-in-{circumcircle, nine-point circle}-inverter of X(858)
X(3) = inverse-in-de-Longchamps-circle of X(3153)
X(3) = inverse-in-Steiner-circumellipse of X(401)
X(3) = inverse-in-Steiner-inellipse of X(441)
X(3) = inverse-in-MacBeath-circumconic of X(3284)
X(3) = radical trace of circumcircle and 8th Lozada circle
X(3) = perspector of medial triangle and polar triangle of the complement of the polar circle
X(3) = pole of line X(6)X(110) wrt Parry circle
X(3) = pole wrt polar circle of trilinear polar of X(2052) (line X(403)X(523))
X(3) = pole wrt {circumcircle, nine-point circle}-inverter of de Longchamps line
X(3) = polar conjugate of X(2052)
X(3) = X(i)-isoconjugate of X(j) for these (i,j): (6,92), (24,91), (25,75), (48,2052), (93,2964), (112,1577), (1101,2970), (2962,3518)
X(3) = X(30)-vertex conjugate of X(523)
X(3) = homothetic center of any 2 of {tangential, Kosnita, 2nd Euler} triangles
X(3) = X(5)-of-excentral-triangle
X(3) = X(26)-of-intouch-triangle
X(3) = antigonal image of X(265)
X(3) = X(2)-of-antipedal-triangle-of-X(6)
X(3) = perspector of the MacBeath Circumconic
X(3) = perspector of ABC and unary cofactor triangle of 5th Euler triangle
X(3) = intersection of trilinear polars of any 2 points on the MacBeath circumconic
X(3) = circumcevian isogonal conjugate of X(1)
X(3) = orthology center of ABC and orthic triangle
X(3) = orthology center of Fuhrmann triangle and ABC
X(3) = orthic isogonal conjugate of X(155)
X(3) = Miquel associate of X(2)
X(3) = X(40)-of-orthic-triangle if ABC is acute
X(3) = X(98)-of-1st-Brocard-triangle
X(3) = X(1380)-of-2nd-Brocard-triangle
X(3) = X(399)-of-orthocentroidal-triangle
X(3) = X(104)-of X(1)-Brocard-triangle
X(3) = X(74)-of X(2)-Brocard-triangle
X(3) = X(74)-of-X(4)-Brocard-triangle
X(3) = X(597)-of-antipedal-triangle-of-X(2)
X(3) = X(182)-of-1st-anti-Brocard-triangle
X(3) = X(381)-of-4th-anti-Brocard-triangle
X(3) = QA-P12 (Orthocenter of the QA-Diagonal Triangle)-of-quadrilateral X(98)X(99)X(110)X(111)
X(3) = orthocenter of X(2)X(9147)X(9149)
X(3) = perspector of ABC and 1st Brocard triangle of 6th Brocard triangle
X(3) = perspector of ABC and 1st Brocard triangle of circumorthic triangle
X(3) = perspector of ABC and 1st Brocard triangle of dual of orthic triangle
X(3) = perspector of ABC and cross-triangle of ABC and half-altitude triangle
X(3) = homothetic center of inner Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(3) = homothetic center of outer Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(3) = anti-Artzt-to-4th-anti-Brocard similarity image of X(6)
X(3) = Thomson-isogonal conjugate of X(2)
X(3) = Lucas-isogonal conjugate of X(2979)
X(3) = X(4)-of-2nd-anti-extouch triangle
X(3) = X(185)-of-A’B’C’, as described in ADGEOM #2697 (8/26/2015, Tran Quang Hung)
X(3) = X(5)-of-3rd-anti-Euler-triangle
X(3) = X(5)-of-4th-anti-Euler-triangle
X(3) = X(671)-of-McCay-triangle
X(3) = centroid of ABC:ref:X(20) <X(20)>
X(3) = Kosnita(X(20),:ref:X(2) <X(2)>) point
X(3) = centroid of incenter and excenters
X(3) = X(265)-of-Fuhrmann-triangle
X(3) = intersection of tangents to 2nd Lemoine circle at intersections with Brocard circle
X(3) = perspector of ABC and antipedal triangle of X(64)
X(3) = trisector nearest X(5) of segment X(5)X(20)
X(3) = Ehrmann-vertex-to-Ehrmann-side similarity image of X(4)
X(3) = Ehrmann-mid-to-ABC similarity image of X(4)
X(3) = Ehrmann-mid-to-Johnson similarity image of X(5)
X(3) = Johnson-to-Ehrmann-mid similarity image of X(20)
X(3) = center of inverse similitude of AAOA triangle and Ehrmann side-triangle
X(3) = X(175)-of-Lucas-central-triangle
X(3) = reflection of X(2080) in the Lemoine axis
X(3) = excentral-isogonal conjugate of X(191)
X(3) = excentral-isotomic conjugate of X(2938)
X(3) = crosssum of foci of orthic inconic
X(3) = crosspoint of foci of orthic inconic
X(3) = similicenter of antipedal triangles of PU(1)
X(3) = excentral-to-ABC functional image of X(40)
X(3) = orthic-to-ABC barycentric image of X(4)
X(3) = orthic-to-ABC functional image of X(5)
X(3) = Feuerbach-to-ABC functional image of X(5)
X(3) = intouch-to-ABC functional image of X(1)
X(3) = ABC-to-excentral barycentric image of X(10)
X(3) = concurrence of Euler lines of intouch triangle and A-, B-, and C-extouch triangles
X(3) = perspector of hexyl triangle and cevian triangle of X(21)
X(3) = perspector of pedal and anticevian triangles of X(1498)
X(3) = perspector of ABC and medial triangle of pedal triangle of X(20)
X(3) = perspector of ABC and the reflection in X(6) of the antipedal triangle of X(6)
X(3) = tangential-isotomic conjugate of tangential-isogonal conjugate of X(35225)
X(3) = Moses-radical-circle-inverse of X(35901)
X(3) = 1st-Brocard-isogonal conjugate of X(2782)
X(3) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,35,55), (1,36,56), (1,46,65), (1,55,3295), (1,56,999), (1,57,942), (1,165,40), (1,171,5711), (1,484,5903), (1,1038,1060), (1,1040,1062), (1,1754,5706), (1,2093,3340), (1,3333,5045), (1,3336,5902), (1,3338,354), (1,3361,3333), (1,3550,5255), (1,3576,1385), (1,3612,2646), (1,3746,3303), (1,5010,35), (1,5119,3057), (1,5131,3336), (1,5264,5710), (1,5563,3304), (1,5697,2098), (1,5903,2099), (2,4,5), (2,5,1656), (2,20,4), (2,21,405), (2,22,25), (2,23,1995), (2,24,6642), (2,25,5020), (2,140,3526), (2,186,6644), (2,377,442), (2,381,5055), (2,382,3851), (2,401,458), (2,404,474), (2,411,3149), (2,418,6638), (2,452,5084), (2,464,440), (2,546,5079), (2,548,1657), (2,549,5054), (2,550,382), (2,631,140), (2,858,5094), (2,859,4245), (2,1010,2049), (2,1113,1344), (2,1114,1345), (2,1370,427), (2,1599,1583), (2,1600,1584), (2,1656,5070), (2,1657,3843), (2,2071,378), (2,2475,2476), (2,2478,4187), (2,2554,2570), (2,2555,2571), (2,2675,2676), (2,3090,3628), (2,3091,3090), (2,3146,3091), (2,3151,469), (2,3152,5125), (2,3522,20), (2,3523,631), (2,3524,549), (2,3525,632), (2,3528,550), (2,3529,546), (2,3534,3830), (2,3543,3545), (2,3545,547), (2,3546,3548), (2,3547,3549), (2,3548,6640), (2,3549,6639), (2,3552,384), (2,3627,5072), (2,3832,5056), (2,3839,5071), (2,4184,1011), (2,4188,404), (2,4189,21), (2,4190,377), (2,4210,4191), (2,4216,859), (2,4226,1316), (2,5046,4193), (2,5056,5067), (2,5059,3832), (2,5189,5169), (2,6636,22), (4,5,381), (4,21,3560), (4,24,25), (4,25,1598), (4,140,1656), (4,186,24), (4,376,20), (4,378,1593), (4,381,3843), (4,382,3830), (4,548,3534), (4,549,3526), (4,550,1657), (4,631,2), (4,632,5079), (4,1006,405), (4,1593,1597), (4,1656,3851), (4,1657,5073), (4,1658,2070), (4,2937,5899), (4,3088,1595), (4,3089,1596), (4,3090,3091), (4,3091,546), (4,3146,3627), (4,3147,3542), (4,3515,3517), (4,3520,378), (4,3522,550), (4,3523,140), (4,3524,631), (4,3525,3090), (4,3526,5055), (4,3528,376), (4,3529,3146), (4,3530,5054), (4,3533,5056), (4,3541,427), (4,3542,235), (4,3543,3853), (4,3545,3832), (4,3548,2072), (4,3627,5076), (4,3628,5072), (4,3832,3845), (4,3839,3861), (4,3855,3839), (4,5054,5070), (4,5056,3850), (4,5067,3545), (4,5068,3858), (4,5071,3855), (4,6353,3089), (4,6621,6624), (4,6622,6623), (5,20,382), (5,26,25), (5,140,2), (5,376,1657), (5,381,3851), (5,382,3843), (5,427,5576), (5,546,3091), (5,547,5056), (5,548,20), (5,549,140), (5,631,3526), (5,632,3628), (5,1656,5055), (5,1657,3830), (5,1658,24), (5,3090,5079), (5,3091,5072), (5,3522,3534), (5,3523,5054), (5,3526,5070), (5,3529,5076), (5,3530,631), (5,3534,5073), (5,3627,546), (5,3628,3090), (5,3845,3850), (5,3850,3545), (5,3853,3832), (5,3858,5066), (5,3861,3855), (5,5066,5068), (5,5498,6143), (5,6642,5020), (5,6644,6642), (6,182,5050), (6,187,1384), (6,371,3311), (6,372,3312), (6,574,5024), (6,1151,371), (6,1152,372), (6,1351,5093), (6,1620,1192), (6,2076,5017), (6,3053,32), (6,3311,6417), (6,3312,6418), (6,3592,6419), (6,3594,6420), (6,4252,58), (6,4255,386), (6,4258,4251), (6,5013,39), (6,5022,4253), (6,5023,3053), (6,5085,182), (6,5102,5097), (6,5210,187), (6,5585,5210), (6,6200,6221), (6,6221,6199), (6,6396,6398), (6,6398,6395), (6,6409,1151), (6,6410,1152), (6,6411,6200), (6,6412,6396), (6,6417,6500), (6,6418,6501), (6,6419,6427), (6,6420,6428), (6,6425,3592), (6,6426,3594), (6,6433,6437), (6,6434,6438), (6,6451,6445), (6,6452,6446), (6,6455,6407), (6,6456,6408), (7,3487,6147), (7,5703,3487), (8,100,5687), (8,2975,956), (8,5657,5690), (8,5731,944), (9,936,5044), (9,1490,5777), (9,5438,936), (10,355,5790), (10,993,958), (10,5267,993), (10,5745,5791), (11,5433,499), (11,6284,1479), (12,5432,498), (15,16,6), (15,62,61), (15,3364,371), (15,3365,372), (15,5237,62), (15,5352,5238), (16,61,62), (16,3389,371), (16,3390,372), (16,5238,61), (16,5351,5237), (20,21,1012), (20,140,381), (20,186,26), (20,376,550), (20,381,5073), (20,404,3149), (20,417,6638), (20,549,1656), (20,550,3534), (20,631,5), (20,1006,3560), (20,1656,3830), (20,1658,2937), (20,2060,3079), (20,3090,3627), (20,3091,3146), (20,3146,3529), (20,3522,376), (20,3523,2), (20,3524,140), (20,3525,546), (20,3526,3843), (20,3528,548), (20,3530,3526), (20,3533,3845), (20,3543,5059), (20,3628,5076), (20,5054,3851), (20,5056,3543), (20,5067,3853), (21,404,2), (21,411,4), (21,416,1982), (21,1816,29), (21,1817,28), (21,3658,3109), (21,4188,474), (21,4203,4195), (21,4225,859), (22,24,26), (22,26,2937), (22,381,5899), (22,426,6638), (22,631,6642), (22,1599,3155), (22,1600,3156), (22,1995,23), (22,6644,2070), (23,1995,25), (24,25,3517), (24,26,2070), (24,186,3515), (24,378,4), (24,1593,1598), (24,1657,5899), (24,3516,1597), (24,3520,1593), (25,378,1597), (25,426,6617), (25,1593,4), (25,3515,24), (25,3516,1593), (26,140,6642), (26,378,382), (26,382,5899), (26,6642,3517), (26,6644,24), (28,4219,4), (29,412,4), (32,39,6), (32,182,3398), (32,187,3053), (32,574,39), (32,3053,1384), (32,5171,2080), (32,5206,187), (33,1753,1872), (35,36,1), (35,56,3295), (35,5010,5217), (35,5204,999), (35,5563,3746), (35,5584,6244), (36,55,999), (36,165,3428), (36,2078,5126), (36,3746,5563), (36,5010,55), (36,5217,3295), (39,187,32), (39,574,5013), (39,5008,5041), (39,5013,5024), (39,5023,1384), (39,5206,3053), (40,57,5709), (40,165,3579), (40,1385,1482), (40,3576,1), (41,672,218), (46,3612,1), (48,71,219), (50,566,6), (52,389,568), (52,569,6), (55,56,1), (55,165,6244), (55,3303,3746), (55,3304,3303), (55,5204,56), (55,5217,35), (55,5584,40), (56,1466,57), (56,3303,3304), (56,3304,5563), (56,5204,36), (56,5217,55), (56,5584,3428), (57,942,5708), (57,1420,1467), (57,3601,1), (58,386,6), (58,580,5398), (58,4256,386), (58,4257,4252), (58,4276,4267), (58,4278,3286), (61,62,6), (61,5238,15), (61,5351,16), (61,5864,1351), (62,5237,16), (62,5352,15), (62,5865,1351), (63,72,3927), (63,78,72), (63,3984,3951), (63,4652,3916), (63,4855,78), (63,5440,3940), (64,154,1498), (65,1155,46), (65,2646,1), (69,3926,3933), (69,6337,3926), (71,1818,3781), (72,78,3940), (72,3916,63), (72,5440,78), (73,255,3157), (73,603,222), (74,1511,399), (74,1614,6241), (76,99,1975), (76,1078,183), (78,1259,1260), (78,3916,3927), (78,3951,3984), (78,4652,63), (78,4855,5440), (84,936,5777), (84,5044,5779), (84,5438,5720), (99,1078,76), (99,5152,5989), (100,2975,8), (100,5303,2975), (101,3730,220), (104,5657,956), (110,1614,156), (140,376,382), (140,381,5070), (140,382,5055), (140,546,3628), (140,549,631), (140,550,4), (140,631,5054), (140,632,3525), (140,1368,3548), (140,1657,3851), (140,1658,6644), (140,3146,5079), (140,3522,1657), (140,3528,3534), (140,3529,5072), (140,3530,549), (140,3534,3843), (140,3627,3090), (140,3628,632), (140,3845,5067), (140,3853,547), (140,5428,1006), (140,6636,2937), (143,5946,3567), (155,1147,3167), (157,160,159), (165,5010,2077), (165,6282,3587), (171,5329,1460), (182,576,575), (182,578,569), (182,1160,6418), (182,1161,6417), (182,1350,1351), (182,5092,5085), (182,5171,32), (183,1975,76), (184,185,1181), (184,394,3167), (184,1092,1147), (184,1147,49), (184,1204,185), (184,3917,394), (184,5562,155), (185,1092,155), (185,3917,5562), (186,376,22), (186,378,25), (186,550,2937), (186,1593,3517), (186,3516,1598), (186,3520,4), (186,3651,2915), (187,574,6), (187,2021,1691), (187,5162,2076), (187,5188,5171), (187,5206,5023), (191,6326,5694), (198,1436,610), (199,1011,25), (199,3145,2915), (212,603,255), (212,4303,3157), (216,577,6), (216,3284,5158), (220,3207,101), (230,5254,3767), (232,1968,2207), (235,468,3542), (235,1885,4), (237,3148,25), (243,1940,158), (255,4303,222), (283,1790,1437), (284,579,6), (371,372,6), (371,1151,6221), (371,1152,3312), (371,1350,1161), (371,2459,6423), (371,3103,6422), (371,3311,6199), (371,3312,6417), (371,3594,6427), (371,6200,1151), (371,6395,6500), (371,6396,372), (371,6398,6418), (371,6409,6449), (371,6410,6398), (371,6411,6455), (371,6412,6450), (371,6419,3592), (371,6420,6419), (371,6425,6447), (371,6426,6428), (371,6449,6407), (371,6450,6395), (371,6452,6408), (371,6453,6425), (371,6454,6420), (371,6455,6445), (371,6481,6432), (371,6484,6429), (371,6486,6480), (371,6497,6446), (372,1151,3311), (372,1152,6398), (372,1350,1160), (372,2460,6424), (372,3102,6421), (372,3311,6418), (372,3312,6395), (372,3592,6428), (372,6199,6501), (372,6200,371), (372,6221,6417), (372,6396,1152), (372,6409,6221), (372,6410,6450), (372,6411,6449), (372,6412,6456), (372,6419,6420), (372,6420,3594), (372,6425,6427), (372,6426,6448), (372,6449,6199), (372,6450,6408), (372,6451,6407), (372,6453,6419), (372,6454,6426), (372,6456,6446), (372,6480,6431), (372,6485,6430), (372,6487,6481), (372,6496,6445), (376,549,381), (376,631,4), (376,1006,1012), (376,3090,3529), (376,3522,548), (376,3523,5), (376,3524,2), (376,3525,3146), (376,3526,5073), (376,3528,3522), (376,3530,1656), (376,5054,3830), (376,5067,5059), (378,2070,3830), (378,2937,5073), (378,3515,1598), (378,3520,3516), (378,6644,381), (381,382,4), (381,1656,5), (381,1657,382), (381,2070,25), (381,3526,1656), (381,5054,2), (381,5072,3091), (381,5079,5072), (382,631,5070), (382,1656,381), (382,3526,5), (382,3534,1657), (382,5054,1656), (382,5076,3627), (382,5079,546), (384,3552,1003), (384,5999,4), (386,573,970), (386,581,5396), (386,991,581), (386,4256,4255), (386,4257,58), (386,5752,5754), (388,3085,495), (388,5218,3085), (389,578,6), (394,1181,155), (394,3796,184), (394,5406,5408), (394,5407,5409), (404,1006,140), (404,4189,405), (404,6636,2915), (405,474,2), (405,1012,3560), (405,2915,25), (405,3149,5), (408,4189,6638), (411,1006,5), (411,3523,474), (411,4189,1012), (417,1593,6617), (418,6641,25), (426,3148,441), (426,6641,2), (427,3575,4), (428,1907,4), (454,3548,6617), (465,466,2), (468,1885,235), (474,1012,5), (474,3560,1656), (485,5418,590), (485,6560,3070), (486,5420,615), (486,6561,3071), (487,488,69), (489,492,637), (490,491,638), (497,3086,496), (498,1478,12), (498,4299,1478), (499,1479,11), (499,4302,1479), (500,582,6), (500,5396,581), (546,549,3525), (546,550,3529), (546,632,3090), (546,3090,5072), (546,3091,381), (546,3146,5076), (546,3525,1656), (546,3529,382), (546,3627,4), (546,3628,5), (546,5079,3851), (547,3543,381), (547,3845,3545), (547,3850,5), (547,3853,3850), (547,5067,1656), (548,549,4), (548,550,376), (548,631,382), (548,632,3529), (548,3523,381), (548,3524,1656), (548,3530,5), (548,5054,5073), (549,550,5), (549,1657,5070), (549,3522,382), (549,3528,1657), (549,3530,3523), (549,3534,5055), (549,3627,632), (549,3853,3533), (549,6636,2070), (550,631,381), (550,632,3627), (550,1656,5073), (550,1658,22), (550,3523,1656), (550,3524,3526), (550,3525,5076), (550,3526,3830), (550,3530,2), (550,3628,3146), (550,3850,5059), (550,5054,3843), (550,5498,3153), (551,5493,4301), (567,568,6), (567,3581,568), (568,6243,52), (569,578,567), (570,571,6), (572,573,6), (572,3430,581), (573,579,5755), (573,581,5752), (574,5171,3095), (574,5206,32), (574,5210,1384), (575,576,6), (577,578,2055), (577,5158,3284), (579,991,5751), (579,5751,5753), (580,581,6), (580,3430,5752), (581,991,500), (582,5398,580), (583,584,6), (590,3070,485), (595,995,1191), (601,602,31), (615,3071,486), (616,628,634), (617,627,633), (620,626,3788), (627,633,298), (628,634,299), (631,1657,5055), (631,3090,3525), (631,3091,632), (631,3146,3628), (631,3523,549), (631,3524,3523), (631,3528,20), (631,3529,3090), (631,3534,3851), (631,3545,3533), (631,3651,3149), (631,5059,547), (631,6636,26), (631,6643,3548), (632,3091,1656), (632,3146,5072), (632,3525,3526), (632,3529,381), (632,3627,5), (632,3628,2), (632,5079,5070), (800,5065,6), (902,1201,3915), (910,1212,169), (936,1490,5720), (936,5732,1490), (936,5777,5780), (938,4313,3488), (940,5706,5707), (942,5709,2095), (943,3487,954), (944,5657,8), (946,1125,5886), (950,1210,5722), (950,3911,1210), (956,5687,8), (958,1376,10), (962,3616,5603), (965,5776,5778), (970,5396,5754), (980,5337,940), (997,1158,5887), (999,3295,1), (1006,3651,4), (1011,4191,2), (1012,3149,4), (1030,5096,5132), (1030,5124,6), (1038,1040,1), (1060,1062,1), (1074,1076,225), (1092,1181,3167), (1092,5562,394), (1106,1253,1496), (1125,5248,1001), (1147,1216,394), (1150,5767,5769), (1151,1152,6), (1151,3312,6199), (1151,3592,6425), (1151,3594,3592), (1151,6200,6449), (1151,6221,6407), (1151,6396,3312), (1151,6398,6417), (1151,6408,6500), (1151,6409,6200), (1151,6410,372), (1151,6411,6409), (1151,6412,1152), (1151,6419,6447), (1151,6425,6453), (1151,6426,6419), (1151,6429,6480), (1151,6431,6437), (1151,6433,6484), (1151,6437,6429), (1151,6438,6431), (1151,6446,6501), (1151,6449,6445), (1151,6450,6418), (1151,6453,6519), (1151,6454,6427), (1151,6456,6395), (1151,6497,6408), (1152,3311,6395), (1152,3592,3594), (1152,3594,6426), (1152,6200,3311), (1152,6221,6418), (1152,6396,6450), (1152,6398,6408), (1152,6407,6501), (1152,6409,371), (1152,6410,6396), (1152,6411,1151), (1152,6412,6410), (1152,6420,6448), (1152,6425,6420), (1152,6426,6454), (1152,6430,6481), (1152,6432,6438), (1152,6434,6485), (1152,6437,6432), (1152,6438,6430), (1152,6445,6500), (1152,6449,6417), (1152,6450,6446), (1152,6453,6428), (1152,6454,6522), (1152,6455,6199), (1152,6496,6407), (1155,2646,65), (1160,3311,1351), (1161,3312,1351), (1180,1627,5359), (1191,3052,595), (1193,4300,1064), (1210,4304,950), (1216,5447,3917), (1312,1313,2072), (1319,3057,1), (1333,4261,6), (1340,1341,182), (1340,1380,6), (1341,1379,6), (1342,1343,182), (1342,1670,6), (1343,1671,6), (1344,1345,381), (1350,5023,5171), (1350,5085,6), (1350,5092,5050), (1351,5050,6), (1368,3547,1656), (1384,5024,6), (1385,6585,1617), (1388,2098,1), (1399,2361,47), (1420,1697,1), (1428,3056,613), (1469,2330,611), (1470,5172,1617), (1479,4302,6284), (1495,5650,5651), (1504,1505,1570), (1504,5062,6), (1505,5058,6), (1532,4187,5), (1578,1579,6), (1583,1584,2), (1583,3156,5020), (1584,3155,5020), (1589,1590,2), (1593,3515,25), (1593,3516,378), (1593,6642,381), (1594,6240,4), (1597,1598,4), (1597,3517,1598), (1597,5020,381), (1598,3517,25), (1599,1600,2), (1614,5876,399), (1621,5253,3616), (1656,1657,4), (1656,2937,25), (1656,3526,2), (1656,3534,382), (1656,5054,3526), (1656,5072,5079), (1656,5076,3091), (1656,5079,3090), (1657,3526,381), (1657,3534,20), (1657,5054,5), (1657,5072,3627), (1657,5079,5076), (1658,2071,1657), (1658,3520,382), (1658,6644,3515), (1662,1663,2456), (1662,1664,6), (1663,1665,6), (1666,1667,1570), (1666,1668,6), (1667,1669,6), (1685,1686,2092), (1687,1688,32), (1687,1689,6), (1688,1690,6), (1689,1690,39), (1691,3094,6), (1692,5028,6), (1698,5691,5587), (1724,3216,4383), (1805,1806,2193), (1995,3146,5198), (2028,2029,1570), (2041,2042,5), (2041,2044,4), (2041,2045,2), (2042,2043,4), (2042,2046,2), (2043,2045,5), (2043,2046,2041), (2044,2045,2042), (2044,2046,5), (2045,2046,140), (2047,2048,5), (2049,2050,5), (2066,6502,1124), (2067,5414,1335), (2070,2937,26), (2070,3526,6642), (2071,6636,376), (2072,6639,1656), (2076,5116,6), (2077,3428,6244), (2077,3576,55), (2080,3398,32), (2080,5024,5093), (2092,5019,6), (2220,5069,6), (2245,2278,6), (2271,5021,6), (2305,5110,6), (2446,2447,5570), (2448,2449,5535), (2454,2455,441), (2459,2460,1691), (2459,3103,372), (2460,3102,371), (2479,2480,401), (2482,3455,2936), (2536,2537,10), (2549,3767,5254), (2558,2559,182), (2560,2561,6), (2562,2563,39), (2564,2565,1), (2566,2567,5), (2568,2569,10), (2570,2571,381), (2572,2573,40), (2673,2674,6), (2937,5054,6642), (2979,5012,1993), (3003,5063,6), (3053,5013,6), (3053,5023,187), (3053,5210,5023), (3060,3567,143), (3068,6460,1587), (3069,6459,1588), (3085,4293,388), (3086,4294,497), (3090,3091,5), (3090,3146,546), (3090,3518,1995), (3090,3525,2), (3090,3529,4), (3090,3627,381), (3090,3628,1656), (3090,5079,5055), (3091,3146,4), (3091,3525,3628), (3091,3529,3627), (3091,3628,5079), (3091,5072,3851), (3091,5076,3843), (3095,3398,6), (3098,5085,1351), (3098,5092,6), (3098,5171,5188), (3102,3103,3094), (3111,5118,6), (3129,3130,25), (3129,3132,3130), (3130,3131,3129), (3131,3132,25), (3146,3523,3525), (3146,3525,5), (3146,3627,382), (3146,3628,381), (3146,5079,3843), (3147,3542,468), (3149,3560,381), (3155,3156,25), (3278,3279,356), (3280,3281,3276), (3282,3283,3277), (3284,5158,6), (3285,4286,6), (3286,4267,58), (3286,5132,6), (3303,3304,1), (3303,3746,3295), (3304,5563,999), (3311,3312,6), (3311,6200,6407), (3311,6221,371), (3311,6395,6501), (3311,6398,3312), (3311,6409,6445), (3311,6410,6408), (3311,6418,6500), (3311,6427,6419), (3311,6428,6427), (3311,6448,6420), (3311,6449,6221), (3311,6450,372), (3311,6451,6449), (3311,6452,1152), (3311,6455,1151), (3311,6456,6398), (3311,6495,6499), (3311,6496,6200), (3311,6497,6450), (3311,6498,6435), (3311,6519,6447), (3311,6522,3594), (3312,6199,6500), (3312,6221,3311), (3312,6396,6408), (3312,6398,372), (3312,6409,6407), (3312,6410,6446), (3312,6417,6501), (3312,6427,6428), (3312,6428,6420), (3312,6447,6419), (3312,6449,371), (3312,6450,6398), (3312,6451,1151), (3312,6452,6450), (3312,6455,6221), (3312,6456,1152), (3312,6494,6498), (3312,6496,6449), (3312,6497,6396), (3312,6499,6436), (3312,6519,3592), (3312,6522,6448), (3313,5157,6), (3334,3335,3272), (3336,5902,5221), (3340,5128,2093), (3364,3365,16), (3364,3389,1151), (3364,3390,6), (3365,3389,6), (3365,3390,1152), (3368,3369,3393), (3368,3395,6), (3369,3396,6), (3371,3372,372), (3371,3386,6), (3372,3385,6), (3379,3380,3368), (3379,3393,6), (3380,3394,6), (3385,3386,371), (3389,3390,15), (3393,3394,3395), (3395,3396,3379), (3428,3576,999), (3474,3485,4295), (3487,5703,5719), (3487,5759,5758), (3513,3514,1617), (3515,3516,4), (3515,3520,1597), (3520,6636,550), (3522,3523,4), (3522,3524,5), (3522,3530,381), (3522,4188,411), (3523,3524,3530), (3523,3529,632), (3523,3534,5070), (3523,4189,1006), (3523,5059,3533), (3523,6636,24), (3524,3528,4), (3524,3651,474), (3524,6636,6644), (3525,3529,3091), (3525,3627,5079), (3525,5072,5070), (3525,5076,5055), (3526,3534,4), (3526,5054,140), (3526,5072,3628), (3526,5076,5079), (3528,3530,382), (3529,5072,3830), (3529,5076,5073), (3533,3543,5), (3533,3545,5067), (3533,3832,547), (3533,3850,1656), (3533,5059,3850), (3533,5067,2), (3534,5054,381), (3534,5076,3529), (3534,5079,3146), (3534,6644,5899), (3537,3538,4), (3543,3545,3845), (3543,3832,4), (3543,5056,3832), (3543,5067,3850), (3545,3832,3850), (3545,3845,381), (3545,5056,5), (3545,5059,3853), (3545,5067,5056), (3546,3547,2), (3546,3549,6640), (3546,6643,1368), (3547,3548,6639), (3547,6643,5), (3548,3549,2), (3549,6643,2072), (3557,3558,576), (3576,3579,1482), (3579,5482,1764), (3582,4330,4857), (3584,4325,5270), (3587,5709,40), (3592,3594,6), (3592,6200,6519), (3592,6396,6448), (3592,6410,6454), (3592,6419,3311), (3592,6420,6427), (3592,6425,371), (3592,6426,6420), (3592,6428,6417), (3592,6448,6418), (3592,6453,6447), (3592,6454,3312), (3594,6200,6447), (3594,6396,6522), (3594,6409,6453), (3594,6419,6428), (3594,6420,3312), (3594,6425,6419), (3594,6426,372), (3594,6427,6418), (3594,6447,6417), (3594,6453,3311), (3594,6454,6448), (3601,5122,5708), (3616,5603,5901), (3624,5259,4423), (3627,3628,3091), (3627,5072,3843), (3627,5076,3830), (3628,5072,5055), (3628,5076,3851), (3653,3656,551), (3683,5918,1709), (3736,5156,6), (3746,5563,1), (3784,3955,222), (3785,3926,69), (3785,6337,3933), (3796,3917,3167), (3830,3843,4), (3830,3851,3843), (3830,5055,381), (3830,5070,3851), (3830,5073,382), (3832,3850,381), (3832,5056,3545), (3832,5059,3543), (3832,5067,5), (3839,3855,3858), (3839,5066,381), (3839,5068,3855), (3839,5071,5066), (3843,3851,381), (3843,5055,3851), (3843,5070,5), (3843,5073,3830), (3845,3850,3832), (3845,3853,4), (3845,5059,382), (3850,3853,3845), (3851,5055,5), (3851,5070,5055), (3851,5073,4), (3851,5899,1598), (3853,5056,381), (3854,3856,381), (3855,3858,381), (3855,5068,5066), (3855,5071,5068), (3858,3861,3839), (3858,5066,3855), (3861,5066,3858), (3861,5068,381), (3869,4511,5730), (3911,4304,5722), (3916,4855,3940), (3916,5440,72), (3917,5562,1216), (3926,6337,6390), (3927,3940,72), (3933,6390,3926), (3951,3984,72), (4184,4210,2), (4184,4225,21), (4188,4189,2), (4188,4225,4191), (4188,5428,5054), (4210,6636,199), (4218,6636,859), (4251,4253,6), (4251,4262,4258), (4251,5030,4253), (4252,4255,6), (4253,4262,4251), (4253,5030,5022), (4254,5120,6), (4256,4257,6), (4256,4276,5132), (4258,5022,6), (4259,5135,6), (4260,5138,6), (4262,5030,6), (4263,5042,6), (4264,5105,6), (4265,5096,6), (4265,5124,3286), (4266,5053,6), (4268,4271,6), (4272,5115,6), (4273,5165,6), (4274,5114,6), (4275,5153,6), (4276,4278,58), (4277,5035,6), (4279,5145,6), (4283,5009,6), (4284,5037,6), (4287,5036,6), (4289,5043,6), (4290,5109,6), (4293,5218,495), (4297,5267,5450), (4297,5745,5787), (4313,5435,938), (4652,4855,72), (4652,5440,3927), (5004,5005,2), (5010,5204,3295), (5012,6101,195), (5013,5023,32), (5013,5171,1351), (5013,5206,1384), (5013,5210,3053), (5013,5585,5206), (5023,5210,5206), (5028,5033,1692), (5034,5052,6), (5044,5720,5780), (5054,5076,3628), (5054,5079,632), (5055,5070,1656), (5055,5073,3843), (5056,5059,4), (5056,5067,547), (5058,6567,371), (5059,5067,3845), (5062,6566,372), (5068,5071,5), (5070,5073,381), (5072,5076,546), (5072,5079,5), (5076,5079,381), (5092,5188,3398), (5131,5538,5535), (5204,5217,1), (5237,5238,6), (5237,5352,61), (5238,5351,62), (5277,5283,5275), (5351,5352,6), (5396,5398,6), (5406,5407,394), (5408,5409,394), (5418,6560,485), (5420,6561,486), (5433,6284,11), (5446,5462,51), (5446,5892,5462), (5463,5464,599), (5538,5885,1482), (5590,5870,6214), (5591,5871,6215), (5597,5598,354), (5603,6361,962), (5611,5615,1351), (5657,5744,5771), (5692,5693,5694), (5699,5700,5695), (5703,5758,5761), (5703,5759,5763), (5719,5763,5761), (5719,6147,3487), (5731,5744,5768), (5736,5757,5760), (5737,5786,5788), (5744,5768,5770), (5745,5787,5789), (5791,6245,5789), (5876,5944,156), (5889,5890,6102), (5980,5981,183), (6039,6040,5999), (6120,6123,358), (6121,6124,1135), (6122,6125,1137), (6199,6395,6), (6199,6408,3312), (6199,6417,3311), (6199,6418,6417), (6199,6445,6221), (6199,6446,6395), (6200,6221,6445), (6200,6396,6), (6200,6398,6199), (6200,6409,6455), (6200,6410,3312), (6200,6411,6451), (6200,6412,6398), (6200,6419,6453), (6200,6420,6425), (6200,6450,6417), (6200,6452,6395), (6200,6454,3592), (6200,6456,6418), (6200,6480,6433), (6200,6484,6486), (6200,6485,6431), (6200,6500,6472), (6200,6501,6474), (6221,6396,6395), (6221,6398,6), (6221,6408,6501), (6221,6412,6446), (6221,6428,3592), (6221,6447,6425), (6221,6448,6427), (6221,6449,1151), (6221,6450,3312), (6221,6451,6200), (6221,6452,6398), (6221,6455,6449), (6221,6456,372), (6221,6496,6455), (6221,6497,1152), (6221,6519,6453), (6221,6522,6428), (6395,6407,3311), (6395,6417,6418), (6395,6418,3312), (6395,6445,6199), (6395,6446,6398), (6396,6398,6446), (6396,6409,3311), (6396,6410,6456), (6396,6411,6221), (6396,6412,6452), (6396,6419,6426), (6396,6420,6454), (6396,6449,6418), (6396,6451,6199), (6396,6453,3594), (6396,6455,6417), (6396,6481,6434), (6396,6484,6432), (6396,6485,6487), (6396,6500,6475), (6396,6501,6473), (6398,6407,6500), (6398,6411,6445), (6398,6427,3594), (6398,6447,6428), (6398,6448,6426), (6398,6449,3311), (6398,6450,1152), (6398,6451,6221), (6398,6452,6396), (6398,6455,371), (6398,6456,6450), (6398,6496,1151), (6398,6497,6456), (6398,6519,6427), (6398,6522,6454), (6407,6408,6), (6407,6417,371), (6407,6418,6199), (6407,6445,1151), (6407,6446,6418), (6408,6417,6395), (6408,6418,372), (6408,6445,6417), (6408,6446,1152), (6409,6410,6), (6409,6412,372), (6409,6420,6519), (6409,6426,6425), (6409,6430,6437), (6409,6431,6484), (6409,6432,6480), (6409,6437,6486), (6409,6450,6199), (6409,6452,6418), (6409,6454,6447), (6409,6456,6417), (6409,6471,6468), (6409,6497,6395), (6410,6411,371), (6410,6419,6522), (6410,6425,6426), (6410,6429,6438), (6410,6431,6481), (6410,6432,6485), (6410,6438,6487), (6410,6449,6395), (6410,6451,6417), (6410,6453,6448), (6410,6455,6418), (6410,6470,6469), (6410,6496,6199), (6411,6412,6), (6411,6430,6484), (6411,6434,6480), (6411,6438,6433), (6411,6450,6407), (6411,6452,6199), (6411,6497,6417), (6412,6429,6485), (6412,6433,6481), (6412,6437,6434), (6412,6449,6408), (6412,6451,6395), (6412,6496,6418), (6417,6418,6), (6417,6446,372), (6418,6445,371), (6419,6420,6), (6419,6426,3312), (6419,6427,6417), (6419,6447,6199), (6419,6453,371), (6419,6454,3594), (6419,6522,6395), (6420,6425,3311), (6420,6428,6418), (6420,6448,6395), (6420,6453,3592), (6420,6454,372), (6420,6519,6199), (6421,6424,6), (6422,6423,6), (6425,6426,6), (6425,6427,6199), (6425,6453,6221), (6425,6454,6428), (6425,6522,6418), (6426,6428,6395), (6426,6453,6427), (6426,6454,6398), (6426,6519,6417), (6427,6428,6), (6427,6447,3592), (6427,6448,3312), (6427,6449,6425), (6427,6450,6448), (6427,6454,6395), (6427,6455,6519), (6427,6456,6454), (6427,6519,371), (6428,6447,3311), (6428,6448,3594), (6428,6449,6447), (6428,6450,6426), (6428,6453,6199), (6428,6455,6453), (6428,6456,6522), (6428,6522,372), (6429,6430,6), (6429,6431,371), (6429,6433,1151), (6429,6434,6432), (6429,6483,6428), (6429,6487,3312), (6430,6432,372), (6430,6433,6431), (6430,6434,1152), (6430,6482,6427), (6430,6486,3311), (6431,6432,6), (6431,6434,372), (6431,6438,6432), (6431,6482,6447), (6431,6486,6221), (6432,6433,371), (6432,6437,6431), (6432,6483,6448), (6432,6487,6398), (6433,6434,6), (6433,6437,6480), (6433,6486,6449), (6434,6438,6481), (6434,6487,6450), (6435,6436,6), (6437,6438,6), (6437,6480,6221), (6437,6485,3312), (6438,6481,6398), (6438,6484,3311), (6439,6440,6), (6441,6442,6), (6441,6479,3312), (6442,6478,3311), (6445,6446,6), (6445,6450,6501), (6446,6449,6500), (6447,6448,6), (6447,6449,6519), (6447,6450,3594), (6447,6452,6454), (6447,6454,6418), (6447,6456,6448), (6447,6519,6221), (6447,6522,6420), (6448,6449,3592), (6448,6450,6522), (6448,6451,6453), (6448,6453,6417), (6448,6455,6447), (6448,6519,6419), (6448,6522,6398), (6449,6450,6), (6449,6451,6455), (6449,6452,372), (6449,6455,6200), (6449,6456,3312), (6449,6496,6409), (6449,6497,6398), (6449,6522,6419), (6450,6451,371), (6450,6452,6456), (6450,6455,3311), (6450,6456,6396), (6450,6496,6221), (6450,6497,6410), (6450,6519,6420), (6451,6452,6), (6451,6455,6409), (6451,6456,3311), (6451,6497,3312), (6451,6522,6425), (6452,6455,3312), (6452,6456,6410), (6452,6496,3311), (6452,6519,6426), (6453,6454,6), (6453,6482,6480), (6453,6484,6482), (6453,6519,6407), (6454,6483,6481), (6454,6485,6483), (6454,6522,6408), (6455,6456,6), (6455,6496,6451), (6455,6497,372), (6456,6496,371), (6456,6497,6452), (6465,6466,6467), (6468,6469,6), (6468,6471,6470), (6469,6470,6471), (6470,6471,6), (6472,6473,6), (6472,6474,6221), (6473,6475,6398), (6474,6475,6), (6476,6477,6), (6478,6479,6), (6480,6481,6), (6480,6484,1151), (6480,6486,6484), (6480,6487,6432), (6481,6485,1152), (6481,6486,6431), (6481,6487,6485), (6482,6483,6), (6482,6487,6420), (6483,6486,6419), (6484,6485,6), (6484,6486,6433), (6485,6487,6434), (6486,6487,6), (6488,6489,6), (6490,6491,6), (6492,6493,6), (6494,6495,6), (6494,6499,6435), (6495,6498,6436), (6496,6497,6), (6496,6522,6519), (6497,6519,6522), (6498,6499,6), (6500,6501,6), (6519,6522,6), (6566,6567,1570), (6639,6640,2)