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

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)).

X(3) lies on 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    47,1399    48,71    49,155    54,97    60,1175    63,72    64,154    66,141    67,542    68,343    69,332    73,212    74,110    76,98    77,1410    83,262    86,1246    90,1898    95,264    101,103    102,109    105,277    106,1293    107,1294    108,1295    111,1296    112,1297    113,122    114,127    119,123    125,131    128,1601    142,516    143,1173    145,1483    149,1484    158,243    161,1209    169,910    191,1768    193,1353    194,385    200,963    201,1807    207,1767    223,1035    225,1074    227,1455    238,978    252,930    256,987    269,939    296,820    298,617    299,616    302,621    303,622    305,1799    315,325    345,1791    347,1119    348,1565    352,353    388,495    390,1058    393,1217    395,398    396,397    476,477    485,590    486,615    489,492    490,491    496,497    525,878    595,995    611,1469    613,1428    618,635    619,636    623,629    624,630    639,641    640,642    653,1148    662,1098    667,1083    691,842    695,1613    847,925    901,953    902,1201    917,1305    920,1858    934,972    945,1457    950,1210    951,1407    955,1170    960,997    962,1621    1000,1476    1033,1249    1037,1066    1054,1283    1055,1334    1057,1450    1093,1105    1167,1413    1177,1576    1180,1627    1184,1194    1196,1611    1298,1303    1331,1797    1364,1795    1397,1682    1398,1870    1406,1464    1411,1772    1427,1448    1452,1905    1728,1864    1737,1837    1770,1836    1779,1780

X(3) is the {X(2),X(4)}-harmonic conjugate of X(5).

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) = inverse-in-nine-point-circle of X(2072)
X(3) = inverse-in-orthocentroidal-circle of X(5)
X(3) = inverse-in-1st-Lemoine-circle of X(2456)
X(3) = inverse-in-2nd-Lemoine-circle of X(1570)
X(3) = complement of X(4)
X(3) = anticomplement of X(5)
X(3) = complementary conjugate of X(5)
X(3) = eigencenter of the medial triangle
X(3) = eigencenter of the tangential triangle

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 any two points on 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)