Trilinears           1 : 1 : 1
Barycentrics    a : b : c

X(1) is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by

r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1,       B-excenter = 1 : -1 : 1,       C-excenter = 1 : 1 : -1.

The radii of the excircles are

2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as r_a, r_b, r_c, then 1/r = 1/r_a + 1/r_b + 1/r_c. Moreover,

area(ABC) = sqrt(r*r_a*r_b*r_c) and r_a + r_b + r_c = r + 4R,

where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under "trilinear multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Paul Yiu, Introduction to the Geometry of the Triangle, 2002;
Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;
Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

X(1) lies on these lines:
2,8    3,35    4,33    5,11    6,9    7,20    15,1251    16,1250    19,28    21,31    24,1061    25,1036    29,92    30,79    32,172    39,291    41,101    49,215    60,110    61,203    62,202    64,1439    69,1245    71,579    75,86    76,350    82,560    84,221    87,192    88,100    90,155    99,741    102,108    104,109    142,277    147,150    159,1486    163,293    164,258    166,1488    167,174    168,173    179,1142    181,970    182,983    184,1726    185,296    188,361    190,537    196,207    201,212    204,1712    224,377    227,1465    228,1730    229,267    256,511    257,385    280,1256    281,282    289,363    312,1089    318,1897    320,752    321,964    329,452    335,384    336,811    341,1050    344,1265    346,1219    357,1508    358,1507    364,365    371,1702    372,1703    376,553    378,1063    393,836    394,1711    409,1247    410,1248    411,1254    442,1834    474,1339    475,1861    512,875    513,764    514,663    522,1459    528,1086    561,718    563,1820    564,1048    572,604    573,941    574,1571    594,1224    607,949    631,1000    644,1280    647,1021    650,1643    651,1156    659,891    662,897    672,1002    689,719    704,1502    727,932    731,789    748,756    761,825    765,1052    810,1577    840,1308    905,1734    908,998    921,1800    939,1260    945,1875    947,1753    951,1435    969,1444    971,1419    989,1397    1013,1430    1037,1041    1053,1110    1057,1598    1059,1597    1075,1148    1106,1476    1168,1318    1170,1253    1185,1206    1197,1613    1292,1477    1333,1761    1342,1700    1343,1701    1361,1364    1389,1393    1399,1727    1406,1480    1409,1765    1437,1710    1472,1791    1719,1790    1855,1886    1859,1871    1872,1887

X(1) is the {X(2),X(8)}-harmonic conjugate of X(10).

X(1) = midpoint of X(I) and X(J) for these (I,J): (7,390), (8,145)

X(1) = reflection of X(I) in X(J) for these (I,J): (2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)

X(1) = isogonal conjugate of X(1)
X(1) = isotomic conjugate of X(75)
X(1) = cyclocevian conjugate of X(1029)
X(1) = inverse-in-circumcircle of X(36)
X(1) = inverse-in-Fuhrmann-circle of X(80)
X(1) = inverse-in-Bevan-circle of X(484)
X(1) = complement of X(8)
X(1) = anticomplement of X(10)
X(1) = anticomplementary conjugate of X(1330)
X(1) = complementary conjugate at X(1329)
X(1) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(1) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(1) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)

X(1) = cevapoint of X(I) and X(J) for these (I,J):
(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (78,1490), (244,513)

X(1) = X(I)-cross conjugate of X(J) for these (I,J):
(2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)

X(1) = crosspoint of X(I) and X(J) for these (I,J):
(2,7), (8,280), (21,29), (59,110), (75,92), (81,86)

X(1) = crosssum of X(I) and X(J) for these (I,J):
(2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (207,1490), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)

X(1) = crossdifference of any two points on line X(44)X(513)

X(1) = X(I)-Hirst inverse of X(J) for these (I,J):
(2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292).

X(1) = X(6)-line conjugate of X(44)

X(1) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651,651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)

X(1) = X(I)-beth conjugate of X(J) for these (I,J):
(1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)