Trilinears           a/(b + c - a) : b/(c + a - b) : c/(a + b - c)
                                    = 1 - cos A : 1 - cos B : 1 - cos C
                                    = sin²(A/2) : sin²(B/2) : sin²(C/2)

Barycentrics    a²/(b + c - a) : b²/(c + a - b) : c²/(a + b - c)

The perspector of the tangential triangle and the reflection of the intangents triangle in X(1).

X(56) lies on these lines:
1,3    2,12    4,11    5,499    6,41    7,21    8,404    10,474    19,207    20,497    22,977    25,34    28,278    30,496    31,154    32,1015    33,963    38,201    58,222    61,202    62,203    63,960    72,997    77,1036    78,480    81,959    85,870    87,238    100,145    101,218    105,279    106,109    140,495    181,386    182,611    197,227    212,939    219,579    220,672    223,937    226,405    255,602    266,289    269,738    330,385    376,1058    411,938    511,613    551,553    607,911    631,1056    667,764    946,1012    978,979    1025,1083    1070,1074    1072,1076

X(56) is the {X(1),X(3)}-harmonic conjugate of X(55).

X(56) = midpoint of X(1) and X(46)
X(56) = reflection of X(I) in X(J) for these (I,J): (1479,496)
X(56) = isogonal conjugate of X(8)
X(56) = anticomplement of X(1329)
X(56) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,221), (7,222), (28,34), (57,6), (59,109), (108,513)
X(56) = X(31)-cross conjugate of X(6)
X(56) = crosspoint of X(I) and X(J) for these (I,J): (1,84), (7,278), (28,58), (57,269), (59,109)

X(56) = crosssum of X(I) and X(J) for these (I,J): (1,40), (2,144), (6,197), (9,200), (10,72), (11,522), (55,219), (175,176), (519,1145)

X(56) = crossdifference of any two points on line X(522)X(650)
X(56) = X(I)-Hirst inverse of X(J) for these (I,J): (6,1458), (34,1430), (57,1429), (604,1428), (1416,1438)
X(56) = X(266)-aleph conjugate of X(1050)
X(56) = X(I)-beth conjugate of X(J) for these (I,J):
(1,1), (21,3), (56,1106), (58,56), and (P,57) for all P on the circumcircle