Trilinears           a(b + c - a) : b(c + a - b) : c(a + b - c)
                                    = 1 + cos A : 1 + cos B : 1 + cos C
                                    = cos²(A/2) : cos²(B/2) : cos²(B/2)
                                    = tan(B/2) + tan(C/2) : tan(C/2) + tan(A/2) : tan(A/2) + tan(B/2)

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

X(55) = center of homothety of three triangles:    tangential, intangents, and extangents. Also, X(55) is the pole-with-respect-to-the-circumcircle of the trilinear polar of X(1). These propreties and others are given in

O. Bottema and J. T. Groenman, "De gemene raaklijnen van de vier raakcirkels van een driehoek, twee aan twee," Nieuw Tijdschrift voor Wiskunde 67 (1979-80) 177-182.

X(55) lies on these lines:
1,3    2,11    4,12    5,498    6,31    8,21    9,200    10,405    15,203    16,202    19,25    20,388    30,495    34,227    41,220    43,238    45,678    48,154    63,518    64,73    77,1037    78,960    81,1002    92,243    103,109    104,1000    108,196    140,496    181,573    182,613    183,350    184,215    192,385    199,1030    201,774    204,1033    219,284    226,516    255,601    256,983    329,1005    376,1056    386,595    392,997    411,962    511,611    515,1012    519,956    574,1015    603,963    631,1058    650,884    654,926    748,899    840,901    846,984    869,893    1026,1083    1070,1076    1072,1074

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

X(55) = reflection of X(1478) in X(495)
X(55) = isogonal conjugate of X(7)
X(55) = inverse-in-circumcircle of X(1155)

X(55) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,6), (3,198), (7,218), (8,219), (9,220), (21,9), (59,101), (104,44), (260,259), (284,41)

X(55) = cevapoint of X(42) and X(228) for these (I,J)
X(55) = X(I)-cross conjugate of X(J) for these (I,J): (41,6), (42,33), (228,212)
X(55) = crosspoint of X(I) and X(J) for these (I,J): (1,9), (3,268), (7,277), (8,281), (21,284), (59,101)

X(55) = crosssum of X(I) and X(J) for these (I,J): (1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234), (241,1362), (513,1086), (905,1364), (1361,1465)

X(55) = crossdifference of any two points on line X(241)X(514)
X(55) = X(I)-Hirst inverse of X(J) for these (I,J): (6,672), (43,241)
X(55) = X(1)-line conjugate of X(241)
X(55) = X(I)-beth conjugate of X(J) for these (I,J): (21,999), (55,31), (100,55), (200,200), (643,2), (1021,1024)