Trilinears           sin A tan A : sin B tan B : sin C tan C = cos A - sec A : cos B - sec B : cos C - sec C
                                    = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b² + c² - a²)

Barycentrics    sin 2A - 2 tan A : sin 2B - 2 tan B : sin 2C - 2 tan C
                                    = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²/(b² + c² - a²)

Constructed as indicated by the name; also X(25) is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle. Also, X(25) is X(57)-of-the-tangential triangle.

X(25) lies on these lines:
1,1036    2,3    6,51    19,33    31,608    32,1184    34,56    35,1900    36,1878    40,1902    41,42    52,155    53,157    57,1473    58,967    64,1192    65,1452    76,1241    92,242    98,107    100,1862    105,108    110,1112    111,112    114,135    125,1853    132,136    143,156    183,264    185,1498    221,1425    225,1842    226,1892    262,275    273,1447    286,1218    317,325    339,1289    343,1352    371,493    372,494    389,1181    393,1033    394,511    669,878    692,913    694,1613    842,1304    847,1179    941,1172    958,1891    999,1870    1001,1848    1073,1297    1096,1402    1235,1239    1300,1302    1324,1785    1376,1861    1470,1877    1503,1619    1604,1863    1631,1826    1726,1736    1730,1754

X(25) is the {X(5),X(26)}-harmonic conjugate of X(3).

X(25) = reflection of X(I) in X(J) for these (I,J): (4,1596), (1370,1368)
X(25) = isogonal conjugate of X(69)
X(25) = isotomic conjugate of X(305)
X(25) = inverse-in-circumcircle of X(468)
X(25) = inverse-in-orthocentroidal-circle of X(427)
X(25) = complement of X(1370)
X(25) = anticomplement of X(1368)
X(25) = X(I)-Ceva conjugate of X(J) for these (I,J): (4,6), (28,19), (250,112)
X(25) = X(32)-cross conjugate of X(6)
X(25) = crosspoint of X(I) and X(J) for these (I,J): (4,393), (6,64), (19,34), (112,250)
X(25) = crosssum of X(I) and X(J) for these (I,J): (2,20), (3,394), (6,159), (8,329), (63,78), (72,306), (125,525)
X(25) = crossdifference of any two points on line X(441)X(525)
X(25) = X(I)-Hirst inverse of X(J) for these (I,J): (4,419), (6,232)
X(25) = X(I)-beth conjugate of X(J) for these (I,J): (33,33), (108,25), (162,278)