Trilinears           1 - cos(B - C) : 1 - cos(C - A) : 1 - cos(A - B)
                                    = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin²(B/2 - C/2)
                                    = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc(b + c - a)(b - c)²

Barycentrics    a(1 - cos(B - C)) : b(1 - cos(C - A)) : c(1 - cos(A - B))
                                    = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(A,B,C) = (b + c - a)(b - c)²

X(11) is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle. Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles.

X(11) is the {X(1),X(5)}-harmonic conjugate of X(12).

X(11) lies on these lines:
1,5    2,55    3,499    4,56    7,658    8,1320    10,121    13,202    14,203    28,1852    30,36    33,427    34,235    35,140    57,1360    65,117    68,1069    110,215    113,942    115,1015    118,226    124,1364    182,1848    133,1838    153,388    212,748    214,442    244,867    278,1857    325,350    381,999    403,1870    429,1104    485,1124    486,1335    498,1656    515,1319    516,1155    517,1737    518,908    523,1090    613,1352    650,1566    774,1393    944,1388    962,1788    971,1538    1012,1470    1040,1368    1111,1358    1146,1639    1193,1834    1427,1856    1428,1503    1455,1877    1500,1506    1697,1698

X(11) = midpoint of X(I) and X(J) for these (I,J): (1,80), (4,104), (5,1484), (100,149)
X(11) = reflection of X(I) in X(J) for these (I,J): (1,1387), (119,5), (214,1125), (1145,10), (1317,1), (1537,946)
X(11) = isogonal conjugate of X(59)
X(11) = inverse-in-Furhmann-circle of X(1837)
X(11) = complement of X(100)
X(11) = anticomplement of X(3035)
X(11) = complementary conjugate of X(513)
X(11) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,523), (4,513), (7,514), (8,522), (262,1491)
X(11) = crosspoint of X(I) and X(J) for these (I,J): (7,514), (8,522)
X(11) = crosssum of X(I) and X(J) for these (I,J): (6,692), (55,101), (56,109), (1381,1382), (1397,1415)
X(11) = crossdifference of any two points on line X(101)X(109)
X(11) = X(I)-beth conjugate of X(J) for these (I,J): (11,244), (522,11), (693,11)