Trilinears           cos(B - C) : cos(C - A) : cos(A - B)
                                    = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C
                                    = g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A - 2 sin B sin C
                                    = h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = bc[a²(b² + c²) - (b² - c²)²]

Barycentrics    a cos(B - C) : b cos(C - A) : c cos(A - B)
                                    = h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a²(b² + c²) - (b² - c²)²

X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

X(5) lies on these lines:
1,11    2,3    6,68    8,1389    9,1729    10,517    13,18    14,17    15,2913    16,2912    32,230    33,1062    34,1060    39,114    40,1698    46,1836    49,54    51,52    53,216    55,498    56,499    57,1728    65,1737    69,1351    72,908    76,262    79,1749    83,98    85,1565    96,1166    113,125    116,118    117,124    122,133    127,132    128,137    129,130    131,136    141,211    142,971    156,184    182,206    183,315    217,1625    225,1465    226,912    252,1157    264,1093    298,634    299,633    302,622    303,621    311,1225    316,1078    339,1235    371,590    372,615    386,1834    388,999    392,1512    491,637    492,638    515,1125    524,576    539,1493    542,575    570,1879    573,1213    578,1147    579,1901    582,1754    601,750    602,748    618,629    619,630    842,1287    920,1454    1073,1217    1090,1091    1155,1770    1173,1487    1181,1899    1214,1838    1498,1853    1848,1871    1861,1872

X(5) is the {X(2),X(4)}-harmonic conjugate of X(3).

X(5) = midpoint of X(I) and X(J) for these (I,J):
(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136)

X(5) = reflection of X(I) in X(J) for these (I,J): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)

X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = inverse-in-circumcircle of X(2070)
X(5) = inverse-in-orthocentroidal-circle of X(3)
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
X(5) = complementary conjugate of X(3)
X(5) = eigencenter of anticevian triangle of X(523)

X(5) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,216), (4,52), (110,523), (264,324), (265,30), (311,343), (324,53)

X(5) = cevapoint of X(I) and X(J) for these (I,J): (3,195), (51,216)

X(5) = X(I)-cross conjugate of X(J) for these (I,J): (51,53), (216,343), (233,2)
X(5) = crosspoint of X(I) and X(J) for these (I,J): (2,264), (311,324)
X(5) = crosssum of X(I) and X(J) for these (I,J): (3,1147), (6,184)
X(5) = crossdifference of any two points on line X(50)X(647)
X(5) = X(1)-aleph conjugate of X(1048)