Trilinears           sec A : sec B : sec C
                                    = cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB
                                    = cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)
                                    = sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)

Barycentrics    tan A : tan B : tan C

X(4) is the point of concurrence of the altitudes of ABC.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

X(4) lies on these lines:
1,33    2,3    6,53    7,273    8,72    9,10    11,56    12,55    13,61    14,62    15,17    16,18    32,98    35,498    36,499    37,1841    39,232    42,1860    46,90    48,1881    49,156    51,185    52,68    54,184    57,84    64,1853    65,158    67,338    69,76    74,107    78,908    79,1784    80,1825    83,182    93,562    94,143    96,231    99,114    100,119    101,118    102,124    103,116    109,117    110,113    111,1560    120,1292    121,1293    122,1294    123,1295    126,1296    127,1289    128,930    129,1303    130,1298    131,135    137,933    141,1350    145,149    147,148    150,152    155,254    162,270    165,1698    171,601    193,1351    195,399    204,1453    218,294    238,602    240,256    250,1553    252,1487    276,327    279,1565    371,485    372,486    390,495    394,1217    477,1304    487,489    488,490    496,999    512,879    523,1552    542,576    569,1179    572,1474    574,1506    575,598    579,1713    580,1714    590,1151    608,1518    615,1152    616,627    617,628    653,1156    774,1254    801,1092    842,935    937,1534    940,1396    941,1880    953,1309    1036,1065    1037,1067    1038,1076    1039,1096    1040,1074    1160,1162    1161,1163    1251,1832    1329,1376    1340,1348    1341,1349    1385,1538    1430,1468    1499,1550    1715,1730    1716,1721    1717,1718    1726,1782

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

X(4) = midpoint of X(I) and X(J) for these (I,J):
(3,382), (147,148), (149,153), (150,152)

X(4) = reflection of X(I) in X(J) for these (I,J): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303,129), (1350,141), (1593,1595)

X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = inverse-in-circumcircle of X(186)
X(4) = inverse-in-nine-point-circle of X(403)
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
X(4) = anticomplementary conjugate of X(20)
X(4) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(4) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(4) = X(I)-Ceva conjugate of X(J) for these (I,J):
(7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393,459)

X(4) = cevapoint of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)

X(4) = X(I)-cross conjugate of X(J) for these (I,J):
(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)

X(4) = crosspoint of X(I) and X(J) for these (I,J): (2,253), (7,189), (27,286), (92,273)

X(4) = crosssum of X(I) and X(J) for these (I,J):
(4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)

X(4) = crossdifference of any two points on line X(520)X(647)

X(4) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)

X(4) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1047), (29,4)

X(4) = X(I)-beth conjugate of X(J) for these (I,J):
(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)