Trilinears           b + c - a : c + a - b : a + b - c
                                    = cot(A/2) : cot(B/2) : cot(C/2)

Barycentrics    a(b + c - a) : b(c + a - b) : c(a + b - c)

X(9) is the symmedian point of the excentral triangle.

X(9) lies on these lines:
1,6    2,7    3,84    4,10    5,1729    8,346    21,41    31,612    32,987    33,212    34,201    35,90    38,614       39,978    42,941    43,256    46,79    48,101    55,200    56,1696    58,975    100,1005    164,168    165,910    171,1707    173,177    192,239    223,1073    228,1011    241,269    261,645    294,1253    312,314    318,1896    321,1751    342,653    348,738    364,366    374,517    393,1785    440,1211    478,1038    498,920    522,657    604,1420    607,1039    608,1041    609,1333    644,1320    654,1639    750,896    943,1802    986,1722    991,1818    1088,1223    1125,1732    1174,1621    1249,1712    1377,1703    1378,1702    1479,1752    1571,1574    1572,1573    1678,1705    1679,1704    1680,1701    1681,1700

X(9) is the {X(44),X(45)}-harmonic conjugate of X(1).

X(9) = midpoint of X(I) and X(J) for these (I,J): (7,144), (8,390)
X(9) = reflection of X(I) in X(J) for these (I,J): (1,1001), (7,142)
X(9) = isogonal conjugate of X(57)
X(9) = isotomic conjugate of X(85)
X(9) = complement of X(7)
X(9) = anticomplement of X(142)
X(9) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,1), (8,200), (21,55), (63,40), (190,522), (312,78), (318,33), (329,1490), (333,8)

X(9) = cevapoint of X(I) and X(J) for these (I,J): (1,165), (6,198), (31,205), (37,71), (41,212), (55,220)

X(9) = X(I)-cross conjugate of X(J) for these (I,J):
(6,282), (37,281), (41,33), (55,1), (71,219), (210,8), (212,78), (220,200)

X(9) = crosspoint of X(I) and X(J) for these (I,J): (2,8), (21,333), (63,271), (312,318)
X(9) = crosssum of X(I) and X(J) for these (I,J): (6,56), (19,208), (65,1400), (244,649), (603,604), (1418,1475)
X(9) = crossdifference of any two points on line X(513)X(663)
X(9) = X(I)-Hirst inverse of X(J) for these (I,J): (1,518), (192,239)

X(9) = X(I)-aleph conjugate of X(J) for these (I,J):
(1,43), (2,9), (9,170), (188,165), (190,1018), (366,1), (507,361), (508,57), (509,978)

X(9) = X(I)-beth conjugate of X(J) for these (I,J):
(9,6), (190,6), (346,346), (644,9), (645,75)