Trilinears           a : b : c
                                    = sin A : sin B : sin C

Barycentrics    a² : b² : c²

X(6) is the point of concurrence of the symmedians (reflections of medians in corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x² + y² + z².

Let A'B'C' be the pedal triangle of an arbitrary point X, and let S(X) be the vector sum XA' + XB' + XC'. Then

S(X) = (0 vector) if and only if X = X(6).

The "if" implication is equivalent to the well known fact that X(6) is the centroid of its pedal triangle, and the converse was proved by Barry Wolk (Hyacinthos #19, Dec. 23, 1999).

X(6) is the radical trace of the 1st and 2nd Lemoine circles. (Peter J. C. Moses, 8/24/03)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 7: The Symmedian Point.

X(6) lies on these lines:
1,9    2,69    3,15    4,53    5,68    7,294    8,594    10,1377    13,14    17,18    19,34    21,941    22,251    23,353    24,54    25,51    26,143    27,1246    31,42    33,204    36,609    40,380    41,48    43,87    57,222    60,1169    64,185    66,427    67,125    70,1594    74,112    75,239    76,83    77,241    88,89    98,262    99,729    100,739    101,106    105,1002    110,111    145,346    157,248    160,237    162,1013    169,942    181,197    190,192    194,384    210,612    226,1751    256,1580    264,287    274,1218    279,1170    281,1146    282,1256    291,985    292,869    297,317    305,1241    314,981    330,1258    344,1332    354,374    442,1714    493,1583    494,1584    513,1024    517,998    519,996    523,879    560,1631    561,720    588,1599    589,1600    593,1171    595,1126    598,671    603,1035    644,1120    657,1459    662,757    688,882    689,703    691,843    692,1438    694,1084    706,1502    717,789    750,899    753,825    755,827    840,919    846,1051    893,1403    909,1415    911,1461    939,1802    943,1612    947,1622    959,961    963,1208    967,1790    971,990    986,1046    1096,1859    1112,1177    1131,1132    1139,1140    1166,1601    1173,1614    1174,1617    1195,1399    1201,1696    1214,1708    1327,1328    1362,1416    1399,1425    1423,1429    1718,1781    1826,1837    1836,1839    1854,1858

X(6) is the {X(15),X(16)}-harmonic conjugate of X(3).

X(6) = midpoint of X(69) and X(193)
X(6) = reflection of X(I) in X(J) for these (I,J): (1,1386), (2,597), (3,182), (67,125), (69,141), (159,206), (182,575), (592,2), (694,1084), (1350,3), (1351,576), (1352,5)

X(6) = isogonal conjugate of X(2)
X(6) = isotomic conjugate of X(76)
X(6) = cyclocevian conjugate of X(1031)
X(6) = inverse-in-circumcircle of X(187)
X(6) = inverse-in-orthocentroidal-circle of X(115)
X(6) = inverse-in-1st-Lemoine-circle of X(1691)
X(6) = complement of X(69)
X(6) = anticomplement of X(141)
X(6) = anticomplementary conjugate of X(1369)
X(6) = complementary conjugate of X(1368)

X(6) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,55), (2,3), (3,154), (4,25), (7,1486), (8,197), (9,198), (10,199), (54,184), (57,56), (58,31), (68,161), (69,159), (74,1495), (76,22), (81,1), (83,2), (88,36), (95,160), (98,237), (110,512), (111,187), (222,221), (249,110), (251,32), (264,157), (275,4), (284,48), (287,1503), (288,54), (323,399), (394,1498)

X(6) = cevapoint of X(I) and X(J) for these (I,J): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217)

X(6) = X(I)-cross conjugate of X(J) for these (I,J):
(25,64), (31,56), (32,25), (39,2), (41,55), (42,1), (51,4), (184,3), (187,111), (213,31), (217,184), (237,98), (512,110)

X(6) = crosspoint of X(I) and X(J) for these (I,J):
(1,57), (2,4), (9,282), (54,275), (58,81), (83,251), (110,249), (266,289)

X(6) = crosssum of X(I) and X(J) for these (I,J):
(1,9), (3,6), (4,1249), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (56,478), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (142,1212), (188,236), (226,1214), (244,661), (395,619), (396,618), (512,1084), (513,1015), (514,1086), (522,1146), (570,1209), (577,1147), (590,641), (615,642), (1125,1213), (1196,1368)

X(6) = crossdifference of any two points on line X(30)X(511)

X(6) = X(I)-Hirst inverse of X(J) for these (I,J): (1,238), (2,385), (3,511), (15,16), (25,232), (56,1458), (58,1326), (523,1316), (1423,1429)

X(6) = X(I)-line conjugate of X(J) for these (I,J): (1,518), (2,524), (3,511)
X(6) = X(I)-aleph conjugate of X(J) for these (I,J): (1,846), (81,6), (365,1045), (366,191), (509,1046)

X(6) = X(I)-beth conjugate of X(J) for these (I,J):
(6,604), (9,9), (21,1001), (101,6), (284,6), (294,6), (644,6), (645,6), (651,6), (652,7), (666,6)