ENCYCLOPEDIA OF TRIANGLE CENTERS

PART 6

X(1001)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a² - a(b + c) - 2bc
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1001) lies on these lines:
1,6   2,11   3,142   7,21   8,344   31,940   35,474   42,748   63,354   182,692   388,452   527,551   529,1056   614,968   750,902   846,982   943,1058
X(1001) = isogonal conjugate of X(1002)

X(1002)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a² - a(b + c) - 2bc]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1002) lies on these lines:
1,672   2,210   6,105   8,274   28,607   42,57   55,81   65,279   145,330   277,942
X(1002) = isogonal conjugate of X(1001)

X(1003)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc(3a⁴ - a²b² - a²c² + 2b²c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1003) lies on these lines: 2,3   6,99   32,538   183,187

X(1004)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁵ - 2a⁴(b + c) + 2a²(b³ + c³) - a(b² + c²)² + 2bc(b - c)(b² - c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1004) lies on these lines: 2,3   7,100   46,200   63,210   65,224

X(1005)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁵ - 2a⁴(b + c) - a³bc + a²(2b³ + 2c³ + b²c + bc²) - a(b⁴ + c⁴ - b³c - bc³ - 4b²c²) + bc(b - c)(b² - c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1005) lies on these lines: 2,3   9,100   55,329   108,342

X(1006)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁶ - a⁵(b + c) - a⁴(2b² + bc + 2c²)
                                              + 2a³(b³ + c³) + a²[b⁴ + c⁴ + 2bc(b² + c²) + 2b²c²]
                                              - a[(b⁵ + c⁵) - bc(b³ + c³)] - bc(b² - c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1006) lies on these lines:
1,201   2,3   9,48   35,950   36,226   54,72   238,1064   944,958   954,999

X(1007)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc[a⁴ - 4a²(b² + c²) + 3b⁴ - 2b²c² + 3c⁴]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1007) lies on these lines: 2,6   4,99   305,311   315,631   316,376   317,459

X(1008)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc[a⁵(b + c) + a⁴(b + c)² + a³(b + c)(b² + bc + c²)
                                              + a²(b² + c² + bc)² + abc(b + c)³ + b²c²(b + c)²]
                                              - a(b⁵ + c⁵) - bc(b³ + c³)) - bc(b² - c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1008) lies on these lines: 1,76   2,3

X(1009)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁴(b + c) + 2a³bc - a²(b³ + c³) + bc(b + c)(b² + c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1009) lies on these lines: 1,39   2,3   72,894   283,1065   518,583

X(1010)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc[a² + (b + c)²]/(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1010) lies on these lines:
1,75   2,3   8,81   10,58   72,894   283,1065   312,975   759,833

X(1011)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a[a³(b + c) + a²bc - a(b + c)(b² + c²) - bc(b + c)²]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1011) lies on these lines: 2,3   6,31   9,228   35,43   51,573   184,572

X(1012)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁷ - a⁵r + 2a⁴bc(b + c) + a³s - a(b + c)²(b² - c²)² - 2bc(b + c)(b² - c²)²,
                        where r = 3b² - 2bc + 3c² and s = 3b⁴ + 2b²c² + 3c⁴
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1012) lies on these lines:
1,84   2,3   7,104   40,958   55,515   56,946   63,517   268,281   516,993   954,971

X(1013)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁸ - a⁷u₁ - a⁶u₂ + a⁵u₁u₂ - a⁴(b⁴ - 4b²c² + c⁴)
                                       + a³u₁u₃ + a²u₂u₃ - au₁u₂u₃ - 2b²c²u₃,
                                       u₁ = b + c, u₂ = b² + c², u₃ = (b² - c²)²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1013) lies on these lines: 2,3   6,162   7,108   33,63   55,92   100,281

X(1014)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[(b + c)(b + c - a)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1014) lies on these lines: 7,21   28,279   57,77   58,269   60,757   69,404   261,552   272,1088   274,961   332,1037   759,934

X(1015)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin A sin²(A/2) [1 - cos(B - C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A
X(1015) lies on these lines:
1,39   2,668   6,101   11,115   32,56   36,187   37,537   55,574   76,330   214,1100   216,1060   244,665   350,538   812,1086
X(1015) = isogonal conjugate of X(1016)
X(1015) = complement of X(668)

X(1016)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/[sin A sin²(A/2) [1 - cos(B - C)]]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = f(A,B,C) sin A
X(1016) lies on these lines:
8,1083   99,813   100,667   190,514   238,519   512,660   644,666
X(1016) = isogonal conjugate of X(1015)
X(1016) = complement of X(1086)

X(1017)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(b + c - 2a)²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1017) lies on these lines: 6,101   44,214

X(1018)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b + c)/(b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1018) lies on these lines:
1,39   9,80   40,728   63,544   99,813   100,101   163,643   190,646   346,573   519,672   664,1025
X(1018) = isogonal conjugate of X(1019)

X(1019)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b - c)/(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1019) lies on these lines:
1,512   36,238   58,1027   81,1022   99,813   239,514   759,840
X(1019) = isogonal conjugate of X(1018)

X(1020)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (cos B + cos C)/(cos B - cos C)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1020) lies on these lines:
1,185   57,1986   101,651   108,109   190,658   269,292   347,573   648,1021
X(1020) = isogonal conjugate of X(1021)

X(1021)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (cos B - cos C)/(cos B + cos C)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1021) lies on these lines:
1,647   239,514   243,522   333,1024   521,650   648,1020
X(1021) = isogonal conjugate of X(1020)

X(1022)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b - c)/(2a - b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1022) lies on these lines:
1,513   2,514   81,1019   89,649   105,106   291,876   812,903
X(1022) = isogonal conjugate of X(1023)

X(1023)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (2a - b - c)/(b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1023) lies on these lines: 1,6   100,101   813,898
X(1023) = isogonal conjugate of X(1022)

X(1024)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b - c)(b + c - a)/[b² + c² - a(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1024) lies on these lines:
6,513   9,522   55,650   57,649   333,1021   673,812
X(1024) = isogonal conjugate of X(1025)

X(1025)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = [b² + c² - a(b + c)]/[(b - c)(b + c - a)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1025) lies on these lines:
2,7   56,1083   100,109   190,658   644,934   664,1018   813,927
X(1025) = isogonal conjugate of X(1024)

X(1026)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = [b² + c² - a(b + c)]/(b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1026) lies on these lines: 1,2   55,1083   100,101   664,668   666,1027
X(1026) = isogonal conjugate of X(1027)

X(1027)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (b - c)/[b² + c² - a(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1027) lies on these lines: 1,514   6,513   56,667   58,1019   105,106   292,659   666,1026
X(1027) = isogonal conjugate of X(1026)

X(1028)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 2a²bc + [(b² - c²)² - a⁴]/(b + c - a)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1028) is the center of the conic that passes through the six points of tangency of the excircles with the sidelines of triangle ABC. (Paul Yiu, "The Clawson Point and Excircles," Dec., 1999.)
X(1028) lies on this line: 6,19

X(1029)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a + 2(a + b + c) cos A]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1029) lies on these lines: 10,191   115,593   319,321
X(1029) = isogonal conjugate of X(1030)
X(1029) = cyclocevian conjugate of X(1)

X(1030)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a + 2(a + b + c) cos A
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1030) lies on these lines:
3,6   35,37   36,1100   45,198   55,199   100,594
X(1030) = isogonal conjugate of X(1029)

X(1031)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a(w²u² + u²v² - v²w²) + 2uvw(au + bv + cw) cos A],
                        where u : v : w = X(6)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1031) lies on this line: 141,384
X(1031) = cyclocevian conjugate of X(6)

X(1032)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a(w²u² + u²v² - v²w²) + 2uvw(au + bv + cw) cos A],
                        where u : v : w = X(20)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1032) lies on this line: 20,394
X(1032) = isogonal conjugate of X(1033)
X(1032) = cyclocevian conjugate of X(20)

X(1033)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(w²u² + u²v² - v²w²) + 2uvw(au + bv + cw) cos A,
                        where u : v : w = X(20)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1033) lies on these lines: 6,64   19,56   25,393   55,204
X(1033) = isogonal conjugate of X(1032)

X(1034)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 1/[a(w²u² + u²v² - v²w²) + 2uvw(au + bv + cw) cos A],
                        where u : v : w = X(329)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1034) lies on these lines: 2,271   20,78
X(1034) = isogonal conjugate of X(1035)
X(1034) = cyclocevian conjugate of X(329)

X(1035)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(w²u² + u²v² - v²w²) + 2uvw(au + bv + cw) cos A,
                        where u : v : w = X(329)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1035) lies on these lines: 3,223   6,603   25,34   55,64   222,581
X(1035) = isogonal conjugate of X(1034)

X(1036)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(1 + cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1036) lies on these lines:
1,25   3,31   4,1065   21,332   29,497   41,219   55,78   56,77   73,1037   282,380   581,947   1058,1067   1059,1066
X(1036) = isogonal conjugate of X(388)

X(1037)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(1 - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1037) lies on these lines:
1,1041   3,1066   4,1067   29,388   48,949   55,77   56,78   73,1036   219,604   332,1014   916,1069   1056,1065   1057,1064
X(1037) = isogonal conjugate of X(497)

X(1038)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A + cos A cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1038) lies on these lines:
1,3   2,34   4,1076   9,478   20,33   21,1041   38,1106   63,210   69,73   72,222   172,577   221,960   223,936   225,377   226,975   278,443   388,612   1068,1074
X(1038) = isogonal conjugate of X(1039)

X(1039)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(cos A + cos A cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1039) lies on these lines:
1,25   4,1096   7,34   8,33   9,607   21,1040   29,314   65,1041   943,968
X(1039) = isogonal conjugate of X(1038)

X(1040)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A - cos A cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1040) lies on these lines:
1,3   2,33   4,1074   20,34   21,1039   63,212   78,345   226,990   243,1096   497,614   1068,1076
X(1040) = isogonal conjugate of X(1041)

X(1041)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(cos A - cos A cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1041) lies on these lines:
1,1037   7,33   8,34   9,608   19,294   21,1038   65,1039
X(1041) = isogonal conjugate of X(1040)

X(1042)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (1 - cos A)(cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1042) lies on these lines:
1,7   31,56   34,207   42,65   57,959   241,960   517,1066   604,608   741,934   942,1064
X(1042) = isogonal conjugate of X(1043)

X(1043)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/[(1 - cos A)(cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1043) lies on these lines:
1,75   8,21   20,64   27,306   29,33   58,519   72,190   81,145   99,103   200,341   220,346   239,1104   280,285   283,643   286,322
X(1043) = isogonal conjugate of X(1042)

X(1044)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos B + cos C - cos A + cos B cos C - cos A cos B - cos A cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1044) lies on these lines: 1,7   43,46

X(1045)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1045) lies on these lines: 1,75   9,43   40,511   42,894   190,872   192,869

X(1046)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(3)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1046) lies on these lines:
1,21   6,986   10,894   40,511   43,46   57,978   72,171   238,942   484,1048

X(1047)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1047) lies on these lines: 1,29   43,46

X(1048)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(5)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1048) lies on these lines: 1,564   484,1046

X(1049)
Trilinears        A : B : C
Barycentrics  A sin A : B sin B : C sin C
X(1049) = isogonal conjugate of X(1085)

X(1050)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(8)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1050) lies on these lines: 1,341   40,978

X(1051)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(37)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1051) lies on these lines: 1,748   6,846   81,1054   165,572

X(1052)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(100)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1052) lies on these lines: 1,765   238,517   513,1054

X(1053)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(101)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1053) lies on these lines: 1,1110   238,517   905,1054

X(1054)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = -u² + v² + w² + vw + wu + uv, u : v : w = X(513)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1054) lies on these lines:
1,88   2,846   43,57   46,978   81,1051   105,165   474,986   513,1052   905,1053

X(1055)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(b² + c² - 2a² + ab + ac - 2bc)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1055) lies on these lines: 6,41   36,101   106,919   187,237   609,995

X(1056)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 2 + cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1056) lies on these lines:
1,4   2,495   7,517   8,443   29,1059   30,390   55,376   56,631   145,377   329,392   355,938   529,1001   1037,1065
X(1056) = isogonal conjugate of X(1057)

X(1057)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(2 + cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1057) lies on these lines:
29,1058   73,1059   77,999   78,392   497,1065   1037,1064
X(1057) = isogonal conjugate of X(1056)

X(1058)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 2 - cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1058) lies on these lines:
1,4   2,496   3,390   8,392   20,999   20,1057   55,631   56,376   149,377   452,956   517,938   942,962   943,1001   1036,1067
X(1058) = isogonal conjugate of X(1059)

X(1059)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(2 - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1059) lies on these lines: 29,1056   73,1057   78,999   388,1067   1036,1066
X(1059) = isogonal conjugate of X(1058)

X(1060)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 2 + sec B sec C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1060) lies on these lines:
1,3   5,34   21,1063   30,33   68,73   72,394   141,997   201,255   216,1015   222,912   377,1068   495,612   601,774   976,1066
X(1060) = isogonal conjugate of X(1061)

X(1061)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(2 + sec B sec C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1061) lies on these lines:
1,24   8,406   21,1062   33,80   34,79   65,1063
X(1061) = isogonal conjugate of X(1060)

X(1062)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 2 - sec B sec C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1062) lies on these lines:
1,3   5,33   21,1061   30,34   394,1069   496,614   602,774
X(1062) = isogonal conjugate of X(1063)

X(1063)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(2 - sec B sec C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1063) lies on these lines:
1,378   8,475   21,1060   33,79   34,80   65,1061
X(1063) = isogonal conjugate of X(1062)

X(1064)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1 + cos A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1064) lies on these lines:
1,4   3,31   38,912   40,386   42,517   102,112   104,256   238,1006   631,978   942,1042   991,995   1037,1057
X(1064) = isogonal conjugate of X(1065)

X(1065)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/[1 + cos A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1065) lies on these lines:
3,388   4,1036   102,226   283,1010   284,515   497,1057   1037,1056
X(1065) = isogonal conjugate of X(1064)

X(1066)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1 - cos A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1066) lies on these lines: 1,4   3,1037   42,942   222,601   517,1042   774,912   947,951   976,1060   1036,1059
X(1066) = isogonal conjugate of X(1067)

X(1067)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/[1 - cos A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1067) lies on these lines:
3,496   4,1037   388,1059   946,951   947,950   1036,1058
X(1067) = isogonal conjugate of X(1066)

X(1068)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1 - sec A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1068) lies on these lines:
1,4   8,860   24,108   92,406   155,651   281,451   318,475   377,1060   429,495   1038,1074   1040,1076
X(1068) = isogonal conjugate of X(1069)

X(1069)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/[1 - sec A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1069) lies on these lines: 1,90   11,68   394,1062   496,613   916,1037
X(1069) = isogonal conjugate of X(1068)

X(1070)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1 + cos B cos C (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1070) lies on these lines: 1,4   55,1076   56,1074

X(1071)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a(b² + c² - a²)[a³(b + c) - a²(b - c)² + (b² - c²)²]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1071) appears in Hyacinthos message #3849, Paul Yiu, Sept. 19, 2001.

X(1072)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1 - cos²B cos C - cos B cos²C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1072) lies on these lines: 1,4   55,1074   56,1076

X(1073)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (cot A)/(cos A - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1073) lies on these lines: 2,253   3,64   9,223   222,268

X(1074)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A + cos²B cos C + cos B cos²C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1074) lies on these lines: 1,224   3,225   4,1040   55,1072   56,1070   1038,1068

X(1075)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos B cos C (cos²C cos²A + cos²A cos²B - cos²B cos²C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1075) lies on these lines: 4,15   155,450   216,631   243,920   648,1092

X(1076)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A - cos²B cos C - cos B cos²C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1076) lies on these lines: 3,225   4,1038   55,1070   56,1072   1040,1068

X(1077)
Trilinears        1/A : 1/B : 1/C
Barycentrics  (sin A)/A : (sin B)/B : (sin C)/C

X(1078) = REFLECTION OF X(64) IN X(3)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A - 1/(cos A - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1078) lies on these lines:
1,84   3,64   4,6   20,394   25,185   30,155   40,219   195,382

X(1079) = REFLECTION OF X(84) IN X(3)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos A - 1/(cos B + cos C - cos A - 1)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1079) lies on these lines: 1,4   3,9   20,78   40,64   63,411   165,191   224,908   386,990   975,991   1045,1047

X(1080)
Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where
                        f(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C - π/3) - sin 2C cos(B - ω) sin(B - π/3)]
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Coordinates for X(1080) are obtained from those of X(383) by changing π/3 to - π/3; contributed by Edward Brisse.
X(1080) lies on these lines: 2,3   13,98   14,262   183,622   298,511   325,621
X(1080) = inverse of X(383) in the orthocentroidal circle

X(1081)
Trilinears       sec(A/2) csc(A/2 - π/3) : sec(B/2) csc(B/2 - π/3) : sec(C/2) csc(C/2 - π/3)
Barycentrics  sin A sec(A/2) csc(A/2 - π/3) : sin B sec(B/2) csc(B/2 - π/3) : sin C sec(C/2) csc(C/2 - π/3)
Coordinates for X(1081) are obtained from those of X(554) by changing π/3 to - π/3; contributed by Edward Brisse.
X(1081) lies on these lines: 1,30   7,559   13,226   75,298

X(1082)
Trilinears       (sec A/2) sin(A/2 - π/3) : (sec B/2) sin(B/2 - π/3) : (sec C/2) sin(C/2 - π/3)
Barycentrics  (sin A/2) sin(A/2 - π/3) : (sin B/2) sin(B/2 - π/3) : (sin C/2) sin(C/2 - π/3)
Coordinates for X(1082) are obtained from those of X(559) by changing π/3 to - π/3; contributed by Edward Brisse.
X(1082) lies on these lines: 1,3   7,554   13,226   298,319

X(1083)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a⁴ - a³(b + c) - a²bc + 2abc(b + c) - bc(b² + c²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1083) lies on a circle related to the 1st and 2nd Brocard points; Hyacinthos message #4053, Paul Yiu, Oct. 4, 2001.
X(1083) lies on these lines:
1,6   3,667   8,1016   55,1026   56,1025   105,644   840,898
X(1083) = inverse of X(667) in the orthocentroidal circle

X(1084)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a³(b² - c²)²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(99).
X(1084) lies on these lines: 2,670   6,694   39,597   115,804   351,865

X(1085)
Trilinears        A² : B² : C²
Barycentrics  A² sin A : B² sin B : C² sin C

X(1086)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = bc(b - c)²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)²
The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(101).
X(1086) lies on these lines:
1,528   2,45   6,7   8,599   10,537   11,244   37,142   44,527   53,273   57,1020   75,141   115,116   220,277   239,320   812,1015   918,1111
X(1086) = isotomic conjugate of X(1016)
X(1086) = complement of X(190)

X(1087) = TRILINEAR SQUARE OF X(5)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos²(B - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1087) lies on these lines: 1,564   31,91   92,255

X(1088) = TRILINEAR SQUARE OF X(7)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sec⁴(A/2)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1088) lies on these lines:
2,85   7,354   57,658   86,269   234,555   272,1014   305,341   675,934
X(1088) = isotomic conjugate of X(200)

X(1089) = TRILINEAR SQUARE OF X(10)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = b²c²(b + c)²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1089) lies on these lines:
1,312   8,80   10,321   76,334   190,191   200,318   244,596   345,498   594,762   740,872
X(1089) = isogonal conjugate of X(849)
X(1089) = isotomic conjugate of X(757)

X(1090) = TRILINEAR SQUARE OF X(11)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = [1 - cos(B - C)]²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1090) lies on these lines: 5,1091   11,523

X(1091) = TRILINEAR SQUARE OF X(121)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = [1 + cos(B - C)]²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1091) lies on these lines: 5,1090   12,1109

X(1092) = TRILINEAR CUBE OF X(3)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cos³A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1092) lies on these lines:
2,578   3,49   4,801   20,110   24,511   54,69   68,125   140,343   156,550   450,1093   648,1075
X(1092) = isogonal conjugate of X(1093)

X(1093) = TRILINEAR CUBE OF X(4)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sec³A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1093) lies on these lines:
3,1105   4,51   5,264   24,107   155,648   158,225   393,800   403,847   436,578   450,1092
X(1093) = isogonal conjugate of X(1092)

X(1094) = TRILINEAR SQUARE OF X(15)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin²(A + π/3)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1094) lies on these lines: 15,36   48,163

X(1095) = TRILINEAR SQUARE OF X(16)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin²(A - π/3)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1095) lies on these lines: 16,36   48,163

X(1096) = TRILINEAR SQUARE OF X(19)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = tan²A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1096) lies on these lines:
1,29   4,1039   19,31   33,42   34,207   63,240   107,741   213,607   243,1040   278,614   281,612
X(1096) = isogonal conjugate of X(326)

X(1097) = TRILINEAR SQUARE OF X(20)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (cos A - cos B cos C)²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1097) lies on these lines: 1,75   31,775

X(1098) = TRILINEAR SQUARE OF X(21)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = 1/(cos B - cos C)²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1098) lies on these lines:
3,662   8,643   21,60   29,270   58,86   65,409   81,1104

X(1099) = TRILINEAR SQUARE OF X(30)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (cos A - 2 cos B cos C)²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1099) lies on these lines: 1,564   75,811   162,255

X(1100)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 2avw + cv² + bw² + u(bv + cw), u : v : w = X(37)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1100) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(37)
X(1100) lies on these lines:
1,6   2,319   36,1030   48,354   65,604   71,583   81,593   86,239   214,1015   284,942   517,572   519,594   536,894   820,836
X(1100) = complement of X(319)

X(1101) = TRILINEAR SQUARE OF X(110)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = csc²(B - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1101) lies on these lines: 59,60   163,798   656,662
X(1101) = isogonal conjugate of X(1109)

X(1102) = TRILINEAR CUBE OF X(63)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = cot³A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1102) lies on these lines: 63,304   255,326

X(1103) = TRILINEAR SQUARE OF X(40)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (cos B + cos C - cos A - 1)²A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1103) lies on these lines: 1,2   31,937   40,221   46,269   165,255

X(1104)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = 2avw + cv² + bw² + u(bv + cw), u : v : w = X(72)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1104) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(72)
X(1104) lies on these lines:
1,6   11,429   25,34   31,65   32,910   58,942   81,1098   105,961   210,976   229,593   239,1043   440,950   517,580   581,995

X(1105)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (sec A)/(cos²B + cos²C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1105) lies on these lines: 3,1093   4,801   20,393   185,648   225,412   243,411   378,847
X(1105) = isogonal conjugate of X(185)

X(1106) = TRILINEAR SQUARE OF X(56)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = (1 + cos A)²
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1106) lies on these lines:
3,1037   7,987   31,56   32,604   34,244   36,255   38,1038   57,961   58,269   77,988   279,985   388,750   601,999   651,978   727,934
X(1106) = isogonal conjugate of X(341)

X(1107)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2avw + cv² + bw² + u(bv + cw),
                        where u : v : w = X(213)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1107) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(213)
X(1107) lies on these lines: 1,6   2,330   10,39   32,993   75,194   210,869   239,257

X(1108)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2avw + cv² + bw² + u(bv + cw),
                        where u : v : w = X(219)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1108) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(219)
X(1108) lies on these lines: 1,6   2,322   19,56   104,112   241,347   278,393   517,579
X(1108) = complement of X(322)

X(1109) = TRILINEAR SQUARE OF X(523)
Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
                        where f(A,B,C) = sin²(B - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1109) lies on these lines:
1,564   11,523   12,1091   31,92   75,799   91,255
X(1109) = isogonal conjugate of X(1101)

X(1110) = TRILINEAR SQUARE OF X(101)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = [a/(b - c)]²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1110) lies on these lines:
1,1053   36,59   101,663   249,849   667,692
X(1110) = isogonal conjugate of X(1111)

X(1111) = TRILINEAR SQUARE OF X(514)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = [(b - c)/a]²
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1111) lies on these lines:
1,85   7,80   75,537   76,334   269,273   348,499   918,1086
X(1111) = isogonal conjugate of X(1110)
X(1111) = isotomic conjugate of X(765)

X(1112)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = a[a⁴(b² + c²) - 2a²(b⁴ + c⁴) + b⁶ + c⁶ ]/(b² + c² - a²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1112) is the center of the conic that passes through the vertices of the cevian triangles of X(4) and X(648), and also through the centers X(i) for i = 4, 113, 155, 193. (Paul Yiu, Oct. 16, 2001, as contributing editor for "Conics associated with a cevian nest," Forum Geometricorum 1 (2001) 141-150; see Example 2.)
X(1112) lies on these lines:
4,94   25,110   51,125   52,113   389,974   428,542   468,511

X(1113)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (a'/a")^1/3,
                        a' = [ca(a² - c² + bd][ba(a² - b²) + cd],
                        a" = [(b² - c²)(b² + c² - a²)]²,
                        d = (a⁶ + b⁶ + c⁶ + 3a²b²c² - S)^1/2,
                        S = b²c²(b² + c²) + c²a²(c² + a²) + a²b²(a² + b²)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1113) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1114).
X(1113) lies on this line: 2,3

X(1114)
Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
                        where f(a,b,c) = (a'/a")^1/3, where
                        a' = [ca(a² - c² - bd][ba(a² - b²] - cd],
                        a" and d as for X(1113)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)
X(1114) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1113).
X(1114) lies on this line: 2,3

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