Clark Kimberling's

Encyclopedia of Triangle Centers - ETC

X(1001) = MIDPOINT OF X(1) AND X(9)

Let O_A be the circle tangent of side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_B and O_C cyclically. X(1001) is the radical center of the circles O_A, O_B, O_C. (Randy Hutson, 9/23/2011)

Let L_A and M_A be the external tangents to circles O_B and O_C, with L_A being the farther from O_A. Define L_B, L_C, M_B, and M_C cyclically. Let A' = L_B∩L_C and A'' = M_B∩M_C, and define B', B'', C', and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(1001).

Let A* be the tangency point of O_A and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of O_B and O_C. The lines A*A**, B*B**, C*C** concur in X(1001).

For details and relationships among X(1001), X(1), X(145), X(3361), X(3616), X(3913), and X(4719), see

Luis González, "On a Triad of Circles Tangent to the Circumcircle and the Sides at Their Midpoints," Forum Geometricorum 11 (2011) 145-154.

Let A' be the line through X(1) parallel to line BC. Let A_B = A'∩AB and A_C = A'∩AC. Define B_C and C_A cyclically, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_B, B_C, C_A lie on a conic whose center is X(1001). (Angel Montesdeoca, April 27, 2016)

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) is the {X(1),X(238)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(1001), click Tables at the top of this page.

X(1001) = midpoint of X(1) and X(9)
X(1001) = reflection of X(142) in X(1125)
X(1001) = isogonal conjugate of X(1002)
X(1001) = complement of X(2550)
X(1001) = crosssum of X(i) and X(j) for these (i,j): (116,824), (788,1015)
X(1001) = crossdifference of every pair of points on line X(513)X(665)
X(1001) = anticomplement of X(3826)
X(1001) = X(6)-of-2nd-circumperp-triangle
X(1001) = X(141)-of-hexyl-triangle
X(1001) = X(5480)-of-excentral-triangle
X(1001) = inverse-in-Feuerbach-hyperbola of X(55)
X(1001) = polar conjugate of isotomic conjugate of X(23151)

X(1002) = ISOGONAL CONJUGATE OF X(1001)

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(1002) = isotomic conjugate of X(4441)
X(1002) = trilinear pole of line X(513)X(665)
X(1002) = X(19)-isoconjugate of X(23151)

X(1003) = INTERCEPT OF EULER LINE AND LINE X(6)X(99)

As a point on the Euler line, X(1003) has Shinagawa coefficients ((E + F)² - 2S², 3S²).

X(1003) lies on these lines: 2,3   6,99   32,538   183,187

X(1003) = midpoint of X(2) and X(33007)
X(1003) = reflection of X(7841) in X(2)
X(1003) = complement of X(33017)
X(1003) = anticomplement of X(33184)
X(1003) = circumcircle-inverse of X(37927)
X(1003) = orthocentroidal-circle-inverse of X(33228)
X(1003) = crossdifference of every pair of points on line X(647)X(888)
X(1003) = {X(2),X(4)}-harmonic conjugate of X(33228)
X(1003) = {X(2),X(20)}-harmonic conjugate of X(32986)

X(1004) = INTERCEPT OF EULER LINE AND LINE X(7)X(100)

As a point on the Euler line, X(1004) has Shinagawa coefficients ((E+F)S²-$abS_AS_B$, -(E+F)S² +$ab$S²).

X(1004) lies on these lines: 2,3   7,100   46,200   63,210   65,224

X(1005) = INTERCEPT OF EULER LINE AND LINE X(9)X(100)

X(1005) lies on these lines: 2,3   9,100   55,329   108,342

X(1005) = isogonal conjugate of X(1242)

X(1006) = INTERCEPT OF EULER LINE AND LINEX(9)X(48)

As a point on the Euler line, X(1006) has Shinagawa coefficients ($aS_A$ + abc, - $aS_A$).

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(1006) = isogonal conjugate of X(1243)

X(1007) = INTERCEPT OF LINES X(2)X(6) AND X(4)X(99)

X(1007) lies on these lines: 2,6   4,99   305,311   315,631   316,376   317,6353

X(1007) = isotomic conjugate of X(7612)
X(1007) = complement of X(37667)
X(1007) = anticomplement of X(37637)
X(1007) = {X(7752),X(7763)}-harmonic conjugate of X(4)

X(1008) = INTERCEPT OF EULER LINE AND LINE X(1)X(76)

As a point on the Euler line, X(1008) has Shinagawa coefficients ((E + F)³ + 2$bc$(E + F)² - $abS_C$(E + F) + 2(E + F)S², $bc$S²).

X(1008) lies on these lines: 1,76   2,3

X(1008) = anticomplement of X(37148)

X(1009) = INTERCEPT OF EULER LINE AND LINE X(1)X(39)

As a point on the Euler line, X(1009) has Shinagawa coefficients (2$bc$(E + F)² + $bcS_BS_C$ - 2$bc(S_A)²$ - 4$bc$S², $bc$S²).

X(1009) lies on these lines: 1,39   2,3   72,672   283,1065   518,583

X(1009) = isogonal conjugate of X(1244)
X(1009) = crossdifference of every pair of points on line X(647)X(659)

X(1010) = INTERCEPT OF EULER LINE AND LINE X(1)X(75)

As a point on the Euler line, X(1010) has Shinagawa coefficients ((E + F)² + $bc$(E + F) + abc$a$, S²).

X(1010) lies on these lines: 1,75   2,3   8,81   10,58   72,894   283,1065   312,975   759,833

X(1010) = isogonal conjugate of X(1245)
X(1010) = crossdifference of every pair of points on line X(647)X(798)

X(1011) = INTERCEPT OF EULER LINE AND LINE X(6)X(31)

As a point on the Euler line, X(1011) has Shinagawa coefficients (E + $bc$, -$bc$).

X(1011) lies on these lines: 2,3   6,31   9,228   35,43   51,573   184,572

X(1011) = isogonal conjugate of X(1246)
X(1011) = inverse-in-orthocentroidal circle of X(3136)
X(1011) = crosssum of X(834) and X(1086)
X(1011) = crossdifference of every pair of points on line X(514)X(647)

X(1012) = INTERCEPT OF EULER LINE AND LINE X(1)X(84)

As a point on the Euler line, X(1012) has Shinagawa coefficients (S², abc$a$ - S²).

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(1012) = center of circle {X(1),X(1709),PU(4)}

X(1013) = INTERCEPT OF EULER LINE AND LINE X(7)X(108)

As a point on the Euler line, X(1013) has Shinagawa coefficients ($aS_A$F, $aS_BS_C$).

X(1013) lies on these lines: 2,3   6,162   7,108   33,63   55,92   100,281

X(1014) = ISOGONAL CONJUGATE OF X(210)

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(1014) = isogonal conjugate of X(210)
X(1014) = cevapoint of X(56) and X(57)
X(1014) = X(58)-cross conjugate of X(81)
X(1014) = isotomic conjugate of X(3701)
X(1014) = trilinear pole of line X(1019)X(1429)
X(1014) = crossdifference of every pair of points on line X(3709)X(4041)

X(1015) = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X(39) and radius 2R sin²ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X(115), and its internal and external centers of similitude with the incircle are X(1500) and X(1015), respectively. (based on notes from Peter J. C. Moses, 5/29/03)

X(1015) is the center of the hyperbola that passes through the points A, B, C, X(1), X(2), and X(28); also, X(1015) lies on the ellipse described at X(1125). (Randy Hutson, Hyacinthos #20179, 8/13/2011)

X(1015) lies on the Steiner inellipse, the Brocard inellipse, and 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   1960,3122

X(1015) = midpoint of X(i) and X(j) for these (i,j): (1,291), (2,3227)
X(1015) = isogonal conjugate of X(1016)
X(1015) = isotomic conjugate of X(31625)
X(1015) = complement of X(668)
X(1015) = crosspoint of X(2) and X(513)
X(1015) = crosssum of X(i) and X(j) for these (i,j): (1,1018), (2,190), (6,100), (8,644), (101,595), (345,1332)
X(1015) = crosssum of circumcircle-intercepts of line X(1)X(6)
X(1015) = crossdifference of every pair of points on line X(100)X(190)
X(1015) = PU(25)-harmonic conjugate of X(1960)
X(1015) = polar conjugate of isogonal conjugate of X(22096)
X(1015) = bicentric difference of PU(25)
X(1015) = bicentric sum of PU(27)
X(1015) = midpoint of PU(27)
X(1015) = center of circumconic that is locus of trilinear poles of lines through X(513)
X(1015) = perspector of circumparabola centered at X(513)
X(1015) = X(2)-Ceva conjugate of X(513)
X(1015) = projection from Steiner circumellipse to Steiner inellipse of X(3227)
X(1015) = barycentric square of X(513)
X(1015) = center of {ABC, Gemini 7}-circumconic

X(1016) = X(1)-CROSS CONJUGATE OF X(99)

X(1016) lies on these lines: 8,1083   99,813   100,667   190,514   238,519   512,660   644,666

X(1016) is the trilinear pole of line X(100)X(190), which is the tangent to the circumcircle at X(100) and to the Steiner circumellipse at X(190). This line is also the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(2),P}}, as P moves on the Nagel line, and the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(6),P}}, as P moves on line X(1)X(6). (Randy Hutson, October 15, 2018)

X(1016) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)

Let A₅B₅C₅ and A₆B₆C₆ be the Gemini triangles 5 and 6. Let A' be the barycentric product A₅*A₆ and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1016). (Randy Hutson, November 30, 2018)

X(1016) = isogonal conjugate of X(1015)
X(1016) = isotomic conjugate of X(1086)
X(1016) = anticomplement of X(6547)
X(1016) = cevapoint of X(i) and X(j) for these (i,j): (2,190), (6,100)
X(1016) = X(i)-cross conjugate of X(j) for these (i,j): (1,99), (2,190), (6,100)
X(1016) = polar conjugate of X(2969)
X(1016) = X(92)-isoconjugate of X(22096)

X(1017) = POINT ALFIRK

X(1017) lies on the Brocard inellipse and these lines: 6,101   44,214

X(1017) = isogonal conjugate of isotomic conjugate of X(4370)
X(1017) = polar conjugate of isotomic conjugate of X(22371)
X(1017) = crosssum of X(2) and X(903)
X(1017) = crossdifference of every pair of points on line X(900)X(903)
X(1017) = bicentric sum of PU(99)
X(1017) = PU(99)-harmonic conjugate of X(1960)
X(1017) = barycentric square of X(44)

X(1018) = X(512)-CROSS CONJUGATE OF X(1)

X(1018) is the intersection, other than the vertices of the Gemini 18 triangle, of the {ABC, Gemini 18}-circumconic and {Gemini 17, Gemini 18}-circumconic. (Randy Hutson, November 30, 2018)

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(1018) = isotomic conjugate of X(7199)
X(1018) = X(512)-cross conjugate of X(1)
X(1018) = crosspoint of X(100) and X(190)
X(1018) = crosssum of X(513) and X(649)
X(1018) = crossdifference of every pair of points on line X(244)X(659)
X(1018) = trilinear pole of line X(37)X(42)

X(1019) = X(99)-CEVA CONJUGATE OF X(1)

Evans conjectured that X(1), X(484), X(1276), X(1277) are concyclic, and reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X(1019). (Lawrence Evans, 2/24/2003)

The circle is now known as the Evans circle. (September 2020)

X(1019) lies on thje cubics K035 and K10174, and on these lines: {1, 512}, {2, 4129}, {3, 39577}, {8, 4807}, {21, 35355}, {36, 238}, {39, 14991}, {40, 28473}, {57, 7178}, {58, 1027}, {81, 1022}, {83, 4444}, {86, 23892}, {99, 813}, {110, 1308}, {163, 1414}, {239, 514}, {522, 5214}, {523, 20369}, {525, 4897}, {659, 6372}, {661, 1931}, {662, 3257}, {663, 6005}, {693, 29013}, {741, 14665}, {759, 840}, {798, 2669}, {799, 4607}, {812, 4509}, {824, 21389}, {830, 2254}, {918, 2483}, {1020, 1262}, {1025, 4237}, {1424, 7031}, {1429, 3669}, {1434, 23599}, {1577, 4369}, {1634, 35338}, {1698, 21051}, {1734, 8678}, {1924, 20981}, {2084, 3572}, {2484, 15411}, {2533, 2787}, {2786, 7265}, {2832, 23738}, {3249, 4375}, {3293, 22320}, {3309, 7659}, {3333, 34958}, {3337, 29136}, {3624, 4806}, {3667, 7253}, {3676, 17925}, {3708, 7266}, {3751, 9040}, {3762, 29402}, {3800, 39545}, {3801, 29029}, {3835, 25510}, {3907, 4761}, {3960, 4979}, {4010, 29150}, {4033, 37205}, {4041, 4160}, {4049, 27003}, {4083, 4378}, {4122, 29090}, {4151, 17166}, {4379, 4823}, {4380, 4801}, {4382, 29270}, {4391, 24601}, {4401, 4724}, {4449, 29350}, {4458, 29118}, {4467, 23879}, {4705, 9508}, {4762, 18196}, {4774, 29268}, {4782, 29198}, {4785, 17179}, {4794, 8643}, {4813, 27646}, {4817, 16737}, {4874, 29170}, {4922, 29298}, {5307, 14618}, {5539, 9424}, {6003, 6909}, {6006, 28591}, {6626, 27929}, {7153, 9336}, {7255, 16549}, {8027, 18169}, {8034, 29821}, {8690, 28520}, {8712, 18199}, {9013, 33844}, {10455, 21211}, {10566, 20950}, {16744, 23355}, {16755, 28859}, {16874, 16877}, {16887, 26843}, {17096, 30723}, {17174, 20295}, {17205, 31647}, {17210, 25381}, {17761, 26845}, {18827, 35172}, {20517, 29132}, {21146, 29070}, {21188, 23788}, {21392, 23875}, {22108, 28902}, {23729, 30724}, {23815, 24719}, {24019, 32668}, {24237, 26856}, {24624, 37222}, {26702, 28838}, {26775, 27013}, {26821, 26853}, {26823, 26825}, {27168, 27169}, {27527, 31286}, {29458, 29459}, {29770, 29772}, {32678, 35049}, {32944, 35353}, {39568, 39600}

X(1019) = midpoint of X(i) and X(j) for these {i,j}: {3669, 4790}, {3733, 4840}, {4367, 4784}, {4378, 4834}, {4380, 4801}, {4560, 7192}
X(1019) = reflection of X(i) in X(j) for these {i,j}: {1, 4367}, {8, 4807}, {661, 14838}, {1577, 4369}, {3737, 3733}, {4040, 667}, {4063, 649}, {4705, 9508}, {4724, 4401}, {4960, 7192}, {7265, 8045}, {14349, 905}, {21124, 21192}, {21385, 4063}, {24719, 23815}
X(1019) = isogonal conjugate of X(1018)
X(1019) = isotomic conjugate of X(4033)
X(1019) = anticomplement of X(4129)
X(1019) = anticomplement of the isotomic conjugate of X(37205)
X(1019) = isogonal conjugate of the anticomplement of X(17761)
X(1019) = isogonal conjugate of the isotomic conjugate of X(7199)
X(1019) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {110, 18133}, {163, 24068}, {596, 21294}, {8050, 21287}, {34594, 69}, {37205, 6327}
X(1019) = X(i)-Ceva conjugate of X(j) for these (i,j): {27, 17197}, {28, 18184}, {81, 16726}, {99, 1}, {100, 18166}, {110, 18164}, {190, 8025}, {662, 81}, {757, 244}, {799, 86}, {873, 3248}, {932, 18171}, {1014, 18191}, {1414, 58}, {1434, 17205}, {3733, 18197}, {4565, 57}, {4584, 18206}, {4589, 18792}, {4598, 16738}, {4603, 17207}, {4614, 4658}, {4623, 18169}, {4637, 1014}, {7192, 3737}, {8690, 18186}, {17096, 7203}, {34594, 16696}, {37205, 2}
X(1019) = X(i)-cross conjugate of X(j) for these (i,j): {244, 757}, {513, 7192}, {649, 3733}, {1015, 1}, {1086, 57}, {1977, 87}, {3123, 32010}, {3248, 873}, {3669, 17925}, {3733, 7203}, {3937, 269}, {3960, 1022}, {4979, 513}, {7252, 3737}, {8042, 16726}, {8632, 1027}, {16726, 81}, {16742, 274}, {17217, 18197}, {18191, 1014}, {23470, 36598}, {38346, 6}
X(1019) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1018}, {2, 4557}, {4, 4574}, {6, 3952}, {8, 4559}, {9, 4551}, {10, 101}, {12, 5546}, {21, 21859}, {31, 4033}, {32, 27808}, {37, 100}, {42, 190}, {55, 4552}, {56, 30730}, {57, 4069}, {58, 4103}, {59, 3700}, {65, 644}, {71, 1897}, {72, 1783}, {82, 35309}, {99, 1500}, {106, 4169}, {108, 3694}, {109, 2321}, {110, 594}, {112, 3695}, {162, 3949}, {163, 1089}, {181, 645}, {200, 1020}, {210, 651}, {213, 668}, {220, 4566}, {225, 4587}, {226, 3939}, {228, 6335}, {281, 23067}, {306, 8750}, {313, 32739}, {321, 692}, {512, 1016}, {523, 1252}, {643, 2171}, {646, 1402}, {647, 15742}, {648, 3690}, {653, 2318}, {660, 2238}, {661, 765}, {662, 756}, {664, 1334}, {666, 20683}, {669, 31625}, {670, 7109}, {677, 17747}, {740, 813}, {787, 28593}, {789, 3774}, {798, 7035}, {799, 872}, {825, 3773}, {831, 28594}, {850, 23990}, {901, 3943}, {919, 3932}, {932, 20691}, {934, 4515}, {983, 7239}, {1023, 4674}, {1026, 18785}, {1042, 6558}, {1110, 1577}, {1126, 4115}, {1211, 32736}, {1213, 8701}, {1254, 7259}, {1275, 4524}, {1292, 3991}, {1293, 3950}, {1331, 1826}, {1332, 1824}, {1400, 3699}, {1415, 3701}, {1427, 4578}, {1461, 4082}, {1576, 28654}, {1834, 29163}, {1880, 4571}, {1918, 1978}, {1962, 37212}, {1983, 15065}, {2092, 8707}, {2149, 4086}, {2197, 36797}, {2205, 6386}, {2276, 4613}, {2284, 13576}, {2292, 36147}, {2295, 3903}, {2328, 4605}, {2333, 4561}, {2346, 35310}, {2427, 38955}, {2702, 6541}, {2748, 16611}, {2901, 29014}, {3122, 6632}, {3257, 21805}, {3610, 32691}, {3696, 8693}, {3704, 8687}, {3709, 4998}, {3710, 32674}, {3714, 32693}, {3724, 36804}, {3747, 4562}, {3778, 4621}, {3842, 28841}, {3930, 36086}, {3948, 34067}, {3971, 34071}, {3986, 28226}, {3992, 32665}, {3994, 34075}, {4024, 4570}, {4029, 6014}, {4041, 4564}, {4058, 28162}, {4072, 8699}, {4076, 7180}, {4078, 28847}, {4079, 4600}, {4080, 23344}, {4085, 28883}, {4095, 29055}, {4120, 9268}, {4125, 34073}, {4136, 8685}, {4158, 6529}, {4171, 7045}, {4516, 31615}, {4553, 18098}, {4556, 6535}, {4558, 7140}, {4565, 6057}, {4567, 4705}, {4573, 7064}, {4585, 34857}, {4595, 23493}, {4596, 21816}, {4603, 21803}, {4606, 37593}, {4618, 21821}, {4628, 15523}, {4629, 8013}, {4646, 6574}, {4729, 5382}, {4730, 5376}, {4756, 28625}, {4767, 28658}, {4770, 5385}, {4849, 27834}, {4878, 37206}, {5257, 8694}, {5295, 36080}, {5377, 24290}, {5378, 21832}, {5380, 21839}, {5381, 14404}, {6065, 7178}, {6539, 35327}, {6540, 20970}, {6543, 17943}, {6577, 21070}, {6606, 21795}, {6725, 6733}, {7012, 8611}, {7141, 32661}, {8684, 18904}, {8706, 21796}, {8708, 16589}, {8709, 21830}, {9271, 21885}, {13138, 21871}, {17757, 32641}, {20491, 20640}, {20501, 20696}, {20680, 39272}, {20693, 37135}, {20964, 27805}, {21033, 36098}, {21045, 31616}, {21075, 36049}, {21078, 36050}, {21759, 36863}, {21801, 36037}, {21868, 29227}, {29127, 37715}
X(1019) = cevapoint of X(i) and X(j) for these (i,j): {6, 23404}, {512, 14991}, {513, 649}, {661, 4132}, {3733, 7252}, {3768, 38349}, {7192, 17217}, {8042, 16726}
X(1019) = crosspoint of X(i) and X(j) for these (i,j): {81, 662}, {86, 799}, {99, 1509}, {190, 1255}, {593, 4565}, {823, 837}, {1014, 4637}, {1414, 1434}, {7192, 17096}
X(1019) = crosssum of X(i) and X(j) for these (i,j): {10, 4151}, {37, 661}, {42, 798}, {71, 8611}, {210, 4171}, {512, 1500}, {513, 3720}, {523, 3925}, {594, 3700}, {649, 1100}, {650, 2269}, {656, 17441}, {822, 836}, {1334, 4041}, {4024, 21675}, {4120, 21942}, {4705, 21816}, {4730, 21821}, {4770, 21822}, {4931, 21943}, {8061, 17456}, {20680, 24290}
X(1019) = trilinear pole of line {244, 659}
X(1019) = crossdifference of every pair of points on line {37, 42}
X(1019) = barycentric product X(i)*X(j) for these {i,j}: {1, 7192}, {6, 7199}, {7, 3737}, {8, 7203}, {9, 17096}, {11, 1414}, {19, 15419}, {21, 3676}, {27, 905}, {28, 4025}, {56, 18155}, {57, 4560}, {58, 693}, {60, 4077}, {63, 17925}, {75, 3733}, {81, 514}, {85, 7252}, {86, 513}, {87, 17217}, {92, 7254}, {99, 244}, {100, 17205}, {101, 16727}, {104, 23788}, {105, 23829}, {108, 17219}, {110, 1111}, {141, 39179}, {162, 1565}, {163, 23989}, {190, 16726}, {256, 17212}, {257, 18200}, {261, 4017}, {269, 7253}, {270, 17094}, {272, 23800}, {273, 23189}, {274, 649}, {279, 1021}, {284, 24002}, {286, 1459}, {310, 667}, {330, 18197}, {333, 3669}, {512, 873}, {522, 1014}, {523, 757}, {552, 4041}, {593, 1577}, {643, 1358}, {648, 3942}, {650, 1434}, {651, 17197}, {659, 18827}, {661, 1509}, {662, 1086}, {664, 18191}, {670, 3248}, {741, 3766}, {759, 4453}, {763, 4024}, {764, 4600}, {799, 1015}, {811, 3937}, {812, 37128}, {849, 850}, {876, 33295}, {893, 16737}, {932, 23824}, {982, 7255}, {1016, 8042}, {1020, 26856}, {1022, 16704}, {1027, 30941}, {1088, 21789}, {1146, 4637}, {1169, 4509}, {1171, 4978}, {1178, 4374}, {1333, 3261}, {1357, 7257}, {1396, 6332}, {1408, 35519}, {1412, 4391}, {1435, 15411}, {1444, 7649}, {1474, 15413}, {1647, 4622}, {1790, 17924}, {1847, 23090}, {1919, 6385}, {1977, 4602}, {2087, 4615}, {2160, 16755}, {2170, 4573}, {2185, 7178}, {2215, 15417}, {2310, 4616}, {2363, 3004}, {2533, 7303}, {2969, 4592}, {2973, 4575}, {3122, 4623}, {3125, 4610}, {3271, 4625}, {3337, 7372}, {3572, 30940}, {3801, 7305}, {3960, 24624}, {4049, 30576}, {4086, 7341}, {4131, 8747}, {4228, 26721}, {4367, 32010}, {4481, 14621}, {4556, 16732}, {4565, 4858}, {4567, 6545}, {4584, 27918}, {4589, 27846}, {4598, 16742}, {4601, 21143}, {4603, 7200}, {4635, 14936}, {4705, 6628}, {4840, 30598}, {4960, 25417}, {4979, 32014}, {5317, 30805}, {6384, 16695}, {6591, 17206}, {7058, 7216}, {7153, 27527}, {7177, 17926}, {7215, 36126}, {8034, 24037}, {9309, 17218}, {9311, 18199}, {10566, 16696}, {13478, 16754}, {14377, 16751}, {16887, 18108}, {20028, 21173}, {21208, 34594}, {23345, 30939}, {24018, 36419}, {30581, 31010}
X(1019) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3952}, {2, 4033}, {6, 1018}, {9, 30730}, {11, 4086}, {21, 3699}, {27, 6335}, {28, 1897}, {31, 4557}, {37, 4103}, {39, 35309}, {44, 4169}, {48, 4574}, {55, 4069}, {56, 4551}, {57, 4552}, {58, 100}, {60, 643}, {75, 27808}, {81, 190}, {86, 668}, {99, 7035}, {110, 765}, {162, 15742}, {163, 1252}, {244, 523}, {261, 7257}, {269, 4566}, {270, 36797}, {274, 1978}, {283, 4571}, {284, 644}, {310, 6386}, {333, 646}, {512, 756}, {513, 10}, {514, 321}, {521, 3710}, {522, 3701}, {523, 1089}, {552, 4625}, {593, 662}, {603, 23067}, {604, 4559}, {643, 4076}, {647, 3949}, {649, 37}, {650, 2321}, {652, 3694}, {656, 3695}, {657, 4515}, {659, 740}, {661, 594}, {662, 1016}, {663, 210}, {665, 3930}, {667, 42}, {669, 872}, {693, 313}, {741, 660}, {757, 99}, {763, 4610}, {764, 3120}, {798, 1500}, {799, 31625}, {810, 3690}, {812, 3948}, {849, 110}, {873, 670}, {891, 3994}, {900, 3992}, {905, 306}, {985, 4613}, {1014, 664}, {1015, 661}, {1021, 346}, {1022, 4080}, {1027, 13576}, {1086, 1577}, {1098, 7256}, {1100, 4115}, {1111, 850}, {1169, 36147}, {1171, 37212}, {1178, 3903}, {1333, 101}, {1357, 4017}, {1358, 4077}, {1396, 653}, {1400, 21859}, {1407, 1020}, {1408, 109}, {1412, 651}, {1414, 4998}, {1427, 4605}, {1434, 4554}, {1437, 1331}, {1444, 4561}, {1459, 72}, {1474, 1783}, {1475, 35310}, {1491, 3773}, {1509, 799}, {1565, 14208}, {1576, 1110}, {1577, 28654}, {1635, 3943}, {1769, 17757}, {1790, 1332}, {1919, 213}, {1924, 7109}, {1946, 2318}, {1960, 21805}, {1977, 798}, {1980, 1918}, {2087, 4120}, {2150, 5546}, {2170, 3700}, {2185, 645}, {2193, 4587}, {2194, 3939}, {2203, 8750}, {2206, 692}, {2254, 3932}, {2275, 7239}, {2287, 6558}, {2328, 4578}, {2363, 8707}, {2423, 2250}, {2483, 28594}, {2488, 21039}, {2516, 4072}, {2522, 3610}, {2530, 15523}, {2605, 3678}, {2832, 4442}, {2969, 24006}, {3004, 18697}, {3063, 1334}, {3120, 4036}, {3121, 4079}, {3122, 4705}, {3123, 21051}, {3125, 4024}, {3248, 512}, {3249, 3121}, {3261, 27801}, {3271, 4041}, {3285, 1023}, {3286, 1026}, {3287, 4095}, {3310, 21801}, {3337, 6758}, {3669, 226}, {3675, 4088}, {3676, 1441}, {3733, 1}, {3736, 3799}, {3737, 8}, {3756, 4404}, {3766, 35544}, {3776, 20234}, {3777, 2887}, {3900, 4082}, {3937, 656}, {3942, 525}, {3960, 3936}, {4014, 21052}, {4017, 12}, {4025, 20336}, {4040, 4651}, {4041, 6057}, {4057, 3293}, {4063, 3995}, {4077, 34388}, {4079, 762}, {4083, 3971}, {4091, 3998}, {4132, 4075}, {4164, 4039}, {4273, 4752}, {4367, 1215}, {4369, 3963}, {4374, 1237}, {4379, 4377}, {4391, 30713}, {4394, 3950}, {4401, 3896}, {4435, 3985}, {4448, 4783}, {4449, 3967}, {4453, 35550}, {4475, 4122}, {4481, 3661}, {4491, 31855}, {4498, 3175}, {4509, 1228}, {4556, 4567}, {4560, 312}, {4565, 4564}, {4567, 6632}, {4591, 5376}, {4610, 4601}, {4637, 1275}, {4653, 4767}, {4658, 4756}, {4705, 6535}, {4724, 3696}, {4762, 4044}, {4777, 4125}, {4782, 3993}, {4784, 3842}, {4790, 5257}, {4802, 4066}, {4833, 3679}, {4840, 1698}, {4874, 4710}, {4879, 4096}, {4960, 28605}, {4977, 4647}, {4978, 1230}, {4979, 1213}, {4983, 8013}, {5009, 3573}, {5029, 20693}, {5323, 14594}, {6085, 4695}, {6129, 21075}, {6363, 4642}, {6371, 2292}, {6372, 21020}, {6377, 21834}, {6545, 16732}, {6586, 4006}, {6588, 21074}, {6589, 21078}, {6591, 1826}, {6615, 21031}, {6628, 4623}, {6729, 6725}, {7054, 7259}, {7058, 7258}, {7117, 8611}, {7178, 6358}, {7180, 2171}, {7192, 75}, {7199, 76}, {7202, 7265}, {7203, 7}, {7216, 6354}, {7234, 21803}, {7250, 1254}, {7252, 9}, {7253, 341}, {7254, 63}, {7255, 7033}, {7303, 4594}, {7304, 36860}, {7341, 1414}, {8027, 3122}, {8034, 2643}, {8042, 1086}, {8054, 4132}, {8578, 22321}, {8632, 2238}, {8642, 4878}, {8643, 4849}, {8656, 21870}, {8712, 4656}, {9002, 4424}, {9508, 6541}, {14419, 4062}, {14438, 4144}, {14838, 3969}, {14936, 4171}, {15419, 304}, {16695, 43}, {16696, 4568}, {16704, 24004}, {16726, 514}, {16727, 3261}, {16737, 1920}, {16742, 3835}, {16751, 17233}, {16754, 4417}, {16755, 33939}, {16757, 4150}, {16947, 1415}, {17096, 85}, {17187, 4553}, {17197, 4391}, {17205, 693}, {17212, 1909}, {17214, 4106}, {17217, 6376}, {17219, 35518}, {17302, 21604}, {17418, 3714}, {17420, 3704}, {17477, 4139}, {17494, 4043}, {17925, 92}, {17926, 7101}, {18108, 18082}, {18155, 3596}, {18163, 25268}, {18184, 25259}, {18186, 25272}, {18191, 522}, {18196, 25264}, {18197, 192}, {18199, 3729}, {18200, 894}, {18210, 4064}, {18211, 3667}, {18268, 813}, {18600, 21580}, {18792, 23354}, {18827, 4583}, {19945, 14431}, {20979, 20691}, {20981, 2295}, {21007, 3294}, {21123, 3954}, {21143, 3125}, {21173, 17751}, {21191, 22028}, {21196, 27569}, {21758, 2245}, {21789, 200}, {21828, 4053}, {21832, 4037}, {22096, 810}, {22383, 71}, {23090, 3692}, {23092, 22370}, {23189, 78}, {23224, 3682}, {23345, 4674}, {23355, 18793}, {23572, 21877}, {23788, 3262}, {23824, 20906}, {23829, 3263}, {23989, 20948}, {24002, 349}, {24006, 7141}, {24624, 36804}, {26822, 18040}, {27527, 4110}, {27644, 4595}, {27846, 4010}, {28209, 4714}, {29226, 4135}, {29545, 26772}, {29821, 21295}, {30940, 27853}, {30966, 4505}, {31947, 21081}, {33295, 874}, {33296, 36863}, {36419, 823}, {36420, 24019}, {37128, 4562}, {38238, 21100}, {39179, 83}

X(1020) = POINT ALGEDI

Let X be a point on the Euler line. Let P and U be the 1st and 2nd bicentrics of X. As X varies, the bicentric sum of P and U trace the line X(42)X(65), of which X(1020) is the trilinear pole. (Randy Hutson, March 25, 2016)

X(1020) lies on these lines: 1,185   57,1086   101,651   108,109   190,658   269,292   347,573   648,1021   653,2637  

X(1020) = isogonal conjugate of X(1021)
X(1020) = crosssum of X(650) and X(652)
X(1020) = trilinear pole of line X(42)X(65)
X(1020) = crosspoint of X(651) and X(653)
X(1020) = cevapoint of Jerabek hyperbola intercepts of antiorthic axis

X(1021) = ISOGONAL CONJUGATE OF X(1020)

X(1021) lies on these lines: 1,647   239,514   243,522   333,1024   521,650   654,2596   648,1020

X(1021) = isogonal conjugate of X(1020)
X(1021) = crosspoint of X(81) and X(162)
X(1021) = crosssum of X(i) and X(j) for these (i,j): (37,656), (65,661), (73,822), (647,1425), (649,1104)
X(1021) = crosssum of Jerabek hyperbola intercepts of antiorthic axis
X(1021) = crossdifference of every pair of points on line X(42)X(65)
X(1021) = cevapoint of X(650) and X(652)
X(1021) = perspector of excentral triangle and tangential triangle, wrt anticevian triangle of X(4), of bianticevian conic of X(1) and X(4)

X(1022) = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514)

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(1022) = crossdifference of every pair of points on line X(44)X(678)
X(1022) = isotomic conjugate of X(24004)
X(1022) = trilinear pole of line X(244)X(513)
X(1022) = barycentric quotient X(106)/X(100)

X(1023) = INTERSECTION OF LINES X(1)X(6) AND X(100)X(101)

X(1023) lies on these lines: 1,6   100,101   813,898

X(1023) = isogonal conjugate of X(1022)
X(1023) = crossdifference of every pair of points on line X(244)X(513)
X(1023) = bicentric difference of PU(28)
X(1023) = PU(28)-harmonic conjugate of X(1)
X(1023) = trilinear pole of line X(44)X(678)

X(1024) = INTERSECTION OF LINES X(6)X(513) ANDX(9)X(522)

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(1024) = crossdifference of every pair of points on line X(241)X(518)

X(1025) = INTERSECTION OF LINES X(2)X(7) AND X(100)X(109)

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(1025) = trilinear pole of line X(241)X(518)

X(1026) = INTERSECTION OF LINES X(1)X(2) AND X(100)X(101)

X(1026) lies on these lines: 1,2   55,1083   100,101   664,668   666,1027

X(1026) = isogonal conjugate of X(1027)
X(1026) = crosssum of X(513) and X(659)
X(1026) = crossdifference of every pair of points on line X(244)X(649)
X(1026) = trilinear pole of line X(518)X(672)

X(1027) = INTERSECTION OF LINES X(1)X(514) AND X(6)X(513)

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(1027) = crosssum of X(659) and X(1279)
X(1027) = crossdifference of every pair of points on line X(518)X(672)
X(1027) = trilinear pole of line X(244)X(649)

X(1028) = POINT ALGENIB

X(1028) = isogonal conjugate of X(1085)

X(1029) = CYCLOCEVIAN CONJUGATE OF X(1)

X(1029) lies on the Kiepert hyperbola and these lines: 10,191   115,593   319,321

X(1029) = isogonal conjugate of X(1030)
X(1029) = isotomic conjugate of X(2895)
X(1029) = X(i)-cross conjugate of X(j) for these (i,j): (79,7), (81,2)
X(1029) = cyclocevian conjugate of X(1)
X(1029) = polar conjugate of X(451)

X(1030) = ISOGONAL CONJUGATE OF X(1029)

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(1030) = X(i)-Ceva conjugate of X(j) for these (i,j): (35,55), (37,6)
X(1030) = crosspoint of X(100) and X(249)
X(1030) = crosssum of X(115) and X(513)
X(1030) = Brocard-circle-inverse of X(5124)
X(1030) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36750)
X(1030) = {X(3),X(6)}-harmonic conjugate of X(5124)
X(1030) = {X(371),X(372)}-harmonic conjugate of X(36750)

X(1031) = CYCLOCEVIAN CONJUGATE OF X(6)

X(1031) lies on this line: 141,384

X(1031) = isotomic conjugate of X(2896)
X(1031) = X(83)-cross conjugate of X(2)
X(1031) = cyclocevian conjugate of X(6)
X(1031) = barycentric product of PU(137)

X(1032) = CYCLOCEVIAN CONJUGATE OF X(20)

X(1032) lies on the Lucas cubic and this line: 20,394

X(1032) = isogonal conjugate of X(1033)
X(1032) = X(4)-cross conjugate of X(69)
X(1032) = cyclocevian conjugate of X(20)
X(1032) = anticomplement of X(3343)
X(1032) = polar conjugate of X(6523)

X(1033) = X(3)-CEVA CONJUGATE OF X(25)

X(1033) lies on these lines: 6,64   19,56   25,393   55,204

X(1033) = isogonal conjugate of X(1032)
X(1033) = X(3)-Ceva conjugate of X(25)

X(1034) = X(4)-CROSS CONJUGATE OF X(8)

X(1034) lies on the Lucas cubic and these lines: 2,271   20,78

X(1034) = perpsector of triangle ABC and the pedal triangle of X(3345)

X(1034) = isogonal conjugate of X(1035)
X(1034) = isotomic conjugate of X(5932)
X(1034) = trilinear pole of line X(8058)X(14331)
X(1034) = anticomplement of X(3342)
X(1034) = anticomplementary conjugate of X(34162)
X(1034) = X(i)-cross conjugate of X(j) for these (i,j): (4,8), (282,2)
X(1034) = cyclocevian conjugate of X(329)

X(1035) = ISOGONAL CONJUGATE OF X(1034)

X(1035) lies on these lines: 3,223   6,603   25,34   55,64   222,581

X(1035) = isogonal conjugate of X(1034)
X(1035) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,56), (223,6)

X(1036) = ISOGONAL CONJUGATE OF X(388)

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) = CEVAPOINT OF X(55) AND X(56)

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(1037) = cevapoint of X(55) and X(56)
X(1037) = trilinear pole of line X(652)X(665)

X(1038) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(34)

X(1038) lies on these lines: 1,3   2,34   4,1076   9,478   20,33   21,1041   38,1106   63,201   69,73   72,222   172,577   221,960   223,936   225,377   226,975   278,443   388,612   1068,1074

X(1038) is the {X(1),X(3)}-harmonic conjugate of X(1040). For a list of other harmonic conjugates of X(1038), click Tables at the top of this page.

X(1038) = isogonal conjugate of X(1039)
X(1038) = homothetic center of 6th anti-mixtilinear triangle and anti-tangential midarc triangle

X(1039) = ISOGONAL CONJUGATE OF X(1038)

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) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(33)

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(1040) = crosspoint of X(i) and X(j) for these (i,j): (21,332), (77,78)
X(1040) = crosssum of X(33) and X(34)
X(1040) = homothetic center of intangents triangle and mid-triangle of orthic and dual of orthic triangles

X(1041) = ISOGONAL CONJUGATE OF X(1040)

The trilinear polar of X(1041) passes through X(650).

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(1041) = cevapoint of X(33) and X(34)

X(1042) = CROSSPOINT OF X(1) AND X(64)

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(1042) = crosspoint of X(i) and X(j) for these (i,j): (1,64), (34,56)
X(1042) = crosssum of X(i) and X(j) for these (i,j): (1,20), (8,78), (200,346)
X(1042) = crossdifference of every pair of points on line X(657)X(1021)
X(1042) = trilinear pole of line X(798)X(7180)

X(1043) = CEVAPOINT OF X(1) AND X(20)

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(1043) = isotomic conjugate of X(3668)
X(1043) = trilinear pole of line X(657)X(1021)
X(1043) = crosspoint of X(1) and X(20) wrt both the excentral and anticomplementary triangles
X(1043) = crossdifference of every pair of points on line X(798)X(7180)
X(1043) = trilinear product of X(8) and X(21)
X(1043) = anticomplement of X(1834)
X(1043) = X(314)-Ceva conjugate of X(333)
X(1043) = cevapoint of X(i) and X(j) for these (i,j): (1,20), (8,78), (200,346)

X(1044) = X(64)-CEVA CONJUGATE OF X(1)

X(1044) lies on these lines: 1,7   43,46

X(1044) = X(64)-Ceva conjugate of X(1)

X(1045) = X(42)-CEVA CONJUGATE OF X(1)

X(1045) lies on these lines: 1,75   6,2665   9,43   40,511   42,894   190,872   192,869

X(1045) = X(42)-Ceva conjugate of X(1)
X(1045) = excentral-isogonal conjugate of X(1764)
X(1045) = excentral-isotomic conjugate of X(20)
X(1045) = perspector of trilinear obverse triangle of X(2) and unary cofactor triangle of trilinear N-obverse triangle of X(2)

X(1046) = X(65)-CEVA CONJUGATE OF X(1)

Let La be the line parallel to the Brocard axis of BCI and passing through the A-excenter. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(1046). (Randy Hutson, January 29, 2018)

X(1046) = reflection of X(i) in X(j) for these (i,j): (1,58), (1330,10)
X(1046) = isogonal conjugate of X(1247)
X(1046) = X(65)-Ceva conjugate of X(1)
X(1046) = excentral-isogonal conjugate of X(20)
X(1046) = {X(1),X(58)}-harmonic conjugate of X(5429)

X(1047) = X(73)-CEVA CONJUGATE OF X(1)

X(1047) lies on these lines: 1,29   43,46

X(1047) = isogonal conjugate of X(1248)
X(1047) = X(73)-Ceva conjugate of X(1)

X(1048) = POINT ALGIEBA

X(1048) lies on these lines: 1,564   484,1046

X(1049) = TRILINEAR PURE ANGLES CENTER

X(1049) = isogonal conjugate of X(1077)

X(1050) = POINT ALGOL

X(1050) lies on these lines: 1,341   40,978

X(1051) = POINT ALGORAB

X(1051) lies on these lines: 1,748   6,846   81,1054   165,572

X(1052) = X(244)-CEVA CONJUGATE OF X(1)

X(1052) lies on these lines: 1,765   238,517   513,1054

X(1052) = X(244)-Ceva conjugate of X(1)

X(1053) = POINT ALHENA

X(1053) lies on these lines: 1,1110   238,517   905,1054

X(1054) = 6th SHARYGIN POINT

See the description at X(1281).

X(1054) lies on the Bevan circle and these lines: 1,88   2,846   43,57   46,978   81,1051   105,165   474,986   513,1052   905,1053

X(1054) = reflection of X(1) in X(106)
X(1054) = isogonal conjugate of X(9282)
X(1054) = inverse-in-circumcircle of X(1283)
X(1054) = X(1)-Hirst inverse of X(244)
X(1054) = crosssum of PU(33)
X(1054) = intersection of tangents at PU(34) to conic {{A,B,C,PU(34)}}
X(1054) = crosspoint of PU(34)
X(1054) = trilinear pole wrt excentral triangle of line X(1)X(6)
X(1054) = excentral isogonal conjugate of X(3667)
X(1054) = X(107) of excentral triangle
X(1054) = Conway-circle-inverse of X(38484)

X(1055) = POINT ALIOTH

X(1055) lies on these lines: 6,41   36,101   106,919   187,237   609,995

X(1055) = isogonal conjugate of X(1121)
X(1055) = crosssum of X(2) and X(527)
X(1055) = complement of anticomplementary conjugate of X(39357)
X(1055) = crossdifference of every pair of points on line X(2)X(522)
X(1055) = inverse-in-Parry-isodynamic-circle of X(5075); see X(2)

X(1056) = POINT ALCYONE

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(1056) = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X(4)

X(1057) = ISOGONAL CONJUGATE OF X(1056)

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) = POINT ALKALUROPS

X(1058) lies on these lines: 1,4   2,496   3,390   8,392   20,999   29,1057   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) = ISOGONAL CONJUGATE OF X(1058)

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) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(34)

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(1060) = homothetic center of 2nd Euler triangle and anti-tangential midarc triangle

X(1061) = ISOGONAL CONJUGATE OF X(1060)

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) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(33)

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(1062) = homothetic center of intangents triangle and 2nd Euler triangle

X(1063) = ISOGONAL CONJUGATE OF X(1062)

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) = INTERSECTION OF LINES X(1)X(4) AND X(3)X(31)

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(1064) = crosssum of X(1) and X(1478)

X(1065) = ISOGONAL CONJUGATE OF X(1064)

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(1065) = polar conjugate of X(30687)

X(1066) = HASTINGS POINT

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(1066) = crosssum of X(1) and X(1479)

X(1067) = ISOGONAL CONJUGATE OF X(1066)

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) = X(46)-CROSS CONJUGATE OF X(4)

X(1068) lies on these lines: 1,4   8,860   24,108   92,406   155,651   158,3542   281,451   318,475   377,1060   429,495   1038,1074   1040,1076   3157,3193  

X(1068) = isogonal conjugate of X(1069)
X(1068) = X(158)-Ceva conjugate of X(4)
X(1068) = X(46)-cross conjugate of X(4)
X(1068) = polar conjugate of X(2994)

X(1069) = X(255)-CROSS CONJUGATE OF X(3)

X(1069) lies on these lines: 1,90   11,68   394,1062   496,613   916,1037

X(1069) = isogonal conjugate of X(1068)
X(1069) = X(90)-Ceva conjugate of X(3)
X(1069) = X(255)-cross conjugate of X(3)
X(1069) = X(92)-isoconjugate of X(2178)

X(1070) = POINT ALKES

X(1070) lies on these lines: 1,4   55,1076   56,1074

X(1071) = INTERSECTION OF LINES X(1)X(84) AND X(4)X(7)

X(1071) appears in Hyacinthos message #3849, Paul Yiu, Sept. 19, 2001.

If you have The Geometer's Sketchpad, you can view X(1071).

X(1071) lies on these lines: 1,84   4,7   6,63   10,2801   20,145   21,104   27,1871   198,1741   227,1735   355,377   412,1872   496,1519   774,1458   910,1729   1210,1532   1317,1364  

X(1071) = reflection of X(i) in X(j) for these (i,j): (4,942), (72,3)
X(1071) = isotomic conjugate of isogonal conjugate of X(23204)
X(1071) = crosspoint of X(7) and X(63)
X(1071) = crosssum of X(i) and X(j) for these (i,j): (1,1777), (19,55), (25,2331)
X(1071) = X(68)-of-intouch triangle
X(1071) = X(20)-of-X(1)-Brocard triangle
X(1071) = intouch isogonal conjugate of X(54)
X(1071) = intouch isotomic conjugate of X(12723)

X(1072) = POINT ALNILAM

X(1072) lies on these lines: 1,4   55,1074   56,1076

X(1073) = X(6)-CROSS CONJUGATE OF X(3)

X(1073) lies on the Thomson cubic and these lines: 1,3341   2,253   3,64   4,3350   6,3343   9,223   57,3351   222,268

X(1073) = isogonal conjugate of X(1249)
X(1073) = complement of X(14361)
X(1073) = anticomplement of X(20207)
X(1073) = X(253)-Ceva conjugate of X(64)
X(1073) = cevapoint of X(6) and X(64)
X(1073) = X(i)-cross conjugate of X(j) for these (i,j): (6,3), (185,69)
X(1073) = crosssum of X(6) and X(1033)
X(1073) = isotomic conjugate of X(15466)
X(1073) = perspector of ABC and antipedal triangle of X(1498)
X(1073) = perspector of pedal and anticevian triangles of X(64)
X(1073) = perspector of ABC and medial triangle of pedal triangle of X(3346)
X(1073) = perspector of circumconic centered at X(3343)
X(1073) = center of circumconic that is locus of trilinear poles of lines passing through X(3343)
X(1073) = X(2)-Ceva conjugate of X(3343)

X(1074) = POINT ALNITAK

X(1074) lies on these lines: 1,224   3,225   4,1040   55,1072   56,1070   1038,1068

X(1075) = X(3)-CEVA CONJUGATE OF X(4)

X(1075) lies on the McCay cubic and these lines: 4,51   155,450   216,631   243,920   648,1092

X(1075) = isogonal conjugate of X(13855)
X(1075) = polar conjugate of X(34287)
X(1075) = eigencenter of cevian triangle of X(3)
X(1075) = eigencenter of anticevian triangle of X(4)
X(1075) = X(3)-Ceva conjugate of X(4)
X(1075) = X(155)-Hirst inverse of X(450)

X(1076) = POINT ALPHARD

X(1076) lies on these lines: 3,225   4,1038   55,1070   56,1072   1040,1068

X(1077) = ISOGONAL CONJUGATE OF X(1049)

X(1077) = isogonal conjugate of X(1049).

X(1078) = INTERSECTION OF LINES X(2)X(32) AND X(3)X(76)

X(1078) lies on these lines: 2,32   3,76   5,316   24,264   35,350   39,385   54,69   140,325   186,1235   187,384   194,574   274,404   298,619   279,618   302,635   303,636   7603,7843

X(1078) = isotomic conjugate of X(3613)
X(1078) = anticomplement of X(1506)
X(1078) = X(249)-Ceva conjugate of X(99)
X(1078) = complement of X(7785)
X(1078) = X(5038)-of-6th-Brocard-triangle
X(1078) = X(5116)-of-1st-anti-Brocard-triangle

X(1079) = TRILINEAR SQUARE OF X(46)

X(1079) lies on these lines: 1,4   46,1406   77,498   484,1103   651,920

X(1080) = INTERCEPT OF EULER LINE AND LINE (13)X(98)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1080) has Shinagawa coefficients (3^1/2S, 3E + 3F).

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-in-orthocentroidal-circle of X(383)

X(1081) = POINT ALPHECCA

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 the cubics K134 and K419a and these lines: {1, 30}, {2, 2306}, {7, 559}, {13, 226}, {14, 43682}, {55, 10651}, {57, 3179}, {75, 298}, {395, 1653}, {497, 30345}, {553, 37773}, {675, 36072}, {1086, 11072}, {1365, 18974}, {2153, 41889}, {3475, 37833}, {3638, 3982}, {5239, 5905}, {5240, 5249}, {6186, 10647}, {7026, 11078}, {11706, 26700}, {30328, 37640}

X(1081) = isogonal conjugate of X(1250)
X(1081) = isotomic conjugate of X(40713)
X(1081) = X(i)-cross conjugate of X(j) for these (i,j): {553, 554}, {30383, 85}
X(1081) = cevapoint of X(i) and X(j) for these (i,j): {1, 1653}, {1251, 2306}
X(1081) = X(i)-Dao conjugate of X(j) for these (i,j): {2, 40713}, {3, 1250}, {223, 1082}, {478, 2307}, {7026, 40578}
X(1081) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1250}, {9, 2307}, {15, 19551}, {31, 40713}, {35, 33653}, {55, 1082}, {1251, 7006}, {2151, 7026}, {5353, 7126}, {7127, 46077}, {10638, 42680}, {35057, 36073}
X(1081) = barycentric product X(i)*X(j) for these {i,j}: {75, 2306}, {85, 1251}, {300, 19373}, {559, 30690}, {3261, 36072}
X(1081) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 40713}, {6, 1250}, {13, 7026}, {56, 2307}, {57, 1082}, {559, 3219}, {1251, 9}, {2153, 19551}, {2160, 33653}, {2306, 1}, {2307, 7006}, {3179, 5240}, {5240, 44688}, {7051, 5353}, {7052, 46077}, {11072, 7126}, {19373, 15}, {33654, 42680}, {36072, 101}, {39153, 5239}, {40714, 42033}
X(1081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 4654, 554}, {226, 3639, 1082}, {3649, 37631, 554}, {3782, 5434, 554}

X(1082) = POINT ALPHERATZ

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 the cubics K134 and K341a and these lines: {1, 3}, {2, 5240}, {6, 7089}, {7, 554}, {13, 226}, {15, 16577}, {63, 5239}, {81, 33655}, {222, 7060}, {298, 319}, {466, 17043}, {497, 37833}, {553, 3638}, {651, 19551}, {1100, 1653}, {1250, 1442}, {1255, 2306}, {1276, 21476}, {1652, 16777}, {1836, 10651}, {1962, 10648}, {2003, 5353}, {2307, 3219}, {3474, 37830}, {3475, 30345}, {4336, 30300}, {7006, 42680}, {7051, 28606}, {9778, 30339}, {10391, 10649}, {10580, 30338}, {10647, 17017}, {10652, 11246}, {14100, 30356}, {17011, 19373}, {17778, 37795}, {25417, 33654}

X(1082) = isogonal conjugate of X(1251)
X(1082) = X(7345)-complementary conjugate of X(141)
X(1082) = X(1255)-Ceva conjugate of X(559)
X(1082) = cevapoint of X(1250) and X(2307)
X(1082) = barycentric product X(i)*X(j) for these {i,j}: {57, 40713}, {75, 2307}, {85, 1250}, {298, 33655}, {319, 33654}, {554, 3219}, {17095, 33653}, {18160, 36073}
X(1082) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 1251}, {223, 1081}, {478, 2306}, {5240, 40580}
X(1082) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1251}, {9, 2306}, {55, 1081}, {79, 10638}, {522, 36072}, {559, 7073}, {2153, 5240}, {3179, 19551}, {5239, 11072}, {6186, 40714}, {7026, 42623}, {7126, 39153}, {11081, 36932}, {33653, 42677}
X(1082) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1251}, {15, 5240}, {56, 2306}, {57, 1081}, {554, 30690}, {1250, 9}, {1415, 36072}, {2003, 559}, {2174, 10638}, {2307, 1}, {3219, 40714}, {5353, 5239}, {7026, 44690}, {7051, 39153}, {19373, 3179}, {33653, 7110}, {33654, 79}, {33655, 13}, {39151, 36933}, {40713, 312}, {46077, 7043}
X(1082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 57, 559}, {1, 37772, 37773}, {56, 20182, 559}, {57, 559, 37773}, {65, 37595, 559}, {226, 3639, 1081}, {241, 3748, 559}, {559, 37772, 57}, {940, 2099, 559}, {1214, 24929, 559}, {1319, 3666, 559}, {1429, 17598, 559}, {7146, 17716, 559}, {13388, 13389, 37772}, {15934, 37543, 559}

X(1083) = MIDPOINT OF X(105) AND X(644)

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 the Brocard circle.

X(1083) lies on the Brocard circle, the circle O(1,3), and these lines: 1,6   3,667   8,1016   55,1026   56,1025   105,644   840,898

X(1083) = midpoint of X(105) and X(644)
X(1083) = circumcircle-inverse of X(667)
X(1083) = Conway-circle-inverse of X(38485)
X(1083) = X(6)-Hirst inverse of X(518)
X(1083) = X(105)-of-1st-Brocard triangle
X(1083) = X(105)-of-X(1)-Brocard triangle
X(1083) = X(112)-of-1st-Montesdeoca-bisector-triangle
X(1083) = X(112)-of-2nd-Montesdeoca-bisector-triangle
X(1083) = similicenter of 1st and 2nd Montesdeoca bisector triangles
X(1083) = intersection, other than X(6), of the Brocard circle and line X(1)X(6)
X(1083) = 1st-Brocard-isogonal conjugate of X(2795)

X(1084) = CROSSPOINT OF X(2) AND X(512)

Let f(a,b,c) = a³(b² - c²)². Then 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).

Randy Hutson observed (9/23/2011) that the centers of homothety of the Lucas(L:W) homothetic triangles and triangle ABC form a circumconic which passes through the points X(493), X(494), X(588), and X(589). Indeed the perspectors are given by barycentric coordinates

a²/[a² + (L/W)S] : b²/[b² + (L/W)S] : c²/[c² + (L/W)S],

and the conic is the isogonal conjugate of the line X(2)X(6). Thus, X(2), X(6), and dozens of other named points lie on the conic; click Tables at the top of ETC, select CENTRAL LINES, and scroll to #15.

X(1084) is the center of the hyperbola H = {{A,B,C,X(2),X(6)}}, which is tangent to Brocard axis at X(6) and to line X(2)X(39) at X(2). Also, H is the locus of the trilinear pole of a line parallel to Lemoine axis (i.e. lines that pass through X(512)), and H is the isotomic conjugate of line the X(2)X(39). (Randy Hutson, July 20, 2016)

X(1084) lies on the Steiner inellipse and these lines: 2,670   6,694   39,597   115,804   351,865

X(1084) = midpoint of X(i) and X(j) for these (i,j): (6,694),(2,3228)
X(1084) = reflection of X(35073) in X(2)
X(1084) = isogonal conjugate of X(34537)
X(1084) = complement of X(670)
X(1084) = anticomplement of X(36950)
X(1084) = crosspoint of X(i) and X(j) for these {i,j}: {2, 512}, {6, 18105}, {32, 669}, {523, 6664}, {1974, 2489}, {798, 872}, {2395, 34238}
X(1084) = crosssum of X(i) and X(j) for these {i,j}: {2, 4576}, {6, 99}, {75, 21604}, {76, 670}, {110, 1627}, {305, 4563}, {799, 873}, {1509, 4623}, {2421, 5976}, {4631, 18021}, {5468, 31128}
X(1084) = crosssum of circumcircle intercepts of line X(2)X(6)
X(1084) = cevapoint of X(9427) and X(23216)
X(1084) = trilinear pole of line X(1645)X(23099)
X(1084) = crossdifference of every pair of points on line X(99)X(670)
X(1084) = center of the circumconic {{A,B,C,X(2),X(6)}}
X(1084) = projection from Steiner circumellipse to Steiner inellipse of X(3228)
X(1084) = Steiner-inellipse-antipode of X(35073)
X(1084) = perspector of circumconic centered at X(512) (parabola {{A,B,C,X(512),X(669)}})
X(1084) = intersection of trilinear polars of X(512) and X(669)
X(1084) = X(2)-Ceva conjugate of X(512)
X(1084) = perspector of ABC and the medial triangle of the cevian triangle of X(512)
X(1084) = perspector of unary cofactor triangles of 3rd, 5th and 6th Brocard triangles
X(1084) = barycentric square of X(512)
X(1084) = polar conjugate of isogonal conjugate of X(23216)
X(1084) = barycentric product X(32)*X(115)

X(1085) = ISOGONAL CONJUGATE OF X(1028)

X(1085) = isogonal conjugate of X(1028)

X(1086) = CENTER OF HYPERBOLA {{A,B,C,X(2),X(7)|}

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). Also, X(1086) is the point of tangency of the Steiner inscribed ellipse with the line tangent to the nine-point circle and the incircle. (Paul Yiu, #4197, 11/24/01).

X(1086) = center of circumconic that is locus of trilinear poles of lines parallel to Gergonne line (i.e. lines that pass through X(514)). This conic is the isotomic conjugate of the Nagel line. (Randy Hutson, September 14, 2016)

Let A₇B₇C₇ and A₈B₈C₈ be the Gemini triangles 7 and 8. Let A' be the barycentric product A₇*A₈ and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1086). (Randy Hutson, November 30, 2018)

X(1086) lies on the Steiner inellipse and 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) = midpoint of X(i) and X(j) for these (i,j): (2,903), (7,673), (75,335), (239,320)
X(1086) = isogonal conjugate of X(1252)
X(1086) = isotomic conjugate of X(1016)
X(1086) = complement of X(190)
X(1086) = crosspoint of X(2) and X(514)
X(1086) = crosssum of X(i) and X(j) for these (i,j): (6,101), (9,1018), (32,692), (219,906)
X(1086) = crossdifference of every pair of points on line X(101)X(692)
X(1086) = perspector of circumconic centered at X(514)
X(1086) = X(2)-Ceva conjugate of X(514)
X(1086) = projection from Steiner circumellipse to Steiner inellipse of X(903)
X(1086) = trilinear pole of line X(764)X(1647)
X(1086) = trilinear pole wrt medial triangle of Nagel line
X(1086) = anticomplement of X(4422)
X(1086) = barycentric product X(5997)*X(5998)
X(1086) = barycentric square of X(514)
X(1086) = {X(3661),X(3662)}-harmonic conjugate of X(17227)

X(1087) = TRILINEAR SQUARE OF X(5)

X(1087) lies on these lines: 1,564   5,2599   31,91   54,2595    92,255

X(1087) = {X(1),X(564)}-harmonic conjugate of X(1109)
X(1087) = {X(2595),X(2596)}-harmonic conjugate of X(54)

X(1088) = TRILINEAR SQUARE OF X(7)

Let A'B'C' be the cross-triangle of the inner and outer Soddy triangles. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1088). (Randy Hutson, December 10, 2016)

X(1088) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X(11019). (Randy Hutson, October 15, 2018)

X(1088) = isogonal conjugate of X(1253)
X(1088) = isotomic conjugate of X(200)
X(1088) = X(7)-cross conjugate of X(85)
X(1088) = cevapoint of X(7) and X(279)

X(1089) = TRILINEAR SQUARE OF X(10)

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(1089) = crosspoint of X(313) and X(321)
X(1089) = trilinear product of vertices of outer Garcia triangle

X(1090) = TRILINEAR SQUARE OF X(11)

X(1090) lies on these lines: 5,1091   11,523

X(1091) = TRILINEAR SQUARE OF X(12)

X(1091) lies on these lines: 5,1090   12,1109

X(1092) = TRILINEAR CUBE OF X(3)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the circumcevian triangle of X(3) at X(1092). (Randy Hutson, June 7, 2019)

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) = reflection of X(1204) in X(3)
X(1092) = isogonal conjugate of X(1093)
X(1092) = X(92)-isoconjugate of X(393)

X(1093) = TRILINEAR CUBE OF X(4)

X(1093) = isogonal conjugate of X(1092)
X(1093) = isotomic conjugate of X(3964)
X(1093) = X(235)-cross conjugate of X(4)
X(1093) = polar conjugate of X(394)
X(1093) = X(978)-of-orthic-triangle if ABC is acute

X(1094) = TRILINEAR SQUARE OF X(15)

X(1094) lies on these lines: 15,36   48,163

X(1095) = TRILINEAR SQUARE OF X(16)

X(1095) lies on these lines: 16,36   48,163

X(1096) = TRILINEAR SQUARE OF X(19)

X(1096) lies on these lines: 1,29   4,1039   19,31   25,1402   33,42   34,207   63,240   107,741   213,607   243,1040   278,614   281,612

X(1096) = isogonal conjugate of X(326)
X(1096) = X(158)-Ceva conjugate of X(19)
X(1096) = crosssum of X(394) and X(1259)
X(1096) = polar conjugate of X(304)
X(1096) = crossdifference of every pair of points on line X(822)X(4131)
X(1096) = trilinear product X(33)*X(34)

X(1097) = TRILINEAR SQUARE OF X(20)

X(1097) lies on these lines: 1,75   31,775

X(1098) = TRILINEAR SQUARE OF X(21)

X(1098) lies on these lines: 3,662   8,643   21,60   29,270   58,86   65,409   81,1104

X(1098) = isogonal conjugate of X(1254)
X(1098) = cevapoint of X(i) and X(j) for these (i,j): (1,411), (21,283)

X(1099) = TRILINEAR SQUARE OF X(30)

X(1099) lies on the inellipse centered at X(10) and on these lines: 1,564   75,811   162,255

X(1100) = COMPLEMENT OF X(319)

X(1100) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(37)

X(1100) = QA-P16 (QA-Harmonic Center) of quadrangle ABCX(1); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/44-qa-p16.html

X(1100) lies on these lines: 1,6   2,319   36,1030   48,354   65,604   71,583   81,593   86,239   214,1015   284,501   517,572   519,594   536,894   820,836

X(1100) is the {X(1),X(6)}-harmonic conjugate of X(37). For a list of other harmonic conjugates of X(1100), click Tables at the top of this page.

X(1100) = isogonal conjugate of X(1255)
X(1100) = isotomic conjugate of X(32018)
X(1100) = complement of X(319)
X(1100) = anticomplement of X(17239)
X(1100) = crosspoint of X(i) and X(j) for these (i,j): (1,81), (2,79)
X(1100) = crosssum of X(i) and X(j) for these (i,j): (1,37), (6,35), (559,1082)
X(1100) = crossdifference of every pair of points on line X(484)X(513)
X(1100) = bicentric sum of PU(31)
X(1100) = midpoint of PU(31)
X(1100) = reflection of X(3775) in X(1125)
X(1100) = X(1)-Ceva conjugate of X(1962)
X(1100) = X(1962)-Ceva conjugate of X(1125) wrt incentral triangle
X(1100) = perspector wrt incentral triangle of bicevian conic of X(1) and X(2)
X(1100) = polar conjugate of isogonal conjugate X(23201)
X(1100) = polar conjugate of isotomic conjugate of X(3916)

X(1101) = TRILINEAR SQUARE OF X(110)

X(1101) lies on these lines: 59,60   163,798   656,662

X(1101) = isogonal conjugate of X(1109)
X(1101) = cevapoint of X(i) and X(j) for these (i,j): (31,163), (60,110)
X(1101) = X(i)-cross conjugate of X(j) for these (i,j): (31,163), (47,162)
X(1101) = isotomic conjugate of X(23994)
X(1101) = trilinear pole of line X(163)X(1983)
X(1101) = X(92)-isoconjugate of X(3708)

X(1102) = TRILINEAR CUBE OF X(63)

X(1102) lies on these lines: 63,304   255,326

X(1103) = TRILINEAR SQUARE OF X(40)

X(1103) lies on these lines: 1,2   31,937   40,221   46,269   165,255

X(1103) = isogonal conjugate of X(1256)

X(1104) = CROSSPOINT OF X(1) AND X(28)

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(1104) = isogonal conjugate of X(1257)
X(1104) = crosspoint of X(i) and X(j) for these (i,j): (1,28), (81,269)
X(1104) = crosssum of X(i) and X(j) for these (i,j): (1,72), (37,200)

X(1105) = CEVAPOINT OF X(3) AND X(4)

Let A'B'C' be the cevian triangle of X(3). Let L_A be the reflection of the line B'C' in the line BC, and define L_B and L_C cyclically. Let A'' = L_B∩L_C, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(1105). (Randy Hutson, 9/23/2011)

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(1105) = cevapoint of X(3) and X(4)
X(1105) = trilinear pole of line X(450)X(2451)

X(1106) = TRILINEAR SQUARE OF X(56)

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(1106) = complement of anticomplementary conjugate of X(17480)
X(1106) = X(92)-isoconjugate of X(3692)
X(1106) = trilinear product of vertices of tangential mid-arc triangle

X(1107) = CROSSPOINT OF X(1) AND X(274)

X(1107) =(tan ω sin 2ω)R*X(10) + r*X(39)
X(1107) = X(1) + X(8) + (cot ω csc 2ω)(2r/R)*X(39)

X(1107) = midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(213)
X(1107) = insimilicenter of the Spieker and (1/2)-Moses circle. [The (1/2)-Moses circle is described at X(1575).]

X(1107) lies on these lines: 1,6   2,330   10,39   32,993   75,194   210,869   239,257

X(1107) = isogonal conjugate of X(1258)
X(1107) = isotomic conjugate of X(1221)
X(1107) = complement of X(1909)
X(1107) = crosspoint of X(i) and X(j) for these (i,j): (1,274), (2,256), (81,87)
X(1107) = crosssum of X(i) and X(j) for these (i,j): (1,213), (6,171), (37,43)
X(1107) = polar conjugate of isogonal conjugate of X(22389)
X(1107) = {X(1),X(9)}-harmonic conjugate of X(2176)
X(1107) = {X(10),X(39)}-harmonic conjugate of X(1575)

X(1108) = CROSSPOINT OF X(2) AND X(84)

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(1108) = crosspoint of X(i) and X(j) for these (i,j): (1,278), (2,84)
X(1108) = crosssum of X(i) and X(j) for these (i,j): (1,219), (6,40)
X(1108) = polar conjugate of isogonal conjugate of X(23204)
X(1108) = {X(1),X(9)}-harmonic conjugate of X(2256)

X(1109) = TRILINEAR SQUARE OF X(523)

X(1109): Let A'B'C' be the Feuerbach triangle. Let La be the trilinear polar of A', and define Lb, Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let Ma be the tangent to conic {{A,B,C,B',C'}} at A, and define Mb and Mc cyclically. Let A* = Mb∩Mc, B* = Mc∩Ma, C* = Ma∩Mb. The lines AA*, BB*, CC* concur in X(1109); see also X(523). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let Ab = BC∩C'A', and define Bc and Ca cyclically. Let Ac = BC∩A'B', and define Ba and Cb cyclically. The points Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse, denoted by E. Let A" be the intersection of the tangents to E at Ba and Ca, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let La be the A-extraversion of line X(2610)X(4024) (the trilinear polar of X(12)), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1109). (Randy Hutson, January 29, 2018)

Let F be the Feuerbach point, X(11), and FaFbFc be the Feuerbach triangle (the extraversion triangle of X(11)). Let A' be the trilinear product F*Fa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1109). (Randy Hutson, January 29, 2018)

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(1109) = complement of X(6758)
X(1109) = anticomplement of X(16598)
X(1109) = crosspoint of X(12) and X(523)
X(1109) = crosssum of X(i) and X(j) for these (i,j): (31,163), (60,110)
X(1109) = trilinear product X(11)*X(12)
X(1109) = polar conjugate of isogonal conjugate of X(3708)
X(1109) = antipode of X(4736) in the inellipse centered at X(10)
X(1109) = reflection of X(4736) in X(10)
X(1109) = crossdifference of every pair of points on line X(163)X(1983)
X(1109) = bicentric difference of PU(73)
X(1109) = PU(73)-harmonic conjugate of X(2624)
X(1109) = polar conjugate of isotomic conjugate of X(20902)
X(1109) = {X(1),X(564)}-harmonic conjugate of X(1087)

X(1110) = TRILINEAR SQUARE OF X(101)

X(1110) lies on these lines: 1,1053   36,59   101,663   249,849   667,692

X(1110) = isogonal conjugate of X(1111)
X(1110) = X(i)-cross conjugate of X(j) for these (i,j): (32,163), (41,101)
X(1110) = crosssum of X(11) and X(1086)

X(1111) = TRILINEAR SQUARE OF X(514)

Let A₇B₇C₇ and A₈B₈C₈ be the Gemini triangles 7 and 8. Let A' be the trilinear product A₇*A₈ and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1111). (Randy Hutson, November 30, 2018)

Let A₇B₇C₇ and A₈B₈C₈ be Gemini triangles 7 and 8, resp. Let A' be the intersection of the tangent to the {ABC, Gemini 7}-circumconic at A₇ and the tangent to the {ABC, Gemini 8}-circumconic at A₈. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1111). (Randy Hutson, January 15, 2019)

X(1111) lies the inellipse centered at X(10) and 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(1111) = crosssum of X(31) and X(692)
X(1111) = antipode of X(4712) in inellipse centered at X(10)
X(1111) = reflection of X(4712) in X(10)
X(1111) = trilinear product X(5997)*X(5998)
X(1111) = perspector of side- and vertex-triangles of Gemini triangles 7 and 8
X(1111) = trilinear product of vertices of Gemini triangle 7
X(1111) = trilinear product of vertices of Gemini triangle 8

X(1112) = CROSSPOINT OF X(4) AND X(250)

If you have The Geometer's Sketchpad, you can view X(1112).

X(1112) is the center of the hyperbola 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 Clark Kimberling, "Conics associated with a cevian nest," Forum Geometricorum 1 (2001) 141-150; see Example 2.)

X(1112) is X(11)-of-the-orthic-triangle if ABC is acute. (Peter Moses, July 7, 2009)

X(1112) lies on these lines: 4,94   6,1177   25,110   51,125   52,113   389,974   428,542   468,511

X(1112) = reflection of X(974) in X(389)
X(1112) = crosspoint of X(4) and X(250)
X(1112) = crosssum of X(3) and X(125)
X(1112) = inverse-in-polar-circle of X(3448)
X(1112) = polar conjugate of isotomic conjugate of X(34990)
X(1112) = inverse-in-orthosymmedial-circle of X(427)
X(1112) = excentral-to-ABC functional image of X(11)
X(1112) = intersection of tangents to Walsmith rectangular hyperbola at X(110) and X(125)

X(1113) = 1st EULER-LINE-CIRCUMCIRCLE INTERSECTION

As a point on the Euler line, X(1113) has Shinagawa coefficients (R - |OH|, -3R + |OH|).

X(1113) is a point of intersection of the Euler line and the circumcircle. The other is X(1114). Of the two, X(1113) is the one closer to X(4).

X(1113) is one of 2 points P such that P is the circumcircle-antipode of Λ (trilinear polar of P); the other is X(1114). (Randy Hutson, November 2, 2017)

If you have The Geometer's Sketchpad, you can view X(1113) and X(1114).

X(1113) lies on these lines: {1, 2100}, {2, 3}, {6, 2104}, {11, 10781}, {54, 14374}, {74, 2575}, {98, 2593}, {100, 2580}, {101, 2576}, {108, 2586}, {109, 1822}, {110, 2574}, {111, 8106}, {112, 8105}, {165, 2101}, {187, 8426}, {511, 2105}, {517, 2103}, {691, 9173}, {759, 2589}, {1495, 13415}, {2249, 2579}, {2777, 14500}, {5840, 10782}, {5972, 14499}, {10287, 10686}

X(1113) = reflection of X(i) in X(j) for these (i,j): (4,1312), (1114,3)
X(1113) = isogonal conjugate of X(2574)
X(1113) = isotomic conjugate of X(22339)
X(1113) = anticomplement of X(1313)
X(1113) = X(250)-Ceva conjugate of X(1114)
X(1113) = trilinear product X(110)*X(1823)
X(1113) = trilinear pole of line X(6)X(1345) (the major axis of the orthic inconic)
X(1113) = Ψ(X(6), X(1345))
X(1113) = pole wrt polar circle of trilinear polar of X(2592) (line X(523)X(1313))
X(1113) = polar conjugate of X(2592)
X(1113) = inverse-in-polar-circle of X(1313)
X(1113) = Thomson isogonal conjugate of X(2575)
X(1113) = Lucas isogonal conjugate of X(2575)

X(1114) = 2nd EULER-LINE-CIRCUMCIRCLE INTERSECTION

X(1114) = 3X(2) - (3 + |OH|/R)*X(3) = - (1 + |OH|/R)*X(3) + X(4)

As a point on the Euler line, X(1114) has Shinagawa coefficients (R + |OH|, -3R - |OH|).

X(1114) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1113).

X(1114) is one of 2 points P such that P is the circumcircle-antipode of Λ (trilinear polar of P); the other is X(1113). (Randy Hutson, November 2, 2017)

X(1114) lies on these lines: {1, 2101}, {2, 3}, {6, 2105}, {11, 10782}, {54, 14375}, {74, 2574}, {98, 2592}, {100, 2581}, {101, 2577}, {108, 2587}, {109, 1823}, {110, 2575}, {111, 8105}, {112, 8106}, {165, 2100}, {187, 8427}, {511, 2104}, {517, 2102}, {691, 9174}, {759, 2588}, {1379, 14899}, {1495, 13414}, {2249, 2578}, {2777, 14499}, {5840, 10781}, {5972, 14500}, {10288, 10687}

X(1114) = reflection of X(i) in X(j) for these (i,j): (4,1313), (1113,3)
X(1114) = isogonal conjugate of X(2575)
X(1114) = isotomic conjugate of X(22340)
X(1114) = anticomplement of X(1312)
X(1114) = X(250)-Ceva conjugate of X(1113)
X(1114) = trilinear product X(110)*X(1822) X(1114) = trilinear pole of line X(6)X(1344) (the minor axis of the orthic inconic)
X(1114) = Ψ(X(6), X(1344))
X(1114) = pole wrt polar circle of trilinear polar of X(2593) (line X(523)X(1312))
X(1114) = polar conjugate of X(2593)
X(1114) = inverse-in-polar-circle of X(1312)
X(1114) = Thomson isogonal conjugate of X(2574)
X(1114) = Lucas isogonal conjugate of X(2574)

Centers 1115-1150

X(1115) = EXTERIOR-ANGLE CURVATURE CENTROID

X(1115) = 9 X[2] - 5 X[18295], 3 X[360] - 5 X[18295], 6 X[18294] - 5 X[18295]

X(1115) is the center of mass of a point-mass system obtained by placing at vertex A a mass equal to the magnitude of the exterior angle (that's π - A) at A, and cyclically for B and C. (Peter Scales, Hyacinthos #5528, 5/22/02) This description is also found in Honsberger, Episodes in Nineteenth and Twentieth Century Eulidean Geometry, p. 120, where it is mistakenly attributed to the Steiner point, X(99).

X(1115) should not be confused with Jakob Steiner's actual "Curvature Centroid" (discovered in 1825), applicable to all polygons, obtained as the weighted sum of vertices with weights determined by sines of doubled vertex angles [1]. Remarkably, the pedal polygon of a polygon with respect to this point has extremal area: it is minimal (resp. maximal) if the sign of the sum of sines of double angles is negative (resp. positive) [1]. For a triangle, this point is the circumcenter X(3) and the extremal pedal triangle is the medial triangle. (Dan Reznik, July 7, 2020).

[1] Jakob Steiner, "Krümmungs Schwerpunkt ebener Curven", Abhandlungen der Königlich Preussischen Akademie der Wissenschaften, 1838. (Dan Reznick, July 7, 2020)

X(1115) lies on this line: 2,360

X(1115) = reflection of X(360) in X(18294)
X(1115) = isogonal conjugate of X(7021)
X(1115) = complement of X(360)
X(1115) = anticomplement of X(18294)
X(1115) = complement of the isogonal conjugate of X(359)
X(1115) = X(i)-complementary conjugate of X(j) for these (i,j): {359, 10}, {1077, 141}
X(1115) = X(i)-isoconjugate of X(j) for these (i,j): {1, 7021}, {6, 7041}
X(1115) = barycentric product X(75)*X(7039)
X(1115) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7041}, {6, 7021}, {7039, 1}, {7041, 7036}, {7044, 7039}
X(1115) = {X(2),X(360)}-harmonic conjugate of X(18294)

X(1116) = CENTER OF THE LESTER CIRCLE

Barycentrics    (b^2 - c^2)*(-2*a^8 + 5*a^6*b^2 - 3*a^4*b^4 - a^2*b^6 + b^8 + 5*a^6*c^2 - 8*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 4*b^6*c^2 - 3*a^4*c^4 + 4*a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) : :
X(1116) = 2 X[15543] + X[18308]

The Lester circle passes through the points X(3), X(5), X(13), X(14). Coordinates of the center were determined by Milorad Stevanovic (#5895, 9/20/02). The circle is described in

June Lester, "Triangles III: complex centre functions and Ceva's theorem," Aequationes Mathematicae 53 (1997) 4-35.

The appearance of i in the following list means that X(i) lies on the Lester circle: 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365

If you have The Geometer's Sketchpad, you can view X(1116).

X(1116) lies on these lines: {{3, 15475}, {4, 39606}, {5, 15543}, {115, 125}, {140, 523}, {381, 32478}, {512, 5892}, {1510, 13363}, {3566, 39482}, {5664, 23105}, {6644, 39481}, {20184, 20299}, {38609, 38611}, {39504, 39512}

X(1116) = midpoint of X(i) and X(j) for these {i,j}: {3, 15475}, {5, 15543}, {5664, 23105}
X(1116) = reflection of X(18308) in X(5)
X(1116) = tripolar centroid of X(13582)
X(1116) = crossdifference of every pair of points on line {110, 11063}
X(1116) = pole wrt orthocentroidal circle of Napoleon axis (line X(6)X(17))

X(1117) = POINT ALRESCHA

Barycentrics   (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (a^6-a^4 b^2-a^2 b^4+b^6-3 a^4 c^2+a^2 b^2 c^2-3 b^4 c^2+3 a^2 c^4+3 b^2 c^4-c^6) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-a^4 c^2+a^2 b^2 c^2+3 b^4 c^2-a^2 c^4-3 b^2 c^4+c^6) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-4 a^6 c^2+a^4 b^2 c^2+a^2 b^4 c^2+2 b^6 c^2+6 a^4 c^4+a^2 b^2 c^4-6 b^4 c^4-4 a^2 c^6+2 b^2 c^6+c^8) : :

X(1117) lies on the Lester circle. See Bernard Gibert's message, Hyacinthos #5613, 5/31/02.

X(1117) on the Lester circle, the cubic K060, and these lines: {5,3470}, {30,5671}, {265,13582}, {11071,11581}

X(1117) = X(13582)-Ceva conjugate of X(11071)
X(1117) = cevapoint of X(3471) and X(5671)
X(1117) = isogonal conjugate of inverse-in-circumcircle of isogonal conjugate of X(399)
X(1117) = antigonal conjugate of X(399)
X(1117) = syngonal conjugate of X(10264)
X(1117) = barycentric product X(1272)*X(11071)
X(1117) = barycentric product X(1989)X(1272)X(13582)
X(1117) = barycentric quotient X(i)/X(j) for these {i,j}: {11071, 1138}, {11074, 3470}

X(1118) = 1st HATZIPOLAKIS PERSPECTOR

Let A'B'C' be the intouch triangle of ABC. Let C_A be the point other than C' in which the perpendicular to BC from C' meets the incircle, let B_A be the point other than B' in which the perpendicular to BC from B' meets the incircle, and let A₀ be the point of intersection of lines BC_A and CB_A. Define B₀ and C₀ cyclically. Then triangle A₀B₀C₀ is perspective to ABC, and the perspector is X(1118). (Antreas Hatzipolakis, #5321, 4/30/02)

X(1118) is the trilinear product A₀*B₀*C₀, where A₀, B₀, C₀ are as defined above. (Randy Hutson, January 15, 2019)

X(1118) lies on the hyperbola {{A,B,C,X(4),X(19)}} these lines: 4,65   7,286   12,281   19,208   20,243   24,108   28,56   34,207   92,388

X(1118) = isogonal conjugate of X(1259)
X(1118) = isotomic conjugate of X(1264)
X(1118) = X(63)-isoconjugate of X(219)
X(1118) = polar conjugate of X(345)

X(1119) = 2nd HATZIPOLAKIS PERSPECTOR

Let triangle A₀B₀C₀ be as defined for X(1118). Let A₁ be the orthogonal projection of A₀ onto line BC, and define B₁ and C₁ cyclically. Then triangle A₁B₁C₁ is perspective to ABC, and the perspector is X(1119). (Antreas Hatzipolakis, #5321, 4/30/02)

X(1119) lies on the hyperbola {{A,B,C,X(4),X(19)}} these lines: 3,347   4,7   19,57   28,279   34,269   142,281   393,1086   579,1020   915,934

X(1119) = isogonal conjugate of X(1260)
X(1119) = isotomic conjugate of X(1265)
X(1119) = X(34)-cross conjugate of X(278)
X(1119) = polar conjugate of X(346)

X(1120) = X(2)-BLAIKIE TRANSFORM OF X(1)

For the definition of Blaikie transform, see X(903).

X(1120) lies on these lines: 6,644   56,100   58,643   106,519  269,664

X(1120) = isogonal conjugate of X(1149)
X(1120) = isotomic conjugate of X(1266)
X(1120) = trilinear pole of line X(9)X(649)

X(1121) = X(7)-BLAIKIE TRANSFORM OF X(2)

For the definition of Blaikie transform, see X(903).

X(1121) lies on the Steiner circumellipse and these lines: 2,664   8,190   29,648   99,333   312,668   519,666   903,918

X(1121) = reflection of X(i) in X(j) for these (i,j): (2,1146), (664,2)
X(1121) = isogonal conjugate of X(1055)
X(1121) = isotomic conjugate of X(527)
X(1121) = polar conjugate of X(23710)
X(1121) = complement of X(39357)
X(1121) = anticomplement of X(35110)
X(1121) = Steiner-circumellipse-antipode of X(664)
X(1121) = projection from Steiner inellipse to Steiner circumellipse of X(1146)
X(1121) = antipode of X(8) in hyperbola {{A,B,C,X(2),X(8)}}
X(1121) = trilinear pole of line X(2)X(522)

X(1122) = 1st GRINBERG POINT

Let A_B be the touchpoint of the A-excircle and line AB, let A_C be the touchpoint of the A-excircle and line AC, and let M_A be the midpoint of segment A_BA_C. Define M_B and M_C cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles M_AM_BM_C and A'B'C' are perspective, and the perspector is X(1122). (Darij Grinberg, 12/28/02)

If you have The Geometer's Sketchpad, you can view X(1122).

X(1122) lies on these lines: 7,8   56,269

X(1122) = isogonal conjugate of X(1261)
X(1122) = crosspoint of X(7) and X(269)
X(1122) = crosssum of X(55) and X(200)

X(1123) = PAASCHE POINT

Let D and E be the congruent circles each tangent to the other and to line BC, with D also tangent to line AB and E also tangent to line CA, meeting in a point A' lying outside triangle ABC. Define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(1123). See

Ivan Paasche, Aufgabe P 933, Praxis der Mathematik 1 (1990), page 40.

Let P_A be the parabola with focus A and directrix BC, and let L_A be the line of the points of intersection of P_A with the segments AB and AC. Define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1123). (Randy Hutson, 9/23/2011)

Suppose that ABC is an acute triangle. Let o_A be the circle with diameter BC. Let w_A be the circle tangent to segments AB and AC and also externally tangent to o_A, in point X_A. Define X_B and X_C cyclically. The lines AX_A, BX_B, CX_C concur in X(1123). (Tomasz Cieśla, 19 January 2013) For a related construction, see X(1336).

In Hutson's construction of X(1123) given above, the parabola P_A meets segments AB and AC in two points, here denoted by Ab and Ac. Cyclically, the parabolas P_B and P_C determine four more points; the six points are then Ab = 0 : 2R : c, Ac = 0 : b : 2R, Bc = a : 0 : 2R, Ba = 2R : 0 : c, Ca = 2R : b : 0, Cb = a : 2R : 0. These points lie on a conic, named the Paasche conic at X(37861). (Vijay Krishna, April 14, 2020)

For a construction and relationships to other points, see the preamble just before X(37994).

If you have The Geometer's Sketchpad, you can view X(1123)

X(1123) lies on these lines: 1,3069   2,586   37,158   57,482   81,1335   498,3302   499,3300   920,3068

X(1123) = isogonal conjugate of X(1124)
X(1123) = isotomic conjugate of X(1267)
X(1123) = polar conjugate of isogonal conjugate of X(34121)

X(1124) = ISOGONAL CONJUGATE OF X(1123)

X(1124) lies on these lines: 1,6   3,2066   11,485   12,486   35,1152   36,1151   42,494   55,372   56,371   176,651   255,605   498,615   499,590

X(1124) = isogonal conjugate of X(1123)
X(1124) = isotomic conjugate of polar conjugate of X(34125)
X(1124) = {X(1),X(6)}-harmonic conjugate of X(1335)
X(1124) = X(19)-isoconjugate of X(13387)
X(1124) = insimilicenter of incircle and 2nd Lemoine circle

X(1125) = COMPLEMENT OF X(10)

The centroid of four points A,B,C,P is the complement of the complement of P with respect to triangle ABC. As an example, X(1125) is the centroid of {A,B,C,X(1)}. (Darij Grinberg, 12/28/02)

Let A' the midpoint of segment BC and let A'' be the midpoint of segment AA'. Define B'' and C'' cyclically. The triangle A''B''C'' is homothetic to ABC, and the center of homothety is X(1125).

Let I be the incenter of triangle ABC and A' the centroid of triangle BCI, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC, and the center of homothety is X(1125).

X(1125) is the center of the ellipse which is the locus of centers of the conics passing through A, B, C, and X(1). This ellipse is also the locus of crosssums of the intersections of the circumcircle and lines through X(1). Furthmore, this ellipse is the bicevian conic of X(1) and X(2) (i.e. the conic which passes through the vertices of the incentral and medial triangles). The ellipse passes through X(11), X(214), X(244), X(1015) and the midpoints of the sides of ABC. (Randy Hutson, 8/13/2011, Hyacinthos #20179; see also #20181, by Chris van Tienhoven.)

A construction of X(1125) is given by Antreas Hatipolakis and Angel Montesdeoca at 24185.

Let A'B'C' be the incentral triangle. Let A" be the reflection of A in A', and define B" and C" cyclically. Let A* be the trilinear pole of line B"C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(1125). (Randy Hutson, July 21, 2017)

Let A'B'C' be the 2nd circumperp triangle. Let A"B"C" be the triangle bounded by the Simson lines of A', B', C'. A"B"C" is homothetic to A'B'C' at X(1125). (Randy Hutson, July 21, 2017)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

For an other construction of the point X(1125) see Antreas Hatzipolakis and Peter Moses, euclid 5220.

X(1125) lies on these lines: 1,2   3,142   5,515   11,214   21,36   33,475   34,406   35,404   37,39   40,631   55,474   56,226   58,86   65,392   72,354   105,831   114,116   140,517   165,962   171,595   274,350   409,759   443,497   749,984   758,942   958,999   1015,1107

X(1125) is the {X(1),X(2)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1125), click Tables at the top of this page.

X(1125) = midpoint of X(i) and X(j) for these (i,j): (1,10), (2,551), (3,946), (5,1385), (8,3244), (11,214), (142,1001), (226,993), (942,960), (999,3452), (1100,3775), (4065,4647)
X(1125) = isogonal conjugate of X(1126)
X(1125) = isotomic conjugate of X(1268)
X(1125) = complement of X(10)
X(1125) = anticomplement of X(3634)
X(1125) = crosspoint of X(2) and X(86)
X(1125) = crosssum of X(6) and X(42)
X(1125) = perspector of circumconic centered at X(1213)
X(1125) = center of circumconic that is locus of trilinear poles of lines passing through X(1213)
X(1125) = center of bicevian conic of X(1) and X(2)
X(1125) = Kosnita(X(1),X(2)) point
X(1125) = X(1)-Ceva conjugate of X(4065)
X(1125) = X(2)-Ceva conjugate of X(1213)
X(1125) = X(214)-of-X(1)-Brocard triangle
X(1125) = complement of X(4065) wrt incentral triangle
X(1125) = trilinear product of vertices of anti-Aquila triangle
X(1125) = X(10110)-of-excentral-triangle
X(1125) = {X(1),X(8)}-harmonic conjugate of X(3244)
X(1125) = {X(2),X(10)}-harmonic conjugate of X(3634)
X(1125) = {X(8),X(10)}-harmonic conjugate of X(4691)
X(1125) = perspector of Gemini triangle 11 and cross-triangle of ABC and Gemini triangle 11
X(1125) = homothetic center of anticomplementary triangle and cross-triangle of Aquila and anti-Aquila triangles
X(1125) = trilinear pole of line X(4969)X(4976) (the perspectrix of ABC and Gemini triangle 12)
X(1125) = polar conjugate of isogonal conjugate of X(22054)
X(1125) = perspector of medial triangle and n(Medial)*n(Incentral) triangle

X(1126) = ISOGONAL CONJUGATE OF X(1125)

X(1126) lies on these lines: 1,748   6,595   10,86   35,42   56,181   145,996   830,1027

X(1126) = isogonal conjugate of X(1125)
X(1126) = isotomic conjugate of X(1269)
X(1126) = cevapoint of X(6) and X(42)
X(1126) = X(512)-cross conjugate of X(101)
X(1126) = X(92)-isoconjugate of X(22054)
X(1126) = perspector of ABC and unary cofactor triangle of Gemini triangle 12

X(1127) = 1st DE VILLIERS POINT

Let A', B', C' be the incenters of triangles XBC, XCA, XAB, respectively, where X is the incenter, X(1). The triangle A'B'C' is perspective to ABC, and the perspector is X(1127). Coordinates found by Darij Grinberg, 8/22/02. (The triangle A'B'C' is the BCI triangle.)

Michael de Villiers,A dual to Kosnita's theorem, reprinted from Mathematics & Informatics Quarterly 6 (1996) 1996.

X(1127) lies on this line: 174,481

X(1127) = isogonal conjugate of X(1129)

X(1128) = 2nd DE VILLIERS POINT

Let A",B",C" be the excenters of ABC, and let A', B', C' be the respective incenters of triangles A"BC, B"CA, C"AB, respectively. The triangle A'B'C' is perspective to ABC, and the perspector is X(1128). (Darij Grinberg, 8/22/02). See references at X(1127).

X(1128) lies on these lines: 164,173   188,519   258,505

X(1128) = isogonal conjugate of X(1130)

X(1129) = ISOGONAL CONJUGATE OF X(1127)

Let A', B', C' be as at X(1127). Let A" = BC' ∩ CB', B" = CA' ∩ AC', C" = AB' ∩ BA'. The lines AA", BB", CC" concur in X(1129). Note: A'B'C' and A"B"C" are analogous to the 1st Morley triangle and adjunct Morley triangle, substituting angle quadrisectors for angle trisectors. (Randy Hutson, January 29, 2018)

X(1129) lies on this line: 1,168

X(1129) = isogonal conjugate of X(1127)
X(1129) = Hofstadter 3/4 point
X(1129) = perspector of ABC and cross-triangle of ABC and BCI triangle

X(1130) = 1st STEVANOVIC POINT

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B),
                        where g(A,B,C) = 1 + 2 sin(A/2)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let A" = BC'∩CB', and define B", C" cyclically. The lines AA", BB", CC" concur in X(1130). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1488) and X(1489).)

X(1130) lies on these lines: 1,164   173,505

X(1130) = isogonal conjugate of X(1128)

X(1131) = ARCTAN(2) KIEPERT POINT

To construct the Vecten point, X(485), squares are erected outward on the sides of ABC. If A', B', C' are the centers of these squares, then triangle A'B'C' is perspective to ABC with perspector X(485). Now let A" be the midpoint of the side of the A-square that does not touch line BC, and define B" and C" cyclically. Then triangle A"B"C" is perspective to ABC with perspector X(1131). Angle(A"BC) = angle(B"CA) = angle(C"AB), so that X(1131) lies on the Kiepert hyperbola. Here, the common angle is arctan(2). (Darij Grinberg 9/22/02)

If you have The Geometer's Sketchpad, you can view X(1131).

X(1131) lies on these lines: 2,490   6,1132   4,3311   20,485   175,226

X(1131) = isogonal conjugate of X(1151)
X(1131) = isotomic conjugate of X(1270)
X(1131) = anticomplement of X(33364)
X(1131) = polar conjugate of X(3535)
X(1131) = {X(6),X(3832)}-harmonic conjugate of X(1132)

X(1132) = ARCTAN(-2) KIEPERT POINT

Barycentrics    1/(2 SA - S) : :

X(1132) is constructed in the manner for X(1131), using squares erected inward, so that the three equal angles have common measure arctan(-2), and X(1132) lies on the Kiepert hyperbola. (Darij Grinberg 9/22/02)

If you have The Geometer's Sketchpad, you can view X(1132).

X(1132) lies on these lines: 2,489   4,3312   6,1131   20,486   176,226

X(1132) = isogonal conjugate of X(1152)
X(1132) = isotomic conjugate of X(1271)
X(1132) = anticomplement of X(33365)
X(1132) = polar conjugate of X(3536)
X(1132) = {X(6),X(3832)}-harmonic conjugate of X(1131)

X(1133) = BURGESS POINT

Rotate line BC about B away from A through angle B/3, and rotate line BC about C away from A through angle C/3; let A' be the point in which the two rotated lines meet. Define B' and C' cyclically. Let A" be the point of intersection of lines BC' and B'C, and define B" and C" cyclically. The lines AA', BB', CC' concur in X(357), and AA", BB", CC" concur in X(358). The first of these with reference to triangle A'B'C' is X(1133); i.e., X(1133) = X(357)-of-A'B'C'.

A. G. Burgess, "Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides,"Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 58-64; page 63.

X(1133) = X(3273)-isoconjugate of X(3602)

X(1134) = 3rd MORLEY-TAYLOR-MARR CENTER

F. Glanville Taylor and W. L. Marr, "The six trisectors of each of the angles of a triangle," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 119-131; especially item 9, p. 127.

If you have The Geometer's Sketchpad, you can view X(1134).

X(1134) lies on these lines: 356,1135   357,3275

X(1134) = isogonal conjugate of X(1135)
X(1134) = perspector of ABC and 3rd Morley triangle
X(1134) = trilinear product of vertices of 3rd Morley triangle

X(1135) = 4th MORLEY-TAYLOR-MARR CENTER

See the reference at X(1134).

If you have The Geometer's Sketchpad, you can view X(1135).

X(1135) lies on these lines: 16,358   356,1134

X(1135) = isogonal conjugate of X(1134)
X(1135) = perspector of ABC and 3rd Morley adjunct triangle
X(1135) = trilinear product of vertices of 3rd Morley adjunct triangle
X(1135) = {X(357),X(3603)}-harmonic conjugate of X(3272)

X(1136) = 5th MORLEY-TAYLOR-MARR CENTER

See the reference at X(1134).

Another construction, by Seiichi Kirikami, appears at Hyacinthos 21423 (January 16, 2013), posted by Chris van Tienhoven.

If you have The Geometer's Sketchpad, you can view X(1136).

X(1136) = isogonal conjugate of X(1137)
X(1136) = perspector of ABC and 2nd Morley triangle
X(1136) = trilinear product of vertices of 2nd Morley triangle

X(1137) = 6th MORLEY-TAYLOR-MARR CENTER

Trilinears       cos(A/3 + π/3) : cos(B/3 + π/3) : cos(C/3 + π/3)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

See the reference at X(1134).

If you have The Geometer's Sketchpad, you can view X(1137).

X(1137) lies on this line: 16,358

X(1137) = isogonal conjugate of X(1136)
X(1137) = perspector of ABC and 2nd Morley adjunct triangle
X(1137) = trilinear product of vertices of 2nd Morley adjunct triangle

X(1138) = ISOGONAL CONJUGATE OF X(399)

There are only two points X such that the pedal triangle of X is similar to the cevian triangle of X. They are X(4) and X(1138). (Jean-Pierre Ehrmann, January 4, 2003)

Let A'B'C' be the anticomplementary triangle of a triangle ABC, and let E_A be the line through A parallel to the Euler line. Let A" be the point of intersection, other than A, of E_A and the circumcircle. Define E_B and E_C cyclically. The locus of a point P such that the Euler line of PBC is parallel to the Euler line of ABC is a conic ABCA'A'' having the midpoint of segment BC as center. The conics, ABCA'A'', ABCB'B'', ABCC'C'' pass through X(1138). (Francisco Javier García Capitán, April 3, 2015: ADGEOM 2458)

X(1138) lies on the following curves and lines: K001 (Neuberg Cubic), K279, K449, K490, K515, K528, K614, Q066, Q105, {1,5677}, {3,3471}, {4,2132}, {15,5624}, {16,5623}, {30,146}, {74,5670}, {186,1990}, {484,3464}, {616,5675}, {617,5674}, {1157,5667}, {1272,3260}, {3258,5627}, {3465,5685}, {3479,5679}, {3480,5678}, {3484,5684}, {5672,5673}

X(1138) = reflection of X(5627) in X(3258)
X(1138) = isogonal conjugate of X(399)
X(1138) = isotomic conjugate of X(1272)
X(1138) = X(30)-Ceva conjugate of X(5670)
X(1138) = X(523)-cevapoint of X(3258)
X(1138) = X(i)-cross conjugate of X(j) for these (i,j): (74,4), (1989,2)
X(1138) = X(i)-vertex conjugate of X(j) for these (i,j): (4,3447), (30,186)
X(1138) = trilinear pole of the line X(526)X(1637)

X(1139) = OUTER PENTAGON POINT

Let A' be the outermost vertex of the regular pentagon erected outward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X(1139). (Steve Sigur and Antreas Hatzipolakis, #5246, 12/31/02; generalizations to n-gons, for odd n, by Milorad R. Stevanovic, #5253, 5256)

If you have The Geometer's Sketchpad, you can view Outer Pentagon Point.

X(1139) lies on these lines: {1,3369}, {3,3370}, {4,3368}, {5,3393}, {6,1140}

X(1139) = isogonal conjugate of X(3396)
X(1139) = X(3394)-cross conjugate of X(3397)

X(1140) = INNER PENTAGON POINT

Barycentrics  1/(cot A - cot 2π/5) : 1/(cot B - cot 2π/5) : 1/(cot C - cot 2π/5)

Let A' be the innermost vertex of the regular pentagon erected inward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X(1140). See references at X(1140).

If you have The Geometer's Sketchpad, you can view Inner Pentagon Point.

X(1140) lies on these lines: {2,3396}, {3,3397}, {4,3395}, {5,3370}, {6,1139}

X(1140) = isogonal conjugate of X(3369)

X(1141) = GIBERT POINT

X(1141) was first noted (Hyacinthos #1498, September 25, 2000) by Bernard Gibert as a point of intersection of the circumcircle and certain cubic, denoted K_n. To define K_n, note first that the Neuberg cubic is the locus of a point M such that the reflections of M in the sidelines of triangle ABC are the vertices of a triangle perspective to ABC. The locus of the perspector is the cubic K_n, and X(1141) is the point, other than A,B,C, in which K_n meets the circumcircle. Also, X(1141) is the perspector when M = X(1157).

In Jean-Pierre Ehrmann and Bernard Gibert, "Special Isocubics," downloadable from Cubics in the Triangle Plane, the point X(1141) is labeled E, barycentrics are given, and it is established that this point also lies on the line X(5)-to-X(110) [listed below as 5,49], two other cubics, and the hyperbola that passes through the points A, B, C, X(4), X(5).

Let A' be the reflection of A in line BC, and define B' and C' cyclically. Let A_B be the reflection of A' in AB, and define A_C, B_C, B_A, C_A, C_B cyclically. Let

       A₁ = BA_B∩CA_C, and define B₁ and C₁ cyclically,
       A₂ = BA_C∩CA_B, and define B₂ and C₂ cyclically,
       A₃ = BB_A∩CC_A, and define B₃ and C₃ cyclically,
       A₄ = BB_C∩CC_B, and define B₄ and C₄ cyclically,
       A₅ = BC_A∩CB_A, and define B₅ and C₅ cyclically,
       A₆ = BC_B∩CB_C, and define B₆ and C₆ cyclically.

Then triangle A_nB_nC_n is perspective to ABC, for n = 1,2,3,4,5,6. The six perspectors are X(1141), X(186), X(4), X(54), X(265), X(5), respectively. (Keith Dean, #4953, 3/12/02; coordinates by Paul Yiu, #4963; summary by Dean, #4971)

X(1141) lies on the conic of {A, B, C, X(3), X(49)}, the conic of {A, B, C, X(6), X(567)}, and the conic of {A, B, C, X(70), X(253), X(254)}.

X(1141) is the antipode of X(930) on the circumcircle, and X(1141) lies on the line of the nine-point center, X(5), and its isogonal conjugate, X(54).

See X(20212) for an additional comment about X(1141); also, 24183.

X(1141) lies on the circumcircle, and cubics K060, K112, K466, K467, K491, the circumconic {{A,B,C,X(4),X(5)}}, and these lines: {2,128}, {3,252}, {4,137}, {5,49}, {20,11671}, {30,1157}, {53,112}, {55,7159}, {56,3327}, {79,109}, {94,96}, {95,99}, {101,7110}, {107,3518}, {140,11016}, {476,2070}, {549,6592}, {621,10409}, {622,10410}, {1303,5890}, {1304,5627}, {1487,10619}, {2166,2222}, {2413,5966}, {3153,10420}, {3459,12254}, {5994,11582}, {5995,11581}, {6069,11464}, {6240,6799}, {7418,9076}, {7731,9512}, {8800,12225}

X(1141) = midpoint of X(20) and X(11671)
X(1141) = reflection of X(i) in X(j) for these (i,j): (4,137), (5,12026), (930,3)
X(1141) = isogonal conjugate of X(1154)
X(1141) = isotomic conjugate of X(1273)
X(1141) = anticomplement of X(128)
X(1141) = X(231)-cross conjugate of X(2)
X(1141) = polar conjugate of X(14918)
X(1141) = antipode of X(4) in hyperbola {{A,B,C,X(4),X(5)}}
X(1141) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,X(4),X(5)}}
X(1141) = Collings transform of X(137)
X(1141) = cevapoint of X(i) and X(j) for these {i,j}: {3,539}, {13,6104}, {14,6105}, {54,1157}, {265,5961}
X(1141) = X(110)-of-Lucas-triangle (defined at X(95))
X(1141) = X(i)-cross conjugate of X(j) for these (i,j): (4,5627), (231,2), (2070,1166), (10412,476), (11063,288)
X(1141) = isoconjugate of X(j) and X(j) for these (i,j): {1,1154}, {2,2290}, {5,6149}, {31,1273}, {63,11062}, {323,1953}, {526,2617}, {662,2081}, {2179,7799}
X(1141) = trilinear pole of line {6,2623}
X(1141) = inverse of X(54) in the circle having diameter OH
X(1141) = X(110)-of-circumorthic-triangle
X(1141) = trilinear pole, wrt circumorthic triangle, of line X(3)X(95)
X(1141) = barycentric product X(i)*X(j) for these {i,j}: {54,94}, {95,1989}, {97,6344}, {264,11077}, {265,275}, {328,8882}, {930,2413}, {2166,2167}
X(1141) = barycentric quotient X(i)/X(j) for these (i,j): (i,j}: (2,1273), (6,1154), (25,11062), (31,2290), (54,323), (94,311), (95,7799), (231,128), (265,343), (275,340), (512,2081), (1989,5), (2148,6149), (2623,526), (6344,324), (8737,6116), (8738,6117), (8882,186), (11060,51), (11071,1263), (11077,3)

X(1142) = 1st MALFATTI-RABINOWITZ POINT

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A', B', C' be the respective centers of the three Malfatti circles of ABC. Let A" be the point of intersection of lines BC' and CB', and define B" and C" cyclically. Then triangle A"B"C" is perspective to ABC, and the perspector is X(1142). (Stanley Rabinowitz, #4610, 12/29/01; coordinates by Paul Yiu, #4614, 12/30/01)

Let A', B', C' be the centers of the Malfatti circles. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1142). The lines A'A", B'B", C'C" concur in X(179). (Randy Hutson, March 21, 2019)

If you have The Geometer's Sketchpad, you can view X(1142) and X(1142) External.

X(1142) lies on this line: 1,179

X(1142) = trilinear product of centers of Malfatti circles

X(1143) = 2nd MALFATTI-RABINOWITZ POINT

Along each side of ABC there is a segment that is a common tangent to two of the three Malfatti circles of ABC. Let A', B', C' be the midpoints of these respective segments. Then triangle A'B'C' isperspective to ABC, and the perspector is X(1143). (Stanley Rabinowitz, #4611, 12/29/01) For coordinates, see Paul Yiu, #4615, 12/30/01, and

Milorad R. Stevanovic, "Triangle Centers Associated with the Malfatti Circles," Forum Geometricorum 3 (2003) 83-93.

Let A' be the touchpoint of the line BC and the incircle of the triangle BCI, where I = incenter of ABC. Define B' and C' cyclically. The line AA', BB', CC' concur in X(1143). (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view X(1143) and X(1143) External.

X(1143) lies on the cubic K200 and these lines: 8,177   174,175   558,1488

X(1143) = isotomic conjugate of X(1274)
X(1143) = X(1489)-cross conjugate of X(2)
X(1143) = {X(8),X(556)}-harmonic conjugate of X(1274)

X(1144) = EHRMANN CONGRUENT SQUARES POINT

Suppose that P is a point inside triangle ABC. Let S_A be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define S_B and S_C cyclically. Then X(1144) is the unique choice of P for which the three squares are congruent. The function L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies 0 < L(a,b,c) < min{a,b,c}. Also, X(1144) lies on the hyperbola {A,B,C,X(1),X(6)}; indeed, X(1144) lies on the open arc from X(1) to the vertex of ABC opposite the shortest side. L(a,b,c) is the common length of the sides of the three squares. (Jean-Pierre Ehrmann, 12/16/01)

See Jean-Pierre Ehrmann, Congruent Inscribed Rectangles.

If you have GeoGebra, you can view X(1144).

If you have The Geometer's Sketchpad, you can view X(1144).

X(1145) = 3rd EHRMANN POINT

Let A',B',C' be the respective excenters of ABC, and let A_B be the projection of A on A'B', let A_C be the projection of A on A'C', and define B_C, B_A, C_A, C_B cyclically. The Euler lines of the three triangles A'A_BA_C, B'B_CB_A, C'C_AC_B concur in X(1145). Also, X(1145) is X(974) of the excentral triangle. (Analogously, X(442) is X(973) of the excentral triangle; see the note at X(442).) Jean-Pierre Ehrmann (#4200, 10/24/01)

X(1145) lies on the Mandart hyperbola and these lines: 2,1000   3,8   9,80   10,11   119,517   144,153   214,519   484,529

X(1145) = midpoint of X(8) and X(100)
X(1145) = reflection of X(i) in X(j) for these (i,j): (11,10), (72,14740), (1317,214), (1320,1387), (1537,119)
X(1145) = isogonal conjugate of X(10428)
X(1145) = anticomplement of X(1387)
X(1145) = outer-Garcia-to-ABC similarity image of X(11)
X(1145) = excentral-to-ABC barycentric image of X(104)
X(1145) = X(974)-of-excentral-triangle
X(1145) = antipode of X(72) in the Mandart hyperbola

X(1146) = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}}

X(1146) lies on the Steiner inellipse, the inconic having perspector X(2052), and these lines: 2,664   6,281   8,220   9,80   101,952   115,124   116,514   169,355   515,910   918,1086

X(1146) = midpoint of X(2) and X(1121)
X(1146) = reflection of X(i) in X(j) for these (i,j): of (1565,116), (35110,2)
X(1146) = reflection of X(35110) in X(2)
X(1146) = isogonal conjugate of X(1262)
X(1146) = isotomic conjugate of X(1275)
X(1146) = complement of X(664)
X(1146) = anticomplement of X(17044)
X(1146) = crosspoint of X(i) and X(j) for these (i,j): (2,522), (4,514), (9,1021)
X(1146) = crosssum of X(i) and X(j) for these (i,j): (3,101), (6,109), (56,1415), (57,1020), (1407,1461)
X(1146) = crosssum of circumcircle intercepts of line X(6)X(41) (or of circle {{X(1),X(15),X(16)}} (V(X(1)))
X(1146) = crossdifference of every pair of points on line X(109)X(692)
X(1146) = projection from Steiner circumellipse to Steiner inellipse of X(1121)
X(1146) = perspector of circumparabola centered at X(522)
X(1146) = center of circumconic that is locus of trilinear poles of lines passing through X(522)
X(1146) = X(2)-Ceva conjugate of X(522)
X(1146) = Steiner-inellipse antipode of X(35110)
X(1146) = trilinear pole wrt medial triangle of line X(2)X(7)
X(1146) = barycentric square of X(522)

X(1147) = ISOGONAL CONJUGATE OF X(847)

V. Thebault, "Sine-triple-angle-circle," Mathesis 65 (1956) 282-284. (Contributed by Edward Brisse, 3/4/02)

Let A'B'C' be the circumcevian triangle of X(4). Let R_A be the radical axis of the circles (B', |B'C|) and (C',|C'B|), and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(1147). For figures, see Concurrent Radical Axes.    (Antreas Hatzipolakis and Peter Moses, April 10, 2013)

Let A'B'C' be the orthic triangle. Let A'' be the orthogonal projection of A onto line B'C', and define B'' and C'' cyclically; then X(1147) is the circumcenter of A''B''C''. Let L be the reflection of line B'C' in the perpendicular bisector of segment BC, and define M and N cyclically. Let A* = M∩N, and define B* and C* cyclically; then X(1147) is the incenter of A*B*C*. (Randy Hutson, August 26, 2014)

Let A'B'C' be the Kosnita triangle. Let L be the line through A' parallel to the Euler line, and define M and N dyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X(1147). (Randy Hutson, August 26, 2014)

Let DEF be the anticevian triangle of the circumcircle, O; then X(1147) is the centroid of the quadrilateral DEFO. (Randy Hutson, August 26, 2014)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(1147) = X(3)-of-A'B'C'. (Randy Hutson, October 15, 2018)

X(1147) is the insimilicenter of the circumcircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)

X(1147) lies on these lines: 2,54   3,49   4,110   5,578   24,52   26,206   30,156   55,1069   56,215   140,141   143,576   195,568   912,960

X(1147) = midpoint of X(3) and X(155)
X(1147) = isogonal conjugate of X(847)
X(1147) = complement of X(68)
X(1147) = anticomplement of X(5449)
X(1147) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,577), (54,3)
X(1147) = crosspoint of X(i) and X(j) for these (i,j): (2,317), (371,372)
X(1147) = crosssum of X(485) and X(486)
X(1147) = orthic-to-ABC barycentric image of X(5)
X(1147) = perspector of the circumconic centered at X(577)
X(1147) = X(92)-isoconjugate of X(2165)
X(1147) = X(4)-of-Kosnita-triangle
X(1147) = X(91)-isoconjugate of X(4)
X(1147) = X(1158)-of-orthic-triangle if ABC is acute
X(1147) = perspector of 1st Hyacinth triangle and 1st Brocard triangle of 2nd Hyacinth triangle
X(1147) = Dao image of X(3)
X(1147) = {X(3),X(49)}-harmonic conjugate of X(184)

X(1148) = YIU-HATZIPOLAKIS POINT

Suppose L_A, L_B, L_C are lines through a point P, respectively perpendicular to sidelines BC, CA, AB. Let A_B be the point where L_A meets AB, and let A_C be the point where L_A meets AC. Define B_C, B_A, C_A, C_B cyclically. Then X(1148) is the point P, which satisfies

|PA_B| + |PA_C| = |PB_C| + |PB_A| = |PC_A| + |PC_B|.

See Hyacinthos messages #4204-4206, 10/01.

X(1148) lies on these lines: 1,1075   3,653   4,65   46,243   92,942

X(1148) = X(1)-Ceva conjugate of X(4)
X(1148) = X(46)-Hirst inverse of X(243)
X(1148) = polar conjugate of X(7361)

X(1149) = ISOGONAL CONJUGATE OF X(1120)

X(1149) lies on these lines: 1,2   31,999   36,106   38,392   244,517   513,663   672,1015   748,956

X(1149) = isogonal conjugate of X(1120)
X(1149) = crosspoint of X(1) and X(106)
X(1149) = crosssum of X(1) and X(519)
X(1149) = crossdifference of every pair of points on line X(9)X(649)
X(1149) = bicentric sum of PU(98)
X(1149) = PU(98)-harmonic conjugate of X(649)

X(1150) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(8)

X(1150) lies on these lines: 2,6   3,8   10,750   58,964   63,321   76,799   88,330   239,980

X(1150) = complement of X(31034)
X(1150) = anticomplement of X(5718)

X(1151) = ISOGONAL CONJUGATE OF X(1131)

X(1151) is the radical center of the Lucas circles, the incenter of the Lucas central triangle, and the perspector of triangle ABC and the Lucas inner triangle.

Fourteen constructions for X(1151) received from Randy Hutson, September 5, 2015:

(1)-(10): Each pair of the following triangles are perspective, and their perspector is X(1151): tangential triangle, Lucas tangents triangle, Lucas inner tangential triangle, 1st Lucas secondary tangents triangle, Lucas Brocard triangle.

(11) Let A', B', C' be the centers of the Kenmotu squares. Let A" be the reflection of A' in X(371), and define B" and C" cyclically. The triangle A"B"C" is homothetic to ABC at X(1151).

(12) Let A'B'C' be the Lucas tangents triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151).

(13) Let A'B'C' be the Lucas central triangle. Let A" be the pole, wrt the A-Lucas circle, of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151). The point A" is also the intersection of the polars of B' and C' wrt the A-Lucas circle, and likewise for B" and C".

(14) Let A'B'C' be the Lucas central triangle. Let A" be the pole, wrt the A-Lucas circle, of line BC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151).

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas central and Lucas tangents triangles, so that A' = 2a(SA + S) : b(2SB + S) : c(2SC + S). The lines AA', BB', CC' concur in X(1151). (Randy Hutson, September 14, 2016)

Let {a'} be the circle through B and C orthogonal to the circumcircle of ABC, and define {b'} and {c'} cyclically. The circle externally tangent to {a'}, {b'}, {c'} has center X(1151); see X(1152). (César Lozada, July 3, 2019)

X(1151) lies on these lines: 2,489   3,6   4,590   30,485   35,1335   36,1124   140,486   141,487   488,524   615,631

X(1151) is the {X(3),X(6)}-harmonic conjugate of X(1152). For a list of other harmonic conjugates of X(1151), click Tables at the top of this page.

X(1151) = reflection of X(485) in X(8981)
X(1151) = isogonal conjugate of X(1131)
X(1151) = inverse-in-Brocard circle of X(1152)
X(1151) = X(493)-Ceva conjugate of X(6)
X(1151) = crosspoint of X(249) and X(1306)
X(1151) = perspector of Lucas(8) central triangle and circumsymmedial triangle
X(1151) = insimilicenter of circumcircle and Lucas inner circle
X(1151) = inner Soddy center (X(176)) of tangential triangle, if ABC is acute
X(1151) = X(3)-of-Lucas-tangents-triangle
X(1151) = X(20)-of-X(2)-quadsquares-triangle
X(1151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,371,6), (3,3311,372), (371,372,3311), (372,3311,6)

X(1152) = ISOGONAL CONJUGATE OF X(1132)

X(1152) is the radical center of the Lucas(-1:1) circles and the perspector of triangle ABC and the Lucas(-1,1) inner triangle.

X(1152) is the perpsector of each pairs of the following five triangles: tangential triangle, Lucas(-1) tangents triangle, Lucas(-1) inner tangential triangle, 1st Lucas(-1) secondary tangents triangle, Lucas(-1) Brocard triangle. Also, X(1152) perspector of Lucas(-8) central triangle and circumsymmedial triangle. (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) tangents triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A" be the pole, wrt the A-Lucas(-1) circle, of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). The pole A" can also be constructed as the intersection of the polars of B' and C' wrt the A-Lucas(-1) circle, and similarly for B" and C". (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A" be the pole, wrt the A-Lucas(-1) circle, of line BC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). (Randy Hutson, October 13, 2015)

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas(-1) central and Lucas(-1) tangents triangles, so that A' = 2a(SA - S) : b(2SB - S) : c(2SC - S). The lines AA', BB', CC' concur in X(1152). (Randy Hutson, September 14, 2016)

Let {a'} be the circle through B and C orthogonal to the circumcircle of ABC, and define {b'} and {c'} cyclically. The circle internally tangent to {a'}, {b'}, {c'} has center X(1152); see X(1151). (César Lozada, July 3, 2019)

X(1152) lies on these lines: 2,490   3,6   4,615   30,486   35,1124   36,1335  140,485   141,488   487,524   590,631

X(1152) = isogonal conjugate of X(1132)
X(1152) = inverse-in-Brocard circle of X(1151)
X(1152) = X(494)-Ceva conjugate of X(6)
X(1152) = crosspoint of X(249) and X(1307)
X(1152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,372,6), (3,3312,371), (371,372,3312), (371,3312,6)
X(1152) = exsimilicenter of circumcircle and Lucas(-1) inner circle; the insimilicenter is X(6398)
X(1152) = outer Soddy center (X(175)) of tangential triangle, if ABC is acute
X(1152) = X(1)-of-Lucas(-1)-central-triangle, if the Lucas(-1) circles are all externally tangent; otherwise, X(1152) is an excenter of the Lucas(-1) central triangle X(1152) = X(3)-of-Lucas(-1)-tangents-triangle

X(1153) = CENTER OF THE VAN LAMOEN CIRCLE

A triangle is divided by its three medians into 6 smaller triangles. The circumcenters of these smaller triangles are concyclic. Their circle, the Van Lamoen circle, is introduced in

Floor van Lamoen Problem 10830, American Mathematical Monthly 107 (2000) 863; solution by the editors, 109 (2002) 396-397.

Numerous messages about this circle and its center can be accessed from the Hyacinthos archive using "Floor's Monthly problem" as search words. M. Stevanovic's message (#5599, 5/28/02) gives coordinates.

If you have The Geometer's Sketchpad, you can view X(1153).

X(1153) lies on the cubic K1258 and these lines: {2, 187}, {3, 7617}, {30, 32414}, {39, 8859}, {140, 524}, {141, 8787}, {404, 7621}, {511, 7606}, {538, 5054}, {543, 549}, {574, 8860}, {599, 10485}, {618, 33475}, {619, 33474}, {620, 11168}, {631, 5485}, {632, 7843}, {671, 8589}, {754, 9771}, {2482, 37688}, {2549, 23053}, {3054, 5461}, {3523, 7620}, {3524, 7615}, {3526, 7775}, {3788, 21356}, {4045, 44401}, {5032, 31401}, {5092, 9830}, {6683, 13330}, {6719, 10354}, {7496, 42008}, {7749, 7817}, {7769, 41136}, {7810, 41133}, {7815, 21358}, {7816, 33274}, {7848, 22110}, {7849, 33000}, {7854, 33204}, {7861, 33215}, {7880, 9167}, {7883, 16923}, {7886, 8359}, {8356, 14971}, {8587, 10302}, {8588, 11317}, {8597, 39601}, {8703, 20112}, {9166, 33273}, {9466, 11152}, {9770, 15709}, {9877, 37455}, {10303, 34511}, {11147, 11151}, {11148, 15721}, {11164, 32456}, {11184, 11842}, {11645, 40278}, {12040, 13468}, {14159, 14160}, {14161, 14162}, {15513, 33013}, {15693, 18546}, {15701, 40727}, {15720, 34505}, {19911, 21163}, {25486, 31274}, {32480, 37512}, {33476, 35304}, {33477, 35303}, {41895, 43619}

X(1153) = midpoint of X(i) and X(j) for these {i,j}: {2, 5569}, {3, 7617}, {549, 15597}, {7610, 7622}, {7619, 34506}, {7620, 34504}, {8176, 8182}, {8703, 20112}, {12040, 13468}
X(1153) = reflection of X(i) in X(j) for these {i,j}: {7619, 140}, {14160, 14159}
X(1153) = complement of X(8176)
X(1153) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 7771, 31173}, {2, 8182, 8176}, {5054, 7610, 7622}, {5569, 8176, 8182}

X(1154) = ISOGONAL CONJUGATE OF X(1141)

As the isogonal conjugate of a point on the circumcircle, X(1154) lies on the line at infinity; X(1154) is, in fact, the point where the Euler line of the orthic triangle meets the line at infinity (Bernard Gibert, Hyacinthos 1498, September 25, 2000).

X(1154) lies on these (parallel) lines: 2,568   3,54   4,93   5,51   26,154   30,511   35,500   140,389   185,550   186,323   403,1112  

X(1154) = isogonal conjugate of X(1141)
X(1154) = complementary conjugate of X(128)
X(1154) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,1511), (4,128)
X(1154) = crosspoint of X(i) and X(j) for these (i,j): (5,1263), (323,340)
X(1154) = crosssum of X(i) and X(j) for these (i,j): (3,539), (54,1157)
X(1154) = X(30)-of-orthic-triangle
X(1154) = X(30)-of-tangential-triangle
X(1154) = excentral-to-ABC functional image of X(30)
X(1154) = infinite point of tangent to hyperbola {{A,B,C,X(4),X(15)}} at X(15) and tangent to hyperbola {{A,B,C,X(4),X(16)}} at X(16)

X(1155) = SCHRÖDER POINT

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
                        = ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

Let XYZ be the intouch triangle of ABC; i.e., the pedal triangle of the incenter, I. The circles AIX, BIY, CIZ concur in two points. One of them is I; the other is X(1155). This result is obtain by inversion in

Heinz Schröder, "Die Inversion und ihre Anwendung im Unterricht der Oberstufe," Der Mathematikunterricht 1 (1957) 59-80.

Each vertex of the tangential triangle of any triangle T is the inverse-in-the-circumcircle-of-T of the midpoints of the sides of T. Applying this to triangle XYZ shows that X(1155) is the inverse-in-the-incircle of the centroid of XYZ; i.e., X(1155) is X(23)-of-the-intouch-triangle. (Darij Grinberg, #6319, 1/11/03; coordinates by Jean-Pierre Ehrmann, #6320, 1/11/03)

X(1155) lies on the Darboux quintic and these lines: 1,3   10,535   11,516   37,750   44,513   47,582   63,210   88,105   89,1002   100,518   227,603   238,1054   243,653   244,902   377,667   404,960

X(1155) = midpoint of X(i) and X(j) for these (i,j): (1,3245), (36,484), (32622, 32623)
X(1155) = reflection of X(1319) in X(36)
X(1155) = isogonal conjugate of X(1156)
X(1155) = inverse-in-circumcircle of X(55)
X(1155) = inverse-in-incircle of X(354)
X(1155) = inverse-in-Bevan-circle of X(57)
X(1155) = crosspoint of X(i) and X(j) for these (i,j): (1,1156), (527,1323)
X(1155) = crosssum of X(1) and X(1155)
X(1155) = crossdifference of every pair of points on line X(1)X(650)
X(1155) = complement of X(5057)
X(1155) = anticomplement of X(5087)
X(1155) = orthogonal projection of X(1) on its trilinear polar
X(1155) = inverse-in-{circumcircle, incircle}-inverter of X(1)
X(1155) = homothetic center of intouch triangle and medial triangle of 1st circumperp triangle
X(1155) = endo-homothetic center of X(2)- and X(4)-Ehrmann triangles; the homothetic center is X(858)
X(1155) = X(468)-of-excentral-triangle
X(1155) = {X(1),X(3)}-harmonic conjugate of X(37600)

X(1156) = ISOGONAL CONJUGATE OF X(1155)

Let B_c be the reflection of B in the internal angle bisector of angle C, and let C_b be the reflection of C in the internal angle bisector of angle B. Let Γ_a be the circle that passes through A, B_c, C_b. Let A' be the pole of BC, with respect to Γ_a and define B 'and C' cyclically. Then X(1156) is the perspector of A'B'C' and ABC, and X(35445) is the perspector of A'B'C' and the excentral triangle. (Angel Montesdeoca, January 13, 2021)

X(1156) lies on the Darboux septic and these lines: 1,651   4,653   7,11   8,190   9,100   21,662   44,294   80,516   90,411   104,971   144,149   314,799   390,952   673,885

X(1156) = midpoint of X(144) and X(149)
X(1156) = reflection of X(i) in X(j) for these (i,j): (7,11), (100,9)
X(1156) = isogonal conjugate of X(1155)
X(1156) = isotomic conjugate of X(30806)
X(1156) = antigonal conjugate of X(7)
X(1156) = symgonal of X(9)
X(1156) = trilinear pole of line X(1)X(650)
X(1156) = polar conjugate of X(37805)
X(1156) = pole wrt polar circle of trilinear polar of X(37805) (line X(6366)X(12831))
X(1156) = intersection of the Feuerbach circumhyperbola and the circumellise centered at X(9)
X(1156) = BSS(a^2 → a) of X(74)

X(1157) = INVERSE-IN-CIRCUMCIRCLE OF X(54)

Trilinears    4 cos A + cos 3A sec A sec(B - C) : :

For any point X, let X_A be the reflection of X in sideline BC, and define X_B and X_C cyclically. Then X(1157) is the unique point X for which the lines AX_A, BX_B, CX_C concur on the circumcircle; the point of concurrence is X(1141).

X(1157) is the tangential of X(3) on the Neuberg cubic.

Let A'B'C' be the reflection triangle. The circumcircles of AB'C', BC'A', CA'B' (i.e., the Yiu circles) concur in X(1157). (Randy Hutson, July 20, 2016)

X(1157) lies on the Neuberg cubic and these lines: 1,3483   3,54   4,3482   5,252   30,1141   74,3484   186,933   1337,1338   3065,3465

X(1157) = isogonal conjugate of X(1263)
X(1157) = inverse-in-circumcircle of X(54)
X(1157) = X(30)-Ceva conjugate of X(3484)
X(1157) = Yiu-isogonal conjugate of X(195)
X(1157) = Cundy-Parry Phi transform of X(195)
X(1157) = Cundy-Parry Psi transform of X(3459)

X(1158) = CIRCUMCENTER OF EXTOUCH TRIANGLE

Trilinears    sin²B/2 cos B + sin²C/2 cos C - sin²A/2 cos A (D. Grinberg, 2/25/04)

X(1158) lies on these lines: 1,104   3,960   4,46   8,20   57,946   65,1012   117,208   165,191

X(1158) = midpoint of X(40) and X(84)
X(1158) = complement of isotomic conjugate of X(34413)
X(1158) = X(318)-Ceva conjugate of X(1)
X(1158) = X(68)-of-Fuhrmann-triangle
X(1158) = excentral isogonal conjugate of X(1745)
X(1158) = X(1147)-of-excentral-triangle
X(1158) = ABC-to-excentral barycentric image of X(3)
X(1158) = X(3)-of-extouch triangle, so that X(210)X(1158) = Euler line of the extouch triangle

X(1159) = GREENHILL POINT

X(1159) lies on these lines: 1,3   7,952

See Hyacinthos #6535 and

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 23.

X(1160) = CIRCUMCENTER OF OUTER GREBE TRIANGLE

See Hyacinthos #6537.

Let O_A be the circle centered at the A-vertex of the anti-inner-Grebe triangle and passing through A; define O_B and O_C cyclically. X(1160) is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)

X(1160) lies on these lines: 3,6   4,1162

X(1160) = reflection of X(1161) in X(3)

X(1161) = CIRCUMCENTER OF INNER GREBE TRIANGLE

X(1161) lies on these lines: 3,6   4,1163

See Hyacinthos #6537.

X(1161) = reflection of X(1160) in X(3)

X(1162) = OUTER GREBE-ORTHIC PERSPECTOR

See Hyacinthos #6537.

X(1162) lies on these lines: 4,1160   428,1163

X(1163) = INNER GREBE-ORTHIC PERSPECTOR

X(1163) lies on these lines: 4,1161   428,1162

See Hyacinthos #6537.

X(1164) = POINT ALTAIR

X(1164) lies on this line: 468,1165

X(1165) = POINT ALTAIS

X(1165) lies on this line: 468,1164

Saragossa Points 1166-1208

X(1166) = 1st SARAGOSSA POINT OF X(5)

X(1166) lies on these lines: 2,252   5,96   52,54

X(1166) = isogonal conjugate of X(1209)
X(1166) = isotomic conjugate of X(1225)

X(1167) = 1st SARAGOSSA POINT OF X(34)

X(1167) lies on these lines: 31,937   34,40   56,580   255,269   271,936   595,998

X(1167) = isogonal conjugate of X(1210)
X(1167) = isotomic conjugate of X(1226)
X(1167) = cevapoint of X(i) and X(j) for these (i,j): (6,212), (31,198)

X(1168) = 1st SARAGOSSA POINT OF X(36)

X(1168) lies on these lines: 36,88   44,517   80,519   484,759   535,903

X(1168) = isogonal conjugate of X(214)
X(1168) = isotomic conjugate of X(1227)
X(1168) = X(6)-cross conjugate of X(88)

X(1169) = 1st SARAGOSSA POINT OF X(37)

X(1169) lies on these lines: 2,261   6,60   21,172   28,961   32,941   42,284   572,849   604,1178

X(1169) = isogonal conjugate of X(1211)
X(1169) = isotomic conjugate of X(1228)
X(1169) = cevapoint of X(6) and X(1333)
X(1169) = X(92)-isoconjugate of X(22076)

X(1170) = 1st SARAGOSSA POINT OF X(41)

X(1170) lies on these lines: 2,220   3,955   6,279   7,218   41,57   56,1002   65,105   81,241   278,607

X(1170) = isogonal conjugate of X(1212)
X(1170) = isotomic conjugate of X(1229)
X(1170) = X(92)-isoconjugate of X(22079)
X(1170) = cevapoint of X(i) and X(j) for these (i,j): (1,218), (6,57)

X(1171) = 1st SARAGOSSA POINT OF X(42)

X(1171) lies on these lines: 6,593   35,58   37,81

X(1171) = isogonal conjugate of X(1213)
X(1171) = isotomic conjugate of X(1230)
X(1171) = cevapoint of X(6) and X(58)
X(1171) = trilinear pole of line X(512)X(1326)
X(1171) = X(92)-isoconjugate of X(22080)

X(1172) = 1st SARAGOSSA POINT OF X(48)

X(1172) lies on the Feuerbach hyperbola and these lines: 1,19   4,6   7,27   8,29   9,33   21,270   25,941   37,943   58,84   104,112   162,1156   186,1030   286,648   406,966

X(1172) = isogonal conjugate of X(1214)
X(1172) = isotomic conjugate of X(1231)
X(1172) = X(27)-Ceva conjugate of X(28)
X(1172) = cevapoint of X(6) and X(19)
X(1172) = X(i)-cross conjugate of X(j) for these (i,j): (6,284), (33,29)
X(1172) = crosspoint of X(i) and X(j) for these (i,j): (27,29), (81,285)
X(1172) = crosssum of X(i) and X(j) for these (i,j): (37,227), (71,73)
X(1172) = crossdifference of every pair of points on line X(520)X(656)
X(1172) = trilinear pole of line X(650)X(1946)
X(1172) = polar conjugate of X(1441)
X(1172) = X(92)-isoconjugate of X(22341)

X(1173) = 1st SARAGOSSA POINT OF X(54)

Let P and Q be the intersections of line BC and the 2nd Lemoine circle. Let X = X(6). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1173); c.f. X(592), where the circle is the 1st Lemoine circle. X(571): Let A'B'C' be the Kosnita triangle. Let A" be the barycentric product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(571). (Randy Hutson, December 2, 2017)

Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let Na be the nine-point center of HaBaCa. Define Nb and Nc cyclically. The lines ANa, BNb, CNc concur in X(1173). (Randy Hutson, December 2, 2017)

X(1173) lies on the the conics {{A, B, C, X(13), X(62)}} and {A, B, C, X(14), X(61)}} and on these lines: 3,143   51,54   69,576   74,389   265,546   575,1176

X(1173) = isogonal conjugate of X(140)
X(1173) = isotomic conjugate of X(1232)
X(1173) = cevapoint of X(i) and X(j) for these (i,j): (6,51), (61,62)
X(1173) = X(6)-cross conjugate of X(288)
X(1173) = X(5506)-of-orthic-triangle if ABC is acute
X(1173) = X(92)-isoconjugate of X(22052)

X(1174) = 1st SARAGOSSA POINT OF X(57)

X(1174) lies on these lines: 41,57   55,218   101,354   226,673   284,672   661,1024

X(1174) = isogonal conjugate of X(142)
X(1174) = isotomic conjugate of X(1233)
X(1174) = cevapoint of X(6) and X(41)
X(1174) = X(513)-cross conjugate of X(101)
X(1174) = X(92)-isoconjugate of X(22053)
X(1174) = crosssum of X(354) and X(1212)

X(1175) = 1st SARAGOSSA POINT OF X(65)

X(1175) lies on these lines: 3,60   21,72   28,65   35,71   58,73   69,261   110,942

X(1175) = isogonal conjugate of X(442)
X(1175) = isotomic conjugate of X(1234)
X(1175) = X(513)-cross conjugate of X(110)
X(1175) = trilinear pole of line X(647)X(2605)

X(1176) = 1st SARAGOSSA POINT OF X(66)

X(1176) lies on these lines: 2,66   4,83   6,22   54,511   65,82   67,110   69,184   74,827   216,248   290,308   575,1173

X(1176) = isogonal conjugate of X(427)
X(1176) = isotomic conjugate of X(1235)
X(1176) = X(83)-Ceva conjugate of X(251)
X(1176) = cevapoint of X(i) and X(j) for these (i,j): (3,184), (6,206)
X(1176) = antigonal conjugate of X(18125)
X(1176) = X(39)-isoconjugate of X(92)

X(1177) = 1st SARAGOSSA POINT OF X(67)

See Angel Montesdeoca, Hyacinthos #21528, 2/12/2013

X(1177) lies on these lines: 6,1112   23,895   66,125   67,468   68,542   69,110   72,692   290,685

X(1177) = reflection of X(i) in X(j) for these (i,j): (66,125), (110,206)
X(1177) = isogonal conjugate of X(858)
X(1177) = isotomic conjugate of X(1236)
X(1177) = cevapoint of X(i) and X(j) for these (i,j): (3,101), (6,109)
X(1177) = trilinear pole of line X(32)X(647)
X(1177) = Jerabek-hyperbola antipode of X(66)
X(1177) = antigonal conjugate of X(66)
X(1177) = barycentric product of circumcircle intercepts of line X(6)X(525)

X(1178) = 1st SARAGOSSA POINT OF X(82)

X(1178) lies on these lines: 6,694   21,238   82,662   284,893   409,1201   604,1169   741,985   759,995   765,872   869,983

X(1178) = isogonal conjugate of X(1215)
X(1178) = isotomic conjugate of X(1237)
X(1178) = X(92)-isoconjugate of X(22061)

X(1179) = 1st SARAGOSSA POINT OF X(96)

X(1179) lies on these lines: 4,569   24,264   25,847   93,324

X(1179) = isogonal conjugate of X(1216)
X(1179) = isotomic conjugate of X(1238)
X(1179) = cevapoint of X(25) and X(53)
X(1179) = polar conjugate of X(37636)
X(1179) = trilinear pole of line X(2501)X(3050)
X(1179) = X(63)-isoconjugate of X(570)

X(1180) = 2nd SARAGOSSA POINT OF X(2)

X(1180) lies on these lines: 2,39,   6,22   111,907

X(1180) = isotomic conjugate of X(1239)
X(1180) = anticomplement of X(8891)
X(1180) = crossdifference of every pair of points on line X(669)X(826)

X(1181) = 2nd SARAGOSSA POINT OF X(3)

X(1181) lies on these lines: 3,49   4,6   5,1899   24,154   25,389   54,64   110,974   125,399   186,1192   1060,1069

X(1181) = reflection of X(1593) in X(578)
X(1181) = isogonal conjugate of X(1217)
X(1181) = crosssum of X(4) and X(631)

X(1182) = 2nd SARAGOSSA POINT OF X(19)

X(1182) lies on these lines: 3,6   9,498   19,208

X(1183) = 2nd SARAGOSSA POINT OF X(21)

X(1183) lies on these lines: 1,41   6,959   8,1036   391,958

X(1184) = 2nd SARAGOSSA POINT OF X(25)

X(1184) lies on these lines: 2,6   3,1194   25,32

X(1185) = 2nd SARAGOSSA POINT OF X(31)

X(1185) lies on these lines: 1,1206   2,6   31,32

X(1185) = isogonal conjugate of X(1218)
X(1185) = crossdifference of every pair of points on line X(512)X(693)

X(1186) = 2nd SARAGOSSA POINT OF X(32)

X(1186) lies on these lines: 2,1207   6,76   32,184

X(1186) = crossdifference of every pair of points on line X(688)X(850)

X(1187) = 2nd SARAGOSSA POINT OF X(37)

X(1187) lies on these lines: 6,60   10,37

X(1188) = 2nd SARAGOSSA POINT OF X(41)

X(1188) lies on these lines: 6,279   31,32

X(1189) = 2nd SARAGOSSA POINT OF X(43)

X(1189) lies on this line: 1,2

X(1190) = 2nd SARAGOSSA POINT OF X(55)

X(1190) lies on these lines: 6,57   41,55   56,1202   165,218   294,940

X(1191) = 2nd SARAGOSSA POINT OF X(56)

X(1191) lies on these lines: 1,6   3,595   28,957   31,56   55,1193   58,999   65,614   105,959   387,1058

X(1191) = isogonal conjugate of X(1219)
X(1191) = crosspoint of X(1016) and X(1310)

X(1192) = 2nd SARAGOSSA POINT OF X(64)

X(1192) lies on these lines: 3,6   25,64

X(1193) = 3rd SARAGOSSA POINT OF X(1)

X(1193) lies on these lines: 1,2   3,31   6,41   21,238   35,595   36,58   37,992   38,72   39,213   57,959   63,988   106,1126   171,404   222,1106   244,942   405,748   474,750   518,872   999,1066   1045,1050

X(1193) is the {X(1),X(43)}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(1193), click Tables at the top of this page.

X(1193) = midpoint of X(1) and X(3293)
X(1193) = isogonal conjugate of X(1220)
X(1193) = isotomic conjugate of X(1240)
X(1193) = crosspoint of X(i) and X(j) for these (i,j): (1,58), (6,893), (57,86)
X(1193) = crosssum of X(i) and X(j) for these (i,j): (1,10), (2,894), (9,42)
X(1193) = crossdifference of every pair of points on line X(522)X(649)
X(1193) = X(92)-isoconjugate of X(2359)
X(1193) = polar conjugate of isotomic conjugate of X(22097)

X(1194) = 3rd SARAGOSSA POINT OF X(2)

Let L be the isogonal conjugate of the isotomic conjugate of line X(2)X(6) (i.e., line X(6)X(25)). Let M be the isotomic conjugate of the isogonal conjugate of line X(2)X(6) (i.e., line X(2)X(39)). Then X(1194) = L∩M. (Randy Hutson, March 21, 2019)

X(1194) lies on these lines: 2,39   6,25   22,32   23,251   230,570

X(1194) = isotomic conjugate of X(1241)
X(1194) = crosspoint of X(i) and X(j) for these (i,j): (2,251), (4,308), (6,893)
X(1194) = crosssum of X(i) and X(j) for these (i,j): (2,384), (6,141)
X(1194) = crossdifference of every pair of points on line X(525)X(669)

X(1195) = 3rd SARAGOSSA POINT OF X(19)

X(1195) lies on these lines: 19,208   41,71   43,165   60,283

X(1195) = crosspoint of X(19) and X(284)
X(1195) = crosssum of X(63) and X(226)

X(1196) = 3rd SARAGOSSA POINT OF X(25)

X(1196) lies on these lines: {2,39}, {3,1611}, {6,2056}, {22,187}, {23,1627}, {25,32}, {51,3051}, {111,251}, {115,427}, {184,1692}, {216,230}, {232,800}, {233,3815}, {394,5028}, {511,1613}, {612,1500}, {614,1015}, {682,3080}, {1084,2493}, {1368,5254}, {1495,1501}, {1570,1993}, {1915,3506}, {1995,5007}, {2092,5275}, {2275,5272}, {2276,5268}, {2670,4263}, {3094,3819}, {3231,3917}, {3796,5033}

X(1196) = complement of X(305)
X(1196) = crosspoint of X(2) and X(25)
X(1196) = crosssum of X(6) and X(96)
X(1196) = perspector of circumconic centered at X(1368)
X(1196) = center of circumconic that is locus of trilinear poles of lines passing through X(1368)
X(1196) = X(2)-Ceva conjugate of X(1368)

X(1197) = 3rd SARAGOSSA POINT OF X(31)

X(1197) lies on these lines: 6,43   31,32   81,239   284,893

X(1197) = isogonal conjugate of X(1221)
X(1197) = crosspoint of X(i) and X(j) for these (i,j): (6,904), (31,81)
X(1197) = crosssum of X(37) and X(75)
X(1197) = polar conjugate of isotomic conjugate of X(22389)

X(1198) = 3rd SARAGOSSA POINT OF X(43)

X(1198) lies on this line: 1,2

X(1199) = 3rd SARAGOSSA POINT OF X(54)

X(1199) lies on these lines: 4,6   5,3410   23,143   54,186   140,195   288,1157   578,1204

X(1199) = crosspoint of X(54) and X(1173)
X(1199) = crosssum of X(5) and X(140)
X(1199) = antipode of X(74) in Moses-Jerabek conic

X(1200) = 3rd SARAGOSSA POINT OF X(55)

X(1200) lies on these lines: 41,55   43,165   57,279   171,294

X(1200) is the {X(55),X(1190)}-harmonic conjugate of X(41). For a list of other harmonic conjugates, click Tables at the top of this page.

X(1200) = crosspoint of X(55) and X(57)
X(1200) = crosssum of X(7) and X(9)

X(1201) = 3rd SARAGOSSA POINT OF X(56)

X(1201) lies on these lines: 1,2   3,902   31,56   32,1055   36,595   38,960   58,106   65,244   73,1104   105,904   205,604   213,1015   409,1178   500,1064   748,958   651, 1476   1279,2293

X(1201) is the {X(56),X(1191)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(1201), click Tables at the top of this page.

X(1201) = isogonal conjugate of X(1222)
X(1201) = crosspoint of X(1) and X(56)
X(1201) = crosssum of X(1) and X(8)
X(1201) = bicentric sum of PU(92)
X(1201) = PU(92)-harmonic conjugate of X(649)

X(1202) = 3rd SARAGOSSA POINT OF X(57)

X(1202) lies on these lines: 6,31   57,279

X(1202) = isogonal conjugate of X(1223)
X(1202) = crosspoint of X(57) and X(1174)
X(1202) = crosssum of X(9) and X(142)

X(1203) = 3rd SARAGOSSA POINT OF X(58)

X(1203) lies on these lines: 1,6   31,35   36,58   42,595   81,1125   580,1064   581,602

X(1203) = isogonal conjugate of X(1224)
X(1203) = crosspoint of X(58) and X(1126)
X(1203) = crosssum of X(10) and X(1125)

X(1204) = 3rd SARAGOSSA POINT OF X(64)

X(1204) lies on these lines: 3,49   4,74   25,64   217,574   378,389   578,1199

X(1204) = reflection of X(1092) in X(3)
X(1204) = crosspoint of X(3) and X(64)
X(1204) = crosssum of X(i) and X(j) for these (i,j): (4,20), (489,490)
X(1204) = inverse-in-Jerabek-hyperbola of X(3357)
X(1204) = {X(4),X(74)}-harmonic conjugate of X(3357)
X(1204) = crossdifference of every pair of points on line X(1636)X(2501)

X(1205) = 3rd SARAGOSSA POINT OF X(67)

X(1205) lies on this line: 74,511

X(1205) = crosspoint of X(67) and X(1177)
X(1205) = crosssum of X(23) and X(858)

X(1206) = 3rd SARAGOSSA POINT OF X(81)

X(1206) lies on these lines: 6,31   81,239

X(1207) = 3rd SARAGOSSA POINT OF X(83)

X(1207) lies on these lines: 3,6   83,3978

X(1208) = 3rd SARAGOSSA POINT OF X(84)

X(1208) lies on these lines: 6,963   56,64

X(1208) = crosspoint of X(84) and X(947)
X(1208) = crosssum of X(40) and X(946)

X(1209) = ISOGONAL CONJUGATE OF X(1166)

In the plane of a triangle ABC, let
A'B'C' = medial triangle;
B_A = reflection of B' in BC, and define C_B and A_C cyclically;
C_A = reflection of C' in BC, and define A_B and B_C cyclically;
O_A = circle {{B', C', B_A, C_A}}, and define O_B and O_C cyclically.
Then X(1209) = radical center of O_A, O_B, O_C. (Dasari Naga Vijay Krishna, September 8, 2021)

X(1209) lies on these lines: 2,54   3,161   5,51   6,17   12,942   125,128   127,129

X(1209) is the {X(17),X(18)}-harmonic conjugate of X(231). For a list of other harmonic conjugates of X(1209), click Tables at the top of this page.

X(1209) = reflection of X(52) in X(973)
X(1209) = isogonal conjugate of X(1166)
X(1209) = complement of X(54)
X(1209) = complementary conjugate of X(140)
X(1209) = crosspoint of X(2) and X(311)
X(1209) = perspector of circumconic centered at X(570)
X(1209) = center of circumconic that is locus of trilinear poles of lines passing through X(570)
X(1209) = X(2)-Ceva conjugate of X(570)
X(1209) = X(54) of X(5)-Brocard triangle

X(1210) = ISOGONAL CONJUGATE OF X(1167)

X(1210) lies on these lines: 1,2   3,950   4,57   5,226   7,3091   11,65   12,354   29,58   36,411   40,497   46,516   56,515   79,1156   142,442   158,273   189,937   355,999   381,553   496,517   1089,1229

X(1210) is the {X(2),X(8)}-harmonic conjugate of X(936). For a list of other harmonic conjugates of X(1210), click Tables at the top of this page.

X(1210) = isogonal conjugate of X(1167)
X(1210) = crosspoint of X(i) and X(j) for these (i,j): (2,273), (75,189)
X(1210) = crosssum of X(i) and X(j) for these (i,j): (6,212), (31,198)
X(1210) = complement of X(78)
X(1210) = complementary conjugate of X(34823)
X(1210) = barycentric product X(1)*X(17862)
X(1210) = homothetic center of 4th Euler triangle and inverse-in-incircle triangle

X(1211) = ISOGONAL CONJUGATE OF X(1169)

X(1211) lies on these lines: 2,6   9,440   10,12   37,306   120,125   223,936   257,312   278,860   313,321   429,960   1086,1227

X(1211) = isogonal conjugate of X(1169)
X(1211) = isotomic conjugate of X(14534)
X(1211) = complement of X(81)
X(1211) = crosspoint of X(i) and X(j) for these (i,j): (2,321), (76,1441)
X(1211) = crosssum of X(6) and X(1333)

X(1212) = ISOGONAL CONJUGATE OF X(1170)

X(1212) lies on these lines: 1,6   2,85   3,169   10,1146   21,294   65,672   281,475

X(1212) = isogonal conjugate of X(1170)
X(1212) = complement of X(85)
X(1212) = X(i)-Ceva conjugate of X(j) for these (i,j) : (2,142), (142,354)
X(1212) = crosspoint of X(i) and X(j) for these (i,j): (1,277), (2,9)
X(1212) = crosssum of X(i) and X(j) for these (i,j): (1,218), (6,57)
X(1212) = isotomic conjugate of X(31618)
X(1212) = polar conjugate of isogonal conjugate of X(22079)
X(1212) = {X(1),X(9)}-harmonic conjugate of X(220)
X(1212) = perspector of circumconic centered at X(142)
X(1212) = center of circumconic that is locus of trilinear poles of lines passing through X(142)

X(1213) = ISOGONAL CONJUGATE OF X(1171)

X(1213) lies on these lines: 2,6   5,573   9,46   10,37   19,429   21,1030   115,121   140,572   190,1268   281,860   440,910   451,1172   1100,1125   1230,1269

X(1213) = isogonal conjugate of X(1171)
X(1213) = isotomic conjugate of X(32014)
X(1213) = complement of X(86)
X(1213) = crosspoint of X(2) and X(10)
X(1213) = crosssum of X(6) and X(58)
X(1213) = crossdifference of every pair of points on line X(512)X(1326)
X(1213) = perspector of circumconic centered at X(1125)
X(1213) = center of circumconic that is locus of trilinear poles of lines passing through X(1125)
X(1213) = X(2)-Ceva conjugate of X(1125)
X(1213) = {X(2),X(6)}-harmonic conjugate of X(17398)
X(1213) = {X(2),X(69)}-harmonic conjugate of X(15668)
X(1213) = {X(2),X(141)}-harmonic conjugate of X(17245)

X(1214) = ISOGONAL CONJUGATE OF X(1172)

X(1214) lies on these lines: 1,3   2,92   5,1838   7,464   9,223   10,227   34,405   37,226   63,77   72,73   216,1108   225,442   304,345   306,307   333,664   343,914

X(1214) = isogonal conjugate of X(1172)
X(1214) = isotomic conjugate of X(31623)
X(1214) = polar conjugate of X(1896)
X(1214) = trilinear pole of line X(520)X(656)
X(1214) = X(6)-isoconjugate of X(29)
X(1214) = X(92)-isoconjugate of X(2194)
X(1214) = complement of X(92)
X(1214) = X(i)-Ceva conjugate of X(j) for these (i,j) : (2,226), (77,73), (307,72), (348, 307)
X(1214) = cevapoint of X(i) and X(j) for these (i,j): (37,227), (71,73)
X(1214) = X(i)-cross conjugate of X(j) for these (i,j): (71,72), (201,307)
X(1214) = crosspoint of X(i) and X(j) for these (i,j): (2,63), (77,348), (1231,1441)
X(1214) = crosssum of X(i) and X(j) for these (i,j): (6,19), (33,607)

X(1215) = ISOGONAL CONJUGATE OF X(1178)

X(1215) lies on these lines: 1,312   2,38   10,12   37,714   42,321   43,75   171,385   190,846   964,976

X(1215) = midpoint of X(42) and X(321)
X(1215) = isogonal conjugate of X(1178)
X(1215) = isotomic conjugate of X(32010)
X(1215) = complement of X(38)
X(1215) = crosssum of X(893) and X(904)
X(1215) = polar conjugate of isogonal conjugate of X(22061)
X(1215) = complementary conjugate of X(21249)
X(1215) = barycentric product X(10)*X(894)

X(1216) = ISOGONAL CONJUGATE OF X(1179)

X(1216) lies on these lines: 2,52   3,49   5,141   54,323   68,69   140,389

X(1216) = midpoint of X(1352) and X(3313)
X(1216) = reflection of X(389) in X(140)
X(1216) = isogonal conjugate of X(1179)
X(1216) = isotomic conjugate of isogonal conjugate of X(23195)
X(1216) = isotomic conjugate of polar conjugate of X(570)
X(1216) = complementary conjugate of X(34835)
X(1216) = complement of X(52)
X(1216) = crosspoint of X(69) and X(97)
X(1216) = crosssum of X(25) and X(53)
X(1216) = anticomplement of X(5462)
X(1216) = X(4) of polar triangle of complement of polar circle

X(1217) = CRETAN PERSPECTOR

Let A'B'C' be the medial triangle of the orthic triangle of triangle ABC. Let A" be the reflection of X(4) in A', and define B" and C" cyclically. Let K_ab and K_ac be the circumcenters of triangles A"BA and A"CA, respectively. Let A''' = BK_ac∩CK_ab, and define B''' and C''' cyclically. The lines AA''', BB''', CC''' concur in X(1217). (Antreas Hatzipolakis, Anopolis #39, 3/19/2002)

Using barycentric coordinates, let P = p : q : r be a point not on a sideline of triangle ABC, and let P' be the isogonal conjugate of P. Let DEF and D'E'F' be the pedal triangles of P and P', respectively. Let

X = PD'∩P'D,     Y = PE'∩P'E,     Z = PF'∩P'F.

Then the triangle XYZ is perspective to ABC (Dominik Burek, June 8, 2012) and the perspector is given by barycentric coordinates k(a,b,c) : k(b,c,a) : k(c,a,b), where k = 1/(pqr(b² + c² - a²) + 2p(b²r² + c²q²))      (Peter Moses, Oct. 22, 2012). The point X(1217) results from taking P = X(3).

X(1217) lies on these lines: 2,1093   3,393   4,394   5,1073   20,97   254,378

X(1217) = trilinear pole of line X(520)X(2501)
X(1217) = isogonal conjugate of X(1181)

X(1218) = ISOGONAL CONJUGATE OF X(1185)

X(1218) lies on these lines: 6,274   25,286   37,76   42,75   767,785

X(1218) = isogonal conjugate of X(1185)
X(1218) = isotomic conjugate of X(5283)
X(1218) = trilinear pole of line X(512)X(693)

X(1219) = ISOGONAL CONJUGATE OF X(1191)

X(1219) lies on these lines: 1,346   2,341   8,57   28,956   72,957   75,279   81,145   105,958   278,318   518,959

X(1219) = isogonal conjugate of X(1191)
X(1219) = isotomic conjugate of X(3672)
X(1219) = trilinear pole of orthic axis of 2nd extouch triangle

X(1220) = ISOGONAL CONJUGATE OF X(1193)

X(1220) lies on these lines: 1,312   2,12   6,8   10,58   34,92   42,1043   65,257   85,269   86,313   106,1125   292,1107   341,612   519,1126

X(1220) = isogonal conjugate of X(1193)
X(1220) = isotomic conjugate of X(4357)
X(1220) = cevapoint of X(i) and X(j) for these (i,j): (1,10), (9,42)
X(1220) = crosspoint of X(1) and X(10) wrt the excentral triangle
X(1220) = trilinear pole of line X(522)X(649)
X(1220) = pole wrt polar circle of trilinear polar of X(1848)
X(1220) = X(19)-isoconjugate of X(22097)
X(1220) = X(48)-isoconjugate (polar conjugate) of X(1848)
X(1220) = intersection of tangents at X(1) and X(10) to hyperbola passing through X(1), X(10) and the excenters

X(1221) = ISOGONAL CONJUGATE OF X(1197)

X(1221) lies on these lines: 43,75   76,192   213,274

X(1221) = isogonal conjugate of X(1197)
X(1221) = isotomic conjugate of X(1107)
X(1221) = cevapoint of X(37) and X(75)

X(1222) = ISOGONAL CONJUGATE OF X(1201)

X(1222) lies on these lines: 1,341   6,145   8,56   10,106   34,318   58,519   75,269   190,3057

X(1222) = isogonal conjugate of X(1201)
X(1222) = cevapoint of X(1) and X(8)
X(1222) = crosssum of X(1) and X(1050)
X(1222) = isotomic conjugate of X(3663)
X(1222) = intersection of tangents at X(1) and X(8) to hyperbola passing through X(1), X(8) and the excenters
X(1222) = crosspoint of X(1) and X(8) wrt the excentral triangle
X(1222) = trilinear pole of line X(649)X(4949)

X(1223) = ISOGONAL CONJUGATE OF X(1202)

X(1223) lies on these lines: 2,480   7,220   9,1038   75,728

X(1223) = isogonal conjugate of X(1202)
X(1223) = cevapoint of X(9) and X(142)

X(1224) = ISOGONAL CONJUGATE OF X(1203)

X(1224) lies on these lines: 1,594   2,1089   10,81   12,57   274,313   1125,1255

X(1224) = isogonal conjugate of X(1203)

X(1225) = ISOTOMIC CONJUGATE OF X(1166)

X(1225) lies on these lines: 5,311   69,2888   76,95   339,1232  

X(1225) = isotomic conjugate of X(1166)

X(1226) = ISOTOMIC CONJUGATE OF X(1167)

X(1226) lies on these lines: 10,75   85,264   86,811   311,349   1233,1234

X(1226) = isotomic conjugate of X(1167)

X(1227) = ISOTOMIC CONJUGATE OF X(1168)

X(1227) lies on these lines: 63,190   75,537   313,1232   320,758   321,545   1086,1211   1234,1269

X(1227) = isotomic conjugate of X(1168)

X(1228) = ISOTOMIC CONJUGATE OF X(1169)

X(1228) lies on these lines: 2,39   12,313   312,857

X(1228) = isotomic conjugate of X(1169)

X(1229) = ISOTOMIC CONJUGATE OF X(1170)

X(1229) lies on these lines: 2,37   294,314   1089,1210

X(1229) = isotomic conjugate of X(1170)
X(1229) = crosspoint of X(76) and X(312)
X(1229) = crosssum of X(32) and X(604)

X(1230) = ISOTOMIC CONJUGATE OF X(1171)

X(1230) lies on these lines: 2,39   312,1234   313,321   339,440   469,1235   1213,1269

X(1230) = isotomic conjugate of X(1171)
X(1230) = crosspoint of X(76) and X(313)

X(1231) = ISOTOMIC CONJUGATE OF X(1172)

X(1231) lies on these lines: 7,8   76,331   201,307   304,345   321,349   664,1043

X(1231) = isogonal conjugate of X(2204)
X(1231) = isotomic conjugate of X(1172)
X(1231) = X(76)-Ceva conjugate of X(349)
X(1231) = cevapoint of X(306) and X(307)
X(1231) = crosspoint of X(76) and X(304)

X(1232) = ISOTOMIC CONJUGATE OF X(1173)

X(1232) lies on these lines: 4,69   95,252   313,1227   339,1225   1238,1273

X(1232) = isotomic conjugate of X(1173)
X(1232) = anticomplement of X(5421)
X(1232) = polar conjugate of X(33631)

X(1233) = ISOTOMIC CONJUGATE OF X(1174)

X(1233) lies on these lines: 69,674   76,85   310,333   1226,1234

X(1233) = isotomic conjugate of X(1174)

X(1234) = ISOTOMIC CONJUGATE OF X(1175)

X(1234) lies on these lines: 4,69   12,313   312,1230   1226,1233   1227,1269

X(1234) = isotomic conjugate of X(1175)

X(1235) = ISOTOMIC CONJUGATE OF X(1176)

X(1235) lies on these lines: 4,69   5,339   24,183   25,1239   54,276   83,648   112,384   297,324   469,1230

X(1235) = isotomic conjugate of X(1176)
X(1235) = X(264)-Ceva conjugate of X(427)
X(1235) = cevapoint of X(141) and X(427)
X(1235) = trilinear product of vertices of 5th Euler triangle
X(1235) = polar conjugate of X(251)

X(1236) = ISOTOMIC CONJUGATE OF X(1177)

X(1236) lies on these lines: 4,69   325,339   826,850

X(1236) = isotomic conjugate of X(1177)

X(1237) = ISOTOMIC CONJUGATE OF X(1178)

X(1237) lies on these lines: 12,313   75,1240   76,334   561,756

X(1237) = isotomic conjugate of X(1178)

X(1238) = ISOTOMIC CONJUGATE OF X(1179)

X(1238) lies on these lines: 3,69   311,325   1232,1273

X(1238) = isotomic conjugate of X(1179)

X(1239) = ISOTOMIC CONJUGATE OF X(1180)

X(1239) lies on these lines: 25,1235   76,251

X(1239) = isotomic conjugate of X(1180)

X(1240) = ISOTOMIC CONJUGATE OF X(1193)

X(1240) lies on these lines: 7,76   75,1237   86,313   903,1269

X(1240) = isotomic conjugate of X(1193)

X(1240) = cevapoint of X(i) and X(j) for these (i,j): (10,312), (75,313)

X(1241) = ISOTOMIC CONJUGATE OF X(1194)

X(1241) lies on these lines: 6,305   25,76   251,384

X(1241) = isotomic conjugate of X(1194)
X(1241) = cevapoint of X(39) and X(69)

X(1242) = ISOGONAL CONJUGATE OF X(1005)

X(1242) lies on these lines: 71,1155   72,527

X(1242) = isogonal conjugate of X(1005)

X(1243) = ISOGONAL CONJUGATE OF X(1006)

X(1243) lies on these lines: 5,72   28,54   71,517   73,942   270,1175

X(1243) = isogonal conjugate of X(1006)

X(1244) = ISOGONAL CONJUGATE OF X(1009)

X(1244) lies on these lines: 71,238   72,239

X(1244) = isogonal conjugate of X(1009)
X(1244) = trilinear pole of line X(647)X(659)

X(1245) = ISOGONAL CONJUGATE OF X(1010)

X(1245) lies on these lines: 1,69   3,31   4,1039   42,72   71,213   895,923   1176,1203

X(1245) = isogonal conjugate of X(1010)
X(1245) = crosspoint of X(1036) and X(1039)
X(1245) = crosssum of X(388) and X(1038)
X(1245) = trilinear pole of line X(647)X(798)

X(1246) = ISOGONAL CONJUGATE OF X(1011)

X(1246) lies on these lines: 2,71   3,86   6,27   7,73   65,273   69,310   72,75

X(1246) = isogonal conjugate of X(1011)
X(1246) = isotomic conjugate of X(10449)
X(1246) = trilinear pole of line X(514)X(647)
X(1246) = X(386)-cross conjugate of X(2)

X(1247) = ISOGONAL CONJUGATE OF X(1046)

X(1247) lies on these lines: 1,409   10,846   65,1046   158,415

X(1247) = isogonal conjugate of X(1046)
X(1247) = X(21)-cross conjugate of X(1)

X(1248) = ISOGONAL CONJUGATE OF X(1047)

X(1248) lies on these lines: 1,410   73,1047   255,416

X(1248) = isogonal conjugate of X(1047)
X(1248) = X(29)-cross conjugate of X(1)

X(1249) = ISOGONAL CONJUGATE OF X(1073)

X(1249) is the perspector of triangle ABC and the tangential triangle of the circumconic centered at X(4). X(1249) is also the perspector of the medial triangle and the triangle formed by the trilinear poles of the sidelines of the orthic triangle. (Randy Hutson, 9/23/2011)

X(1249) is the unique point whose anticomplement is also its polar conjugate, namely X(253). (Randy Hutson, March 14, 2018)

Let Ea be the ellipse with B and C as foci and passing through X(4), and define Eb and Ec cyclically.
Let La be the line tangent to Ea at X(4), and define Lb and Lc cyclically.
Let A' be the trilinear pole of line La, and define B' and C' cyclically.
Then A', B', C' lie on the circumconic centered at X(1249). (Randy Hutson, March 14, 2018)

Let A'B'C' be the medial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B" and C" cyclically. Also, A"B"C" is the triangle formed by trilinear poles of sides of the orthic triangle. The lines A'A", B'B", C'C" concur in X(1249). (Randy Hutson, March 14, 2018)

X(1249) lies on hyperbola {{X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X(2).

X(1249) lies on the Thomson cubic and these lines: {1,281}, {2,253}, {3,1033}, {4,6}, {9,1712}, {19,57}, {20,3172}, {25,5304}, {69,648}, {108,198}, {112,376}, {154,3079}, {186,1609}, {193,297}, {208,2270}, {216,631}, {219,1783}, {223,3352}, {232,800}, {233,3090}, {264,3618}, {273,5222}, {317,1992}, {346,1897}, {347,653}, {461,3192}, {579,1715}, {604,2202}, {608,2122}, {610,3213}, {920,1720}, {1108,1148}, {1118,2264}, {1743,1785}, {1838,2956}, {1870,3554}, {1941,2138}, {1968,5065}, {2165,3018}, {3003,3147}, {3068,3535}, {3069,3536}, {3089,5305}, {3199,5319}, {3284,3529}, {3343,3356}, {5200,5411}}

X(1249) = isogonal conjugate of X(1073)
X(1249) = isotomic conjugate of X(34403)
X(1249) = anticomplement of X(20208)
X(1249) = X(2)-Ceva conjugate of X(4)
X(1249) = X(154)-cross conjugate of X(20)
X(1249) = X(2)-crosspoint of X(20)
X(1249) = crosssum of X(6) and X(64)
X(1249) = X(4)-Hirst inverse of X(1503)
X(1249) = complement of X(253)
X(1249) = polar conjugate of X(253)
X(1249) = perspector of ABC and antipedal triangle of X(3346)
X(1249) = perspector of pedal and anticevian triangles of X(3183)
X(1249) = perspector of ABC and medial triangle of pedal triangle of X(1498)
X(1249) = center of circumconic that is locus of trilinear poles of lines passing through X(4)
X(1249) = trilinear product X(6)*X(661)
X(1249) = perspector, wrt medial triangle, of polar circle

X(1250) = ISOGONAL CONJUGATE OF X(1081)

X(1250) lies on these lines: 1,16   6,31   15,35   37,1251

X(1250) = isogonal conjugate of X(1081)
X(1250) = crosssum of X(1) and X(1277)
X(1250) = {X(6),X(55)}-harmonic conjugate of X(10638)

X(1251) = ISOGONAL CONJUGATE OF X(1082)

X(1251) lies on these lines: 1,15   7,559   13,80   37,1250   55,199

X(1251) = isogonal conjugate of X(1082)

X(1252) = ISOGONAL CONJUGATE OF X(1086)

X(1252) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)

X(1252) lies on these lines: 44,765   59,672   100,650   101,649   110,813   241,1262   644,906   902,1110   1018,1021

X(1252) = isogonal conjugate of X(1086)
X(1252) = isotomic conjugate of X(23989)
X(1252) = cevapoint of X(6) and X(101)
X(1252) = X(i)-cross conjugate of X(j) for these (i,j): (6,101), (31,110), (55,100)
X(1252) = trilinear pole of line X(101)X(692) (the tangent to circumcircle at X(101))
X(1252) = polar conjugate of X(2973)
X(1252) = X(92)-isoconjugate of X(3937)
X(1252) = crossdifference of every pair of points on line X(764)X(1647)

X(1253) = ISOGONAL CONJUGATE OF X(1088)

X(1253) lies on these lines: 1,1170   3,1037   6,31   9,294   33,756   35,255   38,1040   40,1254   48,692   165,269   219,949   220,480   238,390   497,748

X(1253) = isogonal conjugate of X(1088)
X(1253) = X(55)-Ceva conjugate of X(41)
X(1253) = crosspoint of X(55) and X(220)
X(1253) = trilinear square of X(55)
X(1253) = crosssum of X(i) and X(j) for these (i,j): (1,1445), (7,279)

X(1254) = ISOGONAL CONJUGATE OF X(1098)

X(1254) lies on these lines: 1,411   4,774   7,986   10,307   12,201   31,34   38,388   40,1253   42,65   46,255   56,244   57,961   200,1257   208,1096   269,1126   279,291   651,1046   750,1038

X(1254) = isogonal conjugate of X(1098)
X(1254) = crosspoint of X(65) and X(225)
X(1254) = crosssum of X(i) and X(j) for these (i,j): (1,411), (21,283)
X(1254) = trilinear square of X(65)

X(1255) = ISOGONAL CONJUGATE OF X(1100)

Let A₁₂B₁₂C₁₂ be Gemini triangle 12. Let A' be the perspector of conic {{A,B,C,B₁₂,C₁₂}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1255). (Randy Hutson, January 15, 2019)

X(1255) lies on these lines: 1,748   2,594   37,81   89,940   274,321   278,469   1125,1224

X(1255) = isogonal conjugate of X(1100)
X(1255) = isotomic conjugate of X(4359)
X(1255) = cevapoint of X(i) and X(j) for these (i,j): (1,37), (6,35)
X(1255) = crosssum of X(1) and X(1051)
X(1255) = trilinear pole of line X(484)X(513)
X(1255) = X(19)-isoconjugate of X(3916)
X(1255) = X(92)-isoconjugate of X(23201)

X(1256) = ISOGONAL CONJUGATE OF X(1103)

X(1256) lies on these lines: 1,280   6,282   56,84   58,285   189,937   271,936

X(1256) = isogonal conjugate of X(1103)
X(1256) = trilinear square of X(84)

X(1257) = ISOGONAL CONJUGATE OF X(1104)

X(1257) lies on these lines: 2,1265   8,278   28,72   57,78   69,279   105,960   200,1254   518,961

X(1257) = isogonal conjugate of X(1104)
X(1257) = cevapoint of X(i) and X(j) for these (i,j): (1,72), (37,200)

X(1258) = ISOGONAL CONJUGATE OF X(1107)

X(1258) lies on these lines: 6,330   81,172   171,904   213,274   291,1193

X(1258) = isogonal conjugate of X(1107)
X(1258) = cevapoint of X(i) and X(j) for these (i,j): (1,213), (6,171), (37,43)
X(1258) = X(92)-isoconjugate of X(22389)

X(1259) = ISOGONAL CONJUGATE OF X(1118)

X(1259) lies on these lines: 3,63   7,404   8,21   12,377   20,100   35,200   219,283   255,394   268,271   318,1013   329,411   355,1012   651,1035

X(1259) = isogonal conjugate of X(1118)
X(1259) = crosspoint of X(345) and X(1264)
X(1259) = X(92)-isoconjugate of X(608)
X(1259) = isotomic conjugate of isogonal conjugate of X(6056)
X(1259) = isotomic conjugate of polar conjugate of X(219)
X(1259) = X(19)-isoconjugate of X(278)

X(1260) = ISOGONAL CONJUGATE OF X(1119)

X(1260) lies on these lines: 1,939   2,954   3,63   8,405   9,55   31,218   100,329   101,154   212,219   956,1006

X(1260) = isogonal conjugate of X(1119)
X(1260) = crosspoint of X(i) and X(j) for these (i,j): (346,1265), (1252,1331)
X(1260) = crosssum of X(i) and X(j) for these (i,j): (34,1435), (1398,1407)
X(1260) = X(92)-isoconjugate of X(1407)

X(1261) = ISOGONAL CONJUGATE OF X(1122)

X(1261) lies on these lines: 8,56   31,200   41,728

X(1261) = isogonal conjugate of X(1122)
X(1261) = cevapoint of X(55) and X(200)

X(1262) = ISOGONAL CONJUGATE OF X(1146)

X(1262) = isogonal conjugate of X(1146)
X(1262) = isotomic conjugate of X(23978)
X(1262) = anticomplement of complementary conjugate of X(17044)
X(1262) = X(i)-cross conjugate of X(j) for these (i,j): (6,109), (48,110), (198,100)
X(1262) = cevapoint of X(i) and X(j) for these (i,j): (3,101), (6,109)
X(1262) = cevapoint of circumcircle intercepts of line X(6)X(41) (or of circle {{X(1),X(15),X(16)}} (V(X(1)))
X(1262) = trilinear pole of line X(109)X(692) (the tangent to the circumcircle at X(109))
X(1262) = X(92)-isoconjugate of X(3270)

X(1263) = ISOGONAL CONJUGATE OF X(1157)

Let A' be the reflection in BC of the A-vertex of the antipedal triangle of X(5), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X(1263). (Randy Hutson, December 2, 2017)

X(1263) lies on the Neuberg cubic and these lines: 4,195   5,128   30,1141   140,930

X(1263) = reflection of X(i) in X(j) for these (i,j): (5,137), (930,140)
X(1263) = isogonal conjugate of X(1157)
X(1263) = X(399)-of-orthic-triangle
X(1263) = tangential of X(484) on the Neuberg cubic
X(1263) = antigonal conjugate of X(5)
X(1263) = Cundy-Parry Psi transform of X(195)

X(1264) = ISOTOMIC CONJUGATE OF X(1118)

X(1264) lies on these lines: 8,314   69,72   219,332   319,341

X(1264) = isogonal conjugate of X(7337)
X(1264) = isotomic conjugate of X(1118)

X(1265) = ISOTOMIC CONJUGATE OF X(1119)

X(1265) lies on these lines: 1,344   2,1257   8,210   20,190   69,72   78,345   145,1191   220,346

X(1265) = isogonal conjugate of X(1398)
X(1265) = isotomic conjugate of X(1119)
X(1265) = anticomplement of X(17054)

X(1266) = ISOTOMIC CONJUGATE OF X(1120)

X(1266) lies on these lines: 7,145   10,75   44,545   142,192   239,527   320,519   522,693   536,1086

X(1266) = isotomic conjugate of X(1120)
X(1266) = anticomplement of X(2325)
X(1266) = crosspoint of X(75) and X(903)
X(1266) = crosssum of X(31) and X(902)

X(1267) = ISOTOMIC CONJUGATE OF X(1123)

X(1267) lies on these lines: 2,37   7,492   8,491   319,1271   320,1270

X(1267) = isotomic conjugate of X(1123)
X(1267) = {X(2),X(75)}-harmonic conjugate of X(5391)

X(1268) = ISOTOMIC CONJUGATE OF X(1125)

Let A₂₂B₂₂C₂₂ be Gemini triangle 22. Let A' be the perspector of conic {{A,B,C,B₂₂,C₂₂}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1268). (Randy Hutson, January 15, 2019)

X(1268) lies on these lines: 2,594   7,12   10,86   75,1089   190,1213   310,313   333,1171

X(1268) = isogonal conjugate of X(2308)
X(1268) = isotomic conjugate of X(1125)
X(1268) = X(523)-cross conjugate of X(190)
X(1268) = cevapoint of X(2) and X(10)
X(1268) = polar conjugate of X(1839)
X(1268) = X(19)-isoconjugate of X(22054)

X(1269) = ISOTOMIC CONJUGATE OF X(1126)

X(1269) lies on these lines: 7,349   10,75   69,674   79,314   86,310   141,321   903,1240   1213,1230   1227,1234

X(1269) is the {X(75),X(76)}-harmonic conjugate of X(313). For a list of other harmonic conjugates of X(1269), click Tables at the top of this page.

X(1269) = isotomic conjugate of X(1126)
X(1269) = crosspoint of X(76) and X(310)

X(1270) = ISOTOMIC CONJUGATE OF X(1131)

X(1270) lies on these lines: 2,6   4,1160   8,175   20,488   76,1132   320,1267

X(1270) is the {X(2),X(69)}-harmonic conjugate of X(1271). For a list of other harmonic conjugates of X(1270), click Tables at the top of this page.

X(1270) = isotomic conjugate of X(1131)
X(1270) = anticomplement of X(3068)
X(1270) = X(1151)-cross conjugate of X(3535)
X(1270) = homothetic center of anticomplementary triangle and cross-triangle of ABC and outer Grebe triangle
X(1270) = homothetic center of outer Grebe triangle and cross-triangle of ABC and outer Grebe triangle

X(1271) = ISOTOMIC CONJUGATE OF X(1132)

X(1271) lies on these lines: 2,6   4,1161   8,176   20,487   76,1131   319,1267

X(1271) = isotomic conjugate of X(1132)
X(1271) = anticomplement of X(3069)
X(1271) = X(1152)-cross conjugate of X(3536)
X(1271) = homothetic center of anticomplementary triangle and cross-triangle of ABC and inner Grebe triangle
X(1271) = homothetic center of inner Grebe triangle and cross-triangle of ABC and inner Grebe triangle

X(1272) = ISOTOMIC CONJUGATE OF X(1138)

X(1272) lies on these lines: 2,94   69,74

X(1272) = isotomic conjugate of X(1138)
X(1272) = anticomplement of X(1989)

X(1273) = ISOTOMIC CONJUGATE OF X(1141)

X(1273) lies on these lines: 2,231   5,311   54,69   93,264   186,340   325,523   1232,1238

X(1273) = isotomic conjugate of X(1141)
X(1273) = anticomplement of X(231)
X(1273) = X(128)-cross conjugate of X(2)

X(1274) = ISOTOMIC CONJUGATE OF X(1143)

Let I be the incenter of ABC and E_A the excircle of triangle BCI that touches segment BC, and let A' be the touchpoint. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1274). (Randy Hutson, 9/23/2011)

X(1274) lies on the cubic K200 and these lines: 2,1489   8,177   174,176   557,1488

X(1274) = isotomic conjugate of X(1143)
X(1274) = {X(8),X(556)}-harmonic conjugate of X(11143)

X(1275) = ISOTOMIC CONJUGATE OF X(1146)

X(1275) lies on these lines: 7,59   320,765   513,927   522,664   651,666   898,934

X(1275) = isotomic conjugate of X(1146)
X(1275) = isogonal conjugate of X(14936)
X(1275) = cevapoint of X(i) and X(j) for these (i,j): (69,190), (100,220)
X(1275) = X(i)-cross conjugate of X(j) for these (i,j): (63,99), (144,190), (220,100)
X(1275) = trilinear pole of line X(100)X(658) (the tangent to Steiner circumellipse at X(664))
X(1275) = X(2)-cross conjugate of X(664)
X(1275) = barycentric square of X(664)

X(1276) = 2nd EVANS PERSPECTOR

Let T be the excentral triangle, whose vertices are the A-, B-, C- excenters of triangle ABC. Let U be the equilateral triangle having segment BC as base with vertex A' on the side of BC that does not contain vertex A. Define B' and C' cyclically, and let T' be the triangle A'B'C'. Let V be the equilateral triangle having BC as base with vertex A" on the side of BC that contains A. Define B" and C" cyclically, and let T" = A"B"C". Then T and T' are perspective, and X(1276) is their perspector. (Lawrence Evans, 2/4/2003)

Evans conjectured that X(1), X(484), X(1276), X(1277) are concyclic, and he reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X(1019). (Lawrence Evans, 2/24/2003)

X(1276) is the perspector of the excentral triangle and the apices of equilateral triangles constructed outward from the sides, as in the construction of X(13). More generally, the excentral triangle is perspective to every Kiepert triangle. The locus of the perspector K_θ is the line X(4)X(9). Specifically, K_θ divides the segment from X(75) to X(9) in the ratio - ((4R+r)/s) cot θ. (Paul Yiu, 2/27/04).

X(1276) lies on the Neuberg cubic and these lines: 1,15   4,9   14,484   63,616

X(1276) = reflection of X(1277) in X(5011)
X(1276) = anticomplement of X(33397)
X(1276) = X(16)-of-excentral-triangle
X(1276) = inverse-in-Bevan-circle of X(1277) (noted by Peter J. C. Moses, Sept. 8, 2004)

X(1277) = 3rd EVANS PERSPECTOR

Continuing from X(1276), the triangles T and T" are perspective with perspector X(1277). (Lawrence Evans, 2/4/2003).

X(1277) is the perspector of the excentral triangle and the apices of equilateral triangles constructed inward from the sides, as in the construction of X(14); for a generalization, see X(1276). (Paul Yiu, 2/27/04).

X(1277) lies on the Neuberg cubic and these lines: 1,16   4,9   13,484   63,617

X(1277) = reflection of X(1276) in X(5011)
X(1277) = anticomplement of X(33396)
X(1277) = X(15)-of-excentral-triangle
X(1277) = inverse-in-Bevan-circle of X(1276) (noted by Peter J. C. Moses, Sept. 8, 2004)

X(1278) = CONGRUENT MIDWAY-PARALLELIANS POINT

Antreas Hatzipolakis described this point in Hyacinthos #3190 (7/13/2001). See also

X(1278) lies on these lines: 2,37   8,726   145,740   193,742

X(1278) is the {X(75),X(192)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click Tables at the top of this page.

X(1278) = reflection of X(192) in X(75)
X(1278) = isogonal conjugate of X(36614)
X(1278) = isotomic conjugate of X(38247)
X(1278) = crosssum of X(i) and X(j) for these {i,j}: {512, 23571}, {649, 23470}, {667, 23560}
X(1278) = crossdifference of every pair of points on line X(667)X(23472)
X(1278) = anticomplement of X(192)
X(1278) = complement of X(4788)
X(1278) = polar conjugate of isogonal conjugate of X(22149)

X(1279) = MIDPOINT OF X(1) AND X(238)

X(1279) lies on these lines: 1,6   31,354   55,614   105,910   145,344   210,748   244,902   513,663   516,1086   551,752   595,942   56,1418   1201,2293

X(1279) is the {X(1),X(1001)}-harmonic conjugate of X(37). For a list of other harmonic conjugates of X(1279), click Tables at the top of this page.

X(1279) = midpoint of X(1) and X(238)
X(1279) = reflection of X(44) in X(238)
X(1279) = isogonal conjugate of X(1280)
X(1279) = complement of X(32850)
X(1279) = anticomplement of X(3823)
X(1279) = crosspoint of X(i) and X(j) for these (i,j): (1,105), (927,1016)
X(1279) = crosssum of X(i) and X(j) for these (i,j): (1,518), (926,1015)
X(1279) = crossdifference of every pair of points on line X(9)X(513)
X(1279) = {X(1),X(9)}-harmonic conjugate of X(3242)
X(1279) = crossdifference of PU(56)
X(1279) = midpoint of PU(96)
X(1279) = bicentric sum of PU(96)
X(1279) = X(2)-Ceva conjugate of X(39048)
X(1279) = perspector of hyperbola {{A,B,C,X(57),X(100)}}

X(1280) = ISOGONAL CONJUGATE OF X(1279)

X(1280) lies on these lines: 1,644   8,277   57,100   81,643   105,518   145,279   200,244

X(1280) = reflection of X(644) in X(1)
X(1280) = isogonal conjugate of X(1279)
X(1280) = trilinear pole of line X(9)X(513)

X(1281) = 3rd SHARYGIN POINT

The Sharygin points are described in

Darij Grinberg, Sharygin Points Report, Hyacinthos #6293 (1/8/03) and #6315 (1/10/03)

The first of ten sections is an Introduction quoted, in part, here:

We will treat two remarkable triangles: the triangle bounded by the perpendicular bisectors of the internal angle bisectors of a triangle ABC, and the triangle bounded by the perpendicular bisectors of the external angle bisectors of triangle ABC. These two triangles and the triangle ABC are three perspective triangles, having a common perspectrix: the Lemoine axis of ABC. The mutual perspectors of the three triangles will be called the first, second and third Sharygin points of ABC (after a problem of Igor Sharygin - see Section 10).

Let A' be the point where the internal angle bisector of angle CAB meets line BC, and let A" be the point where the external angle bisector of angle CAB meets line BC. Let x be the perpendicular bisector of segment AA', and let x' be the perpendicular bisector of segment AA". Define y, z, y', z' cyclically. Let D be the point where lines y and z meet, and let D' be the point where lines y' and z' meet. Define E, F, E', F' cyclically. Then

X(1281) = points of concurrence of lines DD', EE', FF'
X(846) = homothetic center of the excentral triangle and triangle DEF
X(1054) = center of similitude of the excentral triangle and triangle D'E'F'.

X(1281) lies on the Yff contact circle and these lines: 2,846   21,99   63,147   98,100   256,291   350,1284   385,740   659,804

X(1282) = 5th SHARYGIN POINT

X(1282) is the perspector of the excentral triangle and the triangle D'E'F' constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

Let IaIbIc be the excentral triangle. Let La be the line parallel to the Brocard axis of BCIa and passing through Ia. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(1282). (Randy Hutson, December 2, 2017)

X(1282) lies on the Bevan circle and these lines: 1,41   8,1281   10,150   40,170   43,57   55,846   63,100   152,516   354,1051   518,910   (659,926)

X(1282) = X(98)-of-excentral-triangle

X(1283) = 7th SHARYGIN POINT

X(1283) is the center of similitude of the triangles DEF and D'E'F' constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

X(1283) lies on these lines: 3,1054   10,21   36,244   55,846   242,243

X(1283) = inverse-in-circumcircle of X(1054)
X(1283) = QA-P9 (QA-Miquel Center) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/30-qa-p9.html)

X(1284) = 8th SHARYGIN POINT

X(1284) is the homothetic center of the intouch intriangle and the triangle DEF constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

X(1284) lies on these lines: 1,256   7,21   37,65   57,846   350,1281   513,663

X(1284) = crosspoint of X(i) and X(j) for these (i,j): (1,98), (238,242), (1429,1447)
X(1284) = crosssum of X(i) and X(j) for these (i,j): (1,511), (291,295)
X(1284) = X(65)-Hirst inverse of X(1400)
X(1284) = bicentric sum of PU(88)
X(1284) = PU(88)-harmonic conjugate of X(3287)

X(1285) = LEMOINE HOMOTHETIC CENTER

X(1285) is the homothetic center of the antipedal triangle of X(2) and the pedal triangle of X(6). (Darij Grinberg, Hyacinthos #6577, 2/21/03). See also X(3066).

Let T denote the antipedal triangle of X(2), and let T(m) denote X(m)-of-T. Then T(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that T(m) = X(n). Note the special case m = 1285.

(2,376), (6,2), (115,99), (125,1296), (597,3), (599,20), (1285,1285), (1992,4), (2030,187), (2549,69)

Let U denote the pedal triangle of X(6), and let U(m) denote X(m)-of-U. Then U(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that U(m) = X(n). (This list and the one just above were contributed by Peter Moses, 11/15/2007.)

(2,6), (3,597), (4,1992), (20,599), (69,2549), (99,115), (187,2030), (376,2), (1285, 1285), (1296, 125)

Let A'B'C' be the antipedal triangle of X(2). The centroid of A'B'C' is X(376). Let A"B"C" be the antipedal triangle, wrt A'B'C', of X(376). A"B"C" is homothetic to ABC, and the center of homothety is X(1285). (Randy Hutson, March 21, 2019)

Let A'B'C' be the pedal triangle of X(6). Let K' be X(6) of A'B'C'. Let A"B"C" be the pedal triangle, wrt A'B'C', of K'. A"B"C" is homothetic to ABC, and the center of homothety is X(1285). (Randy Hutson, March 21, 2019)

If you have The Geometer's Sketchpad, you can view X(1285) and X(1285) Intersection.

X(1285) lies on these lines: 2,1384   4,32   6,376   99,1992   172,1058   193,1003   497,609   631,3053   1056,1914

Collings Transforms: 1286-1311

X(1286) = COLLINGS TRANSFORM OF X(22)

X(1286) lies on the circumcircle and this line: 26,98

X(1287) = COLLINGS TRANSFORM OF X(23)

X(1287) lies on the circumcircle and these lines: 5,842   110,826   523,827

X(1288) = COLLINGS TRANSFORM OF X(24)

X(1288) lies on the circumcircle and this line: 70,74

X(1288) = anticomplement of X(35969)
X(1288) = trilinear pole of line X(6)X(70)
X(1288) = Ψ(X(6), X(70))

X(1289) = COLLINGS TRANSFORM OF X(25)

X(1289) lies on the circumcircle and these lines: 4,127   24,98   25,339   66,74   111,459   378,1294   403,842   648,827

X(1289) = isogonal conjugate of X(8673)
X(1289) = trilinear pole of line X(6)X(66)
X(1289) = Ψ(X(3), X(66))
X(1289) = Ψ(X(6), X(66))
X(1289) = polar-circle-inverse of X(127)
X(1289) = polar conjugate of X(33294)
X(1289) = Thomson-isogonal conjugate of X(34146)
X(1289) = Lucas-isogonal conjugate of X(34146)
X(1289) = X(63)-isoconjugate of X(2485)

X(1290) = COLLINGS TRANSFORM OF X(36)

X(1290) lies on the circumcircle and these lines: 23,105   30,104   36,759   74,517   99,693   100,523   101,661   110,513   186,915   354,840

X(1290) = reflection of X(1325) in X(36)
X(1290) = isogonal conjugate of X(8674)
X(1290) = cevapoint of X(36) and X(513)
X(1290) = trilinear pole of line X(6)X(1718)
X(1290) = Ψ(X(6), X(1718))
X(1290) = reflection of X(100) in the Euler line
X(1290) = reflection of X(110) in line X(1)X(3)

X(1291) = COLLINGS TRANSFORM OF X(50)

X(1291) lies on the circumcircle and these lines: 30,1141   74,1154   477,550   523,930

X(1291) = cevapoint of X(50) and X(512)

X(1292) = COLLINGS TRANSFORM OF X(55)

Let L_A be the reflection of the line X(1)X(6) in the line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1292). Let M_A be the reflection of the line X(7)X(8) in the line BC, and define M_B and M_C cyclically. Let A'' = M_B∩M_C, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(1292). (Randy Hutson, 9/23/2011)

Let A'B'C' be the excentral triangle. The van Aubel lines of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(1292). (Randy Hutson, June 27, 2018)

X(1292) lies on the circumcircle and these lines: 3,105   4,120   40,103   104,376   378,915   411,1311   517,840   601,727   906,919

X(1292) = reflection of X(i) in X(j) for these (i,j): (4,120), (105,3)
X(1292) = isogonal conjugate of X(3309)
X(1292) = complement of X(34547)
X(1292) = trilinear pole of line X(6)X(354)
X(1292) = cevapoint of X(55) and X(513)
X(1292) = Ψ(X(6), X(354))
X(1292) = Λ(X(1), X(3309))
X(1292) = Λ(X(4), X(885))
X(1292) = X(127)-of-excentral-triangle
X(1292) = X(132)-of-hexyl-triangle
X(1292) = Cundy-Parry Phi transform of X(14267)
X(1292) = Cundy-Parry Psi transform of X(34159)
X(1292) = Thomson-isogonal conjugate of X(518)
X(1292) = Lucas-isogonal conjugate of X(518)

X(1293) = COLLINGS TRANSFORM OF X(121)

Let L_A be the reflection of the line X(1)X(2) in the line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1293). (Randy Hutson, 9/23/2011)

X(1293) lies on the circumcircle and these lines: 3,106   4,121   40,104   105,165   182,727   572,739

X(1293) = reflection of X(i) in X(j) for these (i,j): (4,121), (106,3)
X(1293) = isogonal conjugate of X(3667)
X(1293) = complement of X(34548)
X(1293) = trilinear pole of line X(6)X(1201)
X(1293) = cevapoint of X(55) and X(649)
X(1293) = Ψ(X(6), X(1201))
X(1293) = trilinear pole wrt 1st circumperp triangle of line X(40)X(518)
X(1293) = X(107)-of-1st-circumperp-triangle
X(1293) = X(122)-of-excentral-triangle
X(1293) = X(133)-of-hexyl-triangle
X(1293) = Λ(X(649), X(4949))
X(1293) = Thomson-isogonal conjugate of X(519)
X(1293) = Lucas-isogonal conjugate of X(519)
X(1293) = Λ(X(2254), X(3667))

X(1294) = COLLINGS TRANSFORM OF X(122)

X(1294) lies on the circumcircle and these lines: 2,133   3,107   4,122   20,110   22,1302   30,1304   112,376   378,1289   550,933

X(1294) = reflection of X(i) in X(j) for these (i,j): (4,122), (107,3)
X(1294) = complement of X(34549)
X(1294) = cevapoint of X(3) and X(30)
X(1294) = X(193)-Hirst inverse of X(297)
X(1294) = isogonal conjugate of X(6000)
X(1294) = anticomplement of X(133)
X(1294) = Λ(X(74), X(186))
X(1294) = Λ(X(5), X(2883))
X(1294) = X(134)-of-hexyl-triangle
X(1294) = eigencenter of circumanticevian triangle of X(4)
X(1294) = Cundy-Parry Phi transform of X(14249)
X(1294) = Cundy-Parry Psi transform of X(14379)
X(1294) = Thomson-isogonal conjugate of X(520)
X(1294) = Lucas-isogonal conjugate of X(520)
X(1294) = trilinear pole, wrt Thomson triangle, of line X(64)X(631)
X(1294) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(20)} (circumconic centered at X(122))
X(1294) = trilinear pole, wrt circumcevian triangle of X(30), of line X(22)X(476)

X(1295) = COLLINGS TRANSFORM OF X(123)

X(1295) lies on the circumcircle and these lines: 3,108   4,123   20,100   21,107   28,1301   40,109   101,610   268,281   347,934

X(1295) = reflection of X(i) in X(j) for these (i,j): (4,123), (108,3)
X(1295) = complement of X(34550)
X(1295) = cevapoint of X(3) and X(517)
X(1295) = isogonal conjugate of X(6001)
X(1295) = Λ(X(3), X(960))
X(1295) = Λ(X(104), X(1319))
X(1295) = X(135)-of-hexyl-triangle
X(1295) = trilinear pole of line X(6)X(2431)
X(1295) = Ψ(X(6), X(2431))
X(1295) = Cundy-Parry Phi transform of X(14257)
X(1295) = Cundy-Parry Psi transform of X(39167)
X(1295) = Thomson-isogonal conjugate of X(521)
X(1295) = Lucas-isogonal conjugate of X(521)

X(1296) = COLLINGS TRANSFORM OF X(126)

Let L_A be the reflection of the line X(2)X(6) in the line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1296). (Randy Hutson, 9/23/2011)

X(1296) lies on the circumcircle and these lines: 3,111   4,126   98,376   182,729   511,843

X(1296) = reflection of X(i) in X(j) for these (i,j): (4,126), (111,3)
X(1296) = isogonal conjugate of X(1499)
X(1296) = cevapoint of X(512) and X(574)
X(1296) = trilinear pole of line X(6)X(373)
X(1296) = Ψ(X(6), X(373))
X(1296) = reflection of X(2696) in the Euler line
X(1296) = reflection of X(2709) in the Brocard axis
X(1296) = reflection of X(2746) in line X(1)X(3)
X(1296) = reflection of X(74) in line X(3)X(351)
X(1296) = X(138)-of-hexyl-triangle
X(1296) = X(74)-of-circumsymmedial-triangle
X(1296) = Λ(trilinear polar of X(1992))
X(1296) = 1st-Parry-to-ABC similarity image of X(111)
X(1296) = X(4) of 4th anti-Brocard triangle
X(1296) = perspector of 4th anti-Brocard and 1st Ehrmann triangles
X(1296) = Cundy-Parry Phi transform of X(14263)
X(1296) = Cundy-Parry Psi transform of X(34161)
X(1296) = Thomson-isogonal conjugate of X(524)
X(1296) = Lucas-isogonal conjugate of X(524)
X(1296) = perspector of ABC and 1st anti-Parry triangle
X(1296) = X(111)-of-1st-anti-Parry-triangle
X(1296) = X(9156)-of-2nd-anti-Parry-triangle

X(1297) = COLLINGS TRANSFORM OF X(127)

X(1297) lies on the circumcircle, the hyperbolas {{A,B,C,X(4),X(22)}} and {{A,B,C,X(2),X(3)}}, and these lines: 2,107   3,112   4,127   20,99   22,110   23,1304   25,1073   30,935   97,933   108,1214   476,858

X(1297) = reflection of X(i) in X(j) for these (i,j): (4,127), (112,3)
X(1297) = X(232)-cross conjugate of X(2)
X(1297) = cevapoint of X(3) and X(511)
X(1297) = crosssum of X(20) and X(147)
X(1297) = isogonal conjugate of X(1503)
X(1297) = isotomic conjugate of X(30737)
X(1297) = complement of X(12384)
X(1297) = anticomplement of X(132)
X(1297) = trilinear pole of line X(6)X(520)
X(1297) = Ψ(X(6), X(520))
X(1297) = Λ(X(4), X(6))
X(1297) = Λ(X(98), X(230))
X(1297) = X(139)-of-hexyl triangle
X(1297) = inverse-in-{circumcircle, nine-point circle}-inverter of X(122)
X(1297) = Cundy-Parry Phi transform of X(8743)
X(1297) = Cundy-Parry Psi transform of X(14376)
X(1297) = Thomson-isogonal conjugate of X(525)
X(1297) = Lucas-isogonal conjugate of X(525)
X(1297) = X(4)-of-1st-anti-orthosymmedial-triangle
X(1297) = SR(P,U), where P and U are the circumcircle intercepts of the van Aubel line
X(1297) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(34186)

X(1298) = COLLINGS TRANSFORM OF X(130)

X(1298) lies on the circumcircle and these lines: 2,129   3,1303   4,130   51,107   54,112   97,110   184,933

X(1298) = reflection of X(i) in X(j) for these (i,j): (4,130), (1303,3)
X(1298) = isogonal conjugate of X(32428)
X(1298) = anticomplement of X(129)
X(1298) = X(107)-of-Lucas-triangle (defined at X(95))
X(1298) = X(99)-of-circumorthic-triangle
X(1298) = trilinear pole, wrt Lucas triangle, of line X(2979)X(33971)
X(1298) = point of intersection, other than A, B, C, of circumcircle and hyperbola {{A,B,C,X(4),X(51)}} (circumconic centered at X(130))

X(1299) = COLLINGS TRANSFORM OF X(135)

X(1299) lies on the circumcircle and these lines: 4,131   24,110   99,317   403,476   459,1302

X(1299) = reflection of X(4) in X(135)
X(1299) = Λ(X(4), X(155))
X(1299) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(4),X(24)}} (circumconic centered at X(135))
X(1299) = isogonal conjugate of crosspoint of X(155) and X(2931) wrt both the excentral and tangential triangles
X(1299) = inverse-in-polar-circle of X(131)

X(1300) = COLLINGS TRANSFORM OF X(136)

X(1300) lies on the circumcircle and these lines: 2,131   3,847   4,110   20,254   24,107   25,1302   93,930   99,264   109,225   112,393   186,476   403,1304   687,691

X(1300) = reflection of X(i) in X(j) for these (i,j): (4,136), (925,3)
X(1300) = isogonal conjugate of X(13754)
X(1300) = cevapoint of X(4) and X(186)
X(1300) = X(i)-cross conjugate of X(j) for these (i,j): (30,4), (50,275)
X(1300) = Λ(X(5), X(389))
X(1300) = anticomplement of X(131)
X(1300) = inverse-in-polar-circle of X(113)
X(1300) = pole wrt polar circle of trilinear polar of X(3580)
X(1300) = X(48)-isoconjugate (polar conjugate) of X(3580)
X(1300) = eigencenter of circumanticevian triangle of X(3)
X(1300) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,X(4),X(93)}} (circumconic centered at X(136))

X(1301) = COLLINGS TRANSFORM OF X(235)

X(1301) lies on the circumcircle and these lines: 4,122   24,64   25,1073   28,1295   98,459   162,934   403,477

X(1301) = anticomplement of X(35968)
X(1301) = X(520)-cross conjugate of X(4)
X(1301) = cevapoint of X(i) and X(j) for these (i,j): (25,647), (235,523)
X(1301) = isogonal conjguate of X(8057)
X(1301) = trilinear pole of line X(6)X(64)
X(1301) = concurrence of reflections in sides of ABC of line X(4)X(64)
X(1301) = Ψ(X(3), X(64))
X(1301) = Ψ(X(4), X(64))
X(1301) = Ψ(X(6), X(64))
X(1301) = Ψ(X(69), X(20))
X(1301) = Λ(X(6587), X(8057)) (line X(6587)X(8057) is the trilinear polar of X(20), which is also perspectrix of ABC and half-altitude triangle)
X(1301) = polar-circle-inverse of X(122)
X(1301) = Moses-radical-circle-inverse of X(32687)
X(1301) = X(63)-isoconjugate of X(6587)

X(1302) = COLLINGS TRANSFORM OF X(381)

X(1302) lies on the circumcircle and these lines: 2,74   22,1294   23,477   25,1300   30,841   459,1299   648,1304

X(1302) = cevapoint of X(381) and X(523)
X(1302) = isogonal conjugate of X(8675)
X(1302) = isotomic conjugate of X(30474)
X(1302) = trilinear pole of line X(6)X(30)
X(1302) = Ψ(X(6), X(30))
X(1302) = Λ(X(6), X(647))
X(1302) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(113)

X(1303) = COLLINGS TRANSFORM OF X(389)

X(1303) lies on the circumcircle and these lines: 2,130   3,1298   4,129   98,185

X(1303) = reflection of X(i) in X(j) for these (i,j): (4,129), (1298,3)
X(1303) = cevapoint of X(389) and X(512)
X(1303) = anticomplement of X(130)
X(1303) = X(1294)-of-Lucas-triangle (defined at X(95))
X(1303) = X(98)-of-circumorthic-triangle

X(1304) = COLLINGS TRANSFORM OF X(403)

Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(1304). (Randy Hutson, February 10, 2016)

X(1304) lies on the circumcircle and these lines: {2, 2697}, {3, 2693}, {4, 477}, {21, 2694}, {23, 1297}, {25, 842}, {27, 2688}, {28, 2687}, {29, 2695}, {30, 1294}, {74, 186}, {98, 468}, {99, 3233}, {100, 5379}, {102, 2075}, {103, 2073}, {104, 2074}, {107, 523}, {110, 250}, {111, 232}, {112, 647}, {378, 841}, {403, 1300}, {476, 4240}, {648, 1302}, {691, 4230}, {877, 2855}, {925, 7471}, {933, 1624}, {935, 2394}, {1141, 5627}, {1290, 4246}, {1292, 7476}, {1295, 1325}, {1296, 7482}, {1305, 7479}, {1494, 2373}, {2071, 5897}, {2159, 2249}, {2328, 2738}, {2360, 2732}, {2370, 7478}, {2433, 2715}, {2689, 7452}, {2690, 4241}, {2691, 4238}, {2696, 4235}, {2734, 3109}, {2752, 4233}, {2758, 4248}, {2766, 7435}, {2770, 4232}, {3470, 3518}, {3565, 7468}

X(1304) = reflection of X(2693) in X(3)
X(1304) = isogonal conjugate of X(9033)
X(1304) = X(186)-cross conjugate of X(250)
X(1304) = cevapoint of X(403) and X(523)
X(1304) = crosssum of X(30) and X(402)
X(1304) = cevapoint of {403, 523}, {647, 1495}
X(1304) = X(i)-cross conjugate of X(j) for these (i,j): (186,250), (526,4), (686,2052), (3003,249),. (5502,110)
X(1304) = trilinear pole of {6, 74}
X(1304) = trilinear product X(i)*X(j) for these {i,j}: {74, 162}, {112, 2349}, {648, 2159}, {662, 8749}
X(1304) = barycentric product X(i)*X(j) for these {i,j}: {74, 648}, {99, 8749}, {112, 1494}, {162, 2349}, {250, 2394}, {811, 2159}
X(1304) = polar-circle-inverse of X(3258)
X(1304) = Moses-radical-circle-inverse of X(112)
X(1304) = trilinear pole of line X(6)X(74)
X(1304) = Ψ(X(i), X(j)) for these (i,j): (3,74), (6,74), (69,74)
X(1304) = Λ(X(i), X(j)) for these (i,j): (74,1294), (107,110), (113,133), (122,125), (1636,1637), (1494,3268)
X(1304) = reflection of X(107) in the Euler line
X(1304) = inverse-in-polar-circle of X(3258)
X(1304) = inverse-in-Moses-radical-circle of X(112)
X(1304) = isotomic conjugate of polar conjugate of X(32695)
X(1304) = polar conjugate of isogonal conjugate of X(32640)
X(1304) = perspector of circumorthic triangle and cross-triangle of ABC and circumcevian triangle of X(186)
X(1304) = X(i)-isoconjugate of X(j) for these {i,j}: {1,9033}, {2,2631}, {30,656}, {63,1637}, {92,1636}, {162,1650}, {520,1784}, {525,2173}, {810,3260}, {1568,2616}, {1577,3284}, {2407,3708}, {2632,4240}, {6357,8611}

X(1305) = COLLINGS TRANSFORM OF X(440)

X(1305) lies on the circumcircle and these lines: 3,917   20,103   22,675   106,347   110,664   112,653   272,759   915,1006

X(1305) = reflection of X(917) in X(3)
X(1305) = isogonal conjugate of X(8676)
X(1305) = anticomplement of X(5190)
X(1305) = trilinear pole of line X(6)X(226)
X(1305) = Ψ(X(6), X(226))
X(1305) = cevapoint of X(i) and X(j) for these (i,j): (3,514), (440,523), (513,1214)

X(1306) = COLLINGS TRANSFORM OF X(639)

X(1306) lies on the circumcircle and these lines: 98,637   111,493

X(1306) = trilinear pole of line X(6)X(493)
X(1306) = Ψ(X(6), X(493))

X(1307) = COLLINGS TRANSFORM OF X(640)

X(1307) lies on the circumcircle and these lines: 98,638   111,494

X(1307) = trilinear pole of line X(6)X(494)
X(1307) = Ψ(X(6), X(494))

X(1308) = COLLINGS TRANSFORM OF X(661)

Let A'B'C' be the excentral triangle. The Fermat axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(1308). (Randy Hutson, June 27, 2018)

X(1308) lies on the circumcircle and these lines: 1,840   36,105   100,514   101,513   103,517   104,516   106,1279   110,1019   813,876   901,1022   919,1027

X(1308) = reflection of X(101) in line X(1)X(3)
X(1308) = isogonal conjugate of X(3887)
X(1308) = cevapoint of X(513) and X(1155)
X(1308) = trilinear pole of line X(6)X(244)
X(1308) = Ψ(X(i), X(j)) for these (i,j): (1, 528), (6, 244), (9, 11)
X(1308) = trilinear product of circumcircle intercepts of line X(1)X(528)

X(1309) = COLLINGS TRANSFORM OF X(860)

Let A', B', C' be the intersections of line X(4)X(8) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(1309). (Randy Hutson, January 29, 2018)

X(1309) lies on the circumcircle and these lines: 4,953   102,515   105,243   109,522   693,934

X(1309) = isogonal conjugate of X(8677)
X(1309) = anticomplement of X(10017)
X(1309) = cevapoint of X(i) and X(j) for these (i,j): (515,522), (523,860)
X(1309) = Λ(X(1459), X(1946))
X(1309) = Λ(trilinear polar of X(905))
X(1309) = trilinear pole of line X(6)X(281)
X(1309) = circumcircle-antipode of X(2734)
X(1309) = Ψ(X(3), X(8))
X(1309) = Ψ(X(6), X(281))
X(1309) = inverse-in-polar-circle of X(3259)
X(1309) = pole wrt polar circle of Sherman line (line X(3259)X(3326)) (see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)
X(1309) = intersection of antipedal lines of X(102) and X(109)
X(1309) = polar conjugate of X(10015)
X(1309) = X(63)-isoconjugate of X(3310)

X(1310) = COLLINGS TRANSFORM OF X(940)

X(1310) lies on the circumcircle and these lines: 98,336   105,1036   107,811   108,664   112,662   741,1245

X(1310) = cevapoint of X(513) and X(940)
X(1310) = isogonal conjugate of X(8678)
X(1310) = isotomic conjugate of X(2517)
X(1310) = trilinear pole of line X(6)X(63)
X(1310) = Ψ(X(1), X(69))
X(1310) = Ψ(X(6), X(63))
X(1310) = Λ(X(661), X(663))
X(1310) = isogonal conjugate of perspector of hyperbola {{A,B,C,X(2),X(19)}}

X(1311) = COLLINGS TRANSFORM OF X(1146)

X(1311) lies on the circumcircle and these lines: 2,109   8,101   29,112   85,934   92,108   100,312   110,333   411,1292

X(1311) = isogonal conjugate of X(8679)
X(1311) = isotomic conjugate of X(33864)
X(1311) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(33650)
X(1311) = trilinear pole of line X(6)X(522)
X(1311) = Ψ(X(6), X(522))
X(1311) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(124)

X(1312) = 1st MOSES INTERSECTION

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B),
                        where g(A,B,C) = (J - 1)cos A + 2(J + 1)cos B cos C, where J = |OH|/R; see X(1113)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (J - 1)a²SA + 2(J + 1) SB SC, where
                        S_A = (b² + c² - a²)/2, and S_B and S_C are defined cyclically.
X(1312) = 3X(2) - (3 + |OH|/R)*X(5) = 3(1 + |OH|/R)*X(2) - (3 + |OH|/R)*X(3)

As a point on the Euler line, X(1312) has Shinagawa coefficients (J - 1, J + 3).

X(1312) is a point of intersection of the Euler line and the nine-point circle. Its antipode on the nine-point circle
is X(1313). Of the two points, X(1312) is the one closer to X(4). The asymptotes of the Jerabek hyperbola meet at X(125) on the nine-point circle. One of the asymptotes meets the circle again at X(1312), and the other, at X(1313). Thus, the points X(125), X(1312), X(1313) form a right triangle of which the midpoint of the hypotenuse X(1312)-to-X(1313) is X(5). (Peter J. C. Moses, 3/14/2003)

Of the points other than X(125) in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X(1312) is the one farther from X(3). (Randy Hutson, December 2, 2017)

X(1312) lies on the nine-point circle, the MacBeath inconic and this line: 2,3

X(1312) = midpoint of X(4) and X(1113)
X(1312) = reflection of X(1313) in X(5)
X(1312) = complement of X(1114)
X(1312) = X(1113)-Ceva conjugate of X(523)
X(1312) = inverse-in-polar-circle of X(1114)
X(1312) = excentral-to-ABC functional image of X(1381)
X(1312) = X(1381)-of-orthic-triangle if ABC is acute
X(1312) = {X(2),X(1113)}-harmonic conjugate of X(468)
X(1312) = {X(2),X(858)}-harmonic conjugate of X(1313)
X(1312) = {X(4),X(403)}-harmonic conjugate of X(1313)
X(1312) = {X(427),X(468)}-harmonic conjugate of X(1313)

For a list of harmonic conjugates, click Tables at the top of this page.

X(1313) = 2nd MOSES INTERSECTION

As a point on the Euler line, X(1313) has Shinagawa coefficients (J + 1, J - 3).

Of the points other than X(125) in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X(1313) is the one nearer to X(3). (Randy Hutson, December 2, 2017)

X(1313) lies on the nine-point circle, the MacBeath inconic, and this line: 2,3

X(1313) = midpoint of X(4) and X(1114)
X(1313) = reflection of X(1312) in X(5)
X(1313) = complement of X(1113)
X(1313) = X(1114)-Ceva conjugate of X(523)
X(1313) = inverse-in-polar-circle of X(1113)
X(1313) = X(1382)-of-orthic-triangle if ABC is acute
X(1313) = excentral-to-ABC functional image of X(1382)
X(1313) = {X(2),X(1114)}-harmonic conjugate of X(468)

X(1314) = 3rd MOSES INTERSECTION

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [d² + (4r - R)R + sqrt(Q)]S_BS_C + (d² + 2r² - R²)a²S_A,
                        where Q = 4d²R(4r - R) + [d² - 3R² + 4r(r + R)]²,
                        d = distance between X(3) and X(4),
                        R = circumradius, r = inradius,
                        S_A = (b² + c² - a²)/2, and S_B and S_C are defined cyclically (Peter J. C. Moses, 3/2003)

As a point on the Euler line, X(1314) has Shinagawa coefficients (d² + 2r² - R²,4rR - 2r² + Q^1/2).

X(1314) is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

X(1314) lies on the incircle and this line: 2,3

X(1315) = 4th MOSES INTERSECTION

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [d² + (4r - R)R - sqrt(Q)]S_BS_C + (d² + 2r² - R²)a²S_A,
                        where Q = 4d²R(4r - R) + [d² - 3R² + 4r(r + R)]²,
                        d = distance between X(3) and X(4),
                        R = circumradius, r = inradius,
                        S_A = (b² + c² - a²)/2, and S_B and S_C are defined cyclically (Peter J. C. Moses, 3/2003)

As a point on the Euler line, X(1315) has Shinagawa coefficients (d² + 2r² - R2, 4rR - 2r² - Q^1/2).

X(1315) is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

X(1315) lies on the incircle and this line: 2,3

X(1316) = 5th MOSES INTERSECTION

Barycentrics    a⁸ + a⁴b²c² - a⁶(b² + c²) + b²c²(b² - c²)² : :
Barycentrics    b²c²(b² - c²)² + a⁴(a² - b²)(a² - c²)
X(1316) = X[2452] + 2 X[2453]

As a point on the Euler line, X(1316) has Shinagawa coefficients (3(E + F)F - S², -(E + F)² + 3S²).

X(1316) is the point of intersection, other than X(3), of the Euler line and Brocard circle.
X(1316) is the point of intersection, other than X(6), of the Brocard circle and orthosymmedial circle.
X(1316) is the point of intersection of the Euler line and the trilinear polar of X(98). (P.J.C. Moses, 6/22/04)
X(1316) is the orthogonal projection of X(6) on the Euler line.

Let A₂B₂C₂, A₃B₃C₃, A₄B₄C₄ be the 2nd, 3rd and 4th Brocard triangles, respectively. Let A' = B₃B₄∩C₃C₄, and define B' and C' cyclically. The lines A₂A', B₂B', C₂C' concur in X(1316). (Randy Hutson, January 29, 2015)

If you have The Geometer's Sketchpad, you can view X(1316).

X(1316) lies on the Brocard circle, the orthosymmedial circle, the cubics K023, K166, K508, K509, K876, K1091, and these lines: {2, 3}, {6, 523}, {51, 31850}, {76, 17941}, {98, 5191}, {99, 9155}, {110, 2782}, {115, 8429}, {125, 2794}, {182, 6795}, {183, 34245}, {184, 18338}, {247, 5972}, {250, 264}, {262, 842}, {287, 10753}, {323, 32515}, {338, 1576}, {373, 3111}, {476, 2698}, {538, 3292}, {542, 16280}, {543, 5465}, {691, 3972}, {1083, 24271}, {1561, 2777}, {1632, 20975}, {1648, 7737}, {1649, 14685}, {1968, 34859}, {1975, 2396}, {2394, 9168}, {2549, 16319}, {2790, 5622}, {3233, 35259}, {3447, 3613}, {3734, 5108}, {3815, 16320}, {3849, 32225}, {5091, 24288}, {5099, 5475}, {5149, 13518}, {5468, 6090}, {5476, 16279}, {5489, 11123}, {6232, 12508}, {6322, 11594}, {6531, 9475}, {6792, 18307}, {6794, 15048}, {7668, 7669}, {7698, 9159}, {7735, 16315}, {7736, 16316}, {7816, 11052}, {8723, 11183}, {8754, 23583}, {9169, 15539}, {11130, 14185}, {11131, 14187}, {11422, 14480}, {11586, 16632}, {12079, 26869}, {13434, 15112}, {14265, 32545}, {14682, 32314}, {15033, 15111}, {15743, 16633}, {16321, 32113}, {19128, 32428}, {19571, 25332}, {22505, 30789}, {22512, 30465}, {22513, 30468}, {23635, 30716}, {30715, 34845}, {32222, 32424}

X(1316) = midpoint of X(6) and X(2453)
X(1316) = reflection of X(i) in X(j) for these {i,j}: {2452, 6}, {5112, 468}, {6795, 182}, {16279, 5476}, {32113, 16321}, {32224, 32217}
X(1316) = isogonal conjugate of X(9513)
X(1316) = complement of X(36163)
X(1316) = anticomplement of X(11007)
X(1316) = crosspoint of X(16070) and X(16071)
X(1316) = crosssum of X(13414) and X(13415)
X(1316) = crossdifference of PU(145)
X(1316) = antigonal conjugate of X(38947)
X(1316) = X(2)-Ceva conjugate of X(39078)
X(1316) = perspector of conic {{A,B,C,X(98),X(648)}}
X(1316) = circumcircle-inverse of X(237)
X(1316) = nine-point-circle-inverse of X(2450)
X(1316) = 2nd-Lemoine-circle-inverse of X(2451)
X(1316) = orthocentroidal-circle-inverse of X(868)
X(1316) = polar-circle-inverse of X(297)
X(1316) = orthoptic-circle-of-Steiner-inellipse-inverse of X(1513)
X(1316) = circumcircle-of-inner-Napoleon-triangle-inverse of X(1080)
X(1316) = circumcircle-of-outer-Napoleon-triangle-inverse-of X(383)
X(1316) = psi-transform of X(98)
X(1316) = X(1)-isoconjugate of X(9513)
X(1316) = crosspoint of X(16070) and X(16071)
X(1316) = crosssum of X(13414) and X(13415)
X(1316) = crossdifference of every pair of points on line X(511)X(647)
X(1316) = intersection, other than X(6), of Brocard circles of ABC and orthocentroidal triangle
X(1316) = X(6)-Hirst inverse of X(523)
X(1316) = X(110) of 1st Brocard triangle
X(1316) = orthocentroidal-to-1st-Brocard similarity image of X(2)
X(1316) = crosssum of S1 and S2 on the Brocard (third) cubic, K019, these being the Brocard-circle intercepts of the line X(2)X(98)
X(1316) = 1st-Brocard-isogonal conjugate of X(690)
X(1316) = 1st-Brocard-isotomic conjugate of X(3569)
X(1316) = barycentric product X(2966)*X(31953)
X(1316) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 9513}, {31953, 2799}
X(1316) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 4, 868}, {2, 20, 35922}, {2, 23, 9832}, {2, 4226, 3}, {2, 10684, 14960}, {2, 26255, 14694}, {3, 25, 21525}, {4, 2409, 25}, {98, 35278, 5191}, {338, 1576, 9512}, {401, 419, 237}, {441, 460, 2450}, {868, 15000, 2}, {1113, 1114, 237}, {1312, 1313, 2450}, {5004, 5005, 15915}, {16179, 16180, 381}, {16181, 16182, 3}, {32460, 32461, 2}, {35606, 35906, 1640}

X(1317) = REFLECTION OF X(11) IN X(1)

X(1317) is the antipode of X(11) on the incircle.

X(1317) lies on these lines: 1,5   7,528   55,104   56,100   149,388   153,497   214,519

X(1317) = midpoint of X(100) and X(145)
X(1317) = reflection of X(i) in X(j) for these (i,j): (11,1), (80,1387), (1145,214)
X(1317) = isogonal conjugate of X(1318)
X(1317) = anticomplement of X(3036)
X(1317) = X(74)-of-intouch-triangle
X(1317) = X(104)-of-Mandart-incircle-triangle
X(1317) = homothetic center of intangents triangle and reflection of extangents triangle in X(104)
X(1317) = inverse-in-Feuerbach-hyperbola of X(1387)
X(1317) = {X(1),X(80)}-harmonic conjugate of X(1387)

X(1318) = ISOGONAL CONJUGATE OF X(1317)

X(1318) lies on these lines: 1,1168   36,106   88,517   679,1319

X(1319) = BEVAN-SCHRÖDER POINT

Let X'Y'Z' be the pedal triangle of the Bevan point, W = X(40); then X(1319) is the point, other than W, in which the circles AWX', BWY', CWZ' concur. (Floor van Lamoen, Hyacinthos #6321, 6352).

X(1319) = intersection of X(1)X(3) and the Helman line (the radical axis of the incircle and circumcircle, i.e., X(513)X(663); see the preamble just before X(51640)).

(i) The Helman line is the radical axis of the Zaslavsky pencil, containing the circumcircle and incenter (A. Zaslavsky, August, 2022)

(ii) the isogonal image of the Helman line is the I-centered circumellipse having semiaxes of lengths 2R(R-r+d) and 2R(R-r-d), where d = |OI. Accordingly, over Chapple's porism (with fixed R an r), the I-circumellipse rotates rigidly about its center. (D. Reznik et al., August, 2022). See GeoGebra sketch

(iii) Over Chapple's porism, the locus of each point on the Helman line is a circle, called a Skutin circle, of radius R. (D. Reznik, August, 2022). This is a special case of a result in A. Skutin, "On Rotation of Isogonal Point", J. Classical Geom., vol 2, 2013.

(iv) The set of Skutin circles corresponding to all the points on a given circle in the Zaslavsky pencil all have the same radius. The locus of the center is a circle. (D. Reznik, August, 2022). See GeoGebra sketch

X(1319) lies on the curves K338, K826, K1160, and Q071, and on these lines: {1, 3}, {2, 3476}, {4, 7704}, {5, 45287}, {7, 2320}, {8, 6049}, {10, 5433}, {11, 515}, {12, 1125}, {19, 37519}, {20, 12701}, {21, 1408}, {30, 1387}, {33, 37391}, {34, 1878}, {37, 604}, {41, 40133}, {42, 1450}, {44, 1404}, {48, 1108}, {59, 518}, {63, 5289}, {72, 8666}, {73, 1104}, {77, 1122}, {78, 12513}, {79, 31776}, {80, 3582}, {85, 24805}, {100, 3880}, {101, 2348}, {104, 2720}, {105, 14733}, {106, 1168}, {108, 953}, {109, 2718}, {140, 10039}, {145, 1788}, {198, 3554}, {210, 956}, {214, 519}, {221, 1616}, {222, 16483}, {223, 16485}, {225, 23675}, {226, 535}, {227, 3924}, {238, 9282}, {269, 35227}, {278, 5146}, {279, 24796}, {329, 34610}, {348, 24798}, {355, 499}, {376, 30305}, {381, 23708}, {388, 2478}, {392, 993}, {404, 4861}, {405, 9850}, {442, 10957}, {474, 3698}, {495, 38028}, {496, 10572}, {497, 5731}, {513, 663}, {516, 15326}, {528, 30379}, {529, 908}, {536, 43135}, {553, 51103}, {572, 38855}, {595, 1399}, {603, 3915}, {611, 38029}, {614, 37366}, {651, 3246}, {664, 1447}, {672, 6603}, {679, 1318}, {751, 1386}, {758, 5083}, {759, 34921}, {840, 934}, {859, 18191}, {901, 1417}, {909, 2161}, {910, 1055}, {912, 6265}, {936, 3983}, {938, 6962}, {943, 15179}, {944, 1837}, {946, 4311}, {950, 37722}, {952, 1737}, {958, 3305}, {959, 25417}, {960, 2975}, {961, 1255}, {1000, 3524}, {1001, 8545}, {1006, 44085}, {1014, 7269}, {1015, 43039}, {1042, 20615}, {1056, 6947}, {1058, 4305}, {1071, 7335}, {1100, 1400}, {1191, 34046}, {1193, 2594}, {1210, 5882}, {1212, 9310}, {1254, 46190}, {1308, 1477}, {1320, 13587}, {1323, 1358}, {1329, 26482}, {1354, 31522}, {1357, 11717}, {1359, 3326}, {1360, 3328}, {1361, 3025}, {1362, 11712}, {1364, 11713}, {1365, 31524}, {1376, 3872}, {1389, 37518}, {1401, 49480}, {1405, 16666}, {1406, 34040}, {1407, 16486}, {1409, 16685}, {1412, 4653}, {1415, 1914}, {1418, 7225}, {1419, 16487}, {1434, 4955}, {1442, 24471}, {1445, 42871}, {1452, 11396}, {1471, 49478}, {1478, 5886}, {1479, 11373}, {1519, 2829}, {1538, 12764}, {1575, 21859}, {1612, 26888}, {1621, 10179}, {1647, 14584}, {1656, 10827}, {1698, 37709}, {1699, 12943}, {1723, 20818}, {1727, 48667}, {1743, 38296}, {1770, 22791}, {1836, 4293}, {1841, 7120}, {1846, 1877}, {1858, 12675}, {1864, 18446}, {1876, 3446}, {1887, 6198}, {1898, 6261}, {1935, 9363}, {1960, 6550}, {2003, 5315}, {2067, 7968}, {2082, 3207}, {2087, 2251}, {2136, 15347}, {2171, 3723}, {2178, 2262}, {2208, 2218}, {2260, 17438}, {2263, 15306}, {2285, 16777}, {2329, 36476}, {2360, 40980}, {2362, 44635}, {2390, 3937}, {2551, 24954}, {2650, 42443}, {2703, 35108}, {2716, 8059}, {2745, 30239}, {2771, 47379}, {2800, 17010}, {2802, 35271}, {3011, 6075}, {3021, 39760}, {3022, 11714}, {3023, 11710}, {3024, 11709}, {3027, 11711}, {3028, 11720}, {3035, 6735}, {3058, 4304}, {3085, 6967}, {3146, 18220}, {3160, 7195}, {3218, 44663}, {3230, 4559}, {3241, 5435}, {3243, 41712}, {3244, 4848}, {3286, 43947}, {3290, 9259}, {3306, 40726}, {3320, 11722}, {3324, 11718}, {3325, 11721}, {3419, 45700}, {3436, 25681}, {3452, 34606}, {3475, 6992}, {3485, 3600}, {3486, 6838}, {3487, 6936}, {3555, 22836}, {3583, 7743}, {3584, 5444}, {3585, 9955}, {3586, 11238}, {3624, 9578}, {3634, 7294}, {3636, 3649}, {3638, 49538}, {3639, 49540}, {3653, 10056}, {3655, 5722}, {3671, 4114}, {3674, 7198}, {3679, 31231}, {3720, 40109}, {3752, 49487}, {3769, 20037}, {3812, 5253}, {3827, 38863}, {3877, 4640}, {3878, 3916}, {3881, 15556}, {3884, 5267}, {3890, 4189}, {3893, 5687}, {3895, 4421}, {3912, 43053}, {3913, 4855}, {3947, 15808}, {3962, 5730}, {4059, 7176}, {4124, 47043}, {4188, 14923}, {4225, 18178}, {4292, 13464}, {4295, 10595}, {4296, 35998}, {4297, 6284}, {4299, 7702}, {4316, 28146}, {4318, 40577}, {4321, 38316}, {4333, 48661}, {4345, 9778}, {4351, 23152}, {4390, 44798}, {4413, 4731}, {4432, 24816}, {4487, 17780}, {4534, 8074}, {4551, 49997}, {4654, 51105}, {4702, 4742}, {4719, 17016}, {4853, 5438}, {4863, 34625}, {4868, 26740}, {4906, 17080}, {4915, 46917}, {4995, 50828}, {4999, 24987}, {5044, 5258}, {5088, 24203}, {5089, 32674}, {5219, 11237}, {5220, 30318}, {5248, 12709}, {5261, 46934}, {5270, 5443}, {5288, 34790}, {5432, 10165}, {5439, 30147}, {5441, 31795}, {5450, 12672}, {5453, 13391}, {5541, 41702}, {5550, 10588}, {5552, 32049}, {5572, 30284}, {5577, 33902}, {5691, 10896}, {5718, 6176}, {5724, 24239}, {5745, 31157}, {5784, 42842}, {5790, 37708}, {5794, 10527}, {5844, 12735}, {5853, 41555}, {5854, 51433}, {5887, 32153}, {5901, 12047}, {5905, 34647}, {6020, 12265}, {6224, 37797}, {6285, 12262}, {6502, 7969}, {6604, 17081}, {6647, 20335}, {6691, 24982}, {6700, 21031}, {6880, 7967}, {6882, 38032}, {6906, 45776}, {7051, 7052}, {7117, 8608}, {7173, 19925}, {7175, 16484}, {7178, 48328}, {7181, 9436}, {7191, 35996}, {7223, 40719}, {7235, 39766}, {7247, 17084}, {7284, 28444}, {7292, 14513}, {7340, 47376}, {7355, 40658}, {7491, 39599}, {7672, 15570}, {7741, 18480}, {7951, 11230}, {7972, 22935}, {8077, 10506}, {8227, 9613}, {8283, 21147}, {8286, 36195}, {8607, 47434}, {8649, 49758}, {8983, 19028}, {9318, 44664}, {9552, 19858}, {9579, 11522}, {9583, 19038}, {9612, 9624}, {9614, 12953}, {9615, 31432}, {9654, 37692}, {9655, 18493}, {9661, 49601}, {9708, 35272}, {9848, 10884}, {9956, 37710}, {10016, 34182}, {10081, 11670}, {10090, 12737}, {10283, 39542}, {10320, 26492}, {10391, 18444}, {10401, 17321}, {10428, 34051}, {10459, 28385}, {10483, 22793}, {10501, 18456}, {10502, 18454}, {10503, 18448}, {10573, 37727}, {10609, 41552}, {10624, 15338}, {10826, 18525}, {10866, 12520}, {10914, 22837}, {10934, 18621}, {10949, 17647}, {10956, 17757}, {11236, 30852}, {11281, 33961}, {11346, 28997}, {11364, 12835}, {11365, 18954}, {11368, 18957}, {11370, 18959}, {11371, 18960}, {11374, 18962}, {11377, 18963}, {11378, 18964}, {11496, 17634}, {11570, 14988}, {11699, 19470}, {11705, 18974}, {11706, 18975}, {11726, 34929}, {11735, 46683}, {11739, 18973}, {11740, 18972}, {11831, 18958}, {12019, 28224}, {12081, 20718}, {12114, 12688}, {12119, 13274}, {12258, 18969}, {12259, 18970}, {12260, 18979}, {12261, 18968}, {12263, 18982}, {12264, 18983}, {12266, 18984}, {12267, 18985}, {12268, 18989}, {12269, 18988}, {12607, 27385}, {12647, 26446}, {12749, 38752}, {12758, 38602}, {12848, 51099}, {13182, 38220}, {13374, 45977}, {13407, 37737}, {13411, 15888}, {13607, 37734}, {13667, 18986}, {13787, 18987}, {13883, 18965}, {13902, 31408}, {13936, 18966}, {13971, 19027}, {14100, 42884}, {14204, 40437}, {14257, 38295}, {14597, 21770}, {14872, 45770}, {15015, 48696}, {15175, 15180}, {15228, 28198}, {15298, 38031}, {15500, 37305}, {15558, 17613}, {15726, 18450}, {15746, 47006}, {15999, 17114}, {16174, 24042}, {16232, 44636}, {16417, 40587}, {16609, 50023}, {16826, 41245}, {16858, 29007}, {17044, 51400}, {17053, 40590}, {17221, 17863}, {17284, 31230}, {17448, 41526}, {17566, 32537}, {17604, 30283}, {17611, 30285}, {17637, 33858}, {17768, 51423}, {18242, 26476}, {18395, 37707}, {18458, 30375}, {18460, 30376}, {18469, 30377}, {18471, 30378}, {18514, 33697}, {18526, 37711}, {18593, 24201}, {18654, 24993}, {18971, 22475}, {18978, 22476}, {18991, 18995}, {18992, 18996}, {19365, 44547}, {19373, 33655}, {19860, 25524}, {21008, 41015}, {21214, 37694}, {21342, 49454}, {21871, 36743}, {22344, 23844}, {22345, 23846}, {23154, 41682}, {23205, 23845}, {23711, 51359}, {24029, 34230}, {24036, 41391}, {24216, 43056}, {24331, 36487}, {24465, 28174}, {24541, 25466}, {24583, 26581}, {24612, 30812}, {24871, 25026}, {25904, 25914}, {26015, 41557}, {26115, 26126}, {26321, 31937}, {26367, 26433}, {26368, 26434}, {26369, 26435}, {26370, 26436}, {26690, 30618}, {26959, 28771}, {28027, 28036}, {28077, 28082}, {28461, 41695}, {28969, 33839}, {29604, 31221}, {29660, 36493}, {29817, 35989}, {30275, 38053}, {30827, 31141}, {31053, 34605}, {31225, 36534}, {31394, 49537}, {32238, 32243}, {32331, 32336}, {32426, 35023}, {33337, 33598}, {33595, 51071}, {33646, 51361}, {33667, 39778}, {34195, 41697}, {34522, 40131}, {34586, 38984}, {34772, 34791}, {35762, 35768}, {35763, 35769}, {36123, 45766}, {36444, 36451}, {36505, 36513}, {36926, 37758}, {36944, 38462}, {37006, 37718}, {37139, 39308}, {37168, 37790}, {37728, 50824}, {37815, 45272}, {39870, 39873}, {39897, 49511}, {41389, 51506}, {43052, 48344}, {43054, 49768}, {43820, 43822}, {45398, 45404}, {45399, 45405}, {45500, 45506}, {45501, 45507}, {48894, 49745}, {50148, 51345}, {50594, 50604}, {51625, 51628}, {51627, 51629}

X(1319) = midpoint of X(i) and X(j) for these {i,j}: {1, 36}, {100, 38460}, {1155, 5048}, {1317, 40663}, {2718, 47622}, {3583, 36975}, {4318, 40577}, {5057, 20067}, {5126, 25405}, {5541, 41702}, {7972, 41684}, {10222, 10225}, {14151, 37787}, {21578, 30384}, {31524, 39751}
X(1319) = reflection of X(i) in X(j) for these {i,j}: {1, 25405}, {3, 18857}, {4, 22835}, {10, 6681}, {11, 44675}, {36, 5126}, {65, 18838}, {484, 5122}, {1155, 36}, {1737, 15325}, {3326, 51616}, {3583, 7743}, {3689, 5440}, {3814, 1125}, {5048, 1}, {5080, 5087}, {5176, 5123}, {5183, 1155}, {5440, 214}, {6735, 3035}, {13528, 3}, {18838, 3660}, {23960, 33179}, {24042, 16174}, {30384, 1387}, {36920, 40663}, {40663, 3911}, {41542, 5427}, {41698, 1538}, {44784, 1145}
X(1319) = isogonal conjugate of X(1320)
X(1319) = complement of X(5176)
X(1319) = anticomplement of X(5123)
X(1319) = circumcircle-inverse of X(56)
X(1319) = incircle-inverse of X(65)
X(1319) = Bevan-circle-inverse of X(5128)
X(1319) = Conway-circle-inverse of X(12435)
X(1319) = isogonal conjugate of the anticomplement of X(1145)
X(1319) = isogonal conjugate of the polar conjugate of X(37790)
X(1319) = X(i)-Ceva conjugate of X(j) for these (i,j): {8, 13539}, {1411, 65}, {1443, 1465}, {1465, 2182}, {2222, 513}, {3911, 44}, {8686, 56}, {24029, 654}, {34051, 6}, {36944, 51422}, {37136, 650}, {37168, 1877}
X(1319) = X(i)-cross conjugate of X(j) for these (i,j): {902, 44}, {2087, 30725}
X(1319) = cevapoint of X(i) and X(j) for these (i,j): {902, 1404}, {1647, 30572}, {1960, 2087}
X(1319) = crosspoint of X(i) and X(j) for these (i,j): {1, 104}, {7, 2006}, {59, 2720}, {7052, 33655}
X(1319) = crosssum of X(i) and X(j) for these (i,j): {1, 517}, {9, 3689}, {11, 2804}, {55, 2323}, {521, 35014}, {2170, 4895}, {3880, 45247}, {4867, 40587}, {5239, 5240}
X(1319) = trilinear pole of line {1635, 20972}
X(1319) = crossdifference of every pair of points on line {9, 650}
X(1319) = X(186)-of-intouch-triangle
X(1319) = X(2077)-of-Mandart-incircle- triangle
X(1319) = homothetic center of intangents triangle and reflection of extangents triangle in X(2077)
X(1319) = orthocenter of pedal triangle of X(36)
X(1319) = perspector of ABC and reflection of extangents triangle in antiorthic axis
X(1319) = X(i)-isoconjugate of X(j) for these (i,j): {1, 1320}, {2, 2316}, {6, 4997}, {8, 106}, {9, 88}, {11, 9268}, {21, 4674}, {36, 36590}, {41, 20568}, {55, 903}, {60, 4013}, {78, 36125}, {100, 23838}, {219, 6336}, {281, 1797}, {284, 4080}, {312, 9456}, {318, 36058}, {341, 1417}, {345, 8752}, {514, 5548}, {519, 1318}, {522, 901}, {644, 1022}, {649, 4582}, {650, 3257}, {663, 4555}, {679, 3689}, {999, 36596}, {1120, 45247}, {1168, 4511}, {1639, 4638}, {2170, 5376}, {2226, 2325}, {2320, 4792}, {2364, 4945}, {3699, 23345}, {3700, 4591}, {3709, 4615}, {3939, 6548}, {4041, 4622}, {4049, 5546}, {4076, 43922}, {4391, 32665}, {4543, 39414}, {4618, 4895}, {4845, 36887}, {5549, 23598}, {6065, 6549}, {6551, 21132}, {6735, 10428}, {7017, 32659}, {7077, 27922}, {8851, 36814}, {14260, 51565}, {14942, 34230}, {32719, 35519}, {36910, 40215}
X(1319) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 1320}, {8, 214}, {9, 4997}, {44, 32851}, {88, 478}, {223, 903}, {312, 4370}, {318, 20619}, {519, 4723}, {522, 38979}, {1319, 5176}, {1647, 4768}, {2316, 32664}, {3160, 20568}, {3262, 3911}, {4080, 40590}, {4391, 35092}, {4397, 51402}, {4582, 5375}, {4674, 40611}, {4858, 6544}, {6548, 40617}, {8054, 23838}, {15898, 36590}
X(1319) = barycentric product X(i)*X(j) for these {i,j}: {1, 3911}, {3, 37790}, {7, 44}, {12, 30576}, {34, 3977}, {36, 14628}, {56, 4358}, {57, 519}, {63, 1877}, {65, 16704}, {75, 1404}, {77, 8756}, {81, 40663}, {85, 902}, {88, 1317}, {89, 36920}, {100, 30725}, {109, 3762}, {214, 2006}, {222, 38462}, {269, 2325}, {273, 22356}, {278, 5440}, {279, 3689}, {331, 23202}, {514, 23703}, {517, 40218}, {603, 46109}, {604, 3264}, {651, 900}, {658, 4895}, {662, 30572}, {664, 1635}, {934, 1639}, {1014, 3943}, {1023, 3676}, {1145, 34051}, {1170, 51463}, {1214, 37168}, {1254, 30606}, {1255, 5298}, {1400, 30939}, {1407, 4723}, {1411, 51583}, {1412, 3992}, {1414, 4120}, {1417, 36791}, {1432, 4434}, {1434, 21805}, {1441, 3285}, {1461, 4768}, {1465, 36944}, {1476, 51415}, {1647, 4564}, {1960, 4554}, {2087, 4998}, {2161, 41801}, {2251, 6063}, {2990, 12832}, {3218, 14584}, {3257, 39771}, {3669, 17780}, {4169, 7203}, {4487, 40151}, {4528, 4617}, {4530, 7045}, {4573, 4730}, {4625, 14407}, {4626, 14427}, {4702, 42290}, {4922, 37137}, {5376, 14027}, {5723, 14191}, {6174, 34056}, {6550, 31615}, {6630, 14122}, {7052, 36668}, {8686, 16594}, {9459, 20567}, {13462, 36915}, {14418, 36118}, {17078, 40172}, {17455, 18815}, {18026, 22086}, {20332, 24816}, {21907, 41541}, {23344, 24002}, {23757, 37136}, {24004, 43924}, {30573, 37139}, {30731, 43932}, {31011, 32636}, {33655, 36669}, {34578, 41553}, {36100, 51422}, {39155, 45273}, {43736, 51406}
X(1319) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 4997}, {6, 1320}, {7, 20568}, {31, 2316}, {34, 6336}, {44, 8}, {56, 88}, {57, 903}, {59, 5376}, {65, 4080}, {100, 4582}, {109, 3257}, {214, 32851}, {519, 312}, {603, 1797}, {604, 106}, {608, 36125}, {649, 23838}, {651, 4555}, {678, 2325}, {692, 5548}, {900, 4391}, {902, 9}, {1017, 3689}, {1023, 3699}, {1317, 4358}, {1395, 8752}, {1397, 9456}, {1400, 4674}, {1404, 1}, {1405, 4792}, {1414, 4615}, {1415, 901}, {1417, 2226}, {1420, 31227}, {1429, 27922}, {1635, 522}, {1639, 4397}, {1647, 4858}, {1877, 92}, {1960, 650}, {2087, 11}, {2099, 4945}, {2149, 9268}, {2161, 36590}, {2171, 4013}, {2251, 55}, {2325, 341}, {2429, 31343}, {3251, 1639}, {3264, 28659}, {3285, 21}, {3669, 6548}, {3689, 346}, {3762, 35519}, {3911, 75}, {3943, 3701}, {3977, 3718}, {3992, 30713}, {4017, 4049}, {4120, 4086}, {4358, 3596}, {4370, 4723}, {4432, 3975}, {4434, 17787}, {4487, 44723}, {4530, 24026}, {4565, 4622}, {4573, 4634}, {4700, 4673}, {4702, 28809}, {4730, 3700}, {4773, 4811}, {4819, 42712}, {4895, 3239}, {4969, 3702}, {4984, 4985}, {5298, 4359}, {5440, 345}, {6544, 4768}, {6550, 40166}, {6610, 36887}, {8756, 318}, {9456, 1318}, {9459, 41}, {12832, 48380}, {14122, 4440}, {14407, 4041}, {14408, 4147}, {14427, 4163}, {14437, 14430}, {14439, 3717}, {14584, 18359}, {14628, 20566}, {16704, 314}, {17455, 4511}, {17780, 646}, {20972, 3880}, {21805, 2321}, {22086, 521}, {22356, 78}, {23202, 219}, {23344, 644}, {23703, 190}, {30572, 1577}, {30576, 261}, {30725, 693}, {30939, 28660}, {31615, 6635}, {36920, 4671}, {36944, 36795}, {37168, 31623}, {37790, 264}, {38462, 7017}, {39251, 3883}, {39771, 3762}, {40172, 36910}, {40218, 18816}, {40663, 321}, {41541, 32849}, {41553, 17264}, {41554, 37758}, {41556, 37788}, {41801, 20924}, {42084, 4530}, {43924, 1022}, {47420, 35014}, {47425, 45269}, {51415, 20895}, {51463, 1229}
X(1319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 3, 3057}, {1, 35, 9957}, {1, 40, 2098}, {1, 46, 1482}, {1, 55, 5919}, {1, 56, 65}, {1, 57, 2099}, {1, 65, 11011}, {1, 165, 7962}, {1, 260, 10508}, {1, 354, 44840}, {1, 988, 37614}, {1, 999, 354}, {1, 1381, 2447}, {1, 1382, 2446}, {1, 1385, 2646}, {1, 1420, 56}, {1, 1482, 33176}, {1, 2093, 16200}, {1, 2646, 37080}, {1, 3304, 17609}, {1, 3336, 11009}, {1, 3361, 3340}, {1, 3428, 17642}, {1, 3576, 55}, {1, 3601, 3303}, {1, 3612, 3295}, {1, 3746, 31792}, {1, 5126, 1155}, {1, 5193, 18838}, {1, 5563, 942}, {1, 5902, 50194}, {1, 5903, 10222}, {1, 7280, 5697}, {1, 7742, 14110}, {1, 7987, 1697}, {1, 10389, 8162}, {1, 10882, 10480}, {1, 11009, 33179}, {1, 13370, 13601}, {1, 13462, 57}, {1, 15803, 7982}, {1, 21842, 1385}, {1, 24928, 20323}, {1, 24929, 3748}, {1, 25415, 10247}, {1, 30282, 31393}, {1, 30389, 3601}, {1, 30392, 13384}, {1, 32612, 25414}, {1, 37524, 11278}, {1, 37525, 24929}, {1, 37552, 37542}, {1, 37587, 5902}, {1, 37602, 5049}, {1, 37605, 37568}, {1, 37616, 3746}, {1, 37617, 3666}, {1, 37618, 3}, {1, 37620, 21334}, {1, 41426, 37566}, {2, 3476, 5252}, {2, 5176, 5123}, {3, 56, 34880}, {3, 999, 22767}, {3, 3057, 37568}, {3, 10680, 40255}, {3, 34489, 37566}, {3, 37618, 37605}, {3, 41426, 56}, {8, 6921, 37828}, {8, 7288, 24914}, {8, 37828, 37829}, {36, 484, 5122}, {36, 2078, 5172}, {36, 3245, 5131}, {36, 5048, 5183}, {36, 5193, 56}, {36, 25405, 5048}, {36, 32760, 3}, {48, 1108, 2264}, {55, 56, 1470}, {55, 3576, 37600}, {56, 65, 32636}, {56, 1388, 1}, {56, 2099, 57}, {56, 3303, 1466}, {56, 3304, 26437}, {56, 5172, 36}, {56, 5221, 3361}, {56, 11510, 3}, {57, 1420, 13462}, {57, 2099, 65}, {57, 13462, 56}, {63, 5289, 31165}, {65, 13751, 942}, {101, 43065, 2348}, {104, 12740, 17638}, {108, 1870, 1875}, {145, 1788, 41687}, {145, 5265, 1788}, {214, 1317, 41541}, {214, 41554, 1317}, {226, 551, 15950}, {226, 4315, 5434}, {226, 15950, 4870}, {348, 30617, 24798}, {355, 499, 17606}, {388, 3616, 11375}, {392, 993, 3683}, {404, 4861, 5836}, {484, 5119, 35460}, {484, 5122, 1155}, {484, 30282, 2077}, {496, 34773, 10572}, {551, 4315, 226}, {551, 5434, 4870}, {559, 1082, 3666}, {664, 1447, 43037}, {672, 17439, 6603}, {934, 38459, 34855}, {942, 15178, 1}, {944, 3086, 1837}, {946, 4311, 7354}, {956, 997, 210}, {958, 19861, 25917}, {999, 1617, 56}, {999, 10246, 1}, {999, 41345, 22765}, {1055, 2170, 910}, {1125, 10106, 12}, {1149, 1458, 1457}, {1155, 2646, 50371}, {1201, 4322, 73}, {1201, 28386, 28389}, {1210, 5882, 10950}, {1279, 1458, 1456}, {1284, 1458, 1463}, {1317, 3911, 36920}, {1317, 5298, 40663}, {1381, 1382, 56}, {1385, 10222, 26287}, {1385, 15178, 24299}, {1385, 20323, 37080}, {1385, 24928, 1}, {1385, 24929, 37525}, {1388, 1420, 65}, {1403, 24806, 65}, {1420, 2078, 5126}, {1420, 34489, 41426}, {1455, 1457, 1456}, {1457, 1458, 1464}, {1467, 3303, 65}, {1467, 3601, 1466}, {1478, 5886, 17605}, {1697, 7987, 5217}, {2078, 3660, 1155}, {2078, 5193, 36}, {2098, 5204, 40}, {2446, 2447, 65}, {2448, 2449, 5128}, {2646, 3748, 24929}, {2646, 20323, 1}, {3057, 37566, 65}, {3057, 37605, 3}, {3304, 34471, 1}, {3336, 11009, 50193}, {3340, 3361, 5221}, {3340, 5221, 65}, {3485, 3600, 10404}, {3513, 3514, 1155}, {3576, 31393, 30282}, {3583, 16173, 7743}, {3585, 37735, 9955}, {3586, 37704, 11238}, {3600, 3622, 3485}, {3616, 4308, 388}, {3660, 5126, 56}, {3748, 24929, 37080}, {3872, 35262, 1376}, {4293, 5603, 1836}, {4297, 12053, 6284}, {4413, 9623, 4731}, {4855, 36846, 3913}, {4881, 38460, 100}, {5172, 18838, 1155}, {5289, 11194, 63}, {5298, 40663, 3911}, {5425, 14798, 5119}, {5433, 10944, 10}, {5434, 15950, 226}, {5537, 11575, 1155}, {5552, 36977, 32049}, {5563, 37583, 56}, {5570, 41345, 1155}, {5597, 5598, 37541}, {5597, 26404, 65}, {5598, 26380, 65}, {5691, 50443, 10896}, {5697, 7280, 3579}, {5901, 18990, 12047}, {5919, 37600, 55}, {6049, 7288, 37738}, {6261, 22760, 1898}, {6265, 10074, 17660}, {7113, 8609, 2182}, {7677, 14151, 37787}, {7967, 18391, 37740}, {7982, 15803, 37567}, {8227, 9613, 10895}, {8666, 30144, 72}, {9957, 13624, 35}, {10090, 12737, 17636}, {10165, 31397, 5432}, {10222, 26287, 33596}, {10222, 37582, 5903}, {10247, 36279, 25415}, {10857, 30389, 3576}, {11011, 32636, 65}, {11014, 37561, 31788}, {11373, 18481, 1479}, {11510, 34489, 65}, {11510, 37566, 37568}, {11510, 41426, 34880}, {13388, 13389, 17595}, {13751, 37583, 32636}, {14792, 14795, 33862}, {15934, 41345, 484}, {16173, 36975, 3583}, {17642, 37578, 7964}, {17728, 37740, 18391}, {18839, 41341, 1155}, {21842, 24928, 2646}, {22765, 41345, 36}, {22837, 25440, 10914}, {24914, 37738, 8}, {24929, 37525, 2646}, {26365, 26366, 37606}, {30282, 31393, 55}, {32760, 34489, 18838}, {32760, 37618, 18857}, {33179, 50193, 11009}, {33812, 41558, 1317}, {34489, 37618, 56}, {34880, 37566, 32636}, {36920, 41541, 3689}, {37566, 37605, 34880}, {37704, 50811, 3586}, {38013, 38014, 50371}, {41556, 50843, 1317}

X(1320) = ISOGONAL CONJUGATE OF X(1319)

Let DEF be the intouch triangle. Let H_a be the orthocenter of IBC. Let A₁ be the point, other than A, where AI meets the circumcircle. Let t_a be the tangent to the circle (DH_aA₁) at H_a, and define t_b and t_c cyclically. The lines t_a, t_b , t_c concur in X(1320). (Angel Montesdeoca, September 3, 2020)

X(1320) lies on the Darboux septic and on these lines: 1,88   2,1000   4,145   7,528   8,11   9,644   21,643   80,519   104,517   518,1156   900,1120   1022,1280

X(1320) = midpoint of X(145) and X(149)
X(1320) = reflection of X(i) in X(j) for these (i,j): (8,11), (100,1), (1145,1387)
X(1320) = isogonal conjugate of X(1319)
X(1320) = anticomplement of X(1145)
X(1320) = cevapoint of X(1) and X(517)
X(1320) = crosssum of X(902) and X(1404)
X(1320) = antigonal conjugate of X(8)
X(1320) = symgonal of X(1)
X(1320) = trilinear pole of line X(9)X(650)
X(1320) = Kirikami concurrent circles image of X(1)
X(1320) = polar conjugate of X(37790)
X(1320) = pole wrt polar circle of trilinear polar of X(37790) (line X(900)X(1846))
X(1320) = excentral-to-ABC barycentric image of X(5541)

X(1321) = 1st YIU SQUARES PERSPECTOR

Let BB_AC_AC be the external square on side BC, and define CC_BA_BA and AA_CB_CB cyclically. Let X = BC_B∩CB_C and X' = BA_B∩CA_C, and define Y, Z and Y', Z' cyclically. The lines AX, BY, CZ concur in X(4), and the lines AX', BY', CZ' concur in X(485). The lines XX', YY', ZZ' concur in X(1321), as shown in

Paul Yiu, On the Squares Erected Externally on the Sides of a Triangle.

X(1321) lies on these lines: 4,371   1322,2165

X(1322) = 2nd YIU SQUARES PERSPECTOR

The construction of X(1322) is like that of X(1321), but using internally positioned squares. See the reference
at X(1321).

X(1322) lies on these lines: 4,372   1321,2165

X(1323) = FLETCHER POINT

X(1323) is the point of intersection of the line X(1)X(7) and the trilinear polar of X(7). These two lines are orthogonal.
X(1323) is named in honor of T. J. Fletcher in

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329.

See also MathWorld, Fletcher Point

X(1323) is the radical trace of the inner and outer Soddy circles.

X(1323) lies on these lines: 1,7   10,348   36,934   40,738   85,1125   106,927   165,479   241,514   519,664   1319,1355

X(1323) = midpoint of X(1155) and X(3328)
X(1323) = isogonal conjugate of X(4845)
X(1323) = inverse-in-incircle of X(7)
X(1323) = X(1260)-cross conjugate of X(527)
X(1323) = crossdifference of every pair of points on line X(55)X(657)
X(1323) = X(187)-of-intouch-triangle

X(1324) = CIRCUMCIRCLE-INVERSE OF X(10)

X(1324) lies on these lines: 3,10   35,228   36,1054   58,181   98,929   759,859

X(1324) = isogonal conjugate of isotomic conjugate of X(21277)
X(1324) = isogonal conjugate of polar conjugate of X(37770)
X(1324) = isogonal conjugate of antigonal image of X(58)
X(1324) = polar conjugate of isotomic conjugate of X(23120)
X(1324) = circumciorcle-inverse of X(10)

X(1325) = CIRCUMCIRCLE-INVERSE OF X(21)

As a point on the Euler line, X(1325) has Shinagawa coefficients ($a$(E + 4F) + 2abc, -4$a$(E + F) - 6abc).

X(1325) lies on these lines: 1,229   2,3   36,759   60,65   104,476   105,691   110,517   1295,1304

X(1325) = reflection of X(1290) in X(36)
X(1325) = circumcircle-inverse of X(21)
X(1325) = nine-point-circle-inverse of X(37983)
X(1325) = crosspoint of X(3) and X(2948) wrt both the excentral and tangential triangles
X(1325) = crossdifference of every pair of points on line X(647)X(2092)
X(1325) = reflection of X(110) in line X(36)X(238) (the polar of X(1) wrt circumcircle)

X(1326) = INVERSE-IN-CIRCUMCIRCLE OF X(58)

X(1326) lies on these lines: 3,6   10,261   35,849   42,593   99,726   106,691   110,902   238,662   249,1101   727,805

X(1326) = isogonal conjugate of X(11599)
X(1326) = complement of X(20558)
X(1326) = anticomplement of X(20546)
X(1326) = circumcircle-inverse of X(58)
X(1326) = X(741)-Ceva conjugate of X(58)
X(1326) = crossdifference of every pair of points on line X(523)X(1213)
X(1326) = X(6)-Hirst inverse of X(58)

X(1327) = ARCTAN(3) KIEPERT POINT

Let BB_AC_AC be the external square on side BC, and define CC_BA_BA and AA_CB_CB cyclically. The lines A_BB_A, B_CC_B, C_AA_C form a triangle perspective to triangle ABC, and the perspector is X(1327).

If you have The Geometer's Sketchpad, you can view X(1327).

X(1327) lies on these lines: 6,1328   30,485   371,1131   381,486   547,1152

X(1327) = isogonal conjugate of X(6200)

X(1328) = ARCTAN(-3) KIEPERT POINT

The construction of X(1328) is like that of X(1327), but using internally positioned squares. See the reference
at X(1321).

X(1328) lies on these lines: 6,1327   30,486   372,1132   381,485   547,1151

X(1328) = isogonal conjugate of X(6396)
X(1328) = X(2)-of-3rd-anti-tri-squares-triangle
X(1328) = perspector of ABC and 3rd anti-tri-squares triangle

X(1329) = COMPLEMENTARY CONJUGATE OF X(1)

X(1329) lies on these lines: 2,12   3,119   5,10   8,11   9,46   65,908   121,124   140,993   355,997   405,498   495,1125   496,519   499,956   518,1210   975,998

X(1329) = isogonal conjugate of X(3450)
X(1329) = isotomic conjugate of isogonal conjugate of X(23638)
X(1329) = complement of X(56)
X(1329) = complementary conjugate of X(1)
X(1329) = crosssum of X(6) and X(1397)
X(1329) = polar conjugate of isogonal conjugate of X(22071)

X(1330) = ANTICOMPLEMENTARY CONJUGATE OF X(1)

X(1330) lies on these lines: 2,58   4,69   8,79   10,894   30,1043   193,387   320,942   333,442   1010,1211

X(1330) = reflection of X(1046) in X(10)
X(1330) = isogonal conjugate of X(3437)
X(1330) = anticomplement of X(58)
X(1330) = anticomplementary conjugate of X(1)
X(1330) = X(313)-Ceva conjugate of X(2)
X(1330) = {X(4),X(69)}-harmonic conjugate of X(10449)

X(1331) = ORTHOCORRESPONDENT OF X(101)

For the definition of orthocorrespondent, see the notes just before X(1992).

X(1331) lies on the MacBeath circumconic and these lines: 63,212   71,895   72,283   78,255   100,109   101,110   145,595   162,190   228,295   287,293   394,1260   677,1252   901,1293   906,4574

X(1331) = isogonal conjugate of X(7649)
X(1331) = isotomic conjugate of polar conjugate of X(101)
X(1331) = trilinear pole of line X(3)X(48)
X(1331) = crossdifference of every pair of points on line X(2170)X(2969)
X(1331) = X(19)-isoconjugate of X(514)
X(1331) = X(92)-isoconjugate of X(649)
X(1331) = X(i)-Ceva conjugate of X(j) for these (i,j): (190,101), (643,100)
X(1331) = X(i)-cross conjugate of X(j) for these (i,j): (521,283), (652,63), (1260,1252)
X(1331) = cevapoint of X(i) and X(j) for these (i,j): (3,1459), (72,521), (212,652)

X(1332) = ORTHOCORRESPONDENT OF X(100)

For the definition of orthocorrespondent, see the notes just before X(1992).

X(1332) lies on the MacBeath circumconic and these lines: 6,344   69,219   72,895   100,110   101,1310   190,644   287,336   345,394   645,648   646,1016   677,765   815,932   4561,4574

X(1332) = reflection of X(2991) in X(6)
X(1332) = isogonal conjugate of X(6591)
X(1332) = isotomic conjugate of X(17924)
X(1332) = MacBeath circumconic antipode of X(2991)
X(1332) = trilinear pole of line X(3)X(63)
X(1332) = X(92)-isoconjugate of X(667)
X(1332) = X(i)-Ceva conjugate of X(j) for these (i,j): (645, 190), (668,100), (1016,345)
X(1332) = X(i)-cross conjugate of X(j) for these (i,j): (521,69), (905,63), (906,100)
X(1332) = cevapoint of X(i) and X(j) for these (i,j): (63,905), (71,1459), (219,521)

X(1333) = POINT ALULA

X(1333) lies on these lines: {2,18744}, {3,6}, {9,609}, {19,2217}, {21,37}, {27,3772}, {28,1104}, {31,48}, {36,16470}, {44,1778}, {45,4877}, {47,22134}, {53,7511}, {56,608}, {65,1950}, {81,593}, {86,3662}, {99,713}, {100,21858}, {104,112}, {110,739}, {141,5337}, {163,9456}, {213,2174}, {219,1780}, {261,27164}, {272,379}, {286,16732}, {292,4628}, {314,19623}, {321,17587}, {333,4386}, {346,17539}, {385,3770}, {536,16046}, {594,5291}, {595,16685}, {603,604}, {662,18274}, {692,1911}, {741,825}, {759,5341}, {849,1437}, {859,2178}, {872,18266}, {896,3958}, {910,5324}, {940,16368}, {963,2332}, {992,27660}, {1010,4426}, {1014,1418}, {1043,17299}, {1086,17189}, {1171,28625}, {1178,3863}, {1191,2255}, {1193,22054}, {1213,5277}, {1396,1427}, {1399,1409}, {1400,1415}, {1407,1412}, {1436,2299}, {1449,7031}, {1575,13588}, {1627,5276}, {1766,15952}, {1790,2221}, {1801,2327}, {1811,5546}, {1817,3752}, {1901,13442}, {1931,4469}, {1951,2264}, {1977,17961}, {2160,3125}, {2162,21769}, {2197,5172}, {2241,4658}, {2242,4653}, {2256,2328}, {2262,18191}, {2268,10457}, {2269,22361}, {2276,4184}, {2277,4225}, {2311,3862}, {2345,11115}, {2361,22074}, {2423,7252}, {2699,2715}, {2991,4558}, {3290,4228}, {3330,28381}, {3454,24935}, {3739,26643}, {3998,4641}, {4000,14953}, {4234,17281}, {4567,5381}, {4749,23868}, {5358,16583}, {5563,16488}, {5839,16704}, {7735,26118}, {8069,22132}, {8822,17276}, {9341,19297}, {10315,21866}, {10789,18194}, {11102,16974}, {11320,18147}, {12194,18170}, {12610,17197}, {13728,17398}, {16047,17263}, {16050,17279}, {16054,17278}, {16350,19701}, {16580,17171}, {16706,21997}, {16917,25457}, {16973,18206}, {17052,24890}, {17053,21773}, {17187,21764}, {18697,24335}, {19308,24530}, {19785,26830}, {21353,21833}

X(1333) = isogonal conjugate of X(321)
X(1333) = isotomic conjugate of X(27801)
X(1333) = complement of X(21287)
X(1333) = anticomplement of X(21245)
X(1333) = X(i)-Ceva conjugate of X(j) for these (i,j): (593,58), (1169,6), (1175,184)
X(1333) = X(31)-cross conjugate of X(58)
X(1333) = cevapoint of X(31) and X(32)
X(1333) = crosspoint of X(i) and X(j) for these (i,j): (28,81), (58,1412), (593,849)
X(1333) = crosssum of X(i) and X(j) for (i,j) = (37,72), (594,1089)
X(1333) = trilinear pole of line X(667)X(838)
X(1333) = X(92)-isoconjugate of X(72)
X(1333) = X(100)-isoconjugate of X(1577)
X(1333) = perspector of ABC and unary cofactor triangle of Gemini triangle 21
X(1333) = perspector of ABC and unary cofactor triangle of Gemini triangle 27

X(1334) = POINT ALYA

X(1334) lies on these lines: 1,672   3,1055   8,9   10,1018   21,644   32,902   35,101   37,65   39,1201   41,55   42,213   607,1253   756,862

X(1334) = isogonal conjugate of X(1434)
X(1334) = complement of X(20244)
X(1334) = anticomplement of X(17050)
X(1334) = X(i)-Ceva conjugate of X(j) for these (i,j): (9, 210), (37,42), (644,663)
X(1334) = crosspoint of X(i) and X(j) for these (i,j): (9,55), (37,210)
X(1334) = crosssum of X(i) and X(j) for (i,j) = (7,57), (81,1014)
X(1334) = crossdifference of every pair of points on line X(1443)X(1447)
X(1334) = trilinear pole of line X(1443)X(1447)

X(1335) = {X(1),X(6)}-HARMONIC CONJUGATE OF X(1124)

X(1335) lies on these lines: 1,6   11,486   12,485   35,1151   36,1152   42,493   55,371   56,372   81,1123   175,651   255,606   498,590   499,615

X(1335) = isogonal conjugate of X(1336)
X(1335) = isotomic conjugate of polar conjugate of X(34121)
X(1335) = X(19)-isoconjugate of X(13386)
X(1335) = exsimilicenter of incircle and 2nd Lemoine circle; the insimilicenter is X(1124)

X(1336) = ISOGONAL CONJUGATE OF X(1335)

The parabola with focus A and directrix BC meets line AB in two points; let A_B be the one further from B, and define A_C similarly. Let L_A be the line A_BA_C, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1336). (Randy Hutson, 9/23/2011)

In the configuration for the Paasche point X(1123), there are 4 circles tangent to the circle with diameter BC and also tangent to the lines AB and AC. Of the 4 circles, there are two pairs, one having X(1123) as perspector, and the other having X(1336). (Peter Moses, 21 January 2013)

If you have The Geometer's Sketchpad, you can view X(1336)

X(1336) lies on these lines: 1,3068   2,585   4,2362   37,158   57,481   81,1124   274,1267   498,3300   499,3302   9203069

X(1336) = isogonal conjugate of X(1335)
X(1336) = isotomic conjugate of X(5391)
X(1336) = polar conjugate of isogonal conjugate of X(34125)

X(1337) = 1st WERNAU POINT

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(2DU - 3^1/2VW)/(4D + 3^1/2a²),
                        D, U, V, W as above; see Hyacinthos #8874

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(2DU - 3^1/2VW)/(4D + 3^1/2a²),

Barycentrics    a^2*(-2*(-a^2+b^2+c^2)*S+sqrt(3)*(a^2-c^2+b^2)*(a^2-b^2+c^2))*(sqrt(3)*b^2+2*S)*(sqrt(3)*c^2+2*S) : :      (César Lozada, December 15, 2019)

Let A'BC be the external equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X(1337). Wernau is a town near Stuttgart, the site of a mathematics olympiad seminar in Spring 2003. (Darij Grinberg; Hyacinthos, April, 2003: #6874, 6881, 6882; coordinates by Jean-Pierre Ehrmann)

X(1337) is the tangential of X(15) on the Neuberg cubic.

X(1337) lies on the Neuberg cubic and these lines: 4,616   15,2981   399,3441   1157,1338

X(1338) = 2nd WERNAU POINT

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(2DU + 3^1/2VW)/(4D - 3^1/2a²),
                        D, U, V, W as at X(1337); see Hyacinthos #8874

Let A'BC be the internal equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X(1338). For details, see X(1337).

X(1338) is the tangential of X(16) on the Neuberg cubic.

X(1338) lies on the Neuberg cubic and these lines: 4,617   16,3458   399,3440   1157,1337

X(1338) = isogonal conjugate of X(3480)
X(1338) = anticomplement of X(33498)
X(1338) = antigonal conjugate of X(661)
X(1338) = symgonal of X(15)

X(1339) = NAGEL-SCHRÖDER POINT

Let X'Y'Z' be the extouch triangle of ABC; viz., X' is where the A-excircle meets line BC, and X'Y'Z' is the pedal triangle of X(40). Let I = incenter of ABC. The circles (AIX'), (BIY'), (CIZ') concur in two points: I and X(1339). (Jean-Pierre Ehrmann, Hyacinthos #6545)

X(1339) lies on this line: 1,474

X(1340) = INSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

If you have The Geometer's Sketchpad, you can view X(1340).

X(1340) lies on the cubics K280, K309, K657, K792, K793, K795, K889, K890, K891, K911, and these lines: {1, 1704}, {2, 1349}, {3, 6}, {4, 1348}, {20, 2542}, {51, 21036}, {55, 1675}, {56, 1674}, {76, 6178}, {110, 6142}, {111, 6141}, {165, 1705}, {262, 6039}, {353, 5638}, {549, 39023}, {958, 1679}, {1083, 11652}, {1344, 2470}, {1345, 2469}, {1376, 1678}, {1503, 19660}, {2039, 43461}, {2040, 7790}, {2549, 31863}, {3413, 7709}, {3589, 19659}, {3972, 46023}, {4045, 14501}, {5091, 36736}, {6189, 7771}, {6190, 7757}, {11650, 11651}, {15048, 39022}, {21032, 22352}

X(1340) = reflection of X(1341) in X(11171)
X(1340) = isogonal conjugate of X(46023)
X(1340) = Brocard-circle-inverse of X(1380)
X(1340) = Schoutte-circle-inverse of X(1341)
X(1340) = 2nd-Brocard-circle-inverse of X(3558)
X(1340) = psi-transform of X(5639)
X(1340) = crossdifference of every pair of points on line {523, 5638}
X(1340) = internal center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)
X(1340) = {X(3),X(182)}-harmonic conjugate of X(1341)
X(1340) = radical center of Lucas(t) circles, for t where circles are tangent to Brocard circle
X(1340) = intersection of Brocard axis and minor axis of Steiner circumellipse
X(1340) = homothetic center of 1st Brocard triangle and circumcevian triangle of X(3414)
X(1340) = barycentric product X(7998)*X(46024)
X(1340) = barycentric quotient X(6)/X(46023)
X(1340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2543, 1349}, {3, 6, 1380}, {3, 182, 1341}, {3, 14630, 3558}, {3, 38596, 3098}, {6, 574, 1341}, {6, 1380, 3558}, {6, 2029, 3557}, {15, 16, 1341}, {32, 5116, 1341}, {39, 2029, 6}, {39, 3094, 1341}, {39, 13325, 3558}, {371, 372, 14631}, {566, 2088, 1341}, {1342, 1343, 2558}, {1380, 14630, 6}, {1670, 1671, 3558}, {1689, 1690, 1341}, {2029, 8589, 44453}, {2080, 39498, 1341}, {3102, 3103, 13326}, {3106, 3107, 1341}, {5013, 5028, 1341}, {5050, 9734, 1341}, {5092, 26316, 1341}, {8589, 10485, 1341}, {13325, 14630, 3557}, {15815, 39764, 1341}, {30260, 30261, 1341}, {35608, 35609, 31862}

X(1341) = EXSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

X(1341) = external center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1341) lies on the cubics K280, K309, K657, K792, K793, K795, K889, K890, K891, K911 and these lines: 1, 1705}, {2, 1348}, {3, 6}, {4, 1349}, {20, 2543}, {51, 21032}, {55, 1674}, {56, 1675}, {76, 6177}, {110, 6141}, {111, 6142}, {165, 1704}, {262, 6040}, {353, 5639}, {549, 39022}, {958, 1678}, {1083, 11651}, {1344, 2469}, {1345, 2470}, {1376, 1679}, {1503, 19659}, {2039, 7790}, {2549, 31862}, {3414, 7709}, {3589, 19660}, {4045, 14502}, {5091, 36735}, {6189, 7757}, {6190, 7771}, {11650, 11652}, {15048, 39023}, {21036, 22352}

X(1341) = reflection of X(1340) in X(11171)
X(1341) = isogonal conjugate of X(46024)
X(1341) = Brocard-circle-inverse of X(1379)
X(1341) = 2nd-Brocard-circle-inverse of X(3557)
X(1341) = Schoutte-circle-inverse of X(1340) X(1341) = psi-transform of X(5638)
X(1341) = crossdifference of every pair of points on line {523, 5639}
X(1341) = intersection of Brocard axis and major axis of Steiner circumellipse
X(1341) = homothetic center of 1st Brocard triangle and circumcevian triangle of X(3413)
X(1341) = barycentric product X(7998)*X(46023)
X(1341) = barycentric quotient X(6)/X(46024)
X(1341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2542, 1348}, {3, 6, 1379}, {3, 182, 1340}, {3, 14631, 3557}, {3, 38597, 3098}, {6, 574, 1340}, {6, 1379, 3557}, {6, 2028, 3558}, {15, 16, 1340}, {32, 5116, 1340}, {39, 2028, 6}, {39, 3094, 1340}, {39, 13326, 3557}, {371, 372, 14630}, {566, 2088, 1340}, {1342, 1343, 2559}, {1379, 14631, 6}, {1670, 1671, 3557}, {1689, 1690, 1340}, {2028, 8589, 44453}, {2080, 39498, 1340}, {3102, 3103, 13325}, {3106, 3107, 1340}, {5013, 5028, 1340}, {5050, 9734, 1340}, {5092, 26316, 1340}, {8589, 10485, 1340}, {13326, 14631, 3558}, {14899, 35607, 31863}, {15815, 39764, 1340}, {30260, 30261, 1340}, {39162, 39163, 5638}

X(1342) = INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

Trilinears    sin A + cos A cot(ω/2) : :
Trilinears    sin A - sin(A - ω) : :
Trilinears    cos A + cos(A - ω) : :
Trilinears    cos(A - ω/2) : :
Trilinears    sin A + (csc ω + cot ω) cos A : :
Trilinears    cos A + (csc ω - cot ω) sin A : :

X(1342) =(sec ω)*X(3) + 2*X(182) = (1 + sec ω)*X(3) + X(6)

X(1342) = internal center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003; cf. X(1343), X(1670), X(1671))

X(1342) lies on this line: 3,6

X(1342) = reflection of X(1343) in X(3398)
X(1342) = isogonal conjugate of X(5403)
X(1342) = Brocard-circle-inverse of X(1670)
X(1342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,1671), (1687,1688,1343)
X(1342) = insimilicenter of 1st and 2nd Brocard circles; the insimilicenter is X(1343)
X(1342) = inner-Montesdeoca-Lemoine-circle-inverse of X(38720)

X(1343) = EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

X(1343) = external center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003)

Let Lbc be the line obtained by rotating line CA through C by an angle of ω/2 toward B. Let Lcb be the line obtained by rotating line AB through B by an angle of ω/2 toward C. Let A' =Lbc∩\Lcb. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1343). (Randy Hutson, October 13, 2015)

X(1343) lies on this line: 3,6

X(1343) = reflection of X(1342) in X(3398)
X(1343) = isogonal conjugate of X(5404)
X(1343) = inverse-in-Brocard-circle of X(1671)
X(1343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,1670), (1687,1688,1342)
X(1343) = exsimilicenter of 1st and 2nd Brocard circles (the insimilicenter is X(1342))
X(1343) = outer-Montesdeoca-Lemoine-circle-inverse of X(38721)

X(1344) = INSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

As a point on the Euler line, X(1344) has Shinagawa coefficients (R + |OH|,3R - |OH|).

X(1344) = internal center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1344) lies on the cubics K281, K297, K708, K843, and on these lines: {2,3}, {6,2574}, {55,2464}, {56,2463}, {64,14375}, {111,8427}, {112,8426}, {183,15165}, {371,2466}, {372,2465}, {958,2468}, {1340,2470}, {1341,2469}, {1342,2472}, {1343,2471}, {1351,8116}, {1376,2467}, {1689,2015}, {1690,2016}, {2575,11472}, {5640,24651}, {5968,16070}, {6090,8115}

X(1344) = orthocentroidal circle inverse of X(1313)
X(1344) = X(11472)-line conjugate of X(2575)
X(1344) = X(1345)-vertex conjugate of X(11181)
X(1344) = crossdifference of every pair of points on line {647, 2575}
X(1344) = homothetic center of orthocentroidal triangle and circumcevian triangle of X(2574)
X(1344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 1313), (2, 1113, 3), (2, 1346, 5094), (2, 1995, 1345), (2, 2553, 1347), (3, 381, 1345), (3, 3830, 15155), (4, 378, 1345), (4, 1113, 25), (4, 1346, 381), (4, 10737, 3830), (4, 14709, 3), (5, 6644, 1345), (5, 20478, 3), (22, 5169, 1345), (23, 10719, 15155), (24, 7577, 1345), (25, 5094, 1345), (458, 4230, 1345), (868, 3148, 1345), (1113, 10719, 23), (1346, 14709, 378), (1347, 2553, 381), (2070, 7579, 1345), (3091, 14709, 3516), (3091, 15078, 1345), (3526, 21308, 1345), (7418, 13860, 1345), (7485, 7533, 1345), (14807, 15158, 20063)

X(1345) = EXSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

As a point on the Euler line, X(1345) has Shinagawa coefficients (R - |OH|,3R + |OH|).

X(1345) lies on the cubics K281, K297, K708, K843, and on these lines: {2,3}, {6,2575}, {55,2463}, {56,2464}, {64,14374}, {111,8426}, {112,8427}, {183,15164}, {371,2465}, {372,2466}, {958,2467}, {1340,2469}, {1341,2470}, {1342,2471}, {1343,2472}, {1351,8115}, {1376,2468}, {1689,2016}, {1690,2015}, {2574,11472}, {5640,24650}, {5968,16071}, {6090,8116}

X(1345) = orthocentroidal circle inverse of X(1312)
X(1345) = X(11472)-line conjugate of X(2574)
X(1345) = X(1344)-vertex conjugate of X(11181)
X(1345) = crossdifference of every pair of points on line {647, 2574}
X(1345) = homothetic center of orthocentroidal triangle and circumcevian triangle of X(2575)
X(1345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 1312), (2, 1114, 3), (2, 1347, 5094), (2, 1995, 1344), (2, 2552, 1346), (3, 381, 1344), (3, 3830, 15154), (4, 378, 1344), (4, 1114, 25), (4, 1347, 381), (4, 10736, 3830), (4, 14710, 3), (5, 6644, 1344), (5, 20479, 3), (22, 5169, 1344), (23, 10720, 15154), (24, 7577, 1344), (25, 5094, 1344), (458, 4230, 1344), (868, 3148, 1344), (1114, 10720, 23), (1346, 2552, 381), (1347, 14710, 378), (2070, 7579, 1344), (3091, 14710, 3516), (3091, 15078, 1344), (3526, 21308, 1344), (7418, 13860, 1344), (7485, 7533, 1344), (14808, 15159, 20063)

X(1346) = INSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

As a point on the Euler line, X(1346) has Shinagawa coefficients (R + |OH|,3R + |OH|).

X(1346) lies on these lines: 2,3   56,2464

X(1346) = internal center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1347) = EXSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

As a point on the Euler line, X(1347) has Shinagawa coefficients (R - |OH|,3R - |OH|).

X(1347) = external center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1347) lies on this line: 2,3

X(1348) = INSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)

X(1348) = internal center of similitude of nine-point circle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1348) lies on these lines: {2, 1341}, {3, 2040}, {4, 1340}, {5, 182}, {6, 2039}, {10, 1694}, {11, 1674}, {12, 1675}, {115, 2033}, {316, 1379}, {485, 1668}, {486, 1669}, {574, 19660}, {1329, 1678}, {1342, 2567}, {1343, 2566}, {1346, 2470}, {1347, 2469}, {1380, 38227}, {1506, 2034}, {1664, 5403}, {1665, 5404}, {1679, 2886}, {1693, 2051}, {1698, 1705}, {1699, 1704}, {2009, 2012}, {2010, 2011}, {2543, 3091}, {2558, 37446}, {2559, 5025}, {3414, 7694}, {3557, 7785}, {6039, 9993}, {6177, 7746}, {7818, 39022}, {7828, 14631}, {13414, 13870}

X(1348) = crosssum of X(1341) and X(3557)
X(1348) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 2542, 1341}, {5, 182, 1349}, {3589, 7844, 1349}

X(1349) = EXSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)

X(1349) = external center of similitude of nine-point circle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1349) lies on these lines: 2,1340   4,1341   5,182

X(1350) = REFLECTION OF X(6) IN X(3)

Let A'B'C' be the reflection of ABC in X(3). Let A_B = BC∩C'A', and define B_C and C_A cyclically. A_C = BC∩A'B', and define B_A and C_B cyclically. The 6 points A_B, B_C, C_A, A_C, B_A, C_B lie on a conic. Let A" be the intersection of the tangents to the conic at A_B and A_C, and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(1350). (Randy Hutson, January 29, 2015)

X(1350) lies on these lines: 2,3066  3,6   4,141   20,64   22,110   30,599   35,611   36,613   40,518   74,1296   103,1293   206,1092   343,1370   376,524   517,990

X(1350) = midpoint of X(20) and X(69)
X(1350) = reflection of X(i) in X(j) for these (i,j): (4,141), (6,3), (1351,182), (1498,159)
X(1350) = isogonal conjugate of X(3424)
X(1350) = X(6) of circumcevian triangle of X(511)
X(1350) = radical center of Lucas(-tan ω) circles
X(1350) = {X(182),X(1351)}-harmonic conjugate of X(6)
X(1350) = inverse-in-2nd-Brocard-circle of X(5188)
X(1350) = antipedal-isogonal conjugate of X(6)
X(1350) = X(53)-of-the-hexyl-triangle
X(1350) = exsimilicenter of circle centered at X(1151) through X(372) and circle centered at X(1152) through X(371); the insimilicenter is X(3053)

X(1351) = REFLECTION OF X(3) IN X(6)

Let T be a triangle inscribed in the circumcircle and circumscribing the orthic inconic. As T varies, its orthocenter traces a circle centered at X(1351) with segment X(4)X(193) as diameter. (Randy Hutson, August 29, 2018)

X(1351) lies on these lines: 3,6   4,193   5,69   25,110   30,1353   49,206   51,394   159,195   183,262   381,524   613,999

X(1351) = midpoint of X(4) and X(193)
X(1351) = reflection of X(i) in X(j) for these (i,j): (3,6), (6,576), (69,5), (1350,182)
X(1351) = isogonal conjugate of X(7612)
X(1351) = inverse-in-2nd-Lemoine-circle of X(1692)
X(1351) = radical center of Lucas(-4 tan ω) circles
X(1351) = intersection of tangents to 2nd Lemoine circle at intersections with circumcircle
X(1351) = inverse-in-{circle centered at X(3) with radius |OK|} of X(182)
X(1351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,1350,182), (1687,1688,5033)
X(1351) = exsimilicenter of circle centered at X(371) through X(1152) and circle centered at X(372) through X(1151); the insimilicenter is X(3053)

X(1352) = REFLECTION OF X(6) IN X(5)

X(1352) lies on these lines: 2,98   3,66   4,69   5,6   11,613   12,611   25,343   30,599   70,1176   193,576   206,1209   298,383   299,1080   355,518   381,524   394,426   2794,3734

X(1352) = midpoint of X(4) and X(69)
X(1352) = reflection of X(i) in X(j) for these (i,j): (3,141), (6,5), (193,576)
X(1352) = isogonal conjugate of X(3425)
X(1352) = complement of X(6776)
X(1352) = anticomplement of X(182)
X(1352) = X(327)-Ceva conjugate of X(2)
X(1352) = X(4)-of-1st-Brocard-triangle
X(1352) = X(3)-of-X(2)-Fuhrmann-triangle
X(1352) = center of the perspeconic of these triangles: Ehrmann side and Johnson
X(1352) = insimilicenter of X(13)- and X(14)-Fuhrmann circles (aka -Hagge circles
X(1352) = 1st-Brocard-isogonal conjugate of X(182)
X(1352) = 1st-Brocard-isotomic conjugate of X(2549)
X(1352) = X(3)-of-obverse-triangle-of-X(69)

X(1353) = REFLECTION OF X(5) IN X(6)

X(1353) lies on these lines: 3,193   5,6   30,1351   69,140   141,575   182,524   511,550   542,3845

X(1353) = midpoint of X(3) and X(193)
X(1353) = reflection of X(i) in X(j) for these (i,j): (5,6), (69,140), (141,575)

Brisse Transforms 1354-1367

X(1354) = BRISSE TRANSFORM OF X(74)

The following ten points lie on a circle: X(i) for i = 11, 36, 65, 80, 108, 759, 1354, 1845, 2588, 2589. (Chris Van Tienhoven, Hyacinthos, January 4, 2011)

X(1354) lies on the incircle and these lines: 7,1367   56,759   942,1364

X(1355) = BRISSE TRANSFORM OF X(98)

X(1355) lies on the incircle and these lines: 56,741   222,1363

X(1356) = BRISSE TRANSFORM OF X(99)

X(1356) lies on the incircle and this line: 56,741

X(1356) = anticomplement of X(3037)
X(1356) = trilinear product X(57)*X(1084)
X(1356) = barycentric product X(7)*X(1084)

X(1357) = BRISSE TRANSFORM OF X(100)

X(1357) lies on the incircle and these lines: 12,121   43,57   55,1293   56,106   65,1317   1086,1365

X(1357) = isogonal conjugate of X(4076)
X(1357) = anticomplement of X(3038)
X(1357) = crosssum of X(i) and X(j) for these (i,j): (55,644), (100,145), (190,344)
X(1357) = X(107)-of-intouch triangle
X(1357) = X(1293)-of-Mandart-incircle-triangle
X(1357) = homothetic center of intangents triangle and reflection of extangents triangle in X(1293)
X(1357) = trilinear pole wrt intouch triangle of line X(7)X(8)
X(1357) = trilinear product of vertices of Mandart-excircles triangle
X(1357) = trilinear product X(57)*X(1015)
X(1357) = barycentric product X(7)*X(1015)

X(1358) = BRISSE TRANSFORM OF X(101)

X(1358) lies on the incircle and these lines: 7,528   11,1111   12,85   55,1292   56,105   65,1362   269,1359   553,1366   1120,1125   1122,1361   1319,1323

X(1358) = isotomic conjugate of X(4076)
X(1358) = anticomplement of X(3039)
X(1358) = X(244)-cross conjugate of X(1086)
X(1358) = crosspoint of X(277) and X(514)
X(1358) = crosssum of X(101) and X(218)
X(1358) = X(112)-of-intouch triangle
X(1358) = X(1292)-of-Mandart-incircle-triangle
X(1358) = homothetic center of intangents triangle and reflection of extangents triangle in X(1292)
X(1358) = trilinear pole wrt intouch triangle of line X(2)X(7)
X(1358) = trilinear product X(57)*X(1086)
X(1358) = barycentric product X(7)*X(1086)

X(1359) = BRISSE TRANSFORM OF X(102)

X(1359) lies on the incircle and these lines: 1,3318   4,11   12,123   55,1295   65,1364   269,1358

X(1359) = X(1299)-of-intouch triangle
X(1359) = X(1295)-of-Mandart-incircle-triangle
X(1359) = homothetic center of intangents triangle and reflection of extangents triangle in X(1295)

X(1360) = BRISSE TRANSFORM OF X(103)

X(1360) lies on the incircle and these lines: 11,57   12,208   55,108   56,105   354,1364

X(1360) = X(3563)-of-intouch triangle (Chris van Tienhoven, Hyacinthos #21096, Jul 14, 2012)

X(1361) = BRISSE TRANSFORM OF X(104)

X(1361) lies on the incircle and these lines: 1,1364   11,65   12,124   55,102   56,106   151,497   181,994   928,1362   962,1118   1122,1358

X(1361) = reflection of X(1364) in X(3)
X(1361) = anticomplement of X(3040)
X(1361) = crosspoint of X(7) and X(1465)
X(1361) = X(1300)-of-intouch triangle
X(1361) = X(102)-of-Mandart-incircle-triangle
X(1361) = homothetic center of intangents triangle and reflection of extangents triangle in X(102)

X(1362) = BRISSE TRANSFORM OF X(105)

X(1362) lies on the incircle and these lines: 7,1002   11,118   12,116   43,57   55,103   56,101   59,840   65,1358   105,651   150,388   152,497   928,1361

X(1362) = anticomplement of X(3041)
X(1362) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,241), (651,665)
X(1362) = crosspoint of X(7) and X(241)
X(1362) = crosssum of X(i) and X(j) for (i,j) = (11,885), (55,294)
X(1362) = crossdifference of every pair of points on line X(294)X(885)
X(1362) = X(672)-Hirst inverse of X(1458)
X(1362) = X(98)-of-intouch triangle
X(1362) = X(103)-of-Mandart-incircle-triangle
X(1362) = homothetic center of intangents triangle and reflection of extangents triangle in X(103)

X(1363) = BRISSE TRANSFORM OF X(107)

X(1363) lies on the incircle and this line:
222,1355

X(1364) = BRISSE TRANSFORM OF X(108)

X(1364) lies on the incircle and these lines: 1,1361   11,124   12,117   55,103   56,102   65,1359   77,296   151,388   185,603   354,1360   942,1354

X(1364) = reflection of X(1361) in X(1)
X(1364) = anticomplement of X(3042)
X(1364) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,905), (189,650), (222,652), (255,520)
X(1364) = crosspoint of X(i) and X(j) for these (i,j): (3,521), (7,905)
X(1364) = crosssum of X(4) and X(108)
X(1364) = X(925)-of-intouch-triangle
X(1364) = X(109)-of-Mandart-incircle-triangle
X(1364) = homothetic center of intangents triangle and reflection of extangents triangle in X(109)
X(1364) = trilinear pole wrt intouch triangle of line X(4)X(7)
X(1364) = intersection, other than X(11), of incircle and Mandart circle
X(1364) = extouch isogonal conjugate of X(522)
X(1364) = crossdifference of every pair of points on line X(1783)X(4559)

X(1365) = BRISSE TRANSFORM OF X(110)

X(1365) lies on the incircle and these lines: 7,1366   56,759   125,1109   1086,1357   1283,1284

X(1365) = crosssum of X(643) and X(1098)
X(1365) = X(933)-of-intouch-triangle
X(1365) = trilinear pole wrt intouch triangle of line X(7)X(21)
X(1365) = X(8)-isoconjugate of X(1101)
X(1365) = trilinear product X(57)*X(115)
X(1365) = barycentric product X(7)*X(115)

X(1366) = BRISSE TRANSFORM OF X(111)

X(1366) lies on the incircle and these lines: 7,1365   222,1367   553,1358

X(1367) = BRISSE TRANSFORM OF X(112)

X(1367) lies on the incircle and these lines: 7,1354   222,1366

X(1368) = COMPLEMENTARY CONJUGATE OF X(6)

As a point on the Euler line, X(1368) has Shinagawa coefficients (E - F, -E - F).

X(1368) lies on these lines: 2,3   11,1040   12,1038   98,801   114,122   120,123   125,343   126,127   230,577   495,612   496,614

X(1368) = midpoint of X(25) and X(1370)
X(1368) = reflection of X(1596) in X(5)
X(1368) = complement of X(25)
X(1368) = complementary conjugate of X(6)
X(1368) = isotomic conjugate of isogonal conjugate of X(6467)
X(1368) = polar conjugate of isogonal conjugate of X(22401)
X(1368) = circumcircle-inverse of X(37928)
X(1368) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1196), (670,525), (222,652), (255,520)
X(1368) = crosspoint of X(2) and X(305)
X(1368) = perspector of circumconic centered at X(1196)
X(1368) = center of circumconic that is locus of trilinear poles of lines passing through X(1196)
X(1368) = X(2)-Ceva conjugate of X(1196)
X(1368) = homothetic center of the medial triangle and the 3rd pedal triangle of X(3)
X(1368) = X(6244)-of-orthic-triangle if ABC is acute

X(1369) = ANTICOMPLEMENTARY CONJUGATE OF X(6)

X(1369) lies on these lines: 2,32   69,3410

X(1369) = anticomplement of X(251)
X(1369) = anticomplementary conjugate of X(6)
X(1369) = isotomic conjugate of isogonal conjugate of X(2916)
X(1369) = isotomic conjugate of cyclocevian conjugate of X(76)
X(1369) = polar conjugate of isogonal conjugate of X(23133)

X(1370) = ANTICOMPLEMENT OF X(25)

As a point on the Euler line, X(1370) has Shinagawa coefficients (E, -2E - 2F).

X(1370) lies on these lines: 2,3   66,69   305,315   343,1350   925,1297

X(1370) = reflection of X(25) in X(1368)
X(1370) = isogonal conjugate of X(34207)
X(1370) = isotomic conjugate of X(13575)
X(1370) = complement of X(7500)
X(1370) = anticomplement of X(25)
X(1370) = crosspoint of X(670) and X(23582)
X(1370) = circumcircle-inverse of X(37929)
X(1370) = cevapoint of X(i) and X(j) for these (i,j): {159, 23115}, {455, 3162}
X(1370) = crosssum of X(669) and X(3269)
X(1370) = anticomplementary conjugate of X(193)
X(1370) = X(i)-Ceva conjugate of X(j) for these (i,j): (305,2), (315,69)
X(1370) = homothetic center of anticomplementary triangle and 3rd antipedal triangle of X(4) (or polar triangle of anticomplementary circle)
X(1370) = pole, wrt polar circle, of the radical axis of any pair of {1st, 2nd and 3rd pedal circles of X(4)}
X(1370) = pole of de Longchamps line wrt anticomplementary circle
X(1370) = inverse-in-de-Longchamps-circle of X(23)

X(1371) = 1st RIGBY POINT

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 326.

See also MathWorld, Rigby Points.

X(1371) lies on this line: 1,7

X(1372) = 2nd RIGBY POINT

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 326.

X(1372) lies on this line: 1,7

X(1373) = 1st GRIFFITHS POINT

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 327.

See also MathWorld, Griffiths Points.

X(1373) lies on these lines: {1,7}, {226,3317}

X(1373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,1374), (7,176,481), (7,482,1), (175,482,1), (176,481,1), (481,482,176)
X(1373) = perspector of inner Soddy triangle and cross-triangle of inner and outer Soddy triangles
X(1373) = perspector of inner Soddy tangential triangle and cross-triangle of inner and outer Soddy tangential triangles

X(1374) = 2nd GRIFFITHS POINT

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 327.

X(1374) lies on these lines: {1,7}, {226,3316}

X(1374) = {X(1),X(7)}-harmonic conjugate of X(1373)
X(1374) = {X(7),X(175)}-harmonic conjugate of X(482)
X(1374) = {X(7),X(481)}-harmonic conjugate of X(1)
X(1374) = {X(481),X(482)}-harmonic conjugate of X(175)
X(1374) = perspector of outer Soddy triangle and cross-triangle of inner and outer Soddy triangles
X(1374) = perspector of outer Soddy tangential triangle and cross-triangle of inner and outer Soddy tangential triangles

X(1375) = EVANS POINT

As a point on the Euler line, X(1375) has Shinagawa coefficients (2$aS_BS_C$ - $a$S², $a$S²). Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 328.

See also MathWorld, Evans Point.

X(1375) lies on these lines: 2,3   241,514

X(1375) = complement of X(857)
X(1375) = crossdifference of every pair of points on line X(55)X(647)

X(1376) = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER CIRCLE)

X(1376) lies on these lines: 1,474   2,11   3,10   4,1329   6,43   7,480   8,56   9,165   12,377   31,899   35,405   36,956   40,936   42,750   45,846   46,72   57,200   63,210   65,78   71,965   75,183   226,1260   227,1038   371,1377   372,1378   442,498   517,997   519,999   748,902   851,1211   978,1191   982,1054

X(1376) = midpoint of X(i) and X(j) for these (i,j): (8,3476), (57,200), (329,3474)
X(1376) = isogonal conjugate of X(9309)
X(1376) = homothetic center of inner-Conway triangle and cross-triangle of Ursa-major and Ursa-minor triangles
X(1376) = complement of X(497)
X(1376) = X(294)-Ceva conjugate of X(518)
X(1376) = cevapoint of X(43) and X(165)
X(1376) = anticomplement of X(3816)
X(1376) = crosssum of PU(46)
X(1376) = crosspoint of PU(112)
X(1376) = crossdifference of every pair of points on line X(665)X(4083)
X(1376) = homothetic center of ABC and cross-triangle of ABC and inner Johnson triangle

X(1377) = INSIMILICENTER(2nd LEMOINE CIRCLE, SPIEKER CIRCLE)

X(1377) lies on these lines: 2,1335   6,10   8,1124   371,1376   372,958   485,1329   993,1152

X(1377) = {X(6),X(10)}-harmonic conjugate of X(1378)

X(1378) = EXSIMILICENTER(2nd LEMOINE CIRCLE, SPIEKER CIRCLE)

X(1378) lies on these lines: 2,1124   6,10   8,1335   371,958   372,1376   486,1329   993,1151

X(1378) = {X(6),X(10)}-harmonic conjugate of X(1377)

X(1379) = 1st BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1379) = (-1 + |OK|/R)*X(3) + X(6)

The Brocard axis, OK, is the line of the circumcenter, O [= X(3)], and the symmedian point, K [= X(6)]. This line meets the circumcircle in two points, X(1379) and X(1380); the closer to X(6) is X(1379.

X(1379) lies on these lines: {2,2039}, {3,6}, {4,2040}, {20,35913}, {23,6141}, {30,31863}, {83,14633}, {98,3414}, {99,3413}, {110,5638}, {111,5639}, {141,19660}, {230,31862}, {316,1348}, {352,6142}, {476,13722}, {513,36735}, {517,36736}, {620,14502}, {667,11651}, {1113,35607}, {1114,14899}, {1349,38227}

X(1379) = isogonal conjugate of X(3413)
X(1379) = reflection of X(i) in X(j) for these (i,j): (4,2040), (1380,3), (3557,2029)
X(1379) = anticomplement of X(2039)
X(1379) = inverse-in-Brocard-circle of X(1341)
X(1379) = X(249)-Ceva conjugate of X(1380)
X(1379) = Ψ(X(2), X(1340))
X(1379) = trilinear pole of line X(6)X(5639)
X(1379) = {X(371),X(372)}-harmonic conjugate of X(3558)
X(1379) = {X(1687),X(1688)}-harmonic conjugate of X(1380)
X(1379) = circumcircle intercept, other than A, B, C, of hyperbola {A,B,C,X(6),PU(118)}
X(1379) = perspector of triangles AiBiCi and (AaBbCc)*, and of triangles AaBbCc and (AiBiCi)*; see preamble before X(11752)
X(1379) = perspector of ABC and (degenerate) cross-triangle of 1st and 2nd anti-circummedial triangles
X(1379) = barycentric product of circumcircle intercepts of line X(2)X(1340)

X(1380) = 2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

X(1380) = (1 + |OK|/R)*X(3) - X(6)

Let A' be the incenter of BCX(15), and define B' and C' cyclically. Let A" be the incenter of BCX(16), and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1380). (Randy Hutson, February 10, 2016)

X(1380) lies on these lines: {2,2040}, {3,6}, {4,2039}, {20,35914}, {23,6142}, {30,31862}, {83,14632}, {98,3413}, {99,3414}, {110,5639}, {111,5638}, {141,19659}, {230,31863}, {316,1349}, {352,6141}, {476,13636}, {513,36736}, {517,36735}, {620,14501}, {667,11652}, {1113,35608}, {1114,35609}, {1348,38227}

In the plane of a triangle ABC, let
F₁ = X(39162) and F₂ = X(39163); these points are the foci of the Steiner inellipse;
L₁ = trilinear polar of F₁
L₂ = trilinear polar of F₁
A₁ = L₁∩BC, and define B₁ and C₁ cyclically
A₂ = L₂∩BC, and define B₂ and C₂ cyclically
The seven circle (ABC), (AB₁C₁), (BC₁A₁), (CA₁B₁), (AB₂C₂), (BC₂A₂), (CA₁B₂) concur in X(1380). See X(1380) (Angel Montesdeoca, February 18, 2023)

X(1380) = isogonal conjugate of X(3414)
X(1380) = reflection of X(i) in X(j) for these (i,j): (4,2039), (1379,3), (3558,2028)
X(1380) = anticomplement of X(2040)
X(1380) = inverse-in-Brocard-circle of X(1340)
X(1380) = X(249)-Ceva conjugate of X(1379)
X(1380) = Ψ(X(2), X(1341))
X(1380) = {X(371),X(372)}-harmonic conjugate of X(3557)
X(1380) = {X(1687),X(1688)}-harmonic conjugate of X(1379)
X(1380) = trilinear pole of line X(6)X(5638) (tangent to hyperbola {{A,B,C,X(6),PU(118)}} at X(6))
X(1380) = barycentric product of circumcircle intercepts of line X(2)X(1341)

X(1381) = 1st INTERCEPT OF LINE X(1)X(3) AND CIRCUMCIRCLE

The line IO, where I = X(1) = incenter and O = X(3) = circumcenter, meets the circumcircle in two points, X(1381) and X(1382), where X(1381) is the nearer of the two to X(1).

X(1381) lies on the circumcircle and this line: 1,3

X(1381) = reflection of X(1382) in X(3)
X(1381) = isogonal conjugate of X(3307)
X(1381) = X(59)-Ceva conjugate of X(1382)
X(1381) = X(513)-cross conjugate of X(1382)

X(1382) = 2nd INTERCEPT OF LINE X(1)X(3) AND CIRCUMCIRCLE

X(1382) lies on the circumcircle and this line: 1,3

X(1382) = reflection of X(1381) in X(3)
X(1382) = isogonal conjugate of X(3308)
X(1382) = X(59)-Ceva conjugate of X(1381)
X(1382) = X(513)-cross conjugate of X(1381)

X(1383) = 1st GRINBERG HOMOTHETIC CENTER

Let A'B'C' be the circumcevian triangle of X(2), and let P(A) be the line through A' parallel to line BC. Define P(B) and P(C) cyclically. Let A" = P(B)∩P(C), and define B" and C" cyclically. Triangle A"B"C" is homothetic to ABC, and X(1383) is the center of the homothety.

The center of the homothety is also X(1383) if "circumcevian triangle of X(6) (circumsymmedial triangle)" or "4th Brocard triangle" is substituted for "circumcevian triangle of X(2)". (Randy Hutson, March 21, 2019)

X(1383) lies on these lines: 2,187   6,23   32,111

X(1383) = isogonal conjugate of X(599)
X(1383) = isotomic conjugate of X(9464)
X(1383) = vertex conjugate of X(2) and X(6)
X(1383) = cevapoint of X(6) and X(1384)

X(1384) = 2nd GRINBERG HOMOTHETIC CENTER

Let G_A be the circumcenter of triangle BCG, where G - centroid(ABC). Define G_B and G_C cyclically. Triangle G_AG_BG_C is homothetic to the pedal triangle of X(6), and X(1384) is the center of the homothety.

Let DEF be the circumsymmedial triangle and let (Oa) be the circle tangent at D to circumcircle and with center Oa=AH∩ OD. Define (Ob) and (Oc) cyclically. The radical center of the circles (Oa), (Ob), (Oc) is X(1384). (Angel Montesdeoca, October 2, 2018)

X(1384) lies on these lines: 3,6   25,111   55,609   230,381   385,1003

X(1384) = X(1383)-Ceva conjugate of X(6)
X(1384) = isogonal conjugate of X(5485)
X(1384) = intersection of tangents at PU(2) to hyperbola {X(6),PU(1),PU(2)}
X(1384) = {X(3),X(6)}-harmonic conjugate of X(5024)
X(1384) = inverse-in-1st-Brocard-circle of X(5024)

X(1385) = MIDPOINT OF INCENTER AND CIRCUMCENTER

Let Na = X(5)-of-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)-of-ABX(1). X(1385) is the antigonal image of X(5) wrt NaNbNc. Also, X(1385) = X(265)-of-NaNbNc. (Randy Hutson, December 10, 2016)

Let A'B'C' be the medial triangle. X(1385) is the radical center of the incircles of AB'C', BC'A', CA'B'. (Randy Hutson, December 10, 2016)

X(1385) lies on the cubics K798 and K981 and these lines: {1, 3}, {2, 355}, {4, 1538}, {5, 515}, {6, 9619}, {7, 38030}, {8, 631}, {9, 20818}, {10, 140}, {11, 6842}, {12, 6882}, {20, 3622}, {21, 104}, {24, 1829}, {28, 1871}, {29, 39529}, {30, 551}, {37, 572}, {41, 43065}, {48, 40937}, {54, 72}, {63, 5730}, {71, 17438}, {73, 34586}, {74, 5606}, {77, 945}, {78, 956}, {79, 4325}, {80, 17606}, {92, 7531}, {98, 29299}, {100, 4861}, {101, 1212}, {102, 7100}, {106, 6011}, {119, 4187}, {142, 37281}, {145, 3523}, {149, 11015}, {150, 17095}, {153, 37162}, {169, 3207}, {182, 518}, {186, 41722}, {191, 28443}, {210, 5258}, {215, 17660}, {224, 41559}, {226, 4311}, {238, 7609}, {284, 1108}, {371, 7968}, {372, 7969}, {376, 962}, {377, 12116}, {378, 1902}, {381, 5691}, {382, 1699}, {386, 37698}, {388, 6827}, {390, 12700}, {399, 33535}, {404, 3753}, {405, 5777}, {442, 24541}, {452, 24558}, {474, 11499}, {495, 6922}, {496, 950}, {497, 4305}, {498, 5252}, {499, 1837}, {500, 1064}, {501, 759}, {511, 1386}, {512, 44811}, {513, 35050}, {516, 550}, {519, 549}, {528, 49600}, {529, 21077}, {537, 51045}, {542, 32238}, {546, 3817}, {547, 19883}, {548, 4301}, {573, 1100}, {574, 31430}, {575, 4663}, {578, 44547}, {581, 995}, {590, 49601}, {595, 37469}, {601, 3915}, {602, 1468}, {612, 16434}, {614, 19544}, {615, 49602}, {632, 3634}, {692, 48694}, {726, 32448}, {730, 49111}, {758, 5428}, {760, 13335}, {891, 44805}, {912, 960}, {936, 5534}, {938, 6988}, {943, 1476}, {947, 1807}, {953, 1290}, {954, 51489}, {958, 997}, {971, 1001}, {978, 37699}, {991, 1279}, {1012, 9856}, {1055, 17451}, {1056, 4308}, {1058, 4313}, {1066, 4322}, {1072, 23675}, {1145, 38760}, {1149, 4300}, {1151, 31439}, {1152, 35774}, {1154, 12266}, {1160, 11371}, {1161, 11370}, {1191, 36746}, {1193, 5396}, {1203, 36750}, {1210, 15325}, {1317, 21154}, {1320, 34474}, {1329, 10942}, {1334, 17439}, {1350, 38315}, {1351, 16475}, {1353, 34379}, {1375, 25935}, {1376, 10156}, {1387, 4304}, {1389, 27003}, {1400, 17440}, {1426, 1870}, {1441, 17221}, {1455, 10571}, {1457, 4303}, {1478, 6928}, {1479, 6923}, {1480, 16486}, {1484, 5499}, {1490, 5436}, {1511, 38612}, {1537, 38761}, {1571, 15815}, {1572, 3053}, {1587, 13902}, {1588, 13959}, {1595, 49542}, {1616, 37501}, {1621, 6906}, {1656, 3624}, {1657, 11522}, {1698, 3526}, {1702, 6221}, {1703, 6398}, {1706, 7966}, {1709, 28444}, {1710, 28445}, {1737, 5433}, {1742, 24661}, {1762, 28446}, {1766, 5356}, {1768, 48667}, {1770, 15326}, {1790, 37227}, {1798, 25713}, {1824, 37117}, {1836, 4299}, {1838, 7510}, {1848, 7511}, {1872, 6198}, {1939, 2649}, {1953, 22054}, {1960, 2826}, {2070, 9626}, {2080, 12194}, {2096, 17576}, {2100, 28447}, {2101, 28448}, {2102, 38708}, {2103, 38709}, {2173, 22357}, {2260, 5755}, {2278, 8609}, {2329, 25066}, {2346, 15179}, {2360, 36011}, {2478, 12115}, {2550, 38122}, {2687, 6584}, {2704, 14665}, {2716, 34921}, {2777, 11723}, {2778, 13293}, {2782, 11710}, {2783, 49484}, {2794, 11724}, {2800, 3884}, {2801, 15254}, {2802, 33814}, {2807, 40647}, {2808, 11712}, {2809, 38599}, {2817, 38600}, {2818, 11700}, {2829, 11729}, {2836, 12584}, {2886, 10943}, {2937, 9625}, {2941, 28449}, {2948, 32609}, {2960, 28450}, {3058, 11826}, {3073, 8235}, {3083, 16433}, {3084, 16432}, {3085, 3476}, {3086, 3486}, {3088, 7718}, {3090, 5550}, {3091, 46934}, {3100, 37404}, {3157, 34046}, {3158, 12629}, {3185, 15654}, {3189, 34625}, {3218, 4018}, {3241, 3524}, {3242, 5085}, {3243, 21153}, {3244, 3530}, {3309, 24286}, {3311, 9583}, {3312, 18991}, {3398, 11364}, {3419, 6889}, {3421, 27383}, {3434, 6897}, {3436, 6947}, {3467, 11279}, {3475, 5761}, {3485, 4293}, {3487, 3600}, {3488, 6908}, {3515, 11396}, {3517, 7713}, {3522, 6361}, {3525, 9780}, {3528, 5734}, {3529, 9812}, {3533, 19877}, {3534, 28202}, {3541, 5090}, {3553, 5120}, {3554, 4254}, {3555, 6986}, {3564, 39870}, {3582, 37702}, {3583, 37735}, {3585, 5443}, {3586, 9669}, {3589, 38167}, {3615, 7424}, {3617, 10303}, {3623, 15717}, {3625, 12108}, {3626, 14869}, {3627, 18483}, {3628, 10175}, {3632, 3689}, {3633, 9588}, {3635, 15712}, {3640, 26348}, {3641, 26341}, {3646, 9845}, {3647, 12104}, {3649, 30264}, {3651, 45977}, {3671, 24470}, {3679, 5054}, {3683, 5693}, {3720, 4192}, {3730, 6603}, {3742, 7686}, {3751, 5050}, {3754, 34353}, {3811, 12513}, {3812, 6796}, {3813, 32214}, {3820, 5795}, {3824, 6917}, {3827, 15577}, {3828, 11539}, {3830, 34628}, {3845, 12571}, {3851, 7988}, {3853, 28190}, {3868, 37106}, {3869, 3916}, {3871, 38460}, {3872, 4855}, {3874, 31806}, {3877, 4189}, {3878, 4640}, {3880, 8715}, {3890, 6950}, {3893, 48696}, {3911, 37728}, {3920, 19649}, {3928, 4930}, {3929, 28451}, {3955, 17625}, {3962, 4867}, {4004, 48363}, {4083, 39227}, {4084, 4973}, {4132, 39210}, {4145, 44812}, {4188, 35271}, {4190, 37000}, {4199, 24550}, {4220, 7191}, {4225, 18180}, {4251, 40133}, {4256, 4646}, {4276, 18178}, {4292, 39542}, {4294, 6948}, {4298, 6147}, {4302, 12701}, {4314, 15172}, {4315, 5719}, {4317, 10404}, {4357, 29287}, {4421, 10912}, {4423, 10157}, {4512, 17571}, {4564, 14887}, {4649, 37510}, {4652, 11682}, {4653, 8143}, {4666, 7580}, {4668, 51515}, {4669, 11812}, {4670, 29069}, {4677, 15701}, {4745, 15713}, {4849, 46822}, {4857, 5441}, {4911, 17084}, {4920, 15903}, {4995, 45081}, {5013, 9620}, {5024, 9593}, {5055, 7989}, {5066, 34648}, {5070, 34595}, {5071, 50864}, {5080, 6902}, {5086, 6224}, {5092, 49465}, {5124, 21853}, {5171, 10800}, {5176, 27529}, {5184, 38225}, {5218, 6961}, {5219, 9613}, {5239, 6192}, {5240, 6191}, {5248, 5450}, {5249, 37468}, {5250, 16370}, {5251, 5506}, {5253, 5439}, {5259, 7489}, {5260, 12738}, {5270, 37701}, {5284, 5927}, {5287, 16435}, {5289, 12514}, {5302, 10176}, {5303, 26877}, {5308, 7397}, {5315, 51340}, {5330, 17549}, {5399, 22350}, {5418, 13911}, {5420, 13973}, {5424, 5557}, {5426, 12688}, {5432, 10039}, {5434, 11827}, {5438, 9623}, {5444, 37710}, {5445, 41684}, {5493, 28212}, {5496, 20718}, {5533, 12743}, {5542, 5762}, {5552, 6967}, {5554, 6921}, {5559, 39781}, {5604, 45553}, {5605, 45552}, {5663, 11699}, {5692, 11935}, {5720, 8583}, {5728, 7677}, {5732, 38316}, {5754, 21363}, {5758, 11037}, {5759, 11038}, {5763, 12577}, {5768, 6857}, {5771, 24391}, {5779, 16866}, {5780, 7308}, {5787, 6824}, {5791, 30478}, {5794, 26363}, {5797, 29833}, {5805, 6869}, {5806, 6985}, {5836, 15813}, {5842, 12609}, {5843, 51090}, {5847, 48876}, {5854, 32157}, {5876, 31751}, {5918, 26088}, {5946, 31760}, {6000, 12262}, {6051, 10448}, {6181, 9374}, {6200, 35642}, {6210, 48908}, {6246, 32557}, {6253, 28452}, {6256, 6929}, {6259, 6930}, {6264, 12331}, {6284, 30384}, {6321, 38220}, {6366, 44819}, {6396, 35641}, {6417, 19003}, {6418, 19004}, {6449, 9616}, {6455, 9582}, {6642, 9798}, {6644, 44662}, {6666, 38179}, {6667, 38182}, {6668, 38183}, {6690, 12616}, {6702, 20107}, {6705, 33899}, {6739, 27687}, {6740, 37158}, {6744, 15935}, {6797, 10090}, {6826, 18517}, {6836, 10532}, {6838, 10586}, {6862, 10198}, {6872, 37002}, {6879, 10585}, {6885, 28629}, {6890, 10587}, {6893, 12667}, {6899, 10597}, {6909, 26200}, {6911, 11500}, {6918, 12650}, {6925, 10531}, {6937, 11680}, {6954, 7288}, {6959, 10200}, {6963, 11681}, {6965, 26127}, {6971, 7951}, {6978, 10588}, {6980, 7741}, {6982, 10591}, {6989, 19843}, {6992, 20076}, {6996, 16826}, {6998, 16823}, {6999, 29586}, {7171, 12705}, {7330, 16418}, {7354, 7491}, {7377, 17397}, {7384, 29592}, {7387, 11365}, {7411, 29817}, {7483, 24987}, {7506, 8185}, {7583, 8983}, {7584, 13971}, {7587, 12491}, {7588, 8100}, {7675, 42884}, {7680, 37356}, {7681, 37406}, {7690, 48740}, {7692, 48741}, {7701, 28453}, {7970, 34473}, {7972, 17663}, {7973, 10606}, {7974, 21157}, {7975, 21156}, {7976, 22712}, {7977, 9751}, {7978, 15055}, {7983, 21166}, {7984, 12778}, {8012, 48263}, {8021, 23204}, {8068, 18976}, {8077, 8099}, {8092, 8130}, {8109, 12488}, {8110, 12489}, {8129, 8351}, {8141, 28454}, {8225, 12490}, {8229, 26230}, {8236, 35514}, {8543, 18450}, {8572, 17054}, {8677, 44807}, {8703, 12512}, {8717, 9943}, {8731, 25941}, {8760, 11247}, {8907, 18732}, {8981, 13883}, {9041, 50983}, {9310, 16601}, {9317, 24774}, {9519, 51531}, {9540, 19066}, {9548, 9567}, {9549, 9566}, {9572, 28455}, {9573, 28456}, {9575, 30435}, {9578, 31479}, {9589, 15696}, {9590, 45735}, {9591, 13564}, {9592, 9605}, {9610, 9642}, {9611, 9641}, {9612, 9655}, {9614, 9668}, {9617, 9691}, {9618, 9690}, {9621, 9704}, {9622, 9703}, {9643, 36984}, {9678, 30556}, {9732, 45399}, {9733, 45398}, {9738, 45501}, {9739, 45500}, {9821, 11368}, {9864, 15561}, {9899, 35450}, {9904, 15041}, {9911, 35243}, {9928, 47391}, {9941, 26316}, {10058, 12740}, {10074, 12739}, {10087, 20586}, {10109, 38076}, {10124, 51082}, {10172, 19878}, {10248, 15682}, {10251, 28457}, {10256, 50772}, {10263, 31757}, {10271, 11719}, {10282, 40660}, {10297, 47469}, {10299, 20057}, {10304, 20070}, {10386, 12575}, {10446, 17394}, {10459, 19514}, {10478, 37869}, {10483, 18393}, {10519, 51192}, {10528, 36977}, {10529, 37112}, {10543, 15908}, {10573, 24914}, {10582, 19541}, {10609, 24390}, {10627, 31737}, {10647, 18469}, {10648, 18471}, {10669, 11377}, {10673, 11378}, {10695, 38690}, {10696, 38691}, {10697, 38692}, {10698, 12515}, {10699, 38694}, {10700, 38695}, {10701, 23239}, {10702, 38696}, {10703, 38697}, {10704, 38698}, {10705, 38699}, {10864, 18540}, {10895, 37692}, {10896, 23708}, {10915, 38455}, {10916, 44669}, {10956, 32554}, {11019, 12433}, {11101, 51420}, {11109, 45766}, {11113, 41012}, {11171, 12782}, {11179, 47358}, {11194, 12635}, {11237, 11929}, {11238, 11928}, {11251, 11831}, {11496, 12520}, {11552, 39782}, {11570, 45288}, {11695, 23841}, {11705, 11739}, {11706, 11740}, {11716, 28915}, {11721, 33962}, {11722, 12265}, {11725, 23698}, {11735, 12261}, {12017, 16496}, {12054, 12197}, {12100, 51071}, {12103, 28178}, {12247, 37291}, {12248, 16128}, {12259, 44665}, {12264, 22475}, {12368, 14643}, {12407, 38724}, {12438, 26451}, {12440, 45623}, {12441, 45624}, {12497, 35248}, {12528, 16865}, {12558, 16160}, {12607, 32213}, {12610, 17045}, {12617, 16617}, {12647, 37738}, {12696, 35241}, {12697, 35246}, {12698, 35247}, {12735, 41554}, {12747, 37718}, {12751, 38752}, {12877, 16159}, {12898, 13211}, {12908, 31791}, {13099, 38717}, {13178, 38224}, {13329, 49478}, {13334, 14839}, {13349, 44660}, {13350, 44659}, {13405, 37364}, {13532, 38776}, {13605, 32423}, {13630, 31728}, {13729, 17618}, {13747, 24982}, {13912, 35255}, {13935, 19065}, {13936, 13966}, {13975, 35256}, {14074, 38451}, {14269, 30308}, {14636, 48882}, {14666, 50926}, {14830, 50881}, {14891, 51077}, {14893, 50862}, {15017, 38755}, {15067, 31752}, {15071, 18515}, {15122, 47476}, {15170, 31777}, {15462, 32278}, {15489, 45955}, {15623, 16374}, {15677, 16116}, {15681, 50865}, {15683, 50819}, {15684, 50806}, {15686, 50815}, {15687, 50802}, {15690, 34638}, {15692, 50810}, {15693, 34718}, {15694, 19875}, {15698, 34631}, {15700, 50805}, {15702, 50818}, {15704, 28150}, {15707, 34747}, {15708, 31145}, {15711, 51107}, {15718, 50817}, {15721, 50804}, {15723, 50871}, {15733, 42842}, {15735, 38668}, {15759, 51104}, {15829, 31424}, {15842, 15843}, {16058, 23168}, {16113, 28460}, {16138, 28461}, {16139, 21161}, {16174, 22938}, {16239, 31399}, {16309, 28462}, {16466, 36742}, {16472, 36749}, {16473, 36753}, {16491, 33878}, {16517, 31468}, {16583, 21008}, {16604, 34460}, {16830, 21554}, {16884, 37499}, {17022, 19517}, {17043, 18589}, {17044, 34847}, {17073, 26130}, {17136, 20880}, {17566, 25005}, {17654, 18861}, {17696, 35274}, {17728, 37724}, {17757, 27385}, {18395, 37706}, {18400, 32331}, {18458, 30385}, {18460, 30386}, {18524, 45976}, {18583, 38049}, {18650, 41007}, {18654, 20895}, {19262, 37482}, {19512, 29571}, {19516, 29821}, {19522, 21352}, {19535, 35258}, {19540, 26102}, {19546, 30950}, {19548, 28082}, {19550, 30116}, {19708, 34632}, {19710, 41150}, {19711, 51091}, {19919, 28463}, {20104, 38114}, {20423, 38023}, {21147, 37697}, {21155, 37734}, {21167, 51147}, {21375, 28464}, {21511, 26639}, {21677, 28465}, {21850, 38040}, {21872, 24047}, {22082, 25416}, {22115, 44782}, {22769, 34381}, {23156, 31825}, {23242, 23243}, {24036, 30618}, {24220, 28639}, {24257, 32941}, {24325, 29010}, {24331, 36477}, {24559, 37086}, {24581, 48381}, {25082, 41391}, {25406, 39898}, {25485, 46684}, {25498, 29109}, {25525, 45630}, {25681, 37713}, {26006, 30810}, {26140, 27006}, {26202, 31649}, {26367, 49378}, {26368, 49377}, {26369, 49038}, {26370, 49039}, {26498, 45718}, {26507, 45717}, {26516, 45719}, {26521, 45720}, {26626, 36698}, {26725, 37230}, {28083, 34461}, {28216, 44245}, {28228, 46853}, {28473, 48328}, {28537, 48344}, {28538, 50977}, {28609, 34740}, {28850, 48932}, {29057, 49482}, {29207, 50290}, {29309, 41430}, {29311, 48886}, {29331, 50023}, {29369, 33682}, {29570, 37416}, {29660, 36530}, {29814, 37400}, {31434, 37709}, {31443, 37512}, {31670, 38035}, {31671, 38036}, {31672, 38037}, {31747, 32424}, {31790, 32183}, {31811, 32046}, {31822, 37411}, {32049, 45701}, {32486, 46362}, {32515, 50775}, {32905, 33812}, {33668, 49113}, {34200, 50808}, {34380, 51196}, {34789, 38753}, {35404, 51080}, {35610, 35811}, {35611, 35810}, {36505, 36561}, {36745, 44414}, {36754, 39523}, {37246, 38396}, {37251, 44425}, {37298, 50843}, {37365, 43223}, {37425, 48903}, {37712, 46219}, {37732, 49997}, {37806, 45272}, {37829, 38762}, {38064, 47359}, {38081, 51069}, {38116, 49688}, {38118, 49529}, {38144, 47355}, {38764, 50903}, {39582, 44545}, {40091, 45219}, {40998, 50241}, {43118, 45714}, {43119, 45713}, {43151, 43179}, {44214, 47321}, {44220, 44661}, {44580, 51096}, {44903, 51075}, {45410, 45427}, {45411, 45426}, {47033, 49176}, {47333, 47593}, {48877, 50420}, {48883, 48907}, {48887, 50418}, {51114, 51117}, {51115, 51116}

X(1385) = midpoint of X(i) and X(j) for these {i,j}: {1, 3}, {2, 3655}, {4, 18481}, {5, 34773}, {8, 37727}, {10, 5882}, {20, 12699}, {21, 33858}, {40, 1482}, {100, 12737}, {104, 6265}, {142, 43175}, {165, 10247}, {214, 11715}, {355, 944}, {376, 3656}, {381, 50811}, {399, 33535}, {500, 9840}, {549, 50824}, {550, 22791}, {942, 31786}, {946, 4297}, {960, 12675}, {991, 31394}, {999, 37611}, {1071, 5887}, {1483, 5690}, {1537, 38761}, {1657, 41869}, {1768, 48667}, {2975, 37733}, {3057, 37562}, {3058, 28458}, {3241, 3654}, {3244, 11362}, {3428, 37533}, {3534, 31162}, {3576, 10246}, {3579, 10222}, {3635, 43174}, {3811, 12513}, {3830, 34628}, {3874, 31806}, {3878, 5884}, {3928, 4930}, {4301, 31730}, {4677, 34748}, {5434, 28459}, {5441, 47032}, {5450, 40257}, {5453, 48930}, {5535, 35457}, {5697, 25413}, {5720, 30283}, {5731, 5886}, {5805, 43161}, {5881, 18526}, {6210, 48908}, {6261, 12114}, {6264, 12331}, {6326, 12773}, {6684, 13607}, {6767, 30503}, {6769, 8158}, {7354, 7491}, {7966, 40587}, {7967, 26446}, {7972, 19914}, {7982, 12702}, {7984, 12778}, {7987, 37624}, {7991, 8148}, {8666, 22836}, {8715, 22837}, {9943, 45776}, {9957, 31788}, {10265, 33337}, {10297, 47469}, {10543, 37401}, {10609, 37726}, {10698, 12515}, {10738, 12119}, {11014, 11849}, {11179, 47358}, {11496, 12520}, {11700, 11713}, {11709, 11720}, {11710, 11711}, {11712, 11714}, {11722, 12265}, {12248, 16128}, {12262, 40658}, {12680, 40263}, {12696, 35241}, {12738, 38669}, {12898, 13211}, {12908, 31791}, {13600, 31798}, {13624, 15178}, {13743, 16132}, {14110, 24474}, {14666, 50926}, {14830, 50881}, {15071, 40266}, {15122, 47476}, {15171, 31775}, {15681, 50865}, {16139, 34195}, {18446, 22758}, {18990, 31789}, {19907, 38602}, {22770, 37531}, {23156, 31825}, {24257, 32941}, {25485, 46684}, {26086, 26087}, {26286, 46920}, {28609, 34740}, {31663, 33179}, {31732, 31738}, {31787, 31792}, {31790, 32183}, {33281, 33862}, {34718, 51093}, {34789, 38753}, {37425, 48903}, {37474, 46475}, {39870, 49511}, {43151, 43179}, {45715, 45716}, {47333, 47593}, {48882, 48909}, {48883, 48907}, {48893, 48894}
X(1385) = reflection of X(i) in X(j) for these {i,j}: {1, 15178}, {3, 13624}, {4, 9955}, {5, 1125}, {10, 140}, {40, 31663}, {65, 5885}, {355, 9956}, {549, 50828}, {942, 13373}, {946, 5901}, {960, 31838}, {1071, 26201}, {1482, 33179}, {1483, 13607}, {3576, 31662}, {3579, 3}, {3627, 18483}, {3647, 12104}, {4640, 7508}, {4663, 575}, {5690, 6684}, {5694, 960}, {5876, 31751}, {6917, 3824}, {10222, 1}, {10225, 23961}, {10263, 31757}, {10284, 9957}, {11230, 38028}, {11231, 10165}, {11278, 10222}, {11500, 40262}, {11567, 33657}, {11699, 11720}, {12261, 11735}, {12611, 11729}, {12619, 6713}, {13464, 3636}, {15686, 50815}, {15687, 50802}, {17502, 3576}, {18357, 3628}, {18480, 5}, {22791, 13464}, {22792, 12608}, {22793, 946}, {22798, 16617}, {22935, 214}, {22936, 21}, {22937, 5428}, {22938, 16174}, {23841, 11695}, {23961, 18857}, {24474, 6583}, {24475, 12005}, {26202, 31649}, {31673, 546}, {31728, 13630}, {31730, 548}, {31737, 10627}, {31788, 40296}, {31811, 32046}, {31828, 31937}, {33592, 11281}, {33697, 4}, {33899, 6705}, {34339, 9940}, {34638, 15690}, {34648, 5066}, {34862, 5450}, {35004, 34339}, {37562, 13145}, {37623, 26286}, {37727, 32900}, {38138, 10172}, {38140, 11230}, {38176, 11231}, {40660, 10282}, {41347, 36}, {48887, 50418}, {48926, 48893}, {50796, 547}, {50808, 34200}, {50821, 549}, {50862, 14893}, {51118, 40273}
X(1385) = isogonal conjugate of X(1389)
X(1385) = complement of X(355)
X(1385) = anticomplement of X(9956)
X(1385) = reflection of X(1385) in the X(1)X(3)
X(1385) = circumcircle-inverse of X(22765)
X(1385) = complement of the isogonal conjugate of X(3417)
X(1385) = X(3417)-complementary conjugate of X(10)
X(1385) = X(i)-Ceva conjugate of X(j) for these (i,j): {27003, 16669}, {33637, 513}
X(1385) = X(1)-isoconjugate of X(1389)
X(1385) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 1389}, {355, 1385}
X(1385) = crosspoint of X(i) and X(j) for these (i,j): {1, 15446}, {59, 43345}
X(1385) = crosssum of X(1) and X(5903)
X(1385) = barycentric product X(75)*X(2317)
X(1385) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 1389}, {2317, 1}
X(1385) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 35, 3057}, {1, 36, 65}, {1, 40, 1482}, {1, 46, 2099}, {1, 55, 9957}, {1, 56, 942}, {1, 65, 50194}, {1, 165, 7982}, {1, 484, 11009}, {1, 999, 5045}, {1, 1319, 24928}, {1, 1388, 25405}, {1, 1420, 999}, {1, 1482, 33179}, {1, 2077, 23340}, {1, 2646, 24929}, {1, 3295, 31792}, {1, 3304, 5049}, {1, 3333, 15934}, {1, 3337, 5425}, {1, 3361, 11529}, {1, 3576, 3}, {1, 3579, 11278}, {1, 3601, 3295}, {1, 3612, 55}, {1, 3746, 5919}, {1, 5010, 5697}, {1, 5119, 2098}, {1, 5204, 50193}, {1, 5563, 354}, {1, 5697, 5048}, {1, 5903, 11011}, {1, 7280, 5903}, {1, 7982, 10247}, {1, 7987, 40}, {1, 7991, 16200}, {1, 8273, 31793}, {1, 10246, 15178}, {1, 10269, 34339}, {1, 10310, 13600}, {1, 10882, 10441}, {1, 11012, 24474}, {1, 13462, 3333}, {1, 13624, 3579}, {1, 15803, 3340}, {1, 15931, 14110}, {1, 16192, 11531}, {1, 16203, 13373}, {1, 18398, 44840}, {1, 21842, 1319}, {1, 22765, 6583}, {1, 22766, 50195}, {1, 22767, 50196}, {1, 26285, 10284}, {1, 30282, 1697}, {1, 30389, 3576}, {1, 31662, 31666}, {1, 32612, 35004}, {1, 35202, 7957}, {1, 35242, 8148}, {1, 37525, 2646}, {1, 37535, 5885}, {1, 37561, 37562}, {1, 37571, 37080}, {1, 37587, 18398}, {1, 37605, 37582}, {1, 37616, 35}, {1, 37617, 37592}, {1, 37618, 56}, {1, 37620, 35631}, {1, 40293, 13601}, {2, 355, 9956}, {2, 944, 355}, {3, 35, 26086}, {3, 40, 31663}, {3, 55, 26285}, {3, 56, 26286}, {3, 942, 37623}, {3, 999, 11249}, {3, 1482, 40}, {3, 3295, 11248}, {3, 3576, 13624}, {3, 5885, 41347}, {3, 6767, 10306}, {3, 7373, 22770}, {3, 10202, 37582}, {3, 10246, 1}, {3, 10247, 12702}, {3, 10267, 32613}, {3, 10269, 32612}, {3, 10306, 35238}, {3, 10679, 10310}, {3, 10680, 3428}, {3, 11849, 2077}, {3, 12000, 35448}, {3, 12702, 165}, {3, 13624, 17502}, {3, 15178, 10222}, {3, 15934, 5709}, {3, 16202, 55}, {3, 16203, 56}, {3, 18443, 9940}, {3, 22765, 11012}, {3, 22770, 35239}, {3, 24299, 24929}, {3, 24926, 11567}, {3, 24927, 24928}, {3, 32612, 23961}, {3, 32613, 33862}, {3, 34471, 46920}, {3, 35004, 10225}, {3, 37525, 26287}, {3, 37533, 31793}, {3, 37535, 36}, {3, 37612, 5122}, {3, 37615, 942}, {3, 37621, 35}, {3, 37624, 1482}, {3, 45923, 46623}, {3, 50317, 37536}, {4, 3616, 5886}, {4, 5731, 18481}, {4, 5886, 9955}, {5, 1125, 11230}, {5, 18480, 38140}, {5, 38028, 1125}, {8, 631, 26446}, {8, 7967, 37727}, {10, 140, 11231}, {10, 10165, 140}, {20, 3622, 5603}, {20, 5603, 12699}, {21, 18444, 1071}, {21, 21740, 5887}, {21, 30285, 9959}, {35, 36, 14792}, {35, 13145, 3579}, {35, 34486, 37621}, {35, 37561, 3}, {35, 37616, 37600}, {36, 65, 37582}, {36, 10902, 3}, {36, 24926, 1}, {36, 33657, 10222}, {40, 3338, 37532}, {40, 3576, 7987}, {40, 7987, 3}, {40, 31663, 3579}, {40, 37571, 33596}, {40, 37624, 33179}, {46, 2099, 50193}, {46, 5204, 5122}, {55, 56, 8071}, {55, 1388, 1}, {55, 40296, 3579}, {56, 2098, 17437}, {56, 31786, 37623}, {56, 34471, 1}, {56, 37615, 13373}, {56, 37618, 5126}, {65, 10202, 5885}, {65, 11567, 10222}, {65, 37605, 36}, {78, 956, 34790}, {100, 4861, 10914}, {104, 21740, 1071}, {104, 33858, 26201}, {145, 3523, 5657}, {165, 7982, 12702}, {214, 51111, 10}, {226, 4311, 18990}, {354, 14110, 24474}, {354, 24474, 6583}, {355, 3655, 944}, {371, 35762, 7968}, {372, 35763, 7969}, {376, 10595, 962}, {376, 38314, 3656}, {377, 12116, 37820}, {382, 18493, 1699}, {388, 6827, 10526}, {392, 1071, 5887}, {405, 18446, 5777}, {496, 950, 18527}, {497, 6850, 10525}, {549, 1483, 5690}, {549, 5690, 6684}, {550, 10283, 22791}, {551, 946, 5901}, {551, 4297, 946}, {602, 1468, 5398}, {631, 7967, 8}, {632, 37705, 38042}, {632, 38042, 3634}, {942, 5126, 56}, {942, 46920, 10222}, {946, 51118, 40273}, {950, 44675, 496}, {958, 997, 5044}, {960, 993, 31445}, {962, 10595, 3656}, {962, 38314, 10595}, {993, 30144, 960}, {1001, 12114, 3560}, {1125, 34773, 18480}, {1151, 35775, 31439}, {1151, 44636, 35775}, {1152, 44635, 35774}, {1155, 11011, 5903}, {1159, 37545, 3339}, {1319, 2646, 1}, {1319, 24299, 15178}, {1319, 26287, 10222}, {1319, 37080, 20323}, {1319, 37525, 24929}, {1381, 1382, 22765}, {1387, 15171, 12053}, {1388, 3576, 31788}, {1388, 3612, 9957}, {1420, 13384, 1}, {1420, 37611, 11249}, {1479, 11376, 7743}, {1482, 7987, 31663}, {1482, 10246, 37624}, {1482, 33179, 10222}, {1482, 37624, 1}, {1483, 50824, 13607}, {1490, 5436, 6913}, {1656, 18525, 5587}, {1697, 37526, 3359}, {1698, 5881, 5790}, {1699, 9624, 18493}, {1702, 9615, 6221}, {2077, 3746, 11849}, {2098, 5217, 5119}, {2099, 5204, 46}, {2478, 12115, 37821}, {2646, 13151, 13624}, {2646, 20323, 37080}, {2646, 21842, 24928}, {2646, 24927, 15178}, {2646, 37080, 37571}, {2975, 4511, 72}, {3057, 26087, 10222}, {3057, 37600, 35}, {3086, 3486, 5722}, {3207, 34522, 169}, {3241, 3524, 3654}, {3244, 10164, 11362}, {3295, 8726, 31787}, {3295, 37606, 3601}, {3303, 10310, 10679}, {3304, 3428, 10680}, {3304, 8273, 3428}, {3333, 15934, 50192}, {3340, 15803, 36279}, {3361, 11529, 5708}, {3428, 8273, 3}, {3487, 6987, 5812}, {3526, 5790, 1698}, {3526, 18526, 5790}, {3530, 11362, 31447}, {3560, 6261, 31937}, {3576, 10269, 18857}, {3576, 13384, 37611}, {3576, 13624, 31666}, {3576, 15178, 3579}, {3576, 18443, 10269}, {3576, 30392, 10246}, {3576, 34471, 31786}, {3576, 34486, 37600}, {3576, 37615, 26286}, {3576, 37624, 31663}, {3579, 17502, 3}, {3579, 31666, 17502}, {3585, 5443, 17605}, {3586, 50443, 9669}, {3601, 8726, 3}, {3616, 5731, 4}, {3616, 18481, 9955}, {3624, 5587, 1656}, {3627, 38034, 18483}, {3628, 18357, 10175}, {3636, 13464, 10283}, {3653, 3655, 2}, {3746, 11014, 23340}, {3816, 18242, 5}, {3817, 31673, 546}, {3872, 4855, 5687}, {3878, 5267, 4640}, {4256, 15955, 4646}, {4297, 5901, 22793}, {4304, 12053, 15171}, {4305, 11373, 31795}, {4308, 5703, 1056}, {4867, 6763, 3962}, {5010, 5697, 37568}, {5048, 37568, 5697}, {5122, 50193, 46}, {5126, 25405, 3660}, {5126, 31786, 26286}, {5219, 9613, 9654}, {5248, 5450, 6914}, {5396, 13731, 34466}, {5426, 16132, 13743}, {5432, 10944, 10039}, {5433, 10950, 1737}, {5438, 9623, 9709}, {5441, 16173, 4857}, {5443, 36975, 3585}, {5563, 11012, 22765}, {5563, 15931, 11012}, {5563, 34890, 36}, {5563, 35597, 10222}, {5690, 6684, 50821}, {5690, 50824, 1483}, {5691, 8227, 381}, {5691, 25055, 8227}, {5790, 18526, 5881}, {5795, 6700, 3820}, {5882, 10165, 10}, {5885, 10902, 3579}, {5885, 11567, 50194}, {5885, 33657, 1}, {5886, 18481, 4}, {5903, 7280, 1155}, {6198, 37305, 1872}, {6200, 35642, 49226}, {6264, 15015, 12331}, {6265, 33858, 21740}, {6396, 35641, 49227}, {6583, 15931, 3579}, {6713, 31659, 140}, {6767, 10306, 37622}, {6883, 37700, 5044}, {6893, 12667, 18516}, {6897, 10806, 3434}, {6914, 13369, 34862}, {6917, 48482, 18407}, {6947, 10805, 3436}, {7354, 15950, 12047}, {7587, 18454, 12491}, {7588, 18456, 8100}, {7677, 30284, 5728}, {7688, 35202, 3}, {7967, 37727, 32900}, {7984, 15035, 12778}, {7987, 10246, 33179}, {7987, 11531, 16192}, {7987, 33179, 3579}, {7987, 37571, 50371}, {7988, 18492, 3851}, {7991, 16200, 8148}, {8071, 31788, 37623}, {8071, 46920, 10284}, {8077, 18448, 8099}, {8227, 50811, 5691}, {8583, 30283, 9947}, {9583, 18992, 3311}, {9940, 10267, 3579}, {9940, 13624, 32612}, {9940, 40296, 18856}, {9943, 10179, 45776}, {9955, 18481, 33697}, {9957, 11227, 31788}, {9957, 18856, 35004}, {9957, 25405, 1}, {10106, 13411, 495}, {10175, 19862, 3628}, {10222, 17502, 3579}, {10222, 31666, 3}, {10246, 10902, 11567}, {10246, 13624, 10222}, {10246, 16203, 37615}, {10246, 30389, 13624}, {10246, 31662, 17502}, {10246, 31666, 11278}, {10246, 32612, 33281}, {10246, 37561, 26087}, {10246, 37618, 13373}, {10247, 12702, 7982}, {10267, 10269, 3}, {10267, 18443, 34339}, {10267, 32612, 33862}, {10269, 32613, 23961}, {10283, 22791, 13464}, {10389, 10857, 6244}, {10698, 38693, 12515}, {10943, 37438, 2886}, {11012, 15931, 3}, {11108, 35272, 8583}, {11227, 31788, 40296}, {11230, 18480, 5}, {11248, 31787, 3579}, {11500, 25524, 6911}, {11510, 22768, 8069}, {11531, 16192, 40}, {11567, 37605, 41347}, {11567, 41347, 11278}, {11707, 11708, 1386}, {12047, 21578, 7354}, {12119, 16173, 10738}, {12667, 26105, 6893}, {12675, 31838, 5694}, {12898, 15061, 13211}, {13145, 15178, 26087}, {13151, 24299, 3}, {13151, 26287, 31666}, {13373, 13624, 26286}, {13607, 50828, 6684}, {13624, 33179, 31663}, {13624, 33281, 10225}, {13624, 34339, 23961}, {14636, 48909, 48882}, {15178, 17502, 11278}, {15178, 18857, 35004}, {15178, 26287, 24929}, {15178, 30389, 31666}, {15178, 31662, 13624}, {15178, 31663, 33179}, {15178, 34339, 33281}, {15325, 37730, 1210}, {15694, 50798, 19875}, {15701, 34748, 38066}, {16200, 35242, 7991}, {17437, 46920, 23960}, {17502, 31666, 13624}, {18444, 21740, 33858}, {18446, 19861, 45770}, {18857, 32613, 17502}, {18857, 34339, 32612}, {18990, 37737, 226}, {19860, 35262, 474}, {19883, 50796, 547}, {20323, 33596, 33179}, {20323, 37080, 1}, {20323, 50371, 1482}, {21161, 34195, 16139}, {21842, 37525, 1}, {22758, 45770, 5777}, {22938, 38044, 16174}, {23961, 33281, 35004}, {23961, 33862, 3}, {24299, 24927, 10246}, {24914, 37740, 10573}, {24926, 37605, 50194}, {24928, 24929, 1}, {25055, 50811, 381}, {25405, 26285, 10222}, {25935, 35290, 1375}, {26086, 37562, 3579}, {26285, 31788, 3579}, {26286, 31786, 3579}, {26365, 26366, 3576}, {26446, 37727, 8}, {26487, 26492, 2}, {30144, 32153, 5694}, {30282, 37526, 3}, {30283, 35272, 5720}, {30389, 30392, 1}, {31393, 37560, 49163}, {31663, 37624, 10222}, {31792, 33574, 31787}, {31838, 32153, 31445}, {32612, 32613, 3}, {32613, 34339, 3579}, {33862, 34339, 10225}, {33862, 35004, 3579}, {34471, 37615, 15178}, {34471, 37618, 942}, {34486, 37561, 35}, {34556, 34557, 17502}, {34628, 38021, 3830}, {34628, 51110, 38021}, {34748, 38066, 4677}, {34773, 38028, 5}, {35238, 37622, 10306}, {37080, 37571, 24929}, {37080, 50371, 33596}, {37561, 37621, 26086}, {37562, 37600, 26086}, {37582, 50194, 65}, {38013, 38014, 24929}, {38053, 43161, 5805}, {38155, 51073, 31399}, {40273, 51118, 22793}, {45620, 45621, 55}, {48460, 48461, 18443}, {50821, 50824, 51087}, {50821, 50828, 51084}, {50821, 51084, 51088}, {50823, 50829, 50821}, {50823, 50833, 50829}, {50824, 50825, 50831}, {50824, 50828, 50821}, {50824, 50832, 50828}, {50824, 50833, 50823}, {50825, 50827, 50821}, {50825, 50831, 50827}, {50827, 50828, 51086}, {50827, 51086, 50825}, {50828, 50829, 50833}, {50828, 51085, 50824}, {50828, 51087, 51088}, {50831, 51086, 50821}, {50832, 51085, 50821}, {51084, 51087, 50821}
X(1385) = {X(1),X(40)}-harmonic conjugate of X(1483)
X(1385) = X(5)-of-2nd circumperp-triangle
X(1385) = X(3)-of-X(1)-Brocard-triangle
X(1385) = X(140)-of-hexyl-triangle
X(1385) = X(26)-of-incircle-circles-triangle
X(1385) = X(3)-of-anti-Aquila-triangle
X(1385) = endo-homothetic center of Ehrmann side-triangle and circumorthic triangle; the homothetic center is X(5)
X(1385) = X(546)-of-excentral-triangle

X(1386) = MIDPOINT OF INCENTER AND SYMMEDIAN POINT

X(1386) lies on these lines: 1,6   7,1456   56,77   65,82   81,105   141,1125   171,1054   182,517   206,942   241,1471   511,1385   519,597   524,551   614,940   751,1319

X(1386) = midpoint of X(1) and X(6)
X(1386) = reflection of X(141) in X(1125)
X(1386) = isogonal conjugate of X(1390)
X(1386) = complement of X(3416)
X(1386) = crosspoint of X(1) and X(985)
X(1386) = crosssum of X(1) and X(984)

X(1387) = MIDPOINT OF INCENTER AND FEUERBACH POINT

X(1387) lies on these lines: 1,5   2,1000   7,104   30,1319   100,474   106,1086   142,214   149,377   153,1056

X(1387) = midpoint of X(i) and X(j) for these (i,j): (1,11), (80,1317), (1145,1320)
X(1387) = isogonal conjugate of X(1391)
X(1387) = inverse-in-incircle of X(80)
X(1387) = complement of X(1145)
X(1387) = crosssum of X(202) and X(203)
X(1387) = inverse-in-Feuerbach-hyperbola of X(1317)
X(1387) = {X(1),X(80)}-harmonic conjugate of X(1317)

X(1388) = POINT ANCHA

X(1388) lies on these lines: 1,3   8,1317   11,944   45,1404   73,1149   499,952   603,1339

X(1388) = isogonal conjugate of X(1392)

X(1389) = ISOGONAL CONJUGATE OF X(1385)

X(1389) lies on these lines: 1,1393   5,8   7,944   21,517   65,104   79,515   80,946   942,1476

X(1389) = isogonal conjugate of X(1385)

X(1390) = ISOGONAL CONJUGATE OF X(1386)

X(1390) lies on these lines: 37,105   38,57   81,518   278,427   279,388   984,985

X(1390) = isogonal conjugate of X(1386)
X(1390) = cevapoint of X(1) and X(984)

X(1391) = ISOGONAL CONJUGATE OF X(1387)

X(1391) lies on this line: 517,1443

X(1391) = isogonal conjugate of X(1387)
X(1391) = cevapoint of X(202) and X(203)

X(1392) = ISOGONAL CONJUGATE OF X(1388)

X(1392) lies on this line: 80,145

X(1392) = isogonal conjugate of X(1388)

Beth Conjugates 1393-1477

X(1393) = X(5)-BETH CONJUGATE OF X(5)

X(1393) lies on these lines: 1,1389   2,201   11,774   12,38   28,34   31,1454   46,602   56,244   65,1193   73,942   225,1210   227,354   278,1148   388,982   595,1421

X(1393) = crosspoint of X(57) and X(273)
X(1393) = crosssum of X(9) and X(212)

X(1394) = X(20)-BETH CONJUGATE OF X(20)

X(1394) lies on these lines: 1,84   3,223   9,478   21,77   28,34   40,109   56,269   73,991   78,651   165,227   614,1106   1104,1407   1398,1473   1420,1457

X(1394) = isogonal conjugate of the isotomic conjugate of X(33673)
X(1394) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,56), (77,57)
X(1394) = X(154)-cross conjugate of X(610)
X(1394) = cevapoint of X(221) and X(1035)
X(1394) = crossdifference of every pair of points on line X(4130)X(8611)

X(1395) = X(25)-BETH CONJUGATE OF X(25)

X(1395) lies on these lines: 4,171   24,602   25,31   28,34   32,1402   56,1472   108,727   212,573   238,459   278,985   427,750   468,748   607,1200   1106,1398   1416,1435

X(1395) = isogonal conjugate of X(3718)
X(1395) = X(34)-Ceva conjugate of X(604)
X(1395) = crosspoint of X(608) and X(1398)
X(1395) = crosssum of X(345) and X(1265)

X(1396) = X(27)-BETH CONJUGATE OF X(27)

X(1396) lies on these lines: 4,940   7,27   21,1214   28,34   108,741   223,284   269,1412   593,1014   1119,1407   1333,1427

X(1396) = X(i)-cross conjugate of X(j) for these (i,j): (1407,1412), (1408,1414), (1474,28)
X(1396) = cevapoint of X(i) and X(j) for these (i,j): (34,608), (1407,1435)

X(1397) = X(31)-BETH CONJUGATE OF X(31)

X(1397) is the vertex conjugate of the foci of the inellipse that is the isogonal conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)

X(1397) lies on these lines: 1,987   6,181   31,184   42,1404   55,572   56,58   57,985   60,959   109,727   171,182   278,1365   392,993   602,1092   603,1472   1257,1407

X(1397) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,1415), (604,32), (1408,604)
X(1397) = anticomplement of complementary conjugate of X(17053)
X(1397) = X(560)-cross conjugate of X(32)
X(1397) = crosspoint of X(i) and X(j) for these (i,j): (56,608), (59,1415), (604,1106)
X(1397) = crosssum of X(i) and X(j) for these (i,j): (8,345), (75,322), (312,341)
X(1397) = isogonal conjugate of X(3596)
X(1397) = X(8)-isoconjugate of X(75)
X(1397) = trilinear product of extraversions of X(55)

X(1398) = X(34)-BETH CONJUGATE OF X(34)

X(1398) lies on these lines: 1,1037   4,496   6,1425   25,34   28,279   184,221   278,961   388,427   475,956   604,608   607,1475   1106,1395   1254,1460   1394,1473   1407,1408

X(1398) = isogonal conjugate of X(1265)
X(1398) = anticomplement of complementary conjugate of X(17054)
X(1398) = X(i)-Ceva conjugate of X(j) for these (i,j): (1119,1407), (1435,608)
X(1398) = X(1395)-cross conjugate of X(608)
X(1398) = homothetic center of anti-Ascella triangle and anti-tangential midarc triangle

X(1399) = X(35)-BETH CONJUGATE OF X(35)

X(1399) lies on these lines: 3,47   6,1195   12,171   31,56   34,1454   35,500   55,255   58,65   201,896   213,1415   580,1155   595,1319   920,1060   1333,1409   1402,1408

X(1400) = X(37)-BETH CONJUGATE OF X(37)

X(1400) lies on these lines: 1,573   2,7   6,41   12,1213   19,208   25,31   36,572   37,65   42,181   44,583   58,1169   85,1218   108,1172   109,111   171,256   213,1042   222,967   292,694   308,349   388,966   478,603   651,1014   910,1200   1100,1319   1122,1418   1171,1412   1254,1426   1258,1432   1333,1415   1420,1449

X(1400) = isogonal conjugate of X(333)
X(1400) = isotomic conjugate of X(28660)
X(1400) = complement of X(20245)
X(1400) = anticomplement of X(21246)
X(1400) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,1409), (56,1402), (57,65), (65,42), (108,663), (226,73), (951,55), (1415,649), (1427,1042)
X(1400) = X(i)-cross conjugate of X(j) for these (i,j): (181,65), (213,42), (1402,1402)
X(1400) = cevapoint of X(213) and X(1402)
X(1400) = crosspoint of X(i) and X(j) for these (i,j): (6,19), (56,57), (65,1427), (225,226)
X(1400) = crosssum of X(i) and X(j) for these (i,j): (1,573), (2,63), (8,9), (283,284)
X(1400) = crossdifference of every pair of points on line X(522)X(663)
X(1400) = X(65)-Hirst inverse of X(1284)
X(1400) = bicentric sum of PU(18)
X(1400) = PU(18)-harmonic conjugate of X(663)
X(1400) = barycentric product of PU(81)
X(1400) = trilinear pole of line X(512)X(810)
X(1400) = X(92)-isoconjugate of X(283)
X(1400) = perspector of ABC and unary cofactor triangle of Gemini triangle 1

X(1401) = X(38)-BETH CONJUGATE OF X(38)

X(1401) lies on these lines: 7,310   43,57   51,244   56,58   65,519   226,1463   354,1122   511,982   1106,1425   1355,1365   1356,1366   1402,1458   1407,1460

X(1401) = anticomplement of complementary conjugate of X(17055)
X(1401) = crosspoint of X(7) and X(56)
X(1401) = crosssum of X(8) and X(55)

X(1402) = X(42)-BETH CONJUGATE OF X(42)

X(1402) lies on these lines: 1,3   21,961   25,1096   31,184   32,1395   42,181   73,1245   98,108   109,741   172,893   226,1284   923,1415   968,1011   1042,1410   1399,1408   1401,1458   1441,1447

X(1402) = isogonal conjugate of X(314)
X(1402) = X(i)-Ceva conjugate of X(j) for these (i,j): (56,1400), (65,1409), (961,6), (1037,73), (1400,213)
X(1402) = crosspoint of X(i) and X(j) for these (i,j): (25,31), (56,604), (1042, 1400)
X(1402) = crosssum of X(i) and X(j) for these (i,j): (8,312),. (69,75), (333,1043)
X(1402) = X(21)-isoconjugate of X(75)
X(1402) = X(92)-isoconjugate of X(1812)

X(1403) = X(43)-BETH CONJUGATE OF X(43)

X(1403) lies on these lines: 1,3   2,1284   6,893   31,1428   42,1469   43,1423   75,183   109,727   1326,1412

X(1403) = isogonal conjugate of X(7155)
X(1403) = complement of X(20557)
X(1403) = anticomplement of X(20545)
X(1403) = X(604)-Ceva conjugate of X(56)
X(1403) = X(1423)-cross conjugate of X(56)
X(1403) = X(31)-Hirst inverse of X(1428)
X(1403) = {X(13388),X(13389)}-harmonic conjugate of X(37596)

X(1404) = X(44)-BETH CONJUGATE OF X(44)

X(1404) lies on these lines: 6,41   35,572   42,1397   44,1319   57,89   59,672   217,1409   649,854   651,1429

X(1404) = X(1319)-Ceva conjugate of X(902)
X(1404) = crosspoint of X(i) and X(j) for these (i,j): (6,909), (57,1411)
X(1404) = crosssum of X(2) and X(908)
X(1404) = crossdifference of every pair of points on line X(8)X(522)

X(1405) = X(45)-BETH CONJUGATE OF X(45)

X(1405) lies on these lines: 6,41   31,51   35,573   44,65   57,88   169,1046

X(1406) = X(46)-BETH CONJUGATE OF X(46)

X(1406) lies on these lines: 3,1464   31,56   55,1066   57,1203   65,222   1411,1413   1427,1454

X(1406) = isogonal conjugate of X(36626)
X(1406) = crossdifference of every pair of points on line X(3239)X(35057)
X(1406) = X(34)-Ceva conjugate of X(56)

X(1407) = X(57)-BETH CONJUGATE OF X(57)

X(1407) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Gergonne line (with center X(4000) and perspector X(279)). (Randy Hutson, October 15, 2018)

X(1407) lies on these lines: 3,951   6,57   7,940   31,56   34,1413   55,1458   63,220   73,1466   81,279   109,1477   189,1146   278,1086   478,1122   479,1462   534,553   608,1435   614,1456   739,934   942,1448   1104,1394   1119,1396   1333,1412   1357,1397   1398,1408   1401,1460   1464,1470

X(1407) = isogonal conjugate of X(346)
X(1407) = complement of isotomic conjugate of X(34546)
X(1407) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,56), (1119,1398), (1262,1461), (1275,934), (1396,1435)
X(1407) = X(i)-cross conjugate of X(j) for these (i,j): (604,56), (608,1413), (1042,269)
X(1407) = cevapoint of X(604) and X(1106)
X(1407) = crosspoint of X(i) and X(j) for these (i,j): (57,1422), (269,738), (279,1119), (934,1275), (1262, 1461), (1396,1412)
X(1407) = crosssum of X(i) and X(j) for these (i,j): (200,728), (220,1260)
X(1407) = trilinear product X(56)*X(57)
X(1407) = X(92)-isoconjugate of X(1260)
X(1407) = perspector of ABC and unary cofactor triangle of Ayme triangle
X(1407) = barycentric square of X(57)

X(1408) = X(58)-BETH CONJUGATE OF X(58)

X(1408) lies on these lines: 21,1319   56,58   60,757   65,81   283,1037   284,1466   603,604   1398,1407   1399,1402   1413,1474

X(1408) = X(1412)-Ceva conjugate of X(1333)
X(1408) = X(604)-cross conjugate of X(1412)
X(1408) = cevapoint of X(604) and X(1397)
X(1408) = crosspoint of X(1014) and X(1396)
X(1408) = X(2)-isoconjugate of X(2321)

X(1409) = X(71)-BETH CONJUGATE OF X(71)

Let V = isotomic conjugate of polar conjugate of line X(1)X(3) and W = polar conjugate of isotomic conjugate of line X(1)X(3); then X(1409) = V∩W. (Randy Hutson, December 26, 2015)

X(1409) lies on these lines: 6,19   31,184   48,577   63,77   71,73   109,284   213,1042   217,1404   287,651   800,1195   1333,1399

X(1409) = isogonal conjugate of X(31623)
X(1409) = complement of anticomplementary conjugate of X(18667)
X(1409) = anticomplement of complementary conjugate of X(18592)
X(1409) = X(63)-isoconjugate of X(1896)
X(1409) = crossdifference of every pair of points on line X(521)X(1948)
X(1409) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,1400), (65,1402), (73,228), (222,73)
X(1409) = crosspoint of X(i) and X(j) for these (i,j): (6,48), (65,1214), (222,603)
X(1409) = crosssum of X(i) and X(j) for these (i,j): (2,92), (21,1172), (281,318)
X(1409) = X(92)-isoconjugate of X(21)
X(1409) = bicentric sum of PU(19)
X(1409) = PU(19)-harmonic conjugate of X(1946)
X(1409) = barycentric product of PU(83)
X(1409) = polar conjugate of isotomic conjugate of X(22341)

X(1410) = X(73)-BETH CONJUGATE OF X(73)

X(1410) lies on these lines: 3,77   25,34   32,604   58,1461   73,228   98,934   184,603   1042,1402

X(1410) = X(56)-Ceva conjugate of X(1042)
X(1410) = crosspoint of X(i) and X(j) for these (i,j): (56,603), (1427,1439)
X(1410) = crosssum of X(8) and X(318)

X(1411) = X(80)-BETH CONJUGATE OF X(80)

X(1411) lies on these lines: 1,5   56,244   58,65   86,664   106,1168   269,1358   388,977   996,1215   1406,1413

X(1411) = X(i)-cross conjugate of X(j) for these (i,j): (1404,57), (1457,56)
X(1411) = cevapoint of X(65) and X(1319)
X(1411) = isogonal conjugate of X(4511)
X(1411) = inverse-in-Feuerbach-hyperbola of X(1807)
X(1411) = {X(1),X(80)}-harmonic conjugate of X(1807)

X(1412) = X(81)-BETH CONJUGATE OF X(81)

X(1412) lies on these lines: 21,1420   28,1422   56,58   57,77   86,226   109,741   110,1477   269,1396   283,951   394,579   552,553   572,940   580,1092   1171,1400   1326,1402   1333,1407   1427,1461

X(1412) = isogonal conjugate of X(2321)
X(1412) = isotomic conjugate of X(30713)
X(1412) = X(1014)-Ceva conjugate of X(58)
X(1412) = X(i)-cross conjugate of X(j) for these (i,j): (56,1014), (604,1408), (1333,58), (1407,1396)
X(1412) = cevapoint of X(i) and X(j) for these (i,j): (56,604), (1333,1408)

X(1413) = X(84)-BETH CONJUGATE OF X(84)

X(1413) lies on these lines: 1,84   3,1167   6,603   34,1407   57,937   64,1364   73,939   86,285   189,1220   280,1222   998,1448   1406,1411   1408,1474

X(1413) = isogonal conjugate of X(7080)
X(1413) = X(i)-Ceva conjugate of X(j) for these (i,j): (84,56), (1422,1436)
X(1413) = X(i)-cross conjugate of X(j) for these (i,j): (608,1407), (1106,56)
X(1413) = crosspoint of X(84) and X(1256)
X(1413) = crosssum of X(40) and X(1103)

X(1414) = X(99)-BETH CONJUGATE OF X(99)

X(1414) lies on these lines: 7,757   58,1416   85,603   99,109   110,934   162,658   163,1019   348,1098   651,662   1014,1122

X(1414) = X(i)-cross conjugate of X(j) for these (i,j): (110,662), (1019,1434)
X(1414) = cevapoint of X(i) and X(j) for these (i,j): (58,1019), (109,651)
X(1414) = isogonal conjugate of X(4041)
X(1414) = isotomic conjugate of X(4086)
X(1414) = trilinear pole of line X(57)X(77)

X(1415) = X(101)-BETH CONJUGATE OF X(101)

X(1415) lies on these lines: 6,909   32,56   41,609   57,609   65,172   101,109   108,112   198,478   213,1399   571,608   604,1417   651,662   910,1455   919,934   923,1402   1055,1457   1333,1400

X(1415) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,1397), (109,692), (1262,56)
X(1415) = X(i)-cross conjugate of X(j) for these (i,j): (649,1333), (667,56), (1397,59)
X(1415) = cevapoint of X(i) and X(j) for these (i,j): (32,667), (649,1400)
X(1415) = crosspoint of X(i) and X(j) for these (i,j): (108,651), (109,1461), (112,163)
X(1415) = crosssum of X(521) and X(650)
X(1415) = crossdifference of every pair of points on line X(11)X(123)
X(1415) = barycentric product of X(1381) and X(1382)
X(1415) = isogonal conjugate of X(4391)
X(1415) = trilinear pole of line X(31)X(184)
X(1415) = X(92)-isoconjugate of X(521)
X(1415) = barycentric product of PU(102)
X(1415) = polar conjugate of isotomic conjugate of X(36059)
X(1415) = perspector of unary cofactor triangles of outer and inner Garcia triangles
X(1415) = barycentric product X(1)*X(109)
X(1415) = barycentric product X(3)*X(108)
X(1415) = trilinear product X(6)*X(109)
X(1415) = trilinear product of circumcircle intercepts of line X(6)X(41) (or of circle {{X(1),X(15),X(16)}} (V(X(1)))

X(1416) = X(105)-BETH CONJUGATE OF X(105)

X(1416) lies on these lines: 6,1362   31,57   32,56   58,1414   294,1468   727,927   738,1106   919,1458   951,1193   1357,1397   1395,1435

X(1416) = isogonal conjugate of X(3717)
X(1416) = X(1462)-Ceva conjugate of X(1438)
X(1416) = cevapoint of X(56) and X(1428)
X(1416) = X(56)-Hirst inverse of X(1438)

X(1417) = X(106)-BETH CONJUGATE OF X(106)

X(1417) lies on these lines: 56,106   65,1320   88,961   604,1415   901,1319   1014,1122   1037,1470

X(1418) = X(142)-BETH CONJUGATE OF X(142)

X(1418) lies on these lines: 6,57   7,37   44,1445   56,1279   65,1458   77,1100   142,1212   354,2293   553,1214   603,1456   942,991   1014,1333   1086,1108   1104,1448   1122,1400   1155,1253

X(1418) = X(57)-Ceva conjugate of X(1475)
X(1418) = X(1475)-cross conjugate of X(354)
X(1418) = crosspoint of X(57) and X(279)
X(1418) = crosssum of X(9) and X(220)

X(1419) = X(144)-BETH CONJUGATE OF X(144)

X(1419) lies on these lines: 1,971   6,57   7,1449   9,77   41,738   48,1461   73,991   109,1253   347,527   1201,1420   1443,1445

X(1419) = isogonal conjugate of X(19605)
X(1419) = X(1)-Ceva conjugate of X(57)
X(1419) = crosspoint of X(1) and X(165)
X(1419) = X(57)-Hirst inverse of X(910)

X(1420) = X(145)-BETH CONJUGATE OF X(145)

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically; A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. The triangle A'B'C' is homothetic to the Hutson intouch triangle at X(1420). (Randy Hutson, July 31 2018)

X(1420) lies on these lines: 1,3   9,604   21,1412   34,106   73,995   84,104   109,1106   222,1191   223,1104   226,452   269,1279   386,1450   388,1125   595,603   610,1108   738,934   936,956   944,1210   1042,1149   1201,1419   1394,1457   1400,1449

X(1420) = isogonal conjugate of X(3680)
X(1420) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,57), (765,109)
X(1420) = {X(1),X(56)}-harmonic conjugate of X(57)
X(1420) = {X(3513),X(3514)}-harmonic conjugate of X(165)
X(1420) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles
X(1420) = {X(1),X(55)}-harmonic conjugate of X(37556)

X(1421) = X(149)-BETH CONJUGATE OF X(149)

X(1421) lies on these lines: 1,5   31,57   34,106   595,1393   1279,1465

X(1422) = X(189)-BETH CONJUGATE OF X(189)

X(1422) lies on these lines: 1,84   2,77   28,1412   34,1256   35,1079   57,1436   268,1214   269,278   271,1257   280,1219

X(1422) = isogonal conjugate of X(2324)
X(1422) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,57), (1440,84)
X(1422) = X(i)-cross conjugate of X(j) for these (i,j): (34,269), (1407,57), (1436,84)
X(1422) = cevapoint of X(1413) and X(1436)

X(1423) = X(192)-BETH CONJUGATE OF X(192)

X(1423) lies on these lines: 1,256   2,7   6,1429   43,1403   56,87   65,984   85,1221   241,1122   269,292   604,651   1201,1419

X(1423) = isogonal conjugate of X(2319)
X(1423) = isotomic conjugate of X(27424)
X(1423) = complement of X(20348)
X(1423) = anticomplement of X(20258)
X(1423) = X(56)-Ceva conjugate of X(57)
X(1423) = crosspoint of X(56) and X(1403)
X(1423) = X(6)-Hirst inverse of X(1429)

X(1424) = X(194)-BETH CONJUGATE OF X(194)

X(1424) lies on these lines: 56,87   57,85   222,1429

X(1424) = X(604)-Ceva conjugate of X(57)

X(1425) = X(201)-BETH CONJUGATE OF X(201)

X(1425) lies on these lines: 1,185   6,1398   12,125   25,221   34,51   55,1204   56,184   65,225   72,307   73,228   181,1254   213,1042   217,1015   999,1181   1093,1148   1106,1401

X(1425) = X(i)-Ceva conjugate of X(j) for these (i,j): (65,1254), (1020,647)
X(1425) = crosspoint of X(65) and X(73)
X(1425) = crosssum of X(21) and X(29)

X(1426) = X(225)-BETH CONJUGATE OF X(225)

X(1426) lies on these lines: 4,7   25,34   65,225   72,860   226,429   227,228   278,959   517,1068   1254,1400

X(1426) = isogonal conjugate of X(1792)
X(1426) = X(34)-Ceva conjugate of X(1042)
X(1426) = crosspoint of X(i) and X(j) for these (i,j): (34,1118), (1119,1435)
X(1426) = crosssum of X(78) and X(1259)

X(1427) = X(226)-BETH CONJUGATE OF X(226)

X(1427) lies on these lines: 2,85   3,1448   6,57   7,941   25,34   31,1456   37,226   42,65   77,940   111,934   212,1155   278,393   307,1211   354,1458   581,942   1014,1169   1106,1451   1333,1396   1406,1454   1412,1461

X(1427) = isogonal conjugate of X(2287)
X(1427) = complement of X(18750)
X(1427) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,1042), (1446,1439)
X(1427) = X(i)-cross conjugate of X(j) for these (i,j): (1400,65), (1410,1439)
X(1427) = cevapoint of X(1042) and X(1400)
X(1427) = crosspoint of X(i) and X(j) for these (i,j): (57,278), (269, 279)
X(1427) = crosssum of X(i) and X(j) for these (i,j): (6,610), (9,219), (200,220)

X(1428) = X(238)-BETH CONJUGATE OF X(238)

X(1428) lies on these lines: 1,182   3,613   6,41   31,1403   36,511   57,985   58,1178   59,518   60,757   65,82   184,614   238,1284   499,1352   611,999   651,1463   692,1279   961,1258   1456,1462

X(1428) = X(1416)-Ceva conjugate of X(56)
X(1428) = crosspoint of X(1014) and X(1462)
X(1428) = X(i)-Hirst inverse of X(j) for these (i,j): (31,1403), (56,604)

X(1429) = X(239)-BETH CONJUGATE OF X(239)

X(1429) lies on these lines: 1,3   6,1423   7,604   73,1244   81,1432   83,226   222,1424   238,1284   239,385   552,553   651,1404   1458,1462

X(1429) = isogonal conjugate of X(4876)
X(1429) = X(i)-Ceva conjugate of X(j) for these (i,j): (1447,238), (1462,57)
X(1429) = X(1284)-cross conjugate of X(1447)
X(1429) = crossdifference of every pair of points on line X(210)X(650)
X(1429) = X(i)-Hirst inverse of X(j) for these (i,j): (6,1423), (56,57)

X(1430) = X(243)-BETH CONJUGATE OF X(243)

X(1430) lies on these lines: 1,1013   4,1468   25,34   27,58   31,278   57,1096   92,171   108,1458   162,238   281,750   603,1118

X(1430) = X(34)-Hirst inverse of X(56)

X(1431) = X(256)-BETH CONJUGATE OF X(256)

X(1431) lies on these lines: 1,256   6,893   7,870   56,904   57,87   58,1178   65,257   86,1447   292,694   518,1222   758,996   979,1046

X(1431) = X(904)-cross conjugate of X(893)

X(1432) = X(257)-BETH CONJUGATE OF X(257)

X(1432) lies on these lines: 1,256   2,257   7,330   28,1178   56,985   57,893   65,291   81,1429   105,904   961,1042   1258,1400

X(1432) = isogonal conjugate of X(2329)
X(1432) = X(893)-cross conjugate of X(256)
X(1432) = cevapoint of X(893) and X(1431)
X(1432) = anticomplement of complementary conjugate of X(17062)

X(1433) = X(271)-BETH CONJUGATE OF X(271)

X(1433) lies on these lines: 1,84   6,282   29,81   55,947   56,102   78,271   145,280   219,255   284,1436   945,999

X(1433) = isogonal conjugate of X(7952)
X(1433) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,1436), (271,3), (285,84)
X(1433) = X(i)-cross conjugate of X(j) for these (i,j): (6,222), (603,3)
X(1433) = cevapoint of X(1364) and X(1459)
X(1433) = X(92)-isoconjugate of X(198)

X(1434) = X(274)-BETH CONJUGATE OF X(274)

X(1434) lies on these lines: 7,21   27,1088   57,85   58,1414   65,664   81,279   99,1477   270,757   310,349   332,951   552,553   658,1446   576,1475\

X(1434) = isogonal conjugate of X(1334)
X(1434) = isotomic conjugate of X(2321)
X(1434) = anticomplement of X(38930)
X(1434) = anticomplementary conjugate of anticomplement of X(38811)
X(1434) = X(552)-Ceva conjugate of X(1014)
X(1434) = X(i)-cross conjugate of X(j) for these (i,j): (57,1014), (81,86), (553,7), (1019,1414)
X(1434) = cevapoint of X(i) and X(j) for these (i,j): (7,57), (81,1014)

X(1435) = X(278)-BETH CONJUGATE OF X(278)

X(1435) lies on these lines: 1,951   19,57   25,34   27,1088   33,354   48,223   108,1477   154,1456   244,1096   269,1396   608,1407   913,1461   1395,1416

X(1435) = isogonal conjugate of X(3692)
X(1435) = polar conjugate of X(341)
X(1435) = X(i)-Ceva conjugate of X(j) for these (i,j): (1119,34), (1396,1407)
X(1435) = X(i)-cross conjugate of X(j) for these (i,j): (608,34), (1106,269), (1426,1119)
X(1435) = cevapoint of X(608) and X(1398)

X(1436) = X(282)-BETH CONJUGATE OF X(282)

X(1436) lies on these lines: 3,9   6,603   19,56   48,55   57,1422   189,333   284,1433   673,1440

X(1436) = isogonal conjugate of X(329)
X(1436) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,1433), (282,6), (1422,1413)
X(1436) = X(i)-cross conjugate of X(j) for these (i,j): (25,56), (604,6)
X(1436) = crosspoint of X(84) and X(1422)
X(1436) = crosssum of X(9) and X(1490)

X(1437) = X(283)-BETH CONJUGATE OF X(283)

X(1437) lies on these lines: 3,49   21,104   28,60   35,692   48,255   56,58   163,911   182,474   215,1364   284,1433   849,1333   1014,1175

X(1437) = X(i)-Ceva conjugate of X(j) for these (i,j): (60,58), (81,1333)
X(1437) = X(603)-cross conjugate of X(58)
X(1437) = cevapoint of X(48) and X(184)
X(1437) = crosspoint of X(81) and X(1444)
X(1437) = crosssum of X(4) and X(451)
X(1437) = X(4)-isoconjugate of X(10)
X(1437) = crosspoint of X(1805) and X(1806)

X(1438) = X(294)-BETH CONJUGATE OF X(294)

X(1438) lies on these lines: 1,41   6,692   32,56   58,163   86,142   87,572   106,919   269,604   665,911   909,1024   950,1220

X(1438) = X(1462)-Ceva conjugate of X(1416)
X(1438) = crosspoint of X(105) and X(1462)
X(1438) = X(i)-Hirst inverse of X(j) for these (i,j): (56,1416)
X(1438) = isogonal conjugate of X(3912)
X(1438) = trilinear pole of PU(48) (line X(31)X(649))
X(1438) = barycentric product of PU(96)
X(1438) = barycentric product X(1)*X(105)
X(1438) = trilinear product X(6)*X(105)
X(1438) = crossdifference of every pair of points on line X(918)X(2254)
X(1438) = polar conjugate of isotomic conjugate of X(36057)
X(1438) = X(63)-isoconjugate of X(1861)

X(1439) = X(307)-BETH CONJUGATE OF X(307)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(1439). (Randy Hutson, September 14, 2016)

X(1439) lies on these lines: 1,64   3,77   4,7   6,57   37,1020   54,1443   71,1214   72,307   74,934   86,658   241,579   284,1461   347,517   1014,1175   1042,1245   1088,1246

X(1439) = X(i)-Ceva conjugate of X(j) for these (i,j): (658,905), (1446,1427)
X(1439) = X(i)-cross conjugate of X(j) for these (i,j): (73,1214), (656,1020), (1410,1427)
X(1439) = crosspoint of X(7) and X(77)
X(1439) = crosssum of X(i) and X(j) for these (i,j): (24,204), (33,55)
X(1439) = isogonal conjugate of X(4183)
X(1439) = perspector of intouch triangle and 3rd extouch triangle
X(1439) = perspector of ABC and cross-triangle of ABC and 3rd extouch triangle
X(1439) = trilinear product of Jerabek hyperbola intercepts of Soddy line

X(1440) = X(309)-BETH CONJUGATE OF X(309)

X(1440) lies on these lines: 2,77   7,84   27,1014   75,280   86,285   269,1256   271,307   273,279   673,1436

X(1440) = isogonal conjugate of X(7074)
X(1440) = isotomic conjugate of X(7080)
X(1440) = polar conjugate of isotomic conjugate of X(34400)
X(1440) = X(i)-cross conjugate of X(j) for these (i,j): (84,189), (269,7), (278,279)
X(1440) = cevapoint of X(84) and X(1422)

X(1441) = X(313)-BETH CONJUGATE OF X(313)

In the plane of a triangle ABC, let
I = incenter = X(1)
O = circumcenter = X(3)
DEF = cevian triangle of I
L_b = line through B perpendicular to AI
L_c = line through C perpendicular to AI
A_b = L_b∩AO
A_c = L_c∩AO)
O_a = circumcircle of DA_bA_c
A' = the point of intersection, other than D of O_a and AI, and define B' and C' cyclically. Then X(1441) = finite fixed point of the affine transformation that carries ABC onto A'B'C". (Angel Montesdeoca, May 30, 2023)

X(1441) lies on these lines: 2,92   7,8   10,307   12,313   19,379   21,286   34,964   57,1150   86,664   95,404   226,306   253,318   269,996   274,961   287,651   305,561   443,1119   1074,1111   1402,1447

X(1441) = isogonal conjugate of X(2194)
X(1441) = isotomic conjugate of X(21)
X(1441) = complement of polar conjugate of isogonal conjugate of X(23171)
X(1441) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,307), (85,226), (349,321), (664,693)
X(1441) = X(i)-cross conjugate of X(j) for these (i,j): (10,321), (12,226), (226,1446), (442,2), (121,76), (1214,1231)
X(1441) = cevapoint of X(i) and X(j) for these (i,j): (10,226), (65,1214)
X(1441) = crosspoint of X(75) and X(264)
X(1441) = crosssum of X(31) and X(184)
X(1441) = polar conjugate of X(1172)

X(1442) = X(319)-BETH CONJUGATE OF X(319)

X(1442) lies on these lines: 1,7   2,914   8,326   37,651   65,1014   74,934   81,1214   86,664   226,1029   241,1100   319,1273   1082,1250   1445,1449

X(1443) = X(320)-BETH CONJUGATE OF X(320)

X(1443) lies on these lines: 1,7   44,241   54,1439   57,89   59,1155   60,757   88,1465   320,1464   679,1318   934,953   1419,1445

X(1443) = anticomplement of X(1489)
X(1443) = cevapoint of X(1319) and X(1465)
X(1443) = crossdifference of every pair of points on line X(657)X(1334)

X(1444) = X(332)-BETH CONJUGATE OF X(332)

X(1444) lies on these lines: 1,969   3,69   7,21   28,242   48,63   58,988   71,1332   77,283   81,593   99,104   100,319   189,333   524,1030   662,911   963,1043

X(1444) = isogonal conjugate of X(1824)
X(1444) = X(i)-Ceva conjugate of X(j) for these (i,j): (261,86), (274,81)
X(1444) = X(i)-cross conjugate of X(j) for these (i,j): (77,86), (1437,58), (1459,1332), (1473,58)
X(1444) = cevapoint of X(3) and X(63)
X(1444) = X(92)-isoconjugate of X(213)

X(1445) = X(344)-BETH CONJUGATE OF X(344)

X(1445) lies on these lines: 1,1170   2,7   6,77   19,273   40,390   44,1418   46,516   56,78   65,1001   169,1446   269,651   942,954   1038,1451   1419,1443   1442,1449

X(1445) = X(i)-Ceva conjugate of X(j) for these (i,j): (765,651), (1088,1)
X(1445) = {X(9),X(57)}-harmonic conjugate of X(7)

X(1446) = X(349)-BETH CONJUGATE OF X(349)

X(1446) lies on these lines: 2,85   4,7   10,307   57,379   76,1229   98,934   169,1445   226,857   294,1170   321,349   658,1434   1111,1210

X(1446) = isotomic conjugate of X(2287)
X(1446) = X(226)-cross conjugate of X(1441)
X(1446) = cevapoint of X(1427) and X(1439)
X(1446) = crosspoint of X(85) and X(331)
X(1446) = polar conjugate of X(4183)

X(1447) = X(350)-BETH CONJUGATE OF X(350)

X(1447) lies on these lines: 2,7   25,273   36,1111   56,85   75,183   77,614   86,1431   87,269   105,927   230,1086   239,385   241,292   261,552   320,325   350,1281   459,1119   664,1319   673,910   1402,1441

X(1447) = isogonal conjugate of X(7077)
X(1447) = isotomic conjugate of X(4518)
X(1447) = X(i)-cross conjugate of X(j) for these (i,j): (238,239), (1284,1429)
X(1447) = cevapoint of X(i) and X(j) for these (i,j): (238,1429), (241,1463)
X(1447) = crossdifference of every pair of points on line X(663)X(1334)
X(1447) = X(7)-Hirst inverse of X(57)

X(1448) = X(377)-BETH CONJUGATE OF X(377)

X(1448) lies on these lines: 1,7   3,1427   28,34   46,255   65,222   85,1010   223,386   226,975   241,405   443,948   942,1407   998,1413   1104,1418   1465,1466

X(1449) = {X(1),X(6)}-HARMONIC CONJUGATE OF X(9)

X(1449) lies on these lines: 1,6   7,1419   32,988   34,1172   40,572   43,1051   57,77   65,380   87,1045   198,999   579,1475   610,942   894,1278   966,1125   1400,1420   1442,1445

X(1449) = isogonal conjugate of X(25430)
X(1449) = anticomplement of X(32099)
X(1449) = {X(1),X(6)}-harmonic conjugate of X(9)
X(1449) = X(391)-beth conjugate of X(391)
X(1449) = {X(1),X(9)}-harmonic conjugate of X(3247)

X(1450) = X(392)-BETH CONJUGATE OF X(392)

X(1450) lies on these lines: 1,631   3,1057   6,41   31,1470   36,1064   42,1319   57,957   65,244   386,1420   388,978   1191,1466

X(1451) = X(405)-BETH CONJUGATE OF X(405)

X(1451) lies on these lines: 1,201   6,41   28,34   31,65   36,581   46,601   255,942   270,273   1038,1445   1106,1427

X(1452) = X(406)-BETH CONJUGATE OF X(406)

X(1452) lies on these lines: 1,24   4,46   19,208   25,65   28,34   33,40   227,607   1038,1039

X(1453) = X(452)-BETH CONJUGATE OF X(452)

X(1453) lies on these lines: 1,6   4,204   28,34   31,40   43,1009   56,223   73,995   84,1039   212,595   222,1467   387,950   581,1193   614,1468

X(1454) = X(498)-BETH CONJUGATE OF X(498)

X(1454) lies on these lines: 1,3   5,920   12,63   31,1393   34,1399   90,381   201,750   208,407   453,1014   603,1254   1406,1427

X(1455) = X(515)-BETH CONJUGATE OF X(515)

X(1455) lies on these lines: 1,84   3,227   25,34   36,1465   37,478   65,603   73,820   109,517   117,515   513,663   608,1108   910,1415   958,1038   993,1214

X(1455) = X(104)-Ceva conjugate of X(56)
X(1455) = crosspoint of X(1) and X(1295)

X(1456) = X(516)-BETH CONJUGATE OF X(516)

X(1456) lies on these lines: 1,971   6,19   7,1386   31,1427   55,223   56,269   77,1001   109,1155   154,1435   222,354   227,1253   238,241   513,663   518,651   603,1418   614,1407   1042,1104   1428,1462

X(1456) = X(105)-Ceva conjugate of X(56)
X(1456) = crosspoint of X(i) and X(j) for these (i,j): (1,972), (269,1462)
X(1456) = crosssum of X(1) and X(971)
X(1456) = crossdifference of every pair of points on line X(9)X(521)

X(1457) = X(517)-BETH CONJUGATE OF X(517)

X(1457) lies on these lines: 1,4   3,945   31,56   36,109   48,608   57,957   65,1193   201,960   222,999   350,664   392,1214   478,604   513,663   517,1465   1055,1415   1394,1420

X(1457) = X(106)-Ceva conjugate of X(56)
X(1457) = crosspoint of X(i) and X(j) for these (i,j): (1,102), (56,1411)
X(1457) = crosssum of X(1) and X(515)
X(1457) = crossdifference of every pair of points on line X(9)X(652)
X(1457) = intersection of tangents at X(1) and X(102) to hyperbola {{A,B,C,X(1),X(3),X(29),X(102)}}

X(1458) = X(518)-BETH CONJUGATE OF X(518)

Let A'B'C' and A"B"C" be the circumcevian triangles of X(3513) and X(3514), resp. Let A* be the trilinear product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(1458). (Randy Hutson, June 7, 2019)

X(1458) lies on these lines: 1,7   3,1037   6,41   31,222   36,59   38,1214   42,57   55,1407   64,963   65,1418   108,1430   109,840   185,1208   223,614   238,651   241,518   244,1465   256,1476   354,1427   513,663   672,1362   919,1416   942,1254   976,1038   999,1064   1201,1419   1401,1402   1429,1462

X(1458) = isogonal conjugate of X(14942)
X(1458) = X(i)-Ceva conjugate of X(j) for these (i,j): (241,672), (1477,56)
X(1458) = crosspoint of X(i) and X(j) for these (i,j): (1,103)
X(1458) = crosssum of X(1) and X(516)
X(1458) = crossdifference of every pair of points on line X(9)X(522)
X(1458) = X(i)-Hirst inverse of X(j) for these (i,j): (6,56), (6,72)
X(1458) = perspector of conic {A,B,C,X(109),PU(48)}
X(1458) = crossdifference of the isogonal conjugates of PU(48)

X(1459) = X(521)-BETH CONJUGATE OF X(521)

X(1459) lies on these lines: 1,522   6,657   106,953   242,514   513,663   520,647   521,656   649,834

X(1459) = reflection of X(656) in X(905)
X(1459) = isogonal conjugate of X(1897)
X(1459) = complement of X(20293)
X(1459) = perspector of hyperbola {{A,B,C,X(3),X(57),X(103)})
X(1459) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39006), (101,1473), (109,603), (514,649), (905,652), (1331,3), (1332,71), (1433,1364), (1461,6)
X(1459) = X(647)-cross conjugate of X(905)
X(1459) = cevapoint of X(647) and X(810)
X(1459) = crosspoint of X(i) and X(j) for these (i,j): (1,109), (3,1331), (81,934), (1332,1444)
X(1459) = crosssum of X(1) and X(522)
X(1459) = crossdifference of every pair of points on line X(4)X(9)
X(1459) = X(92)-isoconjugate of X(101)
X(1459) = orthojoin of X(1146)

X(1460) = X(612)-BETH CONJUGATE OF X(612)

X(1460) lies on these lines: 1,3   6,181   8,961   25,31   42,604   222,1469   332,1014   388,1010   959,1036   1254,1398   1401,1407

X(1460) = isogonal conjugate of X(30479)
X(1460) = crossdifference of every pair of points on line X(650)X(3910)

X(1461) = X(651)-BETH CONJUGATE OF X(651)

X(1461) lies on these lines: 6,911   48,1419   57,909   58,1410   77,572   101,651   109,692   269,604   284,1439   514,653   658,662   913,1435   923,1042   1025,1332   1412,1427

X(1461) = isogonal conjugate of X(3239)
X(1461) = X(i)-Ceva conjugate of X(j) for these (i,j): (934,109), (1262,1407)
X(1461) = X(i)-cross conjugate of X(j) for these (i,j): (649,56), (1407,1262), (1415,109)
X(1461) = cevapoint of X(i) and X(j) for these (i,j): (6,1459), (56,649)
X(1461) = crossdifference of every pair of points on line X(1146)X(2310)
X(1461) = barycentric product of circumcircle intercepts of the Soddy line
X(1461) = trilinear pole of line X(31)X(56) (the isogonal conjugate of the isotomic conjugate of the Soddy line)

X(1462) = X(673)-BETH CONJUGATE OF X(673)

X(1462) lies on these lines: 6,7   31,57   213,1170   241,1279   269,604   479,1407   608,1119   739,927   919,1465   934,1015   1014,1333   1428,1456   1429,1458

X(1462) = isogonal conjugate of X(3693)
X(1462) = X(i)-cross conjugate of X(j) for these (i,j): (1428,1014), (1438,105), (1456,269)
X(1462) = cevapoint of X(i) and X(j) for these (i,j): (6,1279), (57,1429), (1416,1438)
X(1462) = X(i)-Hirst inverse of X(j) for these (i,j): (57,105)
X(1462) = trilinear pole of line X(56)X(667)
X(1462) = barycentric product of circumcircle intercepts of line X(7)X(513)

X(1463) = X(726)-BETH CONJUGATE OF X(726)

X(1463) lies on these lines: 7,8   44,583   56,87   181,553   226,1401   513,663   651,1428

X(1463) = isogonal conjugate of X(8851)
X(1463) = X(1447)-Ceva conjugate of X(241)
X(1463) = {P,Q}-harmonic conjugate of X(65), where P and Q are the incircle intercepts of line X(7)X(8)

X(1464) = X(758)-BETH CONJUGATE OF X(758)

X(1464) lies on these lines: 1,30   3,1406   42,65   56,58   320,1443   354,1064   513,663   1407,1470

X(1464) = crosspoint of X(1) and X(74)
X(1464) = crosssum of X(1) and X(30)
X(1464) = crossdifference of every pair of points on line X(9)X(1021)
X(1464) = PU(86)-harmonic conjugate of X(9404)
X(1464) = isogonal conjugate of X(6740)

X(1465) = X(908)-BETH CONJUGATE OF X(908)

X(1465) lies on these lines: 1,227   2,92   3,34   5,225   6,57   36,1455   46,221   56,998   65,386   73,942   88,1443   106,1168   109,1155   241,514   244,1458   474,1038   517,1457   919,1462   1193,1254   1279,1421   1448,1466

X(1465) = isogonal conjugate of complement of X(36918)
X(1465) = isotomic conjugate of X(36795)
X(1465) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1361), (88,57), (1443,1319)
X(1465) = X(1361)-cross conjugate of X(7)
X(1465) = trilinear pole of line X(1361)X(1769)
X(1465) = pole wrt Stevanovic circle of line X(4)X(9)

X(1466) = X(936)-BETH CONJUGATE OF X(936)

X(1466) lies on these lines: 1,3   6,603   7,404   12,443   42,1106   73,1407   221,1193   222,386   226,474   284,1408   388,1376   1012,1210   1191,1450   1448,1465

X(1466) = {X(1),X(57)}-harmonic conjugate of X(37566)
X(1466) = {X(3),X(57)}-harmonic conjugate of X(56)

X(1467) = X(938)-BETH CONJUGATE OF X(938)

X(1467) lies on these lines: 1,3   7,452   28,1412   34,269   142,388   207,278   222,1453   604,610   614,1042   1066,1103   1104,1394

X(1467) = {X(57),X(1420)}-harmonic conjugate of X(3)

X(1468) = X(958)-BETH CONJUGATE OF X(958)

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_a^2 + s^2), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb ∩ Lc, B' = Lc ∩ La, C' = La ∩ Lb. The lines AA', BB', CC' concur in X(1468). (Randy Hutson, April 9, 2016)

X(1468) lies on these lines: 1,21   3,42   4,1430   6,41   8,171   10,750   36,386   43,404   57,961   65,603   75,757   222,1042   294,1416   330,985   354,1104   474,899   517,601   518,976   602,1385   614,1453   748,1125   756,975   940,958   995,1203   999,1201   1149,1191

X(1468) = isogonal conjugate of X(31359)
X(1468) = crosssum of X(9) and X(612)
X(1468) = crossdifference of every pair of points on line X(522)X(661)

X(1469) = X(984)-BETH CONJUGATE OF X(984)

X(1469) lies on these lines: 1,256   3,611   6,41   7,8   12,141   36,182   42,1403   43,57   51,614   55,1350   109,753   193,330   222,1460   613,999   751,1319   970,888

X(1469) = reflection of X(3056) in X(1)
X(1469) = X(1350)-of-Mandart-incircle-triangle
X(1469) = homothetic center of intangents triangle and reflection of extangents triangle in X(1350)

X(1470) = X(997)-BETH CONJUGATE OF X(997)

X(1470) lies on these lines: 1,3   6,909   11,1012   12,474   31,1450   73,1106   108,378   109,995   388,404   603,1193   1037,1417   1407,1464

X(1470) = {X(55),X(56)}-harmonic conjugate of X(1319)

X(1471) = X(1001)-BETH CONJUGATE OF X(1001)

Let A₁B₁C₁ and A₂B₂C₂ be the 1st and 2nd Auriga triangles. Let A' be the trilinear product A1*A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1471). (Randy Hutson, March 21, 2019)

X(1471) lies on these lines: 1,1170   6,41   7,238   31,57   36,991   58,269   65,1279   212,354   226,748   241,1386   307,1125   602,942   603,1418

X(1472) = X(1036)-BETH CONJUGATE OF X(1036)

X(1472) lies on these lines: 3,31   28,614   32,48   56,1395   58,988   104,1039   238,987   595,997   603,1397   727,1310

X(1472) = isogonal conjugate of X(4385)

X(1473) = X(1040)-BETH CONJUGATE OF X(1040)

X(1473) lies on these lines: 3,63   25,57   31,56   38,55   184,222   197,1155   198,672   988,1036   1394,1398

X(1473) = X(101)-Ceva conjugate of X(1459)
X(1473) = crosspoint of X(58) and X(1444)

X(1474) = X(1172)-BETH CONJUGATE OF X(1172)

X(1474) lies on these lines: 1,19   4,572   6,25   24,573   27,86   29,1220   34,604   56,608   106,112   163,913   198,939   269,1396   281,996   286,870   459,966   468,1213   1408,1413

X(1474) = isogonal conjugate of X(306)
X(1474) = X(27)-Ceva conjugate of X(58)
X(1474) = X(604)-cross conjugate of X(608)
X(1474) = cevapoint of X(604) and X(608)
X(1474) = crosspoint of X(28) and X(1396)
X(1474) = crossdifference of every pair of points on line X(525)X(656)
X(1474) = polar conjugate of X(313)

X(1475) = X(1212)-BETH CONJUGATE OF X(1212)

X(1475) lies on these lines: 1,672   6,41   39,42   57,279   58,163   71,583   213,1015   218,999   354,1212   579,1449   607,1398   649,764   673,1434   934,1170

X(1475) = isogonal conjugate of X(32008)
X(1475) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,1418), (692,649), (934,663)
X(1475) = crosspoint of X(i) and X(j) for these (i,j): (6,57), (354,1418)
X(1475) = crosssum of X(2) and X(9)
X(1475) = crossdifference of every pair of points on line X(522)X(3935)
X(1475) = polar conjugate of isotomic conjugate of X(22053)

X(1476) = X(1222)-BETH CONJUGATE OF X(1222)

X(1476) lies on these lines: 1,1106   3,1000   4,496   8,56   9,604   21,1319   65,1320   80,1210   172,294   256,1458   314,1014   651,1201   942,1389   943,1385

X(1476) = isogonal conjugate of X(3057)
X(1476) = isotomic conjugate of X(20895)
X(1476) = X(i)-cross conjugate of X(j) for these (i,j): (15,1222), (649,651)
X(1476) = cevapoint of X(1) and X(56)

X(1477) = X(1280)-BETH CONJUGATE OF X(1280)

X(1477) lies on the circumcircle and these lines: 1,1292   55,1293   56,101   57,100   99,1434   108,1435   109,1407   110,1412   738,934   919,1416   1308,1319

X(1477) = isogonal conjugate of X(5853)
X(1477) = Λ(X(8), X(9))
X(1477) = Ψ(X(190), X(7))
X(1477) = X(672)-cross conjugate of X(57)
X(1477) = cevapoint of X(56) and X(1458)
X(1477) = trilinear product of circumcircle intercepts of line X(1)X(3309)

X(1478) = CENTER OF JOHNSON-YFF CIRCLE

The Johnson Circle Theorem is the fact that if three congruent circles intersect in a point, then the circle passing through the other three intersections is congruent to them. This fourth circle is the Johnson circle of the three given circles. There are three congruent circles each tangent to two sides of triangle ABC. Peter Yff proved that their Johnson circle has center X(1478). The circle is here named the Johnson-Yff circle of the triangle.

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(1478) is the point C on page 5. See also X(495)-X(499).

X(1478) lies on these lines: 1,4   2,36   3,12   5,56   7,80   8,79   10,46   11,381   13,203   14,202   30,55   65,68   119,1470   148,192   442,958   474,1329   496,546   529,956   612,1370   908,997   975,1076   990,1074   1352,1469

X(1478) = reflection of X(i) in X(j) for these (i,j): (1,226), (55,495), (63,10)
X(1478) = isogonal conjugate of X(3422)
X(1478) = anticomplement of X(993)
X(1478) = X(1065)-Ceva conjugate of X(1)
X(1478) = homothetic center of 2nd isogonal triangle of X(1) and anticomplementary triangle; see X(36)
X(1478) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1478) = homothetic center of 2nd isogonal triangle of X(1) and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1478) = homothetic center of Johnson triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(1478) = homothetic center of Ehrmann mid-triangle and 2nd Johnson-Yff triangle

X(1479) = {X(1),X(4)}-HARMONIC CONJUGATE OF X(1478)

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(1479) is the point C' on page 5. See also X(495)-X(499).

X(1479) lies on these lines: 1,4   2,35   3,11   5,55   7,79   8,80   12,381   20,36   30,56   46,516   63,90   148,330   156,215   315,350   377,1125   382,999   387,1203   442,1001   495,546   528,1329   614,1370   1387,1388

X(1479) = reflection of X(i) in X(j) for these (i,j): (46,1210), (56,496)
X(1479) = X(1067)-Ceva conjugate of X(1)
X(1479) = homothetic center of intangents triangle and reflection of tangential triangle in X(5)
X(1479) = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(1479) = homothetic center of Ehrmann mid-triangle and 1st Johnson-Yff triangle
X(1479) = Ursa-major-to-Ursa-minor similarity image of X(3)
X(1479) = homothetic center of 2nd isogonal triangle of X(1) and the reflection of the anticomplementary triangle in X(4); see X(36)

X(1480) = 1st SHADOW POINT

Antreas P. Hatzipolakis and Paul Yiu, Pedal Triangles and Their Shadows, Forum Geometricorum 1 (2001) 81-90. (X(1480) is point M on page 88. See X(6580) for the 2nd Shadow Point.

X(1480) lies on these lines: 1,1406   3,902   6,517   651,1000

X(1480) = X(1)-Ceva conjugate of X(999)

X(1481) = POINT ASTEROPE

(For the 2nd Shadow Point, see X(6580).

X(1481) lies on these lines: {1,1406}, {2316,4253}

X(1482) = REFLECTION OF CIRCUMCENTER IN INCENTER

X(1482) lies on these lines: 1,3   4,145   5,8   30,944   104,1392   355,381   382,515   518,1351

X(1482) = midpoint of X(i) and X(j) for these (i,j): (4,145), (944,962)
X(1482) = reflection of X(i) in X(j) for these (i,j): (3,1), (8,5), (40,1385), (355,946), (944,1483)
X(1482) = {X(1),X(3)}-harmonic conjugate of X(10246)
X(1482) = {X(1),X(40)}-harmonic conjugate of X(1385)
X(1482) = center of circle that is the poristic locus of X(4)
X(1482) = Johnson-isogonal conjugate of X(37821)

X(1483) = REFLECTION OF X(5) IN X(1)

X(1483) lies on these lines: 1,5   3,145   8,140   10,632   30,944   40,548   517,550   518,1353   519,549

X(1483) = midpoint of X(i) and X(j) for these (i,j): (3,145), (944,1482)
X(1483) = reflection of X(i) in X(j) for these (i,j): (5,1), (8,140)
X(1483) = X(5)-of-5th-mixtilinear-triangle

X(1484) = REFLECTION OF X(5) IN X(11)

X(1484) lies on these lines: 1,5   3,149   30,104   100,140   153,381   528,549

X(1484) = midpoint of X(3) and X(149)
X(1484) = reflection of X(i) in X(j) for these (i,j): (5,11), (100,140)

X(1485) = PERSPECTOR OF ABC AND TANGENTIAL-OF-TANGENTIAL TRIANGLE

X(1485) = X(1486)-of-tangential triangle if ABC is acute. (Darij Grinberg, 5/24/03)

X(1486) = PERSPECTOR OF TANGENTIAL AND INTOUCH TRIANGLES

Let A'B'C' be the intouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1486).

X(1486) lies on these lines: 1,159   3,142   6,692   19,25   56,1279   100,344   219,674   354,1473   513,1037

X(1486) = isogonal conjugate of X(13577)
X(1486) = complement of X(11677)
X(1486) = anticomplement of X(23305)
X(1486) = X(7)-Ceva conjugate of X(6)
X(1486) = crosssum of X(116) and X(522)
X(1486) = crossdifference of every pair of points on line X(905)X(918)
X(1486) = X(173)-of-the-tangential-triangle if ABC is acute; see note at X(1485)
X(1486) = pole, with respect to circumcircle, of the Gergonne line

X(1487) = NAPOLEON CEVAPOINT

Let N denote the nine-point center, X(5). Let N_A = N-of-triangle NBC, and define N_B and N_C cyclically. Triangle N_AN_BN_C is perspective to ABC, and X(1487) is the perspector. X(1487) is the cevapoint of the Napoleon points, X(17) and X(18). (Coordinates found by Paul Yiu.)

The construction just given for X(1487) shows that it is a solution X of the "four-triangle problem" posed in

C. Kimberling, "Triangle centers as functions," Rocky Mountain Journal of Mathematics 23 (1993) 1269-1286. See Section 5; a complete solution to the problem remains to be found.

X(1487) lies on these lines: 4,252   5,1173   140,930

X(1487) = isogonal conjugate of X(1493)
X(1487) = cevapoint of X(17) and X(18)
X(1487) = X(523)-cross conjugate of X(930)
X(1487) = Kosnita(X(5),X(5)) point

X(1488) = 2nd STEVANOVIC POINT

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let A" = IA'∩BC, and define B", C" cyclically. The lines AA", BB", CC" concur in X(1488). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1130) and X(1489).)

X(1488) lies on these lines: 1,166   7,2089   57,173   145,188   557,1274   558,1143

X(1488) = X(1)-cross conjugate of X(174)
X(1488) = SS(A→A') of X(7), where A'B'C' is the excentral triangle
X(1488) = trilinear pole of Monge line of incircles of BCI, CAI, ABI
X(1488) = trilinear pole of Monge line of I-excircles of BCI, CAI, ABI

X(1489) = 3rd STEVANOVIC POINT

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let I be the incenter of ABC, let A'' be the incenter of triangle IBC, and define B'', C'' cyclically. Let X = A'A''∩BC, and define Y, Z cyclically. Then AX, BY, CZ concur in X(1489). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1130) and X(1489).)

X(1489) lies on these lines: 1,188   2,1143   174,558   258,483

X(1489) = complement of X(1143)
X(1489) = cevapoint of X(1) and X(258)
X(1489) = crosspoint of X(2) and X(1274)

X(1490) = X(1)X(4)∩X(3)X(9)

X(1490) is X(68)-of-the-excentral triangle and the reflection of X(84) in X(3). These and other properties itemized here were reported by Darij Grinberg, May 19, 2003.

Let U(A) be the circle with center A having the radius of the A-excircle, and define U(B) and U(C) cyclically. Then X(1490) is the radical center of the three circles. (Hauke Reddmann, Hyacinthos, Jan. 8, 2009)

Let Oa be the circle centered at A and passing through the A-excenter, and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc = X(1490). (Randy Hutson, September 14, 2016)

Let A'B'C' be the cevian triangle of X(1034). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1490). (Randy Hutson, September 14, 2016)

X(1490) lies on the Darboux cubic and these lines: 1,4   3,9   20,78   40,64   63,411   165,191   224,908   386,990   910,1192   975,991   1045,1047   1210,1467   1498,3347   2130,3473   2131,3353   3182,3348   3183,3354

X(1490) = isogonal conjugate of X(3345)
X(1490) = reflection of X(84) in X(3)
X(1490) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,40), (78,1), (329,9)
X(1490) = X(207)-cross conjugate of X(1)
X(1490) = homothetic center of hexyl and 2nd extouch triangles
X(1490) = X(155)-of-hexyl-triangle
X(1490) = X(155)-of-2nd-extouch-triangle
X(1490) = perspector of hexyl triangle and cevian triangle of X(20)
X(1490) = perspector of hexyl triangle and anticevian triangle of X(40)
X(1490) = perspector of ABC and the reflection in X(282) of the antipedal triangle of X(282)
X(1490) = {X(1),X(1745)}-harmonic conjugate of X(223)
X(1490) = excentral-isogonal conjugate of X(46)
X(1490) = excentral-isotomic conjugate of X(1721)
X(1490) = pedal antipodal perspector of X(1) wrt excentral triangle
X(1490) = ABC-to-excentral barycentric image of X(4)

X(1491) = CROSSDIFFERENCE OF X(1) AND X(32)

Let L be the line PU(10) = X(10)X(514); let M be the trilinear polar of the cevapoint of PU(10), so that M = X(522)X(1491). Let V = P(10)-Ceva conjugate of U(10) and let W = U(10)-Ceva conjugate of P(10). The lines L, M, and VW concur in X(1491). (Randy Hutson, December 26, 2015)

X(1491) lies on these lines: 10,514   44,513   325,523   663,1193   667,830

X(1491) = reflection of X(659) in X(650)
X(1491) = isogonal conjugate of X(1492)
X(1491) = isotomic conjugate of X(789)
X(1491) = X(i)-Ceva conjugate of X(j) for these (i,j): (262,11), (1492,1)
X(1491) = cevapoint of X(1) and X(1492)
X(1491) = crosspoint of X(i) and X(j) for these (i,j): (1,1492)
X(1491) = crosssum of X(i) and X(j) for these (i,j): (1,1491), (100,1390)
X(1491) = crossdifference of every pair of points on line X(1)X(32)

X(1492) = COLUMBUS POINT

X(1492) lies on these lines: 88,985   100,825   101,660   110,789   190,692

X(1492) = isogonal conjugate of X(1491)
X(1492) = cevapoint of X(513) and X(1386)
X(1492) = X(i)-cross conjugate of X(j) for these (i,j): (182,59), (1491,1)
X(1492) = trilinear pole of line X(1)X(32)
X(1492) = trilinear product of circumcircle intercepts of line X(2)X(31)

X(1493) = NAPOLEON CROSSSUM

A construction of X(1493) is given by Antreas Hatipolakis and Angel Montesdeoca at 24179.

X(1493) lies on these lines: 3,54   5,539   49,143   110,1173   113,137   141,575   206,576

X(1493) = midpoint of X(54) and X(195)
X(1493) = isogonal conjugate of X(1487)
X(1493) = complement of X(3519)
X(1493) = X(110)-Ceva conjugate of X(1510)
X(1493) = crosspoint of X(61) and X(62)
X(1493) = crosssum of X(17) and X(18)
X(1493) = orthocenter of pedal triangle of X(54)

X(1494) = ISOTOMIC CONJUGATE OF X(30)

X(1494) lies on the Steiner circumellipse and these lines: 2,648   30,340   69,74   95,549   190,306   253,317   264,339   287,524   305,670   307,319   325,892

X(1494) = reflection of X(648) in X(2)
X(1494) = isogonal conjugate of X(1495)
X(1494) = isotomic conjugate of X(30)
X(1494) = complement of X(39358)
X(1494) = anticomplement of X(3163)
X(1494) = cevapoint of X(i) and X(j) for these (i,j): (2,30), (3,323), (298,299)
X(1494) = X(i)-cross conjugate of X(j) for these (i,j): (30,2), (340,95)
X(1494) = antipode of X(2) in hyperbola {A,B,C,X(2),X(69)}
X(1494) = trilinear pole of line X(2)X(525)
X(1494) = crossdifference of PU(87)
X(1494) = pole wrt polar circle of trilinear polar of x(1990)
X(1494) = X(48)-isoconjugate (polar conjugate) of X(1990)

X(1495) = CROSSSUM OF X(2) AND X(30)

Let La be the line through A parallel to the de Longchamps line, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define B', C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the de Longchamps line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(1495). (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, December 26, 2015)

X(1495) lies on the Walsmith rectangular hyperbola and these lines: 3,3426   6,25   23,110   24,185   30,113   52,156   74,186   125,468   182,373   187,237   263,1383   1204,1498

X(1495) = midpoint of X(23) and X(110)
X(1495) = reflection of X(i) in X(j) for these (i,j): (125,468), (1531,113)
X(1495) = isogonal conjugate of X(1494)
X(1495) = isotomic conjugate of isogonal conjugate of X(9407)
X(1495) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,39176), (78,1), (329,9)
X(1495) = crosspoint of X(6) and X(74)
X(1495) = crosssum of X(i) and X(j) for these (i,j): (74,6), (1304,647)
X(1495) = crossdifference of every pair of points on line X(2)X(525)
X(1495) = inverse-in-Moses-radical-circle of X(187)
X(1495) = intersection of tangents to Moses-Jerabek conic at X(6) and X(74)
X(1495) = pole of Brocard axis wrt Moses radical circle
X(1495) = {X(3),X(5651)}-harmonic conjugate of X(5650)
X(1495) = trilinear pole of PU(87)
X(1495) = inverse-in-Parry-isodynamic-circle of X(3569); see X(2)
X(1495) = X(3218)-of-orthic-triangle if ABC is acute
X(1495) = antipode of X(125) in Walsmith rectangular hyperbola
X(1495) = orthocenter of X(110)X(1495)X(3569)
X(1495) = orthocenter of X(125)X(3569)X(3580)
X(1495) = orthic-isogonal conjugate of X(39176)

X(1496) = POINT BETELGEUSE I

X(1496) lies on these lines:    1,21   3,1037   48,820   55,603   56,212   75,775   354,1451   580,1471   602,999

X(1496) = {X(1),X(63)}-harmonic conjugate of X(774)

X(1497) = POINT BETELGEUSE II

X(1497) lies on these lines: 1,21   3,1057   46,1471   55,602   56,601   82,158   498,748   499,750   517,1451   603,999   605,1335   606,1124

X(1497) = {X(1),X(31)}-harmonic conjugate of X(255)

X(1498) = REFLECTION OF X(64) IN X(3)

Let A'B'C' be the cevian triangle of X(1032). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1498). (Randy Hutson, December 10, 2016)

X(1498) is the perspector of the tangential triangle and the reflection of triangle ABC in X(3); also, X(1498) is X(8)-of tangential triangle if ABC is acute. (Darij Grinberg, 6/2/03)

Let (Ba, Ca) be the points where BC is cut by the parallels through A to the b- and c- altitudes, respectively, and define (Cb, Ab), (Ac, Bc) cyclically. Then the perpendicular bisectors of (Ba, Ca), (Cb, Ac) and (Ac, Bc) concur at X(1498). (Hyacinthos #62, #72, #73)

Let O_A be the circle centered at the A-vertex of the anticevian triangle of X(3) and passing through A; define O_B and O_C cyclically. X(1498) is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)

X(1498) lies on the Darboux cubic and these lines: 1,84   3,64   4,6   20,394   24,1192   25,185   30,155   40,219   159,1350   195,382   1158,1214   1490,3347   2131,3183   3182,3354   3353,3473

X(1498) = reflection of X(i) in X(j) for these (i,j): (64,3), (1350,159)
X(1498) = isogonal conjugate of X(3346)
X(1498) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,3), (394,6)
X(1498) = crosssum of X(122) and X(523)
X(1498) = tangential isogonal conjugate of X(24)
X(1498) = tangential isotomic conjugate of X(6
X(1498) = antipedal isogonal conjugate of X(64
X(1498) = Nagel point of tangential triangle if ABC is acute
X(1498) = perspector of antipedal triangle of X(64) and cevian triangle of X(20)
X(1498) = intouch-to-ABC functional image of X(8)
X(1498) = {X(12964),X(12970)}-harmonic conjugate of X(6)
X(1498) = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X(3)), of X(64)
X(1498) = perspector of ABC and the reflection in X(1073) of the antipedal triangle of X(1073)
X(1498) = perspector of excentral triangle and cross-triangle of ABC and hexyl triangle

X(1499) = POINT BIHAM

X(1499) lies on these (parallel) lines: 3,669   4,1550   30,511   74,2770   98,843

X(1499) = isogonal conjugate of X(1296)
X(1499) = isotomic conjugate of X(35179)
X(1499) = X(2)-Ceva conjugate of X(35133)
X(1499) = crosspoint of X(99) and X(598)
X(1499) = crosssum of X(512) and X(574)
X(1499) = crossdifference of every pair of points on line X(6)X(373)
X(1499) = bicentric difference of PU(7)
X(1499) = ideal point of PU(7)
X(1499) = X(2780) of 4th Brocard triangle
X(1499) = X(2780) of orthocentroidal triangle
X(1499) = X(2780) of X(4)-Brocard triangle
X(1499) = Thomson-isogonal conjugate of X(111)
X(1499) = Lucas-isogonal conjugate of X(111)
X(1499) = Cundy-Parry Psi transform of X(14263)

X(1500) = INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X(39) and radius R tan ω sin 2ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X(115), and its internal and external centers of similitude with the incircle are X(1500) and X(1015), respectively. (Peter J. C. Moses, 5/29/03)

X(1500) lies on these lines: 1,39   6,595   10,37   11,1508   12,115   32,55   35,172   41,1017   42,213   56,574   76,192   216,1062   346,941   519,1107   612,1196   756,762   1124,1505   1335,1504

X(1500) = isogonal conjugate of X(1509)
X(1500) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,756), (42,872), (1018,512)
X(1500) = X(872)-cross conjugate of X(181)
X(1500) = crosspoint of X(37) and X(42)
X(1500) = crosssum of X(81) and X(86)
X(1500) = barycentric square of X(37)

X(1501) = TRILINEAR 5th POWER POINT

X(1501) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X(8265) and its perspector is X(32). (Randy Hutson, October 15, 2018)

X(1501) lies on these lines: 6,22   32,184   101,697   110,699   154,1184   701,825   703,827   1196,1495

X(1501) = isogonal conjugate of X(1502)
X(1501) = isotomic conjugate of isogonal conjugate of X(9233)
X(1501) = antigonal conjugate of X(37845)
X(1501) = complement of X(33796)
X(1501) = anticomplement of isogonal conjugate of X(38829)
X(1501) = anticomplementary conjugate of anticomplement of X(38829)
X(1501) = crosssum of X(i) and X(j) for these (i,j): (2,315), (76,305), (115,850)
X(1501) = crossdifference of every pair of points on line X(826)X(850)
X(1501) = trilinear product of PU(12)
X(1501) = X(92)-isoconjugate of X(305)
X(1501) = barycentric product of vertices of circumsymmedial triangle

X(1502) = ISOGONAL CONJUGATE OF X(1501)

X(1502) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the de Longchamps line. The center of this inellipse is X(626). (Randy Hutson, October 15, 2018)

X(1502) lies on these lines: 1,704   2,308   6,706   31,708   32,710   66,315   69,290   75,700   76,141   99,160   264,305   311,327   313,561

X(1502) = isogonal conjugate of X(1501)
X(1502) = isotomic conjugate of X(32)
X(1502) = anticomplement of X(8265)
X(1502) = cevapoint of X(i) and X(j) for these (i,j): (2,315), (76,305), (115,850)
X(1502) = X(i)-cross conjugate of X(j) for these (i,j): (115,850), (626,2)
X(1502) = barycentric product of PU(14)
X(1502) = trilinear pole of line X(826)X(850)
X(1502) = pole wrt polar circle of trilinear polar of X(1974)
X(1502) = X(48)-isoconjugate (polar conjugate) of X(1974)
X(1502) = barycentric square of X(76)

X(1503) = POINT ARKAB

As the isogonal conjugate of a point on the circumcircle, X(1503) lies on the line at infinity.

X(1503) lies on these (parallel) lines: 2,154   3,66   4,6   5,182   11,1428   20,64   22,161   30,511   51,428   67,74   98,230   110,858   125,468   147,325   184,427   221,388   242,1146   265,1177   287,297   376,599   381,597   382,1351   383,395   394,1370   396,1080   546,575   576,1353   611,1478   613,1479   946,1386

X(1503) = isogonal conjugate of X(1297)
X(1503) = isotomic conjugate of X(35140)
X(1503) = complementary conjugate of X(132)
X(1503) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,23976), (4,132), (287,6), (297,230), (685,523)
X(1503) = cevapoint of X(20) and X(147)
X(1503) = crosspoint of X(4) and X(98)
X(1503) = crosssum of X(3) and X(511)
X(1503) = crossdifference of every pair of points on line X(6)X(520)
X(1503) = X(4)-Hirst inverse of X(1249)
X(1503) = intercept of the van Aubel line and the line at infinity
X(1503) = Cundy-Parry Phi transform of X(14376)
X(1503) = Cundy-Parry Psi transform of X(8743)
X(1503) = crosspoint of X(20) and X(147) wrt both the excentral and anticomplementary triangles
X(1503) = infinite point of tangents at X(4) and X(20) to Darboux cubic K004

X(1504) = INSIMILICENTER OF MOSES AND 2nd LEMOINE CIRCLES

X(1504) = internal center of similitude of the Moses circle and the 2nd Lemoine circle. [See X(1500); Peter J. C. Moses, 6/2/03]

X(1504) lies on these lines: 2,588   3,6   115,485   394,493   486,1508   491,626   590,639   1015,1124   1335,1500

X(1504) = X(1306)-Ceva conjugate of X(512)
X(1504) = crosspoint of X(6) and X(493)
X(1504) = isogonal conjugate of the polar conjugate of X(32588)
X(1504) = {X(6),X(39)}-harmonic conjugate of X(1505)
X(1504) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(6) and X(493)

X(1505) = EXSIMILICENTER OF MOSES AND 2nd LEMOINE CIRCLES

X(1505) = external center of similitude of the Moses circle and the 2nd Lemoine circle. [See X(1500); Peter J. C. Moses, 6/2/03]

X(1505) lies on these lines: 2,589   3,6   115,486   394,494   485,1508   492,626   615,640   1015,1335   1124,1500

X(1505) = X(1307)-Ceva conjugate of X(512)
X(1505) = crosspoint of X(6) and X(494)
X(1505) = isogonal conjugate of the polar conjugate of X(32587)
X(1505) = {X(6),X(39)}-harmonic conjugate of X(1504)
X(1505) = intersection of tangents to hyperbola {{A,B,C,X(2),X(6)}} at X(6) and X(494)

X(1506) = INSIMILICENTER OF MOSES AND NINE-POINT CIRCLES

X(1506) = internal center of similitude of the Moses circle and the nine-point circle. [The external center is X(115); see X(1500); Peter J. C. Moses, 6/2/03.]

X(1506) lies on these lines: 2,32   4,574   5,39   6,17   11,1500   12,1015   51,211   125,217   140,187   384,620   485,1505   486,1504

X(1506) = complement of X(1078)
X(1506) = {X(6),X(1656)}-harmonic conjugate of X(7746)
X(1506) = {X(5),X(39)}-harmonic conjugate of X(115)

X(1507) = 1st MORLEY-GIBERT POINT

X(1507) and X(1508) are centers on Morley cubic, which also passes through X(356), X(357), X(358); see Bernard Gibert's site.

X(1507) lies on this line: 1,358

X(1508) = 2nd MORLEY-GIBERT POINT

See the notes at X(1507) and X(356).

X(1508) is the perspector of the excentral tiangle and the 1st Morley adjunct triangle. (César Lozada, January 18, 2015)

X(1508) lies on this line: 1,357

X(1508) = SS(A → A/3) of X(3468)
X(1508) = perspector of excentral and 1st Morley adjunct triangles

X(1509) = ISOGONAL CONJUGATE OF X(1500)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A" be the trilinear pole of the tangent to the circumcircle at A', and define B" and C" cyclically. The lines AA", BB", CC" concur at X(1509). (Randy Hutson, June 27, 2018)

X(1509) lies on these lines: 1,99   2,1171   58,86   76,940   81,239   552,553   593,763

X(1509) = isogonal conjugate of X(1500)
X(1509) = isotomic conjugate of X(594)
X(1509) = X(873)-Ceva conjugate of X(261)
X(1509) = cevapoint of X(81) and X(86)
X(1509) = X(i)-cross conjugate of X(j) for these (i,j): (81,757), (86,873), (757,552), (1019,99)
X(1509) = barycentric square of X(86)

X(1510) = NAPOLEON CROSSDIFFERENCE

X(1510) lies on these lines: 30,511   110,1291

X(1510) = isogonal conjugate of X(930)
X(1510) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39018), (4,137), (110,1493)
X(1510) = X(137)-cross conjugate of X(143)
X(1510) = crosspoint of X(i) and X(j) for these (i,j): (4,933), (110,1173)
X(1510) = crosssum of X(i) and X(j) for these (i,j): (140,523), (512,570)
X(1510) = crossdifference of every pair of points on line X(6)X(17)
X(1510) = X(523)-of-orthic-triangle
X(1510) = perspector of hyperbola {{A,B,C,X(2),X(61),X(62)}}
X(1510) = infinite point of normal to hyperbola {{A,B,C,X(4),X(15)}} at X(15) and normal to hyperbola {{A,B,C,X(4),X(16)}} at X(16)
X(1510) = barycentric square root of X(39018)

X(1511) = FERMAT CROSSSUM

Let A'B'C' be the cevian triangle of X(30). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(1511). Let N_A be the reflection of X(5) in the perpendicular bisector of segment BC, and define N_B and N_C cyclically; then X(1511) = X(186)-of-N_AN_BN_C. (Randy Hutson, August 26, 2014)

Let A'B'C' be the medial triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X(1511). Let (O') be the circle passing through the points X(3), X(4), X(399), X(6069), let Q be the radical axis (O') and the circumcircle; then X(1511) = Q∩X(3)X(74). (Randy Hutson, August 26, 2014)

Let U be the Simson line of X(110), which is the line X(30)X(113). Let V be the line normal to the circumcircle at X(110), which is the line X(3)X(74). Then X(1511) = U∩V. (Randy Hutson, January 29, 2015)

X(1511) lies on (Johnson circumconic of medial triangle), the bicevian conic of X(2) and X(110), and on these lines: 2,265   3,74   24,1112   30,113   36,1464   125,128   141,542   146,376   184,974   186,323   214,960   249,842   389,1493

X(1511) = midpoint of X(i) and X(j) for these (i,j): (3,110), (74,399)
X(1511) = reflection of X(i) in X(j) for these (i,j): (125,140), (1539,113)
X(1511) = complement of X(265)
X(1511) = complementary conjugate of X(2072)
X(1511) = isotomic conjugate of polar conjugate of X(39176)
X(1511) = crosssum of circumcircle-intercepts of line PU(174) (line X(4)X(523))
X(1511) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3284), (3,1154), (110,526)
X(1511) = crosspoint of X(i) and X(j) for these (i,j): (2,340), (15,16)
X(1511) = crosssum of X(13) and X(14)
X(1511) = crossdifference of every pair of points on the line X(1637)X(1989)
X(1511) = inverse-in-circumcircle of X(399)
X(1511) = perspector of the circumconic centered at X(3284)
X(1511) = {X(3),X(399)}-harmonic conjugate of X(74)
X(1511) = orthocentroidal-to-ABC similarity image of X(5)
X(1511) = 4th-Brocard-to-circumsymmedial similarity image of X(5)
X(1511) = reflection of X(5) in X(5972)
X(1511) = antipode of X(5) in the bicevian conic of X(2) and X(110)
X(1511) = X(12515)-of-orthic-triangle if ABC is acute

Orthojoins: X(1512)-X(1568)