| PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
| PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
| PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
| PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
| PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
| PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
| PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
| PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
| PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
| PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
| PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
| PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
Let OA be the circle tangent of side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. X(1001) is the radical center of the circles OA, OB, OC. (Randy Hutson, 9/23/2011)
Let LA and MA be the external tangents to circles OB and OC, with LA being the farther from OA. Define LB, LC, MB, and MC cyclically. Let A' = LB∩LC and A'' = MB∩MC, 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 OA and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of OB and OC. 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 AB = A'∩AB and AC = A'∩AC. Define BC and CA cyclically, and define BA and CB cyclically. The six points AB, BC, CA, AB, BC, CA 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) 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)
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)
As a point on the Euler line, X(1004) has Shinagawa coefficients ((E+F)S2-$abSASB$, -(E+F)S2 +$ab$S2).
X(1004) lies on these lines: 2,3 7,100 46,200 63,210 65,224
X(1005) lies on these lines: 2,3 9,100 55,329 108,342
X(1005) = isogonal conjugate of X(1242)
As a point on the Euler line, X(1006) has Shinagawa coefficients ($aSA$ + abc, - $aSA$).
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) 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)
As a point on the Euler line, X(1008) has Shinagawa coefficients ((E + F)3 + 2$bc$(E + F)2 - $abSC$(E + F) + 2(E + F)S2, $bc$S2).
X(1008) lies on these lines: 1,76 2,3
X(1008) = anticomplement of X(37148)
As a point on the Euler line, X(1009) has Shinagawa coefficients (2$bc$(E + F)2 + $bcSBSC$ - 2$bc(SA)2$ - 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)
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)
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)
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)}
As a point on the Euler line, X(1013) has Shinagawa coefficients ($aSA$F, $aSBSC$).
X(1013) lies on these lines: 2,3 6,162 7,108 33,63 55,92 100,281
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)
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) 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) 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) 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)
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}
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) 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) 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) 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) 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) 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) 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) 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) = isogonal conjugate of X(1085)
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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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
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) 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) 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) lies on these lines: 1,7 43,46
X(1044) = X(64)-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)
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) lies on these lines: 1,21 4,2648 6,986 10,894 40,511 43,46 57,978 72,171 238,942 484,1048
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) 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) lies on these lines: 1,564 484,1046
X(1049) = isogonal conjugate of X(1077)
X(1050) lies on these lines: 1,341 40,978
X(1051) lies on these lines: 1,748 6,846 81,1054 165,572
X(1052) lies on these lines: 1,765 238,517 513,1054
X(1052) = X(244)-Ceva conjugate of X(1)
X(1053) lies on these lines: 1,1110 238,517 905,1054
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) 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) 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) 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) 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) lies on these lines: 29,1056 73,1057 78,999 388,1067 1036,1066
X(1059) = isogonal conjugate of X(1058)
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) 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) 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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 1,4 55,1076 56,1074
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) lies on these lines: 1,4 55,1074 56,1076
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) lies on these lines: 1,224 3,225 4,1040 55,1072 56,1070 1038,1068
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) lies on these lines: 3,225 4,1038 55,1070 56,1072 1040,1068
X(1077) = isogonal conjugate of X(1049).
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) lies on these lines: 1,4 46,1406 77,498 484,1103 651,920
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 (31/2S, 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)
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}
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) 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)
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)
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) 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)
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)
Let A1B1C1 and A2B2C2 be the 1st and 2nd Conway triangles. Let A' be the trilinear product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1088). (Randy Hutson, July 11, 2019) X(1088) lies on these lines: 2,85 7,354 57,658 75,3668 86,269 234,555 272,1014 305,341 675,934
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) 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) lies on these lines: 5,1091 11,523
X(1091) lies on these lines: 5,1090 12,1109
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) = 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) lies on these lines: 15,36 48,163
X(1095) lies on these lines: 16,36 48,163
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) lies on these lines: 1,75 31,775
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) lies on the inellipse centered at X(10) and on these lines: 1,564 75,811 162,255
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) 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) lies on these lines: 63,304 255,326
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) 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)
Let A'B'C' be the cevian triangle of X(3). Let LA be the reflection of the line B'C' in the line BC, and define LB and LC cyclically. Let A'' = LB∩LC, 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)
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) =(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) 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): 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) 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)
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
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)
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) = 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) = 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. (, Hyacinthos #5528, 5/22/02) This description is also found in , 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] , "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)
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))
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}
Let A'B'C' be the intouch triangle of ABC. Let CA be the point other than C' in which the perpendicular to BC from C' meets the incircle, let BA 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 BCA and CBA. 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)
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)
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)
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)
Let AB be the touchpoint of the A-excircle and line AB, let AC be the touchpoint of the A-excircle and line AC, and let MA be the midpoint of segment ABAC. Define MB and MC cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles MAMBMC 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)
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 PA be the parabola with focus A and directrix BC, and let LA be the line of the points of intersection of PA with the segments AB and AC. Define LB and LC cyclically. Let A' = LB∩LC, 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 oA be the circle with diameter BC. Let wA be the circle tangent to segments AB and AC and also externally tangent to oA, in point XA. Define XB and XC cyclically. The lines AXA, BXB, CXC 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 PA meets segments AB and AC in two points, here denoted by Ab and Ac. Cyclically, the parabolas PB and PC 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). See also X(1336).
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) 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
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) 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
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(1127) = Hofstadter 1/4 pointLet 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)
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
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)
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)
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)
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)
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
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)
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
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
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 EA be the line through A parallel to the Euler line. Let A" be the point of intersection, other than A, of EA and the circumcircle. Define EB and EC 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)
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, Hyacinthos 5246, 12/31/02;
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)
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) was first noted (Hyacinthos #1498, September 25, 2000) by Bernard Gibert as a point of intersection of the circumcircle and certain cubic, denoted Kn. To define Kn, 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 Kn, and X(1141) is the point, other than A,B,C, in which Kn 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 AB be the reflection of A' in AB, and define AC, BC, BA, CA, CB cyclically. Let
A1 =
BAB∩CAC, and define B1 and
C1 cyclically,
A2 =
BAC∩CAB, and define B2 and
C2 cyclically,
A3 =
BBA∩CCA, and define B3 and
C3 cyclically,
A4 =
BBC∩CCB, and define B4 and
C4 cyclically,
A5 =
BCA∩CBA, and define B5 and
C5 cyclically,
A6 =
BCB∩CBC, and define B6 and
C6 cyclically.
Then triangle AnBnCn 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)
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
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)
Suppose that P is a point inside triangle ABC. Let SA be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define SB and SC 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).
Let A',B',C' be the respective excenters of ABC, and let AB be the projection of A on A'B', let AC be the projection of A on A'C', and define BC, BA, CA, CB cyclically. The Euler lines of the three triangles A'ABAC, B'BCBA, C'CACB 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) 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)
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 RA be the radical axis of the circles (B', |B'C|) and (C',|C'B|), and define RB and RC cyclically. The lines RA, RB, RC 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)
Suppose LA, LB, LC are lines through a point P, respectively perpendicular to sidelines BC, CA, AB. Let AB be the point where LA meets AB, and let AC be the point where LA meets AC. Define BC, BA, CA, CB cyclically. Then X(1148) is the point P, which satisfies
|PAB| + |PAC| = |PBC| + |PBA| = |PCA| + |PCB|.
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) 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) 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) 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) 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
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}
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)
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)
Let Bc be the reflection of B in the internal angle bisector of angle C, and let Cb be the reflection of C in the internal angle bisector of angle B. Let Γa be the circle that passes through A, Bc, Cb. Let A' be the pole of BC, with respect to Γa 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)
Trilinears 4 cos A + cos 3A sec A sec(B - C) : :
For any point X, let XA be the reflection of X in sideline BC, and define XB and XC cyclically. Then X(1157) is the unique point X for which the lines AXA, BXB, CXC 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)
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) 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(1159) = {X(2099),X(4860)}-harmonic conjugate of X(1)See Hyacinthos #6537.
Let OA be the circle centered at the A-vertex of the anti-inner-Grebe triangle and passing through A; define OB and OC cyclically. X(1160) is the radical center of OA, OB, OC. (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) lies on these lines: 3,6 4,1163
See Hyacinthos #6537.
X(1161) = reflection of X(1160) in X(3)
See Hyacinthos #6537.
X(1162) lies on these lines: 4,1160 428,1163
X(1162) = X(4)-Ceva conjugate of X(3127)X(1163) lies on these lines: 4,1161 428,1162
See Hyacinthos #6537.
X(1163) = X(4)-Ceva conjugate of X(3128)X(1164) lies on this line: 468,1165
X(1165) lies on this line: 468,1164
Saragossa Points 1166-1208
U = B'C"∩B"C' V = C'A"∩C"A' W = A'B"∩A"B'.
Lines AU, BV, CW concur in the 1st Saragossa point of P;
lines A'U, B'V, C'W concur in the 2nd Saragossa point of P;
lines A"U, B"V, C"W concur in the 3rd Saragossa point of P.
These concurrences were presented by Darij Grinberg (Hyacinthos
#6531, February 14, 2003),
with coordinates as follows. Let P = x : y : z (trilinears), and
abbreviate the 1st, 2nd, and 3rd Saragossa points
as Q, Q', Q", respectively; then first trilinears are
for
Q: f(a,b,c) = a/[x(bz + cy)],
for
Q': f(a,b,c) =
ax[(b2z2 + c2y2)x + xyzbc +
ayz(bz + cy)],
for
Q": f(a,b,c) = ax[(b2z2 +
c2y2)x + ayz(bz + cy)].
The name Saragossa refers to the king who proved Ceva's theorem before Ceva did. See
J. B. Hogendijk, "Al-Mu'taman ibn Hud [bar over u], 11th century king of Saragossa and brilliant mathematician," Historia Mathematica, 22 (1995) 1-18.
The points P, Q', Q" are collinear.
The 1st Saragossa point of X(i) is X(j) for these (i,j):
(1,58) (2,251) (3,4)
(4,54) (6,6) (19,284)
(21,961) (24,847) (25,2)
(28,943) (31,81)
(32,83) (51,288) (55,57)
(56,1) (58,1126) (64,3)
(84,947) (154,1073) (184,275)
(198,282) (512,249) (513,59)
(667,1016) (939,937) (1036,959).
The 2nd Saragossa point of X(i) is X(j) for these (i,j):
(1,386) (6,6)
The 3rd Saragossa point of X(i) is X(j) for these (i,j):
(3,185) (4,389) (6,6)
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) 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) 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) 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) 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) 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) 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)
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) 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) 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) 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)
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) 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) 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) 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) 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) lies on these lines: 3,6 9,498 19,208
X(1183) lies on these lines: 1,41 6,959 8,1036 391,958
X(1184) lies on these lines: 2,6 3,1194 25,32
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) 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) lies on these lines: 6,60 10,37
X(1188) lies on these lines: 6,279 31,32
X(1189) lies on this line: 1,2
X(1190) lies on these lines: 6,57 41,55 56,1202 165,218 294,940
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) lies on these lines: 3,6 25,64
X(1192) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(38292)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)
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) 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) 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) 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) lies on this line: 1,2
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) 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) 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) 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) 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) 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) 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) lies on these lines: 6,31 81,239
X(1207) lies on these lines: 3,6 83,3978
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)
In the plane of a triangle ABC, let
A'B'C' = medial triangle;
BA = reflection of B' in BC, and define CB and AC cyclically;
CA = reflection of C' in BC, and define AB and BC cyclically;
OA = circle {{B', C', BA, CA}}, and define OB and OC cyclically.
Then X(1209) = radical center of OA, OB, OC.
(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) 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) 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) 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) 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) 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) 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) 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
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 Kab and Kac be the circumcenters of triangles A"BA and A"CA, respectively. Let A''' = BKac∩CKab, 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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 5,311 69,2888 76,95 339,1232
X(1225) = isotomic conjugate of X(1166)
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) 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) lies on these lines: 2,39 12,313 312,857
X(1228) = isotomic conjugate of X(1169)
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) 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) 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) 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) lies on these lines: 69,674 76,85 310,333 1226,1234
X(1233) = isotomic conjugate of X(1174)
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) 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) lies on these lines: 4,69 325,339 826,850
X(1236) = isotomic conjugate of X(1177)
X(1237) lies on these lines: 12,313 75,1240 76,334 561,756
X(1237) = isotomic conjugate of X(1178)
X(1238) lies on these lines: 3,69 311,325 1232,1273
X(1238) = isotomic conjugate of X(1179)
X(1239) lies on these lines: 25,1235 76,251
X(1239) = isotomic conjugate of X(1180)
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) 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) lies on these lines: 71,1155 72,527
X(1242) = isogonal conjugate of X(1005)
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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 1,15 7,559 13,80 37,1250 55,199
X(1251) = isogonal conjugate of X(1082)
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) 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) 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)
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) 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) 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) 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) 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) 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) 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) = 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)
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) 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) 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) 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) 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)
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) 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) 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) 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) lies on these lines: 2,94 69,74
X(1272) = isotomic conjugate of X(1138)
X(1272) = anticomplement of X(1989)
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)
Let I be the incenter of ABC and EA 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) 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)
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 Kθ is the line X(4)X(9). Specifically, Kθ 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)
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)
Antreas Hatzipolakis described this point in Hyacinthos #3190 (7/13/2001). See also
Paul Yiu, Introduction to the Geometry of the Triangle, 2002, Section 3.3.1, exercise 2.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) 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) 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)
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).The report introduces fifteen Sharygin points, of which the 1st, 2nd, 4th, and 6th are X(256), X(291), X(846),
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(1281) = isogonal conjugate of X(30648)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) 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) 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) 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
S. N. Collings, "Reflections on reflections 2", Mathematical Gazette 1974, page 264.
It was further discussed by Floor van Lamoen and Darij Grinberg, and coordinates were found by Barry Wolk; see Hyacinthos #4547, #4548, #6469, #6538, #6546, #6560. Paul Yiu noted that T(P) is the point, other than A, B, C, in which the circumconic centered at P meets the circumcircle of triangle ABC (#4548).
If P = x : y : z (trilinears), then Q = f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = 1/[bz(ax + by - cz) - cy(ax + cz - by)].
For given Q, the set of points P satisfying T(P) = Q is a conic. Examples follow:
if Q = X(74), the conic passes through X(i) for I = 125;
if Q = X(98), the conic passes through X(i) for I = 115, 868;
if Q = X(99), the conic passes through X(i) for I = 2, 39, 114, 618,
619, 629, 630, 641, 642, 1125;
if Q = X(100), the conic passes through X(i) for I = 1, 9, 10, 119,
142, 214, 442, 1145;
if Q = X(107), the conic passes through X(i) for I = 4, 133, 800,
1249;
if Q = X(110), the conic passes through X(i) for I = 5, 6, 113, 141,
206, 942, 960, 1147, 1209;
if Q = X(476), the conic passes through X(i) for I = 30.
Bernard Gibert (4/02/03) identified T(P) as the trilinear pole of the line of X(6) and the X(2)-Ceva conjugate of P. He identified the locus of P as the rectangular hyperbola that circumscribes the medial triangle and has center W given by the vector equation 4X(2)W = X(2)Q. The anticomplement of this hyperbola is the rectangular ABC-circumhyperbola whose center is the complement of Q. Thus, referring to examples given above:
if Q = X(99), the conic is the Kiepert hyperbola of the medial
triangle;
if Q = X(100), the conic is the Feuerbach hyperbola of the medial
triangle;
if Q = X(110), the conic is the Jerabek hyperbola of the medial
triangle.
X(1286) lies on the circumcircle and this line: 26,98
X(1287) lies on the circumcircle and these lines: 5,842 110,826 523,827
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) 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) 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) lies on the circumcircle and these lines: 30,1141 74,1154 477,550 523,930
X(1291) = cevapoint of X(50) and X(512)
Let LA be the reflection of the line X(1)X(6) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1292). Let MA be the reflection of the line X(7)X(8) in the line BC, and define MB and MC cyclically. Let A'' = MB∩MC, 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)
Let LA be the reflection of the line X(1)X(2) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, 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) 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) 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)
Let LA be the reflection of the line X(2)X(6) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, 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) 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) 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) 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) 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) 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) 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) 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
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) 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) 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) 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))
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)
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) 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) 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)
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
SA
= (b2 + c2 - a2)/2, and SB
and SC 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.
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)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = [d2 + (4r - R)R +
sqrt(Q)]SBSC + (d2 + 2r2 -
R2)a2SA,
where
Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r +
R)]2,
d
= distance between X(3) and X(4),
R
= circumradius, r = inradius,
SA = (b2 + c2 - a2)/2, and SB and SC 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 + Q1/2).
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
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = [d2 + (4r - R)R -
sqrt(Q)]SBSC + (d2 + 2r2 -
R2)a2SA,
where
Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r +
R)]2,
d
= distance between X(3) and X(4),
R
= circumradius, r = inradius,
SA = (b2 + c2 - a2)/2, and SB and SC 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 - Q1/2).
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
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) 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) lies on these lines: 1,1168 36,106 88,517 679,1319
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}
Let DEF be the intouch triangle. Let Ha be the orthocenter of IBC. Let A1 be the point, other than A, where AI meets the circumcircle. Let ta be the tangent to the circle (DHaA1) at Ha, and define tb and tc cyclically. The lines ta, tb , tc 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)
Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically. Let X = BCB∩CBC and X' = BAB∩CAC, 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
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) 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) 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)
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) 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)
Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically. The lines ABBA, BCCB, CAAC 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)
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) 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) 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)
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)
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) 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) 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) 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)
The parabola with focus A and directrix BC meets line AB in two points; let AB be the one further from B, and define AC similarly. Let LA be the line ABAC, and define LB and LC cyclically. Let A' = LB∩LC, 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)
Referring to the parabola described above in Hutson's note, the construction of X(1336) depends on AB being the one further of two points from B; if "further" is changed to "nearer", the resulting point is X(1123). Barycentrics for the nearer of the two points are
a*c : S : 0 = 2*R : b : 0 = csc(B) : 1 : 0 ,
and barycentrics for the further of the two points are
a*c : -S : 0 = -2*R : b : 0 = -csc(B) : 1 : 0 .
If you have GeoGebra, you can view these two points, as well as X(1336) and X(1123). (Peter Moses, December 10, 2024)
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 920,3069
X(1336) = isogonal conjugate of X(1335)
X(1336) = isotomic conjugate of X(5391)
X(1336) = polar conjugate of isogonal conjugate of X(34125)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = a2(2DU - 31/2VW)/(4D +
31/2a2),
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 - 31/2VW)/(4D + 31/2a2),
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(1337) = isogonal conjugate of X(3479)Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where
g(a,b,c) = a2(2DU + 31/2VW)/(4D -
31/2a2),
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)
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
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) = 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}
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) = 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)
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)
As a point on the Euler line, X(1345) has Shinagawa coefficients (R - |OH|,3R + |OH|).
X(1345) = external center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)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)
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)
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) = 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) = 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
Let A'B'C' be the reflection of ABC in X(3). Let AB = BC∩C'A', and define BC and CA cyclically. AC = BC∩A'B', and define BA and CB cyclically. The 6 points AB, BC, CA, AC, BA, CB lie on a conic. Let A" be the intersection of the tangents to the conic at AB and AC, 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)
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) 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) 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
Edward Brisse, Perspective Poristic Triangles: a4/[(b + c - a)u2] : b4/[(c + a - b)v2] : c4/[(a + b - c)w2].
If X is given by trilinears x : y : z, then T(X) has trilinears a/[(b + c - a)x2] : b/[(c + a - b)y2] : c/[(a + b - c)z2].
Examples: X(11) = Feuerbach point = T(X(109))
X(1317) = incircle-antipode of X(11) = T(X(106))
Still open is the question posed in Hyacinthos #6832: to list all polynomial centers on the incircle having low degree and to prove that there are no others. Here, "degree" of X = p(a,b,c) : p(b,c,a) : p(c,a,b) [barycentrics] refers to the degree of homogeneity of p(a,b,c), and "low" means less than 6. (The Feuerbach point, X(11), has degree 3.)
In Hyacinthos #6835, Paul Yiu gives two methods for constructing polynomial centers on the incircle:
(1) If X is a polynomial center on the incircle and W is any other polynomial center, then the line XW meets the incircle in another point that is a polynomial center.
(2) If W is on the line at infinity, then the barycentric square W2 is on the Steiner inscribed ellipse, and the barycentric product X(7)*W2 is on the incircle.
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) lies on the incircle and these lines: 56,741 222,1363
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) 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) 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) 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) 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) 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) 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) lies on the incircle and this line:
222,1355
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) 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) lies on the incircle and these lines: 7,1365 222,1367 553,1358
X(1367) lies on the incircle and these lines: 7,1354 222,1366
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) 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)
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)
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(1371) = {X(1),X(7)}-harmonic conjugate of X(1372)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(1372) = {X(1),X(7)}-harmonic conjugate of X(1371)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
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
As a point on the Euler line, X(1375) has Shinagawa coefficients (2$aSBSC$ - $a$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) 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) 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) 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)
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) = (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)
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) 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)
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)
Let GA be the circumcenter of triangle BCG, where G - centroid(ABC). Define GB and GC cyclically. Triangle GAGBGC 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)
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)
In the plane of a triangle ABC, let
D = the point on line AC such that angle CAB = angle DAB and |AD| = |AB|
E = the point on line AB such that angle CAB = angle CAE and |AE| = |AC|
F = the point on line BC such that angle CBA = angle FBA and |BF| = |AB|
G = the point on line AB such that angle CBA = angle CBG and |BG| = |BC|
H = the point on line BC such that angle ACB = angle ACH and |CH| = |AC|
I = the point on line AC such that angle ACB = angle ICB and |CI| = |BC|
The circumcenters of AGI, BEH, CDF, ABHI, ACFG, and BCDE lie on a conic section centered at X(1385). (Benjamin Warren, December 7, 2024)
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) 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) 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) 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) 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) 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) lies on this line: 517,1443
X(1391) = isogonal conjugate of X(1387)
X(1391) = cevapoint of X(202) and X(203)
X(1392) lies on this line: 80,145
X(1392) = isogonal conjugate of X(1388)
Beth Conjugates 1393-1477
if P = p : q : r, then (P-beth conjugate of P) = up : vq : wr, where
u : v : w = a/(b + c - a) : b/(c + a - b) : c/(a + b - c), or
equivalently,
u : v : w = 1 - cos A : 1 - cos B : 1 - cos C.
Following is a list of pairs (i,j) for which X(i) = X(j)-beth conjugate of X(j):
| 1,8 | 2,312 | 3,78 | 4,318 | 6,9 | 7,75 | 8,341 |
| 9,346 | 12,1089 | 19,281 | 21,1043 | 25,33 | 28,29 | 31,55 |
| 32,41 | 34,4 | 41,220 | 42,210 | 48,219 | 55,200 | 56,1 |
| 57,2 | 58,21 | 59,765 | 60,1098 | 63,345 | 65,10 | 73,72 |
| 77,69 | 78,1265 | 81,333 | 84,280 | 85,76 | 86,314 | 101,644 |
| 109,100 | 110,643 | 142,1229 | 174,556 | 181,756 | 184,212 | 190,646 |
| 212,1260 | 220,728 | 221,40 | 222,63 | 223,329 | 226,321 | 244,11 |
| 255,1259 | 266,188 | 269,7 | 273,264 | 278,92 | 279,85 | 326,1264 |
| 347,322 | 348,304 | 479,1088 | 513,522 | 552,873 | 603,3 | 604,6 |
| 604,6 | 608,19 | 614,497 | 649,650 | 651,190 | 662,645 | 664,668 |
| 667,663 | 738,279 | 757,261 | 849,60 | 934,664 | 951,1257 | 961,1220 |
| 1014,86 | 1027,885 | 1042,65 | 1104,950 | 1106,56 | 1118,158 | 1119,273 |
| 1193,960 | 1214,306 | 1253,480 | 1254,12 | 1284,740 |
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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 6,41 31,51 35,573 44,65 57,88 169,1046
X(1405) = isogonal conjugate of X(30608)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) 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) 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)
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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 56,106 65,1320 88,961 604,1415 901,1319 1014,1122 1037,1470
X(1417) = isogonal conjugate of X(4723)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) 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)
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) lies on these lines: 1,5 31,57 34,106 595,1393 1279,1465
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) 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) lies on these lines: 56,87 57,85 222,1429
X(1424) = X(604)-Ceva conjugate of X(57)
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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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)
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) 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)
In the plane of a triangle ABC, let
I = incenter = X(1)
O = circumcenter = X(3)
DEF = cevian triangle of I
Lb = line through B perpendicular to AI
Lc = line through C perpendicular to AI
Ab = Lb∩AO
Ac = Lc∩AO)
Oa = circumcircle of DAbAc
A' = the point of intersection, other than D of Oa 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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) lies on these lines: 1,24 4,46 19,208 25,65 28,34 33,40 227,607 1038,1039
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) 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) 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) 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) 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)}}
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) 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) 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) 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) 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) 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) 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) 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) 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) 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)
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) 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) 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)
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) 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) 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) 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) 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) 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) 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)
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
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)
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)
(For the 2nd Shadow Point, see X(6580).
X(1481) lies on these lines: {1,1406}, {2316,4253}
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) 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) 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) = X(1486)-of-tangential triangle if ABC is acute. (Darij Grinberg, 5/24/03)
Let A'B'C' be the orthic triangle. Let La be the reflection of line B'C' in the perpedicular bisector of BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(1485). (Randy Hutson, July 31 2018) X(1485) lies on these lines: 22,160 26,206 157,264 232,571Let 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
Let N denote the nine-point center, X(5). Let NA = N-of-triangle NBC, and define NB and NC cyclically. Triangle NANBNC 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
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
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) 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)
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) 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)
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) 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)
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) 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) 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)
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 OA be the circle centered at the A-vertex of the anticevian triangle of X(3) and passing through A; define OB and OC cyclically. X(1498) is the radical center of OA, OB, OC. (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) 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)
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) 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) 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)
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) = 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) = 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) = 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) 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(1507) = SS(A→A/3) of X(3336)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
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) 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)
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 NA be the reflection of X(5) in the perpendicular bisector of segment BC, and define NB and NC cyclically; then X(1511) = X(186)-of-NANBNC. (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)
D(a,b,c) = bc[2abcx + c3y + b3z - bc(by + cz)
- a2(cy + bz)],
E(a,b,c) = [(a4 - (b2 -
c2)2]x - 4a2bc(y cos B + z cos C),
f(a,b,c) = D(a,b,c)E(a,b,c).
Then orthojoin(X) = f(a,b,c) : f(b,c,a) : f(c,a,b). Below, orthojoin(X) is written as H(X).
Suppose X is not X(2) and does not lie on a sideline of triangle
ABC. The crossdifference of X and X(2) has first trilinear
a(b-c)x', where x' = (by - cz)/(b-c).
Let X -1 denote the isogonal conjugate of X. Then
H(X -1) = (by - cz) cos A : (cz - ax) cos B : (ax - by) cos
C.
In other words, if X lies on a line PG through the centroid G, then H(X -1) lies on the line HQ, where H denotes the orthocenter and Q is a point that can be determined from the above formula. Examples:
If X is on the Euler line, L(2,3), then H(X -1) is on the
line L(4,6);
If X is on L(2,6,), then H(X -1) is on the Euler line;
If X is on L(1,2), then H(X -1) is on L(4,9);
If X is on L(2,7), then H(X -1) is on L(1,4).
Suppose P is not X(6), and let
| S = | crossdifference(P, X(6)) | (S lies on the line at infinity) |
| S' = | orthopoint(S) | (S' lies on the line at infinity) |
| S" = | complementary conjugate of S' | (S" lies on the nine-point circle) |
Let X be a point on line PX(6) and not on a sideline of ABC. Then H(X -1) is on line S"S'. Examples:
If X is a center on L(1,6) and X is not X(6), then H(X
-1) is on L(119,517).
If X is a center on L(2,6) and X is not X(6), then H(X -1)
is on L(114,511).
If X is a center on L(3,6) and X is not X(6), then H(X -1)
is on L(113,30).
If X is a center on L(6,31) and X is not X(6), then H(X -1)
is on L(118,516).
If X is a center on L(6,44) and X is not X(6), then H(X -1)
is on L(117,515).
Further,
| H(X(11)) = | L(117,515)∩L(118,516) |
| H(X(37)) = | L(117,515)∩L(119,517) |
| H(X(244)) = | L(118,516)∩L(119,517) |
If X lies on the line L(230,231), then H(X) lies on the nine-point
circle. Examples:
| H(X(230)) = X(114) | H(X(231)) = X(128) | H(X(232)) = X(132) |
| H(X(468)) = X(1560) | H(X(523)) = X(115) | H(X(647)) = X(125) |
| H(X(650)) = X(11) |
X(1512) lies on these lines: 4,9 5,392 6,80 119,517 355,956 944,1210
X(1512) = reflection of X(i) in X(j) for these (i,j): (908,119), (1519,1532), (1537,1538)
X(1512) = orthpole of antiorthic axis
Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b),
where
f(a,b,c) = bc(b4 + c4 -
a2b2 - a2c2)[3a4
+ (b2 - c2)2]
As a point on the Euler line, X(1513) has Shinagawa coefficients (S2, -(E + F)2).
X(1513) lies on these lines: 2,3 98,230 114,325 132,232 147,385 183,1352 1181,1184
X(1513) = reflection of X(i) in X(j) for these (i,j): (98,230), (325,114)
X(1513) = circumcircle-inverse of X(37930)
X(1513) = {X(2),X(3)}-harmonic conjugate of X(37450)
X(1513) = orthopole of Lemoine axis
X(1513) = X(114)-of-1st-anti-Brocard-triangle
X(1514) lies on these lines: 4,6 30,113 74,403 187,1516
X(1514) = orthopole of orthic axisX(1515) lies on the Simson quartic (Q101) and these lines: 4,6 30,1294 133,1559 297,1533
X(1515) = intersection of Simson line of X(107) (line X(133)X(1515)) and trilinear polar of X(107) (line X(4)X(6))X(1516) lies on these lines: 4,96 187,1514
X(1517) lies on this line: 4,218
X(1518) lies on this line: 4,608
X(1519) lies on these lines: 1,4 119,517 499,1158
X(1519) = reflection of X(i) in X(j) for these (i,j): (1512,1532), (1532,1538)
X(1519) = orthopole of PU(96)
X(1520) lies on this line: 4,572
X(1521) lies on these lines: 117,515 118,516
X(1522) lies on these lines: 4,14 1523,1553
X(1523) lies on this line: 4,13 1522,1553
X(1524) lies on these lines: 4,13 30,113
X(1524) = reflection of X(i) in X(j) for these (i,j): (1525,1514), (1546,1525)
X(1525) lies on these lines: 4,14 30,113
X(1525) = reflection of X(i) in X(j) for these (i,j): (1524,1514), (1545,1524)
X(1526) lies on these lines: 4,16 128,1154
X(1527) lies on these lines: 4,15 128,1154
X(1528) lies on this line: 1,4
As a point on the Euler line, X(1529) has Shinagawa coefficients (FS2, -(E + F)[4(E + F)F - S2]).
X(1529) lies on these lines: 2,3 132,1503
X(1530) lies on these lines: 4,8 118,516 152,971 908, 1543
X(1530) = reflection of X(i) in X(j) for these (i,j): (910,118), (1536,1541)
Let MaMbMc be the Ehrmann mid-triangle. Let A' be the crosspoint of Mb and Mc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1531). (Randy Hutson, July 31 2018)
X(1531) lies on these lines: 4,69 30,113 382,1092
X(1531) = reflection of X(i) in X(j) for these (i,j): (1495,113), (1533,1514)
X(1531) = orthopole of de Longchamps line
As a point on the Euler line, X(1532) has Shinagawa coefficients ($a$S2, $a$S2 + 2abc(E + F) + 2$a(SA)2$).
X(1532) lies on these lines: 2,3 11,515 12,946 40,1329 119,517 496,944
X(1532) = reflection of X(i) in X(j) for these (i,j): (1519,1538), (1537,1519)
X(1533) lies on these lines: 4,83 30,113 297,1515
X(1533) = reflection of X(1531) in X(1514)
X(1534) lies on this line: 4,937
X(1535) lies on these lines: 4,7 117,515 151,517
As a point on the Euler line, X(1536) has Shinagawa coefficients (S2, -2(E + F)2 + 2$bcSA$ - S2).
X(1536) lies on these lines: 2,3 118,516
X(1536) = reflection of X(1530) in X(1541)
X(1537) lies on these lines: 4,145 7,104 11,65 100,962 119,517 214,516 515,1317
X(1537) = reflection of X(i) in X(j) for these (i,j): (11,946),
(104,1387), (1145,119), (1512,1538), (1532,1519)
X(1537) = orthopole of PU(55)
X(1538) lies on these lines: 4,1385 11,971 119,517 495,946 515,1387
X(1539) lies on these lines: 4,94 5,2777 30,113 74,381 110,382 125,546
X(1539) = reflection of X(i) in X(j) for these (i,j): (125,546), (1511,113)
X(1539) = orthopole of PU(5)
X(1540) lies on this line: 4,6
X(1541) lies on these lines: 1,4 118,516
X(1541) = orthopole of Gergonne line
X(1542) lies on these lines: 4,9 117,515
X(1543) lies on these lines: 1,4 516,972 908,1530
X(1544) lies on these lines: 4,9 30,113
X(1545) lies on these lines: 4,15 30,113
X(1545) = reflection of X(i) in X(j) for these (i,j): (1525,1524), (1546,1514)
X(1546) lies on these lines: 4,16 30,113
X(1546) = reflection of X(i) in X(j) for these (i,j): (1524,1525), (1545,1514)
X(1547) lies on this line: 4,6 118,516
X(1548) lies on these lines: 4,6 119,517
X(1549) lies on these lines: 4,6 117,515
X(1539) = orthopole of PU(173)
X(1550) lies on these lines: 4,1499 30,74 98,230 542,1551
X(1550) = orthopole of line X(115)X(125)
As a point on the Euler line, X(1551) has Shinagawa coefficients (2(E + F)3 - 9(E - 2F)S2, -3(E + F)[(E +10F)(E + F) - 6S2]).
X(1551) lies on this line: 2,3 542,1550
X(1552) lies on these lines: 4,523 30,1294 74,403
X(1553) lies on these lines: 4,250 30,113 146,476 1522,1523
X(1554) lies on these lines: 30,113 132,1503
X(1555) lies on these lines: 4,39 30,113
X(1555) = reflection of X(1561) in X(1514)
As a point on the Euler line, X(1556) has Shinagawa coefficients (2(E + F)3 + (E + 2F)S2, -(E + F)[5(E + 6F)(E + F) - 2S2]).
X(1556) lies on this line: 2,3
As a point on the Euler line, X(1557) has Shinagawa coefficients ([(E + F)3F + 2(E + F)2S2 + S4] S2, -4(E + F)3F + (E - 11F)(E + F)3S2 - (E + 10F)(E + F)S4 - 3S6).
X(1557) lies on this line: 2,3
X(1558) lies on these lines: 1,4 30,113
As a point on the Euler line, X(1559) has Shinagawa coefficients (F2, -2(2E - F)F + S2).
X(1559) lies on these lines: 2,3 133,1515
X(1559) = inverse-in-circumconic-centered-at-X(4) of X(20)
X(1560) lies on the nine-point circle and these lines: 2,112 4,111 6,67 53,136 115,427 122,216 135,571 187,468
X(1560) 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). (Randy Hutson, June 7, 2019)
X(1560) = complement of X(2373)
X(1560) = X(2)-Ceva conjugate of X(468)
X(1560) = crosspoint of X(2) and X(858)
X(1560) = crosssum of X(6) and X(1177)
X(1560) = perspector of circumconic centered at X(468)
X(1560) = center of circumconic that is locus of trilinear poles of lines passing through X(468)
X(1560) = inverse-in-polar-circle of X(111)
X(1560) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(112)
X(1561) lies on these lines: 4,32 30,113
X(1561) = reflection of X(1555) in X(1514)
X(1561) = orthopole of PU(45)
X(1562) lies on these lines: 115,125 127,525 148,287
X(1562) = X(393)-Ceva conjugate of X(647)
X(1562) = crosspoint of X(4) and X(525)
X(1562) = crosssum of X(393) and X(647)
X(1562) = orthopole of van Aubel line
X(1562) = crossdifference of every pair of points on line X(110)X(1301)
As a point on the Euler line, X(1563) has Shinagawa coefficients ([2(E + F)2 + (9E + 8F)S+10S2]S, -(E + F)2E - 2(5E - F)(E + F)S - 4(7E - 2F)S2 - 22S3).
X(1563) lies on this line: 2,3
As a point on the Euler line, X(1564) has Shinagawa coefficients ([2(E + F)2 - (9E + 8F)S+10S2]S, (E + F)2E - 2(5E - F)(E + F)S + 4(7E - 2F)S2 - 22S3).
X(1564) lies on this line: 2,3
X(1565) lies on these lines: 3,348 4,279 5,85 7,104 11,1111 77,1060 84,738 116,514 150,664 304,337 515,1323 812,1015 1119,1440 1364,1367
X(1565) = midpoint of X(150) and X(664)
X(1565) = reflection of X(1146) in X(116)
X(1565) = X(i)-Ceva conjugate of X(j) for these (i,j): (279,514), (304,525), (348,905)
X(1565) = crosspoint of X(7) and X(693)
X(1565) = crosssum of X(55) and X(692)
X(1565) = orthopole of Soddy line
X(1566) lies on the nine-point circle and these lines: 2,927 11,650 116,514 118,516 125,661 132,242
X(1566) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,676), (4,926)
X(1566) = crosspoint of X(514) and X(516)
X(1566) = crosssum of X(i) and X(j) for these (i,j): (101,103), (651,1814)
X(1566) = perspector of circumconic centered at X(676)
X(1566) = Stevanovic-circle-inverse of X(11)
X(1566) = center of circumconic that is locus of trilinear poles of lines passing through X(676)
As a point on the Euler line, X(1567) has Shinagawa coefficients ((E + F)6 - (5E - 3F)(E + F)3S2 + 3(E + F)2S4 + S6, -(E + 3F)(E + F)5 + 3(E - 3F)(E + F)3S2 + 9(E - F)(E + F)S4 - 3S6).
X(1567) lies on the nine-point circle and this line: 2,3
X(1567) = perspector of circumconic centered at X(694)
X(1567) = center of circumconic that is locus of trilinear poles of lines passing through X(694)
X(1567) = X(2)-Ceva conjugate of X(694)
X(1568) lies on these lines: 4,801 5,51 30,113 265,539 381,394 403,511
X(1568) = reflection of X(125) in X(2072)
X(1569) lies on this line: 5,39 32,99 76,620 98,574 543,598
X(1569) = midpoint of X(99) and X(194)
X(1569) = reflection of X(i) in X(j) for these (i,j): (76,620), (115,39)
X(1569) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(5)
X(1570) lies on these lines: 3,6 193,625
X(1570) = reflection of X(i) in X(j) for these (i,j): (39,2025), (187,1692), (1692,6)
X(1570) = isogonal conjugate of antigonal conjugate of X(38259)
X(1570) = inverse-in-2nd-Lemoine-circle of X(3)
X(1570) = crosssum of X(2) and X(230)
X(1570) = 2nd-Lemoine-circle inverse of X(3)
X(1570) = radical trace of the Brocard and 2nd Lemoine circles
X(1570) = radical trace of 1st Lemoine circle and Ehrmann circle
X(1571) lies on these lines: 1,574 9,1574 32,165 39,40 57,1500
X(1572) lies on these lines: 1,32 6,517 9,1573 39,40 57,1015 165,574
X(1573) lies on these lines: 2,668 8,1500 10,39 32,958 37,519 75,538 187,993 574,1376 1329,1508 1377,1505 1378, 1504
X(1574) lies on these lines: 2,1500 8,1015 10,39 32,1376 38,762 115,1329 213,899 574,993 1377,1504 1378,1505
The Moses circle, M, is introduced at X(1015); the (1/2)-Moses circle is concentric to M with half the radius of M. The insimilicenter of the Spieker and (1/2)-Moses circles is X(1107).
X(1575) lies on these lines: 2,37 6,43 10,39 42,1100 44,513 71,992 172,404 239,292 291,518 519,1015 574,993 1009,1104 1125,1500
X(1575) = isotomic conjugate of X(32020)
X(1575) = complement of X(350)
X(1575) = anticomplement of X(20530)
X(1575) = complementary conjugate of X(20542)
X(1575) = X(i)-Ceva conjugate of X(j) for these (i,j): (239,518), (292,37)
X(1575) = cevapoint of X(43) and X(2108)
X(1575) = crosspoint of X(2) and X(291)
X(1575) = crosssum of X(i) and X(j) for these (i,j): (1,1575), (6,238)
X(1575) = crossdifference of every pair of points on line X(1)X(667)
X(1575) = {X(10),X(39)}-harmonic conjugate of X(1107)
X(1576) is the center of the conic transform of the Stammler quartic (Q066 in Bernard Gibert' catalogue) by X(31)-isoconjugation. This conic is given by the barycentric equation b^4c^4(b^2-c^2)x^2+c^4a^4(c^2-a^2)y^2+a^4b^4(a^2-b^2)z^2 = 0, and it passes through the following triangle centers: X(6), X(31), X(48), X(154), X(1613), X(2578), X(2579), X(5638), X(5639). (Angel Montesdeoca, May 7, 2016)
Let A'B'C' be the circumcevian triangle of X(512). Let A" be the barycentric product B'*C', and define B" and C" cyclically. A", B", C" are collinear on line X(669)X(688). The lines AA", BB", CC" concur in X(1576). (Randy Hutson, August 19, 2019)
X(1576) lies on these lines: 3,1177 6,157 32,1084 50,237 99,827 107,933 110,351 160,206 163,692 250,523 338,1316 662,1492
X(1576) = midpoint of X(648) and X(1632)
X(1576) = isogonal conjugate of X(850)
X(1576) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,1501), (250,6), (827,110), (933,112)
X(1576) = cevapoint of X(i) and X(j) for these (i,j): (32,669), (39,647), (51,512)
X(1576) = X(i)-cross conjugate of X(j) for these (i,j): (669,32), (1501,249)
X(1576) = crosspoint of X(110) and X(112)
X(1576) = crosssum of X(523) and X(525)
X(1576) = crossdifference of every pair of points on line X(115)X(127)
X(1576) = barycentric product of PU(2)
X(1576) = barycentric product of vertices of circumtangential triangle
X(1576) = trilinear pole of line X(32)X(184)
X(1576) = X(92)-isoconjugate of X(525)
X(1576) = X(1577)-isoconjugate of X(2)
X(1576) = barycentric product X(1379)*X(1380)
X(1576) = barycentric product X(3)*X(112)
X(1576) = barycentric product X(6)*X(110)
X(1576) = polar conjugate of isotomic conjugate of X(32661)
X(1576) = vertex conjugate of X(36839) and X(36840)
X(1576) = X(63)-isoconjugate of X(14618)
Let A'B'C' be the excentral triangle, and let U be the bianticevian conic of X(1) and X(4). Let T be the tangential triangle, wrt the anticevian triangle of X(19), of U. Then A'B'C' and T are perspective, and their perspector is X(1577). (Randy Hutson, December 26, 2015)
X(1577) is the trilinear multiplier for the Kiepert hyperbola. (The trilinear product of X(1577) and the circumcircle is the Kiepert hyperbola.) (Randy Hutson, August 19, 2019)
X(1577) lies on these lines: 1,810 115,1111 163,811 240,522 514,661 667,814 784,149 798,812 826,1089
X(1577) = isogonal conjugate of X(163)
X(1577) = isotomic conjugate of X(662)
X(1577) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,1109), (76,1111), (693,523), (799,75), (811,1), (823,92)
X(1577) = cevapoint of X(656) and X(661)
X(1577) = X(i)-cross conjugate of X(j) for these (i,j): (115,1089), (1109,75)
X(1577) = crosspoint of X(i) and X(j) for these (i,j): (75,799), (82,162), (92,823), (662,2167), (811,1969), (1240,1978)
X(1577) = crosssum of X(i) and X(j) for these (i,j): (31,798), (38,656), (48,822), (649,2260), (652,2269), (661,1953), (1923,1924)
X(1577) = crossdifference of every pair of points on line X(31)X(48)
X(1577) = X(i)-aleph conjugate of X(j) for these (i,j): (648,656), (811,1969)
X(1577) = complement of X(4560)
X(1577) = bicentric difference of PU(14)
X(1577) = PU(14)-harmonic conjugate of X(1930)
X(1577) = trilinear product of PU(40)
X(1577) = perspector of hyperbola {{A,B,C,X(75),X(92)}} (centered at X(4858))
X(1577) = center of circumconic that is locus of trilinear poles of lines passing through X(4858)
X(1577) = X(2)-Ceva conjugate of X(4858)
X(1577) = X(6)-isoconjugate of X(110)
X(1577) = pole wrt polar circle of trilinear polar of X(162) (line X(1)X(19))
X(1577) = X(48)-isoconjugate (polar conjugate) of X(162)
X(1577) = X(52)-isoconjugate of X(32692)
X(1577) = trilinear pole of line X(1109)X(2632)
X(1578) lies on these lines: 3,6 394,488 485,1368 1038,1335 1040,1124
X(1578) = inverse-in-Brocard-circle of X(1579)
X(1579) lies on these lines: 3,6 394,487 486,1368 1038,1124 1040,1335
X(1579) = inverse-in-Brocard-circle of X(1578)
X(1580) lies on these lines: 1,21 6,256 41,43 75,560 87,604 171,172 238,1284 239,1281 284,1045 661,830 662,922
X(1580) = isogonal conjugate of X(1581)
X(1580) = isotomic conjugate of X(1934)
X(1580) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39043), (1755,1955), (1910,1), (1967,1582)
X(1580) = X(2236)-cross conjugate of X(1966)
X(1580) = crosssum of X(i) and X(j) for these (i,j): (1755,1964), (1965,1966)
X(1580) = crossdifference of every pair of points on line X(38)X(661)
X(1580) = X(98)-aleph conjugate of X(1755)
X(1580) = perspector of conic {A,B,C,PU(36)}
X(1580) = trilinear product X(171)*X(238)
X(1580) = perspector of unary cofactor triangles of Gemini triangles 32 and 34
X(1581) lies on these lines: 10,257 37,256 65,291 82,662 171,292 733,831 759,805 876,882
Let DEF and D'E'F' be the 1st and 2nd Sharygin triangles. Let A' be the trilinear product D*D', and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1581). (Randy Hutson, December 26, 2015)
X(1581) = isogonal conjugate of X(1580)
X(1581) = isotomic conjugate of X(1966)
X(1581) = cevapoint of X(i) and X(j) for these (i,j): (1755,1964), (1965,1966)
X(1581) = X(i)-cross conjugate of X(j) for these (i,j): (1821,1956), (1959,1), (2227,75)
X(1581) = trilinear pole of PU(35) (line X(38)X(661))
X(1581) = trilinear product of circumcircle intercepts of line PU(11)
X(1581) = trilinear product X(256)*X(291)
X(1581) = trilinear product of Steiner circumellipse intercepts of line PU(1)
X(1581) = areal center of cevian triangles of PU(8)
X(1582) lies on these lines: 1,19 6,291 75,560 82,662 238,992 978,1472
X(1582) = isotomic conjugate of X(9239)
X(1582) = X(1967)-Ceva conjugate of X(1580)
X(1582) = crosssum of PU(35)
X(1582) = crosspoint of PU(36)
X(1582) = intersection of tangents at PU(36) to conic {{A,B,C,PU(36)}}
As a point on the Euler line, X(1583) has Shinagawa coefficients (E + S, -S).
X(1583) lies on these lines: 2,3 6,493 371,394
X(1583) = inverse-in-orthocentroidal-circle of X(1591)
As a point on the Euler line, X(1584) has Shinagawa coefficients (E - S, S).
X(1584) lies on these lines: 2,3 6,494 372,394
X(1584) = inverse-in-orthocentroidal-circle of X(1592)
As a point on the Euler line, X(1585) has Shinagawa coefficients (F, S).
X(1585) lies on these lines: 2,3 53,590 264,491 275,486 317,492 343,638 393,493 394,637 490,1321
X(1585) = inverse-in-orthocentroidal-circle of X(1586)
X(1585) = cevapoint of X(1599) and X(1993)
X(1585) = X(i)-cross conjugate of X(j) for these (i,j): (371,492), (1993,1586)
X(1585) = polar conjugate of X(485)
As a point on the Euler line, X(1586) has Shinagawa coefficients (F, -S).
X(1586) lies on these lines: 2,3 53,615 264,494 317,491 343,637 393,494 394,638 489,1322
X(1586) = polar conjugate of X(486)
X(1586) = inverse-in-orthocentroidal-circle of X(1585)
X(1586) = cevapoint of X(1600) and X(1993)
X(1586) = X(i)-cross conjugate of X(j) for these (i,j): (372,491), (1993,1585)
X(1587) lies on these lines: 2,372 4,6 20,371 193,637 194,487 376,1151 388,1335 394,1587 486,1131 497,1124 590,631 639,1270 1132,1327
X(1588) lies on these lines: 2,371 4,6 20,372 193,638 194,488 376,1152 388,1124 394,1588 485,1132 497,1335 615,631 640,1271 1131,1328
As a point on the Euler line, X(1589) has Shinagawa coefficients (F + S, -S).
X(1589) lies on these lines: 2,3 343,487 394,488
As a point on the Euler line, X(1590) has Shinagawa coefficients (F - S, S).
X(1590) lies on these lines: 2,3 343,488 394,487
As a point on the Euler line, X(1591) has Shinagawa coefficients (E + S, S).
X(1591) lies on these lines: 2,3 343,639 394,485
X(1591) = complement of X(1599)
X(1591) = inverse-in-orthocentroidal-circle of X(1583)
X(1591) = X(1306)-Ceva conjugate of X(523)
As a point on the Euler line, X(1592) has Shinagawa coefficients (E - S, -S).
X(1592) lies on these lines: 2,3 343,640 394,486
X(1592) = complement of X(1600)
X(1592) = inverse-in-orthocentroidal-circle of X(1584)
X(1592) = X(1307)-Ceva conjugate of X(523)
As a point on the Euler line, X(1593) has Shinagawa coefficients (F, E - F).
In the plane of a triangle ABC, let
Γ = circumcircle
DEF = medial triangle
E' = Γ-antipode of E
(A) = circle that passes through E,F,E'
A' = center of (A), and define B' and C' cyclically.
Then the triangles ABC and A'B'C' are homothetic, and their homothetic center is X(1593). (Angel Montesdeoca, February 14, 2023)
X(1593) lies on these lines: 1,1037 2,3 6,64 19,1212 33,56 34,55 51,1204 74,1112 84,1473 184,1498 208,1466 264,1105 578,1181 607,672 1155,1452 1208, 1471
X(1593) = reflection of X(i) in X(j) for these (i,j): (4,1595), (1181,578)
X(1593) = complement of X(37201)
X(1593) = anticomplement of X(6823)
X(1593) = circumcircle-inverse of X(37931)
X(1593) = orthocentroidal-circle-inverse of X(235)
X(1593) = crosspoint of X(4) and X(1595)
X(1593) = crosssum of X(3) and X(1181)
X(1593) = polar conjugate of X(37874)
X(1593) = polar-circle-inverse of complement of X(37944)
X(1593) = homothetic center of orthic triangle and reflection of tangential triangle in X(3)
X(1593) = homothetic center of tangential triangle and reflection of orthic triangle in X(4)
X(1593) = exsimilicenter of circumcircle and incircle of orthic triangle if ABC is acute; the insimilicenter is X(25)
X(1593) = X(1697)-of-orthic-triangle if ABC is acute
X(1593) = {X(3),X(4)}-harmonic conjugate of X(25)
X(1593) = {X(12171),X(12172)}-harmonic conjugate of X(12167)
As a point on the Euler line, X(1594) has Shinagawa coefficients (2F, E + 2F).
Traian Lalescu (Trajan Lalesco) proved in "A Class of Remarcable Triangles," Gazeta Matematica 20 (1915) 213 [in Romanian], that if triangles DEF and D'E'F' are inscribed in a circle and directed arclengths satisfyarc DD' + arc EE' + arc FF' = 0 mod 2π, then the Simson lines of D,E,F with respect to D',E',F' and the Simson lines of D',E',F' with respect to D,E,F concur in the midpoint X of the segment of the orthocenters of DEF and D'E'F'. Daniel Vacaretu considered triangles DEF and D'E'F' associated with left and right isoscelizers and inscribed in the sine-triple-angle circle. He obtained the second set of trilinears shown above for the midpoint X. (See also the bicentric pair PU(61).)
In Episodes in Nineteenth and Twentieth Century Euclidean Geometry,, page 132, Ross Honsberger presents X(1594) as the orthopole of the six sides of two triangles and as the point common to six Simson lines. Honsberger calls this orthopole the Rigby Point. (Notes on Lalescu and Honsberger received from D. Vacaretu, 19/16/03)
X(1594) lies on these lines: 2,3 6,70 50,252 53,566 67,1173 96,275 125,389 128,136 232,1508 264,847 325,1235 933,1166 1209,1216 1225,1238
X(1594) = inverse-in-nine-point-circle of X(186)
X(1594) = circumcircle-inverse of X(37932)
X(1594) = orthocentroidal-circle-inverse of X(24)
X(1594) = complementary conjugate of X(32391)
X(1594) = X(933)-Ceva conjugate of X(523)
X(1594) = crosspoint of X(i) and X(j) for these (i,j): (4,93), (264,275)
X(1594) = crosssum of X(i) and X(j) for these (i,j): (3,49), (184,216)
X(1594) = X(35)-of-orthic-triangle if ABC is acute
X(1594) = excentral-to-ABC functional image of X(35)
X(1594) = polar conjugate of isotomic conjugate of X(37636)
X(1594) = {X(4),X(5)}-harmonic conjugate of X(403)
As a point on the Euler line, X(1595) has Shinagawa coefficients (F, 2E + F).
X(1595) lies on these lines: 2,3 33,496 34,495 39,53 578,1503
X(1595) = midpoint of X(4) and X(1593)
X(1595) = inverse-in-orthocentroidal-circle of X(1598)
X(1595) = {X(4),X(5)}-harmonic conjugate of X(1596)
X(1595) = X(3295)-of-orthic-triangle if ABC is acute
As a point on the Euler line, X(1596) has Shinagawa coefficients (F, -2E + F).
X(1596) lies on these lines: 2,3 33,495 34,496 53,115
X(1596) = midpoint of X(4) and X(25)
X(1596) = reflection of X(1368) in X(5)
X(1596) = circumcircle-inverse of X(37933)
X(1596) = nine-point-circle-inverse of X(37984)
X(1596) = orthocentroidal-circle-inverse of X(1597)
X(1596) = complementary conjugate of complement of X(35512)
X(1596) = {X(4),X(5)}-harmonic conjugate of X(1595)
X(1596) = X(999)-of-orthic-triangle if ABC is acute
X(1596) = center of inverse-in-polar-circle-of-de-Longchamps-line
X(1596) = polar conjugate of isotomic conjugate of X(37648)
X(1596) = homothetic center of Ehrmann mid-triangle and 3rd pedal triangle of X(4)
X(1596) = Ehrmann-side-to-orthic similarity image of X(18531)
As a point on the Euler line, X(1597) has Shinagawa coefficients (F, 2E - F).
X(1597) lies on these lines: 2,3 33,999 64,389 578,1498
X(1597) = complement of X(35513)
X(1597) = circumcircle-inverse of X(37934)
X(1597) = orthocentroidal-circle-inverse of X(1596)
X(1597) = {X(3),X(4)}-harmonic conjugate of X(1598)
X(1597) = polar conjugate of isogonal conjugate of X(33871)
X(1597) = center of conic that is locus of {X(4),P}-harmonic conjugate of Q, where P and Q lie on the circumcircle and are collinear with X(4)
As a point on the Euler line, X(1598) has Shinagawa coefficients (F, -2E - F).
X(1598) lies on these lines: 1,1057 2,3 34,999 51,1181 154,578 155,1351 389,1498 399,1112
X(1598) = circumcircle-inverse of X(37935)
X(1598) = orthocentroidal-circle-inverse of X(1595)
X(1598) = {X(3),X(4)}-harmonic conjugate of X(1597)
X(1598) = X(3333)-of-orthic-triangle if ABC is acute
X(1598) = homothetic center of 3rd pedal triangle of X(4) and 3rd antipedal triangle of X(3)
As a point on the Euler line, X(1599) has Shinagawa coefficients (E + 2S, -2S).
X(1599) lies on these lines: 2,3 6,588 394,1151
X(1599) = anticomplement of X(1591)
X(1599) = X(1585)-Ceva conjugate of X(1993)
As a point on the Euler line, X(1600) has Shinagawa coefficients (E - 2S, 2S).
X(1600) lies on these lines: 2,3 6,589 394,1152
X(1600) = anticomplement of X(1592)
X(1600) = X(i)-Ceva conjugate of X(j) for these (i,j): (346,6), (1586,1993)
TCC Perspectors 1601-1634
The transformation T carries triangle centers to triangle centers. The appearance of i → j in the following list means that T(X(i)) = X(j):
1 → 3, 2 → 22, 3 → 1498, 4 → 24, 5 → 1601, 6 → 6, 7 → 1602, 8 → 1603, 9 → 1604, 13 → 1605, 14 → 1606, 15 → 24303, 17 → 1607, 18 → 1608, 19 → 1609, 21 → 1610, 25 → 1611, 28 → 1612, 31 → 1613, 32 → 33786, 35 → 33669, 54 → 1614, 55 → 1615, 56 → 1616, 57 → 1617, 58 → 595, 59 → 1618, 63 → 1619, 64 → 1620, 75 → 33801, 76 → 33802, 81 → 1621, 83 → 1078, 84 → 1622, 86 → 23374, 88 → 1623, 162 → 1624, 163 → 1625, 174 → 1626, 188 → 2933, 249 → 33803, 251 → 1627, 254 → 1628, 259 → 198, 266, → 56, 275 → 1629, 284 → 1630, 365 → 55, 366 → 1631, 508 → 23852, 509 → 1486, 512 → 33704, 523 → 30715, 648 → 1632, 651 → 1633, 662 → 1634, 1126 → 33771, 1171 → 33774, 2153 → 11142, 2154 → 11141, 3445 → 33804, 5374 → 159, 6727 → 23846, 6733 → 23845, 14085 → 101, 14086 → 30715, 14089 → 99, 14090 → 33704, 18297 → 23849, 18753 → 2176, 20034 → 25
X(1601) lies on these lines: 3,128 6,1166
X(1601) = X(60)-of-tangential-triangle if ABC is acuteX(1602) lies on these lines: 7,1486 22,1626 24,242
X(1603) lies on this line: 8,197 24,1324
X(1604) lies on these lines: 3,9 25,1863
X(1604) = isogonal conjugate of X(34546)
X(1604) = circumcircle-inverse-of X(17112)
X(1604) = X(346)-Ceva conjugate of X(6)
X(1605) lies on this line: 3,618 26,1607
X(1605) = circumcircle-inverse of X(33499)X(1606) lies on this line: 3,619 26,1608
X(1606) = circumcircle-inverse of X(33501)X(1607) lies on this line: 3,619 26,1605
X(1608) lies on this line: 3,618 26,1606
X(1609) lies on these lines: 3,6 24,254 25,53 48,1195 112,1299 159,237 186,1249 590,1583 615,1584
X(1609) = X(i)-Ceva conjugate of X(j) for these (i,j): (24,25), (393,6)
X(1609) = crosspoint of X(107) and X(249)
X(1609) = crosssum of X(115) and X(520)
X(1609) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36747)
X(1609) = {X(371),X(372)}-harmonic conjugate of X(36747)
X(1610) lies on these lines: 1,19 6,959 8,197 20,1633 24,944 172,910 198,958 315,1310
X(1611) lies on these lines: 2,6 3,1196 232,1033 1498,1513
X(1611) = X(i)-Ceva conjugate of X(j) for these (i,j): (459,25), (2207,6)
X(1611) = crosssum of X(520) and X(1084)
X(1611) = center of bicevian conic of PU(4)
X(1612) lies on these lines: 1,21 6,943 55,387 681,1924 1006,1104
X(1612) = crosspoint of X(107) and X(765)
X(1612) = crosssum of X(244) and X(520)
X(1613) lies on these lines: 1,1197 2,6 3,695 25,694 110,699 154,237 305,732 511,1196
X(1613) = isogonal conjugate of X(2998)
X(1613) = isogonal conjugate of the complement of X(32747)
X(1613) = X(i)-Ceva conjugate of X(j) for these (i,j): (32,6)
X(1613) = X(194)-cross conjugate of X(6)
X(1613) = crosspoint of X(i) and X(j) for these (i,j): (1424,1740)
X(1613) = crosssum of X(523) and X(1084)
X(1613) = crossdifference of every pair of points on line X(512)X(625) (complement of Lemoine axis)
X(1613) = X(92)-isoconjugate of X(3504)
X(1613) = trilinear pole of polar wrt 2nd Brocard circle of perspector of 2nd Brocard circle
X(1613) = vertex conjugate of PU(148)
Let P1 and P2 be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. X(1614) is the crosspoint of P1 and P2. (Randy Hutson, August 29, 2018)
Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anti-Euler triangle at X(1614). Also, X(1614) is the exsimilicenter of the circumcircle of ABC and nine-point circle of A'B'C'. (Randy Hutson, August 29, 2018)
> X(1614) lies on these lines: 3,74 4,184 6,1173 20,1147 22,155 23,52 24,154 30,49 51,1199 70,1176 185,186 376,1092 378,1498 389,1495 546,567 1503,1594
X(1614) = isogonal conjugate of X(6662)
X(1614) = Vu tangential transform of X(3)
X(1614) = X(12)-of-tangential-triangle if ABC is acute
X(1614) = tangential isogonal conjugate of X(1601)
X(1614) = {X(3),X(156)}-harmonic conjugate of X(110)
X(1614) = exsimilicenter of circumcircle and nine-point circle of tangential triangle
X(1614) = intouch-to-ABC functional image of X(12)
X(1615) lies on these lines: 6,57 56,1200 165,220
X(1615) = X(i)-Ceva conjugate of X(j) for these (i,j): (165,55), (200,6)
X(1616) lies on these lines: 1,6 42,1293 55,1201 221,1319 595,999 614,3057 902,1392 962,1086 1035,1457 1407,1420
X(1616) = isogonal conjugate of X(6553)
X(1616) = polar conjugate of isotomic conjugate of X(23089)
X(1616) = X(i)-Ceva conjugate of X(j) for these (i,j): (1407,6), (1420,56)
X(1616) = crosspoint of X(934) and X(1016)
X(1617) lies on these lines: 1,3 6,1174 7,1621 22,347 25,105 31,222 41,1202 42,1471 109,1407 219,604 221,595 226,1001 388,405 479,934 518,1260 602,1066 614,1465 910,1108 1006,1056 1055,1200 1279,1427 1362,1397 1384,1415
X(1617) = isogonal conjugate of X(6601)
X(1617) = X(i)-Ceva conjugate of X(j) for these (i,j): (279,6), (1252,109), (1445,218)
X(1617) = crosspoint of X(59) and X(934)
X(1617) = pole wrt circumcircle of line X(513)X(676) (the trilinear polar of X(279))
X(1617) = polar conjugate of isotomic conjugate of X(23144)
X(1617) = {X(55),X(56)}-harmonic conjugate of X(57)
X(1617) = {X(3513),X(3514)}-harmonic conjugate of X(3)
X(1617) = inverse-in-circumcircle of X(3660)
X(1617) = {X(13388),X(13389)}-harmonic conjugate of X(37597)
X(1618) lies on these lines: 109,1459 110,901 513,651
X(1618) = reflection of X(59) in X(692)
X(1619) lies on these lines: 3,64 22,69 25,1503 161,542
X(1619) = crosssum of (122,512)
X(1620) lies on these lines: 3,6 154,1204 186,1498
X(1621) lies on these lines: 1,21 2,11 3,962 7,1617 8,405 9,1174 22,1486 35,404 36,551 37,82 42,238 43,748 99,873 145,958 171,902 213,1206 226,1005 278,1013 329,954 411,946 517,1006 739,932 985,1255
X(1621) = complement of X(33110)
X(1621) = X(i)-Ceva conjugate of X(j) for these (i,j): (1252,100), (1509,6)
X(1621) = crosspoint of X(99) and X(765)
X(1621) = crosssum of (244,512)
X(1621) = tangential-isogonal conjugate of X(35212)
X(1621) = {X(1),X(31)}-harmonic conjugate of X(81)
X(1622) lies on these lines: 1,3 6,947 84,963
X(1623) lies on these lines: 3,8 23,105 36,1168
X(1624) lies on these lines: 3,113 25,132 110,351 112,1301 852,1503 925,1302 933,1304
X(1625) lies on the Johnson circumconic and these lines: 5,217 6,13 110,112
X(1625) = midpoint of X(3289) and X(3331)
X(1625) = isogonal conjugate of X(15412)
X(1625) = X(75)-Ceva conjugate of X(6)
X(1625) = crosspoint of X(i) and X(j) for these (i,j): (110,648), (107,110)
X(1625) = crosssum of (523,647), (520,523)
X(1625) = crossdifference of every pair of points on line X(125)X(526)
X(1625) = barycentric product X(112)*X(343)
X(1626) lies on these lines: 3,10 22,1602 38,55 982,1283
X(1626) = X(85)-Ceva conjugate of X(6)
X(1626) = isogonal conjugate of isotomic conjugate of X(21285)
X(1626) = polar conjugate of isotomic conjugate of X(22125)
X(1627) lies on these lines: 2,32 3,1180 22,1184 23,1196 25,111 110,699 187,1194 571,1370 609,612
X(1627) = isogonal conjugate of X(6664)
X(1627) = Vu tangential transform of X(6)
As a point on the Euler line, X(1628) has Shinagawa coefficients (2(E + F)2E2F - 2(E + 2F)EFS2 + 4FS4, [(E2 + 6EF + 4F2)E - 4(E + F)S2]S2).
X(1628) lies on this line: 2,3
X(1629) lies on these lines: 4,54 22,264 23,324 25,98 251,393 436,1495 1093,1179
X(1629) = polar conjugate of X(36952)
X(1629) = pole wrt polar circle of trilinear polar of X(36952) (line X(684)X(2525))
X(1630) lies on these lines: 1,19 101,102 109,577 1055,1195
X(1631) lies on these lines: 3,142 6,560 22,1602 25,1826 48,674 55,199 198,480 573,692 789,1502
X(1631) = isogonal conjugate of X(7357)
X(1631) = X(75)-Ceva conjugate of X(6)
X(1631) = crossum of X(116) and X(513)
X(1632) lies on these lines: 98,338 99,670 110,925 112,1289 157,264 250,523 476,1302 827,1286 933,1288
X(1632) = reflection of X(648) in X(1576)
X(1632) = cevapoint of X(157) and X(523)
X(1632) = crosspoint of X(99) and X(107)
X(1632) = crosssum of X(512) and X(520)
X(1632) = perspector of ABC and side triangle of circumanticevian triangles of X(2) and X(4)
X(1633) lies on these lines: 7,1486 19,1721 20,1610 28,1770 48,1742 59,1310 99,1310 100,190 101,1292 105,1086 108,109 497,1473
X(1633) = reflection of X(651) in X(692)
X(1633) = X(1275)-Ceva conjugate of X(6)
X(1633) = cevapoint of X(513) and X(1486)
X(1633) = crosspoint of X(i) and X(j) for these (i,j): (99,162),
(100,934)
X(1633) = crosssum of X(512) and X(656)
X(1633) = X(i)-aleph conjugate of X(j) for these (i,j): (100,610),
(1783,1707), (1897,19)
X(1634) lies on these lines: 3,67 6,694 69,160 99,670 110,351 112,907 237,524 660,765 1306,1307
X(1634) = complement of X(25051)
X(1634) = isogonal conjugate of anticomplementary conjugate of X(39346)
X(1634) = crossdifference of every pair of points on line X(115)X(804) (the tangent to the nine-point circle at X(115))
X(1634) = X(19)-isoconjugate of X(4580)
X(1634) = isotomic conjugate of polar conjugate of X(35325)
X(1634) = X(688)-cross conjugate of X(6)
X(1634) = crosspoint of X(99) and X(110)
X(1634) = crosssum of X(512) and X(523)
Tripolar Centroids 1635-1651
Suppose X = x : y : z (trilinears) is a point other than X(2), and define the tripolar centroid of X as the point TG(X) given by
TG(X) = x(by - cz)(by + cz - 2ax) : y(cz - ax)(cz + ax - 2by) : z(ax - by)(ax + by - 2cz).
TG(X) is the centroid of the points BC∩B'C', CA∩C'A', AB∩A'B', where A'B'C' denotes the cevian triangle of X. The notions of centroid and tripolar centroid were contributed by Darij Grinberg, August 24, 2003.
The appearance of (i,j) in the following list means that TG(X(i)) = X(j):
(1,1635), (3,1636), (4,1637), (5,14391), (6,351), (7,1638), (8,1639), (9,14392), (10,4120), (11,14393), (12,14394), (13,9200), (14,9201), (17,14446), (18,14447), (21,14395), (22,14396), (24,14397), (25,14398), (27,11125), (28,14399), (29,14400), (30,14401), (31,14402), (32,14403), (37,14404), (38,14405), (39,14406), (42,14407), (43,14408), (44,14409), (45,14410), (55,14411), (56,14412), (57,14413), (63,14414), (65,14415), (66,14417), (69,14417), (75,4728), (76,9148), (78,14418), (81,14419), (83,14420), (86,4750), (88,14421), (89,14422), (98,1640), (99,1641}, {100,1642), (105,1643), (111,9171), (115,14423), (141,14424), (145,14425), (190,1644), (192,14426), (200,14427), (239,4448), (251,14428), (262,3569), (263,2491), (298,9204), (299,9205), (306,14429), (312,14430), (321,14431), (333,14432), (350,14433), (385,11183), (512,1645), (513,1646), (514,1647}, {519,6544), (523,1648), (524,1649), (525,1650), (536,14434), (551,14435), (648,1651), (671,8371), (869,14436), (899,14437), (957,3310), (985,14438), (1002,665), (1022,244), (1125,4984), (1026,14439), (1267,14440), (1646,14441), (1647,14442), (1648,14443), (1649,14444), (1698,4958), (1916,11182), (1976,6041}, {1992,9125), (2394,125), (2395,6784), (2396,6786), (2403,3756), (2407,5642), (2408,6791), (2409,6793), (2418,12036), (2419,12037), (2421,9155), (3413,13636}, {3414,13722), (3616,4773), (4049,3120), (4240,3163), (5391,14445), (5466,115), (5468,2482), (6548,1086), (9178,3124), (9213,2088), (9221,2081), (13582,1116), (14223,868)
X(1635) lies on these lines: 2,812 44,513 88,1022 100,101 105,1024 244,665 900,1644
X(1635) = reflection of X(1962) in X(351)
X(1635) = isogonal conjugate of X(3257)
X(1635) = complement of X(21297)
X(1635) = anticomplement of X(4928)
X(1635) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,38979), (44,2087), (88,244), (104,2310), (662,214), (1022,513), (1023,44), (2161,2170)
X(1635) = X(2087)-cross conjugate of X(44)
X(1635) = crosspoint of X(i) and X(j) for these (i,j): (44,1023), (57,2222), (88,100), (101,909), (190,1120), (513,1022), (662,759)
X(1635) = crosssum of X(i) and X(j) for these (i,j): (1,1635), (44,513), (88,1022), (100,1023), (514,908), (649,1149), (661,758)
X(1635) = bicentric sum of PU(34)
X(1635) = perspector of hyperbola {{A,B,C,X(1),X(44)}}
X(1635) = PU(34)-harmonic conjugate of X(244)
X(1635) = trilinear pole of line X(2087)X(3251)
X(1635) = centroid of antiorthic axis intercepts with sidelines of ABC
X(1635) = X(647)-of-2nd-extouch-triangle
X(1636) lies on these lines: 110,112 520,647 525,3268 1637,1651
X(1636) = crosspoint of X(648) and X(1294)
X(1636) = crosssum of X(i) and X(j) for these (i,j): (4,1637), (523,1990)
X(1636) = X(2)-Ceva conjugate of X(38999)
X(1636) = perspector of hyperbola {{A,B,C,X(3),X(30)}}
X(1636) = intersection of perspectrix of ABC and orthocentroidal triangle (line X(1636)X(1637)) and perspectrix of ABC and anti-orthocentroidal triangle (line X(520)X(647))
Let A'B'C' and A"B"C" be the cevian and anticevian triangles of X(4), resp. Let OA be the cevian circle of A". Let A* be the intersection, other than A', of OA and line BC. Define B* and C* cyclically. X(1637) is the centroid of triangle A*B*C*. (Randy Hutson, August 28, 2020)
X(1637) lies on these lines: 2,3268 98,111 107,112 115,125 132,1560 230,231 1499,1514 1636,1651
X(1637) = complement of X(3268)
X(1637) = X(1989)-Ceva conjugate of X(115)
X(1637) = crosspoint of X(2) and X(476)
X(1637) = crosssum of X(i) and X(j) for these (i,j): (3,1636), (6,526)
X(1637) = crossdifference of every pair of points on line X(3)X(74)
X(1637) = PU(4)-harmonic conjugate of X(6103)
X(1637) = midpoint of circumcenters of X(13)X(14)X(15) and X(13)X(14)X(16)
X(1637) = perspector of circumconic centered at X(3258)
X(1637) = center of circumconic that is locus of trilinear poles of lines passing through X(3258)
X(1637) = X(2)-Ceva conjugate of X(3258)
X(1637) = centroid of orthic axis intercepts with sidelines of ABC
X(1637) = tripolar centroid of X(4) wrt orthic triangle
X(1637) = center of Dao-Moses-Telv circle
X(1637) = X(115) of 2nd Parry triangle
X(1637) = insimilicenter of circles {{X(98),X(107),X(125),X(132)}} and {{X(6),X(111),X(112),X(115),X(187),X(1560)}}; the exsimilicenter is X(6103)
X(1637) = radical center of orthocentroidal circles of ABC, orthocentroidal triangle, anti-orthocentroidal triangle
X(1637) = inverse-in-Hutson-Parry-circle of X(1640)
X(1637) = {X(13636),X(13722)}-harmonic conjugate of X(1640)
X(1637) = centroid of (degenerate) side triangle of 3rd and 4th isodynamic-Dao triangles
Let Oa be the circle centered at A with radius k(b + c -a) (for some constant k>0), and define Ob and Oc cyclically. Let A' be the exsimilicenter of Ob and Oc, and define B' and C' cyclically. The centroid of (degenerate) triangle A'B'C' = X(1638). This is independent of the choice of k. (Randy Hutson, January 29, 2018)
X(1638) lies on these lines: 2,918 11,244 57,654 88,673 241,514 354,926 651,658
X(1638) = reflection of X(1639) in X(2)
X(1638) = centroid of Gergonne line intercepts with sidelines of ABC
X(1638) = X(351)-of-intouch-triangle
X(1638) = complement of X(30565)
X(1639) lies on these lines: 2,918 9,654 11,1146 210,926 522,650
X(1639) = reflection of X(1638) in X(2)
X(1639) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 15
X(1640) lies on these lines: 2,525 4,1499 6,523 39,647 51,512 115,125
X(1640) = radical center of Brocard circle, orthocentroidal circle, and orthosymmedial circleX(1641) lies on these lines: 2,6 351,690
X(1641) = reflection of X(1648) in X(2)
X(1641) = centroid of line X(2)X(6) intercepts with sidelines of ABC
X(1641) = intersection of tangents to circle {X(2),X(110),X(2770),X(5463),X(5464)} at X(5463) and X(5464)
X(1641) = centroid of degenerate cross-triangle of anticomplementary and Schroeter triangles
X(1642) lies on these lines: 1,6 241,1025
X(1643) lies on these lines: 1,650 6,513 42,663 57,1022 244,665 649,764
X(1644) lies on these lines: {1,2}, {900,1635}
X(1644) = reflection of X(1647) in X(2)
X(1644) = centroid of Nagel line intercepts with sidelines of ABC
X(1645) lies on these lines: 2,39 351,865
X(1645) = X(i)-Ceva conjugate of X(j) for these (i,j): (538,888),
(728,888)
X(1645) = crosspoint of X(i) and X(j) for these (i,j): (538,888),
(728,888)
X(1646) lies on these lines: 2,37 88,292 244,665
X(1646) = isogonal conjugate of X(5381)
X(1646) = X(i)-Ceva conjugate of X(j) for these (i,j): (536,891), (738,891)
X(1646) = crosspoint of X(i) and X(j) for these (i,j): (536,891), (738,891)
X(1646) = crosssum of X(739) and X(898)
X(1647) lies on these lines: 1,2 11,244 80,106 149,1054
X(1647) = reflection of X(1644) in X(2)
X(1647) = X(i)-Ceva conjugate of X(j) for these (i,j): (80,513), (519,900), (903,514), (1120,522)
X(1647) = crosspoint of X(i) and X(j) for these (i,j): (514,903), (519,900)
X(1647) = crosssum of X(i) and X(j) for these (i,j): (101,902), (106,901)
X(1648) lies on these lines: 2,6 115,125 669,865
X(1648) = reflection of X(1641) in X(2)
X(1648) = isotomic conjugate of isogonal conjugate of X(21906)
X(1648) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1649), (67,512), (468,351), (524,690), (671,523)
X(1648) = crosspoint of X(i) and X(j) for these (i,j): (523,671), (524,690)
X(1648) = crosssum of X(i) and X(j) for these (i,j): (110, 187), (111,691)
X(1648) = intersection of tangents to Hutson-Parry circle at X(13) and X(14)
X(1648) = pole wrt Hutson-Parry circle of Fermat axis
X(1648) = inverse-in-Hutson-Parry-circle of X(115)
X(1648) = perspector of circumconic centered at X(1649)
X(1648) = center of circumconic that is locus of trilinear poles of lines passing through X(1649)
X(1648) = intersection of line PU(40) (X(115)X(125)) and trilinear polar of cevapoint of PU(40)
X(1648) = X(2502)-of-4th-Brocard-triangle
X(1648) = X(2502)-of-orthocentroidal-triangle
X(1648) = centroid of (degenerate) cross-triangle of 4th Brocard and orthocentroidal triangle
X(1648) = trilinear pole of line X(2682)X(14443)
X(1648) = {X(13636),X(13722)}-harmonic conjugate of X(115)
X(1649) lies on the Kiepert parabola and these lines: 2,523 3,669 39,647 114,126 351,690
X(1649) = isogonal conjugate of X(34574)
X(1649) = crossdifference of every pair of points on line X(23)X(111)
X(1649) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1648), (99,524), (523,690)
X(1649) = crosspoint of X(i) and X(j) for these (i,j): (99,524), (523,690)
X(1649) = crosssum of X(i) and X(j) for these (i,j): (110, 691), (111,512)
X(1649) = perspector of circumconic centered at X(1648)
X(1649) = center of circumconic that is locus of trilinear poles of lines passing through X(1648)
X(1649) = center of circle {{X(2),X(110),X(2770),X(5463),X(5464)}}
X(1649) = trilinear pole of line X(14444)X(23992)
X(1649) = harmonic center of circles {{X(13),X(15),X(5463),X(5464)}} and {{X(14),X(16),X(5463),X(5464)}}
As a point on the Euler line, X(1650) has Shinagawa coefficients ((4E - 5F)F - S2, -3(E + F)F + S2).
Let W be the circumconic with center X(1650). One of the asymptotes of W is the Euler line. The other is in the direction of X(9033). For a sketch, click X(9033). (Angel Montesdeoca, April 19, 2016)
X(1650) lies on these lines: 2,3 122,125
X(1650) = reflection of X(1651) in X(2)
X(1650) = anticomplement of X(402)
X(1650) = nine-point-circle-inverse of X(37985)
X(1650) = X(i)-Ceva conjugate of X(j) for these (i,j): (265,520), (1294,523), (1494,525)
X(1650) = crosspoint of X(525) and X(1494)
X(1650) = crosssum of X(i) and X(j) for these (i,j): (74,1304), (110,2071), (112,1495)
X(1650) = complement of X(4240)
X(1650) = homothetic center of Gossard and medial triangles
As a point on the Euler line, X(1651) has Shinagawa coefficients (3(2E - 7F)F - S2,9(E + F)F - 3S2).
X(1651) lies on these lines: 2,3 1636,1637
X(1651) = reflection of X(i) in X(j) for these (i,j): (2,402), (1650,2)
X(1651) = Euler line intercept of trilinear polar of X(30)
X(1651) = centroid of Euler line intercepts with sidelines of ABC
X(1651) = X(2)-of-Gossard triangle
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
See also X(1276) and X(1277) for the 2nd and 3rd Evans perspectors.
X(1652) lies on these lines: {1,15}, {2,7}, {13,3464}, {17,3336}, {46,1277}, {56,5240}, {61,3468}, {65,5239}, {396,554}, {559,1100}, {3638,5011}, {4848,5245}
X(1652) = X(i)-Ceva conjugate of X(j) for these (i,j): (554,1),
(2160,1653)
X(1652) = X(554)-aleph conjugate of X(1652)
X(1652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3179,1276), (9,57,1653), (1400,3218,1653), (2285,3306,1653)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin
C)f(C,A,B)
X(1653) lies on these lines: 1,16 2,7 46,1276 395,1081 1082,1100
X(1653) = X(i)-Ceva conjugate of X(j) for these (i,j): (1081,1),
(2160,1652)
X(1653) = X(1081)-aleph conjugate of X(1653)
Let P be a point in the plane of, but not on a sideline of, triangle ABC. Let BA be the point where the line through P parallel to line BC meets line BA, and let CA be the point where the line through P parallel to line BC meets line CA. Define CB, AB, AC, and BC cyclically. If P = X(1654), then
|ABA| + |ACA| = |BCB| + |BAB| = |CAC| + |CBC|(Antreas Hatzipolakis, Anopolis #20, 1/20/02)
X(1654) lies on these lines: 2,6 8,192 10,894 37,319 71,1762 190,594
X(1654) = reflection of X(86) in X(1213)
X(1654) = isotomic conjugate of X(6625)
X(1654) = crosspoint of X(2) and X(8) wrt 2nd Conway triangle
X(1654) = anticomplement of X(86)
X(1654) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,2), (894,192)
X(1654) = anticomplementary isotomic conjugate of X(1)
X(1654) = complement of X(20090)
X(1654) = polar conjugate of isogonal conjugate of X(22139)
X(1654) = {X(2),X(2895)}-harmonic conjugate of X(17778)
X(1654) = perspector of Gemini triangle 39 and cross-triangle of Gemini triangles 39 and 40
X(1654) = {X(2),X(69)}-harmonic conjugate of X(17300)
Continuing from the description of X(1654), let h(B,A) be the distance from the point BA to the line CA, and define five other distances cyclically. If P = X(1655), then
h(B,A) + h(C,A) = h(C,B) + h(A,B) = h(A,C) + h(B,C)(Antreas Hatzipolakis, Anopolis #20, 1/20/02)
X(1655) lies on these lines: 2,39 8,192 21,385 193,452 350,1107 668,1500
X(1655) = anticomplement of X(274)
X(1655) = isotomic conjugate of isogonal conjugate of X(21779)
X(1655) = polar conjugate of isogonal conjugate of X(23079)
X(1655) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,2), (1909,8)
As a point on the Euler line, X(1656) has Shinagawa coefficients (3,1).
Let A' be the reflection of X(3) in A, and define B' and C' cyclically. The triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(1656).
X(1656) lies on these lines: 2,3 6,17 10,1482 11,498 12,499 49,569 51,1216 125,399 141,1351 302,634 303,633 355,1125 373,568 485,615 486,590 517,1698 567,1147 576,599
X(1656) = midpoint of X(5) and X(632)
X(1656) = reflection of X(i) in X(j) for these (i,j): (3,631), (631,632)
X(1656) = complement of X(631)
X(1656) = anticomplement of X(632)
X(1656) = circumcircle-inverse of X(37936)
X(1656) = orthocentroidal-circle-inverse of X(140)
X(1656) = {X(3),X(5)}-harmonic conjugate of X(381)
X(1656) = {X(17),X(18)}-harmonic conjugate of X(6)
X(1656) = {X(1506),X(7746)}-harmonic conjugate of X(6)
X(1656) = homothetic center of 2nd Euler triangle and mid-triangle of orthic and circumorthic triangles
X(1656) = homothetic center of submedial triangle and mid-triangle of orthic and circumorthic triangles
X(1656) = X(5)-of-cross-triangle-of-Euler-and-anti-Euler-triangles
X(1656) = X(1385)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(1656) = homothetic center of X(5)-altimedial and X(3)-anti-altimedial triangles
X(1656) = homothetic center of X(140)-altimedial and X(140)-anti-altimedial triangles
X(1656) = endo-homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle; the homothetic center is X(3843)
X(1656) = radical center of de Longchamps circles of ABC and 1st and 2nd Ehrmann circumscribing triangles
As a point on the Euler line, X(1657) has Shinagawa coefficients (3, -7).
Let La be the polar of X(3) wrt the A-power circle, and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the orthocenter of A'B'C' is X(1657). (Randy Hutson, December 2, 2017)
Let OA be the circle centered at the A-vertex of the Ehrmann mid-triangle and passing through A; define OB and OC cyclically. X(1657) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(1657) lies on these lines: 2,3 195,1181 399,1498 516,1482
X(1657) = reflection of X(i) in X(j) for these (i,j): (3,20), (4,550), (382,3)
X(1657) = complement of X(33703)
X(1657) = {X(381),X(382)}-harmonic conjugate of X(5076)
X(1657) = Ehrmann-mid-to-ABC similarity image of X(382)
X(1657) = circumcircle inverse of X(34152)
X(1657) = endo-homothetic center of Ehrmann mid-triangle and ABC-X3 reflections triangle; the homothetic center is X(382)
As a point on the Euler line, X(1658) has Shinagawa coefficients (E + 8F, -3E - 8F).
The vertices of the Kosnita triangle are the circumcenters of the triangles BOC, COA, AOB, where O is the circumcenter, X(3). (Darij Grinberg, 8/24/03)
Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb, Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A" = Lb∩Lc, and define B" and C" cyclically. Triangle A"B"C" is homothetic to ABC, and its nine-point center is X(1658). (Randy Hutson, July 20, 2016)
X(1658) lies on these lines: 2,3 54,568 143,578 569,973 1092,1511 1147,1154
X(1658) = midpoint of X(3) and X(26)
For a discussion of this point, see Paul Yiu,
Introduction to the Geometry of the Triangle, 2002, Article 3.5.4
Exercise 4d.
(The preceding Exercise 4c presents the Paasche point, X(1123),)
(Contributed by Darij Grinberg, 8/24/03)
See Francisco Javier García Capitán, Hyacinthos #21541, 2/14/2013.
X(1659) lies on these lines: 1,4 2,176 57,482 75,491 92,1585 553,1373
X(1659) = isogonal conjugate of X(2066)
X(1659) = X(482)-cross conjugate of X(7)
X(1659) = crosssum of X(48) and X(605)
Let AB be the point in which the line through A perpendicular to CA meets line BC, and define points AC, BC, BA, CA, CB functionally. Let
XA = midpoint{AB, AC},
YA = midpoint{BA, CA},
ZA = midpoint{BC, CB},
and define XB, XC, YB, YC, ZB, ZC functionally.
The lines AXA, BXB, CXC concur in
X(20).
The lines AYA, BYB, CYC concur in
X(393).
The lines AZA, BZB, CZC concur in
X(6).
The lines XAYA, XBYB,
XCYC concur in X(1660).
The lines YAZA, YBZB,
YCZC concur in X(3).
The lines ZAXA, ZBXB,
ZCXC concur in X(1661).
Contributed by Darij Grinberg, August 24, 2003; see Hyacinthos #7225.
X(1660) lies on these lines: 6,25 30,156 110,1370 394,1619 578,1596 1092,1498 1368,1503
X(1660) = midpoint of X(394) and X(1619)
X(1660) = X(20)-Ceva conjugate of X(577)
X(1660) = X(25)-of-A'B'C', as described by Tran Quang Huyng, ADGEOM #2697 (8/26/2015)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1661) is described at X(1660).
X(1661) lies on these lines: 25,393 154,577 1619,1624
X(1661) = isogonal conjugate of cyclocevian conjugate of X(35510)
X(1661) = X(20)-Ceva conjugate of X(6)
X(1661) = tangential-isogonal conjugate of X(33582)
Circle-related Points 1662-1706
σ = area of triangle ABC
ω = arccot(cot A + cot B + cot C) = arccot[(a2 +
b2 +c2)/(4σ)]; ω is the Brocard angle
of ABC
e = sqrt(1 - 4 sin2ω) (as in Gallatly, p. 96, along
with other formulas involving ω)
s = (a + b + c)/2 = semiperimeter of ABC
r = σ/s = inradius of ABC
Circles mentioned in this section are the following:
| Name | Center | Radius |
|---|---|---|
| circumcircle | X(3) | R |
| incircle | X(1) | r |
| nine-point circle | X(5) | R/2 |
| Brocard circle | X(182) | eR/(2 cos ω) |
| 1st Lemoine circle | X(182) | (1/2)(R sec ω) |
| 2nd Lemoine circle (cosine circle) | X(6) | abc/(a2 + b2 + c2) |
| Spieker circle | X(10) | r/2 |
| Apollonius circle | X(970) | (r2 + s2)/4r |
| Bevan circle | X(40) | 2R |
For further information on many circles, click MathWorld and scroll down to links to various specific circles.
X(1662) lies on these lines: 3,6 1676,2040 1677,2039
X(1662) = reflection of X(1663) in X(182)
X(1662) = inverse-in-Brocard-circle of X(1664)
X(1662) = {X(1687),X(1688)}-harmonic conjugate of X(1663)
X(1663) lies on these lines: 3,6 1676,2039 1677,2040
X(1663) = reflection of X(1662) in X(182)
X(1663) = inverse-in-Brocard-circle of X(1665)
X(1663) = {X(1687),X(1688)}-harmonic conjugate of X(1662)
X(1664) lies on this line: 3,6
X(1664) = reflection of X(1665) in X(182)
X(1664) = inverse-in-Brocard-circle of X(1662)
X(1665) lies on this line: 3,6
X(1665) = reflection of X(1664) in X(182)
X(1665) = inverse-in-Brocard-circle of X(1663)
X(1666) lies on these lines: 3,6 485,2040 486,2039
X(1666) = reflection of X(1667) in X(6)
X(1666) = inverse-in-Brocard-circle of X(1668)
X(1667) lies on these lines: 3,6 485,2039 486,2040
X(1667) = reflection of X(1666) in X(6)
X(1667) = inverse-in-Brocard-circle of X(1669)
X(1667) = inverse-in-2nd-Brocard-circle of X(2562)
X(1668) lies on these lines: 3,6 485,1348 486,1349 1124,1674 1335,1675 1377,1678 1378,1679 1702,1704 1703,1705
X(1668) = inverse-in-Brocard-circle of X(1666)
X(1668) = {X(6),X(182)}-harmonic conjugate of X(1669)
X(1669) lies on these lines: 3,6 485,1349 486,1348 1124,1675 1335,1674 1377,1679 1378,1678 1702,1705 1703,1704
X(1669) = inverse-in-Brocard-circle of X(1667)
X(1669) = {X(6),X(182)}-harmonic conjugate of X(1668)X(1670) is the external center of similitude of the Gallatly circle and the 2nd Lemoine circle. (Peter J. C. Moses, 9/03; cf. X(1342))
X(1670) and X(1671) are the Brocard axis intercepts of the 2nd Brocard circle. (Randy Hutson, August 29, 2018)
X(1670) lies on these lines: 3,6 76,1677 262,1676 485,2009 486,2010 1124,2007 1335,2008 1377,2013 1378,2014 1702,2017 1703,2018
X(1670) = reflection of X(1671) in X(3)
X(1670) = isogonal conjugate of X(1677)
X(1670) = circumcircle-inverse of X(38720)
X(1670) = Brocard-circle-inverse of X(1342)
X(1670) = X(76)-Ceva conjugate of X(1671)
X(1670) = Thomson-isogonal conjugate of X(33707)
X(1670) = {X(6),X(39)}-harmonic conjugate of X(1671)
X(1670) = {X(32),X(3094)}-harmonic conjugate of X(1671)
X(1670) = {X(182),X(3095)}-harmonic conjugate of X(1671)
X(1670) = {X(1379),X(1380)}-harmonic conjugate of X(38720)
X(1671) is the internal center of similitude of the Gallatly circle and the 2nd Lemoine circle. (Peter J. C. Moses, 9/03)
X(1671) lies on these lines: 3,6 76,1676 262,1677 485,2010 486,2009 1124,2008 1335,2007 1377,2014 1378,2013 1702,2018 1703,2017
X(1671) = reflection of X(1670) in X(3)
X(1671) = isogonal conjugate of X(1676)
X(1671) = X(76)-Ceva conjugate of X(1670)
X(1671) = circumcircle-inverse of X(38721)
X(1671) = Brocard-circle-inverse of X(1343)
X(1671) = Thomson-isogonal conjugate of X(33708)
X(1671) = X(1)-of-X(6)PU(1)
X(1671) = {X(6),X(39)}-harmonic conjugate of X(1670)
X(1671) = {X(32),X(3094)}-harmonic conjugate of X(1670)
X(1671) = {X(182),X(3095)}-harmonic conjugate of X(1670)
X(1671) = {X(1379),X(1380)}-harmonic conjugate of X(38721)
X(1672) lies on these lines: 1,182 2,1681 8,1680 11,1676 12,1677 55,1343 56,1342 57,1700 181,1683 371,2008 372,2007 1015,2035 1124,1688 1335,1687 1682,1684 1697,1701
X(1673) lies on these lines: 1,182 2,1680 8,1681 11,1677 12,1676 55,1342 56,1343 57,1701 181,1684 371,2007 372,2008 1015,2036 1124,1687 1335,1688 1500,2035 1682,1683 1697,1700
X(1674) lies on these lines: 1,182 2,1679 8,1678 11,1348 12,1349 55,1341 56,1340 57,1704 181,1693 1015,2033 1124,1668 1335,1669 1500,2034 1682,1694 1697,1705 2007,2012 2008,2011
X(1675) lies on these lines: 1,182 2,1678 8,1679 11,1349 12,1348 55,1340 56,1341 57,1705 181,1694 1015,2034 1124,1669 1335,1668 1500,2033 1682,1693 1697,1704 2007,2011 2008,2012
Let Lab and Lac be the lines obtained by rotating line BC through B and C resp., by an angle of ω/2 away from A. Let A' be Lab∩Lac. Define B' and C'cyclically. The lines AA', BB', CC' concur in X(1676). (Randy Hutson, September 14, 2016)
X(1676) lies on the Kiepert hyperbola and these lines: 2,1343 4,1342 5,182 10,1684 11,1672 12,1673 76,1671 115,2035 262,1670 371,2010 372,2009 485,1688 486,1687 1329,1680 1506,2036 1662,2040 1663,2039 1698,1701 1699,1700
X(1676) = isogonal conjugate of X(1671)
X(1676) = X(32)-Ceva conjugate of X(1677)
Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(1677). (Randy Hutson, July 20, 2016)
X(1677) lies on the Kiepert hyperbola and these lines: 2,1342 4,1343 5,182 10,1683 11,1673 12,1672 76,1670 115,2036 262,1671 371,2009 372,2010 485,1687 486,1688 1329,1681 1506,2035 1662,2039 1663,2040 1698,1700 1699,1701
X(1677) = isogonal conjugate of X(1670)
X(1677) = X(32)-Ceva conjugate of X(1676)
X(1678) lies on these lines: 2,1675 8,1674 9,1705 10,182 958,1341 1329,1348 1340,1376 1377,1668 1378,1669 1573,2034 1574,2033 1704,1706 2011,2014 2012,2013
X(1679) lies on these lines: 2,1674 8,1675 9,1704 10,182 958,1340 1329,1349 1341,1376 1377,1669 1378,1668 1573,2033 1574,2034 1705,1706 2011,2013 2012,2014
X(1680) lies on these lines: 2,1673 8,1672 9,1701 10,182 371,2014 372,2013 958,1343 1329,1676 1342,1376 1377,1688 1378,1687 1573,2036 1574,2035 1700,1706
X(1681) lies on these lines: 2,1672 8,1673 9,1700 10,182 371,2013 372,2014 958,1342 1329,1677 1343,1376 1377,1687 1378,1688 1573,2035 1574,2036 1701,1706
The exsimilicenter of the incircle and Apollonius circle is X(181). Also, the triangle A'B'C' formed (as at X(2092) by the intersections of the Apollonius circle and the excircles is perspective to the cevian triangle of X(1), and the perspector is X(1682). (Paul Yiu, Hyacinthos #8076, 10/01/03)
Let JaJbJc be the excentral triangle and PaPbPc be the Apollonius triangle. Let Pa' = {X(970),Ja}-harmonic conjugate of Pa, and define Pb' and Pc' cyclically. The lines APa', BPb', CPc' concur in X(1682); see also X(11). (Randy Hutson, December 10, 2016)
X(1682) lies on these lines: 1,181 3,1397 10,11 43,1697 55,386 56,573 57,1695 73,1362 212,1472 215,501 988,1401 1124,1686 1335,1685 1672,1684 1673,1683 1674,1694 1675,1693 2007,2020 2008,2019
X(1682) = {X(1),X(970)}-harmonic conjugate of X(181)
X(1682) = perspector of ABC and cross-triangle of ABC and Apollonius triangle
X(1682) = centroid of curvatures of Apollonius circle and excircles
X(1683) lies on these lines: 3,6 10,1677 43,1700 181,1672 1673,1682 1695,1701
X(1684) lies on these lines: 3,6 10,1676 43,1701 181,1673 1672,1682 1695,1700
X(1685) lies on these lines: 3,6 10,486 43,1702 181,1124 1335,1682 1695,1703
X(1686) lies on these lines: 3,6 10,485 43,1703 181,1335 1124,1682 1695,1702
X(1687) lies on these lines: 3,6 83,2010 98,2009 485,1677 486,1676 1124,1672 1335,1672 1377,1681 1378,1680 1700,1703 1701,1702
X(1687) = reflection of X(1688) in X(1691)
X(1687) = isogonal conjugate of X(2009)
X(1687) = circumcircle-inverse of X(1688)
X(1687) = Brocard-circle-inverse of X(1690)
X(1687) = 1st Lemoine-circle-inverse of X(1688)
X(1687) = X(98)-Ceva conjugate of X(1688)
X(1687) = {X(371),X(372)}-harmonic conjugate of X(1690)
X(1687) = insimilicenter of 2nd Brocard circle and circle{{X(371),X(372),PU(1),PU(39)}}; the exsimilicenter is X(1688)
X(1688) lies on these lines: 3,6 83,2009 98,2010 485,1676 486,1677 1124,1672 1335,1673 1377,1680 1378,1681 1700,1702 1701,1703
X(1688) = reflection of X(1687) in X(1691)
X(1688) = isogonal conjugate of X(2010)
X(1688) = circumcircle-inverse of X(1687)
X(1688) = Brocard-circle-inverse of X(1689)
X(1688) = 1st-Lemoine-circle-inverse of X(1687)
X(1688) = X(98)-Ceva conjugate of X(1687)
X(1688) = {X(371),X(372)}-harmonic conjugate of X(1689)
X(1688) = exsimilicenter of 2nd Brocard circle and circle {{X(371),X(372),PU(1),PU(39)}}; the insimilicenter is X(1687)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(1689) is the external center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)
X(1689) lies on these lines: 1,2018 2,2009 3,6 4,2010 55,2007, 56,2008 165,2017 958,2013 1344,2015 1345,2016 1376,2014
X(1689) = reflection of X(1690) in X(6)
X(1689) = inverse-in-Brocard-circle of X(1687)
X(1689) = X(262)-Ceva conjugate of X(1690)
X(1689) = {X(371),X(372)}-harmonic conjugate of X(1688)
X(1690) is the internal center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)
X(1690) lies on these lines: 1,2017 2,2010 3,6 4,2009 55,2008 56,2007 165,2018 958,2014 1344,2016 1345,2015 1376,2013
X(1690) = reflection of X(1689) in X(6)
X(1690) = inverse-in-Brocard-circle of X(1688)
X(1690) = X(262)-Ceva conjugate of X(1689)
X(1690) = {X(371),X(372)}-harmonic conjugate of X(1687)
X(1690) = X(1)-of-X(3)PU(1)
X(1691) is the perspector of ABC and the reflection of the tangential triangle in the Lemoine axis (i.e., the reflection of the anticevian triangle of X(6) in the trilinear polar of X(6)). (Randy Hutson, September 5, 2015)
Let A'B'C' be the 1st anti-Brocard triangle. X(1691) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)
Let A' be the circumcircle intercept, other than A, of the A-Montesdeoca-Lemoine circle. Define B' and C' cyclically. Triangle A'B'C' is perspective to the symmedial triangle at X(1691). (Randy Hutson, July 11, 2019)
X(1691) lies on these lines: 2,1501 3,6 31,893 83,316 98,230 99,698 141,1078 154,1611 184,1613 237,694 249,524 385,732 691,729 695,1176 699,805 1428,1914 1968,1974
X(1691) = midpoint of X(i) and X(j) for these (i,j): (6,2076), (187,1692), (1687,1688)
X(1691) = reflection of X(i) in X(j) for these (i,j): (6,1692), (1692,2030), (2076,187)
X(1691) = isogonal conjugate of X(1916)
X(1691) = isotomic conjugate of X(18896)
X(1691) = complement of X(5207)
X(1691) = inverse-in-circumcircle of X(32)
X(1691) = inverse-in-Brocard circle of X(3094)
X(1691) = inverse-in-1st-Lemoine-circle of X(6)
X(1691) = inverse-in-2nd-Lemoine-circle of X(576)
X(1691) = X(i)-Ceva conjugate of X(j) for these (i,j): (237,1971), (699,32), (1976,6)
X(1691) = crosspoint of X(i) and X(j) for these (i,j): (83,98), (385,419)
X(1691) = crosssum of X(39) and X(511)
X(1691) = reflection of X(2076) in the Lemoine axis
X(1691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,182,6), (371, 372, 3095), (1342,1343,39)
X(1691) = crossdifference of every pair of points on the line PU(11)
X(1691) = center of the circle {{X(1687),X(1688),PU(1),PU(2)}} (the circle orthogonal to the circumcircle and passing through the 1st and 2nd Brocard points)
X(1691) = intersection of tangents to 1st Lemoine circle at intersections with 2nd Lemoine circle
X(1691) = centroid of X(6)X(15)X(16)
X(1691) = X(1691) of circumsymmedial triangle
X(1691) = harmonic center of 1st and 2nd Lemoine circles
X(1691) = harmonic center of 2nd Brocard circle and the circle {{X(371),X(372),PU(1),PU(39)}}
X(1691) = perspector of ABC and the reflection of the 2nd Ehrmann triangle in line X(6)X(512) (the perspectrix of ABC and 2nd Ehrmann triangle)
X(1692) lies on these lines: 3,6 51,1501 114,230 115,1503 184,1196 698,1569 1015,1428 1627,1994
X(1692) = midpoint of X(i) and X(j) for these (i,j): (6,1691), (187,1570)
X(1692) = reflection of X(i) in X(j) for these (i,j): (39,2024), (187,1570), (1570,6), (1691,2030)
X(1692) = isogonal conjugate of X(8781)
X(1692) = circumcircle-inverse of X(3053)
X(1692) = 1st-Lemoine-circle-inverse of X(32)
X(1692) = 2nd-Lemoine-circle-inverse of X(1351)
X(1692) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(9737)
X(1692) = crosspoint of X(i) and X(j) for these (i,j): (6,1976), (230,460)
X(1692) = crosssum of X(2) and X(325)
X(1692) = center of inverse-in-Moses-circle-of-Brocard-circle
X(1692) = radical trace of circumcircle and circle {{X(371),X(372),PU(1),PU(39)}}
X(1692) = radical trace of 2nd Lemoine circle and circle {{X(371),X(372),PU(1),PU(39)}}
X(1692) = radical trace of circles {{P(1),U(2),P(39)}} and {{U(1),P(2),U(39)}}
X(1692) = anticenter of cyclic quadrilateral PU(2)PU(39)
X(1692) = crossdifference of every pair of points on line X(69)X(523)
X(1692) = {X(1687),X(1688)}-harmonic conjugate of X(2456)
X(1692) = X(2)-Ceva conjugate of X(39072)
X(1692) = perspector of conic {{A,B,C,X(25),X(110)}}
X(1692) = {X(371),X(372)}-harmonic conjugate of X(9737)
X(1693) lies on these lines: 3,6 10,1349 43,1704 181,1674 1675,1682 1695,1705
X(1694) lies on these lines: 3,6 10,1348 43,1705 181,1675 1674,1682 1695,1704
The exsimilicenter of the Bevan and Apollonius circles is X(43).
X(1695) lies on these lines: 1,573 10,962 40,43 57,1682 165,386 181,1697 978,1764 1683,1701 1684,1700 1685,1703 1686,1702 1693,1705 1694,1704 2017,2020 2018,2019
X(1695) = X(939)-Ceva conjugate of X(55)
X(1696) is the trilinear product X(6)*X(1706)
X(1696) lies on these lines: 6,1201 9,56 19,25 220,1400 346,1376 999,1743
The exsimilicenter of the Bevan circle and incircle is X(57).
Randy Hutson (January 29, 2015) gives 3 constructions:
(1) Let A'B'C' be the mixtilinear incentral triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1697).
(2) Let A'B'C' be the mixtilinear incentral triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1697).
Let A'B'C' be the intouch triangle, A"B"C" the extouch triangle, and A*B*C* the excentral triangle. Let OA be the circle through A'A"A*, and define OB and OCc cyclically. X(1697) is the radical center of circles OA, OB, OC.
X(1697) lies on these lines: 1,3 2,1706 8,9 10,497 11,1698 12,1699 33,1831 43,1682 63,145 71,1732 84,944 109,1496 181,1695 200,960 212,595 219,380 221,1419 226,962 388,516 392,936 519,1776 580,1497 1015,1571 1058,1210 1124,1703 1317,1768 1335,1702 1500,1572 1672,1701 1673,1700 1674,1705 1675,1704 2007,2018 2008,2017
X(1697) = reflection of X(3340) in X(1)
X(1697) = isogonal conjugate of X(7091)
X(1697) = X(2339)-Ceva conjugate of X(9)
X(1697) = 2nd-extouch-to-excentral similarity image of X(8)
X(1697) = extangents-to-intangents similarity image of X(1)
X(1697) = {X(1),X(40)}-harmonic conjugate of X(57)
X(1697) = homothetic center of excentral and Hutson-intouch triangles
X(1697) = X(1593)-of-excentral triangle
X(1697) = X(1593)-of-Hutson-intouch triangle
X(1697) = X(11414)-of-intouch triangle
X(1697) = centroid of curvatures of Bevan circle and excircles
Let A'B'C' be the Aquila triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. Then A"B"C" is homothetic to ABC at X(1698) and to A'B'C' at X(10). (Randy Hutson, December 10, 2016)
X(1698) lies on these lines: 1,2 4,165 5,40 9,46 11,1697 12,57 33,451 35,405 36,474 58,750 75,1089 115,1571 116,1282 119,1768 121,1054 140,355 171,1724 210,942 226,1788 281,1838 318,1784 320,1757 404,993 406,1861 443,1478 485,1703 486,1702 515,631 517,1656 595,748 632,952 966,1743 986,1739 1348,1705 1349,1704 1506,1572 1676,1701 1677,1700 2009,2018 2010,2017
X(1698) = isotomic conjugate of X(30598)
X(1698) = {X(2),X(10)}-harmonic conjugate of X(1)
X(1698) = homothetic center of excentral and 4th Euler triangles
X(1698) = crossdifference of every pair of points on line X(649)X(2605)
X(1698) = trilinear product of vertices of Aquila triangle
X(1698) = homothetic center of ABC and cross-triangle of ABC and Aquila triangle
X(1698) = {X(1),X(2)}-harmonic conjugate of X(3624)
X(1698) = perspector of Gemini triangle 26 and cross-triangle of ABC and Gemini triangle 26
X(1698) = isogonal conjugate of isotomic conjugate of X(30596)
X(1698) = homothetic center of Ai (aka K798i) triangle and cross-triangle of Fuhrmann and Ai triangles
Let A' be the pole of the Gergonne line wrt the circle with BC as diameter, and define B', C' cyclically. X(1699) is the centroid of A'B'C'. (Randy Hutson, June 27, 2018)
Let A' be the orthocenter of BCX(1), and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X(1), and X(1699) is the centroid of A'B'C'; see also X(3680). (Randy Hutson, June 27, 2018)
X(1699) is the centroid of triangle formed by reflecting vertices of 1st circumperp triangle in corresponding side of ABC. (If 2nd circumperp triangle is substituted, for 1st, the resulting triangle is the Fuhrmann triangle.) (Randy Hutson, June 27, 2018)
Let A'B'C' be the orthic triangle. Let A" be the A-excenter of AB'C', and define B" and C" cyclically. The centroid of triangle A"B"C" is X(1699). (Randy Hutson, July 31 2018)
X(1699) lies on these lines: 1,4 2,165 5,40 10,962 11,57 12,1697 20,1125 36,1012 55,1538 79,84 80,1537 115,1572 118,1282 200,908 210,381 238,1754 354,971 355,546 382,1385 485,1702 486,1703 499,1770 610,1839 614,990 1329,1706 1348,1704 1506,1571 1676,1700 1677,1701 1730,1985 2009,2017 2010,2018
X(1699) = reflection of X(165) in X(2)
X(1699) = crosspoint of X(92) and X(1088)
X(1699) = crosssum of X(48) and X(1253)
X(1699) = centroid of the six touchpoints of the Johnson circles and the sidelines of the inner Johnson triangle
X(1699) = homothetic center of excentral and 3rd Euler triangles
X(1699) = centroid of triangle formed by reflecting excenters in corresponding vertex of ABC
X(1699) = centroid of anticevian triangle, wrt intouch triangle, of X(1)
X(1699) = homothetic center of circumcevian triangle of X(3) and cross-triangle of Aquila and anti-Aquila triangles
X(1699) = centroid of Garcia reflection triangle (aka Gemini triangle 8)
X(1699) = {X(1),X(4)}-harmonic conjugate of X(5691)
X(1700) lies on these lines: 1,1342 9,1681 40,182 43,1683 57,1672 165,1343 371,2018 372,2017 1571,2036 1572,2035 1673,1697 1676,1699 1677,1698 1680,1706 1684,1695 1687,1703 1688,1702
X(1701) lies on these lines: 1,1343 9,1680 40,182 43,1684 57,1673 165,1342 371,2017 372,2018 1571,2035 1572,2036 1672,1697 1676,1698 1677,1699 1681,1706 1683,1695 1687,1702 1688,1703
X(1702) lies on these lines: 1,371 6,40 9,1378 10,1588 43,1685 57,1124 165,372 485,1699 486,1698 516,1587 580,605 1335,1697 1377,1706 1504,1572 1505,1571 1668,1704 1669,1705 1670,2017 1671,2018 1686,1695 1687,1701 1688,1700
X(1703) lies on these lines: 1,372 6,40 9,1377 10,1587 43,1686 57,1335 165,371 485,1698 486,1699 516,1588 580,606 1124,1697 1378,1706 1504,1571 1505,1572 1668,1705 1669,1704 1670,2018 1671,2017 1685,1695 1687,1700 1688,1701
X(1704) lies on these lines: 1,1340 9,1679 40,182 43,1693 57,1674 165,1341 1348,1699 1349,1698 1571,2034 1572,2033 1668,1702 1669,1703 1675,1697 1678,1706 1694,1695 2011,2018 2012,2017
X(1705) lies on these lines: 1,1341 9,1678 40,182 43,1694 57,1675 165,1340 1348,1698 1349,1699 1571,2033 1572,2034 1668,1703 1669,1702 1674,1697 1679,1706 1693,1695 2011,2017 2012,2018
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
The insimilicenter of the Bevan and Spieker circles is X(9).
X(1706) lies on these lines: 1,474 2,1697 4,9 8,57 46,529 65,200 84,355 165,958 404,1420 517,936 1329,1699 1377,1702 1378,1703 1571,1573 1572,1574 1678,1704 1679,1705 1680,1700 1681,1701
X(1706) = {X(10),X(40)}-harmonic conjugate of X(9)
Mimosa Transforms 1707-1788
M(X) = - yz cos A + zx cos B + xy cos C : - zx cos B + xy cos C + yz cos A : - xy cos C + yz cos A + xy cos C
and the inverse Mimosa transform M -1(X), by
M -1(X) = (cos A)/(y + z) : (cos B)/(z + x) : (cos C)/(x + y).
As with other names in ETC, the name Mimosa is that of a star. M(X) is the X*X(4)-Ceva conjugate of X(1), where * denotes trilinear product, and M -1(X) is the trilinear quotient X(3)/P(X), where P(X) is the crosssum of X(1) and X.
Let g(P,X) denote the P-gimel conjugate of X. The Mimosa transform M(X) arises in connection with the equation g(P,X) = X. Referring to the definition of gimel conjugate in the Glossary, if
P = p : q : r and X = x : y : zare triangle centers, then the equation g(P,X) = X is equivalent to the ratio-equation
(2absvS - 8yσ2)/(2abswS - 8zσ2) = y/z,where v = (cos A)/p - (cos B)/q + (cos C)/r and w = (cos A)/p + (cos B)/q - (cos C)/r.
If S = 0, then the ratio-equation holds. As S = x(bq + cr) + y(cr + ap) + z(ap + bq), it follows that if P is given, then g(P,X) = X if X is on the line S = 0 (regarding x : y : z as variable); and that if X is given, then g(P,X) = P if P is on the line S = 0 (regarding p : q : r as variable). There are too many such cases of gimel conjugates for all to be itemized in ETC. For example, if X = X(1), then g(P,X) = X for every P on the line at infinity; if X = X(513), then g(P,X) = X for every P on the line X(1)X(2); and if X = X(656), then g(P,X) = X for every P on the Euler line, X(2)X(3).
If S ≠ 0, the ratio-equation lends itself to easy simplifications and two Tables conclusions: (1) if P is given then X = M(P) is a solution of g(P,X) = X, and (2) if X is given then P = M*(X) is a solution of g(P,X) = X.
Here is a list of pairs (i,j) for which X(j) = M(X(i)):
1,46 2,19 3,1 4,920
8,1158 21,4 48,43 54,47
59,109 60,580 63,9
69,63 71,846 72,191
73,1046 77,57 78,40
81,579 90,90 95,92
96,91 97,48 99,1577
110,656 219,165 228,1045
248,1580 249,163 250,162
252,564 254,921 271,84
283,3 348,169 394,610
603,978 648,822 651,652
662,1021 895,896 1105,158
1176,31 1259,1490 1297,240
1332,649 1444,2 1459,1054
Of course, reversing the pairs gives a list of (J,I) for which X(i) = M*(X(j)).
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1707) lies on these lines: 1,21 9,171 19,1719 33,1776 36,1473 43,165 44,1376 46,1722 57,238 92,1733 109,1395 162,1096 200,1757 204,240 223,1758 326,560 380,1045 484,1774 579,1716 580,1158 610,1740 978,1044 986,1453 1038,1399 1633,1732 1709,1711 1714,1770 1728,1771 1788,1877
X(1707) = isogonal conjugate of X(8769)
X(1707) = {X(31),X(63)}-harmonic conjugate of X(1)
X(1707) = {X(6),X(9)}-harmonic conjugate of X(1)
X(1707) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,1),(1778,1724)
X(1707) = X(i)-aleph conjugate of X(j) for these (i,j): (1,610), (4,19), (19,1707), (108,1783), (162,163), (365,1745), (366,1763),
(509,223), (1778,1724), (1783,1633)
X(1708) lies on these lines: 1,201 2,7 4,46 6,1214 19,1713 33,1736 34,1724 38,1471 40,950 43,1758 44,1427 56,72 58,1038 65,405 109,1395 169,1762 208,860 218,222 223,1743 225,1714 278,1723 354,954 442,1454 518,1260 582,1062 653,1748 1020,1435 115,1864 1396,1778 1412,1812 1711,1738 1712,1715 1750,1768
X(1708) = X(273)-Ceva conjugate of X(1)
X(1708) = cevapoint of X(46) in X(1723)
X(1708) = crosssum of X(652) and X(2170)
X(1708) = {X(9),X(57)}-harmonic conjugate of X(226)
X(1708) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1490), (4,1721), (7,223), (27,580), (92,1158), (174,1745), (273,1708),
(278,1722), (286,1746), (508,610), (653,109)
X(1709) lies on these lines: 1,84 4,46 9,165 11,57 30,40 31,990 33,109 35,1490 55,971 63,516 553,946 774,1448 846,1742 968,991 1707,1711 1719,1744 1730,1889
X(1709) = reflection of X(1) in X(1012)
X(1709) = X(281)-Ceva conjugate of X(1)
X(1709) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1723),
(92,1729), (188,610), (281,1709), (366,223)
X(1709) = excentral-isotomic conjugate of X(2947)
X(1709) = antipode of X(1) in circle {{X(1),X(1709),PU(4)}}
X(1710) lies on these lines: 1,1437 19,91 28,1725 30,40 35,228 46,407 109,1825 1046,1777 1720,1781 1770,1782
X(1710) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1724), (92,1730)
X(1711) lies on these lines: 1,394 9,43 46,407 90,1039 238,1040 920,1714 1707,1709 1708,1738 1720,1722 1756,1763
X(1711) = X(393)-Ceva conjugate of X(1)
X(1711) = X(i)-aleph conjugate of X(j) for these (i,j): (4,9), (19,43),
(33,170), (92,1759), (393,1711), (1897,1018)
X(1712) lies on these lines: 1,204 4,57 9,1249 19,158 63,1895 108,1490 412,1445 774,1096 811,1102 920,1784 1103,1783 1158,1767 1708,1715 1713,1741 1714,1728
X(1712) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,19), (1895,1)
X(1712) = X(i)-aleph conjugate of X(j) for these (i,j): (2,2184), (1895,1712)
X(1712) = eigencenter of cevian triangle of X(63)
X(1712) = eigencenter of anticevian triangle of X(19)
X(1713) lies on these lines: 1,6 4,579 19,1708 71,950 284,1006 379,1445 393,1714 580,1172 583,1901 1712,1741
X(1713) = crosspoint of X(765) and X(823)
X(1713) = crosssum of X(244) and X(822)
X(1713) = X(i)-aleph conjugate of X(j) for these (i,j): (4,846),
(27,6), (92,1761)
X(1714) lies on these lines: 1,2 4,580 6,442 19,46 58,377 100,1612 219,1329 225,1708 238,1479 278,1739 393,1713 405,1834 920,1711 1498,1532 1707,1770 1712,1728 1715,1779
X(1714) = X(i)-aleph conjugate of X(j) for these (i,j): (4,191), (19,1045), (27,2), 29,20)
X(1715) lies on these lines: 1,3 4,1730 19,1158 185,851 412,1896 579,1249 1020,1068 1708,1712 1714,1779 1736,1872
X(1715) = X(i)-Ceva conjugate of X(j) for these (i,j): (412,4),
(1896,1)
X(1715) = crosssum of X(822) and X(2310)
X(1715) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1046),
(29,3), (92,1762), (1896,1715)
X(1716) lies on these lines: 1,69 3,238 4,1721 9,43 579,1707 1402,1423 1745,1756
X(1716) = X(25)-Ceva conjugate of X(1)
X(1716) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1763),
(4,1726), (6,1745), (19,46), (25,1716), (108,1020), (259,1490),
(266,223), (365,610)
X(1717) lies on these lines: 1,30 4,1718 33,46 35,37 90,1172 429,1722 1047,1048
X(1717) = X(1770)-Ceva conjugate of X(46)
X(1718) lies on these lines: 1,5 4,1717 6,1781 34,46 36,1455 90,1720 106,614 244,1468 1723,1783 1727,1735 1737,1870
X(1718) = X(i)-Ceva conjugate of X(j) for these (i,j): (1737,46),
(1870,1)
X(1718) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1727),
(1870,1718)
X(1719) lies on these lines: 1,1790 10,191 19,1707 27,1733 46,407 165,846 1709,1744
X(1719) = X(1826)-Ceva conjugate of X(1)
X(1719) = X(i)-aleph conjugate of X(j) for these (i,j): (4,579),
(1826,1719)
X(1720) lies on these lines: 1,84 46,208 90,1718 846,1047 920,1249 1710,1781 1711,1722 1721,1771
X(1720) = X(1158)-Ceva conjugate of X(46)
X(1720) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1728), (366,282)
X(1720) = trilinear product X(2)*X(22)
X(1721) lies on these lines: 1,7 4,1716 19,1633 40,984 43,1750 46,1736 165,846 294,1743 1039,1885 1040,1836 1045,1047 1707,1709 1720,1771
X(1721) = reflection of X(1) in X(990)
X(1721) = X(33)-Ceva conjugate of X(1)
X(1721) = X(317)-of-excentral-triangle
X(1721) = excentral isotomic conjugate of X(1490)
X(1721) = {X(8947),X(8949)}-harmonic conjugate of X(1722)
X(1721) = X(i)-aleph conjugate of X(j) for these (i,j): (1,223), (4,1708), (9,1490), (19,1722), (29,1746), (33,1721), (188,1763), (259,1745), (281,1158), (1172,580), (1783,109)
X(1722) lies on these lines: 1,2 4,1716 9,986 34,1788 40,238 46,1707 57,1773 87,937 169,1046 171,1453 223,1047 269,979 427,1039 429,1717 920,1772 958,988 1040,1837 1104,1376 1254,1445 1711,1720 1723,1880
X(1722) = X(i)-Ceva conjugate of X(j) for these (i,j): (34,1), (1788,46)
X(1722) = {X(8947),X(8949)}-harmonic conjugate of X(1721)
X(1722) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1490), (4,1158), (19,1721), (27,1746), (28,580), (34,1722), (57,223), (108,109), (174,1763), (266,1745), (278,1708), (509, 610)
X(1723) lies on these lines: 1,6 19,46 35,380 36,610 57,1762 90,1172 169,1400 278,1708 281,1737 672,1766 920,1249 928,1047 1707,1709 1718,1783 1722,1880 1729,1744
X(1723) = X(i)-Ceva conjugate of X(j) for these (i,j): (278,1),
(1708,46)
X(1723) = crosspoint of X(653) and X(765)
X(1723) = crosssum of X(244) and X(652)
X(1723) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1709),
(174,610), (273,1729), (278,1723), (366,1490), (508,1763),
(509,1745)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1724) lies on these lines: 1,6 2,58 4,580 8,595 10,31 21,286 28,579 30,582 32,1009 34,1708 35,43 36,978 46,1707 47,1737 83,1008 90,1039 109,1788 171,1698 191,986 212,950 226,1451 255,1210 270,469 387,452 515,602 581,1006 748,1125 920,1735 985,1224 993,1193 1020,1398 1445,1448 1726,1829 1738,1770
X(1724) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,1),
(579,1754), (1778,1707)
X(1724) = crosspoint of X(162) and X(765)
X(1724) = crosssum of X(244) and X(656)
X(1724) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1710),
(27,1730), (28,1724), (266,1047)
X(1724) = intercept, other than X(1), of line X(1)X(6) and conic {{X(1),X(13),X(14),X(15),X(16)}}
Let L denote the line having trilinears of X(1725) as coefficients. Then L is the line passing through X(3) perpendicular to the Euler line.
X(1725) lies on these lines: {1, 21}, {12, 35194}, {28, 1710}, {33, 46}, {34, 90}, {35, 201}, {36, 5494}, {40, 1775}, {57, 1774}, {79, 45924}, {80, 38945}, {91, 158}, {109, 1727}, {240, 522}, {244, 3582}, {484, 40577}, {495, 24431}, {499, 17063}, {500, 17637}, {564, 45225}, {612, 17699}, {756, 3584}, {912, 8758}, {942, 8143}, {982, 10072}, {984, 10056}, {1172, 1744}, {1210, 36250}, {1214, 4337}, {1254, 3585}, {1393, 7741}, {1399, 18447}, {1406, 7072}, {1454, 37696}, {1464, 2771}, {1478, 24430}, {1479, 37591}, {1755, 3708}, {1758, 3465}, {1776, 1870}, {1779, 5902}, {1807, 5172}, {1858, 37565}, {1973, 21374}, {2083, 2172}, {2159, 2173}, {2166, 18486}, {2181, 17890}, {2247, 9406}, {2310, 3583}, {2312, 16562}, {2349, 36083}, {2361, 18455}, {3649, 5492}, {3670, 24210}, {3911, 16869}, {5348, 37729}, {5497, 32760}, {6198, 7098}, {7069, 7951}, {7073, 45923}, {7082, 37697}, {7100, 8614}, {17757, 24433}, {17879, 18694}, {17881, 40703}, {18180, 42440}, {23580, 26013}, {24028, 41684}, {36034, 36053}, {38336, 41697}
X(1725) = reflection of X(23580) in X(26013)
X(1725) = isogonal conjugate of X(36053)
X(1725) = polar conjugate of the isogonal conjugate of X(2315)
X(1725) = X(i)-Ceva conjugate of X(j) for these (i,j): {18609, 3003}, {32680, 661}, {36034, 656}, {36119, 1}
X(1725) = X(i)-isoconjugate of X(j) for these (i,j): {1, 36053}, {2, 14910}, {3, 1300}, {4, 5504}, {6, 2986}, {30, 10419}, {32, 40832}, {50, 40427}, {74, 15454}, {110, 15328}, {112, 15421}, {113, 39379}, {115, 18879}, {186, 12028}, {254, 15478}, {265, 38936}, {476, 15470}, {477, 39986}, {512, 18878}, {523, 10420}, {525, 32708}, {647, 687}, {656, 36114}, {1495, 40423}, {2501, 43755}, {5627, 39371}, {10733, 39372}, {11064, 40388}, {12383, 35373}, {14385, 39375}, {14911, 34178}, {18315, 35361}
X(1725) = crosspoint of X(i) and X(j) for these (i,j): {1, 2166}, {75, 2349}
X(1725) = crosssum of X(i) and X(j) for these (i,j): {1, 6149}, {31, 2173}, {758, 25440}, {2631, 2643}
X(1725) = crossdifference of every pair of points on line {48, 661}
X(1725) = barycentric product X(i)*X(j) for these {i,j}: {1, 3580}, {10, 18609}, {48, 44138}, {63, 403}, {75, 3003}, {92, 13754}, {113, 2349}, {162, 6334}, {264, 2315}, {304, 44084}, {656, 16237}, {686, 811}, {799, 21731}, {1577, 15329}, {2166, 34834}, {14206, 14264}, {32679, 41512}
X(1725) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 2986}, {6, 36053}, {19, 1300}, {31, 14910}, {48, 5504}, {75, 40832}, {112, 36114}, {113, 14206}, {162, 687}, {163, 10420}, {403, 92}, {656, 15421}, {661, 15328}, {662, 18878}, {686, 656}, {1101, 18879}, {2159, 10419}, {2166, 40427}, {2173, 15454}, {2315, 3}, {2349, 40423}, {2624, 15470}, {3003, 1}, {3580, 75}, {4575, 43755}, {6334, 14208}, {12824, 16568}, {12827, 20884}, {13754, 63}, {14264, 2349}, {15329, 662}, {16237, 811}, {18609, 86}, {21731, 661}, {32676, 32708}, {39985, 36102}, {41512, 32680}, {44084, 19}, {44138, 1969}
X(1725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 920, 47}, {1, 1749, 6149}, {31, 18477, 1}, {774, 44706, 1}, {942, 8143, 11553}, {1406, 7072, 18451}, {1735, 1736, 1737}, {1735, 1737, 1772}, {1749, 6149, 896}, {1755, 3708, 18669}, {1822, 1823, 47}, {3743, 10122, 1}
X(1726) lies on these lines: 1,184 4,1782 9,440 19,1708 25,1736 46,407 57,1020 63,321 92,1746 1158,1753 1724,1829 1754,1824
X(1726) = isogonal conjugate of X(7094)
X(1726) = X(264)-Ceva conjugate of X(1)
X(1726) = polar conjugate of isogonal conjugate of X(36033)
X(1726) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1745), (4,1716), (75,1763), (92,46), (264,1726), (556,1490)
X(1727) lies on these lines: 1,1399 4,46 30,80 35,72 36,1768 63,519 109,1725 1718,1735
X(1727) = X(4)-aleph conjugate of X(1718)
X(1728) lies on these lines: 1,6 3,1864 4,46 5,57 33,580 36,1490 40,1837 63,1210 84,1750 226,499 1707,1771 1711,1720 1712,1714
X(1728) = X(4)-aleph conjugate of X(1720)
X(1729) lies on these lines: 5,9 19,1158 57,1375 63,169 607,1735 920,1752 1723,1744
X(1729) = X(331)-Ceva conjugate of X(1)
X(1729) = X(i)-aleph conjugate of X(j) for these (i,j): (92,1709),
(273,1723), (331,1729), (508,1745)
X(1730) lies on these lines: 1,228 2,573 4,1715 6,57 19,1708 25,1754 27,1746 28,580 40,405 46,1707 51,851 63,169 165,1011 278,1020 572,1817 1709,1889 1735,1905 1736,1824 1786,1787
X(1730) = X(286)-Ceva conjugate of X(1)
X(1730) = X(i)-aleph conjugate of X(j) for these (i,j): (27,1724),
(92,1710), (174,1047), (286,1730)
X(1731) lies on these lines: 8,9 19,46 44,517 63,1266 92,1751 243,522 580,1871
X(1731) = crosspoint of X(21) and X(88)
X(1731) = crosssum of X(44) and X(65)
X(1732) lies on these lines: 9,1125 19,46 44,56 45,354 48,1743 71,1697 1020,1435 1474,1778 1633,1707
X(1731) = crosspoint of X(75) and X(1821)
X(1731) = crosssum of X(31) and X(1755)
X(1731) = X(98)-aleph conjugate of X(1955)
X(1733) lies on these lines: 1,75 19,91 27,1719 92,1707 240,522 896,1109 1580,1821
X(1733) = isogonal conjugate of X(36051)
X(1733) = isotomic conjugate of X(8773)
X(1733) = crosspoint of X(75) and X(1821)
X(1733) = crosssum of X(31) and X(1755)
X(1733) = X(98)-aleph conjugate of X(1955)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1734) lies on these lines: 1,905 100,1110 240,522 484,513 512,1491 649,830
X(1734) = reflection of X(1) in X(905)
X(1734) = X(1783)-Ceva conjugate of X(1)
X(1734) = crosspoint of X(75) in X(100)
X(1734) = crosssum of X(i) and X(j) for these (i,j): (31,513),
(652,2293), (656,1962)
X(1734) = X(1783)-aleph conjugate of X(1734)
X(1735) lies on these lines: 1,3 10,1074 34,1158 109,1870 240,522 607,1729 774,1210 920,1724 946,1393 1711,1720 1718,1727 1730,1905 1765,1880
X(1735) = crosssum of X(31) and X(2182)
X(1736) lies on these lines: 1,6 10,774 25,1726 33,1708 46,1721 90,1041 109,1776 201,950 240,522 241,971 307,1210 920,1771 990,1445 1020,1876 1214,1864 1715,1872 1730,1824
X(1736) = crosssum of X(31) and X(910)
X(1737) lies on these lines: 1,2 3,1837 4,46 5,65 11,517 12,942 29,1780 30,1155 35,950 36,80 40,1479 47,1724 56,355 57,1478 72,1329 91,225 109,1877 117,1845 119,912 150,1447 240,522 281,1723 354,495 381,1836 427,1905 484,516 579,1826 758,908 952,1319 1718,1870 1747,1890 1782,1842
X(1737) = midpoint of X(36) and X(80)
X(1737) = isogonal conjugate of X(36052)
X(1737) = complement of X(4511)
X(1737) = cevapoint of X(46) and X(1718)
X(1737) = X(2252)-cross conjugate of X(914)
X(1737) = crosssum of X(i) and X(j) for these (i,j): (6,2316), (31,2183)
X(1737) = polar conjugate of X(37203)
X(1737) = pole wrt polar circle of trilinear polar of X(37203) (line X(1)X(7649))
X(1738) lies on these lines: 1,142 2,968 4,1716 10,75 19,46 43,226 225,1788 238,516 240,522 518,1086 527,1757 528,1279 899,908 946,978 1054,1758 1708,1711 1724,1770
X(1738) = crosspoint of X(75) and X(673)
X(1738) = crosssum of X(31) and X(672)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1739) lies on these lines: 1,474 10,38 36,1054 46,1707 238,484 240,522 244,519 278,1714 758,899 986,1698
X(1739) = crosspoint of X(75) and X(88)
X(1739) = crosssum of X(31) and X(44)
X(1740) lies on these lines: 1,75 3,238 6,43 19,1581 31,1582 48,1580 560,662 610,1707 869,894
X(1740) = isogonal conjugate of X(3223)
X(1740) = complement of X(21299)
X(1740) = anticomplement of X(21257)
X(1740) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,1), (1958,610)
X(1740) = X(1613)-cross conjugate of X(1424)
X(1740) = X(i)-aleph conjugate of X(j) for these (i,j): (1,63), (2,1760), (4,1748), (6,1), (19,920), (31,1740), (57,1445), (74,2349),
(98,1821), (99,799), (100,190), (101,100), (105,673), (106,88), (107,823), (108,653), (109,651), (110,662), (111,897), (112,162),
(259,40), (266,57), (284,411), (365,9), (366,1759), (509,169), (649,1052), (813,660), (825, 1492), (934,658), (1172,412), (2222, 655),
(2291,1156)
X(1741) lies on these lines: 2,7 19,1158 46,281 920,1249 1712,1713
X(1741) = X(189)-aleph conjugate of X(63)
X(1742) lies on these lines: {{1, 7}, {3, 238}, {6, 9441}, {9, 6184}, {10, 26059}, {31, 7411}, {35, 1745}, {37, 15726}, {38, 11220}, {40, 511}, {42, 9778}, {43, 165}, {45, 16112}, {46, 37409}, {48, 1633}, {55, 6180}, {57, 2942}, {58, 12511}, {75, 28850}, {87, 572}, {103, 7220}, {144, 2340}, {171, 7580}, {200, 4416}, {241, 14100}, {259, 503}, {266, 844}, {371, 31544}, {372, 31545}, {376, 1064}, {386, 12512}, {411, 37603}, {497, 22053}, {500, 37529}, {513, 15624}, {581, 31730}, {601, 3651}, {651, 1253}, {750, 36002}, {846, 1709}, {971, 984}, {982, 10167}, {986, 9943}, {988, 9841}, {1026, 25728}, {1193, 3522}, {1212, 15587}, {1250, 30300}, {1385, 24661}, {1418, 5572}, {1423, 2223}, {1490, 5293}, {1697, 2943}, {1699, 24220}, {1750, 5268}, {1766, 18788}, {1818, 5698}, {1935, 37601}, {1959, 12530}, {2066, 30297}, {2292, 9961}, {2309, 37416}, {2635, 5218}, {2792, 9862}, {2807, 5119}, {2808, 3688}, {2876, 7289}, {3056, 34253}, {3061, 18252}, {3062, 3731}, {3169, 9025}, {3177, 21084}, {3216, 16192}, {3474, 14547}, {3501, 17792}, {3510, 37619}, {3576, 16528}, {3579, 37699}, {3666, 5918}, {3720, 9812}, {3736, 4229}, {3752, 10178}, {3811, 17770}, {3817, 25502}, {3836, 36652}, {3870, 17364}, {3875, 7976}, {3879, 28849}, {3886, 34282}, {3888, 22370}, {3912, 21629}, {3950, 9950}, {4073, 25083}, {4192, 20368}, {4221, 30269}, {4512, 37175}, {4551, 35445}, {4641, 7964}, {5010, 6127}, {5217, 37694}, {5222, 20978}, {5247, 5584}, {5272, 10857}, {5414, 30296}, {5539, 34196}, {5691, 15971}, {5759, 24695}, {5851, 17334}, {5942, 28118}, {6284, 37523}, {6610, 30621}, {6765, 34379}, {7146, 12723}, {7175, 37580}, {7671, 17092}, {7963, 7987}, {7982, 29309}, {7991, 29311}, {8616, 15931}, {8727, 33111}, {8844, 16557}, {9779, 30950}, {9801, 17316}, {10164, 16569}, {10186, 17321}, {10638, 30301}, {10860, 17594}, {10868, 35269}, {11227, 17063}, {12618, 29674}, {13243, 36263}, {13329, 16468}, {15717, 27627}, {16132, 29097}, {17122, 19541}, {17126, 35986}, {17377, 28870}, {17378, 28854}, {17601, 17613}, {17668, 24341}, {17717, 37374}, {20556, 30035}, {20793, 34497}, {20995, 21856}, {21173, 23696}, {22836, 28508}, {24440, 31787}, {24635, 25722}, {26669, 35293}, {28164, 30116}, {29057, 30273}, {30503, 33781}, {31151, 36721}, {31805, 37592}, {35242, 37732}, {35658, 37552}, {37426, 37570}, {37501, 37607}
X(1742) = reflection of X(i) in X(j) for these (i,j): (1,991), (6210,3)
X(1742) = isogonal conjugate of the isotomic conjugate of X(20935)
X(1742) = X(i)-Ceva conjugate of X(j) for these (i,j): (55, 1), (6180, 1743), (31526, 34497)
X(1742) = X(i)-cross conjugate of X(j) for these (i,j): (20995, 34497), (21856, 3177), (31526, 1)
X(1742) = crosspoint of X(i) and X(j) for these (i,j): {1, 36601}, {651, 24011}, {3177, 31526}
X(1742) = crosssum of X(650) and X(24012)
X(1742) = crossdifference of every pair of points on line {657, 4449}
X(1742) = excentral-isogonal conjugate of X(63)
X(1742) = excentral-isotomic conjugate of X(40)
X(1742) = X(264)-of-excentral triangle
X(1742) = Brianchon point of the MacBeacth inconic of the excentral triangle
X(1742) = X(i)-aleph conjugate of X(j) for these (i,j): (1,57), (6,978), (9,40), (55,1742), (174,1445), (188,63), (259,1), (365,1743), (366,169)
X(1742) = barycentric product X(i)*X(j) for these {i,j}: {1, 3177}, {6, 20935}, {8, 34497}, {9, 31526}, {75, 20995}, {81, 21084}, {86, 21856}, {92, 20793}, {100, 21195}
X(1742) = barycentric quotient X(i)/X(j) for these {i,j}: {3177, 75}, {20793, 63}, {20935, 76}, {20995, 1}, {21084, 321}, {21195, 693}, {21856, 10}, {31526, 85}, {34497, 7}
X(1742) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 2951, 1721}, {6, 11495, 9441}, {7, 2293, 1}, {20, 4300, 1}, {55, 6180, 9440}, {77, 4319, 1}, {170, 4335, 1721}, {269, 4326, 1}, {390, 1458, 1}, {651, 7676, 1253}, {1042, 4313, 1}, {1442, 4336, 1}, {1716, 1740, 978}, {2263, 7675, 1}, {2293, 3000, 7}, {3945, 4343, 1}, {4294, 4303, 1}, {4302, 4337, 1}, {4306, 4314, 1}, {4322, 9785, 1}, {7671, 17092, 21346}, {9943, 15852, 986}, {30354, 30355, 4326}, {31573, 31574, 12565}
Let A' be the center of the conic through the contact points of the B- and C-excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the excentral triangle at X(1743). See also X(6), X(25), X(218), X(222), X(940). (Randy Hutson, July 23, 2015)
In the plane of ABC, let a' be the external bisector of A and a" its reflection in BC; define b" and c" cyclically. Then ABC and the triangle bounded by a", b" and c" are perspective with perspector X(15446). (César Lozada, October 10, 2018)
X(1743) lies on these lines: 1,6 10,391 19,1783 31,200 36,198 41,572 43,165 48,1732 57,1122 58,936 71,380 101,604 169,1046 173,266 223,1708 239,1278 241,1419 258,259 269,651 282,1795 284,1778 294,1721 346,519 579,610 580,1490 966,1698 978,1400 999,1696 1249,1785 1750,1754\
X(1743) = isogonal conjugate of X(8056)
X(1743) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,1), (1476,55)
X(1743) = crosspoint of X(i) and X(j) for these (i,j): (57,1420), (651,765)
X(1743) = crosssum of X(244) and X(650)
X(1743) = {X(6),X(9)}-harmonic conjugate of X(1)
X(1743) = trilinear pole wrt excentral triangle of Gergonne line
X(1743) = perspector of ABC and unary cofactor triangle of Triangle T(-1,3)
X(1743) = excentral-isogonal conjugate of X(10860)
X(1743) = X(393)-of-excentral-triangle
X(1743) = X(i)-aleph conjugate of X(j) for these (i,j): (1,165), (2,1766), (7,169), (57,1743), (81,572), (174,9), (259,170), (266,43), (365,1742), (366,40), (507,164), (508,63), (509,1), (513,1053), (651,101)
X(1743) = perspector of excentral triangle and unary cofactor triangle of intangents triangle
X(1744) lies on these lines: 9,46 19,91 846,1754 1046,1409 1172,1725 1709,1719 1723,1729 1770,1826
X(1744) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1754), (92,1765)
X(1745) lies on the McCay cubic and these lines: 1,4 35,1742 36,978 43,46 78,1330 255,411 579,610 908,1076 920,1758 1464,1836 1716,1756
X(1745) = isogonal conjugate of X(3362)
X(1745) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,1), (1935,1046)
X(1745) = X(1148)-cross conjugate of X(1)
X(1745) = {X(4),X(73)}-harmonic conjugate of X(1)
X(1745) = {X(223),X(1490)}-harmonic conjugate of X(1)
X(1745) = excentral isogonal conjugate of X(1158)
X(1745) = X(i)-aleph conjugate of X(j) for these (i,j): (1,46), (2,1726), (3,1745), (6,1716), (63,1763), (174,1708), (188,1158), (259,1721), (266,1722), (365,1707), (366,19), (508,1729), (509,1723), (651,1020)
X(1746) lies on these lines: 2,572 4,580 27,1730 57,1111 92,1726 333,1764 515,1006 946,1203
X(1746) = X(i)-aleph conjugate of X(j) for these (i,j): (27,1722), (29,1721), (86,223), (286,1708), (333,1490), (648,109)
X(1747) lies on these lines: 1,82 19,91 47,240 162,1096 1737,1890
X(1748) lies on these lines: 19,27 31,240 158,920 162,1096 412,1158 653,1708 1013,1859 1445,1767 1776,1857
X(1748) = isogonal conjugate of X(1820)
X(1748) = complement of X(18664)
X(1748) = anticomplement of X(18588)
X(1748) = cevapoint of X(19) and X(920)
X(1748) = X(i)-cross conjugate of X(j) for these (i,j): (563,1), (2180,47)
X(1748) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1740), (92,1), (264,63), (331,1445), (811,662), (823,162), (1969,1760)
X(1748) = SS(A → A') of X(63), where A'B'C' is the orthic triangle
X(1748) = {X(19),X(63)}-harmonic conjugate of X(92)
X(1748) = pole wrt polar circle of trilinear polar of X(91) (line X(661)X(2618))
X(1748) = polar conjugate of X(91)
X(1748) = trilinear product X(2)*X(24)
X(1749) lies on these lines: {1, 21}, {2, 16763}, {3, 16767}, {5, 79}, {30, 80}, {35, 17637}, {36, 2771}, {46, 7701}, {56, 13465}, {65, 22936}, {90, 18514}, {442, 12623}, {499, 14450}, {1087, 2962}, {1733, 16568}, {1755, 16546}, {1768, 5131}, {1776, 3583}, {1858, 14794}, {2099, 28453}, {2475, 18395}, {3086, 31888}, {3218, 3582}, {3219, 3584}, {3337, 3649}, {3375, 3376}, {3383, 3384}, {3585, 7098}, {3648, 5046}, {3820, 16152}, {3929, 17699}, {5427, 6265}, {5441, 10950}, {5444, 31650}, {5445, 5499}, {5692, 19525}, {5694, 14804}, {5844, 10543}, {5902, 7489}, {5903, 13743}, {6701, 7504}, {6949, 16116}, {7161, 13995}, {7280, 16132}, {7308, 17700}, {7491, 16113}, {7741, 16159}, {10266, 23016}, {10573, 15680}, {11263, 27003}, {12535, 13129}, {14526, 18244}, {14527, 17768}, {15079, 16150}, {15950, 16140}, {16155, 26475}, {17757, 18253}
X(1749) = midpoint of X(484) and X(3065)
X(1749) = reflection of X(i) in X(j) for these {i,j}: {3649, 15325}, {17757, 18253}
X(1749) = X(2166)-Ceva conjugate of X(1)
X(1749) = crosspoint of X(24041) and X(32680)
X(1749) = crosssum of X(2624) and X(2643)
X(1749) = crossdifference of every pair of points on line {661, 17438}
X(1749) = barycentric product X(i)*X(j) for these {i,j}: {75, 11063}, {311, 19306}, {799, 6140}, {1157, 14213}, {2349, 10272}, {3470, 14206}, {8562, 32680}, {10413, 24041}
X(1749) = X(i)-isoconjugate of X(j) for these (i,j): {2, 14579}, {6, 13582}, {54, 1263}, {74, 3471}, {186, 15392}, {323, 11071}, {523, 1291}, {3459, 14367}, {15109, 30526}
X(1749) = barycentric quotient X(i)/X(j) for these {i,j}: {{1, 13582}, {31, 14579}, {163, 1291}, {1157, 2167}, {1953, 1263}, {2173, 3471}, {3470, 2349}, {6140, 661}, {8562, 32679}, {10272, 14206}, {10413, 1109}, {11063, 1}, {19306, 54}}
X(1749) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {21, 11684, 3878}, {46, 7701, 16118}, {191, 6763, 11684}, {896, 1725, 6149}, {1725, 6149, 1}, {1755, 16562, 16546}, {1822, 1823, 2964}, {3336, 3467, 5}, {3649, 10021, 5443}, {5441, 16139, 11010}, {16139, 16141, 5441}, {17637, 22937, 35}
X(1750) lies on these lines: 1,4 9,165 20,936 40,210 43,1721 57,971 84,1728 200,329 1708,1768 1743,1754
X(1750) = X(282)-Ceva conjugate of X(1)
X(1750) = X(282)-aleph conjugate of X(1750)
X(1751) lies on these lines: 2,272 4,580 6,226 9,321 10,55 19,1708 27,579 57,379 76,333 92,1731 1479,1794
X(1751) = isogonal conjugate of X(579)
X(1751) = cevapoint of X(11) and X(652)
X(1751) = X(71)-cross conjugate of X(1)
X(1751) = crosssum of X(1724) and X(1754)
X(1751) = trilinear pole of line X(523)X(663) (the polar of X(5125) wrt the polar circle)
X(1751) = polar conjugate of X(5125)
X(1752) lies on these lines: 1,41 9,1479 19,46 920,1729
X(1753) lies on these lines: 1,947 3,33 4,9 34,517 46,208 55,1887 63,318 204,580 225,1217 475,946 1068,1435 1158,1726 1445,1895 1593,1824 1597,1871 1708,1712
X(1753) = X(92)-aleph conjugate of X(1767)
X(1753) = homothetic center of orthic triangle and reflection of intangents triangle in X(3)
X(1754) lies on these lines: 1,3 4,580 5,582 20,58 25,1730 31,516 33,1708 47,1770 63,990 81,991 109,278 184,851 209,916 212,226 219,1376 238,1699 255,1074 283,377 386,411 394,1004 498,1794 579,1172 595,962 602,946 846,1744 950,1451 1707,1709 1726,1824 1743,1750
X(1754) = X(i)-Ceva conjugate of X(j) for these (i,j): (579,1724),
(1172,1)
X(1754) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1744),
(29,1765), (365,1047), (1172,1754)
X(1755) lies on these lines: 6,893 19,27 31,48 44,513 610,1707 1580,1581
X(1755) = isogonal conjugate of X(1821)
X(1755) = X(i)-Ceva conjugate of X(j) for these (i,j): (1581,1964), (1821,1)
X(1755) = cevapoint of X(1580) and X(1955)
X(1755) = crosspoint of X(i) and X(j) for these (i,j): (1,1821), (31,1967), (57,741), (240,1959)
X(1755) = crosssum of X(i) and X(j) for these (i,j): (1,1755), (9,740), (75,1966), (293,1910)
X(1755) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1580), (1821,1755)
X(1755) = X(6)-isoconjugate of X(290)
X(1755) = X(92)-isoconjugate of X(293)
X(1755) = trilinear product X(2)*X(237)
X(1755) = barycentric square root of X(9419)
X(1756) lies on these lines: 1,256 9,46 36,238 1711,1763 1716,1745
X(1756) = reflection of X(1) in X(1284)
X(1756) = isogonal conjugate of X(7095)
X(1756) = X(98)-Ceva conjugate of X(1)
X(1756) = crosspoint of X(86) and X(1821)
X(1756) = crosssum of X(42) and X(1755)
X(1756) = X(98)-aleph conjugate of X(1756)
X(1757) lies on these lines: 1,6 10,894 42,846 43,63 57,1463 81,756 100,896 171,210 190,740 191,1045 200,1707 209,1762 239,726 240,1783 314,1089 320,1698 333,1215 484,513 527,1738 672,1282 765,1110 899,1054
X(1757) = reflection of X(i) in X(j) for these (i,j): (1,238), (238,44)
X(1757) = isogonal conjugate of X(1929)
X(1758) = X(2)-Ceva conjugate of X(39055)
X(1757) = X(291)-Ceva conjugate of X(1)
X(1757) = crosspoint of X(660) and X(765)
X(1757) = crosssum of X(i) and X(j) for these (i,j): (244,659), (1931, 1963)
X(1757) = X(i)-aleph conjugate of X(j) for these (i,j): (291,1757), (660,1026)
X(1757) = crossdifference of PU(31)
X(1757) = perspector of conic {A,B,C,X(100),PU(32)}
X(1757) = intersection of trilinear polars of X(100), P(32), and U(32)
X(1757) = inverse-in-circumconic-centered-at-X(9) of X(37)
X(1758) lies on these lines: 1,3 21,1254 43,1708 73,1046 108,240 223,1707 225,1247 226,846 238,1465 411,774 651,896 920,1745 1044,1158 1054,1738
X(1758) = isogonal conjugate of X(2648)
X(1758) = X(1937)-Ceva conjugate of X(1)
X(1758) = X(1937)-aleph conjugate of X(1758)
X(1758) = crossdifference of PU(80)
X(1758) = perspector of conic {{A,B,C,PU(81)}}
X(1759) lies on these lines: 1,32 9,46 40,728 41,758 63,169 72,910
X(1759) = X(76)-Ceva conjugate of X(1)
X(1759) = excentral-isogonal conjugate of X(32462)
X(1759) = X(i)-aleph conjugate of X(j) for these (i,j): (2,43),
(8,170), (75,9), (76,1759), (92,1711), (366,1740), (508,978),
(556,165), (668,1018)
X(1760) lies on these lines: 1,82 19,27 190,1766 240,255 326,610 1820,1821
X(1760) = isogonal conjugate of X(2156)
X(1760) = isotomic conjugate of isogonal conjugate of X(2172)
X(1760) = isotomic conjugate of complement of X(21215)
X(1760) = isotomic conjugate of anticomplement of X(16582)
X(1760) = isotomic conjugate of X(6)-isoconjugate of X(22)
X(1760) = complement of X(17481)
X(1760) = anticomplement of X(16580)
X(1760) = X(561)-Ceva conjugate of X(1)
X(1760) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1740),
(75,1), (76,63), (264,920), (314,411), (556,1742), (561,1760),
(668,100), (693,1052), (789,1492), (799,662), (811,162), (1969,1748),
(1978,190)
X(1761) lies on these lines: 1,1333 6,986 9,46 19,27 37,171 40,1503 284,758 896,1778 1158,1766
X(1761) = X(321)-Ceva conjugate of X(1)
X(1761) = X(i)-aleph conjugate of X(j) for these (i,j): (10,846), (92,1713), (321,1761), (556,573)
X(1761) = trilinear product X(2)*X(199)
X(1762) lies on these lines: 9,440 10,1782 19,27 30,40 55,846 57,1723 65,1046 71,1654 169,1708 209,1757 1812,1959
X(1762) = anticomplement of X(25361)
X(1762) = X(1441)-Ceva conjugate of X(1)
X(1762) = X(i)-aleph conjugate of X(j) for these (i,j): (2,3),
(75,1764), (92,1715), (226,1046), (508,6), (556,20), (1441,1762)
X(1763) lies on these lines: 1,25 2,169 6,57 9,440 19,226 40,64 43,46 63,573 73,1452 198,1214 329,1766 1711,1756
X(1763) = isogonal conjugate of X(7097)
X(1763) = isotomic conjugate of polar conjugate of X(36103)
X(1763) = anticomplement of X(36850)
X(1763) = crossdifference of every pair of points on line X(2522)X(3900)
X(1763) = X(69)-Ceva conjugate of X(1)
X(1763) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1716),
(2,46), (63,1745), (69,1763), (75,1726), (174,1722), (188,1721),
(366,1707), (508,1723), (556,1158), (664,1020)
X(1764) lies on these lines: 1,3 2,573 63,321 81,572 333,1746 345,1018 978,1695
X(1764) = X(314)-Ceva conjugate of X(1)
X(1764) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1046), (75,1762), (188,1045), (314,1764), (333,3), (556,191)
X(1764) = anticomplement of X(2051)
X(1764) = excentral-isogonal conjugate of X(1045)
X(1764) = X(418)-of-excentral-triangle
X(1764) = homothetic center of excentral triangle and 3rd Conway triangle
X(1764) = perspector of 3rd Conway triangle and (cross-triangle of ABC and 4th Conway triangle)
X(1764) = perspector of 3rd Conway triangle and (cross-triangle of ABC and 5th Conway triangle)
X(1765) lies on these lines: 1,1409 3,9 6,1012 7,1020 19,1158 20,391 21,572 63,321 71,515 580,1778 608,1777 1707,1709 1735,1880 1768,1781
X(1765) = X(i)-aleph conjugate of X(j) for these (i,j): (29,1754), (92,1744), (366,1047)
X(1766) lies on these lines: 1,572 3,37 4,9 6,517 20,346 46,1400 63,321 101,610 165,846 190,1760 329,1763 355,594 672,1723 971,1350 1100,1482 1158,1761
X(1766) = reflection of X(990) in X(3)
X(1766) = X(312)-Ceva conjugate of X(1)
X(1766) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1743), (8,165), (75,169), (188,43), (190,101), (312,1766), (333,572),
(366,978), (522,1053), (556,9)
X(1766) = excentral-isogonal conjugate of X(43)
X(1766) = excentral-isotomic conjugate of X(32462)
X(1767) lies on these lines: 3,207 19,57 46,208 63,653 65,1498 108,165 109,204 1158,1712 1445,1748
X(1767) = X(522)-Ceva conjugate of X(1)
X(1767) = X(i)-aleph conjugate of X(j) for these (i,j): (92,1753),
(342,1767), (508,282), (653,108), (522,1768)
X(1768) lies on the Bevan circle, the excentral-hexyl ellipse, and these lines: 1,104 3,191 10,153 11,57 36,1727 40,550 46,80 63,100 119,1698 149,516 484,515 971,1155 1156,1445 1317,1697 1708,1750 1765,1781
X(1768) = reflection of X(34464) in line X(1)X(3)
X(1768) = isogonal conjugate of X(29374)
X(1768) = reflection of X(i) in X(j) for these (i,j): (1,104), (153,10)
X(1768) = X(i)-aleph conjugate of X(j) for these (i,j): (2,514), (174,905), (366,650), (508,657), (522,1768)
X(1768) = trilinear pole wrt excentral triangle of line X(4)X(9)
X(1768) = excentral-isogonal conjugate of X(513)
X(1768) = X(110)-of-excentral-triangle
X(1768) = intouch-to-excentral similarity image of X(11)
X(1769) lies on these lines: 11,244 104,106 108,109 240,522 513,663
X(1769) = isogonal conjugate of X(36037)
X(1769) = X(162)-Ceva conjugate of X(1845)
X(1769) = crosspoint of X(i) and X(j) for these (i,j): (88,934), (162,759)
X(1769) = crosssum of X(i) and X(j) for these (i,j): (31,1635), (522,1737), (656,758)
X(1769) = trilinear product of circumcircle intercepts of Sherman line
X(1769) = crossdifference of every pair of points on line X(9)X(48) (the Fermat axis of the excentral triangle and of the 2nd extouch triangle)
Let IaIbIc be the reflection triangle of X(1). Let A' be the cevapoint of Ib and Ic, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1770). (Randy Hutson, July 20, 2016)
X(1770) lies on these lines: 1,7 3,1836 4,46 5,1155 10,191 27,1780 28,1633 30,65 35,79 36,946 40,1478 47,1754 57,1479 109,225 165,498 382,1837 1707,1714 1710,1782 1724,1738 1744,1826 1771,1785 1885,1905
X(1770) = cevapoint of X(46) and X(1717)
X(1771) lies on these lines: 1,3 4,109 10,255 31,1210 515,603 516,1076 580,1788 601,950 920,1736 1399,1837 1707,1728 1720,1721 1770,1785
X(1772) lies on these lines: 1,88 3,1411 34,46 40,1421 240,522 498,986 920,1722 1068,1788
X(1773) lies on these lines: 1,25 10,46 40,984 57,1722 169,1400 244,1468 388,1452
X(1773) = X(i)-Ceva conjugate of X(j) for these (i,j): (388,1),
(1452,46)
X(1774) lies on these lines: 4,46 40,47 57,1725 378,1061 484,1707
X(1775) lies on these lines: 4,46 24,1063 40,1725 47,57
X(1776) lies on these lines: 4,46 21,60 33,1707 63,497 109,1736 191,950 243,522 411,1898 519,1697 1005,1864 1155,1156 1725,1870 1748,1857
X(1776) = crosspoint of X(21) and X(1156)
X(1776) = crosssum of X(65) and X(1155)
X(1777) lies on these lines: 1,84 4,109 34,1158 35,1742 36,1044 46,1707 47,1754 90,1041 226,601 255,516 603,946 608,1765 1046,1710 1399,1836
X(1778) lies on these lines: 6,21 9,58 19,1707 28,579 37,81 44,1333 90,1172 284,1743 580,1765 896,1761 966,1010 1396,1708 1474,1732
X(1778) = cevapoint of X(1707) and (1724)
X(1779) lies on these lines: 3,1780 4,46 35,47 43,165 378,580 579,1172 1714,1715
X(1779) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1781), (29,573)
X(1780) lies on these lines: 1,21 3,1779 4,580 27,1770 28,46 29,1737 35,71 90,1172 219,1333 579,1474 1010,1098 1214,1399 1408,1617
X(1780) = X(27)-Ceva conjugate of X(284)
X(1781) lies on these lines: 1,19 6,1718 9,46 35,37 57,1723 71,484 165,846 169,1046 281,1478 1710,1720 1765,1768
X(1781) = X(226)-Ceva conjugate of X(1)
X(1781) = X(i)-aleph conjugate of X(j) for these (i,j): (2,573),
(4,1779), (174,6), (226,1781), (366,3), (508,2)
X(1782) lies on these lines: 4,1726 8,20 10,1762 19,46 58,65 71,191 580,1829 1710,1770 1737,1842
X(1783) lies on these lines: 4,218 6,281 19,1743 28,291 80,1172 100,112 101,108 150,1814 200,204 219,1249 240,1757 644,648 650,1415 651,653 899,1430 905,934 1103,1712 1718,1723 1785,1886
X(1783) = isogonal conjugate of X(905)
X(1783) = isotomic conjugate of X(15413)
X(1783) = trilinear pole of line X(19)X(25) (the tangent to hyperbola {{A,B,C,X(4),X(19)}} at X(19))
X(1783) = pole wrt polar circle of trilinear polar of X(693) (line X(918)X(1086))
X(1783) = polar conjugate of X(693)
X(1783) = crossdifference of every pair of points on line X(1364)X(3270)
X(1783) = perspector of anticevian triangle of X(108) and unary cofactor triangle of intangents triangle
X(1783) = barycentric product of circumcircle intercepts of line X(4)X(8)
X(1783) = barycentric product X(4)*X(100)
X(1783) = X(i)-Ceva conjugate of X(j) for these (i,j): (648,1897), (653,108)
X(1783) = cevapoint of X(i) and X(j) for these (i,j): (1,1734), (6,650), (513,614)
X(1783) = crosspoint of X(i) and X(j) for these (i,j): (162,648), (653,1897)
X(1783) = crosssum of X(i) and X(j) for these (i,j): (647,656), (652,1459), (513,614)
X(1783) = X(i)-aleph conjugate of X(j) for these (i,j): (108,1707), (651,610), (653,19)
X(1783) = X(92)-isoconjugate of X(23224)
X(1783) = trilinear product X(4)*X(101)
X(1783) = trilinear product of circumcircle intercepts of line X(4)X(9)
X(1784) lies on these lines: 1,29 4,79 36,243 240,522 318,1698 451,498 484,653 920,1712 1118,1479 1478,1857
X(1784) = isogonal conjugate of X(35200)
X(1784) = crossdifference of every pair of points on line X(48)X(822)
X(1784) = circle-{{X(11),X(36),X(65)}}-inverse of X(36063)
X(1784) = pole wrt polar circle of trilinear polar of X(2349) (line X(1)X(656))
X(1784) = polar conjugate of X(2349)
X(1784) = X(63)-isoconjugate of X(2159)
X(1785) lies on these lines: 1,4 2,1074 9,393 10,158 20,1076 25,1324 36,108 37,53 40,1118 46,208 65,1872 106,1309 240,522 406,498 407,1844 475,499 517,1361 519,1897 942,1887 1210,1895 1249,1743 1465,1532 1712,1714 1770,1771 1783,1886 1824,1894 1830,1835 1867,1904
X(1785) = reflection of X(1845) in X(1875)
X(1785) = isogonal conjugate of X(1795)
X(1785) = inverse-in-incircle of X(946)
X(1785) = X(4)-Ceva conjugate of X(1845)
X(1785) = Conway-circle-inverse of X(35635)
X(1785) = crossdifference of every pair of points on line X(48)X(652)
X(1785) = inverse-in-polar-circle of X(1)
X(1785) = trilinear product X(8072)*X(8073)
X(1785) = polar conjugate of X(34234)
X(1786) lies on these lines: 33,46 57,77 1730,1787
X(1787) lies on these lines: 34,46 57,88 1020,1435 1730,1786
X(1788) lies on these lines: 1,631 2,65 4,46 7,12 8,56 10,57 11,962 20,1155 34,1722 36,944 40,497 43,73 55,938 109,1724 145,1319 165,950 171,1451 200,1467 201,986 208,1861 225,1738 226,1698 227,241 278,1714 281,579 329,1329 344,1284 345,1403 377,1454 387,1214 412,1857 484,1479 519,1420 580,1771 651,1406 653,1118 899,1042 958,1466 961,1150 978,1457 1068,1772 1707,1877
X(1788) = anticomplement of X(25681)
X(1788) = cevapoint of X(46) and X(1722)
X(1789) lies on these lines: 3,125 21,36
X(1789) = isogonal conjugate of X(1825)
X(1789) = X(19)-isoconjugate of X(16577)
X(1789) = X(92)-isoconjugate of X(21741)
X(1790) lies on these lines: 1,1719 2,572 3,49 6,967 21,84 22,991 27,86 36,58 48,63 57,77 71,1796 73,1798 103,110 199,511 222,1804 228,295 306,332 333,662 1214,1813 1408,1470
X(1790) = isogonal conjugate of X(1826)
X(1790) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,58), (1444,283)
X(1790) = cevapoint of X(3) and X(48)
X(1790) = X(222)-cross conjugate of X(81)
X(1790) = crosssum of X(1) and X(1719)
X(1790) = X(92)-isoconjugate of X(42)
X(1791) lies on these lines: 1,1472 2,12 3,345 8,197 21,37 48,78 63,201 72,1437 228,1792 280,1436 975,993
X(1791) = isogonal conjugate of X(1829)
X(1791) = cevapoint of X(i) and X(j) for these (i,j): (3,72), (37,197), (219,228)
X(1792) lies on these lines: 3,69 8,21 78,212 81,1257 86,939 99,972 228,1791 271,1819 314,943 1260,1265
X(1792) = isogonal conjugate of X(1426)
X(1792) = X(332)-Ceva conjugate of X(1812)
X(1792) = cevapoint of X(i) and X(j) for these (i,j): (78,1259),
(283,1819)
X(1793) lies on these lines: 3,125 10,21 72,283 307,1444
X(1793) = isogonal conjugate of X(1835)
X(1793) = isotomic conjugate of polar conjugate of X(2341)
X(1793) = X(19)-isoconjugate of X(18593)
X(1794) lies on these lines: 1,201 10,29 35,71 36,951 77,255 498,1754 1059,1617 1479,1751
X(1794) = isogonal conjugate of X(1838)
X(1794) = cevapoint of X(71) and X(212)
X(1794) = crosssum of X(1841) and X(1859)
X(1795) lies on these lines: 1,104 3,1364 29,58 35,947 36,102 56,945 57,1845 78,255 117,1478 124,499 163,284 171,1065 219,577 282,1743 912,1807 999,1361
X(1795) = isogonal conjugate of X(1785)
X(1795) = trilinear pole of line X(48)X(652)
X(1796) lies on these lines: 27,1268 35,42 57,1255 71,1790
X(1796) = isogonal conjugate of X(1839)
X(1796) = X(1268)-Ceva conjugate of X(1126)
X(1796) = cevapoint of X(3) and X(71)
X(1796) = X(92)-isoconjugate of X(2308)
X(1797) lies on these lines: 3,1331 27,648 57,88 58,106 63,1332 84,1320 103,677 222,1813 320,908
X(1797) = X(903)-Ceva conjugate of X(106)
X(1797) = isogonal conjugate of X(8756)
X(1797) = X(92)-isoconjugate of X(902)
X(1797) = isotomic conjugate of polar conjugate of X(106)
X(1797) = X(19)-isoconjugate of X(519)
X(1797) = crossdifference of every pair of points on line X(4120)X(4895)
X(1798) lies on these lines: 6,60 54,970 58,1245 65,81 71,283 72,1437 73,1790 110,960 285,1903
X(1798) = isogonal conjugate of X(429)
X(1798) = cevapoint of X(3) and X(1437)
Let A'B'C' be the circummedial triangle (the circumcevian triangle of X(2)). Let A" be the cevapoint of B' and C', and define B" and C" cyclically. AA", BB", and CC" concur in X(1799). (Randy Hutson, December 10, 2016)
X(1799) lies on these lines: 2,32 3,305 22,76 25,183 69,184 95,325 98,689 287,343 385,1194 1402,1441
X(1799) = isogonal conjugate of X(1843)
X(1799) = isotomic conjugate of X(427)
X(1799) = X(308)-Ceva conjugate of X(83)
X(1799) = cevapoint of X(i) and X(j) for these (i,j): (2,22), (3,69)
X(1799) = complement of X(8878)
X(1799) = crosspoint of X(2) and X(22) wrt both the anticomplementary and tangential triangles
X(1799) = X(92)-isoconjugate of X(3051)
X(1799) = perspector of circummedial triangle and cross-triangle of ABC and circummedial triangle
X(1800) lies on these lines: 1,921 3,49 21,90 29,662 46,453 65,1813
X(1800) = X(21)-Ceva conjugate of X(283)
X(1801) lies on these lines: 3,49 58,78 171,306
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1802) lies on these lines: 3,48 6,939 9,943 40,101 41,55 63,1803 255,906
X(1802) = isogonal conjugate of X(1847)
X(1802) = X(i)-Ceva conjugate of X(j) for these (i,j): (200,1253), (219,212)
X(1802) = crosspoint of X(i) and X(j) for these (i,j): (219,1260), (906,1110)
X(1802) = crosssum of X(i) and X(j) for these (i,j): (269,1435), (278,1119)
X(1802) = X(92)-isoconjugate of X(269)
X(1803) lies on these lines: 35,103 41,57 58,1458 63,1802
X(1803) = isogonal conjugate of X(1855)
X(1803) cevapoint of X(48) and X(222)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1804) lies on these lines: 3,77 7,21 20,1440 36,269 55,1442 63,268 69,1809 198,651 219,1813 222,1790 326,1259 347,934 573,1461
X(1804) = isogonal conjugate of X(1857)
X(1804) = X(i)-Ceva conjugate of X(j) for these (i,j): (348,222), (1444,77)
X(1804) = X(92)-isoconjugate of X(607)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1805) lies on this line: 3,6
X(1805) = X(21)-Ceva conjugate of X(1806)
X(1805) = {X(3),X(2193)}-harmonic conjugate of X(1806)
X(1805) = X(1437)-cross conjugate of X(1806)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1806) lies on this line: 3,6
X(1806) = X(21)-Ceva conjugate of X(1805)
X(1806) = {X(3),X(2193)}-harmonic conjugate of X(1805)
X(1806) = X(1437)-cross conjugate of X(1805)
Trilinears 1/(2 - sec A) : 1/(2 - sec B) : 1/(2 - sec C)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1807) lies on these lines: 1,5 3,201 29,1897 37,101 72,283 73,265 77,1060 78,1062 102,517 296,916 912,1795 942,951 945,1482 947,1385 976,1036 999,1037
X(1807) = isogonal conjugate of X(1870)
X(1807) = crosssum of X(i) and X(j) for these (i,j): (1,1718), (1464,1835)
X(1807) = inverse-in-Feuerbach-hyperbola of X(1411)
X(1807) = {X(1),X(80)}-harmonic conjugate of X(1411)
X(1807) = inverse-in-circumconic-centered-at-X(1) of X(80)
X(1808) lies on these lines: 41,60 42,81 228,295
X(1808) = isogonal conjugate of X(1874)
X(1808) = isotomic conjugate of polar conjugate of X(2311)
X(1808) = X(19)-isoconjugate of X(16609)
X(1809) lies on these lines: 3,8 69,1804 78,255 1259,1265 1295,1309
X(1809) = isogonal conjugate of X(1875)
X(1810) lies on these lines: 57,100 222,1331
X(1810) = cevapoint of X(3) and X(1818)
X(1810) = isogonal conjugate of polar conjugate of X(36807)
X(1811) lies on these lines: 56,100 603,1331
X(1811) = isogonal conjugate of X(1878)
X(1812) lies on these lines: 2,6 21,60 48,63 58,997 72,1437 78,212 219,332 222,348 274,1231 280,285 306,1332 314,1172 662,1817 860,1330 1006,1092 1412,1708 1762,1959
X(1812) = isogonal conjugate of X(1880)
X(1812) = X(i)-Ceva conjugate of X(j) for these (i,j): (314,21), (332,1792)
X(1812) = cevapoint of X(i) and X(j) for these (i,j): (63,394), (78,219)
X(1812) = X(92)-isoconjugate of X(1402)
X(1813) lies on the MacBeath circumconic and on these lines: 48,77 59,677 65,1800 73,895 101,651 109,110 219,1804 222,1797 224,1420 283,296 284,1442 287,307 347,1630 604,1445 648,653 1214,1790
X(1813) = isogonal conjugate of X(3064)
X(1813) = X(i)-Ceva conjugate of X(j) for these (i,j): (662,651), (664,109)
X(1813) = cevapoint of X(i) and X(j) for these (i,j): (3,652), (48,1459), (905,1214)
X(1813) = X(219)-cross conjugate of X(59)
X(1813) = trilinear pole of line X(3)X(73)
X(1813) = crossdifference of every pair of points on line X(2310)X(8735)
X(1813) = X(92)-isoconjugate of X(663)
X(1814) lies on these lines: 6,7 48,77 63,212 69,219 81,105 150,1783 286,648 518,677 1438,1449
X(1814) = reflection of X(651) in X(6)
X(1814) = isogonal conjugate of X(5089)
X(1814) = X(92)-isoconjugate of X(2223)
X(1814) = MacBeath circumconic antipode of X(651)
X(1815) lies on these lines: 9,77 86,648 103,110 219,1804 326,1332 394,1260
X(1815) = isogonal conjugate of X(1886)
X(1815) = cevapoint of X(219) and X(1818)
As a point on the Euler line, X(1816) has Shinagawa coefficients (4EF3abc - $aSA$[S2 - (2E + 3F)F][S2 - (2E + F)F] + 2$aSBSC$[S2 - (2E + F)F]F + 4$a(SA)3$F2, [S4 - (2E - F)FS2 + F4]abc + $aSA$[S2 - (2E + 3F)F][S2 - (2E + F)F] - 2$aSBSC$[S2 - (2E + F)F]F - 4$a(SA)3$F2).
X(1816) lies on this line: 2,3
X(1816) = X(283)-Ceva conjugate of X(21)
As a point on the Euler line, X(1817) has Shinagawa coefficients (4EF2S2 - $aSA$(S2 - 4F2)S2 + 3$bcSBSC$S2 + 4$abSC$FS2 - 2$bc(SB)2(SC)2$, [S2 - 4(E + F)F]ES2 + $bc$(S2 - 4F2)S2 - 3$bcSBSC$S2 - 4$abSC$FS2 + 2$bc(SB)2(SC)2$).
X(1817) lies on these lines: 2,3 40,1819 57,77 58,937 63,610 100,306 110,972 189,333 196,347 198,329 572,1730 662,1812 1172,1214 1396,1465
X(1817) = isogonal conjugate of X(1903)
X(1817) = X(i)-Ceva conjugate of X(j) for these (i,j): (333,81), (1444,21)
X(1817) = cevapoint of X(i) and X(j) for these (i,j): (3,610), (40,198)
X(1818) lies on these lines: 1,142 3,48 9,991 42,750 69,73 198,1350 212,394 222,1260 241,518 386,1449 521,656 581,936 603,1259 997,1064 1193,1386
X(1818) = isogonal conjugate of X(36124)
X(1818) = X(i)-Ceva conjugate of X(j) for these (i,j): (1810,3), (1815,219)
X(1818) = crosspoint of X(3) and X(295)
X(1818) = crosssum of X(i) and X(j) for these (i,j): (1,1738), (4,242)
X(1819) lies on these lines: 3,49 9,21 40,1817 58,1167 271,1792
X(1819) = X(1792)-Ceva conjugate of X(283)
X(1820) lies on these lines: 1,563 19,91 68,71 1400,1454 1760,1821
X(1820) = isogonal conjugate of X(1748)
X(1820) = complement of anticomplementary conjugate of X(18664)
X(1820) = crosspoint of X(63) and X(921)
X(1820) = crosssum of X(19) and X(920)
X(1820) = SS(A→A') of X(19), where A'B'C' is the orthic triangle
X(1820) = X(47)-isoconjugate of X(92)
X(1821) lies on these lines: 19,823 31,92 48,75 63,561 71,190 98,100 287,651 653,1400 1580,1733 1760,1820 1934,1959
X(1821) = isogonal conjugate of X(1755)
X(1821) = isotomic conjugate of X(1959)
X(1821) = cevapoint of X(i) and X(j) for these (i,j): (1,1755), (9,740), (75,1966), (293,1910)
X(1821) = crosspoint of X(1581) and X(1956)
X(1821) = crosssum of X(1580) and X(1955)
X(1821) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1740), (290,73), (1821,1)
X(1821) = trilinear pole of line X(1)X(810)
X(1821) = pole wrt polar circle of trilinear polar of X(240)
X(1821) = X(48)-isoconjugate (polar conjugate) of X(240)
X(1821) = X(6)-isoconjugate of X(511)
X(1822) lies on these lines: 1,21 109,1113
X(1822) = isogonal conjugate of X(2588)
X(1822) = X(1101)-Ceva conjugate of X(1823)
X(1822) = X(i)-aleph conjugate of X(j) for these (i,j): (162,1823), (1113,1)
X(1822) = trilinear quotient X(110)/X(1114)
X(1823) = isogonal conjugate of X(2589)
X(1823) = X(1101)-Ceva conjugate of X(1822)
X(1823) = X(i)-aleph conjugate of X(j) for these (i,j): (162,1822), (1114,1)
X(1823) = trilinear quotient X(110)/X(1113)
Zosma Transforms 1824-1907
The Zosma transform of a point X = x : y : z is the isogonal conjugate
of the inverse Mimosa transform of X, given by trilinears
(y + z) sec A : (z + x) sec B : (x + y)sec C.
(Zosma is another star name.)
X(1824) lies on these lines: 4,8 10,429 12,431 19,25 27,295 28,1255 34,1887 42,1880 51,1864 65,225 209,1865 210,430 213,607 240,444 278,1002 427,1848 428,528 518,1889 674,1839 756,862 851,1214 942,1068 989,1039 990,1473 1593,1753 1726,1754 1730,1736 1785,1894\
X(1824) = isogonal conjugate of X(1444)
X(1824) = complement of X(20243)
X(1824) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1826), (33,42), (225,1826)
X(1824) = X(i)-cross conjugate of X(j) for these (i,j): (181,42), (213,37)
X(1824) = crosspoint of X(i) and X(j) for these (i,j): (4,19), (33,1857), (65, 1903), (225,1826)
X(1824) = crosssum of X(i) and X(j) for these (i,j): (3,63), (21,1817), (77,1804), (283,1790)
X(1824) = intersection of tangents to hyperbola {{A,B,C,X(4),X(19)}} at X(4) and X(19)
X(1824) = pole wrt polar circle of trilinear polar of X(274) (line X(320)X(350))
X(1824) = polar conjugate of X(274)
X(1824) = barycentric product of vertices of 2nd extouch triangle
X(1824) = X(174)-of-orthic-triangle if ABC is acute
X(1825) lies on these lines: 4,80 19,41 33,40 34,1126 35,186 65,225 109,1710 250,270 319,340 1829,1877 1859,1902 1872,1905 1875,1900
X(1825) = isogonal conjugate of X(1789)
X(1825) = polar conjugate of isogonal conjugate of X(21741)
X(1826) lies on these lines: 4,9 6,1837 11,1108 12,37 25,1631 27,1268 28,1224 29,1220 33,42 48,515 53,1904 65,1868 80,1172 92,264 101,1300 209,1859 210,430 219,355 286,334 407,1213 427,1841 579,1737 608,1877 857,1441 1744,1770 1836,1853
X(1826) = isogonal conjugate of X(1790)
X(1826) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1824), (281,37)
X(1826) = cevapoint of X(1) and X(1719)
X(1826) = X(42)-cross conjugate of X(10)
X(1826) = crosspoint of X(4) and X(92)
X(1826) = crosssum of X(3) and X(48)
X(1826) = pole wrt polar circle of trilinear polar of X(86) (line X(239)X(514))
X(1826) = polar conjugate of X(86)
X(1827) lies on these lines: 4,7 19,25 46,1721 208,1887 294,1172 430,1856 1425,1547 1828,1843 1845,1905
X(1827) = crosspoint of X(4) and X(33)
X(1827) = crosssum of X(3) and X(77)
X(1827) = polar conjugate of X(31618)
X(1827) = {X(1849),X(1850)}-harmonic conjugate of X(4)
X(1828) lies on these lines: 4,8 10,1883 19,44 25,34 28,88 46,1707 51,65 225,1846 427,1329 428,529 1827,1843 1838,1894 1844,1884 1848,1904
X(1828) = crosspoint of X(4) and X(34)
X(1828) = crosssum of X(3) and X(78)
X(1828) = inverse-in-Fuhrmann-circle of X(5101)
X(1828) = polar conjugate of X(32017)
X(1829) lies on these lines: 1,25 4,8 6,19 10,427 24,1385 27,239 28,60 29,242 40,1593 52,912 56,1452 57,1398 209,1869 225,1866 235,946 278,959 388,1892 392,406 407,1838 428,519 429,960 444,1193 468,1125 516,1885 518,1843 580,1782 1100,1474 1395,1468 1482,1598 1724,1726 1825,1877 1831,1842 1852,1858 1861,1883
X(1829) = reflection of X(1902) in X(4)
X(1829) = isogonal conjugate of X(1791)
X(1829) = anticomplement of X(37613)
X(1829) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,429), (19,444)
X(1829) = crosspoint of X(i) and X(j) for these (i,j): (4,28), (278,286))
X(1829) = crosssum of X(i) and X(j) for these (i,j): (3,72), (37,197), (219,228)
X(1829) = inverse-in-Fuhrmann-circle of X(5090)
X(1829) = X(177)-of-orthic triangle if ABC is acute
X(1829) = polar conjugate of X(30710)
X(1829) = X(1)-of-anti-Ara-triangle
X(1830) lies on these lines: 4,80 33,57 65,1846 225,1872 517,1877 900,1862 908,1861 1785,1835 1887,1902 1888,1900
X(1830) = polar conjugate of isogonal conjugate of X(21742)X(1831) lies on these lines: 4,80 28,501 33,1697 225,1871 1829,1842 1835,1838 1839,1858 1844,1870
X(1831) = crosspoint of X(4) and X(270)
X(1831) = crosssum of X(3) and X(201)
X(1832) lies on these lines: 4,1251 12,37
X(1833) lies on this line: 12,37
X(1834) lies on these lines: 1,442 2,1043 4,6 5,386 8,1211 10,37 11,1193 12,42 30,58 33,429 43,1329 56,851 65,225 115,118 377,940 405,1714 440,950 496,995 497,1191 524,1330 942,1086 1058,1616
X(1834) = complement of X(1043)
X(1834) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1842), (1897,523)
X(1834) = crosspoint of X(4) and X(10)
X(1834) = crosssum of X(i) and X(j) for these (i,j): (3,58), (21,404)
X(1834) = crossdifference of every pair of points on line X(520)X(3733)
X(1835) lies on these lines: 4,79 28,34 36,186 65,225 320,340 513,1874 758,860 1398,1470 1785,1830 1831,1838
X(1835) = isogonal conjugate of X(1793)
X(1835) = X(1870)-Ceva conjugate of X(1464)
X(1835) = polar conjugate of isotomic conjugate of X(18593)
X(1835) = X(63)-isoconjugate of X(2341)
Let A'B'C' be the intangents triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically. A"B"C" is perspective to ABC at X(1836). (Randy Hutson, June 7, 2019)
X(1836) lies on these lines: 1,30 2,1155 3,1770 4,65 5,46 6,1839 7,354 11,57 12,40 19,1901 33,1892 34,1852 55,226 56,946 210,329 221,225 235,1452 278,1456 377,960 381,1737 388,962 517,1478 614,1086 908,1376 942,1479 1040,1721 1158,1454 1399,1777 1470,1519 1826,1853
X(1836) = reflection of X(i) in X(j) for these (i,j): (55,226), (1012,946)
X(1836) = isogonal conjugate of X(37741)
X(1836) = isotomic conjugate of X(34409)
X(1836) = anticomplement of X(4640)
X(1836) = crosspoint of X(4) and X(7)
X(1836) = crosssum of X(i) and X(j) for these {i,j}: {3, 55}, {500, 23207}
X(1836) = pole of orthic axis wrt the incircle
X(1836) = crossdifference of every pair of points on line X(9404)X(34975)
X(1836) = {X(4),X(65)}-harmonic conjugate of X(1837)
X(1836) = orthologic center of these triangles: 1st Johnson-Yff to 3rd extouch
X(1836) = X(40)-of-1st-Johnson-Yff-triangle
X(1836) = outer-Johnson-to-ABC similarity image of X(40)
X(1837) lies on these lines: 1,5 3,1737 4,65 6,1826 8,210 10,55 19,1852 20,1155 30,46 33,429 40,1728 56,515 78,1329 221,1877 354,388 382,1770 499,1385 517,1479 942,1478 944,1319 1040,1722 1399,1771 1464,1745 1853,1854
X(1837) = midpoint of X(65) and X(1898)
X(1837) = reflection of X(i) in X(j) for these (i,j): (1,496), (56,1210), (78,1329)
X(1837) = isotomic conjugate of X(34399)
X(1837) = inverse-in-Fuhrmann-circle of X(11)
X(1837) = crosspoint of X(4) and X(8)
X(1837) = crosssum of X(3) and X(56)
X(1837) = X(24)-of-Fuhrmann-triangle
X(1837) = inverse-in-Feuerbach-hyperbola of X(355)
X(1837) = {X(1),X(80)}-harmonic conjugate of X(355)
X(1837) = {X(4),X(65)}-harmonic conjugate of X(1836)
X(1837) = Ursa-major-to-Ursa-minor similarity image of X(1)
X(1838) lies on these lines: 1,4 2,1076 5,1214 10,92 11,133 19,46 20,1074 27,58 28,36 29,1125 30,1852 47,1754 53,1108 57,1118 65,1243 79,1172 158,273 235,1893 281,1698 403,1873 407,1829 412,516 427,1867 442,1841 517,1888 942,1844 1426,1905 1598,1617 1828,1894 1831,1835
X(1838) = isogonal conjugate of X(1794)
X(1838) = X(4)-Ceva conjugate of X(1844)
X(1838) = cevapoint of X(1841) and X(1859)
X(1838) = crosspoint of X(27) and X(273)
X(1838) = crosssum of X(71) and X(212)
X(1839) lies on these lines: 4,9 6,1836 27,86 48,946 52,916 79,1172 193,3187 225,608 278,1419 430,1213 534,1441 579,1770 610,1699 674,1824 1831,1858 1840,1900 1841,1852 1877,1880
X(1839) = isogonal conjugate of X(1796)
X(1839) = X(4)-Ceva conjugate of X(430)
X(1839) = crosspoint of X(4) and X(27)
X(1839) = crosssum of X(3) and X(71)
X(1839) = polar conjugate of X(1268)
X(1840) lies on these lines: 12,37 19,318 419,1215 1839,1900
X(1840) = polar conjugate of X(32010)X(1841) lies on these lines: 4,37 6,19 28,1104 53,225 71,1888 216,1465 241,273 278,393 281,475 427,1826 442,1838 581,1871 594,1861 1100,1172 1119,1418 1400,1875 1839,1852
X(1841) = X(1838)-Ceva conjugate of X(1859)
X(1841) = crosspoint of X(28) and X(278)
X(1841) = crosssum of X(72) and X(219)
X(1842) lies on these lines: 4,9 25,225 28,36 29,1848 33,976 34,207 51,65 92,1891 278,1420 1737,1782 1829,1831 1878,1888
X(1842) = X(4)-Ceva conjugate of X(1834)
X(1842) = crosspoint of X(27) and X(1119)
X(1842) = crosssum of X(71) and X(1260)
Let E = Euler line of ABC. Let g = isogonal conjugate, and t = isotomic conjugate. Then X(1843) = g(t(E))∩t(g(E)). (Randy Hutson, December 2, 2017)
X(1843) lies on these lines: 4,69 6,25 24,182 34,1469 112,755 113,1596 125,1205 141,427 143,1353 155,1351 157,571 160,570 181,1395 185,1503 216,237 263,393 373,468 428,524 518,1829 674,1824 1350,1593 1827,1828
X(1843) = reflection of X(i) in X(j) for these (i,j): (1205,125), (1289,647)
X(1843) = isogonal conjugate of X(1799)
X(1843) = X(i)-Ceva conjugate for these (i,j): (4,427), (427,39), (1289,647)
X(1843) = crosspoint of X(i) and X(j) for these (i,j): (4,25), (6,66)
X(1843) = crosssum of X(i) and X(j) for these (i,j): (2,22), (3,69)
X(1843) = orthic isotomic conjugate of X(185)
X(1843) = X(7)-of-orthic-triangle if ABC is acute
X(1843) = orthic-isogonal conjugate of X(427)
X(1843) = anticomplement of X(6) wrt orthic triangle
X(1843) = pole wrt polar circle of trilinear polar of X(308) (line X(316)X(512))
X(1843) = X(48)-isoconjugate (polar conjugate) of X(308)
X(1843) = inverse-in-polar-circle of X(316)
X(1843) = excentral-to-ABC functional image of X(7)
X(1843) = Ehrmann-vertex-to-orthic similarity image of X(3818)
X(1843) = perspector of [reflection of symmedial triangle in X(6)] and tangential triangle, wrt symmedial triangle, of circumconic of symmedial triangle centered at X(6) (bicevian conic of X(6) and X(25))
Let A'B'C' be the orthic triangle. Let A" be the incenter of AB'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1844). (Randy Hutson, January 29, 2018)
X(1844) lies on these lines: 1,19 4,79 29,758 33,46 35,186 354,1871 407,1785 942,1838 1828,1884 1831,1870 1846,1887
X(1844) = X(4)-Ceva conjugate of X(1838)
X(1844) = polar circle inverse of X(34301)
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)
Let A'B'C' be the orthic triangle. Let A" be the A-excenter of AB'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1845). (Randy Hutson, January 29, 2018)X(1845) lies on these lines: 1,102 4,80 19,1743 34,46 36,186 57,1795 65,389 92,994 117,1737 162,759 407,1829 517,1361 942,1354 1146,1901 1827,1905
X(1845) = reflection of X(i) in X(j) for these (i,j): (1364,942),
(1785,1875)
X(1845) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1785),
(162,1769)
X(1845) = crosspoint of X(4) and X(1870)
X(1845) = crosssum of X(3) and X(1807)
X(1846) lies on these lines: 4,11 19,53 65,1830 225,1828 517,1361 1319,1877 1844,1887
X(1846) = X(4)-Ceva conjugate of X(1877)
X(1846) = crosspoint of X(4) and X(1785)
X(1846) = crosssum of X(3) and X(1795)
X(1847) lies on these lines: 4,7 27,1088 85,92 158,1111 224,664 278,279 917,934
X(1847) = isogonal conjugate of X(1802)
X(1847) = cevapoint of X(i) and X(j) for these (i,j): (269,1435), (278,1119)
X(1847) = isotomic conjugate of X(3692)
X(1847) = polar conjugate of X(200)
X(1848) lies on these lines: 1,4 2,19 5,1871 7,1435 11,132 25,1001 27,86 28,1125 29,1842 92,264 286,350 427,1824 429,960 608,940 1595,1872 1828,1904 1883,1900
X(1848) = isogonal conjugate of X(2359)
X(1848) = crosspoint of X(27) and X(92)
X(1848) = crosssum of X(i) and X(j) for these (i,j): (42,205), (48,71)
X(1848) = pole wrt polar circle of trilinear polar of X(1220) (line X(522)X(649))
X(1848) = polar conjugate of X(1220)
X(1848) = perspector of Gemini triangle 38 and cross-triangle of Gemini triangles 37 and 38
X(1849) lies on this line: 4,7
X(1850) lies on this line: 4,7
X(1851) lies on these lines: 2,242 4,8 19,672 25,105 34,207 196,1876 281,427 286,310 479,1119 497,1863 1146,1853 1395,1430
X(1851) = X(4)-Ceva conjugate of X(1863)
X(1851) = crosspoint of X(4) and X(1119)
X(1851) = crosssum of X(3) and X(1260)
X(1851) = polar conjugate of X(30701)
X(1852) lies on these lines: 4,12 11,28 19,1837 30,1838 34,1836 516,1888 950,1859 1829,1858 1839,1841
Let A' be the orthocenter of BCX(3), and define B' and C' cyclically; then X(1853) is the centroid of A'B'C'.
X(1853) lies on these lines: 2,154 3,161 4,64 5,1498 6,66 12,221 25,125 122,1073 157,426 343,1350 394,858 1146,1851 1181,1594 1352,1368 1826,1836 1837,1854
X(1853) = reflection of X(154) in X(2)
X(1853) = isotomic conjugate of X(34412)
X(1853) = complement of X(11206)
X(1853) = crosspoint of X(4) and X(253)
X(1853) = crosssum of X(i) and X(j) for these (i,j): (3,154), (206,577)
X(1853) = centroid of pedal triangle of X(64)
X(1854) lies on these lines: 1,84 6,1858 33,64 55,976 56,774 154,968 227,1490 960,1040 1192,1452 1837,1853
X(1854) = reflection of X(221) in X(1)
X(1854) = crosspoint of X(4) and X(280)
X(1854) = crosssum of X(i) and X(j) for these (i,j): (1,1394), (3,221), (55,478)
X(1855) lies on these lines: 1,1886 4,9 55,1856 65,1146 85,92
X(1855) = isogonal conjugate of X(1803)
X(1855) = crosspoint of X(92) and X(281)
X(1855) = crosssum of X(48) and X(222)
X(1856) lies on these lines: 4,57 11,1427 33,42 55,1855 225,235 430,1827
Barycentrics (y + z) tan A : (z + x) tan B : (x + y) tan C, x : y : z = X(223)
X(1857) lies on these lines: 2,243 4,65 8,1896 11,278 33,42 55,281 92,497 189,1364 388,1895 403,1068 412,1788 1478,1784 1748,1776
X(1857) = isogonal conjugate of X(1804)
X(1857) = isotomic conjugate of X(7055)
X(1857) = X(158)-Ceva conjugate of X(393)
X(1857) = pole wrt polar circle of trilinear polar of X(348) (line X(4025)X(4131))
X(1857) = polar conjugate of X(348)
X(1858) lies on these lines: 1,90 3,920 4,65 6,1854 11,113 21,60 52,517 55,72 73,774 144,145 411,1155 758,950 1829,1852 1831,1839
X(1858) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,431), (648,650)
X(1858) = crosspoint of X(4) and X(21)
X(1858) = crosssum of X(i) and X(j) for these (i,j): (1,1935), (3,65), (478,1402)
X(1858) = polar-circle-inverse of X(38949)
X(1859) lies on these lines: 1,1871 4,65 6,1096 11,132 19,25 27,243 29,960 40,1872 92,518 209,1826 210,281 278,354 942,1838 950,1852 1013,1748 1825,1902 1829,1831 1869,1894
X(1859) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1865), (107,650), 1838,1841)
X(1859) = crosspoint of X(i) and X(j) for these (i,j): (4,1172), (281,1896)
X(1859) = crosssum of X(3) and X(1214)
X(1860) lies on these lines: 4,42 6,1836 25,225 27,58 92,984 278,1458
Barycentrics (b^2 + c^2 - a (b + c))/(a^2 - b^2 - c^2) : :
The trilinear polar of X(1861) passes through X(4088).
X(1861) lies on these lines: 1,475 2,33 4,9 5,1872 8,34 12,1887 25,1376 75,225 208,1788 232,1575 235,1329 240,522 378,993 406,1698 427,1824 429,1900 468,1862 518,1876 519,1870 594,1841 765,1877 908,1830 958,1593 960,1902 1528,1532 1595,1871 1829,1883
X(1861) = isogonal conjugate of X(36057)
X(1861) = isotomic conjugate of X(31637)
X(1861) = complement of X(3100)
X(1861) = crossdifference of every pair of points on line X(48)X(1459)
X(1861) = inverse-in-circumconic-centered-at-X(9) of X(19)
X(1861) = pole wrt polar circle of trilinear polar of X(673) (line X(1)X(514))
X(1861) = X(48)-isoconjugate (polar conjugate) of X(673)
X(1861) = perspector of circumconic through the polar conjugates of PU(47) and PU(51)
X(1861) = X(8076)-of-orthic-triangle if ABC is acute
X(1861) = X(63)-isoconjugate of X(1438)
X(1862) lies on these lines: 4,145 11,33 25,100 34,1317 80,1039 104,1593 119,235 428,528 468,1861 519,1878 900,1830 1387,1883 1484,1595
X(1863) lies on these lines: 4,7 25,1604 33,42 242,390 497,1851
X(1863) = X(4)-Ceva conjugate of X(1851)
X(1863) = polar conjugate of X(30705)
X(1864) lies on these lines: 3,1728 4,65 6,33 9,55 11,118 44,212 51,1824 56,1490 57,971 72,519 84,1466 227,774 329,497 381,942 389,1872 405,997 430,1827 452,960 1005,1776 1155,1708 1210,1532 1214,1736
X(1864) = X(i)-Ceva conjugate of X(j) for these (i,j): (1210,1108),
(1897,650)
X(1864) = crosspoint of X(i) and X(j) for these (i,j): (4,9),
(8,282)
X(1864) = crosssum of X(i) and X(j) for these (i,j): (3,57),
(56,223)
X(1865) lies on these lines: 4,6 12,37 19,407 33,430 92,1211 209,1824 281,860 286,297 442,1838 1474,1884
X(1865) = X(4)-Ceva conjugate of X(1859)
X(1865) = crosssum of X(577) and X(1437)
X(1866) lies on these lines: 4,80 19,1405 28,34 51,65 52,1905 225,1829 1878,1887
X(1867) lies on these lines: 4,8 10,407 12,37 427,1838 1118,1892 1785,1904 1884,1891
X(1867) = polar conjugate of X(37870)X(1868) lies on these lines: 4,8 29,894 34,37 65,1826 210,1869 226,429 228,1593
X(1869) lies on these lines: 4,9 27,306 28,35 34,42 65,225 209,1829 210,1868 1710,1770 1859,1894
X(1870) lies on these lines: 1,4 2,1060 3,1398 7,1061 8,475 11,403 12,1594 19,1449 20,1062 24,56 25,999 28,60 36,186 54,65 55,378 59,517 77,1119 104,1455 108,953 109,1735 208,1420 221,1181 232,1015 235,496 242,514 273,1442 354,1905 376,1040 389,1425 427,495 451,1125 459,614 519,1861 631,1038 651,912 982,1395 1000,1041 1006,1214 1100,1172 1318,1878 1385,1426 1718,1737 1725,1776 1831,1844
X(1870) = isogonal conjugate of X(1807)
X(1870) = cevapoint of X(i) and X(j) for these (i,j): (1,1718), (1464,1835)
X(1870) = polar conjugate of X(18359)
X(1870) = pole wrt polar circle of trilinear polar of X(18359) (line X(10)X(522))
X(1870) = homothetic center of circumorthic triangle and anti-tangential midarc triangle
X(1871) lies on these lines: 1,1859 3,19 4,8 5,1848 28,1385 29,392 33,1598 52,916 65,1243 225,1831 273,1148 278,942 354,1844 580,1731 581,1841 952,1891 1595,1861 1597,1753
X(1872) lies on these lines: 1,1887 3,33 4,8 5,1861 19,1598 34,1482 40,1859 65,1785 225,1830 389,1864 1068,1876 1595,1848 1715,1736 1825,1905
X(1873) lies on these lines: 4,79 12,37 403,1838
X(1874) lies on these lines: 4,240 12,37 29,34 238,242 513,1835 862,1284
X(1874) = isogonal conjugate of X(1808)
X(1874) = X(242)-Ceva conjugate of X(1284)
X(1874) = polar conjugate of X(36800)
X(1874) = X(63)-isoconjugate of X(2311)
X(1874) = pole wrt polar circle of trilinear polar of X(36800) (line X(8)X(3907))
X(1875) lies on these lines: 1,945 4,65 25,34 108,953 225,1829 278,957 513,1835 517,1361 859,1465 1119,1122 1400,1841 1452,1454 1825,1900
X(1875) = midpoint of X(1785) and X(1845)
X(1875) = isogonal conjugate of X(1809)
X(1875) = polar conjugate of X(36795)
X(1875) = pole wrt polar circle of trilinear polar of X(36795) (line X(8)X(521))
X(1876) lies on these lines: 1,1037 4,7 6,19 25,57 28,1170 33,354 59,517 72,475 108,840 196,1851 225,1887 226,427 235,1210 242,653 278,1002 428,553 513,1835 518,1861 614,3195 851,1465 950,1885 1011,1214 1020,1736 1068,1872 1458,2356 1471,2212
X(1876) = midpoint of X(65) and X(1456)
X(1876) = polar conjugate of X(36796)
X(1876) = pole wrt polar circle of trilinear polar of X(36796) (line X(8)X(885))
X(1877) lies on these lines: 1,4 11,1455 12,1883 25,1470 30,1465 51,65 109,1737 221,1837 513,1835 517,1830 603,1210 608,1826 751,1890 765,1861 1319,1846 1707,1788 1825,1829 1839,1880
X(1877) = X(4)-Ceva conjugate of X(1846)
X(1877) = pole wrt polar circle of trilinear polar of X(4997) (line X(8)X(522))
X(1877) = polar conjugate of X(4997)
X(1878) lies on these lines: 4,8 25,36 34,1319 428,535 513,1835 519,1862 855,1465 1318,1870 1842,1888 1866,1887
X(1878) = isogonal conjugate of X(1811)
X(1878) = inverse-in-polar-circle of X(8)
X(1878) = polar conjugate of X(36805)
X(1878) = pole wrt polar circle of trilinear polar of X(36805) (line X(8)X(513))
X(1879) lies on these lines: 4,96 5,570 6,13 53,235 230,428 233,566 1598,1609
X(1879) = crosssum of X(6) and X(156)
X(1879) = X(48)-of-orthic-triangle if ABC is acuteX(1880) lies on these lines: 2,92 4,941 6,19 12,37 25,1096 28,961 42,1824 57,967 108,111 331,1218 1171,1396 1254,1400 1411,1474 1722,1723 1735,1765 1839,1877
X(1880) = isogonal conjugate of X(1812)
X(1880) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,1400), (225,1824), (278,225)
X(1880) = crosspoint of X(i) and X(j) for these (i,j): (19,393), (34,278)
X(1880) = crosssum of X(i) and X(j) for these (i,j): (63,394), (78,219)
X(1880) = polar conjugate of X(314)
X(1881) lies on these lines: 4,48 12,37 71,860
X(1882) lies on these lines: 4,65 5,1214 12,37 92,960
As a point on the Euler line, X(1883) has Shinagawa coefficients ($a$F, $a$(E + F) - 2abc).
X(1883) lies on these lines: 2,3 10,1828 12,1877 1387,1862 1829,1861 1848,1900
As a point on the Euler line, X(1884) has Shinagawa coefficients ($a$F, -(E + F)3 + $aSA$ + 3ES2).
X(1884) lies on these lines: 2,3 34,1464 513,1835 1474,1865 1828,1844 1829,1831 1867,1891
X(1884) = polar conjugate of isotomic conjugate of X(35466)
As a point on the Euler line, X(1885) has Shinagawa coefficients (F, E - 3F).
Let A'B'C' be the orthic triangle. X(1885) is the radical center of the 2nd Droz-Farny circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)
X(1885) lies on these lines: 2,3 64,1899 389,974 497,1398 515,1902 516,1829 950,1876 1039,1721 1770,1905
X(1885) = anticomplement of X(31829)
X(1885) = crosspoint of X(4) and X(1105)
X(1885) = crosssum of X(3) and X(185)
X(1885) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(185)
X(1885) = crosspoint, wrt orthic triangle, of X(4) and X(185)
X(1885) = X(20)-of-anti-Ara-triangle
X(1885) = X(3057)-of-orthic-triangle if ABC is acute
X(1886) lies on these lines: 1,1855 6,1836 19,57 33,42 225,607 230,231 910,1360 1783,1785
X(1886) = isogonal conjugate of X(1815)
X(1886) = X(917)-Ceva conjugate of X(25)
X(1886) = crosssum of X(219) and X(1818)
X(1887) lies on these lines: 1,1872 4,65 12,1861 33,56 34,1824 55,1753 208,1827 225,1876 318,518 942,1785 1825,1829 1830,1902 1844,1846 1866,1878
X(1888) lies on these lines: 4,65 19,44 28,1155 33,1426 34,55 71,1841 209,1829 225,1902 516,1852 517,1838 1830,1900 1842,1878
As a point on the Euler line, X(1889) has Shinagawa coefficients (F, -E - F - 2$bc$).
X(1889) lies on these lines: 2,3 6,1836 57,1893 518,1824 1709,1730
X(1889) = inverse-in-orthocentroidal-circle of X(430)
X(1890) lies on these lines: 4,9 7,34 25,1001 27,162 28,142 33,390 82,225 428,528 518,1829 751,1877 1445,1452 1724,1738 1737,1747
X(1890) = X(29)-beth conjugate of X(1826)
X(1890) = polar conjugate of isogonal conjugate of X(21764)
X(1891) lies on these lines: 1,4 8,19 10,28 25,958 27,306 29,1220 65,1503 92,1842 428,529 518,1829 952,1871 1867,1884
X(1891) = polar conjugate of isotomic conjugate of complement of X(27184)X(1892) lies on these lines: 4,7 12,1452 25,226 33,1836 57,427 65,66 79,1041 208,429 225,608 388,1829 1118,1867 1478,1905
X(1892) = X(4)-beth conjugate of X(608)
X(1893) lies on these lines: 4,7 11,1427 12,37 57,1889 226,430 235,1838
X(1893) = X(4)-beth conjugate of X(1880)
As a point on the Euler line, X(1894) has Shinagawa coefficients ($a$F, -2$aSA$ - $a$(E + F)).
X(1894) lies on these lines: 2,3 19,53 225,1829 1785,1824 1828,1838 1859,1869
X(1895) lies on these lines: 1,29 2,280 4,7 8,1054 40,653 48,821 56,243 57,412 63,1712 78,1897 108,411 162,255 196,962 204,1097 240,774 304,811 388,1857 497,1118 517,1148 1210,1785 1445,1753
X(1895) = isogonal conjugate of X(19614)
X(1895) = isotomic conjugate of X(19611)
X(1895) = crosspoint of X(75) and X(18750)
X(1895) = crosssum of X(31) and X(2155)
X(1895) = trilinear pole of line X(14331)X(17898)
X(1895) = X(75)-Ceva conjugate of X(92)
X(1895) = cevapoint of X(i) and X(j) for these (i,j): (1,1712), (204,610), (3176, 7952)
X(1895) = polar conjugate of X(2184)
X(1896) lies on these lines: 1,29 4,51 7,286 8,1857 9,318 21,243 27,84 28,104 393,941 412,1715 823,1156
X(1896) = cevapoint of X(i) and X(j) for these (i,j): (1,1715), (4,158)
X(1896) = X(4)-cross conjugate of X(29)
X(1896) = isogonal conjugate of X(22341)
X(1896) = pole wrt polar circle of trilinear polar of X(1214) (line X(520)X(656))
X(1896) = polar conjugate of X(1214)
X(1896) = X(63)-isoconjugate of X(1409)
X(1897) lies on these lines: 1,318 4,145 27,295 29,1807 33,92 34,1120 78,1895 100,108 101,107 109,522 112,835 162,190 192,1013 243,518 278,1280 346,1249 519,1785 644,1783 726,1430
X(1897) = isogonal conjugate of X(1459)
X(1897) = anticomplement of X(2968)
X(1897) = X(648)-Ceva conjugate of X(1783)
X(1897) = cevapoint of X(i) and X(j) for these (i,j): (1,522),
(523,1834), (650,1864)
X(1897) = X(101)-cross conjugate of X(190)
X(1897) = crosspoint of X(648) and X(811)
X(1897) = crosssum of X(647) and X(810)
X(1897) = isotomic conjugate of X(4025)
X(1897) = trilinear pole of line X(4)X(9) (complement of Soddy line, and Brocard axis of excentral triangle)
X(1897) = pole wrt polar circle of trilinear polar of X(514) (line X(11)X(244), the Feuerbach tangent line)
X(1897) = polar conjugate of X(514)
X(1897) = trilinear product X(4)*X(100)
X(1897) = trilinear product of circumcircle intercepts of line X(4)X(8)
X(1898) lies on these lines: 3,90 4,65 21,662 56,971 84,1470 354,496 411,1776 912,1479 920,1155
X(1898) = reflection of X(65) in X(1837)
X(1898) = crosspoint of X(4) and X(90)
X(1898) = crosssum of X(3) and X(46)
X(1899) lies on these lines: {2,98}, {3,68}, {4,51}, {5,1181}, {6,66}, {20,1204}, {22,3580}, {25,1503}, {54,70}, {64,1885}, {65,5130}, {67,5486}, {69,305}, {154,468}, {217,2548}, {235,1498}, {265,974}, {315,3978}, {388,1425}, {394,1368}, {407,5786}, {429,5706}, {442,5810}, {462,5869}, {463,5868}, {497,3270}, {511,1370}, {578,3541}, {686,804}, {858,1993}, {860,5767}, {940,5820}, {1092,3546}, {1147,3548}, {1321,3070}, {1322,3071}, {1495,6353}, {1587,3127}, {1588,3128}, {1591,6289}, {1592,6290}, {1593,6247}, {1824,5928}, {1864,5101}, {1974,5596}, {2072,5654}, {2450,3767}, {2549,3269}, {2550,3611}, {2888,3523}, {2892,5095}, {2992,2993}, {3134,5877}, {3136,5816}, {3142,5713}, {3332,4207}, {3549,5449}, {3818,5943}, {3851,5644}, {3926,4121}, {5064,5480}, {5133,5422}, {5200,5870}
X(1899) = reflection of X(394) in X(1368)
X(1899) = isotomic conjugate of X(34405)
X(1899) = crosspoint of X(4) and X(69)
X(1899) = crosssum of X(i) and X(j) for these (i,j): (3,25), (52,418), (206,571)
X(1899) = anticomplement of X(9306)
X(1899) = crossdifference of every pair of points on line X(3569)X(6753)
X(1899) = X(1370)-of-1st-Brocard-triangle
X(1899) = X(200)-of-orthic-triangle if ABC is acute
X(1900) lies on these lines: 4,8 10,1904 19,45 25,35 225,1876 407,1785 429,1861 1825,1875 1830,1888 1839,1840 1848,1883 1859,1869
X(1901) lies on these lines: 4,6 5,579 7,857 9,46 12,71 19,1836 30,284 37,226 65,1826 72,594 115,117 198,851 208,429 219,1478 329,1211 377,965 430,1827 583,1713 946,1108 950,1100 1146,1845
X(1901) = X(653)-Ceva conjugate of X(523)
X(1901) = crosspoint of X(4) and X(226)
X(1901) = crosssum of X(3) and X(284)
X(1901) = complement of X(8822)
X(1902) lies on these lines: 1,1037 4,8 10,235 19,220 25,40 33,64 125,429 225,1888 378,1385 392,475 427,946 515,1885 960,1861 1482,1597 1825,1859 1830,1887
X(1902) = reflection of X(1829) in X(4)
Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta' be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb', Tc' cyclically. Let Va = TbTb'∩TcTc', Vb = TcTc'∩TaTa', Vc = TaTa'∩TbTb'. The lines AVa, BVb, CVc concur in X(1903). (See also X(65).) (Randy Hutson, July 20, 2016)
X(1903) lies on these lines: 3,9 6,33 19,64 37,73 65,1826 69,189 71,210 226,1439 285,1798 478,1413 1419,1422
X(1903) = isogonal conjugate of X(1817)
X(1903) = crosspoint of X(84) and X(189)
X(1903) = crosssum of X(i) and X(j) for these (i,j): (3,610), (40,198)
As a point on the Euler line, X(1904) has Shinagawa coefficients (FS2, -(E - F)S2 - 2$bc$S2 + 2$bcSBSC$).
X(1904) lies on these lines: 2,3 10,1900 12,968 53,1826 80,1039 1785,1867 1828,1848
X(1905) lies on these lines: 1,25 3,1452 4,65 33,517 34,222 46,1593 52,1866 169,607 225,1831 354,1870 378,1155 406,960 427,1737 1426,1838 1478,1892 1730,1735 1770,1885 1785,1824 1825,1872 1827,1845 1828,1844
X(1905) = reflection of X(222) in X(942)
X(1905) = crosspoint of X(4) and X(1061)
X(1905) = crosssum of X(3) and X(1060)
As a point on the Euler line, X(1906) has Shinagawa coefficients (F, -2E + F).
X(1906) lies on this line: 2,3
As a point on the Euler line, X(1907) has Shinagawa coefficients (F, 2E + F).
X(1907) lies on this line: 2,3
X(1907) = complement of X(33524)
Centers from Bicentric Pairs, 1908-1982
For a definition of a bicentric pair (e.g., the 1st and 2nd Brocard points) click Tables at the top of this page. Suppose P and U are a bicentric pair. Many operations on P and U result in triangle centers. Among these are trilinear and barycentric product, bicentric sum, bicentric difference, crosssum, and crossdifference. For definitions of these, click Tables. At the time this section is added to ETC (September 15, 2003), bicentric pairs
P(1),U(1); P(2),U(2); ...; P(42),U(42)
are defined in Tables. In this present section, the abbreviation PU(n) means the bicentric pair P(n),U(n).
X(1908) lies on these lines: 39,1155 42,649 43,2235 171,292 551,2666 2243,2276
X(1909) lies on these lines: 1,76 2,330 7,8 10,274 12,325 34,264 35,99 36,1078 37,1655 42,310 56,183 73,290 86,313 172,385 190,1334 226,1432 256,1221 257,335 286,1891 305,612 315,1478 538,1500 732,894 1215,1237 1235,1870
X(1909) = isogonal conjugate of X(904)
X(1909) = isotomic conjugate of X(256)
X(1909) = complement of X(21226)
X(1909) = anticomplement of X(1107)
X(1909) = X(i)-Ceva conjugate of X(j) for these (i,j): (335,350), (1221,2)
X(1909) = cevapoint of X(8) and X(1655)
X(1909) = crosspoint of PU(10)
X(1909) = intersection of tangents at PU(10) to hyperbola {{A,B,C,PU(10)}}
X(1910) lies on these lines: 1,163 10,98 19,560 37,692 48,75 65,172 1580,1581
X(1910) = isogonal conjugate of X(1959)
X(1910) = X(1821)-Ceva conjugate of X(293)
X(1910) = cevapoint of X(i) and X(j) for these (i,j): (1,1580), (240,1957)
X(1910) = barycentric product of PU(88)
X(1910) = trilinear product X(6)*X(98)
X(1911) lies on these lines: 1,335 6,292 42,81 86,334 172,694 238,660 692,1333 739,813 875,890 1403,1407 1429,1458
X(1911) = isogonal conjugate of X(350)
X(1911) = isotomic conjugate of X(18891)
X(1911) = complement of X(20554)
X(1911) = anticomplement of X(20542)
X(1911) = X(741)-Ceva conjugate of X(292)
X(1911) = cevapoint of X(172) and X(1914)
X(1911) = crosspoint of X(727) and X(1438)
X(1911) = trilinear pole of line X(213)X(667)
As the isogonal conjugate of a point on the circumcircle, X(1912) lies on the line at infinity.
X(1912) lies on these (parallel) lines: 30,511 213,667 1166,1203
X(1912) = crossdifference of every pair of points on line X(6)X(350)
X(1913) lies on this line: 213,667
X(1914) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(9) and U(9) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)
X(1914) lies on these lines: 1,32 6,31 9,983 11,230 21,1107 35,39 36,187 37,82 44,765 48,1613 81,593 100,1575 105,910 112,1870 213,595 284,893 292,1438 350,385 577,1040 584,1185 604,1403 649,834 727,813 739,901 741,1326 999,1384 1055,1149 1319,1415 1428,1691
X(1914) = isogonal conjugate of X(335)
X(1914) = isotomic conjugate of X(18895)
X(1914) = complement of X(20553)
X(1914) = anticomplement of X(20541)
X(1914) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39029), (727,31), (1429,1428), (1438,6), (1911,172)
X(1914) = crosspoint of X(i) and X(j) for these (i,j): (81,105), (238,1429), (239,242), (904,1911), (919,1252)
X(1914) = crosssum of X(i) and X(j) for these (i,j): (37,518), (292,295), (350,1909), (918,1086)
X(1914) = {X(1),X(32)}-harmonic conjugate of X(172)
X(1914) = intersection of trilinear polars of P(9) and U(9)
X(1914) = X(92)-isoconjugate of X(295)
X(1914) = perspector of hyperbola {{A,B,C,X(58),X(101),PU(9)}}
X(1914) = barycentric product of PU(134)
X(1914) = homothetic center of intangents triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(1915) lies on these lines: 2,1501 6,25 31,292 32,1613 110,251 141,1799
X(1915) = polar conjugate of isotomic conjugate of X(37893)Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically; then X(1916) is the radical center of the circumcircles of A"BC, B"CA, C"AB. Let A* be the reflection of A in B'C', and define B* and C* cyclically; then X(1916) is the radical center of the circumcircles of A*BC, B*CA, C*AB. The first set of circles equals the second set. (Randy Hutson, February 10, 2016)
X(1916) lies on these lines: 2,694 4,147 10,257 39,83 76,115 98,385 114,262 226,335 256,291 316,736 325,698 538,671 543,598 690,882 804,881
X(1916) = midpoint of X(148) and X(194)
X(1916) = reflection of X(i) in X(j) for these (i,j): (76,115), (99,39)
X(1916) = isogonal conjugate of X(1691)
X(1916) = isotomic conjugate of X(385)
X(1916) = cevapoint of X(39) and X(511)
X(1916) = complement of X(8782)
X(1916) = trilinear pole of line X(141)X(523)
X(1916) = pole wrt polar circle of trilinear polar of X(419)
X(1916) = polar conjugate of isogonal conjugate of X(36214)
X(1916) = X(48)-isoconjugate (polar conjugate) of X(419)
X(1916) = antigonal image of X(76)
X(1916) = X(76) of 1st anti-Brocard triangle
X(1916) = intersection, other than A, B, C, of the 1st and 2nd isobarycs of the circumcircle
X(1916) = perspector of ABC and 1st anti-Brocard triangle
Let A'B'C' and A"B"C" be the 5th Brocard and 5th anti-Brocard triangles, resp. Let A* be the trilinear product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(1917). (Randy Hutson, November 30, 2018)
X(1917) lies on these lines: 1,3409 31,2085 560,9247 922,4020 1501,9448
X(1917) = barycentric product of PU(12)
X(1917) = isogonal conjugate of X(1928)
X(1917) = trilinear product of PU(13)
X(1917) = trilinear square of X(32)
X(1917) = trilinear cube of X(31)
X(1918) lies on these lines: 6,31 9,981 10,82 32,560 86,171 100,715 101,729 213,872 313,983 393,465 692,1333
X(1918) = isogonal conjugate of X(310)
X(1918) = anticomplement of X(17138)
X(1918) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,213), (692,1919), (983,37)
X(1918) = crosspoint of X(i) and X(j) for these (i,j): (31,32), (213,1402)
X(1918) = crosssum of X(i) and X(j) for these (i,j): (75,76), (274,314)
X(1918) = PU(12)-harmonic conjugate of X(1919)
X(1919) lies on these lines: 101,765 649,834 667,788 669,688
X(1919) = isogonal conjugate of X(1978)
X(1919) = complement of X(21304)
X(1919) = anticomplement of X(21262)
X(1919) = X(i)-Ceva conjugate of X(j) for these (i,j): (32,1977), (692,1333)
X(1919) = cevapoint of X(669) and X(1924)
X(1919) = crosspoint of X(i) and X(j) for these (i,j): (31,101), (81,932), (692,1333)
X(1919) = crosssum of X(i) and X(j) for these (i,j): (75,514), (321,693), (646,668), (850,1230)
X(1919) = PU(12)-harmonic conjugate of X(1918)
X(1919) = barycentric product of PU(25)
X(1919) = trilinear product of PU(42)
X(1920) lies on these lines: 2,561 37,1221 75,982 76,85 210,668 310,321 334,1581 1002,1611 1215,1237 1240,1441
X(1920) = isotomic conjugate of X(893)
X(1920) = complement of polar conjugate of isogonal conjugate of X(23192)
X(1920) = perspector of Gemini triangle 31 and cross-triangle of ABC and Gemini triangle 31
X(1920) = trilinear pole of perspectrix of ABC and Gemini triangle 32
X(1920) = X(i)-Ceva conjugaute of X(j) for these (i,j): (334,1921), (1240,76)
X(1921) lies on these lines: 2,561 10,75 37,308 274,1107 350,740 518,668
X(1921) = isogonal conjugate of X(1922)
X(1921) = isotomic conjugate of X(292)
X(1921) = X(334)-Ceva conjugate of X(1920)
X(1921) = anticomplement of polar conjugate of isogonal conjugate of X(23223)
X(1921) = perspector of Gemini triangle 32 and cross-triangle of ABC and Gemini triangle 32
X(1921) = trilinear pole of perspectrix of ABC and Gemini triangle 31
X(1922) lies on these lines: 6,291 58,101 81,335 727,813 1416,1428
X(1922) = isogonal conjugate of X(1921)
X(1922) = isotomic conjugate of isogonal conjugate of X(18897)
X(1922) = perspector of ABC and unary cofactor triangle of Gemini triangle 31
X(1923) lies on these lines: 1,21 110,719
X(1923) = isogonal conjugate of X(18833)
X(1923) = crossdifference of every pair of points on line X(661)X(786)
X(1923) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,1964), (163,1924)
X(1923) = crosspoint of X(31) and X(560)
X(1923) = crosssum of X(75) and X(561)
X(1923) = PU(13)-harmonic conjugate of X(1924)
X(1923) = trilinear product X(i)*X(j) for these {i,j}: {1, 1923}, {6, 3051}, {31, 1964}, {32, 39}, {38, 560}, {99, 9494}, {110, 688}, {141, 1501}, {163, 2084}, {184, 1843}, {427, 14575}, {669, 1634}, {732, 8789}, {826, 14574}, {827, 2531}, {1397, 3688}, {1401, 2175}, {1576, 3005}, {1917, 1930}, {1918, 17187}, {1927, 2236}, {1973, 4020}, {1974, 3917}, {1980, 4553}, {2205, 16696}, {3404, 9417}, {3665, 9448}, {4576, 9426}, {8024, 9233}, {8623, 9468}, {9247, 17442}
X(1924) lies on these lines: 661,830 667,788 681,1612
X(1924) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,1923), (662,31), (1919,669)
X(1924) = crosspoint of X(i) and X(j) for these (i,j): (31,662), (1919, 1980)
X(1924) = crosssum of X(i) and X(j) for these (i,j): (75,661), (1577,1930)
X(1924) = isogonal conjugate of X(4602)
X(1924) = PU(13)-harmonic conjugate of X(1923)
X(1924) = trilinear product of PU(91)
X(1925) lies on these lines: 76,335 92,304 469,1601
X(1924) = complement of X(21305)
X(1924) = anticomplement of X(21263)
X(1925) = X(1934)-Ceva conjugate of X(1926)
X(1925) = crosspoint of PU(14)
X(1925) = intersection of tangents at PU(14) to conic {{A,B,C,PU(14)}}
X(1926) lies on these lines: 38,75 76,257 661,786 799,1755 1590,1636
X(1926) = isogonal conjugate of X(1927)
X(1926) = isotomic conjugate of X(1967)
X(1926) = X(2)-Ceva conjugate of X(39030)
X(1926) = X(1934)-Ceva conjugate of X(1925)
X(1926) = crosssum of X(1932) and X(1933)
X(1926) = perspector of conic {A,B,C,PU(14)}
X(1926) = intersection of trilinear polars of P(14) and U(14)
X(1927) lies on these lines: 82,662 172,694 715,805 733,787
X(1927) = isogonal conjugate of X(1926)
X(1927) = cevapoint of X(1932) and X(1933)
X(1928) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the de Longchamps line. This inellipse has center X(21235). (Randy Hutson, October 15, 2018)
X(1928) = isogonal conjugate of X(1917)
X(1928) = isotomic conjugate of X(560)
X(1928) = anticomplement of isogonal conjugate of X(38812)
X(1928) = anticomplementary conjugate of anticomplement of X(38812)
X(1928) = trilinear product of PU(14)
X(1928) = trilinear product of vertices of Gemini triangle 31
X(1928) = trilinear product of vertices of Gemini triangle 32
X(1929) lies on ths line: 1,3125 2,846 105,2702 1758,2006 2640,3122
X(1929) = isogonal conjugate of X(1757)
X(1929) = cevapoint of X(i) and X(j) for these (i,j): (244,659), (1966,1909)
X(1929) = X(238)-cross conjugate of X(1)
X(1929) = trilinear pole of PU(31) (line X(513)X(1100))
X(1930) lies on these lines: 1,75 8,150 76,334 213,742
X(1930) = isotomic conjugate of X(82)
X(1930) = isogonal conjugate of complement of anticomplementary conjugate of X(17489)
X(1930) = complement of X(17489)
X(1930) = anticomplement of X(16600)
X(1930) = X(75)-Ceva conjugate of X(38)
X(1930) = X(1194)-cross conjugate of X(251)
X(1930) = crosspoint of X(75) and X(561)
X(1930) = crosssum of X(31) and X(560)
X(1931) lies on these lines: 1,21 37,757 44,662 99,239 172,593 241,1414 261,894 661,1019 1014,1423 1326,1757 1444,1778 2641,2642
X(1931) = isogonal conjugate of X(9278)
X(1931) = complement of X(20349)
X(1931) = anticomplement of X(20337)
X(1931) = X(2)-Ceva conjugate of X(39042)
X(1931) = X(1929)-Ceva conjugate of X(1963)
X(1931) = crosssum of X(1757) and X(1961)
X(1931) = perspector of conic {{A,B,C,PU(31)}}
X(1931) = intersection of trilinear polars of P(31) and U(31)
X(1932) lies on this line: 293,774
X(1932) = isogonal conjugate of X(9239)
X(1932) = X(1927)-Ceva conjugate of X(1933)
X(1932) = intersection of tangents at PU(13) to conic {A,B,C,PU(13)}
X(1932) = crosspoint of PU(13)
X(1933) lies on these lines: 31,48 32,904 887,1737 896,1101
X(1933) = isogonal conjugate of X(1934)
X(1933) = X(2)-Ceva conjugate of X(39031)
X(1933) = X(1927)-Ceva conjugate of X(1932)
X(1933) = crosspoint of X(82) and X(1910)
X(1933) = crosssum of X(i) and X(j) for these (i,j): (38,1959), (1925,1926)
X(1933) = perspector of conic {A,B,C,PU(13)}
X(1933) = intersection of trilinear polars of P(13) and U(13)
X(1934) lies on these lines: 38,799 75,1581 257,335 334,1441 561,1109 764,1244 1821,1959
X(1934) = isogonal conjugate of X(1933)
X(1934) = isotomic conjugate of X(1580)
X(1934) = cevapoint of X(i) and X(j) for these (i,j): (38,1959),
(1925,1926)
X(1935) lies on these lines: 1,90 2,603 3,1745 4,255 7,1451 9,478 10,109 12,171 20,212 21,73 31,388 34,63 40,1777 47,1478 56,87 57,1724 58,226 65,1046 84,1040 221,958 222,405 225,283 415,1098 497,1496 748,1106 774,1776 896,1254 940,1806 960,1455 978,1470 1056,1497 1400,1778 1448,1708 1761,1880
X(1935) = X(296)-Ceva conjugate of X(1936)
X(1935) = cevapoint of X(1046) and X(1745)
X(1935) = crosspoint of PU(16)
X(1935) = intersection of tangents at PU(16) to conic {A,B,C,PU(16)}
X(1935) = perspector of ABC and the side-triangle of the 1st and 2nd bicentrics of the orthic triangle
X(1935) = {X(34),X(63)}-harmonic conjugate of X(37591)
Let L be the line X(1)X(3) = trilinear polar of X(651). Let V be the trilinear polar of the cevapoint of X(1) and X(3), so that V = X(521)X(650); let M = X(3157) = X(1)-Ceva conjugate of X(3), and let N = X(1745) = X(3)-Ceva conjugate of X(1). The lines L, V, MN concur in X(1936). (Randy Hutson, December 26, 2015)
Let A1B1C1 and A2B2C2 be the 1st and 2nd bicentrics of the orthic triangle. The six vertices A1, B1, C1, A2, B2, C2 lie on a conic, denoted here by H. Let A' be the intersection of the tangents to H at A1 and A2. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1936). Let A" be the intersection of the tangents to H at B2 and C1. Define B' and C' cyclically. The lines AA", BB", CC" concur in X(1936). (Randy Hutson, March 25, 2016)
X(1936) lies on these lines: 1,3 2,212 4,255 11,238 20,603 29,270 31,497 33,63 47,1479 58,950 73,411 100,1818 109,516 225,412 243,1430 388,1496 495,738 511,1364 521,650 580,1210 750,1253 896,1776 908,1331 938,1451 1044,1406 1046,1858 1058,1497 1762,1859
X(1936) = isogonal conjugate of X(1937)
X(1936) = X(2)-Ceva conjugate of X(39032)
X(1936) = perspector of hyperbola {{A,B,C,X(21),X(651),PU(16)}}
X(1936) = X(296)-Ceva conjugate of X(1935)
X(1936) = crosssum of X(i) and X(j) for these (i,j): (1,1758), (243,1940)
X(1936) = intersection of trilinear polars of P(16) and U(16) (the 1st and 2nd bicentrics of the orthic axis)
X(1936) = crossdifference of every pair of points on line X(65)X(650)
X(1936) = inverse-in-circumconic-centered-at-X(1) of X(55)
X(1936) = X(92)-isoconjugate of X(1949)
X(1936) = perspector of orthic triangle and the side-triangle of the 1st and 2nd bicentrics of the orthic triangle
X(1937) lies on these lines: 1,185 4,774 8,201 21,73 80,1736 90,1745 104,1458 108,1172 225,1896 307,314 515,1694 851,1758 885,1769
X(1937) = isogonal conjugate of X(1936)
X(1937) = cevapoint of X(i) and X(j) for these (i,j): (1,1758), (243,1940)
X(1937) = trilinear pole of line X(65)X(650)
X(1937) = pole wrt polar circle of trilinear polar of X(1948)
X(1937) = X(48)-isoconjugate (polar conjugate) of X(1948)
X(1937) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the MacBeath circumconic
As the isogonal conjugate of a point on the circumcircle, X(1938) lies on the line at infinity.
X(1938) lies on these lines: 30,511 65,650
X(1938) = crossdifference of every pair of points on line X(6)X(1936)
X(1938) = ideal point of PU(i) for these i: 15, 110
X(1939) lies on these lines: 65,650 169,1575 1385,2649
X(1940) lies on these lines: 1,1075 2,1118 3,158 4,46 20,1857 27,1882 29,65 34,87 35,1784 55,1895 56,92 73,1047 162,1399 201,240 225,1247 281,388 318,1376 331,1447 412,1155 425,1098 471,580 1038,1096 1816,1896
X(1940) = X(1937)-Ceva conjugate of X(243)
X(1940) = cevapoint of X(46) and X(1047)
X(1940) = crosspoint of PU(15)
X(1940) = intersection of tangents at PU(15) to conic {A,B,C,PU(15)}
X(1940) = pole wrt polar circle of trilinear polar of X(7108)
X(1940) = X(48)-isoconjugate (polar conjugate) of X(7108)
X(1941) lies on these lines: 3,1075 4,155 185,648 194,1593 450,1092
X(1941) = X(1942)-Ceva conjugate of X(450)
X(1941) = crosspoint of PU(17)
X(1941) = intersection of tangents at PU(17) to conic {A,B,C,PU(17)}
X(1942) lies on these lines: 6,1624 852,895
X(1942) = isogonal conjugate of X(450)
X(1942) = cevapoint of X(450) and X(1941)
X(1942) = trilinear pole of line X(185)X(647)
X(1943) lies on these lines: 2,914 57,239 69,278 75,222 81,1441 92,394 225,1330 321,651 333,664 637,1659 1231,1396
X(1944) lies on these lines: {2, 7}, {69, 281}, {75, 219}, {92, 394}, {220, 4363}, {239, 2323}, {242, 511}, {314, 1172}, {448, 662}, {522, 663}, {524, 1146}, {534, 5195}, {666, 1814}, {960, 1010}, {990, 997}, {1212, 4670}, {1332, 3262}, {1737, 1757}, {1958, 2289}, {2324, 3729}, {2607, 3792}
X(1944) = isogonal conjugate of X(1945)
X(1944) = isotomic conjugate of X(1952)
X(1944) = X(i)-complementary conjugate of X(j) for these (i,j): (2648,141), (2701,4885)
X(1944) = X(i)-cross conjugate of X(j) for these (i,j): (1936,5088), (1951,243)
X(1944) = X(2)-Ceva conjugate of X(39035)
X(1944) = perspector of conic {{A,B,C,X(333),X(664)}}
X(1944) = X(i)-isoconjugate of X(j) for these (i,j): (1,1945), (4,1949), (6,1937), (19,296), (31,1952), (65,2249)
X(1944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (92,394,1943), (2323,4858,239)
X(1945) lies on these lines: 6,1949 19,800 37,1569 109,284 333,664 673,1465
X(1945) = isogonal conjugate of X(1944)
X(1945) = X(1952)-Ceva conjugate of X(296)
X(1946) lies on these lines: 3,905 35,1734 105,2724 110,2714 187,237 650,2202 810,822
X(1946) = isogonal conjugate of X(18026)
X(1946) = bicentric difference of PU(19)
X(1946) = PU(19)-harmonic conjugate of X(1409)
X(1946) = trilinear pole of PU(101)
X(1946) = polar conjugate of isotomic conjugate of X(36054)
X(1946) = crossdifference of every pair of points on line X(2)X(92)
X(1946) = X(92)-isoconjugate of X(651)
X(1946) = polar conjugate of isotomic conjugate of X(36054)
X(1946) = perspector of hyperbola {{A,B,C,X(6),X(48)}}
X(1946) = barycentric product of PU(77)
X(1947) lies on these lines: 57,264 278,330 318,377
X(1947) = X(1952)-Ceva conjugate of X(1948)
X(1947) = crosspoint of PU(20)
X(1947) = intersection of tangents at PU(20) to conic {A,B,C,PU(20)}
X(1947) = pole wrt polar circle of trilinear polar of X(7105)
X(1947) = X(48)-isoconjugate (polar conjugate) of X(7105)
X(1948) lies on these lines: 2,92 9,264 24,547
X(1948) = isogonal conjugate of X(1949)
X(1948) = X(2)-Ceva conjugate of X(39036)
X(1948) = X(1952)-Ceva conjugate of X(1947)
X(1948) = crosssum of X(1950) and X(1951)
X(1948) = perspector of conic {A,B,C,PU(20)}
X(1948) = intersection of trilinear polars of P(20) and U(20)
X(1948) = pole wrt polar circle of line X(65)X(650), PU(15))
X(1948) = X(48)-isoconjugate (polar conjugate) of X(1937)
X(1949) lies on these lines: 6,1945 108,1172 219,296
X(1949) = isogonal conjugate of X(1948)
X(1949) = cevapoint of X(1950) and X(1951)
X(1949) = X(92)-isoconjugate of X(1936)
X(1950) lies on these lines: 3,608 6,1195 19,577 37,1415 65,1333 109,284 604,1403 1011,1395
X(1950) = X(1949)-Ceva conjugate of X(1951)
X(1950) = crosspoint of X(1940) and X(1943)
X(1950) = crosspoint of PU(19)
X(1950) = intersection of tangents at PU(19) to conic {A,B,C,X(109),PU(19)}
X(1951) lies on these lines: 1,1729 3,607 6,41 19,577 21,270 104, 294 517,906 652,663 851,1430 910,1415 1262,1465 1409,1630
X(1951) = isogonal conjugate of X(1952)
X(1951) = X(2)-Ceva conjugate of X(39037)
X(1951) = X(1949)-Ceva conjugate of X(1950)
X(1951) = crosspoint of X(243) and X(1944)
X(1951) = crosssum of X(i) and X(j) for these (i,j): (296,1945), (1947,1948)
X(1951) = perspector of conic {A,B,C,X(109),PU(19)}
X(1951) = intersection of trilinear polars of X(109), P(19), and U(19)
X(1951) = X(92)-isoconjugate of X(296)
X(1952) lies on these lines: 8,201 29,65 92,1146 232,1148 333,664
X(1952) = reflection of X(i) in X(j) for these (i,j): (92,1146), (664,1214)
X(1952) = isogonal conjugate of X(1951)
X(1952) = isotomic conjugate of X(1944)
X(1952) = cevapoint of X(i) and X(j) for these (i,j): (296,1945), (1947,1948)
X(1952) = polar conjugate of X(243)
X(1953) lies on these lines: 1,19 6,1411 9,1389 31,1820 37,2183 38,1755 65,1108 71,517 73,1841 216,1393 219,1482 515,1839 946,1826 991,1414 1457,1880
X(1953) = isogonal conjugate of X(2167)
X(1953) = complement of X(21271)
X(1953) = anticomplement of X(21231)
X(1953) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,661), (823,656)
X(1953) = crosspoint of X(1) and X(92)
X(1953) = crosssum of X(1) and X(48)
X(1953) = crossdifference of every pair of points on line X(656)X(1955)
X(1953) = bicentric sum of PU(21)
X(1953) = PU(21)-harmonic conjugate of X(656)
X(1953) = barycentric product of PU(69)
X(1953) = {X(1),X(19)}-harmonic conjugate of X(48)
X(1954) lies on these lines: 1,21 92,1955
X(1955) lies on these lines: 1,19 47,1740 58,1047 92,1954 293,1755 1580,1733
X(1955) = isogonal conjugate of X(1956)
X(1955) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39038), (293,1), (1755,1580)
X(1955) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1733), (293,1955)
X(1955) = perspector of hyperbola {{A,B,C,X(162),X(2167)}}
X(1956) lies on these lines: 293,1755
X(1956) = isogonal conjugate of X(1955)
X(1956) = X(240)-cross conjugate of X(1)
X(1957) lies on these lines: 1,204 2,1430 19,1707 29,1468 31,92 42,1013 43,1783 63,240 158,255 171,281 212,243 238,278 242,1395 896,1748 1496,1895 1724,1838
X(1957) = X(1910)-Ceva conjugate of X(240)
X(1957) = crosspoint of PU(23)
X(1957) = intersection of tangents at PU(23) to conic {{A,B,C,PU(23)}}
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(1958) lies on these lines: 19,326 31,1582 41,894 48,75 63,610 100,1253 239,604
X(1958) = X(i)-Ceva conjugate of X(j) for these (i,j): (293,1959),
(775,63)
X(1958) = cevapoint of X(610) and X(1740)
X(1959) lies on these lines: 1,21 2,257 19,326 48,1760 92,304 329,1655 514,661 1444,1761 1762,1812 1821,1934
X(1959) = isogonal conjugate of X(1910)
X(1959) = isotomic conjugate of X(1821)
X(1959) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39040), (293,1958), (1934,38)
X(1959) = X(i)-cross conjugate of X(j) for these (i,j): (6,2065), (114,2), (230,98), (1692,1976), (1733,1821)
X(1959) = crosspoint of X(1) and X(1581)
X(1959) = crosssum of X(i) and X(j) for these (i,j): (1,1580), 240,1957)
X(1959) = polar conjugate of X(36120)
X(1959) = pole wrt polar circle of trilinear polar of X(36120) (line X(19)X(798))
X(1959) = X(6)-isoconjugate of X(98)
X(1959) = perspector of hyperbola {{A,B,C,X(75),X(662),PU(22)}}
X(1960) is the center of the circle V(X(101)) = {{15,16,101,106}}; see the preamble to X(6137). (Randy Hutson, December 26, 2015)
X(1960) lies on these lines: 1,659 101,692 187,237 214,900 292,875 660,898 678,1635 884,1438
X(1960) = midpoint of X(i) and X(j) for these (i,j): (1,659), (663,667), (1635,3251)
X(1960) = isogonal conjugate of X(4555)
X(1960) = complement of anticomplementary conjugate of X(39349)
X(1960) = X(i)-Ceva conjugate of X(j) for these (i,j): (101,1017), (106,1015), (901,6), (1319,2087)
X(1960) = crosspoint of X(i) and X(j) for these (i,j): (6,901), (101,106)
X(1960) = crosssum of X(i) and X(j) for these (i,j): (2,900), 514,519)
X(1960) = bicentric sum of PU(25)
X(1960) = PU(25)-harmonic conjugate of X(1015)
X(1960) = bicentric difference of PU(99)
X(1960) = PU(99)-harmonic conjugate of X(1017)
X(1960) = polar conjugate of isotomic conjugate of X(22086)
X(1960) = crossdifference of every pair of points on line X(2)X(45)
X(1961) lies on these lines: 1,2 35,199 37,171 81,756 86,1215 100,1255 111,831 940,984 1051,1100
X(1961) = cevapoint of X(846) and X(1051)
X(1961) = crosspoint of PU(32)
X(1961) = intersection of tangents at PU(32) to conic {{A,B,C,X(100),PU(32)}}
X(1962) lies on these lines: 1,21 2,740 37,42 55,199 100,1255 351,1635
X(1962) = reflection of X(1635) in X(351)
X(1962) = isogonal conjugate of isotomic conjugate of X(4647)
X(1962) = crossdifference of every pair of points on line X(661)X(1019)
X(1962) = homothetic center of incentral triangle and n(Medial)*n(Incentral) triangle
X(1962) = X(2)-of-n(Medial)*n(Incentral)-triangle
X(1962) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1100), (100,661), (1125,1213)
X(1962) = crosspoint of X(i) and X(j) for these (i,j): (1,37), (1100,1125)
X(1962) = crosssum of X(i) and X(j) for these (i,j): (1,81), (1126,1255)
X(1962) = X(2)-of incentral triangle
X(1962) = bicentric sum of PU(32)
X(1962) = PU(32)-harmonic conjugate of X(661)
X(1962) = complement of X(17163)
X(1962) = homothetic center of Gemini triangle 17 and cross-triangle of Gemini triangles 15 and 17
X(1963) lies on these lines: 1,1326 2,6 37,757 662,1100 894,1509
X(1963) = X(1929)-Ceva conjugate of X(1931)
X(1963) = crosspoint of PU(31)
X(1963) = intersection of tangents at PU(31) to conic {{A,B,C,PU(31)}}
X(1964) lies on these lines: 1,75 6,292 31,48 42,1100 82,662 99,719 110,745 214,995 313,730 501,595 741,757 1042,1360 1193,1386 1201,1279
X(1964) = isogonal conjugate of X(3112)
X(1964) = isotomic conjugate of X(18833)
X(1964) = complement of X(21278)
X(1964) = anticomplement of X(21238)
X(1964) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,38), (31,1923), (662,798), (1178,6), (1581,1755)
X(1964) = crosspoint of X(i) and X(j) for these (i,j): (1,31), (39,1401)
X(1964) = crosssum of X(1) and X(75)
X(1964) = PU(36)-harmonic conjugate of X(798)
X(1965) lies on these lines: 2,292 19,27 31,561 38,799 332,375
X(1965) = complement of X(17485)
X(1965) = X(1581)-Ceva conjugate of X(1966)
X(1965) = crosspoint of PU(35)
X(1965) = intersection of tangents at PU(35) to conic {A,B,C,PU(35)}
Let A'B'C' be the 1st anti-Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1966). (Randy Hutson, December 26, 2015)
X(1966) lies on these lines: 1,75 2,893 31,561 240,811 350,1281 668,1757 732,894 798,812 799,896 1821,1934
X(1966) = isogonal conjugate of X(1967)
X(1966) = isotomic conjugate of X(1581)
X(1966) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39044), (874,804), (1581,1965), (1821,75)
X(1966) = X(i)-cross conjugate of X(j) for these (i,j): (698,1916), (1691,699)
X(1966) = crosssum of X(1580) and X(1582)
X(1966) = perspector of conic {{A,B,C,PU(35)}}
X(1966) = intersection of trilinear polars of P(35) and U(35)
X(1966) = trilinear product of PU(133)
X(1967) lies on these lines: 1,1581 38,799 42,694 213,904 256,291 733,813 741,805 875,881
X(1967) = isogonal conjugate of X(1966)
X(1967) = isotomic conjugate of X(1926)
X(1967) = complement of anticomplementary conjugate of X(17493)
X(1967) = cevapoint of X(1580) and X(1582)
X(1967) = X(i)-cross conjugate of X(j) for these (i,j): (32,699), (698,76)
X(1967) = trilinear product of circumcircle intercepts of line PU(1)
X(1967) = trilinear product X(292)*X(893)
X(1968) lies on these lines: 3,232 4,32 6,64 20,393 24,187 25,1611 33,172 39,378 53,571 194,648 217,578 230,235 264,384 1147,1625 1384,1598 1691,1974
X(1968) = X(1976)-Ceva conjugate of X(232)
X(1968) = crosspoint of PU(39)
X(1968) = intersection of tangents at PU(39) to hyperbola {A,B,C,X(4),X(112),PU(39)}
X(1968) = crossdifference of every pair of points on line X(684)X(8057)
X(1969) lies on these lines: 1,336 75,158 76,331 92,304 273,1240
X(1969) = isotomic conjugate of X(48)
X(1969) = cevapoint of X(75) and X(92)
X(1969) = trilinear pole of polar of X(31) wrt polar circle (line X(14208)X(20948))
X(1969) = pole wrt polar circle of trilinear polar of X(31) (line X(667)X(788))
X(1969) = polar conjugate of X(31)
X(1969) = trilinear product of vertices of Gemini triangle 37
X(1969) = trilinear product of vertices of Gemini triangle 38
X(1970) lies on these lines: 3,6 49,1625 54,112
X(1970) = perspector of ABC and 1st Brocard triangle of orthic triangle
X(1971) lies on these lines: 6,25 50,647 53,1629 98,230 217,1614 237,248 571,1613 1609,1619
X(1971) = isogonal conjugate of X(1972)
X(1971) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,39045), (237,1691), (248,6)
X(1971) = crosspoint of X(98) and X(275)
X(1971) = crosssum of X(216) and X(511)
X(1971) = perspector of conic {{A,B,C,X(54),X(112)}}
X(1971) = homothetic center of X(3)-Ehrmann triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles
X(1972) lies on these lines: 95,216 287,401 925,1298
X(1972) = reflection of X(648) in X(216)
X(1972) = isogonal conjugate of X(1971)
X(1972) = isotomic conjugate of X(401)
X(1972) = cevapoint of X(216) and X(511)
X(1972) = antipode of X(264) in hyperbola {{A,B,C,X(2),X(69)}}
X(1973) lies on these lines: 1,19 6,1245 25,41 32,1395 34,1438 47,163 112,741 255,1755 278,1429 604,608 1148,1283 1842,1886
X(1973) = isogonal conjugate of X(304)
X(1973) = complement of anticomplementary conjugate of X(21216)
X(1973) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,31), (608,1395), (1474,25)
X(1973) = X(i)-cross conjuguate of X(j) for these (i,j): (4,683), (682,3), (1196,2), (1368,305)
X(1973) = crosspoint of X(i) and X(j) for these (i,j): (19,1096), (25,608)
X(1973) = crosssum of X(i) and X(j) for these (i,j): (63,326), (69,345), (312,322), (525,1565)
X(1973) = barycentric product of PU(18)
X(1973) = trilinear product of intersections of circumcircle and 2nd Lemoine circle
X(1973) = pole wrt polar circle of trilinear polar of X(561)
X(1973) = X(48)-isoconjugate (polar conjugate) of X(561)
X(1973) = X(75)-isoconjugate of X(63)
X(1973) = X(92)-isoconjugate of X(326)
X(1974) is the X(i)-isoconjugate of X(j) for these (i,j): (48,1502), (92,3926); also X(1974) is the pole with respect to the polar cirle of the trilinear polar of X(1502). Randy Hutson, August 15, 2013
X(1974) lies on these lines: 4,83 6,25 24,511 32,682 34,1428 53,460 66,125 69,459 110,193 112,729 141,468 156,1353 235,1503 237,577 264,419 428,597 571,1576 981,1172 1147,1351 1386,1829 1395,1397 1691,1968
X(1974) = isogonal conjugate of X(305)
X(1974) = X(25)-Ceva conjugate of X(32)
X(1974) = crosspoint of X(i) and X(j) for these {i,j}: {6, 34207}, {1395, 1973}
X(1974) = crosssum of X(i) and X(j) for these (i,j): (2,1370), (304,3718), (339,3267)
X(1974) = crossdifference of every pair of points on the line X(525)X(3267)
X(1974) = X(92)-isoconjugate of X(3926)
X(1974) = trilinear product of vertices of Ara triangle
X(1974) = polar conjugate of X(1502)
X(1974) = barycentric product of intersections of circumcircle and 2nd Lemoine circle
X(1975) lies on these lines: 3,76 4,325 6,194 20,64 25,305 30,315 32,538 56,350 75,958 190,220 221,664 264,1105 274,405 310,1011 316,382 378,1235 394,401 543,626 801,1073
X(1975) = midpoint of X(489) and X(490)
X(1975) = X(i)-Ceva conjugate of X(j) for these (i,j): (287,325), (801,69)
X(1975) = cevapoint of X(20) and X(194)
X(1975) = anticomplement of X(5254)
X(1975) = crosspoint of PU(37)
X(1975) = intersection of tangents at PU(37) to hyperbola {A,B,C,X(99),PU(37)}
X(1975) = crosspoint of X(20) and X(194) wrt excentral triangle
X(1975) = crosspoint of X(20) and X(194) wrt anticomplementary triangle
X(1975) = X(32) of 6th Brocard triangle
X(1975) = 5th-Brocard-to-6th-Brocard similarity image of X(32)
X(1976) lies on these lines: 2,98 6,157 25,1501 32,263 37,692 51,251 111,1495 237,694 290,308 351,878 419,685 879,1177 1492,1821
X(1976) = isogonal conjugate of X(325)
X(1976) = X(i)-Ceva conjugate of X(j) for these (i,j): (98,248), (2065,6)
X(1976) = cevapoint of X(i) and X(j) for these (i,j): (6,1691), (232,1968)
X(1976) = crosssum of X(2) and X(147)
X(1976) = trilinear pole of line X(32)X(512)
X(1976) = crossdifference of every pair of points on line X(2799)X(3569)
X(1976) = barycentric product X(6)*X(98)
X(1976) = barycentric product of circumcircle intercepts of line X(6)X(523)
X(1977) lies on the Brocard inellipse and on these lines: 6,100 213,1017 291,1017 1397,1501
X(1977) = isogonal conjugate of X(31625)
X(1977) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,667), (31,669), (32,1919), (739,890), (1397,1980)
X(1977) = crosspoint of X(i) and X(j) for these (i,j): (6,667), (32,1919), (87,1019)
X(1977) = crosssum of X(i) and X(j) for these (i,j): (2,668), (43,1018), (76,1978)
X(1977) = trilinear pole wrt symmedial triangle of line X(1)X(6)
X(1977) = polar conjugate of isotomic conjugate of X(22096)
X(1977) = crossdifference of every pair of points on line X(668)X(891) (the tangent to the Steiner circumellipse at X(668))
X(1977) = barycentric square of X(649)
As the trilinear product of Steiner circumellipse antipodes, X(1978) lies on conic {{A,B,C,X(668),X(789)}} with center X(6376) and perspector X(75). (Randy Hutson, July 11, 2019)
X(1978) lies on these lines: 75,244 99,835 100,789 101,689 190,670 310,321 312,561 668,891 811,1897
X(1978) = isogonal conjugate of X(1919)
X(1978) = isotomic conjugate of X(649)
X(1978) = complement of X(21224)
X(1978) = anticomplement of X(6377)
X(1978) = X(670)-Ceva conjugate of X(668)
X(1978) = cevapoint of X(i) and X(j) for these (i,j): (75,514), (321,693), (646,668), (850,1230)
X(1978) = crosssum of X(669) and X(1924)
X(1978) = trilinear pole of line X(10)X(75) (the isotomic conjugate of the isogonal conjugate of the Nagel line)
X(1978) = trilinear product of intercepts of Steiner circumellipse and line X(2)X(37)
X(1979) lies on this line: 6,100
X(1979) = isogonal conjugate of X(9295)
X(1979) = X(667)-Ceva conjugate of X(6)
X(1979) = crosspoint of PU(42)
X(1979) = polar conjugate of isotomic conjugate of X(22158)
X(1980) lies on these lines: 667,838 669,688 692,1252 813,929
X(1980) = X(i)-Ceva conjugate of X(j) for these (i,j): (692,32),
(1397,1977)
X(1980) = crosspoint of X(32) and X(692)
X(1980) = crosssum of X(i) and X(j) for these (i,j): (76,693), (850,1228)
X(1980) = isogonal conjugate of X(6386)
X(1980) = crossdifference of every pair of points on line X(76)X(321)
As a point on the Euler line, X(1981) has Shinagawa coefficients ($aSBSC$(E+F)F+$aSA$FS2 -$a$(E-2F)FS2, -$aSBSC$S2 -3$aSA$FS2-$a$[(E+F)F-S2]S2).
If X = x : y : z is a triangle center other than X(1), then the Vega transform of X is defined by trilinears
(y - z)/x : (z - x)/y : (x - y)/z, which is the point of intersection of the line with coefficients x,y,z and the line with coefficients x2, y2, z2. (The first of these lines is the trilinear polar of the isogonal conjugate of X.) Thus, the Vega transform of X(647) lies on the Euler line.X(1981) lies on these lines: 2,3 651,653 662,811
X(1981) = bicentric difference of PU(30)
X(1981) = PU(30)-harmonic conjugate of X(1982)
X(1981) = intersection of lines P(15)U(16) and U(15)P(16)
As a point on the Euler line, X(1982) has Shinagawa coefficients (2$aSA2SB$F -2$aSCSA2$F +$aSC2SA$F -$aSASB2$F, $aSCSA$S2 -$aSASB$S2+3$aSB$FS2 -3$aSC$FS2).
X(1982) lies on these lines: 1,648 2,3 255,1098
X(1982) = bicentric sum of PU(30)
X(1982) = PU(30)-harmonic conjugate of X(1981)
X(1983) lies on these lines: 3,6 101,109 919,1027 1023,1252 1258,1497
X(1983) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,215), (901,692)
X(1983) = X(215)-cross conjugate of X(110)
X(1983) = crosspoint of X(651) and X(1290)
X(1983) = crosssum of X(661) and X1769)
X(1983) = bicentric difference of PU(29)
X(1983) = PU(29)-harmonic conjugate of X(9275)
As a point on the Euler line, X(1984) has Shinagawa coefficients ($a(SA)3$ - $a(SA)2$E + $aSA$(E - F)F - 2abc(E - 2F)F, -$a(SA)3$ + $a(SA)2$E + $aSBSC$(E - 2F) - $aSA$(E + F)F - 2abc[(E + F)F - S2]).
X(1984) lies on this line: 2,3
X(1984) = crosssum of X(851) and X(1020)
As a point on the Euler line, X(1985) has Shinagawa coefficients ($bcSBSC$, $bc$S2).
X(1985) lies on these lines: 2,3 6,11 42,1837 184,1746 1465,1893 1699,1730
X(1985) = inverse-in-orthocentroidal-circle of X(851)
Let A'B'C' be the orthic triangle of triangle ABC. Let AB be the reflection of A in C', and define AC, BC, BA, CA, CB functionally. Then the nine-point circles of the triangles
AABAC, BBCBA, CCACB,
concur in X(1986). (Antreas Hatzipolakis, Hyacinthos 7868, 9/12/03; coordinates by Barry Wolk, Hyacinthos 7876, 9/13/03)
Let A'B'C' = cevian triangle of X(186). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(1986). (Randy Hutson, December 2, 2017)
Let A'B'C' = orthic triangle. Let B'C'A" be the triangle similar to ABC such that segment A'A" crosses the line B'C'. Define B" and C" cyclically. Equivalently, A" is the reflection of A in B'C', and cyclically for B" and C". Equivalently, A" is the isogonal conjugate of A' wrt AB'C', and cyclically for B" and C". The lines A'A", B'B", C'C" concur in X(1986). (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 HaNa, HbNb, HcNc concur in X(1986). (Randy Hutson, December 2, 2017)
Let A'B'C' be the orthic triangle. Let Oa be the A-Johnson circle of triangle AB'C', and define Ob and Oc cyclically. The circles Oa, Ob, Oc concur in X(1986). (Randy Hutson, July 31 2018)
X(1986) lies on these lines: 4,94 6,74 24,110 25,399 113,403 125,389 186,323 542,1843 648,1300 1844,1845
X(1986) = reflection of X(i) in X(j) for these (i,j): (4,1112), (74,974), (125,389)
X(1986) = X(4)-Ceva conjugate of X(403)
X(1986) = crosspoint of X(4) and X(186)
X(1986) = crosssum of X(3) and X(265)
X(1986) = X(80)-of-orthic-triangle if ABC is acute
X(1986) = antigonal conjugate of X(4) wrt orthic triangle
X(1986) = antipode of X(4) in Hatzipolakis-Lozada hyperbola
X(1986) = perspector of orthic triangle and Hatzipolakis-Moses triangle
X(1986) = X(11)-of-circumorthic-triangle if ABC is acute
X(1987) is discussed in Lemoine's paper cited at X(19). Contributed by Darij Grinberg.
X(1987) lies on these lines: 3,1625 54,112 69,1972 72,1956 237,248 290,297
X(1987) = isogonal conjugate of X(401)
X(1987) = cevapoint of X(217) and X(237)
X(1987) = antigonal conjugate of isogonal conjugate of X(37918)
X(1987) = X(232)-cross conjugate of X(6)
X(1987) = trilinear pole of line X(51)X(647)
X(1987) = trilinear pole of PU(157)
X(1987) = polar conjugate of X(16089)
X(1988) is discussed in Lemoine's paper cited at X(19). Contributed by Darij Grinberg.
X(1988) lies on these lines: 6,436 184,1968 394,401 577,1971
X(1988) = isogonal conjugate of X(3164)
X(1988) = X(4)-cross conjugate of X(6)
X(1989) plays a major role in the theory of special isocubics, as presented in Chapter 6 of Jean-Pierre Ehrmann and Bernard Gibert,, "Special Isocubics in the Triangle Plane," downloadable from Bernard Gibert, Cubics in the Triangle Plane.
X(1989) is the barycentric product X(13)*X(14) of the Fermat points. The line through X(50) parallel to the Eular line X(2)X(3) passes through X(1989).
Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. X(1989) is the barycentric product A1*A2 = B1*B2 = C1*C2. (Randy Hutson, June 27, 2018)
Let VaVbVc be the Ehrmann vertex-triangle. Let A' be the barycentric product Vb*Vc, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1989). (Randy Hutson, June 27, 2018)
Let VaVbVc and SaSbSc be the Ehrmann vertex-triangle and Ehrmann side-triangle, resp. Let A' be the barycentric product Va*Sa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1989). (Randy Hutson, June 27, 2018)
X(1989) lies on these lines: 2,94 6,13 30,50 53,112 67,868 111,230 403,1990 1427,2006
X(1989) = isogonal conjugate of X(323)
X(1989) = complement of X(1272)
X(1989) = X(94)-Ceva conjugate of X(265)
X(1989) = cevapoint of X(i) and X(j) for these (i,j): (53,1990), (115,1637), (395,396)
X(1989) = crosspoint of X(2) and X(1138)
X(1989) = crosssum of X(6) and X(399)
X(1989) = barycentric product of X(13) and X(14)
X(1989) = isotomic conjugate of X(7799)
X(1989) = inverse-in-Kiepert-hyperbola of X(265)
X(1989) = {X(13),X(14)}-harmonic conjugate of X(265)
X(1989) = trilinear pole of line X(51)X(512)
X(1989) = pole wrt polar circle of trilinear polar of X(340)
X(1989) = X(48)-isoconjugate (polar conjugate) of X(340)
X(1989) = X(50)-of-orthocentroidal-triangle
X(1989) = perspector of ABC and unary cofactor triangle of Trinh triangle
X(1989) = circumcircle-inverse of X(15550)
X(1989) = cevapoint of X(36298) and X(36299)
X(1989) = barycentric product X(79)*X(80)
X(1989) = barycentric product of circumcircle intercepts of Johnson circle (or line PU(5), X(5)X(523))
X(1989) = Dao-Moses-Telv-circle-inverse of X(34310)
X(1990) is described in section 6.4.2 of the downloadable article cited at X(1989).
X(1990) lies on these lines: 4,6 44,1785 50,112 140,216 186,1138 230,231 297,340 395,471 396,470 403,1989 458,597 550,577 1033,1609
X(1990) = midpoint of X(297) and X(648)
X(1990) = isogonal conjugate of X(14919)
X(1990) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,133), (1300,25), (1989,53)
X(1990) = crosspoint of X(2) and X(1294)
X(1990) = perspector of circumconic centered at X(133)
X(1990) = center of circumconic that is locus of trilinear poles of lines passing through X(133)
X(1990) = pole wrt polar circle of trilinear polar of X(1494) (line X(2)X(525))
X(1990) = polar conjugate of X(1494)
X(1990) = X(44)-of-orthic-triangle if ABC is acute
X(1990) = inverse of X(4) in circumconic centered at X(1249)
X(1990) = PU(4)-harmonic conjugate of X(9209)
X(1990) = excentral-to-ABC functional image of X(44)
Erect squares inwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a' be the perpendicular bisector of PQ, and define b' and c' cyclically. Then a', b', c' concur in X(1991). See X(591) for the 1st Van Lamoen perpendicular bisectors point, constructed from outwardly drawn squares.
If you have GeoGebra, you can view X(591) and X(1991); the label changes with the slider position..
If you have The Geometer's Sketchpad, you can view 2nd Van Lamoen Perpendicular Bisectors Point.
X(1991) lies on these lines: 2,6 371,754 487,3070 638,1151
X(1991) = reflection of X(591) in X(2)
X(1991) = centroid of AbAcBcBaCaCb used in construction of 3rd Lozada circle
X(1991) = perspector of outer Vecten triangle and outer Vecten of inner Vecten triangle
Orthocorrespondents, 1992-2006
Suppose P is a point in the plane of triangle ABC. The perpendiculars through P to the lines AP, BP, CP meet the lines BC, CA, AB, respectively, in collinear points. Let L denote their line. The trilinear pole of L is the orthocorrespondent of P. This definition is introduced in , Orthocorrespondence and Orthopivotal Cubics, Forum Geometricorum 3 (2003) pages 1-27.
If P is given in barycentrics by P = p : q : r, then the
orthocorrespondent of P has barycentrics
f(a,b,c) : f(b,c,a) : f(c,a,b), where
If follows that if X = x : y : z in trilinears, then the
orthocorrespondent of X has trilinears
g(a,b,c) : g(b,c,a) : g(c,a,b), where
The orthocorrespondent of every point on the line at infinity is the centroid. Two orthoassociate points (i.e., an inverse pair in the polar circle, such as X(112) and X(115)) share the same orthocorrespondent.
Let A'B'C' be the 1st Parry triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1992). (Randy Hutson, February 10, 2016)
Let A'B'C' be the anti-Artzt triangle. Let A" be the barycentric product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1992). (Randy Hutson, December 10, 2016)
Let A'B'C' be the orthic triangle. X(1992) is the radical center of the 2nd Lemoine circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)
X(1992) lies on these lines: 2,6 4,542 30,1351 145,190 218,1332 317,1249 344,1743 376,511 575,631
X(1992) = midpoint of X(2) and X(193)
X(1992) = reflection of X(i) in X(j) for these (i,j): (2,6), (69,2), (599, 597)
X(1992) = isogonal conjugate of X(21448)
X(1992) = isotomic conjugate of X(5485)
X(1992) = anticomplement of X(599)
X(1992) = X(598)-Ceva conjugate of X(2)
X(1992) = perspector of ABC and unary cofactor triangle of 4th anti-Brocard triangle
X(1992) = X(4)-of-anti-Artzt-triangle
X(1992) = {X(597),X(599)}-harmonic conjugate of X(2)
X(1992) = {X(37785),X(37786)}-harmonic conjugate of X(2)
X(1992) = trilinear pole of line X(1499)X(8644) (the perspectrix of ABC and 1st Parry triangle, and the orthic axis of the Thomson triangle)
X(1993) lies on these lines: 2,6 3,54 4,155 20,1181 22,184 23,154 24,52 25,110 26,49 51,576 63,2003 68,1594 194,401 219,3219 222,3218 264,275 278,651 283,581 317,467 371,1599 372,1600 389,1092 399,1539 458,1235 493,588 494,589 569,1216 573,1790 631,1199 858,1899 1196,1570 1353,1368
X(1993) = reflection of X(22) in X(184)
X(1993) = isogonal conjugate of X(2165)
X(1993) = isotomic conjugate of X(5392)
X(1993) = polar conjugate of X(847)
X(1993) = {X(2),X(6)}-harmonic conjugate of X(5422)
X(1993) = crosspoint of X(6) and X(155) wrt both the excentral and tangential triangles
X(1993) = X(6)-isoconjugate of X(91)
X(1993) = X(92)-isoconjugate of X(2351)
X(1993) = anticomplement of X(343)
X(1993) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,3),
(275,2), (317,24), (1585,1599), (1586,1600)
X(1993) = cevapoint of X(i) and X(j) for these (i,j): (6,155),
(571,1147)
X(1993) = crosspoint of X(i) and X(j) for these (i,j): (249,648),
(1585,1586)
X(1993) = crosssum of X(115) and X(647)
Let OAOBOC be the Kosnita triangle. Let A' be the trilinear pole of line OBOC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1994). (Randy Hutson, March 21, 2019)
X(1994) lies on these lines: 2,6 3,1199 5,195 22,1351 23,184 49,143 51,110 52,54 94,275 97,216 186,568 427,1353 567,1154 1194,1570 1627,1692
X(1994) = isogonal conjugate of X(2963)
X(1994) = anticomplement of X(37636)
X(1994) = cevapoint of X(6) and X(195)
X(1994) = X(2965)-cross conjugate of X(3518)
X(1994) = crosspoint of X(588) and X(589)
X(1994) = crosssum of X(590) and X(615)
X(1994) = polar conjugate of X(93)
X(1994) = X(6)-isoconjugate of X(2962)
As a point on the Euler line, X(1995) has Shinagawa coefficients (E - 2F, 2E + 2F).
Let P be a point in the plane of ABC and not on the circumcircle. Let (OA) be the circle tangent to the circumcircle at A and passing through P. Let A' be the antipode of A in (OA). Let LA be the tangent to (OA) at A'. Define LB and LC cyclically. Let TA = LB∩LC, and define TB and TC cyclically. Triangle TATBTC is homothetic to the tangential triangle. When P = X(6), the center of homothety is X(1995). (Randy Hutson, June 7, 2019)
X(1995) lies on these lines: 2,3 6,110 51,576 98,1302 100,344 107,264 157,1624 169,2000 182,373 184,575 197,1621 251,1184 323,1351 1383,1384 1611,1627 1915,2001
X(1995) = isogonal conjugate of X(5486)
X(1995) = inverse-in-orthocentroidal-circle of X(858)
X(1995) = X(598)-Ceva conjugate of X(6)
X(1995) = crossdifference of every pair of points on line X(647)X(690)
X(1995) = harmonic center of circumcircle and {circumcircle, nine-point circle}-inverter
X(1995) = Euler line intercept, other than X(378), of circle {X(378),PU(4)}
X(1995) = {X(37775),X(37776)}-harmonic conjugate of X(6)
X(1996) lies on these lines: 2,85 7,11
X(1997) lies on these lines: 2,37 1210,1265
X(1998) lies on these lines: 1,2 57,1004 63,1005 100,1445 224,1467 273,1897 1331,1743
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1999) lies on these lines: 1,2 6,312 27,295 37,333 63,192 75,940 81,314 100,1402 171,740 193,329 226,1943 319,1211 350,1965 553,1266 664,1427
X(1999) = X(65)-Ceva conjugate of X(894)
X(1999) = {X(37794),X(37795)}-harmonic conjugate of X(10)
X(2000) lies on these lines: 1,2 9,1331 33,63 75,1897 169,1995 241,1004
X(2001) lies on these lines: 2,98 1915,1995
X(2002) lies on these lines: 1,1813 2,7 19,77 36,990 85,653 169,651 269,1781 1172,1790
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2003) lies on these lines: 1,90 6,57 9,394 35,500 42,109 56,1203 58,73 63,1993 65,267 77,1708 81,226 84,1181 212,991 255,581 323,1442 354,1421 386,603 648,1947 894,1943 1171,1400 1397,1469 1401,1428
X(2003) = X(1442)-Ceva conjugate of X(35)
X(2003) = {X(58),X(73)}-harmonic conjugate of X(37583)X(2004) lies on these lines: 2,13 32,2005 51,61
X(2005) lies on these lines: 2,14 32,2004 51,62
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2006) lies on these lines: 1,5 7,89 28,108 57,1020 79,1399 81,226 88,655 274,349 1427,1989 1758,1929
X(2006) = isogonal conjugate of X(2323)
X(2006) = isotomic conjugate of X(32851)
X(2006) = cevapoint of X(1400) and X(1457)
X(2006) = X(2)-beth conjugate of X(651)
Gallatly Circle, etc., 2007-2040
In his book The Modern Geometry of the Triangle, 2nd edition (Francis Hodgson, London, 1913), William Gallatly, on page 117, introduces the pedal cicle of the 1st and 2nd Brocard points. The circle is here named the Gallatly circle. Its center is the Brocard midpoint, X(39), and the radius, R sin ω, where ω denotes the Brocard angle, given by
tan ω = 4[area(ABC)]/(a2 + b2 + c2).
Other special symbols used in this section are identified just before X(1662).
Centers X(2007)-X(2040) were contributed with coordinates by Peter J. C. Moses during September, 2003. For centers of similitude of the Gallatly circle and the circumcircle, see X(1689) and X(1690); for those of the Gallatly circle and the 2nd Lemoine circle, see X(1671) and X(1670).
X(2007) lies on these lines: 1,39 11,2009 12,2010 55,1689 56,1690 57,2017 181,2019 371,1673 372,1672 1124,1670 1335,1671 1682,2020 1697,2018
X(2008) lies on these lines: 1,39 11,2010 12,2009 55,1689 56,1690 57,2018 181,2020 371,1672 372,1673 1124,1671 1335,1670 1682,2019 1697,2017
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2009) lies on the Kiepert hyperbola and these lines: 2,1689 4,1690 5,39 10,2020 11,2007 12,2008 83,1688 98,1687 485,1670 486,1671 1346,2016 1347,2015 1348,2012 1349,2011 1698,2018 1699,2017 2026,2040 2027,2039
X(2009) = reflection of X(2010) in X(115)
X(2009) = isogonal conjugate of X(1687)
X(2009) = antigonal conjugate of X(2010)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2010) lies on the Kiepert hyperbola and these lines: 2,1690 4,1689 5,39 10,2019 11,2008 12,2007 83,1687 98,1688 485,1671 486,1670 1346,2015 1347,2016 1348,2011 1349,2012 1698,2017 1699,2018 2026,2039 2027,2040
X(2010) = reflection of X(2009) in X(115)
X(2010) = isogonal conjugate of X(1688)
X(2010) = antigonal conjugate of X(2009)
X(2011) lies on these lines: 3,6 1348,2010 1349,2009
X(2012) lies on these lines: 3,6 1348,2009 1349,2010
X(2013) lies on this line:
2,2008 8,2007 9,2018
10,39 371,1681 372,1680
958,1689 1329,2009 1376,1690
1377,1670 1378,1671 1678,2012
1679,2011 1706,2017
X(2014) lies on these lines: 2,2007 8,2008 9,2017 10,39 371,1680 372,1681 958,1690 1329,2010 1376,1689 1377,1671 1378,1670 1678,2011 1679,2012 1706,2018
X(2015) lies on these lines: 39,381 1344,1689 1345,1690 1346,2010 1347,2009
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2016) lies on these lines: 39,381 1344,1690 1345,1689 1346,2009 1347,2010
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2017) lies on these lines: 1,1690 9,2014 10,2545 39,40 43,2019 57,2007 165,1689 371,1701 372,1700 516,2544 1670,1702 1671,1703 1695,2020 1697,2008 1698,2010 1699,2009 1704,2012 1705,2011 1706,2013 2562,2573 2563,2572
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2018) lies on these lines: 1,1689 9,2013 10,2544 39,40 43,2020 57,2008 165,1690 371,1700 372,1701 516,2545 1670,1703 1671,1702 1695,2019 1697,2007 1698,2009 1699,2010 1704,2011 1705,2012 1706,2014 2562,2572 2563,2573
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2019) lies on these lines: 3,6 10,2010 43,2017 181,2007 1682,2008 1695,2018
X(2019) = reflection of X(2020) in X(2092)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2020) lies on these lines: 3,6 10,2009 43,2018 181,2008 1682,2007 1695,2017
X(2020) = reflection of X(2019) in X(2092)
X(2021) lies on these lines: 3,6 30,2025 115,1513 538,1569 620,736 625,1506
X(2021) = midpoint of X(39) and X(187)
X(2021) = reflection of X(2025) in X(2024)
X(2021) = complement of X(39266)
X(2021) = centroid of PU(1)PU(2)
X(2021) = X(230)-of-X(3)PU(1)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2022) lies on this line: 3,6
Let A'B'C' be the 1st Brocard triangle. Let A" be the orthogonal projection of A on line B'C', and define B" and C" cyclically. Triangle A"B"C" is perpsective to the medial triangle at X(2023). (Note that the A-vertex of the medial triangle is the orthogonal projection of A' on line BC, and cyclically for B' and C'.) (Randy Hutson, July 11, 2019)
X(2023) lies on the half-Moses circle and these lines: 2,694 5,39 6,98 30,2021 194,1007 230,511 325,732 625,736 1503,2024 1575,1738
X(2023) = midpoint of X(39) and X(115)
X(2023) = crosssum of X(1662) and X(1663)
X(2023) = inverse-in-Kiepert-hyperbola of X(114)
X(2023) = {X(2009),X(2010)}-harmonic conjugate of X(114)
X(2024) lies on these lines: 3,6 230,732 1503,2023
X(2024) = midpoint of X(i) and X(j) for these (i,j): (39,1692), (2021,2025)
X(2024) = radical trace of 1st Lemoine circle and circle {{X(371),X(372),PU(1),PU(39)}}
X(2024) = radical trace of Gallatly circle and circle {{X(371),X(372),PU(1),PU(39)}}
X(2024) = Gallatly-circle-inverse of X(35436)
X(2025) lies on this line: 3,6
X(2025) = midpoint of X(39) and X(1570)
X(2025) = reflection of X(2021) in X(2024)
Of the two points of intersection of the Brocard axis, X(3)X(6) with the Gallatly circle, X(2026) is the one nearer to X(3).
X(2026) lies on these lines: 3,6 2009,2040 2010,2039
X(2026) = reflection of X(2027) in X(39)
X(2027) lies on these lines: 3,6 2009,2039 2010,2040
X(2027) = reflection of X(2026) in X(39)
Of the two points of intersection of the Brocard axis, X(3)X(6) with the Moses circle, X(2028) is the one nearer to X(3). Points X(2028) and X(2029) are the points of intersection of the Brocard axis and the asymptotes of the Kiepert hyperbola; see X(2039) and X(2040). For a description of the Moses circle and others, see the notes just above X(1662).
X(2028) lies on the Brocard inellipse and these lines: 3,6 115,2039 1506,2040
X(2028) = isogonal conjugate of isotomic conjugate of X(39023)
X(2028) = crosssum of the minor vertices of the Steiner circumellipse
X(2029) lies on the Brocard inellipse and these lines: 3,6 115,2040 1506,2039
X(2029) = isogonal conjugate of isotomic conjugate of X(39022)
X(2029) = crosssum of the vertices of the Steiner circumellipse
X(2030) lies on these lines: 3,6 111,1495 112,843 230,542 524,620
X(2030) = midpoint of X(6) and X(187)
X(2030) = midpoint of X(1691) and X(1692)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2031) lies on these lines: 3,6 230,625
X(2032) lies on this line: 3,6
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2033) lies on these lines: 3,6 115,1348 1349,1506
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2034) lies on these lines: 3,6 115,1349 1348,1506
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2035) lies on these lines: 3,6 1015,1672 1500,1673 1571,1701 1572,1700 1573,1681 1574,1680
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2036) lies on these lines: 3,6 1015,1673 1500,1672 1571,1700 1572,1701 1573,1680 1574,1681
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Of the two points of intersection of the Brocard axis and Apollonius circle, X(2037) is the one nearer to X(3) and also nearer to X(6).
X(2037) lies on these lines: 3,6 10,2039 2040,2051
X(2037) = reflection of X(2038) in X(970)
X(2038) lies on these lines: 3,6 10,2040 2039,2051
X(2038) = reflection of X(2037) in X(970)
Of the points other than X(115) in which the nine-point circle meets the asymptotes of the Kiepert hyperbola, X(2039) is the one nearer to X(3).
The points X(2039) and X(2040) are endpoints of the diameter of the nine-point circle that is parallel to the line X(25)X(394). (M. Iliev, 5/13/07)
X(2039) lies on the nine-point circle and these lines: {2, 1379}, {3, 1349}, {4, 1380}, {5, 141}, {6, 1348}, {10, 2037}, {114, 3414}, {115, 2028}, {187, 19660}, {381, 38597}, {485, 1667}, {486, 1666}, {1341, 7790}, {1506, 2029}, {1656, 38596}, {1662, 1677}, {1663, 1676}, {1670, 2567}, {1671, 2566}, {2009, 2027}, {2010, 2026}, {2038, 2051}, {2542, 7797}, {2574, 13870}, {3557, 7752}, {5025, 13326}, {5996, 13636}, {6039, 31862}, {6189, 14568}, {13325, 37446}, {14501, 36519}, {18860, 19659}, {22242, 35913}
X(2039) = midpoint of X(4) and X(1380)X(2040) lies on the nine-point circle and these lines: {2, 1380}, {3, 1348}, {4, 1379}, {5, 141}, {6, 1349}, {10, 2038}, {114, 3413}, {115, 2029}, {187, 19659}, {381, 38596}, {485, 1666}, {486, 1667}, {1340, 7790}, {1506, 2028}, {1656, 38597}, {1662, 1676}, {1663, 1677}, {1670, 2566}, {1671, 2567}, {2009, 2026}, {2010, 2027}, {2037, 2051}, {2543, 7797}, {2575, 13870}, {3558, 7752}, {5025, 13325}, {5996, 13722}, {6040, 31863}, {6190, 14568}, {13326, 37446}, {14502, 36519}, {18860, 19660}, {22243, 35914}
X(2040) = midpoint of X(i) and X(j) for these {i,j}: {4, 1379}, {115, 14501}, {6040, 31863}, {6190, 31862}
X(2040) = reflection of X(i) in X(j) for these {i,j}: {2028, 14632}, {2039, 5}
X(2040) = complement of X(1380)
X(2040) = complement of the isogonal conjugate of X(3414)
X(2040) = medial-isogonal conjugate of X(3414)
X(2040) = orthic-isogonal conjugate of X(3414)
X(2040) = X(4)-Ceva conjugate of X(3414)
X(2040) = crosspoint of X(3414) and X(14632)
X(2040) = crosssum of X(1380) and X(14630)
X(2040) = crossdifference of every pair of points on line {3050, 41880}
X(2040) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 3414}, {661, 39023}, {1379, 14838}, {1577, 2039}, {3414, 10}, {5639, 37}, {6190, 4369}, {13722, 8287}
X(2040) = barycentric quotient X(i)/X(j) for these {i,j}: {34981, 2039}, {39022, 6178}
X(2040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {141, 625, 2039}, {623, 624, 2039}, {626, 5031, 2039}, {639, 32432, 2039}, {640, 32435, 2039}, {3454, 20546, 2039}, {3934, 5103, 2039}, {7684, 7685, 2039}, {11675, 39506, 2039}
Euler-Vecten-Gibert Points, 2041 -2046
On August 13, 2003, Bernard Gibert contributed six centers that lie on the Euler line and are related to the Vecten points, X(485) and X(486).
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2041) = 3X(2) - (1 + sqrt(3))*X(3)
As a point on the Euler line, X(2041) has Shinagawa coefficients (31/2 - 1, -31/2 - 1).
X(2041) lies on these lines: 2,3 15,485 16,486 489,622 490,621 491,628 492,627
X(2041) = reflection of X(2042) in X(382)
X(2041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 641, 2042), (3, 4, 2044), (3, 5, 2042), (4, 20, 2042), (376, 3832, 2042), (381, 548, 2042)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2042) = 3X(2) + (-1 + sqrt(3))*X(3)
As a point on the Euler line, X(2042) has Shinagawa coefficients (31/2 + 1, -31/2 + 1).
X(2042) lies on these lines: 2,3 15,486 16,485 489,621 490,622 491,627 492,628
X(2042) = reflection of X(2041) in X(382)
X(2042) = complement of X(35732)
X(2042) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 641, 2041), (3, 4, 2043), (3, 5, 2041), (4, 20, 2041), (376, 3832, 2041), (381, 548, 2041)
As a point on the Euler line, X(2043) has Shinagawa coefficients (31/2 - 1, -3 + 31/2).
X(2043) lies on these lines: 2,3 13,485 14,486 17,1327 18,1328 298,637 299,638 489,616 490,617 491,622 492,621
X(2043) = reflection of X(2044) in X(3)
X(2043) = inverse-in-orthocentroidal-circle of X(2044)
X(2043) = {X(2),X(4)}-harmonic conjugate of X(2044)
X(2043) = {X(3),X(4)}-harmonic conjugate of X(2042)
X(2043) = {X(5),X(381)}-harmonic conjugate of X(2044)
X(2043) = {X(20),X(376)}-harmonic conjugate of X(2044)
X(2043) = {X(382),X(549)}-harmonic conjugate of X(2044)
As a point on the Euler line, X(2044) has Shinagawa coefficients (31/2 + 1,3 + 31/2).
X(2044) lies on these lines: 2,3 13,486 14,485 17,1328 18,1327 298,638 299,637 489,617 490,616 491,621 492,622
X(2044) = reflection of X(2043) in X(3)
X(2044) = inverse-in-orthocentroidal-circle of X(2043)
X(2044) = X(1) of X(381)PU(5)
X(2044) = {X(2),X(4)}-harmonic conjugate of X(2043)
X(2044) = {X(3),X(4)}-harmonic conjugate of X(2041)
X(2044) = {X(5),X(381)}-harmonic conjugate of X(2043)
X(2044) = {X(20),X(376)}-harmonic conjugate of X(2043)
X(2044) = {X(382),X(549)}-harmonic conjugate of X(2043)
As a point on the Euler line, X(2045) has Shinagawa coefficients (3 - 31/2,1 - 31/2).
X(2045) lies on these lines: 2,3 17,485 18,486 302,637 303,638 397,590 398,615 491,634 492,633
X(2045) = inverse-in-orthocentroidal-circle of X(2046)
X(2044) = X(1)-of-X(381)PU(5)
X(2044) = {X(2),X(4)}-harmonic conjugate of X(2043)
X(2044) = {X(3),X(4)}-harmonic conjugate of X(2041)
X(2044) = {X(5),X(381)}-harmonic conjugate of X(2043)
X(2044) = {X(20),X(376)}-harmonic conjugate of X(2043)
X(2044) = {X(382),X(549)}-harmonic conjugate of X(2043)
As a point on the Euler line, X(2046) has Shinagawa coefficients (3 + 31/2,1 + 31/2).
X(2046) lies on these lines: 2,3 17,486 18,485 302,638 303,637 397,615 398,590 491,633 492,634
X(2046) = inverse-in-orthocentroidal-circle of X(2045)
As a point on the Euler line, X(2047) has Shinagawa coefficients (s,r).
X(2047) lies on these lines: 2,3 10,485 86,637 492,1330 590,1834 966,1587
As a point on the Euler line, X(2048) has Shinagawa coefficients (r, s).
X(2048) lies on these lines: 2,3 486,1686
As a point on the Euler line, X(2049) has Shinagawa coefficients (s2,r2).
X(2049) lies on these lines: 2,3 6,10 12,1460 171,1698
As a point on the Euler line, X(2050) has Shinagawa coefficients (r2, s2).
X(2050) lies on these lines: 2,3 11,1460 171,1699 1482,1999
X(2051) is the external center of similitude of the nine-point and Apollonius circles. Trilinears for the internal center, X(10), result from g(a,b,c) by changing the "-" just after "cos A" to "+". (The two circles are described just before X(1662).)
Let A' be the midpoint of BC, A" the midpoint of AX(1), and define B', C' and B", C" cyclically. Let A"'B"'C"' be the incentral triangle. Let A* be the orthocenter of A'A"A"', and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(2051). (Randy Hutson, March 21, 2019)
Let LA be the radical axis of the A-excircle and the excircles radical circle, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(2051). (Randy Hutson, March 21, 2019)
Let A'B'C' be the tangential triangle of the Feuerbach triangle. Let A"B"C" be the tangential triangle of the Apollonius triangle. Let A* be the {A',A"}-harmonic conjugate of X(10), and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(2051). (Randy Hutson, March 21, 2019)
X(2051) lies on these lines: 2,573 4,386 5,10 6,2050 11,181 12,1682 27,275 43,1699 226,1465 321,908 469,2052 485,1685 486,1686 572,2185 1348,1693 1349,1694 1676,1683 1677,1684 1695,1698 1766,2339 2009,2019 2101,2020 2037,2040
X(2051) = isogonal conjugate of X(572)
X(2051) = complement of X(1764)
X(2051) = {X(5),X(970)}-harmonic conjugate of X(10)
X(2051) = perspector of exircles radical circle
X(2051) = Spieker-radical-circle-inverse of X(38472)
X(2051) = pole wrt excircles radical circle of antiorthic axis
Let A'B'C' be the orthic triangle of a triangle ABC, and let
O(A) = circle with center A and radius AA', and define O(B) and O(C) cyclically
p(A) = polar of X(4) wrt O(A), and define p(B) and p(C) cyclically
A'' = p(B)∩p(C), and define B'' and C'' cyclically.
Then A''B''C'' is homothetic to ABC, and the center of homothety is X(2052). (Angel Montesdeoca, September 30, 2016)
X(2052) lies on these lines: 2,216 4,51 6,275 10,158 13,473 14,472 17,470 18,471 24,96 25,98 63,1947 76,297 83,458 92,226 184,436 262,427 331,1446 394,801 401,1968 485,1585 486,1586 648,1993
X(2052) = isogonal conjugate of X(577)
X(2052) = isotomic conjugate of X(394)
X(2052) = cevapoint of X(i) and X(j) for these (i,j): (4,393), (6,24), (324,467), (1585,1586)
X(2052) = X(i)-cross conjugate of X(j) for these (i,j): (4,264), (6,847), (53,4), (393,1093), (686,1304)
X(2052) = trilinear pole of line X(403)X(523) (polar of X(3) wrt polar circle)
X(2052) = pole wrt polar circle of trilinear polar of X(3) (line X(520)X(647))
X(2052) = polar conjugate of X(3)
X(2052) = X(43)-of-orthic-triangle if ABC is acute
X(2052) = complement of isotomic conjugate of X(34287)
X(2052) = barycentric product X(107)*X(850)
X(2053) lies on these lines: 3,2108 6,904 31,172 41,2330 55,2319 56,87 105,330 1403,1580 2115,2144
X(2053) = isogonal conjugate of X(3212)
X(2053) = complement of X(20350)
X(2053) = anticomplement of X(20338)
X(2053) = X(9)-cross conjugate of X(55)
X(2053) = crosssum of X(43) and X(1423)
X(2054) lies on these lines: 2,846 110,1171 111,902 238,1931
X(2054) = complement of X(20351)
X(2054) = anticomplement of X(20339)
Orion Transforms, 2055 -2065
On September 24, 2003, Jean-Pierre Ehrmann suggested in Hyacinthos 7999 a transformation: suppose P = x : y : z (trilinears); the Orion transform of P is given by OT(P) = f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = x[- y2z2 + x2(y2 + z2 + 2yz cos A)].
Among pairs (i,j) for which OT(X(j)) = X(i) are these:
(1,35), (2,69), (4,24), (7,57), (99,249), (100,59), (110,250)
Ehrmann's construction of OT(P) is as the perspector of triangles ABC and A"B"C", where A" is the reflection of P in the sideline B'C' of the cevian triangle A'B'C' of P, and B" and C' are defined cyclically. Alternative and related constructions are given in other Hyacinthos messages prompted by #7999.
Extensions of results in this section were contributed by Ivan Pavlov, October 22, 2023.
X(2055) is the perspector of triangle ABC and the orthic triangle of the cevian triangle of the circumcenter. (Randy Hutson, 9/23/2011)
X(2055) lies on these lines: {3, 6}, {4, 14152}, {5, 275}, {54, 418}, {97, 13434}, {140, 46760}, {185, 45842}, {250, 47304}, {394, 14059}, {417, 43574}, {648, 41481}, {1092, 6760}, {1147, 6638}, {1576, 34782}, {1971, 36433}, {1994, 42441}, {3463, 5562}, {5446, 37081}, {6321, 8800}, {7335, 20764}, {8613, 51888}, {8884, 13322}, {9306, 38281}, {10112, 10600}, {12134, 44231}, {13450, 41202}, {13557, 18404}, {14575, 18925}, {15033, 26897}, {19467, 52435}, {21651, 22143}, {31388, 35921}, {36245, 44076}, {37877, 46219}
X(2055) = reflection of X(i) in X(j) for these {i,j}: {43995, 5}
X(2055) = X(i)-isoconjugate-of-X(j) for these {i, j}: {92, 8612}
X(2055) = X(i)-Dao conjugate of X(j) for these {i, j}: {22391, 8612}, {38976, 14618}
X(2055)= pole of line {23286, 34983} with respect to the Johnson circumconic
X(2055)= pole of line {2, 1075} with respect to the Stammler hyperbola
X(2055) = intersection, other than A, B, C, of circumconics {{A, B, C, X(3), X(8613)}}, {{A, B, C, X(6), X(3463)}}, {{A, B, C, X(52), X(14941)}}, {{A, B, C, X(216), X(40448)}}, {{A, B, C, X(275), X(389)}}, {{A, B, C, X(288), X(578)}}, {{A, B, C, X(1073), X(11432)}}, {{A, B, C, X(19179), X(57686)}}, {{A, B, C, X(32590), X(32592)}}
X(2055) = barycentric product X(i)*X(j) for these (i, j): {3, 8613}, {394, 56298}, {51888, 97}
X(2055) = barycentric quotient X(i)/X(j) for these (i, j): {184, 8612}, {8613, 264}, {38976, 35442}, {51888, 324}, {56298, 2052}
X(2055) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 36749, 30258}, {3, 38292, 11432}, {4, 23606, 14152}, {275, 40448, 5}, {577, 578, 3}, {8613, 56298, 51888}
X(2056) lies on these lines: {2, 5038}, {6, 1196}, {22, 5104}, {25, 13330}, {32, 36615}, {51, 20998}, {110, 251}, {111, 53863}, {141, 58761}, {154, 5017}, {182, 21001}, {184, 1613}, {323, 20859}, {352, 15246}, {394, 3094}, {427, 11646}, {524, 33651}, {576, 34481}, {692, 21792}, {732, 37894}, {1194, 3292}, {1397, 2176}, {1412, 33863}, {1501, 9463}, {1611, 17809}, {1625, 18373}, {1627, 14567}, {1660, 32445}, {1993, 3981}, {1994, 3124}, {2076, 3787}, {2162, 2175}, {2248, 4274}, {2502, 13595}, {3117, 34870}, {3229, 3398}, {3231, 5012}, {3291, 13366}, {3819, 5116}, {3917, 10329}, {3955, 16514}, {5114, 39967}, {5354, 39689}, {6090, 13331}, {6388, 11225}, {6676, 15993}, {6997, 53504}, {8550, 40326}, {8584, 32740}, {9259, 55086}, {10328, 46900}, {10799, 52655}, {11173, 20850}, {12055, 15066}, {12835, 20284}, {14575, 51336}, {17811, 50659}, {20854, 46321}, {20986, 21788}, {21448, 52719}, {21779, 44085}, {21849, 40350}, {23292, 53475}, {32449, 57216}, {35356, 35929}, {35901, 54384}, {36650, 39560}, {39024, 55038}
X(2056) = perspector of ABC and orthic triangle of symmedial triangle
X(2056) = X(i)-isoconjugate-of-X(j) for these {i, j}: {75, 8601}
X(2056) = X(i)-Dao conjugate of X(j) for these {i, j}: {206, 8601}
X(2056) = X(i)-cross conjugate of X(j) for these {i, j}: {31989, 6}
X(2056)= pole of line {1368, 37688} with respect to the Kiepert hyperbola
X(2056)= pole of line {193, 732} with respect to the Stammler hyperbola
X(2056)= pole of line {7777, 35540} with respect to the Wallace hyperbola
X(2056) = intersection, other than A, B, C, of circumconics {{A, B, C, X(733), X(7783)}}, {{A, B, C, X(5943), X(8601)}}
X(2056) = barycentric product X(i)*X(j) for these (i, j): {6, 7783}, {31989, 45857}
X(2056) = barycentric quotient X(i)/X(j) for these (i, j): {32, 8601}, {7783, 76}
X(2056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14153, 5038}, {110, 3051, 1915}, {184, 1613, 1691}, {1196, 34986, 6}, {1915, 3051, 12212}, {1993, 3981, 5111}, {9225, 14153, 2}, {9463, 9544, 1501}, {20998, 45843, 51}
X(2057) lies on these lines: {1, 2}, {3, 51380}, {40, 51379}, {56, 46677}, {63, 10270}, {72, 3359}, {84, 1259}, {100, 1490}, {280, 5423}, {282, 3692}, {474, 17658}, {480, 5784}, {908, 6769}, {944, 55302}, {1260, 5777}, {1320, 56096}, {1476, 35262}, {2956, 35281}, {3436, 6282}, {3681, 21164}, {3689, 26358}, {3699, 52346}, {3711, 22768}, {3940, 37562}, {3965, 5782}, {5176, 12650}, {5193, 6762}, {5223, 16209}, {5440, 5534}, {5587, 56101}, {5687, 5720}, {5780, 10679}, {6256, 21075}, {7994, 11415}, {10269, 34790}, {10310, 18239}, {10388, 41012}, {11010, 52050}, {11682, 39776}, {12751, 55016}, {17757, 37531}, {20588, 25440}, {37244, 58650}, {37561, 57279}, {38271, 45393}, {38901, 52027}
X(2057) = perspector of ABC and orthic triangle of extouch triangle
X(2057) = X(i)-isoconjugate-of-X(j) for these {i, j}: {56, 10309}, {57, 8602}
X(2057) = X(i)-Dao conjugate of X(j) for these {i, j}: {1, 10309}, {268, 1422}, {5452, 8602}, {39026, 30239}
X(2057) = X(i)-cross conjugate of X(j) for these {i, j}: {18239, 8}
X(2057) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(10310)}}, {{A, B, C, X(2), X(56545)}}, {{A, B, C, X(84), X(3086)}}, {{A, B, C, X(280), X(14986)}}, {{A, B, C, X(282), X(1210)}}, {{A, B, C, X(318), X(24982)}}, {{A, B, C, X(519), X(30201)}}, {{A, B, C, X(938), X(1034)}}, {{A, B, C, X(1156), X(5704)}}, {{A, B, C, X(1737), X(38271)}}, {{A, B, C, X(1767), X(40940)}}, {{A, B, C, X(2999), X(57418)}}, {{A, B, C, X(18391), X(44692)}}, {{A, B, C, X(27383), X(45393)}}, {{A, B, C, X(36846), X(51565)}}, {{A, B, C, X(41575), X(44693)}}
X(2057) = barycentric product X(i)*X(j) for these (i, j): {190, 30201}, {1265, 1767}, {4564, 52112}, {10310, 312}, {56545, 8}
X(2057) = barycentric quotient X(i)/X(j) for these (i, j): {9, 10309}, {55, 8602}, {101, 30239}, {1767, 1119}, {10310, 57}, {30201, 514}, {52112, 4858}, {56545, 7}
X(2057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {8, 27383, 14986}, {8, 4511, 36846}, {8, 5552, 24982}, {200, 6745, 1998}, {200, 936, 8}, {3811, 26364, 1}, {5687, 41389, 49163}
X(2058) lies on these lines: {15, 13391}, {8016, 57382}, {18403, 40682}, {37848, 40695}
X(2059) lies on these lines: {16, 13391}, {8017, 57383}, {18403, 40683}, {37850, 40696}
As a point on the Euler line, X(2060) has Shinagawa coefficients (4(E - 2F)F - S2,8F2).
X(2060) lies on these lines: {2, 3}, {2131, 23608}, {3344, 35602}, {3346, 12096}, {3350, 31944}, {5895, 45844}, {5910, 12111}, {8778, 46346}, {14362, 28785}, {14365, 28781}, {23239, 34781}, {31377, 34286}, {34147, 58797}
X(2060) = X(i)-isoconjugate-of-X(j) for these {i, j}: {2184, 31956}, {19614, 42465}
X(2060) = X(i)-Dao conjugate of X(j) for these {i, j}: {4, 42465}, {3350, 31943}, {45248, 3348}
X(2060) = X(i)-Ceva conjugate of X(j) for these {i, j}: {53050, 20}
X(2060) = X(i)-cross conjugate of X(j) for these {i, j}: {3183, 20}
X(2060)= pole of line {3, 2130} with respect to the Stammler hyperbola
X(2060)= pole of line {69, 14362} with respect to the Wallace hyperbola
X(2060) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(4), X(2131)}}, {{A, B, C, X(25), X(28781)}}, {{A, B, C, X(1249), X(6621)}}, {{A, B, C, X(3091), X(56593)}}, {{A, B, C, X(6616), X(36413)}}, {{A, B, C, X(6617), X(35602)}}
X(2060) = barycentric product X(i)*X(j) for these (i, j): {3183, 37669}, {14362, 36413}, {15905, 56593}, {31944, 47633}, {40839, 53050}
X(2060) = barycentric quotient X(i)/X(j) for these (i, j): {154, 31956}, {1249, 42465}, {3183, 459}, {3344, 31943}, {15905, 3348}, {31944, 3343}, {36413, 14365}, {37669, 56594}, {56593, 52581}
X(2060) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 3079, 20}, {20, 1559, 3146}
X(2061) = R(4R2 - s2- S)(4R2 - s2 + S)*X(1) + 2(4rR + 4R2 - s2)(4R3 + rs2 - Rs2)*X(3) (Peter Moses, April 2, 2012)
X(2061) lies on this line: 1,3
X(2062) lies on circumconic {{A, B, C, X(1439), X(40407)}} and on these lines: {3, 77}, {21, 307}, {63, 9119}, {284, 2003}, {1259, 19611}, {3964, 7111}, {5932, 6060}, {7282, 56374}, {7364, 41081}, {18655, 37258}
X(2062)= pole of line {4183, 40942} with respect to the Stammler hyperbola
X(2063) lies on these lines: {2, 6}, {64, 2071}, {110, 1619}, {155, 18913}, {264, 801}, {378, 1092}, {1073, 3964}, {1498, 53050}, {1568, 18418}, {1593, 57648}, {1660, 11413}, {4558, 6617}, {5409, 26936}, {5893, 44440}, {6509, 9723}, {6623, 37498}, {9306, 12294}, {9818, 14128}, {9909, 20772}, {11456, 51394}, {12322, 55444}, {12323, 55443}, {14390, 57483}, {18466, 22115}, {23291, 52077}, {30771, 34966}, {38282, 44492}, {40221, 46639}
X(2063) = X(i)-isoconjugate-of-X(j) for these {i, j}: {19, 43695}, {661, 30249}
X(2063) = X(i)-Dao conjugate of X(j) for these {i, j}: {6, 43695}, {14379, 41489}, {35968, 2501}, {36830, 30249}, {46432, 37197}
X(2063) = X(i)-cross conjugate of X(j) for these {i, j}: {36982, 69}
X(2063)= pole of line {99, 30249} with respect to the Kiepert parabola
X(2063)= pole of line {525, 14939} with respect to the MacBeath circumconic
X(2063)= pole of line {6, 1885} with respect to the Stammler hyperbola
X(2063)= pole of line {2, 14091} with respect to the Wallace hyperbola
X(2063) = intersection, other than A, B, C, of circumconics {{A, B, C, X(2), X(11413)}}, {{A, B, C, X(3), X(53415)}}, {{A, B, C, X(6), X(1660)}}, {{A, B, C, X(64), X(10229)}}, {{A, B, C, X(287), X(40318)}}, {{A, B, C, X(524), X(30211)}}, {{A, B, C, X(1032), X(11433)}}, {{A, B, C, X(1073), X(13567)}}, {{A, B, C, X(2139), X(45302)}}, {{A, B, C, X(6515), X(14919)}}, {{A, B, C, X(18928), X(31371)}}
X(2063) = barycentric product X(i)*X(j) for these (i, j): {394, 46927}, {1660, 305}, {11413, 69}, {30211, 99}, {36982, 57800}, {37669, 57483}
X(2063) = barycentric quotient X(i)/X(j) for these (i, j): {3, 43695}, {110, 30249}, {1660, 25}, {11413, 4}, {14390, 41489}, {30211, 523}, {33583, 31942}, {35602, 51347}, {36982, 235}, {39268, 6526}, {46927, 2052}, {57483, 459}
X(2063) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {394, 11064, 1993}, {394, 17811, 69}, {394, 37669, 20806}
X(2064) lies on these lines: {2, 20234}, {75, 40940}, {76, 85}, {81, 314}, {92, 3596}, {190, 22001}, {278, 345}, {286, 20336}, {305, 7112}, {306, 20929}, {322, 52406}, {329, 1264}, {346, 36428}, {1848, 18835}, {3687, 17788}, {3718, 18750}, {3926, 17076}, {3948, 46828}, {4417, 20444}, {4561, 27398}, {4568, 22020}, {4812, 29841}, {6358, 17787}, {7009, 7283}, {14206, 52369}, {14213, 19811}, {14615, 20928}, {17378, 42034}, {17861, 19806}, {18895, 36800}, {19786, 33940}, {20237, 40875}, {20896, 32779}, {20920, 30713}, {21595, 40071}, {25080, 33116}, {25527, 33930}, {27801, 44129}, {33941, 41258}, {46238, 52421}
X(2064) = perspector of circumconic {{A, B, C, X(4572), X(57976)}}
X(2064) = X(i)-isoconjugate-of-X(j) for these {i, j}: {6, 8615}, {32, 15314}
X(2064) = X(i)-Dao conjugate of X(j) for these {i, j}: {9, 8615}, {6376, 15314}, {16612, 18210}, {34846, 649}, {46878, 2354}, {52355, 53560}
X(2064) = X(i)-anticomplementary conjugate of X(j) for these {i, j}: {56003, 5484}
X(2064) = X(i)-cross conjugate of X(j) for these {i, j}: {48890, 75}
X(2064)= pole of line {2300, 57657} with respect to the Stammler hyperbola
X(2064)= pole of line {832, 4581} with respect to the Steiner circumellipse
X(2064)= pole of line {832, 17072} with respect to the Steiner inellipse
X(2064)= pole of line {284, 3666} with respect to the Wallace hyperbola
X(2064) = intersection, other than A, B, C, of circumconics {{A, B, C, X(81), X(3674)}}, {{A, B, C, X(85), X(7270)}}, {{A, B, C, X(226), X(2298)}}, {{A, B, C, X(349), X(30710)}}, {{A, B, C, X(3912), X(57197)}}, {{A, B, C, X(5285), X(7146)}}, {{A, B, C, X(6063), X(58018)}}, {{A, B, C, X(16612), X(43040)}}, {{A, B, C, X(18033), X(57784)}}, {{A, B, C, X(18134), X(36800)}}, {{A, B, C, X(37445), X(56375)}}, {{A, B, C, X(40940), X(48890)}}
X(2064) = barycentric product X(i)*X(j) for these (i, j): {3596, 4296}, {4554, 57197}, {5279, 76}, {5285, 561}, {7270, 75}, {16612, 1978}, {57186, 6331}
X(2064) = barycentric quotient X(i)/X(j) for these (i, j): {1, 8615}, {75, 15314}, {4296, 56}, {5279, 6}, {5285, 31}, {7270, 1}, {16612, 649}, {34846, 18210}, {48890, 1104}, {56375, 2299}, {57186, 647}, {57197, 650}
X(2064) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {305, 20641, 7112}, {312, 17789, 226}, {20929, 42709, 306}
X(2065) lies on cubic K336 and on these lines: {3, 52091}, {98, 325}, {182, 34157}, {232, 1692}, {248, 1570}, {262, 40820}, {264, 14265}, {460, 685}, {511, 1976}, {523, 47388}, {576, 18873}, {1691, 32654}, {2030, 9474}, {2782, 46648}, {5050, 5968}, {5111, 51542}, {5622, 15407}, {5967, 14356}, {6776, 56572}, {9307, 32545}, {14253, 50684}, {14384, 56688}, {15630, 17974}, {23350, 35364}, {25406, 36891}, {40801, 57493}, {46142, 47385}, {51862, 53766}
X(2065) = reflection of X(i) in X(j) for these {i,j}: {2715, 39085}
X(2065) = isogonal conjugate of X(114)
X(2065) = trilinear pole of line {248, 2422}
X(2065) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 114}, {2, 17462}, {75, 51335}, {92, 47406}, {230, 1959}, {240, 3564}, {325, 8772}, {511, 1733}, {662, 55267}, {1692, 46238}, {1755, 51481}, {14265, 23996}, {40703, 52144}
X(2065) = X(i)-vertex conjugate of X(j) for these {i, j}: {4, 249}, {511, 2065}
X(2065) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 114}, {206, 51335}, {1084, 55267}, {22391, 47406}, {32664, 17462}, {34156, 2974}, {36899, 51481}, {39085, 3564}
X(2065) = X(i)-cross conjugate of X(j) for these {i, j}: {3, 98}, {6, 2987}, {512, 2715}, {34291, 53691}, {47421, 43665}
X(2065)= pole of line {2987, 17974} with respect to the Jerabek hyperbola
X(2065)= pole of line {32654, 44534} with respect to the Kiepert hyperbola
X(2065)= pole of line {114, 51335} with respect to the Stammler hyperbola
X(2065) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(511)
X(2065) = intersection, other than A, B, C, of circumconics, {{A, B, C, X(3), X(460)}}, {{A, B, C, X(4), X(2710)}}, {{A, B, C, X(6), X(232)}}, {{A, B, C, X(25), X(56370)}}, {{A, B, C, X(32), X(2456)}}, {{A, B, C, X(54), X(83)}}, {{A, B, C, X(59), X(2298)}}, {{A, B, C, X(60), X(55018)}}, {{A, B, C, X(67), X(11071)}}, {{A, B, C, X(69), X(57638)}}, {{A, B, C, X(74), X(10630)}}, {{A, B, C, X(96), X(57644)}}, {{A, B, C, X(98), X(1976)}}, {{A, B, C, X(182), X(1691)}}, {{A, B, C, X(184), X(47741)}}, {{A, B, C, X(186), X(50712)}}, {{A, B, C, X(187), X(5050)}}, {{A, B, C, X(287), X(51776)}}, {{A, B, C, X(575), X(2076)}}, {{A, B, C, X(576), X(5111)}}, {{A, B, C, X(685), X(2715)}}, {{A, B, C, X(691), X(10419)}}, {{A, B, C, X(694), X(5966)}}, {{A, B, C, X(755), X(43532)}}, {{A, B, C, X(805), X(44768)}}, {{A, B, C, X(843), X(3431)}}, {{A, B, C, X(879), X(34175)}}, {{A, B, C, X(895), X(40388)}}, {{A, B, C, X(947), X(12031)}}, {{A, B, C, X(1126), X(2700)}}, {{A, B, C, X(1173), X(53894)}}, {{A, B, C, X(1176), X(44174)}}, {{A, B, C, X(1177), X(9161)}}, {{A, B, C, X(1297), X(38873)}}, {{A, B, C, X(1351), X(1570)}}, {{A, B, C, X(1799), X(41271)}}, {{A, B, C, X(1968), X(32545)}}, {{A, B, C, X(2021), X(35429)}}, {{A, B, C, X(2030), X(5085)}}, {{A, B, C, X(2080), X(5034)}}, {{A, B, C, X(2433), X(15398)}}, {{A, B, C, X(2458), X(3398)}}, {{A, B, C, X(2701), X(15379)}}, {{A, B, C, X(2702), X(15380)}}, {{A, B, C, X(2703), X(15381)}}, {{A, B, C, X(2704), X(15382)}}, {{A, B, C, X(2705), X(15383)}}, {{A, B, C, X(2706), X(15384)}}, {{A, B, C, X(2707), X(15385)}}, {{A, B, C, X(2708), X(15386)}}, {{A, B, C, X(2709), X(15387)}}, {{A, B, C, X(2711), X(15402)}}, {{A, B, C, X(2712), X(15403)}}, {{A, B, C, X(2713), X(15404)}}, {{A, B, C, X(2714), X(15405)}}, {{A, B, C, X(2987), X(3563)}}, {{A, B, C, X(3094), X(35377)}}, {{A, B, C, X(3095), X(35388)}}, {{A, B, C, X(3406), X(53966)}}, {{A, B, C, X(3455), X(54998)}}, {{A, B, C, X(5007), X(35458)}}, {{A, B, C, X(5092), X(35006)}}, {{A, B, C, X(5093), X(5107)}}, {{A, B, C, X(5097), X(15514)}}, {{A, B, C, X(5102), X(44496)}}, {{A, B, C, X(5104), X(39561)}}, {{A, B, C, X(5481), X(5970)}}, {{A, B, C, X(5967), X(14355)}}, {{A, B, C, X(7612), X(14659)}}, {{A, B, C, X(8586), X(15520)}}, {{A, B, C, X(9154), X(57742)}}, {{A, B, C, X(9160), X(15396)}}, {{A, B, C, X(9217), X(14528)}}, {{A, B, C, X(13137), X(43665)}}, {{A, B, C, X(14483), X(53973)}}, {{A, B, C, X(14561), X(57268)}}, {{A, B, C, X(14906), X(54869)}}, {{A, B, C, X(14965), X(31850)}}, {{A, B, C, X(15389), X(53059)}}, {{A, B, C, X(15391), X(46237)}}, {{A, B, C, X(18333), X(46427)}}, {{A, B, C, X(19131), X(58312)}}, {{A, B, C, X(29011), X(43702)}}, {{A, B, C, X(30648), X(53873)}}, {{A, B, C, X(34157), X(47736)}}, {{A, B, C, X(35383), X(39764)}}, {{A, B, C, X(38010), X(55703)}}, {{A, B, C, X(44823), X(51456)}}, {{A, B, C, X(46199), X(57771)}}, {{A, B, C, X(52076), X(52451)}}
X(2065) = barycentric product X(i)*X(j) for these (i, j): {248, 35142}, {287, 3563}, {290, 32654}, {512, 55266}, {1821, 36051}, {1910, 8773}, {1976, 8781}, {2966, 35364}, {2987, 98}, {10425, 2395}, {15391, 47736}, {16081, 42065}, {32697, 879}, {34157, 34536}, {40428, 6}, {41932, 52091}, {43705, 6531}, {47388, 57493}, {57260, 57872}
X(2065) = barycentric quotient X(i)/X(j) for these (i, j): {6, 114}, {31, 17462}, {32, 51335}, {98, 51481}, {184, 47406}, {248, 3564}, {512, 55267}, {1910, 1733}, {1976, 230}, {2422, 55122}, {2715, 4226}, {2987, 325}, {3563, 297}, {6531, 44145}, {8773, 46238}, {10425, 2396}, {14600, 52144}, {14601, 1692}, {20975, 41181}, {32654, 511}, {32697, 877}, {34157, 36790}, {34238, 47734}, {35142, 44132}, {35364, 2799}, {36051, 1959}, {40428, 76}, {41932, 14265}, {42065, 36212}, {43705, 6393}, {52091, 32458}, {55266, 670}, {57260, 460}
X(2065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1692, 46237, 2715}, {42065, 51455, 10425}
X(2066) lies on cubic K1273 and on these lines: {1, 371}, {2, 44624}, {3, 1124}, {4, 31472}, {5, 9646}, {6, 31}, {8, 31453}, {9, 7133}, {11, 590}, {12, 3071}, {20, 31408}, {29, 14121}, {30, 9660}, {32, 31471}, {33, 5412}, {34, 11473}, {35, 372}, {36, 6200}, {37, 8576}, {39, 45571}, {40, 2362}, {48, 10132}, {56, 1151}, {57, 9616}, {63, 8393}, {65, 49226}, {77, 13389}, {78, 30556}, {80, 35788}, {102, 54016}, {140, 31499}, {165, 8831}, {172, 6405}, {198, 48308}, {200, 31438}, {203, 51728}, {219, 53065}, {220, 30335}, {221, 19088}, {222, 46376}, {283, 1806}, {332, 56385}, {388, 6459}, {390, 7585}, {405, 1378}, {474, 9679}, {485, 1479}, {486, 498}, {495, 42215}, {496, 8981}, {497, 3068}, {499, 5418}, {601, 3076}, {609, 41410}, {610, 46379}, {613, 19145}, {615, 5432}, {942, 31439}, {950, 13883}, {956, 9678}, {999, 6221}, {1040, 11513}, {1062, 10897}, {1067, 8957}, {1069, 8909}, {1152, 5217}, {1210, 13912}, {1317, 48700}, {1335, 3295}, {1376, 31473}, {1377, 5687}, {1420, 9615}, {1445, 13388}, {1478, 6561}, {1500, 5058}, {1504, 2241}, {1505, 31451}, {1583, 55409}, {1587, 4294}, {1588, 3085}, {1697, 18991}, {1742, 30297}, {1837, 13911}, {2098, 44635}, {2183, 34121}, {2192, 17819}, {2242, 9675}, {2270, 46378}, {2646, 7968}, {2961, 8938}, {3023, 49266}, {3024, 49268}, {3027, 49212}, {3028, 49216}, {3057, 7969}, {3058, 19030}, {3069, 5218}, {3070, 6284}, {3083, 5409}, {3086, 9540}, {3092, 11398}, {3100, 11417}, {3270, 21640}, {3298, 3303}, {3301, 3746}, {3304, 6425}, {3320, 49218}, {3362, 32590}, {3364, 7006}, {3389, 7005}, {3434, 31484}, {3486, 19066}, {3583, 6564}, {3584, 35823}, {3585, 35800}, {3600, 43512}, {3601, 18992}, {3614, 42270}, {4254, 53066}, {4293, 9541}, {4299, 42260}, {4302, 6560}, {4314, 49548}, {4317, 9681}, {4324, 42267}, {4857, 8960}, {4995, 13958}, {5010, 6396}, {5062, 45570}, {5119, 35774}, {5204, 6409}, {5225, 31412}, {5250, 30557}, {5268, 8855}, {5274, 8972}, {5281, 7586}, {5298, 31500}, {5299, 45512}, {5326, 32790}, {5410, 7071}, {5413, 52427}, {5420, 13962}, {5434, 41945}, {5563, 6453}, {5697, 35641}, {5903, 35610}, {6020, 49270}, {6198, 10880}, {6199, 6767}, {6238, 10665}, {6283, 12962}, {6285, 12964}, {6286, 12965}, {6413, 26949}, {6421, 31448}, {6422, 16502}, {6424, 54416}, {6480, 37587}, {6565, 7951}, {6567, 45507}, {7173, 42582}, {7354, 42258}, {7355, 49250}, {7583, 15171}, {7727, 12375}, {7741, 10576}, {7972, 35856}, {8144, 11265}, {8164, 23273}, {8252, 13955}, {8276, 10046}, {8976, 9669}, {8983, 12053}, {9441, 30296}, {9581, 13893}, {9582, 15803}, {9599, 31463}, {9647, 18990}, {9648, 41963}, {9665, 31481}, {9668, 13665}, {9689, 40726}, {9817, 10961}, {9931, 12424}, {10039, 49602}, {10091, 10819}, {10118, 13287}, {10385, 19054}, {10386, 19117}, {10483, 42266}, {10533, 10535}, {10572, 49601}, {10588, 42561}, {10589, 32785}, {10590, 23259}, {10799, 44586}, {10833, 44598}, {10877, 44604}, {10895, 23261}, {10896, 13897}, {10947, 44618}, {10950, 49232}, {10953, 44620}, {10965, 44643}, {10966, 44606}, {11010, 35611}, {11189, 11241}, {11238, 13846}, {11446, 11447}, {11461, 11462}, {11873, 44600}, {11874, 44602}, {11909, 44610}, {11947, 44627}, {11948, 44629}, {12354, 49214}, {12428, 49224}, {12680, 49234}, {12702, 38235}, {12743, 49240}, {12863, 49248}, {12888, 12891}, {12896, 49222}, {12910, 12960}, {12911, 12961}, {12943, 42263}, {12953, 23251}, {12959, 13882}, {12970, 26888}, {12971, 47378}, {13043, 13045}, {13044, 13046}, {13075, 49210}, {13076, 49208}, {13077, 49252}, {13078, 49254}, {13079, 49256}, {13080, 49258}, {13081, 49220}, {13082, 44647}, {13699, 49260}, {13785, 31479}, {13819, 49262}, {13888, 51785}, {13963, 31452}, {15325, 35255}, {15338, 42259}, {15452, 49267}, {16142, 49242}, {17784, 31413}, {18455, 18457}, {18514, 35786}, {18922, 18923}, {18965, 31454}, {18966, 52793}, {19004, 53053}, {19048, 26358}, {19050, 26357}, {19182, 19183}, {19354, 19355}, {19434, 19436}, {19435, 19438}, {19470, 35826}, {22711, 49230}, {22768, 44607}, {22865, 49236}, {22910, 49238}, {22954, 22960}, {22965, 49246}, {26351, 44582}, {26352, 44584}, {26353, 45597}, {26354, 45595}, {26355, 44594}, {26356, 44596}, {30325, 52808}, {30333, 30413}, {31440, 37723}, {32168, 32169}, {32286, 32291}, {32297, 49264}, {32378, 32384}, {32390, 49244}, {34471, 44636}, {35762, 37525}, {35801, 37719}, {35812, 37720}, {35842, 37706}, {36443, 52217}, {36461, 52214}, {37568, 49227}, {39648, 45493}, {39822, 39823}, {39851, 39852}, {39897, 49228}, {46687, 46688}, {55441, 55579}, {55566, 56384}
X(2066) = isogonal conjugate of X(1659)
X(2066) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 1659}, {2, 2362}, {4, 13388}, {7, 7133}, {34, 56386}, {57, 7090}, {92, 2067}, {108, 54017}, {264, 53063}, {273, 5414}, {278, 30557}, {331, 53066}, {651, 58840}, {693, 54018}, {1123, 13389}, {1805, 40149}, {6213, 13390}, {13387, 16232}, {13437, 30556}, {13438, 56385}
X(2066) = X(i)-vertex conjugate of X(j) for these {i, j}: {2066, 48308}
X(2066) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 1659}, {5452, 7090}, {11517, 56386}, {13388, 85}, {22391, 2067}, {32664, 2362}, {36033, 13388}, {38983, 54017}, {38991, 58840}
X(2066) = X(i)-Ceva conjugate of X(j) for these {i, j}: {9, 5414}, {1806, 53065}, {13389, 6502}
X(2066) = X(i)-cross conjugate of X(j) for these {i, j}: {48, 5414}, {53065, 6502}
X(2066)= pole of line {37, 5414} with respect to the Feuerbach hyperbola
X(2066)= pole of line {86, 1659} with respect to the Stammler hyperbola
X(2066)= pole of line {310, 1659} with respect to the Wallace hyperbola
X(2066) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(3)}}, {{A, B, C, X(6), X(6502)}}, {{A, B, C, X(9), X(1124)}}, {{A, B, C, X(31), X(8576)}}, {{A, B, C, X(42), X(56385)}}, {{A, B, C, X(48), X(606)}}, {{A, B, C, X(55), X(13389)}}, {{A, B, C, X(63), X(493)}}, {{A, B, C, X(71), X(14121)}}, {{A, B, C, X(222), X(6415)}}, {{A, B, C, X(268), X(32556)}}, {{A, B, C, X(371), X(2193)}}, {{A, B, C, X(674), X(54019)}}, {{A, B, C, X(1336), X(2335)}}, {{A, B, C, X(1436), X(2362)}}, {{A, B, C, X(3302), X(56232)}}, {{A, B, C, X(6414), X(52431)}}, {{A, B, C, X(13390), X(14547)}}
X(2066) = barycentric product X(i)*X(j) for these (i, j): {1, 30556}, {10, 1806}, {101, 54019}, {312, 53064}, {1124, 7090}, {1331, 58838}, {3083, 7133}, {6502, 8}, {13386, 5414}, {13389, 9}, {13390, 219}, {14121, 3}, {16232, 78}, {30336, 31547}, {30413, 46376}, {30557, 6212}, {32555, 34908}, {34125, 56386}, {42013, 63}, {46744, 53066}, {53065, 75}, {54016, 6332}, {56385, 6}
X(2066) = barycentric quotient X(i)/X(j) for these (i, j): {6, 1659}, {31, 2362}, {41, 7133}, {48, 13388}, {55, 7090}, {184, 2067}, {212, 30557}, {219, 56386}, {605, 13389}, {652, 54017}, {663, 58840}, {1806, 86}, {5414, 13387}, {6502, 7}, {9247, 53063}, {13389, 85}, {13390, 331}, {14121, 264}, {16232, 273}, {30556, 75}, {30557, 46745}, {32739, 54018}, {34125, 13390}, {42013, 92}, {52425, 5414}, {53064, 57}, {53065, 1}, {53066, 6213}, {53069, 6204}, {54016, 653}, {54019, 3261}, {56385, 76}, {58838, 46107}
X(2066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1702, 16232}, {1, 371, 2067}, {3, 1124, 6502}, {3, 31474, 1124}, {6, 44591, 5416}, {11, 13901, 590}, {35, 3299, 372}, {55, 10927, 10387}, {55, 19038, 6}, {371, 35808, 1}, {371, 35817, 45641}, {371, 45643, 35775}, {497, 3068, 44623}, {1151, 3297, 56}, {1250, 7127, 5414}, {1479, 13905, 485}, {1588, 3085, 44622}, {3298, 3592, 18996}, {3301, 3746, 35809}, {5217, 18995, 1152}, {5432, 19029, 615}, {6200, 35769, 36}, {6284, 19028, 3070}, {6419, 35809, 3301}, {10576, 35803, 7741}, {10896, 13897, 42265}, {35800, 35821, 3585}
X(2067) lies on cubic K632 and on these lines: {1, 371}, {2, 31453}, {3, 1335}, {4, 44623}, {5, 9661}, {6, 41}, {11, 3071}, {12, 590}, {27, 1659}, {30, 9647}, {31, 10132}, {33, 11473}, {34, 5412}, {35, 6200}, {36, 372}, {39, 45507}, {46, 35774}, {55, 1151}, {57, 2362}, {63, 3084}, {65, 5415}, {80, 35856}, {84, 7133}, {103, 54018}, {202, 3364}, {203, 3389}, {221, 17819}, {222, 53066}, {226, 8983}, {238, 30385}, {266, 53117}, {377, 31484}, {388, 3068}, {390, 43512}, {405, 9678}, {474, 1377}, {482, 14377}, {484, 35611}, {485, 1478}, {486, 499}, {495, 8981}, {496, 42215}, {497, 6459}, {498, 5418}, {602, 3077}, {603, 606}, {611, 19145}, {615, 5433}, {956, 1378}, {999, 1124}, {1015, 5058}, {1038, 11513}, {1060, 10897}, {1152, 5204}, {1155, 49227}, {1317, 48714}, {1319, 7968}, {1388, 44636}, {1398, 5410}, {1420, 18992}, {1425, 21640}, {1466, 19000}, {1470, 19048}, {1479, 6561}, {1504, 2242}, {1583, 55410}, {1587, 4293}, {1588, 3086}, {1617, 53065}, {1697, 9616}, {1703, 15803}, {1737, 49602}, {1745, 32590}, {1773, 8938}, {1788, 19065}, {1790, 1805}, {1870, 10880}, {1914, 7353}, {2099, 44635}, {2192, 19088}, {2241, 9675}, {3023, 49212}, {3024, 49216}, {3027, 49266}, {3028, 49268}, {3057, 49226}, {3058, 41945}, {3069, 7288}, {3070, 7354}, {3085, 9540}, {3092, 11399}, {3157, 8909}, {3295, 6221}, {3297, 3304}, {3299, 5563}, {3303, 6425}, {3320, 49270}, {3361, 19004}, {3476, 19066}, {3485, 13902}, {3582, 35823}, {3583, 35802}, {3585, 6564}, {3600, 7585}, {3601, 9615}, {3614, 42582}, {3746, 6453}, {3911, 13936}, {3920, 9634}, {4294, 9541}, {4296, 11417}, {4299, 6560}, {4302, 42260}, {4309, 9681}, {4315, 49548}, {4316, 42267}, {4354, 9631}, {4428, 9688}, {4995, 52045}, {5062, 45506}, {5193, 26459}, {5217, 6409}, {5229, 31412}, {5252, 13911}, {5256, 13389}, {5261, 8972}, {5265, 7586}, {5270, 8960}, {5272, 8855}, {5275, 31464}, {5280, 45512}, {5290, 13888}, {5298, 18966}, {5393, 13478}, {5416, 19049}, {5420, 13963}, {5434, 19028}, {5687, 9679}, {5697, 35610}, {5903, 35641}, {6020, 49218}, {6199, 7373}, {6284, 42258}, {6285, 49250}, {6396, 7280}, {6422, 54416}, {6424, 16502}, {6457, 38487}, {6486, 51817}, {6565, 7741}, {6567, 45571}, {6904, 31413}, {7005, 51728}, {7031, 41410}, {7090, 34234}, {7173, 42270}, {7294, 32790}, {7352, 10665}, {7355, 12964}, {7356, 12965}, {7362, 12962}, {7583, 18990}, {7584, 15325}, {7727, 35826}, {7951, 10576}, {7972, 35882}, {8252, 13954}, {8276, 10037}, {8583, 31438}, {8976, 9654}, {9578, 13893}, {9596, 31463}, {9600, 31448}, {9650, 31481}, {9655, 13665}, {9660, 15171}, {9663, 41963}, {9674, 31451}, {9680, 31452}, {9957, 31439}, {10088, 10819}, {10106, 13883}, {10253, 39795}, {10483, 35820}, {10533, 26888}, {10535, 12970}, {10588, 32785}, {10589, 42561}, {10591, 23259}, {10895, 13898}, {10896, 23261}, {10944, 49232}, {10956, 13922}, {10961, 19372}, {10982, 41479}, {11009, 35810}, {11237, 13846}, {11241, 32065}, {11265, 32047}, {11447, 19367}, {11462, 19368}, {11509, 44590}, {11510, 44591}, {12375, 19470}, {12424, 19471}, {12688, 49234}, {12835, 44586}, {12891, 19469}, {12943, 23251}, {12949, 13882}, {12953, 42263}, {12960, 19473}, {12961, 19474}, {13045, 19475}, {13046, 19476}, {13287, 19505}, {13462, 19003}, {13901, 15888}, {13912, 31397}, {13947, 31231}, {13973, 24914}, {15170, 52047}, {15326, 42259}, {16408, 31485}, {16466, 53064}, {18244, 35871}, {18395, 35789}, {18447, 18457}, {18513, 35786}, {18915, 18923}, {18954, 44598}, {18955, 44600}, {18956, 44602}, {18957, 44604}, {18958, 44610}, {18961, 44618}, {18962, 44620}, {18963, 44627}, {18964, 44629}, {18967, 44645}, {18968, 49222}, {18969, 49214}, {18970, 49224}, {18971, 49230}, {18972, 49236}, {18973, 49238}, {18974, 49208}, {18975, 49210}, {18976, 49240}, {18977, 49242}, {18978, 49246}, {18979, 49248}, {18982, 49252}, {18983, 49254}, {18984, 49256}, {18985, 49258}, {18986, 49260}, {18987, 49262}, {18988, 44647}, {18989, 49220}, {19050, 26437}, {19175, 19183}, {19216, 54320}, {19349, 19355}, {19370, 19436}, {19371, 19438}, {19472, 22960}, {19861, 30556}, {20118, 49241}, {21842, 35762}, {23273, 47743}, {25524, 31473}, {26380, 44582}, {26404, 44584}, {26433, 45597}, {26434, 45595}, {26435, 44594}, {26436, 44596}, {26458, 37583}, {31499, 35255}, {32143, 32169}, {32243, 49264}, {32259, 32291}, {32336, 49244}, {32350, 32384}, {35765, 54428}, {35771, 37587}, {35788, 37710}, {35803, 37720}, {35812, 37719}, {35842, 37707}, {35843, 41684}, {35853, 53616}, {36279, 38235}, {36442, 52217}, {36460, 52214}, {39648, 45491}, {39815, 39823}, {39844, 39852}, {39873, 49228}, {40663, 49233}, {45287, 49601}, {46683, 46688}, {49618, 51782}, {55442, 55579}, {55566, 56427}
X(2067) = isogonal conjugate of X(14121)
X(2067) = X(i)-isoconjugate-of-X(j) for these {i, j}: {1, 14121}, {2, 42013}, {4, 30556}, {8, 16232}, {9, 13390}, {19, 56385}, {92, 2066}, {100, 58838}, {264, 53065}, {281, 13389}, {318, 6502}, {1336, 30557}, {1783, 54019}, {1806, 41013}, {4391, 54016}, {6136, 54017}, {6212, 7090}, {7017, 53064}, {7133, 13386}, {13388, 13426}, {34910, 46434}, {40700, 46379}
X(2067) = X(i)-vertex conjugate of X(j) for these {i, j}: {2067, 30336}
X(2067) = X(i)-Dao conjugate of X(j) for these {i, j}: {3, 14121}, {6, 56385}, {478, 13390}, {8054, 58838}, {13388, 46744}, {13389, 75}, {22391, 2066}, {32664, 42013}, {36033, 30556}, {39006, 54019}
X(2067) = X(i)-Ceva conjugate of X(j) for these {i, j}: {1, 6502}, {2066, 8831}, {13388, 5414}, {51841, 8833}
X(2067) = X(i)-cross conjugate of X(j) for these {i, j}: {603, 6502}, {34121, 46377}, {53066, 5414}
X(2067)= pole of line {6458, 6502} with respect to the Feuerbach hyperbola
X(2067)= pole of line {333, 14121} with respect to the Stammler hyperbola
X(2067)= pole of line {14121, 28660} with respect to the Wallace hyperbola
X(2067) = homothetic center of 1st Kenmotu diagonals triangle and anti-tangential midarc triangle
X(2067) = intersection, other than A, B, C, of circumconics {{A, B, C, X(1), X(493)}}, {{A, B, C, X(3), X(27)}}, {{A, B, C, X(6), X(2362)}}, {{A, B, C, X(41), X(53066)}}, {{A, B, C, X(48), X(606)}}, {{A, B, C, X(56), X(13388)}}, {{A, B, C, X(73), X(1659)}}, {{A, B, C, X(102), X(46434)}}, {{A, B, C, X(198), X(7133)}}, {{A, B, C, X(219), X(6415)}}, {{A, B, C, X(604), X(53063)}}, {{A, B, C, X(947), X(32556)}}, {{A, B, C, X(959), X(13387)}}, {{A, B, C, X(998), X(55505)}}, {{A, B, C, X(1036), X(7347)}}, {{A, B, C, X(1037), X(6204)}}, {{A, B, C, X(1124), X(1795)}}, {{A, B, C, X(1193), X(56386)}}, {{A, B, C, X(1437), X(5409)}}, {{A, B, C, X(2183), X(7090)}}, {{A, B, C, X(3207), X(30335)}}, {{A, B, C, X(8576), X(38252)}}, {{A, B, C, X(8679), X(54017)}}
X(2067) = barycentric product X(i)*X(j) for these (i, j): {1, 13388}, {56, 56386}, {109, 54017}, {176, 46377}, {222, 7090}, {1335, 13390}, {1659, 3}, {1805, 226}, {1813, 58840}, {2362, 63}, {4025, 54018}, {5414, 7}, {7133, 77}, {13387, 6502}, {13389, 6213}, {16232, 3084}, {30557, 57}, {42013, 52420}, {46745, 53064}, {53063, 75}, {53066, 85}
X(2067) = barycentric quotient X(i)/X(j) for these (i, j): {3, 56385}, {6, 14121}, {31, 42013}, {48, 30556}, {56, 13390}, {184, 2066}, {603, 13389}, {604, 16232}, {606, 30557}, {649, 58838}, {1335, 56386}, {1459, 54019}, {1659, 264}, {1805, 333}, {2362, 92}, {5414, 8}, {6502, 13386}, {7090, 7017}, {7133, 318}, {9247, 53065}, {13388, 75}, {13389, 46744}, {30557, 312}, {34121, 7090}, {46377, 40700}, {52411, 6502}, {53063, 1}, {53064, 6212}, {53066, 9}, {54017, 35519}, {54018, 1897}, {56386, 3596}, {58840, 46110}
X(2067) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 32556, 42013}, {1, 371, 2066}, {1, 51841, 16232}, {3, 1335, 5414}, {12, 18965, 590}, {36, 3301, 372}, {56, 18996, 6}, {57, 18991, 2362}, {371, 35768, 1}, {371, 35819, 45643}, {371, 45641, 35775}, {388, 3068, 31472}, {495, 8981, 9646}, {999, 3311, 1124}, {1151, 3298, 55}, {1400, 1468, 6502}, {1478, 13904, 485}, {1588, 3086, 44624}, {3297, 3592, 19038}, {3299, 5563, 35769}, {3304, 19038, 3297}, {3361, 19004, 51842}, {3600, 7585, 31408}, {5204, 19037, 1152}, {5433, 19027, 615}, {6199, 7373, 31474}, {6200, 35809, 35}, {6419, 35769, 3299}, {7354, 19030, 3070}, {10576, 35801, 7951}, {10895, 13898, 42265}, {35802, 35821, 3583}
X(2068) lies on cubic K673 and on these lines: {1, 4166}, {6, 366}, {43, 365}, {291, 20458}, {364, 1743}, {4497, 20469}, {17350, 40383}
X(2068) = X(i)-isoconjugate-of-X(j) for these {i, j}: {291, 2069}
X(2068) = X(i)-Dao conjugate of X(j) for these {i, j}: {39029, 2069}
X(2068) = intersection, other than A, B, C, of circumconics {{A, B, C, X(291), X(366)}}, {{A, B, C, X(294), X(4166)}}, {{A, B, C, X(365), X(20332)}}
X(2068) = barycentric quotient X(i)/X(j) for these (i, j): {1914, 2069}
X(2068) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 366, 2069}
X(2069) lies on cubic K673 and on these lines: {1, 364}, {6, 366}, {43, 4166}, {81, 40378}, {256, 20458}, {4393, 40383}, {4471, 20469}
X(2069) = X(i)-isoconjugate-of-X(j) for these {i, j}: {291, 2068}
X(2069) = X(i)-Dao conjugate of X(j) for these {i, j}: {39029, 2068}
X(2069) = intersection, other than A, B, C, of circumconics {{A, B, C, X(81), X(365)}}, {{A, B, C, X(256), X(366)}}
X(2069) = barycentric quotient X(i)/X(j) for these (i, j): {1914, 2068}
X(2069) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 366, 2068}
Inverses in Circumcircle, 2070 -2080
Suppose X is a point in the plane of triangle ABC, other than the circumcenter O and not on the line at infinity. The inverse-in-circumcircle of X is the point X' on ray OX whose distance from O is R2/|OX|2. In normalized trilinears, if
X = (x, y, z) O = (x3, y3, z3), R = circumradius,
then X' is given as (f(a,b,c), f(b,c,a), f(c,a,b)) by f(a,b,c) = x3 + (x - x3)(R/|OX|)2.
In this section, representations using the circumradius R, inradius r, Brocard angle ω and J as defined at X(1113), were contributed by Peter J. C . Moses.
As a point on the Euler line, X(2070) has Shinagawa coefficients (J2 - 2, -J2 + 6).
X(2070) lies on these lines: 2,3 49,52 51,567 54,143 98,1287 110,1154 184,568 187,2079 231,1989 399,1495 476,1141 500,501 827,842
X(2070) = midpoint of X(23) and X(186)
X(2070) = reflection of X(i) in X(j) for these (i,j): (3,186), (2072,468), (18859,3)
X(2070) = isogonal conjugate of X(33565)
X(2070) = anticomplement of X(37938)
X(2070) = circumcircle-inverse of X(5)
X(2070) = X(94)-Ceva conjugate of X(6)
X(2070) = crosspoint of X(i) and X(j) for these (i,j): (250,476), (1141,1166)
X(2070) = crosssum of X(i) and X(j) for these (i,j): (125,526), (1154,1209)
X(2070) = center of circle {{X(23),X(186),PU(4)}}, which is the inverse-in-circumcircle-of-orthocentroidal-circle
X(2070) = inverse-in-tangential-circle of X(3)
X(2070) = pole wrt circumcircle of line PU(5) (line X(5)X(523))
X(2070) = tangential isogonal conjugate of X(399)
X(2070) = X(36)-of-tangential-triangle if ABC is acute
X(2070) = {X(3),X(26)}-harmonic conjugate of X(2937)
X(2070) = {X(3),X(23)}-harmonic conjugate of X(37924)
X(2070) = crossdifference of every pair of points on line X(570)X(647)
As a point on the Euler line, X(2071) has Shinagawa coefficients (J2 + 1, -J2 - 3).
Let A'B'C' be the antipedal triangle of X(64). The circumcircles of AA'X(64), BB'X(64), CC'X(64) concur in two points: X(64) and X(2071). Also, let A'B'C' be the Trinh triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2071). (Randy Hutson, October 13, 2015)
Let P and Q be circumcircle antipodes. X(2071) is the Euler line intercept, other than X(4), of circle {{X(4),P,Q}} for all P, Q. (Randy Hutson, August 28, 2020)
X(2071) lies on these lines: 2,3 64,2063 74,323 99,1236 476,1294 477,925 691,1297 1290,1295
X(2071) = reflection of X(i) in X(j) for these (i,j): (4,2072), (23,186), (186,3)
X(2071) = anticomplement of X(403)
X(2071) = circumcircle-inverse of X(20)
X(2071) = anticomplementary-circle-inverse of X(37444)
X(2071) = {X(3),X(23)}-harmonic conjugate of X(37952)
X(2071) = {X(14807),X(14808)}-harmonic conjugate of X(37444)
X(2071) = crosspoint of X(3) and X(2935) wrt both the excentral and tangential triangles
X(2071) = crossdifference of every pair of points on line X647)X(800)
X(2071) = inverse-in-de-Longchamps-circle of X(4)
X(2071) = inverse-in-first-Droz-Farny-circle of X(5)
X(2071) = X(36) of Trinh triangle if ABC is acute
X(2071) = Trinh-isogonal conjugate of X(74)
X(2071) = antigonal conjugate of X(34170)
X(2071) = insimilicenter of circumcircle and Trinh circle (the exsimilicenter is X(3520))
As a point on the Euler line, X(2072) has Shinagawa coefficients (J2 + 1, J2 - 3).
X(2072) lies on these lines: 2,3 125,1568 127,625 325,339 842,1286
X(2072) = midpoint of X(i) and X(j) for these (i,j): (4,2071), (125,1568), (403,858)
X(2072) = reflection of X(i) in X(j) for these (i,j): (403,5), (2070,468)
X(2072) = isogonal conjugate of X(38534)
X(2072) = complement of X(186)
X(2072) = complementary conjugate of X(1511)
X(2072) = crosssum of X(i) and X(j) for these {i,j}: {6, 34397}, {184, 3003}
X(2072) = crosspoint of X(i) and X(j) for these {i,j}: {2, 328}, {264, 2986}
X(2072) = circumcircle-inverse of X(26)
X(2072) = nine-point-circle-inverse of X(3)
X(2072) = crosspoint of X(2) and X(328)
As a point on the Euler line, X(2073) has Shinagawa coefficients (EF(R/|OX|)2 - ($a$$bc$ - abc)F{1 - (R/|OX|)2}, -[(E + F) + $bc$]E(R/|OX|)2 + ($a$$bc$ - abc)F{1 - (R/|OX|)2}).
X(2073) lies on these lines: 2,3 35,270 103,1304 110,916 476,917 675,935
X(2073) = isogonal conjugate of X(38535)
X(2073) = circumcircle-inverse of X(27)
X(2073) = trilinear quotient X(i)/X(j) for these (i,j): (162, 2690), (2774, 656)
As a point on the Euler line, X(2074) has Shinagawa coefficients (abc$a$F(R/|OX|)2 - [2$a$(E + F) - $a3$ + 2abc]FS{1 - (R/|OX|)2}, -[abc$a$(E+F) + ES2](R/|OX|)2 + [2$a$(E+F) - $a3$ + 2abc]FS{1 - (R/|OX|)2}).
X(2074) lies on these lines: 2,3 104,1304 105,935 110,912 162,1870 476,915
X(2074) = inverse-in-circumcircle of X(28)
As a point on the Euler line, X(2075) has Shinagawa coefficients ([$a$S2 - $aSBSC$ - 2$aSA$F]FS2{1 - (R/|OX|)2} + abc[2F2S2 + (E - 2F)FS2](R/|OX|)2, -[$a$S2 + $aSBSC$ + 2$aSA$F]FS2{1 - (R/|OX|)2} + abc[2F2S2 + {(E - 2F)FS2 - E$bcSBSC$}](R/|OX|)2).
X(2075) lies on these lines: 2,3 36,162 102,1304 250,1101 935,1311
X(2075) = inverse-in-circumcircle of X(29)
Let T1 be the circumcevian triangle of the 1st Brocard point, and let T2 be the circumcevian triangle of the 2nd Brocard point. Let VAVBVC be the vertex triangle of T1 and T2, and let SASBSC be the side triangle of T1 and T2. The lines VASA, VBSB, VCSC concur in X(2076).
Let T be the reflection of the symmedial triangle in the Lemoine axis; i.e., the reflection of the cevian triangle of X(6) in the trilinear polar of X(6). The triangle T is perspective to ABC, and the perspector is X(2076. (Randy Hutson, February 16, 2015)
X(2076) lies on these lines: 3,6 22,1613 99,732 141,384 385,698 599,1003 691,755 733,805 904,1964
X(2076) = reflection of X(i) in X(j) for these (i,j): (6,1691), (1691,187)
X(2076) = inverse-in-circumcircle of X(39)
X(2076) = X(694)-Ceva conjugate of X(6)
X(2076) = crosspoint of X(249) and X(805)
X(2076) = crosssum of X(115) and X(804)
X(2076) = reflection of X(1691) in the Lemoine axis
X(2076) = radical center of Lucas(-csc 2ω) circles
X(2076) = intersection of tangents to circumcircle at intersections with line X(39)X(512)
X(2076) = crossdifference of every pair of points on line X(523)X(3589)
X(2076) = {X(3),X(6)}-harmonic conjugate of X(5116)
X(2076) = perspector of tangential triangle and the vertex triangle of the anticevian triangles of PU(1)
X(2076) = Brocard axis intercept, other than X(3), of circle {X(3),PU(2)}
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Let P and Q be circumcircle antipodes. X(2077) is the line X(1)X(3) intercept, other than X(1), of circle {{X(1),P,Q}}. (Randy Hutson, August 28, 2020)
X(2077) lies on these lines: 1,3 24,1753 30,119 78,1158 84,1259 100,515 102,901 104,519 122,856 376,535 386,601 404,946 516,1519 912,1768 953,1293 972,1308 1012,1376 1593,1878 1618,1818
X(2077) = reflection of X(36) in X(3)
X(2077) = inverse-in-circumcircle of X(40)
X(2077) = X(186) of 1st circumperp triangle
X(2077) = crossdifference of every pair of points on line X(650)X(1108)
X(2077) = {X(1),X(3)}-harmonic conjugate of X(37561)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2078) lies on these lines: 1,3 31,2003 59,672 73,595 105,2006 109,840 226,1005 388,535 581,1497 901,1477 1174,1202 1279,1421 1283,1284 1308,1323
X(2078) = isogonal conjugate of X(3254)
X(2078) = inverse-in-circumcircle of X(57)
X(2078) = crosssum of X(142) and X(527)
X(2078) = inverse-in-{circumcircle, incircle}-inverter of X(65)
X(2078) = {X(3513),X(3514)}-harmonic conjugate of X(36)
X(2079) lies on the tangential circle and these lines: 3,115 6,1511 22,111 24,112 25,1560 186,230 187,2070
X(2079) = inverse-in-circumcircle of X(115)
X(2079) = circumperp conjugate of X(38738)
X(2079) = Stammler-circle-inverse of X(38733)
X(2079) = center of bicevian conic of X(1113) and X(1114)
The locus of X(23) in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(2080) and passing through X(23), X(385) and X(11676). This circle is the reflection of the circumcircle in X(187). (Randy Hutson, August 29, 2018)
X(2080) lies on these lines: 3,6 5,316 30,98 83,140 110,237 114,754 625,1656
X(2080) = reflection of X(i) in X(j) for these (i,j): (3,187), (316,5)
X(2080) = isogonal conjugate of isotomic conjugate of X(39099)
X(2080) = inverse-in-circumcircle of X(182)
X(2080) = reflection of X(3) in the Lemoine axis
X(2080) = pole of line X(23)X(352) wrt Parry circle
X(2080) = crossdifference of every pair of points on line X(523)X(3815)
X(2080) = {X(1687),X(1688)}-harmonic conjugate of X(187)
X(2080) = harmonic center of circumcircle and Ehrmann circle
X(2080) = pole of Lemoine axis wrt circle {{X(1687),X(1688),PU(1),PU(2)}}
X(2080) = radical trace of circumcircle and 9th Lozada circle
X(2080) = intersection of Brocard axes of ABC and Artzt triangle
X(2080) = X(23)-of-X(3)PU(1)
PK and NK Transforms, 2081 - 2088
On October 1, 2003, Bernard Gibert described two transforms, given for X = u : v : w (barycentrics) by
PK(X) = a2(c2v2 - b2w2)u : b2(a2w2 - c2u2)v : c2(b2u2 - a2v2)w;
NK(X) = a2(c2v2 + b2w2)u : b2(a2w2 + c2u2)v : c2(b2u2 + a2v2)w.
Let X -1 denote the isogonal conjugate of X. Then PK(X) is the point of intersection of the trilinear polar of X and the trilinear polar of X -1, and NK(P) is the pole of the line XX -1 with respect to the conic that passes through points A, B, C, X, and X -1.
Also, PK(X) is the crossdifference, and NK(X) the crosssum, of X and X -1.
The transform PK carries any point on the cubic pK(X6,X) to the trilinear polar of X -1, and NK carries any point on the cubic nK0(X6,X) to the trilinear polar of X -1. (The notations for these cubics are given at Bernard Gibert's Cubics in the Triangle Plane.)
PK(X(i)) = X(j) for these (i,j):
(2,512), (3,647), (4,647), (6,512), (9,663), (19,810), (57,663),
(63,810)
NK(X(i)) = X(j) for these (i,j):
(1,1), (2,30), (3,185), (4,185), (6,39), (98,446), (511,446)
In trilinears: suppose X = x : y : z; then
PK(X) = (y2 - z2)x : (z2 - x2)y : (x2 - y2)z;NK(X) = (y2 + z2)x : (z2 + x2)y : (x2 + y2)z.
Z(X2) passes through X(i) for I = 1,2,3,4,6,9,57, 223,
282, 1073, 1249.
Z(X4) passes through X(i) for I =
1,3,4,46,90,155,254,371,372,485,486,487,488.
Z(X5) passes through X(i) for I =
1,3,4,5,17,18,54,61,62,195,627,628.
Z(X20) passes through X(i) for I =
1,3,4,20,40,64,84,1490,1498.
Z(X30) passes through X(i) for I = 1,3,4,13,14,15,16, 30,
74, 399, 484, 616,617,1138,1157,1263,1276,1277,1337,1338.
Z(X98) passes through X(i) for I =
1,98,511,1687,1688,1756,2009,2010.
Z(X99) passes through X(i) for I =
1,39,83,99,512,1018,1019,1379,1380.
Z(X100) passes through X(i) for I = 1,100,513,1381,1382.
Z(X110) passes through X(i) for I =
1,5,54,110,523,1113,1114.
Z+(X1X6) passes through X(i) for I
= 1,44,88,239,241,292,294,1931.
Z+(X3X6) passes through X(i) for I =
2,6,13,14,15,16,111,524
Z+(X1X2) passes through X(i) for I =
1,238,291,899
Z+(X1X3) passes through X(i) for I =
1,105,243,296,518,1155,1156
Additional pass-through points for these and other cubics are given at Bernard Gibert's Cubics in the Triangle Plane.
X(2081) lies on these lines: 50,647 115,125
X(2081) = X(50)-Ceva conjugate of X(2088)
X(2081) = crosspoint of X(94) and X(110)
X(2081) = crosssum of X(50) and X(523)
X(2082) lies on these lines: 1,41 6,19 8,9 32,1951 40,672 55,1212 57,279 63,194 85,673 198,1108 213,1572 218,517 284,1800 354,1190 573,1723 579,1195 604,610 614,1184 1039,1172 1055,1420 1146,1837 1743,1766
X(2082) = isogonal conjugate of X(7131)
X(2082) = X(664)-Ceva conjugate of X(663)
X(2082) = crosspoint of X(i) and X(j) for these (i,j): (4,277), (9,57), (333,1172)
X(2082) = crosssum of X(i) and X(j) for these (i,j): (3,218), (9,57), (664,663)
X(2082) = bicentric sum of PU(115)
X(2082) = PU(115)-harmonic conjugate of X(663)
X(2082) = eigencenter of cevian triangle of X(664)
X(2082) = eigencenter of anticevian triangle of X(663)
X(2083) lies on these lines: 1,163 19,158 48,1820 63,304 774,1973 920,1755 1497,1953 1760,1966 1821,1969
X(2083) = X(811)-Ceva conjugate of X(810)
X(2083) = crosspoint of X(19) and X(63)
X(2083) = crosssum of X(19) and X(63)
Let L be the isogonal conjugate of the isotomic conjugate of the antiorthic axis (i.e., line X(667)X(788)). Let M be the isotomic conjugate of the isogonal conjugate of the antiorthic axis (i.e., line X(514)X(661)). Then X(2084) = L∩M. (Randy Hutson, March 21, 2019)
X(2084) lies on these lines: 512,1500 514,661 667,788
X(2084) = isotomic conjugate of X(37204)
X(2084) = X(i)-cross conjugate of X(j) for these (i,j): (782,880), (881,783)
X(2084) = crosspoint of X(661) and X(798)
X(2084) = crosssum of X(i) and X(j) for these (i,j): (662,799), (2084,2085)
X(2084) = isogonal conjugate of X(4593)
X(2084) = crossdifference of every pair of points on line X(31)X(75)
X(2085) lies on these lines: 1,1581 31,1917 75,1928 1755,1923
X(2085) = isogonal conjugate of X(38847)
X(2085) = complement of X(38844)
X(2085) = crosspoint of X(31) and X(75)
X(2085) = crosssum of X(31) and X(75)
X(2086) lies on these lines: 2,6 351,865 671,729
X(2086) = isogonal conjugate of X(39292)
X(2086) = crossdifference of every pair of points on line X(99)X(512)
X(2086) = X(i)-Ceva conjugate of X(j) for these (i,j): (98,669), (694,512)
X(2086) = crosspoint of X(i) and X(j) for these {i,j}: {6, 2422}, {512, 694}, {804, 3978}, {1691, 5027}, {2489, 6531}
X(2086) = crosssum of X(i) and X(j) for these {i,j}: {2, 2396}, {99, 385}, {805, 9468}, {1916, 18829}, {2086, 14824}, {4563, 36212}
X(2086) = trilinear product X(i)*X(j) for these {i,j}: {115, 1933}, {238, 21823}, {661, 5027}, {740, 21755}, {798, 804}, {1084, 1966}, {1109, 14602}, {1580, 3124}, {1691, 2643}, {1910, 2679}, {1914, 21725}, {1924, 14295}, {1926, 9427}, {1967, 35078}, {2238, 4128}, {3121, 4039}, {3747, 16592}, {3978, 4117}, {4079, 4164}, {4155, 20981}, {7234, 21832}, {18902, 23994}
X(2087) lies on these lines: 1,6 244,665 1914,1983
X(2087) = isogonal conjugate of X(5376)
X(2087) = X(i)-Ceva conjugate of X(j) for these (i,j): (44,1635), (88,513), (104,667), (1022,764), (1319,1960)
X(2087) = crosspoint of X(i) and X(j) for these (i,j): (1,1022), (44,1635), (88,513)
X(2087) = crosssum of X(i) and X(j) for these (i,j): (1,1023), (44,100)
X(2087) = bicentric difference of PU(114)
X(2087) = PU(114)-harmonic conjugate of X(6161)
X(2088) lies on these lines: 3,6 115,125
X(2088) = isogonal conjugate of X(39295)
X(2088) = X(i)-Ceva conjugate of X(j) for these (i,j): (50,2081), (74,512), (94,523), (323,526)
X(2088) = crosspoint of X(i) and X(j) for these (i,j): (94,523), (323,526)
X(2088) = crosssum of X(i) and X(j) for these (i,j): (50,110), (112,403), (476,1989)
X(2088) = Brocard-circle-inverse of X(32761)
Let A', B', C' be the first points of intersection of the angle bisectors of triangle ABC with its incircle. Let A" B" C" be the triangle formed by the lines tangent to the incircle at A', B', C'. Then A"B"C" is perspective to the intouch triangle of ABC, and the perspector is X(2089). (Darij Grinberg, Hyacinthos #8072, 10/01/03)
X(2089) lies on these lines: 1,167 2,178 7,1488
X(2089) = X(7)-Ceva conjugate of X(174)
X(2089) = X(173)-cross conjugate of X(174)
X(2089) = X(56)-of-intouch triangle
X(2089) = X(24)-of-mid-arc-triangle
X(2089) = intouch-isogonal conjugate of X(17641)
X(2089) = SS(A→A')-of-X(8), where A'B'C' is the excentral triangle
X(2089) = trilinear pole of perspectrix of ABC and Yff central triangle
The bisector AX(1) of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the medial triangle, and the perspector is X(2090). (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)
X(2090) lies on these lines: 2,174 8,188 9,362 85,555 177,178 312,556
X(2090) = complement of X(174)
X(2090) = crosspoint of X(2) and X(556)
The bisector AX(1) of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the intouch triangle, and the perspector is X(2091). (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)
X(2091) lies on this line: {7,4146}, {57,173}
X(2091) = X(19)-of-intouch-triangle
Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X(6), and the perspector is X(2092). (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X(1); see X(1682).)
X(2092) lies on these lines: {1,2277}, {2,314}, {3,6}, {4,3597}, {9,43}, {10,37}, {41,2273}, {42,181}, {44,3647}, {69,980}, {71,213}, {100,2298}, {115,119}, {142,3752}, {192,3596}, {214,1015}, {232,1172}, {291,2663}, {407,1880}, {442,1738}, {538,3770}, {656,3126}, {665,2642}, {851,2285}, {869,3688}, {900,2511}, {966,5283}, {968,4204}, {992,3216}, {1017,3031}, {1107,3686}, {1193,1682}, {1194,5276}, {1196,5275}, {1211,3666}, {1449,2275}, {2171,4642}, {2178,2242}, {2183,2653}, {2240,5279}, {2294,3125}, {2303,5277}, {2309,3271}, {2610,3310}, {2667,3122}, {3030,3124}, {3136,3914}, {3247,4050}, {3454,3821}, {3662,3936}, {3728,4111}, {3949,3954}, {4016,4053}
X(2092) = midpoint of X(i) and X(j) for these (i,j): (256, 1045), (2019,2020)
X(2092) = isogonal conjugate of X(14534)
X(2092) = complement of X(314)
X(2092) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,960), (100,512), (1415,647)
X(2092) = crosspoint of X(i) and X(j) for these (i,j): (2,65), (6,37), (429,1211)
X(2092) = crosssum of X(i) and X(j) for these (i,j): (2,81), (6,21), (58,572), (1169,1798)
X(2092) = perspector of circumconic centered at X(960)
X(2092) = polar conjugate of isotomic conjugate of X(22076)
X(2092) = center of circumconic that is locus of trilinear poles of lines passing through X(960)
X(2092) = harmonic center of Apollonius and Gallatly circles
Reflections, 2093 - 2105
(au - bv - cw)x + 2u(by + cz) : (bv - cw - au)y + 2v(cz + ax) : (cw - au - bv)z + 2w(ax + by).
Thus, if x = x(a,b,c), y = x(b,c,a), z = x(c,a,b) and u = u(a,b,c), v = u(b,c,a), w = u(c,a,b), then the reflection of X in U is given by f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (au - bv - cw)x + 2u(by + cz).
In this section, several coordinate representations were contributed by Peter J. C . Moses.
Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the 2nd isogonal triangle of X(1) at X(2093). (Randy Hutson, December 2, 2017)
X(2093) lies on these lines: 1,3 10,329 19,1743 72,1706 196,1785 200,758 515,2096 518,2097 519,2094 527,1478 553,1056 946,1788 962,1210 1046,1719 1103,1254 1699,1737
X(2093) = reflection of X(i) in X(j) for these (i,j) : (1,57), (329,10)
X(2093) = X(25)- of-reflection-triangle-of-X(1)
X(2094) lies on these lines: 2,7 30,2095 189,1121 376,517 519,2093 524,2087 999,1621
X(2094) = reflection of X(i) in X(j) for these (i,j): (2,57), (329,2)
X(2094) = complement of X(31142)
X(2095) lies on these lines: 1,3 5,329 30,2094 381,527 511,2097
X(2095) = reflection of X(i) in X(j) for these (i,j) : (3,57), (329,5)
X(2096) lies on these lines: 3,329 4,57 7,104 20,145 30,2094 376,527 388,1158 515,2093 1478,1768 1503,2097
X(2096) = reflection of X(i) in X(j) for these (i,j): (4,57), (329,3)
X(2096) = anticomplement of X(37822)
X(2097) lies on these lines: 6,57 19,1122 141,329 511,2095 517,990 518,2093 524,2094 527,599 1503,2096
X(2097) = reflection of X(i) in X(j) for these (i,j): (6,57), (329,141)
X(2098) lies on these lines: 1,3 8,11 21,1392 33,1828 145,497 219,1731 499,1387 519,1837 944,1317 952,1479 1000,1389 1108,1732
X(2098) = reflection of X(i) in X(j) for these (i,j): (8,1329),
(56,1)
X(2098) = crosssum of X(1) and X(1420)
X(2098) = {X(1),X(40)}-harmonic conjugate of X(1319)
X(2098) = homothetic center of intangents triangle and reflection of tangential triangle in X(1)
X(2098) = homothetic center of Mandart-incircle triangle and 5th mixtilinear triangle
X(2098) = X(56)-of-Mandart-incircle-triangle
X(2098) = X(56)-of-5th-mixtilinear-triangle
X(2098) = Ursa-major-to-Ursa-minor similarity image of X(8)
X(2099) lies on these lines: 1,3 4,1389 6,1411 7,528 8,12 33,1875 34,1824 42,1457 45,1405 80,381 145,388 226,519 515,1836 674,1469 758,956 946,1837 952,1478 959,1255 1149,1450 1392,1476 1399,1468
X(2099) = reflection of X(55) in X(1)
X(2099) = isogonal conjugate of X(2320)
X(2099) = anticomplement of complementary conjugate of X(17057)
X(2099) = {X(1),X(1155)}-harmonic conjugate of X(3576)
X(2099) = {X(1),X(1159)}-harmonic conjugate of X(4860)
X(2100) lies on the Bevan circle and these lines: 1,1113 30,40 165,1114 1312,1699 1313,1698
X(2100) = reflection of X(i) in X(j) for these (i,j): (1,1113), (2101,40)
X(2100) = X(188)-aleph conjugate of X(2575)
X(2101) lies on the Bevan circle and these lines: 1,1114 30,40 165,1113 1312,1698 1313,1699
X(2101) = reflection of X(i) in X(j) for these (i,j): (1,1114),
(2100,40)
X(2101) = X(188)-aleph conjugate of X(2574)
X(2102) lies on these lines: 1,1113 8,1313 30,944 517,1114 518,2104
X(2102) = reflection of X(i) in X(j) for these (i,j): (8,1313), (1113,1), (2103,1482)
X(2103) lies on these lines: 1,1114 8,1312 30,944 517,1113 518,2105
X(2103) = reflection of X(i) in X(j) for these (i,j): (8,1312), (1114,1), (2102,1482)
X(2104) lies on these lines: 6,1113 30,1351 69,1313 511,1114 518,2102
X(2104) = reflection of X(i) in X(j) for these (i,j) : (69,1313), (1113,6), (2105,1351)
X(2105) lies on these lines: 6,1114 30,1351 69,1312 511,1113 518,2103
X(2105) = reflection of X(i) in X(j) for these (i,j): (69,1312), (1114,6), (2104,1351)
Centers 2106 -2119
For any point P on EAC2, the X(2)-isoconjugate of P is also on EAC2. In the following table, for each (i,j), the centers X(i) and X(j) are on EAC2 and are an X(2)-isoconjugate pair. Each pair are collinear with the pivot X(238).
| I | J |
|---|---|
| 1 | 6 |
| 2 | 31 |
| 105 | 672 |
| 238 | 292 |
| 365 | 365 |
| 1423 | 2053 |
| 1931 | 2054 |
| 2106 | 2107 |
| 2108 | 2109 |
| 2110 | 2111 |
| 2112 | 2113 |
| 2114 | 2115 |
| 2116 | 2117 |
| 2118 | 2119 |
| 2144 | 2145 |
| 2146 | 2147 |
| 3009 | 20332 |
Triangle centers on a cubic yield non-central points on the cubic; e.g., if P and P' are on EAC2, then the line of PP' meets EAC2 in third point, L(P,P'), on EAC2. (Possibly L(P,P') is P or P'.) If A', B', C' are central triangle vertices and X is a triangle center, and all four points A', B', C', and X lie on EAC2, then the points
L(A',X), L(B',X), L(C',X)
are central triangle vertices on EAC2. (A definition of central triangle is given in the Glossary.) In the following table, column 1 indicates the A-vertex of such a triangle; column 2 two points collinear with that vertex; and column 3, the A-vertex of the X(2)-isoconjugate of the triangle. Taking row 1 as an example, - a : c : represents the point -a : c : b, which is the A-vertex of the central triangle with B-vertex c : - b : a and C-vertex b : a : - c. The A-vertex is collinear with A and X(2) (so that the B-vertex is collinear with B and X(2), etc.). Then, in column 3, - bc : a2b : represents the A-vertex of the X(2)-isoconjugate of - a : c : . The vertices - a : c : and - bc : a2b : are collinear with the pivot, X(238).
| A-vertex | Line | A-vertex (iso-) |
|---|---|---|
| - a : c : | A, X(2) | - bc : b2 : |
| - a1/2 : b1/2 : | A, X(365) | - a1/2 : b1/2 : |
| - a(a + b + c) : bc + ca + ab : | A, X(1) | - 1/(a + b + c) : b/(bc + ca + ab) : |
| bc + ab + ac : - (a + b + c)b : | A, X(6) | a/(bc + ab + ac) : - 1/(a + b + c) : |
| - bc : b2 : | A, X(31) | - a : c : |
| a(a + b + c) : a2 + b2 - (a + b)c: | X(6), - a : c : | 1/(a + b + c) : b/[a2 + b2 - (a + b)c] : |
| 2a(b2 + ca)(c2 + ab) : (b2 + ca)(a3 + b3 - c3 - abc) : | X(31), - a : c : | 1/[2a(b2 + ca)(c2 + ab)] : b/[(b2 + ca)(a3 + b3 - c3 - abc)] : |
| (a+b)(a+c)(a2+b2+c2+bc+ca+ab) : b(a+c)(a2+b2+ab-c(a+b+c)) : | X(1), - a : c : |
a/[(a+b)(a+c)(a2+b2+c2+bc+ca+ab)] : 1/[(a+c)(a2+b2+ab-c(a+b+c))] : |
| a(bc + ab + ac) : (c - b)a2 - b2a + b2c : | X(1), - bc : b2 : | 1/[(bc + ab + ac) : b/[(c - b)a2 - b2a + b2c] : |
| 2a2bc(b2 + ca)(c2 + ba) : (b2 + ca)(b3c3 + a3c3 - a3b3 - a2b2c2) : |
X(2), - bc : b2 : | 1/[2abc(b2 + ca)(c2 + ba)] : b/[(b2 + ca)(b3c3 + a3c3 - a3b3 - a2b2c2)] : |
| (a+b)(a+c)[(b2+bc+c2)a2 +
abc(b+c) + b2c2] : b(a+c)[(c2-bc-b2)a2 + abc(c-b) + b2c2] : |
X(6), - bc : b2 : | 1/[(a+b)(a+c)[(b2+bc+c2)a2
+ abc(b+c) + b2c2]] : 1/[(a+c)[(c2-bc-b2)a2 + abc(c-b) + b2c2]] : |
X(2106) lies on these lines: 2,6 105,805 213,1509 238,741 292,1931 662,1914 873,894 1621,1964 2111,2112 2114,2117
X(2106) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,1931),
(741,81)
X(2106) = X(2664)-cross conjugate of X(2669)
X(2107) lies on these lines: 1,1655 31,1979 238,741 672,1967 1423,2116
X(2107) = isogonal conjugate of X(2669)
X(2107) = cevapoint of X(2663) and X(2664)
X(2107) = X(i)-cross conjugate of X(j) for these (i,j): (292,2054),
(2238,42)
X(2108) lies on these lines: 1,39 2,846 3,2053 31,43 42,1051 170,411 238,1575 672,1282 1011,1283
X(2108) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,1), (1575,43)
X(2108) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1757), (100,1026), (238,2108), (365,2664)
X(2109) lies on these lines: 238,1575 1914,2112
X(2109) = isogonal conjugate of X(33888)
X(2109) = trilinear pole of line X(6373)X(8632)
X(2109) = X(i)-cross conjugate of X(j) for these (i,j): (292,6), (727,2162), (1929,2248)
X(2109) = X(92)-isoconjugate of X(20797)
X(2110) lies on these lines: 2,11 6,292 31,1979 238,2111 365,2119 672,2117 1185,1977 1423,1740 1931,2113
X(2110) = isogonal conjugate of anticomplement of X(36906)
X(2110) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,6), (2223,55)
X(2111) lies on these lines: 238,2110 239,672 2106,2112
X(2111) = X(i)-cross conjugate of X(j) for these (i,j): (292,1), (673,57)
X(2112) lies on these lines: 1,41 6,2054 31,292 172,1201 238,2113 604,651 672,2115 1015,1468 1914,2109 2106,2111
X(2112) = complement of X(20353)
X(2112) = anticomplement of X(20341)
X(2112) = X(238)-Ceva conjugate of X(31)
X(2113) lies on these lines: 238,2112 672,1282 1458,2114 1931,2110
X(2113) = X(292)-cross conjugate of X(2)
X(2113) = X(19)-isoconjugate of X(20742)
X(2114) lies on these lines: 9,77 31,57 238,241 269,292 1279,1429 1458,2113 2106,2117
X(2115) lies on these lines: 238,241 672,2112 1931,2116
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2116) lies on these lines: 6,105 238,2117 1423,2107 1931,2115
X(2117) lies on these lines: 238,2116 672,2110 2106,2114
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2118) lies on these lines: 6,365 238,2119 2108,2147 2111,2146
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2119) lies on these lines: 1,2146 238,2118 365,2110
Eigencenters I, 2120 -2143
The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the cevian triangle of a point U = u : v : w, the eigencenter is given by
f(u,v,w) : f(v,w,u) : f(w,u,v), where f(u,v,w) = vw(u2v2 + u2w2 - v2w2).
This point is now named the eigentransform of U, denoted by ET(U). Pairs (i,j) such that X(j) = ET(X(i)) include
(1,1), (2,3), (3,1075), (4,155), (6,194), (7,218), (13,62), (14,61), (63,1712), (88,1), (92,47), (94,49), (99,39), (100,1), (110,5), (162,1), (174,266), (190,1), (648,185), (651,1), (653,1), (655,1), (658,1), (660,1), (662,1), (664,2082), (673,1), (694,384), (771,1), (799,1), (811,2083), (823,1), (897,1), (1113,3), (1114,3), (1156,1), (1492,1), (1821,1), (1942,1941)
For any point P not on a sideline of triangle ABC, let P -1 denote the isogonal conjugate of P. Easily verified properties of eigentransforms include the following.
1. ET(U) = U-Ceva conjugate of U -1.For discussions and generalizations using barycentric coordinates, see Section 1.4 of "Special Isocubics in the Triangle Plane," downloadable from Bernard Gibert's Cubics in the Triangle Plane. The cubic Z(U) is there denoted by pK(X6,U), and ET(U) is = (cevian quotient of U and U*) = (U-Ceva conjugate of U -1); ET(U) is the tangential of U* in pK(X6,U).2. ET(U) = X(1) if and only if U = X(1) or U lies on the Steiner circumellipse: yz + zx + xy = 0.
3. ET(U) = eigencenter of the anticevian triangle of U -1.
4. ET(U) is on the self-isogonal cubic Z(U). That is, if x : y : z denotes a trilinear variable point, then ET(U) is on the cubic
ux(y2 - z2) + vy(z2 - x2) + wz(x2 - y2) = 0.
5. The U-Ceva conjugate of every X on Z(U) is on Z(U); thus, the U-Ceva conjugate, V, of X(1), lies on Z(U); trilinears for V -1 are 1/(1/v + 1/w - 1/u) : 1/(1/w + 1/u - 1/v) : 1/(1/u + 1/v - 1/w);
ET(U) is the third point of intersection of the line X(1)-to-V -1 and the cubic Z(U).6. As U is the pivot of Z(U), the following points on Z(U) are collinear: U, ET(U), [ET(U)] -1 .
This section was added to ETC on 10/(14-17)/03.
X(2120) lies on the Napoleon cubic and these lines: 1,3461 3,3463 4,1157 5,2121 3467,3469
X(2120) = isogonal conjugate of X(2121)
X(2120) = X(5)-Ceva conjugate of X(54)
X(2121) lies on the Napoleon cubic and these lines: 5,2120 195,3462
X(2121) = isogonal conjugate of X(2120)
X(2121) = X(54)-cross conjugate of X(5)
X(2122) lies on the cubic K692 and these lines: 1,84 8,2123 34,1767 109,1035 603,1450 608,1249 1406,1456
X(2122) = isogonal conjugate of X(2123)
X(2122) = X(8)-Ceva conjugate of X(56)
X(2123) lies on the cubic K692 and this line: 8,2122
X(2123) = isogonal conjugate of X(2122)
X(2123) = X(56)-cross conjugate of X(89)
X(2124) lies on the cubic K351 and these lines: 1,971 8,2125 57,279 223,1212 664,728
X(2124) = isogonal conjugate of X(2125)
X(2124) = anticomplement of X(32446)
X(2124) = X(9)-Ceva conjugate of X(57)
X(2124) = perspector of 7th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle
X(2125) lies on the cubic K351 and these lines: 9,2124 165,220
X(2125) = isogonal conjugate of X(2124)
X(2125) = X(57)-cross conjugate of X(9)
X(2126) lies on these lines: 1,229 5,572 10,2127 49,970
X(2126) = isogonal conjugate of X(2127)
X(2126) = X(10)-Ceva conjugate of X(58)
X(2127) lies on this line: 10,2126
X(2127) = isogonal conjugate of X(2126)
X(2127) = X(58)-cross conjugate of X(10)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2128) lies on the cubic Z(X(19)) and these lines: 1,1958 2,2082 19,2129 63,304
X(2128) = isogonal conjugate of X(2129)
X(2128) = X(19)-Ceva conjugate of X(63)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2129) lies on the cubic Z(X(19)) and these lines: 19,2128 1707,1973
X(2129) = isogonal conjugate of X(2128)
X(2129) = X(63)-cross conjugate of X(19)
X(2130) lies on the Darboux cubic (K004) and these lines: 1,3347 3,3348 4,1073 20,2131 40,3354 84,3345 1490,3473
X(2130) = reflection of X(3348) in X(3)
X(2130) = isogonal conjugate of X(2131)
X(2130) = X(20)-Ceva conjugate of X(64)
X(2131) lies on the Darboux cubic (K004) and these lines: 3,3355 20,2130 40,3472 1490,3353 1498,3183
X(2131) = reflection of X(3355) in X(3)
X(2131) = isogonal conjugate of X(2130)
X(2131) = X(64)-cross conjugate of X(20)
X(2132) is the tangential of X(74) on the Neuberg cubic (K001).
X(2132) lies on the Neuberg cubic and these lines: 4,1138 30,2133 3065,3466 3440,3441
X(2132) = isogonal conjugate of X(2133)
X(2132) = X(30)-Ceva conjugate of X(74)
X(2132) = Miquel associate of X(30)
X(2133) lies on the Neuberg cubic (K001) and this line: 30,2132
X(2133) = isogonal conjugate of X(2132)
X(2133) = antigonal conjugate of X(34297)
X(2133) = X(74)-cross conjugate of X(30)
X(2134) lies on the these lines: 1,1326 37,2135 39,1931 81,763
X(2134) = isogonal conjugate of X(2135)
X(2134) = X(37)-Ceva conjugate of X(81)
X(2135) lies on the cubic Z(X(37)) and this line: 37,2134
X(2135) = isogonal conjugate of X(2134)
X(2135) = X(81)-cross conjugate of X(37)
X(2136) lies on the cubics K201 and K372, and on these lines: 1,474 8,9 10,1058 40,376 43,1050 57,145 84,952 100,1420 188,258 517,1490 664,738
X(2136) = isogonal conjugate of X(2137)
X(2136) = X(145)-Ceva conjugate of X(1)
X(2136) = X(145)-aleph conjugate of X(2136)
X(2136) = SS(a→s-a) of X(3)
X(2136) = X(64)-of-excentral-triangle
X(2136) = orthologic center of these triangles: excentral to excenters-midpoints
X(2136) = orthologic center of these triangles: excentral to 2nd Schiffler
X(2136) = intouch-to-ABC barycentric image of X(8)
X(2137) lies on the cubic K830 and these lines: {56, 1743}, {57, 145}, {1407, 1420}
X(2137) = isogonal conjugate of X(2136)
X(2137) = X(9)-cross conjugate of X(57)
X(2138) lies on the cubic K169 and these lines: 6,64 69,2139 1249,1941
X(2138) = isogonal conjugate of X(2139)
X(2138) = X(69)-Ceva conjugate of X(25)
X(2139) lies on the cubic, K169 and these lines: 20,159 69,2138
X(2139) = isogonal conjugate of X(2138)
X(2139) = X(25)-cross conjugate of X(69)
X(2140) lies on the cubic K258 and these lines: 3,142 5,116 39,1086 85,514 101,2141 170,1699 218,226 244,2085
X(2140) = isogonal conjugate of X(2141)
X(2140) = X(101)-Ceva conjugate of X(514)
X(2141) lies on this line: 101, 2140
X(2141) = isogonal conjugate of X(2140)
X(2141) = X(514)-cross conjugate of X(101)
X(2142) lies on the 1st equal-areas cubic (K021) and this line: 512,2143
X(2142) = isogonal conjugate of X(2143)
X(2142) = X(512)-Ceva conjugate of X(99)
X(2143) lies on the 1st equal-areas cubic (K021) and this line: 512,2142
X(2143) = isogonal conjugate of X(2142)
X(2143) = X(99)-cross conjugate of X(512)
Suppose U = u : v : w and P = p : q : r are triangle centers. The P-eigentransform of U, denoted by ET(U,P), is the point given by first trilinear
qrvw(pqu2v2 + pru2w2 - qrv2w2).
X(2144) = complement of X(20354)
X(2144) = anticomplement of X(20342)
X(2144) = X(238)-Ceva conjugate of X(292)
X(2145) lies on the 2nd equal-areas cubic (K155): 238,2144 2108,2110
X(2145) = X(292)-cross conjugate of X(238)
Barycentrics a3/2(b1/2/v + c1/2/w - a1/2/u) : b3/2(c1/2/w + a1/2/u - b1/2/v) : c3/2(a1/2/u + b1/2/v - c1/2/w)
X(2146) lies on the 2nd equal-areas cubic (K155) and these lines: 1,2119 238,2147 292,365 2111,2118
X(2146) = X(238)-Ceva conjugate of X(365)
Barycentrics a2/f(a,b,c) : b2/f(b,c,a) : c2/f(c,a,b)
X(2147) lies on the 2nd equal-areas cubic (K155) and these lines: 238,2146 365,2144 2108,2118
X(2147) = X(292)-cross conjugate of X(365)
X(2148) lies on these lines: 1,563 19,2190 47,48 54,71 63,2167 163,1953 228,2361 933,2249 2083,2158
X(2148) = isogonal conjugate of X(14213)
X(2148) = complement of anticomplementary conjugate of X(17479)
X(2148) = anticomplement of complementary conjugate of X(16577)
X(2148) = X(2167)-Ceva conjugate of X(2169)
X(2148) = cevapoint of X(48) and X(563)
X(2148) = X(i)-cross conjugate of X(j) for these (i,j): (31,2190), (48,2168), (661,163)
X(2148) = crosspoint of X(2167) and X(2190)
X(2149) lies on these lines: 59,672 101,652 109,649 163,1625 661,2222 1055,1262 1110,2223
X(2149) = isogonal conjugate of X(4858)
X(2149) = complement of anticomplementary conjugate of X(4552)
X(2149) = anticomplement of complementary conjugate of X(16578)
X(2149) = X(59)-Ceva conjugate of X(1110)
X(2149) = cevapoint of X(i) and X(j) for these (i,j): (41,692), (101,572), (604,1415)
X(2149) = X(i)-cross conjugate of X(j) for these (i,j): (6,163), (31,109), (41,692), (48,101), (604,1415), (1755,813), (2352,108)
X(2149) = crosspoint of X(59) and X(1262)
X(2149) = crosssum of X(11) and X(1146)
X(2150) lies on these lines: 6,163 9,1098 58,1474 60,283 593,1412 759,1953 849,1333 2193,2194
X(2150) = isogonal conjugate of X(6358)
X(2150) = complement of anticomplementary conjugate of X(18662)
X(2150) = anticomplement of complementary conjugate of X(16579)
X(2150) = crossdifference of every pair of points on line X(4036)X(4064)
X(2150) = X(593)-Ceva conjugate of X(849)
X(2150) = X(i)-cross conjugate of X(j) for these (i,j): (1333,2189), (2194,60)
X(2150) = crosspoint of X(i) and X(j) for these (i,j): (60,593), (270,2185)
X(2150) = crosssum of X(i) and X(j) for these (i,j): (12,594), (201,2171)
X(2151) lies on these lines: 1,1095 31,48 2153,2173
X(2151) = X(2159)-cross conjugate of X(2152)
X(2152) lies on these lines: 1,1094 31,48 2154,2173
X(2152) = X(2159)-cross conjugate of X(2151)
X(2153) lies on these lines: 1,1094 10,13 31,2154 37,1250 65,2306 2151,2173
X(2153) = X(2173)-cross conjugate of X(2154)
X(2153) = trilinear product of circumcircle intercepts of inner Napoleon circle
X(2154) lies on these lines: 1,1095 10,14 31,2153 2152,2173
X(2154) = X(2173)-cross conjugate of X(2153)
X(2154) = trilinear product of circumcircle intercepts of outer Napoleon circle
X(2155) lies on these lines: 19,774 25,2357 41,1409 48,820 63,610 64,71 228,1253 294,2270 607,1400 1301,2249 1820,2173
X(2155) = isogonal conjugate of X(18750)
X(2155) = X(1973)-cross conjugate of X(31)
X(2156) lies on these lines: 31,2083 38,48 63,1930 66,71 228,2353 561,1760 1289,2249 1409,2273 1748,1821 1755,1820
X(2156) = isogonal conjugate of X(1760)
X(2156) = complement of X(21288)
X(2156) = anticomplement of X(21247)
X(2156) = X(i)-cross conjugate of X(j) for these (i,j): (560,1),
(2632,661)
X(2157) lies on these lines: 19,1109 38,163 48,2247 67,71 935,2249 2159,2312
X(2157) = cevapoint of X(38) and X(896)
X(2157) = complement of anticomplementary conjugate of X(17482)
X(2157) = anticomplement of complementary conjugate of X(16581)
X(2157) = X(922)-cross conjugate of X(1)
X(2158) lies on these lines: 70,71 1288,2249 2083,2148
X(2158) = isogonal conjugate of complement of X(18665)
X(2158) = isogonal conjugate of anticomplement of X(18590)
X(2158) = complement of anticomplementary conjugate of X(18665)
X(2158) = anticomplement of complementary conjugate of X(18590)
X(2159) lies on these lines: 48,163 63,662 71,74 228,692 610,1820 1304,2249 1409,1415 1461,2003 1725,2173 2157,2312
X(2159) = isogonal conjugate of X(14206)
X(2159) = cevapoint of X(i) and X(j) for these (i,j): (48,2315), (2151,2152)
X(2159) = X(2624)-cross conjugate of X(163)
X(2159) = polar conjugate of isotomic conjugate of X(35200)
X(2159) = barycentric product of PU(86)
X(2159) = trilinear product X(6)*X(74)
X(2159) = complement of anticomplementary conjugate of X(18668)
X(2159) = anticomplement of complementary conjugate of X(18593)
X(2159) = X(63)-isoconjugate of X(1784)
A construction for X(2160) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.
X(2160) lies on these lines: 6,1406 9,46 19,1990 55,199 56,2164 65,2174 284,501 395,1081 396,554 909,2262 910,1174 1108,2291 1400,1989 1841,2299
X(2160) = isogonal conjugate of X(3219)
X(2160) = cevapoint of X(1652) and X(1653)
X(2160) = X(i)-cross conjugate of X(j) for these (i,j): (2260,6), (2308,1)
X(2160) = crosspoint of X(57) ane X(267)
X(2160) = crosssum of X(9) and X(191)
X(2160) = complement of anticomplementary conjugate of X(17483)
X(2160) = trilinear pole of line X(663)X(2520)
X(2161) lies on these lines: 1,2364 6,1411 9,80 19,53 37,101 44,517 45,55 57,1020 63,545 190,321 198,2164 484,2245 654,900 655,673 909,1319 910,2222 1150,12237 1400,1989 1436,2178 1635,1769 1824,2299 2259,2264
X(2161) = isogonal conjugate of X(3218)
X(2161) = X(2006)-Ceva conjugate of X(1411)
X(2161) = cevapoint of X(i) and X(j) for these (i,j): (37,44), (649,2087), (1635,2170)
X(2161) = X(i)-cross conjugate of X(j) for these (i,j): (902,1), (2183,6)
X(2161) = crosspoint of X(i) and X(j) for these (i,j): (80,2006), (88,104)
X(2161) = crosssum of X(i) and X(j) for these (i,j): (36,2323), (44,517), (758,2245)
X(2161) = trilinear pole of line X(42)X(663)
X(2162) lies on these lines: 2,1977 6,43 31,172 55,1911 81,330 604,1403 608,2201 739,932 1397,1691 1407,1429 2056,2175
X(2162) = isogonal conjugate of X(192)
X(2162) = isotomic conjugate of X(6382)
X(2162) = anticomplement of X(21250)
X(2162) = X(87)-Ceva conjugate of X(2053)
X(2162) = cevapoint of X(i) and X(j) for these (i,j): (1,87), (649,1977)
X(2162) = X(i)-cross conjugate of X(j) for these (i,j): (1,6), (727,2109), (893,2248), (2309,58)
X(2162) = perspector of ABC and unary cofactor triangle of Gemini triangle 16
X(2162) = polar conjugate of isotomic conjugate of X(23086)
X(2162) = complement of polar conjugate of isogonal conjugate of X(23134)
Let IaIbIc be the excentral triangle of a triangle ABC. Let A2 and A3 be the points where the perpendicular bisector of BC meets the internal angle bisector of angle ABC and the internal angle bisector of angle BCA, respectively. Let A'2 and A'3 be the points where the perpendicular bisector of BC meets the external angle bisector of angle ABC and the external angle bisector of angle BCA, respectively. Let Oab, Oac be the circumcenters of triangles IbA2A'3, IcA3A'2, respectively. Define Obc, Oba, Oca, Ocb cyclically. Then the triangle having sidelines OabOac, ObcOba, OcaOcb is perspective to ABC, and the perspector is X(2163). See Angel Montesdeoca, HG090218: El centro del triángulo X(2163).
Let A'B'C' be the circumsymedial triangle. Let (OA) be the circle centered at A' and tangent to line BC. Define (OB) and (OC) cyclically. X(2163) is the trilinear pole of the Monge line of (OA), (OB), (OC). (Randy Hutson, November 30, 2018)
X(2163) lies on these lines: 1,89 3,2334 6,36 31,106 57,1411 996,1150 1055,2279 1126,1468 1220,1698
X(2163) = complement of X(21291)
X(2163) = anticomplement of X(21251)
X(2163) = X(89)-Ceva conjugate of X(2364)
X(2163) = X(995)-cross conjugate of X(1)
X(2163) = isogonal conjugate of X(3679)
X(2163) = vertex conjugate of X(1) and X(6)
X(2164) lies on these lines: 6,1195 9,35 19,1609 55,2174 56,2160 198,2161 284,1069 571,607 573,2316 1630,2291
X(2164) = isogonal conjugate of X(5905)
X(2164) = X(48)-cross conjugate of X(6)
X(2165) lies on these lines: 2,311 4,96 5,6 25,53 37,91 111,925 115,577 206,1976 393,847 493,590 494,615 1263,2079 1321,1322 1400,1454
X(2165) = isogonal conjugate of X(1993)
X(2165) = isotomic conjugate of X(7763)
X(2165) = X(96)-Ceva conjugate of X(2351)
X(2165) = cevapoint of X(115) and X(647)
X(2165) = X(i)-cross conjugate of X(j) for these (i,j): (184,4), (216,6), (2351,68)
X(2165) = crosspoint of X(2) and X(254)
X(2165) = crosssum of X(i) and X(j) for these (i,j): (6,155), (571,1147)
X(2165) = barycentric product of X(485) and X(486)
X(2165) = complement of isogonal conjugate of X(39109)
X(2165) = orthic-isogonal conjugate of X(39111)
X(2165) = X(4)-Ceva conjugate of X(39111)
X(2165) = polar conjugate of X(317)
X(2165) = X(92)-isoconjugate of X(1147)
X(2165) = X(610)-of-orthic-triangle if ABC is acute
Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. X(2166) is the trilinear product A1*A2 = B1*B2 = C1*C2. (Randy Hutson, June 27, 2018)
Let VaVbVc be the Ehrmann vertex-triangle. Let A' be the trilinear product Vb*Vc, and define B', C' cyclically. The lines AA', BB', CC' concur in X(2166). (Randy Hutson, June 27, 2018)
Let VaVbVc and SaSbSc be the Ehrmann vertex-triangle and Ehrmann side-triangle, resp. Let A' be the trilinear product Va*Sa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(2166). (Randy Hutson, June 27, 2018)
X(2166) lies on these lines: 1,564 10,94 37,1989 65,79 162,2190 476,759 897,1733 1141,2222
X(2166) = isogonal conjugate of X(6149)
X(2166) = trilinear pole of line X(661)X(1953)
X(2166) = cevapoint of X(1) and X(1749)
X(2166) = X(i)-cross conjugate of X(j) for these (i,j): (1725,1),
(2173,92)
X(2166) = trilinear product of X(13) and X(14)
X(2166) = trilinear product of circumcircle intercepts of Johnson circle (or line PU(5), X(5)X(523))
X(2167) lies on these lines: 1,1748 38,293 48,92 54,72 63,2148 75,2168 95,306 97,1214 162,1954 226,275
X(2167) = isogonal conjugate of X(1953)
X(2167) = isotomic conjugate of X(14213)
X(2167) = cevapoint of X(i) and X(j) for these (i,j): (1,48), (2148,2169)
X(2167) = X(i)-cross conjugate of X(j) for these (i,j): (48,2169), (822,162), (1577,662), (2148,2190)
X(2167) = X(5)-isoconjugate of X(6)
X(2167) = trilinear pole of line X(656)X(1955)
X(2167) = crosspoint of X(1) and X(48) wrt excentral triangle
X(2168) lies on these lines: 1,563 10,96 48,91 75,2167 921,2169 1953,2216
X(2168) = cevapoint of X(48) and X(1820)
X(2168) = X(48)-cross conjugate of X(2148)
Let A'B'C' be the cevian triangle of I (incenter) and let A'' be the intersection of the perpendicular bisector of AA' and the line through A' perpendicular line to BC; note that A''(lies on AX(3)). Define B'' and C'' cyclically. The finite fixed point of the affine transformation that carries ABC onto A'B'C' is X(2197), and the finite fixed point of the affine transformation that carries A'B'C' onto A''B''C'' is X(2269). (Angel Montesdeoca, December 7, 2021)
X(2169) lies on these lines: 1,1748 36,54 47,48 336,1930 821,1784 921,2168
X(2169) = X(2167)-Ceva conjugate of X(2148)
X(2169) = X(48)-cross conjugate of X(2167)
X(2169) = crosssum of X(1953) and X(2181)
X(2169) = X(4)-isoconjugate of X(5)
X(2170) lies on these lines: 1,41 6,1411 9,644 11,1146 19,604 31,1572 37,374 57,934 65,1475 115,661 163,759 218,1482 220,2098 239,1959 244,665 354,1200 514,1111 517,672 664,673 756,1573 910,1055 926,2310 1086,1358 1100,2264 1107,2292 1108,1400 1195,2260 1212,1334 1404,2182 1731,2323 1870,2202 1914,1951
X(2170) = isogonal conjugate of X(4564)
X(2170) = complement of X(21272)
X(2170) = anticomplement of X(21232)
X(2170) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,663), (6,661), (9,650), (11,2310), (19,649), (57,513), (312,522), (673,2254), (1086,244), (1751,656), (2006,1769), (2161,1635), (2217,667)
X(2170) = cevapoint of X(1) and X(1053)
X(2170) = crosspoint of X(i) and X(j) for these (i,j): (1,514), (9,650), (11,1086), (57,513), (312,522), (1022,1168), (2590,2591)
X(2170) = crosssum of X(i) and X(j) for these (i,j): (1,101), (9,100), (57,651), (59,1252), (63,1813), (78,644), (109,604), (214,1023), (1331,2289)
X(2170) = crossdifference of every pair of points on line X(100)X(3738) (the tangent at X(100) to the circumconic centered at X(1), conic {{A,B,C,X(100),X(664),X(1120),X(1320)}})
X(2170) = intersection of tangents to Steiner inellipse at X(1015) and X(1146)
X(2170) = crosspoint wrt medial triangle of X(1015) and X(1146)
X(2170) = barycentric square of X(6728)
X(2171) lies on these lines: 1,572 6,1411 7,192 9,1405 12,594 19,41 37,65 42,1824 57,1255 101,1781 172,1950 181,756 226,306 312,1240 517,2269 894,1959 1100,1404 1108,1475 1254,1500 1402,1962 1841,1887 2173,2174 2262,2347
X(2171) = isogonal conjugate of X(2185)
X(2171) = complement of X(21273)
X(2171) = anticomplement of X(21233)
X(2171) = X(i)-Ceva conjugate of X(j) for these (i,j): (12,756), (37,2197), (65,181), (226,12)
X(2171) = cevapoint of X(181) and X(1500)
X(2171) = X(i)-cross conjugate of X(j) for these (i,j): (115,661), (181,1254), (1500,756)
X(2171) = crosspoint of X(i) and X(j) for these (i,j): (1,2051), (37,1826), (65,226)
X(2171) = crosssum of X(i) and X(j) for these (i,j): (1,572), (2,284), (81,1790)
X(2172) lies on these lines: 1,19 31,1917 41,386 47,1755 78,101 91,1910 163,255 304,662 379,2140 1479,2201 1725,2083 1930,1958
X(2172) = isogonal conjugate of isotomic conjugate of X(1760)
X(2172) = isogonal conjugate of complement of X(21215)
X(2172) = isogonal conjugate of anticomplement of X(16582)
X(2172) = complement of X(17492)
X(2172) = anticomplement of X(16607)
X(2172) = X(75)-Ceva conjugate of X(31)
X(2173) lies on these lines: 1,19 6,1406 36,1731 44,513 45,198 169,2267 662,1959 897,1910 1405,1454 1725,2159 1760,1958 1761,2287 1762,1817 1820,2155 2151,2153 2152,2154 2171,2174 2260,2264 2261,2270 2262,2317
X(2173) = isogonal conjugate of X(2349)
X(2173) = X(i)-Ceva conjugate of X(j) for these (i,j): (2341,6), (2349,1)
X(2173) = crosspoint of X(i) and X(j) for these (i,j): (1,2349), (57,759), (92,2166), (2153,2154)
X(2173) = crosssum of X(i) and X(j) for these (i,j): (1,2173), (9,758)
X(2173) = X(2349)-aleph conjugate of X(2173)
X(2173) = crossdifference of every pair of points on line X(1)X(656)
X(2173) = isotomic conjugate of X(33805)
X(2173) = bicentric sum of PU(74)
X(2173) = PU(74)-harmonic conjugate of X(656)
X(2174) lies on these lines: 1,584 6,41 9,2278 36,583 37,101 44,572 45,2268 50,1399 53,2202 55,2164 65,2160 71,1030 199,209 213,1333 220,2289 228,2194 560,869 594,2329 662,894 872,922 910,1630 1409,1415 1964,2210 2171,2173 2176,2304 2220,2251 2264,2302
X(2174) = isogonal conjugate of X(30690)
X(2174) = complement of X(21276)
X(2174) = anticomplement of X(21236)
X(2174) = X(i)-Ceva conjugate of X(j) for these (i,j): (80,2361), (1126,31), (2003,1399), (2259,6)
X(2174) = crosspoint of X(35) and X(2003)
X(2174) = crossdifference of every pair of points on line X(522)X(4823) (the polar wrt polar circle of X(6198))
X(2175) lies on these lines: 6,692 9,2330 25,181 31,184 32,560 41,1253 42,2273 48,2223 55,219 154,1460 182,238 213,1973 344,1083 601,1092 760,1760 1036,1682 1037,1362 1401,1473 1631,2245 1691,2176 2056,2162 2209,2210
X(2175) = isogonal conjugate of X(6063)
X(2175) = complement of X(21280)
X(2175) = anticomplement of X(17047)
X(2175) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,6), (31,32), (2194,41)
X(2175) = X(1918)-cross conjugate of X(2212)
X(2175) = crosspoint of X(i) and X(j) for these (i,j): (31,41), (55,607)
X(2175) = crosssum of X(i) and X(j) for these (i,j): (7,348), (8,344), (74,85), (349,1441)
X(2175) = crossdifference of every pair of points on line X(918)X(3261)
X(2175) = X(7)-isoconjugate of X(75)
X(2175) = barycentric product of PU(93)
X(2175) = Lozada perspector of X(6)
X(2175) = perspector of tangential triangle and cross-triangle of intouch and tangential triangles
X(2175) = trilinear product of vertices of Apus triangle (the extraversions of X(56))
X(2176) lies on these lines: 1,6 2,1258 8,2238 31,172 32,101 39,995 41,1914 55,869 56,292 58,2242 71,2277 169,1572 190,194 239,312 304,742 386,1500 672,1201 978,1575 992,2345 1149,1475 1185,1621 1193,1334 1397,2056 1691,2175 2174,2304 2178,2305
X(2176) = isogonal conjugate of X(330)
X(2176) = isotomic conjugate of X(6383)
X(2176) = complement of X(21281)
X(2176) = anticomplement of X(20255)
X(2176) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,6), (983,55), (1423,1403)
X(2176) = X(i)-cross conjugate of X(j) for these (i,j): (43,6), (2209,1403)
X(2176) = crosspoint of X(i) and X(j) for these (i,j): (31,2209), (43,1423), (101,1016)
X(2176) = crosssum of X(i) and X(j) for these (i,j): (2,1278), (87,2319), (514,1015)
X(2176) = X(92)-isoconjugate of X(23086)
X(2176) = {X(1),X(9)}-harmonic conjugate of X(1107)
X(2176) = crossdifference of every pair of points on line X(513)X(3716)
X(2176) = pole wrt circumcircle of line X(667)X(788)
X(2177) lies on these lines: 1,88 6,31 33,2181 35,1468 41,1017 43,748 200,756 519,1150 574,2223 612,1962 751,765 899,1001 1064,1480 1334,2271 1471,2078 1495,2187
X(2177) = complement of X(21283)
X(2177) = anticomplement of X(21242)
X(2177) = X(2099)-Ceva conjugate of X(1405)
X(2177) = crosspoint of X(45) and X(2099)
X(2177) = crosssum of X(89) and X(2320)
X(2178) lies on these lines: 3,37 6,41 9,36 19,1609 25,1841 45,1696 55,199 101,579 108,393 197,1402 218,583 219,2245 230,444 404,2345 571,608 594,1376 859,1333 910,1108 958,1213 999,1100 1182,1630 1319,2262 1420,2270 1436,2161 1457,2199 1470,2285 1486,2223 2092,2242 2176,2305
X(2178) = isogonal conjugate of X(2994)
X(2178) = X(19)-Ceva conjugate of X(6)
X(2178) = crosspoint of X(108) and X(1262)
X(2178) = X(92)-isoconjugate of X(1069)
X(2178) = crosssum of X(i) and X(j) for these (i,j): (521,1146), (1069,2164)
X(2179) lies on these lines: 1,1755 19,158 31,1932 48,1497 902,2198 1334,2225 1953,2180 1964,2085
X(2179) = X(19)-Ceva conjugate of X(2181)
X(2179) = crosspoint of X(i) and X(j) for these (i,j): (19,31),
(1953,2181)
X(2179) = crosssum of X(63) and X(75)
X(2180) lies on these lines: 19,91 31,48 47,563 1195,2183 1953,2179 2253,2355
X(2180) = X(19)-Ceva conjugate of X(1953)
X(2180) = crosspoint of X(47) and X(1748)
X(2180) = crosssum of X(91) and X(1820)
X(2181) lies on these lines: 19,31 33,2177 38,92 42,1859 158,774 244,278 281,756 896,1748 1118,1254 1193,1871 1857,2310 1953,2313
X(2181) = X(19)-Ceva conjugate of X(2179)
X(2181) = X(2179)-cross conjugate of X(1953)
X(2181) = crosspoint of X(19) and X(158)
X(2181) = crosssum of X(63) and X(255)
X(2182) lies on these lines: 3,9 6,19 25,1864 33,154 37,48 44,513 46,1743 51,2355 169,374 184,1824 197,210 219,1766 407,1901 517,2323 572,1630 578,1871 604,1108 909,1319 960,1610 1100,1953 1212,2267 1214,1726 1404,2170 1498,1753 1503,1861
X(2182) = isogonal conjugate of X(36100)
X(2182) = X(i)-Ceva conjugate of X(j) for these (i,j): (1295,55), (1465,1319)
X(2182) = crosspoint of X(i) and X(j) for these (i,j): (4,2006, 57,104)
X(2182) = crosssum of X(i) and X(j) for these (i,j): (1,2182), (3,2323), (9,517)
X(2182) = {X(6),X(19)}-harmonic conjugate of X(2262)
X(2182) = crossdifference of every pair of points on line X(1)X(521)
X(2182) = bicentric sum of PU(100)
X(2182) = PU(100)-harmonic conjugate of X(6129)
X(2183) lies on these lines: 1,957 3,2267 4,9 6,41 25,212 31,197 36,909 37,1953 44,513 45,1334 51,228 101,953 201,1829 213,2288 219,945 226,1730 294,1937 374,1212 478,2199 511,1818 515,2250 579,610 674,2340 937,2215 1018,2325 1167,1474 1195,2180 1253,1486 1696,2256 1708,1763 1738,1756
X(2183) = isogonal conjugate of X(34234)
X(2183) = X(i)-Ceva conjugate of X(j) for these (i,j): (36,902), (80,42), (102,55), (1465,1457), (2222,663), (2316,6)
X(2183) = crosspoint of X(i) and X(j) for these (i,j): (6,2161), (19,913), (57,106), (517,1465), (901,1262), (908,1785)
X(2183) = crosssum of X(i) and X(j) for these (i,j): (1,2183), (2,3218), (9,519), (57,34050), (63,914), (649,1647), (900,1146), (909,1795), (4466,4707
X(2183) = polar conjugate of isotomic conjugate of X(22350)
X(2183) = crossdifference of every pair of points on line X(1)X(522)
X(2183) = X(92)-isoconjugate of X(1795)
X(2184) lies on these lines: 1,204 9,223 40,64 57,282 63,610 84,2130 196,226 253,306 293,1707 1748,2349
X(2184) = isogonal conjugate of X(610)
X(2184) = isotomic conjugate of X(18750)
X(2184) = anticomplement of X(36908)
X(2184) = cevapoint of X(661) and X(2632)
X(2184) = X(i)-cross conjugate of X(j) for these (i,j): (19,1), (774,75), (1427,2), (1903,4)
X(2184) = pole wrt polar circle of trilinear polar of X(1895)
X(2184) = X(48)-isoconjugate (polar conjugate) of X(1895)
X(2184) = X(6)-isoconjugate of X(20)
X(2184) = intersection of tangents at X(189) and X(329) to Lucas cubic K007
Let DEF be the cevian triangle of the incenter. Let OA be the circumcircle of AEF. Let T1 be the line tangent to OA at E, and let T2 be the line tangent to OA at F. Let AB = T1∩BC and AC = T2∩BC, and let A' be the radical center of the circumcircles of AEF, BFAB, CAAC.. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(2185). (Angel Montesdeoca, January 4, 2021)
X(2185) lies on these lines: 1,849 2,662 21,60 27,86 55,643 81,593 110,1621 171,1326 222,1414 261,284 312,2268 501,1125 552,553
X(2185) = isogonal conjugate of X(2171)
X(2185) = isotomic conjugate of X(6358)
X(2185) = complement of anticomplementary conjugate of X(21273)
X(2185) = anticomplement of complementary conjugate of X(21233)
X(2185) = X(6)-isoconjugate of X(12)
X(2185) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,662), (261,1098), (1509,757)
X(2185) = cevapoint of X(i) and X(j) for these (i,j): (1,572), (21,284), (81,1790)
X(2185) = X(i)-cross conjugate of X(j) for these (i,j): (21,261), (60,757), (284,60), (2150,270)
X(2185) = crosspoint of X(261) and X(1509)
X(2185) = crosssum of X(181) and X(1500)
X(2186) lies on these lines: 1,1755 10,262 19,1964 31,1910 37,263 48,82 65,2275 75,1953 1973,2190 2172,2216
X(2186) = isotomic conjugate of X(3403)
X(2187) lies on these lines: 25,41 31,184 40,1817 48,55 84,947 101,200 212,692 228,1253 601,1437 968,2268 1208,1622 1458,1473 1495,2177
X(2187) = isogonal conjugate of X(309)
X(2187) = X(i)-Ceva conjugate of X(j) for these (i,j): (48,41),
(55,31), (221,2199), (947,6), (2360,198)
X(2187) = crosspoint of X(i) and X(j) for these (i,j): (6,963),
(40,2331), (198,221)
X(2187) = crosssum of X(i) and X(j) for these (i,j): (2,962), (7,1440),
(189,280)
X(2188) lies on these lines: 1,1767 3,1433 48,55 57,972 73,972 73,939 84,943 212,577 268,1260 271,1792 282,284 1903,2268
X(2188) = isogonal conjugate of X(342)
X(2188) = X(i)-cross conjugate of X(j) for these (i,j): (41,48), (184,212)
X(2188) = crosspoint of X(i) and X(j) for these (i,j): (268,1433), (271,282)
X(2188) = crosssum of X(i) and X(j) for these (i,j): (1,1767), (208,223)
X(2188) = X(92)-isoconjugate of X(223)
X(2189) lies on these lines: 6,1175 19,112 21,270 25,1169 27,272 28,1104 37,2074 58,1474 284,2299 359,2011 593,1014 2212,2311
X(2189) = cevapoint of X(i) and X(j) for these (i,j): (1333,1474), (2204,2299)
X(2189) = X(i)-cross conjugate of X(j) for these (i,j): (1333,2150), (2299,270)
X(2189) = polar conjugate of X(34388)
Let A'B'C' be the circumorthic triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. A", B", C" are collinear on line X(810)X(2616). The lines AA", BB", CC" concur in X(2190). (Randy Hutson, July 20, 2016)
Let P1 and P2 be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. Let Q1 and Q2 be the respective points of tangency. Q1 and Q2 are also the circumcircle intercepts of line X(186)X(523). X(2190) is the trilinear product of Q1 and Q2. See also X(8882). (Randy Hutson, July 20, 2016)
Let A' be the reflection of A in BC. Let AB, AC be the reflections of A' in CA and AB, resp. Let A" be the trilinear product AB*AC. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(2190). (Randy Hutson, July 11, 2019)
X(2190) lies on these lines: 1,1748 10,275 19,2148 31,158 47,91 54,65 75,255 82,240 162,2166 270,759 1973,2186
X(2190) = isotomic conjugate of X(18695)
X(2190) = cevapoint of X(i) and X(j) for these (i,j): (1,47), (19,31)
X(2190) = X(i)-cross conjugate of X(j) for these (i,j): (31,2148), (1825,4), (2148,2167)
X(2190) = crosspoint of X(1) and X(47) wrt excentral triangle
X(2191) lies on these lines: 1,142 6,354 34,1458 48,1438 56,1279 106,1292 244,1253 1027,1459 1413,1456 1419,1421 2260,2279
X(2191) = isogonal conjugate of X(3870)
X(2191) = cevapoint of X(i) and X(j) for these (i,j): (244,663), (1015,2488)
X(2191) = X(i)-cross conjugate of X(j) for these (i,j): (41,57), (2293,6)
X(2192) lies on these lines: 1,84 6,33 11,1853 48,55 56,64 103,1617 189,1814 200,219 212,220 268,2193 280,285 497,1503 2194,2332
X(2192) = isogonal conjugate of X(347)
X(2192) = anticomplement of X(20307)
X(2192) = X(i)-Ceva conjugate of X(j) for these (i,j): (84,1436), (280,268), (285,282), (1433,6)
X(2192) = cevapoint of X(31) and X(2208)
X(2192) = X(i)-cross conjugate of X(j) for these (i,j): (31,55), (697,6)
X(2192) = crosspoint of X(84) and X(282)
X(2192) = crosssum of X(40) and X(223)
X(2193) lies on these lines: 1,1744 3,6 21,270 30,1865 48,255 71,906 81,1214 112,1295 212,2289 219,283 222,1790 268,2192 286,448 859,1474 1396,1465 1444,1814 1809,2287 1880,1950 2150,2194
X(2193) = complement of isotomic conjugate of X(18123)
X(2193) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,2194), (1790,1437), (1798,184)
X(2193) = cevapoint of X(48) and X(577)
X(2193) = X(i)-cross conjugate of X(j) for these (i,j): (48,284), (652,906)
X(2193) = crosspoint of X(i) and X(j) for these (i,j): (21,1812), (283,1790)
X(2193) = crosssum of X(i) and X(j) for these (i,j): (1,1744), (65,1880), (225,1826)
X(2193) = X(92)-isoconjugate of X(65)
X(2193) = X(108)-isoconjugate of X(1577)
X(2193) = {X(1805),X(1806)}-harmonic conjugate of X(3)
X(2194) lies on these lines: 1,1762 3,1779 6,25 19,2219 21,60 27,1836 28,65 29,1837 31,48 41,212 42,692 55,219 56,58 81,105 199,2245 210,2287 228,2174 283,1036 904,2210 1011,2278 1155,1817 1172,1859 1185,1501 1396,1456 1415,1949 1834,1884 2150,2193 2192,2332 2260,2308
X(2194) = isogonal conjugate of X(1441)
X(2194) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,2193), (1169,32), (1172,2204), (1175,6)
X(2194) = cevapoint of X(i) and X(j) for these (i,j): (31,184), (41,2175)
X(2194) = X(i)-cross conjugate of X(j) for these (i,j): (31,2299), (41,284), (663,692)
X(2194) = crosspoint of X(i) and X(j) for these (i,j): (21,1172), (58,284), (60,2150)
X(2194) = crosssum of X(i) and X(j) for these (i,j): (1,1762), (2,2475), (10,226), (65,1214)
X(2194) = X(92)-isoconjugate of X(1214)
X(2195) lies on these lines: 1,1814 6,692 9,294 19,2212 31,57 42,1174 55,5452 238,516 284,2293 333,643 902,919 1027,1769
X(2195) = isogonal conjugate of X(9436)
X(2195) = X(105)-Ceva conjugate of X(1438)
X(2195) = crosspoint of X(i) and X(j) for these (i,j): (55,2115), (105,294)
X(2195) = crosssum of X(241) and X(518)
X(2195) = trilinear pole of PU(93) (line X(41)X(663))
X(2196) lies on these lines: 3,295 28,291 56,292 63,337 71,1332 104,813 692,1333 906,1437 1472,1922
X(2196) = X(i)-Ceva conjugate of X(j) for these (i,j): (291,1911), (2311,292)
X(2196) = X(92)-isoconjugate of X(238)
X(2196) = crosssum of X(238) and X(2201)
Let A'B'C' be the cevian triangle of I (incenter) and let A'' be the intersection of the perpendicular bisector of AA' and the line through A' perpendicular line to BC; note that A''(lies on AX(3)). Define B'' and C'' cyclically. The finite fixed point of the affine transformation that carries ABC onto A'B'C' is X(2197), and the finite fixed point of the affine transformation that carries A'B'C' onto A''B''C'' is X(2269). (Angel Montesdeoca, December 7, 2021)
X(2197) lies on these lines: 3,2286 6,2259 9,1937 12,37 35,1950 39,604 41,800 42,181 48,216 55,608 71,73 198,607 222,1796 278,2335 306,307 1030,1415 1253,2199 1254,1500 2276,2285
X(2197) = X(i)-Ceva conjugate of X(j) for these (i,j): (12,181), (37,2171), (1214,201)
X(2197) = crosspoint of X(i) and X(j) for these (i,j): (37,71), (73,1214)
X(2197) = X(60)-isoconjugate of X(92)
X(2197) = trilinear product X(12)*X(48)
X(2197) = crosssum of X(i) and X(j) for these (i,j): (27,81), (29,1172), (20,2189), (270,2326)
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X(2198) lies on these lines: 9,964 31,32 37,65 72,672 101,1780 902,2179
X(2198) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,42),
(579,209)
X(2198) = crosspoint of X(579) and X(2352)
X(2199) lies on these lines: 6,603 9,109 25,31 32,604 41,1409 48,1415 198,221 223,1817 255,573 478,2183 651,1958 966,1935 1253,2197 1394,2270 1455,2262 1457,2178 1496,2269 1950,2268 2285,2312
X(2199) = isogonal conjugate of X(34404)
X(2199) = crossdifference of every pair of points on line X(4397)X(6332)
X(2199) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,604), (221,2187), (603,31)
X(2199) = crosspoint of X(i) and X(j) for these (i,j): (6,198), (208,223)
X(2199) = crosssum of X(i) and X(j) for these (i,j): (2,189), (271,283)
X(2200) lies on these lines: 3,48 10,98 25,41 32,560 55,2304 213,1402 306,1799 906,1437 1409,1410 1474,2259 2327,2359
X(2200) = X(i)-Ceva conjugate of X(j) for these (i,j): (41,213), (42,1918), (48,228), (2259,31), (2359,212)
X(2200) = crosspoint of X(i) and X(j) for these (i,j): (42,71), (48,184), (228,1409)
X(2200) = crosssum of X(i) and X(j) for these (i,j): (27,86), (85,331), (92,264)
X(2200) = X(27)-isoconjugate of X(75)
X(2200) = X(86)-isoconjugate of X(92)
X(2201) lies on these lines: 4,1973 19,25 27,86 48,497 242,740 256,1172 513,1430 608,2162 855,1951 862,2238 1284,1914 1479,2172 1755,1936 1842,2332 1851,2280 1884,2202 2310,2312
X(2201) = X(i)-cross conjugate of X(j) for these (i,j): (862,242), (2210,238)
X(2201) = crosssum of X(71) and X(1818)
X(2201) = polar conjugate of X(334)
X(2202) lies on these lines: 4,41 6,19 29,284 48,393 53,2174 101,1785 108,1055 281,2268 318,2329 604,1249 672,1783 851,1430 910,1875 1870,2170 1884,2201
X(2202) = X(1948)-Ceva conjugate of X(1936)
X(2202) = crosssum of X(1943) and X(1944)
X(2202) = perspector of conic {{A,B,C,PU(18)}}
X(2202) = intersection of trilinear polars of P(18) and U(18)
X(2203) lies on these lines: 6,25 19,2214 28,60 31,1932 58,1473 112,739 209,692 468,1211 604,1395 608,1397 1172,1824 1396,1462 1398,1407 2308,2354
X(2203) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,1333), (1474,2204)
X(2203) = cevapoint of X(i) and X(j) for these (i,j): (1395,1397), (1973,1974)
X(2203) = X(i)-cross conjugate of X(j) for these (i,j): (1397,2206), (1973,1474)
X(2203) = X(75)-isoconjugate of X(72)
X(2204) lies on these lines: 19,2218 21,270 25,32 28,105 31,1932 41,2212 55,607 56,608 284,1036 1829,1914 2206,2208
X(2204) = isogonal conjugate of X(1231)
X(2204) = X(i)-Ceva conjugate of X(j) for these (i,j): (1172,2194), (1474,2203), (2189,2299)
X(2204) = X(2212)-cross conjugate of X(2299)
X(2204) = crosspoint of X(1474) and X(2299)
X(2204) = crosssum of X(306) and X(307)
X(2205) lies on these lines: 31,32 37,82 81,172 100,699 101,715 560,1501 1922,2206
X(2205) = isogonal conjugate of X(6385)
X(2205) = X(32)-Ceva conjugate of X(1918)
X(2205) = crosspoint of X(32) and X(560)
X(2205) = crosssum of X(76) and X(561)
X(2206) lies on these lines: 6,199 21,987 31,48 32,184 36,58 42,284 81,982 99,697 110,727 604,1395 902,2328 909,2299 1106,1408 1201,2360 1412,1416 1437,1472 1780,1801 1922,2205 2204,2208
X(2206) = isogonal conjugate of X(313)
X(2206) = X(849)-Ceva conjugate of X(1333)
X(2206) = cevapoint of X(32) and X(560)
X(2206) = X(1397)-cross conjugate of X(2203)
X(2206) = crosspoint of X(i) and X(j) for these (i,j): (58,1474), (1333,1408)
X(2206) = crosssum of X(i) and X(j) for these (i,j): (2,1330), (10,306)
X(2206) = X(10)-isoconjugate of X(75)
X(2206) = X(92)-isoconjugate of X(306)
X(2206) = barycentric product of vertices of 2nd circumperp triangle
X(2207) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the orthic axis. This inellipse has center X(3767) and perspector X(393). (Randy Hutson, October 15, 2018)
X(2207) lies on these lines: 3,232 4,6 19,2281 24,112 25,32 39,1593 76,683 83,458 107,729 155,1625 213,607 235,2138 297,315 459,1611 608,1426 800,1033 981,1896 1015,1398 1572,1829 1918,2212 2271,2332
X(2207) = X(393)-Ceva conjugate of X(25)
X(2207) = cevapoint of X(6) and X(1611)
X(2207) = X(1974)-cross conjugate of X(25)
X(2207) = crosssum of X(6) and X(1619)
X(2207) = isogonal conjugate of X(3926)
X(2207) = pole wrt polar circle of trilinear polar of X(305) (line X(525)X(3267))
X(2207) = polar conjugate of X(305)
X(2207) = X(92)-isoconjugate of X(3964)
X(2207) = crossdifference of every pair of points on line X(520)X(3265)
X(2208) lies on these lines: 21,84 25,604 41,184 48,55 56,1413 105,1422 282,2359 1036,1433 1319,2218 1903,2267 2204,2206
X(2208) = isogonal conjugate of X(322)
X(2208) = X(2192)-Ceva conjugate of X(31)
X(2208) = X(i)-cross conjugate of X(j) for these (i,j): (1397,31), (1973,604)
X(2208) = crosspoint of X(1413) and X(1436)
X(2208) = barycentric product of vertices of hexyl triangle
X(2209) lies on these lines: 1,1258 6,31 8,238 41,904 69,2239 86,750 100,1740 560,692 604,1911 869,2300 941,2344 2175,2210
X(2209) = isogonal conjugate of X(6384)
X(2209) = X(32)-Ceva conjugate of X(31)
X(2209) = X(2176)-cross conjugate of X(31)
X(2209) = crosspoint of X(i) and X(j) for these (i,j): (692,765), (1403,2176)
X(2209) = crosssum of X(244) and X(693)
X(2210) lies on these lines: 6,650 31,32 36,58 42,251 239,1281 284,2309 667,838 894,1582 902,1110 904,2194 1918,2220 1964,2174 2175,2209
X(2210) = isogonal conjugate of X(334)
X(2210) = crosspoint of X(i) and X(j) for these (i,j): (58,1438), (238,2201), (1428,1914)
X(2210) = crosssum of X(1920) and X(1921)
X(2210) = perspector of conic {A,B,C,PU(12)}
X(2210) = intersection of trilinear polars of P(12) and U(12)
X(2211) lies on these lines: 4,6 25,263 32,682 112,1691 182,1968 186,2076 232,511 459,1613 512,1692
X(2211) = X(232)-Ceva conjugate of X(237)
X(2211) = isogonal conjugate of polar conjugate of X(34854)
X(2211) = X(92)-isoconjugate of X(6394)
X(2212) lies on these lines: 4,238 6,2356 9,33 19,2195 24,601 25,31 41,2204 108,1423 171,459 213,1973 427,748 468,750 607,1253 1041,1445 1471,1876 1474,2279 1918,2207 2189,2311
X(2212) = X(i)-Ceva conjugate of X(j) for these (i,j): (25,1973),
(33,41), (2299,607)
X(2212) = X(1918)-cross conjugate of X(2175)
X(2212) = crosspoint of X(i) and X(j) for these (i,j): (25,607),
(2204,2299)
X(2212) = crosssum of X(i) and X(j) for these (i,j): (69,348),
(307,1231)
X(2213) lies on these lines: 3,1412 56,71 57,72 65,1435 69,1434 73,1407 221,1245 738,1439 1903,2285
X(2213) = isogonal conjugate of X(452)
X(2214) lies on these lines: 1,1333 6,10 19,2203 31,37 65,604 75,81 225,608 284,994 534,553 584,2304 739,835 940,2221 1953,2217 2218,2294
X(2214) = cevapoint of X(31) and X(2304)
X(2214) = trilinear pole of line X(661)X(667)
X(2215) lies on these lines: 1,71 6,228 31,1474 34,1400 48,58 56,1409 63,86 937,2183 1220,1732
X(2215) = isogonal conjugate of X(5271)
X(2216) lies on these lines: 31,91 47,75 225,1179 1953,2168 2172,2186
X(2216) = cevapoint of X(31) and X(1953)
X(2216) = X(2618)-cross conjugate of X(162)
Contribution by Angel Montesdeoa, September 10, 2017: (start)
In the plane of a triangle BAC, let
O = X(3) = circumcenter
U(O,q) = circle with center O and radius q
A'B'C' = excentral triangle
A*B*C* = polar triangle of A'B'C' with respect to U(O,q).
There exists a unique q such that the lines AA*, BB*, CC* concur, and the point of concurrence is X(2217). Moreover, the radius and a barycentric equation for U(O,q) are given as follows:
q^2 = (4 r R^2 s(r+R))/(a b c) = (a b c (2 a b c+(a+b-c) (a-b+c)(-a+b+c)))/(2 (a+b-c) (a-b+c) (-a+b+c) (a+b+c));
abc(x^2+y^2+z^2) + 2a (a+b) (a+c)y z + 2b(b+c)(b+a)z x + 2c(c+a)(c+b)x y = 0. (end)
X(2217) lies on these lines: 1,1437 3,10 19,1333 21,1610 28,158 37,48 56,225 58,994 65,603 75,1444 1104,1472 1953,2214
X(2217) = isogonal conjugate of X(3869)
X(2217) = cevapoint of X(667) and X(2170)
X(2217) = X(1402)-cross conjugate of X(6)
X(2217) = X(21)-vertex conjugate of X(21)
X(2218) lies on these lines: 1,1762 10,55 19,2204 21,75 25,225 28,1612 31,65 37,41 105,1305 209,1724 595,994 596,993 1001,1036 1319,2208 1953,2219 2214,2294
X(2218) = isogonal conjugate of X(3868)
X(2218) = X(272)-Ceva conjugate of X(1751)
X(2218) = X(228)-cross conjugate of X(6)
X(2219) lies on these lines: 1,1744 6,225 10,219 19,2194 37,212 48,65 75,1812 158,1172 994,1630 1953,2218
X(2220) lies on these lines: 3,6 31,872 37,82 172,1100 560,692 604,1415 609,1449 906,1743 1918,2210 2174,2251
X(2220) = X(i)-Ceva conjugate of X(j) for these (i,j): (765,692),
(849,31)
X(2220) = crosspoint of X(163) and X(1252)
X(2220) = crosssum of X(1086) and X(1577)
X(2220) = crossdifference of every pair of points on line X(523)X(2530)
X(2220) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37482)
X(2220) = {X(371),X(372)}-harmonic conjugate of X(37482)
X(2221) lies on these lines: 2,2298 3,31 6,63 57,608 58,1473 84,1039 222,604 295,1911 394,2300 739,1310 940,2214 967,2260 1203,1707 1333,1790
X(2221) = isogonal conjugate of X(2345)
X(2221) = cevapoint of X(i) and X(j) for these (i,j): (6,1191), (1245,2281)
X(2221) = X(2281)-cross conjugate of X(1472)
Let A'B'C' be the excentral triangle in the plane of a triangle ABC. Let
A'' = X(110)-of-A'BC, and define B'' and C'' cyclically
(Oa) = circle with diameter AA'', and define (Ob) and (Oc) cyclically
The circles (Oa), (Ob), (Oc), and the circumcircle concur in X(2222). See Hyacinthos 24369. (Antreas Hatzipolakis and Angel Montesdeoca, July 12, 2016)
X(2222) lies on the circumcircle, the cubic K230, and these lines: {1,953}, {3,2716}, {12,3109}, {35,2687}, {36,80}, {40,2745}, {55,2717}, {56,2718}, {57,840}, {59,110}, {74,484}, {98,5143}, {99,4998}, {100,522}, {101,650}, {102,517}, {103,1155}, {105,2006}, {106,1168}, {108,7649}, {109,513}, {112,7115}, {162,933}, {171,2699}, {200,2750}, {243,917}, {651,4588}, {661,2149}, {675,1447}, {741,5061}, {759,859}, {910,2161}, {915,1785}, {919,1024}, {934,3676}, {972,5537}, {1141,2166}, {1295,2077}, {1298,1956}, {1308,2283}, {1309,4242}, {1376,2751}, {1477,3660}, {1793,5080}, {2249,2341}, {2734,6796}, {2739,7580}, {2747,5285}, {2756,5687}, {4552,9070}, {5089,9085}, {5193,8686}
X(2222) = midpoint of X(484) and X(3465)
X(2222) = reflection of X(2716) in X(3)
X(2222) = isogonal conjugate of X(3738)
X(2222) = cevapoint of X(i) and X(j) for these (i,j): (6,8648), (513,1319), (661,3724), (663,2183)
X(2222) = X(i)-cross conjugate of X(j) for these (i,j): (513,1168), (1635,57), (1769,1), (2173,7128), (3724,2149), (5172,59), (8648,6)
X(2222) = inverse-in-Stevanovic-circle of X(101)
X(2222) = trilinear pole of the line X(6)X(1411)
X(2222) = {X(10016,X(10017)}-harmonic conjugate of X(2716)
X(2222) = concurrence of reflections in sides of ABC of line X(4)X(80)
X(2222) = Ψ(X(1), X(5))
X(2222) = Ψ(X(4), X(80))
X(2222) = Λ(X(80), X(3762))
X(2222) = reflection of X(109) in line X(1)X(3)
X(2222) = trilinear product of circumcircle intercepts of line X(1)X(5)
X(2222) = X(i)-isoconjugate of X(j) for these (i,j): {1,3738}, {2,654}, {6,3904}, {9,3960}, {36,522}, {54,6369}, {55,4453}, {60,6370}, {75,8648}, {284,4707}, {320,663}, {513,4511}, {514,2323}, {521,1870}, {526,3615}, {650,3218}, {693,2361}, {758,3737}, {1443,3900}, {1459,5081}, {1464,7253}, {1577,4282}, {1983,4858}, {2167,2600}, {2170,4585}, {2185,2610}, {2245,4560}, {3936,7252}, {3939,4089}, {4242,7004}, {4391,7113}
X(2223) lies on these lines: 1,3 6,2279 10,1009 31,32 39,42 48,2175 100,239 187,237 212,1397 238,2110 292,1438 516,1284 572,2330 574,2177 604,1253 612,2340 674,2245 760,1959 991,1469 1014,2346 1037,1804 1110,2149 1400,2293 1423,1742 1486,2178 1918,1964
X(2223) = isogonal conjugate of X(2481)
X(2223) = complement of X(20556)
X(2223) = anticomplement of X(20544)
X(2223) = X(92)-isoconjugate of X(1814)
X(2223) = trilinear pole of PU(97)
X(2223) = crossdifference of every pair of points on line X(2)X(650)
X(2223) = {X(1),X(3)}-harmonic conjugate of X(37575)
X(2223) = {X(55),X(56)}-harmonic conjugate of X(37580)
X(2224) lies on these lines: 1,692 2,101 36,291 57,609 81,163 274,662 279,1461 1429,2006
X(2224) = X(i)-Ceva conjugate of X(j) for these (i,j): (911,41), (1037,1362)
X(2224) = cevapoint of X(55,2110)
X(2224) = crosspoint of X(i) and X(j) for these (i,j): (6,105), (31,1911), (59,919), (672,1458)
X(2224) = crosssum of X(i) and X(j) for these (i,j): (2,518), (11,918), (75,350), (105,1814)
X(2224) = eigencenter of Gemini triangle 7
X(2225) lies on these lines: 31,32 38,1107 44,513 63,169 1334,2179 2333,2355
X(2225) = crosssum of X(1) and X(2225)
X(2225) = isogonal conjugate of X(37130)
X(2225) = crossdifference of every pair of points on line X(1)X(693)
X(2226) lies on these lines: 36,106 44,88
X(2226) = isogonal conjugate of X(4370)
X(2226) = isotomic conjugate of X(36791)
X(2226) = X(679)-Ceva conjugate of X(1318)
X(2226) = cevapoint of X(6) and X(106)
X(2226) = X(i)-cross conjugate of X(j) for these (i,j): (6,106), (649,901)
X(2226) = X(92)-isoconjugate of X(22371)
X(2226) = crossdifference of every pair of points on line X(3251)X(4543)
X(2226) = trilinear pole of line X(106)X(1960) (the tangent to circumcircle at X(106))
X(2227) lies on these lines: 2,256 31,1582 38,75 44,513 662,1933 1581,1959 1930,2085 1966,1967
X(2227) = X(i)-Ceva conjugate of X(j) for these (i,j): (1966,1959), (1967,38)
X(2227) = crosspoint of X(75) and X(1581)
X(2227) = crosssum of X(i) and X(j) for these (i,j): (1,2227), (31,1580)
X(2228) lies on these lines: 10,75 44,513 291,320 1193,1386
X(2228) = crosssum of X(1) and X(2228)
X(2229) lies on these lines: 2,39 42,2295 44,513
X(2230) lies on these lines: 1,76 42,536 44,513
X(2230) = crosssum of X(1) and X(2230)
X(2231) lies on these lines: 6,76 44,513
X(2232) lies on these lines: 31,76 44,513
X(2232) = crosssum of X(1) and X(2232)
X(2233) lies on these lines: 32,76 44,513
X(2233) = crosssum of X(1) and X(2233)
X(2234) lies on these lines: 1,75 44,513 560,1958 662,922 872,894 897,1581
X(2234) = isogonal conjugate of X(37132)
X(2234) = crossdifference of every pair of points on line X(1)X(798)
X(2234) = crosssum of X(1) and X(2234)
X(2235) lies on these lines: 6,75 9,1740 37,1964 44,513 312,1613 1197,1215
X(2235) = crosssum of X(1) and X(2235)
X(2236) lies on these lines: 31,75 38,1964 44,513 63,1740 171,385 238,1281 1580,1933 1923,1930 1926,1966
X(2236) = crosspoint of X(1580) and X(1966)
X(2236) = crosssum of X(i) and X(j) for these (i,j): (1580,1966), (1581,1967)
X(2237) lies on these lines: 32,75 44,513 1740,1759
X(2237) = crosssum of X(1) and X(2237)
X(2238) lies on these lines: 1,1573 2,6 8,2176 9,43 10,213 32,1009 37,42 44,513 218,442 220,1834 238,1914 239,350 291,1757 294,857 405,2271 429,607 748,2280 800,1713 860,1783 862,2201 978,2275 1107,1193 1778,2305
X(2238) = isogonal conjugate of X(37128)
X(2238) = complement of X(30941)
X(2238) = cevapoint of X(i) and X(j) for these {i,j}: {6, 9509}, {1757, 2664}, {20683, 21830}
X(2238) = crosssum of X(i) and X(j) for these {i,j}: {1, 2238}, {6, 3286}, {9, 35104}, {81, 1931}, {86, 2669}, {291, 292}, {659, 1015}, {741, 2311}, {17103, 33295}
X(2238) = trilinear pole of line X(4094)X(4155)
X(2238) = X(i)-Ceva conjugate conjugate of X(j) for these (i,j): (98,55), (660,512), (666,523), (1929,1962), (2665,2667)
X(2238) = crosspoint of X(i) and X(j) for these (i,j): (1,37128), (2,13576), (37,9278), (42,2107), (57, 35108), (238,239), (660,1016). (740,16609)
X(2238) = bicentric sum of PU(90)
X(2238) = PU(90)-harmonic conjugate of X(4367)
X(2238) = {X(10),X(213)}-harmonic conjugate of X(2295)
X(2239) lies on these lines: 2,31 38,42 43,63 44,513 69,2209 141,1918 239,730
X(2239) = crosssum of X(1) and X(2239)
X(2240) lies on these lines: 2,32 43,1759 44,513
X(2240) = crosssum of X(1) and X(2240)
X(2241) lies on these lines: 1,32 3,1015 6,574 39,55 56,187 99,330 115,1479 213,2280 230,496 405,1573 498,1506 577,1062 613,1692 902,1475 1191,2271 1388,1415 1504,2066 1870,1968
X(2242) lies on these lines: 1,32 3,1500 6,101 36,574 37,993 39,56 42,1055 55,187 58,2176 89,1252 99,192 115,1478 213,1468 230,495 474,1574 499,1506 577,1060 611,1692 713,932 956,1573 1415,2099 1504,2067 2092,2178 2251,2280 2260,2273
X(2243) lies on these lines: 1,32 37,902 44,513
X(2243) = crosssum of X(1) and X(2243)
X(2244) lies on these lines: 1,82 44,513 922,1959
X(2244) = crosssum of X(1) and X(2244)
Let L denote the line having trilinears of X(2245) as coefficients. Then L is the line passing through X(1) perpendicular to the Euler line.
X(2245) lies on these lines: 3,6 9,46 19,407 36,2323 37,65 40,1834 44,513 63,1211 199,2194 209,228 219,2178 377,966 429,1452 440,1708 484,2161 674,2223 1100,2260 1195,2264 1405,2267 1454,2285 1631,2175 1732,2082
X(2245) = isogonal conjugate of X(24624)
X(2245) = inverse-in-Brocard-circle of X(2278)
X(2245) = X(i)-Ceva conjugate conjugate of X(j) for these (i,j): (54,215), (901,512), (1983,654), (2250,37)
X(2245) = cevapoint of X(2088) and X(2624)
X(2245) = X(i)-cross conjugate conjugate of X(j) for these (i,j): (2088,2610), (2624,1983)
X(2245) = crosspoint of X(59) and X(655)
X(2245) = crosssum of X(i) and X(j) for these (i,j): (1,2245), (6,859), (11,654), (80,2161), (759,2341), (2610,2611)
X(2245) = crossdifference of every pair of points on line X(1)X(523) (the perspectrix of Gemini triangles 10 and 27)
X(2245) = perspector of unary cofactor triangles of Gemini triangles 10 and 27
X(2245) = {X(371),X(372)}-harmonic conjugate of X(5398)
X(2246) lies on these lines: 1,41 6,244 9,100 19,1830 44,513 45,55 57,88 1054,1743 1731,2078
X(2246) = isogonal conjugate of X(37131)
X(2246) = crossdifference of every pair of points on line X(1)X(2254)
X(2246) = crosssum of X(1) and X(2246)
X(2247) lies on these lines: 1,163 19,162 31,2153 44,513 48,2157 63,662
X(2247) = crosssum of X(1) and X(2247)
X(2248) lies on these lines: 2,1931 31,2054 37,171 42,172 694,741
X(2248) = X(58)-cross conjugate of X(6)
X(2248) = X(i)-cross conjugate of X(j) for these (i,j): (58,6), (893,2162), (1929,2109)
X(2249) lies on the circumcircle and these lines: 19,107 31,112 48,110 63,99 71,100 81,934 101,228 108,1172 109,284 925,1820 933,2148 935,2157 1288,2158 1289,2156 1301,2155 1304,2159 1305,1952 1309,2250 2222,2341
X(2249) = isogonal conjugate of X(8680)
X(2249) = Ψ(X(6), X(810))
X(2249) = Ψ(X(190), X(72))
X(2249) = trilinear pole of line X(6)X(810)
X(2249) = cevapoint of X(1945) and X(1949)
X(2250) lies on these lines: 9,48 31,33 37,1409 63,190 71,1018 210,228 226,1020 515,2183 654,900 1309,2249 1400,1826 1766,1820
X(2250) = crosspoint of X(37) and X(2245)
X(2250) = crosspoint of X(517) and X(2183)
X(2251) lies on these lines: 6,36 31,32 101,1914 187,672 667,788 758,2243 902,1017 1015,1055 1319,2087 2174,2220 2242,2280
X(2251) = isogonal conjugate of X(20568)
X(2251) = polar conjugate of isotomic conjugate of X(23202)
X(2251) = crosssum of X(i) and X(j) for these {i,j}: {2, 320}, {75, 4358}, {903, 4997}, {1086, 4927}, {1111, 3762}, {4049, 6549}
X(2251) = crosspoint of X(i) and X(j) for these {i,j}: {6, 6187}, {31, 9456}, {902, 1404}, {1110, 32665}
X(2251) = crossdifference of every pair of points on line X(75)X(693)
X(2251) = trilinear product of PU(99)
X(2251) = trilinear product X(i)*X(j) for these {i,j}: {2, 9459}, {6, 902}, {31, 44}, {32, 519}, {41, 1319}, {42, 3285}, {55, 1404}, {101, 1960}, {106, 1017}, {110, 14407}, {163, 4730}, {184, 8756}, {560, 4358}, {604, 3689}, {667, 1023}, {678, 9456}, {692, 1635}, {1110, 2087}, {1397, 2325}, {1415, 4895}, {1492, 14436}, {1501, 3264}, {1576, 4120}, {1918, 16704}, {1919, 17780}, {1922, 4432}, {1973, 5440}, {1974, 3977}, {2175, 3911}, {2206, 3943}, {2429, 8643}, {4434, 7104}, {6187, 17455}
X(2252) lies on these lines: 3,48 6,1195 19,46 35,2302 39,2288 44,513 65,1108 216,1409 563,2269 573,2261 583,2262 909,2077 1182,2285 1400,1454 1765,1826 2180,2354
X(2252) = isogonal conjugate of X(37203)
X(2252) = crossdifference of every pair of points on line X(1)X(7649)
X(2252) = X(i)-Ceva-conjugate of X(j) for these (i,j): (909,48), (2006,73)
X(2252) = crosspoint of X(i) and X(j) for these (i,j): (1,37203), (63,1795), (914,1737)
X(2252) = crosssum of X(i) and X(j) for these (i,j): (1,2252), (19,1785), (652, 35015), (913, 36052)
X(2253) lies on these lines: 19,1708 44,513 48,184 2180,2355
X(2253) = X(911)-Ceva conjugate of X(48)
X(2253) = crosssum of X(1) and X(2253)
The tangent to the nine-point circle at the point where it is tangent to the incircle (X(11), the Feuerbach point) is here named the Feuerbach tangent line. It is line X(11)X(244). The inverse of the incircle in the excircles radical circle is a circle tangent to the Apollonius circle at X(3030). The line tangent to the Apollonius circle at X(3030) is here named the Apollonius tangent line. It is line X(2254)X(3030). X(2254) is the intersection of the Feuerbach tangent line and the Apollonius tangent line. (Randy Hutson, January 15, 2019)
X(2254) lies on these lines: 11,244 44,513 88,1156 100,109 103,105 291,812 294,1027 514,1734 522,693 663,905 665,1642 690,2292 764,891 830,1019 926,1362 1054,1768 3030,6085
X(2254) = reflection of X(i) and X(j) for these (i,j): (661,1491), (663,905)
X(2254) = isogonal conjugate of X(36086)
X(2254) = anticomplement of X(3716)
X(2254) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,38980), (291,244), (673,2170), (813,38), (1025,672), (1026,518)
X(2254) = cevapoint of X(665) and X(926)
X(2254) = crosspoint of X(i) and X(j) for these (i,j): (100,1280), (513,876), (518,1026), (664,673)
X(2254) = crosssum of X(i) and X(j) for these (i,j): (1,2254), (105,1027), (513,1279), (663,672), (1024,2195)
X(2254) = X(666)-aleph conjugate of X(812)
X(2254) = radical center of {incircle, nine-point circle, Apollonius circle}
X(2254) = midpoint of Feuerbach hyperbola intercepts of antiorthic axis
X(2254) = perspector of hyperbola {{A,B,C,X(1),X(514)}}
X(2255) lies on these lines: 6,40 31,198 84,1108 208,608 221,604 1191,1333
X(2256) lies on these lines: 1,6 8,965 48,55 56,71 104,2335 145,2287 198,2269 221,2286 388,1901 579,999 584,1190 604,1334 610,1697 1000,1172 1214,1407 1333,2328 1604,1630 1696,2183 1953,2098 2099,2294
X(2256) = crosssum of X(1) and X(2257)
X(2256) = {X(1),X(9)}-harmonic conjugate of X(1108)
X(2257) lies on these lines: 1,6 3,380 19,57 40,387 48,1420 56,610 58,84 71,1697 198,1617 281,1210 347,1445 393,1838 604,1630 608,1394 800,2277 1400,2082 1475,2285 1696,2348 1699,1901 1778,2328
X(2257) = crosssum of X(i) and X(j) for these (i,j): (9,1706), (939,2343)
Let A3B3C3 be the 3rd Conway triangle. Let La be the trilinear polar of A3, and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(2258). (Randy Hutson, December 10, 2016)
X(2258) lies on these lines: 1,333 6,1402 9,42 31,284 55,213 57,959 200,872 386,1245 741,931 869,2319 1973,2299 2308,2364
X(2258) = X(2484)-cross conjugate of X(101)
X(2258) = crosspoint of X(941) and X(959)
X(2258) = crosssum of X(940) and X(958)
X(2258) = perspector of ABC and unary cofactor triangle of 3rd Conway triangle
X(2259) lies on these lines: 6,2197 9,943 19,41 35,71 42,2299 48,57 55,584 65,2160 101,2294 306,319 572,2364 1474,2200 1826,2332 2161,2264
X(2259) = isogonal conjugate of X(5249)
X(2259) = cevapoint of X(i) and X(j) for these (i,j): (6,2174), (31,2200), (41,42)
X(2259) = X(661)-cross conjugate of X(101)
X(2259) = crosssum of X(442) and X(2294)
X(2260) lies on these lines: 1,71 6,41 9,1125 11,1901 19,57 31,1486 36,284 37,38 42,2352 58,1474 65,1108 219,999 226,1713 238,1778 573,1449 603,608 910,1202 942,2294 946,1765 967,2221 1015,2300 1100,2245 1195,2170 1203,2360 1210,1826 1393,1880 1409,1457 1617,2266 1731,1781 1914,2305 2173,2264 2191,2279 2194,2308 2242,2273
X(2260) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,649),
(1020,513)
X(2260) = crosspoint of X(i) and X(j) for these (i,j): (1,27),
(6,2160), (57,58)
X(2260) = crosssum of X(i) and X(j) for these (i,j): (1,71), (9,10),
(1794,2259)
X(2261) lies on these lines: 1,2317 6,19 9,48 33,184 44,198 154,1864 573,2252 579,610 1181,1753 1449,1953 1630,1723 1728,2360 1766,2323 2173,2270 2268,2302
X(2262) lies on these lines: 1,198 4,1903 6,19 9,374 37,1953 48,354 51,1824 57,1422 71,1212 169,219 184,2355 185,1839 389,1871 393,1875 583,2252 610,942 909,2160 960,966 999,1604 1012,1741 1086,1122 1108,1400 1146,1826 1172,1905 1202,2272 1214,1730 1319,2178 1455,2199 1503,1890 1696,2098 1828,1901 1836,1851 2171,2347 2173,2317
X(2262) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1856), (1461,513)
X(2262) = crosspoint of X(i) and X(j) for these (i,j): (1,189), (4,57)
X(2262) = crosssum of X(i) and X(j) for these (i,j): (1,198), (3,9)
X(2262) = {X(6),X(19)}-harmonic conjugate of X(2182)
X(2262) = intersection of tangents to hyperbola {{A,B,C,X(4),X(40),X(57)}} at X(4) and X(57)
X(2263) lies on these lines: 1,7 4,1041 6,19 31,57 33,1836 40,1253 42,223 55,1427 56,1279 196,1096 226,612 238,1445 241,1001 354,1407 968,1214 1394,1468
X(2264) lies on these lines: 3,1723 6,19 9,55 37,41 40,1743 44,71 48,1108 56,610 218,1766 281,1837 284,1731 294,2298 572,2301 579,1155 910,1200 942,1781 960,1183 1100,2170 1118,1249 1172,1859 1195,2245 1212,2268 1333,1951 1436,1470 1834,1842 1839,1901 1898,1903 2161,2259 2173,2260 2174,2302
X(2264) = crosspoint of X(i) and X(j) for these (i,j): (9,1172),
(21,57)
X(2264) = crosssum of X(i) and X(j) for these (i,j): (9,65),
(57,1214)
X(2265) lies on these lines: 6,1411 9,48 19,1743 37,2317 44,513 163,2341 692,2310
X(2265) = crosssum of X(1) and X(2265)
X(2266) lies on these lines: 1,1802 6,31 19,1174 41,65 48,57 165,284 1617,2260 2093,2301
X(2267) lies on these lines: 3,2183 6,31 9,48 37,604 41,44 169,2173 219,2317 284,1743 374,910 573,2077 1212,2182 1334,1404 1400,1470 1405,2245 1474,2297 1766,1953 1802,2302 1903,2208 2285,2294
X(2268) lies on these lines: 1,572 2,1958 3,1400 6,31 9,21 19,1831 33,1474 35,573 37,48 44,584 45,2174 73,478 219,1334 281,2202 312,2185 346,2329 380,2082 608,1593 748,992 750,851 941,987 950,964 968,2187 1212,2264 1438,2297 1449,1697 1903,2188 1950,2199 2112,2310 2261,2302
X(2268) = X(i)-Ceva conjugate of X(j) for these (i,j): (940,1468), (987,31)
X(2268) = crosspoint of X(i) and X(j) for these (i,j): (1,2339), (940,958)
X(2268) = crosssum of X(i) and X(j) for these (i,j): (1,2285), (941,959)
X(2268) = {X(37),X(48)}-harmonic conjugate of X(9310)
X(2269) lies on these lines: 1,573 3,604 6,31 8,9 11,1213 35,572 37,1953 40,2285 41,219 48,836 60,283 63,193 185,1409 198,2256 497,966 517,2171 563,2252 579,1449 899,992 1100,2245 1193,1682 1201,2277 1496,2199 1829,2292 1936,2303
X(2269) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,652),
(643,663), (1018,650)
X(2269) = crosspoint of X(i) and X(j) for these (i,j): (1,333),
(9,284)
X(2269) = crosssum of X(i) and X(j) for these (i,j): (1,1400),
(2,1999), (57,226), (961,2298)
X(2270) lies on these lines: 1,198 4,9 6,57 37,1697 41,1630 46,1743 48,1449 56,1604 63,391 84,1741 165,374 208,1249 294,2155 517,2324 965,1764 1394,2199 1400,2082 1420,2178 1750,1903 2173,2261 2285,2347
X(2270) = X(189)-Ceva conjugate of X(1)
X(2270) = X(189)-aleph conjugate of X(2270)
X(2270) = pole of Gergonne line wrt Bevan circle
X(2270) = homothetic center of 3rd extouch triangle and polar triangle of the Bevan circle
X(2270) = perspector of any pair of these triangles: {Danneels-Bevan, tangential of excentral, unary cofactor of intangents}
X(2271) lies on these lines: 3,6 25,41 55,213 218,2276 220,1500 405,2238 941,2287 1191,2241 1193,2280 1334,2177 1460,2304 2207,2332
X(2272) lies on these lines: 6,1200 19,57 44,513 48,55 71,165 101,2077 198,1615 354,1953 603,607 909,2291 1202,2262 2280,2317
X(2272) = X(2338)-Ceva conjugate of X(6)
X(2272) = crosspoint of X(57) and X(103)
X(2272) = crosssum of X(i) and X(j) for these (i,j): (1,2272),
(9,516)
X(2273) lies on these lines: 1,6 32,71 39,48 41,2092 42,2175 101,2277 172,579 284,2276 609,2305 1409,2156 1974,2356 2242,2260
X(2274) lies on these lines: 1,75 3,1918 6,41 42,750 58,560 238,993 518,869 873,2296 1001,1201
X(2275) lies on these lines: 1,39 2,330 3,1914 6,41 8,1575 9,1050 32,36 34,232 35,574 46,1572 57,893 65,2186 87,256 194,350 213,995 216,1038 579,2300 614,1194 672,1201 978,2238 1149,1334 1449,2092 1573,1698
X(2275) = isogonal conjugate of X(17743)
X(2275) = crosspoint of X(i) and X(j) for these (i,j): (57,87), (58,274)
X(2275) = crosssum of X(i) and X(j) for these (i,j): (6,1376), (10,213)
X(2276) lies on these lines: 1,39 2,37 3,172 6,31 8,1107 9,43 32,35 33,232 36,574 44,751 45,899 46,1571 100,743 101,753 187,609 194,1909 213,386 216,1040 218,2271 284,2273 612,1194 749,1100 1193,1334 1400,1403 1574,1698 2197,2285
X(2276) = complement of X(4441)
X(2276) = anticomplement of X(21264)
X(2276) = crosspoint of X(2) and X(1002)
X(2276) = crosssum of X(6) and X(1001)
X(2277) lies on these lines: 1,2092 2,37 6,41 9,39 19,232 31,199 71,2176 101,2273 213,579 256,1740 404,2298 478,1470 573,995 800,2257 966,1107 1015,1449 1201,2269 1444,1778
X(2277) = crosspoint of X(2) and X(959)
X(2277) = crosssum of X(6) and X(958)
X(2278) lies on these lines: 2,662 3,6 9,2174 35,2323 37,48 41,44 45,101 46,1449 55,184 65,604 560,2309 563,2252 940,1790 965,1800 1011,2194 1107,2304 1155,2280
X(2278) = inverse-in-Brocard-circle of X(2245)
X(2278) = {X(371),X(372)}-harmonic conjugate of X(5396)
X(2279) lies on these lines: 1,672 6,2223 9,86 31,1438 34,2333 41,58 56,213 87,1743 269,1400 614,2350 649,1027 1055,2163 1474,2212 2191,2260
X(2279) = isogonal conjugate of X(4384)
X(2279) = isotomic conjugate of X(21615)
X(2279) = X(869)-cross conjugate of X(1)
X(2280) lies on these lines: 1,41 3,1475 6,31 9,1174 32,1468 34,2332 37,2348 48,354 57,77 145,2329 165,572 213,2241 218,1334 380,2285 748,2238 985,1002 999,1055 1155,2278 1185,2300 1193,2271 1212,1802 1400,1617 1405,2078 1829,1973 1851,2201 2242,2251 2272,2317 2291,2364
X(2280) = isogonal conjugate of X(27475)
X(2280) = X(985)-Ceva conjugate of X(31)
X(2281) lies on these lines: 6,63 19,2207 31,1974 32,48 71,213 228,1918 729,1310 966,981
X(2281) = X(2221)-Ceva conjugate of X(1245)
X(2281) = crosspoint of X(1472) and X(2221)
X(2282) lies on these lines: 1,228 2,71 28,31 48,81 57,1409 63,274 278,1400 959,1457 961,1451
X(2283) lies on these lines: 1,3 100,658 101,109 108,1292 692,1813 1025,1026 1308,2222
X(2283) = isogonal conjugate of X(885)
X(2283) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,1362), (927,651), (1025,2284)
X(2283) = cevapoint of X(i) and X(j) for these (i,j): (665,2223), (672,926)
X(2283) = X(i)-cross conjugate of X(j) for these (i,j): (665,241), (926,672), (1362,59)
X(2283) = crosspoint of X(i) and X(j) for these (i,j): (100,677), (651,927), (1025,1026)
X(2283) = crosssum of X(i) and X(j) for these (i,j): (513,676), (650,926)
X(2283) = trilinear pole of line X(672)X(1362)
X(2284) lies on these lines: 1,6 101,692 190,644 813,875 874,1016
X(2284) = X(i)-Ceva conjugate of X(j) for these (i,j): (666,100), (1025,2283)
X(2284) = cevapoint of X(665) and X(672)
X(2284) = crosspoint of X(i) and X(j) for these (i,j): (100,666), (101,813), (1025,1026)
X(2284) = crosssum of X(i) and X(j) for these (i,j): (513,665), (514,812), (1042,1027)
X(2284) = trilinear pole of line X(672)X(2223)
X(2285) lies on these lines: 1,572 2,7 6,19 32,1950 37,56 40,2269 41,610 46,573 169,1046 194,2128 198,1466 208,429 380,2280 388,2345 583,1732 612,1460 851,2092 966,1788 968,1011 1038,2303 1041,1172 1100,2099 1182,2252 1404,1449 1454,2245 1470,2178 1475,2257 1903,2213 2197,2276 2199,2312 2267,2294 2270,2347
X(2285) = isogonal conjugate of X(2339)
X(2285) = X(i)-Ceva conjugate of X(j) for these (i,j): (388,612),
(2303,2286)
X(2285) = crosssum of X(9) and X(1697)
X(2286) lies on these lines: 1,608 3,2197 6,41 37,478 55,1950 63,77 71,603 221,2256 284,1037 388,2303 607,610 1413,2336 1496,2199
X(2286) = X(i)-Ceva conjugate of X(j) for these (i,j): (388,1460), (2303,2285)
X(2287) lies on these lines: 2,6 8,29 9,21 27,329 28,72 44,1333 58,936 63,610 71,100 145,2256 200,1253 210,2194 213,2298 218,1010 220,346 271,282 274,1170 294,314 307,651 319,1332 332,949 404,579 411,573 644,2321 662,911 758,1781 941,2271 960,1183 997,1723 1014,1445 1098,1792 1761,2173 1809,2193
X(2287) = isogonal conjugate of X(1427)
X(2287) = isotomic conjugate of X(1446)
X(2287) = X(i)-Ceva conjugate of X(j) for these (i,j): (333,21), (1098,2328)
X(2287) = cevapoint of X(i) and X(j) for these (i,j): (6,610), (9,219), (200,220)
X(2287) = X(i)-cross conjugate of X(j) for these (i,j): (9,2322), (200,1043), (219,2327), (220,2328), (2328,21)
X(2287) = crosspoint of X(333) and X(1043)
X(2287) = crosssum of X(1042) and X(1400)
X(2287) = trilinear pole of line X(1021)X(3900)
X(2288) lies on these lines: 6,19 32,48 39,2252 213,2183 216,1195 394,2339 893,909 1914,2302
X(2289) lies on these lines: 3,48 9,21 40,1630 101,610 212,2193 220,2174 255,577 268,1260 306,1150 394,1804 579,604 584,1212 1944,1958
X(2289) = X(394)-Ceva conjugate of X(255)
X(2289) = crosspoint of X(394) and X(1259)
X(2289) = X(34)-isoconjugate of X(92)
X(2290) lies on these lines: 44,513 47,48 50,1399 1953,2179
X(2290) = crosssum of X(i) and X(j) for these (i,j): (1,2290), (1109,2624)
X(2291) lies on the circumcircle and these lines: 6,109 9,100 19,108 55,101 57,934 99,333 103,654 105,1024 106,665 110,284 112,2299 165,1292 573,1293 649,840 672,901 673,927 902,919 909,2272 910,2161 932,2319 1055,2078 1108,2160 1155,1308 1174,1200 1305,1751 1310,2339 1630,2164 2280,2364
X(2291) = isogonal conjugate of X(527)
X(2291) = anticomplement of X(31844)
X(2291) = circumcircle-antipode of X(28291)
X(2291) = cevapoint of X(6) and X(1055)
X(2291) = X(2078)-cross conjugate of X(1174)
X(2291) = trilinear pole of line X(6)X(663)
X(2291) = Ψ(X(1), X(650))
X(2291) = Ψ(X(6), X(663))
X(2291) = Ψ(X(190), X(8))
Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. The triangle A'B'C' is perspective to the incentral triangle at X(2292). (Randy Hutson, December 2, 2017)
Let FaFbFc be the Fuhrmann triangle, and AbBcCa and AcBaCb be the 1st and 2nd Montesdeoca bisector triangles. X(2292) is the radical center of the circumcircles of FaAbAc, FbBcBa, FcCaCb. (Randy Hutson, December 2, 2017)
X(2292) lies on these lines: 1,21 2,986 8,192 10,321 29,240 37,65 40,612 42,72 46,750 55,976 190,1220 221,2256 226,1254 244,1125 392,1201 690,2254 950,2310 960,1193 1042,1214 1107,2170 1717,1773 1761,2303 1829,2269 1868,1880
X(2292) = isogonal conjugate of X(2363)
X(2292) = reflection of X(2650) in X(1)
X(2292) = complement of X(17164)
X(2292) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1193), (109,656), (190,661)
X(2292) = crosspoint of X(i) and X(j) for these (i,j): (1,10), (75,226)
X(2292) = crosssum of X(i) and X(j) for these (i,j): (1,58), (31,284)
X(2292) = complement wrt incentral triangle of X(2650)
X(2292) = anticomplement wrt incentral triangle of X(1)
X(2292) = incentral isotomic conjugate of X(2667)
X(2293) lies on these lines: 1,7 3,1471 6,31 9,2340 33,1839 37,2310 48,1486 73,1456 200,391 284,2195 354,1418 500,1066 608,2356 651,2346 692,2317 756,1864 1001,1818 1201,1279 1400,2223
X(2293) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,354), (354,1475), (651,657), (1292,649)
X(2293) = crosspoint of X(i) and X(j) for these (i,j): (1,55), (6,2191), (354,1212)
X(2293) = crosssum of X(i) and X(j) for these (i,j): (1,7), (1170,2346)
X(2293) = {X(1),X(1742)}-harmonic Conjugate of X(7)
X(2293) = crossdifference of every pair of points on line X(514)X(657)
X(2293) = bicentric sum of PU(104)
X(2293) = PU(104)-harmonic conjugate of X(657)
X(2293) = isogonal conjugate of [intersection of tangents at X(1) and X(7) to hyperbola passing through X(1), X(7) and the excenters]
X(2293) = isogonal conjugate of [crosspoint of X(1) and X(7) wrt the excentral triangle]
X(2293) = perspector of ABC and unary cofactor triangle of Honsberger triangle
X(2294) lies on these lines: 1,19 9,758 21,1761 37,65 55,199 73,1880 81,1762 86,1959 101,2259 209,756 226,1826 281,1148 354,1108 942,2260 950,1839 1046,1778 1100,2170 1841,1859 2099,2256 2214,2218 2267,2285
X(2294) = X(i)-Ceva conjugate of X(j) for these (i,j): (92,1838),
(101,661), (653,656)
X(2294) = crosspoint of X(i) and X(j) for these (i,j): (1,226),
(10,92)
X(2294) = crosssum of X(i) and X(j) for these (i,j): (1,284), (48,58),
(943,2259)
X(2295) lies on these lines: 1,39 2,1258 6,8 10,213 37,65 42,2229 171,172 190,1655 345,940 672,1107 732,894 1016,1509 1193,1575
X(2295) = complement of X(17152)
X(2295) = crosspoint of X(i) and X(j) for these (i,j): (1,83), (171,894), (1016,1018)
X(2295) = crosssum of X(i) and X(j) for these (i,j): (1,39), (256,893), (1015,1019)
X(2295) = {X(10),X(213)}-harmonic conjugate of X(2238)
X(2296) lies on these lines: 1,310 2,213 7,1402 27,1973 31,86 42,75 675,785 873,2274 1042,1088
Let Oa be the circle passing through B and C, and tangent to the A-excircle. Define Ob and Oc cyclically. Let A' be the point of tangency of Oa and the A-excircle, and define B' and C' cyclically. Then X(2297) is the radical center of circles Oa, Ob, Oc; for the incircle version, see X(57). Note that triangle A'B'C' is perspective to the extouch triangle at X(1). If you have GeoGebra, you can view X(2297) and Perspective2297. (Jianing Song, April 7, 2024)
The point A' in Song's construction is given by
4*a^2*(a + b - c)*(a - b + c) : -((a - b + c)^3*(a + b + c)) : -((a + b - c)^3*(a + b + c))
The radical center of circles Oa, Ob, Oc is X(2297). A'B'C' is perspective to ABC at X(5423), and the tangents at A', B', C' form a triangle that is perspective to ABC at X(51341). (Peter Moses, April 78, 2024)
X(2297) lies on the conic {{A,B,C,X(1),X(6)}} and these lines: {1, 346}, {2, 269}, {6, 200}, {9, 56}, {10, 937}, {34, 281}, {37, 3445}, {58, 936}, {75, 44797}, {86, 17022}, {87, 5268}, {106, 3731}, {282, 1413}, {292, 16517}, {391, 8580}, {610, 31521}, {672, 15479}, {894, 4328}, {939, 25066}, {966, 8568}, {998, 9623}, {1027, 17418}, {1126, 6765}, {1411, 36910}, {1431, 17754}, {1438, 2268}, {1449, 2334}, {1474, 2267}, {1593, 12565}, {2163, 3973}, {2191, 10582}, {2256, 4936}, {2257, 5782}, {2279, 59269}, {2324, 36916}, {2345, 4853}, {2999, 3618}, {3030, 9432}, {3247, 41436}, {3305, 24557}, {3554, 17369}, {4334, 5296}, {4512, 53085}, {5018, 5199}, {5272, 7194}, {5293, 39946}, {5294, 26668}, {5437, 56199}, {5783, 16572}, {7097, 56857}, {7110, 52372}, {7129, 57492}, {7175, 7308}, {7274, 35578}, {10436, 36796}, {16713, 18229}, {17352, 54390}, {17353, 58004}, {18594, 33950}, {19861, 56200}, {25887, 60937}, {31438, 60849}
X(2297) = isogonal conjugate of the complement of X(34255)
X(2297) = isotomic conjugate of the complement of X(27340)
X(2297) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2999}, {2, 1191}, {6, 3672}, {56, 18228}, {57, 1697}, {58, 4656}, {81, 4646}, {100, 8712}, {104, 51413}, {668, 8662}, {934, 40137}, {999, 14556}, {58815, 58985}
X(2297) = X(i)-Dao conjugate of X(j) for these (i,j): {1, 18228}, {3, 2999}, {9, 3672}, {10, 4656}, {5452, 1697}, {8054, 8712}, {14714, 40137}, {32664, 1191}, {40194, 6554}, {40586, 4646}, {40613, 51413}
X(2297) = X(i)-cross conguate of X(j) for these (i,j): (612,1), (7050,7091)
X(2297) = cevapoint of X(i) and X(j) for these (i,j): {1, 8580}, {2, 27340}, {6, 54322}, {9, 1449}, {2484, 3248}, {2522, 34591}
X(2297) = trilinear pole of line {649, 3900}
X(2297) = barycentric product X(i)*X(j) for these {i,j}: {1, 1219}, {8, 7091}, {75, 7050}, {78, 11546}, {514, 6574}, {4397, 58985}, {11051, 44797}, {30705, 40175}
X(2297) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 3672}, {6, 2999}, {9, 18228}, {31, 1191}, {37, 4656}, {42, 4646}, {55, 1697}, {649, 8712}, {657, 40137}, {1219, 75}, {1919, 8662}, {2183, 51413}, {6574, 190}, {7050, 1}, {7091, 7}, {11546, 273}, {40137, 58815}, {40175, 6554}, {57279, 28616}, {58985, 934}
X(2297) = pole of line {3677, 4853} with respect to the ABCGI
X(2297) = pole of line {612, 2297} with respect to the ABCIK
X(2297) = pole of line {1191, 1697} with respect to the Kiepert circumhyperbola of the excentral triangle
X(2297) = pole of line {2999, 3672} with respect to the Jerabek circumhyperbola of the excentral triangle
X(2298) lies on these lines: 1,572 2,2221 4,608 6,8 7,940 9,31 19,1039 21,37 55,941 81,314 100,2092 171,256 213,2287 294,2264 335,1963 388,478 404,2277 1011,2335 1100,1320 1172,1824 1911,2309
X(2298) = isogonal conjugate of X(3666)
X(2298) = isotomic conjugate of X(20911)
X(2298) = cevapoint of X(i) and X(j) for these (i,j): (1,171), (6,37), (55,213)
X(2298) = X(i)-cross conjugate of X(j) for these (i,j): (6,1169), (512,100)
X(2298) = crosssum of X(i) and X(j) for these (i,j): (1193,2269), (2092,2292)
X(2298) = X(92)-isoconjugate of X(22345)
X(2299) lies on these lines: 4,580 6,25 9,33 19,31 21,1039 24,581 27,162 28,34 29,270 42,2259 55,607 73,2360 112,2291 238,1848 284,2189 909,2206 1104,1829 1174,2356 1333,1436 1402,1945 1435,1471 1824,2161 1841,2160 1859,2361 1973,2258
X(2299) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,1474), (29,284), (270,1172), (1172,2332), (2189,2204)
X(2299) = cevapoint of X(i) and X(j) for these (i,j): (23,31), (607,2212)
X(2299) = X(i)-cross conjugate of X(j) for these (i,j): (31,2194), (607,1172), (2204,1474), (2212,2204)
X(2299) = crosspoint of X(i) and X(j) for these (i,j): (28,1172), (270,2189)
X(2299) = crosssum of X(72) and X(1214)
X(2299) = isogonal conjugate of X(307)
X(2299) = polar conjugate of X(349)
X(2299) = crossdifference of every pair of points on line X(525)X(8611)
X(2300) lies on these lines: 1,6 10,992 19,1572 31,184 32,48 36,2305 39,71 81,261 110,1169 239,314 284,893 394,2221 572,595 573,995 579,2275 849,1333 869,2209 1015,2260 1185,2280 1193,1682 1201,1400 1918,1964 2174,2220
X(2300) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,2092), (110,667)
X(2300) = crosspoint of X(i) and X(j) for these (i,j): (6,1333), (31,904), (56,81)
X(2300) = crosssum of X(i) and X(j) for these (i,j): (2,321), (8,37), (75,1909), (1791,2298)
X(2300) = isogonal conjugate of X(30710)
X(2300) = polar conjugate of isotomic conjugate of X(22345)
X(2300) = perspector of unary cofactor triangles of Gemini triangles 39 and 40
X(2300) = crossdifference of every pair of points on line X(513)X(2517) (the perspectrix of Gemini triangles 39 and 40)
X(2301) lies on these lines: 1,19 40,41 55,101 102,949 218,573 572,2264 581,607 2093,2266
X(2302) lies on these lines: 1,19 6,2197 35,2252 54,71 55,184 198,584 1802,2267 1914,2288 2174,2264 2261,2268
X(2303) lies on these lines: 1,19 2,6 9,58 21,37 71,171 226,1396 241,1014 388,2286 1010,2345 1038,2285 1761,2292 1936,2269
X(2304) lies on these lines: 1,19 31,32 55,2200 274,1958 584,2214 1107,2278 1460,2271 2174,2176
X(2305) lies on these lines: 3,6 31,199 36,2300 37,171 71,172 198,1755 230,1901 332,524 404,992 609,2273 1010,1213 1195,1951 1400,1950 1460,1486 1654,1931 1778,2238 1914,2260 2176,2178
X(2305) = X(1400)-Ceva conjugate of X(6)
X(2305) = crosspoint of X(109) and X(239)
X(2305) = crosssum of X(115) and X(522)
X(2306) lies on these lines: 1,15 2,1081 6,1406 13,79 65,2153 1082,1255
X(2306) = X(1081)-Ceva conjugate of X(1251)
X(2307) lies on these lines: 1,61 6,41 11,398 12,396 15,35 36,62 65,2153
X(2307) = X(1082)-Ceva conjugate of X(1250)
X(2308) lies on these lines: 6,31 36,58 38,1386 44,756 56,1392 81,238 110,1171 171,899 184,1475 222,1471 612,1743 748,940 968,1449 999,1201 1042,1451 1100,1962 1197,1977 1397,1400 1402,1404 1405,1460 1458,2003 2194,2260 2203,2354 2258,2364
X(2308) = isogonal conjugate of X(1268)
X(2308) = X(110)-Ceva conjugate of X(649)
X(2308) = crosspoint of X(i) and X(j) for these (i,j): (1,2160), (6,58), (1125,1839)
X(2308) = crosssum of X(i) and X(j) for these (i,j): (2,10), (1126,1796)
X(2308) = {X(31),X(42)}-harmonic conjugate of X(902)
X(2308) = X(92)-isoconjugate of X(1796)
X(2308) = polar conjugate of isotomic conjugate of X(22054)
X(2309) lies on these lines: 1,87 2,1740 6,31 9,869 21,238 37,1964 44,872 86,310 239,1045 284,2210 560,2278 741,1963 1001,1201 1911,2298
X(2309) = X(99)-Ceva conjugate of X(649)
X(2309) = crosspoint of X(i) and X(j) for these (i,j): (1,893), (6,86), (58,2162)
X(2309) = crosssum of X(i) and X(j) for these (i,j): (1,894), (2,42), (10,192)
X(2309) = isogonal conjugate of isotomic conjugate of X(3741)
X(2309) = polar conjugate of isotomic conjugate of X(22065)
Let A'B'C' be the Mandart-incircle triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2310).
X(2310) lies on the incentral inellipse (a.k.a. Hofstadter ellipse E(1/2)) and these lines: 1,651 4,774 9,294 11,244 31,33 37,2293 38,497 42,1864 45,55 73,1898 84,1106 90,255 390,984 516,1736 522,1090 692,2265 748,1040 896,1776 926,2170 950,2292 971,1458 990,1471 1400,1827 1445,1721 1857,2181 2112,2268 2201,2312
X(2310) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,650), (4,661), (11,2170), (33,663), (84,649), ( 90,652), (104,1635), (1088,514, (1098,1021)
X(2310) = isogonal conjugate of X(7045)
X(2310) = crosspoint of X(i) and X(j) for these (i,j): (1,650, (9,522), (11,1146), (19,513), (514,1088), (1021,1098)
X(2310) = crosssum of X(i) and X(j) for these (i,j): (1,651), (57,109), (59,1262), (63,100), (101,1253), (255,1813), (269,934), (412,653), (1020,1254), (1106,1461)
X(2310) = polar conjugate of isotomic conjugate of X(34591)
X(2310) = trilinear square of X(650)
X(2310) = trilinear pole wrt incentral triangle of line X(1)X(3)
X(2310) = X(63)-isoconjugate of X(7128)
X(2311) lies on these lines: 6,694 9,261 21,644 28,291 41,60 58,101 110,2112 272,335 284,2330 651,1014 672,1931 759,813 1580,1581 2189,2212
X(2311) = isogonal conjugate of X(16609)X(2312) lies on these lines: 19,31 38,48 44,513 63,610 198,199 240,293 444,1400 774,1973 1436,1473 1910,1933 2157,2159 2199,2285 2201,2310
X(2312) = X(293)-Ceva conjugate of X(31)
X(2312) = crosspoint of X(19) and X(1910)
X(2312) = crosssum of X(i) and X(j) for these (i,j): (1,2312),
(63,1959)
X(2313) lies on these lines: 44,513 48,92 1953,2181
X(2313) = crosssum of X(1) and X(2313)
X(2314) lies on these lines: 19,47 44,513 48,1820 563,1953
X(2314) = crosssum of X(i) and X(j) for these (i,j): (1,2314), (19,1725)
X(2315) lies on these lines: 19,91 44,513 48,255 774,1953
X(2315) = X(2159)-Ceva conjugate of X(48)
X(2315) = crosssum of X(i) and X(j) for these (i,j): (1,2315),
(19,1784)
X(2316) is the perspector of the anticevian triangle of X(106) and the unary cofactor triangle of the intangents triangle, and also the perspector of ABC and the unary cofactor triangle of Gemini triangle 9. (Randy Hutson, November 30, 2018)
X(2316) lies on these lines: 6,101 9,644 19,1743 36,909 41,2364 44,517 57,88 284,2347 294,1024 329,1751 333,645 527,666 573,2164 579,1436 672,901 1318,2323
X(2316) = isogonal conjugate of X(3911)
X(2316) = X(i)-Ceva conjugate of X(j) for these (i,j): (88,106),
(1318,55)
X(2316) = cevapoint of X(i) and X(j) for these (i,j): (6,2183),
(41,2361)
X(2316) = X(i)-cross conjugate of X(j) for these (i,j): (55,1318),
(654,101), (2323,284), (2342,102)
X(2316) = crosssum of X(i) and X(j) for these (i,j): (44,1319),
(1149,2183), (1635,1647)
X(2317) lies on these lines: 1,2261 6,41 19,1449 37,2265 54,71 163,284 182,1818 219,2267 563,2252 692,2293 1100,1953 2173,2262 2272,2280
X(2318) lies on these lines: 6,2336 9,2335 33,200 37,42 41,55 63,1818 71,228 72,73 78,345 209,1400 212,219 672,2352 997,1450 1808,1812
X(2318) = X(72)-Ceva conjugate of X(71)
X(2319) lies on these lines: 1,893 6,43 55,2053 57,239 385,1423 869,2258 932,2291
X(2319) = isogonal conjugate of X(1423)
X(2319) = complement of X(20537)
X(2319) = anticomplement of X(20528)
X(2319) = X(2053)-Ceva conjugate of X(87)
X(2319) = X(8)-cross conjugate of X(9)
X(2320) lies on these lines: 1,89 2,80 3,1389 4,1385 7,1319 9,2364 55,1320 104,1621 941,1100 1001,1156 1388,1476
X(2320) = isogonal conjugate of X(2099)
X(2320) = X(7)-vertex conjugate of X(56)
X(2321) lies on these lines: 1,2345 6,519 8,9 10,37 33,200 69,527 71,1018 72,1903 75,142 141,536 145,1449 190,319 226,306 284,1043 314,646 515,1766 644,2287 1089,1826 1266,1278
X(2321) = isogonal conjugate of X(1412)
X(2321) = isotomic conjugate of X(1434)
X(2321) = complement of X(3875)
X(2321) = anticomplement of X(3946)
X(2321) = anticomplementary conjugate of anticomplement of X(38825)
X(2321) = X(i)-Ceva conjugate of X(j) for (i,j )= (8,210), (321,10), (646,522)
X(2321) = cevapoint of X(1334) and X(2318)
X(2321) = X(210)-cross conjugate of X(10)
X(2321) = cevapoint of X(1334) and X(2318)
X(2321) = crosssum of X(i) and X(j) for these (i,j): (56,604), (1333,1408)
X(2321) = excentral-to-ABC barycentric image of X(6)
X(2321) = barycentric product X(8)*X(10)
X(2322) lies on these lines: 2,253 4,391 8,29 9,318 19,27 21,268 28,956 37,1897 86,648 297,1654 346,1260 393,966 653,1441 1043,2326 1213,1990 1222,1474
X(2322) = cevapoint of X(i) and X(j) for these (i,j): (9,281), (19,1249), (2328,2332)
X(2322) = X(9)-cross conjugate of X(2287)
X(2322) = crosssum of X(1409) and X(1410)
X(2322) = polar conjugate of X(3668)
X(2323) lies on these lines: 1,6 35,2278 36,2245 40,1181 48,573 54,71 55,2364 57,394 59,672 60,283 63,1993 101,953 155,610 239,1944 282,1069 323,1443 517,2182 521,650 527,651 579,604 644,2325 648,1948 674,692 677,2338 758,1870 908,2006 909,2077 1318,2316 1731,2170 1766,2261
X(2323) = isogonal conjugate of X(2006)
X(2323) = X(i)-Ceva conjugate of X(j) for these (i,j): (908,2077), (1320,55)
X(2323) = X(2361)-cross conjugate of X(36)
X(2323) = crosspoint of X(284) and X(2316)
X(2323) = crosssum of X(i) and X(j) for these (i,j): (73,2252), (1400,1457), (1411,2161), (1769,2170
X(2323) = inverse-in-MacBeath-circumconic of X(1)
Let A'B'C' be the mixtilinear incentral triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" = Lb∩Lc, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2324). (Randy Hutson, January 15, 2019)
X(2324) lies on these lines: 1,6 19,1802 33,200 40,198 41,380 65,1696 77,144 78,280 101,610 190,326 223,329 269,527 394,1422 517,2270 579,1467 936,2345 1103,2331 1195,1334
X(2324) = isogonal conjugate of X(1422)
X(2324) = X(i)-Ceva conjugate of X(j) for these (i,j): (78,200), (329,40), (346,9)
X(2324) = X(198)-cross conjugate of X(9)
X(2324) = crosssum of X(1413) and X(1436)
X(2325) lies on these lines: 2,1266 8,9 10,45 37,39 44,519 142,344 190,320 522,650 644,2323 1018,2183 1698,1738
X(2325) = complement of X(1266)
X(2325) = anticomplement of X(17067)
X(2325) = X(2415)-Ceva conjugate of X(900)
X(2325) = crosspoint of X(2) and X(1120)
X(2325) = crosssum of X(i) and X(j) for these (i,j): (6,1149), (56,1404)
X(2325) = inverse-in-Mandart-inellipse of X(8)
X(2325) = excentral-to-ABC barycentric image of X(44)
X(2326) lies on these lines: 6,162 21,270 27,86 29,284 60,285 163,1765 412,572 1043,2322
X(2326) = cevapoint of X(284) and X(1172)
X(2326) = crosssum of X(1425) and X(2197)
X(2327) lies on these lines: 9,21 27,908 48,63 219,283 306,319 394,1073 758,1744 1043,2322 1333,1801 1789,1800 1792,1802 2200,2359
X(2327) = X(i)-Ceva conjugate of X(j) for these (i,j): (1043,2328),
(1812,283)
X(2327) = cevapoint of X(i) and X(j) for these (i,j): (219,2289),
(1260,1802)
X(2327) = X(i)-cross conjugate of X(j) for these (i,j): (219,2287),
(1260,1792)
X(2327) = crosspoint of X(1792) and X(1812)
X(2327) = crosssum of X(1426) and X(1880)
X(2328) lies on these lines: 1,21 2,1754 3,64 9,33 10,29 25,573 27,516 28,40 35,1819 55,219 71,1474 101,228 103,110 109,1214 165,1817 184,572 199,1495 200,1253 210,2341 220,1260 268,2192 333,643 387,452 394,991 405,580 440,1503 461,966 501,1800 902,2206 963,1437 1043,1098 1167,1785 1333,2256 1412,1617 1778,2257
X(2328) = isogonal conjugate of X(3668)
X(2328) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,284), (643,1021), (1043,2327), (1098,2287), (2322,2332)
X(2328) = X(220)-cross conjugate of X(2287)
X(2328) = crosspoint of X(i) and X(j) for these (i,j): (21,2287), (1043,2322)
X(2328) = crosssum of X(65) and X(1427)
X(2328) = {X(21),X(283)}-harmonic conjugate of X(58)
X(2329) lies on these lines: 1,6 2,1429 8,41 10,98 21,644 35,1018 42,904 48,2345 55,2053 145,2280 171,172 281,1973 284,1043 318,2202 346,2268 404,1055 419,1215 594,2174 1222,1438
X(2329) = isogonal conjugate of X(1432)
X(2329) = X(894)-Ceva conjugate of X(171)
X(2329) = X(2330)-cross conjugate of X(171)
X(2329) = crosssum of X(893) and X(1431)
X(2329) = anticomplement of X(17062)
X(2330) lies on these lines: 1,182 3,611 6,31 9,2175 12,1503 33,1974 35,511 37,692 41,2053 59,1442 172,1691 184,612 210,2194 284,2311 498,1352 572,2223 1376,1958 1500,1692
X(2330) = X(171)-Ceva conjugate of X(172)
X(2330) = crosspoint of X(171) and X(2329)
X(2330) = crosssum of X(256) and X(1432)
X(2331) lies on these lines: 1,281 6,19 9,1783 33,42 55,204 73,207 77,653 196,223 198,208 222,1767 614,1108 1103,2324 1436,1455 1735,1741 1854,1903
X(2331) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,33), (196,208), (281,9)
X(2331) = X(2187)-cross conjugate of X(40)
X(2331) = crosspoint of X(1) and X(223)
X(2331) = crosssum of X(i) and X(j) for these (i,j): (1,282), (268,1433)
X(2331) = trilinear product X(33)*X(223)
X(2331) = polar conjugate of X(309)
X(2332) lies on these lines: 1,19 6,64 27,1803 29,1855 33,41 34,2280 55,607 58,103 200,1802 963,1333 1043,2322 1826,2259 1842,2201 2192,2194 2207,2271
X(2332) = X(2322)-Ceva conjugate of X(2328)
X(2332) = cevapoint of X(41) and X(607)
X(2332) = crosssum of X(1214) and X(1439)
X(2333) lies on these lines: 4,9 25,41 28,291 34,2279 181,213 579,1722 608,1405 756,862 1126,1474 1254,1400 1918,2207 2225,2355
X(2333) = isogonal conjugate of X(17206)
X(2333) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,1824), (607,213), (1826,42)
X(2333) = X(1918)-cross conjugate of X(42)
X(2333) = crosspoint of X(i) and X(j) for these (i,j): (19,25), (1400,2357), (1824,1880)
X(2333) = crosssum of X(i) and X(j) for these (i,j): (63,69), (1444,1812)
X(2333) = polar conjugate of X(310)
X(2333) = crossdifference of every pair of points on line X(1459)X(4025)
Let A'B'C' = circumcevian triangle of X(1), and let
Oa = center of A-mixtilinear incircle
Ta = touchpoint of A-mixtilinear incircle and circumcircle
A1 = circumcenter of A'OaTa, and define B1 and C1 cyclically.
Then AA1, BB1, CC1 concur in X(2334). (Angel Montesdeoca, November 29, 2018)
Let DEF be the cevian triangle of X(8), and let A5 is the QA-P5 center (Isotomic Center) of AEFX(8) (see Chris van Tienhoven, Quadrangle points and other objects). Let B5 be the QA-P5 center of BFDX(8) and C5 the QA-P5 center of CDEX(8). Then the triangles ABC and A5B5C5 are perspective, and X(2334) is their perspector. (Angel Montesdeoca, March 20, 2019)
In the plane of a triangle ABFC, let
DEF = cevian triangle of X(1);
Γ = circumcircle;
Aa = pole of EF wrt Γ
T1 and T2 = lines through Aa tangent to BC;
Ab and Ac = T1∩BC and T2∩BC;
Define Bc and Ca cyclically, and define Ba and Cb cyclically.
The six points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic, 𝒞, given by the barycentric equation
b^2 c^2x^2+a^2 c^2y^2+a^2 b^2z^2+a^2 (a^2+b^2+c^2+2 a (b+c))yz+b^2 (a^2+2 a b+(b+c)^2)zx+c^2 (a^2+2 a c+(b+c)^2)xy= 0.
The polars of Aa, Bb, Cc with respect to 𝒞 form a perspective triangle to ABC, and the perspector is X(2334). (Angel Montesdeoca, November 27, 2022)
X(2334) lies on these lines: 1,210 3,2163 6,1334 8,86 34,1824 42,56 55,58 65,269 106,386 145,1220 607,1474 998,2098 1413,2357 1449,2297
X(2334) = isogonal conjugate of X(3616)
X(2334) = X(1191)-cross conjugate of X(56)
X(2335) lies on these lines: 1,71 4,37 6,943 7,464 9,2318 21,219 55,1172 84,991 104,2256 278,2197 281,1896 314,345 1000,1108 1011,2298
X(2335) = isogonal conjugate of X(37543)
X(2336) lies on these lines: 6,2318 9,937 34,37 55,1474 56,71 58,219 86,345 269,1214 1413,2286
X(2336) = crosssum of X(380) and X(1453)
X(2337) lies on these lines: 6,47 19,24
X(2338) is the perspector of the anticevian triangle of X(103) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, June 7, 2019)
X(2338) lies on these lines:
1,1783 3,101 9,77
78,644 218,1433 219,480
332,645 677,2323
X(2338) = cevapoint of X(i) and X(j) for these (i,j): (6,2272), (220,2340)
X(2338) = X(672)-cross conjugate of X(9)
X(2338) = crosssum of X(i) and X(j) for these (i,j): (910,1456), (1458,2272)
X(2339) lies on these lines: 1,1472 2,19 6,63 9,345 21,1039 55,78 57,348 280,452 284,1812 333,2082 386,1245 394,2288 1310,2291
X(2339) = isogonal conjugate of X(2285)
X(2339) = cevapoint of X(9) and X(1697)
X(2339) = X(2268)-cross conjugate of X(1)
X(2340) lies on these lines: 1,2 6,480 9,2293 31,218 33,1855 41,55 144,1742 210,1212 219,949 241,518 650,663 672,2223 674,2183 677,1815 902,1110 904,1261
X(2340) = reflection of X(1458) in X(1818)
X(2340) = isogonal conjugate of isotomic conjugate of X(3717)
X(2340) = crossdifference of every pair of points on line X(57)X(649)
X(2340) = X(2338)-Ceva conjugate of X(220)
X(2340) = crosspoint of X(677) and X(1252)
X(2340) = crosssum of X(i) and X(j) for these (i,j): (7,1447),
(57,1458), (105,1462), (676,1086)
X(2341) lies on these lines: 6,13 9,1793 37,101 80,1172 81,226 163,2265 210,2328 312,645 644,2287 1725,2159 2222,2249
X(2341) = isogonal conjugate of X(18593)
X(2341) = polar conjugate of isotomic conjugate of X(1793)
X(2341) = X(63)-isoconjugate of X(1835)
X(2341) = crosspoint of X(6) and X(2173)
X(2341) = crosssum of X(1464) and X(2245)
X(2342) lies on these lines: 1,104 31,33 44,2361 55,184 103,2078 200,212 283,643 678,1253
X(2342) = isogonal conjugate of X(22464)
X(2342) = X(104)-Ceva conjugate of X(909)
X(2342) = cevapoint of X(55) and X(2361)
X(2342) = X(109)-cross conjugate of X(55)
X(2342) = crosspoint of X(102) and X(2316)
X(2342) = crosssum of X(517) and X(1465)
X(2343) lies on these lines: 6,939 19,220 57,219 71,1436
X(2343) = crosssum of X(1467) and X(2257)
X(2344) lies on these lines: 1,32 2,2112 4,1973 6,256 7,604 8,41 9,2175 37,983 104,825 284,314 870,1438 941,2209 1156,1492
X(2344) = crosssum of X(1469) and X(2276)
X(2345) lies on these lines: 1,2321 2,37 4,9 6,8 7,141 44,391 45,1213 48,2329 69,894 145,1100 193,319 198,1376 219,1065 220,965 318,393 329,1211 333,1778 388,2285 404,2178 498,1733 519,1449 572,944 936,2324 948,1441 992,2176 1010,2303 1400,1788 1698,1738
X(2345) = isogonal conjugate of X(2221)
X(2345) = complement of X(3672)
X(2345) = anticomplement of X(4657)
X(2345) = X(1010)-Ceva conjugate of X(612)
X(2345) = X(612)-cross conjugate of X(388)
X(2345) = crosspoint of X(2) and X(1219)
X(2345) = crosssum of X(i) and X(j) for these (i,j): (6,1191), (1245,2281)
X(2345) = {X(2),X(75)}-harmonic conjugate of X(4000)
X(2346) lies on these lines: 1,1170 4,390 7,55 8,344 9,1174 21,518 37,294 79,516 84,1803 100,142 523,885 651,2293 1014,2223
X(2346) = isogonal conjugate of X(354)
X(2346) = cevapoint of X(1) and X(55)
X(2346) = X(i)-cross conjugate of X(j) for these (i,j): (514,100), (657,651), (1174,1170)
X(2346) = crosssum of X(i) and X(j) for these (i,j): (2,57), (9,145)
X(2346) = perspector of ABC and triangle formed by orthic axes of BIC, CIA, AIB
X(2346) = perspector of ABC and cross-triangle of ABC and Honsberger triangle
X(2347) lies on these lines: 6,41 8,9 19,1877 37,374 42,51 43,165 44,71 256,294 284,2316 607,2354 2171,2262 2270,2285
X(2347) = isogonal conjugate of isotomic conjugate of X(3452)
X(2347) = X(100)-Ceva conjugate of X(663)
X(2347) = crosspoint of X(6) and X(9)
X(2347) = crosssum of X(i) and X(j) for these (i,j): (2,57), (9,145)
X(2347) = polar conjugate of isotomic conjugate of X(22072)
X(2348) lies on these lines: 6,354 9,55 37,2280 41,1212 44,513 57,1122 65,169 101,1319 105,518 220,2082 238,1282 1617,1723 1696,2257 1783,1875
X(2348) = X(105)-Ceva conjugate of X(55)
X(2348) = crosspoint of X(9) and X(294)
X(2348) = crosssum of X(i) and X(j) for these (i,j): (1,2348),
(57,241), (665,1357)
X(2349) lies on these lines: 1,162 63,662 72,74 92,823 190,306 226,653 293,896 304,799 651,1214 1748,2184
X(2349) = isogonal conjugate of X(2173)
X(2349) = isotomic conjugate of X(14206)
X(2349) = polar conjugate of X(1784)
X(2349) = trilinear pole of line X(1)X(656)
X(2349) = X(6)-isoconjugate of X(30)
X(2349) = cevapoint of X(i) and X(j) for these (i,j): (1,2173), (9,758), (2631,2632)
X(2349) = X(i)-cross conjugate of X(j) for these (i,j): (1725,75), (2173,1), (2631,162)
X(2349) = X(i)-aleph conjugate of X(j) for these (i,j): (74,1740), (1494,63), (2349,1)
X(2350) lies on these lines: 37,38 39,42 58,251 308,310 579,941 614,2279 1400,1401
X(2351) lies on these lines: 3,68 22,98 24,96 25,53 32,51 91,1324 125,426 184,216 228,1820 237,2353
X(2351) = isogonal conjugate of X(317)
X(2351) = X(i)-Ceva conjugate of X(j) for these (i,j): (96,2165), (847,6)
X(2351) = X(418)-cross conjuate of X(184)
X(2351) = crosspoint of X(68) and X(2165)
X(2351) = crosssum of X(i) and X(j) for these (i,j): (24,1993), (52,467)
X(2351) = X(75)-isoconjugate of X(24)
X(2351) = X(92)-isoconjugate of X(1993)
X(2351) = vertex conjugate of polar conjugates of PU(37)
X(2352) lies on these lines: 1,3 6,228 25,1841 28,1612 31,48 37,1011 42,2260 108,917 209,579 212,604 255,1408 595,2360 672,2318 916,1779 1284,1836 1395,1415 1397,2361
X(2352) = isogonal conjugate of X(2997)
X(2352) = X(28)-Ceva conjugate of X(6)
X(2352) = crosspoint of X(i) and X(j) for these (i,j): (59,112),
(108,2149)
X(2352) = crosssum of X(11) and X(525)
X(2353) lies on these lines: 3,66 22,76 24,98 32,1843 39,184 228,2156 237,2351
X(2353) = isogonal conjugate of X(315)
X(2353) = circumcircle-inverse of X(34138)
X(2353) = X(1501)-cross conjugate of X(6)
X(2353) = crosssum of X(69) and X(1370)
X(2353) = vertex conjugate of PU(37)
X(2353) = X(75)-isoconjugate of X(22)
X(2354) lies on these lines: 4,9 6,1245 25,31 28,238 48,1064 58,1474 256,1172 278,1423 579,1707 607,2347 1756,1838 1828,1880 1829,2269 1843,2356 1851,1860 2180,2252 2203,2308
X(2354) = X(1848)-Ceva conjugate of X(1193)
X(2354) = crosspoint of X(i) and X(j) for these (i,j): (19,1474),
(27,34)
X(2354) = crosssum of X(i) and X(j) for these (i,j): (63,306),
(71,78)
X(2355) lies on these lines: 19,25 24,1871 27,242 28,60 46,1707 51,2182 184,2262 407,1842 427,1890 428,1861 430,1213 468,1848 1395,1880 1426,1452 1841,2160 2180,2253 2225,2333
X(2355) = crosspoint of X(19) and X(28)
X(2355) = crosssum of X(63) and X(72)
X(2355) = polar conjugate of X(32018)
X(2356) lies on these lines: 1,4 6,2212 25,41 43,459 55,1395 77,1041 108,1429 468,899 608,2293 661,663 1174,2299 1458,1876 1818,1861 1827,1880 1843,2354 1974,2273
X(2356) = isogonal conjugate of X(31637)
X(2356) = crossdifference of every pair of points on line X(63)X(652)
X(2356) = X(1861)-Ceva conjugate of X(672)
X(2356) = crosssum of X(63) and X(1818)
X(2357) lies on these lines: 8,20 19,1857 25,2155 31,607 42,1409 48,55 71,210 228,1334 1002,1422 1400,1824 1413,2334
X(2357) = isogonal conjugate of X(8822)
X(2357) = trilinear pole of line X(810)X(3709)
X(2357) = X(84)-Ceva conjugate of X(1903)
X(2357) = X(i)-cross conjugate of X(j) for these (i,j): (1402,42), (2333,1400)
X(2357) = crosspoint of X(i) and X(j) for these (i,j): (19,64), (84,1436)
X(2357) = crosssum of X(i) and X(j) for these (i,j): (20,63), (40,329)
X(2358) lies on these lines: 7,92 19,56 65,1826 608,1096 1042,1880 1400,1824
X(2359) lies on these lines: 1,572 3,2197 6,1036 9,205 29,1220 42,284 48,78 71,283 184,219 282,2208 306,332 951,1458 2200,2327
X(2359) = isogonal conjugate of X(1848)
X(2359) = cevapoint of X(i) and X(j) for these (i,j): (42,205), (48,71), (212,2200)
X(2359) = X(810)-cross conjugate of X(1331)
X(2359) = crosssum of X(1193) and X(2354)
X(2359) = X(92)-isoconjugate of X(1193)
See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.
X(2360) lies on these lines: 1,19 3,64 21,84 25,581 27,946 29,515 36,1780 40,1817 56,58 72,101 73,2299 102,110 184,580 208,223 405,572 595,2352 604,1453 662,1043 1191,1333 1201,2206 1203,2260 1214,1782 1728,2261
X(2360) = isogonal conjugate of X(39130)
X(2360) = crossdifference of every pair of points on line X(656)X(3700)
X(2360) = X(21)-Ceva conjugate of X(58)
X(2360) = cevapoint of X(i) and X(j) for these (i,j): (221,7114), (48,154), (198,2187)
X(2360) = X(198)-cross conjugate of X(1817)
X(2360) = trilinear product X(i)*X(j) for these {i,j}: {3, 3194}, {6, 1817}, {21, 221}, {28, 7078}, {29, 7114}, {31, 8822}, {34, 1819}, {40, 58}, {60, 227}, {81, 198}, {86, 2187}, {110, 6129}, {163, 14837}, {196, 2193}, {208, 283}, {223, 284}, {322, 2206}, {329, 1333}, {333, 2199}, {347, 2194}, {593, 21871}, {604, 27398}, {849, 21075}, {859, 15501}, {1014, 7074}, {1172, 7011}, {1408, 7080}, {1412, 2324}, {1437, 7952}, {1444, 3195}, {1576, 17896}, {1790, 2331}, {1812, 3209}, {2287, 6611}, {2299, 7013}, {4565, 14298}
X(2361) lies on these lines: 3,47 6,31 11,238 36,1464 44,2342 46,582 56,255 65,580 109,1155 162,243 228,2148 283,2148 283,960 484,1718 517,1411 518,1331 652,663 920,1062 1040,1707 1110,2149 1397,2352 1724,1837 1754,1836 1780,1858 1859,2299 2150,2193
X(2361) = X(i)-Ceva conjugate of X(j) for these (i,j): (80,2174), (102,48), (2316,41), (2342,55)
X(2361) = crosspoint of X(i) and X(j) for these (i,j): (36,2323), (915,1172)
X(2361) = crosssum of X(i) and X(j) for these (i,j): (36,2003), (65,1465), (80,2006), (912,1214)
Barycentrics a2/f(a,b,c) : b2/f(b,c,a): c2/f(c,a,b)
X(2362) lies on these lines: 1,372 2,176 4,1336 6,19 40,2066 46,371 57,2067 72,1377 81,1805 485,1737 517,1124 606,1451 942,1335 1151,1155 1702,2093
X(2362) = crosspoint of X(4) and X(1123)
X(2362) = crosssum of X(3) and X(1124)
X(2363) lies on these lines: 1,849 10,58 19,270 21,37 27,225 31,1098 60,1610 65,81 75,757 409,1104 662,1193 741,1581
X(2363) = isogonal conjugate of X(2292)
X(2363) = isotomic conjugate of X(18697)
X(2363) = cevapoint of X(i) and X(j) for these (i,j): (1,58),
(1,1220)
X(2363) = X(i)-cross conjugate of X(j) for these (i,j): (1,1220),
(522,162), (649,662)
X(2364) lies on these lines: 1,2161 6,36 9,2320 19,1449 41,2316 55,2323 57,89 572,2259 2258,2308 2280,2291
X(2364) = isogonal conjugate of X(5219)
X(2364) = trilinear pole of line X(654)X(663)
X(2364) = X(89)-Ceva conjugate of X(2163)
X(2364) = crosspoint of X(89) and X(2320)
X(2364) = crosssum of X(45) and X(2099)
More Points on the Circumcircle, 2365-2384
Suppose that U = u : v : w (trilinears) is a triangle center other than X(6), and let CIR(U) be the point given by trilinears
F(u,v,w) : F(v,w,u) : F(w,u,v) = avw/(av2 + aw2 - buv - cuw) : bwu/(bw2 + bu2 - cvw - avu) : cuv/(cu2 + cv2 - awu - bwv)
CIR(U) lies on the circumcircle. Pairs (i,j) such that X(j) = CIR(X(i)) include the following:
(1,106), (2,729), (3,1300), (4,1294), (9,1477), (31,767), (37,741), (41,767), (43,106), (44,106), (74,1294), (75,701), (110,99), (125,827), (194,729), (238,741), (265,1300), (1084,689), (1154,1141), (1279,(1477), (1499,1296), (1503,1297), (1510,930), (1634,689)This section was added to ETC on 10/20/03; the preamble was revised on January 20, 2015, following suggestions by Viktor Kitaysky. Points with trilinears similar to those of CIR(U), called circum-eigentransforms, denoted by CET(U), are given in Section 4.
X(2365) lies on the circumcircle and these lines: 63,108 101,1259 107,333 109,394 112,283
X(2365) = isogonal conjugate of X(2385)
X(2366) lies on the circumcircle and these lines: 69,112 76,107 99,159 108,1231
X(2366) = isogonal conjugate of X(2386)
X(2367) lies on the circumcircle and these lines: 76,110 99,160 101,313 109,349 112,264 276,933 308,827 316,805
X(2367) = isogonal conjugate of X(2387)
X(2367) = isotomic conjugate of X(3001)
X(2368) lies on the circumcircle and these lines: 86,101 100,274 109,1434 110,1509
X(2368) = isogonal conjugate of X(2388)
X(2369) lies on the circumcircle and these lines: 7,101 85,100 108,1847 109,279 110,1434
X(2369) = isogonal conjugate of X(2389)
X(2370) lies on the circumcircle and these lines: 8,109 20,1293 75,934 100,341 101,346 108,318 110,1043 112,2322
X(2370) = isogonal conjugate of X(2390)
X(2370) = isotomic conjugate of X(3007)
X(2370) = cevapoint of X(3) and X(519)
X(2370) = circumcircle antipode of X(32704)
X(2370) = Thomson-isogonal conjugate of X(32475)
X(2370) = Lucas-isogonal conjugate of X(32475)
X(2370) = de-Longchamps-circle-inverse of X(21290)
X(2371) lies on the circumcircle and these lines: 9,934 101,480 109,220
X(2371) = isogonal conjugate of X(2391)
X(2372) lies on the circumcircle and these lines: 10,110 12,109 99,313 100,1089 101,594 112,1826
X(2372) = isogonal conjugate of X(2392)
X(2373) lies on the circumcircle and these lines: 2,112 20,1296 22,99 23,935 25,339 69,110 95,933 101,306 107,264 108,1441 109,307 183,1302 253,1301 316,691 328,476 827,1799 1304,1494
X(2373) = isogonal conjugate of X(2393)
X(2373) = isotomic conjugate of X(858)
X(2373) = anticomplement of X(1560)
X(2373) = cevapoint of X(i) and X(j) for these (i,j): (2,23), (3,524), (669,1648)
X(2373) = X(i)-cross conjugate of X(j) for these (i,j): (67,671), (468,2)
X(2373) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(2),X(69)}}
X(2373) = trilinear pole of line X(6)X(525)
X(2373) = trilinear pole, wrt circummedial triangle, of line X(22)X(183)
X(2373) = Ψ(X(6), X(525))
X(2373) = orthoptic-circle-of-Steiner-inellipse-inverse of X(127)
X(2373) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(13219)
X(2374) lies on the circumcircle and these lines: 4,126 25,99 110,193 112,459 468,691 925,1995
X(2374) = cevapoint of X(25) and X(468)
X(2374) = X(524)-cross conjugate of X(4)
X(2374) = isogonal conjugate of X(8681)
X(2374) = Λ(X(6), X(1196))
X(2374) = Ψ(X(6), X(3566))
X(2374) = inverse-in-polar-circle of X(126)
X(2374) = trilinear pole of line X(6)X(3566)
X(2374) = polar conjugate of anticomplement of X(6390)
X(2374) = X(63)-isoconjugate of X(3291)
X(2375) lies on the circumcircle and these lines: 37,99 100,1500 101,872 110,213
X(2375) = isogonal conjugate of X(8682)
X(2376) lies on the circumcircle and these lines: 34,101 100,278 109,1435 110,1396
X(2377) lies on the circumcircle and these lines: 100,279 101,269
X(2378) lies on the circumcircle and these lines: 14,476 15,110 16,691 99,298 512,2379
X(2378) = isogonal conjugate of X(530)
X(2378) = 2nd-Parry-to-ABC similarity image of X(15)
X(2378) = 3rd-Parry-to-circumsymmedial similarity image of X(16)
X(2379) lies on the circumcircle and these lines: 13,476 15,691 16,110 99,299 512,2378
X(2379) = 2nd-Parry-to-ABC similarity image of X(16)
X(2379) = 3rd-Parry-to-circumsymmedial similarity image of X(15)
X(2379) = isogonal conjugate of X(531)
X(2380) lies on the circumcircle and these lines: 14,99 16,1337 18,930 61,110
X(2380) = isogonal conjugate of X(532)
X(2381) lies on the circumcircle and these lines: 13,99 15,1338 17,930 62,110
X(2381) = isogonal conjugate of X(533)
X(2382) lies on the circumcircle and these lines: 1,898 6,813 31,901 100,238 101,1914 106,667 109,1428 649,739 805,1178
X(2382) = isogonal conjugate of X(537)
X(2382) = Ψ(X(190), X(44))
X(2383) lies on the circumcircle and these lines: 4,128 5,925 24,933 52,110 186,1291
X(2383) = X(1154)-cross conjugate of X(4)
X(2383) = isogonal conjugate of X(539)
X(2383) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(5),X(6)}}
X(2383) = inverse-in-polar-circle of X(128)
X(2384) lies on the circumcircle and these lines: 6,901 44,100 101,902 106,649 109,1404 813,2177 1015,2226 1017,1252
X(2384) = isogonal conjugate of X(545)
As the isogonal conjugate of a point on the circumcircle, X(2385) lies on the line at infinity.
X(2385) lies on these (parallel) lines: 19,208 30,511 774,1842 1886,2312
X(2385) = isogonal conjugate of X(2365)
As the isogonal conjugate of a point on the circumcircle, X(2386) lies on the line at infinity.
X(2386) lies on these (parallel) lines: 25,32 30,511 305,315 626,1368 1968,2353
X(2386) = isogonal conjugate of X(2366)
As the isogonal conjugate of a point on the circumcircle, X(2387) lies on the line at infinity.
X(2387) lies on these lines: 30,511 32,184
X(2387) = isogonal conjugate of X(2367)
As the isogonal conjugate of a point on the circumcircle, X(2388) lies on the line at infinity.
X(2388) lies on these lines: 30,511 42,213
X(2388) = isogonal conjugate of X(2368)
As the isogonal conjugate of a point on the circumcircle, X(2389) lies on the line at infinity.
X(2389) lies on these lines: 30,511 41,55
X(2389) = isogonal conjugate of X(2369)
As the isogonal conjugate of a point on the circumcircle, X(2390) lies on the line at infinity.
X(2390) lies on these (parallel) lines: 30,511 31,56 51,65 1361,1455 1777,2217 1854,2098
X(2390) = isogonal conjugate of X(2370)
X(2390) = X(1797)-Ceva conjugate of X(6)
X(2390) = crosspoint of X(4) and X(106)
X(2390) = crosssum of X(3) and X(519)
X(2390) = 2nd-circumperp-isogonal conjugate of X(32704)
X(2390) = circumorthic-isogonal conjugate of X(32704) if ABC is acute
As the isogonal conjugate of a point on the circumcircle, X(2391) lies on the line at infinity.
X(2391) lies on these (parallel) lines: 30,511 57,279 910,1323
X(2391) = isogonal conjugate of X(2371)
As the isogonal conjugate of a point on the circumcircle, X(2392) lies on the line at infinity.
X(2392) lies on these (parallel) lines: 30,511 36,58 484,1046
X(2392) = isogonal conjugate of X(2372)
As the isogonal conjugate of a point on the circumcircle, X(2393) lies on the line at infinity.
Let A"B"C" be the 2nd Ehrmann triangle. Let Pa be the pole of line B"C" wrt the A-Ehrmann circle, and define Pb and Pc cyclically. The lines A"Pa, B"Pb, C"Pc concur in X(2393). (Randy Hutson, November 18, 2015)
X(2393) lies on these (parallel) lines: 6,25 23,895 30,511 66,69 67,1205 141,1368 157,577 160,216 401,1632 599,1853
X(2393) = isogonal conjugate of X(2373)
X(2393) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1560), (23,187), (895,6), (935,647)
X(2393) = crosspoint of X(i) and X(j) for these (i,j): (4,111), (6,67)
X(2393) = crosssum of X(i) and X(j) for these (i,j): (2,23), (3,524), (669,1648)
X(2393) = homothetic center of X(3)- and X(6)-Ehrmann triangles; see X(25)
GIBERT-SIMSON TRANSFORMS, X(2394) -X(2419)
On October 19, 2003, Bernard Gibert contributed points that lie on the Simson cubic. Indeed, this cubic is the image of the circumcircle under a mapping here named the Gibert-Simson transform. If U = u : v : w (barycentrics) and U is not X(3), then the transform is given by
GS(U) = (b2SB/v - c2SC/w)u/a2 : (c2SC/w - a2SA/u)v/b2 : (a2SA/u - b2SB/v)w/c2.
In case U lies on the circumcircle, these barycentrics are proportional to the respective directed distances from A, B, C, to the Simson line of U. If U* is the antipode of U, then GS(U*) is the isotomic conjugate of GS(U). In the Catalogue at Cubics in the Triangle Plane, the Simson cubic is discussed as K010, and its isogonal transform is the cubic K162. The isogonal conjugates of centers X(2365)-X(2419) are indexed as X(2420)-X(2445).
If U = u : v : w = u(a,b,c) : u(b,c,a) : u(c,a,b) (trilinears), then
GS(U) = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (v cos C - w cos B)(u/a)2,
and the cubic K010 is given by trilinears
(cos A)x(b2y2 + c2z2) + (cos B)y(c2z2 + a2x2) + (cos C)z(a2x2 + b2x2) - (a2 + b2 + c2)xyz = 0,
and the cubic K162 by
(cos A)x(c2y2 + b2z2) + (cos B)y(a2z2 + c2x2) + (cos C)z(b2x2 + a2x2) - (a2 + b2 + c2)xyz = 0,
In the definition of eigentransform before X(2120), if cevian triangle is replaced by circumcevian triangle, the result is CET(U) as discussed before X(2365). Gibert observed that if cevian triangle is replaced by pedal triangle, for U on the circumcircle, the result is the isogonal conjugate of GS(U). That is, [GS(U)] -1 is the eigencenter of the pedal triangle of U (which is a degenerate triangle lying in the Simson line of U).
Regarding triangle centers that do not lie on the circumcircle, GS(X(i)) = X(j) for these (i,j): (32,669), (48,1459), (187,1649), (248,879), (485,850), (486,850).
This section was added to ETC on 10/23/03. For more Gibert-Simson transforms, see X(15411)-X(15423)
X(2394) lies on the Kiepert hyperbola, the cubic K010, and these lines: 2,525 4,523 74,98 671,1494 935,1304 2407,2410
X(2394) = isogonal conjugate of X(2420)
X(2394) = isotomic conjugate of X(2407)
X(2394) = anticomplement of X(5664)
X(2394) = cevapoint of X(i) and X(j) for these (i,j): (523,1637), (526,647)
X(2394) = X(1637)-cross conjugate of X(523)
X(2394) = orthocenter of X(13)X(14)X(98)
X(2394) = trilinear pole of line X(125)X(523) (Simson line of X(74))
X(2394) = pole wrt polar circle of trilinear polar of X(4240) (line X(30)X(1990))
X(2394) = X(48)-isoconjugate (polar conjugate) of X(4240)
X(2394) = barycentric product X(74)*X(850)
X(2395) lies on the cubic K010 and these lines: 2,647 6,523 25,669 98,111 263,512 685,2409 694,804 880,2396 892,2407 1648,2433
X(2395) = isogonal conjugate of X(2421)
X(2395) = isotomic conjugate of X(2396)
X(2395) = anticomplement of isotomic conjugate of X(39291)
X(2395) = X(685)-Ceva conjugate of X(1976)
X(2395) = cevapoint of X(i) and X(j) for these (i,j): (512,2491), (523,804), (878,2422)
X(2395) = (X(i)-cross conjugate of X(j) for these (i,j): (878,879), (2491,512)
X(2395) = pole wrt polar circle of trilinear polar of X(877) (line X(232)X(297))
X(2395) = polar conjugate of X(877)
X(2395) = radical center of circumcircle and circles O(13,15) and O(14,16)
X(2395) = trilinear pole of line X(115)X(512) (the Simson line of X(98), and the polar of X(877) wrt the polar circle)
X(2396) lies on the cubic K010 and these lines: 2,39 99,110 325,868 880,2395 892,2408 1316,1975
X(2396) = isogonal conjugate of X(2422)
X(2396) = isotomic conjugate of X(2395)
X(2396) = anticomplement of isotomic conjugate of X(39292)
X(2396) = anticomplement of anticomplement of X(11052)
X(2396) = cevapoint of X(i) and X(j) for these (i,j): (230,804), (511,2491), (523,2023)
X(2396) = trilinear pole of line X(114)X(325) (the Simson line of X(99))
X(2396) = X(i)-cross conjugate of X(j) for these (i,j): (2421,877), (2491,511)
X(2397) lies on the cubic K010 and these lines: 2,37 190,644 522,1026 666,2402 765,2398
X(2397) = isogonal conjugate of X(2423)
X(2397) = isotomic conjugate of X(2401)
X(2397) = trilinear pole of line X(119)X(517) (the Simson line of X(100))
X(2398) lies on the cubic K010 and these lines: 1,2 59,2406 100,658 644,1783 666,885 677,883 765,2397 901,2403
X(2398) = isogonal conjugate of X(2424)
X(2398) = isotomic conjugate of X(2400)
X(2398) = anticomplement of isotomic conjugate of X(39293)
X(2398) = anticomplement of anticomplement of X(24980)
X(2398) = cevapoint of X(516) and X(676)
X(2398) = X(676)-cross conjugate of X(516)
X(2398) = crosspoint of X(664) and X(666)
X(2398) = crosssum of X(663) and X(665)
X(2398) = trilinear pole of line X(118)X(516) (the Simson line of X(101))
X(2399) lies on the cubic K010 and these lines: 102,1311 189,514 333,2432
X(2399) = isogonal conjugate of X(2425)
X(2399) = isotomic conjugate of X(2406)
X(2400) lies on the cubic K010 and these lines: 2,3239 7,522 86,2424 103,675 673,918 677,883 693,1088
X(2400) = isogonal conjugate of X(2426)
X(2400) = isotomic conjugate of X(2398)
X(2400) = X(676)-cross conjugate of X(514)
X(2400) = cevapoint of X(i) and X(j) for these (i,j): (514,676), (522,918)
X(2400) = trilinear pole of line X(116)X(514) (the Simson line of X(103))
X(2401) lies on the cubic K010 and these lines: 1,522 2,905 57,514 81,2423 104,105 513,957 655,2406 909,2224
X(2401) = isogonal conjugate of X(2427)
X(2401) = isotomic conjugate of X(2397)
X(2401) = cevapoint of X(650) and X(900)
X(2401) = trilinear pole of line X(11)X(513) (the Simson line of X(104))
X(2402) lies on the cubic K010 and these lines: 2,650 514,1024 522,1027 523,885 666,2397 918,1814
X(2402) = isogonal conjugate of X(2428)
X(2402) = isotomic conjugate of X(2414)
X(2402) = anticomplement of isotomic conjugate of X(39272)
X(2402) = trilinear pole of line X(3309)X(4904) (the Simson line of X(105))
X(2403) lies on the cubic K010 and these lines: 2,514 523,1222 901,2398
X(2403) = isogonal conjugate of X(2429)
X(2403) = isotomic conjugate of X(2415)
X(2403) = trilinear pole of line X(3667)X(3756) (the Simson line of X(106))
X(2404) lies on the cubic K010 and this line: 2,216
X(2404) = isogonal conjugate of X(2430)
X(2404) = isotomic conjugate of X(2416)
X(2404) = trilinear pole of line X(133)X(1515) (the Simson line of X(107))
X(2405) lies on the cubic K010 and these lines: 2,92 653,934
X(2405) = isogonal conjugate of X(2431)
X(2405) = isotomic conjugate of X(2417)
X(2405) = trilinear pole of line X(1528)X(6001) (the Simson line of X(108))
X(2406) lies on the cubic K010 and these lines: 2,7 59,2398 651,653 655,2401
X(2406) = isogonal conjugate of X(2432)
X(2406) = isotomic conjugate of X(2399)
X(2406) = anticomplement of isotomic conjugate of X(39294)
X(2406) = trilinear pole of line X(117)X(515) (the Simson line of X(109))
X(2407) lies on the cubic K010 and these lines: 2,6 98,895 99,112 110,1302 249,2411 250,2409 691,2408 892,2395 1723,2358 2394,2410
X(2407) = isogonal conjugate of X(2433)
X(2407) = isotomic conjugate of X(2394)
X(2407) = anticomplement of isotomic conjugate of X(39295)
X(2407) = anticomplement of anticomplement of X(24975)
X(2407) = cevapoint of X(i) and X(j) for these (i,j): (30,1637), (647,974)
X(2407) = X(1637)-cross conjugate of X(30)
X(2407) = crosspoint of X(648) and X(687)
X(2407) = crosssum of X(647) and X(686)
X(2407) = trilinear pole of line X(30)X(113) (the Simson line of X(110))
X(2408) lies on the cubic K010 and these lines: 2,523 690,895 691,2407 892,2396 1499,1992
X(2408) = isogonal conjugate of X(2434)
X(2408) = isotomic conjugate of X(2418)
X(2408) = anticomplement of isotomic conjugate of X(39296)
X(2408) = trilinear pole of line X(1499)X(2686) (the Simson line of X(111))
As a point on the Euler line, X(2409) has Shinagawa coefficients (6(E + F)F2 - 2FS2, -4F(E + F)2 + (E + 4F)S2).
X(2409) lies on the cubic K010 and these lines: 2,3 99,1301 107,112 110,1289 250,2407 685,2395 935,1304
X(2409) = isogonal conjugate of X(2435)
X(2409) = isotomic conjugate of X(2419)
X(2409) = anticomplement of isotomic conjugate of X(39297)
X(2409) = circumcircle-inverse of X(37937)
X(2409) = crosspoint of X(107) and X(685)
X(2409) = crosssum of X(520) and X(684)
X(2409) = polar conjugate of isotomic conjugate of X(34211)
X(2409) = trilinear pole of line X(132)X(1503) (the Simson line of X(112))
X(2410) lies on the cubic K010 and these lines: 2,94 2394,2407
X(2410) = isogonal conjugate of X(2436)
X(2410) = isotomic conjugate of X(2411)
X(2410) = barycentric quotient X(476)/X(477)
X(2410) = trilinear pole of line X(1522)X(1523) (the Simson line of X(476))
X(2411) lies on the cubic K010 and these lines: 249,2407 323,2436 477,842
X(2411) = isogonal conjugate of X(2437)
X(2411) = isotomic conjugate of X(2410)
X(2411) = barycentric quotient X(477)/X(476)
X(2411) = trilinear pole of line X(526)X(3258) (the Simson line of X(477))
X(2412) lies on the cubic K010.
X(2412) = isogonal conjugate of X(2438)
X(2413) lies on the cubic K010.
X(2413) = isogonal conjugate of X(2439)
X(2413) = trilinear pole of line X(137)X(1510) (the Simson line of X(1141))
X(2414) lies on the cubic K010 and these lines: 2,277 100,1292
X(2414) = isogonal conjugate of X(2440)
X(2414) = isotomic conjugate of X(2402)
X(2414) = trilinear pole of line X(120)X(518) (the Simson line of X(1292))
X(2415) lies on the cubic K010.
X(2415) = isogonal conjugate of X(2441)
X(2415) = isotomic conjugate of X(2403)
X(2415) = cevapoint of X(900) and X(2325)
X(2415) = trilinear pole of line X(121)X(519) (the Simson line of X(1293))
X(2416) lies on the cubic K010 and these lines: 394,2430 525,1073 1294,1297
X(2416) = isogonal conjugate of X(2442)
X(2416) = isotomic conjugate of X(2404)
X(2416) = trilinear pole of line X(122)X(520) (the Simson line of X(1294))
X(2417) lies on the cubic K010 and this line: 1812,2431
X(2417) = isogonal conjugate of X(2443)
X(2417) = isotomic conjugate of X(2405)
X(2417) = trilinear pole of line X(123)X(521) (the Simson line of X(1295))
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2418) lies on the cubic K010 and these linea: 99,1296 2,5024
X(2418) = isogonal conjugate of X(2444)
X(2418) = isotomic conjugate of X(2408)
X(2418) = trilinear pole of line X(126)X(524) (the Simson line of X(1296))
X(2419) lies on the cubic K010 and these lines: 2,3265 69,2435 253,523 1297,2373
X(2419) = isogonal conjugate of X(2445)
X(2419) = isotomic conjugate of X(2409)
X(2419) = trilinear pole of line X(127)X(525) (the Simson line of X(1297))
Centers X(2420) to X(2445), on the cubic K162, are isogonal conjugates of centers X(2394) to X(2419), which are on the Simson cubic, K010. For details, see the note just before X(2394).
X(2420) lies on the cubic K162 and these lines: 3,6 110,112 249,2421 691,2422 2433,2437
X(2420) = isogonal conjugate of X(2394)
X(2420) = crosspoint of X(476) and X(648)
X(2420) = crosssum of X(i) and X(j) for these (i,j): (523,1637), (526,647)
X(2420) = X(74)-isoconjugate of X(1577)
X(2420) = crossdifference of every pair of points on line X(125)X(523) (the Simson line of X(74))
X(2421) lies on the cubic K162 and these lines: 2,6 99,1625 110,351 249,2420 250,2445 648,670 691,2444 805,881
X(2421) = isogonal conjugate of X(2395)
X(2421) = crosspoint of X(i) and X(j) for these (i,j): (110,805), (877,2396)
X(2421) = crosssum of X(i) and X(j) for these (i,j): (512,2491), (523,804), (878,2422)
X(2421) = X(92)-isoconjugate of X(878)
X(2421) = crossdifference of every pair of points on line X(115)X(512) (the Simson line of X(98))
X(2422) lies on the cubic K162 and these lines: 6,523 32,512 98,729 525,1975 691,2420 876,1910
X(2422) = isogonal conjugate of X(2396)
X(2422) = X(2086)-cross conjugate of X(6)
X(2422) = crosssum of X(i) and X(j) for these (i,j): (230,804), (511,2491), (523,2023)
X(2422) = crossdifference of every pair of points on line X(114)X(325) (the Simson line of X(99))
X(2423) lies on the cubic K162 and these lines: 6,650 31,663 81,2401 104,739 604,649
X(2423) = isogonal conjugate of X(2397)
X(2423) = X(2087)-cross conjugate of X(6)
X(2423) = crossdifference of every pair of points on line X(119)X(517) (the Simson line of X(100))
X(2424) lies on the cubic K162 and these lines: 1,905 6,657 56,663 86,2400 103,106 269,513 665,911 676,885 2283,2426
X(2424) = isogonal conjugate of X(2398)
X(2424) = X(677)-Ceva conjugate of X(103)
X(2424) = cevapoint of X(663) and X(665)
X(2424) = crosspoint of X(103) and X(677)
X(2424) = crosssum of X(516) and X(676)
X(2424) = crossdifference of every pair of points on line X(118)X(516) (the Simson line of X(101))
X(2425) lies on the cubic K162 and these lines: 6,41 2149,2427
X(2425) = isogonal conjugate of X(2399)
X(2425) = crossdifference of every pair of points on line X(124)X(522) (the Simson line of X(102))
X(2426) lies on the cubic K162 and these lines: 6,31 109,692 884,919 1110,2427 2283,2424
X(2426) = isogonal conjugate of X(2400)
X(2426) = X(677)-Ceva conjugate of X(101)
X(2426) = crosspoint of X(i) and X(j) for these (i,j): (101,677), (109,919)
X(2426) = crosssum of X(i) and X(j) for these (i,j): (514,676), (522,918)
X(2426) = crossdifference of every pair of points on line X(116)X(514) (the Simson line of X(103))
X(2427) lies on the cubic K162 and these lines: 1,6 101,109 644,1783 905,1025 919,2440 1110,2426 2149,2425
X(2427) = isogonal conjugate of X(2401)
X(2427) = crosspoint of X(651) and X(901)
X(2427) = crosssum of X(650) and X(900)
X(2427) = crossdifference of every pair of points on line X(11)X(513) (the Simson line of X(104))
X(2428) lies on the cubic K162 and this line: 6,354
X(2428) = isogonal conjugate of X(2402)
X(2428) = crossdifference of every pair of points on line X(3309)X(4904) (the Simson line of X(105))
X(2429) lies on the cubic K162 and these lines: 6,1201 101,1293
X(2429) = isogonal conjugate of X(2403)
X(2429) = crossdifference of every pair of points on line X(3667)X(3756) (the Simson line of X(106))
X(2430) lies on the cubic K162 and this line: 394,2416
X(2430) = isogonal conjugate of X(2404)
X(2430) = crossdifference of every pair of points on line X(133)X(1515) (the Simson line of X(107))
X(2431) lies on the cubic K162 and this line: 1812,2417
X(2431) = isogonal conjugate of X(2405)
X(2432) lies on the cubic K162 and these lines: 6,652 19,650 57,905 102,2291 333,2399 649,1436 654,909
X(2432) = isogonal conjugate of X(2406)
X(2433) lies on the cubic K162 and these lines: 2,525 6,647 25,512 74,111 351,878 468,879 526,686 1637,1989 1648,2395 2420,2437
X(2433) = isogonal conjugate of X(2407)
X(2433) = cevapoint of X(647) and X(686)
X(2433) = crosssum of X(i) and X(j) for these (i,j): (30,1637), (647,974)
X(2433) = X(2088)-cross conjugate of X(6)
X(2433) = crossdifference of every pair of points on line X(30)X(113) (the Simson line of X(110))
X(2434) lies on the cubic K162 and these lines: 6,373 110,1296
X(2434) = isogonal conjugate of X(2408)
X(2434) = crossdifference of every pair of points on line X(1499)X(2686) (the Simson line of X(111))
X(2435) lies on the Jerabek hyperbola, the cubic K162, and these lines: 4,525 6,520 64,512 66,523 69,2419 74,1297 248,684 526,1177
X(2435) = isogonal conjugate of X(2409)
X(2435) = isotomic conjugate of polar conjugate of X(34212)
X(2435) = orthocenter of X(3)X(6)X(64)
X(2435) = orthocenter of X(3)X(4)X(66)
X(2435) = crossdifference of every pair of points on line X(132)X(1503) (the Simson line of X(112))
X(2435) = antigonal conjugate of isogonal conjugate of X(37937)
X(2436) lies on the cubic K162 and these lines: 6,1637 323,2411
X(2436) = isogonal conjugate of X(2410)
X(2436) = crossdifference of every pair of points on line X(1522)X(1523) (the Simson line of X(476))
X(2437) lies on the cubic K162 and these lines: 6,13 2420,2433
X(2437) = isogonal conjugate of X(2411)
X(2437) = crossdifference of every pair of points on line X(526)X(3258) (the Simson line of X(477))
X(2438) lies on the cubic K162.
X(2438) = isogonal conjugate of X(2412)
X(2439) lies on the cubic K162 and this line: 6,17
X(2439) = isogonal conjugate of X(2413)
X(2439) = crossdifference of every pair of points on line X(137)X(1510) (the Simson line of X(1141))
X(2440) lies on the cubic K162 and these lines: 6,513 512,884 919,2427
X(2440) = isogonal conjugate of X(2414)
X(2440) = crossdifference of every pair of points on line X(120)X(518) (the Simson line of X(1292))
X(2441) lies on the cubic K162 and this line: 6,649
X(2441) = isogonal conjugate of X(2415)
X(2441) = crosssum of X(900) and X(2325)
X(2441) = crossdifference of every pair of points on line X(121)X(519) (the Simson line of X(1293))
X(2442) lies on the cubic K162 and these lines: (4,6), (112,1301)
X(2442) = isogonal conjugate of X(2416)
X(2442) = crossdifference of every pair of points on line X(122)X(520) (the Simson line of X(1294))
X(2443) lies on the cubic K162 and this line: 6,19
X(2443) = isogonal conjugate of X(2417)
X(2443) = crossdifference of every pair of points on line X(123)X(521) (the Simson line of X(1295))
X(2444) lies on the cubic K162 and these lines: 6,512 691,2421 1499,1992
X(2444) = isogonal conjugate of X(2418)
X(2444) = crossdifference of every pair of points on line X(126)X(524) (the Simson line of X(1296))
X(2445) lies on the cubic K162 and these lines: 6,25 110,1301 250,2421
X(2445) = isogonal conjugate of X(2419)
X(2445) = crossdifference of every pair of points on line X(127)X(525) (the Simson line of X(1297))
(Mostly) Circle-related Points, X(2446) - X(2573)
Centers X(2446) to X(2573) were contributed by Peter J. C. Moses during October to November, 2003. See the notes just before X(1662) for an introduction and notation.
Included in this section are centers of similitude of several pairs of circles. However, for some pairs of central circles, the two centers of similitudes are not triangle centers but are instead a pair of bicentric points. Specifically, Moses found that the insimilicenter (i.e., internal center of similitude) and exsimilicenter of the following pairs are bicentric points; as such they are listed in the Bicentric Pairs section (accessed by the Tables button atop this page):
{Parry circle and 2nd Lemoine circle}; see P(62) in Bicentric
Pairs
{Parry circle and circumcircle}; P(63)
{Parry circle and incircle}; P(64)
{Parry circle and nine-point circle}; P(65)
{Parry circle and Spieker circle}; P(66)
{Parry circle and Moses circle}; P(67)
The 2nd Brocard circle, defined in the aforementioned Bicentric Pairs, is mentioned several times [from X(2554) to X(2573)]. A summary of its properties follows.
Definition: The 2nd Brocard circle is the circle having center X(3) and radius eR, where e = (1 - 4 sin2ω)1/2 and R = circumradius. The 2nd Brocard circle and the Brocard circle meet in the 1st and 2nd Brocard points. Peter J. C. Moses found that the 2nd Brocard circle also passes through the points X(i) for I = 1670, 1671, 2554, 2555, 2556, 2557. The square of the radius of the 2nd Brocard circle can be written out as
a2b2c2(a4 + b4 + c 4 - b2c2 - c2a2 - a2b2)/[4(b2c2 + c2a2 + a2b2)(a2 + b2 + c2)]The power of the 2nd Brocard circle with respect to each of the vertices A, B, C is a2b2c2/(b2c2 + c2a2 + a2b2)
Of the two points of intersection, X(2446) is the closer to X(3).
X(2446) lies on the incircle and these lines: 1,3 2170,2590
X(2446) = reflection of X(2447) in X(1)
X(2446) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2590), (1382,513)
X(2446) = crosssum of X(1) and X(1381)
X(2446) = X(1114)-of-intouch-triangle
Of the two points of intersection, X(2447) is the farther from to X(3).
X(2447) lies on the incircle and these lines: 1,3 2170,2591
X(2447) = reflection of X(2446) in X(1)
X(2447) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2591), (1381,513)
X(2447) = crosssum of X(1) and X(1382)
X(2447) = X(1113)-of-intouch-triangle
Of the two points of intersection, X(2448) is the closer to X(1)..
X(2448) lies on the Bevan circle and this line: 1,3
X(2448) = reflection of X(i) in X(j) for these (i,j): (1,1381),
(2449,40)
X(2448) = crosssum of X(2310) and X(2590)
X(2448) = X(366)-aleph conjugate of X(2591)
Of the two points of intersection, X(2449) is the farther from to X(1).
X(2449) lies on the Bevan circle and this line: 1,3
X(2449) = reflection of X(i) in X(j) for these (i,j): (1,1382),
(2448,40)
X(2449) = crosssum of X(2310) and X(2591)
X(2449) = X(366)-aleph conjugate of X(2590)
As a point on the Euler line, X(2450) has Shinagawa coefficients ((E + F)F - S2,(E + F)2 - S2).
X(2450) lies on these lines: 2,3 66,2165 230,248 1503,1976 3566,3569
X(2450) = complement of X(37183)
X(2450) = inverse-in-nine-point-circle of X(1316)
X(2450) = inverse-in-orthocentroidal circle of X(3148)
X(2450) = crosspoint of X(98) and X(264)
X(2450) = crosssum of X(184) and X(511)
X(2451) lies on these lines: 6,523 512,1570 520,2489 647,657
X(2451) = inverse-in-2nd-Lemoine-circle of X(1316)
X(2451) = crosspoint of X(6) and X(648)
X(2451) = crosssum of X(2) and X(647)
X(2451) = inverse-in-2nd-Lemoine-circle of X(1316)
X(2451) = isogonal conjugate of isotomic conjugate of X(30476)
X(2451) = polar conjugate of isotomic conjugate of X(22089)
X(2452) lies on these lines: 3,2407 6,523 30,1351 98,648
X(2452) = reflection of X(1316) in X(6)
X(2453) lies on these lines: 6,523 23,183 30,599 98,338 378,477 476,1995 691,1003
X(2453) = reflection of X(6) in X(1316)
X(2453) = reflection of X(6) in the Euler line
As a point on the Euler line, X(2454) has Shinagawa coefficients (S2 + 2K, -3S2).
Of the two points of intersection, X(2454) is the closer to X(3).
X(2454) lies on the Steiner inscribed ellipse and this line: 2,3
X(2454) = midpoint of X(2) and X(2479)
X(2454) = reflection of X(2455) in X(2)
X(2454) = complement of X(2480)
X(2454) = projection from Steiner circumellipse to Steiner inellipse of X(2479)
As a point on the Euler line, X(2455) has Shinagawa coefficients (S2 - 2K, -3S2).
Of the two points of intersection, X(2455) is the farther from to X(3).
X(2455) lies on the Steiner inscribed ellipse and this line: 2,3
X(2455) = midpoint of X(2) and X(2480)
X(2455) = reflection of X(2454) in X(2)
X(2455) = complement of X(2479)
X(2455) = projection from Steiner circumellipse to Steiner inellipse of X(2480)
X(2456) lies on these lines: 3,6 98,325
X(2456) = reflection of X(i) in X(j) for these (i,j): (1351,1570), (1691,182), (2080,1691)
X(2456) = {X(1687),X(1688)}-harmonic conjugate of X(1692)
X(2456) = harmonic center of circumcircle and 8th Lozada circle
X(2457) lies on these lines: 523,656 650,2529
X(2458) lies on this line: 3,6
X(2458) = midpoint of X(2461) and X(2462)
X(2458) = reflection of X(32) in X(1691)
X(2458) = pole wrt 1st Lemoine circle of line X(39)X(512)
X(2459) lies on these lines: 3,6 485,490
X(2459) = reflection of X(2460) in X(187)
X(2459) = anticomplement of X(32432)
X(2459) = perspector of ABC and the reflection of the Lucas tangents triangle in the Lemoine axis
X(2459) = inverse-in-Lucas(-1)-radical-circle of X(372)
X(2459) = X(2459) of 2nd Brocard triangle
X(2459) = crossdifference of every pair of points on line X(523)X(590)
X(2459) = Brocard axis intercept, other than X(2460), of circle with segment PU(2) as diameter
X(2460) lies on these lines: 3,6 486,489
X(2460) = reflection of X(2459) in X(187)
X(2460) = anticomplement of X(32435)
X(2460) = crossdifference of every pair of points on line X(523)X(615)
X(2460) = perspector of ABC and the reflection of the Lucas(-1) tangents triangle in the Lemoine axis
X(2460) =Lucas-radical-circle-inverse of X(371)
X(2460) = X(2460)-of-2nd-Brocard-triangle
X(2460) = Brocard axis intercept, other than X(2459), of circle with segment PU(2) as diameter
X(2461) lies on this line: 3,6
X(2461) = reflection of X(2462) in X(2458)
X(2461) = inverse-in-1st-Lemoine-circle of X(371)
X(2462) lies on this line: 3,6
X(2462) = reflection of X(2461) in X(2458)
X(2462) = inverse-in-1st-Lemoine-circle of X(372)
X(2463) lies on these lines: 1,381 2,2468 8,2467 388,2553 497,2552 1124,2466 1335,2465 1672,2472 1673,2471 1634,2470 1675,2469
X(2464) lies on these lines: 1,381 2,2467 8,2468 56,1345 388,2552 497,2553 1124,2465 1335,2466 1672,2471 1673,2472 1634,2469 1675,2470
X(2465) lies on these lines: 6,13 1124,2464 1335,2463 1377,2468 1378,2467 1587,2553 1588,2552 1668,2469 1669,2470 1687,2472 1688,2471
X(2465) = {X(6),X(381)}-harmonic conjugate of X(2466)
X(2466) lies on these lines: 6,13 1124,2463 1335,2464 1377,2467 1378,2468 1587,2552 1588,2553 1668,2470 1669,2469 1687,2471 1688,2472
X(2466) = {X(6),X(381)}-harmonic conjugate of X(2465)
X(2467) lies on these lines: 2,2464 8,2463 10,381 1377,2466 1378,2465 1678,2470 1679,2469 1680,2472 1681,2471 2550,2553 2551,2551
X(2468) lies on these lines: 2,2463 8,2464 10,381 1377,2465 1378,2466 1678,2469 1679,2470 1680,2471 1681,2472 2550,2552 2551,2553
X(2469) lies on these lines: 182,381 1668,2465 1669,2466 1674,2464 1675,2463 1678,2468 1679,2467 2542,2553 2543,2552
X(2470) lies on these lines: 182,381 1668,2466 1669,2465 1674,2463 1675,2464 1678,2467 1679,2468 2542,2552 2543,2553
X(2471) lies on these lines: 182,381 1672,2464 1673,2463 1680,2468 1681,2467 1687,2466 1688,2465 2546,2553 2547,2552
X(2472) lies on these lines: 181,381 1672,2463 1673,2464 1680,2467 1681,2468 1687,2465 1688,2466 2546,2552 2547,2553
X(2473 lies on these lines: 513,2487 649,2521 2488,2504
The titular notation (X(4),2R) means the circle with center X(4) and radius 2R, where R denotes the circumradius of triangle ABC.
X(2474) lies on these lines: 512,1570 525,2514 826,2518
As a point on the Euler line, X(2475) has Shinagawa coefficients (abc$a$, 4S2), and also Shinagawa coefficients (R, 2r)..
Let F be the Fuhrmann triangle. Then F is perspective to F-of-orthic-triangle-of-F, and the perspector is X(2475). (Randy Hutson, June 15, 2015)
X(2475) lies on these lines: {1, 149}, {2, 3}, {7, 2893}, {8, 79}, {10, 191}, {11, 5253}, {12, 100}, {35, 3822}, {65, 5086}, {75, 5016}, {80, 3754}, {81, 1834}, {115, 5277}, {144, 1654}, {145, 388}, {148, 1655}, {153, 355}, {193, 5800}, {214, 5443}, {225, 4296}, {274, 316}, {323, 3193}, {495, 3871}, {497, 3622}, {518, 5178}, {519, 5270}, {535, 5258}, {551, 4857}, {950, 5249}, {960, 5057}, {966, 5036}, {1030, 5949}, {1043, 3936}, {1056, 3623}, {1125, 3583}, {1158, 5587}, {1220, 4972}, {1441, 2897}, {1448, 2000}, {1479, 3616}, {1621, 6284}, {1698, 4333}, {1836, 3869}, {1837, 4459}, {1869, 3101}, {1901, 2287}, {1993, 5706}, {2646, 3838}, {2794, 5985}, {2886, 2975}, {2894, 3419}, {3035, 3614}, {3218, 4292}, {3294, 5134}, {3421, 4678}, {3621, 5082}, {3652, 5818}, {3679, 3951}, {3770, 4696}, {3813, 5434}, {3841, 5251}, {3870, 5290}, {3878, 5180}, {3909, 5252}, {3925, 5260}, {4316, 5267}, {4385, 5300}, {4652, 5705}, {4855, 5219}, {4968, 5015}, {4999, 5303}, {5176, 5836}, {5254, 5276}, {5318, 5362}, {5321, 5367}, {5554, 6223}
X(2475) = reflection of X(i) in X(j) for these (i,j): (21,442), (191,10)
X(2475) = isogonal conjugate of X(34435)
X(2475) = inverse-in-Fuhrmann circle of X(3448)
X(2475) = anticomplement of X(21)
X(2475) = X(1441)-Ceva conjugate of X(2)
X(2475) = X(54) of Fuhrmann triangle
X(2475) = {X(2),X(21)}-harmonic conjugate of X(15674)
X(2475) = crosspoint of X(4) and X(8) wrt extraversion triangle of X(8)
X(2476) = inverse-in-orthocentroidal-circle of X(21)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2477) lies on these lines: 1,49 11,54 12,110 55,1069 56,184 156,1478 2595,2606
X(2477) = perspector of ABC and extraversion triangle of X(215)
As a point on the Euler line, X(2478) has Shinagawa coefficients (abc$a$, -2S2).
X(2478) lies on these lines: 1,908 2,3 8,210 10,1479 11,958 12,1001 55,1329 63,1210 78,950 145,1058 149,1145 264,1896 283,1724 329,938 355,392 388,1319 496,956 499,993 1038,1877 1125,1478 1441,1882 1714,2328
X(2478) = complement of X(4190)
X(2478) = anticomplement of X(474)
As a point on the Euler line, X(2479) has Shinagawa coefficients (K + S2, -3S2).
Of the two points of intersection, X(2479) is the one closer to X(3).
X(2479) lies on these lines: 2,3 671,24008
X(2479) = anticomplement of X(2455)
X(2479) = reflection of X(i) in X(j) for these (i,j): (2,2454), (2480,2)
X(2479) = projection from Steiner inellipse to Steiner circumellipse of X(2454)
X(2479) = Steiner-circumellipse-X(3)-antipode of X(2480)
X(2479) = trilinear pole of line X(2)X(24008)
As a point on the Euler line, X(2480) has Shinagawa coefficients (K - S2, 3S2).
Of the two points of intersection, X(2479) is the one farther from X(3).
X(2480) lies on this line: 2,3 671,24007
X(2480) = reflection of X(i) in X(j) for these (i,j): (2,2455), (2479,2)
X(2480) = anticomplement of X(2454)
X(2480) = projection from Steiner inellipse to Steiner circumellipse of X(2455)
X(2480) = Steiner-circumellipse-X(3)-antipode of X(2479)
X(2480) = trilinear pole of line X(2)X(24007)
If you have The Geometer's Sketchpad, you can view X(2481).
X(2481) lies on these lines: 1,85 4,150 8,76 9,75 21,99 104,927 239,294 273,1041 286,648 314,670 334,350 767,919 870,1438 987,1416 1441,2346 1447,2223 1462,2298
X(2481) = isogonal conjugate of X(2223)
X(2481) = isotomic conjugate of X(518)
X(2481) = complement of X(39350)
X(2481) = X(19)-isoconjugate of X(20752)
X(2481) = Steiner-circumellipse-X(1)-antipode of X(664)
X(2481) = Steiner-circumellipse-X(4)-antipode of X(18026)
X(2481) = cevapoint of X(i) and X(j) these (i,j): (2,518), (11,918), (75,350), (105,1814)
X(2481) = X(i)-cross conjugate of X(j) for these (i,j): (239,274), (518,2), (885,666)
X(2481) = crosssum of X(55) and X(211)
X(2481) = trilinear pole of line X(2)X(650)
X(2481) = pole wrt polar circle of trilinear polar of X(5089)
X(2481) = X(48)-isoconjugate (polar conjugate) of X(5089)
X(2481) = crossdifference of PU(97)
X(2481) = areal center of cevian triangles of PU(47)
X(2482) is the midpoint of the centers of the (equilateral) antipedal triangles of X(13) and X(14). (Randy Hutson, 9/23/2011)
Let A'B'C' be the orthic triangle. Then X(2482) is the radical center of the Schoute circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)
X(2482) lies on the Steiner inscribed ellipse, the bicevian conic of X(2) and X(99), and these lines: 2,99 3,67 30,114 32,1992 39,597 187,524 230,5215 351,690 530,619 531,618 538,1569 1086,1125
X(2482) = midpoint of X(2) and X(99)
X(2482) = reflection of X(i) in X(j) for these (i,j): (2,620), (115,2)
X(2482) = isogonal conjugate of X(10630)
X(2482) = complement of X(671)
X(2482) = complementary conjugate of X(625)
X(2482) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,524), (88,690)
X(2482) = crosspoint of X(2) and X(524)
X(2482) = crosssum of X(6) and X(111)
X(2482) = crosssum of circumcircle intercepts of Schoute circle
X(2482) = projection from Steiner circumellipse to Steiner inellipse of X(99)
X(2482) = perspector of circumconic centered at X(524)
X(2482) = center of circumconic that is locus of trilinear poles of lines passing through X(524) (hyperbola {A,B,C,X(2),X(99)})
X(2482) = crossdifference of every pair of points on line X(111)X(351)
X(2482) = isogonal conjugate wrt medial triangle of X(625)
X(2482) = barycentric square of X(524)
X(2482) = midpoint of X(5463) and X(5464)
X(2482) = X(5) of anti-McCay triangle
X(2482) = antipode of X(2) in the bicevian conic of X(2) and X(99)
X(2482) = centroid of mid-triangle of antipedal triangles of X(13) and X(14)
X(2482) = X(2)-of-X(187)-adjunct-anti-altimedial-triangle
X(2482) = QA-P3 (Gergonne-Steiner Point) of quadrangle ABCX(2) (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/18-qa-p3.html)
X(2483) lies on these lines: 44,513 512,1691 918,1019 1919,2605
X(2483) = reflection of X(2484) in X(2515)
X(2483) = midpoint of X(649) and X(2484)
X(2483) = crosssum of X(1) and X(2483)
X(2484) lies on these lines: 6,834 44,513 512,1692 663,1919
X(2484) = reflection of X(i) in X(j) for these (i,j): (649,2483), (2483,2515)
X(2484) = isogonal conjugate of X(37215)
X(2484) = crosssum of X(i) and X(j) for these {i,j}: {1, 2484}, {514, 10436}, {649, 17017}, {2522, 17441}
X(2484) = crosspoint of X(i) and X(j) for these {i,j}: {1, 37215}, {101, 2258}
X(2484) = crossdifference of every pair of points on line X(1)X(69)
X(2485) lies on these lines: 6,520 230,231 512,1691 525,2507 669,2514 2165,2395
X(2485) = midpoint of X(i) and X(j) for these (i,j): (647,2489), (669,2514)
X(2485) = reflection of X(i) in X(j) for these (i,j): (2489,2492), (2506,647)
X(2485) = isogonal conjugate of the isotomic conjugate of X(33294)
X(2485) = complement of X(3267)
X(2485) = X(2)-Ceva conjugate of X(127)
X(2485) = polar conjugate of isotomic conjugate of X(8673)
X(2485) = X(63)-isoconjugate of X(1289)
X(2485) = crosspoint of X(i) and X(j) for these (i,j): (2,112), (83,107)
X(2485) = crosssum of X(i) and X(j) for these (i,j): (6,525), (39,520)
X(2485) = perspector of circumconic centered at X(127) (hyperbola {{A,B,C,X(4),X(22)}})
X(2485) = center of circumconic that is locus of trilinear poles of lines passing through X(127)
X(2485) = intersection of trilinear polars of X(4) and X(22)
X(2485) = crossdifference of every pair of points on line X(3)X(66) (complement of van Aubel line)
X(2485) = PU(4)-harmonic conjugate of X(14580)
X(2486) lies on these lines: 5,2783 11,244 115,804
X(2486) = complement of X(4436)
X(2486) = intersection of tangents to nine-point circle at X(11) and X(115)
X(2486) = pole, wrt nine-point circle, of line X(11)X(115)
X(2486) = perspector of side- and vertex-triangles of tangential triangles of medial and Feuerbach triangles
X(2487) lies on these lines: 514,2516 649,1638 918,2490 1125,1499 2496,2505
X(2487) = X(37)-Ceva conjugate of X(523)
X(2487) = crosspoint of X(10) and X(693)
X(2487) = crosssum of X(58) and X(692)
X(2488) lies on these lines: 187,237 513,676 650,926 884,2194
X(2488) = reflection of X(2495) in X(2494)
X(2488) = crosspoint of X(513) and X(663)
X(2488) = crosssum of X(100) and X(664)
X(2489) lies on these lines: 6,924 19,876 112,250 230,231 393,2395 512,1692 520,2451 882,1843 1974,2422 2507,2524
X(2489) = reflection of X(i) in X(j) for these (i,j): (647,2485), (2485,2492), (2524,2507)
X(2489) = X(112)-Ceva conjugate of X(25)
X(2489) = cevapoint of X(647) and X(2519)
X(2489) = X(i)-cross conjugate of X(j) for these (i,j): (669,512), (1084,1974)
X(2489) = crosspoint of X(25) and X(112)
X(2489) = perspector of hyperbola {{A,B,C,X(4),X(25)}}
X(2489) = intersection of trilinear polars of X(4) and X(25)
X(2489) = isogonal conjugate of X(4563)
X(2489) = pole wrt polar circle of trilinear polar of X(670) (line X(2)X(39))
X(2489) = polar conjugate of X(670)
X(2489) = barycentric product X(8105)*X(8106)
X(2489) = PU(4)-harmonic conjugate of X(3291)
X(2490) lies on these lines: 230,231 513,2505 522,2496 649,1639 661,1213 918,2487
Let A'B'C' and A"B"C" be the circumcevian triangles of X(511) and X(512). Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(2491). (Randy Hutson, October 8, 2019)
X(2491) lies on these lines: 39,690 230,231 351,865 512,2021 669,688 684,3569 804,2023 882,2422
X(2491) = X(i)-Ceva conjugate of X(j) for these (i,j): (882,688), (2395,512), (2396,511)
X(2491) = crosspoint of X(i) and X(j) for these (i,j): (2,805), (112,1976), (511,2396), (512,2395), (669,881)
X(2491) = crosssum of X(i) and X(j) for these (i,j): (6,804), (98,2422), (99,2421), (325,525), (645,874), (670,880)
X(2491) = perspector of circumconic centered at X(2679)
X(2491) = center of circumconic that is locus of trilinear poles of lines passing through X(2679)
X(2491) = X(2)-Ceva conjugate of X(2679)
X(2491) = center of bicevian conic of X(511) and X(512)
X(2491) = barycentric product X(511)*X(512)
X(2492) lies on these lines: 6,526 111,351 112,1576 115,804 230,231 512,2030 1989,2395
X(2492) = midpoint of X(2485) and X(2489)
X(2492) = isogonal conjugate of X(17708)
X(2492) = complement of X(35522)
X(2492) = crosspoint of X(i) and X(j) for these (i,j): (2,691), (111,112)
X(2492) = crosssum of X(i) and X(j) for these (i,j): (6,690), (524,525)
X(2492) = crossdifference of every pair of points on line X(3)X(67)
X(2492) = midpoint of X(8105) and X(8106)
X(2492) = center of circle Moses-Parry circle
X(2492) = intersection of tangents to circumcircle at X(111) and X(112), and to nine-point-circle at X(115) and X(1560)
X(2492) = pole of line X(25)X(111) wrt circumcircle
X(2492) = perspector of hyperbola {{A,B,C,X(4),X(23)}}
X(2492) = polar conjugate of isotomic conjugate of X(9517)
X(2492) = pole wrt polar circle of line X(2)X(339)
X(2492) = intersection of trilinear polars of X(4) and X(23)
X(2492) = X(63)-isoconjugate of X(935)
X(2493) lies on these lines: 2,94 5,39 6,110 23,50 24,112 25,1576 53,136 216,3054 230,231 1084,1196
X(2493) = crosspoint of X(2) and X(842)
X(2493) = crosssum of X(6) and X(542)
X(2493) = PU(4)-harmonic conjugate of polar conjugate of X(35139)
X(2494) lies on these lines: 512,1691 513,676
X(2494) = midpoint of X(2488) and X(2495)
X(2494) = reflection of X(2495) in X(2498)
X(2495) lies on these lines: 512,1692 513,676
X(2495) = refection of X(i) and X(j) for these (i,j): (2488,2494), (2494,2498)
X(2496) lies on these lines: 513,676 522,2490 2487,2505
X(2497) lies on these lines: 512,2021 513,676
X(2498) lies on these lines: 512,2030 513,676
X(2498) = midpoint of X(2494) and X(2495)
X(2499) lies on these lines: 512,650 513,676 661,926
X(2500) lies on these lines: 512,650 513,2532 522,2490
If you have The Geometer's Sketchpad, you can view X(2501).
Let LA be the orthic axis of triangle BCX(4), and define LB, LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. Triangle A'B'C' is here named the orthic axes triangle. A'B'C' is also the cevian triangle of X(4) wrt the orthic triangle. A'B'C' is perspective to ABC at X(4), and homothetic to the polar triangle of the nine-point circle at X(427). The orthic axes of ABC, the orthic triangle, and the orthic axes triangle concur in X(2501). (Randy Hutson, August 19, 2019)
Let A"B"C" be the reflection triangle. Then A' is the polar-circle-inverse of A". (Randy Hutson, January 19, 2020)
X(2501) lies on the inscribed parabola having focus X(112) and on these lines: 4,1499 25,669 107,685 112,476 230,231 297,525 393,2433 419,3288 421,2623 450,2451 460,512 648,892
X(2501) = isogonal conjugate of X(4558)
X(2501) = isotomic conjugate of X(4563)
X(2501) = cevapoint of X(i) and X(j) for these (i,j): (6,2079), (512,2489)
X(2501) = crosspoint of X(i) and X(j) for these (i,j): (2,925), (4,648), (107,2052)
X(2501) = crosssum of X(i) and X(j) for these (i,j): (3,647), (6,924), (343,525), (520,577), (906,1331)
X(2501) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,136), (92,2969), (107,25), (112,53), (115,393), (264,2971), (393,115), (512,523), (648,4), (653,407), (685,460), (687,30), (1783,1865), (1897,430), (2052,2970), (2971,264)
X(2501) = perspector of circumconic centered at X(136)
X(2501) = center of circumconic that is locus of trilinear poles of lines passing through X(136)
X(2501) = X(2)-Ceva conjugate of X(136)
X(2501) = radical center of {circumcircle, nine-point circle, orthosymmedial circle}
X(2501) = radical center of {circumcircle, orthocentroidal circle, orthosymmedial circle}
X(2501) = radical center of {nine-point circle, orthocentroidal circle, orthosymmedial circle}
X(2501) = intersection of orthic axes of ABC and orthic triangle
X(2501) = trilinear pole of line X(115)X(2971) (polar of X(99) wrt polar circle)
X(2501) = pole wrt polar circle of trilinear polar of X(99) (line X(2)X(6))
X(2501) = polar conjugate of X(99)
X(2501) = perspector, wrt anticevian triangle of X(4), of bianticevian conic of X(1) and X(4)
X(2501) = crossdifference of every pair of points on line X(3)X(49)
X(2501) = X(650)-of-orthic-triangle if ABC is acute
X(2501) = excentral-to-ABC functional image of X(650)
X(2501) = PU(4)-harmonic conjugate of X(230)
X(2501) = excentral-to-ABC functional image of X(650)
X(2501) = barycentric product X(4)*X(523)
X(2501) = barycentric product X(24007)*X(24008) (the Kiepert hyperbola intercepts of the orthic axis)
Let L be the trilinear polar of X(6), i.e. the Lemoine axis. Let A' = L∩AX(6), and define B' and C' cyclically; then X(2502) is the centroid of A'B'C'. (Randy Hutson, February 10, 2016)
If you have The Geometer's Sketchpad, you can view X(2502).
X(2502) lies on these lines: 2,353 6,110 23,352 187,237 542,1648 543,1641 694,1383
X(2502) = X(583)-Ceva conjugate of X(6)
X(2502) = crosspoint of X(6) and X(843)
X(2502) = crosssum of X(2) and X(543)
X(2502) = crossdifference of every pair of points on line X(2)X(690)
X(2502) = intersection of tangents to Parry circle at X(15) and X(16)
X(2502) = pole of Brocard axis wrt Parry circle
X(2502) = inverse-in-Parry-circle of X(187)
X(2502) = {X(110),X(111)}-harmonic conjugate of (6)
X(2502) = X(1648)-of-4th-anti-Brocard-triangle
X(2502) = centroid of (degenerate) cross-triangle of ABC and circumsymmedial triangle
X(2502) = perspector of conic {{A,B,C,P,U}}, where P and U are the intersections of circles O(13,14) and O(15,16)
X(2503) lies on these lines: 6,110 37,2611, 44,513
X(2503) = X(842)-Ceva conjugate of X(55)
X(2504) lies on these lines: 11,244 441,525 2473,2488
X(2505) = complement of X(2976)
X(2505) lies on these lines: 11,244 513,2490 2487,2496
X(2506) lies on these lines: 6,512 525,2485 2489,2519
X(2506) = midpoint of X(2489) in X(2519)
X(2507) lies on these lines: 512,2024 525,2485 804,2023 1194,1637 2489,2524
X(2507) = midpoint of X(2489) and X(2524)
X(2508) lies on these lines: 115,804 512,2032 525,2485
X(2509) lies on these lines: 6,521 44,513 525,2485 905,918 926,2494
X(2509) = midpoint of X(661) and X(2484)
X(2509) = crosspoint of X(2) and X(1783)
X(2509) = crosssum of X(i) and X(j) for these (i,j): (1,2509),
(6,905)
X(2510) lies on these lines: 115,804 248,878 441,525 512,2031
X(2511) lies on these lines: {44,513}, {115,804}, {665,1213}, {900,2092}, {4272,4435}
The circle (X(4),2R) is identified at X(2474).
X(2512) lies on these lines: 325,523 512,650
X(2512) = isotomic conjugate of X(35565)
The circle (X(4),2R) is identified at X(2474).
X(2513) lies on these lines: 325,523 512,2021 525,2531 647,688
X(2513) = isotomic conjugate of X(35566)
The circle (X(4),2R) is identified at X(2474).
X(2514) lies on these lines: 325,523 512,1692 525,2474 669,2485
X(2514) = reflection of X(669) in X(2485)
X(2514) = isotomic conjugate of X(35567)
X(2514) = crossdifference of every pair of points on line X(32)X(69)
X(2515) lies on these lines: 44,513 512,2030
X(2515) = midpoint of X(2483) and X(2484)
X(2515) = reflection of X(2529) in X(2527)
X(2516) lies on these lines: 44,513 514,2487 522,2490
X(2516) = complement of X(23813)
X(2516) = anticomplementary isotomic conjugate of trilinear pole, wrt anticomplementary triangle, of Gergonne line
X(2516) = polar conjugate of isogonal conjugate of X(22147)
X(2517) lies on these lines: 240,522 325,523 513,2533
X(2517) = isotomic conjugate of X(1310)
X(2518) lies on these lines: 211,688 512,2031 826,2474
X(2519) lies on these lines: 441,525 512,1570
X(2520) lies on these lines: 460,512 513,676
X(2521) lies on these lines: 512,2031 649,2523
X(2522) lies on these lines: 44,513 441,525 1021,1734
X(2522) = isogonal conjugate of X(36099)
X(2522) = crossdifference of every pair of points on line X(1)X(25)
X(2523) lies on these lines: 441,525 649,2521 650,2527
X(2524) lies on these lines: 39,512 441,525 2489,2507
X(2524) = reflection of X(2489) in X(2507)
X(2524) = crosssum of X(2) and X(2451)
X(2525) lies on these lines: 441,525 523,2526 669,690 826,2474
X(2525) = X(305)-Ceva conjugate of X(125)
X(2525) = crosspoint of X(3) and X(1634)
X(2525) = perspector of hyperbola {{A,B,C,X(69),X(141)}}
X(2525) = intersection of trilinear polars of X(69) and X(141)
X(2526) lies on these lines: 44,513 523,2525 830,905
X(2526) = reflection of X(650) in X(1491)
X(2526) = crosssum of X(1) and X(2526)
X(2527) lies on these lines: 513,2490 514,2487 649,900 650,2523
X(2527) = midpoint of X(2516) and X(2529)
X(2528) lies on the Kiepert parabola and these lines: 39,647 525,669 826,2474
X(2528) = X(99)-Ceva conjugate of X(141)
X(2528) = crosspoint of X(99) and X(141)
X(2528) = crosssum of X(251) and X(512)
X(2529) lies on these lines: 514,2487 650,2457
X(2529) = reflection of X(2516) and X(2537)
X(2530) lies on these lines: 10,514 36,238 512,2254 649,2521 661,665 663,1201 693,784 826,2474 832,1459
X(2530) = reflection of X(667) and X(905)
X(2530) = crosspoint of X(513) and X(693)
X(2530) = crosssum of X(i) and X(j) for these (i,j): (308,850),
(100,692), (650,2330)
X(2531) lies on these lines: 39,512 99,783 525,2513 826,2474
X(2531) = bicentric difference of PU(156)
X(2532) lies on these lines: 513,2500 514,2487
X(2533) lies on these lines: 10,514 65,879 512,1577 523,656 649,814 784,1734
X(2533) = crosspoint of X(10) and X(668)
X(2533) = isotomic conjugate of X(4594)
The Spieker radical circle, with center X(10) and radius d/2, is introduced in
Darij Grinberg and Paul Yiu, "The Apollonius Circle as a Tucker Circle," Forum Geometricorum 2 (2002) 175-182.
X(2534) lies on these lines: 1,2 11,2540 12,2541 55,2537 56,2536 181,2538 1682,2539
See X(2534).
X(2535) lies on these lines: 1,2 11,2541 12,2540 55,2536 56,2537 181,2539 1682,2538
See X(2534).
X(2536) lies on these lines: 2,2541 3,10 4,2540 55,2535 56,2534 386,2538 573,2539
X(2536) = X(573)-Ceva conjugate of X(2537)
See X(2534).
X(2537) lies on these lines: 2,2540 3,10 4,2541 55,2534 56,2535 386,2539 573,2538
X(2537) = X(573)-Ceva conjugate of X(2536)
See X(2534).
X(2538) lies on these lines: 5,10 181,2534 386,2536 573,2537 1682,2535
See X(2534).
X(2539) lies on these lines: 5,10 181,2535 386,2537 573,2536 1682,2534
See X(2534).
X(2540) lies on these lines: 2,2537 4,2536 5,10 11,2534 12,2535
See X(2534).
X(2541) lies on these lines: 2,2536 4,2537 5,10 11,2535 12,2534
The circle (X(4),2R) is identified at X(2474).
X(2542) lies on these lines: {2, 1341}, {4, 83}, {10, 1705}, {20, 1340}, {388, 1675}, {497, 1674}, {516, 1704}, {1349, 3091}, {1379, 14712}, {1587, 1668}, {1588, 1669}, {1678, 2551}, {1679, 2550}, {1693, 9535}, {1694, 9534}, {2011, 2545}, {2012, 2544}, {2033, 2549}, {2034, 2548}, {2039, 7797}, {2469, 2553}, {2470, 2552}, {2559, 6655}, {3557, 20088}, {6039, 7792}, {6190, 14929}
X(2542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 182, 2543}, {1341, 1348, 2}, {3618, 7790, 2543}
The circle (X(4),2R) is identified at X(2474).
X(2543) lies on these lines: 2,1340 4,83 10,1704 20,1341 388,1674 497,1675 516,1705 1587,1669 1588,1668 1678,2550 1679,2551 2011,2544 2012,2545 2033,2548 2034,2549 2469,2552 2470,2553
The circle (X(4),2R) is identified at X(2474).
X(2544) lies on these lines: 2,1689 4,39 10,2018 20,1690 371,2547 372,2546 388,2008 497,2007 516,2017 1587,1670 1588,1671 2011,2543 2012,2542 2013,2551 2014,2550
The circle (X(4),2R) is identified at X(2474).
X(2545) lies on these lines: 2,1690 4,39 10,2017 20,1689 371,2546 372,2547 388,2007 497,2008 516,2018 1587,1671 1588,1670 2011,2542 2012,2543 2013,2550 2014,2551
The circle (X(4),2R) is identified at X(2474).
X(2546) lies on these lines: 2,1343 4,83 10,1701 20,1342 194,1671 371,2545 372,2544 388,1673 497,1672 516,1700 1587,1688 1588,1687 1680,2551 1681,2550 2035,2549 2036,2548 2471,2553 2472,2552
The circle (X(4),2R) is identified at X(2474).
X(2547) lies on these lines: 2,1342 4,83 10,1700 20,1343 194,1670 371,2544 372,2545 388,1672 497,1673 516,1701 1587,1687 1588,1688 1680,2550 1681,2551 2035,2548 2036,2549 2471,2552 2472,2553
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
The circle (X(4),2R) is identified at X(2474).
X(2548) lies on these lines: 2,32 4,39 5,6 10,1572 20,574 115,147 148,1569 172,499 187,631 211,263 217,1899 230,1656 316,2021 388,1015 427,2207 497,1500 498,1914 516,1571 625,1692 1478,2275 1479,2276 1504,1588 1505,1587 1573,2551 1574,2550 2033,2543 2034,2542 2035,2547 2036,2546
X(2548) = complement of X(3785)
X(2548) = anticomplement of X(7815)
X(2548) = {X(5),X(6)}-harmonic conjugate of X(3767)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
The circle (X(4),2R) is identified at X(2474).
X(2549) lies on these lines: 2,99 3,230 4,39 6,30 10,1571 20,32 53,1597 69,538 147,1569 184,1562 187,376 193,754 194,315 388,1500 497,1515 516,1572 625,1007 1194,1370 1478,2276 1479,2275 1504,1587 1505,1588 1573,2550 1574,2551 1885,2207 2033,2542 2034,2543 2035,2546 2036,2547
X(2549) = X(69)-of-1st-Brocard triangle
X(2549) = 1st-Brocard-isogonal conjugate of X(9306)
X(2549) = 1st-Brocard-isotomic conjugate of X(1352)
X(2549) = X(6)-of-obverse-triangle-of-X(69)
X(2549) = {X(10653),X(10654)}-harmonic conjugate of X(11179)
The circle (X(4),2R) is identified at X(2474).
Let A'B'C' be the intangents-to-extangents similarity image of ABC. A'B'C' is homothetic to ABC at X(55) and to the medial triangle at X(2550). (Randy Hutson, December 2, 2017)
Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the anticomplementary triangle at X(2550). (Randy Hutson, December 2, 2017)
X(2550) lies on these lines: 1,142 2,11 3,1602 4,9 7,8 12,480 20,958 144,1654 200,226 210,329 318,1118 355,971 376,993 442,954 519,1056 527,1478 740,2294 936,946 948,2263 960,962 1058,1125 1377,1588 1378,1587 1445,1788 1479,1698 1573,2549 1574,2548 1678,2543 1679,2542 1680,2547 1681,2546 2013,2545 2014,2544 2467,2553 2468,2552
X(2550) = midpoint of X(7) and X(8)
X(2550) = reflection of X(i) in X(j) for these (i,j): (1,142), (9,10), (390,1001)
X(2550) = isogonal conjugate of X(3423)
X(2550) = complement of X(390)
X(2550) = anticomplement of X(1001)
X(2550) = inverse-in-Feuerbach-hyperbola of X(497)
X(2550) = outer-Garcia-to-ABC similarity image of X(9)
The circle (X(4),2R) is identified at X(2474).
X(2551) lies on these lines: 2,12 3,1603 4,9 8,210 20,1376 55,452 63,1788 65,329 200,950 219,387 220,1834 318,1857 377,1155 390,480 443,1478 515,936 518,938 519,1058 631,993 944,997 1056,1125 1377,1587 1378,1588 1573,2548 1574,2549 1678,2542 1679,2543 1680,2546 1681,2547 2013,2544 2014,2545 2467,2552 2468,2553
X(2551) = reflection of X(1706) in X(10)
X(2551) = crosssum of X(56) and X(1466)
As a point on the Euler line, X(2552) has Shinagawa coefficients (1, J + 3).
The circle (X(4),2R) is identified at X(2474).
X(2552) lies on these lines: 2,3 388,2464 497,2463 1587,2466 1588,2465 2467,2551 2468,2550 2469,2543 2470,2542 2471,2547 2472,2546
As a point on the Euler line, X(2553) has Shinagawa coefficients (1, -J + 3).
The circle (X(4),2R) is identified at X(2474).
X(2553) lies on these lines: 2,3 388,2463 497,2464 1587,2465 1588,2466 2467,2550 2468,2551 2469,2542 2470,2543 2471,2546 2472,2547
As a point on the Euler line, X(2554) has Shinagawa coefficients (-J/e + 1, J/e - 3).
Of the two points of intersection, X(2554) is one closer to X(4).
X(2554) lies on this line: 2,3
X(2554) = reflection of X(2555) in X(3)
As a point on the Euler line, X(2555) has Shinagawa coefficients (J/e + 1, -J/e - 3).
Of the two points of intersection, X(2555) is one farther from X(4).
X(2555) lies on this line: 2,3
X(2555) = reflection of X(2554) in X(3)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Of the two points of intersection, X(2556) is one closer to X(1).
X(2556) lies on this line: 1,3
X(2556) = reflection of X(2557) in X(3)
Of the two points of intersection, X(2557) is one farther from X(1).
X(2557) lies on this line: 1,3
X(2557) = reflection of X(2556) in X(3)
X(2558) lies on these lines: 3,6 1672,2565 1673,2564 1676,2567 1677,2566 1680,2569 1681,2568 1700,2573 1701,2572
X(2558) = {X(1687),X(1688)}-harmonic conjugate of X(3557)
X(2558) = homothetic center of 6th anti-Brocard triangle and circumcevian triangle of X(3414)
X(2559) lies on these lines: 3,6 1672,2564 1673,2565 1676,2566 1677,2567 1680,2568 1681,2569 1700,2572 1701,2573
X(2559) = {X(1687),X(1688)}-harmonic conjugate of X(3558)
X(2559) = homothetic center of 6th anti-Brocard triangle and circumcevian triangle of X(3413)
X(2560) lies on these lines: 3,6 485,2567 486,2566 1124,2565 1335,2564 1337,2569 1378,2568 1702,2573 1703,2572
X(2560) = inverse-in-Brocard-circle of X(2561)
X(2560) = radical center of Lucas(2e) circles, e as at X(2556)
X(2561) lies on these lines: 3,6 485,2566 486,2567 1124,2564 1335,2565 1337,2568 1378,2569 1702,2572 1703,2573
X(2561) = inverse-in-Brocard-circle of X(2560)
X(2561) = radical center of Lucas(-2e) circles, e as at X(2556)
X(2562) lies on these lines: 3,6 2007,2565 2008,2564 2009,2567 2010,2566 2013,2569 2014,2568 2017,2573 2018,2572
X(2562) = inverse-in-2nd-Brocard-circle of X(1667)
X(2563) lies on these lines: 3,6 2007,2564 2008,2565 2009,2566 2010,2567 2013,2568 2014,2569 2017,2572 2018,2573
X(2563) = 2nd-Brocard-circle-inverse of X(1666)
X(2564) lies on these lines: 1,3 2,2569 8,2568 11,2566 12,2567 1124,2561 1335,2560 1342,1675 1343,1674 1672,2559 1673,2558 2007,2563 2008,2562
X(2565) lies on these lines: 1,3 2,2568 8,2569 11,2567 12,2566 1124,2560 1335,2561 1342,1674 1343,1675 1672,2558 1673,2559 2007,2562 2008,2563
As a point on the Euler line, X(2566) has Shinagawa coefficients (e + 1, e - 1).
X(2566) lies on these lines: 2,3 11,2564 12,2565 485,2561 486,2560 1329,2568 1342,1349 1343,1348 1670,2040 1671,2039 1676,2559 1677,2558 1698,2573 1699,2572 2009,2563 2010,2562
As a point on the Euler line, X(2567) has Shinagawa coefficients (e - 1, e + 1).
X(2567) lies on these lines: 2,3 11,2565 12,2566 485,2560 486,2561 1329,2569 1342,1348 1343,1349 1670,2039 1671,2040 1676,2558 1677,2559 1698,2572 1699,2573 2009,2562 2010,2563
X(2568) lies on these lines: 2,2565 3,10 8,2564 9,2573 1329,2566 1342,1679 1343,1678 1377,2561 1378,2560 1680,2559 1681,2558 1706,2572 2013,2563 2014,2562
X(2569) lies on these lines: 2,2564 3,10 8,2565 9,2572 1329,2567 1342,1678 1343,1679 1377,2560 1378,2561 1680,2558 1681,2559 1706,2573 2013,2562 2014,2563
As a point on the Euler line, X(2570) has Shinagawa coefficients (J/e + 1, -J/e + 3).
X(2570) lies on this line: 2,3
As a point on the Euler line, X(2571) has Shinagawa coefficients (-J/e + 1, J/e + 3).
X(2571) lies on this line: 2,3
X(2572) lies on these lines: 1,3 9,2569 1342,1705 1343,1704 1698,2567 1699,2566 1700,2559 1701,2558 1702,2561 1703,2560 1706,2568 2017,2563 2018,2562
X(2573) lies on these lines: 1,3 9,2568 1342,1704 1343,1705 1698,2566 1699,2567 1700,2558 1701,2559 1702,2560 1703,2561 1706,2569 2017,2562 2018,2563
As the isogonal conjugate of a point on the circumcircle, X(2574) lies on the line at infinity.
The asymptotes of the Jerabek hyperbola meet the infinity line in X(2574) and X(2575). In general, the infinite points on the right circumhyperbola {{A,B,C, X(4),P}}, where P = p : q : r (barycentrics) are given first barycentrcis u - v and u + v, where
u = p ((a^2-b^2+c^2) q+(-a^2-b^2+c^2) r) (c^2 (a^2-b^2-c^2) p q+b^2 (a^2-b^2-c^2) p r+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) q r
v = (a^2-b^2-c^2) Sqrt[(a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) p q r^2+q^2 ((a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) p r+a^4 r^2)+p^2 (c^4 q^2+(a^4-2 a^2 b^2+b^4-2 a^2 c^2+c^4) q r+b^4 r^2)]).
The infinite points of the asymptotes of the Feuerbach hyperbola at X(3307) and X(3308), and for the Kiepert hyperbola, by X(3413) and X(3414). (Peter Moses, Ocober 8, 2018)
X(2574) lies on the Simson quartic (Q101) and these (parallel) lines: 4,2592 6,1344 30,511 65,2588 71,2578 72,2582 73,2584 74,1114 110,1113 113,1312 125,1313 895,2104
X(2574) = isogonal conjugate of X(1113)
X(2574) = isotomic conjugate of X(15164)
X(2574) = complementary conjugate of X(1313)
X(2574) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,15166), (4,1313), (110,2575), (1113,3)
X(2574) = X(125)-cross conjugate of X(2575)
X(2574) = crosspoint of X(4) and X(1113)
X(2574) = crosssum of X(i) and X(j) for these (i,j): (3,2574), (523,1312)
X(2574) = infinite point of minor axis of orthic inconic
X(2574) = 1st point of intersection of Jerabek hyperbola and line at infinity
X(2574) = Thomson-isogonal conjugate of X(1114)
X(2574) = Lucas-isogonal conjugate of X(1114)
X(2574) = infinite point of major axis of ellipse that is locus of centroids of (degenerate) pedal triangles of points on the circumcircle
X(2574) = infinite point of minor axis of ellipse that is locus of radical center of circles centered at vertices of ABC and tangent to lines through X(3)
As the isogonal conjugate of a point on the circumcircle, X(2575) lies on the line at infinity.
X(2575) lies on the Simson quartic (Q101) and these (parallel) lines: 4,2593 6,1345 30,511 65,2589 71,2579 72,2583 73,2585 74,1113 110,1114 113,1313 125,1312 895,2105
X(2575) = isogonal conjugate of X(1114)
X(2575) = isotomic conjugate of X(15165)
X(2575) = complementary conjugate of X(1312)
X(2575) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,15167), (4,1312), (110,2574), (1114,3)
X(2575) = X(125)-cross conjugate of X(2574)
X(2575) = crosspoint of X(4) and X(1114)
X(2575) = crosssum of X(i) and X(j) for these (i,j): (3,2575), (523,1313)
X(2575) = infinite point of major axis of orthic inconic
X(2575) = 2nd point of intersection of Jerabek hyperbola and line at infinity
X(2575) = Thomson-isogonal conjugate of X(1113)
X(2575) = Lucas-isogonal conjugate of X(1113)
X(2575) = infinite point of minor axis of ellipse that is locus of centroids of (degenerate) pedal triangles of points on the circumcircle
X(2575) = infinite point of major axis of ellipse that is locus of radical center of circles centered at vertices of ABC and tangent to lines through X(3)
X(2576) lies on these lines: 1,19 101,1113 163,1822 662,2580 1755,1823 2159,2579\
X(2576) = isogonal conjugate of X(2582)
X(2576) = X(2580)-Ceva conjugate of X(1822)
X(2576) = cevapoint of X(48) and X(2578)
X(2576) = X(i)-cross conjugate of X(j) for these (i,j): (661,2577), (810,1823), (2578,19)
X(2576) = crosspoint of X(2580) and X(2586)
X(2576) = crosssum of X(2578) and X(2584)
X(2576) = X(i)-aleph conjugate of X(j) for these (i,j): (662,2577), (1113,1707), (2580,19)
X(2577) lies on these lines: 1,19 101,1114 163,1823 662,2581 1755,1822 2159,2578
X(2577) = isogonal conjugate of X(2583)
X(2577) = cevapoint of X(48) and X(2579)
X(2577) = crosspoint of X(2581) and X(2587)
X(2577) = X(2581)-Ceva conjugate of X(1823)
X(2577) = X(i)-cross conjugate of X(j) for these (i,j): (661,2576), (810,1822), (2579,19)
X(2577) = crosssum of X(2579) and X(2585)
X(2577) = X(i)-aleph conjugate of X(j) for these (i,j): (662,2576), (1114,1707), (2581,19)
X(2578) lies on these lines: 19,2588 44,513 48,2584 63,2582 71,2574 163,1822 1114,2249 1821,2581 2159,2577
X(2578) = isogonal conjugate of X(2580)
X(2578) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,2579), (662,2585), (1822,31), (2576,48), (2580,1)
X(2578) = crosspoint of X(i) and X(j) for these (i,j): (1,2580), (19,2576), (63,1822), (2582,2588)
X(2578) = crosssum of X(i) and X(j) for these (i,j): (1,2578), (19,2588), (63,2582), (1822,2576)
X(2578) = X(2580)-aleph conjugate of X(2578)
X(2579) lies on these lines: 19,2589 44,513 48,2585 63,2583 71,2575 163,1823 1113,2249 1821,2580 2159,2576
X(2579) = isogonal conjugate of X(2581)
X(2579) = X(i)-Ceva conjugate of X(j) for these: (i,j): (163,2578), (662,2584), (1823,31), (2577,48), (2581,1), (2583,2585)
X(2579) = cevapoint of X(i) and X(j) for these {i,j}: {1577,2580}, {2577,2587}, {2589,75}
X(2579) = crosspoint of X(i) and X(j) for these (i,j): (1,2581), (19,2577), (63,1823), (2583,2589)
X(2579) = crosssum of X(i) and X(j) for these (i,j): (1,2579), (19,2589), (63,2583), (1823,2577)
X(2579) = X(2581)-aleph conjugate of X(2579)
X(2580) lies on these lines: 19,27 100,1113 162,1822 240,2587 662,2576 897,2589 1821,2579 2349,2583
X(2580) = isogonal conjugate of X(2578)
X(2580) = isotomic conjugate of X(2582)
X(2580) = cevapoint of X(i) and X(j) for these (i,j): (1,2578), (19,2588), (63,2582), (1822,2576)
X(2580) = X(i)-cross conjugate of X(j) for these (i,j): (1577,2581), (2576,2586), (2578,1), (2582,92), (2588,75)
X(2580) = X(i)-aleph conjugate of X(j) for these (i,j): (1113,1740), (2580,1)
X(2581) lies on these lines: 19,27 100,1114 162,1823 240,2586 662,2577 897,2588 1821,2578 2349,2582
X(2581) = isogonal conjugate of X(2579)
X(2581) = isotomic conjugate of X(2583)
X(2581) = cevapoint of X(i) and X(j) for these (i,j): (1,2579), (19,2589), (63,2583), (1823,2577)
X(2581) = crosssum of X(i) and X(j) for these (i,j): (1577,2580), (2577,2587), (2589,75)
X(2581) = X(i)-cross conjugate of X(j) for these (i,j): (2579,1), (2583,92)
X(2581) = X(1114)-aleph conjugate of X(1740)
X(2582) lies on these lines: 1,2584 63,2578 72,2574 226,2592 293,1823 514,661 662,2576 2349,2581
X(2582) = isogonal conjugate of X(2576)
X(2582) = isotomic conjugate of X(2580)
X(2583) lies on these lines: 1,2585 63,2579 72,2575 226,2593 293,1822 514,661 662,2577 2349,2580
X(2583) = isogonal conjugate of X(2577)
X(2583) = isotomic conjugate of X(2581)
X(2584) lies on these lines: 1,2582 48,2578 73,2574 521,656
X(2584) = isogonal conjugate of X(2586)
X(2584) = crosspoint of X(1) and X(1822)
X(2584) = crosssum of X(1) and X(2588)
X(2585) lies on these lines: 1,2583 48,2579 73,2575 521,656
X(2585) = isogonal conjugate of X(2587)
X(2585) = crosssum of X(1) and X(2589)
X(2585) = crosspoint of X(1) and X(1823)
X(2586) lies on these lines: 1,29 108,1113 162,1822 240,2581
X(2586) = isogonal conjugate of X(2584)
X(2586) = cevapoint of X(1) and X(2588)
X(2587) lies on these lines: 1,29 108,1114 162,1823 240,2580
X(2587) = isogonal conjugate of X(2585)
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(2588) lies on these lines: 1,2582 10,2592 11,1313 19,2578 65,2574 162,1822 240,522 759,1114 897,2581 1910,2577
X(2588) = isogonal conjugate of X(1822)
X(2588) = X(2586)-Ceva conjugate of X(1)
X(2588) = Mimosa transform of X(1113)
X(2589) lies on these lines: 1,2583 10,2593 11,1312 19,2579 65,2575 162,1823 240,522 759,1113 897,2580 1910,2576
X(2589) = isogonal conjugate of X(1823)
X(2589) = Mimosa transform of X(1114)
X(2589) = X(2587)-Ceva conjugate of X(1)
X(2590) lies on these lines: 44,513 101,1381 1382,2291 2170,2446
X(2590) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2446),
(101,2591), (1381,55)
X(2590) = X(2170)-cross conjugate of X(2591)
X(2590) = crosspoint of X(57) and X(1381)
X(2590) = crosssum of X(1) and X(2590)
X(2590) = crossdifference of every pair of points on line X(1)X(3308)
X(2591) lies on these lines: 44,513 101,1382 1381,2291 2170,2447
X(2591) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,2447), (101,2590), (1382,55)
X(2591) = crosspoint of X(57) and X(1382)
X(2591) = crosssum of X(1) and X(2591)
X(2591) = X(2170)-cross conjugate of X(2590)
X(2591) = crossdifference of every pair of points on line X(1)X(3307)
X(2592) lies on the Kiepert hyperbola and these lines: 4,2574 10,2588 98,1114 226,2582 262,1346 297,525
X(2592) = isotomic conjugate of X(8115)
X(2592) = polar conjugate of X(1113)
X(2592) = trilinear pole of line X(523)X(1313)
X(2592) = pole wrt polar circle of trilinear polar of X(1113) (line X(6)X(1345))
X(2593) lies on the Kiepert hyperbola and these lines: 4,2575 10,2589 98,1113 226,2583 262,1347 297,525
X(2593) = isotomic conjugate of X(8116)
X(2593) = trilinear pole of line X(523)X(1312)
X(2593) = pole wrt polar circle of trilinear polar of X(1114) (line X(6)X(1344))
X(2593) = polar conjugate of X(1114)
X(2593) = {P*,U*}-harmonic conjugate of X(4), where P* and U* are the polar conjugates of the foci of the orthic inconic
Centers Related to Bicentric Pairs
Most of the centers X(2594) - X(2670) are defined in terms of bicentric pairs of points (e.g., the 1st and 2nd Brocard points).
For a list of bicentric pairs and associated terminology and properties, press the Tables button at the top of this page.
The notation "PU(68)", as described in the Bicentric Pairs section accessible by the Tables button, abbreviates a bicentric pair of points P = p : q : r and U = u : v : w, where p, q, r have the form g(a,b,c), g(b,c,a), g(c,a,b) and u,v,w have the form h(a,b,c), h(b,c,a), h(c,a,b). "Sum of PU(68)" denotes the point
g(a,b,c) + h(a,b,c) : g(b,c,a) + h(b,c,a) : g(c,a,b) + h(c,a,b),X(2594) lies on these lines: 1,5 6,2197 35,500 42,65 55,581 56,181 59,60 319,1273 354,1066 523,2616 654,2598 692,3145 995,1388 1048,2595 1064,3057 1193,1319 1203,2078 1324,1437 1745,1836 1788,3240 2151,2307 2599,2611 2601,2603
X(2594) = isogonal conjugate of X(3615)
X(2594) = X(1)-Ceva conjugate of X(2599)
X(2594) = cevapoint of X(1) and X(1048)
X(2594) = crosspoint of X(1) and X(54)
X(2594) = crosssum of X(1) and X(5)
X(2594) = crossdifference of every pair of points on line X(654)X(1021)
X(2594) = PU(68)-harmonic conjugate of X(654)
X(2595) lies on these lines: 12,2607 54,1087 655,2597 1048,2594 2477,2606
X(2595) = isogonal conjugate of X(7134)
X(2595) = X(2603)-Ceva conjugate of X(2596)
X(2595) = {X(54),X(1087)}-harmonic conjugate of X(2596)
X(2595) = crosspoint of PU(69)
Let L = X(1)X(5), the trilinear polar of X(655). Let M = X(654)X(1012), the trilinear polar of the cevapoint of X(1) and X(5). Let V = X(3615) = X(1)-Ceva conjugate of X(5) and W = X(3460) = X(5)-Ceva conjugate of X(1). The lines L, M, an VW concur in X(2596). (Randy Hutson, December 26, 2015)
X(2596) lies on these lines: 1,5 54,1087 215,2607 654,1021
X(2596) = isogonal conjugate of X(2597)
X(2596) = X(2603)-Ceva conjugate of X(2595)
X(2596) = crosssum of X(2601) and X(2602)
X(2597) lies on these lines: 1,2603 655,2595
X(2597) = isogonal conjugate of X(2596)
X(2597) = cevapoint of X(2601) and X(2602)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2598) lies on this line: 654,2594
X(2599) lies on these lines: 1,54 5,1087 12,523 35,186 201,517 655,2595 2171,2260 2594,2611 2600,2604
X(2599) = X(1)-Ceva conjugate of X(2594)
X(2599) = crosspoint of X(1) and X(5)
X(2599) = crosssum of X(1) and X(54)
X(2599) = intersection of tangents at X(1) and X(5) to hyperbola {{A,B,C,X(1),X(5)}}
X(2599) = PU(69)-harmonic conjugate of X(2600)
X(2600) is the perspector of triangle ABC and the tangential triangle of the hyperbola that passes through the points A, B, C, X(1), and X(5). (Randy Hutson, 9/23/2011)
X(2600) lies on these lines: 44,513 2599,2604
X(2600) = crosssum of X(1) and X(2600)
X(2600) = intersection of antiorthic axis and line X(2081)X(2600) (trilinear polars of X(1) and X(5))
X(2600) = crossdifference of every pair of points on line X(1)X(54)
X(2600) = PU(69)-harmonic conjugate of X(2599)
X(2601) lies on these lines: 1,2120 5,2602 2594,2603
X(2601) = X(2597)-Ceva conjugate of X(2602)
X(2601) = crosspoint of PU(68)
X(2602) lies on these lines: 1,54 5,2601
X(2602) = isogonal conjugate of X(2603)
X(2602) = X(2597)-Ceva conjugate of X(2601)
X(2602) = crosssum of X(2595) and X(2596)
X(2602) = perspector of conic {A,B,C,PU(68)}
X(2603) lies on these lines: 1,2597 2594,2601
X(2603) = isogonal conjugate of X(2602)
X(2603) = cevapoint of X(2595) and X(2596)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2604) lies on this line: 2599,2600
X(2605) lies on these lines: 1,523 11,2616 110,1101 512,1326 513,663 667,834 1919,2483 2245,2609 2612,2614
X(2605) = midpoint of X(663) and X(1459)
X(2605) = isogonal conjugate of X(6742)
X(2605) = X(1)-Ceva conjugate of X(2611)
X(2605) = crosspoint of X(1) and X(110)
X(2605) = crosssum of X(1) and X(523)
X(2605) = crossdifference of every pair of points on line X(9)X(46)
X(2606) lies on these lines: 1,5 110,1109 542,1365 2477,2595 2608,2611
X(2606) = crosspoint of PU(71)
X(2607) lies on these lines: 1,523 9,46 12,2595 110,1109 215,2596 229,409
X(2607) = isogonal conjugate of X(2608)
X(2607) = X(2)-Ceva conjugate of X(39051)
X(2607) = perspector of conic {{A,B,C,PU(71)}}
X(2607) = intersection of trilinear polars of PU(71)
X(2608) lies on these lines: 1,2614 2606,2611
X(2608) = isogonal conjugate of X(2607)
X(2608) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the Kiepert hyperbola
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2609) lies on this line: 2245,2605
X(2610) is the perspector of triangle ABC and the tangential triangle of the hyperbola that passes through the points A, B, C, X(1), and X(12). (Randy Hutson, 9/23/2011)
X(2610) lies on this line: 44,513 115,125 918,1211 1213,1639 1983,2613 2611,2615
X(2610) = isogonal conjugate of X(37140)
X(2610) = cevapoint of X(523) and X(9276)
X(2610) = crossdifference of every pair of points on line X(1)X(60)
X(2610) = crosssum of X(i) and X(j) for these {i,j}: {1, 2610}, {110, 1983}, {2245, 2605}, {3737, 4282}
X(2610) = intersection of trilinear polars of X(1) and X(12)
X(2610) = X(2)-Ceva conjugate of X(38982)
X(2610) = perspector of hyperbola {{A,B,C,X(1),X(12)}}
X(2610) = PU(71)-harmonic conjugate of X(2611)
X(2611) lies on this line: 1,60 11,523 2310,2632 2594,2599 2606,2608 2610,2615
X(2611) = reflection of X(1109) in X(11)
X(2611) = isotomic conjugate of isogonal conjugate of X(17886)
X(2611) = X(1)-Ceva conjugate of X(2605)
X(2611) = crosspoint of X(1) and X(523)
X(2611) = crosssum of X(1) and X(110)
X(2611) = PU(71)-harmonic conjugate of X(2610)
X(2612) lies on these lines: 59,110 523,1101 2605,2614
X(2612) = crosspoint of PU(70)
X(2613) lies on these lines: 1,60 523,1101 1983,2610
X(2613) = reflection of X(2612) in X(1101)
X(2613) = isogonal conjugate of X(2614)
X(2613) = X(2608)-Ceva conjugate of X(2612)
X(2613) = crosssum of X(2606) and X(2607)
X(2613) = perspector of conic {A,B,C,PU(70)}
X(2613) = intersection of trilinear polars of PU(70)
X(2614) lies on these lines: 1,2608 2605,2612
X(2614) = isogonal conjugate of X(2613)
X(2614) = cevapoint of X(2606) and X(2607)
X(2614) = X(245)-cross conjugate of X(1)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2615) lies on these lines: 11,115 2610,2611
X(2715) = anticomplement of X(36471)
X(2616) lies on these lines: 1,2618 11,2605 523,2594 656,1955 1101,2617 2625,2626
X(2616) = isogonal conjugate of X(2617)
X(2616) = X(1109)-cross conjugate of X(1)
X(2616) = trilinear product of PU(73)
X(2617) lies on these lines: 1,564 59,110 63,1956 162,662 1101,2616 2619,2620
X(2617) = isogonal conjugate of X(2616)
X(2617) = X(1101)-Ceva conjugate of X(1)
X(2617) = X(2618)-cross conjugate of X(1)
X(2617) = X(110)-aleph conjugate of X(1)
X(2617) = trilinear product of PU(72)
X(2618) lies on these lines: 1,2616 12,523 240,522 823,2633 1101,2625 2290,2622
X(2618) = isogonal conjugate of X(36134)
X(2618) = X(1)-Ceva conjugate of X(2616)
X(2618) = crosspoint of X(1) and X(2617)
X(2618) = crosssum of X(1) and X(2616)
X(2618) = crossdifference of every pair of points on line X(47)X(48)
X(2618) = perspector of anticevian triangle of X(48) and tangential triangle, wrt anticevian triangle of X(47), of bianticevian conic of X(1) and X(47)
X(2619) lies on these lines: 1109,2621 2617,2620
X(2619) = crosspoint of PU(73)X(2620) lies on these lines: 1,2616 47,48 2617,2619
X(2620) = isogonal conjugate of X(2621)
X(2620) = perspector of conic {{A,B,C,PU(73)}}
X(2621) lies on these lines: 1,2627 1109,2619
X(2621) = isogonal conjugate of X(2620)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2622) lies on this line: 2290,2618
X(2623) lies on these lines: 50,647 323,401
X(2623) = isogonal conjugate of X(14570)
X(2623) = crossdifference of every pair of points on line X(5)X(51)
X(2623) = X(115)-cross conjugate of X(6)
X(2623) = X(92)-isoconjugate of X(23181)
Let A'B'C' be the excentral triangle, and let H be the hyperbola {{X(1),X(5),X(30),A',B',C'}}. Then X(2624) is the perspector wrt A'B'C' of H. (Randy Hutson, December 26, 2015)
Let H be the rectangular hyperbola passing through X(1), X(5), X(30) and the excenters (centered at X(476)); then X(2624) is the isogonal conjugate of the perspector of ABC and the tangential triangle, wrt excentral triangle, of H. (Randy Hutson, February 10, 2016)
X(2624) lies on these lines: 44,513 163,1983 1109,2628
X(2624) = isogonal conjugate of X(32680)
X(2624) = crosssum of X(1) and X(2624)
X(2624) = bicentric sum of PU(73)
X(2624) = crossdifference of every pair of points on line X(1)X(564)
X(2624) = polar conjugate of isotomic conjugate of isogonal conjugate of X(36129)
X(2624) = X(63)-isoconjugate of X(36129)
X(2624) = PU(73)-harmonic conjugate of X(1109)
X(2625) lies on these lines: 1101,2618 2616,2626
X(2625) = X(2621)-Ceva conjugate of X(2626)
X(2625) = crosspoint of PU(72)
X(2626) lies on these lines: 1,564 163,1983 2616,2625
X(2626) = isogonal conjugate of X(2627)
X(2626) = X(2621)-Ceva conjugate of X(2625)
X(2627) lies on these lines: 1,2621 110,1296 1101,2618
X(2627) = isogonal conjugate of X(2626)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2628) lies on this line: 1109,2624
X(2629) lies on the conic TCX(1054) and these lines: {1, 162}, {9, 2634}, {63, 1956}, {191, 1047}, {293, 1707}, {610, 2247}, {656, 9390}, {896, 1955}, {1046, 1771}, {1490, 5974}, {1768, 2636}, {1954, 6508}, {2631, 16562}, {2939, 9324}, {9355, 21381}, {9359, 16575}
X(2629) = isogonal conjugate of X(9390)
X(2629) = X(656)-Ceva conjugate of X(1)
X(2629) = X(2633)-cross conjugate of X(1)
X(2629) = X(i)-aleph conjugate of X(j) for these (i,j): (1,656), (2,1577)
X(2629) = crosspoint of PU(75)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2630) lies on this line: 44,513
X(2631) lies on these lines: 44,513 162,163 2632,2634
X(2631) = crosssum of X(1) and X(2631)
X(2631) = isogonal conjugate of polar conjugate of X(36035)
X(2631) = X(92)-isoconjugate of X(36034)
X(2631) = PU(75)-harmonic conjugate of X(2632)
X(2632) lies on the incentral inellipse (i.e., Hofstadter ellipse E(1/2)) and these lines: 1,162 48,2157 63,293 92,1956 122,1367 336,799 520,1364 1096,2184 2292,2658 2310,2611 2631,2634
X(2632) = isogonal conjugate of X(24000)
X(2632) = isotomic conjugate of X(23999)
X(2632) = X(1)-Ceva conjugate of X(652)
X(2632) = crosspoint of X(i) and X(j) for these (i,j): (1,656), (2584,2585)
X(2632) = crosssum of X(1) and X(162)
X(2632) = trilinear pole wrt incentral triangle of line X(1)X(19)
X(2632) = crossdifference of every pair of points on line X(162)X(163)
X(2632) = trilinear square of X(656)
X(2632) = PU(75)-harmonic conjugate of X(2631)
X(2633) lies on these lines: 162,656 823,2618
X(2633) = isogonal conjugate of X(9392)
X(2633) = X(1)-Ceva conjugate of X(162)
X(2633) = crosspoint of X(1) and X(2629)
X(2633) = crosspoint of PU(74)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2634) lies on these lines: 9,2629 2631,2632
X(2635) lies on these lines: 1,4 11,1458 42,1836 44,513 209,2390 329,2318 411,1935 651,1936 774,1898 823,2659 908,1818 971,1465 1042,1837 1044,1788 1156,1937 1254,1858 1427,1864 1758,1776
X(2635) = crosssum of X(i) and X(j) for these (i,j): (1,2635), (2637,2638)
X(2635) = bicentric sum of PU(76)
X(2635) = PU(76)-harmonic conjugate of X(652)
X(2636) lies on these lines: 1,653 3,1047 46,296 282,1721 1155,2655 1768,2629
X(2636) = X(652)-Ceva conjugate of X(1)
X(2636) = X(1)-aleph conjugate of X(652)
X(2636) = crosspoint of PU(77)
X(2637) lies on these lines: 44,513 653,1020
X(2637) = crosssum of X(1) and X(2637)
X(2637) = PU(77)-harmonic conjugate of X(2638)
X(2637) = isogonal conjugate of isotomic conjugate of isogonal conjugate of X(36140)
X(2638) lies on the incentral inellipse (i.e., Hostadter ellipse E(1/2)) and these lines: 1,653 3,296 29,2656 31,2188 48,692 1984,2310 2646,2658
X(2638) = isogonal conjugate of X(24032)
X(2638) = X(1)-Ceva conjugate of X(652)
X(2638) = crosspoint of X(1) and X(652)
X(2638) = crosssum of X(1) and X(653)
X(2638) = trilinear pole wrt incentral triangle of line X(1)X(4)
X(2638) = PU(77)-harmonic conjugate of X(2637)
X(2639) lies on this line: 652,653
X(2639) = crosspoint of PU(76)
X(2640) lies on these lines: 1,662 3,1247 9,2645 19,1581 45,846 610,1910 1045,1781 1580,2173 1707,2247
X(2640) = crosspoint of PU(79)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2641) lies on these lines: 44,513 1914,2651 1931,2642
X(2642) lies on these lines: 44,513 163,662 665,2092 900,1213 1910,2159 1931,2641
X(2642) = isogonal conjugate of X(36085)
X(2642) = PU(79)-harmonic conjugate of X(2643)
X(2643) lies on the incentral inellipse (i.e., Hofstadter ellipse E(1/2)) and these lines: 1,662 19,560 31,2153 37,2054 65,2652 75,1581 244,2611 267,849 523,1086 678,1962 872,2171 922,2173 1098,1247 1109,2632 1580,2244 1953,1964 1959,2234 1931,2641
X(2643) = reflection of X(4094) in X(37)
X(2643) = complement of X(21295)
X(2643) = trilinear pole wrt incentral triangle of line X(1)X(21)
X(2643) = incentral isotomic conjugate of X(512)
X(2643) = crossdifference of every pair of points on line X(163)X(662)
X(2643) = antipode in the incentral inellipse of X(4094)
X(2643) = trilinear square of X(661)
X(2643) = PU(79)-harmonic conjugate of X(2642)
X(2644) lies on these lines: 661,662 1776,2651 1931,2641
X(2644) = crosspoint of PU(78)
X(2645) lies on these lines: 9,2640 2642,2643
In the place of a triangle ABC, let I = X(1) = incenter, and let
A'B'C' = intouch triangle;
Ma = midpoint of segment AI;
La = MaA', and define Lb and Lc cyclically;
Ra = reflection of La in AI, and define Bb and Rc cyclically.
The lines Ra, Rb, Rc concur in X(2646). (Angel Montesdeoca, October 9, 2021)
X(2646) lies on these lines: 1,3 2,1837 6,1732 8,2320 11,214 12,515 20,1836 21,60 24,1905 28,1859 29,243 33,1900 37,48 41,1212 58,2361 72,993 73,820 78,210 104,943 140,1737 154,968 212,1468 224,1001 284,1731 355,498 377,497 405,997 409,662 501,2360 518,2330 550,1770 650,2649 851,2654 991,1456 1100,2245 1104,1193 1152,2362 1201,1279 1476,2346 1831,2355 1848,1852 2638,2658
X(2646) = midpoint of X(1) and X(35)
X(2646) = X(1)-Ceva conjugate of X(2650)
X(2646) = crosspoint of X(1) and X(21)
X(2646) = crosssum of X(1) and X(65)
X(2646) = {X(1),X(3)}-harmonic conjugate of X(65)
X(2646) = homothetic center of intouch triangle and medial triangle of 2nd circumperp triangle
X(2646) = PU(80)-harmonic conjugate of X(650)
X(2647) lies on these lines: 1,4 21,1254 65,1046 651,2648 894,2263
X(2647) = crosspoint of PU(81)
X(2648) lies on these lines: 1,2652 4,1046 21,2310 80,1757 651,2647 851,1758 896,1156
X(2648) = isogonal conjugate of X(1758)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2649) lies on this line: 650,2646
X(2650) lies on these lines: 1,21 6,2294 40,2177 42,65 72,756 78,750 145,740 221,2099 244,942 326,969 354,1201 500,517 512,764 526,1769 651,2647 661,2653 1002,1432 1104,2308 1409,2171 1419,2263 1480,1482 1858,2310
X(2650) = reflection of X(2292) in X(1)
X(2650) = X(1)-Ceva conjugate of X(2646)
X(2650) = crosspoint of X(1) and X(65)
X(2650) = crosssum of X(1) and X(21)
X(2650) = trilinear pole, wrt incentral triangle, of orthic axis
X(2650) = anticomplement, wrt incentral triangle, of X(2292)
X(2650) = bicentric sum of PU(81)
X(2650) = PU(81)-harmonic conjugate of X(661)
X(2651) lies on these lines: 1,21 65,409 185,411 243,648 296,416 425,653 518,643 661,1021 662,1155
X(2651) = isogonal conjugate of X(2652)
X(2651) = perspector of conic {{A,B,C,PU(80)}}
X(2652) lies on these lines: 1,2648 3,1247 19,2305 243,415 409,662 759,1326 897,1155 1910,2182
X(2652) = isogonal conjugate of X(2651)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2653) lies on these lines: 661,2650 2092,2183
X(2654) lies on these lines: 1,4 8,2318 11,1193 21,1936 37,1953 42,1837 65,774 201,517 212,405 410,823 603,1012 652,2657 851,2646 1042,1836 1046,1776 1065,1067 1858,2310
X(2654) = X(1)-Ceva conjugate of X(2658)
X(2654) = crosspoint of X(1) and X(29)
X(2654) = crosssum of X(1) and X(73)
X(2654) = bicentric sum of PU(82)
X(2654) = PU(82)-harmonic conjugate of X(652)
X(2654) = intersection of tangents at X(1) and X(29) to hyperbola {{A,B,C,X(1),X(3),X(29),X(102)}}
X(2655) lies on these lines: 1,4 65,1047 109,1955 296,851 412,820 1155,2636 1248,1935
X(2655) = isogonal conjugate of X(2656)
X(2655) = X(296)-Ceva conjugate of X(1)
X(2655) = perspector of conic {{A,B,C,PU(83)}}}
X(2656) lies on these lines: 1,2660 3,1047 29,2638 296,851 653,2658 1795,1955
X(2656) = isogonal conjugate of X(2655)
X(2656) = X(243)-cross conjugate of X(1)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2657) lies on these lines: 37,2190 652,2654
X(2658) lies on these lines: 1,29 42,65 653,2656 822,2661 1108,1201 2292,2632 2638,2646
X(2658) = X(1)-Ceva conjugate of X(2654)
X(2658) = crosspoint of X(1) and X(73)
X(2658) = crosssum of X(1) and X(29)
X(2658) = PU(83)-harmonic conjugate of X(822)
X(2659) lies on these lines: 1,29 73,410 185,412 415,1937 648,1936 822,1021 823,2635
X(2659) = isogonal conjugate of X(2660)
X(2659) = perspector of conic {{A,B,C,PU(82)}}
X(2660) lies on these lines: 1,2656 4,1248 410,823 416,1936
X(2660) = isogonal conjugate of X(2659)
Barycentrics (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(2661) lies on this line: 822,2658
X(2662) lies on these lines: 1,3 29,2655 73,1047 653,2656
X(2662) = crosspoint of PU(83)
X(2663) lies on these lines: 1,6 42,894 86,872 190,2665 291,2092 314,1215
X(2663) = crosspoint of PU(85)
X(2664) lies on these lines: 1,2 9,1740 37,1045 86,872 87,1743 101,1580 171,213 190,2234 238,2110 274,1215 292,2238 660,1757 980,984 1376,2176
X(2664) = isogonal conjugate of X(2665)
X(2664) = X(2)-Ceva conjugate of X(39056)
X(2664) = perspector of conic {{A,B,C,PU(85)}}
X(2665) lies on these lines: 1,1655 6,1045 43,2279 190,2663 292,2238 1438,1580
X(2665) = isogonal conjugate of X(2664)
X(2667) lies on these lines: 1,75 37,42 55,2305 190,2663 192,714 284,922 518,2292 798,2670 1100,2309 2293,2650 2294,2643
X(2667) = trilinear pole, wrt incentral triangle, of Lemoine axis
X(2667) = incentral isotomic conjugate of X(2292)
X(2667) = complement, wrt incentral triangle, of X(3728)
X(2667) = anticomplement, wrt incentral triangle, of X(37)
X(2667) = bicentric sum of PU(85)
X(2667) = PU(85)-harmonic conjugate of X(798)
X(2668) lies on these lines: 2,6 42,873 171,1509 799,2107 1403,1434
X(2669) lies on this line: 1,75 42,873 99,238 798,1019 799,899 1434,1469
X(2669) = isogonal conjugate of X(2107)
X(2669) = X(2)-Ceva conjugate of X(39057)
X(2669) = perspector of conic {A,B,C,PU(84)}
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2670) lies on these lines: 511,2653 798,2667 1084,1100
Suppose UV is a segment and 1/2 < s < 1. Let P be the point on UV satisfying |UP|/|VP| = s. Let O be a semicircle having diameter UV. Let H be the semicircle having diameter VP, on the same side of UV as O. Let K be the semicircle having diameter UP, on the same side of UP as O. The three circular arcs form an arbelos, R.
Let L be the line through P perpendicular to BC. The circle tangent to semicircles O and K and line L is the s-Archimedean circle of the arbelos R. Now continuing with the variable s, suppose ABC is an arbitrary triangle. Let A' be the center of the s-Archimedean circle of the outward arbelos on segment BC, and define B' and C' cyclically.
At the Eleventh International Conference on Fibonacci Numbers and Their Applications (TU Braunschweig, Germany, July 2004), Zvonko Cerin established that the lines AA', BB', CC' concur if and only if s = τ. The point of concurrence is X(2671).
Trilinears found by Peter J. C. Moses, July, 2004. For a general account of the arbelos, visit
MathWorld. If you have The Geometer's Sketchpad, you can view 1st Golden Arbelos Point.X(2671) lies on the Kiepert hyperbola and these lines: 2,2674 6,2672
X(2671) = isogonal conjugate of X(2673)
In the construction of X(2671), if A' is the center of the s-Archimedean circle of the inward arbelos on segment BC, and B' and C' are defined cyclically, then A'A, B'B, C'C concur in X(2672).
X(2672) lies on the Kiepert hyperbola and these lines: 2,2673 6,2671
X(2672) = isogonal conjugate of X(2674)
Barycentrics (S-(sqrt(5)-3)*SA)*(SB+SC) : : (C. Lozada, 7/13/19)
X(2673) lies on these lines: 2,2672 3,6
X(2673) = isogonal conjugate of X(2671)
X(2673) = inverse-in-Brocard-circle of X(2674)
X(2673) = radical center of Lucas(τ2) circles, where τ = (1 + sqrt(5))/2 = golden ratio
X(2674) lies on these lines: 2,2671 3,6
X(2674) = isogonal conjugate of X(2672)
X(2674) = inverse-in-Brocard-circle of X(2673)
X(2674) = radical center of Lucas(-τ2) circles, where τ = (1 + sqrt(5))/2 = golden ratio
Barycentrics 12*(S^2+SB*SC)*sqrt(5)+7*S^2+68*SB*SC : : (C. Lozada, 7/13/19)
As a point on the Euler line, X(2675) has Shinagawa coefficients (h, -h - 1).
X(2675) lies on these lines: 2,3 2671,2673 2672,2674
This point of intersection and the fact that it lies on the Euler line were contributed by Peter J. C. Moses, August, 31, 2004.
As a point on the Euler line, X(2676) has Shinagawa coefficients (h - 1, -h - 1).
X(2676) lies on these lines: 2,3 6,2671
Contributed by Peter J. C. Moses, August, 31, 2004.
RS Points and SR Points
Centers X(2677) to X(2770) are examples of Rigby-Simson points and Simson-Rigby points. These two types of points are discussed in
, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 11: The Orthopole, pages 124-136.
Specifically, on pages 132-136, Honsberger defines the "Rigby Point," in honor of John Rigby, as the orthopole of the six sides of two triangles, all of whose vertices lie on a circle, and he notes that this orthopole lies on the six Simson lines of the vertices.
Now suppose that P = p : q : r (trilinears) and U = u : v : w are distinct points on the circumcircle of a triangle ABC. Their Simson lines meet in a point X which is here called the Rigby-Simson point of P and U, denoted by RS(P,U). That is, RS(P,U) is the orthopole of line PU with respect to triangle ABC.
On the circumcircle, there is a unique point K whose Simson line is perpendicular to line PU, and whose Simson line also passes through RS(P,U). The point K is here called the Simson-Rigby point of P and U, denoted by SR(P,U). Note that SR(P,U) = Λ(P,U)
If you have The Geometer's Sketchpad, you can view RS(P,U) and SR(P,U).
Continuing with contributions of Peter Moses during October, 2004, trilinears for RS(P,U) are
f(a,b,c,p,q,r,u,v,w) : f(b,c,a,q,r,p,v,w,u) : f(c,a,b,r,p,q,w,u,v), wheref(a,b,c,p,q,r,u,v,w) = 1/[a2r2v2(q cos A - p cos B)(w cos A - u cos C)] - 1/[a2q2w2(r cos A - p cos C)(v cos A - u cos B)],
74,98,691 74,99,842 74,110,477 74,1113,125 74,1114,125
98,1379,115 98,1380,115
99,110,691
100,101,1308 100,104,953 100,110,1290
101,109,1521 101,929,1521
104,1381 104,1382
110,112,1554 110,476,1553 110,935,1554
112,935,1554
1113,1114,125
1379,1380,115
1381,1382,11
In the next list (from P. Moses, Oct. 15, 2004), the appearance of I, J, K means that SR(X(i),X(j)) = X(K):
74,98,691 74,99,842 74,104,1290 74,110,477 74,841,1302 74,935,1297 74,1113,1114 74,1141,1291 74,1294,1304
98,110,842 98,843,1296 98,1379,1380
99,110,691 99,111,843 99,1379,1379 & 99,1380,1380 99,2378,2379
100,101,1308 100,104,953 100,105,840 100,109,2222 100,110,1290 100,1381,1381 100,1382,1382
101,109,929
102,103,929 102,104,2222 103,104,1308
104,840,1292 104,1381,1382
107,110,1304
110,112,935 110,827,1287 110,930,1291 110,1113,1113 110,1114,1114
X(2677) lies on these lines: 119,517 125,523 673,2481
X(2678) lies on these lines: 118,516 125,523
X(2678) = reflection of X(2685) in X(188)
X(2679) is the center of the rectangular hyperbola that passes through the points A, B, C, and X(32).
X(2679) lies on the nine-point circle, the cevian circle of X(511), and on these lines: 2,805 4,2698 114,325 115,512
X(2679) = midpoint of X(4) and X(2698)
X(2679) = complement of X(805)
X(2679) = complementary conjugate of X(804)
X(2679) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,2491), (4,804)
X(2679) = crosspoint of X(511) and X(512)
X(2679) = crosssum of X(98) and X(99)
X(2679) = perspector of circumconic centered at X(2491)
X(2679) = center of circumconic that is locus of trilinear poles of lines passing through X(2491)
X(2679) = crossdifference of every pair of points on line X(2421)X(2422)
X(2680) lies on these lines: 115,512 119,517
X(2680) = reflection of X(2683) in X(119)
X(2681) lies on these lines: 115,512 118,516
X(2681) = reflection of X(2684) in X(118)
X(2682) lies on these lines: 30,113 115,512 125,1499
X(2682) = orthopole of line X(2)X(98)
X(2683) lies on these lines: 114,325 119,517
X(2683) = reflection of X(2680) in X(119)
X(2684) lies on these lines: 114,325 118,516
X(2684) = reflection of X(2681) in X(118)
X(2685) lies on these lines: 30,113 118,516
X(2685) = reflection of X(2678) in X(118)
X(2686) lies on this line: 30,113
X(2687) lies on the circumcircle and these lines: 3,1290 4,2766 21,476 28,1304 30,100 35,2222 65,2720 74,513 101,2173 104,523 107,2074 108,186 109,484 110,517 376,2691 1006,2690
X(2687) = reflection of X(1290) in X(3)
X(2687) = isogonal conjugate of X(2771)
X(2687) = X(2778)-cross conjugate of X(4)
X(2687) = reflection of X(104) in the Euler line
X(2687) = reflection of X(74) in line X(1)X(3)
X(2688) lies on the circumcircle and these lines: 3,2690 27,1304 30,101 74,514 103,523 107,2073 110,516 112,1886 1305,2071
X(2688) = reflection of X(2690) in X(3)
X(2688) = isogonal conjugate of X(2772)
X(2688) = reflection of X(103) in the Euler line
X(2689) lies on the circumcircle and these lines: 3,2695 23,1311 30,102 74,515 109,523 110,522 946,953 1324,2372
X(2689) = reflection of X(2695) in X(3)
X(2689) = isogonal conjugate of X(2773)
X(2689) = reflection of X(109) in the Euler line
X(2690) lies on the circumcircle and these lines: 3,2688 23,675 80,103 74,516 100,1577 101,523 110,514 186,917 551,953 1006,2687 2453,2758
X(2690) = reflection of X(2688) in X(3)
X(2690) = isogonal conjugate of X(2774)
X(2691) lies on the circumcircle and these lines: 3,2752 30,105 74,518 376,2687 523,1292
X(2691) = reflection of X(2752) in X(3)
X(2691) = isogonal conjugate of X(2775)
X(2692) lies on the circumcircle and these lines: 3,2758 30,106 74,519 523,1293 2071,2370
X(2692) = reflection of X(2758) in X(3)
X(2692) = isogonal conjugate of X(2776)
X(2693) lies on the circumcircle and these lines: 3,1304 20,476 30,107 74,520 110,2071 186,1301 376,935 523,1294 673,2481 858,1302
X(2693) = reflection of X(1304) in X(3)
X(2693) = isogonal conjugate of X(2777)
X(2693) = de-Longchamps-circle-inverse of X(34193)
X(2693) = trilinear pole, wrt circumcevian triangle of X(30), of line X(20)X(2407)
X(2693) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(3),X(30)}}
X(2694) lies on the circumcircle and these lines: 3,2766 20,1290 21,1304 30,108 74,521 100,2071 107,1325 523,1295 1301,2074
X(2694) = reflection of X(2766) in X(3)
X(2694) = isogonal conjugate of X(2778)
X(2694) = cevapoint of X(3) and X(2771)
X(2695) lies on the circumcircle and these lines: 3,2689 29,1304 30,109 74,522 102,523 107,2075 108,1784 110,515 411,1290
X(2695) = reflection of X(2689) in X(3)
X(2695) = isogonal conjugate of X(2779)
X(2695) = reflection of X(102) in the Euler line
X(2696) lies on the circumcircle and these lines: 3,2770 30,111 74,524 110,1499 186,2374 376,842 523,1296 691,2407 2071,2373
X(2696) = reflection of X(2770) in X(3)
X(2696) = isogonal conjugate of X(2780)
X(2696) = Λ(X(74), X(111))
X(2696) = reflection of X(1296) in the Euler line
X(2696) = X(74)-of-circummedial-triangle
X(2697) lies on the circumcircle and these lines: 2,1304 3,935 20,691 22,476 23,107 30,112 74,525 99,1236 110,858 186,1289 468,1301 523,1297 827,3153 850,2373
X(2697) = reflection of X(935) in X(3)
X(2697) = isogonal conjugate of X(2781)
X(2697) = cevapoint of X(3) and X(542)
X(2698) lies on the circumcircle and these lines: 3,805 4,2679 32,2715 98,512 99,511 107,419 110,237 112,1691 182,691 401,925 476,1316
X(2698) = reflection of X(i) in X(j) for these (i,j): (4,2679), (3,805)
X(2698) = isogonal conjugate of X(2782)
X(2698) = anticomplement of X(33330)
X(2698) = reflection of X(98) in the Brocard axis
X(2698) = Collings transform of X(2679)
X(2698) = Cundy-Parry Phi transform of X(14251)
X(2698) = Cundy-Parry Psi transform of X(14382)
X(2698) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(4),X(32)}}
X(2698) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(112)
X(2698) = 3rd-Parry-to-circumsymmedial similarity image of X(2)
X(2699) lies on the circumcircle and these lines: 3,2703 98,513 99,517 100,511 101,1755 105,512 107,422 171,2222 572,2702 1309,1824 1333,2715 1402,2720
X(2699) = reflection of X(2703) in X(3)
X(2699) = isogonal conjugate of X(2783)
X(2699) = reflection of X(104) in the Brocard axis
X(2699) = reflection of X(98) in line X(1)X(3)
X(2699) = Schoute-circle-inverse of X(32722)
X(2700) lies on the circumcircle and these lines: 3,2702 58,2715 98,514 99,516 100,1959 101,511 103,512 107,423 112,1326 991,2701 1350,2705
X(2700) = reflection of X(2702) in X(3)
X(2700) = isogonal conjugate of X(2784)
X(2700) = reflection of X(103) in the Brocard axis
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2701) lies on the circumcircle and these lines: 3,2708 98,515 99,522 102,511 104,2648 109,512 110,663 111,1055 187,2291 759,1326 953,1064 991,2700 1951,2249 1983,2702
X(2701) = reflection of X(2708) in X(3)
X(2701) = isogonal conjugate of X(2785)
X(2701) = cevapoint of X(663) and X(1951)
X(2701) = reflection of X(109) in the Brocard axis
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2702) lies on the circumcircle and these lines: 3,2700 6,2712 98,516 99,514 100,661 101,512 103,511 105,1929 106,187 110,649 111,902 572,2699 573,2708 727,1691 741,1326 1983,2701
X(2702) = reflection of X(2700) in X(3)
X(2702) = isogonal conjugate of X(2786)
X(2702) = cevapoint of X(649) and X(1914)
X(2702) = reflection of X(101) in the Brocard axis
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2703) lies on the circumcircle and these lines: 3,2699 36,741 98,517 99,513 100,512 101,798 104,511 110,667 187,739 238,759 392,2726 713,1691 1083,2752
X(2703) = reflection of X(2699) in X(3)
X(2703) = isogonal conjugate of X(2787)
X(2703) = reflection of X(100) in the Brocard axis
X(2703) = reflection of X(99) in line X(1)X(3)
X(2704) lies on the circumcircle and these lines: 3,2711 98,518 105,511 512,1292
X(2704) = reflection of X(2711) in X(3)
X(2704) = isogonal conjugate of X(2788)
X(2705) lies on the circumcircle and these lines: > 3,2712 98,519 106,511 512,1293 727,2080 1350,2700
X(2705) = reflection of X(2712) in X(3)
X(2705) = isogonal conjugate of X(2789)
X(2706) lies on the circumcircle and these lines: 3,2713 98,520 107,450 512,1294 577,2715
X(2706) = reflection of X(2713) in X(3)
X(2706) = isogonal conjugate of X(2790)
X(2707) lies on the circumcircle and these lines: 3,2714 98,521 107,425 108,511 512,1295 2193,2715
X(2707) = reflection of X(2714) in X(3)
X(2707) = isogonal conjugate of X(2791)
X(2708) lies on the circumcircle and these lines: 3,2701 10,929 98,522 99,515 101,1324 102,512 107,415 108,240 109,511 284,2715 573,2702
X(2708) = reflection of X(2701) in X(3)
X(2708) = isogonal conjugate of X(2792)
X(2708) = reflection of X(102) in the Brocard axis
X(2709) lies on the circumcircle and these lines: 3,843 98,524 99,1499 111,352 512,1296 691,2421 729,2080 842,1350
X(2709) = reflection of X(843) in X(3)
X(2709) = isogonal conjugate of X(2793)
X(2709) = reflection of X(1296) in the Brocard axis
X(2709) = 1st-Parry-to-ABC similarity image of X(352)
X(2709) = X(842)-of-circumsymmedial-triangle
X(2710) lies on the circumcircle and these lines: 3,2715 98,525 99,1503 107,297 112,511 512,1297 691,1350 935,1352
X(2710) = reflection of X(2715) in X(3)
X(2710) = isogonal conjugate of X(2794)
X(2711) lies on the circumcircle and these lines: 3,2704 35,813 65,927 99,518 105,512 110,2223 213,919 511,1292
X(2711) = reflection of X(2704) in X(3)
X(2711) = isogonal conjugate of X(2795)
X(2711) = reflection of X(105) in the Brocard axis
X(2712) lies on the circumcircle and these lines: 3,2705 6,2702 42,901 58,691 99,519 100,896 101,187 106,512 110,902 111,649 511,1293
X(2712) = reflection of X(2705) in X(3)
X(2712) = isogonal conjugate of X(2796)
X(2712) = reflection of X(106) in the Brocard axis
X(2713) lies on the circumcircle and these lines: 3,2706 74,1942 99,520 107,512 477,2452 511,1294
X(2713) = reflection of X(2706) in X(3)
X(2713) = isogonal conjugate of X(2797)
X(2713) = reflection of X(107) in the Brocard axis
X(2714) lies on the circumcircle and these lines: 3,2707 99,521 108,512 109,810 511,1295
X(2714) = reflection of X(2707) in X(3)
X(2714) = isogonal conjugate of X(2798)
X(2714) = reflection of X(108) in the Brocard axis
X(2714) = Schoute-circle-inverse of X(32726)
Let A' = BC∩(Brocard axis), and define CA and AB cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(2715). (Randy Hutson, February 10, 2016)
X(2715) lies on the circumcircle and these lines: 2,2857 3,2710 6,842 32,2698 58,2700 74,187 81,2856 98,230 99,249 103,1326 107,685 110,647 111,1495 112,512 232,1692 284,2708 287,2373 290,2367 476,2395 477,2549 511,1297 577,2706 691,2420 759,1910 827,1625 843,1384 879,935 933,2623 1304,2433 1333,2699 2193,2707 2407,2855
X(2715) = reflection of X(2710) in X(3)
X(2715) = isogonal conjugate of X(2799)
X(2715) = cevapoint of X(i) and X(j) for these (i,j): (512,1692), (523,2450), (1976,2422)
X(2715) = X(i)-cross conjugate of X(j) for these (i,j): (512,2065), (989,98), (1691,249), (2422,1976)
X(2715) = reflection of X(112) in the Brocard axis
X(2715) = concurrence of reflections in sides of ABC of line X(4)X(32)
X(2715) = trilinear pole of line X(6)X(157)
X(2715) = X(92)-isoconjugate of X(684)
X(2715) = Λ(X(i),X(j)) for these (i,j): (2,1637), (99,112)
X(2715) = Ψ(X(i), X(j)) for these (i,j): (2,98), (4,32), (6,157), (76,3)
X(2715) = inverse-in-Moses-radical-circle of X(110)
X(2715) = inverse-in-circle-O(15,16) of X(74)
X(2715) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(98)
X(2715) = polar-circle-inverse of X(38970)
X(2715) = orthoptic-circle-of-Steiner-inellipse-inverse of X(38975)
X(2715) = Brocard-circle-inverse of X(34235)
X(2715) = barycentric product X(98)*X(110) (circumcircle-X(2)-antipodes)
X(2715) = barycentric quotient X(98)/X(850)
X(2716) lies on the circumcircle and these lines: 1,2720 3,2222 36,108 40,901 100,515 101,2182 102,513 104,522 109,517 318,1309 476,1789 929,993 1012,2728
X(2716) = reflection of X(2222) in X(3)
X(2716) = isogonal conjugate of X(2800)
X(2716) = cevapoint of X(55) and X(2265)
X(2716) = reflection of X(102) in line X(1)X(3)
Let A'B'C' be the excentral triangle. Let La be the line X(115)X(125) of the triangle A'BC, and define Lb and Lc cyclically. Let A'' = Lb∩Lc, and define B'' and C'' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is X(2717). (Randy Hutson, July 31 2018)
X(2717) lies on the circumcircle and these lines: 3,1308 36,934 40,2742 55,2222 57,2720 92,1309 100,516 101,517 103,513 104,514 108,2078 109,1155 165,901 573,813 650,2291 1292,2077
X(2717) = reflection of X(1308) in X(3)
X(2717) = isogonal conjugate of X(2801)
X(2717) = cevapoint of X(55) and X(2246)
X(2717) = reflection of X(103) in line X(1)X(3)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2718) lies on the circumcircle and these lines: 1,901 3,2743 36,100 44,101 56,2222 106,513 108,1877 109,1319 214,765 244,1168 517,1293 609,919 898,993 999,1308 1420,2720
X(2718) = reflection of X(2743) in X(3)
X(2718) = isogonal conjugate of X(2802)
X(2718) = X(2265)-cross conjugate of X(57)
X(2718) = reflection of X(106) in line X(1)X(3)
X(2718) = incircle-inverse of X(33645)
X(2718) = trilinear pole, wrt 2nd circumperp triangle, of line X(2)X(11)
X(2718) = trilinear pole of line X(6)X(1635)
X(2718) = Ψ(X(6), X(1635))
X(2718) = trilinear product of circumcircle intercepts of line X(1)X(900)
X(2719) lies on the circumcircle and these lines: 3,2744 100,520, 101,822 107,513 517,1294
X(2719) = reflection of X(2744) in X(3)
X(2719) = isogonal conjugate of X(2803)
X(2719) = reflection of X(107) in line X(1)X(3)
Let A', B', C' be the intersections of line X(1)X(3) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(2720). (Randy Hutson, December 10, 2016)
The tangent lines from X(2720) to the incircle intersect the circumcircle in points on the Sherman Line, so forming a poristic triangle with ABC. (Peter Moses, December 3, 2019)
X(2720) lies on the circumcircle and these lines: 1,2716 3,2745 7,2861 36,102 56,953 57,2717 59,100 65,2687 101,652 103,2078 104,1319 106,1457 108,513 109,1459 497,2723 517,1395 840,1617 883,2865 909,2272 915,1870 919,2423 929,2401 944,2734 972,1155 1214,2747 1397,2726 1402,2699 1420,2718 2283,2742
X(2720) = reflection of X(2745) in X(3)
X(2720) = isogonal conjugate of X(2804)
X(2720) = cevapoint of X(663) and X(1404)
X(2720) = X(i)-cross conjugate of X(j) for these (i,j): (654,57), (1319,59)
X(2720) = concurrence of reflections in sides of ABC of line X(4)X(11)
X(2720) = trilinear pole of line X(6)X(909)
X(2720) = Ψ(X(4), X(11))
X(2720) = Ψ(X(6), X(909))
X(2720) = Ψ(X(9), X(101))
X(2720) = Ψ(X(650), X(6))
X(2720) = isotomic conjugate of polar conjugate of X(32702)
X(2720) = reflection of X(108) in line X(1)X(3)
X(2720) = inverse-in-circle-O(3513,3514)-of-X(105)
X(2720) = perspector of 3rd mixtilinear triangle and cross-triangle of ABC and circumcevian triangle of X(1319)
X(2721) lies on the circumcircle and these lines: 3,2746 81,691 100,524 101,896 104,1499 111,513 517,1296
X(2721) = reflection of X(2746) in X(3)
X(2721) = isogonal conjugate of X(2805)
X(2721) = reflection of X(111) in line X(1)X(3)
X(2722) lies on the circumcircle and these lines: 3,2747 100,525 101,656 104,1503 110,905 112,513 517,1297
X(2722) = reflection of X(2747) in X(3)
X(2722) = isogonal conjugate of X(2806)
X(2722) = reflection of X(112) in line X(1)X(3)
X(2723) lies on the circumcircle and these lines: 3,929 101,515 102,514 103,522 108,243 109,516 497,2720 813,1766 946,2727 1021, 2249
X(2723) = reflection of X(929) in X(3)
X(2723) = isogonal conjugate of X(2807)
X(2724) lies on the circumcircle and these lines: 3,927 4,1566 40,813 101,516 103,514 108,2223
X(2724) = reflection of X(i) in X(j) for these (i,j): (4,1566), (3,927)
X(2724) = isogonal conjugate of X(2808)
X(2724) = anticomplement of X(33331)
X(2725) lies on the circumcircle and these lines: 1,919 3,2736 85,927 101,518 105,514 109,241 516,1292 813,1083 1001,1308
X(2729) = reflection of X(2736) in X(3)
X(2725) = isogonal conjugate of X(2809)
X(2725) = Λ(X(1), X(101))
X(2726) lies on the circumcircle and these lines: 2,901 3,2737 9,813 25,1309 99,859 101,519 104,667 106,514 108,242 109,238 392,2703 516,1293 898,956 934,1447 1016,1145 1083,2742 1397,2720
X(2726) = reflection of X(2737) in X(3)
X(2726) = isogonal conjugate of X(2810)
X(2726) = Λ(X(6), X(101))
X(2727) lies on the circumcircle and these lines: 3,2738, 101,520 107,514 112,1459 516,1294 946,2723
X(2727) = reflection of X(2738) in X(3)
X(2727) = isogonal conjugate of X(2811)
X(2728) lies on the circumcircle and these lines: 3,2739 101,521 108,514 109,905 516,1295 997,2745 1012,2716
X(2728) = reflection of X(2739) in X(3)
X(2728) = isogonal conjugate of X(2812)
X(2729) lies on the circumcircle and these lines: 3,2740 86,691 101,524 103,1499 111,514 516,1296
X(2729) = reflection of X(2740) in X(3)
X(2729) = isogonal conjugate of X(2813)
X(2730) lies on the circumcircle and these lines: 3,2751 102,518 105,515 522,1292 1785,2376
X(2730) = reflection of X(2751) in X(3)
X(2730) = isogonal conjugate of X(2814)
X(2731) lies on the circumcircle and these lines: 3,2757 102,519 106,515 243,2376 522,1293 944,953
X(2731) = reflection of X(2757) in X(3)
X(2731) = isogonal conjugate of X(2815)
X(2732) lies on the circumcircle and these lines: 3,2762 102,520 107,515 522,1294 1304,2360
X(2732) = reflection of X(2762) in X(3)
X(2732) = isogonal conjugate of X(2816)
X(2733) lies on the circumcircle and these lines: 3,2765 102,521 108,515 522,1295
X(2733) = reflection of X(2765) in X(3)
X(2733) = isogonal conjugate of X(2817)
X(2734) lies on the circumcircle and these lines: 3,1309 20,901 102,522 107,859 109,515 944,2720
X(2734) = reflection of X(1309) in X(3)
X(2734) = isogonal conjugate of X(2818)
X(2734) = cevapoint of X(3) and X(952)
X(2735) lies on the circumcircle and these lines: 3,2768 102,524 109,1499 111,515 522,1296
X(2735) = reflection of X(2768) in X(3)
X(2735) = isogonal conjugate of X(2819)
X(2736) lies on the circumcircle and these lines: 3,2725 103,518 105,516 514,1292 1323,2377
X(2736) = reflection of X(2725) in X(3)
X(2736) = isogonal conjugate of X(2820)
X(2736) = cevapoint of X(55) and X(2254)
X(2737) lies on the circumcircle and these lines: 3,2726 103,519 106,516 376,953 514,1293 901,2398
X(2737) = reflection of X(2726) in X(3)
X(2737) = isogonal conjugate of X(2821)
X(2738) lies on the circumcircle and these lines: 3,2727 103,520 107,516 514,1294 1304,2328
X(2738) = reflection of X(2727) in X(3)
X(2738) = isogonal conjugate of X(2822)
X(2739) lies on the circumcircle and these lines: 3,2728 103,521 108,516 514,1295
X(2739) = reflection of X(2728) in X(3)
X(2739) = isogonal conjugate of X(2823)
X(2740) lies on the circumcircle and these lines: 3,2729 101,1499 103,524 111,516 514,1296
X(2740) = reflection of X(2729) in X(3)
X(2740) = isogonal conjugate of X(2824)
X(2741) lies on the circumcircle and these lines: 101,1503 103,525 112,516 514,1297
X(2741) = isogonal conjugate of X(2825)
X(2742) lies on the circumcircle and these lines: 3,840 36,1477 40,2717 59,934 101,1618 103,2077 104,518 105,517 109,1110 513,1292 919,2427 1083,2726 2283,2720
X(2742) = reflection of X(840) in X(3)
X(2742) = isogonal conjugate of X(2826)
X(2742) = X(1155)-cross conjugate of X(59)
X(2742) = perspector of ABC and triangle formed by line X(2)X(11) reflected in sides of ABC
X(2742) = perspector of 4th mixtilinear triangle and cross-triangle of ABC and circumcevian triangle of X(1155)
X(2743) lies on the circumcircle and these lines: 3,2718 40,953 104,519 106,517 165,840 513,1293 1155,1477 2291,2348
X(2743) = reflection of X(2718) in X(3)
X(2743) = isogonal conjugate of X(2827)
X(2743) = cevapoint of X(55) and X(1635)
X(2744) lies on the circumcircle and these lines: 3,2719 72,1309 104,520 107,517 513,1294
X(2744) = reflection of X(2719) in X(3)
X(2744) = isogonal conjugate of X(2828)
X(2745) lies on the circumcircle and these lines: 3,2720 8,1309 40,2222 104,521 108,517 109,2077 513,1295 997,2728 1158,2765
X(2745) = reflection of X(2720) in X(3)
X(2745) = isogonal conjugate of X(2829)
X(2746) lies on the circumcircle and these lines: 3,2721 100,1499 104,525 111,517 513,1296
X(2746) = reflection of X(2721) in X(3)
X(2746) = isogonal conjugate of X(2830)
X(2746) = reflection of X(1296) in line X(1)X(3)
X(2747) lies on the circumcircle and these lines: 3,2722 100,1503 101,2312 104,525 112,517 321,1309 513,1297 1214,2720
X(2747) = reflection of X(2722) in X(3)
X(2747) = isogonal conjugate of X(2831)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2748) lies on the circumcircle and these lines: 105,519 106,518 901,1026 919,1023 1083,2382
X(2748) = isogonal conjugate of X(2832)
X(2749) lies on the circumcircle and these lines: 105,520 107,518
X(2749) = isogonal conjugate of X(2833)
X(2750) lies on the circumcircle and these lines: 69,927 105,521 108,518 109,1818 200,2222 219,919
X(2750) = isogonal conjugate of X(2834)
X(2751) lies on the circumcircle and these lines: 3,2730 9,919 63,901 75,927 105,522 106,905 108,1861 109,518 515,1292 956,1308 1376,2222
X(2751) = reflection of X(2730) in X(3)
X(2751) = isogonal conjugate of X(2835)
X(2752) lies on the circumcircle and these lines: 2,1290 3,2691 21,691 23,100 25,2766 28,935 80,1292 37,919 99,1325 105,523 108,468 109,896 110,518 111,650 112,2074 927,1441 1083,2703
X(2752) = reflection of X(2691) in X(3)
X(2752) = isogonal conjugate of X(2836)
X(2752) = reflection of X(105) in the Euler line
X(2752) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5520)
X(2753) lies on the circumcircle and these lines: 105,524 111,518 352,739 1292,1499
X(2753) = isogonal conjugate of X(2837)
X(2754) lies on these lines: 72,919 105,525 112,518 927,1231 1292,1503
X(2754) = isogonal conjugate of X(2838)
X(2755) lies on the circumcircle and these lines: 106,520 107,519
X(2755) = isogonal conjugate of X(2839)
X(2756) lies on the circumcircle and these lines: 78,901 106,521 108,519
X(2756) = isogonal conjugate of X(2840)
X(2757) lies on the circumcircle and these lines: 3,2731 8,901 101,2325 106,522 109,519 515,1293
X(2757) = reflection of X(2731) in X(3)
X(2757) = isogonal conjugate of X(2841)
X(2758) lies on the circumcircle and these lines: 3,2692 10,901 30,1293 106,523 110,519 404,1290 2453,2690
X(2768) = reflection of X(2692) in X(3)
X(2758) = isogonal conjugate of X(2842)
X(2758) = reflection of X(106) in the Euler line
X(2758) = trilinear pole of line X(6)X(4120)
X(2758) = Ψ(X(6), X(4120))
X(2758) = Stevanovic-circle-inverse of X(111)
X(2759) lies on the circumcircle and these lines: 106,524 111,519 1293,1499
X(2759) = isogonal conjugate of X(2843)
X(2760) lies on the circumcircle and these lines: 106,525 112,519 306,901 691,1043 1293,1503
X(2760) = isogonal conjugate of X(2844)
X(2761) lies on the circumcircle and these lines: 107,521 108,520
X(2761) = isogonal conjugate of X(2845)
X(2762) lies on the circumcircle and these lines: 3,2732 107,522 108,656 109,520 112,652 515,1294
X(2762) = reflection of X(2732) in X(3)
X(2762) = isogonal conjugate of X(2846)
X(2763) lies on the circumcircle and these lines: 107,524 111,520 394,691 1294,1499
X(2763) = isogonal conjugate of X(2847)
X(2764) lies on the circumcircle and these lines: 107,525 112,520 1294,1503 2435,2442
X(2764) = isogonal conjugate of X(2848)
X(2764) = Λ(X(107), X(112))
X(2765) lies on the circumcircle and these lines: 3,2733 108,522 109,521 112,1021 515,1295 1158,2745
X(2765) = reflection of X(2733) in X(3)
X(2765) = isogonal conjugate of X(2849)
X(2765) = Λ(X(108), X(109))
X(2766) lies on the circumcircle and these lines: 3,2694 4,2687 25,2752 30,1295 104,186 105,468 108,523 109,656 110,521 112,650 403,915 759,2074
X(2766) = reflection of X(2694) in X(3)
X(2766) = isogonal conjugate of X(2850)
X(2766) = reflection of X(108) in the Euler line
X(2766) = inverse-in-polar-circle of X(5520)
X(2766) = Stevanovic-circle-inverse of X(112)
X(2767) lies on the circumcircle and these lines: 108,524 111,521 691,1812 1295,1499
X(2767) = isogonal conjugate of X(2851)
X(2768) lies on the circumcircle and these lines: 3,2735 102,1499 109,524 111,522 333,691 515,1296
X(2768) = reflection of X(2735) in X(3)
X(2768) = isogonal conjugate of X(2852)
X(2769) lies on the circumcircle and these lines: 102,1503 108,1577 109,525 112,522 515,1297
X(2769) = isogonal conjugate of X(2853)
X(2770) lies on the circumcircle and these lines: 2,691 3,2696 23,99 25,935 30,1296 74,1499 110,524 111,523 112,468 476,1995
X(2770) = reflection of X(2696) in X(3)
X(2770) = isogonal conjugate of X(2854)
X(2770) = trilinear pole wrt circummedial triangle of line X(3)X(76)
X(2770) = X(110)-of-circummedial-triangle
X(2770) = reflection of X(111) in the Euler line
X(2770) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5099)
As the isogonal conjugate of a point on the circumcircle, X(2771) lies on the line at infinity.
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. Then X(2771) = X(30)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). Let Ae' be the Ai-isogonal conjugate of Ae, and define Be' and Ce' cyclically. Let Ai' be the Ae-isogonal conjugate of Ai, and define Bi' and Ci' cyclically. Triangles Ae'Be'Ce' and Ai'Bi'Ci' are inversely congruent, with similitude center X(2771). Triangle Ai'Bi'Ci' is also the cross-triangle of the Ae and Ai triangles. (Randy Hutson, June 7, 2019)
X(2771) lies on these (parallel) lines: 1,399 3,191 11,113 21,104 30,511 36,1749 65,79 72,74 78,2932 109,1807 119,125 146,149 153,355 214,960 500,2292 946,1484 1464,1725 1829,1986
X(2771) = isogonal conjugate of X(2687)
X(2771) = X(2694)-Ceva conjugate of X(3)
X(2771) = X(1154)-of-Fuhrmann-triangle
X(2771) = X(30)-of-X(1)-Brocard-triangle
X(2771) = X(517)-of-orthocentroidal-triangle
X(2771) = X(517)-of-X(4)-Brocard triangle
As the isogonal conjugate of a point on the circumcircle, X(2772) lies on the line at infinity.
X(2772) lies on these (parallel) lines: 30,511 71,74 102,110 113,116 118,125 146,150 1839,1986 2938,2948
X(2772) = isogonal conjugate of X(2688)
X(2772) = X(516)-of-orthocentroidal-triangle
X(2772) = X(516)-of-X(4)-Brocard-triangle
As the isogonal conjugate of a point on the circumcircle, X(2773) lies on the line at infinity.
X(2773) lies on these (parallel) lines: 30,511 74,102 109,110 113,117 124,125 146,151
X(2773) = isogonal conjugate of X(2689)
X(2773) = X(522)-of-orthocentroidal-triangle
X(2773) = X(522)-of X(4)-Brocard-triangle
As the isogonal conjugate of a point on the circumcircle, X(2774) lies on the line at infinity.
X(2774) lies on these (parallel) lines: 30,511 74,103 101,110 113,118 116,125 146,152 1282,2948
X(2774) = isogonal conjugate of X(2690)
X(2774) = X(514)-of-orthocentroidal-triangle
X(2774) = X(514)-of-X(4)-Brocard-triangle
X(2774) = crossdifference of every pair of points on line X(6)X(3120)
As the isogonal conjugate of a point on the circumcircle, X(2775) lies on the line at infinity.
X(2775) lies on these (parallel) lines: 30,511 74,105 110,1292 113,120
X(2775) = isogonal conjugate of X(2691)
As the isogonal conjugate of a point on the circumcircle, X(2776) lies on the line at infinity.
X(2776) lies on these (parallel) lines: 30,511 74,106 110,1293 113,121
X(2776) = isogonal conjugate of X(2692)
As the isogonal conjugate of a point on the circumcircle, X(2777) lies on the line at infinity.
X(2777) lies on these (parallel) lines: 3,113 4,74 5,1539 20,110 23,1533 30,511 64,265 112,1562 182,1177 185,1986 389,974 399,1498 468,1514 550,1511 858,1531 1316,1561 1352,2892 1568,2071
X(2777) = isogonal conjugate of X(2693)
X(2777) = crosspoint of X(4) and X(477)
As the isogonal conjugate of a point on the circumcircle, X(2778) lies on the line at infinity.
X(2778) lies on these (parallel) lines: 1,2935 30,511 40,2915 65,74 110,1295 113,123 125,429 1498,2948 1717,1854 1858,1986
X(2778) = isogonal conjugate of X(2694)
X(2778) = crosspoint of X(4) and X(2687)
X(2778) = crosssum of X(3) and X(2771)
As the isogonal conjugate of a point on the circumcircle, X(2779) lies on the line at infinity.
X(2779) lies on these (parallel) lines: 30,511 35,73 102,110 113,124 117,125 1364,2646 1844,1845 2939,2948
X(2779) = isogonal conjugate of X(2695)
X(2779) = X(515)-of-orthocentroidal-triangle
X(2779) = X(515)-of-X(4)-Brocard-triangle
As the isogonal conjugate of a point on the circumcircle, X(2780) lies on the line at infinity.
X(2780) lies on these (parallel) lines: 3,351 30,511 74,111 110,1296 113,126 895,2444
X(2780) = isogonal conjugate of X(2696)
X(2780) = X(1499)-of-orthocentroidal-triangle
X(2780) = X(1499)-of-X(4)-Brocard-triangle
X(2780) = circumsymmedial-isogonal conjugate of X(32425)
X(2780) = ideal point of PU(63)
X(2780) = X(1499)-of-4th-anti-Brocard-triangle
As the isogonal conjugate of a point on the circumcircle, X(2781) lies on the line at infinity.
X(2781) lies on these (parallel) lines: 3,1177 4,67 6,74 22,110 30,511 51,125 64,895 66,265 69,146 113,127 159,399 185,1205 206,1511 1498,2930
X(2781) = isogonal conjugate of X(2697)
X(2781) = crosspoint of X(4) and X(842)
X(2781) = crosssum of X(3) and X(542)
As the isogonal conjugate of a point on the circumcircle, X(2782) lies on the line at infinity.
X(2782) lies on these (parallel) lines: 3,76 4,147 5,39 30,511 68,1987 110,1316 140,620 230,2021 262,381 338,1634 385,2080 549,2482 895,2452 1352,2549 2453,2930
X(2782) = isogonal conjugate of X(2698)
X(2782) = X(511)-of-1st-Brocard-triangle
X(2782) = orthic-isogonal conjugate of X(33330)
X(2782) = 1st-Brocard-isogonal conjugate of X(3)
X(2782) = 1st-Brocard-isotomic conjugate of X(34359)
X(2782) = Cundy-Parry Phi transform of X(14382)
X(2782) = Cundy-Parry Psi transform of X(14251)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(2783) = X(511)-of-IaIbIc; "IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2783) lies on these (parallel) lines: 5,2486 11,114 30,511 80,256 98,100 99,104 115,119 147,149 148,153 1284,1733 1764,1768
X(2783) = isogonal conjugate of X(2699)
X(2783) = isotomic conjugate of X(35151)
X(2783) = X(2)-Ceva conjugate of X(35083)
X(2783) = X(517)-of-1st-Brocard-triangle
X(2784) lies on these (parallel) lines: 1,147 8,1281 10,98 30,511 99,103 114,116 115,118 148,152
X(2784) = isogonal conjugate of X(2700)
X(2784) = isotomic conjugate of X(35150)
X(2784) = X(2)-Ceva conjugate of X(35082)
X(2784) = X(516)-of-1st-Brocard-triangle
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2785) lies on these (parallel) lines: 30,511 98,102 99,109 114,117 115,124 147,151 671,1121
X(2785) = isogonal conjugate of X(2701)
X(2785) = isotomic conjugate of X(35154)
X(2785) = X(2)-Ceva conjugate of X(35086)
X(2785) = crosspoint of X(664) and X(1952)
X(2785) = crosssum of X(663) and X(1951)
X(2785) = X(522)-of-1st-Brocard-triangle
X(2785) = 1st-Brocard-isotomic conjugate of X(34361)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2786) lies on these (parallel) lines: 30,511 98,103 99,101 114,118 115,116 147,152 148,150 671,903
X(2786) = isogonal conjugate of X(2702)
X(2786) = isotomic conjugate of X(35148)
X(2786) = X(2)-Ceva conjugate of X(35080)
X(2786) = X(514)-of-1st-Brocard-triangle
X(2786) = 1st-Brocard-isogonal conjugate of X(24279)
X(2786) = 1st-Brocard-isotomic conjugate of X(24281)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(2787) = X(512)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2787) lies on these (parallel) lines: 11,115 30,511 80,291 98,104 99,100 114,119 147,153 148,149 1019,2533
X(2787) = isogonal conjugate of X(2703)
X(2787) = isotomic conjugate of X(35147)
X(2787) = X(2)-Ceva conjugate of X(35079)
X(2787) = X(513)-of-1st-Brocard-triangle
X(2787) = 1st-Brocard-isogonal conjugate of X(5091)
X(2787) = 1st-Brocard-isotomic conjugate of X(24289)
X(2788) lies on these (parallel) lines: 30,511 98,105 99,1292 114,120
X(2788) = isogonal conjugate of X(2704)
X(2789) lies on these (parallel) lines: 30,511 98,106 99,1293 114,121
X(2789) = isogonal conjugate of X(2705)
X(2790) lies on these (parallel) lines: 3,1632 4,1942 25,98 30,511 53,115 99,1294 114,122 147,1370 157,1605
X(2790) = isogonal conjugate of X(2706)
X(2791) lies on these (parallel) lines: 30,511 98,108 99,1295 114,123 115,1865
X(2791) = isogonal conjugate of X(2707)
X(2792) lies on these (parallel) lines: 4,1046 30,511 40,1330 58,946 63,147 98,109 99,102 114,124 115,117 148,151 573,1761
X(2792) = isogonal conjugate of X(2708)
X(2792) = isotomic conjugate of X(35149)
X(2792) = X(2)-Ceva conjugate of X(35081)
X(2792) = X(515)-of-1st-Brocard-triangle
X(2793) lies on these (parallel) lines: 30,511 98,111 99,1296 114,126 671,2408
X(2793) = isogonal conjugate of X(2709)
X(2793) = X(523)-of-McCay-triangle
X(2793) = X(523)-of-anti-McCay-triangle
X(2793) = ideal point of PU(138)
X(2794) lies on these (parallel) lines: 3,114 4,32 20,99 30,511 67,2453 125,1316 187,1513 376,2482 1352,3734
X(2794) = isogonal conjugate of X(2710)
X(2795) lies on these (parallel) lines: 21,99 30,511 98,1292 115,120 148,1655 1358,3027 3021,3023 3029,3034
X(2795) = isogonal conjugate of X(2711)
X(2795) = isotomic conjugate of X(35152)
X(2795) = X(2)-Ceva conjugate of X(35084)
X(2795) = X(518)-of-1st-Brocard-triangle
X(2795) = 1st-Brocard-isogonal conjugate of X(1083)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2796) lies on these (parallel) lines: 2,846 10,190 30,511 86,99 98,1293 115,121 148,1654 553,1357 1086,1125 3029,3030
X(2796) = isogonal conjugate of X(2712)
X(2796) = isotomic conjugate of X(35153)
X(2796) = X(2)-Ceva conjugate of X(35085)
X(2796) = X(519)-of-1st-Brocard-triangle
X(2796) = 1st-Brocard-isotomic conjugate of X(34362)
X(2797) lies on these (parallel) lines: 4,684 30,511 98,1294 99,107 114,133 115,122
X(2797) = isogonal conjugate of X(2713)
X(2797) = X(520)-of-1st-Brocard-triangle
X(2798) lies on these (parallel) lines: 30,511 98,1295 99,108 115,123
X(2798) = isogonal conjugate of X(2714)
X(2798) = X(521)-of-1st-Brocard-triangle
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2799) lies on these (parallel) lines: 2,1637 30,511 39,2507 98,1297 99,112 114,132 115,127 287,2395 620,2492 671,1494 850,2525 2514,2531
X(2799) = isogonal conjugate of X(2715)
X(2799) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,35088), (99,114), (290,125), (325,868), (805,626), (877,511), (1916,115)
X(2799) = isotomic conjugate of X(2966)
X(2799) = pole wrt polar circle of line X(4)X(32) (trilinear polar of X(685)
X(2799) = X(48)-isoconjugate (polar conjugate) of X(685)
X(2799) = ideal point of PU(135)
X(2799) = X(525)-of-1st-Brocard-triangle
X(2799) = bicentric difference of PU(135)
X(2799) = complementary conjugate of X(36471)
X(2799) = X(4)-Ceva conjugate of X(36471)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically; then X(2800) = X(515)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2800) lies on these (parallel) lines: 1,104 3,214 4,80 8,153 10,119 11,65 30,511 40,78 72,1145 84,1320 149,151 942,1387 1012,2099 1317,1364 1457,1735 1519,1737 1765,1953 1862,1902
X(2800) = isogonal conjugate of X(2716)
X(2800) = crosssum of X(55) and X(2265)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2801) = X(516)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2801) lies on these (parallel) lines: 1,651 7,80 9,48 11,118 30,511 63,100 116,119 149,152 480,2932 984,991 990,1814 1458,1736 1776,2078
X(2801) = isogonal conjugate of X(2717)
X(2801) = isotomic conjugate of X(35164)
X(2801) = X(2)-Ceva conjugate of X(35116)
X(2801) = crosssum of X(55) and X(2246)
X(2801) = X(511)-of-Fuhrmann-triangle
X(2801) = endo-homothetic center of Ehrmann vertex-triangle and 2nd Ehrmann triangle; the homothetic center is X(542)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2802) = X(519)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2802) lies on these (parallel) lines: 1,88 8,80 10,11 30,511 40,104 56,2932 65,1317 119,946 153,962 643,759 765,1168 1000,2550 1018,2170 1023,2246 1125,1387 1149,1739 1829,1862 1845,1897
X(2802) = isogonal conjugate of X(2718)
X(2802) = isotomic conjugate of X(35175)
X(2802) = X(2)-Ceva conjugate of X(35129)
X(2802) = crossdifference of every pair of points on line X(6)X(1635)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2803) = X(520)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2803) lies on these (parallel) lines: 11,122 30,511 100,107 104,1294 119,133
X(2803) = isogonal conjugate of X(2719)
X(2804) lies on these (parallel) lines: 11,123 30,511 100,108 104,1295 1145,1769 1317,1359
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2804) = X(521)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2804) = isogonal conjugate of X(2720)
X(2804) = X(i)-Ceva conjugate of X(j) for these (i,j): (100,119), (655,9), (901,1329), (1295,123), (1320,11)
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2805) = X(524)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2805) lies on these (parallel) lines: 11,126 30,511 37,100 75,149 104,1296
X(2805) = isogonal conjugate of X(2721)
X(2806) lies on these (parallel) lines: 11,127 30,511 100,112 104,1297 119,132
Let Ia be the reflection of X(1) in the perpendicular bisector of BC, and define Ib and Ic cyclically. X(2806) = X(525)-of-IaIbIc; IaIbIc is the inner Garcia triangle. (Randy Hutson, February 10, 2016)
X(2806) = isogonal conjugate of X(2722)
X(2807) lies on these (parallel) lines: 1,185 3,692 30,511 51,1699 55,103 101,102 116,117 118,124 150,151 389,946 990,1469 1037,1795 1282,1763 1361,2099 1827,1845
X(2807) = isogonal conjugate of X(2723)
X(2808) lies on these (parallel) lines: 1,1362 3,101 4,150 5,116 30,511 40,170 84,295 970,1490
X(2808) = isogonal conjugate of X(2724)
X(2808) = isogonal conjugate of the anticomplement of X(33331)
X(2808) = orthic-isogonal conjugate of X(33331)
X(2908) = polar conjugate of isotomic conjugate of X(36033)
X(2809) lies on these (parallel) lines: 1,41 8,150 10,116 30,511 40,103 65,1358 118,946 152,962
X(2809) = isogonal conjugate of X(2725)
X(2810) lies on these (parallel) lines: {1,3271}, {6,101}, {3,3939}, {8,3888}, {30,511}, {43,57}, {51,3873}, {100,3937}, {118,5480}, {200,3784}, {210,3819}, {375,3742}, {611,5138}, {1428,5193}, {1843,5185}, {2093,3779}, {2097,4259}, {2979,4661}, {3022,4907}, {3038,3041}, {3060,4430}, {3681,3917}, {3781,5223}
X(2810) = isogonal conjugate of X(2726)
X(2811) lies on these (parallel) lines: 30,511 101,107 103,1294 116,122 118,133
X(2811) = isogonal conjugate of X(2727)
X(2812) lies on these (parallel) lines: 30,511 101,108 103,1295 116,123 1359,1362
X(2812) = isogonal conjugate of X(2728)
X(2813) lies on these (parallel) lines: 30,511 42,101 103,1296 116,126
X(2813) = isogonal conjugate of X(2729)
X(2814) lies on these (parallel) lines: 30,511 40,2254 102,105 109,1292 117,120 1358,1364
X(2814) = isogonal conjugate of X(2730)
X(2815) lies on these (parallel) lines: 30,511 102,106 109,1293 117,121 573,665 1054,2636 1357,1364
X(2815) = isogonal conjugate of X(2731)
X(2816) lies on these (parallel) lines: 29,102 30,511 40,151 109,1294 117,122 124,133 950,1845
X(2816) = isogonal conjugate of X(2732)
X(2817) lies on these (parallel) lines: 1,102 8,151 10,117 30,511 40,109 46,1795 65,1359 124,946
X(2817) = isogonal conjugate of X(2733)
X(2818) lies on these (parallel) lines: 1,1361 3,102 4,151 5,117 30,511 40,1745 56,945 65,389 296,2263 1482,1854
X(2818) = isogonal conjugate of X(2734)
X(2818) = crosspoint of X(4) and X(953)
X(2818) = crosssum of X(3) and X(952)
X(2819) lies on these (parallel) lines: 30,511 102,111 109,1296 117,126
X(2819) = isogonal conjugate of X(2735)
X(2820) lies on these (parallel) lines: 30,511 101,1292 103,105 118,120 165,1635 657,1734
X(2820) = isogonal conjugate of X(2736)
X(2820) = crosssum of X(55) and X(2254)
X(2820) = ideal point of PU(142)
X(2821) lies on these (parallel) lines: 3,1960 30,511 40,659 101,1293 103,106 118,121
X(2821) = isogonal conjugate of X(2737)
X(2822) lies on these (parallel) lines: 20,2939 27,103 30,511 101,1294 116,133 118,122 152,2947
X(2822) = isogonal conjugate of X(2738)
X(2823) lies on these (parallel) lines: 1,1461 30,511 33,57 40,1633 101,610 118,123 152,329 1721,2093
X(2823) = isogonal conjugate of X(2739)
X(2824) lies on these (parallel) lines: 30,511 101,1296 103,111 118,126
X(2824) = isogonal conjugate of X(2740)
X(2825) lies on these (parallel) lines: 30,511 58,103 101,1297 116,132 118,127 152,1330
X(2825) = isogonal conjugate of X(2741)
X(2826) lies on these (parallel) lines: 3,659 11,1111 30,511 100,1292 104,105 119,120 654,1768 1320,2403 1385,1960
X(2826) = isogonal conjugate of X(2742)
X(2826) = X(1156)-Ceva conjugate of X(11)
X(2826) = X(525)-of-Fuhrmann-triangle
X(2827) lies on these (parallel) lines: 11,1357 30,511 100,1293 104,106 119,121 1054,1768
X(2827) = isogonal conjugate of X(2743)
X(2827) = X(520)-of-Fuhrmann-triangle
X(2828) lies on these (parallel) lines: 11,133 28,104 30,511 100,1294 119,122 1715,1768
X(2828) = isogonal conjugate of X(2744)
X(2829) lies on these (parallel) lines: 1,1537 3,119 4,11 20,100 30,511 36,1532 40,1145 46,80 355,1158 944,1317 946,1387 962,1320 990,998 1012,1478 1155,1512 1319,1519 1455,1785 1862,1885
X(2829) = isogonal conjugate of X(2745)
X(2830) lies on these (parallel) lines: 30,511 100,1296 104,111 119,126
X(2830) = isogonal conjugate of X(2746)
X(2831) lies on these (parallel) lines: 11,132 30,511 100,1297 104,112 119,127 153,322 1754,1768
X(2831) = isogonal conjugate of X(2747)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2832) lies on these (parallel) lines: 30,511 105,106 120,121 659,764 1054,1635 1292,1293 1357,1358
X(2832) = isogonal conjugate of X(2748)
X(2833) lies on these (parallel) lines: 30,511 105,107 120,122 1292,1294
X(2833) = isogonal conjugate of X(2749)
X(2834) lies on these (parallel) lines: 25,105 30,511 120,123 269,1358 1292,1295 1723,2270
X(2834) = isogonal conjugate of X(2750)
X(2835) lies on these (parallel) lines: 1,1633 19,1743 30,511 31,57 102,1292 120,124 999,1486 1122,1358
X(2835) = isogonal conjugate of X(2751)
X(2836) lies on these (parallel) lines: 1,2930 6,1718 30,511 65,651 67,72 74,1292 81,105 120,125
X(2836) = isogonal conjugate of X(2752)
X(2836) = X(518)-of-orthocentroidal-triangle
X(2836) = X(518)-of-X(4)-Brocard-triangle
X(2837) lies on these (parallel) lines: 30,511 105,111 120,126 1292,1296
X(2837) = isogonal conjugate of X(2753)
X(2838) lies on these (parallel) lines: 28,105 30,511 120,127 1292,1297
X(2838) = isogonal conjugate of X(2754)
X(2839) lies on these (parallel) lines: 30,511 106,107 121,122 1054,1714 1293,1294
X(2839) = isogonal conjugate of X(2755)
X(2840) lies on these (parallel) lines: 30,511 34,106 121,123 1054,1722 1293,1295 1357,1359
X(2840) = isogonal conjugate of X(2756)
X(2841) lies on these (parallel) lines: 30,511 46,978 56,106 102,1293 121,124 1364,2098 1828,1845
X(2841) = isogonal conjugate of X(2757)
X(2842) lies on these (parallel) lines: 30,511 58,106 74,1293 121,125 1046,1054
X(2842) = isogonal conjugate of X(2758)
X(2842) = X(519)-of-orthocentroidal-triangle
X(2842) = X(519)-of-X(4)-Brocard-triangle
X(2842) = crossdifference of every pair of points on line X(6)X(4120)
X(2843) lies on these (parallel) lines: 30,511 106,111 121,126 1293,1296
X(2843) = isogonal conjugate of X(2759)
X(2844) lies on these (parallel) lines: 30,511 106,112 121,127 923,1042 1293,1297
X(2844) = isogonal conjugate of X(2760)
X(2845) lies on these (parallel) lines: 30,511 107,108 122,123 1294,1295
X(2845) = isogonal conjugate of X(2761)
X(2846) lies on these (parallel) lines: 30,511 102,1294 107,109 117,133 122,124
X(2846) = isogonal conjugate of X(2762)
X(2847) lies on these (parallel) lines: 30,511 107,111 122,126 1294,1296
X(2847) = isogonal conjugate of X(2763)
X(2848) lies on these (parallel) lines: 30,511 107,112 122,127 132,133 1294,1297
X(2848) = isogonal conjugate of X(2764)
X(2849) lies on these (parallel) lines: 30,511 102,1295 108,109 123,124 1359,1361
X(2849) = isogonal conjugate of X(2765)
X(2850) lies on these (parallel) lines: 30,511 74,1295 108,110 123,125 895,1814
X(2850) = isogonal conjugate of X(2766)
X(2850) = X(1290)-Ceva conjugate of X(3)
X(2850) = X(521) of orthocentroidal triangle
X(2850) = X(521) of X(4)-Brocard triangle
X(2851) lies on these (parallel) lines: 30,511 108,111 123,126 1295,1296
X(2851) = isogonal conjugate of X(2767)
X(2852) lies on these (parallel) lines: 30,511 102,1296 109,111 124,126
X(2852) = isogonal conjugate of X(2768)
X(2853) lies on these (parallel) lines: 30,511 102,1297 109,112 117,132 124,127
X(2853) = isogonal conjugate of X(2769)
X(2854) lies on these (parallel) lines: 6,110 30,511 67,69 74,1296 125,126 159,1177 182,1511 265,1352 373,597 399,1351 1112,1843
X(2854) = isogonal conjugate of X(2770)
X(2854) = X(524)-of-orthocentroidal-triangle
X(2854) = X(524)-of-X(4)-Brocard-triangle
X(2854) = X(524)-of-4th-anti-Brocard-triangle
SM Points
In the following list, the appearance of I, J, K means that RS(X(i),X(j)) = X(K):
74,99,477
98,99,842 98,110,2879 98,111,2880 98,112,2881
99,100,1290 99,101,2690
99,102,2695 99,103,2688
99,104,2687 99,105,2752
99,106,2758 99,107,1304
99,108,2766 99,109,2689
99,110,476 99,111,2770
99,112,935
The definitions of SR and SM points depend on two kinds of isoconjugates (i.e., isogonal and isotomic). To generalize, suppose
X = x : y : z, P = p : q : r, U = u : v : wTo conclude, SR(P,U) = S(X(1);P,U) and SM(P,U) = S(X(2);P,U).
X(2855) lies on the circumcircle and these lines: 74,325 877,1304 2407,2715
X(2855) = isogonal conjugate of X(2869)
X(2856) lies on the circumcircle and these lines: 81,2715 98,693 100,325 101,1959
X(2856) = isogonal conjugate of X(2870)
X(2857) lies on the circumcircle and these lines: 2,2715 98,850 110,325 112,297 183,476
X(2857) = isogonal conjugate of X(2871)
X(2857) = anticomplement of X(38975)
X(2857) = orthoptic-circle-of-Steiner-inellipse-inverse of X(36471)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2858) lies on the circumcircle and these lines: 111,325 323,729 691,2396
X(2858) = isogonal conjugate of X(2872)
X(2859) lies on the circumcircle and these lines: {100,3267}, {112,693}
X(2859) = isogonal conjugate of X(2873)
X(2860) lies on the circumcircle and this line: 101,693
X(2860) = isogonal conjugate of X(2874)
X(2861) lies on the circumcircle and these lines: 7,2720 101,908 104,693 264,1309
X(2861) = isogonal conjugate of X(2875)
X(2862) lies on the circumcircle and these lines: 2,919 63,813 105,693 108,1447
X(2862) = isogonal conjugate of X(2876)
X(2863) lies on the circumcircle and these lines: 75,901 106,693 813,1150
X(2863) = isogonal conjugate of X(2877)
X(2864) lies on the circumcircle and these lines: 100,850 101,1577 110,693
X(2864) = isogonal conjugate of X(2878)
X(2865) lies on the circumcircle and these lines: 883,2720 919,2397
X(2865) = isogonal conjugate of X(2879)
X(2866) lies on the circumcircle and these lines: 312,919 561,927
X(2866) = isogonal conjugate of X(2880)
X(2867) lies on the circumcircle and these lines: 107,850 112,525 1297,1503
X(2867) = isogonal conjugate of X(2881)
X(2867) = anticomplement of X(33504)
X(2867) = intersection of antipedal lines of X(112) and X(1297)
X(2868) lies on the circumcircle and these lines: 69,805 76,691 110,3266 111,850 112,385 669,2373
X(2868) = isogonal conjugate of X(2882)
X(2869) lies on these (parallel) lines: 6,2510 30,511 351,878
X(2869) = isogonal conjugate of X(2855)
X(2870) lies on these lines: 30,511 37,692
X(2870) = isogonal conjugate of X(2856)
X(2871) lies on these (parallel) lines: 6,157 30,511 53,1843 263,1989 287,1632
X(2871) = isogonal conjugate of X(2857)
X(2871) = complementary conjugate of X(38975)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2872) lies on these lines: 30,511 1976,2422
X(2872) = isogonal conjugate of X(2858)
X(2873) lies on this line: 30,511
X(2873) = isogonal conjugate of X(2859)
X(2874) lies on this line: 30,511
X(2874) = isogonal conjugate of X(2860)
X(2875) lies on these lines: 30,511 55,184
X(2875) = isogonal conjugate of X(2861)
X(2876) lies on these (parallel) lines: 6,692 19,1843 30,511 1469,2263 1633,1814
X(2876) = isogonal conjugate of X(2862)
X(2877) lies on these lines: 30,511 31,692
X(2877) = isogonal conjugate of X(2863)
X(2878) lies on these lines: 30,511 163,692
X(2878) = isogonal conjugate of X(2864)
X(2879) lies on these lines: 30,511 884,2423
X(2879) = isogonal conjugate of X(2865)
X(2880) lies on these lines: 30,511 560,604
X(2880) = isogonal conjugate of X(2866)
X(2881) lies on these (parallel) lines: 30,511 32,2508 112,1576 154,1636 684,1297 2501,2506
X(2881) = isogonal conjugate of X(2867)
X(2881) = orthic-isogonal conjugate of X(33504)
X(2882) lies on these (parallel) lines: 25,694 30,511 32,1084 315,670
X(2882) = isogonal conjugate of X(2868)
Isogonal Conjugates wrt Special Triangles
Suppose A'B'C' is the medial triangle of ABC and X is a triangle center wrt ABC. Let X' be the isogonal conjugate of X wrt A'B'C'. Then X' is a triangle center wrt ABC. In order to determine its center function g, the method of TCCT, p. 207 applies. Write X = x : y : z (wrt ABC). Then trilinears of X wrt A'B'C' are
(- ax + by + cz)/a : (ax - by + cz)/b : (ax + by - cz)/c,
so that the isogonal conjugate of X wrt A'B'C' is
a/(- ax + by + cz) : b/(ax - by + cz) : c/(ax + by - cz),
which, in trilinears wrt ABC, is g(a,b,c,x,y,z) : g(b,c,a,y,z,x) :
g(c,a,b,z,x,y), where
g(a,b,c,x,y,z) = bc(ax - by - cz)[b2(ax + by - cz) + c2(ax - by + cz)]
This point is the isogonal conjugate of X wrt the medial triangle, or, more simply, the medial isogonal conjugate of X. In like manner, other conjugates of X wrt other triangles are defined. In this section, only isogonal conjugates are considered, and trilinears wrt to selected triangles are given by first trilinear wrt ABC, as follows:
MEDIAL ISOGONAL CONJUGATE OF X = x : y : z
bc(ax - by - cz)[b2(ax + by - cz) + c2(ax - by + cz)]
ANTICOMPLEMENTARY ISOGONAL CONJUGATE OF X = x : y : z
bc[- a2/(by + cz) + b2/(cz + ax) + c2/(ax + by)]
ORTHIC-ISOGONAL CONJUGATE OF X = x : y : z
x(x cos A - y cos B - z cos C)
TANGENTIAL-ISOGONAL CONJUGATE OF X = x : y : z
a[- a3/(cy + bz) + b3/(az + cx) + c3/(bx + ay)]
EXCENTRAL-ISOGONAL CONJUGATE OF X = x : y : z
- a/[(b + c - a)(y + z)] + b/[(c + a - b)(z + x)] + c/[(a + b - c)(x + y)]
In each of the above cases, x : y : z, as a triangle center, is given by trilinears h(a,b,c) : h(b,c,a) : h(c,a,b) for some function h. Accordingly, each of the first trilinears given above defines a triangle center f(a,b,c) : f(b,c,a) : f(c,a,b).
The term "complementary conjugate" already in use at the time of this writing (11/25/04) is a synonym for "medial isogonal conjugate", as is "anticomplementary conjugate" for "anticomplementary isogonal conjugate"; accordingly, the earlier terms will be used in the sequel. Also, "excentral isogonal conjugate" is "X(188)-aleph conjugate" and "orthic isogonal conjugate" is "X(4)-Ceva conjugate"; in these cases, the new terminology will be used.
The subject of isogonal conjugation with respect to a central triangle continues at the preamble just before X(6000).
In the plane of a triangle ABC, let
H = X(4);
L = X(20);
Ab = AL∩BH;
Ac = AL∩CH;
U = line through Ab parallel to AC;
V = line through Ac parallel to AB;
A1 = U∩V;
A2 = U∩BC;
A3 = V∩BC;
O = circle {{A1, A2, A3}};
A' = pole of BC wrt O, and define B' and C' cyclically.
Then A'B'C' = reflection of ABC in X(2883). (Angel Montesdeoca, July 30, 2021)
X(2883) lies on these lines: 2,64 3,1661 4,6 20,154 30,156 51,1906 184,1885 185,235 221,497 388,2192 389,1596 417,1624 459,1192 468,1204 496,942 550,1511 1593,1619
X(2883) = midpoint of X(4) and X(1498)
X(2883) = reflection of X(3357) in X(140)
X(2883) = complement of X(64)
X(2883) = complementary conjugate of X(4)
X(2883) = X(2)-Ceva conjugate of X(800)
X(2883) = X(12513)-of-orthic-triangle if ABC is acute
X(2883) = perspector of circumconic centered at X(800)
X(2883) = center of circumconic that is locus of trilinear poles of lines passing through X(800)
X(2884) lies on these lines: 142,1538 857,948
X(2884) = complementary conjugate of X(7)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2885) lies on these lines: 2,1222 5,121 10,496 45,1213
X(2885) = complement of X(3445)
X(2885) = complementary conjugate of X(8)
X(2886) is the center of the inellipse that is the isotomic conjugate of the isogonal conjugate of the incircle. (Randy Hutson, October 15, 2018)
X(2886) lies on these lines: 1,442 2,11 4,958 5,10 8,12 9,1699 30,993 56,377 63,1836 75,325 115,1573 118,124 141,674 210,908 214,1484 226,518 333,2651 348,2898 405,1479 427,1824 474,499 485,1378 486,1377 495,519 496,1125 529,956 982,1086 1348,1679 1349,1678 1506,1574 1676,1681 1677,1680 1698,1706 2009,2014 2010,2013 2566,2569 2567,2568
X(2886) = midpoint of X(i) and X(j) for these (i,j): (1,3416), (4,3428), (8,2099), (63,1836), (956,1478)
X(2886) = isogonal conjugate of X(3449)
X(2886) = isotomic conjugate of isogonal conjugate of X(21746)
X(2886) = polar conjugate of isogonal conjugate of X(22070)
X(2886) = complement of X(55)
X(2886) = complementary conjugate of X(9)
X(2886) = crosssum of X(6) and X(2175)
X(2887) lies on these lines: 1,977 2,31 10,12 37,744 39,734 120,124 141,674 306,740 334,1581 626,766 1111,1233 1329,2390 2205,2240
X(2887) = isogonal conjugate of X(38813)
X(2887) = isotomic conjugate of isogonal conjugate of X(3778)
X(2887) = complement of X(31)
X(2887) = complementary conjugate of X(37)
X(2887) = crosspoint of X(2) and X(561)
X(2887) = crosssum of X(6) and X(560)
X(2887) = anticomplementary conjugate of anticomplement of X(38831)
X(2887) = crossdifference of every pair of points on line X(3250)X(7252)
X(2887) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 38813}, {32, 38810}, {42, 7305}, {58, 983}, {284, 7132}, {692, 7255}, {788, 33514}, {1333, 17743}, {2206, 7033}
X(2887) = trilinear product X(i)*X(j) for these {i,j}: {2, 3721}, {6, 20234}, {9, 16888}, {10, 982}, {37, 3662}, {42, 33930}, {57, 4136}, {65, 3705}, {75, 3778}, {76, 16584}, {81, 16886}, {85, 20684}, {86, 7237}, {92, 20727}, {226, 3061}, {313, 7032}, {321, 2275}, {349, 20665}, {514, 7239}, {1441, 3056}, {4531, 6063}
X(2887) = trilinear quotient X(i)/X(j) for these (i,j): (1, 38813), (10, 983), (76, 38810), (86, 7305), (226, 7132), (313, 7033), (321, 17743), (693, 7255), (789, 33514), (982, 58), (2275, 1333), (3056, 2194), (3061, 284), (3662, 81), (3705, 21), (3721, 6), (3778, 31), (4136, 9), (4531, 2175), (7032, 2206), (7237, 42), (7239, 101), (16584, 32), (16886, 37), (16888, 57), (20234, 2), (20684, 41), (20727, 48), (33930, 86)
X(2887) = barycentric product X(i)*X(j) for these {i,j}: {1, 20234}, {7, 4136}, {8, 16888}, {10, 3662}, {37, 33930}, {75, 3721}, {76, 3778}, {86, 16886}, {226, 3705}, {264, 20727}, {274, 7237}, {313, 2275}, {321, 982}, {349, 3056}, {561, 16584}, {693, 7239}, {1441, 3061}, {4531, 20567}, {6063, 20684}, {7032, 27801}
X(2887) = barycentric quotient X(i)/X(j) for these (i,j): (6, 38813), (10, 17743), (37, 983), (65, 7132), (75, 38810), (81, 7305), (321, 7033), (514, 7255), (982, 81), (2275, 58), (3056, 284), (3061, 21), (3662, 86), (3705, 333), (3721, 1), (3778, 6), (4136, 8), (4531, 41), (7032, 1333), (7237, 37), (7239, 100), (16584, 31), (16886, 10), (16888, 7), (20234, 75), (20665, 2194), (20684, 55), (20727, 3), (27801, 7034), (33930, 274)
X(2888) lies on these lines: 2,54 4,93 5,195 69,1225 128,252 193,576 323,1594 343,1601 2904,2914
X(2888) = reflection of X(i) in X(j) for these (i,j): (54,1209), (195,5)
X(2888) = isogonal conjugate of X(3432)
X(2888) = anticomplement of X(54)
X(2888) = anticomplementary conjugate of X(3)
X(2888) = X(311)-Ceva conjuagte of X(2)
X(2888) = cevapoint of X(195) and X(2917)
X(2888) = crosspoint of X(195) and X(2917) wrt both the excentral and tangential triangles
X(2889) lies on these lines: 2,1173 182,193
X(2889) = anticomplement of X(1173)
X(2889) = anticomplementary conjugate of X(5))
X(2889) = X(1232)-Ceva conjugate of X(2)
X(2890) lies on this line: 2,1174
X(2890) = anticomplement of X(1174)
X(2890) = anticomplementary conjugate of X(9)
X(2890) = X(1233)-Ceva conjugate of X(2)
X(2891) lies on these lines: 1,1654 2,1126 1051,1125
X(2891) = anticomplement of X(1126)
X(2891) = anticomplementary conjugate of X(10)
X(2891) = X(1269)-Ceva conjugate of X(2)
X(2892) lies on these lines: 2,1177 4,67 20,1632 66,193 1352,2777 1503,2930
X(2892) = anticomplement of X(1177)
X(2892) = anticomplementary conjugate of X(23)
X(2892) = anticomplementary-circle-inverse of X(34163)
X(2892) = X(1236)-Ceva conjugate of X(2)
X(2893) lies on these lines: 2,272 4,69 7,2475 8,2897 9,1654 72,319 75,150 77,2000 86,442 226,1943 322,355 325,332 329,2895 333,440 524,1901 857,2287
X(2893) = anticomplement of X(284)
X(2893) = anticomplementary conjugate of X(63)
X(2893) = X(349)-Ceva conjugate of X(2)
X(2894) lies on these lines: 2,943 4,144 21,149 75,2897 145,377 673,2481
X(2894) = anticomplement of X(943)
X(2894) = anticomplementary conjugate of X(72)
X(2895) lies on these lines: 2,6 8,79 144,2897 314,1230 319,321 329,2893 445,2322 451,2906
X(2895) = reflection of X(81) in X(1211)
X(2895) = isogonal conjugate of X(3444)
X(2895) = isotomic conjugate of 1029
X(2895) = anticomplement of X(81)
X(2895) = anticomplementary conjugate of X(75)
X(2895) = X(i)-Ceva conjugate of X(j) for these (i,j): (319,8), (321,2)
X(2895) = X(1030-cross conjugate of X(451)
X(2895) = {X(1654),X(17778)}-harmonic conjugate of X(2)
Let A'B'C' be the 1st Brocard triangle and A"B"C" the 1st Brocard-reflected triangle. Let A* be the isogonal conjugate of A", and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(2896). (Randy Hutson, June 7, 2019)
Let OA be the circle centered at the A-vertex of the 2nd Neuberg triangle and passing through A; define OB and OC cyclically. X(2896) is the radical center of OA, OB, OC. (Randy Hutson, August 28, 2020)
X(2896) lies on these lines: 2,32 3,147 15,628 16,627 20,1352 69,194 76,148 141,384 599,1975
X(2896) = isogonal conjugate of X(14370)
X(2896) = isotomic conjugate of 1031
X(2896) = anticomplement of X(83)
X(2896) = anticomplementary conjugate of X(76)
X(2896) = X(i)-Ceva conjugate of X(j) for these (i,j): (141,2), (384,194)
X(2896) = anticomplementary isotomic conjugate of X(6)
X(2896) = radical center of reflected Neuberg circles
X(2896) = homothetic center of anticomplementary triangle and 5th Brocard triangle
X(2896) = perspector of 1st and 5th Brocard triangles
X(2896) = X(83) of 5th Brocard triangle
X(2896) = X(83)-of-6th-Brocard-triangle
X(2896) = center of inverse similitude of 5th and 6th Brocard triangles
X(2896) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 5th Brocard triangle
X(2896) = homothetic center of 5th Brocard triangle and cross-triangle of ABC and 5th Brocard triangle
X(2897) lies on these lines: 2,1172 8,2893 20,64 75,2894 144,2895 145,347 153,322 857,2322 1441,2475 1444,2071
X(2897) = anticomplement of X(1172)
X(2897) = anticomplementary conjugate of X(92)
X(2897) = X(1231)-Ceva conjugate of X(2)
X(2898) lies on these lines: 7,354 55,1996 273,1851 348,2886 857,948
X(2898) = X(4)-Ceva conjugate of X(7)
X(2899) lies on these lines: 2,988 8,210 12,344 45,1213 56,1997 190,1788 318,1863 345,1329
X(2899) = X(4)-Ceva conjugate of X(8)
X(2900) lies on these lines: 1,442 9,55 57,1004 78,950 100,1708 517,1490 528,1750
X(2900) = X(4)-Ceva conjugate of X(9)
X(2901) lies on these lines: 1,321 10,37 42,1089 58,1999 72,519 312,386 346,387 354,596 430,1867 536,942 1018,2198
X(2901) = X(4)-Ceva conjugate of X(10)
X(2901) = X(1)-of-inverse(n(Medial)*n(Incentral))-triangle
X(2902) lies on these lines: 5,13 15,1154 16,184 52,2913 571,2903
X(2902) = X(4)-Ceva conjugate of X(15)
X(2903) lies on these lines: 5,14 15,184 16,1154 52,2912 571,2902
X(2903) = X(4)-Ceva conjugate of X(16)
Let O be the circumcircle of a triangle ABC. Let
A' = perpendicular bisector of segment OA, and define B' and C' cyclically
A'' = B'∩C', and define B'' and C'' cyclically
A* = reflection of A* in BC, and define B* and C* cyclically.
Then A*B*C* is perspective to the orthic triangle of ABC, and the perspector is X(2904). (Angel Montesdeoca, November 12, 2016).
Let P be an arbitrary point of the circumcircle , O, and let S(P) be the Simson-Wallace line of P, and P' the reflection of P in S(P). The locus of P' as P ranges through O is a bicircular circumquartic that meets the sidelines of ABC in 6 points other than A, B, C. The 6 points lie on a conic, here called the Montesdeoca conic, M. The center of M is X(2904), and the perspector of M is X(14111). Let u = SA, v = SB, w = SC, f(a,b,c) = u2(S2 - v2)(S2 - w2), g(a,b,c) = 2vw(u2 - S2)(S2 + vw). A barycentric equation for M is
f(a,b,c)x2 + f(b,c,a)y2 + (f,c,a,b)z2 + g(a,b,c)yz + g(b,c,a)zx + g(c,b,a)xy = 0.
X(2904) lies on these lines: 3,1986 4,1994 6,70 24,52 25,195 155,403 156,1112 185,378 576,1843 648,847 2888,2914
X(2904) = X(4)-Ceva conjugate of X(24)
X(2904) = X(90)-of-orthic-triangle if ABC is acute
X(2905) lies on these lines: 6,469 19,423 27,86 29,242 238,270 429,2907 648,1826
X(2905) = X(4)-Ceva conjugate of X(27)
X(2906) lies on these lines: 1,2074 4,1029 28,60 79,1172 163,1782 193,406 451,2895 860,2907 1100,2189
X(2906) = X(4)-Ceva conjugate of X(28)
X(2906) = crosspoints of X(4) and X(451)
X(2907) lies on these lines: 1,415 4,2651 27,239 29,270 185,412 225,648 243,1858 429,2905 860,2906 1896,2648
X(2907) = X(4)-Ceva conjugate of X(29)
X(2908) lies on these lines: 37,48 41,2092 604,1015 1880,1973 2174,2276
X(2908) = X(4)-Ceva conjugate of X(31)
X(2909) lies on these lines:5,182 39,184 76,2001 211,571 512,1968 1692,1974
X(2909) = X(4)-Ceva conjugate of X(32)
X(2909) = isogonal conjugate of isotomic conjugate of X(157)
X(2909) = polar conjugate of isotomic conjugate of X(22391)
X(2910) lies on these lines: 1,1864 9,1181 40,198 101,1753
X(2910) = X(4)-Ceva conjugate of X(40)
X(2911) lies on these lines: 1,6 3,2174 25,209 31,2318 41,71 48,672 55,584 56,583 101,579 193,1332 198,2245 393,1783 571,906 607,1826 692,1974 1333,1801 2262,2348 2287,2345
X(2911) = isogonal conjugate of X(15474)
X(2911) = polar conjugate of isotomic conjugate of X(11517)
X(2911) = X(63)-isoconjugate of X(39267)
X(2911) = X(4)-Ceva conjugate of X(55)
X(2911) = crosspoint of X(1252) and X(1783)
X(2911) = crosssum of X(905) and X(1086)
X(2911) = crossdifference of every pair of points on line X(513)X(5570)
X(2912) lies on these lines: 5,16 17,2004 52,2903 61,143 62,184 571,2913
X(2912) = X(4)-Ceva conjugate of X(61)
X(2913) lies on these lines: 5,15 18,2005 52,2902 61,184 62,143 571,2912
X(2913) = X(4)-Ceva conjugate of X(62)
X(2914) lies on these lines: 4,195 52,110 54,74 113,539 125,1199 186,323 265,1994 2888,2904
X(2914) = X(4)-Ceva conjugate of X(186)
X(2914) = X(3065)-of-orthic-triangle if ABC is acute
As a point on the Euler line, X(2915) has Shinagawa coefficients ($a$*(E + 2F) + abc, -2$a$*(E + F) - abc).
X(2915) lies on these lines: 2,3 35,37 36,1104 40,2778 500,1790 511,1437 1603,1604 1780,2245
X(2915) = midpoint of X(1717) and X(2960)
X(2915) = isogonal conjugate of cyclocevian conjugate of X(35058)
X(2915) = X(321)-Ceva conjugate of X(6)
X(2915) = crosspoint of X(100) and X(250)
X(2915) = crosssum of X(125) and X(513)
X(2915) = isogonal conjugate of isotomic conjugate of X(21287)
X(2915) = polar conjugate of isotomic conjugate of X(23130)
X(2916) lies on these lines: 6,22 69,2930 155,1350 159,599 182,2937 195,511 376,2935 1352,2918 1503,2917
X(2916) = reflection of X(6) in X(1176)
X(2916) = X(141)-Ceva conjugate of X(6)
X(2916) = isogonal conjugate of isotomic conjugate of X(1369)
X(2916) = isogonal conjugate of cyclocevian conjugate of X(76)
X(2916) = polar conjugate of isotomic conjugate of X(23133)
X(2917) lies on these lines: 3,161 6,24 20,2935 22,1498 26,154 49,52 68,1658 159,2934 186,2929 343,1601 399,2937 1503,2916
X(2917) = complement of X(32346)
X(2917) = anticomplement of X(32351)
X(2917) = X(i)-Ceva conjugate of X(j) for these (i,j): (343,6), (2888,195)
X(2917) = X(21)-of-tangential-triangle if ABC is acute
X(2918) lies on these lines: 6,26 22,155 184,195 550,2935 1351,2916 1658,2929
X(2919) lies on this line: 22,1615
X(2920) lies on this line: 22,1616
X(2921) lies on these lines: 3,1603 22,347 36,1104
X(2922) lies on this line: 22,595
X(2923) lies on this line: 15,2070
X(2924) lies on this line: 16,2070
X(2925) lies on these lines: 3,13 22,1605 99,1606
X(2925) = circumcircle-inverse of X(33500)
X(2926) lies on these lines: 3,14 22,1606 99,1605
X(2926) = circumcircle-inverse of X(33498)
X(2927) lies on this line: 15,2937
X(2928) lies on this line: 16,2937
X(2929) lies on these lines: 4,2935 22,1620 24,1192 186,2917 195,389 1658,2918
X(2929) = complement of X(22555)
X(2930) lies on these lines: 1,2836 3,67 6,110 23,524 49,575 69,2916 154,1177 195,576 399,511 518,2948 1498,2781 1503,2892 2453,2782
X(2930) = reflection of X(6) in X(110)
X(2930) = complement of X(32255)
X(2930) = circumcircle-inverse of X(2482)
X(2930) = X(i)-Ceva conjugate of X(j) for these (i,j): (23,3), (524,6)
X(2930) = inverse-in-MacBeath-circumconic of X(6593)
X(2930) = {X(110),X(895)}-harmonic conjugate of X(6593)
X(2931) lies on the tangential circle, the Walsmith rectangular hyperbola, and these lines: 3,125 6,1511 22,74 23,146 24,110 25,113 26,1498 30,2935 68,1658 110,895 159,542 186,3580 195,568 399,1495 912,2948
X(2931) = reflection of X(155) in X(110)
X(2931) = reflection of X(32123) in X(468)
X(2931) = antipode of X(32123) in Walsmith rectangular hyperbola
X(2931) = X(186)-Ceva conjugate of X(3)
X(2931) = inverse-in-circumcircle of X(131)
X(2931) = X(104)-of-tangential-triangle if ABC is acute
X(2932) lies on these lines: 3,8 9,1030 11,474 35,392 55,215 56,2802 72,1768 78,2771 80,1376 119,1012 149,404 480,2801 999,1320 1317,1470 2077,2950
X(2932) = reflection of X(149) in X(496)
X(2932) = crosssum of X(124) and X(513)
X(2932) = inverse-in-circumcircle of X(1145)
X(2933) lies on these lines: 3,10 22,1603 45,198 100,1610 157,1631
X(2933) = isogonal conjugate of isotomic conjugate of X(21286)
X(2933) = polar conjugate of isotomic conjugate of X(23129)
X(2933) = X(312)-Ceva conjugate of X(6)
X(2934) lies on these lines: 22,1225 26,157 159,2917
X(2934) = X(311)-Ceva conjugate of X(6)
X(2935) lies on these lines: 1,2778 3,113 4,2929 6,74 20,2917 30,2931 64,155 110,1498 125,1593 146,2071 154,1511 186,1514 265,1853 376,2916 550,2918 1294,1632 1503,2892
X(2935) = reflection of X(1498) in X(110)
X(2935) = inverse-in-circumcircle of X(3184)
X(2935) = X(i)-Ceva conjugate of X(j) for these (i,j): (146,399), (2071,3)
X(2935) = tangential-triangle symgonal of X(3)
X(2936) lies on these lines: 3,67 22,99 25,543 147,2071 671,1995
As a point on the Euler line, X(2937) has Shinagawa coefficients (27E3 + 144E2F + 240EF2 + 128F3 - 48(3E + 8F)S2, -63E3 - 240E2F - 304EF2 - 128F3 + 48(7E + 8F)S2).
X(2937) lies on these lines: 2,3 15,2927 16,2928 49,511 182,2916 184,195 399,2917 1154,1614 1216,1495
X(2937) = crosspoint of X(250) and X(930)
X(2937) = crosssum of X(125) and X(1510)
X(2937) = circumcircle-inverse of X(37938)
X(2937) = X(35)-of-tangential-triangle if ABC is acute
X(2937) = {X(3),X(26)}-harmonic conjugate of X(2070)
X(2938) lies on these lines: 1,1014 40,511 165,846 170,2939 191,2951 1762,2942 2772,2948
X(2938) = X(210)-Ceva conjugate of X(1)
X(2938) = X(i)-aleph conjugate of X(j) for these (i,j): (8,1764), (9,3), (10,1762), (37,1046), (188,2, (366,579)
X(2938) = X(95)-of-excentral-triangle
X(2938) = excentral isotomic conjugate of X(3)
X(2938) = excentral polar conjugate of X(6)
X(2939) lies on these lines: 1,19 3,1762 20,2822 35,228 40,2947 43,46 72,1761 165,191 170,2938 2779,2948
X(2939) = reflection of X(1) in X(2360)
X(2939) = X(72)-Ceva conjugate of X(1)
X(2939) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1724), (2,1730), (10,1710), (72,2939), (188,4), (366,579)
X(2939) = X(96)-of-excentral-triangle
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 A" be the intersection, other than A, of the circumcircle and Oa; define B" and C" cyclically. The triangle A"B"C" is perspective to the excentral triangle at X(2940). (Randy Hutson, April 9, 2016)
X(2940) lies on these lines: 35,37 40,2948 191,210 484,1046
X(2940) = X(i)-aleph conjugate of X(j) for these (i,j): (100,110), (188,5), (188,6), (366,583)
X(2941) lies on these lines: 1,1412 30,40 165,846 1764,1768 1779,2955
X(2941) = X(2321)-Ceva conjugate of X(1)
X(2941) = X(i)-aleph conjugate of X(j) for these (i,j): (8,573), (10,1781), (188,6), (2321,2941)
X(2941) = X(97)-of-excentral-triangle
X(2942) lies on these lines: 57,1742 63,2951 165,169 1762,2938
X(2942) = X(i)-aleph conjugate of X(j) for these (i,j): (9,55), (188,7)
X(2942) = excentral isotomic conjugate of X(55)
X(2943) lies on these lines: 1,1106 40,978 517,1046 1697,1742 1743,1766
X(2943) = X(i)-aleph conjugate of X(j) for these (i,j): (1,56), (188,8)
X(2943) = X(1105) of excentral triangle
X(2944) lies on these lines: 2,572 3,846 40,43 72,1282 165,411 517,1051 1046,1764
X(2944) = X(960)-Ceva conjugate of X(1)
X(2944) = X(i)-aleph conjugate of X(j) for these (i,j): (188,10),
(366,1400), (960,2944)
X(2945) lies on these lines: 16,846 40,2953 61,1046 573,1761
X(2945) = X(188)-alpha conjugate of X(15)
X(2946) lies on these lines: 15,846 40,2952 62,1046 573,1761
X(2946) = X(188)-alpha conjugate of X(16)
X(2947) lies on these lines: 1,4 9,2954 40,2939 43,1754 71,165 152,2822 846,1709 971,1214 1282,1763
X(2947) = X(219)-Ceva conjugate of X(1)
X(2947) = crosssum of X(513) and X(2638)
X(2947 = X(i)-alpha conjugate of X(j) for these (i,j): (1,1723), (2,1729), (9,1709), (188,19), (365,1722), (366,1708)
X(2947) = excentral isotomic conjugate of X(1709)
X(2948) lies on the Bevan circle and these lines: 1,60 3,191 6,1718 40,2940 74,165 80,2607 113,1699 125,1698 146,516 399,517 518,2930 758,1325 912,2931 1046,1054 1282,2774 1498,2778 2100,2574 2101,2575 2772,2938 2779,2939
X(2948) = reflection of X(1) in X(110)
X(2948) = X(i)-Ceva conjugate of X(j) for these (i,j): (758,1),
(1325,3)
X(2948) = X(i)-alpha conjugate of X(j) for these (i,j): (366,2245),
(758,2948)
X(2949) lies on these lines: 1,201 4,191 5,9 40,1726 63,411 1158,2951
X(2949) = X(i)-aleph conjugate of X(j) for these (i,j): (2,226), (188,75)
X(2950) lies on these lines: 1,104 9,119 40,1145 57,1537 80,1709 84,952 100,1490 484,1512 2077,2932
X(2950) = reflection of X(i) in X(j) for these (i,j): (1490,100),
(1768,1158)
X(2950) = X(i)-aleph conjugate of X(j) for these (i,j): (8,515),
(188,36)
X(2951) lies on these lines: 1,7 9,165 40,971 63,2942 142,1699 144,200 166,167 191,2938 518,2136 610,1633 1050,1740 1158,2949 1292,2371
X(2951) = X(i)-Ceva conjugate of X(j) for these (i,j): (144,9), (200,1)
X(2951) = isogonal conjugate of X(8917)
X(2951) = X(69)-of-excentral-triangle
X(2951) = excentral isotomic conjugate of X(1)
X(2951) = perspector of the excentral inconic with center X(40)
X(2951) = antipedal isotomic conjugate of X(1)
X(2951) = X(7)-of-3rd-antipedal-triangle-of-X(1)
X(2951) = ABC-to-excentral barycentric image of X(7)
X(2951) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1445), (9,1), (55,1740), (100,651), (188,57), (200,2951), (259,978),
(281,920), (312,1760), (318,1748), (643,662), (644,100), (650,1052), (664,658), (1897,653), (2287,411), (2322,412)
X(2952) lies on these lines: 15,2959 40,2946 191,1276 1761,2321
X(2952) = X(188)-aleph conjugate of X(61)
X(2953) lies on these lines: 16,2959 40,2945 191,1277 1761,2321
X(2953) = X(188)-aleph conjugate of X(62)
X(2954) lies on these lines: 1,201 3,1762 9,2947 19,165 43,1723 170,1709 228,1282 846,1744
X(2954) = X(i)-aleph conjugate of X(j) for these (i,j): (188,71), (366,226)
X(2955) lies on these lines: 1,951 4,1781 35,1718 40,1723 165,1722 1779,2941
X(2955) = X(950)-Ceva conjugate of X(1)
X(2955) = X(i)-aleph conjugate of X(j) for these (i,j): (174,1427),
(188,72), (950,2955)
X(2956) lies on these lines: 1,84 46,1743 109,1490 165,1745 223,1158 266,505 651,1103 978,1044 1046,1719 1249,1838 1750,1771
X(2956) = reflection of X(1) in X(1394)
X(2956) = X(40)-Ceva conjugate of X(1)
X(2956) = X(i)-aleph conjugate of X(j) for these (i,j): (40,2956),
(188,84), (366,2270), (651,1461)
X(2957) lies on these lines: 1,59 5,2607 36,1736 238,517 484,516 655,1090
X(2957) = X(11)-Ceva conjugate of X(1)
X(2957) = X(i)-aleph conjugate of X(j) for these (i,j): (11,2957),
(174,651), (188,100), (366,101), (513,978), (514,57), (522,40),
(555,658), (556,190), (650,1742)
X(2958) lies on these lines: 1,1262 484,516 514,1768
X(2958) = X(1146)-Ceva conjugate of X(1)
X(2958) = X(188)-aleph conjugate of X(101)
Barycentrics af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2959) lies on these lines: 1,1326 15,2952 16,2953 19,1247 40,511 238,2640 612,846 1761,2076
X(2959) = reflection of X(1) in X(1326)
X(2959) = X(188)-aleph conjugate of X(147)
X(2960) lies on these lines: 1,1719 30,40 35,37 46,1834 429,1698 1498,2779
X(2960) = reflection of X(1717) in X(2915)
X(2960) = X(188)-aleph conjugate of X(155)
X(2961) lies on these lines: 1,1037 4,1773 40,238 46,516 57,1721 672,1723 990,1717 1253,1718 1699,1781
X(2961) = X(497)-Ceva conjugate of X(1)
X(2961) = X(i)-aleph conjugate of X(j) for these (i,j): (174,269),
(188,200), (497,2961)
X(2962) lies on these lines: 19,2290 37,2963 82,1733 93,225 759,930 1087,1749 2216,2964
X(2962) = isogonal conjugate of X(2964)
X(2962) = X(2)-isoconjugate of X(2965)
Let Q be the pedal curve (a limaçon of Pascal) of the circle with center X(7728) and radius |OH|, with respect to X(265). Let A'B'C' be the triangle formed by the tangents at A,B,C to Q. Then the triangles ABC and A'B'C' are perspective, and their perspector is X(2963). Moreover, X(21975) is the only finite fixed point of the affine transformation that maps a triangle ABC onto A'B'C'. See X(21975). (Angel Montesdeoca, September 10, 2019)
In the plane of a triangle ABC, let
N= Nine point circle,
N'a, A' = reflection of N, A in BC, define N'b, B' and N'c, C' cyclically
Pa = polar of A' with respect to N'a, and define Pb and Pc cyclically;
A" = Pc∩Pb, and define B" and C" cyclically.
The triangle A"B"C" is perspective to ABC, and the perspector is X(2963).
(Dasari Naga Vijay Krishna, June 8, 2021)
Generalization of Vijay's construction of X(2963). Let
C(U) = the circle with powers u, v, w with respect to the vertices A, B, C, respectively;
C'(U) = reflection of C(U) in side BC;
A' = reflection of A in in side BC;
Pa = polar of A in C'(U), and define Pb and Pc cyclically;
A" = Pc∩Pb, and define B" and C" cyclically.
The triangle A"B"C" is perspective to ABC, and the perspector is
a^2 / (a^2*(a^2*b^2 - b^4 + a^2*c^2 + b^2*c^2 - c^4) + a^2*(b^2 + c^2 - u)*u - (a^4 + a^2*b^2 - b^4 + 2*b^2*c^2 - c^4)*v + (a^2 + b^2 - c^2)*v^2 - (a^4 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4)*w + (a^2 - b^2 + c^2)*w^2 - a^2*(u*v + u*w - v*w)) : :
(Peter Moses, June 11, 2021)
X(2963) = isogonal conjugate of X(1994)
X(2963) = cevapoint of X(590) and X(615)
X(2963) = X(570)-cross conjugate of X(6)
X(2963) = crosssum of X(i) and X(j) for these (i,j): (6,195), (49,2965)
X(2963) = isotomic conjugate of X(7769)
X(2963) = {X(17),X(18)}-harmonic conjugate of X(3519)
X(2963) = X(2)-isoconjugate of X(2964)
X(2963) = X(92)-isoconjugate of X(49)
X(2963) = polar conjugate of X(32002)
X(2963) = perspector of ABC and unary cofactor triangle of Kosnita triangle
X(2963) = barycentric product of nine-point centers of 1st and 2nd Ehrmann inscribed triangles
Let A'B'C' be the Kosnita triangle. Let A" be the trilinear product of the circumcircle intercepts of line B'C'. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(2964). (Randy Hutson, July 31 2018)
X(2964) lies on these lines: 1,21 35,2361 36,1399 109,1393 162,2166 163,2179 484,580 2148,2180 2216,2962
X(2964) = isogonal conjugate of X(2962)
X(2964) = X(2216)-Ceva conjugate of X(1)
X(2964) = X(2)-isoconjugate of X(2963)
Let A'B'C' be the Kosnita triangle. Let A" be the barycentric product of the circumcircle intercepts of line B'C'. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(2965). (Randy Hutson, July 31 2018)
X(2965) lies on these lines: 3,6 5,2963 53,112 230,1627 524,1238
X(2965) = X(i)-Ceva conjugate of X(j) for these (i,j): (1173,184), (1994,49)
X(2965) = crosssum of X(6) and X(2937)
X(2965) = X(2)-isoconjugate of X(2962)
Let P(2) and U(2) be the 1st and 2nd Beltrami points (as indexed at Bicentric Pairs, accessible using the Tables button at the top of ETC), and let P(40) and U(40) be the isogonal conjugates of P(2) and U(2), respectively. Then X(2966) is the point of intersection of the lines P(2)U(40) and P(40)U(2). (Peter Moses, July 1, 2009). See also X(3568).
X(2966) lies on the Steiner circumellipse, the hyperbola {{A,B,C,X(2),X(476)}}, the conic {{A,B,C,X(112),X(248)}}, and on these lines: 30,98 99,249 230,297 248,290 250,523 287,524 325,441 448,2481 668,906 879,4266 892,2395 2421,2422
X(2966) = midpoint of X(385) and X(401)
X(2966) = reflection of X(i) and X(j) for these (i,j): (297,230), (325,441)
X(2966) = isogonal conjugate of X(3569)
X(2966) = isotomic conjugate of X(2799)
X(2966) = anticomplement of X(35088)
X(2966) = polar conjugate of X(16230)
X(2966) = complement of X(39359)
X(2966) = trilinear pole of line X(2)X(98)
X(2966) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(476)}}
X(2966) = cevapoint of X(i) and X(j) for these (i,j): (2,2799),
(98,2395), (112,2409), (230,523), (248,879), (441,525), (511,647)
X(2966) = X(i)-cross conjugate of X(j) for these (i,j): (879,290), (2395,98), (2715,685), (2799,2)
X(2966) = barycentric product of Steiner circumellipse intercepts of line X(2)X(98)
MacBeath Points
In the parlance of TCCT, page 238, the MacBeath inconic is W(X(184)), with trilinear equation
U2x2 + V2y2 + W2z2 - 2WVyz - 2WUzx - 2UVxy = 0,Points lying on the MacBeath inconic were contributed by Peter Moses (Hyacinthos, Nov. 12, 2004), and a selection of these are given here as X(2967) to X(2973). Comments written by Alexander Murray MacBeath are given in a Hyacinthos message of Dec. 12, 2004 from Antreas P. Hatzipolakis. See also MathWorld and more than a dozen Hyacinthos messages numbered between 10870 and 10923.
Macbeath authored articles entitled "The Deltoid I," The Deltoid II," and "The Deltoid III," in Eureka. In the second of these, 11 (1949) 26-29, he introduces the conic as follows:
Lemma 1: Let O, H be the common points of a coaxal system of circles. Let a variable circle of the system cut the line of centres at C. Let T be a point on the circumference such that TC=k*OC, where k is a fixed ratio. Then the locus of T is a conic with foci at O,H.In Hyacinthos message 2009, dated Dec. 9, 2000, John Conway wrote
This I know as "the MacBeath ellipse", since Murray MacBeath wrote a nice article about it in "Eureka", the Cambridge University undergraduate mathematical journal.Thus did the conic get its name. Points X(339), X(1312), and X(1313) lie on the MacBeath inconic. Peter Moses (Nov. 12, 2004) noted that if X is on this conic, then the reflection of X in X(5), and also the reflection of X in the Euler line, lie on the conic. He introduced the MacBeath circumconic, dual to the MacBeath inconic. The circumconic is given in barycentrics by (sin 2A)yz + (sin 2B)zx + (sin 2C)xy = 0,
X(2967) lies on these lines: 2,1972 3,112 4,147 5,339 25,110 98,648 114,132 186,2080 232,511 250,842 262,264 324,427 1312,2592 1313,2593
X(2967) = reflection of X(339) in X(5)
X(2967) = reflection of X(38553) in X(3)
X(2967) = reflection of X(38552) in the Euler line
X(2967) = X(274)-Ceva conjugate of X(297)
X(2967) = crosspoint of X(264) and X(297)
X(2967) = crosssum of X(i) and X(j) for these (i,j): (125,879), (184,248)
X(2967) = polar conjugate of X(34536)
X(2967) = pole wrt polar circle of trilinear polar of X(34536) (line X(98)X(804))
X(2968) is the center of circumconic (hyperbola {{A,B,C,X(8),X(280)}}) that is the locus of trilinear poles of lines passing through X(3239). This hyperbola is the isotomic conjugate of the Soddy line. (Randy Hutson, October 15, 2018)
X(2968) lies on these lines: 2,1897 3,8 4,280 5,318 11,123 78,1062 116,122 124,522 189,972 200,1040 216,594 239,441 253,1119 345,1260 521,1364 525,2632 867,2969 1465,1861
X(2968) = reflection of X(38554) in X(3)
X(2968) = complement of X(1897)
X(2968) = anticomplement of X(15252)
X(2968) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,521), (75,525), (253,514), (280,522), (2370,900)
X(2968) = crosspoint of X(189) and X(693)
X(2968) = crosssum of X(i) and X(j) for these (i,j): (184,1415), (198,692)
X(2968) = polar conjugate of X(23984)
X(2968) = perspector of circumconic centered at X(3239)
X(2968) = X(2)-Ceva conjugate of X(3239)
X(2969) lies on the orthic inconic and these lines: 3,915 4,145 19,583 25,105 34,1411 92,427 124,136 125,1146 225,1828 242,468 407,1829 430,1848 867,2968 1118,1398 1565,2973 1785,1878 1824,1830 1870,1884
X(2969) = reflection of X(34332) in X(5)
X(2969) = MacBeath inconic antipode of X(34332)
X(2969) = X(i)-Ceva conjugate of X(j) for these (i,j): (917,676),
(2973,1086)
X(2969) = X(1015)-cross conjugate of X(1086)
X(2969) = crosssum of X(i) and X(j) for these (i,j): (3,1331), (184,906), (692,2911)
X(2969) = trilinear pole wrt orthic triangle of line X(4)X(9)
X(2969) = pole wrt polar circle of trilinear polar of X(1016) (line X(100)X(190))
X(2969) = polar conjugate of X(1016)
X(2970) lies on these lines: 2,2974 3,847 4,94 25,98 115,135 125,136 324,427 339,868
X(2970) = reflection of X(34333) in X(5)
X(2970) = MacBeath-inconic-antipode of X(34333)
X(2970) = X(847)-Ceva conjugate of X(523)
X(2970) = X(115)-cross conjugate of X(338)
X(2970) = crosssum of X(571) and X(1576)
X(2970) = pole wrt polar circle of trilinear polar of X(249) (line X(110)X(351))
X(2970) = polar conjugate of X(249)
X(2970) = X(3)-isoconjugate of X(1101)
X(2970) = {X(39240),X(39241)}-harmonic conjugate of X(35235)
X(2971) lies on these lines: 4,147 5,2974 25,111 127,136
X(2971) = reflection of X(2974) in X(5)
X(2971) = X(25)-Ceva conjugate of X(2489)
X(2971) = crosspoint of X(25) and X(2489)
X(2971) = crosssum of X(110) and X(193)
X(2971) = polar conjugate of X(34537)
X(2971) = pole wrt polar circle of trilinear polar of X(34537) (line X(99)X(670))
X(2972) lies on these lines: 2,1972 3,74 25,1073 69,1942 122,125 127,136 394,426 511,852 1368,2974 1624,2781
X(2972) = reflection of X(34334) in X(5)
X(2972) = isogonal conjugate of X(32230)
X(2972) = complement of X(35360)
X(2972) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,520), (264,525), (1073,647)
X(2972) = crosspoint of X(i) and X(j) for these (i,j): (3,520), (64,523), (264,525)
X(2972) = crosssum of X(i) and X(j) for these (i,j): (4,107), (20,110), (112,184)
X(2972) = crossdifference of every pair of points on line X(107)X(112)
X(2972) = polar conjugate of X(34538)
X(2972) = MacBeath-inconic-antipode of X(34334)
X(2972) = center of conic {{A,B,C,X(3),X(5)}}
X(2972) = X(92)-isoconjugate of X(23964)
X(2973) lies on these lines: 3,917 4,150 25,675 1235,1969 1565,2969
X(2973) = reflection of X(34335) in X(5)
X(2973) = cevapoint of X(1986) and X(2969)
X(2973) = pole wrt polar circle of trilinear polar of X(1252) (line X(101)X(692))
X(2973) = polar conjugate of X(1252)
X(2973) = MacBeath-inconic-antipode of X(34335)
X(2974) lies on these lines: 2,2970 3,76 5,2971 25,925 131, 132 1368,2972
X(2974) = reflection of X(2971) in X(5)
X(2974) = polar conjugate of isogonal conjugate of X(35067)
The term "AC-incircle" is introduced for "anticomplement of the incircle" by Peter Moses (Dec. 2, 2004). Thus, the AC-incircle is the incircle of the anticomplementary triangle; the circle has center X(8) and radius 2r. The exsimilicenter of the circumcircle and AC-incircle is X(100), their touchpoint, and the anticomplement of X(11).
X(2975) lies on these lines: 1,21 2,12 3,8 9,604 10,36 20,2894 28,92 35,519 48,2287 54,72 55,145 75,1444 78,947 105,330 110,1098 144,1001 172,1107 198,391 229,409 238,1201 329,405 333,1610 348,934 411,515 518,2330 593,2363 596,759 668,1078 672,2329 908,1125 943,2320 950,1005 960,1319 962,1012 966,2178 995,1724 1457,1935 1470,1788 1478,2476 1761,1953 2475,2886
X(2975) = isogonal conjugate of X(34434)
X(2975) = isotomic conjugate of isogonal conjugate of X(20986)
X(2975) = complement of X(20060)
X(2975) = anticomplement of X(12)
X(2975) = anticomplementary conjugate of isogonal conjugate of X(36903)
X(2975) = polar conjugate of isogonal conjugate of X(22118)
X(2975) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,100), (261,2)
X(2975) = crosssum of X(512) and X(2170)
X(2975) = X(35)-of-inner-Garcia triangle
See X(2975) for "AC-incircle."
X(2976) lies on these lines: 2,2505 100,190 513,676
X(2976) = anticomplement of X(2505)
See X(2975) for "AC-incircle."
X(2977) lies on these lines: 100,190 230,231
X(2977) = center of circle {{X(100),X(120),X(5521)}}
X(2978) lies on these lines: 187,237 320,350 659,834 661,788
X(2978) = X(785)-Ceva conjugate of X(6)
X(2978) = crosspoint of X(6) and X(785)
X(2978) = crosssum of X(2) and X(784)
Dual triangles, DC and CD Points
Suppose U is a point having cevian triangle DEF and dual triangle D'E'F'. Then there exists a point DC(U) whose circum-anticevian triangle (TCCT, p. 201) is D'E'F'. The mapping DC is given for U = u : v : w (trilinears) by
DC(U) = vw/(bv + cw) : wu/(cw + au) : uv/(au + bv).If you have The Geometer's Sketchpad, you can view DUAL TRIANGLE.
The appearance of (i,j) in the following list means that X(j) = DC(X(i)):
1,81 2,6 3,275 4,2 5,288 6,83 7,1 8,57 9,1170 10,1171 20,1073 63,1172 64,801 66,76 67,671 68,2052 69,4 75,58 76,251 79,1255 80,88 81,2298 83,3108 85,1174 86,1126 92,284 95,1173 99,110 100,651 110,648 189,9 190,101 253,3 254,1993 255,829 256,1258 264,54 273,1167 280,222 286,943 290,98 309,947 311,1166 314,961 317,847 320,1168 321,1169 325,2065 329,282 330,31 335,1438 347,1433 485,588 486,589 513,1016 514,1252 519,2226 522,1262 523,249 596,593 648,112 651,1783 660,666 664,109 666,919 668,100 670,99 671,111 693,59 850,250 889,898 892,691 903,106 927,677 1000,89 1029,37 1031,39 1032,1249 1034,223 1121,2291 1131,493 1132,494 1138,323 1219,2221 1280,1462 1297,287 1305,1331 1441,1175 1494,74 1821,2311 1897,1461 1916,1976 1972,248 2113,239 2184,2287 2349,2341 2370,1797 2373,895 2481,105
The longest chain among the listed points is this one: 69 -→ 4 -→ 2 -→ 6 -→ 83 → 3108
Inversely, the circum-anticevian triangle of a point P is the dual of the cevian triangle of a point CD(P), given for P = p : q : r by the inverse of the DC-mapping; that is:
CD(P) = bc/(-a/p + b/q + c/r) : ca/(a/p - b/q + c/r) : ab/(a/p + b/q - c/r) = isogonal conjugate of X(6)-Ceva conjugate of P.1,7 2,4 3,253 4,69 6,2 9,189 31,330 37,1029 39,1031 54,264 57,8 58,75 59,693 74,1494, etc. (Just transpose the pairs in the list for DC(U).)
The vertices of the dual D'E'F' of the cevian triangle of a point U = u : v : w are given by
D' = 1/(b/w - c/v) : -1/(c/u + a/w) : 1/(a/v + b/u)
E' = 1/(b/w + c/v) : 1/(c/u - a/w) : -1/(a/v + b/u)
F' = -1/(b/w + c/v) : 1/(c/u + a/w) : 1/(a/v - b/u).
Triangle D'E'F' can be obtained from DEF by translation, dilation, and rotation. First translate DEF by vector O'O, where O' = circumcenter of D'E'F' and O = circumcenter of ABC. Then dilate D'E'F' so that its circumradius is that of ABC. Then rotate D'E'F' so that angle AOD' is twice the angle between lines BC and EF.
X(2979) lies on these lines: 2,51 3,54 4,1216 6,1627 22,110 52,631 66,69 182,1994 184,323 343,858 476,2706 549,568 805,2857 930,1298 991,1779 1160,1584 1161,1583 1501,2076 2888,2889
X(2979) = reflection of X(568) in X(549)
X(2979) = isogonal conjugate of X(2980)
X(2979) = isotomic conjugate of anticomplement of complementary conjugate of X(34845)
X(2979) = anticomplement of X(51)
X(2979) = crosspoint of X(249) and X(670)
X(2979) = crosssum of X(115) and X(669)
X(2979) = X(4)-of-Lucas-triangle (defined at X(95))
X(2979) = Lucas-isogonal conjugate of X(3)
X(2980) lies on these lines: 5,182 22,157 53,428 95,160 1485,2353
X(2980) = isogonal conjugate of X(2979)
X(2980) = isotomic conjugate of X(7796)
X(2980) = cevapoint of X(2) and X(784)
X(2980) = X(784)-cross conjugate of X(2)
X(2981) lies on the hyperbola {{A,B,C,X(2),X(6)}} and these lines: 15,1337 61,110 396,1989
Let U be the line tangent to the Neuberg cubic (K001) at X(13), and let V be the line tangent to K001 at X(15). Then X(2981) = U∩V. Let U' be the line tangent to the Napoleon-Feuerbach cubic (K005) at X(17), and let V' be the line tangent to K005 are X(61). Then X(2981) = U'∩V'; see X(6151). Also, X(2981) = trilinear pole of the line X(16)X(512). (Randy Hutson, January 5, 2015)
X(2981) = isogonal conjugate of X(396)
X(2981) = cevapoint of X(i) and X(j) for these (i,j): (6,15), (203,2245)
X(2981) = perspector of ABC and unary cofactor triangle of outer Napoleon triangle
X(2982) lies on these lines: 1,201 2,219 6,278 28,65 48,57 55,955 81,1214 222,279 274,1231 1002,1617 1396,1409 1630,1730
X(2282) = cevapoint of X(i) and X(j) for these (i,j): (6,65), (56,1409), (57,2003)
X(2982) = X(i)-cross conjugate of X(j) for these (i,j): (6,1175), (647,108), (2259,943), (2605,934)
X(2982) = crosssum of X(1) and X(2954)
X(2983) lies on these lines: 1,1257 6,1260 9,34 10,2322 56,219 58,2327 63,269 71,1474 268,1413 937,1743 998,1723 1438,2269
X(2983) = cevapoint of X(i) and X(j) for these (i,j): (6,71),
(42,220)
X(2983) = X(647)-cross conjugate of X(101)
X(2983) = crosssum of X(1104) and X(2264)
X(2984) lies on this line: 97,3199
X(2984) = isogonal conjugate of complement of X(34386)
X(2984) = X(2485)-cross conjugate of X(933)
X(2985) lies on these lines: 8,1397 519,595 1999,2300
X(2985) = isotomic conjugate of X(1029)
X(2985) = cevapoint of X(6) and X(8)
X(2985) = X(2605)-cross conjugate of X(99)
X(2985) = X(19)-isoconjugate of X(23154)
X(2985) = X(92)-isoconjugate of X(23196)
Let A'B'C' be the circumcevian triangle of X(30). Let A" be the trilinear pole of line B'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(2986). (Randy Hutson, June 27, 2018)
X(2986) lies on the MacBeath circumconic and these lines: 4,110 10,1331 94,323 98,858 226,1813 262,1995 321,1332 338,394 340,687 648,1993
X(2986) = isogonal conjugate of X(3003)
X(2986) = isotomic conjugate of X(3580)
X(2986) = cevapoint of X(i) and X(j) for these (i,j): (2,323),
(6,30)
X(2986) = X(i)-cross conjugate of X(j) for these (i,j): (265,1494),
(529,99), (686,925), (2072,264), (2433,476)
X(2986) = trilinear pole of line X(3)X(523) (the line of the Ehrmann cross-triangle)
X(2986) = polar conjugate of X(403)
X(2987) lies on the MacBeath circumconic and these lines: 25,110 37,1332 42,1331 69,2165 111,323 141,2963 193,317 251,1994 263,576 287,2395 511,1976 524,1989 526,895 651,1880 1400,1813
X(2987) = isogonal conjugate of X(230)
X(2987) = isotomic conjugate of anticomplement of the X(36212)
X(2987) = cevapoint of X(6) and X(511)
X(2987) = isotomic conjugate of isogonal conjugate of X(32654)
X(2987) = MacBeath circumconic antipode of X(4558)
X(2987) = reflection of X(4558) in X(6)
X(2987) = trilinear pole of line X(3)X(512)
X(2987) = X(i)-cross conjugate of X(j) for these (i,j): (6,2065), (248,1297), (1570,6), (2422,805)
X(2988) lies on the MacBeath circumconic and these lines: 2,1813 8,1331 29,110 92,651 312,1332
X(2988) = isogonal conjugate of X(8607)
X(2988) = cevapoint of X(6) and X(515)
X(2988) = complement of anticomplementary conjugate of X(35516)
X(2988) = X(2432)-cross conjugate of X(1309)
X(2988) = trilinear pole of line X(3)X(522)
X(2989) lies on the MacBeath circumconic and these lines: 2,1331 7,1813 27,110 75,1332 273,651 1815,2400
X(2989) = isogonal conjugate of X(8608)
X(2989) = isotomic conjugate of polar conjugate of X(917)
X(2989) = X(19)-isoconjugate of X(916)
X(2989) = complement of anticomplementary conjugate of X(35517)
X(2989) = cevapoint of X(6) and X(516)
X(2989) = X(2424)-cross conjugate of X(927)
X(2990) lies on the MacBeath circumconic and these lines: 1,1331 2,1332 28,110 57,1813 278,651 394,1086 908,2006 1022,1797
X(2990) = isogonal conjugate of X(8609)
X(2990) = isotomic conjugate of isogonal conjugate of X(32655)
X(2990) = isotomic conjugate of polar conjugate of X(915)
X(2990) = X(19)-isoconjugate of X(912)
X(2990) = cevapoint of X(i) and X(j) for these (i,j): (1,2323), (6,517)
X(2990) = trilinear pole of line X(3)X(513)
X(2990) = X(i)-cross conjugate of X(j) for these (i,j): (654,100), (2423,901)
X(2991) lies on the MacBeath circumconic and these lines: 6,344 31,1331 110,2203 193,608 239,1462 294,335 604,1445 918,1814 1911,2340
X(2991) = reflection of X(1332) in X(6)
X(2991) = isogonal conjugate of X(3290)
X(2991) = cevapoint of X(6) and X(518)
X(2991) = X(665)-cross conjugate of X(100)
X(2991) = MacBeath circumconic antipode of X(1332)
X(2991) = trilinear pole of PU(44) (line X(3)X(667))
X(2992) lies on these lines: 3,299 6,471 68,633 265,300 1899,2993
X(2992) = isogonal conjugate of X(3129)
X(2992) = isotomic conjugate of X(621)
X(2992) = X(i)-cross conjugate of X(j) for these (i,j): (15,2), (340,2993)
X(2992) = antigonal conjugate of the isogonal conjugate of X(32461)
X(2993) lies on these lines: 3,298 6,470 68,634 265,301 1899,2992
X(2993) = isogonal conjugate of X(3130)
X(2993) = isotomic conjugate of X(622)
X(2993) = antigonal conjugate of the isogonal conjugate of X(32460)
X(2993) = X(i)-cross conjugate of X(j) for these (i,j): (16,2), (340,2992)
X(2994) lies on these lines: 2,914 8,90 29,1069 92,1947 281,1993 312,319 333,2164 394,1146
X(2994) = isogonal conjugate of X(2178)
X(2994) = cevapoint of X(i) and X(j) for these (i,j): (521,1146),
(1069,2164)
X(2994) = isotomic conjugate of X(5905)
X(2994) = X(i)-cross conjugate of X(j) for these (i,j): (63,2), (1479,7)
X(2994) = cyclocevian conjugate of X(7219)
X(2994) = pole wrt polar circle of trilinear polar of X(1068)
X(2994) = X(48)-isoconjugate (polar conjugate) of X(1068)
X(2994) = barycentric product of PU(129)
X(2995) lies on these lines: 63,321 69,313 75,1444 77,1441 81,92
X(2995) = isogonal conjugate of X(3185)
X(2995) = isotomic conjugate of X(3869)
X(2995) = trilinear pole of line X(905)X(1577)
X(2995) = cevapoint of X(123) and X(525)
X(2995) = X(65)-cross conjugate of X(2)
X(2996) is the trilinear pole of the line X(523)X(4885), this line being the complement of orthic axis; the line is also the polar of X(20) wrt {circumcircle, nine-point circle}-inverter). Also, let Ra be the radical trace of the A-McCay circle and A-Neuberg circle, and define Rb and Rc cyclically. Then the triangle RaRbRc is perspective to ABC at X(2996). (Randy Hutson, October 13, 2015)
X(2996) lies on these lines: 2,1975 4,193 20,98 194,262 230,439 315,671 485,487 486,488
X(2996) = reflection of X(i) in X(j) for these (i,j): (487,485), (488,486)
X(2996) = isogonal conjugate of X(3053)
X(2996) = isotomic conjugate of X(193)
X(2996) = cevapoint of X(115) and X(525)
X(2996) = X(i)-cross conjugate of X(j) for these (i,j): (69,2), (1368,264)
X(2996) = antigonal conjugate of X(8781)
X(2996) = Kiepert hyperbola antipode of X(8781)
X(2996) = reflection of X(8781) in X(115)
X(2997) lies on the Feuerbach hyperbola these lines: 1,1441 2,2335 4,916 8,313 9,321 21,75 92,1172 104,1305 149,2897 314,561 322,1320
X(2997) = isogonal conjugate of X(2352)
X(2997) = isotomic conjugate of X(3868)
X(2997) = cevapoint of X(1) and X(525)
X(2997) = X(72)-cross conjugate of X(2)
X(2998) lies on these lines: 6,194 25,385 42,192 69,694 193,263 941,1655 1084,1502
X(2998) = isogonal conjugate of X(1613)
X(2998) = isotomic conjugate of X(194)
X(2998) = complement of X(32747)
X(2998) = anticomplement of X(6374)
X(2998) = polar conjugate of X(3186)
X(2998) = trilinear pole of line X(512)X(625) (the complement of the Lemoine axis)
X(2998) = cyclocevian conjugate of X(66)
X(2998) = cevapoint of X(523) and X(1084)
X(2998) = X(76)-cross conjugate of X(2)
X(2998) = X(19)-isoconjugate of X(20794)
Let RA be the radical axis of the Bevan circle and the A-excircle. Define RB and RC cyclically. The three radical axes form a triangle homothetic to triangle ABC, and X(2999) is the center of homothety. (Eric Danneels, Hyacinthos #10926, Dec. 5, 2004.
Let BA be the inverse-in-the-A-excircle of the B-excenter and CA the inverse-in-the-A-excircle of the C-excenter. Let LA be the line BACA, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is homothetic to triangle ABC, and the center of homothety is X(2999). (Randy Hutson, 9/23/2011)
X(2999) lies on these lines: 1,2 3,1453 6,57 31,165 46,1203 58,937 63,1743 73,1467 244,1282 278,2331 440,1040 673,2258 748,968 940,1449 990,1750 1191,1697 1214,2257 1376,1386 1394,1466
X(2999) = isogonal conjugate of X(2297)
X(2999) = complement of X(34255)
X(2999) = crosssum of X(9) and X(1449)
Intersections of Central Lines
This section presents points X(3000) to X(3019) of intersection of lines listed in MathWorld's Central Line. Contributed by Eric Weisstein, December, 2004.
The antiorthic axis is line X(44)X(513), more simply represented as L(1), meaning the line whose coefficients in trilinear coordinates are 1 : 1 : 1, these also being trilinears for X(1). The Soddy line is X(1)X(7), alias L(657). For such identifications as these, use the MathWorld link just before X(3000).
X(3000) lies on these lines: 1,7 44,513 241,2310 527,2340
X(3000) = reflection of X(2310) in X(241)
X(3000) = crosssum of X(1) and X(3000)
| PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
| PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
| PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
| PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
| PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
| PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
| PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
| PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
| PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
| PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
| PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
| PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
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