| 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) |
As a point on the Euler line, X(5001) has Shinagawa coefficients (1 -k2, -1 + 3k2).
X(5000) and X(5001) are the antiorthocorrespondents of X(6); i.e. they share the same orthocorrespondent, X(6). See Bernard Gibert, Table 55: X(5000, X(5001) and related curves.
For a construction of X(5000) and X(5001) see AdGeom 5185.
Click here for a 3-dimensional representation of X(5001).
For a construction, see Antreas Hatzipolakis and Peter Moses, euclid 7200.
X(5000) lies on 8th Grozdev-Dekov-Parry circle, Dao-Moses-Telv circle, Moses radical circle, Stevanovic circle, Walsmith rectangular hyperbola; cubics K018, K270, K336, K337, K570, K608, K828, K829, K1091, K1092, K1129, K1133a, K1133b; curves Q019, Q021, Q024, Q026, Q037, Q049, Q054, Q098, Q115, Q116, Q117, Q118, Q144, Q146, Q147 and this line: {2,3}
X(5001) = isogonal conjugate of X(32619)
X(5001) = isotomic conjugate of X(42812)
X(5001) = complement of X(5003)
X(5001) = circumcircle-inverse of X(5000)
X(5001) = nine-point-circle-circumcircle-inverse of X(5000)
X(5001) = orthocentrodal-circle-inverse of X(5000)
X(5001) = orthoptic circle of Steiner inellipse-inverse of X(5000)
X(5001) = polar-circle-inverse of X(5000)
X(5001) = tangential-circle-inverse of X(5000)
X(5001) = MacBeath-inconc-inverse of X(5000)
X(5001) = Yff-hypergbola-inverse of X(5000)
X(5001) = Walsmith-rectangular-hyperbola-inverse of X(5000)
X(5001) = polar conjugate of X(41195)
X(5001) = complementary conjugate of the complement of X(34135)
X(5001) = antigonal conjugate of X(42810)
X(5001) = orthoassociate of X(5000)
As a point on the Euler line, X(5002) has Shinagawa coefficients (1 +k2, -1 - 3k2).
X(5002) is the point whose tripolar distances in the plane of triangle ABC are proportional to (a, b, c); the actual tripolar distances in case ABC is acute are ka, kb, kc.
If the reference triangle ABC is obtuse, then the barycentrics of X(5002) are a nonreal complex numbers. Nevertheless, for all triangles ABC, the midpoint of X(5002) and X(5003) is X(858).
A method for converting from homogeneous tripolar coordinates (henceforth simply "tripolars") to homogeneous barycentrics, found by Peter Moses (March, 2012), depends on finding the point of intersection of the radical axes of radical circles centered at A, B, C. Write the tripolars for a point U as u : v : w, and let
da = b2 + c2 - a2, db = c2 + a2 - c2, dc = a2 + b2 - c2.
Then barycentrics x : y : z for U are given by
x = a2da +
k2[dcv2 + dbw2 -
2a2u2]
y = b2db +
k2[daw2 + dcu2 -
2b2v2]
z = c2dc +
k2[dbu2 + dav2 -
2c2w2],
where k2 has two values (as in the quadratic formula): (-f - g)/h or (-f + g)/h, where
f = - a2u2da -
b2v2db -
c2w2dc
g = 2S[(-au + bv + cw)(au - bv + cw)(au + bv - cw)(au + bv +
cw)]1/2
h = 2[a2(u2 - v2)(u2 -
w2) + b2(v2 -
w2)(v2 - u2) +
c2(w2 - u2)(w2 -
v2)]
The meaning of k can be stated thus: starting with tripolars u : v : w, the actual tripolar distances are ku, kv, kw. That is, |UA| = ku, |UB| = kv, |UC| = kw.
X(5002) lies on this line: {2,3}
X(5002) = isogonal conjugate of X(34136)
X(5002) = antigonal conjugate of X(34240)
X(5002) = circumcircle-inverse of X(5003)
As a point on the Euler line, X(5003) has Shinagawa coefficients (1 - k2, -1 - 3k2).
The four points, X(i) for i = 5000, 5001, 5002, 5003, all lie on the Euler line of triangle ABC, and all are nonreal complex-valued if ABC is obtuse.
If the reference triangle ABC is obtuse, then the barycentrics of X(5002) are a nonreal complex numbers. Nevertheless, for all triangles ABC, the midpoint of X(5002) and X(5003) is X(858).
X(5003) lies on this line: {2,3}
X(5003) = isogonal conjugate of X(34135)
X(5003) = antigonal conjugate of X(34239)
X(5003) = circumcircle-inverse of X(5002)
As a point on the Euler line, X(5004) has Shinagawa coefficients ((E + 4F)S + abc(2Sω)1/2, -4(E + F)S - 3abc(2Sω)1/2).
X(5004) is the point in the plane of triangle ABC whose tripolar distances are proportional to ((b2 + c2)1/2, (c2 + a2)1/2, (a2 + b2)1/2). Like X(5000) and X(5002), the point X(5004) lies on the Euler line; unlike X(5000) and X(5002), this point is real-valued when ABC is obtuse.
X(5004) is the circumcircle-inverse of X(5005). The midpoint of X(5004) and X(5005) is X(23). Of the two points, X(5004) is the one outside the circumcircle. (Peter Moses, March 7, 2012)
X(5004) and X(5005) are the two points whose pedal antipodal perspectors (defined at Hyacinthos #20403 and #20405) are both X(6). (Randy Hutson, Febrary 20, 2015)
If you have The Geometer's Sketchpad, you can view X(5004) and X(5005).
X(5004) lies on this line: {2,3}
X(5004) = isogonal conjugate of X(34221)
As a point on the Euler line, X(5005) has Shinagawa coefficients ((E + 4F)S - abc(2Sω)1/2, -4(E + F)S + 3abc(2Sω)1/2).
X(5005) is the circumcircle-inverse of X(5004). The midpoint of X(504) and X(5005) is X(23). (Peter Moses, March 7, 2012)
X(5005) lies on this line: {2,3}
X(5005) = isogonal conjugate of X(34222)
X(5006) lies on these lines: {3, 6}, {60, 213}, {99, 712}, {104, 2715}, {110, 3230}, {112, 2699}, {163, 1914}, {172, 849}, {249, 1931}, {691, 739}, {713, 805}, {1325, 3125}
Let T be the symmedial triangle, and let H be the bicevian conic of X(2) and X(6). Let V be the triangle formed by the lines tangent to H at the vertices of T. Then T and V are perspective, and their perspector is X(5007). (Randy Hutson, February 20, 2015)
X(5007) lies on these lines: {3, 6}, {23, 251}, {44, 3678}, {83, 385}, {115, 546}, {172, 1015}, {194, 3972}, {211, 4173}, {230, 1506}, {232, 3518}, {248, 1173}, {384,538}, {395, 635}, {396, 636}, {609, 2275}, {632, 3815}, {1078, 3329}, {1100, 3881}, {1193, 2251}, {1196, 1995}, {1500, 1914}, {1573, 4426}, {1574, 4386}, {2223, 2308}, {2241, 3303}, {2242, 3304}, {2243, 3670}, {2548, 3090}, {2549, 3529}, {3051, 3229}, {3091, 3767}, {3629,3933}
X(5007) = midpoint of X(61) and X(62)
X(5007) = isogonal conjugate of X(10159)
X(5007) = radical center of Lucas(-6 cot ω) circles
X(5007) = {X(371),X(372)}-harmonic conjugate of X(3098)
X(5007) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(3098)
X(5007) = radical trace of Brocard circle and circle {{X(4),X(194),X(3557),X(3558)}}
X(5007) = inverse-in-1st-Brocard-circle of X(7772)
X(5007) = X(39)-of-5th-anti-Brocard-triangle
X(5007) = X(187)-of-X(6)PU(1)
X(5007) = polar conjugate of isotomic conjugate of X(22352)
X(5007) = {X(32),X(39)}-harmonic conjugate of X(187)
X(5008) lies on these lines: {3, 6}, {111, 251}, {115, 3845}, {230, 547}, {538, 3972}, {609, 1015}, {1285, 2549}, {1506, 3054}, {3589, 3793}, {3767, 3832}
X(5008) = complement of X(7850)X(5009) lies on these lines: {1, 82}, {3, 6}, {21, 976}, {60, 1178}, {81, 982}, {110, 2382}, {238, 2210}, {333, 4438}, {740, 1580}, {741, 919}, {757, 763}, {765, 1110}
X(5009) = isogonal conjugate of the isotomic conjugate of X(33295)
X(5009) = isogonal conjugate of the polar conjugate of X(31905)
X(5010) lies on these lines: {1, 3}, {2, 3583}, {4, 4324}, {9, 1030}, {10, 4189}, {11, 549}, {12, 550}, {20, 498}, {31, 4256}, {33, 186}, {34, 3520}, {42, 4257}, {43, 4184}, {78, 191}, {90, 3467}, {99, 3761}, {100, 993}, {187, 609}, {203, 1250}, {214, 3877}, {376, 1478}, {386, 2308}, {388, 3528}, {404, 3624}, {495, 4995}, {497, 3524}, {499, 3523}, {574, 1914}, {601, 2964}, {631, 1479}, {672, 4262}, {678, 1623}, {750, 4653}, {902, 995}, {956, 4421}, {975, 1719}, {1006, 3586}, {1054, 4218}, {1078, 3760}, {1125, 4188}, {1151, 3301}, {1152, 3299}, {1203, 4255}, {1737, 4304}, {2163, 2177}, {2267, 2316}, {2278, 2364}, {2330, 3098}, {2975, 3632}, {3085, 3522}, {3086, 4309}, {3218, 3894}, {3614, 3627}, {3633, 3871}, {3647, 3876}, {3651, 4333}, {3751, 4265}, {3811, 4652}, {3872, 4996}, {3873, 4973}, {3899, 4511}
X(5011) lies on these lines: {1, 1055}, {2, 5074}, {4, 9}, {8, 1759}, {20, 1729}, {30, 1146}, {32, 3959}, {36, 2170}, {46, 2082}, {57, 1323}, {65, 2301}, {101, 517}, {113, 1566}, {116, 4872}, {163, 1325}, {191, 3691}, {239, 514}, {284, 501}, {484, 672}, {519, 3509}, {572, 2262}, {758, 3684}, {759, 2702}, {995, 1572}, {1155, 1308}, {1212, 3579}, {1404, 3339}, {1475, 3336}, {1482, 3207}, {1652, 3638}, {1653, 3639}, {1730, 3101}, {1731, 2245}, {1761, 3686}, {1781, 2269}, {1845, 2202}, {1914, 3125}, {1951, 1983}, {2173, 2323}, {2246, 3245}, {2249, 2690}, {2328, 2355}, {3735, 4386}, {3871, 3970}, {3916, 4875}, {4165, 4680}, {4316, 4530}
X(5011) = midpoint of X(1276) and X(1277)
X(5011) = reflection of X(101) in antiorthic axis
X(5011) = complement of X(5195)
X(5011) = anticomplement of X(5074)
X(5011) = X(187)-of-excentral-triangle
X(5012) = is the pole of Brocard axis wrt the ellipse {{X(3), X(6), X(24), X(60), X(143), X(1511), X(1986)} (Randy Hutson, February 20, 2015)
X(5012) lies on these lines: {2, 98}, {3, 54}, {4, 569}, {5, 1614}, {6, 22}, {20, 578}, {23, 51}, {26, 3567}, {30, 567}, {49, 140}, {52, 1199}, {60, 386}, {143, 2937}, {154, 1995}, {156, 1656}, {206, 3618}, {237, 3398}, {249, 3111}, {323, 3917}, {404, 1437}, {511, 1994}, {572, 4184}, {580, 4225}, {620, 3044}, {631, 1147}, {692, 1621}, {1078, 3203}, {1092, 3523}, {1194, 1692}, {1627, 1691}, {1790, 4210}, {2056, 3231}, {2206, 4279}, {2330, 3920}, {3035, 3045}, {3218, 3955}, {3292, 3819}
X(5012) = isogonal conjugate of X(3613)
Let U be the circle centered at X(371) and passing through X(1151), and let U′ the circle centered at X(372) and passing through X(1152); then X(5013) is the insimilicenter of U and U′. Let V be the circle centered at X(1151) and passing through X(371), and let V′ be the circle centered at X(371) and passing through X(1151); then X(5013) is the insimilicenter of V and V′. Let W be the circle having diameter X(371)X(372), and let W′ be the circle having diameter X(1151)X(1152); Then X(5013) is the exsimilicenter of W and W′. (Randy Hutson, September 5, 2014)
X(5013) lies on these lines: {2, 1975}, {3, 6}, {4, 3815}, {5, 2549}, {30, 2548}, {37, 988}, {53, 3088}, {55, 2275}, {56, 2276}, {83, 1003}, {99, 2023}, {115, 1656}, {140, 3767}, {141, 3926}, {154, 3148}, {183, 194}, {230, 631}, {232, 1593}, {378, 2207}, {381, 1506}, {517, 1571}, {524, 3785}, {599, 3933}, {958, 1575}, {999, 1500}, {1015, 3295}, {1107, 1376}, {1180, 1184}, {1181, 3269}, {1194, 1611}, {1572, 3579}, {1597, 3199}, {1968, 3516}, {3054, 3525}, {3055, 3090}, {3329, 3552}, {3788, 4045}
X(5013) = isogonal conjugate of X(5395)
X(5013) = radical center of Lucas(cot ω) circles
X(5013) = {(X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3053), (3,39,6), (371,372,5050), (8406,8414,182)
X(5013) = insimilicenter of circumcircle and (1/2)-Moses circle
X(5014) lies on these lines: {1, 4202}, {2, 1279}, {4, 8}, {10, 748}, {38, 4660}, {42, 4865}, {55, 3006}, {63, 4450}, {75, 1369}, {100, 3705}, {149, 312}, {319, 4441}, {320, 4430}, {497, 4358}, {519, 3891}, {528, 3703}, {740, 4137}, {902, 4438}, {1150, 1754}, {1479, 3701}, {2280, 4071}, {2550, 4359}, {2886, 4030}, {2887, 3938}, {3058, 3932}, {3416, 4863}, {3632, 4442}, {3696, 4914}, {3722, 3771}, {3870, 3936}, {3873, 4645}, {3886, 3969}, {3935, 4417}, {3966, 4651}
X(5014) = anticomplement of X(3744)
X(5015) lies on these lines: {1, 977}, {3, 3705}, {4, 8}, {10, 82}, {12, 4030}, {21, 3006}, {69, 3673}, {75, 315}, {76, 319}, {149, 3702}, {312, 1479}, {320, 3874}, {333, 1780}, {345, 4294}, {442, 3757}, {518, 1330}, {528, 3704}, {752, 1046}, {942, 4645}, {986, 4660}, {1089, 3583}, {1930, 4872}, {2475, 4968}, {3178, 3750}, {3496, 4136}, {3585, 4692}, {3684, 4109}, {3685, 3695}, {3811, 4417}, {4153, 4251}
X(5016) lies on these lines: {1, 3454}, {2, 1104}, {4, 8}, {10, 31}, {40, 4450}, {41, 4109}, {44, 391}, {75, 2475}, {141, 4950}, {145, 4514}, {149, 4673}, {306, 950}, {377, 4359}, {518, 4812}, {519, 4101}, {958, 3006}, {1330, 3868}, {1478, 4968}, {1479, 3702}, {1834, 3187}, {1837, 3056}, {2478, 4358}, {2887, 3924}, {2975, 3705}, {3496, 4165}, {3586, 3969}, {4201, 4850}, {4642, 4660}
X(5016) = anticomplement of X(37539)
X(5017) lies on these lines: {3, 6}, {22, 3051}, {24, 2211}, {25, 694}, {69, 384}, {141, 315}, {154, 2056}, {159, 3499}, {172, 3056}, {193, 3552}, {251, 2979}, {263, 3148}, {352, 1383}, {394, 1915}, {524, 1003}, {599, 754}, {626, 3763}, {732, 1975}, {760, 3242}, {1184, 3981}, {1460, 2162}, {1469, 1914}, {1501, 1993}, {1627, 3060}, {1843, 1968}, {1995, 3231}
X(5017) = reflection of X(6) in X(32)
X(5018) lies on these lines: {1, 7}, {34, 87}, {43, 223}, {46, 3468}, {57, 985}, {59, 484}, {65, 4649}, {109, 1758}, {171, 1427}, {226, 1961}, {238, 241}, {296, 3466}, {514, 4581}, {651, 1757}, {664, 740}, {741, 927}, {846, 1214}, {934, 2700}, {982, 1407}, {1020, 1756}, {1046, 1409}, {1054, 1465}, {1386, 1418}, {1404, 3339}, {1419, 3751}, {1735, 2958}
X(5019) lies on these lines: {2, 261}, {3, 6}, {9, 172}, {31, 184}, {36, 2277}, {37, 993}, {48, 213}, {56, 478}, {87, 1716}, {609, 1743}, {672, 2273}, {941, 4189}, {980, 1444}, {992, 1724}, {1100, 2241}, {1172, 1968}, {1449, 1914}, {1468, 2268}, {1572, 3554}, {1631, 4749}, {1761, 3735}, {2298, 2975}, {3169, 3550}, {3686, 4386}
X(5019) = isogonal conjugate of X(34258)
X(5019) = crosspoint of X(i) and X(j) for these {i,j}: {2, 20029}, {56, 2221}, {1252, 34074}
X(5019) = crosssum of X(i) and X(j) for these (i,j): {2, 5739}, {6, 11337}, {8, 2345}, {386, 573}, {1086, 4801}
X(5019) = crossdifference of every pair of points on line X(523)X(4391)
X(5019) = {X(371),X(372)}-harmonic conjugate of X(970)
As a point on the Euler line, X(5020) has Shinagawa coefficients (E - F,E + F).
X(5020) lies on these lines: {2, 3}, {6, 1196}, {32, 1611}, {51, 394}, {111, 907}, {115, 2936}, {154, 182}, {159, 3589}, {184, 373}, {197, 1001}, {238, 1460}, {262, 801}, {612, 3295}, {614, 999}, {1007, 3964}, {1070, 3011}, {1184, 3291}, {1350, 3819}, {1376, 1486}, {1473, 3306}, {1495, 3796}, {1619, 1853}, {3556, 3812}
X(5020) = circumcircle-inverse of X(37897)X(5021) lies on these lines: {2, 967}, {3, 6}, {25, 2350}, {31, 1475}, {56, 213}, {172, 218}, {220, 2242}, {474, 2238}, {604, 2200}, {672, 1468}, {750, 3691}, {956, 2295}, {999, 2176}, {1015, 1191}, {1046, 3061}, {1106, 1400}, {1571, 4646}, {1834, 2549}, {2241, 3052}, {3230, 3304}, {3290, 3338}, {3496, 4650}
X(5021) = isogonal conjugate of X(32022)
X(5021) = crosssum of X(2) and X(391)
X(5022) lies on these lines: {2, 1434}, {3, 6}, {9, 3361}, {36, 218}, {37, 3333}, {55, 1475}, {56, 220}, {57, 1212}, {604, 1802}, {999, 3730}, {1011, 2350}, {1015, 1616}, {1146, 1788}, {1155, 2082}, {1191, 2275}, {1334, 3304}, {1732, 2182}, {2332, 3516}, {3230, 3445}, {3691, 4413}
X(5023) lies on these lines: {3, 6}, {20, 230}, {22, 1611}, {26, 2079}, {64, 1971}, {69, 439}, {115, 1657}, {160, 682}, {183, 3552}, {186, 2207}, {248, 3532}, {548, 2549}, {549, 2548}, {550, 3767}, {599, 3785}, {1003, 1078}, {1968, 3515}, {3054, 3091}, {3523, 3815}
X(5023) = midpoint of X(1151) and X(1152)
X(5023) = isogonal conjugate of X(38259)
X(5023) = center of inverse-in-circumcircle-of-Moses-circle
X(5023) = center of inverse-in-circumcircle-of-2nd-Lemoine-circle
X(5023) = {X(371),X(372)}-harmonic conjugate of X(5093)
X(5023) = radical center of Lucas(-1/2 cot ω) circles
Let A′B′C′ be the circumcevian triangle of the symmedian point (Lemoine point), X(6). The sidelines BC, CA, AB meet the sidelines of B′C′, C′A′, A′B′ in 9 points, of which 6 do not lie on the trilinear polar of K; barycentrics for the 6 points are 0 : b2 : 2c2, 0 : 2b2 : c2, 2a2 : 0 : c2, a2 : 0 : 2c2, a2 : 2b2 : 0, 2a2 : b2 : 0. The 6 points lie on a conic with center X(5024) and equation
2(b4c4x2 + c4a4y2 + a4b4z2) -5a2b2c2(a2yz + b2zx + c2xy) = 0.The conic described above is an ellipse with major axis X(512)X(5024). (Randy Hutson, February 20, 2015)
X(5024) lies on these lines: {2, 2418}, {3, 6}, {22, 1383}, {232, 1597}, {353, 3148}, {381, 2549}, {382, 2548}, {988, 3731}, {999, 2276}, {1003, 3329}, {1506, 3851}, {1656, 3055}, {1992, 3793}, {2275, 3295}, {3054, 3526}, {3172, 3520}, {3331, 3426}, {3619, 3926}, {3620, 3933}
X(5024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1384), (3,39,9605), (6,574,3)
X(5024) = radical center of Lucas(4/3 cot ω) circles
X(5024) = insimilicenter of (1/2)-Moses and Stammler circles; the exsimilicenter is X(9605)
As a point on the Euler line, X(5025) has Shinagawa coefficients ((E + F)2 - S2, -2S2).
X(5025) lies on these lines: {2, 3}, {32, 316}, {39, 625}, {76, 115}, {83, 3407}, {99, 3788}, {148, 1975}, {183, 2896}, {194, 325}, {315, 385}, {623, 3104}, {624, 3105}, {1348, 2559}, {1349, 2558}, {1479, 4366}, {1506, 4045}, {2548, 3329}, {3096, 3934}
X(5025) = midpoint of X(7807) and X(33229)
X(5025) = reflection of X(3552) in X(7807)
X(5025) = reflection of X(33246) in X(2)
X(5025) = complement of X(3552)
X(5025) = anticomplement of X(7807)
X(5025) = {X(2),X(3)}-harmonic conjugate of X(7907)
X(5025) = {X(2),X(4)}-harmonic conjugate of X(384)
X(5025) = {X(2),X(5)}-harmonic conjugate of X(16921)
X(5025) = {X(2),X(20)}-harmonic conjugate of X(16925)
X(5026) lies on these lines: {2, 353}, {6, 99}, {114, 1503}, {115, 3589}, {141, 542}, {148, 3618}, {182, 2782}, {187, 524}, {385, 732}, {538, 2030}, {543, 597}, {698, 1569}, {804, 4107}, {1428, 3027}, {2330, 3023}, {2796, 3946}, {2854, 3111}
X(5026) = X(187)-of-1st-Brocard-triangle
X(5026) = 1st-Brocard-isogonal conjugate of X(599)
X(5027) lies on these lines: {6, 888}, {99, 110}, {111, 729}, {182, 2793}, {187, 237}, {688, 3050}, {707, 737}, {804, 4107}, {808, 3267}, {882, 2422}, {1511, 2780}, {1580, 4367}, {1976, 2395}, {2492, 2872}, {2799, 3506}, {3049, 3221}, {4155, 4435}
X(5027) = isogonal conjugate of X(18829)
X(5027) = cevapoint of X(2491) and X(9429)
X(5027) = crosspoint of X(i) and X(j) for these {i,j}: {6, 805}, {25, 685}, {99, 3225}, {110, 1976}, {385, 880}, {733, 4577}, {2966, 14382}
X(5027) = crosssum of X(i) and X(j) for these {i,j}: {2, 804}, {69, 684}, {325, 523}, {512, 3229}, {694, 881}, {732, 3005}, {812, 6682}, {1634, 2421}, {3569, 14251}
X(5027) = crossdifference of every pair of points on line X(2)X(694)
X(5027) = trilinear pole of line X(2086)X(2679)
X(5027) = inverse-in-Parry-circle of X(669)
X(5027) = inverse-in-2nd-Lemoine-circle of X(2456)
X(5027) = radical center of circumcircle, Brocard circle, Brocard circle of 1st Brocard triangle
X(5027) = radical center of Brocard circles of ABC, 1st Brocard triangle, 1st anti-Brocard triangle
X(5027) = (Lemoine axis of ABC)∩(Lemoine axis of the 1st Brocard triangle)
X(5027) = X(182)-of-1st-Parry-triangle
X(5027) = inverse-in-Parry-isodynamic-circle of X(3231); see X(2)
X(5027) = X(669)-of-1st-Brocard-triangle
X(5027) = 1st-Brocard-isogonal conjugate of X(30229)
X(5027) = intersection of perspectrix of ABC and 1st Brocard triangle (line X(804)X(4107)) and perspectrix of ABC and 1st anti-Brocard triangle (line X(187)X(237))
X(5027) = trilinear product X(i)*X(j) for these {i,j}: {31, 804}, {42, 4164}, {171, 4455}, {213, 4107}, {238, 7234}, {385, 798}, {419, 810}, {512, 1580}, {523, 1933}, {560, 14295}, {661, 1691}, {662, 2086}, {667, 4039}, {669, 1966}, {875, 4154}, {880, 4117}, {923, 11183}, {1577, 14602}, {1918, 14296}, {1924, 3978}, {1926, 9426}, {2210, 2533}, {2236, 18105}, {2295, 8632}, {3573, 4128}, {3747, 4367}, {4010, 7122}
Let P1′ and U1′ be the 2nd-Lemoine-circle-inverses of P(1) and U(1), resp. Then X(5028) = P(1)U1′∩U(1)P1′. (Randy Hutson, January 17, 2020)
X(5028) lies on these lines: {2, 2987}, {3, 6}, {69, 626}, {115, 1352}, {193, 315}, {394, 1196}, {611, 1500}, {613, 1015}, {754, 1992}, {760, 3751}, {1180, 1994}, {1184, 3787}, {1194, 1993}, {1469, 2242}, {2241, 3056}, {2549, 2794}
X(5028) = reflection of X(32) in X(6)
X(5029) lies on these lines: {2, 4107}, {101, 110}, {106, 111}, {187, 237}, {245, 3708}, {661, 4367}, {798, 2605}, {1015, 3124}, {1635, 3722}, {2054, 3572}, {2183, 2609}, {3723, 4145}, {3733, 4079}, {3960, 4813}, {4024, 4560}, {4160, 4893}
X(5029) = isogonal conjugate of X(35148)X(5030) lies on these lines: {3, 6}, {35, 1475}, {36, 101}, {56, 3730}, {106, 292}, {484, 2170}, {595, 2275}, {661, 1019}, {1155, 1308}, {2285, 3361}, {2332, 3520}, {2350, 4184}, {3247, 3333}, {3509, 4973}
X(5031) lies on these lines: {2,1501}, {5,141}, {6,7752}, {69,5111}, {114,1503}, {115,698}, {125,9152}, {182,7862}, {187,6292}, {316,2076}, {325,732}, {524,1570}, {542,13196}, {599,15514}, {1352,2456}, {1506,1692}, {2021,8362}, {2024,3815}, {2080,7800}, {2458,7867}, {3094,5025}, {3098,7825}, {3619,5104}, {3620,8586}, {3631,5107}, {3788,3818}, {4074,5133}, {5017,7773}, {5039,7775}, {5116,7769}, {5162,7822}, {5969,6393}, {6656,10007}, {6680,16385}, {7746,8177}, {7777,13331}, {7778,10516}, {7785,12212}, {7789,18860}, {7810,10631}, {7815,14693}, {7821,14994}, {7832,24273}, {7842,14810}, {7925,11646}, {14603,18896}, {14712,16898}, {17047,20255}
X(5031) = crosspoint of X(2) and X(18896)X(5032) lies on these lines: {2, 6}, {20, 576}, {145, 4663}, {376, 1351}, {381, 1353}, {542, 3839}, {575, 3523}, {598, 2996}, {1570, 3849}, {3241, 3751}, {3545, 3564}
X(5033) lies on these lines: {3, 6}, {69, 620}, {184, 3231}, {206, 3016}, {729, 3565}, {1078, 3620}, {1196, 3796}, {1428, 2241}, {2242, 2330}, {3618, 4045}
X(5033) = {X(1687),X(1688)}-harmonic conjugate of X(1351)
X(5034) lies on these lines: {3, 6}, {83, 2996}, {193, 1078}, {611, 1015}, {613, 1500}, {1352, 1506}, {1428, 2242}, {2241, 2330}, {3564, 3815}, {3618, 3767}
X(5034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5052), (1687,1688,5050)
X(5034) = inverse-in-1st-Brocard-circle of X(5052)
X(5034) = pole of Lemoine axis wrt 1st Lemoine circle
X(5035) lies on these lines: {3, 6}, {31, 692}, {37, 2975}, {44, 172}, {45, 2242}, {593, 662}, {1405, 1415}, {1468, 2267}, {1914, 4689}
X(5036) lies on these lines: {3, 6}, {9, 484}, {45, 71}, {966, 2475}, {1213, 2476}, {1400, 2099}, {2209, 4484}, {3196, 3197}
X(5037) lies on these lines: {3, 6}, {9, 976}, {595, 2911}, {609, 2260}, {995, 2174}, {1449, 3509}, {1914, 2273}, {2251, 2277}
X(5038) lies on these lines: {2, 2056}, {3, 6}, {83, 597}, {98, 3815}, {524, 1078}, {542, 1506}, {2023, 3329}
X(5038) = {X(1687),X(1688)}-harmonic conjugate of X(575)
X(5038) = Brocard axis intercept, other than X(2080), of the circle {{X(2080),PU(1)}}
X(5038) = harmonic center of 2nd Lemoine circle and Ehrmann circle
X(5039) lies on these lines: {3, 6}, {69, 83}, {184, 251}, {206, 3203}, {609, 1428}, {732, 3734}, {1078, 3618}
X(5039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,5188), (1687,1688,5085)
X(5040) lies on these lines: {2, 4164}, {31, 4455}, {100, 110}, {111, 739}, {187, 237}, {650, 1980}, {1977, 3124}
X(5040) = isogonal conjugate of X(35147)
X(5040) = crossdifference of every pair of points on line X(2)X(3125)
X(5040) = inverse-in-Parry-isodynamic-circle of X(3230); see X(2)
X(5041) lies on these lines: {3, 6}, {83, 538}, {115, 3850}, {547, 1506}, {597, 3933}, {2548, 3545}, {3329, 3934}
X(5041) = center of inverse-in-Moses-circle-of-circumcircle
X(5041) = radical center of Lucas(10 cot ω) circles
X(5042) lies on these lines: {3, 6}, {9, 2242}, {172, 1743}, {213, 604}, {594, 996}, {1449, 2241}, {4497, 4749}
X(5043) lies on these lines: {3, 6}, {9, 3337}, {31, 4484}, {45, 672}, {1334, 2260}, {2503, 4383}
X(5044) = r*X(3) - (r + 4R)(X(9)
X(5044) = r*X(5) - (r + 2R)*X(10)
X(5044) = X(1) + 3X(210)
X(5044) = 3X(2) + x(72) (Peter Moses, April 3,
2012)
X(5044) lies on these lines: {1, 210}, {2, 72}, {3, 9}, {5, 10}, {6, 975}, {8, 392}, {35, 3683}, {37, 386}, {43, 3931}, {44, 58}, {45, 4255}, {46, 4413}, {56, 3715}, {57, 3927}, {63, 474}, {65, 1698}, {78, 405}, {140, 912}, {191, 1155}, {200, 3295}, {201, 1465}, {226, 3824}, {281, 1871}, {329, 443}, {354, 3624}, {355, 2551}, {404, 3219}, {442, 908}, {496, 4847}, {500, 1818}, {518, 1125}, {519, 4015}, {536, 3159}, {581, 1212}, {631, 1071}, {748, 976}, {756, 1193}, {758, 3634}, {762, 3230}, {899, 2292}, {958, 997}, {966, 3781}, {978, 984}, {1001, 3811}, {1018, 4520}, {1089, 4009}, {1203, 3745}, {1376, 3579}, {1479, 4679}, {1621, 4420}, {1864, 3601}, {2140, 3739}, {2478, 3419}, {2771, 3035}, {2802, 4540}, {3057, 3679}, {3216, 3666}, {3290, 3954}, {3294, 3693}, {3303, 3711}, {3306, 3951}, {3555, 3616}, {3617, 3877}, {3625, 3898}, {3626, 3880}, {3636, 4547}, {3687, 3695}, {3689, 3746}, {3702, 4651}, {3742, 3874}, {3753, 3869}, {3754, 3828}, {3827, 3844}, {3833, 4127}, {3838, 3841}, {3848, 3988}, {3873, 4539}, {3885, 4678}, {3889, 4661}, {3893, 4668}, {3899, 4731}, {3952, 4968}, {4113, 4975}, {4158, 4187}
X(5044) = complement of X(942)
X(5044) = centroid of {A,B,C,X(72)}
Let A′ be the inverse-in-incircle of the A-excenter, and define B′, C′ cyclically. The triangle A′B′C′ is homothetic to ABC, and its circumcenter is X(5045). (Randy Hutson, February 16, 2015)
X(5045) lies on these lines: {1, 3}, {2, 3555}, {5, 3947}, {7, 1058}, {8, 4002}, {10, 3742}, {30, 4298}, {37, 4253}, {58, 1279}, {72, 3616}, {145, 3753}, {210, 3624}, {226, 496}, {355, 938}, {392, 3622}, {404, 3957}, {405, 4666}, {474, 3870}, {495, 1210}, {500, 1458}, {518, 1125}, {519, 3812}, {536, 596}, {550, 4314}, {551, 960}, {582, 1471}, {758, 3636}, {936, 3243}, {946, 971}, {975, 3242}, {1100, 2174}, {1149, 2650}, {1387, 2771}, {1621, 3916}, {1770, 3058}, {2886, 3824}, {2891, 4886}, {2901, 4891}, {3086, 3475}, {3306, 4917}, {3488, 3600}, {3626, 3833}, {3632, 3698}, {3634, 3848}, {3635, 3754}, {3655, 4308}, {3876, 4430}, {3877, 4018}, {3894, 3962}, {3898, 4084}, {3968, 4701}, {4533, 4661}, {4668, 4731}
X(5045) = complement of X(34790)
X(5045) = X(140)-of-intouch-triangle
X(5045) = X(5) of inverse-in-incircle-triangle
As a point on the Euler line, X(5046) has Shinagawa coefficients (abc$a$, - 4S2), and also (R, -2r).
X(5046) lies on these lines: {1, 5080}, {2, 3}, {8, 80}, {10, 3583}, {11, 2975}, {12, 1621}, {35, 3814}, {36, 3825}, {65, 5057}, {78, 3586}, {79, 5883}, {100, 1329}, {115, 5985}, {145, 497}, {153, 944}, {191, 3467}, {210, 5178}, {312, 5016}, {324, 1896}, {341, 5014}, {355, 3877}, {388, 1388}, {519, 4857}, {535, 5563}, {551, 5270}, {908, 950}, {938, 5905}, {952, 5330}, {960, 5086}, {962, 5554}, {1043, 5741}, {1058, 3623}, {1125, 3585}, {1210, 3218}, {1478, 3616}, {1724, 5127}, {1749, 3648}, {1837, 3869}, {1842, 3101}, {1877, 4296}, {1994, 3193}, {2551, 3434}, {2646, 5087}, {2886, 5260}, {3057, 5176}, {3419, 3876}, {3421, 3621}, {3614, 6690}, {3701, 5015}, {3816, 5253}, {3822, 5259}, {3868, 5722}, {3890, 5252}, {3897, 5886}, {3924, 3944}, {4294, 5552}, {4297, 4881}, {4313, 5748}, {4514, 4696}, {4666, 5290}, {4678, 5082}, {4679, 5794}, {4723, 5100}, {5180, 5903}, {5250, 5587}, {5283, 5475}, {5303, 5433}, {5318, 5367}, {5321, 5362}, {5422, 5706}, {5445, 6702}, {5731, 6256}
X(5046) = complement of X(37256)As a point on the Euler line, X(5047) has Shinagawa coefficients (3abc$a$ + 2S2, -2S2).
X(5047) lies on these lines: {1, 748}, {2, 3}, {8, 344}, {9, 3868}, {10, 1621}, {35, 3634}, {81, 1724}, {100, 1698}, {373, 970}, {908, 1125}, {942, 3219}, {956, 3622}, {958, 3304}, {968, 1722}, {993, 3624}, {1320, 3890}, {3074, 3562}, {3214, 3750}, {3216, 4653}, {3295, 3617}, {3336, 3647}, {3583, 3841}, {3683, 3812}, {3697, 3935}, {3701, 3757}, {3740, 4420}, {3748, 4662}, {3889, 4666}, {3915, 4279}
X(5047) = anticomplement of X(17529)
Let I be the incenter of a triangle ABC, let NA be the nine-point circle of the triangle IBC, and define NB and NC cyclically. Let RA be the reflection of NA in the line AI, and define RB and RC cyclically. Then X(5048) is the radical center of the circles RA, RB, RC. See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos #22502, July 7, 2014. For another construction see Antreas Hatzipolakis and Peter Moses, Hyacinthos #28076.
X(5048) lies on these lines: {1, 3}, {8, 1392}, {11, 519}, {33, 1878}, {78, 3893}, {145, 1837}, {210, 3872}, {495, 4870}, {497, 3241}, {513, 4162}, {515, 1317}, {535, 3058}, {950, 3635}, {960, 4861}, {1318, 1320}, {1387, 1737}, {1391, 1870}, {1478, 3656}, {1836, 3476}, {2170, 2348}, {2269, 3723}, {3021, 3328}, {3318, 3319}, {3486, 3623}, {3655, 4302}, {3683, 3877}, {3693, 4919}, {3711, 4915}
X(5048) = reflection of X(1319) in X(1)Let A′ be the midpoint of X(1) and A-intouch point. Define B′, C′ cyclically. The centroid of A′B′C′ is X(5049). (Randy Hutson, February 16, 2015)
X(5049) lies on these lines: {1, 3}, {2, 3921}, {10, 3848}, {72, 3622}, {101, 1100}, {374, 1449}, {392, 3873}, {496, 3817}, {518, 551}, {519, 3742}, {956, 4666}, {960, 3636}, {962, 3296}, {1125, 3740}, {1387, 2801}, {3241, 3753}, {3243, 3940}, {3244, 3812}, {3555, 3616}, {3621, 4002}, {3633, 3698}, {3828, 4711}, {3885, 4004}, {3890, 4018}, {4677, 4731}
X(5049) = X(2)-of-incircle-circles-triangle
Let A′B′C′ be the reflection of ABC in X(6). Let AB = BC∩A′C′ and define BC and CA cyclically. Let AC = BC∩A′B′, and define BA and CB cyclically. The six points AB, BC, CA, AC, BA, CB lie on the 2nd Lemoine circle. Let A″ be the point of intersection of the tangents to the 2nd Lemoine circle at BA and CA, and define B″, C″ cyclically. The centroid of triangle A″B″C″ is X(5050). X(5050) and X(5085) trisect the segment X(3)X(6). Also, X(5050) is the exsimilicenter of the circle with center X(371) and pass-through point X(1151) and the circle with center X(372) and pass-through point X(1152). (Randy Hutson, January 29, 2015)
Continuing, the triangles AABAC, BBCBA, CCACB are pairwise similar and and each inversely similar to ABC. Let SA be the similitude center of BBCBA and CCACB, and define SB and SC cyclically. Then the triangle SASBSC is perspective to ABC at X(6) and homothetic to the circumsymmedial triangle at X(6). Moreover, X(5050) = X(3)-of-SASBSC. (Randy Hutson, October 13, 2015)
X(5050): Let T be a triangle inscribed in the circumcircle and circumscribing the orthic inconic. As T varies, its centroid traces a circle centered at X(5050) with segment X(2)X(14912) as diameter. (Randy Hutson, August 29, 2018)
X(5050) lies on these lines: {2, 3167}, {3, 6}, {5, 3618}, {51, 3796}, {69, 140}, {141, 3526}, {184, 373}, {193, 631}, {381, 597}, {549, 1992}, {611, 999}, {613, 2330}, {632, 3619}, {895, 1511}, {1176, 3527}, {1352, 1656}, {1385, 3751}, {1386, 1482}, {1495, 3066}, {1598, 1974}, {1843, 3517}, {3525, 3620}, {3818, 3851}
X(5050) = reflection of X(5085) in X(182)
X(5050) = radical center of the Lucas(4 tan ω) circles
X(5050) = center of inverse-in-1st-Lemoine-circle-of-circumcircle
X(5050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1351), (371,372,5013), (1687,1688,5034)
As a point on the Euler line, X(5051) has Shinagawa coefficients (2(E + F)2 + 2$bc$(E + F) + abc$a$, -2S2).
X(5051) lies on these lines: {1, 3454}, {2, 3}, {8, 1211}, {10, 321}, {12, 1284}, {45, 1213}, {75, 1228}, {81, 1330}, {115, 1281}, {225, 1441}, {846, 1698}, {984, 4812}, {1046, 4683}, {1193, 3847}, {1230, 4385}, {1962, 3178}, {2298, 4645}, {2901, 3969}, {3017, 3578}, {3214, 4085}, {3695, 3995}, {3704, 4854}
X(5052) lies on these lines: {2, 3787}, {3, 6}, {51, 1196}, {69, 2548}, {76, 193}, {141, 1506}, {251, 1994}, {263, 3117}, {373, 3231}, {538, 1992}, {611, 2241}, {613, 2242}, {726, 4856}, {732, 3629}, {1015, 1469}, {1194, 3060}, {1353, 2782}, {1500, 3056}, {1572, 3751}, {1843, 2211}, {1974, 3202}
X(5052) = reflection of X(39) in X(6)
X(5052) = inverse-in-1st-Brocard-circle of X(5034)
X(5052) = bicentric sum of PU(191)
X(5052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5034), (371,372,5171)
X(5053) lies on these lines: {1, 2267}, {2, 1412}, {3, 6}, {9, 604}, {36, 909}, {44, 101}, {48, 1732}, {59, 672}, {241, 1461}, {527, 1429}, {602, 947}, {661, 3737}, {1174, 2364}, {1449, 1697}, {1474, 4222}, {1630, 1723}, {1724, 2360}, {1731, 2182}, {1766, 3554}, {2223, 3939}, {3684, 4700}
As a point on the Euler line, X(5054) has Shinagawa coefficients (7, -3).
Let T be a triangle inscribed in the circumcircle and circumscribing the Steiner inellipse. As T varies, its centroid traces a circle centered at X(5054) with segment X(2)X(3524) as diameter. (Randy Hutson, August 29, 2018)
X(5054) lies on these lines: {2, 3}, {10, 3655}, {46, 4870}, {55, 3582}, {56, 3584}, {182, 599}, {355, 3828}, {499, 3058}, {519, 3653}, {538, 1153}, {551, 1482}, {568, 3917}, {597, 1351}, {1125, 3656}, {1384, 3815}, {1385, 3679}, {2549, 3054}, {3017, 4255}, {3295, 4995}, {3579, 3624}
X(5054) = complement of X(3545)
X(5054) = anticomplement of X(15699)
X(5054) = circumcircle-inverse of X(37967)
X(5054) = trisector nearest X(2) of segment X(2)X(3)
X(5054) = centroid of X(3)PU(116)
X(5054) = centroid of X(3)PU(117)
X(5054) = centroid of X(3)PU(118
)
X(5054) = centroid of X(3)PU(119)
As a point on the Euler line, X(5055) has Shinagawa coefficients (5, 3).
X(5055) lies on these lines: {2, 3}, {10, 3656}, {355, 551}, {498, 3058}, {499, 3614}, {515, 3653}, {517, 4731}, {539, 3167}, {597, 1352}, {599, 1351}, {946, 3654}, {999, 3582}, {1125, 3655}, {1159, 1737}, {1479, 4995}, {1482, 3679}, {2549, 3055}, {3295, 3584}
X(5055) = homothetic center of Johnson triangle and mid-triangle of Euler and anti-Euler triangles
X(5055) = inverse-in-circle-O(PU(5)) of X(20)
X(5055) = homothetic center of X(140)-altimedial and X(3)-anti-altimedial triangles
X(5055) = {X(2),X(3)}-harmonic conjugate of X(15694)
X(5055) = {X(2043),X(2044)}-harmonic conjugate of X(546)
X(5055) = trisector nearest X(5) of segment X(2)X(5)
X(5055) = trisector nearest X(2) of segment X(2)X(381)
X(5055) = homothetic center of Ehrmann side-triangle and submedial triangle
X(5055) = X(3839)-of-Ehrmann-side-triangle
X(5055) = X(3839)-of-submedial-triangle
As a point on the Euler line, X(5056) has Shinagawa coefficients (3, 2).
X(5056) lies on these lines: {2, 3}, {233, 393}, {355, 3622}, {371, 1132}, {372, 1131}, {388, 3614}, {390, 498}, {485, 3591}, {486, 3590}, {499, 3600}, {637, 3595}, {638, 3593}, {962, 1698}, {1482, 4678}, {1699, 3634}, {3311, 3316}, {3312, 3317}
X(5057) lies on these lines: {1, 535}, {2, 1155}, {4, 8}, {7, 3660}, {10, 3245}, {11, 1776}, {21, 36}, {30, 4511}, {31, 3944}, {46, 4193}, {63, 1699}, {100, 516}, {114, 1281}, {149, 518}, {190, 3006}, {214, 4316}, {226, 1005}, {238, 3120}, {239, 4442}, {243, 3326}, {320, 350}, {388, 3890}, {404, 1770}, {411, 2077}, {484, 1698}, {497, 3873}, {524, 4956}, {527, 1156}, {528, 3935}, {758, 3583}, {901, 1311}, {946, 2975}, {960, 2475}, {1319, 3485}, {1330, 3702}, {1478, 3877}, {1479, 3868}, {1839, 2287}, {2325, 4071}, {2478, 4295}, {2886, 3219}, {2895, 3706}, {3058, 3957}, {3306, 4312}, {3336, 3825}, {3416, 4671}, {3486, 3623}, {3582, 4973}, {3585, 3878}, {3648, 3916}, {3685, 3936}, {3717, 4756}, {3741, 4683}, {3847, 4418}, {3874, 4857}, {3883, 4054}, {3920, 4415}, {4062, 4693}, {4358, 4645}, {4432, 4892}, {4654, 4666}, {4661, 4863}, {4713, 4799}
X(5057) = anticomplement of X(1155)
X(5057) = intersection of antiorthic axes of 1st and 2nd Ehrmann circumscribing triangles
X(5057) = intersection of antiorthic axes of anticevian triangles of PU(4)
X(5058) lies on these lines: {3, 6}, {115, 3071}, {172, 3299}, {251, 588}, {492, 3788}, {590, 1506}, {615, 642}, {1015, 2067}, {1124, 2242}, {1335, 2241}, {1500, 2066}, {1588, 3767}, {1914, 3301}, {2548, 3068}
X(5058) = {X(6),X(32)}-harmonic conjugate of X(5062)
X(5058) = perspector of symmedial triangle and Lucas Brocard triangle
X(5058) = radical center of Lucas(4 tan ω) circles
X(5058) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9738)
X(5058) = {X(371),X(372)}-harmonic conjugate of X(9738)
As a point on the Euler line, X(5059) has Shinagawa coefficients (3, -8).
Let J be the conic {{X(13),X(14),X(15),X(16),X(20)}}. Let U be the line tangent to J at X(15), and let V be the line tangent to J at X(16). Then X(5059) = U∩V. (Randy Hutson, February 16, 2015)
X(5059) lies on these lines: {2, 3}, {40, 4678}, {145, 516}, {323, 1498}, {515, 3621}, {962, 3623}, {1131, 1151}, {1132, 1152}, {3085, 4324}, {3086, 4316}, {3622, 4297}, {4299, 4857}
X(5059) = anticomplement of X(3146)
X(5059) = pole of Brocard axis wrt conic {{X(13),X(14),X(15),X(16),X(20)}}
X(5059) = polar conjugate of isogonal conjugate of X(33636)
X(5060) lies on these lines: {3, 6}, {19, 1247}, {21, 3496}, {36, 163}, {102, 2715}, {110, 1055}, {112, 2708}, {162, 2202}, {691, 2291}, {759, 2702}, {1951, 2249}, {3735, 4653}
X(5061) lies on these lines: {1, 3}, {2, 1397}, {12, 1408}, {59, 4600}, {81, 181}, {98, 2720}, {108, 2699}, {109, 1284}, {604, 750}, {741, 2222}, {899, 1404}, {1428, 3911}
X(5062) lies on these lines: {3, 6}, {115, 3070}, {172, 3301}, {251, 589}, {491, 3788}, {590, 641}, {615, 1506}, {1124, 2241}, {1335, 2242}, {1587, 3767}, {1914, 3299}, {2548, 3069}
X(5062) = {X(6),X(32)}-harmonic conjugate of X(5058)
X(5062) = perspector of symmedial triangle and Lucas(-1) Brocard triangle
X(5062) = radical center of Lucas(-4 - 2 cot ω) circles
X(5062) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9739)
X(5062) = {X(371),X(372)}-harmonic conjugate of X(9739)
Let A′B′C′ be the Trinh triangle. Let A″ be the barycentric product B′*C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5063). (Randy Hutson, October 13, 2015)
X(5063) lies on these lines: {2, 2986}, {3, 6}, {53, 1885}, {160, 1974}, {184, 1576}, {231, 3767}, {468, 3815}, {1968, 1990}, {2393, 3148}, {2549, 3018}, {3087, 3542}
X(5063) = isogonal conjugate of X(34289)
X(5063) = crosssum of X(6) and X(6644)
X(5063) = crossdifference of every pair of points on line X(523)X(11799)
X(5063) = Schoute-circle-inverse of X(37470)
X(5063) = {X(15),X(16)}-harmonic conjugate of X(37470)
As a point on the Euler line, X(5064) has Shinagawa coefficients (F, 3E + 3F).
Let W be the orthocentroidal circle, L the line tangent to W at the P(4)-Ceva conjugate of U(4) and M the line tangent to W at the U(4)-Ceva conjugate of P(4). Then X(5064) = L∩M. (Randy Hutson, February 16, 2015)
X(5064) lies on these lines: {2, 3}, {51, 1853}, {115, 1184}, {394, 3818}, {524, 3867}, {553, 1892}, {599, 1843}, {1498, 3574}, {1829, 3679}, {1876, 4654}
X(5064) = pole wrt orthocentroidal circle of line X(428)X(523)
X(5064) = {X(2),X(4)}-harmonic conjugate of X(428)
X(5064) = homothetic center of orthic triangle and reflection of tangential triangle in X(2)
X(5065) lies on these lines: {2, 801}, {3, 6}, {393, 2549}, {1249, 1968}, {1950, 2082}, {1951, 2285}, {2241, 3554}, {2242, 3553}, {2548, 3087}, {3618, 4558}
X(5065) = isogonal conjugate of X(37874)
X(5065) = crosspoint of X(2) and X(14457)
X(5065) = crosssum of X(i) and X(j) for these {i,j}: {2, 11433}, {6, 17928}, {1146, 2517}
X(5065) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9729)
X(5065) = {X(371),X(372)}-harmonic conjugate of X(9729)
As a point on the Euler line, X(5066) has Shinagawa coefficients (5, 9).
X(5066) lies on these lines: {2, 3}, {517, 3956}, {597, 3818}, {946, 4669}, {952, 3817}, {1699, 3654}, {3583, 4995}, {3584, 3614}, {3656, 4677}
X(5066) = reflection of X(i) in X(j) for these (i,j): (5,11737), (547,5)
X(5066) = complement of X(8703)
X(5066) = {X(2),X(3)}-harmonic conjugate of X(15713)
X(5066) = {X(2),X(4)}-harmonic conjugate of X(3534)
X(5066) = {X(2),X(5)}-harmonic conjugate of X(10109)
X(5066) = midpoint of X(5) and X(381)
As a point on the Euler line, X(5067) has Shinagawa coefficients (4, 1).
X(5067) lies on these lines: {2, 3}, {6, 3316}, {373, 3567}, {498, 1058}, {499, 1056}, {944, 3624}, {3614, 4293}, {3634, 4301}
X(5067) = {X(3316),X(3317)}-harmonic conjugate of X(6)
As a point on the Euler line, X(5068) has Shinagawa coefficients (3, 4).
X(5068) lies on these lines: {2, 3}, {8, 3817}, {355, 3623}, {497, 3614}, {946, 3617}, {1131, 3069}, {1132, 3068}, {3085, 4857}
X(5068) = {X(3544),X(3545)}-harmonic conjugate of X(5)
X(5069) lies on these lines: {2, 3770}, {3, 6}, {37, 2275}, {42, 3941}, {44, 2277}, {749, 1100}, {980, 3589}, {3056, 4735}, {3061, 4016}
As a point on the Euler line, X(5070) has Shinagawa coefficients (7, 1).
X(5070) lies on these lines: {2, 3}, {17, 3411}, {18, 3412}, {373, 1216}, {1351, 3763}, {1482, 1698}, {2548, 3054}, {3055, 3767}
X(5070) = homothetic center of X(140)-altimedial and X(2)-anti-altimedial triangles
As a point on the Euler line, X(5071) has Shinagawa coefficients (4, 3).
X(5071) lies on these lines: {2, 3}, {388, 3582}, {497, 3584}, {542, 3618}, {1587, 3317}, {1588, 3316}, {3086, 3614}, {3817, 3828}
X(5071) = centroid of cross triangle of Euler and anti-Euler triangles
X(5071) = {X(2),X(3)}-harmonic conjugate of X(15709)
X(5071) = {X(3090),X(3545)}-harmonic conjugate of X(2)
As a point on the Euler line, X(5072) has Shinagawa coefficients (5, 7).
X(5072) lies on these lines: {2, 3}, {355, 3635}, {946, 4691}, {1351, 3630}, {1482, 3625}, {3295, 3614}
X(5072) = {X(3090),X(3091)}-harmonic conjugate of X(546)
As a point on the Euler line, X(5073) has Shinagawa coefficients (3, -11).
Let OA be the circle centered at A with radius 2*sqrt(b2 + c2), and define OB and OC cyclically. Then X(5073) is the radical center of OA, OB, OC. (Randy Hutson, February 16, 2015)
X(5073) lies on these lines: {2, 3}, {516, 4701}, {999, 4857}, {2996, 3793}, {3357, 3581}, {3426, 3519}
X(5073) = circumcircle-inverse of X(37968)
X(5073) = {X(381),X(382)}-harmonic conjugate of X(3627)
X(5074) lies on these lines: {2, 5011}, {3, 142}, {101, 4872}, {116, 517}, {226, 1323}, {304, 4153}, {514, 661}
X(5074) = isotomic conjugate of X(37213)
X(5074) = complement of X(5011)
X(5075) = center of this circle: V(X(846)) = {{15,16,846,1054,1283,5197}}; see the preamble to X(6137).
X(5075) lies on these lines: {109, 110}, {111, 2291}, {187, 237}, {659, 1769}, {846, 2786}, {4414, 4750}
X(5075) = isogonal conjugate of X(35154)
X(5075) = crossdifference of every pair of points on line X(2)X(9317)
X(5075) = inverse-in-Parry-isodynamic-circle of X(1055); see X(2)
As a point on the Euler line, X(5076) has Shinagawa coefficients (1, -13).
X(5076) lies on these lines: {2, 3}, {355, 4746}, {517, 4816}, {3303, 3585}, {3304, 3583}
X(5076) = {X(381),X(382)}-harmonic conjugate of X(1657)
As a point on the Euler line, X(5077) has Shinagawa coefficients (2(E + F)2 + 3S2, -9S2).
X(5077) lies on these lines: {2, 3}, {6, 3849}, {183, 671}, {524, 2549}, {543, 599}
X(5077) = Artzt-to-anti-Artzt similarity image of X(3)
X(5078) lies on these lines: {1, 3}, {22, 3052}, {197, 4383}, {595, 2915}, {1979, 2076}
As a point on the Euler line, X(5079) has Shinagawa coefficients (7, 5).
X(5079) lies on these lines: {2, 3}, {355, 3636}, {999, 3614}, {1351, 3631}, {1482, 3626}
X(5079) = complement of X(10299)
X(5079) = homothetic center of X(5)-altimedial and X(20)-anti-altimedial triangles
X(5079) = {X(2),X(4)}-harmonic conjugate of X(3530)
X(5079) = {X(2),X(5)}-harmonic conjugate of X(3851)
X(5080) lies on these lines: {2, 36}, {4, 8}, {5, 2975}, {10, 191}, {11, 529}, {12, 21}, {20, 2077}, {30, 100}, {56, 4193}, {79, 3754}, {80, 758}, {119, 4996}, {145, 1479}, {149, 519}, {153, 515}, {316, 668}, {377, 1155}, {381, 956}, {388, 1319}, {404, 1329}, {452, 2078}, {495, 1621}, {497, 3241}, {498, 4189}, {513, 2517}, {666, 671}, {946, 4861}, {958, 2476}, {1168, 4080}, {1330, 2392}, {1698, 4652}, {1699, 3872}, {1737, 3218}, {1793, 2222}, {1837, 3868}, {1877, 4318}, {3244, 4857}, {3245, 3617}, {3586, 3870}, {3614, 4999}, {4188, 4299}
X(5080) = isogonal conjugate of X(34442)
X(5080) = isotomic conjugate of isogonal conjugate of X(20989)
X(5080) = polar conjugate of isogonal conjugate of X(22123)
X(5080) = anticomplement of X(36)
X(5080) = inverse-in-anticomplementary-circle of X(8)
X(5081) lies on these lines: {4, 8}, {10, 275}, {25, 3705}, {27, 3687}, {29, 270}, {30, 2968}, {33, 3872}, {34, 78}, {36, 4242}, {69, 273}, {75, 317}, {102, 515}, {162, 447}, {186, 4996}, {200, 4680}, {225, 4101}, {239, 297}, {242, 1884}, {264, 319}, {280, 3146}, {281, 391}, {320, 340}, {458, 3661}, {518, 1875}, {519, 1785}, {521, 1948}, {758, 1845}, {765, 1861}, {860, 1870}, {1043, 3559}, {1325, 2766}, {1841, 3965}, {1852, 3704}, {1876, 4645}, {1990, 4969}, {2202, 3684}, {2322, 3686}, {2345, 3087}, {4853, 4894}
X(5081) = anticomplement of isogonal conjugate of X(36121)
X(5082) lies on these lines: {1, 142}, {2, 496}, {4, 8}, {7, 3555}, {10, 497}, {20, 956}, {40, 4847}, {65, 4863}, {69, 2891}, {100, 631}, {145, 377}, {149, 1145}, {200, 946}, {376, 2975}, {388, 519}, {390, 405}, {515, 4853}, {518, 4295}, {528, 958}, {938, 3753}, {944, 3872}, {966, 3294}, {1000, 3885}, {1210, 1706}, {1376, 3086}, {1478, 3632}, {1479, 2551}, {1699, 4882}, {2475, 3621}, {2886, 3085}, {3296, 3889}, {3303, 3925}, {3485, 3811}, {3487, 3870}, {3583, 4668}, {3585, 4677}, {3983, 4679}, {4421, 4999}
X(5083) is the center of the rectangular hyperbola, H, that is the locus of perspectors of the pedal triangle of X(1) (the intouch triangle) and A′B′C′, where AA′/AX(1) = BB′/BX(1) = CC′/CX(1). Also, H is the Jerabek hyperbola of the intouch triangle, and H passes through X(1), X(7), X(65), X(145), X(224), X(1071), X(1317), X(1537), X(3174), X(3649), X(5586), and the vertices of the intouch triangle. (Randy Hutson, February 16, 2015)
X(5083) lies on these lines: {1, 104}, {7, 149}, {11, 118}, {56, 214}, {57, 100}, {65, 1317}, {73, 3953}, {80, 388}, {119, 1210}, {153, 938}, {244, 4551}, {518, 3035}, {528, 553}, {651, 1421}, {758, 1319}, {942, 952}, {950, 2829}, {1071, 1537}, {1145, 3555}, {1320, 3340}, {1387, 2771}, {1388, 3878}, {1420, 3868}, {1457, 4694}, {1465, 3999}, {1466, 2932}, {1787, 3333}, {1862, 1876}, {2078, 3218}, {2099, 3892}, {2835, 3937}, {3036, 3812}, {3256, 3957}, {3738, 4458}
X(5083) = X(125)-of-intouch-triangle
X(5083) = midpoint of X(i) and X(j) for these {i,j}: {65,1317}, {1071, 1537}
As a point on the Euler line, X(5084) has Shinagawa coefficients (abc$a$, -S2).
X(5084) lies on these lines: {1, 2551}, {2, 3}, {8, 392}, {9, 1210}, {10, 497}, {12, 4423}, {65, 4679}, {72, 938}, {78, 3488}, {145, 3940}, {226, 1467}, {329, 942}, {387, 4383}, {388, 1125}, {908, 3487}, {936, 950}, {958, 3086}, {962, 3753}, {966, 4266}, {997, 3486}, {1001, 1329}, {1056, 3436}, {1376, 4294}, {1478, 3624}, {1479, 1698}, {2078, 3814}, {2899, 4385}, {3295, 3820}, {3670, 4419}, {3812, 4295}, {3983, 4863}
X(5085) and X(5050) trisect the segment X(3)X(6). Also, X(5085) is the exsimilicenter of the circle with center X(1151) and pass-through point X(371) and the circle with center X(1152) and pass-through point X(372). (Randy Hutson, January 29, 2015)
X(5085) lies on these lines: {2, 154}, {3, 6}, {4, 3589}, {20, 3618}, {23, 3066}, {25, 373}, {26, 2916}, {35, 613}, {36, 611}, {40, 1386}, {55, 1428}, {56, 2330}, {64, 1176}, {69, 3523}, {140, 1352}, {141, 631}, {206, 1498}, {376, 597}, {518, 3576}, {524, 3524}, {549, 599}, {1177, 2935}, {1385, 3242}, {1407, 3955}, {1511, 2930}, {1593, 1974}, {1656, 3818}, {1843, 3515}, {3167, 3819}, {4220, 4383}
X(5085) = reflection of X(5050) in X(182)
X(5085) = radical center of the Lucas(tan ω) circles
X(5085) = centroid of the triangle X(4)X(6)X(20)
X(5085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1350), (1687,1688,5039)
X(5086) lies on these lines: {1, 2476}, {2, 1837}, {4, 8}, {5, 4511}, {10, 21}, {65, 2475}, {79, 4084}, {145, 3485}, {149, 3057}, {153, 2894}, {224, 4197}, {388, 3873}, {404, 1737}, {411, 515}, {497, 3890}, {758, 3585}, {950, 1621}, {952, 4861}, {997, 4193}, {1441, 2893}, {1478, 3868}, {1479, 3877}, {1698, 4855}, {1788, 4190}, {1826, 2287}, {3583, 3878}, {3621, 4863}, {3884, 4857}, {4325, 4973}
X(5087) lies on these lines: {2, 1155}, {5, 10}, {11, 518}, {36, 405}, {65, 4193}, {120, 3259}, {145, 1837}, {149, 3689}, {226, 3660}, {381, 997}, {388, 1319}, {429, 1878}, {513, 3716}, {516, 1538}, {535, 1125}, {942, 3825}, {1001, 2078}, {1376, 1699}, {1647, 3999}, {1698, 3245}, {2077, 3149}, {3006, 4009}, {3011, 3246}, {3705, 3967}, {3752, 3944}, {3812, 4187}, {3829, 4847}, {4442, 4706}
X(5087) = complement of X(1155)
X(5088) lies on these lines: {1, 7}, {3, 85}, {4, 348}, {28, 242}, {30, 1565}, {36, 1111}, {46, 3212}, {56, 3673}, {75, 956}, {104, 927}, {150, 515}, {169, 3177}, {187, 4403}, {239, 514}, {273, 1804}, {304, 1975}, {411, 1446}, {517, 664}, {675, 1308}, {910, 3732}, {934, 2723}, {942, 1434}, {1366, 3328}, {2369, 2736}, {2646, 4059}, {3665, 4911}
X(5089) lies on these lines: {1, 607}, {2, 92}, {9, 608}, {19, 25}, {28, 1390}, {34, 1212}, {111, 2766}, {112, 2074}, {225, 1855}, {230, 231}, {427, 1826}, {428, 1839}, {614, 1108}, {653, 1447}, {672, 1876}, {976, 1973}, {1172, 2346}, {1334, 1902}, {1729, 1771}, {1783, 1870}, {1829, 2333}, {1861, 3693}, {2299, 3745}, {2322, 3757}, {2340, 2356}
X(5089) = isogonal conjugate of X(1814)
X(5089) = PU(4)-harmonic conjugate of X(650)
X(5089) = pole wrt polar circle of trilinear polar of X(2481) (the line X(2)X(650))
X(5089) = X(48)-isoconjugate (polar conjugate) of X(2481)
X(5090) lies on these lines: {1, 427}, {4, 8}, {10, 25}, {19, 594}, {27, 3661}, {33, 429}, {34, 1883}, {40, 3575}, {65, 66}, {80, 1039}, {239, 469}, {388, 1876}, {428, 3679}, {468, 1698}, {515, 1593}, {607, 1826}, {944, 3088}, {952, 1595}, {1385, 3541}, {1843, 3416}, {1861, 1891}, {2204, 4426}, {3516, 4297}
X(5091) is the point of intersection, other than X(3), of the Brocard circle and the line X(1)X(3). (Randy Hutson, February 16, 2015)
X(5091) lies on the Brocard circle and these lines: {1, 3}, {2, 1083}, {6, 513}, {7, 59}, {81, 3110}, {516, 1428}, {651, 4014}, {692, 1086}, {760, 3218}, {840, 1002}, {1023, 4413}, {1026, 1376}, {1290, 2711}, {1397, 3474}, {1404, 3000}, {1572, 2087}, {1618, 2175}, {1633, 3271}, {2330, 3663}, {2720, 2724}, {3735, 4414}, {4440, 4579}
X(5091) = orthogonal projection of X(6) on line X(1)X(3)
X(5091) = X(100)-of-1st-Brocard-triangle
X(5091) = 1st-Brocard-isogonal conjugate of X(2787)
X(5091) = 1st-Brocard-isotomic conjugate of X(24290)
X(5092) is the centroid of the quadrilateral {{X(3),X(4),X(6),X(20)}}. (Randy Hutson, Febrary 16, 2015)
In the plane of a triangle ABC, let
O = X(3) = circumcenter
K = X(6) = symmedian point
AbAc = line through K parallel to BC, and define BcBa and CaCb cyclically
U = circumcircle of A, Ab, Ac
V = circumcircle of B, C, Cb, Bc
{A1, A2} = U∩V, and define {B1, B2} and {C1, C2} cyclically.
The six points A1, A2, B2, B2, C2, C3 lie on a circle, here named the Dao-symmedial circle, with center X(5092). See Dao-symmedial circle, in which X(5092) is labeled as M. (Dao Thanh Oai, February 11, 2017)
Let u be the radius of the Dao-symmedial circle. Then
u2 = R2/2 + |OK|2/16
u2 = 3R2(3 - (S/Sω)2) /16
u2 = 3R2(3 - tan2ω) /16
(César Lozada, February 13, 2017)
X(5092) lies on these lines: {2, 1495}, {3, 6}, {23, 373}, {30, 3589}, {35, 1428}, {36, 2330}, {69, 3431}, {74, 827}, {140, 1503}, {141, 542}, {184, 3819}, {186, 1843}, {206, 4550}, {323, 3917}, {376, 3618}, {378, 1974}, {631, 1352}, {1386, 3579}, {2070, 2916}, {3523, 3620}, {3530, 3564}, {3934, 4048}
X(5092) = midpoint of X(3) and X(182)
X(5092) = reflection of X(575) in X(182)
X(5092) = {X(3),X(6)}-harmonic conjugate of X(3098)
X(5092) = intersection of Brocard axes of ABC and X(2)-Brocard triangle
X(5092) = {X(15),X(16)}-harmonic conjugate of X(39)
X(5092) = X(575)-Gibert-Moses centroid; see the preamble just before X(21153)
X(5093) lies on these lines: {3, 6}, {4, 1353}, {5, 193}, {25, 1994}, {30, 5032}, {49, 1974}, {51, 3167}, {69, 1656}, {143, 3517}, {155, 3527}, {373, 394}, {381, 1992}, {399, 895}, {518, 4930}, {1352, 3629}, {1482, 3751}, {1503, 3830}, {1993, 5020}, {3066, 3292}, {3526, 3618}, {3620, 3628}
X(5093) = isogonal conjugate of isotomic conjugate of X(34803)
X(5093) = centroid of polar triangle of 2nd Lemoine circle
X(5093) = center of inverse-in-2nd-Lemoine-circle-of-circumcircle
X(5093) = {X(371),X(372)}-harmonic conjugate of X(5023)
X(5093) = pole of Lemoine axis wrt 2nd Lemoine circle
As a point on the Euler line, X(5094) has Shinagawa coefficients (3F, E + F).
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∩BC. The triangle A′B′C′ is homothetic to ABC, and the center of homothety is X(5094). (Randy Hutson, February 16, 2015)
X(5094) lies on these lines:
{2, 3}, {6, 67}, {12, 1398}, {53, 3055}, {126, 136}, {183, 340}, {184,
1853}, {232, 566}, {264, 2970}, {281, 2969}, {1235, 3266}, {1351,
3580}, {1506, 2207}, {1698, 1829}, {1843, 3763}, {1892, 3911}, {1990,
3815}, {2453, 3258}, {2548, 3172}
X(5094) = complement of X(7493)
X(5094) = circumcircle-inverse of X(37969)
X(5094) = intersection of tangents to orthocentroidal circle at PU(4)
X(5094) = pole of orthic axis wrt orthocentroidal circle
X(5094) = pole wrt polar circle of trilinear polar of X(598) (the line X(351)X(523))
X(5094) = X(48)-isoconjugate (polar conjugate) of X(598)
X(5094) = harmonic center of polar circle and {circumcircle, nine-point circle}-inverter
X(5094) = homothetic center of orthic triangle and X(2)-Ehrmann triangle; see X(25)
X(5094) = Euler line intercept, other than X(381), of circle {X(381),PU(4)}
X(5094) = homothetic center of the AOA and AAOA triangles
X(5095) lies on the orthic inconic and these lines: {4, 542}, {6, 67}, {25, 2930}, {110, 193}, {113, 3564}, {114, 2407}, {184, 1177}, {185, 1205}, {468, 524}, {511, 1986}, {868, 3163}, {1112, 1843}, {1829, 2836}, {1839, 2969}, {1858, 3270}, {1899, 2892}, {2452, 2794}
X(5095) = reflection of X(125) in X(6)X(5096) lies on these lines: {3, 6}, {21, 3589}, {22, 4383}, {35, 1386}, {36, 518}, {44, 3220}, {56, 976}, {69, 4188}, {141, 404}, {474, 3763}, {656, 3733}, {674, 1428}, {1155, 3827}, {2915, 2916}, {3618, 4189}
X(5097) lies on these lines: {3, 6}, {5, 3629}, {51, 110}, {323, 373}, {524, 547}, {542, 1353}, {1352, 1992}, {1503, 3853}, {3533, 3618}, {3543, 5032}, {3564, 3850}, {3628, 3631}, {3818, 3832}
X(5097) = midpoint of X(6) and X(576)X(5098) lies on these lines: {110, 919}, {111, 840}, {187, 237}, {244, 661}, {513, 3290}, {518, 650}, {523, 3726}, {672, 3709}, {1638, 4776}, {1914, 2605}, {3509, 3737}, {3700, 4358}, {3797, 4467}
Let ABC be a triangle with orthic triangle DEF, and let L be a line in the plane of ABC. Let A′ be the reflection of A in L, and define B′ and C′ cyclically. The circumcircles of DB′C′, EC′A′, FA′C′ concur. Antreas Hatzipolakis, Anolpolis #816, September 2013.
If L is the Euler line of ABC, then the circumcircles concur in X(5099). Seiichi Kirikami, September 25, 2013.
X(5099) is the center of hyperbola H = {{A,B,C,X(4),X(23)}}. H is the isogonal conjugate of line X(3)X(67) and the isotomic conjugate of line X(67)X(69). Also, H passes through X(316), X(842), X(1383). Moreover, H intersects the circumcircle at X(842) and is tangent to line X(4)X(67) at X(4). (Randy Hutson, January 29, 2015)
As a line L varies through the set of all lines that pass through X(2492), the locus of the trilinear pole of L is a circumconic, and its center is X(5099). (Randy Hutson, January 29, 2014)
Let A′B′C′ be the orthic triangle. Let La be the Fermat axis of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5099), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)
X(5099) lies on the nine-point circle, the 2nd Steiner circle, the cevian circle of X(23), and these lines: {2, 691}, {4, 842}, {23, 316}, {30, 114}, {113, 511}, {115, 523}, {125, 512}, {126, 625}, {132, 403}, {187, 468}, {381, 2453}, {690, 2682}, {868, 1649}
X(5099) = midpoint of X(i) and X(j) for these {i,j}: {4,842}, {23,316}
X(5099) = complement of X(691)
X(5099) = crosssum of circumcircle intercepts of line X(3)X(67)
X(5099) = perspector of the circumconic centered at X(2492)
X(5099) = X(2)-Ceva conjugate of X(2492)
X(5099) = inverse-in-polar-circle of X(935)
X(5099) = inverse-in-{circumcircle, nine-point circle}-inverter of X(2770)
X(5099) = reflection of X(115) in Euler line
X(5099) = Λ(X(115), X(125)) with respect to the orthic triangle
X(5099) = orthopole of line X(3)X(67)
X(5099) = Kirikami-six-circles image of X(23)
X(5100) lies on these lines: {1, 3836}, {4, 8}, {10, 4514}, {149, 3701}, {319, 1269}, {341, 1479}, {3006, 3871}, {3496, 4119}, {3555, 4645}, {3625, 4792}, {3632, 4680}, {3679, 4894}, {3992, 4857}
X(5101) lies on these lines: {1, 1883}, {4, 8}, {10, 4186}, {11, 33}, {19, 428}, {25, 1376}, {429, 1717}, {1709, 1726}, {1753, 3575}, {1830, 1836}, {1837, 1853}, {1864, 1899}, {1891, 4214}
X(5102) lies on these lines: {3, 6}, {4, 3629}, {154, 3060}, {193, 3832}, {323, 3066}, {524, 3545}, {547, 599}, {1352, 3850}, {1503, 1992}, {3090, 3631}, {3533, 3589}, {3564, 3845}
X(5102) = centroid of X(4)X(6)X(193)
X(5103) lies on these lines: {2, 2076}, {4, 4048}, {5, 141}, {6, 5025}, {83, 316}, {115, 732}, {325, 698}, {597, 1692}, {1503, 2456}, {1570, 3629}
X(5104) lies on these lines: {3, 6}, {22, 2056}, {23, 352}, {99, 524}, {111, 694}, {141, 316}, {599, 3734}, {625, 3763}, {1915, 2979}, {1971, 2781}
X(5104) = reflection of X(6) in X(187)X(5105) lies on these lines: {1, 2321}, {3, 6}, {9, 1193}, {37, 995}, {42, 1449}, {43, 3686}, {966, 3216}, {1201, 3247}, {2276, 2300}, {3214, 4034}
X(5106) lies on these lines: {2, 99}, {6, 694}, {32, 110}, {39, 373}, {187, 237}, {1384, 1613}, {1976, 5033}, {2021, 3291}, {3051, 5008}
X(5106) = inverse-in-Parry-isodynamic-circle of X(669); see X(2)
X(5107) lies on these lines: {3, 6}, {69, 625}, {111, 323}, {115, 524}, {193, 316}, {352, 3291}, {843, 3565}, {1992, 2549}, {2502, 3292}
X(5107) = reflection of X(187) in X(6)
X(5107) = inverse-in-Ehrmann-circle of X(574)
X(5107) = {X(33517), X(33518)}-harmonic conjugate of X(13449)
X(5108) lies on the Brocard circle and these lines: {2, 6}, {3, 669}, {99, 2502}, {110, 5026}, {126, 542}, {805, 2770}, {1078, 2142}, {1316, 3734}, {3124, 4563}
X(5108) = anticomplement of X(32525)
X(5108) = X(111)-of-1st-Brocard-triangle
X(5108) = X(111)-of-X(2)-Brocard-triangle
X(5108) = 1st-Brocard-isogonal conjugate of X(543)
X(5108) = intersection, other than X(2), of the Hutson-Parry circles of the inner and outer Vecten triangles
X(5108) = orthogonal projection of X(3) on line X(2)X(6)
X(5108) = intersection of lines X(2)X(6) of antipedal triangles of PU(1)
X(5109) lies on these lines: {1, 3943}, {3, 6}, {37, 1201}, {42, 678}, {44, 1193}, {45, 995}, {1100, 2295}, {1404, 2594}, {3293, 4969}
X(5110) lies on these lines: {3, 6}, {21, 992}, {35, 2300}, {48, 2276}, {55, 1964}, {141, 332}, {171, 1100}, {2268, 2277}
(5111) lies on these lines: {3, 6}, {316, 3629}, {323, 3124}, {385, 2023}, {694, 2987}, {1915, 3060}, {1993, 2056}
X(5111) = radical trace of 2nd Lemoine circle and Ehrmann circle
As a point on the Euler line, X(5112) has Shinagawa coefficients (3(E + F)F + S2, -(E + F)2 - 3S2).
X(5112) lies on these lines: {2, 3}, {373, 4045}, {523, 3569}, {754, 3292}, {1495, 2794}, {1555, 2777}, {2782, 3580}
X(5112) = nine-point-circle-inverse of X(37988)
X(5113) lies on these lines: {110, 827}, {111, 755}, {187, 237}, {620, 690}, {826, 4142}, {888, 2492}, {2485, 3221}
X(5113) = crossdifference of PU(147)
X(5113) = isogonal conjugate of isotomic conjugate of X(9479)
X(5113) = X(2)-Ceva conjugate of X(39079)
X(5113) = perspector of hyperbola {{A,B,C,X(6),X(1031),X(2076),X(34214),X(35511)}}
X(5114) lies on these lines: {3, 6}, {42, 1397}, {44, 993}, {213, 2267}, {2175, 2309}, {2268, 2300}, {2276, 2323}
X(5115) lies on these lines: {2, 757}, {3, 6}, {31, 1100}, {37, 1468}, {172, 3204}, {560, 2308}, {1107, 2214}
X(5116) lies on these lines: {2, 4048}, {3, 6}, {384, 3589}, {732, 1078}, {2056, 3819}, {2211, 3520}, {3552, 3618}
X(5116) = radical center of the Lucas(csc 2ω) circles
X(5116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2076), (1340,1341,32)
As a point on the Euler line, X(5117) has Shinagawa coefficients (2(E + F)F, (E + F)2 - S2).
X(5117) lies on these lines: {2, 3}, {141, 3186}, {275, 3406}, {2052, 3399}, {2887, 3061}, {3096, 3819}
X(5118) lies on these lines: {3, 6}, {99, 110}, {512, 2421}, {691, 805}, {1316, 3734}
X(5119) = (R - r)X(1) + 2r*X(3)
X(5119) = r(r + 4R)*X(9) - R(2r - R)X(80)
X(5119) = 2rR*X(8) + (R2 - 2rR - r2)*X(90)
Let A′B′C′ be the orthic triangle of ABC. Let LA be the antiorthic axis of AB′C′, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. Then A″B″C″ is inversely similar to ABC, with similitude center X(9), and X(5119) = A″B″C″-to-ABC similarity image of X(1). (Randy Hutson, December 4, 2014)
For a construction of X(5119) see Peter Moses, Euclid 962
X(5119) lies on these lines: {1, 3}, {8, 90}, {9, 80}, {10, 1479}, {63, 519}, {71, 1723}, {72, 2900}, {78, 3878}, {100, 997}, {145, 4305}, {169, 1334}, {190, 4737}, {191, 2136}, {376, 1000}, {388, 1770}, {392, 1376}, {404, 3890}, {495, 1836}, {497, 1737}, {498, 946}, {515, 1709}, {516, 1478}, {549, 1387}, {551, 3306}, {674, 3751}, {748, 4695}, {758, 3870}, {846, 855}, {920, 3486}, {944, 1158}, {950, 1728}, {956, 3880}, {962, 3085}, {993, 2802}, {1001, 3753}, {1056, 3474}, {1058, 1788}, {1150, 3902}, {1253, 1718}, {1317, 3655}, {1449, 4268}, {1532, 1699}, {1571, 2275}, {1572, 2276}, {1698, 1706}, {1702, 3301}, {1703, 3299}, {1708, 3488}, {1717, 1773}, {1722, 3987}, {1742, 2807}, {1745, 2943}, {1763, 3465}, {1766, 2269}, {1824, 4186}, {2082, 3730}, {2270, 3731}, {2975, 3885}, {3058, 3654}, {3158, 3899}, {3208, 3496}, {3218, 3241}, {3243, 3894}, {3656, 4995}, {3689, 3940}, {3729, 4692}, {3811, 3869}, {3820, 4679}, {3915, 4642}, {3929, 4677}, {4067, 4917}, {4189, 4861}, {4384, 4714}
X(5119) = reflection of X(1) in X(55)
X(5119) = {X(1),X(40)}-harmonic conjugate of X(46)
X(5119) = X(22)-of-reflection-triangle-of X(1)
X(5119) = {X(1),X(3)}-harmonic conjugate of X(37618)
X(5120) lies on these lines: {2, 1014}, {3, 6}, {9, 56}, {36, 198}, {37, 999}, {44, 2178}, {46, 2262}, {48, 218}, {55, 1449}, {219, 604}, {268, 1741}, {391, 404}, {474, 966}, {517, 3554}, {602, 1622}, {859, 1778}, {956, 2345}, {1055, 3217}, {1100, 3295}, {1108, 1766}, {1172, 1593}, {1260, 2352}, {1376, 3686}, {1385, 3553}, {1402, 4047}, {1420, 2324}, {1436, 2270}, {1444, 3618}, {1445, 1804}, {1475, 2268}, {1486, 4497}, {1604, 2183}, {1723, 2182}, {1728, 1903}, {1732, 2261}, {2256, 3730}, {2257, 3428}, {2260, 2267}, {2343, 3451}, {3247, 3304}
X(5120) = {X(3),X(6)}-harmonic conjugate of X(4254)
X(5121) lies on these lines: {1, 2}, {11, 1738}, {88, 5057}, {105, 2743}, {109, 238}, {244, 908}, {516, 1054}, {518, 3756}, {982, 3452}, {988, 5084}, {1086, 5087}, {1279, 3035}, {1362, 3660}, {2254, 3667}, {3752, 3816}
X(5121) = complement of X(5205)
X(5121) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1)
X(5121) = radical trace of incircle and excircles-radical circle
X(5122) lies on these lines: {1, 3}, {7, 3524}, {28, 1878}, {30, 3911}, {72, 4188}, {140, 4292}, {186, 1876}, {226, 549}, {404, 3219}, {474, 3305}, {513, 4401}, {518, 4973}, {535, 3828}, {548, 950}, {550, 1210}, {582, 603}, {910, 5030}, {938, 3528}, {1439, 3431}, {3476, 3654}, {3530, 3982}, {3534, 3586}, {3752, 4257}, {3897, 4004}, {3928, 3940}, {4742, 4781}
X(5123) lies on these lines: {2, 1319}, {5, 10}, {8, 1392}, {9, 484}, {11, 3880}, {12, 3812}, {36, 474}, {377, 1155}, {495, 3742}, {513, 3823}, {515, 3035}, {518, 1737}, {519, 1387}, {529, 3911}, {535, 3828}, {1012, 1376}, {1878, 1883}, {2476, 3698}, {3057, 4193}, {3586, 4421}, {3634, 4999}, {3753, 3838}, {3992, 4858}, {4711, 4847}
X(5123) = complement of X(1319)
X(5124) lies on these lines: {2, 1029}, {3, 6}, {35, 1100}, {36, 37}, {45, 1696}, {55, 4497}, {56, 2171}, {141, 1444}, {165, 3554}, {198, 2265}, {404, 1213}, {594, 2975}, {672, 2174}, {966, 4188}, {992, 4225}, {1006, 1901}, {1078, 3770}, {1172, 3520}, {1449, 5010}, {2238, 4210}, {3252, 3446}, {3815, 4220}
X(5124) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37509)
X(5124) = {X(371),X(372)}-harmonic conjugate of X(37509)
As a point on the Euler line, X(5125) has Shinagawa coefficients ($a$F, -$aSA$ - abc).
X(5125) lies on these lines: {2, 3}, {6, 2907}, {8, 278}, {10, 92}, {19, 4429}, {34, 78}, {46, 1748}, {75, 225}, {158, 1737}, {162, 1724}, {208, 1445}, {243, 1837}, {608, 4645}, {653, 1118}, {960, 1888}, {1068, 1897}, {1096, 1722}, {1210, 1785}, {1848, 1869}, {1859, 3812}, {1871, 3753}
X(5125) = complement of X(7538)
X(5125) = anticomplement of X(7515)
X(5125) = pole wrt polar circle of trilinear polar of X(1751) (the line X(523)X(663))
X(5125) = X(48)-isoconjugate (polar conjugate) of X(1751)
X(5126) lies on these lines: {1, 3}, {5, 4311}, {44, 101}, {104, 971}, {106, 1279}, {214, 518}, {495, 4315}, {496, 4297}, {513, 1960}, {516, 1387}, {535, 1125}, {631, 4308}, {934, 953}, {952, 3911}, {1055, 2246}, {1483, 4848}, {1538, 2829}, {1878, 4222}, {3474, 3656}, {3634, 4999}, {3935, 4881}
X(5126) = midpoint of X(38013) and X(38014)X(5127) lies on these lines: {1, 21}, {3, 501}, {5, 580}, {10, 1098}, {35, 60}, {36, 110}, {44, 2341}, {71, 2150}, {162, 1785}, {163, 672}, {229, 3336}, {249, 1101}, {409, 3754}, {484, 1325}, {517, 759}, {519, 643}, {656, 3737}, {1323, 1414}, {1437, 4278}, {2194, 4276}, {3286, 3446}
X(5127) = isogonal conjugate of X(5620)
X(5127) = Conway-circle-inverse of X(35637)
X(5128) lies on these lines: {1, 3}, {8, 3928}, {10, 3474}, {12, 4312}, {20, 4848}, {44, 2270}, {63, 1706}, {80, 4333}, {227, 1419}, {516, 1788}, {728, 3509}, {962, 3911}, {1044, 4551}, {1698, 1836}, {3000, 3214}, {3085, 4654}, {3158, 3868}, {3218, 3621}, {3243, 3871}, {3812, 4512}, {4430, 4917}
As a point on the Euler line, X(5129) has Shinagawa coefficients (2abc$a$ + S2, -2S2).
X(5129) lies on these lines: {2, 3}, {8, 3305}, {9, 938}, {10, 390}, {144, 942}, {145, 392}, {388, 4423}, {519, 4866}, {908, 3616}, {936, 4313}, {1001, 2551}, {1125, 3600}, {1330, 4869}, {1698, 4294}, {1788, 3683}, {2899, 3757}, {3189, 3740}, {3241, 3984}, {3485, 4679}, {3624, 4293}
X(5129) = anticomplement of X(17582)
X(5130) lies on these lines: {1, 429}, {4, 8}, {6, 1826}, {10, 4185}, {12, 34}, {25, 958}, {29, 2203}, {33, 1904}, {65, 1899}, {388, 1426}, {407, 1211}, {431, 2886}, {469, 1999}, {996, 4186}, {1861, 4214}, {1869, 1889}, {2333, 3691}, {2355, 4198}, {2975, 4231}
X(5131) lies on these lines: {1, 3}, {10, 4325}, {21, 3833}, {79, 140}, {100, 4973}, {191, 404}, {516, 3582}, {1054, 1325}, {1210, 4324}, {1698, 4652}, {1737, 4316}, {1749, 1768}, {1770, 3817}, {3530, 3649}, {3583, 3911}, {3624, 4338}, {3740, 3916}, {3814, 4197}, {3901, 4855}
X(5132) lies on these lines: {2, 11}, {3, 6}, {35, 238}, {36, 4649}, {81, 4210}, {86, 404}, {228, 3666}, {940, 4191}, {984, 4557}, {1009, 3589}, {1011, 4383}, {1193, 1918}, {1386, 2223}, {1818, 2269}, {2209, 2274}, {2703, 2711}, {3923, 4436}, {4245, 4653}
As a point on the Euler line, X(5133) has Shinagawa coefficients (E + 2F, 2E + 2F).
X(5133) lies on these lines: {2, 3}, {12, 3920}, {51, 3580}, {114, 137}, {115, 1194}, {141, 2979}, {184, 3818}, {230, 1627}, {311, 325}, {316, 1799}, {343, 3060}, {1176, 3589}, {1352, 1993}, {1503, 5012}, {1989, 3108}, {1994, 3410}, {4074, 5031}
X(5133) = inverse-in-orthocentroidal-circle of X(22)
X(5133) = homothetic center of polar triangle of nine-point circle and orthoanticevian triangle of X(2)
X(5134) lies on these lines: {4, 9}, {11, 5030}, {30, 101}, {190, 316}, {220, 382}, {514, 4024}, {672, 3583}, {995, 2549}, {1055, 4316}, {1334, 3585}, {1475, 4857}, {1479, 4253}, {1657, 3207}, {2372, 2702}, {2475, 3294}, {4262, 4302}
X(5135) lies on these lines: {1, 692}, {2, 2194}, {3, 6}, {35, 674}, {60, 404}, {65, 82}, {81, 5012}, {184, 940}, {377, 3618}, {442, 3589}, {518, 2330}, {673, 1492}, {1001, 2175}, {1974, 4185}, {3612, 3751}
As a point on the Euler line, X(5136) has Shinagawa coefficients ($a$F, $aSA$).
X(5136) lies on these lines: {2, 3}, {6, 281}, {10, 212}, {33, 997}, {47, 1724}, {92, 1870}, {225, 1125}, {264, 811}, {318, 4511}, {392, 1824}, {1068, 3616}, {1395, 1877}, {1825, 3878}, {1826, 2267}
X(5137) lies on these lines: {1, 3}, {11, 1428}, {184, 3772}, {513, 1430}, {692, 3011}, {917, 2720}, {1284, 2361}, {1397, 1836}, {1404, 2635}, {1408, 4292}, {1548, 2829}, {3025, 3320}, {3782, 3955}
X(5137) = crossdifference of every pair of points on the line X(72)X(650)
X(5138) lies on these lines: {1, 2175}, {3, 6}, {28, 1974}, {35, 3779}, {57, 985}, {69, 261}, {81, 184}, {206, 942}, {443, 3618}, {518, 993}, {611, 2810}, {940, 2194}, {2330, 3601}
X(5139) is the center of the hyperbola {A,B,C,X(4),X(25)}, which meets the circumcircle at X(3573) (and A,B,C) and tangents to the line X(4)X(69) at X(4) and tangent to the line X(6)(25) at X(25). Moreover, X(5139) is the perspector of the circumconic centered at X(2489). (Randy Hutson, November 22, 2014)
X(5139) lies on the nine-point circle and these lines: {2, 2374}, {4, 99}, {25, 1560}, {113, 1596}, {115, 2971}, {120, 429}, {122, 868}, {123, 3140}, {126, 427}, {127, 3143}, {131, 381}, {132, 235}
X(5139) = midpoint of X(4) and X(3563)
X(5139) = complement of X(3565)
X(5139) = X(2)-Ceva conjugate of X(2489)
X(5139) = crosssum of circumcircle intercepts of line X(3)X(69)
X(5139) = orthopole of line X(3)X(69)
X(5139) = Kirikami-six-circles image of X(25)
X(5140) lies on these lines: {4, 69}, {25, 187}, {115, 2386}, {133, 2679}, {232, 2971}, {427, 625}, {428, 3849}, {460, 512}, {1598, 2080}, {1692, 2207}, {1974, 2030}, {2021, 3199}
As a point on the Euler line, X(5141) has Shinagawa coefficients (abc$a$ + 4S2, 4S2).
X(5141) lies on these lines: {2, 3}, {8, 4867}, {10, 3899}, {11, 3622}, {12, 145}, {149, 3085}, {495, 3623}, {1125, 2320}, {2886, 3614}, {3245, 3814}, {3616, 3822}, {3624, 4881}
X(5141) = complement of X(17548)
As a point on the Euler line, X(5142) has Shinagawa coefficients ($a$F, $a$(E + F) + abc).
X(5142) lies on these lines: {2, 3}, {6, 2906}, {10, 1848}, {12, 278}, {19, 1698}, {34, 975}, {264, 1969}, {281, 1329}, {1125, 1891}, {1172, 1714}, {1228, 1235}, {1826, 1838}
X(5142) = nine-point-circle-inverse of X(37989)
X(5143) lies on these lines: {1, 3}, {31, 5012}, {43, 3185}, {98, 2222}, {100, 740}, {109, 2699}, {172, 1908}, {513, 3510}, {741, 901}, {1756, 4551}, {4225, 4642}, {4276, 4868}
X(5144) lies on these lines: {1, 1055}, {3, 142}, {36, 105}, {56, 1323}, {100, 2725}, {187, 1279}, {238, 5030}, {514, 659}, {910, 2809}, {3361, 5018}, {4251, 4649}, {4471, 4667}
X(5144) = X(187)-of-2nd-circumperp triangle
X(5145) lies on these lines: {1, 87}, {3, 6}, {10, 1740}, {35, 2209}, {42, 3097}, {76, 86}, {81, 4203}, {238, 993}, {869, 1757}, {984, 1964}, {1911, 3864}, {3051, 4476}
X(5146) lies on these lines: {4, 8}, {19, 484}, {28, 36}, {133, 3259}, {225, 2078}, {242, 860}, {278, 1319}, {1168, 1877}, {1869, 3245}, {1870, 1884}, {2077, 4219}
X(5147) lies on these lines: {2, 4154}, {31, 110}, {42, 2054}, {100, 4094}, {111, 2177}, {187, 237}, {662, 3571}, {1402, 2107}, {1911, 4117}, {1962, 3722}, {1976, 2187}
X(5148) lies on these lines: {1, 256}, {11, 625}, {55, 187}, {316, 497}, {512, 4162}, {538, 3023}, {1500, 2021}, {1914, 2031}, {2030, 2330}, {2080, 3295}, {3058, 3849}
X(5148) = X(187)-of-Mandart-incircle triangle
X(5148) = homothetic center of the intangents triangle and the reflection of the extangents triangle in X(187)
Let A′B′C′ be the 1st Brocard triangle. The radical center of the circumcircles of AB′C′, BC′A′, CA′B′ is X(5149). Let A″ be the A′B′C′-isogonal-conjugate of A, and define B″ and C″ cyclically; the lines A′A″, B′B″, C′C″ concur in X(5149). (Randy Hutson, November 22, 2014)
X(5149) lies on these lines: {2, 4159}, {3, 114}, {39, 83}, {76, 4027}, {98, 3934}, {182, 2782}, {538, 1692}, {736, 1691}, {754, 2076}, {1003, 2482}, {1569, 1975}
X(5149) = X(1691)-of-1st-Brocard triangle
X(5149) = 1st-Brocard-triangle-isogonal-conjugate of X(76)
X(5149) = center of the perspeconic of these triangles: 1st Brocard and 6th anti-Brocard
X(5150) lies on these lines: {1, 4579}, {9, 48}, {31, 43}, {182, 2783}, {184, 4011}, {386, 987}, {528, 597}, {692, 4432}, {726, 1428}, {2787, 4164}, {3840, 3955}
X(5150) = X(36)-of-1st-Brocard trangle
X(5150) = inverse of X(32115) in the 1st Lemoine circle
X(5151) lies on these lines: {4, 145}, {11, 1883}, {25, 2932}, {100, 2899}, {900, 1846}, {1145, 4723}, {1317, 1877}, {1811, 1997}, {1828, 2802}, {1878, 3880}
X(5151) = polar conjugate of isogonal conjugate of X(20972)
X(5151) = polar conjugate of isotomic conjugate of X(16594)
X(5152) lies on these lines: {2, 4159}, {3, 76}, {32, 1916}, {39, 4027}, {83, 2023}, {115, 384}, {148, 3552}, {316, 2794}, {671, 1003}, {2854, 4590}
X(5152) = complement of X(32528)
X(5153) lies on these lines: {1, 594}, {3, 6}, {37, 992}, {42, 1100}, {604, 2594}, {1009, 1386}, {1201, 3723}, {1213, 3216}, {2260, 3588}, {2309, 4749}
As a point on the Euler line, X(5154) has Shinagawa coefficients (abc$a$ - 4S2, -4S2).
X(5154) lies on these lines: {2, 3}, {8, 3814}, {11, 145}, {12, 3622}, {496, 3623}, {519, 1392}, {1329, 3617}, {3614, 3816}, {3616, 3825}
X(5154) = complement of X(37307)
X(5154) = anticomplement of X(17566)
X(5155) lies on these lines: {1, 1904}, {4, 8}, {10, 4214}, {25, 993}, {34, 429}, {608, 1826}, {1875, 1892}, {1891, 4186}, {3897, 4194}
X(5156) lies on these lines: {1, 1918}, {2, 31}, {3, 6}, {36, 2274}, {593, 5012}, {595, 1001}, {1010, 1724}, {1468, 2209}, {1740, 2228}
X(5157) lies on these lines: {2, 66}, {3, 6}, {69, 5012}, {110, 3619}, {141, 184}, {159, 3796}, {427, 1974}, {1370, 3618}, {3575, 3867}
Let SaSbSc be the Ehrmann side-triangle. Let A′ be the barycentric product Sb*Sc, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5158). (Randy Hutson, June 27, 2018)
X(5158) lies on these lines: {2, 648}, {3, 6}, {5, 1990}, {53, 546}, {232, 1995}, {233, 1249}, {393, 3091}, {441, 597}, {3087, 3146} X(5158) = isogonal conjugate of isotomic conjugate of X(37638)As a point on the Euler line, X(5159) has Shinagawa coefficients (E - 5F, -E - F).
X(5159) = (inverse-in-de-Longchamps-circle of X(22)) = (radical trace of nine-point circle and de-Longchamps circle) = (radical trace of polar circle and de-Longchamps circle) = (reflection of X(23) in de Longchamps line) (Randy Hutson, August-September, 2013)
X(5159) lies on these lines: {2, 3}, {125, 3292}, {216, 3055}, {230, 3284}, {339, 3266}, {523, 4885}, {577, 3054}, {1007, 2452}
X(5159) = midpoint of X(468) and X(858)
X(5159) = complement of X(468)
X(5159) = inverse-in-{circumcircle, nine-point circle}-inverter of X(20)
X(5159) = inverse-in-complement-of-polar-circle of X(2)
X(5160) lies on these lines: {1, 30}, {11, 858}, {12, 4354}, {23, 55}, {33, 468}, {403, 3614}, {511, 3024}, {523, 4724}
X(5160) = X(23)-of-Mandart-incircle triangle
X(5160) = homothetic center of intangents triangle and reflection of extangents triangle in X(23)
X(5161) lies on these lines: {3, 31}, {81, 849}, {560, 4414}, {649, 834}, {896, 2210}, {902, 1818}, {2206, 3666}, {3218, 5009}
X(5161) = isogonal conjugate of X(37842)
X(5161) = crossdifference of every pair of points on line X(10)X(6590)
X(5162) s the point of intersection of the Lemoine axes of the circumcevian triangles of the 1st and 2nd Brocard points. (Randy Hutson, November 22, 2014)
X(5162) lies on these lines: {3, 6}, {99, 736}, {315, 3552}, {316, 384}, {737, 805}, {754, 2482}, {1003, 3849}, {2387, 3455}
X(5162) = reflection of X(32) in X(187)
X(5163) lies on these lines: {6, 3121}, {37, 100}, {110, 739}, {187, 237}, {574, 4414}, {1977, 2300}, {2092, 3030}
X(5163) = isogonal conjugate of X(35155)
X(5163) = crossdifference of every pair of points on line X(2)X(2787)
X(5163) = inverse-in-Parry-isodynamic-circle of X(667); see X(2)
X(5164) lies on these lines: {3, 6}, {115, 517}, {502, 594}, {512, 661}, {730, 3029}, {1500, 2653}, {3124, 3230}
X(5164) = crossdfference of every pair of points on the line X(81)X(523)
X(5165) lies on these lines: {2, 44}, {3, 6}, {37, 3868}, {45, 3927}, {603, 1405}, {1201, 2260}, {2308, 3764}
X(5166) lies on these lines: {2, 6}, {111, 2393}, {112, 843}, {729, 2696}, {895, 3291}, {1499, 3049}
X(5167) lies on these lines: {4, 69}, {113, 2679}, {115, 2387}, {187, 237}, {206, 1691}, {217, 1692}
X(5167) = anticomplement of X(35060)
X(5168) lies on these lines: {6, 2054}, {42, 101}, {58, 106}, {187, 237}, {1015, 2308}, {1017, 3124}
X(5168) = isogonal conjugate of X(35153)
X(5168) = crossdifference of every pair of points on line X(2)X(2786)
X(5168) = inverse-in-Parry-isodynamic-circle of X(649); see X(2)
As a point on the Euler line, X(5169) has Shinagawa coefficients (E + 4F, 4E + 4F).
X(5169) lies on these lines: {2, 3}, {6, 3448}, {94, 262}, {110, 3818}, {323, 1352}, {1993, 3410}
X(5169) = complement of X(7492)
X(5169) = anticomplement of X(7495)
X(5169) = harmonic center of nine-point circle and {circumcircle, nine-point circle}-inverter
X(5170) lies on these lines: {3, 6}, {31, 512}, {163, 1015}, {249, 593}, {691, 2384}, {953, 2715}
The function p = sin(2ω)/(2 + cos(2ω)) is the Tucker parameter for X(5171); see the preamble to X(13323.)
X(5171) = center of the circumcircle-inverse of the 1st Lemoine circle. (Randy Hutson, November 22, 2014)
Let H be the hyperbola of these five points: X(182), PU(1), PU(2). One vertex of H is X(182); the other is X(5171). (Randy Hutson, November 22, 2014)
X(5171) lies on these lines: {2,732}, {3,6}, {69,10007}, {76,3589}, {83,4048}, {141,7786}, {147,2023}, {194,3618}, {262,1503}, {373,1194}, {597,698}, {694,9155}, {1180,3981}, {1352,11272}, {1386,12782}, {1428,12837}, {1613,5650}, {1915,6800}, {1916,5026}, {2056,6090}, {2330,12836}, {2782,6034}, {3051,7998}, {3108,5012}, {3299,12840}, {3301,12841}, {3329,10334}, {3763,6683}, {5031,7777}, {5103,7790}, {5309,7697}, {5480,9607}, {5965,11261}, {6309,7819}, {7760,8177}, {7829,8149}, {7875,9865}, {10347,12216}
X(5171) = inverse-in-Brocard-circle of X(12212)
X(5171) = inverse-in-second-Brocard-circle of X(7772)
X(5171) = circumperp conjugate of X(35383)
X(5171) = {X(i),X(j)-harmonic conjugate of X(k) for these (i,j,k): (3,6,12212), (6,39,3094), (6,2076,5039), (6,5013,5017), (6,5024,5104), (6,5116,32), (39,2021,5024), (39,7772,3095), (182,7772,6), (371,372,12054), (574,5039,2076), (1670,1671,7772), (1689,1690,3098), (3106,3107,3095), (12055,12212,3)
X(5172) lies on these lines: {1, 3}, {12, 21}, {58, 2594}, {59, 3286}, {73, 1399}, {74, 2720}, {108, 186}, {109, 1464}, {181, 4276}, {187, 1415}, {198, 1609}, {388, 4189}, {404, 2886}, {405, 3814}, {474, 3841}, {513, 1946}, {674, 1428}, {759, 859}, {902, 1457}, {906, 3002}, {1030, 1400}, {1055, 2272}, {1317, 4996}, {1333, 2197}, {1408, 4278}, {1437, 2477}, {1458, 3446}, {1469, 4265}, {1725, 1807}, {1727, 2771}, {2161, 2173}, {2932, 3911}, {3434, 4188}
X(5172) = {X(1),X(3)}-harmonic conjugate of X(37564)
X(5173) lies on these lines: {1, 3}, {7, 3434}, {42, 1465}, {72, 3485}, {81, 4318}, {105, 2982}, {222, 2263}, {226, 518}, {278, 1002}, {388, 3419}, {528, 553}, {672, 2171}, {910, 1630}, {971, 1836}, {1001, 1708}, {1071, 4295}, {1202, 2170}, {1360, 3024}, {1362, 1365}, {1445, 4666}, {1456, 2003}, {1468, 4332}, {1699, 1864}, {1838, 1887}, {2900, 3243}, {3600, 3889}, {3671, 3874}, {3742, 3911}, {3812, 4848}, {3869, 4323}, {3881, 4298}, {3892, 4315}
X(5174) lies on these lines: {4, 8}, {10, 29}, {19, 3692}, {27, 306}, {28, 100}, {34, 3870}, {40, 1748}, {80, 1896}, {145, 278}, {162, 2907}, {225, 1897}, {240, 4642}, {270, 447}, {281, 3617}, {286, 319}, {317, 322}, {412, 515}, {427, 3757}, {518, 1888}, {519, 1838}, {528, 1852}, {653, 4848}, {958, 1013}, {1214, 3152}, {1441, 2475}, {1826, 2322}, {1844, 3754}, {1848, 4514}, {2349, 2816}, {2975, 4219}
X(5175) lies on these lines: {2, 950}, {4, 8}, {7, 2475}, {9, 3617}, {10, 452}, {12, 3189}, {20, 4652}, {63, 3146}, {78, 3091}, {100, 405}, {145, 226}, {377, 938}, {442, 496}, {546, 3940}, {908, 3832}, {958, 1005}, {1490, 3872}, {1750, 4853}, {1837, 2550}, {2000, 4296}, {2094, 4292}, {2551, 3983}, {2886, 3486}, {3241, 3487}, {3476, 3813}, {3529, 3916}, {3627, 3927}, {4084, 4295}
X(5176) lies on these lines: {1, 3814}, {2, 1319}, {4, 8}, {5, 4861}, {10, 36}, {30, 1145}, {63, 484}, {80, 519}, {100, 515}, {145, 1837}, {149, 3880}, {498, 3897}, {513, 4397}, {529, 3036}, {758, 1109}, {901, 2370}, {952, 4511}, {1155, 3617}, {1479, 3885}, {2478, 3890}, {2802, 3583}, {2995, 3596}, {3035, 4881}, {3245, 3626}, {3586, 3895}
X(5176) = anticomplement of X(1319)
X(5176) = excircle-radical-circle-inverse of X(573)
As a point on the Euler line, X(5177) has Shinagawa coefficients (abc$a$ + S2, 2S2).
X(5177) lies on these lines: {2, 3}, {8, 226}, {10, 329}, {12, 480}, {72, 3617}, {145, 3419}, {200, 3947}, {225, 347}, {253, 318}, {388, 2886}, {950, 3616}, {954, 3871}, {966, 1901}, {1125, 3586}, {1655, 2996}, {1698, 1770}, {1864, 3812}, {2551, 3925}, {2893, 3945}, {3011, 4339}, {3085, 3822}, {3485, 3838}, {3488, 3622}, {3614, 4413}
X(5177) = complement of X(17576)
X(5177) = anticomplement of X(6857)
X(5178) lies on these lines: {1, 3841}, {2, 3189}, {4, 8}, {5, 4420}, {10, 1621}, {12, 3935}, {80, 3626}, {145, 3475}, {149, 960}, {377, 3873}, {390, 1837}, {518, 2475}, {1043, 3006}, {1479, 3876}, {1483, 4861}, {1699, 3984}, {1834, 3920}, {2476, 3811}, {2975, 4297}, {3583, 3678}, {3647, 4330}, {4361, 4950}
X(5178) = anticomplement of X(37080)
Let U be the radical circle of the excircles. Then X(5179) is the U-inverse of X(573). (Randy Hutson, November 22, 2014)
X(5179) lies on these lines: {1, 8776}, {2, 5088}, {4, 9}, {5, 1212}, {6, 5722}, {8, 21073}, {12, 16601}, {20, 27541}, {21, 27068}, {30, 910}, {37, 495}, {41, 10572}, {44, 12019}, {72, 22018}, {76, 7112}, {80, 294}, {85, 17671}, {92, 21062}, {101, 515}, {116, 9436}, {119, 1566}, {142, 27471}, {150, 10025}, {218, 1837}, {220, 355}, {225, 25087}, {346, 21074}, {379, 17729}, {514, 661}, {517, 1146}, {519, 4919}, {527, 10708}, {579, 24005}, {644, 5176}, {672, 1737}, {942, 21049}, {946, 24045}, {948, 1323}, {950, 4251}, {952, 6603}, {997, 24247}, {1018, 6735}, {1055, 21578}, {1104, 5305}, {1210, 4253}, {1329, 25066}, {1334, 10039}, {1370, 15487}, {1446, 17758}, {1479, 2082}, {1738, 16611}, {1856, 7070}, {2170, 30384}, {2182, 7359}, {2325, 21066}, {3061, 21616}, {3177, 17181}, {3207, 18481}, {3208, 10915}, {3294, 24987}, {3436, 17742}, {3560, 32561}, {3583, 5540}, {3661, 5195}, {3686, 21065}, {3691, 21029}, {3693, 17757}, {3731, 5726}, {3732, 4872}, {3752, 15048}, {3767, 16968}, {3814, 24036}, {3911, 5030}, {3991, 12607}, {4193, 26690}, {4223, 5144}, {4262, 4304}, {4431, 27492}, {4875, 24390}, {5046, 26793}, {5254, 16583}, {5283, 19754}, {5285, 36010}, {5530, 25092}, {5532, 20683}, {5546, 7424}, {5690, 21872}, {5720, 18328}, {5886, 34522}, {6506, 6882}, {6656, 25994}, {6684, 24047}, {6734, 16552}, {6913, 15288}, {7187, 33837}, {7377, 30854}, {7687, 18327}, {9581, 16572}, {10712, 31160}, {10916, 21384}, {11681, 25082}, {12047, 17451}, {13161, 16600}, {13407, 21808}, {14873, 16589}, {15612, 20623}, {16549, 24982}, {17170, 30694}, {17353, 24279}, {17753, 26531}, {17866, 25002}, {18750, 21621}, {20235, 20236}, {20367, 26001}, {21070, 21078}, {23649, 28096}, {26563, 33839}, {27250, 27254}, {28133, 35270}, {28827, 36698}, {29594, 31142}, {29962, 29967}, {30031, 30063}, {32706, 35182}, {35068, 35122}
X(5179) = midpoint of X(i) and X(j) for these {i,j}: {150, 10025}, {1146, 17747}, {3732, 4872}, {5011, 5134}
X(5179) = reflection of X(i) in X(j) for these {i,j}: {5011, 8074}, {8074, 5199}, {9436, 116}
X(5179) = reflection of X(5011) in the Gergonne line
X(5179) = isotomic conjugate of X(37214)
X(5179) = complement of X(5088)
X(5179) = Spieker-radical-circle-inverse of X(573)
X(5179) = polar-circle-inverse of X(19)
X(5179) = circumcircle-of anticomplementary-triangle-inverse of X(11677)
X(5179) = X(i)-Ceva conjugate of X(j) for these (i,j): {1981, 2811}, {8777, 1}
X(5179) = crosssum of X(6) and X(26884)
X(5179) = crossdifference of every pair of points on line {31, 1459}
X(5179) = barycentric product X(i)*X(j) for these {i,j}: {10, 14956}, {8777, 20623}
X(5179) = barycentric quotient X(14956)/X(86)
X(5179) = intersection of Gergonne lines of 1st and 2nd Ehrmann circumscribing triangles
X(5179) = intersection of Gergonne lines of anticevian triangles of PU(4)
X(5179) = X(i)-complementary conjugate of X(j) for these (i,j): {213, 35075}, {296, 34822}, {1937, 2886}, {1945, 142}, {1949, 17073}, {1952, 17046}, {2249, 3739}, {35145, 21240}
X(5179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6554, 169}, {85, 17671, 34847}, {672, 21044, 1737}, {5046, 26793, 33950}, {5254, 16583, 23537}, {8804, 20262, 573}
X(5180) lies on these lines: {2, 484}, {4, 8}, {7, 1319}, {36, 3616}, {79, 3884}, {149, 758}, {320, 4742}, {513, 4801}, {516, 4511}, {529, 1320}, {535, 3241}, {1537, 4996}, {1727, 3218}, {1836, 3877}, {2475, 3878}, {2975, 3648}, {3245, 3814}, {4084, 4857}, {4301, 4861}
X(5181) lies on the bicevian conic of X(2) and X(110) and on these lines: {2, 895}, {3, 67}, {20, 1632}, {69, 110}, {113, 511}, {125, 126}, {468, 524}, {684, 1649}, {858, 2393}, {960, 2836}, {1176, 3047}, {1205, 3917}, {1350, 2777}, {1352, 4550}, {1511, 3564}, {2781, 2883}, {3448, 3620}
X(5181) = reflection of X(6) in X(5972)
X(5181) = complement of X(895)
X(5181) = antipode of X(6) in the bicevian conic of X(2) and X(110)
X(5182) lies on these lines: {2, 98}, {6, 99}, {30, 2456}, {32, 1992}, {69, 620}, {83, 597}, {115, 3618}, {249, 524}, {384, 575}, {385, 2030}, {538, 1692}, {543, 5034}, {576, 3552}, {599, 1078}, {754, 2458}, {5032, 5039}
X(5182) = X(2)-of-6th-anti-Brocard-triangle
X(5182) = inverse-in-Thomson-Gibert-Moses-hyperbola of X(35279)
X(5182) = {X(2),X(110)}-harmonic conjugate of X(35279)
X(5183) lies on these lines: {1, 3}, {9, 4731}, {19, 1878}, {44, 4695}, {405, 3922}, {513, 4041}, {535, 4669}, {758, 3689}, {1478, 3654}, {1706, 3983}, {2308, 4642}, {2348, 5011}, {3218, 3880}, {3650, 4691}, {3683, 3753}, {3814, 3828}
X(5183) = inverse-in-circumcircle of X(5217)
X(5184) lies on these lines: {1, 187}, {10, 316}, {40, 511}, {238, 5011}, {291, 484}, {512, 659}, {517, 2080}, {625, 1698}, {761, 2702}, {986, 1326}, {1386, 1691}, {1572, 2021}, {2076, 3242}, {3679, 3849}, {4649, 4868}
X(5185) lies on these lines: {4, 150}, {25, 101}, {33, 181}, {34, 1362}, {103, 1593}, {116, 427}, {118, 235}, {428, 544}, {1112, 2774}, {1827, 1845}, {1829, 2809}, {1830, 2821}, {1843, 2810}, {1862, 3887}
X(5186) lies on these lines: {4, 147}, {25, 99}, {33, 3023}, {34, 3027}, {98, 1593}, {114, 235}, {115, 427}, {428, 543}, {468, 620}, {690, 1112}, {1569, 3199}, {1862, 2787}, {1885, 2794}, {1974, 5026}
As a point on the Euler line, X(5187) has Shinagawa coefficients (abc$a$ - 2S2, -4S2).
X(5187) lies on these lines: {2, 3}, {11, 3436}, {145, 1837}, {960, 3617}, {1001, 3614}, {1320, 3621}, {1329, 3434}, {1478, 3825}, {1479, 3814}, {1728, 3218}, {2899, 3006}, {3476, 3622}, {3947, 4666}
Let T1 be the antipedal triangle of the 1st Brocard point, and let T2 be the antipedal triangles of 2nd Brocard point (these points comprising the bicentric pair PU(1)). Then X(5188) is the point in which the Brocard axis meets the line of the circumcenters of T1 and T2. Also, X(5188) is the radical trace of the circumcircles of T1 and T2, as well as the insimilicenter of those circles and the midpoint of their centers. (Randy Hutson, November 22, 2014)
X(5188) lies on these lines: {3, 6}, {4, 3934}, {20, 76}, {194, 3522}, {237, 3917}, {262, 631}, {376, 538}, {550, 2782}, {626, 1513}, {730, 4297}, {827, 1297}, {1092, 3202}, {3117, 3787}
X(5188) = reflection of X(39) in X(3)
X(5188) = inverse-in-2nd-Brocard-crcle of X1350)
X(5188) = (39)-of-circumcevian triangle of X(511)
X(5188) = Brocard-circle-inverse of X(37479)
X(5188) = {X(3),X(6)}-harmonic conjugate of X(37479)
X(5188) = {X(371),X(372)}-harmonic-conjugate of X(5039).
As a point on the Euler line, X(5189) has Shinagawa coefficients (3E, -8E - 8F).
X(5189) lies on these lines: {2, 3}, {98, 1291}, {149, 4442}, {316, 3266}, {323, 1503}, {511, 3448}, {523, 2528}, {842, 930}, {933, 2697}, {2393, 2892}, {2453, 3314}, {2979, 3410}
X(5189) = isogonal conjugate of X(34437)
X(5189) = anticomplement of X(23)
X(5189) = inverse-in-anticomplementary-circle of X(2)
X(5189) = inverse-in-deLongchamps-circle of X(22)
X(5189) = inverse-in-{circumcircle, nine-point circle}-inverter of X(140)
X(5189) = reflection of X(23) in the deLongchamps line
X(5189) = isotomic conjugate of isogonal conjugate of X(19596)
X(5189) = polar conjugate of isogonal conjugate of X(22121)
X(5189) = antigonal conjugate of X(38946)
X(5189) = nine-point-circle-inverse of X(37990)
X(5190) is the center of the hyperbola {A,B,C,X(4),X(27)}, which meets the circumcircle at X(917) and is tangent to the line X(4)X(9) at X(4), and tangent to the line X(27)X(86) at X(27). (Randy Hutson, November 22, 2014)
X(5190) lies on the nine-point circle and these lines: {2, 1305}, {4, 101}, {19, 117}, {113, 1839}, {116, 2973}, {119, 1826}, {120, 1855}, {121, 281}, {122, 3138}, {125, 1146}, {132, 1842}, {1560, 1860}
X(5190) = midpoint of X(4) and X(917)
X(5190) = complement of X(1305)
X(5190) = X(2)-Ceva conjugate of X(7649)
X(5190) = crosssum of circumcircle intercepts of line X(3)X(48)
X(5190) = orthopole of line X(3)X(48)
X(5190) = Kirikami-six-circles image of X(92)
X(5191) lies on these lines: {3, 74}, {6, 157}, {23, 2080}, {25, 111}, {32, 3124}, {51, 5008}, {98, 1316}, {184, 574}, {187, 237}, {868, 2794}, {2782, 4226}, {3098, 3506}
X(5191) = isogonal conjugate of X(5641)
X(5191) = pole of the line X(23)X(110) with respect to the Parry circle
X(5191) = inverse-in-Parry-isodynamic-circle of X(647); see X(2)
As a point on the Euler line, X(5192) has Shinagawa coefficients (2(E + F)2 + 2$bc$(E + F) - abc$a$, 2S2).
X(5192) lies on these lines: {1, 996}, {2, 3}, {8, 1191}, {10, 748}, {31, 3831}, {614, 4968}, {1089, 3891}, {1150, 1724}, {1220, 2899}, {1468, 3840}, {1479, 4972}, {2292, 4011}
X(5193) lies on these lines: {1, 3}, {104, 1519}, {106, 1457}, {108, 1877}, {109, 1149}, {388, 3814}, {995, 2003}, {1398, 1878}, {1404, 2316}, {1421, 1455}, {1428, 2810}, {2975, 3452}
X(5194) lies on these lines: {1, 256}, {12, 625}, {56, 187}, {172, 2031}, {316, 388}, {512, 3669}, {538, 3027}, {999, 2080}, {1015, 2021}, {1357, 1429}, {1428, 2030}
X(5195) lies on these lines: {1, 7}, {2, 5011}, {30, 664}, {72, 319}, {74, 927}, {150, 517}, {514, 4024}, {534, 1944}, {1479, 3212}, {3057, 4911}
X(5195) = anticomplement of X(5011)
As a point on the Euler line, X(5196) has Shinagawa coefficients (E + 4F + 2$bc$, -4E - 4F - 6$bc$).
X(5196) lies on these lines: {2, 3}, {60, 1770}, {99, 3006}, {103, 476}, {110, 516}, {523, 4467}, {593, 3914}, {675, 691}, {759, 4316}, {1326, 3120}
X(5197) lies on these lines: {31, 36}, {43, 3955}, {57, 985}, {81, 1325}, {110, 4414}, {182, 1054}, {986, 1437}, {991, 1283}, {1326, 1790}, {2194, 4650}
As a point on the Euler line, X(5198) has Shinagawa coefficients (F, -3E - F).
X(5198) lies on these lines: {2, 3}, {33, 3303}, {34, 3304}, {51, 1498}, {53, 1033}, {159, 3574}, {1173, 3527}, {1753, 2355}, {2207, 5007}
X(5199) lies on these lines: {2, 1323}, {4, 9}, {121, 1566}, {220, 3626}, {514, 4521}, {519, 1146}, {1212, 3634}, {2297, 5018}
As a point on the Euler line, X(5200) has Shinagawa coefficients (F, -E - F - S).
X(5200) lies on these lines: {2, 3}, {6, 1162}, {51, 1588}, {154, 3070}, {184, 1587}, {615, 1165}, {1164, 3087}
X(5200) = pole wrt polar circle of trilinear polar of X(5490)
X(5200) = X(48)-isoconjugate (polar conjugate) of X(5490)
X(5201) lies on these lines: {3, 6}, {23, 385}, {160, 193}, {183, 1995}, {237, 524}, {340, 4230}, {2930, 3511}
X(5202) lies on these lines: {1, 60}, {31, 4128}, {187, 237}, {213, 3124}, {692, 2643}, {1400, 2054}
X(5203) lies on these lines: {4, 193}, {30, 3565}, {126, 468}, {2501, 3566}
X(5203) = antigonal image of X(468)
X(5204) lies on these lines: {1, 3}, {2, 3614}, {5, 4299}, {11, 20}, {12, 631}, {21, 4423}, {30, 499}, {33, 3516}, {34, 3515}, {41, 5022}, {44, 198}, {45, 1696}, {100, 3621}, {140, 1478}, {145, 4421}, {172, 5013}, {212, 1106}, {215, 1092}, {218, 5030}, {220, 1055}, {376, 3086}, {377, 4999}, {382, 4316}, {388, 3523}, {404, 958}, {474, 993}, {495, 3530}, {496, 548}, {497, 3522}, {498, 549}, {518, 4855}, {550, 1479}, {602, 1399}, {603, 2361}, {611, 5092}, {613, 3098}, {672, 3207}, {859, 4278}, {896, 1473}, {899, 4191}, {936, 3715}, {956, 3626}, {960, 4652}, {997, 3916}, {1001, 4189}, {1125, 1836}, {1152, 2067}, {1193, 4252}, {1201, 3052}, {1259, 4996}, {1350, 1428}, {1376, 2975}, {1436, 2173}, {1443, 1804}, {1450, 4300}, {1468, 4255}, {1469, 5085}, {1475, 4258}, {1656, 3585}, {1657, 3583}, {1837, 3911}, {1914, 5023}, {2071, 5160}, {2275, 3053}, {2886, 4190}, {3035, 3436}, {3085, 3524}, {3240, 4210}, {3286, 4225}, {3474, 3616}, {3526, 4325}, {3528, 4294}, {3534, 3582}, {3622, 4428}, {3624, 3824}, {3869, 4881}, {3928, 3962}, {5087, 5121}
X(5204) = isogonal conjugate of X(7319)
X(5204) = polar conjugate of isotomic conjugate of X(23140)
X(5204) = {X(55),X(56}-harmonic conjugate of X(3304)
X(5204) = {X(3),X(56)}-harmonic conjugate of X(55)
X(5205) lies on these lines: {1, 2}, {20, 2899}, {36, 3992}, {56, 341}, {75, 4413}, {100, 2726}, {125, 3936}, {171, 4672}, {190, 1155}, {238, 4434}, {295, 660}, {312, 1376}, {319, 4023}, {333, 3740}, {404, 3701}, {474, 4385}, {496, 5100}, {497, 1997}, {518, 3699}, {645, 2651}, {649, 3239}, {675, 2748}, {726, 1054}, {750, 894}, {851, 3948}, {908, 4645}, {999, 4737}, {1156, 4607}, {1265, 1788}, {1311, 2743}, {1447, 3263}, {3035, 3932}, {3218, 3952}, {3452, 4388}, {3550, 4011}, {3717, 3911}, {3769, 4383}, {3816, 4514}, {3975, 4447}, {4187, 5015}, {4997, 5087}
X(5205) = complement of X(5211)
X(5205) = anticomplement of X(5121)
X(5205) = inverse-in-{circumcircle, nine-point circle}-inverter of X(10)
X(5205) = crossdifference of PU(92)
X(5205) = crossdifference of every pair of points on line X(649)X(1201)
X(5205) = X(2)-Ceva conjugate of X(39059)
X(5205) = perspector of conic {{A,B,C,PU(59)}}
Let H be the ellipse of these five points: X(574), PU(1), PU(2). One vertex of H is X(574); the other is X(5206). (Randy Hutson, November 22, 2014)
X(5206) lies on these lines: {3, 6}, {20, 115}, {22, 3291}, {35, 2242}, {36, 2241}, {172, 5010}, {186, 1968}, {230, 550}, {315, 620}, {376, 3767}, {439, 3785}, {546, 3054}, {631, 1506}, {1003, 3934}, {1078, 3552}, {1658, 2493}, {1971, 3357}, {2079, 2937}, {2482, 3926}, {2548, 3523}, {2549, 3522}, {3199, 3515}, {3530, 3815}, {3787, 3796}, {5087, 5121}
X(5206) = isogonal conjugate of polar conjugate of X(37453)
X(5206) = {X(371),X(372)}-harmonic conjugate of X(5097)
X(5206) = radical center of Lucas(-2/5 cot ω) circles
X(5207) lies on these lines: {2, 1501}, {4, 69}, {6, 5025}, {141, 384}, {147, 325}, {148, 698}, {187, 3619}, {193, 5111}, {334, 1966}, {512, 3267}, {625, 1692}, {626, 2458}, {732, 1916}, {1570, 1992}, {1965, 4388}, {2080, 3785}, {3620, 5104}, {4576, 5189}, {5087, 5121}
X(5207) = anticomplement of X(1691)
X(5207) = crosspoint of X(147) and X(2896) with respect to the excentral triangle
X(5207) = crosspoint of X(147) and X(2896) with respect to the anticomplementary triangle
X(5208) lies on these lines: {1, 21}, {2, 3786}, {7, 310}, {27, 295}, {65, 1043}, {86, 354}, {228, 4225}, {284, 3509}, {333, 518}, {942, 1010}, {982, 3736}, {1412, 5083}, {3218, 4184}, {5087, 5121}
X(5209) lies on these lines: {1, 75}, {36, 99}, {80, 334}, {313, 757}, {670, 4495}, {730, 741}, {811, 1785}, {1019, 1577}, {1323, 4625}, {1509, 1909}, {1931, 3948}, {1963, 3963}, {4039, 4600}, {5087, 5121}
X(5210) lies on these lines: {3, 6}, {4, 3054}, {22, 111}, {115, 3534}, {154, 5191}, {230, 376}, {439, 3620}, {548, 3767}, {631, 3055}, {2548, 3530}, {3524, 3815}, {3630, 3926}, {3631, 3785}, {5087, 5121}
X(5210) = radical center of Lucas(-1/3 cot ω) circles
X(5210) = harmonic center of circumcircle and circle O(15,16)
X(5211) lies on these lines: {1, 2}, {110, 2726}, {244, 4645}, {320, 3999}, {497, 3210}, {675, 2705}, {982, 4388}, {1330, 3953}, {3315, 3936}, {3667, 4025}, {3752, 4514}, {4440, 5057}, {5087, 5121}
X(5211) = anticomplement of X(5205)
X(5211) = isotomic conjugate of anticomplement of X(39059)
X(5212) lies on these lines: {1, 2}, {518, 3030}, {661, 3667}, {908, 4442}, {1155, 4831}, {1266, 4706}, {1738, 4892}, {3684, 4700}, {3879, 4413}, {4023, 4357}, {4656, 4734}, {5087, 5121}
X(5212) = complement of X(38473)
Let (N) be the nine-point circle of a triangle ABC. Let (IA) be the A-excircle of ABC, and define (IB) and (IC) cyclically. Let A′ be the midpoint of side BC, and define B′ and C′ cyclically. Let (KA) be the circle, other than (N), that passes through B′ and C′ and touches (IA), and define (KB) and (KC) cyclically. Let (K) be the circle externally tangent to (KA), (KB), (KC), and let (L) be the circle externally tangent to (IA), (IB), (IC). Then X(5213) = (K)∩(L). (Tran Quang Hung, July 16, 2014)
If you have The Geometer's Sketchpad, you can view X(5213).
X(5213) lies on the Apollonius circle and these lines: {10, 115}, {181, 1356}, {214, 1015}, {386, 741}, {573, 759}, {1018, 3124}, {1575, 5164}, {2238, 5011}, {5087, 5121}
X(5213) = complement of X(38477)
X(5213) = similitude center of Apollonius triangle and polar triangle of excircles radical circle
X(5214) lies on these lines: {1, 523}, {513, 4960}, {522, 1019}, {3733, 4777}, {4151, 4581}, {4802, 4833}, {4840, 4926}, {5087, 5121}
X(5213) = pole of the Euler line with respect to the Conway circle
X(5215) = centroid of {X(13), X(14), X(15), X(16), X(5463), X(5464)} (Randy Hutson, November 22, 2014)
X(5215) lies on these lines: {2, 187}, {230, 2482}, {511, 5054}, {524, 1692}, {597, 5107}, {599, 2030}, {5087, 5121}
X(5216) lies on these lines: {1, 512}, {513, 4960}, {834, 3737}, {1734, 4481}, {2978, 4040}, {3733, 4834}, {5087, 5121}
X(5216) = pole of the Brocard axis with respect to the Conway circle
Let A″B″C″ be as at X(12512). Then A″B″C″ is homothetic to the intouch triangle at X(5217). (Randy Hutson, January 17, 2020)
X(5217) lies on these lines: {1, 3}, {4, 3614}, {5, 4302}, {8, 4421}, {11, 631}, {12, 20}, {21, 1376}, {30, 498}, {31, 4255}, {33, 3515}, {34, 3516}, {42, 4252}, {45, 198}, {73, 3532}, {78, 4640}, {100, 958}, {140, 1479}, {172, 5023}, {186, 5160}, {191, 3940}, {212, 1399}, {218, 4262}, {376, 3085}, {382, 4324}, {388, 3522}, {404, 1001}, {405, 3634}, {474, 4423}, {480, 1259}, {495, 548}, {496, 3530}, {497, 3523}, {499, 549}, {518, 4652}, {550, 1478}, {601, 2361}, {603, 1253}, {611, 3098}, {613, 5092}, {672, 4258}, {899, 1011}, {902, 1191}, {936, 3683}, {956, 3625}, {960, 4855}, {991, 2594}, {993, 3626}, {1092, 2477}, {1152, 2066}, {1193, 3052}, {1334, 2272}, {1350, 2330}, {1468, 2334}, {1500, 5206}, {1621, 4188}, {1656, 3583}, {1657, 3585}, {1788, 4313}, {1837, 4304}, {1898, 5044}, {1914, 5013}, {2276, 3053}, {2280, 5022}, {2478, 3035}, {2975, 3621}, {3056, 5085}, {3058, 3086}, {3240, 4184}, {3434, 4999}, {3474, 3649}, {3526, 4330}, {3528, 4293}, {3534, 3584}, {3616, 4428}, {3811, 3916}, {3890, 4881}, {3911, 4314}, {3929, 4005}
X(5217) = inverse-in-circumcircle of X(5183)
X(5217) = {X(3),X(55)}-harmonic conjugate of X(56)
X(5218) = 9r*X(2) - 2(2r - R)*X(11)
X(5218) = 2(2r + R)*X(12) + r*X(20)
X(5218) lies on these lines: {1, 631}, {2, 11}, {3, 388}, {4, 35}, {5, 4294}, {7, 1155}, {8, 2320}, {9, 1776}, {10, 3486}, {12, 20}, {21, 2551}, {36, 1056}, {40, 3485}, {46, 3487}, {56, 3523}, {57, 3475}, {69, 2330}, {140, 3086}, {165, 226}, {171, 212}, {197, 4224}, {243, 281}, {329, 4640}, {344, 5205}, {345, 3790}, {346, 3712}, {355, 4305}, {376, 1478}, {391, 4023}, {452, 1329}, {496, 3526}, {499, 1058}, {549, 999}, {601, 3074}, {612, 1040}, {650, 885}, {750, 1253}, {899, 2293}, {944, 3612}, {950, 1698}, {966, 2268}, {991, 4551}, {993, 3421}, {1125, 1697}, {1479, 3090}, {1737, 3488}, {1742, 2635}, {1837, 4313}, {1858, 3876}, {1864, 3740}, {2066, 3069}, {2098, 3622}, {2999, 4989}, {3011, 4000}, {3056, 3618}, {3057, 3616}, {3158, 4847}, {3161, 4009}, {3296, 3337}, {3436, 4189}, {3452, 4512}, {3476, 3576}, {3528, 4299}, {3529, 3585}, {3545, 3583}, {3579, 4295}, {3600, 5204}, {3614, 3832}, {3634, 4314}, {3671, 5128}, {3855, 4330}, {3913, 4999}, {4309, 5067}, {4402, 4706}, {4414, 4419}
X(5219) lies on these lines: {1, 5}, {2, 7}, {4, 3601}, {10, 3340}, {34, 975}, {37, 1465}, {40, 498}, {55, 1538}, {56, 3624}, {65, 1698}, {78, 2476}, {85, 4554}, {109, 750}, {165, 1836}, {191, 1454}, {200, 2886}, {208, 451}, {210, 5173}, {278, 1826}, {319, 4417}, {381, 3586}, {388, 1125}, {442, 936}, {468, 1892}, {497, 3817}, {499, 3333}, {551, 3476}, {631, 4292}, {938, 5056}, {940, 2003}, {942, 1656}, {946, 1697}, {948, 1323}, {950, 3091}, {991, 2635}, {997, 3822}, {1000, 1512}, {1001, 2078}, {1155, 4312}, {1210, 3090}, {1376, 3256}, {1419, 4648}, {1441, 4358}, {1466, 3824}, {1478, 3576}, {1617, 4423}, {1788, 3634}, {1876, 5094}, {2099, 3679}, {2475, 4855}, {2999, 3553}, {3006, 4901}, {3158, 3434}, {3339, 3649}, {3488, 3545}, {3584, 5119}, {3585, 3612}, {3617, 4323}, {3832, 4313}, {4032, 4687}, {4054, 4659}, {4295, 5128}, {4671, 4873}, {5054, 5122}
X(5219) = isogonal conjugate of X(2364)
X(5219) = isotomic conjugate of X(30608)
X(5219) = {X(2),X(7)}-harmonic conjugate of X(3911)
X(5219) = {X(2),X(57)}-harmonic conjugate of X(31231)
X(5219) = endo-homothetic center of the AOA and AAOA triangles
X(5220) lies on these lines: {1, 6}, {2, 3715}, {3, 2801}, {7, 12}, {8, 190}, {10, 527}, {38, 4383}, {40, 4662}, {46, 3697}, {55, 1776}, {56, 3876}, {57, 3740}, {63, 210}, {65, 3951}, {69, 3932}, {78, 4005}, {100, 3711}, {142, 3634}, {144, 1654}, {183, 4518}, {191, 4436}, {200, 3929}, {319, 3790}, {321, 4042}, {329, 2886}, {344, 4966}, {354, 3305}, {355, 382}, {390, 3621}, {480, 1259}, {612, 4641}, {672, 3789}, {726, 4361}, {756, 940}, {758, 1159}, {971, 1158}, {993, 3940}, {997, 5126}, {1150, 3952}, {1621, 4661}, {1706, 4866}, {1890, 5130}, {2246, 4712}, {2646, 3984}, {3052, 3961}, {3218, 4413}, {3245, 3679}, {3286, 3786}, {3416, 3717}, {3625, 4133}, {3632, 4693}, {3683, 3870}, {3696, 3729}, {3773, 4445}, {3873, 4423}, {3883, 4899}, {3916, 4533}, {4078, 4851}, {4671, 4756}
X(5220) = X(67) of Fuhrman triangle
X(5220) = perspector of Fuhrmann and outer Johnson triangles
X(5221) lies on these lines: {1, 3}, {2, 3649}, {6, 1406}, {7, 12}, {10, 553}, {11, 4295}, {34, 2355}, {44, 2285}, {45, 1400}, {63, 3812}, {72, 4413}, {79, 381}, {88, 959}, {89, 961}, {208, 1827}, {221, 1393}, {226, 3634}, {227, 1418}, {244, 1191}, {386, 1464}, {388, 3617}, {405, 3647}, {474, 758}, {936, 3962}, {938, 3474}, {952, 4317}, {956, 3754}, {958, 3218}, {960, 3306}, {997, 4018}, {1046, 4383}, {1210, 1836}, {1254, 1407}, {1317, 4308}, {1376, 3868}, {1399, 1451}, {1417, 4792}, {1434, 3212}, {1452, 1876}, {1469, 3214}, {1475, 2272}, {1698, 3715}, {1722, 4641}, {1835, 4185}, {1837, 4292}, {2334, 4646}, {2594, 4306}, {2650, 4255}, {3125, 5021}, {3600, 3621}, {3624, 4870}, {3626, 4031}, {3671, 3911}, {3740, 3951}, {3873, 3913}, {3901, 3940}, {3924, 4252}, {3947, 3982}
X(5221) = {X(1),X(5708)}-harmonic conjugate of X(4860)
X(5221) = {X(13388),X(13389)}-harmonic conjugate of X(37595)
X(5222) lies on these lines: {1, 2}, {6, 7}, {9, 3672}, {20, 1453}, {27, 3194}, {41, 1429}, {44, 4419}, {57, 279}, {69, 3759}, {75, 3618}, {81, 277}, {142, 1449}, {144, 1743}, {192, 3161}, {193, 3662}, {218, 329}, {220, 4383}, {223, 4350}, {241, 2275}, {273, 1249}, {278, 607}, {319, 3619}, {320, 1992}, {344, 4360}, {346, 3875}, {347, 1445}, {390, 3755}, {391, 4357}, {527, 4346}, {594, 4371}, {597, 4363}, {599, 4969}, {857, 1834}, {966, 4657}, {1100, 4648}, {1104, 4313}, {1203, 4295}, {1212, 3666}, {1266, 4454}, {1386, 2550}, {1423, 2347}, {1442, 3554}, {1468, 4209}, {1738, 4307}, {2345, 3589}, {3664, 4859}, {3729, 4452}, {3731, 4021}, {3739, 4798}, {3751, 4310}, {3879, 4869}, {4460, 4852}, {4470, 4688}
X(5222) = complement of X(29616)
X(5222) = anticomplement of X(17284)
X(5222) = {X(1),X(2)}-harmonic conjugate of X(5308)
X(5223) lies on these lines: {1, 6}, {3, 480}, {7, 10}, {8, 144}, {38, 2999}, {40, 971}, {55, 3929}, {56, 4005}, {57, 210}, {63, 100}, {69, 3717}, {142, 1698}, {190, 3886}, {191, 3174}, {329, 1699}, {344, 4684}, {354, 3715}, {390, 519}, {443, 4355}, {474, 4533}, {517, 4915}, {527, 1478}, {528, 4677}, {537, 673}, {668, 3403}, {936, 1445}, {991, 2340}, {997, 4134}, {1155, 3711}, {1156, 2802}, {1376, 3928}, {1706, 4662}, {1707, 3961}, {1738, 4862}, {2184, 2947}, {2809, 4752}, {2810, 3781}, {2975, 3984}, {3008, 4310}, {3158, 4640}, {3219, 3870}, {3305, 3873}, {3333, 5044}, {3340, 3962}, {3416, 4901}, {3576, 3940}, {3677, 4383}, {3696, 4659}, {3755, 4419}, {3869, 4853}, {3925, 4654}, {4420, 4652}, {4430, 4666}
X(5223) = reflection of X(1) in X(9)
X(5223) = outer-Garcia-to-ABC similarity image of X(7)
X(5224) lies on these lines: {1, 319}, {2, 6}, {3, 3437}, {7, 12}, {8, 4026}, {9, 1760}, {10, 75}, {37, 3661}, {38, 4446}, {85, 307}, {142, 4751}, {190, 2345}, {192, 594}, {239, 4657}, {261, 1078}, {274, 4283}, {286, 5125}, {320, 1698}, {326, 936}, {334, 1218}, {404, 1444}, {405, 2893}, {638, 2047}, {894, 4643}, {1086, 4699}, {1100, 4690}, {1125, 3879}, {1278, 4665}, {1330, 2049}, {1975, 4201}, {2321, 4664}, {3305, 4872}, {3616, 4966}, {3617, 3672}, {3625, 4464}, {3626, 4021}, {3634, 3664}, {3644, 4431}, {3662, 3739}, {3679, 3875}, {3686, 3759}, {3728, 4443}, {3758, 4416}, {3779, 3789}, {3786, 4259}, {3912, 4687}, {3943, 4704}, {4441, 4972}, {4472, 4741}
X(5224) = complement of X(17379)
X(5224) = anticomplement of X(17398)
X(5224) = {X(2),X(6)}-harmonic conjugate of X(17381)
X(5224) = {X(2),X(69)}-harmonic conjugate of X(86)
X(5224) = {X(2),X(141)}-harmonic conjugate of X(17234)
X(5225) lies on these lines: {1, 4}, {5, 4294}, {8, 3967}, {11, 20}, {12, 390}, {30, 3086}, {35, 3090}, {36, 3529}, {55, 3091}, {56, 3146}, {100, 5187}, {149, 3436}, {376, 499}, {381, 3085}, {382, 496}, {452, 2886}, {495, 3843}, {498, 3545}, {516, 1788}, {546, 3295}, {631, 4302}, {908, 3189}, {938, 1836}, {960, 5175}, {962, 1837}, {999, 3627}, {1001, 5177}, {1210, 3474}, {1452, 2961}, {1898, 3868}, {2478, 2550}, {2551, 3434}, {3058, 3839}, {3153, 5160}, {3421, 3625}, {3525, 5010}, {3528, 4324}, {3601, 3817}, {3616, 3838}, {3626, 5082}, {3634, 5084}, {3855, 4309}, {3925, 5129}, {3974, 5015}, {4208, 4423}, {4330, 5067}
X(5226) lies on these lines: {1, 3091}, {2, 7}, {4, 4313}, {5, 938}, {8, 12}, {11, 3475}, {56, 5047}, {65, 3740}, {78, 5177}, {223, 1442}, {278, 469}, {312, 1441}, {381, 3488}, {388, 1319}, {390, 1699}, {479, 1996}, {484, 498}, {495, 1532}, {497, 3748}, {612, 4318}, {631, 5122}, {651, 940}, {857, 948}, {936, 4208}, {942, 3090}, {950, 3832}, {962, 3085}, {975, 4296}, {1000, 3656}, {1125, 3600}, {1210, 5056}, {1456, 4682}, {1698, 3671}, {1788, 3649}, {2550, 3838}, {2900, 5175}, {3146, 3601}, {3241, 4870}, {3339, 3634}, {3340, 3617}, {3523, 4292}, {3543, 4304}, {3585, 4305}, {3586, 3839}, {3624, 4298}, {3681, 5173}
Let H be the homothety that maps the 2nd extouch triangle onto the excentral triangle; then X(5227) = H(X(6)). (Randy Hutson, November 22, 2014)
X(5227) lies on these lines: {1, 6}, {3, 3694}, {8, 19}, {40, 1503}, {48, 78}, {57, 141}, {63, 69}, {84, 1350}, {144, 4329}, {159, 197}, {169, 3686}, {193, 3219}, {198, 3965}, {210, 965}, {281, 3421}, {284, 3811}, {319, 1760}, {329, 1848}, {388, 2285}, {515, 1766}, {524, 3929}, {599, 3928}, {612, 2303}, {988, 4261}, {1038, 2286}, {1474, 2287}, {1781, 3679}, {1792, 4288}, {1826, 3436}, {1839, 3434}, {1953, 3872}, {2171, 4390}, {2182, 3713}, {2268, 3930}, {3169, 3496}, {3218, 3620}, {3305, 3618}, {3306, 3619}, {3927, 4047}, {3951, 3958}
X(5228) lies on these lines: {1, 3}, {2, 220}, {6, 7}, {9, 4328}, {37, 1445}, {63, 1212}, {75, 3713}, {77, 1100}, {81, 279}, {85, 239}, {142, 219}, {175, 3297}, {176, 3298}, {218, 226}, {222, 553}, {269, 1449}, {277, 2982}, {307, 4657}, {481, 1124}, {482, 1335}, {518, 4327}, {664, 4393}, {965, 3739}, {1001, 1471}, {1119, 1172}, {1231, 4359}, {1323, 4031}, {1373, 3301}, {1374, 3299}, {1376, 2340}, {1386, 2263}, {1427, 4350}, {1441, 4361}, {1616, 4323}, {2256, 4648}, {3668, 3946}, {3912, 4513}, {4334, 4649}
X(5228) = crossdifference of every pair of points on the line X(650)X(926)
X(5229) lies on these lines: {1, 4}, {2, 3614}, {5, 4293}, {7, 1837}, {8, 1836}, {10, 3474}, {11, 3600}, {12, 20}, {30, 3085}, {35, 3529}, {36, 3090}, {55, 3146}, {56, 3091}, {144, 1654}, {355, 4295}, {376, 498}, {377, 1155}, {381, 3086}, {382, 495}, {443, 3634}, {496, 3843}, {499, 3545}, {518, 5175}, {546, 999}, {631, 4299}, {958, 5177}, {1420, 3817}, {1788, 4292}, {3295, 3627}, {3421, 3626}, {3434, 3621}, {3528, 4316}, {3601, 3947}, {3625, 5082}, {3855, 4317}, {4312, 4848}, {4325, 5067}
X(5230) lies on these lines: {1, 2}, {4, 31}, {6, 12}, {11, 1191}, {19, 208}, {40, 3914}, {55, 1834}, {58, 1478}, {65, 3772}, {171, 377}, {213, 3767}, {227, 1108}, {235, 3195}, {238, 2478}, {278, 1254}, {318, 4008}, {388, 1468}, {443, 750}, {497, 3915}, {595, 1479}, {748, 5084}, {902, 4294}, {959, 2006}, {1068, 1148}, {1104, 1837}, {1329, 4383}, {1460, 4185}, {1788, 4000}, {2650, 3487}, {3120, 4295}, {4257, 4299}, {4307, 5177}, {4339, 5175}
X(5231) lies on these lines: {1, 2}, {9, 11}, {36, 1004}, {57, 2886}, {63, 1699}, {75, 4554}, {165, 3434}, {244, 4859}, {329, 3817}, {377, 3361}, {442, 3333}, {497, 4512}, {993, 1005}, {1260, 4423}, {1376, 2078}, {1697, 3813}, {1836, 3928}, {2550, 3911}, {3120, 4862}, {3158, 4863}, {3218, 4312}, {3419, 3576}, {3601, 4999}, {3677, 3772}, {3693, 4519}, {3829, 3929}, {3838, 4654}, {4297, 5175}, {4298, 5177}, {4855, 5178}
X(5232) lies on these lines: {2, 6}, {7, 10}, {8, 3672}, {37, 4748}, {75, 3617}, {77, 936}, {144, 2345}, {145, 319}, {279, 307}, {320, 3823}, {346, 3661}, {390, 3775}, {452, 2893}, {474, 1014}, {594, 4419}, {997, 1442}, {1122, 3983}, {1444, 4188}, {1698, 3664}, {3616, 3879}, {3621, 4360}, {3632, 4021}, {3663, 3679}, {3723, 4916}, {3946, 4034}, {4364, 4445}, {4389, 4452}, {4657, 4690}, {4708, 4851}
X(5233) lies on these lines: {2, 6}, {8, 11}, {9, 4070}, {43, 3847}, {75, 908}, {200, 4514}, {210, 3705}, {312, 2321}, {320, 3306}, {345, 3161}, {474, 1330}, {497, 3996}, {899, 4429}, {997, 998}, {1043, 2478}, {1054, 4655}, {1376, 4388}, {3210, 4415}, {3242, 5211}, {3416, 5205}, {3685, 4679}, {3696, 5087}, {3755, 5212}, {3790, 4009}, {3807, 4671}, {3911, 4416}, {4389, 4850}, {4413, 4645}, {4734, 4854}
X(5233) = complement of X(37684)
X(5234) lies on these lines: {1, 6}, {2, 3361}, {8, 4314}, {10, 20}, {21, 200}, {55, 4882}, {63, 3339}, {65, 3929}, {142, 4355}, {144, 3671}, {191, 2093}, {210, 3601}, {443, 1478}, {452, 4847}, {936, 993}, {1697, 3683}, {1706, 4640}, {2646, 3715}, {2975, 3305}, {3158, 4662}, {3452, 3624}, {3576, 5044}, {3698, 5128}, {3812, 3928}, {3885, 4853}, {5123, 5131}
X(5235) lies on these lines: {2, 6}, {10, 21}, {27, 281}, {28, 5130}, {45, 4671}, {58, 750}, {63, 1781}, {88, 274}, {314, 4358}, {899, 3736}, {958, 4225}, {1014, 3911}, {1043, 3617}, {1155, 3846}, {1255, 1999}, {1376, 4184}, {2177, 3679}, {3218, 3739}, {3286, 4413}, {3624, 4658}, {3681, 5208}, {3712, 4733}, {3757, 4981}, {3977, 4967}, {4384, 4850}, {4396, 4708}
X(5235) = isotomic conjugate of X(30588)X(5236) lies on these lines: {1, 4}, {2, 1435}, {7, 19}, {27, 1803}, {28, 4298}, {85, 92}, {108, 2725}, {142, 281}, {241, 5089}, {273, 1826}, {514, 3064}, {518, 1861}, {908, 4564}, {958, 1398}, {1430, 3011}, {1456, 1503}, {1783, 3008}, {1890, 1892}, {2331, 4000}, {3947, 5142}
X(5237) lies on these lines: {3, 6}, {13, 140}, {14, 20}, {17, 631}, {18, 30}, {35, 202}, {203, 5204}, {395, 550}, {396, 3530}, {397, 549}, {398, 548}, {530, 630}, {532, 628}, {616, 636}, {619, 634}, {627, 3642}, {1092, 3201}, {2306, 5131}
X(5237) = Schoutte-circle-inverse of X(34755)
X(5238) lies on these lines: {3, 6}, {13, 20}, {14, 140}, {17, 30}, {18, 631}, {35, 203}, {202, 5204}, {395, 3530}, {396, 550}, {397, 548}, {398, 549}, {531, 629}, {533, 627}, {617, 635}, {618, 633}, {628, 3643}, {1092, 3200}, {2307, 5010}
X(5238) = Schoutte circle inverse of X(34754)
X(5239) lies on these lines: {1, 6}, {2, 559}, {3, 1277}, {8, 1251}, {10, 17}, {16, 214}, {56, 1653}, {61, 3878}, {63, 1082}, {65, 1652}, {142, 3638}, {203, 758}, {471, 1833}, {517, 1276}, {527, 3639}, {2307, 3869}
X(5239) = complement of X(36928)
X(5240) lies on these lines: {1, 6}, {2, 1082}, {3, 1276}, {10, 18}, {15, 214}, {21, 1251}, {36, 3179}, {56, 1652}, {62, 3878}, {63, 559}, {65, 1653}, {142, 3639}, {202, 758}, {470, 1832}, {517, 1277}, {527, 3638}
X(5240) = isogonal conjugate of X(33655)
X(5240) = complement of X(36929)
X(5241) lies on these lines: {1, 4023}, {2, 6}, {10, 11}, {354, 4104}, {594, 4358}, {899, 4026}, {908, 3739}, {3216, 4205}, {3306, 4643}, {3775, 4871}, {3846, 5087}, {3847, 3925}, {4054, 4688}, {4239, 5096}, {4359, 4415}, {4364, 4850}, {4665, 4671}
X(5241) = complement of X(37633)
X(5242) lies on these lines: {2, 7}, {10, 18}, {302, 3912}, {303, 4416}, {395, 1100}, {946, 1277}
X(5243) lies on these lines: {2, 7}, {10, 17}, {302, 4416}, {303, 3912}, {396, 1100}, {946, 1276}
X(5244) lies on these lines: {6, 7}, {10, 12}, {57, 1759}, {241, 3674}, {1386, 1890}, {2295, 4415}
X(5245) lies on these lines: {8, 9}, {10, 17}, {515, 1277}, {1652, 4848}
X(5246) lies on these lines: {8, 9}, {10, 18}, {515, 1276}, {1653, 4848}
X(5247) lies on these lines: {1, 6}, {2, 1468}, {3, 43}, {4, 1957}, {8, 31}, {10, 58}, {21, 42}, {28, 291}, {32, 3684}, {35, 3293}, {36, 3216}, {40, 1707}, {46, 4650}, {56, 978}, {57, 1722}, {63, 986}, {65, 1046}, {71, 1778}, {100, 3214}, {109, 4848}, {145, 3915}, {162, 2907}, {172, 2238}, {191, 4424}, {212, 3486}, {227, 1758}, {239, 384}, {256, 1245}, {341, 3769}, {355, 3072}, {386, 993}, {388, 1451}, {404, 899}, {484, 3987}, {515, 580}, {517, 3073}, {519, 595}, {602, 944}, {603, 1788}, {614, 3976}, {651, 1042}, {744, 4647}, {748, 3616}, {846, 3931}, {896, 4642}, {902, 3871}, {938, 1496}, {959, 1405}, {961, 1400}, {976, 3681}, {988, 2999}, {1009, 3783}, {1043, 1918}, {1126, 4653}, {1183, 2347}, {1193, 2975}, {1253, 4313}, {1330, 2887}, {1376, 4252}, {1430, 5125}, {1445, 4320}, {1471, 3600}, {1478, 1714}, {1572, 4051}, {1610, 2183}, {1737, 3075}, {1738, 4292}, {1739, 3336}, {1777, 2093}, {1837, 1936}, {1891, 2299}, {1914, 3780}, {2239, 4201}, {2292, 3219}, {2650, 4722}, {3008, 4298}, {3052, 3913}, {3240, 4189}, {3436, 5230}, {3647, 4868}, {3686, 4264}, {3720, 5047}, {3868, 3924}, {4234, 4685}, {4355, 4859}, {4362, 4385}, {4640, 4646}, {4673, 4676}
X(5248) lies on these lines: {1, 21}, {2, 35}, {3, 142}, {4, 3822}, {8, 3746}, {9, 943}, {10, 55}, {32, 37}, {36, 3616}, {40, 1006}, {41, 3294}, {42, 1724}, {56, 551}, {72, 3683}, {86, 4278}, {100, 1698}, {101, 2304}, {140, 3816}, {165, 3833}, {198, 3986}, {200, 4015}, {214, 3612}, {238, 386}, {354, 3916}, {377, 4302}, {388, 535}, {392, 2646}, {404, 3624}, {411, 1699}, {452, 3085}, {474, 4423}, {496, 4999}, {498, 2478}, {515, 3560}, {519, 958}, {581, 3073}, {631, 2077}, {748, 3216}, {759, 931}, {936, 4326}, {942, 4640}, {956, 3244}, {976, 2210}, {978, 4256}, {997, 3601}, {999, 3636}, {1012, 4297}, {1013, 1838}, {1100, 4047}, {1104, 3931}, {1107, 2241}, {1214, 4347}, {1259, 4847}, {1376, 3634}, {1500, 4426}, {1617, 4298}, {1697, 2802}, {1706, 3968}, {1748, 1844}, {1777, 4303}, {1788, 3256}, {1792, 3886}, {2177, 3293}, {2293, 3682}, {2346, 5223}, {2476, 3583}, {2901, 4362}, {2922, 3145}, {3006, 4894}, {3149, 3817}, {3158, 3956}, {3246, 4719}, {3338, 4652}, {3428, 4301}, {3434, 4309}, {3454, 3771}, {3555, 3748}, {3579, 3812}, {3626, 3913}, {3670, 4414}, {3679, 3871}, {3689, 3697}, {3828, 4421}, {3924, 4424}, {4004, 5183}, {4197, 4330}, {5084, 5218}
X(5248) = excentral-to-2nd-circumperp similarity image of X(12514)
X(5249) lies on these lines: {1, 224}, {2, 7}, {5, 1071}, {8, 4208}, {10, 3681}, {11, 3742}, {12, 3812}, {20, 946}, {21, 36}, {27, 86}, {37, 3782}, {42, 1738}, {55, 1004}, {75, 306}, {77, 278}, {78, 443}, {81, 3664}, {85, 92}, {171, 3011}, {210, 3826}, {239, 2890}, {312, 1269}, {320, 333}, {321, 1930}, {354, 2886}, {379, 2140}, {442, 942}, {474, 1259}, {495, 3753}, {497, 4666}, {516, 1621}, {518, 3925}, {528, 3748}, {551, 4304}, {554, 5239}, {914, 1441}, {938, 5177}, {940, 3772}, {948, 4350}, {950, 2475}, {960, 3649}, {1001, 1836}, {1012, 1519}, {1056, 3872}, {1081, 5240}, {1086, 3666}, {1210, 2476}, {1211, 3739}, {1215, 3836}, {1659, 3084}, {1737, 3822}, {1838, 4303}, {1959, 3674}, {2550, 3475}, {2895, 3686}, {2975, 4298}, {2999, 4859}, {3075, 3561}, {3120, 3720}, {3187, 3879}, {3220, 4228}, {3601, 4190}, {3622, 4313}, {3671, 3869}, {3687, 3936}, {3706, 4966}, {3741, 5208}, {3757, 4645}, {3771, 3980}, {3814, 3833}, {3841, 3874}, {3847, 4892}, {3848, 5087}, {3873, 4847}, {3890, 4301}, {3897, 4311}, {3969, 4431}, {4312, 4512}
X(5249) = isogonal conjugate of X(2259)
X(5249) = complement of X(3219)
In the plane of a triangle ABC, let
I = X(1) = incenter;
DEF = intouch triangle;
(O) = circumcircle;
A' = the point, other than A, where the circle {{A,I,D}} meets (O), and define B' and C' cyclically;
Oa = center of the circle {{B',E,F,C'}}, and define Ob and Oc cyclically;
Oaa = A-extraversion of Oa, and define Obb and Occ cyclically;
T = affine transfomation that maps ABC onto OaObOc;
TT = affine transformation that maps ABC onto OaaObbOcc.
Then X(4350) = finite fixed point of T, and X(5250) = finite fixed point of TT. (Angel Montesdeoca, December 29, 2023)
X(5250) lies on these lines: {1, 21}, {2, 40}, {3, 392}, {8, 9}, {10, 1479}, {19, 29}, {35, 997}, {46, 1125}, {55, 78}, {56, 4640}, {57, 3616}, {65, 1001}, {72, 3295}, {77, 221}, {100, 936}, {145, 3219}, {165, 404}, {169, 3294}, {200, 3871}, {210, 3913}, {220, 4520}, {333, 4673}, {377, 516}, {380, 2287}, {405, 517}, {443, 3587}, {474, 3579}, {484, 3624}, {518, 3303}, {551, 3338}, {614, 986}, {631, 3359}, {748, 1722}, {908, 3085}, {942, 4666}, {958, 3057}, {964, 1766}, {976, 3749}, {988, 1201}, {999, 3916}, {1005, 1490}, {1039, 2212}, {1158, 3576}, {1191, 3666}, {1220, 4676}, {1329, 4679}, {1698, 4193}, {1699, 2476}, {1708, 3340}, {1709, 4297}, {2255, 2256}, {2334, 4663}, {2944, 4203}, {3158, 4420}, {3218, 3333}, {3241, 3929}, {3555, 3927}, {3586, 5086}, {3601, 4511}, {3652, 3655}, {3678, 4917}, {3679, 5178}, {3704, 3966}, {3714, 4387}, {3715, 4662}, {3729, 4968}, {3742, 5221}, {3746, 3811}, {3748, 3962}, {3812, 4423}, {3885, 4853}, {4255, 4689}, {4329, 4357}, {4383, 4646}
X(5251) lies on these lines: {1, 6}, {2, 36}, {3, 1698}, {8, 3746}, {10, 21}, {28, 1224}, {30, 3925}, {40, 3560}, {42, 4653}, {55, 3679}, {56, 3624}, {63, 4880}, {65, 191}, {71, 4877}, {119, 140}, {165, 1012}, {261, 5209}, {404, 3634}, {442, 3585}, {443, 4299}, {452, 1479}, {484, 3753}, {498, 2551}, {499, 5084}, {515, 1006}, {517, 3683}, {519, 1621}, {748, 995}, {750, 4257}, {758, 3219}, {846, 4424}, {899, 4256}, {908, 1125}, {936, 3612}, {997, 3305}, {999, 4423}, {1308, 2752}, {1334, 4752}, {1376, 5010}, {1573, 1914}, {1699, 3428}, {2099, 3899}, {2475, 3841}, {2550, 4302}, {2646, 5044}, {2886, 3583}, {3086, 5129}, {3295, 3632}, {3303, 3633}, {3336, 3812}, {3579, 3698}, {3582, 3816}, {3626, 3871}, {3647, 3754}, {3691, 4251}, {3715, 3940}, {3757, 4692}, {3826, 4316}, {3833, 4973}, {3844, 4265}, {3884, 4861}, {3901, 3927}, {3913, 4668}, {4015, 4420}, {4187, 4999}, {4223, 5144}, {4309, 5082}, {4428, 4677}, {4512, 5119}
X(5252) lies on these lines: {1, 5}, {2, 1319}, {4, 1000}, {7, 8}, {10, 56}, {30, 5119}, {34, 1883}, {55, 515}, {57, 3679}, {63, 529}, {145, 3485}, {210, 3421}, {225, 5130}, {226, 519}, {354, 1056}, {392, 4679}, {443, 3698}, {484, 3654}, {498, 1385}, {517, 1478}, {528, 3895}, {553, 4669}, {594, 2285}, {899, 1450}, {944, 2646}, {946, 2098}, {950, 954}, {960, 3436}, {962, 5229}, {993, 5172}, {999, 1737}, {1010, 1408}, {1125, 1388}, {1155, 4293}, {1210, 3304}, {1376, 1470}, {1415, 4386}, {1420, 1698}, {1788, 3600}, {1826, 2256}, {1877, 5101}, {2475, 3909}, {2476, 4861}, {2886, 3872}, {3036, 3306}, {3058, 3586}, {3241, 4870}, {3244, 3947}, {3339, 4668}, {3340, 3632}, {3434, 3880}, {3474, 5183}, {3488, 3748}, {3579, 4299}, {3584, 3655}, {3621, 5178}, {3625, 3671}, {3626, 4031}, {3877, 5080}, {3890, 5046}, {3893, 5082}, {4297, 5217}, {4311, 5204}, {4415, 5155}, {4654, 4677}
X(5252) = outer-Johnson-to-ABC similarity image of X(1)
X(5253) lies on these lines: {1, 88}, {2, 12}, {3, 962}, {5, 104}, {8, 474}, {11, 2475}, {21, 36}, {35, 551}, {40, 3890}, {46, 3877}, {55, 3622}, {57, 3869}, {63, 3361}, {78, 3333}, {81, 1193}, {85, 934}, {86, 4225}, {145, 1376}, {171, 1201}, {191, 4973}, {377, 3086}, {411, 3576}, {484, 3884}, {497, 4190}, {499, 2476}, {758, 3337}, {908, 4298}, {936, 3681}, {942, 4511}, {960, 3218}, {976, 3976}, {978, 1468}, {993, 3624}, {997, 3338}, {1001, 4189}, {1004, 4313}, {1014, 4357}, {1104, 4239}, {1210, 5086}, {1290, 3109}, {1319, 3812}, {1470, 3485}, {1476, 5176}, {1478, 4193}, {2260, 2287}, {2306, 5240}, {2478, 4293}, {2646, 3742}, {3294, 5030}, {3336, 3878}, {3428, 3523}, {3555, 4420}, {3585, 3825}, {3601, 4666}, {3617, 4413}, {3623, 3913}, {3636, 3746}, {3753, 4861}, {3811, 3889}, {3816, 5046}, {4187, 5080}, {4696, 5205}, {5187, 5229}
Let U be the circle obtained as the inverse-in-the-polar-circle of the 2nd Lemoine circle. The center of U is X(5254). (Randy Hutson, November 22, 2014)
X(5254) lies on these lines: {2, 1975}, {3, 230}, {4, 6}, {5, 39}, {11, 2275}, {12, 2276}, {20, 3053}, {30, 32}, {76, 141}, {83, 597}, {140, 574}, {148, 384}, {184, 460}, {185, 1562}, {187, 550}, {194, 325}, {232, 235}, {290, 695}, {297, 3981}, {315, 524}, {316, 3629}, {338, 1235}, {376, 5023}, {381, 2548}, {395, 616}, {396, 617}, {427, 1194}, {489, 3068}, {490, 3069}, {495, 1500}, {496, 1015}, {538, 626}, {548, 5206}, {594, 4385}, {595, 5134}, {726, 4136}, {1086, 3673}, {1105, 1970}, {1107, 2886}, {1146, 3959}, {1180, 5133}, {1184, 1370}, {1196, 1368}, {1329, 1575}, {1353, 1570}, {1384, 1657}, {1574, 3820}, {1596, 3199}, {1656, 3055}, {1885, 1968}, {3061, 3944}, {3522, 5210}, {3564, 5028}, {3589, 4048}, {3627, 5007}, {3721, 3782}, {3845, 5041}, {3934, 4045}, {4173, 5167}
X(5254) = midpoint of X(3070) and X(3071)
X(5254) = complement of X(1975)
X(5254) = anticomplement of X(7789)
X(5254) = exsimilicenter of nine-point and (1/2)-Moses circles; the insimilicenter is X(3815)
X(5254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,39,3815), (39,115,5)
X(5255) lies on these lines: {1, 3}, {2, 3915}, {4, 983}, {6, 979}, {8, 31}, {9, 989}, {10, 82}, {21, 902}, {32, 2329}, {37, 3496}, {42, 3871}, {44, 4662}, {58, 519}, {72, 3961}, {100, 1193}, {145, 1468}, {213, 3684}, {341, 4676}, {355, 3073}, {404, 1201}, {518, 1046}, {528, 1834}, {582, 3654}, {601, 944}, {603, 3476}, {643, 2363}, {750, 3616}, {752, 1330}, {958, 3052}, {976, 3869}, {978, 1191}, {1106, 4308}, {1203, 3293}, {1253, 4344}, {1254, 4318}, {1279, 3812}, {1386, 4646}, {1572, 3061}, {1706, 1722}, {1724, 3679}, {1743, 3713}, {1914, 2295}, {2176, 4386}, {2269, 2298}, {2292, 3920}, {2321, 4264}, {2650, 3722}, {2901, 4693}, {3434, 5230}, {3743, 5184}, {3769, 4673}, {3868, 3938}, {3923, 4385}, {3973, 4866}, {3997, 4251}, {4255, 4421}, {4418, 4968}, {4649, 5145}
X(5255) = {X(1),X(3)}-harmonic conjugate of X(37617)
X(5256) lies on these lines: {1, 2}, {6, 63}, {7, 223}, {21, 1453}, {27, 34}, {33, 469}, {37, 3305}, {38, 3751}, {55, 1386}, {56, 4719}, {57, 77}, {58, 4652}, {92, 2331}, {193, 4001}, {204, 1013}, {226, 3946}, {238, 968}, {312, 4360}, {321, 3875}, {329, 3672}, {333, 3759}, {345, 3618}, {380, 3101}, {440, 1062}, {464, 1040}, {553, 4667}, {748, 1962}, {894, 3210}, {908, 3553}, {940, 1100}, {982, 4649}, {988, 1468}, {1211, 4272}, {1214, 1445}, {1230, 3760}, {1376, 3745}, {1427, 4350}, {1707, 2308}, {1743, 3219}, {1763, 2172}, {2177, 3749}, {2352, 5132}, {3052, 4689}, {3247, 3930}, {3434, 3755}, {3677, 3873}, {3886, 3896}, {3923, 4970}, {3966, 4026}, {3993, 4011}, {4021, 4656}, {4085, 4865}, {4255, 4855}, {4270, 4357}, {4285, 4643}, {4413, 4682}, {4868, 5119}, {4886, 5224}
X(5256) = {X(1),X(2)}-harmonic conjugate of X(5287)
X(5257) lies on these lines: {1, 966}, {2, 7}, {6, 1125}, {8, 3247}, {10, 37}, {19, 406}, {45, 3634}, {71, 3294}, {75, 4044}, {86, 4416}, {141, 4698}, {145, 4034}, {192, 4967}, {198, 405}, {225, 281}, {228, 4204}, {238, 4264}, {391, 1449}, {392, 2262}, {461, 4512}, {551, 1100}, {573, 946}, {756, 3778}, {860, 1826}, {978, 5105}, {993, 2178}, {1001, 4254}, {1010, 4877}, {1211, 4035}, {1266, 4699}, {1654, 3879}, {1698, 1738}, {1743, 3624}, {2171, 4848}, {2238, 4104}, {3008, 4657}, {3244, 3723}, {3617, 4007}, {3622, 4982}, {3632, 4545}, {3636, 4856}, {3663, 3739}, {3664, 4643}, {3671, 4047}, {3679, 4060}, {3912, 4687}, {3946, 4384}, {3949, 3970}, {3965, 4847}, {3985, 4656}, {4021, 4361}, {4061, 4771}, {4260, 5044}, {4389, 4751}, {4431, 4664}, {4648, 4748}, {4665, 4681}, {4668, 4898}
X(5258) lies on these lines: {1, 6}, {3, 3679}, {8, 35}, {10, 36}, {21, 519}, {55, 3632}, {56, 1698}, {65, 4880}, {100, 3626}, {101, 3691}, {172, 1573}, {191, 517}, {200, 3612}, {210, 1385}, {214, 4015}, {442, 529}, {443, 4317}, {484, 3916}, {498, 3421}, {499, 2551}, {515, 3651}, {528, 4330}, {535, 2475}, {551, 5047}, {961, 1224}, {999, 3624}, {1005, 4847}, {1319, 5044}, {1388, 3715}, {1444, 4967}, {1478, 5177}, {1482, 3899}, {1621, 3244}, {2099, 3927}, {2550, 4299}, {2802, 3647}, {2886, 3585}, {3214, 4256}, {3218, 3754}, {3219, 3878}, {3295, 3633}, {3336, 3753}, {3337, 3812}, {3560, 3929}, {3582, 4187}, {3625, 3871}, {3678, 4511}, {3681, 3897}, {3730, 4390}, {3813, 4857}, {3820, 5193}, {3913, 4677}, {3918, 4973}, {3956, 4881}, {4302, 5082}, {4668, 5010}, {4853, 5119}
X(5259) lies on these lines: {1, 6}, {2, 35}, {3, 1699}, {10, 1621}, {12, 2078}, {21, 36}, {28, 1839}, {46, 4512}, {55, 1698}, {56, 4355}, {58, 3720}, {100, 3634}, {105, 1224}, {140, 2077}, {142, 1770}, {191, 942}, {386, 748}, {411, 3817}, {442, 3583}, {443, 4302}, {452, 1478}, {474, 5010}, {484, 3812}, {498, 5084}, {551, 2975}, {846, 3670}, {946, 1006}, {993, 3616}, {1089, 3757}, {1193, 4653}, {1259, 5231}, {1283, 5051}, {1329, 3584}, {1838, 4183}, {2260, 4877}, {2308, 4658}, {2550, 4309}, {2886, 4857}, {3085, 5129}, {3218, 3647}, {3219, 3874}, {3245, 3754}, {3293, 3750}, {3295, 3679}, {3303, 3632}, {3305, 3811}, {3336, 4640}, {3337, 3742}, {3560, 3576}, {3582, 4999}, {3685, 4647}, {3822, 5046}, {3848, 5131}, {3894, 3927}, {3898, 4861}, {3935, 4015}, {4068, 4716}
X(5260) lies on these lines: {1, 748}, {2, 12}, {8, 405}, {9, 1405}, {10, 21}, {36, 3634}, {55, 3617}, {63, 3339}, {65, 3219}, {104, 140}, {145, 1001}, {191, 3754}, {355, 1006}, {392, 4861}, {404, 993}, {442, 5080}, {452, 3434}, {484, 3647}, {502, 1224}, {644, 3294}, {846, 4642}, {950, 5178}, {956, 3616}, {984, 3924}, {997, 3897}, {1043, 4651}, {1104, 3920}, {1320, 3884}, {1376, 4189}, {1478, 4197}, {1722, 4850}, {1757, 2650}, {1891, 4233}, {2078, 5176}, {2475, 3925}, {2646, 3740}, {2886, 5046}, {3091, 3428}, {3218, 3812}, {3293, 4653}, {3303, 3621}, {3337, 3833}, {3579, 4002}, {3585, 3841}, {3622, 4423}, {3626, 3746}, {3679, 3871}, {3697, 4420}, {3698, 4640}, {3757, 4696}, {3872, 3890}, {3913, 4678}, {3935, 4662}, {4183, 5174}, {4188, 4413}, {4511, 5044}
X(5261) lies on these lines: {1, 3091}, {2, 12}, {4, 390}, {5, 1056}, {7, 10}, {8, 226}, {11, 5068}, {20, 35}, {34, 3920}, {55, 3146}, {65, 3617}, {85, 341}, {145, 3485}, {192, 2996}, {355, 3487}, {381, 1058}, {387, 1126}, {391, 1405}, {442, 3421}, {452, 2078}, {496, 3545}, {497, 3832}, {498, 3523}, {519, 4323}, {612, 4296}, {976, 2647}, {984, 1254}, {986, 4346}, {999, 3090}, {1125, 4308}, {1219, 3705}, {1393, 4392}, {1441, 4385}, {1469, 3620}, {1479, 3839}, {1617, 5047}, {1698, 4298}, {1722, 4327}, {1837, 3475}, {2099, 3621}, {3086, 5056}, {3303, 5225}, {3304, 3614}, {3361, 3634}, {3476, 3622}, {3522, 5218}, {3543, 3585}, {3584, 4299}, {3616, 5219}, {3624, 4315}, {3649, 4678}, {3671, 3679}, {3704, 4461}, {3870, 5175}, {3961, 4332}, {4654, 4848}
X(5262) lies on these lines: {1, 2}, {3, 4850}, {6, 977}, {7, 34}, {21, 1104}, {28, 60}, {31, 986}, {37, 5047}, {57, 4296}, {58, 3218}, {63, 1453}, {65, 82}, {75, 964}, {77, 1467}, {238, 2292}, {257, 1178}, {312, 5192}, {350, 1228}, {377, 4000}, {404, 3752}, {452, 3672}, {595, 4424}, {758, 1203}, {950, 3100}, {982, 1468}, {990, 3146}, {1010, 4359}, {1040, 4313}, {1046, 2308}, {1062, 3488}, {1100, 2303}, {1191, 3877}, {1220, 4968}, {1245, 4388}, {1325, 2363}, {1442, 3212}, {1449, 2082}, {1621, 3931}, {1724, 3219}, {1743, 3951}, {2476, 3772}, {2646, 4719}, {3210, 4195}, {3315, 5045}, {3337, 4351}, {3339, 4347}, {3744, 3871}, {3745, 3812}, {3746, 4868}, {3876, 4383}, {3891, 4385}, {4972, 5015}, {5090, 5142}
X(5263) lies on these lines: {1, 75}, {2, 11}, {6, 8}, {9, 4676}, {10, 82}, {19, 29}, {31, 333}, {37, 3685}, {38, 4418}, {42, 3996}, {69, 4307}, {85, 2263}, {87, 1222}, {141, 4645}, {171, 3741}, {190, 984}, {239, 1386}, {312, 612}, {321, 3920}, {516, 4357}, {518, 894}, {519, 4649}, {752, 3775}, {958, 4195}, {982, 3980}, {993, 4234}, {1008, 5224}, {1125, 1738}, {1211, 4388}, {1215, 3961}, {1266, 4353}, {1279, 3739}, {1441, 4318}, {1757, 4672}, {1861, 5174}, {1999, 3706}, {2049, 3295}, {2607, 3878}, {2975, 3286}, {3219, 4981}, {3241, 4499}, {3242, 4363}, {3246, 3846}, {3416, 3661}, {3616, 4000}, {3664, 4684}, {3744, 3757}, {3751, 3758}, {3842, 4432}, {3879, 4349}, {3993, 4693}, {4709, 4716}, {4732, 4974}
X(5263) = anticomplement of X(4026)
X(5264) lies on these lines: {1, 3}, {2, 595}, {6, 3293}, {8, 58}, {10, 31}, {32, 2295}, {37, 1759}, {41, 3997}, {43, 1203}, {44, 3697}, {79, 983}, {80, 987}, {81, 3871}, {82, 4429}, {90, 989}, {100, 386}, {109, 388}, {191, 984}, {213, 4386}, {238, 1698}, {404, 995}, {405, 3052}, {474, 1191}, {515, 601}, {519, 1468}, {573, 2298}, {594, 4275}, {609, 2329}, {748, 3634}, {750, 1125}, {758, 976}, {956, 4252}, {1046, 3961}, {1089, 3923}, {1104, 3753}, {1106, 4315}, {1210, 1497}, {1253, 4349}, {1254, 4347}, {1376, 3216}, {1451, 4848}, {1453, 1706}, {1478, 1777}, {1714, 2550}, {2308, 3214}, {2345, 4264}, {2975, 4257}, {3085, 4307}, {3754, 3924}, {3874, 3938}, {4362, 4647}, {4450, 5051}
X(5265) lies on these lines: {1, 3523}, {2, 12}, {3, 390}, {7, 1125}, {8, 1420}, {10, 4308}, {11, 3146}, {20, 36}, {34, 4232}, {43, 4322}, {57, 3616}, {65, 3622}, {108, 4200}, {140, 1056}, {145, 1319}, {193, 1428}, {201, 4392}, {238, 1106}, {279, 1447}, {348, 3598}, {376, 496}, {391, 604}, {404, 1617}, {439, 4366}, {495, 3525}, {497, 3522}, {499, 3091}, {551, 3339}, {614, 4296}, {631, 999}, {938, 3576}, {944, 5126}, {956, 1476}, {978, 1458}, {988, 3672}, {993, 5129}, {1388, 3623}, {1445, 3333}, {1466, 1621}, {1470, 4189}, {1471, 3945}, {1478, 5056}, {1698, 4315}, {3241, 4848}, {3295, 3524}, {3304, 5218}, {3476, 3617}, {3543, 3582}, {3624, 4298}, {3660, 3868}, {5059, 5225}, {5068, 5229}
X(5266) lies on these lines: {1, 3}, {2, 5015}, {4, 4339}, {6, 3694}, {10, 1104}, {21, 3920}, {31, 72}, {32, 37}, {38, 3916}, {39, 1100}, {42, 1009}, {44, 3678}, {58, 518}, {187, 3723}, {200, 1453}, {210, 1724}, {238, 5044}, {386, 1386}, {387, 3189}, {392, 1472}, {405, 612}, {442, 3011}, {474, 614}, {519, 3704}, {595, 960}, {601, 1071}, {902, 2292}, {943, 2298}, {975, 1001}, {983, 987}, {997, 1191}, {1010, 3757}, {1125, 1279}, {1384, 3247}, {1427, 4347}, {1468, 3555}, {1707, 3927}, {1770, 3782}, {1785, 1852}, {2204, 5089}, {3242, 4252}, {3293, 3689}, {3419, 5230}, {3475, 4340}, {3487, 4307}, {3753, 3924}, {3831, 4434}, {3879, 3933}, {3881, 4864}, {4195, 4385}, {4256, 4719}
X(5266) = {X(1),X(3)}-harmonic conjugate of X(37592)
X(5267) lies on these lines: {1, 89}, {2, 3585}, {3, 10}, {8, 5010}, {12, 535}, {21, 36}, {30, 4999}, {35, 519}, {46, 3919}, {55, 3244}, {56, 551}, {58, 2185}, {63, 3612}, {78, 4134}, {100, 3626}, {140, 3814}, {149, 4330}, {187, 1107}, {191, 4511}, {214, 960}, {404, 3634}, {405, 5204}, {501, 1098}, {549, 1329}, {550, 2886}, {574, 4426}, {758, 2646}, {942, 4973}, {956, 3625}, {1011, 3840}, {1030, 3686}, {1055, 3294}, {1155, 3754}, {1319, 3884}, {1385, 3878}, {1444, 3664}, {1621, 3636}, {1698, 4188}, {1861, 3520}, {2178, 3986}, {2475, 4316}, {2550, 3528}, {2551, 3524}, {3035, 3530}, {3560, 3817}, {3635, 3746}, {3741, 4184}, {3812, 5122}, {3927, 4525}, {3940, 4537}, {4386, 5206}
X(5268) lies on these lines: {1, 2}, {6, 3740}, {9, 171}, {12, 1038}, {22, 5010}, {25, 35}, {31, 3305}, {33, 5218}, {37, 1376}, {38, 3306}, {45, 4640}, {55, 5020}, {57, 984}, {63, 750}, {69, 4104}, {87, 2297}, {100, 968}, {165, 846}, {181, 3781}, {210, 940}, {230, 3553}, {305, 3761}, {345, 4078}, {427, 5155}, {474, 988}, {1001, 3749}, {1196, 2276}, {1215, 3718}, {1370, 3585}, {1447, 4328}, {1448, 3947}, {1469, 3819}, {1742, 1750}, {2263, 5226}, {2650, 3984}, {3158, 3750}, {3242, 3742}, {3247, 3290}, {3550, 4512}, {3554, 3815}, {3666, 4413}, {3715, 4641}, {3729, 3971}, {3744, 4423}, {3745, 4383}, {3772, 3826}, {3929, 4650}, {3966, 5241}, {3967, 4363}, {4339, 5129}
X(5268) = orthoptic-circle-of-Steiner-inellipse-inverse of X(38471)
X(5268) = {X(1),X(2)}-harmonic conjugate of X(5272)
X(5269) lies on these lines: {1, 3}, {2, 3883}, {6, 200}, {9, 31}, {10, 1453}, {33, 1395}, {37, 3052}, {38, 3928}, {42, 1449}, {63, 3920}, {81, 3870}, {84, 601}, {181, 3056}, {197, 2270}, {204, 281}, {210, 1743}, {226, 3424}, {380, 3198}, {388, 1394}, {553, 4310}, {595, 975}, {611, 2003}, {614, 750}, {869, 2258}, {902, 968}, {950, 4339}, {984, 1707}, {985, 1961}, {987, 989}, {1001, 4682}, {1254, 4348}, {1376, 1386}, {1397, 2330}, {1407, 4321}, {1706, 4695}, {1999, 3886}, {2303, 2328}, {2318, 3997}, {3243, 3938}, {3474, 3663}, {3475, 3664}, {3632, 4046}, {3683, 3731}, {3715, 3973}, {3751, 3961}, {3782, 4312}, {3791, 4457}, {3923, 4135}, {4418, 4659}, {4641, 5223}
X(5270) lies on these lines: {1, 4}, {2, 4317}, {3, 3584}, {5, 3582}, {8, 3901}, {10, 3218}, {11, 3850}, {12, 36}, {21, 535}, {30, 3746}, {35, 495}, {55, 1657}, {56, 1656}, {65, 2962}, {79, 517}, {80, 942}, {149, 3635}, {377, 3679}, {381, 3304}, {382, 3303}, {442, 529}, {484, 4292}, {496, 3858}, {498, 3523}, {499, 3600}, {519, 2475}, {548, 4995}, {551, 5046}, {952, 3649}, {999, 3851}, {1125, 5080}, {1698, 3436}, {1737, 3337}, {1935, 2964}, {2550, 4668}, {2975, 3822}, {3058, 3627}, {3085, 3522}, {3086, 5068}, {3146, 4309}, {3295, 5073}, {3434, 3633}, {3754, 5176}, {3874, 5086}, {3884, 5057}, {3920, 5189}, {3947, 4311}, {4302, 5059}
X(5270) = {X(1),X(4)}-harmonic conjugate of X(4857)
X(5271) lies on these lines: {1, 2}, {7, 4001}, {9, 321}, {19, 27}, {45, 3175}, {55, 3696}, {57, 1150}, {77, 1943}, {226, 3686}, {278, 307}, {312, 3305}, {322, 3306}, {329, 391}, {344, 3610}, {379, 4968}, {440, 3419}, {469, 5174}, {518, 4042}, {740, 968}, {850, 1021}, {940, 3739}, {964, 1453}, {1001, 3706}, {1211, 3772}, {1376, 2352}, {1621, 3886}, {1707, 4418}, {1746, 1766}, {1790, 1958}, {1817, 2975}, {2886, 3966}, {3219, 3729}, {3416, 3925}, {3434, 3883}, {3487, 4101}, {3578, 4654}, {3666, 4361}, {3715, 3967}, {3731, 3995}, {3782, 4643}, {3846, 4682}, {3875, 5235}, {3891, 4981}, {3929, 4659}, {3936, 4034}, {3969, 4007}, {4363, 4641}, {4417, 4886}
X(5271) = isogonal conjugate of X(2215)
X(5272) lies on these lines: {1, 2}, {6, 3742}, {9, 982}, {11, 1040}, {25, 36}, {31, 3306}, {38, 3305}, {56, 5020}, {57, 238}, {63, 244}, {87, 269}, {105, 165}, {142, 1716}, {230, 3554}, {305, 3760}, {354, 3751}, {405, 988}, {497, 1738}, {968, 4850}, {984, 3677}, {990, 3817}, {1001, 3752}, {1191, 3812}, {1196, 2275}, {1279, 1376}, {1370, 3583}, {1386, 3848}, {1435, 1957}, {1449, 4038}, {1699, 1721}, {1724, 3338}, {1739, 5119}, {3052, 3246}, {3056, 3819}, {3242, 3740}, {3271, 3784}, {3315, 3681}, {3361, 4223}, {3553, 3815}, {3666, 4423}, {3729, 4011}, {3744, 4413}, {3772, 3816}, {3782, 4679}, {3895, 4695}, {4327, 5226}, {4641, 4860}
X(5272) = {X(1),X(2)}-harmonic conjugate of X(5268)
X(5273) lies on these lines: {2, 7}, {8, 21}, {10, 20}, {27, 281}, {31, 4344}, {81, 219}, {189, 268}, {191, 4295}, {210, 5218}, {220, 940}, {261, 1264}, {312, 3161}, {348, 479}, {354, 960}, {377, 1155}, {390, 4512}, {391, 3687}, {405, 938}, {443, 3916}, {497, 3683}, {631, 1071}, {910, 966}, {936, 3523}, {1002, 5208}, {1200, 3691}, {1210, 5129}, {1212, 3666}, {1214, 3160}, {1329, 4197}, {1479, 2894}, {1617, 2975}, {1698, 4208}, {1707, 4307}, {1764, 3730}, {2096, 3820}, {2550, 4640}, {3187, 4460}, {3210, 4402}, {3241, 3748}, {3474, 3925}, {3487, 3927}, {3679, 4304}, {3711, 4995}, {3772, 4419}, {3869, 4323}, {3877, 4345}, {4860, 4999}
X(5273) = {X(2),X(63)}-harmonic conjugate of X(7)
X(5273) = complement of isotomic conjugate of X(30711)
X(5274) lies on these lines: {1, 3091}, {2, 11}, {4, 496}, {5, 1058}, {7, 1699}, {8, 3452}, {12, 5068}, {20, 36}, {56, 3146}, {145, 1837}, {150, 4845}, {279, 2898}, {330, 2996}, {346, 3705}, {381, 1056}, {388, 3832}, {495, 3545}, {499, 3523}, {519, 4345}, {614, 3100}, {938, 946}, {950, 3616}, {962, 1210}, {982, 2310}, {1125, 4208}, {1478, 3839}, {1788, 5183}, {1864, 3873}, {2098, 3621}, {2551, 3813}, {2900, 4511}, {3056, 3620}, {3057, 3617}, {3085, 5056}, {3090, 3295}, {3304, 5229}, {3486, 3622}, {3543, 3583}, {3582, 4302}, {3598, 4872}, {3624, 4314}, {3679, 4342}, {3741, 5232}, {3944, 4310}, {3945, 4038}, {4187, 5082}
X(5275) lies on these lines: {1, 2271}, {2, 6}, {9, 171}, {19, 25}, {21, 3053}, {22, 1030}, {32, 405}, {39, 474}, {45, 2243}, {56, 1107}, {169, 975}, {172, 958}, {220, 2295}, {305, 3770}, {392, 1572}, {404, 5013}, {406, 2207}, {442, 3767}, {614, 1100}, {672, 750}, {956, 1573}, {984, 3509}, {1001, 1914}, {1194, 4261}, {1196, 2092}, {1376, 2276}, {1447, 5228}, {1449, 4038}, {1468, 3691}, {1575, 4413}, {1610, 3207}, {1655, 1975}, {2235, 5205}, {2280, 3720}, {2548, 4187}, {3242, 3726}, {3247, 3750}, {3263, 4363}, {3291, 4277}, {3550, 3731}, {3923, 3985}, {4189, 5023}, {4223, 4258}, {4254, 5020}, {4262, 4653}, {4655, 4987}
X(5275) = {X(2),X(385)}-harmonic conjugate of X(16992)
X(5276) lies on these lines: {1, 41}, {2, 6}, {9, 31}, {21, 32}, {25, 941}, {37, 82}, {38, 3509}, {39, 404}, {42, 3684}, {100, 743}, {171, 672}, {172, 1107}, {284, 4224}, {384, 1655}, {573, 1754}, {584, 4228}, {595, 3294}, {609, 993}, {614, 1449}, {894, 3263}, {910, 3666}, {984, 985}, {1100, 3290}, {1180, 4261}, {1194, 2092}, {1196, 2670}, {1206, 3757}, {1333, 1627}, {1500, 3871}, {1572, 3877}, {1778, 4275}, {1922, 4518}, {2207, 4194}, {2292, 3496}, {2348, 3745}, {2476, 3767}, {2548, 4193}, {2651, 4274}, {3053, 4189}, {3598, 5228}, {3930, 3961}, {4188, 5013}, {4209, 4352}, {4239, 4277}, {4424, 5011}, {5007, 5047}
X(5277) lies on these lines: {1, 1929}, {2, 32}, {6, 474}, {8, 2242}, {9, 2305}, {10, 172}, {12, 1415}, {21, 187}, {35, 37}, {36, 1107}, {39, 404}, {41, 750}, {58, 2238}, {99, 1655}, {100, 1500}, {101, 2295}, {112, 451}, {115, 2475}, {171, 213}, {199, 612}, {230, 442}, {274, 385}, {377, 3767}, {391, 5042}, {405, 3053}, {406, 1968}, {468, 2204}, {574, 4188}, {609, 1698}, {762, 2248}, {763, 1654}, {846, 2135}, {940, 2271}, {966, 5019}, {992, 4264}, {1125, 1914}, {1213, 1333}, {1573, 2975}, {2092, 2303}, {2160, 4016}, {2241, 3616}, {2549, 4190}, {3291, 4239}, {3509, 3954}, {3727, 5011}, {4189, 5206}
X(5278) lies on these lines: {1, 4981}, {2, 6}, {8, 405}, {9, 321}, {10, 31}, {37, 3187}, {45, 3995}, {55, 4651}, {63, 169}, {75, 3219}, {100, 1011}, {142, 4001}, {226, 1405}, {306, 2280}, {317, 445}, {573, 1746}, {748, 3741}, {756, 4362}, {896, 3980}, {956, 4245}, {968, 3896}, {984, 3891}, {1001, 4042}, {1125, 4101}, {1212, 3998}, {1229, 3719}, {1330, 4197}, {1441, 1708}, {1714, 5051}, {2177, 4685}, {2205, 4426}, {2476, 2651}, {2550, 4450}, {3006, 3966}, {3011, 4104}, {3120, 4703}, {3305, 4358}, {3681, 3757}, {3683, 3696}, {3691, 3765}, {3715, 3952}, {3729, 4980}, {3739, 4641}, {3791, 3842}, {3883, 5014}, {5081, 5136}
X(5279) lies on these lines: {2, 7}, {6, 977}, {8, 19}, {10, 1781}, {20, 346}, {21, 37}, {27, 321}, {28, 72}, {40, 3692}, {48, 4511}, {69, 1760}, {71, 1761}, {75, 379}, {78, 610}, {100, 3694}, {101, 2327}, {169, 391}, {198, 1259}, {219, 608}, {272, 335}, {281, 3436}, {306, 2897}, {377, 2345}, {380, 3870}, {518, 2264}, {573, 1759}, {604, 3061}, {910, 3965}, {965, 3876}, {975, 3731}, {1172, 4463}, {1330, 4456}, {1442, 1959}, {1723, 4310}, {1817, 3998}, {1826, 5080}, {1953, 4861}, {2092, 2240}, {2171, 2329}, {2173, 3949}, {2174, 4053}, {2256, 3877}, {2269, 3496}, {2354, 4388}, {3950, 4304}
X(5280) lies on these lines: {1, 6}, {3, 609}, {31, 3730}, {32, 35}, {36, 39}, {41, 386}, {42, 251}, {48, 5105}, {58, 672}, {71, 4264}, {81, 3912}, {83, 350}, {101, 1193}, {304, 3758}, {595, 1334}, {651, 3674}, {894, 1930}, {986, 1759}, {1015, 5041}, {1126, 1438}, {1174, 2299}, {1197, 3507}, {1384, 5217}, {1448, 2285}, {1468, 4253}, {1500, 1914}, {1890, 3755}, {1922, 3864}, {1973, 4270}, {2174, 5153}, {2242, 2275}, {2260, 4284}, {2503, 2653}, {3053, 5010}, {3056, 5039}, {3293, 3684}, {3496, 4424}, {3509, 3670}, {3685, 4099}, {3710, 3997}, {3744, 3991}, {3934, 4396}, {3961, 4006}, {4642, 5011}, {5024, 5204}
X(5281) lies on these lines: {1, 3523}, {2, 11}, {3, 1056}, {7, 165}, {8, 3158}, {9, 1200}, {10, 4313}, {12, 3146}, {20, 35}, {33, 4232}, {43, 2293}, {140, 1058}, {144, 4640}, {145, 2646}, {171, 1253}, {193, 2330}, {376, 495}, {388, 3522}, {391, 2268}, {496, 3525}, {498, 3091}, {516, 5226}, {551, 4345}, {612, 3100}, {631, 3295}, {999, 3524}, {1040, 3920}, {1155, 3475}, {1447, 3672}, {1479, 5056}, {1697, 3616}, {1698, 4314}, {1961, 4336}, {3057, 3622}, {3086, 3746}, {3486, 3617}, {3487, 3579}, {3543, 3584}, {3550, 4307}, {3614, 3854}, {3712, 3974}, {4293, 5010}, {5059, 5229}, {5068, 5225}
X(5282) lies on these lines: {2, 7}, {6, 38}, {8, 3496}, {10, 1759}, {31, 37}, {32, 976}, {41, 72}, {44, 4003}, {45, 896}, {55, 3930}, {66, 71}, {141, 4376}, {169, 3691}, {191, 3730}, {198, 199}, {201, 220}, {210, 910}, {218, 3927}, {517, 4390}, {518, 2280}, {748, 3290}, {956, 2170}, {984, 985}, {997, 1055}, {1212, 1451}, {1395, 5089}, {1707, 1961}, {1709, 1766}, {1761, 2345}, {1914, 3938}, {2235, 3116}, {2239, 2276}, {2243, 4386}, {2246, 4712}, {2269, 5227}, {2329, 3869}, {2911, 3958}, {2975, 3061}, {3679, 5011}, {3681, 3684}, {3693, 4640}, {3721, 3924}, {4119, 5014}, {4136, 5016}
X(5283) lies on these lines: {1, 6}, {2, 39}, {8, 1500}, {10, 2276}, {21, 32}, {35, 4386}, {42, 3691}, {115, 2476}, {172, 993}, {187, 4189}, {232, 406}, {377, 2549}, {386, 2238}, {391, 941}, {404, 574}, {474, 5013}, {612, 1011}, {756, 869}, {846, 3496}, {940, 5021}, {966, 2092}, {968, 2082}, {986, 3125}, {992, 5105}, {1015, 3616}, {1125, 2275}, {1213, 4261}, {1475, 3720}, {1506, 4193}, {1575, 1698}, {1621, 2241}, {2242, 2975}, {2268, 2304}, {2292, 3735}, {2295, 3730}, {2303, 5019}, {2478, 2548}, {3199, 4194}, {3666, 4384}, {3815, 4187}, {3959, 4424}, {4185, 5089}, {4251, 4653}, {4264, 4877}
X(5283) = isotomic conjugate of X(1218)
X(5283) = complement of X(34284)
X(5283) = {X(1),X(9)}-harmonic conjugate of X(213)
X(5284) lies on these lines: {1, 748}, {2, 11}, {9, 3873}, {21, 36}, {37, 3108}, {38, 3315}, {44, 4883}, {81, 238}, {210, 3957}, {244, 846}, {329, 405}, {354, 3219}, {404, 3624}, {484, 3833}, {496, 943}, {899, 3750}, {958, 3622}, {968, 4850}, {1155, 3848}, {1279, 3920}, {1320, 3898}, {1479, 4197}, {1617, 5226}, {1698, 3871}, {1848, 4233}, {2895, 4966}, {3218, 3683}, {3246, 3745}, {3303, 3617}, {3306, 4512}, {3337, 3647}, {3436, 5129}, {3634, 3746}, {3685, 4359}, {3715, 4661}, {3740, 3748}, {3741, 5235}, {3757, 4358}, {3812, 5183}, {3841, 4857}, {3936, 4204}, {4228, 4872}, {4418, 4432}, {4430, 5220}
X(5285) lies on these lines: {1, 3}, {9, 25}, {10, 28}, {22, 63}, {23, 3219}, {31, 579}, {33, 1766}, {42, 284}, {48, 3190}, {71, 1474}, {72, 2915}, {73, 3430}, {100, 306}, {101, 2318}, {109, 1297}, {154, 219}, {159, 197}, {181, 2330}, {184, 2323}, {198, 1260}, {199, 228}, {209, 2194}, {212, 573}, {222, 1350}, {226, 4220}, {291, 1283}, {511, 2003}, {516, 1848}, {951, 1042}, {1376, 3844}, {1397, 3056}, {1473, 3928}, {1486, 4512}, {1495, 3690}, {1631, 3185}, {1995, 3305}, {2187, 2289}, {2222, 2747}, {2299, 4456}, {2360, 3682}, {3098, 3784}, {3752, 5096}, {4221, 4304}
X(5286) lies on these lines: {2, 39}, {4, 6}, {20, 32}, {83, 2996}, {115, 147}, {140, 5024}, {148, 4027}, {172, 4293}, {187, 3522}, {193, 315}, {230, 631}, {232, 3089}, {316, 1570}, {376, 3053}, {385, 3785}, {390, 2241}, {487, 3068}, {488, 3069}, {550, 1384}, {574, 3523}, {578, 1217}, {609, 4299}, {672, 5230}, {962, 1572}, {1212, 3772}, {1285, 3529}, {1506, 5056}, {1851, 2082}, {1885, 3172}, {1914, 4294}, {2242, 3600}, {2275, 3086}, {2276, 3085}, {2345, 4385}, {3054, 3533}, {3090, 3815}, {3096, 3620}, {3146, 5007}, {3528, 5023}, {3673, 4000}, {3832, 5041}, {4644, 4911}, {5008, 5059}
X(5286) = anticomplement of X(7795)
X(5286) = barycentric product X(2345)*X(4000)
X(5287) lies on these lines: {1, 2}, {6, 3305}, {9, 81}, {27, 33}, {34, 469}, {37, 63}, {45, 4641}, {46, 3743}, {55, 4682}, {57, 1255}, {77, 226}, {86, 312}, {171, 968}, {223, 1442}, {329, 3945}, {333, 4687}, {440, 1060}, {464, 1038}, {750, 1962}, {756, 3751}, {984, 4038}, {1001, 3745}, {1100, 4383}, {1211, 4851}, {1230, 3761}, {1386, 4423}, {1453, 5047}, {1790, 2268}, {1817, 3601}, {2334, 4662}, {3175, 4363}, {3219, 3731}, {3242, 4883}, {3306, 3666}, {3664, 4656}, {3715, 4663}, {3723, 3752}, {3729, 3995}, {3737, 4789}, {3782, 4675}, {3875, 4359}, {3980, 3993}
X(5287) = {X(1),X(2)}-harmonic conjugate of X(5256)
X(5288) lies on these lines: {1, 6}, {3, 3632}, {8, 36}, {21, 3244}, {35, 519}, {46, 4853}, {55, 3633}, {56, 3679}, {100, 3625}, {145, 993}, {191, 3057}, {214, 4420}, {404, 3626}, {499, 3421}, {528, 4324}, {529, 3585}, {758, 4861}, {999, 1698}, {1329, 3582}, {1376, 4668}, {1388, 3940}, {1621, 3635}, {1759, 4051}, {2098, 3899}, {2099, 3901}, {2178, 4034}, {2550, 4317}, {3219, 3884}, {3304, 3624}, {3337, 3753}, {3579, 3893}, {3583, 3813}, {3584, 4999}, {3636, 5047}, {3872, 4880}, {3880, 3916}, {3913, 5010}, {4253, 4390}, {4278, 4720}, {4299, 5082}, {4816, 5204}
X(5289) lies on these lines: {1, 6}, {2, 2099}, {3, 214}, {8, 11}, {10, 1482}, {21, 2320}, {36, 3899}, {55, 3877}, {56, 3218}, {63, 1319}, {65, 3306}, {78, 3057}, {145, 2551}, {200, 3880}, {210, 3872}, {329, 529}, {517, 997}, {519, 3452}, {527, 4315}, {551, 4930}, {758, 999}, {965, 1953}, {1388, 2975}, {1389, 3090}, {2390, 3784}, {3207, 3496}, {3295, 3884}, {3303, 3890}, {3304, 3868}, {3338, 4018}, {3340, 3812}, {3445, 3976}, {3576, 4640}, {3616, 4999}, {3679, 5123}, {3680, 4882}, {3876, 4861}, {3885, 4420}, {4421, 5119}, {4662, 4853}, {4711, 4915}
X(5290) lies on these lines: {1, 4}, {2, 3361}, {5, 3333}, {7, 10}, {8, 3671}, {12, 57}, {40, 495}, {56, 3624}, {65, 3679}, {79, 5119}, {85, 1930}, {142, 2551}, {165, 3085}, {200, 377}, {381, 5045}, {551, 4308}, {553, 1788}, {612, 1448}, {975, 4320}, {986, 4862}, {1074, 1103}, {1125, 3600}, {1388, 4870}, {1435, 5142}, {1697, 1836}, {1722, 4859}, {1773, 1781}, {2099, 3633}, {2475, 3870}, {2476, 5231}, {2550, 4882}, {3146, 4314}, {3244, 4323}, {3340, 3632}, {3616, 4315}, {3704, 4659}, {3920, 4347}, {3982, 4848}, {4666, 5046}, {4847, 5177}
X(5291) lies on these lines: {1, 6}, {2, 2242}, {8, 32}, {10, 172}, {21, 1500}, {31, 4390}, {36, 1575}, {39, 2975}, {58, 2295}, {100, 187}, {101, 2238}, {111, 898}, {115, 5080}, {145, 2241}, {232, 1783}, {385, 668}, {404, 1574}, {519, 1914}, {594, 1333}, {609, 3679}, {650, 667}, {759, 813}, {899, 1055}, {993, 2276}, {1016, 1252}, {1150, 3661}, {1571, 4652}, {1572, 3872}, {1759, 3959}, {2239, 5091}, {2243, 5011}, {2251, 3684}, {2345, 5019}, {2703, 5164}, {3125, 3509}, {3436, 3767}, {3734, 4441}, {3780, 4251}, {4112, 4362}
X(5292) lies on these lines: {1, 2}, {3, 1834}, {4, 58}, {5, 6}, {20, 4257}, {30, 4252}, {31, 1479}, {46, 1076}, {57, 225}, {69, 3454}, {81, 2476}, {140, 4255}, {230, 2271}, {283, 1724}, {345, 2901}, {442, 940}, {496, 1191}, {497, 595}, {579, 1766}, {631, 4256}, {902, 4309}, {942, 3772}, {959, 994}, {967, 1889}, {1046, 3944}, {1068, 4000}, {1150, 5051}, {1468, 1478}, {1719, 3336}, {2163, 4325}, {3072, 5156}, {3192, 3542}, {3193, 4193}, {3769, 5015}, {3824, 4675}, {3927, 4415}, {4187, 4383}, {4340, 5177}
X(5293) lies on these lines: {1, 2}, {3, 984}, {9, 32}, {12, 2647}, {21, 756}, {31, 3876}, {35, 228}, {37, 1247}, {38, 404}, {58, 1757}, {72, 171}, {100, 2292}, {201, 1758}, {238, 5044}, {474, 982}, {750, 3868}, {872, 4281}, {943, 2648}, {970, 3688}, {986, 1376}, {1010, 1215}, {1054, 3670}, {1104, 3740}, {1220, 3699}, {1468, 3681}, {1490, 1742}, {2303, 3949}, {3242, 3976}, {3496, 4386}, {3509, 3954}, {3731, 4262}, {3847, 5015}, {3927, 4650}, {4005, 4641}, {4096, 4234}, {4252, 5220}, {4267, 4557}, {4332, 5226}
X(5294) lies on these lines: {1, 3710}, {2, 7}, {6, 306}, {8, 1453}, {10, 31}, {38, 1125}, {44, 1211}, {81, 3912}, {141, 4001}, {228, 1009}, {345, 3618}, {474, 1473}, {516, 4972}, {519, 3969}, {896, 3634}, {1210, 5192}, {1215, 3011}, {1386, 3703}, {1698, 1707}, {1730, 4456}, {1738, 4418}, {1861, 2299}, {1890, 4429}, {2221, 4383}, {2321, 3187}, {2325, 3995}, {2887, 4672}, {3008, 4359}, {3586, 4217}, {3589, 3666}, {3683, 4026}, {3717, 3920}, {3745, 3932}, {3772, 4054}, {3773, 3791}, {3836, 4697}, {3914, 3923}, {4202, 4292}
X(5294) = complement of X(17184)
X(5295) lies on these lines: {1, 2049}, {4, 8}, {5, 3687}, {10, 37}, {12, 4046}, {65, 4647}, {75, 942}, {200, 3191}, {210, 1089}, {306, 442}, {312, 5044}, {319, 1330}, {341, 4043}, {387, 2345}, {392, 3702}, {728, 3294}, {964, 3187}, {1010, 1999}, {1150, 3916}, {1479, 3966}, {2292, 4365}, {3159, 3626}, {3175, 3679}, {3295, 3886}, {3555, 4968}, {3617, 3995}, {3678, 3967}, {3697, 3701}, {3698, 4714}, {3729, 3927}, {3876, 4671}, {3878, 4717}, {3952, 4533}, {3983, 3992}, {4015, 4125}, {4054, 4101}, {4658, 4670}, {4894, 4914}
X(5296) lies on these lines: {1, 391}, {2, 7}, {6, 3616}, {8, 37}, {10, 346}, {19, 4194}, {21, 198}, {45, 1213}, {69, 4687}, {141, 4748}, {145, 3247}, {200, 4343}, {344, 5224}, {573, 962}, {958, 1696}, {1125, 1743}, {1449, 3622}, {1621, 4254}, {2262, 3877}, {2297, 4334}, {2321, 3617}, {3621, 4034}, {3624, 3973}, {3625, 4898}, {3626, 4098}, {3672, 4384}, {3679, 3950}, {3739, 4419}, {3912, 5232}, {3945, 4416}, {4000, 4364}, {4007, 4029}, {4072, 4691}, {4363, 4488}, {4461, 4967}, {4643, 4648}, {4755, 4851}
X(5297) lies on these lines: {1, 2}, {12, 858}, {22, 5217}, {23, 35}, {33, 4232}, {37, 100}, {45, 2243}, {55, 1995}, {81, 210}, {86, 3699}, {88, 1390}, {110, 2330}, {171, 756}, {741, 4518}, {750, 984}, {894, 3952}, {940, 3681}, {1010, 3701}, {1370, 5229}, {1442, 4551}, {1500, 3291}, {1870, 5094}, {1909, 3266}, {2895, 4104}, {3100, 5218}, {3306, 4392}, {3579, 4220}, {3585, 5189}, {3614, 5133}, {3740, 3745}, {3842, 4434}, {3971, 4418}, {4096, 4697}, {4318, 5219}, {4413, 4850}, {4670, 4767}, {4995, 5160}
X(5297) = isogonal conjugate of X(34916)
X(5298) lies on these lines: {1, 549}, {2, 12}, {3, 3058}, {11, 30}, {46, 3656}, {55, 3524}, {65, 551}, {140, 3584}, {214, 519}, {376, 3086}, {381, 499}, {484, 1387}, {524, 1428}, {528, 5172}, {546, 4325}, {547, 3614}, {548, 4857}, {553, 1125}, {597, 1469}, {631, 3304}, {999, 5054}, {1358, 1447}, {1388, 1788}, {1420, 3679}, {1478, 5055}, {1479, 3534}, {1656, 4317}, {1737, 5126}, {2482, 3027}, {3303, 3523}, {3361, 4654}, {3530, 3746}, {3545, 4293}, {3585, 5066}, {3616, 5221}, {3813, 4188}, {3830, 4299}
X(5299) lies on these lines: {1, 6}, {31, 4253}, {32, 36}, {35, 39}, {41, 995}, {42, 3108}, {48, 5037}, {56, 609}, {58, 163}, {71, 4284}, {83, 1909}, {101, 1201}, {169, 614}, {172, 1015}, {239, 1930}, {304, 3759}, {386, 2280}, {572, 4300}, {595, 672}, {604, 2172}, {982, 1759}, {1193, 4251}, {1384, 5204}, {1429, 2003}, {1432, 2224}, {1469, 5039}, {1500, 5041}, {2241, 2276}, {2260, 4264}, {3216, 3684}, {3496, 3670}, {3509, 3953}, {3730, 3915}, {3934, 4400}, {5010, 5013}, {5024, 5217}
X(5300) lies on these lines: {1, 4202}, {2, 5015}, {3, 3006}, {4, 3701}, {7, 8}, {10, 31}, {41, 4071}, {145, 5100}, {306, 379}, {315, 3263}, {341, 5080}, {404, 3705}, {516, 3710}, {540, 1046}, {612, 5051}, {976, 2887}, {1125, 4894}, {1193, 4865}, {1330, 3681}, {1478, 4696}, {1479, 4358}, {1839, 3610}, {2177, 3178}, {2292, 4660}, {2475, 4385}, {3434, 3702}, {3436, 4723}, {3616, 4514}, {3757, 4197}, {3811, 3936}, {3876, 4388}, {3902, 5082}, {4193, 5205}, {4198, 5174}, {4200, 5081}, {4417, 4420}
X(5301) lies on these lines: {1, 1333}, {6, 31}, {9, 2220}, {19, 2204}, {32, 37}, {35, 4261}, {44, 3694}, {48, 3285}, {53, 1852}, {56, 1950}, {213, 584}, {284, 595}, {560, 3747}, {577, 1108}, {594, 4426}, {609, 3247}, {906, 1723}, {1030, 2277}, {1100, 2241}, {1172, 1612}, {1213, 4386}, {1449, 5035}, {1474, 2352}, {1621, 2303}, {1839, 3011}, {1841, 1968}, {2174, 2176}, {2178, 3053}, {2242, 3723}, {2251, 3204}, {2275, 5124}, {2278, 2300}, {3730, 5037}, {3749, 5227}, {4026, 4660}
X(5302) lies on these lines: {1, 6}, {3, 3740}, {8, 3683}, {10, 30}, {21, 210}, {35, 3697}, {55, 4662}, {56, 3305}, {58, 4682}, {63, 3812}, {65, 3219}, {78, 3715}, {100, 3983}, {191, 3753}, {333, 3714}, {354, 5047}, {375, 970}, {377, 1155}, {484, 4002}, {846, 4646}, {993, 5044}, {1329, 3634}, {1698, 3916}, {2646, 3876}, {3158, 4866}, {3214, 4689}, {3338, 3848}, {3452, 4999}, {3617, 5086}, {3694, 4877}, {3826, 4292}, {3913, 4512}, {4383, 4719}, {4390, 4520}, {4413, 4652}
X(5303) lies on these lines: {1, 4757}, {2, 3614}, {3, 8}, {21, 36}, {35, 3244}, {46, 3897}, {55, 3623}, {56, 1621}, {140, 5080}, {145, 5217}, {191, 214}, {320, 1444}, {404, 993}, {958, 4188}, {960, 4881}, {1030, 4969}, {1420, 3890}, {1476, 2078}, {2475, 4999}, {2476, 4299}, {2646, 3218}, {3434, 3522}, {3436, 3523}, {3576, 3869}, {3579, 4861}, {3601, 3873}, {3612, 3868}, {3621, 4421}, {3633, 3871}, {3681, 4855}, {3754, 5131}, {3822, 4325}, {3916, 4511}, {4297, 5086}
X(5304) lies on these lines: {2, 6}, {4, 3172}, {20, 32}, {25, 1249}, {30, 1285}, {39, 3523}, {98, 5039}, {111, 3163}, {115, 3839}, {172, 3600}, {216, 1180}, {232, 4232}, {251, 393}, {376, 1384}, {387, 4251}, {390, 1914}, {577, 1627}, {609, 4293}, {800, 1194}, {910, 3598}, {1202, 2257}, {1447, 5222}, {2243, 4346}, {2548, 5056}, {2996, 3407}, {3053, 3522}, {3091, 3767}, {3509, 4310}, {3524, 5024}, {3543, 5008}, {3553, 3920}, {4220, 4254}
X(5304) = midpoint of X(37640) and X(37641)
X(5304) = {X(2),X(6)}-harmonic conjugate of X(37665)
X(5304) = {X(3068),X(3069)}-harmonic conjugate of X(141)
X(5305) lies on these lines: {2, 3933}, {4, 3172}, {5, 6}, {20, 1384}, {30, 32}, {39, 140}, {112, 1885}, {115, 546}, {169, 3772}, {187, 548}, {218, 5230}, {251, 428}, {385, 2896}, {393, 1598}, {524, 626}, {547, 1506}, {549, 5013}, {550, 2549}, {574, 3530}, {631, 5024}, {732, 3589}, {1104, 5179}, {1184, 1368}, {1249, 3089}, {1285, 3146}, {1596, 2207}, {1759, 3782}, {1834, 4251}, {1901, 4264}, {1990, 3199}, {3628, 3815}, {3853, 5008}
X(5305) = midpoint of X(7583) and X(7584)
X(5305) = complement of X(3933)
X(5306) lies on these lines: {2, 6}, {5, 5007}, {25, 1990}, {30, 32}, {39, 549}, {50, 1627}, {51, 2871}, {53, 428}, {114, 1353}, {115, 3845}, {251, 1989}, {376, 3053}, {381, 3767}, {383, 398}, {397, 1080}, {519, 4136}, {566, 1180}, {1084, 1196}, {1194, 3003}, {1368, 3284}, {1384, 2549}, {1572, 3656}, {1914, 3058}, {2023, 5052}, {2031, 3849}, {2243, 3782}, {2276, 4995}, {2548, 5055}, {3017, 4251}, {3524, 5013}, {3705, 4969}
X(5306) = {X(395),X(396)}-harmonic conjugate of X(141)
X(5306) = complement of X(7788)
X(5307) lies on these lines: {1, 4}, {2, 1826}, {8, 1869}, {19, 27}, {28, 993}, {56, 1882}, {193, 1839}, {273, 1435}, {312, 1840}, {321, 5227}, {407, 1211}, {518, 1824}, {535, 5146}, {912, 1871}, {958, 1867}, {960, 1868}, {1465, 2050}, {1503, 1836}, {1708, 1746}, {1723, 1751}, {1842, 4198}, {1851, 1890}, {1861, 4196}, {1865, 3772}, {1880, 3666}, {1894, 5155}, {1957, 2299}, {2250, 2282}, {2333, 4384}, {2501, 4897}, {3822, 5142}
X(5308) lies on these lines: {1, 2}, {7, 37}, {9, 3945}, {45, 4644}, {57, 1334}, {69, 4687}, {81, 218}, {86, 344}, {142, 3247}, {144, 3664}, {220, 940}, {226, 279}, {277, 1255}, {354, 4517}, {379, 4313}, {391, 3879}, {599, 4748}, {857, 948}, {894, 3161}, {966, 4690}, {1001, 4344}, {2295, 5228}, {2345, 4472}, {3950, 4461}, {4021, 4859}, {4029, 4659}, {4357, 4869}, {4360, 4402}, {4361, 4460}, {4413, 4433}, {4643, 4755}
X(5308) = anticomplement of X(16832)
X(5308) = {X(1),X(2)}-harmonic conjugate of X(5222)
X(5309) lies on these lines: {2, 39}, {4, 5007}, {6, 13}, {30, 32}, {148, 3972}, {183, 4045}, {187, 376}, {230, 549}, {395, 3643}, {396, 3642}, {519, 4153}, {524, 5028}, {543, 1003}, {547, 3815}, {597, 5034}, {671, 3407}, {1506, 5055}, {1570, 1992}, {1596, 1990}, {2241, 3058}, {2275, 3582}, {2276, 3584}, {2452, 5099}, {2548, 3545}, {3053, 3534}, {3162, 5064}, {3543, 5008}, {5013, 5054}
X(5309) = complement of X(32833)
X(5309) = anticomplement of X(7880)
X(5309) = X(32)-of-4th-Brocard-triangle
X(5309 = X(32)-of orthocentroidal-triangle
X(5309 = inverse-in-Kiepert-hyperbola of X(3818)
X(5309) = centroid of reflection triangle of X(32)
X(5309 = {X(13),X(14)}-harmonic conjugate of X(3818)
X(5309) = {X(6),X(381)}-harmonic conjugate of X(7753)
X(5310) lies on these lines: {1, 22}, {2, 35}, {3, 614}, {12, 428}, {19, 25}, {23, 3743}, {31, 579}, {38, 3220}, {42, 251}, {51, 2330}, {56, 4348}, {184, 3056}, {199, 2223}, {350, 1799}, {354, 4265}, {613, 3796}, {674, 2194}, {858, 4330}, {1030, 3290}, {1194, 1914}, {1281, 1283}, {1370, 4302}, {1631, 2352}, {2920, 3057}, {2922, 3670}, {3011, 4220}, {3583, 5133}, {4228, 4276}
X(5311) lies on these lines: {1, 2}, {6, 756}, {9, 2308}, {31, 37}, {33, 1839}, {38, 940}, {55, 199}, {63, 3989}, {81, 984}, {171, 4414}, {192, 4418}, {197, 1953}, {210, 1100}, {748, 1386}, {750, 3666}, {902, 968}, {985, 1255}, {1460, 2171}, {2177, 3723}, {2206, 2303}, {3681, 4649}, {3791, 3842}, {3873, 4038}, {3923, 3995}, {4349, 4656}, {4722, 5220}
X(5312) lies on these lines: {1, 2}, {6, 35}, {9, 4272}, {36, 4255}, {55, 1203}, {57, 2594}, {58, 5010}, {73, 3339}, {165, 581}, {595, 2177}, {749, 3736}, {750, 4658}, {986, 3901}, {999, 2334}, {1126, 1468}, {1449, 5153}, {1743, 4270}, {1745, 4312}, {3555, 4719}, {3670, 3894}, {3743, 3876}, {3869, 4868}, {3874, 4850}, {3916, 4663}, {4023, 4205}
X(5313) lies on these lines: {1, 2}, {3, 1203}, {6, 36}, {9, 5153}, {31, 4256}, {35, 3052}, {57, 1464}, {72, 4719}, {73, 3361}, {165, 1064}, {748, 4653}, {751, 3736}, {758, 4850}, {982, 3894}, {1191, 3746}, {1420, 2594}, {1449, 4272}, {1453, 3612}, {1470, 2003}, {1743, 5105}, {2308, 4257}, {3670, 3901}, {3792, 4277}, {3877, 4868}, {3899, 4424}
X(5314) lies on these lines: {3, 63}, {9, 22}, {25, 3305}, {31, 35}, {36, 38}, {55, 1386}, {71, 1176}, {100, 3687}, {184, 3781}, {209, 5135}, {219, 3796}, {284, 672}, {378, 3587}, {908, 4220}, {1707, 5010}, {1790, 1818}, {2003, 2979}, {2172, 3730}, {2221, 4255}, {2323, 5012}, {2915, 5044}, {3219, 3220}, {3666, 5096}, {3917, 3955}, {4265, 4641}
X(5315) lies on these lines: {1, 6}, {31, 36}, {35, 595}, {40, 1480}, {56, 2163}, {58, 106}, {65, 1421}, {81, 551}, {109, 1450}, {221, 3361}, {386, 2177}, {484, 3752}, {651, 4315}, {982, 4880}, {1017, 5007}, {1046, 3953}, {1149, 2308}, {1319, 2003}, {1834, 4857}, {1999, 4975}, {2382, 2703}, {2999, 5119}, {3052, 5010}, {3679, 4383}, {3792, 4749}
X(5316) lies on these lines: {2, 7}, {10, 11}, {312, 4431}, {516, 4413}, {519, 3711}, {899, 3755}, {936, 950}, {946, 1698}, {956, 1125}, {960, 4848}, {984, 5121}, {1000, 3679}, {1150, 3707}, {1210, 5044}, {2321, 4358}, {3601, 5129}, {3698, 4301}, {3740, 3816}, {3752, 4656}, {3817, 3925}, {3826, 5087}, {3840, 4104}, {3883, 5205}, {3912, 5233}
X(5316) = complement of X(3306)
X(5317) lies on these lines: {4, 6}, {19, 31}, {27, 2221}, {28, 1104}, {29, 2303}, {34, 604}, {37, 4183}, {81, 286}, {107, 739}, {112, 915}, {158, 2214}, {162, 1778}, {232, 4220}, {240, 1761}, {608, 1118}, {648, 2991}, {1119, 1396}, {1430, 2260}, {1880, 2204}, {1896, 2298}, {2287, 5016}, {2322, 2345}, {2331, 2332}, {4219, 4261}
X(5317) = isogonal conjugate of X(3998)
X(5318) lies on these lines: {4, 6}, {5, 16}, {12, 1250}, {13, 15}, {14, 3845}, {17, 550}, {18, 3850}, {61, 3627}, {62, 546}, {141, 622}, {230, 1080}, {381, 395}, {383, 3815}, {463, 1495}, {524, 621}, {530, 623}, {590, 2043}, {615, 2044}, {633, 3630}, {634, 3631}, {1546, 3003}, {3411, 3856}, {3628, 5237}
X(5318) = crosssum of X(3) and X(15)
X(5318) = crosspoint of X(4) and X(13)
X(5319) lies on these lines: {2, 3108}, {4, 5007}, {5, 6}, {20, 32}, {39, 631}, {115, 3832}, {172, 4317}, {187, 3528}, {193, 626}, {230, 3526}, {548, 3053}, {609, 4325}, {1249, 3199}, {1572, 4301}, {1598, 1990}, {1906, 2207}, {1914, 4309}, {3530, 5013}, {3547, 5158}, {3618, 3934}, {3785, 4045}, {3815, 5070}, {5041, 5067}
X(5320) lies on these lines: {2, 5138}, {4, 1175}, {6, 25}, {22, 4260}, {31, 32}, {42, 2175}, {55, 584}, {65, 2355}, {81, 4223}, {182, 4220}, {198, 4275}, {199, 579}, {284, 1011}, {386, 3145}, {1200, 2357}, {1395, 1409}, {1397, 1400}, {1751, 3136}, {1824, 2264}, {2174, 2352}, {2206, 5019}, {2328, 4251}, {4383, 5135}
X(5320) = crosssum of X(2) and X(377)
X(5321) lies on these lines: {4, 6}, {5, 15}, {13, 3845}, {14, 16}, {17, 3850}, {18, 550}, {61, 546}, {62, 3627}, {141, 621}, {230, 383}, {381, 396}, {462, 1495}, {524, 622}, {531, 624}, {590, 2044}, {615, 2043}, {633, 3631}, {634, 3630}, {1080, 3815}, {1545, 3003}, {3412, 3856}, {3628, 5238}
X(5321) = crosssum of X(3) and X(16)
X(5321) = crosspoint of X(4) and X(14)
X(5322) lies on these lines: {1, 22}, {2, 36}, {3, 612}, {11, 428}, {25, 34}, {31, 3220}, {35, 3920}, {51, 1428}, {104, 4231}, {172, 1194}, {184, 1469}, {210, 5096}, {611, 3796}, {858, 4325}, {1370, 4299}, {1460, 1473}, {1626, 2352}, {1799, 1909}, {3011, 4224}, {3585, 5133}, {3745, 4265}, {4640, 5078}
X(5323) lies on these lines: {1, 1412}, {3, 4340}, {7, 21}, {28, 34}, {65, 81}, {73, 3736}, {229, 4228}, {333, 1788}, {388, 1010}, {404, 4417}, {894, 1791}, {1038, 2285}, {1043, 3476}, {1325, 5221}, {1350, 2213}, {1400, 1778}, {1420, 4653}, {1466, 1817}, {1470, 4225}, {1848, 4292}, {3340, 4658}, {4224, 4252}
X(5324) lies on these lines: {6, 4224}, {8, 21}, {27, 3423}, {28, 34}, {81, 105}, {165, 4221}, {479, 1014}, {672, 1778}, {759, 3256}, {859, 1617}, {910, 1333}, {940, 4223}, {1040, 2082}, {1183, 2646}, {1350, 4383}, {1437, 3660}, {1473, 1851}, {1633, 3914}, {1812, 3794}, {1817, 3286}, {2287, 2348}, {3060, 4259}
X(5325) lies on these lines: {2, 7}, {10, 30}, {210, 4995}, {306, 3578}, {333, 2321}, {345, 3686}, {519, 958}, {549, 5044}, {551, 960}, {846, 3755}, {936, 3524}, {971, 3740}, {1125, 3927}, {1999, 4029}, {2551, 3585}, {3058, 3683}, {3679, 5234}, {3687, 3707}, {3712, 4061}, {4035, 4416}, {4042, 4923}, {4641, 4667}
X(5326) lies on these lines: {1, 632}, {2, 11}, {3, 3614}, {5, 5010}, {12, 36}, {35, 3628}, {56, 3525}, {498, 999}, {547, 3583}, {1125, 5048}, {1478, 5054}, {1479, 5070}, {1914, 3055}, {2276, 3054}, {2646, 3634}, {3057, 3918}, {3085, 3533}, {3090, 5217}, {3530, 3585}, {3850, 4324}, {4302, 5055}
X(5327) lies on these lines: {4, 6}, {7, 21}, {27, 1836}, {28, 3556}, {29, 65}, {58, 946}, {81, 497}, {226, 2328}, {238, 1780}, {284, 516}, {333, 2651}, {411, 5132}, {960, 1010}, {990, 3736}, {1430, 1848}, {1817, 3474}, {1858, 2905}, {2287, 2550}, {2303, 4307}, {2360, 4292}, {5057, 5137}
X(5328) lies on these lines: {2, 7}, {8, 11}, {10, 5056}, {153, 214}, {497, 3689}, {936, 3091}, {938, 4187}, {960, 3698}, {1997, 4417}, {2478, 4313}, {2550, 5087}, {2551, 3616}, {3061, 3119}, {3090, 5044}, {3436, 4308}, {4310, 5121}, {4671, 4858}, {4679, 5218}, {5175, 5187}
X(5328) = anticomplement of X(31190)
X(5329) lies on these lines: {1, 3}, {22, 31}, {24, 602}, {25, 238}, {43, 197}, {159, 1740}, {181, 182}, {199, 985}, {394, 3792}, {511, 1397}, {748, 1995}, {1376, 5096}, {1469, 3955}, {1473, 4650}, {1626, 3286}, {1707, 3220}, {1790, 3736}, {2076, 2162}, {2178, 3509}
X(5330) lies on these lines: {1, 21}, {2, 1482}, {8, 11}, {78, 2136}, {145, 1058}, {392, 5047}, {404, 517}, {452, 3623}, {644, 3061}, {952, 5046}, {960, 4861}, {1788, 2099}, {3057, 3871}, {3244, 4867}, {3579, 4881}, {3621, 3940}, {3872, 3876}, {3880, 4420}, {4673, 4720}
X(5331) lies on these lines: {1, 333}, {6, 21}, {27, 34}, {29, 3192}, {42, 1043}, {56, 81}, {58, 2185}, {86, 1193}, {87, 3736}, {106, 931}, {269, 1434}, {270, 1474}, {284, 2363}, {386, 1010}, {958, 2334}, {1126, 4653}, {2215, 4269}, {2279, 3601}
X(5332) lies on these lines: {1, 5007}, {6, 31}, {32, 36}, {39, 5010}, {44, 3681}, {172, 999}, {238, 3789}, {239, 4376}, {609, 1015}, {893, 2364}, {982, 2243}, {995, 2251}, {1040, 3284}, {1100, 3873}, {1403, 1404}, {2220, 2277}, {2300, 5037}, {3703, 4969}
X(5333) lies on these lines: {1, 4720}, {2, 6}, {21, 36}, {58, 748}, {142, 1817}, {226, 1014}, {274, 321}, {314, 4359}, {1001, 4184}, {1010, 3616}, {1043, 3622}, {1412, 5219}, {1698, 4658}, {3219, 4670}, {3286, 4423}, {3720, 3736}, {3786, 3873}, {4654, 4877}
X(5333) = trilinear pole of line X(4716)X(4802) (the perspectrix of ABC and Gemini triangle 24)
X(5333) = perspector of Gemini triangle 23 and cross-triangle of ABC and Gemini triangle 23
X(5334) lies on these lines: {2, 14}, {4, 6}, {13, 3839}, {16, 20}, {17, 5068}, {18, 3523}, {61, 3091}, {62, 3146}, {193, 622}, {376, 395}, {396, 3545}, {633, 3620}, {1131, 3367}, {1132, 3366}, {1250, 4294}, {2043, 3069}, {2044, 3068}
X(5334) = {X(4),X(6)}-harmonic conjugate of X(5335)
X(5335) lies on these lines: {2, 13}, {4, 6}, {14, 3839}, {15, 20}, {17, 3523}, {18, 5068}, {61, 3146}, {62, 3091}, {193, 621}, {376, 396}, {395, 3545}, {634, 3620}, {1131, 3392}, {1132, 3391}, {1250, 3085}, {2043, 3068}, {2044, 3069}
X(5335) = {X(4),X(6)}-harmonic conjugate of X(5334)
X(5336) lies on these lines: {1, 6}, {19, 32}, {25, 1096}, {31, 2171}, {46, 2305}, {609, 1781}, {800, 2331}, {992, 997}, {1184, 5089}, {1400, 3924}, {1572, 1953}, {1731, 5037}, {1826, 3767}, {2285, 5019}, {2321, 4362}, {3290, 5020}, {3612, 5110}
X(5337) lies on these lines: {1, 3}, {2, 32}, {6, 3882}, {39, 81}, {58, 1009}, {63, 3954}, {69, 5019}, {141, 1333}, {172, 3912}, {193, 5042}, {524, 5035}, {1150, 3661}, {2220, 3589}, {3793, 5241}, {4044, 4396}, {4220, 5188}, {4384, 4386}
X(5338) lies on these lines: {2, 1890}, {4, 165}, {19, 25}, {28, 34}, {51, 2261}, {154, 2262}, {204, 1841}, {212, 2270}, {354, 1829}, {461, 4512}, {607, 1190}, {1155, 1878}, {1474, 2280}, {1598, 1753}, {1839, 4207}, {1871, 3517}
X(5339) lies on these lines: {3, 14}, {4, 6}, {13, 3843}, {15, 1656}, {16, 1657}, {17, 3851}, {20, 395}, {61, 381}, {62, 382}, {154, 462}, {396, 3091}, {599, 633}, {621, 3763}, {3526, 5238}, {3534, 5237}
X(5339) = {X(4),X(6)}-harmonic conjugate of X(5340)
X(5340) lies on these lines: {3, 13}, {4, 6}, {14, 3843}, {15, 1657}, {16, 1656}, {18, 3851}, {20, 396}, {61, 382}, {62, 381}, {154, 463}, {395, 3091}, {599, 634}, {622, 3763}, {3526, 5237}, {3534, 5238}
X(5340) = {X(4),X(6)}-harmonic conjugate of X(5339)
X(5341) lies on these lines: {6, 19}, {9, 484}, {35, 37}, {45, 1766}, {50, 1950}, {583, 1731}, {759, 1333}, {910, 3256}, {1400, 1989}, {1719, 4640}, {1723, 5043}, {1760, 4363}, {2171, 2173}, {2178, 5172}, {4271, 5011}
X(5342) lies on these lines: {4, 8}, {27, 4384}, {29, 34}, {75, 1890}, {242, 4185}, {278, 4194}, {281, 4200}, {391, 4047}, {452, 1441}, {461, 3616}, {469, 3912}, {1039, 2481}, {1904, 2969}, {4101, 4673}
X(5343) lies on these lines: {2, 5238}, {4, 6}, {14, 20}, {15, 5056}, {16, 5059}, {17, 3091}, {18, 3522}, {61, 3832}, {62, 3543}, {395, 3529}, {396, 3855}, {1131, 3364}, {1132, 3365}
X(5343) = {X(4),X(6)}-harmonic conjugate of X(5344)
X(5344) lies on these lines: {2, 5237}, {4, 6}, {13, 20}, {15, 5059}, {16, 5056}, {17, 3522}, {18, 3091}, {61, 3543}, {62, 3832}, {395, 3855}, {396, 3529}, {1131, 3389}, {1132, 3390}
X(5344) = {X(4),X(6)}-harmonic conjugate of X(5343)
X(5345) lies on these lines: {1, 22}, {2, 3585}, {23, 614}, {25, 36}, {609, 1194}, {612, 5010}, {846, 3415}, {988, 2915}, {990, 1719}, {1370, 4316}, {1707, 3220}, {1799, 3761}, {5020, 5204}
X(5346) lies on these lines: {2, 5041}, {4, 5008}, {6, 17}, {30, 32}, {39, 631}, {115, 3843}, {187, 3522}, {230, 632}, {385, 3096}, {1186, 2086}, {2548, 5071}, {3091, 3767}
X(5347) lies on these lines: {1, 3}, {2, 5096}, {6, 22}, {25, 4383}, {81, 4265}, {184, 4259}, {199, 5132}, {386, 2915}, {1626, 4497}, {2194, 4260}, {3220, 4641}, {4184, 5124}
X(5348) lies on these lines: {1, 3}, {2, 2361}, {4, 1399}, {5, 47}, {11, 31}, {12, 255}, {58, 1837}, {109, 1836}, {181, 1364}, {212, 750}, {394, 1376}, {1253, 4995}
X(5349) lies on these lines: {4, 6}, {5, 5238}, {13, 3861}, {14, 3627}, {15, 3850}, {17, 3858}, {18, 30}, {61, 3845}, {62, 3853}, {382, 395}, {396, 546}
X(5349) = {X(4),X(6)}-harmonic conjugate of X(5350)
X(5350) lies on these lines: {4, 6}, {5, 5237}, {13, 3627}, {14, 3861}, {16, 3850}, {17, 30}, {18, 3858}, {61, 3853}, {62, 3845}, {382, 396}, {395, 546}
X(5350) = {X(4),X(6)}-harmonic conjugate of X(5349)
X(5351) lies on these lines: {3, 6}, {13, 631}, {14, 550}, {17, 549}, {18, 20}, {202, 5217}, {395, 548}, {397, 3530}, {622, 630}, {1092, 3206}, {3411, 3528}
X(5352) lies on these lines: {3, 6}, {13, 550}, {14, 631}, {17, 20}, {18, 549}, {203, 5217}, {396, 548}, {398, 3530}, {621, 629}, {1092, 3205}, {3412, 3528}
X(5353) lies on these lines: {1, 6}, {15, 35}, {16, 36}, {42, 2981}, {61, 3746}, {395, 3582}, {396, 3584}, {398, 4857}, {651, 3639}, {1082, 2003}, {1094, 3170}
X(5353) = {X(1),X(6)}-harmonic conjugate of X(5357)
X(5354) lies on these lines: {2, 6}, {22, 1384}, {23, 32}, {25, 1383}, {111, 251}, {187, 1194}, {574, 1180}, {1915, 2502}, {2030, 5012}, {3291, 5007}, {3767, 5169}
X(5355) lies on these lines: {5, 5041}, {6, 13}, {20, 32}, {30, 5008}, {39, 140}, {385, 4045}, {543, 3972}, {574, 3524}, {1506, 3090}, {2548, 5068}, {3627, 5007}
X(5356) lies on these lines: {1, 4287}, {6, 19}, {9, 3336}, {36, 37}, {44, 1781}, {46, 5036}, {1385, 1766}, {1400, 2963}, {1950, 2965}, {2160, 2183}, {2161, 2260}
X(5357) lies on these lines: {1, 6}, {15, 36}, {16, 35}, {62, 1250}, {395, 3584}, {396, 3582}, {397, 4857}, {559, 2003}, {651, 3638}, {1095, 3171}
X(5357) = {X(1),X(6)}-harmonic conjugate of X(5353)
X(5358) lies on these lines: {1, 4228}, {10, 21}, {22, 1714}, {28, 34}, {169, 284}, {386, 4224}, {1210, 4233}, {1817, 3008}, {1842, 3220}, {4269, 4456}
X(5359) lies on these lines: {2, 6}, {3, 1180}, {4, 3162}, {22, 32}, {25, 251}, {51, 5039}, {169, 614}, {1196, 1995}, {3767, 5133}
X(5360) lies on these lines: {4, 8}, {31, 32}, {37, 263}, {42, 4531}, {100, 2698}, {237, 1755}, {511, 1959}, {512, 661}, {674, 4053}
X(5361) lies on these lines: {2, 6}, {8, 35}, {63, 4659}, {100, 4042}, {956, 4216}, {3219, 4671}, {3679, 4257}, {3757, 4430}, {3769, 4981}
X(5362) lies on these lines: {2, 6}, {15, 21}, {16, 404}, {37, 2981}, {61, 5047}, {100, 1250}, {470, 1172}, {2323, 5243}
X(5362) = {X(2),X(6)}-harmonic conjugate of X(5367)
X(5363) lies on these lines: {1, 3}, {23, 31}, {181, 575}, {238, 1995}, {576, 1397}, {1283, 3941}, {1395, 3518}, {1740, 2930}
X(5364) lies on these lines: {2, 7}, {31, 32}, {198, 1755}, {292, 1613}, {846, 3730}, {968, 1334}, {1707, 2664}, {4020, 5021}
X(5365) lies on these lines: {4, 6}, {14, 3146}, {15, 5068}, {17, 3854}, {18, 20}, {61, 3839}, {3412, 3832}
X(5365) = {X(4),X(6)}-harmonic conjugate of X(5366)
X(5366) lies on these lines: {4, 6}, {13, 3146}, {16, 5068}, {17, 20}, {18, 3854}, {62, 3839}, {3411, 3832}
X(5366) = {X(4),X(6)}-harmonic conjugate of X(5365)
X(5367) lies on these lines: {2, 6}, {15, 404}, {16, 21}, {62, 5047}, {471, 1172}, {1250, 1621}, {2323, 5242}
X(5367) = {X(2),X(6)}-harmonic conjugate of X(5362)
X(5368) lies on these lines: {6, 17}, {20, 32}, {39, 549}, {115, 546}, {230, 5041}, {3545, 3767}
X(5369) lies on these lines: {7, 8}, {31, 32}, {674, 3721}, {1046, 1282}, {1193, 4531}, {2292, 3688}
X(5370) lies on these lines: {1, 22}, {23, 36}, {25, 5204}, {612, 5217}, {858, 4316}, {896, 3220}
X(5371) lies on these lines: {6, 22}, {31, 32}, {81, 1915}, {584, 2276}, {2194, 3051}, {2277, 4275}
X(5372) lies on these lines: {2, 6}, {3, 4720}, {8, 36}, {63, 4671}, {1330, 5141}, {4362, 4392}
For any point X inside an acute triangle ABC, let A′ B′ C′ denote the pedal triangle of X. Then X(5373) is the point X for which the quadrilaterals AC′XB′, BA′XC′, CB′XA′ all have the same area.
X(5373) is discussed in the following articles:
Apoloniusz Tyszka, "Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 181-185: Tyszka article.
Jean Pierre Ehrmann, "Constructive solution of a generalization of Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 187-190: Ehrmann article.
Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.
X(5373) is the incenter of the Thomson triangle, as proved in Thomson Triangle.
For any point P on segment BC of an acute triangle ABC, let Q be the point on AB nearest to P and let R be the point on AC nearest to P. Let A′ be the choice of P for which area(A′QB) = area(A′RC). Define B′ and C′ cyclically. Then the lines AA′, BB′, CC′ concur in X(5374).
X(5374) is introduced in Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.
X(5375) is the center of the circumconic with perspector X(100). This conic passes through the bicentric pairs P(26), U(26), and P(33), U(33) (Randy Hutson, 9/10/2012) and is a hyperbola (Peter Moses, 10/10/2012). It is introduced here as the Hutson-Moses hyperbola, discussed in the preamble to X(5376).
X(5375) lies on these lines: {44, 3290}, {100, 650}, {101, 661}, {190, 4467}, {644, 3239}, {651, 3676}, {666, 693}, {901, 4394}, {908, 3008}, {2323, 4700}
X(5375) = complement of X(8047)
X(5375) = X(2)-Ceva conjugate of X(100)
X(5375) = complementary conjugate of complement of X(16686)
X(5375) = crosssum of circumcircle intercepts of line PU(27) (line X(244)X(665))
Points on the Hutson-Moses hyperbola: X(5376)-X(5389)
The Hutson-Moses hyperbola, introduced at X(5375), is given by the following barycentric equation:
a(a - b)(a - c)yz + b(b - c)(b - a)zx + c(c - a)(c - b)xy = 0.
The hyperbola has perspector X(100), center X(5375), meets the circumcircle in X(898) and the Steiner circumellipse in X(666), and is the isogonal conjugate of the line X(244)X(665). If X = x : y : z (barycentrics) is a point on the circumcircle, then the point
H(X) = x/(a(b - c)) : y/(b(c - a)) : z/(c(a - b))
is on the Hutson-Moses hyperbola. Examples are shown in the following table:
| X | H(X) |
| X(99) | X(4601) |
| X(100) | X(1016) |
| X(101) | X(765) |
| X(105) | X(666) |
| X(106) | X(3257) |
| X(109) | X(4564) |
| X(110) | X(4567) |
| X(739) | X(898) |
| X(741) | X(4584) |
| X(934) | X(1275) |
| X(901) | X(5376) |
| X(919) | X(5377) |
| X(813) | X(5378) |
| X(112) | X(5379) |
| X(111) | X(5380) |
| X(898) | X(5381) |
| X(1293) | X(5382) |
| X(932) | X(5383) |
| X(825) | X(5384) |
| X(4588) | X(5385) |
| X(753) | X(5386) |
| X(2748) | X(5387) |
| X(789) | X(5388) |
| X(755) | X(5389) |
The acute angle Ψ between the asymptotes of the circumhyperbola with perspector X = x : y : z is given by
tan(Ψ) = S*T/(x*SA + y*SB + z*SC),
where T = (x2 + y2 + z2 - 2(yz + zx +
xy))1/2, and the eccentricity e is then given by e =
sec(Ψ/2). (Peter Moses, 10/11/12). For the Hutson-Moses hyperbola,
(x, y, z) = a2/(b2 - c2),
b2/(c2 - a2),
c2/(a2 - b2).
X(5376) lies on the Hutson-Moses hyperbola and these lines: {1, 765}, {2, 1016}, {57, 4564}, {81, 4567}, {89, 1252}, {100, 3251}, {105, 1320}, {106, 291}, {274, 4601}, {279, 1275}, {666, 4555}, {898, 901}, {1022, 1023}, {1929, 4674}, {2006, 4997}, {2397, 2401}, {4584, 4622}
X(5376) = isogonal conjugate of X(2087)
X(5376) = trilinear product of PU(28)
X(5377) is the trilinear pole of the line X(100)X(650), this line being tangent at X(100) to the conic {A, B, C, X(2), X(100), PU(112)}, where PU(112) are the isogonal conjugates of the bicentric pair PU(46). (Randy Hutson, September 29, 2014)
X(5377) lies on the Hutson-Moses hyperbola, the Feuerbach circumhyperbola, and these lines: {1, 1053}, {2, 38310}, {4, 6074}, {7, 59}, {8, 1016}, {9, 765}, {21, 4567}, {55, 14947}, {79, 40724}, {80, 14942}, {100, 3126}, {104, 59101}, {105, 1320}, {294, 1642}, {314, 4601}, {497, 31633}, {666, 885}, {673, 3254}, {692, 57018}, {898, 919}, {927, 2742}, {1023, 23893}, {1025, 35355}, {1027, 3257}, {1041, 7012}, {1156, 28071}, {1172, 5379}, {1252, 40779}, {1438, 4876}, {2195, 9365}, {2320, 5385}, {2344, 5384}, {2737, 59133}, {3680, 5382}, {4584, 54353}, {4998, 5218}, {5381, 36798}, {5383, 7155}, {6601, 56850}, {11124, 31628}, {11604, 13576}, {11609, 56853}, {13136, 36802}, {23836, 32735}, {43740, 52456}
X(5377) = reflection of X(100) in X(52884)
X(5377) = isogonal conjugate of X(3675)
X(5377) = symgonal image of X(52884)
X(5377) = isogonal conjugate of the complement of X(53358)
X(57536)-Ceva conjugate of X(1252)
X(5377) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3675}, {11, 1458}, {57, 17435}, {109, 52305}, {241, 2170}, {244, 518}, {291, 38989}, {512, 23829}, {513, 2254}, {514, 665}, {522, 53539}, {649, 918}, {650, 53544}, {663, 43042}, {672, 1086}, {673, 35505}, {764, 1026}, {909, 42770}, {926, 3676}, {1015, 3912}, {1019, 24290}, {1027, 3126}, {1111, 2223}, {1357, 3717}, {1358, 2340}, {1438, 35094}, {1565, 2356}, {1643, 52228}, {1647, 34230}, {1769, 57468}, {1818, 2969}, {1861, 3937}, {1876, 7004}, {2162, 23773}, {2195, 3323}, {2283, 21132}, {2284, 6545}, {2310, 34855}, {2428, 23760}, {3120, 3286}, {3121, 18157}, {3122, 30941}, {3125, 18206}, {3248, 3263}, {3252, 27918}, {3271, 9436}, {3693, 53538}, {3733, 4088}, {3737, 53551}, {3930, 16726}, {3942, 5089}, {4712, 43921}, {4858, 52635}, {5236, 7117}, {7649, 53550}, {8638, 52621}, {9454, 23989}, {14942, 61056}, {16727, 39258}, {17205, 20683}, {18210, 54407}, {21143, 42720}, {22116, 27846}, {23225, 46107}, {24002, 46388}, {36819, 42753}, {43035, 56787}, {43924, 50333}, {43929, 53583}, {52614, 58817}
X(5377) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3675}, {11, 52305}, {3126, 52304}, {5375, 918}, {5452, 17435}, {6184, 35094}, {23980, 42770}, {33675, 23989}, {39026, 2254}, {39029, 38989}, {39054, 23829}, {39063, 3323}
X(5377) = cevapoint of X(i) and X(j) for these (i,j): {55, 2284}, {100, 518}, {105, 36086}, {294, 52927}, {692, 1914}, {11124, 17435}
X(5377) = trilinear pole of line {100, 650}
X(5377) = barycentric product X(i)*X(j) for these {i,j}: {9, 39293}, {59, 36796}, {100, 666}, {101, 51560}, {105, 1016}, {190, 36086}, {294, 4998}, {518, 57536}, {644, 927}, {646, 32735}, {651, 36802}, {668, 919}, {673, 765}, {692, 36803}, {885, 31615}, {1027, 6632}, {1110, 18031}, {1252, 2481}, {1275, 28071}, {1438, 7035}, {1462, 4076}, {1814, 15742}, {1978, 32666}, {3699, 36146}, {3939, 34085}, {4391, 59101}, {4554, 52927}, {4564, 14942}, {4567, 13576}, {4600, 18785}, {4601, 56853}, {5378, 6654}, {5381, 52902}, {6065, 34018}, {6559, 7045}, {31628, 35313}, {39272, 53337}, {42722, 59021}, {43929, 57950}
X(5377) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3675}, {43, 23773}, {55, 17435}, {59, 241}, {100, 918}, {101, 2254}, {105, 1086}, {109, 53544}, {241, 3323}, {294, 11}, {517, 42770}, {518, 35094}, {644, 50333}, {650, 52305}, {651, 43042}, {662, 23829}, {666, 693}, {673, 1111}, {692, 665}, {765, 3912}, {885, 40166}, {906, 53550}, {919, 513}, {927, 24002}, {1016, 3263}, {1018, 4088}, {1024, 21132}, {1026, 53583}, {1027, 6545}, {1110, 672}, {1252, 518}, {1262, 34855}, {1415, 53539}, {1416, 53538}, {1438, 244}, {1462, 1358}, {1814, 1565}, {1914, 38989}, {1983, 53555}, {2149, 1458}, {2195, 2170}, {2223, 35505}, {2284, 3126}, {2427, 42758}, {2481, 23989}, {4557, 24290}, {4559, 53551}, {4564, 9436}, {4567, 30941}, {4570, 18206}, {4600, 18157}, {4619, 41353}, {4998, 40704}, {5378, 40217}, {5379, 15149}, {6065, 3693}, {6559, 24026}, {7012, 5236}, {7115, 1876}, {8751, 2969}, {9503, 15634}, {13576, 16732}, {14942, 4858}, {15742, 46108}, {17435, 52304}, {18785, 3120}, {23990, 2223}, {28071, 1146}, {28132, 42455}, {31615, 883}, {32641, 57468}, {32658, 3937}, {32666, 649}, {32735, 3669}, {34085, 52621}, {35333, 16892}, {36057, 3942}, {36086, 514}, {36146, 3676}, {36796, 34387}, {36802, 4391}, {36803, 40495}, {39293, 85}, {41339, 1566}, {41934, 43921}, {43929, 764}, {46163, 2530}, {51560, 3261}, {51987, 42753}, {52635, 61056}, {52902, 52626}, {52927, 650}, {54235, 2973}, {54364, 42754}, {56853, 3125}, {57192, 4925}, {57536, 2481}, {57731, 42720}, {59101, 651}, {59149, 1026}
X(5377) = {X(885),X(35313)}-harmonic conjugate of X(666)
X(5378) is the trilinear pole of the line X(100)X(649), this line being tangent at X(100) to the hyperbola {A, B, C, X(81), X(100), PU(8)}. (Randy Hutson, September 29, 2014)
X(5378) lies on the Hutson-Moses hyperbola and these lines: {1, 1016}, {6, 765}, {56, 4564}, {58, 4567}, {86, 4601}, {87, 4076}, {106, 291}, {269, 1275}, {660, 876}, {666, 1026}, {813, 898}, {1411, 4518}, {1438, 4876}
X(5378) = isogonal conjugate of X(27846)
X(5379) lies on the Hutson-Moses hyperbola and these lines: {59, 5080}, {100, 1304}, {110, 1309}, {112, 898}, {162, 3257}, {250, 2074}, {422, 4601}, {648, 666}, {685, 692}, {2397, 2409}, {4564, 4570}
X(5379) = isogonal conjugate of X(18210)
X(5379) = polar conjugate of X(16732)
X(5379) = X(63)-isoconjugate of X(3125)
X(5380) lies on the Hutson-Moses hyperbola and these lines: {100, 691}, {111, 898}, {291, 4584}, {666, 671}, {668, 892}, {765, 1018}, {897, 1757}, {1016, 3952}, {1275, 4566}, {2397, 2408}, {4551, 4564}
X(5380) = isogonal conjugate of X(14419)X(5381) lies on the Hutson-Moses hyperbola and these lines: {6, 1016}, {31, 765}, {81, 4601}, {604, 4564}, {666, 889}, {1275, 1407}, {1333, 4567}, {3257, 3570}
X(5381) = isogonal conjugate of X(1646)
X(5382) lies on the Hutson-Moses hyperbola and these lines: {644, 3669}, {765, 1279}, {898, 1293}, {1016, 3008}, {1332, 3257}, {2397, 2415}
X(5382) = isotomic conjugate of complement of X(25268)
X(5383) lies on the Hutson-Moses hyperbola and these lines: {87, 4076}, {190, 1919}, {645, 4584}, {898, 932}, {3257, 4598}, {3287, 4583}
X(5383) = isogonal conjugate of X(6377)
X(5383) = isotomic conjugate of X(21138)
X(5384) lies on the Hutson-Moses hyperbola and these lines: {110, 4584}, {666, 4586}, {825, 898}, {1016, 1110}, {1492, 3257}, {4570, 4601}
X(5385) lies on the Hutson-Moses hyperbola and these lines: {89, 1252}, {100, 4825}, {101, 3257}, {666, 4597}, {898, 4588}
X(5385) = isotomic conjugate of X(4957)
X(5386) lies on the Hutson-Moses hyperbola and these lines: {753, 898}, {765, 3799}, {1016, 3807}
X(5387) lies on the Hutson-Moses hyperbola and these lines: {898, 2748}, {1016, 3759}
X(5388) lies on the Hutson-Moses hyperbola and these lines: {789, 898}, {799, 4584}
X(5389) lies on the Hutson-Moses hyperbola and these lines: {755, 898}, {4553, 4567}
Barycentrics a*f(A/3, B/3, C/3) : b*f(B/3, C/3, A/3) : c*f(C/3, A/3, B/3), where f(x,y,z) is defined using the abbreviations [m,n] for sin(x + 2my + 2nz) + sin(x + 2ny + 2mz) and [m] for [m,m]/2, as follows:
f(x,y,z) = [-3,-1] + 5[-2,-1] + [-2,0] - 5[-1,1] - 3[-1,2] - [-1,3] + [0,1] + 2[0,2] - [0,3] - 2[1,3] - 2[2,3] + [-3] - 2[-2] + 3[-1] + 3[0] + 3[1] + 3[2] - 2[3] - [4] (Barry Wolk)
Let DEF be the classical Morley triangle. The Euler lines of the three triangles AEF, BFD, CDE to concur in X(5390), as discovered by Zhao Yong of Anhui, China, October 2, 2012. For a construction and derivation of barycentric coordinates by Shi Yong, see Problem 20 at Unsolved Problems and Rewards. For further developments, including the development of trilinear and barycentric coordinates as shown above, type X(5390) into Search at Hyacinthos.
If you have The Geometer's Sketchpad, you can view X(5390).
X(5390) lies on this line: {357, 1136}
X(5391) lies on these lines: {2,37}, {7,491}, {8,492}, {10,5490}, {69,13388}, {100,9099}, {239,3069}, {274,1123}, {319,1270}
X(5391) = isotomic conjugate of X(1336)
X(5391) = {X(2),X(75)}-harmonic conjugate of X(1267)
X(5392) lies on the Euler perspective cubic (K045), corresponding to the point X(68) on the Euler central cubic (K044). (Randy Hutson, November 22, 2014)
X(5392) lies on the Kiepert hyperbola and these lines: {2, 311}, {3, 96}, {4, 52}, {10, 91}, {22, 98}, {226, 914}, {262, 5133}, {264, 275}, {338, 394}, {467, 2052}
X(5392) = isogonal conjugate of X(571)
X(5392) = isotomic conjugate of X(1993)
X(5392) = pole wrt polar circle of trilinear polar of X(24)
X(5392) = X(48)-isoconjugate (polar conjugate) of X(24)
X(5392) = Cundy-Parry Phi transform of X(96)
X(5392) = Cundy-Parry Psi transform of X(52)
X(5392) = trilinear pole of line X(523)X(2072) (the line of the degenerate cross-triangle of medial and 2nd Euler triangles)
Let W(BA) and W(CA) be the two congruent circles, within triangle ABC, each tangent to the other and to sideline BC of triangle ABC, with W(BA) also tangent to sideline AB and W(CA) also tangent to sideline AC; cf. the Paache configuration at X(1123). Let BA and CA be the touchpoints of these circles with sideline BC. Define the points CB, AC cyclically and define the points AB, BC cyclically. The six points lie on an ellipse having center X(5393) and equation
d(2 + d)x2 + e(2 + e)y2 + f(2 + f)z2 - 2(2 + e + f + ef)yz - 2(2 + f + d + fd)zx - 2(2 + d + e + de)xy = 0,
where d = cot(A/2), e = cot(B/2), f = cot(C/2). Let X = X(5393). Then |GX|/|IX| = s/(3r), where G = centroid, I = incenter, r = inradius, and s = semiperimeter. (Alexei Myakishev, December 25, 2012).
An associated conic, the Paache-Myakishev-Moses conic, is introduced at X(5405). This conic results from the two congruent circles that do not lie within triangle ABC.
If you have The Geometer's Sketchpad, you can view
X(5393), including the ellipse. You can also view the configuration for pairs of circles used in the constructions of X(5393) and X(5405): Pairs of Circles.X(5393) lies on these lines: {1, 2 }, {9, 3068}, {37, 590}, {57, 482}, {81, 3300}, {175, 5226}, {226, 481}, {491, 4357}, {492, 3879}, {515, 2048}, {615, 1100}, {642, 3666}, {940, 1335}, {1124, 4383}, {1255, 3302}, {1267, 3875}, {1449, 3069}, {1585, 1785}, {1991, 4643}
X(5393) = {X(1),X(2)}-harmonic conjugate of X(5405)
X(5394) is the point X for which the three triangles AXB, BXC, CXA have congruent incircles. The existence of this point is proved by Noam Elkies in Mathematics Magazine 60 (1987) 117. His proof applies to a much wider range of functions (with the inradius replaced by the area, semiperimeter, etc., or any positive combination thereof).
Following is a copy-and-run Mathematica program that computes actual trilinear distances (1.7916..., 1.7057..., 1.6328...) of X(5394) for the triangle given by (a,b,c) = (6,9,13).
(1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)] {x/a, y/b, z/c} /. #1 /.
NSolve[{x + y + z == 1, (a + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2])/
x == (b + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
y == (c + Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
z /. #1}, {x, y, z}, WorkingPrecision → 40][[1]] &)[Thread[{a, b, c} → {6, 9,13}]]
(* Code by Peter Moses, October 23, 2012. *)
Let S = 2 area ABC, s1 = a + b + c, s2 = b c + c a + a c, s3 = a b c. Then the radius of the congruent incircles is the least positive solution of the following 8th-degree polynomial equation:
1 -
8 s1 x / S +
(23 s1^2 + 8 s2) x^2 / S^2 -
2 s1 (13 s1^2 + 24 s2) x^3 / S^3 +
2 (3 s1^4 + 42 s1^2 s2 + 8 s2^2 + 12 s1 s3) x^4 / S^4 -
16 s1 (s1^2 s2 + 4 s2^2 + 6 s1 s3) x^5 / S^5 +
4 s1 (2 s1^5 - 9 s1^3 s2 + 4 s1 s2^2 + 24 s1^2 s3 + 24 s2 s3) x^6 / S^6 +
96 s1 s2 x^7 / S^5 -
288 s1 s3 x^8 / S^6 = 0.
For example, if {a,b,c} = {6,9,13} the equation is as follows:
1-4 Sqrt[7/5] x+(2503 x^2)/280-(2021 x^3)/(40 Sqrt[35])+(269373 x^4)/62720-(140289 x^5)/(22400 Sqrt[35])+(187 x^6)/1960+(747 x^7)/(44800 Sqrt[35])-(3159 x^8)/6272000 = 0,
of which the least positive solution is x = 0.803384896325630173615878150981...
Here is a Mathematica code for computing the radius:
testtriangle = {6, 9, 13};(Notes and code from Peter Moses, July 24, 2019)
N[First[Sort[Select[x/.Solve[(1 - 8 s1 x / S + (23 s1^2 + 8 s2) x^2 / S^2 - 2 s1 (13 s1^2 + 24 s2) x^3 / S^3 + 2 (3 s1^4 + 42 s1^2 s2 + 8 s2^2 + 12 s1 s3) x^4 / S^4 - 16 s1 (s1^2 s2 + 4 s2^2 + 6 s1 s3) x^5 / S^5 + 4 s1 (2 s1^5 - 9 s1^3 s2 + 4 s1 s2^2 + 24 s1^2 s3 + 24 s2 s3) x^6 / S^6 + 96 s1 s2 x^7 / S^5 - 288 s1 s3 x^8 / S^6 /.Thread[{s1, s2, s3, S} → {a + b + c, b c + c a + a b, a b c, 1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)]}] /.Thread[{a, b, c} → testtriangle]) == 0, x], #>0&]]], 30]
Following is another approach to formulating and computing the congruent incircles radius. Let
k0 = s1^2 (s1^3-4 s1 s2+12 s3)^2
k1 = -s1 (11 s1^5-60 s1^3 s2+80 s1 s2^2+48 s1^2 s3-96 s2 s3)
k2 = 4 (11 s1^4-36 s1^2 s2+16 s2^2+24 s1 s3)
k3 = -16 (5 s1^2-8 s2)
k4 = 64, where
s1 = a + b + c
s2 = b c + c a + a b
s3 = a b c.
Next, let x be the maximal root of k0 + k1 x + k2 x^2 + k3 x^3 + k4 x^4 = 0. Then the
desired radius is S / (s1 + 2 Sqrt[x]) where S = 2 * Area ABC.
Here is a Mathematica code for computing the radius:
testtriangle={6,9,13}; N[S/(s1+2 Sqrt[Last[Sort[x/.Solve[ (s1^2 (s1^3-4 s1 s2+12 s3)^2-s1 (11 s1^5-60 s1^3 s2+80 s1 s2^2+48 s1^2 s3-96 s2 s3)x+4 (11 s1^4-36 s1^2 s2+16 s2^2+24 s1 s3) x^2-16 (5 s1^2-8 s2) x^3+64 x^4/.#)==0,x]]]])/.#&[Thread[{s1,s2,s3,S} → {a+b+c,b c+c a+a b,a b c,1/2 Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]}]/.Thread[{a,b,c} → testtriangle]],30](Notes and code from Peter Moses, August 13, 2019)
X(5394) lies on no line X(i)X(j) for 1 <= i < j <= 33000.
Let X, Y, Z be the points defined by Dominik Burek as at X(1217). If the initial point P is the centroid, then the perspector of the triangles XYZ and ABC is X(5395). (Peter Moses, June 9, 2012)
X(5395) lies on these lines: {2, 3053}, {4, 5050}, {6, 2996}, {20, 262}, {76, 193}, {83, 5033}, {98, 3091}, {439, 3815}, {458, 459}, {620, 2548}, {671, 5286}, {3146, 3329}, {3424, 3832}
X5395) = isogonal conjugate of X(5013)
X(5395) = polar conjugate of X(8889)
X(5396)-X(5400) were submitted with trilinears by Randy Hutson, December 12, 2012.
X(5396) = {X(3),X(6)}-harmonic conjugate of X(5398)
X(5396) lies on these lines: {1, 5}, {3, 6}, {35, 2361}, {40, 5312}, {42, 517}, {51, 859}, {54, 60}, {73, 942}, {140, 3216}, {515, 2051}, {912, 3666}, {1066, 5045}, {1155, 4337}, {1193, 1385}, {1450, 5126}, {1871, 1880}, {2800, 4868}, {3060, 4216}, {3190, 3940}, {3576, 5313}, {3579, 4300}, {3682, 5044}}
X(5396) = isogonal conjugate of X(5397)
X(5396) = crossdifference of every pair of points on line
X(523)X(654)
X(5396) = {X(371),X(372)}-harmonic conjugate of X(2278)
X(5397) lies on the hyperbola that passes through the points A, B, C, X(1), X(36), as well as the Kiepert hyperbola. X(5397) is the trilinear pole of the line X(523)X(654). (Randy Hutson, Dec. 31, 2012)
X(5397) lies on the Kiepert hyperbola and these lines: {4, 2278}, {5, 60}, {10, 2323}, {12,54}, {36, 226}, {59, 495}, {94, 3615}, {275, 860}, {321, 4511}, {1443, 1446}, {2051, 4276}, {2052, 5136}, {2618, 3737}
X(5397) = isogonal conjugate of X(5396)
X(5398) lies on these lines: {1, 2361}, {3, 6}, {4, 162}, {5, 1724}, {30, 1754}, {31, 517}, {36, 2003}, {46, 1399}, {47, 65}, {56, 215}, {81, 1006}, {184, 859}, {255, 942}, {283, 405}, {355, 3072}, {595, 1482}, {601, 3579}, {602, 1385}, {603, 1465}, {912, 4641}, {1060, 1708}, {1064, 2308}, {1411, 2964}, {1496, 5045}, {1718, 3336}, {1737, 5348}, {1780, 3560}, {2979, 4218}, {4216, 5012}
X(5398) = {X(371),X(372)}-harmonic conjugate of X(2245)
X(5399) = {X(1), X(2594)}-harmonic conjugate of X(5396)
X(5399) lies on these lines: {1, 5}, {3, 947}, {42, 942}, {54, 59}, {55, 500}, {73, 517}, {386, 999}, {581, 3295}, {1048, 2607}, {1060, 3811}, {1870, 5174}, {3333, 5312}, {3579, 4303}, {4322, 5126}}
X(5400) = trilinear pole, with respect to the excentral triangle, of the Brocard axis (Randy Hutson, Dec. 31, 2012)
X(5400) lies on the hexyl-excentral ellipse and on these lines: {1, 5}, {2, 991}, {4, 3216}, {42, 3817}, {43, 1699}, {118, 2999}, {165, 2108}, {200, 5014}, {244, 2801}, {386, 3091}, {500, 3628}, {516, 899}, {581, 3090}, {946, 3293}, {970, 3030}, {1054, 1768}, {1465, 1736}, {1724, 3149}, {1754, 4383}, {2635, 3911}, {2800, 4674}, {3214, 4301}, {3634, 4300}
On the sides of ABC construct external regular pentagons ABC1C2C3, BCA1A2A3, CAB1B2B3. The Aubert lines of [ABA1B3], [BCB1C3], [CAC1A3] concur in X(5401). These lines pass through A, B, C. (Aubert lines are defined at X(45010); see also A construction of X(3381) and X(5401) using Aubert lines). For the case of external pentagons, see X(3381). (Ivan Pavlov, April 15, 2022)
X(5401) lies on the Kiepert hyperbola and these lines: {2, 3379}, {3, 3382}, {4, 3380}, {5, 3368}, {6, 3381}, {1139, 3395}, {1140, 3393}, {3370, 3394}, {3396, 3397}
X(5401) = isogonal conjugate of X(3393)
X(5401) = X(3369)-cross conjugate of X(5402)
X(5401) = X(3)-Dao conjugate of X(3393)
X(5401) = barycentric quotient X(6)/X(3393)
X(5401) = {X(5),X(3368)}-harmonic conjugate of X(5402)
X(5402) lies on the Kiepert hyperbola and these lines: {2, 3380}, {3, 3381}, {4, 3379}, {5, 3368}, {6, 3382}, {1139, 3396}, {1140, 3394}, {3370, 3393}, {3395, 3397}
X(5402) = isogonal conjugate of X(3394)
Let A′ be the apex of the isosceles triangle BA′C constructed outward 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(5403). (Randy Hutson, July 20, 2016)
X(5403) lies on the Kiepert hyperbola and these lines: {2, 1670}, {3, 1676}, {4, 1671}, {5, 141}, {6, 1677}, {11, 3238}, {12, 3237}, {83, 1342}, {98, 1343}, {485, 1690}, {486, 1689}, {1348, 1664}, {1349, 1665}, {2009, 3102}, {2010, 3103}
X(5403) = isogonal conjugate of X(1342)
X(5403) = circumcircle-inverse of X(34134)
Let B* be a point such that (angle CBB*) = ω/2, and let C* be a point such that (angle CBC*) = ω/2. Let A′ = (line BB*)∩(line CC*), and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5404). (Randy Hutson, September 5, 2014)
X(5404) lies on the Kiepert hyperbola and these lines: {2, 1671}, {3, 1677}, {4, 1670}, {5, 141}, {6, 1676}, {11, 3237}, {12, 3238}, {83, 1343}, {98, 1342}, {485, 1689}, {486, 1690}, {1348, 1665}, {1349, 1664}, {2009, 3103}, {2010, 3102}
X(5404) = isogonal conjugate of X(1343)
X(5404) = circumcircle-inverse of X(34133)For the construction of this conic, see X(5393), where the associated Paache-Myakishev ellipse is introduced.
If you have The Geometer's Sketchpad, you can view X(5405), including the conic.
X(5405) lies on these lines: {1, 2}, {9, 3069}, {37, 615}, {57, 481}, {81, 3299}, {176, 5226}, {226, 482}, {491, 3879}, {492, 4357}, {590, 1100}, {591, 4643}, {641, 3666}, {940, 1124}, {946, 2048}, {1255, 3300}, {1335, 4383}, {1449, 3068}, {1586, 1785}, {1659, 5219}
X(5405) = {X(1),X(2)}-harmonic conjugate of X(5393)
Let A′B′C′ be the Lucas central 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(5406), which is the {X(3), X(394)}-harmonic conjugate of X(5407). (Randy Hutson, February 9, 2013)
X(5406) lies on these lines: {2, 490}, {3, 49}, {6, 588}, {25, 9739}, {323, 6411}, {343, 488}, {372, 1583}, {493, 5062}, {1151, 1993}, {1584, 6396}, {1591, 6560}, {1592, 5420}, {1600, 6410}, {1994, 3592}, {3155, 9733}, {3311, 8910}, {3594, 5422}, {3785, 8223}, {6423, 8962}
Let A′B′C′ be the Lucas(-1) central 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(5407), which is the {X(3), X(394)}-harmonic conjugate of X(5406). (Randy Hutson, February 9, 2013)
X(5407) lies on these lines: {2, 489}, {3, 49}, {6, 589}, {25, 9738}, {323, 6412}, {343, 487}, {371, 1584}, {494, 5058}, {1152, 1993}, {1583, 6200}, {1591, 5418}, {1592, 6561}, {1599, 6409}, {1994, 3594}, {3156, 9732}, {3592, 5422}, {3785, 8222}, {6347, 9678}, {6348, 9679}, {6805, 9540}, {6806, 9541}, {8962, 9600}
Let A′B′C′ be the Lucas tangents 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(5408). The same is true if A′B′C′ is the Lucas(2) central triangle. X(5408) = {X(3), X(394)}-harmonic conjugate of X(5409). (Randy Hutson, February 9, 2013)
X(5408) lies on the conic {A, B, C, X(69), X(97)} and these lines: {2, 372}, {3, 49}, {6, 493}, {25, 9733}, {63, 3083}, {69, 1590}, {317, 492}, {323, 6200}, {371, 1599}, {486, 6504}, {490, 1586}, {494, 8770}, {511, 3155}, {615, 1592}, {1152, 1584}, {1194, 6421}, {1589, 6458}, {1591, 3070}, {1600, 6396}, {1994, 6419}, {2987, 7598}, {3069, 6806}, {3084, 5414}, {3156, 9306}, {5422, 6420}, {6337, 8222}, {6460, 6805}, {8909, 8910}, {8911, 9723}
X(5408) = X(2)-Ceva conjugate of X(5409)
Let A′B′C′ be the Lucas(-1) tangents 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(5409). The same is true if A′B′C′ is the Lucas(-2) central triangle. X(5409) = {X(3), X(394)}-harmonic conjugate of X(5408). (Randy Hutson, February 9, 2013)
X(5409) lies on the conic {A, B, C, X(69), X(97)} and these lines: {2, 371}, {3, 49}, {6, 494}, {22, 8989}, {25, 9732}, {63, 2067}, {69, 1589}, {317, 491}, {323, 6396}, {372, 1600}, {485, 6504}, {489, 1585}, {493, 8770}, {511, 3156}, {590, 1591}, {1151, 1583}, {1194, 6422}, {1590, 6457}, {1592, 3071}, {1599, 6200}, {1994, 6420}, {2066, 3083}, {2987, 7599}, {3068, 6805}, {3155, 9306}, {5422, 6419}, {6337, 8223}, {6459, 6806}, {6503, 8909}
X(5409) = X(2)-Ceva conjugate of X(5408)
Let A′ be the perspector of the A-Lucas circle, and define B′ and C′ cyclically. (The lines AA′, BB′, CC′ concur in X(1151).) Let LA be the polar of A′ with respect to the A-Lucas circle, and define LB and LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. The lines AA*, BB*, CC* concur in X(5410). X(5410) = {X(6), X(25)}-harmonic conjugate of X(5411). (Randy Hutson, February 10, 2013)
X(5410) lies on these lines: {4, 1131}, {6, 25}, {24, 3312}, {186, 6398}, {235, 1588}, {371, 1593}, {372, 3515}, {378, 6221}, {427, 3068}, {468, 3069}, {485, 7507}, {590, 5094}, {1151, 3516}, {1271, 3536}, {1398, 2067}, {1587, 3575}, {1594, 8976}, {1597, 6199}, {1598, 6417}, {1702, 1902}, {1885, 6459}, {2066, 7071}, {2207, 5058}, {3089, 7582}, {3092, 5198}, {3127, 8975}, {3172, 6424}, {3517, 6418}, {3518, 6428}, {3520, 6449}, {3541, 8981}, {3542, 7584}, {6353, 7586}, {7487, 7581}, {8889, 8972}
Let A′ be the perspector of the A-Lucas(-1) circle, and define B′ and C′ cyclically. (The lines AA′, BB′, CC′ concur in X(1152).) Let LA be the polar of A′ with respect to the A-Lucas(-1) circle, and define LB and LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. The lines AA*, BB*, CC* concur in X(5411). X(5411) = {X(6), X(25)}-harmonic conjugate of X(5410). (Randy Hutson, February 10, 2013)
X(5411) lies on these lines: {4, 1132}, {6, 25}, {24, 3311}, {186, 6221}, {235, 1587}, {371, 3515}, {372, 1593}, {378, 6398}, {427, 3069}, {468, 3068}, {486, 7507}, {615, 5094}, {1152, 3516}, {1249, 5200}, {1270, 3535}, {1398, 6502}, {1588, 3575}, {1597, 6395}, {1598, 6418}, {1703, 1902}, {1885, 6460}, {2207, 5062}, {3089, 7581}, {3093, 5198}, {3147, 8981}, {3172, 6423}, {3517, 6417}, {3518, 6427}, {3520, 6450}, {3542, 7583}, {5414, 7071}, {6353, 7585}, {7487, 7582}, {7505, 8976}
Let U, V, W be the congruent squares described at X(371). Let LA be the extended diagonal of U that does not contain X(371), and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. The triangle A′B′C′ is homothetic to the orthic triangle, and the center of homothety is X(5412). Also, A′B′C′ is homothetic to the tangential triangle at X(6), to the intangents triangle at X(2066), and to the extangents triangle at X(5415); A′B′C′ is named the 1st Kenmotu diagonals triangle at X(31). (Randy Hutson, February 9, 2013)
X(5412) lies on the conic {A, B, C, X(4), X(24)} and these lines: {3, 3093}, {4, 371}, {6, 25}, {19, 5415}, {24, 372}, {33, 2066}, {34, 2067}, {186, 6396}, {216, 3155}, {235, 3071}, {317, 491}, {378, 6200}, {393, 8576}, {403, 6565}, {427, 590}, {468, 615}, {486, 3542}, {577, 3156}, {605, 2212}, {606, 1395}, {1151, 1593}, {1152, 3515}, {1164, 3087}, {1452, 2362}, {1587, 7487}, {1588, 3089}, {1595, 8981}, {1597, 6221}, {1598, 3092}, {1829, 7969}, {1968, 6406}, {2207, 6424}, {3069, 6353}, {3070, 3575}, {3088, 9540}, {3147, 5420}, {3199, 5058}, {3312, 3517}, {3516, 6409}, {3518, 6420}, {3536, 5591}, {3541, 5418}, {3592, 5198}, {4232, 7586}, {5094, 8253}, {5417, 6419}, {6756, 7583}, {6995, 7585}, {7378, 8972}, {8577, 8882}
X(5412) = isogonal conjugate of X(11091)
X(5412) = cevapoint of X(i) and X(j) for these {i,j}: {25, 6424}, {371, 8855}
X(5412) = X(393)-Ceva conjugate of X(5413)
X(5412) = X(571)-cross conjugate of X(5413)
X(5412) = polar conjugate of X(34392)
X(5412) = {X(4),X(371)}-harmonic conjugate of X(11473)
X(5412) = {X(6), X(25)}-harmonic conjugate of X(5413)
X(5412) = X(6203)-of-orthic-triangle if ABC is acute
X(5412) = trilinear product X(i)*X(j) for these {i,j}: {19, 372}, {31, 1586}, {491, 1973}, {1096, 5409}, {1748, 8577}, {3377, 5413}
Let U′, V′, W′ be the congruent squares as described at X(371), but with two vertices each on the extended sides of triangle ABC, and having common vertex X(372). Let LA be the extended diagonal of U′ that does not contain X(372), and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. The triangle A′B′C′ is homothetic to the orthic triangle, and the center of homothety is X(5413). Also, A′B′C′ is homothetic to the tangential triangle at X(6), to the intangents triangle at X(5414), and to the extangents triangle at X(5416); A′B′C′ is named the 2nd Kenmotu diagonals triangle at X(31). (Randy Hutson, February 9, 2013)
X(5413) lies on the conic {{A, B, C, X(4), X(24)}} and these lines: {3, 3092}, {4, 372}, {6, 25}, {19, 5416}, {24, 371}, {33, 5414}, {34, 6502}, {186, 6200}, {216, 3156}, {235, 3070}, {317, 492}, {378, 6396}, {393, 5200}, {403, 6564}, {427, 615}, {468, 590}, {485, 3542}, {577, 3155}, {605, 1395}, {606, 2212}, {1151, 3515}, {1152, 1593}, {1165, 3087}, {1587, 3089}, {1588, 7487}, {1597, 6398}, {1598, 3093}, {1829, 7968}, {1968, 6291}, {2207, 6423}, {3068, 6353}, {3071, 3575}, {3147, 5418}, {3199, 5062}, {3311, 3517}, {3516, 6410}, {3518, 6419}, {3535, 5590}, {3541, 5420}, {3594, 5198}, {4232, 7585}, {5094, 8252}, {5419, 6420}, {6756, 7584}, {6995, 7586}, {8576, 8882}
X(5413) = isogonal conjugate of X(11090)
X(5413) = cevapoint of X(i) and X(j) for these {i,j}: {25, 6423}, {372, 8854}
X(5413) = X(393)-Ceva conjugate of X(5412)
X(5413) = X(571)-cross conjugate of X(5412)
X(5413) = polar conjugate of X(34391)
X(5413) = {X(4),X(372)}-harmonic conjugate of X(11472)
X(5413) = {X(6), X(25)}-harmonic conjugate of X(5412)
X(5413) = X(6204)-of-orthic-triangle if ABC is acute
X(5413) = trilinear product X(i)*X(j) for these {i,j}: {19, 371}, {31, 1585}, {158, 8911}, {492, 1973}, {1096, 5408}, {1748, 8576}, {3378, 5412}
The A′B′C′ defined at X(5413) is homothetic to the intangents triangle, and the center of homothety is X(5414). Also, X(5414) = {X(6), X(55)}-harmonic conjugate of X(2066) and X(5414) = {X(3), X(1335)}-harmonic conjugate of X(2067). (Randy Hutson, February 9, 2013)
X(5414) lies on the conic {A, B, C, X(1), X(3)} and these lines: {1, 372}, {3, 1335}, {6, 31}, {11, 615}, {12, 3070}, {29, 7090}, {33, 5413}, {35, 371}, {36, 6396}, {37, 8577}, {56, 1152}, {63, 8394}, {140, 9661}, {165, 8833}, {172, 6283}, {259, 7014}, {283, 1805}, {388, 6460}, {390, 7586}, {405, 1377}, {485, 498}, {486, 1479}, {497, 3069}, {499, 5420}, {550, 9647}, {590, 5432}, {601, 3077}, {999, 6398}, {1124, 3295}, {1151, 5217}, {1378, 5687}, {1478, 6560}, {1500, 5062}, {1505, 2241}, {1587, 3085}, {1588, 4294}, {2646, 7969}, {2961, 8942}, {3057, 7968}, {3068, 5218}, {3071, 6284}, {3084, 5408}, {3297, 3303}, {3299, 3746}, {3304, 6426}, {3365, 7006}, {3390, 7005}, {3583, 6565}, {4302, 6561}, {5010, 6200}, {5204, 6410}, {5268, 8854}, {5281, 7585}, {5411, 7071}, {5563, 6454}, {6395, 6767}, {6564, 7951}, {7583, 9646}
X(5414) = isogonal conjugate of X(13390)
X(5414) = X(9)-Ceva conjugate of X(2066)
X(5414) = cevapoint of X(48) and X(606)
The A′B′C′ defined at X(5412) is homothetic to the extangents triangle, and the center of homothety is X(5415). Also, X(5415) = {X(6), X(55)}-harmonic conjugate of X(5416). (Randy Hutson, February 9, 2013)
X(5415) lies on these lines: {6, 31}, {19, 5412}, {40, 371}, {65, 2067}, {590, 3925}, {1151, 5584}, {1702, 6769}, {2550, 3068}, {3070, 6253}, {6200, 7688}, {7133, 8557}
The A′B′C′ defined at X(5413) is homothetic to the extangents triangle, and the center of homothety is X(5416). Also, X(5416) = {X(6), X(55)}-harmonic conjugate of X(5415). (Randy Hutson, February 9, 2013)
X(5416) lies on these lines: {6, 31}, {19, 5413}, {40, 372}, {65, 6414}, {615, 3925}, {1152, 5584}, {1703, 6769}, {2550, 3069}, {3071, 6253}, {3553, 7133}, {6396, 7688}
The 1st Kenmotu circle is defined at MathWorld. Let A′ be the pole of line BC with respect to the 1st Kenmotu circle, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5417). (Randy Hutson, February 9, 2013)
X(5417) lies on the conic {A, B, C, X(2), X(1173)} and these lines: {371, 5446}, {372, 1599}, {491, 639}, {5412, 6419}, {5419, 5421}
X(5417) = isogonal conjugate of X(5418)
X(5418) lies on these lines: {2, 371}, {3, 485}, {4, 6200}, {5, 1151}, {6, 140}, {10, 9557}, {15, 2046}, {16, 2045}, {20, 6564}, {25, 9683}, {30, 6409}, {33, 9631}, {54, 9676}, {55, 9661}, {56, 9646}, {69, 641}, {110, 9677}, {115, 9674}, {230, 6422}, {316, 2460}, {372, 631}, {376, 1327}, {381, 6449}, {382, 6455}, {468, 3092}, {491, 1078}, {492, 7769}, {498, 2067}, {499, 2066}, {547, 6429}, {549, 1152}, {550, 6411}, {597, 9975}, {615, 3311}, {620, 8980}, {632, 3592}, {1062, 9632}, {1124, 5433}, {1328, 5055}, {1329, 9678}, {1335, 5432}, {1377, 3035}, {1378, 4999}, {1504, 7749}, {1506, 9675}, {1579, 6676}, {1587, 3523}, {1590, 8968}, {1591, 5407}, {1656, 3071}, {1657, 6451}, {1698, 9583}, {1699, 9582}, {1702, 3624}, {2041, 3389}, {2042, 3364}, {2043, 3391}, {2044, 3366}, {2051, 9556}, {2886, 9679}, {3055, 8375}, {3069, 3525}, {3090, 6453}, {3091, 9541}, {3102, 7786}, {3147, 5413}, {3312, 5054}, {3524, 6460}, {3530, 6410}, {3533, 7582}, {3534, 6496}, {3541, 5412}, {3545, 6484}, {3628, 6425}, {3815, 6424}, {3832, 6486}, {3850, 6433}, {3851, 6445}, {3857, 6488}, {5056, 6480}, {5079, 6519}, {5254, 9600}, {5587, 9615}, {5972, 8994}, {6036, 8997}, {6289, 8414}, {6420, 7585}, {6684, 8983}, {6699, 8998}, {6813, 9993}, {7386, 8280}, {7389, 7790}, {7486, 9542}, {7494, 8854}, {8227, 9616}, {8964, 8967}
X(5418) = isogonal conjugate of X(5417)
X(5418) = {X(6),X(140}-harmonic conjugate of X(5420)
The 2nd Kenmotu circle has center X(372) and passes through the six contact points of the congruent squares in the construction of the 2nd Kenmotu point. Let Let A′ be the pole of line BC with respect to the 2nd Kenmotu circle, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5419). (Randy Hutson, February 9, 2013)
X(5419) lies on the conic {A, B, C, X(2), X(1173)} and these lines: {371, 1600}, {372, 5446}, {492, 640}, {5413, 6420}, {5417, 5421}
X(5419) = isogonal conjugate of X(5420)
X(5420) = {X(6),X(140}-harmonic conjugate of X(5418). (Randy Hutson, February 9, 2013)
X(5420) lies on these lines: {2, 372}, {3, 486}, {4, 6396}, {5, 1152}, {6, 140}, {15, 2045}, {16, 2046}, {20, 6565}, {30, 6410}, {69, 642}, {230, 6421}, {316, 2459}, {371, 631}, {376, 1328}, {381, 6450}, {382, 6456}, {468, 3093}, {491, 7769}, {492, 1078}, {498, 6502}, {499, 5414}, {547, 6430}, {549, 1151}, {550, 6412}, {590, 3312}, {597, 9974}, {632, 3594}, {1124, 5432}, {1327, 5055}, {1335, 5433}, {1377, 4999}, {1378, 3035}, {1505, 7749}, {1578, 6676}, {1588, 3523}, {1592, 5406}, {1656, 3070}, {1657, 6452}, {1703, 3624}, {2041, 3365}, {2042, 3390}, {2043, 3367}, {2044, 3392}, {3055, 8376}, {3068, 3525}, {3090, 6454}, {3103, 7786}, {3147, 5412}, {3311, 5054}, {3524, 6459}, {3530, 6409}, {3533, 7581}, {3534, 6497}, {3541, 5413}, {3545, 6485}, {3628, 6426}, {3815, 6423}, {3832, 6487}, {3850, 6434}, {3851, 6446}, {3857, 6489}, {5056, 6481}, {5079, 6522}, {6290, 8406}, {6395, 8976}, {6419, 7586}, {6478, 9692}, {6811, 9993}, {7386, 8281}, {7388, 7790}, {7494, 8855}
X(5420) = isogonal conjugate of X(5419)
X(5421) lies on these lines: {2, 1232}, {3, 6}, {51, 160}, {53, 1907}, {232, 428}, {233, 9698}, {615, 8962}, {1180, 6997}, {1194, 3815}, {1634, 9822}, {1879, 5254}, {2165, 5286}, {3055, 3291}, {5417, 5419}, {5422, 9723}
X(5421) = complement of X(1232)
X(5421) = the center of the bicevian conic of X(61) and X(62)
X(5421) = {X(6),X(39)}-harmonic conjugate of X(570)
X(5422) is the point of intersection of the following pairs of
lines:
(1) the line joining the center of the 1st Kenmotu circle and the
perspector of the 2nd Kenmotu circle, these two points being X(371) and
X(5419);
(2) the line joining the center of the 2nd Kenmotu circle and the
perspector of the 1st, these being X(372) and X(5417). See X(5446).
Also, X(5422) = {X(2), X(6)}-harmonic conjugate of X(1993).
(Randy Hutson, April 8, 2013)
X(5422) lies on these lines: {2, 6}, {3, 143}, {5, 7592}, {22, 51}, {23, 3796}, {24, 569}, {25, 5012}, {52, 7509}, {54, 6642}, {83, 5392}, {110, 5020}, {154, 3066}, {155, 1199}, {184, 575}, {195, 5070}, {324, 458}, {371, 1600}, {372, 1599}, {373, 9306}, {378, 9730}, {389, 7503}, {401, 7787}, {493, 589}, {494, 588}, {511, 7485}, {567, 6644}, {568, 7514}, {576, 3917}, {611, 7191}, {613, 3920}, {1181, 3091}, {1194, 5034}, {1249, 6819}, {1351, 2979}, {1498, 3832}, {1503, 7394}, {1505, 8962}, {1583, 3312}, {1584, 3311}, {1591, 7584}, {1592, 7583}, {1614, 7529}, {1853, 5169}, {1899, 5133}, {2003, 3306}, {2052, 8745}, {2323, 3305}, {3083, 3301}, {3084, 3299}, {3148, 3398}, {3167, 5643}, {3193, 5084}, {3410, 7605}, {3592, 5407}, {3594, 5406}, {3819, 5097}, {3981, 5038}, {4193, 5707}, {5046, 5706}, {5085, 6636}, {5093, 7998}, {5395, 8796}, {5408, 6420}, {5409, 6419}, {5421, 9723}, {5449, 7569}, {5480, 7391}, {5651, 6688}, {5889, 7395}, {5890, 9818}, {6146, 7544}, {6243, 7516}, {6505, 8257}, {6776, 6997}, {6805, 7582}, {6806, 7581}, {7387, 9781}, {7506, 9707}
Let A′ be the point in which the A-excircle is tangent to the circle OA that passes through vertices B and C, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5423); for the incircle version, see X(479). (Peter Moses, December 10, 2015)
A′ = -4 a^2 (a+b-c) (a-b+c) : (a-b+c)^3 (a+b+c) : (a+b-c)^3 (a+b+c) (barycentrics)
(center of circle OA) = -2 a^2 (5 a^3+3 a^2 b-a b^2+b^3+3 a^2 c+2 a b c-b^2 c-a c^2-b c^2+c^3)
: a^5+3 a^4 b+12 a^3 b^2+4 a^2 b^3-5 a b^4+b^5+3 a^4 c+2 a^3 b c+2 a^2 b^2 c+2 a b^3 c-b^4 c+2 a^3 c^2-4 a^2 b c^2+8 a b^2 c^2-2 b^3 c^2-2 a^2 c^3-2 a b c^3+2 b^2 c^3-3 a c^4+b c^4-c^5
: a^5+3 a^4 b+2 a^3 b^2-2 a^2 b^3-3 a b^4-b^5+3 a^4 c+2 a^3 b c-4 a^2 b^2 c-2 a b^3 c+b^4 c+12 a^3 c^2+2 a^2 b c^2+8 a b^2 c^2+2 b^3 c^2+4 a^2 c^3+2 a b c^3-2 b^2 c^3-5 a c^4-b c^4+c^5
(power of A wrt OA) = {((a+b+c) (a^2+2 a b+b^2+2 a c-2 b c+c^2))/(8 a)
(radius of OA) = (3 a^3+5 a^2 b+a b^2-b^3+5 a^2 c-2 a b c+b^2 c+a c^2+b c^2-c^3)/(16 S)
The radical center of the circles OA, OB, OC is X(2297). (Peter Moses, December 10, 2015)
If you have The Geometer's Sketchpad, you can view X(5423).
X(5423) lies on these lines: {2, 3677}, {7, 3263}, {8, 210}, {9, 7172}, {11, 6557}, {55, 1261}, {190, 9778}, {200, 346}, {280, 2057}, {329, 2835}, {345, 3699}, {612, 5749}, {644, 7074}, {756, 5296}, {1219, 8583}, {1260, 4578}, {1357, 8051}, {1863, 7046}, {2098, 8834}, {2325, 3158}, {2550, 3967}, {3006, 5748}, {3021, 4387}, {3452, 4901}, {3474, 4488}, {3596, 4441}, {3705, 5328}, {3710, 7080}, {3711, 6057}, {4076, 6632}, {4308, 9369}, {4847, 10005}, {5205, 5435}, {5273, 7081}, {6552, 6736}
X(5423) = isogonal conjugate of X(7023)
X(5423) = isotomic conjugate of X(479)
X(5423) = anticomplement of X(5573)
X(5423) = X(i)-Ceva conjugate of X(j) for these (i,j): (341,346), (4076,6558)
X(5423) = cevapoint of X(i) and X(j) for these {i,j}: {3022,4130}, {3900,4953}, {4081,4163}
X(5423) = crosspoint of X(i) and X(j) for these {i,j}: {346,6556}, {4076,6558}
X(5423) = crosssum of X(1106) and X(7366)
X(5423) = trilinear pole of the line X(4130)X(4163)
X(5423) = trilinear square of X(6731)
X(5423) = barycentric cube of X(8)
X(5423) = polar conjugate of isotomic conjugate of X(30681)
X(5423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,8055,497), (200,4082,346), (210,3974,8), (341,1265,8), (346,6555,200), (497,4009,8055)
X(5423) = X(i)-cross conjugate of X(j) for these (i,j): (728,346), (3022,4130}, (4012,8), (4081,4163), (5574,2)
X(5423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,8055,497), (200,4082,346), (210,3974,8), (341,1265,8), (346,6555,200), (497,4009,8055)
X(5423) = X(i)-isoconjugate of X(j) for these {i,j}: {1,7023}, {2,7366}, {6,738}, {7,1106}, {31,479}, {34,7053}, {56,269}, {57,1407}, {77,1398}, {222,1435}, {223,6612}, {244,7339}, {278,7099}, {279,604}, {513,6614}, {593,7147}, {603,1119}, {608,7177}, {649,4617}, {667,4626}, {757,7143}, {849,6046}, {1014,1042}, {1088,1397}, {1254,7341}, {1357,7045}, {1395,7056}, {1408,3668}, {1412,1427}, {1414,7250}, {1422,6611}, {1461,3669}, {1472,7197}, {4565,7216}, {4637,7180}
Let I be the incenter and L the Euler line of triangle ABC. Let LA be the Euler line of IBC, and define LB and LC cyclically. (The four Euler lines concur in the Schiffler point, X(21).) Let OA be the circumcenter of IBC, and define OB and OC cyclically.
Continuing, let AB, AC be the orthogonal projections of OA on LB and LC, respecitively, and define BC, BA and CA, CB cyclically. Let A′ be the circumcenter of OAABAC, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5424), and the circumcenter X(5428) of A′B′C′ lies on L. (Antreas Hatzipolakis, February 8, 2013)
A summary of Hyacinthos discussions of centers X(5424)-X(5429) is presented at Euler Lines, Circumcircles.
X(5424) lies on the Feuerbach hyperbola and these lines: {1, 5427}, {4, 5441}, {9, 5426}, {21, 4867}, {30, 5561}, {79, 2646}, {80, 3584}, {758, 2320}, {1385, 5557}, {1389, 3746}, {3612, 5665}, {5560, 7951}
X(5425) lies on these lines: {1, 3}, {2, 4867}, {8, 3841}, {21, 4084}, {79, 3671}, {80, 226}, {81, 759}, {100, 3919}, {191, 4018}, {515, 3982}, {519, 5249}, {758, 3219}, {956, 3894}, {958, 3901}, {993, 4880}, {1001, 3899}, {1100, 5341}, {1203, 3924}, {1210, 5443}, {1389, 5557}, {1411, 2003}, {1770, 5441}, {1835, 6198}, {2802, 3957}, {3305, 5692}, {3485, 6873}, {3584, 5719}, {3585, 3649}, {3624, 5730}, {3636, 5330}, {3830, 5561}, {3868, 5258}, {3869, 5259}, {3874, 5288}, {3881, 4861}, {3911, 5444}, {3918, 4420}, {4067, 5260}, {4511, 5883}, {4640, 5426}, {5226, 7951}, {5270, 6147}, {5357, 7052}, {5542, 7972}
The triangle A′B′C′ of circumcenters at X(5424) is perspective to the excentral triangle, and the perspector is X(5426). (Peter Moses, February 8, 2013)
X(5426) lies on these lines: {1, 21}, {9, 5424}, {30, 1699}, {35, 3753}, {36, 3742}, {40, 5428}, {55, 5541}, {57, 5427}, {80, 6690}, {100, 3968}, {210, 5251}, {214, 5284}, {355, 10021}, {405, 5506}, {442, 3586}, {484, 3919}, {1006, 5538}, {1125, 2475}, {1420, 3649}, {1698, 1837}, {1768, 6914}, {2320, 3065}, {2646, 5259}, {3158, 3679}, {3219, 4525}, {3336, 4189}, {3337, 5267}, {3616, 4299}, {3636, 3648}, {3651, 7987}, {3683, 4867}, {3746, 3880}, {3956, 5260}, {4316, 5249}, {4539, 5302}, {4640, 5425}, {4677, 4933}, {5131, 5883}, {5535, 7508}, {5691, 6841}, {5903, 8261}, {6326, 7489}
X(5426) = {X(1),X(21)}-harmonic conjugate of X(191)The triangle A′B′C′ of circumcenters at X(5424) is perspective to the intouch triangle, and the perspector is X(5427). (Peter Moses, February 8, 2013)
X(5427) lies on these lines: {1, 5424}, {7, 21}, {11, 30}, {12, 5251}, {57, 5426}, {79, 5886}, {100, 5172}, {191, 1420}, {392, 3647}, {442, 5433}, {758, 1319}, {993, 5434}, {1317, 2078}, {1411, 1758}, {1749, 6265}, {2475, 7288}, {2771, 5126}, {3651, 5204}, {4189, 5221}, {5441, 5722}, {5902, 7508}, {6841, 7354}
As a point on the Euler line, X(5428) has Shinagawa coefficients (4$aSA$ + abc, -4$aSA$ + abc).
X(5428) is the circumcenter of the triangle A′B′C′ defined at X(5424). X(5428) lies on the Euler line of ABC. (Antreas Hatzipolakis, February 8, 2013)
X(5428) lies on these lines: {1, 5424}, {2, 3}, {36, 3649}, {40, 5426}, {58, 5453}, {79, 5444}, {191, 3576}, {214, 960}, {517, 8261}, {758, 1385}, {952, 5258}, {970, 5946}, {1837, 5010}, {3579, 3754}, {3650, 5303}, {3652, 7987}, {5690, 8715}, {7171, 7701}
In the construction at X(5424), if L is taken to be the Brocard axis instead of the Euler line, then the resulting triangle A′B′C′ of circumcenters is not perspective to ABC. However, it is perspective to the excentral triangle, and X(5429) is the perspector, and its center lies on the line L. The triangle is also perspective to the intouch, hexyl, Yff, and 1st and 2nd cirumperp triangles. (Peter Moses, February 8, 2013)
For more, see Four Concurrent Lines, Circumcircles.
X(5429) lies on these lines: {1, 21}, {8, 8258}, {36, 199}, {171, 3753}, {210, 5247}, {511, 3576}, {740, 4234}, {976, 4661}, {978, 1453}, {986, 4252}, {995, 7032}, {999, 7083}, {1104, 3742}, {1125, 1330}, {1193, 4881}, {1247, 2363}, {1698, 6693}, {1757, 4134}, {1961, 5251}, {2308, 4511}, {2792, 5603}, {2938, 4221}, {3430, 7987}, {3454, 3624}, {3616, 6536}, {3880, 5255}, {5691, 7683}
X(5429) = {X(1),X(58)}-harmonic conjugate of X(1046)
Let LA be the line parallel to line BC and tangent to the circumcircle of triangle ABC on the negative side of BC (the region that does not contain A) , and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. Let AB be the reflection of A in line A′B′, and let AC be the reflection of A in line A′C′. Let C1 be the touchpoint of the incircle of triangle ABAB and line AB, and let B2 be the touchpoint of the incircle of ACAC and line AC. Define A1, B1 and C2, A2 cyclically. Then |AB2| = |CB1|, |BA1| = |CA2|, |AC1| = |BC2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic. A barycentric equation for this 1st Grigoriev conic follows:
x2 + y2 + z2 - 2 csc(A/2) yz - 2 csc(B/2) zx - 2 csc(C/2) xy = 0The perspector of the 1st Grigoriev conic is X(188). (Randy Hutson, March 30, 2013)
If you have The Geometer's Sketchpad, you can view X(5430)
X(5430) lies on these lines: {8, 188}, {178, 6557}, {236, 3161}, {346, 7027}
Let LA be the line parallel to line BC and tangent to the circumcircle of triangle ABC on the positive side of BC (the region that includes A), and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. Let AB be the reflection of A in line A′B′, and let AC be the reflection of A in line A′C′. Let C1 be the touchpoint of the incircle of triangle ABAB and line AB, and let B2 be the touchpoint of the incircle of ACAC and line AC. Define A1, B1 and C2, A2 cyclically. Then |AB2| = |CB1|, |BA1| = |CA2|, |AC1| = |BC2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic. A barycentric equation for this 2nd Grigoriev conic follows:
x2 + y2 + z2 - 2 sec(A/2) yz - 2 sec(B/2) zx - 2 sec(C/2) xy = 0The perspector of the 2nd Grigoriev conic is X(5451).
(Randy Hutson, April 8, 2013)
See also X(5452).
If you have The Geometer's Sketchpad, you can view X(5431)
X(5431) lies on this line: {178, 5451}
The coefficients in the combo shown just above are (R, 3r, r); for comparison, the coefficients for X(5433), X(11), and X(12) are, respectively, (R, -3r, -r), (R, -3r, r), and (R, 3r, -r).
X(5432) lies on these lines: {1, 140}, {2, 11}, {3, 12}, {4, 3614}, {5, 35}, {8, 4999}, {9, 3255}, {10, 2646}, {21, 1329}, {30, 5010}, {33, 468}, {36, 495}, {46, 3649}, {56, 631}, {65, 6684}, {119, 6914}, {141, 2330}, {165, 1836}, {171, 2361}, {210, 5745}, {212, 750}, {215, 5012}, {226, 1155}, {230, 2276}, {312, 3712}, {333, 4023}, {345, 6057}, {354, 3911}, {355, 3612}, {371, 9648}, {372, 9646}, {381, 4302}, {388, 3523}, {396, 7127}, {496, 632}, {499, 3295}, {550, 3585}, {551, 5048}, {569, 9666}, {590, 5414}, {597, 8540}, {601, 7299}, {612, 7499}, {615, 2066}, {620, 3023}, {756, 7004}, {846, 2607}, {908, 4640}, {950, 3634}, {958, 5552}, {999, 5054}, {1006, 5172}, {1040, 5268}, {1058, 3533}, {1062, 7542}, {1092, 9653}, {1124, 5420}, {1125, 3057}, {1213, 2268}, {1317, 6713}, {1335, 5418}, {1361, 6711}, {1362, 6712}, {1364, 6718}, {1399, 3074}, {1479, 1656}, {1500, 7749}, {1697, 3624}, {1698, 1837}, {1788, 5703}, {1852, 5142}, {1858, 5044}, {1914, 3815}, {1995, 9673}, {2077, 6907}, {2098, 3616}, {2099, 5657}, {2320, 3036}, {2476, 6668}, {2829, 6950}, {3011, 3752}, {3022, 6710}, {3024, 5972}, {3027, 6036}, {3028, 6699}, {3053, 9596}, {3056, 3589}, {3086, 3303}, {3090, 4294}, {3158, 4863}, {3301, 8981}, {3304, 7288}, {3305, 7082}, {3318, 6717}, {3321, 7056}, {3336, 6147}, {3337, 5442}, {3340, 9588}, {3428, 6954}, {3452, 3683}, {3474, 5226}, {3475, 4860}, {3486, 9780}, {3487, 5221}, {3522, 5229}, {3524, 4293}, {3528, 9656}, {3530, 7280}, {3576, 5252}, {3580, 9637}, {3627, 4324}, {3628, 7741}, {3689, 4847}, {3699, 4126}, {3703, 7081}, {3705, 4030}, {3715, 5273}, {3761, 6390}, {3813, 3871}, {3820, 5251}, {3967, 3977}, {4187, 5248}, {4255, 5230}, {4305, 5818}, {4309, 5070}, {4316, 8703}, {4414, 4415}, {4512, 4679}, {4855, 5794}, {4870, 5183}, {4998, 6066}, {5055, 9668}, {5056, 5225}, {5067, 9670}, {5119, 5886}, {5206, 9650}, {5260, 9711}, {5261, 9657}, {5297, 7495}, {5332, 9300}, {5441, 10021}, {5584, 6988}, {5697, 5901}, {5698, 5748}, {5719, 5902}, {5840, 6980}, {5842, 6830}, {6018, 6715}, {6019, 6719}, {6020, 6720}, {6067, 6600}, {6198, 10018}, {6238, 9820}, {6253, 6796}, {6286, 8254}, {6449, 9649}, {6459, 9662}, {6565, 9660}, {6677, 9817}, {6696, 7355}, {6716, 7158}, {6883, 8069}, {6905, 7680}, {6918, 7958}, {6926, 8273}, {6949, 7681}, {7486, 9671}, {7509, 9672}, {7987, 9578}, {7988, 9580}, {8144, 10020}, {9306, 9667}
The coefficients in the combo shown just above are (R, -3r, r); for comparison, the coefficients for X(5432), X(11), and X(12) are, respectively, (R, 3r, r), (R, -3r, r), and (R, 3r, -r).
X(5433) lies on these lines: {1, 140}, {2, 12}, {3, 11}, {4, 5204}, {5, 36}, {8, 1317}, {10, 1319}, {21, 3816}, {30, 7280}, {34, 468}, {35, 496}, {46, 5886}, {55, 631}, {57, 191}, {65, 392}, {79, 10021}, {85, 7181}, {100, 3813}, {104, 6949}, {141, 1428}, {172, 3815}, {201, 244}, {210, 6700}, {230, 2275}, {238, 1399}, {261, 7342}, {348, 1358}, {371, 9663}, {372, 9661}, {381, 4299}, {395, 2307}, {404, 2886}, {405, 1470}, {442, 5427}, {474, 3925}, {484, 5442}, {495, 632}, {497, 3523}, {498, 999}, {550, 3583}, {551, 4848}, {569, 9653}, {590, 6502}, {602, 5348}, {603, 748}, {604, 1213}, {614, 7499}, {615, 2067}, {620, 3027}, {946, 1155}, {993, 4187}, {1001, 6910}, {1015, 7749}, {1038, 5272}, {1056, 3533}, {1060, 7542}, {1092, 9666}, {1124, 5418}, {1210, 2646}, {1214, 7561}, {1335, 5420}, {1357, 6715}, {1359, 6717}, {1361, 6718}, {1362, 6710}, {1364, 6711}, {1376, 6921}, {1385, 1737}, {1387, 5697}, {1420, 1698}, {1447, 3665}, {1454, 3306}, {1466, 4423}, {1469, 3589}, {1478, 1656}, {1532, 5450}, {1770, 5122}, {1788, 2099}, {1836, 8227}, {1837, 3576}, {1852, 7501}, {1858, 9940}, {1870, 10018}, {1995, 9658}, {2093, 9624}, {2098, 5657}, {2361, 3075}, {2477, 5012}, {2594, 3216}, {2829, 6941}, {3022, 6712}, {3023, 6036}, {3024, 6699}, {3028, 5972}, {3053, 9599}, {3057, 6684}, {3085, 3304}, {3090, 4293}, {3295, 4995}, {3299, 8981}, {3303, 5218}, {3320, 6720}, {3324, 6716}, {3325, 6719}, {3336, 5443}, {3337, 6147}, {3361, 5219}, {3428, 6891}, {3476, 9780}, {3485, 5221}, {3486, 5704}, {3487, 4860}, {3522, 5225}, {3524, 4294}, {3528, 9671}, {3530, 5010}, {3612, 5722}, {3627, 4316}, {3628, 7951}, {3660, 5044}, {3671, 4870}, {3678, 5083}, {3760, 6390}, {3820, 5193}, {3825, 5267}, {3826, 7677}, {3847, 5046}, {4193, 6667}, {4301, 5183}, {4317, 5070}, {4324, 8703}, {4652, 7702}, {4861, 8256}, {4881, 5086}, {5055, 9655}, {5056, 5229}, {5067, 9657}, {5126, 9956}, {5206, 9665}, {5231, 5438}, {5274, 9670}, {5536, 5763}, {5584, 6926}, {5687, 6174}, {5841, 6971}, {5842, 6942}, {5901, 5903}, {6253, 6905}, {6285, 6696}, {6358, 6533}, {6449, 9662}, {6459, 9649}, {6565, 9647}, {6666, 8581}, {6824, 7958}, {6847, 7965}, {6883, 8071}, {6906, 7681}, {6911, 7742}, {6952, 7680}, {6988, 8273}, {7179, 7198}, {7292, 7495}, {7296, 9300}, {7352, 9820}, {7356, 8254}, {7486, 9656}, {7509, 9659}, {7753, 9341}, {7962, 9588}, {7987, 9581}, {7988, 9579}, {9306, 9652}
X(5434) lies on these lines: {1, 30}, {2, 12}, {3, 4317}, {4, 3304}, {5, 3582}, {7, 528}, {8, 5221}, {11, 381}, {20, 3303}, {34, 428}, {35, 8703}, {36, 495}, {46, 3654}, {55, 376}, {57, 3679}, {65, 519}, {85, 7198}, {104, 7680}, {172, 5306}, {226, 535}, {229, 7478}, {330, 7837}, {354, 515}, {355, 3338}, {390, 8162}, {396, 7051}, {442, 8666}, {496, 3585}, {497, 3543}, {498, 5054}, {499, 3614}, {516, 5919}, {524, 1469}, {527, 8581}, {537, 4032}, {541, 3024}, {542, 3023}, {543, 3027}, {544, 1362}, {547, 7951}, {550, 3746}, {597, 1428}, {752, 1463}, {944, 6253}, {952, 5902}, {956, 3925}, {982, 5724}, {993, 5427}, {1015, 7753}, {1388, 3485}, {1398, 5064}, {1406, 5710}, {1411, 7194}, {1420, 5290}, {1470, 6174}, {1479, 3830}, {1565, 7272}, {1652, 7043}, {1653, 7026}, {1657, 4309}, {1770, 9957}, {1837, 3333}, {1870, 7576}, {2098, 4295}, {2242, 5309}, {2275, 9300}, {2475, 3813}, {2646, 4311}, {2829, 5603}, {2886, 6175}, {3057, 4292}, {3085, 3524}, {3086, 3545}, {3091, 9656}, {3146, 9670}, {3295, 3534}, {3324, 6020}, {3336, 5690}, {3339, 4677}, {3340, 4355}, {3361, 9578}, {3421, 4413}, {3475, 5731}, {3627, 4857}, {3633, 5586}, {3665, 7176}, {3748, 4304}, {3816, 5080}, {3828, 3911}, {3839, 5229}, {3849, 5194}, {3872, 5880}, {3913, 4190}, {4302, 6767}, {4312, 7962}, {4320, 7667}, {4321, 6173}, {4669, 4848}, {4846, 6580}, {5066, 7741}, {5258, 8728}, {5289, 5905}, {5691, 9845}, {5842, 7967}, {7179, 7181}
X(5434) = reflection of X(3058) in X(1)
X(5435) lies on these lines: {1, 3523}, {2, 7}, {3, 938}, {4, 5704}, {8, 56}, {10, 3361}, {11, 3474}, {20, 1210}, {21, 1466}, {31, 9364}, {36, 5731}, {40, 9785}, {43, 1458}, {46, 962}, {55, 7677}, {65, 3616}, {77, 2999}, {78, 1467}, {88, 278}, {100, 1617}, {109, 9083}, {140, 3487}, {145, 1420}, {165, 390}, {171, 1471}, {174, 7002}, {175, 5405}, {176, 5393}, {190, 1997}, {208, 4200}, {223, 1443}, {238, 9316}, {241, 2275}, {269, 8051}, {273, 7490}, {279, 3008}, {333, 1014}, {354, 5218}, {376, 5122}, {388, 9780}, {452, 4652}, {479, 658}, {496, 6361}, {497, 1155}, {498, 3337}, {499, 3336}, {516, 5274}, {517, 4345}, {604, 3684}, {614, 4318}, {631, 942}, {651, 1407}, {673, 2898}, {910, 5838}, {912, 6970}, {940, 5933}, {950, 3522}, {978, 1042}, {999, 5657}, {1000, 3654}, {1038, 5262}, {1058, 3579}, {1071, 6927}, {1106, 5247}, {1125, 3339}, {1214, 4850}, {1319, 3241}, {1403, 8299}, {1442, 5256}, {1532, 2096}, {1656, 5714}, {1698, 4298}, {1707, 5121}, {1722, 4320}, {1728, 6953}, {1737, 4293}, {1750, 8544}, {1836, 9779}, {1876, 6353}, {1892, 8889}, {1999, 4460}, {2099, 5298}, {2263, 5272}, {2295, 5228}, {3052, 3756}, {3085, 3338}, {3091, 4292}, {3144, 7103}, {3146, 9581}, {3149, 9799}, {3210, 4552}, {3216, 4306}, {3333, 6684}, {3340, 3622}, {3434, 9352}, {3475, 4860}, {3485, 5221}, {3486, 5204}, {3526, 6147}, {3624, 3671}, {3634, 5290}, {3660, 3873}, {3673, 7397}, {3679, 4315}, {3740, 8581}, {3816, 5698}, {3817, 4312}, {3828, 5726}, {3832, 9579}, {3868, 6921}, {3913, 9797}, {3916, 5084}, {3947, 4355}, {4032, 4699}, {4190, 5175}, {4208, 5705}, {4220, 5807}, {4302, 5131}, {4305, 7280}, {4307, 9746}, {4321, 5686}, {4327, 5268}, {4384, 7176}, {4423, 8543}, {4430, 5083}, {4488, 6557}, {4662, 9850}, {4899, 6555}, {4911, 7402}, {4998, 6632}, {5022, 6554}, {5054, 5719}, {5056, 9612}, {5126, 7967}, {5129, 9843}, {5205, 5423}, {5287, 7269}, {5439, 6857}, {5444, 5902}, {5556, 7173}, {5658, 5729}, {5687, 6764}, {5691, 7319}, {5709, 6926}, {5734, 5903}, {5758, 6891}, {5768, 6905}, {5770, 6911}, {5804, 6906}, {5809, 7580}, {5811, 6944}, {5825, 5927}, {6223, 6848}, {6350, 7011}, {6654, 9358}, {6734, 6904}, {6738, 7987}, {6964, 7330}, {6988, 9940}, {7191, 8270}, {7678, 7965}
X(5435) = {X(8),X(56)}-harmonic conjugate of X(4308)
X(5436) lies on these lines: {1, 6}, {2, 950}, {3, 5437}, {4, 1125}, {10, 3158}, {20, 142}, {21, 57}, {34, 4183}, {40, 1006}, {55, 1706}, {65, 4512}, {78, 5047}, {84, 3560}, {165, 3812}, {200, 3983}, {226, 452}, {329, 3622}, {382, 3824}, {442, 3586}, {443, 4304}, {515, 6846}, {551, 3487}, {631, 9843}, {936, 2900}, {938, 5745}, {942, 3928}, {943, 3680}, {946, 6987}, {968, 3924}, {976, 7322}, {988, 5573}, {993, 3333}, {997, 3646}, {1005, 5253}, {1012, 8726}, {1043, 4384}, {1210, 6857}, {1260, 3303}, {1385, 1490}, {1451, 2328}, {1621, 1697}, {1698, 3419}, {1708, 3340}, {2136, 3295}, {2478, 5219}, {2550, 4314}, {2646, 4423}, {2647, 7273}, {2654, 7070}, {2975, 4666}, {3306, 4189}, {3339, 4640}, {3361, 3742}, {3452, 5129}, {3523, 6692}, {3671, 5698}, {3811, 4015}, {3822, 6990}, {3825, 6829}, {3868, 3929}, {3870, 5260}, {4292, 6173}, {4301, 5759}, {4308, 8232}, {4428, 5836}, {4659, 7283}, {4678, 4917}, {5177, 5550}, {5218, 8582}, {5249, 6872}, {5257, 5802}, {5443, 9612}, {5587, 6832}, {5691, 8226}, {5705, 5722}, {5715, 5886}, {5812, 5901}, {6765, 9708}, {6920, 9845}, {6936, 9624}, {6988, 7682}, {7330, 7489}, {7580, 7987}, {7675, 9859}
X(5437) lies on these lines: {1, 474}, {2, 7}, {3, 5436}, {4, 9841}, {5, 84}, {10, 1056}, {37, 9574}, {38, 7322}, {40, 631}, {46, 3624}, {65, 8583}, {85, 738}, {88, 4606}, {100, 4666}, {140, 5709}, {165, 1001}, {171, 5272}, {173, 7028}, {200, 354}, {210, 4860}, {236, 258}, {244, 612}, {281, 1435}, {312, 4659}, {377, 9581}, {381, 7171}, {388, 7091}, {392, 2093}, {404, 3601}, {443, 1210}, {518, 8580}, {549, 3587}, {614, 750}, {936, 942}, {940, 1449}, {946, 6926}, {950, 6904}, {958, 3361}, {960, 3339}, {982, 5268}, {997, 5883}, {999, 9623}, {1054, 8299}, {1155, 4423}, {1329, 5290}, {1375, 5834}, {1420, 5253}, {1490, 6918}, {1519, 6833}, {1656, 3824}, {1697, 3616}, {1698, 3338}, {1699, 3816}, {1709, 3838}, {1768, 6667}, {1995, 7293}, {2098, 3922}, {2478, 9579}, {2551, 4298}, {3149, 8726}, {3182, 7532}, {3189, 6744}, {3208, 5308}, {3220, 5020}, {3247, 3666}, {3304, 3698}, {3359, 5886}, {3475, 6745}, {3487, 6700}, {3576, 3833}, {3589, 7289}, {3600, 5795}, {3740, 5223}, {3754, 7982}, {3763, 5227}, {3772, 4859}, {3784, 5943}, {3825, 6845}, {3873, 9342}, {3920, 9335}, {3925, 5231}, {3980, 4871}, {4035, 4869}, {4187, 9612}, {4208, 5704}, {4292, 5084}, {4415, 4862}, {4454, 8055}, {4640, 8167}, {4652, 5047}, {4850, 5287}, {5044, 5708}, {5045, 6765}, {5119, 5444}, {5123, 5726}, {5128, 5250}, {5284, 9352}, {5285, 7484}, {5535, 6681}, {5587, 6854}, {5705, 8728}, {5715, 6922}, {5737, 6706}, {6223, 9842}, {6245, 6864}, {6260, 6964}, {6668, 6763}, {6701, 7701}, {6705, 6846}, {6916, 7682}, {7521, 7713}
X(5438) lies on these lines: {1, 474}, {2, 950}, {3, 9}, {4, 6700}, {8, 1420}, {10, 631}, {20, 3452}, {21, 7308}, {40, 997}, {44, 8951}, {55, 8583}, {56, 200}, {57, 78}, {63, 4188}, {72, 3928}, {80, 1698}, {100, 1697}, {140, 5705}, {142, 5703}, {165, 960}, {210, 5204}, {214, 6264}, {226, 6904}, {377, 5219}, {386, 1449}, {388, 6745}, {452, 5316}, {480, 4321}, {515, 6926}, {518, 3361}, {549, 5791}, {728, 9310}, {908, 4190}, {938, 6692}, {958, 7987}, {975, 3247}, {976, 3677}, {978, 7290}, {988, 5293}, {999, 6765}, {1058, 1125}, {1193, 5269}, {1260, 1466}, {1319, 4853}, {1329, 5691}, {1377, 9583}, {1385, 9623}, {1453, 3216}, {1496, 3939}, {1702, 9679}, {1743, 4252}, {1788, 6737}, {1861, 7521}, {2093, 5730}, {2270, 3430}, {2551, 4297}, {2646, 4413}, {2886, 3624}, {3146, 5328}, {3149, 6282}, {3218, 3984}, {3243, 3333}, {3304, 3689}, {3305, 4189}, {3340, 4511}, {3359, 7971}, {3421, 4311}, {3476, 6736}, {3486, 8582}, {3487, 6173}, {3488, 9843}, {3523, 5745}, {3586, 4187}, {3617, 4881}, {3623, 4917}, {3632, 8256}, {3646, 5248}, {3679, 5445}, {3740, 5234}, {3814, 6845}, {3869, 5128}, {3870, 5253}, {3876, 3929}, {3927, 5122}, {4304, 5084}, {4386, 9575}, {4512, 5217}, {4847, 7288}, {5096, 5227}, {5231, 5433}, {5289, 7991}, {5552, 9578}, {5587, 6833}, {5709, 6924}, {5719, 5832}, {5731, 5795}, {5735, 5763}, {6734, 6921}, {6854, 8227}, {7091, 8828}
X(5439) lies on these lines: {1, 474}, {2, 72}, {3, 3306}, {4, 9776}, {5, 1071}, {7, 5084}, {8, 4002}, {10, 354}, {20, 5806}, {37, 3670}, {46, 1001}, {57, 405}, {63, 5708}, {65, 392}, {142, 442}, {145, 5049}, {210, 3634}, {226, 4187}, {355, 6854}, {377, 5722}, {388, 3660}, {406, 1876}, {443, 938}, {498, 5570}, {517, 631}, {518, 1698}, {519, 3698}, {551, 3057}, {614, 5711}, {750, 5266}, {908, 6147}, {912, 1656}, {956, 3333}, {958, 3338}, {960, 3624}, {971, 3091}, {986, 6051}, {1155, 5248}, {1214, 1393}, {1279, 5264}, {1385, 5253}, {1426, 5136}, {1439, 7532}, {1479, 5880}, {1621, 3579}, {1699, 9943}, {1788, 5173}, {1829, 7521}, {1871, 7543}, {1898, 7173}, {2476, 3824}, {3090, 5777}, {3218, 5047}, {3244, 3918}, {3295, 4666}, {3305, 3927}, {3336, 4640}, {3337, 5251}, {3488, 6904}, {3576, 7686}, {3617, 3889}, {3622, 9957}, {3625, 3968}, {3626, 3892}, {3635, 3893}, {3636, 3922}, {3720, 3931}, {3740, 4533}, {3748, 8715}, {3811, 4413}, {3828, 3983}, {3838, 7741}, {3869, 5550}, {3870, 9709}, {3872, 7373}, {3873, 3921}, {3884, 3919}, {3894, 4539}, {3897, 5126}, {3947, 8581}, {4189, 5122}, {4359, 5295}, {4420, 9342}, {4423, 5221}, {5302, 6763}, {5435, 6857}, {5603, 6926}, {5704, 6856}, {5714, 6919}, {5730, 8583}, {5768, 6864}, {5770, 6887}, {5787, 6835}, {5804, 6916}, {5805, 6836}, {5812, 6947}, {5885, 5887}, {5886, 6833}, {6001, 8227}, {6173, 9612}, {6245, 8226}, {6259, 6957}, {6734, 8728}, {6845, 9955}, {7580, 8726}, {8100, 8126}, {9581, 9844}, {9779, 9961}
X(5440) lies on these lines: {1, 474}, {2, 3419}, {3, 63}, {4, 5748}, {8, 631}, {10, 2646}, {20, 5658}, {21, 5044}, {30, 908}, {35, 960}, {36, 518}, {40, 5730}, {44, 2251}, {46, 4018}, {48, 3694}, {55, 392}, {56, 3555}, {80, 5123}, {100, 517}, {101, 2751}, {109, 2756}, {140, 6734}, {149, 7743}, {200, 956}, {210, 993}, {214, 519}, {238, 5529}, {306, 7536}, {318, 7531}, {329, 376}, {355, 5552}, {404, 942}, {405, 936}, {443, 5703}, {484, 4867}, {498, 5794}, {515, 6745}, {521, 656}, {551, 3748}, {572, 3965}, {758, 1155}, {914, 6699}, {944, 6926}, {950, 4187}, {952, 6735}, {958, 3612}, {971, 6909}, {995, 3744}, {999, 3870}, {1012, 5720}, {1018, 6603}, {1055, 3930}, {1104, 3216}, {1125, 3925}, {1149, 3722}, {1193, 5266}, {1375, 3912}, {1386, 5313}, {1420, 6765}, {1437, 1792}, {1455, 4551}, {1737, 3035}, {2057, 5534}, {2077, 2932}, {2551, 4305}, {2802, 5048}, {2975, 4420}, {3057, 8715}, {3086, 3189}, {3090, 5175}, {3218, 5122}, {3421, 5731}, {3434, 5886}, {3436, 6899}, {3452, 4304}, {3487, 6904}, {3524, 5744}, {3579, 3869}, {3583, 5087}, {3616, 5082}, {3617, 3897}, {3666, 4256}, {3678, 5267}, {3740, 5251}, {3868, 4188}, {3871, 9957}, {3876, 4189}, {3921, 9708}, {3935, 4881}, {3957, 5049}, {3991, 9310}, {4002, 9709}, {4257, 4641}, {4313, 5084}, {4421, 5119}, {4539, 5220}, {4640, 5010}, {4662, 5258}, {4694, 4864}, {4702, 4975}, {4880, 5131}, {5045, 5253}, {5086, 6852}, {5174, 7543}, {5176, 6224}, {5249, 5719}, {5728, 8257}, {5761, 6885}, {5777, 6906}, {5787, 6890}, {5791, 6910}, {5806, 6915}, {5812, 6934}, {5882, 6736}, {5904, 7280}, {6282, 7580}, {6684, 6737}, {6900, 9955}, {6940, 9940}
X(5440) = isogonal conjugate of X(36125)
X(5440) = isotomic conjugate of isogonal conjugate of X(23202)
X(5440) = isotomic conjugate of polar conjugate of X(44)
X(5440) = X(19)-isoconjugate of X(88)
X(5440) = crossdifference of every pair of points on the line X(19)X(4394)
X(5440) = inner-Garcia-to-ABC similarity image of X(11)
X(5441) lies on these lines: {1, 30}, {3, 5442}, {4, 5424}, {8, 3647}, {10, 21}, {11, 5499}, {20, 5902}, {36, 950}, {65, 4324}, {145, 758}, {191, 2136}, {354, 4325}, {442, 3586}, {515, 3746}, {517, 4330}, {548, 5131}, {550, 3336}, {942, 4316}, {944, 4309}, {952, 3065}, {1125, 6175}, {1385, 4857}, {1478, 4313}, {1479, 2475}, {1697, 7701}, {1770, 5425}, {1837, 5010}, {2646, 3583}, {2771, 3057}, {3486, 4302}, {3488, 4299}, {3534, 5221}, {3601, 6841}, {3633, 3650}, {3884, 6224}, {4004, 8261}, {4297, 5563}, {5427, 5722}, {5432, 10021}, {5691, 7680}, {5692, 6872}
X(5442) lies on these lines: {1, 549}, {2, 79}, {3, 5441}, {5, 5131}, {10, 36}, {35, 3911}, {40, 6713}, {46, 3624}, {65, 5444}, {80, 7280}, {140, 3336}, {484, 5433}, {499, 6361}, {631, 5902}, {1145, 3632}, {1155, 9955}, {3035, 6763}, {3337, 5432}, {3579, 3582}, {3585, 5122}, {3616, 3884}, {4325, 9956}, {5054, 5221}, {5559, 5657}, {5563, 6684}, {5692, 6921}, {5697, 7288}, {7951, 8728}
X(5443) is the QA-P7 center of quadrangle ABCX(1); see QA-Nine-point Center Homothetic Center.
X(5443) lies on these lines: {1, 5}, {2, 3754}, {3, 5444}, {10, 7504}, {17, 7052}, {21, 36}, {35, 946}, {46, 3624}, {56, 7489}, {140, 484}, {191, 4999}, {214, 2475}, {226, 5563}, {388, 6965}, {451, 1845}, {497, 6900}, {498, 5603}, {499, 3485}, {908, 5258}, {942, 3582}, {1210, 5425}, {1319, 5270}, {1385, 3585}, {1388, 9654}, {1389, 5559}, {1478, 3616}, {1479, 4313}, {1656, 2099}, {1699, 3612}, {1749, 3337}, {1836, 7280}, {2646, 3583}, {2800, 6952}, {3057, 3584}, {3085, 6979}, {3086, 6884}, {3245, 6684}, {3336, 5433}, {3467, 5557}, {3576, 7491}, {3746, 6915}, {3817, 7548}, {4295, 5550}, {4305, 9779}, {4317, 5714}, {4867, 6734}, {5436, 9612}
X(5443) = {X(1),X(5)}-harmonic conjugate of X(80)
X(5444) lies on these lines: {1, 140}, {2, 80}, {3, 5443}, {35, 404}, {36, 226}, {65, 5442}, {79, 5428}, {90, 3646}, {442, 3586}, {484, 549}, {498, 3476}, {499, 3488}, {631, 5903}, {952, 5326}, {1319, 3584}, {1387, 4995}, {1479, 5550}, {2099, 5054}, {3487, 5557}, {3576, 6882}, {3616, 3754}, {3653, 5252}, {3822, 4881}, {3911, 5425}, {4324, 9955}, {4870, 5122}, {5010, 5886}, {5119, 5437}, {5298, 5719}, {5435, 5902}
X(5445) lies on these lines: {1, 140}, {2, 3754}, {3, 80}, {5, 484}, {8, 214}, {9, 46}, {10, 36}, {12, 3336}, {35, 950}, {40, 6882}, {201, 1772}, {355, 7280}, {495, 3337}, {498, 1788}, {499, 5657}, {942, 3584}, {946, 3245}, {1125, 5330}, {1155, 3585}, {1210, 3746}, {1478, 9780}, {1727, 6907}, {1749, 5499}, {1837, 5010}, {1838, 8756}, {2099, 3526}, {2800, 6949}, {3057, 3582}, {3579, 3583}, {3679, 5438}, {3813, 5541}, {3828, 4292}, {3911, 5563}, {3916, 5123}, {4299, 5818}, {4325, 5122}, {5046, 6702}, {5119, 9588}, {5131, 7354}, {5183, 9955}, {5204, 5790}, {5288, 6735}, {5552, 5904}, {5557, 5708}, {5660, 5693}, {5901, 7294}, {7742, 9709}
Continuing the discussion at X(5422), the point X(5446) lies on the
following pairs of lines:
(1) the line joining the center of the 1st Kenmotu circle and its
perspector, these two points being X(371) and X(5417);
(2) the line joining the center of the 2nd Kenmotu circle and its
perspector, these two points being X(372) and X(5419).
Also, X(5446) is the complement of X(3) with respect to the orthic
triangle. (Randy Hutson, April 8, 2013)
X(5446) lies on these lines: {2, 5447}, {3, 51}, {4, 52}, {5, 141}, {6, 7387}, {20, 3567}, {22, 569}, {23, 54}, {25, 1147}, {26, 578}, {30, 143}, {49, 1495}, {113, 6152}, {131, 134}, {140, 5943}, {155, 1351}, {184, 7517}, {185, 382}, {193, 9936}, {235, 5448}, {343, 7403}, {371, 5417}, {372, 5419}, {373, 3526}, {381, 5562}, {394, 7529}, {427, 5449}, {428, 539}, {546, 1154}, {550, 5946}, {567, 2937}, {576, 2393}, {631, 5640}, {632, 6688}, {970, 6914}, {1092, 7506}, {1112, 3575}, {1209, 5133}, {1350, 7393}, {1593, 7689}, {1614, 1994}, {1656, 3917}, {1885, 6746}, {2979, 3090}, {3089, 5654}, {3091, 5891}, {3098, 7516}, {3146, 5890}, {3292, 7545}, {3543, 6241}, {3547, 9967}, {3560, 5752}, {3574, 10024}, {3627, 6000}, {3628, 3819}, {3845, 5876}, {3853, 5663}, {5056, 7999}, {5067, 7998}, {5070, 5650}, {5093, 6467}, {5102, 9973}, {6146, 7553}, {6193, 6995}, {6676, 6689}, {7713, 9928}, {7718, 9933}
X(5446) = midpoint of X(4) and X(52)
X(5446) = reflection of X(i) in X(j) for these (i,j): (389, 143), (1216,5)
X(5446) = anticomplement of X(5447)
Let A′B′C′ be the tangential triangle of triangle ABC. Let OA be the circle with center A′ that is tangent to line BC. Define OB and OC cyclically. Then X(5447) is the radical center of the three circles. (Randy Hutson, April 8, 2013)
X(5447) lies on these lines: {2, 5446}, {3, 49}, {4, 7998}, {5, 3819}, {20, 5891}, {26, 3098}, {51, 3526}, {52, 631}, {68, 7386}, {140, 143}, {389, 549}, {548, 6000}, {550, 5907}, {569, 7485}, {578, 7516}, {632, 5943}, {858, 1209}, {1154, 3530}, {1350, 6642}, {1368, 5449}, {1498, 8717}, {1656, 5650}, {3060, 3525}, {3523, 9730}, {3524, 5889}, {3533, 5640}, {3546, 9967}, {5054, 6243}, {5448, 6823}, {5651, 7517}, {5654, 7400}, {5876, 8703}, {6643, 9927}, {6689, 7499}
X(5447) = complement of X(5446)
X(5447) = midpoint of X(3) and X(1216)
Let A′B′C′ be the pedal triangle of the orthocenter, X(4), and let A″B″C″ be the circumcevian triangle of X(4) with respect to A′B′C′. 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(5448). For figures, see Concurrent Radical Axes. (Antreas Hatzipolakis, April 10, 2013)
Let A′B′C′ be the orthic triangle. Let (A) be the circle centered at A′ with radius a/2, and define (B), (C) cyclically. X(5448) is the radical center of circles (A), (B), (C). (Randy Hutson, January 29, 2018)
X(5448) lies on these lines: {2, 7689}, {3, 1568}, {4, 110}, {5, 389}, {30, 5893}, {52, 403}, {68, 1173}, {155, 195}, {185, 2072}, {235, 5446}, {525, 7764}, {541, 3357}, {912, 6583}, {1204, 6640}, {1533, 5073}, {1614, 3153}, {1699, 9928}, {2929, 6642}, {3060, 6242}, {3167, 3843}, {3546, 4846}, {3564, 3850}, {3818, 7564}, {3832, 6193}, {3855, 9936}, {5447, 6823}, {5476, 8548}, {5562, 10024}, {6238, 7951}, {6689, 7503}, {7352, 7741}, {7552, 7691}
X(5448) = midpoint of X(4) and X(1147)
X(5448) = complement of X(7689)
Let A′B′C′ be the pedal triangle of the circumcenter, X(3), and let A″B″C″ be the circumcevian triangle of X(3) with respect to A′B′C′. 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(5449). The midpoint of X(5448) and X(5449) is X(5). For figures, see Concurrent Radical Axes. (Antreas Hatzipolakis, April 10, 2013)
Let D = X(68); then X(5449) is the centroid of ABCD. (Randy Hutson, August 25, 2014)
Let NANBNC be the reflection triangle of X(5). Let OA be the circumcenter of ANBNC, and define OB and OC cyclically. X(5449) is the orthocenter of OAOBOC. (Randy Hutson, June 7, 2019)
X(5449) lies on these lines: {2, 54}, {3, 125}, {4, 7689}, {5, 389}, {30, 6696}, {51, 5576}, {52, 1594}, {69, 8538}, {113, 7722}, {115, 8571}, {136, 847}, {155, 1656}, {156, 542}, {184, 6639}, {185, 10024}, {343, 1216}, {427, 5446}, {511, 6697}, {568, 3574}, {575, 3564}, {912, 3812}, {1092, 6640}, {1368, 5447}, {1614, 3448}, {1698, 9928}, {1853, 7387}, {1899, 3549}, {2072, 5562}, {3090, 5643}, {3091, 7693}, {3167, 5070}, {3616, 9933}, {3624, 9896}, {5020, 9908}, {5067, 9936}, {5169, 9781}, {5422, 7569}, {5889, 7577}, {5892, 7399}, {6146, 7542}, {6238, 7741}, {7352, 7951}, {7393, 9937}, {7514, 9932}, {7552, 9140}, {7706, 9786}, {7846, 9923}, {8253, 8909}
X(5449) = midpoint of X(68) and X(1147)
X(5449) = complement of X(1147)
X(5449) = X(6796)-of-orthic-triangle if ABC is acute
Let A′B′C′ be the circumcevian triangle of X(1). Let RA be the radical axis of the circles (B, |BC′|) and (C,|CB′|), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5450). For figures, see Concurrent Radical Axes. See also X(1147). (Antreas Hatzipolakis, April 10, 2013)
Let M be the isogonal conjugate of the trilinear polar of X(57) with respect to the circumcevian triangle of X(1). Then M is a conic, and its center is X(5450). (Angel Montesdeoca, August 9, 2019)
X(5450) lies on these lines: {1, 104}, {2, 6256}, {3, 10}, {4, 36}, {5, 2829}, {8, 2077}, {21, 84}, {30, 3829}, {35, 944}, {40, 2975}, {48, 1765}, {55, 5882}, {56, 946}, {100, 5881}, {318, 1309}, {388, 6935}, {404, 5587}, {405, 6260}, {411, 5303}, {498, 6977}, {517, 8666}, {519, 8668}, {550, 5842}, {631, 5251}, {950, 8071}, {952, 8715}, {995, 3073}, {997, 7330}, {999, 3671}, {1006, 1490}, {1071, 2646}, {1125, 3560}, {1210, 1470}, {1385, 5248}, {1457, 1777}, {1478, 6833}, {1479, 6938}, {1482, 4084}, {1532, 5433}, {1621, 7971}, {1698, 6940}, {1699, 7704}, {2096, 3485}, {3072, 4257}, {3149, 5204}, {3436, 6966}, {3522, 7688}, {3585, 6830}, {3624, 6920}, {3746, 7967}, {3814, 6958}, {3822, 6862}, {3825, 6929}, {3885, 6264}, {4189, 5731}, {4231, 5345}, {4293, 6847}, {4305, 5768}, {4325, 6845}, {4511, 5693}, {4999, 6907}, {5080, 6972}, {5126, 9856}, {5229, 6956}, {5253, 6912}, {5258, 5657}, {5288, 5537}, {5538, 6763}, {5563, 5603}, {5691, 6905}, {6259, 7489}, {6681, 6959}, {6831, 7354}, {6946, 7989}, {6952, 7951}, {7677, 8544}
X(5450) = midpoint of X(1) and X(1158)
The 1st and 2nd Grigoriev conics are presented at X(5430) and X(5431). Their perspectors are X(188) and X(5451), respectively, (Randy Hutson, April 2013)
X(5451) lies on this line: {178, 5431}
Barycentrics a2(b + c - a)[a3 - a2b - a2c + ab2 + ac2 - (b + c)(b - c)2]
Let A′B′C′ be the anticomplementary triangle of triangle ABC. Let A″ be the reflection of A′ in the perpendicular bisector of segtment BC, and define B″ and C″ cyclically. Let A1 be the touchpoint of the incircle of A″BC and line BC, and let A2 be the touchpoint of the incircle of A′BC and line BC. Define the points B1, B2, C1, C2 cyclically. Then |AC2| = |BC1|, |BA1| = |CA2|, |CB1| = |AB2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic, named in honor of Alexander Privalov. A barycentric equation for the Privalov conic follows:
x2 + y2 + z2 + f(a,b,c)yz + f(b,c,a)zx + f(c,a,b)xy = 0, where f(a,b,c) = 2[(b - c)2 + a2]/[(b - c)2 - a2], or, equivalently, by x2 + y2 + z2 - g(A,B,C)yz - g(B,C,A)zx - g(C,A,B)xy = 0, where g(A,B,C) = tan(B/2) tan(C/2) [cot2(B/2) + cot2(C/2)]. (Dmitry Grigoriev, April 15, 2013.)The Privalov conic is the bicevian conic of X(7) and X(8) - that is, the conic through the vertices of the intouch and extouch triangles. Its center X(5452) is also the center of the conic through A, B, C, X(101), X(294), X(651), X(666), which is the isogonal conjugate of the Gergonne line. Also, X(5452) = crossdifference of every pair of points on the polar of X(6) with respect to the incircle. See also X(5545). (Randy Hutson, April 19, 2013)
As a line L varies through the set of all lines that pass through X(55), the locus of the trilinear pole of L is a circumconic, and its center is X(5452). (Randy Hutson, April 19, 2013)
Let L be a line tangent to the incircle. Let P and U be the circumcircle intercepts of L. Let X be the crossdifference of P and U. As L varies, X traces the circumellipse centered at X(5452). This ellipse shares a major axis with, and is homothetic to, the Privalov conic. Therefore, the Privalov conic is an ellipse for every triangle ABC. (Randy Hutson, July 20, 2016)
Let A′B′C′ be the orthic triangle. Let La be the Gergonne line of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X(5452). (Randy Hutson, July 31 2018)
Let P be a point on the circumcircle. Let A′B′C′ be the anticevian triangle of P. A′B′C′ is perspective to the unary cofactor triangle of the intangents triangle for all P. As P varies, the perspector traces the circumconic centered at X(5452). (Randy Hutson, July 11, 2019)
For another conic with center X(5452), see X(175) and X(46417.
If you have The Geometer's Sketchpad, you can view X(5452)
X(5452) lies on these lines: {2, 1814}, {6, 354}, {9, 1040}, {33, 210}, {55, 2195}, {212, 8012}, {218, 226}, {219, 3686}, {294, 497}, {650, 1376}, {651, 7056}, {666, 6063}, {1212, 7124}, {2238, 2911}
X(5452) = X(2)-Ceva conjugate of X(55)
Let A′B′C′ be the cevian triangle of I (the incenter, X(1)). Let NA be the nine-point center of triangle IB′C′, and define NB and NC cyclically. The points I, NA, NB, NC are concyclic, and their circle, described by Antreas Hatzipolakis, April 17, 2013.
X(5453) lies on these lines: {1, 30}, {3, 81}, {5, 581}, {21, 323}, {55, 6097}, {58, 5428}, {73, 5719}, {140, 3216}, {155, 6914}, {186, 2906}, {386, 549}, {511, 1385}, {550, 991}, {1064, 5901}, {1154, 2646}, {1442, 6356}, {1834, 5499}, {1962, 5492}, {2771, 3743}, {3945, 6869}
X(5453) = midpoint of X(1) and X(500)
Let DEF be the 1st Morley triangle of triangle ABC. The Newton lines of the quadrilaterals AEDF, BFED, CDFE concur in X(5454). (Seiichi Kirikami, April 26, 2013)
If you have The Geometer's Sketchpad, you can view X(5454).
X(5454) lies on these lines: {356, 1134}, {357, 3280}
Let DEF be the 1st Morley triangle of triangle ABC. Let LA be the line of the centroid of AEF and the centroid of BCD, and define LB and LC cyclically. The lines LA, LB, LC concur in X(5455). (Seiichi Kirikami, April 26, 2013)
If you have The Geometer's Sketchpad, you can view X(5455).
X(5455) lies on this line: {2, 356}
Let DEF be the 1st Morley triangle of triangle ABC, and let D1E1F1 be the 1st Morley adjunct triangle (defined at MathWorld). Let D2 = DD1∩BC, and define E2 and F2 cyclically. The lines AD2, BE2, CF2 concur in X(5456). (Seiichi Kirikami, April 26, 2013)
Let DEF be the 1st Morley triangle. Let D' be the trilinear pole of line EF, and define E', F' cyclically. Let D" be the trilinear pole of line E'F', and define E", F" cyclically. The lines AD", BE", CF" concur in X(5456). (Randy Hutson, September 29, 2014)
If you have The Geometer's Sketchpad, you can view X(5456).
X(5456) lies on these lines: {356, 3605}, {3274, 3602}
Let DEF be the 1st Morley triangle of triangle ABC, and let D3 be the reflection of D in line BC, and define E3 and F3 cyclically. The lines AD3, BE3, CF3 concur in X(5457). (Seiichi Kirikami, April 26, 2013)
X(5457) is the Hofstadter -1/3 point; see X(359). (Randy Hutson, August 9, 2014)
If you have The Geometer's Sketchpad, you can view X(5457) and X(5458).
X(5457) lies on this line: {357, 5628}
X(5457) = isogonal conjugate of X(6123)
Let DEF be the 1st Morley triangle of triangle ABC, and let D1E1F1 be the 1st Morley adjunct triangle (defined at MathWorld). Let D4 be the reflection of D1 in line BC, and define E4 and F4 cyclically. The lines AD4, BE4, CF4 concur in X(5458). (Seiichi Kirikami, April 26, 2013)
X(5458) is the Hofstadter -2/3 point; see X(359). (Randy Hutson, August 9, 2014)
D4 is the BCD-isogonal conjugate of A, and cyclically for E4 and F4(Randy Hutson, January 29, 2015
X(5458) lies on these lines: {}
X(5459) is the center of the equilateral triangle formed by the nine-point centers of the triangles BCF, CAF, ABF, where F is the Fermat point, X(13). Also, X(5459) is the center of the circle through X(2), X(13), and the previously mentioned nine-point centers, and X(5459) = {X(5),X(597)}-harmonic conjugate of X(5460). (Randy Hutson, April 25, 2013)
The circle just defined is here given the name 1st Hutson circle. The 2nd Hutson circle and Hutson-Parry circles are defined at X(5460) and X(5466); see also X(8371). (CK, October 31, 2015)
X(5459) lies on these lines: {2, 13}, {5, 542}, {14, 9166}, {17, 671}, {30, 5478}, {61, 5469}, {115, 396}, {148, 9114}, {395, 5472}, {524, 623}, {543, 619}, {599, 635}, {630, 2482}, {3524, 5473}, {3545, 6770}, {3642, 9763}, {5055, 5617}, {5318, 6671}, {5476, 7685}, {8371, 9194}
X(5459) = reflection of X(5460) in X(5461)
X(5459) = complement of X(5463)
X(5459) = radical center of the polar circles of triangles BCX(13), CAX(13), ABX(13)
X(5459) = radical trace of 1st Hutson and Hutson-Parry circles
X(5460) is the center of the equilateral triangle formed by the nine-point centers of the triangles BCF', CAF', ABF', where F' = X(14). Also, X(5460) is the center of the circle through X(2), X(14), and the previously mentioned nine-point centers. (Randy Hutson, April 25, 2013)
The circle just described is here given the name 2nd Hutson circle. The 1st Hutson and Hutson-Parry circles are defined at X(5459) and X(5466); see also X(8371). (CK, October 31, 2015)
X(5460) lies on these lines: {2, 14}, {5, 542}, {13, 9166}, {18, 671}, {30, 5479}, {62, 5470}, {115, 395}, {148, 9116}, {396, 5471}, {524, 624}, {543, 618}, {599, 636}, {629, 2482}, {3524, 5474}, {3545, 6773}, {3643, 9761}, {5055, 5613}, {5321, 6672}, {5476, 7684}, {8371, 9195}
X(5460) = reflection of X(5459) in X(5461)
X(5460) = complement of X(5464)
X(5460) = {X(5),X(597)}-harmonic conjugate of X(5459)
X(5460) = radical center of the polar circles of triangles BCX(14), CAX(14), ABX(14)
X(5460) = radical trace of 2nd Hutson and Hutson-Parry circles
X(5461) is the center of the rectangle having vertices X(2), X(115), X(125), and X(5465). (Paul Yiu, Anopolis #170, April 25, 2013)
X(5461) lies on these lines: {2, 99}, {3, 9880}, {5, 542}, {6, 8176}, {30, 6036}, {98, 3545}, {114, 5055}, {125, 5465}, {187, 8352}, {230, 3849}, {316, 8859}, {376, 9754}, {381, 2794}, {523, 9165}, {524, 625}, {530, 6670}, {531, 6669}, {538, 2023}, {547, 2782}, {598, 7806}, {599, 626}, {690, 9183}, {754, 9478}, {1153, 3054}, {1506, 7827}, {1656, 8724}, {1698, 9881}, {1992, 3767}, {1995, 3455}, {2796, 3634}, {3090, 7902}, {3363, 7804}, {3589, 9830}, {3616, 9884}, {3618, 8593}, {3624, 9875}, {3934, 5969}, {5020, 9876}, {5025, 7810}, {5054, 6321}, {5071, 6054}, {5077, 5569}, {5182, 7808}, {5215, 8598}, {5463, 5470}, {5464, 5469}, {6680, 8370}, {6704, 8367}, {6781, 8597}, {7610, 7761}, {7746, 7830}, {7749, 7833}, {7755, 7812}, {7798, 9770}, {7801, 7887}, {7846, 9878}, {7861, 8359}, {7886, 8369}
X(5461) = midpoint of X(i) and X(j) for these (i,j): (2,115),
(125,5465)
X(5461) = complement of X(2482)
X(5462) = (X(i),X(j))-harmonic conjugate of X(k) for these (i,j,k): (2,52,1216), (3,51,5446), (24,5422,569). (Randy Hutson, April 24, 2013)
X(5462) lies on these lines: {2, 52}, {3, 51}, {4, 4846}, {5, 389}, {6, 1147}, {20, 9781}, {24, 569}, {26, 182}, {30, 9729}, {54, 6153}, {68, 7401}, {110, 1199}, {125, 5576}, {140, 143}, {155, 5020}, {184, 7506}, {185, 381}, {195, 3292}, {343, 7405}, {373, 568}, {468, 6746}, {539, 9827}, {546, 5893}, {570, 3133}, {575, 2393}, {578, 6644}, {631, 3060}, {632, 3819}, {973, 6689}, {1112, 6699}, {1154, 3628}, {1181, 3066}, {1209, 3580}, {1843, 3517}, {1899, 7528}, {1995, 7592}, {2070, 2918}, {2072, 3574}, {2807, 9955}, {2979, 3525}, {3090, 5889}, {3091, 5890}, {3518, 5012}, {3526, 3917}, {3533, 7998}, {3564, 9822}, {3618, 9967}, {3796, 9714}, {3832, 6241}, {3850, 5663}, {5092, 7525}, {5643, 7550}, {5752, 6883}, {6237, 9816}, {6238, 9817}, {6677, 9820}, {7689, 9786}, {8548, 9813}, {9815, 9927}
X(5462) = midpoint of X(i) and X(j) for these (i,j): (5,389), (140,143)
X(5462) = complement of X(1216)
X(5462) = centroid of {A,B,C,X(52)}
Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463)}; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014). X(5463) is the center of the equilateral antipedal triangle of X(13), and X(5463) = (X(3), X(599))-harmonic conjugate of X(5464). (Randy Hutson, May 2, 2013)
The circle {{X(2), X(110), X(5463), X(5464)}} has center X(1649) and passes through X(2770). It is tangent to the Euler line at X(2) and is the reflection in the Euler line of the circle {{X(2), X(13), X(14), X(111), X(476)}}. (Randy Hutson, August 26, 2014)
Let NANBNC and N′AN′BN′C be the inner and outer Napoleon triangles, resp. Let A′ be the reflection of NA in line N′BN′C, and define B′ and C′ cyclically. Triangle A′B′C′ is equilateral, the reflection of N′AN′BN′C in X(618), and X(5463) is its center. (Randy Hutson, January 17, 2020)
X(5463) lies on these lines: {2, 13}, {3, 67}, {6, 9115}, {14, 543}, {15, 524}, {18, 671}, {30, 5473}, {61, 1992}, {62, 597}, {99, 298}, {299, 7771}, {395, 6772}, {396, 9112}, {519, 7975}, {620, 6778}, {627, 8591}, {2770, 9203}, {2796, 5699}, {3104, 5969}, {3106, 7757}, {3524, 6770}, {3545, 5478}, {5054, 6771}, {5461, 5470}, {5476, 5615}, {5569, 9763}, {5978, 8592}, {6054, 9749}, {6670, 9166}, {6782, 9741}, {7840, 8291}, {7865, 9982}, {9168, 9205}, {9830, 9886}
X(5463) = midpoint of X(2) and X(616)
X(5463) = reflection of X(2) in X(618)
X(5463) = reflection of X(5464) in X(2482)
X(5463) = anticomplement of X(5459)
X(5463) = Thomson-isogonal-conjugate-of-X(15)
Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463):; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014).
X(5464) is the center of the equilateral antipedal triangle of X(14), and X(5464) = (X(3), X(599))-harmonic conjugate of X(5463). (Randy Hutson, May 2, 2013)
Let NANBNC and N′AN′BN′C be the inner and outer Napoleon triangles, resp. Let A″ be the reflection of N′A in line NBNC, and define B″ and C″ cyclically. Triangle A″B″C″ is equilateral, the reflection of NANBNC in X(619), and X(5464) is its center. (Randy Hutson, January 17, 2020)
X(5464) lies on these lines: {2, 14}, {3, 67}, {6, 9117}, {13, 543}, {16, 524}, {17, 671}, {30, 5474}, {61, 597}, {62, 1992}, {99, 299}, {298, 7771}, {395, 9113}, {396, 6775}, {519, 7974}, {620, 6777}, {628, 8591}, {2770, 9202}, {2796, 5700}, {3105, 5969}, {3107, 7757}, {3524, 6773}, {3545, 5479}, {5054, 6774}, {5461, 5469}, {5476, 5611}, {5569, 9761}, {5979, 8592}, {6054, 9750}, {6669, 9166}, {6783, 9741}, {7840, 8292}, {7865, 9981}, {9168, 9204}, {9830, 9885}
X(5464) = midpoint of X(2) and X(617)
X(5464) = reflection of X(2) in X(619)
X(5464) = reflection of X(5463) in X(2482)
X(5464) = anticomplement of X(5460)
X(5464) = Thomson-isogonal-conjugate-of-X(16)
X(5465) is the point, other than X(2), of intersection of the circle defined at X(5459) and the circle defined at X(5460). (Randy Hutson, May 2, 2013)
X(5465) is the fourth vertex of a rectangle determined by three vertices X(2), X(115), and X(125); the center of this rectangle is X(5461). (Paul Yiu, Anopolis #170, April 25, 2013)
X(5465) lies on the circle O(2,6) (with diameter X(2)X(6)) and these lines: {2, 690}, {6, 13}, {30, 9181}, {110, 671}, {125, 5461}, {541, 6055}, {543, 1316}, {2482, 5972}, {2780, 3111}, {6593, 9830}, {9140, 9166}
X(5465) = midpoint of X(110) and X(671)
X(5465) = reflection of X(125) and X(5461)
X(5465) = complement of X(11006)
Let L be the line tangent at X(13) to the circle defined at X(5459), and let L′ be the line tangent to X(14) to the circle defined at X(5460). Then X(5466) = L∩L′. Also, on the circle passing through X(2), X(13), X(14), X(111), and X(476), the antipode of X(2) is X(5466). (Randy Hutson, May 3 2013)
The circle just described is here given the name Hutson-Parry circle. (CK, October 31, 2015). The circle is mentioned briefly in TCCT, page 228.
In the plane of a triangle ABC, let
Γ = circumcircle;
D = line tangent to $Gamma; at A;
Ta = D∩BC;
Oab = center of the circle through Ta that is tangent to Γ at B;
Oac = center of the circle through Ta that is tangent to Γ at C;
Ab = orthogonal projection of Oab onto AC;
Ac = orthogonal projection of Oac onto AB;
A' = AbAc∩BC, and define B' and C' cyclically.
The lines AA', BB', CC' concur in X(5466). See X(5466) (Angel Montesdeoca, August 21, 2022)
X(5466) lies on the Kiepert hyperbola and these lines: {2, 523}, {4, 1499}, {10, 4024}, {13, 9201}, {14, 9200}, {76, 850}, {83, 5643}, {98, 111}, {115, 9180}, {262, 5996}, {321, 4036}, {351, 8587}, {476, 691}, {512, 598}, {525, 5485}, {647, 7607}, {671, 690}, {685, 4240}, {868, 2394}, {879, 9154}, {892, 5468}, {895, 2986}, {1916, 9148}, {2799, 5503}, {3268, 8781}, {3288, 7708}, {5067, 8151}, {7612, 9209}, {9123, 9189}, {9125, 9131}
X(5466) = isogonal conjugate of X(5467)
X(5466) = isotomic conjugate of X(5468)
X(5466) = pole wrt polar circle of trilinear polar of X(4235)
X(5466) = X(48)-isoconjugate (polar conjugate) of X(4235)
X(5466) = barycentric product X(111)*X(850)
X(5467) = {X(1576),X(4558)}-harmonic conjugate of X(1634) (Peter Moses, May 7, 2013)
X(5467) lies on these lines: {3, 6}, {23, 5968}, {110, 351}, {112, 1296}, {250, 4230}, {523, 2407}, {524, 5967}, {691, 9178}, {895, 9142}, {1649, 5468}, {1989, 6321}, {2709, 2715}, {2794, 3014}, {2854, 5191}, {3292, 9717}, {4436, 4612}, {5994, 9203}, {5995, 9202}, {6036, 6128}, {6593, 9155}
X(5467) = midpoint of X(2407) and X(4226)
X(5467) = crossdifference of every pair of points on line
X(115)X(523)
X(5467) = X(111)-isoconjugate of X(1577)
X(5468) is the unique point on line X(2)X(6) whose trilinear polar is parallel to line X(2)X(6). (Randy Hutson, March 21, 2019)
X(5468) lies on these lines: {2, 6}, {99, 110}, {126, 5477}, {316, 9141}, {511, 7417}, {691, 6082}, {805, 9150}, {877, 4240}, {892, 5466}, {1316, 6090}, {1649, 5467}, {2418, 2434}, {2502, 5969}, {2709, 9080}, {2715, 2858}, {3111, 7771}, {3266, 3292}, {4590, 9170}, {4615, 6548}, {5642, 7664}, {6390, 9717}
X(5468) = isogonal conjugate of X(9178)
X(5468) = crossdifference of every pair of points on line X(512)X(3124)
X(5468) = trilinear pole of line X(187)X(524)
X(5468) = {X(110),X(4563)}-harmonic conjugate of X(4576)
Let A′B′C′ be the antipedal triangle of X(13), let A″ be the nine-point center of the triangle BCX(14), and define B″ and C″ cyclically. Then X(5469) is the homothetic center of A′B′C′ and A″B″C″. (Randy Hutson, May 3, 2013)
X(5469) = reflection of X(5470) in X(115) (Randy Hutson,
May 3, 2013)
X(5469) = {X(14),X(115)}-harmonic conjugate of X(13) (Peter
Moses, May 7, 2013)
X(5469) lies on these lines: {2, 5982}, {6, 13}, {18, 671}, {61, 5459}, {98, 5479}, {99, 6670}, {148, 618}, {395, 6779}, {531, 9166}, {617, 6669}, {5461, 5464}, {5473, 6321}, {5474, 6036}, {5478, 6773}
Let A′B′C′ be the antipedal triangle of X(14), let A″ be the nine-point center of the triangle BCX(13), and define B″ and C″ cyclically. Then X(5470) is the homothetic center of A′B′C′ and A″B″C″. (Randy Hutson, May 3, 2013)
X(5470) = reflection of X(5469) in X(115) (Randy Hutson,
May 3, 2013)
X(5470) = {X(13),X(115)}-harmonic conjugate of X(14) (Peter
Moses, May 7, 2013)
X(5470) lies on these lines: {2, 5983}, {6, 13}, {17, 671}, {62, 5460}, {98, 5478}, {99, 6669}, {148, 619}, {396, 6780}, {530, 9166}, {616, 6670}, {5461, 5463}, {5473, 6036}, {5474, 6321}, {5479, 6770}
Let A′B′C′ be the pedal triangle of X(15), let A″ be the nine-point center of the triangle BCX(14), and define B″ and C″ cyclically. Then X(5471) is the homothetic center of A′B′C′ and A″B″C″. (Peter Moses, May 7, 2013)
Let U be the pedal circle of X(14) (and of X(16). Then X(5471) is the U-inverse of X(187). (Randy Hutson, January 29,2105)
X(5471) lies on these lines: {2, 9117}, {5, 6783}, {6, 13}, {15, 6774}, {16, 6781}, {18, 7749}, {39, 398}, {61, 1506}, {62, 7747}, {187, 395}, {230, 6114}, {233, 2903}, {302, 620}, {396, 5460}, {543, 8595}, {590, 6303}, {615, 6307}, {1569, 3106}, {2482, 9761}, {2549, 5334}, {2782, 6782}, {3815, 6109}, {5318, 5479}, {5339, 7748}, {6775, 7737}
X(5471) = isogonal conjugate (and isotomic conjugate) of X(16) w.r.t to the pedal triangle of X(16)
X(5471) = X(14)-antipedal-to-X(16)-pedal similarity image of X(14)
X(5471) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5474,5472), (115,5477,5472)
X(5471) = X(14)-antipedal-to-X(16)-pedal similarity image of X(14)
Let A′B′C′ be the pedal triangle of X(16), let A″ be the nine-point center of the triangle BCX(13), and define B″ and C″ cyclically. Then X(5472) is the homothetic center of A′B′C′ and A″B″C″. (Peter Moses, May 7, 2013)
Let V be the pedal circle of X(13) (and of X(15). Then X(5472) is the V-inverse of X(187). (Randy Hutson, January 29,2105)
X(5472) lies on these lines: {2, 9115}, {5, 6782}, {6, 13}, {15, 6781}, {16, 6771}, {17, 7749}, {39, 397}, {61, 7747}, {62, 1506}, {187, 396}, {230, 6115}, {233, 2902}, {303, 620}, {395, 5459}, {543, 8594}, {590, 6302}, {615, 6306}, {1569, 3107}, {2482, 9763}, {2549, 5335}, {2782, 6783}, {3815, 6108}, {5321, 5478}, {5340, 7748}, {6772, 7737}
X(5472) = isogonal conjugate (and isotomic conjugate) of X(15) w.r.t. the pedal triangle of X(15)
X(5472) = X(13)-antipedal-to-X(15)-pedal similarity image of X(13)
X(5472) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5475,5471), (115,5477,5471)
X(5472) = X(13)-antipedal-to-X(15)-pedal similarity image of X(13)
X(5473) lies on these lines: {2, 5478}, {3, 13}, {4, 618}, {15, 9112}, {16, 2549}, {20, 616}, {30, 5463}, {62, 9607}, {165, 9901}, {376, 530}, {477, 9203}, {511, 6779}, {517, 7975}, {542, 1350}, {631, 6669}, {3098, 9982}, {3524, 5459}, {5469, 6321}, {5470, 6036}, {5979, 9749}
X(5473) = reflection of X(13) in X(3)
X(5473) = anticomplement of X(5478)
X(5473) = isogonal conjugate (and isotomic conjugate) of X(13) wrt the antipedal triangle of X(13)
X(5473) = {X(1350), X(3534)}-harmonic conjugate of X(5474)
X(5474) is the isogonal conjugate (and isotomic conjugate) of X(14) with respect to the antipedal triangle of X(14). (Randy Hutson, May 7, 2013)
X(5474) lies on these lines: {2, 5479}, {3, 14}, {4, 619}, {15, 2549}, {16, 9113}, {20, 617}, {30, 5464}, {61, 9607}, {165, 9900}, {376, 531}, {477, 9202}, {511, 6780}, {517, 7974}, {542, 1350}, {631, 6670}, {3098, 9981}, {3524, 5460}, {5469, 6036}, {5470, 6321}, {5978, 9750}
X(5474) = reflection of X(14) in X(3)
X(5474) = anticomplement of X(5479)
X(5474) = {X(1350), X(3534)}-harmonic conjugate of X(5473)
X(5475) is the {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5471,5472,6), (13,14,5476), and X(5475) is the inverse-in-Kiepert-hyperbola of X(5476). (Randy Hutson, May 7, 2013)
X(5475) is the {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,2548,39), (6,115,5309), (6,381,115), (3767,5007,5346) (Peter Moses, May 12, 2013)
X(5475) lies on these lines: {1, 9650}, {2, 187}, {3, 1506}, {4, 39}, {5, 32}, {6, 13}, {11, 2242}, {12, 2241}, {30, 574}, {51, 5167}, {69, 7845}, {76, 7759}, {83, 3407}, {99, 7777}, {112, 7577}, {140, 5206}, {141, 7818}, {148, 7757}, {156, 9697}, {172, 7741}, {183, 754}, {194, 7858}, {233, 1609}, {315, 3934}, {325, 3734}, {376, 8589}, {382, 5013}, {384, 3788}, {385, 7812}, {427, 9515}, {485, 5058}, {486, 5062}, {524, 3363}, {538, 7774}, {546, 5254}, {547, 3054}, {549, 3055}, {566, 7574}, {590, 9675}, {620, 1003}, {626, 7770}, {1015, 1478}, {1078, 7823}, {1194, 7394}, {1196, 6997}, {1285, 5071}, {1316, 5099}, {1352, 5052}, {1384, 5055}, {1479, 1500}, {1503, 5034}, {1504, 3071}, {1505, 3070}, {1569, 6321}, {1572, 5587}, {1594, 1968}, {1596, 5065}, {1656, 3053}, {1699, 9620}, {1914, 7951}, {1975, 7764}, {1992, 7615}, {1995, 9745}, {2088, 7706}, {2207, 7507}, {2275, 3585}, {2276, 3583}, {2458, 5103}, {2896, 7860}, {3066, 6388}, {3087, 6623}, {3091, 3767}, {3096, 7885}, {3314, 7809}, {3329, 7790}, {3526, 5023}, {3545, 5008}, {3552, 7769}, {3589, 5033}, {3814, 4386}, {3830, 5024}, {3832, 5041}, {3839, 7739}, {3843, 7765}, {3845, 9300}, {3850, 5305}, {3851, 6287}, {3853, 9606}, {3855, 5319}, {3861, 9607}, {3933, 7903}, {4045, 7841}, {4193, 5277}, {5028, 5480}, {5046, 5283}, {5054, 5210}, {5066, 5306}, {5640, 6787}, {5691, 9619}, {5913, 8585}, {6292, 7784}, {6655, 7786}, {6656, 7808}, {6658, 7782}, {6680, 7887}, {6683, 7791}, {7517, 9700}, {7526, 9608}, {7530, 9609}, {7533, 9465}, {7547, 8743}, {7622, 8598}, {7750, 7815}, {7751, 7762}, {7754, 7838}, {7760, 7921}, {7763, 7816}, {7768, 7900}, {7776, 7794}, {7778, 7820}, {7787, 7828}, {7789, 7888}, {7792, 7844}, {7795, 7821}, {7796, 7941}, {7797, 7878}, {7800, 7873}, {7802, 7824}, {7803, 7861}, {7807, 7862}, {7813, 9766}, {7814, 7836}, {7819, 7867}, {7829, 7851}, {7832, 7912}, {7846, 7901}, {7859, 7933}, {7866, 7889}, {7876, 7911}, {7892, 7899}, {7935, 8362}, {8035, 8036}, {9115, 9761}, {9117, 9763}
X(5475) = reflection of X(574) in X(3815)
Let A′ be the intersection, other than X(4), of the A-altitude and the orthocentroidal circle, and define B′ and C′ cyclically. The triangle A′B′C′, introduced here as the orthocentroidal triangle, is inversely similar to ABC, with center X(6) of similitude. If, in that definition, X(4) replaced by X(2) and A-altitude by A-median, the resulting triangle is the 4th Brocard triangle. Regarding a point X as a function of a triangle, X(A′B′C′) - that is, X of A′B′C′ - is the reflection of X(ABC) in the centroid of the pedal triangle of X. X(5476) = X(182) of the orthocentroidal triangle, and X(5476) = X(182) of the 4th Brocard triangle. Also, X(5476) = inverse-in-Kiepert-hyperbola of X(5475), and X(5476) = {X(13),X(14)}-harmonic conjugate of X(5475). (Randy Hutson, May 7, 2013)
Balrycenjtrics for the vertices of the orthocentroidal triangle are given by Peter
Moses (June 17, 2014): A′ = a2 : a2 +
b2 - c2 : a2 - b2 +
c2
B′ = b2 - c2 + a2 : b2 :
b2 + c2 - a2
C′ = c2 + a2 - b2 : c2 -
a2 + b2 : c2
The orthocentroidal triangle is the circumsymmedial triangle of the 4th Brocard triangle. (Randy Hutson, November 22, 2014)
Another construction of the orthocentroidal triangle: Let NaNbNc and Na'Nb'Nc' be the inner and outer Napoleon triangles, resp. Then A′ = NbNc'∩NcNb', and B′ and C′ are constructed cyclically; i.e, A′B′C′ is the cross-triangle of the inner and outer Napoleon triangles. Also, A′ is the center of inverse similitude of ABC and the A-altimedial triangle, and cyclically for B′ and C′. (Randy Hutson, December 10, 2016)
X(5476) lies on these lines: {2, 51}, {4, 575}, {5, 524}, {6, 13}, {30, 182}, {67, 7579}, {69, 5071}, {141, 547}, {376, 3618}, {403, 8541}, {546, 8550}, {549, 3098}, {569, 7540}, {599, 1351}, {1350, 5054}, {1352, 1992}, {1469, 3582}, {1503, 3845}, {1974, 7576}, {1995, 5642}, {2030, 7737}, {2781, 5946}, {3056, 3584}, {3066, 5972}, {3090, 7922}, {3091, 5032}, {3095, 7801}, {3329, 9993}, {3534, 5085}, {3564, 5066}, {3574, 9977}, {3830, 5050}, {3839, 6776}, {4663, 9955}, {5038, 7748}, {5039, 5306}, {5093, 5965}, {5107, 7603}, {5169, 9140}, {5171, 8359}, {5448, 8548}, {5459, 7685}, {5460, 7684}, {5463, 5615}, {5464, 5611}, {5569, 7606}, {5654, 8681}, {6032, 6792}, {6791, 9745}, {7533, 9143}, {7552, 9781}, {7951, 8540}, {8369, 9737}, {8538, 10024}, {9760, 9763}, {9761, 9762}, {9771, 10011}
X(5476) = centroid of reflection triangle of X(182)
X(5477) lies on these lines: {6, 13}, {30, 5107}, {39, 8550}, {69, 620}, {98, 5034}, {99, 193}, {110, 6388}, {111, 9143}, {114, 230}, {126, 5468}, {147, 5304}, {187, 524}, {511, 1569}, {543, 1992}, {575, 1506}, {576, 7747}, {597, 7603}, {599, 9167}, {671, 5032}, {690, 5095}, {1353, 2782}, {1384, 8724}, {1503, 1570}, {1648, 5642}, {1691, 5965}, {2458, 4027}, {2502, 6791}, {2549, 2794}, {2796, 4856}, {3618, 6722}, {3629, 5969}, {3815, 6055}, {5038, 9698}, {5050, 6036}, {5093, 6321}, {6054, 7735}, {7837, 8289}, {8584, 9830}
X(5477) = {X(5471), X(5472)}-harmonic conjugate of X(115) (Randy Hutson, May 7, 2013)New pair of equilateral triangles (Vu Thanh Tung, November 25, 2019):
Let A0B0C0 be the orthic triangle of ABC. Let A15 = X(15)-of-A0B0C0, and define B15 and C15 cyclically, so that . A15B15C15 is the 3rd isodynamic-Dao equilateral triangle of ABC.
Let A′15 be the point, other than A, in which the line AA15 meets the circle {{A,B0,C0}}, and define B′15 and C′15 cyclically. The triangle A′15B′15C′15 is here named the Vu-Dao-X(15)-isodynamic equilateral triangle.
If X(15) is replaced by X(16) in the above construction, the resulting triangle, A′16B′16C′16, is here named the Vu-Dao-X(16)-isodynamic equilateral triangle. The centers of the two equilateral triangles are X(5478) and X(5479), respectively, and the diameters of the circumcircles of the two triangles are the segments X(4)X(13) and X(4)X(14), respectively.
Barycentrics and properties (Peter Moses, November 27, 2019):
A′15 = -2*(a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) + 2*Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) + 2*S) : :
A′16 = -2*(a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) + 2*Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) - 2*S) : :
With perspector X(13), the triangle A′15B′15C′15 is perspective to the following triangles: ABC, outer Fermat (TCCT p178), 1st isodynamic-Dao equilateral (see X(16802)), 3st isodynamic-Dao equilateral (see X(31683)), 1st half-diamonds triangle (see X(33338)), and 2nd Lemoine-Dao equilateral triangle (see X(16940)).
X(5478) lies on these lines: {2, 5473}, {3, 6669}, {4, 13}, {5, 618}, {30, 5459}, {98, 5470}, {107, 473}, {115, 5318}, {381, 530}, {383, 6115}, {531, 9880}, {542, 1353}, {616, 3091}, {624, 3734}, {1080, 6108}, {1598, 9916}, {1699, 9901}, {2043, 6306}, {2044, 6302}, {3545, 5463}, {5321, 5472}, {5334, 9112}, {5469, 6773}, {5603, 7975}, {5613, 6321}, {6201, 6268}, {6202, 6270}, {9982, 9993}
X(5478) = midpoint of X(4) and X(13)
X(5478) = complement of X(5473)
X(5478) = X(13)-of-Euler-triangle
X(5478) = center of circle {{A′15, B′15, C′15, X(4), X(13), X(16179)}}
X(5478) = {X(3845),X(5480)}-harmonic conjugate of X(5479)
See X(5478).
X(5479) lies on these lines: {2, 5474}, {3, 6670}, {4, 14}, {5, 619}, {30, 5460}, {98, 5469}, {107, 472}, {115, 5321}, {381, 531}, {383, 6109}, {530, 9880}, {542, 1353}, {617, 3091}, {623, 3734}, {1080, 6114}, {1598, 9915}, {1699, 9900}, {2043, 6303}, {2044, 6307}, {3545, 5464}, {5318, 5471}, {5335, 9113}, {5470, 6770}, {5603, 7974}, {5617, 6321}, {6201, 6269}, {6202, 6271}, {9981, 9993}
X(5479) = midpoint of X(4) and X(14)
X(5479) = complement of X(5474)
X(5479) = X(14)-of-Euler-triangle
X(5479) = {X(3845),X(5480)}-harmonic conjugate of X(5478)
Let O′ denote the orthosymmedial circle, introduced here as the circle having segment X(4)X(6) as diameter, so that X(5480) is the center of O′. Note that O′∩(Euler line) = {X(4), X(1316)} and O′∩(Brocard circle) = {X(6), X(1316)}; X(5480) = X(6) of Euler triangle; X(5480) = {X(5478),X(5479)}-harmonic conjugate of X(3845); The orthosymmedial circle is the inverse-in-polar-circle of the line X(297)X(525). (Randy Hutson, August 26, 2014)
X(5480) lies on these lines: {2, 1350}, {3, 3589}, {4, 6}, {5, 141}, {11, 1469}, {12, 3056}, {20, 3618}, {30, 182}, {51, 125}, {52, 7403}, {66, 3527}, {69, 3091}, {86, 7385}, {98, 5306}, {113, 2854}, {114, 5969}, {115, 5052}, {118, 2810}, {119, 9024}, {140, 3098}, {154, 6995}, {159, 1598}, {184, 428}, {185, 1907}, {193, 3832}, {206, 578}, {230, 5017}, {235, 1843}, {262, 1513}, {265, 9970}, {323, 7533}, {343, 3060}, {355, 5846}, {381, 524}, {382, 5050}, {383, 396}, {389, 1595}, {394, 6997}, {395, 1080}, {403, 6403}, {515, 1386}, {516, 4085}, {518, 946}, {542, 1353}, {546, 576}, {550, 5092}, {567, 7540}, {569, 7553}, {575, 3627}, {590, 6813}, {599, 3545}, {611, 1479}, {613, 1478}, {615, 6811}, {674, 7680}, {698, 3095}, {732, 6248}, {858, 5640}, {966, 7407}, {1213, 7380}, {1368, 5943}, {1428, 7354}, {1482, 9053}, {1593, 5894}, {1594, 9781}, {1596, 2393}, {1692, 7747}, {1699, 3751}, {1848, 1864}, {1853, 7378}, {1861, 2262}, {1890, 2182}, {1899, 5064}, {1906, 6467}, {1974, 3575}, {1992, 3839}, {1993, 7394}, {2051, 4260}, {2330, 6284}, {2550, 5782}, {2552, 8115}, {2553, 8116}, {3054, 5104}, {3068, 7000}, {3069, 7374}, {3088, 6696}, {3089, 7716}, {3090, 3763}, {3146, 7864}, {3242, 5603}, {3313, 7399}, {3416, 5587}, {3531, 5486}, {3580, 5169}, {3619, 5056}, {3620, 5068}, {3630, 3850}, {3631, 3851}, {3656, 9041}, {3796, 7500}, {3827, 7686}, {3843, 5093}, {3858, 5965}, {4026, 6210}, {4259, 6831}, {4265, 6905}, {4549, 9818}, {5028, 5475}, {5034, 7748}, {5039, 5305}, {5096, 6906}, {5188, 8362}, {5422, 7391}, {5718, 8229}, {5805, 5845}, {5999, 7792}, {6393, 7752}, {6747, 6755}, {6800, 7519}, {7496, 7605}, {7681, 8679}, {7735, 9748}, {8359, 8722}, {8721, 9605}, {9019, 9967}, {9300, 9744}, {9830, 9880}
X(5480) = isogonal conjugate of X(5481)
X(5480) = complement of X(1350)
X(5480) = crosspoint of X(4) and X(262)
X(5480) = crosssum of X(3) and X(182)
X(5480) = inverse-in-Jerabek-hyperbola of X(51)
X(5480) = Johnson-to-Ehrmann-mid similarity image of X(6)
X(5480) = X(1001)-of-orthic-triangle if ABC is acute
X(5480) = {X(34221),X(34222)}-harmonic conjugate of X(51)
X(5481) lies on the hyperbola {A,B,C,X(2),X(3)} and these lines: {2, 1629}, {216, 1297}, {394, 5012}, {1073, 7484}, {1078, 3523}
X(5481) = isogonal conjugate of X(5480)
X(5481) = cevapoint of X(3) and X(182)
Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). Let R be the radical center of the circles (A′, |A′B|, {B′,|B′C|), (C′, |C′A|), and let S be the radical center of the circles (A′,|A′C|), (B′,|B′A|), (C′,|C′B|). X(5482) is the midpoint of the segment RS. (Antreas Hatzipolakis, May 4, 2013)
X(5482) is the {X(3),X(1764)}-harmonic conjugate of X(3579) (Peter Moses, May 13, 2013)
For the construction and generalizations, see Hechos Geométricos en el Triángulo.
X(5482) lies on these lines: {1, 3}, {140, 143}, {549, 970}, {631, 5752}, {916, 6705}, {3819, 6675}, {3834, 9955}, {3917, 7483}{1,3}, {140,143}, {549,970}
Let A′B′C′ be the cevian triangle of the incenter. Let AB = (reflection of A′ in BB′), and define BC and CA cyclically. Let AC = (reflection of A′ in CC′), and define BA and CB cyclically. The ellipse passing through the points AB, AC, BC, BA, CA, CB is here introduced as the Hutson Ellipse, and X(5483) is its center. (Antreas Hatzipolakis, May 17, 2013)
X(5483) lies on these lines: {1, 5180}, {81, 593}, {226, 1029}, {3920, 5988}
Let A′B′C′ be the cevian triangle of point X. Let AB = (reflection of A′ in BB′), and define BC and CA cyclically. Let AC = (reflection of A′ in CC′), and define BA and CB cyclically. Let HA = (orthocenter of A′ABAC), and define HB and HC cyclically. The orthocentric triangle of X is here introduced as the (central) triangle HAHBHC. (Antreas Hatzipolakis, May 17, 2013)
For X = X(1), the orthocentric triangle HAHBHC is perspective to the anticomplementary triangle, and X(5484) is the perspector. (Peter Moses, May 17, 2013)
Also, HAHBHC is perspective to ABC at X(10). (Randy Hutson, May 18, 2013)
X(5484) is the crosspoint of X(1) and X(8) with respect to the extraversion triangle of X(8). (Randy Hutson, August 26, 2014)
X(5484) lies on these lines: {1, 1330}, {2, 12}, {8, 38}, {10, 1054}, {69, 145}, {519, 2891}, {1469, 3869}, {1626, 4189}, {2345, 9597}, {3622, 5712}, {3662, 4327}, {5263, 7354}
X(5484) = anticomplement of X(1220)
Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The Euler lines of the triangles AHBHC, BHCHA, CHAHB concur in the Kirikami-Euler image of P. Let P be given by barycentrics p : q : r. Then Q is given by
Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/[(a2 - b2 - c2)p2 + (a2 - b2 + c2)pq + (a2 + b2 - c2)pr + 2a2qr]
If P = X(2), then Q = X(5485). (Seichii Kirikami, May 20, 2013)
The Kirikami-Euler image K(P) of a point P is related to the mapping H(P) called "pedal antipodal perspector", defined in Hyacinthos #20403 and #20405, November 2011, by Randy Hutson, with general coordinates given in #20404 by Francisco Javier. X(5485) = H(X(i)) for I = 6 and I = 187. In general, K(P) = H(P′) = H(P*), where P′ denotes the isogonal conjugate of P, and P* = (inverse-in-circumcircle of P′); for example, K(X(1)) = X(8), K(X(3)) = X(68), and K(X(6)) = X(5486). (Randy Hutson, May 22, 2013)
If you have The Geometer's Sketchpad, you can view X(5485).
X(5485) lies on the Kiepert hyperbola and these lines: {2, 2418}, {4, 524}, {10, 4419}, {30, 3424}, {69, 671}, {98, 376}, {115, 5503}, {262, 538}, {459, 5523}, {525, 5466}, {598, 1992}, {631, 1153}, {1327, 5861}, {1328, 5860}, {2799, 9180}, {2996, 7841}, {3090, 7608}, {3524, 7610}, {3533, 7619}, {3590, 7389}, {3591, 7388}, {3855, 7775}, {5032, 5395}, {8587, 8591}, {8781, 9166}
X(5485) = isogonal conjugate of X(1384)
X(5485) = isotomic conjugate of X(1992)
X(5485) = pole wrt polar circle of trilinear polar of X(4232)
X(5485) = X(48)-isoconjugate (polar conjugate) of X(4232)
Using the notation at X(5485) for Kirikami-Euler image and pedal antipodal perspector, X(5486) = K(X(6)) = H(X(2)) = H(X(23)). Also, X(5485) is the trilinear pole of the line X(647)X(690). (Randy Hutson, May 22, 2013)
X(5486) lies on the Jerabek hyperbola and these lines: {2, 895}, {3, 524}, {4, 2393}, {6, 468}, {66, 6467}, {67, 1899}, {68, 8681}, {69, 3266}, {71, 4062}, {72, 9004}, {74, 6776}, {125, 5505}, {140, 8548}, {141, 6391}, {182, 5504}, {184, 1177}, {193, 1176}, {248, 5063}, {265, 1352}, {511, 4846}, {523, 2549}, {879, 8675}, {1173, 3542}, {1503, 3426}, {1992, 7493}, {2435, 9007}, {3521, 9019}, {3531, 5480}, {9145, 9516}
X(5486) = isogonal conjugate of X(1995)
X(5487) lies on these lines: {13, 633}, {14, 627}, {3620, 5488}
X(5487) = isotomic conjugate of anticomplement of X(34540)X(5488) lies on these lines: {13, 628}, {14, 634}, {3620, 5487}
X(5488) = isotomic conjugate of anticomplement of X(34541)X(5489) lies on these lines: {3, 525}, {4, 523}, {39, 647}, {140, 5664}, {520, 5562}, {669, 2353}, {684, 6334}, {690, 9409}, {826, 3574}, {868, 6070}, {2409, 5502}, {3265, 3926}
X(5489) = crossdifference of every pair of pints on line X(23)X(232)
X(5490) lies on these lines: {2, 493}, {4, 488}, {10, 5391}, {69, 485}, {76, 5590}, {83, 3069}, {98, 637}, {99, 6568}, {141, 5491}, {486, 641}, {487, 7612}, {491, 3316}, {1131, 1270}, {1132, 3593}, {1271, 3590}, {6504, 8223}
X(5490) = pole wrt polar circle of trilinear polar of X(5200)
X(5490) = X(48)-isoconjugate (polar conjugate) of X(5200)
X(5491) lies on these lines: {2, 494}, {4, 487}, {10, 1267}, {69, 486}, {76, 5591}, {83, 3068}, {98, 638}, {99, 6569}, {141, 5490}, {485, 642}, {488, 7612}, {492, 3317}, {1131, 3595}, {1132, 1271}, {1270, 3591}, {6504, 8222}
X(5490) = pole wrt polar circle of trilinear polar of X(5200)
X(5490) = X(48)-isoconjugate (polar conjugate) of X(5200)
Let T be the orthocentric triangle HAHBHC of X(1), as defined at X(5484). X(5492) is the orthocenter of T, and T is perspective to the Fuhrmann triangle with perspector X(1), and T is perspective to the anticomplementary triangle, with perspector X(5484). (Peter Moses, May 17, 2013)
T is similar to the incentral triangle, with center of similitude I, the incenter. Let A″B″C″ be the antipedal triangle of X(1) with repect to the incentral triangle. X(5492) is the nine-point center of A″B″C″. (The triangle A″B″C″ is also the triangle formed by the lines LA, LB, LC, where LA is the polar of A with respect to the circle BCI, and LB and LC are defined cyclically.) (Randy Hutson, May 18, 2013)
X(5492) lies on these lines: {1, 399}, {3, 846}, {5, 3120}, {30, 2292}, {58, 3652}, {355, 2783}, {381, 986}, {405, 7986}, {500, 3743}, {774, 6147}, {1725, 3649}, {1772, 3614}, {1962, 5453}, {3670, 9955}, {3724, 6097}
X(5492) = reflection of X(3743) in X(500)
X(5493) is the center of the circle Y that bisects each of the three excircles of ABC. Let J be the radius of Y; then 4J2 = r2 + 16rR + 64R2 - 7s2. (Paul Yiu, Francisco Javier, AdvPlGeom, May 17, 2013)
X(5493) lies on these lines: {1, 3522}, {2, 9589}, {3, 551}, {4, 9}, {8, 5059}, {20, 519}, {30, 4669}, {55, 3671}, {56, 4342}, {65, 4314}, {140, 946}, {144, 4882}, {165, 962}, {355, 5073}, {376, 7982}, {382, 3654}, {390, 3339}, {411, 5537}, {484, 1210}, {497, 5128}, {515, 1657}, {517, 550}, {527, 3913}, {553, 3303}, {595, 9441}, {758, 7957}, {1656, 3817}, {1697, 3474}, {1698, 5068}, {1699, 3634}, {1770, 5270}, {1788, 9580}, {1836, 3947}, {2093, 4294}, {2951, 5850}, {3057, 4315}, {3091, 3828}, {3146, 3679}, {3295, 5542}, {3361, 9785}, {3428, 5267}, {3516, 8193}, {3517, 9911}, {3524, 9624}, {3529, 5881}, {3533, 8227}, {3543, 4745}, {3555, 5918}, {3600, 7320}, {3626, 5691}, {3635, 5731}, {3636, 7987}, {3663, 5255}, {3746, 7411}, {3811, 7994}, {3854, 9780}, {3858, 9956}, {3878, 9858}, {3931, 4349}, {3962, 6154}, {4015, 5927}, {4067, 6001}, {4192, 9569}, {4229, 4658}, {4292, 5119}, {4304, 5903}, {4311, 5697}, {4356, 5711}, {4525, 5693}, {4848, 5183}, {4866, 6172}, {5248, 5584}, {5325, 9710}, {5904, 9961}, {6766, 9841}, {7580, 8715}
X(5493) = reflection of X(i) in X(j) for these (i,j): (10,40), (946,3579), (962,1125), (3244,4297), (3543,4745), (4301,3)
X(5493) = complement of X(9589)
Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). Let AB be the reflection of A′ in line BB′, and define BC and CA cyclically. Let AC be the reflection of A′ in line CC′, and define BA and CB cyclically. Let L be the Euler line of ABC, let LA be the Euler line of AABAC, and define LB and LC cyclically. Let MA be the reflection of LA in AA′, and define MB and MC cyclically. The lines MA, MB, MC concur in X(5494). Moreover, the four Euler lines L, LA, LB, LC are parallel, concurring in X(30). (Antreas Hatzipolakis, May 25, 2013)
For the construction and discussion, see
Hechos Geométricos en el Triángulo.X(5494) lies on these lines: {1,2779}, {21,104}, {36,1725}, {65,74}, {125,860}
Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). Let LA be the line through A′ perpendicular to line AA′, and define LB add LC cyclically. Let
UA = reflection of LA in AA′
UB = reflection of LA in BB′
UC = reflection of LA in CC′
VA = reflection of LB in AA′
VB = reflection of LB in BB′
VC = reflection of LB in CC′
WA = reflection of LC in AA′
WB = reflection of LC in BB′
WC = reflection of LC in CC′
TA = triangle formed by the lines in UA,
UB, UC
TB = triangle formed by the lines in VA,
VB, VC
TC = triangle formed by the lines in WA,
WB, WC
OA = circumcenter of TA, OB = circumcenter of TA, OC = circumcenter of TA, O = X(3) = circumcenter of ABC. The points O, OA, OB, OC are concyclic. The center of their circle is X(5495). (Antreas Hatzipolakis, May 28, 2013)
For the construction and discussion, see Concyclic Circumcenters.
X(5495) lies on these lines: {1, 30}, {3, 143}, {511, 6097}, {6102, 7416}
Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). Let LA be the line through A′ perpendicular to line AA′, and define LB add LC cyclically. Using the notation at X(5495), let MAB be the line parallel to UB through B′, and let MAC be the line parallel to UC through C′. Let A″ = MAB∩MAC, and define B″ and C″ cyclically. Let OA = circumcenter of A″B′C′, and define OB and OC cyclically. Then the points X(1), OA, OB, OC are concyclic, and the center of their circle is X(5496). (Antreas Hatzipolakis, May 29, 2013)
For a discussion, see Concurrent Circles.
X(5496) lies on these lines: {1, 21}, {3191, 3678}, {3682, 3841}, {3724, 5903}, {4511, 4647}, {6757, 7073}
Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). The circles OA, OB, OC defined at X(5496) concur in X(5497). (Antreas Hatzipolakis, May 29, 2013)
For a discussion, see Hechos Geométricos en el Triángulo.
X(5497) lies on these lines: {1, 149}, {37, 101}, {1283, 2292}, {1331, 1780}, {3326, 9629}
Let ABC be a triangle, let NA be the nine-point center of the triangle BCO, where O = X(3), and define NB and NC cyclically. The nine-point center of the triangle NANBNC is X(5498), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)
X(5498) lies on this line: {2,3}
Let IA be the A-excenter of a triangle ABC and let NA be the nine-point center of IABC. Define NB and NC cyclically. The circumcenter of NANBNC is X(5499), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)
Let A′B′C′ be the Feuerbach triangle, and let A″ be the reflection of X(11) in line B′C′; define B″ and C″ cyclically. Then A″, B″, C″ are collinear, and their line, X(12)X(79) is here named the Feuerbach line. X(5499) is the point of intersection of the Feuerbach line and the Euler line. (Randy Hutson, August 26, 2014)
X(5499) lies on these lines: {2, 3}, {10, 2771}, {11, 5441}, {12, 79}, {119, 3652}, {495, 3649}, {758, 5690}, {1329, 3647}, {1385, 1484}, {1698, 7701}, {1749, 5445}, {1834, 5453}, {3579, 3822}, {3884, 6701}, {9943, 9956}
Let A′B′C′ be the antipedal triangle of the nine-point center, N = X(5) of a triangle ABC. Let NA be the nine-point center of NB′C′, and define NB and NC cyclically. The nine-point center of NANBNC is X(5500), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)
X(5500) lies on this line: (2,3}
X(5500) = reflection of X(5) in X(10286)As a point on the Euler line, X(5501) has Shinagawa coefficients (18E2 + 128EF + 192F2 + 64S2, -54E2 - 128EF - 64F2 + 320S2).
Let N be a the nine-point center of triangle ABC. Let NA be the nine-point center of NBC, and define NB and NC cyclically. The circumcenter of NANBNC is X(5501), which lies on the Euler line of ABC. (Antreas Hatzipolakis, June 2, 2013)
See For a discussion, see Hechos Geométricos en el Triángulo.
X(5501) lies on these lines: X(5501) lies on these lines: {2, 3}, {137, 8254}, {1154, 31879}, {6150, 23280}, {6592, 31376}, {10095, 20327}, {13372, 23281}, {13856, 15425}, {14140, 15307}, {20414, 32551}, {21230, 24573}, {24385, 36842}, {25150, 34598}, {25340, 32423}, {34597, 34768}
X(5501) = midpoint of X(i) and X(j) for these {i, j}: {3, 20120}, {4, 14142}, {5, 10285}, {10095, 20327}, {10205, 28237}, {20030, 36837}, {20414, 32551}
X(5501) = reflection of X(i) in X(j) for these (i, j): (2, 25403), (3, 15327), (4, 25404), (5, 15957), (546, 19940), (10126, 3628), (10289, 13469), (13856, 15425)
X(5501) = complement of X(10205)
X(5501) = anticomplement of the anticomplement of X(12056)
X(5501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3628, 27090, 140), (10289, 13469, 547)
Let L be the Euler line of a triangle ABC. Let LA be the reflection of L in line BC, and define LB and LC cyclically. Let A′ = L∩BC, and define B′ and C′ cyclically. The circles whose diameters are the segments AA′, BB′, CC′ are coaxial. Let D be their coaxial axis (the line X(4)X(74)); let DA be the reflection of D in line BC, and define DB and DC cyclically. Let HA = LB∩DC, and define HB and HC cyclically. Let MA = LC∩DB, and define MB and MC cyclically. The triangles HAHBHC and MAMBMC are perspective, and their perspector is X(5502). (Antreas Hatzipolakis, June 3, 2013)
For a discussion, see Hechos Geométricos en el Triángulo.
X(5502) lies on these lines: {3,64}, {110, 351}
Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. Let OA be the circle of the points A, HB, HC, and define OB and OC cyclically. The circles OA, OB, OC concur in a point Q, the Kirikami concurrent circles image of P. Let P be given by barycentrics p : q : r. Then Q given by
Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/(2 a^4 q r(p+q) (p+r)+a^2 p (c^2 q (p+q-3 r) (p+r)+b^2 (p+q) r (p-3 q+r))-p (b^4 (p+q) (p-q-r) r-c^4 q (p+r) (-p+q+r)+b^2 c^2 (p^2 (q+r)+2 q r (q+r)+p (q^2+r^2)))) .
If P = X(2), then Q = X(5503). (Seichii Kirikami, June 2, 2013)
If P is on the circumcircle, then Q(P) = P. This follows from the fact that the denominators of g(a,b,c,p,q,r) and g(b,c,a,q,r,p} are polynomial multiples of a2qr + b2rp+c2pq. (Seichii Kirikami, July 27, 2013)
If you have The Geometer's Sketchpad, you can view X(5503).
X(5503) lies on these lines: {4,543}, {98,524}, {99,598}, {115,5485}, {325,671}, {542,3424}, {2799,5466}, {3407,5182}
X(5503) = reflection of X(5485) in X(115)
X(5503) = isotomic conjugate of X(22329)
See X(5503) for the definition of Kirikami concurrent circles image Q of a point P and X(5509) for an occurrence of X(5504) as a point of concurrence given by the Hatzipolakis-Moses Theorem.
If P = X(3), then Q = X(5504). (Seichii Kirikami, June 2, 2013)
Let A′B′C′ be the tangential triangle. Let LA be the line through A′ parallel to the Euler line, and define LB and LC cyclically. Let RA be the reflection of LA in BC, and define RB and RC cyclically. The lines RA, RB, RC concur in X(5504); see X(399). (Randy Hutson, August 17, 2014)
Continuing, let A″ be the reflection of A′ in line BC, and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5504). Morevoer, X(5504) is the antigonal image of X(68), the trilinear pole of line X(577)X(647), and the X(92)-isoconjugate of X(3003). (Randy Hutson, August 17, 2014)
X(5504) lies on the Jerabek hyperbola and these lines: {3,974}, {4,110}, {6,1511}, {20,3047}, {49,3521}, {64,155}, {66,542}, {67,3564}, {68,125}, {70,3448}, {74,323}, {182,5486}, {184,4846}, {265,2072}, {290,1236}, {399,3167}, {511,1177}, {1069,3024}, {1986,1993}, {2850,3657}, {3028,3157}, {3431,5012}
X(5504) = reflection of X(i) in X(j) for these (i,j): (110,1147),
(68,125), (2931, 1511)
X(5504) = isogonal conjugate of X(403)
If P = X(6), then Q = X(5505). (Seichii Kirikami, June 2, 2013)
X(5505) lies on these lines: {3,2854}, {72,3908}, {74,2393}, {125,5486}, {265,524}, {323,895}, {542,4846}, {1177,1495}, {2781,3426}
X(5505) = reflection of X(5486) in X(125)
Let I be the incenter of a triangle ABC. Let NA be the nine-point center of IBC, and define NB and NC cyclically. The triangle NANBNC is X(5501)is both similar to and perspective to the excentral triangle of ABC. The perspector is X(5506). (Barry Wolk, June 1, 2013)
X(5506) lies on these lines: {1,748}, {2,191}, {5,40}, {9,583}, {10,149}, {140,1768}, {405,5426}, {411,2951}, {484,3634}, {1006,1490}, {1045,3216}, {1385,5251}, {2136,3679}, {2950,5316}, {3219,3337}, {3647,5131}, {3740,3746}, {5044,5259}
X(5506) = X(1173)-of-excentral-triangle
Let OA be the circle with center A and radius R, the circumradius of triangle ABC. Let BA be the point where OA meets line AB nearest to B. Define CB and AC cyclically. Let CA be the point where OA meets line AC nearest to C. Define AB and BC cyclically. X(5507) is the radical center of the circles ABACA, BCBAB, CACBC. If "nearest to" is changed to "farthest from" in the construction, the resulting point is X(600). See also X(600). (Peter Moses, June 19, 2013)
If you have The Geometer's Sketchpad, you can view X(5507).
X(5507) lies on this line: {600, 4640}
Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The circumcircles of HABC, HBCA, HCAB concur in a point Q, the Kirikami concurrent circles image (2nd kind) of P; see X(5503). Let P be given by barycentrics p : q : r. Then Q is given by
Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/[(a2 - b2 - c2)p2 + a2qr + (a2 - b2)pq + (a2 - c2)pr].
If P = X(31), then Q = X(5508). (Seichii Kirikami, July 2, 2013)
The barycentrics for Q show that "concurrent circles image (2nd kind)" is the same as "antigonal image". (Randy Hutson, July 15, 2013)
If you have The Geometer's Sketchpad, you can view X(5508).
X(5508) lies on these lines: {31, 5509}, {815, 2887}
The Kirikami six circles image of P is simply the center of the rectangular hyperbola {{A, B, C, X(4), P}}.
Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The nine-point circles of the six triangles HABC, HBCA, HCAB, AHBHC, BHCHA, CHAHB concur in a point, Q = Q(P), the Kirikami six-circles-image of P. Let P be given by barycentrics p : q : r. Then Q(P) is given by
Q = h(a,b,c,p,q,r) : h(b,c,a,q,r,p) : h(c,a,b,r,p,q), where h(a,b,c,p,q,r) = p[(a2 - b2 + c2)q - (a2 + b2 - c2)r][(pr + qr)b2 - (pq + rq)c2].
The point Q lies on the nine-point circle of ABC. If P = X(31), then Q = X(5509). If P = X(1), then Q = X(11); if P = X(2), then Q = X(115); if P = X(3), then Q = X(125). (Seichii Kirikami, July 2, 2013)
Q maps each right circumhyperbola onto its center. Special cases: Q maps the Feuerbach hyperbola onto X(11), the Kiepert hyperbola onto X(115), and the Jerabek hyperbola onto X(125). (Peter Moses, July 7, 2013)
The Kirikami six-circles-image, Q(P), of a point P is also the point of concurrence of the nine-point circles of BCP, CAP, ABP (these being the same as the nine-point circles of BCHA, CAHB, ABHC). Also, Q(P) is the center of the rectangular circumhyperbola passing through P, and Q(P) lies on the cevian circle of P. (Randy Hutson, July 15, 2013)
The Kirikami six-circles-image of P is also the QA-P2 center (Euler-Poncelet Point) of the quadrangle ABCP; see Encyclopedia of Quadri-Figures.
If you have The Geometer's Sketchpad, you can view X(5509).
The Kirikami six circles configuration led to a conjecture by Antreas Hatzipolakis (July 5, 2013), proved by Peter Moses, and stated here as the Hatzipolakis-Moses Theorem: Suppose that P and P* are an isogonal conjugate pair of points in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. Let H'A be the orthocenter of triangle P*BC, and define H'B and H'C cyclically. Then circumcircles of HAHBHC concur and the circumcircles of H'AH'BH'C concur.
The known proof of the theorem depends on a Mathematica program that runs for several minutes. Barycentrics for most choices of P are too long to be included here. An exception is P = X(3), for which P* = X(4) and the two points of concurrence are H(3) = X(265) and H(4) = X(5504).
See Hyacinthos #21992.
Let P′ be the isogonal conjugate of P. Then the Kirikami-six-circles image of P is the orthopole of line X(3)P′, which is also the crosssum of the circumcircle intercepts of line X(3)P′. (Randy Hutson, March 29, 2020)
X(5509) lies on these lines: {2,185}, {31,5508}, {115,3271}
X(5509) = crosssum of circumcircle intercepts of line X(3)X(75)
X(5509) = orthopole of line X(3)X(75)
X(5509) = center of hyperbola {{A,B,C,X(4),X(31)}}
Kirikami six-circles-images are introduced at X(5509).
X(5510) is the homothetic center of the cyclic quadrilateral ABCX(106) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5510) is the anticenter of ABCX(106) (Randy Hutson, July 15, 2013)
Let A′B′C′ be the orthic triangle. Let LA be the Nagel line of AB′C′, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5510). Also, X(5510) = X(3)-of-A″B″C″. See X(8054). (Randy Hutson, August 12, 2015)
X(5510) lies on these lines: {2, 1293}, {4, 106}, {5, 121}, {11, 1357}, {113, 2842}, {114, 2796}, {115, 2789}, {116, 2821}, {117, 2841}, {118, 2810}, {119, 946}, {120, 3817}, {124, 2815}, {125, 2776}, {132, 2844}, {133, 2839}, {1054, 1699}, {2051, 3030}, {2886, 3038}, {3667, 3756}
X(5510) = midpoint of X(4) and X(106)
X(5510) = reflection of X(121) in X(5)
X(5510) = complement of X(1293)
X(5510) = crosssum of circumcircle intercepts of line X(3)X(519)
X(5510) = orthopole of line X(3)X(519)
X(5510) = center of hyperbola {{A,B,C,X(4),X(106)}}
X(5510) = X(106)-of-Euler-triangle
X(5510) = polar-circle-inverse of X(32704)
Kirikami six-circles-images are introduced at X(5509).
X(5511) is the homothetic center of the cyclic quadrilateral ABCX(105) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5511) is the anticenter of ABCX(105) (Randy Hutson, July 15, 2013)
X(5511) = X(105)-of-Euler-triangle (Randy Hutson, August 17, 2014)
X(5511) = reflection of X(120) in X(5)
X(5511) = midpoint of X(4) and X(105)
X(5511) = complement of X(1292)
X(5511) lies on these lines: {2, 1292}, {4, 105}, {5, 120}, {11, 1111}, {12, 3021}, {113, 2836}, {114, 2795}, {115, 2788}, {116, 2820}, {117, 2835}, {118, 946}, {119, 381}, {124, 2814}, {125, 2775}, {132, 2838}, {133, 2833}, {1596, 2834}, {2051, 3034}, {2886, 3039}, {3309, 4904}
X(5511) = crosssum of circumcircle intercepts of line X(3)X(518)
X(5511) = orthopole of line X(3)X(518)
X(5511) = center of hyperbola {{A,B,C,X(4),X(105)}}
Kirikami six-circles-images are introduced at X(5509).
X(5512) is the homothetic center of the cyclic quadrilateral ABCX(111) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5512) is the anticenter of ABCX(111) (Randy Hutson, July 15, 2013)
Let A′B′C′ be the orthic triangle. Let La be line X(2)X(6) of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5512), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)
X(5512) lies on these lines: {2, 1296}, {4, 111}, {5, 126}, {11, 2830}, {54, 3048}, {113, 2854}, {114, 381}, {115, 2793}, {116, 2824}, {117, 2852}, {118, 2813}, {119, 2805}, {124, 2819}, {125, 2780}, {132, 1596}, {133, 2847}, {1499, 2686}
X(5512) = X(111)-of-Euler-triangleKirikami six-circles-images are introduced at X(5509).
Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5513) is the inverse-in-O* of X(101). (Randy Hutson, August 17, 2014)
X(5513) lies on the nine-point circle, the Yff contact circle, and these lines: {2, 101}, {9, 124}, {11, 37}, {115, 3136}, {118, 4120}, {125, 1213}, {127, 440}, {427, 5190}, {430, 5139}, {3259, 4370}, {3690, 5509}
X(5513) = complement of X(675)Kirikami six-circles-images are introduced at X(5509).
X(5514) is the point of intersection, other than X(11), of the nine-point circle and the Mandart circle. (Randy Hutson, July 15, 2013)
X(5514) is the center of the hyperbola {A,B,C,X(4),X(40)}, and X(5514) = X(972)-of-Euler-triangle. (Randy Hutson, August 17, 2014)
X(5514) = midpoint of X(4) and X(972)
X(5514) = complement of X(934)
X(5514) lies on these lines: {2, 934}, {4, 972}, {9, 119}, {10, 118}, {11, 1146}, {12, 208}, {117, 374}, {120, 1329}, {3814, 5199}
X(5514) = Spieker-radical-circle-inverse of X(34457)
X(5514) = crosssum of the circumcircle intercepts of line X(3)X(9)
X(5514) = orthopole of line X(3)X(9)
X(5514) = point of concurrence of cevian circles of vertices of anticevian triangle of X(8)
Kirikami six-circles-images are introduced at X(5509).
X(5515) lies on these lines: {2, 835}, {116, 244}, {117, 5230}, {118, 2999}, {121, 1054}, {124, 3120}, {125, 1086}, {127, 2968}
X(5515) = X(2)-Ceva conjugate of X(6590)
X(5515) = polar-circle-inverse of X(32691)
X(5515) = crosssum of circumcircle intercepts of line X(3)X(31)
X(5515) = orthopole of line X(3)X(31)
X(5515) = center of hyperbola {{A,B,C,X(4),X(75)}}
Kirikami six-circles-images are introduced at X(5509).
X(5516) lies on these lines: {120, 5121}, {121, 519}, {1647, 3259}, {3667, 3756}
X(5516) = complement of X(6079)
X(5516) = crosspoint of X(519) and X(3667)
X(5516) = crosssum of X(106) and X(1293)
X(5516) = orthopole of line X(3)X(106)
X(5516) = center of hyperbola {{A,B,C,X(4),X(145)}}
Kirikami six-circles-images are introduced at X(5509).
X(5517) lies on these lines: {2, 1310}, {11, 3125}, {120, 1698}, {123, 1146}
X(5517) = crosssum of circumcircle intercepts of line X(3)X(37)
X(5517) = orthopole of line X(3)X(37)
X(5517) = center of hyperbola {{A,B,C,X(4),X(81)}}
Kirikami six-circles-images are introduced at X(5509).
X(5518) is the touchpoint, other than X(11), of the line through X(1086) tangent to the nine-point circle. (Randy Hutson, July 15, 2013)
X(5518) lies on these lines: {2, 932}, {12, 85}, {121, 3822}
X(5518) = complement of X(932)
X(5518) = crosssum of circumcircle intercepts of line X(3)X(238)Kirikami six-circles-images are introduced at X(5509).
X(5519) lies on these lines: {120, 518}, {1566, 3323}, {3309, 4904}
X(5519) = complement of X(6078)
X(5519) = crosspoint of X(518) and X(3309)
X(5519) = crosssum of X(105) and X(1292)
X(5519) = orthopole of line X(3)X(105)
X(5519) = center of hyperbola {{A,B,C,X(4),X(218)}}
Kirikami six-circles-images are introduced at X(5509).
X(5520) is the touchpoint, other than X(11), of the line through X(867) tangent to the nine-point circle. Also, X(5520) is the reflection of X(11) in the Euler line. (Randy Hutson, July 15, 2013)
Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5520) is the inverse-in-O* of X(2752). Also, X(5520) = inverse-in-polar-circle of X(2766). (Randy Hutson, August 17, 2014)
X(5520) lies on these lines: {2, 1290}, {4, 2687}, {11, 523}, {12, 2222}, {30, 119}, {113, 517}, {115, 650}, {116, 4369}, {120, 858}, {125, 513}, {1325, 5080}, {1560, 5089}, {1985, 2453}, {2074, 5172}, {3139, 3258}, {3140, 5099}
X(5520) = complement of X(1290)
X(5520) = inverse-in-Stevanovic-circle of X(115)
X(5520) = crosssum of circumcircle intercepts of line X(3)X(191)
X(5520) = orthopole of line X(3)X(191)
X(5520) = center of hyperbola {{A,B,C,X(4),X(267)}}
Kirikami six-circles-images are introduced at X(5509).
X(5521) lies on these lines: {4, 100}, {11, 2969}, {113, 1829}, {117, 1828}, {118, 1824}, {120, 427}, {121, 1883}, {122, 3139}, {123, 867}, {127, 3140}, {403, 5146}, {431, 1842}, {1560, 1841}
X(5521) = midpoint of X(4) and X(915)
X(5521) = X(2)-Ceva conjugate of X(6591)
X(5521) = crosssum of circumcircle intercepts of line X(3)X(63)
X(5521) = inverse-in-polar-circle of X(100)
X(5521) = center of the hyperbola {A,B,C,X(4),X(19)}
X(5521) = X(915)-of-Euler-triangle
X(5521) = orthopole of line X(3)X(63)
Kirikami six-circles-images are introduced at X(5509).
X(5522) lies on these lines: {113, 3091}, {132, 5064}, {133, 1906}, {2970, 5139}
X(5522) = crosssum of circumcircle intercepts of line X(3)X(51)
X(5522) = orthopole of line X(3)X(51)
X(5522) = center of hyperbola {{A,B,C,X(4),X(95)}}
Let P be a point in the plane of a triangle ABC with orthocenter H. Let DEF be the orthic triangle of ABC. The circumcircles of APD, BPE, CPF concur in two points, P and Q. The point Q = Q(P) is the orthoassociate, or Burek concurrent circles image, of P. Examples: Q(X(1)) = X(1785), Q(X(2)) = X(468), Q(X(3)) = X(403), Q(X(4)) = X(4), Q(X(5)) = X(186). (Dominic Burek, July 15, 2013)
The mapping Q is included as an orthoassociate of P in Bernard Gibert's paper, "Orthocorrespondence and Orthopivotal Cubics," Forum Geometricorum 3 (2003) 1-27. If P is given by barycentrics p : q : r, then Q is given by
Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = SBSC[(pq + pr)SA - q2SB - r2SC].
If you have The Geometer's Sketchpad, you can view X(5523) and X(5523) generalized. The latter has a movable point P.
X(5523) lies on these lines: {4,6}, {24,3767}, {30,112}, {39,1594}, {111,468}, {115,232}, {186,230}, {297,525}, {316,648}, {378,2549}, {382,3172}, {427,1180}, {459,5485}, {858,1560}, {1300,2715}, {1783,5080}, {3575,5305},
X(5523) = reflection of X(112) in the orthic axis
X(5523) = isogonal conjugate of complement of X(34163)
X(5523) = anticomplementary-circle-inverse of X(36851)
X(5523) = inverse-in-polar-circle of X(6)
X(5523) = radical trace of the polar circle and the orthosymmedial circle
X(5523) = pole with respect to the polar circle of the line X(6)X(525)
X(5523) = X(48)-isoconjugate of X(2373)
X(5523) = {5024,5094}
X(5523) = inverse-in-circle-O(PU(4)) of X(111)
Let P be a point in the plane of a triangle ABC with orthocenter H. Let DEF be the excentral triangle of ABC. The circumcircles of APD, BPE, CPF concur in two points, P and Q. The point Q = Q(P) is the Gibert-Burek-Moses concurrent circles image of P. The points X(1), P, Q(P) are collinear, and circumcenters of APD, BPE, CPF are collinear. Let L denote the line of the circumcenters; then Q is the reflection of P in L. Examples: Q(X(3)) = X(484), Q(X(4)) = X(3465), Q(X(15)) = X(1276), Q(X(16)) = X(1277), Q(X(20)) = X(5018). (Peter Moses, June 16, 2013)
The appearance of (i,j) in the following list means that Q(X(i)) =
X(j):
(36, 40), (46, 2077), (74, 3464), (100,1054), (105,1282), (109, 1768),
(165, 1155), (759, 2948), (1381, 2449), (1382, 2448)
(Randy Hutson, July 19, 2013)
Let O denote the imaginary circle with center X(1) and squared radius -4rR. Then Q(P) is the O-inverse of P; see Bernard Gibert's "Antiorthocorrespondents of Circumconics," Forum Geometricorum 3 (2003) 231-249. Accordingly, if U is an arbitrary circle, then Q(U) is a circle; here "circle" includes lines, regarded as circles of infinite radius. Examples: Q(circumcircle) = Bevan circle; Q(Euler line) is a circle with center X(3737); Q(antiorthic axis) is a circle with center X(3476); If P is a point on the circumcircle, then Q(P) is the Brisse transform of P with respect to the tangential triangle of the excentral triangle of ABC. (This paragraph is based on notes received from Bernard Gibert and Randy Hutson, July 17-19, 2013.)
Let P be given by barycentrics p : q : r. Then Q is given by
Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = a[bcp2 - caq2 - abr2 + (a - b - c)(aqr - bpr - cpq)] (Peter Moses, June 16, 2013)
If you have The Geometer's Sketchpad, you can view X(5524) and X(5524) generalized. The latter has a movable point P.
X(5524) lies on these lines: X(5524) lies on these lines: {1, 2}, {44, 3684}, {100, 896}, {111, 2748}, {171, 4663}, {210, 846}, {238, 3689}, {518, 1054}, {740, 3699}, {984, 3711}, {1051, 3745}, {1155, 1282}, {3667, 4724}, {3740, 3750}, {3956, 4653}, {3994, 4767}, {4009, 4693}, {4551, 5018}, {4557, 5143}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524).
X(5525) lies on these lines: {1,6}, {35,3991}, {36,3693}, {46,728}, {101,2752}, {111,2748}, {169,3632}, {191,1334}, {346,4293}, {484,1018}, {644,758}, {1759,3208}, {1781,2321}, {2082,3633}, {3065,4876}, {3218,3912}, {3309,4790}, {3336,3501}, {3950,4304}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524).
X(5526) lies on these lines: {1,6}, {35,41}, {36,101}, {71,2301}, {80,294}, {115,2238}, {187,1017}, {484,910}, {517,2348}, {519,644}, {573,3217}, {645,5209}, {650,1734}, {651,1323}, {739,2748}, {902,1110}, {908,3008}, {1018,3684}, {1334,3746}, {1783,1785}, {1795,2338}, {2246,3245}, {2291,2742}, {2503,5164}, {3509,4880}, {3632,4513}, {3997,5276}, {5219,5228}
X(5526) = isogonal conjugate of X(34578)
X(5526) = crosspoint of X(4845) and X(10482)
X(5526) = crosssum of X(i) and X(j) for these {i,j}: {1086, 1638}, {1323, 10481}
X(5526) = crossdifference of every pair of points on line X(354)X(513)
X(5526) = trilinear pole of line X(8645)X(22108)
X(5526) = Conway-circle-inverse of X(35892)
Gibert-Burek-Moses concurrent circles images are introduced at X(5524).
X(5527) = X(316)-of-excentral triangle; also, X(5527) is the excentral isotomic conjugate of X(5528). (Randy Hutson, July 18, 2013)
X(5527) lies on these lines: {1, 7}, {165, 5011}, {514,4105}, {1053, 2958}, {1308, 5536}, {1699,5074}
X(5527) = reflection of X(5536) in X(1308)
X(5527) = excentral-isogonal conjugate of X(34925)
X(5528) is the antipode of X(9) in the rectangular hyperbola that passes through X(1), X(9), and the 3 excenters. Also, X(5528) is X(67)-of-the-excentral triangle, the excentral isogonal conjugate of X(5536), and the excentral isotomic conjugate of X(5527). (Randy Hutson, July 18, 2013)
X(5528) lies on these lines: {1, 528}, {9, 100}, {11, 4329}, {2951, 5531}
X(5528) = reflection of X(i) in X(j) for these (i,j):
{1,528}, {9,100}, {11,4326}, {35,5506}, {40,2801}, {142,149},
{191,4436}, {518,3245}, {527,3935}, {971,2950}, {1317,4321},
{2136,3868}, {2802,3243}, {2949,3579}, {2951,5531}, {3020,3340},
{3646,5248}
X(5528) = Bevan-circle-inverse of X(34925)
X(5529) is the inverse-in-excircles-radical-circle of X(5530). (Randy Hutson, July 18, 2013)
X(5529) lies on these lines: {1,2}, {9,5110}, {36,1757}, {238,5440}, {404,1046}, {609,1743}, {758,1054}, {846,4256}, {982,3940}, {1326,5150}, {1739,4867}, {2948,5131}, {3667,4040}, {5400,5538}
X(5529) = Conway-circle-inverse of X(35633)
Let A′ be the inverse-in-excircles-radical-circle of A, and define B′ and C′ cyclically. Let IA be the inverse-in-excircles-radical-circle of the A-excenter, and define IB and IC cyclically. The lines A′IA, B′IB, C′IC concur in X(5530). (Randy Hutson, July 18, 2013)
X(5530) is the inverse-in-excircles-radical-circle of X(5529). (Randy Hutson, July 18, 2013)
X(5530) lies on these lines: {1,2}, {5,3931}, {12,3666}, {36,961}, {37,1329}, {46,573}, {65,970}, {171,580}, {181,942}, {226,986}, {388,988}, {429,1785}, {442,1738}, {517,1682}, {908,2292}, {968,2478}, {1686,2362}, {1695,2093}, {1838,1880}, {2051,4424}, {2476,3914}, {2886,4646}, {3596,4078}, {3614,4854}, {3663,3947}, {3743,3814}, {4339,5281}
Four of the circles that are tangent to two of the sidelines BC, CA, AB pass through X(11), namely, the incircle and 3 others. The centers of those 3 are collinear. (See Barry Wolk's Hyacinthos messages #21431, #21433, etc., January 2013). Let A′B′C′ be the triangle formed by the radical axes of these circles and the corresponding mixtilinear excircle. A′B′C′ is homothetic to the hexyl triangle, and the center of homothety is X(5531). Moreover, X(5531) is the Fuhrmann-triangle-to-excentral triangle similarity image of X(40). Further, in the definition of X(5495), if A′B′C′ is the excentral triangle, then the circumcircles of TA, TB, TC concur in X(5531). Also, X(5531) is the inverse of X(1) in the circumcircle of OA, OB, OC. (Randy Hutson, July 18, 2013)
X(5531) lies on these lines: {1,5}, {3,3711}, {40,2771}, {63,100}, {101,3119}, {104,4866}, {149,1699}, {153,3811}, {214,936}, {484,912}, {515,5538}, {516,3935}, {518,5536}, {528,1750}, {971,3689}, {1145,4882}, {1156,4326}, {1490,2800}, {1709,3158}, {2951,5528}, {3062,3174}, {3817,3957}, {4297,4420}
X(5531) = reflection of X(i) in X(j) for these (i,j): (1768,100), (2951,5528), (5537,3689)
X(5532) is the point of intersection, other than X(11), of the three collinear circles described at X(5531). (Barry Wolk, Hyacinthos #21433, January 18, 2013)
X(5532) lies on these lines: {11,514}, {516,5183}, {1111,3323}, {1146,3022}, {2310,4041}, {3689,5199}, {4081,4163}
X(5533) is the Gibert-Burek-Moses concurrent circles image of X(5534). (Randy Hutson, July 18, 2013)
X(5533) = inverse-in-incircle of X(5), and X(5533) = {X(11),X(1317)}-harmonic conjugate of X(5).
X(5533) lies on these lines: {1,5}, {100,499}, {104,1479}, {149,3086}, {528,3582}, {1145,3813}, {1647,1772}, {1737,2802}, {2829,3583}, {3036,4187}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). In the definition of X(5495), if A′B′C′ is the excentral triangle, then X(5534) is the center of the circumcircle of OA, OB, OC. In this case, unlike that of X(5495), the circle does not also pass through O. Also, X(5534) = X(5)-of-3rd-antipedal-triangle-of-X(1). (Randy Hutson, July 18, 2013)
X(5534) lies on these lines: {1,5}, {3,200}, {4,3870}, {20,3935}, {40,912}, {78,944}, {84,3158}, {104,4855}, {515,3811}, {517,1490}, {936,1385}, {971,3174}, {1062,1103}, {1158,2801}, {1728,2078}, {1998,3149}, {2057,5440}, {3072,3751}, {3073,3749}, {3090,4666}, {3091,3957}, {3576,5258}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5535) = inverse-in-Bevan-circle-of- X(3) = X(2070)-of-excentral-triangle = X(36)-of-tangential-triangle-of-excentral triangle. (Randy Hutson, July 18, 2013)
X(5535) = midpoint of X(484) and X(5536)
X(5535) = reflection of X(i) in X(j) for these (i,j): (40,484),
(104,4973), (2077,1155), (5180,946), (5538,3)
X(5535) = inverse-in-Bevan-circle of X(3)
X(5535) lies on these lines: {1,3}, {5,191}, {9,3814}, {30,1768}, {63,5080}, {104,4973}, {442,2949}, {515,3218}, {535,3928}, {546,3652}, {912,4880}, {946,5180}, {1727,3583}, {2272,5011}, {3628,5506}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5536) = inverse-in-Bevan-circle-of-X(165) = X(23)-of-excentral-triangle = X(1155)-of-tangential-triangle-of-excentral-triangle = excentral isogonal conjugate of X(5528). (Randy Hutson, July 18, 2013)
X(5536) = reflection of X(i) in X(j) for these (i,j): (484,5535),
(1768,3218), (5527,1308), (5537,1155), (5538,36)
X(5536) = inverse-in-Bevan-circle of X(165)
X(5536) lies on these lines: {1,3}, {9,5087}, {63,1699}, {103,1290}, {110,2717}, {149,516}, {191,946}, {411,3874}, {518,5531}, {672,2957}, {910,2323}, {1308,5527}, {1421,2361}, {1709,3928}, {1757,5400}, {2949,5506}, {3219,3817}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5537) is the radical trace of each pair of the 1st, 2nd, and 3rd antipedal circles of X(1); also, X(5537) = X(23)-of-1st-circumperp-triangle = X(858)-of-excentral-triangle. (Randy Hutson, July 18, 2013)
X(5537) lies on these lines: {1,3}, {20,535}, {100,516}, {103,677}, {105,2743}, {200,1709}, {404,4301}, {411,5493}, {480,3062}, {518,1768}, {840,1293}, {971,3689}, {972,2222}, {991,2177}, {1012,3679}, {1260,1750}, {1376,1699}, {1618,2272}, {2291,2742}, {2800,4867}, {2801,3935}, {3091,3814}, {3146,5080}, {3871,4297}, {5288,5450}
X(5537) = reflection of X(i) in X(j) for these (i,j): (36,2077), (3245,40), (5526,2742), (5531,3689), (5536,1155)
X(5537) = circumcircle-inverse of X(165)
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5538) = X(2070)-of-hexyl-triangle. (Randy Hutson, July 18, 2013)
X(5538) = reflection of X(i) in X(j) for these (i,j): (484,2077),
(5535,3), (5536,36)
X(5538) = inverse-in-hexyl-circle of X(3)
X(5538) lies on these lines: {1,3}, {78,5080}, {200,5176}, {515,5531}, {516,4511}, {758,1768}, {936,3814}, {997,1699}, {1006,5426}, {5400,5529}
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5539) = X(1356)-of-tangential-triangle-of-excentral-triangle. (Randy Hutson, July 18, 2013)
X(5539) = reflection of X(1) in X(741)
X(5539) lies on the Bevan circle and these lines: {1,99}, {9,3037}, {43,5213}, {57,1356}, {484,3510}, {1015,3571}, {1045,5541}, {1046,1282}, {1716,3464}, {1740,2948}, {2640,5540)
Gibert-Burek-Moses concurrent circles images are introduced at X(5524).
X(5540) is the point of concurrence of the reflections of the line X(1)X(6) in the sides of the excentral triangle. (Randy Hutson, August 14, 2013)
The 1st and 2nd Montesdeoca bisector triangles are inversely similar to the excentral triangle. Let s1 and s2 be the similarity mappings. Then there is a unique point X such that s1(X) = s2(X), and X = X(5540). See HGT2017 and AdvGeom3769
X(5540) lies on the Bevan circle and these lines: {1,41}, {6,1718}, {9,80}, {19,1743}, {35,1212}, {36,910}, {37,3196}, {43,3034}, {44,3245}, {57,1358}, {115,2503}, {120,1698}, {165,1292}, {190,4986}, {191,2795}, {484,672}, {517,2348}, {519,5525}, {579,3464}, {610,909}, {614,5354}, {644,2802}, {654,1768}, {657,2957}, {673,1111}, {952,4534}, {1023,4919}, {1053,4040}, {1054,1635}, {1475,3337}, {1697,3021}, {1699,5511}, {1713,2833}, {1723,2270}, {1724,2838}, {1731,2183}, {1766,3973}, {2173,5053}, {2238,5164}, {2448,2591}, {2449,2590}, {2640,5539}, {2814,5400}, {3336,4253}, {3583,5179}, {4875,5258}, {5030,5131}
X(5540) = reflection of X(i) in X(j) for these (i,j): (1,105), (5526,2348)
X(5540) = X(112)-of-excentral-triangle
X(5540) = Stevanovic-circle-inverse of X(34464)
X(5540) = X(1358)-of-tangential-triangle-of-excentral-triangle
X(5540) = excentral isogonal conjugate of X(3309)
X(5540) = trilinear-pole-with-respect-to-excentral-triangle-of-the-line-X(2)X(7)
Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5541) = Bevan-circle-antipode-of-X(1768) = X(74)-of-excentral-triangle = {X(100), X(1320)}-harmonic-conjugate-of-X(214) = X(1317)-of-tangential-triangle-of-excentral-triangle. Also, X(5541) is the antipode of X(1) in the rectangular hyperbola that passes through X(1), X(9) and the 3 excenters, and X(5541) is the inverse of X(214) in the circumconic centered at X(1). (Randy Hutson, July 18, 2013)
Let A′B′C′ be the excentral triangle. Let EA be the Euler line of BCA′. Let LA be the line through A′ parallel to EA, and define LB and LC cyclically. The lines LA, LB, LC concur in X(5541). (Randy Hutson, August 14, 2013)
X(5541) = reflection of X(i) in X(j) for these (i,j): (1,100), (80,1145), (149,10), (1320,214), (1768,40), (4867,3689), (4880,5183)
X(5541) lies on the Bevan circle and these lines: {1,88}, {8,191}, {9,80}, {10,149}, {11,1697}, {36,2932}, {40,550}, {43,3032}, {46,2136}, {55,5426}, {57,1317}, {63,4677}, {104,165}, {119,1699}, {145,3336}, {153,516}, {190,4738}, {200,3899}, {484,519}, {515,2950}, {517,3689}, {518,3245}, {984,2805}, {1045,5539}, {1050,3216}, {1282,3887}, {1490,2800}, {1706,3035}, {1759,4050}, {2093,3174}, {2246,4752}, {2448,3307}, {2449,3308}, {2801,2951}, {2948,4730}, {3219,4669}, {3244,3337}, {3339,5083}, {3464,4707}, {3579,3893}, {3654,4863}, {3813,5445}, {3919,3957}, {3968,5284}, {4880,5183}, {5011,5525}
X(5541) = {X(10),X(149)}-harmonic conjugate of X(37718)
X(5542) = X(182)-of-intouch-triangle = {X(175), X(176)}-harmonic conjugate of X(5543). (Randy Hutson, July 18, 2013)
X(5542) lies on these lines: {1,7}, {2,5223}, {6,4989}, {9,1125}, {10,141}, {11,118}, {35,2346}, {55,553}, {56,954}, {57,3475}, {75,4684}, {144,3616}, {320,3883}, {474,480}, {497,4654}, {519,1056}, {527,551}, {537,4078}, {673,4649}, {726,3950}, {938,5290}, {946,971}, {1086,3755}, {1155,4031}, {1210,3947}, {1386,4667}, {1445,3338}, {1836,3982}, {1870,1890}, {2321,4966}, {3008,3751}, {3059,3555}, {3242,4675}, {3244,4780}, {3295,5493}, {3452,3742}, {3649,4890}, {3720,4656}, {3748,4114}, {3782,4883}, {3790,3912}, {3873,4847}, {3911,4860}, {4061,4359}
X(5542) = midpoint of X(i) and X(j) for these (i,j): (1,7), (390,4312), (962,2951), (2550,3243), (3059,3555), (4295,4326)
X(5542) = reflection of X(i) on X(j) for these (i,j): (9,1125), (10,142)
X(5542) = complement of X(5223)
X(5542) = harmonic center of of inner and outer Soddy circles
X(5542) = X(6)-of-incircle-circles-triangle
X(5543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,3160), (1,4350,1442), (175,176,5542), (3945,4328,7), (5228,5308,5435). (Randy Hutson, July 18, 2013 and Peter Moses, August 13, 2013)
X(5543) lies on these lines: {1,7}, {85,3241}, {354,3599}, {1441,4460}, {2295,5228}, {3340,3598}, {3772,5222}
Let X be a point in the plane of a triangle ABC, and let A′B′C′ be the pedal triangle of X. The sum |AX|2 + |BX|2 + |CX|2 + |A′X|2 + |B′X|2 + |C′X|2 is minimized by X = X(5544). (Jean-Baptiste Hiriart-Urruty; Toulouse, France; August 30, 2013)
The minimal value is (4S4 - 12PT + 15S2T2)/(20S2T - 18P), where P = SASBSC and T = SA + SB + SC. (Peter Moses, August 30, 2013)
X(5544) is the only point whose polar conic in the Thomson cubic. (K002) is a circle. (Bernard Gibert, June 22, 2014)
If you have The Geometer's Sketchpad, you can view X(5544).
The following notes pertaining to minimal sums were contributed by Peter Moses, March 5, 2024.
Let X be a point in the plane of a triangle ABC, and let A'B'C' be the pedal triangle of X. The sum
|AX|^2 + |A'X|^2 + |BX|^2 + |B'X|^2 + |CX|^2 + |C'X|^2
is minimized by X(5544). More generally, the sum
|AX|^2 + k*|A'X|^2 + |BX|^2 +k*|B'X|^2 + |CX|^2 + k*|C'X|^2
is minimized by the following point:
X
or, equivalentley,
X
The point X
m
There are 3 values of k for which m
16*k*(1 + k)^2*S^4 - 96*(1 + k)*SA*SB*SC*SW + 16*(2 + k)*(3 + 2*k)*S^2*SW^2 = 0.
The corresponding points X
The point X
The appearance of (k,X(n)) in the following list means that X
(-6,55038), (-4,9716), (-3,3167), (-2,110), (-5/3,55157), (-9/7,61771), (-14/9,55156), (-3/2,154), (-4/3,7712), (-6/5,6030), (-1,3), (-6/7,61772), (-3/4,61773), (-2/3,5888), (-1/2,5646), (-1/3,61774), (0,2), (1/2,14924), (1,5544), (2,5643), (3,5644), (4,5645), (6,61775)
X(5544) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2, 1351}, {3, 373}, {5, 5656}, {6, 16187}, {23, 55682}, {25, 6030}, {51, 55584}, {110, 5050}, {125, 5055}, {140, 21970}, {154, 182}, {323, 5645}, {354, 3751}, {381, 11820}, {382, 35253}, {392, 1482}, {394, 5644}, {474, 48921}, {511, 5646}, {567, 12309}, {576, 12045}, {1350, 5943}, {1495, 55697}, {1656, 5654}, {1995, 7712}, {3030, 37502}, {3090, 32605}, {3124, 5024}, {3167, 5651}, {3357, 11479}, {3426, 40280}, {3526, 3527}, {3545, 15431}, {3589, 23326}, {3618, 6391}, {3628, 11432}, {3819, 55722}, {3843, 44300}, {5054, 32269}, {5070, 12316}, {5079, 45303}, {5085, 32237}, {5097, 10219}, {5422, 55038}, {5640, 5888}, {5643, 11482}, {5648, 5972}, {5650, 44456}, {5652, 8371}, {5653, 11637}, {5892, 11472}, {5921, 45298}, {6090, 9716}, {6244, 16058}, {6388, 31489}, {6642, 9920}, {6723, 15131}, {6800, 30734}, {7392, 39884}, {7393, 32205}, {7395, 11465}, {7484, 11451}, {7485, 55648}, {7496, 55639}, {7529, 13339}, {7998, 55724}, {8547, 40670}, {8717, 18535}, {8780, 11003}, {9171, 34291}, {9306, 55711}, {9818, 12041}, {9909, 53094}, {10545, 55678}, {10546, 55156}, {10620, 41670}, {11002, 55580}, {11328, 34099}, {11433, 61545}, {11477, 15082}, {11484, 15805}, {11579, 40917}, {12099, 38396}, {12167, 52290}, {12174, 15022}, {12310, 15040}, {13361, 14826}, {13364, 58764}, {14810, 17810}, {15107, 55643}, {15448, 38064}, {15693, 20192}, {15703, 58891}, {16042, 26864}, {17508, 31860}, {17809, 55709}, {18928, 40330}, {19124, 21313}, {19347, 19360}, {20190, 41424}, {20998, 55165}, {25555, 59767}, {33586, 55616}, {33750, 37910}, {35259, 55157}, {35283, 39899}, {37493, 55857}, {37672, 55714}, {38110, 40132}, {40916, 55610}, {41462, 55602}, {43650, 55692}, {45311, 51941}, {45578, 55579}, {45579, 55577}, {46219, 61644}, {55587, 58470}
X(5544) = midpoint of X(3) and X(3531)
X(5544) = reflection of X(i) in X(j) for these {i,j}: {3531, 52163}, {18489, 5}, {40912, 44833}
X(5544) = complement of X(44833)
X(5544) = Thomson-isogonal conjugate of X(10304)
X(5544)-Dao conjugate of X(44833)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {373, 22112, 3066}, {3066, 22112, 3}, {5651, 10601, 53091}, {5651, 53091, 3167}, {6090, 15018, 53092}, {6688, 17825, 5020}
X(5544) = pole of line {3524, 5032} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5544) = pole of line {5093, 5544} with respect to the Thomson-Gibert-Moses hyperbola
X(5544) = pole of line {5646, 54439} with respect to the Parry circle
Suppose that P is a point in the plane of triangle ABC. Let A′B′C′ be the anticevian triangle of P and let A″B″C″ be the 1st circumperp triangle. The locus of P for which the lines A′A″, B′B″, C′C″ concur is the union of the line X(1)X(6) and the conic U = {A, B, C, X(66), X(101), X(294), X(651)}; i.e., the isogonal conjugate of the Gergonne line. The conic U has center X5452) and is given by the trilinear equation
a(b + c - a)yz + b(c + a - b)zx + c(a + b - c)xy = 0.
For X on X(1)X(6)∪U, let F(X) be the point of concurrence. Then if X is on X(1)X(6), the image F(X) is on the line X(1)X(3); a pair (i,j) in the following list indicates that F(X(i)) = X(j): (1,165), (6,3), (9,40), (37,3579), (44,517), 281,55), 1713,1715), (1723,46), (1724,1754), (1743,1), (2323,2077), (5247,171), 5526,5537). On the other hand, if X is on U, the image F(X) is on the circumcircle; a pair (i,j) in the following list indicates that F(X(i)) = X(j): (101,109), (110,5543), (111,5543), (294,105), (644,100), (645,99), (651,934), (666,927), (1783,108), (2311,741), (2316,106), (4627,5545). (César Lozada; August 29, 2013)
Suppose that P is on X(1)X(6). If P = p : q : r (trilinears), then
F(P) = a(b + c - a)/[(b - c)p] : b(c + a - b)/[(c - a)q] : c(a + b -
c)/[(a - b)r];
If P = p : q : r (barycentrics), then F(P) = a3(b + c -
a)/[(b - c)p] : b3(c + a - b)/[(c - a)q] : c3(a +
b - c)/[(a - b)r]
Suppose that P is on U. If P = p : q : r (trilinears), then F(P) = (b +
c - a)p : (c + a - b)q : (a + b - c)r];
If P = p : q : r (barycentrics), then F(P) = then F(P) = (b + c - a)p :
(c + a - b)q : (a + b - c)r]. (Peter Moses; September 2,
2013)
X(5545) lies on the circumcircle and these lines: {100,1414}, {101,4565}, {105,5323}, {835,4624}
X(5545) = trilinear pole of the line X(6)X(1412)
X(5545) = Ψ(X(6), X(1412))
X(5544) = Thomson-isogonal conjugate of X(10304)
X(5546) = F(X(110)), where F is the mapping defined at X(5545). (Peter Moses; September 2, 2013)
X(5546) is the perspector of the anticevian triangle of X(110) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, July 31 2018)
X(5546) lies on these lines: {9,1793}, {21,294}, {41,60}, {58,1810}, {99,666}, {100,112}, {101,110}, {283,2338}, {284,2316}, {345,4548}, {593,609}, {643,644}, {645,4612}, {648,4552}, {651,662}, {672,5060}, {910,1325}, {1018,1021}, {1333,1811}, {1576,4557}, {1625,2427}, {1809,2193}, {1951,4511}, {1983,2610}, {2251,5006}, {2328,4845}, {3732,4237}, {3939,4587}, {4556,4627}, {5127,5526}
X(5546) = isogonal conjugate of X(7178)
X(5546) = X(2)-Ceva conjugate of-X(34961)
X(5546) = X(19)-isoconjugate of X(17094)
X(5547) = F(X(111)), where F is the mapping defined at X(5545). (Peter Moses; September 2, 2013)
X(5547) is the perspector of the anticevian triangle of X(111) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, July 31 2018)
X(5547) lies on these lines: {8,645}, {42,101}, {65,651}, {210,644}, {666,671}, {1334,3939}, {1783,1824}, {2334,4627}
X(5547) = isogonal conjugate of X(7181)
X(5548) lies on these lines: {101,649}, {106,5526}, {294,1320}, {644,650}, {645,4560}, {651,3257}, {663,3939}, {666,4555}, {1318,2316}, {1783,5375}, {2340,4845}, {2423,2427}, {2429,2441}, {4591,4627}
X(5548) = isogonal conjugate of X(30725)
X(5548) = F(X(44)), where F is the mapping defined at X(5545) (Peter Moses; September 3, 2013)
X(5549) = F(X(45)), where F is the mapping defined at X(5545). (Peter Moses; September 3, 2013)
X(5549) lies on these lines: {21,2341}, {41,2316}, {101,4588}, {294,2320}, {651,4604}, {666,4597}, {1783,4242}, {4558,4627}
Let O(A,t) be the circle with center A and radius t*r, where r is the inradius of triangle ABC. Define O(B,t) and O(C,t) cyclically. Of the two parallel lines tangent to O(B,t) and O(C,t), let TA be the closer to A, and define TB and TC cyclically. Let D = TB∩TC, and define E and F cyclically. Let A′ be the midpoint of segment BC, and define B′ and C′ cyclically. Let A″ be the touchpoint of TA and the incircle of DEF. Then AA′, BB′, CC′ concur and AA″, BB″, CC″ concur. (Emmanuel José Garcia; September 11, 2013)
The triangle DEF has incenter X(1) and is similar to ABC with dilation factor 1-t. Let G(t) = AA′∩BB′∩ CC′ and GF(t) = AA″∩BB″∩ CC″. The point G(t) lies on the line X(1)X(2) and has barycentric coordinates given by
G(t) = a + b + c - (b + c - a)t : a + b + c - (c + a - b)t : a + b + c - (a + b - c)t
and satisfies |X(1)G(t)|/|X(2)G(T)| = 3(1 - t)/(2t). (Peter Moses; September 12, 2013)
The point GF(t) lies on the Feuerbach hyperbola (the isogonal conjugate of the line X(1)X(3)) and has barycentric coordinates given (Peter Moses; September 12, 2013) by
GF(t) = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)t]
In the following list, the appearance of {n,t} indicates that X(n) = G(t) :
{1,1}, {2,0}, {10,-1}, {78,1+(2 R)/r}, {145,2}, {200,1+(4 R)/r}, {498,R/(2 r+R)}, {499,-(R/(2 r-R))}, {551,3/5}, {936,(r+2 R)/(r-2 R)}, {938,1+r/(2 R)}, {997,(r+R)/(r-R)}, {1125,1/3}, {1210,-((r+2 R)/(r-2 R))}, {1698,-(1/3)}, {1737,-((r+R)/(r-R))}, {3085,R/(r+R)}, {3086,-(R/(r-R))}, {3241,3/2}, {3244,5/3}, {3582,-((3 R)/(4 r-3 R))}, {3584,(3 R)/(4 r+3 R)}, {361,1/2}, {3617,-2}, {3621,4}, {3622,2/3}, {3623,4/3}, {3624,1/5}, {3625,7}, {3626,-5}, {3632,5}, {3633,7/3}, {3634,-(1/5)}, {3635,7/5}, {3636,5/7}, {3679,-3}, {3811,(r+3 R)/(r+R)}, {3828,-(3/7)}, {3870,(r+4 R)/(r+2 R)}, {3872,1-(2 R)/r}, {3935,(r+4 R)/(r+R)}, {3957,(r+4 R)/(r+3 R)}, {4420,1+(3 R)/r}, {4511,(r+R)/r}, {4666,(r+4 R)/(r+6 R)}, {4668,-7}, {4678,-4},v{4691,-(7/3)}, {4847,-1-(4 R)/r}, {4853,1-(4 R)/r}, {4861,1-R/r}, {4882,1+(8 R)/r}, {4915,1-(8 R)/r}, {5231,-((r+4 R)/(3 r)) (Peter Moses; September 14, 2013)
In the next list, the appearance of {n,t} indicates that X(n) = GF(t):
{1,1}, {7,0}, {8,2}, {9,(r+4 R)/(r+2 R)}, {21,(r+2 R)/(r+R)}, {79,-1}, {80,3}, {84,1+(2 R)/r}, {90,(r+3 R)/(r+R)}, {104,(r+R)/r}, {943,(r+3 R)/(r+2 R)}, {1000,3/2}, {1156,(r+4 R)/(r+R)}, {1320,(r-2 R)/(r-R)}, {1389,1-R/r}, {1392,(2 (r-R))/(2 r-R)}, {1476,r/(r-R)}, {2320,(2 (r+R))/(2 r+R)}, {2346,(r+4 R)/(r+3 R)}, {3062,1+(4 R)/r}, {3065,(2 r+5 R)/(2 r+R)}, {3254,-((r+4 R)/(r-2 R))}, {3255,(r+4 R)/(3 r+2 R)}, {3296,1/2}, {3427,(2 (r+R))/r}, {3467,(2 r+7 R)/(2 r+3 R)}, {3577,1-(2 R)/r}, {3680,(r-4 R)/(r-2 R)}, {4866,(r+8 R)/(r+4 R)}, {4900,(r-8 R)/(r-4 R)}, {5424,(4 r+7 R)/(4 r+5 R)} (Peter Moses; September 14, 2013)
X(5550) lies on these lines: {1,2}, {3,5284}, {11,4197}, {12,4308}, {21,4423}, {44,5296}, {56,5047}, {63,3646}, {65,3848}, {210,3889}, {226,5265}, {354,3876}, {355,5067}, {377,5225}, {404,1001}, {405,5253}, {474,1621}, {515,5056}, {517,3525}, {631,962}, {632,1482}, {756,3976}, {944,1656}, {946,3523}, {952,5070}, {958,5328}, {999,5260}, {1155,5180}, {1385,3090}, {1386,3619}, {1420,5261}, {1479,5444}, {1699,3522}, {1788,4323}, {2098,5326}, {2476,3816}, {2478,5229}, {3091,3576}, {3146,3817}, {3219,3338}, {3246,4645}, {3305,3333}, {3485,5221}, {3600,5219}, {3601,5274}, {3614,4193}, {3618,4663}, {3653,5071}, {3678,4430}, {3681,5045}, {3697,5049}, {3698,3885}, {3742,3868}, {3753,3890}, {3812,3877}, {3822,5154}, {3825,5141}, {3832,4297}, {3869,5439}, {3871,4413}, {3873,4539}, {3874,4532}, {3881,4661}, {3993,4772}, {4188,5248}, {4189,5259}, {4295,5443}, {4419,4798}, {4747,4758}, {4860,4999}, {5080,5084}, {5128,5250}, {5177,5436}
X(5550) = {X(1),X(2)}-harmonic conjugate of X(9780)
Garcia-Feuerbach points are defined at X(5550).
X(5551) lies on these lines: {1,4114}, {8,4004}, {943,5204}
Garcia points are defined at X(5550).
X(5552) lies on these lines: {1,2}, {3,3436}, {4,100}, {5,3434}, {9,1195}, {11,3913}, {12,377}, {20,2077}, {21,2551}, {40,908}, {55,1329}, {56,3035}, {140,956}, {149,5154}, {318,406}, {329,3359}, {345,3701}, {355,5440}, {388,404}, {405,3820}, {442,1260}, {452,5281}, {474,495}, {475,5081}, {480,3826}, {497,3871}, {515,4855}, {529,5204}, {631,2975}, {944,5176}, {958,5432}, {962,1519}, {1056,5253}, {1145,1482}, {1213,3713}, {1331,1771}, {1478,4190}, {1479,3814}, {1621,5084}, {1706,5219}, {1788,3868}, {1837,5123}, {1877,4200}, {2476,2550}, {2899,4194}, {3090,5082}, {3256,5177}, {3295,4187}, {3303,3816}, {3452,5250}, {3524,5303}, {4188,4293}, {4294,5046}, {5193,5265}
Garcia-Feuerbach points are defined at X(5550).
X(5553) lies on these lines: {8,912}, {9,2252}, {21,2096}, {84,1519}, {90,499}, {944,1320}, {962,1392}, {1389,4295}
X(5553) = perspector of ABC and mid-triangle of hexyl triangle and reflection triangle of X(1)
Garcia points are defined at X(5550).
X(5554) lies on these lines: {1,2}, {20,3359}, {63,4848}, {65,3436}, {100,3486}, {119,2476}, {355,377}, {388,5176}, {404,944}, {474,952}, {515,4190}, {517,2478}, {529,5221}, {631,3897}, {908,3340}, {946,5187}, {962,5046}, {1058,3885}, {1145,3295}, {1220,2994}, {1329,2099}, {1470,1788}, {1478,3754}, {1482,4187}, {1519,3091}, {1837,3434}, {2077,4189}, {2098,3816}, {2550,5086}, {2551,3869}, {3256,5273}, {3421,3868}, {3476,5253}, {3488,3871}, {3812,5252}, {3877,5084}, {4295,5080}, {4308,5193}
Garcia-Feuerbach points are defined at X(5550).
X(5555) lies on these lines: {21,1470}, {90,1210}, {388,1320}, {497,1476}, {943,5281}, {1039,1877}, {1392,4323}, {3434,3680}
Garcia-Feuerbach points are defined at X(5550).
X(5556) lies on these lines: {1,3146}, {7,5225}, {8,1836}, {9,5128}, {21,4423}, {79,938}, {80,4295}, {962,1000}, {1156,5221}, {3474,3614}, {3617,4866}, {3621,4900}, {3832,4312}, {5217,5226}
X(5556) = isotomic conjugate of X(32099)
X(5556) = orthocenter of triangle X(1)X(4)X(8) (Randy Hutson, November 22, 2014 )
Garcia-Feuerbach points are defined at X(5550).
X(5557) lies on these lines: {1, 550}, {8, 2891}, {9, 583}, {21, 551}, {35, 2346}, {36, 943}, {79, 354}, {80, 942}, {90, 3333}, {140, 3337}, {256, 3953}, {553, 3746}, {1320, 3635}, {1385, 5424}, {1389, 5425}, {1476, 3671}, {1656, 4860}, {3065, 3649}, {3467, 5443}, {3487, 5444}
X(5557) = isogonal conjugate of X(3746)
X(5557) = perspector of ABC and mid-triangle of 2nd circumperp and 1st Conway triangles
Garcia-Feuerbach points are defined at X(5550).
X(5558) lies on these lines: {1, 3522}, {2, 4866}, {4, 5045}, {8, 354}, {9, 1475}, {21, 3304}, {56, 2346}, {80, 938}, {145, 4900}, {942, 1000}, {943, 999}, {962, 3296}, {1156, 3485}, {1476, 4323}, {3062, 5542}, {3241, 3680}, {3333, 3523}, {3854, 5290}, {4298, 5059}
X(5558) = isogonal conjugate of X(3303)
X(5558) = isotomic conjugate of X(32087)
Garcia-Feuerbach points are defined at X(5550).
Let Pa be the isotomic conjugate of X(1222) with respect to AX1)X(8), and define Pb and Pc cyclically. Then ABC and PaPbPc are perspective at X(5559). (Angel Montesdeoca, September 23, 2018)
X(5559) lies on these lines: {1, 140}, {2, 1392}, {8, 3884}, {9, 3632}, {10, 1320}, {21, 519}, {35, 104}, {36, 1476}, {79, 517}, {80, 3057}, {84, 5119}, {90, 1697}, {145, 2320}, {314, 3264}, {518, 3255}, {952, 3065}, {1389, 5443}, {1656, 2098}, {3254, 4553}, {3679, 3680}, {4668, 4900}, {4677, 4866}
Garcia-Feuerbach points are defined at X(5550).
X(5560) lies on these lines: {1, 381}, {7, 3585}, {8, 3583}, {21, 1698}, {40, 3467}, {46, 3065}, {79, 1837}, {84, 3336}, {90, 484}, {943, 3586}, {1000, 1479}, {1125, 2320}, {1320, 3633}, {1389, 1699}, {1392, 3244}, {1478, 3296}
Garcia-Feuerbach points are defined at X(5550).
Let A′B′C′ be the cevian triangle of X(1) with respect to the incentral triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5561). (Randy Hutson, August 17, 2014)
X(5561) lies on these lines: {1, 382}, {7, 3583}, {8, 3585}, {9, 484}, {21, 3624}, {30, 5424}, {46, 3467}, {57, 3065}, {80, 1836}, {84, 3337}, {90, 3336}, {104, 1699}, {551, 2320}, {1000, 1478}, {1156, 4312}, {1392, 3635}, {1479, 3296}, {1770, 3634}, {3830, 5425}, {5010, 5219}
X(5561) = isogonal conjugate of X(5010)
X(5561) = isotomic conjugate of X(17360)
X(5561) = perspector of the circle centered at X(1) with radius 2r
X(5561) = X(15110)-of-excentral-triangle
Let A′ be the point, other than A, in which the line parallel to BC meets the circumcircle of ABC, and define B′ and C′ cyclically. Let PA be the point in which the line through A′ perpendicular to BC meets BC, and define PB and PC cyclically; the points PA, PB, PC are collinear, forming the so-called Simson line of A′. The Simson lines of A′, B′, C′ concur in X(5562). (Dao Thanh Oai, October 2, 2013)
X(5562) = isotomic conjugate of isogonal conjugate of X(418), and A′B′C′ is homothetic to the orthic triangle of ABC from X(2) with ratio -2. (Peter Moses, October 4, 2013)
Let A′B′C′ be the cevian triangle of X(3). Let A″B″C″ be the reflection of A′B′C′ in X(3). Let A*B*C be the tangential triangle, with respect to A′B′C′, of the circumconic of A′B′C′ centered at X(3) (that is, the bicevian conic of X(3) and X(394)). The lines A″B*, B″B*, C″C* concur in X(5562). (Randy Hutson, August 17, 2014)
The following items were contributed by Randy Hutson, August 17,
2014:
X(5562) = X(20)-of-orthic-triangle
X(5562)-of-excentral-triangle = X(20)
X(5562)-of-hexyl-triangle = X(4)
X(5562)-of-intouch-triangle = X(950)
X(5562) is the QA-P5 center (Isotomic Center) of the quadrangle
ABCX(4); see
Isotomic Center
.
X(5562) is the QA-P37 center of quadrangle ABCX(4); see
QA-P37. (For certain special cases of
quadrangles, such as orthocentric systems, some QA points coincide.)
The tangents at A, B, C to the Euler central cubic (K044) concur in X(5562), which lies on the Euler central cubic. (Randy Hutson, November 22, 2014)
Let L be the line X(2)X(6). Let U = isogonal conjugate of polar conjugate of L, and let V = polar conjugate of isogonal conjugate of L. Then X(5562) = U∩V. (U = X(3)X(49) and V = X(4)X(69).) (Randuy Hutson, February 16, 2015)
If you have The Geometer's Sketchpad, you can view X(5562).
X(5562) lies on the hyperbola {A,B,C,X(4),X(51)} and these lines: {2,389}, {3,49}, {4,69}, {5,51}, {20,2979}, {26,1495}, {39,3289}, {40,2807}, {99,1298}, {146,2889}, {159,1350}, {195,567}, {216,217}, {255,1364}, {265,3519}, {373,568}, {381,5446}, {399,2918}, {417,2972}, {520,5489}, {542,1205}, {575,1199}, {578,1993}, {631,3819}, {916,1071}, {970,1812}, {1060,1425}, {1062,3270}, {1503,3313}, {2055,3463}, {2072,5449}, {2781,2883}, {2818,3869}, {2888,3153}, {3060,3091}, {3090,3567}, {3564,4173}, {3719,4158}
X(5562) = reflection of X(i) in X(j) for these (i,j): (185,3), (52,5), (3,1216), (1843,1352)
X(5562) = isogonal conjugate of X(8884)
X(5562) = isotomic conjugate of X(8795)
X(5562) = complement of X(5889)
X(5562) = anticomplement of X(389)
X(5662) = X(343)-Ceva conjugate of X(216)
X(5662) = crosspoint of X(3) and X(68)
X(5662) = crosssum of X(4) and X(24)
X(5662) = crossdifference of every pair of points on the line X(421)X(2501)
X(5662) = orthocenter-of-2nd-Euler-triangle
X(5562) = pole wrt polar circle of trilinear polar of X(8794)
X(5562) = X(48)-isoconjugate (polar conjugate) of X(8794)
X(5563) lies on these lines: {1,3}, {2,5258}, {4,4317}, {5,3582}, {10,5253}, {11,546}, {12,3628}, {21,551}, {23,5322}, {30,4325}, {58,106}, {61,5357}, {62,5353}, {73,1173}, {79,104}, {80,1210}, {100,3244}, {101,1475}, {140,3584}, {172,1015}, {191,392}, {202,2307}, {214,3881}, {226,5443}, {229,759}, {376,4309}, {388,499}, {404,519}, {474,3679}, {495,632}, {496,3583}, {497,3529}, {498,1056}, {529,4187}, {535,5046}, {550,3058}, {575,1428}, {576,1469}, {595,1149}, {614,1995}, {908,1125}, {956,1698}, {958,3624}, {993,3616}, {995,1203}, {997,3984}, {1014,3663}, {1054,3987}, {1055,4251}, {1058,4302}, {1066,1450}, {1106,1497}, {1108,1781}, {1124,3592}, {1250,5237}, {1283,1623}, {1290,2718}, {1334,5030}, {1335,3594}, {1376,3632}, {1398,5198}, {1449,2178}, {1478,3086}, {1479,3146}, {1621,3636}, {1696,3973}, {1804,4328}, {1866,1870}, {2067,3299}, {2163,3445}, {2242,2275}, {3085,5265}, {3218,3878}, {3241,4188}, {3530,4995}, {3560,4654}, {3622,5248}, {3635,3871}, {3723,5124}, {3731,5120}, {3754,4861}, {3825,5080}, {3868,4867}, {3869,4880}, {3874,4511}, {3884,4973}, {3892,4881}, {3911,5445}, {3915,4257}, {4225,4658}, {4253,5526}, {4297,5441}, {5302,5506}
X(5563) = {X(1),X(36)}-harmonic conjugate of X(35)
X(5564) lies on these lines: {2,3723}, {7,8}, {10,4360}, {86,519}, {190,3686}, {239,594}, {314,3264}, {316,5015}, {321,4886}, {326,4853}, {333,3977}, {350,4651}, {527,4545}, {536,1654}, {668,1269}, {872,4489}, {894,3629}, {966,4664}, {1086,4478}, {1125,1268}, {1213,4971}, {1266,4746}, {1267,3595}, {1278,4643}, {2345,3759}, {2895,4980}, {3593,5391}, {3619,4402}, {3625,3879}, {3626,4357}, {3661,3763}, {3662,4445}, {3663,4669}, {3664,4701}, {3672,4678}, {3679,3875}, {3729,4034}, {3757,4046}, {3912,4060}, {3975,4043}, {4007,4384}, {4021,4691}, {4389,4668}, {4419,4764}, {4440,4726}, {4675,4772}, {4686,4690}, {4698,4727}, {4699,4851}, {4741,4821}
X(5564) = anticomplement of X(3723)
Let X be a point in the plane of a triangle ABC. Let A* = AX∩BC, and define B* and C* cyclically. Let O(BA*) be the circle having diameter BA* and O(A*C) the circle having diameter A*C. There are two lines tangent to the circles O(BA*) and O(A*C). Let UA be the inner one (i.e., closer to A) and VA the outer. Define UB and UC cyclically and VB and VC cyclically. Let A′ = VB∩VC, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in the outer Dominican image of X, denoted by D(X). Let A″ = UB∩UC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in the inner Dominican image of X, denoted by E(X). (Emmanuel José GarcÃÂa, September 28, 2013)
Suppose that X = x : y : z (barycentrics). Let
f(a,b,c,x,y,z) =
a2[a2y2z2 -
x2(2(y + z)(yz)1/2S + y2SB
+ z2SC)] and
g(a,b,c,x,y,z) = a2[a2y2z2
- x2(2(y + z)(yz)1/2(- S) +
y2SB + z2SC)].
Then D(X) = f(a,b,c,x,y,z) : f(b,c,a,y,z,x) : f(c,a,b,z,x,y) and E(X) = g(a,b,c,x,y,z) : g(b,c,a,y,z,x) : g(c,a,b,z,x,y).
Note that D(X) and E(X) lie in the real plane of ABC if and only if X lies inside ABC; equivalently, yz >0, zx > 0, xy > 0. (Peter Moses, September 30, 2013)
If the construction is modified by using the A-internal tangent and the B- and C- external tangents, the resulting triangle is perspective to ABC, and likewise for 5 other perspectivities, for a total of 8 perspectors, of which only two (D(X) and E(X)) are central if X is central. The 8 perspectors are given by barycentrics
a2[a2y2z2 - x2(2(y + z)(yz)1/2S*i + y2SB + z2SC )] : b2[b2z2x2 - y2(2(z + x)(zx)1/2S*j + z2SB + x2SC )] : c2[c2x2y2 - z2(2(x + y)(xy)1/2S*k + x2SB + y2SC)],
where (i,j,k) ranges through 8 3-tuples listed here as additive-inverse pairs: (-1,-1,-1) & (1,1,1), (-1,-1,1) & (1,1,-1), (-1,1,-1) & (1,-1,1), (-1,1,1) & (1,-1,-1). Each pair determines a line, and the four lines concur in the point having 1st barycentric
a2t/(t2 - w2), where t = x2(y2SB + z2SC - a2y2z2, w = 2x2(y + z)(yz)1/2S.
The 4 lines determined by pairs differing only in the first coordinate, such as (-1,1,1) & (1,1,1), concur in A; those 4 differing only in the 2nd coordinate concur in B, and those 4 differing only in the 3rd coordinate concur in C. (Peter Moses, October 1, 2013)
X(5565) lies on these lines: {}
For definitions and discussion, see X(5565).
X(5566) lies on these lines: {}
For definitions and discussion, see X(5565).
X(5567) lies on these lines: {}
For definitions and discussion, see X(5565).
X(5568) lies on these lines: {}
Let ABC be a triangle, and let AB be the center of the circle through A and tangent to the B-median at X(2), and define BC and CA cyclically. Let AC be the center of the circle through A and tangent to the C-median at X(2), and define BA and CB cyclically. The points AB, BA, BC, CB, CA, AC lie on a circle, of which X(5569) is the center. (Dao Thanh Oai, Nov. 3, 2013)
The following properties were communicated by Peter Moses, November 4, 2013. Let Δ = area of ABC, r = radius of the Dao 6-point circle, and ω = Brocard angle of ABC. Let fa = 2b2 + 2c2 - a2, and define fb and fc cyclically. Then
r = [fafbfc(b2c2 + c2a2 + a2b2)]1/2/(144Δ)2
|ABBA| = |BCCB| = |CAAC| = [fafbfc]1/2/(36Δ)
Let X = X(5569). Then angle(ABXBA) = angle(BCXCB) = angle(CAXAC) = Tan-1[(a2 + b2 + c2)/(4Δ)]
angle(ABBAX) = angle(BCCBX) = angle(CAACX) = π/2 - ω
If you have The Geometer's Sketchpad, you can view X(5569).
X(5569) lies on these lines: {2,187}, {3,543}, {182,524}, {183,2482}, {538,3524}, {599,620}, {754, 5054}, {3406, 5503}, {5077,5461}
X(5569) = midpoint of X(2) and X(8182)
X(5569) = reflection of X(2) in X(1153)
X(5569) = harmonic center of medial-van Lamoen and anticomplementary-van Lamoen circles
X(5570) lies on these lines: {1,3}, {11,912}, {72,499}, {496,1858}, {498,5439}, {515,5083}, {518,1737}, {938,5080}, {971,3583}, {1066,1393}, {1071,1479}, {1210,3814}, {1785,1876}, {2771,5533}, {3086,3868}, {3873,5176}
X(5570) = inverse-in-de-Longchamps-ellipse of X(3)
X(5570) = {X(2446),X(2447)}-harmonic conjugate of X(3)
X(5570) = X(2072)-of-intouch-triangle. (Randy Hutson, July 18, 2014)
Let ABC be a triangle. Let A′ = (inverse-in-incircle) of A, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the inverse-in-incircle triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 9, 2013:
Barycentric coordinates for vertices:
A′ = (b - c)2 - ab - ac : b(a - b - c) : c(a - b - c)
B′ = a(b - c - a) : (c - a)2 - bc - ba : c(b - c - a)
C′ = a(c - a - b) : b(c - a - b) : (a - b)2 : a(c - a - b).
|B′C′|2 = (a - b - c)2[a2 - (b -
c)2]/(16bc)
area(A′B′C′)/area(ABC) = (b + c - a)(c + a - b)(a + b - c)/(16abc)
X(354) = centroid of A′B′C′
X(942) = circumcenter of A′B′C′
X(1) = orthocenter of A′B′C′
X(5045) = nine-point center of A′B′C′
The following triangles are perspective to A′B′C′, with perspector X(1): reflection of T in X(3), excentral, incentral, mid-arc, 2nd circumperp, mixtilinear, 1st circumperp. Also, A′B′C′ is perspective to other central triangles, with perspectors as shown:
medial, X(142)
intouch, X(354)
hexyl, X(3333)
2nd circumperp, X(57)
X(5571) lies on these lines: {1,164}, {65,209}, {177,354}
X(5571) = X(10)-of-intouch-triangle. (Randy Hutson, July 18, 2014)
See X(5571) for the inverse-in-incircle triangle.
X(5572) = X(7) - 3X(354) (Peter Moses, November 9, 2013)
X(5572) = X(141)-of-intouch-triangle. Let A′ be the inverse-in-incircle of the A-excenter, and define B′ and C′ cyclically. Then X(5572) = X(9)-of-A′B′C′. (Randy Hutson, July 18, 2014)
X(5572) lies on these lines: {1,6}, {2,3059}, {7,354}, {55,1445}, {57,4326}, {65,390}, {105,2264}, {142,2886}, {144,3873}, {241,2293}, {480,3870}, {516,942}, {938,2550}, {946,971}, {982,4335}, {1210,3826}, {1376,3174}, {1387,2801}
X(5572) = complement of X(3059)
X(5572) = X(6)-of-inverse-in-incircle triangle
Let A′ be the center of the inverse-in-incircle of the A-excircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Andromeda triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 9, 2013:
Barycentric coordinates for vertices:
A′ = a[a2 + 3(b - c)2] : b[3a2 + (b -
c)2] : c[3a2 + (b - c)2]
B′ = a[3b2 + (c - a)2] : b[b2 + 3(c -
a)2] : c[3b2 + (c - a)2]
C′ = a[3c2 + (a - b)2] : b[3c2 + (a -
b)2] : c[c2 + 3(a - b)2]
The triangle A′B′C′ is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, mixtilinear, 2nd circumperp. Also, A′B′C′ is perspective to the intouch triangle at X(4907).
X(5573) lies on these lines: {1,474}, {2,3677}, {9,982}, {31,57}, {43,3243}, {165,1279}, {223,3660}, {238,3928}, {354,2999}, {748,3929}, {988,5436}, {1054,3749}, {1086,1699}, {1104,3361}, {1191,3339}, {1201,3340}, {1261,3872}, {1420,3924}, {1453,3338}, {1722,3976}, {2276,3247}, {2886,4859}, {3305,4392}, {3306,5269}, {3315,3870}, {3452,4310}, {3756,3772}, {3915,5128}, {3999,4383}, {4003,4423}, {4666,4850}, {4907,5274}
X(5573) = complement of X(5423)
Let A′ be the center of the inverse-in-A-excircle of the incircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Antlia triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 10, 2013:
Barycentric coordinates for vertices:
A′ = a[a2 + 3(b - c)2] : - b[3a2 + (b
- c)2] : - c[3a2 + (b - c)2]
B′ = - a[3b2 + (c - a)2] : b[b2 + 3(c
- a)2] : - c[3b2 + (c - a)2]
C′ = - a[3c2 + (a - b)2] : - b[3c2 +
(a - b)2] : c[c2 + 3(a - b)2]
The triangle A′B′C′ is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, mixtilinear, 2nd circumperp.
X(5574) lies on these lines: {2,479}, {9,165}, {200,3119}, {2391,3452}, {3041,5223}, {3817,5199}
See X(5574) for the Antlia triangle.
X(5575) lies on these lines: {7,346}, {57,1122}, {269,604}, {1463,5223}, {1469,3339}, {3062,4014}
As a point on the Euler line, X(5576) has Shinagawa coefficients (E + 4F, 3E + 4F).
X(5576) lies on these lines: {2,3}, {51,5449}, {125,5462}, {143,3580}, {195,3564}, {511,1209}, {524,3519}, {570,1506}, {1199,3448}
X(5576) = polar-circle-inverse of X(37932)
X(5576) = center of inverse-in-polar-circle-of-tangential-circle
X(5576) = center of inverse-in-{circumcircle, nine-point circle}-inverter-of-tangential-circle
X(5576) = inverse-in-orthocentroidal-circle of X(26)
The inverter of circles (U,u) and (V,v) is introduced here as the circle (W,w) such that (V,v) is the inverse-in-(W,w) of (U,u), where W is the insimilicenter of (U,u) and (V,v).
Peter Moses (Nov. 12, 2013) found representations for W and w, as follows. The center W of (W,w) is the combo uV + vU; that is, barycentrics for W are given by u(vA, vB, vC) + v(uA, uB, uC), where (uA, uB, uC) are normalized barycentrics for U, and (vA, vB, vC) are normalized barycentrics for V. The radius of (W,w) is w = sqrt[uv(1 - (|UV|/(u + v))2], so that the inverter is real if and only if u + v >= |UV|. Moses also gave properties for the case that (U,u) = (O,R) = circumcircle and (V,v) = (I,r) = incircle, for which the inverter is given by (W,w) = (X(55), (r/(r + R))sqrt(rR + 4R2)). The power of A with respect to (W,w) is
- abc(b + c - a)2/D, where D = 2(a3 + b3 + c2 - a2b - a2c - ab2 - ac2 - b2c - bc2); likewise, (power of B) = - abc(a - b + c)2/D and (power of C) = - abc(a + b - c)2/D.
The alternate inverter of circles (U,u) and (V,v) is introduced here as the circle (W′,w') such that (V,v) is the inverse-in-(W,w) of (U,u), where W′ is the exsimilicenter of (U,u) and (V,v). The center W′ is the combo uV - vU, and the radius w' of W′ is given by sqrt[uv(- 1 + (|UV|/(u - v))2], so that the alternate inverter is real if and only if |u - v| <= |UV|. (Peter Moses, September 3, 2014)
The appearance of (i,j) in the following list means that X(i) is on the circumcircle, X(j) is on the incircle, and each is the inverse-in-(W,w) of the other: (98, 5578), (99, 5579), (100, 3021), (101, 5580), (103, 1364), (105, 11), (106, 5577), (108, 1360), (109, 1362), (840, 3025), (934, 3321), (939, 5582), (972, 3318), (1381, 2447), (1382, 2446), (1477, 1357), (2222, 3322), (2291, 3022), (2384, 5583), (2717, 3326).
The barycentrics for X(5577) are of the form g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b + c - a)(b - c)2[b2 + c2 - a2 - kbc]2, where k is a homogeneous of degree zero and symmetric in (a,b,c); every such point lies on the incircle. (Peter Moses, August 28, 2014)
If you have The Geometer's Sketchpad, you can view Inverter.
X(5577) lies on the incircle and these lines: {55,106}, {57,1361}, {244,1364}, {354,1317}, {1086,3326}, {1362,4860}, {3025,3271}, {3319,3660}
Inverters are dicussed at X(5577).
X(5578) lies on the incircle and these lines: {55,98},
{354,1355}
Inverters are discussed at X(5577).
X(5579) lies on the incircle and these lines: {11,4357}, {55,99}, {354,1356}, {1357,3664}, {1358,3666}, {1365,3663}
Inverters are discussed at X(5577).
X(5580) lies on the incircle and these lines: {11,142}, {55,101}, {354,1358}, {1357,4860}, {1364,3056}, {1365,4890}
Inverters are discussed at X(5577).
X(5581) lies on the incircle and this line: {55,739}
Inverters are discussed at X(5577).
X(5582) lies on the incircle and this line: {55, 2384}
Inverters are discussed at X(5577).
X(5583) lies on these lines: {55, 523}, {1946, 6362}, {3737, 10383}
Let A′ be the insimilicenter of the circumcircle and A-excircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Apus triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 12, 2013:
Barycentric coordinates for vertices:
A′ = a2(a - b + c)(a + b - c) : b2(b - c - a)(a +
b + c) : c2(c - b - a)(a + b + c)
B′ = a2(a - c - b)(a + b + c) : b2(b - c + a)(b +
c - a) : c2(c - a - b)(a + b + c)
C′ = a2(a - b - c)(a + b + c) : b2(b - a - c)(a +
b + c) : c2(c - a + b)(c + a - b)
The triangle A′B′C′ is perspective to ABC at X(55), the excentral and hexyl triangles at X(3), the incentral triangle at X(56), the tangential triangle at X(198), the Feuerbach triangle at X(4), and the Apollonius triangle at X(573).
The Apus triangle is the extraversion triangle of X(56). (Randy Hutson, July 18, 2014)
Let A′B′C′ be the intouch triangle of the extangents triangle, if ABC is acute. Then A′B′C′ is homothetic to the cevian triangle of X(3) at X(5584). (Randy Hutson, December 2, 2017)
X(5584) lies on these lines: {1,3}, {4,3925}, {6,4300}, {19,1212}, {20,958}, {64,71}, {72,480}, {104,3528}, {201,1854}, {210,1490}, {212,221}, {218,573}, {227,1035}, {380,5120}, {405,516}, {411,1376}, {946,4423}, {954,3671}, {956,4297}, {962,1001}, {1042,1253}, {1151,5415}, {1152,5416}, {1204,3611}, {1350,3779}, {1407,1496}, {1742,5247}, {1753,1859}, {1802,3207}, {1804,3160}, {2266,4258}, {2951,5234}, {2975,3522}, {3146,5260}, {3149,4413}, {5248,5493}
Let A′B′C′ be the tangential triangle, so that A′ is the center of the circle OA through B and C that is orthogonal to the circumcircle (whence OA is self-inverse with respect to the circumcircle). Define OB and OC cyclically. Let O(A,B) be the circle which is the inverse-in-OA of OB; define O(B,C) and O(C,A) cyclically. Let O(A,C) be the circle which is the inverse-in-OA of OC; define O(B,A) and O(C,B) cyclically. The centers of these six circles lie on a conic, introduced here as the Aquarius conic, of which X(5585) is the center. The following properties were found by Peter Moses (Nov. 18, 2013).
The centers of the 6 circles are given by the following barycentrics:
- a2 : b2 :
3c2, 3a2 : -
b2 : c2
a2 : 3b2 : - c2;
- a2 : 3b2 :
c2, a2 : -
b2 : 3c2
3a2 : b2 : - c2
The radius of O(A,B) is abc/(-a2 + b2 + 3c2); the remaining 5 radii are found by cyclical and bicentric modifications. The Aquarius conic has equation
b4c4x2 + c4a4y2 + a4b4z2 + 11a2b2c2(a2yz + b2zx + c2xy) = 0
The major axis of the Aquarius conic is the Brocard axis, and the perspector is X(6). (Randy Hutson, November 30, 2013)
X(5585) lies on these lines: {3,6}, {20,3054}, {439,3619}, {3055,3523}
Let A′ = reflection of the incenter in A, and define B′ and C′ cyclically. The triangle A′B′C′ is introduced here as the Aquila triangle (in TCCT, , p. 173, as T(1,2)). The following properties were found by Peter Moses (Nov. 18, 2013).
A′ = a + 2b + 2c : -a : -a (trilinears)
B′ = -b : b + 2c + 2a : -b
C′ = -c : -c : c + 2a + 2b
area(A′B′C′) = 4*area(ABC).
The Aquila triangle is perspective to the following triangles with perspector X(1): ABC, excentral, incentral, mid arc, 2nd circumperp, and mixtilinear. The Aquila triangle is perspective to other triangles with perspectors as listed here: medial, X(1698); anticomplementary, X(10); intouch, X(5586); Euler, X(1699); hexyl, X(1768); tangential 1st circumperp, X(35); tangential 2nd circumperp, X(36); Carnot, X(5587); outer Grebe, X(5588); inner Grebe, (X5589).
X(5586) lies on these lines: {1,376}, {7,10}, {57,191}, {65,3632}, {145,4298}, {388,4114}, {942,4312}, {986,4888}, {1046,4859}, {1317,3340}, {1537,1768}, {1698,3715}, {1788,3982}, {3361,3616}, {3485,4031}, {3600,3635}, {3633,5434}
The Aquila triangle is introduced at X(5586). Not only is X(5587) the perspector of the Aquila and Carnot triangles, but also, X(5587) is also the perspector of the Euler triangle and the outer Garcia triangle, defined as follows. Let TATBTC be the extouch triangle of a triangle ABC, and let LA be the line perpendicular to line BC at TA. Of the two points on LA at distance r from TA, let A′ be the one farther from A and let A″ be the closer. Define B′, C′ and B″, C″ cyclically. We call A′B′C′ the outer Garcia triangle and A″B″C″ the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014). In subsequent postings, Paul Yiu reports that A′B′C′ is oppositely congruent to ABC at X(10), the Euler lines of the four triangles AB′C′, BC′A′, CA′B′, ABC concur in X(2475), and the circles (BCA′), (CAB′), (ABC′) concur in X(80). Peter Moses reports that
A′ = -a : a + c : a + b, B′ = b + c : -b : b + a, C′ = c + b : c + a : -c
A″ = a2 : ca + c2 - b2 : ab + b2 - c2, B″ = bc + c2 - a2 : b2 : ab + a2 - c2 , C″ = bc + b2 - a2 : ca + a2 - b2 : c2
Randy Hutson notes that the inner Garcia triangle is the anticomplement of orthic-triangle-of-Fuhrmann-triangle, that X(5587) = X(2)-of-Fuhrmann triangle, and X(5587) = {X(355), X(5)}-harmonic conjugate of X(1).
Let MA be the midpoint of segment BC and A′ the reflection of X(1) in MA; define B′ and C′ cyclically. Then A′B′C′ is the outer Garcia triangle. See Emmanuel José GarcÃa, A Note on Reflections.
Triangle A′B′C′ is perspective to the cevian triangle of the points on the cubic (K366) and to the anticevian triangles of points on the cubic (K345), as well at the following triangles, with perspectors:
medial, X(1)
excentral, extouch, extangents, X(40)
anticomplementary, Fuhrmann, X(8)
circumcircle midard, 2nd circumperp, X(100)
tangential 1st circumperp, X(5587)
tangential 2nd circumperp, X(956)
Euler, X(5587)
outer Grebe, X(5688)
inner Grebe, X(5689)
The appearance of (i,j) is the following list means that (X(i) of A′B′C′) = X(j): (1,8), (2,3679), (3,355), (4,40), (5, 5690), (6,3416), (7,5223), (8,1), (9,2550), (10,10), (11,1145), (20,5691). (Peter Moses, June 21, 2014)
The point A′ is also the orthocenter of BCIA and cyclically for B′ and C′, where IA, IB, IC are the excenters. (Randy Hutson, October 8, 2019)
X(5587) lies on these lines: {1,5}, {2,515}, {3,1698}, {4,9}, {8,908}, {30,165}, {35,3560}, {46,3585}, {55,3586}, {57,1478}, {63,5080}, {78,5086}, {84,377}, {145,5068}, {149,3895}, {153,3306}, {200,3419}, {210,381}, {235,5090}, {262,730}, {265,2948}, {282,1549}, {376,3828}, {382,3579}, {388,1210}, {404,5450}, {411,5260}, {442,1490}, {498,3601}, {499,1420}, {519,3545}, {547,3655}, {551,5071}, {631,3634}, {912,4654}, {936,1329}, {938,5261}, {942,5290}, {944,1125}, {950,3085}, {956,5231}, {958,3149}, {962,3617}, {997,3814}, {1000,4342}, {1012,1376}, {1071,3812}, {1158,2475}, {1350,3844}, {1352,3751}, {1385,1656}, {1453,5230}, {1479,1697}, {1482,3632}, {1532,2886}, {1537,3036}, {1572,5475}, {1702,3071}, {1703,3070}, {1709,3359}, {1724,3072}, {1750,3925}, {1770,5128}, {1771,1935}, {1785,1857}, {1788,4292}, {1834,2910}, {1836,2093}, {1853,3753}, {2364,5397}, {2782,3097}, {3073,5264}, {3338,5270}, {3416,5480}, {3421,4847}, {3487,3947}, {3583,5119}, {3616,5056}, {3626,3855}, {3633,5072}, {3646,5084}, {3654,3845}, {3656,4677}, {3850,4668}, {3854,4678}, {3872,5176}, {3911,4293}, {3949,4007}, {4295,4848}, {4304,5218}, {5046,5250}, {5087,5289}
X(5587) = midpoint of X(1699) and X(3679)
X(5587) = reflection of X(i) in X(j) for these (i,j): (1699, 381), (3576,2)
X(5587) = crossdifference of every pair of points on the line X(654)X(1459)
X(5587) = homothetic center of hexyl and 4th Euler triangles
X(5587) = centroid of the six touchpoints of the Johnson circles and the outer Johnson triangle
The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.
X(5588) lies on these lines: {1,6}, {10,1270}, {40,1160}, {1374,1738}, {1698,5590}
The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.
X(5589) lies on these lines: {1,6}, {10,1271}, {40,1161}, {1373,1738}, {1698,5591}
The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.
X(5590) = {X(2), X(141)}-harmonic conjugate of X(5591)
X(5590) lies on these lines: {2,6}, {3,5594}, {4,639}, {5,1160}, {8,5604}, {10,3640}, {76,5490}, {626,638}, {631,641}, {640,3090}, {642,3525}, {1162,1165}, {1267,3662}, {1698,5588}, {3535,5413}, {3661,5391}
X(5590) = homothetic center of ABC and cross-triangle of ABC and outer Grebe triangle
The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.
X(5591) = {X(2), X(141)}-harmonic conjugate of X(5590)
X(5591) lies on these lines: {2,6}, {3,5595}, {4,640}, {5,1161}, {8,5605}, {10,3641}, {76,5491}, {626,637}, {631,642}, {639,3090}, {641,3525}, {1163,1164}, {1267,3661}, {1698,5589}, {3536,5412}, {3662,5391}
X(5591) = homothetic center of ABC and cross-triangle of ABC and inner Grebe triangle
If you have The Geometer's Sketchpad, you can view X(5592).
X(5592) lies on these lines: {1,514}, {20,3667}, {513,960}, {659,2785}, {661,5051}, {764,4778}, {1960,4458}, {2789,3762}, {2899,3239}
X(5592) = Yff-contact-isogonal conjugate of X(8)
The Yiu conic is presented at X(478); it passes through the 6 of the nine touch-points of the sidelines of a triangle and the excircles of the triangle. When the triangle is the tangential, the conic has center X(5593).
Let u(a,b,c) = 4a2b4c4 and v(a,b,c) = a2(a8 + b8 + c8 - 2a6b2 - 2a6c2 + 2a4b4 + 2a4c4 - 2a2b6 - 2a2c6 + 6a2b4c2 + 6a2b2c4 - 4b6c2 - 4b6c2 + 6b4c4).
The Yiu conic of the tangential triangle of a triangle ABC is given by
u(a,b,c)x + u(b,c,a)y + u(c,a,b) z + v(a,b,c)yz + v(b,c,a)zx + v(c,a,b)xy = 0. (Peter Moses, Nov. 19, 2013)
If you have The Geometer's Sketchpad, you can view X(5593).
X(5593) lies on these lines: {4,157}, {184,216}
Let A′B′C′ be the tangential triangle of triangle ABC. Let A″ be the center of the A′-excircle of A′B′C′, unless this is also the circumcircle of ABC, in which case let A″ be the incenter of A′B′C′. Define B″, C″ cyclically. The triangle A″B″C″ is introduced here as the Ara triangle, which appears in the sketch at X(5593). The vertices of the Ara triangle are given by Peter Moses (Nov. 19, 2013):
A″ = - a2(a2 + b2 + c2)
: b2(a2 + b2 - c2) :
c2(a2 - b2 + c2)
B″ = a2(b2 - c2 + a2) : -
b2(b2 + c2 + a2) :
c2(b2 + c2 - a2)
C″ = a2(c2 + a2 - b2) :
b2(c2 - a2 + b2) : -
c2(c2 + a2 + b2)
The Ara triangle is perspective to triangles as listed here with perspectors: ABC, X(25); anticomplementary, X(22); Euler, X(1598); tangential 1st circumperp, X(197). (Peter Moses, Nov. 19, 2013)
The Ara triangle is homothetic to triangle ABC. If ABC is acute then the Ara triangle is the excentral triangle of the tangential triangle. (Randy Hutson, August 17, 2014)
X(5594) lies on these lines: {6,25} et al
X(5594) = {X(25),X(159)}-harmonic conjugate of X(5595)
The Ara triangle is introduced at X(5594).
X(5595) lies on these lines: {6,25} et al
X(5595) = {X(25),X(159)}-harmonic conjugate of X(5594)
Let A′B′C′ be the tangential triangle of an acute triangle ABC. Let A″ be the touchpoint of the A-excircle of A′B′C′ and the line B′C′; define B″ and C″ cyclically. The triangle A″B″C″ is introduced here as the Aries triangle, with vertices given by Peter Moses (Nov. 21, 2013).
A″ = a4 + b4 + c4 -
2b2c2 : 2b2(c2 -
b2) : 2c2(b2 - c2)
B″ = 2a2(c2 - a2) : b4 +
c4 + a4 - 2c2a2 :
2c2(a2 - c2)
C″ = 2a2(b2 - a2) :
2b2(a2 - b2) : c4 +
a4 + b4 - 2a2b2
The Aries triangle is perspective to the tangential triangle, with perspector X(1498).
If ABC is acute, the Aries triangle is the extouch triangle of the tangential triangle. (Randy Hutson, July 18, 2014)
If ABC is acute, then the vertices of the Aries triangle lie on the cubic K075. (César Lozada, October 21, 2015)
If you have The Geometer's Sketchpad, you can view X(5596).
X(5596) lies on these lines: {2,66}, {4,6}, {20,3313}, {22,69}, {110,2892}
X(5596) = isogonal conjugate of X(34427)
X(5596) = isotomic conjugate of isogonal conjugate of X(20993)
X(5596) = complement of X(20079)
X(5596) = anticomplement of X(66)
X(5596) = polar conjugate of isogonal conjugate of X(22135)
Let U be the inverter of the circumcircle and incircle, as described at X(5577). There are two triangles that circumscribe U and are homothetic to triangle ABC, one of which has A-vertex on the same side of line BC as A. This triangle, A′B′C′, is introduced here as the 1st Auriga triangle, and the other, as the 2nd Auriga triangle. Each is the reflection of the other in the center, X(55), of the inverter. The six points A′, B′, C′, A″, B″, C″ lie on a conic introduced here as the Auriga conic. Let D = (rR + 4R2)1/2; barycentrics for the six points and conic were found by Peter Moses (Nov. 21, 2013):
A′ = a4 - a2(b + c)2 - 4(b + c)SD : b4 - b2(c + a)2 + 4bSD : c4 - c2(a + b)2 + 4cSD
A″ = a4 - a2(b + c)2 + 4(b + c)SD : b4 - b2(c + a)2 - 4bSD : c4 - c2(a + b)2 - 4cSD
where B′, C′, B″, C″ are determined cyclically.
The Auriga conic is given by {cyclic sum[g(a,b,c)x2 + h(a,b,c)yz} = 0, where
g(a,b,c) = bc(b + c - a)(b5 + c5 + 3a3bc + a2b3 + a2c3 - a2b2c - a2bc2 + ab3c + abc3 - 2ab2c2 + 3b4c + 3bc4 - 2b3c2 - 2b2c3)
h(a,b,c) = a[a7 - 2a6(b + c) -
a5(b2 + c2 - 4a5bc) +
4a4(b3 + c3)
- a3(b4 + c4 + 4b3c +
4bc3 - 8b2c2)
- 2a2(b5 + c5 - b4c -
bc4 + 5b3c2 +
5b2c3)
+ a(b6 + c6 + b4c2 +
b2c4 - 4b3c3)
+ 2(b5c2 + b2c5 -
b4c3 - b3c4)
The two Auriga triangles are perspective with perpsector X(55), which is the center of the Auriga conic.
If you have The Geometer's Sketchpad, you can view X(5597).X(5597) lies on these lines: {1,3}, {2,5599}, {8,5600}, {145,5602}
X(5597) = {X(1), X(55)}-harmonic conjugate of X(5598)
X(5597) = exsimilicenter of circumcircles of ABC and 1st Auriga triangle; the insimilicenter is X(11822)
The 1st and 2nd Auriga triangles are introduced at X(5597).
If you have The Geometer's Sketchpad, you can view X(5598).
X(5598) lies on these lines: {1,3}, {2,5600}, {8,5599}, {145,5601}
X(5598) = {X(1), X(55)}-harmonic conjugate of X(5597)
X(5598) = insimilicenter of circumcircles of ABC and 2nd Auriga triangle; the exsimilicenter is X(11823)
The 1st and 2nd Auriga triangles are introduced at X(5597).
X(5599) = {X(10), X(55)}-harmonic conjugate of X(5600)
X(5599) lies on these lines: {2,5597}, {8,5598}, {10,55}, {3617,5602}
The 1st and 2nd Auriga triangles are introduced at X(5597).
X(5600) = {X(10), X(55)}-harmonic conjugate of X(5599)
X(5600) lies on these lines: {2,5598}, {8,5597}, {10,55}, {3617,5601}
The 1st and 2nd Auriga triangles are introduced at X(5597).
X(5601) = {X(8), X(55)}-harmonic conjugate of X(5602)
X(5601) lies on these lines: {{2,5597}, {8,21}, {145,5598}, {3617,5600}
The 1st and 2nd Auriga triangles are introduced at X(5597).
X(5602) = {X(8), X(55)}-harmonic conjugate of X(5601)
X(5602) lies on these lines: {2,5598}, {8,21}, {145,5597}, {3617,5599}
The reflection of triangle ABC in the incenter, here called the Caelum triangle, is perspective to the medial triangle with perspector X(8), to the anticomplementary and intouch triangles with perspector X(145), and to the Euler triangle at X(5603). See also X(5604) and X(5605). The vertices of the Caelum triangle were found by Peter Moses (Nov. 21, 2013):
A′ = a - b - c : 2a : 2a B′ = 2b : b - c - a : 2b C′ = 2c : 2c : c - a - b.
Let A′ be the orthocenter of triangle BCX(7), and define B′ and C′ cyclically. Then X(5603) is the centroid of A′B′C′. (Randy Hutson, November 22, 2014)
X(5603) lies on these lines: {1,4}, {2,392}, {3,962}, {5,8}, {7,104}, {10,3090}, {11,2099}, {12,2098}, {20,1385}, {29,945}, {36,3474}, {40,631}, {56,4295}, {65,3086}, {78,5082}, {79,4317}, {84,3296}, {86,4221}, {119,1320}, {140,5550}, {145,355}, {165,3524}, {281,1953}, {329,956}, {376,516}, {381,952}, {495,1532}, {496,938}, {498,5443}, {499,1788}, {519,3545}, {546,1483}, {908,3421}, {912,3873}, {929,953}, {971,5049}, {995,4000}, {997,2550}, {1000,1512}, {1001,1006}, {1060,4318}, {1065,3478}, {1071,5045}, {1158,3338}, {1210,3340}, {1279,3332}, {1312,2102}, {1313,2103}, {1319,1836}, {1420,4292}, {1468,3073}, {1476,5553}, {1698,5067}, {1829,3089}, {1872,4200}, {1902,3088}, {2093,3911}, {2476,5330}, {2646,4294}, {2792,5429}, {2801,3892}, {2829,5434}, {2886,5289}, {2975,3560}, {3057,3085}, {3072,3915}, {3149,3295}, {3242,5480}, {3244,3855}, {3304,3649}, {3306,3359}, {3333,3671}, {3434,4511}, {3436,4861}, {3523,3579}, {3525,3624}, {3529,3636}, {3543,3655}, {3617,5056}, {3621,5068}, {3623,3832}, {3679,5071}, {3820,5328}, {4193,5554}, {5048,5252}, {5119,5218}, {5450,5563}
X(5603) = midpoint of X(1) and X(1699)
X(5603) = relection of X(i) in X(j) for these (i,j): (4,1699), (376,3576), (1699,946), (3576,551)
X(5603) = {X(1),X(4)}-harmonic conjugate of X(944)
The Caelum triangle is defined at X(5603).
X(5604) = {X(1), X(3242)}-harmonic conjugate of X(5605)
X(5604) lies on these lines: {1,6}, {8,5590}, {145,1270}, {1160,1482}
The Caelum triangle is defined at X(5603).
X(5605) = {X(1), X(3242)}-harmonic conjugate of X(5604)
X(5605) lies on these lines: {1,6}, {8,5591}, {145,1271}, {1161,1482}
Let I be the incenter and A′ the nine-point center of triangle IBC. Define B′ and C′ cyclically. The circles AB′C′, BC′A′, CA′B′ concur in X(5606). (Antreas Hatzipolakis, June 2, 2013: see Concurrent Circumcircles)
Let L be the Euler line of the incentral triangle of ABC, so that L is the line X(500)X(1962). Let LA be the reflection of L in line 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(5606). (Randy Hutson, July 18, 2014)
X(5606) lies on the circumcircle and these lines: {74,1385}, {229,759}, {2372,5253}
X(5606) = isogonal conjugate of X(8702)
X(5606) = anticomplement of X(5952)
X(5606) = cevapoint of X(513) and X(3337)
X(5606) = trilinear pole of line X(6)X(3336)
X(5606) = Ψ(X(6), X(3336))
A construction of the 1st Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617). (Dao Thanh Oai and Peter Moses, Nov., 2013)
X(5607) lies on this line: {526, 5608}
A construction of the 2nd Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617). (Dao Thanh Oai and Peter Moses, Nov., 2013)
X(5608) lies on this line: {526, 5607}
See X(5607) and X(5608). (Dao Thanh Oai and Peter Moses, Nov., 2013)
Let A′B′C′ be the Euler 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 line BC, and define M′ and N′ cyclically. The lines L′, M′, N′ concur in X(5609); c.f., X(113), X(399), and X(1511). Let NA be the reflection of X(5) in the perpendicular bisector of BC, and define NB and NC cyclically. Then X(5609) = X(23)-of-NANBNC. (Randy Hutson, July 18, 2014)
X(5609) lies on these lines: {3,74}, {5,542}, {23,1154}, {30,3292}, {113,137}, {125,3628}, {526,5607}
X(5609) = circumcircle- inverse of X(32609)
X(5610) lies on these lines: {13,531}, {15,110}, {511,2379}
X(5610) = reflection of X(2378) in X(15)
X(5611) lies on these lines: {3,6}, {5,303}, {110,3129}, {147,1080}, {381,531}, {623,1656}, {1993,3131}, {3060,3132}, {5464,5476}
X(5611) = reflection of X(i) in X(j) for these (i,j): (3,15), (621,5)
X(5612) lies on these lines: {3,3201}, {5,14}, {15,399}, {16,323}, {62,195}, {3166,5238}
X(5612) = X(13)-Ceva conjugate of X(16)
X(5612) = trilinear product X(16)*X(1749)
Lengths of segments: |X(5613)X(13)| = |X(5617)X(14)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier GarcÃa Capitán), ADGEOM #1256, April 23, 2014)
The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco GarcÃa Capitán), ADGEOM #1259, April 23, 2014)
The Euler line is tangent to the circle {{X(3), X(5613), X(5617)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)
Suppose that P is a point. The P-Fuhrmann triangle is the triangle A″B″C″, where A″ is the reflection in line BC of the A-vertex of the cercumcevian triangle of P, and B″ and C″ are defined cyclically (so that taking P to be X(1), X(4), and X(6) yields the classical Fuhrmann triangle, the single-point triangle X(4), and the 4th Brocard triangle, respectively. The X(16)-Fuhrmann triangle is equilateral, and X(5613) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5617). (Randy Hutson, July 7, 2014)
Let BA′C be the equilateral triangle with side BC and A′ on the side of BC that includes A. Let La be the line through A′ parallel to BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″and C″ cyclically. The Euler lines of B′A″C′, C′B″A′, A′C″B′ concur in X(5613). The acute angle between each pair of these Euler lines is π/6. See Dao Thanh Oai, ADGEOM 4030), and Francisco Javier GarcÃa Capitán, ADGEOM 4031). See also X(5617) and X(14144).
X(5613) lies on these lines: {2,98}, {3,619}, {4,617}, {5,14}, {13,2782}, {30,5464}, {99,622}, {299,383}, {303,1080}, {381,531}, {395,3564}, {576,3180}, {5055,5460}
X(5613) = midpoint of X(i) and X(j) for these (i,j): (4,617), (299,383)
X(5613) = reflection of X(i) in X(j) for these (i,j): (14,5), (3,619), (5617,114)
X(5613) = inner-Napoleon-to-outer-Napoleon similarity image of X(3)
X(5613) = outer-Napoleon-isogonal conjugate of X(16)
X(5614) lies on these lines: {14,530}, {16,110}, {511,2378}
X(5614) = reflection of X(2379) in X(16)
X(5615) lies on these lines: {3,6}, {5,302}, {110,3130}, {147,383}, {381,530}, {624,1656}, {1993,3132}, {3060,3131}, {5463,5476}
X(5615) = reflection of X(i) in X(j) for these (i,j): (3,16), (622,5)
X(5616) lies on these lines: {3, 3200}, {5, 13}, {15, 323}, {16, 399}, {61, 195}, {3165, 5237}
X(5616) = X(14)-Ceva conjugate of X(15)
X(5616) = trilinear product X(15)*X(1749)
Lengths of segments: |X(5617)X(14)| = |X(5613)X(13)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier GarcÃa Capitán), ADGEOM #1256, April 23, 2014)
The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco Javier GarcÃa Capitán), ADGEOM #1259, April 23, 2014)
The Euler line is tangent to the circle {{X(3), X(5617), X(5661)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)
The X(15)-Fuhrmann triangle, defined at X(5613), is equilateral, and X(5617) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5613). (Randy Hutson, July 7, 2014)
Let BA′C be the equilateral triangle having A′ on the side of BC that does not include C. Let LA be the line through A′ parallel to BC, and define Lb and Lc cyclicalloy. Let A″ = Lb∩Lc, and define B″ and C″ cyclically.The Euler lines of B′A″C′, C′B″A′, A′C″B′ concur in X(5617), and the acute angle between each pair of these lines is π/2. See Dao Thanh Oai, ADGEOM 4030), and Francisco Javier GarcÃa Capitán, ADGEOM 4031). See also X(5613) and X(14144).
X(5617) lies on these lines: {2,98}, {3,618}, {4,616}, {5,13}, {14,2782}, {30,5463}, {99,621}, {298,511}, {302,383}, {381,530}, {396,3564}, {576,3181}, {5055,5459}
X(5617) = midpoint of X(i) and X(j) for these (i,j): (4,616), (298,1080)
X(5617) = reflection of X(i) in X(j) for these (i,j): (13,5), (3,618), (5613,114)
Let AP, BP, CP be the cevians of a point P in the plane of a triangle ABC. The A-positive Montesdeoca equilateral triangle is constructed as follows: let LA be the line through A perpendicular to CP; let UA be the 30-degree rotation of LA, where the angle BAC, for present purposes, defines the positive direction of rotation, and angle CAB, the negative (used in X(5619)). Let A1 = UA∩CP, let VA be the - 60 degree rotation of CP about A1, let AB = VA∩BP, let AC be the - 60 degree rotation of segment AAB about A. Then AABAC is an equilateral triangle. Define BBCBA and CCACB cyclically. These are the positive Montesdeoca equilateral triangles. X(5618) is the unique choice of P on the circumcircle of ABC for which the lines ABAC, BCBA, CBCA concur. For arbitrary P, the centers of the three equilateral triangles are collinear with P; denote their line by L(P). If P is on the circumcircle of ABC, then L(P) passes through X(110). (Angel Montesdeoca, November 3, 2013)
For details, see Hechos Geométricos en el Triángulo.
If you have The Geometer's Sketchpad, you can view Montesdeoca Equilateral Triangles.
X(5618) lies on the circumcircle and these lines: {13,74}, {115,2378}, {1989,2380}
X(5618) = intersection, other than X(111), of circumcircle and Parry circle of pedal triangle of X(13)
The negative Montesdeoca equilateral triangles for a point P are constructed as follows: in the construction of the positive Montesdeoca equilateral triangles atX(5618), replace the rotation angles (30, -60, -60) by (-30, 60, 60). Barycentrics for X(5619) are obtained from those of X(5618) by replacing S by - S. (Peter Moses, November 8, 2013)
If you have The Geometer's Sketchpad, you can view Montesdeoca Equilateral Triangles.
X(5619) lies on the circumcircle and these lines: {14,74}, {115,2379}, {1989,2381}
X(5619) = intersection, other than X(111), of circumcircle and Parry circle of pedal triangle of X(14)
Let A′B′C′ be the excentral triangle of ABC. Let NA be the nine-point center of A′BC, and let OA be the circumcircle of NABC. Define OB and OC cyclically. The circles OA, OB, OC concur in X(5620). (Angel Montesdoca, Anapolis #1120, November 2013: see Concurrent Circumcircles)
Let A′B′C′ be the incentral triangle and let A″ be the point such that triangle A″BC is similar to A′B′C′ and A″ is on the same side of line BC as A. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5620); see also X(502). (Randy Hutson, July 7, 2014)
X(5620) lies on these lines: {1,149}, {10,1109}, {36,759}, {37,115}, {65,1365}, {162,1838}, {267,3336}, {897,1738}, {1054,1247}, {1737,2166}, {2218,2915}
X(5620) = isogonal conjugate of X(5127)
X(5620) = X(2245)-cross conjugate of X(226)
X(5620) = incircle-inverse-of X(33593)
X(5620) = X(i)-isoconjugate of X(j) for these (i,j): (3,2074), (21,5172)
X(5620) = trilinear pole of line X(661)X(2294)
X(5620) = trilinear product X(523)*X(1290)
X(5620) = barycentric product X(1290)*X(1577)
X(5620) = X(10214)-of-excentral-triangle
Let L be the 1st Lemoine circle of a triangle ABC. Let AB be the point nearer A where line AB meets L, and define BC and CA cyclically. Let AC be the point nearer A where line AC meets L, and define BA and CB cyclically. The Euler lines of the triangles AABAC, BBCBA, CCACB concur in X(5621). (Seiichi Kirikami, November 7, 2013)
X(5621) lies on these lines: {3, 67}, {6, 74}, {25, 125}, {64, 1177}, {110, 3796}, {146, 3589}, {165, 2836}, {186, 1503}, {246, 1976}, {343, 3448}, {399, 5092}, {524, 2071}, {895, 3532}, {1204, 1205}, {1597, 2777}, {2453, 2790}, {2916, 2931}, {3516, 5095}
Continuing from the configuration in X(5621), let K denote the symmedian point (Lemoine point, X(6)) of ABC. The Euler lines of the triangles KABAC, KBCBA, KCACB concur in X(5622). (Seiichi Kirikami, November 7, 2013)
X(5622) lies on these lines: {2, 98}, {3, 895}, {4, 1177}, {6, 74}, {54, 67}, {69, 5504}, {113, 3618}, {185, 575}, {186, 2393}, {217, 5038}, {265, 1176}, {389, 1205}, {403, 1503}, {511, 2071}, {576, 1204}, {578, 5095}, {631, 5181}, {1316, 2790}, {2854, 5085}, {2892, 3541}, {3431, 5505}
Referring to the construction of X(5618), the lines ABAC, BCBA, CACB concur in X(5623). (Peter Moses, December 4, 2013)
X(5623) lies on the Neuberg cubic and these lines: {13,74}, {14,3440}, {16,1138}, {3065,3383}
X(5623) = reflection of X(13) in X(5618)
X(5623) = X(30)-Ceva conjugate of X(13)
Referring to the construction of X(5619), the lines ABAC, BCBA, CACB concur in X(5624). (Peter Moses, December 4, 2013)
X(5624) lies on the Neuberg cubic and these lines: {14,74}, {13,3441}, {15,1138}, {3065,3376}
X(5624) = reflection of X(14) in X(5619)
X(5624) = X(30)-Ceva conjugate of X(14)
Suppose that P is a point in the plane of a triangle ABC. Let LA be the line through P parallel to BC, and let BA = LA∩AB and CA = LA∩CA. Define AB and CB cyclically, and define AC and BC cyclically. Let UA be the line of the midpoints of segments ABAC and BCCB, and define UB and UC cyclically. The lines UA, UB, UC concur in a point Q = Q(P). If P is given by barycentrics p : q : r, then Q = g(p, q, r) : g(q, r, p) : g(r, p, q), where g(p,q,r) = (2p + q + r)(p2 + 2pq + 2pr + qr). If P = X(1), then Q = X(5625). (Seiichi Kirikami, December 8, 2013)
X(5625) lies on these lines: {1,75}, {10,4478}, {519,4733}, {524,551}, {726,3723}, {1100,1125}, {1255,4756}, {1279,3636}, {1654,3616}, {1961,3699}, {1962,4427}, {2796,4353}, {3244,4923}, {3624,3759}, {3842,4649}, {3945,4655}, {3993,4670}
X(5625) = midpoint of X(1) and X(86)
X(5625) = reflection of X(1213) in X(1125)
X(5625) = trilinear product X(i)*X(j) for these (i,j): (1125,4649), (4427,4784)
X(5625) = barycentric product X(4359)*X(4649)
ga = coth(aλ/(a + b + c)), gb = coth(bλ/(a + b + c)), gc = coth(cλ/(a + b + c)), where λ is the unique positive solution of the equation
[(u2 - a2) (a2 - (v - w)2)]1/2 + [(v2 - b2) (b2 - (w - u)2)]1/2 + [(w2 - c2) (c2 - (u - v)2)]1/2 = [2(b2c2 + c2a2 + a2b2) - a4 - b4 - c4)]1/2,
where u = aga, v = bgb, w = cgc
X(5626) is the point of maximal electrostatic potential inside a triangle ABC having a homogeneous surface charge distribution. (Hrvoje Abraham and Vjekoslav Kovac, December 11, 2013)
Download From electrostatic potentials to yet another triangle center.
Here is a Mathematica program that gives λ = 4.6547... for the (user-chosen) triangle ABC as testTriangle = {6,9,13), followed by the normalized barycentric coordinates and then normalized trilinear coordinates for X(5626).
\[Lambda] =.; testTriangle = {6, 9, 13}; {u, v, w} = Map[# Coth[# \[Lambda]/(a + b + c)] &, {a, b, c}]; {lhs, area} = {Sqrt[(u^2 - a^2) (a^2 - (v - w)^2)] + Sqrt[(v^2 - b^2) (b^2 - (w - u)^2)] + Sqrt[(w^2 - c^2) (c^2 - (u - v)^2)], Sqrt[(-a + b + c) (a + b - c) (a - b + c) (a + b + c)]/4} /. Thread[{a, b, c} → testTriangle]; \[Lambda] = \[Lambda] /. FindRoot[lhs == 4 area, {\[Lambda], 1}, WorkingPrecision → 50]; {\[Lambda], #, 2 area #/testTriangle} &[#/Apply[Plus, #] &[ Sqrt[(u^2 - a^2) (a^2 - (v - w)^2)] /. Map[Thread[{a, b, c} → #] &, NestList[RotateLeft, testTriangle, 2]]]] (* Peter Moses, December 20, 2013 *)
Geogebra sketch by David Fernández-De la Cruz: X(5626). (November 21, 2021)
If you have The Geometer's Sketchpad (version 5.05 or later), you can view X(5626).
Paul Yiu introduced this point on New Year's Day, January 1, 2014. He noted that X(74) is the unique point whose reflections in the sidelines of triangle ABC are collinear and perspective to ABC. The perspector is X(5627). The line of the reflections is perpendicular to the Euler line at X(4), and the rectangular circumhyperbola through X(5627), here called the Yiu hyperbola, YH, has asymptotes parallel and perpendicular to the Euler line. The center of YH is X(3258), and the perspector of YH is X(1637); YH meets the circumcircle in X(477), which is the reflection of X(74) in the Euler line. (Paul Yiu, ADGEOM, "An easy new year puzzle", January 1, 2014)
The line tangent to YH at X(5627) is parallel to the line X(74)X(1138). The axes of YH are the Wallace-Simson lines of X(74) and X(110). The Steiner circumellipse meets YH in four points: A, B, C, and X(5641). The isogonal conjugate of YH is the line X(3)X(74). X(5627) is the cevapoint of the 1st and 2nd Fermat Points. (Peter Moses, January 2, 2014)
X(5627) is the perspector of ABC and the reflection of the Euler triangle in the Euler line. (Randy Hutson, Jan. 8, 2014)
Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Let A″ be the intersection, other than X(3258) of the nine-point circle and the line A′X(3258); define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5627). (Randy Hutson, Jan. 8, 2014)
Let P = X(110) and A1A2, B1B2, C1C2 be the diameters of the circumcircle of ABC, parallel to AP, BP, CP, respectively. The Simson-Wallace lines of the points A1 and A2 intersect orthogonally in Pa, on the nine-point circle. The points Pb and Pc are defined similarly. The triangles ABC and PaPbPc are perspective and inversely similar; the perspector is X(5627). (Angel Montesdeoca, October 21, 2018)
X(5627) lies on these lines: {5,1117}, {30,74}, {125,477}, {328,1494}, {403,1989}, {1138,3258}, {1141,1304}
X(5627) = reflection of X(1138) in X(3258)
X(5627) = isogonal conjugate of X(1511)
X(5627) = isotomic conjugate of X(6148)
X(5627) = cevapoint of X(i) and X(j) for these (i,j): (13,14), (74,3470)
X(5627) = crossconjugate of X(i) and X(j) for these (i,j): (4,1141), (115,2394), (523, 476)
X(5627) = isoconjugate of X(i) and X(j) for these (i,j): (1,1511), (63,39176), (323,2173), (1101,3258), (2407,2624)
X(5627) = trilinear pole of line X(1637) X(1989)
X(5627) = trilinear product X(i)*X(j) for these (74,94), (476,2394), (1494,1989), (2166,2349)
X(5627) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(186)
X(5628) and several other triangle centers are perspectors of each
pair of the following triangles in the plane of a triangle ABC:
T1 = 1st Morley triangle; MathWorld: First Morley triangle
T2 = 2nd Morley triangle; MathWorld: Second Morely triangle, etc.
T3 = 3rd Morley triangle>
T4 = 1st adjunct Morley triangle; MathWorld: First adjunct Morley
triangle, etc.)
T5 = 2nd adjunct Morley triangle
T6 = 3rd adjunct Morley triangle
T7 = 1st p-Morley triangle (defined below)
T8 = 2nd p-Morley triangle
T9 = 3rd p-Morley triangle
T10 = 1st p-adjunct Morley triangle (defined below)
T11 = 2nd p-adjunct Morley triangle
T12 = 3rd p-adjunct Morley triangle
The p-Morley triangles T7, T8, T9 have as vertices the points of intersection of pairs of perpendiculars to trisectors at corresponding vertices that form T1, T2, T3, respectively. For example, T7 is formed as follows from the 1st Morley triangle A′B′C′: let LB be the line perpendicular to BA′ at B, let LC be the line perpendicular to CA′ at C; then the A-vertex of T7 is LB∩LC, and the B-vertex and C-vertex are defined cyclically. Similarly, the p-adjunct Morley triangles T10, T11, T12 are defined from T4, T5, T6.
X(5628) = 1st Morley-van Tienhoven point = perspector of ABC and
T7
X(5629) = 2nd Morley-van Tienhoven point = perspector of ABC and
T10
X(5630) = 3rd Morley-van Tienhoven point = perspector of ABC and T8
X(5631) = 4th Morley-van Tienhoven point = perspector of ABC and
T11
X(5632) = 5th Morley-van Tienhoven point = perspector of ABC and T9
X(5633) = 6th Morley-van Tienhoven point = perspector of ABC and
T12
X(356) = 7th Morley-van Tienhoven point = perspector of each pair of
T1, T4, T7
X(3276) = 8th Morley-van Tienhoven point = perspector of each pair of
T2, T5, T8
X(3277) = 9th Morley-van Tienhoven point = perspector of each pair of
T3, T6, T9
X(5634) = 10th Morley-van Tienhoven point = perspector of T7 and
T10
X(5635) = 11th Morley-van Tienhoven point = perspector of T8 and
T11
X(5636) = 12th Morley-van Tienhoven point = perspector of T9 and
T12
X(5637) = 13th Morley-van Tienhoven point
X(5628)-X(5633) were found in connection with Chris van Tienhoven's rotations of Morley trisectors and subsequent collaborations with Bernard Gibert, including is the cubic K587 in Gibert's catalog of cubics: Morley - van Tienhoven cubic.
X(5628) lies on these lines: {357,5457}, {3272,3274}, {3602,3606}
X(5628) = isogonal conjugate of X(5629)
X(5629) = perspector of ABC and T10; see X(5628)
X(5629) lies on these lines: {356,357}, {3273,3281}, {3274,3606}
X(5629) = isogonal conjugate of X(5628)
X(5630) = perspector of ABC and T8; see X(5628)
X(5630) lies on these lines: {3603,3607}, {3272,3275}
X(5630) = isogonal conjugate of X(5631)
X(5631) = perspector of ABC and T11; see X(5628)
X(5631) lies on these lines: {1136,1137}, {3274,3283}, {3275,3607}
X(5631) = isogonal conjugate of X(5630)
X(5632) = perspector of ABC and T9; see X(5628)
X(5632) lies on these lines: {356,3272}, {3604,3605}
X(5632) = isogonal conjugate of X(5633)
X(5633) = perspector of ABC and T12; see X(5628)
X(5633) lies on these lines: {356,1134}, {3273,3605}, {3275,3279}
X(5633) = isogonal conjugate of X(5632)
X(5634) = perspector of the triangles T7 and T10 listed at X(5628).
X(5634) lies on these lines: {356,3605}, {3276,3606}
X(5635) = perspector of the triangles T8 and T11 listed at X(5628).
X(5635) lies on these lines: {3276,3606}, {3277,3607}
X(5636) = perspector of the triangles T9 and T12 as listed at X(5628).
X(5636) lies on these lines: {356,3605}, {3277,3607}
Using the notation at X(5628), let
L1 = perspectrix of each pair of the triangles ABC, T1, T4
L2 = perspectrix of each pair of the triangles ABC, T2, T5
L3 = perspectrix of each pair of the triangles ABC, T3, T6
The lines L1, L2, L3 concur in X(5637). (Chris van Tienhoven, January 3, 2014)
X(5637) lies on this line: {396,523}
X(5637) = isogonal conjugate of X(14146)
X(5637) = crossdifference of every pair of points on line X(16)X(358)
X(5637) = perspector of Morley circumconic
The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5638) is the farther from X(2); also, X(5638) = intersection farther from X(2) of the Parry circle and Lemoine axis. X(5638) = (F1, F2)-harmonic conjugate of X(1341), where F1 and F2 are the foci of the Steiner inellipse. (Randy Hutson, June 16, 2014)
X(5638) lies on the Steiner major axis, the circumconic {{A,B,C,X(2),X(6)}}, the Parry circle, the Thomson-Gibert-Moses hyperbola, the cubics K889, K890, K891, K1067, the curve Q090, and these lines: {2, 1341}, {3, 6141}, {6, 6142}, {100, 11651}, {110, 1379}, {111, 1380}, {154, 21032}, {187, 237}, {353, 1340}, {739, 11652}, {2028, 3124}, {2029, 39689}, {2395, 13636}, {3228, 6189}, {3414, 9147}, {3557, 9716}, {5643, 14631}, {5648, 46463}, {5652, 52723}, {16081, 57013}, {21001, 21036}, {36213, 39067}
X(5638) = reflection of X(5639) in X(351)
X(5638) = isogonal conjugate of X(6190)
X(5638) = reflection of X(5638) in the Lemoine axis
X(5638) = Parry-isodynamic-circle-inverse of X(5639)
X(5638) = isogonal conjugate of the anticomplement of X(39023)
X(5638) = isogonal conjugate of the complement of X(39366)
X(5638) = isogonal conjugate of the isotomic conjugate of X(3413)
X(5638) = Thomson-isogonal conjugate of X(6039)
X(5638) = psi-transform of X(1341)
X(5638) = X(31)-complementary conjugate of X(39068)
X(5638) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39068}, {6, 2028}, {110, 5639}, {1379, 6}, {6177, 39023}, {6190, 3558}, {41880, 512}, {57013, 13636}
X(5638) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6190}, {63, 57014}, {75, 1379}, {662, 3414}, {799, 5639}, {13722, 24041}, {36085, 52723}
X(5638) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6190}, {206, 1379}, {1084, 3414}, {3005, 13722}, {3162, 57014}, {13722, 850}, {21905, 46463}, {38988, 52723}, {38996, 5639}, {39023, 76}, {39067, 30508}, {39068, 2}
X(5638) = trilinear pole of line {512, 2028}
X(5638) = complement of anticomplementary conjugate of X(39366)
X(5638) = anticomplement of complementary conjugate of X(39023)
X(5638) = X(2)-Ceva conjugate of X(39068)
X(5638) = X(110)-Ceva conjugate of X(5639)
X(5638) = X(3124)-cross conjugate of X(5639)
X(5638) = X(6)-vertex conjugate of X(5639)
X(5638) = inverse-in-circumcircle of X(6141)
X(5638) = antipode-in-Parry-circle of X(5639)
X(5638) = trilinear pole of the line X(512)X(2028)
X(5638) = crossdifference of every pair of points on the line X(2)X(1340)
X(5638) = X(1380)-of-1st-Parry-triangle
X(5638) = X(1379)-of-2nd-Parry-triangle
X(5638) = crossdifference of PU(i) for these i: 117, 119
X(5638) = PU(118)-harmonic conjugate of X(1341)
X(5638) = X(i)-line conjugate of X(j) for these (i,j): {187, 5639}, {353, 1340}, {1341, 2}, {2395, 13636}, {5652, 52723}, {9147, 3414}
X(5638) = barycentric product X(i)*X(j) for these {i,j}: {6, 3413}, {110, 13636}, {111, 52722}, {512, 6189}, {523, 1380}, {647, 57013}, {691, 46462}, {1379, 39023}, {2028, 6190}, {3414, 41880}, {5639, 30509}
X(5638) = pole of line {7668, 39023} with respect to the Kiepert circumhyperbola
X(5638) = pole of line {2028, 3269} with respect to the Jerabek circumhyperbola
X(5638) = pole of line {2028, 3124} with respect to ABCGK
X(5638) = pole of line {39068, 44312} with respect to ABCGGe
X(5638) = pole of line {1015, 2028} with respect to ABCIK
X(5638) = pole of line {670, 6190} with respect to the Steiner-Wallace right hyperbola
X(5638) = pole of line {99, 1379} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5638) = pole of line {190, 6190} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5638) = pole of line {668, 6190} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5638) = pole of line {670, 6190} with respect to the Kiepert circumhyperbola of the anticomplementary triangle
X(5638) = pole of line {194, 3414} with respect to the Steiner circumellipse
X(5638) = pole of line {3167, 5639} with respect to the MacBeath circumconic
X(5638) = pole of line {39, 3414} with respect to the Steiner inellipse
X(5638) = pole of line {6, 5639} with respect to the Brocard inellipse
X(5638) = pole of line {669, 30509} with respect to the Kiepert parabola
X(5638) = pole of line {5638, 9155} with respect to the Thomson-Gibert-Moses hyperbola
X(5638) = pole of line {6, 5639} with respect to the Aquarius conic (see X(5585))
X(5638) = pole of line {6, 5639} with respect to the circumcircle
X(5638) = pole of line {574, 5639} with respect to the Brocard circle
X(5638) = pole of line {511, 5638} with respect to the Parry circle
X(5638) = pole of line {262, 3414} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5638) = pole of line {3414, 44434} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5638) = pole of line {3414, 44460} with respect to the circumcircle of the inner Napoleon triangle
X(5638) = pole of line {3414, 44464} with respect to the circumcircle of the outer Napoleon triangle
X(5638) = pole of line {5639, 6468} with respect to the Lucas inner circle
X(5638) = pole of line {5639, 6221} with respect to the Lucas circles radical circle
X(5638) = pole of line {5639, 50663} with respect to the outer Montesdeoca-Lemoine circle
X(5638) = pole of line {5639, 50662} with respect to the inner Montesdeoca-Lemoine circle
X(5638) = pole of line {511, 5639} with respect to the Parry isodynamic circle
X(5638) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6190}, {25, 57014}, {32, 1379}, {351, 52723}, {512, 3414}, {669, 5639}, {1379, 57576}, {1380, 99}, {2028, 3413}, {3124, 13722}, {3413, 76}, {5639, 30508}, {6189, 670}, {13636, 850}, {21906, 46463}, {41880, 6189}, {46462, 35522}, {52722, 3266}, {57013, 6331}
X(5638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 2502, 5639}, {647, 3569, 5639}, {649, 5029, 5639}, {663, 5075, 5639}, {665, 5098, 5639}, {667, 5040, 5639}, {669, 5027, 5639}, {890, 42655, 5639}, {902, 5168, 5639}, {1495, 5191, 5639}, {3005, 5113, 5639}, {3230, 5163, 5639}, {3231, 5106, 5639}, {3724, 5202, 5639}, {3747, 5147, 5639}, {6137, 6138, 5639}, {8644, 9135, 5639}, {8651, 42663, 5639}, {9208, 17414, 5639}, {9409, 42654, 5639}, {14899, 35607, 13722}, {15451, 42651, 5639}, {39162, 39163, 1341}, {39202, 39203, 51493}
The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5639) is the closer to X(2); also, X(5639) = the intersection closer to X(2) of the Parry circle and Lemoine axis. X(5639) = perspector of the hyperbola {{A,B,C,X(6), F1, F2}}, where F1 and F2 are the foci of the Steiner inellipse; also X(5639) = intersection of the trilinear polars of X(6), F1, and F2. (Randy Hutson, June 16, 2014)
X(5639) lies on the Steiner minor axis, the circumconic {{A,B,C,X(2), X(6)}}, the Parry circle, the Thomson-Gibert-Moses hyperbola, K889, K890, K891, K1067, the curve Q090, and these lines: {2, 1340}, {3, 6142}, {6, 6141}, {100, 11652}, {110, 1380}, {111, 1379}, {154, 21036}, {187, 237}, {353, 1341}, {739, 11651}, {2028, 39689}, {2029, 3124}, {2395, 13722}, {3228, 6190}, {3413, 9147}, {3558, 9716}, {5643, 14630}, {5648, 46462}, {5652, 52722}, {16081, 57014}, {21001, 21032}, {36213, 39068}
X(5639) = reflection of X(5638) in X(351)
X(5639) = isogonal conjugate of X(6189)
X(5639) = complement of anticomplementary conjugate of X(39365)
X(5639) = anticomplement of complementary conjugate of X(39022)
X(5639) = X(3124)-cross conjugate of X(5638)
X(5639) = X(6)-vertex conjugate of X(5638)
X(5639) = circumcircle-inverse of X(6142)
X(5639) = Parry-circle-antipode of X(5638)
X(5639) = trilinear pole of the line X(512)X(2029)
X(5639) = crossdifference of every pair of points on the line X(2)X(1341)
X(5639) = X(1379)-of-1st-Parry-triangle
X(5639) = X(1380)-of-2nd-Parry-triangle
X(5639) = crossdifference of PU(i) for these i: 116, 118
X(5639) = perspector of hyperbola {{A,B,C,X(6),PU(118)}}
X(5639) = isogonal conjugate of the anticomplement of X(39022)
X(5639) = isogonal conjugate of the complement of X(39365)
X(5639) = isogonal conjugate of the isotomic conjugate of X(3414)
X(5639) = Thomson-isogonal conjugate of X(6040)
X(5639) = psi-transform of X(1340)
X(5639) = X(31)-complementary conjugate of X(39067)
X(5639) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39067}, {6, 2029}, {110, 5638}, {1380, 6}, {6178, 39022}, {6189, 3557}, {41881, 512}, {57014, 13722}
X(5639) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6189}, {63, 57013}, {75, 1380}, {662, 3413}, {799, 5638}, {13636, 24041}, {36085, 52722}
X(5639) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6189}, {206, 1380}, {1084, 3413}, {3005, 13636}, {3162, 57013}, {13636, 850}, {21905, 46462}, {38988, 52722}, {38996, 5638}, {39022, 76}, {39067, 2}, {39068, 30509}
X(5639) = X(i)-line conjugate of X(j) for these (i,j): {187, 5638}, {353, 1341}, {1340, 2}, {2395, 13722}, {5652, 52722}, {9147, 3413}
X(5639) = pole of line {7668, 39022} with respect to the Kiepert circumhyperbola
X(5639) = pole of line {2029, 3269} with respect to the Jerabek circumhyperbola
X(5639) = pole of line {2029, 3124} with respect to ABCGK
X(5639) = pole of line {39067, 44312} with respect to ABCGGe
X(5639) = pole of line {1015, 2029} with respect to ABCIK
X(5639) = pole of line {670, 6189} with respect to the Steiner / Wallace right hyperbola
X(5639) = pole of line {99, 1380} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5639) = pole of line {190, 6189} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5639) = pole of line {668, 6189} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5639) = pole of line {670, 6189} with respect to the Kiepert circumhyperbola of the anticomplementary triangle
X(5639) = pole of line {194, 3413} with respect to the Steiner circumellipse
X(5639) = pole of line {3167, 5638} with respect to the MacBeath circumconic
X(5639) = pole of line {39, 3413} with respect to the Steiner inellipse
X(5639) = pole of line {6, 5638} with respect to the Brocard inellipse
X(5639) = pole of line {669, 30508} with respect to the Kiepert parabola
X(5639) = pole of line {5639, 9155} with respect to the Thomson-Gibert-Moses hyperbola
X(5639) = pole of line {6, 5638} with respect to the Aquarius conic (see X(5585))
X(5639) = pole of line {6, 5638} with respect to the circumcircle
X(5639) = pole of line {574, 5638} with respect to the Brocard circle
X(5639) = pole of line {511, 5639} with respect to the Parry circle
X(5639) = pole of line {262, 3413} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5639) = pole of line {3413, 44434} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5639) = pole of line {3413, 44460} with respect to the circumcircle of the inner Napoleon triangle
X(5639) = pole of line {3413, 44464} with respect to the circumcircle of the outer Napoleon triangle
X(5639) = pole of line {5638, 6468} with respect to the Lucas inner circle
X(5639) = pole of line {5638, 6221} with respect to the Lucas circles radical circle
X(5639) = pole of line {5638, 50663} with respect to the outer Montesdeoca-Lemoine circle
X(5639) = pole of line {5638, 50662} with respect to the inner Montesdeoca-Lemoine circle
X(5639) = pole of line {511, 5638} with respect to the Parry isodynamic circle
X(5639) = barycentric product X(i)*X(j) for these {i,j}: {6, 3414}, {110, 13722}, {111, 52723}, {512, 6190}, {523, 1379}, {647, 57014}, {691, 46463}, {1380, 39022}, {2029, 6189}, {3413, 41881}, {5638, 30508}
X(5639) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6189}, {25, 57013}, {32, 1380}, {351, 52722}, {512, 3413}, {669, 5638}, {1379, 99}, {1380, 57575}, {2029, 3414}, {3124, 13636}, {3414, 76}, {5638, 30509}, {6190, 670}, {13722, 850}, {21906, 46462}, {41881, 6190}, {46463, 35522}, {52723, 3266}, {57014, 6331}
X(5639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 2502, 5638}, {647, 3569, 5638}, {649, 5029, 5638}, {663, 5075, 5638}, {665, 5098, 5638}, {667, 5040, 5638}, {669, 5027, 5638}, {890, 42655, 5638}, {902, 5168, 5638}, {1495, 5191, 5638}, {3005, 5113, 5638}, {3230, 5163, 5638}, {3231, 5106, 5638}, {3724, 5202, 5638}, {3747, 5147, 5638}, {6137, 6138, 5638}, {8644, 9135, 5638}, {8651, 42663, 5638}, {9208, 17414, 5638}, {9409, 42654, 5638}, {15451, 42651, 5638}, {35608, 35609, 13636}, {39204, 39205, 51492}
The orthocentroidal triangle is introduced at X(5476). For another construction of X(5640), let A′B′C′ be the orthic triangle, let A″ be the centroid of AB′C′, and define B″ and C″ cyclically. Then X(5640) is the centroid of A″B″C″. Also, X(5640) is the trilinear pole of the polar, with respect to the Moses circle, of the perspector of the Moses circle. (Randy Hutson, June 16, 2014)
Vertices of the central triangle A″B″C″ are given by barycentrics as follows:
A″ = b4 + c4 - a2b2 - a2c2 - 4b2c2 : b2(a2 - b2 - c2) : c2(a2 - b2 - c2) (Peter Moses, June 20, 2014)
A″B″C″ is perspective to ABC, the tangential triangle and the second Brocard triangle, all with perspector X(6), and perspective to the Euler triangle at X(125). The Euler line of A″B″C″ passes througth X(i) for these I: 6,110,111,895,1995,2493,2502,2503,2854,2930,3066,3124 (Peter Moses, June 20, 2014)
Let A′B′C′ be the Artzt triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically; then X(5640) = centroid of A″B″C″; see X(6032). (Randy Hutson, December 10, 2016) Let A′B′C′ be the anti-Artzt triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically; then X(5640) = centroid of A″B″C″. (Randy Hutson, December 10, 2016) X(5640) lies on these lines: {2,51}, {4,4846}, {5,568}, {6,110}, {22,5085}, {23,182}, {25,5012}, {52,3090}, {125,5169}, {143,1656}, {185,3832}, {323,576}, {324,3168}, {375,3681}, {381,5663}, {389,3091}, {394,5102}, {476,1316}, {512,598}, {569,3518}, {575,1495}, {631,5446}, {858,5480}, {1112,5094}, {1154,5055}, {1180,3981}, {1216,5067}, {1383,2030}, {1843,4232}, {1993,5020}, {3056,5297}, {3111,3972}, {3291,5052}, {3292,5097}, {3448,3818}, {3533,5447}, {5039,5354}
X(5640) = midpoint of X(51) and X(373)
X(5640) = reflection of X(2) in X(373)
X(5640) = isotomic conjugate of polar conjugate of X(33885)
X(5640) = complement of X(33884)
X(5640) = anticomplement of X(5650)
X(5640) = crossdifference of every pair of points on line X(690)X(3288)
X(5640) = intercept, other than X(110), of line X(6)X(110) and conic {{X(13),X(14),X(15),X(16),X(110)}}
X(5640) = SS(A/3 → A) of X(8065) (symbolic trilinear substitution)
X(5640) = X(2)-of-reflection-triangle of X(2)
X(5640) = isogonal conjugate of X(2) with respect to the pedal triangle of X(2)
X(5640) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(2)
X(5640) = {X(11624),X(11626)}-harmonic conjugate of X(6)
X(5640) = homothetic center of X(2)-altimedial and X(2)-adjunct anti-altimedial triangles
X(5640) = circummedial-to-X(2)-pedal similarity image of X(2)
X(5640) = {X(2),X(51)}-harmonic conjugate of X(3060)
The Steiner circumellipse meets the Yiu Hyperbola, defined at X(5627), in four points: A, B, C, and X(5641). (Peter Moses, January 2, 2014)
X(5641) lies on these lines: {2,2966}, {30,99}, {69,892}, {290,850}, {297,340}, {523,1494}, {525,671}, {670,3260}
X(5641) = reflection of X(2966) in X(2)
X(5641) = reflection of X(2) in X(35088)
X(5641) = isogonal conjugate of X(5191)
X(5641) = isotomic conjugate of X(542)
X(5641) = cevapoint of X(2) and X(542)
X(5641) = X(542)-cross conjugate of X(2)
X(5641) = isoconjugate of X(i) and X(j) for these (i,j): (6,2247), (163,1640)
X(5641) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(325)}}
X(5641) = trilinear pole of line X(2)X(1637)
X(5641) = trilinear product X(75)*X(842)
X(5641) = barycentric product X(76)*X(842)
The point X(5542) minimizes a certain sum involving the pedal triangle A′B′C′ of a triangle ABC. During May, 2014, Peter Moses generalized the sum to include a parameter t, as follows:
|XA|2 + |XB|2 + |XC|2 + t(|XA′|2 + |XB′|2 + |XC′|2),
and found the solution to the extremal problem to be the point H(t) = f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = a2[(t2 + 3t + 3)S2 + (2t + 3) S2A + tSBSC].
Let u = 3abc[-3abc + (9a2b2c2 - 8S2S2W)1/2/(4S2S2W) and v = 3abc[-3abc - (9a2b2c2 - 8S2S2W)1/2/(4S2S2W)
Moses found that if t > u, then H(t) minimizes the sum, and if t < v, then H(t) maximizes the sum, and if t > -1, then H(t) minimizes the sum. The extreme value in all cases is
[(6t + 6)SASBSCSW - (2t + 3)(t + 2)S2S2W - t(t + 1)2S4]/[(9t + 9)SASBSC - (2t + 3)(t + 3)S2SW].
The locus of H(t) as t ranges through the extended real number line is a rectangular hyperbola, here named the Thomson-Gibert-Moses hyperbola, which passes through the triangle centers X(i) for I = 2, 3, 6, 110, 154, 354, 392, 1201, 2574, 2575, 3167, 5544, 5638, 5639, 5643, 5644, 5645, 5646, 5648, 5652-5656. The axes of this hyperbola are parallel to the Simson-Wallace lines of X(1113) and X(1114), these being the points of intersection of the Euler line and the circumcircle. See X(5643). (based on notes from Peter Moses, June 7, 2014)
X(5642) is the radical trace of the Parry circles of ABC and the 1st Brocard triangle, and also the centroid of the (degenerate) pedal triangle of X(110). The Thomson-Gibert-Moses hyperbola intersects the circumcircle in X(110) and the vertices of the Thomson triangle (see Thomson Triangle. The 4 points of intersection form an X(74)-centric system; i.e., each is X(74) of the triangle of the other three. Moreover, the Thomson-Gibert-Moses hyperbola is the Thomson isogonal conjugate (i.e., isogonal-conjugate-with-respect-to-the-Thomson-triangle) of the Euler line. In general, the Thomson isogonal conjugate of a point P is the centroid of the antipedal triangle of the isogonal conjugate of P; consequently, the Thomson-Gibert-Moses hyperbola is the locus of the centroid of the antipedal triangle of a point P that traverses the Jerebek hyperbola. Indeed, the Thomson-Gibert-Moses hyperbola is the Jerabek hyperbola of the Thomson triangle, as noted at Thomson Triangle (Randy Hutson, June 16-17, 2014)
Let O = X(3) and suppose that P is a point other than O. Let OP be the circle with segment PO as diameter. Let A′ be the point of intersection, other than O, of OP and the perpendicular bisector of segment BC, and define B′ and C′ cyclically. Triangle A′B′C′ is called the P-Brocard triangle, and X(5642) is X(23)-of-the-X(2)-Brocard triangle. (Randy Hutson, June 16-17, 2014)
The Thomson-Gibert-Moses hyperbola is the image of the Euler line under a mapping T discussed in connection with the third Deaux cubic (K609). (Bernard Gibert, June 22, 2014)
Let O* be the circle with segment X(13)X(14) as diameter (and center X(115). Let P be the perspector of O*. Then X(5642) is the trilinear pole of the polar of P with respect to O*. See X(3292) for a similar property involving the segment X(15)X(16). (Randy Hutson, July 18, 2014)
X(5642) lies on these lines: {2,98}, {3,541}, {30,113}, {74,3524}, {115,2502}, {126,5026}, {265,5055}, {351,690}, {373,597}, {376,2777}, {399,5054}, {468,524}, {543,1316}, {620,5108}, {1636,1637}, {1648,5477}, {1995,5476}, {2781,3917}, {2836,3742}, {3024,4995}, {3028,5298}, {3849,5112}
X(5642) = midpoint of X(2) and X(110)
X(5642) = reflection of X(i) in X(j) for these (i,j): (2,5972), (125,2),
X(5642) = crossdifference of every pair of points on the line X(74)X(111)
X(5642) = X(3524)-line-conjugate of X(74)
X(5642) = radical trace of Parry circles of ABC and anti-Artzt triangle
X(5643) is the only point whose polar conic in the Napoleon cubic (K005) is a circle. (Bernard Gibert, June 22, 2014)
The Thomson-Gibert-Moses hyperbola is defined at X(5642). Points H(t) on this hyperbola include the following:
| t | H(t) |
| -3 | X(3167) |
| -2 | X(110) |
| -3/2 | X(154) |
| -1/2 | X(5646) |
| -1 | X(3) |
| 0 | X(2) |
| 1 | X(5544) |
| 2 | X(5643) |
| 3 | X(5644) |
| 4 | X(5645) |
| infinity | X(6) |
| 3R/r | X(354) |
| 6rR/(r2 - 2rR - s2 | X(354) |
Let A′ be the centroid of the A-altimedial triangle, and define B′ and C′ cyclically; then X(5643) is the center of similitude of ABC and A′B′C′. (Randy Hutson, July 7, 2014)
X(5643) is the only finite fixed point of the affine transformation that maps a triangle ABC onto the pedal triangle of X(5). (Angel Montesdeoca, August 19, 2016)
X(5643) lies on these lines: {2,576}, {83,5466}, {110,373}, {111,5038}, {154,1995}, {354,4663}, {392,5047}, {597,895}, {632,1173}, {3090,5449}, {3167,5422}, {3580,3628}
X(5643) = Thomson-isogonal conjugate of X(8703)
X(5644) is the Thomson isogonal conjugate of X(3522); see X(5642).
X(5644) lies on these lines: {2,5093}, {3,51}, {110,5020}, {154,5050}, {343,5070}, {373,3167}, {394,5544}, {1351,3819}, {1899,3851}
X(5644) = Thomson-isogonal conjugate of X(3522)
The Thomson-Gibert-Moses hyperbola is defined at X(5642).
X(5645) lies on these lines: {2,5097}, {154,3066}, {323,5544}, {2889,3533}, {3448,3545}
The Thomson-Gibert-Moses hyperbola is defined at X(5642). X(5646) = (X(6) of the Thomson triangle); see Thomson Triangle (Randy Hutson, June 16, 2014)
X(5646) lies on these lines: {2,1350}, {40,392}, {64,631}, {110,5085}, {182,3167}, {354,612}, {511,5544}, {1201,2177}, {1351,3819}
Let A′B′C′ be the cevian triangle of the circumcenter. Let
LAB = reflection of AA′ in AB, LAC = reflection of AA′ in AC; HAB = orthogonal projection of A′ on LAB, HAC = orthogonal projection of A′ on LAC, and define LBC, LCA, LBA, LCB, and HBC, HCA, HBA, HCB cyclically,
M11 = midpoint(HAB, HAC),
M12 = midpoint(HBC,
HBA), M13 =
midpoint(HCA, HCB)
M21 = midpoint(HBA, HCA),
M22 = midpoint(HCB,
HAB), M23 =
midpoint(HAC, HBC)
M31 = midpoint(HBC, HCB),
M32 = midpoint(HCA,
HAC), M33 =
midpoint(HAB, HBA).
Antreas Hatzipolakis (Anopolis , May 17, 2014) asked if the Euler lines of the triangles M11M12M13, M21M22M23, M31M32M33 concur, and César Lozada established that they do. The point of concurrence is X(5647).
X(5647) lies on these lines: {5,51}, {154,157}
X(5648) is the radical center of the the circumcircle, the circle {{X(13), X(15), X(5463), X(5464)}} and the circle {{X(14), X(16), X(5463), X(5464)}}. (Randy Hutson, August 26, 2014)
X(5648) lies on these lines: {2,2854}, {3,67}, {6,5642}, {110,524}, {125,5646}, {141,5888}, {392,2836}, {511,5655}, {523,5653}, {526,5652}, {541,1350}, {543,2453}, {597,895}, {2781,5656}
X(5648) = crossdifference of every pair of points on the line X(2492)X(2780)
X(5648) = circumcircle-inverse of X(34013)
X(5648) = Thomson-isogonal conjugate of X(7464)
X(5649) is the trilinear pole of the line X(23)X(110); at X(110), this line is tangent to the Thomson-Gibert-Moses hyperbola and parallel to the trilinear polar of X(110). (Randy Hutson, June 16, 2014)
X(5649) lies on the hyperbola {A, B, C, X(2), X(110)} and this line: {250, 4230}
X(5649) = isogonal conjugate of X(1640)
X(5649) = isotomic conjugate of X(18312)
Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(2), and let V be the tangent at X(3); then X(5650) = U∩V. Also, X(5650) is the centroid of the X(2)-Brocard triangle; see X(5642). (Randy Hutson, June 16, 2014)
X(5650) = pole of the Euler line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)
X(5650) lies on these lines: {2,51}, {3,1495}, {6,5646}, {110,5092}, {549,5642}
X(5650) = reflection of X(373) in X(2)
X(5650) = complement of X(5640)
X(5650) = isotomic conjugate of polar conjugate of X(33843)
X(5650) = {X(3),X(5651)}-harmonic conjugate of X(1495)
Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(3), and let V be the tangent at X(6); then X(5651) = U∩V. Also, X(5651) is the inverse-in-Thomson-Gibert-Moses hyperbola of X(182). (Randy Hutson, June 16, 2014)
X(5651) = pole of the Brocard axis line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)
X(5651) lies on these lines: {2,98}, {3,1495}, {6,373}
X(5651) = crossdifference of every pair of points on the line X(1499)X(3569)
X(5651) = {X(1495),X(5650)}-harmonic conjugate of X(3)
X(5652) = antipode of X(5653) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)
X(5652) lies on the circle {{X(3), X(6), X(110), X(353), X(843)}}. (Randy Hutson, November 22, 2014)
X(5652) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2,3111}, {3,669}, {6,523}, {83,5466}, {99,110}
X(5652) = reflection of X(5653) in X(5642)
X(5652) = Thomson-isogonal conjugate of X(7418)
X(5653) = antipode of X(5652) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)
X(5653) lies on the Thomson-Gibert-Moses hyperbola, the circle {{X(2), X(3), X(6), X(111), X(691)}}, and these lines: {2,690}, {3,351}, {6,526}, {110,249}
X(5653) is denoted QA-P9 (QA-Miquel Center) of the quadrangle X(13)X(14)X(15)X(16); see Chris van Tienhoven's site.
X(5653) = crossdifference of every pair of points on the line
X(542)X(1648)
X(5653) = reflection of X(5652) in X(5642)
X(5654) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(22). (Peter Moses, June 18, 2014)
X(5654) lies on these lines: {3,4549}, {4,110}, {5,6}, {11,3157}, {12,1069}, {30,154}, {52,3542}, {140,5646}, {184,1568}, {185,3548}, {354,912}, {381,3167}, {382,1514}, {403,1993}, {539,3545}, {1216,3547}, {1656,5544}, {1899,2072}, {3089,5446}, {3090,5449}, {3549,5562}, {5055,5644}, {5056,5645}
Let A′B′C′ be the orthocentroidal triangle. Let L be the lines 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 L′ cyclically. The lines L′,M′,N′ concur in X(5655); c.f. X(i) for i = 74, 113, 265, 399, 1147, 1511, 5504, 5609. (Randy Hutson, August 26, 2014)
X(5655) is the center of the perspeconic of the orthocentroidal and anti-orthocentroidal triangles (see preamble before X(15254)). This conic is a circle. (Randy Hutson, December 2, 2017)
X(5655) lies on these lines: {3,541}, {4,5609}, {5,5643}, {6,13}, {30,110}, {74,549}, {125,5055}, {146,376}, {154,2777}, {354,2771}, {1539,3543}, {3167,3830}, {3448,3545}, {5054,5646}
X(5655) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(23). (Peter Moses, June 18, 2014)
X(5655) = antipode of X(3) in Thomson-Gibert-Moses hyperbola
X(5656) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(25). (Peter Moses, June 18, 2014)
X(5656) lies on these lines: {4,6}, {5,5544}, {20,110}, {30,3167}, {64,631}, {154,376}, {185,3089}, {193,1533}, {221,1058}, {354,5603}, {378,1619}, {381,5644}, {1056,2192}, {1853,3545}, {3091,5643}, {3357,3523}, {3832,5645}
Let A* be the parabola with focus A and directrix BC, and let A** be the polar of X(1) with respect to A*. Define B** and C** cyclically, and let A′ = B**∩C**, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(1000), and the centroid of A′B′C′ is X(5657). (Randy Hutson, July 7, 2014)
X(5657) lies on these lines: {1,631}, {2,392}, {3,8}, {4,9}, {5,962}, {7,495}, {12,4295}, {20,355}, {21,5554}, {35,3486}, {36,3476}, {43,1064}, {46,388}, {55,1006}, {57,1056}, {63,2096}, {65,3085}, {80,4302}, {140,1482}, {145,1385}, {165,376}, {191,2950}, {201,1148}, {226,2093}, {333,4221}, {387,1108}, {484,1478}, {497,1737}, {498,3485}, {499,5445}, {519,3158}, {549,3241}, {580,5264}, {581,3293}, {601,5247}, {602,5255}, {912,3681}, {938,3295}, {946,1698}, {993,2077}, {999,5435}, {1012,5273}, {1058,1210}, {1072,1738}, {1083,2726}, {1125,3525}, {1155,4293}, {1376,3428}, {1387,4345}, {1483,3530}, {1519,3452}, {1532,3820}, {1537,5328}, {1699,3545}, {1714,3987}, {1770,5229}, {1829,3088}, {1836,5183}, {1837,4294}, {1872,4194}, {1902,3089}, {2098,5433}, {2099,5432}, {3035,5289}, {3057,3086}, {3214,4300}, {3240,5396}, {3522,4678}, {3526,5550}, {3528,3626}, {3533,3624}, {3560,5260}, {3634,4301}, {3651,5584}, {3817,3828}, {3869,5552}, {4292,5128}, {4305,5217}, {4642,5230}, {5084,5250}, {5251,5537}, {5258,5450}, {5442,5559}
X(5657) = midpoint of X(165) and X(3679)
X(5657) = reflection of X(i) in X(j) for these (i,j): (376, 165), (3817, 3828)
X(5657) = anticomplement of X(5886)
X(5657) = crossdifference of every pair of points on the line X(1459)X(3310)
X(5657) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(55). (Peter Moses, June 18, 2014)
X(5657) = {X(10),X(40)}-harmonic conjugate of X(4)
X(5657) = centroid of antipedal triangle of X(7)
X(5658) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(56). (Peter Moses, June 18, 2014)
X(5658) is the centroid of the antipedal triangle of X(8). Let AAABAC be the orthic triangle of the A-extouch triangle, let BABBBC be the orthic triangle of the B-extouch triangle, and let CACBCC be the orthic triangle of the C-extouch triangle. Let A′ be the centroid of AABACA, let B′ be the centroid of ABBBCB, and let C′ be the centroid of ACBCCC. Then triangle A′B′C′ is homothetic to ABC with center of homothety X(7308), and X(5658) = X(84)-of-A′B′C′. (Randy Hutson, July 7, 2014)
X(5658) lies on these lines: {1,4}, {2,971}, {9,2272}, {20,5440}, {84,631}, {100,329}, {516,3158}, {1538,5274}, {1709,5218}, {3305,3358}
X(5659) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(35). (Peter Moses, June 18, 2014)
X(5659) lies on these lines: {1,140}, {9,1699}, {100,4847}, {515,3651}, {3925,5536}
X(5660) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(36). (Peter Moses, June 18, 2014)
Let A′B′C′ be the orthic triangle. Let A* be the antiorthic axis of triangle AB′C′, and define B* and C* cyclically. Let A″ = B*∩C*, and define B″ and C″ cyclically. Then X(5660) = X(165)-of-A″B″C″. (Randy Hutson, July 7, 2014)
X(5660) lies on these lines: {1,5}, {2,2801}, {9,1768}, {100,516}, {104,5251}, {153,214}, {528,1699}, {1512,4867}, {1537,5541}, {1538,3689}, {1639,2826}
Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given (Peter Moses, June 23, 2014) by
T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)
X(5661) is the minimizer T when P = X(3) and Q = X(6), so that PU is the Brocard axis. Other examples follow:
If PU = Lemoine axis (P = X(187), Q = X(237)), then T = X(3229)
If PU = orthic axis (P = X(230), Q = X(223)), then T = X(230)
If PU = anti-orthic axis (P = X(44), Q = X(513)), then T = X(1575)
If PU = De Longchamps line (P = X(325), Q = X(523)), then T =
X(325)
If PU = Gergonne line (P = X(241), Q = X(514)), then T = X(3008)
If P = X(1) and Q = X(3), then T = X(5662)
If P = X(1) and Q = X(6), then T = X(5701)
X(5661) lies on these lines: {2,647}, {3,6}, {3672,4235}
X(5661) = crossdifference of every pair of points on the line
X(237)X(523)
X(5661) = crosssum of X(6) and X(5091)
Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given (Peter Moses, June 23, 2014) by
T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)
X(5662) is the minimizer T when P = X(1) and Q = X(3).
X(5662) lies on these lines: {1,3}, {2,905}, {63,2427}
Let A′B′C′ be the orthocentroidal triangle. Let A″ be the reflection of A in line B′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5663). (Randy Hutson, July 11, 2019)
X(5663) lies on the line at infinity and these lines: {1,3024}, {3,74}, {4,94}, {5,113}, {23,3581}, {26,1498}, {30,511}, {40,2940}, {49,3043}, {51,3845}, {52,3627}, {67,1352}, {64,155}, {182,4550}, {389,546}, {500,3746}, {548,1216}, {895,1351}, {1147,3357}, {1350,2930}, {1353,5095}, {1597,5093}, {1625,3269}, {2088,2493}, {3567,3843}, {3850,5462}
X(5663) = isogonal conjugate of X(477)
X(5663) = X(2693)-Ceva conjugate of X(3)
X(5663) = anticomplementary conjugate of X(34193)
X(5663) = X(477)-anticomplementary conjugate of X(8)
X(5663) = complementary conjugate of X(10)
X(5663) = crossdifference of X(6) and X(1637)
X(5663) = crosssum of X(i) and X(j) for these {I,J}: {55, 3013}, {523,3154}
X(5663) = X(3)-vertex conjugate of X(526)
X(5663) = trilinear product X(2410)*X(2624)
X(5663) = barycentric product X(i)*X(j) for these {I,J}: {526,2410}, {2437,3268}
X(5663) = Thomson-isogonal conjugate of X(476)
X(5663) = Lucas-isogonal conjugate of X(476)
X(5663) = X(952)-of-orthic-triangle if ABC is acute
X(5663) = Cundy-Parry Phi transform of X(14385)
X(5663) = Cundy-Parry Psi transform of X(14254)
X(5663) = X(92)-isoconjugate of X(32663)
Let U be the circle (X(5617), R) and V the circle (X(5613),R), so that U passes through X(14) and V passes through X(13), and let Γ be the circumcircle of ABC. Then X(5664) is the radical center of the circles U, V, and Γ. (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)
X(5664) is the isotomic conjugate of the trilinear pole of line X(30)X(74). (Randy Hutson, July 7, 2014)
X(5664) lies on these lines: {2, 525}, {3, 523}, {39, 2485}, {99, 5649}, {113, 114}, {140, 5489}, {323, 2411}, {2407, 2420}, {2482, 2799}, {2848, 3184}
X(5664) = isotomic conjugate of X(39290)
X(5664) = complement of X(2394)
X(5664) = anticomplement of X(14566)
Let A′B′C′ be the intouch triangle of a triangle ABC. Let (BA′) be the circle having diameter BA′, and likewise for (A′C); define circles (CB′) and (AC′) cyclically, and define circles (B′A) and (C′B) cyclically. Let U be the point, other than A′, in which (BA′) meets the incircle, and define V and W cyclically. Let U′ be the point, other than A′, in which (A′C) meets the incircle, and define V′ and W′ cyclically. Let A″B″C″ be the triangle formed by the lines UU′, VV′, WW′. Let A″' be the point, other than A, in which the circles (AB′) and (AC′) meet, and define B‴ and C‴ cyclically. Then A″B″C″ is perspective to ABC, and the perspector is X(5665). Moreover, A″B″C″ is perspective to A″B″C″, and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)
X(5665) lies on the Feuerbach hyperbola and these lines: (1, 1427), (4, 3671), (7, 950), (8, 226), (9, 65), (21, 57), (34, 1172), (40, 943), (72, 4866), (79, 3586), (84, 942), (85, 314), (104, 3333), (388, 3243), (405, 3339), (728, 2171), (946, 3427), (1000, 3487), (1420, 2320), (1490, 3577), (1697, 2346), (1728, 3467), (1896, 5342), (2099, 2900), (2263, 2298), (2335, 3247), (3296, 3488), (3419, 5290), (3600, 5558), (3612, 5424), (4332, 5269), (4355, 5557)
X(5665) = isogonal conjugate of X(3601)
Let A′B′C′ be the intouch triangle of a triangle ABC. Let (BA′) be the circle having diameter BA′, and likewise for (A′C); define circles (CB′) and (AC′) cyclically, and define circles (B′A) and (C′B) cyclically. Let U be the point, other than A′, in which (BA′) meets the incircle, and define V and W cyclically. Let U′ be the point, other than A′, in which (A′C) meets the incircle, and define V′ and W′ cyclically. Let A″B″C″ be the triangle formed by the lines UU′, VV′, WW′. Let A‴ be the point, other than A, in which the circles (AB′) and (AC′) meet, and define B‴ and C‴ cyclically. Then A″B″C″ is perspective to ABC, and the perspector is X(5665). Moreover, A″B″C″ is perspective to A″B″C″, and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)
X(5666) lies on these lines: (1, 1427), (1054, 3339), (1707, 3361)
As X(30)-Ceva conjugates of points on the Neuberg cubic (K001), the points X(5667) - X(5685) are also on the Neuberg cubic. Specifically, if P is a point on the Neuberg cubic, then the points X(74), P, and the X(30)-Ceva conjugate of P are collinear, since X(74) is the isopivot (or secondary pivot) of the cubic. See Table 9: Points on the Neuberg Cubic.
X(5667) inverse-in-circumconic-centered-at-X(4) of X(133); also, X(5667) is the antipode of X(4) in the bianticevian conic of X(1) and X(4). (Randy Hutson, July 7, 2014)
X(5667) lies on the Neuberg cubic and these lines: {1,2816}, {3,3462}, {4,74}, {19,2822}, {20,1075}, {112,376}, {122,631}, {146,4240}, {399,2133}, {1138,1157}, {1148,4302}, {1263,3481}, {2790,3186}, {3087,3269}, {3183,3529}, {3324,4293}, {3440,3479}, {3441,3480}
X(5667) = reflection of X(I() in X(j) for these (i,j): (4,107), (1294,3184)
X(5667) = antigonal conjugate of X(34298)
X(5668) lies on the Neuberg cubic and these lines: {3,3166}, {4,14}, {13,3479}, {15,74}, {16,1495}, {61,185}, {1277,3466}, {2133,5623}, {3284,5669}
X(5668) = reflection of X(5669) in X(3284)
X(5668) = anticomplement of X(33501)
X(5668) = polar-circle-inverse of X(35715)
X(5668) = homothetic center of X(14)- and X(16)-Ehrmann triangles; see X(25)
X(5669) lies on the Neuberg cubic and these lines: {3,3165}, {4,13}, {14,3480}, {15,1495}, {16,74}, {62,185}, {1276,3466}, {2133,5624}, {3284,5668}
X(5669) = reflection of X(5668) in X(3284)
X(5669) = anticomplement of X(33499)
X(5669) = polar-circle-inverse of X(35714)
X(5669) = homothetic center of X(13)-Ehrmann triangle and X(15) triangle; see X(25)
X(5670) lies on the Neuberg cubic and these lines: {3,2133}, {74,1138}
X(5671) is the tangential of X(3065) on the Neuberg cubic.
X(5671) lies on the Neuberg cubic, the Lester circle, and these lines: {3,1138}, {30,1117}, {74,1263}, {2133,3484}>
X(5671) = X(1117)-Ceva conjugate of X(3471)
X(5672) is the Gibert-Burek-Moses-concurrent-circles image of X(13). (Randy Hutson, July 7, 2014)
X(5672) lies on the Neuberg cubic and these lines: {1,13}, {16,3065}, {74,1276}, {1277,3440}, {1138,5673}, {2940,2953}
X(5673) is the Gibert-Burek-Moses-concurrent-circles image of X(14). (Randy Hutson, July 7, 2014)
X(5673) lies on the Neuberg cubic and these lines: {1,14}, {15,3065}, {74,1277}, {1276,3441}, {1138,5672}, {2940,2952}
X(5674) is the tangential of X(13) on the Neuberg cubic.
X(5674) lies on the Neuberg cubic and these lines: {3,3440}, {13,2981}, {74,1337}, {617,1138}, {1263,3480}
X(5675) is the tangential of X(14) on the Neuberg cubic.
X(5675) lies on the Neuberg cubic and these lines: {3,3441}, {74,1338}, {616,1138}, {1263,3479}
X(5676) lies on the Neuberg cubic and this line: {74,2133}
X(5677) lies on the Neuberg cubic and these lines: {1,1138}, {74,3065}, {1263,3466}, {2133,3465}
X(5678) lies on the Neuberg cubic and these lines: {74,3440}, {617,2133}, {1138,3480}
X(5679) lies on the Neuberg cubic and these lines: {74,3441}, {616,2133}, {1138,3479}
X(5680) lies on the Neuberg cubic and these lines: {74,3466}, {484,2133}
X(5681) lies on the Neuberg cubic and these lines: {4,3441}, {74,3479}
X(5682) lies on the Neuberg cubic and these lines: {4,3440}, {74,3480}
X(5683) lies on the Neuberg cubic and these lines: {4,3463}, {74,3481}
X(5684) lies on the Neuberg cubic and these lines: {3,1263}, {74,3482}, {1138,3484}, {3065,3483}
X(5685) lies on the Neuberg cubic and these lines: {1,1263}, {3,3065}, {74,3483}, {1138,3465}, {3466,3482}
X(5686) lies on these lines: {1, 4924}, {2, 210}, {7, 10}, {8, 9}, {144, 1654}, {145, 1001}, {200, 5273}, {405, 2346}, {480, 958}, {497, 3715}, {516, 3543}, {936, 5265}, {971, 5657}, {984, 3672}, {1145, 1156}, {1445, 3600}, {1698, 5542}, {1738, 4346}, {1743, 4344}, {1757, 4307}, {1788, 3983}, {3059, 4662}, {3158, 5325}, {3189, 5302}, {3243, 3616}, {3485, 4005}, {3696, 4461}, {3711, 5218}, {3751, 3945}, {3974, 4042}, {4313, 5234}, {4321, 5435}, {4326, 4882}, {4384, 4899}, {4678, 5086}, {4847, 5274}, {5231, 5328}
X(5686) = anticomplement of X(38053)
The outer Garcia triangle is defined at X(5587). The definition is re-stated here. Let TATBTC be the extouch triangle of a triangle ABC, and let LA be the line perpendicular to line BC at TA. Of the two points on LA at distance r from TA, let A′ be the one farther from A and let A″ be the closer. Define B′, C′ and B″, C″ cyclically. Then A′B′C′ is the outer Garcia triangle, and A″B″C″ the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014).
X(5687) lies on the cubic K844 and these lines: {1, 474}, {2, 496}, {3, 8}, {4, 1260}, {5, 3434}, {6, 3293}, {9, 3697}, {10, 55}, {19, 3694}, {20, 3421}, {21, 3617}, {25, 3695}, {30, 3436}, {31, 3214}, {35, 958}, {36, 3632}, {40, 64}, {43, 5255}, {46, 518}, {56, 519}, {57, 3555}, {63, 3579}, {65, 3689}, {78, 517}, {101, 4513}, {144, 3650}, {145, 404}, {149, 4193}, {165, 3916}, {169, 3693}, {191, 4436}, {197, 2915}, {198, 2321}, {210, 1898}, {214, 1388}, {218, 3501}, {219, 3362}, {220, 1018}, {221, 4551}, {228, 5295}, {346, 4222}, {355, 1012}, {377, 495}, {382, 5080}, {390, 5084}, {392, 936}, {442, 954}, {480, 516}, {497, 4187}, {498, 2886}, {499, 3035}, {528, 1329}, {529, 4299}, {573, 3713}, {595, 4383}, {612, 3931}, {668, 1975}, {859, 1043}, {899, 3915}, {910, 4515}, {942, 3870}, {960, 5119}, {976, 4642}, {986, 3961}, {993, 3626}, {997, 3057}, {1001, 1698}, {1004, 3868}, {1011, 4651}, {1054, 3976}, {1071, 3359}, {1089, 4557}, {1125, 3303}, {1148, 1897}, {1191, 3216}, {1319, 3893}, {1377, 2066}, {1378, 5414}, {1385, 3872}, {1482, 4511}, {1486, 3932}, {1500, 5275}, {1574, 2241}, {1617, 1788}, {1696, 3950}, {1722, 3749}, {1724, 3052}, {1759, 4006}, {1766, 3965}, {1783, 3172}, {2093, 4018}, {2098, 2802}, {2222, 2756}, {2271, 2295}, {2345, 4254}, {2478, 3820}, {2551, 4294}, {2894, 4197}, {3053, 5291}, {3185, 3714}, {3220, 4901}, {3241, 5253}, {3242, 3670}, {3244, 3304}, {3306, 4917}, {3550, 5247}, {3560, 5086}, {3576, 4853}, {3621, 4188}, {3625, 5204}, {3633, 5563}, {3634, 4423}, {3678, 3711}, {3681, 3927}, {3683, 3983}, {3701, 4186}, {3705, 5100}, {3715, 4015}, {3780, 5021}, {3869, 3940}, {3874, 5221}, {3881, 4860}, {3921, 4512}, {3924, 4695}, {3962, 5183}, {4185, 5300}, {4189, 4678}, {4225, 4720}, {4428, 5259}, {4640, 4662}, {4668, 5010}, {4669, 5267}, {4677, 5288}, {5044, 5250}, {5289, 5541}
X(5687) = anticomplement of X(496)The outer Garcia triangle is defined at X(5587).
X(5688) lies on these lines: {1, 5590}, {6, 10}, {8, 175}, {355, 1160}, {519, 5604}, {3679, 5588}
The outer Garcia triangle is defined at X(5587).
X(5689) lies on these lines: {{1, 5591}, {6, 10}, {8, 176}, {355, 1161}, {519, 5605}, {3679, 5589}
The outer Garcia triangle is defined at X(5587).
X(5690) lies on these lines: {1, 140}, {2, 1482}, {3, 8}, {4, 3617}, {5, 10}, {6, 38116}, {7, 38121}, {9, 38126}, {11, 5697}, {12, 5903}, {20, 4678}, {21, 11849}, {24, 12135}, {26, 8193}, {30, 40}, {35, 7508}, {36, 10944}, {46, 5252}, {55, 10573}, {56, 12647}, {63, 31775}, {65, 495}, {72, 6735}, {78, 52265}, {80, 3467}, {119, 5692}, {142, 38041}, {145, 631}, {149, 6902}, {153, 12849}, {165, 548}, {182, 5846}, {200, 37700}, {201, 24028}, {210, 5887}, {219, 59594}, {226, 50193}, {281, 7524}, {333, 15952}, {371, 49233}, {372, 49232}, {376, 50798}, {381, 962}, {382, 6361}, {388, 24470}, {390, 59381}, {392, 17527}, {404, 22765}, {405, 5554}, {411, 18524}, {427, 41722}, {443, 2095}, {474, 10680}, {484, 7354}, {496, 1737}, {498, 2099}, {499, 1387}, {511, 49524}, {515, 550}, {516, 3627}, {518, 10915}, {519, 549}, {528, 60912}, {537, 51048}, {542, 50949}, {546, 5587}, {547, 3656}, {551, 11539}, {572, 17362}, {573, 594}, {590, 35641}, {615, 35642}, {632, 1125}, {730, 32448}, {740, 51046}, {758, 5499}, {900, 59898}, {912, 6736}, {936, 45080}, {938, 6767}, {942, 4848}, {950, 10386}, {958, 6914}, {971, 24393}, {993, 26285}, {997, 37828}, {999, 1788}, {1000, 14986}, {1001, 37622}, {1006, 3871}, {1012, 35448}, {1018, 40997}, {1056, 5708}, {1064, 3214}, {1146, 3730}, {1155, 45287}, {1159, 3487}, {1210, 9957}, {1211, 30449}, {1317, 21842}, {1320, 57298}, {1350, 59407}, {1351, 59406}, {1353, 5847}, {1376, 6924}, {1386, 38110}, {1388, 12735}, {1478, 37567}, {1479, 12019}, {1484, 2802}, {1512, 3697}, {1532, 3876}, {1537, 6941}, {1565, 33298}, {1574, 34460}, {1595, 1829}, {1596, 1902}, {1656, 5603}, {1657, 9778}, {1658, 15177}, {1697, 5722}, {1698, 3628}, {1699, 3850}, {1706, 5709}, {1766, 17275}, {1770, 5183}, {1836, 10827}, {1837, 5119}, {1872, 46878}, {1953, 21012}, {2077, 5258}, {2080, 12195}, {2093, 9578}, {2102, 57323}, {2103, 57322}, {2136, 61122}, {2361, 54350}, {2475, 16150}, {2476, 38058}, {2550, 5762}, {2551, 6929}, {2646, 36920}, {2800, 3678}, {2807, 5876}, {2826, 24093}, {2829, 33559}, {3035, 19907}, {3036, 5840}, {3058, 37563}, {3070, 35611}, {3071, 35610}, {3085, 5719}, {3090, 18493}, {3091, 61262}, {3097, 61625}, {3109, 35193}, {3146, 28182}, {3189, 28466}, {3219, 35460}, {3241, 5054}, {3243, 38122}, {3244, 10165}, {3245, 3585}, {3293, 5396}, {3295, 6883}, {3311, 19065}, {3312, 19066}, {3336, 5434}, {3340, 11374}, {3359, 24467}, {3416, 3564}, {3419, 31789}, {3421, 3927}, {3428, 11499}, {3434, 6928}, {3436, 6923}, {3474, 9655}, {3485, 31479}, {3517, 7718}, {3523, 3621}, {3524, 31145}, {3525, 3622}, {3526, 3616}, {3530, 3576}, {3533, 46934}, {3534, 34627}, {3541, 11396}, {3555, 10202}, {3560, 9708}, {3577, 11530}, {3587, 5787}, {3589, 38040}, {3600, 37545}, {3601, 37739}, {3612, 37740}, {3614, 18393}, {3623, 10303}, {3624, 16200}, {3625, 5882}, {3633, 12108}, {3634, 11230}, {3635, 58441}, {3636, 61853}, {3649, 37719}, {3653, 11812}, {3655, 4677}, {3681, 37401}, {3696, 29010}, {3703, 37619}, {3704, 48930}, {3717, 29327}, {3751, 34380}, {3753, 8728}, {3754, 25466}, {3756, 56804}, {3816, 3884}, {3826, 3918}, {3828, 15699}, {3830, 34632}, {3843, 9812}, {3845, 4745}, {3847, 6702}, {3853, 28216}, {3857, 61260}, {3858, 18483}, {3861, 9589}, {3869, 6842}, {3870, 37615}, {3874, 5885}, {3877, 4187}, {3880, 10916}, {3897, 37298}, {3911, 24928}, {3913, 10267}, {3925, 15104}, {3932, 31394}, {3940, 6825}, {4004, 5249}, {4015, 20117}, {4018, 10273}, {4026, 31395}, {4046, 10434}, {4192, 4651}, {4193, 34122}, {4220, 33091}, {4295, 9654}, {4297, 4669}, {4300, 49984}, {4311, 5122}, {4314, 51787}, {4324, 37006}, {4384, 19512}, {4420, 21740}, {4424, 50067}, {4437, 29697}, {4512, 50243}, {4646, 48847}, {4662, 6001}, {4665, 29069}, {4701, 17502}, {4711, 9943}, {4732, 29054}, {4746, 28236}, {4769, 50772}, {4816, 61792}, {4847, 31786}, {4853, 37611}, {4882, 5534}, {4995, 37571}, {4999, 46920}, {5010, 37706}, {5046, 10738}, {5050, 51192}, {5066, 7989}, {5067, 46932}, {5070, 5734}, {5071, 50872}, {5072, 9779}, {5079, 61267}, {5082, 6827}, {5086, 7491}, {5090, 6756}, {5128, 9613}, {5174, 7510}, {5176, 56288}, {5178, 28459}, {5179, 21872}, {5220, 6256}, {5223, 5843}, {5260, 7489}, {5263, 37510}, {5264, 5398}, {5267, 26086}, {5270, 11246}, {5288, 37561}, {5289, 11729}, {5305, 9620}, {5399, 37558}, {5418, 44635}, {5420, 44636}, {5426, 44254}, {5428, 8715}, {5439, 58561}, {5443, 11280}, {5446, 23841}, {5453, 37698}, {5493, 28146}, {5531, 16132}, {5541, 62354}, {5542, 38111}, {5550, 46219}, {5552, 5730}, {5599, 11253}, {5600, 11252}, {5686, 5779}, {5688, 5874}, {5689, 5875}, {5693, 37725}, {5698, 38211}, {5703, 11041}, {5710, 36754}, {5711, 44414}, {5727, 61763}, {5732, 59414}, {5752, 31785}, {5759, 59413}, {5777, 31798}, {5780, 6848}, {5791, 9623}, {5794, 54286}, {5795, 31445}, {5805, 38200}, {5853, 31658}, {5854, 6713}, {5855, 22836}, {5883, 6583}, {5884, 13145}, {5902, 15888}, {5946, 58487}, {6003, 62323}, {6175, 61552}, {6191, 7026}, {6192, 7043}, {6197, 7511}, {6200, 35843}, {6210, 33165}, {6211, 33076}, {6246, 38213}, {6259, 54156}, {6265, 11014}, {6396, 35842}, {6642, 12410}, {6666, 38043}, {6667, 38044}, {6668, 38045}, {6675, 19860}, {6690, 30147}, {6734, 6922}, {6738, 15935}, {6762, 37534}, {6765, 18443}, {6789, 53799}, {6853, 59382}, {6862, 19843}, {6868, 17784}, {6882, 14923}, {6889, 10528}, {6892, 18231}, {6906, 35000}, {6909, 26321}, {6911, 9709}, {6916, 54398}, {6918, 8158}, {6936, 20075}, {6940, 37535}, {6949, 10698}, {6951, 20060}, {6952, 35457}, {6954, 59591}, {6955, 20076}, {6958, 10527}, {6960, 48667}, {6967, 10529}, {6971, 11680}, {6973, 8165}, {6980, 11681}, {6988, 20007}, {6989, 15934}, {6999, 51353}, {7280, 37707}, {7377, 29593}, {7385, 24808}, {7412, 56877}, {7483, 61520}, {7525, 37557}, {7530, 9911}, {7534, 54294}, {7555, 9626}, {7580, 18518}, {7583, 13911}, {7584, 13973}, {7609, 61621}, {7705, 17533}, {7715, 49542}, {7962, 11373}, {7968, 13966}, {7969, 8981}, {7970, 15561}, {7971, 62218}, {7976, 11171}, {7978, 14643}, {7983, 38224}, {7984, 15061}, {7988, 12812}, {8144, 54295}, {8185, 17714}, {8197, 32146}, {8200, 11823}, {8204, 32147}, {8207, 11822}, {8214, 32177}, {8215, 32178}, {8229, 31079}, {8270, 32047}, {8582, 13600}, {8666, 32612}, {8727, 25006}, {9041, 50977}, {9534, 19543}, {9548, 59313}, {9590, 12107}, {9624, 11224}, {9669, 30305}, {9834, 35245}, {9835, 35244}, {9857, 32151}, {9864, 12783}, {9905, 50708}, {9933, 47391}, {9940, 24391}, {9947, 9949}, {10106, 37582}, {10107, 12609}, {10109, 38021}, {10124, 25055}, {10171, 61907}, {10172, 61270}, {10197, 11281}, {10248, 14269}, {10269, 12513}, {10310, 22758}, {10327, 19544}, {10389, 31436}, {10525, 22938}, {10526, 38178}, {10543, 12104}, {10572, 37568}, {10592, 12047}, {10593, 17606}, {10597, 37462}, {10625, 16980}, {10695, 57297}, {10696, 57303}, {10697, 38764}, {10699, 57327}, {10700, 57328}, {10701, 57329}, {10702, 57330}, {10703, 38776}, {10704, 57331}, {10705, 57332}, {10742, 37437}, {10791, 32134}, {10826, 12701}, {10902, 48696}, {10912, 26492}, {10955, 41686}, {10993, 15338}, {11009, 15950}, {11015, 59347}, {11019, 31792}, {11113, 26878}, {11179, 50783}, {11251, 16210}, {11277, 33858}, {11375, 25415}, {11376, 30323}, {11500, 35239}, {11501, 59317}, {11522, 35018}, {11529, 51784}, {11540, 51105}, {11684, 47032}, {11695, 58535}, {11715, 54176}, {11725, 34127}, {11735, 34128}, {11900, 32162}, {11928, 17556}, {11929, 17532}, {12102, 61257}, {12103, 37712}, {12106, 37546}, {12114, 35238}, {12115, 56879}, {12454, 45620}, {12455, 45621}, {12495, 26316}, {12514, 37290}, {12515, 12751}, {12565, 18528}, {12571, 38071}, {12575, 18527}, {12610, 17239}, {12616, 58637}, {12626, 26451}, {12627, 26341}, {12628, 26348}, {12635, 26487}, {12636, 45623}, {12637, 45624}, {12648, 16203}, {12649, 16202}, {12675, 40296}, {12737, 61566}, {12747, 13199}, {12778, 12785}, {12780, 22896}, {12781, 22851}, {12782, 22697}, {12811, 61263}, {12898, 15035}, {13099, 57304}, {13369, 31787}, {13373, 34791}, {13411, 50194}, {13463, 24387}, {13731, 17751}, {13743, 61622}, {13747, 61521}, {13883, 19117}, {13893, 13925}, {13936, 19116}, {13947, 13993}, {13996, 37726}, {14077, 15584}, {14110, 37356}, {14647, 61556}, {14666, 50925}, {14731, 36154}, {14830, 50880}, {14839, 49111}, {14891, 50804}, {14892, 30308}, {14893, 50865}, {14940, 31948}, {15026, 58469}, {15174, 37724}, {15310, 49693}, {15326, 37572}, {15462, 32298}, {15681, 50864}, {15683, 50809}, {15684, 50797}, {15686, 28208}, {15687, 28198}, {15690, 34628}, {15692, 50818}, {15693, 34748}, {15694, 38314}, {15704, 28160}, {15708, 20049}, {15713, 38068}, {15717, 20052}, {15720, 20050}, {15803, 37709}, {15844, 34353}, {15901, 30329}, {15955, 37646}, {15971, 48928}, {15973, 31778}, {16128, 61605}, {16137, 17718}, {16174, 38182}, {16189, 34595}, {16192, 34200}, {16408, 61535}, {16417, 26062}, {16475, 51732}, {16608, 21231}, {16616, 38139}, {17051, 61852}, {17057, 38109}, {17504, 34641}, {17531, 45977}, {17555, 21664}, {17564, 17614}, {17566, 34123}, {17567, 35272}, {17619, 41012}, {18253, 31649}, {18519, 37022}, {18583, 38047}, {18802, 32554}, {18857, 33956}, {18908, 40263}, {19540, 59296}, {19542, 56810}, {19547, 54433}, {19649, 33090}, {19710, 51070}, {19861, 52264}, {19862, 55859}, {19872, 61877}, {19876, 47599}, {19878, 61876}, {19883, 51077}, {20014, 61820}, {20053, 58230}, {20054, 61814}, {20057, 55863}, {20107, 23960}, {20328, 21258}, {20368, 33169}, {20420, 37584}, {20423, 38087}, {20653, 30272}, {21013, 33299}, {21075, 51362}, {21077, 44663}, {21271, 40999}, {21620, 31794}, {21850, 38165}, {22651, 51688}, {22652, 51690}, {22799, 37821}, {22937, 33856}, {24299, 41575}, {25030, 30196}, {25416, 38032}, {25440, 26286}, {25444, 61699}, {25557, 33815}, {26006, 31186}, {26129, 38084}, {26287, 51111}, {26382, 48519}, {26398, 48493}, {26406, 48520}, {26422, 48494}, {26442, 49428}, {26443, 49427}, {26444, 49086}, {26445, 49087}, {26482, 45288}, {26498, 49402}, {26507, 49401}, {26516, 49060}, {26521, 49061}, {26629, 30136}, {26938, 56960}, {27076, 59703}, {27529, 62826}, {27870, 49600}, {28150, 33697}, {28154, 62044}, {28158, 62047}, {28164, 62155}, {28168, 62159}, {28172, 62162}, {28202, 33699}, {28232, 62006}, {28538, 50979}, {28870, 48932}, {28905, 48944}, {29207, 50308}, {29298, 44824}, {29309, 48888}, {29311, 48934}, {29331, 49772}, {29667, 37360}, {30286, 53053}, {30312, 38055}, {30315, 44904}, {30331, 38130}, {30389, 61291}, {30392, 61288}, {30810, 48381}, {31330, 37365}, {31398, 31406}, {31425, 58221}, {31472, 38235}, {31650, 35016}, {31666, 61802}, {31670, 38144}, {31671, 38149}, {31672, 38154}, {31793, 51755}, {31803, 56762}, {31847, 53800}, {31849, 32142}, {31855, 37732}, {31937, 58631}, {32049, 62858}, {32900, 61293}, {33923, 35242}, {34351, 51696}, {34638, 62154}, {34720, 49176}, {34747, 61827}, {34847, 59615}, {35203, 44039}, {35251, 55868}, {35404, 50814}, {36155, 52200}, {37299, 50890}, {37440, 49553}, {37468, 48363}, {37509, 57280}, {37525, 37734}, {37529, 50418}, {37536, 45955}, {37598, 37715}, {37612, 62874}, {37618, 37738}, {37623, 57284}, {37699, 59294}, {37708, 58887}, {37711, 59316}, {37718, 61601}, {37950, 47492}, {38029, 49681}, {38052, 61509}, {38057, 61511}, {38064, 51000}, {38076, 61956}, {38083, 51069}, {38107, 40333}, {38108, 43166}, {38118, 49684}, {38569, 51631}, {39004, 40944}, {39523, 62805}, {41723, 47515}, {42215, 49226}, {42216, 49227}, {42450, 58647}, {43118, 49330}, {43119, 49329}, {43575, 43827}, {43830, 50476}, {44245, 61247}, {44266, 47496}, {44452, 51725}, {44903, 50801}, {45444, 49355}, {45445, 49356}, {45546, 48772}, {45547, 48773}, {45976, 62318}, {46704, 48917}, {46930, 60781}, {46931, 61886}, {47749, 53283}, {48894, 62185}, {49697, 59620}, {50190, 58605}, {50799, 61978}, {50800, 61985}, {50802, 61942}, {50806, 61936}, {50812, 62114}, {50813, 62116}, {50819, 62088}, {50841, 51717}, {50871, 62089}, {51073, 55861}, {51092, 61833}, {51103, 61851}, {51107, 61845}, {51110, 61860}, {51120, 61949}, {52354, 59586}, {55866, 58238}, {58247, 61919}, {59412, 60922}, {59420, 62126}, {61248, 62123}, {61250, 62144}, {61252, 62151}, {61254, 62034}, {61256, 62026}, {61279, 61858}, {61282, 61835}, {61294, 61801}
X(5690) = midpoint of X(i) and X(j) for these {i,j}: {2, 34718}, {3, 8}, {4, 12702}, {10, 11362}, {20, 18525}, {40, 355}, {72, 37562}, {100, 19914}, {376, 50798}, {381, 50810}, {382, 6361}, {549, 50823}, {550, 37705}, {944, 12645}, {1482, 12245}, {3534, 34627}, {3625, 5882}, {3626, 43174}, {3632, 37727}, {3654, 3679}, {3655, 4677}, {3830, 34632}, {3869, 25413}, {3913, 49168}, {3927, 6850}, {4297, 47745}, {5493, 31673}, {5541, 62354}, {5657, 59503}, {5731, 51515}, {5752, 31785}, {5777, 31798}, {5779, 35514}, {5790, 59417}, {5881, 18481}, {6259, 54156}, {7991, 12699}, {9947, 31797}, {10625, 16980}, {10993, 62616}, {11179, 50783}, {11684, 47032}, {12247, 12331}, {12513, 49169}, {12515, 12751}, {12747, 13199}, {12778, 13211}, {13996, 37726}, {14666, 50925}, {14830, 50880}, {15681, 50864}, {15971, 48928}, {16139, 47033}, {18802, 32554}, {20070, 48661}, {26446, 63143}, {31788, 34790}, {32049, 62858}, {34641, 51705}, {46704, 48917}, {48746, 48747}, {48915, 48937}
X(5690) = reflection of X(i) in X(j) for these {i,j}: {1, 140}, {3, 61524}, {4, 18357}, {5, 10}, {145, 61286}, {355, 61510}, {549, 50821}, {550, 3579}, {946, 9956}, {960, 58630}, {962, 40273}, {1385, 6684}, {1482, 5901}, {1483, 1385}, {1484, 12619}, {1537, 61580}, {3244, 15178}, {3579, 43174}, {3627, 18480}, {3655, 12100}, {3656, 547}, {3874, 5885}, {4297, 31663}, {4301, 9955}, {5446, 23841}, {5694, 3678}, {5882, 13624}, {5884, 13145}, {5887, 31835}, {6147, 37438}, {6265, 61562}, {8715, 32157}, {9957, 31838}, {10222, 1125}, {10246, 61614}, {10283, 11231}, {10284, 3884}, {10543, 12104}, {10680, 61530}, {10738, 61553}, {11278, 13464}, {12675, 40296}, {12699, 546}, {12737, 61566}, {13369, 31787}, {13463, 24387}, {13464, 3634}, {13743, 61622}, {15686, 50808}, {15687, 50796}, {15704, 31730}, {16128, 61605}, {18481, 548}, {18525, 61249}, {19907, 3035}, {20117, 4015}, {20330, 3826}, {22758, 61539}, {22791, 5}, {22793, 19925}, {24474, 61541}, {24475, 34339}, {31162, 5066}, {31649, 18253}, {31803, 56762}, {31837, 58643}, {31870, 3918}, {31937, 58631}, {33668, 5499}, {33699, 34648}, {33858, 11277}, {34628, 15690}, {34773, 3}, {34791, 13373}, {37533, 61533}, {38028, 26446}, {38034, 38042}, {38042, 38112}, {38112, 38127}, {38138, 38176}, {40273, 61259}, {41869, 3853}, {42450, 58647}, {44266, 47496}, {48933, 15973}, {50811, 34200}, {50824, 549}, {50865, 14893}, {51709, 3828}, {52200, 36155}, {58535, 11695}, {61148, 4999}, {61245, 47745}, {61283, 10165}, {61295, 5882}, {61597, 51700}, {62036, 31673}, {62041, 33697}, {62154, 34638}
X(5690) = complement of X(1482)
X(5690) = anticomplement of X(5901)
X(5690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 140, 38028}, {1, 5445, 5433}, {1, 24914, 15325}, {1, 26446, 140}, {2, 1482, 5901}, {2, 12245, 1482}, {3, 956, 32153}, {3, 5657, 61524}, {3, 5687, 32141}, {3, 12331, 11491}, {3, 12645, 944}, {3, 18526, 5731}, {3, 32153, 38602}, {3, 51515, 18526}, {3, 59503, 8}, {4, 3617, 5790}, {4, 5790, 18357}, {4, 20070, 48661}, {4, 59417, 12702}, {5, 10, 38042}, {5, 22791, 38034}, {5, 38112, 10}, {8, 944, 12645}, {8, 5657, 3}, {8, 61524, 34773}, {10, 946, 9956}, {10, 960, 3820}, {10, 3817, 31399}, {10, 3878, 1329}, {10, 4301, 10175}, {10, 5836, 31419}, {10, 5837, 5044}, {10, 10176, 9711}, {10, 21616, 5123}, {12, 5903, 39542}, {20, 4678, 59388}, {20, 59388, 18525}, {35, 41684, 10950}, {40, 3679, 355}, {46, 5252, 18990}, {55, 10573, 37730}, {65, 495, 6147}, {65, 10039, 495}, {80, 11010, 6284}, {140, 61597, 51700}, {145, 631, 10246}, {145, 10246, 61286}, {165, 4668, 5881}, {165, 5881, 18481}, {165, 18481, 548}, {210, 5887, 31835}, {219, 59671, 59594}, {355, 3654, 40}, {355, 3679, 61510}, {355, 16139, 11827}, {381, 962, 40273}, {381, 5818, 61259}, {388, 36279, 24470}, {392, 24982, 17527}, {484, 37710, 7354}, {498, 2099, 37737}, {499, 2098, 1387}, {549, 1483, 1385}, {550, 59400, 37705}, {632, 10283, 1125}, {946, 9956, 5}, {958, 11248, 6914}, {962, 5818, 381}, {962, 53620, 5818}, {997, 37828, 47742}, {999, 1788, 34753}, {1006, 3871, 37621}, {1125, 10222, 10283}, {1125, 11231, 632}, {1376, 11249, 6924}, {1385, 1483, 50824}, {1385, 6684, 549}, {1385, 50821, 6684}, {1482, 34718, 12245}, {1656, 5603, 61272}, {1656, 8148, 5603}, {1697, 5722, 15172}, {1698, 5886, 3628}, {1698, 7982, 5886}, {1699, 61261, 3850}, {1737, 3057, 496}, {1837, 5119, 15171}, {2093, 9578, 57282}, {3090, 18493, 61269}, {3241, 37624, 61281}, {3244, 10165, 15178}, {3244, 15178, 61283}, {3295, 18391, 12433}, {3340, 31434, 11374}, {3359, 57279, 24467}, {3419, 55104, 31789}, {3523, 3621, 7967}, {3526, 10247, 3616}, {3576, 3632, 37727}, {3579, 3626, 37705}, {3616, 10247, 61278}, {3617, 12702, 18357}, {3617, 59417, 4}, {3624, 16200, 61276}, {3625, 10164, 5882}, {3625, 13624, 61295}, {3627, 38138, 18480}, {3632, 9588, 3576}, {3634, 11230, 55856}, {3634, 13464, 11230}, {3656, 19875, 547}, {3753, 24474, 61541}, {3753, 24987, 8728}, {3826, 20330, 38171}, {3828, 51709, 15699}, {3877, 25005, 4187}, {4297, 4669, 47745}, {4297, 31663, 8703}, {4301, 10175, 9955}, {4669, 31663, 61245}, {4677, 7987, 61296}, {4701, 17502, 61297}, {4848, 31397, 942}, {4882, 30503, 5534}, {5303, 34474, 3}, {5493, 38155, 31673}, {5587, 7991, 12699}, {5587, 12699, 546}, {5603, 9780, 1656}, {5686, 35514, 5779}, {5687, 32141, 51525}, {5697, 18395, 11}, {5790, 12702, 4}, {5818, 50810, 962}, {5882, 10164, 13624}, {6361, 59387, 382}, {6734, 51433, 10914}, {6735, 37562, 10942}, {6940, 54391, 37535}, {7987, 61296, 3655}, {8148, 9780, 61272}, {8227, 11531, 3656}, {9588, 37727, 3530}, {9589, 61258, 3861}, {9708, 10306, 3560}, {9709, 22770, 6911}, {9955, 10175, 5}, {10164, 13624, 15712}, {10222, 11231, 1125}, {10738, 59415, 61553}, {10915, 34339, 32213}, {11230, 11278, 13464}, {11362, 38112, 22791}, {11362, 38127, 10}, {11522, 54447, 61268}, {11531, 19875, 8227}, {11545, 15171, 1837}, {12702, 48661, 20070}, {12787, 12788, 3416}, {13911, 35774, 7583}, {13973, 35775, 7584}, {15694, 50805, 38314}, {15908, 21031, 119}, {16189, 34595, 61275}, {17606, 30384, 10593}, {18483, 38140, 3858}, {18525, 59388, 61249}, {19925, 22793, 3845}, {20053, 58230, 61290}, {21271, 40999, 41007}, {22791, 38042, 5}, {24475, 44222, 31657}, {30305, 54361, 9669}, {31423, 61597, 38028}, {34632, 38074, 3830}, {34632, 51068, 38074}, {34718, 38066, 2}, {35610, 35789, 3071}, {35611, 35788, 3070}, {37563, 37702, 3058}, {37724, 59337, 15174}, {37734, 52793, 37525}, {40273, 61259, 381}, {49226, 49602, 42215}, {49227, 49601, 42216}, {50810, 53620, 381}, {50821, 50823, 50824}, {50821, 50824, 50825}, {50821, 50827, 50823}, {50821, 50828, 50826}, {50821, 50830, 50833}, {50821, 51087, 50829}, {50822, 50823, 50821}, {50822, 50827, 50824}, {50823, 50824, 50830}, {50823, 50826, 50831}, {50824, 50825, 50833}, {50825, 50830, 50824}, {50826, 50831, 50828}, {50828, 50831, 50824}, {50829, 51087, 50832}, {50832, 51087, 50824}, {51073, 58240, 61273}, {51111, 54192, 26287}, {51700, 61597, 1}, {54447, 61268, 35018}
X(5690) = pole of line {9957, 10950} with respect to the Feuerbach circumhyperbola
X(5690) = pole of line {10, 10943} with respect to the Feuerbach circumhyperbola of the medial triangle
X(5690) = pole of line {859, 37535} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5690) = pole of line {572, 2183} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5690) = pole of line {517, 2975} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5690) = pole of line {34589, 35015} with respect to the Mandart parabola
X(5690) = pole of line {900, 48391} with respect to the circumcircle
X(5690) = pole of line {28217, 39540} with respect to the incircle
X(5690) = pole of line {513, 39508} with respect to the nine point circle
X(5690) = pole of line {28217, 39547} with respect to the Conway circle
X(5690) = pole of line {4977, 39534} with respect to the polar circle
X(5690) = pole of line {4784, 39386} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5690) = pole of line {18004, 39386} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5690) = pole of line {28217, 44409} with respect to the Suppa Cucoanes circle (see X(53614))
The outer Garcia triangle is defined at X(5587).
Let I = X(1) and O = X(3). Let A″ be the reflection of I in line AO and let IA be the reflection of A″ in line AI. Define IB and IC cyclically. Then ABC and IAIB IC are orthologic triangles, and X(5691) is the ABC-orthology center of IAIB IC. (Angel Montesdeoca, January 21, 2015)
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. A′B′C′ is homothetic to the 4th Euler triangle at X(5691). (Randy Hutson, July 31 2018)
X(5691) lies on these lines: {1,4}, {2,4297}, {3,1698}, {5,3576}, {8,144}, {10,20}, {11,1420}, {12,3601}, {30,40}, {35,1012}, {36,3149}, {46,80}, {57,1837}, {63,5086}, {65,971}, {78,5080}, {79,3577}, {5692,5777}
X(5691) = reflection of X(20) in X(10)
X(5691) = {X(1),X(4)}-harmonic conjugate of X(1699)
The inner Garcia triangle A″B″C″ is defined at X(5587). Another construction of A″ is as the reflection of X(1) in the perpendicular bisector of side BC, so that |OA″| = |X(1)X(3)|, and A″, B″, C″, and X(1) are on a circle with center X(3). (Paul Yiu, ADGEOM #1214, April 3, 2014)
A″ = a2 : c2 - b2 + ac :
b2 - c2 + ab
B″ = c2 - a2 + bc : b2 :
a2 - c2 + ba
C″ = b2 - a2 + cb : a2 -
b2 + ca : c2
(Peter Moses, April 4, 2014)
The appearance of (i,j) in the following list means that (X(i) of A″B″C′) = X(j): (1,8), (3,3), (10,3878), (11,72), (21,191), (35,2975), (36,100), (40,944), (55,956), (63,4302), (78,1479), (80,3869), (100,1), (104,40), (214,10), (238,190), (662,2607), (663,3762), (667,659), (976,4894), (1001,5220), (1125,3678), (1145,3057), (1149,4738), (1193,1089), (1319,1145), (1320,3632), (1325,2948), (1376,5289), (1459,4768), (1734,3904), (1768,20), (1818,4858), (2077,104), (2932,56), (3032,2901), (3035,960), (3065,3648), (3220,1633), (3286,4436), (4057,4491), (4367,4730), (4511,80), (4855,499), (4996,35), (5150,3923), (5313,4671), (5440,11), (5531,962), (5541,145). (Peter Moses, April 3, 2014)
X(5692) lies on these lines: {1,6}, {2,758}, {3,191}, {8,80}, {10,908}, {119,5690}, {5691,5777}, {2805,5695}, {4302,5696} et al.
X(5692) = reflection of X(1) in X(392)
X(5692) = anticomplement of X(5883)
X(5692) = {X(1),X(72)}-harmonic conjugate of X(5904)
X(5692) = X(1)-of-X(2)-anti-altimedial-triangle
X(5692) = X(2)-of-X(1)-adjunct-anti-altimedial-triangle
X(5692) = X(12022)-of-excentral-triangle
X(5692) = homothetic center of inner Garcia triangle and X(1)-adjunct anti-altimedial triangle
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5693) is the incenter of the X(3)-Fuhrmann triangle, defined at X(5613).
X(5693) lies on these lines: {1,90}, {2,5884}, {3,191}, {4,758}, {5,3649}, {8,153}, {10,6937}, {944,2801}, {952,5697} et al.
X(5693) = anticomplement of X(5884)
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
Let A* be the reflection of X(5) in the perpendicular bisector of segment BC, and define B* and C* cyclically. Triangle A*B*C* is inversely similar to ABC, with similitude center X(3); also, A*B*C* is perspective to ABC, with perspector X(3519), and X(5694) is the incenter of A*B*C*.
X(5694) lies on these lines: {1,195}, {2,5885}, {3,191}, {4,8}, {5,758} et al.
X(5694) = anticomplement of X(5885)
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5695) lies on these lines: {1,536}, {2,3712}, {3,2783}, {4,3704}, {6,740}, {8,190}, {9,3696}, {10,45}, {2805,5692} et al.
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5696) lies on these lines: {1,5784}, {7,2894}, {8,2801}, {9,35}, {528,5697}, {4302,5692} et al.
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5697) lies on these lines: {1,3}, {2,3884}, {4,5559}, {7,7278}, {8,80}, {10,4193}, {11,5690}, {528,5696}, {952,5693} et al.
X(5697) = X(20)-of-reflection-triangle-of-X(1)
X(5697) = {X(1),X(40)}-harmonic conjugate of X(36)
X(5697) = X(1)-of-X(1)-anti-altimedial-triangle
X(5697) = endo-homothetic center of Ehrmann vertex-triangle and anti-Hutson intouch triangle; the homothetic center is X(382)
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5698) lies on these lines: {1,527}, {2,1155}, {3,1633}, {4,9}, {7,21}, {8,190}, {11,5744}, {944,2801}, {4302,5692} et al.
X(5698) = anticomplement of X(5880)
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5699) lies on these lines: X(5699) lies on line {3,2783}, {10,13}, {15,740}, {16,3923} et al.
The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).
X(5700) lies on these lines: {3,2783}, {10,14}, {15,3923}, {16,740} et al.
Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given by
T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv) (Peter Moses, June 23, 2014)
X(5701) is the minimizer T when P = X(1) and Q = X(6). See also X(5661) and X(5662).
X(5701) lies on these lines: {1,6}, {2,650}, {1252,1621}, {3693,4702}, {5284,5375}
Let ABC be a triangle, let PA be the polar of A with respect to the circle with diameter BC, and define PB and PC cyclically. Let AB = PA∩AB and AC = PA∩AC, and define BC, CA, BA, and CB cyclically. The six points AB, AC, BC, BA, CA, CB lie on, and define, the Montesdeoca conic. (Angel Montesdeoca, June 23, 2014)
A barycentric equation for the Montesdeoca conic is found from AB = SC : 0 : 2SA and AC = SB : 2SA : 0 to be as follows:
2(S2Ax2 + S2By2 + S2Cz2) - 5(SBSCyz + SCSAzx + SASBxy) = 0 (Peter Moses, June 23, 2014)
The Montesdeoca conic is the anticevian-intersection conic when P = X(4); this conic is defined by Francisco J. GarcÃa Capitán (The Anticevian Intersection Conic and Hyacinthos #20547 (December 19, 2011). Also, the perspector of the Montesdeoca conic is X(4).
X(5702) lies on these lines: {4,6}, {297,5032}, {340,1992}, {376,3284}, {468,5304}, {578,3183}, {631,5158}, {3163,3545}
X(5703) lies on these lines: {1, 2}, {3, 7}, {4, 4313}, {5, 3488}, {12, 3486}, {20, 226}, {21, 329}, {35, 4295}, {40, 5281}, {55, 411}, {56, 3475}, {57, 3523}, {65, 5218}, {72, 5273}, {86, 939}, {142, 5438}, {165, 3671}, {307, 3945}, {388, 2646}, {390, 946}, {443, 5440}, {452, 908}, {495, 944}, {515, 5261}, {517, 4323}, {631, 942}, {940, 3562}, {950, 3091}, {988, 4310}, {1056, 1385}, {1445, 3333}, {1446, 3160}, {1478, 4305}, {1699, 4314}, {1788, 5432}, {2287, 5296}, {2476, 5175}, {2886, 3189}, {3146, 4304}, {3149, 3295}, {3361, 5542}, {3452, 5129}, {3474, 3649}, {3522, 4292}, {3576, 3600}, {3586, 3832}, {3612, 4293}, {4000, 4255}, {4252, 4644}, {4297, 5290}, {4344, 5266}, {4855, 5249}, {5084, 5328}
X(5704) lies on these lines: {1, 2}, {4, 5435}, {5, 7}, {11, 962}, {20, 3911}, {40, 5274}, {57, 3091}, {72, 5328}, {88, 5125}, {90, 5556}, {104, 3149}, {140, 3488}, {226, 5056}, {307, 4346}, {329, 4193}, {355, 4308}, {404, 5175}, {411, 5204}, {496, 5657}, {515, 5265}, {631, 4313}, {942, 3090}, {950, 3523}, {1155, 5225}, {1158, 1445}, {1656, 3487}, {1728, 3218}, {3035, 3189}, {3306, 5177}, {3333, 5261}, {3339, 3817}, {3486, 5433}, {3522, 3586}, {3529, 5122}, {3562, 4383}, {3600, 5587}, {3614, 4860}, {3832, 4292}, {4208, 5437}, {4306, 5400}, {5084, 5273}
X(5705) lies on these lines: {1, 2}, {5, 9}, {21, 3586}, {40, 2886}, {57, 442}, {63, 2476}, {72, 5219}, {140, 5438}, {283, 5235}, {411, 993}, {443, 3911}, {958, 3149}, {965, 2323}, {1445, 3841}, {1479, 4512}, {1656, 5044}, {2475, 4652}, {3090, 3452}, {3091, 5273}, {3219, 5141}, {3305, 4193}, {3306, 4197}, {3419, 3601}, {3545, 5325}, {3576, 4999}, {3646, 3816}, {3814, 5536}, {3822, 5290}, {4208, 5435}, {4292, 5177}, {4304, 5175}
X(5706) lies on these lines: {1, 3}, {4, 6}, {5, 1714}, {7, 3562}, {10, 219}, {20, 81}, {28, 154}, {33, 1712}, {51, 4186}, {58, 1012}, {64, 4219}, {98, 3597}, {184, 4185}, {221, 278}, {222, 4292}, {377, 394}, {386, 3149}, {405, 580}, {429, 1899}, {602, 1001}, {774, 4336}, {990, 1071}, {991, 4658}, {1191, 5603}, {1203, 1699}, {1260, 3191}, {1376, 3682}, {1451, 2654}, {1478, 3173}, {1612, 3052}, {1753, 2285}, {1765, 2257}, {1780, 3560}, {1836, 1838}, {1853, 5142}, {1993, 2475}, {2192, 2982}, {2256, 5657}, {3695, 4513}, {3713, 5295}, {4259, 5562}, {5046, 5422}
X(5706) = {X(1),X(40)}-harmonic conjugate of X(37528)
X(5707) lies on these lines: {1, 3}, {2, 3193}, {4, 81}, {5, 6}, {7, 1068}, {58, 3560}, {222, 225}, {226, 3157}, {283, 405}, {394, 442}, {581, 4658}, {602, 3720}, {965, 2323}, {1069, 1210}, {1216, 4259}, {1437, 4185}, {1480, 4301}, {1656, 4383}, {1993, 2476}, {1994, 5141}, {3149, 5396}, {3487, 3562}, {4193, 5422}
X(5707) = {X(1),X(57)}-harmonic conjugate of X(37565)
X(5708) lies on these lines: {1, 3}, {2, 3927}, {5, 7}, {28, 89}, {30, 938}, {45, 579}, {63, 5439}, {72, 3306}, {140, 3487}, {142, 3634}, {191, 4423}, {222, 1393}, {226, 1656}, {355, 4298}, {381, 553}, {382, 4031}, {405, 3218}, {443, 3617}, {474, 3868}, {495, 1788}, {496, 4295}, {499, 3649}, {548, 4313}, {550, 3488}, {950, 1657}, {952, 3600}, {1086, 5292}, {1376, 3874}, {1435, 1871}, {1439, 3527}, {1483, 4308}, {1598, 1876}, {3526, 3911}, {3586, 5073}, {3624, 4880}, {3628, 5226}, {3872, 4004}, {3881, 3913}, {3982, 5079}, {4084, 5289}, {4114, 5072}, {4306, 5396}, {4321, 5534}, {4355, 5587}, {4654, 5055}, {5044, 5437}, {5070, 5219}, {5445, 5557}
X(5708) = {X(4860),X(5221)}-harmonic conjugate of X(1)
X(5709) = X(26)-of-excentral-triangle.
X(5709) lies on these lines: {1, 3}, {4, 63}, {5, 9}, {20, 3218}, {28, 283}, {30, 84}, {34, 255}, {72, 3149}, {90, 3583}, {140, 5437}, {155, 610}, {191, 1699}, {212, 1393}, {223, 3157}, {225, 1217}, {381, 3929}, {411, 3868}, {443, 5657}, {516, 1158}, {518, 5534}, {578, 3955}, {579, 1766}, {602, 614}, {631, 3306}, {912, 1490}, {920, 1479}, {1012, 3916}, {1068, 1119}, {1069, 3345}, {1070, 4331}, {1071, 1998}, {1072, 5230}, {1093, 1948}, {1210, 1708}, {1254, 1496}, {1352, 5227}, {1453, 5398}, {1512, 3436}, {1707, 3073}, {1776, 5225}, {1817, 3193}, {2000, 4219}, {3090, 3305}, {3091, 3219}, {5250, 5603}
X(5709) = intouch-to-excentral similarity image of X(3)
X(5710) lies on these lines: {1, 3}, {2, 1191}, {6, 8}, {10, 3966}, {21, 3052}, {31, 958}, {37, 5250}, {42, 3913}, {58, 956}, {81, 145}, {87, 2334}, {100, 4255}, {197, 1036}, {218, 3997}, {220, 5276}, {221, 388}, {387, 5082}, {392, 975}, {405, 595}, {474, 995}, {608, 1891}, {611, 5252}, {612, 960}, {614, 3812}, {750, 1201}, {962, 5244}, {978, 4413}, {1001, 1918}, {1056, 4340}, {1100, 3895}, {1193, 1376}, {1203, 3679}, {1406, 5434}, {1407, 3600}, {1449, 2136}, {1616, 3616}, {1698, 5315}, {1706, 2999}, {1722, 3698}, {1834, 3434}, {1999, 4673}, {2176, 5275}, {2256, 2303}, {2650, 3938}, {2886, 5230}, {2975, 4252}, {3242, 3868}, {3486, 4339}, {3782, 4295}, {3869, 3920}, {4363, 4968}, {4646, 5256}
X(5711) lies on these lines: {1, 3}, {4, 608}, {6, 10}, {8, 81}, {31, 405}, {58, 958}, {72, 612}, {213, 5275}, {218, 5276}, {219, 2303}, {220, 3997}, {221, 226}, {222, 388}, {281, 3194}, {341, 3758}, {386, 1376}, {387, 2550}, {406, 3195}, {442, 5230}, {474, 750}, {495, 611}, {551, 1616}, {595, 1001}, {601, 1012}, {614, 5439}, {651, 5261}, {894, 4385}, {938, 4344}, {946, 2050}, {956, 1468}, {960, 975}, {976, 2650}, {984, 1046}, {993, 4252}, {1064, 3149}, {1065, 1433}, {1100, 4646}, {1107, 5021}, {1125, 1191}, {1203, 1698}, {1386, 3812}, {1407, 4298}, {1449, 1706}, {1714, 3925}, {1740, 4649}, {2271, 4386}, {2292, 5311}, {2295, 3695}, {2305, 3743}, {2886, 5292}, {3052, 5248}, {3216, 4413}, {3242, 3874}, {3488, 4339}, {3562, 3945}, {3624, 5315}, {3720, 3915}, {3736, 3913}, {3868, 3920}, {3876, 5297}, {3940, 5293}, {4356, 5493}, {4662, 4663}, {4868, 5110}, {5044, 5268}, {5250, 5287}
X(5712) lies on these lines: {1, 4}, {2, 6}, {3, 4340}, {7, 941}, {37, 329}, {42, 2550}, {55, 4307}, {57, 573}, {63, 4644}, {142, 2999}, {171, 212}, {306, 2345}, {312, 1909}, {345, 894}, {354, 1469}, {386, 443}, {387, 442}, {406, 3194}, {553, 4888}, {580, 631}, {908, 5287}, {1100, 3772}, {1104, 3616}, {1125, 1453}, {1212, 5308}, {1215, 3974}, {1730, 4266}, {1788, 5530}, {1834, 5177}, {3247, 4656}, {3296, 3953}, {3622, 5484}, {3663, 4654}, {3672, 3782}, {3677, 5542}, {3744, 4344}, {3752, 4277}, {3931, 4295}, {3982, 4862}, {4000, 5249}, {4349, 5269}, {4641, 5273}, {4658, 5292}
X(5712) = isotomic conjugate of polar conjugate of X(37384)
X(5712) = complement of X(14552)
X(5712) = anticomplement of X(5737)
X(5713) lies on these lines: {1, 4}, {2, 283}, {5, 6}, {212, 498}, {499, 1451}, {1899, 3142}, {2299, 3542}, {2476, 3193}
X(5714) lies on these lines: {1, 4}, {2, 3824}, {3, 5226}, {5, 7}, {9, 3634}, {12, 4295}, {40, 3947}, {45, 1901}, {57, 3090}, {72, 3617}, {79, 498}, {329, 442}, {381, 938}, {382, 4313}, {405, 5253}, {443, 908}, {452, 5126}, {495, 962}, {517, 5261}, {553, 5071}, {631, 4292}, {942, 3091}, {943, 5556}, {952, 4323}, {1000, 4301}, {1006, 5204}, {1210, 3545}, {1656, 5435}, {1770, 5218}, {1836, 3085}, {1892, 3089}, {2345, 3454}, {3333, 3817}, {3419, 3621}, {3529, 3601}, {3614, 5221}, {3651, 5217}, {3671, 5587}, {3911, 5067}, {4208, 5044}, {4317, 5443}, {4330, 5561}, {4644, 5292}, {5045, 5274}, {5084, 5249}
X(5715) lies on these lines: {1, 4}, {3, 3824}, {5, 9}, {40, 442}, {72, 5587}, {79, 1709}, {329, 3091}, {355, 3577}, {962, 5177}, {1006, 3624}, {1071, 4654}, {1158, 4312}, {3149, 5219}
X(5716) lies on these lines: {1, 4}, {2, 1104}, {6, 8}, {10, 1453}, {20, 3666}, {29, 2303}, {37, 452}, {42, 3189}, {55, 4339}, {56, 4220}, {65, 3056}, {85, 3945}, {145, 321}, {171, 1451}, {212, 5255}, {345, 4195}, {377, 4000}, {387, 3419}, {405, 1612}, {580, 5264}, {612, 2551}, {938, 940}, {942, 3784}, {975, 5084}, {986, 3474}, {1036, 1610}, {1427, 3600}, {1697, 1766}, {1834, 5175}, {1837, 3745}, {1841, 4198}, {3085, 5266}, {3146, 3672}, {3436, 3920}, {3616, 5051}, {3677, 4298}, {3772, 5177}, {3868, 4644}, {3931, 4294}, {4190, 4850}, {5218, 5530}
X(5717) lies on these lines: {1, 4}, {2, 1453}, {6, 10}, {12, 3745}, {40, 2269}, {57, 4340}, {171, 580}, {204, 406}, {212, 5264}, {306, 964}, {377, 5256}, {387, 1449}, {443, 2999}, {511, 942}, {516, 3931}, {519, 5295}, {553, 3670}, {937, 2551}, {938, 3945}, {940, 1210}, {975, 3452}, {1010, 3687}, {1100, 1834}, {1104, 1125}, {1329, 4682}, {1330, 4357}, {1427, 4298}, {1451, 3911}, {1842, 2294}, {2047, 5405}, {2303, 3194}, {2334, 4863}, {2478, 5287}, {3085, 5269}, {3666, 4292}, {4208, 5222}, {5129, 5308}, {5249, 5262}
X(5718) lies on these lines: {1, 5}, {2, 6}, {10, 4023}, {37, 908}, {42, 2886}, {43, 3925}, {55, 4192}, {57, 4888}, {65, 970}, {140, 5398}, {171, 2361}, {226, 1465}, {312, 3963}, {313, 4358}, {377, 4255}, {386, 442}, {469, 1865}, {516, 4689}, {528, 2177}, {536, 4054}, {631, 4340}, {750, 3035}, {851, 5132}, {899, 3826}, {986, 3649}, {1086, 4850}, {1215, 3703}, {1386, 3011}, {1468, 4999}, {1834, 2476}, {1848, 1880}, {3058, 3750}, {3306, 4675}, {3550, 4995}, {3664, 3911}, {3687, 4967}, {3706, 4028}, {3712, 3923}, {3720, 3816}, {3752, 5249}, {3772, 5256}, {3821, 4892}, {3838, 3914}, {3943, 4671}, {3944, 4854}, {3999, 5542}, {4009, 4078}, {4030, 4865}, {4031, 4896}, {4090, 4126}, {4220, 5347}, {4307, 5218}, {4654, 4902}, {5308, 5328}
X(5718) = complement of X(1150)
X(5719) lies on these lines: {1, 5}, {2, 3940}, {3, 7}, {30, 226}, {35, 3649}, {37, 3002}, {57, 549}, {73, 5453}, {140, 942}, {381, 3488}, {382, 4313}, {484, 4995}, {518, 1125}, {546, 950}, {548, 4292}, {550, 3601}, {551, 3452}, {553, 5122}, {938, 1656}, {956, 3616}, {999, 3475}, {1000, 1482}, {1086, 4256}, {1159, 5657}, {1210, 3628}, {1385, 4315}, {3295, 3485}, {3296, 5265}, {3530, 4031}, {3579, 3671}, {3584, 5425}, {3586, 3845}, {3622, 5084}, {3748, 4870}, {3874, 4999}, {4415, 4653}, {5054, 5435}, {5249, 5440}, {5298, 5444}
X(5719) = {X(1),X(12)}-harmonic conjugate of X(37730)
X(5720) lies on these lines: {1, 5}, {3, 9}, {4, 78}, {8, 1512}, {30, 1750}, {57, 912}, {72, 3149}, {200, 517}, {210, 3428}, {223, 1060}, {381, 2900}, {386, 3553}, {411, 3876}, {474, 1071}, {515, 997}, {581, 975}, {612, 1064}, {944, 5084}, {946, 3811}, {962, 4420}, {1006, 3305}, {1012, 5440}, {1038, 1745}, {1040, 3465}, {1217, 1826}, {1376, 3359}, {1482, 3577}, {1519, 3434}, {1532, 3419}, {1709, 2077}, {1743, 5398}, {3560, 3601}, {3576, 5251}, {3870, 5603}
X(5721) lies on these lines: {1, 5}, {3, 1714}, {4, 6}, {30, 1754}, {40, 1723}, {51, 1894}, {65, 1243}, {184, 1884}, {209, 517}, {442, 581}, {518, 1072}, {912, 3782}, {1064, 2886}, {1108, 1512}, {1210, 1465}, {1214, 1737}, {1329, 3682}, {1785, 1864}, {2361, 3073}, {3149, 5292}, {3428, 4192}
X(5722) lies on these lines: {1, 5}, {2, 3419}, {3, 950}, {4, 7}, {6, 5179}, {8, 392}, {10, 1001}, {30, 57}, {55, 1737}, {65, 1479}, {72, 2478}, {78, 4187}, {79, 5561}, {81, 5155}, {90, 3652}, {140, 3601}, {200, 3820}, {222, 1877}, {224, 442}, {226, 381}, {354, 1478}, {376, 5122}, {377, 5439}, {382, 4031}, {388, 5045}, {390, 5657}, {405, 1259}, {443, 5175}, {497, 517}, {499, 2646}, {515, 999}, {519, 3452}, {553, 3830}, {631, 4313}, {912, 1864}, {943, 5047}, {997, 3816}, {1056, 5049}, {1062, 1834}, {1104, 5292}, {1145, 3895}, {1155, 4302}, {1319, 3655}, {1329, 3811}, {1385, 3086}, {1770, 5221}, {1788, 3579}, {1836, 3583}, {1936, 5398}, {2099, 3656}, {3058, 3654}, {3091, 3487}, {3241, 5176}, {3434, 3753}, {3436, 3555}, {3545, 5226}, {3582, 3653}, {3612, 5433}, {3616, 5086}, {3679, 4863}, {3824, 5177}, {3845, 4654}, {3868, 5046}, {3873, 5080}, {4295, 5225}, {5090, 5142}, {5274, 5603}, {5427, 5441}
X(5722) = midpoint of X(i) in X(j) for these {i,j}: {1,5727}, {4,5768}
X(5722) = X(25)-of-Fuhrmann-triangle
X(5722) = inverse-in-Feuerbach-hyperbola of X(5252)
X(5722) = {X(1),X(80)}-harmonic conjugate of X(5252)
X(5722) = {X(1),X(1837)}-harmonic conjugate of X(355)
X(5723) lies on these lines: {1, 5}, {2, 664}, {6, 7}, {57, 1358}, {88, 279}, {226, 544}, {241, 514}, {278, 1783}, {1419, 4859}, {1441, 3589}, {1456, 1738}, {4422, 4552}
X(5723) = crossdifference of every pair of points on the line X(55)X(654)
X(5724) lies on these lines: {1, 5}, {6, 8}, {10, 4434}, {30, 4424}, {38, 529}, {65, 511}, {388, 4310}, {515, 3666}, {519, 1215}, {982, 5434}, {1146, 5276}, {1478, 3782}, {1834, 5086}, {1880, 1891}, {2361, 5255}, {2646, 5530}, {3679, 5269}, {3920, 5176}, {4304, 4689}, {4415, 5080}, {5264, 5398}
X(5725) lies on these lines: {1, 5}, {3, 5530}, {4, 941}, {6, 10}, {42, 3419}, {171, 5398}, {388, 1465}, {515, 2050}, {940, 1737}, {942, 1469}, {975, 998}, {1478, 3666}, {1783, 5276}, {1788, 4340}, {1836, 4424}, {2361, 5264}, {3085, 5266}, {3772, 3822}, {3820, 5268}, {4205, 5336}, {4302, 4689}, {4307, 5657}, {4682, 5123}
X(5726) lies on these lines: {1, 5}, {2, 4315}, {7, 10}, {8, 3947}, {165, 1478}, {226, 3679}, {388, 1698}, {519, 5226}, {946, 1000}, {1788, 4031}, {2099, 4677}, {2476, 4853}, {2886, 4915}, {3085, 4304}, {3340, 4668}, {3436, 5234}, {3485, 3632}, {3600, 3634}, {3617, 3671}, {3625, 4323}, {3731, 5179}, {3828, 5435}, {4312, 5657}, {4512, 5080}, {5123, 5437}
X(5727) lies on these lines: {1, 5}, {4, 3340}, {8, 9}, {10, 3486}, {20, 4848}, {30, 2093}, {40, 920}, {46, 4316}, {55, 3679}, {57, 515}, {65, 971}, {145, 908}, {388, 5542}, {497, 519}, {517, 1864}, {944, 1210}, {1012, 3256}, {1249, 1826}, {1478, 4654}, {1698, 2646}, {1699, 2099}, {1706, 5554}, {1737, 3576}, {1788, 4297}, {2098, 3633}, {3057, 3632}, {3058, 4677}, {3241, 5274}, {3617, 4313}, {3626, 4314}, {3671, 5229}, {3832, 4323}, {3870, 5176}, {4301, 5225}, {4304, 5657}, {4863, 4915}
X(5727) = reflection of X(1) in X(5722)
X(5728) = {X(1),X(9)}-harmonic conjugate of X(954)
X(5728) lies on these lines: {1, 6}, {2, 955}, {3, 1445}, {4, 7}, {10, 3059}, {11, 118}, {40, 4326}, {55, 1708}, {65, 516}, {81, 162}, {142, 442}, {144, 452}, {241, 991}, {329, 3873}, {390, 517}, {480, 3811}, {497, 5173}, {774, 3931}, {943, 2346}, {986, 4335}, {990, 5228}, {1005, 3218}, {1012, 3358}, {1156, 2771}, {1260, 3870}, {1376, 2900}, {1490, 3333}, {1730, 3198}, {1737, 3826}, {1890, 1905}, {1898, 3649}, {2550, 3419}, {2951, 3339}, {3062, 5665}, {3085, 3697}, {3487, 5045}, {3586, 4312}
X(5729) lies on these lines: {1, 6}, {4, 653}, {5, 7}, {56, 2801}, {144, 2478}, {226, 4860}, {516, 1837}, {527, 1210}, {938, 3927}, {971, 1445}, {1155, 1708}, {1260, 3935}, {3245, 3586}, {5435, 5658}
X(5730) lies on these lines: {1, 6}, {2, 4930}, {3, 3417}, {5, 8}, {10, 2099}, {35, 3899}, {40, 5440}, {55, 3878}, {56, 758}, {57, 4018}, {63, 1385}, {65, 474}, {78, 517}, {145, 1058}, {214, 5204}, {329, 944}, {355, 908}, {381, 5086}, {382, 5057}, {442, 3485}, {519, 1837}, {527, 4311}, {936, 3340}, {946, 3419}, {952, 3436}, {957, 1257}, {999, 3868}, {1319, 3962}, {1320, 3621}, {1388, 4067}, {1457, 3682}, {1759, 3207}, {2093, 5438}, {2271, 3727}, {2800, 2932}, {2975, 3927}, {3057, 3811}, {3219, 3897}, {3244, 4679}, {3295, 3877}, {3303, 3884}, {3304, 3874}, {3445, 4694}, {3576, 3916}, {3579, 4855}, {3612, 4640}, {3624, 5425}, {3626, 3711}, {3632, 5087}, {3681, 4861}, {3754, 4413}, {3820, 5554}, {3872, 3984}, {3885, 3935}, {3901, 5563}, {4084, 5221}, {4255, 4424}
X(5731) = {X(1),X(20)}-harmonic conjugate of X(962). Let A′ be the antipode of the A-extouch point in the A-excircle, and define B′ and C′ cyclically, and let A″ be the antipode of the A-intouch point in the incircle, and define B″ and C″ cyclically. Then X(5731) is the centroid of {A′,B′,C′,A″,B″,C″}. (Randy Hutson, July 7, 2014)
The triangle A′B′C′ is here named the Hutson-extouch triangle, and A″B″C″, the Hutson-intouch triangle - not to be confused with the outer and inner Hutson triangles defined at X(363). Hutson established (July 10, 2014) that A′B′C′ and A″B″C″ are orthologic with orthology center X(3555); also that A″B″C″ and A′B′C′ are orthologic with orthology center X(5920). X(5731) is the midpoint of the centroids of A′B′C′ and A″B″C″; see X(5918) and X(5919).
Peter Moses (July 15, 2014) gives barycentrics for Hutson-extouch
triangle,
-4a2 : (a + b + c)(a + b - c) : (a + b + c)(a - b + c)
(a + b + c)(b - c + a) : -4b2 : (a + b + c)(b + c - a)
(a + b + c)(c + a - b) : (a + b + c)(c - a + b) : -4c2
and for the Hutson-intouch triangle,
4a2 : (-a + b + c)(a - b + c) : (-a + b + c)(a + b - c)
(-b + c + a)(b + c - a) : 4b2 : (-b + c + a)(b - c + a)
(-c + a + b)(c - a + b) : (-c + a + b)(c + a - b) : 4c2
X(5731) lies on these lines: {1, 7}, {2, 515}, {3, 8}, {4, 1385}, {5, 5550}, {10, 3523}, {21, 3427}, {30, 5603}, {36, 5435}, {40, 145}, {55, 3476}, {56, 411}, {84, 5250}, {153, 214}, {165, 519}, {329, 4511}, {355, 631}, {376, 517}, {377, 3897}, {388, 2646}, {392, 971}, {452, 1490}, {497, 1319}, {548, 1483}, {550, 1482}, {551, 1699}, {840, 2737}, {946, 3146}, {950, 1420}, {963, 1043}, {993, 5273}, {999, 3488}, {1012, 1621}, {1071, 3869}, {1125, 3091}, {1210, 5265}, {1478, 5226}, {1602, 1610}, {1788, 5204}, {2099, 3474}, {3085, 3612}, {3149, 5253}, {3189, 5584}, {3421, 5440}, {3475, 5434}, {3528, 3579}, {3545, 3653}, {3586, 5274}, {3624, 5056}, {3635, 5493}, {3817, 3839}, {4188, 5554}, {4189, 5450}, {4420, 5534}, {5218, 5252}
X(5731) = anticomplement of X(5587)
X(5731) = X(15030)-of-excentral-triangle
X(5731) = endo-homothetic center of Ehrmann side-triangle and 2nd anti-Conway triangle; the homothetic center is X(568)
X(5732) = (X(6) of hexyl triangle) = (X(69) of 2nd circumperp triangle) = (X(1352) of excentral triangle) = (isogonal conjugate of X(3)-vertex conjugate of X(57) = isotomic conjugate with respect to X(1)-circumcevian triangle) of X(1) (Randy Hutson, July 7, 2014)
X(5732) lies on these lines: {1, 7}, {2, 1750}, {3, 9}, {4, 142}, {21, 3062}, {40, 518}, {63, 100}, {78, 144}, {223, 1040}, {376, 527}, {411, 1445}, {464, 2947}, {515, 2550}, {517, 3243}, {912, 3587}, {950, 1467}, {952, 5528}, {954, 3601}, {1001, 1012}, {1699, 5249}, {1709, 4512}, {1818, 2324}, {1998, 3218}, {2808, 3781}, {2900, 3928}, {3059, 5584}, {3333, 5572}, {3452, 5658}, {3579, 5534}, {3826, 5587}, {3868, 3895}
X(5732) = midpoint of X(i) and X(j) for these (i,j): (1,2951), (7,20)
X(5732) = reflection of X(i) in X(j) for these (i,j): (4,142), (9,3)
X(5732) = complement of X(36991)
X(5732) = midpoint of Mandart hyperbola intercepts of Soddy line
X(5733) lies on these lines: {1, 7}, {4, 4658}, {5, 6}, {225, 1419}, {631, 4648}, {3193, 4197}
X(5734) lies on these lines: {1, 7}, {4, 1392}, {5, 8}, {40, 3622}, {145, 946}, {165, 3636}, {329, 4861}, {355, 3855}, {382, 944}, {388, 5048}, {411, 3303}, {515, 3623}, {517, 631}, {519, 3091}, {551, 3523}, {938, 2099}, {952, 3843}, {1385, 3528}, {1388, 3474}, {1483, 3853}, {1699, 3244}, {2093, 5265}, {2098, 3485}, {3262, 4673}, {3525, 3654}, {3526, 5550}, {3529, 3655}, {3621, 5587}, {3632, 3817}, {3679, 5056}, {3878, 5273}, {4197, 5330}
X(5735) lies on these lines: {1, 7}, {4, 527}, {5, 9}, {63, 1699}, {84, 3254}, {142, 631}, {144, 3832}, {165, 5249}, {382, 971}, {1004, 5537}, {1750, 1998}, {2801, 3868}, {3436, 5223}, {3817, 5273}, {5220, 5587}
X(5736) lies on these lines: {1, 1441}, {2, 6}, {3, 7}, {77, 1446}, {226, 2268}, {255, 307}, {273, 1442}, {284, 379}, {1253, 1754}, {1958, 5249}, {2476, 2893}, {3485, 4329}
Let A4B4C4 be the 4th Conway triangle. Let A′ be the barycentric product B4*C4, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5737). (Randy Hutson, December 10, 2016)
Let A′B′C′ be the tangential triangle, wrt the excentral triangle, of the excentral-hexyl ellipse. A′B′C′ is homothetic to the polar triangle of the Spieker circle at X(5737). (Randy Hutson, August 19, 2019)
X(5737) lies on these lines: {2, 6}, {3, 10}, {9, 1764}, {42, 4042}, {45, 312}, {57, 3739}, {58, 2049}, {63, 4363}, {226, 4643}, {306, 4445}, {345, 594}, {405, 4267}, {573, 2050}, {968, 3706}, {980, 1107}, {1001, 3741}, {1010, 4252}, {1215, 5220}, {1698, 5247}, {2345, 5273}, {3052, 5263}, {3242, 3757}, {3666, 4361}, {3771, 3775}, {3772, 4357}, {4205, 5292}, {4426, 5337}
X(5737) = complement of X(5712)
X(5738) lies on these lines: {1, 307}, {2, 6}, {4, 7}, {65, 4329}, {77, 581}, {78, 3879}, {322, 5554}, {377, 2893}, {411, 1014}, {573, 1445}, {1060, 1442}, {1210, 3664}
Let A′B′C′ be the extangents triangle, and let AB be the touch point of the A-excircle and the line A′B′, and define BC and CA cyclically. Let AC be the touch point of the A-excircle and the line A′C′, and define BA and CB cyclically. Let A″ = BCBA∩CACB, and define B″ and C″ cyclically. The A″B″C″ is homothetic to ABC and to the outer and inner Grebe triangles at X(6), to the medial triangle at X(1211), and to the anticomplementary triangle at X(5739). (Randy Hutson, July 7, 2014)
X(5739) lies on these lines: {1, 4101}, {2, 6}, {4, 8}, {7, 4359}, {9, 306}, {57, 4001}, {63, 573}, {75, 4886}, {78, 581}, {209, 2550}, {210, 3416}, {223, 307}, {226, 3686}, {312, 319}, {345, 3219}, {346, 3969}, {377, 1330}, {387, 5051}, {388, 959}, {516, 4061}, {518, 3966}, {519, 4656}, {612, 4104}, {740, 4703}, {968, 4028}, {1376, 4023}, {1479, 4044}, {1714, 3454}, {1743, 5294}, {1836, 3696}, {2478, 3948}, {2886, 4042}, {3305, 3912}, {3666, 4277}, {3703, 5220}, {3707, 4035}, {3715, 3932}, {3782, 4361}, {3870, 3883}, {3879, 5287}, {3929, 3977}, {3952, 3974}, {3965, 3998}, {4034, 4054}, {4270, 4357}, {4384, 5249}, {4423, 4966}, {4660, 4685}, {4666, 4684}
X(5739) = anticomplement of X(940)
X(5739) = isotomic conjugate of isogonal conjugate of X(36744)
X(5740) lies on these lines: {2, 6}, {5, 7}, {269, 5400}, {273, 2973}, {307, 1210}, {404, 2893}, {579, 857}, {1441, 1737}, {1442, 5396}, {1788, 4329}
X(5741) lies on these lines: {2, 6}, {5, 8}, {42, 3847}, {43, 4972}, {78, 5016}, {100, 4192}, {149, 3996}, {200, 5014}, {210, 3006}, {226, 4359}, {306, 3452}, {312, 3969}, {321, 908}, {329, 4488}, {386, 5051}, {404, 1330}, {748, 3771}, {899, 2887}, {970, 3869}, {1043, 5046}, {1210, 4101}, {2886, 4023}, {3216, 3454}, {3681, 3705}, {3703, 3952}, {3706, 5087}, {3817, 4061}, {3909, 3917}, {3911, 4001}, {3935, 4514}, {3944, 4442}, {4035, 5316}, {4054, 4980}, {4104, 4981}, {4414, 4703}, {4420, 5015}, {4511, 5396}, {5219, 5271}
X(5741) = complement of X(37639)
X(5741) = anticomplement of X(37634)
X(5742) lies on these lines: {2, 6}, {5, 9}, {48, 4999}, {71, 2886}, {307, 3739}, {442, 579}, {936, 5396}, {970, 2262}, {1210, 5257}, {1698, 1723}, {1839, 4640}, {1865, 5125}, {1901, 2476}
X(5743) lies on these lines: {2, 6}, {5, 10}, {37, 3687}, {42, 4023}, {43, 4026}, {45, 345}, {57, 4643}, {75, 4415}, {226, 3739}, {312, 594}, {321, 3264}, {329, 4363}, {386, 4205}, {518, 4104}, {536, 4656}, {612, 3966}, {756, 3703}, {984, 4884}, {997, 5396}, {1376, 4192}, {1999, 4886}, {2161, 2339}, {2887, 3826}, {2999, 4657}, {3035, 5150}, {3416, 5268}, {3666, 4364}, {3741, 3816}, {3752, 4357}, {3772, 4384}, {3775, 3840}, {3782, 4359}, {3838, 3846}, {3980, 4703}, {4199, 5132}, {4239, 5347}
X(5743) = complement of X(940)
X(5744) = {X(2),X(63)}-harmonic conjugate of X(329)
X(5744) lies on these lines: {2, 7}, {3, 8}, {4, 3916}, {10, 4293}, {20, 4652}, {21, 938}, {72, 631}, {78, 3523}, {88, 277}, {92, 4359}, {140, 3927}, {145, 3601}, {165, 4847}, {189, 333}, {191, 499}, {346, 3977}, {348, 658}, {376, 3419}, {391, 610}, {392, 942}, {443, 3436}, {452, 1210}, {497, 4640}, {516, 5231}, {518, 5218}, {549, 3940}, {940, 2256}, {958, 1466}, {1000, 2320}, {1108, 3666}, {1155, 2550}, {1467, 5265}, {1473, 4220}, {2000, 3100}, {2095, 5603}, {2886, 3474}, {3011, 4310}, {3035, 5220}, {3161, 4358}, {3189, 5217}, {3427, 3428}, {3485, 4999}, {3524, 5440}, {3579, 5082}, {3752, 5069}, {3870, 5281}, {4054, 4454}, {4189, 4313}, {4292, 5177}, {4305, 5267}, {4384, 5088}, {4850, 5222}, {5122, 5176}
X(5744) = anticomplement of X(5219)
X(5745) = {X(2),X(63)}-harmonic conjugate of X(226)
X(5745) lies on these lines: {2, 7}, {3, 10}, {8, 3158}, {11, 3683}, {21, 950}, {38, 3011}, {39, 1212}, {55, 4847}, {69, 4035}, {71, 1764}, {81, 2323}, {114, 124}, {140, 912}, {165, 2550}, {200, 5218}, {210, 5432}, {219, 940}, {261, 284}, {281, 5307}, {306, 1150}, {312, 2325}, {321, 3977}, {345, 2321}, {377, 4652}, {405, 1210}, {442, 3916}, {443, 1478}, {497, 4512}, {516, 2886}, {535, 3828}, {551, 4930}, {610, 966}, {631, 936}, {758, 942}, {938, 5436}, {1146, 2482}, {1155, 3925}, {1329, 3634}, {1737, 5251}, {1817, 5235}, {1861, 4219}, {1936, 2328}, {2329, 3912}, {2801, 3035}, {2999, 5105}, {3036, 4745}, {3220, 4220}, {3419, 4304}, {3523, 5438}, {3663, 3772}, {3666, 3946}, {3689, 4995}, {3705, 3883}, {3706, 3712}, {3741, 4154}, {3914, 4414}, {3936, 4001}, {4042, 4061}, {4138, 4655}, {4224, 5285}, {4359, 4858}, {4416, 4417}, {5247, 5530}, {5537, 5659}
X(5745) = isotomic conjugate of isogonal conjugate of X(21748)
X(5745) = complement of X(226)
X(5745) = complementary conjugate of X(17052)
X(5745) = polar conjugate of isogonal conjugate of X(22361)
X(5745) = centroid of {A,B,C,X(63)}
X(5746) lies on these lines: {2, 7}, {4, 6}, {19, 4295}, {20, 284}, {37, 3487}, {48, 4293}, {65, 281}, {71, 3085}, {72, 2345}, {193, 2893}, {219, 388}, {377, 2287}, {380, 516}, {386, 990}, {391, 5177}, {405, 5120}, {442, 966}, {443, 965}, {452, 5053}, {573, 1715}, {610, 4292}, {946, 2257}, {950, 1449}, {1056, 2256}, {1100, 3488}, {1108, 5603}, {1713, 4253}, {1714, 1743}, {1836, 2264}, {2260, 3086}, {2303, 4340}, {4264, 5304}
X(5747) lies on these lines: {1, 1826}, {2, 7}, {3, 1901}, {4, 284}, {5, 6}, {12, 219}, {48, 1478}, {71, 498}, {281, 3485}, {377, 2327}, {380, 1699}, {386, 3553}, {387, 5587}, {442, 965}, {495, 2256}, {499, 2260}, {594, 3940}, {2287, 2476}, {2303, 5142}, {2549, 5110}, {5053, 5084}, {5105, 5286}
X(5748) lies on these lines: {2, 7}, {4, 5440}, {5, 8}, {72, 3090}, {78, 3091}, {92, 4358}, {189, 1997}, {200, 3817}, {312, 3262}, {497, 5087}, {936, 5177}, {938, 4193}, {962, 1519}, {1056, 3436}, {1329, 3485}, {2975, 5550}, {3006, 5423}, {3035, 3474}, {3146, 4855}, {3190, 5400}, {3241, 5176}, {3419, 3545}, {3475, 3816}, {3487, 4187}, {3525, 3916}, {3628, 3927}, {3753, 3869}, {3870, 5274}, {4313, 5046}, {4323, 5554}
X(5748) = anticomplement of X(31231)X(5749) lies on these lines: {1, 346}, {2, 7}, {6, 8}, {10, 391}, {19, 4200}, {37, 2275}, {44, 966}, {45, 5550}, {69, 3758}, {75, 3618}, {86, 344}, {100, 4254}, {141, 4644}, {145, 1449}, {193, 3661}, {198, 404}, {218, 1010}, {281, 608}, {318, 1249}, {319, 1992}, {320, 3619}, {335, 4473}, {377, 2182}, {597, 4361}, {604, 2329}, {612, 5423}, {644, 2256}, {941, 2276}, {962, 1766}, {1100, 3241}, {1125, 3731}, {1698, 3973}, {1901, 5051}, {2092, 2229}, {2171, 3061}, {2264, 2550}, {2268, 4195}, {2269, 3501}, {2322, 3194}, {2325, 3247}, {2975, 5120}, {3240, 4270}, {3553, 4511}, {3554, 4861}, {3589, 4000}, {3617, 3686}, {3621, 4007}, {3623, 4873}, {3624, 3986}, {3629, 4445}, {3632, 4058}, {3635, 4072}, {3636, 4098}, {3663, 4454}, {3664, 4747}, {3672, 3729}, {3739, 4470}, {3745, 3974}, {3875, 4461}, {3912, 3945}, {3946, 4452}, {4034, 4678}, {4371, 4665}, {4393, 4460}, {4416, 5232}, {4419, 4488}, {4648, 4670}, {4698, 4798}, {5042, 5291}
X(5749) = anticomplement of X(17306)
X(5750) = {X(6),X(10)}-harmonic conjugate of X(3686)
X(5750) lies on these lines: {1, 2321}, {2, 7}, {6, 10}, {8, 1449}, {19, 475}, {34, 281}, {37, 39}, {43, 4270}, {44, 1213}, {45, 3986}, {69, 4667}, {75, 3946}, {86, 3912}, {141, 3664}, {145, 4007}, {171, 4264}, {198, 474}, {213, 992}, {218, 965}, {239, 4967}, {284, 1010}, {346, 3247}, {380, 2550}, {442, 2182}, {443, 610}, {478, 2122}, {515, 572}, {516, 4026}, {519, 594}, {536, 4021}, {551, 3950}, {742, 3008}, {946, 1766}, {950, 964}, {958, 5120}, {966, 1698}, {997, 3553}, {1172, 1861}, {1203, 1224}, {1220, 5053}, {1376, 4254}, {1450, 2324}, {1574, 4263}, {1575, 2092}, {1826, 2267}, {1901, 4205}, {2262, 3753}, {2264, 3925}, {2295, 2300}, {2298, 4071}, {2303, 5280}, {2663, 3783}, {3161, 5550}, {3244, 4058}, {3617, 4034}, {3618, 4384}, {3622, 4873}, {3624, 3731}, {3626, 4545}, {3629, 4690}, {3636, 3723}, {3661, 3879}, {3663, 4363}, {3672, 4659}, {3713, 4847}, {3758, 4416}, {3763, 4675}, {4000, 4470}, {4360, 4431}, {4395, 4739}, {4422, 4698}, {4426, 5019}, {4478, 4725}, {4665, 4852}, {4691, 4969}, {5124, 5267}
X(5750) = complement of X(4357)
X(5751) lies on these lines: {1, 916}, {3, 6}, {4, 7}, {55, 1779}, {81, 4219}, {1817, 3060}
X(5752) = circumcenter of the triangle A″B″C″ be as defined at X(5739); also X(5752) = {X(371)X(372)}-harmonic conjugate of X(1333). Let Γ be the circle of the points X(371), X(372), PU(1), PU(39); then X(5752) is the inverse-in-Γ of X(1333). (Randy Hutson, July 7, 2014)
X(5752) lies on these lines: {3, 6}, {4, 8}, {5, 1211}, {21, 3060}, {24, 2203}, {40, 209}, {51, 405}, {184, 2915}, {404, 2979}, {474, 3917}, {631, 5482}, {674, 3811}, {916, 1490}, {966, 3781}, {978, 3792}, {1006, 3567}, {1437, 1993}, {2183, 3682}, {3056, 5266}, {3149, 5562}, {3240, 3579}, {3560, 5446}, {5047, 5640}
X(5752) = anticomplement of X(37536)
X(5752) = anticomplement of anticomplement of X(34466)
X(5753) lies on these lines: {3, 6}, {5, 7}, {57, 5400}, {916, 2260}, {942, 1736}
X(5754) lies on these lines: {3, 6}, {5, 8}, {355, 2051}, {517, 3293}, {3240, 4192}
X(5755) lies on these lines: {3, 6}, {5, 9}, {30, 1765}, {40, 1723}, {57, 4888}, {71, 517}, {198, 3211}, {672, 4192}, {942, 1400}, {1385, 2260}, {1766, 2161}, {1781, 5535}, {2197, 5399}, {2361, 5285}, {3973, 5400}
X(5756) lies on these lines: {3, 6}, {7, 10}
X(5757) lies on these lines: {3, 7}, {4, 6}, {212, 226}
X(5758) lies on these lines: {3, 7}, {4, 8}, {9, 946}, {40, 226}, {84, 527}, {405, 5603}, {442, 5657}, {499, 5536}, {516, 1490}, {1006, 3616}, {1260, 3149}, {1482, 3488}, {1708, 3086}, {3428, 3485}, {3649, 5584}, {4299, 5538}
X(5759) lies on these lines: {3, 7}, {4, 9}, {20, 72}, {37, 3332}, {63, 3358}, {100, 329}, {142, 631}, {165, 226}, {212, 278}, {376, 527}, {390, 517}, {405, 962}, {497, 1708}, {515, 5223}, {518, 944}, {990, 4419}, {991, 4644}, {1001, 1006}, {1253, 4331}, {1490, 2951}, {2318, 2947}, {2724, 2742}, {3576, 5542}, {4295, 5584}, {4301, 5436}
X(5759) = reflection of X(7) in X(3)
X(5759) = isogonal conjugate of the X(3)-vertex conjugate of X(55)
X(5759) = {X(4),X(9)}-harmonic conjugate of X(5817)
X(5760) lies on these lines: {3, 7}, {5, 6}
X(5761) lies on these lines: {3, 7}, {5, 8}, {140, 2095}, {329, 3560}, {382, 5658}, {517, 3085}, {946, 3811}, {1385, 3475}, {3359, 3671}
X(5762) lies on these lines: {3, 7}, {4, 144}, {5, 9}, {30, 511}, {40, 495}, {140, 142}, {144, 2894}, {165, 4654}, {355, 5223}, {390, 1482}, {495, 4312}, {1385, 5542}, {1483, 3243}, {1484, 3254}, {1699, 3929}, {1754, 3782}, {3332, 4419}, {3817, 5325}, {4654, 4995}
X(5763) lies on these lines: {3, 7}, {4, 3940}, {5, 10}, {30, 1490}, {140, 5437}, {165, 3649}, {1058, 1482}, {5433, 5536}
X(5764) lies on these lines: {1, 4552}, {3, 7}, {6, 8}, {3006, 5294}, {3085, 4307}
X(5765) lies on these lines: {3, 7}, {6, 10}
X(5766) lies on these lines: {3, 7}, {8, 9}, {55, 329}, {72, 4313}, {226, 5281}, {516, 3085}, {527, 3601}, {528, 5175}, {2801, 4305}, {3486, 5220}, {3811, 4326}
X(5767) lies on these lines: {3, 8}, {4, 6}, {10, 48}, {184, 5136}, {515, 1754}, {517, 3187}, {860, 1899}, {912, 4463}, {940, 1056}
X(5768) lies on these lines: {1, 3427}, {3, 8}, {4, 7}, {20, 3218}, {30, 2094}, {57, 515}, {84, 950}, {142, 5587}, {329, 912}, {355, 443}, {390, 3358}, {601, 4339}, {1006, 5273}, {1012, 3488}, {1072, 4310}, {1158, 4294}, {1181, 3562}, {1210, 1467}, {1519, 5274}, {1532, 5658}, {1768, 4302}, {4305, 5450}
X(5768) = reflection of X(4) in X(5722)
X(5769) lies on these lines: {3, 8}, {5, 6}, {495, 940}
X(5770) lies on these lines: {2, 912}, {3, 8}, {4, 3218}, {5, 7}, {57, 1478}, {355, 1788}, {381, 2094}, {516, 1158}, {938, 3560}, {942, 3086}, {3359, 4847}
X(5770) = anticomplement of X(37713)
X(5771) lies on these lines: {2, 2095}, {3, 8}, {5, 9}, {57, 495}, {140, 942}, {484, 5659}, {1532, 3219}, {3628, 5316}, {3925, 5535}
X(5772) lies on these lines: {2, 3677}, {6, 8}, {7, 10}, {894, 3617}, {1215, 5226}, {1698, 4310}, {3679, 4307}, {3755, 4461}, {3932, 5308}, {4645, 4715}
X(5773) lies on these lines: {2, 101}, {3, 8}, {6, 7}, {57, 4566}, {239, 514}, {1055, 3911}, {1647, 5168}, {2398, 2809}
X(5774) lies on these lines: {3, 8}, {6, 10}, {40, 5295}, {69, 495}, {171, 3679}, {381, 4388}, {517, 2050}, {996, 3626}, {1010, 3617}, {1737, 3966}, {3706, 5119}, {3715, 3992}, {3753, 5271}, {3927, 4385}
X(5775) lies on these lines: {3, 8}, {7, 10}, {144, 5587}, {519, 5281}, {758, 5226}, {938, 1001}, {1788, 4413}, {2094, 3421}, {3218, 3617}, {3679, 4293}, {5251, 5273}
The B-excircle meets the sidelines of ABC in 3 points, and likewise for the C-excircle. The 6 points lie on a conic, denoted by A*. Let A′ be the center of A*, and define B′ and C′ cyclically. Then X(5776) is the perspector of A′B′C′ and the 2nd extouch triangle (defined at X(5927). (Randy Hutson, July 7, 2014)
A′B′C′ is also the unary cofactor triangle of the intangents triangle, which is also the cevian triangle of X(1743) wrt excentral triangle. (Randy Hutson, August 29, 2018)
X(5776) lies on these lines: {3, 9}, {4, 6}, {20, 2287}, {40, 4047}, {72, 1766}, {154, 4183}, {219, 515}, {222, 226}, {281, 3197}, {284, 1012}, {405, 572}, {579, 3149}, {944, 2256}, {1713, 5120}, {1715, 2270}, {1743, 1750}
X(5776) = pole of Gergonne line wrt excentral-hexyl ellipse
Let P be a point in the plane of a triangle ABC and let A′B′C′ be the cevian triangle of P. Let HA be the orthocenter of triangle AB′C′, and define HB and HC cyclically. Then the points A′, B′, C′, HA, HB, HC lie on a conic. (Dominik Burek (ADGEOM #424, Aug 2, 2013)
The conic {A′, B′, C′, HA, HB, HC} is here named the Burek-cevian conic of P. If P = X(8), the conic has center X(5777). More generally, if P is on the Lucas cubic, then the triangles A′B′C′ and HAHBHC are homothetic, and HAHBHC is perspective to ABC at a point on the Darboux cubic. (ADGEOM #431, August 3, 2013, and related postings)
X(5777) = (X(5) of 2nd extouch triangle); see X(5776). Also, X(5777) lies on the Burek-Hutson central cubic, K645.
X(5777) lies on these lines: {1, 1864}, {2, 1071}, {3, 9}, {4, 8}, {5, 226}, {12, 1858}, {20, 3876}, {37, 581}, {40, 210}, {44, 580}, {55, 1898}, {56, 1728}, {63, 3149}, {65, 5587}, {78, 1012}, {119, 125}, {201, 2635}, {342, 1148}, {389, 916}, {392, 452}, {405, 1385}, {411, 3219}, {499, 3660}, {515, 960}, {516, 3678}, {518, 946}, {756, 4300}, {943, 1156}, {950, 952}, {1125, 2801}, {1158, 1376}, {1159, 5665}, {1214, 1745}, {1260, 2057}, {2800, 3036}, {3057, 3586}, {3073, 5266}, {3074, 3465}, {3090, 5439}, {3091, 3868}, {3295, 5534}, {3487, 5045}, {3555, 5603}, {3651, 3652}, {3670, 5400}, {3697, 5657}, {3715, 5584}, {3746, 5531}, {3753, 5177}, {3817, 3874}
X(5777) = midpoint of X(4) and X(72)
X(5777) = reflection of X(942) in X(5)
X(5777) = complement of X(1071)
X(5778) lies on these lines: {3, 9}, {4, 2287}, {5, 6}, {219, 355}, {284, 3560}, {940, 2003}, {952, 2256}, {1012, 2327}, {3713, 3940}
Let T be a triangle inscribed in the circumcircle and circumscribing the Mandart inellipse. As T varies, its orthocenter traces a circle centered at X(5779) with segment X(4)X(144) as diameter. (Randy Hutson, August 29, 2018)
X(5779) lies on these lines: {3, 9}, {4, 144}, {5, 7}, {40, 3062}, {44, 990}, {45, 991}, {55, 5531}, {119, 3826}, {142, 1656}, {165, 3715}, {210, 1709}, {355, 382}, {381, 527}, {390, 952}, {517, 4915}, {518, 1351}, {954, 3560}, {1001, 2801}, {1012, 3940}, {1538, 5231}, {1750, 3929}, {1768, 4413}, {2951, 3579}, {3711, 5537}, {4312, 5587}, {4326, 5534}, {5273, 5658}
X(5779) = midpoint of X(4) and X(144)
X(5779) = complement of X(36996)
X(5779) = Johnson-isogonal conjugate of X(37820)
X(5780) lies on these lines: {3, 9}, {5, 8}, {72, 2095}, {355, 3452}, {952, 5084}, {1210, 1656}, {3149, 3876}
X(5781) lies on these lines: {3, 9}, {6, 7}, {19, 518}, {20, 220}, {21, 3207}, {48, 1001}, {63, 910}, {101, 1012}, {144, 2287}, {169, 1071}, {218, 4292}, {219, 516}, {284, 954}, {390, 2256}, {1376, 2272}, {1503, 2550}, {1615, 5273}, {2173, 5220}, {2257, 4321}, {3059, 5227}, {3713, 5279}} SEARCH: -0.33541830108619423321
X(5782) lies on these lines: {3, 9}, {6, 8}, {44, 4386}, {346, 2256}, {391, 2255}, {940, 4670}, {956, 5053}, {958, 2267}, {1211, 3330}, {1376, 2183}, {1404, 4390}, {1743, 5264}, {2057, 3965}, {2221, 4383}, {2235, 5205}, {2257, 2297}, {2550, 5480}, {4363, 5228}
X(5783) lies on these lines: {1, 3713}, {3, 9}, {6, 10}, {37, 997}, {45, 5110}, {72, 2285}, {171, 1743}, {210, 1460}, {218, 1010}, {219, 1065}, {332, 344}, {405, 2268}, {474, 1400}, {475, 608}, {478, 1211}, {572, 958}, {573, 1376}, {604, 956}, {651, 5232}, {960, 1766}, {2050, 3452}, {2256, 2321}
X(5784) = X(7) of the X(1)-Brocard triangle (see X(5642)).
X(5784) lies on these lines: {2, 1864}, {3, 9}, {7, 8}, {10, 1071}, {19, 1350}, {20, 960}, {21, 662}, {37, 1818}, {46, 5223}, {63, 210}, {72, 527}, {141, 1861}, {142, 442}, {144, 4190}, {219, 990}, {224, 1001}, {354, 2886}, {390, 3890}, {392, 4304}, {480, 2057}, {511, 2262}, {516, 3878}, {528, 3057}, {997, 1012}, {1824, 3917}, {2261, 5085}, {2348, 3423}, {3660, 5231}, {3740, 5273}, {3812, 4208}, {3881, 5542}
X(5785) lies on these lines: {3, 9}, {7, 10}, {20, 3062}, {144, 4292}, {377, 4312}
X(5786) lies on these lines: {3, 10}, {4, 6}, {20, 333}, {29, 154}, {40, 1765}, {64, 412}, {65, 5307}, {84, 1715}, {243, 1854}, {386, 2050}, {388, 940}, {405, 1746}, {407, 1899}, {572, 2049}, {965, 2551}, {1012, 4267}, {1754, 5247}, {1766, 5295}, {1837, 1891}, {1853, 5125}, {3714, 5227}
Let A′B′C′ be the excentral triangle. X(5787) is the radical center of the anticomplementary circles of triangles A′BC, B′CA, C′AB. (Randy Hutson, June 27, 2018)
X(5787) lies on these lines: {3, 10}, {4, 7}, {5, 1490}, {20, 3419}, {30, 84}, {40, 3358}, {57, 1837}, {382, 2095}, {962, 4018}, {990, 1834}, {1467, 1750}, {1699, 3649}, {3091, 5658}, {3601, 5252}, {4219, 5090}X(5788) lies on these lines: {3, 10}, {4, 333}, {5, 6}, {12, 940}, {63, 1867}, {394, 3142}, {970, 2050}, {1150, 3436}, {3560, 4267}, {5247, 5587}
X(5789) lies on these lines: {3, 10}, {5, 7}, {153, 443}, {381, 3928}
X(5790) lies on these lines: {1, 1656}, {2, 952}, {3, 10}, {4, 3617}, {5, 8}, {30, 5657}, {40, 382}, {55, 80}, {119, 2886}, {140, 944}, {145, 3090}, {165, 3534}, {210, 381}, {226, 1159}, {442, 5554}, {495, 3475}, {516, 3654}, {519, 5055}, {546, 962}, {547, 3241}, {912, 3753}, {946, 3626}, {956, 5176}, {997, 5123}, {999, 1737}, {1000, 5274}, {1071, 4002}, {1125, 5070}, {1145, 3434}, {1260, 3419}, {1351, 3416}, {1385, 1698}, {1483, 3616}, {1598, 5090}, {1657, 3579}, {1837, 3295}, {2095, 3421}, {2801, 3968}, {3091, 4678}, {3428, 5659}, {3436, 3927}, {3560, 5086}, {3576, 5054}, {3621, 5056}, {3622, 5067}, {3632, 5079}, {3655, 3828}, {3656, 3817}, {3814, 5289}, {3843, 4691}, {4668, 5072}, {5154, 5330}, {5204, 5445}, {5221, 5270}
X(5790) = anticomplement of X(38028)
X(5790) = centroid of X(3)X(4)X(8)
X(5790) = orthocenter of cross-triangle of Fuhrmann and Ai (aka K798i) triangles
X(5791) lies on these lines: {2, 72}, {3, 10}, {4, 5273}, {5, 9}, {7, 3824}, {12, 57}, {21, 3419}, {28, 5130}, {37, 5292}, {46, 3925}, {63, 442}, {140, 936}, {142, 3634}, {191, 1836}, {210, 498}, {226, 3927}, {345, 5295}, {377, 3916}, {381, 5325}, {405, 1259}, {443, 3436}, {496, 5231}, {549, 5438}, {965, 3211}, {997, 4999}, {1479, 3683}, {1656, 2095}, {1706, 3587}, {1714, 3666}, {1837, 5251}, {2476, 3219}, {2550, 3579}, {3218, 4197}, {3295, 4847}, {3305, 4187}, {3416, 5138}, {3454, 4643}, {3601, 3679}, {3656, 3878}, {3697, 5552}, {3746, 4863}, {5067, 5328}, {5234, 5587}, {5252, 5258}
X(5792) lies on these lines: {2, 3207}, {3, 10}, {6, 7}, {19, 4361}, {610, 3739}, {910, 4384}, {2182, 4363}
X(5793) lies on these lines: {1, 3714}, {3, 10}, {6, 8}, {65, 4363}, {141, 388}, {333, 3617}, {1211, 3436}, {3052, 4195}, {3679, 5247}, {4720, 5331}, {5078, 5176}
X(5794) lies on these lines: {1, 442}, {2, 1837}, {3, 10}, {4, 960}, {5, 997}, {7, 8}, {12, 78}, {20, 4640}, {46, 529}, {72, 1478}, {80, 1698}, {145, 3475}, {149, 3890}, {210, 3436}, {224, 3925}, {329, 5229}, {392, 1479}, {407, 1211}, {443, 3812}, {474, 1737}, {495, 3811}, {497, 5175}, {498, 5440}, {528, 1697}, {594, 5227}, {936, 1329}, {938, 3742}, {946, 5289}, {950, 1001}, {965, 1826}, {1155, 3617}, {1159, 3625}, {1220, 5135}, {1265, 3967}, {1420, 5231}, {1610, 4220}, {1836, 2475}, {1861, 1891}, {2182, 2345}, {2476, 4511}, {2551, 3740}, {3057, 3434}, {3091, 5087}, {3421, 4662}, {3485, 3838}, {3576, 4999}, {3698, 5554}, {3876, 5080}, {3880, 5082}, {3916, 4299}, {3966, 5016}, {4255, 5530}, {4314, 4428}, {4679, 5046}, {4855, 5432}
X(5795) lies on these lines: {1, 2551}, {2, 1420}, {3, 10}, {8, 9}, {20, 1706}, {63, 4848}, {65, 527}, {142, 388}, {145, 3984}, {200, 3486}, {226, 3436}, {281, 1891}, {329, 3340}, {474, 4311}, {495, 1125}, {497, 4853}, {519, 960}, {529, 3812}, {535, 3918}, {936, 944}, {952, 5044}, {956, 1210}, {997, 5534}, {1005, 5086}, {1212, 1573}, {1220, 5053}, {1385, 3820}, {1716, 3755}, {1737, 5258}, {1837, 4847}, {2078, 5176}, {2098, 4679}, {2324, 4270}, {2478, 3872}, {2550, 2951}, {2784, 3041}, {2975, 3911}, {3036, 4691}, {3058, 3893}, {3158, 4313}, {3189, 4882}, {3244, 5289}, {3586, 5082}, {3600, 5437}, {3617, 5273}, {3622, 5328}, {3626, 5302}, {3634, 4999}, {3679, 5234}, {3753, 4292}, {3913, 4314}, {4858, 4968}
X(5796) lies on these lines: {4, 6}, {5, 7}, {226, 1736}, {3100, 5396}
X(5797) lies on these lines: {4, 6}, {5, 8}, {10, 1953}, {355, 3187}
X(5798) lies on these lines: {4, 6}, {5, 9}, {226, 1465}, {442, 573}, {1243, 1903}, {1490, 5396}, {1699, 1723}
X(5799) lies on these lines: {4, 6}, {5, 10}, {40, 4026}, {51, 1904}, {65, 1848}, {154, 4198}, {573, 4205}, {2050, 5292}
X(5800) lies on these lines: {4, 6}, {7, 8}, {10, 5227}, {28, 159}, {55, 464}, {81, 1370}, {141, 443}, {193, 2475}, {376, 4265}, {497, 1386}, {515, 990}, {631, 5096}, {958, 4026}, {1056, 3242}, {1352, 4260}, {1478, 3751}, {1861, 2285}, {1890, 2082}, {2263, 5236}, {2478, 3618}, {3100, 3486}, {3101, 3474}, {3187, 3434}, {3475, 3920}, {3589, 5084}, {3827, 4295}, {3914, 5307}, {4663, 5229}
X(5801) lies on these lines: {4, 6}, {7, 10}
X(5802) lies on these lines: {2, 272}, {4, 6}, {8, 9}, {10, 380}, {20, 579}, {37, 3488}, {48, 3086}, {71, 4294}, {219, 497}, {226, 1449}, {281, 1837}, {329, 3187}, {405, 966}, {515, 2257}, {573, 1713}, {610, 1210}, {944, 1108}, {965, 5084}, {1058, 2256}, {1100, 3487}, {1213, 4258}, {1708, 3101}, {1743, 3586}, {1839, 4295}, {2260, 4293}, {2287, 2478}, {2345, 3419}, {3189, 3694}, {4207, 5320}, {5037, 5304}, {5257, 5436}
X(5803) lies on these lines: {4, 7}, {5, 6}
X(5804) lies on these lines: {4, 7}, {5, 8}, {517, 5084}, {946, 3340}, {1532, 3487}, {3149, 3488}
X(5805) is the Gergonne point of the Euler triangle.
X(5805) lies on these lines: {3, 142}, {4, 7}, {5, 9}, {11, 57}, {79, 3062}, {144, 3091}, {355, 518}, {381, 527}, {390, 5603}, {392, 443}, {515, 5542}, {517, 2550}, {528, 3656}, {952, 3243}, {954, 3149}, {990, 1086}, {991, 4675}, {1750, 4654}, {3332, 4000}, {4301, 5289}, {5223, 5587}
X(5805) = midpoint of X(4) and X(7)
X(5805) = Johnson-isogonal conjugate of X(37822)
X(5805) = center of the perspeconic of these triangles: inner and outer Johnson
X(5806) lies on these lines: {3, 5436}, {4, 7}, {5, 10}, {20, 5439}, {65, 1699}, {72, 3091}, {515, 5045}, {516, 3812}, {546, 912}, {944, 5049}, {962, 3753}, {1465, 2654}, {1482, 3577}, {1709, 5221}, {1837, 5173}, {1902, 5142}, {3057, 5219}, {3585, 5570}, {3742, 4297}, {3832, 3868}, {3876, 5068}, {3940, 4882}
X(5806) = complement of X(31793)
X(5807) lies on these lines: {4, 7}, {6, 8}, {390, 1766}, {452, 5279}, {950, 2285}, {4200, 5262}, {4220, 5435}, {5269, 5294}
X(5808) lies on these lines: {4, 7}, {6, 10}
X(5809) lies on these lines: {4, 7}, {8, 9}, {10, 4326}, {20, 1445}, {142, 5177}, {226, 5274}, {329, 497}, {388, 5572}, {405, 4313}, {480, 3189}, {516, 2093}, {952, 954}, {1001, 3486}, {1728, 4294}, {1750, 4321}, {1837, 2550}, {2551, 3059}, {3100, 5222}, {3755, 4907}
X(5810) lies on these lines: {2, 1437}, {3, 1211}, {4, 8}, {5, 6}, {442, 1899}, {2203, 3542}, {3410, 5141}
X(5811) lies on these lines: {3, 5658}, {4, 8}, {5, 7}, {9, 1158}, {84, 3452}, {104, 405}, {226, 3086}, {390, 5534}, {474, 2096}, {912, 938}, {997, 1490}, {1071, 5084}, {1483, 3488}, {1737, 3339}, {4295, 4848}, {4309, 5531}
X(5811) = anticomplement of X(37534)
X(5812) lies on these lines: {3, 226}, {4, 8}, {5, 9}, {11, 1728}, {12, 40}, {30, 1490}, {68, 1903}, {79, 165}, {222, 1076}, {452, 5603}, {580, 3772}, {908, 1259}, {946, 958}, {950, 1482}, {1006, 5253}, {1210, 2095}, {1385, 3487}, {1479, 1864}, {1766, 1901}, {5177, 5657}
X(5812) = anticomplement of X(37623)
X(5812) = X(26)-of-2nd-extouch-triangle
X(5812) = X(40)-of-outer-Johnson-triangle
X(5812) = orthologic center of these triangles: outer-Johnson to 3rd extouch
X(5813) lies on these lines: {2, 169}, {4, 8}, {6, 7}, {9, 4329}, {226, 2082}, {307, 2270}, {857, 1211}, {3616, 4223}, {3661, 5179}
The triangle A″B″C″ defined at X(5739) is homothetic to the outer Garcia triangle at X(5814).
X(5814) lies on these lines: {1, 1211}, {3, 3687}, {4, 8}, {6, 10}, {9, 3695}, {69, 942}, {75, 1330}, {78, 5396}, {209, 4680}, {306, 405}, {442, 5271}, {1125, 4035}, {1479, 3706}, {1836, 4647}, {2895, 3868}, {3187, 5051}, {3295, 3883}, {3454, 3772}, {3927, 4416}, {4651, 5300}, {4863, 4894}, {5244, 5290}
X(5814) = anticomplement of X(37594)
X(5815) lies on these lines: {1, 5129}, {2, 3333}, {4, 8}, {7, 10}, {20, 200}, {40, 144}, {69, 341}, {210, 388}, {443, 3697}, {452, 3870}, {516, 4882}, {518, 938}, {527, 1706}, {936, 3600}, {944, 3940}, {956, 3616}, {997, 4308}, {1056, 5044}, {1722, 4310}, {2550, 4662}, {3085, 5273}, {3086, 5328}, {3091, 4847}, {3555, 5084}, {3679, 4295}, {3811, 4313}, {3927, 5657}, {3961, 4339}, {4005, 5252}, {4301, 4915}, {4419, 4646}, {4863, 5225}, {5056, 5231}
X(5815) = anticomplement of X(3333)
X(5815) = isotomic conjugate of X(30501)
X(5816) lies on these lines: {2, 572}, {3, 1213}, {4, 9}, {5, 6}, {37, 355}, {80, 941}, {119, 5517}, {377, 1765}, {391, 3091}, {498, 2268}, {499, 604}, {515, 5257}, {581, 975}, {946, 3686}, {1400, 1478}, {1474, 3542}, {1479, 2269}, {1737, 2285}, {1899, 3136}, {2328, 4207}, {3419, 3965}, {3553, 4270}
X(5817) lies on these lines: {2, 971}, {4, 9}, {5, 7}, {44, 3332}, {119, 1156}, {142, 3090}, {144, 3091}, {355, 390}, {443, 3358}, {518, 5603}, {527, 3545}, {944, 1001}, {946, 5223}, {948, 1736}, {952, 954}, {1698, 3062}, {1788, 4312}, {3487, 5045}
X(5817) = anticomplement of X(38122)
X(5817) = {X(4),X(9)}-harmonic conjugate of X(5759)
X(5818) lies on these lines: {1, 3090}, {2, 355}, {3, 5260}, {4, 9}, {5, 8}, {12, 3487}, {46, 5229}, {80, 498}, {100, 3560}, {104, 474}, {119, 2476}, {145, 5056}, {165, 3529}, {377, 2096}, {381, 962}, {388, 1737}, {495, 938}, {499, 3476}, {515, 631}, {517, 3091}, {519, 5071}, {547, 1483}, {942, 5261}, {946, 3545}, {952, 1656}, {986, 4947}, {1056, 1210}, {1071, 4208}, {1125, 5067}, {1478, 1788}, {1699, 3855}, {1837, 3085}, {2099, 3614}, {2475, 3652}, {3086, 5252}, {3089, 5090}, {3146, 3579}, {3241, 5055}, {3474, 3585}, {3524, 3828}, {3525, 3576}, {3628, 5550}, {3654, 3839}, {3753, 5177}, {3877, 5187}, {3925, 5658}, {4299, 5445}, {4301, 4691}, {4305, 5432}, {4330, 5560}, {4678, 5068}, {5086, 5552}, {5119, 5225}
X(5818) = incenter of cross triangle of Euler and anti-Euler triangles
X(5819) lies on these lines: {2, 910}, {4, 9}, {6, 7}, {20, 1212}, {37, 390}, {41, 3485}, {75, 144}, {101, 5603}, {142, 610}, {198, 1001}, {218, 4295}, {220, 962}, {348, 4209}, {388, 2082}, {518, 2262}, {672, 3474}, {857, 1213}, {954, 4254}, {1449, 5542}, {1478, 5540}, {1738, 1743}, {1836, 2348}, {2170, 3476}, {2280, 3475}, {3207, 3616}, {3487, 4251}, {3686, 5223}, {3772, 5304}, {5263, 5296}
X(5820) lies on these lines: {5, 6}, {7, 8}, {141, 474}, {542, 5138}, {940, 1899}, {1012, 1503}
X(5821) lies on these lines: {5, 6}, {7, 10}
X(5822) lies on these lines: {5, 6}, {8, 9}, {284, 966}, {499, 2317}, {1737, 2261}, {1743, 1826}
X(5823) lies on these lines: {5, 7}, {6, 8}, {1736, 3086}
X(5824) lies on these lines: {5, 7}, {6, 10}, {982, 1736}
X(5825) lies on these lines: {5, 7}, {8, 9}, {11, 329}, {72, 4345}, {1728, 4293}, {1737, 4312}, {1864, 3740}, {3086, 5542}
X(5826) lies on these lines: {5, 8}, {6, 7}
X(5827) lies on these lines: {5, 8}, {6, 10}, {355, 2050}, {2551, 3695}, {5295, 5587}
X(5828) lies on these lines: {5, 8}, {7, 10}, {341, 3262}, {4853, 5056}
X(5829) lies on these lines: {5, 9}, {6, 7}, {142, 1375}, {511, 2262}, {528, 1953}, {910, 5249}, {1723, 4312}
X(5830) lies on these lines: {5, 9}, {6, 8}
X(5831) lies on these lines: {5, 9}, {6, 10}, {442, 2285}, {475, 1880}, {498, 3965}, {1766, 2886}, {2268, 3419}
X(5832) lies on these lines: {5, 9}, {7, 8}, {63, 1836}, {142, 474}, {495, 1706}, {516, 993}, {611, 1738}, {956, 4292}, {1376, 5249}, {1387, 3254}, {1818, 4675}, {1861, 4363}
X(5833) lies on these lines: {5, 9}, {7, 10}, {63, 4312}, {142, 936}, {515, 2550}, {1699, 5273}
X(5834) lies on these lines: {5, 10}, {6, 7}, {19, 3589}, {141, 2262}, {597, 2182}, {1375, 5437}, {2183, 4364}, {2270, 4657}, {3674, 3752}
X(5835) lies on these lines: {1, 3704}, {5, 10}, {6, 8}, {65, 141}, {388, 4363}, {1211, 3869}, {1213, 3959}, {2292, 4364}, {2975, 5078}
X(5835) = complement of X(37614)
A construction of X(5836) is given by Antreas Hatipolakis and Angel Montesdeoca at 24129.
X(5836) lies on these lines: {1, 474}, {2, 3057}, {5, 10}, {7, 8}, {12, 3838}, {37, 3208}, {40, 958}, {46, 956}, {56, 3872}, {57, 4853}, {72, 3679}, {78, 2099}, {92, 1888}, {100, 2646}, {145, 354}, {191, 3245}, {200, 3340}, {210, 3617}, {318, 1875}, {341, 3967}, {392, 1698}, {404, 1319}, {405, 5119}, {409, 643}, {442, 1145}, {484, 3916}, {519, 942}, {528, 950}, {529, 4292}, {672, 4875}, {758, 3626}, {891, 4925}, {910, 2329}, {936, 5289}, {962, 2551}, {993, 3579}, {997, 1482}, {1001, 1697}, {1104, 5255}, {1125, 1387}, {1155, 2975}, {1191, 1722}, {1193, 4695}, {1210, 3813}, {1212, 3501}, {1616, 5272}, {1829, 1861}, {1836, 3436}, {1837, 3434}, {1858, 5086}, {1859, 5174}, {1864, 5175}, {1887, 5081}, {2098, 4413}, {2171, 3965}, {2262, 2345}, {2292, 2643}, {2475, 5176}, {2800, 3036}, {2817, 3040}, {3244, 5045}, {3246, 3915}, {3303, 3895}, {3304, 3306}, {3336, 5288}, {3339, 4915}, {3421, 4295}, {3555, 3632}, {3601, 4421}, {3616, 3848}, {3621, 3873}, {3625, 3874}, {3634, 3884}, {3635, 5049}, {3636, 3833}, {3666, 4642}, {3670, 4674}, {3678, 4691}, {3681, 3962}, {3683, 5260}, {3711, 3984}, {3744, 3924}, {3746, 5541}, {3876, 3983}, {3877, 4731}, {3894, 4816}, {3899, 3921}, {3900, 4142}, {4015, 4745}, {4018, 4668}, {4071, 4167}, {4084, 4669}, {4428, 5436}, {4746, 4757}, {4847, 4848}
X(5836) = midpoint of X(8) and X(65)
X(5836) = complement of X(3057)
X(5837) lies on these lines: {2, 3340}, {5, 10}, {8, 9}, {65, 142}, {145, 5273}, {226, 3869}, {388, 527}, {392, 1210}, {443, 2093}, {519, 958}, {551, 4999}, {936, 5657}, {1125, 5289}, {1145, 3697}, {1212, 1500}, {1479, 2551}, {2078, 2975}, {3057, 4847}, {3212, 4357}, {3305, 5554}, {3486, 4512}, {3600, 3928}, {3625, 5302}, {3632, 5234}, {3813, 4342}, {3916, 4311}, {4297, 4640}
X(5838) lies on these lines: {6, 7}, {8, 9}, {41, 3616}, {144, 239}, {169, 938}, {218, 962}, {497, 2348}, {516, 1743}, {910, 5435}, {1001, 5296}, {1212, 4313}, {1445, 2270}, {2170, 3241}, {2264, 2550}
X(5839) lies on these lines: {1,966}, {2,319}, {6,8}, {7,524}, {9,519}, {10,1449}, {37,145}, {44,346}, {48,3684}, {55,4819}, {69,239}, {71,3169}, {72,5802}, {75,193}, {78,3554}, {141,5222}, {144,536}, {219,1067}, {281,2323}, {318,3087}, {345,1914}, {348,4372}, {387,5814}, {393,5081}, {518,2262}, {572,5657}, {573,944}, {597,4478}, {604,1788}, {740,3958}, {894,1992}, {938,965}, {956,4254}, {982,4771}, {1043,1778}, {1086,4402}, {1108,3965}, {1213,3616}, {1400,3476}, {1654,4393}, {1698,4982}, {1740,4489}, {1743,2321}, {1901,5175}, {1953,4051}, {2082,5227}, {2269,3486}, {2325,3973}, {3061,3949}, {3161,3943}, {3187,5739}, {3240,5153}, {3241,5296}, {3244,3247}, {3293,5105}, {3419,5746}, {3553,3872}, {3589,4445}, {3618,3661}, {3623,3723}, {3625,4007}, {3629,4363}, {3630,4395}, {3633,3707}, {3635,3986}, {3672,4643}, {3679,5750}, {3696,4307}, {3739,3945}, {3758,5564}, {3770,4441}, {3875,4416}, {3879,4384}, {4029,4898}, {4047,4294}, {4058,4701}, {4060,4677}, {4061,5269}, {4454,4686}, {4470,4967}, {4545,4668}, {4657,4690}, {5015,5286}, {5120,5687}, {5271,5712}, {5603,5816}, {5703,5742}
X(5839) = anticomplement of X(4851)
X(5840) lies on these lines: {3, 11}, {4, 100}, {5, 3035}, {20, 104}, {30, 511}, {36, 5533}, {40, 80}, {140, 3825}, {153, 3146}, {214, 946}, {224, 1537}, {355, 1145}, {550, 1484}, {944, 1320}, {1156, 5759}, {1317, 1482}, {1385, 1387}, {1614, 3045}, {1768, 5709}, {1862, 1872}, {2077, 3583}, {2932, 3149}, {3036, 5690}, {3254, 5732}, {3359, 3586}, {4292, 5083}, {5541, 5691}
X(5841) lies on these lines: {3, 12}, {4, 2975}, {5, 993}, {30, 511}, {40, 4333}, {63, 355}, {80, 5535}, {119, 4996}, {140, 3822}, {226, 1385}, {1872, 1885}, {2077, 4316}, {4305, 5761}, {5691, 5709}
X(5842) lies on these lines: {3, 2886}, {4, 12}, {5, 5248}, {11, 5172}, {20, 2894}, {30, 511}, {31, 5721}, {40, 1726}, {140, 3841}, {550, 5450}, {944, 2099}, {946, 4314}, {1006, 3925}, {1012, 4302}, {1071, 1770}, {1072, 3744}, {1158, 5787}, {1479, 3149}, {1532, 3583}, {1824, 3575}, {1834, 3072}, {3058, 5603}, {3189, 5758}, {3474, 5768}, {3811, 5812}, {4292, 5173}, {4512, 5587}, {5119, 5691}
X(5843) lies on these lines: {3, 144}, {5, 7}, {9, 140}, {30, 511}, {84, 5763}, {142, 3628}, {355, 4312}, {390, 1483}, {546, 5805}, {548, 5732}, {550, 5759}, {1156, 1484}, {2096, 3940}, {2951, 5534}, {3339, 5587}, {3853, 5735}, {3927, 5657}, {5223, 5690}, {5730, 5731}
X(5844) lies on these lines: {1, 140}, {3, 145}, {4, 3621}, {5, 8}, {10, 3628}, {30, 511}, {36, 1317}, {40, 548}, {119, 4867}, {165, 3655}, {355, 546}, {495, 2099}, {496, 2098}, {547, 3679}, {549, 3241}, {550, 944}, {631, 3623}, {632, 3616}, {946, 3625}, {962, 3627}, {1000, 2346}, {1056, 1159}, {1145, 4511}, {1320, 1484}, {1385, 3244}, {1387, 1737}, {1656, 3617}, {2136, 5709}, {3036, 3814}, {3090, 4678}, {3488, 5729}, {3526, 3622}, {3576, 3654}, {3656, 4677}, {3680, 5763}, {3820, 5289}, {3861, 4301}, {4187, 5330}, {4534, 5526}, {5535, 5541}
X(5845) lies on these lines: {6, 7}, {9, 141}, {30, 511}, {41, 3665}, {69, 144}, {101, 1565}, {142, 3589}, {150, 1146}, {193, 4440}, {348, 3207}, {390, 3242}, {597, 4795}, {599, 4370}, {903, 1992}, {1001, 4364}, {1350, 5759}, {1352, 5779}, {1386, 4667}, {2550, 4363}, {3416, 4901}, {3618, 4747}, {3620, 4473}, {3751, 4312}, {3763, 4748}, {3826, 4472}, {4904, 5540}, {5480, 5805}
X(5845) = isotomic conjugate of X(35158)
X(5845) = X(2)-Ceva conjugate of X(35093)
X(5845) = crossdifference of every pair of points on line X(6)X(926)
X(5846) lies on these lines: {1, 141}, {6, 8}, {10, 1386}, {30, 511}, {31, 3703}, {37, 3883}, {42, 4030}, {44, 3717}, {63, 4884}, {69, 145}, {100, 5078}, {182, 5690}, {193, 3621}, {238, 3932}, {306, 3744}, {345, 3052}, {355, 5480}, {595, 3695}, {597, 3679}, {599, 3241}, {612, 3966}, {902, 3712}, {944, 1350}, {1086, 4645}, {1125, 3844}, {1211, 3920}, {1279, 3912}, {1352, 1482}, {1697, 5227}, {1738, 4395}, {1743, 4901}, {1834, 5015}, {1999, 4514}, {2550, 4361}, {2886, 4362}, {2975, 4265}, {3008, 3823}, {3035, 4434}, {3187, 5014}, {3244, 3631}, {3616, 3763}, {3617, 3618}, {3619, 3622}, {3620, 3623}, {3625, 4663}, {3629, 3632}, {3630, 3633}, {3685, 3943}, {3696, 4399}, {3704, 5255}, {3705, 3769}, {3722, 4062}, {3745, 4914}, {3755, 4852}, {3756, 5211}, {3782, 3891}, {3790, 4676}, {3811, 5396}, {3867, 5090}, {4153, 5305}, {4307, 4363}, {4349, 4670}, {4366, 4437}, {4388, 4415}, {4684, 4864}, {5082, 5800}, {5085, 5657}, {5241, 5297}
X(5847) lies on these lines: {1, 69}, {6, 10}, {8, 193}, {9, 4078}, {30, 511}, {31, 306}, {38, 4001}, {40, 3169}, {44, 3932}, {55, 4028}, {72, 3688}, {141, 1125}, {171, 3687}, {226, 4362}, {238, 3912}, {239, 1738}, {319, 5263}, {322, 4008}, {345, 1707}, {355, 1351}, {551, 599}, {581, 3811}, {597, 3828}, {612, 4104}, {613, 1210}, {896, 3977}, {902, 4062}, {940, 3966}, {946, 1352}, {950, 3056}, {976, 4101}, {984, 4416}, {1001, 4851}, {1100, 4026}, {1211, 3745}, {1279, 4966}, {1350, 4297}, {1353, 5690}, {1428, 3911}, {1698, 3618}, {1733, 3262}, {1757, 3717}, {1992, 3679}, {1999, 4388}, {2308, 5294}, {2321, 3923}, {2887, 3791}, {2895, 3920}, {3008, 3836}, {3011, 3936}, {3187, 3914}, {3242, 3244}, {3510, 3783}, {3578, 4981}, {3589, 3634}, {3616, 3620}, {3619, 3624}, {3626, 3629}, {3630, 3635}, {3631, 3636}, {3663, 4655}, {3703, 4641}, {3722, 4938}, {3755, 4660}, {3759, 4429}, {3769, 4417}, {3771, 4035}, {3772, 4138}, {3773, 4672}, {3821, 3946}, {3896, 4450}, {4085, 4856}, {4133, 5695}, {4265, 5267}, {4432, 4437}, {4656, 4703}, {4682, 5743}, {4769, 5028}, {4847, 4865}, {4909, 5625}, {5093, 5790}, {5138, 5745}
X(5847) = isogonal conjugate of X(28476)
X(5847) = crossdifference of every pair of points on line X(6)X(834)
X(5848) lies on these lines: {6, 11}, {30, 511}, {69, 100}, {80, 3751}, {119, 1352}, {141, 3035}, {149, 193}, {611, 5820}, {1145, 3416}, {1317, 3242}, {1353, 1484}, {1386, 1387}, {3013, 3140}
X(5849) lies on these lines: {6, 12}, {30, 511}, {69, 2975}, {141, 4999}
X(5850) lies on these lines: {1, 144}, {6, 4353}, {7, 10}, {8, 4312}, {9, 1125}, {30, 511}, {72, 4298}, {142, 3634}, {190, 4684}, {210, 553}, {320, 3717}, {390, 3244}, {946, 5779}, {954, 993}, {962, 3062}, {984, 3664}, {997, 4321}, {1001, 3636}, {1738, 4887}, {1743, 4310}, {1757, 3008}, {2325, 4966}, {2550, 3626}, {2951, 5493}, {3243, 3635}, {3475, 3929}, {3663, 3751}, {3685, 4480}, {3811, 5732}, {3817, 5817}, {3874, 5728}, {3881, 5572}, {3925, 3982}, {3946, 4663}, {4021, 4649}, {4031, 4413}, {4297, 5759}, {4349, 4644}, {4356, 4419}, {4645, 4899}, {4860, 5316}, {5715, 5811}
X(5851) lies on these lines: {7, 11}, {9, 1768}, {30, 511}, {80, 4312}, {100, 144}, {104, 1001}, {119, 3826}, {153, 2550}, {390, 1317}, {1145, 5223}, {1387, 5542}, {2951, 5528}, {3062, 3254}, {5083, 5572}, {5220, 5657}, {5289, 5698}
X(5852) lies on these lines: {7, 12}, {9, 583}, {30, 511}, {144, 1001}, {190, 4966}, {320, 3932}, {329, 3816}, {345, 3632}, {553, 3740}, {1086, 1757}, {2550, 4678}, {3035, 3218}, {3058, 4430}, {3244, 3772}, {3631, 3773}, {3650, 3746}, {3663, 4663}, {4480, 4684}
X(5853) lies on these lines: {1, 142}, {2, 3158}, {7, 145}, {8, 9}, {10, 1001}, {11, 3689}, {30, 511}, {40, 5768}, {55, 4847}, {100, 2078}, {144, 3621}, {149, 908}, {190, 4899}, {200, 497}, {210, 3058}, {226, 2900}, {238, 3939}, {306, 5014}, {480, 1837}, {553, 3873}, {673, 3912}, {936, 1058}, {938, 1706}, {944, 5732}, {946, 3811}, {954, 3419}, {956, 4304}, {958, 4314}, {1000, 5785}, {1043, 4483}, {1086, 4864}, {1125, 3813}, {1210, 5687}, {1279, 3008}, {1320, 3254}, {1445, 4848}, {1449, 4344}, {1482, 5805}, {2346, 5178}, {2348, 3021}, {2551, 4882}, {2885, 3622}, {3011, 3722}, {3057, 3059}, {3242, 3663}, {3244, 4780}, {3421, 3586}, {3445, 4678}, {3476, 4321}, {3486, 4326}, {3555, 4292}, {3625, 4133}, {3626, 3773}, {3632, 5223}, {3633, 4312}, {3635, 4743}, {3687, 3996}, {3706, 4030}, {3711, 4679}, {3748, 3925}, {3878, 4523}, {3914, 3938}, {3932, 4702}, {3957, 5249}, {3966, 4061}, {3985, 4541}, {4001, 4450}, {4028, 4865}, {4046, 4914}, {4082, 4387}, {4307, 4667}, {4342, 5289}, {4358, 4939}, {4512, 5325}, {4527, 4701}, {4535, 4746}, {4538, 4662}, {4645, 4684}, {5218, 5231}, {5263, 5750}, {5572, 5836}
X(5853) = isogonal conjugate of X(1477)
X(5853) = isotomic conjugate of X(35160)
X(5853) = X(2)-Ceva conjugate of X(35111)
X(5854) lies on these lines: {1, 1145}, {8, 11}, {10, 1387}, {30, 511}, {46, 2136}, {56, 100}, {80, 3632}, {119, 1482}, {149, 3436}, {214, 3244}, {644, 3039}, {1000, 1001}, {1146, 4919}, {1828, 1862}, {1846, 1897}, {3254, 4900}, {3829, 5790}, {3871, 4996}, {4738, 4939}, {4861, 4999}
X(5855) lies on these lines: {1, 4999}, {8, 12}, {30, 511}, {55, 145}, {1317, 4996}, {1329, 5730}, {1482, 3813}, {2161, 4969}, {3035, 4511}, {3036, 4867}, {3039, 5526}, {3340, 5794}, {3419, 3632}, {3428, 3913}, {3434, 3621}, {3633, 5119}, {3816, 5289}, {3829, 5603}, {4930, 5790}, {5173, 5836}
X(5856) lies on these lines: {7, 100}, {9, 11}, {30, 511}, {80, 5223}, {104, 5759}, {119, 5805}, {142, 3035}, {144, 149}, {214, 5542}, {390, 1320}, {956, 5698}, {1001, 1387}, {1086, 3939}, {1145, 2550}, {1317, 3243}, {3174, 5528}, {4312, 5541}
X(5857) lies on these lines: {7, 2975}, {9, 12}, {30, 511}, {142, 4999}, {219, 4331}, {954, 5698}, {956, 4295}, {4312, 5832}, {5248, 5719}
Peter Moses (June 27, 2014) constructed families of triangles T(k,n) and U(k,n) as follows. On side BC of a triangle ABC, erect a regular n-sided polygon, externally. Starting at B and going around the polygon until reaching C, label the vertices v(0), v(1), ..., v(n-1), so that line v(0)v(n-1) = BC, and if n is even, then the lines L(A,k,n) = v(k)v(n-k-1), for k = 1, ..., (n-2)/n, are parallel to BC. If n is odd, let L(A,k,n), for k = (n-1)/2, be the line through v((n-1)/2)) that is parallel to BC. Define lines L(B,k,n) and L(C,k,n) cyclically. Then for each k from 1 to floor((n-1)/2), the lines L(A,k,n), L(B,k,n), L(C,k,n) form a triangle T(k,n) homothetic to ABC. T(1,4) and U(1,4) are the outer and inner Grebe triangles, respectively. Triangle centers defined from T(k,n) include the following, given by 1st barycentrics:
centroid of T(k,n): -2a2 + b2 + c2
+ S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of T(k,n): a2[2S2A -
2SBSC + SSA csc(kπ/n)
csc(π/n+kπ/n) sin(π/n)]
orthocenter of T(k,n): a2SBSC -
SA(S2B + S2C) +
SSBSC[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of T(k,n): 2SA(a2SA
- S2B - S2C) +
S(SBSC + S2)[cot(kπ/n)) -
cot(kπ/n+π/n)]
If the polygons are erected internally instead of externally, the
resulting triangles are denoted by U(k,n), with triangle centers given
by changing S to - S in the above first barycentrics:
centroid of U(k,n): -2a2 + b2 + c2 -
S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of U(k,n): a2[2S2A -
2SBSC - SSA csc(kπ/n)
csc(π/n+kπ/n) sin(π/n)]
orthocenter of U(k,n): a2SBSC -
SA(S2B + S2C) -
SSBSC[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of U(k,n): 2SA(a2SA
- S2B - S2C) -
S(SBSC + S2)[cot(kπ/n)) -
cot(kπ/n+π/n)]
Note that T(1,6) = T(1,3) and U(1,6) = U(1,3).
X(5858) lies on these lines: {2, 6}, {3, 533}, {381, 532}, {530, 3830}, {531, 3534}, {538, 3104}, {1351, 5617}
X(5858) = reflection of X(5859) in X(2)
X(5858) = anticomplement of X(33458)
The triangle U(1,3) is defined at X(5858).
X(5859) lies on these lines: {2, 6}, {3, 532}, {381, 533}, {530, 3534}, {531, 3830}, {538, 3105}, {1351, 5613}
X(5859) = reflection of X(5858) in X(2)
X(5859) = anticomplement of X(33459)
The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle..
X(5860) lies on these lines: {2, 6}, {30, 1160}, {519, 3640}, {637, 754}, {1328, 5485}, {3241, 5604}, {3679, 5588}
X(5860) = reflection of X(5861) in X(2)
The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle.
X(5861) lies on these lines: {2, 6}, {30, 1161}, {519, 3641}, {638, 754}, {1327, 5485}, {3241, 5605}, {3679, 5589}
X(5861) = reflection of X(5860) in X(2)
The triangle T(2,6) is defined at X(5858).
X(5862) lies on these lines: {2, 6}, {4, 532}, {376, 533}
X(5862) = reflection of X(5863) in X(2)
The triangle U(2,6) is defined at X(5858).
X(5863) lies on these lines: {2, 6}, {4, 533}, {376, 532}
X(5863) = reflection of X(5862) in X(2)
The triangle T(1,3) is defined at X(5858).
X(5864) lies on these lines: {3, 6}, {4, 298}, {20, 3181}, {383, 634}, {394, 3130}, {627, 1080}
X(5864) = reflection of X(5865) in X(3)
The triangle U(1,3) is defined at X(5858).
X(5865) lies on these lines: {3, 6}, {4, 299}, {20, 3180}, {383, 628}, {394, 3129}, {633, 1080}
X(5865) = reflection of X(5864) in X(3)
X(5866) lies on these lines: {3,69}, {25,5203}, {99,186}, {187,4558}, {325,2071}, {378,1007}, {669,3265}, {2373,3266}
X(5867) lies on these lines: {3,81}, {31,501}, {669,2106}
The triangle U(1,3) is defined at X(5858).
X(5868) lies on these lines: {3, 618}, {4, 6}, {20, 298}, {64, 2993}, {154, 470}, {463, 1899}, {471, 1853}, {633, 1350}, {3146, 3181}
X(5868) = reflection of X(5869) in X(4)
The triangle U(1,3) is defined at X(5858).
X(5869) = reflection of X(5868) in X(4)
X(5869) lies on these lines: {3, 619}, {4, 6}, {20, 299}, {64, 2992}, {154, 471}, {462, 1899}, {470, 1853}, {634, 1350}, {3146, 3180}
The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle. X(5870) is the outer-Grebe-triangle-orthologic center of the following triangles: ABC, Euler, anticomplementary, and inner Grebe. (César Lozada, ADGEOM #978, Dec. 20, 2013)
X(5870) lies on these lines: {3, 5590}, {4, 6}, {20, 488}, {30, 1160}, {40, 5688}, {147, 487}, {154, 3535}, {184, 3127}, {185, 1162}, {486, 3424}, {489, 3926}, {515, 3640}, {944, 5604}, {1853, 3536}, {1899, 5200}, {5588, 5691}
X(5870) = reflection of X(5871) in X(4)
The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle. X(5871) is the inner-Grebe-triangle-orthologic center of the following triangles: ABC, Euler triangle, anticomplementary triangle, and outer Grebe triangle. (César Lozada, ADGEOM #978, Dec. 20, 2013)
X(5871) lies on these lines: {3, 5591}, {4, 6}, {20, 487}, {30, 1161}, {40, 5689}, {147, 488}, {154, 3536}, {184, 3128}, {185, 1163}, {485, 3424}, {490, 3926}, {515, 3641}, {944, 5605}, {1853, 3535}, {5589, 5691}
X(5871) = reflection of X(5870) in X(4)
The triangle T(1,3) is defined at X(5858).
X(5872) lies on these lines: {3, 298}, {4, 3181}, {5, 6}, {18, 5613}, {61, 5617}, {182, 635}, {2782, 3104}
X(5872) = reflection of X(5873) in X(5).
The triangle U(1,3) is defined at X(5858).
X(5873) lies on these lines: {3, 299}, {4, 3180}, {5, 6}, {17, 5617}, {62, 5613}, {182, 636}, {2782, 3105}
X(5873) = reflection of X(5872) in X(5).
The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle.
X(5874) lies on these lines: {3, 1270}, {5, 6}, {26, 5594}, {30, 1160}, {140, 5590}, {355, 5588}, {952, 3640}, {1483, 5604}, {5688, 5690}
X(5874) = reflection of X(5875) in X(5)
The triangle U(1,4) is defined at X(5858), is also the inner Grebe triangle.
X(5875) lies on these lines: {3, 1271}, {5, 6}, {26, 5595}, {30, 1161}, {140, 5591}, {355, 5589}, {952, 3641}, {1483, 5605}, {5689, 5690}
X(5875) = reflection of X(5874) in X(5)
X(5876) is the orthocenter of the triangle A*B*C* defined at X(5694), and X(5876) is the nine-point center of the X(3)-Fuhrmann triangle; see X(5613). (Randy Hutson, July 7, 2014)
X(5876) is the radical center of the orthocentroidal circles of the adjunct anti-altimedial triangles. (Randy Hutson, November 2, 2017)
X(5876) lies on these lines: {3,74}, {4,93}, {5,389}, {30,5562}, {51,3850}, {52,546}, {140,185}, {143,381}, {511,3627}, {548,3917}, {550,1216}, {568,3091}, {578,1493}, {1498,2918}, {1657,2979}, {2779,5694}, {2807,5690}, {3060,3843}, {3518,3581}, {3567,3851}, {3845,5446}, {5072,5640}
X(5876) = complement of X(34783)
X(5877) is the similitude center of these equilateral triangles: X(15)-Fuhrmann and X(16)-Fuhrmann. (Randy Hutson, July 7, 2014)
X(5877) lies on these lines: {4,523}, {5,6}, {1899,3134}
X(5878) = (X(3) of X(20)-Fuhrmann triangle). (Randy Hutson, July 7, 2014)
X(5878) is the orthologic center of the Carnot (aka Johnson) and half-altitude (midheight) triangles. (César Lozada, Perspective-Orthologic-Parallelogic.pdf, ADGEOM #978, December 20, 2013)
X(5878) lies on these lines: {2,3357}, {3,1661}, {4,51}, {5,64}, {20,110}, {30,155}, {66,3521}, {68,5663}, {113,3548}, {154,550}, {382,1351}, {546,1853}, {1181,1885}, {1204,3542}, {3292,3529}
X(5878) = isogonal conjugate of X(5879)
X(5878) = anticomplement of X(3357)
Let A′B′C′ be the half-altitude triangle of ABC. Let LA be the reflection of line B′C′ in line 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(5879). (Randy Hutson, July 18, 2014)
X(5879) lies on this line: {1093,1294}
X(5879) = isogonal conjugate of X(5878)
X(5880) lies on these lines: {1,528}, {2,1155}, {3,142}, {4,3812}, {5,1158}, {6,1738}, {7,8}, {9,46}, {10,527}, {11,3306}, {19,5829}, {40,5735}, {55,1004}, {57,2886}, {63,3925}, {78,3649}, {171,3772}, {200,4654}, {226,1260}, {241,4331}, {329,3740}, {354,3434}, {355,2801}, {390,2646}, {405,1770}, {443,960}, {495,5856}, {497,3742}, {519,1159}, {553,4847}, {612,3782}, {673,1492}, {740,4851}, {750,3120}, {894,4429}, {908,4413}, {940,3914}, {958,4292}, {966,3846}, {1056,3880}, {1373,3641}, {1374,3640}, {1386,4000}, {1445,1454}, {1478,3753}, {1479,5439}, {1633,4223}, {1699,3816}, {1706,5290}, {1714,5165}, {1737,5729}, {1788,5177}, {1837,2475}, {1861,1892}, {1890,4185}, {2182,5819}, {2345,3844}, {2887,3980}, {3035,5219}, {3058,4666}, {3243,3633}, {3244,4780}, {3255,5560}, {3333,3813}, {3419,5696}, {3436,3698}, {3579,3824}, {3617,4741}, {3662,5263}, {3664,3755}, {3729,3932}, {3836,3923}, {3841,5791}, {3872,5434}, {3873,4863}, {3886,4966}, {3912,5695}, {3922,5554}, {3946,4349}, {3966,4359}, {4001,4042}, {4082,4942}, {4361,5847}, {4415,5268}, {4644,4663}, {4691,5850}, {4854,5287}, {5218,5766}, {5223,5852}, {5587,5851}
X(5880) = complement of X(5698)
Let A′ be the reflection of X(40) in A and let A″ be the reflection of X(40) in line BC. Define B′, C′, B″, and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5881). Also, X(5881) = (X(20) of the Fuhrmann triangle). (Randy Hutson, July 7, 2014)
More generally, suppose that X is a point in the plane of triangle ABC, and let A′ be the reflection of X in A and A″ be the reflection of X in line BC. Define B′, C′, B″, and C″ cyclically. The triangles A′B′C′ and A″B″C″ are perspective if X lies on the Darboux cubic. The perspector is here called the Darboux image of X. The appearance of (i,j) in the following list means that X(j) is the Darboux image of the point X(i) on the Darboux cubic: (1,4312), (3,381), (4,4), (20,5921), (40,5881), (64,5922), (80,5923), (1490,5924), (1498,5925).
If X is on the Darboux cubic and P is the perspector of ABC and the pedal triangle of X, then the Darboux image of X is the reflection of X in P. (Randy Hutson, July 18, 2014)
X(5881) lies on these lines: {1,5}, {3,3679}, {4,519}, {8,20}, {10,631}, {30,4677}, {46,4325}, {57,4317}, {78,5176}, {100,5450}, {140,3655}, {145,946}, {165,548}, {376,4669}, {382,517}, {516,3625}, {518,5735}, {546,3656}, {550,3654}, {551,3090}, {573,4034}, {632,3653}, {912,4338}, {962,3621}, {996,5767}, {1012,3913}, {1071,5836}, {1125,5067}, {1210,3476}, {1385,1698}, {1420,1737}, {1478,3340}, {1482,1699}, {1490,3419}, {1532,3813}, {1697,4309}, {1750,4863}, {1766,4007}, {1788,4311}, {1907,5090}, {2077,5687}, {3057,3586}, {3091,3241}, {3244,3855}, {3245,4333}, {3247,5816}, {3524,4745}, {3525,3828}, {3528,3626}, {3529,5493}, {3560,3746}, {3617,5731}, {3624,5070}, {3635,3817}, {3853,5844}, {3872,5086}, {3899,5694}, {4293,4848}, {4299,5128}, {4330,5119}, {4654,5270}, {4882,5787}
X(5881) = reflection of X(i) in X(j) for these (i,j); (1,355), (40,8)
X(5881) = anticomplement of X(5882)
X(5881) = exsimilicenter of hexyl and 1st Steiner circles; the insimilicenter is X(9624)
X(5881) = {X(1),X(5)}-harmonic conjugate of X(9624)
Let A′B′C′ be the Euler triangle. Let LA be the reflection of line B′C′ in line BC, and define LB and LC cyclically. Let A″ = LB∩LC, B″ = LC∩LA, C″ = LA∩LB. Triangle A″B″C″ is homothetic to ABC at X(4), and X(5882) = X(1)-of-A″B″C″.
X(5882) lies on these lines: {1,4}, {3,519}, {5,551}, {8,3523}, {10,140}, {20,3241}, {30,4301}, {35,104}, {36,4848}, {40,145}, {55,5450}, {65,4311}, {84,4313}, {119,3825}, {165,3633}, {355,1125}, {382,3656}, {516,1482}, {517,550}, {549,4669}, {553,4317}, {572,2321}, {631,3679}, {912,3878}, {942,4315}, {962,3623}, {993,5837}, {997,5534}, {1006,5258}, {1012,3303}, {1071,1317}, {1158,1697}, {1210,1319}, {1388,1837}, {1389,5425}, {1698,3533}, {2077,3871}, {2099,4292}, {2360,4248}, {2801,3884}, {2829,4342}, {2894,4861}, {3086,5727}, {3146,5734}, {3149,3304}, {3333,4308}, {3340,4293}, {3524,4677}, {3526,3653}, {3577,5558}, {3579,5844}, {3601,5768}, {3616,5056}, {3622,5068}, {3624,5818}, {3625,5690}, {3632,5657}, {3634,5790}, {3636,3851}, {3671,5842}, {3817,3850}, {3877,5693}, {4745,5054}, {5316,5531}
X(5882) = X(40) of X(1)-Brocard triangleLet T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5883) = centroid of T. Let T' be the triangle whose vertices are the centroids of the altimedial triangles; then X(5883) = incenter of T'. (Randy Hutson, July 18, 2014)
X(5883) lies on these lines: {1,88}, {2,758}, {4,3255}, {5,2771}, {8,3881}, {10,141}, {21,3336}, {42,1739}, {46,5248}, {51,2392}, {57,993}, {58,409}, {65,392}, {72,3634}, {79,5046}, {191,5047}, {210,3828}, {226,3814}, {354,519}, {373,2842}, {405,3647}, {484,1621}, {513,4049}, {517,549}, {597,2836}, {956,4860}, {958,5708}, {960,4084}, {986,3743}, {997,5437}, {1006,5535}, {1159,5289}, {1445,3339}, {1698,3678}, {1737,3822}, {1835,5136}, {1844,5125}, {1963,4658}, {2650,3216}, {2690,2699}, {2801,5587}, {2975,3337}, {3035,5719}, {3057,3636}, {3090,5693}, {3218,5251}, {3219,4880}, {3244,5045}, {3290,3997}, {3555,3626}, {3616,3884}, {3622,5697}, {3624,3869}, {3628,5694}, {3632,3889}, {3635,3922}, {3649,4187}, {3660,4315}, {3679,3873}, {3681,3894}, {3720,4424}, {3740,4134}, {3848,4744}, {3876,3901}, {3880,5049}, {3887,4809}, {4002,4691}, {4067,5044}, {4511,5425}, {4666,5119}, {4675,5725}, {4731,4745}, {5083,5252}, {5131,5426}
X(5883) = complement of X(5692)
X(5883) = centroid of the six touchpoints of the Odehnal tritangent circles and the sidelines of ABC
Let P be a point on the circumcircle. Let A′ be the orthogonal projection of P on the A-altitude, and define B′ and C′ cyclically. As P traces the circumcircle, the locus of the incenter of A′B′C′ is an ellipse with center X(5884). (Antreas Hatzipolkis, Hyancinthos #20792, February 6, 2012, and subsequent postings)
Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5884) = orthocenter of T. Let T' be the triangle whose vertices are the orthocenters of the altimedial triangles; then X(5884) = incenter of T'. (Randy Hutson, July 18, 2014)
X(5884) lies on these lines: {1,104}, {2,5693}, {3,758}, {4,79}, {5,2771}, {10,912}, {40,3868}, {52,2392}, {65,515}, {73,1735}, {117,1425}, {140,5694}, {165,3901}, {185,2779}, {191,1006}, {355,2801}, {411,5535}, {496,942}, {517,550}, {572,1761}, {580,1046}, {581,986}, {631,5692}, {944,3474}, {1064,3670}, {1210,1858}, {1385,3878}, {1482,3881}, {1490,3339}, {1656,3833}, {1765,2294}, {2096,3486}, {2695,2719}, {3149,5221}, {3359,3811}, {3576,3869}, {3812,5777}, {3918,5790}, {4295,5768}
X(5884) = complement of X(5693)
X(5885) is the nine-point center of the Fuhrmann triangle of the orthic triangle of the Fuhrmann triangle. (Randy Hutson, July 7, 2014)
Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5885) = X(5)-of-T. Let T' be the triangle whose vertices are the nine-point centers of the altimedial triangles; then X(5885) = incenter of T'. (Randy Hutson, July 18, 2014)
X(5885) lies on these lines: {1,3}, {2,5694}, {5,2771}, {140,758}, {143,2392}, {575,2836}, {912,3812}, {952,3754}, {1656,5693}, {1772,2594}, {3526,5692}, {3628,3833}, {3874,5690}, {3881,5844}
X(5885) = complement of X(5694)
Let A′ be the nine-point center of the triangle IBC, where I = X(1), and define B′ and C′ cyclically. The triangle A′B′C′ is homothetic to the Fuhrmann triangle at X(1), and X(5886) is the centroid of A′B′C′.
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 centroids of ABC, AFH, BDI, and CEG are concyclic about X(5886).
(Benjamin Warren, December 16, 2024)
X(5886) lies on these lines: {{1,5}, {2,392}, {3,142}, {4,1385}, {8,3090}, {10,1482}, {30,1699}, {36,1836}, {40,140}, {46,5433}, {56,3560}, {65,499}, {79,5427}, {84,3255}, {145,5056}, {165,549}, {226,999}, {230,1572}, {238,5398}, {354,912}, {381,515}, {382,4297}, {405,5812}, {475,1872}, {498,3057}, {519,5055}, {546,5691}, {547,3679}, {631,962}, {908,956}, {942,3086}, {944,3091}, {960,5791}, {995,3772}, {997,2886}, {1000,4345}, {1006,5284}, {1012,1519}, {1056,5226}, {1058,5703}, {1064,3720}, {1100,5816}, {1104,5713}, {1108,5747}, {1319,1478}, {1352,1386}, {1389,5330}, {1479,2646}, {1537,3306}, {1594,5090}, {1698,3628}, {1737,2099}, {1770,5204}, {1829,3542}, {1902,3541}, {2095,5745}, {3241,5071}, {3244,5079}, {3333,5843}, {3338,3649}, {3359,5437}, {3417,3615}, {3419,4511}, {3421,5748}, {3428,4423}, {3434,5440}, {3474,5122}, {3475,5049}, {3487,5045}, {3488,5274}, {3526,4301}, {3600,5714}, {3634,5070}, {3636,3851}, {3646,5763}, {3671,5708}, {3811,3813}, {3868,5694}, {3897,5046}, {3940,4847}, {4221,5333}, {4293,5126}, {4305,5225}, {4323,5704}, {4679,5251}, {5010,5444}, {5044,5761}, {5067,5734}, {5119,5432}, {5436,5715}, {5542,5779}
X(5886) = complement of X(5657)
X(5886) = {X(1),X(5)}-harmonic conjugate of X(355)
X(5886) = {X(1),X(80)}-harmonic conjugate of X(37740)
X(5886) = perspector of [cross-triangle of Fuhrmann and Ai (aka K798i) triangles] and [cross-triangle of 2nd Fuhrmann and Ae (aka K798e) triangles]
X(5886) = endo-homothetic center of Ehrmann side-triangle and 3rd anti-Euler triangle; the homothetic center is X(11459)
X(5887) lies on these lines: {1,90}, {3,960}, {4,8}, {5,65}, {10,119}, {19,5778}, {21,104}, {40,5692}, {56,920}, {210,5690}, {221,1060}, {411,3579}, {515,3878}, {518,1351}, {758,946}, {936,3359}, {942,3086}, {944,3877}, {952,1898}, {971,5698}, {1012,5730}, {1062,1854}, {1064,2292}, {1352,3827}, {1656,3812}, {1697,5534}, {2476,3753}, {2745,2766}, {2801,3884}, {3817,4084}, {3868,5603}, {3876,5657}, {3899,5691}, {3931,5396}, {4047,5755}, {4067,4301}, {5790,5836}
X(5887) = X(4)-of-X(1)-Brocard triangle)
X(5887) = anticomplement of X(34339)
X(5888) is the similitude center of ABC and the X(2)-Brocard triangle.
X(5888) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2,3098}, {74,549}, {110,5092}, {140,3581}, {141,5648}, {323,3819}, {354,3920}, {392,404}, {511,5643}, {631,5654}, {1201,3746}, {1995,5646}, {2979,5644}, {3167,5012}, {3357,3523}, {3524,4550}, {5113,5653}, {5544,5640}
X(5888) = Thomson-isogonal conjugate of X(549)
X(5889) is the circumorthic-triangle-orthologic center of these triangles: extangents, intangents, orthic, and tangential. (César Lozada, ADGEOM #978, Dec. 20, 2013)
Let O =- X(3) and let A′ be the isogonal conjugate of A with respect to OBC, and define B′ and C′ cyclically. Let A″ be the isogonal conjugate of A′ with respect to OB′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5889). (Randy Hutson, July 7, 2014)
X(5889) lies on these lines: {2,389}, {3,54}, {4,52}, {5,568}, {20,185}, {22,1181}, {24,110}, {26,1614}, {49,1658}, {51,3091}, {64,895}, {143,381}, {156,2070}, {186,1147}, {323,1092}, {382,5663}, {411,5752}, {569,1199}, {578,1994}, {631,1216}, {962,2807}, {1204,2071}, {1351,1593}, {3090,5462}, {3167,3515}, {3523,3917}, {3524,5447}, {3564,3575}
X(5889) = reflection of X(i) in X(j) for these (i,j): (3,6102), (4,52)
X(5889) = anticomplement of X(5562)
X(5889) = homothetic center of X(4)-altimedial and X(2)-adjunct anti-altimedial triangles
X(5890) lies on these lines: {2,5654}, {3,54}, {4,51}, {6,74}, {20,52}, {24,154}, {30,568}, {64,1173}, {143,382}, {184,186}, {373,5071}, {376,511}, {381,5640}, {477,2452}, {578,1199}, {631,3819}, {1141,1303}, {1216,3523}, {1994,2071}, {2807,5603}, {3091,5462}, {3146,5446}, {3524,3917}, {3651,5752}}
X(5890) = reflection of X(i) in X(j) for these (i,j): (4,51), (2979,3)
X(5890) = anticomplement of X(5891)
X(5890) = X(2)-of-circumorthic-triangle
X(5890) = X(4)-of-orthocentroidal-triangle
X(5890) = X(20)-of-Lucas-triangle (defined at X(95))
X(5890) = homothetic center of X(4)-altimedial and X(4)-adjunct anti-altimedial triangles
X(5891) = (X(376) of orthic triangle) = (X(4) of X(2)-Brocard triangle); also, (X(5891) of hexyl triangle) = X(2) and (X(5891) of excentral triangle) = X(376). (Randy Hutson, July 7, 2014)
X(5891) lies on these lines: {2,5654}, {3,64}, {4,1216}, {5,51}, {20,5447}, {30,3917}, {113,127}, {128,130}, {140,185}, {155,569}, {216,1625}, {373,547}, {378,4550}, {381,511}, {389,1656}, {399,5092}, {549,5642}, {568,5055}, {1352,2393}, {3060,3545}, {3090,5462}, {3091,5446}, {3313,3818}, {3567,5056}, {5071,5640}
X(5891) = reflection of X(51) in X(5)
X(5891) = complement of X(5890)
X(5891) = anticomplement of X(5892)
X(5892) = (X(376) of polar triangle of complement of polar circle) = {X(52),X(631)}-harmonic conjugate of X(5447) (Randy Hutson, July 7, 2014)
X(5892) lies on these lines: {2,5654}, {3,51}, {5,2883}, {52,631}, {140,389}, {143,3530}, {182,2393}, {185,1656}, {373,381}, {376,5640}, {511,549}, {512,1116}, {547,5663}, {568,3917}, {2779,3833}, {2781,3589}, {3060,3524}, {3523,3567}, {3526,5562}
X(5892) = midpoint of X(i) and X(j) for these {I,J}: {3,51},
{389,3819}
X(5892) = reflection of X(i) in X(j) for these (i,j): (1216,3819),
(3819,140)
X(5892) = complement of X(5891)
The half-altitude circle is the circumcircle of the half-altitude
triangle. Its radius is
[R/(16SASBSC)][2(b2c2J2
+ S2 -
S2A)(c2a2J2 +
S2 -
S2B)(a2b2J2 +
S2 - S2C)]1/2, where J =
|OH|/R (as at X(1113).
X(5893) lies on these lines: {2,5894}, {4,6}, {5,3357}, {30,5448}, {64,3091}, {140,2777}, {154,3146}, {221,5225}, {546,5462}, {1853,3832}, {2192,5229}
X(5893) = midpoint of X(4) and X(2883)
X(5893) = complement of X(5894)
X(5893) = X(11260)-of-orthic-triangle if ABC is acute
X(5894) is the center of the pedal circle of X(20) and of X(64), and the center of the cevian circle of X(69) and of X(253); X(5864) is also (X(64) of X(4)-Brocard triangle. (Randy Hutson, July 7, 2014)
X(5894) lies on these lines: {2,5893}, {3,1661}, {4,1192}, {5,1539}, {20,64}, {30,3357}, {154,3522}, {185,1205}, {376,1498}, {550,1216}, {1204,1885}, {1593,5480}, {1853,3146}, {1854,3474}, {2935,3520}, {3528,5656}, {4219,5799}
X(5894) = midpoint of X(20) and X(64)
X(5894) = complement of X(5895)
X(5894) = anticomplement of X(5893)
Let H be the hyperbola {A,B,C,X(4),X(20)}. Let L(X) denote the line tangent to H at a point X on H. Then X(5895) is the point of intersection of L(X(4)) and L(X(20)). (Randy Hutson, July 7, 2014)
Let A′ be the trilinear pole of the perpendicular bisector of BC, and define B′ and C′ cyclically. A′B′C′ is also the anticomplement of the anticomplement of the midheight triangle. X(5895) = X(3)-of-A′B′C′. (Randy Hutson, January 29, 2018)
X(5895) lies on these lines: {2,5893}, {3,113}, {4,64}, {6,1885}, {20,154}, {25,2929}, {30,155}, {52,382}, {193,1503}, {235,1192}, {381,3357}, {468,1620}, {1181,2904}, {1514,3542}, {1562,3172}, {1593,3574}, {1836,1854}, {2778,5693}, {2906,5706}, {2907,5786}, {3529,5656}
X(5895) = isotomic conjugate of X(34410)
X(5895) = anticomplement of X(5894)
X(5895) = crosssum of X(3) and X(64)
X(5895) = crosspoint of X(4) and X(20)
Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5896) = Λ(X(3), X(5893) = Λ(X(20), X(154)). (Randy Hutson, July 7, 2014)
X(5896) lies on the circumcircle and these lines: {64,110}, {99,253}, {107,3146}, {1301,3515}}
Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5897) = Λ(X(3), X(1661) = Λ(X(4), X(64) = Λ(X(5), X(5893)) = Λ(X(20), X(394) = Λ(X(146), X(2071). X(5897) is the point of intersection, other than A,B,C, of the circumcircle and the hyperbola {A,B,C,X(3),X(20)}; also, X(5897) is the antipode of X(1301) on the circumcircle. (Randy Hutson, July 7, 2014)
X(5897) lies on the circumcircle and these lines: {3,1301}, {20,107}, {110,1498}, {112,1033}, {376,1289}, {393,3344}, {1302,1370}, {1304,2071}
X(5897) = reflection of X(1301) in X(3)
Let H be the Stammler hyperbola, and let T be the tangential triangle. X(5898) is the antipode of X(195) in H. The conic H is a rectangular hyperbola passing through X(i) for these I: 1,3,6,155,159,195,399,1498, 2916,2917,2918,2929,2930,2931,2935,2948,3511, the excenters and the vertices of T; the center of H is X(110). H is the isogonal conjugate of the Euler line with respect to T, and H is also the isogonal conjugate of the line X(30)X(40) with respect to the excentral triangle. Also, H is the locus of P for which the P-Brocard triangle is perspective to ABC; see X(5642). X(5898) is the isogonal conjugate of X(5899) with respect to T. (Randy Hutson, July 7, 2014)
X(5898) lies on these lines: {3,2888}, {6,3200}, {25,2914}, {110,143}, {399,1154}, {539,2931}, {542,2916}, {2918,3519}
X(5899) = X(3) - 4X(23)
X(5899) = 12X(2) + (J2 - 12)X(3) where J = |OH|/R
X(5899) = 4R2X(2) + (3|OG|2 - 4R2)X(3)
(Peter Moses, July 11, 2014)
As a point on the Euler line, X(5899) has Shinagawa coefficients (3E + 8F, -11E - 8F).
X(5899) is the isogonal conjugate of X(5898) with respect to the tangential triangle, and X(5899) is the pole with respect to the circumcircle of the line X(140)X(523). (Randy Hutson, July 7, 2014)
X(5899) lies on these lines: {2,3}, {195,1614}, {399,1154}, {1533,2931}, {2918,3574}
X(5899) = isogonal conjugate of X(5900)
X(5899) = crossdifference of every pair of points on the line X(647)X(5421)
X(5899) = {X(3),X(23)}-harmonic conjugate of X(37923)
X(5899) = {X(15154),X(15155)}-harmonic conjugate of X(4)
X(5900) is the trilinear pole of the line X(647)X(5421), and X(5900) is the the antipode-in-Jerabek-hyperbola of X(1173). Also, X(5900) is the antigonal image of X(1173). (Randy Hutson, July 7, 2014)
X(5900) lies on these lines: {125,1173}, {146,3521}, {2889,3448}
X(5900) = isogonal conjugate of X(5899)
X(5900) = reflection of X(1173) in X(125)
As at X(5886), let A′ be the nine-point center of the triangle IBC, where I = X(1), and define B′ and C′ cyclically; then X(5901) = (X(5) of A′B′C′), as well as the complement of X(5) with respect to A′B′C′. Let A* be the circle with center A and diameter b + c, and define B* and C* cyclically; then X(5901) is the radical center of A*, B*, C*. Let A″ be the nine-point center of triangle IBC, and define B″ and C″ cyclically; then I, A″, B″, C″ comprise an orthocentric system whose common nine-point circle has center X(5901). (Hyacinthos #21518, February 10, 2013, and following posts by Antreas Hatzipolakis and Randy Hutson)
X(5901) lies on these lines: {1,5}, {2,1482}, {3,962}, {4,3622}, {8,1656}, {10,3628}, {30,551}, {40,549}, {104,5606}, {140,517}, {145,3090}, {381,944}, {382,5731}, {392,5771}, {476,953}, {498,2098}, {499,2099}, {515,546}, {516,548}, {519,547}, {550,3576}, {632,3624}, {912,5045}, {999,3485}, {1001,5762}, {1064,5453}, {1159,4323}, {1191,5707}, {1386,3564}, {1388,1478}, {1699,3627}, {3241,5055}, {3336,5298}, {3487,5811}, {3526,5550}, {3530,3579}, {3617,5067}, {3623,5056}, {3649,5563}, {3655,3845}, {3817,3850}, {3874,5694}, {3878,4999}, {4292,5126}, {4308,5714}, {4511,5178}, {5049,5777}, {5180,5303}, {5432,5697}, {5436,5812}, {5542,5843}
X(5901) = midpoint of X(1) and X(5)
X(5901) = {X(1),X(11)}-harmonic conjugate of X(37730)
X(5901) = {X(1),X(80)}-harmonic conjugate of X(37734)
X(5902) lies on these lines: {1,3}, {2,758}, {4,79}, {5,3649}, {6,1718}, {7,80}, {8,2891}, {10,3681}, {20,5441}, {43,1739}, {47,1451}, {58,3924}, {61,2306}, {63,4880}, {72,1698}, {81,1325}, {90,5665}, {145,3881}, {191,405}, {226,1737}, {244,995}, {355,5270}, {374,1743}, {381,2771}, {386,2650}, {392,3742}, {498,1788}, {499,3485}, {515,553}, {518,599}, {519,3873}, {551,3877}, {579,2294}, {584,2160}, {614,5315}, {631,5442}, {912,4654}, {938,1479}, {944,4317}, {950,1770}, {952,5434}, {960,3624}, {985,2224}, {993,3218}, {994,4850}, {997,3306}, {1002,2809}, {1012,1768}, {1046,1724}, {1051,2939}, {1068,1825}, {1071,5586}, {1125,3869}, {1210,3671}, {1254,4306}, {1464,5396}, {1656,5694}, {1717,2955}, {1725,1779}, {1790,4658}, {1836,3583}, {1837,3585}, {1876,1905}, {2280,5011}, {2362,3301}, {2392,3060}, {2800,5603}, {2802,3241}, {2842,5640}, {3244,3889}, {3296,5559}, {3419,5696}, {3474,3488}, {3475,5657}, {3476,5083}, {3486,4299}, {3501,3970}, {3555,3632}, {3586,4312}, {3616,3878}, {3617,3918}, {3622,3884}, {3634,3876}, {3635,3885}, {3636,3890}, {3752,5313}, {3792,4675}, {3828,4134}, {3922,4668}, {3940,4413}, {3962,5044}, {3968,4661}, {3980,5208}, {4002,4662}, {4116,4128}, {4414,4653}, {4645,4680}, {5432,5719}, {5435,5444}
X(5902) = midpoint of X(65) and X(354)
X(5902) = reflection of X(1) in X(354)
X(5902) = isogonal conjugate of X(15175)
X(5902) = X(1)-of-orthocentroidal-triangle
X(5902) = X(381)-of-intouch-triangle
X(5902) = {X(1),X(65)}-harmonic conjugate of X(5903)
X(5902) = anticomplement of X(10176)
X(5902) = X(2)-of-reflection-triangle-of-X(1)
X(5902) = Cundy-Parry Phi transform of X(35)
X(5902) = Cundy-Parry Psi transform of X(79)
X(5902) = homothetic center of X(1)-altimedial and X(1)-adjunct anti-altimedial triangles
X(5902) = X(7576)-of-excentral-triangle
X(5902) = {X(1),X(3)}-harmonic conjugate of X(37571)
X(5902) = endo-homothetic center of Ehrmann vertex-triangle and tangential triangle; the homothetic center is X(381)
X(5903) = {X(1),X(40)}-harmonic conjugate of X(35); X(5903) = {X(1),X(65)}- harmonic conjugate of X(5902). Let A′ be the isogonal conjugate of A with respect to triangle IBC, where I = X(1), and define B′ and C′ cyclically. Let A″ be the isogonal conjugate of A′ with respect to IB′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5903). X(5903) is the incenter of the triangle denoted by A″B″C″ at X(5905) and the triangle of the same notation at X(5906). (Randy Hutson, July 14, 2014)
Let PA be the reflection of X(1) in line BC, and define PB and PC cyclically; then X(5903) is the isogonal conjugate of X(1) with respect to PAPBPC. (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)
Let Ja be the A-excenter of the A-adjunct anti-altimedial triangle, and define Jb and Jc cyclically. The lines AJa, BJb, CJc concur in X(5903). (Randy Hutson, November 2, 2017)
Let A4B4C4 and A6B6C6 be the Gemini triangles 4 and 6, resp. Let LA and MA be the lines through A4 and A6, resp., parallel to BC. Define LB, LC, MB, MC cyclically. Let A′4 = LB∩LC and define B′4 and C′4 cyclically. Let A′6 = MB∩MC and define B′6 and C′6 cyclically. Triangles A′4B′4C′4 and A′6B′6C′6 are homothetic at X(5903). (Randy Hutson, November 30, 2018)
X(5903) lies on these lines: {1,3}, {2,3754}, {4,80}, {7,5559}, {8,79}, {10,908}, {12,5690}, {37,5036}, {41,5011}, {43,3987}, {63,5258}, {72,3679}, {78,4867}, {90,3577}, {145,2802}, {150,4056}, {169,5526}, {181,3944}, {191,958}, {203,2306}, {213,3959}, {214,4188}, {218,5540}, {219,1781}, {267,2948}, {355,1836}, {386,4642}, {392,3624}, {474,5289}, {495,3649}, {498,3485}, {499,1788}, {515,1770}, {518,3632}, {519,3868}, {551,3890}, {573,2171}, {579,1953}, {595,3924}, {631,5444}, {764,4083}, {912,4338}, {944,3474}, {946,1737}, {960,1698}, {962,1479}, {978,1739}, {984,1756}, {1000,5557}, {1012,1727}, {1046,1710}, {1068,1835}, {1100,4287}, {1111,3212}, {1122,4902}, {1125,3877}, {1148,1784}, {1210,4301}, {1376,5730}, {1393,1772}, {1411,2964}, {1464,5399}, {1572,5299}, {1717,1854}, {1724,3460}, {1743,2262}, {1759,2329}, {1797,4792}, {1829,4214}, {1837,3583}, {1838,1869}, {1858,5727}, {1871,1888}, {1872,1875}, {1902,1905}, {2170,4253}, {2176,3125}, {2295,3735}, {2362,3299}, {3179,5239}, {3208,3970}, {3216,4674}, {3218,4861}, {3241,3881}, {3244,3873}, {3476,4317}, {3486,4302}, {3488,4309}, {3555,3633}, {3582,3656}, {3584,3654}, {3616,3884}, {3617,3678}, {3622,3898}, {3623,3892}, {3626,3681}, {3635,3889}, {3698,5044}, {3724,5496}, {3740,4002}, {3751,3827}, {3833,5550}, {3872,4880}, {3897,5267}, {3913,5541}, {3962,4668}, {4127,4678}, {4134,4691}, {4153,4165}, {4304,5493}, {4513,5525}, {4646,5312}, {4857,5722}, {5046,5180}, {5250,5259}, {5252,5270}, {5253,5330}, {5435,5734}, {5694,5790}
X(5903) = reflection of X(i) in X(j) for these (i,j): (1,65), (5904,8)
X(5903) = isogonal conjugate of X(15446)
X(5903) = anticomplement of X(3878)
X(5903) = X(4)-of-reflection-triangle-of-X(1)
X(5903) = perspector of reflection triangle of X(1) and 2nd isogonal triangle of X(1)
X(5903) = Cundy-Parry Phi transform of X(36)
X(5903) = Cundy-Parry Psi transform of X(80)
X(5903) = X(6240)-of-excentral-triangle
X(5903) = {X(1),X(3)}-harmonic conjugate of X(37525)
X(5904) is the incenter of the triangle A*B*C* described at X(5905) and also the incenter of the triangle A*B*C* described at X(5906). Also, X(5904) = {(X(1),X(9)}-harmonic conjugate of X(5259), and X(5904) = {(X(1),X(72)}-harmonic conjugate of X(5692). (Randy Hutson, July 7, 2014)
X(5904) lies on these lines: {1,6}, {2,3678}, {8,79}, {10,3681}, {20,2801}, {35,63}, {36,78}, {38,386}, {40,912}, {43,3670}, {46,200}, {55,191}, {56,3940}, {58,976}, {65,3679}, {69,1930}, {80,3436}, {144,4294}, {145,3878}, {165,1071}, {210,942}, {281,1844}, {329,1479}, {354,3624}, {382,517}, {474,3337}, {484,5687}, {519,3869}, {527,1770}, {551,3889}, {579,3949}, {595,3938}, {651,4347}, {936,3338}, {978,3953}, {982,3216}, {986,3293}, {997,3984}, {1046,3961}, {1066,2318}, {1125,3873}, {1158,5537}, {1282,2939}, {1376,3336}, {1482,5694}, {1697,1858}, {1699,5777}, {1756,4073}, {1759,3684}, {2093,4882}, {2340,4303}, {2771,5541}, {2774,4088}, {2802,3621}, {3057,3633}, {3059,4312}, {3149,5536}, {3189,4302}, {3218,4420}, {3219,5248}, {3241,3884}, {3244,3877}, {3419,3585}, {3501,4006}, {3579,3689}, {3616,3881}, {3617,3754}, {3622,3892}, {3623,3898}, {3626,4084}, {3635,3890}, {3666,5312}, {3697,3812}, {3711,5221}, {3730,3930}, {3735,3780}, {3740,4533}, {3742,4539}, {3746,3870}, {3753,4662}, {3831,4090}, {3833,4547}, {3875,4523}, {3916,5010}, {3919,4691}, {4018,4668}, {4188,4973}, {4251,5282}, {4292,5850}, {4309,5698}, {4388,4894}, {4413,5708}, {4423,5506}, {4641,5266}, {4658,5311}, {4678,4757}, {5270,5794}, {5445,5552}
X(5904) = reflection of X(i) in X(j) for these (i,j): (1,72), (5903,8)
X(5904) = anticomplement of X(3874)
Let A′B′C′ be the orthic triangle, and let LA be the reflection of line B′C′ in the internal bisector of angle A, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC at X(57), homothetic to the medial triangle at X(908), and to the anticomplementary triangle at X(5905). Let MA be the reflection of the line B′C′ in the external bisector of angle A, and define MB and MC cyclically. Let A* = MB∩MC, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC at X(9), homothetic to the medial triangle at X(5249), and to the anticomplementary triangle at X(5905). (Randy Hutson, July 7, 2014)
X(5905) lies on these lines: {2,7}, {4,912}, {6,3782}, {8,79}, {10,3951}, {20,5758}, {21,3487}, {46,5552}, {65,3436}, {69,321}, {72,377}, {75,4886}, {78,4190}, {81,4644}, {92,1947}, {100,3474}, {145,515}, {149,152}, {192,3151}, {193,1839}, {200,4312}, {239,5813}, {278,651}, {281,445}, {306,3729}, {312,320}, {345,3936}, {355,4018}, {388,3869}, {390,3957}, {442,3927}, {443,3876}, {481,3084}, {482,3083}, {497,3873}, {516,3870}, {518,1836}, {529,2099}, {535,3241}, {537,4865}, {554,5240}, {938,5046}, {940,4415}, {942,2478}, {944,5841}, {958,3649}, {993,3616}, {1004,1260}, {1046,5230}, {1056,3877}, {1058,3889}, {1068,3157}, {1071,5812}, {1081,5239}, {1086,4383}, {1210,5187}, {1211,4363}, {1215,4655}, {1329,5221}, {1331,1754}, {1351,2969}, {1479,3874}, {1532,2095}, {1621,3475}, {1707,3011}, {1750,1998}, {1770,3811}, {1797,4080}, {1851,3060}, {2476,5714}, {2550,3681}, {2886,5852}, {2975,3485}, {2999,4862}, {3091,5811}, {3175,4851}, {3210,4440}, {3583,3894}, {3585,3901}, {3617,5815}, {3663,5256}, {3664,4656}, {3715,3826}, {3742,4679}, {3751,3914}, {3772,4641}, {3816,4860}, {3920,4307}, {3925,5220}, {3962,5794}, {3970,3995}, {4001,4054}, {4187,5708}, {4189,5703}, {4293,4511}, {4387,4966}, {4416,5271}, {4419,5712}, {4438,4892}, {4463,5800}, {4666,5542}, {4847,5850}, {5086,5229}, {5154,5704}, {5289,5434}
X(5905) = isogonal conjugate of X(2164)
X(5905) = isotomic conjugate of X(2994)
X(5905) = anticomplement of X(63)
X(5905) = pole wrt polar circle of trilinear polar of X(7040)
X(5905) = X(48)-isoconjugate (polar conjugate) of X(7040)
Let A′B′C′ be the circumorthic triangle, and let LA be the reflection of line B′C′ in the internal bisector of angle A, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC at X(3075) and homothetic to the anticomplementary triangle at X(5906). Let MA be the reflection of the line B′C′ in the external bisector of angle A, and define MB and MC cyclically. Let A* = MB∩MC, and define B* and C* cyclically. The triangle A*B*C*' is homothetic to ABC at X(3074) and to homothetic to the anticomplementary triangle at X(5906). (Randy Hutson, July 7, 2014)
X(5906) lies on these lines: {2,255}, {8,79}, {69,349}, {78,1448}, {651,5125}, {860,3157}, {962,2817}, {1259,3936}, {1788,2406}, {3868,5081}
Let A′B′C′ be the half-altitude triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of the line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5). Let A* be the trilinear pole, with respect to A′B′C′, of the line B″C″, and define B* and C* cyclically. The lines A′A*, B′B*, C′C* concur in X(5907). Also, X(5907) is the center of the conic described at X(5777) for P = X(69). In this case, triangle HAHBHC is perspective to ABC at X(64). (Randy Hutson, July 7, 2014)
X(5907) lies on the Burek-Hutson central cubic (K645) and these lines: {2,185}, {3,64}, {4,69}, {5,389}, {10,2807}, {20,3917}, {30,1216}, {40,3781}, {51,3091}, {52,381}, {84,3784}, {113,1209}, {114,130}, {140,5663}, {141,2883}, {143,3850}, {155,578}, {182,1181}, {235,343}, {373,5056}, {378,1092}, {394,1593}, {546,1154}, {550,5447}, {568,3851}, {916,942}, {970,3149}, {1071,2808}, {1147,4550}, {1204,5651}, {1364,1935}, {1568,1594}, {2979,3146}, {3060,3832}, {3523,5650}, {3545,3567}, {3574,5133}, {4260,5706}, {5068,5640}, {5777,5908}
X(5907) = reflection of X(389) in X(5)
X(5907) = complement of X(185)
X(5907) = X(4)-of-X(5)-Brocard-triangle
X(5907) = anticomplement of X(9729)
X(5908) is the center of the conic described at X(5777) for P = X(189). In this case, triangle HAHBHC is perspective to ABC at X(40). (Randy Hutson, July 7, 2014)
X(5908) lies on the Burek-Hutson central cubic (K645) and these lines: {1,3}, {4,189}, {5,5909}, {222,1753}, {282,2262}, {971,1872}, {1364,1887}, {1535,5174}, {5777,5907}
X(5908) = reflection of X(5909) in X(5)
X(5909) is the center of the conic described at X(5777) for P = X(329). In this case, triangle HAHBHC is perspective to ABC at X(3345). (Randy Hutson, July 7, 2014)
X(5909) lies on the Burek-Hutson central cubic (K645) and these lines: {3,223}, {4,8}, {5,5908}, {389,942}, {960,2817}, {2262,5715}, {2270,5709}
X(5909) = reflection of X(5908) in X(5)
X(5910) is the center of the conic described at X(5777) for P = X(1032). In this case, triangle HAHBHC is perspective to ABC at X(1498). (Randy Hutson, July 7, 2014)
X(5910) lies on the Burek-Hutson central cubic (K645) and these lines: {3,64}, {4,1032}, {3079,5562}
X(5911) is the center of the conic described at X(5777) for P = X(1034). In this case, triangle HAHBHC is perspective to ABC at X(1490). (Randy Hutson, July 7, 2014)
X(5911) lies on the Burek-Hutson central cubic (K645) and these lines: {3,9}, {4,1034}
X(5912) is the point QA-P2 center of quadrangle X(13)X(14)X(15)X(16); see Euler-Poncelet Point.
X(5912) is the point common to the nine-points circles of these 4 triangles: X(14)X(15)X(16), X(13)X(15)X(16), X(13)X(14)X(16), X(13)X(14)X(15); also, X(5912) is the center of the rectangular hyperbola that passes through the points X(13), X(14), X(15), X(16). (Randy Hutson, July 7, 2014)
Let O(13,15) be the circle with segment X(13)X(15) as diameter (and center X(396)), and let O(14,16) be the circle with segment X(14)X(16) as diameter (and center X(395)); then X(5912) is the radical trace of O(13,15) and O(14,16). (Randy Hutson, August 17, 2014)
X(5912) lies on these lines: {2,6}, {98,843}, {111,523}
X(5912) = reflection of X(i) in X(j) for these (i,j): (5913,230), (111,5914)
X(5913) is the QA-P3 center of the quadrangle X(13)X(14)X(15)X(16); see Gergonne-Steiner Point.
X(5913) = inverse-in-{circumcircle, nine-point circle}-inverter of X(6); see X(5577) for the definition of inverter.
The {circumcircle, nine-point circle}-inverter is the orthopic circle of the Steiner inscribed ellipse; its center is X(2), its radius is [(a2 + b2 + c2)/18]1/2, and the powers of A,B,C with respect to this circle are (-a2 + b2 + c2)/6, (a2 - b2 + c2)/6, (a2 + b2 - c2)/6. (Peter Moses, July 16, 2014)T
X(5913) lies on these lines: {2,6}, {23,2079}, {30,111}, {112,468}, {115,858}, {403,1560}, {843,1302}, {1499,1513}, {2030,5642}
X(5913) = reflection of X(5912) in X(230)
X(5913) = isogonal conjugate of X(6096)
X(5913) = complement of X(5971)
X(5914) is the QA-P6 center of the quadrangle X(13)X(14)X(15)X(16); see Parabola Axes Crosspoint.
X(5914) lies on these lines: {30,115}, {111,523}
X(5914) = midpoint of X(111) and X(5912)
X(5915) is the QA-P23 center of the quadrangle X(13)X(14)X(15)X(16); see Inscribed Square Axes Crosspoint
X(5915) is the centroid of the trapezoid X(2378)X(5916)X(2379)X(5917), which is similar to and orthogonal to the trapezoid X(13)X(15)X(14)X(16), with similitude center X(111); see X(5916). (Randy Hutson, July 7, 2014)
X(5915) lies on these lines: {30,115}, {98,843}, {111,477}
X(5916) lies on these lines: {14,530}, {98,2379}, {523,2378}, {690,5917}
X(5916) = isogonal conjugate of X(15) with respect to the triangle X(13)X(14)X(16)
X(5916) = 2nd-Parry-to-ABC similarity image of X(13)
X(5917) lies on these lines: {13,531}, {98,2378}, {523,2379}, {690,5916}
X(5917) = isogonal conjugate of X(16) with respect to the triangle X(13)X(14)X(15); see X(5915)
X(5917) = 2nd-Parry-to-ABC similarity image of X(14)
The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).
X(5918) lies on these lines: {3,1709}, {20,65}, {55,5732}, {57,2951}, {63,3059}, {84,5584}, {165,210}, {170,241}, {354,516}, {376,6001}, {411,1776}, {517,3534}, {548,5887}, {940,1721}, {960,3522}, {990,3745}, {1040,1456}, {1155,1708}, {1407,4319}, {1427,3000}, {1742,3666}, {1750,4413}, {3057,4297}, {3146,3812}, {3555,5493}, {3689,6244}, {3698,5691}, {4640,5784}, {5731,5919}
X(5918) = reflection of X(i) in X(j) for these (i,j): (210,165), (5919,5731)
The Hutson-intouch and Hutson-extouch triangles are defined at X(5731).
X(5919) lies on these lines: {1,3}, {2,3880}, {8,3740}, {10,3893}, {12,3817}, {37,374}, {72,3244}, {145,960}, {210,392}, {226,4342}, {390,3476}, {497,5252}, {515,3058}, {516,5434}, {518,1992}, {551,2802}, {950,5927}, {956,3683}, {997,3689}, {1001,3872}, {1056,1836}, {1058,1837}, {1100,2267}, {1122,3672}, {1125,3698}, {1149,3752}, {1201,4646}, {1317,2801}, {1320,2346}, {1358,3663}, {1376,3895}, {1483,5887}, {1864,3488}, {1898,3486}, {2256,2264}, {3421,4679}, {3555,3635}, {3616,3848}, {3621,4662}, {3622,3812}, {3623,3869}, {3625,3697}, {3632,5044}, {3636,3922}, {3649,4301}, {3696,3902}, {3870,5289}, {3881,4018}, {3921,4669}, {4009,4737}, {4870,5603}, {5731,5918}
X(5919) = reflection of X(i) in X(j) for these (i,j): (354,1), (5918,5731)
The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).
The reciprocal orthologic center is X(3555). (Randy Hutson, November 2, 2017)
X(5920) lies on these lines: {1,7160}, {1697,12658}, {3601,12439}, {9785,9804}, {9953,10866}
Darboux images are discussed at X(5881).
X(5921) lies on these lines: {2,98}, {3,3620}, {4,193}, {6,3091}, {20,64}, {68,3089}, {141,3523}, {153,5848}, {381,1353}, {511,3146}, {524,3543}, {546,5093}, {611,5261}, {613,5274}, {962,5847}, {1992,3839}, {2888,5596}, {3090,5050}, {3580,4232}, {3618,5056}, {3619,5085}, {3818,3832}
X(5921) = reflection of X(20) in X(69)
X(5921) = X(20)-of-obverse-triangle-of-X(69)
Darboux images are discussed at X(5881).
X(5922) lies on these lines: {20,64}, {122,1073}, {154,459}
X(5922) = reflection of X(64) in X(253)
Darboux images are discussed at X(5881).
X(5923) lies on these lines: {8,20}, {282,5514}, {1256,1837}
X(5923) = reflection of X(84) in X(189)
Darboux images are discussed at X(5881).
X(5924) lies on these lines: {4,2093}, {9,119}, {20,78}, {57,5715}, {84,5812}, {226,2096}, {2095,5735}, {2800,3586}
X(5924) = reflection of X(1490) in X(329)
Darboux images are discussed at X(5881).
X(5925) lies on these lines: {3,113}, {4,1192}, {20,394}, {30,64}, {154,550}, {221,4302}, {376,2883}, {382,1853}, {599,2892}, {1503,3529}, {1514,3147}, {1620,3542}, {1770,1854}, {2192,4299}, {3146,3580}
X(5925) = reflection of X(1498) in X(20)
Barycentrics a2(b2 - c2)(2a6 - 4a4b2 -
4a4c2 + 2a2b4 + 2a2c4 + a2b2c2 - b4c2 - b2c4) : :
X(5926) = 7 X[3523] + X[31299], 7 X[3526] - 5 X[31279], 3 X[5054] - X[31176], X[11616] + 3 X[39214], 3 X[15724] - X[19912], 3 X[39228] - 4 X[39477], X[39228] - 4 X[39481], X[39477] - 3 X[39481]
The circumcircle-inverse of a line is a circle; if the line does not pass through X(3), then the radius of the circle is finite. The appearance if (i,j) in the following list means that X(i) is on the line X(2)X(6) and that X(j) is the inverse of X(i): (2,23), (6,187), (69, 5866), (81,5867), (86,5937), (141,5938), (183, 5939), (193,5940), (230,5941), (352,353), (524,3), (5108,669).
X(5926) lies on these lines: {2, 39511}, {3, 669}, {24, 2501}, {25, 39533}, {32, 39518}, {140, 23301}, {182, 9009}, {351, 9128}, {512, 6132}, {523, 7575}, {525, 34952}, {549, 25423}, {574, 39501}, {924, 8552}, {1658, 8151}, {1995, 39492}, {2780, 8644}, {3523, 31299}, {3526, 31279}, {3800, 39201}, {4897, 39578}, {5054, 31176}, {6563, 7488}, {6642, 14341}, {7669, 14657}, {8651, 11615}, {9494, 30217}, {11186, 37184}, {14809, 37813}, {15724, 19912}, {16235, 32472}, {16678, 39545}, {19357, 30451}, {19543, 25537}, {34513, 39495}, {41357, 44272}
X(5926) = midpoint of X(3) and X(669)
X(5926) = reflection of X(i) in X(j) for these {i,j}: {6563, 32204}, {11615, 8651}, {23301, 140}, {32231, 9126}
X(5926) = anticomplement of X(39511)
X(5926) = crossdifference of every pair of points on line {566, 3291}
The classical extouch triangle is regarded as the 1st extouch triangle. The 2nd, 3rd, 4th, and 5th extouch triangles are defined by Randy Hutson (July 11, 2014) as follows. Let AA, AB, AC be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define BB, BC, BA and CC, CA, CB cyclically.
Let A1 = AA, B1 = BB, C1 = CC; 1st extouch triangle = A1B1C1
Let A2 = BCBA∩CACB, and define B2 and C2 cyclically; 2nd extouch triangle = A2B2C2
Let A3 = CAAC∩ABBA, and define B3 and C3 cyclically; 3rd extouch triangle = A3B3C3
Let A4 = BCCA∩BACB, and define B4 and C4 cyclically; 4th extouch triangle = A4B4C4
Let D1 = BBBC∩CCCA, and define D2 and D3 cyclically. Let E1 = BBBA∩CCCB, and define E2 and E3 cyclically. Define A5 = D2E2∩D3E3, and define B5 and C5 cyclically; 5th extouch triangle = A5B5C5
Barycentric coordinates for A-vertices of the the five
triangles:
A1 = 0 : a - b + c : a + b - c
A2 = 2a(b + c) : - a2 - b2 +
c2 : - a2 + b2 - c2
A3 = 2a(b + c)(a - b + c)(a + b - c) : (a + b + c)(a - b -
c)(a2 + b2 - c2) : (a + b + c)(a - b -
c)(a2 - b2 + c2)
A4 = 2a(b + c)(a + b + c) : (a - b - c)(a2 -
b2 + c2) : (a - b - c)(a2 +
b2 - c2)
A5 = 2a(b + c)(a - b + c)(a + b - c) : (a - b - c)(a + b -
c)[b2 + (a + c)2] : (a - b - c)(a - b +
c)[c2 + (a + b)2]
A2B2C2 is perspective to ABC and
A3B3C3 at X(4).
A2B2C2 is homothetic to the excentral
triangle at X(9).
A2B2C2 is homothetic to the intouch
triangle at X(226).
A2B2C2 is perspective to the extouch
triangle and (extraversion triangle of X(65)) at X(72).
A2B2C2 is perspective to the
anticevian triangle of X(8) at X(329).
A2B2C2 is homothetic to the hexyl
triangle at X(1490).
A2B2C2 is perspective to the Feuerbach
triangle at X(442).
A2B2C2 is perspective to
A4B4C4 at X(5928).
A2B2C2 is homothetic to the inner
Hutson triangle at X(5934).
A2B2C2 is homothetic to the outer
Hutson triangle at X(5935).
A2B2C2 is homothetic to the 2nd
circumperp triangle at X(405).
A2B2C2 is homothetic to the
inverse-in-incircle triangle (see X(5571) at X(5728).
A2B2C2 is homothetic to the
Hutson-intouch triangle at X(950).
A2B2C2 is perspective to the
Hutson-extouch triangle at X(442).
In the following list, the appearance of (i,j) means that (X(i) of
the 2nd extouch triangle) = X(j):
(3,4), (4,72), (5,5777), (6,9), (25,329), (26,5812), (54, 442),
(184,226), (185,950), (195,3651), (578, 10), (647,1635), (1181,1)
A3B3C3 is perspective to the
intouch triangle and (extraversion triangle of X(65)) at X(1439).
A3B3C3 is perspective to the extouch
triangle and A5B5C5 at X(5930).
A3B3C3 is perspective to
A4B4C4 at X(5929).
A3B3C3 is perspective to the
anticevian triangle of X(7) at X(5932).
A4B4C4 is perspective to ABC at
X(69).
A4B4C4 is perspective to the intouch
triangle and A5B5C5 at X(65).
A4B4C4 is perspective to the
anticevian triangle of X(7) at X(5933).
A5B5C5 is perspective to ABC at
X(388).
A5B5C5 is perspective to the
anticevian triangle of X(7) at X(8).
X(5927) is also the centroid of the triangle formed by the polars of the incenter with respect to the excircles. (Randy Hutson, July 11, 2014) Note added (11/25/2015): for more about this triangle, named the Atik triangle, see the preamble to X(8580).
X(5927) lies on these lines: {2,971}, {3,3305}, {4,8}, {5,1071}, {9,165}, {11,118}, {12,1898}, {20,5044}, {40,3697}, {44,1754}, {51,916}, {57,5729}, {63,5779}, {84,474}, {210,516}, {374,1903}, {377,6259}, {381,912}, {392,515}, {405,1490}, {442,6260}, {443,6223}, {518,1699}, {942,3091}, {946,3555}, {950,5919}, {960,5691}, {990,4383}, {1012,5440}, {1214,2635}, {1385,5284}, {1427,1736}, {1853,3753}, {2478,5787}, {2808,5943}, {3146,3876}, {3149,3916}, {3452,5784}, {3487,5049}, {3752,5400}, {3832,3868}, {3921,5657}, {4002,5818}, {4015,5493}, {4018,5693}, {4187,6245}, {5435,5825}, {5805,5905}
X(5927) = reflection of X(i) in X(j) for these (i,j): (165,3740), (354,3817), (3753,5587)
X(5927) = complement of X(11220)
X(5928) is also the perspector of the 2nd (and 4th) extouch triangle and the polar triangle of the Yiu conic; see X(5927).
X(5928) lies on these lines: {2,2182}, {4,65}, {6,1848}, {9,440}, {33,1503}, {69,189}, {222,226}, {225,5786}, {355,6358}, {1038,1490}, {1753,6247}, {1824,1899}, {1853,1861}, {4329,5739}, {5307,6354}
X(5928) = 2nd-extouch-isogonal conjugate of X(12689)
See X(5927).
X(5929) lies on these lines: {4,69}, {65,1439}, {77,851}, {284,940}, {1211,3452}, {2898,5932}
X(5929) = perspector of [cross-triangle of ABC and 3rd extouch triangle] and [cross-triangle of ABC and 4th extouch triangle]
See X(5927).
X(5930) lies on these lines: {1,4}, {8,253}, {10,227}, {20,1394}, {40,3182}, {57,387}, {65,1439}, {77,377}, {109,1294}, {208,1763}, {221,516}, {222,4292}, {603,1754}, {610,1249}, {851,1410}, {1042,3914}, {1074,4303}, {1210,1465}, {1400,2082}, {1419,3332}, {1427,1834}, {1455,4297}, {1456,6284}, {1467,4000}, {1612,2078}, {1630,2202}, {1714,3911}, {1935,2328}, {3682,4551}, {3710,4552}, {3987,4848}, {4304,6357}
X(5930) = isotomic conjugate of X(5931)
X(5930) = X(8)-Ceva conjugate of X(65)
X(5931) is the trilinear pole of the perspectrix of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). (Randy Hutson, July 11, 2014) See X(5927).
X(5931) lies on these lines: {20,64}, {75,1895}
X(5931) = isotomic conjugate of X(5930)
X(5932) is the perspector of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). Also, X(5932) is the perspector of ABC and the pedal triangle of X(3182), as well as the perspector of ABC and the triangle obtained by reflecting the pedal triangle of X(223) in X(223). (Randy Hutson, July 11, 2014)
X(5932) lies on the Lucas cubic and these lines: {2,77}, {4,7}, {8,253}, {20,3182}, {57,5802}, {69,1034}, {151,4329}, {269,1210}, {329,1032}, {411,1804}, {1264,4554}, {1442,5703}, {1443,5704}, {1445,2270}, {2062,6060}, {2898,5929}, {3086,4341}
X(5932) = isotomic conjugate of X(1034)
X(5932) = complement of X(20212)
X(5932) = anticomplement of X(282)
X(5932) = X(69)-Ceva conjugate of X(7)
X(5932) = perspector of 3rd extouch triangle and cross-triangle of ABC and 3rd extouch triangle
X(5933) lies on these lines: {7,8}, {57,3879} et al
X(5933) = perspector of 4th extouch triangle and cross-triangle of ABC and 4th extouch triangle
X(5934) = X(222)-of-2nd-extouch-triangle. (Inner Hutson triangle is defined at X(363).)
X(5934) lies on these lines: {9,363}, {503,1750}
X(5935) = X(219)-of-2nd-extouch-triangle. (The outer Hutson triangle is defined at X(363).)
X(5935) lies on these lines: {9,164}, {503,1750}
Let A21B21C21 be the Gemini triangle 21. Let LA be the line through A21 parallel to BC, and define LB and LC cyclically. Let A′21 = LB∩LC, and define B′21, C′21 cyclically. Triangle A′21B′21C′21 is homothetic to ABC at X(5936). (Randy Hutson, November 30, 2018)
X(5936) lies on these lines: {2,2321}, {7,10}, {8,86}, {27,281}, {75,3701}, {272,4313}, {310,3596}, {335,4699}, {594,5308}, {673,2345}, {903,5224}, {936,306}, {966,6172}, {1002,4111}, {1088,1441}, {1440,3160}, {1698,3672}, {3241,5564}, {3616,4460}, {3679,3945}, {3879,4678}, {4357,4373}, {4360,5550}, {4409,4748}, {4461,5257}, {4472,5839}, {4677,4909}
X(5936) = trilinear pole of the line X(514)X(1635) (which is the Lemoine axis of the 2nd extouch triangle)
X(5937) lies on these lines: {3,86}, {669,4367}
X(5938) lies on these lines: {3,66}, {25,5523}, {353,3148}, {525,669}, {755,2715}
X(5939) = X(187)-of-circummedial triangle. Let A′B′C′ be the triangle of which ABC is the 1st Brocard triangle; here called the 1st anti-Brocard triangle, with circumcircle called the anti-Brocard circle; then X(5939) = X(187)-of-A′B′C′, and X(5939) = inverse-in-anti-Brocard-circle of X(99). (Randy Hutson, July 18, 2014)
Barycentrics for the vertices of the 1st anti-Brocard triangle are as follows (Peter Moses, August 21, 2014):
A′ = a4 - b2c2 : c4 -
a2b2 : b4 -
a2c2
B′ = c4 - b2a2 : b4 -
c2a2 : a4 -
b2c2
C′ = b4 - c2a2 : a4 -
c2b2 : c4 -
a2b2
For more properties of A′B′C′, see X(5976).
X(5939) lies on these lines: {2,353}, {3,76}, {147,1007}, {187,543}, {325,542}, {385,5104}, {669,804}, {671,3972}, {2023,3329}
X(5940) lies on these lines: {3,193}, {353,5585}
X(5941) lies on these lines: {3,230}, {25,669}
X(5942) lies on these lines: {2,77}, {7,4858}, {8,144}, {63,3686}, {69,1229}, {92,1947}, {281,651}, {329,2893}, {894,5554}, {1654,3152}, {3416,3436}
The locus of the centroid of the pedal triangle of P as P varies around the nine-point circle is an ellipse with center X(5943). Also, X(5943) is the centroid of the pedal triangle of X(5), as well as the centroid of the 6 points of intersection of the nine-point circle and the sidelines of ABC. (Randy Hutson, July 18, 2014). The ellipse is here named the Hutson centroidal ellipse.
Let A′B′C′ be any one of the following nine triangles: medial, 1st Brocard, McCay, 1st Neuberg, 2nd Neuberg, inner Napoleon, outer Napoleon, inner Vecten, outer Vecten; and let Ba and Ca be the orthogonal projections of B′ and C′ on BC, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. X(5943) is the centroid of {Ba, Ca, Cb, Ab, Ac, Bc}. (Randy Hutson, March 25, 2016)
If you have The Geometer's Sketchpad, you can view X(5943), with the Hutson centroidal ellipse.
X(5943) lies on these lines: {2,51}, {5,389}, {6,1196}, {22,5092}, {23,5643}, {25,182}, {30,5892}
X(5943) = midpoint of X(i) and X(j) for these (i,j): (2,51), (5,5946)
X(5943) = complement of X(3917)
X(5943) = {X(51),X(373)}-isoconjugate of X(2)
Let d be a line tangent to the nine-point circle of a triangle ABC. Let DA be the reflection of d in line BC, and define DB and DC cyclically. Let XYZ be the triangle formed by the lines DA, DB, DC. The envelope of the circumcircle of the variable triangle XYZ is a circle. (Tran Quang Hung, ADGEOM #1387, July 9, 2014). The circle is here named the Hung circle.
The diameter of the Hung circle is the segment X(3)X(1614), so that X(5944) is the midpoint of this segment. The radius of the circle is (R/2)(3 + k)/(1 + k), where k = 2(cos 2A + cos 2B + cos 2C). The lines AX, BY, CZ concur in a point Q = Q(P) on the circumcircle of ABC. The appearance of (i,j) in the following list means that X(j) = Q(X(i)): (11,953), (113,477), (114,2698), (115,2698), (116,2724), (117,2734), (118,2724), (119,953), (124,2734), (125,477), (1312,74), (1313,74), (2039,98), (2040,98). (César Lozada, ADGEOM #1388 et al, July 9, 2014)
The triangles XYZ form a family of similar triangles, and Q is the incenter of XYZ. If P = p : q : r is a point on the nine-point circle, then
Q = Q(P) = a2/[p(v + w) - u(q + r)] : b2/[q(w + u) - v(r + p)] : c2/[r(u + v) - w(p + q)],
where u : v : w = X(5). (Peter Moses, August 1, 2014)
If ABC is acute, then Q is the incenter of XYZ, and XYZ has the orientation opposite that of ABC. Let J = |OH|/R and σ = area(ABC). Maximal area(XYZ) = σ(J - 1)/(J + 1) occurs with P = X(1312) and minimal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313). If ABC is not acute, then Q is an excenter of XYZ, and XYZ has the same orientation as ABC. In this case, maximal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313) and minimal area(XYZ) = 0, which occurs when angle X(5)-to-P-to-X(4) is a right angle and XYZ is a one of the points of intersection of the circumcircle and the Hung circle. Also, there is a local maximum when P = X(1312), and in this case, area(XYZ) = σ(J - 1)/(J + 1). (Peter Moses, August 9, 2014)
Let A′ be the reflection of X(5) in line BC, and define B′ and C′ cyclically. Let A″ be the circumcenter of triangle BCA′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5944). (Randy Hutson, August 17, 2014)
If you have The Geometer's Sketchpad 5, you can view X(5944).
X(5944) lies on these lines: {3,74}, {24,5946}, {49,1154}, {52,1493}, {54,143}, {184,1658}, {546,1495}, {567,3518}, {3146,3431}
The family of Hofstadter ellipses are introduced at X(359) and further described at MathWorld. The ellipses are indexed as E(r) for 0 <= r <= 1, and E(1 - r) = E(r). Thus, the Hofstadter 0-ellipse and the Hofstadter 1-ellipse are identical. (Submitted by Valery Nemychnikova, Moscow Chemical Lyceum, July 28, 2014.)
X(5945) = X(2)-Ceva conjugate of X(359)
X(5946) lies on these lines: {2,568}, {3,143}, {4,3521}, {5,389}, {6,1511}, {24,5944}, {26,3796}, {30,51}, {49,1199}, {52,140}, {185,546}, {186,567}, {373,547}, {378,1112}, {381,5640}, {511,549}, {550,5446}, {569,973}, {632,1216}, {970,5428}, {974,1539}, {1147,1493}, {1656,5889}, {1995,5609}, {2070,5012}, {2781,5476}, {2979,5054}, {3628,5562}
X(5946) = midpoint of X(2) and X(568)
X(5946) = reflection of X(5) in X(5943)
X(5946) = orthocentroidal-circle-inverse of X(38724)
X(5946) = X(2)-of-reflection-triangle-of-X(5)
X(5947) is the Feuerbach-isogonal conjugate of X(5949); i.e., the hisogonal-conjugate-with-respect-to-Feuerbach-triangle of X(5949).
X(5947) lies on these lines: {3,31750}, {4,31764}, {5,5948}, {10,31756}, {11,10276}, {52,31754}, {119,10277}, {442,10209}, {946,31759}, {952,10281}, {3614,5953}, {5562,31765}, {14866,31761}, {38109,44847}
X(5948) is the Feuerbach-isogonal conjugate of X(5) and also the anticomplement of X(5) with respect to the Feuerbach triangle. (Randy Hutson, August 5, 2014)
X(5948) lies on these lines: {3,31764}, {4,31750}, {5,5947}, {10,31759}, {11,10277}, {12,79}, {30,10209}, {52,31765}, {119,5953}, {946,31756}, {5164,14132}, {5562,31754}, {10276,23513}, {12506,31761}
X(5949) is the Feuerbach-isogonal conjugate of X(5947).
X(5949) lies on these lines: {2,1029}, {5,572}, {6,2476}, {9,46}, {12,594}, {37,115}, {338,1441}, {1030,2475}, {1834,5725}, {2908,3136}, {3841,4047}
X(5950) lies on the nine-point circle and these lines: {2,5951}, {4,5606}, {5,5952}, {11,79}
X(5950) = reflection of X(5952) in X(5)
X(5950) = complement of X(5951)
X(5951) lies on the circumcircle and these lines: {2,5950}, {3,5606}, {4,5952}, {100,3648}, {110,3579}
X(5951) = reflection of X(5606) in X(3)
X(5951) = anticomplement of X(5950)
X(5951) = cevapoint of X(35) and X(484)
X(5952) lies on the nine-point circle and these lines: {2,5606}, {4,5951}, {5,5950}, {119,3652}
X(5952) = reflection of X(5950) in X(5)
X(5952) = complement of X(5606)
Let A′B′C′ be the Feuerbach triangle. Let A″ be the isogonal conjugate of A with respect to A′B′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5953). (Randy Hutson, August 5, 2014)
X(5953) lies on these lines: {5,191}, {119,5948}, {3614,5947}
Let A′B′C′ be the Feuerbach triangle, L the line through A and X(11), and A″ = L∩B′C′; and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5954). See also X(3614) and X(3615). (Randy Hutson, August 5, 2014)
Continuing, let A* be the trilinear pole, with respect to A′B′C′, of line BC, and define B* and C* cyclically. Let A** be the trilinear pole, with respect to A′B′C′, of line B*C*, and define B** and C** cyclically. The lines A′A**, B′B**, C′C** concur in X(5954). (Randy Hutson, August 5, 2014)
X(5954) lies on these lines: {5,2607}, {11,523}, {12,59}
In the configuration for X(5213), the circle internally tangent to the circles (KA), (KB), (KC) is here named the inner Hung circle.
If you have The Geometer's Sketchpad, you can view X(5955).
X(5955) lies on these lines: {2,3702}, {3,10}, {46,1211}, {171,5814}, {899,1245}, {975,3704}, {997,5835}, {1010,5725}, {1574,2092}, {1698,3712}, {2049,5530}, {3454,5880}, {3687,5711}, {3695,5268}, {3696,5292}, {3927,4104}, {3966,5264}
In the configuration for X(5213), the circle externally tangent to the circles (KA), (KB), (KC) is here named the outer Hung circle; see X(5955) and X(5213).
If you have The Geometer's Sketchpad, you can view X(5955), which includes X(5956).
X(5956) lies on these lines: {2,3702}, {171,1203}, {386,1100}, {404,593}, {970,5213}, {1575,5044}
For I = 5957, 5958, 5959, 5948, 5949, 5953, X(i) is the isogonal conjugate of a point with respect to the Feuerbach triangle, called a Feuerbach isogonal conjugate.
Follwing is the first barycentric of the Feuerbach isogonal conjugate of an arbitrary point p : q : r. This lengthy function of a,b,c,p,q,r (degree 11 in a,b,c and degree 2 in p,q,r) can be cut-and-pasted into Mathematica. (Peter Moses, August 10, 2014)
(a^9 b^2+a^8 b^3-4 a^7 b^4-4 a^6 b^5+6 a^5 b^6+6 a^4 b^7-4 a^3 b^8-4 a^2 b^9+a b^10+b^11+2 a^9 b c+3 a^8 b^2 c-6 a^7 b^3 c-12 a^6 b^4 c+4 a^5 b^5 c+16 a^4 b^6 c+2 a^3 b^7 c-8 a^2 b^8 c-2 a b^9 c+b^10 c+a^9 c^2+3 a^8 b c^2-8 a^7 b^2 c^2-16 a^6 b^3 c^2+3 a^5 b^4 c^2+15 a^4 b^5 c^2+9 a^3 b^6 c^2+a^2 b^7 c^2-5 a b^8 c^2-3 b^9 c^2+a^8 c^3-6 a^7 b c^3-16 a^6 b^2 c^3+2 a^5 b^3 c^3+23 a^4 b^4 c^3+14 a^3 b^5 c^3+5 a^2 b^6 c^3-3 b^8 c^3-4 a^7 c^4-12 a^6 b c^4+3 a^5 b^2 c^4+23 a^4 b^3 c^4+18 a^3 b^4 c^4+6 a^2 b^5 c^4+4 a b^6 c^4+2 b^7 c^4-4 a^6 c^5+4 a^5 b c^5+15 a^4 b^2 c^5+14 a^3 b^3 c^5+6 a^2 b^4 c^5+4 a b^5 c^5+2 b^6 c^5+6 a^5 c^6+16 a^4 b c^6+9 a^3 b^2 c^6+5 a^2 b^3 c^6+4 a b^4 c^6+2 b^5 c^6+6 a^4 c^7+2 a^3 b c^7+a^2 b^2 c^7+2 b^4 c^7-4 a^3 c^8-8 a^2 b c^8-5 a b^2 c^8-3 b^3 c^8-4 a^2 c^9-2 a b c^9-3 b^2 c^9+a c^10+b c^10+c^11) p^2+2 c (2 a^8 b^2+a^7 b^3-6 a^6 b^4-3 a^5 b^5+6 a^4 b^6+3 a^3 b^7-2 a^2 b^8-a b^9-2 a^8 b c+a^7 b^2 c+a^6 b^3 c-8 a^5 b^4 c-2 a^4 b^5 c+8 a^3 b^6 c+4 a^2 b^7 c-a b^8 c-b^9 c+a^7 b c^2+5 a^6 b^2 c^2-3 a^5 b^3 c^2-12 a^4 b^4 c^2-a^3 b^5 c^2+8 a^2 b^6 c^2+3 a b^7 c^2-b^8 c^2+a^7 c^3+7 a^6 b c^3+3 a^5 b^2 c^3-9 a^4 b^3 c^3-11 a^3 b^4 c^3-a^2 b^5 c^3+6 a b^6 c^3+4 b^7 c^3+a^6 c^4-2 a^5 b c^4-9 a^4 b^2 c^4-11 a^3 b^3 c^4-7 a^2 b^4 c^4-a b^5 c^4+4 b^6 c^4-3 a^5 c^5-11 a^4 b c^5-10 a^3 b^2 c^5-10 a^2 b^3 c^5-10 a b^4 c^5-6 b^5 c^5-3 a^4 c^6-a^3 b c^6-2 a^2 b^2 c^6-3 a b^3 c^6-6 b^4 c^6+3 a^3 c^7+7 a^2 b c^7+6 a b^2 c^7+4 b^3 c^7+3 a^2 c^8+2 a b c^8+4 b^2 c^8-a c^9-b c^9-c^10) p q-(b-c) (a+b+c) (a^8 b-4 a^6 b^3+6 a^4 b^5-4 a^2 b^7+b^9-a^8 c+4 a^4 b^4 c-4 a^2 b^6 c+b^8 c+2 a^6 b c^2-4 a^5 b^2 c^2-a^4 b^3 c^2+6 a^3 b^4 c^2+3 a^2 b^5 c^2-2 a b^6 c^2-4 b^7 c^2+2 a^6 c^3-4 a^5 b c^3+3 a^4 b^2 c^3+10 a^3 b^3 c^3+5 a^2 b^4 c^3-2 a b^5 c^3-4 b^6 c^3+4 a^4 b c^4+6 a^3 b^2 c^4+5 a^2 b^3 c^4+4 a b^4 c^4+6 b^5 c^4+2 a^3 b c^5+a^2 b^2 c^5+4 a b^3 c^5+6 b^4 c^5-4 a^2 b c^6-2 a b^2 c^6-4 b^3 c^6-2 a^2 c^7-2 a b c^7-4 b^2 c^7+b c^8+c^9) q^2-2 b (-a^7 b^3-a^6 b^4+3 a^5 b^5+3 a^4 b^6-3 a^3 b^7-3 a^2 b^8+a b^9+b^10+2 a^8 b c-a^7 b^2 c-7 a^6 b^3 c+2 a^5 b^4 c+11 a^4 b^5 c+a^3 b^6 c-7 a^2 b^7 c-2 a b^8 c+b^9 c-2 a^8 c^2-a^7 b c^2-5 a^6 b^2 c^2-3 a^5 b^3 c^2+9 a^4 b^4 c^2+10 a^3 b^5 c^2+2 a^2 b^6 c^2-6 a b^7 c^2-4 b^8 c^2-a^7 c^3-a^6 b c^3+3 a^5 b^2 c^3+9 a^4 b^3 c^3+11 a^3 b^4 c^3+10 a^2 b^5 c^3+3 a b^6 c^3-4 b^7 c^3+6 a^6 c^4+8 a^5 b c^4+12 a^4 b^2 c^4+11 a^3 b^3 c^4+7 a^2 b^4 c^4+10 a b^5 c^4+6 b^6 c^4+3 a^5 c^5+2 a^4 b c^5+a^3 b^2 c^5+a^2 b^3 c^5+a b^4 c^5+6 b^5 c^5-6 a^4 c^6-8 a^3 b c^6-8 a^2 b^2 c^6-6 a b^3 c^6-4 b^4 c^6-3 a^3 c^7-4 a^2 b c^7-3 a b^2 c^7-4 b^3 c^7+2 a^2 c^8+a b c^8+b^2 c^8+a c^9+b c^9) p r+2 a (a-b) (a-c) (b-c)^2 (a+b+c)^2 (a^4-2 a^2 b^2+b^4-3 a^2 b c+a b^2 c-2 a^2 c^2+a b c^2-2 b^2 c^2+c^4) q r-(b-c) (a+b+c) (a^8 b-2 a^6 b^3+2 a^2 b^7-b^9-a^8 c-2 a^6 b^2 c+4 a^5 b^3 c-4 a^4 b^4 c-2 a^3 b^5 c+4 a^2 b^6 c+2 a b^7 c-b^8 c+4 a^5 b^2 c^2-3 a^4 b^3 c^2-6 a^3 b^4 c^2-a^2 b^5 c^2+2 a b^6 c^2+4 b^7 c^2+4 a^6 c^3+a^4 b^2 c^3-10 a^3 b^3 c^3-5 a^2 b^4 c^3-4 a b^5 c^3+4 b^6 c^3-4 a^4 b c^4-6 a^3 b^2 c^4-5 a^2 b^3 c^4-4 a b^4 c^4-6 b^5 c^4-6 a^4 c^5-3 a^2 b^2 c^5+2 a b^3 c^5-6 b^4 c^5+4 a^2 b c^6+2 a b^2 c^6+4 b^3 c^6+4 a^2 c^7+4 b^2 c^7-b c^8-c^9) r^2
X(5957) lies on these lines: {30, 511}, {901, 20189}, {953, 13597}, {3259, 11792}
X(5958) lies on these lines: {30, 511}, {2699, 13597}, {2703, 20189}, {11246, 15544}, {11792, 46671}
X(5959) lies on tthese lines: {30, 511}, {1290, 20189}, {2605, 11553}, {2687, 13597}, {5520, 11792}
X(5961) is the Hofstadter 4 point and the antigonal image of X(5964), in accord with the following conjecture and corollary: If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point; thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1, then the antigonal image of the Hofstadter r point is the Hofstadter -r point. (Randy Hutson, August 10, 2014)
The Hofstadter r point is defined at X(359), where further examples are given.
Let A′B′C′ be the Kosnita triangle. The circumcircles of A′BC, B′CA, C′AB concur in X(5961). (Randy Hutson, August 10, 2014)
X(5961) lies on these lines: {3,125}, {24,136}, {94,96}, {186,476}, {1989,2079}
X(5961) = isogonal conjugate of X(5962)
X(5961) = complement of X(39118)
X(5961) = circumcircle-inverse of X(265)
X(5962) is the Hofstadter -3 point; see the conjecture at X(5961).
X(5962) lies on these lines: {4,52}, {30,925}, {96,275}, {128,186}, {136,539}, {250,403}, {378,2351}, {485,1322}, {486,1321}, {2165,3087}
X(5962) = isogonal conjugate of X(5961)
X(5962) = inverse-in-circumcircle of X(5963)
X(5962) = inverse-in-polar-circle of X(52)
X(5963) is the antigonal image of the Hofstadter -5 point; see the conjecture at X(5961).
X(5963) lies on these lines: {3,5962}, {186,847}, {925,1658}
X(5963) = isogonal conjugate of X(5964)
X(5963) = inverse-in-circumcircle of X(5962)
X(5964) is the antigonal image of the Hofstadter 4 point, X(5961), and the inverse-in-circumcircle of the Hofstadter 6 point; see the conjecture at X(5961).
X(5964) lies on these lines: {4, 14104}, {9927, 18403}
X(5964) = isogonal conjugate of X(5963)
The Napoleon line is the line X(6)X(17), which passes through the Napoleon points, X(17) and X(18). X(5965) is the point in which the Napoleon line meets the line X(30)X(511) at infinity.
X(5965) lies on these lines: {5,3629}, {6,17}, {30,511}, {54,69}, {114,385}, {115,5111}, {125,323}, {140,3631}, {141,575}, {193,576}, {381,5102}, {599,5050}, {1204,3098}, {1351,3818}, {1691,5477}, {1992,5071}, {2914,5095}, {2930,5898}, {3180,5617}, {3181,5613}, {3292,3580}, {3630,5092}, {3858,5480}, {5093,5476}
X(5965) = isogonal conjugate of X(5966)
Let O* denote the inverter of the circumcircle and nine-point circle, as defined at X(5577). X(5966) is the inverse-in-O* of X(137). (Randy Hutson, August 13, 2014)
X(5966) lies on the circumcircle, the hyperbola {{A,B,C,X(2),X(61),X(62)}}, and these lines: {2,137}, {5,99}, {23,1291}, {25,933}, {51,110}, {112,3199}, {691,2070}, {1141,2413}
X(5966) = isogonal conjugate of X(5965)
X(5966) = anticomplement of X(31843)
X(5966) = Thomson-isogonal conjugate of X(32478)
X(5966) = Lucas-isogonal conjugate of X(32478)
X(5967) lies on these lines: {2,98}, {4,685}, {6,523}, {23,5968}, {69,4590}, {248,5063}, {263,2698}, {468,1648}, {511,4226}, {524,5467}, {868,1503}, {1641,2434}, {1992,2966}, {2715,2770}, {3266,3292}
X(5967) = isogonal conjugate of X(5968)
X(5967) = perspector of unary cofactor triangles of 2nd and 3rd Parry triangles
Let P denote the antipode of X(23) in the Parry circle. Then X(5968) is the vertex conjugate of X(23) and P, and X(5968) is the perspector of the circumcevian triangle of X(23) and the circumcevian triangle of P. (Randy Hutson, August 13, 2014)
X(5968) lies on these lines: {2,523}, {3,691}, {6,110}, {22,3447}, {23,5967}, {25,250}, {183,892}, {232,4230}, {262,381}, {264,2970}, {325,868}, {511,2421}, {956,5380}, {2065,5050}, {3613,5169}
X(5968) = isogonal conjugate of X(5967)
X(5968) = trilinear pole of the line X(511)X(3569)
X(5969) lies on these lines: {2,694}, {6,99}, {30,511}, {39,597}, {69,148}, {76,338}, {98,1350}, {111,5108}, {114,5480}, {115,141}, {194,1992}, {262,5503}, {385,5104}, {620,3589}, {1469,3027}, {1569,5052}, {1843,5186}, {2076,5152}, {2502,5468}, {3023,3056}, {3029,4260}, {3104,5463}, {3105,5464}, {3629,5477}, {3934,5461}, {4048,5028}
X(5969) = isogonal conjugate of X(5970)
X(5969) = isotomic conjugate of X(35146)
X(5969) = X(2)-Ceva conjugate of X(35077)
X(5970) is the singular focus of the cubic K688. (Bernard Gibert, August 17, 2014)
X(5970) lies on the circumcircle and these lines: {6,805}, {32,691}, {99,187}, {110,1691}, {111,669}, {182,2709}, {512,729}, {1296,2080}, {4027,4590}
X(5970) = isogonal conjugate of X(5969)
X(5970) = Ψ(X(2), X(351))
X(5970) = Λ(X(76), X(338))
X(5970) = barycentric product of circumcircle intercepts of line X(2)X(351)
X(5970) = inverse-in-O(15,16) of X(99), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(5970) = inverse in O* of X(110), where O* is the circle {{X(1687, X(1688), PU(1), PU(2)}}
X(5970) is the point, other than X(99), in which the circumcircle meets the circle {{X(3), X(6), X(99)}}
See X(5577) for the definition of inverter.
X(5971) lies on these lines: {2,6}, {23,99}, {111,538}, {126,754}, {675,2759}, {1003,1383}, {2071,2373}
X(5971) = anticomplement of X(5913)
X(5971) = X(111)-of-1st-anti-Brocard-triangle
X(5971) = intersection, other than X(23), of the line X(2)X(6) and circle {{X(2),X(98),X(99)}}
Let (A′) be the circle centered at A and tangent to the Euler line, and define (B′) and (C′) cyclically; then X(5972) is the radical center of (A′), (B′), (C′). Let D = X(110), let Q be the cyclic quadrilateral ABCD, of which the centroid is X(5972). Let Q′ be the quadrilateral formed by the centroids of the triangles ABC, BCD, CAD, ABD; then Q and Q′ are homothetic at X(5972), which is the {X(2),X(110)}-harmonic conjugate of X(125) and also the inverse-in-Thomson-Gibert-Moses-hyperbola of X(125). Also, X(5972) = X(125)-of-medial-triangle = X(468)-of-1st-Brocard-triangle. (Randy Hutson, August 20, 2014)
The X(2)-Ceva conjugate of P, as P traces the Brocard axis, is a hyperbola, H, with center X(5972); H is the Jerabek hyperbola of the medial triangle, and it passes through X(i) for i = 3,5,6,113, 141, 206, 942, 960, 1147, 1209, 2883. H is also the bicevian conic of X(2) and X(110). (Randy Hutson, August 20, 2014)
X(5972) lies on these lines: {2,98}, {3,113}, {5,1511}, {6,5181}, {67,3763}, {69,5095}, {74,631}, {140,5663}, {146,3523}, {186,1568}, {247,1316}, {265,1656}, {399,3526}, {468,511}, {541,549}, {550,1539}, {578,5504}, {620,690}, {632,5609}, {858,1495}, {895,3618}, {1365,2607}, {1503,5159}, {1986,5562}, {2482,5465}, {2606,4092}, {2781,3819}, {2836,3848}, {2854,3589}, {2948,3624}, {3024,5432}, {3028,5433}, {3066,5476}, {3154,3233}, {3292,3580}, {3818,5094}, {5054,5646}, {5544,5648}
X(5972) = midpoint of X(i) and X(j) for these {i,j}: {2,5642}, {3,113}, {5,1511}, {6,5181}, {110, 125}
X(5972) = complement of X(125)
X(5972) = inverse-in-{circumcircle, nine-point circle}-inverter of X(147)
X(5972) = X(186)-of-X(5)-Brocard-triangle
Let FAFBFC be the Feuerbach triangle of a triangle ABC with excircles (IA), (IB), (IC). Let (KA) be the circle, other than the nine-point circle, which passes through FB and FC and is tangent to (IA), and let D be the touchpoint. Define (KB, (KC) and E, F cyclically. Let TA be the line through D tangent to (IA), and define TB and TC cyclically. The lines TA, TB, TC form a triangle [here named the Hung-Feuerbach triangle] that is perspective to FAFBFC, and X(5973) is the perspector. (Tran Quang Hung, ADGEOM, August 19, 2014) See also X(5974) and X(5975).
Continuing from X(5973), the Hung-Feuerbach circle is here defined as the circle (K) tangent to each of the following 6 circles: (KA, (KB, (KC), (IA), (IB), (IC), so that (K) is also tangent to the Apollonius circle. (Tran Quang Hung and Peter Moses, ADGEOM, August 19, 2014)
X(5974) lies on these lines: {72,171}, {191,1045}, {846,3931}, {970,5975}, {1054,5956}, {1490,2629}, {2959,5687}, {3579,5524}}
The Hung-Feuerbach circle is defined at X(5974).
X(5975) is the inverse-in-Speiker-radical-circle of X(5993). (Peter Moses, August 22, 2014)
If you have The Geometer's Sketchpad, you can view X(5975).
X(5975) lies on the Apollonius circle and this line: {970,5974}
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin B csc(B - 2ω) + sin C csc(C - 2ω) (Randy Hutson, August 22, 2014)
The 1st anti-Brocard triangle is defined at X(5939). Suppose that P = p : q : r (barycentrics) is a point. Then P-of-1st-anti-Brocard-triangle is the point
T(P) = (a4 - b2c2)p + (c4 - a2b2)q + (b4 - a2c2)r : (b4 - c2a2)q + (a4 - b2c2)r + (c4 - b2a2)p : (c4 - a2b2)r + (b4 - c2a2)p + (a4 - c2b2)q
The inverse mapping T*(P), satisfying T*(T(P)) = T(T*(P)) = P, is the given by
T*(P) = a2p + c2q + b2r : c2p + b2q + a2r : b2p + a2q + c2r
The formulas for T and T*, and the following list, were contributed by Peter Moses, August 21, 2014. The appearance of (i,j) in the list means that X(i)-of-1st-anti-Brocard-triangle = X(j); i.e., X(j) = T(X(i)):
(1,1281), (2,2), (3,98), (4,147), (5,114), (6,99), (30,542), (69,148), (76,1916), (99,385), (110,23), (111,5971), (114,1513), (115,325), (125,858), (126,5913), (141,115), (182,3), (184,22), (187,5939), (287,401), (351,4108), (384,4027), (511,2782), (512,804), (513,2787), (514,2786), (515,2792), (516,2784), (517,2783), (518,2795), (519,2796), (520,2797), (521,2798), (522,2785), (523,690), (524,543), (525,2799), (530,531), (531,530), (538,5969), (542,30), (543,524), (574,183), (597,2482), (599,671), (620,230), (690,523), (694,3978), (804,512), (846,3757), (1054,5205), (1083,105), (1316,110), (1352,4), (1499,2793), (1503,2794), (1640,3268), (1691,5152), (1899,1370), (2549,69), (2782,511), (2783,517), (2784,516), (2785,522), (2786,514), (2787,513), (2788,3309), (2789,3667), (2792,515), (2793,1499), (2794,1503), (2795,518), (2796,519), (2797,520), (2798,521), (2799,525), (3094,76), (3120,3006), (3124,3266), (3125,3263), (3309,2788), (3413,3413), (3414,3414), (3448,5189), (3569,850), (3589,620), (3642,13), (3643,14), (3667,2789), (3734,6), (3735,75), (3821,10), (3923,1), (3934,2023), (3944,3705), (3980,612), (3981,305), (4011,614), (4045,141), (4048,32), (4074,1194), (4107,649), (4112,31), (4154,3747), (4159,1501), (4164,667), (4172,560), (4418,3920), (5026,187), (5027,669), (5028,1975), (5091,100), (5108,111), (5116,1078), (5149,1691), (5150,36), (5613,383), (5617,1080), (5622,2071), (5651,1995), (5921,3146), (5967,4226), (5969,538), (5972,468)
Let H be the hyperbola {{A,B,C,PU(37)}}, which is the isogonal conjugate of the line PU(39), this being the line X(32)X(512); also H is the isotomic conjugate of the line PU(45), which is the line X(6)X(523). This hyperbola passes through the points X(76), X(99), X(877), and X(2396). The center of H is X(5976). Let RA be the radical axis of the nine-point circle of triangle BCP(1) and the nine-point circle of BCU(1), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5976). (Randy Hutson, August 22, 2014; see Hyacinthos #21938-21940, 4/12/2013)
Let L be the Simson line of X(99), and let L′ the line normal to the circumcircle at X(99). Then X(5976) = L∩L′. (L = X(114)X(325) and L′ = X(3)X(76).) (Randy Hutson, December 4, 2014)
X(5976) lies on the bicevian conic of X(2) and X(99) and on these lines: {2,694}, {3,76}, {32,5149}, {39,620}, {69,147}, {114,325}, {115,3934}, {187,736}, {230,698}, {315,6033}, {350,1281}, {385,732}, {538,1569}, {618,6109}, {619,6108}, {641,3102}, {642,3103}, {877,2967}, {1125,5977}, {1649,3268}, {1909,3023}, {2491,2799}, {2794,5188}, {4357,5988}
X(5976) = midpoint of X(76) and X(88)
X(5976) = reflection of X(i) in X(j) for these (i,j): (39,620), (115,3934), (1916,2023)
X(5976) = isogonal conjugate of X(34238)
X(5976) = isotomic conjugate of X(36897)
X(5976) = isotomic conjugate of isogonal conjugate of X(36213)
X(5976) = crosspoint of X(2) and X(385)
X(5976) = crosssum of circumcircle intercepts of line PU(39) (line X(32)X(512))
X(5976) = crosssum of X(i) and X(j) for these (i,j): {6, 694}, {98, 8870}, {512, 15630}
X(5976) = crossdifference of every pair of points on line X(882)X(2422)
X(5976) = circumcircle-inverse of X(5989)
X(5976) = complement of X(1916)
X(5976) = anticomplement of X(2023)
X(5976) = midpoint of PU(133)
X(5976) = X(385)-daleth conjugate of X(732)
X(5976) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,325), (76,732), (99,804)
X(5976) = antipode of X(39) in the bicevian conic of X(2) and X(99)
X(5976) = crosssum of circumcircle intercepts of circle {{X(1687),X(1688),PU(1),PU(2)}}
X(5976) = X(i)-isoconjugate of X(j) for these (i,j): {19,15391}, {98,1967}, {290,1927}, {694,1910}, {733,3404}, {1581,1976}
X(5976) = X(i)-complementary conjugate of X(j) for these (i,j): (1,5031), (31,325), (163,804), (172,3836), (385,2887), (560,3229), (1580,141), (1691,10), (1914,3847), (1933,2), (1966,626), (2210,4357), (4164,116)
X(5976) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1916,2023), (98,99,5989), (98,5989,5939), (99,183,5939), (99,1078,5152), (183,5989,98)
See X(5976).
X(5977) lies on these lines: {2,3125}, {21,99}, {75,2783}, {98,336}, {114,5509}, {325,758}, {1125,5976}, {2787,3766}, {5971,5990}
See X(5976).
X(5978) lies on these lines: {2,14}, {13,1916}, {30,99}, {98,5982}, {114,1080}, {147,616}, {298,542}, {299,383}, {302,5092}, {385,533}, {395,1691}, {396,6034}, {3314,3643}
X(5978) = anticomplement of X(6109)
See X(5976).
X(5979) lies on these lines: {2,13}, {14,1916}, {30,99}, {98,5983}, {114,383}, {147,617}, {298,511}, {299,542}, {303,5092}, {385,532}, {395,6034}, {396,1691}, {3314,3642}
X(5979) = anticomplement of X(6108)
See X(5976).
X(5980) lies on these lines: {2,13}, {3,76}, {15,385}, {22,1605}, {62,3329}, {147,627}, {262,5615}, {298,542}, {302,383}, {633,5984}, {1799,3438}, {1916,3104}, {5983,6036}
X(5980) = circumcircle-inverse of X(5981)
X(5980) = anticomplement of X(6115)
See X(5976).
X(5981) lies on these lines: {2,14}, {3,76}, {16,385}, {22,1606}, {61,3329}, {147,628}, {262,5611}, {299,542}, {303,1080}, {634,5984}, {1799,3439}, {1916,3105}, {5982,6036}
X(5981) = circumcircle-inverse of X(5980)
X(5981) = anticomplement of X(6114)
See X(5976).
X(5982) lies on these lines: {2, 5469}, {5, 99}, {17, 1916}, {114, 383}, {147, 627}
See X(5976).
X(5983) lies on these lines: {2, 5470}, {5, 99}, {18, 1916}, {114, 1080}, {147, 628}
See X(5976).
X(5984) lies on these lines: {2, 98}, {8, 1281}, {20, 2782}, {99, 3522}, {115, 3832}, {148, 2794}, {193, 1916}, {385, 1503}, {390, 3027}, {2792, 5905}, {3023, 3600}, {3926, 5152}
X(5984) = anticomplement of X(147)
See X(5976).
X(5985) lies on these lines: {2, 98}, {21, 2782}, {99, 4189}, {115, 5046}, {274, 5152}, {385, 518}, {1281, 1283}, {1621, 3027}, {2475, 2794}, {2975, 3023}
See X(5976).
X(5986) lies on these lines: {2, 98}, {22, 2782}, {99, 1799}, {305, 5152}, {385, 2393}
See X(5976).
X(5987) lies on these lines: {2, 98}, {23, 2782}, {67, 3314}, {183, 2930}, {385, 2854}, {2794, 5189}, {3266, 5152}, {5939, 5971}
X(5987) = inverse-in-1st-anti-Brocard-circle of X(2)
See X(5976).
X5988) is the anticenter of the four points of intersection of the incircle and the Steiner inellipse; see X(5997) and X(5998). (Randy Hutson, August 22, 2014)
X(5988) lies on these lines: {1, 147}, {2, 846}, {10, 257}, {11, 114}, {98, 109}, {99, 1010}, {115, 120}, {183, 4655}, {325, 740}, {542, 3745}, {804, 3837}, {908, 2239}, {1575, 1738}, {2254, 2786}, {2793, 4927}, {3699, 4104}, {3920, 5483}
X5988) = complement of X(1281)
See X(5976).
X(5989) lies on these lines: {2, 4048}, {3, 76}, {6, 1916}, {75, 1281}, {114, 3818}, {115, 5149}, {147, 325}, {148, 384}, {385, 698}, {538, 5162}, {543, 1003}, {2023, 5026}, {5017, 5969}
X(5989) = isogonal conjugate of X(34214)
X(5989) = cevapoint of X(i) and X(j) for these (i,j): {147, 8782}, {4027, 8784}
X(5989) = crosssum of X(512) and X(2679)
X(5989) = crossdifference of every pair of points on line X(2491)X(5113)
X(5989) = circumcircle-inverse of X(5976)
X(5989) = crosspoint of PU(133)
X(5989) = perspector of 1st anti-Brocard triangle and cross-triangle of ABC and 1st anti-Brocard triangle
See X(5976).
X(5990) lies on these lines: {2, 1083}, {98, 901}, {115, 2240}, {385, 513}, {1281, 5205}, {4613, 5143}
See X(5976).
X(5991) lies on these lines: {36, 1111}, {98, 927}, {99, 859}, {385, 514}
b See X(5976).
X(5992) lies on these lines: {2, 846}, {98, 901}, {145, 2784}, {147, 149}, {148, 1655}, {2403, 2789}, {2792, 5905}, {3314, 5695}, {4440, 4459}
X(5992) = anticomplement of X(1281)
See X(5975) and X(5976).
X(5993) lies on the nine-point circle and these lines: {10,5975}, {11,2643}, {3259,5954}
X(5993) = complement of X(38470)Let P be a point of the line X(2)X(14), other than X(2). Let A′B′C′ be the cevian triangle of P. Let A″ by the {B,C}-harmonic conjugate of A′ (i.e., A″ = BC∩B′C′), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5994). Let L,M,N be the lines obtained by reflecting the line X(4)X(14) in the sidelines of ABC; then L,M,N concur in X(5994). Also, X(5994) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(14),X(15)}}. Further, X(5994) is the trilinear pole of the line X(6)X(3129), and X(5994) = Ψ(X(i), X(j)) for these (i,j): (2,14), (4,14), (6,3458), (16,6), (76,14). (Randy Hutson, August 22, 2014)
X(5994) lies on the circumcircle and these lines: {6,2379}, {14,98}, {15,842}, {16,74}, {99,17402}, {110,6137}, {111,3458}, {187,2378}, {249,10409}, {301,2367}, {476,20579}, {477,15743}, {512,1576}, {691,9207}, {759,2154}, {805,14184}, {1141,11582}, {1297,14539}, {1300,6110}, {1625,16806}, {2380,11141}, {2381,11060}, {2710,14538}, {5467,9203}, {6772,18776}, {9181,9202}, {10410,17403}, {11083,15544}, {11085,11135}
X(5994) = midpoint of X(16) and X(5669)
X(5994) = reflection of X(5995) in Brocard
X(5994) = Schoutte-circle-inverse of X(2378)
X(5994) = polar conjugate of isotomic conjugate of X(38413)
X(5994) = X(i)-cross conjugate of X(j) for these (i,j): {512, 16460}, {2420, 5995}, {3130, 250}, {6138, 6}, {11135, 23357}, {19781, 249}
X(5994) = X(5995)-Hirst inverse of X(14560)
X(5994) = cevapoint of X(6) and X(6138)
X(5994) = trilinear pole of line {6, 3129}
X(5994) = barycentric product of circumcircle intercepts of line X(2)X(14)
X(5994) = X(i)-isoconjugate of X(j) for these (i,j): {16, 1577}, {75, 6138}, {299, 661}, {300, 2624}, {471, 656}, {850, 2152}, {897, 9205}, {1109, 17403}, {2153, 3268}, {8740, 14208}
X(5994) = barycentric product X(i)X(j) for these {i,j}: {14, 110}, {15, 476}, {99, 3458}, {249, 20579}, {298, 14560}, {301, 1576}, {524, 9207}, {662, 2154}, {1989, 17402}, {4558, 8738}, {5619, 11130}, {5995, 11092}, {8015, 10410}, {8836, 16807}, {9203, 21467}, {11085, 17403}, {16771, 16806}
X(5994) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 850}, {15, 3268}, {32, 6138}, {110, 299}, {112, 471}, {187, 9205}, {476, 300}, {1576, 16}, {2154, 1577}, {3457, 23283}, {3458, 523}, {5995, 11078}, {8738, 14618}, {9207, 671}, {11060, 20578}, {14560, 13}, {16806, 19779}, {17402, 7799}, {17403, 11128}, {20579, 338}, {23347, 6111}, {23357, 17403}
Let P be a point of the line X(2)X(13), other than X(2). Let A′B′C′ be the cevian triangle of P. Let A″ by the {B,C}-harmonic conjugate of A′ (i.e., A″ = BC∩B′C′), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5995). Let L,M,N be the lines obtained by reflecting the line X(4)X(13) in the sidelines of ABC; then L,M,N concur in X(5995). Also, X(5995) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(13),X(16)}}. Further, X(5995) is the trilinear pole of the line X(6)X(3130), and X(5995) = Ψ(X(i), X(j)) for these (i,j): (2,13), (4,13), (6,3457), (15,6), (76,13). (Randy Hutson, August 22, 2014)
X(5995) lies on the circumcircle and these lines: {6,2378}, {13,98}, {15,74}, {16,842}, {99,17403}, {110,6138}, {111,3457}, {187,2379}, {249,10410}, {300,2367}, {476,20578}, {477,11586}, {512,1576}, {691,9206}, {759,2153}, {805,14183}, {1141,11581}, {1297,14538}, {1300,6111}, {1625,16807}, {2380,11060}, {2381,11142}, {2710,14539}, {5467,9202}, {6775,18777}, {9181,9203}, {10409,17402}, {11080,11136}, {11088,15544}
X(5995) = midpoint of X(15) and X(5668)
X(5995) = reflection of X(5995) in Brocard axis
X(5995) = Schoutte-circle-inverse of X(2379)
X(5995) = X(i)-cross conjugate of X(j) for these (i,j): {512, 16459}, {2420, 5994}, {3129, 250}, {6137, 6}, {11136, 23357}, {19780, 249}
X(5995) = X(5994)-Hirst inverse of X(14560)
X(5995) = cevapoint of X(6) and X(6137)
X(5995) = trilinear pole of line {6, 3130}
X(5995) = polar conjugate of isotomic conjugate of X(38414)
X(5995) = X(i)-isoconjugate of X(j) for these (i,j): {15, 1577}, {75, 6137}, {298, 661}, {301, 2624}, {470, 656}, {850, 2151}, {897, 9204}, {1109, 17402}, {2154, 3268}, {8739, 14208}
X(5995) = barycentric product of circumcircle intercepts of line X(2)X(13)
X(5995) = barycentric product X(i)X(j) for these {i,j}: {13, 110}, {16, 476}, {99, 3457}, {249, 20578}, {299, 14560}, {300, 1576}, {524, 9206}, {662, 2153}, {1989, 17403}, {4558, 8737}, {5618, 11131}, {5994, 11078}, {8014, 10409}, {8838, 16806}, {9202, 21466}, {11080, 17402}, {16770, 16807}
X(5995) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 850}, {16, 3268}, {32, 6137}, {110, 298}, {112, 470}, {187, 9204}, {476, 301}, {1576, 15}, {2153, 1577}, {3457, 523}, {3458, 23284}, {5994, 11092}, {8737, 14618}, {9206, 671}, {11060, 20579}, {14560, 14}, {16807, 19778}, {17402, 11129}, {17403, 7799}, {20578, 338}, {23347, 6110}, {23357, 17402}
Let L be the de Longchamps line of ABC, and let L′ be the de Longchamps line of the 1st anti-Brocard triangle. Then X(5996) = L∩L′. Let O′ be the inverter of the circumcircle and the nine-point circle; then X(5996) is the pole of the Brocard axis with respect to O′. (Randy Hutson, August 22, 2014)
X(5996) lies on these lines: {2,512}, {262,5466}, {325,523}, {1180,2524}, {1499,1513}
X(5997) and X(5998) are described by Randy Hutson (Hyacinthos #21045) and Barry Wolk (Hyacinthos #21047, June 1, 2012). One of the 4 points of intersection of the incircle and Steiner inellipse is a triangle center, X(5997), and the other 3 are the vertices of a central triangle; of those 3, let A′ be the one farthest from A, and define B′ and C′ cyclically. The triangle A′B′C′ is perspective to ABC at X(5998). Barycentrics for A′, B′, C′ follow:
A′ = a - [(a - b + c)(a + b - c)]1/2 : b + [(b - c + a)(b
+ c - a)]1/2 : c + [(c - a + b)(c + a -
b)]1/2
B′ = a + [(a - b + c)(a + b - c)]1/2 : b - [(b - c + a)(b +
c - a)]1/2 : c + [(c - a + b)(c + a - b)]1/2
C′ = a + [(a - b + c)(a + b - c)]1/2 : b + [(b - c + a)(b +
c - a)]1/2 : c - [(c - a + b)(c + a - b)]1/2
X(5997) is the {X(1),X(508)}-harmonic conjugate of X(5998). (Randy Hutson, August 22, 2014)
X(5997) lies on the incircle, the Steiner inellipse, and this line: {1,508)
Continuing from X(5997), the lines AA′, BB′, CC′ concur in X(5998), and X(5998) is the {X(1),X(508)}-harmonic conjugate of X(5997). (Randy Hutson, August 22, 2014)
X(5998) lies on this line: {1,508}
As a point on the Euler line, X(5999) has Shinagawa coefficients ((E + F)2 - S2, -2(E + F)2).
Let A′B′C′ be the cross-triangle of the 1st and 2nd Neuberg triangles. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5999). (Randy Hutson, July 31 2018)
X(5999) lies on these lines: {2,3}, {98,385}, {115,5162}, {147,325}, {182,262}, {183,1350}, {230,2076}, {316,2794}, {516,5988}, {517,5990}, {625,5149}, {1078,5188}, {1352,3314}, {1499,5971}, {1691,2023}, {2456,4027}, {3424,5921}, {3564,5984}, {3815,5116}, {4518,6211}, {5663,5987}
X(5999) = reflection of X(385) in X(98)
X(5999) = anticomplement of X(1513)
X(5999) = {X(6039),X(6040)}-harmonic conjugate of X(3)
X(5999) = pole of Brocard axis wrt conic {{X(13),X(14),X(15),X(16),X(98)}}
X(5999) = {X(2),X(3)}-harmonic conjugate of X(37455)
Isogonal Conjugates With Respect To Various Triangles
Many triangle centers are constructible as isogonal conjugates with respect to a central triangle T other than ABC. See, for example, the preamble to X(2883). Other examples are X(5642), for which T is the Thomson triangle, and X(5957), for which T is the Feuerbach triangle. In the latter case, a long formula for barycentric coordinates is given for the Feuerbach isogonal conjugate of an arbitrary point p : q : r is given. Peter Moses (August 11, 2014) developed a much longer formula for the isogonal conjugate of a point p : q : r with respect to a triangle having vertices d1 : d2 : d3, e1 : e2 : e3, f1 : f2 : f3. The total number of characters in the formula is (pending). Although too long to be stated here, the formula served as a basis for finding triangle centers X(i) for i = 6000-6030.
For well-established central triangles T, (e.g., Thomson, Feuerbach, intouch, extouch), the phrase "isogonal conjugate with respect to T can be shortened to "T isogonal conjugate" or "T-I C″, as in many of the next triangle centers.
It T is a triangle, then T-isogonal conjugation carries points on the circumcircle of T to the line at infinity. For example, taking T = Brocard triangle, the appearance of (X(i),X(j)) in the following list means that T-I C of X(i) is X(j): (3,2782), (6,804), (1083,2795), (1316,690), (5091,2787), (5108,543).
In each of the next examples (i,j), the point X(i) lies on the incircle, and X(j), the intouch-IC-of-X(i), lies on the line at infinity: (1317,517), (1356,6002), (1357,3667), (1358,3309), (1360,971), (1361,5151), (1362,516), (1364,522), (1365,6003), (2446, 3307), (2447,3308), (3020,6004), (3021,518), (3022,514), (3023,512), (3024,523), (3025,900), (3026,6005), (3027,511), (3028,30), (3318,521), (3319,2800), (3320,1503), (3322,2801), (3323,2820), (3325,1499), (3326,3738), (3327,1510), (3328,3887), (5577,6006), (5579,6007), (5580,520), (5581,6008), (5582,6009)
X(6000) lies on these lines: {2,5656}, {3,64}, {4,51}, {5,2883}, {6,1597}, {20,2979}, {24,1204}, {30,511}, {40,2939}, {52,382}, {65,1844}, {66,3818}, {74,186}, {98,2713}, {99,2706}, {100,2744}, {101,2738}, {102,2762}, {103,2727}, {104,2719}, {109,2732}, {110,2071}, {113,2072}, {125,403}, {133,1515}, {143,3853}, {146,1531}, {159,3098}, {184,378}, {187,1971}, {206,4550}, {221,500}, {232,3269}, {373,3545}, {376,3917}, {381,1853}, {399,2935}, {546,5462}, {548,5447}, {550,1216}, {568,3830}, {578,1181}, {858,1568}, {933,3484}, {974,1514}, {999,2192}, {1092,5879}, {1192,3517}, {1292,2749}, {1293,2755}, {1295,2761}, {1296,2763}, {1297,2764}, {1533,3580}, {1562,5523}, {1614,3520}, {1657,5925}, {1870,3270}, {3060,3543}, {3146,5889}, {3284,5668}, {3524,5650}, {3581,5899}, {3587,3781}, {3627,5446}, {3839,5640}, {3845,5946}
X(6000) = isogonal conjugate of X(1294)
X(6000) = crosssum of X(4) and X(74)
X(6000) = orthic-IC-of-X(133) = medial-IC-of-X(133)
X(6000) = intouch-IC-of-X(3324)
X(6000) = Euler-IC-of-X(122)
X(6000) = 1st-circumperp-IC-of-X(1294)
X(6000) = circumtangential-IC-of-X(1294)
X(6000) = circumcircle-midarc-IC-of-X(107)
X(6000) = circumorthic-IC-of-X(107)
X(6000) = 2nd-circumperp-IC-of-X(107)
X(6000) = circumnormal-IC-of-X(107)
X(6000) = Thomson-IC-of-X(107)
X(6000) = Thomson-isogonal conjugate of X(107)
X(6000) = Lucas-isogonal conjugate of X(107)
X(6000) = Cundy-Parry Phi transform of X(14379)
X(6000) = Cundy-Parry Psi transform of X(14249)
X(6000) = intersection of tangents to Moses-Jerabek conic at X(74) and X(1199)
X(6001) lies on these lines: {1,84}, {3,960}, {4,65}, {5,3812}, {7,3427}, {9,3197}, {10,5777}, {11,1519}, {19,5776}, {20,3869}, {30,511}, {36,1727}, {40,64}, {46,3149}, {63,3428}, {74,2766}, {98,2714}, {99,2707}, {100,2745}, {101,2739}, {102,2765}, {103,2728}, {104,1319}, {109,2733}, {110,2694}, {119,5123}, {153,5176}, {154,392}, {165,5692}, {185,1829}, {207,1712}, {210,5657}, {227,1745}, {281,1903}, {354,5603}, {355,5836}, {376,5918}, {496,942}, {497,5768}, {573,4047}, {581,3931}, {774,1042}, {944,3057}, {962,3868}, {1001,3358}, {1006,3683}, {1064,3666}, {1072,3782}, {1104,3073}, {1292,2750}, {1293,2756}, {1294,2761}, {1296,2767}, {1376,3359}, {1385,5248}, {1430,5706}, {1456,1870}, {1465,1735}, {1532,1737}, {1537,5570}, {1699,5902}, {1739,5400}, {1750,2093}, {1853,3753}, {1871,5786}, {2077,2932}, {2096,4293}, {2194,4227}, {2292,4300}, {2551,5811}, {2935,2948}, {3062,3577}, {3357,3579}, {3698,5818}, {3742,5886}, {3817,5883}, {3874,4301}, {3877,5731}, {3878,4297}, {3889,5734}, {3913,5534}, {3914,5721}, {4018,5895}, {4067,5493}, {4314,5882}, {4662,5690}, {4867,5538}, {4880,5536}, {5691,5903}, {5787,5878}, {5806,5893}
X(6001) = crosssum of X(4) and X(104)
X(6001) = midarc-of-X(3318)
X(6001) = intouch-IC-of-X(1359)
X(6001) = Euler-IC-of-X(123)
X(6001) = 1st-circumperp-IC-of-X(1295)
X(6001) = circumtangential-IC-of-X(1295)
X(6001) = circumcircle-midarc-IC-of-X(108)
X(6001) = circumorthic-IC-of-X(108)
X(6001) = 2nd-circumperp-IC-of-X(108)
X(6001) = circumnormal-IC-of-X(108)
X(6001) = Thomson-IC-of-X(108)
X(6001) = Thomson-isogonal conjugate of X(108)
X(6001) = Lucas-isogonal conjugate of X(108)
X(6001) = Cundy-Parry Psi transform of X(14257)
X(6002) lies on these lines: {1, 4170}, {4, 4444}, {8, 4729}, {30, 511}, {649, 4391}, {661, 4560}, {667, 3716}, {905, 3835}, {1019, 1577}, {2533, 4784}, {3669, 4106}, {3762, 4063}, {4010, 4367}, {4380, 4462}, {4382, 4801}, {4705, 4913}, {4879, 4922}
X(6002) = isogonal conjugate of X(6010)
X(6003) lies on these lines: {1, 4017}, {3, 3733}, {8, 4404}, {30, 511}, {573, 798}, {656, 3737}, {661, 1021}, {1459, 3960}, {1532, 4129}, {4581, 4761}
X(6003) = isogonal conjugate of X(6011)
X(6004) lies on these lines: {1, 3777}, {30, 511}, {659, 1734}, {663, 1201}, {667, 2254}, {764, 4449}, {905, 1960}, {1016, 3888}, {1491, 3216}, {2473, 2488}, {3214, 4705}, {4367, 4905}, {4498, 4730}
X(6004) = isogonal conjugate of X(6012)
X(6005) lies on these lines: {30, 511}, {649, 2664}, {659, 4834}, {661, 1734}, {663, 1019}, {665, 3803}, {667, 4784}, {693, 4170}, {1491, 4983}, {2254, 4822}, {2533, 4791}, {2978, 4932}, {3250, 4979}, {4010, 4823}, {4063, 4724}, {4147, 4807}, {4367, 4775}, {4378, 4879}, {4391, 4761}, {4489, 4813}, {4490, 4730}
X(6005) = isogonal conjugate of X(6013)
X(6006) lies on these lines: {7, 3676}, {9, 649}, {30, 511}, {142, 3835}, {144, 4468}, {1769, 4905}, {2254, 4776}, {3239, 4790}, {4401, 4491}, {4406, 4828}, {4462, 4768}
X(6006) = isogonal conjugate of X(6014)
X(6007) lies on these lines: {7, 310}, {9, 43}, {30, 511}, {39, 4443}, {86, 4890}, {142, 3840}, {192, 3688}, {239, 3271}, {320, 4014}, {995, 1001}, {1463, 4684}, {1469, 3886}, {1654, 4111}, {1959, 4516}, {2234, 3122}, {2245, 4436}, {3056, 3875}, {3729, 3779}, {3792, 4693}, {3882, 4433}, {3923, 4260}, {3943, 4553}, {4259, 5695}
X(6007) = isogonal conjugate of X(6015)
X(6008) lies on these lines: {7, 3669}, {9, 4063}, {30, 511}, {144, 4462}, {390, 4162}, {649, 4106}, {650, 4380}, {667, 1001}, {693, 4790}, {1638, 4786}, {3768, 4498}, {3803, 4170}, {3835, 4394}, {4312, 4905}, {4375, 4782}, {4382, 4979}
X(6008) = isogonal conjugate of X(6016)
X(6009) lies on these lines: {30, 511}, {88, 673}, {659, 1001}, {676, 4830}, {1086, 2087}, {1635, 4927}, {2550, 4925}, {3826, 3837}, {4453, 4773}
X(6009) = isogonal conjugate of X(6017)
X(6010) lies on the circumcircle and these lines: {3, 741}, {40, 98}, {55, 1356}, {99, 3882}, {103, 3430}, {165, 5539}, {573, 759}, {813, 4574}, {1376, 3037}, {2077, 2699}
X(6010) = isogonal conjugate of X(6002)
X(6011) lies on the circumcircle and these lines: {3, 759}, {40, 74}, {55, 1365}, {102, 3430}, {104, 3651}, {105, 1283}, {106, 1385}, {107, 4242}, {112, 1983}, {354, 1477}, {573, 2249}, {741, 991}, {840, 5536}, {2077, 2687}
X(6011) = isogonal conjugate of X(6003)
X(6011) = reflection of X(759) in X(3)
X(6011) = circumcircle-antipode of X(759)
X(6012) lies on the circumcircle and these lines: {55, 3020}, {105, 404}, {759, 4234}
X(6012) = isogonal conjugate of X(6004)
X(6013) lies on the circumcircle and these lines: {55, 3026}, {101, 4436}, {105, 5248}, {741, 4658}
X(6013) = isogonal conjugate of X(6005)
X(6014) lies on the circumcircle and these lines: {55, 106}, {104, 165}, {672, 2384}, {739, 2280}, {901, 3939}, {953, 5537}
X(6014) = isogonal conjugate of X(6006)
X(6015) lies on the circumcircle and these lines: {55, 99}, {57, 4128}, {100, 894}, {101, 171}, {109, 1918}, {110, 2175}, {165, 5539}, {934, 1402}, {1155, 2703}, {1308, 5143}
X(6015) = isogonal conjugate of X(6007)
X(6016) lies on the circumcircle and these lines: {55, 739}, {644, 898}, {672, 2382}, {727, 2280}
X(6016) = isogonal conjugate of X(6008)
X(6017) lies on the circumcircle and these lines: {44, 105}, {55, 2384}, {106, 672}, {901, 2284}, {1252, 4588}, {2280, 2382}
X(6017) = isogonal conjugate of X(6009)
Let A′ be the point of intersection, other than X(11), of the Mandart inellipse and the incircle that is farthest from A; define B′ and C′ cyclically. A′B′C′ is here named the Mandart-incircle triangle. It is homothetic to ABC at X(55), to the medial triangle at X(11), to the Euler triangle at X(12), and to the anticomplementary triangle at X(497). X(6018) = X(106)-of-Mandart-incircle triangle. (Randy Hutson, August 26, 2014)
X(6018) lies on the incircle and these lines: {1,1357}, {8,3038}, {10,11}, {12,5510}, {55,106}, {56,1293}, {65,4661}, {354,4701}, {988,3242}, {1054,1697}, {1317,2827}, {1320,3271}, {1356,4890}, {1358,3663}, {1361,2815}, {1362,2821}, {1364,2098}, {2776,3028}, {2789,3027}, {2796,3023}, {2810,3022}, {2832,3021}, {2840,3318}, {2842,3024}, {2843,6019}, {2844,6020}, {2899,5178}, {3025,5048}, {3616,4731}, {3624,3893}, {3626,4004}, {3632,3698}, {3635,4002}, {3679,3962}, {3753,4746}, {3922,4668}, {4005,5836}, {4533,4691}
X(6018) = reflection of X(1357) in X(1)
X(6018) = X(1294)-of-intouch-triangle
X(6019) lies on the incircle and these lines: {1,3325}, {11,126}, {12,5512}, {55,111}, {56,1296}, {215,3048}, {543,3023}, {1317,2830}, {1357,4890}, {1358,4854}, {1361,2819}, {1362,2824}, {1364,2852}, {2780,3028}, {2793,3027}, {2813,3022}, {2837,3021}, {2843,6018}, {2851,3318}, {2854,3024}, {5148,5160}
X(6019) = reflection of X(3325) in X(1)
X(6019) = X(111) of Mandart-incrcle triangle (see X(6018))
X(6020) lies on the incircle and these lines: {1,3320}, {11,127}, {12,132}, {55,112}, {56,1297}, {1317,2831}, {1362,2825}, {1364,2853}, {2781,3028}, {2794,3027}, {2799,3023}, {2838,3021}, {2844,6018}, {3324,5434}, {5148,6023}
X(6020) = reflection of X(3320) in X(1)
X(6020) = X(112) of Mandart-incrcle triangle (see X(6018))
X(6021) lies on the incircle and these lines: {11,141}, {55,739}, {1357,4003}
X(6022) lies on the incircle and these lines: {55,729}, {3023,3056}
X(6023) lies on the incircle and these lines: {1,6027}, {12,5099}, {30,3023}, {55,842}, {56,691}, {511,3024}, {512,3028}, {523,3027}, {3325,5194}, {5148,6020}
X(6023) = reflection of X(6027) in X(1)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^2 b-4 a b^2+b^3+a^2 c+4 a b c-4 a c^2+c^3)^2
X(6024) lies on the incircle and these lines:
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (a^3 b-a^2 b^2+a^3 c+b^3 c-a^2 c^2-2 b^2 c^2+b c^3)^2
X(6025) lies on the incircle and these lines: {11,37}, {55,675}, {544,3022}
X(6026) lies on the incircle and this line: {55,689}
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c)^2 (a^4-b^4+2 a^2 b c-b^3 c+b^2 c^2-b c^3-c^4)^2
X(6027) lies on the incircle and these lines: {1,6023}, {11,5099}, {30,3027}, {55,691}, {56,842}, {511,3028}, {512,3024}, {523,3023}, {3320,5194}, {5148,5160}
X(6027) = reflection of X(i) in X(j) for these (i,j): (6023,1), (5160,5148)
X(6028) lies on the incircle and this line: {55,697}
X(6029) lies on the incircle and these lines: {11,626}, {55,713}
X(6030) lies on the Thomson-Gibert-Moses hyperbola; see X(5642). X(6030) = {X(22),X(3796)}-harmonic conjugate of X(3060; also, X(6030) = {X(3060),X(3796)}-harmonic conjugate of X(5012). (Peter Moses, August 16, 2014)
X(6030) lies on these lines: {6,22}, {23,5643}, {25,5544}, {36,595}, {110,3917}, {182,5645}, {376,5654}, {392,4881}, {550,3521}, {1495,5888}, {2937,5946}, {2979,3167}, {5640,5644}
X(6031) lies on these lines: {2,187}, {3,5971}, {22,5938}, {23,183}, {69,5648}, {99,5987}, {353,524}, {385,6322}, {2373,6236}
X(6031) = isogonal conjugate of X(30488)
X(6031) = isotomic conjugate of X(34213)
X(6031) = anticomplement of X(6032)
X(6031) = circumcevian-isogonal conjugate of X(2)
X(6031) = X(2)-pedal-to-circummedial similarity image of X(2)
Let A′B′C′ be the Artzt triangle. Let A″ be the reflection of A in B′C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(9744), and X(6032) = centroid of A″B″C″; see X(5640). (Randy Hutson, December 10, 2016)
X(6032) lies on these lines: {2,187}, {5,5913}, {111,381}, {112,5094}, {115,5169}, {262,5466}, {427,1180}, {566,858}, {1648,5640}, {1995,2079}
X(6032) = complement of X(6031)
X(6032) = inverse-in-orthocentroidal circle of X(111)
X(6032) = X(353)-of-orthocentroidal triangle
X(6032) = anti-Artzt-to-4th-Brocard similarity image of X(2)
X(6033) = Johnson-triangle-orthologic center of the1st Brocard triangle. (César Lozada, August 7, 2013)
Let D = X(99), let HA = X(4)-of-BCD, and define HB, HC, HD cyclically. Then HAHBHCHD is a cyclical quadrilateral, of which the circumcenter is X(6033). (Randy Hutson, August 22, 2014)
X(6033) lies on these lines: {2, 5191}, {3, 114}, {4, 147}, {5, 83}, {6, 13}, {30, 99}, {110, 868}, {315, 5976}, {543, 3830}, {623, 5617}, {624, 5613}, {671, 3845}, {946, 2784}, {1478, 3023}, {1479, 3027}, {1503, 2456}, {1513, 2080}, {1656, 6036}, {2023, 2548}, {2476, 5985}, {2482, 3534}, {3091, 5984}, {4027, 5025}, {5133, 5986}, {5169, 5987}
X(6033) = midpoint of X(4) and X(147)
X(6033) = reflection of X(3) in X(114)
X(6033) = complement of X(9862)
X(6033) = circumcircle-inverse of X(34217)
X(6033) = X(3)-of-X(511)-Fuhrmann-triangle
X(6033) = inverse-in-Kiepert-hyperbola of X(6034)
X(6033) = X(35002)-of-orthocentroidal-triangle
X(6033) = {X(13),X(14)}-harmonic conjugate of X(6034)
X(6034) lies on these lines: {2,694}, {6,13}, {30,1691}, {83,597}, {98,5306}, {99,3589}, {148,3618}, {182,6321}, {230,5104}, {395,5979}, {396,5978}, {524,5103}, {543,5149}, {599,626}, {1350,6036}, {2482,5013}, {5017,6055}
X(6034) = midpoint of X(671) and X(5182)
X(6034) = reflection of X(5182) in X(597)
X(6034) = inverse-in-Kiepert-hyperbola of X(6033)
X(6034) = {X(13),X(14)}-harmonic conjugate of X(6033)
X(6035) lies on the hyperbola {A, B, C, PU(37)} and these lines: {69, 892}, {877, 4240}, {2396, 2407}
X(6035) = isogonal conjugate of X(6041)
X(6035) = isotomic conjugate of X(1640)
X(6035) = trilinear pole of line X(30)X(99)
Let D = X(98), let GA = X(2)-of-BCD, and define GB, GC, GD cyclically. Then GAGBGCGD is a cyclic quadrilateral which is homothetic to the cyclic quadrilateral ABCD, and X(6036) is the homothetic center, as well as the centroid of ABCD and the anticenter of the cyclic quadrilateral whose vertices are X(115) and the vertices of the medial triangle. Let (OA) be the circle centered at the A-vertex of the 1st Brocard triangle and tangent to line BC; define (OB) and (OC) cyclically; then X(6036) is the radical center of (OA), (OB), (OC). Also, X(6036) lies on the axis of the parabola {A, B, C, X(511), X(805}. (Randy Hutson, August 22, 2014)
X(6036) is the QA-P36 center of the quardrangle ABCX(2). See Complement of QA-P30 wrt the QA-Diagonal Triangle.
X(6036) lies on these lines: {2, 98}, {3, 115}, {5, 2794}, {30, 5461}, {99, 631}, {140, 620}, {148, 3523}, {230, 511}, {231, 3001}, {325, 5965}, {338, 2974}, {543, 549}, {575, 3815}, {671, 3524}, {1155, 5988}, {1350, 6034}, {1506, 3398}, {1624, 5020}, {1656, 6033}, {2482, 5054}, {2783, 3035}, {2784, 3634}, {3023, 5433}, {3027, 5432}, {3054, 5092}, {5050, 5477}, {5097, 5306}, {5469, 5474}, {5470, 5473}, {5980, 5983}, {5981, 5982}
X(6036) = midpoint of X(98) and X(114)
X(6036) = complement of X(114)
X(6036) = anticomplement of X(6721)
X(6036) = X(140)-of-1st-anti-Brocard-triangle
X(6036) = center of bicevian conic of X(2) and X(98)
Barycentrics 1/((b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) (a^2 b^2 + a^2 c^2 - b^4 - c^4)) :
X(6037) lies on the circumcircle and these lines: {98,237}, {110,2966}, {112,685}, {262,842}, {263,2698}, {327,2857}, {401,1297}, {419,3563}, {805,4226}
X(6037) = inverse-in-1st-anti-Brocard-circle of X(6038)
X(6037) = Ψ(X(6),X(98)
X(6037) = Λ(X(290),X(879)
X(6037) = Λ(X(684),X(2491)
Let A′ be the inverse-in-1st-anti-Brocard-circle of A, and define B′ and C′ cyclically; then X(6038) is the centroid of A′B′C′.
X(6038) lies on these lines: {2, 154}, {98, 237}, {669, 804}
Let O′ be the circle with center X(2) that passes through the points X(98) and X(842); then X(6039) and X(6040) are the points of intersection of O′ and the Euler line. Also, ({X(6039),X(6040}-harmonic conjugate of X(4)) = X(1513), and ({X(6039),X(6040}-harmonic conjugate of X(383)) = X(1080). (Peter Moses, August 24, 2014)
X(6039) lies on these lines: {2,3}, {98,1380}, {262,1340}
X(6039) = reflection of X(6040) in X(2)
Let O′ be the circle with center X(2) that passes through the points X(98) and X(842); then X(6039) and X(6040) are the points of intersection of O′ and the Euler line. Also, ({X(6039),X(6040}-harmonic conjugate of X(4)) = X(1513), and ({X(6039),X(6040}-harmonic conjugate of X(383)) = X(1080). (Peter Moses, August 24, 2014)
X(6040) lies on these lines: {2,3}, {98,1379}, {262,1341}
X(6040) = reflection of X(6039) in X(2)
(See X(1635) for the definition of tripolar centroid.)
X(6041) lies on these lines: {6,647}, {25,2489}, {32,512}, {351,865}, {523,5306}, {2510,2780}, {3049,3051}
X(6041) = isogonal conjugate of X(6035)
X(6041) = crossdifference of every pair of points on the line
X(30)X(99)
Let (Ap) be the Apollonius circle, and let (KA), (KB), (KC) be the circles described at X(5973) in association with the Hung-Feuerbach circle at X(5974). Let LA be the radical axis of (Ap) and (KA), and define LB and LC cyclically. The lines LA, LB, (LC form a triangle T (here named the Montesdeoca-Hung triangle) that is perspective to ABC, and the perspector is X(6042). (Tran Quang Hung, ADGEOM #1506; Angel Montesdeoca, ADGEOM #1525, August 24, 2014)
The triangle T is a central triangle of type 1, perpsective to each of the following triangles: ABC, excentral (see X(6043), Apollonius, incentral, Feuerbach, and extangents. (Peter Moses, August 24, 2014).
The A-vertex of the central triangle T has these barycentrics:
g(a,b,c) : b (a+c)^2 (a^2+a b+b c+c^2)^2 : (a+b)^2 c (a^2+b^2+a c+b
c)^2, where
g(a,b,c) = -a (2 a^6+4 a^5 b+4 a^4 b^2+4 a^3 b^3+3 a^2 b^4+2 a
b^5+b^6+4 a^5 c+12 a^4 b c+16 a^3 b^2 c+12 a^2 b^3 c+6 a b^4 c+2 b^5
c+4 a^4 c^2+16 a^3 b c^2+20 a^2 b^2 c^2+8 a b^3 c^2+b^4 c^2+4 a^3
c^3+12 a^2 b c^3+8 a b^2 c^3+3 a^2 c^4+6 a b c^4+b^2 c^4+2 a c^5+2 b
c^5+c^6) (Peter Moses, August 24, 2014)
X(6042) lies on these lines: {1,5974}, {10,2643}, {73,2632}, {191,849}, {238,3743}, {757,1046}, {846,1098}, {960,1193}, {1104,1962}, {1682,5975}, {3708,3954}, {3847,4647}
Continuing from X(6042), the triangle T is perspective to the excentral triangle, and the perspector is X(6043). (Peter Moses, August 24, 2014)
X(6043) lies on these lines: {1,5974}, {8,2363}, {31,643}, {33,162}, {42,2185}, {43,662}, {58,519}, {81,518}, {82,3791}, {100,593}, {171,319}, {181,5975}, {270,1891}, {849,3293}, {1098,5247}, {1931,4640}, {1963,4682}, {2651,4641}, {4203,5035}
Following constructions at X(5973), Peter Moses (August 24, 2014) constructs a triangle DEF and circle (J), beginning with the orthic triangle HAHBHC and the excircles (IA), (IB), (IC). Let (JA) be the circle, other than the nine-point circle, that passes through HB and HC and is tangent to (IA). Let D be the touchpoint. Define (JB) and (JC) cyclically, and define E and F cyclically. Let (J) be the circle (here named the Moses-Hung circle) tangent to (JA), (JB), (JC), and let (Ap) be the Apollonius circle. The circles (J) and (Ap) are tangent, and their touchpoint is X(6044). The triangle DEF, here named the Moses-Hung triangle, has vertices given by these barycentrics:
D = 2a3 + b3 + c3 + a2b
+ a2c - b2c - bc2 : -(a + b + c)(a +
c)2(a + b - c)3 : -(a + b + c)(a +
b)2(a - b + c)3
E = -(a + b + c)(b + a)2(b + c - a)3 :
2b3 + c3 + a3 + b2c +
b2a - c2a - ca2 : -(a + b + c)(b +
a)2(b + c - a)3
F = -(a + b + c)(c + b)2(c + a - b)3 : -(a + b +
c)(c + a)2(c + a - b)3 : 2c3 +
a3 + b3 + c2a + c2b -
a2b - ab2
X(6044) lies on the Apollonius circle and these lines: {10,125}, {43,1726}, {181,1365}, {386,759}, {407,1829}, {440,2968}, {573,2249}, {2092,2264}, {2610,5213}
X(6044) = complement of X(38480)
X(6044) = inverse-in-Spieker-radical-circle of X(125)
The Moses-Hung circle is defined at X(6044).
X(6045) lies on these lines: {4,8}, {442,511}, {573,2264}
The Moses-Hung triangle is defined at X(6044).
X(6046) lies on these lines: {7,3486}, {11,273}, {55,347}, {56,1119}, {65,1439}, {269,1358}, {279,961}, {1427,1880}, {1441,3925}
X(6046) = isogonal conjugate of X(6061)
The Moses-Hung triangle is defined at X(6044).
X(6047) lies on these lines: {10,12}, {40,1723}, {44,1842}, {55,387}, {71,1834}, {1865,1869}
As at X(3596), let A′, B′, C′ denote the respective excircles of a triangle ABC. Let A″ be the circle tangent to A′, B′, C′ whose interior includes A′, and define B″ and C″ cyclically. Let (M) be the circle externally tangent to all three of the circles A′, B′, C′; that is, the circular hull of A′, B′, C′, of which the center is X(6048). Also, (M) is tangent to the Apollonius circle at X(3030), and the tangent line through X(3030) passes through X(2254). The radius of (M) is r(r2 + s2)/(16rR - r2 - s2). (Peter Moses, August 26, 2014) The circle (M) is here named the Moses hull circle.
The Moses hull circle is the inverse of the incircle in the excircles-radical circle. (Randy Hutson, September 5, 2014)
Let (Ia), (Ib), (Ic) be the excircles of ABC, and let (AP) be the classical Apollonius circle (e.g., TCCT, p. 102). Let (Ja) be the circle internally tangent to (Ia) and externally tangent to (Ib) and (Ic), and define (Jb) and Jc) cyclically. Let (J) be the circle internally tangent to (Ja), (Jb), (Jc). Then X(6048) = center of (J). See ADGEOM #1541 and Problema de Apolonio relativo a las circunferencias exinscritas . (Angel Montesdeoca, February 25, 2017)
X(6048) lies on these lines: {1,2}, {46,1757}, {181,3339}, {210,986}, {238,5687}, {573,3973}, {748,3871}, {970,3030}, {984,3697}, {992,3169}, {1376,4252}, {1724,3550}, {1739,5904}, {1743,4274}, {2092,3731}, {2177,5047}, {2238,3501}, {2640,5974}, {3230,4050}, {3596,3760}, {3666,3983}, {3740,4646}, {3752,4662}, {3812,4849}, {3869,4695}, {3876,4642}, {3987,5692}, {4383,5255}, {4659,4721}
X(6048) = isogonal conjugate of X(36602)
X(6048) = X(979)-Ceva conjugate of X(1)
X(6048) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(10,43,1), (8,899, 978), (8,978,1), (200,1722,1), (1698,3293,1),
(3216,3679,1)
X(6049) is the perspector of ABC and a triangle denoted by A2B2C2 in Nikolaos Dergiades, Antirhombi.
X(6049) lies on these lines: {1,7}, {8,1319}, {12,1388}, {55,1476}, {56,3241}, {57,3623}, {145,1420}, {388,4870}, {496,944}, {519,5265}, {551,5261}, {651,1616}, {952,5704}, {1058,3655}, {1317,1788}, {3361,3635}, {3621,3911}, {3622,5226}, {4188,5193}, {5252,5550}
X(6049) = isogonal conjugate of X(33963)
X(6049) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(1,20,4345), (1,3600,4323), (1,4293,5734), (1,4308,7), (145,1420,5435),
(175,176,4862), (1388,3476,3616), (3600,4323,7), (4308,4323,3600),
(4315,4355,3600)
The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).
X(6050) lies on these lines: {512,4394}, {513,4401}, {649,4822}, {650,667}, {659,905}, {663,1635}, {1491,3803}, {1960,2516}, {4162,4730}, {4790,4983}
The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).
X(6051) lies on these lines: {1,6}, {2,3702}, {3,968}, {10,3706}, {21,593}, {38,5045}, {42,5044}, {55,975}, {58,3683}, {201,5173}, {227,5219}, {241,3671}, {495,1070}, {501,2360}, {595,3745}, {612,3295}, {750,3579}, {846,3916}, {942,2292}, {962,5308}, {986,5439}, {1010,3685}, {1046,4038}, {1125,3666}, {1193,1962}, {1214,3485}, {1621,5266}, {1698,4646}, {1961,5255}, {2334,3715}, {2478,5725}, {2999,3646}, {3120,3824}, {3178,3847}, {3293,3740}, {3624,3752}, {3634,4868}, {3670,3742}, {3739,4647}, {3750,5293}, {3812,4424}, {3871,5297}, {3874,4883}, {3915,5311}, {3989,5049}, {3995,4968}, {3997,4520}, {4187,5530}, {4295,4648}, {4340,5698}, {4673,4687}, {4682,5264}, {4850,5550}, {5250,5287}, {5262,5284}, {5268,5687}
X(6051) = X(4627)-Ceva conjugate of X(650)
X(6051) = radical center of circles (Ka), (Kb), (Kc) described at X(5213)
X(6051) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,984,3555), (1,1203,1100), (1,5259,1104), (2292,3720,942), (1125,3743, 3666), (5250,5287,5711)
The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).
X(6052) lies on this line: {2,3702}
Let A′B′C′ be the half-altitude 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(6053); see also X(i) for i = 74, 113, 265, 399, 1147, 1511, 5504, 5609, 5655. (Randy Hutson, August 26, 2014)
X(6053) lies on these lines: {6,13}, {20,110}, {74,3524}, {125,3090}, {133,648}, {140,5663}, {323,1533}, {541,1511}, {1387,2771}, {1539,3627}, {1596,3629}, {1619,2935}, {3448,5068}
Let P be a point on the Steiner circumellipse. Let A′ be the orthocenter of BCP, and define B′, C′ cyclically. Let Q be the centroid of A′B′C′. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(99), Q = X(6054). See also X(98) and X(24808). (Randy Hutson, October 15, 2018)
X(6054) lies on these lines: {2,98}, {4,543}, {13,6114}, {14,6115}, {30,99}, {115,3545}, {148,3839}, {262,381}, {376,2482}, {378,2936}, {383,530}, {523,1551}, {524,1513}, {531,1080}, {551,2784}, {620,3524}, {3845,6321}, {5071,5461}
X(6054) = midpoint of X(2) and X(147)
X(6054) = reflection of X(2) in X(114)
X(6054) = anticomplement of X(6055)
X(6054) = Thomson isogonal conjugate of X(187)
X(6054) = centroid of X(511)-Fuhrmann triangle
X(6054) = X(381)-of-1st-anti-Brocard-triangle
X(6054) = antipode of X(98) in the circle O′ described at X(6039)
X(6054) = McCay-to-Artzt similarity image of X(3)
X(6054) = X(671)-of-Artzt-triangle
X(6054) = 1st-tri-squares-to-Artzt similarity image of X(13640)
X(6054) = orthologic center of these triangles: Artzt to 1st Brocard
X(6055) lies on these lines: {2,98}, {3,543}, {30,115}, {99,3524}, {351,2793}, {376,671}, {381,2794}, {541,5465}, {549,2482}, {620,5054}, {2023,5052}, {2783,6174}, {2784,3828}, {3023,5298}, {3027,4995}, {3534,6321}, {3815,5477}, {5017,6034}, {5055,6033}, {5092,5939}
X(6055) = complement of X(6054)
X(6055) = circumcircle-inverse of X(33900)
X(6055) = {X(6108),X(6109)}-harmonic conjugate of X(115)
X(6055) = center of inverse-in-O′-of-line X(115)X(125), where O′ = inverter of circumcircle and nine-point circle
X(6055) = harmonic center of circles O(13,16) and O(14,15)
X(6055) = McCay-to-Artzt similarity image of X(5)
Lozada Perspectors
Suppose that A′B′C′ be the cevian triangle of a point P. Let L be the line through A′ tangent to the incircle, and let A* be the touchpoint; define B* and C* cyclically. Then A*B*C* is perspective to ABC for all P. (César Lozada, May 7, 2014, Anopolis #1466)
If P = p : q : r (trilinears), then the perspector is P* = a(b + c - a)p2 : b(a + c - b)q2 : c(a + b - c)r2.
In the following list, the appearance of (i,j) means that the Lozada perspector of X(i) is X(j):
{1,55}, {2,8}, {4,1857}, {6,2175}, {7,7}, {8,5423}, {9,480}, {11,5532}, {36,215}, {56,1397}, {57,56}, {63,1259}, {65,181}, {69,1264}, {75,3596}, {77,1804}, {81,60}, {86,261}, {88,1318}, {174,1}, {188,200}, {190,4076}, {226,12}, {241,1362}, {259,1253}, {266,31}, {278,1118}, {279,479}, {365,41}, {366,9}, {507,259}, {508,2}, {509,6}, {513,3271}, {514,11}, {519,4152}, {522,4081}, {523,4092}, {555,1088}, {556,341}, {650,3022}, {651,59}, {664,4998}, {900,4542}, {905,1364}, {1143,179}, {1274,400}, {1323,3321}, {1434,552}, {1465,1361}, {1638,3328}, {1659,1123}, {2003,2477}, {3008,3021}, {3218,4996}, {3666,1682}, {3668,6046}, {3669,1357}, {3676,1358}, {3776,3020}, {3911,1317}, {3960,3025}, {4146,75}, {4182,728}, {4369,3023}, {5374,219}, {5723,3322}
Let P = p : q : r (barycentrics). Let A′B′C′ be the cevian triangle of P, and let A*B*C* be the triangle defined in the preamble just above. The points A′, B′, C′, A*, B*, C* lie on a conic. (Vu Thanh Tung, February 15, 2021)
Peter Moses showed (February 16, 2021) that the conic just defined is given by the equation
f(a,b,c,p,q,r,x,y,z) + f(b,c,a,q,r,p,y,z,x) + f(c,a,b,r,p,q,z,x,y) = 0, where
f(a,b,c,p,q,r,x,y,z) = (a-b-c) q r ((a-b-c)^2 p^2+2 (a-b-c) (a-b+c) p q+(a-b+c)^2 q^2+2 (a-b-c) (a+b-c) p r+(a+b-c)^2 r^2) x^2+p ((a-b-c) p+(a-b+c) q+(a+b-c) r) ((a-b-c) (a-b+c) p q+(a-b+c)^2 q^2+(a-b-c) (a+b-c) p r-2 (a+b-c) (a-b+c) q r+(a+b-c)^2 r^2) y z
Continuing, Moses showed (February 17, 2021) gave the following generalization. Let T be the inscribed conic having perspector U = u : v : w. Let L be the line through A′ tangent to T, and let A*, given by
A* = w*(r*v - q*w^2 : q^2*v*w^2 : r^2*v^2*w,
be the touchpoint. Define B* and C* cyclically. Then A*B*C* is perspective to ABC with perspector v*w*p^2 : w*u*q^2 : u*v*r^2, and the points A′, B′, C′, A*, B*, C* lie on a conic, given by
g(a,b,c,p,q,r,u,v,w,x,y,z) + g(b,c,a,q,r,p,v,w,u,y,z,x) + g(c,a,b,r,p,q,u,u,v,z,x,y) = 0, where
g(a,b,c,p,q,r,u,v,w, x,y,z) = q*r*v*w*(r^2*u^2*v^2 - 2*p*r*u*v^2*w + q^2*u^2*w^2 - 2*p*q*u*v*w^2 + p^2*v^2*w^2)*x^2 - p*u*(-(r*u*v) - q*u*w + p*v*w)*(-(r^2*u*v^2) + 2*q*r*u*v*w + p*r*v^2*w - q^2*u*w^2 + p*q*v*w^2)*y*z
Trilinears (sin22A)/(1 - cos A) : (sin22B)/(1 - cos B) : (sin22C)/(1 - cos C)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4*(-a + b + c)*(a^2 - b^2 - c^2)^2
X(6056) lies on these lines: {3,1794}, {11,1751}, {48,184}, {55,219}, {56,580}, {72,993}, {154,197}, {255,1092}, {394,1364}, {577,4055}, {578,3074}, {1397,2352}, {1802,2188}, {2330,5227}
X(6056) = isogonal conjugate of isotomic conjugate of X(1259)
X(6056) = isogonal conjugate of polar conjugate of X(219)
X(6056) = isotomic conjugate of polar conjugate of isogonal conjugate of X(331)
X(6056) = X(19)-isoconjugate of X(331)
X(6056) = X(92)-isoconjugate of X(278)
X(6056) = crossdifference of every pair of points on line X(7178)X(17924)
In the consturciton of X(5213), let D be the touchpoints of the circles (KA) and (IA), and define E and F cyclically. Then triangle DEF is perspective to ABC at X(6057). (Randy Hutson, August 26, 2014)
X(6057) lies on these lines: {8,3058}, {10,3175}, {11,312}, {12,1089}, {42,3943}, {55,346}, {200,4069}, {210,2321}, {306,3967}, {321,3925}, {345,5432}, {442,4066}, {594,756}, {1211,3773}, {2325,3683}, {2886,4671}, {2887,4135}, {2895,4756}, {3021,3886}, {3685,4030}, {3687,4009}, {3701,3704}, {3706,3717}, {3710,3714}, {3711,5423}, {3712,4995}, {3741,4439}, {3952,3969}, {3994,4415}, {3995,4026}, {4096,4535}, {4420,5160}, {4519,4847}, {4527,4685}, {4819,4849}, {4863,4901}, {4903,5233}, {4942,5905}
X(6057) = isotomic conjugate of X(552)
X(6057) = trilinear square of X(6725)
X(6058) lies on these lines: {181,594}, {756,4092}, {1397,2345}
X(6059) lies on these lines: {25,1096}, {55,281}, {107,6015}, {158,242}, {181,3195}, {204,1460}, {607,2175}, {608,3271}, {1918,2207}
X(6059) = isogonal conjugate of X(7055)
X(6059) = isogonal conjugate of isotomic conjugate of X(1857)
X(6060) lies on these lines: {{8,21}, {20,1394}, {479,3188}, {2062,5932}, {3474,6046}
X(6061) lies on these lines: {21,60}, {27,5057}, {63,110}, {72,2074}, {212,5546}, {219,2189}, {405,1175}, {2287,2326}, {2327,2328}
X(6061) = isogonal conjugate of X(6046)
X(6062) lies on these lines: {11,214}, {55,4092}, {243,3326}, {1099,1354}, {1731,4542}, {3683,4081}
Let Q denote isotomic conjugate of isogonal conjugate; e.g., the line X(918)X(3261) = Q(Gergonne line), of which X(6063) is the trilinear pole. Also, X(6063) = Brianchon point (perspector) of the inellipse that is Q(incircle), of which the center is X(2886). (Randy Hutson, September 5, 2014)
X(6063) lies on these lines: {2,4554}, {7,310}, {75,1088}, {76,85}, {181,3212}, {222,4573}, {274,278}, {305,561}, {497,2481}, {666,5452}, {873,4625}, {927,2862}, {1111,3944}, {1376,4998}
X(6063) = isogonal conjugate of X(2175)
X(6063) = isotomic conjugate of X(55)
X(6063) = complement of X(21218)
X(6063) = anticomplement of X(16588)
X(6063) = X(75)-cross conjugate of X(76)
X(6063) = pole with respect to the polar circle of the triilinear polar of X(607)
X(6063) = X(48)-isoconjugate of X(607)
X(6063) = trilinear product of the extraversions of X(8)
X(6063) = trilinear product of vertices of Gemini triangle 40
X(6064) lies on these lines: {99,4897}, {249,1016}, {422,4601}, {645,3700}, {4062,4570}
X(6065) lies on these lines: {8,1016}, {59,518}, {100,2742}, {110,2748}, {644,3900}, {663,3939}, {840,4996}, {902,1110}
X(6065) = isogonal conjugate of X(1358)
X(6065) = perpsector of ABC and the reflection of the intouch triangle
in the line X(1)X(6)
X(6066) lies on these lines: {55,1252}, {1110,2149}, {4998,5432}
X(6067) lies on these lines: {2,480}, {5,5223}, {7,2886}, {8,3826}, {9,11}, {12,518}, {56,2550}, {75,4081}, {142,354}, {390,3813}, {442,5542}, {516,3916}, {528,4996}, {1001,1259}, {1317,3872}, {1329,5686}, {2646,5853}, {3174,4863}
X(6068) lies on Mandart hyperbola and these lines: {7,3035}, {8,190}, {9,11}, {12,5880}, {72,2801}, {100,144}, {119,5762}, {200,5528}, {214,5850}, {390,5854}, {516,1145}, {518,1317}, {527,1155}, {952,5223}, {2829,5759}, {3036,5686}, {3057,3271}, {4542,5853}, {4996,5852}, {5779,5840}
X(6068) = reflection of X(11) in X(9)
X(6068) = reflection of X(3059) in X(14740)
X(6068) = antipode of X(3059) in the Mandart hyperbola
X(6068) = antipode in Mandart inellipse of X(11)
Let ABC be a triangle. Let L be a line through A, let M be a line through B parallel to L, and let N be a line through C parallel to L. Let L′ be the reflection of L in line BC, let M′ be the reflection of M in CA, and let N′ be the reflection of N in AB. Let A′B′C′ be the triangle formed by the lines L′, M′, N′. As L rotates about A, the Euler line of A′B′C′ rotates about a fixed point, X(6069).
X(6069) is described at http://anthrakitis.blogspot.com/2014/04/euler-lines-of-triagles-bounded-by.html (and misrepresented there as the Parry-Pohoata point, X(8157)).
X(6069) lies on this line: {68,1658}
Points Associated with the Steiner Deltoid
Suppose that ABC is a triangle. Its Steiner deltoid, SD, is the envelope of the Simson lines of ABC, as shown dynamically at Simson line. Suppose the P = p : q : r (barycentrics) is a point, other than A, B, C, on the circumcircle, denoted by (O), of ABC. Peter Moses (September 14, 2014) defines a mapping from (O) to SD; the mapping is here denoted by M and named the Moses-Steiner image, as follows:
M(P) = b4c4p6(b2rSB - c2qSC)2(c2q2 + b2 r2 + 2qrSA)[SA(c2q2 + b2r2) + 2qr(S2 + S2A)].
The Simson line of P is tangent to SD at M(P), and the Simson line intersects SD, in two other points, which are the points of intersection of the Simson line with the circle that has radius R and center the midpoint of P and the orthocenter, H. The SD is given by the equation s(a,b,c,x,y,z) + s(b,c,a,y,z,x) + s(c,a,b,z,x,y) = 0, where
s(a,b,c,x,y,z) = (a^2 + b^2 - c^2)^2 (a^2 - b^2 + c^2)^2 x^4 - 4 (a^2 + b^2 - c^2) (a^6 - 2 a^4 b^2 + a^2 b^4 - 2 a^4 c^2 - 2 a^2 b^2 c^2 - 4 b^4 c^2 + a^2 c^4 + 4 b^2 c^4) y^3 z - 4 (a^2 - b^2 + c^2) (a^6 - 2 a^4 b^2 + a^2 b^4 - 2 a^4 c^2 - 2 a^2 b^2 c^2 + 4 b^4 c^2 + a^2 c^4 - 4 b^2 c^4) z^3 y + 2 (3 a^8 - 6 a^6 b^2 + 2 a^4 b^4 + 2 a^2 b^6 - b^8 - 6 a^6 c^2 - 8 a^4 b^2 c^2 - 2 a^2 b^4 c^2 + 16 b^6 c^2 + 2 a^4 c^4 - 2 a^2 b^2 c^4 - 30 b^4 c^4 + 2 a^2 c^6 + 16 b^2 c^6 - c^8) y^2 z^2 + 4 (a^8 - 9 a^6 b^2 + 15 a^4 b^4 - 7 a^2 b^6 - 9 a^6 c^2 + 48 a^4 b^2 c^2 + 7 a^2 b^4 c^2 + 2 b^6 c^2 + 15 a^4 c^4 + 7 a^2 b^2 c^4 - 4 b^4 c^4 - 7 a^2 c^6 + 2 b^2 c^6) x^2 y z.
Let L1 be the line of the midpoint of P and H and the circumcircle-antipode of the isogonal conjugate of P. Let W be the point of intersection of the line of P and the isogonal conjugate of P with the line of the circumcircle-antipode of P and the orthopoint of the isogonal conjugate of P. Let L2 be the line of W and the point of intersection of the anticomplement of P and the circumcircle-antipode of P. Then M(P) = L1∩L2. For example, if P = X(110), then L1 = X(30)X(113) and L2 = X(146)X(476), and M(P) = X(1553), the only point X(i), for i < 6070, that lies on SD. (Peter Moses, September 14, 2014).
Let P* denote the circumcircle-antipode of P, and let Q= M(P*); then |P*Q| = 2R. Let U(P) be the reflection of X(4) in the midpoint of segment P*Q. Then U(P) lies on the circumcircle, so that the point T = M(U(P)) is on the SD; indeed, the line M(P)Q is tangent to SD at T (Moses, September 20, 2014). The mapping U is the Moses-Steiner ortho-image, and T, the Moses-Steiner tangential-image. Examples:
U(X(74)) = U(X(110)) = X(901)
U(X(98)) = U(X(99)) = X(805)
U(X(100)) = U(X(104)) = X(476)
U(X(101)) = U(X(103)) = X(927)
U(X(102)) = U(X(109)) = X(1309)
U(X(112)) = U(X(1297)) = X(2867)
U(X(105)) = U(X(1292)) = X(6078)
U(X(106)) = U(X(1293)) = X(6079)
U(X(107)) = U(X(1294)) = X(6080)
U(X(108)) = U(X(1295)) = X(6081)
U(X(111)) = U(X(1296)) = X(6082)
U(X(759)) = U(X(6011)) = X(6083)
Another construction of U(P), along with barycentrics, follows (Moses, September 20, 2014). Let L3 be the line through P perpendicular to the Simson line of P, and let L4 be the line through P perpendicular to the Simson line of M(P). Then U = L3∩L4. Moreover, L3 and L4 both meet the nine-point circle in the midpoint, U′, of U and P; thus, a third construction of U(P) is the -2 dilation of U′ in G. Barycentrics for U are u(a,b,c,p,q,r) : u(b,c,a,q,r,p) : u(c,a,b,r,p,q), where
u(a,b,c,p,q,r) = a6q6r6(a2q2 - b2p2)(a2r2 - c2p2)/[(b2 + c2 - a2)qr) + b2r2 + c2q2]2,
and barycentrics for the point T = M(U(P)) are t(a,b,c,p,q,r) : t(b,c,a,q,r,p) : t(c,a,b,r,p,q), where
t(a,b,c,p,q,r) = a^4 (2 b^2 c^2 q r+c^2 q^2 SA+b^2 r^2 SA)^2 (6 b^4 c^4 q^2 r^2 (S^2+SB SC)+c^6 q^4 (4 S^2 SA-b^2 (3 S^2-SB SC))+b^6 r^4 (4 S^2 SA-c^2 (3 S^2-SB SC))+4 b^2 c^2 q r (b^4 r^2 (S^2-SB^2)+c^4 q^2 (S^2-SC^2))))/(c^2 q^2+b^2 r^2+2 q r SA)^4.
The vertices of the SD form a central triangle A′B′C′, here named the 1st Steiner equilateral triangle, for which Moses (September 14, 2014) gives these trilinears:
cos(B - C) + 3 cos(B/3 - C/3) : cos(C - A) - 3 cos(B/3 + 2C/3) : cos(A - B) - 3 cos(2B/3 + C/3).
The SD touches the nine-point circle in points that form another central triangle A*B*C*, here named the 2nd Steiner equilateral triangle, for which A* has these trilinears:
cos(B - C) - cos(B/3 - C/3) : cos(C - A) + cos(B/3 + 2C/3) : cos(A - B) + cos(2B/3 + C/3).
Properties of these triangles and the SD are given in C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications (Dover, 1963), and H. Dorrie, Great Problems of Elementary Mathematics (Dover, 1963), p. 226. Such properties are as follows: The triangles A′B′C′ and A*B*C* are homothetic at X(5), with ratio -1/3. The circumcircle of A′B′C′ is centered at X(5) with radius 3R/2. The SD is tangent to the sidelines of ABC at the vertices of the pedal triangle of X(20). The triangles ABC, BCH, CAH, and ABH have the same SD.
A construction of A* is given in H. S. M. Coxeter, Introduction to Geometry, 2nd ed., p. 115: Let DEF be the medial triangle and UVW the orthic triangle; on the ninepoint circle, A* is the point of trisection of the arc DU.
The radical axes of the Stammler circles are tangent to the SD. These axes are the Simson lines of the circumtangential triangle.
The triangle A*B*C* is the complement of the triangle A′B′C′, which is homothetic to the circumtangential triangle at X(20) with ratio 3/2. The circumtangential is homothetic to A*B*C* at X(2) with ratio -1/2. The triangle A*B*C* is homothetic to the circumnormal triangle at X(4) with ratio 1/2, and the circumnormal triangle is homothetic to A′B′C′ at X(631) with ratio -3/2. The homothetic centers X(2), X(4), X(20), X(631) all lie on the Euler line.
The Stammler triangle is homothetic to A′B′C′ at X(3526) with ratio -3/4, and also homothetic to A*B*C* at X(381) with ratio 1/4. The homothetic centers X(3526) and X(381) lie on the Euler line.
Let LA be the line tangent to the SD at A*, and define LB and LC cyclically. The lines LA, LB, LC form an equilateral triangle which is the complement of the Stammler triangle, homothetic to A′B′C′ at X(5) with ratio 2/3. Trilinears for the A-vertex are given by Moses as follows:
cos(B - C) + 2 cos(B/3 - C/3) : cos(C - A) - 2 cos(B/3 + 2C/3) : cos(A - B) - 2 cos(2B/3 + C/3).
As a point P traverses the SD, the locus of the orthopoles of N to P on the SD is an ellipse inscribed to SD; its center is X(546) and it passes through X(125) and Z(1539). (Peter Moses, September 14, 2014). A barycentric equation for this Moses-Steiner ellipse is
f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0, where
f(a,b,c,x,y,z) = (a^2-b^2-2 c^2) (a^2-2 b^2-c^2) (4 a^4-3 a^2 b^2-b^4-3 a^2 c^2+2 b^2 c^2-c^4) x^2 + 2 (2 a^8+2 a^6 b^2-9 a^4 b^4+4 a^2 b^6+b^8+2 a^6 c^2+20 a^4 b^2 c^2-4 a^2 b^4 c^2-9 b^6 c^2-9 a^4 c^4-4 a^2 b^2 c^4+16 b^4 c^4+4 a^2 c^6-9 b^2 c^6+c^8) y z
The Simson line of X(74) is parallel to the orthic axis.
X(6070) lies on the Steiner deltoid and these lines: {4,5627}, {125,523}, {137,2970}, {476,3448}, {868,5489}, {1522,1523}
The Simson line of X(98) is perpendicular to the Brocard axis and parallel to the Lemoine axis.
X(6071) lies on the Steiner deltoid and these lines: {115,512}, {148,805}, {1567,2782}
The Simson line of X(99) is parallel to the Brocard axis.
X(6072) lies on these lines: {114,325}, {147,805}, {1567,2782}
The Simson line of X(100) is parallel to the line X(1)X(3).
X(6073) lies on the Steiner deltoid and these lines: {119,517}, {153,901}
The Simson line of X(101) is parallel to the Soddy line.
X(6074) lies on the Steiner deltoid and these lines: {4,5377}, {118,516}, {152,927}, {1517,2808}
The Simson line of X(104) is parallel to the antiorthic axis and perpendicular to X(1)X(3).
X(6075) lies on the Steiner deltoid and these lines: {11,513}, {36,851}, {149,901}, {244,1365}, {764,1647}, {1319,3011}, {2841,5533}, {2969,3756}, {3814,4871}, {5176,5205}
The Simson line of X(111) is perpendicular to X(2)X(6).
X(6076) lies on the Steiner deltoid and these lines: {1499,2686}, {6077,6092}
X(6076) = reflection of X(6077) in X(6092)
X(6076) = orthojoin of X(2482)
The Simson line of X(1296) is parallel to X(2)X(6).
X(6077) lies on the Steiner deltoid and these lines: {126,524}, {6076,6092}
Let A′ = BC∩X(8)X(9), and define B′ and C′ cyclically. The circumcircles of AB′C′, BC′A′, CA′B′ concur in X(6078). (Randy Hutson, September 29, 2014)
X(6078) lies on the circumcircle and these lines: {2,5519}, {100,4394}, {101,6065}, {103,1810}, {105,518}, {106,5526}, {109,1252}, {644,1292}, {649,1293}, {672,1477}, {759,5525}, {883,927}, {901,2441}, {919,2284}, {934,4564}, {1023,1308}
X(6078) = isogonal conjugate of X(6084)
X(6078) = anticomplement of X(5519)
X(6078) = trilinear pole of the line X(6)X(3939)
X(6078) = intersection of antipedal lines of X(105) and X(1292)
X(6078) = Ψ(X(6), X(3939)
From Randy Hutson (September 28, 2014):
X(6079) = Ψ(X(6), X(644))
X(6079) = Λ(X(244), X(1357)); the line X(244)X(1357) is the
trilinear polar of X(3669)
X(6079) = Λ(X(121), X(6085)); the line X(121)X(6085) is tangent
to the nine-point circle at X(121)
X(6079) = Λ(X(2254), X(3030)); the line X(2254)X(3030) is
tangent to the Apollonius circle at X(3030)
X(6079) lies on the circumcircle and these lines: {2,5516}, {100,4076}, {104,1811}, {105,5205}, {106,519}, {109,765}, {190,6014}, {741,5524}, {934,4998}, {1293,3667}
X(6079) = isogonal conjugate of X(6085)
X(6079) = isotomic conjugate of X(4927)
X(6079) = anticomplement of X(5516)
X(6079) = intersection of antipedal lines of X(106) and X(1293)
X(6079) = trilinear pole of the line X(6)X(644)
X(6080) lies on the circumcircle and these lines: {107,520}, {1294,6000}, {2693,3357}
X(6080) = isogonal conjugate of X(6086)
X(6080) = anticomplement of X(35579)
X(6080) = trilinear pole of line X(6)X(35071)
X(6080) = Ψ(X(6), X(35071))
X(6080) = intersection of antipedal lines of X(107) and X(1294)
X(6081) lies on the circumcircle and these lines: {84,2716}, {108,521}, {189,2723}, {271,2733}, {280,2734}, {934,4131}, {1295,6001}, {2357,2708}
X(6081) = isogonal conjugate of X(6087)
X(6081) = anticomplement of X(35580)
X(6081) = trilinear pole of the line X(6) X(268)
X(6081) = intersection of antipedal lines of X(108) and X(1295)
X(6081) = Ψ(X(6),X(268))
Let H be the rectangular circumhyperbola that has center X(6092) and has major axis parallel to the line X(2)X(6). Then X(6082) is the point, other than A, B, C, in which H intersects the circumcircle. (Randy Hutson, September 29, 2014)
X(6082) lies on the circumcircle and these lines: {98,5971}, {111,524}, {352,729}, {691,5468}, {1296,1499}, {5108,5970}
X(6082) = reflection of X(6093) in X(3)
X(6082) = isogonal conjugate of X(6088)
X(6082) = anticomplement of X(31654)
X(6082) = trilinear pole of the line X(6)X(2482)
X(6082) = Ψ(X(96), X(2482))
X(6082) = Λ(X(111), X(351))
X(6082) = Collings transform of X(6092)
X(6082) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(138)}}
X(6082) = intersection of antipedal lines of X(111) and X(1296)
X(6083) lies on the circumcircle and these lines: {99,4897}, {105,2651}, {106,5127}, {108,5379}, {109,4570}, {643,6003}, {758,759}
X(6083) = isogonal conjugate of X(6089)
X(6083) = anticomplement of X(35583)
X(6083) = trilinear pole of the line X(6)X(5546)
X(6083) = Ψ(X(6), X(5546))
X(6084) lies on these lines: {2,4927}, {30,511}, {105,659}, {120,2977}, {190,2415}, {673,2402}, {903,2403}, {1086,1358}, {1635,1638}, {1639,4728}, {2487,3676}, {2490,4885}, {2505,4925}, {2527,4369}, {2976,3021}, {3039,4422}, {3700,4382}, {4106,4468}, {4380,4897}, {4458,4830}, {4750,4773}
X(6084) = crossdifference of every pair of points on the line X(6)X(3939)
X(6085) lies on these lines: {30,511}, {106,1960}, {244,1357}, {659,1054}, {665,3768}, {764,1769}, {1293,3939}, {2254,3030}, {2316,2441}, {3888,3952}
X(6085) = crossdifference of every pair of points on the line X(6)X(644)
X(6086) lies on these lines: {30,511}, {107,1624}
X(6086) = crossdifference of every pair of points on line X(6)X(35071)
X(6086) = complementary conjugate of X(35579)
X(6087) lies on these lines: {30,511}, {108,676}
X(6087) = isogonal conjugate of X(6081)
X(6087) = complementary conjugate of X(35580)
X(6087) = crossdifference of every pair of points on the line X(6)X(268)
X(6088) lies on these lines: {30,511}, {111,351}
X(6088) = crossdifference of every pair of points on the line X(6)X(2482)
X(6088) = X(523)-of-4th-anti-Brocard-triangle
X(6088) = X(526)-of-circumsymmedial-triangle
X(6089) lies on these lines: {30,511}, {659,1283}, {1365,2611}, {1635,1637}
X(6089) = crossdifference of every pair of points on the line X(6)X(5546)
X(6089) = complementary conjugate of X(35583)
X(6090) occurs in connection with the Steinhaus cubic: (K698)
Let H be the Thomson-Gibert-Moses hyperbola. Let U be the line tangent at X(6) to H, and let V the the line tangent at X(154) to H. Then X(6090) = U∩V. (Randy Hutson, September 29, 2014)
X(6090) lies on these lines: {2,3167}, {3,74}, {6,373}, {25,394}, {51,5102}, {69,468}, {154,3917}, {184,5085}, {323,1351}, {352,1384}, {426,1073}, {520,2433}, {576,3066}, {599,5642}, {1092,1593}, {1316,5468}, {1350,1495}, {1352,5094}, {1975,4563}, {1993,5020}, {2715,2763}, {3515,5562}, {3516,5907}, {3796,3819}, {5108,5967}
X(6090) = crossdifference of every pair of points on the line
X(1499)X(1514)
X(6090) = pole with respect to the Thomson-Gibert-Moses hyperbola of
the line X(6)X(25)
X6090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(184,5650,5085), (323,1995,1351), (1993,5640,5093), (3292,5651,6),
(5020,5093,5640)
X(6091) occurs in connection with the Steinhaus cubic: (K698)
X(6091) lies on these lines: {3,895}, {22,111}, {186,691}, {187,5166}, {376,671}
X(6091) = X(111)-Ceva conjugate of X(895)
X(6091) = inverse-in-circumcircle of X(895)
X(6092) is the center of the rectangular hyperbola {A, B, C, X(4), X(6082)}. This hyperbola is the isogonal conjugate of the line X(3)X(6088); its major axis is parallel to the line X(2)X(6). (Randy Hutson, September 29, 2014)
For a construction, see, Antreas Hatzipolakis and Peter Moses, Hyacinthos 29114.
X(6092) lies on the nine-point circle and these lines: {2, 6093}, {4, 6082}, {5, 31654}, {125, 8355}, {126, 1499}, {524, 5512}, {599, 5099}, {5139, 16183}, {6076, 6077}, {20388, 20389}
X(6092) = midpoint of X(i) and X(j) for these {i,j}: {4, 6082}, {6076, 6077}X(6093) lies on these lines: {2,6092}, {3,6082}, {111,1499}, {524,1296}, {691,1992}
X(6093) = isogonal conjugate of X(33962)
X(6093) = anticomplement of X(6092)
X(6093) = trilinear pole of line X(6)X(9125)
X(6093) = circumcircle-antipode of X(6082)
X(6093) = Λ(X(3),X(111))
Suppose that P = p : q : r (barycentrics) and P* are a pair of isogonal conjugate points in the plane of a triangle ABC. Let A′B′C′ be the pedal triangle of P and A″B″C″ the pedal triangle of P*. Let A* be the reflection of A′ in line B″C″, and define B* and C* cyclically. In Hyacinthos (October 3, 2014), Antreas Hatzipolakis asks for the locus of P for which the circumcircles of A*BC, B*CA, C*AB concur. Angel Montesceoca responds that the three circumcircles concur for all choices of P. He further notes that if P is not on the circumcircle and not on the line at infinity, then the point Q of conurrence of the three circles has barycentrics Q(a,b,c,p,q,r) : Q(b,c,a,q,r,p) : Q(c,a,b,r,p,q) given by
Q(a,b,c,p,q,r) = (q + r)/[a4qr(p + q)(p + r) - 2a2(q + r)(r + p)(p + q)(c2q + b2r) + p(q + r)(b4r(p + q) + c4q(q + r) + 2b2c2(q2 + r2 + pq + qr + rp))]
Writing Q as Q(P), the occurrence of (i,j) in the following list means that Q(X(i)) = X(j): (1,5620), (2,6094), (3, 1263), (8, 6095), (20, 265), (69, 6096), (1138, 1138). In particular, Q(X(2)) = X(6094).
X(6094) lies on these lines: {6,543}, {263,2854}
X(6094) = isogonal conjugate of X(352)
X(6094) = trilinear pole of the line X(373)X(512)
X(6095) = Q(X(8)); see X(6094).
X(6095) lies on these lines: {1,121}, {56,2802}, {106,1739}
X(6096) = Q(X(69)); see X(6094).
X(6096) lies on this line: {25,2854}
X(6096) = isogonal conjugate of X(5913)
Let ABC be a triangle and A′B′C′ the cevian triangle of X(1). Let OAB be the circumcenter of ABA′, and define OBC and OCA cyclically; let OAC be the circumcenter of ACA′, and define OBA and OCB cyclically. Let OA be the circumcenter of triangle AOABOAC, and define OB and OC cyclically. Hatzipolakis proposed, and Montesdeoca proved, that the Euler lines concur, in X(186), and that the orthocenter of triangle OAOBOC, which is X(6097), lies on the Euler line. See Anthrakitis (October 3, 2014)
X(6097) lies on these lines: {2,3}, {35,500}, {55,5453}, {511,5495}, {3724,5492}
Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let OA be the circumcenter of A′BC, and define OB and OC cyclically. The circumcircles of the four triangles A″B″C″, A″BC, AB″C, ABC″ concur in X(6098). See Hyacinthos 22617, October 8, 2014.
X(6098) lies on this line: {119,1845}
Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let OA be the circumcenter of A′BC, and define OB and OC cyclically. The circumcircles of the four triangles ABC, AB″C″, A″BC″, A″B″C concur in X(6099). The lines OAA″, OBB″, OCC″ also concur in X(6099). See Hyacinthos 22617, October 8, 2014.
Let T be the triangle whose sidelines are the reflections of the line X(3)X(11) in the sidelines of ABC. Then T is perspective to ABC, and X(6099) is the perpsector. (Randy Hutson, October 16, 2014)
X(6099) lies on the circumcircle and these lines: {59,108}, {100,1618}, {104,912}, {105,2990}, {107,5379}, {517,915}, {917,5057}, {929,3573}, {1290,3657}, {1300,5080}, {1311,3006}, {3326,6056}
X(6099) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(11)Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let OA be the circumcenter of A′BC, and define OB and OC cyclically. The points A″, B″, C″, X(6098) lie on a circle, of which the center is X(6100). See Hyacinthos 22617, October 8, 2014.
X(6100) lies on this line: {6001, 12515}
Let AB be the reflection of A in line OB, where O = circumcenter of ABC, and define BC and CA cyclically. Let AC be the reflection of A in line OC, and define BA and CB cyclically. Let A′ be the nine-point center of triangle AABAC, and define B′ and C′ cyclically. The circumcircles of the four triangles A′B′C′, A′BC, AB′C, ABC′ concur in X(6101). (Also, the circumcircles of the four triangles ABC, AB′C′, A′BC′, A′B′C concur in X(930)). See Hyacinthos 22624, October 10, 2014.
X(6101) lies on these lines: {2,143}, {3,54}, {4,2889}, {5,141}, {20,5663}, {22,156}, {26,394}, {30,5562}, {49,323}, {51,3628}, {52,140}, {67,68}, {110,2937}, {155,1350}, {185,548}, {389,549}, {546,5891}, {568,631}, {632,3819}, {1092,1511}, {1147,5944}, {1656,3060}, {2392,5694}, {2781,5609}, {3313,3564}, {3526,3567}, {3627,5907}, {5070,5640}
X(6101) = reflection of X(i) in X(j) for these (i,j): (5,1216), (52,140), (185,548), (389,5447), (3627,5907), (5946,3917)
X(6101) = complement of X(6243)
X(6101) = anticomplement of X(143)
X(6101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,195,5012), (52,140,5946), (52,3917,140), (389,5447,549), (1092,1658,1511), (3819,5462,632)
Let AB be the reflection of A in line OB, where O = circumcenter of ABC, and define BC and CA cyclically. Let AC be the reflection of A in line OC, and define BA and CB cyclically. Let A′ be the nine-point center of triangle AABAC, and define B′ and C′ cyclically. Then X(6102) is the ABC-orthology center of A′B′C′ on the circumcircle of A′B′C′. (Also, X(1141) is the ABC-orthology center of ABC on the circumcircle of ABC.) See Hyacinthos 22628 and Hechos Geometricos 121014, October 12, 2014.
Let NA be the reflection of X(5) in the A-altitude, and define NB and NC cyclically. Then X(6102) is the orthocenter of NANBNC. Let A′B′C′ be the reflection triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of the line BC, and define B″ and C″ cyclically. Let A* be the trilinear pole, with respect to A′B′C′, of line B″C″, and define B* and C* cyclically. The lines A′A*, B′B*,C′C* concur in X(6102), and the lines A′A″, B′B″, C′C″ concur in X(382). (Randy Hutson, October 16, 2014)
X(6102) lies on these lines: {3,54}, {4,94}, {5,389}, {24,156}, {26,1181}, {30,52}, {49,186}, {51,546}, {140,5562}, {155,2929}, {184,1658}, {381,3567}, {382,3060}, {511,550}, {549,1216}, {567,1199}, {576,2781}, {632,5892}, {974,1204}, {1147,1511}, {1614,2070}, {1994,3520}, {3530,3917}, {3627,5446}, {3628,5891}, {3851,5640}
X(6102) = midpoint of X(i) and X(j) for these (i,j): (52,185), (3,5889)
X(6102) = reflection of X(i) in X(j) for these (i,j): (4,143), (5,389), (5562,140), (3627,5446), (5876,5), (5907,5462)
X(6102) = X(5)-of-circumorthic-triangle
X(6102) = X(355)-of-orthic-triangle if ABC is acute
X(6102) = X(80)-of-1st-Hyacinth-triangle if ABC is acute
X(6102) = homothetic center of X(4)-altimedial and X(5)-adjunct anti-altimedial triangles
X(6102) = Ehrmann-side-to-orthic similarity image of X(12111)
X(6102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(4,568,143), (5,389,5946), (184,1658,5944), (389,5907,5462),
(5462,5907,5), (5876,5946,5), (5889,5890,3)
Dao-Moses-Telv Circle and Associated Points
The Dao-Moses-Telv circle is defined as the circle passing through these six points:
F1 = X(13), the 1st Fermat point
F2 = X(14), the 2nd Fermat point
F3 = inverse of F1 in the circumcircle
F4 = inverse of F2 in the circumcircle
F5 = inverse of F1 in the nine-point circle
F6 = inverse of F2 in the nine-point circle
Dao Thanh Oai noted that these points are concyclic, Telv Cohl gave a proof, and Peter Moses discovered properties described in this preamble, along with centers X(6103)-X(6111). The circle is first noted at ADGEOM 971. See also ADGEOM 971 DB.
The center of the Dao-Moses-Telv circle is X(1637). (Francisco Javier, ADGEOM 1977, November 3, 2014)
The Dao-Moses-Telv circle is orthogonal to the circumcircle, the nine-point circle, and all the other circles in their coaxal family. The A-power of the circle is
(b2 + c2 - a2)[((b2 + c2 - a2)2 - b2c2]/[6(a2 - b2)(a2 - c2)]
The circle passes through X(5000) and X(5001), these being the Walsmith point and its inverse in the circumcircle. (Peter Moses, November 6, 2014)
The Dao-Moses-Telv circle is defined in the preamble just before X(6103).
X(6103) = exsimilicenter of the circles {{X(98), X(107), X(125), X(132)}} and {{X(6), X(111), X(112), X(115), X(187), X(1560)}}. The trilinear polar of X(6103) passes through X(1640). (Randy Hutson, November 30, 2014)
X(6103) lies on the Dao-Moses-Telv circle, the cubics K483 and K1353, and these lines: {2, 648}, {4, 32}, {5, 52951}, {6, 67}, {23, 52945}, {25, 1989}, {39, 37118}, {53, 10301}, {74, 6794}, {107, 111}, {113, 40354}, {183, 11331}, {187, 5523}, {216, 7495}, {230, 231}, {233, 7570}, {264, 7806}, {297, 22329}, {317, 63048}, {325, 56021}, {339, 6720}, {340, 385}, {378, 5309}, {403, 14581}, {427, 5306}, {542, 52469}, {543, 4235}, {577, 16063}, {620, 41676}, {671, 40890}, {858, 3284}, {1112, 61675}, {1235, 6680}, {1249, 52290}, {1562, 10990}, {1594, 5007}, {1609, 8428}, {1625, 16534}, {1627, 36423}, {1640, 60509}, {1691, 51434}, {1971, 51363}, {1986, 15544}, {2072, 52950}, {2079, 3515}, {2165, 51936}, {2420, 17702}, {2549, 35485}, {2697, 59108}, {2967, 6036}, {3016, 61209}, {3053, 37196}, {3087, 63097}, {3172, 13881}, {3269, 20417}, {3516, 34866}, {3520, 7765}, {3541, 5319}, {3569, 32119}, {3788, 56015}, {3815, 52293}, {4230, 9149}, {5063, 59768}, {5139, 15546}, {5189, 9380}, {5254, 35903}, {5304, 52284}, {5305, 44468}, {5477, 11005}, {5702, 7736}, {6055, 16279}, {6108, 6110}, {6109, 6111}, {6143, 9698}, {6240, 35007}, {6353, 36898}, {6699, 64645}, {6721, 10414}, {6781, 56369}, {7426, 18487}, {7473, 16092}, {7533, 36412}, {7574, 10317}, {7577, 7753}, {7687, 9408}, {7746, 8743}, {7749, 39575}, {7772, 37119}, {7792, 39931}, {7813, 56016}, {7829, 37125}, {7856, 37337}, {8739, 62199}, {8740, 62200}, {8744, 52192}, {8778, 44518}, {8859, 37765}, {8889, 63006}, {9300, 62958}, {9412, 13202}, {9475, 15000}, {10018, 41366}, {10104, 39604}, {10314, 61327}, {10316, 14791}, {10766, 32247}, {11063, 21284}, {11646, 32250}, {11648, 35481}, {12173, 22331}, {13114, 53345}, {13473, 63543}, {13638, 62956}, {13758, 62957}, {13854, 14910}, {14002, 61314}, {14357, 60507}, {14568, 15014}, {14590, 62298}, {14961, 15122}, {15595, 63464}, {16188, 51456}, {17008, 17907}, {17968, 52477}, {17986, 34366}, {18312, 60513}, {22240, 52300}, {23969, 40118}, {25555, 59208}, {27376, 37458}, {28343, 32274}, {30497, 57608}, {30714, 32661}, {30745, 52058}, {33843, 35484}, {34369, 51428}, {34570, 65060}, {35486, 41361}, {36794, 63019}, {37070, 62512}, {37637, 45141}, {37690, 56013}, {37981, 52952}, {37984, 43291}, {38227, 41204}, {38552, 43087}, {39170, 41392}, {39563, 57584}, {40347, 41489}, {40938, 63846}, {41370, 43620}, {41512, 56399}, {43448, 49670}, {43457, 50718}, {44534, 57262}, {45280, 48981}, {46942, 53723}, {47322, 47327}, {48453, 50188}, {51437, 53475}, {52283, 63029}, {52299, 63024}, {52301, 62195}, {57655, 58908}, {59229, 62981}, {60502, 60511}, {61304, 62950}
X(6103) = isogonal conjugate of X(65308)
X(6103) = Dao-Moses-Telv-circle-inverse of X(468)
X(6103) = Moses-radical-circle-inverse of X(47223)
X(6103) = Moses-Parry-circle-inverse of X(232)
X(6103) = polar conjugate of X(5641)
X(6103) = complement of the isotomic conjugate of X(2697)
X(6103) = isogonal conjugate of the isotomic conjugate of X(60502)
X(6103) = polar conjugate of the isotomic conjugate of X(542)
X(6103) = polar conjugate of the isogonal conjugate of X(5191)
X(6103) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 42426}, {2697, 2887}
X(6103) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 42426}, {648, 60509}, {16080, 52469}, {35907, 1640}, {40118, 25}, {50188, 36201}, {60502, 542}
X(6103) = X(i)-cross conjugate of X(j) for these (i,j): {1640, 35907}, {5191, 542}, {47427, 6}
X(6103) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65308}, {48, 5641}, {63, 842}, {162, 35911}, {293, 46787}, {336, 52199}, {656, 5649}, {662, 35909}, {810, 6035}, {4575, 14223}, {4592, 14998}, {8552, 36096}, {34055, 46157}, {35200, 51228}, {36060, 52094}
X(6103) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65308}, {125, 35911}, {132, 46787}, {133, 51228}, {136, 14223}, {1084, 35909}, {1249, 5641}, {1560, 52094}, {3162, 842}, {5139, 14998}, {23967, 69}, {35582, 14417}, {38970, 34765}, {39062, 6035}, {40596, 5649}, {42426, 2}, {48317, 50942}, {61503, 57481}
X(6103) = cevapoint of X(6041) and X(51428)
X(6103) = crosspoint of X(2) and X(2697)
X(6103) = crosssum of X(6) and X(2781)
X(6103) = crossdifference of every pair of points on line {3, 684}
X(6103) = pole of the Euler line with respect to the Dao-Moses-Telv circle
X(6103) = pole of the trilinear polar of X(5641) with respect to the polar circle
X(6103) = PU(4)-harmonic conjugate of X(1637)
X(6103) = barycentric product X(i)*X(j) for these {i,j}: {4, 542}, {6, 60502}, {30, 17986}, {92, 2247}, {98, 54380}, {112, 18312}, {186, 43087}, {232, 46786}, {264, 5191}, {297, 34369}, {403, 51456}, {468, 16092}, {511, 52491}, {523, 7473}, {525, 35907}, {648, 1640}, {690, 53155}, {842, 38552}, {935, 55142}, {1503, 47105}, {1990, 51227}, {2501, 14999}, {2697, 42426}, {2781, 58087}, {6041, 6331}, {14223, 60505}, {14273, 50941}, {16188, 40118}, {16230, 34761}, {17983, 45662}, {18020, 51428}, {18808, 64607}, {23968, 44427}, {44145, 61446}, {45321, 65349}, {46106, 48451}, {51405, 60428}, {52469, 54527}, {60508, 60590}, {60512, 60591}
X(6103) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5641}, {6, 65308}, {25, 842}, {112, 5649}, {232, 46787}, {419, 57452}, {460, 34174}, {468, 52094}, {512, 35909}, {542, 69}, {647, 35911}, {648, 6035}, {1640, 525}, {1843, 46157}, {1990, 51228}, {2211, 52199}, {2247, 63}, {2489, 14998}, {2501, 14223}, {5191, 3}, {6041, 647}, {7473, 99}, {14273, 50942}, {14581, 48453}, {14999, 4563}, {16092, 30786}, {16230, 34765}, {17986, 1494}, {17994, 23350}, {18312, 3267}, {18384, 54554}, {23347, 51263}, {23968, 60053}, {32696, 53691}, {34369, 287}, {34397, 52179}, {34761, 17932}, {34854, 52492}, {35907, 648}, {42671, 40080}, {43087, 328}, {45662, 6390}, {46786, 57799}, {47105, 35140}, {48451, 14919}, {51428, 125}, {51456, 57829}, {52491, 290}, {53155, 892}, {54380, 325}, {58348, 16163}, {60502, 76}, {60505, 14999}, {61446, 43705}
X(6103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 115, 36204}, {125, 6793, 6}, {230, 1990, 468}, {230, 3018, 10418}, {230, 16306, 3003}, {230, 16310, 3291}, {230, 16318, 232}, {232, 41358, 1990}, {468, 1990, 232}, {468, 16318, 1990}, {468, 47202, 47204}, {468, 56390, 41360}, {1637, 14273, 16230}, {1637, 47200, 10418}, {1990, 16318, 41358}, {2492, 6130, 1637}, {3003, 14580, 232}, {5304, 52284, 62213}, {5702, 62960, 7736}, {6108, 6110, 57593}, {6109, 6111, 57585}, {8105, 8106, 232}, {8737, 8738, 8754}, {8791, 64213, 1560}, {10314, 61327, 62937}, {16303, 47187, 232}, {16317, 16318, 47162}, {16318, 44467, 3018}, {24007, 24008, 468}, {34366, 35907, 42426}, {41200, 41201, 115}, {44467, 47202, 1637}, {44467, 47228, 232}, {47200, 47204, 468}, {52604, 53264, 25}
X(6104) lies on the Dao-Moses-Telv circle and these lines: {3,13}, {14,1606}, {15,1337}, {16,1511}, {54,62}, {61,143}, {110,5616}, {231,1989}, {300,1078}, {1154,5612}
X(6104) = X(532)-crosssum of X(623)
X(6104) = trilinear product X(2166)*X(3201)
X(6104) = barycentric product X(94)*X(3201)
X(6105) lies on the Dao-Moses-Telv circle, K073, K112, K261b, K947, K1133b, the curve Q049, and these lines: {3, 14}, {13, 1605}, {15, 1511}, {16, 1338}, {23, 6108}, {24, 8738}, {30, 6106}, {35, 42680}, {36, 46075}, {54, 61}, {62, 143}, {94, 40104}, {110, 5612}, {186, 6110}, {187, 32906}, {231, 1989}, {265, 51274}, {301, 1078}, {476, 39424}, {533, 14368}, {618, 11092}, {1154, 5616}, {2380, 36515}, {3130, 15442}, {3200, 11127}, {3439, 46059}, {3518, 52670}, {5961, 45778}, {5994, 39555}, {6107, 32460}, {6117, 11587}, {6672, 8836}, {7575, 11549}, {11060, 19780}, {11085, 48366}, {11087, 60070}, {11126, 36981}, {11133, 52221}, {11142, 43967}, {13349, 54473}, {13859, 38431}, {14169, 30440}, {14179, 51203}, {14270, 23284}, {16257, 36759}, {16462, 21159}, {32223, 51267}, {34008, 61371}, {35470, 36970}, {44879, 46926}
X(6105) = reflection of X(i) in X(j) for these {i,j}: {14, 15930}, {5616, 52971}, {18864, 3}
X(6105) = isogonal conjugate of X(11601)
X(6105) = reflection of X(6106) in the orthic axis
X(6105) = circumcircle-inverse of X(14)
X(6105) = antigonal image of X(42618)
X(6105) = isogonal conjugate of the anticomplement of X(33527)
X(6105) = isogonal conjugate of the isotomic conjugate of X(52221)
X(6105) = X(i)-Ceva conjugate of X(j) for these (i,j): {14, 36209}, {1141, 14}
X(6105) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11601}, {18, 51805}, {2166, 37850}, {2962, 6104}, {3384, 36211}
X(6105) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11601}, {10639, 11078}, {11131, 299}, {11597, 37850}, {19295, 14921}, {33529, 1273}
X(6105) = cevapoint of X(3129) and X(11063)
X(6105) = crosspoint of X(i) and X(j) for these (i,j): {62, 8456}, {2381, 57383}
X(6105) = crosssum of X(i) and X(j) for these (i,j): {18, 8173}, {526, 30468}, {533, 624}
X(6105) = crossdifference of every pair of points on line {8562, 23283}
X(6105) = barycentric product X(i)*X(j) for these {i,j}: {6, 52221}, {14, 11127}, {15, 8836}, {62, 11092}, {94, 3200}, {298, 11088}, {301, 11136}, {303, 11086}, {323, 11582}, {470, 50469}, {472, 50466}, {1994, 11600}, {3458, 11133}, {7799, 16464}, {10633, 40710}, {10677, 16771}, {11145, 36210}, {16770, 36209}, {20412, 65326}, {23284, 52606}, {30529, 37848}
X(6105) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11601}, {50, 37850}, {62, 11078}, {1994, 52220}, {2965, 6104}, {3200, 323}, {3458, 11082}, {8836, 300}, {10633, 471}, {10677, 19779}, {11086, 18}, {11088, 13}, {11092, 34390}, {11127, 299}, {11134, 36208}, {11136, 16}, {11137, 10678}, {11142, 36211}, {11582, 94}, {11600, 11140}, {16464, 1989}, {20412, 14918}, {33527, 14921}, {34394, 8604}, {36209, 19778}, {36297, 52204}, {50466, 40711}, {50469, 40709}, {52221, 76}, {55223, 23283}, {64464, 44719}
X(6105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11141, 14}, {3, 32628, 31940}, {14, 8930, 11138}, {14, 51269, 18}, {15, 11086, 36209}, {2070, 11063, 6104}, {5474, 21157, 44800}, {8173, 16242, 33498}, {11088, 50469, 62}
X(6106) lies on the Dao-Moses-Telv circle and these lines: {2, 14368}, {5, 13}, {14, 3480}, {16, 51269}, {30, 6105}, {113, 15610}, {231, 2072}, {403, 6111}, {623, 3580}, {858, 6109}, {6104, 32461}, {6108, 47119}, {6112, 11543}, {6670, 33527}, {8839, 16965}, {15778, 37850}, {18581, 22648}, {18912, 40694}, {22850, 64253}
X(6106) = midpoint of X(5616) and X(11601)
X(6106) = complement of X(14368)
X(6106) = reflection of X(6105) in the Orthic axis
X(6106) = nine-point-circle-inverse of X(13)
X(6106) = complement of the isogonal conjugate of X(14372)
X(6106) = X(14372)-complementary conjugate of X(10)
X(6106) = X(6106)-Dao conjugate of X(14368)
X(6106) = crosssum of X(16) and X(3200)
X(6106) = {X(13),X(18)}-harmonic conjugate of X(40580)
X(6107) lies on the Dao-Moses-Telv circle and these lines: {2, 14369}, {5, 14}, {13, 3479}, {15, 51276}, {30, 6104}, {113, 15609}, {231, 2072}, {403, 6110}, {624, 3580}, {858, 6108}, {6105, 32460}, {6109, 47118}, {6113, 11542}, {6669, 33526}, {8837, 16964}, {15802, 37848}, {16267, 41889}, {18582, 22649}, {18912, 40693}, {22894, 64254}
X(6107) = midpoint of X(5612) and X(11600)
X(6107) = complement of X(14369)
X(6107) = reflection of X(6104) in the Orthic axis
X(6107) = nine-point-circle-inverse of X(14)
X(6107) = complement of the isogonal conjugate of X(14373)
X(6107) = X(14373)-complementary conjugate of X(10)
X(6107) = X(6107)-Dao conjugate of X(14369)
X(6107) = crosssum of X(15) and X(3201)
X(6107) = {X(14),X(17)}-harmonic conjugate of X(40581)
midpoint of X(5612) and X(11600)
X(6107) = complement of X(14369)
X(6107) = reflection of X(6104) in the Orthic axis
X(6107) = nine-point-circle-inverse of X(14)
X(6107) = complement of the isogonal conjugate of X(14373)
X(6107) = X(14373)-complementary conjugate of X(10)
X(6107) = X(6107)-Dao conjugate of X(14369)
X(6107) = crosssum of X(15) and X(3201)
X(6107) = {X(14),X(17)}-harmonic conjugate of X(40581)
X(6108) lies on the Dao-Moses-Telv circle, the cubics K892 and K1274, and these lines:
{2, 13}, {3, 6772}, {4, 16634}, {14, 98}, {23, 6105}, {25, 1605}, {30, 115}, {62, 37464}, {111, 61472}, {114, 9115}, {147, 16530}, {183, 3642}, {325, 532}, {381, 22513}, {385, 533}, {395, 542}, {396, 511}, {397, 20415}, {427, 31687}, {468, 6111}, {524, 6783}, {531, 22329}, {543, 35303}, {619, 5976}, {671, 35932}, {842, 5618}, {858, 6107}, {1078, 11303}, {1080, 5478}, {1302, 2379}, {1368, 40682}, {1513, 41023}, {1691, 62199}, {2076, 62232}, {2782, 44223}, {2794, 41017}, {3054, 42118}, {3055, 63197}, {3815, 5472}, {3849, 31693}, {3934, 37341}, {5025, 9989}, {5031, 44383}, {5104, 62198}, {5152, 35918}, {5306, 47610}, {5318, 10617}, {5461, 31694}, {5469, 36970}, {5470, 23005}, {5473, 21159}, {5607, 6137}, {5615, 42974}, {5617, 10516}, {5916, 11658}, {5999, 22510}, {6034, 62200}, {6036, 13349}, {6054, 6778}, {6103, 6110}, {6106, 47119}, {6670, 9478}, {6680, 37340}, {6770, 9749}, {6775, 11632}, {6777, 11177}, {7610, 11296}, {7735, 10654}, {7736, 9112}, {7784, 11305}, {7793, 9988}, {7828, 11304}, {8280, 31689}, {8281, 31692}, {8859, 23004}, {9752, 44459}, {9753, 33388}, {9760, 37785}, {9763, 22580}, {10033, 54589}, {10352, 12214}, {10616, 36756}, {11298, 15271}, {11486, 41745}, {11549, 62490}, {11580, 34313}, {11623, 36252}, {11645, 53442}, {11646, 62197}, {11705, 44660}, {12829, 52647}, {13873, 13875}, {13926, 13928}, {14137, 41071}, {14539, 21156}, {14693, 52650}, {14830, 22512}, {15246, 37848}, {16001, 21402}, {16092, 61474}, {16644, 59410}, {16965, 35230}, {17830, 31685}, {18581, 41042}, {19781, 42154}, {20252, 43416}, {20426, 41746}, {20429, 59401}, {22496, 43274}, {22572, 36330}, {22576, 23055}, {22712, 43538}, {22796, 22847}, {23234, 25235}, {23698, 44250}, {25154, 37637}, {25166, 63047}, {26244, 37834}, {26613, 35931}, {29012, 53465}, {31707, 47638}, {32461, 47200}, {33498, 48442}, {35021, 42940}, {36318, 49812}, {36344, 49861}, {36363, 49906}, {36383, 49948}, {36755, 53428}, {36764, 42092}, {36765, 42910}, {36961, 41037}, {36967, 39555}, {36994, 59393}, {36995, 59394}, {37640, 51207}, {37689, 42085}, {37786, 51206}, {40694, 41020}, {41151, 42108}, {42035, 63029}, {42086, 62992}, {43228, 44497}, {44382, 51388}, {47857, 61719}, {53015, 54569}
X(6108) = midpoint of X(i) and X(j) for these {i,j}: {13, 16}, {98, 383}, {385, 5978}, {395, 53430}, {671, 35932}, {6778, 22998}, {8595, 51482}, {23005, 36968}, {42943, 53435}
X(6108) = reflection of X(i) in X(j) for these {i,j}: {114, 52263}, {618, 6672}, {624, 6669}, {6109, 230}, {6782, 395}, {9115, 42913}, {31694, 5461}, {31706, 22847}, {31710, 115}, {33517, 13}, {51388, 44382}, {52021, 45880}, {52650, 14693}
X(6108) = complement of X(5979)
X(6108) = reflection of X(6109) in the orthic axis
X(6108) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(13)
X(6108) = circumcircle-of-outer-Napoleon-triangle-inverse of X(10653)
X(6108) = psi-transform of X(10654)
X(6108) = crossdifference of every pair of points on line {6137, 34291}
X(6108) = X(5607)-line conjugate of X(6137)
X(6108) = radical trace of the circles {{X(13), X(14), X(16)}} and {{X(13), X(15), X(16)}}
X(6108) = X(15)-pedal-to-X(16)-pedal similarity image of X(13)
X(6108) = {X(115),X(6055)}-harmonic conjugate of X(6109)
X(6108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13, 6115}, {2, 5980, 618}, {2, 51482, 9762}, {13, 22489, 18582}, {13, 23006, 51482}, {13, 41100, 47859}, {13, 41107, 47861}, {13, 46054, 5459}, {13, 47859, 47865}, {115, 6055, 6109}, {395, 44219, 6114}, {5335, 59378, 13}, {5470, 36968, 23005}, {5915, 46980, 6109}, {6103, 57593, 6110}, {6770, 37641, 51200}, {6778, 16963, 22998}, {10614, 61514, 14139}, {47363, 47364, 10653}
X(6109) lies on the Dao-Moses-Telv circle the cubics K892 and K1274, and these lines: {2, 14}, {3, 6775}, {4, 16635}, {13, 98}, {23, 6104}, {25, 1606}, {30, 115}, {61, 37463}, {111, 61474}, {114, 9117}, {147, 16529}, {183, 3643}, {325, 533}, {381, 22512}, {383, 5479}, {385, 532}, {395, 511}, {396, 542}, {398, 20416}, {427, 31688}, {468, 6110}, {524, 6782}, {530, 22329}, {543, 35304}, {618, 5976}, {671, 35931}, {842, 5619}, {858, 6106}, {1078, 11304}, {1302, 2378}, {1368, 40683}, {1513, 41022}, {1691, 62200}, {2076, 62233}, {2782, 52650}, {2794, 41016}, {3054, 42117}, {3055, 63196}, {3815, 5471}, {3849, 31694}, {3934, 37340}, {5025, 9988}, {5031, 44382}, {5104, 62197}, {5152, 35917}, {5306, 47611}, {5321, 10616}, {5461, 31693}, {5469, 23004}, {5470, 36969}, {5474, 21158}, {5608, 6138}, {5611, 42975}, {5613, 10516}, {5917, 11659}, {5999, 22511}, {6034, 62199}, {6036, 13350}, {6054, 6777}, {6103, 6111}, {6107, 47118}, {6669, 9478}, {6680, 37341}, {6772, 11632}, {6773, 9750}, {6778, 11177}, {7610, 11295}, {7735, 10653}, {7736, 9113}, {7784, 11306}, {7793, 9989}, {7828, 11303}, {8280, 31691}, {8281, 31690}, {8859, 23005}, {9752, 44463}, {9753, 33389}, {9761, 22579}, {9762, 37786}, {10033, 54590}, {10352, 12213}, {10617, 36755}, {11297, 15271}, {11485, 41746}, {11537, 62490}, {11580, 34314}, {11623, 36251}, {11645, 53430}, {11646, 62198}, {11706, 44659}, {12829, 52648}, {13873, 13876}, {13926, 13929}, {14136, 41070}, {14538, 21157}, {14693, 44223}, {14830, 22513}, {15246, 37850}, {16002, 21401}, {16092, 61472}, {16964, 35229}, {17830, 31686}, {18582, 41043}, {19780, 42155}, {20253, 43417}, {20425, 41745}, {20428, 59402}, {22495, 43275}, {22571, 35752}, {22575, 23055}, {22712, 43539}, {22797, 22893}, {23234, 25236}, {25156, 63047}, {25164, 37637}, {26244, 37831}, {26613, 35932}, {29012, 53454}, {31708, 47638}, {32460, 47200}, {33500, 48441}, {35021, 42941}, {35731, 35742}, {36319, 49862}, {36320, 49813}, {36362, 49905}, {36382, 49947}, {36756, 53440}, {36764, 36776}, {36962, 41036}, {36968, 39554}, {36992, 59395}, {36993, 59396}, {37641, 51206}, {37689, 42086}, {37785, 51207}, {40693, 41021}, {41151, 42109}, {42036, 63029}, {42085, 62992}, {43229, 44498}, {44250, 47113}, {44383, 51387}, {47864, 61719}, {53015, 54570}
X(6109) = midpoint of X(i) and X(j) for these {i,j}: {14, 15}, {98, 1080}, {385, 5979}, {396, 53442}, {671, 35931}, {6777, 22997}, {8594, 51483}, {23004, 36967}, {42942, 53447}
X(6109) = reflection of X(i) in X(j) for these {i,j}: {114, 52266}, {619, 6671}, {623, 6670}, {6108, 230}, {6783, 396}, {9117, 42912}, {31693, 5461}, {31705, 22893}, {31709, 115}, {33518, 14}, {44223, 14693}, {44250, 47113}, {51387, 44383}, {52022, 45879}
X(6109) = complement of X(5978)
X(6109) = reflection of X(6108) in the Orthic axis
X(6109) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(14)
X(6109) = circumcircle-of-inner-Napoleon-triangle-inverse of X(10654)
X(6109) = psi-transform of X(10653)
X(6109) = crossdifference of every pair of points on line {6138, 34291}
X(6109) = X(5608)-line conjugate of X(6138)
X(6109) = radical trace of the circles {{X(13), X(14), X(15)}} and {{X(14), X(15), X(16)}}
X(6109) = X(16)-pedal-to-X(15)-pedal similarity image of X(14)
X(6109) = reflection of X(6108) in X(230)
X(6109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14, 6114}, {2, 5981, 619}, {2, 51483, 9760}, {14, 22490, 18581}, {14, 23013, 51483}, {14, 41101, 47860}, {14, 41108, 47862}, {14, 46053, 5460}, {14, 47860, 47866}, {115, 6055, 6108}, {5334, 59379, 14}, {5469, 36967, 23004}, {5915, 46980, 6108}, {6103, 57585, 6111}, {6773, 37640, 51203}, {6777, 16962, 22997}, {10613, 61513, 14138}, {47361, 47362, 10654}
X(6110) lies on the Dao-Moses-Telv circle and these lines: {4,13}, {14,471}, {15,470}, {16,5667}, {30,1990}, {297,531}, {340,533}, {389,398}, {530,648}, {1300,5994}
X(6110) = reflection of X(6111) in X(1990)
X(6110) = trilinear product X(i)*X(j) for these {i,j}: {15,1784}, {470,2173}
X(6110) = isogonal conjugate of X(39377)
X(6110) = crossdifference of every pair of points on line X(14380)X(36296)
X(6110) = polar conjugate of X(36308)
X(6110) = pole wrt polar circle of trilinear polar of X(36308) (line X(13)X(2394))
X(6110) = barycentric product X(i)*X(j) for these {i,j}: {14, 35201}, {30,470}, {298,1990}, {2154, 14920}, {6137, 24001}, {8739, 14206}
X(6110) = X(i)-Ceva conjugate of X(j) for these {i,j}: {4,6111}, {14,6117}
X(6111) lies on the Dao-Moses-Telv circle and these lines: {4,14}, {13,470}, {15,5667}, {16,471}, {30,1990}, {297,530}, {340,532}, {389,397}, {531,648}, {1300,5995}.
X(6111) = reflection of X(6110) in X(1990)
X(6111) = isogonal conjugate of X(39378)
X(6111) = crossdifference of every pair of points on line X(14380)X(36297)
X(6111) = trilinear product X(i)*X(j) for these {i,j}: {13, 35201}, {16,1784}, {471,2173}, {2153, 14920}, {6138, 24001}, {8740, 14206}
X(6111) = barycentric product X(i)*X(j) for these {i,j}: {30,471}, {299,1990}
X(6111) = X(i)-Ceva conjugate of X(j) for these {i,j}: {4,6110}, {13,6116}
X(6111) = polar conjugate of X(36311)
X(6111) = pole wrt polar circle of trilinear polar of X(36311) (line X(14)X(2394))
Moses Radical Circle and Associated Points
Summarizing notes from Peter Moses (November 7, 2014), the circle here named the Moses radical circle, MRC, is defined as the radical circle of the circumcircle, nine-point circle, and Brocard circle. The center of MRC is X(647), and, like the Dao-Moses-Telv circle (see X(6103), MRC passes through X(5000) and X(5001). The radical center of {circumcircle, nine-point circle, MRC} is X(232), and the power of vertex A with respect to MRC is
-b2c2(b2 + c2 - a2)/[2(a2 - b2)(a2 - c2)]
If U is any circle in the coaxal family of the circumcircle and nine-point circle, then the inverses of X(15) and X(16) in U lie on MRC. Examples of such inverses include X(6112)-X(6117).
If you have The Geometer's Sketchpad, you can view Moses Radical Circle, including X(6112)-X(6117).
X(6112) lies on the Moses radical circle and these lines: {5,15}, {570,1506}.
X(6113) lies on the Moses radical circle and these lines: {5,16}, {570,1506}.
X(6114) lies on the Moses radical circle and these lines: {2,14}, {5,39}, {6,5613}, {13,6054}, {16,383}, {18,98}, {62,147}, {230,5471}, {325,624}, {395,542}, {629,5982}, {636,3104}, {1080,5479}
X(6114) = complement of X(5981)
X(6115) lies on the Moses radical circle and these lines: {2,13}, {5,39}, {6,5617}, {14,6054}, {15,1080}, {17,98}, {61,147}, {230,5472}, {325,623}, {383,5478}, {396,542}, {630,5983}, {635,3105}
X(6115) = complement of X(5980)
X(6116) lies on the Moses radical circle and these lines: {4,15}, {5,53}, {13,275}, {16,471}, {62,3462}, {264,623}, {297,624}, {1833,1838}, {2903,5962}
X(6116) =X(13)-Ceva conjugate of X(6111)
X(6116) = X(53)-Hirst inverse of X(6117)
X(6116) = trilinear product X(i)*X(j) for these {(i,j}: {299,2181}, {324,2152}, {471,1953}
X(6116) = barycentric product X(i)*X(j) for these {(i,j}: {5,471}, {16,324}, {53,299}
X(6116) = X(i)-isoconjugate of X(j) for these {i,j}: {14,2169}, {97,2154}
X(6116) = {X(5),X(53)}-harmonic conjugate of X(6117)
X(6116) = X(15)-of-orthic-triangle
X(6116) = nine-point-circle-inverse of X(6117)
X(6116) = polar conjugate of the isotomic conjugate of X(33530)
X(6117) lies on the Moses radical circle and these lines: {4,16}, {5,53}, {14,275}, {15,470}, {61,3462}, {264,624}, {297,623}, {1832,1838}, {2902,5962}
X(6117) =X(14)-Ceva conjugate of X(6110)
X(6117) = X(53)-Hirst inverse of X(6116)
X(6117) = trilinear product X(i)*X(j) for these {(i,j}: {298,2181}, {324,2151}, {470,1953}
X(6117) = barycentric product X(i)*X(j) for these {(i,j}: {5,470}, {15,324}, {53,298}
X(6117) = X(i)-isoconjugate of X(j) for these {i,j}: {13,2169}, {97,2153}
X(6117) = {X(5),X(53)}-harmonic conjugate of X(6116)
X(6117) = X(16)-of-orthic-triangle
X(6117) = nine-point-circle-inverse of X(6116)
X(6117) = polar conjugate of the isotomic conjugate of X(33529)
In the plane of a triangle ABC, let A′B′C′ be the outer Vecten triangle, and let MAMBMC be the medial triangle. Let OA be the circumcenter of triangle B′C′MA, and define OB and OC cyclically. Let O′A be the point, other than A′, in which the circles A′B′MC and C′A′MB meet, and define O′B and O′C cyclically. The six points OA, OB, OC, O′A, O′B, O′C lie on a circle, here named the 1st Dao-Vecten circle. If the construction is carried out using the inner Vecten triangle for A′B′C′, the resulting six points line on the 2nd Dao-Vecten circle. (Thanh Oai Dao, November 4, 2014).
The line X(6118)X(6119) passes through X(575), X(3564), X(3589), X(3628), X(5449), and the radical axis of the 1st and 2nd Dao-Vecten circles passes through X(2501), X(3566), X(5203). X(6118) = {X(3589),X(3628)}-harmonic conjugate of X(6119). (Francisco Javier, ADGEOM, November 4-5, 2014).
The 1st Dao-Vecten circle is the nine-point circle of the outer Vecten triangle. Also, let (O″A) be the circle centered at A′ and tangent to BC at MA, and define (O″B) and (O″C) cyclically. Then X(6118) is the radical center of (O″A), (O″B), (O″C). (Randy Hutson, January 29, 2015)
If you have The Geometer's Sketchpad, you can view X(6118).
X(6118) lies on these lines: {2,372}, {575,3564}, {590,639}, {3316,5590}
X(6118) = midpoint of X(485) and X(641)
X(6118) = {X(2),X(485)}-harmonic conjugate of X(645)
X(6118) = X(5)-of-outer-Vecten-triangle
See X(6118) for a construction of the 2nd Dao-Vecten circle and some of its properties.
The 2nd Dao-Vecten circle is the nine-point circle of the inner Vecten triangle. Let A″B″C″ be the inner Vecten triangle. Also, let (O*A) be the circle centered at A″ and tangent to BC at MA, and define (O*B) and (O*C) cyclically. Then X(6119) is the radical center of (O*A), (O*B), (O*C). (Randy Hutson, January 29, 2015)
X(6119) lies on these lines: {2,371}, {575,3564}, {615,640}, {3317,5591}
X(6119) = midpoint of X(486) and X(642)
X(6119) = {X(2),X(486)}-harmonic conjugate of X(642)
X(6119) = X(5)-of-inner-Vecten-triangle
In the plane of ABC, let AB be the internal trisector at A that is closer to B, and define BC and CA cyclically. Let AC be the internal trisector of A that is closer to C, and define BA and CB cyclically. Let PA be the Miquel point (QL-P1) of the quadrilateral having vertices BC, BA, CA, CB, and define PB and PC cyclically. The points PA, PB, PC lie on the circumcircle of ABC, and the triangle PA, PB, PC, here named the 1st Hatzipolakis-Van Tienhoven triangle, is perspective to ABC. The perspector is X(6120). There are 3 sets of trisectors, so that there are 2nd and 3rd triangles and 2nd and 3rd points, as found at X(6121) and X(6122). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014: 22717, 22719)
Let LA be the line through the circumcircle intercepts of the interior trisector of angle A, and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. Then A′B′C′ is homothetic to ABC at X(6120). (Randy Hutson, September 18, 2014; see ADGEOM 1733
X(6120) lies on these lines: {3,358}, {357,3279}, {1135,6125}, {1136,3335}, {1137,6124}, {3273,6121}, {3274,6122}
X(6120) = {X(3),X(358)}-harmonic conjugate of X(6123)
X(6121) is the perspector of the 2nd Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6120). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014; see 22717, 22719)
X(6121) lies on these lines: {3,1135}, {357,3335}, {358,6125}, {1134,3283}, {1137,6123}, {3273,6120}, {3275,6122}, {3278,3604}, {3334,3603}
X(6121) = {X(3),X(1135)}-harmonic conjugate of X(6124)
X(6122) is the perspector of the 3nd Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6120). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014; see 22717, 22719)
X(6122) lies on these lines: {3,1137}, {358,6124}, {1134, 3335}, {1135,6123}, {1136,3281}, {3274,6120}, {3275,6121}, {3282,3603}, {3334,3602}
X(6122) = {X(3),X(1137)}-harmonic conjugate of X(6125)
In the plane of ABC, let AB be the line perpendicular to the internal trisector at A that is closer to B, and define BC and CA cyclically. Let AC be the line perpendicular to the internal trisector of A that is closer to C, and define BA and CB cyclically. Let QA be the Miquel point (QL-P1) of the quadrilateral having vertices BC, BA, CA, CB, and define QB and QC cyclically. The points QA, QB, QC lie on the circumcircle of ABC, and the triangle QA, QB, QC, here named the 4th Hatzipolakis-Van Tienhoven triangle, is perspective to ABC. The perspector is X(6123). There are 3 sets of trisectors, so that there are 5th and 6th triangles and 5th and 6th points, as found at X(6124) and X(6125). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)
X(6123) lies on these lines: {3,358}, {357,3278}, {1135,6122}, {1136,3334}, {1137,6121}, {3273,6124}, {3274,6125}, {3281,3702}, {3335,3604}
X(6123) = isogonal conjugate of X(5457)
X(6123) = Hofstadter 4/3 point
X(6123) = {X(3),X(358)}-harmonic conjugate of X(6120)
X(6124) is the perspector of the 5th Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6123). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)
X(6124) lies on these lines: {3,1135}, {357,3334}, {358,6122}, {1134,3282}, {1137,6120}, {3273,6123}, {3275,6125}, {3279,3604}, {3335,3603}
X(6124) = {X(3),X(1135)}-harmonic conjugate of X(6121)
X(6125) is the perspector of the 6th Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6123). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)
X(6125) lies on these lines: {3,1137}, {358,6121}, {1134,3334}, {1135,6120}, {1136,3280}, {3274,6123}, {3275,6124}, {3283,3603}, {3335, 3602}
X(6125) = {X(3),X(1137)}-harmonic conjugate of X(6122)
X(6126) is the point QA-P41 ('Involutary Conjugate of QA-P4') of the quadrangle ABCX(1). X(6126)-of-orthocentroidal-triangle = X(1). These properties and others are presented in Hyacinthos messages 21651 and 22707-22710 by Shapi Topor (real name of Stefan Dominte), A. Montesdeoca, R. Hutson, and A. Hatzipolakis; see 22710)
X(6126) lies on these lines: {1,399}, {6,1718}, {35,73}, {36,1464}, {58,106}, {80,651}, {221,2778}, {244,1203}, {323,758}, {500,3746}, {1411,2003}, {1419,5119}, {1870,2914}, {2293,2772}, {3157,5903}
X(6126) = X(80)-isoconjugate of X(3065)
X(6126) = orthocentroidal-to-ABC similarity image of X(1)
X(6126) = 4th-Brocard-to-circumsymmedial similarity image of X(1)
X(6126) = perspector of anti-orthocentroidal and incentral triangles
X(6127) lies on these lines: {1,5}, {43,994}, {386,3120}, {583,1743}, {1046,1054}, {1064,3584}, {1193,5270}, {1742,5010}, {1787,3337}, {2635,4316}
X(6128) and related radical center of three circles are listed here:
X(231) = radical center of circumcircle, nine-point circle, Lester circle
X(1989) = radical center of circumcircle, Dao-Moses-Telv circle, Lester circle
X(6103) = radical center of circumcircle, nine-point circle, Dao-Moses-Telv circle
X(6128) = radical center of Dao-Moses-Telv circle, nine-point circle, Lester circle.
(Antreas Hatzipolakis, November 30, 2014; see
22823)
X(6128) = X(2407)-Ceva conjugate of X(523)
X(6128) = center of the bicevian conic of X(13) and X(14)
X(6128) = complement X(6148)
X(6128) = inverse-in-Kiepert-hyperbola of X(113)
X(6128) = polar conjugate of cevapoint of X(2) and X(113)
X(6128) = polar conjugate of isotomic conjugate of X(6699)
X(6128) = {X(i),X(j)}-harmonic conjugate of X(k), for these (i,j,k): (6,115,3018), (6,1989,3163), (13,14,113), (115,3163,1989), (1989,3,3018)
X(6128) lies on these lines: {2,2986}, {6,13}, {30,3003}, {44,1737}, {112,3087}, {231,2072}, {427,5306}, {526,1637}, {1194,2493}, {1524,5321}, {1525,5318}, {1634,5020}, {5467,6036}
Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Nine-Point Circle
Continuing Peter Moses's discussion in the preamble to X(6112), suppose that P = p : q : r (barycentrics) is a point other than the circumcenter. Let U(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the coaxal system of the circumcircle and nine-point circle. Then X(5000) and X(5001) lie on U, and the center of U is the point M(U) = f(a,b,c) : f(b,c,a) : f(c,a,b) given by
(b2 - c2)(a2 - b2 - c2)p2 + (a2 + b2 - c2)(a2 - b2 + c2)pq + a2(a2 - b2 + c2)q2 - (a2 + b2 - c2)(a2 - b2 + c2)pr - a2(a2 + b2 - c2)r2
Following is a list of centers M(U) for selected points P and circles U(P):
X(6129) = center of this circle: U(X(1)) = {{1,36,1785,5000,5001,5121}}
X(6130) = center of this circle: U(X(98)) = {{98,107,125,132,5000,5001}}
X(6131) = center of this circle: U(X(99)) = {{99,126,2374,5000,5001,5139}}
X(6132) = center of this circle: U(X(110)) = {{110,114,136,3563,5000,5001}}
X(6133) = center of this circle: U(X(10)) = {{10,242,1324,3814,5000,5001,5205}}
X(6134) = center of this circle: U(X(141)) = {{141,625,5000,5001,5938,5971,6031}}
X(676) = center of this circle: U(X(105)) = {{11,105,108,1360,3513,3514,4667,5000,5001}}
X(2492) = center of this circle: U(X(111)) = {{6,111,112,115,187,1560,2079,3569,5000,5001,5523,5913,6032}}}
X(6129) is the center of this circle: U(X(1)) = {{1,36,1785,5000,5001,5121}}; see the preamble to X(6129).
X(6129) lies on these lines: {1,521}, {2,4397}, {37,2509}, {106,2716}, {230,231}, {513,663}, {522,905}, {656,3900}, {2191,2424}, {2517,4885}, {3667,3960}
X(6129) = midpoint of X(i) and X(j) for these (i,j): {663,4017}, {1459,1769}
X(6129) = reflection of X(2517) in X(4885)
X(6129) = complement of X(4397)
X(6129) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5514), (108,208), (522,513), (905,650)
X(6129) = crossdifference of every pair of points on the line X(3)X(9)
X(6129) = {X(37),X(2509)}-harmonic conjugate of X(4130)
X(6129) = perspector of the hyperbola {{A,B,C,X(4),X(40),X(57),X(972)}}, this being the isogonal conjugate of the line X(3)X(9)
X(6129) = bicentric difference of PU(100)
X(6129) = PU(100)-harmonic conjugate of X(2182)
X(6130) is the center of this circle: U(X(98)) = {{98,107,125,132,5000,5001}}; see the preamble to X(6129).
Let (O) be the circumcircle and (N) the nine-point circle. Let L be the line tangent to (O) at X(98), and let L′ be the line tangent to (O) at X(107); then X(6130) = L∩L′. Let M be the line tangent to (N) at X(125) and let M′ be the line tangent to (N) at X(132). Then X(6130) = M∩M′. (Randy Hutson, December 4, 2014)
X(6130) lies on these lines: {2,684}, {3,2797}, {98,804}, {107,1624}, {125,526}, {132,2881}, {230,231}, {402,5972}, {879,1987}, {2799,6036}
X(6130) = midpoint of X(879) and X(3569)
X(6130) = complement of X(684)
X(6130) = crosspoint of X(98) and X(107)
X(6130) = crosssum of X(511) and X(520)
X(6130) = crossifference of every pair of points on the line X(3)X(1625)
X(6130) = inverse-in-Dao-Moses-Telv-circle of X(647)
X(6130) = X(i)-Ceva conjugate of X(j) for these (i,j): (879,523), (3569,804)
X(6130) = midpoint of Jerabek hyperbola intercepts of orthic axis
X(6131) is the center of this circle: U(X(99)) = {{99,126,2374,5000,5001,5139}}; see the preamble to X(6129).
X(6131) lies on these lines: {99,670}, {126,6088}, {141,2872}, {230,231}
X(6132) is the center of this circle: U(X(110)) = {{110,114,136,3563,5000,5001}}; see the preamble to X(6129).
X(6132) lies on these lines: {39,2510}, {110,351}, {114,804}, {182,2869}, {230,231}, {512,5926}, {669,1510}
X(6132) = X(2)-Ceva conjugate of X(36472)X(6133) is the center of this circle: U(X(10)) = {{10,242,1324,3814,5000,5001,5205}}; see the preamble to X(6129).
X(6133) lies on these lines: {10,832}, {230,231}, {513,3823}, {659,2517}, {667,4086}, {814,4036}, {3716,4507}
X(6133) = midpoint of X(i) and X(j) for these {i,j}: {659,2517}, {667,4086}X(6134) is the center of this circle: U(X(141)) = {{141,625,5000,5001,5938,5971,6031}}; see the preamble to X(6129).
X(6134) lies on this line: {230,231}
The incircle of the antipedal triangle of the inner Soddy point, X(175), of a triangle ABC is tangent to the circumcircle of ABC, and X(6135) is the touchpoint. (Based on a problem by Tran Quang Hung, with contributions by T. Cohl ane P. Moses, Art of Problem Solving: Tangent to circumcircle, October 16, 2014),
X(6135) lies on these lines: {105,1123}, {662,1307}, {4557,6136}
X(6135) = isogonal conjugate of X(6364)
The incircle or one of the excircles of the antipedal triangle of the outer Soddy point, X(176), of a triangle ABC is tangent to the circumcircle of ABC, and X(6136) is the touchpoint. (Based on a problem by Tran Quang Hung, with contributions by T. Cohl and P. Moses, Art of Problem Solving: Tangent to circumcircle, October 16, 2014),
X(6136) lies on these lines: {105,1336}, {662,1306}, {4557,6135}
X(6136) = isogonal conjugate of X(6365)
Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Brocard Circle
Continuing Peter Moses's discussion in the preambles to X(6112) and X(6129), suppose that P = p : q : r (barycentrics) is a point other than the circumcenter. Let V(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the Schoute coaxal system; that is, the coaxal system of the circumcircle and Brocard circle. Then X(15) and X(16) lie on V, and the center of V is the point M(V) = f(a,b,c) : f(b,c,a) : f(c,a,b) given by
a2[c(2(a2 + b2 - c2)pq - b2(a2 - b2 + c2)pr + a2c2q2 - a2b2r2]
The power of A with respect to V(P) is
b2c4p/[a2(b2 - c2)r - c2(a2 + b2)p]
Following is a list of centers M(V) for selected points P and circles V(P):
X(351) = center of this circle: V(X(2)) = Parry circle = {{2,15,16,23,110,111,352,353,5638,5639}}
X(649) = center of this circle: V(X(1276)) = {{15,16,1276,1277}}
X(663) = center of this circle: V(X(1)) = {{1,15,16,36,3465,4040,5526,5529}}
X(669) = center of this circle: V(X(5004)) = {{15,16,5004,5005,5980,5981}}
X(887) = center of this circle: V(X(99)) = {{15,16,99,729}}
X(890) = center of this circle: V(X(100)) = {{15,16,100,739}}
X(1960) = center of this circle: V(X(101)) = {{15,16,101,106,214}}
X(2488) = center of this circle: V(X(100)) = {{15,16,3513,3514}}
X(3005) = center of this circle: V(X(5002)) = {{15,16,5002,5003}}
X(5075) = center of this circle: V(X(846)) = {{15,16,846,1054,1283,5197}}
X(6137) = center of this circle: V(X(13)) = {{13,15,16,3165,5616,5669}}
X(6138) = center of this circle: V(X(14)) = {{14,15,16,3166,5612,5668}}
X(6139) = center of this circle: V(X(55)) = {{15,16,55,109,654,1155,2291}}
X(6140) = center of this circle: V(X(115)) = {{15,16,115,128,399,1263,1511,2079}}
X(6137) is the center of this circle: V(X(13)) = {{13,15,16,3165,5616,5669}}; see the preamble to X(6137).
X(6137) lies on these lines: {110,5994}, {111,2379}, {187,237}, {396,523}, {2151,2605}, {2433,3457}, {5607,6108}
X(6137) = reflection of X(6138) in X(647)
X(6137) = X(5995)-Ceva conjugate of X(6)
X(6137) = crosspoint of X(i) and X(j) for these (i,j): (6,5995), (100,2981)
X(6137) = crossdifference of every pair of points on the line X(2)X(13)
X(6137) = crosssum of X(396) and X(523)
X(6137) = X(512)-Hirst inverse of X(6138)
X(6137) = X(2)-Ceva conjugate of X(38993)
X(6137) = perspector of hyperbola {{A,B,C,X(6),X(14),X(15)}}
X(6137) = trilinear product X(i)*X(j) for these {i,j}: {14,2624}, {15,661}, {298,798}, {470,810}, {523,2151}, {526,2154}
X(6137) = barycentric product X(i)*X(j) for these {i,j}: {14,526}, {15,523}, {298,512}, {470,647}, {1577,2151}, {3268,3458}
X(6137) = X(i)-isoconjugate of X(j) for these (i,j): {13,662}, {75,5995}, {99,2153}, {163,300}, {799,3457}
X(6137) = X(i)-line conjugate of X(j) for these (i,j): (2433,3457), (5607,6108)
X(6137) = X(16)-of-2nd-Parry-triangle
X(6137) = X(16)-of-3rd-Parry-triangle
X(6137) = {X(351),X(3569)}-harmonic conjugate of X(6138)
X(6138) is the center of this circle: V(X(14)) = {{14,15,16,3166,5612,5668}}; see the preamble to X(6137).
X(6138) lies on these lines: {110,5995}, {111,2378}, {187,237}, {395,523}, {2152,2605}, {2433,3458}, {5608,6109}
X(6138) = reflection of X(6137) in X(647)
X(6138) = X(5994)-Ceva conjugate of X(6)
X(6138) = crosspoint of X(6) and X(5994)
X(6138) = crosssum of X(396) and X(523)
X(6138) = crossdifference of every pair of points on the line X(2)X(14)
X(6138) = X(2)-Ceva conjugate of X(38994)
X(6138) = perspector of hyperbola {{A,B,C,X(6),X(13),X(16)}}
X(6138) = X(512)-Hirst inverse of X(6138)
X(6138) = trilinear product X(i)*X(j) for these {i,j}: {13,2624}, {16,661}, {299,798}, {471,810}, {523,2152}, {526,2153}
X(6138) = barycentric product X(i)*X(j) for these {i,j}: {13,526}, {16,523}, {299,512}, {471,647}, {1577,2152}, {3268,3457}
X(6138) = {X(351,X(3569)}-harmonic conjugate of X(6137)
X(6138) = X(i)-isoconjugate of X(j) for these (i,j): {14,662}, {75,5994}, {99,2154}, {163,301}, {799,3458}
X(6138) = X(i)-line conjugate of X(j) for these (i,j): (2433,3458), (5608,6109)
X(6138) = X(15)-of-2nd-Parry-triangle
X(6138) = X(15)-of-3rd-Parry-triangle
X(6139) is the center of this circle: V(X(55)) = {{15,16,55,109,654,1155,2291}}; see the preamble to X(6137).
X(6139) lies on these lines: {31,3310}, {55,654}, {109,692}, {187,237}, {650,2520}, {652,4524}, {884,2195}, {918,4640}, {1155,1638}, {1639,3683}
X(6139) = midpoint of X(55) and X(654)
X(6139) = isogonal conjugate of X(35157)
X(6139) = crossdifference of every pair of points on line X(2)X(664)
X(6139) = X(i)-isoconjugate of X(j) for these (i,j): {651,1121}, {664,1156}, {2291,4554}, {4569,4845}
X(6139) = bicentric sum of PU(103)
X(6140) is the center of this circle: V(X(115)) = {{15,16,115,128,399,1263,1511,2079}}; see the preamble to X(6137).
X(6140) lies on these lines: {187,237}, {690,6132}, {2380,2381}
X(6140) = X(1291)-Ceva conjugate of X(6)
X(6140) = X(75)-isoconjugate of X(1291)
X(6141) lies on the Parry circle, the Brocard circle, and these lines: {3,5638}, {6,5639}, {23,1379}, {110,1341}, {111,1340}, {352,1380}, {574,2502}, {2028,3292}
X(6142) lies on the Parry circle, the Brocard circle, and these lines: {3,5639}, {6,5638}, {23,1380}, {110,1340}, {111,1341}, {352,1379}, {574,2502}, {2029,3292}
As a point on the Euler line, X(6143) has Shinagawa coefficients (8F, E).
Let A′B′C′ be the orthic triangle of ABC. Let A′ be the nine-point center of OBC, where O = X(3), and define B′ and C′ cyclically. Let A″ be the orthogonal projection of A′ on the altitude A′, and define B″ and C″ cyclically. The Euler lines of the jfour triangles ABC, A″BC, AB″C, ABC″ concur in X(6143). (Antreas Hatzipolakis and Peter Moses, November 18, 2014; see 22767)
X(6143) lies on these lines: {2,3}, {49,3448}, {54,125}, {93,2970}, {112,1506}, {252,562}
X(6143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3548,631), (5,3520,4), (5,5498,3), (140,1594,186), (186,1594,4), (427,3518,4), (3526,5094,24)
Let A′B′C′ be the orthic triangle and A″B″C″ the medial triangle of ABC. Let A*B*C* be the mid-altitude triangle (midpoints of altitudes). Let U be the reflection of line A″A* in A′, and define V and W cyclically. The lines U,V,W concur in X(6144). (Antreas Hatzipolakis and Peter Moses, November 18, 2014; see 22765)
X(6144) lies on these lines: {2,6}, {7,4969}, {45,3879}, {511,1657}, {518,3633}, {548,1350}, {576,5072}, {621,5340}, {622,5339}, {742,4764}, {1351,3818}, {1352,3850}, {1353,5085}, {3242,3635}, {3416,4691}, {3564,3627}, {3625,5847}, {3729,4725}, {3751,4668}, {3758,4445}, {3793,5210}, {3875,4715}, {4363,5564}, {4409,5845}, {4795,4967}
X(6144) = reflection of X(6) in X(193)
Let A′B′C′ be the orthic triangle of ABC. Let AB be the orthogonal projection of A′ onto line AB, and define BC and CA cyclically. Let AC be the orthogonal projection of A′ onto line AC, and define BA and CB cyclically. Let A′B be the reflection of A in AB, and let A′C be the reflection of A in AC . Let NA be the nine-point center of the triangle AA′BA′C, and define NB and NC cyclically. The triangle NANBNC, here named the Hatzipolakis-Moses triangle, is orthologic to ABC and perpective to ABC. The A-vertex of this triangle is given by
NA = -a^4 b^2+2 a^2 b^4-b^6-a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6 : b^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) : c^2 (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4)The ABC-to-NANBNC center of orthology is X(6145), and the NANBNC-to-ABC center of orthology is X(6146). The perspector of ABC and NANBNC is X(54). (Based on notes , Antreas Hatzipolakis and Peter Moses, November 19, 2014; see 22774)
X(6145) lies on the Jerabek hyperbola and these lines: {3,161}, {4,973}, {6,3574}, {52,265}, {54,1594}, {68,1154}, {69,1225}, {235,1177}, {826,2435}, {1176,1503}, {3426,5895}, {3521,6000}
X(6145) = reflection of X(2917) in X(1209)
X(6145) = complement of X(32354)
X(6145) = anticomplement of X(32391)
The Hatzipolakis-Moses triangle is defined at X(6145).
X(6146) lies on these lines: {3,68}, {4,6}, {5,156}, {24,161}, {30,52}, {49,2072}, {54,1594}, {125,128}, {154,3542}, {186,2917}, {389,973}, {403,1614}, {427,578}, {539,1216}, {542,5907}, {550,1204}, {567,5576}, {1092,1368}, {1853,3541}, {1885,6000}, {3547,3796}, {3564,4173}
X(6146) = reflection of X(3575) in X(389)
X(6146) = X(72)-of-orthic-triangle if ABC is acute
X(6146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,68,343), (4,2883,1514)
Let A′B′C′ be the pedal triangle of the incenter of ABC. Then X(6147) is the radical center of the nine-point circles of the triangles AA′I, BB′I, CC′I. (Antreas Hatzipolakis and Randy Hutson, December 2, 2014; see 22830)
X(6147) lies on these lines: {1,30}, {2,3927}, {3,7}, {5,226}, {8,1159}, {10,3824}, {12,5902}, {57,140}, {65,495}, {72,5249}, {142,5044}, {354,496}, {355,5290}, {381,938}, {382,3488}, {386,1086}, {388,952}, {405,5905}, {442,3868}, {443,3940}, {498,5221}, {499,4860}, {517,3671}, {527,1125}, {546,5722}, {548,3601}, {549,553}, {550,3982}, {632,3911}, {774,5492}, {908,5439}, {946,971}, {950,3627}, {975,4675}, {999,3485}, {1056,1482}, {1156,3296}, {1329,5883}, {1385,4298}, {1387,3304}, {1484,5083}, {1490,5805}, {1595,1876}, {1656,5226}, {1657,4313}, {1871,5236}, {2886,3874}, {3295,3475}, {3333,5843}, {3336,5432}, {3337,5433}, {3340,5844}, {3467,5443}, {3526,5435}, {3576,4355}, {3586,3853}, {3624,3929}, {3628,5219}, {3664,4920}, {3670,5718}, {3678,3826}, {3811,5880}, {3812,3820}, {3813,3881}, {3925,5904}, {4114,5122}, {5055,5704}, {5244,5752}, {5261,5790}, {5270,5425}, {5715,5787}
X(6147) = midpoint of X(3295) and X(4295)
X(6147) = reflection of X(i) in X(j) for these (i,j): (10,3824), (3560,5901)
X(6147) = complement of X(3927)
X(6147) = crosssum of X(55) and X(584)
X(6147) = (2nd extouch triangle)-to-(intouch triangle) similarity image of X(5)
X(6148) lies on these lines: {2,2986}, {22,1634}, {30,3260}, {69,74}, {76,328}, {186,340}, {316,1553}, {320,1443}, {325,3233}, {339,1225}, {526,3268}, {2407,3163}, {4226,5181}, {4590,5641}
X(6148) = anticomplement of X(6128)
X(6148) = isotomic conjugate of X(5627)
X(6148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,1272,1494), (99,1494,1272), (616,617,74)
Let DEF be the circumtangential triangle. Let D' be the trilinear product E*F, and define E' and F' cyclically. Then D', E', F' all lie on the line X(1)X(21). Let UVW be the circumnormal triangle. Let U′ be the trilinear product V*W, and define V′ and W′ cyclically. Then U′, V′, W′ all line on the line X(656)X(1955). Moreover, X(6149) = X(1)X(21)∩X(656)X(1955), and the trilinear polar of X(6149) passes through X(2624). (Randy Hutson, December 12, 2014)
Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the trilinear product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(6149). (Randy Hutson, December 14, 2014)
Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the barycentric product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(50) (for which trilinears are sin 3A : sin 3B : sin 3C). In the special case that DEF is the circumtangential triangle, the points D',E',F' lie on the Brocard axis, and in case DEF is the circumnormal triangle, the points D',E'F' lie on the line X(50)X(647). (Randy Hutson, December 15, 2014)
X(6149) is a major center, as noted by César Lozada ( Major Centers)
Let E be the inellipse with perspector X(1101); then X(6149) is the intersection of the tangents to E at X(1094) and X(1095). Let A′B′C′ and A″B″C″ be the (equilateral) circumcevian triangles of X(15) and X(16), let A* be the trilinear product A′*A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(6149). (Randy Hutson, January 29, 2015)
Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A′ be the trilinear product A1*A2, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6149); see also X(50). (Randy Hutson, October 13, 2015)
Let A′B′C′ be the anti-orthocentroidal triangle. Let A″ be the trilinear product B′*C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6149). (Randy Hutson, December 10, 2016)
X(6149) lies on these lines: {1,21}, {35,500}, {36,1464}, {109,484}, {162,1784}, {163,1755}, {171,3584}, {212,5010}, {223,3215}, {238,3582}, {526,2605}, {580,3336}, {656,1955}, {922,1101}, {1094,2151}, {1095,2152}, {1577,2627}, {1718,5535}, {1935,3585}, {1936,3583}, {2159,2315}, {2190,2962}, {3072,5270}, {3073,4857}, {5398,5902}
X(6149) = isogonal conjugate of X(2166)
X(6149) = crossdifference of every pair of points on the line X(661)X(1953)
X(6149) = trilinear product of X(15) and X(16)
X(6149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,47,2964), (1,1749,1725), (47,255,1), (896,1725,1749), (1822,1823,1)
X(6149) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2166}, {2,1989}, {4,265}, {5,1141}, {6,94}, {13,14}, {25,328}, {30,5627}, {79,80}, {300,3458}, {301,3457}, {476,523}, {847,5961}
Let N be the nine-point center of ABC, and let A′ be the nine-point center of the triangle NBC. Define B′ and C′ cyclically. Let U be the reflection of the line NA′ in line BC, and define V and W cyclically. The lines U,V,W concur in X(6150). (A. Hatzipolakis, C. Lozada, P. Moses, December 15, 2014; see 22886)
X(6150) is the orthocenter of the pedal triangle of X(1157). (Randy Hutson, December 16, 2014)
X(6150) lies on these lines: {3,54}, {30,137}, {1291,2070}
X(6150) = midpoint of X(i) and X(j) for these (i,j): (3,1157), (1291,2070)
X(6150) = complementary conjugate of complement of X(34418)
X(6150) = X(1291)-Ceva conjugate of X(1510)
X(6150) = circumcircle-inverse of X(195)
Let U be the line tangent to the Neuberg cubic (K001) at X(14), and let V be the line tangent to K001 at X(16). Then X(6151) = U∩V. Let U′ be the line tangent to the Napoleon-Feuerbach cubic (K005) at X(18), and let V′ be the line tangent to K005 are X(62). Then X(6151) = U′∩V′; see X(2981). Also, X(6151) = DC(X(14)) (see the preamble to X(2979). (Randy Hutson, January 5, 2014)
X(6151) lies on the hyperbola {{A,B,C,X(2),X(6)}} and these lines: {14,5675}, {16,1338}, {37,5367}, {39,2981}, {42,5357}, {62,110}, {395,1989}
X(6151) = isogonal conjugate of X(395)
X(6151) = cevapoint of X(i) and X(j) for these (i,j): (6,16, (202, 2245)
X(6151) = X(i)-cross conjugate of X(j) for these (i,j): (323,2981), (6138,110)
X(6151) = trilinear pole of the line X(15)X(512)
X(6151) = X(i)-isoconjugate of X(j) for these {i,j}: {1,395}, {63,462}, {533,2153}, {619,2154}
X(6151) = X(i)-vertex conjugate of X(j) for these {i,j}: {1976,2981}, {2981,1976}
X(6151) = perspector of ABC and unary cofactor triangle of inner Napoleon triangle
X(6152) is introduced by Randy Hutson, César Lozada, Peter Moses, December 17, 2014; see 22900
X(6152) is the orthologic center of the pedal triangles of X(4) and X(5). This point is one of several found in connection with the following configuration: Let ABC be a triangle, and let T1, T2, T3 be the pedal triangles of X(3), X(4), X(5), respectively. The three triangles are pairwise orthologic. Let Pi,j be the orthologic center of Ti, Tj. Then the orthocenter of T3 is the midpoint of P1,3 and P2,3. (Antreas Hatzipolakis and César Lozada, December 17, 2014; see 22907
See also Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25814.
X(6152) lies on these lines: {4,93}, {6,24}, {25,195}, {49,143}, {52,539}, {70,6145}, {113,5446}, {155,3060}, {403,3574}, {648,1179}, {1112,2914}, {1209,1216}, {1843,5965}, {1986,3575}
X(6152) = reflection of X(i) in X(j) for these (i,j); (54,973), (1493,143), (1209,6153), (2914,1112)
X(6152) = X(4)-Ceva conjugate of X(1594)
X(6152) = X(4)-crosspoint of X(3518)
X(6152) = X(3)-crosssum of X(3519)
X(6152) = X(2216)-isoconjugate of X(3519)
X(6152) = X(79)-of-orthic-triangle if ABC is acute
X(6152) = trilinear product X(1594)*X(2964)
X(6152) = barycentric product X(1594)*X(1994)
X(6152) = X(10122) of tangential triangle, if ABC is acute
X(6152) = Ehrmann-vertex-to-orthic similarity image of X(6288)
X(6153) is the orthocenter of the pedal triangle of X(5); see X(6154). (Antreas Hatzipolakis and César Lozada, December 17, 2014; see 22907
See also Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25814.
X(6153) lies on these lines: {51,195}, {52,2888}, {54,5462}, {539,973}, {546,1154}, {1209,1216}
X(6153) = midpoint of X(i) and X(j) for these {i,j}: {52,2888}, {1209,6152}
X(6153) = reflection of X(i) in X(j) for these (i,j); (54,5462), (1216,1209)
X(6154) lies on these lines: {2,11}, {19,1862}, {40,550}, {65,1317}, {104,3528}, {119,546}, {165,4863}, {214,3636}, {382,5840}, {516,3689}, {519,5183}, {548,5288}, {678,3120}, {900,4088}, {1086,3722}, {1145,3626}, {1155,5853}, {1320,5558}, {1836,3158}, {1842,5151}, {2805,4681}, {2829,3529}, {3059,6068}, {3174,5528}, {3189,5854}, {3629,3779}, {3698,4314}, {3711,5698}, {3893,4297}, {3952,4152}, {3962,5493}, {4819,5847}, {5082,5217}
X(6154) = anticomplement of anticomplement of X(35023)
X(6154) = intangents-to-extangents similarity image of X(11)
X(6154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,100,6174), (100,149,3035), (149,3035,11), (3434,4421,5432)
Suppose that X = x : y : z (trilinears) is a triangle center. The points P(X) = y/z : z/x : x/y and U(X) = z/y : x/z : y/x are introduced at Bicentric Pairs as the 1st and 2nd bicentric quotients of X, respectively. The pair is both a bicentric pair and an isogonal conjugate pair. As an example, P(X(2)) and U(X(2)) are the 1st and 2nd Brocard points. The points PU(113) are the bicentric quotients of X(37) = b + c : c + a : a + b, so that PU(113) consists of the following bicentric pair: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). The bicentric sum of two points p : q : r and u : v : w is the point p + u : q + v : r + w. For the bicentric difference of PU(113), see X(4983).
X(6155) lies on these lines: {1,1929}, {6,191}, {37,762}, {39,3121}, {42,3954}, {81,763}, {213,3931}, {1100,3285}, {2238,3743}, {2295,4868}, {4037,4065}
X(6155) = PU(113)-harmonic conjugate of X(4983)
X(6156) is the crosssum of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)
X(6156) lies on these lines: {37,2134}, {81,1126}, {762,763}
X(6156) = crosspoint of PU(113)
X(6157) is the crossdifference of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)
X(6157) lies on these lines: {37,81}, {762,763}, {1654,2248}, {1961,1963}, {2134,3293}
X(6157) = isogonal conjugate of X(6158)
X(6157) = crossdifference of every pair of points on the line X(4983)X(6155)
X(6157) = X(2)-Ceva conjugate of X(39064)
X(6157) = perspector of the conic {{A,B,C,PU(113)}}
X(6158) is the trilinear pole of the line of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)
X(6158) lies on this line: {1962, 2054}
X(6158) = isogonal conjugate of X(6157)
X(6158) = trilinear pole of the line X(4983)X(6155)
X(6159) is the ideal point of the line of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)
X(6159) lies on these lines: {30,511}, {4983,6155}
X(6159) = crossdifference of every pair of points on the line X(6)X(6157)
X(6160) is the midpoint of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155).
X(6160) lies on this line: {4983,6155}
Suppose that X = x : y : z (trilinears) is a triangle center. The points P(X) = y/z : z/x : x/y and U(X) = z/y : x/z : y/x are introduced at Bicentric Pairs as the 1st and 2nd bicentric quotients of X, respectively. The pair is both a bicentric pair and an isogonal conjugate pair. As an example, P(X(2)) and U(X(2)) are the 1st and 2nd Brocard points. The points PU(114) are the bicentric quotients of X(513) = b - c : c - a : a - b, so that PU(114) consists of the following bicentric pair: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). For the bicentric difference of PU(114), see X(2087).
X(6161) lies on these lines: {1,513}, {3,667}, {10,4448}, {512,2292}, {514,3244}, {522,5592}, {649,1334}, {659,3887}, {661,1643}, {663,1201}, {830,4983}, {832,1245}, {876,6005}, {891,4895}, {1027,2334}, {1388,3669}, {1491,4794}, {1960,2254}, {2098,4162}, {2136,3900}, {3293,4040}, {3667,4297}, {3803,4834}, {4083,5697}
X(6161) = PU(114)-harmonic conjugate of X(2087)
See X(6155) and X(6156).
X(6162) lies on these lines: {1,88}, {764,6163}, {5376,6161}
X(6162) = crosspoint of PU(114)
See X(6155) and X(6156).
Let P′ be the line tangent to the Hutson-Moses hyperbola H = {{A,B,C,PU(26),PU(33)}} at P(33) , and let U′ be the line tangent to H at U(33). Then X(6163) = P′∩U′. Also, the trilinear polar of X(6163) passes through X(1054). (Randy Hutson, January 6, 2015)
X(6163) lies on these lines: {44,3290}, {100,513}, {190,523}, {238,1149}, {244,1052}, {385,4831}, {518,5048}, {651,660}, {758,1757}
X(6163) = isogonal conjugate of X(6164)
X(6163) = X(2)-Ceva conjugate of X(39065)
X(6163) = crosspoint of PU(33)
X(6163) = crosssum of PU(34)
X(6163) = crossdifference of every pair of points on the line X(2087)X(6166)
X(6163) = perspector of the conic {{A,B,C,PU(114)}}
X(6164) is the trilinear pole of the line of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6158).
X(6158) lies on these lines: {513,1052}, {900,3035}, {1635,3722}
X(6164) = isogonal conjugate of X(6163)
X(6164) = cevapoint of PU(34)
X(6164) = trilinear pole of the line X(2087)X(6166)
X(6165) is the ideal point of the line of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6159).
X(6165) lies on these lines: {30,511}, {238,1960}, {659,1757}
X(6165) = crossdifference of every pair of points on the line X(6)X(6163)
X(6166) is the midpoint of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6160).
X(6166) lies on these lines: {1,39}, {2087,6161}
X(6167) = crosssum of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).
X(6167) lies on these lines: {1,1462}, {9,2124}, {57,145}, {77,2329}, {194,2128}, {664,2082}, {728,738}
X(6167) = crosspoint of PU(115)
X(6168) = crossdifference of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).
X(6168) lies on these lines: {2,7}, {279,3501}, {348,1334}, {521,2254}, {651,3684}, {728,738}, {1018,1323}, {1419,3158}, {2124,2136}, {2272,5845}, {3160,3208}
X(6168) = isogonal conjugate of X(6169)
X(6168) = crossdifference of every pair of points on the line X(663)X(2082)
X(6168) = X(2)-Ceva conjugate of X(39066)
X(6168) = perspector of the conic {{A,B,C,PU(115)}}
X(6169) = trilinear pole of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).
X(6169) lies on these lines: {6,1633}, {1438,3451}
X(6169) = isogonal conjugate of X(6168)
X(6169) = trilinear pole of the line X(663)X(2082)
X(6170) = ideal point of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).
X(6170) lies on these lines: {30,511}, {663,2082}
X(6170) = crossdifference of every pair of points on the line X(6)X(6168)
X(6171) = midpoint of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).
X(6171) lies on this line: {663, 2082}
X(6172) lies on these lines: {2,7}, {8,190}, {30,5759}, {44,4419}, {45,4644}, {69,3161}, {72,4313}, {75,4488}, {192,4460}, {220,651}, {346,4416}, {376,971}, {381,5762}, {390,519}, {391,3729}, {452,3951}, {480,4421}, {516,3543}, {518,1992}, {536,5838}, {545,673}, {549,5843}, {551,5850}, {599,4370}, {938,3927}, {956,4345}, {958,4323}, {960,4308}, {966,5936}, {984,4344}, {1014,1696}, {1212,5543}, {1442,2324}, {1743,3672}, {2346,4428}, {2550,5080}, {2801,5692}, {3008,4346}, {3091,5735}, {3177,4552}, {3474,3715}, {3545,5805}, {3616,3758}, {3663,3973}, {3678,5696}, {3686,4461}, {3707,4659}, {3731,3945}, {3829,6067}, {3876,5784}, {4384,4454}, {4473,4741}, {4755,4795}, {4866,5493}, {4916,6144}, {4971,5839}, {5258,5734}, {5851,6174}
X(6172) = midpoint of X(2) and X(144)
X(6172) = reflection of X(i) in X(j) for these (i,j): (7,2), (2,9)
X(6172) = anticomplement of X(6173)
X(6172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,144,7), (44,4419,5222), (45,4644,5308), (329,3219,5273), (329,5273,5226), (1992,4664,3241), (4384,4480,4454), (5220,5698,8)
X(6173) lies on these lines: {1,528}, {2,7}, {3,5735}, {6,4859}, {30,5732}, {36,1001}, {37,4862}, {69,4034}, {75,4007}, {85,1121}, {320,4384}, {376,516}, {381,971}, {518,599}, {519,1056}, {547,5843}, {549,5762}, {597,4795}, {610,5829}, {673,2364}, {903,4664}, {936,6147}, {942,5784}, {958,4355}, {1125,5698}, {1449,3664}, {1470,4870}, {1698,5220}, {1699,3742}, {2321,4869}, {2323,5228}, {2325,4454}, {2801,5587}, {3008,4644}, {3058,4326}, {3062,3255}, {3158,3475}, {3241,5853}, {3247,3663}, {3487,5438}, {3524,5759}, {3620,4967}, {3624,3916}, {3731,4902}, {3812,5290}, {3824,5705}, {3826,5223}, {3828,5850}, {3834,4363}, {3912,4659}, {3945,3946}, {4292,5436}, {4321,5434}, {4346,5308}, {4361,4725}, {4419,4887}, {4445,4739}, {4667,5222}, {4851,4971}, {4860,5231}, {5055,5779}, {5071,5817}, {5856,6174}
X(6173) = midpoint of X(2) and X(7)
X(6173) = reflection of X(i) in X(j) for these (i,j): (9,2), (2,142)
X(6173) = complement of X(6172)
X(6173) = trilinear product X(i)*X(j) for these {i,j}: {2,4860}, {57,5231}
X(6173) = barycentric product X(i)*X(j) for these ({i,j}: {7,5231}, {75,4860}
X(6173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,144,7), (44,4419,5222), (45,4644,5308), (329,3219,5273), (329,5273,5226), (1992,4664,3241), (4384,4480,4454), (5220,5698,8).
X(6174) lies on these lines: {2,11}, {30,119}, {104,3524}, {165,5660}, {210,2801}, {214,519}, {376,2829}, {381,5840}, {527,1155}, {529,4996}, {549,952}, {551,2802}, {599,5848}, {678,1647}, {900,1635}, {1387,5541}, {1470,5434}, {1768,3929}, {2482,2787}, {2783,6055}, {2805,4755}, {3241,5854}, {3530,5258}, {3711,5744}, {3712,5205}, {3722,3756}, {3820,5010}, {3887,4763}, {5433,5687}, {5856,6173}
X(6174) = midpoint of X(i) and X(j) for these {i,j}: {2,100}, {165,5660)
X(6174) = reflection of X(i) in X(j) for these (i,j): (11,2), (2,3035)
X(6174) = crossdifference of every pair of points on the line X(106)X(665)
X(6174) = trilinear product X(i)*X(j) for these {i,j}: {44,527}, {519,1155}, {1023,1638}, {1055,4358}, {1323,3689}
X(6174) = barycentric product {519,527}, {1055,3264}, {1155,4358}, {1323,2325}
X(6174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4421,3058), (100,3035,11), (214,1145,1317), (1376,5432,3925)
X(6174) = X(i)-isoconjugate of X(j) for these (i,j): {88,2291}, {106,1156}
As a point on the Euler line, X(6175) has Shinagawa coefficients (3abc$a$ + 2S2, 6S2).
X(6175) lies on these lines: {2,3}, {8,3649}, {10,79}, {13,5362}, {14,5367}, {81,3017}, {100,3584}, {519,5178}, {528,2346}, {540,4921}, {671,2795}, {758,3679}, {1125,5441}, {1330,3578}, {1441,1494}, {1698,3647}, {2771,3753}, {2886,5434}, {3241,3475}, {3474,3648}, {3582,5253}, {3583,5284}, {3585,3841}, {3656,5330}, {3838,4511}, {3868,4654}, {3925,5080}, {3936,4720}, {5276,5309}
X(6175) = midpoint of X(2) and X(2475)
X(6175) = reflection of X(i) in X(j) for these (i,j): (21,2), (2,442)
X(6175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,21,15671), (2,3545,4193), (2,3839,2478), (2,4188,5054), (2,4190,3524), (2,5141,5055), (4,4197,5047), (377,2476,404), (377,5177,2476), (442,2475,21), (443,3545,2), (474,5055,2), (3585,3841,5260), (3839,4208,2)
Let ABC be an acute triangle, and let LA be the circle that is externally tangent to the nine-point circle and to the sidelines AB and AC. Define LB and LC cyclically. The three circles are named the Odehnal tritangent circles at X(3822). The center of their outer Apollonian circle is X(6176). (Boris Odehnal, "A Triad of Tritangent Circles, Journal for Geometry and Graphics 18 (2014) no. 1, 61-71)
X(6176) lies on these lines: {1,181}, {3,4653}, {5,515}, {405,1437}, {551,2051}, {995,5396}, {1319,5718}, {3576,4192}
See also X(3557) and X(6178). Contributed by Bernard Gibert, January 4, 2015.
The angle θ for X(6177) as a Kiepert perspector is given by cot(θ) = - cot(ω) - e csc(ω). (Peter Moses, January 6, 2015)
In the plane of a triangle ABC, let
F1 and F2 be the real foci of the Steiner inellipse
A1 = AF1∩BC
A2 = AF2∩BC
Ga = conic tangent to AF1 at A1, to AF2 at A2, and to F1F2
Ta = Ga∩F1F2
Ao = center of Ga, and define Bo and Co cyclically
The lines AAo, BBo, CCo concur in X(14633), and the lines ATa, BTb, CTc concur in X(6177). See X(6177) and X(14633). (Angel Montesdeoca, July 21, 2023)
X(6177) lies on the cubics K009, K704, the Kiepert hyperbola, and these lines: {3,3414}, {4,1380}, {76,1341}, {98,2558}, {140,141}, {262,3558}, {1078,2559}, {3557,6179}
X(6177) = isogonal conjugate of X(3557)
X(6177) = {X(141,3788)}-harmonic conjugate of X(6178)
X(6177) = cevapoint of PU(118)
See also X(3558) and X(6177). Contributed by Bernard Gibert, January 4, 2015.
The angle θ for X(6178) as a Kiepert perspector is given by cot(θ) = - cot(ω) + e csc(ω). (Peter Moses, January 6, 2015)
X(6178) lies on the cubics K009, K704, the Kiepert hyperbola, and these lines: {3,3413}, {4,1379}, {76,1340}, {98,2559}, {140,141}, {262,3557}, {1078,2558}, {3558,6179}X(6178) = isogonal conjugate of X(3558)
X(6178) = {X(141,3788)}-harmonic conjugate of X(6177)
Contributed by Bernard Gibert, January 4, 2015. See K704.
X(6179) lies on these lines: {2,5007}, {6,1078}, {24,648}, {32,76}, {83,183}, {99,3053}, {187,194}, {193,1692}, {305,1627}, {316,3767}, {350,n+52}, {371,6312}, {372,6316}, {382,671}, {538,3552}, {575,631}, {609,1909}, {754,5025}, {1384,1975}, {1724,3570}, {1799,5359}, {3557,6177}, {3558,6178}, {3785,5304}, {3793,5305}, {3934,5008}, {4027,6309}, {4045,5368}
X(6179) = complement of X(7946)
X(6179) = anticomplement of X(7821)
X(6179) = (X(i),X(i))-harmonic conjugate of X(k) for these (i,,j,k): (32,76,3972), (32,385,76)
Contributed by Randy Hutson, January 6, 2015.
X(6180) lies on these lines: {1,971}, {2,1407}, {6,7}, {9,241}, {12,1406}, {37,77}, {44,1418}, {45,1443}, {55,1742}, {56,87}, {57,1122}, {63,1427}, {65,3751}, {69,3713}, {72,1448}, {85,894}, {141,5782}, {144,220}, {175,3298}, {176,3297}, {192,664}, {219,527}, {221,388}, {222,226}, {223,3666}, {278,3782}, {307,965}, {347,2256}, {394,5905}, {405,4306}, {481,1335}, {482,1124}, {513,1037}, {518,2263}, {608,5236}, {644,4488}, {954,991}, {958,1042}, {960,4320}, {984,5018}, {1001,1458}, {1004,1331}, {1014,1778}, {1044,5584}, {1146,5942}, {1191,3600}, {1203,4355}, {1212,4350}, {1253,3000}, {1373,3299}, {1374,3301}, {1376,6168}, {1386,4327}, {1441,4363}, {1449,4328}, {1476,3445}, {1615,3599}, {1616,4308}, {1736,5779}, {1804,2178}, {2003,4654}, {3157,5706}, {3242,4318}, {3729,4513}, {4032,4559}, {5290,5711}
X(6180) = crosspoint of PU(46)
X(6180) = crosssum of PU(112)
X(6180) = {X(222),X(226)}-harmonic conjugate of X(940)
Contributed by Randy Hutson, January 6, 2015.
X(6181) lies on these lines: {3,1939}, {55,650}, {1376,6184}, {3119,4414}
Contributed by Randy Hutson, January 6, 2015.
X(6182) lies on these lines: {30,511}, {55,650}, {657,4041}, {661,4105}, {693,3434}, {1824,4024}, {2520,4524}, {2886,4885}
X(6182) = isogonal conjugate of X(6183)
X(6182) = crossdifference of every pair of points on the line X(6)X(77)
X(6183) is the point of intersection, other than A, B, C, of the circumcircle and the hyperbola {{A, B, C, PU(46)}}. Contributed by Randy Hutson, January 6, 2015.
X(6183) lies on the circumcircle and these lines: {7,105}, {100,883}, {101,1025}, {108,658}, {112,1414}, {651,919}, {664,1292}, {927,1633}, {2724,5088}
X(6183) = isogonal conjugate of X(6182)
X(6183) = trilinear pole of the line X(6)X(77)
X(6183) = Ψ(X(6), X(77))
X(6183) = Λ(X(657), X(4041))
X(6183) = Λ(PU(112))
Contributed by Randy Hutson, January 6, 2015.
X(6184) lies on the Steiner inellipse and these lines: {1,39}, {2,2481}, {3,101}, {6,3939}, {9,1742}, {10,1146}, {32,218}, {37,142}, {55,1438}, {100,294}, {106,5024}, {115,120}, {119,1566}, {187,1017}, {241,3693}, {528,5701}, {650,6174}, {665,1642}, {672,2223}, {867,5513}, {80,5308}, {1084,2092}, {1376,6181}, {1575,3008}, {2311,3110}
X(6184) = isogonal conjugate of X(6185)
X(6184) = complement of X(2481)
X(6184) = X(2)-Ceva conjugate of X(518)
X(6184) = crossdifference of every pair of points on the line X(105)X(659)
X(6184) = barycentric square of X(518)
X(6184) = complementary conjugate of X(20544)
X(6184) = polar conjugate of isogonal conjugate of X(20776)
X(6184) = center of the hyperbola {{A,B,C,X(2),X(100),PU(112)}}
X(6184) = crosssum of circumcircle intercepts of line PU(46) (line X(6)X(513))
Contributed by Randy Hutson, January 6, 2015.
X(6185) lies on these lines: {105,2223}, {238,516}, {239,294}, {241,927}, {1429,1438}, {5091,5222}
X(6185) = isogonal conjugate of X(6184)
X(6185) = isotomic conjugate of X(4437)
X(6185) = cevapoint of X(6) and X(105)
X(6185) = X(6)-cross conjugate of X(105)
X(6185) = polar conjugate of X(34337)
X(6185) = barycentric square of X(673)
X(6185) = trilinear pole of the line X(105)X(659) (tangent to circumcircle at X(105))
X(6186) lies on these lines: {21,36}, {31,1495}, {35,1255}, {55,199}, {105,5322}, {614,3415}, {999,1036}, {2194,2260}, {2206,3122}, {2306,3129}
X(6186) = cevapoint of X(667) and X(3122)
X(6186) = crosspoint of X(56) and X(3444)
X(6186) = crosssum of X(i) and X(j) for these {i,j}: {2,3648}, {8,2895}, {3219,4420}, {3678,3969}
X(6186) = trilinear pole of X(3063) and X(4834)
X(6186) = trilinear product X(i)*X(j) for these {i,j}: {6,2160}, {31,79}, {1402,3615}
X(6186) = barycentric product X(i)*X(j) for these {i,j}: {1,2160}, {6,79}, {36,1989}
X(6186) = barycentric product of vertices of reflection triangle of X(1)
X(6186) = perspector of ABC and unary cofactor triangle of Gemini triangle 25
X(6186) = X(i)-isoconjugate of X(j) for these (i,j): {1,319}, {2,3219}, {7,4420}, {8,1442}, {35,75}, {76,2174}, {81,3969}, {86,3678}, {100,4467}, {312,2003}, {314,2594}, {332,1825}, {340,1807}, {668,2605}, {1255,3578}, {1268,3647}, {1399,3596}, {2611,4600}
X(6187) lies on the cubic K312 and these lines: {10,21}, {23,5143}, {25,2181}, {31,51}, {36,88}, {41,1017}, {42,692}, {45,55}, {56,244}, {105,2006}, {902,2183}, {976,1036}, {1911,5040}, {1960,3310}, {2204,2333}, {3145,4642}
X(6187) = X(i)-Ceva conjugate of X(j) for these (i,j): (759,2161), (1168,6)
X(6187) = X(2251)-cross conjugate of X(6)
X(6187) = X(i)-vertex conjugate of X(j) for these (i,j): (291,1929), (292,649), (1635,2161)
X(6187) = X(55)-beth conjugate of X(692)
X(6187) = cevapoint of X(1960) and X(3271)
X(6187) = crosspoint of X(i) and X(j) for these {i,j}: {106,909}, {1411,2161}
X(6187) = crossdifference of every pair of points on the line X(3904)X(3960)
X(6187) = crosssum of X(i) and X(j) for these {i,j}: {519,908}, {758,3936}, {3218,4511}, {4089,4453}
X(6187) = trilinear pole of X(213) and X(3063)
X(6187) = vertex conjugate of PU(34)
X(6187) = trilinear product X(i)*X(j) for these {i,j}: {6,2161}, {25,1807}, {31,80}, {41,2006}, {42,759}, {55,1411}, {655,3063}, {663,2222}, {902,1168}, {1400,2341}, {1989,2174}
X(6187) = barycentric product X(i)*X(j) for these {i,j}: {1,2161}, {6,80}, {9,1411}, {19,1807}, {35,1989}, {37,759}, {44,1168}, {55,2006}, {65,2341}, {650,2222}, {655,663}, {1793,1880}, {2166,2174}
X(6187) = X(i)-isoconjugate of X(j) for these (i,j): {1,320}, {2,3218}, {7,4511}, {8,1443}, {36,75}, {69,1870}, {77,5081}, {81,3936}, {85,2323}, {86,758}, {100,4453}, {106,1227}, {190,3960}, {214,903}, {274,2245}, {310,3724}, {314,1464}, {332,1835}, {349,4282}, {514,4585}, {651,3904}, {654,4554}, {662,4707}, {664,3738}, {765,4089}, {860,1444}, {870,3792}, {1268,4973}, {1509,4053}, {1983,3261}, {2361,6063}, {2610,4610}, {4025,4242}, {4373,4881}
Let rA be a positive valued function of a,b,c, and define rB and rC cyclically. Let (A) denote the circle with center A and radius rA, and define (B) and (C) cyclically. Let P be the radical center of (A), (B), (C). Let rP be a positive valued function of a,b,c, let LA be the radical axis of (A), (B), (P), and define LB and LC cyclically. Let A′ = LB∩LC, and define B′ and C′ cyclically. Then the lines AA′, BB′, CC″ concur in a point, D. Moreover, the six points A, B, C, X(4), P, D lie on a hyperbola. (Dao Thanh Oai, ADGEOM 893, November 26, 2013)
The hyperbola is here called the Dao (rA,rB,rC,rP) hyperbola. The triangle A′B′C′ is the Dao (rA,rB,rC,rP) triangle, and the perspector D of ABC and A′B′C′ is the Dao (rA,rB,rC,rP) perspector.
For details and examples, including coordinates and equations, see Angel Montesdeoca's presentation at Hechos Geometricos.
X(6188) lies on these lines: {20, 399}, {1272, 14615}, {14249, 14940}
X(6189) is the tripole of the axis through the real foci of the Steiner circumellipse. The barycentric product X(6189)*X(6190) is the Steiner point, X(99). Contributed by Bernard Gibert, January 7, 2015.
X(6189) lies on the curves K106, K242, K482, Q012, Q017, Q053 and these lines: {2,6}, {76,3558}, {99,1380}, {511,6040}, {538,2028}, {542,6039}, {671,3413}, {892,13636}, {1078,14630}, {1340,7771}, {1341,7757}, {2039,14568}, {2543,14929}, {3228,5638}, {3557,6177}, {3933,19659}, {6178,7871}, {7760,14631}
X(6189) = reflection of X(6190) in X(2)
X(6189) = isogonal conjugate of X(5639)
X(6189) = isotomic conjugate of X(3414)
X(6189) = complement of X(39365)
X(6189) = anticomplement of X(39022)
X(6189) = cevapoint of X(i) and X(j) for these (i,j): {2,3414}, {3557,5639}
X(6189) = X(i)-cross conjugate of X(j) for these (i,j): (3414,2), (5639,6177)
X(6189) = crossdifference of every pair of points on the line X(512)X(2029)
X(6189) = trilinear pole of line X(2)X(1341)
X(6189) = X(i)-isoconjugate of X(j) for these (i,j): (1,5639), (31,3414), (661,1379)
X(6189) = trilinear product X(799)*X(5638)
X(6189) = barycentric product X(i)*X(j) for these {i,j}: {76,1380}, {99,3413}, {670,5638}
X(6189) = X(4590)-Ceva conjugate of X(6190)
X(6189) = trilinear pole of PU(i) for these i: 116, 118
X(6189) = {X(6),X(385)}-harmonic conjugate of X(6190)
X(6190) is the tripole of the axis through the imaginary foci of the Steiner circumellipse. The barycentric product X(6189)*X(6190) is the Steiner point, X(99). Contributed by Bernard Gibert, January 7, 2015.
X(6190) lies on the curves K106, K242, K482, Q012, Q017, Q053 and these lines: {2,6}, {76,3557}, {99,1379}, {511,6039}, {538,2029}, {542,6040}, {671,3414}, {892,13722}, {1078,14631}, {1340,7757}, {1341,7771}, {2040,14568}, {2542,14929}, {3228,5639}, {3558,6178}, {3933,19660}, {6177,7871}, {7760,14630}
X(6190) = reflection of X(6189) in X(2)
X(6190) = isogonal conjugate of X(5638)
X(6190) = isotomic conjugate of X(3413)
X(6190) = complement of X(39366)
X(6190) = anticomplement of X(39023)
X(6190) = cevapoint of X(i) and X(j) for these (i,j): {2,3413}, {3557,5638}
X(6190) = X(i)-cross conjugate of X(j) for these (i,j): (3413,2), (5638,6179)
X(6190) = crossdifference of every pair of points on the line X(512)X(2028)
X(6190) = trilinear pole of line X(2)X(1340)
X(6190) = X(i)-isoconjugate of X(j) for these (i,j): (1,5638), (31,3413), (661,1380)
X(6190) = trilinear product X(799)*X(5639)
X(6190) = barycentric product X(i)*X(j) for these {i,j}: {76,1379}, {99,3414}, {670,5639}
X(6190) = X(4590)-Ceva conjugate of X(6189)
X(6190) = trilinear pole of PU(i) for these i: 117, 119
X(6190) = {X(6),X(385)}-harmonic conjugate of X(6189)
X(6191) lies on these lines: {1,61}, {3,5672}, {4,9}, {14,2946}, {18,1653}, {63,627}, {191,2945}, {1385,5240}, {1483,5239}, {1652,3338}, {3383,3467}
X(6191) = anticomplement of X(33428)
X(6191) = X(62)-of-excentral-triangle
X(6191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6192), (9,1276,1277)
X(6192) lies on these lines: {1,62}, {3,5673}, {4,9}, {13,2945}, {17,1652}, {63,628}, {191,2946}, {1385,5239}, {1482,5240}, {1653,3338}, {3376,3467}
X(6192) = anticomplement of X(33429)
X(6192) = X(61)-of-excentral-triangle
X(6192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6191), (9,1277,1276)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let P be a point in the plane of triangle ABC. Let A′B′C′ be the circumcevian triangle of P, and let A″B″C″ be the reflection of A′B′C′ in X(3). Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines A′A*, B′B*, C′C* concur in a point Q, here introduced as the circumcevian antipodal perspector of P. X(6193) is the circumcevian antipodal perspector of X(4). (Randy Hutson, January 29, 2015)
X(6193) lies on these lines: {2, 54}, {3, 69}, {4, 155}, {5, 3167}, {20, 6241}, {30, 6225}, {49, 3549}, {52, 193}, {70, 3448}, {110, 3542}, {146, 3146}, {184, 3547}, {388, 3157}, {497, 1069}, {511, 5596}, {542, 2892}, {578, 1352}, {912, 944}, {1092, 1899}, {2889, 3522}, {3088, 5921}, {3091, 5654}, {3147, 3580}, {3832, 5448}, {4294, 6238}
X(6193) = reflection of X(i) in X(j) for these (i,j): (4, 155), (68, 1147)
X(6193) = isogonal conjugate of X(34428)
X(6193) = X(317)-Ceva conjugate of X(2)
X(6193) = anticomplement of X(68)
X(6193) = anticomplementary conjugate of X(37444)
X(6193) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (24, 8), (47, 20), (158, 68), (162, 924), (317, 6327), (1101, 925), (1748, 69), (1993, 4329)
X(6193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (68,1147,2), (1092,1899,3546)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6194) lies on these lines: {1, 1447}, {2, 51}, {3, 194}, {4, 2896}, {20, 76}, {39, 3523}, {63, 6211}, {69, 147}, {98, 3098}, {165, 726}, {183, 1350}, {376, 2782}, {631, 3095}, {730, 5731}, {1351, 3329}, {1513, 3314}, {2080, 3407}, {3091, 3934}, {3146, 6248}, {3552, 5171} {6195, {{6, 538}, {574, 3117}
X(6194) = X(i)-Ceva conjugate of X(j) for these (i,j): (183, 2), (1350, 20), (5999, 147)
X(6194) = anticomplement of X(262)
X(6194) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 1352), (182, 8), (183, 6327), (3403, 315)
X(6194) = Thomson-isogonal conjugate of X(32524)
X(6194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,5188,20), (183,1350,5999)
X(6194) = perspector of 1st Neuberg triangle and cross-triangle of ABC and 2nd Neuberg triangle
X(6194) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(33873)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6195) lies on these lines: {6, 538}, {574, 3117}
X(6195) = vertex conjugate of PU(139)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6196) lies on these lines: {1, 76}, {3, 238}, {39, 256}, {43, 213}, {83, 3494}, {384, 904}, {386, 1045}, {695, 3496}, {1492, 1932}, {2108, 3216}, {2896, 4388}, {3336, 5539}, {3493, 3497}
X(6196) = X(i)-Ceva conjugate of X(j) for these (i,j): (384, 3496), (904, 1)
X(6196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,4116,1), (1909,4161,1), (2230,4161,1909)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6197) lies on these lines: {3, 3101}, {4, 9}, {8, 1748}, {24, 55}, {28, 517}, {34, 2093}, {46, 278}, {54, 65}, {196, 1068}, {240, 5255}, {378, 5584}, {389, 3611}, {484, 1838}, {1181, 3197}, {1594, 3925}, {1715, 1782}, {1844, 3746}, {1871, 3579}, {1872, 4222}, {1888, 5183}, {1902, 2355}, {5889, 6237}, {6239, 6252}, {6240, 6253}, {6241, 6254}, {6242, 6255}
X(6197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19,40,4), (1871,3579,4219)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6198) lies on these lines: {1, 4}, {2, 1062}, {3, 3100}, {7, 1063}, {8, 406}, {10, 451}, {11, 1594}, {12, 403}, {19, 3247}, {20, 1060}, {24, 55}, {25, 3295}, {28, 1255}, {29, 1807}, {30, 4296}, {35, 186}, {36, 3520}, {37, 943}, {42, 4213}, {47, 1776}, {56, 378}, {65, 74}, {72, 4183}, {101, 2332}, {112, 172}, {145, 4194}, {162, 2906}, {208, 3340}, {232, 1500}, {235, 495}, {270, 2074}, {281, 3811}, {318, 4511}, {350, 1235}, {376, 1038}, {389, 3270}, {427, 496}, {461, 3870}, {468, 5297}, {475, 3616}, {500, 1442}, {502, 1826}, {580, 1736}, {631, 1040}, {774, 3072}, {860, 5174}, {912, 3562}, {942, 1902}, {984, 2212}, {999, 1593}, {1000, 1039}, {1013, 3868}, {1041, 3296}, {1069, 1993}, {1125, 1861}, {1148, 4336}, {1181, 2192}, {1214, 3651}, {1249, 3553}, {1319, 1887}, {1385, 1872}, {1387, 1862}, {1398, 1597}, {1425, 6000}, {1452, 5119}, {1717, 1770}, {1753, 3576}, {1827, 1871}, {1829, 4222}, {1830, 5563}, {1835, 5425}, {1841, 3723}, {1876, 5045}, {1892, 4318}, {1905, 3057}, {1968, 2242}, {1986, 3024}, {2310, 3073}, {2326, 5279}, {2915, 3101}, {3083, 3536}, {3084, 3535}, {3085, 3542}, {3086, 3541}, {3092, 3298}, {3093, 3297}, {3147, 5218}, {3518, 3746}, {3622, 4200}, {3720, 4212}, {3931, 4231}, {3969, 4420}, {4291, 5248}, {4861, 5081}, {5090, 5142}, {5889, 6238}, {6239, 6283}, {6240, 6284}, {6241, 6285}, {6242, 6286}
X(6198) = X(i)-cevapoint of X(j) for these (i,j): (1, 1717), (1825, 2594)
X(6198) = X(i)-cross conjugate of X(j) for these (i,j): (1844, 4), (2174, 3219), (2594, 35)
X(6198) = crosspoint of X(i) and X(j) for these (i,j): (1, 3469), (1897, 5379)
X(6198) = crosssum of X(1) and X(3468)
X(6198) = crossdifference of every two points on the line X(652)X(2523)
X(6198) = polar conjugate of X(30690)
X(6198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,1870), (1,33,4), (1,3465,73)
X(6198) = X(i)-isoconjugate of X(j) for these (i,j): (3, 79), (36, 265), (63, 2160), (65, 1789), (69, 6186), (73, 3615)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let (LA), (LB), (LC) be the Lucas circles. Let (L′A) be the circle, other than (LA), tangent to (LB), (LC), and the circumcircle. Define (L′B), (L′C) cyclically. The circles (L′A), (L′B), (L′C) are here introduced as the Lucas secondary circles, and the triangle formed by their centers as the Lucas secondary central triangle. Let MA be the line through the touch points of (LA) and the adjacent Lucas secondary circles ((L′B) and (L′C)). Define MB and MC cyclically. The 1st Lucas secondary tangents triangle is the triangle having sidelines MA, MB, MC. Let NA be the line through the touch points of (L′A) and the adjacent Lucas circles ((LB) and (LC)). Define NB and NC cyclically. The 2nd Lucas secondary tangents triangle is the triangle having sidelines NA, NB, NC. The 1st and 2nd Lucas secondary tangents triangles are perspective, and their perspector is X(6199). Moreover, the six touch points of the Lucas circles and Lucas secondary circles lie on a circle, here introduced as the Lucas secondary tangents circle, with center X(6409). X(6199) is the inverse-in-Lucas-secondary-tangents-circle of X(187). (Randy Hutson, January 29, 2015)
X(6199) lies on these lines: {3, 6}, {5, 1132}, {323, 1583}, {381, 3068}, {485, 3843}, {486, 5070}, {590, 5055}, {1131, 3853}, {1327, 3830}, {1482, 1702}, {1587, 1657}, {1588, 1656}, {1597, 5410}, {3069, 5054}, {3070, 5073}, {3071, 3851}
X(6199) = radical center of the Lucas(8/3) circles
X(6199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,371,6221), (6,6221,3), (371,3311,3), (371,3592,3311), (1132,3316,5), (1151,3312,3), (3311,6221,6)
See César Lozada, Perspective-Orthologic-Parallelogic.
The circumsymmedial, Lucas tangents, and Lucas(4) central triangles are pairwise perspective, and the perspector of each pair is X(6200). Also, X(6200) is the perspector of Lucas central triangle and circumsymmedial tangential triangle. Let A′, B′, C′ be the free vertices of the Kenmotu squares, and let A″, B″, C″ be the reflections of A′, B′, C′ in X(371). The triangle A″B″C″ is homothetic to ABC at X(6200). (see also X(6396)). (Randy Hutson, February 16, 2015)
Let A′B′C′ and A″B″C″ be the outer and inner Vecten triangles, respectively. Let A* be the trilinear pole, wrt A′B′C′, of the line B″C″. Define B* and C* cyclically. The lines A′A*, B′B*, C′C* concur in X(6200). (Randy Hutson, February 16, 2015)
X(6200) is the trilinear pole, with respect to the inner Vecten triangle, of the line X(2501)X(3566), this being the perspectrix of the outer and inner Vecten triangles. (Randy Hutson, February 20, 2015)
X(6200) lies on these lines: {2, 1328}, {3, 6}, {4, 5418}, {20, 485}, {30, 590}, {35, 2067}, {36, 2066}, {76, 6316}, {140, 3071}, {186, 5413}, {323, 5408}, {376, 3068}, {378, 5412}, {486, 631}, {487, 3620}, {489, 639}, {549, 615}, {550, 3070}, {637, 641}, {1124, 5204}, {1335, 5217}, {1495, 3155}, {1583, 5407}, {1587, 3522}, {1588, 3523}, {1599, 5409}, {369, 3524}, {3092, 3515}, {3093, 3516}, {5010, 5414}
X(6200) = isogonal conjugate of X(1327)
X(6200) = crosssum of X(1327) and X(1327)
X(6200) = insimilicenter of circumcircle and Lucas circles radical circle
X(6200) = radical center of Lucas(2/3) circles
X(6200) = X(1)-isoconjugate of X(1327)
X(6200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6396), (3,371,372), (3,1151,371), (3,3311,1152), (3,6221,6), (6,1151,6221), (6,6221,371), (15,16,371), (1588,3523,5420), (3371,3372,3312), (3385,3386,1151)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6201) lies on these lines: {4, 6}, {5, 1160}, {51, 3127}, {114, 6320}, {115, 6226}, {381, 5860}, {546, 5874}, {638, 1270}, {946, 3640}, {1162, 3574}, {1598, 5594}, {1699, 5588}, {3832, 6250}, {3839, 6251}, {5478, 6268}, {5479, 6269}, {5587, 5688}, {5603, 5604}, {6245, 6257}, {6246, 6262}, {6247, 6266}, {6248, 6272}, {6249, 6274}
X(6201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,5870), (4,1587,5871), (4,5480,6202), (5,1160,5590)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6202) lies on these lines: {4, 6}, {5, 1161}, {51, 3128}, {114, 6319}, {115, 6227}, {381, 5861}, {546, 5875}, {637, 1271}, {946, 3641}, {1163, 3574}, {1598, 5595}, {1699, 5589}, {3832, 6251}, {3839, 6250}, {5478, 6270}, {5479, 6271}, {5587, 5689}, {5603, 5605}, {6245, 6258}, {6246, 6263}, {6247, 6267}, {6248, 6273}, {6249, 6275}
X(6202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,5871), (4,1588,5870), (4,5480,6201), (5,1161,5591)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6203) lies on these lines: {1, 372}, {2, 7}, {19, 3069}, {46, 486}, {169, 481}, {175, 2082}, {494, 1880}, {604, 3084}, {1372, 5540}, {1766, 5405}, {2171, 3083}
X(6203) = X(19)-Ceva conjugate of X(6204)
X(6203) = X(5412)-of-excentral-triangle
X(6203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2285,6204), (9,57,6204), (63,1400,6204), (579,1708,6204), (672,1445,6204), (1423,3509,6204)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6204) lies on these lines: {1, 371}, {2, 7}, {19, 1659}, {46, 485}, {169, 482}, {176, 2082}, {493, 1880}, {604, 3083}, {1371, 5540}, {1722, 2362}, {1766, 5393}, {2171, 3084}
X(6204) = X(i)-Ceva conjugate of X(j) for these (i,j): (19, 6203), (1659, 1)
X(6204) = X(5413)-of-excentral-triangle
X(6204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2285,6203), (9,57,6203), (63,1400,6203), (579,1708,6203), (672,1445,6203), (1423,3509,6203)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6205) lies on these lines: {1, 574}, {9, 46}, {43, 5213}, {57, 1018}, {1100, 5264}, {3208, 3337}, {3218, 3661}, {3336, 3501}, {3970, 5221}, {3987, 5021}, {4390, 4973}
X(6205) = X(598)-Ceva conjugate of X(1)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6206) lies on the 3rd Morley cubic (K301) and these lines: {1,1136}, {3606,6209)
X(6206) = X(3276)-Ceva conjugate of X(6208)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6207) lies on the 2nd Morley cubic (K030) and these lines: {1,1134}, {3607,6209}
X(6207) = X(3277)-Ceva conjugate of X(6209)
The perspector of the excentral triangle and the classical (i.e. 1st) Morley triangle is X(1507). See César Lozada, Perspective-Orthologic-Parallelogic.
X(6208) lies on these lines: {1, 1137}, {357, 1507}, {1136, 6209}, {3606, 6206}
X(6208) = X(3276)-Ceva conjugate of X(6206)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6209) lies on these lines: {1, 1135}, {1134, 1507}, {1136, 6208}, {3607, 6207}
X(6209) = X(3277)-Ceva conjugate of X(6207)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6210) lies on these lines: {1, 256}, {3, 238}, {4, 9}, {31, 4220}, {63, 147}, {72, 4073}, {98, 2319}, {165, 2108}, {262, 3402}, {515, 3883}, {517, 984}, {613, 1429}, {740, 3169}, {946, 4357}, {991, 995}, {1001, 1350}, {1245, 1453}, {1351, 4649}, {1400, 4307}, {1699, 1764}, {3208, 3508}, {3333, 3664}, {3430, 5248}, {3755, 4266}, {3882, 3886}, {4026, 5480}, {4050, 4133}, {4199, 4512}, {4269, 5327}
X(6210) = reflection of X(i) in X(j) for these (i,j): (1742, 3), (40, 573)
X(6210) = crossdifference of every two points on the line X(1459)X(3287)
X(6210) = X(264)-of-hexyl-triangle
X(6210) = hexyl-isotomic conjugate of X(1)
X(6210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1756,1423), (9,40,6211), (1284,3056,1), (6212,6213,3496)
X(6210) = X(21)-beth conjugate of X(1423)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6211) lies on these lines: {1, 182}, {3, 984}, {4, 9}, {46, 1423}, {63, 6194}, {98, 813}, {100, 2708}, {200, 1726}, {210, 1762}, {238, 517}, {484, 1756}, {511, 1757}, {515, 3717}, {756, 4220}, {759, 2742}, {1324, 2077}, {1350, 5220}, {1503, 3932}, {2792, 4645}, {2938, 3579}, {3430, 3678}, {4073, 5687}, {4518, 5999}, {4901, 5881}
X(6211) = X(4518)-Ceva conjugate of X(1)
X(6211) = excentral-isogonal conjugate of X(2108)
X(6211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6210), (6212,6213,3501)
X(6211) = X(188)-aleph conjugate of X(2108)
See César Lozada, Perspective-Orthologic-Parallelogic.
It is a well known property of a cyclic quadrilateral that the incenters and excenters of the component triangles form a rectangular grid. Let PA be the unique point on circumcircle arc BC such that the rectangle formed by the incenter of PABC and the A-excenters of ABC, PACA, and PAAB is a square; define PB and PC cyclically. Let JA, JB, JC be the excenters of ABC. Let IA be the incenter of PABC, IB the incenter of APBC, Ic the incenter of ABPC.
Let (AB) = A-excenter of PABC and (AC) = A-excenter of PAAB. Define (BC) and (CA) cyclically, and define (BA) and (CB) cyclically.
Let LA be the extended diagonal (AB)(AC) of square IA(AB)JA(AC), and define LB and LC cyclically.
Let A′ = LB∩LC, B′ = LC∩LA, C′ = LA∩LB. The lines AA′, BB′, CC′ concur in X(6212). Also, X(6212) is the orthocenter of A′B′C′. See also X(6213). The trilinear polar of X(6212) passes through X(1459). (Randy Hutson, January 29, 2015)
Let ABC be a triangle with orthocenter H and orthic triangle A'B'C'. Let Ia, Ib, Ic be the incenters of HB'C', HC'A', HA'B'. Lines AIa, BIb, CIc concur in X(6212). (César Lozada - July 9, 2023)
X(6212) lies on these lines: {1, 371}, {4, 9}, {46, 486}, {57, 481}, {58, 605}, {63, 488}, {90, 3377}, {103, 6136}, {222, 1124}, {372, 1707}, {487, 2128}, {517, 1378}, {607, 3093}, {608, 3092}, {1587, 2082}, {1588, 2285}, {1703, 1743}, {1790, 3083}, {2809, 3641}
X(6212) = reflection of X(6213) in X(169)
X(6212) = isogonal conjugate of X(6213)
X(6212) = X(i)-cross conjugate of X(j) for these (i,j): (605, 3083)
X(6212) = X(372)-of-excentral-triangle
X(6212) = excentral-isogonal conjugate of X(32556)
X(6212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,19,6213), (9,40,6213), (10,1766,6213), (281,1753,6213), (1276,6191,6213), (1277,6192,6213), (3496,6210,6213), (3501,6211,6213)
X(6212) = X(i)-isoconjugate of X(j) for these (i,j): (1, 6213), (3, 1123), (4, 1335), (19, 3084), (25, 5391), (92, 606), (905, 6135), (1659, 5414)
See César Lozada, Perspective-Orthologic-Parallelogic.
It is a well known property of a cyclic quadrilateral that the incenters of the component triangles form a rectangle. Let QA be the unique point on circumcircle arc BC such that the rectangle formed by the incenters of ABC, QABC, QACA, QAAB is a square. Define QB and QC cyclically. Let I = X(1), and let
[AA] = incenter of QABC, [AB] = incenter of QACA, [AC] = incenter of QAAB,
[BA] = incenter of QBCA, [BB] = incenter of QBAB, [BC] = incenter of QBBC,
[CA] = incenter of QCAB, [CB] = incenter of QCBC, [CC] = incenter of QCCA.
Let MA be the extended diagonal [AB][AC] of square I[AA][AB][AC]], and define MB and MC cyclically. Let A″ = MB∩MC, B″ = MC∩MA, C″ = MA∩MB. The lines AA″, BB″, CC″ concur in X(6213), and the trilinear polar of X(6213) passes through X(1459). See also X(6212). (Randy Hutson, January 29, 2015)
X(6213) is the radical center of the circumcircles of the squares squares IA(AB)JA(AC), IB(BC)JB(BA), IC(CA)JC(CB). (Randy Hutson, April 11, 2015)
Let ABC be a triangle with orthocenter H and orthic triangle A'B'C'. Let Ja, Jb, Jc be the excenters of HB'C', HC'A', HA'B', corresponding to H. Lines AJa, BJb, CJc concur in X(6213). (César Lozada - July 9, 2023)
X(6213) lies on these lines: {1, 372}, {4, 9}, {46, 485}, {57, 482}, {58, 606}, {63, 487}, {90, 3378}, {103, 6135}, {164, 3645}, {222, 1335}, {371, 1707}, {488, 2128}, {517, 1377}, {607, 3092}, {608, 3093}, {1587, 2285}, {1588, 2082}, {1702, 1743}, {1790, 3084}, {2809, 3640}
X(6213) = isogonal conjugate of X(6212)
X(6213) = reflection of X(6212) in X(169)
X(6213) = X(i)-cross conjugate of X(j) for these (i,j): (606, 3084), (2067, 1)
X(6213) = X(371)-of-excentral-triangle
X(6213) = excentral-isogonal conjugate of X(32555)
X(6213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,19,6212), (9,40,6212), (10,1766,6212), (281,1753,6212), (1276,6191,6212), (1277,6192,6212), (3496,6210,6212), (3501,6211,6212)
X(6213) = X(i)-isoconjugate of X(j) for these (i,j): (1, 6212), (3, 1336), (4, 1124), (19, 3083), (25, 1267), (92, 605), (905, 6136)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6214) lies on these lines: {3, 5590}, {4, 1160}, {5, 6}, {355, 3640}, {381, 5860}, {517, 5688}, {637, 3933}, {639, 1503}, {952, 5604}, {3095, 6272}, {5587, 5588}, {5613, 6269}, {5617, 6268}, {5878, 6266}, {6033, 6226}, {6257, 6259}, {6262, 6265}, {6274, 6287}, {6276, 6288}, {6320, 6321}
X(6214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1270,1160), (5,1352,6215), (5,5874,6), (5,5875,485), (6,6278,5874), (1352,6289,5), (5590,5870,3)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6215) lies on these lines: {3, 5591}, {4, 1161}, {5, 6}, {355, 3641}, {381, 5861}, {517, 5689}, {638, 3933}, {640, 1503}, {952, 5605}, {3095, 6273}, {5587, 5589}, {5613, 6271}, {5617, 6270}, {5878, 6267}, {6033, 6227}, {6258, 6259}, {6263, 6265}, {6275, 6287}, {6277, 6288}, {6319, 6321}
X(6215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1271,1161), (5,1352,6214), (5,5874,486), (5,5875,6), (6,6281,5875), (1352,6290,5), (5591,5871,3)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6216) lies on this line: {371, 5860}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6217) lies on these lines: {4, 6266}, {5, 1160}, {51, 1162}, {389, 5870}
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-18 a^6 b^2+18 a^2 b^6-b^8-18 a^6 c^2-32 a^4 b^2 c^2-18 a^2 b^4 c^2+4 b^6 c^2-18 a^2 b^2 c^4-6 b^4 c^4+18 a^2 c^6+4 b^2 c^6-c^8+4 (2 a^6+3 a^4 b^2-4 a^2 b^4-b^6+3 a^4 c^2+8 a^2 b^2 c^2+b^4 c^2-4 a^2 c^4+b^2 c^4-c^6) S
See César Lozada,
Perspective-Orthologic-Parallelogic.
X(6218) lies on these lines: {4, 6267}, {5, 1161}, {51, 1163}, {389, 5871}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6219) lies on these lines: {4, 6276}, {382, 1160}, {5870, 6239}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6220) lies on these lines: {4, 6277}, {382, 1161}, {5871, 6241}
See César Lozada, Perspective-Orthologic-Parallelogic.
Let A′B′C′ be the Lucas central triangle. Let A″ be the trilinear pole, wrt A′B′C′, of BC; define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6221). Let A″B″C: be the Lucas tangents triangle. Let A* be the trilinear pole, wrt A″B″C″, of line BC; define B*, C* cyclically. The lines A″A*, B″B*, C″C* concur in X(6221). The following triangles are perspective from X(6221): circumsymmedial triangle, Lucas inner triangle, and Lucas(2) central triangle. (Randy Hutson, January 29, 2015)
X(6221) lies on these lines: {3, 6}, {4, 3590}, {30, 3068}, {140, 1588}, {186, 5411}, {323, 1599}, {378, 5410}, {381, 590}, {382, 485}, {486, 3526}, {487, 3619}, {549, 3069}, {550, 1587}, {615, 5054}, {999, 2066}, {1132, 5067}, {1385, 1702}, {1597, 5412}, {1656, 3071}, {1657, 3070}, {2067, 3295}, {3092, 3517}, {3299, 5204}, {3301, 5217}, {3316, 3832}
X(6221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,371,3311), (3,3311,3312), (3,6199,6), (6,371,6199), (6,1151,6200), (6,6199,3311), (6,6200,3), (15,16,1151), (371,372,3592), (371,1151,3), (371,6200,6), (3071,5418,1656)
X(6221) = radical center of Lucas(4/3) circles
X(6221) = inverse-in-Lucas-radical-circle of X(187)
X(6221) = exsimilicenter of circumcircle and Lucas inner circle
X(6221) = trilinear pole wrt Lucas central triangle of Lemoine axis
X(6221) = trilinear pole wrt Lucas tangents triangle of Lemoine axis
X(6221) = X(6)-of-Lucas-tangents-triangle
X(6221) = X(7)-of-Lucas-central-triangle
X(6221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,Y(155) (below)), (3,371,3311), (3,3311,3312), (371,372,3592), (371,1151,3)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6222) lies on these lines: {2, 6289}, {1350, 6312}, {1991, 6279}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6223) lies on these lines: {2, 84}, {3, 5658}, {4, 7}, {8, 6001}, {20, 78}, {30, 5758}, {40, 144}, {145, 515}, {189, 318}, {443, 5927}, {651, 1498}, {1012, 5703}, {1158, 3219}, {1270, 6257}, {1271, 6258}, {1330, 2897}, {1532, 5704}, {1709, 3085}, {1750, 4292}, {2096, 3149}, {2475, 5554}, {2829, 6224}, {2894, 5175}, {3062, 5290}, {3091, 6245}, {4295, 5691}, {5731, 6261}
X(6223) = reflection of X(i) in X(j) for these (i,j): (20,1490), (4,6250)
X(6223) = isogonal conjugate of X(34432)
X(6223) = X(322)-Ceva conjugate of X(2)
X(6223) = anticomplement of X(84)
X(6223) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 962), (3, 280), (9, 189), (40, 8), (100, 4397), (101, 6332), (198, 2), (221, 145), (223, 7), (227, 2475), (322, 6327), (329, 69), (347, 3434), (651, 4131), (1103, 6223), (1167, 84), (1262, 934), (1817, 75), (2187, 192), (2199, 3210), (2324, 329), (2331, 5905), (2360, 1), (3194, 3868), (3195, 193), (6129, 149)
X(6223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1071,938), (84,6260,2), (3146,5905,962)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6224) lies on these lines: {1, 149}, {2, 80}, {3, 8}, {4, 6265}, {7, 528}, {11, 2476}, {20, 2800}, {30, 5180}, {59, 3319}, {78, 5531}, {144, 2801}, {145, 2802}, {153, 515}, {484, 519}, {535, 4867}, {900, 3904}, {962, 5840}, {1270, 6262}, {1271, 6263}, {1385, 5086}, {1387, 3488}, {1444, 4720}, {1737, 4881}, {1768, 4297}, {1862, 4214}, {2345, 4287}, {2771, 3648}, {2829, 6223}, {2894, 4861}, {3036, 6174}, {3091, 6246}, {3189, 5854}, {3419, 3655}, {3600, 5083}, {3884, 5441}, {3897, 5794}, {4084, 4325}, {4855, 5881}, {5036, 5839}, {5176, 5440}, {5330, 6284}, {5528, 5853}
X(6224) = reflection of X(i) in X(j) for these (i,j): (1320,1317), (149,1), (1523,6326), (1768,4297), (4,6265), (5080,4511), (5176,5440), (6264,5882), (80,214), (8,100)
X(6224) = isogonal conjugate of X(34431)
X(6224) = isotomic conjugate of X(36917)
X(6224) = complement of X(20085)
X(6224) = X(320)-Ceva conjugate of X(2)
X(6224) = X(i)-cevapoint of X(j) for these (i,j): (5541, 6326)
X(6224) = anticomplement of X(80)
X(6224) = X(484)-of-inner-Garcia-triangle
X(6224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (80,214,2), (100,104,4996), (104,1145,5744), (1317,1320,3241)
X(6224) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 5080), (36, 8), (58, 758), (101, 3762), (106, 80), (109, 3738), (320, 6327), (758, 1330), (1443, 3434), (1464, 2475), (1870, 4), (1983, 514), (2245, 2895), (2323, 329), (2361, 144), (3218, 69), (3724, 1654), (3960, 150), (4282, 63), (4511, 3436), (4973, 2891), (6149, 3648)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let U be the line tangent to K007 (the Lucas cubic) at X(4), and let V be the line tangent to K007 at X(20). Then X6225) = U∩V. (Randy Hutson, May 5, 2015)
X(6225) lies on these lines: {2, 64}, {3, 5656}, {4, 51}, {20, 394}, {74, 3147}, {107, 3183}, {154, 3522}, {193, 1503}, {221, 390}, {388, 6285}, {631, 3357}, {962, 3868}, {1192, 4232}, {1270, 6266}, {1271, 6267}, {1595, 3426}, {1853, 3832}, {2192, 3600}, {2777, 3529}, {3091, 6247}
X(6225) = reflection of X(i) in X(j) for these (i,j): (X[3146], X[5895]), (X[4], X[5878]), (X[64], X[2883])
X(6225) = isotomic conjugate of cyclocevian conjugate of X(35510)
X(6225) = anticomplement of X(64)
X(6225) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 3146), (20, 8), (63, 253), (154, 192), (204, 193), (610, 2), (662, 3265), (775, 64), (1097, 6225), (1249, 5905), (1394, 145), (1895, 4), (2287, 2184), (3198, 1654), (5930, 2475)
X(6225): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64,2883,2), (154,5894,3522), (1853,5893,3832)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6226) lies on these lines: {6, 98}, {114, 5590}, {115, 6201}, {147, 1270}, {542, 5860}, {1160, 2782}, {2794, 5870}, {5874, 6274}, {6033, 6214}
X(6226) = reflection of X(i) in X(j) for these (i,j): (6227, 98), (6320, 1160)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6227) lies on these lines: {6, 98}, {114, 5591}, {115, 6202}, {147, 1271}, {542, 5861}, {1161, 2782}, {2794, 5871}, {5875, 6275}, {6033, 6215}
X(6227) = reflection of X(i) in X(j) for these (i,j): (6226, 98), (6319, 1161)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6228) lies on these lines: {2, 372}, {3, 6230}, {639, 6250}
X(6228) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (485,488,6312)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6229) lies on these lines: {2, 371}, {3, 6231}, {640, 6251}
X(6229) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (486,487,6316)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6230) lies on these lines: {2, 98}, {3, 6228}, {99, 638}, {115, 485}, {1991, 6321}, {2782, 3070}, {6033, 6290}
X(6230) = reflection of X(6231) in X(114)
X(6230) = complement of X(33430)
X(6230) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (98,1352,6231)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6231) lies on these lines: {2, 98}, {3, 6229}, {99, 637}, {115, 486}, {591, 6321}, {2782, 3071}, {6033, 6289}
X(6231) = reflection of X(6230) in X(114)
X(6231) = complement of X(33431)
X(6231) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (98,1352,6230)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6232) lies on the Brocard circle and these lines: {3, 3849}, {182, 6322}
X(6232) = reflection of X(6322) in X(182)
X(6232) = Brocard circle antipode of X(6322)
X(6232) = 5th-Euler-to-2nd-Brocard similarity image of X(12494)
X(6232) = X(6233)-of-1st-Brocard-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
Let AB, AC, BC, BA, CA, CB be the points on the Dao 6-point circle as defined at X(5569). The triangles BACBAC and CAABBC are perspective from X(2), and X(6233) is the isogonal conjugate of the infinite point of their perspectrix. (Randy Hutson, January 29, 2015)
X(6233) lies on the circumcircle and these lines: {3, 6323}, {111, 182}, {542, 6325}, {690, 6236}, {843, 2080}
X(6233) = reflection of X(6323) in X(3)
X(6233) = isogonal conjugate of X(8704)
X(6233) = X(98)-of-circumsymmedial-triangle
X(6233) = Λ(radical axis of orthocentroidal circle and McCay circumcircle)
X(6233) = Λ(polar of X(2) wrt Dao 6-point circle)
X(6233) = 1st-Parry-to-ABC similarity image of X(353)
X(6233) = X(6785)-of-4th-anti-Brocard-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6234) lies on these lines: {3, 3224}, {4, 147}, {98, 3225}, {511, 694}, {805, 2080}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6235) lies on the orthocentroidal circle and this line: {381, 6324}
X(6235) = reflection of X(6324) in X(381)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6236) lies on the circumcircle and these lines: {3, 6325}, {111, 381}, {542, 6323}, {690, 6233}, {842, 3849}, {2373, 6031}
X(6236) = reflection of X(6325) in X(3)
X(6236) = anticomplement of X(34113)
X(6237) is also the extangents-triangle-orthologic center of these triangles: intangents and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.
X(6237) lies on these lines: {3, 48}, {19, 52}, {30, 6254}, {55, 155}, {65, 68}, {539, 6255}, {3564, 3779}, {3611, 5562}, {5777, 5816}, {5889, 6197}
X(6237) = reflection of X(6238) in X(155)
X(6237) = orthocenter-of-extangents-triangle
X(6238) is also the intangents-triangle-orthologic center of these triangles: extangents, tangential, and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.
X(6238) lies on these lines: {3, 1069}, {30, 6285}, {33, 52}, {35, 1147}, {55, 155}, {68, 1479}, {185, 1060}, {498, 5654}, {539, 6286}, {912, 3057}, {916, 2293}, {942, 5733}, {1040, 1216}, {1062, 3270}, {3056, 3564}, {4296, 6241}, {5889, 6198}
X(6238) = reflection of X(6237) in X(155)
X(6238) = orthocenter-of-intangents triangle
X(6238) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (3270,5562,1062)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6239) lies on these lines: {4, 69}, {24, 1151}, {1306, 3563}, {1588, 3567}, {5870, 6219}, {6197, 6252}, {6198, 6283}
X(6239) = reflection of X(4) in X(6291)
X(6239) = isogonal conjugate of X(6401)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let MaMbMc = medial triangle and DEF = circumorthic triangle. Let (Oa) be the circle with center A that passes through E and F (and X(4)). Let E' be the point, other than E, in which (Oa) meets EMa, and let F' be the point, other than F, in which (Oa) meets FMa. Let A′ = EF'∩FE', and define B′ and C′ cyclically. The lines A′DF, B′E, C′F concur in X(6240). (Angel Montesdeoca, September 25, 2020)
X(6240) lies on these lines: {2, 3}, {32, 5523}, {50, 53}, {52, 1986}, {64, 70}, {74, 6145}, {185, 6152}, {930, 1299}, {933, 1300}, {1105, 1179}, {1503, 6241}, {6197, 6253}, {6198, 6284}
X(6240) = reflection of X(4) in X(3575)
X(6240) = circumcircle-inverse of X(37970)
X(6240) = Kosnita-to-orthic similarity image of X(4)
X(6240) = Ehrmann-vertex-to-orthic similarity image of X(3)
X(6240) = X(5903)-of-orthic-triangle if ABC is acute
X(6240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,1594), (4,20,378), (4,24,403), (4,186,5), (4,376,3541), (4,3147,3091), (4,3518,235), (4,3520,427), (25,382,4), (235,3627,4), (427,550,3520), (3089,3543,4)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6241) lies on these lines: {3, 74}, {4, 51}, {24, 1192}, {30, 5889}, {52, 3146}, {54, 64}, {70, 4846}, {143, 3830}, {184, 3357}, {186, 1204}, {376, 5562}, {382, 3060}, {403, 2883}, {511, 3529}, {546, 5640}, {550, 2979}, {568, 3627}, {631, 5907}, {944, 2807}, {1147, 2071}, {1154, 1657}, {1173, 3426}, {1216, 3522}, {1503, 6240}, {1594, 6247}, {1986, 5895}, {2772, 5693}, {2781, 5925}, {3523, 5891}, {3528, 3917}, {3534, 6101}, {3542, 5656}, {3543, 5446}, {3832, 5462}, {3843, 5946}, {3855, 5943}, {4296, 6238}, {5056, 5892}, {5870, 6219}, {5871, 6220}, {6197, 6254}, {6198, 6285}
X(6241) = reflection of X(i) in X(j) for these (i,j): (3146,52), (382,6102), (4,185)
X(6241) = crosssum of X(3) and X(382)
X(6241) = X(20)-of-X(3)-Fuhrmann-triangle
X(6241) = X(4)-of-X(4)-anti-altimedial-triangle
X(6241) = X(4)-of-X(4)-adjunct-anti-altimedial-triangle
X(6241) = X(8)-of-circumorthic-triangle if ABC is acute
X(6241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,185,5890), (4,5890,3567), (64,1181,378), (74,1614,3), (184,3357,3520), (378,1181,54), (382,6102,3060), (1899,5878,4), (12287,12288, 12283)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6242) lies on the circumcircle and these lines: {4, 93}, {24, 195}, {52, 110}, {54, 186}, {539, 5889}, {568, 1493}, {3060, 5448}, {6197, 6255}, {6198, 6286}
X(6242) = reflection of X(4) in X(6152)
X(6243) is also the reflection-triangle-orthologic center of these triangles: intangents, extangents, tangential, and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.
Let A′B′C′ be the reflection triangle. Let OA be the circle with center A and pass-through point A′, and define OB and OC cyclically. The radical center of the circles OA, OB, OC is X(6243). If O′A has center A′ and pass-through point A, and cyclically for O′B, O′C, the radical center is X(3519). (Randy Hutson, January 29, 2015)
Let A′B′C′ be the reflection triangle. Let AB, AC be the orthogonal projections of A′ on CA, AB, resp. Let A″ = CAAC∩ABBA, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X(6). The lines A′A″, B′B″, C′C″ concur in X(6243), which is also X(3)-of-A″B″C″. (Randy Hutson, March 29, 2020)
X(6243) lies on these lines: {2, 143}, {3, 6}, {4, 93}, {5, 3060}, {20, 6102}, {23, 156}, {26, 49}, {30, 5889}, {51, 1216}, {70, 265}, {140, 2979}, {184, 195}, {185, 1657}, {323, 3518}, {343, 5576}, {381, 5446}, {550, 5890}, {631, 5946}, {1112, 3542}, {1147, 2070}, {3146, 5663}, {3526, 3917}, {3628, 5640}, {3843, 5907}, {3851, 5891}, {5054, 5447}, {5070, 5943}, {5073, 6000}
X(6243) = reflection of X(i) in X(j) for these (i,j): (1657, 185), (20,6102), (3519,6152), (3,52), (5562,5446), (6101),143)
X(6243) = anticomplement of X(6101)
X(6243) = inverse-in-circle-{{X(371), X(372), PU(1), PU(39)}} of X(2965)
X(6243) = polar-circle-inverse of X(35718)
X(6243) = Brocard-circle-inverse of X(13353)
X(6243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13353), (3,52,568), (26,1993,49), (51,1216,1656), (143,6101,2), (195,2937,184), (371,372,2965), (2979,3567,140), (3917,5462,3526), (5446,5562,381)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6244) lies on these lines: {1, 3}, {4, 3820}, {20, 3421}, {100, 329}, {101, 1615}, {102, 6014}, {103, 1293}, {197, 2823}, {200, 971}, {210, 1709}, {218, 1200}, {219, 2272}, {474, 962}, {480, 2951}, {516, 1376}, {527, 4421}, {910, 1766}, {954, 5281}, {1012, 5273}, {1158, 3927}, {1598, 1753}, {1699, 4413}, {2717, 2743}, {3158, 5732}, {3474, 5762}, {3522, 3871}, {3689, 5918}, {3913, 4297}, {3940, 6001}, {4295, 5763}
X(6244) = midpoint of X(i) and X(j) for these {i,j}: {20, 3421}, {40, 6282}
X(6244) = reflection of X(i) in X(j) for these (i,j): (3359, 3579), (4,3820), (999,3)
X(6244) = anticomplement of X(7956)
X(6244) = X(144)-Ceva conjugate of X(220)
X(6244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (35,5584,3), (40,2077,3428), (55,165,3), (55,1155,1617), (165,5537,55), (210,1709,5779), (2077,3428,3)
X(6244) = X(1368)-of-excentral-triangle
X(6244) = X(78)-gimel conjugate of X(3428)
X(6244) = radical center of mixtilinear excircles (Forum Geometricorum, FG200601
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6245) lies on these lines: {1, 3427}, {2, 1490}, {3, 10}, {4, 57}, {5, 142}, {7, 5715}, {8, 6282}, {20, 4652}, {40, 4847}, {104, 4311}, {149, 2950}, {189, 3345}, {226, 1071}, {381, 6259}, {443, 5587}, {496, 942}, {516, 1158}, {527, 5812}, {944, 3601}, {950, 1012}, {990, 5292}, {1125, 6261}, {1155, 6253}, {1466, 1837}, {1467, 3086}, {1479, 1709}, {1699, 5586}, {1737, 4299}, {1768, 1770}, {1855, 2272}, {1872, 2823}, {2800, 4084}, {2829, 6246}, {3090, 5658}, {3091, 6223}, {3149, 3911}, {3452, 5777}, {3579, 5771}, {3664, 5713}, {4187, 5927}, {5705, 5732}, {5708, 5805}, {6201, 6257}, {6202, 6258}
X(6245) = midpoint of X(i) and X(j) for these {i,j}: {149,2950}, {3,5787}, {4,84}
X(6245) = reflection of X(i) in X(j) for these (i,j): (4297,5450), (6260,5), (6261,1125)
X(6245) = crosspoint of X(189) anmd X(273)
X(6245) = crosssum of X(198) and X(212)
X(6245) = complement of X(1490)
X(6245) = X(i)-complementary conjugate of X(j) for these (i,j): (56, 3341), (1034, 1329), (3342, 6260), (3345, 10)
X(6245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5789,5791), (946,5884,3671), (3358,5709,1158)
X(6245) = X(521)-gimel conjugate of X(1770)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let OA, OB, OC be the circles with collinear centers described at X(5531) and Hyacinthos #21433 (Barry Wolk, January 2013). Let A′B′C′ be the triangle formed by the radical axes of these circles and the corresponding excircle. Then X(6246) is the circumcenter of A′B′C′. (Randy Hutson, January 29, 2015)
X(6246) lies on these lines: {4, 80}, {5, 214}, {10, 5840}, {11, 515}, {100, 5587}, {104, 5560}, {355, 2802}, {381, 6265}, {546, 946}, {1320, 5881}, {1387, 5882}, {1478, 5083}, {1837, 5884}, {2801, 5805}, {2829, 6245}, {3091, 6224}, {5768, 6256}, {6201, 6262}, {6202, 6263}
X(6246) = midpoint of X(i) and X(j) for these {i,j}: {104,5691}, {1320,5881}, {4,80}
X(6246) = reflection of X(i) in X(j) for these (i,j): (214,5), (5882,1387)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6247) lies on these lines: {2, 1498}, {3, 66}, {4, 64}, {5, 2883}, {6, 3088}, {10, 5777}, {11, 6285}, {20, 343}, {30, 3357}, {51, 1907}, {52, 2781}, {74, 6145}, {125, 235}, {154, 631}, {185, 427}, {221, 3085}, {253, 3346}, {381, 5878}, {389, 1595}, {1181, 3541}, {1204, 3575}, {1368, 5907}, {1593, 1899}, {1594, 6241}, {1753, 5928}, {2192, 3086}, {2777, 3627}, {2935, 3448}, {3090, 5656}, {3091, 6225}, {3146, 3580}, {3925, 6254}, {5085, 5596}, {5890, 6293}, {6201, 6266}, {6202, 6267}
X(6247) = midpoint of X(i) and X(j) for these {i,j}: {2935, 3448}, {3146,5925}, {4,64}
X(6247) = reflection of X(i) in X(j) for these (i,j): (2883,5), (5878,5893), (5894,3357)
X(6247) = crosspoint of X(i) and X(j) for these (i,j): (253, 2052)
X(6247) = crosssum of X(154) and X(577)
X(6247) = complement of X(1498)
X(6247) = X(i)-complementary conjugate of X(j) for these (i,j): (19, 3343), (3346, 10)
X(6247) = X(3913)-of-orthic-triangle if ABC is acute
X(6247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64,1853,4), (381,5878,5893), (389,1595,5480)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6248) lies on these lines: {3, 3734}, {4, 69}, {5, 39}, {30, 5188}, {83, 575}, {98, 384}, {183, 5171}, {194, 262}, {381, 538}, {671, 3399}, {730, 946}, {732, 5480}, {3102, 6289}, {3103, 6290}, {3146, 6194}, {3564, 5052}, {6201, 6272}, {6202, 6273}, {6287, 6321}
X(6248) = midpoint of X(4) and X(76)
X(6248) = reflection of X(i) in X(j) for these (i,j): (39,5), (3,3934)
X(6248) = X(76)-of-Euler-triangle
X(6248) = midpoint of orthocenters of antipedal triangles of PU(1)
X(6248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,3091,262), (2009,2010,5254)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6249) lies on these lines: {4, 83}, {5, 5188}, {115, 546}, {381, 754}, {382, 4045}, {732, 5480}, {2896, 3091}, {6201, 6274}, {6202, 6275}
X(6249) = midpoint of X(4) and X(83)
X(6249) = reflection of X(6292) in X(5)
X(6249) = X(83)-of-Euler-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6250) lies on these lines: {3, 6118}, {4, 371}, {5, 641}, {115, 3070}, {381, 591}, {488, 3091}, {546, 576}, {639, 6228}, {1131, 5870}, {2043, 6304}, {2044, 6305}, {3832, 6201}, {3839, 6202}
X(6250) = midpoint of X(4) and X(485)
X(6250) = reflection of X(i) in X(j) for these (i,j): (3,6118), (641,5)
X(6250) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (546,5480,6251)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6251) lies on these lines: {3, 6119}, {4, 372}, {5, 642}, {115, 3071}, {381, 1991}, {487, 3091}, {546, 576}, {640, 6229}, {1132, 5871}, {2043, 6301}, {2044, 6300}, {3832, 6202}, {3839, 6201}
X(6251) = midpoint of X(4) and X(486)
X(6251) = reflection of X(i) in X(j) for these (i,j): (3,6119), (642,5)
X(6251) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (546,5480,6250)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6252) lies on these lines: {19, 6291}, {40, 511}, {55, 1151}, {65, 175}, {6197, 6239}
X(6252) = reflection of X(6283) in X(1151)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6253) lies on these lines: {1, 5805}, {3, 3925}, {4, 12}, {5, 5259}, {11, 3149}, {19, 3575}, {20, 958}, {30, 40}, {46, 5787}, {65, 515}, {72, 516}, {165, 5791}, {382, 5840}, {411, 2886}, {528, 962}, {910, 1855}, {944, 5434}, {946, 3058}, {971, 1770}, {1155, 6245}, {1490, 1836}, {1503, 3779}, {1750, 5812}, {1867, 3198}, {1885, 5130}, {3070, 5415}, {3071, 5416}, {3072, 5721}, {3146, 3436}, {5221, 5768}, {5762, 5904}, {5786, 6047}, {6197, 6240}
X(6253) = reflection of X(6284) in X(4)
X(6253) = {X(20),X(2550)}-harmonic conjugate of X(5584)
X(6253) = homothetic center of extangents triangle and reflection of intangents triangle in X(4)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6254) lies on these lines: {4, 65}, {19, 185}, {30, 6237}, {40, 2939}, {55, 1498}, {64, 71}, {1503, 3779}, {3925, 6247}, {6197, 6241}
X(6254) = reflection of X(6285) in X(1498)
X(6254) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (64,3197,5584)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6255) lies on these lines: {19, 6152}, {40, 1154}, {54, 71}, {55, 195}, {65, 2962}, {539, 6237}, {3779, 5965}, {6197, 6242}
X(6255) = reflection of X(6286) in X(195)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6256) lies on these lines: {1, 4}, {2, 5450}, {3, 119}, {8, 153}, {10, 1158}, {12, 1012}, {20, 2077}, {30, 4421}, {40, 3436}, {46, 1512}, {56, 1532}, {80, 5553}, {84, 377}, {104, 499}, {191, 2950}, {355, 5836}, {382, 5842}, {971, 5880}, {1071, 1837}, {1470, 3149}, {1537, 2098}, {1788, 2096}, {2475, 5554}, {2478, 3576}, {3086, 5193}, {3434, 5881}, {3577, 5555}, {5046, 5731}, {5220, 5690}, {5768, 6246}, {5777, 5794}
X(6256) = midpoint of X(i) and X(j) for these (i,j): (1490,5691), (355,6259)
X(6256) = reflection of X(i) and X(j) for these (i,j): (1158,10), (6261, 6260)
X(6256) = anticomplement of X(5450)
X(6256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,388,946), (4,944,1479), (20,5552,2077), (3585,5691,4)
X(6256) = X(8)-beth conjugate of X(1158)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6257) lies on these lines: {6, 84}, {971, 1160}, {1270, 6223}, {2829, 6262}, {3640, 6001}, {5590, 6260}, {6201, 6245}, {6214, 6259}
X(6257) = reflection of X(6258) in X(84)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6258) lies on these lines: {6, 84}, {971, 1161}, {1271, 6223}, {2829, 6263}, {3641, 6001}, {5591, 6260}, {6202, 6245}, {6215, 6259}
X(6258) = reflection of X(6257) in X(84)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6259) lies on these lines: {3, 3452}, {4, 7}, {5, 84}, {10, 5779}, {12, 1709}, {20, 5440}, {30, 1490}, {40, 6068}, {355, 5836}, {377, 5927}, {381, 6245}, {382, 515}, {516, 3913}, {952, 3680}, {1158, 3652}, {1538, 3086}, {1836, 3340}, {2829, 6261}, {3587, 5924}, {4208, 5817}, {5044, 5811}, {6214, 6257}, {6215, 6258}
X(6259) = midpoint of X(4) and X(6223)
X(6259) = reflection of X(i) in X(j) for these (i,j): (355, 6256), (3,6260), (5787,4), (5,84)
X(6259) = anticomplement of X(34862)
X(6259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7,5806), (4,1071,5722)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6260) lies on these lines: {1, 4}, {2, 84}, {3, 3452}, {5, 142}, {9, 1158}, {10, 5777}, {20, 908}, {40, 329}, {72, 1145}, {165, 5924}, {214, 2829}, {381, 5787}, {405, 5450}, {442, 5927}, {452, 3576}, {496, 1538}, {498, 1709}, {516, 5812}, {527, 5709}, {631, 5316}, {962, 3895}, {1071, 1210}, {1466, 3149}, {1512, 4848}, {1715, 2183}, {1728, 3911}, {1753, 1763}, {1901, 2092}, {3091, 5249}, {3651, 5660}, {4667, 5707}, {5175, 5881}, {5177, 5587}, {5534, 5853}, {5590, 6257}, {5591, 6258}, {5779, 5791}, {5806, 6147}, {5837, 5887}
X(6260) = midpoint of X(i) and X(j) for these {i,j}: (3,6259), (4,1490), (6256,6261), (84,6223)
X(6260) = reflection of X(6245) in X(5)
X(6260) = X(2)-Ceva conjugate of X(1108)
X(6260) = crosspoint of X(i) and X(j) for these (i,j): (2, 322), (329, 342)
X(6260) = crosssum of X(i) and X(j) for these (i,j): (6, 2208), (1436, 2188)
X(6260) = complement of X(84)
X(6260) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 946), (6, 57), (31, 1108), (40, 10), (42, 1901), (55, 281), (56, 3086), (198, 2), (208, 1210), (221, 1), (223, 142), (227, 442), (322, 2887), (329, 141), (347, 2886), (1103, 6260), (1817, 3739), (2187, 37), (2199, 3752), (2324, 3452), (2331, 226), (2334, 4295), (2360, 1125), (3194, 942), (3195, 6), (3209, 3772), (3342, 6245), (6129, 11)
X(6260) = X(84)-isoconjugate of X(1167)
X(6261) is also the 2nd-circumperp-triangle-orthologic center of the extouch triangle. See César Lozada, Perspective-Orthologic-Parallelogic.
X(6261) lies on these lines: {1, 4}, {3, 960}, {9, 1630}, {10, 5720}, {20, 224}, {21, 84}, {36, 920}, {40, 78}, {56, 1071}, {57, 5884}, {63, 5693}, {65, 3149}, {72, 3428}, {90, 104}, {355, 2886}, {517, 3811}, {519, 5534}, {602, 1780}, {758, 5709}, {958, 5777}, {971, 1001}, {990, 3736}, {1012, 2646}, {1062, 2883}, {1125, 6245}, {1319, 1898}, {1532, 1837}, {1709, 3612}, {2077, 4855}, {2476, 5587}, {2720, 2733}, {2829, 6259}, {3086, 5768}, {3632, 5531}, {3740, 5780}, {3870, 3885}, {3872, 5086}, {3901, 5536}, {3957, 5734}, {5396, 5799}, {5698, 5732}, {5731, 6223}, {5787, 5886}
X(6261) = midpoint of X(1) in X(1490)
X(6261) = reflection of X(i) in X(j) for these {i,j}: {1158,3}, (6245,1125), (6256,6260), (84,5450)
X(6261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3485,946), (40,6326,78), (84,3576,5450), (411,3869,40)
X(6261) = X(21)-beth conjugate of X(34)
X(6261) = X(9927)-of-excentral-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6262) lies on these lines: {6, 80}, {100, 5688}, {214, 5590}, {952, 3640}, {1270, 6224}, {2800, 5870}, {2829, 6257}, {6201, 6246}, {6214, 6265}
X(6262) = reflection of X(6263) in X(80)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6263) lies on these lines: {6, 80}, {100, 5689}, {214, 5591}, {952, 3641}, {1271, 6224}, {2800, 5871}, {2829, 6258}, {6202, 6246}, {6215, 6265}
X(6263) = reflection of X(6262) in X(80)
X(6263) = X(21)-beth conjugate of X(1411)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6264) lies on the hexyl circle and these lines: {1, 5}, {3, 5541}, {40, 104}, {84, 1320}, {100, 3576}, {149, 515}, {153, 946}, {214, 5438}, {517, 1768}, {528, 5732}, {936, 3036}, {1145, 4853}, {1389, 3881}, {1482, 2771}, {2077, 3880}, {2098, 5693}, {2801, 3243}, {2894, 4861}, {3885, 5450}, {5538, 5844}
X(6264) = reflection of X(i) in X(j) for these (i,j): (153,946), (355,1484), (40,104), (5531, 6265), (5541,3), (5881,80), (6224,5882), (6326,1)
X(6264) = X(74)-of-hexyl-triangle
X(6264) = hexyl-isogonal conjugate of X(517)
X(6264) = X(7728)-of-excentral-triangle
X(6264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5531,6265), (5531,6265,6326)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6265) lies on these lines: {1, 5}, {3, 214}, {4, 6224}, {21, 104}, {78, 1145}, {100, 517}, {149, 5603}, {153, 944}, {224, 1537}, {381, 6246}, {528, 3656}, {912, 1319}, {997, 3035}, {999, 5083}, {1001, 2801}, {1259, 5730}, {1320, 1389}, {1482, 2802}, {1749, 5427}, {1768, 3576}, {2095, 4930}, {2783, 3736}, {2829, 6259}, {2975, 5694}, {3654, 6174}, {3811, 5854}, {3869, 4996}, {5253, 5885}, {6214, 6262}, {6215, 6263}
X(6265) = midpoint of X(i) and X(j) for these {i,j}: {153,944), (1,6326), (4,6224), (5531,6264)
X(6265) = reflection of X(i) in X(j) for these (i,j): (104,1385), (1484, 5901), (355,119), (3654,6174), (3,214), (80,5)
X(6265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5531,6264), (6264,6326,5531)
X(6265) = inner-Garcia-to-ABC similarity image of X(355)
X(6265) = X(10113)-of-excentral-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6266) lies on these lines: {4, 6217}, {6, 64}, {1160, 6000}, {1162, 5895}, {1270, 6225}, {2883, 5590}, {5878, 6214}, {6201, 6247}
X(6266) = reflection of X(6267) in X(64)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6267) lies on these lines: {4, 6218}, {6, 64}, {1161, 6000}, {1163, 5895}, {1271, 6225}, {2883, 5591}, {5878, 6215}, {6202, 6247}
X(6267) = reflection of X(6266) in X(64)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6268) lies on these lines: {6, 13}, {530, 5860}, {616, 1270}, {618, 5590}, {2041, 6278}, {5478, 6201}, {5617, 6214}
X(6268) = reflection of X(6270) in X(13)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6269) lies on these lines: {6, 13}, {531, 5860}, {617, 1270}, {619, 5590}, {2042, 6278}, {5479, 6201}, {5613, 6214}
X(6269) = reflection of X(6271) in X(14)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6270) lies on these lines: {6, 13}, {530, 5861}, {616, 1271}, {618, 5591}, {2042, 6281}, {5478, 6202}, {5617, 6215}
X(6270) = reflection of X(6268) in X(13)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6271) lies on these lines: {{6, 13}, {531, 5861}, {617, 1271}, {619, 5591}, {2041, 6281}, {5479, 6202}, {5613, 6215}
X(6271) = reflection of X(6269) in X(14)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6272) lies on these lines: {{6, 76}, {39, 5590}, {194, 1270}, {511, 5870}, {538, 5860}, {637, 6318}, {730, 3640}, {1160, 2782}, {3095, 6214}, {6201, 6248}
X(6272) = reflection of X(6273) in X(76)
X(6272) = X(76)-of-outer-Grebe-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6273) lies on these lines: {6, 76}, {39, 5591}, {194, 1271}, {511, 5871}, {538, 5861}, {638, 6314}, {730, 3641}, {1161, 2782}, {3095, 6215}, {6202, 6248}
X(6273) = reflection of X(6272) in X(76)
X(6273) = X(76)-of-inner-Grebe-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6274) lies on these lines: {6, 76}, {637, 754}, {1270, 2896}, {5590, 6292}, {5874, 6226}, {6201, 6249}, {6214, 6287}
X(6274) = reflection of X(6275) in X(83)
X(6274) = X(83)-of-outer-Grebe-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6275) lies on these lines: {6, 76}, {638, 754}, {1271, 2896}, {5591, 6292}, {5875, 6227}, {6202, 6249}, {6215, 6287}
X(6275) = reflection of X(6274) in X(83)
X(6275) = X(83)-of-inner-Grebe-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6276) lies on these lines: {4, 6219}, {6, 24}, {539, 5860}, {1154, 1160}, {1162, 3574}, {1209, 5590}, {1270, 2888}, {6214, 6288}
X(6276) = reflection of X(6277) in X(54)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6277) lies on these lines: {4, 6220}, {6, 24}, {539, 5861}, {1154, 1161}, {1163, 3574}, {1209, 5591}, {1271, 2888}, {6215, 6288}
X(6277) = reflection of X(6276) in X(54)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6278) lies on these lines: {4, 5860}, {5, 6}, {20, 488}, {382, 1160}, {631, 641}, {2041, 6268}, {2042, 6269}, {3640, 5881}, {3832, 6201}, {5067, 6118}
X(6278) = reflection of X(i) in X(j) for these (i,j): (485,6289), (6279,485)
X(6278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5874,6280), (5874,6214,6)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6279) lies on these lines: {5, 6}, {376, 5861}, {488, 1271}, {641, 3525}, {1161, 1657}, {1991, 6222}, {3146, 5871}, {3839, 6202}
X(6279) = reflection of X(6278) in X(485)
X(6279) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6,5875,6281)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6280) lies on these lines: {5, 6}, {376, 5860}, {487, 1270}, {642, 3525}, {1160, 1657}, {3146, 5870}, {3839, 6201}
X(6280) = reflection of X(6281) in X(486)
X(6280) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6,5874,6278)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6281) lies on these lines: {4, 5861}, {5, 6}, {20, 487}, {382, 1161}, {631, 642}, {2041, 6271}, {2042, 6270}, {3641, 5881}, {3832, 6202}, {5067, 6119}
X(6281) = reflection of X(i) in X(j) for these (i,j): (486,6290), (6280,486)
X(6281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5875,6279), (5875,6215,6)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6282) lies on these lines: {1, 3}, {4, 936}, {8, 6245}, {9, 1012}, {20, 78}, {28, 1753}, {30, 1750}, {72, 84}, {101, 610}, {102, 1292}, {142, 5603}, {200, 515}, {284, 4221}, {376, 527}, {377, 5715}, {411, 4855}, {443, 946}, {516, 997}, {519, 5768}, {971, 3940}, {1308, 2745}, {1709, 5692}, {2057, 3436}, {2716, 2742}, {2951, 6326}, {3149, 5438}, {3358, 5223}, {3577, 3753}, {3654, 5771}, {3811, 4297}, {3820, 5587}, {3870, 5731}, {3872, 5744}, {4292, 5758}, {4882, 5787}, {5657, 5745}
X(6282) = midpoint of X(20) and X(329)
X(6282) = reflection of X(i) in X(j) for these (i,j): (1750,5720), (2093,3359), (40,6244), (4, 3452), (57,3)
X(6282) = anticomplement of X(7682)
X(6282) = X(25)-of-hexyl-triangle
X(6282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3587,165), (20,78,1490), (40,2077,165), (40,3576,3428), (165,2093,3359), (165,5538,1)
X(6282) = X(78)-gimel conjugate of X(165)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6283) lies on these lines: {1, 256}, {33, 6291}, {55, 1151}, {172, 5414}, {1335, 2330}, {6198, 6239}
X(6283) = reflection of X(6252) in X(1151)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6284) lies on these lines: {1, 30}, {2, 5217}, {3, 11}, {4, 12}, {5, 35}, {8, 190}, {10, 3683}, {20, 56}, {21, 2886}, {31, 1834}, {33, 3575}, {34, 1885}, {36, 496}, {37, 1839}, {40, 1728}, {46, 5722}, {52, 517}, {65, 516}, {72, 1898}, {80, 3467}, {100, 1329}, {140, 5010}, {145, 529}, {149, 2975}, {197, 4186}, {201, 2310}, {215, 1614}, {221, 5895}, {226, 4314}, {227, 1877}, {229, 5196}, {329, 3189}, {354, 4292}, {355, 5119}, {376, 3086}, {377, 1001}, {381, 498}, {382, 1478}, {388, 390}, {404, 3816}, {405, 3925}, {411, 5172}, {427, 5310}, {428, 612}, {440, 2218}, {442, 5248}, {443, 4423}, {452, 2550}, {495, 3585}, {515, 3057}, {519, 3962}, {535, 3244}, {601, 5348}, {936, 4679}, {938, 3474}, {942, 1770}, {944, 1317}, {946, 2646}, {952, 5693}, {958, 3434}, {962, 2099}, {964, 4026}, {976, 4415}, {999, 1657}, {1043, 4388}, {1058, 3304}, {1104, 3914}, {1155, 1210}, {1250, 5321}, {1319, 4297}, {1376, 2478}, {1388, 5731}, {1399, 1936}, {1456, 5930}, {1467, 2951}, {1482, 5841}, {1486, 4185}, {1503, 3056}, {1621, 2475}, {1656, 5326}, {1697, 5252}, {1699, 3601}, {1709, 5787}, {1737, 3579}, {1842, 3198}, {1884, 3185}, {1914, 5254}, {2066, 3070}, {2330, 5480}, {2361, 3073}, {2654, 4300}, {2777, 3028}, {2794, 3027}, {3035, 4193}, {3052, 5230}, {3071, 5414}, {3091, 5218}, {3219, 5178}, {3338, 4333}, {3436, 3913}, {3485, 4313}, {3488, 4295}, {3522, 5274}, {3543, 5229}, {3584, 3845}, {3600, 5059}, {3612, 5886}, {3665, 4872}, {3695, 4680}, {3703, 5015}, {3704, 5016}, {3826, 5047}, {3832, 5281}, {3871, 5080}, {3932, 5300}, {4030, 4385}, {4046, 5814}, {4189, 4999}, {4251, 5134}, {4305, 5603}, {4316, 5563}, {4413, 5084}, {4421, 5552}, {4689, 5530}, {4848, 5183}, {5048, 5882}, {5250, 5794}, {5330, 6224}, {5687, 6154}, {6198, 6240}
X(6284) - reflection of X(i) in X(j) for these (i,j): (1770,942), (5434,3058), (6253,4), (65,950)
X(6284) = crosspoint of X(i) and X(j) for these (i,j): (7, 1751)
X(6284) = crosssum of X(55) and X(579)
X(6284) = orthocenter of Mandart-incircle triangle
X(6284) = homothetic center of intangents triangle and reflection of extangents triangle in X(4)
X(6284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,79,6147), (1,1836,3649), (3,11,5433), (3,1479,11), (4,55,12), (4,4294,55), (5,35,5432), (20,497,56), (35,3583,5), (36,4324,550), (36,4857,496), (40,3586,1837), (100,5046,1329), (149,2975,3813), (376,3086,5204), (381,498,3614), (382,3295,1478), (388,390,3303), (390,3146,388), (495,3627,3585), (496,550,36), (938,3474,5221), (944,2098,1317), (946,4304,2646), (962,3486,2099), (999,1657,4299), (1058,3529,4293), (1058,4293,3304), (1478,4309,3295), (1479,4302,3), (1697,5691,5252), (3086,5204,5298), (3583,4330,35), (3585,3746,495), (3614,4995,498), (4324,4857,36), (4848,5493,5183)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6285) lies on these lines: {1, 6000}, {11, 6247}, {12, 2883}, {30, 6238}, {33, 185}, {34, 3270}, {36, 3357}, {55, 1498}, {56, 64}, {65, 3332}, {84, 1364}, {154, 5217}, {221, 2293}, {388, 6225}, {944, 3057}, {1040, 5907}, {1478, 5878}, {1503, 3056}, {1854, 2099}, {3085, 5656}, {6198, 6241}
X(6285) = reflection of X(6254) in X(1498)
X(6285) = crosspoint of X(i) and X(j) for these (i,j): (84, 3362)
X(6285) = crosssum of X(40) and X(1745)
X(6285) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (64,2192,56)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6286) lies on these lines: {1, 1154}, {33, 6152}, {35, 54}, {55, 195}, {539, 6238}, {1479, 2888}, {2293, 3746}, {3056, 5965}, {3519, 4857}, {3583, 6288}, {6198, 6242}
X(6286) = reflection of X(6255) in X(195)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6287) lies on these lines: {3, 2916}, {4, 2896}, {5, 83}, {316, 546}, {381, 754}, {732, 1352}, {2080, 6308}, {2782, 3399}, {3851, 5475}, {5613, 6295}, {5617, 6297}, {6214, 6274}, {6215, 6275}, {6248, 6321}, {6289, 6312}, {6290, 6313}
X(6287) = midpoint of X(4) and X(2896)
X(6287) = reflection of X(i) in X(j) for these (i,k): (3,6292), (83,5)
X(6287) = X(83)-of-Johnson-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
Let (Ha) be the hyperbola having diameter X(3)X(4), passing through A, with an asymptote parallel to the internal bisector of angle A. Let A′ be the reflecttion of A in X(5), and let Ta be the tangent to (Ha) at A″. Define Tb and Tc cyclically. The lines Ta, Tb, Tc concur in X(6288). See X(6288). (Angel Montesdeoca, December 9, 2019)
X(6288) lies on these lines: {3, 161}, {4, 93}, {5, 49}, {68, 568}, {155, 195}, {1351, 3818}, {1493, 3091}, {1511, 6143}, {3521, 5663}, {3575, 3581}, {3583, 6286}, {6214, 6276}, {6215, 6277}
X(6288) = midpoint of X(4) and X(2888)
X(6288) = reflection of X(i) in X(j) for these (i,k): (195,3574), (3519,2888), (3,1209), (54,5)
X(6288) = X(54)-of-X(4)-Brocard-triangle
X(6288) = X(54)-of-Johnson-triangle
X(6288) = X(79)-of-Ehrmann-vertex-triangle if ABC is acute
X(6288) = homothetic center of 2nd Euler triangle and cross-triangle of Ehrmann side-and Ehrmann vertex-triangles
X(6288) = perspector of Ehrmann vertex-triangle and Johnson triangle
X(6288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3410,5876), (195,381,3574)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6289) lies on these lines: {2, 6222}, {3, 639}, {4, 488}, {5, 6}, {114, 371}, {325, 637}, {381, 591}, {1591, 1899}, {1656, 6118}, {3095, 6311}, {3102, 6248}, {5613, 6303}, {5617, 6302}, {6033, 6231}, {6287, 6312}
X(6289) = midpoint of X(i) and X(j) for these {i,k}: {485,6278}, {4,488}
X(6289) = reflection of X(i) in X(j) for these (i,k): (3,641), (485,5)
X(6289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1352,6290), (5,6214,1352)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6290) lies on these lines: {3, 640}, {4, 487}, {5, 6}, {114, 372}, {325, 638}, {381, 1991}, {1592, 1899}, {1656, 6119}, {3095, 6314}, {3103, 6248}, {5613, 6300}, {5617, 6301}, {6033, 6230}, {6287, 6313}
X(6290) = midpoint of X(i) and X(j) for these {i,k}: {486,6281}, {4,487}
X(6290) = reflection of X(i) in X(j) for these (i,k): (3,642), (486,5)
X(6290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1352,6289), (5,6215,1352)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6291) lies on these lines: {4, 69}, {19, 6252}, {25, 1151}, {33, 6283}, {51, 3071}, {1968, 5413}, {1974, 3092}
X(6291) = midpoint of X(4) and X(6239)
X(6291) = crosssum of X(3) and X(488)
X(6291) = X(176)-of-orthic-triangle if ABC is acute
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6292) lies on these lines: {2, 32}, {3, 2916}, {5, 5188}, {39, 141}, {76, 4045}, {114, 140}, {115, 3934}, {187, 5031}, {211, 3917}, {385, 5368}, {524, 5041}, {574, 3619}, {620, 5152}, {1125, 1279}, {1500, 3666}, {1649, 3005}, {3589, 5007}, {5590, 6274}, {5591, 6275}
X(6292) = midpoint of X(i) and X(j) for these {i,j}: (3,6287), (6296, 6297), (6313,6317), (83,2896)
X(6292) = reflection of X(6249) in X(5)
X(6292) = isotomic conjugate of isogonal conjugate of X(11205)
X(6292) = anticomplement of X(6704)
X(6292) = X(i)-Ceva conjugate of X(j) for these (i,j): (2, 3589), (99, 826)
X(6292) = crosspoint of X(i) and X(j) for these (i,j): (2, 141)
X(6292) = crosssum of X(6) and X(251)
X(6292) = complement of X(83)
X(6292) = polar conjugate of isogonal conjugate of X(22078)
X(6292) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 3934), (6, 1215), (19, 5943), (31, 3589), (38, 141), (39, 10), (141, 2887), (163, 826), (560, 1194), (798, 3124), (799, 688), (810, 339), (1401, 142), (1415, 4142), (1634, 4369), (1843, 226), (1923, 39), (1930, 626), (1964, 2), (1967, 732), (1973, 5305), (2084, 115), (2530, 116), (3051, 37), (3404, 511), (3688, 3452), (3954, 3454), (4020, 3), (4553, 3835)
X(6292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,626,1506), (2,2896,83), (2,3096,626), (83,6308,32)
X(6292) = X(i)-isoconjugate of X(j) for these (i,j): (82, 3108)
X(6292) = trilinear pole, wrt medial triangle, of de Longchamps line
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6293) lies on these lines: {4, 973}, {6, 64}, {20, 2781}, {52, 382}, {68, 5663}, {154, 5562}, {161, 1498}, {343, 2883}, {389, 1853}, {399, 2917}, {569, 3357}, {1503, 5889}, {5890, 6247}
X(6293) = reflection of X(64) in X(185)
X(6293) = X(12625)-of-orthic-triangle if ABC is acute
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6294) lies on these lines: {2, 39}, {3, 6295}, {524, 3105}, {698, 3106}, {732, 3107}, {3095, 5617}, {3104, 5463}, {3643, 6297}
X(6294) = anticomplement of X(33483)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6295) lies on these lines: {2, 14}, {3, 6294}, {542, 6298}, {3098, 5969}, {5613, 6287}
X(6295) = reflection of X(6299) in X(619)
X(6295) = circumtangential-isogonal conjugate of X(16)
X(6295) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6303,6307,3642)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6296) lies on these lines: {2, 32}, {3, 6298}, {732, 3107}, {5613, 6287}
X(6296) = reflection of X(6297) in X(6292)
X(6296) = anticomplement of X(33484)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6297) lies on these lines: {2, 32}, {3, 6299}, {732, 3106}, {3643, 6294}, {5617, 6287}
X(6297) = reflection of X(6296) in X(6292)
X(6297) = anticomplement of X(33485)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6298) lies on these lines: {2, 13}, {3, 6296}, {542, 6295}, {3095, 5617}, {3818, 6299}
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6299) lies on these lines: {2, 14}, {3, 6297}, {3095, 5613}, {3818, 6298}
X(6299) = reflection of X(6295) in X(619)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6300) lies on these lines: {2, 371}, {3, 6302}, {2044, 6251}, {3564, 6305}, {3642, 6304}, {5613, 6290}
X(6300) = reflection of X(6301) in X(642)
X(6300) = anticomplement of X(33445)
X(6300): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,486,6301)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6301) lies on these lines: {2, 371}, {3, 6303}, {2043, 6251}, {3564, 6304}, {3643, 6305}, {5617, 6290}
X(6301) = reflection of X(6300) in X(642)
X(6301) = anticomplement of X(33447)
X(6301) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,486,6300)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6302) lies on these lines: {2, 13}, {3, 6300}, {115, 615}, {542, 6307}, {590, 5472}, {2044, 5478}, {5617, 6289}
X(6302) = reflection of X(6306) in X(618)
X(6302) = complement of X(33441)
X(6302) = inverse of X(33440) in the inner-Napoleon circle
X(6302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13,6306), (616,5463,6306)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6303) lies on these lines: {2, 14}, {3, 6301}, {115, 615}, {542, 6306}, {590, 5471}, {2043, 5479}, {5613, 6289}
X(6303) = reflection of X(6307) in X(619)
X(6303) = complement of X(33443)
X(6303) = outer-Napoleon-circle-inverse of X(33442)
X(6303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,14,6307), (617,5464,6307), (3642,6295,6307)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6304) lies on these lines: {2, 372}, {3, 6306}, {2043, 6250}, {3564, 6301}, {3642, 6300}, {5613, 6289}
X(6304) = reflection of X(6305) in X(641)
X(6304) = anticomplement of X(33444)
X(6304) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,485,6305)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6305) lies on these lines: {2, 372}, {3, 6307}, {2044, 6250}, {3564, 6300}, {3643, 6301}, {5617, 6289}
X(6305) = reflection of X(6304) in X(641)
X(6305) = anticomplement of X(33446)
X(6305) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,485,6304)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6306) lies on these lines: {2, 13}, {3, 6304}, {115, 590}, {542, 6303}, {615, 5472}, {2043, 5478}, {5617, 6290}
X(6306) = reflection of X(6302) in X(618)
X(6306) = complement of X(33440)
X(6306) = inverse of X(33441) in the inner-Napoleon circle
X(6306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13,6302), (616,5463,6302)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6307) lies on these lines: {2, 14}, {3, 6305}, {115, 590}, {542, 6302}, {615, 5471}, {2044, 5479}, {5613, 6290}
X(6307) = reflection of X(6303) in X(619)
X(6307) = complement of X(33442)
X(6307) = inverse of X(33443) in the outer-Napoleon circle
X(6307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,14,6303), (617,5464,6303), (3642,6295,6303)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6308) lies on these lines: {2, 32}, {3, 732}, {98, 5188}, {2080, 6287}
X(6308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,6292,83), (6313,6317,2896)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6309) lies on these lines: {2, 39}, {3, 732}, {698, 3095}, {2021, 6337}, {3094, 3933}, {4027, 6179}
X(6309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,3926,39), (6314,6318,194)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6310) lies on these lines: {3, 3229}, {4, 69}, {695, 1196}
X(6310) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (76,5167,3491)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6311) lies on these lines: {2, 372}, {3, 6313}, {3095, 6289}, {3564, 6316}, {3818, 6315}
X(6311) = reflection of X(6312) in X(641)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6312) lies on these lines: {2, 372}, {3, 6314}, {98, 6320}, {371, 6179}, {1350, 6222}, {3098, 6316}, {3564, 6315}, {6287, 6289}
X(6312) = reflection of X(6311) in X(641)
X(6312) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (485,488,6228)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6313) lies on these lines: {2, 32}, {3, 6311}, {732, 3103}, {6287, 6290}
X(6313) = reflection of X(6317) in X(6292)
X(6313) = complement of X(33454)
X(6313) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2896,6308,6317)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6314) lies on these lines: {2, 39}, {3, 6312}, {638, 6273}, {698, 3103}, {732, 3102}, {3095, 6290}
X(6314) = reflection of X(6318) in X(39)
X(6314) = complement of X(33452)
X(6314) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (194,6309,6318)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6315) lies on these lines: {2, 371}, {3, 6317}, {3095, 6290}, {3564, 6312}, {3818, 6311}
X(6315) = reflection of X(6316) in X(642)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6316) lies on these lines: {2, 371}, {3, 6318}, {76, 6200}, {98, 6319}, {372, 6179}, {3098, 6312}, {3564, 6311}, {6287, 6290}
X(6316) = reflection of X(6315) in X(642)
X(6316) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (486,487,6229)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6317) lies on these lines: {2, 32}, {3, 6315}, {732, 3102}, {6287, 6289}
X(6317) = reflection of X(6313) in X(6292)
X(6317) = complement of X(33455)
X(6317): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2896,6308,6313)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6318) lies on these lines: {2, 39}, {3, 6316}, {637, 6272}, {698, 3102}, {732, 3103}, {3095, 6289}
X(6318) = reflection of X(6314) in X(39)
X(6318) = complement of X(33453)
X(6318): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (194,6309,6314)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6319) lies on these lines: {6, 99}, {98, 6316}, {114, 6202}, {115, 5591}, {148, 1271}, {543, 5861}, {1161, 2782}, {6215, 6321}
X(6319) = reflection of X(i) in X(j) for these (i,j): (6227,1161), (6320,99)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6320) lies on these lines: {6, 99}, {98, 6312}, {114, 6201}, {115, 5590}, {148, 1270}, {543, 5860}, {1160, 2782}, {6214, 6321}
X(6320) = reflection of X(i) in X(j) for these (i,j): (6226,1160), (6319,99)
See César Lozada, Perspective-Orthologic-Parallelogic.
Let Q be the quadrilateral ABCX(98). Taking the vertices 3 at a time yields four triangles whose orthocenters are the vertices of a cyclic quadrilateral whose circumcenter is X(6321). (Randy Hutson, January 29, 2015)
X(6321) lies on these lines: {3, 115}, {4, 147}, {5, 99}, {13, 5611}, {14, 5615}, {30, 98}, {114, 381}, {156, 3044}, {182, 6034}, {265, 690}, {382, 2794}, {542, 1351}, {591, 6231}, {620, 1656}, {1352, 5969}, {1478, 3027}, {1479, 3023}, {1569, 5475}, {1989, 5467}, {1991, 6230}, {2023, 2549}, {2482, 5055}, {3398, 5254}, {3455, 5899}, {3534, 6055}, {3543, 5984}, {3845, 6054}, {5054, 5461}, {5093, 5477}, {5469, 5473}, {5470, 5474}, {5478, 5613}, {5479, 5617}, {6214, 6320}, {6215, 6319}, {6248, 6287}
X(6321) = midpoint of X(4) and X(148)
X(6321) = reflection of X(i) in X(j) for these (i,j): (3534,6055), (3,115), (5613,5478), (5617,5479), (6033,4), (6054,3845), (99,5)
X(6321) = anticomplement of X(33813)
X(6321) = crosssum of X(2459) and X(2460)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6322) lies on the Brocard circle and these lines: {6, 3849}, {182, 6232}, {385, 6031}
X(6322) = reflection of X(6232) in X(182)
X(6322) = Brocard-circle-antipode of X(6232)
X(6322) = X(6323)-of-1st-Brocard-triangle
See César Lozada, Perspective-Orthologic-Parallelogic.
Let A′B′C′ be the circumsymmedial triangle. Let LA be the reflection of the Brocard axis in line B′C′, and define LB and LC cyclically. Let A″ = LB∩LC, B″ = LC∩LA, C″ = LA∩LB. The lines A′A″, B′B″, C′C″ concur in X(6323). Also, X(6323)-of-orthocentroidal-triangle = X(99)-of-4th-Brocard-triangle. (Randy Hutson, January 29, 2015)
X(6323) lies on the circumcircle and these lines: {3, 6233}, {99, 599}, {110, 353}, {542, 6236}, {690, 6325}, {691, 5104}, {1296, 3098}
X(6323) = reflection of X(6233) in X(3)
X(6323) = isogonal conjugate of X(3849)
X(6323) = crosssum of X(3849) and X(3849)
X(6323) = X(99)-of-circumsymmedial-triangle
X(6323) = X(1)-isoconjugate of X(3849)
X(6323) = Λ(Brocard axis of the 3rd pedal triangle of X(6))
X(6323) = 2nd-Parry-to-ABC similarity image of X(353)
X(6323) = X(6787)-of-4th-anti-Brocard-triangle
X(6323) = Schoute-circle-inverse of X(32694)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6324) lies on these lines: {353, 1495}, {381, 6235}
X(6324) = reflection of X(6235) in X(381)
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6325) lies on the circumcircle and these lines: {3, 6236}, {99, 5987}, {110, 599}, {112, 5094}, {542, 6233}, {690, 6323}, {691, 3849}, {1296, 3534}
X(6325) = reflection of X(6236) in X(3)
X(6325) = isogonal conjugate of X(8705)
X(6325) = trilinear pole of the line X(6)X(3906)
X(6325) = Λ(Euler line of 3rd pedal triangle of X(2))
X(6325) = Λ(Euler line of 3rd antipedal triangle of X(6))
X(6325) = trilinear pole of line X(6)X(3906)
X(6325) = Ψ(X(6), X(3906))
See César Lozada, Perspective-Orthologic-Parallelogic.
X(6326) lies on the hexyl circle, the excentral-hexyl ellipse, and these lines:
{1, 5}, {3, 191}, {9, 48}, {30, 5538}, {35, 5887}, {36, 912}, {40, 78}, {63, 4996}, {72, 2949}, {84, 224}, {110, 1793}, {149, 946}, {153, 515}, {200, 1145}, {404, 5884}, {480, 3428}, {516, 5528}, {517, 3689}, {519, 1512}, {528, 1537}, {758, 5535}, {936, 3035}, {1045, 2783}, {1158, 4855}, {1320, 3577}, {1385, 5251}, {1490, 2829}, {1787, 3333}, {2077, 2932}, {2136, 2802}, {2646, 5777}, {2951, 6282}, {3553, 5153}, {3646, 3897}, {4305, 5811}, {5732, 5851}
Let A′B′C′ be the orthic triangle. Let LA be the antiorthic axis of AB′C′, and define LB and LC cyclically. Let A″ = LB∩LC, B″ = LC∩LA, C″ = LA∩LB. Then the triangle A″B″C″ is inversely similar to ABC, with similitude center X(9), and X(6326) = X(40)-of-A″B″C″. (Randy Hutson, January 29, 2015)
X(6326) = midpoint of X(i) and X(j) for these {i,j}: {153,6224}, {1,5531}
X(6326) = reflection of X(i) in X(j) for these (i,j): (104, 214), (149,946), (1768,3), (1,6265), (2077,5440), (40,100), (5587,5660), (6264,1), (80,119)
X(6326) = X(i)-Ceva conjugate of X(j) for these (i,j): (153, 2950), (515, 40), (908, 9), (4511, 1), (5080, 191), (6224, 5541)
X(6326) = crossdifference of every two points on the line X(654)X(1769)
X(6326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5720,5587), (1,6127,1718), (3,5694,191), (78,6261,40), (80,119,5587), (80,5660,119), (104,214,3576), (5531,6265,6264)
X(6326) = excentral-isogonal conjugate of X(484)
X(6326) = X(188)-aleph conjugate of X(484)
X(6326) = X(21)-beth conjugate of X(1421)
X(6326) = antipode in excentral-hexyl ellipse of X(1768)
X(6326) = X(110)-of-hexyl triangle
X(6326) = hexyl-isogonal conjugate of X(513)
X(6326) = Fuhrmann-to-excentral similarity image of X(3)
X(6326) = inner-Garcia-to-ABC similarity image of X(4)
X(6326) = X(265)-of-excentral-triangle
X(6326) = intersection, other than vertices of hexyl triangle, of hexyl circle and excentral-hexyl ellipse
Contributed by Peter Moses, January 15, 2015.
X(6327) lies on these lines: {2, 31}, {6, 4972}, {8, 79}, {38, 4655}, {42, 4660}, {55, 3936}, {63, 3006}, {65, 5016}, {69, 674}, {72, 5300}, {100, 4417}, {192, 744}, {194, 734}, {209, 2550}, {306, 516}, {312, 5057}, {315, 766}, {320, 3873}, {321, 1836}, {329, 2835}, {345, 4427}, {518, 5014}, {742, 4799}, {756, 4703}, {896, 4438}, {902, 3771}, {984, 4683}, {1150, 2886}, {1376, 5741}, {1759, 4153}, {2390, 3436}, {2979, 3888}, {3011, 4138}, {3058, 4966}, {3120, 4362}, {3187, 3914}, {3218, 3705}, {3454, 5264}, {3578, 4042}, {3782, 3891}, {3827, 4463}, {3868, 5015}, {3874, 4894}, {3883, 5249}, {3925, 5278}, {3966, 4359}, {3969, 5695}, {4001, 4847}, {4071, 5282}, {4425, 5311}, {4643, 4981}, {5051, 5711}
X(6327) = reflection of X(i) in X(j) for these (i,j): (8,4680), (31,2887), (3187,3914), (3891,3782)
X(6327) = anticomplement of X(31)
X(6327) = X(561)-Ceva conjugate of X(2)
X(6327) = X(3156)-cevapoint of X(193)
X(6327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31,2887,2), (320,4514,3873), (748,3836,2), (750,3847,2), (1836,3416,321), (2550,5739,4651), (3936,4450,55), (3966,5880,4359), (4388,4645,2), (4655,4865,38)
X(6327) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 192), (3, 3164), (4, 193), (6, 194), (7, 145), (8, 144), (9, 3177), (10, 1654), (13, 3180), (14, 3181), (27, 3187), (37, 1655), (57, 3210), (69, 20), (75, 8), (76, 69), (83, 6), (85, 7), (86, 1), (87, 330), (92, 5905), (95, 3), (98, 385), (99, 523), (141, 2896), (183, 6194), (190, 514), (253, 3146), (261, 2975), (264, 4), (274, 75), (275, 1993), (276, 264), (279, 4452), (286, 3868), (287, 401), (290, 511), (298, 616), (299, 617), (300, 621), (301, 622), (302, 627), (303, 628), (304, 4329), (305, 1370), (306, 3151), (307, 3152), (308, 76), (309, 962), (311, 2888), (312, 329), (313, 1330), (314, 3869), (315, 5596), (317, 6193), (318, 5942), (319, 3648), (320, 6224), (321, 2895), (322, 6223), (325, 147), (327, 1352), (328, 3153), (330, 1278), (333, 63), (334, 4645), (348, 347), (349, 2893), (491, 487), (492, 488), (514, 4440), (523, 148), (561, 6327), (598, 1992), (646, 4462), (648, 525), (658, 4025), (662, 4560), (664, 522), (666, 918), (668, 513), (670, 512), (671, 524), (673, 239), (693, 149), (801, 394), (827, 4580), (850, 3448), (886, 888), (889, 891), (892, 690), (903, 519), (1016, 190), (1121, 527), (1219, 4461), (1220, 894), (1221, 1909), (1222, 3729), (1231, 2897), (1232, 2889), (1233, 2890), (1236, 2892), (1255, 3995), (1268, 10), (1269, 2891), (1275, 664), (1434, 3875), (1441, 2475), (1494, 30), (1502, 315), (1509, 4360), (1799, 22), (2373, 23), (2481, 518), (2966, 2799), (2986, 323), (3222, 669), (3224, 2998), (3225, 698), (3226, 726), (3227, 536), (3228, 538), (3260, 146), (3261, 150), (3262, 153), (3596, 3436), (3699, 4468), (4357, 5484), (4373, 3621), (4554, 693), (4555, 900), (4562, 812), (4564, 4552), (4569, 3900), (4573, 4467), (4577, 826), (4586, 824), (4590, 99), (4597, 4777), (4598, 649), (4600, 4427), (4632, 4608), (4997, 908), (4998, 100), (5376, 2397), (5490, 1270), (5491, 1271), (5641, 542), (5936, 3617), (6063, 3434), (6189, 3414), (6190, 3413), (6330, 297), (6331, 850), (6335, 4391)
X(6327) = isogonal conjugate of X(7087)
X(6327) = trilinear product X(i)*X(j) for these (i,j): {2,1759}, {7,4149}}, {75,1631}, {81,4153}
X(6327) = barycentric product X(i)*X(j) for these (i,j): {75,1759}, {76,1631}, {85,4149}, {86,4153}
X(6327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31,2887,2), (320,4514,3873), (748,3836,2), (750,3847,2), (1836,3416,321), (2550,5739,4651), (3936,4450,55), (3966,5880,4359), (4388,4645,2), (4655,4865,38)
The A′B′C′ be the orthic triangle, and let A″ be the intersection of the A-altitude and line B′C′. Define B″ and C″ cyclically. The circumcircles of A″BC, AB″C, ABC″ concur in X(6328). (Antreas Hatzipolakis, César Lozada, January 16, 2014, Hyacinthos 23027, 23029)
Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Then X(6328) is the radical center of the circumcircles of A′BC, AB′C, ABC′. (Randy Hutson, January 16, 2014, Hyacinthos 23032)
X(6328) lies on the curves K072, K474, K567 and this line: {3, 3447}
X(6328) = X(i)-cross conjugate of X(j) for these (i,j): (512, 115)
X(6328) = antigonal conjugate of X(115)
X(6328) = circumcircle-inverse of X(3447)
X(6328) = X(1101)-isoconjugate of X(3448)
X(6328) = similitude center of tangential and normal triangles of Kiepert hyperbola
Let P be a point in the plane of a triangle ABC, and let A′B′C′ be the pedal triangle of P. Let A″B″C″ be the pedal triangle of the circumcenter O′ of A′B′C′. Let A* be the reflection of A′ in O′, and define B* and C* cyclically, so that A*B*C* is the antipodal triangle of A′B′C′. Let D be the midpoint of segment A″A*, and define E and F cyclically. If P = X(2), then the circumcenter of DEF is X(6329). See A. Hatzipolakis and R. Hutson, Hyacinthos 22995, January 10, 2015.
X(6329) lies on these lines: {2, 6}, {140, 5097}, {182, 550}, {382, 5050}, {511, 3530}, {518, 3636}, {545, 3946}, {546, 575}, {631, 5102}, {742, 4739}, {894, 4395}, {1100, 4422}, {1352, 5079}, {1386, 3244}, {1743, 4364}, {2261, 5834}, {3528, 5085}, {3626, 5846}, {3628, 5965}, {3759, 4399}, {3793, 5007}, {4363, 4402}, {4371, 4665}, {4478, 4969}, {4795, 4859}, {4796, 5845}, {4873, 4910}, {4889, 4982}
X(6329) = midpoint of X(i) and X(j) for these {i,j}: (140,a5097), (3629,3631), (6,3589)
X(6329) = complement of X(3631)
X(6329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,3629), (2,3629,3631), (6,597,3589), (6,3618,141), (6,3763,1992), (6,6144,5032), (141,597,3618), (141,3618,3589), (1992,3763,3630), (3589,3631,2), (3619,5032,6144)
Let H be the circumhyperbola {{A,B,C,X(2),X(69)}}, and let J be the circumconic centered at X(1249); that is, the conic {{A,B,C,X(107),X(648)}}. Then H and J intersect in four points: A, B, C, and X(6330). (Randy Hutson, January 29, 2015)
X(6330) lies on these lines: {2, 107}, {69, 648}, {253, 393}, {287, 297}, {305, 6331}, {306, 1897}, {307, 653}, {438, 5895}, {2419, 2799}
X(6330) = reflection of X(648) in X(1249)
X(6330) = isogonal conjugate of X(8779)
X(6330) = X(i)-cevapoint of X(j) for these (i,j): (2, 297), (868, 2501), (2972, 3569)
X(6330) = X(i)-cross conjugate of X(j) for these (i,j): (2799, 648)
X(6330) = trilinear pole of the line X(4)X(525)
X(6330) = X(i)-isoconjugate of X(j) for these (i,j): (3, 2312), (31, 441), (48, 1503), (822, 2409)
X(6331) = isogonal conjugate of X(3049)
X(6331) = isotomic conjugate of X(647)
X(6331) = trilinear pole of the line X(4)X(69)
X(6331) = X(i)-cevapoint of X(j) for these (i,j): (2, 850), (4, 2489), (76, 3267), (99, 648), (343, 3265), (427, 2501), (523, 5254), (647, 1899)
X(6331) = X(i)-cross conjugate of X(j) for these (i,j): (99, 670), (523, 683), (2489, 4), (2799, 290), (3267, 76), (4554, 799), (6335, 811)
X(6331) = X(4)-cross conjugate of isogonal conjugate of X(23216)
X(6331) = crosssum of X(i) and X(j) for these (i,j): (647, 2524), (3049, 3049)
X(6331) = X(i)-isoconjugate of X(j) for these (i,j): (1, 3049), (3, 798), (6, 810), (25, 822), (31, 647), (32, 656), (48, 512), (63, 669), (69, 1924), (71, 667), (72, 1919), (73, 3063), (184, 661), (213, 1459), (217, 2616), (228, 649), (255, 2489), (293, 2491), (306, 1980), (513, 2200), (520, 1973), (525, 560), (603, 3709), (652, 1402), (657, 1410), (663, 1409), (878, 1755), (905, 1918), (906, 3122), (1084, 4592), (1176, 2084), (1331, 3121), (1400, 1946), (1437, 4079), (1576, 3708), (1917, 3267), (1923, 4580), (2196, 4455), (2205, 4025), (3124, 4575), (3248, 4574), (4117, 4563)
X(6332) lies on these lines: {2, 2399}, {8, 4163}, {63, 4091}, {100, 2728}, {110, 2769}, {190, 1813}, {441, 525}, {514, 661}, {522, 663}, {644, 4568}, {650, 3910}, {824, 4529}, {918, 3669}, {1459, 4064}, {3716, 3810}, {3907, 4522}, {4088, 4449}, {4561, 4587}
X(6332) = midpoint of X(i) and X(j) for these {i,j}: {1459,4064}, {3904,4391}, {4088,4449}
X(6332) = reflection of X(i) in X(j) for these (i,j): (4025, 905), (4391, 3239), (8,4163)
X(6332) = isogonal conjugate of X(32674)
X(6332) = isotomic conjugate of X(653)
X(6332) = X(i)-Ceva conjugate of X(j) for these (i,j): (69, 2968), (190, 63), (662, 3687), (664, 8), (1332, 306), (4554, 307), (4561, 78)
X(6332) = trilinear pole of line X(2968)X(4082)
X(6332) = polar conjugate of X(36127)
X(6332) = pole wrt polar circle of trilinear polar of X(36127) (line X(19)X(208))
X(6332) = X(i)-cross conjugate of X(j) for these (i,j): (521, 4025), (525, 4391), (652, 522), (2968, 69)
X(6332) = crosspoint of X(i) and X(j) for these (i,j): (190, 312), (304, 4561), (314, 4554), (348, 664), (1332, 1812)
X(6332) = crosssum of X(i) and X(j) for these (i,j): (604, 649), (607, 663), (1402, 3063)
X(6332) = crossdifference of every two points on the line X(25)X(31)
X(6332) = X(i)-complementary conjugate of X(j) for these (i,j): (2217, 116)
X(6332) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (84, 150), (101, 6223), (109, 5932), (112, 1895), (1436, 149), (1903, 3448), (2208, 4440)
X(6332) = X(i)-isoconjugate of X(j) for these (i,j): (4, 1415), (6, 108), (19, 109), (25, 651), (28, 4559), (31, 653), (33, 1461), (34, 101), (56, 1783), (65, 112), (100, 608), (107, 1409), (110, 1880), (162, 1400), (163, 225), (190, 1395), (278, 692), (604, 1897), (607, 934), (644, 1398), (648, 1402), (658, 2212), (664, 1973), (906, 1118), (919, 1876), (1020, 2299), (1096, 1813), (1295, 2443), (1396, 4557), (1397, 6335), (1414, 2333), (1426, 5546), (1435, 3939), (1474, 4551), (1824, 4565), (1974, 4554), (2203, 4552), (2204, 4566)
X(6332) = X(1812)-beth conjugate of X(4091)
X(6333) lies on these lines: {99, 110}, {125, 339}, {287, 2419}, {441, 525}, {523, 4143}, {826, 850}, {2799, 3569}
X(6333) = isogonal conjugate of X(32696)
X(6333) = X(i)-Ceva conjugate of X(j) for these (i,j): (877, 325), (2419, 3265)
X(6333) = X(i)-cross conjugate of X(j) for these (i,j): (684, 2799)
X(6333) = crosspoint of X(i) and X(j) for these (i,j): (325, 877)
X(6333) = crosssum of X(i) and X(j) for these (i,j): (669, 2211), (878, 1976)
X(6333) = crossdifference of every two points on the line X(25)X(1501)
X(6333) = X(i)-isoconjugate of X(j) for these (i,j): (19, 2715), (31, 685), (112, 1910), (162, 1976), (1973, 2966)
X(6334) lies on these lines: {3, 690}, {113, 131}, {115, 127}, {441, 525}, {684, 5489}, {2394, 2986}
X(6334) = midpoint of X(i) and X(j) for these {i,j}: {2394,3268}, {684,5489}
X(6334) = isogonal conjugate of X(32708)
X(6334) = X(i)-Ceva conjugate of X(j) for these (i,j): (328, 125), (2394, 525)
X(6334) = crosspoint of X(1494) and X(4563)
X(6334) = crosssum of X(1495) and X(2489)
X(6334) = crossdifference of every two points on the line X(25)X(1576)
X(6334) = X(i)-complementary conjugate of X(j) for these (i,j): (48, 3258), (163, 1511)
X(6334) = X(i)-isoconjugate of X(j) for these (i,j): (19,10420), (31, 687), (163, 1300)
X(6335) lies on these lines: {92, 4997}, {100, 107}, {112, 839}, {190, 653}, {243, 5205}, {264, 281}, {278, 1997}, {286, 334}, {341, 1895}, {645, 648}, {646, 2397}, {651, 4391}, {685, 692}, {687, 4567}, {1118, 2899}, {1265, 3176}, {1633, 2517}, {1784, 3992}, {1897, 3699}, {1948, 3912}, {3239, 4605}
X(6335) = isogonal conjugate of X(22383)
X(6335) = isotomic conjugate of X(905)
X(6335) = X(811)-Ceva conjugate of X(1897)
X(6335) = X(i)-cevapoint of X(j) for these (i,j): (2, 4391), (10, 3239), (37, 4036), (100, 1783), (429, 2501), (650, 1837), (693, 3673)
X(6335) = X(i)-cross conjugate of X(j) for these (i,j): (37, 5379), (100, 668), (4397, 75), (4552, 190), (5552, 4998)
X(6335) = X(6591)-cross conjugate of X(4)
X(6335) = crosspoint of X(811) and X(6331)
X(6335) = crosssum of X(810) and X(3049)
X(6335) = trilinear pole of the line X(4)X(8)
X(6335) = polar conjugate of X(513)
X(6335) = pole wrt polar circle of trilinear polar of X(513) (line X(244)X(665))
X(6335) = X(i)-isoconjugate of X(j) for these (i,j): (3, 649), (6, 1459), (19,23224), (25, 4091), (28, 822), (31, 905), (32, 4025), (48, 513), (56, 652), (57, 1946), (58, 647), (63, 667), (69, 1919), (71, 3733), (77, 3063), (81, 810), (86, 3049), (101, 3937), (184, 514), (212, 3669), (222, 663), (228, 1019), (244, 906), (304, 1980), (512, 1790), (520, 1474), (521, 604), (525, 2206), (603, 650), (656, 1333), (659, 2196), (661, 1437), (692, 3942), (798, 1444), (1015, 1331), (1021, 1410), (1332, 3248), (1357, 4587), (1397, 6332), (1409, 3737), (1461, 3270), (1472, 2522), (1576, 4466), (1795, 3310), (1797, 1960), (1803, 2488), (1813, 3271), (1973, 4131), (1977, 4561), (2193, 4017), (3121, 4592), (3122, 4558), (3125, 4575)
X(6336) lies on these lines: {4, 145}, {27, 648}, {88, 278}, {92, 4997}, {106, 107}, {469, 4945}, {687, 4591}, {901, 917}, {1222, 4054}, {1838, 4792}
X(6336) = isogonal conjugate of X(22356)
X(6336) = isotomic conjugate of X(3977)
X(6336) = X(i)-cevapoint of X(j) for these (i,j): (1, 1731), (19, 1870)
X(6336) = X(i)-cross conjugate of X(j) for these (i,j): (106, 903), (1785, 273)
X(6336) = trilinear pole of the line X(4)X(2457)
X(6336) = polar conjugate of X(519)
X(6336) = pole wrt polar circle of trilinear polar of X(519) (line X(900)X(1635))
X(6336) = X(i)-isoconjugate of X(j) for these (i,j): (3, 44), (6, 5440), (31, 3977), (48, 519), (63, 902), (69, 2251), (72, 3285), (78, 1404), (184, 4358), (212, 3911), (219, 1319), (222, 3689), (603, 2325), (678, 1797), (900, 906), (1023, 1459), (1331, 1635), (1332, 1960), (1437, 3943), (1813, 4895), (1877, 2289), (2196, 4432), (4120, 4575), (4558, 4730)
x(S2Cy2 - S2Bz2) + y(S2Az2 - S2Cx2) + z(S2Ax2 - S2Ay2) = 0.
The following four circles intersect in two points, and their crossum is X(6337): (1) circumcircle, (2) 2nd Lemoine circle, (3) {{X(371),X(372),PU(1),PU(39)}} (whose center is X(32)), and (4) {{X(4),X(194),X(3557),X(3558)}} (whose center is X(3095). (Randy Hutson, January 29, 2015)
As a line L varies through the set of all lines that pass through X(69), the locus of the trilinear pole of L is a circumconic, and its center is X(6337). (Randy Hutson, January 29, 2015)
Let A′ be the trilinear product of the vertices of the A-adjunct anti-altimedial triangle, and define B′ and C′ cyclically. Triangle A′B′C′ is the anticomplementary triangle of the 1st Brocard triangle, and is perspective to ABC at X(4), and to the medial triangle at X(6337). (Randy Hutson, November 2, 2017)
X(6337) lies on the curves K009, K164, K168, Q052 and these lines: {2, 1975}, {3, 69}, {4, 99}, {6, 6339}, {20, 325}, {32, 1992}, {39, 3618}, {76, 631}, {183, 3523}, {184, 4176}, {193, 439}, {194, 5976}, {315, 376}, {316, 3529}, {485, 642}, {486, 641}, {524, 5023}, {574, 3619}, {620, 3767}, {988, 1125}, {1078, 3524}, {1649, 3265}, {1909, 5218}, {2021, 6309}, {2275, 3666}, {2549, 3788}, {3630, 5585}, {4384, 5745}
X(6337) = midpoint of X(i) and X(j) for these {i,j}: {99,8781}, 487,488}
X(6337) = reflection of X(i) in X(j) for these (i,j): (485,642), (486,641)
X(6337) = isogonal conjugate of X(14248)
X(6337) = isotomic conjugate of X(34208)
X(6337) = complement of X(2996)
X(6337) = anticomplement of X(13881)
X(6337) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,69), (99,3566)
X(6337) = X(3167)-cross conjugate of X(193)
X(6337) = crosspoint of X(2) and X(193)
X(6337) = crosssum of X(512) and X(2971)
X(6337) = X(i)-complementary conjugate of X(j) for these (i,j): (6,4138), (31,69), (163,3566), (193,2887), (1707,141), (3053,10)
X(6337) = X(4592)-beth conjugate of X(56)
X(6337) = X(i)-Hirst inverse of X(j) for these (i,j): (20,235), (69,3564)
X(6337) = X(i)-isoconjugate of X(j) for these (i,j): {923,5203}, {1973,2996}
X(6337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3926,69), (3,3933,3785), (193,439,3053), (3785,3926,3933), (3785,3933,69)
X(6337) = trilinear product X(i)*X(j) for these (i,j): {63,193}, {69,1707}, {75,3167}, {304,3053}, {1332,3798}, {1444,4028}, {3566,4592}
X(6337) = barycentric product X(i)*X(j) for these (i,j): {69,193}, {76,3167}, {304,1707}, {305,3053}, {3266,6091}, {3566,4563}, {3798,4561}
X(6337) = barycentric quotient X(i)/X(j) for these (i,j): {69,2996}, {193,4}, {524,5203}, {1707,19}, {3053,25}, {3167,6}, {3566,2501}, {3787,1843}, {4028,1826}, {4558,3565}, {6091,111}
X(6337) = X(4592)-beth conjugate of X(56)
X(6337) = perspector of the circumconic centered at X(69)
X(6337) = center of the conic {{A,B,C,X(99),X(4554),X(4563)}}, the isotomic conjugate of the orthic axis)
X(6337) = isogonal conjugate of X(12248)
X(6337) = Cundy-Parry Phi transform of X(3564)
X(6337) = Cundy-Parry Psi transform of X(3563)
"SS(a → SA)" denotes barycentric symbolic substitution; see X(6337).
X(6338) lies on these lines: {2, 6339}, {69, 1368}, {126, 193}, {141, 5490}, {393, 670}
X(6338) = complement of X(6339)
X(6338) = X(2)-Ceva conjugate of X(3926)
X(6338) = X(i)-cross conjugate of X(j) for these (i,j): (3926, 6341)
X(6338) = X(i)-complementary conjugate of X(j) for these (i,j): (31, 3926), (1611, 10), (2128, 1368)
X(6338) = trilinear product X(69)*X(2128)
X(6338) = barycentric product X(304)*X(2128)
X(6338) = barycentric quotient X(i)/X(j) for these (i,j): (63,2129), (1611,2207), (2128,19), (2519,2489)
X(6338) = X(25)-isoconjugate of X(2129)
"SS(a → SA)" denotes barycentric symbolic substitution; see X(6337). Also, X(6339) = SS(a → S2A) of X(7).
X(6339) lies on these lines: {2, 6338}, {6, 6337}, {25, 193}, {3926, 6342}
X(6339) = reflection of X(6338) in X(6387)
X(6339) = isogonal conjugate of X(1611)
X(6339) = isotomic conjugate of X(6392)
X(6339) = anticomplement of X(6338)
X(6339) = X(2)-Ceva conjugate of X(6342)
X(6339) = cevapoint of X(520) and X(1084)
X(6339) = crosssum of X(1611) and X(1611)
X(6339) = X(i)-cross conjugate of X(j) for these (i,j): (3296,2), (6391,69)
X(6339) = X(i)-complementary conjugate of X(j) for this (i,j): (31, 6342)
X(6339) = X(i)-anticomplementary conjugate of X(j) for this (i,j): (2129,1370)
X(6339) = X(i)-isoconjugate of X(j) for these (i,j): (1,1611), (25,2128), (31,6392), (162,2519)
X(6339) = {X(6338),X(6387)}-harmonic conjugate of X(2)
X(6339) = trilinear product X(69)*X(2129)
X(6339) = barycentric product X(304)*X(2129)
X(6339) = barycentric quotient X(i)/X(j) for these (i,j): (6,1611), (63,2128), (647,2519), (2129,19)
"SS(a → SA)" denotes barycentric symbolic substitution; see X(6337).
X(6340) lies on these lines: {2, 1975}, {25, 5203}, {69, 1368}, {125, 4176}, {253, 325}, {264, 1007}, {1370, 2373}
X(6340) = isogonal conjugate of X(19118)
X(6340) = isotomic conjugate of X(6353)
X(6340) = X(2996)-Ceva conjugate of X(69)
X(6340) = X(125)-cevapoint of X(3265)
X(6340) = X(i)-cross conjugate of X(j) for these (i,j): (115, 3267), (3926, 69)
"SS(a → SA)" denotes barycentric symbolic substitution; see X(6337).
X(6341) lies on this line: {2, 6342}
X(6341) = X(3926)-cross conjugate of X(6338)
"SS(a → SA)" denotes barycentric symbolic substitution; see X(6337).
X(6342) lies on these lines: {2, 6341}, {3926, 6339}
X(6342) = X(2)-Ceva conjugate of X(6339)
X(6342) = X(i)-complementary conjugate of X(j) for this (i,j): (31, 6339)
Let N the the nine-point center of a triangle ABC, and let NA the reflection of N in line BC. Define NB and NC cyclically. Let NAB be the reflection of NA in line NB, and define NBC and NCA cyclically. Let NAC be the reflection of NA in line NC, and define NBA and NCB cyclically. Let LA be the Euler line of triangle NANABNAC, and define LB and LC cyclically. Let L′A be the reflection of LA in line BC, and define L′B and L′C cyclically. The lines L′A, L′B, L′C concur in X(6343). (Antreas Hatzipolakis and César Lozada, January 22, 2015; see Hyacinthos 23050)
X(6343) lies on these lines: {3,2888}, {54,1263}X(6343) = reflection of X(1263) in X(54)
Suppose that P is a point in the plane of a triangle ABC. Let PA be the reflection of P in line BC, and define Let PB and Let PC cyclically. Let PAB be the reflection of PA in PB, and let PAC be the reflection of PA in PC. Let NA be the nine-point center of the triangle PAPABPAC, and define NB and NC cyclically. The triangle NANBNC is the P-Hatzipolakis-Lozada triangle. For P = X(3), the ABC-orthologic center of this triangle is X(6344). See also X(6345) and X(5346). These four orthologic centers, and also X(1), X(4), X(5), X(1113), and X(1114) lie on the excentral-circum-septic curve described at Hyacinthos 23055. (Antreas Hatzipolakis and César Lozada, January 23, 2015)
X(6344) lies on these lines: {4,94}, {5,93}, {107,1141}, {186,476}, {225,2166}, {264,328}, {393,1989}, {648,2914}, {1105,3520}
X(6344) = isogonal conjugate of X(22115)
X(6344) = X(48)-isoconjugate (polar conjugate) of X(323)
X(6344) = X(63)-isoconjugate of X(50)
X(6344) = X(403)-cross conjugate of X(4)
X(6344) = trilinear pole of line X(53)X(2501)
X(6344) = trilinear product of vertices of Ehrmann vertex-triangle
See X(6344) and X(6346). (Antreas Hatzipolakis and César Lozada, January 23, 2015)
X(6345) lies on the circumcircle and these lines: {930, 32744}, {1291, 15345}, {1510, 32749}
See X(6344) and X(6345). (Antreas Hatzipolakis and César Lozada, January 23, 2015)
X(6346) lies on these lines: {546, 11572}, {11801, 20030}
X(6347) lies on these lines: {1, 2}, {394, 1377}, {958, 1584}, {993, 1600}, {1329, 1591}, {1376, 1583}, {1512, 2048}, {1585, 1861}, {1592, 2886}
X(6347) = {X(2),X(10)}-harmonic conjugate of X(6348)
X(6348) lies on these lines: {1, 2}, {394, 1378}, {958, 1583}, {993, 1599}, {1329, 1592}, {1376, 1584}, {1586, 1861}, {1591, 2886}
X(6348) = {X(2),X(10)}-harmonic conjugate of X(6347)
Let EA be the ellipse that passes through A and has foci B and C. Call EA the A-Soddy ellipse. Let LA be the polar of X(3) with respect to EA. Define LB and LB cyclically. Let A′ = LB∩LC, B′ = LC∩LA, C′ = LA∩LB. Then triangle A′B′C′ is homothetic to the medial triangle, with homothethic center X(6349). Also, A′B′C′ is homothetic to ABC, with homthetic center X(281). The orthocenter of A′B′C′ is X(6361). (Randy Hutson, January 30, 2015)
X(6349) lies on these lines: {2, 92}, {3, 962}, {63, 348}, {189, 3160}, {464, 5249}, {1038, 1457}, {1040, 4666}, {1060, 4511}, {1817, 4329}
X(6349) = isotomic conjugate of polar conjugate of X(4295)
X(6349) = {X(2),X(1214)}-harmonic conjugate of X(6350)
Let HA be the hyperbola that passes through A and has foci B and C. HA is the A-Soddy hyperbola, as in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let MA be the polar of X(3) with respect to HA. Define MB and MC cyclically. Let A″ = MB∩MC, B″ = MC∩MA, C″ = MA∩MB. Triangle A″B″C″ is homothetic to the medial triangle, with homothetic center X(6350). Also, triangle A″B″C″ is homothetic to ABC, with homothetic center X(278). The orthocenter of A′B′C′ is X(944). (Randy Hutson, January 30, 2015)
X(6350) lies on these lines: {2, 92}, {3, 8}, {20, 5174}, {63, 69}, {189, 268}, {329, 440}, {344, 908}, {1040, 3870}, {1748, 3101}
X(6350) = anticomplement of X(37695)
X(6350) = isotomic conjugate of polar conjugate of X(18391)
X(6350) = {X(2),X(1214)}-harmonic conjugate of X(6349)
X(6351) lies on these lines: {1, 1123}, {2, 37}, {9, 3068}, {45, 590}, {498, 1336}, {1270, 4851}, {1271, 4643}, {1659, 6203}, {3247, 5405}, {3879, 5860}, {3912, 5590}, {4357, 5591}, {4416, 5861}
X(6351) = complement of X(32793)
X(6351) = {X(2),X(37)}-harmonic conjugate of X(6352)
X(6352) lies on these lines: {1, 1336}, {2, 37}, {9, 3069}, {45, 615}, {498, 1123}, {1270, 4643}, {1271, 4851}, {1659, 1826}, {3247, 5393}, {3879, 5861}, {3912, 5591}, {4357, 5590}, {4416, 5860}
X(6352) = complement of X(32794)
X(6352) = {X(2),X(37)}-harmonic conjugate of X(6351)
As a point on the Euler line, X(6353) has Shinagawa coefficients (2F, -E - F).
X(6353) lies on the cubic K163 and these lines: {2,3}, {33,5218}, {34,5272}, {43,2356}, {69,1974}, {98,459}, {107,3563}, {111,1289}, {112,2374}, {171,2212}, {193,3167}, {230,393}, {232,800}, {238,1395}, {242,278}, {317,1007}, {612,6198}, {614,1870}, {966,1474}, {1068,3011}, {1119,1447}, {1172,5275}, {1192,2883}, {1204,6225}, {1299,1302}, {1398,5265}, {1452,3485}, {1495,1899}, {1611,2207}, {1613,2211}, {1620,5894}, {1829,3616}, {1843,3618}, {1848,5338}, {1876,5435}, {1892,5226}, {2203,5739}, {2299,5712}, {3068,5413}, {3069,5412}, {3087,3815}, {5139,6091}
X(6353) = isogonal conjugate of X(6391)
X(6353) = isotomic conjugate of X(6340)
X(6353) = complement of X(7396)
X(6353) = anticomplement of X(30771)
X(6353) = cevapoint of X(i) and X(j) for these {i,j}: {25,1611}, {193,439}
X(6353) = X(193)-cross conjugate of X(4)
X(6353) = X(4)-Hirst inverse of X(460)
X(6353) = inverse-in-polar-circle of X(5159)
X(6353) = pole wrt polar circle of trilinear polar of X(2996) (line X(523)X(4885))
X(6353) = X(48)-isoconjugate (polar conjugate) of X(2996)
X(6353) = X(i)-Ceva conjugate of X(j) for these {i,j}: {393,4}, {4590,112}
X(6353) = X(i)-cross conjugate of X(j) for these {i,j}: {193,4}, {3053,193}
X(6353) = X(i)-isoconjugate of X(j) for these {i,j}: {31,6340}, {48,2996}, {656,3565}
X(6353) = Euler line intercept, other than X(20), of circle {X(20),PU(4)}
X(6353) = perspector of circumcevian triangle of X(25) and (cross-triangle of ABC and circumcevian triangle of X(25))
X(6353) = cyclocevian conjugate of X(4) wrt anticevian triangle of X(4)
X(6353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,20,1368), (2,23,1370), (2,25,4), (2,4232,25), (3,3089,4), (4,186,376), (4,3147,631), (4,3524,378), (4,3525,3541), (4,5067,1594), (20,235,4), (24,3542,4), (25,468,2), (25,5094,428), (27,4207,4), (28,406,4), (140,1598,3088), (235,3515,20), (429,4198,4), (468,4232,4), (475,4222,4), (1585,5200,4), (1598,3088,4), (3091,3575,4), (4185,4194,4), (4186,4200,4)
X(6353) = trilinear product of X(i)*X(j) for these {i,j}: {4,1707}, {19,193}, {28,4028}, {92,3053}, {158,3167}, {162,3566}, {1096,6337}, {1783,3798}
X(6353) = barycentric product X(i)*X(j) for these {i,j}: {4,193}, {27,4028}, {92,1707}, {264,3053}, {393,6337}, {648,3566}, {1897,3798}, {2052,3167}, {4590,5139}
X(6353) = barycentric quotient X(i)/X(j){i/j} for these {i,j}: {2,6340}, {4,2996}, {112,3565}, {193,69}, {439,6337}, {1707,63}, {3053,3}, {3167,394}, {3566,525}, {3787,3917}, {3798,4025}, {4028,306}, {5139,115}, {6337,3926}
Question A is copied here from X(554): Suppose that X and Y are triangle centers. Let
YA = (Y of the triangle XBC), YB = (Y of the triangle XCA), YC = (Y of the triangle XAB).
Let A′ = XYA∩, and define B′ and C′ cyclically. In Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438, Question A is this: for what choices of X and Y do the lines AA′, BB′, CC′ concur? A solution (X,Y) is called the (X,Y)-answer to Question A. Answers X(6354)-X(6359) were contributed by Peter Moses, January 28, 2015, in conjunction questions raised by Viktor Kitaisky.
See X(554) for a list of major centers that provide answers to Question A. In the following list, the appearance of (i,j) means that X(j) is a non-major center which is the (X(1),X(i))-answer to Question A: (115,6354), (125,12), (135,6355), (136,6356), (155,84), (231,6357), (249,2185), (250,757), (254,347), (338,6358), (339,1089), (401,894), (421,6359), (426,612), (436,144), (454,1256) (Peter Moses, January 28, 2015)
X(6354) lies on these lines: {6,278}, {7,940}, {12,201}, {33,1836}, {37,226}, {51,2969}, {55,4331}, {57,1020}, {65,225}, {92,1146}, {181,1365}, {196,393}, {220,329}, {223,3553}, {241,5249}, {269,4654}, {279,1255}, {321,349}, {335,1088}, {347,5712}, {394,5905}, {524,1943}, {1042,3649}, {1060,1448}, {1068,5706}, {1211,1441}, {1435,2285}, {1659,3070}, {2171,6046}, {4306,6147}, {4332,6284}, {5307,5928}
Regarding Question A, see X(6354).
X(6355) lies on these lines: {226,1439}, {269,282}, {1367,6046}, {1433,1440}
Regarding Question A, see X(6354).
X(6356) lies on these lines: {2,1119}, {3,7}, {4,347}, {5,273}, {9,1020}, {12,6046}, {37,226}, {72,307}, {75,2968}, {77,1060}, {216,1086}, {269,1038}, {348,1791}, {405,1398}, {441,894}, {442,1441}, {583,1708}, {594,4605}, {664,2893}, {1040,4328}, {1426,4205}, {1442,5453}, {1536,1827}
Regarding Question A, see X(6354).
X(6357) lies on these lines: {7,27}, {34,5722}, {56,5358}, {57,2160}, {73,5453}, {109,2688}, {223,5219}, {241,514}, {319,1943}, {323,651}, {942,1831}, {1387,1457}, {3649,4658}, {4304,5930}
X(6357) = isogonal conjugate of X(15627)
Regarding Question A, see X(6354).
X(6358) lies on these lines: {2,2006}, {9,92}, {10,201}, {12,1089}, {57,75}, {65,4647}, {85,4102}, {109,4418}, {171,1733}, {196,2550}, {222,4363}, {226,306}, {278,2345}, {312,5219}, {355,5928}, {553,4980}, {756,1109}, {894,1943}, {1091,1254}, {1111,3782}, {1215,4551}, {1365,6058}, {1708,5271}, {1766,5307}, {2078,3757}, {3175,3991}, {3262,3687}, {3706,5173}, {3719,3729}, {3911,4359}, {3947,4066}, {4008,5269}, {4671,5226}, {4692,5252}
X(6358) = isogonal conjugate of X(2150)
X(6358) = isotomic conjugate of X(2185)
X(6358) = complement of X(18662)
X(6358) = anticomplement of X(16579)
X(6358) = trilinear pole of line X(4036)X(4064)
X(6358) = pole wrt polar circle of trilinear polar of X(270)
X(6358) = X(48)-isoconjugate (polar conjugate) of X(270)
Regarding Question A, see X(6354).
X(6359) lies on these lines: {7,90}, {77,3362}, {86,658}, {87,269}, {273,1804}, {348,1119}, {894,1020}, {934,1441}, {1014,1446}, {1247,3668}, {1442,4566}
Let A′B′C′ be as at X(6349) and A″B″C″ as at X(6350). Then A′B′C′ and A″B″C″ are homothetic to each other and to the anticomplementary triangle. The homothetic center is X(6360). (Randy Hutson, January 30, 2015)
X(6360) lies on these lines: {2,92}, {8,3152}, {20,145}, {63,1943}, {192,3151}, {322,3998}, {329,4552}, {394,664}, {653,n+31}, {1148,1816}, {1944,6505}, {1952,2994}, {3100,3957}, {3101,3164}, {3177,3219}
X(6360) = isogonal conjugate of X(8761)
X(6360) = anticomplement of X(92)
X(6360) = anticomplementary conjugate of X(21270)
X(6360) = pole wrt polar circle of trilinear polar of X(7049)
X(6360) = X(48)-isoconjugate (polar conjugate) of X(7049)
X(6361) is the orthocenter of the triangle A′B′C′ described at X(6349). (Randy Hutson, January 30, 2015)
Let P(k) = M(A,B,C) : M(B,C,A) : M(C,A,B), where
M(A,B,C) = (1 + Cos[A] - Cos [B] - Cos[C] + k Cos[B] Cos [C])
Then P(k) lies on X(4)X(9). The appearance of (k,m) in the following list means that P(k) = X(m) (-2,516), (-1,6361), (-2/3,5493), (0,40), (1,5657), (2,10), (3,5818), (4,5587) ((2r - 2R)/r, 1512), (4R/(2R - r), 1706), (r + 2R)/R, 2550), (2R - r)/R, 2551), (2R - r)/(r + R), 5698), (-r/(2R - r), 5759). (Peter Moses, February 1, 2015)
X(6361) lies on these lines: {1,376}, {2,3579}, {3,962}, {4,9}, {7,3295}, {8,30}, {20,145}, {35,3485}, {46,497}, {55,3487}, {57,1058}, {63,5082}, {65,3488}, {165,631}, {329,5687}, {348,5195}, {355,3146}, {382,5690}, {387,5165}, {388,1770}, {390,942}, {443,3587}, {452,3753}, {484,1479}, {496,5435}, {499,5442}, {515,3529}, {548,5734}, {549,5550}, {550,1482}, {595,4000}, {758,3189}, {950,2093}, {952,1657}, {1006,5584}, {1044,1066}, {1056,1697}, {1062,4318}, {1125,3524}, {1155,3086}, {1210,5128}, {1385,3522}, {1698,3545}, {1699,3090}, {1737,5225}, {1836,3085}, {1837,5183}, {2099,4305}, {3057,4293}, {3058,5221}, {3149,6244}, {3241,3534}, {3340,4304}, {3475,3746}, {3476,4299}, {3486,4302}, {3523,5886}, {3528,3576}, {3543,3617}, {3586,4848}, {3622,3656}, {3623,3655}, {3627,5790}, {3634,5071}, {3635,4297}, {3817,5067}, {3871,5905}, {3877,4190}, {3931,4307}, {3962,6001}, {4127,5693}, {4309,5902}, {4424,5716}, {4533,5777}, {4668,5691}, {4701,5881}, {5057,5552}, {5122,5265}
X(6361) = reflection of X(i) in X(j) for these (i,j): (4,40), (40,5493), (382,5690), (944,20), (962,3), (1482,550), (3146,355), (3241,3534), (3543,3654)
X(6361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,553,3296), (3,962,5603), (4,40,5657), (4,5657,5818), (55,4295,3487), (65,4294,3488), (165,946,631), (484,1479,1788), (550,1482,5731), (1697,4292,1056), (1770,5119,388), (1836,3085,5714), (4299,5697,3476), (4302,5903,3486)
X(6362) lies on these lines: {30, 511}, {442, 1577}, {652, 4976}, {676, 905}, {1946, 5583}, {3649, 4804}, {4990, 6332}
X(6362) = crossdifference of every pair of points on line X(6)X(1174)
X(6363) lies on these lines: {30, 511}, {58, 3733}, {764, 4017}, {1027, 1431}, {1459, 1960}, {2605, 4491}, {3709, 3768}
X(6363) = isogonal conjugate of X(8706)
X(6363) = crossdifference of every pair of points on line X(6)X(145)
X(6364) lies on this line: {30, 511} (the infinity line)
X(6364) = isogonal conjugate of X(6135)
X(6365) lies on this line: {30, 511} (the infinity line)
X(6365) = isogonal conjugate of X(6136)
X(6366) lies on these lines: {1, 676}, {8, 3904}, {11, 1146}, {30, 511}, {100, 658}, {104, 972}, {885, 1320}, {1145, 3126}, {1317, 1360}, {1638, 6174}, {2976, 6161}, {3004, 4477}, {3700, 4474}, {4391, 4990}
X(6366) = isogonal conjugate of X(14733)
X(6366) = isotomic conjugate of X(35157)
X(6366) = X(2)-Ceva conjugate of X(35091)
X(6366) = crossdifference of every pair of points on line X(6)X(109)
X(6367) lies on these lines: {30, 511}, {4024, 4705}, {4041, 4838}, {4647, 4978}, {4983, 4988}
X(6367) = isogonal conjugate of X(6578)
X(6367) = crossdifference of every pair of points on line X(6)X(593)
Let U be the anticomplementary triangle, and let V be the triangle is formed by that tangents to the bianticevian conic of X(2) and X(3) at the vertices of U. Then U and V are perspective, and X(6368) is their perspector.
X(6368) lies on these lines: {30, 511}, {201, 656}, {684, 2525}, {879, 1176}, {2081, 2600}, {3267, 6333}
X(6368) = isogonal conjugate of X(933)
X(6368) = isotomic conjugate of polar conjugate of X(12077)
X(6368) = X(2)-Ceva conjugate of X(39019)
X(6368) = barycentric square root of X(39019)
X(6368) = crossdifference of every pair of points on line X(6)X(24)
X(6368) = perspector of hyperbola {{A,B,C,X(2),X(5)}}
X(6368) = X(163)-isoconjugate of X(275)
X(6369) lies on this line: {30, 511} (the infinity line)
X(6369) = isogonal conjugate of X(36078)
X(6369) = crossdifference of every pair of points on line X(6)X(2148)
X(6370) lies on these lines: {30, 511}, {351, 4809}, {643, 4427}, {1109, 2632}, {1637, 1639}, {1769, 2292}, {3268, 4453}, {3569, 4016}, {4036, 4064}, {4647, 4768}, {4707, 4736}
X(6370) = isogonal conjugate of X(36069)
X(6370) = crossdifference of every pair of points on line X(6)X(163)
X(6371) lies on these lines: {6,2483}, {30,511}, {649,854}, {650,2532}, {656,2530}, {659,3737}, {665,798}, {667,1459}, {1960,2605}, {2499,2978}, {2526,4524}, {3250,3709}, {3271,4475}, {3733,5009}, {3768,4079}, {4010,4985}
X(6371) = crossdifference of every pair of points on line X(6)X(8)
X(6371) = isogonal conjugate of X(8707)
X(6371) = isotomic conjugate of anticomplement of X(39015)
X(6371) = X(2)-Ceva conjugate of X(39015)
X(6371) = perspector of hyperbola {{A,B,C,X(2),X(56)}}
X(6371) = barycentric square root of X(39015)
X(6372) lies on these lines: {30,511}, {659,1019}, {661,665}, {663,4378}, {667,4724}, {764,4983}, {1491,4905}, {1734,4490}, {1960,4040}, {2254,4705}, {2533,3762}, {3250,4813}, {3271,4403}, {3837,4129}, {4010,4978}, {4063,4784}, {4449,4775}, {4498,4834}
X(6372) = crossdifference of every pair of points on line X(6)X(1621)
X(6372) = isogonal conjugate of X(8708)
X(6373) lies on these lines: {30,511}, {42,890}, {291,659}, {665,3572}, {668,3888}, {1015,1960}, {2978,4813}, {4507,4790}
X(6373) = crossdifference of every pair of points on line X(6)X(190)
X(6373) = isogonal conjugate of X(8709)
X(6373) = polar conjugate of isotomic conjugate of X(22092)
X(6373) = ideal point of PU(25)
"SS(a → bc)" denotes the barycentric symbolic substitution a → bc, b →ca, c → ab, which carries lines to lines, conics to conics, cubics to cubics, etc.; see X(6337).
X(6374) is the center of conic {{A,B,C,X(670),X(689),X(1978)}}. (This conic is the isogonal conjugate of line X(669)X(688, which is the trilinear polar of X(32)), and it is also the isotomic conjugate of the Lemoine axis.
As a line L varies through the set of all lines that pass through X(76), the locus of the trilinear pole of L is a circumconic, and its center is X(6374). As W varies through the set of all conics that pass through the isotomic conjugates of the incenter and the excenters, the locus of the center of W is X(6374). (Randy Hutson, February 16, 2015)
X(6374) lies on these lines: {2,2998}, {6,670}, {69,3978}, {75,982}, {76,141}, {157,5152}, {192,1978}, {264,5117}, {304,1921}, {305,3314}, {308,3763}, {325,1368}, {5286,6338}
X(6374) = isotomic conjugate of X(3224)
X(6374) = complement of X(2998)
X(6374) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,76), (670,3221)
X(6374) = anticomplement of X(6375)
X(6374) = crosspoint of X(2) and X(194)
X(6374) = crosssum of X(6) and X(3224)
X(6374) = perspector of circumconic centered at X(76)
X(6374) = centroid of X(75) and its three extraversions
X(6374) = X(i)-isoconjugate of X(j) for these {i,j}: {31,3224}, {32,3223}, {560,2998}, {1924,3222}, {1973,3504}
X(6374) = X(i)-complementary conjugate of X(j) for these (i,j): (31,76), (194,2887), (662,3221), (1424,2886), (1613,10), (1740,141)
See X(6374).
X(6375) lies on these lines: {2,2998}, {6,3229}, {39,698}, {141,1084}, {230,800}, {3117,3618}
X(6375) = complement of X(6374)
X(6375) = isotomic conjugate of isogonal conjugate of X(9490)
X(6375) = polar conjugate of isogonal conjugate of X(23221)
X(6375) = centroid of X(37) and its three extraversions
X(6375) = X(4609)-Ceva conjugate of X(512)
X(6375) = X(i)-complementary conjugate of X(j) for these (i,j): (3223,626), (3224,2887)
X(6375) = bicentric sum of PU(154)
X(6375) = midpoint of PU(154)
See X(6374).
X(6376) is the center of the conic W = {{A,B,C,X(668), X(789), X(799), X(811), X(1978), X(4554), X(4593), X(4602)}. W is the isotomic conjugate of the antiorthic axis, as well as the the isogonal conjugate of the line X(667)X(788). (Randy Hutson, February 16, 2015)
The conic W is also the locus of the trilinear pole of a line that varies through the set of all lines through X(75), as well as the locus of the trilinear product of the two Steiner circumellipse intercepts of a line that varies through the set of all the lines through X(2). The trilinear polar of X(6376) passes through X(3835). (Randy Hutson, February 16, 2015)
X(6376) lies on these lines: 1,668}, {2,330}, {8,350}, {9,1966}, {10,75}, {12,85}, {37,2998}, {41,3570}, {69,2551}, {86,979}, {183,958}, {190,3501}, {192,4110}, {194,1575}, {257,312}, {274,1698}, {319,5722}, {325,1329}, {384,4386}, {385,4426}, {538,1574}, {561,756}, {612,1965}, {750,799}, {993,1078}, {1376,1975}, {1500,4033}, {1573,3934}, {1655,2276}, {1930,3992}, {3061,3452}, {3208,4595}, {3507,3905}, {3617,4441}, {3679,3760}, {3820,3933}, {4377,4708}
X(6376) = isotomic conjugate of X(87)
X(6376) = complement of X(330)
X(6376) = anticomplement of X(16604)
X(6376) = complementary conjugate of X(20255)
X(6376) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,75), (668,4083), (1221,321)
X(6376) = X(i)-cross conjugate of X(j) for these (i,j): (3123,3835), (3971,192), (4147,4595)
X(6376) = perspector of circumconic centered at X(75)
X(6376) = polar conjugate of isogonal conjugate of X(22370)
X(6376) = X(i)-complementary conjugate of X(j) for these (i,j): (6,3840), (31,75), (41,3061), (43,141), (101,4083), (192,2887), (1403,142), (1423,2886), (2176,10), (2209,2), (3208,1329), (4083,116)
X(6376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,76,75), (313,5224,75), (984,1921,75), (1698,3761,274), (3264,4389,75), (3596,4357,75), (3661,3948,312)
X(6376) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2162}, {31,87}, {32,330}, {56,2053}, {604,2319}, {667,932}, {1919,4598}, {1977,5383}
See X(6374).
X(6377) lies on these lines: 2,1978}, {39,4850}, {115,5515}, {239,3229}, {244,665}, {292,1054}, {512,4117}, {649,1977}, {1084,1086}, {1386,1908}, {2666,5625}, {3117,5222}, {3226,4598}
X(6377) = isogonal conjugate of X(5383)
X(6377) = isotomic conjugate of isogonal conjugate of X(21762)
X(6377) = polar conjugate of isogonal conjugate of X(22386)
X(6377) = complementary conjugate of X(21262)
X(6377) = complement of X(1978)
X(6377) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3835), (75,512), (192,4083), (330,513), (2998,514), (3248,1015)
X(6377) = X(3123)-cross conjugate of X(1015)
X(6377) = X(i)-complementary conjugate of X(j) for these (i,j): (31,3835), (32,513), (513,626), (560,514), (649,2887), (667,141), (669,1211), (798,3454), (1333,512), (1397,4885), (1501,650), (1918,4129), (1919,10), (1922,3837), (1924,1213), (1977,11), (1980,2), (2205,661), (2206,4369), (3063,1329), (3121,125), (3248,116), (4612,3037)
X(6377) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2162}, {31,87}, {32,330}, {56,2053}, {604,2319}, {667,932}, {1919,4598}, {1977,5383}
See X(6374).
X(6378) lies on these lines: 2,1221}, {9,87}, {37,714}, {330,5283}, {594,1084}, {894,3229}, {932,2375}, {2162,4275}, {3117,5749}
X(6378) = complement of X(34086)
X(6378) = X(3123)-cross conjugate of X(1015)
X(6378) = X(i)-isoconjugate of X(j) for these {i,j}: {{43,1509}, {192,757}, {261,1423}, {552,3208}, {763,3971}, {873,2176}, {2185,3212}, {3123,4590}, {4083,4610}
See X(6374).
X(6379) lies on these lines: {2,2998}, {30,511}, {670,3229}
X(6379) = isogonal conjugate of X(6380)
See X(6374).
X(6380) lies on the circumcircle and these lines: {6,3222}, {99,1613}
See X(6374).
X(6381) lies on these lines: {2,3761}, {8,3760}, {10,75}, {37,716}, {85,3947}, {183,993}, {190,4495}, {274,3634}, {325,3814}, {334,4013}, {350,519}, {514,661}, {538,1575}, {545,4506}, {561,3971}, {799,5209}, {1078,5267}, {1107,3934}, {1111,3263}, {1125,1909}, {1237,4075}, {1323,4554}, {1329,3933}, {1500,4681}, {1930,3701}, {2295,4721}, {3008,3975}, {3230,4465}, {3661,4044}, {3679,4441}, {3734,4386}, {3770,5750}, {4364,4377}, {4396,5291}, {4410,4472}, {4419,4494}, {4479,4669}, {4723,4986}, {5088,5205}
X(6381) = midpoint of X(350) and X(668)
X(6381) = isotomic conjugate of X(37129)
X(6381) = X(3994)-cross conjugate of X(536)
X(6381) = X(715)-complementary conjugate of X(3739)
X(6381) = {X(1111),X(3992)}-harmonic conjugate of X(3263)
X(6381) = X(i)-isoconjugate of X(j) for these {i,j}: {6,739}, {32,3227}, {667,898}, {889,1980}, {1919,4607}, {1977,5381}
See X(6374).
X(6382) lies on these lines: {2,1978}, {8,3978}, {55,874}, {75,982}, {76,321}, {312,1921}, {594,1502}, {670,4363}, {1965,3923}, {1966,4362}, {3263,3705}, {6063,6358}
X(6382) = isotomic conjugate of X(2162)
X(6382) = complement of polar conjugate of isogonal conjugate of X(23183)
X(6382) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,76), (1978,3835)
X(6382) = {X(321),X(561)}-harmonic conjugate of X(76)
X(6382) = X(i)-isoconjugate of X(j) for these {i,j}: {31,2162}, {32,87}, {330,560}, {604,2053}, {932,1919}, {1397,2319}, {1980,4598}
See X(6374).
X(6383) lies on these lines: {2,1221}, {7,3978}, {75,982}, {76,3662}, {85,1921}, {87,870}, {274,330}, {334,3596}, {670,4361}, {767,932}, {1086,1502}, {1218,3739}, {1278,1978}
X(6383) = isotomic conjugate of X(2176)
X(6383) = X(561)-cross conjugate of X(76)
X(6383) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2209}, {31,2176}, {32,43}, {41,1403}, {192,560}, {1397,3208}, {1423,2175}, {1980,4595}
See X(6374).
X(6384) lies on these lines: {2,330}, {7,350}, {57,1966}, {75,982}, {76,3840}, {86,87}, {244,561}, {312,335}, {334,3705}, {614,1965}, {673,2319}, {675,932}, {748,799}, {903,4479}, {940,2162}, {2296,3720}, {4373,4441}
X(6384) = isogonal conjugate of X(2209)
X(6384) = isotomic conjugate of X(43)
X(6384) = complement of polar conjugate of isogonal conjugate of X(23177)
X(6384) = anticomplement of polar conjugate of isogonal conjugate of X(23213)
X(6384) = X(330)-Ceva conjugate of X(75)
X(6384) = X(i)-cross conjugate of X(j) for these {i,j}: (76,75), (3123,514), (3662,85), (3840,2), (3944,92)
X(6384) = {X(982),X(1920)}-harmonic conjugate of X(75)
X(6384) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2209}, {6,2176}, {31,43}, {32,192}, {41,1423}, {55,1403}, {604,3208}, {692,4083}, {1110,3123}, {1919,4595}, {2175,3212}, {2206,3971}
See X(6374).
X(6385) lies on these lines: {2,1218}, {37,1221}, {38,75}, {76,141}, {86,870}, {99,767}, {274,1107}, {313,334}, {314,670}, {1444,4623}, {1978,4043}, {3770,3978}
X(6385) = isogonal conjugate of X(2205)
X(6385) = isotomic conjugate of X(213)
X(6385) = X(i)-cross conjugate of X(j) for these {i,j}: (76,310), (3125,693), (3662,86)
X(6385) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2205}, {6,1918}, {10,1501}, {25,2200}, {31,213}, {32,42}, {37,560}, {41,1402}, {71,1974}, {100,1924}, {101,669}, {184,2333}, {228,1973}, {321,1917}, {688,4628}, {692,798}, {872,1333}, {1018,1980}, {1084,4570}, {1110,3121}, {1334,1397}, {1400,2175}, {1409,2212}, {1500,2206}, {1576,4079}, {1919,4557}, {1922,3747}, {2207,4055}, {4117,4567}
See X(6374).
X(6386) lies on these lines: {75,3123}, {76,1086}, {99,839}, {100,689}, {313,334}, {668,670}, {1332,4601}, {1502,3596}, {1920,4377}, {1978,3807}, {6331,6335}
X(6386) = midpoint of X(6338) and X(6339)
X(6386) = isogonal conjugate of X(1980)
X(6386) = isotomic conjugate of X(667)
X(6386) = complement of polar conjugate of isogonal conjugate of X(23190)
X(6386) = anticomplement of polar conjugate of isogonal conjugate of X(23227)
X(6386) = X(4602)-Ceva conjugate of X(1978)
X(6386) = X(i)-cross conjugate of X(j) for these {i,j}: (693,76), (4705,321)
X(6386) = trilinear pole of line X(76)X(321)
X(6386) = X(i)-isoconjugate of X(j) for these {i,j}: {1,1980}, {6,1919}, {31,667}, {32,649}, {58,669}, {81,1924}, {101,1977}, {163,3121}, {512,2206}, {513,560}, {514,1501}, {604,3063}, {663,1397}, {692,3248}, {693,1917}, {798,1333}, {810,2203}, {875,2210}, {1019,2205}, {1084,4556}, {1252,3249}, {1356,4636}, {1395,1946}, {1459,1974}, {1474,3049}, {1576,3122}, {1918,3733}, {1927,4164}
See X(6374).
X(6387) lies on this line: {2,6338}
X(6387) = complement of X(6338)
See X(6374).
X(6388) lies on these lines: {2,2987}, {6,5181}, {110,5477}, {111,3448}, {115,125}, {126,4576}, {136,2501}, {373,1506}, {468,1692}, {1112,1560}, {3066,5475}, {3291,3580}
X(6388) = complement of X(4563)
X(6388) = trilinear pole, wrt medial triangle, of the line X(5)X(6)
See X(6374).
X(6389) lies on these lines: {2,216}, {3,66}, {6,441}, {69,248}, {122,126}, {127,131}, {193,3284}, {317,401}, {338,2165}, {426,1899}, {1589,5590}, {1590,5591}, {2968,4361}, {3546,3788}, {3547,3934}, {3618,5158}, {3926,6338}, {4363,6356}
X(6389) = complement of X(393)
See X(6374).
X(6390) lies on these lines: {2,2418}, {3,69}, {5,1975}, {30,99}, {32,3629}, {39,698}, {76,140}, {141,574}, {183,549}, {187,524}, {193,1384}, {194,5305}, {230,538}, {315,550}, {381,1007}, {441,525}, {468,3266}, {543,625}, {732,2021}, {1078,3530}, {3035,6382}, {3053,6144}, {3167,4176}, {3734,3815}, {3760,5433}, {3761,5432}, {3763,5013}, {3788,5254}
X(6390) = midpoint of X(99) and X(325)
X(6390) = reflection of X(i) in X(j) for these (i,j): (230,620), (3793,187)
X(6390) = isotomic conjugate of X(17983)
X(6390) = isogonal conjugate of X(8753)
X(6390) = complement of polar conjugate of X(2374)
X(6390) = crossdifference of every pair of points on line X(25)X(2489)
See X(6374).
X(6391) lies on these lines: {4,193}, {6,1196}, {54,5050}, {64,511}, {65,3751}, {66,524}, {69,1368}, {74,3565}, {141,5486}, {155,3527}, {159,1177}, {1350,3532}, {1353,6193}, {2213,4260}, {3618,5544}, {5965,6145}
X(6391) = isogonal conjugate of X(6353)
X(6391) = reflection of X(6193) in X(1353)
See X(6374). X(6392) = SS(a → S2A) OF X(8)
X(6392) lies on these lines: {2,39}, {4,193}, {20,385}, {69,5254}, {99,439}, {148,2794}, {230,6337}, {384,5304}, {393,6339}, {1657,3793}, {2549,3785}, {3734,5319}, {5032,5395}
X(6392) = reflection of X(3926) in X(3767)
X(6392) = isotomic conjugate of X(6339)
X(6392) = anticomplement of X(3926)
X(6392) = X(393)-Ceva conjugate of X(2)
X(6392) = X(6338)-cross conjugate of X(2)
X(6392) = polar conjugate of cyclocevian conjugate of X(38259)
X(6392) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (19,1370), (25,4329), (158,315), (393,6327), (1096,69), (1395,347), (1973,20), (1974,6360), (2207,8), (6059,329)
X(6392) = X(i)-isoconjugate of X(j) for these {i,j}: {3,2129}, {31,6339}
X(6392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,5286,2), (193,2996,4), (3767,3926,2)
See X(6374).
X(6393) lies on these lines: {2,4176}, {3,69}, {30,5207}, {76,141}, {99,1503}, {114,325}, {249,524}, {305,343}, {315,1350}, {525,3267}, {620,1692}, {858,4576}, {1352,1975}, {3266,3580}, {3788,5028}, {3917,4121}, {5031,5969}
X(6393) = reflection of X(1692) in X(620)
X(6393) = isotomic conjugate of X(6531)
X(6393) = isotomic conjugate of isogonal conjugate of X(36212)
See X(6374).
X(6394) lies on these lines: {3,76}, {69,248}, {264,3148}, {305,426}, {325,441}, {336,1231}, {418,1799}, {865,2374}, {879,6333}, {1092,3926}, {2366,2715}, {2972,4563}
X(6394) = isogonal conjugate of X(34854)
X(6394) = isotomic conjugate of X(6530)
X(6394) = X(19)-isoconjugate of X(232)
X(6394) = X(92)-isoconjugate of X(2211)
X(6394) = crossdifference of every pair of points on line X(2491)X(17994) (the tangent to the inellipse that is the barycentric square of the orthic axis, at the barycentric square of X(232))
Centers associated with Lucas triangles and Lucas circles: X(6395)-X(6502)
Contributed by Randy Hutson, February 3, 2015
Definitions are given at MathWorld:
Lucas Central Triangle
Lucas Circles
Lucas Central Circle
Lucas Radical Circle
Lucas Inner Triangle
Lucas Inner Circle
Lucas Tangents Triangle
Additional definitions are given in ETC:
Lucas(t) circles, defined at X(371), where t = L/W
Lucas(t) homothetic triangle, defined at X(493)
Lucas(t) reflection triangle, defined at X(6401)
Lucas secondary circles, defined at X(6199)
Lucas secondary central triangle, defined at X(6199)
1st Lucas secondary tangents triangle, defined at X(6199)
2nd Lucas secondary tangents triangle, defined at X(6199)
Lucas secondary tangents circle, defined at X(6199)
Lucas(t) Brocard triangle, defined at X(6421)
Lucas(t) antipodal triangle, defined at X(6457)
Lucas(t) inner tangential triangle = tangential triangle of the Lucas(t) inner triangle
circumsymmedial tangential triangle = tangential triangle of the circumsymmedial triangle
An effort is being made to identify a publication in which Édouard Lucas (1842-1891) introduced the circles that are now known as Lucas circles. They are described in at least two articles:
Let A′, B′, C′ be the free vertices of the 2nd Kenmotu squares. Let A″, B″, C″ be the reflections of A′, B′, C′ in X(372). Triangle A″B″C″ is homothetic to ABC at X(6396). The following 4 triangles are pairwise perspective: (1) circumsymmedial, (2) Lucas(-1) tangents, (3) Lucas(-4) central, and (4) 1st Lucas secondary tangents triangles. Their common perspector is X(6396). (Randy Hutson, January 29, 2015)
Let A′B′C′ and A″B″C″ be the outer and inner Vecten triangles, resp. Let A* be the trilinear pole, wrt A′B′C′, of line B″C″. Define B*, C* cyclically. The lines A′A*, B′B*, C′C* concur in X(6396).
X(6396) lies on these lines: {2,1327}, {3,6}, {4,5420}, {20,486}, {30,615}, {35,6502}, {36,5414}, {76,6312}, {140,3070}, {186,5412}, {323,5409}, {376,3069}, {485,631}, {488,3620}, {490,640}, {549,590}, {550,3071}, {638,642}, {1124,5217}, {1335,5204}, {1495,3156}, {1584,5406}, {1587,3523}, {1588,3522}, {1600,5408}, {2066,5010}, {2362,3612}, {3068,3524}, {3092,3516}, {3093,3515}
X(6396) = isogonal conjugate of X(1328)
X(6396) = perspector of Lucas(-1) central triangle and circumsymmedial tangential triangle
X(6396) = exsimilicenter of circumcircle and Lucas(-1) radical circle
X(6396) = radical center of Lucas(-2/3) circles
X(6396) = trilinear pole, wrt outer Vecten triangle, of line X(2501)X(3566) (the perspectrix of the outer and inner Vecten triangles)
X(6396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6200), (3,372,371), (3,1152,372), (3,3312,1151), (6,1151,6199), (6,6200,371), (15,16,372), (187,3098,6200), (372,6200,6),
(574,5092,6200), (1587,3523,5418), (3312,6199,6), (3371,3372,1152), (3385,3386,3311)
X(6397) lies on this line: {372, 5861}
Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let KA = 1 if O(A) is internally tangent to Γ, and let KA = -1 if O(A) is externally tangent to Γ. Define KB and KC cyclically. Let LA and RA be the center and radius of O(A), and define LB and LC, and also RB and RC, cyclically. Then X(6398) = (KA/RA)*LA + (KB/RB)*LB + (KC/RC)*LC. Also, the circumsymmedial triangle, Lucas(-1) inner triangle, and Lucas(-2) central triangle are pairwise perspective, and their common perspector is X(6398). Let A′B′C′ be the Lucas(-1) central triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6398). Let A*B*C* be the Lucas(-1) tangents triangle. Let A** be the trilinear pole, wrt A*B*C*, of line BC. Define B**, C** cyclically. The lines A*A**, B*B**, C*C** concur in X(6398). (Randy Hutson, January 29, 2015)
X(6398) lies on these lines: {3,6}, {4,3591}, {30,3069}, {140,1587}, {186,5410}, {323,1600}, {378,5411}, {381,615}, {382,486}, {485,3526}, {488,3619}, {549,3068}, {550,1588}, {590,5054}, {999,5414}, {1131,5067}, {1385,1703}, {1597,5413}, {1656,3070}, {1657,3071}, {3093,3517}, {3295,6502}, {3299,5217}, {3301,5204}, {3317,3832}
X(6398) = radical center of Lucas(-4/3) circles
X(6398) = X(6) of Lucas(-1) tangents triangle
X(6398) = inverse-in-Lucas(-1)-radical-circle of X(187)
X(6398) = insimilicenter of circumcircle and Lucas(-1) inner circle
X(6398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6221), (3,372,3312), (3,3312,3311), (3,6199,6200), (6,6200,6199), (6,6221,3311), (15,16,1152), (371,372,3594), (372,1152,3), (3070,5420,1656), (3312,6221,6), (6199,6200,6221)
X(6398) = trilinear pole with respect to Lucas(-1) central triangle of Lemoine axis
X(6398) = trilinear pole with respect to Lucas(-1) tangents triangle of Lemoine axis
X(6398) = X(7) of Lucas(-1) central triangle, if all Lucas(-1) circles are externally tangent
X(6399) lies on these lines: {2,6290}, {511,6222}, {591,6280}, {1350,6316}.
X(6400) lies on these lines: {4, 69}, {24, 1152}, {1307, 3563}, {1587, 3567}, {5871, 6220}, {6197, 6404}, {6198, 6405}
X(6400) = isogonal conjugate of X(6402)
X(6400) = {X(4),X(6403)}-harmonic conjugate of X(6239)
Barycentrics SA/[(2*S-Sω)*SA^2+2*SA*S^2+S^2*(2*S+Sω)] : :
Barycentrics (b^2 + c^2 - a^2)/(a^4b^2 + a^4c^2 - 2a^2b^4 - 2a^2c^4 - 3a^2b^2c^2 + b^6 + c^6 - 4b^2c^2S) : :
Let A′B′C′ be the Lucas(t) central triangle. Let LA be the reflection of line B′C′ in line BC, and define LB, LC cyclically. Let A″ = LB∩LC, B″ = LC∩LA, C″ = LA∩LB. The triangle A″B″C″ is introduced here as the Lucas(t) reflection triangle. The Lucas(t) reflection triangle is perspective to ABC, the Lucas(u) central triangle, the Lucas(u) tangents triangle, the Lucas(u) inner triangle, and the Lucas(u) reflection triangle, for all t, u.
X(6401) lies on the conic {{A,B,C,X(3),X(25),X(32),X(98)}} and these lines: {3, 11941}, {6, 11984}, {32, 7583}, {67, 19390}, {262, 14167}, {371, 11977}, {372, 11975}, {488, 42065}, {1151, 11959}, {1152, 11960}, {1916, 22499}, {2782, 13029}, {3564, 10670}, {6402, 11983}, {6417, 11967}, {6418, 11969}, {6419, 11971}, {6420, 11973}, {6421, 11937}, {6422, 11938}, {6425, 11963}, {6426, 11964}, {6776, 39801}
X(6401) = isogonal conjugate of X(6239)
X(6402) lies on the conic {{A,B,C,X(3),X(25),X(32),X(98)}} and these lines: {3, 11943}, {6, 11985}, {32, 7584}, {67, 19391}, {184, 8964}, {262, 14168}, {371, 11976}, {372, 11978}, {487, 42065}, {1151, 11961}, {1152, 11962}, {1916, 22500}, {2782, 13031}, {3564, 10674}, {6401, 11983}, {6417, 11970}, {6418, 11968}, {6419, 11974}, {6420, 11972}, {6421, 11939}, {6422, 11940}, {6425, 11965}, {6426, 11966}, {6776, 39802}
X(6402) = isogonal conjugate of X(6400)
X(6403) lies on these lines: {4,69}, {6,24}, {25,110}, {51,6353}, {52,193}, {66,70}, {112,5017}, {141,1594}, {143,3517}, {159,1614}, {182,186}, {232,5052}, {378,1350}, {389,6467}, {403,5480}, {427,2979}, {468,5640}, {568,1353}, {576,1974}, {935,2698}, {1216,3620}, {1469,1870}, {1503,6240}, {1986,2854}, {2393,5890}, {3056,6198}, {3089,5446}, {3098,3520}, {3147,3618}, {3313,3541}, {3515,5050}, {3564,3575}, {3779,6197}, {5965,6242}
X(6403) = reflection of X(i) in X(j) for these (i,j): (193,52) (6467,389) (4,1843)
X(6403) = {X(6239),X(6400)}-harmonic conjugate of X(4)
X(6403) = anticomplement of X(9967)
X(6403) = X(7)-of-circumorthic-triangle if ABC is acute
X(6404) lies on these lines: {19,6406}, {40,511}, {55,1152}, {65,176}, {6197,6400}
X(6404) = {X(40),X(3779)}-harmonic conjugate of X(6252)
X(6405) lies on these lines: {1,256}, {55,1152}, {172,2066}, {1124,2330}, {6198,6400}
X(6405) = {X(1),X(3056)}-harmonic conjugate of X(6283)
X(6406) lies on these lines: {4,69}, {19,6404}, {25,1152}, {51,3070}, {1968,5412}, {1974,3093}
X(6406) = {X(4),X(1843)}-harmonic conjugate of X(6291)
X(6406) = crosssum of X(3) and X(487)
X(6406) = X(175)-of-orthic-triangle if ABC is acute
X(6407) lies on these lines: {3,6}, {30,1131}, {140,3317}, {485,5073}, {590,3843}, {1328,5055}, {1588,5054}, {1656,6459}, {1657,3068}, {3071,5070}, {3316,3845}, {3793,6462}, {3851,6561}
X(6407) = perspector of Lucas central triangle and Lucas inner triangle
X(6407) = radical center of Lucas(8/7) circles
X(6407) = {X(3),X(6)}-harmonic conjugate of X(6408)
X(6407) = {X(371),X(1152)}-harmonic conjugate of X(3311)
X(6408) lies on these lines: {3,6}, {30,1132}, {140,3316}, {486,5073}, {615,3843}, {1327,5055}, {1587,5054}, {1656,6460}, {1657,3069}, {3070,5070}, {3317,3845}, {3793,6463}, {3851,6560}
X(6408) = perspector of Lucas(-1) central triangle and Lucas(-1) inner triangle
X(6408) = radical center of Lucas(-8/7) circles
X(6408) = {X(3),X(6)}-harmonic conjugate of X(6407)
X(6408) = {X(372),X(1151)}-harmonic conjugate of X(3312)
The Lucas secondary tangents circle is introduced at X(6199).
X(6409) lies on these lines: {3,6}, {20,590}, {30,5418}, {485,550}, {486,549}, {2066,5204}, {3516,5412} .
X(6409) = perspector of symmedial triangle and Lucas inner tangential triangle
X(6409) = radical center of Lucas(1/2) circles
X(6409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6410), (3,371,1152), (3,1151,6), (371,1152,6), (1151,1152,371)
X(6410) lies on these lines: {3,6}, {20,615}, {30,5420}, {485,549}, {486,550}, {3516,5413}, {5204,5414} .
X(6410) = perspector of symmedial triangle and Lucas(-1) inner tangential triangle
X(6410) = radical center of Lucas(-1/2) circles
X(6410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6409), (3,372,1151), (3,1152,6), (372,1151,6), (1151,1152,372)
X(6411) lies on these lines: {3,6}, {323,5406}, {485,548}, {486,3530}, {487,3631}, {488,3630} (at least).
X(6411) = radical center of Lucas(1/3) circles
X(6411) = perspector of Lucas(-1) tangents triangle and Lucas inner tangential triangle
X(6411) = perspector of 2nd Lucas secondary tangents triangle and circumsymmedial tangential triangle
X(6411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6412), (3,1151,1152)
X(6412) lies on these lines: {3,6}, {323,5704}, {485,3530}, {486,548}, {487,3630}, {488,3631} .
X(6412) = radical center of Lucas(-1/3) circles
X(6412) = perspector of Lucas tangents triangle and Lucas(-1) inner tangential triangle
X(6412) = perspector of 2nd Lucas(-1) secondary tangents triangle and circumsymmedial tangential triangle
X(6412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6411), (3,1152,1151)
Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(-1) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let PA be the polar of A′ with respect to the circumcircle. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. The lines AA″, BB″, CC″ concur in X(6413).
X(6413) Lies on the Jerabek hyperbola and lines {4,371}, {6,3156}, {54,372}, {69,5409}, {184,216} .
X(6413) = X(92)-isoconjugate of X(371)
X(6413) = {X(184),X(216)}-harmonic conjugate of X(6414)
X(6413) = perspector of 1st Kenmotu diagonals triangle and cross-triangle of ABC and 1st Kenmotu diagonals triangle
Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let PA be the polar of A′ with respect to the circumcircle. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. The lines AA″, BB″, CC″ concur in X(6414).
X(6414) Lies on the Jerabek hyperbola and these lines: {4,372}, {6,3155}, {54,371}, {69,5408}, {184,216} .
X(6414) = X(92)-isoconjugate of X(372)
X(6414) = {X(184),X(216)}-harmonic conjugate of X(6413)
X(6414) = perspector of 2nd Kenmotu diagonals triangle and cross-triangle of ABC and 2nd Kenmotu diagonals triangle
Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(-1/2) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let PA be the polar of A′ with respect to the circumcircle. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. The lines AA″, BB″, CC″ concur in (X(6415).
X(6415) lies on the Jerabek hyperbola and these lines: {4,1131}, {54,3312}, {64,371}, {184,6416}, {1151,3532}
X(6415) = X(92)-isoconjugate of X(1151)
Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(1/2) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let PA be the polar of A′ with respect to the circumcircle. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. The lines AA″, BB″, CC″ concur in X(6416).
X(6416) lies on the Jerabek hyperbola and these lines: {4,1132}, {54,3311}, {64,372}, {184,6415}, {1152,3532}
X(6416) = X(92)-isoconjugate of X(1152)
The Lucas secondary central triangle is introduced at X(6199).
X(6417) lies on these lines: {3,6}, {485,3851}, {486,5055}, {1131,3845}, {1132,3850}
X(6417) = isogonal conjugate of X(34089)
X(6417) = radical center of Lucas(8) circles
X(6417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6418), (6,371,3312), (6,3311,3), (371,3312,3), (3311,3312,371)
X(6418) lies on these lines: {3,6}, {485,5055}, {486,3851}, {1131,3850}, {1132,3845} .
X(6418) = isogonal conjugate of X(34091)
X(6418) = radical center of Lucas(-8) circles
X(6418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6417), (6,372,3311), (6,3312,3), (372,3311,3), (3311,3312,372)
The 1st Lucas secondary tangents triangle is introduced at X(6199).
X(6419) lies on these lines: {3,6}, {4,1327}, {485,1132}, {486,3068}, {1994,5408}, {2066,3301}, {2067,3299}, {3518,5413}, {5198,5410}, {5409,5422}, {5412,5417} .
X(6419) = perspector of Lucas secondary central triangle and circumsymmedial tangential triangle
X(6419) = radical center of Lucas(6) circles
X(6419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6420), (6,371,372), (6,1151,3312), (6,3311,371), (61,62,371)
X(6420) lies on these lines: {3,6}, {486,1131}
X(6420) = perspector of Lucas(-1) secondary central triangle and circumsymmedial tangential triangle
X(6420) = radical center of Lucas(-6) circles
X(6420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6419), (6,372,371), (6,1152,3311), (6,3312,372), (61,62,372)
Let RA be the radical axis of the A-Lucas(t) circle and Brocard circle, and define RB and RC cyclically. Let A′ = RB∩RC, B′ = RC∩RA, C′ = RA∩RB. Triangle A′B′C′ is here introduced as the Lucas(t) Brocard triangle.
X(6421) lies on these lines: {2,494}, {3,6}, {493,589}
X(6421) = radical center of Lucas(2 cot ω - 2) circles
X(6421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6424), (6,39,6422), (8396,8400,3)
X(6421) = trilinear product of vertices of inner Grebe triangle
X(6421) = exsimilicenter of 2nd Lemoine and (1/2)-Moses circles; the insimilicenter is X(6422)
X(6422) lies on these lines: {2,493}, {3,6}, {494,588}
X(6422) = radical center of Lucas(2 cot ω + 2) circles
X(6422) = {X(3),X(6)}-harmonic conjugate of X(6243)
X(6422) = {X(6),X(39)}-harmoninc conjugate of X(6421)
X(6422) = trilinear product of vertices of outer Grebe triangle
X(6422) = insimilicenter of 2nd Lemoine and (1/2)-Moses circles; then exsimilicenter is X(6421)
Let (OA) be the circumcircle of the A-1st-Kenmotu square, and define (OB) and (OC) cyclically. Let RA be the radical axis of the circumcircle and (OA), and define RB and RC cyclically. Let A′ = RB∩RC, and define B′, C′ cyclically. The lines AA′, BB′, CC′ concur in X(6423). (Randy Hutson, February 3, 2015)
X(6423) lies on these lines: {3,6}, {251,494}
X(6423) = isogonal conjugate of X(5490)
X(6423) = radical center of Lucas(-2 cot ω - 2) circles
X(6423) = {X(3),X(6)}-harmonic conjugate of X(6422)
X(6423) = {X(3),X(32)}-harmonic conjugate of X(6424)
X(6423) = trilinear product of vertices of anti-outer-Grebe triangle
Let (OA) be the circumcircle of the A-2nd-Kenmotu square, and define (OB) and (OC) cyclically. Let RA be the radical axis of the circumcircle and (OA), and define RB and RC cyclically. Let A′ = RB∩RC, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6424) (Randy Hutson, February 3, 2015)
X(6424) lies on these lines: {3,6}, {251,493}
X(6424) = isogonal conjugate of X(5491)
X(6424) = radical center of Lucas(-2 cot ω + 2) circles
X(6424) = {X(3),X(6)}-harmonic conjugate of X(6421)
X(6424) = {X(3),X(32)}-harmonic conjugate of X(6423)
X(6424) = exsimilicenter of (1/2)-Moses and Lucas radical circles; the insimilicenter is X(9600)
X(6424) = {X(39),X(1151)}-harmonic conjugate of X(9600)
X(6424) = trilinear product of vertices of anti-inner-Grebe triangle
Let LA and RA be the center and radius of the A-Lucas circle, and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas radical circle and R* its radius. Then X(6425) = LA/RA + LB/RB + LC/RC + L*/R* - X(3)/R.
X(6425) lies on these lines: {3,6}, {485,3627}, {486,632}, {1131,3068}
X(6425) = perspector of ABC and Lucas inner tangential triangle
X(6425) = perspector of Lucas tangents triangle and 2nd Lucas secondary tangents triangle
X(6425) = radical center of Lucas(3/2) circles
X(6425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6426), (371,1151,6), (1152,3311,6)
Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let KA = 1 if O(A) is internally tangent to Γ, and let KA = -1 if O(A) is externally tangent to Γ. Define KB and KC cyclically. Let LA and RA be the center and radius of O(A), and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas(-1) radical circle and R* its radius. Then X(6426) = (KA/RA)*LA + (KB/RB)*LB + (KC/RC)*LC - L*/R* - X(3)/R; see also X(6449). (Randy Hutson, February 3, 2015)
X(6426) lies on these lines: {3,6}, {485,632}, {486,3627}, {1132,3069}
X(6426) = perspector of ABC and Lucas(-1) inner tangential triangle
X(6426) = perspector of Lucas(-1) tangents triangle and 2nd Lucas(-1) secondary tangents triangle
X(6426) = radical center of Lucas(-3/2) circles
X(6426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6425), (372,1152,6), (1151,3312,6)
X(6427) lies on these lines: {3,6}, {485,5072}, {486,5079}, {546,1588}, {632,3069}, {1173,6415}, {1587,3627}, {3068,3628}, {3070,5076}, {3299,3304}, {3301,3303}, {3518,5411}
X(6427) = radical center of Lucas(12) circles
X(6427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6428), (6,3311,3312), (61,62,1151)
X(6428) lies on these lines: {3,6}, {485,5079}, {486,5072}, {546,1587}, {632,3068}, {1173,6416}, {1588,3627}, {3069,3628}, {3071,5076}, {3299,3303}, {3301,3304}, {3518,5410}
X(6428) = radical center of Lucas(-12) circles
X(6428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6427), (6,3312,3311), (61,62,1152)
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(6429). Also, let A′B′C′ be the Lucas inner triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6429).
X(6429) lies on these lines: {3,6}, {547,5418}, {590,3832}, {3068,5059}, {3071,5067}, {3850,6561}, {5056,6459}
X(6429) = radical center of Lucas(5/4) circles
X(6429) = {X(3),X(6)}-harmonic conjugate of X(6430)
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(6430). Also, let A′B′C′ be the Lucas(-1) inner triangle, let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6430).
X(6430) lies on these lines: {3,6}, {547,5420}, {615,3832}, {3069,5059}, {3070,5067}, {3850,6560}, {5056,6460}
X(6430) = radical center of Lucas(-5/4) circles
X(6430) = {X(3),X(6)}-harmonic conjugate of X(6429)
Let A′B′C′ be the 2nd Lucas secondary 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(6431).
X(6431) lies on these lines: {3,6}, {485,3850}, {486,547}, {487,597}, {590,5067}, {615,3533}, {1588,3545}, {3068,5056}, {3070,3543}, {3071,3832}, {5059,6459}
X(6431) = radical center of Lucas(5) circles
X(6431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6432), (6,371,1152), (371,1152,1151)
Let A′B′C′ be the 2nd Lucas(-1) secondary tangents triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6432).
X(6432) lies on these lines: {3,6}, {485,547}, {486,3850}, {488,597}, {590,3533}, {615,5067}, {1587,3545}, {3069,5056}, {3070,3832}, {3071,3543}, {5059,6460}
X(6432) = radical center of Lucas(-5) circles
X(6432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6431), (6,372,1151), (372,1151,1152)
X(6433) lies on these lines: {3,6}, {547,6561}, {590,3543}, {3071,3533}, {3850,5418}
X(6433) = radical center of Lucas(5/6) circles
X(6433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6434), (6456,6458,3)
X(6434) lies on these lines: {3,6}, {547,6560}, {615,3543}, {3070,3533}, {3850,5420}
X(6434) = radical center of Lucas(-5/6) circles
X(6434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6433), (6457,6459,3)
X(6435) lies on these lines: {3,6}, {1328,6564}, {1588,3854}
X(6435) = radical center of Lucas(22/3) circles
X(6435) = {X(3),X(6)}-harmonic conjugate of X(6436)
X(6436) lies on these lines: {3,6}, {1327,6565}, {1587,3854}
X(6436) = radical center of Lucas(-22/3) circles
X(6436) = {X(3),X(6)}-harmonic conjugate of X(6435)
X(6437) lies on these lines: {3,6}, {485,3853}, {590,3545}, {1588,3533}, {3068,3543}, {3070,5059}, {3071,5056}, {3832,6459}, {3845,6561}
X(6437) = radical center of Lucas(5/3) circles
X(6437) = {X(3),X(6)}-harmonic conjugate of X(6438)
X(6438) lies on these lines: {3,6}, {486,3853}, {615,3545}, {1587,3533}, {3069,3543}, {3070,5056}, {3071,5059}, {3832,6460}, {3845,6560}
X(6438) = radical center of Lucas(-5/3) circles
X(6438) = {X(3),X(6)}-harmonic conjugate of X(6437)
X(6439) lies on line {3,6}
X(6439) = radical center of Lucas(13/12) circles
X(6439) = {X(3),X(6)}-harmonic conjugate of X(6440)
X(6440) lies on line {3,6}
X(6440) = radical center of Lucas(-13/12) circles
X(6440) = {X(3),X(6)}-harmonic conjugate of X(6439)
X(6441) lies on these lines: {3,6}, {485,3859}
X(6441) = radical center of Lucas(13/3) circles
X(6441) = {X(3),X(6)}-harmonic conjugate of X(6442)
X(6442) lies on these lines: {3,6}, {486,3859}
X(6442) = radical center of Lucas(-13/3) circles
X(6442) = {X(3),X(6)}-harmonic conjugate of X(6441)
X(6443) lies on line {3,6}
X(6443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,574,1151), (6,5024,6444)
X(6444) lies on line {3,6}
X(6444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,574,1152), (6,5024,6443)
X(6445) lies on these lines: {3,6}, {590,3830}, {3068,3534}, {3316,3853}, {3526,6459}, {3851,5418}, {5055,6561}
X(6445) = perspector of circumsymmedial triangle and Lucas(3) central triangle
X(6445) = radical center of Lucas(8/9) circles
X(6445) = inverse-in-Lucas(-1)-secondary-radical-circle of X(6509)
X(6445) = {X(3),X(6)}-harmonic conjugate of X(6446)
X(6446) lies on these lines: {3,6}, {615,3830}, {3069,3534}, {3317,3853}, {3526,6460}, {3851,5420}, {5055,6560}
X(6446) = perspector of circumsymmedial triangle and Lucas(-3) central triangle
X(6446) = radical center of Lucas(-8/9) circles
X(6446) = inverse-in-Lucas-secondary-radical-circle of X(6510)
X(6446) = {X(3),X(6)}-harmonic conjugate of X(6445)
X(6447) lies on these lines: {3,6}, {382,1327}, {485,5076}, {546,6459}, {590,5072}, {632,1588}, {1132,3090}, {3068,3627}, {3071,5079}, {3091,3316}
X(6447) = radical center of Lucas(12/7) circles
X(6447) = {X(3),X(6)}-harmonic conjugate of X(6448)
X(6448) lies on these lines: {3,6}, {382,1328}, {486,5076}, {546,6460}, {615,5072}, {632,1587}, {1131,3090}, {3069,3627}, {3070,5079}, {3091,3317}
X(6448) = radical center of Lucas(-12/7) circles
X(6448) = {X(3),X(6)}-harmonic conjugate of X(6447)
Let LA and RA be the center and radius of the A-Lucas circle, and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas inner circle and R* its radius. Then X(6449) = LA/RA + LB/RB + LC/RC - L*/R* - X(3)/R; see also X(6426). (Randy Hutson, February 3, 2015)
Let AB = inverse-in-A-Lucas-circle of B, and define BC and CA cyclically. Let AC = inverse-in-A-Lucas-circle of C, and define BA and CB cyclically. The six points AB, BC, CA, AC, BA, CB lie on a circle with center X(6449). See X(6519). (Peter Moses, February 11, 2015)
If you have The Geometer's Sketchpad, you can view X(6449).
X(6449) lies on these lines: {3,6}, {140,6459}, {381,5418}, {382,590}, {485,1657}, {486,5054}, {548,1587}, {549,1588}, {550,3068}, {1132,3533}, {1656,6561}, {3069,3530}, {3070,3534}, {3071,3526}, {3316,3543}, {3520,5410}, {5070,6565}, {5073,6564}
X(6449) = perspector of Lucas tangents triangle and Lucas secondary central triangle
X(6449) = radical center of Lucas(4/5) circles
X(6449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6450), (3,371,3312), (371,3312,3311)
Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let KA = 1 if O(A) is internally tangent to Γ, and let KA = -1 if O(A) is externally tangent to Γ. Define KB and KC cyclically. Let LA and RA be the center and radius of O(A), and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas(-1) inner circle and R* its radius. Then X(6450) = (KA/RA)*LA + (KB/RB)*LB + (KC/RC)*LC + L*/R* - X(3)/R.
X(6450) lies on these lines: {3,6}, {140,6460}, {381,5420}, {382,615}, {485,5054}, {486,1657}, {548,1588}, {549,1587}, {550,3069}, {1131,3533}, {1656,6560}, {3068,3530}, {3070,3526}, {3071,3534}, {3317,3543}, {3520,5411}, {5070,6564}
X(6450) = perspector of Lucas(-1) tangents triangle and Lucas(-1) secondary central triangle
X(6450) = radical center of Lucas(-4/5) circles
X(6450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6449), (3,372,3311), (372,3311,3312)
X(6451) lies on these lines: {3,6}, {20,3316}, {590,1327}, {631,1132}, {1657,5418}, {3530,6459}, {5054,6561}
X(6451) = radical center of Lucas(4/9) circles
X(6451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6452), (3,1151,3312)
X(6452) lies on these lines: {3,6}, {20,3317}, {615,1328}, {631,1131}, {1657,5420}, {3530,6460}, {5054,6560}
X(6452) = radical center of Lucas(-4/9) circles
X(6452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6451), (3,1152,3311)
Combos by Randy Hutson, February 3, 2015
1) Let LA and RA be the center and radius of the A-Lucas circle, and define LB and LC, and also RB and RC, cyclically.
Let L* be the center of the Lucas inner circle and R* its radius. Then X(6453) = LA/RA + LB/RB + LC/RC + L*/R*
(2) Let L** and R** be the center and radius of the Lucas radical circle. Then X(6453) = LA/RA + LB/RB + LC/RC + L**/R**.
X(6453) lies on these lines: {3,6}, {485,3146}, {486,3525}
X(6453) = perspector of Lucas central triangle and Lucas inner tangential triangle
X(6453) = exsimilicenter of Lucas radical circle and Lucas inner circle
X(6453) = radical center of Lucas(6/5) circles
X(6453) = {X(3),X(6)}-harmonic conjugate of X(6454)
Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let KA = 1 if O(A) is internally tangent to Γ, and let KA = -1 if O(A) is externally tangent to Γ. Define KB and KC cyclically. Let LA and RA be the center and radius of O(A), and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas(-1) inner circle and R* its radius. Let L** be the center of the Lucas(-1) radical circle and R** its radius. Then X(6454) = (KA/RA)*LA + (KB/RB)*LB + (KC/RC)*LC + L*/R* = (KA/RA)*LA + (KB/RB)*LB + (KC/RC)*LC - L**/R**.
X(6454) lies on these lines: {3,6}, {485,3525}, {486,3146}
X(6454) = perspector of Lucas(-1) central triangle and Lucas(-1) inner tangential triangle
X(6454) = exsimilicenter of Lucas(-1) radical circle and Lucas(-1) inner circle
X(6454) = radical center of Lucas(-6/5) circles
X(6454) = {X(3),X(6)}-harmonic conjugate of X(6453)
X(6455) lies on these lines: {3,6}, {382,5418}, {485,3534}, {548,3068}, {549,6459}, {590,1657}, {1588,3530}, {3071,5054}, {3316,5059}, {3526,6561}
X(6455) = radical center of Lucas(4/7) circles
X(6455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6456), (3,1151,3311)
X(6456) lies on these lines: {3,6}, {382,5420}, {486,3534}, {548,3069}, {549,6460}, {615,1657}, {1587,3530}, {3070,5054}, {3317,5059}, {3526,6560}
X(6456) = radical center of Lucas(-4/7) circles
X(6456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6455), (3,1152,3312)
Let A′ be the antipode of A in the A-Lucas(t) circle, and define B′ and C′ cyclically. The triangle A′B′C′ is here introduced as the Lucas(t) antipodal triangle.
X(6457) lies on these lines: {4,371}, {185,577}
X(6457) = {X(185),X(577)}-harmonic conjugate of X(6458)
X(6458) lies on these lines: {4,372}, {185,577}
X(6458) = {X(185),X(577)}-harmonic conjugate of X(6457)
Let A′B′C′ be the Lucas antipodal triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6459).
X(6459) lies on these lines: {2,489}, {3,1588}, {4,371}, {5,6221}, {6,20}, {30,1587}, {193,490}, {372,376}, {1152,3522}, {3070,3146}
X(6459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1588,3069), (4,371,3068), (6,20,6460)
Let A′B′C′ be the Lucas(-1) antipodal triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6460).
X(6460) lies on these lines: {2,490}, {3,1587}, {4,372}, {5,6398}, {6,20}, {30,1588}, {193,489}, {371,376}, {1151,3522}, {3071,3146}
X(6460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1587,3068), (4,372,3069). (6,20,6459)
X(6461) lies on these lines: {6,6337}, {39,493}, {63,7124}, {394,3926}, {1181,8220}, {3796,8194}, {8188,8189}, {8201,8202}, {8208,8209}, {8210,8211}, {8212,8213}, {8214,8215}, {8216,8217}, {8218,8219}
X(6461) = perspector of symmedial, Lucas homothetic, and Lucas(-1) homothetic triangles (see X(493) and X(494))
X(6461) = X(6464)-of-Lucas-homothetic-triangle
X(6461) = X(6464)-of -Lucas(-1)-homothetic-triangle
X(6461) = barycentric product X(6)*X(6338)
X(6461) = {X(6462),X(6463)}-harmonic conjugate of X(6339)
X(6461) = insimilicenter of circumcircles of Lucas homothetic and Lucas(-1) homothetic triangles
Let A′B′C′ be the Lucas homothetic triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6462).
X(6462) lies on these lines: {2,493}, {6,6337}, {20,6465}, {148,1131}, {193,371}
X(6462) = {X(6339),X(6461)}-harmonic conjugate of X(6463)
Let A′B′C′ be the Lucas(-1) homothetic triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6463).
X(6463) lies on these lines: {2,494}, {6,6337}, {20,6466}, {148,1132}, {193,372}
X(6463) = {X(6339),X(6461)}-harmonic conjugate of X(6462)
X(6464) lies on these lines: {39,493}, {141,5490}, {155,1351}, {6392,6515}
X(6464) = Lucas(-1)-homothetic-to-ABC similarity image of X(6461)
X(6464) = endo-homothetic center of Lucas homothetic and Lucas(-1) homothetic triangles
Let A′ be the midpoint of the intersections, other than A, of the A-Lucas circle and AB and AC. Define B′ and C′ cyclically. Let PA be the polar with respect to the A-Lucas circle of A′. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. Triangle A″B″C″ is homothetic to the Lucas homothetic triangle at X(6465).
X(6465) lies on these lines: {3,6466}, {20,6462}, {25,371}
X(6465) = {X(3),X(6467)}-harmonic conjugate of X(6466)
Let A′ be the midpoint of the intersections, other than A, of the A-Lucas(-1) circle and AB and AC. Define B′ and C′ cyclically. Let PA be the polar with respect to the A-Lucas(-1) circle of A′. Define PB and PC cyclically. Let A″ = PB∩PC, B″ = PC∩PA, C″ = PA∩PB. The triangle A″B″C″ is homothetic to the Lucas(-1) homothetic triangle at X(6466).
X(6466) lies on these lines: {3,6465}, {20,6463}, {25,372}
X(6466) = {X(3),X(6467)}-harmonic conjugate of X(6465)
X(6467) lies on these lines: {3,6465}, {6,25}, {20,185}, {69,305}
X(6467) = isogonal conjugate of isotomic conjugate of X(1368)
X(6467) = isotomic conjugate of X(683)
X(6467) = crosssum of X(2) and X(25)
X(6467) = crosspoint of X(6) and X(69)
X(6467) = crossdifference of every pair of points on line X(525)X(2451)
X(6467) = polar conjugate of isotomic conjugate of X(22401)
X(6467) = {X(6465), X(6466)}-harmonic conjugate of X(3)
Let A′B′C′ be the Lucas inner triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6468).
X(6468) lies on these lines: {3,6}, {590,3839}, {5066,6561}
X(6468) = perspector of Lucas inner triangle and Lucas inner tangential triangle
X(6468) = radical center of Lucas(7/6) circles
X(6468) = inverse-in-Lucas-inner-circle of X(187)
X(6468) = trilinear pole, with respect to Lucas inner triangle, of Lemoine axis
X(6468) = {X(3),X(6)}-harmonic conjugate of X(6469)
Let A′B′C′ be the Lucas(-1) inner triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6469).
X(6469) lies on these lines: {3,6}, {615,3839}, {5066,6560}
X(6469) = perspector of Lucas(-1) inner triangle and Lucas(-1) inner tangential triangle
X(6469) = radical center of Lucas(-7/6) circles
X(6469) = inverse-in-Lucas(-1)-inner-circle of X(187)
X(6469) = trilinear pole, with respect to Lucas(-1) inner triangle, of Lemoine axis
X(6469) = {X(3),X(6)}-harmonic conjugate of X(6468)
X(6470) lies on these lines: {3,6}, {485,3858}, {1132,3068}, {1588,3316}, {3071,3855}
X(6470) = radical center of Lucas(7/2) circles
X(6470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6471), (371,3312,1151)
X(6471) lies on these lines: {3,6}, {486,3858}, {1131,3069}, {1587,3317}, {3070,3855}
X(6471) = radical center of Lucas(-7/2) circles
X(6471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6476), (372,3311,1152)
X(6472) lies on line {3,6}
X(6472) = radical center of Lucas(32/25) circles
X(6472) = {X(3),X(6)}-harmonic conjugate of X(6473)
X(6473) lies on line {3,6}
X(6473) = radical center of Lucas(-32/25) circles
X(6473) = {X(3),X(6)}-harmonic conjugate of X(6472)
X(6474) lies on these lines: {3,6}, {1131,5073}
X(6474) = radical center of Lucas(32/23) circles
X(6474) = {X(3),X(6)}-harmonic conjugate of X(6475)
X(6475) lies on these lines: {3,6}, {1132,5073}
X(6475) = radical center of Lucas(-32/23) circles
X(6475) = {X(3),X(6)}-harmonic conjugate of X(6474)
X(6476) lies on line {3,6}
X(6476) = radical center of Lucas(26/21) circles
X(6476) = {X(3),X(6)}-harmonic conjugate of X(6477)
X(6477) lies on line {3,6}
X(6477) = radical center of Lucas(-26/21) circles
X(6477) = {X(3),X(6)}-harmonic conjugate of X(6476)
X(6478) lies on these lines: {3,6}, {590,3859}
X(6478) = radical center of Lucas(26/19) circles
X(6478) = {X(3),X(6)}-harmonic conjugate of X(6479)
X(6479) lies on these lines: {3,6}, {615,3859}
X(6479) = radical center of Lucas(-26/19) circles
X(6479) = {X(3),X(6)}-harmonic conjugate of X(6478)
X(6480) lies on these lines: {3,6}, {547,6565}, {590,3845}, {3543,6564}, {3545,6561}, {5056,5418}, {5067,6459}
X(6480) = radical center of Lucas(10/9) circles
X(6480) = insimilicenter of Lucas radical circle and Lucas inner circle
X(6480) = {X(3),X(6)}-harmonic conjugate of X(6481)
X(6481) lies on these lines: {3,6}, {547,6564}, {615,3845}, {3545,6560}, {5056,5420}, {5067,6460}
X(6481) = radical center of Lucas(-10/9) circles
X(6481) = insimilicenter of Lucas(-1) radical circle and Lucas(-1) inner circle
X(6481) = {X(3),X(6)}-harmonic conjugate of X(6480)
X(6482) lies on these lines: {3,6}, {3543,3590}
X(6482) = radical center of Lucas(-30/29) circles
X(6482) = {X(3),X(6)}-harmonic conjugate of X(6483)
X(6483) lies on these lines: {3,6}, {3543,3591}
X(6483) = radical center of Lucas(30/29) circles
X(6483) = {X(3),X(6)}-harmonic conjugate of X(6482)
X(6484) lies on these lines: {3,6}, {485,5059}, {590,3853}, {3533,6459}, {3545,5418}, {5056,6561}, {5067,6565}
X(6484) = radical center of Lucas(10/11) circles
X(6484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6485), (3,6433,6486)
X(6485) lies on these lines: {3,6}, {486,5059}, {615,3853}, {3533,6460}, {3545,5420}, {5056,6560}, {5067,6564}
X(6485) = radical center of Lucas(-10/11) circles
X(6485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6484), (3,6433,6487)
X(6486) lies on these lines: {3,6}, {3832,5418}, {5067,6561}
X(6486) = radical center of Lucas(10/13) circles
X(6486) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6487), (3,6433,6484)
X(6487) lies on these lines: {3,6}, {3832,5420}, {5067,6560}
X(6487) = radical center of Lucas(-10/13) circles
X(6487) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6486), (3,6434,6485)
X(6488) lies on these lines: {3,6}, {1328,3628}, {3857,5418}
X(6488) = radical center of Lucas(9/11) circles
X(6488) = {X(3),X(6)}-harmonic conjugate of X(6489)
X(6489) lies on these lines: {3,6}, {1327,3628}, {3857,5420}
X(6489) = radical center of Lucas(-9/11) circles
X(6489) = {X(3),X(6)}-harmonic conjugate of X(6488)
X(6490) lies on these lines:
X(6490) = radical center of Lucas(19/15) circles
X(6490) = {X(3),X(6)}-harmonic conjugate of X(6491)
X(6491) lies on line {3,6}
X(6491) = radical center of Lucas(-19/15) circles
X(6491) = {X(3),X(6)}-harmonic conjugate of X(6490)
X(6492) lies on line {3,6}
X(6492) = radical center of Lucas(19/13) circles
X(6492) = {X(3),X(6)}-harmonic conjugate of X(6493)
X(6493) lies on line {3,6}
X(6493) = radical center of Lucas(-19/13) circles
X(6493) = {X(3),X(6)}-harmonic conjugate of X(6492)
X(6494) lies on line {3,6}
X(6494) = radical center of Lucas(44/13) circles
X(6494) = {X(3),X(6)}-harmonic conjugate of X(6495)
X(6495) lies on line {3,6}
X(6495) = radical center of Lucas(-44/13) circles
X(6495) = {X(3),X(6)}-harmonic conjugate of X(6494)
X(6496) lies on these lines: {3,6}, {376,1131}, {1328,5054}, {3317,3523}, {3534,5418}
X(6496) = radical center of Lucas(4/11) circles
X(6496) = {X(3),X(6)}-harmonic conjugate of X(6497)
X(6496) = {X(1152),X(3311)}-harmonic conjugate of X(3312)
X(6497) lies on these lines: {3,6}, {376,1132}, {1327,5054}, {3316,3523}, {3534,5420}
X(6497) = radical center of Lucas(-4/11) circles
X(6497) = {X(3),X(6)}-harmonic conjugate of X(6496)
X(6497) = {X(1151),X(3312)}-harmonic conjugate of X(3311)
X(6498) lies on line {3,6}
X(6498) = radical center of Lucas(44) circles
X(6498) = {X(3),X(6)}-harmonic conjugate of X(6499)
X(6499) lies on line {3,6}
X(6499) = radical center of Lucas(-44) circles
X(6499) = {X(3),X(6)}-harmonic conjugate of X(6498)
X(6500) lies on these lines: {3,6}, {381,1132}, {1327,3071}, {1587,3830}, {1588,3843}, {1656,3316}, {3068,5070}
X(6500) = radical center of Lucas(16) circles
X(6500) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6501), (3311,3312,1151)
X(6501) lies on these lines: {3,6}, {381,1131}, {1328,3070}, {1587,3843}, {1588,3830}, {1656,3317}, {3069,5070}
X(6501) = radical center of Lucas(-16) circles
X(6501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6500), (3311,3312,1152)
Points Caroli I and II are inexed as X(2066) and X(2067).
X(6502) lies on these lines: {1,372}, {3,1124}, {6,41}, {12,615}, {35,6396)}, {36,371}, {55,1152}, {63,3083}, {65,5416}, {485,499}, {486,1478}, {603,605}, {1151,5204}, {3295,6398}, {3301,5563}
X(6502) = homothetic center of 2nd Kenmotu diagonals triangle and anti-tangential midarc triangle
X(6502) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,372,5414), (3,1124,2066), (6,56,2067), (48,73,2067)
See X(6374).
X(6503) lies on these lines: {3, 68}, {25, 114}, {39, 493}, {99, 2052}, {155, 454}, {394, 577}, {641, 1583}, {642, 1584}, {1993, 4558}
X(6503) = complement of X(6504)
X(6503) = X(2)-Ceva conjugate of X(394)
X(6503) = X(i)-complementary conjugate of X(j) for these (i,j): (31,394), (48,3548), (920,141), (1609,10), (1973,2165)
X(6503) = X(i)-isoconjugate of X(j) for these (i,j): {19,254}, {393,921}
See X(6374).
X(6504) lies on the Kiepert hyperbola and these lines: {4, 155}, {10, 921}, {69, 5392}, {96, 97}, {98, 1370}, {317, 2052}, {459, 3580}, {485, 5409}, {486, 5408}
X(6504) = isogonal conjugate of X(1609)
X(6504) = isotomic conjugate of X(6515)
X(6504) = anticomplement of X(6503)
X(6504) = cyclocevian conjugate of X(3346)
X(6504) = polar conjugate of X(3542)
X(6504) = X(i)-cross conjugate of X(j) for these (i,j): (68,69), (136,850), (394,2)
X(6504) = X(254)-anticomplementary conjugate of X(4329)
X(6504) = X(i)-isoconjugate of X(j) for these {i,j}: {{1,1609}, {6,920}, {19,155}, {48,3542}}
See X(6374).
Let L be a line through X(3). Let P and U be the intersections of L and the Steiner circumellipse. Let Q be the trilinear product P*U. The locus of Q as L varies is the circumconic centered at X(6505); its perspector is X(63).
X(6505) lies on these lines: {1, 224}, {2, 914}, {46, 453}, {57, 1813}, {63, 77}, {69, 6349}, {78, 1060}, {92, 664}, {223, 908}, {306, 326}, {329, 3160}, {940, 1100}, {1212, 3305}, {1708, 1993}, {1944, 6360}, {5287, 5949}
X(6505) = complement of X(2994)
X(6505) = X(2)-Ceva conjugate of X(63)
X(6505) = X(i)-complementary conjugate of X(j) for these (i,j): (31,63), (46,141), (604,499), (1406,142), (2178,10), (5905,2887)
X(6505) = {X(394),X(1214)}-harmonic conjugate of X(63)
X(6505) = X(i)-isoconjugate of X(j) for these (i,j): {4,2164}, {19,90}, {25,2994}, {393,1069}
See X(6374).
X(6506) lies on these lines: {5, 169}, {11, 1146}, {101, 119}, {115, 123}, {124, 5517}, {219, 2165}, {220, 1329}, {910, 1532}, {1212, 4187}, {1566, 5511}, {5190, 5521}
X(6506) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,523), (2990,2804), (2994,522)
X(6506) = X(90)-isoconjugate of X(1262)
X(6506) = {X(11),X(5514)}-harmonic conjugate of X(1146)
X(6506) = X(i)-complementary conjugate of X(j) for these (i,j): (19,4885), (25,522), (29,512), (33,513), (34,3900), (281,3835), (522,1368), (607,514), (652,6389), (663,3), (1096,521), (1172,4369), (1395,6129), (1973,905), (2212,650), (2299,523), (2310,123), (2333,1577), (2356,3126), (3063,1214), (3064,141), (3271,2968), (3709,440), (6059,3239)
See X(6374).
X(6507) lies on these lines: {48, 63}, {92, 1958}, {255, 820}, {304, 2167}, {394, 1804}, {1748, 1959}
X(6507) = isogonal conjugate of X(6520)
X(6507) = isotomic conjugate of X(6521)
X(6507) = X(326)-Ceva conjugate of X(255)
X(6507) = X(4100)-cross conjugate of X(255)
X(6507) = barycentric cube of X(63)
X(6507) = X(i)-isoconjugate of X(j) for these {i,j}: {4,393}, {6,1093}, {19,158}, {25,2052}, {92,1096}, {107,2501}, {264,2207}, {278,1857}, {281,1118}, {331,6059}, {1880,1896}
See X(6374).
X(6508) lies on these lines: {1, 204}, {2, 1952}, {38, 2632}, {48, 63}, {73, 960}, {92, 1953}, {336, 1965}, {774, 820}, {1748, 2173}, {1763, 6261}, {1954, 2629}, {2167, 2349}
X(6508) = X(811)-Ceva conjugate of X(656)
X(6508) = polar conjugate of X(821)
X(6508) = X(i)-isoconjugate of X(j) for these {i,j}: {6,1105}, {19,775}, {25,801}, {48,821}
See X(6374).
X(6509) lies on these lines: {2, 216}, {3, 64}, {39, 441}, {51, 852}, {99, 801}, {114, 122}, {184, 426}, {185, 417}, {394, 577}, {418, 2972}, {465, 618}, {466, 619}, {828, 3666}, {1214, 5745}, {1993, 3284}
X(6509) = complement of X(2052)
X(6509) = X(99)-Ceva conjugate of X(520)
X(6509) = {X(418),X(2972)}-harmonic conjugate of X(3917)
X(6509) = X(i)-isoconjugate of X(j) for these (i,j): {6,821}, {19,1105}, {393,775}, {801,1096}
X(6509) = X(i)-complementary conjugate of X(j) for these (i,j): (48,5), (163,520), (184,226), (255,141), (326,626), (394,2887), (560,3767), (563,1147), (577,10), (822,125), (1820,5449), (2148,389), (2149,3042), (2289,1329), (3990,3454), (4055,1211), (4100,3), (6056,3452)
See X(6374).
X(6510) lies on these lines: {1, 5696}, {63, 77}, {142, 1100}, {241, 2323}, {326, 3694}, {521, 656}, {527, 1323}, {664, 1944}, {908, 6357}, {940, 1449}, {995, 1386}, {1060, 3940}, {1419, 2324}, {1439, 2289}, {3812, 4658}, {5733, 5880}
X(6510) = midpoint of X(664) and X(1944)
X(6510) = X(i)-isoconjugate of X(j) for these (i,j): {4,2291}, {19,1156}, {25,1121}, {278,4845}
See X(6374).
X(6511) lies on these lines: {394, 1804}, {1800, 3157}, {2178, 5905}, {3964, 3998}
X(6511) = X(63)-Ceva conjugate of X(394)
X(6511) = X(i)-isoconjugate of X(j) for these (i,j): {90,393}, {158,2164}, {1096,2994}
See X(6374).
X(6512) lies on these lines: {283, 1069}, {333, 2164}
X(6512) = X(i)-isoconjugate of X(j) for these (i,j): {19,1068}, {46,393}, {158,2178}, {1096,5905}, {1880,3559}
See X(6374).
X(6513) lies on these lines: {2, 914}, {21, 90}, {78, 1069}, {280, 4511}, {2164, 2339}
X(6513) = X(2994)-Ceva conjugate of X(63)
X(6513) = X(i)-cross conjugate of X(j) for these (i,j): (394,63), (1062,77)
X(6513) =X(i)-isoconjugate of X(j) for these (i,j): {4,2178}, {6,1068}, {19,46}, {25,5905}, {281,1406}, {393,3157}, {608,5552}, {1400,3559}, {1880,3193}}.
See X(6374).
X(6514) lies on these lines: {48, 63}, {78, 1800}, {110, 2365}, {261, 284}, {271, 1792}, {394, 577}, {1259, 6056}, {1801, 4575}, {1958, 5307}, {2289, 3719}
X(6314) = X(394)-cross conjugate of X(1812)
X(6314) = X(i)-isoconjugate of X(j) for these (i,j): {4,1880}, {12,5317}, {19,225}, {34,1826}, {37,1118}, {65,393}, {108,2501}, {158,1400}, {226,1096}, {273,2333}, {278,1824}, {281,1426}, {1093,1409}, {1402,2052}, {1427,1857}, {1441,2207}, {1446,6059}
See X(6374).
X(6515) lies on the cubic K621 and these lines: {2, 6}, {4, 52}, {23, 161}, {24, 6193}, {25, 3564}, {51, 1352}, {92, 1947}, {110, 6353}, {155, 3542}, {317, 2052}, {393, 467}, {406, 3193}, {427, 1351}, {459, 2986}, {468, 3167}, {487, 1600}, {488, 1599}, {511, 1370}, {569, 631}, {914, 1708}, {973, 2888}, {1147, 3147}, {1209, 3090}, {1895, 5081}, {2781, 3448}, {3266, 4176}, {6225, 6293}
X(6515) = isotomic conjugate of X(6504)
X(6515) = reflection of X(1370) in X(1899)
X(6515) = anticomplement of X(394)
X(6515) = complement of polar conjugate of X(36612)
X(6515) = X(i)-Ceva conjugate of X(j) for these (i,j): (317,4), (2052,2)
X(6515) = X(1609)-cross conjugate of X(3542)
X(6515) = polar conjugate of X(254)
X(6515) = X(i)-isoconjugate of X(j) for these (i,j): {6,921}, {19,39109}, {48,254}
X(6515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,193,1993), (6,343,2), (52,68,4), (1993,3580,2), (2994,5905,92)
X(6515) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (4,4329), (19,20}, (25,6360},(34,347}, (92,1370}, (158,69}, (225,2897}, (393,8}, (823,512}, (1096,2}, (1118,7}, (1824,3151}, (1857,329}, (1880,3152}, (1973,3164}, (2052,6327}, (2207,192}, (5317,1}, (6059,3177)
See X(6374).
X(6516) lies on these lines: {3, 348}, {7, 1470}, {56, 6337}, {69, 1804}, {73, 1808}, {85, 404}, {99, 108}, {100, 658}, {101, 1025}, {104, 150}, {109, 1310}, {190, 2406}, {279, 4188}, {307, 1444}, {336, 1231}, {644, 4564}, {651, 662}, {668, 4998}, {905, 906}, {927, 1292}, {1004, 1996}, {1055, 6168}, {1332, 1813}, {1461, 3882}, {1810, 4855}, {1812, 1949}, {4561, 4571}, {5172, 5866}
X(6516) = isogonal conjugate of X(18344)
X(6516) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,664), (1275,394), (4554,651), (4620,1812), (4998,69)
X(6516) = X(i)-cross conjugate of X(j) for these (i,j): (78,4564), (394,1275), (521,63), (905,348), (1331,1332), (4025,1444), (4131,69)
X(6516) = {X(100),X(934)}-harmonic conjugate of X(664)
X(6516) = trilinear pole of line X(63)X(77)
X(6516) = X(i)-isoconjugate of X(j) for these (i,j): {4,663}, {6,3064}, {19,650}, {25,522}, {27,3709}, {28,4041}, {29,512}, {33,513}, {34,3900}, {92,3063}, {108,2310}, {158,1946}, {162,4516}, {270,4705}, {278,657}, {281,649}, {284,2501}, {318,667}, {333,2489}, {393,652}, {514,607}, {521,1096}, {523,2299}, {608,3239}, {661,1172}, {693,2212}, {810,1896}, {884,1861}, {885,2356}, {1021,1880}, {1024,5089}, {1119,4105}, {1395,4397}, {1396,4171}, {1398,4163}, {1435,4130}, {1459,1857}, {1474,3700}, {1577,2204}, {1783,2170}, {1824,3737}, {1897,3271}, {1973,4391}, {2189,4024}, {2203,4086}, {2207,6332}, {2333,4560}, {2969,3939}, {4017,4183}, {4025,6059}
See X(6374).
X(6517) lies on these lines: {99, 109}, {1813, 4558}
X(6517) = X(i)-isoconjugate of X(j) for these (i,j): {19,3064}, {107,4516}, {158,663}, {393,650}, {512,1896}, {513,1857}, {522,1096}, {693,6059}, {1093,1946}, {1118,3900}, {1172,2501}, {2052,3063}, {2207,4391}, {3700,5317}
See X(6374).
X(6518) lies on these lines: {9, 326}, {63, 77}, {69, 2289}, {226, 801}, {261, 284}, {281, 1958}, {304, 2329}, {521, 1946}, {527, 1275}, {991, 993}, {1944, 5088}, {1948, 2202}
X(6518) = X(i)-isoconjugate of X(j) for these (i,j): {4,1945}, {19,1937}, {25,1952}, {158,1949}, {225,2249}, {296,393}
X(6519) lies on these lines: {3,6}, {3628,6459}, {5072,6561}, {5079,5418}
Let (R) be the radical circle of the Lucas circles, and let (L) = inverse in (R) of the circumcircle. Then (L) tangent to each of the 3 Lucas circles. Let A′ be the touchpoint of (L) and the A-Lucas circle, and define B′ and C′ cyclically. Then
A′ = a2(3S + 4SA) : 2b2 (S + 2SB) : 2c2 (S + 2SC)
B′ = 2a2 (S + 2SA) : b2(3S + 4SB) : 2c2 (S + 2SC )
C′ = 2a2 (S + 2SA) : 2b2 (S + 2SB) : c2 (3S + 4SC).
Let A′B = inverse-in-A-Lucas-circle of B′, and define B′C and C′A cyclically. Let A′C = inverse-in-A-Lucas-circle of C′, and define B′A and C′B cyclically. The six points A′B, B′C, C′A, A′C, B′A, C′B lie on a circle with center X(6519). The circle is here called the Moses-Lucas Circle. It and the circle constructed from 6 points at X(6449) are members of the Schoute coaxal system; the Schoute circle with center
a2(pSA + qS) : b2(pSB + qS) : c2(pSC + qS) = pS*X(3) + qSω*X(6)
has radius R(S(p2 - 3q2)1/2) / (pS + qSω). See X(6449) and X(6522). (Peter Moses, February 11, 2015)
A combo for X(6519) is found as follows: Let LA and RA be the center and radius of the A-Lucas circle, and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas inner circle and R* its radius. Then X(6519) = LA/RA + LB/RB + LC/RC + L*/R* + X(3)/R. (Circle (L) is the Lucas inner circle, and A′B′C′ is the Lucas inner triangle.) (Randy Hutson, February 12, 2015)
If you have The Geometer's Sketchpad, you can view X(6519).
X(6519) lies on these lines: {3,6}, {3628,6459}, {5072,6561}, {5079,5418}
X(6519) = radical center of the Lucas(12/11) circles
X(6519) = inverse-in-Brocard-circle of X(6522)
X(6519) = inverse-in-Lucas-secondary-tangents-circle of X(6566)
X(6519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6522), (3,371,6427), (3,3592,3312), (3,6199,6420), (3,6221,6447), (3,6407,6453), (3,6417,6426), (3,6419,6448), (3,6420,6450), (3,6426,6452), (3,6427,6398), (3,6447,3311), (3,6453,6221), (3,6522,6497), (6,1151,6484), (6,6496,6497), (371,1151,6445), (371,6200,6410), (371,6398,3311), (371,6411,6418), (371,6445,6455), (371,6451,6495), (371,6455,6398), (1151,6221,6449), (1151,6407,6221), (1151,6429,6200), (1151,6453,3), (1151,6468,371), (1151,6480,6407), (3311,6449,6451), (3311,6451,6456), (3312,6410,6398), (3312,6450,6438), (3592,6200,3), (3592,6489,6), (3594,6410,6454), (3594,6425,371), (3594,6427,6428), (6199,6409,6450), (6221,6445,6398), (6221,6449,3311), (6221,6452,6437), (6221,6455,371), (6221,6484,6497), (6398,6428,3594), (6398,6449,6455), (6409,6420,3), (6410,6468,6429), (6417,6474,6437), (6417,6487,3312), (6419,6448,6428), (6427,6455,3), (6440,6500,3312), (6445,6455,6449), (6445,6468,6221), (6447,6449,3), (6456,6495,6398), (6484,6496,6449), (6496,6522,3)
X(6520) lies on these lines: {19, 158}, {48, 821}, {92, 1953}, {393, 1880}, {610, 1784}, {1093, 1826}, {1118, 2358}, {1824, 1857}
X(6520) = isotomic conjugate of X(1102)
X(6520) = pole wrt polar circle of trilinear polar of X(326) (the line X(822)X(4131))
X(6520) = polar conjugate of X(326)
X(6520) = X(1096)-cross conjugate of X(158)
X(6520) = X(i)-isoconjugate of X(j) for these (i,j): {2,1092}, {3,394}, {6,3964}, {31,1102}, {32,4176}, {48,326}, {63,255}, {69,577}, {75,4100}, {77,2289}, {97,5562}, {184,3926}, {219,1804}, {222,1259}, {249,2972}, {348,6056}, {417,801}, {520,4558}, {593,4158}, {603,3719}, {822,4592}, {906,4131}, {1331,4091}, {1437,3998}, {1444,3990}, {1576,4143}, {1790,3682}, {3289,6394}
X(6521) lies on these lines: {92, 1953}, {158, 774}, {823, 1748}
X(6521) = isogonal conjugate of X(4100)
X(6521) = isotomic conjugate of X(6507)
X(6521) = X(1881)-cross conjugate of X(4)
X(6521) = barycentric cube of X(92)
X(6521) = pole wrt polar circle of trilinear polar of X(255) (line X(680)X(822))
X(6521) = polar conjugate of X(255)
X(6521) = X(i)-isoconjugate of X(j) for these (i,j): {1,4100}, {3,577}, {6,1092}, {32,3964}, {48,255}, {97,418}, {184,394}, {222,6056}, {560,1102}, {603,2289}, {822,4575}, {1437,3990}, {1501,4176}, {1790,4055}
Let (R) be the radical circle of the Lucas(-1) circles, and let (L) = inverse in (R) of the circumcircle. Then (L) tangent to each of the 3 Lucas(-1) circles. Let A′ be the touchpoint of (L) and the A-Lucas(-1) circle, and define B′ and C′ cyclically. Then
A′ = a2(3S - 4SA) : 2b2(S - 2SB) : 2c2(S - 2SC)
B′ = 2a2(S - 2SA) : b2(3S - 4SB) : 2c2(S - 2SC )
C′ = 2a2(S - 2SA) : 2b2(S - 2SB) : c2(3S - 4SC)
Let A′B = inverse-in-A-Lucas(-1)-circle of B′, and define B′C and C′A cyclically. Let A′C = inverse-in-A-Lucas(-1)-circle of C′, and define B′A and C′B cyclically. The six points A′B, B′C, C′A, A′C, B′A, C′B lie on a circle with center X(6522). The circle is here called the Moses-Lucas(-1) Circle. It is a member of the Schoute coaxal system; see X(6519). Note that (L) is the Lucas(-1) inner circle and A′B′C′ is the Lucas(-1) inner triangle. (Randy Hutson, February 12, 2015)
Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let KA = 1 if O(A) is internally tangent to Γ, and let KA = -1 if O(A) is externally tangent to Γ. Define KB and KC cyclically. Let LA and RA be the center and radius of O(A), and define LB and LC, and also RB and RC, cyclically. Let L* be the center of the Lucas(-1) inner circle and R* its radius. Then X(6522) = (KA/RA) *LA + (KB/RB)*LB + (KC/RC)*LC + L*/R* + X(3)/R. (Randy Hutson, February 12, 2015)
X(6522) lies on these lines: {3,6}, {3628,6460}, {5072,6560}, {5079,5420}
X(6522) = radical center of the Lucas(-12/11) circles
X(6522) = inverse-in-Lucas(-1)-secondary-tangents-circle of X(6567)
X(6522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6519), (3,372,6428), (3,3594,3311), (3,6395,6419), (3,6398,6448), (3,6408,6454), (3,6418,6425), (3,6419,6449), (3,6420,6447), (3,6425,6451), (3,6428,6221), (3,6448,3312), (3,6454,6398), (3,6519,6496), (6,1152,6485), (6,6497,6496), (372,1152,6446), (372,6221,3312), (372,6396,6409), (372,6412,6417), (372,6446,6456), (372,6452,6494), (372,6456,6221), (1152,6398,6450), (1152,6408,6398), (1152,6430,6396), (1152,6454,3), (1152,6469,372), (1152,6481,6408), (3311,6409,6221), (3311,6449,6437), (3312,6450,6452), (3312,6452,6455), (3592,6409,6453), (3592,6426,372), (3592,6428,6427), (3594,6396,3), (3594,6488,6), (6221,6427,3592), (6221,6450,6456), (6395,6410,6449), (6398,6446,6221), (6398,6450,3312), (6398,6451,6438), (6398,6456,372), (6398,6485,6496), (6409,6469,6430), (6410,6419,3), (6418,6475,6438), (6418,6486,3311), (6420,6447,6427), (6428,6456,3), (6439,6501,3311), (6446,6456,6450), (6446,6469,6398), (6448,6450,3), (6455,6494,6221), (6485,6497,6450), (6497,6519,3)
See X(6374).
X(6523) lies on the cubic pK(X393,X2), this being the polar conjugate of the Lucas cubic; see Hyacinthos #28116. Also, X(6523) is the centoid of the quadrilateral whose vertices are X(4) together with the vertices of the anticevian triangle of X(4). (Randy Hutson, February 16, 2015)
Let A′B′C′ be the medial triangle. Let A″ be the center of the circle which is the inverse-in-polar-circle of line B′C′. Define B″and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(6523). (Randy Hutson, July 11, 2019)
X(6523) lies on these lines: {2, 3346}, {4, 64}, {20, 107}, {133, 5878}, {158, 278}, {393, 800}, {1075, 5656}, {1249, 3349}, {1559, 6225}
X(6523) = midpoint of X(4) and X(3183)
X(6523) = complement of X(3346)
X(6523) = X(2)-Ceva conjugate of X(393)
X(6523) = X(i)-complementary conjugate of X(j) for these (i,j): (1,6247), (31,393), (48,1073), (1033,226), (1498,10), (1712,5)
X(6523) = polar conjugate of X(1032)
X(6523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,6247), (1093,3089,393)
X(6523) = X(i)-isoconjugate of X(j) for these {i,j}: {48,1032}, {255,3346}
See X(6374).
X(6524) lies on these lines: {4, 51}, {25, 393}, {107, 3563}, {125, 459}, {154, 1990}, {184, 1249}, {1118, 1426}, {1204, 3183}, {1495, 3079}, {1824, 1857}
X(6524) = X(1093)-Ceva conjugate of X(393)
X(6524) = X(2207)-cross conjugate of X(393)
X(6524) = X(i)-isoconjugate of X(j) for these (i,j): {1,3964}, {3,326}, {6,1102}, {31,4176}, {48,3926}, {63,394}, {69,255}, {75,1092}, {76,4100}, {77,1259}, {78,1804}, {163,4143}, {222,3719}, {304,577}, {348,2289}, {520,4592}, {603,1264}, {757,4158}, {822,4563}, {1331,4131}, {1332,4091}, {1444,3682}, {1790,3998}, {3265,4575}
X(6524) = isogonal conjugate of X(3964)
X(6524) = isotomic conjugate of X(4176)
X(6524) = barycentric cube of X(4)
X(6524) = trilinear pole of line X(2489)X(2508)
X(6524) = pole wrt polar circle of trilinear polar of X(3926) (the line X(520)X(3265))
X(6524) = polar conjugate of X(3926)
See X(6374).
Let DEF be the orthic triangle and D' the midpoint of AD . Let A′ be the reflection of A in D'X(3), Let La be the perpendicular bisector of A′D, and define Lb and Lc cyclically. Let A″ = Lb?Lc, B″= Lc?La, C″ = La?Lb. The lines DA″, EB″, FC″ concur in X(6525). (Angel Montesdeoca, August 21, 2019)
X(6525) lies on these lines: {2, 107}, {4, 64}, {19, 1857}, {25, 393}, {34, 207}, {51, 3087}, {154, 1249}, {1859, 2262}, {1896, 4198}, {2052, 3424}
X(6525) = X(4)-Ceva conjugate of X(393)
X(6525) = X(3172)-cross conjugate of X(1249)
X(6525) = polar conjugate of X(34403)
X(6525) = Danneels point of X(107)
X(6525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,1853), (4,3183,64), (1249,3079,154)
X(6525) = X(i)-isoconjugate of X(j) for these (i,j): {63,1073}, {64,326}, {253,255}, {394,2184}, {2155,3926}
See X(6374).
Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Let A″ be the intersection, other than X(122), of the nine-point circle and the line A′X(122); define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6526). (Randy Hutson, February 16, 2015)
Let A′B′C′ be the reflection of the orthic triangle in X(4). X(6526) is the trilinear product A′*B′*C′. Moreover, the trilinear polar of X(6526) passes through X(2501). (Randy Hutson, February 16, 2015)
X(6526) lies on these lines: {2, 1105}, {4, 64}, {5, 1073}, {107, 3146}, {122, 3346}, {225, 1857}, {235, 393}, {253, 264}, {254, 403}, {1249, 5922}, {1300, 1301}, {2184, 5715}
X(6526) = isogonal conjugate of X(35602)
X(6526) = cevapoint of X(4) and X(6622)
X(6526) = X(459)-Ceva conjugate of X(393)
X(6526) = polar conjugate of X(37669)
X(6526) = perspector of ABC and the 2nd pedal triangle of X(64)
X(6526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,64), ( 4,3183,5895)
X(6526) = X(i)-isoconjugate of X(j) for these (i,j): {63,15905}, {20,255}, {122,1101}, {154,326}, {204,3964}, {394,610}, {1092,1895}, {1102,3172}, {1259,1394}
See X(6374).
X(6527) lies on these lines: {2, 216}, {20, 64}, {75, 280}, {99, 5897}, {193, 401}, {317, 3146}, {322, 345}, {340, 5059}, {441, 1249}, {1632, 5596}, {2385, 4329}, {3346, 3926}
X(6527) = reflection of X(393) in X(6389)
X(6527) = isotomic conjugate of X(3346)
X(6527) = anticomplement of X(393)
X(6527) = X(3926)-Ceva conjugate of X(2)
X(6527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20,253,69), (280,347,75), (393,6389,2)
X(6527) = X(i)-isoconjugate of X(j) for these {i,j}: {31,3346}, {1032,1973}, {2155,3344}
X(6527) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (2,5906), (3,5905), (31,6392), (48,193), (63,4), (75,317), (219,5942), (255,2), (283,92), (326,69), (394,8), (577,192), (662,520), (775,2052), (822,148), (1069,2994), (1092,6360), (1101,648), (1102,1370), (1259,329), (1331,4391), (1437,3187), (1790,3868), (1804,7), (1813,521), (1822,2592), (1823,2593), (2167,5889), (2169,1993), (2289,144), (3682,2895), (3719,3436), (3926,6327), (3964,4329), (3990,1654), (3998,1330), (4055,1655), (4091,149), (4100,3164), (4131,150), (4575,525), (4592,850), (6056,3177)
See X(6374).
X(6528) lies on the Steiner circumellipse and these lines: {4, 290}, {5, 276}, {99, 107}, {112, 2966}, {190, 823}, {264, 339}, {393, 3228}, {648, 1625}, {664, 811}, {670, 877}, {671, 2052}, {1896, 2481}, {2207, 3225}
X(6528) = isogonal conjugate of X(39201)
X(6528) = isotomic conjugate of X(520)
X(6528) = anticomplement of X(35071)
X(6528) = anticomplement of isogonal conjugate of X(34538)
X(6528) = center of bianticevian conic of X(2) and X(4)
X(6528) = trilinear pole of line X(2)X(216)
X(6528) = Brianchon point (perspector) of inscribed parabola with focus X(107)
X(6528) = polar conjugate of X(647)
X(6528) = X(i)-cross conjugate of X(j) for these (i,j): (520,2}, (525,276}, (648,6331}, (850,264}, (3060,250}, (5889,249)
X(6528) = X(i)-isoconjugate of X(j) for these (i,j): {3,810}, {6,822}, {31,520}, {48,647}, {63,3049}, {73,1946}, {163,3269}, {184,656}, {213,4091}, {228,1459}, {255,512}, {326,669}, {394,798}, {418,2616}, {513,4055}, {560,3265}, {577,661}, {649,3990}, {652,1409}, {667,3682}, {905,2200}, {1576,2632}, {1636,2159}, {1918,4131}, {1919,3998}, {1924,3926}, {2501,4100}, {4017,6056}
See X(6374).
X(6529) lies on these lines: {4, 1562}, {107, 112}, {115, 393}, {127, 6330}, {133, 1249}, {648, 1625}, {1093, 2207}
X(6529) = isotomic conjugate of X(4143)
X(6529) = trilinear pole of line X(25)X(393)
X(6529) = polar conjugate of X(3265)
X(6529) = X(i)-cross conjugate of X(j) for these (i,j): (2501,393), (3049,25)
X(6529) = trilinear product of vertices of circumanticevian triangle of X(4)
X(6529) = barycentric product X(4)*X(107)
X(6529) = X(i)-isoconjugate of X(j) for these (i,j): {31,4143}, {48,3265}, {63,520}, {69,822}, {71,4131}, {72,4091}, {255,525}, {326,647}, {394,656}, {512,1102}, {661,3964}, {662,2972}, {798,4176}, {810,3926}, {850,4100}, {905,3682}, {1019,4158}, {1092,1577}, {1459,3998}, {2632,4558}, {3269,4592}, {3990,4025}
See X(6374).
X(6530) lies on these lines: {4, 6}, {5, 264}, {24, 1485}, {25, 3425}, {30, 250}, {107, 468}, {132, 232}, {186, 3447}, {262, 427}, {297, 511}, {317, 1351}, {324, 5133}, {325, 2967}, {403, 523}, {428, 1629}, {429, 1896}, {460, 685}, {467, 3060}, {648, 3564}, {1594, 3613}
X(6530) = isotomic conjugate of X(6394)
X(6530) = polar conjugate of X(287)
X(6530) = perspector of circumconic through the polar conjugates of PU(37)
X(6530) = X(i)-cross conjugate of X(j) for these (i,j): (132,4), (232,297), (2450,325)
X(6530) = pole wrt polar circle of trilinear polar of X(287) (the line PU(37) = X(3)X(525))
X(6530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (53,5480,4), (132,232,1513)
X(6530) = X(238)-of-orthic-triangle if ABC is acute
X(6530) = radical center of circumcircle and P(39)- and U(39)-Fuhrmann circles (aka -Hagge circles)
X(6530) = X(i)-isoconjugate of X(j) for these (i,j): {3,293}, {31,6394}, {48,287}, {63,248}, {98,255}, {184,336}, {326,1976}, {394,1910}, {577,1821}, {822,2966}, {878,4592}, {879,4575}
See X(6374).
X(6531) lies on these lines: {2, 6394}, {4, 32}, {6, 264}, {107, 3124}, {111, 4240}, {213, 1783}, {225, 1910}, {230, 297}, {232, 419}, {275, 3051}, {393, 1974}, {436, 1196}, {450, 3291}, {578, 1217}, {685, 1990}, {1093, 2207}, {1105, 1970}, {1300, 2715}, {1501, 1629}, {1821, 2281}, {1826, 1918}
X(6531) = isogonal conjugate of X(36212)
X(6531) = isotomic conjugate of X(6393)
X(6531) = trilinear pole of line X(25)X(669) (the line through the polar conjugates of PU(37))
X(6531) = pole wrt polar circle of trilinear polar of X(325) (the line X(2799)X(3569))
X(6531) = polar conjugate of X(325)
X(6531) = barycentric product X(4)*X(98)
X(6531) = cevapoint of X(i) and X(j) for these {i,j}: {4, 419}, {6, 230}, {25, 2211}, {607, 862}, {3124, 17994}
X(6531) = X(i)-cross conjugate of X(j) for these (i,j): (460,4), (1976,98), (2086,2489), (2211,25)
X(6531) = X(i)-isoconjugate of X(j) for these (i,j): {3,1959}, {31,6393}, {48,325}, {63,511}, {69,1755}, {75,3289}, {163,6333}, {232,326}, {237,304}, {240,394}, {255,297}, {656,2421}, {662,684}, {810,2396}, {822,877}, {2799,4575}, {3405,3917}, {3569,4592}
See X(6374).
X(6532) lies on these lines: {2, 596}, {10, 244}, {519, 3812}, {726, 4698}, {1125, 3666}
X(6532) = midpoint of X(596) and X(4075)
X(6532) = complement of X(4075)
X(6532) = X(849)-complementary conjugate of X(4075)
X(6532) = {X(2),X(596)}-harmonic conjugate of X(4075)
See X(6374).
X(6533) lies on these lines: {1, 3996}, {2, 1089}, {8, 3892}, {10, 244}, {75, 3624}, {190, 5506}, {274, 1111}, {333, 3337}, {341, 1698}, {443, 4680}, {596, 756}, {872, 3216}, {1125, 1962}, {3338, 4384}, {3634, 3992}, {3828, 4696}, {3881, 4651}, {4858, 4999}, {5433, 6358}
X(6533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1125,4359,4647), (1125,4647,4975), (3634,4968,3992)
See X(6374).
X(6534) lies on these lines: {2, 596}, {30, 511}, {551, 3159}
See X(6375).
X(6535) lies on these lines: {10, 3995}, {42, 2321}, {321, 2887}, {594, 756}, {968, 4873}, {1211, 3994}, {1215, 3969}, {1962, 3943}, {2345, 5311}, {3836, 4980}, {3896, 4527}, {4058, 4082}, {4368, 4651}, {4439, 4981}
X(6535) = X(594)-Ceva conjugate of X(762)
X(6535) = barycentric cube of X(10)
X(6535) = X(i)-isoconjugate of X(j) for these {i,j}: {6,763}, {58,757}, {60,1014}, {81,593}, {86,849}, {261,1408}, {552,2194}, {873,2206}, {1019,4556}, {1333,1509}, {1412,2185}, {1434,2150}}
X(6535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (594,6057,756), (1215,3969,4062), (1215,4535,3969)
See X(6375).
X(6536) lies on these lines: {1, 2895}, {2, 846}, {10, 3995}, {38, 3122}, {42, 4104}, {86, 4683}, {748, 4657}, {756, 4026}, {1211, 1962}, {1213, 4037}, {1284, 4204}, {1473, 4423}, {2292, 4205}, {2486, 3925}, {3616, 5429}, {3624, 3648}, {3706, 4708}, {3720, 4357}, {3724, 4199}, {3842, 4972}, {3914, 5257}
X(6536) = X(4632)-Ceva conjugate of X(4988)
X(6536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4425,3120), (1211,1962,4062)
See X(6375).
X(6537) lies on these lines: {2, 1171}, {10, 115}, {39, 5743}, {44, 1213}, {594, 4075}, {626, 5224}, {960, 5164}, {966, 3767}, {1211, 3912}, {2238, 5280}, {3124, 3954}, {3841, 4047}
X(6537) = complement of X(1509)
X(6537) = X(i)-complementary conjugate of X(j) for these (i,j): (37,3741), (41,4999), (42,3739), (181,142), (213,1125), (594,2887), (756,141), (762,3454), (798,244), (813,4155), (872,2), (1018,512), (1089,626), (1110,620), (1334,960), (1400,3742), (1402,3946), (1500,10), (1918,3666), (2171,2886), (2333,942), (3949,1368), (4079,11), (4557,4369), (4705,116), (6378,3840)
See X(6375).
X(6538) lies on these lines: {10, 3995}, {12, 6058}, {80, 3626}, {313, 4066}, {519, 1126}, {594, 4075}, {1125, 1224}, {1268, 3634}, {1826, 4058}, {3773, 3841}
X(6538) = isotomic conjugate of X(30593)
X(6538) = X(i)-isoconjugate of X(j) for these {i,j}: {553,2150}, {593,1100}, {757,2308}, {849,1125}, {4556,4979}
See X(6375).
Let A13B13C13 be the Gemini triangle 13. Let LA be the line through A13 parallel to BC, and define LB and LC cyclically. Let A′13 = LB∩LC, and define B′13 and C′13 cyclically. Triangle A′13B′13C′13 is homothetic to ABC at X(6539). (Randy Hutson, November 30, 2018)
X(6539) lies on the Kiepert hyperbola and these lines: {2, 594}, {4, 3617}, {8, 1126}, {10, 3995}, {1029, 1654}, {1211, 4080}, {4033, 4359}, {4444, 4608}
X(6539) = isotomic conjugate of X(8025)
X(6539) = X(i)-cross conjugate of X(j) for these (i,j): (10,1268), (514,4033), (4024,3952)
X(i)-isoconjugate of X(j) for these {i,j}: {58,1100}, {81,2308}, {110,4979}, {163,4977}, {553,2194}, {593,1962}, {849,1213}, {1125,1333}, {1408,3686}, {1412,3683}, {1437,1839}, {1474,3916}, {1576,4978}, {1790,2355}, {2150,3649}, {2203,4001}, {2206,4359}, {4556,4983}
X(6539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1255,1268,2), (1268,4102,1255)
See X(6375).
X(6540) lies on the Steiner circumellipse and these lines: {99, 3952}, {190, 4103}, {645, 4629}, {668, 4756}, {903, 1268}, {1016, 4115}, {1121, 4102}, {1126, 3226}, {1255, 3227}, {1509, 4075}, {4555, 4608}
X(6540) = isotomic conjugate of X(4977)
X(6540) = complement of X(39348)
X(6540) = anticomplement of X(35076)
X(6540) = X(i)-cross conjugate of X(j) for these (i,j): (10,1016), (190,4632), (319,4998), (4608,1268), (4977,2)
X(6540) = X(i)-isoconjugate of X(j) for these (i,j): {6,4979}, {31,4977}, {32,4978}, {58,4983}, {513,2308}, {553,3063}, {604,4976}, {649,1100}, {667,1125}, {849,6367}, {875,4974}, {1106,4990}, {1269,1980}, {1333,4988}, {1397,4985}, {1459,2355}, {1919,4359}, {1962,3733}, {3248,4427}
X(6540) = {X(4596),X(4632)}-harmonic conjugate of X(99)
See X(6375).
X(6541) lies on the Steiner circumellipse and these lines: {1, 3790}, {10, 37}, {192, 3821}, {226, 4135}, {238, 519}, {306, 3971}, {335, 726}, {346, 3923}, {518, 4439}, {523, 4129}, {536, 3836}, {537, 4966}, {752, 4908}, {756, 3969}, {1089, 3178}, {1215, 6057}, {1268, 3634}, {1386, 3244}, {2325, 5847}, {2784, 6211}, {2796, 4645}, {2887, 3175}, {3701, 4710}, {3712, 4434}, {3844, 4681}, {3936, 3994}, {3952, 4062}, {3995, 4425}, {4015, 4111}, {4028, 4082}, {4037, 4071}, {4387, 4865}, {4422, 4974}, {4432, 5846}, {4669, 4923
X(6541) = midpoint of X(3932) and X(3943)
X(6541) = reflection of X(i) in X(j) for these (i,j): (10,3932), (4974,4422)
X(6541) = X(i)-isoconjugate of X(j) for these (i,j): {58,1929}, {757,2054}, {1019,2702}
X(6541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,3950,3993), (10,4072,4133), (10,4133,4709), (37,3773,10), (594,3842,10), (2321,4078,10), (3842,4535,594), (4028,4082,4090)
See X(6375).
X(6542) lies on the cubic K185 and these lines:
{1, 2}, {7, 1278}, {37, 319}, {44, 4473}, {57, 4050}, {63, 3208}, {69, 192}, {75, 4675}, {81, 2295}, {86, 594}, {100, 4433}, {141, 4360}, {190, 524}, {193, 346}, {287, 677}, {297, 1897}, {312, 3765}, {314, 3963}, {320, 536}, {321, 1909}, {335, 740}, {344, 5839}, {514, 4024}, {518, 2113}, {599, 4389}, {668, 3948}, {671, 4080}, {752, 4693}, {894, 2321}, {1016, 1252}, {1086, 4971}, {1100, 4889}, {1213, 4478}, {1330, 2901}, {1334, 3219}, {1655, 2895}, {1931, 6157}, {1959, 3930}, {1978, 3978}, {2112, 3684}, {2345, 4916}, {3620, 3672}, {3662, 3875}, {3664, 4431}, {3681, 4517}, {3685, 5847}, {3739, 5564}, {3751, 3790}, {3770, 4043}, {3773, 4649}, {3836, 4716}, {3888, 6007}, {3946, 4464}, {3950, 4416}, {3975, 4358}, {3994, 4938}, {4060, 4967}, {4366, 4437}, {4422, 4969}, {4445, 5224}, {4643, 4664}, {4648, 4699}, {4774, 4789}
X(6542) = 9 X[2] - 8 X[3008] = 3 X[239] - 4 X[3008] = 3 X[2] - 4 X[3912] = 2 X[3008] - 3 X[3912] = 4 X[44] - 5 X[4473] = X[320] + 2 X[4727] = X[4440] + 4 X[4727]
X(6542) = reflection of X(i) in X(j) for these (i,j): (190,3943), (239,3912), (4440,320), (4716,3836), (4969,4422)
X(6542) = isotomic conjugate of X(6650)
X(6542) = complement of X(20016)
X(6542) = anticomplement of X(239)
X(6542) = anticomplementary conjugate of X(20345)
X(6542) = X(335)-Ceva conjugate of X(2)
X(6542) = inverse-in-Steiner-circumellipse of X(10)
X(6542) = X(2)-Hirst inverse of X(10)
X(6542) = trilinear pole of line X(2786)X(9508) (the perspectrix of ABC and Gemini triangle 19)
X(6542) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (291,69), (292,8), (295,4329), (334,315), (335,6327), (694,4388), (741,75), (813,513), (875,4440), (876,150), (1911,2), (1922,192), (2196,20), (2311,3869), (3572,149), (4584,512), (4876,3436), (5378,668)
X(6542) = X(i)-isoconjugate of X(j) for these (i,j): (6,1929), (81,2054), (513,2702)
X(6542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3661,2), (2,145,4393), (37,319,1654), (69,4419,4741), (192,4741,4419), (239,3912,2), (306,1999,2), (2321,3879,894), (4433,4447,100)
See X(6375).
X(6543) lies on these lines: {10, 115}, {190, 1213}, {313, 338}, {334, 3948}, {1224, 1929}, {1648, 4080}, {2372, 2702}
X(6543) = X(4037)-cross conjugate of X(594)
X(6543) = X(i)-isoconjugate of X(j) for these {i,j}: {58,1931}, {81,1326}, {423,1437}, {593,1757}
See X(6375).
The trilinear polar of X(6544) passes through X(4542). (Randy Hutson, February 16, 2015)
X(6544) lies on the Yff parabola, the cubic K219, and these lines: {2, 514}, {9, 649}, {37, 650}, {121, 5513}, {661, 1213}, {900, 1635}, {1023, 4169}, {1566, 3259}, {1647, 2087}, {3161, 3239}, {3251, 4543}, {4088, 4809}, {4728, 6009}, {4750, 4763}
X(6544) = isogonal conjugate of X(4638)
X(6544) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1647), (190,519), (514,900), (1317,4542)
X(6544) = X(4542)-cross conjugate of X(1317)
X(6544) = tripolar centroid of X(519)
X(6544) = X(i)-complementary conjugate of X(j) for these (i,j):} (31,1647), (44,116), (101,3834), (163,4395), (678,3259), (692,519), (902,11), (1023,141), (1110,900), (1252,4928), (1404,4904), (2251,1086), (3689,124), (3939,5123}
X(6544) = X(i)-isoconjugate of X(j) for these {i,j}: {1,4638), (6,4618}, {88,901}, {100,2226}, {101,679}, {106,3257}, {651,1318}, {1417,4582}, {4591,4674}}
See X(6375).
X(6545) lies on the cubic K656 and these lines: {2, 514}, {57, 649}, {321, 693}, {354, 513}, {561, 3261}, {614, 1027}, {764, 1647}, {812, 4453}, {918, 4120}, {1230, 1577}, {1427, 3669}, {1635, 1638}, {3004, 4988}, {3616, 5592}, {3837, 4088}, {3938, 4449}, {4025, 4382}, {4448, 4977}
X(6545) = reflection of X(i) in X(j) for these (i,j): (1635, 1638), (4120,4728), (4728,4927), (4750, 4453), (4984, 4750)
X(6545) = isotomic conjugate of X(6632)
X(6545) = barycentric cube of X(514)
X(6545) = X(i)-Ceva conjugate of X(j) for these (i,j): {514,1086}, {693,3120}, {3261,1111}, {3676,244}
X(6545) = X(i)-isoconjugate of X(j) for these (i,j): {59,644}, {100,1252}, {101,765}, {190,1110}, {200,4619}, {651,6065}, {692,1016}, {1018,4570}, {1101,4103}, {1262,4578}, {1415,4076}, {2149,3699}, {2284,5377}, {3939,4564}, {4554,6066}, {4557,4567}, {4574,5379}
See X(6375).
X(6546) lies on these lines: {2, 514}, {8, 5592}, {63, 649}, {210, 513}, {522, 3158}, {523, 1962}, {612, 1027}, {650, 3752}, {659, 4088}, {661, 1211}, {663, 3938}, {812, 4120}, {918, 1635}, {1639, 4728}, {1647, 4124}, {1763, 4498}, {2254, 2977}, {2786, 4984}, {3239, 4382}, {3961, 4040}, {3995, 4024}, {4453, 4763}
X(6546) = reflection of X(i) in X(j) for these (i,j): (4453,4763), (4728,1639), (4750,1635)
See X(6375).
X(6547) lies on these lines: {2, 1016}, {11, 4162}, {115, 4129}, {239, 3936}, {244, 4041}, {514, 1086}, {519, 3836}, {905, 1015}, {3120, 4983}, {3271, 6004}
X(6547) = complement of X(1016)
X(6547) = X(2)-Ceva-conjugate of X(4422)
X(6547) = X(i)-complementary conjugate of X(j) for these {i,j}: {9,3038}, {31,4422}, {41,3039}, {244,141}, {512,4129}, {513,3835}, {604,3035}, {649,513}, {667,514}, {764,116}, {798,661}, {849,620}, {875,812}, {1015,10}, {1019,512}, {1086,2887}, {1111,626}, {1357,142}, {1919,650}, {1977,37}, {2087,121}, {2170,1329}, {3063,4521}, {3121,1213}, {3122,1211}, {3125,3454}, {3248,2}, {3249,1015}, {3271,3452}, {3572,3837}, {3733,4369}, {3942,1368}, {4817,788}
See X(6375).
X(6548) lies on the cubic K015), the conic {{A, B, C, X(2), X(7)}, and these lines: {2, 514}, {7, 3676}, {75, 693}, {86, 4833}, {88, 673}, {106, 675}, {335, 4080}, {900, 903}, {901, 927}, {918, 4945}, {1268, 3004}, {1647, 4089}, {1797, 2989}, {4025, 4373}, {4555, 4618}, {4615, 5468}
X(6548) = isogonal conjugate of X(23344)
X(6548) = isotomic conjugate of X(17780)
X(6548) = barycentric quotient X(106)/X(101)
X(6548) = X(4555)-Ceva conjugate of X(903)
X(6548) = X(i)-cross conjugate of X(j) for these (i,j): (900,514), (1647,2), (4089,7), (4927,693)
X(6548) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (679,150), (2226,149), (4618,69), (4638,8)
X(6548) = X(i)-isoconjugate of X(j) for these (i,j): {6,1023}, {44,101}, {59,4895}, {100,902}, {109,3689}, {163,3943}, {190,2251}, {519,692}, {644,1404}, {678,901}, {765,1960}, {900,1110}, {1017,3257}, {1018,3285}, {1252,1635}, {1319,3939}, {1333,4169}, {1415,2325}, {1576,3992}, {1639,2149}, {1743,2429}, {4570,4730}
See X(6375).
X(6549) lies on these lines: {88, 3008}, {106, 927}, {320, 519}, {334, 4013}, {514, 1086}, {693, 1111}, {1358, 3676}, {1647, 4089}, {3912, 4080}, {4049, 4444}
X(6549) = X(1647)-cross conjugate of X(1086)
X(6549) = X(i)-isoconjugate of X(j) for these (i,j): {44,1252}, {59,3689}, {101,1023}, {163,4169}, X(6549) = {519,1110}, {765,902}, {1016,2251}, {1017,5376}, {1319,6065}, {2149,2325}
X(6549) = barycentric product X(903)*X(1086)
See X(6375).
X(6550) lies on these lines: {30, 511}, {36, 659}, {764, 1647}, {1319, 1960}, {2254, 4695}, {3762, 3992}, {3814, 3837}
X(6550) = isogonal conjugate of X(6551)
X(6550) = isotomic conjugate of X(6635)
X(6550) = crossdifference of every pair of points on line X(6)X(1252)
X(6550) = X(i)-Ceva conjugate of X(j) for these (i,j): (513,3259), (900,1647), (2401,1015)
See X(6375).
X(6551) lies on the circumcircle and these lines: {100, 6161}, {105, 1320}, {214, 765}, {840, 4996}, {919, 5548}, {927, 4555}, {929, 4582}, {1016, 1145}, {1017, 1252}, {1110, 2382}, {1308, 3257}, {2757, 4076}
X(6551) = trilinear pole of line X(6)X(1252)
X(6551) = Ψ(X(6), X(1252))
X(6551) = X(i)-cross conjugate of X(j) for these (i,j): (2427,1016), (5548,5376)
X(6551) = X(i)-isoconjugate of X(j) for these (i,j): {244,900}, {513,1647}, {514,2087}, {519,764}, {1015,3762}, {1086,1635}, {1111,1960}, {1357,4768}, {1358,4895}, {3669,4530}
See X(6375).
X(6552) is the centroid of the quadrilateral whose vertices are X(8) together with the vertices of the anticevian triangle of X(8). (Randy Hutson, February 16, 2015)
X(6552) lies on these lines: {2, 6553}, {8, 3452}, {10, 4310}, {145, 3699}, {279, 668}, {341, 346}, {1211, 3617}, {1219, 4737}, {1706, 4454}, {3086, 4738}, {3672, 6376}, {3913, 4578}
X(6552) = complement of X(6553)
X(6552) = X(i)-complementary conjugate of X(j) for these (i,j): (31,346), (1616,10), (2136,1329), (4452,2887)
See X(6375).
X(6553) lies on the cubic K201, the conic {{A,B,C,X(1), X(2)}}, and these lines: {1, 3161}, {2, 6552}, {57, 145}, {81, 3623}, {88, 3621}, {279, 4452}, {330, 4461}, {957, 3555}, {1022, 4962}
X(6553) = isogonal conjugate of X(1616)
X(6553) = isotomic conjugate of X(4452)
X(6553) = anticomplement of X(6552)
X(6553) = X(i)-cross conjugate of X(j) for these (i,j): (346,2), (3680,8)
X(6553) = X(2137)-anticomplementary conjugate of X(3436)
X(6553) = X(i)-isoconjugate of X(j) for these (i,j): {1,1616}, {19,23089}, {31,4452}, {56,2136}
See X(6375).
X(6554) lies on these lines: {2, 85}, {4, 9}, {6, 938}, {8, 220}, {20, 910}, {41, 3486}, {101, 944}, {120, 1329}, {219, 1067}, {277, 1111}, {341, 346}, {344, 6376}, {345, 3975}, {379, 5273}, {497, 2082}, {672, 1788}, {857, 1211}, {958, 4223}, {960, 3789}, {1104, 5304}, {1479, 5540}, {1802, 3189}, {1837, 2348}, {1944, 4644}, {2098, 4534}, {2321, 4882}, {2391, 3452}, {3207, 5731}, {3488, 4251}, {3673, 4000}, {4012, 4319}, {4258, 4313}, {5022, 5435}, {5276, 5716}
X(6554) = isotomic conjugate of X(30705)
X(6554) = complement of X(279)
X(6554) = complementary conjugate of X(21258)
X(6554) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,4000), (668,3900)
X(6554) = X(4319)-cross conjugate of X(497)
X(6554) = X(i)-isoconjugate of X(j) for these (i,j): {57,1037}, {222,1041}
X(6554) = X(i)-complementary conjugate of X(j) for these (i,j): {9,2886}, {31,4000}, {41,1}, {55,142}, {101,3900}, {200,141}, {213,1834}, {220,10}, {284,3742}, {341,626}, {346,2887}, {480,3452}, {604,5573}, {607,1210}, {657,11}, {663,4904}, {728,1329}, {1174,5572}, {1253,2}, {1334,442}, {1802,3}, {2175,3752}, {2194,3946}, {2212,3772}, {2287,3741}, {2328,3739}, {2332,942}, {3063,3756}, {3692,1368}, {3900,116}, {3939,4885}, {4130,124}, {4171,125}, {4515,3454}, {4578,3835}, {4936,2885}
X(6554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3177,348), (4,169,5819), (9,281,2345), (169,5179,4), (220,1146,8).
See X(6375).
X(6555) lies on these lines: {2, 1280}, {8, 3452}, {55, 4152}, {145, 4487}, {200, 346}, {210, 391}, {936, 1219}, {3158, 3161}, {3434, 4767}, {3711, 3974}, {4090, 4307}, {4126, 5218}, {4163, 4543}, {4899, 5435}
X(6555) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,346), (3699,4521)
X(6555) = X(i)-cross conjugate of X(j) for these (i,j): (4162,4578), (4936,3161), (4953,4546)
X(6555) = X(i)-isoconjugate of X(j) for these {i,j}: {269,3445}, {1106,4373}
X(6555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (200,5423,346)
See X(6375).
X(6556) lies on these lines: {2, 1222}, {8, 3452}, {10, 1219}, {20, 1293}, {75, 3617}, {145, 1997}, {253, 3007}, {318, 4723}, {3621, 3699}, {3679, 4052}
X(6556) = X(i)-cross conjugate of X(j) for these (i,j): (11,4397), (4953,3239), (5423,346)
X(6556) = X(i)-isoconjugate of X(j) for these {i,j}: {56,1420}, {145,1106}, {269,3052}, {604,5435}, {1398,4855}, {1407,1743}, {1408,4848}, {1410,4248}, {1461,4394}
See X(6375).
Let A27B27C27 be the Gemini triangle 27. Let LA be the line through A27 parallel to BC, and define LB and LC cyclically. Let A′27 = LB∩LC, and define B′27 and C′27 cyclically. Triangle A′27B′27C′27 is homothetic to ABC at X(6557). (Randy Hutson, November 30, 2018)
X(6557) lies on these lines: {2, 2415}, {7, 1997}, {8, 3452}, {11, 5423}, {85, 5226}, {92, 4358}, {178, 5430}, {189, 908}, {312, 5328}, {345, 4997}, {1121, 4417}, {1220, 2899}, {1293, 1311}, {4102, 5233}, {4488, 5435}, {4518, 4903}
X(6557) = isotomic conjugate of X(5435)
X(6557) = X(4373)-Ceva conjugate of X(8)
X(6557) = X(i)-cross conjugate of X(j) for these (i,j): (346,8), (1086,4391), (3680,4373), (4534,522)
X(6557) = X(i)-isoconjugate of X(j) for these (i,j): {6,1420}, {31,5435}, {56,1743}, {57,3052}, {109,4394}, {145,604}, {608,4855}, {1106,3161}, {1333,4848}, {1407,3158}, {1408,3950}, {1409,4248}, {1412,4849}, {1415,3667}, {1461,4162}, {2149,3756}, {4565,4729}}
See X(6375).
X(6558) lies on these lines: {8, 4534}, {101, 4169}, {190, 646}, {341, 728}, {346, 1146}, {644, 1639}, {1016, 4561}, {1043, 4515}, {3039, 3161}, {4103, 4752}, {4528, 4578}
X(6558) = X(i)-Ceva conjugate of X(j) for these (i,j): (646,3699), (1016,1265), (4076,5423)
X(6558) = X(i)-cross conjugate of X(j) for these (i,j): (657,200), (3239,346), (3900,1043), (4163,341), (4546,8), (4578,3699), (5423,4076}
X(6558) = X(i)-isoconjugate of X(j) for these (i,j): {56,3669}, {244,1461}, {269,649}, {279,667}, {479,3063}, {513,1407}, {514,1106}, {604,3676}, {651,1357}, {658,3248}, {663,738}, {764,1262}, {905,1398}, {934,1015}, {1019,1042}, {1088,1919}, {1358,1415}, {1412,4017}, {1427,3733}, {1435,1459}, {1977,4569}, {3121,4616}, {3122,4637}, {3271,4617}
See X(6375).
X(6558) lies on these lines: {8, 220}, {9, 75}, {105, 958}, {341, 728}, {346, 480}, {666, 2338}, {919, 2370}, {927, 2371}, {1222, 1438}, {1462, 5749}, {3119, 3699}
X(6559) = X(i)-isoconjugate of X(j) for these (i,j): {56,241}, {57,1458}, {222,1876}, {269,672}, {279,2223}, {518,1407}, {603,5236}, {665,934}, {738,2340}, {926,4617}, {1106,3912}, {1262,3675}, {1362,1462}, {1427,3286}, {1435,1818}, {1461,2254}, {2283,3669}
Contributed by Randy Hutson, February 9, 2015.
X(6560) lies on these lines: {2,1327}, {3,485}, {4,372}, {5,1152}, {6,30}, {20,371}, {490,3096}, {550,1151}, {1131,3523}
X(6560) = reflection of X(6561) in X(6)
Contributed by Randy Hutson, February 9, 2015.
X(6561) lies on these lines: {2,1328}, {3,486}, {4,371}, {5,1151}, {6,30}, {20,372}, {489,3096}, {550,1152}, {1132,3523}
X(6561) = reflection of X(6560) in X(6)
X(6561) = homothetic center of inner Vecten and 3rd anti-tri-squares triangles
The perspectrix of any pair of {ABC, outer Vecten triangle, Lucas(-1) antipodal triangle} is the trilinear polar of X(3068). The perspectrix of any pair of {ABC, inner Vecten triangle, Lucas antipodal triangle} is the trilinear polar of X(3069). The intersection of these two perspectrices is X(6562). (Randy Hutson, February 9, 2015)
X(6562) lies on these lines: {23,385}, {804,3265}, {3566,6563}
X(6562) = radical center of {nine-point circle, outer Vecten circle, inner Vecten circle}
X(6562) = crossdifference of every pair of points on line X(39)X(493)
Contributed by Randy Hutson, February 9, 2015.
X(6563) lies on these lines: {2,2501}, {323,401}, {325,523}, {3566,6562}
X(6563) = isogonal conjugate of X(32734)
X(6563) = isotomic conjugate of X(925)
X(6563) = anticomplement of X(2501)
X(6563) = crossdifference of every pair of points on line X(32)X(51)
X(6563) = radical center of {circumcircle, inner Vecten circle, outer Vecten circle}
X(6563) = pole, wrt de Longchamps circle, of line X(2)X(6)
X(6563) = de-Longchamps-circle-inverse of X(38940)
X(6563) = barycentric product X(523)*X(7763)
Let A′B′C′ be the outer Vecten triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6564). (Randy Hutson, February 9, 2015)
X(6564) lies on these lines: {2,1327}, {3,3366}, {4,371}, {5,372}, {6,13}, {20,5418}, {30,590}, {382,1151}, {486,1131}, {1152,1656}
X(6564) = {X(6),X(381)}-harmonic conjugate of X(6565)
X(6564) = homothetic center of 1st Kenmotu diagonals triangle and Ehrmann vertex-triangle
Let A′B′C′ be the inner Vecten triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6565). (Randy Hutson, February 9, 2015)
X(6565) lies on these lines: {2,1328}, {3,3367}, {4,372}, {5,371}, {6,13}, {20,5420}, {30,615}, {382,1152}, {485,1132}, {1151,1656}
X(6565) = homothetic center of 2nd Kenmotu diagonals triangle and Ehrmann vertex-triangle
X(6565) = {X(6),X(381)}-harmonic conjugate of X(6564)
X(6566) is the intersection of the following lines: (1) Brocard axis, (2) the trilinear polar of the cevapoint of X(6) and X(1151) (i.e., X(493)), (3) the line through the X(6)-Ceva conjugate of X(1151) and the X(1151)-Ceva conjugate of X(6). (Randy Hutson, February 9, 2015)
X(6566) lies on these lines: {3,6}, {316,3593}, {625,6460}, {638,3788}, {641,3070}
X(6566) = isogonal conjugate of X(6568)
X(6566) = perspector of ABC and the reflection of the Lucas inner triangle in the Lemoine axis
X(6566) = perspector of the Lucas(-1) central triangle and the reflection of the Lucas central triangle in the Lemoine axis
X(6566) = perspector of the Lucas(-1) tangents triangle and the reflection of the Lucas tangents triangle in the Lemoine axis
X(6566) = inverse-in-Lucas(-1)-radical-circle of X(3)
X(6566) = inverse-in-Lucas(-1)-inner-circle of X(1152)
X(6566) = inverse-in-Lucas(-1)-secondary-radical-circle of X(6445)
X(6566) = inverse-in-Lucas-secondary-tangents-circle of X(6519)
X(6566) = {X(3),X(1570)}-harmonic conjguate of X(6567)
X(6566) = {X(1379),X(1380)}-harmonic conjugate of X(1151)
X(6566) = reflection of X(6567) in X(187)
X(6566) = crossdifference of every pair of points on line X(523)X(3068)
X(6566) = radical trace of circumcircle and 4th Lozada circle
X(6567) is the intersection of the following lines: (1) Brocard axis, (2) the trilinear polar of the cevapoint of X(6) and X(1152) (i.e., X(494)), (3) the line through the X(6)-Ceva conjugate of X(1152) and the X(1152)-Ceva conjugate of X(6). (Randy Hutson, February 9, 2015)
X(6567) lies on these lines: {3,6}, {316,3595}, {625,6459}, {637,3788}, {642,3071}
X(6567) = isogonal conjugate of X(6569)
X(6567) = perspector of ABC and the reflection of the Lucas(-1) inner triangle in the Lemoine axis
X(6567) = perspector of the Lucas central triangle and the reflection of the Lucas(-1) central triangle in the Lemoine axis
X(6567) = perspector of the Lucas tangents triangle and the reflection of the Lucas(-1) tangents triangle in the Lemoine axis
X(6567) = inverse-in-Lucas-radical-circle of X(3)
X(6567) = inverse-in-Lucas-inner-circle of X(1151)
X(6567) = inverse-in-Lucas-secondary-radical-circle of X(6446)
X(6567) = inverse-in-Lucas(-1)-secondary-tangents-circle of X(6522)
X(6567) = {X(3),X(1570)}-harmonic conjguate of X(6566)
X(6567) = {X(1379),X(1380)}-harmonic conjugate of X(1152)
X(6567) = reflection of X(6566) in X(187)
X(6567) = crossdifference of every pair of points on line X(523)X(3069)
X(6567) = radical trace of circumcircle and 3rd Lozada circle
X(6568) lies on the Kiepert hyperbola and these lines: {99,5490}, {115,1131}
Contributed by Randy Hutson, February 9, 2015.
X(6568) = isogonal conjugate of X(6566)
X(6568) = antipode in Kiepert hyperbola of X(1131)
X(6568) = reflection of X(1131) in X(115)
X(6568) = trilinear pole of line X(523)X(3068)
Contributed by Randy Hutson, February 9, 2015.
X(6569) lies on the Kiepert hyperbola and these lines: {99,5491}, {115,1132}
X(6569) = isogonal conjugate of X(6567)
X(6569) = antipode in Kiepert hyperbola of X(1132)
X(6569) = reflection of X(1132) in X(115)
X(6569) = trilinear pole of line X(523)X(3069)
Circumeigencenters (CET): X(6570)-X(6577)
The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the circumcevian triangle of a point U = u : v : w (trilinears), the eigencenter is given by
avw/(av2 - aw2 + buv - cuw) : bwu/(bw2 - bu2 + cvw - avu) : cuv/(cu2 - cv2 + awu - bwv)
This point is here named the circum-eigentransform of U, denoted by CET(U). The point CET(U) lies on the circumcircle. Pairs (i,j) such that CET(X(i)) = X(j) include the following:
(1, 101), (2, 99), (3, 112), (4, 6570), (6, 99), (9, 6571), (15, 5618), (16, 5619), (25, 99), (31, 839), (32, 6572), (37, 99), (39, 6573), (42, 99), (55, 934), (56, 6574), (57, 6575), (75, 773), (76, 6576), (111, 99), (187, 99), (237, 99), (241, 934), (251, 99), (263, 99), (308, 99), (351, 99), (393, 99), (493, 99), (494, 99), (512, 99), (513, 101), (514, 6577), (523, 112), (588, 99), (589, 99), (647, 99), (649, 99), (650, 934), (663, 99), (665, 99), (667, 99), (669, 99), (694, 99), (887, 99), (890, 99), (902, 99), (941, 99), (967, 99), (1055, 99), (1169, 99), (1171, 99), (1218, 99), (1239, 99), (1241, 99), (1383, 99), (1400, 99), (1427, 99), (1495, 99), (1880, 99), (1946, 99), (1960, 99), (1976, 99), (1989, 99), (2054, 99), (2165, 99), (2223, 99), (2248, 99), (2350, 99), (2395, 99), (2433, 99), (2488, 99), (2502, 99), (2574, 112), (2575, 112), (2590, 934), (2591, 934), (2963, 99), (2978, 99), (2981, 99), (2987, 99), (2998, 99), (3005, 99), (3009, 99), (3010, 99), (3016, 99), (3108, 99), (3228, 99), (3229, 99), (3230, 99), (3231, 99), (3250, 99), (3288, 99), (3331, 99), (3444, 99), (3457, 99), (3458, 99), (3513, 101), (3514, 101), (3569, 99), (3572, 99), (3724, 99), (3747, 99), (3804, 99), (4775, 99), (4834, 99), (5027, 99), (5029, 99), (5040, 99), (5075, 99), (5098, 99), (5106, 99), (5113, 99), (5147, 99), (5163, 99), (5167, 99), (5168, 99), (5191, 99), (5202, 99), (5638, 99), (5639, 99), (6094, 99), (6096, 99), (6137, 99), (6138, 99), (6139, 99), (6140, 99), (6151, 99), (6339, 99)
Peter Moses observes (January, 2015) that CET(U) and CIR(U), discussed in the preamble to X(2365), are special cases of points defined by 1st trilinear of the form
avw/(jav2 + kaw2 - kbuv - jcuw).
Taking (j, k) = (1,1) gives CIR(U), and taking (j, k) = (1,-1) gives CET(U). Also, the locus of X for which CET(X) = X(99) is the union of the line X(187)X(237) and the conic {{A,B,C,X(2),X(6)}}.
If U is given by barycentrics u : v : w, then barycentrics for CET(U) are f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a2vw/(a2c2v2 - a2b2w2 + b2c2uv - b2c2uw).
X(6570) lies on the circumcircle and these lines: {98, 1181}, {107, 1625}
X(6570) = isoconjugate of X(3900) and X(1577)
X(6570) = trilinear pole of the line X(6)X(418)
X(6570) = Ψ(X(6), X(418))
X(6571) lies on the circumcircle.
X(6571) = isogonal conjugate of X(8710)
X(6571) = isoconjugate of X(3900) and X(4308)
X(6572) lies on the circumcircle and these lines: {111, 1239}, {670, 827}
X(6572) = Ψ(X(6), X(1239))
X(6572) = trilinear pole of the line X(6)X(1239)
X(6572) = X(i)-isoconjugate of X(j) for these (i,j): {798,1180}, {1924,3096}
X(6573) lies on the circumcircle and these lines: {83, 699}, {111, 1799}, {733, 1078}
X(6573) = isogonal conjugate of X(8711)
X(6573) = X(4576)-cross conjugate of X(4577)
X(6573) = trilinear pole of line X(6)X(6664)
X(6573) = Ψ(X(6), X(6664))
X(6574) lies on the circumcircle and these lines: {101, 4578}, {105, 958}, {106, 2297}, {109, 644}, {190, 934}, {1018, 1293}
X(6574) = isogonal conjugate of X(8712)
X(6574) = trilinear pole of the line X(6)X(200)
X(6574) = X(4512)-cross conjugate of X(765)
X(6574) = Ψ(X(6), X(200))
X(6574) = eigencenter of 3rd mixtilinear triangle
X(6574) = X(i)-isoconjugate of X(j) for these {i,j}: {513,2999}, {514,1191}, {649,3672}, {1019,4646}, {1697,3669}, {3733,4656}
X(6575) lies on the circumcircle and these lines: {103,5584}, {105,3601}
X(6575) = isogonal conjugate of X(8713)
X(6575) = trilinear pole of line X(6)X(8012)
X(6575) = Ψ(X(6), X(8012))
X(6576) lies on the circumcircle.
X(6577) lies on the circumcircle and these lines: {105, 595}, {1918, 2368}, {3939, 6013}
X(6577) = isogonal conjugate of X(8714)
X(6577) = X(4040)-cross conjugate of X(58)
X(6578) lies on the circumcircle and these lines: {100, 4596}, {101, 4556}, {111, 1171}, {249, 2702}, {476, 4608}, {835, 4632}, {1268, 2372}
X(6578) = isogonal conjugate of X(6367)
X(6578) = trilinear pole of line X(6)X(593)
X(6578) = Ψ(X(6), X(593))
X(6578) = X(i)-cross conjugate of X(j) for these (i,j): (58,249), (110,4629), (4184,250))
X(i)-isoconjugate of X(j) for these (i,j): {1,6367}, {10,4983}, {37,4988}, {181,4985}, {430,656}, {512,4647}, {523,1962}, {594,4979}, {661,1213}, {756,4977}, {798,1230}, {1100,4024}, {1125,4705}, {1254,4990}, {1500,4978}, {2171,4976}, {2308,4036}, {2355,4064}, {2501,3958}, {2643,4427}, {3125,4115}, {3649,4041}, {4017,4046}, {4079,4359}
X(6579) lies on the circumcircle and these lines: {32, 689}, {83, 3222}
The 1st Shadow Point is X(1480). See Antreas P. Hatzipolakis and Paul Yiu, Pedal Triangles and Their Shadows, Forum Geometricorum 1 (2001) 81-90. (X(6580) is point M′ on page 88; the trilinears have been corrected here.)
Suppose that t is a number (or symmetric function of A,B,C) and let f(A,B,C,k) = (t + cos A)/(k + cos A - cos B - cos C). Let P(j) = f(A,B,C,j) : f(B,C,A,j) : f(C,A,B,j) (trilinears). The appearance of (j,k) in the following list means that P(j) lies on a line X(1)X(k): (-3,5920), (-1,6), (0,90), (1,84), (2,1406), (3,1407). (Peter Moses, February 9, 2015)
X(6580) lies on these lines: {1,1406}, {56,4550}, {999,3426}, {4846,5434}
Barycentrics for X(6581) are obtained from X(6294) by the transformation S → -S. (Richard Hilton, February 9, 2015)
X(6581) lies on these lines: {2, 39}, {524, 3104}, {698, 3107}, {732, 3106}, {3095, 5613}, {3105, 5464}, {3642, 6296}
X(6581) = reflection of X(6294) in X(39)
X(6581) = anticomplement of X(33482)
Barycentrics for X(6582) are obtained from X(6295) by the transformation S → -S. (Richard Hilton, February 9, 2015)
X(6582) lies on these lines: {2, 13}, {542, 6299}, {3098, 5969}, {5617, 6287}
X(6582) = reflection of X(6298) in X(618)
X(6582) = circumtangential-isogonal conjugate of X(15)
X(6582) = {X(6302),X(6306)}-harmonic conjugate of X(3643)
In the plane of ABC, let P be a point, and let A′B′C′ be the pedal triangle of P. Let AB be the reflection of A in B′, and define BC and CA cyclically. Let AC be the reflection of A in C′, and define BA and CB cyclically. Let NA be the nine-point center of AABAC, and define NB and NC cyclically. The nine-point center of NANBNC is here named the Hatzipolakis-Lozada Nine-Point Image of P. See Hyacinthos 23083. (Antreas Hatzipolakis and César Lozada, February 2, 2015)
X(6583) lies on these lines: {1,3}, {5,3874}, {355,3873}, {546,2801}, {758,5901}, {912,5448}, {946,1484}, {952,3881}, {1393,5399}, {1483,3892}, {1656,5904}, {3628,3678}, {3754,5844}, {3868,5694}, {4430,5818}, {5690,5883}
X(6583) = midpoint of X(i) and X(j) for these {i,j}: {5,3874}, {3868,5694}
X(6583) = reflection of X(i) in X(j) for these (i,j): (3678, 3628), (5885,942)
Let I = X(1), and let NA be the nine-point center of IBC. Define NB and NC cyclically. Let KA = X(54)-of-IBC; i.e,, the Kosnita point of IBC, and define KB and KC cyclically. The circumcircles of the four triangles ABC, AKBKC, BKCKA, ACAKB concur in X(6584); their radical center is X(484). Also, KA = IBC-isogonal conjugate of NA, and the points KA, KB, KC lie on the line X(1)X(3). See Hyacinthos 23086. (Antreas Hatzipolakis and César Lozada, February 1, 2015)
X(6584) lies on the circumcircle and this line: {1385,2687}
The construction below for X(6585) is a tangential triangle variation of the construction described by Pavel Dolgirev: Interest circle: open problem. Let A′B′C′ be the Gergonne triangle of ABC; that is, the cevian triangle of X(7)). Draw tangent lines from the vertices A,B,C to the incircle of A′B′C′ and let A1, A2, B1, B2, C1, C2 be the points of intersection of these tangent lines with the sides of the triangle ABC. These 6 points lie on a circle, of which the center is X(6585). See Hyacinthos 23112. (Antreas Hatzipolakis and César Lozada, February 1, 2015)
X(6585) lies on this line: {1,167}, {40,1130}
X(6585) = midpoint of X(8091) and X(8351)X(6586) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
X(6586) lies on these lines: {2, 2412}, {6, 657}, {37, 522}, {230, 231}, {513, 665}, {649, 4057}, {667, 2483}, {798, 834}, {905, 918}, {1919, 5029}, {2484, 3733}, {2605, 3063}, {4526, 4926}
X(6586) = isotomic conjugate of X(31624)X(6587) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
Let A′B′C′ be the orthic triangle. Let LA be the orthic axis of AB′C′, and define LB and LC cyclically. Let A″ = LB∩ LC, B″ = LC∩ LA, C″ = LA∩ LB. The triangle A″B″C″ is inversely similar to ABC, with similitude center X(6), and X(6587) = (orthic axis of ABC)∩(orthic axis of A″B″C″. Also, the trilinear polar of X(6587) passes through X(1562). (Randy Hutson, February 20, 2015) X(6587):
X(6587) lies on these lines: {2,2419}, {6,2430}, {230,231} et al.
X(6587) = complement of X(3265)X(6588) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
As a line L varies through the set of all lines that pass through X(123), the locus of the trilinear pole of L is a circumconic, and its center is X(6588). (Randy Hutson, February 20, 2014)
X(6588) lies on these lines: {6,2431}, {230,231} et al.
X(6588) = complement of X(35518)X(6589) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
As a line L varies through the set of all lines that pass through X(124), the locus of the trilinear pole of L is a circumconic, and its center is X(6589). (Randy Hutson, February 20, 2014)
X(6589) lies on these lines: {230,231}, {649,834} et al.
X(6589) = X(2)-Ceva conjugate of X(124)X(6590) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
X(6590) lies on these lines: {230,231} et al.
X(6590) = isotomic conjugate of X(37215)X(6591) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
As a line L varies through the set of all lines that pass through X(5521), the locus of the trilinear pole of L is a circumconic, and its center is X(6591). (Randy Hutson, February 20, 2014)
X(6591) lies on these lines: {19,4394}, {230,231}, {513,2201}, {661,663}, {693,905}
X(6591) = isogonal conjugate of X(1332)Let U = A′B′C′ be the medial triangle and X = X(5). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6592) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(137) . See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
X(6592) lies on the cubics K038, K067 and these lines: {2,1263}, {3,2888}, {5,930}, {30,128}, {137,3628} et al.
X(6592) = midpoint of X(5) and X(930)Let U = A′B′C′ be the medial triangle and X = X(6). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6593) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(125) . See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
Let A′B′C′ be the cevian triangle of X(23). Let A″ be the inverse-in-circumcircle of A′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6593). Let P and U be the circumcircle intercepts of the perpendicular to the Euler line through X(2); then X(6593) = crosssum of P and U. Also, X(6593) is the QA-P41 center (Involutary Conjugate of QA-P4) of quadrangle ABCX(2). (Randy Hutson, February 20, 2015)
X(6593) lies on the bicevian conic of X(2) and X(110), on the cubics K042, K043, K565, and on these lines: {2,67}, {3,1177}, {5,542}, {6,110}, {74,5085}, {113,1503}, {125,3589}, {141,5972} et al.
X(6593) = midpoint of X(6) and X(110)Let U = A′B′C′ be the medial triangle and X = X(9). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6594) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(11). See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.
X(6594) lies on these lines: {1,3939}, {2,3254}, {3,2801}, {9,100}, {10,528}, {11,6666}, {142,3035} et al.
X(6594) = midpoint of X(9) and X(100)Let I = X(1), the incenter of a triangle ABC. Let LA be the Euler line of triangle IBC, and define LB and LC cyclically. These lines concur in S = X(21), the Schiffler point of ABC. Let C2 be the point, other than S, of intersection of the line LB and the circle (C, |CS|). Let B3 be the point, other than S, of intersection of the line LC and the circle (B, |BS|). Let O1 be the circumcenter of SC2B3, and define O2 and O3 cyclically. The triangle O1O2O3 is here named the 1st Schiffler triangle. The triangles ABC and O1O2O3 are cyclologic, and X(6595) is the (ABC, O1O2O3)-cyclologic center. See X(6596)-X(6599) and Hyacinthos 23098 and Hyacinthos 23101. (S. Kirikami, A. Hatzipolakis and A.Montesdeoca, February 4-5, 2015)
The points O1, O2, O3 lie on the Feuerbach hyperbola. (Randy Hutson, September 14, 2016)
X(6595) lies on the Feuerbach hyperbola and these lines: {104, 6584}, {1389, 2771}
Continuing from X(6595), let W = X(3065). Let C′2 be the point, other than S, of intersection of the Euler line of the triangle WCA and the circle (C, |CS|). Let B′3 be the point, other than S, of intersection of the Euler line of the triangle WAB and the circle (B, |BS|). Let O′1 be the circumcenter of SC2B3, and define O′2 and O′3 cyclically. The triangle O′1O′2O′3 is here named the 2nd Schiffler triangle.The triangles ABC, O1O2O3, and O′1O′2O′3 and pairwise cyclologic:
X(6595) = (ABC, O1O2O3)-cyclologic center
X(3065) = (O1O2O3, ABC)-cyclologic center
X(1320) = (ABC, O′1O′2O′3)-cyclologic center
X(1) = (O′1O′2O′3, ABC)-cyclologic center
X(6596) = (O1O2O3, O′1O′2O′3)-cyclologic center
X(6597) = (O′1O′2O′3, O1O2O3)-cyclologic center
The points O′1, O′2, O′3 lie on the Feuerbach hyperbola. (Randy Hutson, September 14, 2016)
See Hyacinthos 23101. (S. Kirikami, A. Hatzipolakis and A.Montesdeoca, February 4-5, 2015)
The 2nd Schiffler triangle is also the reflection of X(100) in every point of the excenters-midpoints triangle. (Randy Hutson, July 11, 2019)
X(6596) lies on the Feuerbach hyperbola and these lines: {4, 6326}, {11, 6598}, {84, 2771}, {104, 758}, {1389, 2802}, {1476, 5083}, {3065, 4867}, {3158, 3577}, {3467, 5692}, {5559, 5854}
See X(6596).
X(6597) lies on the Feuerbach hyperbola and these lines: {4, 5535}, {11, 6599}, {80, 191}, {758, 1389}
See X(6596).
X(6598) lies on the Feuerbach hyperbola and these lines: {1, 442}, {4, 758}, {7, 2475}, {9, 1837}, {10, 943}, {11, 6596}, {21, 950}, {30, 84}, {72, 80}, {90, 191}, {104, 3651}, {226, 5086}, {355, 3577}, {519, 1389}, {1172, 2323}, {1896, 5081}, {2346, 5178}, {2997, 5016}, {3649, 5665}, {3680, 4863}, {3880, 5559}, {5436, 5705}
In addition to the notes at X(6596), Angel Montesdeoca notes in Hyacinthos 23101 that the following 15 points lie on the Feuerbach hyperbola: O1, O2, O3, O′1, O′2, O′3, the cyclologic ceners X(1), X(1320), X(3065), X(6595)-X(6599), and S.
X(6599) lies on the Feuerbach hyperbola and these lines: {5, 12660}, {11, 6597}, {90, 12769}, {100, 12639}, {5046, 12543}, {5443, 12524}, {6265, 12600}, {7161, 12623}, {7491, 12845}, {10950, 12657}
Let V = X(9), and let VA,VB,VC be the vertices of the anticevian triangle of V. Then X(6600) is the centroid of {V, VA, VB, VC}. (Randy Hutson, February 20, 2015)
X(6600) lies on these lines: {2, 2346}, {3, 518}, {6, 3939}, {7, 100}, {9, 55}, {10, 1001}, {21, 5686}, {35, 5223}, {56, 3243}, {119, 381}, {142, 1376}, {214, 999}, {218, 4878}, {219, 949}, {281, 1863}, {390, 2478}, {405, 3189}, {442, 954}, {516, 5812}, {521, 3126}, {527, 4421}, {1445, 1617}, {1466, 4321}, {1486, 4557}, {2136, 3303}, {2177, 4343}, {2223, 5120}, {2271, 3774}, {2951, 5537}, {3161, 4578}, {3358, 5534}, {3423, 4712}, {3647, 5220}, {5432, 6067}, {5759, 5857}
X(6600) = complement of X(6601)
Let AkBkCk be the cevian triangle of X(k) for k = 7, 8, 9, respectively. Let A′ = {A7,A8}-harmonic conjugate of A9, and define B′ and C′ cyclically. Then A′B′C′ is the cevian triangle of X(6601). (Randy Hutson, February 20, 2015)
X(6601) lies on these lines: {1, 142}, {2, 2346}, {4, 518}, {7, 3434}, {8, 1229}, {9, 497}, {11, 480}, {21, 390}, {55, 6067}, {69, 2481}, {80, 3421}, {84, 516}, {90, 5698}, {104, 376}, {144, 149}, {219, 294}, {278, 1041}, {388, 3243}, {517, 3427}, {519, 3577}, {521, 885}, {527, 3062}, {943, 1001}, {1000, 3880}, {1476, 4190}, {1479, 5223}, {2478, 5686}, {2551, 4866}, {2900, 3475}, {3059, 4863}, {3296, 5880}, {3826, 3913}, {4012, 4858}
X(6601) = isogonal conjugate of X(1617)
X(6601) = isotomic conjugate of X(6604)
X(6601) = complement of X(7674)
X(6601) = anticomplement of X(6600)
X(6601) = X(19)-isoconjugate of X(23144)
X(6601) = perspector of ABC and the reflection of the cevian triangle of X(7) in the centroid of {A,B,C,X(7)}
X(6602) lies on these lines: {9, 1174}, {37, 2266}, {41, 55}, {101, 165}, {200, 3119}, {218, 1202}, {219, 604}, {1212, 3748}, {2171, 2324}, {2187, 3690}, {2329, 5273}, {3996, 6559}
X(6602) = barycentric cube of X(9)
X(6603) is the point of intersection of the line X(1)X(9) and the trilinear polar of X(9); c.f. X(1323). (Randy Hutson, April 11, 2015)
X(6603) lies on these lines: {1, 6}, {21, 4520}, {40, 3207}, {41, 3057}, {65, 1802}, {78, 4513}, {101, 517}, {145, 6554}, {169, 1482}, {210, 4390}, {214, 6184}, {241, 1252}, {294, 1320}, {519, 1146}, {527, 1323}, {536, 1944}, {544, 5074}, {644, 3693}, {650, 663}, {672, 1319}, {908, 5723}, {952, 5179}, {1018, 5440}, {1055, 1155}, {1334, 2646}, {1385, 3730}, {1420, 5022}, {1455, 4559}, {1697, 4258}, {1807, 2338}, {2082, 2098}, {2170, 2348}, {2174, 2301}, {2280, 5919}, {2389, 3022}, {3008, 6547}, {3059, 4336}, {3306, 5228}, {3684, 3880}, {4345, 5838}, {4861, 4875}, {5030, 5126}, {5222, 5328}
X(6603) = isogonal conjugate of X(34056)
X(6603) = inverse-in-circumconic-centered-at-X(1) of X(9)
See X(6600).
X(6604) lies on these lines: {1, 348}, {2, 220}, {4, 150}, {7, 8}, {56, 6337}, {57, 345}, {145, 279}, {193, 6180}, {218, 4904}, {226, 4384}, {239, 948}, {269, 3879}, {344, 1445}, {347, 4360}, {349, 4441}, {607, 1814}, {651, 1992}, {944, 5088}, {962, 4872}, {1025, 4253}, {1043, 1434}, {1265, 3263}, {1323, 3244}, {1418, 4851}, {1447, 1788}, {1475, 6168}, {1482, 1565}, {1847, 5174}, {2099, 3665}, {2295, 4648}, {2389, 3434}, {3160, 3241}, {3340, 3674}, {3616, 5543}, {3668, 3875}, {3672, 5738}, {3721, 4419}, {3870, 4350}, {3945, 5710}, {4295, 4911}, {4321, 4684}, {4328, 4357}, {5244, 5739}
X(6604) = isotomic conjugate of X(6601)
X(6604) = polar conjugate of isogonal conjugate of X(23144)
X(6604) = {X(7),X(8)}-harmonic conjugate of X(85)
X(6605) lies on these lines: {2, 220}, {9, 1174}, {346, 3996}, {644, 4847}, {1212, 3957}, {2287, 3693}, {4651, 6559}
X(6606) lies on these lines: {99, 883}, {100, 4569}, {664, 3939}, {666, 4552}, {668, 4578}, {927, 4557}, {1170, 3227}, {1441, 2346}, {2283, 4616}
X(6607) lies on these lines: {30, 511}, {657, 4105}
X(6607) = complementary conjugate of X(38973)
X(6608) lies on these lines: {521, 4724}, {522, 693}, {647, 1962}, {649, 6182}, {650, 663}, {656, 676}, {661, 926}, {1021, 2328}, {3939, 5375}
X(6608) = complement of isotomic conjugate of X(35312)
X(6609) lies on these lines: {57, 1422}, {329, 934}, {946, 999}, {1398, 1851}
X(6609) = complement of X(34546)
X(6610) lies on these lines: {6, 57}, {7, 1100}, {37, 77}, {44, 241}, {73, 1242}, {85, 4670}, {227, 1406}, {279, 4644}, {348, 4643}, {349, 4410}, {513, 663}, {518, 5018}, {527, 1323}, {536, 664}, {604, 1122}, {674, 1362}, {738, 3207}, {948, 4675}, {1104, 4306}, {1108, 4341}, {1386, 4334}, {1442, 3723}, {2099, 2263}, {2182, 3942}, {2293, 3748}, {3160, 4419}, {3945, 4872}, {4318, 4864}, {4648, 5226}
X(6610) = isogonal conjugate of isotomic conjugate of X(37780)
X(6610) = crossdifference of every pair of points on line X(9)X(3900)
X(6611) lies on these lines: {2, 934}, {6, 2155}, {25, 34}, {57, 1422}, {198, 223}, {221, 2187}, {222, 1461}, {604, 1407}, {2654, 3304}
X(6612) lies on these lines: {56, 1413}, {57, 1422}, {84, 999}, {951, 1433}, {2137, 5193}
X(6613) lies on these lines: {190, 1461}, {668, 934}, {670, 4616}, {903, 3668}, {1476, 2481}, {4555, 4566}
X(6614) lies on these lines: {109, 934}, {738, 1106}, {1015, 1407}, {1414, 4616}, {1415, 1461}, {3939, 6516}
X(6614) = isogonal conjugate of X(4163)
X(6615) lies on these lines: {21, 3737}, {244, 3259}, {256, 885}, {513, 663}, {521, 4895}, {522, 3717}, {523, 2292}, {656, 900}, {661, 3310}, {1734, 4962}, {2254, 3667}, {4171, 4526}
As a point on the Euler line, X(6616) has Shinagawa coefficients (2F2, -4EF + S2).
Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "SS(a → tan A) of X" is the point f(tan A, tan B, tan C) : f(tan B, tan C, tan A) : f(tan C, tan A, tan B). Symbolic substitutions carry lines to lines, conics to conics, cubics to cubics, etc. In particular, SS(a → tan A) carries the line X(1)X(2) to the line X(2)X(3), the Euler line.
X(6616) lies on these lines: {2, 3}, {107, 3183}, {1498, 6523}, {1503, 6526}, {2883, 6525}
As a point on the Euler line, X(6617) has Shinagawa coefficients (2EF - S2, S2).
See X(6616).
X(6617) lies on these lines: {2, 3}, {6, 6509}, {63, 268}, {100, 6060}, {394, 1073}, {1032, 3964}, {1033, 6527}, {1619, 1624}, {2063, 4558}
As a point on the Euler line, X(6618) has Shinagawa coefficients (2F2, -2(E + F)F + S2).
See X(6616).
X(6618) lies on these lines: {2, 3}, {6, 6525}, {154, 393}, {184, 1249}, {185, 3183}, {459, 1899}, {1857, 2182}, {2187, 2202}
As a point on the Euler line, X(6619) has Shinagawa coefficients (2F2, 2(E + F)F - S2).
See X(6616).
X(6619) lies on these lines: {2, 3}, {125, 459}, {393, 1853}, {1249, 1899}
As a point on the Euler line, X(6620) has Shinagawa coefficients ((E + F)F, -(E + F)2 + S2).
See X(6616).
X(6620) lies on these lines: {2, 3}, {154, 5254}, {184, 5286}, {193, 3186}, {393, 1974}, {1843, 3087}, {2207, 6524}
As a point on the Euler line, X(6621) has Shinagawa coefficients (4F2, -4EF + S2).
See X(6616).
X(6621) lies on these lines: {2, 3}, {154, 6526}, {459, 1498}, {1249, 3349}, {2883, 3183}
As a point on the Euler line, X(6622) has Shinagawa coefficients (2F, -E + F).
See X(6616).
X(6622) lies on these lines: {2, 3}, {1068, 3012}, {1192, 5893}, {1249, 3767}, {1876, 5704}, {5319, 5702}, {6523, 6530}
X(6622) = center of inverse-in-polar-circle-of-de-Longchamps-circle
As a point on the Euler line, X(6623) has Shinagawa coefficients (F, -E + 2F).
See X(6616).
X(6623) lies on these lines: {2, 3}, {113, 193}, {115, 393}, {133, 5139}, {253, 339}, {1093, 6526}, {1870, 5274}, {1899, 5656}, {1902, 5818}, {1974, 5622}, {3087, 5475}, {3620, 5891}, {5261, 6198}, {2, 3}, {1068, 3012}, {1192, 5893}, {1249, 3767}, {1876, 5704}, {5319, 5702}, {6523, 6530}
As a point on the Euler line, X(6624) has Shinagawa coefficients (2F2, -2(2E - F)F + S2).
See X(6616).
X(6624) lies on these lines: {2, 3}, {133, 1249}, {459, 6000}, {1498, 6526}, {2883, 6523}, {3183, 5878}
X(6625) lies on the Kiepert hyperbola and these lines: {1, 148}, {2, 1931}, {4, 2905}, {10, 894}, {115, 1509}, {321, 1909}, {2895, 6539}, {2996, 3945}, {4567, 5277}
X(6625) = isogonal conjugate of X(18755)
X(6625) = isotomic conjugate of X(1654)
X(6625) = trilinear pole of line X(523)X(2487)
X(6625) = pole wrt polar circle of trilinear polar of X(4213)
X(6625) = X(19)-isoconjugate of X(22139)
X(6625) = X(48)-isoconjugate (polar conjugate) of X(4213)
X(6626) lies on these lines: {2, 1931}, {3, 3437}, {10, 99}, {39, 2669}, {58, 86}, {75, 1247}, {239, 257}, {274, 1111}, {502, 1268}, {620, 6537}, {799, 1909}, {966, 6337}, {1414, 1935}, {1434, 1447}, {1975, 5737}
X(6626) = X(2)-Ceva conjugate of X(86)
X(6626) = perspector of circumconic centered at X(86)
X(6627) lies on these lines: {2, 4610}, {115, 125}, {2511, 2615}
X(6627) = complement of X(4610)
X(6628) lies on these lines: {81, 4610}, {86, 4683}, {171, 4600}, {593, 763}, {757, 873}, {3995, 4632}
X(6628) = barycentric cube of X(86)
X(6629) lies on these lines: {58, 86}, {69, 4257}, {99, 519}, {110, 2729}, {187, 524}, {193, 4262}, {239, 514}, {325, 540}, {518, 3110}, {593, 4001}, {799, 5209}, {1323, 4573}, {1366, 4831}, {1444, 4276}, {1931, 3912}, {4592, 5127}
X(6630) lies on these lines: {239, 3911}, {514, 4440}, {519, 1757}, {1016, 4473}, {3975, 4358}, {4555, 6547}
X(6630) = isogonal conjugate of X(9259)
X(6630) = isotomic conjugate of X(4440)
X(6630) = cevapoint of PU(24)
X(6630) = X(19)-isoconjugate of X(22148)
X(6631) lies on these lines: {100, 4367}, {190, 514}, {519, 1738}, {660, 4083}, {664, 3669}, {666, 5382}, {668, 1577}, {898, 4401}, {3936, 6542}
X(6631) = crosssum of PU(25)
X(6631) = X(2)-Ceva conjugate of X(190)
X(6631) = perspector of circumconic centered at X(190)
X(6631) = trilinear pole of line X(1054)X(4440)
X(6631) = center of hyperbola {{A,B,C,PU(24),PU(58)}}
X(6631) = crosspoint of PU(24)
X(6632) lies on these lines: {2, 1016}, {31, 765}, {42, 5378}, {81, 4600}, {190, 6546}, {4076, 5423}, {4998, 5435}
X(6632) = isotomic conjugate of X(6545)
X(6632) = barycentric cube of X(190)
X(6633) lies on these lines: {1, 2}, {190, 514}, {668, 4791}, {4440, 6549}, {4562, 4597}
X(6634) lies on these lines: {2, 1016}, {43, 5378}, {57, 4998}, {171, 765}, {2319, 5383}, {4555, 6545}
X(6635) lies on the Steiner circumellipse and these lines: {99, 6551}, {190, 6546}, {666, 4582}, {889, 901}, {903, 1644}, {1016, 4370}, {3227, 5376}, {3257, 4562}
X(6635) = isotomic conjugate of X(6550)
X(6635) = trilinear pole of line X(2)X(1016)
As a point on the Euler line, X(6636) has Shinagawa coefficients (3E + 4F, -4E - 4F).
X(6636) lies on these lines: {2,3}, {32,1180}, {35,3920}, {36,5310}, {39,251}, {51,5092}, {81,4265}, {97,933}, {98,930}, {99,1799}, {100,3703}, {110,3917}, {141,2916}, {159,3620}, {160,3314}, {182,3060}, {184,323}, {187,1194}, {343,3448}, {511,1994}, {612,5010}, {842,1291}, {1030,5276}, {1176,3313}, {1184,5023}, {1199,6243}, {1216,1614}, {1350,1993}, {1495,3819}, {1501,3094}, {1503,3410}, {1609,5304}, {2076,3051}, {2917,6247}, {3053,5359}, {3218,5285}, {3219,3220}, {3705,4996}, {5085,5422}, {5297,5370}, {5866,6031}
X(6636) = reflection of X(1994) in X(5012)
X(6636) = anticomplement X(5133)
X(6636) = circumcircle-inverse of X(5189)
X(6636) = crosspoint, wrt the excentral triangle, of X(3) and X(2916)
X(6636) = crosspoint, wrt the tangential triangle, of X(3) and X(2916)
X(6636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,22,23), (2,5189,427), (3,4,37126), (3,22,2), (3,24,3523), (3,26,631), (3,199,4210), (3,376,2071), (3,550,3520), (3,859,4218), (3,2070,549), (3,2915,404), (3,2937,140), (20,3523,3088), (35,5322,3920), (140,2937,3518), (184,2979,323), (184,3098,2979), (187,1194,1627), (550,6240,20), (1113,1114,5189), (1194,1627,5354), (1350,3796,1993), (3131,3132,237), (3220,5314,3219), (4265,5347,81), (5004,5005,5), (5010,5345,612).
See Hyacinthos #23129.
X(6637) lies on this line: X(1)X(4)
As a point on the Euler line, X(6638) has Shinagawa coefficients ((E - F)F - S2, (E + F)F + S2).
See X(6337). Centers X(6638)-X(6644) exemplify of the fact that the symbolic substitution (SS: a → sin 2A) takes the Nagel line, X(1)X(2), to the Euler line, X(2)X(3).
X(6638) lies on these lines: {2, 3}, {154, 1624}, {216, 5943}, {511, 6509}, {577, 1971}, {1147, 2055}, {2972, 2979}, {3164, 3168}, {3167, 3289}
X(6638) = isotomic conjugate of the polar conjugate of X(32445)
X(6638) = X(92)-isoconjugate of X(1988)
X(6638) = X(577)-Ceva conjugate of X(3)
As a point on the Euler line, X(6639) has Shinagawa coefficients (E + 8F, -E).
See X(6338).
X(6639) lies on these lines: {2, 3}, {49, 68}, {184, 5449}, {195, 6515}, {231, 5158}
X(6639) = complement of X(37119)
As a point on the Euler line, X(6640) has Shinagawa coefficients (E - 8F, -E).
See X(6338).
X(6640) lies on these lines: {2, 3}, {49, 1899}, {113, 3357}, {125, 1147}, {974, 5654}, {1092, 5449}, {1204, 5448}
X(6640) = complement of X(7505)
As a point on the Euler line, X(6641) has Shinagawa coefficients (F2 - S2, (E + F)F - S2).
See X(6338).
X(6641) lies on these lines: {2, 3}, {51, 577}, {97, 3060}, {154, 157}, {160, 161}, {184, 216}, {5651, 6509}
X(6641) = orthocentroidal-circle-inverse of X(34965)
As a point on the Euler line, X(6642) has Shinagawa coefficients (E - 2F, E + 2F).
See X(6338).
X(6642) lies on these lines: {2, 3}, {6, 1147}, {51, 1092}, {52, 394}, {54, 5422}, {143, 1351}, {154, 5892}, {155, 389}, {159, 182}, {568, 6090}, {578, 5943}, {1217, 6525}, {1350, 5447}, {1993, 3567}, {2883, 4846}, {2929, 5448}, {2931, 5972}, {3167, 5946}, {5562, 5651}
X(6642) = circumcircle-inverse of X(37971)
As a point on the Euler line, X(6643) has Shinagawa coefficients (E, -E - 2F).
See X(6338).
X(6643) lies on these lines: {2, 3}, {68, 69}, {96, 97}, {216, 2548}, {388, 1060}, {394, 6193}, {497, 1062}, {577, 3767}, {626, 6389}, {1038, 1478}, {1040, 1479}, {1578, 3070}, {1579, 3071}, {1899, 5562}, {3284, 5319}, {5707, 5800}
X(6643) = orthocentroidal-circle-inverse of X(7401)
X(6643) = homothetic center of 3rd pedal triangle of X(3) and 3rd pedal triangle of X(4)
X(6643) = {X(2),X(4)}-harmonic conjugate of X(7401)
As a point on the Euler line, X(6644) has Shinagawa coefficients (E - 4F, E + 4F).
See X(6338).
X(6644) lies on these lines: {2, 3}, {6, 1511}, {52, 1092}, {110, 5890}, {155, 2929}, {156, 1181}, {157, 1605}, {182, 2393}, {183, 1236}, {389, 1147}, {394, 1154}, {567, 5422}, {568, 1993}, {569, 6146}, {578, 5462}, {1192, 5876}, {1485, 5961}, {1539, 2935}, {2782, 2936}, {3425, 5968}, {3455, 6055}, {5651, 5891}
X(6644) = midpoint of X(3) and X(25)
X(6644) = inverse-in-circumcircle of [reflection of X(3) in the orthic axis]
X(6644) = harmonic center of nine-point circle and tangential circle
X(6644) = tangential isogonal conjugate of X(35237)
X(6644) = X(999)-of-tangential-triangle if ABC is acute
X(6644) = center of circle that is inverse-in-circumcircle of orthic axis (circle with segment X(3)X(25) as diameter)
X(6644) = center of circle {X(3),X(25),PU(4)} (the inverse-in-circumcircle of the orthic axis)
See X(6338).
X(6645) lies on these lines: {1, 335}, {2, 12}, {6, 330}, {55, 3552}, {76, 2242}, {83, 1015}, {99, 1500}, {172, 385}, {192, 1975}, {257, 3509}, {274, 5291}, {287, 1425}, {401, 4296}, {439, 5281}, {458, 1398}, {668, 5277}, {894, 2329}, {904, 3510}, {1003, 3295}, {1043, 6542}, {1478, 5025}, {2241, 3972}, {2275, 3329}, {3023, 4027}
X(6645) = barycentric square of X(894)
See X(6338).
X(6646) lies on these lines: {2, 7}, {6, 4389}, {8, 726}, {37, 320}, {38, 256}, {69, 192}, {72, 4201}, {75, 1654}, {86, 1931}, {141, 190}, {145, 5847}, {193, 3672}, {239, 3663}, {319, 536}, {333, 3782}, {346, 3620}, {391, 4346}, {524, 4360}, {545, 594}, {960, 1463}, {966, 4699}, {982, 4703}, {984, 4645}, {1100, 4715}, {1266, 3686}, {1284, 2975}, {1469, 3869}, {1757, 3821}, {1999, 4001}, {2896, 3954}, {3210, 5739}, {3631, 3943}, {3661, 3729}, {3688, 3888}, {3758, 4657}, {3870, 4335}, {3957, 4343}, {3986, 4896}, {4026, 5852}, {4361, 4398}, {4363, 5224}, {4366, 5845}, {4384, 4862}, {4392, 5211}, {4429, 5220}, {4454, 5232}, {4664, 4851}, {4675, 4687}, {4686, 4690}
X(6646) = complement of X(31300)
X(6646) = anticomplement of X(894)
X(6646) = isotomic conjugate of isogonal conjugate of X(21008)
X(6646) = polar conjugate of isogonal conjugate of X(22161)
See X(6338).
X(6647) lies on these lines: {142, 4315}, {527, 1323}, {535, 5074}, {664, 3509}, {894, 2329}, {2533, 3907}, {4051, 4209}
See X(6338).
X(6648) lies on the Steiner circumellipse and these lines: {99, 4552}, {109, 190}, {478, 3596}, {553, 903}, {645, 4559}, {651, 668}, {664, 1461}, {666, 4581}, {670, 4573}, {961, 3227}, {1121, 1220}, {1462, 2298}, {4569, 4617}
X(6648) = isotomic conjugate of X(3910)
X(6648) = trilinear pole of line X(2)X(12)
See X(6338).
X(6649) lies on these lines: {85, 750}, {100, 658}, {109, 789}, {651, 3570}, {799, 1414}, {4551, 4573}
See X(6338).
X(6650) lies on these lines: {2, 846}, {7, 4393}, {27, 2969}, {75, 1654}, {86, 1086}, {148, 3125}, {190, 1213}, {335, 740}, {524, 903}, {675, 2702}, {4359, 6158}, {4750, 6548}
X(6650) = isotomic conjugate of X(6542)
X(6650) = anticomplement of X(6651)
X(6650) = polar conjugate of X(17927)
See X(6338).
X(6651) lies on these lines: {2, 846}, {9, 1654}, {37, 86}, {238, 239}, {385, 3985}, {524, 4370}, {528, 4733}, {537, 5625}, {1213, 4422}, {1331, 1999}, {1757, 6541}, {1966, 3948}, {2786, 5029}, {3570, 4760}, {4375, 6544}
X(6651) = complement of X(6650)
X(6651) = center of hyperbola {{A,B,C,X(789),PU(6)}}
X(6651) = X(2)-Ceva conjugate of X(239)
X(6651) = crosssum of circumcircle intercepts of line PU(8) (line X(42)X(649))
See X(6338).
X(6652) lies on these lines: {2, 31}, {105, 335}, {239, 3573}, {1914, 3253}, {2112, 2145}, {4366, 4368}
See X(6338).
X(6653) lies on these lines: {{1,17565}, {2,11}, {8,6655}, {10,6651}, {69,1278}, {148, 668}, {190, 594}, {335, 740}, {903, 4971}, {966, 4473}, {1086, 4360}, {3661, 4660}, {5025, 5687}
X(6653) = anticomplement of X(4366)
See X(6338).
X(6654) lies on the hyperbola {{A,B,C,X(81),X(100),PU(8)}} and these lines: {1, 6185}, {2, 11}, {57, 927}, {239, 3573}, {312, 6632}, {350, 3570}, {870, 1438}, {873, 2185}, {1447, 1914}, {1462, 2162}
X(6654) = isogonal conjugate of X(3252)
X(6654) = trilinear pole of line X(238)X(812)
Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "(SS: a → a4 + b2c2) of X" is the point
f(a4 + b2c2, b4 + c2a2, c4 + a2b2) : f(b4 + c2a2, c4 + a2b2, a4 + b2c2) : f(c4 + a2b2, a4 + b2c2, b4 + c2a2).
As a point on the Euler line, X(6655) has Shinagawa coefficients ((E + F)2 + S2, -4S2).
Symbolic substitutions carry lines to lines, conics to conics, cubics to cubics, etc. In particular, SS(a → a4 + b2c2) carries the line X(1)X(2) to the line X(2)X(3), the Euler line. See X(6616).
X(6655) lies on these lines: {2, 3}, {8, 6653}, {39, 316}, {69, 698}, {76, 148}, {83, 4045}, {99, 626}, {115, 1078}, {194, 315}, {385, 5254}, {621, 3104}, {622, 3105}, {1975, 3314}, {2542, 2559}, {2543, 2558}, {2794, 4027}, {3094, 5207}, {3096, 3734}, {3849, 5007}, {3917, 6310}, {4366, 6284}, {5103, 5116}, {5309, 6179}, {7762,7839}
X(6655) = reflection of X(2) in X(7924)
X(6655) = reflection of X(19686) in X(2)
X(6655) = complement of X(6658)
X(6655) = anticomplement of X(384)
X(6655) = anticomplementary conjugate of X(37889)
X(6655) = 2nd-Brocard-Circle-inverse of X(37896)
X(6655) = orthocentroidal-circle-inverse of X(16044)
X(6655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,33259), (2,4,16044), (2,20,3552), (3,5025,2), (148,2896,76), (315,2549,194)
As a point on the Euler line, X(6656) has Shinagawa coefficients ((E + F)2, -S2).
See X(6655).
X(6656) lies on these lines: {2, 3}, {6, 315}, {39, 325}, {69, 5286}, {76, 141}, {83, 316}, {115, 3934}, {183, 3767}, {194, 3314}, {230, 1078}, {239, 5015}, {385, 2896}, {491, 6422}, {492, 6421}, {574, 3788}, {625, 1506}, {639, 3102}, {640, 3103}, {695, 3978}, {754, 5007}, {894, 4911}, {1193, 4766}, {1235, 5523}, {1975, 2549}, {3491, 4173}, {3620, 6392}, {3661, 4385}, {3662, 3673}, {3695, 3797}, {5306, 6179}
X(6656) = midpoint of X(2) and X(7924)
X(6656) = reflection of X(6661) in X(2)
X(6656) = complement of X(384)
X(6656) = anticomplement of X(7819)
X(6656) = crosssum of X(32) and X(39)
X(6656) = crosspoint of X(76) and X(83)
X(6656) = pole of line X(5)X(6) wrt conic {{X(13),X(14),X(15),X(16),X(141)}}
X(6656) = orthocentroidal-circle-inverse of X(7770)
X(6656) = complementary conjugate of X(37890)
X(6656) = {X(2),X(3)}-harmonic conjugate of X(7807)
X(6656) = {X(2),X(4)}-harmonic conjugate of X(7770)
X(6656) = {X(2),X(5)}-harmonic conjugate of X(32992)
X(6656) = {X(2),X(20)}-harmonic conjugate of X(14001)
X(6656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5025,5), (3,5,1513), (39,626,325), (76,3096,141), (115,6292,3934), (141,3094,6393), (141,5254,76), (194,3314,3933), (626,4045,39)
As a point on the Euler line, X(6657) has Shinagawa coefficients ((E + F)6 + (E + 3F)(E - F)S4 - 2(E + F)4S2, [3(E + F)4 - 2(E + F)(E - F)S2 - S4]S2).
See X(6655).
X(6657) lies on this line: {2,3}
As a point on the Euler line, X(6658) has Shinagawa coefficients ((E + F)2 - 3S2,8S2).
See X(6655).
X(6658) lies on these lines: {2, 3}, {32, 148}, {193, 698}, {2896, 3734}, {3924, 6650}, {6284, 6645}
X(6658) = reflection of X(2) in X(19686)
X(6658) = complement of X(19691)
X(6658) = anticomplement of X(6655)
X(6658) = orthocentroidal-circle-inverse of X(32993)
X(6658) = {X(2),X(4)}-harmonic conjugate of X(32993)
X(6658) = {X(2),X(20)}-harmonic conjugate of X(33260)
X(6658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3552,2), (382,1003,5025), (439,3832,2)
As a point on the Euler line, X(6659) has Shinagawa coefficients ((E + F)6 - (E2 + 2EF - 31F2)S4 + (E + F)4S2 - S6, -[6(E + F)4 - 4(E + F)(E - 7F)S2 - 10S4)]S2).
See X(6655).
X(6659) lies on this line: {2,3}
As a point on the Euler line, X(6660) has Shinagawa coefficients ((E + F)(E + 3F) - S2, -3(E + F)2 + S2).
See X(6655).
X(6660) lies on these lines: {2, 3}, {32, 3981}, {39, 1915}, {51, 3398}, {184, 3095}, {256, 1582}, {291, 1580}, {511, 3506}, {525, 669}, {694, 2076}, {1283, 2223}, {1605, 6582}, {1606, 6295}, {1631, 4363}, {2353, 3504}, {3229, 5162}, {3978, 5989}
X(6660) = anticomplement of X(21536)
X(6660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22,3148,3), (2479,2480,384)
X(6660) = tangential-isogonal conjugate of X(3511)
As a point on the Euler line, X(6661) has Shinagawa coefficients (3(E + F)2 - 2S2, 3S2).
See X(6655).
X(6661) lies on these lines: {2, 3}, {76, 5306}, {99, 3589}, {141, 3972}, {597, 698}, {1285, 3620}, {3329, 6390}, {3734, 5309}
X(6661) = midpoint of X(2) and X(384)
X(6661) = reflection of X(2) in X(7819)
X(6661) = reflection of X(6656) in X(2)
X(6661) = complement of X(7924)
X(6661) = {X(2),X(4)}-harmonic conjugate of X(33219)
X(6661) = orthocentroidal circle inverse of X(33219)
Let G' be the centroid of the quadrilateral {A,B,C,X(3)}, and let U be the reflection of the cevian triangle of X(3) in G'. Then ABC and U are perspective, and X(6662) is their perspector. The trilinear polar of X(6662) passes through X(1637). (Randy Hutson, February 24, 2015)
X(6662) lies on these lines: {2,6663}, {30,5562}, {140,216}
X(6662) = anticomplement of X(6663)
X(6662) = isotomic conjugate of anticomplement of X(36412)
Let P and Q be the points on the circumcircle whose Steiner lines are tangent to the circumcircle. Then X(6663) = crosssum of P and Q. Let A′B′C′ be the anticevian triangle of X(5). Then X(6663) is the centroid of {A′, B′, C′, X(5)}. (Randy Hutson, February 24, 2015)
X(6663) lies on these lines: {2,6662}, {5,324}, {30,143}
X(6663) = complement of X(6662)
Let G' be the centroid of the quadrilateral {A,B,C,X(6)}, and let U be the reflection of the cevian triangle of X(6) in G'. Then ABC and U are perspective, and X(6664) is their perspector. The trilinear polar of X(6664) passes through X(3005). (Randy Hutson, February 24, 2015)
X(6665) lies on these lines: {2,6665}, {39,698}, {428,524}, {755,6573}
X(6664) = isotomic conjugate of X(7760)
X(6664) = anticomplement of X(6665)
Let A′B′C′ be the anticevian triangle of X(141). Then X(6665) is the centroid of {A′,B′,C′,X(141). (Randy Hutson, February 24, 2015)
X(6665) lies on these lines: {2,6664}, {6,4576}, {698,3934}, {3589,4074}
X(6665) = complement of X(6664)
Centroids of Quadrilaterals {A,B,C,X}: X(6666)-X(6723)
Suppose that X is a point in the plane of a triangle ABC. The centroid, G,′ of {A,B,C,X} is the complement of the complement of X, hence also the midpoint of X and the complement of X, as well as the Kosnita(X,X(2)) point. If X = x : y : z (barycentrics), then G' = 2x + y + z : x + 2y : z : x + y + 2z.
Randy Hutson, February 24, 2015, notes that properties described at X(6600) and X(6601) hold generally, as follows. Let G″ be the complement of X, and let A′B′C′ be the anticevian triangle of G″. The reflection of the cevian triangle of X in G' is perspective to ABC, and the perspector is the anticomplement of the centroid of {A′,B′,C′,G″}. This perspector is also the isogonal conjugate of the TCC-perspector of the isogonal conjugate of G″. (The TCC-perspector of a point is defined in the preamble to X(1601).)
Example 1. X = X(1)
X(596) = perspector of ABC and reflection of cevian triangle of X(1) in centroid of {A,B,C,X(1)}
X(596) = anticomplement of X(4075)
X(596) = isogonal conjugate of TCC-perspector of X(58)
X(4075) = complement of X(596) = centroid of {A′,B′,C′,X(10)), where A′B′C′ = anticevian triangle of X(10)
Example 2. X = X(4)
X(68) = perspector of ABC and reflection of cevian triangle of X(4) in centroid of {A,B,C,X(4)}
X(68) = anticomplement of X(1147)
X(68) = isogonal conjugate of TCC-perspector of X(58)
X(1147) = complement of X(68) = centroid of {A′,B′,C′,X(3)), where A′B′C′ = anticevian triangle of X(3)
Example 8. X = X(8)
X(4) = perspector of ABC and reflection of cevian triangle of X(8) in centroid of {A,B,C,X(8)}
X(4) = anticomplement of X(3)
X(4) = isogonal conjugate of TCC-perspector of X(1)
X(3) = complement of X(4) = centroid of {A′,B′,C′,X(1)), where A′B′C′ = anticevian triangle of X(1)
The appearance of (i,j) in the following list means that the centroid of {A,B,C,X(i)} is X(j):
(1,1125), (2,2), (3,140), (4,5), (5,3628), (6,3589), (7,142), (8,10), (9,6666), (10,3634), (11,6667), (12,6668), (13,6669), (14,6690), (15,6671), (16,6672), (17,6673), (18,6674), (20,3), (21, 6675), (22,6676), (23,468), (25,6677), (27,6678), (31,6679), (32,6680), (36,6681), (37,4698), (38,6682), (39,6683), (40,6684), (42,6685), (43,6686), (44,6687), (51,6688), (52,5462), (54,6689), (55,6690), (56,6691), (57,6692), (58,6693), (61,6694), (62,6695), (63,5745), (64,6696), (65,3812), (66,6697), (67,6698), (68,5449), (69,141), (72,5044), (74,6699), (75,3739), (76,3934), (78,6700), (79,6701), (80,6702), (81,6703), (83,2704), (84,6705), (85,6706), (86,6707), (92,6708), (95,6709), (98,6036), (99,620), (100,3035), (101,6710), (102,6711), (103,6712), (104,6713), (105,6714), (106,6715), (107,6716), (108,6717), (109,6718), (110,5972), (111,6719), (112,6720), (114,6721), (115,6722), (125,6723)
X(6666) lies on these lines: {1, 4878}, {2, 7}, {5, 516}, {10, 1001}, {11, 6594}, {37, 3008}, {44, 3664}, {45, 3663}, {69, 3707}, {75, 2325}, {140, 971}, {239, 4464}, {319, 3686}, {344, 2321}, {405, 4304}, {480, 4423}, {518, 1125}, {528, 3828}, {551, 3940}, {631, 5732}, {673, 1268}, {899, 4343}, {950, 5047}, {954, 1210}, {958, 4315}, {1212, 1323}, {1449, 5308}, {1479, 1698}, {1656, 5805}, {1743, 4648}, {1890, 5142}, {2346, 5284}, {2551, 5726}, {3090, 5759}, {3161, 4659}, {3243, 3616}, {3247, 5222}, {3358, 6260}, {3526, 5779}, {3601, 5809}, {3624, 5223}, {3628, 5762}, {3731, 4000}, {3739, 4422}, {3740, 5572}, {3875, 4029}, {3879, 4700}, {3950, 4361}, {3973, 4644}, {3986, 4657}, {4298, 5302}, {4326, 5218}, {4395, 4681}, {4419, 4859}, {4708, 5845}, {5067, 5735}, {5220, 5542}, {5252, 5795}
X(6666) = midpoint of X(i) and X(j) for these {i,j}: {9,142}, {10,100}, {11,6594}, {3358, 6260}, {3950,4361}, {5220,5542}
X(6666) = reflection of X(3826) in X(3634)
X(6666) = isotomic conjugate of X(32015)
X(6666) = complement of X(142)
X(6666) = {X(2),X(9)}-harmonic conjugate of X(142)
X(6667) lies on these lines: {1, 3036}, {2, 11}, {5, 2829}, {10, 1387}, {37, 3055}, {80, 3624}, {104, 3090}, {119, 1656}, {140, 3825}, {142, 5851}, {499, 956}, {529, 3814}, {547, 3822}, {632, 5248}, {908, 5852}, {952, 1125}, {1145, 1698}, {1317, 3616}, {1737, 5855}, {1768, 5437}, {2800, 3812}, {2801, 3848}, {2802, 3634}, {3589, 5848}, {3614, 5253}, {3742, 5083}, {3911, 5087}, {4187, 4999}, {4193, 5433}, {4996, 5047}, {5080, 5298}, {5187, 5204}, {5220, 5328}
X(6667) = midpoint of X(i) and X(j) for these {i,j}: {1,3036}, {5,6713}, {10,1387}, {11,3035}, {1125,6702}, {3911,5087}
X(6667) = complement of X(3035)
X(6668) lies on these lines: {2, 12}, {5, 5248}, {10, 5855}, {21, 3614}, {140, 3822}, {142, 5852}, {404, 5326}, {442, 3035}, {498, 2886}, {547, 3825}, {758, 3634}, {952, 1125}, {1001, 3090}, {1656, 3816}, {2476, 5432}, {3085, 3813}, {3295, 3829}, {3589, 5849}, {5141, 6284}
X(6668) = midpoint of X(12) and X(4999)
X(6668) = complement of X(4999)
X(6669) lies on these lines: {2, 13}, {3, 5478}, {17, 299}, {99, 5470}, {115, 619}, {395, 1506}, {396, 533}, {397, 629}, {531, 5461}, {542, 547}, {617, 5469}, {631, 5473}, {1656, 5617}, {3642, 6300}
X(6669) = midpoint of X(i) and X(j) for these {i,j}: {2,5459}, {3,5478}, {13,618}, {115,619}, {396,623}, {624,6108}
X(6669) = reflection of X(6670) in X(67822)
X(6669) = complement of X(618)
X(6670) lies on these lines: {2, 14}, {3, 5479}, {18, 298}, {99, 5469}, {115, 618}, {395, 532}, {396, 1506}, {398, 630}, {530, 5461}, {542, 547}, {616, 5470}, {631, 5474}, {1656, 5613}, {3643, 6301}
X(6670) = midpoint of X(i) and X(j) for these {i,j}: {2, 5469), {3,5479}, {14,619}, {115,618}, {395,624}, {623,6109}
X(6670) = reflection of X(6669) in X(6722)
X(6670) = complement of X(619)
X(6671) lies on these lines: {2, 14}, {17, 622}, {61, 302}, {140, 143}, {187, 624}, {303, 636}, {396, 532}, {627, 3412}, {630, 6292}, {3526, 5611}, {5318, 5459}
X(6671) = midpoint of X(i) and X(j) for these {i,j}: {15,623}, {187,624}, {396,618}, {619,6109}
X(6671) = complement of X(623)
X(6671) = isogonal conjugate of X(34321)
X(6671) = cevapoint of X(396) and X(15802)
X(6671) = crosspoint of X(302) and X(8838)
X(6671) = crosssum of X(8603) and X(21461)
X(6671) = Kosnita(X(15),X(13)) point
X(6672) lies on these lines: {2, 13}, {18, 621}, {62, 303}, {140, 143}, {187, 623}, {302, 635}, {395, 533}, {628, 3411}, {629, 6292}, {3526, 5615}, {5321, 5460}
X(6672) = midpoint of X(i) and X(j) for these {i,j}: {16,624}, {187,623}, {395,619}, {618,6108}
X(6672) = complement of X(624)
X(6672) = isogonal conjugate of X(34322)
X(6672) = cevapoint of X(395) and X(15778)
X(6672) = crosspoint of X(303) and X(8836)
X(6672) = crosssum of X(8604) and X(21462)
X(6672) = Kosnita(X(16),X(14)) point
X(6673) lies on these lines: {{2, 17}, {3589, 5965}
X(6673) = midpoint of X(17) and X(629)
X(6673) = complement of X(629)
X(6674) lies on these lines: {2, 18}, {3589, 5965}
X(6674) = midpoint of X(18) and X(630)
X(6674) = complement of X(630)
X(6675) lies on these lines: {1, 5791}, {2, 3}, {11, 5259}, {12, 5251}, {35, 3925}, {57, 191}, {63, 6147}, {72, 5719}, {79, 4679}, {141, 5138}, {142, 3647}, {274, 6390}, {284, 1213}, {392, 5771}, {495, 958}, {496, 1001}, {498, 3820}, {499, 4423}, {620, 2795}, {758, 942}, {1212, 3002}, {1698, 1837}, {1724, 5718}, {1834, 4653}, {1901, 4877}, {2771, 5972}, {2886, 5248}, {3035, 3634}, {3487, 3927}, {3576, 5787}, {3589, 4260}, {3616, 5730}, {3712, 4647}, {3819, 5482}, {3824, 4292}, {3916, 5249}, {3940, 5703}, {4369, 5592}, {5087, 5122}, {5283, 5305}, {5436, 5705}, {5550, 5708}, {5709, 5886}, {5768, 5789}
X(6675) = midpoint of X(i) and X(j) for these {i,j}: {5,5428}, {21,442}, {191,3649}
X(6675) = complement of X(442)
X(6675) = center of bicevian ellipse of X(2) and X(21)
As a point on the Euler line, X(6676) has Shinagawa coefficients (E + 3F, -E - F).
Three constructions of X(6676) are given at Hyacinthos 24128, Hyacinthos 24100, and Hyacinthos 24098. (Antreas Hatzipolakis, August 29, 2016)
X(6676) lies on these lines: {2, 3}, {69, 3167}, {141, 206}, {154, 1352}, {184, 343}, {216, 230}, {305, 6390}, {577, 3815}, {612, 1062}, {614, 1060}, {1038, 5272}, {1040, 5268}, {1194, 5305}, {1353, 6515}, {1447, 6356}, {1578, 5420}, {1579, 5418}, {1899, 3796}, {2781, 3819}, {3580, 5012}, {3589, 5943}, {5158, 5306}
X(6676) = midpoint of X(i) and X(j) for these {i,j}: {22,427}, {184,343}
X(6676) = complement of X(427)
X(6676) = isotomic conjugate of polar conjugate of X(7745)
X(6676) = crossdifference of every pair of points on the radical axis of the Brocard circle and tangential circle
X(6676) = circumcircle-inverse of X(37972)
X(6676) = homothetic center of orthocevian triangle of X(2) and medial triangle
As a point on the Euler line, X(6677) has Shinagawa coefficients (E - 3F, E + F).
X(6677) lies on these lines: {2, 3}, {6, 6387}, {182, 1660}, {230, 800}, {343, 5651}, {1196, 5305}, {1353, 3167}, {1627, 5913}, {2393, 3589}, {2790, 6036}, {3815, 5065}, {5943, 5972}, {6090, 6515}
X(6677) = midpoint of X(i) and X(j) for these {i,j}: {{3,1596}, {5,6644}, 25,1368}
X(6677) = complement of X(1368)
X(6677) = circumcircle-inverse of X(37973)
X(6677) = X(57)-of-submedial-triangle if ABC is acute
As a point on the Euler line, X(6678) has Shinagawa coefficients (E - 3F + $bc$, E + F + $bc$).
X(6678) lies on these lines: {2, 3}, {57, 1723}, {1427, 3002}, {1730, 5755}, {3739, 5745}, {3925, 5285}
X(6678) = midpoint of X(27) and X(440)
X(6678) = complement of X(440)
X(6679) lies on these lines: {1, 4438}, {2, 31}, {6, 3771}, {10, 1104}, {55, 4085}, {226, 4672}, {306, 3791}, {333, 3775}, {674, 3589}, {734, 3934}, {744, 3739}, {758, 942}, {902, 4972}, {1215, 3011}, {1698, 4680}, {1707, 4655}, {2308, 3936}, {3052, 4660}, {3550, 4429}, {3662, 4650}, {3683, 4425}, {3687, 4974}, {3712, 4970}, {3772, 3923}, {3773, 4362}, {3792, 3794}, {3821, 4640}, {3944, 4676}, {4697, 5249}
X(6675) = midpoint of X(i) and X(j) for these {i,j}: {31,2887}, {306,3791}
X(6679) = isogonal conjugate of X(38831)
X(6679) = complement of X(2887)
X(6679) = complementary conjugate of complement of X(38813)
X(6679) = crosssum of X(6) and X(3778)
X(6679) = trilinear product X(57)*X(4168)
X(6679) = trilinear quotient X(4168)/X(9)
X(6679) = barycentric product X(7)*X(4168)
X(6679) = barycentric quotient X(4168)/X(8)
X(6680) lies on these lines: {2, 32}, {3, 4045}, {5, 2794}, {6, 3788}, {39, 620}, {114, 3398}, {115, 384}, {140, 143}, {194, 5355}, {230, 736}, {325, 5007}, {538, 5305}, {543, 5254}, {746, 3739}, {760, 1125}, {1235, 6103}, {1698, 4769}, {1975, 5309}, {2031, 3055}, {3314, 6179}, {3491, 5972}, {3618, 5028}, {3734, 3767}, {3926, 5319}, {3933, 5306}, {3972, 5025}
X(6680) = midpoint of X(32) and X(626j
X(6680) = complement of X(626)
X(6680) = complementary conjugate of complement of X(38826)
X(6681) lies on these lines: {2, 36}, {3, 3825}, {10, 1319}, {80, 4881}, {100, 3582}, {140, 517}, {214, 1737}, {468, 1878}, {474, 3841}, {484, 3624}, {499, 3434}, {519, 3035}, {549, 3816}, {551, 5048}, {631, 2077}, {758, 3911}, {908, 4973}, {1001, 5054}, {1698, 5176}, {3245, 5550}, {3421, 5193}, {3634, 4999}, {3660, 3678}, {4187, 5267}, {5057, 5131}, {5087, 5122}, {5437, 5535}
X(6681) = midpoint of X(i) and X(j) for these {i,j}: {10,1319}, {36,3814}, {214,1737}, {908,4973}, {5087,5122}, {5128,5126}
X(6681) = reflection of X(5123) in X(3634)
X(6681) = complement of X(3814)
X(6682) lies on these lines: {1, 3769}, {2, 38}, {10, 3752}, {11, 4425}, {37, 3840}, {63, 4672}, {325, 3847}, {333, 4974}, {341, 1698}, {714, 3739}, {740, 3666}, {758, 942}, {846, 4432}, {899, 4981}, {1107, 3229}, {1150, 3791}, {1647, 6536}, {2886, 3821}, {3005, 4369}, {3218, 4697}, {3634, 6532}, {3687, 3775}, {3816, 4364}, {3848, 4698}, {3920, 4434}, {3944, 4389}, {3989, 4358}, {4038, 5625}, {4085, 4847}, {4104, 4407}, {4732, 4850}
X(6682) = midpoint of X(i) and X(j) for these {i,j}: {38,1215}, {3666,3741}, {3706,4970}
X(6682) = complement of X(1215)
X(6683) lies on these lines: {2, 39}, {5, 4045}, {83, 187}, {140, 143}, {182, 3202}, {262, 631}, {325, 6292}, {385, 5041}, {620, 2023}, {625, 1506}, {626, 3815}, {726, 4698}, {730, 3634}, {1078, 3329}, {1656, 6248}, {2782, 3628}, {3095, 3526}, {3618, 5052}, {3734, 5013}
X(6683) = midpoint of X(i) and X(j) for these {i,j}: {39,3934}, {620,2023}, {625,2021}
X(6683) = complement of X(3934)
X(6684) lies on these lines: {1, 631}, {2, 40}, {3, 10}, {4, 165}, {5, 516}, {8, 3523}, {9, 1158}, {12, 1155}, {20, 5587}, {21, 2077}, {30, 3828}, {35, 950}, {43, 581}, {46, 226}, {55, 1210}, {57, 3085}, {63, 5552}, {65, 5432}, {71, 1715}, {72, 5884}, {84, 5273}, {100, 5178}, {104, 5258}, {109, 3074}, {117, 122}, {119, 3647}, {140, 517}, {142, 5709}, {171, 580}, {182, 5847}, {201, 1735}, {210, 1071}, {212, 1771}, {376, 5691}, {390, 5704}, {406, 1753}, {468, 1902}, {474, 3428}, {495, 4298}, {499, 5119}, {518, 5771}, {519, 549}, {551, 1482}, {553, 3336}, {572, 3686}, {573, 5750}, {601, 1724}, {602, 5264}, {899, 4300}, {912, 3678}, {936, 6261}, {938, 5281}, {944, 3524}, {952, 3530}, {960, 2800}, {997, 5837}, {1056, 3361}, {1064, 3216}, {1276, 5242}, {1277, 5243}, {1329, 4640}, {1503, 3844}, {1512, 5251}, {1532, 3683}, {1571, 3767}, {1656, 3817}, {1697, 3086}, {1699, 3090}, {1702, 3069}, {1703, 3068}, {1766, 5257}, {1785, 1940}, {1837, 4304}, {1838, 2954}, {1848, 6197}, {2093, 3485}, {2550, 5705}, {2551, 6256}, {2792, 3454}, {2801, 4015}, {2951, 5817}, {3057, 5433}, {3149, 4413}, {3245, 5443}, {3333, 5435}, {3339, 3487}, {3359, 3452}, {3416, 5085}, {3436, 4652}, {3515, 5090}, {3525, 3624}, {3526, 4301}, {3617, 5731}, {3635, 5844}, {3655, 4669}, {3701, 3977}, {3740, 5777}, {3755, 5292}, {3824, 5762}, {3876, 5693}, {4295, 5128}, {4303, 4551}, {4305, 5727}, {4311, 5204}, {4312, 5714}, {4314, 5722}, {4357, 6211}, {4512, 5084}, {4847, 5687}, {4999, 5836}, {5044, 6001}, {5082, 5231}, {5131, 5270}, {5176, 5303}, {5183, 5326}, {5249, 5535}, {5259, 5537}, {5442, 5563}, {5534, 5770}, {5542, 5708}, {5715, 5759}, {5812, 5880}
X(6684) = midpoint of X(i) and X(j) for these {i,j}: {3,10}, {5,3579}, {8,5882}, {40,946}, {72,5884}, {355,4297}, {551,3654}, {1158,6260}, {1385,5690}, {3359,3452}, {3655,4669}
X(6684) = reflection of X(i) in X(j) for thesse (i,j): (5,3634), (1125,140)
X(6684) = complement of X(946)
X(6684) = centroid of {X(4),A′,B′,C′}, where A′B′C′ = excentral triangle
X(6684) = homothetic center of anti-Euler triangle and cross-triangle of ABC and Aquila triangle
X(6685) lies on these lines: {1, 2}, {35, 4203}, {37, 6375}, {171, 4279}, {181, 3911}, {192, 4135}, {226, 1403}, {312, 3993}, {321, 4970}, {350, 4021}, {516, 2051}, {674, 3589}, {726, 1215}, {908, 4425}, {968, 4011}, {984, 4090}, {994, 3919}, {1155, 4697}, {1575, 2092}, {1962, 4358}, {2886, 4085}, {2887, 5718}, {3683, 4759}, {3740, 3842}, {3745, 4434}, {3758, 4650}, {3791, 4991}, {3847, 4026}, {3952, 3989}, {4260, 5745}, {4640, 4672}, {4704, 4903}
X(6685) = midpoint of X(i) and X(j) for these {i,j}: {42,3741}, {321,4970}, {1215,3666}
X(6685) = complement of X(3741)
X(6686) lies on these lines: {1, 2}, {726, 3752}, {982, 4090}, {1401, 3911}, {3210, 4135}, {3452, 3821}, {3816, 4085}, {3971, 4850}, {4358, 4970}, {4640, 4759}
X(6686) = midpoint of X(i) and X(j) for these {i,j}: {10,995}, {43,3840}, {982,4090}, {3210,4135}
X(6686) = complement of X(3840)
X(6687) lies on these lines: {2, 44}, {8, 1279}, {10, 3246}, {141, 3707}, {238, 1698}, {239, 4727}, {344, 4852}, {518, 1125}, {536, 2325}, {752, 3634}, {1086, 4480}, {1266, 4370}, {2321, 4405}, {3622, 4864}, {3759, 4889}, {3912, 4725}, {4361, 4873}
X(6687) = midpoint of X(i) and X(j) for these {i,j}: {10,3246}, {44,3834}, {238,3823}, {2325,4395}, {3008,4422}
X(6687) = complement of X(3834)
X(6688) lies on these lines: {2, 51}, {5, 2883}, {6, 5644}, {25, 5092}, {52, 5070}, {154, 182}, {185, 5056}, {375, 2810}, {389, 1656}, {394, 5097}, {632, 5446}, {1154, 3628}, {1611, 5039}, {1915, 2030}, {2390, 3812}, {2393, 3589}, {3030, 3750}, {3090, 5890}, {3292, 5643}, {5067, 5562}, {5422, 5651}
X(6688) = midpoint of X(i) and X(j) for these {i,j}: {2,5943}, {5,5892}, {51, 3819}, {375,3742}, {389,5891}, {5890,5907}
X(6688) = complement of X(3819)
X(6689) lies on these lines: {2, 54}, {3, 3574}, {5, 5944}, {140, 389}, {141, 575}, {195, 394}, {973, 5462}, {1656, 6288}, {2917, 6642}, {3589, 6153}, {3628, 5972}, {4999, 5044}
X(6689) = midpoint of X(i) and X(j) for these {i,j}: {3,3574}, {54,1209}, {973,5462}
X(6689) = complement of X(1209)
X(6690) lies on these lines: {1, 4999}, {2, 11}, {5, 5248}, {12, 21}, {30, 3822}, {31, 5718}, {35, 442}, {37, 230}, {63, 5852}, {80, 5426}, {140, 517}, {141, 3771}, {165, 5880}, {226, 4640}, {321, 3712}, {377, 5217}, {405, 498}, {468, 1824}, {495, 529}, {516, 3838}, {518, 5745}, {631, 3428}, {674, 3589}, {758, 5719}, {846, 4415}, {908, 3683}, {958, 3085}, {1104, 5530}, {1155, 5249}, {1387, 3898}, {1698, 3419}, {1758, 6354}, {1788, 2099}, {2346, 6067}, {2476, 6284}, {3006, 4030}, {3011, 3666}, {3295, 3813}, {3475, 5744}, {3584, 5251}, {3601, 5794}, {3614, 5046}, {3624, 5119}, {3628, 3825}, {3742, 3911}, {3782, 4414}, {3811, 5791}, {3914, 4689}, {4023, 5278}, {4187, 5259}, {4512, 5219}, {5220, 5273}, {5226, 5698}
X(6690) = midpoint of X(i) and X(j) for these {i,j}: {55,2886}, {226,4640}, {495,993}
X(6690) = complement of X(2886)
X(6690) = center of bicevian ellipse of X(2) and X(55)
X(6691) lies on these lines: {1, 1145}, {2, 12}, {3, 3816}, {5, 2829}, {11, 404}, {30, 3825}, {36, 4187}, {46, 3624}, {140, 517}, {142, 3647}, {468, 1828}, {474, 499}, {496, 528}, {549, 5248}, {631, 1001}, {960, 3911}, {1058, 4421}, {1104, 5121}, {1376, 3086}, {1388, 5554}, {1788, 5289}, {2098, 3616}, {2478, 5204}, {3304, 5552}, {3628, 3822}, {3871, 6174}, {4188, 6284}, {4292, 5087}, {4652, 4679}, {5302, 5316}
X(6691) = midpoint of X(56) and X(1329)
X(6691) = complement of X(1329)
X(6691) = center of bicevian ellipse of X(2) and X(56)
X(6692) lies on these lines: {2, 7}, {10, 999}, {56, 5795}, {140, 517}, {171, 5121}, {404, 950}, {474, 1210}, {516, 3816}, {631, 6282}, {938, 5438}, {946, 3359}, {1001, 6244}, {1329, 4298}, {1376, 5853}, {1476, 5176}, {1698, 3421}, {1788, 5837}, {1997, 3729}, {2093, 3624}, {2095, 3526}, {2096, 3090}, {2551, 3361}, {3035, 3742}, {3523, 5436}, {3628, 3824}, {3634, 3820}, {3748, 6174}, {3752, 3946}, {3817, 5880}, {4187, 4292}, {4413, 4847}, {4454, 6557}
X(6692) = midpoint of X(i) and X(j) for these {i,j}: {10,999}, {57,3452}, {946,3359}, {3820,3634}
X(6692) = complement of X(3452)
X(6693) lies on these lines: {2, 58}, {10, 4434}, {140, 143}, {631, 3430}, {758, 942}, {1043, 3017}, {1046, 3624}, {1698, 5429}, {2842, 5972}, {3647, 4425}, {3712, 4065}
X(6693) = midpoint of X(58) and X(3454)
X(6693) = complement of X(3454)
X(6693) = center of bicevian ellipse of X(2) and X(58)
X(6694) lies on these lines: {2, 18}, {17, 83}, {62, 618}, {140, 143}, {299, 3412}, {395, 629}, {396, 636}, {397, 530}, {398, 623}
X(6694) = midpoint of X(i) and X(j) for these {i,j}: {61,635}, {636,5007}
X(6694) = complement of X(635)
X(6695) lies on these lines: {2, 17}, {18, 83}, {61, 619}, {140, 143}, {298, 3411}, {395, 635}, {396, 630}, {397, 624}, {398, 531}
X(6695) = midpoint of X(i) and X(j) for these {i,j}: {62,636}, {635,5007}
X(6695) = complement of X(636)
X(6696) lies on these lines: {2, 64}, {3, 66}, {4, 1192}, {5, 3357}, {20, 1853}, {30, 5449}, {74, 1594}, {125, 1885}, {140, 6000}, {154, 3523}, {221, 5218}, {389, 2781}, {427, 1204}, {546, 2777}, {631, 1498}, {1656, 5878}, {1788, 1854}, {1899, 3516}, {2071, 2888}, {2892, 5621}, {3088, 5480}, {3091, 5895}, {3525, 5656}, {5044, 6001}, {5433, 6285}
X(6696) = midpoint of X(i) and X(j) for these {i,j}: {3,6247}, {4,5894}, {5,3357}, {64,2883}
X(6696) = reflection of X(5893) in X(5)
X(6696) = complement of X(2883)
X(6697) lies on these lines: {2, 66}, {6, 67}, {140, 1503}, {141, 1368}, {159, 1853}, {511, 5449}, {626, 3852}, {858, 3313}, {868, 1879}, {1352, 3548}, {3619, 5888}, {3818, 6644}, {3827, 3844}
X(6697) = midpoint of X(66) and X(206)
X(6697) = complement of X(206)
X(6697) = homothetic center of orthocevian triangle of X(2) and 3rd antipedal triangle of X(4)
X(6698) lies on these lines: {2, 67}, {5, 2781}, {110, 3763}, {125, 126}, {140, 542}, {524, 5159}, {597, 5095}, {599, 895}, {2836, 5044}, {3448, 3619}
X(6698) = midpoint of X(i) and X(j) for these {i,j}: {67,6593}, {125,141}
X(6698) = complement of X(6593)
X(6699) lies on these lines: {2, 74}, {3, 125}, {5, 1539}, {67, 5085}, {69, 5504}, {110, 631}, {140, 5663}, {141, 542}, {399, 5054}, {690, 6036}, {914, 5440}, {1112, 5462}, {1204, 5448}, {1531, 2072}, {2771, 3035}, {2778, 3812}, {2781, 3589}, {3024, 5433}, {3028, 5432}, {3043, 5012}, {3448, 3523}, {3763, 5621}, {3818, 6644}, {5050, 5095}
X(6699) = midpoint of X(i) and X(j) for these {i,j}: {3,125}, {74,113}, {1112,5462}
X(6699) = reflection of X(i) in X(jk) for these (i,j): (5,6723), (5972,140)
X(6699) = complement of X(113)
X(6699) = isotomic conjugate of polar conjugate of X(6128)
X(6700) lies on these lines: {1, 2}, {3, 3452}, {4, 5438}, {9, 631}, {20, 5328}, {72, 3911}, {140, 912}, {210, 5433}, {226, 474}, {404, 908}, {405, 5316}, {443, 5219}, {496, 5853}, {515, 1329}, {516, 3149}, {946, 1376}, {950, 4187}, {960, 2800}, {1058, 3158}, {1385, 3820}, {1420, 3421}, {1445, 5850}, {1706, 5603}, {1795, 3074}, {2478, 4304}, {2551, 3576}, {3159, 3998}, {3436, 4311}, {3487, 5437}, {3526, 5791}, {3601, 5084}, {3660, 3678}, {3740, 4999}, {4679, 5217}, {4848, 5730}, {5054, 5325}, {5257, 5783}, {5265, 5815}, {5267, 5660}, {5720, 6245}
X(6700) = midpoint of X(i) and X(j) for these {i,j}: {78,1210}, {3436,4311}, {4848,5730}
X(6700) = complement of X(1210)
X(6701) lies on these lines: {1, 6175}, {2, 79}, {5, 3833}, {10, 12}, {21, 3624}, {30, 1125}, {142, 3825}, {191, 3305}, {1387, 3636}, {1479, 2475}, {1656, 3652}, {2476, 5883}, {2771, 3812}, {2886, 3881}, {3120, 3743}, {3651, 5248}, {3884, 5499}, {5087, 5122}, {5586, 5705}
X(6701) = midpoint of X(i) and X(j) for these {i,j}: {10,3649}, {79,3647}
X(6701) = complement of X(3647)
Let A′B′C′ be the Fuhrmann triangle. Let A″ be the orthogonal projection of A on line B′C′, and define B″, C″ cyclically. Then A″B″C″ is perspective to the medial triangle at X(6702). (Randy Hutson, April 11, 2015)
X(6702) lies on these lines: {2, 80}, {3, 6246}, {5, 2800}, {10, 11}, {12, 5083}, {100, 1698}, {104, 5587}, {116, 119}, {519, 1387}, {528, 3828}, {535, 3911}, {537, 4013}, {551, 1317}, {758, 908}, {952, 1125}, {1210, 3881}, {1320, 3679}, {1329, 3678}, {1484, 3816}, {1537, 3817}, {1656, 6265}, {2771, 3812}, {2805, 3846}, {2886, 3968}, {2932, 4413}, {3035, 3634}, {3065, 6175}, {3582, 5176}, {3626, 5854}, {3820, 3956}, {3878, 4193}, {4973, 5080}, {4996, 5251}, {5046, 5445}, {5154, 5903}
X(6702) = midpoint of X(i) and X(j) for these {i,j}: {3,6246}, {10,11}, {80,214}, {1387,3036}, {1737,3814}, {4973,5080}
X(6702) = reflection of X(i) in X(j) for these (i,j): (1125,6667), (3035,3634)
X(6702) = complement of X(214)
X(6703) lies on these lines: {1, 3704}, {2, 6}, {3, 5799}, {10, 4682}, {57, 1761}, {58, 4205}, {63, 4364}, {140, 970}, {171, 4026}, {226, 4670}, {551, 4906}, {594, 1999}, {758, 942}, {894, 4415}, {896, 6536}, {1010, 1834}, {1100, 3687}, {1375, 5437}, {1961, 3932}, {1962, 3712}, {2049, 5292}, {2160, 2339}, {2836, 3848}, {3187, 4399}, {3286, 4199}, {3703, 5311}, {3745, 4914}, {3986, 5325}, {4038, 4966}, {4104, 4663}, {4359, 4395}, {4418, 4854}, {4422, 5294}, {4425, 4697}, {4886, 4969}
X(6703) = midpoint of X(i) and X(j) for these {i,j}: {81,1211}, {4418,4854}, {4425,4697}
X(6703) = complement of X(1211)
X(6704) lies on these lines: {2, 32}, {3, 6249}, {5, 5092}, {620, 2023}, {732, 3589}, {1656, 6287}, {3246, 3634}, {3628, 6036}
X(6704) = midpoint of X(i) and X(j) for these {i,j}: {3,6249}, {83,6292}
X(6704) = complement of X(6292)
X(6705) lies on these lines: {2, 84}, {3, 10}, {4, 3911}, {40, 5744}, {57, 946}, {140, 971}, {142, 3358}, {443, 6256}, {499, 1709}, {631, 1490}, {916, 5482}, {942, 1387}, {1012, 1210}, {1125, 6001}, {1656, 6259}, {2095, 4301}, {3525, 5658}, {3601, 5768}, {3928, 5758}, {4853, 6282}
X(6705) = midpoint of X(i) and X(j) for these {i,j}: {3,6245}, {84,6260}, {142,3358}, {946,1158}
X(6705) = complement of X(6260)
X(6706) lies on these lines: {2, 85}, {10, 141}, {37, 3673}, {75, 4515}, {277, 3085}, {517, 2140}, {536, 3991}, {672, 4059}, {938, 4648}, {4000, 4646}, {5437, 5737}
X(6706) = midpoint of X(85) and X(1212)
X(6706) = complement of X(1212)
X(6707) lies on these lines: {1, 4399}, {2, 6}, {9, 4798}, {10, 4478}, {37, 4472}, {142, 3647}, {551, 4852}, {620, 2796}, {740, 1125}, {1086, 6651}, {1268, 6542}, {1698, 4851}, {3616, 4361}, {3622, 4405}, {3624, 4657}, {3626, 4889}, {3664, 4708}, {3723, 4967}, {3729, 4499}, {4000, 5550}, {4021, 4739}, {4422, 4698}, {4670, 5257}, {4725, 4909}
X(6707) = midpoint of X(i) and X(j) for these {i,j}: {1,4733}, {10,5625}, {86,1213}, {1086,6651}, {8723,4967}
X(6707) = complement of X(1213)
X(6708) lies on these lines: {2, 92}, {5, 10}, {9, 1730}, {123, 132}, {124, 3838}, {200, 3706}, {219, 5271}, {312, 3694}, {333, 1944}, {1746, 2182}, {1824, 1985}, {1829, 3142}, {1943, 6510}, {3085, 6051}, {3666, 4858}, {3739, 5745}, {3767, 3772}
X(6708) = midpoint of X(92) and X(1214)
X(6708) = complement of X(1214)
X(6709) lies on these lines: {2, 95}, {141, 575}, {3525, 6389}
X(6709) = midpoint of X(95) and X(233)
X(6709) = complement of X(233)
X(6709) = isotomic conjugate of polar conjugate of X(35884)
X(6710) lies on these lines: {2, 101}, {3, 118}, {103, 631}, {140, 2808}, {152, 3523}, {468, 5185}, {620, 2786}, {910, 5074}, {1125, 2809}, {1282, 3624}, {1362, 5433}, {2774, 5972}, {2784, 3634}, {2810, 3589}, {3022, 5432}, {3035, 3887}, {3041, 4999}, {3046, 5012}
X(6710) = midpoint of X(i) and X(j) for these {i,j}: {3,118}, {101,116}, {910,5074}
X(6710) = reflection of X(6712) in X(140)
X(6710) = complement of X(116)
X(6711) lies on these lines: {2, 102}, {3, 124}, {109, 631}, {140, 2818}, {620, 2792}, {960, 2800}, {1125, 2817}, {1361, 5432}, {1364, 5433}, {2779, 5972}, {2785, 6036}, {3042, 4999}
X(6711) = midpoint of X(i) and X(j) for these {i,j}: {3,124}, {102,117}
X(6711) = refelction of X(6718) in X(140)
X(6711) = complement of X(117)
X(6712) lies on these lines: {2, 103}, {3, 116}, {101, 631}, {140, 2808}, {150, 3523}, {544, 549}, {620, 2784}, {1362, 5432}, {2772, 5972}, {2786, 6036}, {2801, 3035}, {3022, 5433}
X(6712) = midpoint of X(i) and X(j) for these {i,j}: {3,116}, {103,118}
X(6712) = reflection of X(6710) in X(140)
X(6712) = complement of X(118)
X(6713) lies on these lines: {2, 104}, {3, 11}, {5, 2829}, {10, 140}, {35, 5533}, {36, 5841}, {40, 5442}, {80, 3576}, {100, 631}, {149, 3523}, {182, 5848}, {517, 1387}, {528, 549}, {620, 2783}, {1006, 4996}, {1125, 2800}, {1145, 3872}, {1317, 5432}, {1320, 5657}, {1537, 3306}, {1768, 3624}, {2077, 3582}, {2771, 5972}, {2787, 6036}, {4297, 6246}, {5054, 6174}, {5690, 5854}, {5883, 5901}
X(6713) = midpoint of X(i) and X(j) for these {i,j}: {3,11}, {104,119}, {4297,6246}
X(6713) = reflection of of X(i) in X(j) for these (i,j): (5,6667), (3035,140)
X(6713) = complement of X(119)
X(6714) lies on these lines: {2, 11}, {3, 5511}, {348, 1358}, {620, 2795}, {631, 1292}, {1083, 4904}, {1125, 2809}, {1212, 3039}, {2788, 6036}, {2836, 3848}, {3624, 5540}, {3756, 5272}
X(6714) = midpoint of X(i) and X(j) for these {i,j}: {3, 5511}, {105, 120}, {1083,4904}
X(6714) = complement of X(120)
X(6715) lies on these lines: {2, 106}, {3, 5510}, {620, 2796}, {631, 1293}, {1054, 3624}, {1125, 1387}, {1357, 5433}, {2789, 6036}, {2810, 3589}, {2842, 5972}, {3038, 4999}, {3872, 6095}, {5432, 6018}
X(6715) = midpoint of X(i) and X(j) for these {i,j}: {3,5510}, {106,121}
X(6715) = complement of X(121)
X(6716) lies on these lines: {2, 107}, {3, 133}, {4, 3184}, {5, 1539}, {402, 5972}, {620, 2797}, {631, 1294}, {2790, 6036}, {2803, 3035}, {3090, 5667}, {3324, 5433}
X(6716) = midpoint of X(i) and X(j) for these {i,j}: {3,133}, {4,3184}, {107,122}
X(6716) = complement of X(122)
X(6717) lies on these lines: {2, 108}, {5, 2829}, {620, 2798}, {631, 1295}, {676, 2804}, {1125, 2817}, {1359, 5433}, {2778, 3812}, {2791, 6036}, {2850, 5972}, {3318, 5432}
X(6717) = midpoint of X(108) and X(123)
X(6717) = complement of X(123)
X(6718) lies on these lines: {{2, 109}, {3, 117}, {102, 631}, {140, 2818}, {151, 3523}, {498, 1795}, {620, 2785}, {1125, 2800}, {1361, 5433}, {1364, 5432}, {2773, 5972}, {2792, 6036}, {3035, 3042}, {3040, 4999}
X(6718) = midpoint of X(i) and X(j) for these {i,j}: {3,117}, {109,124}
X(6718) = reflection of X(6711) in X(140)
X(6718) = complement of X(124)
X(6719) lies on these lines: {2, 99}, {3, 5512}, {69, 6387}, {468, 5140}, {631, 1296}, {754, 5913}, {2793, 6036}, {2805, 3035}, {2854, 3589}, {3048, 5012}, {3325, 5433}, {5432, 6019}
X(6719) = midpoint of X(i) and X(j) for these {i,j}: {3,5512}, {111,126}
X(6719) = complement of X(126)
X(6719) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(35369)
X(6720) lies on these lines: {2, 112}, {3, 132}, {5, 2794}, {339, 6103}, {620, 2492}, {631, 1297}, {1485, 5961}, {2781, 3589}, {2806, 3035}, {3320, 5433}, {5432, 6020}
X(6720) = midpoint of X(i) and X(j) for these {i,j}: {3,132}, {112,127}
X(6720) = complement of X(127)
X(6721) lies on these lines: {2, 98}, {5, 620}, {99, 3090}, {115, 1656}, {140, 2794}, {230, 5965}, {543, 547}, {576, 1007}, {2023, 3055}, {2482, 5055}, {2782, 3628}, {3526, 6033}
X(6721) = midpoint of X(i) and X(j) for these {i,j}: {5,620}, {114,6036}
X(6721) = reflection of X(6722) in X(3628)
X(6721) = complement of X(6036)
Let θ be a variable angle. Construct isosceles triangle BA′C on BC (outwardly if θ is positive, inwardly if θ is negative), such that ∠A′BC = ∠A′CB = θ. Define B′ and C′ cyclically. Let (Ia) be the incircle of BA′C, and define (Ib) and (Ic) cyclically. As θ varies, the locus of the radical center of (Ia), (Ib), (Ic) is the Kiepert hyperbola of the medial-of-medial triangle, with center X(6722). The same locus is obtained if nine-point circle is substituted for incircle. (Randy Hutson, July 20, 2016)
X(6722) lies on these lines: {2, 99}, {3, 23514}, {5, 2794}, {10, 11725}, {30, 15092}, {32, 32961}, {39, 33249}, {98, 3090}, {114, 1656}, {140, 7861}, {141, 14645}, {147, 7486}, {183, 626}, {187, 33228}, {230, 625}, {315, 33277}, {325, 31275}, {485, 13967}, {486, 8980}, {524, 9165}, {542, 547}, {549, 22515}, {597, 19662}, {631, 7872}, {632, 33813}, {690, 6723}, {1003, 18424}, {1007, 3767}, {1506, 3329}, {1569, 7786}, {1916, 14065}, {2023, 3934}, {2548, 32988}, {2782, 3628}, {2787, 6667}, {3054, 33184}, {3091, 34473}, {3523, 10723}, {3525, 21166}, {3526, 6321}, {3533, 13172}, {3545, 10722}, {3618, 5477}, {3619, 10754}, {3624, 13178}, {3763, 6034}, {3788, 13881}, {3815, 7817}, {3845, 26614}, {3849, 8355}, {5025, 7749}, {5054, 9880}, {5055, 6033}, {5056, 10991}, {5067, 14651}, {5070, 15561}, {5071, 9862}, {5206, 14063}, {5215, 8352}, {5326, 15452}, {5355, 7777}, {5368, 7858}, {5459, 32553}, {5460, 32552}, {5939, 6704}, {5969, 33213}, {5972, 15359}, {5976, 8363}, {6292, 7901}, {6390, 32457}, {6777, 22490}, {6778, 22489}, {6781, 14041}, {7603, 7792}, {7735, 7775}, {7737, 32984}, {7739, 34803}, {7747, 7857}, {7748, 33233}, {7752, 7755}, {7753, 7806}, {7756, 7907}, {7761, 11318}, {7765, 7769}, {7780, 14929}, {7781, 32958}, {7794, 7899}, {7795, 32955}, {7797, 9698}, {7800, 33199}, {7803, 32998}, {7810, 7934}, {7813, 7925}, {7814, 7890}, {7815, 14064}, {7818, 17008}, {7822, 33218}, {7825, 32972}, {7826, 7912}, {7827, 17005}, {7831, 14046}, {7845, 22329}, {7846, 33002}, {7847, 16923}, {7848, 13468}, {7851, 31455}, {7859, 16922}, {7863, 7940}, {7865, 34229}, {7867, 32458}, {7869, 8781}, {7889, 7942}, {7902, 31401}, {7914, 32951}, {7915, 33186}, {7918, 33015}, {7935, 33283}, {7983, 9780}, {8029, 19598}, {8176, 15484}, {8252, 13989}, {8253, 8997}, {8588, 33017}, {8722, 9754}, {9182, 23991}, {9861, 11484}, {10150, 22110}, {10175, 11710}, {10278, 12076}, {10352, 32999}, {11230, 11724}, {11313, 13873}, {11314, 13926}, {11632, 15703}, {11711, 19862}, {12117, 15709}, {13174, 19872}, {13175, 16419}, {13846, 13968}, {13847, 13908}, {13916, 13982}, {13917, 13981}, {15031, 33225}, {15059, 15357}, {15271, 33240}, {15491, 15699}, {15513, 33229}, {15515, 33000}, {16041, 21843}, {16509, 33896}, {19108, 32786}, {19109, 32785}, {35018, 35021
X(6722) = midpoint of X(i) and X(j) for these {i,j}: {2, 5461}, {5, 6036}, {10, 11725}, {114, 11623}, {115, 620}, {230, 625}, {597, 19662}, {2023, 3934}, {5972, 15359}, {6390, 32457}, {6669, 6670}, {6704, 9478}, {6721, 20398}
X(6722) = reflection of X(i) in X(j) for these {i,j}: {6721, 3628}, {20399, 6721}, {22247, 2}, {35022, 620}
X(6722) = complement of X(620)
X(6722) = X(14728)-Ceva conjugate of X(690)
X(6722) = crosssum of X(6) and X(20976)
X(6722) = orthoptic-circle-of-Steiner-inellipse-inverse of X(20099)
X(6722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 31274}, {2, 115, 620}, {2, 671, 9167}, {2, 7844, 4045}, {2, 9166, 2482}, {2, 14061, 115}, {2, 14971, 5461}, {5, 7886, 6680}, {5, 34127, 6036}, {99, 31274, 620}, {115, 2482, 148}, {115, 14061, 5461}, {115, 14971, 14061}, {115, 31274, 99}, {148, 9166, 115}, {620, 5461, 115}, {1007, 3767, 7798}, {1007, 7798, 7764}, {1506, 7828, 7829}, {3628, 20398, 20399}, {3767, 7862, 7764}, {3767, 32969, 7862}, {5025, 7749, 7830}, {7746, 7887, 626}, {7752, 7755, 7838}, {7798, 7862, 1007}, {7828, 32967, 1506}, {7852, 32992, 6704}, {7857, 32966, 7747}, {7925, 14568, 7813}, {7934, 17004, 7810}, {7942, 16921, 7889}, {14046, 17006, 7831}, {22247, 35022, 620}, {32832, 33248, 7867}
X(6723) lies on these lines: {2, 98}, {5, 1539}, {74, 3090}, {113, 1656}, {265, 3526}, {511, 5159}, {541, 547}, {632, 1511}, {895, 3619}, {974, 5907}, {1112, 5943}, {3066, 5094}, {3618, 5095}, {3628, 5663}, {3763, 5181}, {5070, 6053}
X(6723) = midpoint of X(i) and X(j) for these {i,j}: {5,6699}, {125,5972}, {974,5907}
X(6723) = complement of X(5972)
X(6723) = X(5159)-of-1st-Brocard-triangle
X(6723) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5984)
Homologous images: X(6724)-X(6796)
This preamble and centers X(6724)-X(6796) were contributed by Randy Hutson, March 15, 2015.
The term "homologous" means "having the same relation or relative position to". Let T1 and T2 be two triangles, and P a point in the plane of triangle T1. The point P′ in the plane of triangle T2 is a T1-to-T2 homologous image of P if P′ has the same relation or relative position to T2 as P has to T1. If T1 and T2 are similar triangles, this meaning is unambiguous, and we call P′ the T1-to-T2 similarity image of P. If, however, T1 and T2 are not similar, the term "homologous images" can apply to several kinds of mappings. We present three such mappings here.
Trilinear image:
Let P be a point with trilinears x : y : z wrt T1. Then the T1-to-T2 trilinear image of P is the point P′ with trilinears x : y : z wrt T2. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points. Parallel lines do not remain parallel under this mapping, so that points on the line at infinity map to finite points. Circumconics of T1 map to circumconics of T2 and inconics of T1 map to inconics of T2.
Barycentric image:
Let P be a point with barycentrics x : y : z wrt T1. Then the T1-to-T2 barycentric image of P is the point P′ with barycentrics x : y : z wrt T2. This is an affine collineation, preserving collinearities as well as ratios of distances between collinear points. Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Conics map to conics of the same type (ellipses, including circles, map to ellipses, parabolas to parabolas, hyperbolas to hyperbolas). Circumconics of T1 map to circumconics of T2 and inconics of T1 map to inconics of T2.
Functional image:
Let the sidelengths of T1 be denoted a1, b1, c1 and the sidelengths of T2 be denoted a2, b2, c2. Let P be a point with coordinates f(a1,b1,c1) : g(b1,c1,a1) : h(c1,a1,b1) with respect to T1, where f, g, h either have the same degree of homogeneity, or else one is the zero function and the other two have the same degree of homogeneity. (If P is a center of T1, then f = g = h.) Then the T1-to-T2-functional image of P is the point P′ with coordinates f(a2,b2,c2) : g(b2,c2,a2) : h(c2,a2,b2) with respect to T2. Coordinates here can be trilinears or barycentrics, as long as the same system is used for T1 and T2. P′ serves the same 'function' wrt T2 (e.g., centroid, circumcenter, 1st Brocard point, A-vertex of orthic triangle, etc.) as P serves wrt T1. For example, the orthic-to-excentral functional image of X(5) = X(40), since X(5) and X(40) are the respective circumcenters of the orthic and excentral triangles. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points (except where these distance ratios define the point, such as midpoints, reflections, etc.). Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Circles map to circles. Other conics map to conics, but not necessarily of the same type (e.g., the MacBeath circumconic of T1 may be an ellipse, but maps to the MacBeath circumconic of T2, which may be a hyperbola.) Circumconics of T1 map to circumconics of T2 and inconics of T1 map to inconics of T2.
Let A′B′C′ be the incentral triangle. Let OA be the circle centered at A and passing through A′. Define OB and OC cyclically. X(6724) is the trilinear pole of the Monge line of OA, OB, OC.
X(6724) lies on these lines: {1,6732}, {174,188}, {6727,6733}
X(6724) = isogonal conjugate of X(6727)
X(6724) = SS(a → a') of X(5), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6724) = trilinear square root of X(12)
X(6724) = {X(174),X(188)}-harmonic conjugate of X(266)
X(6725) lies on line {188,259}
X(6725) = SS(a → a') of X(53), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6725) = trilinear square root of X(6057)
X(6726) lies on line {174,188}
X(6726) = SS(a → a') of X(25), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6726) = trilinear square root of X(480)
The trilinear polar of X(6727) passes through X(6729). (Randy Hutson, April 11, 2015)
X(6727) lies on these lines: {6724,6733}
X(6727) = isogonal conjugate of X(6724)
X(6727) = SS(a → a') of X(54), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6727) = trilinear square root of X(60)
X(6728) lies on these lines: {6729,6730}
X(6728) = isogonal conjugate of X(6733)
X(6728) = SS(a → a') of X(523), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6728) = trilinear square root of X(11)
X(6728) = (trilinear polar of X(174))∩(trilinear polar of X(188))
X(6728) = crossdifference of every pair of points on line X(259)X(260)
X(6729) lies on these lines: {6728,6730}
X(6729) = SS(a → a') of X(647), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6729) = trilinear square root of X(3271)
X(6729) = crossdifference of every pair of points on line X(174)X(188)
X(6730) lies on these lines: {6728,6729}
X(6730) = SS(a → a') of X(2501), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6730) = trilinear square root of X(4081)
The trilinear polar of X(6731) passes through X(6730).
X(6731) lies on these lines: {8,178}, {174,556}, {188,259}
X(6731) = SS(a → a') of X(393), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6731) = {X(483),X(3082)}-harmonic conjugate of X(8)
Let A′,B′,C′ be the second points of intersection of the angle bisectors of ABC with its incircle. The tangents at A′, B′, C′ form a triangle A″B″C″, introduced here as the outer tangential mid-arc triangle. A″B″C″ is the reflection in X(1) of the (inner) tangential mid-arc triangle. A″B″C″ is homothetic to the inner Hutson triangle (defined at X(363)), and the center of homothety is X(6732).
X(6732) lies on these lines: {1,6724}, {174,7670}, {266,8092}, {5934,8080}, {7588,8109}, {8076,8107}, {8082,8111}, {8090,8138}
X(6732) = SS(a → a') of X(11), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6733) lies on these lines: {6724,6727}
X(6733) = isogonal conjugate of X(6728)
X(6733) = excentral-to-ABC trilinear image of X(1768)
X(6733) = SS(a → a') of X(110), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6733) = trilinear square root of X(59)
X(6733) = trilinear pole of line X(259)X(260)
X(6734) lies on these lines: {1,2}, {3,3419}, {4,63}, {5,72}, {11,960}, {12,518}, {20,5175}, {21,950}, {29,270}, {37,5742}, {40,3434}, {56,5792}, {57,377}, {75,225}
X(6734) = complement of X(34772)
X(6734) = {X(2),X(8)}-harmonic conjugate of X(78)
X(6734) = {X(8),X(10)}-harmonic conjugate of X(6735)
The trilinear polar of X(6735) passes through X(2804).
X(6735) lies on these lines: {1,2}, {11,3880}, {12,3838}, {21,5795}, {40,3436}, {63,2096}, {72,5690}, {100,515}, {119,517}, {153,2950}, {224,5534}, {280,6556}, {281,3692}, {307,322}, {318,341}, {355,1012}, {377,1706}, {392,3820}, {495,3753}, {497,3895}, {516,5080}, {522,3717}, {529,1155}, {594,3965}, {765,1861}, {944,4855}, {950,3871}, {952,5440}, {962,5828}, {1018,5179}, {1056,3306}, {1146,3693}, {1260,3419}, {1319,3035}, {1329,3057}, {1376,1470}, {1697,2478}, {1738,4695}, {1837,3913}, {2093,5905}, {2321,4165}, {2551,5250}, {2802,3814}, {2975,6684}, {3036,3689}, {3158,5727}, {3434,5587}, {3452,3877}, {3583,5541}, {3813,3893}, {3816,5919}, {3826,4731}, {3868,4848}, {3876,5837}, {3885,4193}, {3911,5193}, {3951,5815}, {4004,6147}, {4095,4167}, {4188,4311}, {5048,5854}, {5082,5818}, {5288,5445}
X(6735) = midpoint of X(i) and X(j) for these {i,j}: {8,4511}, {100,5176}, {3583,5541}
X(6735) = reflection of X(i) ini X(j) for these (i,j): (11,5123), (1319,3035), (1519,119), (1737,10), (4511, 6745)
X(6735) = AC-incircle-inverse of X(6790)
X(6735) = circumconic-centered-at-X(1)-inverse of X(78)
X(6735) = X(2743)-complementary conjugate of X(513)
X(6735) = X(3262)-Ceva conjugate of X(908)
X(6735) = X(1145)-cross conjugate of X(8)
X(6735) = crossdifference of every pair of points on line X(604)X(649)
X(6735) = X(i)-beth conjugate of X(j) for these (i,j): (8,1737}, (341,6735}, (643,2077}, (1043,519}, (3699,6735}, (6735,517}
X(6735) = X(i)-gimel conjugate of X(j) for these (i,j): (513,6735), (3536,6735)
X(6735) = X(i)-isoconjugate of X(j) for these (i,j): {34,1795}, {56,104}, {57,909}, {104,56}, {269,2342}, {513,2720}, {651,2423}, {909,57}, {1398,1809}, {1412,2250}, {1415,2401}, {1795,34}, {1809,1398}, {2250,1412}, {2342,269}, {2401,1415}, {2423,651}, {2720,513}
X(6735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2057,78), (2,8,3872), (8,10,6734), (8,4420,6737), (8,5552,1), (10,6736,8), (200,3679,8), (495,3753,5249), (3421,5657,63), (3626,6737,8)
X(6736) lies on these lines: {1,2}, {11,3893}, {40,2123}, {55,5795}, {72,1145}, {100,4297}, {142,3698}, {210,5837}, {226,5836}, {322,3668}, {341,4082}, {346,6556}, {388,1706}, {404,4315}, {480,1837}, {495,3824}, {497,2136}, {515,5687}, {516,3436}, {518,4848}
X(6736) = complement of X(36846)
X(6736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10,8582), (2,8,4853), (8,10,4847)
X(6737) lies on these lines: {1,2}, {72,515}, {210,5793}, {219,2321}, {950,960}, {3057,3059}
X(6737) = {X(1),X(8)}-harmonic conjugate of X(4847)
X(6737) = anticomplement of X(6738)
X(6738) lies on these lines: {1,2}, {4,3671}, {7,5691}, {20,3339}, {40,3488}, {46,4304}, {57,3486}, {65,516}
X(6738) = midpoint of X(65) and X(950)
X(6738) = reflection of X(4298) in X(942)
X(6738) = complement of X(6737)
X(6738) = X(8)-of-3rd-pedal-triangle-of-X(1)
The locus of the trilinear square of a point on the line at infinity is the inellipse centered at X(10). This inellipse passes through X(244), X(1099), X(1109), X(1111), X(4712), X(4736) and X(4738), and its Brianchon point (perspector) is X(75). The tangents to this inellipse at X(1099) and X(4736) intersect at X(6739).
X(6739) lies on these lines: {2,6740}, {8,6742}, {10,6741}, {11,214}, {30,113}, {4707,4736}
X(6739) = reflection of X(3109) in X(5972)
X(6739) = crossdifference of every pair of points on line X(2159)X(2433)
X(6739) = complement of X(6740)
X(6740) lies on these lines: {1,564}, {2,6739}, {8,643}, {10,21}, {30,74}
X(6740) = isogonal conjugate of X(1464)
X(6740) = anticomplement of X(6739)
X(6740) = trilinear pole of line X(9)X(1021)
X(6741) lies on these lines: {2,6742}, {8,643}, {10,6739}, {11,4092}, {125,523}
X(6741) = center of hyperbola {A,B,C,X(8),X(10)}
X(6741) = crossdifference of every pair of points on line X(163)X(2420)
X(6741) = complement of X(6742)
Line X(9)X(46) is the trilinear polar of the Feuerbach point, X(11), wrt the Feuerbach triangle.
X(6742) lies on these lines: {1,564}, {2,6741}, {8,6739}, {110,476}
X(6742) = isogonal conjugate of X(2605)
X(6742) = isotomic conjugate of X(4467)
X(6742) = anticomplement of X(6741)
X(6743) = anticomplement of X(6744)
X(6744) lies on these lines: {1,2}, {4,5542}, {516,942}, {3812,5853}
X(6744) = complement of X(6743)
X(6744) = incircle-inverse of X(5529)
X(6744) = X(10) of 3rd pedal triangle of X(1)
X(6745) is the point of intersection of the line X(1)X(8) and the trilinear polar of X(8); c.f. X(1323).
X(6745) lies on these lines: {1,2}, {9,1776}, {11,3689}, {55,3452}, {72,5884}, {100,516}, {522,656}
X(6745) = midpoint of X(i) and X(j) for these (i,j): (11,3689), (100,908)
X(6745) = reflection of X(3911) in X(3035)
X(6745) = inverse-in-circumconic-centered-at-X(1) of X(200)
X(6745) = trilinear pole of line X(6068)X(6366) (the tangent to the Mandart inellipse at the antipode of X(11), i.e. X(6068))
X(6745) = crossdifference of every pair of points on line X(56)X(649)
X(6745) = centroid of {A,B,C,X(3935)}
X(6745) = perspector of ABC and side-triangle of extouch and Yff contact triangles
Let A′B′C′ be the reflection triangle. Let A″ be the orthogonal projection of A on line B′C′, and define B″, C″ cyclically. Triangle A″B″C″ is perspective to the orthic triangle, and the perspector is X(6746).
X(6746) lies on these lines: {4,94}, {24,5944}, {25,1614}, {51,235}, {52,427}, {1593,3060}
X(6746) = X(12) of orthic triangle if ABC is acute
X(6746) = {X(4),X(143)}-harmonic conjugate of X(1112)
X(6747) lies on these lines: {4,54}, {53,232}, {125,2052}
X(6747) = X(31) of orthic triangle if ABC is acute
X(6748) lies on these lines: {4,6}, {5,577}, {25,160}, {30,216}, {32,1595}, {39,6756}, {50,252}, {51,6752}, {125,138}, {140,233}, {141,317}, {184,6755}, {230,427}, {232,428}, {235,5063}, {264,524}, {275,1971}, {297,3589}, {340,3631}, {395,473}, {396,472}, {460,1843}, {468,3055}, {546,3284}, {566,6240}, {570,3575}, {590,1586}, {594,5081}, {615,1585}, {1100,1785}, {1321,1322}, {1593,1609}, {1596,5065}, {1598,2548}, {1841,1877}, {1885,3003}, {1907,1968}, {3053,3088}, {3054,5094}, {3627,5158}, {5064,5306}
Let A′B′C′ be the orthic triangle. Let OA be the circle with center A passing through A′. Let RA be the radical axis of OA and the nine-point circle; define RB, RC cyclically. Let A″ = RB∩RC, B″ = RC∩RA, C″ = RA∩RB. The lines A′A″, B′B″, C′C′ concur in X(6748).
X(6748) = isogonal conjugate of X(31626)
X(6748) = crosssum of X(3) and X(216)
X(6748) = crosspoint of X(4) and X(275)
X(6748) = {X(4),X(6)}-harmonic conjugate of X(53)
X(6748) = inverse-in-orthosymmedial-circle of X(53)
X(6748) = X(37) of orthic triangle if ABC is acute
X(6748) = {X(1321),X(1322)}-harmonic conjugate of X(7507)
X(6749) lies on these lines: {4,6}, {5,3284}, {140,577}, {216,550}, {239,5094}
X(6749) = {X(4),X(6)}-harmonic conjugate of X(1990)
X(6749) = inverse-in-orthosymmedial-circle of X(1990)
X(6749) = X(45) of orthic triangle if ABC is acute
X(6750) lies on these lines: {4,54}, {5,53}
X(6750) = X(58) of orthic triangle if ABC is acute
X(6750) = orthic isogonal conjugate of X(389)
X(6750) = X(4)-Ceva conjugate of X(389)
X(6750) = inverse-in-polar-circle of X(3484)
X(6751) is the QA-P19 center (AntiComplement of QA-P16 wrt the QA-Diagonal Triangle) of quadrangle ABCX(4); see QA-P19.
X(6751) lies on these lines: {51,53}, {185,1503}
X(6751) = X(69) of orthic triangle
X(6751) = orthic isotomic conjugate of X(52)
X(6751) = anticomplement of X(53) wrt orthic triangle
X(6751) = {X(185),X(1843)}-harmonic conjugate of X(6752)
X(6752) lies on these lines: {6,1987}, {185,1503}
X(6752) = orthic isotomic conjugate of X(4)
X(6752) = X(75) of orthic triangle if ABC is acute
X(6752) = {X(185),X(1843)}-harmonic conjugate of X(6751)
X(6753) is the perspector of the circumconic centered at X(135), which is hyperbola {A,B,C,X(4),X(24)}. This hyperbola is the isogonal conjugate of line X(3)X(68), and intersects the circumcircle (other than at A, B, and C) at X(1299).
The trilinear polar of X(6753) passes through X(6754).
X(6753) lies on these lines: {230,231}, {512,2623}
X(6753) = X(652) of orthic triangle if ABC is acute
X(6753) = (trilinear polar of X(4))∩(trilinear polar of X(24))
X(6753) = crossdifference of every pair of points on line X(3)X(68)
X(6753) = X(2)-Ceva conjugate of X(135)
X(6753) = X(4)-Ceva conjugate of X(6754)
X(6753) = polar conjugate of isogonal conjugate of X(34952)
X(6753) = polar conjugate of anticomplement of X(39013)
X(6753) = orthic-isogonal conjugate of X(6754)
X(6753) = X(63)-isoconjugate of X(925)
X(6754) lies on these lines: {52,5095} et al.
X(6754) = isogonal conjugate of isotomic conjugate of X(34338)
X(6754) = X(653) of orthic triangle if ABC is acute
X(6754) = trilinear pole, wrt orthic triangle, of line X(4)X(52)
X(6754) = X(4)-Ceva conjugate of X(6753)
X(6754) = orthic isogonal conjugate of X(6753)
X(6754) = polar conjugate of isotomic conjugate of X(39013)
Let A′B′C′ be the Euler triangle. Let A″B″C″ be the triangle bounded by the trilinear polars of A′, B′, C′. The lines AA″, BB″, CC″ concur in X(53). Let A*B*C* be the triangle bounded by the trilinear polars of A″, B″, C″. The lines AA*, BB*, CC* concur in X(6755).
X(6755) lies on these lines: {2,3}, {51,53}
X(6755) = X(940) of orthic triangle if ABC is acute
As a point of the Euler line, X(6756) has Shingawa coefficients (F, -2E - 3F).
X(6756) lies on these lines: {2,3}, {52,1843}
X(6756) = X(942) of orthic triangle if ABC is acute
X(6756) = X(5) of 2nd pedal triangle of X(4)
X(6756) = X(3) of 3rd pedal triangle of X(4)
X(6756) = {X(3),X(4)}-harmonic conjugate of X(1595)
X(6756) = homothetic center of orthocevian triangle of X(2) and Euler triangle
X(6756) = {X(4),X(24)}-harmonic conjugate of X(427)
A construction for X(6757) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.
X(6757) lies on these lines: {1,564}, {5,523}, {8,79}, {92,5248}, {265,355}, {281,451}, {993,1789}, {1441,3841}, {2321,4053}, {3678,6358}
X(6757) = Feuerbach-to-ABC barycentric image of X(5948)
X(6757) = trilinear product of vertices of X(1)-altimedial triangle
X(6758) lies on these lines: {2,1109}, {8,4736}, {100,523}, {6327,6360}
X(6758) = isotomic conjugate of X(7372)
X(6758) = anticomplement of X(1109)
X(6758) = polar conjugate of isogonal conjugate of X(23084)
Let A′B′C′ be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); X(6759) = X(4)-of-A′B′C′. (Randy Hutson, July 21, 2017)
X(6759) lies on these lines: {3,64}, {4,54}, {5,182}, {6,1598}, {20,110}, {22,5562}, {23,5889}, {24,185}, {25,389}, {30,156}, {52,161}, {155,159}
X(6759) = X(10) of tangential triangle if ABC is acute
X(6759) = inverse-in-circumcircle of X(6760)
Let A′B′C′ be the reflection of the anticevian triangle of X(3) in the trilinear polar of X(3). The lines AA′, BB′, CC′ concur in X(6760).
X(6760) lies on these lines: {2,6761}, {3,64}, {30,1294}
X(6760) = complement of X(6761)
X(6760) = inverse-in-circumcircle of X(6759)
X(6760) = crosssum of X(36302) and X(36303)
X(6760) = crossdifference of every pair of points on line X(53)X(6587)
X(6761) lies on these lines: {2,6760}, {4,51}, {5,3462}, {30,5667}
X(6761) = anticomplement of X(6760)
X(6761) = inverse-in-polar-circle of X(389)
X(6762) lies on these lines: {1,6}, {3,3158}, {8,57}, {10,1056}, {12,5231}, {20,5853}, {40,376}, {56,200}, {63,145}, {65,4853}, {84,517}
X(6762) = mixtilinear-incentral-to-mixtilinear-excentral similarity image of X(9)
X(6762) = {X(8),X(57)}-harmonic conjugate of X(1706)
X(6762) = X(8) of tangential triangle of excentral triangle
X(6762) = X(1498) of excentral triangle
X(6763) lies on these lines: {1,21}, {2,3337}, {3,5904}, {4,5536}, {8,484}, {9,583}, {10,3218}, {12,57}, {35,518}, {36,72}, {40,550}, {56,3927}, {65,5258}, {75,267}, {79,2886}
X(6763) = {X(1),X(63)}-harmonic conjugate of X(191)
X(6763) = X(12) of tangential triangle of excentral triangle
X(6764) lies on these lines: {1,2}, {7,3555}, {20,5853}, {518,962}
X(6764) = anticomplement of X(6765)
X(6765) lies on these lines: {1,2}, {3,3158}, {4,5853}, {6,4515}, {9,3295}, {40,518}, {56,3689}, {57,3555}, {1058,3452}
X(6765) = complement of X(6764)
X(6765) = perspector of pedal and antipedal triangles of X(84)
X(6765) = X(10) of 3rd antipedal triangle of X(1)
Let A′B′C′ be the Hutson extouch triangle (defined at X(5731)). Let LA be the tangent to the A-excircle at A′, and define LB, LC cyclically. Let A″ = LB∩LC, B″ = LC∩LA, C″ = LA∩LB. Triangle A″B″C″ is homothetic to ABC at X(57), and X(6766) = X(40) of A″B″C″.
X(6766) lies on these lines: {1,3}, {9,4301}
Let A′B′C′ be a triangle in the poristic system of triangles (triangles that share the same incircle and circumcircle with ABC). Let X be the trilinear pole of line X(1)X(3) wrt A′B′C′. The locus of X as A′B′C′ varies is an ellipse, E, centered at X(3) with vertices X(1381) and X(1382). E is the poristic locus of X(651). Let F1 and F2 be the foci of E. Then X(6767) is the {F1,F2}-harmonic conjugate of X(1).
Let L be a line through X(1), and let P and Q be the circumcircle intercepts of L. Let U be the {X(1),P}-harmonic conjugate of Q. The locus of U as L varies is an ellipse centered at X(6767).
Let (Ap) denote the Apollonian circle of the mixtilinear circles. The touch point of (Ap) and the A-mixtilinear circle is the point with barycentrics a(a^2+2 a b-3 b^2+2 a c+6 b c-3 c^2) : -2 b^2 (-a+b-c) : 2 c^2 (a+b-c)}. (Peter Moses, October 4, 2015)
The vertices of the 8th mixtilinear triangle are the touchpoints of the inner Apollonian circle of the mixtilinear incircles, with A-vertex, A′, given by these trilinears:
A′ = a^2 + 2*a*b - 3*b^2 + 2*a*c + 6*b*c - 3*c^2 : -2*b*(-a + b - c) : 2*(a + b - c)*c (Dan Reznik and Peter Moses, September 10, 2021).
If you have The Geometer's Sketchpad, you can view X(6767).
X(6767) lies on these lines: {1,3}, {5,1058}, {8,5284}, {11,5055}, {12,3851}, {21,3623}, {30,390}, {145,405}, {381,495}, {382,388}, {386,1616}, {392,3870}, {474,3622}, {480,4915}, {496,1656}, {498,5070}, {499,5326}, {519,1001}, {546,5261}, {549,5281}, {550,3600}, {551,1376}, {611,5093}, {758,4068}, {938,5690}, {943,7320}, {952,954}, {956,1621}, {958,3244}, {962,6147}, {993,4428}, {1000,2346}, {1012,7967}, {1015,5024}, {1057,1807}, {1059,7100}, {1124,6418}, {1125,3913}, {1149,2177}, {1203,2334}, {1260,3872}, {1335,6417}, {1384,2242}, {1478,3058}, {1479,3843}, {1480,2293}, {1483,3560}, {1597,1870}, {1598,6198}, {1657,4294}, {2066,6199}, {2308,3915}, {3086,3526}, {3297,3312}, {3298,3311}, {3299,6501}, {3301,6500}, {3530,5265}, {3534,4293}, {3555,3927}, {3616,5687}, {3621,5047}, {3633,5259}, {3635,5248}, {3646,4882}, {3656,4342}, {3679,4423}, {3753,3895}, {3877,3957}, {3880,6600}, {3890,5730}, {3898,5289}, {3920,5020}, {4302,5434}, {4309,7354}, {4387,4692}, {5044,6765}, {5054,5218}, {5073,6284}, {5076,5229}, {5219,7743}, {5414,6395}, {5603,5719}, {5703,5901}, {5722,5790}
X(6767) = midpoint of X(1) and X(6766)
X(6767) = reflection of X(6769) in X(1385)
X(6767) = X(3)-of-8th-mixtilinear-triangle
X(6767) = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X(3)
X(6767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,7373), (1,35,3304), (1,40,5045), (1,55,999), (1,57,5049), (1,1697,942), (1,3295,3), (1,3303,3295), (1,3746,56), (1,5119,354), (40,5045,5708), (55,56,5010), (55,999,3), (392,3870,3940), (495,497,381), (496,3085,1656), (954,3488,6913), (999,3295,55), (1621,3241,956), (1870,7071,1597), (3555,5250,3927), (3576,6244,3), (3622,3871,474), (3746,5010,55), (3748,5919,1), (3895,4666,3753)
Trilinears are 1/|A′A″| : 1/|B′B″| : 1/|C′C″| , where A′B′C′ and A″B″C″ are the pedal and antipedal triangles of X(1)
X(6768) lies on these lines: {}
X(6768) = mixtilinear-excentral-to-ABC trilinear image of X(1)
X(6769) lies on these lines: {1,3}, {4,200}, {19,1802}
X(6769) = X(3) of 3rd antipedal triangle of X(1)
X(6770) lies on these lines: {2,98}, {3,299}, {4,13}, {6,383}, {15,6778}, {30,5611}, {69,5980}, {115,5334}, {298,3564}, {376,530}, {396,1080}, {511,3180}, {617,2782}, {618,631}, {1350,5859}, {2549,5335}, {2794,6772}, {3090,6669}, {3107,7709}, {3524,5463}, {3545,5459}, {5470,5479}, {6268,6306}, {6270,6302}, {7967,7975}
X(6770) = reflection of X(i) in X(j) for these (i,j): (4, 13), (147, 5613), (616, 3), (1080, 396), (5617, 6771), (6773, 98)
X(6770) = anticomplement of X(5617)
X(6770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6776,6773), (3,5873,634), (182,5613,2), (5617,6771,2)
X(6771) lies on these lines: {2,98}, {3,13}, {5,6669}, {15,115}, {16,5472}, {30,5459}, {298,5965}, {396,511}
X(6771) = complement of X(5617)
X(6771) = midpoint of X(5617) and X(6770)
X(6771) = reflection of X(6774) in X(6036)
X(6771) = {X(2),X(182)}-harmonic conjugate of X(6774)
Let A′ be the orthocenter of BCX(13), and define B′, C′ cyclically. X(6772) is X(6) of A′B′C′.
X(6772) lies on these lines: {2,99}, {3,6108}, {6,530}, {13,15}, {14,2782}, {62,6779}, {298,538}, {2794,6770}
X(6772) = X(298) of 1st Brocard triangle
X(6772) = reflection of X(6775) in X(115)
X(6772) = {X(2),X(2549)}-harmonic conjugate of X(6775)
X(6772) = intersection, other than X(15), of line X(13)X(15) and circle {{X(6), X(14), X(15)}}
X(6773) lies on these lines: {2,98}, {3,298}, {4,14}, {6,1080}, {16,6777}, {30,5615}, {69,5981}, {299,3564}, {511,3181}, {2794,6775}
X(6773) = anticomplement of X(5613)
X(6773) = reflection of X(5613) in X(6774)
X(6773) = reflection of X(6770) in X(98)
X(6773) = {X(2),X(6776)}-harmonic conjugate of X(6770)
X(6774) lies on these lines: {2,98}, {3,14}, {5,6670}, {15,5471}, {16,115}, {30,5460}, {299,5965}, {395,511}
X(6774) = complement of X(5613)
X(6774) = midpoint of X(5613) and X(6773)
X(6774) = reflection of X(6771) in X(6036)
X(6774) = {X(2),X(182)}-harmonic conjugate of X(6771)
Let A′ be the orthocenter of BCX(14), and define B′, C′ cyclically. X(6775) is X(6) of A′B′C′.
X(6775) lies on these lines: {2,99}, {3,6109}, {6,531}, {13,2782}, {14,16}, {61,6780}, {299,538}, {2794,6773}
X(6775) = X(299) of 1st Brocard triangle
X(6775) = reflection of X(6772) in X(115)
X(6775) = {X(2),X(2549)}-harmonic conjugate of X(6772)
X(6775) = intersection, other than X(16), of line X(14)X(16) and circle {{X(6), X(13),X(16)}}
Let LA be the polar of X(4) wrt the circle centered at A and passing through X(6). Define LB, LC cyclically. (X(4) is the perspector of any circle centered at a vertex of ABC.) Let A′ = LB∩LC, and define B′, C′ cyclically. The orthocenter of triangle A′B′C′ is X(6776).
From Hyacinthos #20428, Bui Quang Tuan, 11/26/2011:
Let A*B*C* be the orthic triangle and L the de Longchamps point, X(20). Let A″, B″, C″ be the midpoints of LA*, LB*, LC*. Triangle A″B″C″ is perspective to the circumcevian triangle of X(3), and the perspector is X(6776).
X(6776) is the perspector of the Artzt triangle and the intriangle of X(2). (The intriangle of a point given by trilinears x : y : z is the central triangle having A-vertex 0 : y + z cos A : z + y cos A; see TCCT, p. 196. Thus, the A-vertex of the intriangle of X(2) is 0 : c + b cos A : b + c cos A.) Contributed by César Lozada, February 11, 2017.
X(6776) lies on these lines: {2,98}, {3,69}, {4,6}, {20,185}, {30,1351}
X(6776) = complement of X(5921)
X(6776) = anticomplement of X(1352)
X(6776) = reflection of X(i) in X(j) for these (i,j): (4,6), (69,3), (1352,182)
X(6776) = X(20) of 1st Brocard triangle
X(6776) = crossdifference of every pair of points on line X(520)X(2451)
X(6776) = {X(6770),X(6773)}-harmonic conjugate of X(2)
X(6776) = homothetic center of 2nd Hyacinth triangle and cross-triangle of Aries and 2nd Hyacinth triangles
X(6776) = X(9)-of-circumorthic-triangle if ABC is acute
X(6776) = X(4)-of-obverse-triangle-of-X(69)
X(6776) = Ehrmann-mid-to-Johnson similarity image of X(6)
X(6776) = second-Lemoine-circle-inverse of X(34137)
X(6777) lies on these lines: {6,13}, {15,5617}, {16,6773}, {18,98}, {61,147}, {99,298}
X(6777) = reflection of X(i) in X(j) for these (i,j): (13,14), (6778,115)
X(6777) = anticomplement of X(32552)
X(6777) = X(16)-Fuhrmann-to-X(15)-Fuhrmann similarity image of X(16)
X(6777) = inverse-in-X(15)-Fuhrmann-circle of X(15)
X(6777) = {X(13),X(14)}-harmonic conjugate of X(5469)
X(6778) lies on these lines: {6,13}, {15,6770}, {16,5613}, {17,98}, {62,147}, {99,299}
X(6778) = reflection of X(i) in X(j) for these (i,j): (14,13), (6777,115)
X(6778) = anticomplement of X(32553)
X(6778) = X(15)-Fuhrmann-to-X(16)-Fuhrmann similarity image of X(15)
X(6778) = inverse-in-X(16)-Fuhrmann-circle of X(16)
X(6778) = {X(13),X(14)}-harmonic conjugate of X(5470)
X(6779) lies on these lines: {2,13}, {14,5615}, {62,6772}, {99,532}, {6780,6781}
X(6779) = reflection of X(i) in X(j) for these (i,j): (13,16), (6780,6781)
X(6779) = X(14)-antipedal-to-X(13)-antipedal similarity image of X(14)
X(6779) = inverse-in-antipedal-circle-of-X(13) of X(13)
X(6780) lies on these lines: {2,14}, {13,5611}, {61,6775}, {99,533}, {6779,6781}
X(6780) = reflection of X(i) in X(j) for these (i,j): (14,15), (6779,6781)
X(6780) = X(13)-antipedal-to-X(14)-antipedal similarity image of X(13)
X(6780) = inverse-in-antipedal-circle-of-X(14) of X(14)
X(6781) lies on these lines: {3,1506}, {4,5206}, {6,3534}, {15,5472}, {16,5471}, {30,115}, {99,754}, {511,1569}, {6779,6780}
X(6781) = midpoint of X(6779) and X(6780)
Let O* be the circle with segment X(13)X(16) as diameter (and center X(6108)). Let P be the perspector of O*. Then X(6782) is the trilinear pole of the polar of P with respect to O*. See also X(3292), X(5642), and X(6783).
X(6782) lies on these lines: {6,5617}, {13,3545}, {14,530}, {15,298}, {16,6773}, {114,230}, {395,542}, {524,6109}
X(6782) = reflection of X(6783) in X(230)
X(6782) = {X(114),X(5477)}-harmonic conjugate of X(6783)
Let O* be the circle with segment X(14)X(15) as diameter (and center X(6109)). Let P be the perspector of O*. Then X(6783) is the trilinear pole of the polar of P with respect to O*. See also X(3292), X(5642), and X(6782).
X(6783) lies on these lines: {6,5613}, {13,531}, {14,3545}, {15,6770}, {16,299}, {114,230}, {396,542}, {524,6108}
X(6783) = reflection of X(6782) in X(230)
X(6783) = {X(114),X(5477)}-harmonic conjugate of X(6782)
Let P and P* be antipodes on the circumcircle. Let Q be the intersection, other than P, of the circumcircle and line X(6)P. Let G' be the centroid of the (degenerate) pedal triangle of P. Then G' is the crossdifference of P* and Q. The locus of G' as P varies is an ellipse centered at X(2), with major axis line X(2)X(2574), and passing through X(125), X(3756), X(5642), X(6784), X(6786), X(6791) and X(6793). This ellipse is homothetic to the Hutson centroidal ellipse (defined at X(5943)), with insimilicenter X(373) and exsimilicenter X(51).
The reflection of P in G' lies on the orthocentroidal circle, and is P of the orthocentroidal triangle.
X(6784) lies on the above described ellipse and these lines: {2,6786}, {51,2871}, {98,6785}, {115,512}, {125,526}, {373,597}, {511,6055}
X(6784) = reflection of X(6786) in X(2)
X(6784) = midpoint of X(98) and X(6785)
X(6784) = crossdifference of every pair of points on line X(99)X(1625)
X(6785) lies on the orthocentroidal circle and these lines: {2,51}, {4,512}, {381,6787}, {98,6784}
X(6785) = reflection of X(98) in X(6784)
X(6785) = 1st-Brocard-to-orthocentroidal similarity image of X(3)
X(6785) = X(6233)-of-4th-Brocard-triangle
X(6785) = Λ(X(6), X(13)) wrt orthocentroidal triangle
X(6786) lies on the ellipse described at X(6784) and these lines: {2,6784}, {99,6787}, {114,325}, {125,126}, {373,3815}, {512,2482}, {526,5642}
X(6786) = reflection of X(6784) in X(2)
X(6786) = midpoint of X(99) and X(6787)
X(6786) = crossdifference of every pair of points on line X(98)X(729)
X(6786) = pole of orthic axis wrt Thomson-Gibert-Moses hyperbola
X(6787) lies on the orthocentroidal circle and these lines: {2,512}, {4,69}, {99,6786}, {381,6785}
X(6787) = reflection of X(99) in X(6786)
X(6787) = 1st-Brocard-to-orthocentroidal similarity image of X(6)
X(6787) = Λ(X(115), X(125)) wrt orthocentroidal triangle
X(6788) lies on the orthocentroidal circle, the Fuhrmann circle, the circle O(1,4), and these lines: {1,2}, {4,2457}, {106,952}, {121,3699}
X(6788) = complement of X(6790)
X(6788) = anticomplement of X(6789)
X(6788) = X(107) of Fuhrmann triangle
X(6788) = reflection of X(106) in X(3756)
X(6788) = inverse-in-polar-circle of X(7649)
X(6788) = intersection, other than X(4), of the orthocentroidal and Fuhrmann circles
X(6788) = intersection of Nagel lines of 1st and 2nd Ehrmann circumscribing triangles
X(6788) = intersection of Nagel lines of anticevian triangles of PU(4)
X(6789) lies on circles O(1,3), O(2,3), and these lines: {1,2}, {3,3667}, {121,952}, {3756,6715}
X(6789) = complement of X(6788)
X(6789) = X(106) of X(1)-Brocard triangle
X(6789) = X(106) of X(2)-Brocard triangle
X(6789) = intersection, other than X(2), of Nagel line and circle O(2,3) (circle {{X(2), X(3), X(3111), X(5108)}})
X(6789) = intersection, other than X(3), of circles O(1,3) and O(2,3)
X(6789) = midpoint of segment cut off by intersection of circumcircle and Nagel line
X(6789) = inverse-in-circumcircle of X(4057)
X(6789) = inverse-in-{circumcircle, nine-point circle}-inverter of X(3006)
X(6790) lies on circle O(1,20) and these lines: {1,2}, {20,3667}, {952,3699}
X(6790) = anticomplement of X(6788)
X(6790) = AC-incircle-inverse of X(6735)
X(6790) = intersection, other than X(1), of Nagel line and circle O(1,20) (circle {{X(1), X(20), X(3109)}})
Let LA, LB, LC be the lines through A, B, C, resp. parallel to line X(2)X(6). Let MA, MB, MC be the reflections of lines BC, CA, AB in LA, LB, LC, resp. Let A′ = MB∩MC, and define B′, C′ cyclically. 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 line X(2)X(6). The triangle A″B″C″ is homothetic to ABC, with center of homothety X(6791). (See Hyacinthos #16741/16782, Sep 2008.)
X(6791) lies on the ellipse described at X(6784) and these lines: {2,5503), {6,5642}, {111,542}, {115,125}, {126,5969}, {373,3815}, {1499,2686}
X(6791) = midpoint of X(111) and X(6792)
X(6791) = crossdifference of every pair of points on line X(110)X(1296)
Let A′ be the intersection, other than X(4), of the A-altitude and the orthosymmedial circle, and define B′ and C′ cyclically. Triangle A′B′C′ is here introduced as the 1st orthosymmedial triangle. A′B′C′ is inversely similar to ABC, with similitude center X(251). X(6792) is X(111) of A′B′C′.
Let A″ be the intersection, other than X(6), of the A-symmedian and the orthosymmedial circle, and define B″ and C″ cyclically. Triangle A″B″C″ is here introduced as the 2nd orthosymmedial triangle. A″B″C″ is the circummedial triangle of A′B′C′. A′B′C′ and A″B″C″ are perspective at X(51).
Let A′ be the trilinear product of the vertices of the A-adjunct anti-altimedial triangle, and define B′ and C′ cyclically. Triangle A′B′C′ is the anticomplementary triangle of the 1st Brocard triangle and is similar to the orthocentroidal triangle, with similitude center X(6792). (Randy Hutson, November 2, 2017)
X(6792) lies on the orthocentroidal circle, the orthosymmedial circle, the Hutson-Parry circle, and these lines: {2,6}, {4,1499}, {110,5477}, {111,542}
X(6792) = complement of X(38940)
X(6792) = orthogonal projection of X(4) on line X(2)X(6)
X(6792) = intersection, other than X(2), of line X(2)X(6) and orthocentroidal circle
X(6792) = intersection, other than X(6), of line X(2)X(6) and orthosymmedial circle
X(6792) = intersection, other than X(4), of orthocentroidal and orthosymmedial circles
X(6792) = anticomplement of X(5108)
X(6792) = reflection of X(111) in X(6791)
X(6792) = reflection of X(111) in line X(115)X(125)
X(6792) = X(110) of 4th Brocard triangle
X(6792) = X(111) of orthocentroidal triangle
X(6792) = X(111) of 1st orthosymmedial triangle
X(6792) = inverse-in-polar-circle of X(2501)
X(6792) = inverse-in-{circumcircle, nine-point circle}-inverter of X(230)
X(6792) = 1st-Brocard-to-orthocentroidal similarity image of X(5108)
X(6792) = Λ(X(6), X(110)) wrt orthocentroidal triangle
X(6792) = intersection, other than X(5640), of Hutson-Parry circles of ABC and orthocentroidal triangle
X(6792) = intersection, other than X(4), of orthocentroidal circle and X(2)-Fuhrmann circle (aka X(2)-Hagge circle)
X(6792) = circummedial-to-X(2)-Fuhrmann similarity image of X(111)
X(6792) = intersection of lines X(2)X(6) of 1st and 2nd Ehrmann circumscribing triangles
X(6792) = intersection of lines X(2)X(6) of anticevian triangles of PU(4)
X(6792) = trilinear pole, wrt 4th Brocard triangle, of Fermat axis
X(6793) lies on the ellipse described at X(6784) and these lines: {6,67}, {51,2871}, {112,1562}, {132,1503}
X(6793) = midpoint of X(112) and X(6794)
X(6793) = crossdifference of every pair of points on line X(74)X(1297)
X(6794) lies on the orthocentroidal circle and these lines: {2, 525}, {4, 6}, {5, 18337}, {20, 2420}, {39, 18304}, {54, 7765}, {74, 6103}, {112, 1562}, {115, 11005}, {247, 39024}, {477, 2549}, {868, 6792}, {1316, 15048}, {2088, 3767}, {2132, 5158}, {2409, 35260}, {2697, 4846}, {3163, 10706}, {4235, 30227}, {5024, 15000}, {5309, 5890}, {6324, 11226}, {7422, 7735}, {10413, 26917}, {16278, 36696}, {18809, 33504}, {18911, 38971}, {22146, 32423}, {32640, 39008}, {34211, 35923}
X(6794) = midpoint of X(1562) and X(6793)
X(6794) = reflection of X(112) in X(6793)
X(6794) = intersection, other than X(4), of van Aubel line and orthocentroidal circle
X(6794) = polar-circle-inversee of X(1990)
X(6794) = orthoptic-circle-of-Steiner-inellipse-inverse of X(9209)
X(6794) = psi-transform of X(1637)
X(6794) = crossdifference of every pair of points on line {520, 1495}
X(6794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5523, 13509, 15340}, {5523, 15341, 13509}
X(6795) lies on the Brocard circle and these lines: {3,523}, {6,30}, {182,1316}
X(6795) = Brocard circle antipode of X(1316)
X(6795) = reflection of X(1316) in X(182)
X(6795) = X(74) of 1st Brocard triangle
X(6795) = 1st-Brocard-isogonal-conjugate of X(542)
X(6795) = intersection, other than X(6), of Brocard circle and line X(6)X(30)
X(6796) lies on these lines: {1,1389}, {3,10}, {4,35}, {5,5248}, {21,5587}, {36,944}, {40,78}, {55,946}, {56,5882}, {73,1771}, {165,191}
X(6796) = excentral isogonal conjugate of X(3460)
X(6796) = {X(165),X(1490)}-harmonic conjugate of X(1158)
X(6796) = X(5449)-of-excentral-triangle
Let I be the incenter of a triangle ABC. Let NA be the nine-point center of IBC, and define NB and NC cyclically. Let
A′ = reflection of NA in AI;
B′ = reflection of NB in BI;
C′ = reflection of NC in CI.
Let (NA) be the nine-point circle of A′B′C′, and define (NB) and (NC) cyclically. The circles (NA), (NB), (NC) concur in X(6797). See Hyacinthos 23071 (January-March 2015, Antreas Hatzipolakis, Peter Moses, César Lozada)
X(6797) lies on these lines: {11,517}, {65,79}, {100,3753}, {149,5722}, {214,3812}, {942,952}, {960,6702}, {999,6264}, {1125,1387}, {1210,1484}, {1317,5045}, {1537,5806}, {3295,5541}, {6001,6246}
Let HA be the reflection of the orthocenter in BC, and define HB and HC cyclically. Let HAB be the reflection of HA in HB, and define HBC and HCA cyclically. Let HAC be the reflection of HA in HC, and define HBA and HCB cyclically. Let NA be the nine-point center of triangle HAHABHAC, and define NB and NC cyclically. Then ABC and NANBNC are cyclologic triangles. The (ABC, NANBNC)-cyclologic center is X(6798), and the (NANBNC, ABC)-cyclologic center is X(6799). See Hyacinthos 23138 (March 2, 2015, Antreas Hatzipolakis, César Lozada)
X(6798) lies on the cubic K464 and these lines: {49,52}, {389,3481}
See X(6798) and Hyacinthos 23138 (March 2, 2015, Antreas Hatzipolakis, César Lozada)
X(6799) lies on the circumcircle and this line: {1141,6240}
Let P be a point in the plane of a triangle ABC. Let PA be the reflection of P in BC, and define PB and PC cyclically. Let PAB be the reflection of P in BPA, and define PBC and PCA cyclically. Let PAC be the reflection of P in CPA, and define PBA and PCB cyclically. At Hyacinthos 23148 (March 6, 2015) Antreas Hatzipolakis asks for the locus of P such that the perpendicular bisectors of the segements PABPAC, PBCPBA, PCAPCB concur and for the locus of the point Z(P) of concurrence. César Lozada responds that the first locus is the the union of {A,B,C} and the Euler line of ABC, and that the second locus is the line of the points X(3), X(74), and X(110). The appearance of (i,j) in the following list means that Z(X(i)) = X(j): (2,6800), (3,3), (4,3), (5,5944), (25,6090). The point Z(P), for P = t*|OH| and X = X(110), is given by Z(P) = t(t - 1)|OQ|/[t - (r/|OH|)]2.
For another construction see Antreas P. Hatzipolakis and César Lozada, euclid 2823.
X(6800) lies on these lines: 2, 154}, {3, 74}, {4, 14389}, {6, 23}, {20, 19357}, {22, 184}, {24, 9730}, {25, 5012}, {26, 568}, {39, 44116}, {49, 13340}, {51, 39561}, {54, 7387}, {155, 7512}, {159, 1176}, {182, 373}, {183, 35356}, {185, 38444}, {206, 26206}, {323, 1350}, {352, 5210}, {376, 40112}, {378, 14915}, {394, 6636}, {428, 38136}, {468, 18911}, {567, 7530}, {569, 10594}, {575, 32237}, {576, 44109}, {599, 9143}, {1112, 15074}, {1147, 10323}, {1181, 7488}, {1204, 38438}, {1351, 11422}, {1352, 7495}, {1498, 14118}, {1501, 5359}, {1598, 13434}, {1599, 35299}, {1600, 35300}, {1656, 34514}, {1915, 13331}, {1975, 10330}, {1976, 44420}, {1994, 5102}, {2360, 37301}, {2502, 20481}, {2715, 35901}, {2854, 16165}, {2916, 18882}, {2937, 12161}, {2979, 3167}, {3053, 9463}, {3060, 5093}, {3066, 14002}, {3091, 37476}, {3098, 3292}, {3146, 11425}, {3148, 11171}, {3291, 5033}, {3398, 20897}, {3431, 7464}, {3448, 37638}, {3455, 30541}, {3515, 10574}, {3516, 12279}, {3517, 15043}, {3518, 36752}, {3522, 35602}, {3547, 14516}, {3549, 34224}, {3564, 44210}, {3567, 9714}, {3580, 6776}, {3917, 44108}, {4189, 26637}, {4550, 41450}, {4563, 7782}, {4846, 10295}, {5017, 8627}, {5059, 14528}, {5092, 5651}, {5133, 31383}, {5169, 36990}, {5408, 13617}, {5409, 13616}, {5480, 7519}, {5486, 32220}, {5544, 30734}, {5650, 7485}, {5878, 34005}, {5889, 9715}, {5890, 14070}, {5899, 39522}, {5943, 44082}, {5967, 9832}, {5968, 32729}, {6515, 10565}, {6642, 26882}, {6644, 40114}, {6660, 32447}, {6676, 11442}, {6759, 7503}, {6795, 7471}, {7391, 23292}, {7393, 43598}, {7394, 37649}, {7395, 14530}, {7426, 11179}, {7484, 8780}, {7494, 37636}, {7500, 11427}, {7502, 18445}, {7506, 13363}, {7509, 10170}, {7514, 10540}, {7516, 18350}, {7517, 32046}, {7529, 43651}, {7542, 11457}, {7547, 11750}, {7552, 14852}, {7555, 37494}, {7556, 15032}, {7558, 12134}, {7566, 13419}, {7691, 12164}, {8548, 12310}, {8550, 32269}, {8717, 10564}, {8718, 12085}, {8719, 35933}, {8836, 41041}, {8838, 41040}, {9140, 45082}, {9183, 21163}, {9465, 40825}, {9545, 37498}, {9704, 13564}, {9777, 20850}, {9818, 14157}, {9833, 13160}, {10117, 34117}, {10154, 11245}, {10282, 10984}, {10298, 10605}, {10299, 44833}, {10301, 18583}, {10541, 16042}, {10546, 11284}, {10575, 35477}, {10601, 13595}, {11004, 11477}, {11064, 16063}, {11202, 15078}, {11317, 35706}, {11410, 13445}, {11413, 13367}, {11414, 34148}, {11417, 19356}, {11418, 19355}, {11420, 19364}, {11421, 19363}, {11423, 37493}, {11472, 12112}, {11550, 31236}, {11842, 37914}, {11935, 37496}, {12045, 20190}, {12082, 13352}, {12088, 36747}, {12107, 37490}, {12173, 41482}, {12203, 41238}, {12220, 19125}, {12233, 31304}, {12254, 12293}, {12315, 15062}, {12824, 19153}, {13198, 45237}, {13321, 37956}, {13346, 33524}, {13353, 13861}, {13366, 15520}, {13383, 18912}, {13754, 44837}, {14683, 15069}, {14805, 31861}, {14927, 31099}, {15033, 18534}, {15139, 44883}, {15246, 17811}, {15305, 32063}, {15329, 15920}, {15448, 37648}, {15531, 19121}, {15647, 41670}, {15750, 43601}, {16776, 20987}, {17714, 36749}, {17810, 34545}, {17814, 37126}, {17821, 22467}, {17834, 38435}, {17845, 34007}, {18437, 44888}, {18451, 32620}, {20998, 39560}, {21525, 41330}, {21663, 38446}, {21734, 40911}, {21850, 37899}, {22234, 44107}, {23061, 33878}, {23515, 44795}, {24981, 34507}, {26257, 39141}, {26913, 37453}, {29012, 31133}, {31608, 35890}, {31670, 37900}, {32235, 32305}, {32321, 32379}, {32423, 44262}, {32534, 40647}, {33532, 37477}, {35006, 42295}, {35228, 37978}, {35243, 43574}, {36181, 44526}, {36753, 37440}, {37454, 39884}, {37514, 44802}, {37902, 44415}, {37925, 44413}, {38010, 41394}, {38225, 41275}, {39899, 41724}, {40130, 41412}, {40709, 41021}, {40710, 41020}, {42154, 44462}, {42155, 44466}
X(6800) = midpoint of X(184) and X(35268)
X(6800) = reflection of X(i) in X(j) for these (i, j): (2, 13394), (3, 34513), (22, 35268), (378, 39242), (39242, 18475), (39588, 5050), (44795, 23515)
X(6800) = anticomplement of X(45303)
X(6800) = isogonal conjugate of the isotomic conjugate of X(14907)
X(6800) = crossdifference of every pair of points on line {X(1637), X(3906)}
X(6800) = X(206)-Dao conjugate of-X(14906)
X(6800) = X(75)-isoconjugate-of-X(14906)
X(6800) = X(32)-reciprocal conjugate of-X(14906)
X(6800) = perspector of the circumconic {{A, B, C, X(11636), X(44769)}}
X(6800) = intersection, other than A, B, C, of circumconics {{A, B, C, X(23), X(30541)}} and {{A, B, C, X(74), X(1383)}}
X(6800) = X(20481)-of-circumsymmedial triangle
X(6800) = X(22111)-of-anti-McCay triangle
X(6800) = X(34513)-of-X3-ABC reflections triangle
X(6800) = barycentric product X(6)*X(14907)
X(6800) = barycentric quotient X(32)/X(14906)
X(6800) = trilinear product X(31)*X(14907)
X(6800) = trilinear quotient X(31)/X(14906)
X(6800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 154, 35264), (2, 35265, 35259), (3, 110, 15066), (3, 1614, 11441), (3, 6090, 7998), (3, 12174, 11440), (3, 15066, 21766), (3, 26864, 110), (22, 184, 1993), (23, 11003, 6), (25, 5012, 5422), (110, 7998, 6090), (110, 15080, 3), (154, 3796, 2), (154, 5085, 35259), (154, 23041, 35260), (154, 35259, 35265), (3796, 35259, 5085), (5012, 5640, 5050), (5012, 26881, 25), (5050, 5640, 5422), (5085, 35259, 2), (6090, 7998, 15066), (6636, 9544, 394), (6776, 7493, 3580), (7712, 11003, 23), (7712, 43697, 32124), (8546, 32217, 6), (9306, 17508, 5650), (9715, 19347, 5889), (9909, 11402, 3060), (11422, 15107, 1351), (14002, 15018, 3066), (14169, 14170, 3), (15080, 26864, 15066), (17809, 33586, 1994), (25406, 35260, 2), (35259, 35265, 35264)
Let A′B′C′ be the anticevian triangle of X(54). Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The triangles ABC and A″B″C″ are cyclologic. The (ABC, A″B″C″)-cyclologic center is X(6801), and the (A″B″C″, ABC)-cyclologic center is X(6802). See Hyacinthos 23162 (March 19, 2015, Randy Hutson, César Lozada)
If you have The Geometer's Sketchpad, you can view X(6801) and X(6802).
X(6801) lies on these lines: {54,136}, {1986,14111}
Let A′B′C′ be the anticevian triangle of X(54). Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The triangles A″B″C″ are cyclologic. The (ABC, A″B″C″)-cyclologic center is X(6801), and the (A″B″C″, ABC)-cyclologic center is X(6802). See X(6801) and Hyacinthos 23162 (March 19, 2015, Randy Hutson, César Lozada)
X(6802) lies on these lines: {}
As a point on the Euler line, X(6803) has Shinagawa coefficients (E, F).
X(6803) lies on these lines: {2, 3}, {68, 5892}, {69, 389}, {578, 3618}, {1192, 3763}, {1217, 6530}
As a point on the Euler line, X(6804) has Shinagawa coefficients (E, -F).
X(6804) lies on these lines: {2, 3}, {800, 2548}, {3767, 5065}
As a point on the Euler line, X(6805) has Shinagawa coefficients (E, S).
X(6805) lies on these lines:
{2, 3}, {388, 3084}, {394, 1587}, {497, 3083}, {2550, 6348}, {2551, 6347}, {3068, 5409}, {3316, 6504}, {5408, 6460}
X(6805) = orthocentroidal-circle-inverse of X(6806)
X(6805) = {X(2),X(4)}-harmonic conjugate of X(6806)
As a point on the Euler line, X(6806) has Shinagawa coefficients (E, -S).
X(6806) lies on these lines:
{2, 3}, {388, 3083}, {394, 1588}, {497, 3084}, {2550, 6347}, {2551, 6348}, {3069, 5408}, {3317, 6504}, {5409, 6459}
X(6806) = orthocentroidal-circle-inverse of X(6805)
X(6806) = {X(2),X(4)}-harmonic conjugate of X(6805)
As a point on the Euler line, X(6807) has Shinagawa coefficients (S, E).
X(6807) lies on these lines: {2, 3}, {590, 1498}, {1181, 3068}
As a point on the Euler line, X(6808) has Shinagawa coefficients (S, -E).
X(6808) lies on these lines: {2, 3}, {615, 1498}, {1181, 3069}
As a point on the Euler line, X(6809) has Shinagawa coefficients (S, F).
X(6809) lies on these lines: {2, 3}, {492, 5562}
As a point on the Euler line, X(6810) has Shinagawa coefficients (S, -F).
X(6810) lies on these lines: {2, 3}, {491, 5562}
As a point on the Euler line, X(6811) has Shinagawa coefficients (S, E + F), and also (1, cot ω).
X(6811) lies on these lines: {2, 3}, {98, 485}, {114, 489}, {183, 638}, {230, 3070}, {262, 486}, {325, 637}, {491, 1352}, {492, 511}, {590, 1503}, {615, 5480}, {1587, 6423}, {1588, 6422}, {2459, 6560}, {3071, 3815}, {3316, 3424}
As a point on the Euler line, X(6812) has Shinagawa coefficients (S, E - F).
X(6812) lies on these lines: {2, 3}, {491, 5907}, {590, 2883}
As a point on the Euler line, X(6813) has Shinagawa coefficients (S, -E - F), and also (1, -cot ω).
X(6813) lies on these lines: {2, 3}, {98, 486}, {114, 490}, {183, 637}, {230, 3071}, {262, 485}, {325, 638}, {491, 511}, {492, 1352}, {590, 5480}, {615, 1503}, {1587, 6421}, {1588, 6424}, {2460, 6561}, {3070, 3815}, {3317, 3424}
As a point on the Euler line, X(6814) has Shinagawa coefficients (S, -E + F).
X(6814) lies on these lines: {2, 3}, {492, 5907}, {615, 2883}
As a point on the Euler line, X(6815) has Shinagawa coefficients (E, 2F).
X(6815) lies on these lines: {2, 3}, {69, 5889}, {185, 1352}, {389, 6515}
X(6815) = {X(3),X(4)}-harmonic conjugate of X(1370)
As a point on the Euler line, X(6816) has Shinagawa coefficients (E, -2F).
X(6816) lies on these lines: {2, 3}, {68, 5891}, {569, 5654}, {1352, 6467}, {1478, 4320}, {1479, 4319}, {1899, 5907}, {2165, 5063}, {5562, 6515}
As a point on the Euler line, X(6817) has Shinagawa coefficients (E, $bc$).
X(6817) lies on these lines: {2, 3}, {42, 388}, {43, 1478}, {69, 310}, {497, 2293}, {899, 5229}, {3421, 4651}, {3434, 4645}
As a point on the Euler line, X(6818) has Shinagawa coefficients (E, -$bc$).
X(6818) lies on these lines: {2, 3}, {42, 497}, {43, 1479}, {388, 3720}, {899, 5225}, {4651, 5082}
As a point on the Euler line, X(6819) has Shinagawa coefficients (EF, S2).
X(6819) lies on these lines: {2, 3}, {394, 3087}, {1249, 5422}, {3066, 6525}, {5943, 6524}
As a point on the Euler line, X(6820) has Shinagawa coefficients (EF, -S2).
X(6820) lies on these lines: {2, 3}, {69, 2052}, {196, 5905}, {275, 3618}, {393, 394}, {459,3580}, {511,6524}, {1249,1993}
As a point on the Euler line, X(6821) has Shinagawa coefficients (2E, $bc$).
X(6821) lies on these lines: {2, 3}, {42, 1056}, {43, 388}, {1058, 3720}, {2550, 3741}
As a point on the Euler line, X(6822) has Shinagawa coefficients (2E, -$bc$).
X(6822) lies on these lines: {2, 3}, {42, 1058}, {43, 497}, {1056, 3720}, {2551, 3741}
As a point on the Euler line, X(6823) has Shinagawa coefficients (E + F, -E + F).
X(6823) lies on these lines: {2, 3}, {11, 1038}, {12, 1040}, {131, 5522}, {141, 2883}, {185, 343}, {216, 5254}, {485, 1579}, {486, 1578}, {495, 1062}, {496, 1060}, {1181, 3564}, {1352, 1498}, {2968, 4385}, {3673, 6356}, {5447, 5448}
X(6823) = complement of X(1593)
As a point on the Euler line, X(6824) has Shinagawa coefficients ($aSA$, abc).
X(6824) lies on these lines: {2, 3}, {57, 499}, {58, 5713}, {68, 1175}, {72, 5761}, {81, 155}, {142, 3358}, {284, 5816}, {355, 3085}, {498, 3601}, {517, 5791}, {912, 3487}, {942, 3086}, {946, 5709}, {1125, 6245}, {1352, 5138}, {1385, 5787}, {1698, 6282}, {3587, 6684}, {3616, 5768}, {3822, 6256}, {3869, 5603}, {5086, 5552}, {5226, 5811}, {5273, 5758}, {5698, 5805}, {5789, 5901}, {6523, 6708}
As a point on the Euler line, X(6825) has Shinagawa coefficients ($aSA$, -abc).
X(6825) lies on these lines: {2, 3}, {10, 5720}, {12, 3428}, {40, 498}, {119, 2551}, {226, 5709}, {387, 5396}, {499, 3576}, {517, 3085}, {573, 5747}, {581, 5292}, {944, 5086}, {966, 5778}, {993, 6256}, {1000, 1482}, {1064, 5230}, {1071, 5770}, {1214, 1217}, {1385, 3086}, {1490, 5705}, {1519, 5250}, {2095, 6147}, {2193, 3087}, {3359, 3452}, {3616, 5804}, {3869, 5552}, {3890, 5603}, {3927, 5771}, {3940, 5690}, {4847, 5534}, {5044, 5887}, {5226, 5758}, {5229, 5841}, {5273, 5811}, {5706, 5718}, {5707, 5712}, {5742, 5776}, {5743, 6247}, {5745, 6260}, {5746, 5755}, {5777, 5791}, {5806, 5886}
X(6825) = complement of X(6847)
X(6825) = anticomplement of X(6862)
As a point on the Euler line, X(6826) has Shinagawa coefficients (abc, $aSA$).
X(6826) lies on these lines: {2, 3}, {7, 912}, {10, 5709}, {33, 1074}, {57, 1478}, {69, 1243}, {78, 5761}, {142, 515}, {226, 5720}, {278, 1060}, {355, 388}, {386, 5713}, {387, 5707}, {497, 5886}, {516, 3587}, {517, 2550}, {579, 5816}, {612, 1072}, {936, 5715}, {940, 5721}, {946, 997}, {952, 1056}, {965, 5798}, {966, 5755}, {1001, 5842}, {1038, 1838}, {1058, 5901}, {1352, 4260}, {1479, 3601}, {1482, 5082}, {1699, 6282}, {1708, 4292}, {2095, 3421}, {2551, 5791}, {3428, 3925}, {3434, 4511}, {3436, 5818}, {4295, 5887}, {5044, 5812}, {5080, 5744}, {5175, 5804}, {5396, 5712}, {5746, 5778}, {5880, 6001}, {6245, 6256}
As a point on the Euler line, X(6827) has Shinagawa coefficients (abc, -$aSA$).
X(6827) lies on these lines: {1, 5761}, {2, 3}, {11, 3428}, {34, 1076}, {40, 1479}, {63, 5770}, {165, 3583}, {182, 5800}, {329, 912}, {355, 2551}, {388, 1385}, {497, 517}, {515, 997}, {516, 3359}, {572, 5747}, {580, 5292}, {602, 5230}, {614, 1072}, {938, 5758}, {942, 5812}, {944, 3436}, {952, 3421}, {962, 5804}, {1040, 1785}, {1056, 5719}, {1058, 1482}, {1210, 1708}, {1376, 5842}, {1478, 3576}, {2077, 4302}, {2095, 5762}, {3434, 5657}, {3579, 5225}, {3586, 6282}, {4293, 5841}, {4297, 6256}, {4383, 5721}, {5080, 5731}, {5082, 5690}, {5777, 5787}
As a point on the Euler line, X(6828) has Shinagawa coefficients ($aSA$abc, abc + $aSA$).
X(6828) lies on these lines: {2, 3}, {11, 938}, {12, 3486}, {63, 5715}, {78, 5086}, {81, 5713}, {165, 3841}, {920, 1445}, {936, 3814}, {946, 3869}, {962, 2886}, {1210, 3671}, {1699, 5705}, {2287, 5816}, {3219, 5812}, {3813, 5734}, {3822, 5691}, {4018, 5887}, {5057, 5535}, {5221, 5704}, {5249, 6245}, {6253, 6690}
As a point on the Euler line, X(6829) has Shinagawa coefficients (abc + $aSA$, $aSA$).
X(6829) lies on these lines: {2, 3}, {9, 3814}, {11, 3488}, {12, 3487}, {40, 3841}, {226, 1737}, {498, 943}, {1071, 3824}, {1213, 5798}, {1698, 5715}, {1708, 3336}, {1714, 5713}, {1751, 5397}, {2886, 5289}, {3419, 4511}, {3614, 5221}, {3822, 5587}, {3825, 5436}, {3826, 5759}, {3925, 5657}, {5776, 5949}, {5777, 5885}, {5817, 5851}
As a point on the Euler line, X(6830) has Shinagawa coefficients (-abc + $aSA$, $aSA$).
X(6830) lies on these lines: {2, 3}, {11, 2099}, {12, 944}, {104, 1478}, {119, 6224}, {355, 4511}, {484, 1699}, {946, 1737}, {997, 3814}, {1329, 5818}, {1512, 5316}, {1519, 3817}, {1708, 5715}, {2886, 5657}, {3576, 3822}, {3585, 5450}, {5226, 5768}, {5432, 5842}, {5770, 5905}
As a point on the Euler line, X(6831) has Shinagawa coefficients (-abc + $aSA$, abc + $aSA$).
X(6831) lies on these lines: {2, 3}, {11, 65}, {12, 515}, {35, 5842}, {40, 2886}, {46, 1699}, {57, 5715}, {63, 5812}, {78, 355}, {79, 1768}, {216, 1865}, {226, 1071}, {386, 5721}, {495, 944}, {496, 938}, {573, 5742}, {579, 5798}, {581, 5718}, {908, 5777}, {936, 1329}, {940, 5713}, {942, 1425}, {965, 5816}, {1076, 1214}, {1158, 1454}, {1181, 5707}, {1445, 5805}, {1446, 1565}, {1465, 1838}, {1490, 5219}, {1503, 5135}, {1765, 1901}, {2829, 3585}, {3468, 6357}, {3487, 5768}, {3612, 5691}, {3649, 5884}, {3814, 6700}, {3817, 3825}, {3820, 5818}, {3822, 4297}, {3925, 6684}, {3927, 5789}, {4259, 5480}, {4292, 6705}, {5292, 5706}, {5432, 6253}, {5747, 5776}
X(6831) = complement of X(411)
As a point on the Euler line, X(6832) has Shinagawa coefficients (abc + $aSA$, abc).
X(6832) lies on these lines: {2, 3}, {72, 5886}, {226, 499}, {354, 3086}, {496, 954}, {497, 943}, {498, 950}, {583, 5746}, {584, 5802}, {960, 5603}, {1490, 3624}, {1713, 5747}, {1724, 5713}, {1837, 3085}, {3419, 5552}, {3698, 5657}, {3876, 5761}, {5436, 5587}
X(6832) = complement of X(37112)
As a point on the Euler line, X(6833) has Shinagawa coefficients (-abc + $aSA$, abc).
X(6833) lies on these lines: {2, 3}, {46, 499}, {65, 3086}, {84, 5219}, {104, 388}, {224, 5720}, {226, 6705}, {355, 5440}, {498, 515}, {940, 1181}, {944, 2646}, {988, 1072}, {1454, 4295}, {1478, 5450}, {1512, 1698}, {1519, 5437}, {1765, 5747}, {2096, 5714}, {3075, 3215}, {3428, 4999}, {3868, 5761}, {3916, 5812}, {3940, 5789}, {4255, 5721}, {5123, 5794}, {5135, 5820}, {5217, 5842}, {5438, 5587}, {5439, 5886}, {5657, 5836}, {5703, 5768}, {5704, 5804}, {5705, 6282}, {5744, 5758}, {5748, 5811}, {5759, 5832}, {5784, 5817}
As a point on the Euler line, X(6834) has Shinagawa coefficients (-abc + $aSA$, -abc).
X(6834) lies on these lines: {1, 1512}, {2, 3}, {36, 6256}, {40, 1519}, {119, 3436}, {498, 946}, {499, 515}, {517, 5552}, {908, 5709}, {944, 1319}, {960, 5657}, {1068, 1465}, {1145, 5730}, {1181, 4383}, {1329, 3428}, {1538, 3579}, {1728, 3911}, {1737, 6261}, {2829, 5204}, {3057, 3085}, {5087, 6361}, {5316, 6684}, {5660, 5904}, {5703, 5804}, {5704, 5768}, {5744, 5811}, {5748, 5758}
As a point on the Euler line, X(6835) has Shinagawa coefficients (abc, abc + $aSA$).
X(6835) lies on these lines: {2, 3}, {72, 5805}, {78, 946}, {354, 388}, {355, 3555}, {497, 5703}, {908, 5715}, {936, 1699}, {960, 962}, {1001, 6253}, {1210, 1478}, {1445, 1728}, {1479, 3817}, {1490, 5249}, {3306, 6245}, {3419, 5806}, {3436, 5587}, {3826, 5584}, {3876, 5758}, {5082, 5730}, {5229, 5704}, {5234, 5705}, {5439, 5787}, {5777, 5905}
As a point on the Euler line, X(6836) has Shinagawa coefficients (abc, -abc - $aSA$).
X(6836) lies on these lines: {1, 1076}, {2, 3}, {8, 3427}, {40, 3434}, {46, 516}, {63, 6245}, {65, 497}, {72, 5787}, {78, 515}, {165, 5705}, {224, 908}, {225, 1040}, {273, 4329}, {355, 3697}, {388, 2646}, {936, 5691}, {1062, 1068}, {1071, 5812}, {1155, 5225}, {1181, 3193}, {1376, 6253}, {1454, 3474}, {1478, 3612}, {1754, 5292}, {1936, 3215}, {2551, 3740}, {2886, 5584}, {3868, 5758}, {4652, 6705}, {5135, 5800}, {5249, 5715}, {5439, 5805}
As a point on the Euler line, X(6837) has Shinagawa coefficients ($aSA$, 2abc).
X(6837) lies on these lines: {2, 3}, {7, 90}, {63, 946}, {84, 5249}, {153, 5261}, {354, 1858}, {498, 4304}, {499, 3817}, {920, 4295}, {962, 5273}, {1071, 5886}, {1259, 3434}, {1780, 3332}, {2327, 5816}, {3085, 4313}, {3219, 5758}, {3306, 6705}, {3486, 5252}, {3616, 6261}, {3624, 5732}, {3868, 5603}, {3916, 5805}, {5552, 5587}, {5698, 5832}, {5820, 5921}
As a point on the Euler line, X(6838) has Shinagawa coefficients ($aSA$, -2abc).
X(6838) lies on these lines: {2, 3}, {8, 6261}, {40, 908}, {63, 6260}, {498, 516}, {499, 4297}, {962, 3085}, {1319, 3486}, {1329, 5584}, {1512, 5554}, {1750, 5705}, {1788, 1858}, {3057, 3485}, {3086, 5731}, {3219, 5811}, {3305, 6684}, {3428, 3436}, {3876, 5657}, {3916, 6259}, {5709, 5905}, {5744, 6223}, {5791, 5927}
As a point on the Euler line, X(6839) has Shinagawa coefficients (abc, 2$aSA$).
X(6839) lies on these lines: {2, 3}, {7, 80}, {8, 2894}, {63, 5080}, {78, 5715}, {81, 5721}, {145, 1389}, {149, 5603}, {355, 2888}, {515, 5249}, {946, 4511}, {962, 3878}, {997, 1699}, {1071, 5885}, {1072, 3920}, {1074, 3100}, {1146, 5829}, {1479, 4313}, {1621, 5842}, {1737, 3336}, {1754, 5127}, {1812, 5799}, {1838, 4296}, {2287, 5798}, {3419, 5805}, {3583, 3817}, {3679, 5735}, {3876, 5812}, {5221, 5229}, {5800, 5921}
As a point on the Euler line, X(6840) has Shinagawa coefficients (abc, -2$aSA$).
X(6840) lies on these lines: {2, 3}, {100, 5842}, {104, 5841}, {149, 517}, {153, 515}, {264, 2897}, {355, 3876}, {484, 516}, {497, 2099}, {962, 1479}, {997, 5691}, {1029, 5397}, {1076, 4296}, {1329, 6253}, {1478, 5226}, {1577, 6003}, {1785, 3100}, {2800, 5180}, {2894, 2949}, {3305, 5587}, {3585, 4297}, {3868, 5812}, {3984, 5881}, {4301, 4857}, {5057, 6001}, {5768, 5905}
As a point on the Euler line, X(6841) has Shinagawa coefficients ($aSA$, 2abc + $aSA$).
X(6841) lies on these lines: {2, 3}, {11, 113}, {57, 79}, {114, 2795}, {119, 6246}, {123, 133}, {131, 5521}, {142, 3825}, {191, 1699}, {265, 1175}, {355, 3811}, {495, 3486}, {496, 3485}, {758, 946}, {920, 1836}, {1698, 3587}, {2095, 5789}, {2886, 5791}, {3579, 3925}, {3601, 5441}, {3647, 5745}, {3648, 5744}, {3650, 5057}, {3652, 5805}, {3656, 3813}, {3817, 6245}, {3818, 5138}, {4420, 5086}, {5426, 5691}, {5787, 5886}
As a point on the Euler line, X(6842) has Shinagawa coefficients ($aSA$, -2abc + $aSA$), and also (r + R, r - R).
X(6842) lies on these lines: {2, 3}, {10, 119}, {11, 1385}, {12, 517}, {35, 5840}, {79, 5535}, {355, 2886}, {495, 1482}, {496, 3486}, {946, 3822}, {952, 4861}, {1698, 3359}, {1737, 1858}, {1834, 5396}, {1901, 5755}, {2829, 4999}, {3579, 3614}, {3585, 5841}, {3655, 3829}, {3814, 6684}, {3869, 5690}, {5226, 5761}
As a point on the Euler line, X(6843) has Shinagawa coefficients ($aSA$, -2abc + $aSA$).
X(6843) lies on these lines: {2, 3}, {10, 5715}, {72, 5818}, {226, 5587}, {329, 5775}, {355, 3487}, {387, 5713}, {966, 5798}, {997, 3817}, {1737, 3339}, {3017, 5733}, {3419, 5603}, {3488, 5274}, {3824, 5787}, {4511, 5175}, {5226, 5720}, {5249, 5768}, {5712, 5721}, {5714, 5777}, {5746, 5816}
As a point on the Euler line, X(6844) has Shinagawa coefficients (-abc + $aSA$, 2$aSA$).
X(6844) lies on these lines: {2, 3}, {145, 5761}, {226, 5768}, {515, 5219}, {944, 5261}, {946, 3340}, {1071, 5714}, {1210, 5715}, {1699, 1737}, {3452, 5587}, {4511, 5720}, {5044, 5818}, {5218, 5842}, {5274, 5603}
As a point on the Euler line, X(6845) has Shinagawa coefficients (-abc + $aSA$, 2abc + $aSA$).
X(6845) lies on these lines: {2, 3}, {11, 4295}, {12, 4305}, {104, 4317}, {355, 4420}, {946, 5902}, {1158, 1699}, {1770, 3911}, {2886, 6361}, {3811, 5881}, {3814, 5438}, {3825, 5437}, {4325, 5450}
As a point on the Euler line, X(6846) has Shinagawa coefficients (abc + $aSA$, 2abc).
X(6846) lies on these lines: {2, 3}, {9, 946}, {72, 5603}, {84, 142}, {226, 3086}, {355, 3488}, {390, 943}, {498, 3586}, {499, 3361}, {515, 5436}, {580, 3332}, {950, 3085}, {954, 1058}, {1125, 1490}, {1260, 5082}, {1708, 4295}, {1713, 4253}, {1750, 3624}, {1901, 5022}, {3073, 4307}, {3419, 5818}, {3487, 5045}, {3577, 5837}, {3817, 5715}, {3824, 6259}, {4002, 5657}, {4251, 5802}, {5175, 5552}, {5273, 5709}, {5342, 6350}, {5437, 6705}, {5703, 5720}, {5779, 6147}, {5791, 5806}
X(6846) = complement of X(37108)
X(6846) = anticomplement of X(6989)
As a point on the Euler line, X(6847) has Shinagawa coefficients (-abc + $aSA$, 2abc).
X(6847) lies on these lines: {1, 3427}, {2, 3}, {7, 3358}, {10, 6282}, {11, 1466}, {40, 5745}, {57, 946}, {63, 5758}, {81, 1181}, {84, 226}, {104, 3600}, {498, 5691}, {499, 1699}, {515, 3085}, {601, 4307}, {908, 5811}, {912, 5761}, {940, 1498}, {942, 5603}, {944, 5787}, {962, 5709}, {1071, 3487}, {1210, 5804}, {1439, 1440}, {1537, 5708}, {1765, 5746}, {2829, 5229}, {3485, 6001}, {3577, 4848}, {3582, 4338}, {3927, 5763}, {4292, 5715}, {4293, 5450}, {4340, 5713}, {5217, 6253}, {5219, 6260}, {5226, 6223}, {5657, 5791}
X(6847) = complement of X(37421)
X(6847) = anticomplement of X(6825)
As a point on the Euler line, X(6848) has Shinagawa coefficients (-abc + $aSA$, -2abc).
X(6848) lies on these lines: {1, 5804}, {2, 3}, {8, 1512}, {40, 3452}, {57, 6260}, {63, 5811}, {84, 3911}, {104, 5265}, {329, 5709}, {391, 5778}, {498, 1699}, {499, 5691}, {515, 1420}, {908, 5758}, {944, 5722}, {946, 1697}, {962, 1519}, {1071, 5658}, {1210, 1467}, {1498, 4383}, {1528, 1753}, {1537, 5763}, {1714, 5400}, {1750, 6245}, {1788, 6001}, {2096, 6259}, {2551, 3428}, {4266, 5747}, {4293, 6256}, {5044, 5657}, {5225, 5842}, {5435, 6223}, {5587, 5795}, {5603, 5806}, {5690, 5780}, {6282, 6700}
As a point on the Euler line, X(6849) has Shinagawa coefficients (abc, 2abc + $aSA$).
X(6849) lies on these lines: {2, 3}, {355, 5806}, {388, 5045}, {500, 4648}, {946, 3811}, {1058, 5719}, {1478, 3333}, {1479, 5219}, {2550, 5044}, {3361, 3585}, {3434, 4420}, {3818, 5800}, {3940, 5082}, {4251, 5747}, {5763, 5780}, {5777, 5805}
As a point on the Euler line, X(6850) has Shinagawa coefficients (abc, -2abc + $aSA$).
X(6850) lies on these lines: {2, 3}, {8, 912}, {10, 1158}, {34, 1074}, {40, 1478}, {153, 3617}, {165, 3585}, {226, 5761}, {355, 971}, {388, 517}, {497, 1385}, {498, 2077}, {511, 5800}, {601, 5230}, {944, 3434}, {952, 5082}, {958, 2829}, {1038, 1785}, {1056, 1482}, {1068, 4296}, {1070, 4320}, {1071, 3419}, {1479, 3576}, {3421, 3927}, {3436, 5657}, {3579, 5229}, {3824, 5886}, {4292, 5709}, {4294, 5840}, {4338, 5270}, {4340, 5707}, {5175, 5768}, {5720, 6260}, {5777, 6259}, {5794, 6001}
X(6850) = anticomplement of X(3560)
As a point on the Euler line, X(6851) has Shinagawa coefficients (abc, -2abc - $aSA$).
X(6851) lies on these lines: {2, 3}, {10, 3587}, {33, 1076}, {57, 1479}, {278, 1062}, {388, 4305}, {497, 942}, {500, 5712}, {515, 3811}, {516, 1158}, {517, 5787}, {912, 5758}, {962, 5768}, {971, 5812}, {991, 5713}, {1040, 1838}, {1068, 3100}, {1070, 4319}, {1478, 3601}, {2550, 3579}, {3332, 5707}, {3434, 6361}, {3436, 4420}, {3583, 4333}, {4338, 4857}, {5138, 5800}, {5691, 6282}, {5715, 5732}, {5771, 5789}
X(6851) = anticomplement of X(6985)
As a point on the Euler line, X(6852) has Shinagawa coefficients (2$aSA$, abc).
X(6852) lies on these lines: {2, 3}, {80, 498}, {499, 3485}, {1698, 5538}, {2077, 3841}, {3336, 3911}, {3616, 6265}, {3624, 6261}, {3869, 5886}, {3878, 5603}, {5086, 5440}, {5439, 5885}
As a point on the Euler line, X(6853) has Shinagawa coefficients (2$aSA$, -abc).
X(6853) lies on these lines: {2, 3}, {10, 6326}, {104, 4999}, {119, 5260}, {226, 2949}, {484, 6684}, {498, 3485}, {499, 3486}, {1385, 5086}, {1698, 6261}, {2099, 3085}, {2950, 5316}, {3884, 5603}
As a point on the Euler line, X(6854) has Shinagawa coefficients (2abc, $aSA$).
X(6854) lies on these lines: {2, 3}, {354, 1056}, {355, 5439}, {388, 1737}, {997, 2550}, {1072, 5268}, {1478, 3911}, {2096, 5817}, {3216, 5713}, {3428, 3826}, {3434, 5440}, {4423, 5842}, {4511, 5082}, {5249, 5720}, {5437, 5587}
As a point on the Euler line, X(6855) has Shinagawa coefficients (2$aSA$, abc + $aSA$).
X(6855) lies on these lines: {2, 3}, {10, 3577}, {78, 5818}, {355, 5703}, {938, 5886}, {946, 5705}, {1210, 3485}, {3486, 5587}, {3869, 5775}, {5226, 5777}, {5273, 5812}, {5704, 5887}, {5715, 5745}, {5758, 5791}, {5789, 6147}
As a point on the Euler line, X(6856) has Shinagawa coefficients (2$aSA$, -abc + $aSA$).
X(6856) lies on these lines: {2, 3}, {10, 3340}, {12, 3421}, {63, 5714}, {72, 5226}, {145, 5719}, {226, 5705}, {274, 1007}, {329, 5791}, {387, 5718}, {388, 3822}, {392, 5806}, {498, 2550}, {956, 5261}, {966, 5747}, {993, 5229}, {1125, 3486}, {1376, 6668}, {1698, 2093}, {2886, 3085}, {3256, 3841}, {3419, 5703}, {3616, 5086}, {3617, 3940}, {3634, 5128}, {3753, 3869}, {3838, 4295}, {4018, 5775}, {4293, 4999}, {4294, 6690}, {4658, 5292}, {5225, 5248}, {5439, 5704}, {5698, 6666}, {5720, 5818}, {5742, 5746}, {5804, 5886}
As a point on the Euler line, X(6857) has Shinagawa coefficients (2$aSA$, abc - $aSA$).
X(6857) lies on these lines: {1, 5745}, {2, 3}, {7, 3916}, {8, 5791}, {10, 3486}, {35, 2550}, {55, 5082}, {57, 1125}, {58, 5712}, {63, 3487}, {69, 261}, {72, 5273}, {142, 3624}, {274, 6337}, {284, 966}, {388, 993}, {392, 942}, {497, 5248}, {498, 2551}, {499, 5259}, {610, 5257}, {943, 1259}, {946, 4512}, {950, 5705}, {958, 3085}, {1000, 4861}, {1001, 3086}, {1056, 2975}, {1058, 1621}, {1210, 5436}, {1385, 5768}, {1466, 4423}, {1482, 5771}, {1780, 4648}, {2095, 5901}, {2886, 4294}, {3419, 4313}, {3576, 6245}, {3618, 4260}, {3813, 4428}, {3822, 5229}, {3925, 5217}, {3927, 5719}, {4252, 4340}, {4295, 4640}, {4305, 5794}, {4652, 5249}, {4653, 5292}, {4877, 5747}, {5057, 5122}, {5250, 5603}, {5260, 5552}, {5281, 5687}, {5435, 5439}, {5731, 5787}, {5742, 5802}, {6261, 6705}, {6282, 6684}
X(6857) = complement of X(5177)
As a point on the Euler line, X(6858) has Shinagawa coefficients (abc + 2$aSA$, $aSA$).
X(6858) lies on these lines: {2, 3}, {10, 5761}, {1737, 5219}, {5249, 5770}, {5719, 5790}, {5748, 5775}
As a point on the Euler line, X(6859) has Shinagawa coefficients (-abc + 2$aSA$, $aSA$).
X(6859) lies on these lines: {2, 3}, {226, 5770}, {912, 5226}, {1737, 3340}, {4511, 5818}
As a point on the Euler line, X(6860) has Shinagawa coefficients (-abc + 2$aSA$, abc + $aSA$).
X(6860) lies on these lines: {2, 3}, {46, 3817}, {65, 5704}, {5252, 5703}
As a point on the Euler line, X(6861) has Shinagawa coefficients (abc + 2$aSA$, abc).
X(6861) lies on these lines: {2, 3}, {498, 1837}, {499, 942}, {960, 5791}, {1728, 5219}, {3085, 5790}, {5398, 5713}, {5550, 5768}, {5730, 5901}
As a point on the Euler line, X(6862) has Shinagawa coefficients (-abc + 2$aSA$, abc).
X(6862) lies on these lines: {2, 3}, {65, 499}, {155, 940}, {355, 498}, {944, 2320}, {952, 3085}, {1058, 1484}, {1352, 5135}, {2278, 5816}, {3086, 5901}, {3487, 5770}, {3584, 5881}, {3612, 5587}, {3822, 5450}, {5552, 5790}, {5763, 5771}
X(6862) = complement of X(6825)
As a point on the Euler line, X(6863) has Shinagawa coefficients (-abc + 2$aSA$, -abc).
X(6863) lies on these lines: {2, 3}, {119, 958}, {498, 517}, {499, 1385}, {1482, 3085}, {5217, 5840}, {5219, 5709}, {5231, 5534}, {5292, 5396}, {5552, 5690}, {5705, 5720}, {5707, 5718}, {5742, 5778}, {5747, 5755}
As a point on the Euler line, X(6864) has Shinagawa coefficients (2abc, abc + $aSA$).
X(6864) lies on these lines: {2, 3}, {7, 5777}, {78, 5082}, {142, 1490}, {355, 938}, {388, 1210}, {581, 4648}, {936, 946}, {1058, 5703}, {1445, 5817}, {1478, 3361}, {2551, 5705}, {3419, 5804}, {3421, 5818}, {3452, 5715}, {3487, 5720}, {3811, 6601}, {3817, 6700}, {4253, 5816}, {4423, 6253}, {5044, 5758}, {5437, 6245}, {5439, 5768}
X(6864) = complement of X(37423)
As a point on the Euler line, X(6865) has Shinagawa coefficients (2abc, -abc - $aSA$).
X(6865) lies on these lines: {2, 3}, {7, 5812}, {9, 6245}, {11, 5584}, {40, 497}, {72, 5768}, {78, 944}, {142, 5715}, {165, 1479}, {278, 1076}, {329, 1071}, {388, 3576}, {515, 936}, {517, 938}, {942, 5758}, {950, 6282}, {971, 5811}, {1056, 1385}, {1158, 5698}, {1445, 5709}, {1490, 3452}, {2550, 5705}, {3086, 3428}, {3359, 6361}, {3436, 5731}, {3579, 5704}, {4297, 6700}, {4309, 5537}, {4413, 6253}, {4648, 5713}, {5044, 5787}, {5082, 5657}, {5085, 5800}, {5708, 5762}, {5732, 6260}, {5909, 5932}
As a point on the Euler line, X(6866) has Shinagawa coefficients ($aSA$, abc + 2$aSA$).
X(6866) lies on these lines: {2, 3}, {355, 5761}, {3485, 5722}, {3817, 6261}, {4658, 5713}, {5261, 5719}, {5763, 5790}, {5806, 5887}
As a point on the Euler line, X(6867) has Shinagawa coefficients ($aSA$, -abc + 2$aSA$).
X(6867) lies on these lines: {2, 3}, {355, 3485}, {912, 5714}, {952, 5261}, {3340, 5587}, {3419, 5761}, {3486, 5886}, {3869, 5818}, {3947, 5534}, {5086, 5603}, {5274, 5901}
As a point on the Euler line, X(6868) has Shinagawa coefficients ($aSA$, abc - 2$aSA$).
X(6868) lies on these lines: {2, 3}, {40, 920}, {165, 4324}, {387, 5398}, {388, 5841}, {390, 1482}, {393, 2193}, {515, 5837}, {517, 3486}, {527, 5882}, {573, 5822}, {912, 944}, {950, 5709}, {952, 3927}, {958, 5842}, {971, 5698}, {1385, 3485}, {3428, 6284}, {3576, 4299}, {3600, 6147}, {3916, 5770}, {3929, 5881}, {4297, 6261}, {4313, 5758}, {4317, 4654}, {4653, 5713}, {4877, 5816}, {5086, 5657}, {5755, 5802}, {5761, 5812}
As a point on the Euler line, X(6869) has Shinagawa coefficients ($aSA$, -abc - 2$aSA$).
X(6869) lies on these lines: {2, 3}, {57, 4299}, {500, 4340}, {515, 5709}, {516, 6261}, {517, 3189}, {942, 3486}, {944, 3873}, {1175, 4846}, {1385, 5805}, {3296, 3600}, {3428, 6253}, {3485, 4294}, {3601, 4302}, {3869, 6361}, {3945, 5453}, {5770, 5787}
As a point on the Euler line, X(6870) has Shinagawa coefficients (a$SA$, 2abc + 2$aSA$).
X(6870) lies on these lines: {2, 3}, {90, 5556}, {3485, 5274}, {3486, 5261}, {5715, 5905}
As a point on the Euler line, X(6871) has Shinagawa coefficients ($aSA$, -2abc + 2$aSA$).
X(6871) lies on these lines: {2, 3}, {12, 3434}, {144, 5832}, {145, 3485}, {193, 5820}, {210, 3617}, {1376, 3614}, {1479, 3822}, {1621, 5225}, {1837, 3838}, {1898, 3812}, {2886, 3436}, {2900, 5175}, {2975, 5229}, {3304, 3829}, {3486, 3622}, {3868, 5714}, {3870, 3947}, {4881, 5550}, {5057, 5123}, {5217, 6668}, {5554, 5587}, {5818, 5887}
As a point on the Euler line, X(6872) has Shinagawa coefficients ($aSA$, 2abc - 2$aSA$).
X(6872) lies on these lines: {1, 5905}, {2, 3}, {8, 90}, {10, 4302}, {35, 5552}, {40, 5554}, {55, 3436}, {63, 950}, {78, 4304}, {100, 2551}, {144, 145}, {148, 5985}, {329, 4313}, {345, 5016}, {387, 1780}, {388, 1621}, {391, 4271}, {497, 2975}, {499, 5267}, {515, 5250}, {519, 3951}, {529, 3303}, {551, 4317}, {908, 3601}, {938, 3218}, {944, 3877}, {958, 3434}, {960, 1898}, {993, 1479}, {1043, 5739}, {1125, 4299}, {1145, 4678}, {1210, 4652}, {1319, 3485}, {1329, 5217}, {1478, 5248}, {1698, 4324}, {1837, 4640}, {2550, 5260}, {2647, 4331}, {3085, 5080}, {3189, 3681}, {3421, 3871}, {3452, 4855}, {3476, 3890}, {3488, 3868}, {3616, 4293}, {3617, 5086}, {3624, 4316}, {3679, 4330}, {3683, 5794}, {3816, 5204}, {3870, 4314}, {3897, 5603}, {3916, 5722}, {3920, 4339}, {3935, 5815}, {4298, 4666}, {4305, 4511}, {4512, 5691}, {5175, 5273}, {5249, 5436}, {5441, 5692}, {5731, 6223}
X(6872) = complement of X(31295)
As a point on the Euler line, X(6873) has Shinagawa coefficients (2$aSA$, abc + 2$aSA$).
X(6873) lies on these lines: {2, 3}, {946, 3899}, {3485, 5425}
As a point on the Euler line, X(6874) has Shinagawa coefficients (2$aSA$, -abc + 2$aSA$).
X(6874) lies on these lines: {2, 3}, {944, 3822}, {3485, 5818}, {4004, 5887}, {5086, 5886}
As a point on the Euler line, X(6875) has Shinagawa coefficients (2$aSA$, abc - 2$aSA$).
X(6875) lies on these lines: {2, 3}, {35, 3486}, {36, 3485}, {920, 3612}, {944, 993}, {970, 3567}, {1385, 3869}, {3085, 5172}, {3576, 5267}, {3647, 5693}, {5010, 6684}, {5248, 5603}, {5251, 5818}
X(6875) = {X(3),X(4)}-harmonic conjugate of X(6876)
As a point on the Euler line, X(6876) has Shinagawa coefficients (2$aSA$, -abc - 2$aSA$).
X(6876) lies on these lines: {2, 3}, {35, 3485}, {36, 3486}, {74, 931}, {165, 6261}, {970, 5890}, {3428, 3913}, {3579, 3869}, {3587, 4855}, {4294, 5172}
X(6876) = {X(3),X(4)}-harmonic conjugate of X(6875)
As a point on the Euler line, X(6877) has Shinagawa coefficients (2abc + 2$aSA$, $aSA$).
X(6877) lies on these lines: {2, 3}, {1708, 5714}, {1737, 3487}
As a point on the Euler line, X(6878) has Shinagawa coefficients (2abc + 2$aSA$, -$aSA$).
X(6878) lies on these lines: {2, 3}, {943, 3086}, {1708, 3338}, {1737, 3488}, {5550, 5758}
As a point on the Euler line, X(6879 has Shinagawa coefficients (-2abc + 2$aSA$, $aSA$).
X(6879) lies on these lines: {2, 3}, {997, 5818}, {1737, 5603}
As a point on the Euler line, X(6880) has Shinagawa coefficients (-2abc + 2$aSA$, -$aSA$).
X(6880) lies on these lines: {2, 3}, {165, 1519}, {944, 1737}, {997, 5657}, {1512, 3576}, {3035, 3428}, {5119, 5218}
As a point on the Euler line, X(6881) has Shinagawa coefficients (2abc + $aSA$, $aSA$).
X(6881) lies on these lines: {2, 3}, {12, 942}, {116, 119}, {495, 3475}, {517, 3925}, {912, 5249}, {946, 3841}, {997, 2886}, {1213, 5755}, {1329, 5791}, {1441, 2973}, {1698, 5709}, {1699, 3587}, {1714, 5707}, {2095, 3820}, {3649, 5694}, {3814, 5745}, {3824, 5777}, {3826, 5805}, {4511, 5178}, {5251, 5841}
As a point on the Euler line, X(6882) has Shinagawa coefficients (-2abc + $aSA$, $aSA$)and also (r - R, r + R).
X(6882) lies on these lines: {2, 3}, {11, 517}, {12, 1385}, {36, 5841}, {40, 5445}, {104, 5080}, {119, 214}, {329, 5770}, {355, 997}, {495, 3476}, {496, 1482}, {908, 912}, {938, 5761}, {946, 3754}, {952, 4511}, {1320, 1484}, {1699, 3359}, {1708, 5812}, {2077, 3583}, {3035, 5842}, {3576, 5444}, {3649, 5885}, {3654, 3829}, {3816, 5886}, {3820, 5790}, {5087, 6001}, {5179, 6506}, {5704, 5758}, {5748, 5768}
As a point on the Euler line, X(6883) has Shinagawa coefficients (2abc + $aSA$, -$aSA$).
X(6883) lies on these lines: {2, 3}, {9, 912}, {36, 5219}, {40, 5259}, {55, 1737}, {182, 6176}, {495, 1617}, {515, 6666}, {517, 1001}, {572, 5778}, {580, 5707}, {582, 5706}, {748, 1064}, {938, 943}, {940, 5398}, {942, 1708}, {944, 5260}, {956, 3681}, {958, 997}, {993, 3452}, {999, 3475}, {1000, 2346}, {1482, 3890}, {1621, 5657}, {3074, 3157}, {3295, 5690}, {3359, 4512}, {3428, 4423}, {3576, 5251}, {3579, 5806}, {3616, 5761}, {3826, 5842}, {4383, 5396}, {5178, 5687}, {5248, 6684}, {5273, 5770}, {5284, 5603}, {5462, 5752}, {5763, 5901}
As a point on the Euler line, X(6884) has Shinagawa coefficients (abc + 2$aSA$, 2abc).
X(6884) lies on these lines: {2, 3}, {7, 499}, {80, 3085}, {498, 4313}, {1259, 2894}, {3086, 5443}, {3622, 6265}, {3868, 5694}, {5273, 5536}
As a point on the Euler line, X(6885) has Shinagawa coefficients ($aSA$ - abc, abc - 2$aSA$).
X(6885) lies on these lines: {2,3}, {57,4317}, {119,5229}, {355,1788}, {390,5901}, {942,3476}, {952,3600}, {2551,5841}, {3474,5887}, {3601,4309}, {4256,5713}, {4292,5720}, {4294,5886}, {4298,5534}, {4299,5587}, {4340,5396}, {5440,5761}
As a point on the Euler line, X(6886) has Shinagawa coefficients (2abc + $aSA$, 2abc).
X(6886) lies on these lines: {2, 3}, {7, 1728}, {498, 4314}, {499, 4298}, {946, 3305}, {1837, 3748}, {3086, 5226}, {3555, 5886}, {3876, 5603}
As a point on the Euler line, X(6887) has Shinagawa coefficients (2abc + $aSA$, abc).
X(6887) lies on these lines: {2, 3}, {499, 3333}, {582, 3332}, {946, 6666}, {1058, 2346}, {1125, 5720}, {3085, 5722}, {3086, 3475}, {3940, 5901}, {4253, 5747}, {5044, 5761}, {5178, 5552}, {5439, 5770}, {5719, 5780}
X(6887) = complement of X(37407)
As a point on the Euler line, X(6888) has Shinagawa coefficients (-abc + 2$aSA$, 2abc).
X(6888) lies on these lines: {2, 3}, {10, 5538}, {12, 153}, {498, 4305}, {499, 3336}, {946, 5180}, {1158, 3306}, {1770, 3817}, {3086, 5902}, {4652, 5715}, {4855, 5587}, {5249, 6705}, {5885, 5886}
As a point on the Euler line, X(6889) has Shinagawa coefficients (abc + $aSA$, -abc).
X(6889) lies on these lines: {2,3}, {9,2252}, {46,226}, {65,3085}, {72,5552}, {165,5715}, {224,6734}, {499,950}, {581,1714}, {943,5218}, {944,5794}, {1068,1214}, {1071,5791}, {1155,5714}, {1213,5776}, {1385,3419}, {1490,1698}, {1754,5713}, {2245,5746}, {2278,5802}, {2646,3086}, {2949,3336}, {3074,3215}, {5249,5709}, {5251,6256}, {5759,5880}
As a point on the Euler line, X(6890) has Shinagawa coefficients (-2abc + $aSA$, 2abc).
X(6890) lies on these lines: {2, 3}, {46, 962}, {63, 6705}, {78, 6245}, {84, 908}, {498, 4297}, {499, 516}, {515, 4855}, {946, 3306}, {2646, 3476}, {3085, 3612}, {3218, 5758}, {4999, 5584}, {5440, 5787}, {5748, 6223}
As a point on the Euler line, X(6891) has Shinagawa coefficients (-2abc + $aSA$, abc).
X(6891) lies on these lines: {2, 3}, {40, 499}, {72, 5770}, {104, 3436}, {498, 3576}, {517, 1788}, {942, 5761}, {944, 5176}, {946, 3359}, {1056, 1476}, {1385, 3085}, {1479, 2077}, {2095, 5763}, {3428, 5433}, {3452, 6705}, {3814, 6256}, {3911, 5709}, {5225, 5840}, {5328, 5811}, {5435, 5758}, {5720, 6245}
As a point on the Euler line, X(6892) has Shinagawa coefficients (2$aSA$ - abc, 2abc - $aSA$).
X(6892) lies on these lines: {1,5770}, {2,3}, {63,5761}, {355,4305}, {912,5703}, {952,5789}, {1125,1158}, {2320,5768}, {3601,5881}, {4257,5713}, {4295,5886}, {4301,5709}, {5708,5901}, {5734,5744}
As a point on the Euler line, X(6893) has Shinagawa coefficients (abc, 2abc - $aSA$).
X(6893) lies on these lines: {2, 3}, {119, 3085}, {153, 3622}, {329, 5804}, {355, 497}, {388, 5886}, {517, 2551}, {908, 5761}, {912, 938}, {946, 5795}, {950, 5720}, {952, 1058}, {1056, 5901}, {1125, 6256}, {1210, 5770}, {1420, 1478}, {1479, 1697}, {1482, 3421}, {1512, 5250}, {3434, 5818}, {3436, 4861}, {4266, 5816}, {5082, 5790}, {5126, 5229}, {5722, 5777}, {5778, 5802}, {5806, 5812}
As a point on the Euler line, X(6894) has Shinagawa coefficients (abc, 2abc + 2$aSA$).
X(6894) lies on these lines: {2, 3}, {72, 2894}, {78, 1699}, {149, 946}, {153, 6583}, {938, 1478}, {1210, 3337}, {1479, 5703}, {1621, 6253}, {2949, 3219}, {3868, 5805}, {3951, 5735}, {4860, 5229}, {5080, 5536}
As a point on the Euler line, X(6895) has Shinagawa coefficients (abc, -2abc - 2$aSA$).
X(6895) lies on these lines: {2, 3}, {78, 5080}, {100, 6253}, {149, 151}, {225, 3100}, {938, 1479}, {1158, 5535}, {1210, 1770}, {1446, 4872}, {1478, 4305}, {3218, 6245}, {3868, 5787}, {4333, 5131}, {5221, 5225}
X(6895) = anticomplement of X(411)
As a point on the Euler line, X(6896) has Shinagawa coefficients (2abc, 2abc + $aSA$).
X(6896) lies on these lines: {2, 3}, {1056, 1837}, {1058, 3748}, {3811, 5603}, {4420, 5082}, {5400, 5713}
As a point on the Euler line, X(6897) has Shinagawa coefficients (2abc, -2abc + $aSA$).
X(6897) lies on these lines: {2, 3}, {46, 388}, {65, 1056}, {355, 4002}, {497, 3612}, {944, 2550}, {1038, 1068}, {1058, 2646}, {1385, 3434}, {1478, 6684}, {1698, 6256}, {3436, 3916}, {4861, 5082}
As a point on the Euler line, X(6898) has Shinagawa coefficients (2abc, 2abc - $aSA$).
X(6898) lies on these lines: {2, 3}, {497, 5818}, {1058, 1837}, {2551, 5603}, {3421, 4861}, {3436, 5886}, {3624, 6256}
As a point on the Euler line, X(6899) has Shinagawa coefficients (2abc, -2abc - $aSA$).
X(6899) lies on these lines: {2, 3}, {46, 497}, {65, 1058}, {355, 3921}, {388, 3612}, {944, 3811}, {1040, 1068}, {1056, 2646}, {1479, 3911}, {3421, 4420}, {3434, 3579}, {3436, 5440}
As a point on the Euler line, X(6900) has Shinagawa coefficients (2abc, abc + 2$aSA$).
X(6900) lies on these lines: {2, 3}, {80, 388}, {355, 3873}, {497, 5443}, {1389, 6601}, {1478, 3337}, {2550, 3878}, {2894, 3940}, {3189, 5603}, {3585, 3911}, {5536, 5818}
As a point on the Euler line, X(6901) has Shinagawa coefficients (2abc, -abc + 2$aSA$).
X(6901) lies on these lines: {2, 3}, {10, 5535}, {149, 5901}, {355, 5885}, {388, 5902}, {553, 5270}, {946, 5538}, {1074, 6198}, {1478, 1788}, {3585, 5131}, {3916, 5080}, {5260, 5841}, {5881, 6173}
As a point on the Euler line, X(6902) has Shinagawa coefficients (2abc, abc - 2$aSA$).
X(6902) lies on these lines: {2, 3}, {149, 5690}, {497, 5697}, {944, 6326}, {1056, 1388}, {1058, 2098}, {1199, 3193}, {1385, 5080}, {1479, 5657}, {3583, 6684}, {5253, 5841}, {5303, 6713}, {5506, 5818}
As a point on the Euler line, X(6903) has Shinagawa coefficients (2abc, -abc -2$aSA$).
X(6903) lies on these lines: {2, 3}, {484, 1479}, {497, 5903}, {1058, 2099}, {1076, 1870}, {3436, 6224}
As a point on the Euler line, X(6904) has Shinagawa coefficients (2R - r, 2r).
X(6904) lies on these lines: {2, 3}, {7, 78}, {8, 57}, {10, 4293}, {56, 2550}, {69, 1434}, {142, 390}, {144, 3876}, {145, 942}, {193, 4260}, {200, 4298}, {226, 5438}, {274, 3785}, {329, 936}, {347, 1038}, {354, 3189}, {386, 4340}, {388, 1376}, {391, 579}, {392, 6361}, {610, 5749}, {614, 4339}, {938, 3306}, {944, 3753}, {950, 5437}, {960, 3474}, {962, 6282}, {976, 4310}, {997, 4295}, {999, 5082}, {1056, 5687}, {1125, 4294}, {1193, 4307}, {1210, 5175}, {1329, 5229}, {1467, 3872}, {1698, 4299}, {1788, 5794}, {2096, 5777}, {2551, 4413}, {3241, 3680}, {3339, 6737}, {3361, 4847}, {3434, 5253}, {3476, 5836}, {3485, 5880}, {3486, 3812}, {3487, 5440}, {3488, 5439}, {3621, 5708}, {3624, 4302}, {3679, 4317}, {3752, 5716}, {3816, 5225}, {3925, 5204}, {4255, 5712}, {4315, 4853}, {4652, 5273}, {4855, 5249}, {5122, 5791}, {5128, 5837}, {5261, 5552}, {5277, 5286}, {5290, 6745}, {5435, 6734}, {5554, 5768}, {5587, 6705}, {5748, 6700}
As a point on the Euler line, X(6905) has Shinagawa coefficients (r, -R - r).
X(6905) lies on these lines: {1, 1389}, {2, 3}, {11, 5172}, {35, 946}, {36, 80}, {40, 997}, {46, 6261}, {54, 1437}, {55, 5603}, {56, 944}, {63, 5720}, {73, 3075}, {78, 5709}, {81, 5396}, {95, 286}, {100, 517}, {108, 2734}, {119, 4996}, {171, 1064}, {216, 1172}, {355, 2975}, {484, 2800}, {516, 1519}, {580, 3216}, {602, 978}, {603, 1745}, {656, 3737}, {758, 5535}, {912, 3218}, {958, 5818}, {970, 1812}, {971, 5122}, {972, 2728}, {993, 5587}, {1155, 6001}, {1193, 3072}, {1292, 2726}, {1376, 3428}, {1385, 5253}, {1465, 1870}, {1470, 4293}, {1482, 3871}, {1490, 1708}, {1503, 5096}, {1610, 3417}, {1621, 5886}, {1699, 5010}, {1749, 1768}, {1783, 1951}, {1809, 5081}, {2051, 4276}, {2096, 5658}, {2287, 5755}, {2801, 4973}, {3336, 5884}, {3576, 3833}, {3916, 5777}, {4255, 5706}, {4265, 5480}, {4299, 6256}, {4313, 5804}, {5088, 6516}, {5433, 6253}, {5435, 5768}, {5450, 5691}, {5563, 5882}, {6127, 6149}
X(6905) = inverse-in-excentral-hexyl-ellipse of X(4)
X(6905) = Thomson-isogonal conjugate of X(33883)
As a point on the Euler line, X(6906) has Shinagawa coefficients (r, R - r).
X(6906) lies on these lines: {1, 104}, {2, 3}, {10, 2077}, {12, 2829}, {35, 515}, {36, 946}, {40, 993}, {55, 944}, {56, 4295}, {84, 943}, {100, 355}, {191, 5538}, {284, 1765}, {318, 1809}, {498, 6256}, {516, 5267}, {517, 2975}, {553, 5563}, {577, 1172}, {950, 6705}, {952, 3871}, {958, 5657}, {1125, 1519}, {1193, 3073}, {1376, 5818}, {1385, 1621}, {1389, 5903}, {1437, 1614}, {1470, 3086}, {1503, 4265}, {1512, 5251}, {1537, 5901}, {1633, 3417}, {1699, 4333}, {1709, 3612}, {2078, 4311}, {2094, 5734}, {2096, 3487}, {2646, 6001}, {2654, 3075}, {2687, 2689}, {3428, 6361}, {3576, 5248}, {3577, 5128}, {3652, 5694}, {3746, 5882}, {4252, 5706}, {4304, 6245}, {4313, 5768}, {4511, 5887}, {4652, 5709}, {4855, 5720}, {4996, 5840}, {5010, 5691}, {5096, 5480}, {5122, 5806}, {5253, 5886}, {5258, 5537}, {5435, 5804}, {5440, 5777}, {5761, 5905}
As a point on the Euler line, X(6907) has Shinagawa coefficients (r + 2R, r - 2R).
X(6907) lies on these lines: {2, 3}, {7, 2095}, {9, 119}, {10, 5777}, {11, 3576}, {12, 40}, {52, 6045}, {63, 5771}, {72, 5690}, {84, 5705}, {226, 495}, {329, 5657}, {342, 6356}, {355, 1490}, {392, 1519}, {496, 950}, {515, 2886}, {516, 3822}, {573, 1901}, {581, 1834}, {944, 5175}, {946, 3838}, {952, 3419}, {958, 6256}, {993, 2829}, {1071, 6734}, {1074, 1465}, {1214, 1785}, {1329, 4640}, {1478, 3428}, {1482, 3487}, {1512, 3753}, {1537, 3877}, {1698, 1709}, {1727, 5445}, {1737, 1864}, {1750, 3925}, {1765, 5742}, {2077, 5432}, {2096, 5744}, {3584, 5537}, {3813, 5882}, {3824, 5806}, {4999, 5450}, {5219, 6282}, {5658, 5790}, {5714, 5758}, {5791, 6259}, {5794, 6261}
As a point on the Euler line, X(6908) has Shinagawa coefficients (r + 2R, -2R).
X(6908) lies on these lines: {2, 3}, {7, 5709}, {8, 5534}, {9, 1158}, {10, 1490}, {12, 5584}, {40, 226}, {72, 5657}, {84, 5745}, {165, 498}, {318, 6350}, {329, 3359}, {347, 1068}, {387, 581}, {388, 3428}, {499, 3586}, {516, 5715}, {517, 3487}, {572, 5802}, {573, 1715}, {943, 5281}, {944, 3419}, {950, 3086}, {966, 5776}, {971, 5791}, {991, 5292}, {1211, 6247}, {1214, 3346}, {1385, 3488}, {1698, 1750}, {2096, 3916}, {3072, 4307}, {3332, 5713}, {3579, 5714}, {3584, 4338}, {3587, 5226}, {3824, 5805}, {3945, 5707}, {4300, 5230}, {5175, 5731}, {5273, 6223}, {5658, 5777}, {5704, 5809}, {5705, 5732}, {5706, 5712}, {5768, 6734}
As a point on the Euler line, X(6909) has Shinagawa coefficients (- r + R, r - 2R).
X(6909) lies on these lines: {1, 1106}, {2, 3}, {35, 4297}, {36, 516}, {40, 2975}, {55, 3476}, {56, 962}, {63, 6282}, {78, 84}, {100, 515}, {104, 517}, {105, 2737}, {153, 2932}, {165, 993}, {497, 1470}, {519, 5537}, {603, 3562}, {758, 1768}, {934, 2739}, {938, 1466}, {944, 3871}, {946, 5253}, {956, 6244}, {971, 5440}, {997, 1709}, {1019, 6003}, {1158, 3869}, {1261, 4737}, {1262, 3100}, {1295, 1309}, {1490, 4855}, {1621, 3576}, {1754, 4257}, {1765, 2287}, {2096, 5905}, {2829, 5080}, {3304, 5734}, {4301, 5563}, {4511, 6001}, {4872, 6516}, {4973, 5536}, {4996, 5842}, {5260, 6684}, {6705, 6734}
As a point on the Euler line, X(6910) has Shinagawa coefficients (2 r + R, -r).
X(6910) lies on these lines: {2, 3}, {8, 2320}, {10, 3612}, {35, 3434}, {46, 1125}, {55, 3813}, {56, 6690}, {65, 3616}, {69, 5135}, {78, 5745}, {224, 936}, {226, 4652}, {498, 993}, {499, 5248}, {940, 3193}, {958, 5432}, {966, 2278}, {988, 3011}, {1001, 5433}, {1155, 5180}, {1454, 3485}, {1478, 5267}, {1621, 3086}, {1714, 4256}, {2182, 5296}, {2886, 5217}, {2975, 3085}, {3218, 3487}, {3305, 6700}, {3601, 6734}, {3618, 4259}, {3622, 5330}, {3624, 4512}, {3822, 4299}, {3868, 5703}, {3871, 5281}, {3876, 5273}, {3913, 4995}, {3916, 5905}, {4252, 5718}, {4293, 5303}, {4305, 5086}, {4423, 6691}, {5440, 5791}
As a point on the Euler line, X(6911) has Shinagawa coefficients (R - r, R + r), and also (2abc - $aSA$, $aSA$).
X(6911) lies on these lines: {2, 3}, {36, 5587}, {46, 5887}, {55, 5886}, {56, 355}, {57, 912}, {100, 5603}, {119, 1470}, {155, 3216}, {386, 5707}, {515, 6692}, {517, 997}, {569, 1437}, {579, 5778}, {750, 1064}, {936, 5709}, {940, 5396}, {944, 5253}, {952, 999}, {956, 5176}, {965, 5755}, {978, 3072}, {1060, 1465}, {1125, 6796}, {1216, 5752}, {1482, 4511}, {1699, 2077}, {1708, 5777}, {2095, 3940}, {2099, 6265}, {2975, 5818}, {3075, 3157}, {3295, 5901}, {3333, 5534}, {3336, 5693}, {3428, 4413}, {3816, 5842}, {3927, 5780}, {4383, 5398}, {5435, 5770}, {5535, 5692}, {5563, 5881}, {5902, 6326}
As a point on the Euler line, X(6912) has Shinagawa coefficients (R + r, 2R - r).
X(6912) lies on these lines: {1, 651}, {2, 3}, {10, 5537}, {36, 3817}, {40, 5260}, {65, 1776}, {100, 5587}, {104, 5886}, {144, 956}, {153, 495}, {355, 3871}, {515, 1621}, {516, 5251}, {517, 3219}, {774, 2647}, {946, 2975}, {958, 962}, {993, 1699}, {1001, 5731}, {1259, 5175}, {1466, 5704}, {1768, 5883}, {1935, 2654}, {3303, 3486}, {3304, 3485}, {3305, 6282}, {3555, 5887}, {3576, 5284}, {3616, 6223}, {3746, 4314}, {3869, 3872}, {3916, 5806}, {4256, 5400}, {4297, 5259}, {4298, 5563}, {4301, 5258}, {5086, 6735}, {5248, 5691}, {5253, 5450}, {5603, 5905}
As a point on the Euler line, X(6913) has Shinagawa coefficients (2R + r, 2R - r).
X(6913) lies on these lines: {1, 1864}, {2, 3}, {9, 374}, {40, 3683}, {55, 3586}, {63, 2095}, {65, 1728}, {72, 1482}, {78, 5780}, {226, 999}, {329, 956}, {355, 950}, {515, 1001}, {912, 5728}, {946, 958}, {952, 954}, {962, 5260}, {993, 3817}, {1125, 6260}, {1158, 3812}, {1159, 5729}, {1260, 3419}, {1385, 1490}, {1478, 1617}, {1699, 3428}, {1724, 5706}, {1750, 3576}, {1901, 5120}, {2077, 4413}, {3073, 5711}, {3303, 5881}, {3304, 4870}, {3487, 5811}, {4254, 5816}, {5175, 5687}, {5259, 5691}, {5273, 5771}, {5284, 5731}, {5709, 5806}
As a point on the Euler line, X(6914) has Shinagawa coefficients (R + 2r, R - 2r).
X(6914) lies on these lines: {1, 1399}, {2, 3}, {35, 355}, {36, 1836}, {55, 952}, {56, 5901}, {100, 5790}, {104, 1621}, {119, 5432}, {155, 5453}, {156, 1437}, {497, 1484}, {517, 993}, {946, 5267}, {954, 5843}, {956, 5844}, {958, 5690}, {970, 5446}, {1030, 5816}, {1352, 4265}, {1385, 5248}, {1478, 5172}, {1482, 2975}, {1483, 3295}, {1709, 3576}, {1768, 5426}, {2077, 5251}, {2646, 5887}, {2829, 6690}, {2886, 5840}, {3065, 6326}, {3488, 5770}, {3652, 5693}, {3654, 5537}, {3816, 6713}, {4252, 5707}, {5010, 5587}, {5180, 5603}
X(6914) = circumcircle-inverse of X(37976)
As a point on the Euler line, X(6915) has Shinagawa coefficients (R - r, 2R + r).
X(6915) lies on these lines: {2, 3}, {10, 5659}, {35, 3817}, {56, 5704}, {58, 5400}, {80, 1210}, {100, 946}, {515, 5253}, {651, 3075}, {936, 3878}, {938, 3304}, {962, 1376}, {1014, 5740}, {1259, 5748}, {1261, 5100}, {1320, 1389}, {1470, 5229}, {1490, 3306}, {1621, 6796}, {2801, 3337}, {2975, 5587}, {3218, 5777}, {3303, 5703}, {3678, 5536}, {3746, 5443}, {3816, 6253}, {3868, 5720}, {3871, 5603}, {3876, 5709}, {3881, 5531}, {3889, 5534}, {3913, 5734}, {5176, 6734}, {5440, 5806}, {5537, 6700}
As a point on the Euler line, X(6916) has Shinagawa coefficients (2R, R - 2r).
X(6916) lies on these lines: {2, 3}, {7, 517}, {8, 1071}, {10, 84}, {40, 388}, {63, 2096}, {165, 1478}, {226, 6282}, {278, 1074}, {495, 6244}, {497, 3576}, {515, 2550}, {936, 6260}, {944, 3872}, {962, 3890}, {966, 1765}, {1058, 1385}, {1350, 5800}, {1422, 5930}, {2077, 5218}, {2551, 6256}, {3419, 5768}, {3428, 4293}, {3434, 5731}, {3436, 5828}, {3587, 5759}, {4340, 5706}, {5044, 5811}, {5249, 5603}, {5439, 5804}, {5658, 5720}, {5705, 6705}, {5777, 6223}, {5784, 6001}
As a point on the Euler line, X(6917) has Shinagawa coefficients (R, R + 2r).
X(6917) lies on these lines: {2, 3}, {46, 3585}, {65, 68}, {79, 5693}, {155, 5706}, {225, 1060}, {388, 952}, {497, 5901}, {517, 5794}, {958, 5841}, {971, 5880}, {1056, 1483}, {1062, 1074}, {1352, 4259}, {1385, 3824}, {1454, 1737}, {1479, 2646}, {1482, 3434}, {1714, 5398}, {1834, 5707}, {1836, 5887}, {1901, 5778}, {2245, 5816}, {2550, 5690}, {3436, 3927}, {3564, 5800}, {3583, 3612}, {3822, 6796}, {4654, 5270}, {5080, 5818}, {5082, 5844}, {5290, 5534}, {5784, 5805}
As a point on the Euler line, X(6918) has Shinagawa coefficients (2R - r, 2R + r).
X(6918) lies on these lines: {2, 3}, {40, 4413}, {56, 5587}, {57, 5777}, {72, 2095}, {78, 1482}, {355, 999}, {499, 1617}, {517, 936}, {912, 5708}, {938, 952}, {942, 5720}, {946, 1376}, {956, 5818}, {1001, 6796}, {1071, 3306}, {1445, 5779}, {1490, 5437}, {1698, 3428}, {3216, 5706}, {3295, 5886}, {3304, 5881}, {3452, 5812}, {3812, 6261}, {5044, 5709}, {5045, 5534}, {5120, 5816}, {5221, 5693}, {5603, 5687}, {5703, 5901}, {5790, 6734}, {6245, 6692}
As a point on the Euler line, X(6919) has Shinagawa coefficients (2R - r, -2r).
X(6919) lies on these lines: {1, 5748}, {2, 3}, {8, 3452}, {11, 2551}, {63, 5704}, {78, 5328}, {145, 5722}, {329, 1210}, {388, 3816}, {390, 5552}, {392, 5818}, {496, 3421}, {497, 1329}, {908, 938}, {936, 5175}, {1376, 5225}, {1706, 5250}, {2899, 3705}, {3085, 3814}, {3086, 3825}, {3485, 5087}, {3586, 6700}, {3600, 5080}, {3614, 4423}, {3616, 5219}, {3617, 3877}, {3621, 3940}, {3634, 4512}, {3820, 5082}, {4847, 4866}, {5439, 5714}, {5554, 5804}
X(6919) = complement of X(37267)
X(6919) = anticomplement of X(17567)
As a point on the Euler line, X(6920) has Shinagawa coefficients (2R + r, R - r).
X(6920) lies on these lines: {2, 3}, {9, 1389}, {40, 3918}, {55, 5818}, {56, 5714}, {72, 1173}, {80, 943}, {104, 1125}, {226, 5443}, {355, 1621}, {515, 5259}, {517, 5260}, {944, 1001}, {946, 5251}, {958, 5603}, {1385, 5284}, {2077, 3634}, {2654, 3074}, {2975, 5886}, {3303, 3488}, {3304, 3487}, {3614, 5172}, {3616, 5811}, {3624, 5450}, {3647, 5535}, {3652, 5885}, {3871, 5790}, {4423, 5658}, {5248, 5587}, {5273, 5804}, {5506, 5538}, {5777, 6265}
X(6920) = complement of X(37163)
As a point on the Euler line, X(6921) has Shinagawa coefficients (R - 2r, r).
X(6921) lies on these lines: {2, 3}, {8, 1319}, {36, 3436}, {55, 6691}, {56, 3035}, {63, 6700}, {78, 3911}, {100, 3086}, {145, 1145}, {499, 3434}, {944, 4881}, {966, 4268}, {1125, 5119}, {1210, 4855}, {1329, 5204}, {1376, 5433}, {1385, 5554}, {1420, 6735}, {1788, 4511}, {3057, 3616}, {3085, 5253}, {3452, 4652}, {3814, 4299}, {3816, 5217}, {3825, 4302}, {3868, 5435}, {3876, 5744}, {3913, 6174}, {4413, 4999}, {5438, 6734}, {5442, 5692}
As a point on the Euler line, X(6922) has Shinagawa coefficients (r - 2R, r + 2R).
X(6922) lies on these lines: {2, 3}, {11, 40}, {12, 3576}, {57, 5812}, {78, 952}, {355, 936}, {495, 1385}, {496, 517}, {499, 3428}, {515, 1329}, {516, 3825}, {908, 1071}, {938, 1482}, {946, 3812}, {1076, 1465}, {1445, 5762}, {1706, 5705}, {2077, 6284}, {2095, 5758}, {2886, 6684}, {3035, 6796}, {3216, 5721}, {3452, 5777}, {3814, 4297}, {3927, 5770}, {4857, 5537}, {5437, 5715}, {5690, 6734}, {5703, 6049}, {5720, 5787}
As a point on the Euler line, X(6923) has Shinagawa coefficients (R, 2r - R).
X(6923) lies on these lines: {2, 3}, {40, 3585}, {55, 5840}, {119, 1376}, {355, 5836}, {388, 1482}, {497, 1387}, {511, 5820}, {517, 1478}, {912, 3419}, {952, 3434}, {971, 5832}, {1060, 1785}, {1074, 1877}, {1351, 5800}, {1385, 1479}, {1709, 3359}, {2096, 5770}, {2550, 5779}, {2829, 2886}, {3428, 5841}, {3436, 5690}, {3576, 3583}, {3838, 5886}, {4640, 5123}, {5080, 5657}, {5714, 5761}, {5794, 5887}
As a point on the Euler line, X(6924) has Shinagawa coefficients (R - 2r, R + 2r).
X(6924) lies on these lines: {2, 3}, {35, 5886}, {36, 355}, {55, 5901}, {56, 952}, {100, 1482}, {499, 5172}, {970, 1216}, {999, 1483}, {1155, 5887}, {1329, 5841}, {1352, 5096}, {1376, 5690}, {1385, 3812}, {1484, 3086}, {2975, 5790}, {3216, 5398}, {3336, 6326}, {3361, 5534}, {4255, 5707}, {5122, 5777}, {5124, 5816}, {5438, 5709}, {5687, 5844}, {5752, 6101}, {5842, 6691}, {5903, 6265}
As a point on the Euler line, X(6925) has Shinagawa coefficients (R, r - 2R).
X(6925) lies on these lines: {1, 5555}, {2, 3}, {8, 6001}, {10, 1709}, {40, 3436}, {72, 6259}, {78, 6260}, {84, 6734}, {144, 153}, {388, 962}, {497, 1319}, {515, 3434}, {516, 1478}, {517, 5905}, {908, 6282}, {971, 3419}, {1040, 1877}, {1479, 4297}, {1512, 3359}, {2096, 3218}, {2551, 4640}, {2829, 3428}, {3219, 5657}, {3586, 5732}, {3876, 5811}
As a point on the Euler line, X(6926) has Shinagawa coefficients (r - 2R, 2R).
X(6926) lies on these lines: {2, 3}, {9, 6705}, {40, 3086}, {57, 5758}, {78, 5768}, {84, 3452}, {165, 499}, {496, 6244}, {498, 5726}, {515, 5438}, {936, 6245}, {944, 5440}, {946, 5437}, {962, 3359}, {1210, 6282}, {1490, 6700}, {1706, 6684}, {2077, 4294}, {3085, 3576}, {5328, 6223}, {5433, 5584}, {5435, 5709}, {5439, 5603}, {5552, 5731}, {5708, 5763}
As a point on the Euler line, X(6927) has Shinagawa coefficients (2r, -r - 2R).
X(6927) lies on these lines: {2, 3}, {40, 6700}, {497, 6796}, {936, 5657}, {938, 6049}, {944, 1210}, {946, 5218}, {971, 5825}, {1071, 5435}, {1490, 3911}, {1519, 6361}, {1697, 5603}, {1750, 6705}, {1788, 6261}, {2096, 6260}, {3916, 5811}, {4345, 5703}, {5053, 5822}, {5122, 6259}, {5126, 5704}, {5705, 5795}, {5744, 5777}, {5748, 5812}, {5766, 5805}, {5771, 5780}
As a point on the Euler line, X(6928) has Shinagawa coefficients (R, -R - 2r).
X(6928) lies on these lines: {2, 3}, {40, 3583}, {56, 5841}, {355, 960}, {497, 1482}, {517, 1479}, {944, 5080}, {952, 3436}, {1062, 1785}, {1076, 1877}, {1329, 5842}, {1385, 1478}, {1728, 5709}, {2551, 5790}, {3434, 5690}, {3488, 5761}, {3576, 3585}, {3814, 6796}, {5050, 5800}, {5087, 6256}, {5204, 6713}, {5292, 5398}, {5722, 5812}, {5729, 5762}
As a point on the Euler line, X(6929) has Shinagawa coefficients (R, R - 2r).
X(6929) lies on these lines: {2, 3}, {55, 119}, {355, 1479}, {388, 5901}, {497, 952}, {912, 1864}, {1058, 1483}, {1060, 1877}, {1145, 3434}, {1319, 1478}, {1376, 5840}, {1385, 6256}, {1482, 3436}, {1484, 5274}, {1837, 5887}, {2551, 5690}, {2829, 3816}, {3421, 5844}, {3583, 5119}, {3586, 5720}, {3825, 5450}, {4271, 5816}, {4857, 5881}, {5080, 5603}
As a point on the Euler line, X(6930) has Shinagawa coefficients (R + r, R - 2r).
X(6930) lies on these lines: {2, 3}, {119, 5218}, {355, 4294}, {390, 952}, {517, 5698}, {912, 3488}, {944, 3890}, {1001, 2829}, {1385, 6259}, {1420, 4317}, {1478, 2078}, {1697, 4309}, {2550, 5840}, {3486, 5887}, {3600, 5901}, {4293, 5126}, {4302, 5587}, {4304, 5720}, {4313, 5811}, {4314, 5534}, {5057, 5603}, {5248, 6256}, {5722, 5770}
As a point on the Euler line, X(6931) has Shinagawa coefficients (R - 2r, -r).
X(6931) lies on these lines: {2, 3}, {8, 1392}, {11, 3913}, {56, 6667}, {145, 1387}, {498, 3825}, {499, 3436}, {3193, 4383}, {3616, 5252}, {3617, 5330}, {3618, 5820}, {3634, 5250}, {3680, 6735}, {3868, 5704}, {3871, 5274}, {3876, 5328}, {3877, 4731}, {3897, 5550}, {4299, 6681}, {4423, 6668}, {4866, 5231}, {5554, 5886}
As a point on the Euler line, X(6932) has Shinagawa coefficients (r + R, r - 2R).
X(6932) lies on these lines: {2, 3}, {11, 5731}, {12, 962}, {40, 5057}, {119, 5657}, {153, 956}, {165, 3814}, {392, 1538}, {495, 1000}, {946, 3890}, {1329, 5698}, {1388, 3486}, {1519, 3877}, {1699, 3822}, {2098, 3485}, {2975, 6256}, {3428, 5080}, {3697, 5887}, {3869, 6735}, {3872, 5086}, {3947, 4301}, {6260, 6734}
As a point on the Euler line, X(6933) has Shinagawa coefficients (R - 2r, r).
X(6933) lies on these lines: {2, 3}, {55, 6668}, {498, 3434}, {499, 3822}, {908, 5705}, {958, 3614}, {960, 3698}, {1728, 3306}, {1837, 3616}, {2320, 5550}, {2886, 5552}, {3218, 5714}, {3303, 3829}, {3436, 5258}, {3617, 5730}, {3817, 5250}, {3868, 5226}, {3876, 5748}, {5087, 5180}, {5219, 6734}
As a point on the Euler line, X(6934) has Shinagawa coefficients (r, -R - 2r).
X(6934) lies on these lines: {2, 3}, {46, 515}, {56, 5842}, {65, 944}, {355, 3916}, {553, 4317}, {946, 3612}, {1478, 6796}, {1512, 4316}, {1699, 4324}, {1770, 6261}, {2646, 4294}, {3436, 5841}, {3928, 5881}, {4252, 5721}, {4259, 6776}, {5440, 5812}, {5657, 5794}, {5759, 5784}
As a point on the Euler line, X(6935) has Shinagawa coefficients (2r, 2R - r).
X(6935) lies on these lines: {1, 6705}, {2, 3}, {57, 5603}, {104, 1056}, {142, 1519}, {226, 2096}, {388, 5450}, {515, 5218}, {517, 5744}, {942, 4323}, {944, 3601}, {946, 3474}, {1071, 5703}, {1158, 3485}, {1466, 3086}, {2077, 2550}, {3916, 5758}, {5122, 5805}, {5657, 5745}
As a point on the Euler line, X(6936) has Shinagawa coefficients (2R + r, -R - 2r).
X(6936) lies on these lines: {2, 3}, {9, 5881}, {226, 4317}, {390, 5729}, {944, 960}, {950, 1728}, {1319, 3487}, {1837, 4294}, {3057, 3488}, {3586, 4330}, {3922, 6361}, {4268, 5746}, {4271, 5802}, {4293, 5714}, {4302, 6684}, {4679, 5658}, {5731, 5811}, {5734, 5758}
As a point on the Euler line, X(6937) has Shinagawa coefficients (2R + r, -R + r).
X(6937) lies on these lines: {2, 3}, {10, 5693}, {11, 4305}, {12, 4295}, {40, 3822}, {119, 5811}, {226, 5903}, {484, 4338}, {944, 2886}, {1158, 1698}, {1770, 6684}, {1901, 5036}, {2099, 3487}, {3419, 4861}, {3826, 5817}, {3841, 5587}, {3925, 5658}, {5175, 6224}
As a point on the Euler line, X(6938) has Shinagawa coefficients (r, R - 2r).
X(6938) lies on these lines: {1, 5553}, {2, 3}, {35, 6256}, {55, 2829}, {104, 497}, {165, 1512}, {515, 1709}, {944, 3057}, {946, 4299}, {1319, 1836}, {1479, 5450}, {1519, 3576}, {1699, 4316}, {2096, 3488}, {3434, 5840}, {4309, 5882}, {4324, 5691}, {4640, 5657}
As a point on the Euler line, X(6939) has Shinagawa coefficients (2R, 2R - r).
X(6939) lies on these lines: {2, 3}, {72, 5804}, {144, 2095}, {355, 1058}, {497, 5587}, {908, 3421}, {938, 5777}, {942, 5811}, {946, 2551}, {1056, 5226}, {2096, 3306}, {3305, 5657}, {3488, 5720}, {5082, 5818}, {5316, 6282}, {5758, 5806}, {5768, 5927}
As a point on the Euler line, X(6940) has Shinagawa coefficients (2R - r, r - R).
X(6940) lies on these lines: {2, 3}, {10, 104}, {36, 6684}, {40, 3884}, {56, 5657}, {100, 1385}, {517, 5253}, {601, 978}, {944, 1376}, {1125, 2077}, {1389, 3754}, {1466, 3487}, {1470, 3085}, {1698, 5450}, {1706, 3576}, {4413, 5818}, {5559, 5563}
As a point on the Euler line, X(6941) has Shinagawa coefficients (r, r - R).
X(6941) lies on these lines: {2, 3}, {8, 119}, {10, 1519}, {11, 944}, {12, 2098}, {40, 3814}, {104, 499}, {355, 4861}, {946, 1512}, {1329, 5657}, {1537, 5690}, {1538, 4002}, {2829, 5433}, {2886, 5818}, {3576, 3825}, {3583, 6796}, {5226, 5804}
As a point on the Euler line, X(6942) has Shinagawa coefficients (2r, -R - 2r).
X(6942) lies on these lines: {2, 3}, {35, 5603}, {36, 944}, {104, 5204}, {946, 5010}, {993, 5818}, {1071, 5122}, {1512, 4297}, {2077, 6361}, {3086, 5172}, {3436, 4996}, {4652, 5720}, {4855, 5709}, {5082, 6585}, {5096, 6776}, {5267, 5587}, {5433, 5842}
As a point on the Euler line, X(6943) has Shinagawa coefficients (R - r, -2R - r).
X(6943) lies on these lines: {2, 3}, {11, 962}, {12, 5731}, {78, 5176}, {273, 3007}, {908, 6245}, {938, 2099}, {1158, 5057}, {1210, 4301}, {1445, 5735}, {1699, 3825}, {3035, 6253}, {3218, 5812}, {3306, 5715}, {3476, 5703}, {3814, 5691}
As a point on the Euler line, X(6944) has Shinagawa coefficients (R - r, R).
X(6944) lies on these lines: {2, 3}, {119, 388}, {355, 3086}, {498, 1697}, {499, 1420}, {1210, 5720}, {1788, 5887}, {3085, 5886}, {3452, 5709}, {5053, 5816}, {5176, 5818}, {5328, 5758}, {5435, 5811}, {5552, 5603}, {5770, 5777}, {6260, 6692}
As a point on the Euler line, X(6945) has Shinagawa coefficients (R - r, 2R - r).
X(6945) lies on these lines: {2, 3}, {11, 3476}, {119, 1320}, {153, 999}, {946, 6735}, {962, 1329}, {1476, 3086}, {1512, 3877}, {1538, 3753}, {1699, 3814}, {3816, 5731}, {3817, 4342}, {3825, 5691}, {3872, 5176}, {5253, 6256}, {5819, 6506}
As a point on the Euler line, X(6946) has Shinagawa coefficients (2R - r, R + r).
X(6946) lies on these lines: {2, 3}, {56, 5818}, {100, 5886}, {104, 5587}, {355, 5253}, {946, 5537}, {997, 1706}, {1376, 5603}, {1466, 5714}, {1737, 5563}, {2077, 3817}, {3306, 5720}, {3624, 6796}, {3871, 5901}, {4413, 5657}, {5883, 6326}
As a point on the Euler line, X(6947) has Shinagawa coefficients (2R, -R - r).
X(6947) lies on these lines: {2, 3}, {497, 1737}, {515, 5316}, {944, 997}, {1056, 1319}, {1058, 3057}, {1072, 5272}, {1385, 3436}, {1479, 6684}, {3219, 5770}, {3421, 4511}, {3428, 3816}, {4413, 5842}, {4679, 6001}, {5439, 5812}
As a point on the Euler line, X(6948) has Shinagawa coefficients (R - r, 2r - R).
X(6948) lies on these lines: {2, 3}, {40, 4299}, {104, 3434}, {165, 4316}, {497, 5840}, {515, 3359}, {517, 3474}, {912, 2096}, {1376, 2829}, {1385, 4294}, {1478, 2077}, {1482, 3600}, {3419, 5770}, {3576, 4302}, {5176, 5657}
As a point on the Euler line, X(6949) has Shinagawa coefficients (2r, -R), and also (4S2, -abc$a$).
X(6949) lies on these lines: {2, 3}, {8, 6265}, {80, 499}, {104, 5433}, {119, 2975}, {498, 5443}, {1000, 1389}, {1125, 1512}, {1388, 3086}, {1519, 6684}, {1768, 3467}, {2800, 5445}, {3582, 5882}, {3878, 5657}, {3890, 5886}
As a point on the Euler line, X(6950) has Shinagawa coefficients (2r, R - 2r).
X(6950) lies on these lines: {2, 3}, {35, 944}, {36, 3474}, {40, 5267}, {55, 104}, {497, 5533}, {515, 5010}, {993, 2077}, {1158, 3612}, {1385, 3890}, {1946, 2401}, {2829, 5432}, {3434, 4996}, {4265, 6776}, {4293, 5172}
As a point on the Euler line, X(6951) has Shinagawa coefficients (2R, 2r - R).
X(6951) lies on these lines: {2, 3}, {104, 2886}, {153, 5790}, {388, 5903}, {484, 1478}, {1056, 2099}, {1074, 1870}, {1621, 5840}, {2077, 3822}, {2550, 2801}, {2829, 3925}, {3434, 6224}, {3585, 6684}, {5818, 6256}
As a point on the Euler line, X(6952) has Shinagawa coefficients (2r, R), and also (4S2, abc$a$).
X(6952) lies on these lines: {2, 3}, {12, 104}, {81, 1199}, {355, 6224}, {484, 946}, {498, 944}, {499, 1788}, {1512, 3634}, {2099, 3086}, {2800, 5443}, {2829, 3614}, {3584, 5882}, {5253, 6713}, {5303, 5841}
As a point on the Euler line, X(6953) has Shinagawa coefficients (R - r, 2R).
X(6953) lies on these lines: {2, 3}, {153, 3600}, {498, 3817}, {499, 4311}, {946, 5552}, {962, 5328}, {1728, 5435}, {1837, 3476}, {3086, 4308}, {3218, 5811}, {3306, 6260}, {3885, 5603}, {5292, 5400}
As a point on the Euler line, X(6954) has Shinagawa coefficients (R + 2r, -R - r).
X(6954) lies on these lines: {2, 3}, {517, 5218}, {572, 5822}, {912, 5744}, {997, 5837}, {1737, 3576}, {2095, 5719}, {3428, 5432}, {3940, 5771}, {4511, 5657}, {5267, 6256}, {5709, 5761}, {5720, 5745}
As a point on the Euler line, X(6955) has Shinagawa coefficients (2R - r, -R + 2r).
X(6955) lies on these lines: {2, 3}, {46, 4317}, {104, 2550}, {390, 1387}, {944, 5836}, {1155, 4293}, {1706, 5881}, {2096, 5784}, {2829, 4413}, {3612, 4309}, {4299, 6684}, {5603, 5880}, {5735, 6282}
As a point on the Euler line, X(6956) has Shinagawa coefficients (2r, r + 2R).
X(6956) lies on these lines: {2, 3}, {936, 5818}, {946, 1788}, {1071, 5226}, {1210, 3340}, {2096, 6705}, {3911, 5715}, {5219, 6245}, {5229, 5450}, {5587, 6700}, {5657, 5705}, {5744, 5812}, {5748, 5777}
As a point on the Euler line, X(6957) has Shinagawa coefficients (R, 2R - r).
X(6957) lies on these lines: {2, 3}, {153, 1056}, {497, 5252}, {938, 1864}, {946, 3436}, {962, 2551}, {1478, 3817}, {2550, 5123}, {3434, 5587}, {3868, 5804}, {5439, 6259}, {5722, 5927}
As a point on the Euler line, X(6958) has Shinagawa coefficients (2r - R, R).
X(6958) lies on these lines: {2, 3}, {56, 6713}, {182, 5820}, {355, 5123}, {498, 1385}, {499, 517}, {952, 5552}, {1387, 1482}, {3812, 5886}, {3814, 5450}, {5204, 5841}, {5780, 5789}
As a point on the Euler line, X(6959) has Shinagawa coefficients (R - 2r, R).
X(6959) lies on these lines: {2, 3}, {56, 119}, {155, 4383}, {355, 499}, {498, 3057}, {952, 3086}, {1145, 1482}, {3085, 5901}, {3582, 5881}, {3825, 6796}, {4268, 5816}, {5450, 6681}
As a point on the Euler line, X(6960) has Shinagawa coefficients (R + 2r, -2R).
X(6960) lies on these lines: {2, 3}, {8, 6326}, {40, 5180}, {153, 2975}, {329, 2949}, {498, 962}, {499, 5731}, {2829, 5303}, {3085, 5697}, {3584, 4301}, {3678, 5660}, {4015, 5659}
As a point on the Euler line, X(6961) has Shinagawa coefficients (R - 2r, r - R).
X(6961) lies on these lines: {2, 3}, {57, 5761}, {78, 5770}, {104, 5552}, {499, 2077}, {1125, 3359}, {1385, 5218}, {3086, 6713}, {4861, 5657}, {5122, 5812}, {5720, 6705}
As a point on the Euler line, X(6962) has Shinagawa coefficients (R + 2r, -2R - r).
X(6962) lies on these lines: {2, 3}, {938, 1319}, {962, 5218}, {1837, 5704}, {3035, 5584}, {3057, 5703}, {3428, 5552}, {3434, 6796}, {4301, 5119}, {4652, 6260}, {5881, 6734}
As a point on the Euler line, X(6963) has Shinagawa coefficients (r - 2R, r + R).
X(6963) lies on these lines: {2, 3}, {11, 5657}, {40, 3825}, {119, 5731}, {944, 1329}, {997, 5881}, {1072, 5121}, {1737, 5697}, {3576, 3814}, {3816, 5603}, {5328, 5768}
As a point on the Euler line, X(6964) has Shinagawa coefficients (2R - r, 2R).
X(6964) lies on these lines: {2, 3}, {40, 5316}, {57, 5811}, {78, 5804}, {84, 6692}, {119, 5261}, {938, 5720}, {946, 1706}, {3086, 5587}, {3452, 5758}, {5437, 6260}
As a point on the Euler line, X(6965) has Shinagawa coefficients (2R, R - 2r).
X(6965) lies on these lines: {2, 3}, {80, 497}, {104, 3816}, {119, 1621}, {149, 5790}, {355, 3890}, {388, 5443}, {1479, 5818}, {2551, 3878}, {5080, 5886}
As a point on the Euler line, X(6966) has Shinagawa coefficients (R - 2r, r - 2R).
X(6966) lies on these lines: {2, 3}, {46, 4301}, {65, 5734}, {78, 6705}, {962, 1155}, {2077, 3434}, {3436, 5450}, {4855, 6245}, {5218, 5252}, {5881, 6735}
As a point on the Euler line, X(6967) has Shinagawa coefficients (r - 2Rr, R).
X(6967) lies on these lines: {2, 3}, {104, 2551}, {499, 5119}, {1319, 3085}, {1385, 5552}, {3057, 3086}, {3428, 6691}, {3812, 5603}, {3876, 5770}, {5316, 6705}
As a point on the Euler line, X(6968) has Shinagawa coefficients (r, 2r - R).
X(6968) lies on these lines: {2, 3}, {119, 3434}, {1387, 5274}, {1512, 1699}, {1519, 5587}, {1537, 5790}, {5048, 5252}, {5123, 5657}, {5817, 5832}, {5818, 5836}
As a point on the Euler line, X(6969) has Shinagawa coefficients (2r, r - 2R).
X(6969) lies on these lines: {2, 3}, {119, 3421}, {517, 5748}, {1000, 1512}, {1071, 5704}, {1519, 3452}, {2096, 3911}, {3617, 5780}, {3872, 5720}, {5225, 6796}
As a point on the Euler line, X(6970) has Shinagawa coefficients (R - 2r, R + r).
X(6970) lies on these lines: {2, 3}, {119, 4293}, {355, 5126}, {499, 2078}, {912, 5435}, {1420, 1737}, {3911, 5720}, {5218, 5886}, {5709, 6700}
As a point on the Euler line, X(6971) has Shinagawa coefficients (2r - R, 2r + R).
X(6971) lies on these lines: {2, 3}, {11, 1482}, {145, 1484}, {355, 3814}, {1329, 5790}, {3825, 5886}, {5087, 5887}, {5433, 5841}
X(6971) = {X(3),X(5)}-harmonic conjugate of X(6980)
As a point on the Euler line, X(6972) has Shinagawa coefficients (2r - R, 2R).
X(6972) lies on these lines: {2, 3}, {484, 499}, {498, 5731}, {908, 6705}, {3086, 5903}, {3582, 4301}, {5080, 5450}, {5552, 6224}
As a point on the Euler line, X(6973) has Shinagawa coefficients (R - r, R - 2r).
X(6973) lies on these lines: {2, 3}, {119, 497}, {952, 5274}, {1478, 5193}, {3476, 5886}, {3825, 6256}, {5176, 5603}, {5261, 5901}
As a point on the Euler line, X(6974) has Shinagawa coefficients (R + 2r, 2R - r).
X(6974) lies on these lines: {2, 3}, {946, 4652}, {962, 4640}, {1125, 1709}, {1727, 3338}, {2320, 3427}, {3218, 5603}, {3616, 6001}
As a point on the Euler line, X(6975) has Shinagawa coefficients (2R - r, R - r).
X(6975) lies on these lines: {2, 3}, {11, 5818}, {119, 3616}, {944, 3816}, {1329, 5603}, {3825, 5587}, {4861, 5886}, {5328, 5804}
As a point on the Euler line, X(6976) has Shinagawa coefficients (2R + r, R - 2r).
X(6976) lies on these lines: {2, 3}, {1512, 4512}, {1864, 3488}, {2829, 4423}, {3683, 5657}, {4294, 5818}, {5259, 6256}
As a point on the Euler line, X(6977) has Shinagawa coefficients (2r, R - r).
X(6977) lies on these lines: {2, 3}, {46, 5603}, {104, 3085}, {498, 5450}, {944, 3612}, {1519, 3624}, {3218, 5761}
As a point on the Euler line, X(6978) has Shinagawa coefficients (2r - R, r + R).
X(6978) lies on these lines: {2, 3}, {908, 5770}, {912, 5748}, {1210, 5761}, {3359, 3817}, {4293, 6713}
As a point on the Euler line, X(6979) has Shinagawa coefficients (R - 2r, 2R).
X(6979) lies on these lines: {2, 3}, {56, 153}, {80, 3086}, {145, 6265}, {3085, 5443}, {3874, 5660}
As a point on the Euler line, X(6980) has Shinagawa coefficients (2r + R, 2r - R).
X(6980) lies on these lines: {2, 3}, {12, 1482}, {119, 2886}, {3822, 5886}, {5180, 5657}, {5432, 5840}
X(6980) = {X(3),X(5)}-harmonic conjugate of X(6971)
As a point on the Euler line, X(6981) has Shinagawa coefficients (R - 2r, R - r).
X(6981) lies on these lines: {2, 3}, {119, 3086}, {499, 5193}, {912, 5704}, {4861, 5818}
As a point on the Euler line, X(6982) has Shinagawa coefficients (R + r, 2r - R).
X(6982) lies on these lines: {2, 3}, {119, 2550}, {1482, 5261}, {5057, 5657}, {5218, 5840}
As a point on the Euler line, X(6983) has Shinagawa coefficients (2R - r, R).
X(6983) lies on these lines: {2, 3}, {3085, 5919}, {3086, 5252}, {5552, 5886}, {5603, 5836}
As a point on the Euler line, X(6984) has Shinagawa coefficients (2R + r, R + 2r).
X(6984) lies on these lines: {2, 3}, {65, 5714}, {3487, 5252}, {5603, 5794}, {5817, 5880}
As a point on the Euler line, X(6985) has Shinagawa coefficients ($aSA$, -2abc - $aSA$).
X(6985) lies on these lines: {2, 3}, {35, 5219}, {40, 5692}, {46, 1858}, {56, 5722}, {100, 6361}, {255, 2635}, {355, 3428}, {496, 1617}, {500, 940}, {516, 6796}, {517, 3811}, {573, 5778}, {581, 4658}, {582, 1780}, {912, 1490}, {920, 1155}, {936, 3587}, {943, 5226}, {956, 5086}, {962, 5761}, {970, 6000}, {999, 3486}, {1062, 1465}, {1376, 3579}, {1385, 5806}, {1470, 4299}, {1480, 5255}, {1745, 1936}, {3295, 3485}, {3303, 3656}, {3304, 3655}, {3869, 3940}, {5396, 5706}, {5731, 5804}, {5755, 5776}, {5841, 6256}
X(6985) = complement of X(6851)
X(6985) = anticomplement of X(37356)
As a point on the Euler line, X(6986) has Shinagawa coefficients (2abc + $aSA$, -abc - $aSA$).
X(6986) lies on these lines: {1, 1170}, {2, 3}, {35, 1210}, {36, 4298}, {40, 1621}, {55, 938}, {56, 3475}, {78, 947}, {81, 580}, {100, 5178}, {165, 3833}, {212, 3562}, {238, 4300}, {515, 5260}, {516, 5259}, {572, 2287}, {651, 3074}, {936, 993}, {942, 943}, {946, 5284}, {958, 5731}, {962, 1001}, {991, 1724}, {1014, 5736}, {1071, 3219}, {1257, 2224}, {1259, 5744}, {1385, 3555}, {1490, 3305}, {1768, 3647}, {3428, 3616}, {3826, 6253}, {3871, 5657}, {4015, 5531}, {4297, 5251}, {5217, 5704}, {5267, 5660}
As a point on the Euler line, X(6987) has Shinagawa coefficients (abc + $aSA$, -2$aSA$).
X(6987) lies on these lines: {1, 5758}, {2, 3}, {9, 515}, {40, 950}, {63, 5768}, {72, 944}, {144, 912}, {165, 1737}, {226, 3576}, {329, 4511}, {347, 1870}, {387, 580}, {390, 517}, {497, 3428}, {572, 5746}, {573, 1713}, {938, 5709}, {946, 5436}, {954, 1056}, {997, 1490}, {1125, 5715}, {1260, 3421}, {1385, 3487}, {2550, 5842}, {3419, 5657}, {3587, 5809}, {4004, 6361}, {4304, 6282}, {4314, 6769}, {5081, 6350}, {5534, 5815}, {5584, 6284}, {5698, 6001}
As a point on the Euler line, X(6988) has Shinagawa coefficients (2$aSA$, -abc - $aSA$).
X(6988) lies on these lines: {2, 3}, {40, 3485}, {78, 5657}, {515, 5705}, {517, 4323}, {936, 6261}, {938, 1385}, {944, 6734}, {1071, 5744}, {1210, 3486}, {1490, 5745}, {2096, 4652}, {2550, 6796}, {3085, 3428}, {3452, 5924}, {3487, 5709}, {5086, 5731}, {5226, 5812}, {5273, 5777}, {5432, 5584}, {5732, 6705}
As a point on the Euler line, X(6989) has Shinagawa coefficients (2abc + $aSA$, -abc).
X(6989) lies on these lines: {2, 3}, {10, 5534}, {57, 498}, {142, 5709}, {499, 3601}, {942, 1788}, {946, 3587}, {3358, 6260}, {3579, 5805}, {3624, 6282}, {3634, 6245}, {3681, 5552}, {3824, 5812}, {4340, 5398}, {4648, 5707}, {5708, 5771}, {5770, 5791}
X(6989) = complement of X(6846)
As a point on the Euler line, X(6990) has Shinagawa coefficients (abc + $aSA$, 2abc + $aSA$).
X(6990) lies on these lines: {2, 3}, {11, 3487}, {12, 3488}, {943, 1479}, {946, 5692}, {1901, 5043}, {3419, 4420}, {3811, 5587}, {3813, 5603}, {3822, 5436}, {3925, 6361}, {4860, 5714}, {5536, 5715}, {5777, 6583}, {5817, 5852}
As a point on the Euler line, X(6991) has Shinagawa coefficients (2abc + $aSA$, abc + $aSA$).
X(6991) lies on these lines: {2, 3}, {11, 5703}, {12, 938}, {78, 5178}, {962, 3925}, {1210, 3947}, {1260, 2894}, {1699, 3841}, {2346, 3085}, {3305, 5715}, {3337, 4355}, {3614, 4860}, {3681, 6734}, {3814, 5536}, {3824, 5927}
As a point on the Euler line, X(6992) has Shinagawa coefficients (2abc + $aSA$, -2$aSA$).
X(6992) lies on these lines: {2, 3}, {390, 5119}, {515, 3305}, {908, 3576}, {938, 1708}, {944, 3876}, {997, 5731}, {1319, 3475}, {1737, 4294}, {3219, 5768}, {3984, 5882}, {4293, 5226}
As a point on the Euler line, X(6993) has Shinagawa coefficients (2abc + $aSA$, 2$aSA$).
X(6993) lies on these lines: {2, 3}, {7, 1737}, {3475, 5252}, {3868, 5818}, {5123, 5832}, {5249, 5587}
As a point on the Euler line, X(6994) has Shinagawa coefficients (-E, $a2$).
X(6994) lies on these lines: {2, 3}, {19, 3219}, {92, 144}, {193, 1839}, {393, 1171}
As a point on the Euler line, X(6995) has Shinagawa coefficients (-F, $a2$).
The trilinear polar of X(6995) meets the line at infinity at X(3800). (Randy Hutson, April 11, 2015)
X(6995) lies on these lines: {2, 3}, {19, 346}, {33, 390}, {34, 3600}, {51, 6776}, {145, 1829}, {154, 5480}, {193, 1843}, {232, 1180}, {251, 393}, {273, 3598}, {499, 5345}, {612, 4294}, {614, 4293}, {1194, 3199}, {1219, 1891}, {1627, 1968}, {1824, 3995}, {1861, 5338}, {1974, 5012}, {2052, 3424}, {2207, 5359}, {2333, 4651}, {2355, 5101}, {3085, 5310}, {3086, 5322}, {3162, 5354}, {3617, 5090}, {3618, 3796}, {4299, 5272}, {4302, 5268}, {5446, 6193}, {5921, 6515}
X(6995) = isogonal conjugate of X(34817)
X(6995) = anticomplement of X(7386)
X(6995) = circumcircle-inverse of X(37977)
X(6995) = polar conjugate of X(18840)
X(6995) = pole wrt polar circle of trilinear polar of X(18840) (line X(523)X(2525))
X(6995) = orthocentroidal-circle-inverse of X(7378)
X(6995) = {X(2),X(4)}-harmonic conjugate of X(7386)
As a point on the Euler line, X(6996) has Shinagawa coefficients ($a2$, -(a + b + c)2).
X(6996) lies on these lines: {2, 3}, {40, 4384}, {57, 3673}, {75, 1766}, {83, 2051}, {86, 572}, {220, 5792}, {226, 4911}, {238, 516}, {239, 517}, {241, 5088}, {315, 4417}, {333, 1746}, {355, 3661}, {515, 3912}, {952, 6542}, {962, 5222}, {1019, 1577}, {1229, 5279}, {1482, 4393}, {1944, 2182}, {1999, 3555}, {3687, 5015}, {5021, 5286}, {5135, 5327}, {5195, 5723}, {5224, 5816}, {5308, 5731}
As a point on the Euler line, X(6997) has Shinagawa coefficients (E, $a2$).
X(6997) lies on these lines: {2, 3}, {51, 1352}, {69, 3060}, {206, 3618}, {251, 2165}, {262, 6504}, {312, 3434}, {324, 6524}, {394, 5480}, {497, 3920}, {498, 5310}, {499, 5322}, {612, 1479}, {614, 1478}, {1180, 5421}, {1194, 2548}, {1196, 5475}, {1899, 3818}, {3583, 5268}, {3585, 5272}, {3589, 3796}, {5225, 5297}, {5422, 6776}
X(6997) = anticomplement of X(7484)
X(6997) = orthocentroidal circle inverse of X(1370)
X(6997) = {X(2),X(4)}-harmonic conjugate of X(1370)
As a point on the Euler line, X(6998) has Shinagawa coefficients (-(a + b + c)2, $a2$).
X(6998) lies on these lines: {2, 3}, {10, 98}, {86, 511}, {114, 6626}, {230, 1834}, {325, 1330}, {387, 2271}, {842, 2690}, {942, 1447}, {966, 6776}, {1125, 3430}, {1213, 1503}, {1305, 3563}, {1352, 5224}, {1654, 3564}, {2245, 5327}, {3705, 5814}, {3842, 6211}, {5275, 5706}
X(6998) = Euler line intercept, other than X(27), of circle {X(27),PU(4)}
As a point on the Euler line, X(6999) has Shinagawa coefficients (-$bc$, (a + b + c)2).
X(6999) lies on these lines: {2, 3}, {40, 3661}, {239, 515}, {241, 4872}, {516, 3685}, {517, 6542}, {573, 1654}, {661, 4560}, {944, 4393}, {1764, 2896}, {1975, 4417}, {2051, 6625}, {2356, 3100}, {3177, 5813}, {4352, 5712}, {4384, 5691}
As a point on the Euler line, X(7000) has Shinagawa coefficients (-S, $a2$).
X(7000) lies on these lines: {2, 3}, {98, 1132}, {147, 6463}, {262, 1131}, {371, 6202}, {372, 5871}, {485, 6201}, {486, 3424}, {511, 1271}, {1160, 6215}, {1270, 1352}, {1350, 5591}, {1503, 3069}, {1588, 5304}, {3068, 5480}, {3593, 3818}
X(7000) = orthocentroidal-circle-inverse of X(7374)
X(7000) = {X(2),X(4)}-harmonic conjugate of X(7374)
| 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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