| 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) |
Centers associated with extra-triangles: X(7001)-X(7373)
This preamble and centers X(7001)-X(7373) were contributed by Richard Hilton, March 19, 2015.
Suppose that f(a,b,c) is a polynomial and that a triangle center X has barycentric coordinates f(a,b,c) : f(b,c,a) : f(c,a,b). Then the A-extraversion of X is obtained by replacing a by -a in all three of its barycentric coordinates. Likewise, the B-extraversion is obtained by the substitution b → -b, and the C-extraversion by c → -c. The three extraversions are the vertices of a central triangle which we shall call the extra-triangle of X. In the notation introduced in the preamble to X(3758), the extra-triangle of X is T(f(-a,b,c), f(b,c,-a)).
For centers that have no such polynomial representation, the three extraversions are defined "by construction". Examples are centers based on the Morley triangles or the Malfatti circles, as, in both cases, figures can readily be constructed in the A-, B-, C- exterior regions of the triangle, following the same procedures as for the interior figures. Since every center can be represented by barycentric coordinates in terms of A, B, C, the following definition is adopted here:
If the barycentric coordinates if triangle center X are f(A,B,C) : f(B,C,A) : f(C,A,B) then the coordinates of the A-extraversion of X are obtained by the substitutions A→ -A, B → π - B, C→ π - C. Coordinates for the B- and C-extraversions are obtained by the corresponding cyclic substitutions.
Clearly, each of the substitutions a → -a, b → -b, c → -c yields Δ → -Δ, S → -S, and cot(ω) → -cot(ω). Stipulating that R remains positive, then the substitutions a → -a, b → -b, c → -c, as in the first paragraph, give the three extraversions, and, equivalently, these subsitutions: R → -R, a →, b → -b, c → -c. The results of these substitutions on expressions involving the inradius, r, as well as exressions in Conway notation, follow from those just stated.
Centers with barycentric coordinates involving square roots of non-numeric expressions, and centers for which the transformations ω → - ω and ω → π - ω lead to undesirable results, are excluded from consideration here.
If a pair of points (such as the Fermat points) have barycentric coordinates of the form f(a2,b2,c2,S) and f(a2,b2,c2,-S), they may be regarded as intermediate between strong and weak points, in that the extraversions of one of the pair coincide at the other. A geometric interpretation is that for points on the Kiepert hyperbola, outward rotations from the edges of the reference triangle are equivalent to inward rotations from one of the exterior edges.
The focus here is on centers for which the extra-triangle is perspective with ABC.
Assuming that the barycentric coordinates of a center X are f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,C,B) = k*f(A,B,C) for k = ±1, the extra-triangle of X is perspective to ABC if f(π - A, π - B, -C) can be expressed as u(A,B,C)*v(B,C,A) where u(A,C,B) = k*u(A,B,C) and v(A,C,B) = v(A,B,C). The first barycentric coordinate of the perspector is then u(A,B,C) / v(A,B,C).
When the trilinear coordinates of X are polynomials of the form f(a2,b2,c2) : f(b2,c2,a2) : f(c2,a2,b2), the extra-triangle of X is simply the anticevian triangle of X. These points include the triangle centers X(n) for the following choices of n:
1, 19, 31, 38, 47, 48, 63, 75, 82, 91, 92, 158, 162, 163, 204, 240, 255, 293, 304, 326, 336, 560, 561, 563, 564, 605, 606, 610, 656, 661, 662, 774, 775, 798, 799, 810. 811, 820, 821, 822, 823, 896, 897, 920, 921, 922, 923, 1087, 1096, 1097, 1099, 1101, 1102, 1109, 1496, 1497, 1577, 1580, 1581, 1582, 1707, 1712, 1725, 1733, 1740, 1747, 1748, 1749, 1755, 1760, 1784, 1820, 1821, 1822, 1823, 1895, 1910, 1917, 1923, 1924, 1925, 1926, 1927, 1928, 1930, 1932, 1933, 1934, 1953, 1954, 1955, 1956, 1957, 1958, 1959, 1964, 1965, 1966, 1967, 1969, 1973, 2083, 2084, 2085, 2128, 2129, 2148, 2155, 2156, 2157, 2158, 2159, 2166, 2167, 2168, 2169, 2172, 2173, 2179, 2180, 2181, 2184, 2186, 2190, 2216, 2227, 2234, 2236, 2244, 2247, 2290, 2312, 2313, 2314, 2315, 2349, 2564, 2565, 2576, 2577, 2578, 2579, 2580, 2581, 2582, 2583, 2584, 2585, 2586, 2587, 2588, 2589, 2616, 2617, 2618, 2619, 2620, 2621, 2624, 2625, 2626, 2627, 2629, 2631, 2632, 2633, 2640, 2642, 2643, 2644, 2962, 2964, 3112, 3113, 3116, 3223, 3400, 3401, 3402, 3403, 3404, 3405, 3408, 3409, 3604, 3708, 4008, 4020, 4100, 4117, 4118, 4575, 4592, 4593, 4599, 4602, 6149, 6507, 6508, 6520, 6521.
Other centers X for which the extraversions lie on the cevians of X are X(n) for the following values of n:
1123, 1124, 1136, 1137, 1267, 1335, 1336, 1489, 3076, 3077, 3083, 3084, 3237, 3238, 3273, 3297, 3298, 3299, 3300, 3301, 3302, 5353, 5357, 5391, 5565, 5566, 5567, 5568, 5630, 5631, 6122, 6125, 6212, 6213.
For certain other centers, the extra-triangle of X is perspective to ABC and the perspector is a center Y other than X. In this case, we call X and Y twin centers, as the extra-triangle of Y is also perspective with ABC, with perspector X. Twin pairs include {X(i), X(j)} for the following pairs {i, j}:
{7, 8}, {9, 57}, {11, 12}, {33, 34}, {35, 36}, {40, 84}, {41, 604}, {43, 87}, {46, 90}, {55, 56}, {59, 60}, {77, 78}, {79, 80}, {85, 312}, {171, 238}, {172, 1914}, {174, 188}, {181, 3271}, {189, 329}, {192, 330}, {198, 1436}, {200, 269}, {212, 603}, {215, 2477}, {219, 222}, {220, 1407}, {221, 2192}, {223, 282}, {236, 557}, {239, 894}, {243, 1940}, {244, 756}, {256, 291}, {257, 335}, {259, 266}, {261, 4998}, {273, 318}, {278, 281}, {279, 346}, {280, 347}, {292, 893}, {319, 320}, {341, 1088}, {345, 348}, {350, 1909}, {357, 1134}, {358, 1135}, {360, 1115}, {388, 497}, {390, 3600}, {479, 5423}, {483, 1488}, {484, 3065}, {495, 496}, {498, 499}, {552, 4076}, {556, 4146}, {559, 5240}, {593, 1252}, {594, 1086}, {601, 602}, {607, 608}, {611, 613}, {612, 614}, {643, 1414}, {645, 4573}, {728, 738}, {748, 750}, {749, 751}, {757, 765}, {849, 1110}, {872, 3248}, {904, 1911}, {999, 3295}, {1000, 3296}, {1015, 1500}, {1016, 1509}, {1018, 1019}, {1020, 1021}, {1034, 5932}, {1056, 1058}, {1057, 1059}, {1060, 1062}, {1061, 1063}, {1069, 3157}, {1082, 5239}, {1089, 1111}, {1090, 1091}, {1094, 1095}, {1103, 1256}, {1106, 1253}, {1118, 1857}, {1127, 1128}, {1129, 1130}, {1146, 6354}, {1219, 3672}, {1254, 2310}, {1259, 1804}, {1354, 6062}, {1358, 6057}, {1365, 4092}, {1395, 2212}, {1397, 2175}, {1399, 2361}, {1401, 3688}, {1403, 2053}, {1422, 2324}, {1423, 2319}, {1425, 3270}, {1428, 2330}, {1429, 2329}, {1432, 4876}, {1442, 4511}, {1443, 4420}, {1469, 3056}, {1478, 1479}, {1480, 6580}, {1490, 3345}, {1565, 3695}, {1672, 1673}, {1674, 1675}, {1745, 3362}, {1870, 6198}, {1920, 1921}, {1935, 1936}, {1943, 1944}, {1947, 1948}, {1950, 1951}, {2003, 2323}, {2007, 2008}, {2066, 6502}, {2067, 5414}, {2082, 2285}, {2089, 3082}, {2149, 2150}, {2151, 2152}, {2153, 2154}, {2160, 2161}, {2162, 2176}, {2164, 2178}, {2170, 2171}, {2185, 4564}, {2187, 2208}, {2241, 2242}, {2275, 2276}, {2297, 2999}, {2345, 4000}, {2463, 2464}, {2533, 4010}, {2595, 2596}, {2601, 2602}, {2606, 2607}, {2612, 2613}, {2968, 6356}, {2994, 5905}, {3023, 3027}, {3024, 3028}, {3058, 5434}, {3072, 3073}, {3074, 3075}, {3085, 3086}, {3100, 4296}, {3182, 3347}, {3218, 3219}, {3220, 5285}, {3235, 3236}, {3274, 3275}, {3303, 3304}, {3305, 3306}, {3320, 6020}, {3325, 6019}, {3336, 3467}, {3341, 3342}, {3351, 3352}, {3353, 3354}, {3375, 3384}, {3376, 3383}, {3377, 3378}, {3460, 3461}, {3465, 3466}, {3468, 3469}, {3472, 3473}, {3494, 3502}, {3495, 3503}, {3496, 3497}, {3500, 3501}, {3509, 3512}, {3553, 3554}, {3582, 3584}, {3583, 3585}, {3596, 6063}, {3602, 3603}, {3662, 3662}, {3665, 3703}, {3673, 4385}, {3690, 3937}, {3733, 4557}, {3737, 4551}, {3746, 5563}, {3758, 3759}, {3760, 3761}, {3781, 3784}, {3801, 4122}, {3862, 3863}, {3864, 3865}, {3928, 3929}, {3942, 3949}, {4017, 4041}, {4056, 4680}, {4077, 4086}, {4081, 6046}, {4170, 4761}, {4293, 4294}, {4299, 4302}, {4309, 4317}, {4316, 4324}, {4319, 4320}, {4325, 4330}, {4328, 4853}, {4351, 4354}, {4361, 4363}, {4366, 6645}, {4372, 4376}, {4396, 4400}, {4552, 4560}, {4565, 5546}, {4584, 4603}, {4589, 4594}, {4644, 5839}, {4857, 5270}, {4858, 6358}, {4911, 5015}, {4995, 5298}, {5148, 5194}, {5204, 5217}, {5222, 5749}, {5225, 5229}, {5261, 5274}, {5265, 5281}, {5268, 5272}, {5280, 5299}, {5310, 5322}, {5432, 5433}, {5557, 5559}, {5560, 5561}, {5628, 5632}, {5629, 5633}, {6023, 6027}, {6120, 6121}, {6123, 6124}, {6186, 6187}, {6376, 6384}, {6377, 6378}, {6382, 6383}, {6505, 6513}, {6511, 6512}.
Note that 'twinning' preserves isogonal and isotomic conjugacies and, indeed, any isoconjugacies for which the pole is a strong point.
Let A32B32C32 be Gemini triangle 32. Let A' be the perspector of conic {{A,B,C,B32,C32}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(7018). (Randy Hutson, January 15, 2019)
X(7018) lies on these lines: {2,893}, {69,1431}, {75,325}, {76,3865}, {141,3863}, {226,1432}, {256,314}, {257,312}, {310,3120}, {321,1916}, {333,4603}, {334,1581}, {799,4683}, {1921,3847}, {1965,4388}, {3596,6382}, {3662,6384}, {3903,4514} X(7018) = isogonal conjugate of X(7122)
X(7021) = isogonal conjugate of X(1115)
X(7021) = 1st Saragossa point of X(359)
X(7022) lies on the line {7,177}
X(7022) = trilinear square of X(2089)
X(7023) lies on these lines: {7,1476}, {11,1440}, {55,77}, {56,269}, {198,6610}, {279,961}, {479,1014}, {604,1407}, {1119,1358}, {1417,6614}, {1435,6612}, {1436,3942}, {1439,1466}, {1443,1804}, {1696,6180}, {2751,3323}
X(7023) = isogonal conjugate of X(5423)
X(7023) = X(738)-Ceva conjugate of X(1407)
X(7023) = X(i)-cross conjugate of X(j) for these (i,j): (1106,1407), (1398,6612)
X(7023) = crosssum of X(i) and X(j) for (i,j) = (3022,4130), (3900,4953), (4081,4163)
X(7023) = crossdifference of any pair of centers on the line through X(4130) and X(4163)
X(7026) lies on these lines: {1,395}, {2,559}, {9,80}, {10,13}, {30,1277}, {92,472}, {355,6192}, {519,5240}, {551,5242}, {1276,3654}, {1653,5434}, {3828,5243}, {4669,5246}, {5690,6191}
X(7026) = X(5245)-cross conjugate of X(8)
X(7026) = cevapoint of X(1) and X(1277)
X(7027) lies on these lines: {178,312}, {188,556}, {346,5430}
X(7027) = isotomic conjugate of X(7371)
X(7028) lies on these lines: {1,188}, {2,174}, {9,258}, {173,5437}, {289,1488}
X(7028) = X(2)-Ceva conjugate of X(39121)
X(7028) = X(9)-cross conjugate of X(188)
X(7028) = {X(2),X(174)}-harmonic conjugate of X(236)
X(7029) lies on these lines: {894,3083}, {1267,1909}
X(7029) = isotomic conjugate of X(586)
X(7029) = trilinear pole of the line through X(4369) and X(6364)
X(7030) lies on these lines: {894,3084}, {1909,5391}
X(7030) = isotomic conjugate of X(585)
X(7030) = trilinear pole of the line through X(4369) and X(6365)
X(7031) lies on these lines: {1,32}, {3,5299}, {6,35}, {9,2220}, {31,4251}, {34,112}, {36,3053}, {39,5010}, {41,595}, {55,5280}, {56,1384}, {58,2280}, {71,5037}, {101,3915}, {187,2275}, {218,3052}, {251,612}, {350,6179}, {384,3761}, {385,3760}, {388,1285}, {571,3554}, {614,1627}, {902,3730}, {1019,1424}, {1193,4262}, {1203,2271}, {1333,1449}, {1415,1420}, {1475,4257}, {1500,5008}, {1698,4386}, {1724,3684}, {1909,3972}, {1923,2209}, {2176,2251}, {2205,3294}, {2276,5007}, {2330,5039}, {2549,4324}, {3299,6423}, {3301,6424}, {3583,3767}, {3624,5277}, {3632,5291}, {3679,4426}, {3734,4400}, {3924,5011}, {4294,5304}, {4302,5286}, {4330,5319}, {5248,5276}, {5259,5275}, {5305,6284}, {5310,5359}
X(7031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,32,609), (32,1914,1), (32,2241,172), (172,1914,2241), (172,2241,1), (2220,5301,9)
X(7031) = crosssum of X(1086) and X(3700)
X(7031) = crossdifference of any pair of centers on the line through X(1491) and X(4802)
X(7031) = crosspoint of X(1252) and X(4565)
X(7032) lies on these lines: {1,87}, {2,3510}, {6,292}, {9,3009}, {31,184}, {42,1449}, {48,2210}, {56,904}, {77,614}, {86,870}, {239,1740}, {263,1400}, {741,985}, {893,2162}, {982,3794}, {983,4579}, {995,5429}, {1015,4116}, {1045,4393}, {1201,1419}, {1386,2274}, {1918,3941}, {2209,2223}, {2234,4361}, {2275,3056}, {3051,5364}
X(7032) = isogonal conjugate of X(7033)
X(7032) = isotomic conjugate of X(7034)
X(7032) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,3662), (664,649)
X(7032) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,87,894), (6,1964,869), (1964,3248,6), (2275,3056,3778), (3056,3778,4787)
X(7032) = crosssum of X(i) and X(j) for (i,j) = (1,3729), (8,192), (42,321)
X(7032) = crossdifference of any pair of centers on the line through X(812) and X(4391)
X(7032) = crosspoint of X(i) and X(j) for (i,j) = (56,2162), (86,1333)
Let A'B'C' be the trilinear N-obverse triangle of X(2). 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. Triangle A"B"C" is homothetic to ABC at X(7033). (Randy Hutson, November 30, 2018)
X(7033) lies on these lines: {43,350}, {75,183}, {76,3502}, {192,893}, {239,312}, {314,983}, {321,2205}, {727,3923}, {870,1215}, {894,2162}, {1909,6063}, {1978,3938}, {2319,3729}, {3961,6382}, {4621,6654}
X(7033) = isogonal conjugate of X(7032)
X(7033) = isotomic conjugate of X(982)
X(7033) = X(663)-cross conjugate of X(190)
X(7033) = cevapoint of X(i) and X(j) for (i,j) = (1,3729), (8,192), (42,321)
X(7033) = trilinear pole of the line through X(812) and X(4391)
X(7034) lies on these lines: {192,1921}, {213,3114}, {257,6382}, {330,1920}, {561,3212}, {871,1237}, {1925,4518}, {1978,3061}, {3226,4485}
X(7034) = isotomic conjugate of X(7032)
X(7034) = X(522)-cross conjugate of X(1978)
X(7034) = cevapoint of X(3596) and X(6382)
X(7035) lies on these lines: {190,649}, {239,1016}, {312,6632}, {321,6634}, {350,899}, {660,799}, {661,4562}, {668,693}, {765,4600}, {870,5378}, {873,1215}, {1227,6635}, {1447,3263}, {1978,3699}, {2382,4432}, {2863,6551}, {3758,5381}
X(7035) is the trilinear pole of the line X(190)X(646), which is the locus of the trilinear pole of the tangent at P to the hyperbola {{A,B,C,X(1),P}}, as P moves on the Nagel line. (Randy Hutson, April 11, 2015)
Let A5B5C5 and A6B6C6 be the Gemini triangles 5 and 6. Let A' be the trilinear product A5*A6 and define B', C' cyclically. The lines AA', BB', CC' concur in X(7035). (Randy Hutson, November 30, 2018)
X(7035) = isogonal conjugate of X(3248)
X(7035) = isotomic conjugate of X(244)
X(7035) = X(4601)-Ceva conjugate of X(1016)
X(7035) = X(i)-cross conjugate of X(j) for these (i,j): (1,190), (2,799), (43,100), (75,668), (86,6540), (312,1978), (341,646), (740,4562), (872,1018), (899,4607), (978,651), (1089,4033), (1215,3952), (1714,823), (1722,653), (2664,660), (3216,662), (3699,6632), (3875,664), (4360,99), (4986,75)
X(7035) = crosssum of X(1015) and X(6377)
X(7035) = crosspoint of X(1016) and X(5383)
X(7035) = trilinear square of X(190)
X(7035) = X(649)-isoconjugate of X(649)
X(7036) = isogonal conjugate of X(7044)
X(7037) lies on these lines: {3,223}, {6,2188}, {55,204}, {154,198}, {1034,1792}, {1260,2324}
X(7037) = isogonal conjugate of X(5932)
X(7037) = complement of anticomplementary conjugate of X(20212)
X(7037) = X(3342)-Ceva conjugate of X(6)
X(7037) = X(25)-cross conjugate of X(55)
X(7037) = X(223)-vertex conjugate of X(223)
X(7037) = crosssum of X(223) and X(3182)
X(7037) = crosspoint of X(282) and X(3347)
X(7038) lies on these lines: {1,6359}, {33,1940}, {55,1935}, {64,1044}
X(7038) = isogonal conjugate of X(1044)
X(7038) = X(20)-cross conjugate of X(1)
X(7038) = crosspoint of X(282) and X(3347)
X(7039) lies on these lines: {1,1049}, {1028,1077}
X(7039) = isogonal conjugate of X(7041)
X(7039) = crosssum of X(1) and X(1049)
X(7039) = crosspoint of X(1) and X(1077)
X(7040): Let A'B'C' be the orthic triangle. Let OA be the circle centered at A' and passing through A. Define OB, OC cyclically. X(7040) is the trilinear pole of the Monge line of OA, OB, OC. This line is also the trilinear polar, with respect to the orthic triangle, of X(65). (Randy Hutson, April 11, 2015)
X(7040) lies on these lines: {4,46}, {24,1857}, {29,1069}, {158,1068}, {243,3147}, {281,3811}, {318,406}, {403,1118}
X(7040) = isogonal conjugate of X(3157)
X(7040) = X(i)-cross conjugate of X(j) for these (i,j): (1,4), (1858,1896), (2164,2994)
X(7040) = {X(158),X(3542)}-harmonic conjugate of X(1068)
X(7040) = cevapoint of X(1) and X(90)
X(7040) = pole wrt polar circle of trilinear polar of X(5905)
X(7040) = X(48)-isoconjugate (polar conjugate) of X(5905)
X(7040) = SS(a → cos A) of X(7) (trilinear substitution)
X(7040) = trilinear pole of line X(3064)X(15313)
X(7041) lies on these lines: {1,1085}, {1049,1077}
X(7041) = isogonal conjugate of X(7039)
X(7041) = cevapoint of X(1) and X(1049)
X(7042) lies on these lines: {3,90}, {77,499}, {78,4354}, {1079,3542}, {3345,3583}
X(7042) = isogonal conjugate of X(1079)
X(7042) = X(1069)-cross conjugate of X(90)
X(7042) = trilinear square of X(90)
X(7043) lies on these lines: {1,396}, {2,1082}, {9,80}, {10,14}, {30,1276}, {92,473}, {355,6191}, {519,5239}, {551,5243}, {1277,3654}, {1652,5434}, {3828,5242}, {4669,5245}, {5690,6192}
X(7043) = isogonal conjugate of X(7051)
X(7043) = X(5246)-cross conjugate of X(8)
X(7043) = cevapoint of X(1) and X(1276)
X(7044) = isogonal conjugate of X(7036)
X(7045) is the trilinear pole of the line X(101)X(651), which is the locus of the trilinear pole of the tangent at P to hyperbola {{A,B,C,X(1),P}}, as P moves on line X(1)X(3). (Randy Hutson, April 11, 2015)
X(7045) lies on these lines: {57,2149}, {59,1155}, {100,677}, {109,658}, {241,1252}, {320,765}, {650,651}, {664,4025}, {901,934}, {1414,4566}, {1758,4570}
X(7045) = isogonal conjugate of X(2310)
X(7045) = isotomic conjugate of X(24026)
X(7045) = X(1275)-Ceva conjugate of X(4564)
X(7045) = X(i)-cross conjugate of X(j) for these (i,j): (1,651), (3,662), (7,1414), (40,190), (46,653), (57,658), (59,4564), (77,664), (109,4619), (165,100), (255,1813), (269,934), (484,655), (517,3257), (1106,1461), (1253,101), (1254,1020), (1407,4637), (1715,823), (1754,162), (1764,799), (3561,4558), (3562,648), (3576,4604), (4350,4626)
X(7045) = cevapoint of X(i) and X(j) for (i,j) = (1,651), (7,4566), (57,109), (59,1262), (63,100), (101,1253), (255,1813), (269,934), (412,653), (1020,1254), (1106,1461)
X(7045) = trilinear square of X(691)
X(7045) = X(690)-isoconjugate of X(690)
X(7046) lies on these lines: {2,1897}, {3,280}, {4,8}, {10,459}, {25,1261}, {29,4720}, {33,200}, {34,4853}, {55,4081}, {75,1119}, {78,6198}, {108,1376}, {189,971}, {196,2550}, {208,1706}, {210,1857}, {253,6356}, {278,1861}, {346,1260}, {393,594}, {406,3695}, {451,5552}, {480,6057}, {728,4082}, {1172,3713}, {1219,1398}, {1249,2345}, {1783,3195}, {1785,3679}, {1826,4061}, {1859,3059}, {1863,5423}, {1870,3872}, {2202,4390}
X(7046) = isogonal conjugate of X(7053)
X(7046) = isotomic conjugate of X(7056)
X(7046) = X(i)-Ceva conjugate of X(j) for these (i,j): (318,281), (1897,3239)
X(7046) = X(i)-cross conjugate of X(j) for these (i,j): (210,200), (220,346), (1863,4)
X(7046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,318,4), (33,281,461), (1851,5101,4), (3974,4012,200)
X(7046) = pole wrt polar circle of trilinear polar of X(279) (line X(513)X(676))
X(7046) = X(48)-isoconjugate (polar conjugate) of X(279)
X(7047) lies on the line: {1,1133}
X(7048) is the perspector of the inconic with center X(188), this inconic being the excentral-to-ABC barycentric image of the incircle of the excentral triangle. (Randy Hutson, April 11, 2015)
X(7048) lies on these lines: {2,174}, {7,2091}, {145,188}
X(7048) = isotomic conjugate of X(7057)
X(7048) = X(188)-cross conjugate of X(2)
X(7048) = anticomplement of X(236)
X(7049) lies on the Feuerbach hyperbola and these lines: {1,1075}, {4,6285}, {90,243}, {196,3296}, {450,1069}, {498,3462}, {1000,3176}, {1068,1937}, {1093,3270}, {1249,2335}, {3168,6198}, {4299,5667}
X(7049) = X(3362)-Ceva conjugate of X(4)
X(7049) = X(158)-cross conjugate of X(4)
X(7049) = {X(1),X(1075)}-harmonic conjugate of X(1148)
X(7049) = cevapoint of X(3064) and X(3270)
X(7049) = pole wrt polar circle of trilinear polar of X(6360)
X(7049) = X(48)-isoconjugate (polar conjugate) of X(6360)
X(7050) lies on these lines: {1,1407}, {6,200}, {31,220}, {33,608}, {37,2255}, {55,604}, {81,145}, {739,6574}, {940,1462}, {1191,2221}, {1333,2256}, {2203,2332}
X(7050) = isogonal conjugate of X(3672)
X(7050) = X(2221)-vertex conjugate of X(2221)
X(7050) = crosssum of X(i) and X(j) for (i,j) = (1697,2999), (4646,4656)
X(7050) = trilinear pole of the line through X(657) and X(667)
X(7050) = perspector of ABC and unary cofactor triangle of inverse-in-excircles triangle
X(7051) lies on these lines: {1,15}, {3,1250}, {6,41}, {11,5321}, {14,3582}, {16,36}, {17,5270}, {61,5357}, {395,5298}, {396,5434}, {1464,2152}, {2975,5362}, {3086,5334}, {3218,5239}, {3746,5238}, {4293,5335}, {5253,5367}
X(7051) = isogonal conjugate of X(7043)
X(7051) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16,203,5353), (36,5353,16), (2067,6502,2307)
X(7051) = crosssum of X(1) and X(1276)
X(7051) = homothetic center of inner tri-equilateral triangle and anti-tangential midarc triangle
X(7052) lies on these lines: {1,61}, {2,1082}, {6,1411}, {14,80}, {16,484}, {17,5443}, {56,2306}, {62,5903}, {65,2154}, {81,559}, {202,5902}, {1250,5119}, {3130,6187}, {5357,5425}
X(7052) = isogonal conjugate of X(5239)
X(7053) lies on these lines: {1,963}, {3,77}, {4,1440}, {6,911}, {7,104}, {28,279}, {48,222}, {56,269}, {57,1422}, {85,6359}, {198,1419}, {241,5120}, {326,1260}, {348,1791}, {757,4637}, {1020,6180}, {1106,1472}, {1333,1407}, {1442,3295}, {1443,3417}, {1617,3433}, {1811,6516}, {2178,6610}, {2217,2385}, {2283,6600}, {3149,5932}, {3304,4328}
X(7053) = isogonal conjugate of X(7046)
X(7053) = X(i)-Ceva conjugate of X(j) for these (i,j): (279,1407), (1014,269)
X(7053) = X(i)-cross conjugate of X(j) for these (i,j): (603,222), (1459,1461), (1473,3)
X(7053) = {X(77),X(1804)}-harmonic conjugate of X(3)
X(7053) = cevapoint of X(56) and X(6611)
X(7053) = crosssum of X(3900) and X(4081)
X(7053) = X(92)-isoconjugate of X(220)
X(7054) lies on these lines: {9,1793}, {21,270}, {32,941}, {48,110}, {60,283}, {77,4565}, {81,593}, {162,1005}, {163,572}, {261,4612}, {448,1441}, {577,4189}, {1098,1792}, {1400,5060}, {1474,4225}, {1789,4282}, {1950,4296}, {2327,2328}
X(7054) = isogonal conjugate of X(6354)
X(7054) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,4636), (1098,6061), (2185,60)
X(7054) = X(i)-cross conjugate of X(j) for these (i,j): (284,2326), (1021,5546), (2328,1098)
X(7054) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (284,2150,60), (1474,4288,4225)
X(7054) = cevapoint of X(i) and X(j) for (i,j) = (6,1630), (284,2193)
X(7054) = crosssum of X(i) and X(j) for (i,j) = (125,4024), (1254,2171)
X(7054) = crosspoint of X(i) and X(j) for (i,j) = (249,4636), (250,4556), (1098,2185)
X(7054) = barycentric square of X(21)
X(7055) lies on these lines: {7,310}, {69,1439}, {222,348}, {320,1088}, {329,4554}, {658,4417}, {1102,3719}, {1264,4176}
X(7055) = isogonal conjugate of X(6059)
X(7055) = isotomic conjugate of X(1857)
X(7055) = X(i)-cross conjugate of X(j) for these (i,j): (326,3926), (1804,348)
X(7056) lies on these lines: {2,658}, {7,354}, {63,348}, {69,1439}, {77,1040}, {81,279}, {85,189}, {222,1814}, {226,1996}, {286,1119}, {651,5452}, {738,3674}, {873,4635}, {934,1621}, {969,3668}, {2185,4637}, {3321,5432}, {3873,6604}, {4569,6063}
X(7056) = isogonal conjugate of X(7071)
X(7056) = isotomic conjugate of X(7046)
X(7056) = X(i)-cross conjugate of X(j) for these (i,j): (77,348), 3270,905), (4025,658)
X(7056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,479,1088), (7,2898,1836)
X(7056) = cevapoint of X(905) and X(3270)
Let IA, IB, IC be the excenters. Let A' be the IA-extouch point of triangle IABC, and define B' and C' cyclically. Then the triangle A'B'C' is the cevian triangle of X(7057). (Randy Hutson, April 11, 2015)
X(7057) lies on these lines: {2,178}, {7,2091}, {8,177}, {145,174}
X(7057) = isotomic conjugate of X(7048)
X(7057) = reflection of X(188) in X(178)
X(7057) = X(4146)-Ceva conjugate of X(2)
X(7057) = X(i)-cross conjugate of X(j) for these (i,j): (177,2089), (236,2)
X(7057) = {X(178),X(188)}-harmonic conjugate of X(2)
X(7057) = anticomplement of X(188)
X(7057) = perspector of the inconic with center X(236)
X(7057) = cevapoint of X(177) and X(178)
X(7058) lies on these lines: {8,60}, {21,1183}, {63,99}, {81,239}, {261,284}, {312,645}, {314,1172}, {643,3996}, {931,3185}, {981,1185}, {1016,3969}, {1043,1098}
X(7058) = isotomic conjugate of X(6354)
X(7058) = X(i)-cross conjugate of X(j) for these (i,j): (1098,261), (2287,1098)
X(7058) = {X(333),X(2185)}-harmonic conjugate of X(261)
X(7058) = cevapoint of X(i) and X(j) for (i,j) = (6,1610), (333,1812), (1043,2287)
X(7058) = crosspoint of X(4631) and X(6064)
X(7059) lies on the Neuberg cubic and these lines: {1,5669}, {3,5672}, {14,1653}, {15,3465}, {19,2822}, {30,1277}, {63,616}, {222,559}, {484,3384}, {1338,5673}, {3065,3383}
X(7059) = isogonal conjugate of X(1277)
X(7060) lies on the Neuberg cubic and these lines: {1,5668}, {3,5673}, {13,1652}, {16,3465}, {19,2822}, {30,1276}, {63,617}, {222,1082}, {484,3375}, {1337,5672}, {3065,3376}
X(7060) = isogonal conjugate of X(1276)
X(7061) lies on the Pelletier strophoid (K040), the cubic K323 and these lines: {1,147}, {7,5984}, {63,2319}, {75,1281}, {98,1447}, {239,1916}, {894,4027}
X(7061) = X(1580)-cross conjugate of X(894)
X(7061) = {X(75),X(5989)}-harmonic conjugate of X(1281)
X(7061) = cevapoint of X(4107) and X(4459)
X(7062) lies on the Mandart inellipse and these lines: {11,1211}, {3688,4092}, {4548,6056}
X(7063) lies on the Mandart inellipse and the line {1084,4117}
X(7063) = crosssum of X(56) and X(4573)
X(7063) = crossdifference of any pair of centers on the line through X(4573) and X(4631)
X(7063) = crosspoint of X(8) and X(3709)
X(7063) = extouch isotomic conjugate of X(3700)
X(7064) lies on these lines: {9,3056}, {37,4890}, {72,4078}, {181,756}, {210,2321}, {220,2175}, {594,4092}, {872,1500}, {960,3717}, {1253,3022}, {1357,5650}, {1654,3799}, {1682,3790}, {2329,5148}, {3678,6541}, {3731,3779}, {3740,4967}, {3877,6018}, {3952,3963}
X(7064) = isogonal conjugate of X(552)
X(7064) = X(i)-Ceva conjugate of X(j) for these (i,j): (756,1500), (4069,3709)
X(7064) = crosssum of X(1014) and X(1434)
X(7064) = crosspoint of X(210) and X(1334)
X(7065) lies on the Mandart inellipse and the line {219, 6862}
See Tran Quang Hung and Randy Hutson, AdGeom 2047.
See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28731.
X(7066) lies on these lines: {3,1794}, {9,19366}, {10,12}, {34,26893}, {40,1745}, {55,581}, {56,219}, {64,7074}, {71,73}, {78,296}, {185,212}, {201,1425}, {227,22276}, {255,1364}, {329,1118}, {388,26872}, {389,3074}, {394,7335}, {511,1935}, {517,1838}, {603,3917}, {756,7324}, {970,24310}, {1038,3781}, {1361,3869}, {1397,19762}, {1469,5227}, {1490,6254}, {1682,22134}, {1762,29958}, {1802,7114}, {1859,5777}, {1936,5907}, {2175,10831}, {2218,3271}, {2323,19365}, {3075,11793}, {3611,18673}, {3682,22341}, {3695,7068}, {5285,26888}, {6285,7070}, {6354,15443}, {7085,19349}, {7352,26921}, {7957,10374}, {12835,23150}, {14059,20764}, {18915,26939}, {21015,26955}, {23154,26934}
X(7066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {255, 5562, 1364}, {1425, 3690, 201}
X(7066) = isotomic of the polar conjugate of X(2197)
X(7066) = isogonal of the polar conjugate of X(26942)
X(7066) = X(i)-Ceva conjugate of X(j) for these (i,j): {59, 23067}, {72, 201}, {26942, 2197}
X(7066) = X(2632)-cross conjugate of X(520)
X(7066) = X(i)-isoconjugate of X(j) for these (i,j): {4, 270}, {11, 24000}, {21, 8747}, {27, 1172}, {28, 29}, {58, 1896}, {60, 158}, {81, 8748}, {92, 2189}, {107, 3737}, {261, 1096}, {278, 2326}, {286, 2299}, {333, 5317}, {393, 2185}, {757, 1857}, {823, 7252}, {873, 6059}, {1098, 1118}, {1364, 24021}, {1396, 2322}, {2052, 2150}, {2170, 23582}, {3271, 23999}, {4560, 24019}, {4858, 23964}
X(7066) = crosspoint of X(i) and X(j) for these (i,j): {59, 23067}, {72, 3682}, {1214, 28786}
X(7066) = crosssum of X(28) and X(8747)
X(7066) = trilinear square of X(7591)
X(7066) = barycentric product X(i) X(j) for these {i,j}: {3, 26942}, {12, 394}, {59, 15526}, {63, 201}, {65, 3998}, {69, 2197}, {71, 307}, {72, 1214}, {73, 306}, {77, 3949}, {181, 3926}, {219, 6356}, {222, 3695}, {226, 3682}, {228, 1231}, {255, 6358}, {278, 4158}, {312, 7138}, {321, 22341}, {326, 2171}, {345, 1425}, {348, 3690}, {349, 4055}, {520, 4552}, {525, 23067}, {594, 1804}, {756, 7183}, {1089, 7125}, {1252, 1367}, {1254, 3719}, {1259, 6354}, {1260, 20618}, {1262, 7068}, {1409, 20336}, {1439, 3694}, {1441, 3990}, {1500, 7055}, {1813, 4064}, {2149, 17879}, {2632, 4564}, {3265, 4559}, {3269, 4998}, {3964, 8736}, {4024, 6517}, {4131, 21859}, {4551, 24018}, {4574, 17094}, {7335, 28654}
X(7066) = barycentric quotient X(i) / X(j) for these {i,j}: {12, 2052}, {37, 1896}, {42, 8748}, {48, 270}, {59, 23582}, {71, 29}, {73, 27}, {181, 393}, {184, 2189}, {201, 92}, {212, 2326}, {228, 1172}, {255, 2185}, {394, 261}, {520, 4560}, {577, 60}, {822, 3737}, {1214, 286}, {1259, 7058}, {1364, 26856}, {1367, 23989}, {1400, 8747}, {1402, 5317}, {1409, 28}, {1410, 1396}, {1425, 278}, {1500, 1857}, {1804, 1509}, {2149, 24000}, {2171, 158}, {2197, 4}, {2200, 2299}, {2289, 1098}, {2318, 2322}, {2632, 4858}, {2972, 26932}, {3269, 11}, {3682, 333}, {3690, 281}, {3695, 7017}, {3926, 18021}, {3949, 318}, {3990, 21}, {3998, 314}, {4055, 284}, {4158, 345}, {4551, 823}, {4552, 6528}, {4559, 107}, {4564, 23999}, {6056, 7054}, {6356, 331}, {6517, 4610}, {7068, 23978}, {7109, 6059}, {7125, 757}, {7138, 57}, {7183, 873}, {7335, 593}, {8736, 1093}, {20975, 8735}, {22061, 14006}, {22341, 81}, {23067, 648}, {24018, 18155}, {26942, 264}
X(7067) lies on the Mandart inellipse and these lines: {8,645}, {11,3686}, {960,3271}, {3685,4542}
X(7067) = X(8)-Ceva conjugate of X(3712)
X(7067) = crosspoint of X(8) and X(3712)
X(7068) lies on the Mandart inellipse and these lines: {8,6062}, {219,6867}, {345,6056}, {3695, 6866}, {3703,6862}
X(7069) lies on these lines: {4,201}, {5,1393}, {9,33}, {11,38}, {12,774}, {37,1864}, {45,55}, {51,1953}, {53,2181}, {71,1859}, {72,2654}, {73,5777}, {90,601}, {171,1776}, {184,2265}, {222,5779}, {226,1736}, {278,5817}, {497,984}, {896,5348}, {1006,3465}, {1040,3305}, {1214,2635}, {1451,1728}, {1824,2183}, {1837,2292}, {1898,4300}, {1936,3219}, {2964,3467}, {3074,6198}, {3974,4073}, {5532,6058}
X(7069) = X(5)-Ceva conjugate of X(1953)
X(7069) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,33,212), (756,2310,55), (1214,5927,2635)
X(7069) = crosssum of X(57) and X(603)
X(7069) = crosspoint of X(9) and X(318)
X(7070) lies on these lines: X(7070) lies on these lines: {1,3}, {9,33}, {20,1394}, {25,2270}, {31,2257}, {63,3100}, {84,255}, {154,610}, {200,219}, {204,1249}, {222,5732}, {226,3332}, {278,516}, {387,950}, {612,1253}, {643,3719}, {920,4354}, {968,4336}, {1108,3052}, {1260,2324}, {1490,1498}, {1630,2187}, {1723,2361}, {1743,1864}, {1856,5179}, {2323,2900}, {2654,5436}, {2947,4551}, {3158,3190}, {3474,3668}, {3586,5721}
X(7070) = isogonal conjugate of X(8809)
X(7070) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,610), (78,9)
X(7070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,1214), (1,1754,57), (33,212,9), (154,3198,610), (643,4123,3719), (1040,1936,57)
X(7071) lies on these lines: {1,1037}, {3,3100}, {4,390}, {6,3270}, {11,5094}, {19,25}, {34,3303}, {35,3515}, {42,3195}, {56,3516}, {64,1425}, {184,2192}, {192,1013}, {220,2332}, {235,3085}, {281,1863}, {318,3685}, {346,1260}, {378,999}, {388,1885}, {406,5687}, {427,497}, {458,4366}, {468,5218}, {480,4515}, {496,3541}, {516,1892}, {607,1253}, {608,2293}, {950,5090}, {1001,1861}, {1033,3553}, {1058,3088}, {1119,3672}, {1500,2207}, {1597,1870}, {1697,1829}, {1905,5119}, {2066,5410}, {2171,3209}, {2175,3022}, {3058,5064}, {3575,4294}, {3746,5198}, {3871,4194}, {4329,6356}, {5281,6353}, {5411,5414}
X(7071) = isogonal conjugate of X(7056)
X(7071) = X(33)-Ceva conjugate of X(607)
X(7071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1593,1398), (19,33,1827), (33,55,25)
X(7071) = crosssum of X(905) and X(3270)
X(7071) = crosspoint of X(2332) and X(4183)
X(7072) lies on these lines: {1,90}, {33,2911}, {55,2164}, {1406,1725}, {2192,2361}, {3022,6056}
X(7072) = X(90)-Ceva conjugate of X(2164)
X(7072) = X(212)-cross conjugate of X(55)
X(7072) = isogonal conjugate of isotomic conjugate of X(36626)
X(7073) lies on these lines: {1,30}, {33,430}, {42,1989}, {55,199}, {103,354}, {200,4007}, {220,3715}, {265,1411}, {963,3304}, {1043,3615}, {1807,2166}, {1859,2332}, {1864,4845}, {1961,4995}, {2194,4516}, {2328,2361}, {4102,4420}
X(7073) = isogonal conjugate of X(1442)
X(7073) = X(79)-Ceva conjugate of X(2160)
X(7073) = {X(1),X(1717)}-harmonic conjugate of X(500)
X(7073) = cevapoint of X(663) and X(4516)
X(7073) = crosssum of X(35) and X(2003)
X(7074) lies on these lines: {3,947}, {6,31}, {33,210}, {40,221}, {72,1854}, {73,5584}, {81,5281}, {100,394}, {109,6244}, {154,197}, {165,222}, {198,2187}, {200,219}, {255,1413}, {480,2318}, {497,4383}, {518,1040}, {612,2256}, {613,3749}, {644,5423}, {756,4336}, {940,5218}, {1155,1407}, {1191,3057}, {1260,3939}, {1376,1936}, {1455,6282}, {1466,1496}, {1498,2947}, {1616,2098}, {1864,4319}, {2299,2343}, {2323,3158}, {2331,3195}, {2911,4849}, {3085,5706}, {3100,3681}, {3157,3579}, {3190,6600}, {3197,3198}, {3474,6180}, {3475,5228}, {3974,4513}
X(7074) = isogonal conjugate of X(1440)
X(7074) = X(i)-Ceva conjugate of X(j) for these (i,j): (40,198), (200,55), (219,220)
X(7074) = X(2187)-cross conjugate of X(55)
X(7074) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40,1103,227), (42,1253,55), (55,2361,3052), (197,692,154)
X(7074) = crosssum of X(i) and X(j) for (i,j) = (11,3669), (84,1422), (1465,1537), (1565,3676)
X(7074) = crosspoint of X(i) and X(j) for (i,j) = (40,2324), (59,644)
X(7074) = trilinear product X(40)*X(55)
X(7075) lies on these lines: {9,312}, {55,2053}, {63,3797}, {171,3501}, {194,1424}, {219,3169}, {609,1018}, {1613,1740}, {1999,5364}, {3996,4050}
X(7075) = X(i)-Ceva conjugate of X(j) for these (i,j): (41,9), (194,1740)
X(7075) = {X(2319),X(3208)}-harmonic conjugate of X(55)
X(7076) lies on these lines: {2,1430}, {9,1096}, {25,2053}, {29,5247}, {31,281}, {33,2911}, {42,1783}, {43,1013}, {44,1859}, {92,238}, {158,3074}, {162,171}, {204,612}, {212,1857}, {240,3219}, {278,748}, {450,1935}, {899,4219}, {968,2331}, {1011,2202}, {1826,2299}, {1897,3971}, {2333,4206}
X(7076) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1957,1430), (1783,4183,42)
X(7077) lies on the hyperbola {{A, B, C, X(6), X(9)}} and these lines: {6,292}, {9,3056}, {19,1843}, {42,694}, {43,57}, {69,334}, {200,2319}, {210,333}, {239,335}, {284,2311}, {511,1757}, {674,2161}, {813,2291}, {894,3888}, {926,1024}, {1197,1922}, {1431,1581}, {1436,2196}, {1751,4362}, {2329,4531}, {3799,6651}
X(7077) = isogonal conjugate of X(1447)
X(7077) = reflection of X(3056) in X(3271)
X(7077) = X(291)-Ceva conjugate of X(292)
X(7077) = X(2340)-cross conjugate of X(55)
X(7077) = X(926)-line conjugate of X(4435)
X(7077) = {X(6),X(3862)}-harmonic conjugate of X(292)
X(7077) = cevapoint of X(i) and X(j) for (i,j) = (42,5360), (926,3271), (2329,3684)
X(7077) = crosssum of X(i) and X(j) for (i,j) = (238,1429), (241,1463), (812,4124)
X(7077) = crossdifference of any pair of centers on the line through X(812) and X(4107)
X(7077) = crosspoint of X(291) and X(4876)
X(7077) = trilinear pole of the line through X(663) and X(1334)
X(7078) lies on these lines: {1,6}, {2,3562}, {3,73}, {12,5713}, {20,651}, {33,5777}, {34,517}, {40,221}, {46,1427}, {55,581}, {56,580}, {58,939}, {78,271}, {81,5703}, {109,1035}, {198,2360}, {225,5812}, {226,5706}, {278,5758}, {329,3194}, {474,3075}, {602,1066}, {692,3556}, {912,1062}, {948,4295}, {999,1451}, {1012,1935}, {1040,1071}, {1069,1807}, {1073,1260}, {1155,1406}, {1181,3173}, {1193,1496}, {1210,4383}, {1253,4300}, {1259,1331}, {1265,1332}, {1376,1771}, {1393,2095}, {1394,6282}, {1465,5709}, {1490,1498}, {1783,3176}, {1854,5693}, {1936,3149}, {2003,3601}, {3085,5711}, {4292,6180}, {5710,5717}
X(7078) = X(i)-Ceva conjugate of X(j) for these (i,j): (78,3), (329,198), (394,219), (1262,906)
X(7078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3074,405), (3,3157,222), (6,1191,1453), (6,3990,219), (73,212,3), (392,5299,5234), (602,1066,1617), (1124,1335,219)
X(7078) = crosssum of X(1) and X(1728)
X(7078) = crossdifference of any pair of centers on the line through X(513) and X(3064)
X(7078) = crosspoint of X(59) and X(1331)
X(7079) lies on these lines: {1,1783}, {2,1435}, {4,9}, {12,208}, {28,5234}, {33,210}, {34,1212}, {37,2331}, {41,3119}, {48,282}, {84,2272}, {92,3305}, {108,2371}, {200,1802}, {204,612}, {273,1223}, {318,6559}, {341,2322}, {480,4515}, {728,4082}, {756,1096}, {1146,1837}, {1172,4866}, {1334,1857}, {1452,5282}, {1696,3209}, {1712,3085}, {1859,3715}, {1903,3197}, {1957,5268}, {1973,5573}, {2324,3949}, {5227,5815}, {6335,6376}
X(7079) = isogonal conjugate of X(7177)
X(7079) = X(i)-Ceva conjugate of X(j) for these (i,j): (281,33), (1783,3900)
X(7079) = X(i)-cross conjugate of X(j) for these (i,j): (1253,200), (1334,220)
X(7079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,281,1855), (9,281,19)
X(7079) = cevapoint of X(657) and X(3119)
X(7079) = crosspoint of X(2322) and X(4183)
X(7079) = trilinear pole of the line through X(4105) and X(4171)
X(7080) lies on these lines: {1,2}, {3,3421}, {4,1260}, {5,5082}, {12,480}, {20,100}, {21,5281}, {40,329}, {55,452}, {63,5815}, {72,5657}, {149,5187}, {210,1858}, {226,1706}, {227,322}, {280,341}, {281,318}, {326,1440}, {377,5261}, {388,1376}, {390,2478}, {404,3600}, {406,3695}, {443,495}, {474,1056}, {497,1329}, {518,1788}, {527,5128}, {528,5225}, {631,956}, {651,2122}, {668,3926}, {908,962}, {944,5440}, {946,5748}, {950,3158}, {958,5218}, {966,3713}, {1034,3692}, {1058,4187}, {1145,5730}, {1265,3699}, {1621,5129}, {1697,3452}, {1837,3189}, {1897,3176}, {2345,3965}, {2899,3685}, {2975,3523}, {3059,3983}, {3091,3434}, {3146,5080}, {3174,5809}, {3295,3820}, {3346,3998}, {3419,5818}, {3475,3812}, {3485,5836}, {3487,3753}, {3601,5795}, {3693,6554}, {3702,6708}, {3704,3974}, {3710,5423}, {3895,5328}, {3932,4012}, {3940,5690}, {4193,5274}, {4200,5081}, {4645,5906}, {4723,6552}, {4855,5731}, {5175,5587}, {5204,6174}, {5534,5768}, {5744,6684}
X(7080) = isogonal conjugate of X(1413)
X(7080) = isotomic conjugate of X(1440)
X(7080) = X(i)-Ceva conjugate of X(j) for these (i,j): (322,329), (341,8), (345,346)
X(7080) = X(i)-cross conjugate of X(j) for these (i,j): (40,8), (2324,329)
X(7080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,3616,3872), (8,4511,145), (8,5552,2), (10,200,8), (10,3085,2), (12,2550,5177), (55,2551,452), (100,3436,20), (281,3694,346), (1329,3913,497), (1698,4882,4847), (1837,3689,3189), (2478,3871,390), (3295,3820,5084), (4847,4882,8)
X(7080) = anticomplement of X(3086)
X(7080) = cevapoint of X(40) and X(1103)
X(7080) = crosspoint of X(646) and X(4998)
X(7081) lies on these lines: {1,2}, {3,4385}, {5,5015}, {6,3769}, {9,2319}, {11,4030}, {21,3701}, {22,1324}, {25,318}, {35,1089}, {36,4692}, {37,893}, {55,312}, {63,6194}, {75,183}, {86,4682}, {98,100}, {105,1261}, {165,3729}, {171,385}, {190,3967}, {210,333}, {226,4645}, {230,594}, {319,325}, {341,958}, {345,3790}, {346,5281}, {404,4968}, {427,5081}, {444,3963}, {452,2899}, {846,3971}, {908,4388}, {956,4737}, {1043,3714}, {1150,3681}, {1197,5276}, {1219,5265}, {1621,4358}, {1654,4104}, {1757,4090}, {1909,4447}, {1943,4551}, {2223,4203}, {2329,4095}, {2476,5300}, {2968,6676}, {2975,4696}, {3052,4676}, {3158,3886}, {3175,4689}, {3219,3952}, {3329,3791}, {3416,4417}, {3452,3883}, {3550,3923}, {3683,4009}, {3689,3706}, {3702,3871}, {3703,5432}, {3711,4042}, {3712,4995}, {3713,5275}, {3717,5745}, {3761,5088}, {3772,4429}, {3875,4734}, {3891,4850}, {3913,4673}, {3932,6690}, {3940,5774}, {3944,4660}, {3948,4199}, {3955,4579}, {3966,5233}, {3992,5251}, {4023,4886}, {4421,5695}, {4450,5057}, {4512,4903}, {4661,5372}, {5273,5423}, {5304,5749}, {5686,6555}, {5699,5981}, {5700,5980}
X(7081) = isogonal conjugate of X(1431)
X(7081) = isotomic conjugate of X(7049)
X(7081) = X(i)-Ceva conjugate of X(j) for these (i,j): (1909,894), (4601,644), (4876,3685)
X(7081) = X(i)-cross conjugate of X(j) for these (i,j): (2329,894), (4459,3907)
X(7081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,8,3705), (11,4030,4514), (43,4362,239), (55,312,3685), (75,183,1447), (171,1215,894), (333,3699,210), (345,3974,3790), (1215,4434,171), (3507,5293,869), (3689,3706,3996), (3757,4816,5205), (3757,5205,2), (3967,4640,190), (3974,5218,345), (4995,6057,3712)
X(7081) = cevapoint of X(3907) and X(4459)
X(7081) = crosssum of X(3271) and X(6371)
X(7081) = trilinear pole of the line through X(3287) and X(3907)
X(7082) lies on these lines: {1,195}, {2,1776}, {3,90}, {5,920}, {9,55}, {11,63}, {33,2361}, {46,381}, {65,1728}, {84,5204}, {212,2310}, {405,1858}, {430,2245}, {497,3219}, {1155,1709}, {1158,1532}, {1617,5779}, {1697,4677}, {1707,5348}, {1708,1836}, {1711,4383}, {2187,2265}, {3305,5432}, {3333,4870}, {3719,4387}, {5172,5720}
X(7082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,920,1454), (1864,3683,55)
X(7082) = crosssum of X(46) and X(57)
X(7082) = crosspoint of X(9) and X(90)
X(7083) lies on these lines: {3,238}, {6,692}, {7,105}, {9,55}, {19,6059}, {21,332}, {25,31}, {41,2293}, {56,269}, {154,1397}, {171,5020}, {197,3052}, {198,2223}, {218,3779}, {219,3056}, {220,3688}, {242,4008}, {497,5324}, {513,3433}, {614,1473}, {651,1037}, {674,2911}, {958,3883}, {984,3295}, {999,5429}, {1001,4357}, {1104,3556}, {1183,4313}, {1253,2347}, {1423,1617}, {1598,3072}, {1633,4000}, {2082,4319}, {2170,4336}, {2176,5017}, {2178,3941}, {2280,4343}, {2340,3217}, {3685,3718}, {3717,3913}, {4186,5230}, {4223,4307}
X(7083) = isogonal conjugate of X(8817)
X(7083) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,1040), (651,3063), (5324,2082)
X(7083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,31,1460), (2175,3271,6)
X(7083) = crosssum of X(i) and X(j) for (i,j) = (7,8), (63,3870), (65,307)
X(7083) = crossdifference of any pair of centers on the line through X(918) and X(3669)
X(7083) = crosspoint of X(i) and X(j) for (i,j) = (19,2191), (21,2299), (55,56), (614,2082)
X(7084) lies on these lines: {1,1416}, {3,1037}, {31,218}, {32,1802}, {55,1395}, {58,1792}, {212,1397}, {603,1362}, {612,4183}, {663,2440}, {943,1041}, {985,5255}, {1472,1918}, {5266,5728}
X(7084) = isogonal conjugate of X(3673)
X(7084) = cevapoint of X(i) and X(j) for (i,j) = (31,1253), (228,1918)
X(7084) = crosssum of X(497) and X(4000)
X(7085) lies on these lines: {3,63}, {6,31}, {9,25}, {10,4185}, {22,3219}, {35,1707}, {38,56}, {40,1593}, {48,2318}, {100,5739}, {154,205}, {165,1763}, {169,2355}, {184,219}, {197,210}, {198,199}, {218,5320}, {222,3917}, {329,4220}, {386,2221}, {394,3781}, {405,5294}, {572,3190}, {612,1460}, {851,1211}, {896,5217}, {984,5329}, {1265,1791}, {1397,3688}, {1486,3683}, {1766,1824}, {2194,2911}, {2203,2328}, {2345,4206}, {2550,4196}, {3220,3929}, {3305,5020}, {4219,5759}, {4224,5273}, {5687,5814}
X(7085) = X(1038)-ceva conjugate of X(2286)
X(7085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,63,1473), (3,1260,228), (9,5285,25), (63,5314,3), (184,3690,219), (3781,3955,394)
X(7085) = crossdifference of any pair of centers on the line through X(514) and X(6591)
X(7085) = crosspoint of X(1038) and X(5227)
X(7086) lies on these lines: {3,4322}, {55,5398}, {1407,3579}, {1480,5119}
X(7087) lies on these lines: {35,984}, {75,1631}, {256,4471}, {291,4497}, {1399,1469}, {1626,4184}, {2174,2276}, {4361,6660}
X(7087) = isogonal conjugate of X(6327)
X(7087) = X(560)-cross conjugate of X(6)
X(7087) = X(75)-vertex conjugate of X(75)
X(7087) = crosssum of X(1759) and X(4149)
X(7088) lies on these lines: {6,559}, {14,1277}, {62,6191}, {395,1081}, {2082,3305}
X(7088) = isogonal conjugate of X(1653)
X(7088) = X(1250)-cross conjugate of X(1)
X(7089) lies on these lines: {6,1082}, {13,1276}, {61,6192}, {396,554}, {2082,3305}
X(7089) = isogonal conjugate of X(1652)
X(7090) lies on these lines: {1,1123}, {2,176}, {4,9}, {29,5414}, {63,6347}, {92,1586}, {219,1377}, {278,3536}, {388,6203}, {638,1944}, {1146,3071}, {1336,1785}, {1489,2090}, {1788,6204}, {1851,3128}, {3305,6348}
X(7090) = isogonal conjugate of X(6502)
X(7090) = complement of X(176)
X(7090) = cevapoint of X(1) and X(6213)
X(7091) lies on the Feuerbach hyperbola and these lines: {1,1407}, {4,1435}, {7,738}, {8,57}, {9,56}, {21,1412}, {40,1000}, {46,5559}, {65,3680}, {79,4355}, {80,3338}, {84,999}, {90,5563}, {256,4334}, {294,1416}, {314,1434}, {354,5665}, {388,5437}, {474,3361}, {941,1458}, {942,3577}, {943,3576}, {1106,5269}, {1320,3340}, {1466,3158}, {1477,6574}, {1697,3522}, {2136,3476}, {2298,4327}, {2346,3601}, {3255,3649}, {3296,3671}, {3339,4900}, {3677,4320}, {4187,5290}
X(7091) = isogonal conjugate of X(1697)
X(7091) = X(i)-cross conjugate of X(j) for these (i,j): (2285,57), (3304,1), (4790,651)
X(7091) = cevapoint of X(1) and X(3361)
X(7092) lies on these lines: {171,3553}, {393,1711}, {894,3085}
X(7092) = isogonal conjugate of X(1711)
X(7092) = X(394)-cross conjugate of X(1)
X(7092) = cevapoint of X(822) and X(4128)
X(7093) lies on these lines: {25,1716}, {171,3501}, {612,894}, {1460,1740}
X(7093) = isogonal conjugate of X(1716)
X(7093) = X(69)-cross conjugate of X(1)
X(7093) = cevapoint of X(656) and X(4128)
X(7093) = trilinear pole of the line through X(2484) and X(4367)
X(7094) lies on these lines: {264,1726}, {573,1759}, {1631,3185}
X(7094) = isogonal conjugate of X(1726)
X(7094) = isotomic conjugate of X(20926)
X(7094) = X(184)-cross conjugate of X(1)
X(7095) lies on these lines: {35,1018}, {57,2606}, {98,1756}, {171,2003}, {3219,3952}
X(7095) = isogonal conjugate of X(1756)
X(7095) = X(511)-cross conjugate of X(1)
X(7095) = cevapoint of X(42) and X(1755)
X(7095) = trilinear pole of the line through X(37) and X(2605)
X(7096) lies on these lines: {35,984}, {76,1759}, {609,3497}, {1726,3219}, {2003,5280}
X(7096) = isogonal conjugate of X(1759)
X(7096) = isotomic conjugate of X(20444)
X(7096) = X(32)-cross conjugate of X(1)
X(7096) = cevapoint of X(798) and X(2611)
X(7096) = trilinear pole of the line through X(1491) and X(2605)
X(7097) lies on the cubic K169 and on these lines: {1,2138}, {2,6359}, {9,478}, {20,346}, {69,1763}, {159,197}, {269,2184}, {281,388}, {2303,4183}
X(7097) = isogonal conjugate of X(1763)
X(7097) = isotomic conjugate of X(20914)
X(7097) = X(25)-cross conjugate of X(1)
X(7097) = trilinear pole of the line through X(2522) and X(3900)
X(7098) lies on these lines: {2,1454}, {4,46}, {12,3219}, {21,65}, {34,1707}, {36,5884}, {40,3486}, {47,1870}, {56,3218}, {57,1125}, {63,388}, {73,1046}, {109,1780}, {171,201}, {191,226}, {227,4641}, {238,1393}, {243,1715}, {411,1155}, {484,4324}, {497,5709}, {580,1735}, {603,4650}, {653,3559}, {757,1442}, {774,1936}, {896,1254}, {986,1451}, {1118,1748}, {1210,5535}, {1399,4296}, {1400,1761}, {1445,5698}, {1725,6198}, {1749,3585}, {1778,1880}, {2078,3874}, {2476,5880}, {3336,3911}
X(7098) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,920,1776), (46,920,4), (46,1158,3474), (46,1708,1788), (46,1727,1770), (896,1254,1935), (1046,1758,73), (1155,1858,411)
X(7098) = crosssum of X(652) and X(4516)
X(7099) lies on these lines: {19,1422}, {48,222}, {56,1413}, {57,909}, {77,2359}, {184,603}, {269,1396}, {394,1802}, {604,1407}, {1088,4637}, {1106,1408}, {1364,2188}
X(7099) = isogonal conjugate of X(7101)
X(7099) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,1106), (1412,1407)
Let I be the incenter of triangle ABC. Let LB be the line through I perpendiculat to AC, and let AB = LB∩BC and BA = LB∩BA. Define BC and CA cyclically, and define CB and AC cyclically. Let A' be the circumcenter of IABAC, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC, and the perspector is X(7100). (Angel Montesdeoca, June 11, 2016)
X(7100) lies on these lines: {1,30}, {5,3468}, {29,1870}, {56,3422}, {73,265}, {77,1062}, {78,1060}, {102,1385}, {219,3157}, {222,1069}, {283,1789}, {284,501}, {517,947}, {999,1036}, {1037,3295}, {1212,2338}, {1214,1794}, {1411,2166}, {1442,1446}
X(7100) = isogonal conjugate of X(6198)
X(7100) = X(3615)-ceva conjugate of X(79)
X(7100) = X(4303)-cross conjugate of X(3)
X(7100) = cevapoint of X(1) and X(3468)
X(7100) = crosssum of X(i) and X(j) for (i,j) = (1,1717), (1825,2594)
X(7100) = crosspoint of X(1789) and X(3615)
X(7100) = trilinear pole of the line through X(652) and X(2523)
X(7101) lies on these lines: {8, 7003}, {75, 6335}, {76, 1847}, {92, 264}, {158, 1089}, {273, 1229}, {281, 318}, {321, 459}, {341, 2322}, {653, 3729}, {1249, 4696}, {1783, 4737}, {1857, 3974}, {1863, 5423}, {1895, 4385}, {1897, 2331}
X(7101) = isogonal conjugate of X(7099)
X(7101) = isotomic conjugate of X(7177)
X(7101) = X(i)-Ceva conjugate of X(j) for these (i,j): {6335, 4397}, {7017, 318}
X(7101) = X(i)-cross conjugate of X(j) for these (i,j): (200, 341), (2321, 346), (7046, 318)
X(7101) = polar conjugate of X(269)
X(7101) = X(i)-beth conjugate of X(j) for these (i,j): {6335, 342}, {7101, 281}
X(7101) = X(1)-zayin conjugate of X(7099)
X(7101) = cevapoint of X(200) and X(7079)
X(7101) = isoconjugate of X(j) and X(j) for these (i,j): {1, 7099}, {3, 1407}, {6, 7053}, {31, 7177}, {32, 7056}, {34, 7125}, {48, 269}, {56, 222}, {57, 603}, {63, 1106}, {73, 1412}, {77, 604}, {78, 7366}, {81, 1410}, {184, 279}, {212, 738}, {219, 7023}, {255, 1435}, {278, 7335}, {348, 1397}, {394, 1398}, {577, 1119}, {593, 1425}, {608, 1804}, {652, 6614}, {810, 4637}, {1014, 1409}, {1042, 1790}, {1088, 9247}, {1214, 1408}, {1262, 3937}, {1333, 1439}, {1395, 7183}, {1413, 7011}, {1422, 7114}, {1427, 1437}, {1433, 6611}, {1459, 1461}, {1946, 4617}, {2197, 7341}, {3049, 4616}, {4558, 7250}, {4575, 7216}, {6612, 7078}, {7117, 7339}
X(7101) = barycentric product X(i)*X(j) for these {i,j}: {4, 341}, {8, 318}, {9, 7017}, {29, 3701}, {33, 3596}, {75, 7046}, {76, 7079}, {92, 346}, {158, 1265}, {200, 264}, {220, 1969}, {273, 5423}, {281, 312}, {286, 4082}, {313, 4183}, {321, 2322}, {331, 728}, {561, 7071}, {646, 3064}, {1098, 7141}, {1857, 3718}, {1896, 3710}, {1897, 4397}, {2052, 3692}, {2501, 7258}, {3239, 6335}, {4171, 6331}, {7020, 7080}
X(7101) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7053}, {2, 7177}, {4, 269}, {6, 7099}, {8, 77}, {9, 222}, {10, 1439}, {19, 1407}, {25, 1106}, {29, 1014}, {33, 56}, {34, 7023}, {42, 1410}, {55, 603}, {75, 7056}, {78, 1804}, {92, 279}, {108, 6614}, {158, 1119}, {200, 3}, {210, 73}, {212, 7335}, {219, 7125}, {220, 48}, {264, 1088}, {270, 7341}, {273, 479}, {278, 738}, {281, 57}, {312, 348}, {318, 7}, {341, 69}, {345, 7183}, {346, 63}, {393, 1435}, {461, 3361}, {480, 212}, {607, 604}, {608, 7366}, {644, 1813}, {648, 4637}, {653, 4617}, {728, 219}, {756, 1425}, {811, 4616}, {1021, 7254}, {1043, 1444}, {1089, 6356}, {1096, 1398}, {1146, 3942}, {1172, 1412}, {1253, 184}, {1260, 255}, {1265, 326}, {1334, 1409}, {1783, 1461}, {1802, 577}, {1824, 1042}, {1826, 1427}, {1855, 1418}, {1857, 34}, {1863, 614}, {1897, 934}, {2052, 1847}, {2212, 1397}, {2287, 1790}, {2299, 1408}, {2310, 3937}, {2321, 1214}, {2322, 81}, {2324, 7011}, {2326, 593}, {2328, 1437}, {2331, 6611}, {2332, 1333}, {2501, 7216}, {3064, 3669}, {3119, 7117}, {3239, 905}, {3596, 7182}, {3690, 7138}, {3692, 394}, {3699, 6516}, {3701, 307}, {3718, 7055}, {3900, 1459}, {3974, 1038}, {4012, 1040}, {4073, 3784}, {4081, 7004}, {4082, 72}, {4105, 1946}, {4130, 652}, {4163, 521}, {4171, 647}, {4183, 58}, {4319, 1473}, {4397, 4025}, {4515, 71}, {4524, 810}, {4571, 6517}, {4578, 1331}, {5081, 1443}, {5423, 78}, {6057, 201}, {6059, 1395}, {6331, 4635}, {6335, 658}, {6554, 7289}, {6555, 4855}, {6558, 1332}, {6559, 1814}, {6605, 1803}, {7003, 1422}, {7008, 1413}, {7012, 7339}, {7017, 85}, {7020, 1440}, {7046, 1}, {7071, 31}, {7074, 7114}, {7079, 6}, {7080, 7013}, {7102, 4320}, {7129, 6612}, {7140, 1254}, {7256, 4592}, {7258, 4563}, {7259, 4558}, {8736, 7147}, {8748, 1396}
X(7101) = pole wrt polar circle of trilinear polar of X(269)
X(7102) lies on these lines: {4,8}, {19,5282}, {25,281}, {33,42}, {196,1892}, {278,427}, {469,1897}, {756,1840}, {1068,5142}, {1659,3127}, {1853,6354}, {1859,3779}, {1861,4196}, {2050,2968}, {2345,4206}, {2969,5064}, {2994,3060}
X(7102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,92,1851), (33,1826,4207), (33,1857,1863), (1861,5307,4196), (1867,5090,4)
X(7103) lies on these lines: {4,7}, {34,207}, {196,1829}, {225,1435}, {278,961}, {2049,6356}, {2285,5286}, {3144,5435}, {4206,5323}
X(7103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1119,1426), (4,1895,1863), (34,1118,1851)
X(7104) lies on these lines: {6,256}, {41,904}, {213,2330}, {284,893}, {983,2176}, {1431,1438}, {1918,1927}, {1922,1967}, {1973,2211}, {3865,5299}
X(7104) = isogonal conjugate of X(1920)
X(7104) = X(2210)-cross conjugate of X(1922)
X(7104) = crosssum of X(1237) and X(3963)
X(7104) = crossdifference of any pair of centers on the line through X(2533) and X(3805)
X(7104) = cevapoint of PU(12)
X(7104) = perspector of ABC and unary cofactor triangle of Gemini triangle 32
X(7105) lies on these lines: {43,46}, {65,1942}, {915,3073}, {1068,1148}, {1195,2202}, {1816,1936}, {1937,1940}
X(7105) = isogonal conjugate of X(1935)
X(7105) = X(i)-cross conjugate of X(j) for these (i,j): (243,1937), (1858,1)
X(7105) = crosssum of X(1046) and X(1745)
X(7105) = crosspoint of X(1247) and X(3362)
X(7105) = pole wrt polar circle of trilinear polar of X(1947)
X(7105) = X(48)-isoconjugate (polar conjugate) of X(1947)
X(7105) = cevapoint of PU(15)
X(7106) lies on the line {2176,2178}
X(7106) = isogonal conjugate of X(1943)
X(7106) = X(i)-cross conjugate of X(j) for these (i,j): (1195,6), (2202,1945)
X(7106) = cevapoint of PU(18)
X(7107) lies on these lines: {1195,2202}, {1949,1950}, {2176,2178}
X(7107) = isogonal conjugate of X(1947)
X(7107) = X(1951)-cross conjugate of X(1949)
X(7107) = cevapoint of PU(19)
X(7108) lies on these lines: {192,3151}, {243,1858}, {1947,1952}
X(7108) = isogonal conjugate of X(1950)
X(7108) = isotomic conjugate of X(1943)
X(7108) = X(i)-cross conjugate of X(j) for these (i,j): (1948, 1952), (1951, 1949)
X(7108) = pole wrt polar circle of trilinear polar of X(1940)
X(7108) = X(48)-isoconjugate (polar conjugate) of X(1940)
X(7108) = cevapoint of PU(20)
X(7109) lies on these lines: {2,1258}, {6,1621}, {31,1911}, {42,213}, {43,1018}, {55,1185}, {171,3231}, {181,3124}, {594,2238}, {672,2300}, {902,1197}, {1015,2350}, {1206,3750}, {1501,2175}, {1918,2205}, {2276,5153}, {3230,3720}
X(7109) = X(i)-Ceva conjugate of X(j) for these (i,j): (213,872), (1016,4557)
X(7109) = X(4117)-cross conjugate of X(669)
X(7109) = {X(31),X(3051)}-harmonic conjugate of X(1977)
X(7109) = crosssum of X(274) and X(310)
X(7109) = crossdifference of any pair of centers on the line through X(3766) and X(6372)
X(7109) = crosspoint of X(i) and X(j) for (i,j) = (213,1918), (1016,4557)
X(7109) = barycentric square of X(42)
X(7110) lies on these lines: {9,46}, {19,403}, {37,1989}, {80,2174}, {101,1141}, {200,4007}, {281,451}, {346,5552}, {1224,5302}, {2287,2323}
X(7110) = isogonal conjugate of X(2003)
X(7110) = isotomic conjugate of X(17095)
X(7110) = X(i)-cross conjugate of X(j) for these (i,j): (2328,6598), (2361,80), (3683,8)
X(7110) = cevapoint of X(1146) and X(4976)
X(7110) = crosssum of X(1399) and X(2174)
X(7110) = trilinear pole of the line through X(3900) and X(4820)
X(7111) lies on these lines: {3,271}, {63,3428}, {283,6061}, {521,1946}, {1259,6617}, {1817,3687}, {2062,3964}
X(7111) = foot of the perpendicular to the line X(i)X(j) from X(k) for (i,j,k) = (63,3428,1946), (521,1946,63)
X(7112) lies on these lines: {28,242}, {75,516}, {76,5179}, {85,142}, {92,6063}, {281,348}, {304,309}, {305,2064}, {514,1921}, {5082,5195}
X(7112) = X(857)-cross conjugate of X(4872)
X(7112) = foot of the perpendicular to the line X(i)X(j) from X(k) for (i,j,k) = (75,516,1921), (514,1921,75)
X(7113) lies on these lines: {1,2278}, {6,41}, {9,4268}, {36,2245}, {37,572}, {44,101}, {45,2267}, {65,2302}, {71,5124}, {184,2352}, {213,5035}, {215,2361}, {239,662}, {241,1813}, {284,501}, {570,2197}, {584,1449}, {609,4290}, {610,3554}, {649,834}, {672,3446}, {692,2223}, {849,1333}, {851,5137}, {909,1319}, {910,3660}, {922,2210}, {1030,2269}, {1086,1429}, {1104,2360}, {1108,1630}, {1213,4999}, {1397,3185}, {1409,2148}, {1436,2164}, {1461,6610}, {1474,1841}, {1631,3056}, {1743,3204}, {1790,3666}, {1818,5096}, {1953,5341}, {1958,4361}, {1990,2202}, {2170,2173}, {2171,5356}, {2175,3941}, {2187,3052}, {2242,5114}, {2252,5172}, {2268,4287}, {2273,5069}, {2280,4289}, {2911,5120}, {3684,4969}, {3779,4497}, {4053,4511}, {5109,5280}
X(7113) = complement of X(21277)
X(7113) = anticomplement of X(21237)
X(7113) = X(i)-Ceva conjugate of X(j) for these (i,j): (36,2361), (106,31), (909,6), (2006,1399)
X(7113) = X(3724)-cross conjugate of X(36)
X(7113) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,48,2174), (36,2323,2245), (48,604,6), (101,5053,44), (922,3248,2210), (1055,1404,2183), (1400,2317,6), (1404,2183,6)
X(7113) = crosssum of X(i) and X(j) for (i,j) = (6,1324), (10,4053), (35,2323), (37,517), (321,4358), (1146,2804), (1807,2161), (3762,4858)
X(7113) = crossdifference of any pair of centers on the line through X(10) and X(522)
X(7113) = crosspoint of X(i) and X(j) for (i,j) = (79,2006), (81,104), (1262,2720), (1870,3218)
X(7113) = X(92)-isoconjugate of X(1870)
X(7114) lies on these lines: {1,947}, {3,1433}, {6,41}, {78,1813}, {108,1745}, {154,1035}, {184,603}, {208,223}, {221,2187}, {1394,1461}, {1398,1457}
X(7114) = isogonal conjugate of X(7020)
X(7114) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,603), (223,2199), (2360,221)
X(7114) = X(2187)-cross conjugate of X(48)
X(7114) = {X(184),X(1410)}-harmonic conjugate of X(603)
X(7114) = crosssum of X(1) and X(1753)
X(7115) lies on these lines: {4,1521}, {108,919}, {109,652}, {112,2222}, {650,1415}, {1110,2356}, {1252,5089}, {1262,1465}, {2149,2183}, {2427,2443}, {4567,4998}
X(7115) = isogonal conjugate of X(26932)
X(7115) = cevapoint of circumcircle intercepts of Stevanovic circle
X(7115) = X(i)-cross conjugate of X(j) for these (i,j): (6,1783), (19,112), (25,108), (55,109), (197,100), (198,101), (910,919), (1486,934), (2197,4559), (3185,110), (3192,1897)
X(7115) = cevapoint of X(i) and X(j) for (i,j) = (6,1415), (101,573), (2197,4559)
X(7115) = trilinear pole of the line through X(692) and X(2498)
X(7115) = X(63)-isoconjugate of X(11)
X(7115) = polar conjugate of X(34387)
X(7116) lies on these lines: {6,893}, {48,3289}, {256,1172}, {257,1762}, {333,4603}, {904,2194}, {1432,2982}, {2196,3955}
X(7116) = X(256)-ceva conjugate of X(904)
X(7117) lies on these lines: {3,906}, {6,909}, {11,5190}, {36,1951}, {39,41}, {48,216}, {56,607}, {57,1945}, {104,1783}, {219,4587}, {232,2202}, {244,665}, {570,2174}, {604,800}, {608,1436}, {647,3708}, {650,1146}, {663,3022}, {905,1565}, {1195,2300}, {1410,1475}, {1457,2272}, {1814,6516}, {2082,2275}, {2637,2638}, {4996,5546}
X(7117) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,1946), (11,3271), (48,647), (56,663), (219,652), (222,1459), (278,513), (345,521), (911,665), (1436,649), (2217,512), (3435,667), (3942,3937)
X(7117) = {X(48),X(216)}-harmonic conjugate of X(2197)
X(7117) = crosssum of X(i) and X(j) for (i,j) = (2,651), (4,1783), (9,4551), (12,4559), (100,219), (108,608), (190,4417), (278,653), (281,1897)
X(7117) = crossdifference of any pair of centers on the line through X(100) and X(108)
X(7117) = crosspoint of X(i) and X(j) for (i,j) = (3,905), (6,650), (57,3737), (60,4560), (219,652), (222,1459), (278,513), (345,521), (1413,3669)
X(7117) = X(92)-isoconjugate of X(59)
X(7117) = trilinear product X(11)*X(48)
X(7118) lies on these lines: {6,603}, {31,607}, {41,184}, {44,1903}, {84,294}, {212,220}, {271,282}, {604,2155}, {949,1433}, {1170,1422}, {2082,2312}
X(7118) = isogonal conjugate of isotomic conjugate of X(282)
X(7118) = X(i)-Ceva conjugate of X(j) for these (i,j): (282,2188), (1436,2208)
X(7118) = X(i)-cross conjugate of X(j) for these (i,j): (32,41), (2212,31)
X(7118) = crosssum of X(i) and X(j) for (i,j) = (2,5932), (329,347)
X(7118) = crosspoint of X(1436) and X(2192)
X(7119) lies on these lines: {4,1973}, {6,19}, {25,2053}, {28,291}, {29,1220}, {42,4206}, {48,388}, {172,444}, {225,1910}, {419,1215}, {1107,5089}, {1254,2312}, {1478,2172}, {1755,1935}, {1869,2332}, {2179,3073}, {2295,2330}
X(7119) = X(444)-cross conjugate of X(19)
X(7119) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1973,2201), (28,1783,2333), (607,4185,19)
X(7120) lies on these lines: {4,604}, {19,1609}, {33,2285}, {41,1249}, {48,393}, {73,5317}, {108,1172}, {225,1474}, {232,2197}, {273,1429}, {572,1785}, {608,2162}, {1100,1875}, {1319,1841}, {1990,2174}, {2171,6198}, {2207,2286}, {2208,6618}
X(7120) = X(i)-Ceva conjugate of X(j) for these (i,j): (1945,2202), (1947,1935)
X(7120) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (33,3213,2285), (48,393,2202), (108,1172,1400)
X(7120) = crosspoint of PU(18)
X(7121) lies on these lines: {1,727}, {31,172}, {32,2209}, {41,1922}, {58,87}, {330,985}, {609,1923}, {750,4598}, {1106,1428}, {1397,2210}, {3123,3500}
X(7121) = isogonal conjugate of X(6376)
X(7121) = X(i)-cross conjugate of X(j) for these (i,j): (6,31), (667,932), (1197,1333)
X(7121) = cevapoint of X(6) and X(2162)
X(7121) = crosssum of X(3123) and X(3835)
X(7122) lies on these lines: {6,560}, {31,184}, {32,2209}, {35,849}, {42,284}, {48,869}, {75,4412}, {77,1758}, {172,1691}, {239,1582}, {572,2309}, {692,1333}, {872,922}, {894,1580}, {1400,1976}, {1458,3449}, {1468,5138}, {1922,1967}, {195,3451}, {2643,5341}, {4649,5009}
X(7122) = isogonal conjugate of X(7018)
X(7122) = X(1922)-ceva conjugate of X(2210)
X(7122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,560,2210), (6,1631,3764), (692,1333,1918), (872,922,2174), (2175,5019,31)
X(7122) = crosssum of X(i) and X(j) for (i,j) = (2,4388), (257,4451), (693,3120)
X(7122) = crossdifference of any pair of centers on the line through X(824) and X(4391)
X(7122) = crosspoint of X(692) and X(4570)
X(7122) = crosspoint of PU(12)
X(7123) lies on these lines: {2,1462}, {6,3692}, {9,608}, {31,218}, {63,220}, {101,1473}, {213,2221}, {219,604}, {607,3501}, {650,1376}, {949,6184}, {1333,1801}, {1707,5526}, {1818,3423}, {2203,2328}, {2322,2345}
X(7123) = isogonal conjugate of X(4000)
X(7123) = X(i)-cross conjugate of X(j) for these (i,j): (1459,101), (3126,2284)
X(7123) = cevapoint of X(i) and X(j) for (i,j) = (6,220), (9,3501), (71,213)
X(7123) = crosssum of X(614) and X(2082)
X(7123) = crosspoint of X(1275) and X(6012)
X(7123) = trilinear pole of the line through X(667) and X(926)
X(7124) lies on these lines: {1,607}, {3,906}, {6,41}, {63,6461}, {78,219}, {284,1036}, {348,1814}, {608,610}, {614,1184}, {672,1208}, {944,1783}, {976,2256}, {1212,5452}, {1436,1950}, {2202,2207}
X(7124) = X(i)-Ceva conjugate of X(j) for these (i,j): (3939,652), (6516,1946)
X(7124) = {X(6),X(48)}-harmonic conjugate of X(2286)
X(7124) = crosssum of X(i) and X(j) for (i,j) = (226,1880), (278,281)
X(7124) = crosspoint of X(i) and X(j) for (i,j) = (219,222), (284,1812)
X(7125) lies on these lines: {3,1433}, {41,2003}, {48,222}, {56,1064}, {57,77}, {63,1813}, {109,2187}, {198,6612}, {223,1461}, {255,1092}, {394,1804}, {603,1437}, {610,1422}, {1394,2360}, {1815,6602}, {1943,1958}, {2208,3220}, {6340,6518}
X(7125) = X(i)-Ceva conjugate of X(j) for these (i,j): (77,603), (1790,222), (1804,255), (1813,4091)
X(7125) = X(577)-cross conjugate of X(255)
X(7125) = {X(394),X(6507)}-harmonic conjugate of X(2289)
X(7125) = crosssum of X(3700) and X(5514)
X(7125) = X(92)-isoconjugate of X(33)
X(7126) lies on the Feuerbach hyperbola and these lines: {1,61}, {6,1251}, {9,1250}, {14,79}, {16,3065}, {37,2154}, {45,55}, {559,651}, {1320,5239}, {2173,3130}, {2320,5240}
X(7127) lies on these lines: {1,62}, {3,2307}, {6,31}, {9,1251}, {11,395}, {12,397}, {14,3583}, {15,5010}, {16,36}, {35,61}, {46,2306}, {396,5432}, {398,6284}, {3299,3390}, {3301,3389}, {3877,5240}, {4511,5239}, {5284,5367}
X(7127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (16,203,36), (36,5353,203), (2066,5414,1250)
X(7127) = crossdifference of any pair of centers on the line through X(514) and X(3638)
X(7128) lies on these lines: {59,517}, {162,1624}, {514,653}, {905,934}, {908,4564}, {1262,1465}, {1845,2717}, {2405,2406}
X(7128) = isogonal conjugate of X(34591)
X(7128) = polar conjugate of X(24026)
X(7128) = X(63)-isoconjugate of X(2310)
X(7128) = X(i)-cross conjugate of X(j) for these (i,j): (1,934), (6,162), (19,108), (48,109), (57,653), (223,651), (610,100), (1610,99), (1630,110), (1730,823), (1763,190), (2173,2222), (2331,1897)
X(7128) = cevapoint of X(i) and X(j) for (i,j) = (1,1783), (19,108), (34, 32674), (40,101), (48,109), (57,1461), (651, 17080), (1435, 32714)
X(7128) = trilinear pole of the line through X(108) and X(109)
X(7129) lies on these lines: {1,281}, {4,937}, {6,33}, {9,1167}, {19,56}, {34,393}, {37,939}, {58,84}, {86,309}, {108,2270}, {208,2262}, {269,278}, {608,1413}, {612,2336}, {1474,2208}, {1753,5120}, {1886,2191}, {2215,2357}
X(7129) = X(1422)-ceva conjugate of X(34)
X(7129) = X(i)-cross conjugate of X(j) for these (i,j): (608,19), (1096,34), (2208,84)
X(7129) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1249,2331), (393,3554,34)
X(7129) = cevapoint of X(2170) and X(6591)
X(7129) = crosssum of X(1103) and X(2324)
X(7129) = crosspoint of X(1256) and X(1422)
X(7129) = pole wrt polar circle of trilinear polar of X(322)
X(7129) = X(48)-isoconjugate (polar conjugate) of X(322)
X(7129) = X(63)-isoconjugate of X(40)
X(7130) = cevapoint of X(56) and X(2178)
X(7131) lies on these lines: {1,1416}, {2,1435}, {9,348}, {19,4209}, {21,1038}, {56,78}, {57,345}, {63,220}, {85,6559}, {169,514}, {664,2082}, {1170,3873}, {1214,2339}, {1412,1708}, {1434,2285}
X(7131) = isogonal conjugate of X(2082)
X(7131) = X(663)-cross conjugate of X(664)
X(7131) = {X(2082),X(6167)}-harmonic conjugate of X(664)
X(7131) = cevapoint of X(i) and X(j) for (i,j) = (3,218), (9,57), (1214,1400)
X(7131) = trilinear pole of the line through X(521) and X(2254)
X(7132) lies on these lines: {1,182}, {2,1429}, {6,1432}, {32,3503}, {56,291}, {57,172}, {65,985}, {105,3924}, {330,604}, {349,3114}, {1403,1580}, {2344,3407}
X(7132) = isogonal conjugate of X(3061)
X(7132) = X(1919)-cross conjugate of X(109)
X(7132) = cevapoint of X(i) and X(j) for (i,j) = (6,1403), (41,3915)
Let DEF be the cevian triangle of X(176). Let OAB be the A-excenter of triangle ABD, and define OBC and OCA cyclically. Let OAC be the A-excenter of triangle ACD, and define OBA and OCB cyclically. Let A' = OBCOBA∩OCAOCB, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(7133). Also, X(7133) lies on the trilinear polar of X(6135). (Randy Hutson, April 11, 2015)
X(7133) lies on the Feuerbach hyperbola, the cubic K233 and these lines: {1,372}, {4,1123}, {7,1659}, {9,2066}, {19,25}, {21,1806}, {84,2067}, {90,371}, {497,6351}, {1721,6204}, {3553,5416}, {5218,6352}
X(7133) = X(i)-Ceva conjugate of X(j) for these (i,j): (1659,2362), (6135,650)
X(7133) = X(650)-cross conjugate of X(6135)
X(7133) = {X(1),X(6213)}-harmonic conjugate of X(2362)
X(7133) = crosssum of X(2066) and X(6502)
X(7133) = crossdifference of any pair of centers on the line through X(905) and X(6364)
X(7134) lies on the line {2597,2601}
X(7134) = isogonal conjugate of X(2595)
X(7134) = X(2602)-cross conjugate of X(2597)
X(7134) = crosssum of X(1048) and X(3460)
X(7134) = cevapoint of PU(68)
X(7135) lies on these lines: {1048,3460}, {2595,2603}
X(7135) = isogonal conjugate of X(2601)
X(7135) = X(2596)-cross conjugate of X(2603)
X(7135) = cevapoint of PU(69)
X(7136) lies on these lines: {54,2620}, {2607,3615}, {2608,2612}
X(7136) = isogonal conjugate of X(2606)
X(7136) = X(2613)-cross conjugate of X(2608)
X(7136) = cevapoint of PU(70)
X(7137) lies on the line {2606,2614}
X(7137) = isogonal conjugate of X(2612)
X(7137) = X(2607)-cross conjugate of X(2614)
X(7137) = cevapoint of PU(71)
X(7138) lies on these lines: {1,412}, {3,296}, {29,2655}, {42,65}, {48,1106}, {158,1745}, {201,2632}, {653,1047}, {1409,1410}
X(7138) = X(73)-ceva conjugate of X(1425)
X(7138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,820,2638), (65,73,2658), (2655,2662,29)
X(7138) = crosssum of X(i) and X(j) for (i,j) = (1,412), (29,1896), (1021,2638)
X(7138) = trilinear square of X(73)
X(7139) lies on the line {4,2908}
X(7139) = isogonal conjugate of X(20926)
X(7140) lies on these lines: {4,3617}, {10,407}, {25,281}, {51,1146}, {92,427}, {125,6354}, {181,4092}, {210,430}, {242,428}, {278,5094}, {318,1904}, {429,3695}, {1351,2994}, {1851,5064}, {1897,4213}
X(7140) = X(1500)-cross conjugate of X(594)
X(7140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,1867,407), (92,427,2969), (1824,1826,430)
X(7140) = cevapoint of X(4092) and X(4705)
X(7140) = crosssum of X(1437) and X(1790)
X(7141) lies on these lines: {429,3695}, {451,6335}, {1068,3963}, {1235,1969}, {1826,4153}
X(7141) = {X(1235),X(1969)}-harmonic conjugate of X(2973)
X(7142) lies on the line {594,3027}
X(7143) lies on these lines: {34,3271}, {55,951}, {65,1439}, {181,1254}, {221,2175}, {1042,1402}, {1106,1357}, {1362,3868}, {1397,1398}, {1401,4320}
X(7143) = {X(1254),X(1425)}-harmonic conjugate of X(181)
X(7143) = crosssum of X(1043) and X(1792)
X(7143) = crosspoint of X(1042) and X(1426)
See ADGEOM #2158, 1/8/2015 by Tran Quang Hung, and related postings.
X(7144) lies on these lines: {12,3690}, {201,3028}, {502,6058}
X(7145) lies on these lines: {87,3271}, {269,3784}, {1958,3033}
X(7145) = {X(1958),X(6467)}-harmonic conjugate of X(3033)
X(7146) lies on these lines: {1,3}, {2,257}, {6,2114}, {7,192}, {37,1423}, {63,2329}, {76,85}, {77,2285}, {109,761}, {222,3497}, {239,4051}, {599,4053}, {604,1442}, {611,6211}, {869,4475}, {984,1469}, {1014,1963}, {1580,4650}, {1930,6358}, {1953,4000}, {1999,3905}, {2003,5280}, {2092,5929}, {2170,5222}, {2294,4648}, {3175,3970}, {3314,3661}, {3752,3959}, {3942,4644}
X(7146) = isogonal conjugate of X(2344)
X(7146) = X(2276)-cross conjugate of X(984)
X(7146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,57,1429), (2,1959,3061), (65,241,57), (3674,3912,226)
X(7146) = cevapoint of X(i) and X(j) for (i,j) = (1469,2276), (3250,4475)
X(7146) = trilinear pole of the line through X(1491) and X(3805)
X(7147) lies on these lines: {41,223}, {181,1254}, {226,857}, {278,2170}, {604,1435}, {1042,1426}, {1400,1427}, {2171,6046}, {4220,5018}
X(7147) = X(6046)-ceva conjugate of X(1254)
X(7147) = crosssum of X(2287) and X(2327)
X(7148) lies on these lines: {8,291}, {10,3728}, {42,2229}, {43,1258}, {76,3123}, {87,1126}, {1500,6378}, {1909,2227}, {2053,6187}, {2085,4692}, {2209,3501}
X(7148) = X(6378)-ceva conjugate of X(756)
X(7148) = X(i)-cross conjugate of X(j) for these (i,j): (594,756), (1084,661)
X(7149) lies on the Feuerbach hyperbola and these lines: {1,196}, {8,1034}, {9,1249}, {84,278}, {1000,1148}, {1838,3062}
X(7149) = X(1034)-ceva conjugate of X(4)
X(7149) = X(1118)-cross conjugate of X(4)
X(7149) = cevapoint of X(3270) and X(6591)
X(7150) lies on these lines: {1,61}, {9,80}, {14,484}, {57,1081}, {559,2003}
X(7150) = X(16)-cross conjugate of X(1)
X(7151) lies on these lines: {6,33}, {19,2255}, {25,604}, {29,81}, {31,607}, {34,1407}, {84,1039}, {280,4195}, {608,1096}, {613,1957}, {1333,1436}, {1397,6059}
X(7151) = X(1413)-ceva conjugate of X(608)
X(7151) = X(i)-cross conjugate of X(j) for these (i,j): (1395,25), (2207,608)
X(7151) = {X(6),X(204)}-harmonic conjugate of X(3195)
X(7152) lies on the cubic K179 and these lines: {6,208}, {19,2192}, {40,219}, {48,221}, {64,1436}, {154,198}, {2193,2360}
X(7152) = X(i)-cross conjugate of X(j) for these (i,j): (608,6), (2155,1436), (2208,56)
X(7152) = X(219)-vertex conjugate of X(219)
X(7152) = {X(610),X(1498)}-harmonic conjugate of X(3197)
X(7153) lies on these lines: {56,87}, {57,239}, {932,1477}, {1015,3500}, {1016,3501}, {1407,1429}
X(7153) = isogonal conjugate of X(3208)
X(7153) = isotomic conjugate of X(4110)
X(7153) = X(i)-cross conjugate of X(j) for these (i,j): (7,57), (2162,87)
X(7153) = {X(56),X(1424)}-harmonic conjugate of X(1423)
X(7154) lies on these lines: {19,56}, {21,268}, {31,607}, {84,3423}, {282,380}, {1398,2170}, {1593,2082}, {1898,1903}, {1973,2208}, {2192,2194}
X(7154) = X(1436)-ceva conjugate of X(25)
X(7154) = X(i)-cross conjugate of X(j) for these (i,j): (1973,607), (6059,25)
X(7154) = {X(19),X(1033)}-harmonic conjugate of X(3209)
X(7155) lies on the Feuerbach hyperbola and these lines: {1,87}, {2,256}, {4,4645}, {7,350}, {8,3056}, {9,2319}, {21,2053}, {31,983}, {80,4680}, {104,932}, {314,3794}, {346,4876}, {941,2276}, {1045,4734}, {1156,4598}, {1403,4203}, {1431,1966}, {2162,2298}, {2268,2344}, {2481,6383}, {3061,4451}, {3271,3596}, {3551,3662}, {3680,3886}, {4073,4518}, {5377,5383}
X(7155) = isogonal conjugate of X(1403)
X(7155) = isotomic conjugate of X(3212)
X(7155) = X(i)-Ceva conjugate of X(j) for these (i,j): (2319,8), (6384,330)
X(7155) = X(i)-cross conjugate of X(j) for these (i,j): (312,8), (2319,330), (3061,2)
X(7155) = perspector of the inconic with center X(3061)
X(7155) = cevapoint of X(i) and X(j) for (i,j) = (8,4903), (11,3810), (522,3271)
X(7155) = trilinear pole of the line through X(650) and X(3907)
X(7156) lies on these lines: {6,19}, {33,41}, {40,1783}, {198,1033}, {204,3172}, {208,910}, {218,1753}, {281,380}, {610,1249}, {1118,1886}, {1200,5338}, {4200,5838}
X(7156) = X(i)-Ceva conjugate of X(j) for these (i,j): (9,33), (1249,204)
X(7156) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (610,1249,3213), (2082,2202,34)
X(7157) lies on these lines: {56,7003}, {1256,1411}
X(7158) lies on the incircle and these lines: {1,3324}, {11,122}, {12,133}, {55,107}, {56,1294}, {1317,2828}, {1354,4304}, {1361,2816}, {1362,2822}, {1364,2846}, {2777,3028}, {2790,3027}, {2797,3023}, {2811,3022}, {2833,3021}, {2839,6018}, {2845,3318}, {2847,6019}, {2848,6020}, {3058,3320}, {4294,5667}, {5432,6716}
X(7158) = reflection of X(3324) in X(1)
X(7158) = foot of the perpendicular to the line X(i)X(j) from X(k) for (i,j,k) = (11,122,1317), (1317,2828,11), (1361,2816,1364), (1362,2822,3022), (1364,2846,1361), (2777,3028,3024), (2790,3027,3023), (2797,3023,3027), (2811,3022,1362), (2833,3021,1358), (2839,6018,1357), (2845,3318,1359), (2847,6019,3325), (2848,6020,3058), (3058,3320,6020)
X(7158) = X(107) of Mandart-incircle triangle
X(7159) lies on the incircle and these lines: {1,3327}, {11,128}, {12,137}, {55,1141}, {56,930}, {495,1263}
X(7159) = reflection of X(3327) in X(1)
X(7159) = foot of the perpendicular to the line X(11)X(128) from X(1317)
X(7160) lies on the Feuerbach hyperbola and these lines: {1,5920}, {2,9874}, {3,7091}, {4,1697}, {7,40}, {8,3305}, {9,3295}, {10,6601}, {21,3870}, {46,5557}, {55,84}, {57,3296}, {79,5119}, {90,3746}, {104,3601}, {517,5665}, {987,3749}, {1476,3576}, {1706,3254}, {2136,6598}, {2257,2335}, {3057,3577}, {3333,3523}, {5128,5551}, {5250,5815}
X(7160) = reflection of X(8000) in X(1)
X(7160) = isogonal conjugate of X(3333)
X(7160) = X(3303)-cross conjugate of X(1)
X(7160) = cevapoint of X(55) and X(2256)
X(7160) = orthologic center of ABC to Hutson-extouch triangle
X(7161) lies on the Feuerbach hyperbola and these lines: {7,498}, {12,79}, {21,3678}, {35,3065}, {40,5561}, {55,3467}, {84,5010}, {100,6595}, {140,3337}, {1320,3884}, {3254,4187}, {5119,5560}, {5251,6596}
X(7161) = isogonal conjugate of X(3337)
X(7161) = X(3746)-cross conjugate of X(1)
X(7162) lies on the Feuerbach hyperbola and these lines: {4,5119}, {7,46}, {9,3746}, {21,3681}, {35,84}, {40,79}, {55,90}, {57,5557}, {80,1697}, {104,3612}, {191,3255}, {631,3296}, {943,1728}, {1320,3890}, {1698,3254}, {1723,2335}, {3065,5531}, {3158,6597}, {3577,5697}, {3584,5709}, {3587,5270}, {3679,6598}, {5084,6601}, {5558,5703}, {5561,5726}, {5665,5903}
X(7162) = isogonal conjugate of X(3338)
X(7162) = X(3295)-cross conjugate of X(1)
X(7163) lies on these lines: {1,378}, {29,1478}, {35,77}, {36,78}, {56,1807}, {73,3422}, {283,4278}, {1067,1479}, {1777,4905}
X(7163) = isogonal conjugate of X(1479)
X(7163) = X(1066)-cross conjugate of X(1)
X(7163) = X(34)-vertex conjugate of X(90)
X(7163) = cevapoint of X(55) and X(2178)
X(7164) lies on the Neuberg cubic and these lines: {1,2132}, {3,5677}, {4,5680}, {30,3464}, {399,3465}, {484,5667}, {1276,5623}, {1277,5624}, {3065,5670}, {3483,5671}, {3484,5685}, {5668,5672}, {5669,5673}
X(7164) = isogonal conjugate of X(3464)
X(7164) = X(74)-cross conjugate of X(1)
X(7164) = antigonal conjugate of X(34299)
X(7165) lies on the Neuberg cubic and these lines: {1,3484}, {3,3460}, {4,3461}, {30,3483}, {484,3482}, {1138,5680}, {1263,3464}, {3065,5667}, {3466,5683}
X(7165) = isogonal conjugate of X(3483)
X(7166) lies on the cubic K128 and these lines: {1,1281}, {32,3502}, {76,3494}, {87,3662}, {292,3229}, {385,3507}, {979,1330}, {1015,3865}, {1423,3506}
X(7166) = isogonal conjugate of X(3507)
X(7166) = cevapoint of X(659) and X(3123)
X(7167) lies on the cubic K128 and these lines: {1,3511}, {32,3503}, {76,3495}, {291,511}, {385,3508}, {1015,1432}, {1423,3402}, {2319,3403}, {3404,3512}, {3405,3509}
X(7167) = isogonal conjugate of X(3508)
X(7167) = reflection of X(1432) in X(1015)
X(7168) lies on the cubic K128 and these lines: {32,6196}, {192,869}, {385,1911}, {904,4366}, {1423,1740}, {2319,3223}, {2664,3229}
X(7168) = isogonal conjugate of X(3510)
X(7168) = X(i)-cross conjugate of X(j) for these (i,j): (350,1), (694,3512)
X(7168) = trilinear pole of the line through X(1107) and X(4083)
X(7169) lies on these lines: {33,2285}, {55,1950}, {63,1619}, {64,1407}, {154,205}, {159,197}, {219,1660}, {268,1661}
X(7169) = isogonal conjugate of X(4329)
X(7169) = X(1973)-cross conjugate of X(6)
X(7169) = X(63)-vertex conjugate of X(63)
X(7169) = X(92)-isoconjugate of X(22119)
X(7169) = {X(63),X(1619)}-harmonic conjugate of X(3556)
X(7169) = cevapoint of X(i) and X(j) for (i,j) = (667,3270)
X(7170) lies on these lines: {1,39}, {99,3571}, {512,5539}, {3903,4128}
X(7170) = crosssum of PU(90)
X(7171) lies on these lines: {1,1406}, {3,9}, {4,3306}, {20,3218}, {30,57}, {40,550}, {46,4316}, {63,376}, {90,3065}, {214,6261}, {381,5437}, {515,3359}, {912,6282}, {944,3895}, {1062,1394}, {1158,4297}, {1709,3576}, {3098,5227}, {3305,3524}, {3333,4312}, {3338,4857}, {3428,5918}, {3534,3928}, {3651,4652}, {3689,5534}, {3784,6000}, {5289,6001}, {5328,6223}
X(7171) = midpoint of X(20)X(5768)
X(7171) = reflection of X(5720) in X(3)
X(7171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,971,5720), (63,376,3587)
X(7172) lies on these lines: {1,2}, {9,5423}, {20,4385}, {22,280}, {25,1261}, {55,346}, {56,1219}, {75,3598}, {210,391}, {312,390}, {321,3198}, {333,5686}, {345,5281}, {452,3701}, {497,4030}, {756,4073}, {1089,4294}, {1215,4307}, {2321,3158}, {2345,5304}, {3091,5015}, {3161,4082}, {3189,3714}, {3474,4454}, {3475,4869}, {3703,5218}, {3713,5276}, {3717,5273}, {3967,5698}, {4220,5687}, {4293,4692}, {4387,4779}, {4451,4704}, {4514,5274}, {4849,5839}, {4901,5745}, {5177,5300}, {5269,5749}
X(7172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55,3974,346), (391,6555,210), (4082,4512,3161)
X(7173) lies on these lines: {1,5}, {2,5217}, {4,5204}, {8,3829}, {35,3628}, {36,546}, {55,3090}, {56,3091}, {65,3817}, {140,3583}, {320,4902}, {381,499}, {388,5068}, {404,6667}, {428,5370}, {442,3825}, {497,5056}, {498,3058}, {547,4995}, {632,5010}, {958,5187}, {999,5072}, {1210,3649}, {1329,3617}, {1478,3851}, {1479,1656}, {1621,6668}, {1699,5128}, {1898,5439}, {2072,5160}, {2098,5818}, {2475,6691}, {2476,3816}, {2886,4193}, {3085,5071}, {3086,3545}, {3295,5079}, {3303,5274}, {3304,3544}, {3526,4302}, {3530,4324}, {3582,5066}, {3585,3850}, {3621,3813}, {3626,3814}, {3634,3925}, {3843,4299}, {3853,4316}, {3854,5265}, {3855,4293}, {3856,4325}, {4294,5067}, {4679,5705}, {4860,5714}, {4999,5046}, {5220,6067}, {5221,5704}, {5435,5556}
X(7173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,3614), (1,3614,12), (2,5225,5217), (5,11,12), (11,12,37722), (11,3614,1), (35,3628,5326), (1479,1656,5432), (5217,5225,6284)
X(7174) lies on these lines: {1,6}, {2,3677}, {8,3672}, {10,4000}, {31,3929}, {38,57}, {40,990}, {43,3789}, {55,3220}, {63,3920}, {65,4328}, {142,4310}, {144,4344}, {145,3883}, {171,3928}, {192,3886}, {200,3666}, {210,2999}, {241,4321}, {256,3680}, {386,4878}, {388,3668}, {390,4907}, {497,4656}, {516,4419}, {519,4356}, {527,4307}, {614,756}, {726,4659}, {846,3749}, {968,3938}, {975,3333}, {976,3601}, {982,5268}, {986,1706}, {988,5293}, {1002,3720}, {1072,5587}, {1423,3340}, {1469,3688}, {1697,2292}, {1699,4415}, {2550,3663}, {3158,3961}, {3306,4392}, {3576,6211}, {3681,5256}, {3729,5263}, {3744,4512}, {3826,4859}, {3873,5287}, {3891,4981}, {4003,4413}, {4026,4929}, {4073,4853}, {4349,4644}, {4643,5846}, {4646,4882}, {4648,5542}, {4854,4863}, {4862,5880}, {5222,5686}
X(7174) = reflection of X(4644) in X(4349)
X(7174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,984,9), (1,1743,1386), (1,3731,1001), (1,3751,1449), (1,5223,6), (1,5234,1104), (2,3677,5574), (8,3672,3755), (10,4353,4000), (37,3242,1), (38,612,57), (63,3920,5269), (982,5268,5437), (988,5293,5438), (1386,5220,1743), (3243,3247,1), (3891,4981,5271), (3938,3989,968), (4349,5850,4644), (4392,5297,3306)
X(7175) lies on these lines: {6,57}, {7,604}, {48,4644}, {56,87}, {65,4649}, {73,3736}, {77,2285}, {86,226}, {109,1918}, {142,5053}, {171,2330}, {193,1958}, {284,4667}, {572,3664}, {603,5156}, {651,1014}, {673,3451}, {894,2329}, {961,1042}, {1001,1420}, {1442,2171}, {1458,2309}, {1460,1740}, {1461,1910}, {2267,4648}, {2268,3945}, {4268,4675}, {4306,5145}
X(7175) = X(961)-ceva conjugate of X(57)
X(7175) = X(172)-cross conjugate of X(171)
X(7175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,604,1429), (56,6180,1423), (193,1958,3684), (651,1014,1400)
X(7175) = crosspoint of X(651) and X(4620)
X(7176) lies on these lines: {1,7}, {56,85}, {57,239}, {65,664}, {98,934}, {194,2128}, {213,651}, {226,6625}, {241,1107}, {273,1398}, {274,961}, {348,388}, {894,2329}, {999,3673}, {1111,5563}, {1231,5323}, {1319,4059}, {1399,1414}, {1476,2481}, {1565,4911}, {1909,4447}, {1975,3685}, {2082,4209}, {2176,6180}, {3476,6604}, {3665,5434}, {4384,5435}
X(7176) = isotomic conjugate of X(4451)
X(7176) = X(i)-cross conjugate of X(j) for these (i,j): (171,894), (4459,4369)
X(7176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,6049,5543), (56,85,1447), (279,3600,7), (664,1434,65), (1323,4298,3674), (3674,4298,7)
X(7176) = cevapoint of X(4369) and X(4459)
X(7176) = trilinear pole of the line through X(3287) and X(4369)
X(7177) lies on these lines: {1,103}, {3,77}, {7,84}, {27,1088}, {40,3160}, {46,1323}, {48,1803}, {56,3423}, {57,279}, {58,269}, {63,348}, {85,658}, {241,5022}, {295,1425}, {967,1427}, {1407,2221}, {1445,4253}, {1461,4251}, {1509,4616}, {1810,4855}, {2285,3497}, {3501,6167}, {3692,3926}, {5256,6611}
X(7177) = isogonal conjugate of X(7079)
X(7177) = isotomic conjugate of X(7101)
X(7177) = X(i)-Ceva conjugate of X(j) for these (i,j): (1088,269), (1434,279)
X(7177) = X(i)-cross conjugate of X(j) for these (i,j): (222,77), (905,934)
X(7177) = {X(57),X(738)}-harmonic conjugate of X(279)
X(7177) = crosssum of X(657) and X(3119)
X(7177) = crossdifference of any pair of centers on the line through X(4105) and X(4171)
X(7178) lies on these lines: {56,4367}, {57,1019}, {65,512}, {108,2722}, {109,2690}, {226,4049}, {241,514}, {513,1835}, {523,656}, {525,1577}, {649,2504}, {661,6587}, {663,676}, {693,3910}, {918,3261}, {1499,4170}, {2099,4879}, {2254,6362}, {2789,4504}, {2826,4905}, {3566,4010}, {3649,4806}, {3665,4444}, {3800,4761}, {3907,4458}, {4449,6366}, {4498,6084}, {4559,4566}, {4784,5221}, {4804,4843}, {4807,4848}, {4885,6332}, {4897,6002}
X(7178) = midpoint of X(i)X(j) for these (i,j): (1577,4707), (2533,3801)
X(7178) = reflection of X(i) in X(j) for these (i,j): (663,676), (3700,1577), (3669,3676), (6332,4885)
X(7178) = isogonal conjugate of X(5546)
X(7178) = isotomic conjugate of X(645)
X(7178) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1365), (57,1086), (85,1358), (273,11), (664,3649), (927,1284), (1020,6354), (3668,3120), (3676,4017), (4077,523), (4552,226), (4566,65), (4573,7)
X(7178) = X(i)-cross conjugate of X(j) for these (i,j): (1365,7), (3120,3668), (3125,65), (3569,876)
X(7178) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (514,3676,3669), (525,1577,3700), (1577,4707,525), (2533,3801,523)
X(7178) = cevapoint of X(661) and X(4017)
X(7178) = crosssum of X(i) and X(j) for (i,j) = (9,1021), (55,3709), (101,906)
X(7178) = crossdifference of any pair of centers on the line through X(55) and X(219)
X(7178) = crosspoint of X(i) and X(j) for (i,j) = (7,4573), (57,1020), (226,4552), (651,2982), (1446,4566)
X(7178) = trilinear pole of the line through X(1365) and X(2611)
X(7178) = polar conjugate of X(36797)
X(7178) = polar conjugate of isotomic conjugate of X(17094)
X(7178) = barycentric product of Kiepert hyperbola intercepts of Gergonne line
X(7179) lies on these lines: {1,147}, {2,7}, {3,4911}, {5,3673}, {10,3212}, {12,85}, {35,4056}, {55,4872}, {75,325}, {77,612}, {183,320}, {269,5268}, {273,427}, {279,5261}, {305,561}, {347,2898}, {348,388}, {495,1565}, {651,5276}, {664,5252}, {986,4920}, {1086,3815}, {1368,6356}, {1442,3920}, {1443,5297}, {1469,3786}, {1478,5088}, {2893,2900}, {3314,3661}, {3485,6604}, {3663,3817}, {3672,5274}, {3740,5224}, {3933,4385}, {4104,5232}, {4328,5272}, {4389,5087}, {5119,5195}, {5275,6180}
X(7179) = X(984)-cross conjugate of X(3661)
X(7179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7,1447), (10,3674,3212), (12,3665,85), (75,325,3705)
X(7180) lies on these lines: {109,2702}, {241,514}, {512,810}, {513,6589}, {647,661}, {649,854}, {663,2499}, {669,1402}, {1357,6377}, {1401,3572}, {1458,5098}, {1499,3931}, {1880,2501}, {2487,3752}, {3666,4897}
X(7180) = isogonal conjugate of X(645)
X(7180) = anticomplement of complementary conjugate of X(17058)
X(7180) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1356), (34,3271), (57,1357), (278,1365), (604,1015), (1042,3122), (1880,3125), (3669,4017), (4017,512), (4551,181), (4552,65), (4559,1400), (4565,56), (4573,1401)
X(7180) = X(i)-cross conjugate of X(j) for these (i,j): (798,512), (1356,7), (2491,875), (3121,1402), (3122,1042)
X(7180) = {X(647),X(661)}-harmonic conjugate of X(3709)
X(7180) = crosssum of X(i) and X(j) for (i,j) = (8,3700), (9,3737), (190,1332), (220,4477), (333,4560), (522,3686), (644,3699), (650,960), (3688,3709)
X(7180) = crossdifference of any pair of centers on the line through X(8) and X(21)
X(7180) = crosspoint of X(i) and X(j) for (i,j) = (56,4565), (57,4551), (65,4552), (649,6591), (651,961), (1400,4559)
X(7180) = trilinear pole of the line through X(1356) and X(3122)
X(7181) lies on these lines: {7,21}, {11,5088}, {36,1565}, {65,5194}, {85,5433}, {109,2729}, {241,514}, {934,2752}, {1055,5845}, {1125,4059}, {1358,1447}, {1366,4831}, {1388,6604}, {1788,3160}, {3712,6390}, {5172,5866}, {6046,6359}
X(7181) = isogonal conjugate of X(5547)
X(7181) = X(7)-ceva conjugate of X(1366)
X(7181) = X(i)-cross conjugate of X(j) for these (i,j): (896,524), (1366,7)
X(7181) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56,348,3665), (1358,5298,1447)
X(7181) = crossdifference of any pair of centers on the line through X(55) and X(3709)
X(7181) = trilinear pole of the line through X(1366) and X(4750)
See ADGEOM #1893, by César Lozada, 10/13/2014.
X(7182) lies on these lines: {7,4388}, {8,479}, {57,85}, {69,1439}, {75,1088}, {77,332}, {273,310}, {304,345}, {305,307}, {312,4554}, {314,5931}, {561,4572}, {664,3996}, {883,3681}, {982,3673}, {5173,6604}
X(7182) = isogonal conjugate of X(2212)
X(7182) = isotomic conjugate of X(33)
X(7182) = X(310)-ceva conjugate of X(6063)
X(7182) = X(i)-cross conjugate of X(j) for these (i,j): (69,304), (77,85), (307,348)
X(7182) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75,1088,6063), (348,1231,304)
X(7182) = cevapoint of X(i) and X(j) for (i,j) = (69,348), (307,1231)
X(7182) = X(1973)-isoconjugate of X(9)
X(7183) lies on these lines: {40,664}, {57,85}, {63,348}, {69,271}, {77,283}, {78,6516}, {84,4872}, {255,6517}, {279,3218}, {326,1259}, {738,3928}, {934,2365}, {1102,3719}, {1394,1414}, {5088,5709}
X(7183) = X(i)-cross conjugate of X(j) for these (i,j): (394,326), (4091,6517)
X(7183) = {X(1102),X(3926)}-harmonic conjugate of X(3719)
X(7183) = cevapoint of X(394) and X(1804)
X(7184) lies on these lines: {1,7}, {2,87}, {43,193}, {69,1740}, {86,741}, {171,2330}, {319,2234}, {320,1964}, {982,3056}, {1045,3879}, {2663,4667}, {2664,4416}, {3009,6646}, {3778,3888}
X(7184) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,1740,3783), (4334,5018,1042)
X(7184) = crosssum of X(43) and X(3961)
X(7185) lies on these lines: {1,7}, {12,85}, {262,1446}, {348,1447}, {496,1565}, {1408,1434}, {3061,3662}, {3212,4848}
X(7185) = X(982)-cross conjugate of X(3662)
X(7185) = {X(1358),X(3665)}-harmonic conjugate of X(85)
X(7186) lies on these lines: {31,2979}, {35,500}, {171,181}, {238,3917}, {750,3060}, {982,3056}, {1216,3073}, {1350,5329}, {1397,3098}, {1943,2606}, {2887,3794}, {3271,3819}, {4324,5697}
X(7186) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31,2979,3792), (3056,3784,982), (3794,3888,2887)
X(7187) lies on these lines: {2,85}, {63,1424}, {75,330}, {194,304}, {384,5088}, {894,2329}, {960,1463}, {1278,4673}, {1565,6656}, {3061,3662}, {3888,4531}, {3934,4403}, {4872,6655}
X(7187) = {X(194),X(304)}-harmonic conjugate of X(3797)
X(7188) lies on these lines: {1,7153}, {171,7176}, {256,7145}
X(7189) lies on these lines: {1,3790}, {3589,3809}
X(7190) lies on these lines: {1,7}, {2,2324}, {33,273}, {37,1445}, {57,1255}, {69,3872}, {75,78}, {85,4360}, {226,3946}, {307,6604}, {612,1447}, {651,1449}, {999,1804}, {1014,1420}, {1062,6356}, {1100,6180}, {1119,6198}, {1418,3723}, {1429,2285}, {1439,3426}, {1441,3870}, {2999,5226}, {3553,4000}, {3554,4644}, {3598,3920}
X(7190) = X(3295)-cross conjugate of X(3305)
X(7190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,77), (1,269,1442), (1,1721,2293), (1,4328,7), (7,1442,269), (37,5228,1445), (269,1442,77)
X(7190) = cevapoint of X(3305) and X(4917)
X(7191) lies on these lines: {1,2}, {6,3726}, {11,5133}, {22,56}, {23,5322}, {25,999}, {31,982}, {33,5274}, {34,3600}, {36,5310}, {37,3108}, {38,238}, {55,4850}, {57,4318}, {58,3953}, {63,3677}, {77,3598}, {81,105}, {100,3744}, {149,3914}, {171,244}, {172,251}, {226,1421}, {229,2363}, {312,3891}, {320,4749}, {390,1040}, {404,5266}, {427,496}, {497,1370}, {595,3670}, {611,5422}, {613,1993}, {748,984}, {758,5315}, {942,4224}, {986,3915}, {988,4189}, {1015,1194}, {1038,5265}, {1058,1062}, {1100,3290}, {1104,2975}, {1180,2275}, {1191,3869}, {1203,3874}, {1255,1390}, {1279,1621}, {1385,4220}, {1428,5012}, {1442,1447}, {1468,3976}, {1469,3060}, {1616,3890}, {1627,1914}, {1920,3112}, {1995,3304}, {2003,5083}, {2191,3945}, {2223,4210}, {2979,3056}, {3242,3681}, {3246,3683}, {3263,4360}, {3306,5269}, {3361,4347}, {3434,4000}, {3554,5304}, {3662,6327}, {3742,3745}, {3751,4430}, {3782,5057}, {3844,4914}, {4003,4640}, {4190,4339}, {4202,5015}, {4223,5045}, {4310,5905}, {4351,5345}, {4359,5263}, {4385,5192}, {4429,5014}, {4514,4972}, {4857,5189}
X(7191) = complement of X(33091)
X(7191) = crosssum of X(i) and X(j) for (i,j) = (1,3961), (42,3954)
X(7191) = crossdifference of any pair of centers on the line through X(649) and X(6004)
X(7191) = foot of the perpendicular to the line X(i)X(j) from X(k) for (i,j,k) = (390,1040,4307), (986,3915,5145)
X(7191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,3920), (1,42,3957), (1,43,3938), (1,614,2), (1,978,976), (1,995,4511), (1,2999,3870), (1,5272,612), (2,3920,5297), (10,5311,498), (31,982,3218), (36,5310,6636), (38,238,3219), (43,3938,3935), (63,3677,4392), (81,3315,354), (354,1386,81), (354,4906,3315), (612,614,5272), (612,5272,2), (1100,3290,5276), (1279,3666,1621), (1386,4906,354), (2999,3870,3240), (3242,4383,3681), (3744,3752,100), (5269,5574,3306)
X(7192) is the perspector, with respect to the extraversion triangle of X(8), of the conic through X(4), X(8), and the extraversions of X(8). This conic is a rectangular hyperbola centered at X(3448). (Randy Hutson, April 11, 2015)
Let A17B17C17 and A18B18C18 be the Gemini triangles 17 and 18, resp. Let E17 and E18 be the {ABC, Gemini 17}-circumconic and {ABC, Gemini 18}-circumconic, resp. Let A' be the intersection of the tangent to E17 at A17 and the tangent to E18 at A18. Define B' and C' cyclically. The lines AA', BB', CC" concur in X(7192). (Randy Hutson, November 30, 2018)
X(7192) lies on the Kiepert hyperbola and these lines: {2,661}, {7,4077}, {8,4160}, {27,2400}, {81,6654}, {86,4833}, {99,901}, {110,927}, {239,514}, {320,350}, {523,4467}, {659,3004}, {660,799}, {669,4367}, {670,889}, {812,4979}, {850,4374}, {875,3112}, {876,3005}, {885,2488}, {1305,6517}, {1414,4566}, {1443,1447}, {1649,6626}, {2398,4236}, {2786,4024}, {3667,5214}, {3700,4789}, {3766,6372}, {3776,4817}, {3835,4379}, {4375,6545}, {4380,4762}, {4382,4785}, {4776,4885}, {5216,6005}
X(7192) = midpoint of X(1019)X(4960)
X(7192) = reflection of X(i) in X(j) for these (i,j): (4560,1019), (4813,3835), (661,4369), (8,4761), (4380,4790), (4467,4897), (649,4932)
X(7192) = isogonal conjugate of X(4557)
X(7192) = isotomic conjugate of X(3952)
X(7192) = complement of X(31290)
X(7192) = anticomplement of X(661)
X(7192) = anticomplementary conjugate of X(21221)
X(7192) = cevapoint of X(i) and X(j) for (i,j) = (513,514), (523,4129), (1019,3737)
X(7192) = crosssum of X(i) and X(j) for (i,j) = (42,512), (213,669), (667,2308), (756,4041), (798,872), (1334,4524)
X(7192) = crossdifference of any pair of centers on the line through X(42) and X(213)
X(7192) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,86), (670,274), (799,2), (1414,7), (1509,1086), (4573,81), (4616,1434), (4633,5333)
X(7192) = X(i)-cross conjugate of X(j) for these (i,j): (244,2), (513,1019), (1086,1509), (1111,7), (3004,693), (3248,330), (3737,4560), (3937,593), (3942,279), (4453,6548), (4965,8), (6371,3669)
X(7192) = crosspoint of X(i) and X(j) for (i,j) = (86,99), (274,670), (668,1268), (757,1414), (799,873), (1434,4616)
X(7192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (514,1019,4560), (514,4932,649), (523,4897,4467), (661,4369,2), (799,4576,3952), (4160,4761,8), (4379,4813,3835), (4762,4790,4380)
X(7192) = perspector of the inconic with center X(244)
X(7192) = trilinear pole of the line through X(812) and X(1015)
X(7192) = perspector of side- and vertex-triangles of Gemini triangles 17 and 18
X(7193) lies on these lines: {1,2175}, {3,48}, {9,182}, {35,3688}, {37,5135}, {58,2300}, {63,184}, {72,1176}, {110,2651}, {172,5156}, {209,5347}, {220,5085}, {222,3167}, {238,1284}, {239,242}, {386,2273}, {394,1473}, {436,1947}, {511,2323}, {517,1618}, {518,692}, {984,2330}, {1040,6056}, {1397,1707}, {1437,1444}, {1459,4091}, {1738,5091}, {1914,5009}, {2174,5132}, {2194,3666}, {3219,5012}, {3292,3937}, {3306,5651}, {3573,3685}, {3690,5314}, {5256,5320}
X(7193) = midpoint of X(2323)X(3220)
X(7193) = X(i)-Ceva conjugate of X(j) for these (i,j): (239,1914), (2196,3955)
X(7193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,219,3781), (63,184,3955), (394,1473,3784), (2323,3220,511)
X(7193) = crosssum of X(1824) and X(5089)
X(7193) = crosspoint of X(1444) and X(1814)
X(7194) lies on these lines: {1,2896}, {6,982}, {34,4334}, {56,5018}, {87,614}, {141,3961}, {1126,3874}, {1401,1431}, {1411,5434}, {2297,5272}
X(7194) = isogonal conjugate of X(3961)
X(7194) = X(172)-cross conjugate of X(57)
X(7194) = cevapoint of X(244) and X(4367)
X(7194) = trilinear pole of the line through X(649) and X(3777)
X(7195) lies on these lines: {2,3665}, {4,1111}, {7,8}, {34,269}, {56,105}, {57,169}, {196,5236}, {348,1447}, {497,3673}, {1086,2097}, {1118,1847}, {1319,3160}, {1323,1420}, {1334,4419}, {1350,4310}, {1565,3086}, {1697,3663}, {1930,3974}, {2082,4000}, {3303,3672}, {3485,3674}, {3664,5716}, {4872,5225}, {4911,5229}
X(7195) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,497), (653,3676)
X(7195) = X(614)-cross conjugate of X(4000)
X(7195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,85,388), (7,3212,6604), (56,1358,279), (279,3598,56)
X(7195) = crosssum of X(55) and X(480)
X(7195) = crosspoint of X(7) and X(479)
X(7196) lies on these lines: {2,85}, {7,350}, {42,664}, {57,6063}, {171,6649}, {226,4554}, {310,349}, {331,1435}, {658,1821}, {1008,1448}, {1323,6685}, {1909,4447}, {1920,4032}, {2003,4573}, {4192,5088}
X(7196) = X(i)-cross conjugate of X(j) for these (i,j): (894,1909), (4369,6649)
X(7196) = trilinear pole of the line through X(3907) and X(4374)
X(7197) lies on these lines: {7,354}, {34,269}, {279,961}, {738,3668}
X(7198) lies on these lines: {7,21}, {11,4911}, {12,1447}, {85,5434}, {172,1086}, {241,553}, {388,3598}, {496,4056}, {1038,4328}, {1319,3674}, {1429,5244}, {1475,5845}, {1565, 4059,4298}, {5221,6604}
X(7198) = {X(7),X(56)}-harmonic conjugate of X(3665)
X(7198) = crosssum of X(55) and X(3688)
X(7199) lies on these lines: {75,523}, {86,1027}, {99,1308}, {110,2860}, {274,1022}, {308,3572}, {312,4789}, {314,5214}, {320,350}, {514,1921}, {522,4406}, {662,4620}, {670,4033}, {798,4369}, {799,3257}, {812,1019}, {1275,4573}, {1577,4960}, {3287,3758}, {3669,4560}, {3709,4687}, {3766,4977}, {4411,4802}
X(7199) = isotomic conjugate of X(1018)
X(7199) = reflection of X(i) in X(j) for these (i,j): (798,4369), (75,4374)
X(7199) = X(i)-Ceva conjugate of X(j) for these (i,j): (670,75), (799,274), (873,1111), (4573,85), (4602,310), (4625,86)
X(7199) = X(i)-cross conjugate of X(j) for these (i,j): (1015,6384), (1086,75), (1111,873), (1565,1088), (3942,757), (4509,3261), (4978,693)
X(7199) = cevapoint of X(i) and X(j) for (i,j) = (514,693), (661,4151)
X(7199) = crosssum of X(i) and X(j) for (i,j) = (213,798), (1500,3709), (1918,1924)
X(7199) = crossdifference of any pair of centers on the line through X(213) and X(872)
X(7199) = crosspoint of X(i) and X(j) for (i,j) = (274,799), (310,4602), (1509,4573)
X(7199) = trilinear pole of the line through X(244) and X(1111)
X(7200) lies on these lines: {1,2795}, {85,2275}, {99,4760}, {244,4124}, {514,3125}, {538,4037}, {804,3023}, {1015,1111}, {1086,1358}, {1914,5088}, {3902,4686}
X(7200) = X(i)-Ceva conjugate of X(j) for these (i,j): (894,4369), (1909,2533), (1920,4374)
X(7200) = {X(1015),X(4403)}-harmonic conjugate of X(1111)
X(7200) = crosssum of X(101) and X(213)
X(7200) = crosspoint of X(i) and X(j) for (i,j) = (274,514), (894,4369), (1920,4374)
X(7201) lies on these lines: {7,192}, {37,57}, {65,984}, {75,226}, {193,4051}, {388,740}, {518,3340}, {536,4654}, {553,4664}, {726,3671}, {894,3061}, {1219,1432}, {1429,2285}, {1788,3842}, {1953,4644}, {2099,6180}, {2294,4419}, {3644,3982}, {3739,5219}, {3911,4687}, {3930,4461}, {3993,4298}, {4699,5226}
X(7201) = {X(7),X(192)}-harmonic conjugate of X(4032)
X(7202) lies on these lines: {1,2836}, {6,2114}, {81,4556}, {513,4516}, {523,4459}, {524,1959}, {526,2611}, {599,3061}, {651,2161}, {1014,2160}, {1086,1358}, {1442,2174}, {1804,2164}, {2486,4977}, {3248,4475}
X(7202) = reflection of X(4053) in X(1959)
X(7202) = X(i)-Ceva conjugate of X(j) for these (i,j): (1014,513), (1442,2605)
X(7202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (524,1959,4053), (2170,3942,1086)
X(7202) = crosssum of X(37) and X(101)
X(7202) = crosspoint of X(81) and X(514)
X(7203) lies on these lines: {1,1499}, {1019,1429}, {1414,4622}, {1443,1447}, {2487,2999}, {4551,4573}
X(7203) = isogonal conjugate of X(4069)
X(7203) = X(i)-Ceva conjugate of X(j) for these (i,j): (1414,1014), (4573,57), (4625,1434), (4637,1412)
X(7203) = X(i)-cross conjugate of X(j) for these (i,j): (1357,57), (1358,269), (3733,1019)
X(7203) = cevapoint of X(513) and X(4498)
X(7203) = crosssum of X(i) and X(j) for (i,j) = (42,4729), (210,4041), (650,3691), (663,3683)
X(7203) = crossdifference of any pair of centers on the line through X(210) and X(1334)
X(7203) = crosspoint of X(i) and X(j) for (i,j) = (552,4573), (1014,1414), (1434,4625)
X(7204) lies on these lines: {6,57}, {7,256}, {75,1088}, {77,171}, {244,3598}, {279,291}, {753,934}, {986,3674}, {1423,3290}, {1804,5329}, {3509,6180}, {3677,4328}
X(7205) lies on these lines: {7,871}, {37,4554}, {75,1088}, {85,1921}, {290,1439}, {1441,4572}, {1442,4625}, {1920,4032}
X(7205) = X(i)-cross conjugate of X(j) for these (i,j): (1909,1920), (2533,4554)
X(7206) lies on these lines: {1,3790}, {10,3995}, {12,1089}, {321,3841}, {346,4294}, {496,3703}, {594,4099}, {2321,3294}, {3454,3994}, {3678,3969}, {3704,3992}, {3932,4647}, {4354,4420}, {4387,4894}
X(7206) = {X(3695),X(6057)}-harmonic conjugate of X(1089)
X(7207) lies on these lines: {31,1414}, {99,4094}, {4128,4367}
X(7208) lies on these lines: {75,3227}, {194,4099}, {244,514}, {330,1930}, {538,4975}, {604,2224}, {1015,1111}, {1086,2087}
X(7209) lies on these lines: {7,350}, {85,1921}, {279,330}, {932,2369}
X(7209) = isotomic conjugate of X(3208)
X(7209) = X(i)-cross conjugate of X(j) for these (i,j): (330,6384), (6063,85)
X(7209) = trilinear pole of the line through X(3676) and X(3766)
X(7210) lies on these lines: {1,4056}, {29,34}, {269,977}, {304,664}, {315,4123}, {348,4296}, {388,1442}, {1394,1414}, {1434,1448}, {4318,6604}
X(7210) = X(22)-cross conjugate of X(1760)
X(7210) = {X(34),X(77)}-harmonic conjugate of X(85)
X(7211) lies on these lines: {12,201}, {65,321}, {171,4459}, {181,6358}, {226,3971}, {388,1265}, {1215,4032}, {1840,2295}, {2171,4037}, {3649,3994}, {3671,4135}, {3681,5252}, {4124,5943}
X(7212) lies on these lines: {1,2788}, {523,656}, {804,3027}, {812,4107}, {1086,1358}, {1284,4455}, {1428,4164}, {1577,4140}, {2799,4707}, {3669,4560}, {4033,4552}
X(7212) = reflection of X(4140) in X(1577)
X(7212) = crossdifference of any pair of centers on the line through X(284) and X(2311)
X(7213) lies on these lines: {1442,3920}, {2003,5280}
X(7213) = isogonal conjugate of X(4149)
X(7213) = X(1397)-cross conjugate of X(57)
X(7214) lies on these lines: {7,1397}, {241,514}
X(7214) = X(2244)-cross conjugate of X(754)
X(7215) lies on these lines: {102,6614}, {934,2818}, {1102,4158}, {1565,3937}
X(7215) = crosspoint of X(1804) and X(4131)
X(7216) lies on these lines: {7,6002}, {65,4729}, {512,4017}, {525,4171}, {649,3669}, {661,6587}, {934,2702}, {3600,4504}, {3671,4170}, {4394,4941}
X(7216) = isogonal conjugate of X(7259)
X(7216) = isotomic conjugate of X(7258)
X(7216) = X(i)-Ceva conjugate of X(j) for these (i,j): (1020,1427), (1119,244), (4566,1254), (4637,269)
X(7216) = X(3125)-cross conjugate of X(1426)
X(7216) = crosssum of X(i) and X(j) for (i,j) = (200,4171), (644,4587), (1021,2287)
X(7216) = crossdifference of any pair of centers on the line through X(200) and X(1253)
X(7216) = crosspoint of X(i) and X(j) for (i,j) = (269,4637), (279,4566), (1020,1427)
X(7217) lies on these lines: {7,1397}, {1365,6063}, {4121,4178}
X(7217) = X(4118)-cross conjugate of X(626)
X(7218) lies on these lines: {1,3598}, {55,2999}, {200,390}, {220,1697}, {1282,4845}
X(7218) = isogonal conjugate of X(4321)
X(7219) lies on these lines: {8,1943 20,346 253,279 280,3600 304,4329 318,377 341,1370 1010,2322 1398,2968}
X(7219) = isogonal conjugate of X(3556)
X(7219) = isotomic conjugate of X(4329)
X(7219) = anticomplement of X(36103)
X(7219) = cyclocevian conjugate of X(2994)
X(7219) = polar conjugate of X(17903)
X(7219) = X(i)-cross conjugate of X(j) for these (i,j): (19,2), (4320,1219)
X(7219) = perspector of the inconic with center X(19)
X(7219) = cevapoint of X(i) and X(j) for (i,j) = (123,522), (513,2968)
X(7219) = trilinear pole of the line through X(2509) and X(3239)
X(7219) = X(19)-isoconjugate of X(22119)
X(7220) lies on these lines: {1,1447}, {33,242}, {43,55}, {87,291}, {103,1742}, {200,3685}, {220,3208}, {390,3783}, {1350,4334}
X(7220) = isogonal conjugate of X(4334)
X(7220) = X(i)-cross conjugate of X(j) for these (i,j): (390,1), (4517,9)
X(7220) = trilinear pole of the line through X(657) and X(4435)
X(7221) lies on these lines: {1,7}, {11,33}, {55,5322}, {56,3516}, {78,3790}, {495,1062}, {612,1040}, {1469,3270}, {3617,4480}
X(7221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,20,4348), (1,990,2263), (1,1721,4318), (1,3100,4319)
X(7222) lies on these lines: {2,4912}, {6,4402}, {7,141}, {37,4454}, {45,4488}, {75,193}, {144,3739}, {145,4686}, {329,5241}, {346,4675}, {391,4688}, {527,966}, {536,3945}, {894,3618}, {1086,5749}, {1100,4452}, {2321,4888}, {3475,4418}, {3664,4659}, {3672,4670}, {3729,4648}, {4346,4657}, {4357,4470}, {4461,4851}, {4748,6646}, {4795,4852}, {4862,5750}, {5222,6329}
X(7222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,4363,2345), (75,193,4371), (75,4644,5839), (193,4371,5839), (4371,4644,193), (4452,4747,1100)
X(7223) lies on these lines: {1,4059}, {7,528}, {12,348}, {55,5088}, {56,85}, {226,1323}, {279,388}, {999,1111}, {1434,3212}, {1478,1565}, {2242,4403}, {3160,3485}, {3304,3673}, {3340,4955}, {3676,4049}, {4032,4559}, {4363,4390}
X(7223) = X(750)-cross conjugate of X(4363)
X(7223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,664,2099), (279,388,3665), (1358,5434,7), (1434,3212,5221)
X(7223) = cevapoint of X(4378) and X(4403)
X(7224) lies on these lines: {63,2896}, {69,3974}, {77,612}, {561,4645}, {1965,4388}
X(7224) = isogonal conjugate of X(23868)
X(7224) = isotomic conjugate of X(4388)
X(7224) = X(i)-cross conjugate of X(j) for these (i,j): (171,2), (4911,7)
X(7224) = {X(1965),X(5207)}-harmonic conjugate of X(4388)
X(7224) = perspector of the inconic with center X(171)
X(7224) = trilinear pole of the line through X(824) and X(905)
X(7224) = X(19)-isoconjugate of X(23150)
X(7225) lies on these lines: {7,604}, {41,4000}, {48,1086}, {57,1255}, {101,4859}, {141,4390}, {572,4862}, {1100,1122}, {1253,5091}, {1284,1471}, {1319,1418}, {1400,5228}, {1404,6180}, {1405,1423}, {2268,3663}, {2280,3946}, {2285,4328}, {3008,3217}, {3684,4402}, {4361,4400}
X(7225) = X(2241)-cross conjugate of X(748)
X(7225) = {X(7),X(1429)}-harmonic conjugate of X(604)
X(7226) lies on these lines: {1,2308}, {2,38}, {8,4424}, {37,3873}, {42,4661}, {45,5284}, {57,5297}, {63,3920}, {75,4981}, {145,2292}, {201,3600}, {210,4850}, {333,3891}, {612,3218}, {846,3938}, {968,3957}, {976,4189}, {986,3617}, {1211,4884}, {1278,3728}, {1621,3242}, {2979,3688}, {3210,4651}, {3240,3666}, {3305,3677}, {3315,4423}, {3434,4419}, {3616,4694}, {3731,4666}, {3740,4003}, {3741,4135}, {3889,6051}, {3953,5550}, {3961,4414}, {4005,4719}, {4134,5313}, {4188,5293}, {4362,5361}, {4389,4972}, {4642,4678}, {4683,4865}, {5223,5256}, {6327,6646}
X(7226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,38,4392), (38,756,982), (38,984,2), (756,982,2), (982,984,756), (3666,3681,3240)
X(7227) lies on these lines: {2,4398}, {6,4399}, {7,141}, {8,3629}, {37,4472}, {44,4967}, {75,3589}, {86,3943}, {190,1213}, {239,6329}, {319,524}, {321,6703}, {346,4470}, {536,4021}, {545,4357}, {597,4361}, {1100,4431}, {1125,4681}, {1757,4733}, {2321,4670}, {2325,4698}, {3008,4739}, {3219,5341}, {3247,4798}, {3578,6539}, {3630,4445}, {3631,3661}, {3729,4364}, {3739,4422}, {3775,5852}, {3946,4726}, {4058,4667}, {4060,4725}, {4488,4748}, {4657,4659}, {4969,5564}, {5252,5835}, {5723,5830}, {6357,6358}
X(7227) = midpoint of X(i)X(j) for these (i,j): (594,894), (1100,4431)
X(7227) = reflection of X(4478) in X(594)
X(7227) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,4665,4399), (37,4472,6707), (75,3589,4395), (524,594,4478), (594,894,524), (1100,4431,4971), (2345,4363,141), (4361,5749,597), (4445,4644,3630)
X(7228) lies on these lines: {1, 28530}, {2, 17255}, {6, 4395}, {7, 141}, {8, 3630}, {9, 34824}, {10, 5852}, {37, 545}, {44, 24199}, {69, 4478}, {75, 524}, {86, 4440}, {142, 4422}, {144, 17259}, {190, 17245}, {192, 17392}, {193, 17119}, {239, 7277}, {320, 594}, {346, 17313}, {513, 17049}, {519, 4726}, {527, 3739}, {536, 3664}, {591, 32793}, {597, 4000}, {894, 1086}, {903, 17302}, {966, 20059}, {1100, 1266}, {1125, 17767}, {1211, 17483}, {1213, 6646}, {1278, 17378}, {1449, 4795}, {1991, 32794}, {2245, 29382}, {2321, 4896}, {3218, 5356}, {3598, 15271}, {3629, 4361}, {3644, 17391}, {3662, 17369}, {3663, 4670}, {3686, 4715}, {3729, 4675}, {3758, 6329}, {3782, 6703}, {3826, 32935}, {3834, 17355}, {3879, 4686}, {3923, 25557}, {3925, 32940}, {3943, 17300}, {3945, 17318}, {3993, 28556}, {4026, 32857}, {4054, 37520}, {4346, 17323}, {4357, 4472}, {4364, 6707}, {4370, 17263}, {4373, 4747}, {4389, 17398}, {4398, 17379}, {4405, 5839}, {4409, 16826}, {4416, 4688}, {4419, 15668}, {4431, 17374}, {4445, 21296}, {4454, 4648}, {4461, 17309}, {4470, 17327}, {4480, 16814}, {4643, 25590}, {4657, 4862}, {4659, 4851}, {4667, 4852}, {4681, 17132}, {4697, 17061}, {4698, 4912}, {4699, 17330}, {4718, 29574}, {4740, 17377}, {4751, 17333}, {4764, 17389}, {4772, 17346}, {4869, 17269}, {4887, 5750}, {4889, 17133}, {4902, 17306}, {4967, 17344}, {4969, 17117}, {5241, 17484}, {5737, 9965}, {5743, 5905}, {5749, 17290}, {5835, 10404}, {5836, 9026}, {6007, 13476}, {6173, 17279}, {6356, 34828}, {7198, 16720}, {7202, 17868}, {7240, 18170}, {7745, 33940}, {9053, 24349}, {9300, 33891}, {10022, 17274}, {11246, 32771}, {15569, 28526}, {16675, 20073}, {16885, 31139}, {17056, 32939}, {17120, 37756}, {17121, 20583}, {17147, 37631}, {17160, 20090}, {17234, 17340}, {17254, 28653}, {17273, 28604}, {17277, 31300}, {17281, 17298}, {17289, 20582}, {17329, 29576}, {17336, 27147}, {17337, 17350}, {17359, 21255}, {17382, 36525}, {17768, 24325}, {19732, 20078}, {25584, 26651}, {26976, 26979}
X(7229) lies on these lines: {1,4461}, {2,2415}, {6,4371}, {7,141}, {8,193}, {10,144}, {37,4470}, {75,3618}, {145,4431}, {190,5296}, {192,3616}, {346,5308}, {391,4967}, {527,5232}, {594,4644}, {966,5936}, {1100,4460}, {1992,5564}, {2321,3945}, {2550,5772}, {2938,4418}, {3617,4416}, {3672,4659}, {3879,4747}, {4007,4667}, {4357,4454}, {4361,6329}, {4665,5839}, {4681,4798}, {4898,4909}
X(7229) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75,3618,4402), (75,5749,5222), (2345,4363,7), (3618,4402,5222), (4402,5749,3618), (4659,5750,3672), (5936,6172,966)
X(7230) lies on these lines: {37,4066}, {39,312}, {115,3695}, {304,4403}, {346,3767}, {594,4075}, {1089,1500}, {2241,4387}, {3797,3934}, {3954,3994}, {4671,5283}
X(7230) = {X(1089),X(4037)}-harmonic conjugate of X(1500)
X(7231) lies on these lines: {7, 141}, {75, 3629}, {319, 3630}, {524, 17118}, {545, 10436}, {594, 17361}, {597, 894}, {1086, 17368}, {1268, 6646}, {4021, 4670}, {4357, 10022}, {4361, 8584}, {4370, 27147}, {4395, 31995}, {4399, 4644}, {4409, 17247}, {4418, 37703}, {4419, 6707}, {4422, 20195}, {4440, 17398}, {4454, 15668}, {4464, 4686}, {4470, 17255}, {4472, 17276}, {4480, 31238}, {4659, 4898}, {4667, 4726}, {4747, 32105}, {4795, 17151}, {4887, 17385}, {4896, 17229}, {4908, 17243}, {4912, 5257}, {5743, 17484}, {6666, 17351}, {7321, 17291}, {16777, 28297}, {17119, 32455}, {17132, 28639}, {17251, 20059}, {17275, 28333}, {17330, 31300}, {17332, 25590}, {17340, 26806}, {25728, 31285}, {26685, 31139}
X(7232) lies on these lines: {6,320}, {7,141}, {8,3631}, {9,3834}, {45,6646}, {69,1086}, {75,599}, {142,4643}, {144,4422}, {346,545}, {524,4000}, {536,4862}, {594,3620}, {894,3763}, {903,1278}, {1001,4655}, {1743,4715}, {2321,4887}, {3242,4645}, {3454,5708}, {3589,4644}, {3629,5222}, {3630,4395}, {3663,4851}, {3664,4657}, {3679,4739}, {3739,6173}, {3759,6144}, {3823,5223}, {3836,5220}, {3973,6687}, {4007,4726}, {4310,5846}, {4357,4675}, {4364,4648}, {4398,6542}, {4419,4869}, {4423,4683}, {4452,4971}, {4659,4902}, {4670,4888}, {4896,5750}, {5249,5737}
X(7232) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,141,4363), (69,1086,4361), (75,599,4445), (320,3662,6), (3630,4395,5839)
X(7233) lies on these lines: {7,192}, {12,85}, {65,664}, {77,1911}, {241,292}, {279,291}, {330,4051}, {653,1880}, {741,1305}, {813,2369}, {1441,4572}, {1581,3668}, {2481,3675}
X(7233) = isotomic conjugate of X(3685)
X(7233) = X(i)-cross conjugate of X(j) for these (i,j): (291,335), (1738,2)
X(7233) = {X(334),X(337)}-harmonic conjugate of X(4518)
X(7233) = perspector of the inconic with center X(1738)
X(7233) = cevapoint of X(i) and X(j) for (i,j) = (57,5018), (65,241), (514,3675)
X(7233) = trilinear pole of the line through X(226) and X(3676)
X(7234) lies on these lines: {42,649}, {100,4589}, {512,810}, {513,3510}, {650,667}, {659,4824}, {661,669}, {798,3221}, {878,1402}, {890,4979}, {2533,3907}, {4036,4874}
X(7234) = isogonal conjugate of X(4594)
X(7234) = X(i)-Ceva conjugate of X(j) for these (i,j): (100,171), (651,213), (932,37)>BR>
X(7234) = {X(661),X(669)}-harmonic conjugate of X(4455)
X(7234) = crosssum of X(i) and X(j) for (i,j) = (86,513), (190,4553)
X(7234) = crossdifference of any pair of centers on the line through X(239) and X(257)
X(7234) = crosspoint of X(42) and X(100)
X(7235) lies on these lines: {7,8}, {12,594}, {56,4361}, {181,6358}, {238,4124}, {239,1428}, {511,1733}, {523,656}, {664,2669}, {740,1284}, {1402,4362}, {1429,4716}, {1756,2783}, {1874,2238}, {2277,5230}, {3027,4037}, {3056,4008}
X(7235) = reflection of X(4459) in X(1733)
X(7235) = crossdifference of any pair of centers on the line through X(284) and X(3063)
X(7236) lies on these lines: {36,984}, {75,4497}, {256,1631}, {751,4471}
X(7236) = X(751)-vertex conjugate of X(751)
X(7237) lies on these lines: {37,1918}, {141,4118}, {181,756}, {594,2643}, {872,4053}, {3721,3778}, {3728,3954}, {4073,4787}
X(7237) = isogonal conjugate of X(7305)
X(7237) = {X(141),X(4118)}-harmonic conjugate of X(4475)
X(7237) = crosssum of X(81) and X(2206)
X(7237) = crosspoint of X(i) and X(j) for (i,j) = (37,313), (2887,3721)
X(7238) lies on these lines: {7,141 69,4399 75,3631 239,320 527,3834 536,4887 545,3912 597,4644 599,4665 903,6542 1266,4971 2325,4912 3008,4715 3361,3617 3589,3662 3629,4000 3630,4361 3823,5850 3836,5852 3943,4440 4357,6707 4364,4675 4643,6173 4657,4888 4670,4896 4693,4966 4851,4862}
X(7238) = midpoint of X(i)X(j) for these (i,j): (320,1086), (3943,4440)
X(7238) = reflection of X(i) in X(j) for these (i,j): (4395,1086), (4422,3834)
X(7238) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75,3631,4478), (320,1086,524), (524,1086,4395), (527,3834,4422)
X(7239) lies on these lines: {100,825}, {101,833}, {661,3952}, {982,3094}, {1018,4551}, {4481,4576}
X(7239) = X(5388)-ceva conjugate of X(984)
X(7239) = crosssum of X(513) and X(1919)
X(7239) = crosspoint of X(100) and X(1978)
X(7239) = trilinear pole of the line through X(3721) and X(3778)
X(7240) lies on these lines: {7,87}, {193,4489}, {1045,4667}, {1054,2347}, {1740,4644}, {3598,5272}
X(7241) lies on these lines: {75,4484}, {239,749}, {256,4363}, {291,4361}, {751,894}, {984,1698}, {1469,3214}, {2276,3720}, {3736,3913}, {3778,4492}
X(7241) = isogonal conjugate of X(17127)
X(7241) = X(i)-cross conjugate of X(j) for these (i,j): (1574,2), (4111,37)
X(7241) = perspector of the inconic with center X(1574)
X(7241) = cevapoint of X(244) and X(4041)
X(7241) = crosssum of X(4170) and X(4965)
X(7241) = trilinear pole of the line through X(3250) and X(4813)
X(7242) lies on these lines: {76,2085}, {1909,3116}, {4429,4446}
X(7242) = {X(76),X(2085)}-harmonic conjugate of X(4493)
X(7243) lies on these lines: {57,6063}, {85,4102}, {551,5686}, {1233,3729}, {3760,4387}
X(7243) = X(4361)-cross conjugate of X(3760)
X(7244) lies on these lines: {2,4495}, {171,1920}, {238,561}, {982,3403}, {3760,4387}
X(7244) = {X(1920),X(1966)}-harmonic conjugate of X(171)
X(7245) lies on these lines: {2,292}, {291,3679}, {295,544}, {335,536}, {538,3864}, {599,3862}, {660,4715}, {1581,4496}, {3761,4403}
X(7246) lies on these lines: {75,4471}, {291,1631}, {749,4497}, {984,3746}
X(7246) = X(749)-vertex conjugate of X(749)
X(7247) lies on these lines: {1,4056}, {7,8}, {12,1447}, {27,1803}, {83,226}, {150,942}, {239,5244}, {315,4514}, {348,3600}, {664,3674}, {1010,1434}, {1088,1370}, {1111,5270}, {1220,4357}, {1414,2363}, {1478,3673}, {3146,3672}, {3598,5261}, {3665,5434}
X(7247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4911,4872), (7,388,85)
X(7248) lies on these lines: {2,1463}, {7,350}, {31,56}, {43,57}, {65,145}, {222,1428}, {226,4871}, {263,1427}, {269,1431}, {354,4307}, {604,5332}, {614,3937}, {982,3056}, {1122,3598}, {1403,1458}, {1699,4014}, {3271,5574}, {3819,4517}
X(7248) = X(4569)-ceva conjugate of X(3669)
X(7248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,1401,1469), (982,3784,3056), (1357,1401,57)
X(7248) = crosssum of X(i) and X(j) for (i,j) = (55,4513), (200,3208)
X(7248) = crossdifference of any pair of centers on the line through X(3239) and X(4435)
X(7249) lies on these lines: {1,5999}, {2,257}, {7,256}, {75,325}, {85,6384}, {86,1431}, {226,335}, {272,1178}, {614,904}, {673,893}, {871,6063}, {1240,1441}, {1934,4518}, {3674,3865}
X(7249) = isogonal conjugate of X(2330)
X(7249) = isotomic conjugate of X(7081)
X(7249) = X(i)-cross conjugate of X(j) for these (i,j): (256,257), (2530,651), (3674,7), (4142,653), (4459,514)
X(7249) = cevapoint of X(i) and X(j) for (i,j) = (11,3004), (256,1432), (514,4459)
X(7250) lies on these lines: {65,4139}, {512,4017}, {513,676}, {656,4524}, {667,6363}, {934,2703}, {3669,6371}
X(7250) = isogonal conjugate of X(7256)
X(7250) = reflection of X(4524) in X(656)
X(7250) = X(i)-Ceva conjugate of X(j) for these (i,j): (1106,1357), (1435,1015), (4566,1427)
X(7250) = crosssum of X(i) and X(j) for (i,j) = (200,1021), (3699,4571), (3900,3965), (4578,6558)
X(7250) = crossdifference of any pair of centers on the line through X(220) and X(346)
X(7250) = crosspoint of X(i) and X(j) for (i,j) = (269,1020), (1427,4566)
X(7251) lies on these lines: {3,1415}, {56,608}, {348,4565}, {2172,4548}
X(7251) = X(7)-ceva conjugate of X(1397)
X(7252) lies on these lines: {6,661}, {81,6654}, {110,919}, {112,2720}, {163,1625}, {512,5006}, {513,1430}, {521,650}, {647,2605}, {649,834}, {652,663}, {654,4282}, {884,2194}, {940,4369}, {1019,1429}, {1333,2423}, {1980,2978}, {2148,2623}, {3287,3700}, {3288,4879}, {3904,3910}, {4761,5711}, {4840,4979}, {5546,5548}
X(7252) = isogonal conjugate of X(4552)
X(7252) = X(i)-Ceva conjugate of X(j) for these (i,j): (60,3271), (110,2194), (112,1333), (163,6), (643,55), (645,21), (1019,3733), (4565,58), (4573,5324), (4612,4267), (5546,284)
X(7252) = X(i)-cross conjugate of X(j) for these (i,j): (663,3737), (2170,6), (3271,60)
X(7252) = X(2149)-vertex conjugate of X(2149)
X(7252) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (163,1625,4559), (1021,3737,650)
X(7252) = cevapoint of X(663) and X(3063)
X(7252) = crosssum of X(i) and X(j) for (i,j) = (10,3700), (37,523), (514,3666), (520,828), (525,1214), (553,3669), (650,950), (661,4642), (1018,4551), (1577,6358), (2171,4017), (3676,4059)
X(7252) = crossdifference of any pair of centers on the line through X(10) and X(12)
X(7252) = crosspoint of X(i) and X(j) for (i,j) = (21,645), (58,4565), (81,110), (101,2298), (107,829), (109,2982), (112,1172), (163,2150), (284,5546), (643,2185), (651,951), (1019,3737)
X(7252) = X(92)-isoconjugate of X(23067)
X(7253) lies on the parabola inscribed in ABC that has focus X(107). (Randy Hutson, April 15, 2015)
X(7253) lies on the Kiepert parabola and these lines: {2,656}, {8,4086}, {29,2399}, {86,2400}, {99,4620}, {107,6081}, {110,1309}, {314,885}, {320,350}, {346,4171}, {447,525}, {512,4581}, {514,5214}, {521,1948}, {522,663}, {523,4833}, {643,765}, {648,677}, {657,1021}, {811,4566}, {900,3733}, {1019,3667}, {1577,6003}, {2406,4238}, {3265,4467}, {3287,3700}, {3738,4985}, {3900,4397}
X(7253) = isotomic conjugate of X(4566)
X(7253) = reflection of X(i) in X(j) for these (i,j): (4560,3737), (8,4086)
X(7253) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,333), (643,8), (645,2287), (648,2322), (811,2)
X(7253) = X(i)-cross conjugate of X(j) for these (i,j): (1021,4560), (2310,346), (3900,1021), (4990,3239)
X(7253) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (522,3737,4560), (4171,4529,346)
X(7253) = anticomplement of X(656)
X(7253) = cevapoint of X(i) and X(j) for (i,j) = (521,522), (3239,3900)
X(7253) = crosssum of X(i) and X(j) for (i,j) = (42,647), (512,1400), (1254,4017)
X(7253) = crossdifference of any pair of centers on the line through X(213) and X(1042)
X(7253) = crosspoint of X(i) and X(j) for (i,j) = (86,648), (99,333), (314,645), (643,1098)
X(7253) = trilinear pole of the line through X(1146) and X(2968)
X(7253) = polar conjugate of isogonal conjugate of X(23090)
X(7254) lies on these lines: {3,810}, {58,2424}, {81,2401}, {905,4131}, {940,1577}, {1019,1429}, {1414,1625}, {1459,4091}, {3733,5009}, {3907,5711}, {4574,4592}
X(7254) = X(i)-Ceva conjugate of X(j) for these (i,j): (1509,1565), (4558,1790), (4563,1444), (4592,3)
X(7254) = X(3942)-cross conjugate of X(222)
X(7254) = crosssum of X(i) and X(j) for (i,j) = (37,3700), (523,3914), (756,4171), (1824,2489), (1826,2501), (1839,6591)
X(7254) = crossdifference of any pair of centers on the line through X(210) and X(430)
X(7254) = crosspoint of X(i) and X(j) for (i,j) = (81,4565), (757,4637), (1332,1796), (1444,4563), (1790,4558)
X(7255) lies on these lines: {824,4560}, {3113,4367}
X(7255) = cevapoint of X(514) and X(667)
X(7256) lies on these lines: {99,100}, {643,645}, {646,4571}, {670,6606}, {677,4563}, {931,4557}, {3733,6079}
X(7256) = isogonal conjugate of X(7250)
X(7256) = X(i)-cross conjugate of X(j) for these (i,j): (341,4076), (3692,1016), (4477,200)
X(7256) = {X(643),X(3699)}-harmonic conjugate of X(645)
X(7256) = cevapoint of X(i) and X(j) for (i,j) = (200,1021), (3699,4571), (3900,3965), (4578,6558)
X(7256) = trilinear pole of the line through X(220) and X(346)
X(7257) lies on these lines: {86,1120}, {99,100}, {274,1280}, {314,1320}, {519,5209}, {643,4631}, {644,645}, {662,4033}, {664,670}, {811,1897}, {874,3903}, {4087,4511}, {4110,4390}, {4592,4600}, {4614,4623}, {6002,6010}
X(7257) = isotomic conjugate of X(4017)
X(7257) = X(i)-Ceva conjugate of X(j) for these (i,j): (670,799), (4631,645)
X(7257) = X(i)-cross conjugate of X(j) for these (i,j): (332,4600), (645,799), (3699,645), (3737,333), (3907,8), (3996,4076), (4086,312)
X(7257) = {X(99),X(668)}-harmonic conjugate of X(799)
X(7257) = cevapoint of X(i) and X(j) for (i,j) = (1,6002), (8,4560), (200,4529), (312,4086), (333,3737), (522,3687), (646,3699), (663,3691), (668,4561), (1193,4498), (3702,4391)
X(7257) = trilinear pole of the line through X(9) and X(312)
X(7258) lies on these lines: {99,6574}, {190,670}, {644,645}
X(7258) = isotomic conjugate of X(7216)
X(7258) = X(i)-cross conjugate of X(j) for these (i,j): (1021,1043), (4529,346)
X(7258) = cevapoint of X(1021) and X(1043)
X(7258) = trilinear pole of the line through X(200) and X(341)
X(7259) lies on these lines: {99,101}, {110,6574}, {163,1023}, {220,1098}, {341,1802}, {643,644}, {765,4574}, {798,6010}, {1043,6559}, {3699,4587}, {4563,4637}, {4567,4592}
X(7259) = isogonal conjugate of X(7216)
X(7259) = X(i)-Ceva conjugate of X(j) for these (i,j): (645,643), (4567,1792)
X(7259) = X(i)-cross conjugate of X(j) for these (i,j): (1021,2287), (1260,765), (4171,200)
X(7259) = {X(644),X(5546)}-harmonic conjugate of X(643)
X(7259) = cevapoint of X(i) and X(j) for (i,j) = (200,4171), (644,4587), (1021,2287)
X(7259) = trilinear pole of the line through X(200) and X(1253)
X(7260) lies on these lines: {257,274}, {668,4705}, {799,2396}, {874,3903}, {4603,4631}
X(7260) = X(i)-cross conjugate of X(j) for these (i,j): (514,257), (522,310), (3835,86)
X(7260) = cevapoint of X(i) and X(j) for (i,j) = (274,514), (668,4568)
X(7260) = trilinear pole of the line through X(256) and X(314)
X(7261) lies on the Feuerbach hyperbola, the cubic K251 and these lines: {1,147}, {2,2112}, {4,3212}, {7,4459}, {9,1654}, {21,6626}, {79,1111}, {80,5195}, {256,2310}, {294,857}, {334,1966}, {561,4388}, {651,2298}, {885,4010}, {1172,2905}, {1959,3930}, {2896,3495}
X(7261) = isotomic conjugate of X(4645)
X(7261) = X(i)-cross conjugate of X(j) for these (i,j): (238,2), (4872,7)
X(7261) = {X(1966),X(5207)}-harmonic conjugate of X(4645)
X(7261) = perspector of the inconic with center X(238)
X(7261) = cevapoint of X(i) and X(j) for (i,j) = (11,812), (740,1211), (4155,6627)
X(7261) = trilinear pole of the line through X(650) and X(824)
X(7261) = X(19)-isoconjugate of X(20741)
X(7262) lies on these lines: {1,3683}, {2,896}, {3,5529}, {6,846}, {9,171}, {31,984}, {43,44}, {45,1961}, {55,1757}, {63,238}, {165,2348}, {190,4362}, {191,986}, {192,3791}, {210,3550}, {212,1776}, {333,3923}, {386,3647}, {405,1046}, {582,3652}, {748,3218}, {902,3681}, {920,3074}, {968,4649}, {978,3916}, {1580,3061}, {2941,5540}, {3052,3961}, {3210,4974}, {3579,6048}, {3626,6681}, {3722,4661}, {3725,5145}, {3741,4676}, {3749,5223}, {3750,3751}, {3759,4970}, {3769,3971}, {3840,4759}, {3928,5272}, {3979,4428}, {4388,4438}, {4418,5278}, {4421,5524}, {4643,4797}
X(7262) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,896,4650), (9,1707,171), (31,3219,984), (44,4640,43), (63,238,982), (191,1724,986), (3052,5220,3961), (3683,4641,1), (3751,4512,3750)
X(7263) lies on these lines: {2,4398}, {6,4395}, {7,524}, {8,3631}, {9,545}, {10,4739}, {37,1266}, {69,4399}, {75,141}, {142,536}, {239,3629}, {320,3630}, {597,894}, {599,4478}, {726,3826}, {903,6646}, {966,4346}, {1213,4389}, {1278,3943}, {2321,3834}, {3218,5341}, {3589,4000}, {3663,3739}, {3664,4852}, {3686,4887}, {3729,4422}, {3782,4359}, {3820,4013}, {3875,4675}, {3912,4686}, {3925,4884}, {3946,4670}, {4052,4748}, {4357,4688}, {4373,4419}, {4402,4644}, {4452,4648}, {4472,4657}, {4499,5750}, {4643,4862}, {4659,4859}, {4772,5224}, {4851,4971}, {5222,6329}, {5846,5880}
X(7263) = midpoint of X(7)X(4361)
X(7263) = complement of X(17262)
X(7263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,4361,524), (75,141,4665), (75,1086,141), (75,3662,594), (594,1086,3662), (594,3662,141), (3663,3739,4364), (3782,4359,5743), (3834,4726,2321), (4000,4363,3589), (4389,4699,1213)
X(7264) lies on these lines: {1,85}, {7,79}, {35,1447}, {69,4894}, {75,1089}, {76,4692}, {273,4328}, {304,4975}, {307,1210}, {331,1784}, {350,1930}, {496,3665}, {942,4955}, {1441,4021}, {2085,3123}, {3085,3672}, {3212,5697}, {3583,4911}, {3598,4294}, {4044,4359}, {4059,5045}, {4441,4647}, {4857,4872}, {4986,6376}, {5088,5563}
X(7264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3673,1111), (7,1479,4056), (75,3760,1089)
X(7265) lies on these lines: {37,905}, {72,3900}, {190,4567}, {226,2394}, {321,4391}, {512,4122}, {514,4024}, {522,3465}, {523,4170}, {525,1577}, {690,2533}, {824,4079}, {826,4010}, {918,4978}, {1019,2786}, {1734,4522}, {2799,4140}, {3566,4761}, {3762,3910}, {3801,3906}, {4088,4151}, {4120,4129}, {4824,6367}
X(7265) = reflection of X(i) in X(j) for these (i,j): (4707,1577), (1577,3700), (1734,4522)
X(7265) = X(i)-Ceva conjugate of X(j) for these (i,j): (190,3219), (664,10)
X(7265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (525,1577,4707), (525,3700,1577)
X(7265) = crosssum of X(649) and X(1333)
X(7265) = crossdifference of any pair of centers on the line through X(2194) and X(2260)
X(7265) = crosspoint of X(i) and X(j) for (i,j) = (190,321), (4552,6539)
X(7266) lies on these lines: {1,1414}, {99,4736}, {1019,3708}, {1565,4897}
X(7267) lies on these lines: {56,4361}, {99,4037}, {172,894}, {187,4760}, {292,4589}, {742,1055}, {1279,3616}, {2275,3759}, {2533,3907}, {4393,4850}, {4396,5088}
X(7268) lies on these lines: {1,7}, {145,1434}, {956,1014}
Let A' be the isogonal conjugate of A with respect ot the incentral triangle, and define B' and C' cyclically. The triangle A'B'C' is perspective to the incentral triangle at X(3746) and to the intouch triangle at X(7269). (Randy Hutson, April 11, 2015)
X(7269) lies on these lines: {1,7}, {69,4861}, {75,4511}, {241,3723}, {273,6198}, {651,1100}, {934,5049}, {936,5936}, {1014,1319}, {1429,2171}, {1441,4360}, {1445,3247}, {1447,3920}, {1804,3304}, {1963,4565}, {3553,5222}, {3870,4460}, {5226,5256}, {5287,5435}
X(7269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,1442), (1,4328,77), (7,1442,1443), (77,4328,7)
X(7270) lies on these lines: {1,977}, {2,1104}, {4,312}, {7,8}, {10,58}, {20,345}, {27,306}, {29,4150}, {30,3695}, {34,78}, {40,3719}, {56,3705}, {72,1330}, {100,1792}, {101,4153}, {145,5014}, {190,3710}, {286,313}, {304,315}, {321,2475}, {329,1265}, {341,1370}, {344,452}, {346,3146}, {519,5100}, {664,5930}, {744,2292}, {936,5233}, {950,3912}, {960,4388}, {1089,3585}, {1329,5205}, {1478,4385}, {1791,4220}, {1834,1999}, {1930,4911}, {2329,4071}, {2975,3006}, {3152,3998}, {3434,4673}, {3509,4136}, {3666,4201}, {3685,6284}, {3701,5080}, {3769,5230}, {3797,6655}, {3869,6327}, {3933,5088}, {3974,5229}, {4202,5262}, {4358,5046}, {4692,5270}, {4857,4975}, {4886,5814}
X(7270) = reflection of X(5247) in X(10)
X(7270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4680,5015), (1,5015,4514), (8,377,75), (8,4645,65), (304,315,4872), (3416,5794,8)
X(7270) = anticomplement of X(1104)
X(7271) lies on these lines: {1,7}, {9,1418}, {56,5666}, {57,1122}, {69,4882}, {75,4915}, {223,553}, {238,3361}, {241,3731}, {948,4859}, {1419,5228}, {1445,3973}, {1449,6610}, {3339,3751}, {3598,5272}
X(7271) = X(3304)-cross conjugate of X(5437)
X(7271) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,77,4328), (7,269,1), (7,279,3663), (7,3668,4862), (57,6180,1743), (77,4328,1), (269,4328,77), (481,482,962), (2263,4321,1)
X(7271) = crossdifference of any pair of centers on the line through X(657) and X(4162)
X(7272) lies on these lines: {1,4056}, {7,80}, {69,4692}, {75,4680}, {85,5270}, {315,4894}, {320,3761}, {348,4317}, {1565,5434}, {3585,3673}
X(7272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4911,4056), (7,150,5902), (7,1478,1111)
X(7272) = crosssum of X(55) and X(4471)
X(7273) lies on these lines: {1,1427}, {8,279}, {34,2212}, {40,1448}, {56,5574}, {57,961}, {65,269}, {388,3668}, {1042,3340}, {1394,1399}, {1407,3339}, {1697,2263}, {2647,5436}, {3146,4907}, {3600,3677}, {4296,5269}, {5290,6354}
X(7273) = {X(1254),X(4320)}-harmonic conjugate of X(57)
X(7274) lies on these lines: {1,7}, {57,3731}, {69,4915}, {75,4882}, {223,3982}, {226,4859}, {354,4907}, {984,3339}, {1122,3340}, {1418,3247}, {1738,5290}, {1743,5228}, {2324,6173}, {3598,5268}, {3870,4373}
X(7274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,4328,1), (7,4909,4308)
X(7275) lies on these lines: {8,291}, {10,6384}, {87,2334}
X(7276) lies on these lines: {7,3952}, {57,1215}, {65,1089}, {181,6358}, {226,756}, {388,3678}, {872,4032}, {3671,4075}, {3842,5219}, {4096,4654}, {4965,5640}
X(7277) lies on these lines: {1,5852}, {6,7}, {8,6144}, {37,4667}, {44,3664}, {45,3945}, {58,5719}, {75,3629}, {81,4415}, {141,3758}, {193,4363}, {319,524}, {320,3589}, {527,1100}, {536,4464}, {545,4360}, {597,3662}, {599,5749}, {966,4747}, {1213,4416}, {1268,1654}, {1743,4675}, {1992,4361}, {2003,6354}, {3244,4718}, {3284,6356}, {3623,3752}, {3630,3661}, {3739,4796}, {3879,3943}, {3925,4722}, {4357,4715}, {4393,4409}, {4431,4725}, {4480,4681}, {4672,4966}, {5252,5849}, {5733,5779}
X(7277) = reflection of X(594) in X(894)
X(7277) = barycentric product of PU(51)
X(7277) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (75,3629,4969), (524,894,594), (4416,4670,1213)
X(7278) lies on these lines: {1,85}, {7,5697}, {75,3633}, {76,4975}, {79,5195}, {274,4714}, {304,4692}, {388,4056}, {484,1434}, {517,4955}, {538,4099}, {1089,1909}, {3746,5088}, {4872,5270}
X(7278) = trilinear product of PU(51)
X(7279) lies on these lines: {3,7}, {35,1442}, {75,4996}, {77,5010}, {273,3520}, {651,1030}, {934,5951}, {1014,5172}, {1804,5217}, {2197,4565}
X(7280) lies on these lines: {1,3}, {2,3585}, {4,4316}, {9,5124}, {10,4188}, {11,550}, {12,549}, {16,2307}, {20,499}, {21,3624}, {22,5272}, {30,5433}, {33,3520}, {34,186}, {39,609}, {41,5030}, {43,4210}, {58,5313}, {73,3431}, {78,4996}, {79,5428}, {80,5442}, {84,3467}, {90,3065}, {99,3760}, {100,3632}, {172,574}, {187,2275}, {191,997}, {198,3973}, {202,5352}, {203,5351}, {214,3869}, {355,5445}, {376,1479}, {388,3524}, {404,993}, {474,5251}, {496,5298}, {497,3528}, {498,3523}, {548,6284}, {572,1405}, {573,1404}, {602,2964}, {603,6149}, {614,6636}, {631,1478}, {632,3614}, {956,4668}, {978,4225}, {1030,1449}, {1054,6187}, {1055,3730}, {1078,3761}, {1124,6409}, {1125,1770}, {1126,1468}, {1151,3299}, {1152,3301}, {1193,4257}, {1203,4252}, {1335,6410}, {1376,5258}, {1428,3098}, {1469,5092}, {1475,4262}, {1699,4333}, {1737,4297}, {1745,6127}, {1768,6261}, {1836,5443}, {1914,5206}, {1995,5370}, {2067,6396}, {2178,3731}, {2975,3679}, {3053,5299}, {3085,4317}, {3086,3522}, {3207,5526}, {3218,3901}, {3525,5229}, {3530,5432}, {3586,3651}, {3751,5096}, {3754,3897}, {3868,4973}, {3899,4881}, {3916,5692}, {4193,6681}, {4294,5265}, {4305,5435}, {4677,5288}, {5013,5280}, {5248,5253}, {5268,5322}, {5427,5441}, {5440,5904}, {5450,5691}, {6200,6502}
X(7280) = isogonal conjugate of X(5560)
X(7280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,5010), (1,5131,46), (2,4299,3585), (3,36,1), (3,56,35), (3,999,5217), (3,5204,36), (20,499,3583), (35,36,56), (35,56,1), (46,3576,1), (55,5563,1), (57,3612,1), (376,1479,4324), (404,993,1698), (404,5303,993), (498,4293,5270), (997,4652,191), (999,3746,1), (999,5217,3746), (1155,1385,5903), (1319,3579,5697), (1319,5697,1), (1385,5903,1), (1420,5119,1), (1468,4256,5312), (2163,5312,1468), (2646,5902,1), (3086,3522,4302), (3086,4302,4857), (3338,3601,1), (3523,4293,498), (3582,4324,1479), (5288,5687,4677)
X(7280) = crossdifference of any pair of centers on the line through X(650) and X(4838)
X(7281) lies on these lines: {1,147}, {33,43}, {55,846}, {101,5293}, {200,3790}, {1503,5018}, {1682,3022}, {2328,3786}, {3100,3783}, {5160,5524}
X(7281) = isogonal conjugate of X(5018)
X(7281) = X(3100)-cross conjugate of X(1)
X(7281) = cevapoint of X(2310) and X(4435)
X(7281) = trilinear pole of the line through X(657) and X(2269)
X(7282) lies on these lines: {4,7}, {27,226}, {29,307}, {30,6356}, {33,77}, {57,469}, {69,318}, {75,317}, {86,3559}, {92,1947}, {108,1014}, {144,281}, {150,1905}, {264,320}, {269,4056}, {270,1935}, {297,894}, {319,340}, {347,3146}, {393,4644}, {427,1447}, {445,3219}, {458,3662}, {500,1442}, {651,1172}, {653,1826}, {1214,3151}, {1441,2475}, {1785,3664}, {1839,5236}, {1897,3879}, {2322,4416}, {2822,3668}, {3087,4000}, {3212,5090}, {5342,6604}
X(7282) = isogonal conjugate of X(8606)
X(7282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7,273), (75,317,5081)
X(7282) = cevapoint of X(i) and X(j) for (i,j) = (1,1786), (226,1770), (2475,5905)
X(7283) lies on these lines: {1,87}, {3,312}, {4,345}, {8,90}, {10,846}, {12,3712}, {20,346}, {21,321}, {29,3998}, {30,3695}, {35,1089}, {37,1010}, {55,4385}, {56,4387}, {58,1999}, {72,190}, {75,405}, {78,3362}, {100,3701}, {172,4037}, {239,1724}, {242,3596}, {304,1975}, {306,1330}, {318,1013}, {333,5295}, {341,5687}, {344,443}, {350,1009}, {384,3797}, {404,4358}, {528,5100}, {536,1104}, {740,5247}, {956,4673}, {958,5695}, {978,4011}, {1008,2276}, {1219,4779}, {1220,3931}, {1248,3682}, {1447,3760}, {1453,3875}, {1479,3705}, {1621,4968}, {1770,4645}, {1791,4221}, {1935,1943}, {1940,1947}, {2944,4297}, {2975,3702}, {3175,4234}, {3263,4223}, {3696,5302}, {3703,5015}, {3714,4640}, {3746,4692}, {3757,5248}, {3790,4302}, {3871,4696}, {3912,4292}, {3913,4737}, {3933,4872}, {3952,4420}, {3971,5293}, {4153,5134}, {4189,4671}, {4296,4552}, {4359,5047}, {4400,4760}, {4647,5251}, {4659,5436}, {4850,5192}, {4918,5724}, {4975,5563}
X(7283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (58,2901,1999), (190,1043,72), (192,4195,1), (304,1975,5088), (3703,6284,5015)
X(7283) = cevapoint of X(192) and X(3151)
X(7283) = anticevian isogonal conjugate of X(10)
X(7284) lies on the Feuerbach hyperbola and these lines: {1,1406}, {4,3338}, {8,46}, {9,36}, {40,5559}, {56,90}, {57,80}, {79,3333}, {84,5563}, {104,1709}, {376,1000}, {943,3612}, {946,5553}, {1320,3873}, {1478,3306}, {1479,5555}, {1937,4334}, {2093,4900}, {3254,4312}, {3337,5560}, {3577,5902}, {3680,5903}, {4325,5709}
X(7284) = isogonal conjugate of X(5119)
X(7284) = X(999)-cross conjugate of X(1)
X(7284) = X(i)-vertex conjugate of X(j) for these (i,j) or (J,I): (1,3433), (2320,3418)
X(7284) = cevapoint of X(999) and X(1481)
X(7285) lies on the Feuerbach hyperbola and these lines: {8,3929}, {9,5217}, {57,5556}, {1000,4314}, {3146,5128}, {3333,5551}, {3928,5225}, {4866,5687}
X(7285) = isogonal conjugate of X(5128)
X(7285) = X(5204)-cross conjugate of X(1)
Let A'B'C' be the reflection of the Mandart-incircle triangle in X(1); then X(7286) = X(23) of A'B'C'. (Randy Hutson, April 11, 2015)
X(7286) lies on these lines: {1,30}, {11,4351}, {12,858}, {23,56}, {34,468}, {186,5204}, {388,5189}, {511,3028}, {1038,5159}, {1317,4318}, {1325,5221}, {1358,1443}, {1411,5018}, {2071,5217}, {2072,3614}, {3153,5229}, {3325,5194}
X(7286) = inverse in the incircle of X(5434)
X(7286) = reflection of X(i) in X(j) for these (i,j): (5160,1), (6023,5194)
X(7287) lies on these lines: {1,295}, {512,1018}
X(7288) lies on these lines: {1,631}, {2,12}, {3,496}, {4,36}, {5,4293}, {7,5550}, {8,1319}, {10,1420}, {11,20}, {21,1470}, {34,5272}, {35,1058}, {46,5603}, {55,3523}, {57,1125}, {65,3616}, {69,1428}, {72,3660}, {73,978}, {108,475}, {140,999}, {145,1388}, {201,982}, {226,3361}, {238,603}, {278,1940}, {348,1447}, {350,6337}, {354,5703}, {355,5126}, {376,1479}, {390,5217}, {404,2550}, {443,3841}, {452,3816}, {468,1398}, {474,1617}, {495,3526}, {498,1056}, {549,3295}, {551,3340}, {602,3075}, {604,966}, {614,1038}, {748,1106}, {899,4322}, {938,2646}, {944,1737}, {946,3474}, {960,5744}, {962,1155}, {993,5084}, {1001,1466}, {1210,3486}, {1317,3621}, {1421,4347}, {1451,5712}, {1467,5745}, {1469,3618}, {1471,4648}, {1478,3090}, {1708,3338}, {1837,5704}, {1870,3147}, {2067,3069}, {2078,5082}, {2099,3622}, {2192,6696}, {2475,5427}, {3068,6502}, {3189,4855}, {3303,5281}, {3304,5432}, {3333,3475}, {3421,5193}, {3434,4188}, {3488,3612}, {3522,5274}, {3528,4302}, {3529,3583}, {3545,3585}, {3598,3665}, {3634,4315}, {3855,4325}, {4295,5886}, {4298,5219}, {4301,5128}, {4305,5722}, {4308,5252}, {4311,5587}, {4317,5067}, {4321,6666}, {4652,5698}, {4847,5438}, {5083,5904}, {5234,5316}, {5442,5697}
X(7288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,631,5218), (1,3911,1788), (2,56,388), (2,2975,2551), (2,3600,12), (2,5265,56), (3,496,4294), (3,3086,497), (5,4293,5229), (10,1420,3476), (11,20,5225), (11,5204,20), (12,56,3600), (12,3600,388), (36,499,4), (56,5298,5265), (56,5433,2), (57,1125,3485), (140,999,3085), (496,4294,497), (498,5563,1056), (748,1106,1935), (958,6691,2), (1056,3525,498), (1058,3524,35), (1210,3576,3486), (3086,4294,496), (3361,3624,226), (3522,5274,6284), (3616,5435,65), (5261,5434,388), (5265,5433,388), (5298,5433,56), (5704,5731,1837)
X(7289) lies on these lines: {1,159}, {6,57}, {7,19}, {9,141}, {40,518}, {46,3293}, {48,77}, {63,69}, {81,1474}, {84,1503}, {105,2191}, {142,169}, {193,3218}, {198,241}, {320,1760}, {511,5709}, {513,2961}, {524,3928}, {527,1766}, {599,3929}, {614,5324}, {651,2261}, {988,4267}, {1040,1473}, {1122,2264}, {1364,3056}, {1386,3333}, {1429,3554}, {1437,3338}, {1444,4288}, {1445,2183}, {1610,4320}, {1630,4341}, {1633,4319}, {1697,3242}, {1742,2876}, {1768,5848}, {1781,4888}, {2082,4000}, {2182,6180}, {2262,5228}, {2285,4644}, {2391,3946}, {2810,3359}, {2836,3576}, {3098,3587}, {3219,3620}, {3305,3619}, {3306,3618}, {3589,5437}, {3601,4265}, {3937,6467}, {4292,5800}, {4859,5540}
X(7289) = X(i)-Ceva conjugate of X(j) for these (i,j): (100,905), (3673,614)
X(7289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (48,3942,77), (63,69,5227)
X(7289) = crosssum of X(37) and X(2333)
X(7289) = crossdifference of any pair of centers on the line through X(2509) and X(3900)
X(7289) = X(6)-of-tangential-triangle-of-excentral-triangle
X(7289) = X(157)-of-excentral triangle
X(7290) lies on these lines: {1,6}, {2,3883}, {31,57}, {34,2212}, {38,3929}, {40,595}, {43,3158}, {55,2999}, {56,269}, {58,2191}, {63,3677}, {81,4666}, {84,3073}, {142,4307}, {145,3717}, {165,3052}, {171,5272}, {193,4684}, {200,3744}, {204,278}, {221,1467}, {223,1617}, {239,3886}, {390,3755}, {516,4000}, {519,4901}, {527,4310}, {612,748}, {936,5266}, {946,3332}, {975,3646}, {978,5438}, {982,1707}, {983,3680}, {991,995}, {1086,4312}, {1125,4349}, {1193,2293}, {1201,1419}, {1253,1697}, {1282,5332}, {1418,3361}, {1428,2175}, {1445,4318}, {1469,3271}, {1621,5256}, {1699,3772}, {1706,1722}, {2550,3008}, {3011,5219}, {3100,4907}, {3305,3920}, {3340,3924}, {3616,3945}, {3663,5698}, {3666,4512}, {3685,3875}, {3729,4676}, {3745,4423}, {3923,4659}, {4008,4858}, {4223,4264}, {4257,5144}, {4321,6180}, {4353,4419}, {4384,5263}, {4644,5542}, {4859,5880}, {5250,5262}, {5284,5287}
X(7290) = midpoint of X(1)X(1743)
X(7290) = crosssum of X(i) and X(j) for (i,j) = (1,5223), (3729,4384)
X(7290) = crossdifference of any pair of centers on the line through X(513) and X(4130)
X(7290) = crosspoint of X(3598) and X(5222)
X(7290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,238,9), (1,3751,3243), (1,5223,3242), (6,1279,1), (31,614,57), (43,3749,3158), (44,3242,5223), (56,1456,269), (57,614,5574), (171,5272,5437), (390,5222,3755), (982,1707,3928), (1001,1386,1), (1104,1191,1), (1125,4349,4648), (1386,3246,1001), (1419,1420,1458), (1471,2263,57), (1722,5255,1706), (3052,3752,165), (3744,4383,200), (3755,4989,5222)
X(7291) lies on these lines: {2,169}, {7,19}, {8,20}, {9,5232}, {28,60}, {48,1442}, {57,279}, {69,1760}, {77,610}, {85,379}, {144,1766}, {222,607}, {239,514}, {241,294}, {651,2182}, {857,4872}, {1071,6197}, {1282,2340}, {1375,1565}, {1429,2170}, {1436,1804}, {1443,2173}, {1445,2270}, {1448,3339}, {1726,3219}, {1753,6223}, {1781,3664}, {1814,3827}, {1959,4511}, {2210,4475}, {2939,4303}, {3008,5540}, {3100,3220}, {3188,3212}
X(7291) = reflection of X(i) in X(j) for these (i,j): (651,2182), (3100,3220)
X(7291) = X(4872)-ceva conjugate of X(3100)
X(7291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,2082,5222), (69,1760,5279)
X(7291) = cevapoint of X(910) and X(3827)
X(7291) = crosssum of X(i) and X(j) for (i,j) = (9,3509), (37,910), (518,4640)
X(7291) = crossdifference of any pair of centers on the line through X(42) and X(4105)
X(7291) = crosspoint of X(57) and X(3512)
X(7292) lies on these lines: {1,2}, {9,4392}, {11,858}, {22,5204}, {23,36}, {34,4232}, {44,3290}, {56,1995}, {63,5574}, {81,3742}, {88,105}, {100,1279}, {110,1428}, {149,1738}, {210,4906}, {229,4228}, {238,244}, {350,3266}, {354,4663}, {468,1870}, {518,3315}, {651,3660}, {748,982}, {902,1054}, {1001,4850}, {1015,3291}, {1027,6548}, {1040,5274}, {1086,5057}, {1104,4239}, {1370,5225}, {1421,3911}, {1443,1447}, {1469,5640}, {1621,3752}, {3305,3677}, {3583,5189}, {3666,5284}, {3745,3848}, {3873,4383}, {4442,4956}, {4702,4706}, {5094,6198}, {5299,5354}, {5315,5883}
X(7292) = isogonal conjugate of X(34893)
X(7292) = X(5380)-ceva conjugate of X(513)
X(7292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,5297), (1,899,3935), (1,5297,3920), (2,5211,3006), (238,244,3218), (614,5272,2), (748,982,3219), (1421,3911,4318), (3011,5121,2)
X(7292) = crosssum of X(1) and X(5524)
X(7292) = crossdifference of any pair of centers on the line through X(649) and X(1334)
X(7292) = perspector, wrt Gemini triangle 29, of {ABC, Gemini 29}-circumconic
X(7293) lies on these lines: {2,3220}, {3,63}, {22,57}, {25,3306}, {31,36}, {35,38}, {100,1261}, {142,4228}, {171,5322}, {184,3784}, {222,3796}, {404,5294}, {672,4210}, {982,5310}, {1486,4666}, {1790,4575}, {1995,5437}, {2003,5012}, {2221,4252}, {2323,2979}, {3218,5285}, {3666,4265}, {3937,3955}, {4224,5249}, {4641,5096}
X(7293) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,63,5314), (3,1473,63), (3218,6636,5285)
X(7294) lies on these lines: {1,632}, {2,12}, {10,1317}, {11,35}, {21,6667}, {36,3614}, {55,3525}, {57,5506}, {65,3833}, {172,3055}, {442,6681}, {496,4995}, {499,3295}, {547,3585}, {631,6284}, {1213,1404}, {1319,3634}, {1478,5070}, {1479,5054}, {2099,5550}, {2275,3054}, {3086,3533}, {3090,5204}, {3340,3624}, {3530,3583}, {3621,4035}, {3649,3911}, {3850,4316}, {4195,6049}, {4299,5055}, {5445,5901}
X(7294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,632,5326), (2,5253,6668), (2,5433,12), (12,5433,5298), (36,3628,3614), (56,5261,5434), (499,3526,5432)
X(7295) lies on these lines: {1,159}, {3,238}, {9,35}, {22,31}, {24,601}, {25,171}, {55,846}, {56,5018}, {63,5310}, {182,3271}, {197,3550}, {256,3145}, {511,2175}, {750,1995}, {982,1473}, {991,2195}, {993,3883}, {1001,4265}, {1259,4073}, {1350,3792}, {1631,3286}, {1707,5285}, {1780,4269}, {2076,2176}, {2305,2915}, {4357,5248}, {4471,5132}
X(7295) = {X(22),X(31)}-harmonic conjugate of X(5329)
X(7296) lies on these lines: {1,5007}, {6,41}, {12,5306}, {31,4517}, {32,35}, {39,609}, {44,3876}, {386,2251}, {894,4372}, {986,2243}, {1038,3284}, {1100,3889}, {1126,4251}, {1478,5319}, {1500,5008}, {1914,3295}, {2242,5299}, {2273,4264}, {3585,5309}, {4426,5260}, {5261,5304}
X(7296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5007,5332), (6,172,2275), (32,5280,2276)
X(7296) = crossdifference of any pair of centers on the line through X(522) and X(4810)
X(7297) lies on these lines: {1,4289}, {6,19}, {9,5560}, {37,3746}, {44,3245}, {45,169}, {46,5043}, {50,1951}, {910,2078}, {1100,1781}, {1731,2245}, {1760,4361}, {1953,2174}, {2160,2260}, {2161,2183}, {2170,2173}, {2210,2643}, {2503,5057}, {3218,4395}, {3219,4665}, {3684,4053}, {4727,5525}
X(7297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,19,5341), (6,5341,5356), (1731,5011,2245)
X(7297) = crosssum of X(i) and X(j) for (i,j) = (9,484), (3218,3219)
X(7297) = crosspoint of X(i) and X(j) for (i,j) = (57,3065), (2160,2161)
X(7298) lies on these lines: {1,22}, {2,3583}, {3,5272}, {23,612}, {25,35}, {614,6636}, {1370,4324}, {1707,5285}, {1799,3760}, {3920,4354}, {5020,5217}, {6284,6676}
X(7298) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,22,5345), (22,5310,1), (25,35,5268)
X(7299) lies on these lines: {1,195}, {2,1399}, {4,2361}, {5,47}, {11,255}, {12,31}, {55,3073}, {56,87}, {65,1724}, {90,1062}, {212,6284}, {580,1836}, {582,1770}, {595,5252}, {601,5432}, {603,748}, {896,1393}, {1106,5298}, {1155,1777}, {1411,3869}, {1451,3649}, {1454,1707}, {2003,5259}, {2099,5247}, {2218,3271}, {2594,5248}, {4426,4559}
X(7299) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,47,5348), (238,1935,56), (603,748,5433), (3073,3074,55)
X(7300) lies on these lines: {6,19}, {9,4668}, {37,3196}, {583,5011}, {1475,2160}, {1723,5036}, {1731,4271}, {1951,2965}, {2161,2347}, {2170,2174}, {3219,4399}, {4969,5279}
X(7300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,19,5356), (19,5356,5341)
X(7300) = crosssum of X(9) and X(3336)
X(7300) = crosspoint of X(57) and X(3467)
X(7301) lies on these lines: {1,2836}, {3,238}, {9,3467}, {23,31}, {35,4471}, {171,1995}, {575,3271}, {576,2175}, {984,3746}, {1486,4649}, {2212,3518}, {3220,5563}
X(7301) = {X(23),X(31)}-harmonic conjugate of X(5363)
X(7302) lies on these lines: {1,22}, {23,35}, {25,5217}, {428,3614}, {614,5204}, {858,4324}, {896,5285}, {1995,5010}
X(7302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,22,5370), (1,5370,5322), (22,5310,5322), (5310,5370,1)
X(7303) lies on these lines: {42,4600}, {81,893}, {261,1178}, {1412,1432}
X(7303) = {X(81),X(893)}-harmonic conjugate of X(4603)
X(7304) lies on these lines: {31,99}, {57,552}, {81,239}, {86,3741}, {171,2669}, {261,1178}, {593,4610}, {2668,4649}
X(7304) = isogonal conjugate of X(6378)
X(7304) = X(757)-ceva conjugate of X(1509)
X(7304) = {X(81),X(873)}-harmonic conjugate of X(1509)
X(7304) = crosssum of X(1084) and X(4079)
X(7305) lies on the line {310,2206}
X(7305) = isogonal conjugate of X(7237)
X(7305) = cevapoint of X(81) and X(2206)
X(7307) lies on the line {310,2206}
X(7307) = isogonal conjugate of X(21815)
X(7307) = cevapoint of X(81) and X(310)
Let A'B'C' be as defined at X(5658). A'B'C' is homothetic to ABC with center of homothety X(7308). (Randy Hutson, April 11, 2015)
X(7308) lies on these lines: {1,210}, {2,7}, {3,1750}, {5,40}, {10,497}, {21,5438}, {37,2999}, {38,5574}, {45,3752}, {56,5234}, {78,5047}, {84,631}, {165,3683}, {200,1001}, {223,1212}, {238,5268}, {312,728}, {344,3687}, {354,3715}, {381,3587}, {405,936}, {612,748}, {614,756}, {899,968}, {940,1743}, {950,5129}, {958,1420}, {960,3340}, {975,1453}, {984,3677}, {1125,3475}, {1213,2270}, {1376,4512}, {1385,5780}, {1449,5287}, {1621,3158}, {1656,5709}, {1706,5250}, {1723,5718}, {1728,6675}, {1817,4877}, {1995,5314}, {2136,3617}, {3062,5918}, {3243,3681}, {3247,3930}, {3303,3983}, {3333,3624}, {3576,5251}, {3589,5227}, {3634,5128}, {3666,3731}, {3679,4863}, {3680,3890}, {3691,5308}, {3698,5806}, {3711,3748}, {3742,5220}, {3781,5943}, {3782,4859}, {3816,5231}, {3870,5284}, {3973,4641}, {4000,4656}, {4187,5705}, {4358,5271}, {4359,4659}, {4915,5919}, {5020,5285}, {5732,5927}
X(7308) = X(3303)-cross conjugate of X(4328)
X(7308) = complement of X(9776)
X(7308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9,57), (2,63,5437), (2,329,142), (2,3219,3306), (2,3305,9), (2,3452,5219), (2,5273,3911), (2,5744,6692), (9,57,3929), (9,3928,3219), (9,5437,63), (63,5437,57), (78,5047,5436), (142,329,4654), (210,4423,1), (238,5268,5269), (354,3715,5223), (405,936,3601), (984,5272,3677), (1001,3740,200), (1698,1699,3925), (3219,3306,3928), (3303,3983,4882), (3306,3928,57), (3452,6666,2), (3681,4666,3243), (3683,4413,165), (3925,4679,1699), (5325,6692,5744)
X(7308) = crossdifference of any pair of centers on the line through X(663) and X(4790)
X(7309) lies on these lines: {3273,3603}, {3277,3607}
X(7309) = cevapoint of X(3277) and X(3604)
X(7310) lies on the line {1134,5632}
X(7310) = isogonal conjugate of X(6124)
X(7312) lies on these lines: {1,2796}, {6,1054}, {524,5524}, {740,1120}, {1357,1431}, {1411,5018}
X(7312) = isogonal conjugate of X(5524)
X(7313) lies on these lines: {1,2836}, {2,5540}, {291,484}, {524,5525}, {758,1280}, {1219,4294}, {1390,2809}
X(7313) = isogonal conjugate of X(5525)
In a variation of the Lozada perspector, let A'B'C' be the cevian triangle of a point P. Let L be the line (other than BC) through A' tangent to the A-excircle, and let A* be the touchpoint; define B* and C* cyclically. The triangle A*B*C* is perspective to ABC for all P. 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. (See ADGEOM #1560, #1572, #1574, September 1-2, 2014); if P = X(12) then P* = X(7314). (Randy Hutson, April 11, 1015)
X(7314) lies on these lines: {1091,6058}, {1254,1365}
X(7315) lies on these lines: {1,2142}, {512,5539}, {1500,3571}
X(7315) = isogonal conjugate of X(5539)
X(7315) = X(99)-cross conjugate of X(1)
X(7316) lies on these lines: {7,1365}, {56,4565}, {65,651}, {109,111}, {671,6648}, {923,1042}
X(7316) = isogonal conjugate of X(3712)
X(7316) = X(923)-cross conjugate of X(111)
X(7316) = {X(895),X(897)}-harmonic conjugate of X(5547)
X(7317) lies on the Feuerbach hyperbola and these lines: {1,3525}, {9,3625}, {21,3621}, {65,5551}, {104,5217}, {517,5556}, {1056,5557}, {1058,5559}, {1320,3617}, {2320,5775}, {3626,3680}, {4816,4866}, {5225,5560}, {5229,5561}
X(7318) lies on these lines: {2,914}, {7,90}, {11,1804}, {77,499}, {269,3582}, {673,2164}, {1069,5738}, {1440,1443}
X(7318) = isotomic conjugate of X(5552)
X(7318) = X(i)-cross conjugate of X(j) for these (i,j): (77,7), (90,2994), (499,2)
X(7318) = perspector of the inconic with center X(499)
X(7318) = cevapoint of X(11) and X(905)
Let DEF be the extouch triangle of ABC. Let D' be the point, other than A, in which the line AD meets the A-excircle, and define E' and F' cyclically. Let La be the line tangent to the A-excircle at D', and define Lb and Lc cyclically. Let A' = LbLc, and define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(7319). (Angel Montesdeoca, July 14, 2018)
X(7319) lies on the Feuerbach hyperbola and these lines: {1,3091}, {7,1837}, {8,3967}, {9,3617}, {11,6049}, {21,1376}, {65,5556}, {80,962}, {104,3149}, {329,6598}, {355,1000}, {388,5558}, {938,3296}, {942,5551}, {1478,5557}, {1479,5559}, {2320,5550}, {2475,3255}, {3146,5128}, {3436,6601}, {3486,3614}, {3621,3680}, {3625,4900}, {3626,4866}, {3749,4355}, {3832,4323}, {4295,5561}, {4313,5587}, {4345,5881}, {5435,5691}, {5553,6223}
X(7319) = isogonal conjugate of X(5204)
X(7319) = X(1743)-cross conjugate of X(2)
X(7319) = {X(3832),X(5727)}-harmonic conjugate of X(4323)
X(7319) = perspector of the inconic with center X(1743)
X(7319) = cevapoint of X(i) and X(j) for (i,j) = (1,5128), (11,3667)
X(7319) = X(19)-isoconjugate of X(23140)
X(7320): Let A1B1C1, A7B7C7, A8B8C8 be the cevian triangles of X(1), X(7), X(8). Let A' = {A1, A8}-harmonic conjugate of A7, and define B' and C' cyclically. Then A'B'C' is the cevian triangle of X(7320). (Randy Hutson, April 11, 2015)
Let A"B"C" be the intouch triangle. Let HA be the hyperbola with foci A and A'' that passes through B and C. Let LA be the line through the intersections of HA with CA and AB (other than B and C). Define LB and LC cyclically. Let A* = LB∩\LC, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(7320). (Randy Hutson, April 11, 2015)
X(7320) lies on the Feuerbach hyperbola and these lines: {1,3523}, {2,3680}, {7,3057}, {8,3740}, {9,145}, {10,4900}, {21,3241}, {55,1476}, {65,5558}, {79,962}, {84,4313}, {104,3295}, {388,5556}, {390,3062}, {517,3296}, {519,4866}, {938,1000}, {986,4691}, {1156,3486}, {1172,4248}, {1222,3161}, {1320,3035}, {1389,5703}, {1697,3522}, {2403,4778}, {3577,4345}, {3600,5493}, {3622,6692}, {3884,5815}, {4051,5296}, {4323,5665}, {4342,5261}, {4857,5560}, {5270,5561}, {5557,5697}
X(7320) = isogonal conjugate of X(3304)
X(7320) = X(3731)-cross conjugate of X(2)
X(7320) = perspector of the inconic with center X(3731)
X(7320) = cevapoint of X(1) and X(1697)
X(7320) = trilinear pole of the line through X(650) and X(3667)
X(7321) lies on these lines: {1,4398}, {2,4912}, {7,8}, {37,4440}, {86,99}, {142,190}, {192,4675}, {239,3629}, {314,5557}, {316,4911}, {326,4328}, {344,4454}, {894,1086}, {1266,3635}, {1267,3593}, {1278,4851}, {1654,4688}, {1992,4402}, {2481,3255}, {3595,5391}, {3624,4389}, {3634,4357}, {3662,3763}, {3687,3982}, {3729,6173}, {3739,6646}, {3758,4000}, {3759,4644}, {3875,4888}, {3879,4896}, {3945,4373}, {3976,5559}, {3977,5249}, {4361,6144}, {4417,4654}, {4419,4687}, {4480,6666}, {4643,4699}, {4648,4664}, {4686,6542}, {4741,4772}, {4745,4967}, {4902,5224}
X(7321) = isotomic conjugate of X(5559)
X(7321) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,75,320), (69,75,5564), (69,5564,319), (75,320,319), (86,903,3663), (320,5564,69), (1266,3664,4360)
X(7322) lies on these lines: {1,210}, {2,3677}, {9,31}, {10,3974}, {37,200}, {38,5437}, {42,3247}, {45,4512}, {55,3731}, {57,984}, {63,5297}, {171,3929}, {181,3340}, {750,3928}, {940,5223}, {968,3158}, {976,5436}, {1449,5311}, {1706,2292}, {1743,3715}, {1961,3751}, {2345,4082}, {2550,4656}, {2999,3740}, {3243,3720}, {3305,3920}, {3601,5293}, {3681,5287}, {4682,5220}, {6051,6765}
X(7322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,612,5269), (612,756,9), (984,5268,57), (3715,3745,1743)
X(7323) lies on these lines: {1,4006}, {9,3913}, {41,200}, {210,728}, {220,5573}, {1500,3731}, {2321,2551}, {3061,4915}, {3501,5223}, {3620,5575}
X(7324) lies on these lines: {181,7144}, {756,7066 }
X(7325) lies on the Neuberg cubic and these lines: {1,5624}, {13,5677}, {16,3464}, {30,5673}, {399,1276}, {484,5668}, {617,5672}, {1277,5675}, {5670,7326}
X(7325) = isogonal conjugate of X(5673)
X(7326) lies on the Neuberg cubic and these lines: {1,5623}, {14,5677}, {15,3464}, {30,5672}, {399,1277}, {484,5669}, {616,5673}, {1276,5674}, {5670,7325}
X(7326) = isogonal conjugate of X(5672)
X(7327) lies on the Neuberg cubic and these lines: {1,5670}, {30,5677}, {399,3464}, {484,2132}, {1157,5680}, {3465,5671}, {3466,5676}, {5623,5673}, {5624,5672}, {5667,5685}
X(7327) = isogonal conjugate of X(5677)
X(7327) = X(74)-cross conjugate of X(484)
X(7328) lies on the Neuberg cubic and these lines: {30,5680}, {2132,3465}, {3065,5676}, {3464,5667}, {3483,5670}, {3484,5677}, {5683,5685}
X(7328) = isogonal conjugate of X(5680)
X(7328) = X(74)-cross conjugate of X(3465)
X(7329) lies on the Neuberg cubic and these lines: {1,5671}, {4,5677}, {30,5685}, {399,484}, {1157,3464}, {3465,5684}, {3466,5670}, {3481,5680}
X(7329) = isogonal conjugate of X(5685)
X(7330) lies on these lines: {1,90}, {2,5811}, {3,9}, {4,63}, {5,57}, {10,1158}, {20,3219}, {30,40}, {33,255}, {46,3585}, {55,5534}, {72,1012}, {119,1698}, {144,5758}, {381,3928}, {388,1776}, {405,1071}, {443,2096}, {452,5768}, {511,5227}, {515,5837}, {517,3927}, {527,946}, {580,990}, {601,612}, {631,3305}, {920,1478}, {944,5250}, {952,1697}, {958,6001}, {993,6261}, {997,5450}, {1040,3074}, {1060,1394}, {1093,1947}, {1210,5770}, {1445,5817}, {1482,6762}, {1496,2310}, {1656,3824}, {1699,6763}, {1706,5790}, {1707,3072}, {1708,4292}, {1711,5247}, {1723,5398}, {1763,5810}, {2095,5806}, {2801,5248}, {2829,5794}, {3065,5531}, {3090,3306}, {3091,3218}, {3149,3916}, {3333,5843}, {3338,3582}, {3452,6705}, {3577,6597}, {3612,6326}, {4330,5119}, {4641,5706}, {5223,6769}, {5273,6223}, {5325,6684}, {5745,6260}, {5791,6259}
X(7330) = reflection of X(1) in X(3560)
X(7330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5777,5720), (3,5779,5777), (4,63,5709), (9,84,3), (10,1158,3359), (191,5691,40), (912,3560,1), (3916,5927,3149)
X(7330) = 2nd-extouch-to-excentral similarity image of X(3)
X(7330) = center of circle that is the locus of crosssums of Bevan circle antipodes
X(7331) lies on the line {581,1442}
X(7331) = isotomic conjugate of X(5906)
X(7331) = X(255)-cross conjugate of X(2)
X(7331) = perspector of the inconic with center X(255)
X(7331) = foot of the perpendicular from X(2307) to the line X(581)X(1442)
Let A'B'C' be the Feuerbach triangle. 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. Triangle A"B"C" is homothetic to ABC, and the center of homothety is X(7332). (Randy Hutson, April 11, 2015)
X(7332) lies on these lines: {11,5952}, {662,5949}, {2486,5954}, {3254,4187}
X(7332) = cevapoint of X(115) and X(2611)
X(7333) lies on the incircle and these lines: {1,6022}, {56,729}, {1469,3027}
X(7333) = reflection of X(6022) in X(1)
X(7333) = incircle transform of X(2234)
X(7333) = foot of the perpendicular from X(3023) to the line X(1469)X(3027)
X(7334) lies on the incircle and these lines: {1,6026}, {56,689}, {1356,4032}
X(7334) = reflection of X(6026) in X(1)
X(7334) = Brisse transform of X(719)
X(7335) lies on these lines: {3,1364}, {55,947}, {56,58}, {57,3468}, {154,1413}, {184,603}, {221,1361}, {255,1092}, {348,1367}, {577,4100}, {578,3075}, {604,2288}, {944,2720}, {1038,3955}, {1071,1319}, {1399,2175}
X(7335) = X(1437)-ceva conjugate of X(603)
X(7335) = {X(255),X(1092)}-harmonic conjugate of X(6056)
X(7335) = crosssum of X(281) and X(1857)
X(7335) = crosspoint of X(222) and X(1804)
X(7335) = excircles variation of Lozada perspector of X(3); see X(7314)
X(7336) lies on these lines: {11,522}, {105,3322}, {244,1365}, {513,1086}, {517,1738}, {726,3814}, {1090,5532}, {1266,5087}, {1357,2969}, {1411,3319}, {1618,2175}, {2170,3328}, {3120,3259}, {3717,5123}, {4086,4092}, {5048,5853}
X(7336) = reflection of X(3717) in X(5123)
X(7336) = X(i)-Ceva conjugate of X(j) for these (i,j): (1358,6545), (2969,764)
X(7336) = crosspoint of X(1358) and X(6545)
X(7336) = foot of the perpendicular to the line X(i)X(j) from X(k) for (i,j,k) = (513,1086,1738), (517,1738,1086)
X(7336) = excircles variation of Lozada perspector of X(11); see X(7314)
X(7337) lies on these lines: {12,406}, {19,1460}, {25,1096}, {28,56}, {108,6353}, {181,607}, {608,1397}, {961,4198}, {1430,1473}, {1973,2207}, {2175,3195}
X(7337) = isogonal conjugate of X(1264)
X(7337) = X(1118)-ceva conjugate of X(608)
X(7337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,1096,6059), (25,3209,1402)
X(7337) = excircles variation of Lozada perspector of X(19); see X(7314)
X(7338) lies on these lines: {7,21}, {108,6223}, {1097,6060}
X(7338) = excircles variation of Lozada perspector of X(20); see X(7314)
X(7339) lies on these lines: {59,1155}, {649,1461}, {934,2720}, {1055,1262}
X(7339) = isogonal conjugate of X(4081)
X(7339) = X(i)-cross conjugate of X(j) for these (i,j): (56,1461), (221,651), (603,4565), (1407,4617), (2175,1415)
X(7339) = cevapoint of X(i) and X(j) for (i,j) = (56,1461), (109,222), (692,3207), (1415,2175)
X(7339) = trilinear pole of the line through X(1415) and X(1461)
X(7339) = excircles variation of Lozada perspector of X(651); see X(7314)
X(7340) lies on these lines: {249,1509}, {4590,4620}
X(7340) = isotomic conjugate of X(4092)
X(7340) = X(i)-cross conjugate of X(j) for these (i,j): (7,4573), (261,4610), (1397,4565), (6061,4612)
X(7340) = cevapoint of X(i) and X(j) for (i,j) = (7,4573), (261,4610), (873,4625), (1397,4565), (4612,6061)
X(7340) = trilinear pole of the line through X(4565) and X(4573)
X(7340) = excircles variation of Lozada perspector of X(99); see X(7314)
X(7341) lies on these lines: {7,1509}, {60,757}, {552,6628}, {593,1412}, {604,4565}, {3217,4627}
X(7341) = isogonal conjugate of X(6057)
X(7341) = X(849)-cross conjugate of X(593)
X(7341) = cevapoint of X(1408) and X(1412)
X(7341) = excircles variation of Lozada perspector of X(81); see X(7314)
X(7342) lies on these lines: {56,593}, {261,5433}, {849,1408}
X(7342) = excircles variation of Lozada perspector of X(58); see X(7314)
X(7343) lies on the Gergonne strophoid (K086) and these lines: {1,399}, {35,1511}, {36,74}, {110,3746}, {519,6740}, {1094,1250}, {2914,6198}, {5563,5663}
X(7343) = X(6149)-cross conjugate of X(35)
X(7343) = {X(1),X(399)}-harmonic conjugate of X(6126)
X(7344) lies on these lines: {3,1276}, {5,6192}, {18,3460}, {19,3462}, {61,1652}, {63,627}, {3336,3376}, {3375,3467}
X(7344) = isogonal conjugate of X(6192)
X(7344) = Kosnita(X(1276),X(3)) point
X(7345) lies on these lines: {3,1277}, {5,6191}, {17,3460}, {19,3462}, {62,1653}, {63,628}, {3336,3383}, {3384,3467}
X(7345) = isogonal conjugate of X(6191)
X(7345) = X(2307)-cross conjugate of X(1)
X(7345) = Kosnita(X(1277),X(3)) point
X(7346) lies on the Brocard 4th cubic (K020) and these lines: {32,3510}, {194,869}, {384,904}, {1740,3499}, {3223,3500}, {3491,3503}
X(7346) = isogonal conjugate of X(6196)
X(7346) = X(i)-cross conjugate of X(j) for these (i,j): (695,3497), (1909,1)
X(7347) lies on these lines: {2,2082}, {6,6203}, {19,1659}, {41,3084}, {63,6462}, {169,5393}, {371,1707}, {485,6212}, {2170,3083}
X(7347) = isogonal conjugate of X(6204)
X(7347) = X(2066)-cross conjugate of X(1)
X(7347) = {X(19),X(3068)}-harmonic conjugate of X(6204)
X(7348) lies on these lines: {2,2082}, {6,6204}, {19,3069}, {41,3083}, {63,6463}, {169,5405}, {372,1707}, {486,6213}, {2170,3084}
X(7348) = isogonal conjugate of X(6203)
X(7348) = X(5414)-cross conjugate of X(1)
X(7348) = {X(19),X(3069)}-harmonic conjugate of X(6203)
X(7349) lies on these lines: {35,4471}, {598,6205}
X(7349) = isogonal conjugate of X(6205)
X(7349) = X(574)-cross conjugate of X(1)
X(7350) lies on these lines: {3,2329}, {4,3500}, {63,6194}, {98,1423}, {171,222}, {511,2319}, {2221,3195}, {5999,6210}
X(7350) = isogonal conjugate of X(6210)
X(7350) = X(1469)-cross conjugate of X(1)
X(7350) = trilinear pole of the line through X(1459) and X(3287)
X(7351) lies on these lines: {3,3061}, {4,3497}, {63,147}, {98,3512}, {222,613}, {295,511}, {1790,3794}, {4518,5999}
X(7351) = isogonal conjugate of X(6211)
X(7351) = X(1428)-cross conjugate of X(1)
X(7352) lies on these lines: {1,6238}, {3,73}, {34,52}, {36,1147}, {46,4551}, {56,155}, {65,68}, {185,1062}, {499,5654}, {942,5713}, {999,1069}, {1038,1216}, {1060,1425}, {1469,3564}, {1870,5889}, {3100,6241}, {4293,6193}, {5663,6285}
X(7352) = reflection of X(6238) in X(1)
X(7352) = X(4)-of-anti-tangential-midarc-triangle
X(7352) = anti-tangential-midarc-isogonal conjugate of X(32047)
X(7352) = {X(1425),X(5562)}-harmonic conjugate of X(1060)
X(7353) lies on these lines: {1,256}, {34,6406}, {56,1152}, {65,176}, {175,1463}, {1335,1428}, {1870,6400}, {1914,2067}
X(7353) = reflection of X(6405) in X(1)
X(7354) lies on these lines: {1,30}, {2,3614}, {3,12}, {4,11}, {5,36}, {7,3486}, {8,529}, {10,535}, {20,55}, {33,1885}, {34,3575}, {35,495}, {40,5252}, {46,355}, {48,1901}, {57,1837}, {63,5794}, {65,515}, {80,3336}, {145,528}, {172,5254}, {225,1455}, {226,2646}, {278,1852}, {354,950}, {376,3085}, {377,958}, {381,499}, {382,999}, {390,5059}, {397,2307}, {404,1329}, {407,2217}, {427,5322}, {428,614}, {442,993}, {452,4423}, {484,5690}, {496,3583}, {497,3146}, {516,3057}, {517,1770}, {519,4018}, {527,3962}, {548,5010}, {553,6738}, {603,5348}, {609,5305}, {631,5326}, {938,4860}, {944,2099}, {946,1319}, {952,5903}, {962,2098}, {971,1858}, {986,5724}, {1056,3303}, {1086,3924}, {1108,1839}, {1124,6561}, {1193,2635}, {1220,4201}, {1317,1482}, {1335,6560}, {1376,3436}, {1388,5603}, {1399,3072}, {1420,1699}, {1428,5480}, {1458,2654}, {1468,1834}, {1469,1503}, {1470,3149}, {1565,4056}, {1614,2477}, {1657,3295}, {1854,4331}, {1870,6240}, {1888,1891}, {1935,2361}, {2067,3070}, {2093,5881}, {2192,5895}, {2475,2886}, {2476,4999}, {2551,4413}, {2777,3024}, {2794,3023}, {3035,4188}, {3071,6502}, {3218,5086}, {3333,3586}, {3338,5722}, {3339,5727}, {3340,4312}, {3475,4313}, {3485,5731}, {3487,4305}, {3522,5218}, {3543,5225}, {3582,3845}, {3601,5290}, {3660,5806}, {3665,4911}, {3679,5128}, {3698,5795}, {3711,5815}, {3746,4324}, {3748,4314}, {3816,5046}, {3822,5267}, {3826,5260}, {3832,5265}, {3913,6154}, {4189,6690}, {4193,6691}, {4252,5230}, {4301,5048}, {4309,6767}, {4333,5119}, {4366,6658}, {4418,5835}, {4863,6762}, {5131,5445}, {5137,5799}, {5154,6667}, {5180,5330}, {5263,5484}, {5345,6676}, {5538,5763}, {5552,6174}, {6645,6655}
X(7354) = reflection of X(i) in X(j) for these (i,j): (6284,1), (65,4292), (950,4298), (3058,5434), (3962,6737)
X(7354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,30,6284), (1,6284,3058), (3,12,5432), (3,1478,12), (4,56,11), (4,4293,56), (5,36,5433), (20,388,55), (30,5434,3058), (35,4316,550), (35,5270,495), (36,3585,5), (57,5691,1837), (226,4297,2646), (376,3085,5217), (377,958,3925), (382,999,1479), (404,5080,1329), (495,550,35), (496,3627,3583), (497,3600,3304), (515,4292,65), (527,6737,3962), (944,4295,2099), (946,4311,1319), (950,4298,354), (962,3476,2098), (1056,3529,4294), (1056,4294,3303), (1478,4299,3), (1479,4317,999), (1657,3295,4302), (2475,2975,2886), (3085,5217,4995), (3146,3600,497), (3436,4190,1376), (3522,5261,5218), (3583,5563,496), (3585,4325,36), (4316,5270,35), (4911,5088,3665), (5046,5253,3816), (5204,5229,3614), (5434,6284,1)
X(7354) = crosssum of X(55) and X(573)
X(7354) = X(20) of Mandart-incircle triangle
X(7355) lies on these lines: {1,6000}, {4,65}, {11,2883}, {12,6247}, {19,3330}, {33,1425}, {34,185}, {35,3357}, {40,1745}, {55,64}, {56,1498}, {154,5204}, {209,2390}, {408,3185}, {497,6225}, {1038,5907}, {1364,1394}, {1469,1503}, {1479,5878}, {1854,2654}, {1870,6241}, {2183,3197}, {2192,3304}, {3086,5656}, {3152,3869}, {3779,3827}, {5432,6696}, {5663,6238}
X(7355) = reflection of X(6285) in X(1)
X(7355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,6000,6285), (64,221,55)
X(7355) = crosssum of X(55) and X(5776)
X(7356) lies on these lines: {1,1154}, {34,6152}, {36,54}, {56,195}, {65,2962}, {1469,5965}, {1478,2888}, {1870,6242}, {3336,4551}, {3585,6288}
X(7356) = reflection of X(6286) in X(1)
X(7356) = {X(1),X(1154)}-harmonic conjugate of X(6286)
X(7357) lies on these lines: {150,4430}, {319,3681}, {561,6327}, {1442,3920}, {1626,4184}, {1726,3219}
X(7357) = isogonal conjugate of X(1631)
X(7357) = isotomic conjugate of X(6327)
X(7357) = anticomplement of X(32664)
X(7357) = polar conjugate of X(17904)
X(7357) = cyclocevian conjugate of X(330)
X(7357) = X(i)-cross conjugate of X(j) for these (i,j): (31,2), (4056,7)
X(7357) = perspector of the inconic with center X(31)
X(7357) = cevapoint of X(116) and X(513)
X(7357) = trilinear pole of the line through X(824) and X(6586)
X(7357) = X(19)-isoconjugate of X(20739)
X(7358) lies on these lines: {8,1897}, {72,1145}, {123,521}, {200,223}, {517,1528}, {4081,6741}, {6737,6739}
X(7358) = X(1034)-ceva conjugate of X(522)
X(7358) = crosssum of X(154) and X(1415)
X(7358) = crosspoint of X(253) and X(4391)
X(7359) lies on these lines: {8,29}, {9,46}, {11,1731}, {30,2173}, {37,3003}, {44,1737}, {45,498}, {72,1844}, {101,2695}, {320,1944}, {522,650}, {1125,3002}, {1146,2323}, {1784,1990}, {2182,5179}, {2324,4873}, {4395,4858}, {4727,6603}
X(7359) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6739) (8,6062)
X(7359) = X(6062)-cross conjugate of X(8)
X(7359) = center of the inconic with perspector X(6740)
X(7359) = crosssum of X(6) and X(1464)
X(7359) = crossdifference of any pair of centers on the line through X(56) and X(2605)
X(7359) = crosspoint of X(2) and X(6740)
X(7360) lies on these lines: {1,2}, {100,2723}, {243,1948}, {312,1260}, {318,1013}, {657,1021}, {851,5088}, {1105,4219}, {1792,4183}, {1936,1944}, {3262,4998}, {3712,4081}, {4012,5218}
X(7360) = crossdifference of any pair of centers on the line through X(649) and X(1042)
The trilinear polar of X(7361) meets the line at infinity at X(521). (Randy Hutson, April 15, 2015)
X(7361) lies on these lines: {2,1947}, {63,1943}, {78,3362}, {1944,6513}, {3164,3219}
X(7361) = isotomic conjugate of X(6360)
X(7361) = X(92)-cross conjugate of X(2)
X(7361) = perspector of the inconic with center X(92)
X(7361) = pole wrt polar circle of trilinear polar of X(1148)
X(7361) = X(48)-isoconjugate (polar conjugate) of X(1148)
X(7362) lies on these lines: {1,256}, {34,6291}, {56,1151}, {65,175}, {176,1463}, {1124,1428}, {1870,6239}, {1914,6502}
X(7362) = reflection of X(6283) in X(1)
X(7362) = {X(1),X(511)}-harmonic conjugate of X(6283)
X(7363) lies on these lines: {2,6512}, {63,6506}, {222,2165}, {223,2006}, {225,431}, {1069,5713}, {6504,6511}
X(7364) lies on these lines: {2,6507}, {57,326}, {63,6503}, {226,6514}, {278,1958}, {304,1429}, {306,1813}, {307,1790}, {1812,1949}, {1943,6359}, {2289,6340}
X(7364) = {X(2),X(6507)}-harmonic conjugate of X(6518)
X(7364) = crosspoint of X(4620) and X(6516)
X(7365) lies on these lines: {2,85}, {4,1448}, {7,940}, {19,57}, {56,4224}, {77,5712}, {222,4644}, {226,269}, {329,6180}, {347,3666}, {388,612}, {497,2263}, {1042,3485}, {1254,1788}, {1418,3772}, {1458,3475}, {1565,2050}, {1936,3474}, {1943,5839}, {1999,6604}, {3476,3938}, {3487,4306}, {4206,5323}, {4296,5716}, {4350,5287}
X(7365) = X(i)-cross conjugate of X(j) for these (i,j): (2285,388), (5286,2345)
X(7365) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,279,1427), (57,278,4000), (1407,6354,7)
X(7365) = cevapoint of X(2285) and X(4320)
X(7366) lies on these lines: {41,222}, {57,3451}, {604,1407}, {738,1412}, {1106,1397}, {1422,2170}, {2208,3937}
X(7366) = crosssum of X(3119) and X(4163)
X(7366) = crosspoint of X(1407) and X(6612)
X(7367) lies on these lines: {3,9}, {4,972}, {37,939}, {41,2188}, {212,220}, {218,1433}, {280,6559}, {480,1802}, {728,1260}, {1223,1440}, {2184,6611}
X(7367) = X(i)-Ceva conjugate of X(j) for these (i,j): (268,55), (282,2192)
X(7367) = X(41)-cross conjugate of X(220)
X(7367) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,282,1903), (268,282,1436)
X(7367) = isogonal conjugate of X(14256)
X(7367) = Cundy-Parry Phi transform of X(971)
X(7367) = Cundy-Parry Psi transform of X(972)
X(7368) lies on these lines: {9,3295}, {40,198}, {41,55}, {56,6603}, {219,572}, {480,4515}, {644,1259}, {728,1260}, {1035,4559}, {1212,3303}, {1436,2289}, {1696,2171}, {3913,6554}
X(7368) = X(i)-Ceva conjugate of X(j) for these (i,j): (728,220), (1260,480)
X(7368) = {X(1334),X(6602)}-harmonic conjugate of X(220)
X(7368) = crosspoint of X(1252) and X(4578)
X(7369) lies on the line {894,7211}
X(7370) lies on these lines: {1,845}, {57,289}, {174,259}
X(7370) = isogonal conjugate of X(6731)
X(7370) = crosssum of X(1) and X(845)
X(7371) lies on these lines: {1,844}, {7,1488}, {174,259}, {188,555}, {269,2091}
X(7371) = isogonal conjugate of X(6726)
X(7371) = isotomic conjugate of X(7027)
X(7371) = X(555)-ceva conjugate of X(174)
X(7371) = X(i)-cross conjugate of X(j) for these (i,j): (266,174), (2091,7)
X(7371) = {X(1488),X(2089)}-harmonic conjugate of X(7)
X(7371) = crosssum of X(1) and X(844)
X(7372) lies on the line {4560,7265}
X(7372) = isogonal conjugate of X(21784)
X(7372) = isotomic conjugate of X(6758)
X(7372) = X(1109)-cross conjugate of X(2)
X(7372) = perspector of the inconic with center X(1109)
X(7372) = X(19)-isoconjugate of X(23084)
X(7373) lies on these lines: {1,3}, {5,1056}, {11,3851}, {12,5055}, {30,1058}, {58,1616}, {73,3531}, {104,5558}, {145,474}, {381,388}, {382,497}, {390,550}, {392,3927}, {404,3623}, {405,3622}, {495,1656}, {499,5070}, {546,5274}, {549,5265}, {551,958}, {613,5093}, {934,5543}, {938,952}, {944,5804}, {946,6259}, {956,3616}, {995,1126}, {1001,3636}, {1059,1807}, {1124,6417}, {1201,5644}, {1210,5790}, {1335,6418}, {1376,3244}, {1384,2241}, {1387,3485}, {1398,1597}, {1476,3296}, {1478,3843}, {1479,3830}, {1500,5024}, {1598,1870}, {1657,4293}, {2067,6199}, {2334,5313}, {3058,4299}, {3085,3526}, {3241,5253}, {3297,3311}, {3298,3312}, {3299,6500}, {3301,6501}, {3487,5811}, {3488,4308}, {3530,5281}, {3534,4294}, {3555,3940}, {3560,5843}, {3632,4413}, {3635,3913}, {3656,3671}, {3828,4307}, {3872,5439}, {3873,5730}, {3874,5289}, {4317,6284}, {4423,5258}, {4428,5267}, {5044,6762}, {5076,5225}, {5572,6261}, {5603,6147}, {5882,6744}, {6395,6502}
X(7373) = midpoint of X(i)X(j) for these (i,j): (1,3333), (1058,3600)
X(7373) = reflection of X(5708) in X(3333)
X(7373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,6767), (1,36,3303), (1,46,5919), (1,56,3295), (1,942,1482), (1,999,3), (1,3304,999), (1,3333,517), (1,3338,3057), (1,3612,3748), (1,5563,55), (1,5902,2098), (56,3295,3), (388,496,381), (495,3086,1656), (517,3333,5708), (942,1482,1159), (999,3295,56), (1058,3600,30), (1398,6198,1597), (3241,5253,5687)
As a point on the Euler line, X(7374) has Shinagawa coefficients (S, $a2$).
X(7374) lies on these lines: {2, 3}, {98, 1131}, {147, 6462}, {262, 1132}, {371, 5870}, {372, 6201}, {485, 3424}, {486, 6202}, {511, 1270}, {1161, 6214}, {1271, 1352}, {1350, 5590}, {1503, 3068}, {1587, 5304}, {3069, 5480}, {3595, 3818}
X(7374) = {X(2),X(4)}-harmonic conjugate of X(7000)
As a point on the Euler line, X(7375) has Shinagawa coefficients ($a2$, S).
X(7375) lies on these lines: {2, 3}, {76, 3317}, {83, 3316}, {141, 1588}, {371, 5591}, {486, 5590}, {615, 5286}, {637, 3619}, {638, 3618}, {640, 3068}, {1271, 3311}, {1350, 6202}, {1587, 3589}, {3071, 3763}, {5085, 5871}, {5861, 6419}
As a point on the Euler line, X(7376) has Shinagawa coefficients ($a2$, -S).
X(7376) lies on these lines: {2, 3}, {76, 3316}, {83, 3317}, {141, 1587}, {372, 5590}, {485, 5591}, {590, 5286}, {637, 3618}, {638, 3619}, {639, 3069}, {1270, 3312}, {1350, 6201}, {1588, 3589}, {3070, 3763}, {5085, 5870}, {5860, 6420}
As a point on the Euler line, X(7377) has Shinagawa coefficients ($a2$, (a + b + c)2).
X(7377) lies on these lines: {2, 3}, {57, 4911}, {76, 2051}, {150, 5228}, {226, 3673}, {239, 355}, {517, 3661}, {573, 5224}, {946, 3912}, {952, 4393}, {1146, 5834}, {1482, 6542}, {1699, 2887}, {3687, 4385}, {3732, 5813}, {4384, 5587}
X(7377) = complement of X(37416)
As a point on the Euler line, X(7378) has Shinagawa coefficients (F, $a2$).
X(7378) lies on these lines: {2, 3}, {33, 5274}, {34, 3920}, {69, 3867}, {145, 5090}, {251, 1968}, {275, 3424}, {393, 3108}, {1829, 3617}, {1843, 2979}, {1853, 5480}, {1993, 5921}, {3087, 5304}
X(7378) = anticomplement of X(7494)
X(7378 = {X(2),X(4)}-harmonic conjugate of X(6995)
As a point on the Euler line, X(7379) has Shinagawa coefficients ($bc$, $a2$.
X(7379) lies on these lines: {1, 147}, {2, 3}, {8, 1959}, {86, 1503}, {98, 6625}, {325, 1043}, {511, 1654}, {516, 2938}, {1350, 5224}, {1447, 4292}, {3945, 5921}
As a point on the Euler line, X(7380) has Shinagawa coefficients ((a + b + c)2, $a2$).
X(7380) lies on these lines: {2, 3}, {10, 262}, {86, 1352}, {183, 1330}, {511, 5224}, {986, 5988}, {1213, 5480}, {1351, 1654}, {1834, 3815}, {3705, 5295}
As a point on the Euler line, X(7381) has Shinagawa coefficients (-E, (a + b + c)2).
X(7381) lies on these lines: {2, 3}, {69, 1230}, {1478, 5287}, {1479, 5256}, {2999, 3583}, {3434, 3696}, {3436, 3714}, {5905, 5928}
As a point on the Euler line, X(7382) has Shinagawa coefficients (E, (a + b + c)2).
X(7382) lies on these lines: {2, 3}, {1478, 5256}, {1479, 5287}, {2051, 6504}, {2262, 2994}, {2999, 3585}, {3416, 3434}, {3770, 5739}
As a point on the Euler line, X(7383) has Shinagawa coefficients ($a2$, -E).
X(7383) lies on these lines: {2, 3}, {95, 315}, {141, 1181}, {193, 1199}, {570, 5286}, {1498, 3763}, {5012, 6193}, {5157, 6776}
As a point on the Euler line, X(7384) has Shinagawa coefficients ($bc$, (a + b + c)2).
X(7384) lies on these lines: {2, 3}, {239, 946}, {355, 6542}, {1654, 5816}, {1699, 4384}, {3661, 5587}, {4393, 5603}
As a point on the Euler line, X(7385) has Shinagawa coefficients (-$bc$, $a2$).
X(7385) lies on these lines: {2, 3}, {86, 5480}, {262, 6625}, {391, 5921}, {1210, 1447}, {1352, 1654}, {4645, 6210}
X(7385) = anticomplement of X(21554)
As a point on the Euler line, X(7386) has Shinagawa coefficients (-2E, $a2$).
X(7386) is the homothetic center of complement of the orthic triangle and anticomplement of the tangential triangle. (Randy Hutson, April 11, 2015)
X(7386) lies on these lines: {2, 3}, {68, 5447}, {69, 305}, {141, 1853}, {147, 2972}, {251, 1285}, {388, 612}, {394, 6776}, {487, 6465}, {488, 6466}, {497, 614}, {940, 5800}, {1056, 1060}, {1058, 1062}, {1184, 5286}, {1196, 2549}, {1249, 3162}, {1352, 3819}, {1478, 5268}, {1479, 5272}, {1578, 1587}, {1579, 1588}, {1611, 5254}, {1799, 6340}, {2974, 3448}, {2979, 6515}, {3421, 4723}, {3434, 4359}, {3909, 5739}, {5784, 5928}
X(7386) = complement of X(6995)
X(7386) = anticomplement of X(5020)
X(7386) = homothetic center of medial triangle and 3rd antipedal triangle of X(4)
X(7386) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,7494), (2,4,7392)
As a point on the Euler line, X(7387) has Shinagawa coefficients (E - $a2$, E + $a2$).
X(7387) lies on these lines: {2, 3}, {6, 5446}, {52, 1181}, {54, 6800}, {68, 1503}, {154, 1147}, {155, 159}, {156, 3167}, {161, 1498}, {517, 3556}, {569, 3796}, {577, 3199}, {1092, 1495}, {1160, 5594}, {1161, 5595}, {1176, 3527}, {1216, 1350}, {1609, 3767}, {1614, 1993}, {1853, 5449}, {2777, 2931}, {3073, 5329}, {3220, 5709}, {3564, 5596}
X(7387) = complement of X(34938)
X(7387) = X(5) of 3rd antipedal triangle of X(3)
X(7387) = {X(3),X(5)}-harmonic conjugate of X(7393)
X(7387) = {X(3),X(4)}-harmonic conjugate of X(9818)
X(7387) = {X(12978),X(12979)}-harmonic conjugate of X(37491)
As a point on the Euler line, X(7388) has Shinagawa coefficients ($a2$, -2S) and also (1, - tan ω)
X(7388) lies on these lines: {2, 3}, {6, 638}, {69, 1588}, {76, 486}, {83, 485}, {141, 637}, {315, 371}, {316, 2460}, {372, 490}, {487, 5591}, {488, 3069}, {489, 3096}, {615, 5254}, {639, 6565}, {642, 6200}, {1587, 3618}, {1991, 3592}, {2996, 3317}, {3068, 6424}, {3070, 3589}, {3316, 5395}, {3591, 5485}
X(7388) = perspector of Kosnita triangle and cross-triangle of ABC and Kosnita triangle
As a point on the Euler line, X(7389) has Shinagawa coefficients ($a2$, 2S), and also (1, tan ω).
X(7389) lies on these lines: {2, 3}, {6, 637}, {69, 1587}, {76, 485}, {83, 486}, {141, 638}, {315, 372}, {316, 2459}, {371, 489}, {487, 3068}, {488, 5590}, {490, 3096}, {590, 5254}, {591, 3594}, {640, 6564}, {641, 6396}, {1588, 3618}, {2996, 3316}, {3069, 6423}, {3071, 3589}, {3317, 5395}, {3590, 5485}
As a point on the Euler line, X(7390) has Shinagawa coefficients ((a + b + c)2, -2$a2$).
X(7390) lies on these lines: {2, 3}, {10, 3424}, {40, 1334}, {198, 2550}, {387, 4251}, {391, 6776}, {511, 3945}, {516, 3986}, {573, 3332}, {612, 6769}, {938, 1447}, {942, 3598}, {966, 1503}, {1350, 4648}, {1352, 5232}, {1400, 4307}, {1654, 5921}, {1834, 4258}, {2551, 5514}, {3333, 5717}, {5276, 5706}
As a point on the Euler line, X(7391) has Shinagawa coefficients (E, -2$a2$).
X(7391) lies on these lines: {2, 3}, {69, 1369}, {305, 316}, {612, 3585}, {614, 3583}, {1180, 2549}, {1288, 1297}, {1352, 2979}, {1478, 3920}, {1503, 1993}, {1627, 3767}, {1853, 3580}, {1899, 3060}, {1994, 6776}, {2781, 3448}, {2794, 5986}, {3162, 5523}, {3424, 6504}, {3818, 3917}, {5254, 5359}, {5422, 5480}
X(7391) = isogonal conjugate of X(34436)
X(7391) = anticomplement of X(22)
X(7391) = inverse-in-anticomplementary-circle of X(23)
X(7391) = isotomic conjugate of isogonal conjugate of X(20987)
X(7391) = polar conjugate of isogonal conjugate of X(22120)
X(7391) = circumcircle-inverse of X(37978)
As a point on the Euler line, X(7392) has Shinagawa coefficients (2E, $a2$).
X(7392) lies on these lines: {2, 3}, {51, 69}, {154, 3589}, {184, 3618}, {264, 6524}, {343, 3066}, {373, 1899}, {388, 614}, {497, 612}, {1058, 3920}, {1196, 2548}, {1285, 1627}, {1352, 5943}, {1478, 5272}, {1479, 5268}, {3434, 4358}, {3702, 5082}, {3818, 6688}, {4383, 5800}, {5218, 5310}, {5640, 6515}
X(7392) = orthocentroidal-circle-inverse of X(7386)
X(7392) = {X(2),X(4)}-harmonic conjugate of X(7386)
As a point on the Euler line, X(7393) has Shinagawa coefficients (E + $a2$, E - $a2$).
X(7393) lies on these lines: {2, 3}, {6, 1216}, {49, 6090}, {64, 4550}, {68, 141}, {155, 182}, {394, 569}, {578, 3819}, {1092, 5650}, {1181, 5891}, {1232, 3964}, {1350, 5446}, {1351, 6101}, {1609, 2548}, {2931, 6723}, {5092, 6759}
X(7393) = {X(3),X(5)}-harmonic conjugate of X(7387)
As a point on the Euler line, X(7394) has Shinagawa coefficients (E, 2$a2$).
X(7394) lies on these lines: {2, 3}, {51, 3818}, {115, 5986}, {251, 3767}, {612, 3583}, {614, 3585}, {1180, 2548}, {1194, 5475}, {1352, 3060}, {1479, 3920}, {1503, 5422}, {1853, 3066}, {1899, 5640}, {1993, 5480}, {3410, 6515}, {3434, 3974}
X(7394) = anticomplement of X(7485)
As a point on the Euler line, X(7395) has Shinagawa coefficients ($a2$, -2F).
X(7395) lies on these lines: {2, 3}, {6, 5562}, {54, 3167}, {76, 3964}, {155, 569}, {182, 1181}, {206, 1498}, {216, 2207}, {394, 578}, {1060, 1398}, {1147, 6090}, {2165, 5254}, {3060, 3527}, {3796, 6759}, {5422, 5889}
X(7395) = homothetic center of Euler triangle and cross-triangle of ABC and Ara triangle
As a point on the Euler line, X(7396) has Shinagawa coefficients (E - F, -$a2$).
X(7396) lies on these lines: {2, 3}, {69, 1853}, {193, 1899}, {253, 305}, {280, 3705}, {325, 6527}, {347, 2898}, {394, 5921}, {612, 4296}, {614, 3100}, {801, 3424}, {1568, 5656}, {3620, 3917}
X(7396) = anticomplement of X(6353)
X(7396) = pole wrt de Longchamps circle of orthic axis
As a point on the Euler line, X(7397) has Shinagawa coefficients (-2$a2$, (a + b + c)2).
X(7397) lies on these lines: {2, 3}, {40, 3008}, {517, 5222}, {572, 4648}, {944, 3912}, {1385, 5308}, {1766, 4000}, {2999, 6769}, {3332, 5085}, {3673, 5435}, {4384, 5657}, {4911, 5226}, {5022, 5286}
As a point on the Euler line, X(7398) has Shinagawa coefficients (E - F, $a2$).
X(7398) lies on these lines: {2, 3}, {51, 193}, {154, 3618}, {264, 6525}, {390, 612}, {614, 3600}, {3066, 5921}, {4293, 5272}, {4294, 5268}, {5265, 5322}, {5281, 5310}, {5943, 6776}
As a point on the Euler line, X(7399) has Shinagawa coefficients ($a2$, 2F).
X(7399) lies on these lines: {2, 3}, {66, 1498}, {141, 5562}, {343, 389}, {570, 5254}, {1181, 1352}, {1199, 1353}, {1503, 5157}, {3313, 5480}, {3574, 3917}, {5449, 5892}
As a point on the Euler line, X(7400) has Shinagawa coefficients ($a2$, -2E).
X(7400) lies on these lines: {2, 3}, {69, 1181}, {141, 1498}, {216, 5286}, {1038, 3086}, {1040, 3085}, {1235, 6527}, {1578, 3069}, {1579, 3068}, {5447, 5654}, {5656, 5907}
As a point on the Euler line, X(7401) has Shinagawa coefficients (E, -E + $a2$).
X(7401) lies on these lines: {2, 3}, {6, 6193}, {52, 69}, {68, 5462}, {389, 1352}, {569, 3618}, {3567, 6515}, {3574, 5651}, {4846, 6225}
X(7401) = orthocentroidal circle inverse of X(6643)
X(7401) = {X(2),X(4)}-harmonic conjugate of X(6643)
As a point on the Euler line, X(7402) has Shinagawa coefficients (2$a2$, (a + b + c)2).
X(7402) lies on these lines: {2, 3}, {355, 5222}, {3008, 5587}, {3673, 5226}, {3912, 5603}, {4384, 5818}, {4911, 5435}, {5256, 5534}, {5308, 5886}
As a point on the Euler line, X(7403) has Shinagawa coefficients ($a2$ - E, $a2$ + E).
X(7403) lies on these lines: {2, 3}, {52, 5480}, {311, 3933}, {343, 5446}, {569, 1503}, {578, 3818}, {5050, 5596}
As a point on the Euler line, X(7404) has Shinagawa coefficients ($a2$ - E, E).
X(7404) lies on these lines: {2, 3}, {182, 5596}, {311, 3926}, {567, 5921}, {569, 6776}, {578, 1352}, {3589, 6247}
As a point on the Euler line, X(7405) has Shinagawa coefficients ($a2$ + E, $a2$ - E).
X(7405) lies on these lines: {2, 3}, {39, 233}, {52, 141}, {343, 5462}, {569, 3589}, {1232, 3933}, {3618, 6193}
As a point on the Euler line, X(7406) has Shinagawa coefficients ($a2$, -2(a + b + c)2).
X(7406) lies on these lines: {2, 3}, {239, 962}, {516, 4384}, {948, 4872}, {2051, 5395}, {3912, 5691}, {3914, 5222}
As a point on the Euler line, X(7407) has Shinagawa coefficients ((a + b + c)2, 2$a2$).
X(7407) lies on these lines: {2, 3}, {511, 5232}, {966, 5480}, {1352, 3945}, {3487, 5808}, {3920, 5534}, {4253, 5801}
As a point on the Euler line, X(7408) has Shinagawa coefficients (F, -2$a2$).
X(7408) lies on these lines: {2, 3}, {53, 5304}, {1180, 3199}, {1829, 3621}, {3060, 5921}, {4678, 5090}
As a point on the Euler line, X(7409) has Shinagawa coefficients (F, 2$a2$).
X(7409) lies on these lines: {2, 3}, {193, 3867}, {317, 1369}, {1829, 4678}, {3621, 5090}, {5304, 6748}
As a point on the Euler line, X(7410) has Shinagawa coefficients (-2(a + b + c)2, $a2$).
X(7410) lies on these lines: {2, 3}, {230, 387}, {1007, 1330}, {1213, 6776}, {1350, 6707}
Let AB, AC, BC, BA, CA, CB be as in the construction of the Conway circle; see http://mathworld.wolfram.com/ConwayCircle.html. Let A' = BCBA∩ CACB, and define B' and C' cyclically. Triangle A'B'C,' here named the Conway triangle, is the anticomplement of the 2nd extouch triangle. Also, the Conway triangle is perspective to ABC, homothetic to the intouch triangle at X(7), and homothetic to the 1st circumperp triangle at X(7411). (Randy Hutson, April 11, 2015)
X(7411) lies on these lines: {2, 3}, {7, 55}, {8, 5584}, {31, 1742}, {35, 4292}, {36, 4304}, {40, 3868}, {56, 4313}, {63, 100}, {77, 7070}, {81, 991}, {84, 1796}, {144, 1260}, {212, 651}, {224, 3869}, {390, 1617}, {516, 1621}, {517, 3957}, {662, 6061}, {675, 1292}, {968, 1721}, {971, 3219}, {1071, 3579}, {1088, 3188}, {1156, 7082}, {1214, 3100}, {1290, 2688}, {1376, 5273}, {1490, 3876}, {1633, 3185}, {1699, 5284}, {1750, 3305}, {1764, 5208}, {1804, 6060}, {2808, 3690}, {2951, 4512}, {2975, 4297}, {3198, 7291}, {3428, 5731}, {3562, 4303}, {3576, 4666}, {3693, 5279}, {3730, 6605}, {3746, 5493}, {4640, 5784}, {5231, 5303}, {5260, 5691}, {5759, 5905}
X(7411) = {X(3),X(411)}-harmonic conjugate of X(404)(Randy Hutson, April 9, 2016)
X(7412) lies on these lines: {1, 102}, {2, 3}, {33, 40}, {34, 3576}, {35, 1785}, {41, 1783}, {52, 2906}, {53, 1030}, {55, 1118}, {60, 162}, {100, 318}, {104, 3417}, {165, 1753}, {198, 281}, {317, 1444}, {515, 1610}, {517, 6198}, {573, 1172}, {578, 5320}, {581, 3192}, {1155, 1887}, {1158, 1633}, {1249, 4254}, {1385, 1426}, {1400, 3072}, {1425, 2818}, {1824, 6197}, {1827, 1872}, {1861, 6684}, {1875, 2646}, {1892, 5812}, {1897, 3871}, {2212, 6210}, {2269, 7120}, {2695, 2766}, {2975, 5081}, {3193, 5889}, {3220, 6245}, {5124, 6748}, {5687, 7046}
X(7413) lies on these lines: {2, 3}, {9, 1755}, {10, 3430}, {72, 7081}, {98, 109}, {230, 1901}, {242, 6708}, {262, 1751}, {325, 332}, {333, 511}, {517, 3757}, {842, 2689}, {1214, 7009}, {1215, 6211}, {1350, 5737}, {1490, 5268}, {3011, 3072}, {3419, 3705}, {3487, 5711}, {3815, 5110}, {5205, 5777}, {5275, 5776}, {5712, 6776}, {5774, 7172}, {6350, 7102}
X(7413) = complement of X(37443)
X(7413) = Euler line intercept, other than X(29), of circle {{X(29),PU(4)}}
X(7414) lies on these lines: {1, 1835}, {2, 3}, {33, 46}, {34, 3612}, {35, 225}, {55, 1068}, {65, 74}, {100, 1300}, {165, 2960}, {477, 2766}, {484, 1825}, {915, 6011}, {1030, 1865}, {1172, 2245}, {1292, 3563}, {1824, 3579}, {1826, 7110}, {1870, 2646}, {1986, 2906}, {3562, 7352}
X(7414) = isogonal conjugate of X(34800)
X(7414) = crossdifference of every pair of points on line X(647)X(14395)
X(7414) = circumcircle-inverse of X(37979)
X(7414) = {X(3),X(4)}-harmonic conjugate of X(37117)
X(7415) lies on these lines: {2, 3}, {40, 1043}, {58, 4297}, {81, 5731}, {86, 3576}, {99, 102}, {171, 4304}, {333, 515}, {516, 4653}, {517, 5208}, {581, 5331}, {691, 2695}, {1296, 1311}, {3736, 5732}, {4313, 5711}
X(7416) lies on these lines: {2, 3}, {55, 103}, {185, 500}, {228, 971}, {990, 2352}, {1617, 3332}, {1709, 3185}, {1754, 3286}, {1779, 4259}, {2688, 2689}, {5495, 6102}
X(7417) lies on these lines: {2, 3}, {98, 111}, {511, 5468}, {842, 2770}, {1297, 2374}, {1350, 5108}, {1495, 5967}, {1503, 1648}, {2373, 3563}, {2407, 5968}, {6776, 6792}
X(7418) lies on these lines: {2, 3}, {6, 6785}, {74, 111}, {98, 804}, {477, 2770}, {511, 2421}, {1294, 2374}, {1297, 3563}, {1300, 2373}, {1634, 6054}, {2493, 2781}
X(7418) = Thomson isogonal conjugate of X(5652)
X(7418) = crosspoint of X(98) and X(842)
X(7419) lies on these lines: {2, 3}, {58, 106}, {81, 3304}, {107, 2370}, {333, 1222}, {386, 5640}, {476, 2758}, {1623, 1624}, {3066, 4255}, {3303, 4267}, {3746, 4653}
X(7420) lies on these lines: {2, 3}, {40, 3185}, {56, 581}, {101, 102}, {228, 517}, {580, 2194}, {1064, 1400}, {1214, 1905}, {2690, 2695}, {5320, 5398}
X(7421) lies on these lines: {2, 3}, {35, 73}, {36, 2654}, {74, 109}, {477, 2689}, {581, 5890}, {1030, 3330}, {1745, 5010}, {2077, 2392}
X(7421) = {X(3),X(4)}-harmonic conjugate of X(37115)
Let O' be the circle described at X(6039). Let U be the tangent to O' at X(98) and V the tangent to O' at X(842); then X(7422) = U∩V.
X(7422) lies on these lines: {2, 3}, {74, 98}, {477, 842}, {511, 2407}, {841, 2770}, {1294, 3563}, {1297, 1300}, {1640, 5915}
X(7422) = Thomson-isogonal conjugate of X(34291)
X(7423) lies on these lines: {2, 3}, {98, 105}, {104, 111}, {842, 2752}, {915, 2373}, {1295, 2374}, {2687, 2770}
X(7424) lies on these lines: {2,3}, {80,5127}, {102,476}, {110,515}, {229,7354}, {517,6740}, {523,4833}, {643,5176}, {691,1311}, {759,3583}, {1098,5086}, {1385,3615}, {1826,7054}, {5179,5546}
X(7424) = antigonal conjugate of X(38945)
X(7425) lies on these lines: {2, 3}, {74, 105}, {98, 104}, {477, 2752}, {842, 2687}, {915, 1297}, {1295, 3563}
Let QA-Co3 be the Gergonne-Steiner Conic of quadrangle the X(13)X(14)X(15)X(16) Let U be the tangent to QA-Co3 at X(15) and V the tangent to QA-Co3 at X(16). Then X(7426) = U∩V. See Gergonne-Steiner Conic.
X(7426) lies on these lines: {2,3}, {110,524}, {111,230}, {187,5913}, {232,3163}, {325,3233}, {351,523}, {511,5642}, {542,1495}, {597,5640}, {842,1302}, {1304,1494}, {2030,6791}, {3581,5655}, {3849,5099}, {4995,5160}, {5298,7286}, {5971,6390}
X(7426) = midpoint of X(2) and X(23)
X(7426) = reflection of X(2) in X(468)
X(7426) = reflection of X(858) in X(2)
X(7426) = isogonal conjugate of X(5505)
X(7426) = reflection of X(2) in the orthic axis
X(7426) = inverse-in-{circumcircle, nine-point circle}-inverter of X(381)
X(7426) = inverse-in-circle-O(PU(4)) of X(4)
X(7426) = antigonal conjugate of X(38951)
X(7427) lies on these lines: {2, 3}, {98, 759}, {104, 105}, {511, 4585}, {517, 3573}, {2687, 2752}, {2716, 2726}
X(7428) lies on these lines: {2, 3}, {56, 106}, {160, 1486}, {956, 1222}, {2689, 2758}, {3060, 5754}, {3185, 3612}
X(7429) lies on these lines: {2, 3}, {74, 104}, {477, 2687}, {841, 2752}, {915, 1294}, {1295, 1300}
X(7430) lies on these lines: {2, 3}, {71, 74}, {477, 2690}, {573, 5890}, {1300, 1305}, {2178, 2335}
X(7431) lies on these lines: {2, 3}, {55, 1396}, {58, 103}, {99, 917}, {935, 2688}, {1172, 3286}
X(7432) lies on these lines: {2, 3}, {98, 675}, {103, 111}, {917, 2373}, {2688, 2770}
X(7433) lies on these lines: {2, 3}, {74, 675}, {98, 103}, {842, 2688}, {917, 1297}
X(7434) lies on these lines: {2, 3}, {98, 106}, {842, 2758}, {2370, 3563}, {3430, 6789}
X(7435) lies on these lines: {2, 3}, {100, 1301}, {107, 108}, {162, 4565}, {1304, 2766}
X(7436) lies on these lines: {2, 3}, {102, 112}, {935, 2695}, {1172, 3428}, {3194, 4267}
X(7437) lies on these lines: {2, 3}, {100, 101}, {108, 1305}, {885, 2283}, {1290, 2690}
X(7438) lies on these lines: {2, 3}, {100, 2374}, {108, 111}, {1824, 5297}, {2766, 2770}
X(7439) lies on these lines: {2, 3}, {98, 1311}, {102, 111}, {2695, 2770}
X(7440) lies on these lines: {2, 3}, {74, 103}, {477, 2688}, {917, 1294}
X(7441) lies on these lines: {2, 3}, {74, 1311}, {98, 102}, {842, 2695}
X(7442) lies on these lines: {2, 3}, {103, 104}, {917, 1295}, {2687, 2688}
X(7443) lies on these lines: {2, 3}, {102, 105}, {104, 1311}, {2695, 2752}
X(7444) lies on these lines: {2, 3}, {74, 106}, {477, 2758}, {1300, 2370}
X(7445) lies on these lines: {2, 3}, {103, 105}, {104, 675}, {2688, 2752}
X(7446) lies on these lines: {2, 3}, {103, 106}, {917, 2370}, {2688, 2758}
X(7447) lies on these lines: {2, 3}, {104, 106}, {915, 2370}, {2687, 2758}
X(7448) lies on these lines: {2, 3}, {106, 111}, {2370, 2374}, {2758, 2770}
X(7449) lies on these lines: {2, 3}, {109, 111}, {228, 5297}, {2689, 2770}
X(7450) lies on these lines: {2, 3}, {109, 110}, {476, 2689}, {1624, 1633}
X(7451) lies on these lines: {2, 3}, {100, 109}, {1290, 2689}, {3871, 5399}
X(7452) lies on these lines: {2, 3}, {107, 109}, {110, 1897}, {1304, 2689}
X(7453) lies on these lines: {2, 3}, {42, 101}, {1305, 2374}, {2690, 2770}
X(7454) lies on these lines: {2, 3}, {74, 102}, {477, 2695}
X(7455) lies on these lines: {2, 3}, {102, 103}, {2688, 2695}
X(7456) lies on these lines: {2, 3}, {102, 104}, {2687, 2695}
X(7457) lies on these lines: {2, 3}, {102, 106}, {2695, 2758}
X(7458) lies on these lines: {2, 3}, {105, 111}, {2752, 2770}
X(7459) lies on these lines: {2, 3}, {105, 106}, {2752, 2758}
X(7460) lies on these lines: {2, 3}, {101, 109}, {2689, 2690}
X(7461) lies on these lines: {2, 3}, {108, 109}, {2689, 2766}
X(7462) lies on these lines: {2, 3}, {99, 109}, {691, 2689}
X(7463) lies on these lines: {2, 3}, {109, 112}, {935, 2689}
As a point on the Euler line, X(7464) has Shinagawa coefficients (-2 (E - 2 F), 5 E - 4 F).
Let P and Q be circumcircle antipodes. X(7464) is the Euler line intercept, other than X(2), of circle {{X(2),P,Q}} for all P, Q. (Randy Hutson, August 17, 2020)
X(7464) lies on these lines: {2, 3}, {55, 7286}, {56, 5160}, {74, 511}, {98, 2696}, {99, 477}, {104, 2691}, {110, 841}, {112, 2693}, {316, 5866}, {323, 5663}, {576, 5890}, {842, 1296}, {935, 1294}, {1292, 2687}, {1503, 2892}, {1533, 5972}, {3292, 6000}, {3426, 6090}, {3431, 6800}, {5643, 5892}
X(7464) = midpoint of X(20) and X(5189)
X(7464) = reflection of X(4) in X(858)
X(7464) = reflection of X(23) in X(3)
X(7464) = reflection of X(4) in the de Longchamps line
X(7464) = Thomson-isogonal conjugate of X(5648)
X(7464) = circumcircle-inverse of X(376)
X(7464) = {X(3),X(23)}-harmonic conjugate of X(184)
X(7465) lies on these lines: {2, 3}, {35, 3011}, {46, 612}, {65, 3920}, {75, 100}, {81, 2979}, {226, 5314}, {614, 3612}, {1155, 5297}, {1796, 5268}, {2245, 5276}, {2646, 7191}, {2975, 3006}, {5096, 5718}, {5249, 5285}, {5745, 7293}, {5905, 7085}
X(7465) = anticomplement of X(37315)
X(7465) = {X(2),X(3)}-harmonic conjugate of X(37449)
X(7466) lies on these lines: {2, 3}, {19, 100}, {108, 273}, {162, 2189}, {607, 3240}, {1257, 3869}, {1612, 1838}, {1621, 1848}, {1891, 2975}, {2203, 5012}, {3006, 5174}, {5089, 5297}
X(7467) lies on these lines: {2, 3}, {98, 689}, {182, 1501}, {251, 3398}, {511, 1194}, {1180, 3095}, {1196, 5188}, {1350, 1613}, {1627, 2080}, {2456, 5012}, {3098, 3231}, {5201, 5306}
X(7468) lies on these lines: {2, 3}, {99, 476}, {110, 249}, {112, 6753}, {523, 1634}, {669, 3233}, {827, 1291}, {925, 935}, {930, 1287}, {1302, 2696}, {1304, 3565}
X(7468) = antigonal conjugate of X(34175)
X(7468) = isogonal conjugate of antigonal conjugate of X(35364)
X(7469) lies on these lines: {2, 3}, {105, 476}, {110,518}, {1302,2687}
X(7469) = isogonal conjugate of X(10100)
X(7470) lies on these lines: {2, 3}, {74, 689}, {76, 3098}, {83, 5092}, {98, 5188}, {698, 1350}, {736, 6309}, {2076, 5254}, {3096, 3818}, {5017, 5286}
X(7471) lies on these lines: {2, 3}, {107, 250}, {110, 476}, {542, 6070}, {691, 1302}, {925, 1304}, {1553, 2777}, {3258, 5972}, {6795, 6800}
X(7471) = anticomplement of X(3154)
X(7471) = reflection of X(110) in X(3233)
X(7471) = reflection of X(110) in its Simson line (line X(30)X(113))
X(7471) = antigonal conjugate of X(34150)
X(7471) = polar-circle-inverse of X(35235)
X(7471) = Euler line intercept of axis of Kiepert parabola
X(7472) lies on these lines: {2, 3}, {99, 523}, {110, 1499}, {187, 5912}, {249, 3566}, {476, 1296}, {620, 5099}, {935, 3565}, {2682, 5972}
X(7472) = antigonal conjugate of X(34169)
X(7473) lies on these lines: {2, 3}, {99, 1304}, {107, 691}, {110, 525}, {112, 476}, {250, 523}, {933, 1287}, {1552, 1553}
X(7473) = isogonal conjugate of X(35909)
X(7473) = isotomic conjugate of polar conjugate of X(35907)
X(7473) = anticomplement of X(37987)
X(7473) = polar conjugate of X(14223)
X(7473) = pole wrt polar circle of trilinear polar of X(14223) (line X(523)X(868))
X(7473) = cevapoint of X(1640) and X(5191)
X(7473) = crossdifference of every pair of points on line X(647)X(16186)
X(7473) = orthogonal projection of X(648) on its trilinear polar
X(7473) = trilinear pole of line X(542)X(6103)
X(7473) = X(19)-isoconjugate of X(35911)
X(7473) = X(63)-isoconjugate of X(14998)
X(7474) lies on these lines: {2, 3}, {58, 3011}, {81, 3475}, {86, 110}, {103, 1302}, {333, 3681}, {498, 5358}, {1043, 3006}
X(7374) = orthocentroidal circle inverse of X(7000)
X(7475) lies on these lines: {2, 3}, {99, 693}, {100, 691}, {110, 2691}, {476, 1292}, {523, 4436}, {2766, 3565}
X(7475) = anticomplement of X(37986)
X(7475) = antigonal conjugate of X(34173)
X(7476) lies on these lines: {2, 3}, {99, 2766}, {100, 935}, {107, 2691}, {108, 691}, {112, 1290}, {1292, 1304}
X(7477) lies on these lines: {2, 3}, {100, 476}, {110, 513}, {925, 2766}, {1302, 2691}, {3233, 3733}
X(7477) = antigonal conjugate of X(38952)
X(7478) lies on these lines: {2, 3}, {106, 476}, {110, 519}, {229, 5434}, {759, 3582}, {1304, 2370}
X(7378) = orthocentroidal circle inverse of X(6995)
X(7479) lies on these lines: 2, 3}, {101, 476}, {110, 514}, {1304, 1305}, {3233, 7192}
X(7480) lies on these lines: {2, 3}, {107, 476}, {110, 250}, {523, 1624}, {935, 1302}
X(7480) = isogonal conjugate of X(14220)
X(7480) = anticomplement of X(37985)
X(7481) lies on these lines: {2, 3}, {99, 2758}, {106, 691}, {935, 2370}, {1403, 7286}
X(7482) lies on these lines: {2, 3}, {99, 935}, {107, 2696}, {112, 250}, {1296, 1304}
X(7483) lies on these lines: {1, 4999}, {2, 3}, {10, 2646}, {11, 5248}, {12, 993}, {35, 2886}, {46, 3624}, {58, 5718}, {65, 392}, {72, 5745}, {78, 5791}, {80, 1698}, {141, 5135}, {216, 828}, {226, 3916}, {230, 5283}, {284, 5742}, {495, 2975}, {496, 1621}, {498, 958}, {499, 1001}, {952, 3897}, {956, 3085}, {1155, 6681}, {1213, 2278}, {1329, 5251}, {1387, 3890}, {1389, 5657}, {1714, 4255}, {1770, 3838}, {2182, 5257}, {2950, 3646}, {3218, 6147}, {3255, 3467}, {3419, 3601}, {3487, 5744}, {3584, 5258}, {3589, 4259}, {3697, 6745}, {3746, 3813}, {3753, 6684}, {3816, 5259}, {3820, 5260}, {3822, 5267}, {3824, 5122}, {3829, 4857}, {3868, 5719}, {3877, 5901}, {3917, 5482}, {5082, 5281}, {5218, 5687}, {5250, 5886}, {5784, 6666}, {6265, 6713}
X(7484) lies on these lines: {2, 3}, {6, 3917}, {9, 1473}, {35, 5272}, {36, 5268}, {39, 1184}, {51, 1350}, {55, 614}, {56, 612}, {57, 7085}, {95, 183}, {141, 1899}, {154, 5646}, {159, 1853}, {182, 394}, {184, 5085}, {197, 4413}, {216, 1033}, {251, 1384}, {493, 3103}, {494, 3102}, {574, 1196}, {620, 2936}, {748, 7083}, {750, 1460}, {978, 1036}, {999, 3920}, {1038, 1398}, {1040, 7071}, {1073, 5481}, {1180, 5024}, {1194, 1611}, {1351, 2979}, {1486, 4423}, {1609, 3815}, {1993, 5050}, {3066, 6688}, {3098, 5943}, {3167, 5012}, {3220, 7308}, {3295, 7191}, {3305, 7293}, {3306, 5314}, {3787, 5034}, {3796, 5092}, {5120, 5276}, {5204, 5322}, {5217, 5310}, {5285, 5437}, {5621, 5642}
X(7484) = complement of X(6997)
X(7484) = circumcircle-inverse of X(37899)
X(7484) = orthocentroidal-circle-inverse of X(37439)
X(7484) = {X(2),X(4)}-harmonic conjugate of X(37439)
As a point on the Euler line, X(7485) has Shinagawa coefficients (-3 E - 2 F, 2 Sω).
X(7485) lies on these lines: {2, 3}, {6, 1627}, {9, 7293}, {35, 614}, {36, 612}, {39, 5359}, {51, 3098}, {55, 4850}, {56, 3920}, {57, 5314}, {110, 3796}, {154, 5888}, {159, 3619}, {182, 1993}, {183, 1232}, {184, 3819}, {251, 3053}, {305, 1078}, {394, 5012}, {511, 5422}, {569, 5447}, {574, 1194}, {620, 3455}, {748, 7295}, {750, 5329}, {940, 5096}, {1180, 1184}, {1350, 3060}, {1473, 3219}, {1486, 5284}, {1613, 5116}, {1994, 5050}, {3218, 7085}, {3220, 3305}, {3306, 5285}, {4265, 4383}, {5010, 5272}, {5024, 5354}, {5124, 5275}, {5204, 5297}, {5217, 7292}, {5268, 5322}, {5646, 6030}, {5650, 6800}
X(7485) = complement of X(7394)
X(7485) = anticomplement of X(37439)
X(7485) = circumcircle-inverse of X(37900)
As a point on the Euler line, X(7486) has Shinagawa coefficients (5,2).
X(7486) lies on these lines: {2, 3}, {10, 5734}, {146, 6723}, {147, 6722}, {148, 6721}, {153, 6667}, {373, 5889}, {498, 5274}, {499, 5261}, {590, 6470}, {615, 6471}, {962, 3634}, {1506, 5319}, {1698, 4301}, {3589, 5921}, {3614, 7288}, {3616, 5881}, {3617, 5886}, {3621, 5901}, {3622, 5818}, {3623, 5790}, {4309, 5281}, {4317, 5265}, {5218, 7173}, {5219, 5704}, {5237, 5366}, {5238, 5365}, {5328, 5705}, {5550, 5587}, {6459, 6468}, {6460, 6469}
X(7386) = orthocentroidal circle inverse of X(7392)
As a point on the Euler line, X(7487) has Shinagawa coefficients (-F, E + 2 F).
X(7487) lies on these lines: {2, 3}, {32, 393}, {33, 4294}, {34, 4293}, {39, 3087}, {52, 193}, {53, 3053}, {254, 1179}, {264, 3785}, {317, 3926}, {389, 1843}, {390, 6198}, {578, 1974}, {944, 1829}, {1093, 6525}, {1192, 6247}, {1285, 3172}, {1395, 3072}, {1587, 5412}, {1588, 5413}, {1870, 3600}, {1892, 3487}, {1902, 6361}, {1905, 3486}, {2212, 3073}, {3092, 6459}, {3093, 6460}, {3867, 5085}, {5013, 6748}, {5090, 5657}
X(7487) = {X(4),X(24)}-harmonic conjugate of X(2)
As a point on the Euler line, X(7488) has Shinagawa coefficients (-E - 4 F, 2 (E + 2 F)).
Let A'B'C' be the Kosnita triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(7488).
X(7488) lies on these lines: {2, 3}, {35, 3100}, {36, 4296}, {49, 1154}, {52, 54}, {94, 96}, {110, 5562}, {143, 567}, {146, 2883}, {159, 5921}, {184, 5889}, {323, 1147}, {343, 1601}, {389, 5012}, {568, 1199}, {569, 3567}, {578, 3060}, {827, 1297}, {1092, 2979}, {1181, 6800}, {1287, 2697}, {1495, 5907}, {2916, 2929}, {2918, 2931}, {3565, 5966}, {3581, 6102}, {5926, 6563}, {6146, 6515}
X(7488) = isogonal conjugate of X(6145)
X(7488) = complement of anticomplementary conjugate of X(32354)
X(7488) = anticomplement of X(1594)
X(7488) = anticevian isogonal conjugate of X(5)
X(7488) = circumcircle-inverse of X(3153)
X(7488) = circumtangential-isogonal conjugate of X(32401)
X(7488) = crosspoint, wrt both the excentral and tangential triangles, of X(3) and X(2917)
X(7488) = center of inverse-in-circumcircle-of-de-Longchamps-circle
X(7488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,24,2), (3,26,4)
X(7488) = isogonal conjugate, wrt Kosnita triangle, of the Kosnita point (X(54))
X(7489) lies on these lines: {1, 195}, {2, 3}, {55, 80}, {56, 5443}, {104, 5284}, {119, 6690}, {355, 5248}, {517, 3683}, {952, 1621}, {958, 1482}, {993, 5886}, {1001, 2801}, {1385, 5259}, {1749, 5902}, {1807, 7069}, {2975, 5901}, {3652, 5884}, {3925, 5840}, {4653, 5396}, {4870, 5563}, {4877, 5755}, {5127, 5398}, {5260, 5690}, {5426, 6326}, {5436, 7330}, {5450, 6259}, {6583, 6763}
X(7490) lies on these lines: {2, 3}, {6, 1396}, {19, 57}, {34, 2999}, {69, 7058}, {92, 4359}, {142, 1848}, {154, 3332}, {226, 610}, {273, 5435}, {281, 5307}, {284, 5712}, {579, 1730}, {940, 1172}, {948, 1763}, {1474, 4648}, {1829, 5222}, {1841, 3752}, {1851, 2355}, {1869, 3601}, {1870, 5256}, {2550, 5285}, {5122, 5146}, {5226, 7282}, {5287, 6198}, {5709, 6197}, {7009, 7046}
X(7491) lies on these lines: {1, 5841}, {2, 3}, {40, 80}, {52, 517}, {53, 2193}, {119, 6796}, {355, 5842}, {515, 3878}, {535, 5882}, {920, 1837}, {952, 3869}, {962, 1389}, {1385, 7354}, {1482, 3486}, {1780, 5721}, {1834, 5398}, {2077, 4324}, {2829, 6259}, {3576, 5443}, {3586, 5709}, {4313, 5761}, {5086, 5690}, {5698, 5779}, {6713, 7280}
As a point on the Euler line, X(7492) has Shinagawa coefficients (-5 E - 8 F, 8 Sω).
X(7492) lies on these lines: {2, 3}, {35, 5370}, {36, 7302}, {69, 2916}, {99, 5987}, {110, 3098}, {146, 4549}, {184, 6030}, {323, 1350}, {353, 5104}, {574, 1383}, {575, 3060}, {576, 5012}, {1180, 5007}, {1994, 3796}, {2979, 3292}, {3053, 5354}, {3746, 5322}, {3920, 5345}, {5010, 5297}, {5092, 5640}, {5310, 5563}, {7191, 7298}, {7280, 7292}
X(7492) = anticomplement of X(5169)
As a point on the Euler line, X(7493) has Shinagawa coefficients (-E - 4 F, 2 Sω).
Let LA be the polar of X(4) wrt the circle centered at A and passing through X(2), and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X(7493). The orthocenter of A'B'C' is X(376). (Note: the orthocenter is the perspector of every circle centered at a vertex of ABC.) (Randy Hutson, April 15, 2015)
X(7493) lies on these lines: {2, 3}, {69, 110}, {111, 925}, {113, 4549}, {154, 343}, {184, 6515}, {1194, 5319}, {1297, 1302}, {1352, 1495}, {1992, 5486}, {3066, 3589}, {3098, 5972}, {3100, 5218}, {3266, 6337}, {3580, 6776}, {3618, 5640}, {4296, 7288}, {4299, 5370}, {4302, 7302}, {4309, 5310}, {4317, 5322}, {4325, 5345}, {4330, 7298}, {5033, 6388}
X(7493) = complement of X(31099)
X(7493) = anticomplement of X(5094)
X(7493) = circumcircle-inverse of X(37980)
X(7493) = trilinear pole, wrt the circummedial triangle, of the de Longchamps line
X(7493) = homothetic center of orthocevian triangle of X(2) and anticomplementary triangle
As a point on the Euler line, X(7494) has Shinagawa coefficients (-E - 2 F, Sω).
Let DEF = medial triangle and D'E'F' = circummedial triangle. Let Γ = circumcircle, and let Ab be the point, other than E', in which the line E'F intersects Γ. Define Bc and Ca cyclically. Let Ac be the point, other than F', in which the line F'E intersects Γ. Let Oa be the center of the conic tangent to the five lines BC, AE', AF', Bab, CAc, and define Ob and Oc cyclically. The finite fixed point of the affine transformation that carries ABC onto OaObOC is X(7494). (Angel Montesdeoca, May 3, 2020)
X(7494) lies on these lines: {2, 3}, {51, 3618}, {69, 184}, {141, 154}, {216, 1194}, {305, 6337}, {343, 3796}, {388, 5322}, {497, 5310}, {612, 1040}, {614, 1038}, {1060, 7191}, {1062, 3920}, {1478, 5345}, {1479, 7298}, {1495, 3619}, {2968, 7172}, {3598, 6356}, {3757, 6350}, {5012, 6515}, {5225, 7302}, {5229, 5370}, {5347, 5800}
X(7494) = complement of X(7378)
X(7494) = {X(2),X(3)}-harmonic conjugate of X(7386)
As a point on the Euler line, X(7495) has Shinagawa coefficients (-3 (E + 2 F), 2 Sω).
Let LA be the tangent at A to the A-Yiu circle, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Triangle A'B'C' is here named the Yiu tangents triangle. A'B'C' is homothetic to the polar triangle of the nine-point circle at X(427), and to the orthoanticevian triangle of X(2) at X(7495). (Randy Hutson, August 19, 2019)
A'B'C' is also the antipedal triangle of X(5). (Randy Hutson, January 17, 2020)
X(7495) lies on these lines: {2, 3}, {50, 3815}, {67, 110}, {95, 933}, {111, 930}, {125, 5092}, {182, 3580}, {216, 6103}, {230, 566}, {343, 5012}, {1291, 2770}, {1352, 6800}, {3583, 7302}, {3585, 5370}, {3589, 5640}, {3619, 5596}, {4857, 5310}, {5270, 5322}, {5297, 5432}, {5433, 7292}, {5650, 5972}
X(7495) = complement of X(5169)
X(7495) = anticomplement of X(37454)
As a point on the Euler line, X(7496) has Shinagawa coefficients (-7 E - 4 F, 4 Sω).
X(7496) lies on these lines: {2, 3}, {35, 7292}, {36, 5297}, {39, 5354}, {51, 5643}, {110, 5092}, {141, 2930}, {182, 323}, {575, 1994}, {576, 2979}, {748, 7301}, {750, 5363}, {1078, 3266}, {1383, 5210}, {1627, 5007}, {3098, 5640}, {3231, 5116}, {3292, 3819}, {3329, 5201}, {3746, 7191}, {3920, 5563}
X(7497) lies on these lines: {1, 1859}, {2, 3}, {6, 1243}, {19, 219}, {34, 222}, {46, 1888}, {56, 1838}, {57, 1875}, {92, 956}, {155, 1829}, {278, 999}, {355, 1891}, {1479, 1852}, {1486, 5842}, {1848, 5886}, {1890, 5805}, {2194, 5706}, {2299, 5398}, {5174, 5687}, {5285, 5587}
X(7498) lies on these lines: {1, 281}, {2, 3}, {10, 7070}, {33, 936}, {78, 6198}, {92, 3616}, {196, 1940}, {273, 3160}, {278, 1125}, {318, 4358}, {940, 3194}, {965, 1172}, {1210, 1453}, {1468, 7076}, {1838, 3624}, {1844, 5692}, {1857, 2646}, {1895, 5703}, {5250, 6197}, {6513, 7040}
As a point on the Euler line, X(7499) has Shinagawa coefficients (-2 E - 3 F, Sω).
X(7499) lies on these lines: {2, 3}, {11, 5310}, {12, 5322}, {51, 3313}, {66, 154}, {95, 325}, {141, 184}, {182, 343}, {230, 570}, {571, 3815}, {612, 5432}, {614, 5433}, {1180, 5305}, {1352, 3796}, {1899, 5085}, {3564, 5012}, {3614, 5370}, {5050, 6515}, {5447, 6689}, {7173, 7302}
As a point on the Euler line, X(7500) has Shinagawa coefficients (-E - 2 F, 4 Sω).
X(7500) lies on these lines: {2, 3}, {148, 5986}, {193, 2393}, {251, 5286}, {612, 4302}, {614, 4299}, {1478, 5310}, {1479, 5322}, {1503, 6515}, {2790, 5984}, {3060, 6776}, {3424, 5392}, {3583, 5345}, {3585, 7298}, {3796, 5480}, {3920, 4294}, {4293, 7191}, {4316, 5272}, {4324, 5268}
X(7500) = anticomplement of X(1370)
X(7501) lies on these lines: {1, 6197}, {2, 3}, {19, 3576}, {36, 278}, {56, 196}, {57, 1870}, {104, 1436}, {281, 993}, {915, 934}, {1243, 3431}, {1748, 4511}, {1838, 7280}, {1852, 5433}, {1891, 6684}, {2096, 3220}, {3194, 4252}, {3601, 6198}, {4260, 6403}, {5285, 5657}
As a point on the Euler line, X(7502) has Shinagawa coefficients (-3 E - 8 F, 5 E + 8 F).
Let A'B'C' be the Kosnita triangle. Let LA be the reflection of line B'C' in the perpendicular bisector of BC, and define LB, LC cyclically. Let A" = LB∩LC, B" = LC∩LA, C" = LA∩LB. The intouch triangle of A"B"C" is homothetic to the Kosnita triangle at X(7502).
X(7502) lies on these lines: {2, 3}, {54, 6243}, {143, 569}, {156, 5562}, {160, 2934}, {182, 5946}, {184, 1154}, {206, 1511}, {567, 3060}, {568, 5012}, {1147, 5944}, {1263, 2079}, {1495, 5891}, {2782, 3455}, {2916, 2931}, {2917, 2918}, {3581, 5890}, {5092, 5892}, {5876, 6759}
X(7502) = midpoint of X(3) and X(22)
X(7502) = center of the circle that is inverse-in-circumcircle of the de Longchamps line
X(7502) = inverse-in-circumcircle of X(7574)
X(7502) = X(37584)-of-orthic-triangle if ABC is acute
As a point on the Euler line, X(7503) has Shinagawa coefficients (-E - 2 F, 2 F).
X(7503) lies on these lines: {2, 3}, {6, 5889}, {40, 5314}, {54, 155}, {64, 1176}, {84, 7293}, {182, 185}, {184, 5907}, {216, 1968}, {311, 1975}, {389, 5422}, {578, 1993}, {973, 3060}, {1147, 5891}, {1181, 5012}, {1498, 3796}, {5448, 6689}, {5621, 6593}, {6759, 6800}
X(7503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,22), (3,5,24)
X(7504) lies on these lines: {2, 3}, {10, 5443}, {11, 6668}, {80, 1125}, {498, 3871}, {1389, 5330}, {1698, 3878}, {1749, 6701}, {3245, 3634}, {3585, 5303}, {3614, 4999}, {3814, 5260}, {3822, 5253}, {3825, 5284}, {3868, 5219}, {3876, 5705}, {3897, 5587}, {6690, 7173}
X(7504) = complement of X(37291)
As a point on the Euler line, X(7505) has Shinagawa coefficients (-4 F, E).
Let LA be the polar of X(4) with respect to the circle centered at A and passing through X(3), and define LB and LC cyclically. (Note: X(4) is the perspector of every circle centered at a vertex of ABC.) Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. Then A'B'C' is homothetic to ABC, and the center of homothety is X(7505). Also, A'B'C' is homothetic to the medial triangle, and the center of homothety is X(3548), and A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X(20). The orthocenter of A'B'C' is X(20) and the nine-point center of A'B'C' is X(1658). (Randy Hutson, April15, 2015)
X(7505) lies on these lines: {2, 3}, {68, 110}, {74, 5878}, {93, 393}, {125, 6759}, {155, 3580}, {254, 6344}, {498, 6198}, {499, 1870}, {973, 6242}, {1092, 5972}, {1112, 6101}, {1286, 3563}, {1514, 5894}, {1614, 1899}, {3168, 3462}, {5654, 5889}
X(7505) = anticomplement of X(6640)
As a point on the Euler line, X(7506) has Shinagawa coefficients (E - 4 F, 3 E + 4 F).
X(7506) lies on these lines: {2, 3}, {6, 49}, {51, 1147}, {54, 5640}, {110, 3567}, {143, 1993}, {155, 568}, {156, 5946}, {159, 5050}, {161, 569}, {184, 5462}, {195, 973}, {394, 6243}, {567, 3066}, {1092, 5446}, {1216, 5651}, {3527, 5504}
X(7506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,26,3), (5,24,3)
As a point on the Euler line, X(7507) has Shinagawa coefficients (F, E + 3 F).
X(7507) lies on these lines: {2, 3}, {6, 3574}, {70, 3527}, {185, 1853}, {485, 5410}, {486, 5411}, {946, 5090}, {1210, 1892}, {1321, 1322}, {1398, 1478}, {1479, 7071}, {1699, 1902}, {1829, 5587}, {2207, 5475}, {3092, 6565}, {3093, 6564}, {5889, 6746}
X(7507) = {X(1321),X(1322)}-harmonic conjugate of X(6748)
X(7508) lies on these lines: {2, 3}, {35, 5690}, {55, 5844}, {80, 5010}, {143, 970}, {495, 5172}, {952, 993}, {1385, 3878}, {1483, 2975}, {1484, 4996}, {1768, 3576}, {1836, 5443}, {3647, 5694}, {5248, 5901}, {5426, 5535}, {5427, 5902}, {5841, 6690}
As a point on the Euler line, X(7509) has Shinagawa coefficients (-2 Sω, E + 2 F.
X(7509) lies on these lines: {2, 3}, {52, 5422}, {54, 394}, {76, 95}, {155, 5012}, {182, 5562}, {569, 1216}, {578, 3917}, {626, 3425}, {1092, 3819}, {1181, 5085}, {1199, 5050}, {1614, 3796}, {5092, 5907}, {5096, 5706}, {5314, 5709}, {7293, 7330}
X(7509) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,24), (3,5,22)
X(7510) lies on these lines: {2, 3}, {33, 1807}, {51, 1243}, {52, 1866}, {53, 284}, {92, 952}, {196, 1159}, {281, 5790}, {517, 1859}, {579, 6748}, {942, 1875}, {1385, 1838}, {1842, 1871}, {2194, 5721}, {3185, 5842}, {5174, 5690}
X(7511) lies on these lines: {1, 1852}, {2, 3}, {19, 355}, {52, 912}, {53, 1333}, {515, 1871}, {517, 1891}, {971, 1890}, {1385, 1848}, {1770, 1888}, {1838, 7354}, {1870, 6147}, {4261, 6748}, {5307, 5841}, {5690, 6197}
As a point on the Euler line, X(7512) has Shinagawa coefficients (-2 E - 4 F), 3 E + 4 F).
X(7512) lies on these lines: {2, 3}, {49, 323}, {52, 1199}, {54, 511}, {110, 1216}, {155, 6800}, {182, 3567}, {311, 1078}, {569, 3060}, {1092, 3098}, {1147, 2979}, {1503, 2916}, {1614, 5562}, {1994, 6243}, {6146, 6776}
X(7512) = crosspoint, with respect to the excentral triangle, of X(3) and X(2918)
X(7512) = crosspoint, with respect to the tangential triangle, of X(3) and X(2918)
X(7512) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,22,4), (3,26,2)
X(7513) lies on these lines: {1, 273}, {2, 3}, {33, 1895}, {64, 1246}, {78, 92}, {185, 2905}, {243, 1882}, {264, 1043}, {270, 580}, {278, 5703}, {1105, 1826}, {1859, 1940}, {4292, 7282}, {5174, 6734}
As a point on the Euler line, X(7514) has Shinagawa coefficients (-3 E - 4 F, E + 4 F).
X(7514) lies on these lines: {2, 3}, {6, 1154}, {114, 3455}, {184, 5891}, {206, 4550}, {567, 1993}, {568, 5422}, {569, 5562}, {578, 1216}, {1181, 5876}, {1209, 6146}, {2165, 2549}, {5085, 5621}
X(7515) lies on these lines: {2, 3}, {11, 2218}, {37, 216}, {78, 1062}, {283, 343}, {307, 3916}, {936, 1040}, {1104, 1210}, {1125, 1214}, {2193, 5742}, {3616, 6350}, {5550, 6349}, {7011, 7288}
X(7515) = complement of X(5125)
As a point on the Euler line, X(7516) has Shinagawa coefficients (-5 E - 4 F, 3 E + 4 F).
X(7516) lies on these lines: {2, 3}, {6, 6101}, {125, 6146}, {155, 5085}, {156, 3796}, {182, 1216}, {569, 3917}, {578, 5447}, {1147, 3819}, {3098, 5446}, {5096, 5707}, {5422, 6243}, {5609, 5621}
As a point on the Euler line, X(7517) has Shinagawa coefficients (-E - 4 F, 5 E + 4 F).
X(7517) lies on these lines: {2, 3}, {49, 154}, {52, 161}, {155, 6243}, {156, 1993}, {159, 195}, {184, 5446}, {568, 1181}, {1147, 1495}, {1209, 3818}, {1614, 3060}, {5447, 5651}
X(7517) = crosspoint of circumcircle intercepts of 1st Droz-Farny circle
X(7517) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,26,3), (5,22,3)
X(7518) lies on these lines: {2, 3}, {92, 145}, {193, 5942}, {278, 3622}, {281, 3617}, {286, 3945}, {318, 4671}, {1785, 5703}, {1838, 3616}, {1859, 3869}, {1871, 3877}, {1895, 5342}
X(7518) = anticomplement of X(37180)
As a point on the Euler line, X(7519) has Shinagawa coefficients (-E - 4 F, 8 Sω).
X(7519) lies on these lines: {2, 3}, {94, 3424}, {148, 5987}, {193, 2854}, {251, 5319}, {612, 4330}, {614, 4325}, {3920, 4309}, {4299, 7292}, {4302, 5297}, {4317, 7191}, {5480, 6800}
X(7520) lies on these lines: {1, 3101}, {2, 3}, {8, 5285}, {56, 347}, {57, 4296}, {78, 610}, {197, 7080}, {280, 1436}, {1055, 3100}, {1394, 7013}, {3616, 4329}, {4339, 5310}
X(7521) lies on these lines: {2, 3}, {19, 1125}, {34, 3911}, {278, 1940}, {975, 6198}, {1108, 1148}, {1698, 1891}, {1829, 5439}, {1848, 3624}, {1861, 5438}, {3361, 5236}, {5603, 6197}
X(7522) lies on these lines: {2, 3}, {6, 226}, {9, 1730}, {57, 1713}, {72, 5271}, {278, 6356}, {306, 3419}, {329, 5278}, {1214, 1841}, {1746, 5776}, {4269, 5737}, {5712, 5802}
X(7523) lies on these lines: {2, 3}, {35, 1738}, {57, 201}, {284, 3216}, {579, 2303}, {1125, 5285}, {1175, 5135}, {1745, 2267}, {1791, 3418}, {3487, 7085}, {3601, 4256}, {3916, 5279}
X(7524) lies on these lines: {2, 3}, {53, 5747}, {92, 1482}, {278, 5901}, {281, 5690}, {318, 3940}, {1148, 1159}, {1838, 5886}, {1859, 5887}, {5174, 5790}, {5786, 6759}
As a point on the Euler line, X(7525) has Shinagawa coefficients (-5 E - 8 F, 7 E + 8 F).
X(7525) lies on these lines: {2, 3}, {49, 2979}, {143, 182}, {156, 1216}, {184, 6101}, {1092, 5944}, {1147, 3098}, {1352, 2916}, {1614, 6030}, {5012, 6243}, {5092, 5462}
As a point on the Euler line, X(7526) has Shinagawa coefficients (E + 4 F, E - 4 F).
X(7526) lies on these lines: {2, 3}, {6, 6102}, {154, 5944}, {155, 5876}, {182, 3357}, {185, 569}, {399, 3047}, {1092, 5891}, {1147, 4550}, {1181, 5663}, {5012, 6241}
As a point on the Euler line, X(7527) has Shinagawa coefficients (E + 4 F, 2 E - 4 F).
X(7527) lies on these lines: {2, 3}, {49, 5609}, {185, 575}, {323, 4550}, {567, 5663}, {569, 6241}, {576, 5889}, {597, 5621}, {1968, 5158}, {3292, 5907}, {5012, 6000}
As a point on the Euler line, X(7528) has Shinagawa coefficients (E, 3 E + 4 F).
X(7528) lies on these lines: {2, 3}, {32, 2165}, {51, 68}, {52, 1352}, {69, 6243}, {143, 6515}, {206, 569}, {389, 3818}, {1899, 5462}, {3574, 5654}
X(7528) = {X(2),X(4)}-harmonic conjugate of X(14790)
X(7528) = orthocentroidal circle inverse of X(14790)
As a point on the Euler line, X(7529) has Shinagawa coefficients (E - 2 F, 3 E + 2 F).
X(7529) lies on these lines: {2, 3}, {51, 155}, {154, 569}, {195, 5093}, {206, 5050}, {394, 5446}, {1181, 3066}, {1614, 5422}, {3167, 3527}, {5943, 6759}
As a point on the Euler line, X(7530) has Shinagawa coefficients (-E - 4 F, 7 E + 4 F).
X(7530) lies on these lines: {2, 3}, {143, 1181}, {155, 2930}, {567, 6800}, {576, 2393}, {1498, 6102}, {1539, 2931}, {3072, 7301}, {3073, 5363}, {3199, 3284}
X(7531) lies on these lines: {1, 1075}, {2, 3}, {48, 281}, {92, 1385}, {158, 2646}, {243, 3612}, {318, 5440}, {1451, 3075}, {1857, 4305}, {2188, 3488}
X(7532) lies on these lines: {1, 6708}, {2, 3}, {6, 1210}, {10, 7074}, {1125, 5930}, {1439, 5439}, {1724, 3075}, {1936, 5705}, {2360, 5786}, {3182, 5437}
X(7532) = complement of X(37180)
As a point on the Euler line, X(7533) has Shinagawa coefficients (3 E, 8 Sω).
X(7533) lies on these lines: {2, 3}, {51, 3410}, {115, 5987}, {323, 5480}, {3448, 3818}, {3583, 5297}, {3585, 7292}, {3920, 4857}, {5270, 7191}
X(7534) lies on these lines: {2, 3}, {53, 5292}, {92, 3927}, {155, 1839}, {273, 5708}, {278, 6147}, {912, 1871}, {1246, 3527}, {4269, 5788}
X(7535) lies on these lines: {2, 3}, {6, 169}, {57, 1724}, {958, 6708}, {1210, 1751}, {1698, 5285}, {1730, 5709}, {3927, 5279}, {4340, 5324}
X(7535) = complement of X(37179)
X(7536) lies on these lines: {2, 3}, {57, 6356}, {216, 1108}, {306, 5440}, {343, 1790}, {1038, 2999}, {1060, 5256}, {1062, 5287}, {2193, 5718}
X(7537) lies on these lines: {2, 3}, {278, 499}, {946, 6197}, {1068, 2006}, {1210, 1870}, {1852, 7173}, {1861, 6700}, {3086, 3176}
X(7538) lies on these lines: {1, 6360}, {2, 3}, {78, 3362}, {192, 3100}, {224, 1944}, {347, 3622}, {938, 3075}, {1210, 4257}
X(7538) = anticomplement of X(5125)
As a point on the Euler line, X(7539) has Shinagawa coefficients (2 E + 3 F, Sω).
X(7539) lies on these lines: {2, 3}, {206, 1853}, {233, 3162}, {1007, 1238}, {1899, 3589}, {2165, 3815}, {3763, 3917}, {3796, 3818}
X(7540) lies on these lines: {2, 3}, {52, 542}, {187, 1879}, {524, 6243}, {567, 5480}, {568, 1503}, {569, 5476}
X(7541) lies on these lines: {2, 3}, {92, 5587}, {653, 1737}, {908, 5081}, {946, 5174}, {1807, 1870}, {2907, 3574}
As a point on the Euler line, X(7542) has Shinagawa coefficients (-E - 6 F, E + 2 F).
X(7542) lies on these lines: {2, 3}, {49, 3564}, {54, 3580}, {343, 1147}, {973, 5462}, {1060, 5433}, {1062, 5432}
X(7543) lies on these lines: {2, 3}, {19, 3462}, {278, 1148}, {1838, 1940}, {1871, 5439}, {5174, 5440}
As a point on the Euler line, X(7544) has Shinagawa coefficients (E, 2 E + 4 F).
X(7544) lies on these lines: {2, 3}, {68, 3567}, {185, 3818}, {973, 2888}, {1352, 5889}, {1994, 6193}
X(7544) = orthocentroidal-circle-inverse of X(37444)
X(7544) = {X(2),X(4)}-harmonic conjugate of X(37444)
As a point on the Euler line, X(7545) has Shinagawa coefficients (2 F, E + 6 F).
X(7545) lies on these lines: {2, 3}, {143, 5609}, {195, 576}, {399, 568}, {567, 1495}, {3292, 5446}
X(7546) lies on these lines: {2, 3}, {52, 916}, {53, 58}, {386, 6748}, {1865, 5398}, {6147, 7282}
X(7547) lies on these lines: {2, 3}, {1853, 6241}, {1986, 3567}, {2548, 5523}, {2904, 3574}, {5876, 6746}
X(7548) lies on these lines: {2, 3}, {355, 1389}, {946, 5086}, {1699, 3878}, {3817, 5443}, {3869, 5587}
X(7549) lies on these lines: {2, 3}, {81, 5562}, {216, 5317}, {946, 5285}, {1871, 3101}, {5758, 7085}
As a point on the Euler line, X(7550) has Shinagawa coefficients (-4 Sω, E + 4 F).
X(7550) lies on these lines: {2, 3}, {54, 3292}, {323, 567}, {575, 1199}, {5012, 5891}, {5462, 5643}
X(7551) lies on these lines: {1, 3462}, {2, 3}, {92, 5886}, {281, 1953}, {1148, 3485}
As a point on the Euler line, X(7552) has Shinagawa coefficients (-2 E - 8 F, 3 E).
X(7552) lies on these lines: {2, 3}, {156, 2888}, {542, 1614}, {1494, 3470}, {5655, 5876}
As a point on the Euler line, X(7553) has Shinagawa coefficients (-E - 2 F, 5 E + 6 F).
X(7553) lies on these lines: {2, 3}, {52, 1503}, {569, 5480}, {1351, 5596}, {3564, 6243}
X(7554) lies on these lines: {2, 3}, {19, 1075}, {54, 1246}, {92, 3916}, {278, 603}
X(7555) lies on these lines: {2, 3}, {143, 575}, {2918, 2930}, {3292, 6101}, {5562, 5609}
As a point on the Euler line, X(7556) has Shinagawa coefficients (-2 E - 8 F, 5 E + 8 F).
X(7556) lies on these lines: {2, 3}, {54, 576}, {575, 3567}, {2917, 2930}, {5944, 6243}
X(7557) lies on these lines: {2, 3}, {226, 5262}, {975, 3586}, {5279, 6734}
As a point on the Euler line, X(7558) has Shinagawa coefficients (-2 E - 4 F, E).
X(7558) lies on these lines: {2, 3}, {68, 5012}, {70, 1176}, {1199, 6515}
X(7559) lies on these lines: {2, 3}, {1852, 3614}, {1891, 3817}, {2906, 3574}
X(7560) lies on these lines: {2, 3}, {278, 1950}, {3101, 3164}, {4652, 5271}
X(7561) lies on these lines: {2, 3}, {343, 1437}, {975, 1062}, {1214, 5433}
X(7562) lies on these lines: {2, 3}, {942, 2003}, {2262, 2323}
X(7563) lies on these lines: {2, 3}, {1210, 7282}, {2905, 3574}
X(7564) lies on these lines: {2, 3}, {1993, 6288}, {3818, 5448}
X(7565) lies on these lines: {2, 3}, {524, 2888}, {542, 3574}
As a point on the Euler line, X(7566) has Shinagawa coefficients (E + 2 F, 4 E + 6 F).
X(7566) lies on these lines: {2, 3}, {1351, 2888}, {3574, 3818}
X(7567) lies on these lines: {2, 3}, {226, 3075}, {333, 5562}
X(7568) lies on these lines: {2, 3}, {511, 6689}, {524, 1493}
As a point on the Euler line, X(7569) has Shinagawa coefficients (2 E + 6 F, E + 2 F).
X(7569) lies on these lines: {2, 3}, {1209, 1993}, {5422, 5449}
X(7570) lies on these lines: {2, 3}, {233, 6103}, {3448, 3589}
X(7570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5,23), (2,5169,7496), (2,5189,140), (2,7533,7495), (5,7495,7533), (5,7550,7565), (140,5133,5189), (858,3628,2), (7495,7533,23), (7539,7571,2)
As a point on the Euler line, X(7571) has Shinagawa coefficients (5 E + 6 F, 2 Sω).
X(7571) lies on these lines: {2, 3}, {2979, 3763}, {3410, 5050}
X(7571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5,22), (2,5056,6997), (2,5133,7485), (2,5169,7484), (2,6997,7495), (2,7391,140), (2,7394,7499), (2,7570,7539), (5,7499,7394), (5,7509,7566), (427,3628,2), (1656,7539,2), (3090,6804,5056), (3090,7577,5055), (5070,7484,2), (7394,7499,22)
X(7572) lies on these lines: {2, 3}, {95, 307}, {1210, 4256}
X(7572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,29), (2,3152,5), (2,7538,7532), (3,7532,7538), (3,7567,412), (631,7523,7573), (7532,7538,29),
X(7573) lies on these lines: {2, 3}, {95, 306}, {4855, 5271}
X(7573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,27), (2,464,469), (2,3151,5), (2,7560,7522), (3,7522,7560), (140,440,2), (631,7523,7572), (7522,7560,27),
X(7574) = 3 X(3) + (J^2 - 3) X(4)
As a point on the Euler line, X(7574) has Shinagawa coefficients (-3E, -7E + 16F).
X(7574) lies on these lines: {2, 3}, {50, 115}, {67, 265}, {125, 3581}, {128, 5099}, {339, 340}, {399, 1503}, {566, 5475}, {1216, 6288}, {1478, 7286}, {1479, 5160}, {3001, 6033}, {3519, 6145}}.{2,3}, {50,115}, {67,265}, {125,3581}
X(7574) = midpoint of X(4) and X(5189)
X(7574) = reflection of X(i) in X(j) for these (i,j): (2070,2072), (23,5), 3581,125), (3,858), (5899,403)
X(7574) = reflection of X(3) in the de Longchamps line
X(7574) = anticomplement of X(7575)
X(7574) = inverse-in-circumcircle of X(7502)
X(7574) = inverse-in-{circumcircle, nine-point-circle}-inverter of X(7495)
X(7574) = inverse-in-polar-circle of X(7576)
X(7574) = orthologic center of these triangles: AAOA to ABC
X(7574) = homothetic center of Ehrmann vertex-triangle and X(2)-Ehrmann triangle
X(7574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,7579,2), (5,7540,7545), (5,7555,7552), (20,7552,7555), (23,2071,7512), (382,7545,7540), (468,2072,1656), (1113,1114,7502), (1656,2070,468), (3153,5189,4), (7550,7565,5)
As a point on the Euler line, X(7575) has Shinagawa coefficients (E + 16F, -7E - 16F).
X(7575) lies on these lines: {2, 3}, {35, 5160}, {36, 7286}, {110, 3581}, {187, 2493}, {389, 5944}, {511, 1511}, {523, 5926}, {575, 5946}, {842, 2080}, {1154, 3292}, {1495, 5663}, {2770, 6236}, {2930, 2931}{2,3}, {110,3581}, {187,2493}
X(7575) = midpoint of X(i) and X(j) for these {i,j}: {3,23}, {110,3581}, {186,2070}, {842,2080}, {2071,5899}
X(7575) = reflection of X(i) in X(j) for these (i,j): (5,468), (858,140)
X(7575) = complement of X(7574)
X(7575) = reflection of X(5) in the orthic axis
X(7575) = inverse-in-circumcircle of X(381)
X(7575) = inverse-in-{circumcircle, nine-point-circle}-inverter of X(5169)
X(7575) = inverse-in-polar-circle of X(7577)
X(7575) = inverse-in-orthocentroidal-circle of X(7579)
X(7575) = radical trace of Kosnita and tangential circles
X(7575) = crossdifference of every pair of points on line X(566)X(647)
X(7575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,7579), (3,2070,23), (3,3518,546), (3,7545,7527), (3,7556,7555), (23,186,3), (24,186,468), (24,1658,5), (403,858,5576), (468,3575,403), (1113,1114,381), (3518,7527,7545), (6644,7502,549), (7527,7545,546)
As a point on the Euler line, X(7576) has Shinagawa coefficients (-2 F, 3 E + 6 F).
X(7576) = inverse-in-polar-circle of X(7574)
X(7576) lies on these lines:
{2, 3}, {52, 539}, {53, 112}, {99, 2383}, {317, 1238}, {524, 6403}, {542, 1843}, {827, 1300}, {1166, 1179}, {1503, 5890}, {1870, 5434}, {1974, 5476}, {3058, 6198}, {3654, 5090}, {5309, 5523}
X(7576) = midpoint of X(428) and X(3575)
X(7576) = reflection of X(i) in X(j) for these (i,j): (428,6756), (4,428)
X(7576) = Ehrmann-vertex-to-orthic similarity image of X(381)
X(7576) = X(5902)-of-orthic-triangle if ABC is acute
X(7576) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,24,1594), (4,25,403), (4,186,427), (4,3518,5), (4,3520,1595), (4,3542,7547), (4,3575,6240), (4,7487,24), (4,7505,7507), (20,7401,7509), (25,2070,3518), (382,5198,4), (472,473,467), (1906,3627,4), (2043,2044,7503), (3517,7507,7505), (3518,7488,24), (3575,6756,4)
As a point on the Euler line, X(7577) has Shinagawa coefficients (4 F, E + 4 F).
Let Sa be the similitude center of the orthocentroidal triangle and the A-altimedial triangle. Define Sb, Sc cyclically. Triangle SaSbSc is perspective to the orthic triangle at X(7577). (Randy Hutson, March 9, 2017)
X(7577) lies on these lines: {2, 3}, {6, 2914}, {93, 847}, {112, 5475}, {125, 5890}, {264, 328}, {265, 3043}, {1209, 6242}, {1560, 6032}, {1986, 5946}, {3567, 3574}, {3815, 5523}, {5449, 5889}{2,3}, {6,2914}, {112,5475}, {125,5890}
X(7577) = inverse-in-polar-circle of X(7575)
X(7577) = pole wrt polar circle of trilinear polar of X(7578)
X(7577) = X(48)-isoconjugate (polar conjugate) of X(7578)
X(7577) = harmonic center of nine-point circle and polar circle
X(7577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,186), (2,3153,3), (3,7547,4), (4,3090,7505), (4,5067,3147), (4,6143,3), (4,7505,3518), (5,427,403), (5,1594,4), (5,2072,2), (5,5133,3545), (5,5576,3091), (5,7579,5169), (24,7507,4), (378,381,4), (381,5094,378), (403,427,4), (403,1594,427), (1344,1345,24), (1346,1347,4), (1656,7507,24), (1656,7514,2), (2552,2553,7528), (3090,3545,7392), (3091,3541,4), (5055,7571,3090), (5094,7579,1594), (6829,6990,7522)
X(7578) lies on the Kiepert hyperbola and these lines: {2, 50}, {4, 567}, {6, 94}, {23, 262}, {76, 323}, {98, 5169}, {1994, 5392}
X(7578) = trilinear pole of line X(523)X(5926) (the radical axis of Kosnita and tangential circles)
X(7578) = X(48)-isoconjugate (polar conjugate) of X(7577)
X(7578) = pole wrt polar circle of trilinear polar of X(7577)
As a point on the Euler line, X(7579) has Shinagawa coefficients (16F + E, 16F + 7E) = (2|OH|2 - 3R2, 2|OH|2 - 9R2)
Let Sa be the similitude center of the orthocentroidal triangle and the A-altimedial triangle. Define Sb, Sc cyclically. X(7579) = X(3)-of-SaSbSc. (Randy Hutson, March 9, 2017)
X(7579) lies on these lines: {2,3}, {67,5476}
X(7579) = midpoint of X(1346) and X(1347)
X(7579) = harmonic center of nine-point circle and orthocentroidal circle
X(7579) = inverse-in-orthocentroidal-circle of X(7575)
X(7579) = center of inverse-in-orthocentroidal-circle-of-circumcircle
X(7579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,7575), (2,7574,3), (5,5169,381), (378,1995,26), (381,1656,1995), (381,5094,3), (431,6874,3506), (1344,1345,2070), (1594,7577,5094), (1995,7509,6644), (5094,7507,378), (5169,7577,5)
As a point on the Euler line, X(7580) has Shinagawa coefficients (2R + r, -4R - r).
Let A'B'C' be the 1st circumperp triangle. Let A* be the polar of A with respect to the A-excircle, and define B* and C* cyclically. Let A'' = B*∩C*, B'' = C*∩A*, C'' = A*∩B*. The triangles A'B'C' and A''B''C'' are homothetic at X(7580). Also, X(7580) = X(1993)-of-A'B'C' = X(1993)-of-A''B''C''. César Lozada, Anopolis, April 16, 2015.
Note that triangle A"B"C" is the 2nd extouch triangle.
X(7580) lies on these lines: {1, 1427}, {2, 3}, {6, 1754}, {9, 165}, {10, 5584}, {33, 1214}, {36, 3586}, {40, 64}, {43, 170}, {55, 226}, {56, 950}, {57, 5728}, {63, 971}, {84, 3916}, {100, 329}, {108, 7011}, {171, 1742}, {185, 5752}, {212, 2635}, {219, 2947}, {222, 1936}, {223, 7070}, {497, 1617}, {500, 5707}, {515, 956}, {517, 3870}, {518, 2900}, {573, 5776}, {581, 5706}, {940, 991}, {943, 5556}, {958, 5691}, {962, 3295}, {990, 3666}, {999, 3488}, {1001, 1699}, {1035, 5930}, {1040, 1465}, {1071, 1998}, {1088, 5088}, {1155, 1708}, {1259, 6259}, {1350, 1764}, {1385, 4666}, {1482, 3957}, {1745, 7078}, {1766, 3693}, {1779, 2245}, {1897, 6360}, {1961, 5527}, {2222, 2739}, {2941, 4436}, {2975, 5175}, {3219, 5779}, {3303, 4301}, {3332, 5712}, {3579, 5777}, {3587, 5720}, {3817, 4423}, {4254, 5746}, {4300, 5711}, {4329, 6356}, {4421, 5537}, {4551, 7074}, {4654, 5735}, {5120, 5802}, {5249, 5805}, {5435, 5809}, {5440, 6282}, {5730, 6261}, {5758, 6361}, {5762, 5905}, {5787, 6734}, {6260, 6745}, {7589, 7593}
X(7580) = midpoint of X(20) and X(6925)
X(7580) = reflection of X(i) in X(j) for these (i,j): (4,6907), (956,3428), (1012,3), (1709,4640)
X(7580) = anticomplement of X(8727)
X(7580) = X(343)-of-excentral-triangle
X(7580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7411,3), (3,4,405), (3,381,6883), (3,382,3560), (3,3149,474), (3,3830,7489), (3,6913,1006), (3,6918,631), (3,6985,3149), (4,376,6987), (4,631,6846), (4,1006,6913), (4,3651,3), (4,6829,381), (4,6889,5), (4,6908,442), (9,1750,5927), (20,411,3), (20,6838,6836), (25,440,405), (40,1490,72), (55,226,954), (100,329,1260), (165,1709,4640), (165,1750,9), (376,6905,3), (404,3522,3), (548,6924,3), (1006,6913,405), (1011,7522,405), (1155,1864,1708), (1708,1864,5729), (3146,5177,4), (3528,6940,3), (3529,6876,6906), (5658,5759,329), (6825,6851,6831), (6834,6899,6922), (6836,6838,5), (6840,6932,381), (6847,6988,7483), (6848,6865,4187), (6876,6906,3), (6890,6962,140), (6943,6960,1656)
Endo-homothetic centers: X(7581)-X(7588)
This preamble and centers X(7581)-X(7586) were contributed by César Eliud Lozada, April 22-27, 2015.
Suppose that U and V are a pair of homothetic triangles. There is a well-known point, X, called the homothetic center of U and V. For example, the homothetic center of the 1st circumperp and 2nd extouch triangles is X = X(7580). Now, we can "view" X from ABC as X(7580), or we can "view" X from U, in which case X is X(1993)-of-U, and by homothety, X is also X(1993)-of-V. In general, if X = X'-of-U (or equivalently, X = X'-of-V), then the point X' (as a function of the reference triangle ABC) is introduced here as the endo-homothetic center of U and V.
Following are examples using pairs of homothetic triangles (all homothethic to ABC):
{U, V} = {anticomplementary, Euler}; X = X(3091), X' = X(631)
{U, V} = {anticomplementary, Johnson}; X = X(4), X' = X(3)
{U, V} = {anticomplementary, medial}; X = X(4), X' = X(2)
{U, V} = {Euler, Johnson}; X = X(381), X' = X(2)
{U, V} = {Euler, medial}; X = X(5), X' = X(3)
{U, V} = {Johnson, medial}; X = X(5), X' = X(3)
{U, V} = {outer Grebe, inner Grebe}; X = X(6), X' = X(6)
{U, V} = {outer Grebe, anticomplementary}; X = X(1270), X' = X(3068)
{U, V} = {inner Grebe, anticomplementary}; X = X(1271), X' = X(3069)
{U, V} = {outer Grebe, anticomplementary}; X = X(1270), X' = X(7581)
{U, V} = {inner Grebe, anticomplementary}; X = X(1271), X' = X(7582)
{U, V} = {outer Grebe, Euler}; X = X(1271), X' = X(7583)
{U, V} = {outer Grebe, Euler}; X = X(6201), X' = X(7584)
{U, V} = {outer Grebe, Johnson}; X = X(6214), X' = X(7585)
{U, V} = {outer Grebe, Johnson}; X = X(6215), X' = X(7586)
{U, V} = {outer Grebe, Johnson}; X = X(5590), X' = X(7587)
{U, V} = {outer Grebe, Johnson}; X = X(5591), X' = X(7588)
Following are examples using pairs of homothetic triangles: 1st circumperp, 2nd circumperp, excentral, hexyl, intouch, and Yff-central:
{U, V} = {circumorthic, extangents}; X = X(6197), X' = X(7587)
{U, V} = {circumorthic, intangents}; X = X(6198), X' = X(7588)
{U, V} = {circumorthic, orthic}; X = X(4), X' = X(1)
{U, V} = {circumorthic, tangential}; X = X(24), X' = X(56)
{U, V} = {extangents, intangents}; X = X(55), X' = X(174)
{U, V} = {extangents, orthic}; X = X(19), X' = X(173)
{U, V} = {extangents, tangential}; X = X(55), X' = X(174)
{U, V} = {intangents, orthic}; X = X(33), X' = X(258)
{U, V} = {intangents, tangential}; X = X(55), X' = X(174)
{U, V} = {orthic, tangential}; X = X(25), X' = X(57)
{U, V} = {2nd extouch, 1st circumperp}; X = X(7580), X' = X(1993)
{U, V} = {2nd extouch, 2nd circumperp}; X = X(405), X' = X(7592)
{U, V} = {2nd extouch, excentral}; X = X(9), X' = X(6)
{U, V} = {2nd extouch, hexyl}; X = X(1490), X' = X(155)
{U, V} = {2nd extouch, intouch}; X = X(226), X' = X(184)
{U, V} = {2nd extouch, Yff central}; X = X(7593), X' = (pending)
Following are examples using pairs of homothetic triangles: circumorthic, extangents, intangents, orthic, and tangential:
{U, V} = {1st circumperp, 2nd circumperp}; X = X(3), X' = X(3)
{U, V} = {1st circumperp, excentral}; X = X(165), X' = X(2)
{U, V} = {1st circumperp, hexyl}; X = X(40), X' = X(4)
{U, V} = {1st circumperp, intouch}; X = X(55), X' = X(22)
{U, V} = {2nd circumperp, excentral}; X = X(1), X' = X(4)
{U, V} = {2nd circumperp, hexyl}; X = X(3576), X' = X(2)
{U, V} = {2nd circumperp, intouch}; X = X(56), X' = X(24)
{U, V} = {excentral, hexyl}; X = X(3), X' = X(5)
{U, V} = {excentral, intouch}; X = X(57), X' = X(25)
{U, V} = {hexyl, intouch}; X = X(1), X' = X(3)
{U, V} = {Yff-central, 1st circumperp}; X = X(7589), X' = (pending)
{U, V} = {Yff-central, 2nd circumperp}; X = X(7587), X' = (pending)
{U, V} = {Yff-central, excentral}; X = X(173), X' = X(19)
{U, V} = {Yff-central, hexyl}; X = X(7590), X' = (pending)
{U, V} = {Yff-central, intouch}; X = X(174), X' = X(55)
The homothetic center of the outer Grebe and Euler triangles is X(6201). See the preamble to X(7581).
X(7581) lies on these lines: {2, 3312}, {3, 7585}, {4, 6}, {5, 6418}, {20, 3311}, {30, 6417}, {69, 7376}, {140, 6395}, {193, 7389}, {371, 376}, {372, 631}, {381, 1131}, {382, 6500}, {388, 3301}, {391, 2047}, {394, 3539}, {485, 3069}, {486, 3545}, {497, 3299}, {546, 1132}, {550, 6199}, {590, 3525}, {615, 5067}, {637, 1992}, {638, 3618}, {639, 5860}, {1056, 1335}, {1058, 1124}, {1151, 3528}, {1152, 3524}, {1285, 6424}, {1578, 3538}, {1579, 3537}, {1702, 6361}, {1703, 5657}, {1993, 6805}, {3089, 5411}, {3091, 6428}, {3146, 6427}, {3522, 6221}, {3523, 6398}, {3529, 6419}, {3530, 6408}, {3533, 5420}, {3855, 6564}, {5304, 6811}, {5410, 7487}, {5422, 6806}
X(7581) = reflection of X(4) in X(7601)
X(7581) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,7582), (5,6418,7586), (6,1587,4), (6,3070,1588), (371,6460,376), (372,3068,631), (485,3069,3090), (485,6420,3069), (638,3618,7375), (1587,1588,3070), (1588,3070,4), (3069,3090,3317), (3312,7583,2), (5480,5871,4), (6419,6560,6459), (6459,6560,3529)
The homothetic center of the outer Grebe and Euler triangles is X(6201). See the preamble to X(7581).
X(7582) lies on these lines: {2, 3311}, {3, 7586}, {4, 6}, {5, 6417}, {20, 3312}, {30, 6418}, {69, 7375}, {140, 6199}, {193, 7388}, {371, 631}, {372, 376}, {381, 1132}, {382, 6501}, {388, 3299}, {394, 3540}, {485, 3545}, {486, 3068}, {497, 3301}, {546, 1131}, {550, 6395}, {590, 5067}, {615, 3525}, {637, 3618}, {638, 1992}, {640, 5861}, {1056, 1124}, {1058, 1335}, {1151, 3524}, {1152, 3528}, {1285, 6423}, {1578, 3537}, {1579, 3538}, {1702, 5657}, {1703, 6361}, {1993, 6806}, {3089, 5410}, {3091, 6427}, {3146, 6428}, {3522, 6398}, {3523, 6221}, {3529, 6420}, {3530, 6407}, {3533, 5418}, {3855, 6565}, {5304, 6813}, {5411, 7487}, {5422, 6805}
X(7582) = reflection of X(4) in X(7602)
X(7582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,7581), (5,6417,7585), (6,1588,4), (6,3071,1587), (371,3069,631), (372,6459,376), (486,3068,3090), (486,6419,3068), (637,3618,7376), (1587,1588,3071), (1587,3071,4), (3068,3090,3316), (3311,7584,2), (5480,5870,4), (6420,6561,6460), (6460,6561,3529)
The homothetic center of the outer Grebe and Euler triangles is X(6214). See the preamble to X(7581).
X(7583) lies on these lines: {2, 3312}, {3, 1587}, {4, 1131}, {5, 6}, {11, 3299}, {12, 3301}, {20, 6221}, {30, 371}, {32, 6401}, {61, 3391}, {62, 3366}, {140, 372}, {230, 5062}, {376, 6449}, {381, 1588}, {382, 6199}, {395, 3390}, {396, 3365}, {397, 3389}, {398, 3364}, {491, 3933}, {495, 1335}, {496, 1124}, {524, 639}, {546, 3071}, {548, 6200}, {549, 1152}, {550, 1151}, {615, 3628}, {631, 6398}, {632, 3594}, {640, 3589}, {1132, 3855}, {1271, 7376}, {1327, 6470}, {1377, 3820}, {1504, 5254}, {1505, 3815}, {1591, 1993}, {1592, 5422}, {1595, 3093}, {1596, 3092}, {1656, 3069}, {3090, 6428}, {3091, 6427}, {3317, 7486}, {3522, 6455}, {3523, 6450}, {3524, 6456}, {3526, 6395}, {3528, 6451}, {3529, 6447}, {3530, 6396}, {3534, 6407}, {3542, 5411}, {3592, 3627}, {3845, 6431}, {3850, 6565}, {3851, 6500}, {3856, 6435}, {5055, 6501}, {5412, 6756}
X(7583) = midpoint of X(371) and X(3070)
X(7583) = reflection of X(i) in X(j) for these (i,j): (6214,5), (7584,5305)
X(7583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7581,3312), (4,7585,3311), (5,6,7584), (5,5874,6289), (5,5875,1352), (6,485,5), (372,590,140), (381,6417,1588), (382,6199,6459), (1151,6560,550), (1152,5418,549), (1353,3767,7584), (1587,3068,3), (1656,6418,3069), (3071,6564,546), (6419,6564,3071)
The homothetic center of the inner Grebe and Euler triangles is X(6215). See the preamble to X(7581).
X(7584) lies on these lines: {2, 3311}, {3, 1588}, {4, 1132}, {5, 6}, {11, 3301}, {12, 3299}, {20, 6398}, {30, 372}, {32, 6402}, {61, 3392}, {62, 3367}, {140, 371}, {230, 5058}, {376, 6450}, {381, 1587}, {382, 6395}, {395, 3389}, {396, 3364}, {397, 3390}, {398, 3365}, {492, 3933}, {495, 1124}, {496, 1335}, {524, 640}, {546, 3070}, {548, 6396}, {549, 1151}, {550, 1152}, {590, 3628}, {631, 6221}, {632, 3592}, {639, 3589}, {1131, 3855}, {1270, 7375}, {1328, 6471}, {1378, 3820}, {1504, 3815}, {1505, 5254}, {1591, 5422}, {1592, 1993}, {1595, 3092}, {1596, 3093}, {1656, 3068}, {3090, 6427}, {3091, 6428}, {3316, 7486}, {3522, 6456}, {3523, 6449}, {3524, 6455}, {3526, 6199}, {3528, 6452}, {3529, 6448}, {3530, 6200}, {3534, 6408}, {3542, 5410}, {3594, 3627}, {3845, 6432}, {3850, 6564}, {3851, 6501}, {3856, 6436}, {5055, 6500}, {5413, 6756}
X(7584) = midpoint of X(372) and X(3071)
X(7584) = reflection of X(i) in X(j) for these (i,j): (6215,5), (7583,5305)
X(7584) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7582,3311), (4,7586,3312), (5,6,7583), (5,5874,1352), (5,5875,6290), (6,486,5), (371,615,140), (381,6418,1587), (382,6395,6460), (1151,5420,549), (1152,6561,550), (1353,3767,7583), (1588,3069,3), (1656,6417,3068), (3070,6565,546), (6420,6565,3070)
The homothetic center of the outer Grebe and medial triangles is X(5590). See the preamble to X(7581).
X(7585) lies on these lines: {2, 6}, {3, 7581}, {4, 1131}, {5, 6417}, {20, 371}, {30, 6199}, {44, 6351}, {140, 6418}, {176, 2082}, {194, 6462}, {372, 3523}, {376, 6221}, {390, 2066}, {393, 588}, {485, 1132}, {486, 3590}, {548, 6407}, {549, 6395}, {631, 3312}, {962, 1702}, {1100, 6352}, {1151, 3522}, {1162, 1164}, {1249, 1585}, {1267, 3759}, {1504, 5286}, {1656, 3316}, {1743, 5393}, {2067, 3600}, {2285, 7347}, {3070, 3146}, {3071, 3832}, {3085, 3301}, {3086, 3299}, {3090, 6427}, {3317, 3628}, {3524, 6398}, {3525, 6428}, {3526, 6501}, {3528, 6449}, {3543, 6561}, {3758, 5391}, {3839, 6564}, {4232, 5413}, {5265, 6502}, {5281, 5414}, {5411, 6353}, {5412, 6995}, {5418, 6420}, {6776, 7374}, {6807, 7592}
X(7585) = anticomplement of X(5590)
X(7585) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,7586), (2,193,1270), (5,6417,7582), (6,590,3069), (6,3068,2), (193,5304,7586), (371,1587,20), (485,1588,3091), (485,6419,1588), (491,3618,2), (590,3069,2), (1151,6460,3522), (1588,3091,1132), (1991,3589,5591), (3068,3069,590), (3070,3592,6459), (3070,6459,3146), (3311,7583,4), (3589,5591,2)
The homothetic center of the inner Grebe and medial triangles is X(5591). See the preamble to X(7581).
X(7586) lies on these lines: {2, 6}, {3, 7582}, {4, 1132}, {5, 6418}, {20, 372}, {30, 6395}, {44, 6352}, {140, 6417}, {175, 2082}, {194, 6463}, {371, 3523}, {376, 6398}, {390, 5414}, {393, 589}, {485, 3591}, {486, 1131}, {548, 6408}, {549, 6199}, {631, 3311}, {962, 1703}, {1100, 6351}, {1152, 3522}, {1163, 1165}, {1249, 1586}, {1267, 3758}, {1505, 5286}, {1656, 3317}, {1743, 5405}, {2066, 5281}, {2067, 5265}, {2285, 7348}, {3070, 3832}, {3071, 3146}, {3085, 3299}, {3086, 3301}, {3090, 6428}, {3316, 3628}, {3524, 6221}, {3525, 6427}, {3526, 6500}, {3528, 6450}, {3543, 6560}, {3600, 6502}, {3759, 5391}, {3839, 6565}, {4232, 5412}, {5410, 6353}, {5413, 6995}, {5420, 6419}, {6776, 7000}, {6808, 7592}
X(7586) = anticomplement of X(5591)
X(7586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,7585), (2,193,1271), (5,6418,7581), (6,615,3068), (6,3069,2), (193,5304,7585), (372,1588,20), (486,1587,3091), (486,6420,1587), (492,3618,2), (591,3589,5590), (615,3068,2), (1152,6459,3522), (1587,3091,1131), (3068,3069,615), (3071,3594,6460), (3071,6460,3146), (3312,7584,4), (3589,5590,2)
The homothetic center of the circumorthic and extangents triangles is X(6197). See the preamble to X(7581).
X(7587) lies on these lines: {1, 168}, {3, 7589}, {21, 177}, {56, 174}, {236, 958}, {258, 3361}, {266, 361}, {405, 7593}, {3576, 7590}
X(7587) = homothetic center of the Yff-central triangle and 2nd circumperp triangle
X(7587) = {X(56),X(174)}-harmonic conjugate of X(7588)
The homothetic center of the circumorthic and intangents triangles is X(6198). See the preamble to X(7581).
X(7588) lies on these lines: {1, 164}, {56, 174}, {100, 260}, {173, 3361}, {259, 978}, {289, 361}, {958, 7028}, {3333, 7590}{1,164}, {56,174}
X(7588) = {X(56),X(174)}-harmonic conjugate of X(7587)
X(7589) lies on these lines: {1, 164}, {3, 7587}, {40, 7590}, {55, 174}, {105, 3659}, {165, 173}, {177, 260}, {236, 1376}, {259, 503}, {1001, 7028}, {7580, 7593}
X(7590) lies on these lines: {1, 167}, {3, 173}, {40, 7589}, {236, 936}, {258, 942}, {1490, 7593}, {3333, 7588}, {3576, 7587}
X(7591) lies on this line: {259,260}
X(7591) = trilinear square root of X(7066)
X(7592) lies on these lines: {2, 155}, {3, 54}, {4, 6}, {5, 5422}, {20, 1994}, {22, 52}, {24, 184}, {25, 1614}, {26, 568}, {49, 6644}, {51, 6759}, {69, 7383}, {74, 3516}, {81, 6833}, {110, 6642}, {143, 7517}, {154, 3518}, {156, 5946}, {161, 973}, {182, 5562}, {185, 378}, {193, 7400}, {323, 3523}, {343, 7558}, {394, 631}, {399, 3851}, {436, 1075}, {567, 7526}, {569, 7503}, {575, 5907}, {940, 6952}, {1173, 3527}, {1216, 7485}, {1353, 6823}, {1513, 5359}, {1593, 6241}, {1594, 1899}, {1995, 5462}, {3060, 7387}, {3193, 6827}, {3547, 6515}, {3549, 3580}, {3564, 7399}, {3796, 7512}, {4383, 6949}, {5050, 7395}, {5640, 7529}, {5707, 6830}, {6193, 6815}, {6807, 7585}, {6808, 7586}
X(7592) = reflection of X(7503) in X(569)
X(7592) = crossdifference of every pair of points on line X(520)X(12077)
X(7592) = Cundy-Parry Phi transform of X(97)
X(7592) = Cundy-Parry Psi transform of X(53)
X(7592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1199,6), (6,1181,4), (54,5890,3), (156,5946,7506), (182,5562,7509), (184,389,24), (185,578,378), (1614,3567,25), (5012,5889,3)
X(7593) lies on these lines: {4, 5935}, {9, 173}, {174, 226}, {405, 7587}, {1490, 7590}, {7580, 7589}
A = 1 : 0 : 0
AA = -(S2 + SBSC)/a : cS + bSB : bS + cSC
AB = -(cSC - bS)/a : S + bc : -SA
AC = -(bSB - cS)/a : -SA : S + bc
The Pamfilos-Zhou B- and C-rectangles, RB and RC, are defined cyclically.
Let A' = BCBA∩CACB, and define B' and C' cyclically. The 1st Pamfilos-Zhou triangle, A'B'C', is perspective to ABC, and the perspector is X(7594).
Let A'' = CAAC∩ABBA, and define B'' and C'' cyclically. The 2nd Pamfilos-Zhou triangle, A''B''C'', is perspective to ABC, and the perspector is X(7595).
See Paris Pamfilos, Li Zhou, César Lozada, Randy Hutson, ADGEOM 2497 (May 2015)
Trilinears for the A-vertex of the 1st Pamfilos-Zhou triangle:
A' = a^6+b*c*(a^4-4*S*(b+c)*a+(b^2-c^2)^2)-(b^2+c^2)*(b+c)^2*a^2 :
-(a^2-b^2+c^2)*(a^3*b+b*(b^2+c^2)*a+2*c^2*S) :
-(a^2+b^2-c^2)*(a^3*c+c*(b^2+c^2)*a+2*b^2*S)
Trilinears for the A-vertex of the 2nd Pamfilos-Zhou triangle:
A" = -2*(b+c)*S+(a-b-c)*((b+c)*a+(b-c)^2):
(2*(a-c)*a*S-(a-b-c)*(a^2*c+(b-c)*(b^2+a*b+c^2)))/b :
(2*(a-b)*a*S-(a-c-b)*(a^2*b+(c-b)*(c^2+a*c+b^2)))/c
X(7594) lies on these lines: {1, 3103}, {43, 7347}, {1659, 7146}
X(7595) lies on the Feuerbach hyperbola and these lines: {1, 7596}, {8, 637}
X(7596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8224,8225,3), (8228,8230,5)
X(7596) = X(3)-of-2nd-Pamfilos-Zhou-triangle
Touchpoints of pairs of circles: X(7597)-X(7602)
Tran Quang Hung found that the incircle of the hexyl triangle is tangent to the circumcircle of ABC. The point of tangency, or touchpoint, is X(7597). Let S be the set of incircles, circumcircles, and nine-point circles of 60 central triangles listed in MathWorld. César Lozada checked pairs of circles in S for tangency and found several new points, X(7598)-X(7602). He notes (May 10, 2015) that these six points together with points already in ETC account for all touchpoints of pairs of circles in S.
See X(7606) and ADGEOM #2541, Randy Hutson, May 19, 2015.
Let PA be the parabola with focus A and directrix BC. Let LA be the polar of X(3) with respect to PA, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The lines AA', BB', CC' concur in X(262), and X(7610) = X(3)-of-A'B'C'. (Randy Hutson, May 27, 2015) X(7610) lies on the McCay circumcircle and these lines: {2,6}, {3,543}, {30,7615}, {98,6233}, {115,5077}, {376,7620}, {381,2080}, {538,1153}, {549,7618}, {754,5055}, {2021,5215}, {2453,7426}, {3524,5485}, {5503,7607} X(7610) = reflection of X(i) in X(j) for these (i,j): (3,5569), (381,7617)Let A'' be the insimiliceter of the A-McCay circle and the A-Neuberg circle, and define B'' and C'' cyclically. The triangle A''B''C'' is perspective to ABC, and the perspector is X(7612). Let A* be the exsimilicenter of the A-McCay circle and the A-Neuberg circle, and define B* and C* cyclically. Then A*B*C* is the medial triangle of ABC.
See X(7606) and ADGEOM #2541, Randy Hutson, May 19, 2015. X(7612) lies on the Kiepert hyperbola, the cubic K698, and these lines: {2,3167}, {3,2996}, {4,230}, {5,5395}, {10,7410}, {69,6036}, {76,631}, {83,3090}, {94,7493}, {98,5033}, {262,5052}, {376,671}, {487,5490}, {488,5491}, {598,3545}, {1131,6811}, {1132,6813}, {1513,3424}, {1916,6194}, {2052,6353}, {3524,5485}, {5392,7494}, {6504,7386} X(7612) = isogonal conjugate of X(1351)The bisector circle of the McCay circles (which bisects all three McCay circles) has center X(7615) and radius-squared (3 - cot2ω)[27R2 - (S cot ω)(6 + cot2ω)]/81.
See ADGEOM #2542, César Lozada, May 20, 2015. If you have The Geometer's Sketchpad, you can view X(7615). X(7615) lies on these lines: {2,99}, {4,3849}, {6,3363}, {30,7610}, {262,538}, {376,5569}, {381,524}, {754,3839}, {1153,3524}, {1506,2996}, {1992,5475}, {5368,5395} X(7615) = reflection of X(376) in X(5569)See X(7617).
Let G be the centroid of a triangle ABC, andSee X(7617).
X(7622) lies on these lines: {2,99}, {3,3849}, {39,5215}, {182,524}, {538,1153}, {625,5077}, {754,3524}, {3055,3363}, {5969,7606} X(7622) = midpoint of X(2) and X(7618)
X(7664) = complement of X(31125)
X(7664) = barycentric product X(316)*X(524)
X(7665) = anticomplement of X(30786)
X(7667) = (|OH|2 - 5 R2)*X(2) + 2Sω*X(3) = 2 X[20] + X[1885] = 4 X[3] - X[3575] = 7 X[3528] - X[6240] = 5 X[631] - 2 X[6756] = 3 X[5054] - X[7540] = 4 X[140] - X[7553] = 3 X[3524] - X[7576] = (2R2 - Sω)*X(2) + Sω*X(3)
As a point on the Euler line, X(7667) has Shinagawa coefficients (2E + F, -3E - 3F). (César Lozada, June 4, 2015)
Let A'B'C' be the medial triangle of a triangle ABC, and let O = X(3). Let A'' = reflection of A in line OA', and define B'' and C'' cyclically. Let A* be the reflection of A' in line OA, and define B* and C* cyclically. Let MA = midpoint of segment A''A*, and define MB and MC cyclically. Then X(7667) = centroid of MAMBMC = (Euler line of ABC)∩(Euler line of MAMBMC). Barycentrics for MA are as follows:
a^2 b^4-b^6-4 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6 : b^2 (-a^2 b^2+b^4+2 b^2 c^2-3 c^4) : c^2 (-3 b^4-a^2 c^2+2 b^2 c^2+c^4) See Hyacinthos 23268 and X(7734). X(7667) lies on these lines: {2,3}, {66,599}, {338,523}, {343,3098}, {524,3313}, {571,5306}, {597,5157}, {612,7354}, {614,6284}, {1184,2549}, {1350,1899}, {1503,3917}, {1627,5305}, {2979,3564}, {3058,4319}, {4320,5434} X(7667) = reflection of X(428) in X(2)The Honsberger triangle is defined at X(7671).
X(7671) lies on these lines: {1,651}, {7,354}, {9,1174}, {144,4430}, {165,1445}, {374,5838}, {390,517}, {516,5902}, {518,1992}, {527,3873}, {971,5049}, {1001,4511}, {1125,5696}, {3059,3740}, {3295,5729}, {3303,5220}, {3576,7675}, {3616,5784}, {3817,7678}, {3868,5698}, {4907,7190} X(7671) = midpoint X(144) and X(4430)The Honsberger triangle is defined at X(7671).
X(7672) lies on these lines: {1,1170}, {2,5173}, {6,4318}, {7,8}, {9,1405}, {10,7679}, {40,7675}, {56,3889}, {57,100}, {72,5686}, {142,4848}, {145,7674}, {210,5226}, {226,3681}, {354,5218}, {390,517}, {516,5903}, {651,2263}, {758,5223}, {942,5657}, {946,7672}, {960,4323}, {962,5809}, {1001,2099}, {1156,2800}, {1456,4663}, {1465,3240}, {1617,3957}, {1621,1708}, {1758,2177}, {2093,5732}, {2262,5838}, {2801,4312}, {3057,5572}, {3242,5228}, {3339,3874}, {3361,3881}, {3485,3876}, {3555,3600}, {3671,5904}, {4084,5850}, {4332,5247}, {5045,5265}, {5542,5902}, {5805,6797}, {5817,5887}, {7174,7190} X(7672) = reflection of X(i) in X(j) for these (i,j): (3057, 5572), (3059,5836), (3869,9), (390,5728), (7,65)The Honsberger triangle is defined at X(7671).
X(7673) lies on these lines: {1,7676}, {7,3057}, {10,7678}, {40,7677}, {142,3890}, {390,517}, {516,5697}, {518,3644}, {946,7679}, {1156,2802}, {1697,2346}, {2550,3877}, {3869,5853} X(7673) = reflection of X(7) in X(3057)The Honsberger triangle is defined at X(7671).
X(7674) lies on these lines: {2, 2346}, {7, 3174}, {8, 9}, {20, 518}, {142, 3158}, {144, 4661}, {145, 7672}, {153, 528}, {480, 497}, {516, 6223}, {954, 5082}, {2550, 2894}, {3243, 3600}, {3880, 5572}, {4294, 5223}, {5218, 6067} X(7674) = reflection of X(i) and X(j) for these (i,j): (7, 3174), (2550, 3913), (6601, 6600)The Honsberger triangle is defined at X(7671).
X(7675) lies on these lines: {1, 7}, {2, 5809}, {3, 1445}, {8, 3174}, {9, 21}, {27, 33}, {37, 5781}, {40, 7672}, {55, 63}, {56, 5572}, {57, 7411}, {81, 7070}, {84, 1803}, {142, 377}, {144, 5766}, {200, 5273}, {224, 1001}, {464, 1040}, {497, 4666}, {528, 5832}, {601, 3561}, {943, 7330}, {954, 971}, {958, 3059}, {997, 5785}, {1004, 3306}, {1071, 3295}, {1156, 6326}, {1253, 3751}, {1259, 6600}, {1490, 5703}, {1697, 3243}, {1750, 5226}, {1837, 3826}, {1864, 3305}, {1998, 5744}, {2550, 3486}, {2900, 5745}, {3358, 6906}, {3488, 6916}, {3576, 7671}, {3586, 6839}, {3811, 5223}, {3872, 5853}, {5587, 7679}, {5720, 5817}X(7675) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,990,7190), (1,991,77), (1,1742,2263), (1,4326,390), (1,5732,7), (3,5728,1445), (7,4313,390), (4292,4304,4302), (4292,5542,7), (7672,7676,40).
The Honsberger triangle is defined at X(7671).
X(7676) lies on these lines: {2, 7678}, {3, 390}, {4, 7679}, {7, 55}, {9, 100}, {21, 2550}, {35, 411}, {40, 7672}, {63, 3174}, {142, 1621}, {144, 6600}, {165, 1445}, {171, 4343}, {404, 1001}, {480, 4421}, {518, 3871}, {528, 4996}, {651, 1253}, {954, 3651}, {1030, 7437}, {1155, 5572}, {1292, 2369}, {2975, 5853}, {3059, 4640}, {3550, 4335}, {3579, 5728}, {3746, 5542}, {3826, 5047}, {4294, 6986}, {4302, 6912}, {4313, 5584}, {5281, 7580}, {5686, 5687}{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,390,7677), (7,55,2346), (40,7675,7672), (165,4326,1445), (1253,1742,651), (1445,4326,7671)
The Honsberger triangle is defined at X(7671). The Honsberger triangle, defined at X(7671), is also perspective to the circumcircle-midarc triangle, with perspector X(7677).
X(7677) lies on these lines: X(7677) lies on these lines: {1, 1170}, {2, 1617}, {3, 390}, {4, 7678}, {7, 21}, {9, 604}, {36, 516}, {55, 5435}, {57, 1621}, {59, 518}, {77, 7290}, {100, 2078}, {104, 971}, {105, 927}, {142, 5253}, {145, 6600}, {198, 5838}, {226, 5284}, {238, 651}, {241, 1279}, {244, 1758}, {278, 4233}, {388, 5047}, {404, 2550}, {405, 3600}, {411, 3086}, {480, 6049}, {496, 3651}, {497, 7411}, {499, 6915}, {527, 5193}, {528, 5172}, {602, 3562}, {934, 2725}, {943, 5045}, {954, 999}, {956, 5686}, {958, 4308}, {993, 4321}, {1056, 6883}, {1214, 7191}, {1385, 5728}, {1386, 1442}, {1443, 1456}, {1465, 7292}, {1467, 5250}, {1708, 3873}, {1788, 3871}, {2178, 5819}, {2646, 5572}, {3174, 4855}, {3218, 3660}, {3361, 5248}, {3576, 7671}, {3737, 4017}, {3826, 5433}, {3988, 5223}, {4293, 6912}, {4298, 5259}, {4315, 5251}, {4322, 5247}, {4423, 5226}, {4578, 4899}, {5260, 6666}, {5274, 7580}, {5542, 5563}, {5731, 5809}X(7677) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1445,7672), (3,390,7676), (56,1001,7), (238,1458,651), (241,1279,4318), (1420,2975,1476), (2078,3911,100)
The Honsberger triangle is defined at X(7671).
X(7678) lies on these lines: X(7678) lies on these lines: {2, 7676}, {4, 7677}, {5, 390}, {7, 11}, {149, 6600}, {497, 2346}, {516, 6943}, {946, 7672}, {954, 6990}, {1001, 2476}, {1445, 1699}, {1479, 6991}, {2550, 4193}, {3059, 5087}, {3817, 7671}, {3826, 7173}, {3829, 6067}, {4197, 5225}{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,390,7679)
The Honsberger triangle is defined at X(7671).
X(7679) lies on these lines: X(7679) lies on these lines: {2, 1617}, {4, 7676}, {5, 390}, {7, 12}, {10, 7672}, {119, 1156}, {498, 6828}, {516, 6932}, {954, 6829}, {1001, 4193}, {1445, 1698}, {2346, 3085}, {2476, 2550}, {3925, 5226}, {4321, 5726}, {5587, 7675}X(7679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,390,7678), (12,3826,7)
Let P be a point in the plane of a triangle ABC, and let A'B'C' be the circumcevian triangle of P. Let AB be the reflection of A' is line AC, and define BC and CA cyclically. Let AC be the reflection of A' is line AB, and define BA and CB cyclically. Let MA be the midpoint of segment ABAC, and define MB and MC cyclically. (The circumcircle of MAMBMC passes through X(4).) Antreas Hatzipolakis calls attention to the circumcenter of MAMBMC, here named the Hatzipolakis-Moses image of P and denoted by HM(P). If barycentrics for P are given by P = p : q : r, then
HM(P) = -2a2(b4 + c4 - a2b2 - a2c2 - 2b2c2)qr + b2(a2 - b2 + c2)(a2 + b2 - c2)pr + c2(a2 - b2 + c2)(a2 + b2 - c2)pq : : The appearance of (i,j) in the following list indicates that HM(X(i)) = X(j): (1,946), (2,5480), (3,4), (4,5), (5,3574), (6,381), (7,7680), (8,7681), (9,7682), (10,7683), (13,7684), (14,7685), (15,5478), (16,5479), (21,7686). Hatzipolakis asks further: as P moves on the Euler line, what is the locus of HM(P). Moses answers that if P on the Euler line is parameterized by P(k) = a2SA + kSBSC : : , then HM(P(k)) = 8 a^2 b^2 c^2 (a^2+b^2-c^2) (a^2-b^2+c^2)
X(7680) = midpoint of X(i) and X(j) for these {i,j}: {4, 55}, {1012, 1478}
X(7680) = reflection of X(i) and X(j) for these (i,j): (3, 6690), (2886, 5), (6907, 3822)
X(7680) = complement of X(3428)
X(7680) = X(3427)-complementary conjugate of X(10)
X(7680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,946,7681), (10,946,7686), (3814,3817,5), (5603,6830,11), (5657,6829,3925), (5886,6882,3816)
Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7681) lies on these lines: {1, 1532}, {3, 3816}, {4, 11}, {5, 10}, {8, 6945}, {12, 2098}, {40, 4187}, {46, 1699}, {55, 6834}, {65, 1519}, {100, 6979}, {117, 2841}, {119, 1482}, {235, 1828}, {355, 3813}, {381, 529}, {496, 515}, {497, 6848}, {499, 1012}, {516, 3825}, {942, 1538}, {958, 6893}, {962, 4193}, {999, 6256}, {1001, 6825}, {1125, 6907}, {1210, 6001}, {1376, 6944}, {1479, 3149}, {1484, 6246}, {1512, 3057}, {1537, 5903}, {1621, 6960}, {1656, 3826}, {2478, 3428}, {2550, 6964}, {3035, 6959}, {3085, 6969}, {3090, 3925}, {3091, 3436}, {3434, 6953}, {3485, 5804}, {3560, 4999}, {3577, 5559}, {3616, 6932}, {3652, 5805}, {4294, 6927}, {4413, 6983}, {4423, 6889}, {4853, 5587}, {5204, 6938}, {5217, 6880}, {5231, 5715}, {5432, 6949}, {5433, 6906}, {5584, 6947}, {5657, 6975}, {5722, 6261}, {5817, 6067}, {5840, 6924}, {5886, 6842}, {6284, 6905}, {6361, 6963}, {6667, 6958}, {6690, 6863}, {6830, 7173}
X(7681) = midpoint of X(i) and X(j) for these {i,j}: {4, 56}, {1479, 3149}
X(7681) = reflection of X(i) in X(j) for these (i,j): (3,6691), (1329,5), (6922,3825)
X(7681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,946,7680), (946,7682,7686), (5603,6941,12)
Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7682) lies on these lines: {1, 5804}, {2, 6282}, {3, 6692}, {4, 57}, {5, 10}, {9, 6939}, {40, 5084}, {63, 6957}, {78, 6953}, {116, 2823}, {117, 2835}, {142, 6907}, {226, 1532}, {329, 3091}, {381, 527}, {515, 999}, {516, 3359}, {519, 5720}, {908, 6945}, {936, 6964}, {938, 1490}, {942, 6260}, {950, 3149}, {962, 6919}, {1000, 1512}, {1012, 3911}, {1125, 6825}, {1699, 1737}, {1750, 5768}, {2094, 3839}, {2262, 5514}, {3306, 6925}, {3421, 4847}, {3601, 6927}, {3634, 6887}, {4297, 6985}, {4298, 6256}, {4304, 6905}, {4311, 5193}, {4314, 6796}, {4317, 5691}, {5249, 6932}, {5325, 5771}, {5436, 6988}, {5437, 6916}, {5657, 7308}, {5705, 6846}, {5708, 6259}, {5709, 6893}, {5745, 6913}, {5748, 6735}, {5785, 6843}, {6244, 6684}, {6261, 6738}, {6700, 6944}
X(7682) = midpoint of X(i) and X(j) for these {i,j}: {4, 57}, {1750, 5768}
X(7682) = reflection of X(i) in X(j) for these (i,j): (3,6692), (3452,5), (6244,6684)
X(7682) = complement of X(6282)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1210,6245), (5,5806,946), (5603,6969,5219), (5804,6848,1), (7681,7686,946)
Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7683) lies on these lines: {2, 3430}, {3, 6693}, {4, 58}, {5, 141}, {113, 2842}, {116, 132}, {117, 7686}, {381, 540}, {758, 946}, {1046, 1699}, {1210, 1905}, {1330, 3091}, {2051, 6831}, {2476, 3794}, {4231, 5358}, {5429, 5691}
X(7683) = midpoint X(4) and X(58)
X(7683) = reflection of X(i) and X(j) for these (i,j): (3,6693), (3454,5)
X(7683) = complement X(3430)
X(7683) = X(3429)-complementary conjugate of X(10)
Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7684) lies on these lines: {3,6671}, {4,15}, {5,141}, {13,98}, {30,5459}, {107,470}, {187,5318}, {262,6114}, {381,531}, {532,5617}, {621,3091}, {1513,6115}, {5460,5476}, {6117,6530} X(7684) = midpoint of X(i) and X(j) for these {i,j}: {3, 6671}, {4,15}, {13,1080}, {623,5}
Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7685) lies on these lines: {3,6672}, {4,16}, {5,141}, {14,98}, {30,5460}, {107,471}, {187,5321}, {262,6115}, {381,530}, {533,5613}, {622,3091}, {1513,6114}, {5459,5476}, {6116,6530} X(7685) = midpoint of X(i) and X(j) for these {i,j}: {3,6672}, {4,16}Hatzipolakis-Moses image is defined at X(7680). See Hyacinthos 23282, June 4, 2015
X(7686) lies on these lines: {1, 227}, {3, 3812}, {4, 65}, {5, 10}, {8, 6835}, {12, 1512}, {40, 405}, {46, 1012}, {72, 5587}, {84, 3339}, {117, 7683}, {125, 429}, {200, 5730}, {210, 5818}, {281, 2262}, {354, 944}, {355, 518}, {381, 5887}, {497, 5804}, {515, 942}, {516, 3754}, {758, 5777}, {950, 5842}, {958, 5709}, {962, 2478}, {971, 5884}, {997, 6918}, {1071, 5586}, {1104, 3072}, {1155, 6906}, {1385, 3742}, {1455, 3075}, {1482, 3811}, {1598, 3556}, {1699, 5903}, {1706, 6769}, {1728, 2093}, {1737, 6831}, {1788, 6847}, {1826, 5798}, {1829, 3574}, {2551, 5758}, {2646, 6905}, {2771, 6246}, {2800, 6797}, {2817, 5908}, {2829, 4292}, {3057, 3085}, {3091, 3869}, {3485, 6848}, {3529, 5918}, {3555, 5881}, {3560, 4640}, {3576, 5439}, {3616, 6962}, {3660, 4311}, {3671, 6260}, {3683, 6920}, {3698, 5657}, {3827, 5480}, {3838, 6842}, {3877, 6933}, {3885, 5734}, {3916, 5535}, {3922, 6361}, {4018, 5693}, {4297, 5883}, {4511, 6915}, {4662, 5790}, {4679, 6898}, {5045, 5882}, {5086, 6839}, {5258, 5536}, {5713, 5725}, {5794, 6826}, {5880, 6850}, {5886, 6863}, {6675, 6684}, {6953, 7080}
X(7686) = midpoint of X(i) and X(j) for these {i,j}: {4, 65}, {1071, 5691}, {3555, 5881}, {4018, 5693}
X(7686) = reflection of X(i) and X(j) for these (i,j): (3,3812), (946,5806), (960,5), (5882,5045)
X(7686) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,946,7680), (946,7682,7681), (4018,5927,5693), (5691,5902,1071)
The Jerabek hyperbola of the Euler triangle is the Hatzipolakis-Moses image of the Euler line; see X(7680) and Hyacinthos 23282, June 4, 2015
X(7687) lies on these lines: {3, 6723}, {4, 74}, {5, 1511}, {6, 13}, {30, 6699}, {51, 1986}, {67, 3531}, {110, 578}, {146, 3839}, {323, 1568}, {389, 546}, {403, 1495}, {541, 1539}, {973, 1112}, {974, 1514}, {1192, 5076}, {1531, 3580}, {1550, 2682}, {1597, 2935}, {1853, 3426}, {1974, 5622}, {2914, 3574}, {3448, 3832}, {3545, 5642}, {3857, 5609}, {5318, 6111}, {5321, 6110} X(7687) = midpoint of X(i) and X(j) for these {i,j}: {4,125}, {113, 265}There is a unique equilateral triangle AA1A2 inscribed in the circumcircle of triangle ABC, where, for concreteness, the labels are fixed so that AA1A2 has the same orientation as ABC. Let BB1B2 and CC1C2 be the corresponding equilateral triangles. Let A' = B1B2∩C1C2, B' = C1C2∩A1A2 and C' = A1A2∩B1B2. The triangle A'B'C' is here named the Trinh triangle of ABC after Trinh Xuân Minh. César Lozada found perspectivities and orthologies, as follows.
The Trinh triangle is the reflection in X(3) of the Kosnita triangle. (Randy Hutson, Octover 13, 2015) The appearance of (T,i) in the following list means that the triangle T is perspective to A'B'C', with perspector X(i), and T* indicates that T is homothetic to A'B'C'. (ABC, 74), (1st Brocard, 3), (circumorthic*, 3520), (1st circumperp, 3), (2nd circumperp, 3), (extangents*, 7688), (Fuhrmann, 3), (intangents*, 36), (Johnson, 3), (medial, 3), (inner Napoleon, 3), (outer Napoleon, 3), (1st Neuberg, 3), (2nd Neuberg, 3), (orthic*, 378), (tangential*, 3), (inner Vecten, 3), outer Vecten, 3) The appearance of (T,i,j) in the next list means that the triangle T is orthologic to A'B'C' with orthologic centers X(i) and X(j). (ABC, 3, 3), (anticomplementary, 3, 4), (circumorthic, 7689,6237), (Euler, 3, 5), (2nd Euler*, 3), (extangents, 7689, 6237), (inner Grebe, 3, 1161), (outer Grebe, 3, 1160), (intangents, 7689, 6238), (Johnson, 3, 4), (Kosnita*, 3), (Lucas central, 7690, 3), (MacBeath, 550,4), (medial, 3, 5), (midheight, 3357, 389), (orthic, 7689, 52), (orthocentroidal, 74, 568), (reflection, 7691, 6243), (tangential, 7689, 155) See Hyacinthos 23292, June 5, 2015 X(7688) lies on these lines: {1, 3}, {4, 3841}, {10, 3651}, {19, 378}, {30, 3925}, {71, 74}, {376, 993}, {411, 6684}, {515, 7411}, {516, 1006}, {550, 6253}, {580, 2308}, {582, 1203}, {672, 2301}, {759, 1292}, {943, 3671}, {946, 5284}, {1308, 2687}, {1698, 6985}, {1699, 6883}, {1869, 7414}, {2003, 4337}, {2071, 3101}, {2266, 4262}, {3098, 3779}, {3357, 6254}, {3520, 6197}, {3522, 5450}, {5248, 6361}, {5415, 6200}, {5416, 6396}, {5587, 7580}, {6237, 7689}, {6252, 7690}, {6255, 7691}X(7688) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,3587), (3,165,2077), (3,3428,3576), (3,3579,35), (3,5584,40), (35,484,3256), (65,3579,40), (165,484,3579), (2093,5010,55)
The Trinh triangle is defined at X(7688). X(7689) is also the orthologic center of the Trinh triangle and these triangles: intangents, orthic, and tangential.
Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let (Oa) be the circle with segment BaCa as diameter, and define (Ob) and (Oc) cyclically. X(7689) is the radical center of (Oa), (Ob), (Oc). (Randy Hutson, November 2, 2017) X(7689) lies on these lines: {2, 5448}, {3, 49}, {4, 5449}, {5, 4550}, {20, 68}, {26, 6000}, {30, 3357}, {35, 7352}, {36, 6238}, {52, 378}, {64, 7387}, {113, 7505}, {376, 539}, {382, 3581}, {389, 7526}, {541, 5878}, {548, 3098}, {569, 5890}, {578, 6102}, {631, 5654}, {912, 3579}, {1069, 5204}, {1192, 6642}, {1568, 6640}, {1593, 5446}, {1658, 5663}, {2889, 3522}, {2931, 2937}, {3157, 5217}, {3520, 5889}, {3547, 4846}, {3548, 6699}, {3567, 7527}, {4549, 6643}, {5892, 7395}, {5907, 6644}, {6237, 7688}, {6241, 7488} X(7689) = midpoint of X(i) and X(j) for these {i,j}: {20,68}, {64,7387}X(7690) is closely related to X(7692). The Trinh triangle is defined at X(7688).
X(7690) lies on these lines: {3, 6}, {30, 641}, {35, 7362}, {36, 6283}, {378, 6291}, {488, 542}, {538, 6312}, {3520, 6239}, {6252, 7688} X(7690) = {X(3),X(3098)}-harmonic conjugate of X(7692)The Trinh triangle is defined at X(7688).
X(54) = X(7691)-of-X(4)-Brocard triangle (Randy Hutson, June 6, 2015) X(7691) lies on the cubic K566 and these lines: {2,3574}, {3,54}, {4,1209}, {20,2888}, {22,1498}, {23,5907}, {35,7356}, {74,550} et al
X(7691) = midpoint of X(20) and X(2888)
X(7691) = reflection of X(i) in X(j) for these (i,j): (4,1209), (54,3)
X(7691) = anticomplement of X(3574)
X(7691) = isogonal conjugate of X(3)-vertex conjugate of X(5)
X(7691) = X(79)-of-Trinh-triangle if ABC is acute
X(7691) = Trinh-isogonal conjugate of X(3520)
X(7691) = circumnormal-isogonal conjugate of X(35720)
X(7692) is closely related to X(7690). The Trinh triangle is defined at X(7688).
X(7692) lies on these lines: {3,6}, {30,642}, {35,7353}, {36,6405}, {378,6406}, {487,542}, {538,6316}, {3520,6400}, {6404,7688}
X(7692) = {X(3),X(3098)}-harmonic conjugate of X(7690)
Similar triangles and centers of similitude: X(7693)-X(7709)
César Lozada (June 8, 2015) contributes the following introduction and triangle centers X(7693)-X(7706) and X(7708). Centers X(7707) and X(7709), are contributed by Randy Hutson.
If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, also called a self-homologous point, which occupies the same homologous position with respect to the two figures (Johnson 1929, p. 16). See Similitude Center at MathWorld. Algebraically, the center of similitude of two similar triangles U and V is the invariant triangle center under the affine transformation that maps U into V. If U and V are homothetic then their center of similitude coincides with their homothetic center. Following is a list of similar triangles U and V, their center of similitude, and the type of similarity.| U | V | center | type |
|---|---|---|---|
| ABC | 1st Brocard | X(2) | inverse |
| ABC | orthocentroidal | X(6) | inverse |
| outer Hutson | Yff central | X(7707) | (homothetic) |
| anticomplementary | 1st Brocard | X(2) | inverse |
| anticomplementary | orthocentroidal | X(7693) | inverse |
| 1st Brocard | Euler | X(7694) | inverse |
| 1st Brocard | inner Grebe | X(7695) | inverse |
| 1st Brocard | outer Grebe | X(7696) | inverse |
| 1st Brocard | Johnson | X(7697) | inverse |
| 1st Brocard | medial | X(2) | inverse |
| 1st Brocard | orthocentroidal | X(7698) | direct |
| 2nd Brocard | circummedial | X(2) | inverse |
| 2nd Brocard | 5th Euler | X(2) | inverse |
| 4th Brocard | circumsymmedial | X(6) | inverse |
| 4th Brocard | McCay | X(111) | direct |
| circumnormal | inner Napoleon | (pending) | inverse |
| circumnormal | outer Napoleon | (pending) | direct |
| 1st circumperp | Fuhrmann | X(1158) | inverse |
| 2nd circumperp | Fuhrmann | X(1) | inverse |
| Euler | orthocentroidal | X(7699) | inverse |
| excentral | Fuhrmann | X(1) | inverse |
| 2nd extouch | Fuhrmann | X(7700) | inverse |
| Feuerbach | incentral | X(115) | direct |
| Fuhrmann | hexyl | X(7701) | inverse |
| Fuhrmann | intouch | X(7702) | inverse |
| inner Grebe | orthocentroidal | X(6) | inverse |
| outer Grebe | orthocentroidal | X(6) | inverse |
| medial | orthocentroidal | X(7703) | inverse |
| 1st Morley | inner Napoleon | (pending) | inverse |
| 1st Morley | outer Napoleon | (pending) | direct |
| 2nd Morley | inner Napoleon | (pending) | inverse |
| 2nd Morley | outer Napoleon | (pending) | direct |
| 3rd Morley | inner Napoleon | (pending) | inverse |
| 3rd Morley | outer Napoleon | (pending) | direct |
| inner Napoleon | outer Napoleon | X(2) | inverse |
| inner Napoleon | Stammler | (pending) | inverse |
| outer Napoleon | Stammler | (pending) | direct |
| 3rd Euler | Fuhrmann | X(7704) | inverse |
| 4th Euler | Fuhrmann | X(7705) | inverse |
| Johnson | orthocentroidal | X(7706) | inverse |
| circumsymmedial | McCay | X(7708) | inverse |
| 1st Brocard | reflection-of-X(3)-in-ABC | X(7709) | inverse |
See the preamble to X(7693).
X(7693) lies on these lines: {2,3098}, {23,2916}, {146,381}, {373,5189}, {2552,2575}, {2553,2574}, {3066,5169}, {3091,5449}, {3410,6997}, {3448,3818}, {3681,3966}, {3832,5878}, {3839,4846} X(7693) = anticomplement of X(5888)Let Sa be the similitude center of the orthocentroidal triangle and the A-altimedial triangle. Define Sb and Sc cyclically. Triangle SaSbSc is perspective to the orthocentroidal triangle at X(7699). (Randy Hutson, December 10, 2016)
X(7699) lies on these lines: {2,4549}, {4,1495}, {5,568}, {54,7547}, {74,5094}, {110,381}, {113,5169}, {125,5890}, {195,3851}, {262,2394}, {403,5480}, {1173,5072}, {1346,2574}, {1347,2575}, {1352,1992}, {1614,7507}, {1995,2931}, {2452,5627}, {3091,5654}, {3569,6032}, {5663,7579}, {5777,6583} X(7699) = midpoint of X(4) and X(3431)X(7712)-of-medial-triangle = X(7712)-of-orthocentroidal-triangle = X(7703)-of-ABC.
X(7712) lies on the Thomson-Gibert-Moses hyperbola, the cubics K727, K914, K922, K923, K1123, K1249, the curve Q136, and these lines: {2, 1495}, {3, 11738}, {4, 10610}, {6, 23}, {20, 5654}, {22, 323}, {25, 5644}, {26, 15032}, {30, 3431}, {69, 5648}, {74, 10298}, {110, 3098}, {111, 5033}, {146, 376}, {154, 6636}, {160, 669}, {182, 5643}, {184, 11004}, {354, 14996}, {392, 4189}, {399, 7502}, {511, 9716}, {550, 38942}, {1201, 4257}, {1614, 15083}, {1658, 43807}, {1994, 9909}, {1995, 5544}, {2071, 35237}, {2502, 8617}, {2937, 12316}, {2979, 44110}, {3060, 44109}, {3091, 15432}, {3146, 11430}, {3164, 31296}, {3288, 37918}, {3410, 11206}, {3448, 7493}, {3522, 5656}, {3523, 26882}, {3524, 44834}, {3529, 5944}, {3534, 44786}, {3543, 18475}, {3581, 7556}, {3618, 7693}, {3620, 5596}, {3743, 37571}, {3796, 13595}, {4550, 14157}, {5012, 34417}, {5027, 5653}, {5059, 13367}, {5085, 14924}, {5169, 13394}, {5191, 37184}, {5422, 31860}, {5640, 5645}, {5642, 48892}, {5646, 7496}, {5651, 5888}, {5663, 41398}, {6030, 9306}, {7426, 48906}, {7488, 11456}, {7495, 18358}, {7512, 15068}, {7520, 37585}, {8627, 9463}, {8717, 15035}, {9140, 40291}, {9464, 10330}, {9545, 12088}, {9707, 37483}, {9715, 43605}, {9934, 11202}, {10192, 31101}, {10201, 15081}, {10540, 33533}, {10605, 13620}, {10645, 14170}, {10646, 14169}, {10984, 43584}, {10989, 48905}, {11179, 37909}, {11422, 16981}, {11444, 50414}, {11449, 32605}, {11817, 15026}, {12041, 20421}, {12087, 19357}, {12319, 44440}, {12367, 19127}, {12824, 17710}, {13203, 15131}, {13451, 14491}, {13857, 48891}, {15037, 37440}, {15246, 35264}, {15690, 44747}, {17538, 32171}, {18440, 47596}, {18911, 37760}, {20063, 43621}, {21850, 47313}, {26316, 37465}, {29012, 31857}, {31670, 37901}, {32609, 33544}, {33534, 37944}, {34099, 42671}, {36181, 43619}, {37470, 43804}, {37907, 43273}, {39899, 44555}, {40112, 48881}, {41450, 44837}, {42085, 44462}, {42086, 44466}, {43394, 49138}, {44210, 46818}
X(7712) = reflection of X(i) in X(j) for these {i,j}: {23, 32124}, {43720, 12041}
X(7712) = isogonal conjugate of X(11058)
X(7712) = anticomplement of X(7703)
X(7712) = isogonal conjugate of the anticomplement of X(19601)
X(7712) = isogonal conjugate of the isotomic conjugate of X(11057)
X(7712) = Thomson-isogonal conjugate of X(381)
X(7712) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11058}, {75, 14479}
X(7712) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11058}, {206, 14479}
X(7712) = crosssum of X(2) and X(19569)
X(7712) = crossdifference of every pair of points on line {3906, 9210}
X(7712) = barycentric product X(6)*X(11057)
X(7712) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11058}, {32, 14479}, {11057, 76}
X(7712) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 23, 48912}, {6, 48912, 11002}, {22, 26864, 323}, {23, 6800, 11003}, {23, 11003, 11002}, {110, 7492, 33884}, {110, 35268, 7492}, {184, 15107, 11004}, {323, 26864, 9544}, {1495, 5092, 10546}, {1495, 15080, 2}, {5092, 10546, 2}, {6800, 32124, 43697}, {10546, 15080, 5092}, {11003, 48912, 6}, {11004, 37913, 15107}, {15080, 26881, 1495}
Hatzipolakis-Lozada Homothetic Centers X(7713)-X(7718)
Antreas Hatzipolakis poses the following construction and related questions (June 10, 2015). Let A'B'C' be the orthic triangle of a triangle ABC and H = X(4). Let AB be the orthogonal projection of A' on line HB', and define Bc and Ca cyclically. Let AC be the orthogonal projection of A' on line HC', and define BA and CB cyclically. Let P be a point (in the usual sense of a function defined on a subset of the set of points in the plane of an abstract triangle with variable side-lengths a,b,c). Let PA = P-of-A'ABAC, PB = P-of-B'BCBA, PC = P-of-C'CACB. Let PAB be the reflection of PA in line HB', and define PBC and PCA cyclically. Let PAC be the reflection of PA in line HC', and define PBC and PCA cyclically. Let MA be the midpoint of PAB and PAC, and define MB and MC cyclically. Then MAMBMC is homothetic to ABC, and the Euler line of MAMBMC is parallel to the Euler line of ABC.
César Lozada (June 10, 2015) finds that if P = p : q : r (trilinears), then the homothetic center of MAMBMC and ABC is the point given by
HL(P) = [(a2 + b2 + c2)p + 2abq + 2acr]/((a2 - b2 - c2) : : = (Sωp + abq + acr)/SA : :
If P is on the infinity line, then HL(P) = P, and if P lies on the Euler line of ABC, then Euler-line-of-MAMBMC = Euler-line-of-ABC. The appearance of (i,j) in the following list indicates that HL(X(i)) = X(j): (1, 7713), (2,7714), (3,1598), (4,7487), (5,7715), (6,7716), (7,7717), (8,7718), (20,4), (25,25), (1885,3575), (4319,19), (6467,1843), (7386,6995), (7387,6642), (7667,428). See
Hyacinthos 23304.As a point on the Euler line, X(7714) has Sinagawa coefficients (-2F, 3E + 3F).
See the preamble to X(7713). X(7714) lies on these lines: {2,3}, {393,5306}, {519,7713}, {524,7716}, {527,7717}, {1629,6524}, {1829,3241}, {1843,1992}, {2333,4685}, {5218,7298), {5345,7288}As a point on the Euler line, X(7715) has Sinagawa coefficients (-3F, 4E + 5F).
See the preamble to X(7713). X(7715) lies on these lines: {2,3}, {143,1353}, {1483,1829}, {952,7713}, {3564,7716}, {5843,7717}, {5844,7718} et al X(7715) = midpoint of X(1598) and X(7487)Trilinears (sec A cos2 A/2)(cos4 B/2 + cos4 C/2 - cos4 A/2) : :
Trilinears (1 + sec A)[(1 + cos B)2 + (1 + cos C)2 - (1 + cos A)2] : :
Let A' be the perspector of the A-mixtilinear excircle, and define B' and C' cyclically. (The lines AA', BB', CC' concur in X(9).) Let LA be the polar of A' with respect to the A-mixtilinear excircle, and define LB and LC cyclically. Let A'' = LB∩LC, B''' = LC∩LA, C'' = LA∩LB. The lines AA'', BB'', CC'' concur in X(7719); see also X(7713). (Randy Hutson, June 16, 2015)
X(7719) lies on these lines: {1,607}, {4,9}, {25,200}, {33,7322}, {34,1783}, {46,1729}, {57,5236}, {92,4384}, {6743,7718} et al
X(7719) = trilinear product X(4)*X(218)
X(7719) = trilinear product X(57)*X(3870)
Centers related to the orthocentroidal triangle: X(7720)-X(7733)
César Lozada (June 18, 2015) finds perspectors, orthologic centers, and parallelogic centers associated with the orthocentroidal triangle. This triangle is defined at X(5476) as follows. Let A' be the intersection, other than X(4), of the A-altitude and the orthocentroidal circle, and define B' and C' cyclically. The orthocentroidal triangle, A'B'C', is inversely similar to ABC, with center X(6) of similitude.
In the following table, each triangle in column 1 is perspective to the orthocentroidal triangle, with perspector shown in column 2.
| triangle | perspector |
|---|---|
| ABC | X(4) |
| 4th Brocard | X(6) |
| circumorthic | X(4) |
| Euler | X(4) |
| excentral | X(3336) |
| 2nd extouch | X(4) |
| 3rd extouch | X(4) |
| inner Grebe | X(7720) |
| outer Grebe | X(7721) |
| midheight | X(4) |
| inner Napoleon | X(61) |
| outer Napoleon | X(62) |
In the next table, the appearance of T, X(i), X(j) in a row indicates that the triangle T is orthologic to the orthocentroidal triangle A'B'C', that X(i) = A'B'C'-to-T orthologic center, and that X(j) = T-to-A'B'C' orthologic center.
| triangle | X(i) | X(j) |
|---|---|---|
| ABC | X(4) | X(74) |
| anticomplementary | X(4) | X(146) |
| circumorthic | X(568) | X( ) |
| Euler | X(4) | X(125) |
| 2nd Euler | X(568) | X(7723) |
| extangents | X(568) | X(7724) |
| inner Grebe | X(4) | X(7725) |
| outer Grebe | X(4) | X(7726) |
| intangents | X(568) | X(7727) |
| Johnson | X(4) | X(7728) |
| Lucas homothetic | X(4) | (pending) |
| Lucas(-1) homothetic | X(4) | (pending) |
| medial | X(4) | X(113) |
| midheight | X(7729) | X(974) |
| orthic | X(568) | X(1986) |
| reflection | X(7730) | X(7731) |
| tangential | X(568) | X(399) |
In the next table, the appearance of T, X(i), X(j) in a row indicates that the triangles T is parallelogic to the orthocentroidal triangle A'B'C', that X(i) = A'B'C'-to-T parallelogic center, and X(j) = T-to-A'B'C' parallelogic center.
| triangle | X(i) | X(j) |
| ABC | X(2) | X(110) |
| anticomplementary | X(2) | X(3448) |
| Euler | X(2) | X(113) |
| inner Grebe | X(2) | X(7732) |
| outer Grebe | X(2) | X(7733) |
| Johnson | X(2) | X(265) |
| Lucas homothetic | X(2) | (pending) |
| Lucas(-1) homothetic | X(2) | (pending) |
| medial | X(2) | X(125) |
Two of the less-well-known triangles in the above tables, the Lucas homothetic and the Lucas(-1) homothetic, are discussed in the premable to X(6395).
As a point on the Euler line, X(7734) has Shinagawa coefficients (5E + F, -3E - 3F). (Peter Moses, June 16, 2015)
Let A'B'C' be the pedal triangle of X(5). Let AB be the orthogonal projecton of A' on line AC, and define BC and CA cyclically. Let AC be the orthogonal projecton of A' on line AB, and define BA and CB cyclically. Let U be the orthogonal projection of B on line A'AB, let V be the orthogonal projection of C on line A'AC, and let MA be the midpoint of U and V. Define MB and MC cyclically. Then the X(7734) = centroid of MAMBMC = (Euler line of ABC)∩(Euler line of MAMBMC).
See Hyacinthos 23333 and X(7667). X(7734) lies on these lines: {2,3}, {3564,3819}, {5065,5306} X(7734) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1368,7484,140), (6676,7485,3530)X(7735) is the perspector of the circumconic described by Francisco Javier Garcia Capitan in Hyacinthos #21466, 1/30/2013 in response to a problem by Antreas P. Hatzipolakis, January 29, 2013 (http://anthrakitis.blogspot.gr/2013/01/conics-centered-at-o.html).
Let A'B'C' be the Artzt triangle. Let A" be the barycentric product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(7735). (Randy Hutson, April 9, 2016) X(7735) lies on these lines: {2,6}, {3,5286}, {4,32}, {20,3053}, {22,1609}, {25,393}, {30,1384}, {39,631}, {41,5230}, {53,6995}, {111,1302}, {148,5939}, {172,388}, {187,376}, {194,5976}, {216,1194}, {232,800}, {235,3172}, {251,2165}, {262,5052}, {263,6784}, {281,5336}, {315,6179}, {316,2031}, {383,5334}, {387,2271}, {427,3087}, {443,5277}, {468,2452}, {497,1914}, {498,5280}, {499,5299}, {549,5024}, {570,1180}, {571,1370}, {574,3524}, {576,6036}, {577,7386}, {594,7172}, {609,1478}, {612,2318}, {614,3554}, {910,3772}, {1056,2242}, {1058,2241}, {1080,5335}, {1086,3598}, {1108,3290}, {1383,1989}, {1447,4000}, {1479,7031}, {1501,1971}, {1506,5067}, {1513,6776}, {1572,5603}, {1587,6423}, {1588,6424}, {1834,4258}, {1975,6392}, {1990,4232}, {2021,7709}, {2207,3089}, {2275,7288}, {2276,5218}, {2345,7081}, {2395,6041}, {2548,3090}, {2550,4386}, {2551,4426}, {2965,7391}, {3003,7493}, {3070,7374}, {3071,7000}, {3094,6194}, {3421,5291}, {3522,5023}, {3523,5013}, {3525,5368}, {3528,5206}, {3533,5041}, {3545,5008}, {3552,5989}, {3705,5839}, {3785,6656}, {4251,5292}, {4264,5747}, {4644,7179}, {5017,5999}, {5058,7582}, {5062,7581}, {5283,6857}, {5477,6054}, {5746,7413}, {6748,7378}, {6791,6793}, {6794,7422} X(7735) = complement of X(37668)X(7736) lies on these lines: {2,6}, {4,39}, {5,5286}, {20,5013}, {25,3087}, {30,5024}, {32,631}, {53,7378}, {98,5034}, {111,6128}, {115,3545}, {147,2023}, {172,7288}, {187,1285}, {216,7386}, {251,571}, {376,574}, {383,5335}, {384,6337}, {387,7380}, {388,2275}, {393,427}, {497,2276}, {498,5299}, {499,5280}, {549,1384}, {570,1370}, {577,7494}, {612,3554}, {614,3553}, {1015,1056}, {1058,1500}, {1080,5334}, {1107,2551}, {1180,5421}, {1194,7392}, {1447,4644}, {1504,7582}, {1505,7581}, {1506,3090}, {1571,6361}, {1572,5657}, {1575,2550}, {1587,6421}, {1588,6422}, {1609,7485}, {1656,5305}, {1834,7407}, {1914,5218}, {2024,5207}, {2165,3108}, {2207,3088}, {2345,3705}, {2482,5503}, {3053,3523}, {3070,7000}, {3071,7374}, {3091,5254}, {3363,7620}, {3525,5007}, {4000,7179}, {4232,6749}, {4284,5747}, {5021,6998}, {5022,7390}, {5041,5067}, {5063,7493}, {5065,6353}, {5071,5309}, {5084,5283}, {5702,6103}, {5802,7413}, {5839,7081}, {6748,6995}, {7608,7612}
X(7736) = crosssum of X(6) and X(5085)Let PA be the reflection of X(6) in line BC, and define PB and PC cyclically; then X(7737) is the isogonal conjugate of X(6) with respect to PAPBPC. (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011); see also X(5903). Also, X(7737) = X(6) of PAPBPC.
X(7737) lies on these lines: {2,187}, {3,2548}, {4,32}, {5,3053}, {6,30}, {20,39}, {69,754}, {140,5023}, {172,1479}, {193,538}, {194,6658}, {230,381}, {251,7391}, {262,2021}, {263,512}, {315,384}, {325,1003}, {376,574}, {382,5254}, {388,2241}, {428,1184}, {476,843}, {497,2242}, {515,1572}, {543,1992}, {549,5210}, {550,5013}, {597,5077}, {609,3583}, {620,1007}, {631,1506}, {800,3087}, {1015,4293}, {1194,7500}, {1196,6995}, {1316,1648}, {1352,5017}, {1478,1914}, {1500,4294}, {1504,6459}, {1505,6460}, {1513,7694}, {1587,5058}, {1588,5062}, {1597,6748}, {1627,7394}, {1839,5336}, {1843,2386}, {1995,5913}, {2030,5476}, {2165,2965}, {2207,3575}, {2275,4299}, {2276,4302}, {2478,5277}, {3051,3331}, {3054,5055}, {3055,5054}, {3070,6424}, {3071,6423}, {3146,5007}, {3199,7487}, {3363,7610}, {3434,5291}, {3534,5024}, {3543,5008}, {3552,5149}, {3585,7031}, {3618,4045}, {3627,5305}, {3785,3934}, {3830,5306}, {5041,5059}, {5052,6776}, {5283,6872}, {5395,6683}, {5471,6775}, {5472,6772} X(7737) = reflection of X(i) in X(j) for these (i,j): (69, 3734), (2549, 6), (5077, 597)X(7738) lies on these lines: {2,1975}, {3,5286}, {4,39}, {5,5024}, {6,20}, {32,376}, {69,194}, {115,3090}, {148,2023}, {183,6392}, {187,3528}, {230,3523}, {384,3618}, {387,5021}, {388,2276}, {393,1593}, {443,5283}, {497,2275}, {548,1384}, {570,6815}, {574,631}, {966,4201}, {1007,5025}, {1015,1058}, {1056,1500}, {1100,4339}, {1107,2550}, {1180,1370}, {1194,7386}, {1249,1968}, {1285,5007}, {1504,7581}, {1505,7582}, {1506,3545}, {1571,5657}, {1572,6361}, {1575,2551}, {1587,6422}, {1588,6421}, {1834,5022}, {3053,3522}, {3055,7486}, {3087,3575}, {3091,3815}, {3524,5309}, {3541,5523}, {3734,6704}, {3926,6656}, {4190,5276}, {4299,5280}, {4302,5299}, {4352,4648}, {5023,5306}, {5030,5292}, {5206,5355}, {5275,6904}
X(7738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5286,7735), (4,39,7736), (39,2549,4), (574,3767,631), (3522,5304,3053), (5013,5254,2)X(7739) lies on these lines: {2,39}, {3,5306}, {6,30}, {20,5007}, {32,376}, {69,4045}, {115,3545}, {187,5304}, {230,5024}, {381,2548}, {543,5034}, {549,5013}, {574,3524}, {597,4048}, {616,6772}, {617,6775}, {754,1992}, {1285,6781}, {1506,5071}, {1570,3849}, {1597,1990}, {1975,6661}, {2782,6034}, {3017,4253}, {3108,7394}, {3543,5041}, {3618,3734}, {3815,5055}, {3839,5475}, {4299,7296}, {4302,5332}, {5206,5368}, {6337,6680}
X(7739) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5286,5309), (2,5309,3767), (6,2549,7737), (39,5286,3767), (39,5309,2), (574,5355,7735)X(7740) = center of the circle D = {{X(3), X(110), X(4240}}. Dao Thanh Oai finds seven other points on D, as follows. Let A' = (Euler line)∩BC, and define B' and C' cyclically. The points X(3)-of-AB'C', X(3)-of-A'BC', X(3)-of-A'B'C, X(110)-of-AB'C', X(110)-of-A'BC', X(110)-of-A'B'C all lie on D. Let P be the paralogic triangle whose perspectrix is the Euler line of ABC. Then X(3)-of-P lies on D. See ADGEOM 2342 (February 2015) and Centro X(5502).
X(7740) lies on these lines: {3,64}, {526,1511} X(7740) = X(2166)-isoconjugate of X(2693)Each of the points X(7741)-X(7744) is the center of a circle obtained by inverting one circle in another. The following list was provided by César Lozada (July 4, 2015):
X(26) = center of circumcircle-inverse of nine-point circle
X(496) = center of incircle-inverse of nine-point circle
X(942) = center of circumcircle-inverse of circumcircle
X(2070) = center of circumcircle-inverse of orthocentroidal circle
X(2072) = center of nine-point-circle-inverse of orthocentroidal circle
X(5576) = center of nine-point-circle-inverse of circumcircle
X(7579) = center of orthocentroidal-circle-inverse of circumcircle
X(7741) = center of nine-point-circle-inverse of incircle
X(7742) = center of circumcircle-inverse of incircle
X(7743) = center of incircle-inverse of orthocentroidal circle
X(7744) = center of orthocentroidal-circle-inverse of incircle
Let L be a line tangent to the incircle. Let P be the pole of L with respect to the circumcircle. The locus of P as L varies is a conic with center X(7742), and one focus at X(3). (Randy Hutson, July 23, 2015)
X(7742) lies on these lines: {1,3}, {11,6985}, {12,6883}, {21,4293}, {22,1612}, {24,278}, {25,1838}, {31,4303), {47,222}, {90,971}, {197,1714}, {198,1723}, {212,1066}, {219,7113}, {238,1745}, {255,1458}, {347,7488}, {388,1006}, {405,1478}, {474,3826}, {497,3651}, {582,5399}, {910,1752}, {943,3475}, {993,4311}, {1012,4299}, {1037,1794}, {1076,3011}, {1108,1609}, {1147,3173}, {1260,5904}, {1437,1780}, {1479,7580}, {1497,4300}, {1621,4295}, {1756,7083}, {1804,4341}, {2256,5124}, {2328,4278}, {2361,3157}, {3002,7124}, {3560,7354}, {3585,6913}, {3911,6796}, {4292,5248}, {4294,7411}, {5229,6920}, {5433,6911}, {6905,7288}
X(7742) = circumcircle-inverse of X(5560)X(7743) lies on these lines: {1,381}, {4,4308}, {10,3829}, {11,517}, {30,5126}, {80,5048}, {149,5440}, {226,5049}, {355,6973}, {382,1420}, {495,3817}, {496,942}, {515,1387}, {516,5122}, {519,5087}, {551,3838}, {912,1484}, {944,7704}, {950,5901}, {999,1699}, {1319,3583}, {1385,1479}, {1656,1697}, {1858,6583}, {2646,4857}, {2771,5533}, {2802,5123}, {3616,3824}, {3627,4311}, {3814,3880}, {3825,5836}, {3885,5154}, {5219,6767}
X(7743) = midpoint of X(i) and X(j) for these {i,j}: {80,5048}, {149,5440}, {1319,3583}X(7743) = {X(2463),X(2464)}-harmonic conjugate of X(7744)
X(7744) lies on these lines: {1,381}, {5094,5121}
X(7744) = {X(2463),X(2464)}-harmonic conjugate of X(7743)
The form k1a4 + k2a2(b2 + c2) + k3b2c2 + k4(b4 + c4)
Peter J. C. Moses contributes (July 6, 2015) triangle centers X(7745)-X(7954). All except X(7927) and X(7950) have barycentrics of the form
k1a4 + k2a2(b2 + c2) + k3b2c2 + k4(b4 + c4) : :
where |ki| is 0, 1, or 2. Included are 16 points on the Euler line: X(i) for i = 7770, 7791, 7807, 7819, 7824, 7833, 7841, 7866, 7876, 7887, 7892, 7901, 7907, 7924, 7933, 7948.
X(7745) lies on these lines: {2, 3053}, {3, 2548}, {4, 6}, {5, 32}, {11, 172}, {12, 1914}, {20, 5013}, {30, 39}, {51, 460}, {76, 524}, {83, 316}, {112, 1594}, {115, 546}, {140, 187}, {141, 315}, {232, 3575}, {251, 5133}, {297, 1915}, {325, 384}, {355, 1572}, {381, 3767}, {382, 2549}, {427, 1968}, {428, 1194}, {485, 6424}, {486, 6423}, {495, 2241}, {496, 2242}, {548, 6781}, {549, 5206}, {550, 574}, {571, 7399}, {577, 6823}, {590, 7388}, {594, 5015}, {615, 7389}, {625, 6680}, {631, 5023}, {754, 3934}, {1086, 4911}, {1184, 6997}, {1285, 3090}, {1329, 4386}, {1384, 1656}, {1609, 7395}, {1611, 7392}, {1657, 5024}, {1992, 6392}, {2023, 2794}, {2275, 7354}, {2276, 6284}, {2478, 5275}, {2886, 4426}, {3091, 7735}, {3146, 7738}, {3172, 7507}, {3199, 6756}, {3329, 6655}, {3523, 5210}, {3564, 5052}, {3583, 5280}, {3585, 5299}, {3628, 7603}, {3727, 5724}, {3734, 3933}, {3818, 5039}, {3830, 7739}, {3832, 5304}, {3843, 5319}, {3845, 5309}, {3849, 6683}, {3850, 5008}, {3853, 5041}, {3858, 5346}, {3861, 5355}, {4187, 5277}, {5046, 5276}, {5058, 7583}, {5062, 7584}, {5116, 7470}, {5354, 7533}, {5359, 7394}, {6421, 6560}, {6422, 6561} X(7745) = complement of X(7750)
X(7746) lies on these lines: {2, 39}, {3, 115}, {4, 187}, {5, 32}, {6, 17}, {11, 2241}, {12, 2242}, {30, 5206}, {69, 1570}, {98, 3406}, {140, 574}, {141, 5028}, {183, 626}, {216, 2165}, {232, 7505}, {315, 625}, {345, 7230}, {348, 4403}, {381, 3053}, {382, 5023}, {403, 1968}, {485, 5062}, {486, 5058}, {498, 1500}, {499, 1015}, {547, 5306}, {571, 5576}, {590, 639}, {615, 640}, {620, 1975}, {631, 2549}, {1078, 5025}, {1184, 7539}, {1348, 6177}, {1349, 6178}, {1352, 1692}, {1384, 3851}, {1503, 5033}, {1609, 7529}, {1657, 5210}, {1691, 3818}, {1879, 7517}, {2021, 6248}, {2476, 5277}, {2548, 3090}, {3091, 7737}, {3199, 3542}, {3525, 7738}, {3526, 5013}, {3589, 5034}, {3628, 3815}, {3814, 4426}, {4173, 6784}, {5008, 5056}, {5041, 5067}, {5070, 5355}, {5215, 7615}, {5276, 7504}, {5304, 7486}, {5354, 7570}, {5359, 7571}, {5737, 6537}, {5872, 6783}, {5873, 6782}, {7612, 7694}
X(7746) = complement of X(7763)X(7747) lies on these lines: {2, 5206}, {3, 1506}, {4, 32}, {5, 187}, {6, 382}, {20, 574}, {30, 39}, {61, 5472}, {62, 5471}, {76, 754}, {83, 4045}, {99, 6658}, {140, 7603}, {172, 3583}, {194, 543}, {211, 5167}, {230, 546}, {232, 6240}, {315, 3734}, {316, 384}, {381, 3053}, {428, 1196}, {550, 3815}, {576, 5477}, {620, 3552}, {800, 6748}, {1003, 3788}, {1015, 7354}, {1384, 3843}, {1478, 2241}, {1479, 2242}, {1500, 6284}, {1503, 5052}, {1504, 6561}, {1505, 6560}, {1569, 3095}, {1572, 5691}, {1656, 5023}, {1657, 5013}, {1692, 5480}, {1879, 2965}, {1914, 3585}, {2549, 3146}, {3055, 3530}, {3070, 5058}, {3071, 5062}, {3199, 3575}, {3526, 5210}, {3529, 7736}, {3543, 5286}, {3627, 5007}, {3818, 5017}, {3830, 5309}, {3849, 3934}, {3853, 5008}, {3972, 5025}, {5046, 5277}, {5076, 5346}
X(7748) lies on these lines: {3, 115}, {4, 39}, {5, 574}, {6, 382}, {20, 187}, {30, 32}, {53, 5065}, {76, 148}, {99, 3788}, {194, 316}, {230, 550}, {315, 538}, {381, 1506}, {543, 626}, {546, 3815}, {671, 1078}, {1003, 6680}, {1015, 1479}, {1194, 7391}, {1196, 1370}, {1478, 1500}, {1503, 5028}, {1504, 3070}, {1505, 3071}, {1569, 6033}, {1570, 6776}, {1571, 5587}, {1657, 3053}, {1968, 5523}, {2176, 5134}, {2241, 6284}, {2242, 7354}, {2275, 3583}, {2276, 3585}, {2475, 5283}, {2996, 3785}, {3054, 3530}, {3091, 7603}, {3094, 3818}, {3146, 5007}, {3529, 7735}, {3534, 5023}, {3543, 5041}, {3734, 6656}, {3843, 5024}, {3972, 6658}, {4045, 6704}, {5008, 5319}, {5034, 5480}, {5038, 5476}, {5058, 6561}, {5062, 6560}, {5073, 5355}, {5339, 5471}, {5340, 5472}
X(7749) lies on these lines: {2, 32}, {3, 115}, {4, 5206}, {5, 187}, {6, 3411}, {17, 5472}, {18, 5471}, {39, 140}, {76, 620}, {141, 1692}, {183, 3788}, {216, 7542}, {231, 566}, {233, 571}, {381, 5023}, {382, 5210}, {468, 3199}, {498, 2242}, {499, 2241}, {549, 5254}, {574, 631}, {577, 3548}, {590, 641}, {615, 642}, {632, 3815}, {1015, 5433}, {1196, 7499}, {1352, 5033}, {1384, 5070}, {1500, 5432}, {1504, 5418}, {1505, 5420}, {1572, 3624}, {1573, 4999}, {1574, 3035}, {1656, 3053}, {1879, 2937}, {1968, 7505}, {1975, 2482}, {2021, 3934}, {2549, 3523}, {3055, 5008}, {3090, 7737}, {3291, 7495}, {3525, 5368}, {3533, 7736}, {3589, 5052}, {3628, 7603}, {5013, 5054}, {5025, 6722}, {5041, 5306}, {5319, 7616}
X(7749) = X(7749) = complement of X(7752)X(7750) lies on these lines: {2, 3053}, {3, 315}, {4, 183}, {5, 316}, {20, 64}, {30, 76}, {32, 6656}, {39, 754}, {86, 4201}, {99, 550}, {141, 384}, {187, 626}, {193, 7738}, {194, 524}, {230, 5025}, {264, 3575}, {297, 1968}, {305, 7667}, {317, 1593}, {343, 401}, {350, 6284}, {376, 3926}, {385, 5254}, {427, 1799}, {491, 1151}, {492, 1152}, {548, 6390}, {639, 2459}, {640, 2460}, {682, 7467}, {1007, 3523}, {1235, 6240}, {1369, 6636}, {1513, 5171}, {1909, 7354}, {2794, 5188}, {3096, 3972}, {3098, 6393}, {3313, 3852}, {3314, 3552}, {3522, 6337}, {3788, 5206}, {3793, 5305}, {3849, 3934}, {4045, 5007}, {4195, 5224}, {5989, 7470}
X(7750) = anticomplement of X(7745)X(7751) lies on these lines: {1, 4396}, {2, 3108}, {3, 538}, {4, 754}, {5, 524}, {6, 3934}, {20, 543}, {31, 4721}, {32, 76}, {39, 183}, {69, 626}, {99, 5206}, {115, 315}, {141, 5305}, {172, 3761}, {182, 732}, {187, 1975}, {193, 2548}, {194, 574}, {230, 3788}, {350, 2241}, {382, 3849}, {525, 7689}, {595, 4713}, {620, 3926}, {625, 5111}, {631, 7622}, {698, 3098}, {1089, 4376}, {1111, 4372}, {1909, 2242}, {1914, 3760}, {2549, 3785}, {2782, 5171}, {2980, 6664}, {3526, 7610}, {3530, 5569}, {3618, 6704}, {3770, 5019}, {3832, 7615}, {4045, 5286}, {4426, 6381}, {5309, 6656}, {5355, 6292}, {6194, 6308}, {6680, 7735}
X(7752) lies on these lines: {2, 32}, {3, 316}, {4, 99}, {5, 76}, {6, 5031}, {20, 7694}, {39, 625}, {61, 303}, {62, 302}, {69, 576}, {95, 7558}, {115, 194}, {182, 5207}, {183, 1656}, {211, 3060}, {230, 6179}, {264, 847}, {274, 2476}, {305, 5133}, {317, 3542}, {381, 1975}, {384, 3788}, {485, 492}, {486, 491}, {546, 6390}, {574, 6655}, {620, 3552}, {623, 3105}, {624, 3104}, {1235, 7577}, {1502, 3613}, {2039, 3557}, {2040, 3558}, {3091, 3926}, {3094, 5103}, {3118, 3981}, {3266, 5169}, {3314, 3934}, {3814, 6376}, {3815, 6656}, {5152, 6033}, {5480, 6393}, {6311, 6564}, {6315, 6565}
X(7752) = exsimilicenter of nine-point circle and circle {{X(4),X(194),X(3557),X(3558)}}X(7753) lies on these lines: {2, 32}, {4, 7739}, {5, 5007}, {6, 13}, {30, 39}, {51, 2387}, {172, 3582}, {187, 549}, {230, 547}, {232, 7576}, {262, 2794}, {316, 3329}, {325, 6661}, {376, 574}, {395, 623}, {396, 624}, {428, 3199}, {524, 5052}, {543, 598}, {597, 1692}, {620, 3972}, {1003, 2482}, {1015, 5434}, {1500, 3058}, {1572, 3679}, {1596, 6749}, {1914, 3584}, {2021, 3849}, {2207, 5064}, {2549, 3543}, {3053, 5054}, {3091, 5319}, {3524, 5206}, {3534, 5013}, {3545, 3767}, {3839, 5286}, {3845, 5041}, {3926, 5395}, {5032, 7615}, {5066, 5305}, {5071, 7735}, {5965, 7697}, {6103, 7577}, {6792, 7698}
X(7753) = X(39)-of-4th-Brocard-triangleX(7754) lies on these lines: {2, 3933}, {3, 194}, {4, 193}, {6, 76}, {32, 538}, {39, 183}, {69, 5286}, {99, 3053}, {148, 382}, {192, 3295}, {239, 3673}, {274, 5275}, {297, 6515}, {305, 1184}, {315, 524}, {316, 6144}, {325, 3767}, {330, 999}, {405, 1655}, {419, 3167}, {458, 1235}, {550, 3793}, {576, 6248}, {599, 3096}, {626, 5309}, {648, 2207}, {698, 5017}, {894, 4385}, {1078, 5013}, {1384, 3552}, {1613, 3978}, {2275, 4396}, {2276, 4400}, {3225, 3360}, {3729, 5255}, {3734, 5007}, {3760, 5299}, {3761, 5280}, {3785, 7738}, {3926, 7735}, {5346, 6680}, {5976, 6309}, {7388, 7584}, {7389, 7583}
X(7754) = anticomplement of X(3933)X(7755) lies on these lines: {2, 3108}, {3, 5309}, {4, 32}, {5, 5007}, {6, 17}, {25, 3456}, {39, 140}, {61, 6771}, {62, 6774}, {76, 6680}, {172, 5270}, {183, 6292}, {187, 550}, {194, 620}, {251, 7533}, {385, 626}, {395, 636}, {396, 635}, {397, 7684}, {398, 7685}, {468, 1196}, {543, 3552}, {574, 3523}, {631, 7739}, {671, 6658}, {754, 5025}, {800, 1990}, {1078, 4045}, {1184, 5094}, {1194, 7495}, {1384, 5073}, {1627, 5189}, {1657, 3053}, {1914, 4857}, {1989, 7545}, {2548, 5056}, {2549, 3522}, {3095, 6036}, {3399, 7607}, {3549, 5158}, {3815, 5041}, {3850, 5008}, {3851, 5475}
X(7756) lies on these lines: {3, 115}, {4, 574}, {6, 1657}, {20, 32}, {30, 39}, {76, 543}, {83, 6658}, {99, 626}, {148, 1078}, {172, 4316}, {187, 550}, {194, 754}, {230, 548}, {376, 3767}, {382, 5013}, {384, 4045}, {512, 4173}, {546, 7603}, {620, 5025}, {1015, 6284}, {1196, 7667}, {1384, 5346}, {1500, 7354}, {1504, 6560}, {1505, 6561}, {1569, 2794}, {1571, 5691}, {1885, 3199}, {1914, 4324}, {2241, 4302}, {2242, 4299}, {2482, 3788}, {2548, 3146}, {3053, 3534}, {3055, 3850}, {3529, 7737}, {3552, 6680}, {3627, 3815}, {3734, 6292}, {5024, 5073}, {5162, 7470}
X(7756) = X(141) of 6th Brocard triangleX(7757) = center of ellipse that is locus of radical centers of parametrized circles used in construction of Brocard points (continued below). (Randy Hutson, July 23, 2015)
X(7757) is the center of the ellipse that is the locus of radical centers of parametrized circles used in construction of the Brocard points: Let A', B', C' be points on lines BC, CA, AB, resp., such that BA'/BC = CB'/CA = AC'/AB = t. Let (Oa) be the circle passing through A and tangent to line BC at A'. Define (Ob), (Oc) cyclically. As t varies, the radical center of (Oa), (Ob), (Oc) traces an arc of an ellipse with center X(7757). The arc has as endpoints the 1st and 2nd Brocard points (for t = 0 and 1), and passes through X(194) (for t = 1/2 and infinity). The non-traced section of the ellipse passes through X(2) (the antipode of X(194)). The ellipse is the Steiner circumellipse of triangle X(194)PU(1). The tangents to the ellipse at the 1st and 2nd Brocard points (PU(1)) intersect at the 3rd Brocard point, X(76). Bernard Gibert notes that the ellipse also circumscribes the 3rd Brocard triangle. (Randy Hutson, January 26, 2016) X(7757) lies on these lines: {2, 39}, {3, 6179}, {6, 99}, {13, 6299}, {14, 6298}, {30, 3095}, {83, 1975}, {148, 5475}, {183, 5024}, {190, 995}, {192, 1015}, {262, 381}, {315, 7738}, {316, 2549}, {330, 1500}, {376, 511}, {378, 648}, {385, 574}, {524, 3094}, {543, 598}, {547, 7697}, {551, 726}, {597, 698}, {599, 732}, {620, 5355}, {691, 2452}, {730, 3097}, {1002, 3227}, {1078, 5013}, {1340, 6190}, {1341, 6189}, {2021, 7618}, {3096, 3933}, {3106, 5463}, {3107, 5464}, {3314, 4045}, {3329, 3734}, {3545, 6248}, {3552, 5007}, {4234, 5145}, {5032, 5052} X(7757) = midpoint of X(2) and X(194)X(7758) lies on these lines: {2, 3108}, {3, 524}, {4, 538}, {6, 3933}, {20, 754}, {32, 193}, {39, 69}, {76, 2548}, {115, 6392}, {187, 6337}, {194, 315}, {325, 3767}, {439, 2482}, {498, 4400}, {499, 4396}, {511, 6309}, {543, 3146}, {574, 3785}, {626, 5286}, {632, 7610}, {633, 6294}, {634, 6581}, {637, 6272}, {638, 6273}, {732, 1352}, {980, 5739}, {1975, 7737}, {1992, 5007}, {3053, 6144}, {3091, 7615}, {3529, 3849}, {3618, 5041}, {3620, 6292}, {3630, 5024}, {3787, 4176}, {3788, 7735}, {3793, 5023}, {3934, 7736}, {5304, 6680}, {6656, 7739}
X(7759) lies on these lines: {2, 5007}, {3, 754}, {4, 538}, {5, 524}, {6, 626}, {20, 3849}, {32, 325}, {39, 315}, {69, 2548}, {76, 5475}, {83, 3314}, {141, 5039}, {147, 6309}, {183, 1506}, {193, 625}, {194, 316}, {317, 3199}, {382, 543}, {491, 5058}, {492, 5062}, {591, 6420}, {620, 3053}, {631, 5569}, {732, 3818}, {736, 3095}, {1193, 4805}, {1991, 6419}, {1992, 5319}, {3096, 3329}, {3528, 7618}, {3530, 7622}, {3629, 5305}, {3670, 4799}, {3734, 3933}, {3763, 6704}, {3855, 7615}, {3926, 7737}, {5025, 5309}, {5070, 7610}, {6683, 7736}
X(7760) lies on these lines: {2, 3108}, {3, 6179}, {4, 542}, {6, 76}, {32, 99}, {39, 385}, {69, 3096}, {98, 3095}, {190, 595}, {192, 2241}, {193, 315}, {274, 5276}, {305, 5359}, {316, 3629}, {325, 5305}, {330, 2242}, {350, 5299}, {384, 538}, {511, 7470}, {524, 6656}, {543, 6658}, {626, 5355}, {754, 6655}, {1180, 1799}, {1185, 7304}, {1909, 5280}, {1975, 3972}, {2896, 4045}, {3051, 3978}, {3216, 3570}, {3266, 5354}, {3329, 3934}, {3673, 3759}, {3758, 4385}, {3788, 5346}, {3926, 5304}, {5025, 5309}, {5097, 6248}, {5171, 7709}, {5368, 6680}
X(7760) = isotomic conjugate of X(6664)X(7761) lies on these lines: {2, 187}, {3, 114}, {4, 3934}, {5, 5171}, {6, 754}, {30, 141}, {32, 6656}, {39, 315}, {69, 538}, {76, 148}, {99, 3314}, {115, 183}, {193, 7739}, {298, 6775}, {299, 6772}, {325, 574}, {382, 6287}, {384, 3096}, {385, 5309}, {543, 599}, {672, 4805}, {736, 3094}, {1003, 6781}, {1078, 5025}, {1180, 1369}, {2548, 6683}, {2795, 4655}, {3053, 6680}, {3258, 5108}, {3329,7812}, {3670, 4950}, {3767, 3785}, {3793, 5306}, {3917, 5167}, {4159, 6636}, {4713, 5134}, {5112, 5651}, {5346, 6179}, {5461, 7610}, {6228, 6229}
X(7761) = complement of X(7737)X(7762) lies on these lines: {4, 193}, {5, 385}, {6, 315}, {30, 194}, {32, 325}, {39, 754}, {76, 524}, {83, 141}, {140, 3793}, {148, 3627}, {183, 2548}, {230, 6179}, {239, 4911}, {297, 1993}, {316, 3629}, {317, 2207}, {384, 3933}, {458, 6515}, {491, 6424}, {492, 6423}, {511, 4173}, {626, 5007}, {894, 5015}, {1003, 3926}, {1078, 3815}, {1975, 7737}, {1992, 5286}, {2896, 3329}, {3096, 3589}, {3311, 7388}, {3312, 7389}, {3552, 6390}, {3785, 7736}, {4045, 5041}, {5008, 6680}, {5017, 6393}, {5025, 5305}, {5965, 6248}, {6655,7839}
X(7763) lies on these lines: {2, 39}, {3, 315}, {4, 99}, {5, 1975}, {6, 6393}, {20, 316}, {24, 317}, {30,7773}, {32, 620}, {54, 69}, {83, 7736}, {110, 2909}, {140, 183}, {147, 5152}, {193, 1692}, {254, 264}, {331, 4554}, {339, 6640}, {350, 499}, {371, 492}, {372, 491}, {384, 2548}, {498, 1909}, {574, 626}, {668, 5552}, {693, 7626}, {754, 5206}, {1236, 3548}, {1506, 3734}, {1509, 5712}, {2549, 5025}, {3095, 5976}, {3267, 7630}, {3520, 5866}, {3523, 3785}, {3552, 5149}, {5013, 6656}, {6374, 6389}
X(7763) = isotomic conjugate of X(2165)X(7764) lies on these lines: {2, 3108}, {3, 754}, {4, 543}, {5, 538}, {6, 3788}, {32, 620}, {39, 325}, {69, 5034}, {76, 1506}, {99, 6658}, {114, 3095}, {115, 194}, {140, 524}, {141, 6683}, {262, 6309}, {315, 574}, {491, 1505}, {492, 1504}, {525, 5448}, {550, 3849}, {591, 6419}, {625, 5254}, {1007, 3767}, {1975, 5475}, {1991, 6420}, {2482, 3552}, {2548, 3734}, {3314, 6292}, {3522, 7618}, {3523, 7622}, {3815, 3933}, {3818, 6311}, {3854, 7620}, {5052, 6393}, {5056, 7617}, {5068, 7615}, {6337, 7737}
X(7765) lies on these lines: {2,7781}, {3, 5309}, {4, 7739}, {5, 39}, {6, 382}, {20, 32}, {30, 5007}, {54, 6794}, {76, 4045}, {83, 148}, {99, 6680}, {172, 4325}, {187, 548}, {194, 626}, {217, 1562}, {230, 3530}, {384, 543}, {538, 6656}, {550, 5306}, {574, 631}, {754, 6655}, {858, 1194}, {1180, 5169}, {1906, 3199}, {1914, 4330}, {2241, 4309}, {2242, 4317}, {2548, 3832}, {3053, 5346}, {3520, 6103}, {3526, 5013}, {3528, 5206}, {3843, 5475}, {3853, 5041}, {3855, 7736}, {4197, 5283}, {5024, 5070}
X(7766) lies on these lines: {2, 6}, {4, 5984}, {23, 2452}, {31, 983}, {32, 99}, {39, 6179}, {61, 5980}, {62, 5981}, {76, 5007}, {98, 576}, {148, 7737}, {172, 330}, {182, 6194}, {192, 1914}, {251, 2998}, {262, 5097}, {315, 5319}, {316, 5309}, {350, 5332}, {538, 3972}, {626, 5368}, {729, 6195}, {754, 5355}, {895, 5987}, {1351, 5999}, {1353, 1513}, {1383, 3228}, {1621, 4704}, {1909, 7296}, {1916, 5939}, {2080, 7709}, {2280, 2344}, {3212, 7132}, {5025, 5305}, {5201, 7492}, {5286, 6655}
X(7766) = anticomplement of X(3314)X(7767) lies on these lines: {3, 69}, {5, 183}, {30, 76}, {32, 141}, {39, 524}, {99, 548}, {140, 325}, {187, 3631}, {230, 626}, {264, 6756}, {311, 7553}, {316, 546}, {317, 1595}, {343, 441}, {385, 2896}, {550, 1975}, {574, 3630}, {599, 3053}, {754, 3934}, {1007, 3526}, {1211, 5337}, {1235, 3575}, {1369, 5133}, {1384, 3620}, {1447, 5015}, {1503, 5188}, {1691, 6308}, {1799, 6676}, {3096, 6179}, {3589, 5007}, {3760, 6284}, {3761, 7354}, {3917, 4173}, {4030, 7198}, {4357, 5266}, {4911, 7081}
X(7767) = X(7767) = isotomic conjugate of isogonal conjugate of X(22352)X(7768) lies on these lines: {2, 5007}, {4, 69}, {6, 3096}, {32, 3314}, {39, 2896}, {61, 299}, {62, 298}, {67, 670}, {83, 141}, {99, 550}, {140, 325}, {147, 5188}, {183, 1656}, {290, 3519}, {302, 6695}, {303, 6694}, {319, 4911}, {320, 3874}, {350, 4857}, {384, 754}, {385, 626}, {524, 6656}, {538, 6655}, {542, 7470}, {1007, 3533}, {1369, 5189}, {1657, 1975}, {1799, 7495}, {1909, 5270}, {3329, 6292}, {3471, 5641}, {3522, 3926}, {3523, 3785}, {3630, 5254}, {3849, 6658}
X(7768) = isogonal conjugate of X(3456)The trilinear polar of X(7769) passes through X(1510). (Randy Hutson, July 23, 2015)
X(7769) lies on these lines: {2, 39}, {3, 316}, {5, 99}, {61, 302}, {62, 303}, {69, 575}, {83, 3815}, {95, 1166}, {114, 5152}, {140, 325}, {141, 5038}, {183, 3526}, {249, 1970}, {276, 6331}, {288, 343}, {311, 3459}, {315, 631}, {317, 3147}, {384, 620}, {491, 5420}, {492, 5418}, {574, 5025}, {625, 6655}, {632, 3933}, {1235, 6143}, {1509, 5718}, {1656, 1975}, {2548, 3972}, {3090, 6337}, {3329, 6680}, {3552, 5475}, {3589, 6393}, {3628, 6390}, {5031, 5116}, {5092, 5207} X(7769) = isotomic conjugate of X(2963)As a point on the Euler line, X(7770) has Shinagawa coefficients ((E + F)2, S2).
X(7770) lies on these lines: {2, 3}, {6, 76}, {32, 183}, {39, 1975}, {99, 2023}, {115, 5149}, {141, 315}, {182, 6248}, {194, 3329}, {239, 4385}, {264, 2207}, {287, 1181}, {316, 2076}, {325, 2548}, {574, 6683}, {625, 5162}, {626, 5475}, {894, 3673}, {999, 6645}, {1078, 3053}, {1506, 3788}, {3114, 3224}, {3295, 4366}, {3398, 7697}, {3589, 4048}, {3618, 5286}, {3620, 5395}, {3661, 5015}, {3662, 4911}, {3760, 5280}, {3761, 5299}, {3926, 7736}, {4045, 6704}, {4396, 7296}, {4400, 5332}
X(7770) = complement of X(7791)X(7771) lies on these lines: {2, 187}, {3, 76}, {32, 3329}, {39, 6179}, {69, 3431}, {83, 3053}, {186, 264}, {262, 2080}, {274, 4188}, {298, 5464}, {299, 5463}, {302, 3642}, {303, 3643}, {315, 631}, {325, 549}, {350, 5010}, {384, 5206}, {385, 574}, {599, 5026}, {620, 3314}, {671, 7610}, {843, 5108}, {1003, 5210}, {1340, 6189}, {1341, 6190}, {1799, 7485}, {1909, 7280}, {2407, 5661}, {2896, 3788}, {3111, 5468}, {3523, 3785}, {3552, 3934}, {5025, 6722}, {5267, 6376}, {5971, 7496}
X(7771) = anticomplement of X(7603)Let P1' and U1' be the circle-O(61,62)-inverses of P(1) and U(1), resp. Then X(7772) = P(1)U1'∩U(1)P1'. (Randy Hutson, January 17, 2020)
X(7772) lies on these lines: {2, 3108}, {3, 6}, {4, 7739}, {5, 5309}, {23, 1180}, {35, 5332}, {36, 7296}, {69, 6292}, {76, 3329}, {83, 194}, {115, 147}, {140, 5306}, {183, 6683}, {184, 3456}, {230, 632}, {251, 7492}, {315, 4045}, {439, 7618}, {546, 5254}, {1015, 3304}, {1194, 1995}, {1500, 3303}, {1506, 3090}, {1595, 1990}, {2241, 2276}, {2242, 2275}, {2549, 3146}, {3199, 5198}, {3525, 5368}, {3529, 7737}, {3589, 3933}, {3628, 3815}, {5047, 5283}, {5079, 7603}, {6749, 6756}
X(7772) = inverse-in-1st-Brocard-circle of X(5007)X(7773) lies on these lines: {2,3053}, {3,316}, {4,325}, {5,183}, {6,5025}, {30,7763}, {32,625} and others
X(7774) lies on these lines: {2, 6}, {4, 147}, {32, 620}, {39, 315}, {76, 2548}, {99, 7737}, {114, 576}, {192, 497}, {232, 317}, {239, 7179}, {262, 1352}, {316, 2549}, {330, 388}, {384, 3926}, {538, 5475}, {549, 3793}, {574, 754}, {625, 5309}, {631, 3398}, {637, 3103}, {638, 3102}, {858, 2452}, {894, 3705}, {1003, 6390}, {1351, 1513}, {1370, 3164}, {1655, 2478}, {2996, 3832}, {3091, 6392}, {3552, 6337}, {3788, 5007}, {5025, 5286}, {5999, 6776}, {6655, 7738}
X(7774) = anticomplement of X(183)X(7775) lies on these lines: {2, 32}, {3, 3849}, {4, 543}, {5, 524}, {6, 625}, {20, 7618}, {140, 5569}, {183, 7603}, {194, 671}, {262, 736}, {316, 574}, {325, 3734}, {381, 538}, {384, 598}, {599, 3934}, {620, 1007}, {631, 7619}, {1153, 3526}, {1656, 7610}, {1992, 3767}, {3054, 3793}, {3091, 7615}, {3363, 3933}, {3832, 7620}, {3855, 5485}, {4045, 7736}, {5013, 5077}, {5031, 5039}, {5032, 5319}, {5034, 5207}, {5971, 6032}, {6722, 7735}
X(7776) lies on these lines: {3, 315}, {4, 3933}, {5, 69}, {6, 626}, {20, 6390}, {30, 3926}, {76, 381}, {99, 1657}, {140, 1007}, {141, 2548}, {183, 1656}, {193, 5305}, {218, 4766}, {316, 382}, {317, 1598}, {491, 3311}, {492, 3312}, {524, 3767}, {550, 6337}, {599, 3934}, {620, 5023}, {625, 5111}, {637, 1161}, {638, 1160}, {754, 3053}, {1078, 3526}, {1235, 7507}, {1369, 7485}, {3629, 5319}, {3705, 4911}, {3964, 7387}, {5015, 7179}
X(7777) lies on these lines: {2, 6}, {5, 194}, {11, 192}, {12, 330}, {23, 160}, {39, 625}, {76, 1506}, {83, 3788}, {99, 5475}, {114, 262}, {126, 6374}, {140, 3793}, {148, 381}, {316, 574}, {384, 2548}, {538, 7603}, {598, 2482}, {620, 3972}, {623, 3107}, {624, 3106}, {858, 3164}, {1502, 3266}, {1655, 4193}, {1995, 7665}, {2996, 5068}, {3096, 6683}, {3613, 5169}, {3818, 6054}, {5013, 6655}, {5056, 6392}, {5355, 6722}, {5640, 6786}
X(7777) = anticomplement of X(37688)
X(7777) = {X(2),X(193)}-harmonic conjugate of X(17008)
X(7777) = {X(2),X(385)}-harmonic conjugate of X(17004)
X(7778) lies on these lines: {2, 6}, {3, 114}, {76, 2023}, {126, 3014}, {148, 1975}, {160, 7467}, {315, 3053}, {316, 1003}, {381, 625}, {754, 1384}, {858, 2453}, {1350, 1513}, {1368, 6389}, {1447, 7232}, {1656, 3095}, {2482, 5077}, {2549, 6390}, {3001, 5094}, {3398, 3526}, {3598, 7238}, {3705, 4361}, {3767, 3933}, {3926, 5254}, {4045, 5024}, {4048, 5999}, {4363, 7179}, {4445, 7081}, {4478, 7172}, {5013, 6656}, {5026, 6054}, {5695, 5988}
X(7778) = complement of X(7735)X(7779) lies on these lines: {2, 6}, {38, 256}, {39, 2896}, {76, 5475}, {98, 5965}, {99, 754}, {147, 511}, {148, 316}, {187,7799}, {194, 315}, {232, 340}, {291, 2227}, {384, 3933}, {523, 2528}, {532, 5978}, {533, 5979}, {626, 5355}, {633, 3105}, {634, 3104}, {732, 1916}, {740, 5992}, {894, 4071}, {1278, 3434}, {1959, 3930}, {1975, 6658}, {2458, 4027}, {3552, 3926}, {3564, 5984}, {3788, 6179}, {3797, 4872}, {4754, 6625}, {7187, 7270}
X(7779) = anticomplement of X(385)X(7780) lies on these lines: {2, 5007}, {3, 538}, {4, 3849}, {5, 754}, {6, 6683}, {32, 183}, {35, 4396}, {36, 4400}, {39, 385}, {69, 1692}, {76, 187}, {83, 5008}, {98, 5188}, {140, 524}, {141, 6680}, {187,7799}, {230, 626}, {315, 625}, {512, 6310}, {543, 550}, {620, 3933}, {732, 5092}, {1196, 1799}, {1656, 7610}, {1975, 5206}, {2080, 6248}, {3053, 3734}, {3098, 6312}, {3523, 5569}, {3767, 3785}, {3851, 7617}, {4045, 5305}
X(7781) lies on these lines: {2,7765}, {3, 538}, {4, 543}, {20, 754}, {32, 99}, {39, 1975}, {76, 574}, {140, 7622}, {182, 698}, {385, 5206}, {524, 550}, {525, 1147}, {576, 5969}, {620, 3767}, {626, 2549}, {732, 3098}, {736, 6309}, {1003, 5007}, {1656, 7617}, {1657, 3849}, {3499, 6195}, {3523, 7616}, {3533, 5485}, {3788, 5254}, {3934, 5013}, {4045, 7738}, {4396, 7280}, {4400, 5010}, {5024, 6683}, {5056, 7615}, {5068, 7620}, {5286, 6680}
X(7782) lies on these lines: {3, 76}, {20, 316}, {39, 3552}, {69, 3528}, {75, 5267}, {83, 1003}, {160, 6374}, {187, 194}, {264, 3520}, {274, 4189}, {305, 6636}, {315, 376}, {325, 550}, {350, 7280}, {384, 574}, {385, 5206}, {439, 5286}, {489, 6278}, {490, 6281}, {548, 6390}, {598, 2548}, {620, 5025}, {626, 2482}, {1007, 3529}, {1909, 5010}, {3266, 7492}, {3522, 3926}, {3788, 6655}, {4563, 6800}, {5475, 6658}, {5866, 7512}
X(7783) lies on these lines: {2, 1975}, {3, 194}, {5, 148}, {6, 3552}, {39, 83}, {55, 330}, {56, 192}, {76, 574}, {193, 1350}, {325, 6655}, {404, 1655}, {439, 5304}, {538, 1078}, {543, 1506}, {698, 5116}, {1569, 5152}, {2275, 4366}, {2276, 6645}, {2482, 6680}, {2549, 5025}, {2896, 3933}, {3146, 7710}, {3203, 5118}, {3208, 7153}, {3314, 3926}, {3523, 6392}, {3802, 7188}, {5038, 5969}, {5206, 6179}, {5718, 6625}, {6390, 6656}
X(7784) lies on these lines: {2, 3053}, {3, 114}, {4, 141}, {6, 315}, {69, 5254}, {76, 338}, {183, 2896}, {218, 4805}, {230, 3785}, {316, 2076}, {325, 5013}, {381, 3934}, {382, 3734}, {489, 5591}, {490, 5590}, {524, 5286}, {625, 1656}, {1151, 7388}, {1152, 7389}, {1369, 5359}, {1384, 6680}, {1975, 3314}, {2549, 3933}, {3673, 7232}, {3851, 6249}, {4048, 7470}, {4361, 5015}, {4363, 4911}, {4385, 4445}, {5475, 6292}
X(7785) lies on these lines: {2, 32}, {4, 147}, {5, 385}, {6, 5025}, {13, 3181}, {14, 3180}, {39, 316}, {51, 3491}, {61, 624}, {62, 623}, {76, 5475}, {99, 6658}, {192, 1479}, {193, 576}, {325, 384}, {330, 1478}, {621, 3105}, {622, 3104}, {625, 5007}, {1031, 1502}, {1348, 3557}, {1349, 3558}, {1655, 5046}, {1992, 6034}, {2996, 3839}, {3329, 6656}, {3552, 5149}, {3628, 3793}, {3788, 3972}, {3832, 6392}
X(7785) = inverse-in-circle-{{X(4),X(194),X(3557),X(3558)}} of X(148)X(7786) lies on these lines: {2, 39}, {3, 83}, {6, 1078}, {32, 3329}, {99, 2023}, {140, 3095}, {315, 7736}, {316, 2021}, {325, 3096}, {384, 574}, {511, 631}, {620, 1916}, {726, 4687}, {730, 1698}, {732, 3763}, {1506, 4045}, {1569, 6722}, {1656, 2782}, {1975, 5024}, {3090, 6248}, {3094, 3589}, {3097, 3624}, {3102, 5418}, {3103, 5420}, {3202, 5012}, {3314, 6292}, {3523, 5188}, {3628, 7697}, {3815, 6656}, {5475, 6655}
X(7786) = isogonal conjugate of X(10014)X(7787) lies on these lines: {2, 32}, {3, 3329}, {4, 3398}, {6, 194}, {20, 182}, {39, 3552}, {66, 1031}, {76, 5007}, {98, 3091}, {148, 4027}, {192, 5280}, {193, 5039}, {330, 5299}, {350, 7296}, {401, 5422}, {458, 3172}, {631, 2080}, {727, 5264}, {1687, 2545}, {1688, 2544}, {1691, 3618}, {1724, 4195}, {1909, 5332}, {2549, 6658}, {3523, 5171}, {3933, 6661}, {3934, 5008}, {5038, 7738}, {6655, 7737}
X(7787) = complement of X(7929)X(7788) lies on these lines: {2, 6}, {22, 1634}, {25, 340}, {30, 315}, {76, 381}, {99, 3534}, {147, 1350}, {264, 5064}, {305, 670}, {316, 3830}, {317, 428}, {319, 7179}, {320, 3705}, {376, 3926}, {383, 634}, {519, 3905}, {542, 5989}, {626, 5309}, {633, 1080}, {754, 1003}, {1078, 5054}, {1272, 1369}, {1273, 7485}, {2871, 2979}, {2896, 5013}, {3524, 3785}, {5976, 6054}, {6656, 7739}
X(7788) = anticomplement of X(5306)X(7789) lies on these lines: {2, 1975}, {3, 66}, {5, 3734}, {6, 3926}, {30, 626}, {32, 524}, {39, 698}, {69, 3053}, {76, 230}, {83, 6661}, {99, 6656}, {140, 620}, {187, 3631}, {315, 1003}, {325, 384}, {439, 3620}, {489, 5590}, {490, 5591}, {538, 5305}, {546, 625}, {599, 3785}, {980, 6703}, {988, 4657}, {1384, 3630}, {2482, 6292}, {3314, 3552}, {3665, 4376}, {3703, 4372}
X(7789) = complement of X(5254)X(7790) lies on these lines: {2, 99}, {3,7828}, {4, 83}, {6, 316}, {30, 3972}, {32, 6655}, {39, 625}, {76, 141}, {114, 7709}, {193, 315}, {194, 626}, {264, 5523}, {385, 5309}, {538, 3314}, {597, 598}, {623, 3106}, {624, 3107}, {754, 5355}, {1078, 3767}, {1340, 2040}, {1341, 2039}, {1384, 5077}, {3329, 5475}, {3552, 6680}, {3793, 5305}, {3849, 5008}, {5418, 7389}, {5420, 7388}, {6071, 6787}
X(7790) = isotomic conjugate of X(9516)As a point on the Euler line, X(7791) has Shinagawa coefficients ((E + F)2 + S2 , -2S2).
X(7791) lies on these lines: {2, 3}, {32, 4045}, {39, 315}, {69, 194}, {76, 2549}, {83, 7737}, {99, 3096}, {141, 1975}, {148, 5976}, {183, 5254}, {316, 2021}, {325, 5013}, {385, 3785}, {574, 626}, {637, 3102}, {638, 3103}, {1078, 3767}, {1691, 3618}, {3314, 3926}, {3620, 6393}, {3734, 6292}, {3819, 6310}, {4293, 6645}, {4294, 4366}, {5082, 6653}, {5206, 6680}, {5319, 6179}, {5475, 6683}
X(7791) = midpoint of X(2) and X(33263)X(7792) lies on these lines: {2, 6}, {4,7851}, {5, 83}, {30, 3972}, {32, 6656}, {39, 620}, {76, 5305}, {114, 575}, {115, 5939}, {140, 3095}, {148, 384}, {182, 1513}, {187, 4045}, {538, 5355}, {626, 5007}, {754, 5008}, {1003, 2549}, {1196, 6375}, {1975, 5286}, {2542, 6039}, {2543, 6040}, {3001, 7495}, {3096, 6179}, {3705, 3759}, {3734, 5309}, {3758, 7179}, {4672, 5988}, {5480, 5999}, {6722, 7603}
X(7792) = complement of X(3314)X(7793) lies on these lines: {2, 32}, {3, 194}, {4, 2080}, {20, 98}, {35, 192}, {36, 330}, {39, 6179}, {69, 1691}, {76, 187}, {99, 5206}, {140, 3793}, {182, 193}, {183, 384}, {230, 5025}, {631, 3398}, {1379, 6177}, {1380, 6178}, {1655, 4189}, {1975, 5023}, {1992, 5038}, {3164, 7488}, {3522, 6392}, {3767, 6655}, {3926, 4027}, {3934, 3972}, {5008, 6683}, {6200, 6312}, {6316, 6396}
X(7793) = complement of X(7900)X(7794) lies on these lines: {2, 3108}, {3, 67}, {32, 69}, {39, 141}, {76, 115}, {99, 2896}, {183, 3788}, {187, 3631}, {194, 3096}, {315, 3734}, {325, 1506}, {384, 754}, {385, 6680}, {439, 3785}, {524, 5007}, {538, 6656}, {543, 6655}, {574, 3620}, {593, 5337}, {596, 1086}, {620, 1078}, {3117, 4121}, {3329, 6704}, {3398, 5965}, {3456, 6660}, {3589, 5041}, {3630, 5008}, {4721, 4766}
X(7795) lies on these lines: {2, 39}, {3, 66}, {4, 626}, {6, 3933}, {32, 69}, {98, 620}, {99, 3096}, {147, 5149}, {187, 3620}, {193, 5007}, {315, 384}, {325, 2548}, {574, 3619}, {599, 3053}, {625, 3091}, {637, 5590}, {638, 5591}, {1007, 1506}, {1384, 3631}, {1975, 2549}, {2896, 3552}, {3763, 5013}, {4045, 7738}, {4851, 5266}, {5162, 5207}, {6375, 6387}, {6680, 7735}
X(7795) = complement of X(5286)X(7796) lies on these lines: {2, 3108}, {5, 76}, {20, 99}, {24, 340}, {39, 3096}, {54, 69}, {61, 298}, {62, 299}, {183, 3526}, {194, 626}, {264, 1238}, {274, 4197}, {305, 858}, {316, 382}, {385, 3788}, {524, 6179}, {538, 5025}, {548, 6390}, {574, 2896}, {633, 5617}, {634, 5613}, {635, 3107}, {636, 3106}, {754, 3552}, {1007, 5067}, {3528, 6337}, {6148, 7556}
X(7796) = isotomic conjugate of X(2980)X(7797) lies on these lines: {2, 39}, {4, 3398}, {5, 147}, {6, 5025}, {32, 6655}, {61, 621}, {62, 622}, {83, 115}, {99, 6680}, {148, 384}, {239, 4109}, {315, 5319}, {316, 5007}, {385, 2896}, {575, 3091}, {625, 5041}, {626, 5355}, {754, 5368}, {1078, 4045}, {1746, 7384}, {2039, 2542}, {2040, 2543}, {2549, 3552}, {3618, 5038}, {3839, 5395}, {3972, 6658}, {5346, 6179}
X(7797) = anticomplement of X(7832)X(7798) lies on these lines: {2, 5355}, {6, 538}, {30, 3629}, {32, 99}, {39, 183}, {69, 4045}, {76, 3329}, {193, 754}, {325, 5309}, {385, 574}, {543, 1992}, {576, 2782}, {620, 7735}, {626, 5286}, {698, 5039}, {736, 5028}, {1003, 5008}, {1007, 3767}, {1975, 5007}, {2548, 6392}, {3180, 6775}, {3181, 6772}, {3788, 5305}, {3926, 5319}, {5206, 6179}, {5306, 6390}, {7603, 7617}
X(7799) lies on these lines: {2, 39}, {15, 298}, {16, 299}, {30, 99}, {69, 3431}, {83, 6661}, {95, 1238}, {148, 625}, {183, 5054}, {186, 340}, {187,7779}, {249, 524}, {315, 376}, {320, 4973}, {350, 3582}, {381, 1975}, {385, 620}, {542, 5152}, {549, 1078}, {574, 3314}, {599, 5116}, {698, 6034}, {754, 2482}, {1007, 3545}, {1138, 1272}, {1909, 3584}, {3096, 5013}, {5971, 7664}
X(7799) = isogonal conjugate of X(11060)X(7800) lies on these lines: {2, 32}, {3, 66}, {4, 3934}, {20, 3734}, {39, 69}, {76, 2549}, {127, 3549}, {147, 620}, {183, 3767}, {187, 3619}, {385, 5319}, {538, 7738}, {574, 3620}, {599, 3933}, {625, 3090}, {631, 3788}, {1992, 5041}, {2080, 5031}, {3053, 3763}, {3491, 3819}, {3618, 5007}, {3631, 5024}, {4045, 5286}, {4657, 5266}, {6337, 7618}, {6683, 7736}
X(7800) = anticomplement of X(7808)X(7801) lies on these lines: {2, 39}, {3, 67}, {32, 524}, {69, 187}, {99, 3314}, {141, 574}, {183, 620}, {193, 5008}, {315, 3849}, {325, 3734}, {543, 626}, {671, 5025}, {754, 1003}, {1007, 7603}, {1078, 5569}, {1992, 5007}, {3095, 5476}, {3525, 7607}, {3630, 3793}, {3642, 6298}, {3643, 6299}, {3763, 5024}, {4048, 5162}, {5013, 6292}, {5346, 6680}, {6337, 7618}
X(7801) = anticomplement of X(7817)X(7802) lies on these lines: {2, 5206}, {3, 316}, {4, 1078}, {20, 99}, {30, 76}, {32, 6655}, {39, 3849}, {69, 3529}, {83, 7737}, {183, 382}, {187, 5025}, {194, 754}, {264, 6240}, {302, 5237}, {303, 5238}, {325, 550}, {384, 3096}, {626, 3552}, {1007, 3528}, {1657, 1975}, {1799, 7391}, {2896, 3734}, {3098, 5207}, {3146, 3785}, {3972, 6656}, {5254, 6179}
X(7803) lies on these lines: {2, 39}, {4, 83}, {5,7851}, {6, 315}, {20, 3972}, {32, 4045}, {69, 3096}, {99, 7738}, {114, 3090}, {148, 5149}, {183, 5305}, {316, 1692}, {371, 489}, {372, 490}, {384, 2549}, {385, 5319}, {574, 6680}, {598, 5395}, {1078, 7735}, {2548, 3329}, {2909, 5012}, {3589, 4048}, {3758, 4911}, {3759, 5015}, {3785, 5304}, {5355, 6292}, {6655, 7737}
X(7803) = anticomplement of X(7822)X(7804) lies on these lines: {2, 187}, {3, 6683}, {5, 2794}, {6, 538}, {30, 3589}, {32, 183}, {39, 83}, {76, 5007}, {115, 5939}, {141, 754}, {194, 5041}, {325, 6661}, {373, 1316}, {385, 5008}, {543, 597}, {574, 1003}, {575, 2782}, {620, 3815}, {1007, 2548}, {1194, 4159}, {2549, 3618}, {2795, 4672}, {3363, 5461}, {3398, 6248}, {3760, 7296}, {3761, 5332}
X(7804) = complement of X(7761)X(7805) lies on these lines: {2, 5041}, {5, 3629}, {6, 3934}, {32, 538}, {39, 385}, {69, 5319}, {76, 5007}, {183, 6683}, {187, 194}, {193, 625}, {315, 5309}, {384, 5008}, {524, 626}, {597, 6704}, {648, 3199}, {754, 5254}, {1992, 2548}, {3760, 5332}, {3761, 7296}, {3785, 7739}, {3788, 7735}, {3933, 5306}, {4396, 5299}, {4400, 5280}, {5355, 6656}, {6392, 7737}
X(7806) lies on these lines: {2, 6}, {32, 316}, {76, 6680}, {98, 3407}, {99, 5309}, {115, 3972}, {148, 1003}, {194, 5305}, {262, 6036}, {264, 6103}, {384, 3767}, {468, 3186}, {598, 5461}, {620, 5355}, {625, 5008}, {626, 6179}, {850, 6041}, {3053, 6655}, {3106, 6672}, {3107, 6671}, {3291, 6375}, {3526, 7616}, {3552, 5254}, {3788, 5346}, {5068, 5395}
X(7806) = complement of X(7897)As a point on the Euler line, X(7807) has Shinagawa coefficients ((E + F)2 - 2S2, S2).
X(7807) lies on these lines: {2, 3}, {6, 6393}, {32, 325}, {39, 620}, {76, 230}, {83, 3815}, {99, 5254}, {141, 1078}, {187, 626}, {194, 5305}, {315, 3053}, {385, 3933}, {491, 6423}, {492, 6424}, {495, 6645}, {496, 4366}, {524, 6179}, {639, 2460}, {640, 2459}, {1975, 3767}, {2021, 3934}, {3094, 3589}, {3926, 7735}, {5286, 6337}, {6036, 6248}
X(7807) = midpoint of X(2) and X(33246)X(7808) lies on these lines: {2, 32}, {3, 6683}, {4, 4045}, {5, 182}, {6, 3934}, {39, 1975}, {76, 3329}, {98, 3090}, {140, 5171}, {141, 5039}, {183, 5007}, {384, 574}, {458, 3199}, {543, 7738}, {597, 5305}, {625, 1691}, {1656, 3398}, {2023, 5149}, {2080, 3526}, {3618, 3767}, {3788, 3815}, {3972, 5206}, {5038, 7617}, {5182, 5461}, {5475, 6656}
X(7808) = complement of X(7800)X(7809) lies on these lines: {2, 32}, {13, 298}, {14, 299}, {30, 99}, {69, 1568}, {76, 381}, {95, 1157}, {183, 5055}, {264, 1225}, {274, 6175}, {328, 1494}, {340, 403}, {385, 625}, {524, 5103}, {538, 671}, {542, 5207}, {1007, 3524}, {1272, 1273}, {1975, 3830}, {3314, 5475}, {3543, 3926}, {3845, 3933}, {3849, 5149}, {5025, 5309}
X(7810) lies on these lines: {2, 32}, {3, 67}, {30, 5188}, {39, 524}, {69, 574}, {76, 543}, {115, 183}, {141, 187}, {385, 4045}, {597, 5007}, {620, 3314}, {671, 6655}, {1384, 3763}, {2387, 3917}, {3091, 7616}, {3589, 3793}, {3631, 6390}, {3642, 6582}, {3643, 6295}, {3734, 6781}, {3788, 5569}, {3849, 3934}, {3926, 7618}, {5025, 5461}
X(7810) = complement of X(7812)Let LA be the line through the circumcircle intercepts of lines AP(1) and AU(1); define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The triangle A'B'C' (the 5th Brocard triangle, defined at X(32)) is homothetic to ABC at X(32), and X(7811) is the centroid of A'B'C'. (Randy Hutson, July 23, 2015)
X(7811) lies on these lines: {2, 32}, {15, 299}, {16, 298}, {30, 76}, {69, 74}, {141, 3972}, {183, 316}, {187, 3314}, {264, 7576}, {325, 549}, {340, 378}, {385, 5309}, {524, 3094}, {532, 3105}, {533, 3104}, {599, 1003}, {1285, 3619}, {1975, 3534}, {2387, 2979}, {2794, 6194}, {3099, 3679}, {5306, 6179}, {5965, 7709}, {6031, 7664} X(7811) = anticomplement of X(7753)X(7812) lies on these lines: {2, 32}, {4, 542}, {5, 6179}, {6, 316}, {30, 3095}, {39, 3849}, {76, 524}, {99, 7737}, {194, 543}, {325, 3972}, {385, 5475}, {530, 3104}, {531, 3105}, {597, 6656}, {625, 5008}, {1007, 1285}, {1383, 7664}, {2387, 3060}, {2482, 3552}, {3329,7761}, {5007, 5025}, {5032, 5286}, {5039, 5207}, {5206, 7622}, {6392, 7620}
X(7812) = anticomplement of X(7810)X(7813) lies on these lines: {2, 5355}, {3, 5965}, {32, 193}, {39, 141}, {69, 574}, {76, 1506}, {99, 754}, {115, 325}, {126, 3266}, {187, 524}, {194, 626}, {316, 543}, {385, 620}, {599, 5024}, {736, 1569}, {826, 2474}, {1384, 6144}, {2021, 6393}, {3051, 4175}, {3314, 4045}, {3629, 5008}, {5041, 6329}, {5206, 6337}, {5368, 6680}
X(7814) lies on these lines: {2, 5007}, {5, 76}, {17, 298}, {18, 299}, {20, 316}, {69, 5067}, {93, 264}, {99, 382}, {183, 5070}, {194, 625}, {302, 3411}, {303, 3412}, {305, 5169}, {315, 631}, {340, 7505}, {384, 598}, {1078, 3526}, {1506, 3314}, {1975, 3843}, {3096, 3815}, {3788, 3972}, {3832, 3926}, {3853, 6390}
X(7815) lies on these lines: {2, 32}, {3, 3734}, {5, 5171}, {6, 6683}, {39, 183}, {69, 5034}, {76, 574}, {98, 620}, {127, 6639}, {140, 141}, {384, 5206}, {538, 5013}, {599, 5038}, {625, 1656}, {1186, 3231}, {1207, 1613}, {1691, 3763}, {2549, 2996}, {3329, 6179}, {3398, 3526}, {3589, 5039}, {3619, 5033}, {3767, 4045}
X(7815) = complement of X(2548)X(7816) lies on these lines: {3, 3734}, {4, 625}, {5, 620}, {30, 626}, {32, 538}, {39, 83}, {76, 187}, {141, 550}, {183, 5206}, {194, 3972}, {315, 3849}, {316, 6658}, {511, 4048}, {512, 3491}, {543, 5254}, {574, 6683}, {575, 5026}, {754, 3933}, {1235, 4235}, {1506, 2482}, {2548, 6337}, {2996, 3767}, {3926, 7737}
X(7817) lies on these lines: {2, 39}, {5, 542}, {6, 625}, {32, 3849}, {115, 5939}, {230, 4045}, {315, 5346}, {316, 5008}, {325, 5355}, {381, 3398}, {384, 671}, {524, 626}, {543, 5254}, {754, 5306}, {1992, 5319}, {3053, 5077}, {3329, 7603}, {3815, 6722}, {5007, 5025}, {5013, 7622}, {5038, 7617}, {7618, 7738}
X(7817) = complement of X(7801)Let A', B', C' be the reflections of X(32) in A, B, C, resp. X(7818) is the centroid of A'B'C'. (Randy Hutson, July 11, 2019)
X(7818) lies on these lines: {2, 32}, {69, 115}, {127, 577}, {141, 5475}, {183, 625}, {193, 5355}, {316, 3314}, {325, 574}, {376, 2482}, {381, 511}, {519, 4769}, {524, 5028}, {591, 1505}, {639, 3070}, {640, 3071}, {746, 4664}, {760, 3679}, {1003, 3849}, {1504, 1991}, {3098, 6033}, {3788, 5206}, {4766, 4805} X(7818) = reflection of X(32) in X(2)As a point on the Euler line, X(7819) has Shinagawa coefficients (2(E + F)2 - S2 , S2).
X(7819) lies on these lines: {2, 3}, {6, 3933}, {32, 141}, {39, 698}, {76, 5305}, {83, 325}, {187, 5031}, {230, 736}, {524, 5007}, {620, 2023}, {1384, 3619}, {3053, 3763}, {3096, 3972}, {3398, 3564}, {3618, 3926}, {3631, 5008}, {3734, 5254}, {3788, 3815}, {3912, 5266}, {5024, 6337}, {5041, 6329}, {5103, 5162}
X(7819) = midpoint of X(2) and X(6661)X(7820) lies on these lines: {2, 99}, {3, 2916}, {32, 69}, {39, 698}, {76, 6680}, {140, 6248}, {141, 187}, {316, 384}, {325, 6661}, {524, 5008}, {538, 5355}, {599, 1384}, {623, 5321}, {624, 5318}, {754, 3314}, {1003, 6781}, {1506, 3788}, {2021, 3934}, {3096, 3552}, {3629, 3933}, {3631, 3793}, {6036, 7697}
X(7820) = complement of X(7790)X(7821) lies on these lines: {2, 5007}, {39, 325}, {69, 1570}, {76, 625}, {114, 5188}, {115, 3933}, {141, 1506}, {187, 315}, {193, 5346}, {316, 6658}, {538, 5025}, {550, 2482}, {576, 599}, {3096, 6683}, {3314, 3934}, {3552, 3849}, {3629, 5368}, {3763, 5039}, {3815, 6292}, {5008, 6680}, {6054, 7470}
X(7821) = complement of X(6179)X(7822) lies on these lines: {2, 39}, {3, 2916}, {32, 141}, {69, 5007}, {83, 3314}, {183, 6680}, {187, 3619}, {384, 3096}, {620, 5989}, {626, 5475}, {639, 3071}, {640, 3070}, {1078, 3407}, {1975, 4045}, {2896, 3972}, {3526, 6036}, {3589, 3933}, {3618, 5041}, {3620, 5008}, {3734, 6656}, {5031, 5162}
X(7822) = complement of X(7803)X(7823) lies on these lines: {2, 3053}, {4, 385}, {6, 6655}, {30, 194}, {32, 316}, {39, 3849}, {76, 754}, {115, 6179}, {148, 382}, {192, 6284}, {193, 1503}, {315, 384}, {325, 3552}, {330, 7354}, {546, 3793}, {626, 3972}, {1078, 5475}, {1916, 2794}, {1975, 6658}, {3543, 6392}, {5017, 5207}
As a point on the Euler line, X(7824) has Shinagawa coefficients ((E + F)2 + 3S2 , -2S2).
X(7824) lies on these lines: {2, 3}, {32, 3329}, {35, 4366}, {36, 6645}, {39, 385}, {76, 574}, {83, 187}, {99, 3934}, {141, 5116}, {183, 194}, {262, 5171}, {316, 1506}, {325, 2896}, {620, 5152}, {2076, 3589}, {3096, 3788}, {3231, 3499}, {3618, 5017}, {3763, 4048}, {3972, 5206}, {5162, 6680}
X(7824) = complement of X(16044)X(7825) lies on these lines: {2, 5206}, {3, 625}, {4, 626}, {5, 5171}, {20, 620}, {30, 3788}, {32, 316}, {115, 315}, {141, 546}, {182, 5103}, {381, 3934}, {543, 3926}, {574, 6655}, {754, 3767}, {2548, 4045}, {3053, 3849}, {3098, 5031}, {5007,7851}, {5028, 5207}, {5475, 6656}, {6680, 7737}
X(7826) lies on these lines: {3, 5965}, {6, 6292}, {32, 69}, {39, 524}, {76, 754}, {115, 315}, {141, 5007}, {183, 1506}, {187, 3630}, {385, 626}, {574, 3785}, {1975, 6781}, {2482, 3926}, {2895, 5337}, {2896, 4045}, {3314, 6179}, {3564, 5188}, {3629, 5041}, {3631, 5008}, {5355, 6656}
X(7827) lies on these lines: {2, 39}, {4, 575}, {5, 6054}, {6, 316}, {30, 3398}, {61, 531}, {62, 530}, {83, 597}, {115, 3329}, {315, 1992}, {384, 543}, {385, 4045}, {524, 6656}, {599, 3096}, {1078, 5305}, {1506, 5461}, {2482, 6680}, {2549, 3972}, {3849, 5007}, {5319, 6179}
X(7828) lies on these lines: {2, 39}, {3,7790}, {4, 3972}, {5, 83}, {6, 5031}, {32, 316}, {61, 623}, {62, 624}, {99, 5254}, {115, 384}, {183, 3096}, {187, 6655}, {230, 1078}, {315, 6179}, {325, 5305}, {385, 626}, {575, 1352}, {625, 5007}, {1506, 3329}, {1995, 2353}, {3589, 5038}
X(7828) = complement of X(7836)X(7829) lies on these lines: {2, 3108}, {5, 542}, {6, 626}, {32, 4045}, {39, 620}, {76, 5355}, {83, 115}, {183, 5346}, {230, 6683}, {325, 5041}, {384, 543}, {385, 5368}, {625, 6329}, {732, 3589}, {754, 5007}, {1506, 3329}, {2794, 3398}, {3618, 3767}, {3734, 5286}, {4075, 4422}
X(7830) lies on these lines: {2, 5206}, {3, 114}, {20, 3734}, {30, 3934}, {32, 4045}, {39, 754}, {76, 543}, {99, 2896}, {115, 1078}, {140, 625}, {141, 550}, {187, 6656}, {315, 574}, {316, 1506}, {384, 6292}, {2549, 3785}, {3096, 3552}, {3849, 6683}, {5025, 6722}, {5355, 6179}
X(7831) lies on these lines: {2, 187}, {3, 3096}, {39, 2896}, {76, 2549}, {99, 141}, {230, 1078}, {315, 7736}, {376, 3619}, {384, 6292}, {385, 4045}, {549, 6033}, {550, 6287}, {574, 3314}, {754, 3329}, {1003, 3763}, {3785, 5304}, {3819, 5167}, {3934, 6655}, {5650, 6787}
X(7832) lies on these lines: {2, 39}, {3, 3096}, {15, 635}, {16, 636}, {32, 3314}, {69, 6179}, {83, 325}, {98, 140}, {99, 6656}, {141, 1078}, {187, 2896}, {315, 3972}, {316, 384}, {385, 6680}, {620, 5152}, {631, 1352}, {2353, 7485}, {3734, 5025}, {3763, 5116}
X(7832) = complement of X(7797)As a point on the Euler line, X(7833) has Shinagawa coefficients ((E + F)2 + 3S2 , -6S2).
X(7833) lies on these lines: {2, 3}, {39, 3849}, {76, 543}, {99, 3314}, {148, 183}, {194, 524}, {316, 574}, {385, 2549}, {530, 3105}, {531, 3104}, {599, 1975}, {626, 2482}, {671, 1078}, {956, 6653}, {1992, 7738}, {3329, 7737}, {3972, 4045}, {4299, 6645}, {4302, 4366}
X(7833) = reflection of X(2) in X(8356)X(7834) lies on these lines: {2, 39}, {3, 4045}, {5, 182}, {6, 626}, {32, 6656}, {69, 5319}, {83, 3407}, {114, 1656}, {115, 5149}, {141, 5305}, {183, 6292}, {315, 5007}, {385, 3096}, {620, 5013}, {625, 1692}, {2896, 6179}, {3734, 5254}, {3972, 6655}, {6720, 7526}
X(7834) = complement of X(7795)X(7835) lies on these lines: {2, 99}, {3, 3096}, {76, 230}, {83, 7736}, {140, 7697}, {187, 3314}, {194, 5355}, {316, 1003}, {325, 3972}, {384, 3788}, {626, 3552}, {2896, 5206}, {3424, 3523}, {3524, 3619}, {3815, 6661}, {3926, 5304}, {3933, 6179}, {4048, 5152}
X(7836) lies on these lines: {2, 39}, {3, 147}, {15, 633}, {16, 634}, {69, 1691}, {99, 626}, {141, 5116}, {148, 1975}, {315, 3552}, {316, 6658}, {325, 384}, {385, 3933}, {574, 3096}, {620, 1078}, {3523, 3620}, {4048, 5207}, {5368, 6680}, {6390, 6656}
X(7836) = anticomplement of X(7828)X(7837) lies on these lines: {2, 6}, {30, 194}, {147, 1351}, {148, 3830}, {192, 3058}, {262, 5965}, {315, 7739}, {330, 5434}, {383, 5873}, {532, 3104}, {533, 3105}, {542, 1916}, {894, 4865}, {1080, 5872}, {3096, 5041}, {3839, 6392}, {3933, 6661}, {5025, 5309}
X(7837) = anticomplement of X(37671)X(7838) lies on these lines: {5, 3629}, {6, 626}, {32, 620}, {39, 754}, {141, 6704}, {193, 2548}, {194, 543}, {315, 4045}, {325, 5007}, {385, 1506}, {524, 3934}, {625, 5305}, {1369, 3108}, {1992, 3767}, {2794, 3095}, {3329, 6292}, {5025, 5355}, {5041, 6656}
X(7839) lies on these lines: {2, 3933}, {6, 194}, {20, 1351}, {39, 385}, {76, 3329}, {83, 538}, {99, 5007}, {315, 7739}, {401, 1994}, {524, 2896}, {574, 6179}, {620, 5368}, {1992, 7738}, {2548, 7615}, {3095, 5999}, {4366, 5299}, {5025, 5286}, {5280, 6645}, {6655,7762}
X(7840) lies on these lines: {2, 6}, {23, 1634}, {30, 147}, {99, 3849}, {316, 543}, {511, 6054}, {519, 5988}, {530, 5978}, {531, 5979}, {532, 5982}, {533, 5983}, {538, 671}, {542, 5999}, {598, 3734}, {670, 3266}, {754, 2482}, {5965, 6055}
X(7840) = anticomplement of X(22329)As a point on the Euler line, X(7841) has Shinagawa coefficients ((E + F)2 , -3S2).
X(7841) lies on these lines: {2, 3}, {6, 316}, {32, 3849}, {76, 338}, {83, 598}, {115, 183}, {148, 3314}, {315, 524}, {325, 2549}, {543, 626}, {574, 625}, {754, 5309}, {1078, 7610}, {1992, 5286}, {2482, 3788}, {2996, 5485}, {4045, 5475}
X(7841) = reflection of X(2) in X(33184)X(7842) lies on these lines: {3, 625}, {4, 3934}, {20, 3788}, {30, 626}, {32, 3849}, {39, 316}, {141, 3627}, {187, 5025}, {315, 538}, {382, 3734}, {543, 3933}, {550, 620}, {754, 5254}, {5013, 5077}, {5092, 5103}, {5149, 7470}, {5475, 6683}
X(7843) lies on these lines: {3, 3849}, {4, 538}, {5, 754}, {32, 625}, {39, 316}, {315, 3934}, {524, 546}, {543, 3627}, {632, 1153}, {1078, 7603}, {2548, 6683}, {3525, 5569}, {3788, 7737}, {5007, 5025}, {5052, 5207}, {5072, 7617}, {5079, 7610}
X(7844) lies on these lines: {2, 99}, {4, 6680}, {5, 182}, {6, 625}, {32, 316}, {39,7851}, {69, 626}, {325, 5309}, {754, 7735}, {1007, 7739}, {1384, 3849}, {1656, 6248}, {3094, 3763}, {3629, 5305}, {3788, 5254}, {5039, 5103}, {5077, 5210}, {5206, 6655}
X(7845) lies on these lines: {2, 5008}, {39, 315}, {69, 5475}, {99, 3849}, {115, 524}, {127, 3284}, {148, 316}, {183, 7603}, {187, 325}, {193, 5309}, {385, 625}, {511, 6033}, {626, 5007}, {2896, 6683}, {3629, 5355}, {5041, 6656}, {6390, 6781}
X(7846) lies on these lines: {2, 32}, {76, 5305}, {99, 7738}, {140, 262}, {141, 6179}, {631, 3098}, {3090, 6036}, {3094, 3589}, {3099, 3624}, {3104, 6694}, {3105, 6695}, {3314, 5007}, {3329, 3788}, {3552, 4045}, {3972, 6656}, {5056, 7694}, {5254, 6661}
X(7846) = complement of X(7938)X(7847) lies on these lines: {3,7790}, {20, 3972}, {30, 83}, {39, 316}, {76, 2549}, {99, 6656}, {140, 6321}, {148, 3934}, {315, 7738}, {384, 4045}, {538, 2896}, {543, 6292}, {574, 5025}, {1078, 5254}, {1975, 3096}, {3529, 3618}, {3849, 5041}, {5286, 6179}
X(7848) lies on these lines: {2, 5008}, {30, 3631}, {39, 2896}, {69, 538}, {141, 754}, {183, 625}, {187, 3314}, {230, 626}, {315, 3934}, {524, 4045}, {599, 3734}, {3096, 5007}, {3620, 7737}, {3785, 3788}, {5355, 6656}, {6683, 7736}
X(7849) lies on these lines: {2, 5007}, {5, 141}, {39, 3096}, {69, 5319}, {187, 2896}, {325, 6292}, {382, 3734}, {384, 3849}, {538, 6656}, {620, 3530}, {631, 3788}, {2548, 3619}, {3398, 3526}, {3620, 3767}, {3631, 5305}, {3933, 4045}
X(7849) = complement of X(5007)X(7850) lies on these lines: {2, 5008}, {4, 69}, {83, 3763}, {99, 3534}, {183, 5055}, {305, 1369}, {325, 549}, {548, 6390}, {598, 599}, {626, 6179}, {754, 3314}, {1078, 3526}, {3096, 3589}, {3629, 6656}, {4911, 5564}, {5015, 7321}
X(7850) = anticomplement of X(5008)X(7851) lies on these lines: {2, 1975}, {3,7790}, {4,7792}, {5,7803}, {6, 5025}, {32, 3849}, {39,7844}, {83, 381}, {115, 5149}, {183, 3767}, {315, 5305}, {325, 5286}, {382, 3972}, {626, 5309}, {698, 3763}, {754, 5346}, {1003, 6680}, {1503, 3091}, {1656, 2782}, {3053, 6655}, {5007,7825}
X(7852) lies on these lines: {2, 39}, {69, 5346}, {83, 625}, {114, 3628}, {182, 1656}, {187, 6656}, {230, 6292}, {315, 5008}, {325, 5041}, {397, 624}, {398, 623}, {524, 5368}, {626, 5007}, {1506, 1692}, {3933, 5355}, {5939, 6704}
X(7852) = complement of X(7832)X(7853) lies on these lines: {2, 187}, {5, 5188}, {39, 325}, {69, 5309}, {115, 141}, {127, 216}, {315, 5007}, {381, 3098}, {524, 5355}, {538, 3314}, {631, 7694}, {754, 5008}, {868, 5650}, {1656, 5171}, {3096, 3934}, {5092, 6033}
X(7854) lies on these lines: {2, 5007}, {3, 67}, {6, 6292}, {32, 141}, {39, 69}, {76, 148}, {183, 626}, {187, 3620}, {193, 5041}, {315, 3934}, {385, 3096}, {574, 3631}, {1078, 3314}, {1352, 5188}, {3619, 5008}, {5309, 6656}
X(7855) lies on these lines: {2, 5041}, {3, 5965}, {32, 524}, {39, 69}, {76, 5475}, {187, 439}, {193, 5007}, {315, 538}, {385, 3788}, {574, 3630}, {599, 6292}, {626, 5309}, {754, 1975}, {980, 2895}, {2482, 5023}, {5206, 6390}
X(7856) lies on these lines: {2, 3108}, {6, 5031}, {32, 6655}, {76, 5305}, {83, 3767}, {99, 5286}, {194, 5355}, {315, 5304}, {384, 5309}, {385, 3096}, {546, 598}, {626, 5368}, {1078, 7735}, {3972, 5254}, {5007, 5025}, {5306, 6179}
X(7857) lies on these lines: {2, 32}, {4, 6036}, {5, 3972}, {76, 230}, {99, 3767}, {115, 3552}, {140, 3095}, {187, 5025}, {194, 620}, {316, 3053}, {325, 6179}, {385, 3788}, {576, 3525}, {3104, 6672}, {3105, 6671}, {5206, 6655}
X(7858) lies on these lines: {2, 5007}, {6, 5031}, {39, 316}, {76, 2548}, {83, 325}, {194, 5475}, {315, 7736}, {385, 1506}, {546, 671}, {625, 5041}, {626, 3329}, {648, 1594}, {1078, 3815}, {1992, 3090}, {2896, 6683}, {5368, 6722}
X(7859) lies on these lines: {2, 39}, {4, 5092}, {6, 3096}, {15, 6694}, {16, 6695}, {83, 316}, {115, 6704}, {315, 3618}, {384, 4045}, {385, 5368}, {420, 6688}, {626, 3329}, {2896, 5007}, {5162, 6680}, {6560, 7388}, {6561, 7389}
X(7859) = isotomic conjugate of isogonal conjugate of X(34482)X(7860) lies on these lines: {4, 69}, {32,7885}, {99, 1657}, {183, 3851}, {305, 5189}, {325, 550}, {597, 6656}, {626, 3972}, {754, 5025}, {1078, 1656}, {1369, 7533}, {1975, 5073}, {2896, 5475}, {3552, 3849}, {3785, 5056}, {3926, 5059}
X(7861) lies on these lines: {5, 4045}, {30, 6680}, {32, 3849}, {39, 625}, {115, 3934}, {140, 6722}, {187, 6655}, {315, 5309}, {316, 5007}, {538, 626}, {546, 3589}, {754, 5305}, {2549, 3788}, {3767, 3785}, {5023, 5077}
X(7862) lies on these lines: {2, 32}, {3, 625}, {4, 620}, {5, 3734}, {127, 6640}, {141, 576}, {182, 5031}, {316, 5206}, {543, 6337}, {574, 5025}, {1007, 3767}, {1656, 3095}, {3098, 5103}, {3849, 5023}, {5033, 5207}
X(7863) lies on these lines: {3, 67}, {32, 193}, {39, 698}, {69, 5206}, {76, 620}, {99, 626}, {115, 1975}, {187, 3630}, {194, 5355}, {315, 6781}, {543, 5025}, {574, 3619}, {754, 3552}, {1501, 4175}, {1506, 3734}
X(7864) lies on these lines: {2, 1975}, {4, 3329}, {5, 7709}, {6, 6655}, {39, 625}, {76, 4045}, {194, 3314}, {315, 7739}, {384, 2549}, {385, 3785}, {538, 3096}, {1078, 5309}, {3146, 5480}, {3832, 7710}, {5355, 6179}
X(7865) lies on these lines: {2, 32}, {30, 141}, {69, 4045}, {381, 3934}, {538, 599}, {549, 3788}, {574, 3314}, {620, 3524}, {625, 5055}, {2076, 3849}, {2386, 3819}, {2549, 3620}, {3619, 7737}, {5309, 6656}
As a point on the Euler line, X(7866) has Shinagawa coefficients (2(E + F)2 - S2 , -S2).
X(7866) lies on these lines: {2, 3}, {6, 626}, {69, 5305}, {141, 3767}, {183, 3096}, {524, 5319}, {625, 1691}, {2548, 3589}, {3053, 6680}, {3094, 3763}, {3619, 6393}, {3788, 4045}, {3933, 5286}, {6390, 7738}
X(7866) = midpoint of X(2) and X(33223)X(7867) lies on these lines: {2, 32}, {141, 5028}, {193, 5368}, {511, 1656}, {524, 5346}, {574, 3788}, {625, 5162}, {631, 2794}, {746, 4687}, {760, 1698}, {1125, 4769}, {2458, 5031}, {3734, 5025}, {3933, 5309}
X(7868) lies on these lines: {2, 6}, {3, 3096}, {147, 5085}, {157, 7485}, {262, 1656}, {626, 5475}, {1003, 6781}, {1975, 2549}, {2896, 3053}, {3523, 7710}, {3788, 6292}, {5054, 6054}, {6661, 7737}
X(7868) = complement of X(16989)X(7869) lies on these lines: {2, 3108}, {4, 626}, {32, 3314}, {69, 6680}, {140, 141}, {147, 620}, {574, 3096}, {625, 3851}, {1656, 3095}, {2896, 5206}, {3763, 6683}, {3926, 4045}, {6704, 7736}
X(7870) lies on these lines: {2, 39}, {3, 6054}, {69, 2030}, {325, 3972}, {384, 598}, {524, 6179}, {543, 5025}, {599, 1078}, {620, 3314}, {626, 2482}, {671, 1975}, {3526, 7607}, {3552, 3849}
X(7871) lies on these lines: {2, 5041}, {5, 76}, {69, 575}, {99, 1657}, {315, 376}, {316, 3146}, {1078, 5054}, {1975, 3830}, {3314, 6292}, {3523, 3785}, {3788, 6179}, {6177, 6190}, {6178, 6189}
X(7872) lies on these lines: {4, 4045}, {20, 6680}, {32, 6655}, {148, 3096}, {381, 6683}, {574, 5025}, {625, 5013}, {626, 2549}, {631, 6722}, {754, 5286}, {3053, 5077}, {3589, 3627}, {3734, 6656}
X(7873) lies on these lines: {39, 315}, {187, 626}, {316, 2896}, {382, 599}, {384, 3849}, {538, 6655}, {548, 2482}, {625, 1078}, {639, 6566}, {640, 6567}, {754, 5007}, {1194, 1369}, {4045, 5041}
X(7874) lies on these lines: {2, 39}, {49, 182}, {114, 140}, {141, 1692}, {187, 626}, {325, 5007}, {384, 625}, {620, 6656}, {635, 6671}, {636, 6672}, {639, 6567}, {640, 6566}, {2909, 5651}
X(7874) = complement of X(7828)X(7875) lies on these lines: {2, 6}, {76, 5355}, {83, 3407}, {140, 6194}, {147, 5050}, {384, 2549}, {1194, 6375}, {3096, 5007}, {3104, 6695}, {3105, 6694}, {3424, 5068}, {3972, 4045}, {6179, 6292}
X(7875) = {X(2),X(385)}-harmonic conjugate of X(16986)As a point on the Euler line, X(7876) has Shinagawa coefficients (3(E + F)2 - S2 , -2S2).
X(7876) lies on these lines: {2, 3}, {6, 2896}, {39, 3096}, {76, 4045}, {141, 194}, {315, 3329}, {385, 5319}, {635, 3106}, {636, 3107}, {1975, 3763}, {3619, 7738}, {4309, 4366}, {4317, 6645}
X(7876) = {X(2),X(3)}-harmonic conjugate of X(7892)X(7877) lies on these lines: {4, 5965}, {6, 3096}, {69, 83}, {76, 524}, {193, 315}, {194, 754}, {316, 6144}, {325, 6179}, {626, 5368}, {671, 6392}, {3314, 5007}, {3629, 6656}, {3933, 3972}
X(7878) lies on these lines: {2, 5007}, {4, 575}, {6, 76}, {32, 3329}, {39, 3552}, {182, 7470}, {194, 5041}, {262, 3398}, {315, 3618}, {316, 1692}, {597, 6656}, {3096, 3589}, {5008, 6683}
X(7879) lies on these lines: {3, 147}, {4, 3620}, {6, 3096}, {69, 5286}, {76, 338}, {83, 3763}, {141, 315}, {183, 626}, {3631, 5254}, {3661, 4911}, {3662, 5015}, {3788, 5569}
X(7880) lies on these lines: {2, 39}, {30, 626}, {141, 542}, {187, 3314}, {325, 6661}, {381, 625}, {599, 1691}, {1003, 3849}, {3933, 5306}, {4045, 6390}, {5116, 7622}, {5149, 6054}
X(7880) = complement of X(5309)X(7881) lies on these lines: {2, 3933}, {3, 147}, {76, 2023}, {183, 3788}, {315, 1003}, {325, 2548}, {543, 626}, {599, 1078}, {631, 3564}, {732, 3763}, {3096, 5013}, {3926, 6656}
X(7882) lies on these lines: {5, 3630}, {39, 2896}, {69, 2548}, {141, 6704}, {315, 538}, {340, 3199}, {524, 626}, {625, 5111}, {754, 3933}, {1975, 3849}, {3096, 5041}, {3314, 5007}
X(7883) lies on these lines: {2, 32}, {3, 6054}, {76, 338}, {99, 3314}, {141, 316}, {384, 3849}, {524, 6656}, {530, 635}, {531, 636}, {543, 6655}, {1975, 5077}, {5056, 7616}
X(7884) lies on these lines: {2, 39}, {30, 3972}, {83, 381}, {598, 3407}, {671, 5989}, {2896, 5346}, {3096, 5305}, {3314, 5355}, {3545, 3618}, {5055, 6054}, {5254, 6661}, {5306, 6179}
X(7885) lies on these lines: {2, 3053}, {4, 3314}, {5, 2896}, {32,7860}, {148, 3933}, {315, 385}, {316, 384}, {325, 6655}, {625, 1078}, {3096, 5475}, {3329, 6656}, {3620, 3832}, {6033, 7470}
X(7886) lies on these lines: {2, 39}, {5, 2794}, {32, 625}, {83, 7603}, {140, 4045}, {187, 5025}, {230, 626}, {575, 3564}, {620, 5254}, {1007, 5319}, {1656, 3398}, {3053, 3849}
X(7886) = complement of X(3788)As a point on the Euler line, X(7887) has Shinagawa coefficients ((E + F)2 - 2S2 , -S2).
X(7887) lies on these lines: {2, 3}, {6, 5031}, {32, 625}, {76, 2023}, {115, 1975}, {183, 626}, {230, 315}, {316, 3053}, {325, 3767}, {1007, 5286}, {5017, 5103}, {5475, 6680}
X(7887) = complement of X(16925)X(7888) lies on these lines: {2, 3108}, {20, 2482}, {32, 325}, {115, 2996}, {141, 5034}, {315, 620}, {384, 598}, {574, 626}, {575, 599}, {625, 1975}, {1007, 1506}, {5067, 7608}
X(7889) lies on these lines: {2, 32}, {39, 698}, {76, 5355}, {115, 5149}, {140, 5188}, {141, 5007}, {187, 5103}, {384, 4045}, {597, 3933}, {620, 1916}, {1656, 6033}, {2482, 5013}
X(7890) lies on these lines: {2, 5368}, {3, 6144}, {32, 193}, {39, 524}, {69, 6292}, {141, 5041}, {194, 754}, {620, 6179}, {626, 5355}, {2482, 3053}, {3095, 5965}, {3629, 3933}
X(7891) lies on these lines: {2, 1975}, {3, 147}, {76, 620}, {99, 3788}, {140, 7709}, {194, 5305}, {325, 3552}, {384, 2548}, {385, 3926}, {626, 2482}, {1078, 5569}, {3522, 7710}
As a point on the Euler line, X(7892) has Shinagawa coefficients (3(E + F)2 - 3S2 , 2S2).
X(7892) lies on these lines: {2, 3}, {32, 3314}, {76, 6680}, {83, 3788}, {187, 3096}, {620, 1916}, {626, 3972}, {1078, 3407}, {2896, 3053}, {3106, 6694}, {3107, 6695}, {3763, 5207}
X(7892) = complement of X(7933)X(7893) lies on these lines: {6, 2896}, {20, 3564}, {32, 3314}, {69, 384}, {76, 754}, {194, 524}, {315, 385}, {340, 1968}, {626, 6179}, {2979, 4173}, {3096, 5007}, {3552, 3933}
X(7894) lies on these lines: {2, 5041}, {4, 5097}, {6, 76}, {39, 6179}, {194, 3972}, {315, 1992}, {316, 1570}, {524, 3096}, {598, 2996}, {3552, 5008}, {3629, 6656}, {5025, 5355}
X(7895) lies on these lines: {2, 5041}, {39, 3096}, {69, 1692}, {76, 625}, {140, 3631}, {141, 6683}, {315, 3849}, {325, 1506}, {524, 6680}, {538, 626}, {599, 5038}
X(7896) lies on these lines: {5, 3631}, {32, 3314}, {69, 626}, {141, 5039}, {315, 3734}, {574, 2896}, {599, 3934}, {620, 3785}, {2548, 3620}, {3619, 6704}, {3630, 5305}
X(7897) lies on these lines: {2, 6}, {76, 625}, {114, 6194}, {160, 7492}, {194, 626}, {315, 3552}, {1502, 4609}, {3098, 6054}, {3266, 6374}, {3926, 6655}, {3933, 5025}
X(7897) = anticomplement of X(7806)X(7898) lies on these lines: {2, 187}, {4, 2896}, {30, 3314}, {69, 148}, {194, 315}, {230, 5025}, {376, 6033}, {626, 3552}, {754, 5355}, {2979, 5167}, {3543, 3620}
X(7899) lies on these lines: {2, 32}, {76, 2023}, {99, 3788}, {141, 5111}, {262, 1656}, {325, 5305}, {384, 625}, {620, 6655}, {671, 1975}, {3523, 7694}, {3619, 5067}
X(7900) lies on these lines: {2, 32}, {147, 3146}, {193, 5111}, {194, 316}, {325, 3552}, {625, 6179}, {1352, 3832}, {3060, 3491}, {3926, 6658}, {5025, 5305}, {6655, 7738}
X(7900) = anticomplement of X(7793)As a point on the Euler line, X(7901) has Shinagawa coefficients (3(E + F)2 - 3S2 , -2S2).
X(7901) lies on these lines: {2, 3}, {83, 625}, {230, 2896}, {316, 6680}, {385, 626}, {1916, 3934}, {3314, 3767}, {3589, 5207}, {4366, 4857}, {5270, 6645}, {6292, 6722}
X(7901) = complement of X(33225)X(7902) lies on these lines: {32, 6655}, {315, 5355}, {546, 597}, {620, 7738}, {626, 5286}, {754, 5319}, {2549, 6680}, {3090, 5461}, {3734, 5254}, {3767, 4045}, {5309, 6656}
X(7903) lies on these lines: {32, 325}, {69, 1506}, {115, 6392}, {315, 574}, {754, 5206}, {1656, 5097}, {1992, 5368}, {3629, 5346}, {3933, 5475}, {6292, 7736}, {6337, 6781}
X(7904) lies on these lines: {2, 3053}, {3, 147}, {76, 543}, {141, 3552}, {183, 6655}, {187, 3096}, {385, 3785}, {1078, 5025}, {1503, 3522}, {3972, 6292}, {4045, 5368}
X(7905) lies on these lines: {2, 5041}, {39, 2896}, {76, 2548}, {83, 3933}, {193, 1692}, {194, 316}, {315, 7738}, {325, 5305}, {524, 1078}, {3329, 6704}, {3926, 3972}
X(7906) lies on these lines: {2, 3933}, {39, 3096}, {76, 1506}, {193, 6393}, {194, 325}, {384, 3926}, {620, 6179}, {1916, 6309}, {2896, 5013}, {3552, 6390}, {3788, 5346}
As a point on the Euler line, X(7907) has Shinagawa coefficients ((E + F)2 - 5S2 , 2S2).
X(7907) lies on these lines: {2, 3}, {76, 620}, {194, 230}, {316, 5206}, {498, 6645}, {499, 4366}, {1078, 3314}, {1506, 3972}, {3104, 6671}, {3105, 6672}, {5007, 5215}
X(7907) = complement of X(32966)X(7908) lies on these lines: {2, 5355}, {69, 620}, {230, 3788}, {315, 6781}, {325, 3734}, {549, 3631}, {574, 3314}, {599, 7622}, {626, 2549}, {5304, 6680}
X(7909) lies on these lines: {2, 3108}, {3, 6054}, {76, 2023}, {83, 325}, {99, 626}, {141, 5038}, {620, 2896}, {671, 5025}, {1078, 3314}, {3628, 7608}
X(7910) lies on these lines: {30, 3096}, {76, 148}, {315, 7738}, {316, 2021}, {626, 2482}, {1078, 7610}, {1975, 5077}, {3793, 5305}, {3972, 6656}, {5025, 6722}
X(7911) lies on these lines: {2, 5206}, {4, 3096}, {76, 338}, {83, 316}, {99, 626}, {115, 2896}, {193, 315}, {754, 5368}, {1078, 5025}, {3630, 5254}
X(7912) lies on these lines: {2, 32}, {5, 3314}, {20, 114}, {69, 5031}, {76, 625}, {148, 3926}, {194, 325}, {316, 3552}, {2996, 5503}, {3620, 5056}
X(7913) lies on these lines: {2, 99}, {32, 6656}, {69, 5355}, {127, 5158}, {141, 5309}, {182, 6033}, {381, 5092}, {3589, 5033}, {3767, 6292}, {5206, 6680}
X(7914) lies on these lines: {2, 32}, {5, 3098}, {141, 5305}, {625, 2076}, {3094, 3763}, {3619, 3767}, {3620, 5319}, {3734, 6656}, {4045, 7738}
X(7915) lies on these lines: {2, 39}, {140, 1503}, {141, 6680}, {187, 3096}, {625, 5162}, {632, 6036}, {1691, 3763}, {3314, 5007}, {3815, 6704}
X(7915) = complement of X(7834)X(7916) lies on these lines: {2, 5368}, {69, 5034}, {140, 3630}, {193, 6680}, {524, 3788}, {599, 6683}, {626, 5286}, {754, 3926}, {3734, 3933}
X(7917) lies on these lines: {20, 99}, {69, 576}, {76, 381}, {83, 3314}, {140, 325}, {183, 5070}, {316, 3627}, {626, 5355}, {1975, 5073}
X(7918) lies on these lines: {4, 5092}, {76, 141}, {83, 598}, {315, 1992}, {316, 1692}, {1506, 4045}, {2896, 5309}, {3972, 6655}, {5346, 6179}
X(7919) lies on these lines: {2, 99}, {83, 3407}, {230, 1078}, {315, 5304}, {625, 3329}, {626, 5355}, {3096, 3767}, {3314, 5309}, {6655, 6680}
X(7920) lies on these lines: {2, 3933}, {6, 5025}, {76, 5355}, {83, 5309}, {384, 5286}, {385, 5319}, {1078, 5346}, {3329, 3767}, {4045, 5368}
X(7921) lies on these lines: {6, 5025}, {39, 3849}, {83, 3314}, {315, 3329}, {384, 3926}, {385, 2548}, {1506, 6179}, {3060, 4173}, {3091, 3564}
X(7922) lies on these lines: {2, 5007}, {3, 6054}, {76, 115}, {315, 3972}, {325, 3096}, {2896, 3788}, {3090, 5476}, {3620, 5107}
X(7923) lies on these lines: {2, 1975}, {115, 6704}, {385, 2896}, {1656, 7709}, {2548, 3329}, {3096, 5309}, {3314, 5286}, {5068, 7710}
As a point on the Euler line, X(7924) has Shinagawa coefficients (3(E + F)2 + S2 , -6S2).
X(7924) lies on these lines: {2, 3}, {141, 148}, {315, 7739}, {316, 3329}, {385, 5309}, {599, 698}, {2549, 3314}, {2896, 5254}
X(7924) = midpoint of X(2) and X(6655)X(7925) lies on these lines: {2, 6}, {99, 625}, {114, 5999}, {140, 2896}, {148, 6390}, {316, 620}, {384, 3788}, {2549, 5025}
X(7925) = {X(99),X(625)}-harmonic conjugate of X(14041)X(7926) lies on these lines: {2, 5008}, {76, 5475}, {230, 6179}, {315, 7736}, {316, 2549}, {325, 3972}, {598, 3734}, {5025, 5355}
X(7927) lies on these lines: {30, 511}, {661, 4808}, {669, 3806}, {879, 1173}, {2643, 6547}, {3801, 4761}, {4088, 4983}, {4122, 4170}
X(7927) = isogonal conjugate of X(7953)X(7928) lies on these lines: {2, 3053}, {141, 6655}, {315, 3329}, {316, 6292}, {384, 3096}, {385, 2896}, {3314, 3926}
X(7929) lies on these lines: {2, 32}, {69, 698}, {147, 3522}, {1352, 3146}, {2979, 3491}, {3314, 3552}, {3620, 5104}
X(7929) = anticomplement of X(7787)X(7930) lies on these lines: {2, 39}, {98, 3526}, {315, 1285}, {626, 3972}, {1352, 3525}, {2030, 3619}, {3314, 6179}
X(7931) lies on these lines: {2, 6}, {316, 384}, {3096, 3788}, {3186, 5094}, {3818, 5999}, {5092, 6054}, {6390, 6656}
X(7931) = {X(2),X(385)}-harmonic conjugate of X(16984)X(7932) lies on these lines: {2, 39}, {147, 3090}, {626, 5368}, {2896, 7735}, {3314, 5305}, {3552, 6680}, {5056, 6776}
As a point on the Euler line, X(7933) has Shinagawa coefficients (3(E + F)2 - S2 , -4S2).
X(7933) lies on these lines: {2, 3}, {115, 3096}, {194, 626}, {315, 5319}, {2896, 3767}, {3314, 5254}, {3618, 5103}
X(7933) = anticomplement of X(7892)X(7934) lies on these lines: {2, 187}, {5, 3096}, {76, 115}, {315, 6179}, {3545, 3619}, {3788, 6655}, {3815, 6656}
X(7935) lies on these lines: {2, 5206}, {4, 6292}, {32, 6656}, {315, 4045}, {382, 3763}, {574, 626}, {3096, 3734}
X(7936) lies on these lines: {3, 6054}, {76, 148}, {315, 7736}, {3096, 3972}, {5025, 5461}, {5306, 6179}
X(7937) lies on these lines: {2, 187}, {76, 141}, {315, 3618}, {2896, 6179}, {3314, 4045}, {5025, 6292}
X(7938) lies on these lines: {2, 32}, {5, 6194}, {114, 3523}, {141, 5025}, {194, 3314}, {3619, 5031}
X(7938) = anticomplement of X(7846)X(7939) lies on these lines: {69, 5025}, {315, 384}, {325, 2896}, {385, 626}, {3096, 3329}, {3933, 6655}
X(7940) lies on these lines: {2, 39}, {140, 3096}, {325, 6179}, {620, 5025}, {625, 3552}, {3533, 3619}
X(7941) lies on these lines: {325, 384}, {626, 3329}, {2548, 3314}, {2896, 3815}, {5025, 5286}, {6390, 6658}
X(7942) lies on these lines: {2, 39}, {98, 1656}, {230, 3096}, {626, 6179}, {1352, 5067}, {3972, 5025}
X(7943) lies on these lines: {2, 39}, {2030, 5207}, {3090, 3818}, {3096, 6179}, {3972, 6656}
X(7944) lies on these lines: {2, 32}, {99, 6656}, {140, 6033}, {1916, 3934}, {5103, 5104}
X(7945) lies on these lines: {2, 39}, {147, 631}, {620, 3096}, {626, 3552}, {2030, 3620}
X(7946) lies on these lines: {2, 5007}, {194, 315}, {524, 5025}, {626, 5368}, {754, 3552}
X(7946) = anticomplement of X(6179)X(7947) lies on these lines: {2, 3933}, {325, 384}, {385, 3788}, {3926, 5025}, {6390, 6655}
As a point on the Euler line, X(7948) has Shinagawa coefficients (5(E + F)2 - S2 , -6S2).
X(7948) lies on these lines: {2, 3}, {385, 3096}, {626, 3329}, {698, 3763}
X(7949) lies on these lines: {76, 5475}, {315, 7738}, {325, 6179}
X(7950) lies on these lines: {30, 511}, {4770, 4808}
X(7950) = isogonal conjugate of X(7954)Let A' be the center of the nine-point-circle-inverse of A-excircle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(7745). (Hyacinthos #21600, César Lozada, 2/20/2013)
X(7951) lies on these lines: {1,5}, {2,36}, {3,3585}, {4,35}, {8,4867}, {9,46}, {10,908}, {13,7127}, {17,2307}, {30,5010}, {33,403}, {34,1594}, {40,6842}, {43,3136}, {55,381}, {56,1656}, {57,6881}, {65,5694}, {85,4089}, {93,2599}, {113,7727}, {115,2276}, {140,7280}, {165,6907}, {215,567}, {226,1737}, {230,609}, {325,3761}, {382,4324}, {388,499}, {390,1479}, {392,5087}, {427,5155}, {428,7298}, {484,1836}, {485,3301}, {486,3299}, {497,3545}, {515,6830}, {516,6932}, {517,6980}, {546,6284}, {547,5434}, {549,5326}, {612,5133}, {631,4299}, {946,1512}, {950,6990}, {999,3582}, {1015,7603}, {1056,5071}, {1060,2072}, {1087,1091}, {1089,3790}, {1111,7179}, {1125,3897}, {1203,5230}, {1209,7356}, {1210,3947}, {1385,6971}, {1447,7272}, {1506,2275}, {1532,1699}, {1714,5747}, {1725,7069}, {1745,3142}, {1770,6684}, {1785,1856}, {1788,5714}, {1826,1838}, {1834,5312}, {1870,7577}, {1914,5475}, {2066,6565}, {2077,6923}, {2093,3820}, {2099,5790}, {2330,3818}, {2362,3302}, {2478,5259}, {2548,5299}, {2551,6856}, {2635,4337}, {2886,3679}, {2964,7299}, {2975,7504}, {3058,5066}, {3086,5056}, {3245,5657}, {3295,3851}, {3303,5072}, {3304,5079}, {3338,5290}, {3436,5258}, {3485,5818}, {3486,6873}, {3526,4325}, {3574,6286}, {3576,5444}, {3577,5559}, {3600,7486}, {3601,5441}, {3612,5691}, {3616,3825}, {3624,4187}, {3628,5433}, {3633,3813}, {3634,4197}, {3705,4692}, {3753,3838}, {3767,5280}, {3817,4342}, {3832,4294}, {3839,5281}, {3841,4295}, {3843,4330}, {3845,4995}, {3855,4309}, {3944,4424}, {4297,6943}, {4317,5067}, {4338,5128}, {4680,7081}, {4880,5905}, {5046,5248}, {5169,5297}, {5172,7489}, {5226,5425}, {5310,7394}, {5345,7499}, {5348,6149}, {5414,6564}, {5424,5560}, {5448,6238}, {5449,7352}, {5450,6952}, {5537,6982}, {5552,6871}, {5883,6702}, {5904,6734}, {5919,7743}, {6256,6833}, {6668,7483}, {7302,7519} X(7951) = reflection of X(5010) in X(5432)X(7952) was contributed by Peter Moses, July 6, 2015, who notes that the center of the inverse-in-incirce of the polar circle is X(946).
Let LA be the polar of X(4) wrt the circle centered at A and passing through X(1), and define LB and LC cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = LB∩LC cyclically, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC, and the center of homothety is X(7952). The orthocenter of A'B'C' is X(944). (Randy Hutson, July 23, 2015)
X(7952) lies on the cubic pK(X393,X2) (the polar conjugate of the Lucas cubic) and these lines: {1,4}, {2,280}, {3,108}, {7,412}, {8,1897}, {9,1249}, {10,459}, {12,1857}, {29,5703}, {37,158}, {40,196}, {45,1990}, {55,1118}, {57,1753}, {81,3559}, {92,4194}, {145,5081}, {201,1148}, {220,1783}, {235,1863}, {240,3144}, {273,3672}, {329,3194}, {342,347}, {354,1887}, {429,7102}, {443,1074}, {451,498}, {475,3086}, {603,2096}, {774,5230}, {860,1834}, {938,5125}, {939,4183}, {942,1872}, {1038,6916}, {1040,6865}, {1060,6850}, {1062,6827}, {1100,3087}, {1103,2324}, {1119,3663}, {1158,1720}, {1210,1861}, {1214,3346}, {1426,1902}, {1430,1496}, {1433,6223}, {1452,6197}, {1465,6848}, {1593,7103}, {1714,1736}, {1735,1788}, {1771,3474}, {1845,5697}, {1846,2098}, {1851,4186}, {1867,4207}, {1875,3057}, {1876,4310}, {1886,7079}, {1892,4307}, {1900,4196}, {1940,5218}, {2268,7120}, {2322,5296}, {3100,6836}, {3191,3192}, {3535,5393}, {3536,5405}, {3562,5905}, {3575,4339}, {3945,7282}, {4292,7365}, {4295,6354}, {4296,6925}, {5266,7487}, {5719,7524}, {5932,6355}
X(7952) = isogonal conjugate of X(1433)X(7953) lies on the circumcircle and these lines: {98,140}, {111,1194}, {827,1634}, {842,5899}, {4577,6573}
X(7953) = isogonal conjugate of X(7927)Barycentrics a2/[(b2 - c2)(a2 + 2b2 + 2c2)] : :
X(7954) lies on the circumcircle and this line: {98, 1656}
X(7954) = isogonal conjugate of X(7950)
Centers associated with mixtilinear triangles: X(7955)-X(7999)
César Lozada (July 6, 2015) introduces the 2nd-to-6th mixtilinear triangles and contributes associated triangle centers. Notation:
{a'} = A-mixtilinear-incircle, with center I'a; {a''} = A-mixtilinear-excircle, with center I''a
A'c = {a'}∩ AB; A'b = {a'}∩ AC; A"c = {a''}∩ AB; A"b = {a''}∩ AC
A'b+ = the closer to B of the 2 points in {a'}∩BC; A'c+ = the other point in {a'}∩BC.
The following objects are defined cyclically: {b'}, {c'}; {b''}, {c''}; B'a, C'b; B''a, B''c; C'b, A'c; C''b, C''a; B'c+, C'a+; B'a+, C'b+ .
Theorems:
1) Points A'b, A'c, B'c, B'a, C'a, C'b lie on an conic with center X(1).
2) Points A"b, A"c, B"c, B"a, C"a,C"b lie on a conic with center X(7955).
3) Points A'b+, A'c+, B'c+, B'a+, C'a+, C'b+ lie on a conic with center X(221).
4) Lines (A'b-A'c), (B'c-B'a), (C'a-C'b) concur in X(1).
Definitions of triangles (all coordinates are trilinears):
1st mixtilinear Triangle (the classical mixtilinear triangle, also called the mixtilinear incentral triangle): I'aI'bI'c, where I'a = 1 + cos A - cos B - cos C : 2 : 2. I'aI'bI'c is perspective at X(1) to these triangles: ABC, 2nd circumperp, excentral, incentral, midarc.
In the following list, the appearance of (T,i,j) means that I'aI'bI'c is orthologic to triangle T and the orthologic centers are X(i) and X(j): (1st circumperp,1,6244), (2nd circumperp,1,999), (excentral,1,57), (hexyl,1,6282), (intouch,1,57), (midheight,221,57), (3rd Euler,1,7956), (4th Euler, 1,3820), (2nd extouch,1,329)
2nd mixtilinear triangle (also called the mixtilinear excentral triangle): I''aI"bI"c, where I''a = 1 + cos A + cos B + cos C : -2 : -2.
Next, (T,i) means that I''aI"bI"c is perspective to triangle T and the perspector is X(i): (ABC,1), (2nd circumperp,1), (excentral,1), (extangents, 7957}, (Feuerbach , 7958), (incentral, 1), (midarc,1), (1st mixtilinear,1)
Next, (T,i,j) means that I''aI"bI"c is orthologic to T and the orthologic centers are X(i) and X(j) and (T,i,j)* means that I''aI"bI"c and T are homothetic: (1st circumperp,1,6244), (2nd circumperp,1,999), (excentral,1,57), (hexyl,1,6282), (intouch,1,57), (midheight,7969,57), (1st mixtilinear,7970,7971)*, (3rd Euler,1,7972), (4th Euler,1,3820), (2nd extouch,1,329)
3rd mixtilinear triangle: A'tB'tC't, where A't = {a'}∩{circumcircle} = 1/2 : -b/(a - b + c) : -c/(a + b - c)]; also, A'tB'tC't is the circumcevian triangle of X(56). Next, (T,i) means that A'tB'tC't is perspective to T and the perspector is X(i): (ABC,56), (1st circumperp,1), (excentral, 7963}, (1st mixtilinear, 3), (tangential,1616).
4th mixtilinear triangle: A"tB"tC"t, where A''t = {a''}∩{circumcircle} = 1/2 : -b/(a + b - c) : -c/(a - b +c); also, A''tB''tC''t is the circumcevian triangle of X(55). Next, (T,i) means that A"tB"tC"t is perspective to T and the perspector is X(i), and (T,i)* means that A"tB"tC"t is homothetic to T:
(ABC, 55), (1st circumperp,165), (excentral,165), (extangent,7964), (Feuerbach,7965), (incentral,57), (2nd mixtilinear,3), (tangential, 1615)
5th mixtilinear triangle: A5B5C5 (called the Caelum triangle at X(5603), where it is defined as the reflection of ABC in X(1)), where A5= (C'a-A'c)∩(A'b-B'a) = a - b - c : 2a : 2a, and B5 and C5 are defined cyclically. Next, (T,i) means that A5B5C5 is perspective to T and the perspector is X(i), and (T,i)* means that A5B5C5 is homothetic to T:
(ABC,1)*, (anticomplementary,145)*, (2nd circumperp,1), (Euler 5603)*, (excentral,1), (inner Grebe,5605)*, (outer Grebe,5604)*, (hexyl,7966), (incentral, 1), (intangents, 3057), (intouch,145), (Johnson,952)*, (medial,8)*, (midarc,1), (1st mixtilinear,1), (2nd mixtilinear,1), (3rd mixtilinear,100)
6th mixtilinear triangle: A6B6C6, where A6 = (B''c-B''a)∩(C''a-C''b) = a2 - 2a(b + c) + (b - c)2 : a2 + (b - c)(3c + b - 2a) : a2 - (b - c)(3b + c - 2a), and B6 and C6 are defined cyclically.
A6B6C6 is the anticomplementary triangle of the excentral triangle of ABC. Also, A6 is the midpoint of the A-vertex of the inner Hutson triangle and the A-vertex of the outer Hutson triangle, and cyclically for B6 and C6.
7th mixtilinear triangle: (See X(8916)
X(7955) lies on these lines: (220,2124), (223,3160)
X(7955) = center of the perspeconic of these triangles: 6th and 7th mixtilinear
X(7956) = midpoint of X(i) and X(j) for these (i,j): (4,999), (946,7682)
X(7956) = reflection of X(3820) in X(5)
X(7956) = complement of X(6244)
X(7956) = X(999)-of-Euler-triangle
X(7957) lies on these lines: (1,3), (4,210), (19,220), (20,518), (34,7074), (64,3827), (71,1212), (72,516), (185,674), (218,1766), (392,4301), (758,5493), (910,1802), (946,3925), (960,962), (971,5904), (1253,4332), (1456,7078), (1698,5806), (1699,5044), (1709,3927), (1770,5762), (1836,5758), (1864,6284), (2340,3198), (2800,6154), (3091,3740), (3146,3681), (3522,3873), (3523,3742), (3555,4297), (3678,5927), (3698,5657), (3811,7580), (3880,6764), (3922,6878), (3962,6001), (3983,5587), (4005,5777), (4292,5920), (4294,5759), (4313,7672), (4314,5728), (4661,5059), (5178,6895), (5302,6912), (5836,6837)
X(7957) = reflection of X(i) in X(j) for these (i,j): (65,40), (962,960), (3555,4297)
X(7958) lies on these lines: (1,5), (2,5584), (4,4423), (55,6864), (56,6846), (72,6067), (946,3925), (4313,7678), (4679,5715)
X(7959) lies on these lines: (1,84), (64,71), (774,1407), (1490,7074), (2256,4300), (2883,5228)
X(7959) = reflection of X(i) in X(j) for these (i,j): (64,3556), (221,1498)
X(7960) lies on these lines: (1,527), (57,169), (218,279), (329,5308), (1212,4675), (1323,1419), (2093,2391), (2094,5222), (4293,5845)
X(7960) = X(4)-of-2nd-mixtilinear-triangle
X(7961) lies on these lines: (1,527), (57,1766), (984,3421), (999,4310), (1210,4659), (3452,3731), (4859,6692)
X(7961) = X(4)-of-1st-mixtilinear-triangle
X(7962) is also the orthologic center of the Hutson intouch (defined at X(5731) and 2nd mixtilinear triangles. In both cases, the other orthologic center is X(1).
X(7962) lies on these lines: (1,3), (2,4345), (8,3452), (9,644), (11,3679), (78,2136), (145,329), (200,3880), (210,4915), (388,4301), (390,527), (392,7308), (497,519), (516,3476), (551,5218), (728,3061), (952,3586), (956,3929), (960,4853), (997,2802), (1000,1512), (1056,4654), (1149,5573), (1358,4862), (1389,7160), (1419,2823), (1479,5881), (1699,5252), (1837,3632), (2096,4304), (2269,3247), (2809,4845), (2810,3022), (2835,4319), (3158,3895), (3244,3486), (3474,4315), (3522,6049), (3616,6692), (3621,3984), (3623,4313), (3624,5326), (3635,4314), (3711,4900), (3869,6762), (3890,5284), (3893,4882), (4294,5882), (4311,6361), (4312,5434), (4323,5665), (4861,5250), (5559,7741), (5587,6973), (5722,5844), (5730,6765), (5790,7743)
X(7962) = midpoint of X(145) and X(329)
X(7962) = reflection of X(i) in X(j) for these (i,j): (8,3452), (57,1), (200,5289), (497,4342), (2093,999), (3474/,315), (5727,497)
X(7963) lies on these lines: (1,474), (36,2956), (56,1743), (106,6765), (165,2943), (1046,3361), (3616,4859)
X(7963) = isogonal conjugate of X(39123)
X(7964) lies on these lines: (1,3), (63,3059), (71,910), (210,7580), (212,1456), (516,3683), (518,7411), (573,2348), (584,2266), (672,2264), (1253,1427), (1490,4005), (1615,3197), (1742,4641), (1750,3715), (1762,2938), (1836,5759), (2550,4640), (2951,3929), (3146,5302), (3174,3928), (3189,3522), (6361,6832)
X(7964) = midpoint of X(40) and X7688)
X(7965) lies on these lines: (4,12), (5,165), (11,57), (63,6067), (92,4081), (119,3845), (142,5918), (235,5338), (354,946), (381,6244), (515,3748), (516,3683), (546,5537), (910,1839), (1155,6831), (1329,3832), (1617,7354), (1858,5173), (2310,6354), (2717,5520), (2886,5273), (3062,4654), (3652,5536), (3715,5817), (3962,4301), (5435,7678)
X(7966) lies on these lines: (1,227), (3,2136), (9,952), (40,145), (84,944), (100,3576), (390,515), (517,3243), (1385,1706), (1483,5709), (1768,5119), (3359,3655), (3427,7160), (3587,5844)
X(7966) = midpoint of X(944) and X(1000)
X(7966) = reflection of X(3577) in X(1)
X(7966) = perspector of hexyl triangle and antipedal triangle of X(9)
X(7967) lies on these lines: (1,4), (2,952), (3,145), (5,3622), (8,631), (10,3525), (20,1482), (40,3244), (51,957), (55,104), (56,6942), (140,3617), (149,6923), (153,6929), (355,3090), (376,517), (390,6938), (495,6830), (496,6941), (519,3158), (551,5071), (912,3877), (938,6049), (942,4308), (956,1006), (962,3529), (999,6905), (1012,6767), (1125,5067), (1319,6880), (1320,6948), (1387,5274), (1389,3296), (1484,6980), (2096,4304), (2098,4294), (2099,4293), (2550,6264), (2646,6977), (2801,3898), (2829,3058), (2975,6875), (3057,4305), (3091,5901), (3149,7373), (3242,6776), (3243,5759), (3295,6906), (3340,4311), (3359,3895), (3421,4511), (3434,6224), (3436,6902), (3523,3621), (3544,3636), (3600,6934), (3632,6684), (3635,4297), (3746,5450), (3884,5693), (3890,5887), (3897,6857), (4421,5854), (4861,5082), (5126,5435), (5252,6879), (5330,6872), (5434,5842), (5563,6796), (5687,6940), (5697,5884), (5703,6956), (5719,6844), (5722,6969), (5730,6936), (5768,6935), (5919,6001), (6967,7080)
X(7967) = midpoint of X(3241) and X(5731)
X(7967) = reflection of x(i) in X(j) for these (i,j): (376,5731), (5587,551), (5657,3576), (5731,3655)
X(7967) = anticomplement of X(5790)
X(7967) = {X(1),X(4)}-harmonic conjugate of X(10595)
X(7968) lies on these lines: (1,6), (8,3069), (10,615), (40,1152), (65,5416), (145,7586), (175,4000), (176,4644), (355,486), (371,1385), (372,517), (481,1086), (515,3071), (590,1125), (605,1468), (606,3915), (940,3083), (944,1588), (946,3070), (952,7584), (997,1378), (1151,1702), (1319,2067), (1482,3312), (1703,3594), (1829,5413), (2066,2646), (2099,2362), (3057,5414), (3068,3616), (3084,4383), (3622,7585), (5901,7583)
X(7968) = reflection of X(5605) in X(1)
X(7968) = {X(1),X(6)}-harmonic conjugate of X(7969)
X(7969) lies on these lines: (1,6), (8,3068), (10,590), (40,1151), (56,2362), (65,2067), (145,7585), (175,4644), (176,4000), (355,485), (371,517), (372,1385), (482,1086), (515,3070), (605,3915), (606,1468), (615,1125), (940,3084), (944,1587), (946,3071), (952,7583), (997,1377), (1152,1703), (1482,3311), (1702,3592), (1829,5412), (2066,3057), (2646,5414), (3069,3616), (3083,4383), (3579,6200), (3622,7586), (3779,6283), (5901,7584)
X(7969) = reflection of X(5604) in X(1)
X(7969) = {X(1),X(6)}-harmonic conjugate of X(7968)
X(7970) lies on these lines: (1,98), (8,114), (99,517), (145,147), (511,3903), (519,6054), (542,3241), (620,5657), (944,2794), (952,6033), (1320,2783), (1482,2782), (2098,3027), (2099,3023), (2784,3244), (3616,6036), (3623,5984)
X(7970) = midpoint of X(145) and X(147)
X(7970) = reflection of X(i) in X(j) for these (i,j): (8,114), (98,1)
Let A' = reflection of A in X(1), and define B' and C' cyclically. Let A" be the orthogonal projection of A' on BC. Let Ta be the tangent at A' to the circle (IA'A"), and define Tb and Tc cyclically. The lines Ta, Tb, Tc concur in X(7971). (Angel Montedeoca, April 20, 2020)
X(7971) lies on these lines: (1,84), (4,3340), (9,5887), (40,78), (145,515), (390,5882), (517,1490), (912,6762), (938,946), (952,3680), (971,1482), (1158,3576), (1159,5806), (1361,7355), (1621,5450), (1706,5720), (2093,3149), (2096,4311), (3333,5884), (3359,5438), (3434,5881), (3616,6705), (4848,6848), (5128,6905), (5554,5587), (5604,6257), (5605,6258), (5665,5715), (5730,6282), (5795,5811), (5837,6908)
X7971) = reflection of X(i) in X(j) for these (i,j): (84,1), (2136,5534), (5881,6256)
X(7972) lies on these lines: (1,5), (8,214), (35,104), (36,100), (46,2136), (79,1320), (101,4530), (145,2802), (149,1478), (153,1479), (354,6797), (390,2801), (484,5844), (517,4316), (528,3243), (664,4089), (758,6758), (944,2800), (1145,3632), (1537,5691), (1698,3036), (1768,5119), (2093,6154), (2771,3057), (2932,3913), (3035,3679), (3476,5083), (3583,5048), (3616,6702), (3655,5010), (3790,5150), (4677,6174), (4738,6790), (5425,5542), (5604,6262), (5605,6263)
X(7972) = midpoint of X(i) and X(j) for these (i,j): (145,6224), (3633,5541)
X(7972) = reflection of X(i) in X(j) for these (i,j): (1,1317), (8,214), (80,1), (104,5882), (1320,3244), (3583,5048), (3632,1145),
(4677,6174), (5691,1537), (5881,119)
X(7973) lies on these lines: (1,64), (6,1902), (8,2883), (20,664), (40,154), (65,2192), (221,3057), (515,5895), (517,1498), (946,1853), (952,5878), (962,1503), (1482,6000), (1854,2099), (2098,7355), (3555,6001), (3616,6696), (5731,5894)
X(7973) = reflection of X(i) in X(j) for these (i,j): (8,2883), (64,1)
X(7974) lies on these lines: (1,14), (8,619), (145,617), (517,5474), (519,5464), (531,3241), (542,3242), (952,5613), (3616,6670)
X(7975) lies on these lines: (1,13), (8,618), (145,616), (517,5473), (519,5463), (530,3241), (542,3242), (952,5617), (3616,6669)
X(7975) = reflection of X(13) in X(1)
X(7976) lies on these lines: (1,76), (8,39), (145,194), (262,355), (511,944), (538,3241), (726,3244), (732,3242), (952,3095), (1482,2782), (1742,3875), (1964,3596), (3094,5846), (3097,3632), (3616,3934), (5145,5263), (5188,5731), (5604,6272), (5605,6273)
X(7976) = midpoint of X(145) and X(194)
X(7976) = reflection of X(i) in X(j) for these (i,j): (8,39), (76,1)
X(7976) = X(76)-of-5th-mixtilinear-triangle
X(7977) lies on these lines: (1,83), (8,6292), (145,2896), (320,3244), (732,3242), (754,3241), (3616,6704), (5604,6274), (5605,6275)
X7977) = midpoint of X(145) and X(2897)
X(7977) = reflection of X(i) in X(j) for these (i,j): (8,6292), (83,1)
X(7977) = X(83)-of-5th-mixtilinear-triangle
X(7978) lies on these lines: (1,74), (8,113), (30,6742), (110,517), (145,146), (541,3241), (944,2777), (952,7728), (1320,2771), (1386,5622), (1482,5663), (2098,3028), (2102,2575), (2103,2574), (2778,3057), (2779,7727), (2781,3242), (2807,7722), (3616,6699), (5604,7726), (5605,7725), (5657,5972)
X7978) = midpoint of X(145) and X(146)
X(7978) = reflection of X(i) in X(j) for these (i,j): (8,113), (74,1)
X(7979) lies on these lines:
(1,54), (8,1209), (145,2888), (517,7691), (539,3241), (1154,1482), (3555,5887), (3616,6689)
X7979) = midpoint of X(145) and X(2888)
X(7979) = reflection of X(i) in X(j) for these (i,j): (8,1209), (54,1)
X(7980) lies on these lines: (1,486), (8,642), (145,487), (1483,3242), (3616,6119)
X(7980) = reflection of X(486) in X(1)
X(7980) = {X(1483),X(3242)}-harmonic conjugate of X(7981)
X(7981) lies on these lines: (1,485), (8,641), (145,488), (1483,3242), (3616,6118)
X(7981) = reflection of X(485) in X(1)
X(7981) = {X(1483),X(3242)}-harmonic conjugate of X(7980)
X(7982) = {P,Q}-harmonic conjugate of X(40), where P = mixtilinear-incentral-to-mixtilinear-excentral similarity image of X(40) (i.e., X(6766)), and Q = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X(40) X(7982). Let A'B'C' be the excentral triangle and A"B"C" the hexyl triangle. Let OA be the circle centered at A' and passing through A", and define OB and OC cyclically. X(7982) is the radical center of the circles OA, OB, OC. (Randy Hutson, July 23, 2015)
X(7982) lies on these lines: (1,3), (2,5734), (4,519), (8,908), (9,1389), (10,3090), (20,3241), (33,1866), (34,1830), (63,4861), (72,4853), (78,6915), (84,1320), (145,515), (200,5730), (355,546), (376,5493), (381,4677), (392,3646), (516,944), (518,5693), (528,5735), (550,3655), (551,631), (573,3247), (576,3751), (580,3915), (632,3624), (758,6762), (936,5289), (950,5758), (952,3627), (997,1706), (1056,3671), (1058,4342), (1066,7273), (1125,3525), (1449,1766), (1479,5727), (1537,5854), (1572,5007), (1698,3628), (1702,3592), (1703,3594), (1709,3901), (1737,6978), (1753,1870), (1829,5198), (2136,2802), (2262,2324), (2814,4895), (2948,5609), (3149,3913), (3419,5715), (3452,5804), (3474,4311), (3476,4292), (3530,3653), (3544,3626), (3545,4669), (3555,6001), (3560,3929), (3586,5812), (3616,6684), (3623,5731), (3635,4297), (3711,5780), (3754,5437), (3813,6831), (3828,5067), (3857,4816), (3870,3885), (3871,6796), (3873,5884), (3877,5047), (3880,6765), (3928,6906), (3940,4882), (4034,5816), (4221,4658), (4309,6868), (4317,6948), (4654,6850), (4668,5072), (4745,5071), (4857,6928), (4867,5720), (5082,6737), (5219,5761), (5270,6923), (5400,5754), (5559,6842), (5722,5763), (5887,5904), (6173,6897), (6210,7174), (6211,7290), (6734,6860), (6736,7682), (7673,7675)
X(7982) = midpoint of X(i) and X(j) for these (i,j): (145,962), (3633,5691)
X(7982) = anticomplement of X(11362)
X(7982) = X(20)-of-hexyl-triangle
X(7982) = reflection of X(i) in X(j) for these (i,j): (1,1482), (4,4301), (8,946), (20,5882), (40,1), (944,3244),
(2077,5048), (2136,3811), (3632,355), (4297,3635), (4677,381), (5881,4), (5904,5887), (6264,1320), (6361,4297)
X(7983) lies on these lines: (1,99), (8,115), (98,517), (145,148), (519,671), (543,3241), (620,3616), (645,2643), (952,6321), (962,2794), (1320,2787), (1386,5182), (1482,2782), (1916,3903), (2098,3023), (2099,3027), (2784,4301), (3242,5969), (5604,6320), (5605,6319), (5657,6036)
X(7984) lies on these lines: (1,60), (8,125), (67,5846), (74,517), (145,3448), (265,952), (518,895), (523,6740), (542,3241), (962,2777), (1482,5663), (2099,3028), (2102,2574), (2103,2575), (2836,5919), (2854,3242), (3616,5972), (3708,5546), (5604,7733), (5605,7732), (5657,6699)
X(7984) = midpoint of X(145) and X(3448)
X(7984) = reflection of X(i) in X(j) for these (i,j): (8,125), (110,1)
X(7985) lies on this line: (726,3241)
X(7986) lies on these lines: (1,1406), (3,2292), (6,2771), (381,3120), (405,5492), (517,990), (912,3751), (942,1854), (986,6985), (999,7004), (1054,6911), (1386,6001), (3649,4846), (4550,5221), (5707,5884), (6826,7613)
X(7986) = reflection of X(1480) in X(1)
Let P be a point in the plane of ABC. Let OA be the circumcircle of BCP, and define OB and OC cyclically. Let A' be the intersection, other than P, of OA and AP, and define B' and C' cyclically. Let LA be the tangent to OA at A', and define LB and LC cyclically. Let A" = LB∩LC and define B" and C" cyclically. The lines AA", BB", CC" concur for all P. If P = X(1), the lines AA", BB" CC" concur in X(7987). (Randy Hutson, July 23, 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 2nd circumperp triangle at X(7987). (Randy Hutson, July 31 2018)
X(7987) lies on these lines: (1,3), (2,4297), (4,3624), (9,3207), (10,3523), (20,1125), (21,3062), (41,572), (63,5303), (78,5223), (100,4853), (104,4866), (140,5587), (200,2975), (214,1768), (355,549), (376,946), (405,1750), (515,631), (516,3522), (551,962), (573,1475), (581,5313), (936,993), (944,3524), (950,7288), (952,4668), (956,4882), (958,5438), (991,1193), (995,4300), (997,5267), (1001,2951), (1006,1490), (1012,5259), (1055,3731), (1210,4305), (1478,6865), (1479,6916), (1572,5206), (1702,6200), (2136,4421), (2801,3876), (3146,3817), (3430,5429), (3485,4312), (3486,3911), (3515,7713), (3530,5881), (3583,6850), (3585,6827), (3622,4301), (3632,5657), (3636,5493), (3651,5426), (3652,5428), (3653,5901), (3655,4677), (3751,5085), (3822,6943), (3825,6932), (4189,4512), (4220,5272), (4293,5290), (4298,5703), (4299,6987), (4308,5281), (4313,5265), (4316,6868), (4324,6948), (4326,7677), (4511,4652), (4915,5687), (5144,5527), (5219,7354), (5248,6909), (5253,7411), (5435,6738), (5436,7580), (5744,6737), (6256,6947), (6796,6940), (6907,7741)
X(7987) = midpoint of X(3522) and X(3616)
X(7987) = reflection of X(1698) in X(631)
X(7987) = {X(1),X(3)}-harmonic conjugate of X(165)
X(7987) = homothetic center of excentral triangle and medial triangle of 2nd circumperp triangle
X(7987) = X(4)-of-cross-triangle of these triangles: Aquila and anti-Aquila
X(7987) = X(3091)-of-excentral-triangle
X(7988) lies on these lines: (1,5), (2,165), (4,3624), (9,5087), (10,5056), (35,6918), (36,6913), (40,1656), (57,7082), (142,3062), (226,5817), (517,4731), (908,5223), (946,1698), (1125,5691), (1482,3711), (1709,3838), (1750,3816), (1768,3306), (3337,7330), (3626,5734), (3632,5818), (3671,5704), (3742,5927), (3911,4312), (4297,5550), (4326,7678), (4355,5714), (4413,5537), (4654,5843), (4677,5790), (4679,5805), (4847,5748), (4859,5121), (4860,5779), (5851,6173), (5880,6667), (8140,8378)
X(7988) = {X(1),X(5)}-harmonic conjugate of X(7989)
X(7988) = homothetic center of Euler triangle and cross-triangle of these triangles: Aquila and anti-Aquila
X(7989) lies on these lines: (1,5), (2,4297), (4,165), (8,3817), (10,962), (19,7559), (20,3634), (35,6913), (36,6918), (40,381), (57,5789), (200,5178), (210,5806), (442,1750), (515,3090), (516,3832), (517,3851), (936,3814), (938,3947), (944,5071), (946,3545), (993,6915), (1125,5056), (1210,5290), (1385,5055), (1478,3361), (1479,6939), (1482,4677), (1490,6829), (1656,3576), (1706,5123), (1737,3339), (1743,5816), (1768,6702), (1788,4312), (2093,6867), (2951,3826), (3062,5177), (3149,5251), (3336,7330), (3436,5231), (3544,3632), (3560,5010), (3579,3843), (3583,6893), (3585,6826), (3617,4301), (3625,5734), (3812,5927), (3822,6991), (3828,3839), (3841,6932), (3855,5657), (3911,5229), (4298,5704), (4326,7679), (4668,5790), (4866,7682), (5175,6745), (5221,5779), (5223,6734), (5226,6738), (5234,5705), (5692,7686), (5748,6737), (5777,5902), (8140,8380)
X(7989) = reflection of X(3624) in X(3090)
X(7989) = {X(1),X(5)}-harmonic conjugate of X(7988)
X(7989) = homothetic center of Euler triangle and cross-triangle of ABC and Aquila triangle
X(7990) lies on these lines: (1,5806), (100,4853), (145,4297), (165,2136), (390,3062), (952,4866), (2951,3243), (3339,5882)
X(7991) is the orthocenter of the 6th. mixtilinear triangle and is also the orthologic center of this triangle and following: 2nd CIRCUMPERP, EXCENTRAL, HEXYL, INTOUCH, 3rd EULER, 4th EULER, 2nd EXTOUCH
Let T be the triangular hull of mixtilinear excircles (i.e., the mixtilinear excircles extangents triangle). Then T is perspective to the mixtilinear excentral triangle, and the perspector is X(7991). Let A"B"C" be as defined at X(6766); then X(7991) = X(1)-of A"B"C". (Randy Hutson, July 23, 2015)
X(7991) lies on the cubic K077 and these lines: (1,3), (2,4301), (4,3679), (8,144), (9,5836), (10,962), (20,519), (30,4677), (63,4853), (72,1750), (84,4900), (145,4297), (200,3869), (329,6736), (355,3627), (376,5882), (377,5735), (388,4312), (390,6738), (497,4848), (515,3529), (518,2136), (546,5587), (548,3655), (551,3523), (573,1334), (758,6765), (936,3878), (944,3633), (946,1698), (950,5759), (954,5665), (960,1706), (970,3030), (1012,5258), (1046,2941), (1056,4355), (1064,5312), (1158,6763), (1445,7673), (1490,2800), (1616,5573), (1709,4915), (1743,1766), (1768,2802), (1902,5198), (2550,5837), (2817,2956), (2999,4642), (3241,3522), (3244,5731), (3306,3890), (3525,3624), (3543,4669), (3577,5251), (3680,3928), (3698,5806), (3839,4745), (3868,3895), (3880,6762), (3894,5884), (3913,7580), (3922,4423), (4295,5290), (4309,6987), (4323,5281), (4325,6948), (4326,7672), (4330,6868), (4338,5270), (4345,5265), (4652,4861), (4662,5927), (4857,6827), (4862,7195), (4863,5787), (5047,5250), (5219,5763), (5234,6912), (5252,5762), (5289,5438), (5698,5795), (5705,6860), (5715,6984), (5726,5758), (5727,6284), (5904,6001)
X(7991) reflection of X(i) in X(j) for these (i,j): (1,40), (20,5493), (145,4297), (962,10), (1482,3579), (3062,5223), (3543,4669), (3633,944), (5531,5541), (5536,3245), (5691,8)
X(7991) = anticomplement of X(4301)
X(7991) = X(20)-of-excentral-triangle
X(7991) = {X(1),X(40)}-harmonic conjugate of X(165)
X(7992) lies on these lines: (1,84), (4,3062), (20,6737), (40,971), (46,1750), (165,191), (269,774), (515,3529), (912,6769), (920,1768), (1046,1721), (1699,5586), (1770,2093), (2800,3901), (2801,6765), (2950,5531), (3624,6705), (3929,5584), (4866,5657), (5234,7330), (5588,6257), (5589,6258), (5658,6684), (5693,6282), (5887,7171)
X(7992) = reflection of X(i) in X(j) for these (i,j): (1,84), (1490,1158), (5531,2950)
X(7992) = excentral isogonal conjugate of X(3182)
X(7993) lies on these lines: (1,5), (100,4853), (104,165), (149,5691), (153,1699), (519,5538), (528,2951), (1145,4915), (1320,2801), (1768,2802), (2800,3901), (3333,6797)
X(7993) = reflection of X(i) in X(j) for these (i,j): (1,6264), (5531,1), (5541,104), (5691,149)
X(7993) = Gibert-Burek-Moses concurrent circles image of X(1317)
X(7994) = X(25)-of-6th-mixtilinear-triangle. Also X(7994) = orthologic center of 6th mixtilinear triangle and 2nd. mixtilinear triangle
X(7994) lies on these lines: (1,3), (20,6765), (200,329), (380,2266), (527,2951), (910,2324), (936,962), (1282,2823), (1490,6361), (1615,6603), (1698,7682), (1699,2550), (1709,5223), (1721,3961), (3059,3062), (3158,7580), (3421,4882), (3811,5493), (3870,5732), (5531,6154)
X(7994) = reflection of X(i) in X(j) for these (i,j): (1,6282), (57,6244), (1750,200), (2093,40), (2095,3579), (5691,3421)
X(7994) = intangents-to-extangents similarity image of X(57)
X(7995) lies on these lines: (1,84), (4,2093), (8,144), (40,210), (165,411), (607,1743), (971,1697), (1210,1699), (1467,3358), (1706,5927), (1768,3361), (2951,5784), (3579,5780), (4189,4512), (5250,5732), (5693,6769), (5887,6282)
X(7996) lies on these lines: (1,7175), (8,144), (165,846), (2961,5536)
X(7996) = reflection of X(1721) in X(1766)
X(7997) lies on these lines: (165,7701), (355,3627), (1006,1490), (3339,7702), (5538,5777), (5557,7741)
X(7998) lies on these lines: (2,51), (3,74), (4,5447), (6,5888), (23,3098), (52,3525), (54,7516), (69,3266), (76,4576), (97,426), (140,568), (141,858), (154,6030), (182,323), (352,574), (376,5891), (394,5012), (549,5890), (599,2854), (631,1216), (748,7186), (750,3792), (805,2770), (1078,4563), (1154,5054), (1180,1613), (1350,1995), (1401,7226), (1469,5297), (1495,7492), (1993,5050), (2842,5692), (3056,7292), (3094,3231), (3111,5468), (3218,3781), (3219,3784), (3292,5092), (3313,3619), (3314,6786), (3522,5907), (3523,5562), (3526,3567), (3533,5462), (3688,4392), (3818,5189), (3909,5233), (4550,7464), (5067,5446), (5093,5422), (5094,6403), (5107,7708), (5643,5646)
X(7998) = midpoint of X(i) and X(j) for these (i,j): (2979,5640), (3917,5650)
X(7998) = reflection of X(i) in X(j) for these (i,j): (2,5650), (3060,5640), (5640,2), (5650,3819)
X(7998) = anticomplement of X(373)
X(7998) = X(2)-of-X(2)-anti-altimedial triangle
X(7998) = centroid of the nine vertices of the anti-altimedial triangles
X(7999) lies on these lines: (2,52), (3,74), (4,3917), (5,2979), (20,5447), (51,5067), (54,394), (140,5889), (141,1594), (143,5070), (155,7485), (185,3524), (323,569), (376,5907), (389,3525), (458,4994), (511,3090), (568,632), (578,7550), (631,3819), (1154,3526), (1173,1351), (1656,3060), (1993,7393), (3518,5651), (3528,6000), (3628,5640), (5012,7516), (5054,6102), (5056,5446), (5752,6946), (5972,7731), (6045,6829), (6644,7691), (7484,7592)
X(8000) lies on these lines: {1,5920}, {56,12333}, {145,9874}, {517,12120}, {5597,12465}, {5598,12464}, {7967,12249}, {8192,12411}, {9804,11036}, {10800,12200}, {11396,12139}
X(8000) = reflection of X(7160) in X(1)
Trilinears p^6-2*p^4+(3*p^2-2)*p*q^3+(p^4-10*p^2+12)*p*q-(5*p^4-14*p^2+4)*q^2-4 : : where p = sin(A/2) and q = cos(B/2 - C/2)
X(8001) lies on these lines: {1, 5920}, {3633, 7990}
Centers associated with the Stammler triangle: X(8002) -X(8011)
César Lozada (July 22, 2015) introduces centers associated with the Stammler triangle. A'B'C', an equilateral triangle whose vertices are the centers of the Stammler circles; see MathWorld: Stammler Circles. The A-vertex of Stammler triangle is given by trilinears
A' = cos A - 2 cos(B/3 - C/3) : cos B + 2 cos(B/3 + 2C/3) : cos C + 2 cos(2*B/3 + C/3)
A'B'C' is homothetic to these triangles:
• circumnormal triangle, at X(3)
• 1st Morley triangle, at X(8002)
• 2nd Morley triangle, at X(8003)
• 3rd Morley triangle, at X(8004)
A'B'C' is orthologic to these triangles:
• 1st Morley-Adjunct triangle, at X(356) and X(8005)
• 2nd Morley-Adjunct triangle, at X(3276) and X(8006)
• 3rd Morley-Adjunct triangle, at X(3277) and X(8007)
• cnner-Napoleon triangle, at X(8009) and X(8011)
Also, A'B'C' is parallelogic to the inner Napoleon triangle, at X(8008) and X(8010).
X(8002) lies on these lines: (3,356), (357,3280), (358,8003), (1135,8004), (3605,8005)
X(8003) lies on these lines: (3,3276), (358,8002), (1137,8004), (3606,8006)
X(8004) lies on these lines: (3,3277), (1135,8002), (1137,8003), (3607,8007)
X(8005) lies on these lines: (3,3276), (3605,8002)
X(8005) = reflection of X(3) in X(3281)
Trilinears 2(au + bv + cw) cos A + (cos A - 4u)(S/R) : : , where u : v : w = cos(A/3 + π/3) + cos(B/3 + π/3) cos(C/3 + π/3) : :
X(8006) lies on these lines: (3,3277), (3606,8003)
X(8006) = reflection of X(3) in X(3283)
Trilinears 2(au + bv + cw) cos A + (cos A - 4u)(S/R) : : , where u : v : w = cos(A/3 - π/3) + cos(B/3 - π/3) cos(C/3 - π/3) : :
X(8007) lies on these lines: (3,356), (3607,8004)
X(8007) = reflection of X(3) in X(3279)
X(8008) is the antipode of X(8009) in the Stammler circle.
X(8008) lies on the Stammler circle and this line: (3,8009)
X(8008) = reflection of X(8009) in X(3)
X(8009) is the antipode of X(8008) in the Stammler circle.
X(8009) lies on the Stammler circle and this line: (3,8008)
X(8009) = reflection of X(8008) in X(3)
X(8010) is the antipode of X(8011) in the inner Napoleon circle.
X(8010) lies on the inner Napoleon circle and these lines: (2,8011), (1135,5390)
X(8010) = reflection of X(8011) in X(2)
X(8010) = anticomplement of X(33493)
X(8011) is the antipode of X(8010) in the inner Napoleon circle.
Let A'B'C' be the 1st Morley triangle. Let Ma be the line through A parallel to B'C', and define Mb, Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. Then X(8011) is the center of (equilateral) triangle A"B"C". Also, A"B"C" is homothetic to A'B'C' at X(8065). (Randy Hutson, August 12, 2015)
Let OaObOc be the Stammler triangle. Let A* be the orthogonal projection of Oa on line BC, and define B* and C* cyclically. X(8011) = X(2)-of-A*B*C*. (Randy Hutson, August 12, 2015)
X(8011) lies on the inner Napoleon circle and these lines: {2,8010}, {356,8065}
X(8011) = reflection of X(8010) in X(2)
X(8011) = anticomplement of X(33492)
Danneels points: X(8012) -X(8042)
Danneels points are introduced at X(3078). The definition is restated here for a point U = u : v : w (barycentrics): D(U) = u2(v + w) : v2(w + u) : w2(u + v).
It is proved at X(3078) that if U is on the Euler line of a triangle ABC, then D(U) is also on the Euler line. Also,
• D(Steiner circumellipse) = X(2)
• D(P) is on the line P-to-X(2) for every point P
• D(line at infinity) = X(2)-of-T(X), where T(X) is the cevian triangle of X
• D(X) = X-Ceva conjugate of (X-crosspoint of X(2) (Peter Moses, July 29, 2015)
The appearance of (i, j) in the following list means that D(X(i)) = X(j):
(1,42), (2,2), (3,418), (4,25), (5,3078), (6,3051), (7,57), (8,200), (9,8012), (10,8013), (13,8014), (14,8015), (15,8016), (16,8017), (17,8018), (18,8019), (19,8020), (20,3079), (21,8021), (25,3080), (30,3081), (31,8022), (32,8023), (69,394), (75,321), (76,8024), (86,8025), (99,2), (100,55), (107,6525), (110,184), (145,3635), (189,1422), (190,2), (192,8026), (253,459), (264,324), (290,2), (330,6384), (366,367), (384,6657), (485,8035), (486,8036), (513, 8027), (514,6545), (519,8028), (523,8029), (524,8030), (536,8031), (648,2), (651,222), (653,196), (664,2), (666,2), (668,2), (670,2), (671,2), (752,8032), (799,8023), (876,8034), (886,2), (889,2), (892,2), (903,2), (934,6611), (1113,25), (1114,25), (1121,2), (1131,8037), (1132,8038), (1370,455), (1494,2), (1502,8039), (1897,7046), (1978,6382), (2479,2), (2480,2), (2481,2), (2966,2), (2994,6513), (2996,6340), (3225,2), (3226,2), (3227,2), (3228,2), (3625,8), (3699,6555), (3952,756), (4240,3081), (4373,6557), (4427,8040), (4552,6358), (4555,2), (4562,2), (4569,2), (4576,8041), (4577,2), (4586,2), (4597,2), (5641,2), (6189,2), (6190,2), (6516,6511), (6528,2), (6540,2), (6548,6545), (6606,2), (6613,2), (6635,2), (6648,2), (6655,6659), (7048,7028), (7192,8042)
X(8012) lies on these lines: {1, 1202}, {2, 7}, {41, 55}, {71, 910}, {100, 6605}, {165, 170}, {198, 1615}, {210, 3119}, {212, 5452}, {219, 2280}, {333, 6559}, {354, 1212}, {2266, 2911}, {2347, 2348}, {3691, 6554}, {3748, 6603}
X(8012) = X(i)-Ceva conjugate of X(j) for these (i,j): (9,1212), (100,4105), (1212,2293), (1223,1)
X(8012) = X(i)-isoconjugate of X(j) for these (i,j): (7,1170), (273,1803), (279,2346), (479,6605), (1088,1174), (3669,6606)
X(8012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55,220,6602), (1200,1334,55)
X(8013) lies on these lines: {1, 2}, {594, 756}, {740, 6536}, {1126, 1224}, {1211, 3120}, {1213, 1962}, {1230, 4647}, {1654, 4418}, {2308, 3686}, {3578, 4697}, {3775, 4359}, {3842, 3969}, {3952, 6539}, {4024, 8029}, {4204, 4433}, {4705, 8034}, {4732, 4972}
X(8013) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,1213), (1224,37), (3952,4024)
X(8013) = X(i)-isoconjugate of X(j) for these (i,j): (81,1171), (513,6578), (593,1255), (757,1126), (849,1268), (1019,4629), (3733,4596)
X(8013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (594,756,6535), (1213,1962,8040), (1213,4046,1962).
X(8014) lies on these lines: {2, 13}, {51, 512}, {2153, 7126}
X(8014) = isogonal conjugate of X(38403)
X(8014) = crosspoint of X(13) and X(11080)
X(8014) = crosssum of X(15) and X(11131)
X(8014) = crossdifference of every pair of points on line X(323)X(6137)
X(8014) = trilinear product X(396)*X(2153)
X(8014) = X(1094)-isoconjugate-of-X(11119)
X(8014) = X(13)-Ceva conjugate of X(396)
X(8015) lies on these lines: {2, 14}, {51, 512}
X(8015) = isogonal conjugate of X(38404)
X(8015) = crosspoint of X(14) and X(11085)
X(8015) = crosssum X(16) and X(11130)
X(8015) = crossdifference of every pair of points on line X(323)X(6138)
X(8015) = trilinear product X(395)*X(2154)
X(8015) = X(1095)-isoconjugate-of-X(11120)
X(8015) = X(14)-Ceva conjugate of X(395)
X(8016) lies on this line: {2, 14}
X(8017) lies on this line: {2, 13}
X(8018) lies on these lines: {2, 17}, {7603, 8019}
X(8019) lies on these lines: {2, 18}, {7603, 8018}
X(8020) lies on these lines: {2, 19}, {756, 862}, {1851, 2082}, {2201, 5276}, {2489, 8034}
X(8020) = X(332)-isoconjugate of X(7131)
As a point on the Euler line, X(8021) has Shinagawa coefficients ($a$S2+$aSBSC$+[$aSA$+abc]E, -$a$S2-$aSBSC$).
X(8021) lies on these lines: {2, 3}, {51, 5755}, {55, 219}, {57, 3286}, {81, 955}, {610, 3185}, {1260, 2287}, {2193, 2299}
X(8021) = X(100)-Ceva conjugate of X(1021)
X(8021) = X(i)-isoconjugate of X(j) for these (i,j): (226,2982), (943,3668), (1446,2259)
X(8022) lies on these lines: {2, 31}, {707, 825}, {2232, 8024}
X(8022) = X(i)-isoconjugate of X(j) for these (i,j): (86,7034), (310,7033), (321,7307), (6386,7255)
X(8023) lies on these lines: {2, 32}, {710, 8039}
The trilinear polar of X(8024) passes through X(826). (Randy Hutson, August 19, 2015)
X(8024) lies on these lines: {2, 39}, {6, 1239}, {22, 1975}, {66, 69}, {75, 4972}, {99, 1799}, {141, 6665}, {183, 1232}, {251, 384}, {264, 7378}, {308, 3108}, {311, 325}, {313, 1233}, {315, 7391}, {339, 1368}, {350, 7191}, {385, 1627}, {427, 1235}, {612, 3761}, {614, 3760}, {732, 3051}, {850, 2528}, {1031, 7779}, {1369, 5189}, {1501, 4048}, {1502, 3314}, {1909, 3920}, {1916, 4609}, {2232, 8022}, {2531, 5996}, {2782, 7467}, {3260, 7788}, {3770, 5276}, {3917, 4576}, {4494, 7243}, {5064, 7776}, {5169, 7796}, {5359, 7754}, {6390, 7499}, {7667, 7767}
X(8024) = reflection of X(3051) in X(4074)
X(8024) = isotomic conjugate of X(251)
X(8024) = anticomplement X(1194)
X(8024) = X(1241)-complementary conjugate of X(6237)
X(8024) = X(i)-Ceva conjugate of X(j) for these (i,j): (76,141), (99,3267), (1241,2), (4609,850)
X(8024) = X(i)-cross conjugate of X(j) for these (i,j): (141,1235), (2525,4576), (7794,141)
X(8024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,194,1180), (2,305,3266), (76,305,2), (99,1799,6636)
X(8024) = X(i)-isoconjugate of X(j) for these (i,j): (31,251), (32,82), (83,560), (308,1917), (661,4630), (667,4628), (669,4599), (733,1933), (798,827), (1176,1973), (1501,3112), (1924,4577)
X(8025) lies on these lines: {1, 596}, {2, 6}, {7, 1412}, {8, 4658}, {21, 999}, {58, 3616}, {145, 1010}, {190, 1255}, {274, 4393}, {321, 4670}, {593, 763}, {894, 3995}, {1029, 6625}, {1043, 3623}, {1100, 4359}, {1125, 2308}, {1408, 3485}, {1621, 3286}, {1961, 3952}, {1962, 4427}, {2193, 6349}, {3219, 3294}, {3666, 4760}, {3720, 4368}, {3786, 4661}, {3842, 4722}, {4267, 5253}, {4375, 6545}, {4600, 6634}, {4610, 6650}, {4649, 4651}
X(8025) = isotomic conjugate of X(6539)
X(8025) =X(4979)-cross conjugate of X(4427)
X(8025) =X(i)-Ceva conjugate of X(j) for these (i,j): (86,1125), (190,1019), (4610,7192)
X(8025) =X(i)-isoconjugate of X(j) for these (i,j): (31,6539), (37,1126), (42,1255), (213,1268), (756,1171), (798,6540), (1333,6538), (1402,4102), (1
796,1824), (4079,4596), (4629,4705)
X(8025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (81,86,2), (81,5333,333), (86,333,5333), (333,5333,2), (1962,4697,4427), (3936,6703,2), (4697,5625,1962)
X(8026) lies on these lines: {2, 37}, {76, 4135}, {190, 7075}, {561, 3994}, {726, 6384}, {3685, 7033}, {3790, 7018}, {3971, 6376}
X(8026) =X(92)-Ceva conjugate of X(6376)
X(8026) =X(87)-isoconjugate of X(7121)
X(8026) = {X(3971),X6382)}-harmonic conjugate of X(6376)
X(8027) lies on these lines: {2, 513}, {31, 667}, {42, 649}, {512, 1962}, {693, 6384}, {764, 8042}, {1397, 1980}, {1635, 6373}, {1646, 8034}, {2978, 4790}, {3669, 7248}, {3808, 4453}, {3873, 4083}, {4155, 4984}
X(8027) = isogonal conjugate of isotomic conjugate of X(764)
X(8027) = barycentric cube of X(513)
X(8027) =X(i)-Ceva conjugate of X(j) for these (i,j): (513,1015), (649,3121), (667,3248)
X(8027) =X(i)-isoconjugate of X(j) for these (i,j): (2,6632), (100,7035), (190,1016), (519,6635), (646,4564), (664,4076), (668,765), (874,5378), (1018,4601), (1110,6386), (1252,1978), (1275,6558), (3264,6551), (3699,4998), (3952,4600), (4033,4567), (4103,4590), (4505,5384), (4572,6065), (4595,5383)
X(8028) lies on these lines: {1, 2}, {678, 4152}, {900, 6546}, {3251, 4543}, {3689, 4908}
X(8028) = reflection of X(1647) in X(1644)
X(8028) = X(519)-Ceva conjugate of X(4370)
X(8028) = barycentric cube of X(519)
X(8028) = X(i)-isoconjugate of X(j) for these (i,j): (88,2226), (06,679), (1022,4638)
X(8029) lies on these lines: {2, 523}, {25, 669}, {51, 512}, {351, 1637}, {850, 2528}, {868, 5489}, {1499, 3830}, {3265, 6340}, {3800, 5644}, {4024, 8013}
X(8029) = reflection of X(351) in X(1637)
X(8029) = isotomic conjugate of X(31614)
X(8029) = X(i)-Ceva conjugate of X(j) for these (i,j): (523,115), (2501,3124), (5466,1648)
X(8029) = X(i)-isoconjugate of X(j) for these (i,j): (99,1101), (163,4590), (249,662), (250,4592), (4556,4567)
X(8029) = barycentric cube of X(523)
X(8029) = barycentric product of vertices of Schroeter triangle
X(8029) = X(2)-of-X-parabola-tangential-triangle
X(8025) = isotomic conjugate of X(6539)
X(8030) lies on this line: {2,6}
X(8030) = reflection of X(i) in X(j) for these (i,j): (1641,5468), (1648,1641)
X(8030) = X(i)-Ceva conjugate of X(j) for these (i,j): (524,2482), (5468,1649)
X(8030) = barycentric cube of X(524)
X(8031) lies on this line: {2,37}
X(8031) = barycentric cube of X(536)
X(8032) lies on this line: {2,31}
X(8032) = barycentric cube of X(752)
X(8033) lies on these lines: {1, 2668}, {2, 799}, {6, 7304}, {43, 2669}, {55, 99}, {57, 85}, {76, 940}, {81, 310}, {86, 87}, {171, 1909}, {222, 4573}, {350, 4038}, {670, 4363}, {870, 982}, {874, 4418}, {875, 3112}, {894, 1920}, {1010, 7093}, {1966, 4697}, {3114, 7307}, {3761, 5209}, {3978, 4754}, {7175, 7196}
X(8033) = X(i)-cross conjugate of X(j) for these (i,j): (3287,99), (4754,894)
X(8033) = X(799)-Ceva conjugate of X(4369)
X(8033) = {X(799),X(873)}-harmonic conjugate of X(2)
X(8033) = X(i)-isoconjugate of X(j) for these (i,j): (10,7104), (37,904), (42,893), (213,256), (257,1918), (694,3747), (733,4093), (798,3903), (881,3570), (1178,1500), (1334,1431), (1824,7116), (1927,3948), (1967,2238), (2205,7018), (2333,7015)
X(8034) lies on these lines: {2, 876}, {42, 512}, {244, 8042}, {321, 523}, {351, 4455}, {513, 3666}, {514, 3741}, {593, 3733}, {649, 2308}, {661, 756}, {669, 1402}, {764, 1647}, {890, 3310}, {1646, 8027}, {2489, 8020}, {3572, 5098}, {3957, 4879}, {4367, 7191}, {4705, 8013}
X(8034) = reflection of X(890) in X(3310)
X(8034) = X(i)-Ceva conjugate of X(j) for these (i,j): (512,3122), (523,3125), (661,3124), (3733,1015), (7180,3121)
X(8034) = X(i)-isoconjugate of X(j) for these (i,j): (59,7257), (81,6632), (99,765), (100,4600), (101,4601), (110,7035), (190,4567), (249,4033), (643,4998), (644,4620), (645,4564), (662,1016), (668,4570), (670,1110), (799,1252), (1018,4590), (1262,7258), (1275,7259), (1414,4076), (4069,7340), (4551,6064), (4561,5379), (4625,6065), (7045,7256)
X(8035) lies on these lines: {2, 372}, {5475, 8036}
X(8035) = X(485)-Ceva conjugate of X(590)
X(8036) lies on these lines: {2, 371}, {5475, 8035}
X(8036) =X(486)-Ceva conjugate of X(615)
X(8037) lies on this line: {2, 490}
X(8037) =X(1131)-Ceva conjugate of X(3068)
X(8038) lies on this line: {2,489}
X(8038) = X(1132)-Ceva conjugate of X(3069)
X(8039) lies on these lines: {2, 308}, {710, 8023}
X(8039) = X(1502)-Ceva conjugate of X(626)
X(8040) lies on these lines: {2, 846}, {37, 6535}, {42, 5257}, {756, 3122}, {1125, 2308}, {1213, 1962}, {3578, 5625}, {3724, 4204}, {3986, 4082}, {6626, 6628}
X(8040) = X(4427)-Ceva conjugate of X(4988) for this (i,j): {4427,4988}
X(8040) = X(1171)-isoconjugate of X(1255)
X(8040) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6536,3120), (1213,1962,8013)
In the plane of a triangle ABC, let
G = centroid = X(2);
DEF = cevian triangle of X(6);
TA = line tangent to circumcircle at A;
LAB = line through B parallel to AG;
LAC = line through C parallel to AG;
A2 = TA∩LAB;
A3 = TA∩LAC;
Ab = DA2∩AG;
Ac = DA3∩AG;
A' = {Ab,ACb}-harmonic conjugate of A, and define B' and C' cyclically.
The finite fixed point of the affine transformation that carries ABC onto A'B'C' is X(8041). See
X(8041). (Angel Montesdeoca, February 23, 2021.)
X(8041) lies on these lines: {2, 694}, {3, 1501}, {6, 1627}, {39, 3051}, {141, 6665}, {184, 574}, {251, 2076}, {394, 5013}, {570, 3289}, {741, 4283}, {1180, 1613}, {1194, 3231}, {1196, 5650}, {1915, 6636}, {1994, 5038}, {3118, 6292}, {4175, 7794}, {4265, 5371}, {5012, 5116}, {5110, 7054}
X(8041) = X(i)-Ceva conjugate of X(j) for these (i,j): (141,7794), (249,1634), (4576,3005)
X(8041) = X(i)-isoconjugate of X(j) for these (i,j): (82,83), (251,3112)
X(8041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4576,4074), (39,3917,3051), (1180,7998,1613), (1194,3819,3231)
X(8041) = barycentric square of X(38)
X(8042) lies on these lines: {2, 661}, {38, 876}, {81, 1019}, {244, 8034}, {310, 7199}, {513, 3720}, {514, 4359}, {764, 8027}, {798, 2350}, {1401, 4017}, {1407, 7216}, {1412, 7203}
X(8042) = X(7192)-Ceva conjugate of X(244)
X(8042) = X(i)-isoconjugate of X(j) for these (i,j): (42,6632), (765,1018), (1016,4557), (1110,4033), (1252,3952), (3943,6551), (4069,4564), (4076,4559), (4082,4619), (4103,4570), (4552,6065)
Suppose that P is a point in the plane of a triangle ABC. Let DEF be the cevian triangle of P, and let OP be the circumcircle of DEF. Let D' be the point, other than D, in AD∩OP, and define E' and F' cyclically. Let PA = E'F'∩BC, and define PB and PC cyclically. Then PA, PB, PC are collinear. Let P'A = E'F'∩EF, and define P'B and P'C cyclically. Then P'A, P'B, P'C are collinear. Let L(P) be the line PAPBPC and let L'(P) be the line P'AP'BP'C. Next, suppose that Q is the cyclocevian conjugate (also called Terquem conjugate) of P. The four lines L(P), L'(P), L(Q), L'(Q) concur in a point, introduced here as the Farhangi-cyclocevian image of P, denoted by f(P). If P = X(2), then Q = X(4), and the four lines are perpendicular to the Euler line; they concur in X(523). (Based on notes from Sohail Farhangi, August 4, 2015).
Peter Moses finds (August 5, 2015) that if P = p: q : r (barycentrics), then
f(P) = p (p q r (p ((a^2-b^2+2 c^2) q-(a^2+2 b^2-c^2) r)+ ((a^2-b^2+c^2) q^2-(a^2+b^2-c^2) r^2))-q^2 r^2((2 b^2-2 c^2) p-a^2 (q-r)) +p^2 (c^2 q^3-b^2 r^3)) : :
f(P) = p[p2q2r(a2 - b2 + 2c2) - p2qr2(a2 + 2b2 - c2) + pq3r(a2 - b2 + c2) - pqr3(a2 + b2 - c2) - 2pq2r2(b2 - c2) + q2r2(q - r)a2 + p2q3c2 - p2r3b2] : :
The following table shows, for 11 choices of P, the cyclocevian conjugate Q and the Farhangi-cyclocevian image of P.
| P | Q | f(P) |
|---|---|---|
| X(2) | X(4) | X(523) |
| X(6) | X(1031) | X(5113) |
| X(8) | X(189) | X(522) |
| X(69) | X(253) | X(525) |
| X(1) | X(1029) | X(8043) |
| X(75) | X(8044) | X(8045) |
| X(20) | X(1032) | X(8057) |
| X(329) | X(1034) | X(8058) |
| X(330) | X(7357) | X(8060) |
| X(2994) | X(8062) | |
| X(5932) | X(8063) |
f(P) = perspector of ABC and the side-triangle of the cevian triangles of P and Q. (Randy Hutson, August 12, 2015)
If P lies on the Lucas cubic (other than X(7)), then f(P) lies on the line at infinity, and is, in fact, the infinite point of the trilinear polar of P. (Randy Hutson, August 12, 2015)
X(8043) is the trilinear pole of line X(23063)X(23064), the tangent to the incentral inellipse at X(23063). (Randy Hutson, October 15, 2018)
X(8043) lies on these lines: {44,513}, {905,4802}, {2605,4041}, {3733,4705}, {4036,4560}
X(8043) = midpoint of X(i) and X(j) for these {i,j}: {2605, 4041}, {3733, 4705}, {4036, 4560}
X(8043) = X(4729)-Ceva conjugate of X(37)
X(8043) = complement of X(30591)
X(8043) = X(i)-complementary conjugate of X(j) for these (i,j): (1126,125), (1171,116), (1796,127), (4596,2887), (4629,141), (4632,626), (6578,3741)
X(8044) lies on the Jerabek hyperbola and these lines: {3,3437}, {6,469}, {65,7282}, {71,1654}, {72,319}, {73,1442}, {313,1330}, {1246,1899}
X(8044) = isogonal conjugate of X(199)
X(8044) = isotomic conjugate of X(1330)
X(8044) = X(58)-cross conjugate of X(2)
X(8044) =X(i)-isoconjugate of X(j) for these (i,j): (1,199), (6,1761), (31,1330)
See X(8043)
X(8045) lies on these lines: {514,661}, {522,667}, {525,4369}, {824,905}, {826,4458}, {1019,2786}, {2533,2785}, {3700,6002}, {4024,4560}, {4122,4367}, {4142,4874}
X(8045) = isotomic conjugate of X(8052)
X(8046) lies on these lines: {320,4358}, {484,519}, {1443,3911}
X(8046) = isogonal conjugate of X(3196)
X(8046) = isotomic conjugate of X(30578)
X(8046) = trilinear pole of line X(900)X(1387)
X(8047) lies on these lines: {2,5375}, {149,693}, {150,7192}, {320,3263}, {1443,3935}, {3218,3912}, {3446,5078}
X(8047) = isotomic conjugate of X(149)
X(8047) = anticomplement of X(5375)
X(8047) = trilinear pole of line X(918)X(4422)
X(8047) = perspector of conic through X(7), X(8), and the extraversions of X(8) (with center X(149))
X(8048) lies on these lines: {2,478}, {4,2995}, {63,573}, {69,3827}, {77,4357}, {193,1814}, {969,5738}, {1444,3435}, {3436,3596}
X(8048) = isogonal conjugate of X(197)
X(8048) = isotomic conjugate of X(3436)
X(8048) = anticomplement of X(478)
X(8048) = trilinear pole of line X(905)X(3910)
X(8049) lies on these lines: {2,2140}, {75,3681}, {86,1621}, {310,2388}, {675,6577}, {1086,7109}
X(8049) = trilinear pole of line X(514)X(6586)
X(8049) = isogonal conjugate of X(8053)
X(8049) = X(19)-isoconjugate of X(22126)
X(8050) = center of hyperbola {{X(8),X(69),X(1330),X(1654),X(2891),X(3436)}} (See Hyacinthos #21238, 10/4/2012, by Nikolaos Dergiades)
X(8050) lies on these lines: {8,596}, {100,1634}, {291,4651}, {668,4576}, {1018,4427}, {3416,4863}, {3909,3952}, {4596,4610}
X(8050) = isogonal conjugate of X(4057)
X(8050) = isotomic conjugate of anticomplement of X(649)
X(8050) = X(19)-isoconjugate of X(22154)
X(8050) = trilinear pole of line X(37)X(39)
X(8050) = anticomplement of X(8054)
Let Na be the Nagel point. The incircle intersects ANa at two points, the closer of which to the vertex A is denoted by A1 and the other by A2. Define B1, C1 and B2, C2 cyclically. X(8051) is the trilinear pole of the perspectrix of the triangles A1B1C1 and A2B2C2, line X(3667)X(3669). (Angel Montesdeoca, November 17, 2021)
X(8051) lies on these lines: {57,145}, {269,5435}, {1357,5423}, {1396,4248}
X(8051) = isotomic conjugate of X(8055)
X(8051) = X(8)-cross conjugate of X(7)
X(8051) = trilinear pole of line X(3667)X(3669)
X(8052) = trilinear pole of line X(1)X(1330) (the line tangent at X(1) to the rectangular hyperbola passing through X(1), X(8), and the extraversions of X(8))
X(8052) = isotomic conjugate of X(8045)
X(8052) = trilinear pole of line X(1)X(1330) (the line tangent at X(1) to the rectangular hyperbola passing through X(1), X(8), and the extraversions of X(8))
X(8053) lies on these lines: {1,3286}, {3,142}, {6,31}, {9,4557}, {21,5263}, {56,7225} et al
X(8053) = crossdifference of every pair of points on line X(514)X(6586)
X(8053) = perspector, wrt excentral triangle, of the circumcircle
X(8053) = X(3613)-of-excentral-triangle
X(8053) = polar conjugate of isotomic conjugate of X(22126)
X(8053) = tangential isogonal conjugate of X(199)
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, B" = Lc∩La, C" = La∩Lb. Triangle A"B"C" is inversely similar to ABC, with similitude center X(8054). See also X(5510). (Randy Hutson, August 13, 2015)
X(8054) lies on the bicevian conic of X(1) and X(2), and also on these lines: {1,3952}, {2,8050}, {11,3141}, {42,1386}, {244,659}, {1015,3124} et al
X(8054) = anticomplement of X(36951)
X(8054) = X(2)-Ceva conjugate of X(649)
X(8054) = crosssum of circumcircle intercepts of Nagel line
X(8054) = crossdifference of every pair of points on line X(1018)X(4427)
X(8054) = perspector of circumconic centered at X(649)
X(8054) = center of hyperbola {{A,B,C,X(1),X(6)}} (the isogonal conjugate of the Nagel line and the isotomic conjugate of line X(10)X(75))
Let A' be the inverse of A in the A-excircle, and define B' and C' cyclically. Triangle A'B'C', here introduced as the inverse-in-excircles triangle (c.f. X(5571)), is perspective to the anticomplementary triangle at X(8055). (Randy Hutson, August 13, 2015)
X(8055) lies on these lines: {2,2415}, {7,8051}, {8,210}, {57,4488} et al
X(8055) = X(7)-Ceva conjugate of X(8)
X(8055) = anticomplement of X(8056)
X(8055) = anticomplementary conjugate of X(21296)
X(8056) lies on these lines: {1,474}, {2,2415}, {8,6553}, {10,1219}, {57,1122}, {63,88} et al
X(8056) = isotomic conjugate of X(18743)
X(8056) = complement of X(8055)
X(8056) = trilinear pole of line X(513)X(4162)
X(8056) = X(9)-cross conjugate of X(1)
X(8056) = anticomplement of complementary conjugate of X(21255)
X(8056) = homothetic center of ABC and vertex-triangle of Gemini triangles 5 and 7
X(8057) lies on these lines: {3,2416}, {30,511}, {69,2419} et al
X(8057) = isogonal conjugate of X(1301)
X(8057) = isotomic conjugate of polar conjugate of X(6587)
X(8057) = isotomic conjugate of anticomplement of X(39020)
X(8057) = X(2)-Ceva conjugate of X(39020)
X(8057) = barycentric square root of X(39020)
X(8057) = complementary conjugate of X(35968)
X(8057) = crossdifference of every pair of points on line X(6)X(64)
X(8057) = Farhangi-cyclocevian image of X(i) for these i: 20, 1032
X(8057) = infinite point of the trilinear polars of X(20) and X(1032)
X(8057) = perspector of the hyperbola {{A,B,C,X(2),X(20)}}
X(8058) lies on these lines: {8,2399}, {30,511}, {3064,3239} et al
X(8058) = isogonal conjugate of X(8059)
X(8058) = crossdifference of every pair of points on line X(6)X(603)
X(8058) = Farhangi-cyclocevian image of X(i) for these i: 329, 1034
X(8058) = infinite point of the trilinear polars of these points: X(318), X(329), X(1034)
X(8059) lies on the circumcircle and these lines: {1,1295}, {56,102}, {57,972}, {84,104}, {100,1813}, {101,2425}, {103,1617}, {105,1422}, {106,1413}, {108,1461} et al.
X(8059) = trilinear pole of the line X(6)X(603)
X(8059) = Ψ(X(i),X(j)) for these (i,j): (1,84), (2,77), (4,57), (6,603), (8,20)
X(8059) = barycentric product of circumcircle intercepts of line X(2)X(77)
X(8059) = Λ(3064,3239). The line X(3064)X(3239) is the trilinear polar of X(318), also the radical axis of Mandart circle and excircles radical circle; also the polar of X(57) wrt polar circle.
X(8060) lies on these lines: {2,8061}, {513,3716}, {824,6586} et al
X(8060) = Farhangi-cyclocevian image of X(i) for these i: 330, 7357
X(8060) = complement of X(8061)
X(8060) = crossdifference of every pair of points on the line X(1631)X(2176)
X(8061) lies on these lines: {2,8060}, {44,513}, {523,594}, {824,1577} et al
X(8061) = isogonal conjugate of X(4599)
X(8061) = isotomic conjugate of X(4593)
X(8061) = anticomplement of X(8060)
X(8061) = crossdifference of every pair of points on the line X(1)X(82)
X(8061) = perspector, wrt excentral triangle, of bianticevian conic of X(1) and X(31)
The trilinear polar of X(8062) passes through X(2797). (Randy Hutson, August 12, 2015)
X(8062) lies on these lines: {1,4086}, {2,656}, {513,3716} et al
X(8062) = isotomic conjugate of isogonal conjugate of X(21761)
X(8062) = complement of X(656)
X(8062) = complementary conjugate of X(34846)
X(8062) = crossdifference of every pair of points on the line X(2176)X(2178)
X(8062) = polar conjugate of isogonal conjugate of X(22382)
X(8063) lies on these lines: {30,511} et al
X(8063) = isogonal conjugate of X(8064)
X(8063) = crossdifference of every pair of points on the line X(6)X(2188)
X(8063) = infinite point of trilinear polar of X(5932)
X(8064) lies on the circumcircle and these lines: {107,7152} et al
X(8064) = trilinear pole of line X(6)X(2188)
Let A'B'C' be the 1st Morley triangle. Let Ma be the line through A parallel to B'C', and define Mb, Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. The triangle A''B''C'' is equilateral with center X(8011), and A''B''C'' is homothetic to A'B'C' at X(8065). (Randy Hutson, August 12, 2015; Seiichi Kirikami, ADGEOM #1515, 8/19/2014)
(The 1st Morley-Kirikami point is X(5454).)
X(8065) lies on these lines: {16,358}, {61,5631}, {356,8011}
X(8065) = {X(358),X(3275)}-harmonic conjugate of X(3280)
Let A'B'C' be the 2nd Morley triangle. Let Ma be the line through A parallel to B'C', and define Mb, Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. The triangle A''B''C'' is equilateral with center X(8011), and A''B''C'' is homothetic to A'B'C' at X(8066). (Peter Moses, August 18, 2015; see X(8065))
X(8066) lies on these lines: {16,358}, {61,5633}, {3276,8011}
X(8066) = {X(1137),X(3273)}-harmonic conjugate of X(3282)
Let A'B'C' be the 3rd Morley triangle. Let Ma be the line through A parallel to B'C', and define Mb, Mc cyclically. Let A" = Mb∩Mc, B" = Mc∩Ma, C" = Ma∩Mb. The triangle A''B''C'' is equilateral with center X(8011), and A''B''C'' is homothetic to A'B'C' at X(8067). (Peter Moses, August 18, 2015; see X(8065))
X(8067) lies on these lines: {16,358}, {61,5629}, {3277,8011}
X(8067) = {1135,3274}-harmonic conjugate of X(3278)
Let A'B'C' be the orthic triangle. Let A"B"C" the the triangle bounded by the antiorthic axes of AB'C', BC'A', CA'B'. Then A"B"C" is inversely similar to ABC, with similitude center X(9). Also, A"B"C" is perspective to ABC at X(80) and to the orthic triangle at X(119). Finally, X(8068) = X(35)-of-A"B"C". (Randy Hutson, August 15, 2015)
X(8068) lies on these lines: {1,5}, {9,6506}, {35,5840}, {36,5841}, {55,6980}, {56,6971}, {100,498}, {104,1478}, {381,8069}, {1656,8071} et al.
X(8068) = {X(1),X(5)}-harmonic conjugate of X(8070)
X(8068) = harmonic center of incircle and nine-point circle
X(8068) = center of hyperbola passing through X(11), X(12) and the extraversions of X(11) (i.e. the vertices of the Feuerbach triangle)
X(8069) lies on these lines: {1,3}, {6,906}, {11,6911}, {12,3560}, {72,920}, {381,8068}, {1656,8070} et al.
X(8069) = {X(1),X(3)}-harmonic conjugate of X(8071)
X(8069) = harmonic center of circumcircle and incircle
X(8069) = isogonal conjugate of isotomic conjugate of {X(7),X(8)}-harmonic conjugate of X(3262)
X(8070) lies on these lines: {1,5}, {35,6882}, {36,6842}, {55,6971}, {56,6980}, {381,8071}, {1656, 8069} et al
X(8070) = {X(11),X(12)}-harmonic conjugate of X(5901)
X(8070) = {X(1),X(5)}-harmonic conjugate of X(8068)
X(8071) lies on these lines: {1,3}, {11,3560}, {12,6911}, {381,8070}, {920,3916} {1656,8068} et al
X(8071) = {X(55),X(56)}-harmonic conjugate of X(1385)
X(8071) = {X(1),X(3)}-harmonic conjugate of X(8069)
X(8071) = homothetic center of medial triangle and mid-triangle of 1st and 2nd Johnson-Yff triangles
Suppose that (U,u) and (V,v) are circles. Let (U,t*u) be the dilation from U of (U,u) by factor t, and likewise for (V,t*v). For t in some interval, the circles (U,t*u) and (V,t*v) meet in 1 or 2 points. The locus of the points of intersection is the similitude circle (or circle of similitude) of (U,u) and (V,v), denoted here by (W,w). Let V' = dilation of V from U by factor v/(u + v), and let U' = dilation of U from V by factor u/(u - v). Then (W,w) is the circle with diameter U'V', and V' = {U,V}-harmonic conjugate of U'. For many choices of 3 circles, each pair gives a similitude circle and the three similitude circles meet in two points. If they have barycentrics of the form P(a,b,c) + Q(a,b,c)*R(a,b,c) : : and P(a,b,c) - Q(a,b,c)*R(a,b,c) : : , where P and Q are polynomials with positive leading coefficients when regarded as polynomials in a, and R has a power series expansion in a such that the coefficient of the first nonzero term is positive, then the first of these is here introduced as the 1st harmonic trace of the 3 circles, and the second, as the 2nd harmonic trace., then the first of these is here introduced as the 1st harmonic trace of the 3 circles, and the second, as the 2nd harmonic trace. (Clark Kimberling, August 7, 2015; revised, November 27, 2016)
Suppose (A',u}, (B',v), (C',w) are circles in the plane of a triangle ABC. It appears that for an arbitary triangle center X of an acute triangle ABC, the following choices of (A',f}, (B',g), (C',h) yield harmonic traces:
f = |AA'|, g = |BB'|, h = |CC'|, where A'B'C' = cevian triangle of X
f = |A'X|, g = |B'X|, h = |C'X|, where A'B'C' = cevian triangle of X
f = |A'X|, g = |B'X|, h = |C'X|, where A'B'C' = pedal triangle of X
f = |AA'|, g = |BB'|, h = |CC'|, where A'B'C' = anticevian triangle of X
The centers of the three intersecting similitude circles lie on the radical axis of the circumcircle of the centers of the original circles and their radical circle. The circles are coaxal, with common radical trace = radical trace of the circumcircle of the centers of the original circles and their radical circle. (Randy Hutson, August 15, 2015)
Soon after the appearance of X(8072) in ETC, Randy Hutson introduced the name harmonic center for the center W of the similitude circle of (U,u) and (V,v), and found examples shown in the table below.
A combo for harmonic centers is given by W = v2U - u2V, and W is the midpoint of the insimilicenter(U,V) and exsimilicenter(U,V). (Randy Hutson, January 17, 2016)
| k | Circles of which X(k) is the harmonic center |
|---|---|
| 2 | pedal circle of X(13) and pedal circle of X(14) |
| 5 | 1st Hutson circle and 2nd Hutson circle |
| 6 | circumcircle and Gallatly circle |
| 115 | nine-point circle and Gallatly circle |
| 182 | 1st Kenmotu circle and 2nd Kenmotu circle |
| 182 | Lucas radical circle and Lucas (-1) radical circle |
| 182 | Lucas inner circle and Lucas (-1) inner circle |
| 182 | 2nd Lemoine circle and {{X(1687), X(1688), PU(1), PU(2)}} |
| 351 | {{X(14),X(15),X(16)}} and {{X(13), X(15), X(16)}} |
| 378 | circumcircle and polar circle |
| 381 | circumcircle and nine-point circle |
| 512 | antipedal circle of P(1) and antipedal circle of U(1) |
| 516 | Bevan circle and anticomplementary circle |
| 574 | Gallatly circle and Ehrmann circle |
| 597 | O(13,15) and O(14,16) |
| 946 | 1st Johnson-Yff circle and 2nd Johnson-Yff circle |
| 1691 | 1st Lemoine circle and 2nd Lemoine circle |
| 1691 | 2nd Brocard circle and {{X(371),X(372),PU(1),PU(39)}} |
| 1995 | circumcircle and {circumrcircle, nine-point circle}-inverter |
| 2080 | Apollonius circle and Gallatly circle |
| 3095 | 2nd Lemoine circle and {{X(371),X(372),PU(1),PU(39)}} |
| 3241 | incircle and AC-incircle |
| 3398 | circumcircle and 1st Lemoine circle |
| 3398 | 1st Brocard circle and 2nd Brocard circle |
| 3543 | polar circle and de Longchamps circle |
| 3679 | incircle and Spieker circle |
| 3679 | Conway circle and excircles-radical circle |
| 4521 | Spieker circle and AC-incircle |
| 4996 | circumcircle and AC-incircle |
| 5038 | 2nd Lemoine circle and Ehrmann circle |
| 5094 | polar circle and {circumcircle, nine-point circle}-inverter |
| 5169 | nine-point circle and {circumcircle, nine-point circle}-inverter |
| 5210 | circumcircle and O(15,16) |
| 5542 | inner Soddy circle and outer Soddy circle |
| 5569 | medial-van Lamoen circle and anticomplementary-van Lamoen circle |
| 6055 | O(13,16) and O(14,15) |
| 6644 | nine-point circle and tangential circle |
| 7577 | nine-point circle and polar circle |
| 7579 | nine-point circle and orthocentroidal circle |
| 7617 | nine-point circle and Ehrmann circle |
| 8068 | nine-point circle and incircle |
| 8069 | circumcircle and incircle |
| 8724 | antipedal circle of X(13) and antipedal circle of X(14) |
| 9130 | circumcircle and Parry circle |
| 9175 | Parry circle and Hutson-Parry circle |
Barycentrics found by Peter Moses, August 18, 2015; note that X(8072) is real if and only if s^2-4 r R-4 R^2 ≥ 0.
The center of the A-similitude circle (i.e., the similitude circle of the B- and C- excircles) has barycentrics b-c : b : c; also,
(radius squared) = bc(c + a - b)(a + b - c)/(4(b - c)2
(power of A wrt A-similude circle) = 0
(power of B wrt A-similude circle) = c(-a^2+b^2-c^2))/(2(b-c))
(power of C wrt A-similude circle) = b(a^2+b^2-c^2))/(2 (b-c))
The A-, B-, C- similitude circles concur in two points: X(8072) and X(8073).
For a related triple of circles, see X(11065). (Peter Moses, November 30, 2016)
X(8072) lies on the curves K058, K269, K352, Q039, the Stevanovic circle, and this line: {4,9}
X(8072) = reflection of X(8073) in the Gergonne line
X(8072) = Bevan-circle-inverse of X(8073)
X(8072) = Spieker-radical-circle-inverse of X(8073)
X(8072) = polar-circle-inverse of X(8073)
X(8072) = isogonal conjugate of X(39146)
X(8072) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,281,8073), (10,10445,8073), (40,2270,8073)
X(8073) is real if and only if s^2-4 r R-4 R^2 >= 0. See X(8072).
X(8073) lies on the curves K058, K269, K352, Q039, the Stevanovic circle, and this line: {4,9}
X(8073) = reflection of X(8072) in the Gergonne line
X(8073) = isogonal conjugate of X(39147)
X(8073) = Bevan-circle-inverse of X(8072)
X(8073) = Spieker-radical-circle-inverse of X(8072)
X(8073) = polar-circle-inverse of X(8072)
X(8073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,281,8072), (10,10445,8072), (40,2270,8072)
X(8074) lies on these lines: {4,9}, {39,1939}, {101,519}, {117,1566, {241,514} et al
X(8074) = reflection of X(5179 in X(5199)
X(8074) = midpoint of X(i) and X(j) for these {i,j}: {5011,5179}, {8072,8073}
X(8074) = radical trace of Bevan circle and excircles radical circle
X(8074) = inverse-in-Bevan-circle of X(2270)
Points associated with tangential-midarc triangles: X(8075) -X(8104)
This section was contributed by César Eliud Lozada, August 18, 2015.
Let ABC be a triangle with incenter I. Let D' and D'' be the points in which the line AI meets the incircle, where D′ is the closer of the two points to A. Define E' and F' cyclically, and defined E'' and F'' cyclically. Let A'B'C' be the triangle whose sidelines are the tangents to the incircle at D', E', F'. The triangle A'B'C' has been called the tangential-midarc triangle of ABC (e.g., MathWorld), but here it is the 1st tangential-midarc triangle of ABC. Let A''B''C'' be the triangle whose sidelines are the tangents to the incircle at D'', E'', F''. Triangle A''B''C'' is here introduced as the 2nd tangential-midarc triangle of ABC.
Trilinears
A' = -cos(B/2)*cos(C/2) : (cos(A/2)+cos(C/2))*cos(C/2) : (cos(A/2)+cos(B/2))*cos(B/2)
A' = c*sin(B/2)+sin(C/2)*b+(a-s) : -c*sin(B/2)-sin(C/2)*(a-c)+(a-s) : -sin(C/2)*b -sin(B/2)*(a-b)+(a-s)
A'' = -cos(B/2)*cos(C/2) : (-cos(A/2)+cos(C/2))*cos(C/2) : (-cos(A/2)+cos(B/2))*cos(B/2)
A'' = c*sin(B/2)+sin(C/2)*b-(a-s) : -c*sin(B/2)-sin(C/2)*(a-c)-(a-s) : -sin(C/2)*b -sin(B/2)*(a-b)-(a-s)
The following table lists triangles perspective to the 1st and 2nd tangential-midarc triangles, with perspectors labeled "1st perspector" for the perspector of the triangle and the 1st tangential mid-arc triangle, and likewise for the 2nd perspector. An asterisk (*) indicates that the triangles are homothetic. A blank space indicates that the triangles are not perspective.
| triangle | 1st perspector | 2nd perspector |
|---|---|---|
| ABC | 177 | |
| anticomplementary | 177 | |
| BCI | 1 | 1 |
| 1st circumperp* | 8075 | 8076 |
| 2nd circumperp* | 8077 | 7588 |
| excentral* | 8078 | 258 |
| 2nd extouch* | 8079 | 8080 |
| hexyl | 8081 | 8082 |
| intouch | 2089 | 174 |
| midarc | 8083 | 8084 |
| 3rd Euler* | 8085 | 8086 |
| 4th Euler* | 8087 | 8088 |
| 6th mixtilinear | 8089 | 8090 |
| triangle | 1st | 2nd |
|---|---|---|
| ABC | 1, 8091 | 1, 8092 |
| anticomplementary | 8, 8091 | 8, 8092 |
| 1st circumperp | 40, 8093 | 40, 8094 |
| 2nd circumperp | 1, 8093 | 1, 8094 |
| excentral | 1, 8093 | 1, 8094 |
| extouch | 72, 8095 | 72, 8096 |
| 2nd extouch | 72, 8093 | 72, 8096 |
| Fuhrmann | 8, 8097 | 8, 8098 |
| inner Grebe | 3641,8091 | 3641, 8092 |
| outer Grebe | 3640,8091 | 3640,8092 |
| hexyl | 40, 8093 | 40, 8084 |
| incentral | 1, 8099 | 1,8100 |
| intouch | 65, 8093 | 65, 8094 |
| Johnson | 355, 8091 | 355, 8092 |
| medial | 10, 8091 | 10, 8092 |
| midarc | 1, 1 | 1, 1 |
| 3rd Euler | 946, 8093 | 946, 8094 |
| 4th Euler | 10, 8093 | 10, 8094 |
| mixtilinear | 1, 8101 | 1, 8102 |
| 6th mixtilinear | 7991, 8093 | 7991, 8094 |
The 1st tangential-midarc and Fuhrmann triangles are parallelogic with centers X(4) and X(8103).
The 2nd tangential-midarc and Fuhrmann triangles are parallelogic with centers X(4) and X(8104).
Centers X(8075)-X(8104) occur in pairs with trilinears of the form: f(a,b,c)sin(A/2) + g(a,b,c)sin(B/2) + g(a,c,b)sin(C/2) ± h(a,b,c) : : .
X(8075) lies on these lines: (1, 168), (2, 8085), (3, 8077), (4, 8087), (40, 8081), (55, 2089), (100, 8103), (104, 8097), (165, 8078), (167, 258), (177, 260), (188, 1376), (266, 503), (1158, 8095)
X(8076) lies on these lines: (1, 168), (2, 8086), (3, 7588), (4, 8088), (40, 8082), (55, 174), (100, 8104), (104, 8098), (164, 8084), (165, 258), (167, 8078), (177, 2346), (236, 1001), (266, 844), (1158, 8096), (1376, 7028), (3579, 8100), (6244, 8102), (7580, 8080)
X(8076) = X(1861)-of-excentral-triangle
X(8077) lies on these lines: (1,164), (2, 8087), (3, 8075), (4, 8085), (21, 177), (56, 2089), (100, 8097), (104, 8103), (188, 958), (405, 8079), (999, 8101), (1385, 8099), (3576, 8081), (6261, 8095), (7987, 8089), (8082, 8084)
Let Iab and Iac be the excenters of IBC respective to sidelines CI and IB; let Ibc and Iba be the excenters of ICA respective to sidelines AI and IC; let Ica and Icb be the excenters of IAB respective to sidelines BI, IA. The points Iab, Iac, Ibc, Iba, Ica, Icb lie on an ellipse, E. Let A' be the intersection of the tangents to E at Iab and Iac, and define B' and C' cyclically. The triangle A'B'C' is perspective to the excentral triangle at X(8078). (See Anopolis #402, 6/12/2013, Antreas Hatzipolakis and Randy Hutson for other related centers).
X(8078) lies on these lines: (1,164), (3, 8081), (9, 173), (40, 8091), (56, 3659), (57, 2089), (165, 8075), (166, 8083), (167, 8076), (1490, 8095), (1698, 8087), (1699, 8085), (1768, 8103), (5541, 8097), (8084, 8090)
X(8078) = X(34) of excentral triangle
X(8078) = {X(1),X(164)}-harmonic conjugate of X(258)
X(8078) = exsimilicenter of incircle and incircle of excentral triangle
X(8079) lies on these lines: (1, 8080), (4, 5934), (9, 173), (72, 8093), (226, 2089), (329, 8101), (405, 8077), (442, 8087), (1490, 8081), (1750, 8089), (5777, 8099), (7580, 8075)
X(8080) lies on these lines: (1, 8079), (4, 8092), (9, 258), (72, 8094), (174, 226), (329, 8102), (405, 7588), (442, 8088), (954, 7589), (1490, 8082), (1750, 8090), (5777, 8100), (5934, 6732), (7580, 8076)
X(8081) lies on these lines: (1, 167), (3, 8078), (40, 8075), (84, 8095), (188, 936), (1490, 8079), (3576, 8077), (5587, 8087), (6264, 8097), (6282, 8101), (6326, 8103)
X(8082) lies on these lines: (1, 167), (3, 258), (40, 8076), (56, 7597), (84, 8096), (173, 942), (936, 7028), (1490, 8080), (3333, 7587), (3576, 7588), (5587, 8088), (6264, 8098), (6282, 8102), (6326, 8104), (8077, 8084)
X(8083) lies on these lines: (1, 168), (7, 177), (57, 7589), (166, 8078), (174, 354), (236, 518), (942, 8094), (3333, 7588), (3742, 7028), (5045, 8092), (5571, 8084)
X(8083) = X(6) of mid-arc triangle
X(8084) lies on these lines: (164, 8076), (167, 258), (5571, 8083), (8077, 8082), (8078, 8090), (8091, 8100), (8092, 8093)
X(8085) lies on these lines: (1,8086), (2,8075), (4,8077), (5,8087), (11, 2089), (188, 2886) , (946,8093), (1699,8078), (7956,8101), (7988,8089)
X(8086) lies on these lines: (1,8085) , (2,8076), (5, 8088), (11, 174), (258, 1699), (497, 7589), (946, 8094), (3086, 7587), (7956, 8102), (7988, 8090)
X(8087) lies on these lines: (1, 8088), (2, 8077), (4, 8075), (5, 8085), (10, 8093), (11, 8097), (12, 2089), (119, 8103), (188, 1329), (442, 8079), (1698, 8078), (3820, 8101), (5587, 8081), (7989, 8089)
X(8088) lies on these lines: (1, 8087), (2, 7588), (4, 8076), (5, 8086), (10, 8094), (11, 8098), (12, 174), (119, 8104), (173, 5290), (258, 1698), (388, 7587), (442, 8080), (1329, 7028), (3085, 7589), (3820, 8102), (5587, 8082), (7989, 8090)
X(8089) lies on these lines: (1, 167), (40, 8099), (165, 8075), (1750, 8079 (5531,8103), (7987,8077), (7988,8085), (7989,8087), (7991,8093), (7992,8095), (7994,8101)
X(8090) lies on these lines: (1, 167), (40, 8100), (165, 258), (1750, 8080), (5531,8104), (7988,8086), (7989,8088), (7991,8094), (7992,8096), (7994,8102)
X(8091) is also the orthologic center from the 1st tangential midarc triangle to every triangle homothetic to ABC.
X(8091) lies on these lines: (1, 167), (3, 8075), (4, 5934), (5, 8085), (10, 188), (40, 8078), (259, 7593), (517, 8093), (952, 8097), (6001, 8095), (8084, 8100)
X(8091) = midpoint of X(8097) and X(8103)
X(8091) = reflection of X(i) in X(j) for these (i,j): (8084,8100), (8092,1), (8093,8099)
X(8091) = radical center of circles centered at A, B, C, and externally tangent to the incircle
X(8092) is also the orthologic center from the 2nd tangential midarc triangle to every triangle homothetic to ABC.
X(8092) lies on these lines: (1, 167), (3, 7588), (4, 8080), (5, 8086), (10, 7028), (40, 258), (173, 3333), (236, 1125), (266, 6732), (483, 1127), (517, 8094), (952, 8098), (999, 7587), (3295, 7589), (3296, 7707), (3487, 7593), (5045, 8083), (6001, 8096), (8084, 8093)
X(8092) = midpoint of X(i) and X(j) for these {i,j}: {8084,8093}, {8098,8104}
X(8092) = reflection of X(i) in X(j) for these (i,j): (8091,1), (8094,8100)
X(8092) = {X(1),X(174)}-harmonic conjugate of X(8352)
X(8092) = radical center of circles centered at A, B, C, and internally tangent to the incircle
X(8093) is also the orthologic center from the 1st tangential midarc triangle to every triangle homothetic to the excentral triangle.
X(8093) lies on these lines: (1, 164), (8, 177), (10, 8087), (40,8075) , (65, 2089), (72, 8079), (188, 960), (515, 8095), (517, 8091), (946, 8085), (2800, 8103), (2802, 8097), (7991, 8089), (8084, 8092)
X(8093) = reflection of X(i) in X(j) for these (i,j): (8091,8099), (8094,1)
X(8093) = orthocenter of 1st tangential midarc triangle
X(8093) = X(155)-of-mid-arc-triangle
X(8093) = radical center of circles centered at the excenters and internally tangent to the incircle
X(8093) = radical center of circles centered at the vertices of the 2nd circumperp triangle and internally tangent to the incircle
X(8094) is also the orthologic center from the 2nd tangential midarc triangle to every triangle homothetic to the excentral triangle.
X(8094) lies on these lines: (1, 164), (10, 8088), (40, 8076), (57, 7587), (65, 174), (72, 8080), (173, 3339), (236, 3812), (259, 1046), (515, 8096), (517, 8092), (942, 8083), (946, 8086), (960, 7028), (2800, 8104), (2802, 8098), (7991, 8090)
X(8094) = reflection of X(i) in X(j) for these (i,j): (8092,8100), (8093,1)
X(8094) = orthocenter of 2nd tangential midarc triangle
X(8094) = radical center of circles centered at the excenters and externally tangent to the incircle
X(8094) = radical center of circles centered at the vertices of the 2nd circumperp triangle and externally tangent to the incircle
X(8095) lies on these lines: (1, 8096), (4, 177), (84, 8081), (188, 5777), (515, 8093) , (971,8099), (1071, 2089), (1158, 8075), (1490, 8078), (2800, 8097), (6001, 8091), (6261, 8077), (7992, 8089)
X(8095) = reflection of X(8096) in X(1)
X(8095) = tangential-midarc-isotomic conjugate of X(12726)
X(8096) lies on these lines: (1, 8095), (84, 8082), (174, 1071), (258, 1490), (515, 8094), (971, 8100), (1158, 8076), (2800, 8098), (5777, 7028), (6001, 8092), (6261, 7588), (7992, 8090)
X(8096) = reflection of X(8095) in X(1)
X(8096) = 2nd-tangential-midarc-isotomic conjugate of X(12727)
X(8097) lies on these lines: (1, 8098), (11, 8087), (80, 1128), (100, 8077), (104, 8075), (119, 8085), (177, 1320), (188, 3036), (952, 8091), (1317, 2089), (2800, 8095), (2802, 8093), (5541, 8078), (6264, 8081), (7993, 8089)
X(8097) = reflection of X(8098) in X(1)
X(8098) lies on these lines: (1, 8097), (11, 8088), (100, 260), (104, 8076), (119, 8086), (174, 1317), (258, 5541), (952, 8092), (2800, 8096), (2802, 8094), (3036, 7028), (6264, 8082), (7993, 8090)
X(8098) = reflection of X(8097) in X(1)
X(8099) lies on these lines: {1,8100}, {3,8078}, {4,9793}, {40,8089}, {188,5044}, {517,8091}, {942,2089}, {971,8095}, {1385,8077}, {2771,8103}, {3579,8075}, {5728,8387}, {5777,8079}, {8085,9955}, {8087,9956}, {8241,9957}, {8249,9959}, {8733,9940}
X(8099) = reflection of X(8100) in X(1)
X(8099) = radical center of circles centered at the vertices of the incentral triangle and internally tangent to the incircle
X(8100) lies on these lines: (1, 8099), (3, 258), (40, 8090), (173, 5708), (174, 942), (517, 8092), (971, 8096), (1385, 7588), (2771, 8104), (3579, 8076), (5044, 7028), (5777, 8080), (8084, 8091)
X(8100) = midpoint of X(i) and X(j) for these {i,j}: {8084,8091}, {8092,8094}
X(8100) = reflection of X(8099) in X(1)
X(8100) = radical center of circles centered at the vertices of the incentral triangle and externally tangent to the incircle
X(8101) lies on these lines: (1, 8102), (57, 2089), (188, 3452), (329, 8079), (517, 8091), (999, 8077), (3820, 8087), (6244, 8075), (6282, 8081), (7956, 8085), (7994, 8089)
X(8101) = reflection of X(8102) in X(1)
X(8101) = radical center of circles centered at the mixtilinear incenters and internally tangent to the incircle
X(8101) = radical center of circles centered at the mixtilinear excenters and internally tangent to the incircle
X(8102) lies on these lines: (1, 8101), (57, 173), (236, 6692), (329, 8080), (517, 8092), (999, 7588), (3452, 7028), (3820, 8088), (6244, 8076), (6282, 8082), (7956, 8086), (7994, 8090)
X(8102) = reflection of X(8101) in X(1)
X(8102) = radical center of circles centered at the mixtilinear incenters and externally tangent to the incircle
X(8102) = radical center of circles centered at the mixtilinear excenters and externally tangent to the incircle
X(8103) lies on these lines: (1, 8104), (11, 2089), (100, 8075), (104, 8077), (119, 8087), (188, 3035), (952, 8091), (1768, 8078), (2771, 8099), (2800, 8093), (5531, 8089), (6326, 8081)
X(8103) = reflection of X(i) in X(j) for these (i,j): (8097,8091), (8104,1)
X(8104) lies on these lines: (1, 8103), (11, 174), (100, 8076), (104, 7588), (119, 8088), (236, 6667), (258, 1768), (952, 8092), (2771, 8100), (2800, 8094), (3035, 7028), (5531, 8090), (6326, 8082)
X(8104) = reflection of X(i) in X(j) for these (i,j): (8098,8092), (8103,1)
The trilinear polar of X(8105) meets the line at infinity at X(512).
Let MaMbMc = medial triangle and HaHbHc = orthic triangle. Let A' = reflection of A in X(3), and let A'' be the point, other than A', where the line A'Ma meets the circumcircle. Define B'' and C'' cyclically. The points D=BC∩HbHc, E=CA∩HcHa, F=AB∩HaHb, D'=B"C"∩MbMc, E'=C"A"∩McMa, F'=A"B"∩MaMb lie on orthic axis (common perspectrix of ABC and HaHbHc, MaMbMc and A"B"C" ). The fixed points of the projectivity that maps D, E, F onto D', E', F', respectively, are X(8105) and X(8106). (Angel Montesdeoca, May 17, 2019)
X(8105) and X(8106) lie on a remarkable number of curves, listed below. Contributed by Bernard Gibert, August 28, 2015. See X(8115) and X(8116).
X(8105) lies on the circumconic {{A, B, C, X(2), X(6)}}, the circle {{X(6), X(111), X(112), X(115), X(187)}}, the cubics K237, K511, K606, K624, the quartic Q081, and these lines: {2,2592}, {6,1344}, {37,2588}, {111,1114}, {112,1113}, {115,1313}, {230,231}, {1312,1560}, {1346,6032}, {2575,3569}, {2987,8116}
X(8105) = reflection of X(8106) in X(2492)
X(8105) = isogonal conjugate of X(8115)
X(8105) = crosspoint of X(2) and X(1113)
X(8105) = crosssum of X(6) and X(2574)
X(8105) = X(2489)/X(8106) (barycentric quotient)
X(8105) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1313), (648,2575), (1113,25), (2592,2574)
X(8105) = X(i)-complementary conjugate of X(j) for these (i,j): (31,1313), (1113,2887), (1822,1368), (2576,141), (2579,127), (2580,626)
X(8105) = X(i)-isoconjugate of X(j) for these {i,j}: {2,1822}, {3,2580}, {63,1113}, {69,2576}, {99,2579}, {110,2583}, {394,2586}, {648,2585}, {662,2575}, {2589,4558}, {2593,4575}
X(8105) = perspector of circumconic centered at X(1313) (the hyperbola {{A,B,C,X(4),X(1114)}})
X(8105) = intersection of trilinear polars of X(4) and X(1114)
X(8105) = crossdifference of every pair of points on line X(3)X(2575)
X(8105) = PU(4)-harmonic conjugate of X(8106)
X(8105) = {X(i),X(j)}-harmonic conjugate of X(8106) for these (i,j): (230,2493), (232,6103), (647,1637), (3003,3018)
The trilinear polar of X(8106) meets the line at infinity at X(512).
X(8106) and X(8105) lie on a remarkable number of curves, listed below. Contributed by Bernard Gibert, August 28, 2015.
X(8106) lies on the circumconic {{A, B, C, X(2), X(6)}}, the circle {{X(6), X(111), X(112), X(115), X(187)}}, the cubics K237, K511, K606, K624, the quartic Q081, and these lines: {2,2593}, {6,1345}, {37,2589}, {111,1113}, {112,1114}, {115,1312}, {230,231}, {1313,1560}, {1347,6032}, {2574,3569}, {2987,8115}
X(8106) = reflection of X(8105) in X(2492)
X(8106) = isogonal conjugate of X(8116)
X(8106) = crosspoint of X(2) and X(1114)
X(8106) = crosssum of X(6) and X(2575)
X(8106) = X(2489)/X(8105) (barycentric quotient)
X(8106) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1312), (648,2574), (1114,25), (2593,2575)
X(8106) = X(i)-complementary conjugate of X(j) for these (i,j): (31,1312), (1114,2887), (1823,1368), (2577,141), (2578,127), (2581,626)
X(8106) = X(i)-isoconjugate of X(j) for these {i,j}: {2,1823}, {3,2581}, {63,1114}, {69,2577}, {99,2578}, {110,2582}, {394,2587}, {648,2584}, {662,2574}, {2588,4558}, {2592,4575}
X(8106) = perspector of circumconic centered at X(1312) (hyperbola {A,B,C,X(4),X(1113)})
X(8106) = intersection of trilinear polars of X(4) and X(1113)
X(8106) = crossdifference of every pair of points on line X(3)X(2574)
X(8106) = PU(4)-harmonic conjugate of X(8105)
X(8106) = {X(i),X(j)}-harmonic conjugate of X(8105) for these (i,j): (230,2493), (232,6103), (647,1637), (3003,3018)
X(8106) = {P,U}-harmonic conjugate of X(1345), where P and U are the foci of the orthic inconic
X(8107) lies on these lines: (3, 8109), (40, 8111), (55, 8113), (165, 166), (167, 7987), (259, 503), (5934, 7580), (6732, 8076)
X(8108) lies on these lines: (3, 8110), (40, 8112), (55, 8114), (164, 6726), (165, 166), (5935, 7580), (7589, 7707)
X(8109) lies on these lines: (1, 289), (3, 8107), (56, 8113), (3576, 8111), (6732, 7588), (7987, 8110)
X(8110) lies on these lines: (1, 168), (3, 8108), (56, 8114), (405, 5935), (3576, 8112), (7987, 8109)
X(8111) lies on these lines: (1, 8112), (3, 363), (40, 8107), (84, 266), (505, 7597), (1490, 5934), (3576, 8109), (6732, 8082), (7590, 8095), (7971, 8096)
X(8112) lies on these lines: (1, 8111), (3, 168), (40, 8108), (84, 7590), (164, 517), (1490, 5935), (3576, 8110)
X(8113) lies on these lines: (1, 8111), (2, 178), (55, 8107), (56, 8109), (57, 363), (174, 6732), (226, 5934)
X(8114) lies on these lines: (1, 8111), (7, 174), (55, 8108), (56, 8110), (57, 168), (188, 3870), (226, 5935), (289, 1488)
Contributed by Bernard Gibert, August 30, 2015. See X(8105) and X(8106).
X(8115) is the trilinear pole of line X(3)X(2575). This line is the major axis of the ellipse that is the locus of radical centers of circles centered at A, B, C and tangent to lines through X(3). This ellipse is centered at X(3) and passes through (complement of X(125)) = X(5972). (Randy Hutson, September 5, 2015)
X(8115) lies on the MacBeath circumconic, the orthic inconic, and these lines: {2,6}, {110,1113}, {511,1114}, {648,2592}, {651,2580}, {895,2105}, {1312,3564}, {1331,1822}, {1344,6090}, {1345,1351}, {1346,1352}, {2552,5480}, {2593,2986}, {2987,8106}
X(8115) = reflection of X(8116) in X(6)
X(8115) = isotomic conjugate of X(2592)
X(8115) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,323,8116), (1993,3580,8116)
X(8115) = X(249)-Ceva conjugate of X(8116)
X(8115) = cevapoint of X(6) and X(2574)
X(8115) = X(i)-cross conjugate of X(j) for these (i,j): (525,8116), (647,1114), (2574,69)
X(8115) = X(i)-iso conjugate of X(j) for these {i,j}: {1,8105}, {4,2578}, {6,2588}, {19,2574}, {25,2582}, {31,2592}, {393,2584}, {512,2581}, {523,2577}, {647,2587}, {661,1114}, {1313,2576}, {1823,2501}
X(8116) is the trilinear pole of line X(3)X(2574). This line is the minor axis of the ellipse described at X(8115). (Randy Hutson, September 5, 2015)
Contributed by Bernard Gibert, August 30, 2015. See X(8105) and X(8106).
X(8116) lies on the MacBeath circumconic, the orthic inconic, and these lines: {2,6}, {110,1114}, {511,1113}, {648,2593}, {651,2581}, {895,2104}, {1313,3564}, {1331,1823}, {1344,1351}, {1345,6090}, {1347,1352}, {2553,5480}, {2592,2986}, {2987,8105}
X(8116) = reflection of X(8115) in X(6)
X(8116) = isotomic conjugate of X(2593)
X(8116) = cevapoint of X(6) and X(2575)
X(8116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,323,8115), (1993,3580,8115)
X(8116) = X(249)-Ceva conjugate of X(8115)
X(8116) = X(i)-cross conjugate of X(j) for these (i,j): (525,8115), (647,1113), (2575,69)
X(8116) = X(i)-iso conjugate of X(j) for these {i,j}: {1,8106}, {4,2579}, {6,2589}, {19,2575}, {25,2583}, {31,2593}, {393,2585}, {512,2580}, {523,2576}, {647,2586}, {661,1113}, {1312,2577}, {1822,2501}
Endo-homothetic centers: X(8117) -X(8140)
This section was contributed by César Eliud Lozada, September 7, 2015.
X(8117) lies on these lines: (2, 8121), (3, 8119), (4, 8123), (20, 8118), (30, 8120), (376, 8124), (3146, 8122)
X(8118) lies on these lines: (2, 8122), (3, 8120), (4, 8124), (20, 8117), (30, 8119), (376, 8123), (3146, 8121)
X(8119) lies on these lines: (2, 8123), (3, 8117), (4, 5934), (20, 8124), (30, 8118)
X(8120) lies on these lines: (2, 8124), (3, 8118), (4, 5934), (20, 8123), (30, 8117)
X(8121) lies on these lines: (2, 8117), (4, 5934), (5, 8123), (30, 8124), (3146, 8118)
X(8122) lies on these lines: (2, 8118), (4, 5934), (5, 8124), (30, 8123), (3146, 8117)
X(8123) lies on these lines: (2, 8119), (3, 8124), (4, 8117), (5, 8121), (20, 8120), (30, 8122), (376, 8118)
X(8123) = {X(8134),X(8136)}-harmonic conjugate of X(174)
X(8124) lies on these lines: (2, 8120), (3, 8123), (4, 8118), (5, 8122), (20, 8119), (30, 8121), (376, 8117)
X(8124) = {X(8137),X(8139)}-harmonic conjugate of X(174)
X(8125) lies on these lines: (2, 174), (3, 8127), (4, 8129), (8, 8092), (22, 8131), (63, 258), (72, 8100), (78, 8082), (100, 8076), (140, 8128), (173, 3306), (200, 8090), (329, 8080), (631, 8130), (1621, 7589), (2975, 7588), (3869, 8094), (5253, 7587), (7485, 8132)
X(8125) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,174,8126), (174,7028,2)
X(8126) lies on these lines: (2, 174), (3, 8128), (4, 8130), (22, 8132), (63, 173), (78, 7590), (100, 7589), (140, 8127), (258, 3306), (329, 7593), (631, 8129), (2975, 7587), (3616, 8092), (3873, 8083), (5253, 7588), (5439, 8100), (7485, 8131)
X(8126) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,174,8125), (174,236,2)
X(8127) lies on these lines: (2, 8129), (3, 8125), (4, 7028), (24, 8131), (140, 8126), (174, 631), (236, 3525), (3523, 8130), (5657, 8092)
X(8127) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (174,631,8128)
X(8128) lies on these lines: (2, 8130), (3, 8126), (4, 236), (24, 8132), (140, 8125), (174, 631), (3523, 8129), (3525, 7028)
X(8128) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (174,631,8127)
X(8129) lies on these lines: (2, 8127), (3, 174), (4, 8125), (5, 7028), (140, 236), (258, 5709), (517, 8092), (631, 8126), (912, 8096), (3523, 8128), (5812, 8080), (5840, 8104), (6769, 8090)
X(8129) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,174,8130), (8092,8102,8100)
X(8130) lies on these lines: (2, 8128), (3, 174), (4, 8126), (5, 236), (140, 7028), (173, 5709), (631, 8125), (1385, 8092), (3523, 8127), (5812, 7593), (6713, 8104)
X(8130) = {X(3),X(174)}-harmonic conjugate of X(8129)
X(8131) lies on these lines: (3, 174), (22, 8125), (24, 8127), (25, 7028), (236, 7484), (258, 5285), (7485, 8126)
X(8131) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,174,8132)
X(8132) lies on these lines: (3, 174), (24, 8128), (25, 236), (173, 5285), (7028, 7484), (7485, 8125)
X(8132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,174,8131)
X(8133) lies on these lines: (1, 6724), (2, 178), (4, 5934), (177, 266), (363, 8078), (8077, 8109), (8081, 8111), (8089, 8135)
X(8134) lies on these lines: (3,8139), (174, 8136), (260, 289), (7028, 8121), (8117, 8125), (8119, 8127), (8123, 8129)
X(8134) = {X(174),X(8123)}-harmonic conjugate of X(8136)
X(8135) lies on these lines: (1, 8138), (168, 8078), (177, 7707), (503, 6724), (2089, 8114), (5935, 8079), (8075, 8108), (8077, 8110), (8081, 8112), (8089, 8133)
X(8136) lies on these lines: (3, 8137), (174, 8134), (236, 8121), (8117, 8126), (8119, 8128), (8123, 8130)
X(8136) = {X(174),X(8123)}-harmonic conjugate of X(8134)
The homothetic center of the two triangles is X(6732). X(8137) is also the endo-homothetic center of these two triangles: 2nd tangential-midarc and outer Hutson.
X(8137) lies on these lines: (3, 8136), (174,8124), (7028, 8122), (8118, 8125), (8120, 8127), (8124, 8129)
X(8137) = {X(174),X(8124)}-harmonic conjugate of X(8139)
X(8138) lies on these lines: (1, 8135), (7, 174), (168, 258), (944, 8092), (5935, 8080), (6732, 8090), (7588, 8110), (8076, 8108), (8082, 8112)
X(8138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (174,8114,7707)
X(8139) lies on these lines: (3, 8134), (174, 8124), (236, 8122), (8118, 8126), (8120, 8128), (8124, 8130)
X(8139) = {X(174),X(8124)}-harmonic conjugate of X(8137)
The endo-homothetic center of the inner Hutson and outer Hutson triangles is X(174).
X(8140) lies on these lines: (1, 8111), (165, 166), (503, 1750), (3062, 7707), (6732, 8090), (7987, 8110), (7988,8378), (7989,8380), (8089, 8133)
X(8140) = anticomplement wrt excentral triangle of X(845)
Centers of central circles: X(8141) -X(8155)
This section was contributed by César Eliud Lozada, September 22, 2015.
There are more than sixty central circles included at MathWorld. Triangles centers X(8141)-X(8155) are centers of such circles that are not among the points X(i) for i < 8141.
| circle | center |
|---|---|
| Adams | X(1) |
| anticomplementary | X(4) |
| Apollonius | X(970) |
| Bevan | X(40) |
| Brocard | X(182) |
| 2nd Brocard | X(3) |
| circumcircle | X(3) |
| Conway | X(1) |
| Dao-Moses-Telv | X(1637) |
| 1st Droz-Farny | X(4) |
| EGS | X(8153) |
| excircles radical circle | X(10) |
| extangents | X(8141) |
| Fuhrmann | X(355) |
| Gallatly | X(39) |
| GEOS | X(8142) |
| half-Moses | X(39) |
| hexyl | X(1) |
| incentral | X(8143) |
| incircle | X(1) |
| intangents | X(8144) |
| 1st Johnson-Yff | X(1478) |
| 2nd Johnson-Yff | X(1479) |
| Kenmotu | X(371) |
| 1st Lemoine | X(182) |
| 2nd Lemoine | X(6) |
| 3rd Lemoine | X(8345) |
| Lester | X(1116) |
| Longuet-Higgins | X(962) |
| Lucas circles radical circle | X(1151) |
| Inner Lucas | X(6407) |
| Macbeath | X(8146) |
| Mandart | X(1158) |
| McCay | X(7617) |
| midheight | X(5893) |
| mixtilinear | X(8147) |
| Morley | X(356) |
| Moses | X(39) |
| Moses-Longuet-Higgins | X(8148) |
| Inner Napoleon | X(2) |
| Outer Napoleon | X(2) |
| Neuberg circles radical circle | X(194) |
| 1st Neuberg | X(8149) |
| 2nd Neuberg | X(8150) |
| nine points circle | X(5) |
| orthocentroidal | X(381) |
| orthoptic circle of the Steiner inellipse | X(2) |
| Parry | X(351) |
| polar | X(4) |
| reflection | X(195) |
| sine triple-angle | X(49) |
| inner Soddy | X(176) |
| outer Soddy | X(175) |
| Spieker | X(10) |
| Stammler | X(3) |
| Stammler circles radical circle | X(5) |
| Steiner | X(5) |
| 2nd Steiner | X(8151) |
| Stevanovic | X(650) |
| symmedial | X(8152) |
| tangential | X(26) |
| Taylor | X(389) |
| Van Lamoen | X(1153) |
| inner Vecten | X(642) |
| outer Vecten | X(641) |
| Yff central | X(8351) |
| Yff contact | X(5592) |
| Yiu | X(8154) |
X(8141) lies on these lines: (3,3101), (5,19), (26,55), (30,40), (52,3611), (155,3197), (517,6759), (534,6684), (1154,6237), (5341,5713), (5663,6254)
X(8141) = reflection of X(8144) in X(26)
X(8141) = {X(3101),X(6197)}-harmonic conjugate of X(3)
X(8142) lies on these lines: (3,4885), (20,650), (376,4762) , (514,8153), (693,3522)
X(8142) = midpoint of X(20) and X(650)
X(8142) = reflection of X(4885) in X(3)
X(8143) is the QA-P11 center (Circumcenter of the QA-Diagonal Triangle) of quadrangle ABCX(1).
Let La be the trilinear polar of A wrt BCX(1), and define Lb and Lc cyclically. Let Li be the trilinear polar of X(1) (i.e., the antiorthic axis). Then X(8143) is the QL-P9 center (Circumcenter QL-Diagonal Triangle) of quadrilateral LaLbLcLi. (Randy Hutson, October 13, 2015)
X(8143) lies on these lines: (1,399), (3,2941), (5,2486), (30,3743), (37,2160), (55,1717), (81,3652), (500,1962), (517,2292), (942,1725), (1385,4653), (4854,6841)
X(8143) = midpoint of X(1) and X(5492)
X(8143) = circumcenter of X(11)X(115)X(3024)
X(8143) = incentral-isogonal conjugate of X(500)
X(8143) = incentral-triangle-complement of X(500)
X(8144) lies on these lines: (1,30), (3,3100), (5,33), (26,55), (34,3627), (35,1658), (52,3270), (140,1040), (155,2192), (382,1870), (548,1038), (550,1060), (582,1736), (1154,6238), (1657,4296), (1807,3362), (3295,7387), (3695,4123), (5663,6285)
X(8144) = midpoint of X(6285) and X(7352)
X(8144) = reflection of X(8141) in X(26)
X(8144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1717,1836), (33,1062,5), (3100,6198,3)
See Third Lemoine Circle at MathWorld.
X(8145) lies on this line: {5,6054}
See MacBeath Circle at MathWorld.
X(8146) lies on these lines: (4,2917), (5,5961)
X(8147) lies on these lines: (6,6244), (991,999), (4648,7956)
See Moses-Longuet-Higgins Circle at MathWorld.
X(8148) lies on these lines: (1,3), (4,3621), (5,3617), (8,381), (10,3656), (20,1483), (30,145), (72,3531), (355,3625), (376,3623), (382,952), (399,7978), (474,5330), (515,5073), (519,3830), (549,3622), (550,7967), (944,1657), (946,3626), (1125,3654), (1317,4299), (1387,1788), (1389,7489), (1484,6943), (1656,5603), (1699,4816), (3241,3534), (3526,5550), (3545,4678), (3616,5054), (3634,5070), (3635,3655), (3811,4930), (4323,5719), (4420,5730), (4663,5093), (5072,5818), (5780,5806), (6199,7969), (6395,7968), (6982,7317)
X(8148) = reflection of X(i) in X(j) for these (i,j): (3,1482), (20,1483), (355,4301), (382,962), (399,7978), (1482,7982), (1657,944), (3534,3241)
X(8148) = Stammler-circle-inverse of X(36)
X(8148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3245,5204), (1,5128,5126), (1,5221,999), (1, 3245, 5204), (1, 5128, 5126), (1, 5221, 999), (1, 5903, 5221), (946, 5790, 3851), (2098, 5221, 1), (2098, 5903, 999), (2099, 5697, 3295), (5603, 5690, 1656), (5657, 5734, 5901), (5657, 5901, 3526)
X(8149) lies on these lines: (2,39), (3,736), (5,698), (32,5149), (114,3095), (182,732), (511,7759), (626,3094), (1692,7805), (1916,7752), (2023,7862), (2782,7781), (2909,3506), (5969,7775)
X(8149) = midpoint of X(76) and X(6309)
X(8149) = reflection of X(3095) in X(7764)
X(8149) = anticomplement of X(32189)
X(8149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,3934,7834), (39,7874,7786), (194,7763,39)
X(8150) lies on these lines: (2,32), (182,732), (1691,3934), (5038,7805), (7709,7781)
X(8150) = midpoint of X(83) and X(6308)
X(8150) = anticomplement of X(32190)
See Second Steiner Circle at MathWorld.
X(8151) lies on the cubic K462 and these lines: (5,523), (20,1499), (155,525), (512,1216), (669,2937), (1649,3526), (1656,8029), (1658,5926), (2501,7505), (5067,5466)
X(8152) lies on these lines: (39,6234), (185,575)
As a point of the Euler line, X(8153) has Shinagawa coefficients (E-7*F-3*r^2-4*R*r, -E+11*F+5*r^2+8*R*r)
See Euler-Gergonne-Soddy Circle at MathWorld.
X(8153) lies on these lines: (2,3), (514,8142)
X(8153) = midpoint of X(20) and X(1375)
X(8154) lies on this line: {1157,8157}
See Lucas Central Circle at MathWorld.
Contributed by Randy Hutson, September 26, 2015.
Barycentrics for X(8157) were found by Francisco Javier García Capitán (Hyacinthos #15827, November 19, 2007) and are included in Cosmin Pohoata's article, On the Parry reflection point; after a correction of removing (b + c)^2 from f16,a(b,c).
See X(399).
The two harmonic traces of the Yiu circles are X(8157) and X(1157). (Randy Hutson, September 26, 2015)
X(8157) lies on these lines: {4,137}, {1157,8154}
Let (Aem) denote the Apollonian circle of the external mixtilinear circles. The touch point of (Aem) and the A-mixtilinear external circle has 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) See X(6767).
The vertices of the 9th mixtilinear triangle are the touchpoints of the outer 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*c*(a - b + c) (Dan Reznik and Peter Moses, September 10, 2021).
If you have The Geometer's Sketchpad, you can view X(8158) and X(8159).
X(8158) lies on these lines: {1,3}, {145,7580}, {952,6764}, {954,4323}, {956,962}, {958,4301}, {971,6762}, {2551,7956}, {2818,7959}, {2943,4650}, {3623,7411}, {5055,7958}, {5073,5841}, {5603,5763}, {5690,6918}, {5758,6913}, {5844,6985}
X(8158) = midpoint of X(1) and X(6766)
X(8158) = reflection of X(6769) in X(1385)
X(8158) = X(3)-of-9th-mixtilinear-triangle
X(8158) = mixtilinear-incentral-triangle-to-mixtilinear-excentral-triangle similarity image of X(3)
X(8158) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1482,6767), (40,999,3), (56,6244,3), (56,7991,6244), (3295,3428,3), (3428,7982,3295), (8165,8166,5)
Contributed by Peter Moses, October 4, 2015. See X(8147). If you have The Geometer's Sketchpad, you can view X(8158) and X(8159).
X(8159) lies on these lines: {1,8147}, {220,6244}
Properties of the two circles are given by Peter Moses, November 12, 2015:
outer Montesdeoca-Lemoine circle: radius = (1 + 2 cos ω)R/2; A-power = - b2c2(sin ω)/(2S)
inner Montesdeoca-Lemoine circle: radius = (-1 + 2 cos ω)R/2; A-power = b2c2(sin ω)/(2S)
The points X(8160) and X(8161) are labeled Z1 and Z2, respectively, in this sketch: X(8160) and X(8161).
X(8160) lies on these lines: {3,6}, {30,5404}, {35,3238}, {36,3237}, {140,5403}, {1676,6683}, {2546,7786}
X(8160) = midpoint of X(3) and X(1670)
X(8160) = reflection of X(8161) in X(3)
X(8160) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1343,5092), (39,5092,8161)
X(8161) lies on these lines: {3,6}, {30,5403}, {35,3237}, {36,3238}, {140,5404}, {1677,6683}, {2547,7786}
X(8161) = midpoint of X(3) and X(1671)
X(8161) = reflection of X(8160) in X(3)
X(8161) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1342,5092), (39,5092,8160)
Similicenters of pairs of circles: X(8162)-X(8171)
This section was contributed by Peter Moses, October 5, 2015.
Abbreviations:
AIMC = Apollonian circle of the internal mixtilinear circles
AEMC = Apollonian circle of the external mixtilinear circles
internal = insimilicenter (internal center of similtude)
external = exsimilicenter (external center of similtude)
| circle | center | radius | A-power |
|---|---|---|---|
| AIMC | X(6767) | 3rR/(r + 4R) | 8ab2c2 (a-b-c) / [(a+b+c)2(a2+b2+c2-2bc-2ca-2ab)] |
| AEMC | X(8158) | R/(4R - 3r)/r | 8ab2c2 / [(-a+b-c) (a+b-c) (-a+b+c)] |
| circles | internal | external |
|---|---|---|
| AIMC, incircle | X(8162) | X(55) |
| AEMC, incircle | X(8163) | X(56) |
| AIMC, circumcircle | X(55) | X(999) |
| AEMC, circumcircle | X(6244) | X(56) |
| AIMC, nine-point circle | X(8164) | X(5274) |
| AEMC, nine-point circle | X(8165) | X(8166) |
| AIMC, Spieker | X(8167) | X(8168) |
| AEMC, Spieker | X(8169) | X(8170) |
| AIMC, AEMC | X(8171) | X(7962) |
X(8162) lies on these lines: {1,3}, {11,5071}, {12,5068}, {145,5284}, {390,5434}, {405,3635}, {480,3872}, {495,5066}, {497,3839}, {519,4423}, {551,4413}, {958,3623}, {1001,3241}, {1056,3058}, {1058,3855}, {1191,2334}, {1358,3672}, {1479,3861}, {2170,3196}, {3086,5326}, {3622,3913}, {3636,5687}, {3742,3895}, {3880,4666}, {3957,5289}, {5281,5298}
X(8163) lies on these lines: {1,3}, {958,4345}
X(8163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8158,56), (3057,3304,55), (7962,7991,3057)
X(8164) lies on these lines: {1,3090}, {2,495}, {3,5261}, {4,12}, {5,1058}, {8,6856}, {10,3487}, {11,5071}, {30,5281}, {35,3529}, {36,388}, {40,3947}, {56,3525}, {116,4648}, {119,6939}, {140,3600}, {145,6933}, {153,6974}, {226,2093}, {355,5703}, {376,1478}, {381,390}, {392,5748}, {442,7080}, {443,5552}, {459,1148}, {496,5056}, {497,3545}, {515,5726}, {517,5226}, {612,1870}, {942,3681}, {944,6956}, {952,6859}, {954,6843}, {1000,1512}, {1060,5297}, {1479,3855}, {1698,3296}, {1737,3475}, {2476,5082}, {2550,3822}, {3086,5067}, {3091,3295}, {3333,3634}, {3436,6857}, {3488,5587}, {3524,4293}, {3526,5265}, {3528,7354}, {3533,7288}, {3616,5176}, {3622,6931}, {3628,7373}, {3746,5225}, {3828,5542}, {3871,6871}, {4295,5183}, {5045,5704}, {5084,5284}, {5177,5687}, {5252,6879}, {5290,6684}, {5719,5790}, {5791,5815}, {5901,6981}, {6198,6622}, {6623,7071}, {6881,7679}
X(8164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,495,1056), (5,6767,5274), (12,3085,4), (35,5229,3529), (40,3947,5714), (388,498,631), (497,7951,3545), (1478,3584,5218), (1478,5218,376), (4293,5432,3524), (5274,6767,1058)
X(8165) lies on these lines: {1,5328}, {2,12}, {4,3820}, {5,8158}, {8,3452}, {9,5128}, {10,962}, {119,6908}, {144,1788}, {346,2899}, {390,2478}, {452,5281}, {960,3617}, {1210,5815}, {1376,3146}, {1698,4208}, {2550,3832}, {2886,5068}, {3085,5129}, {3086,5288}, {3421,4187}, {3523,5267}, {3616,5795}, {3621,5289}, {3634,5234}, {3740,6870}, {3814,5056}, {4313,6745}, {4413,5229}, {5044,5818}, {5057,5123}, {5080,6904}, {5686,6734}, {5690,6973}, {5731,6700}, {5812,6843}
X(8165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3436,3600), (5,8158,8166), (8,6919,5274), (452,5552,5281), (1329,2551,2), (2478,7080,390)
X(8166) lies on these lines: {2,6244}, {4,11}, {5,8158}, {7,1538}, {9,3817}, {392,5806}, {517,5328}, {631,5259}, {946,1698}, {956,3091}, {1000,1512}, {1056,1532}, {1058,6848}, {1699,3474}, {3421,6945}, {5057,5744}, {5071,7680}, {5082,6953}, {5128,6956}, {5183,6879}, {5231,5817}
X(8166) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,8158,8165), (1512,5603,1000)
X(8167) lies on these lines: {1,3697}, {2,11}, {3,3817}, {6,4038}, {9,3742}, {10,6767}, {36,405}, {37,5272}, {45,982}, {56,5047}, {57,3848}, {140,7956}, {210,4666}, {354,3305}, {474,5010}, {518,7308}, {748,940}, {958,999}, {960,3646}, {1279,5268}, {1698,3913}, {2093,3812}, {3219,4860}, {3246,5269}, {3295,3634}, {3304,5260}, {3306,3683}, {3333,5302}, {3711,3957}, {3715,3873}, {3720,3789}, {3731,5573}, {3840,5737}, {3895,4731}, {4011,4363}, {4359,4387}, {4640,5437}, {4679,5249}, {4682,7290}, {4703,7232}, {4906,7174}, {5183,5250}, {5275,5332}, {6691,6857}, {6836,7958}, {6989,7681}, {7580,7988}
X(8167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,497,3826), (2,1001,1376), (2,1621,4413), (2,4423,1001), (2,5284,55), (10,6767,8168), (55,4423,5284), (55,5284,1001), (354,3305,5220), (1001,1376,4428), (1001,4421,1621), (1621,4413,4421), (4413,4421,1376), (5047,5550,56)
X(8168) lies on these lines: {1,3848}, {3,3625}, {8,21}, {9,4711}, {10,6767}, {35,4816}, {36,3632}, {56,3621}, {78,3893}, {200,3880}, {210,3895}, {220,4050}, {346,3039}, {405,4668}, {474,3633}, {480,3036}, {518,2093}, {519,999}, {528,3421}, {956,4421}, {960,2136}, {1001,3679}, {1329,5274}, {1616,6048}, {1697,4662}, {2098,4420}, {2099,3935}, {2802,3940}, {3158,4915}, {3241,4413}, {3295,3626}, {3303,3617}, {3689,3872}, {3711,3877}, {3813,7080}, {4060,4254}, {4387,4723}, {4428,4669}, {4666,4731}, {4737,5695}, {4746,5248}, {4863,6735}, {5119,5220}, {5836,6765}
X(8168) = reflection of X(5289) in X(200)
X(8168) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,3913,958), (10,6767,8167), (2136,4882,960)
X(8169) lies on these lines: {2,12}, {10,8158}, {55,5328}, {200,3880}, {516,1376}, {960,1706}, {3474,4413}
X(8169) = {X(10),X(8158)}-harmonic conjugate of X(8170)
X(8170) lies on these lines: {8,56}, {9,5836}, {10,8158}, {958,6244}, {2886,5068}, {6600,6738}
X(8170) = {X(10),X(8158)}-harmonic conjugate of X(8169)
X(8171) lies on these lines: {1,3}, {4428,5542}, {5284,5748}
X(8172) = circumcircle-inverse of X(37848)
X(8173) = circumcircle-inverse of X(37850)
X(8174) = anticomplement of X(33496)
X(8174) = antigonal conjugate of X(38943)
Barycentrics Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (a^14-4 a^12 b^2+6 a^10 b^4-5 a^8 b^6+5 a^6 b^8-6 a^4 b^10+4 a^2 b^12-b^14-4 a^12 c^2+9 a^10 b^2 c^2-6 a^8 b^4 c^2+7 a^4 b^8 c^2-11 a^2 b^10 c^2+5 b^12 c^2+6 a^10 c^4-6 a^8 b^2 c^4+2 a^6 b^4 c^4-a^4 b^6 c^4+8 a^2 b^8 c^4-9 b^10 c^4-5 a^8 c^6-a^4 b^4 c^6-2 a^2 b^6 c^6+5 b^8 c^6+5 a^6 c^8+7 a^4 b^2 c^8+8 a^2 b^4 c^8+5 b^6 c^8-6 a^4 c^10-11 a^2 b^2 c^10-9 b^4 c^10+4 a^2 c^12+5 b^2 c^12-c^14)-2 (3 a^16-19 a^14 b^2+50 a^12 b^4-69 a^10 b^6+50 a^8 b^8-13 a^6 b^10-6 a^4 b^12+5 a^2 b^14-b^16-19 a^14 c^2+69 a^12 b^2 c^2-85 a^10 b^4 c^2+29 a^8 b^6 c^2+10 a^6 b^8 c^2+10 a^4 b^10 c^2-22 a^2 b^12 c^2+8 b^14 c^2+50 a^12 c^4-85 a^10 b^2 c^4+28 a^8 b^4 c^4+3 a^6 b^6 c^4-4 a^4 b^8 c^4+36 a^2 b^10 c^4-28 b^12 c^4-69 a^10 c^6+29 a^8 b^2 c^6+3 a^6 b^4 c^6-19 a^2 b^8 c^6+56 b^10 c^6+50 a^8 c^8+10 a^6 b^2 c^8-4 a^4 b^4 c^8-19 a^2 b^6 c^8-70 b^8 c^8-13 a^6 c^10+10 a^4 b^2 c^10+36 a^2 b^4 c^10+56 b^6 c^10-6 a^4 c^12-22 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16) S : :
X(8175) is the perspector of triangles 4 and 6 in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane.
X(8175) lies on the Neuberg cubic K001 and these lines: {4,16}, {13,1157}, {15,3480}, {74,8173}, {484,3384}, {1276,7165}, {3440,8172}, {3466,5673}, {3481,8174}, {5685,7326}
X(8175) = anticomplement of X(33497)
X(8175) = antigonal conjugate of X(38944)
Centers associated with Van Lamoen circles: X(8176)-X(8182)
This section was contributed by César Eliud Lozada, October 10, 2015.
Let A′B′C′ be the medial triangle of ABC, and let G = X(2), the centroid of ABC (and of A'B'C'). It is well known that the circumcenters of triangles GBC', GCA', GAB', GCB', GAC', GBA' lie on a circle, called the van Lamoen circle of ABC. See van Lamoen Circle at MathWorld.
More generally, let U be any triangle perspective to ABC with perspector P such that the six circumcenters all lie on a circle. That circle is introduced here as the U-van Lamoen circle.| circle | perspector, P | center |
|---|---|---|
| medial-van Lamoen (the classical van Lamoen circle) | X(2) | X(1153) |
| anticomplementary-van Lamoen | X(2) | X(8176) |
| 1st Brocard-van Lamoen | X(76) | X(8177) |
| 1st anti-Brocard-van Lamoen | X(1916) | X(8178) |
| 2nd Neuberg-van Lamoen | X(262) | X(8179) |
| outer vecten-van Lamoen | X(485) | X(8180) |
| McCay-van Lamoen | X(7607) | X(8181) |
The squared radius of the anticomplementary-van Lamoen circle is sqrt(2)ρ1
The squared radius of the 1st Brocard-van Lamoen circle is 21/2ρ = -1/[64S2ωS4(S2ω + S2)]h(A,B,C)h(B,C,A)h(C,A,B), where h(A,B,C) = (SA + 3Sω)S2 - (3SA + Sω)S2ω
The squared radius of the 1st anti-Brocard-van Lamoen circle is [(S2ω + 9S2)/(5184S6(S2ω + S2)2]j(A,B,C)j(B,C,A)j(C,A,B), where j(A,B,C) = (3SA + 5Sω)S2 + (Sω - SA)S2ω
The squared radius of the 1st anti-Brocard-van Lamoen circle is [(S2ω + 9S2)(81S4 + 14S2S2ω + S4ω)/(576S6(5S2ω - 27S2)4]k(A,B,C)k(B,C,A)k(C,A,B), where k(A,B,C) = 3S2(9SA - Sω) + (Sω - SA)S2ω
X(8176) = midpoint of X(i) and X(j) for these {i,j}: {4,7618}, {7617,7775}
X(8176) = reflection of X(i) in X(j) for these (i,j): (3,7619), (5569,2), (7617,5), (8182,1153)
X(8176) = complement of X(8182)
X(8176) = anticomplement of X(1153)
X(8176) = {X(2),X(8182)}-harmonic conjugate of X(1153)
Definition: Suppose that T = TaTbTc is a triangle in the plane of ABC that is perspective to ABC. Let P be the perspector. If the circumcenters of PBTc, PCTa, PATb, PCTb, PATc, PBTa lie on a circle, that circle is the T-van Lamoen circle.
If you have The Geometer's Sketchpad, you can view X(8181).
X(8183) = perspector of conic {{A,B,C,PU(124)}}
Barycentrics (SB + SC -2 S) (SA (SW - S) - S (3 SW - 5 S)) : :
X(8184) lies on these lines: {5,6119}, {486,489}, {615,642}
Centers associated with homothetic pairs of triangles: X(8185)-X(8253)
This section was contributed by César Eliud Lozada, October 13, 2015.
| Triangle U | The appearance of (V,i) in this column means that triangles U and V are homothetic with center X(i) |
|---|---|
| Aquila defined at X(5586) | (ABC, 1), (anticomplementary, 10), (Ara, 8185), (1st Auriga, 8186), (2nd Auriga, 8187), (Caelum, 1), (Euler, 1699), (outer-Garcia, 3679), (inner-Grebe, 5589), (outer-Grebe, 5588), (Johnson, 5587), (Lucas homothetic, 8188), (Lucas(-1) homothetic, 8189), (medial, 1698) |
| Ara defined at X(5594) | (ABC, 25), (anticomplementary, 22), (Aquila, 8185), (1st Auriga, 8190), (2nd Auriga, 8191), (Caelum, 8192), (Euler, 1598), (outer-Garcia, 8193), (inner-Grebe, 5595), (outer-Grebe, 5594), (Johnson, 3), (Lucas homothetic, 8194), (Lucas(-1) homothetic, 8195), (medial, 3) |
| 1st Auriga defined at X(5597) | (ABC, 5597), (anticomplementary, 5601), (Aquila, 8186), (Ara, 8190), (2nd Auriga, 55), (Caelum, 5598), (Euler, 8196), (outer-Garcia, 8197), (inner-Grebe, 8198), (outer-Grebe, 8199), (Johnson, 8200), (Lucas homothetic, 8201), (Lucas(-1) homothetic, 8202), (medial, 5599) |
| 2nd Auriga defined at X(5597) | (ABC, 5598), (anticomplementary, 5602), (Aquila, 8187), (Ara, 8191), (1st Auriga, 55), (Caelum, 5597), (Euler, 8203), (outer-Garcia, 8204), (inner-Grebe, 8205), (outer-Grebe, 8206), (Johnson, 8207), (Lucas homothetic, 8208), (Lucas(-1) homothetic, 8209), (medial, 5600) |
| Caelum (*) defined at X(5603) | (ABC, 1), (anticomplementary, 145), (Aquila, 1), (Ara, 8192), (1st Auriga, 5598), (2nd Auriga, 5597), (Euler, 5603), (outer-Garcia, 519), (inner-Grebe, 5605), (outer-Grebe, 5604), (Johnson, 952), (Lucas homothetic, 8210), (Lucas(-1) homothetic, 8211), (medial, 8) |
| outer-Garcia defined at X(5587) | (ABC, 10), (anticomplementary, 8), (Aquila, 3679), (Ara, 8193), (1st Auriga, 8197), (2nd Auriga, 8204), (Caelum, 519), (Euler, 5587), (inner-Grebe, 5689), (outer-Grebe, 5688), (Johnson, 517), (Lucas homothetic, 8214), (Lucas(-1) homothetic, 8215), (medial, 1) |
| Lucas homothetic defined at X(493) | (ABC, 493), (anticomplementary, 6462), (Aquila, 8188), (Ara, 8194), (1st Auriga, 8201), (2nd Auriga, 8208), (Caelum, 8210), (Euler, 8212), (outer-Garcia, 8214), (inner-Grebe, 8216), (outer-Grebe, 8218), (Johnson, 8220), (Lucas(-1) homothetic, 6461), (medial, 8222) |
| Lucas(-1) homothetic defined at X(493) | (ABC, 494), (anticomplementary, 6463), (Aquila, 8189), (Ara, 8195), (1st Auriga, 8202), (2nd Auriga, 8209), (Caelum, 8211), (Euler, 8213), (outer-Garcia, 8215), (inner-Grebe, 8217), (outer-Grebe, 8219), (Johnson, 8221), (Lucas homothetic, 6461), (medial, 8223) |
| 3rd Euler defined at X(3758) | (1st circumperp, 2), (2nd circumperp, 4), (4th Euler, 5), (excentral, 1699), (2nd extouch, 8226), (hexyl, 8227), (Honsberger, 7678), (Hutson-intouch, 12), (intouch, 11), (6th mixtilinear, 7988), (2nd Pamfilos-Zhou, 8228), (1st Sharygin, 8229), (tangential-midarc, 8085), (2nd tangential-midarc, 8086) |
| 4th Euler defined at X(3758) | (1st circumperp, 4), (2nd circumperp, 2), (3rd Euler, 5), (excentral, 1698), (2nd extouch, 442), (hexyl, 5587), (Honsberger, 7679), (Hutson-intouch, 11), (intouch, 12), (6th mixtilinear, 7989), (2nd Pamfilos-Zhou, 8230), (1st Sharygin, 5051), (tangential-midarc, 8087), (2nd tangential-midarc, 8088) |
| 2nd extouch defined at X(5927) | (1st circumperp, 7580), (2nd circumperp, 405), (3rd Euler, 8226), (4th Euler, 442), (excentral, 9), (hexyl, 1490), (Honsberger, 8232), (Hutson-intouch, 950), (intouch, 226), (6th mixtilinear, 1750), (2nd Pamfilos-Zhou, 8233), (1st Sharygin, 4199), (tangential-midarc, 8079), (2nd tangential-midarc, 8080) |
| Honsberger defined at X(7670) | (1st circumperp, 7676), (2nd circumperp, 7677), (3rd Euler, 7678), (4th Euler, 7679), (excentral, 1445), (2nd extouch, 8232), (hexyl, 7675), (Hutson-intouch, 8236), (intouch, 7), (6th mixtilinear, 4326), (2nd Pamfilos-Zhou, 8237), (1st Sharygin, 8238), (tangential-midarc, pending), (2nd tangential-midarc, pending) |
| Hutson-intouch defined at X(5731) | (1st circumperp, 56), (2nd circumperp, 55), (3rd Euler, 12), (4th Euler, 11), (excentral, 1697), (2nd extouch, 950), (hexyl, 1), (Honsberger, 8236), (intouch, 1), (6th mixtilinear, 1), (2nd Pamfilos-Zhou, 8239), (1st Sharygin, 8240), (tangential-midarc, 8241), (2nd tangential-midarc, 8242) |
| 6th mixtilinear defined at X(7955) | (1st circumperp, 165), (2nd circumperp, 7987), (3rd Euler, 7988), (4th Euler, 7989), (excentral, 165), (2nd extouch, 1750), (hexyl, 1), (Honsberger, 4326), (Hutson-intouch, 1), (intouch, 1), (2nd Pamfilos-Zhou, 8244), (1st Sharygin, 8245), (tangential-midarc, 8089), (2nd tangential-midarc, 8090) |
| 2nd Pamfilos-Zhou defined at X(7594) | (1st circumperp, 8224), (2nd circumperp, 8225), (3rd Euler, 8228), (4th Euler, 8230), (excentral, 8231), (2nd extouch, 8233), (hexyl, 8234), (Honsberger, 8237), (Hutson-intouch, 8239), (intouch, 8243), (6th mixtilinear, 8244), (1st Sharygin, 8246), (tangential-midarc, 8247), (2nd tangential-midarc, 8248) |
| 1st Sharygin defined at X(8229) | (1st circumperp, 4220), (2nd circumperp, 21), (3rd Euler, 8229), (4th Euler, 5051), (excentral, 846), (2nd extouch, 4199), (hexyl, 8235), (Honsberger, 8238), (Hutson-intouch, 8240), (intouch, 1284), (6th mixtilinear, 8245), (2nd Pamfilos-Zhou, 8246), (tangential-midarc, 8249), (2nd tangential-midarc, 8250) |
| 2nd tangential-midarc defined at X(8075) | (1st circumperp, 8076), (2nd circumperp, 7588), (3rd Euler, 8086), (4th Euler, 8088), (excentral, 258), (2nd extouch, 8080), (hexyl, 8082), (Honsberger, pending), (Hutson-intouch, 8242), (intouch, 174), (6th mixtilinear, 8090), (2nd Pamfilos-Zhou, 8248), (1st Sharygin, 8250), (tangential-midarc, 1) |
| 2nd Euler defined at X(3758) | (circumorthic, 2), (extangents, 8251), (intangents, 1062), (orthic, 5), (tangential, 3), (Trinh, 3) |
| Trinh defined at X(7688) | (circumorthic, 3520), (2nd Euler, 3), (extangents, 7688), (intangents, 36), (orthic, 378), (tangential, 3) |
Note: the Caelum triangle is also called the 5th mixtilinear triangle; see the preamble to X(7955)-X(7999).
X(8185) lies on these lines: {1,25}, {3,1698}, {8,23}, {10,22}, {24,515}, {26,355}, {28,1478}, {35,197}, {40,7387}, {46,3220}, {159,3751}, {498,4224}, {517,7517}, {944,3518}, {958,2915}, {1125,1995}, {1385,7506}, {1473,3336}, {1479,4222}, {1486,3746}, {1598,1699}, {1722,5345}, {1724,5329}, {2172,2333}, {2933,7428}, {2937,5790}, {3556,5903}, {3576,6642}, {3583,4186}, {3585,4185}, {3624,5020}, {3634,7485}, {3679,8193}, {5230,5358}, {5264,7295}, {5588,5594}, {5589,5595}, {5818,7512}, {7395,7989}, {7529,8227}, {8186,8190}, {8187,8191}, {8188,8194}, {8189,8195}
X(8186) lies on these lines: {1,3}, {10,5601}, {519,5602}, {1698,5599}, {1699,8196}, {3632,8204}, {3679,5600}, {5587,8200}, {5588,8199}, {5589,8198}, {5881,8207},8176,8190}, {8188,8201}, {8189,8202}
X(8187) lies on these lines: {1,3}, {10,5602}, {519,5601}, {1698,5600}, {1699,8203}, {3632,8197}, {3679,5599}, {5587,8207}, {5588,8206}, {5589,8205}, {5881,8200},8176,8191}, {8188,8208}, {8189,8209}
X(8188) lies on these lines: {1,493}, {10,6462}, {1698,8222}, {1699,8212}, {3679,8214}, {5587,8220}, {5588,8218}, {5589,8216}, {6461,8189},8176,8194}, {8186,8201}, {8187,8208}
X(8189) lies on these lines: {1,494}, {10,6463}, {1698,8223}, {1699,8213}, {3679,8215}, {5587,8221}, {5588,8219}, {5589,8217}, {6461,8188},8176,8195}, {8186,8202}, {8187,8209}
X(8190) lies on these lines: {3,5599}, {22,5601}, {25,5597}, {55,8191}, {1598,8196}, {5594,8199}, {5595,8198}, {5598,8192},8176,8186}, {8193,8197}, {8194,8201}, {8195,8202}
X(8191) lies on these lines: {3,5600}, {22,5602}, {25,5598}, {55,8190}, {1598,8203}, {5594,8206}, {5595,8205}, {5597,8192},8176,8187}, {8193,8204}, {8194,8208}, {8195,8209}
X(8192) lies on these lines: {1,25}, {3,8}, {10,7484}, {22,145}, {23,3623}, {24,7967}, {26,1483}, {28,1056}, {40,1473}, {48,607}, {56,197}, {159,3242}, {198,1108}, {355,7395}, {388,4185}, {497,4186}, {515,1593}, {519,8193}, {946,5198}, {999,5262}, {1058,4222}, {1324,8071}, {1460,1468}, {1470,2933}, {1478,4214}, {1482,7387}, {1486,3303}, {1598,5603}, {1610,3476}, {1697,3220}, {1995,3622}, {3057,3556}, {3295,7291}, {3515,5882}, {3616,5020}, {3617,7485}, {3621,6636}, {3915,7083}, {5285,6762}, {5594,5604}, {5595,5605}, {5597,8191}, {5598,8190}, {5790,7393}, {5901,7529}, {8194,8210}, {8195,8211}
X(8192) = homothetic center of tangential triangle and reflection of orthic triangle in X(1)
X(8193) lies on these lines: {1,3}, {8,22}, {10,25}, {23,3617}, {24,5657}, {26,5690}, {28,2550}, {71,1973}, {72,3556}, {145,6636}, {159,3416}, {197,2915}, {198,3694}, {219,2172}, {355,7387}, {378,6361}, {386,1036}, {405,1486}, {516,1593}, {519,8192}, {607,4456}, {946,7395}, {962,7503}, {1037,4306}, {1125,7484}, {1191,5096}, {1398,4347}, {1598,5587}, {1698,5020}, {1724,7083}, {2187,3682}, {2551,4222}, {2933,3435}, {3074,6210}, {3085,4220}, {3516,5493}, {3616,7485}, {3621,7492}, {3679,8185}, {3925,7535}, {4254,5280}, {5120,5299}, {5247,7295}, {5250,5314}, {5594,5688}, {5595,5689}, {5603,7509}, {5790,7517}, {5844,7525}, {5886,7393}, {5901,7516}, {8092,8131}, {8190,8197}, {8191,8204}, {8194,8214}, {8195,8215}
X(8193) = reflection of X(i) in X(j) for these (i,j): (5090,10)
X(8194) lies on these lines: {3,8220}, {22,6462}, {25,371}, {1598,8212}, {3796,6461}, {5594,8218}, {5595,8216},8176,8188}, {8190,8201}, {8191,8208}, {8192,8210}, {8193,8214}
X(8195) lies on these lines: {3,8221}, {22,6463}, {25,372}, {1598,8213}, {3796,6461}, {5594,8219}, {5595,8217},8176,8189}, {8190,8202}, {8191,8209}, {8192,8211}, {8193,8215}
X(8196) lies on these lines: {4,5597}, {5,5599}, {55,946}, {381,8200}, {517,5600}, {1482,8207}, {1598,8190}, {1699,8186}, {3091,5601}, {5587,8197}, {5598,5603}, {6201,8199}, {6202,8198}, {7982,8204}, {8201,8212}, {8202,8213}
X(8196) = {X(55),X(946)}-harmonic conjugate of X(8203)
X(8197) lies on these lines: {1,5599}, {8,21}, {10,5597}, {517,8200}, {519,5598}, {3632,8187}, {3679,5600}, {5587,8196}, {5688,8199}, {5689,8198}, {7982,8203}, {8190,8193}, {8201,8214}, {8202,8215}
X(8197) = {X(8),X(55)}-harmonic conjugate of X(8204)
X(8198) lies on these lines: {6,5597}, {55,3641}, {1271,5601}, {5589,8186}, {5591,5599}, {5595,8190}, {5598,5605}, {5689,8197}, {6202,8196}, {6215,8200}, {8201,8216}, {8202,8217}
X(8198) = {X(55),X(3641)}-harmonic conjugate of X(8205)
X(8199) lies on these lines: {6,5597}, {55,3640}, {1270,5601}, {5588,8186}, {5590,5599}, {5594,8190}, {5598,5604}, {5688,8197}, {6201,8196}, {6214,8200}, {8201,8218}, {8202,8219}
X(8199) = {X(55),X(3640)}-harmonic conjugate of X(8206)
X(8200) lies on these lines: {3,5599}, {4,5601}, {5,5597}, {55,355}, {381,8196}, {517,8197}, {952,5598}, {1482,8203}, {5587,8186}, {5600,5790}, {5881,8187}, {6215,8198}, {8201,8220}, {8202,8221}
X(8200) = {X(55),X(355)}-harmonic conjugate of X(8207)
X(8201) lies on these lines: {55,8208}, {493,5597}, {5599,8222}, {5601,6462}, {6461,8202}, {8186,8188}, {8190,8194}, {8196,8212}, {8197,8214}, {8198,8216}, {8199,8218}, {8200,8220}
X(8202) lies on these lines: {55,8209}, {494,5597}, {5598,8211}, {5599,8223}, {5601,6463}, {6461,8201}, {8186,8189}, {8190,8195}, {8196,8213}, {8197,8215}, {8198,8217}, {8199,8219}, {8200,8221}
X(8203) lies on these lines: {4,5598}, {5,5600}, {55,946}, {381,8207}, {517,5599}, {1482,8200}, {1598,8191}, {1699,8187}, {3091,5602}, {5587,8204}, {5597,5603}, {6201,8206}, {6202,8205}, {7982,8197}, {8208,8212}, {8209,8213}
X(8203) = {X(55),X(946)}-harmonic conjugate of X(8196)
X(8204) lies on these lines: {1,5600}, {8,21}, {10,5598}, {517,8207}, {519,5597}, {3632,8186}, {3679,5599}, {5587,8203}, {5688,8206}, {5689,8205}, {7982,8196}, {8191,8193}, {8208,8214}, {8209,8215}
X(8204) = {X(8),X(55)}-harmonic conjugate of X(8197)
X(8205) lies on these lines: {6,5598}, {55,3641}, {1271,5602}, {5589,8187}, {5591,5600}, {5595,8191}, {5597,5605}, {5689,8204}, {6202,8203}, {6215,8207}, {8208,8216}, {8209,8217}
X(8205) = {X(55),X(3641)}-harmonic conjugate of X(8198)
X(8206) lies on these lines: {6,5598}, {55,3640}, {1270,5602}, {5588,8187}, {5590,5600}, {5594,8191}, {5597,5604}, {5688,8204}, {6201,8203}, {6214,8207}, {8208,8218}, {8209,8219}
X(8206) = {X(55),X(3640)}-harmonic conjugate of X(8199)
X(8207) lies on these lines: {3,5600}, {4,5602}, {5,5598}, {55,355}, {381,8203}, {517,8204}, {952,5597}, {1482,8196}, {5587,8187}, {5599,5790}, {5881,8186}, {6214,8206}, {6215,8205}, {8208,8220}, {8209,8221}
X(8207) = {X(55),X(355)}-harmonic conjugate of X(8200)
X(8208) lies on these lines: {55,8201}, {493,5598}, {5597,8210}, {5600,8222}, {5602,6462}, {6461,8209}, {8187,8188}, {8191,8194}, {8203,8212}, {8204,8214}, {8205,8216}, {8206,8218}, {8207,8220}
X(8209) lies on these lines: {55,8202}, {494,5598}, {5597,8211}, {5600,8223}, {5602,6463}, {6461,8208}, {8187,8189}, {8191,8195}, {8203,8213}, {8204,8215}, {8205,8217}, {8206,8219}, {8207,8221}
X(8210) lies on these lines: {1,493}, {8,8222}, {145,6462}, {519,8214}, {952,8220}, {5597,8208}, {5598,8201}, {5603,8212}, {5604,8218}, {5605,8216}, {6461,8211}, {8192,8194}
X(8211) lies on these lines: {1,494}, {8,8223}, {145,6463}, {519,8215}, {952,8221}, {5597,8209}, {5598,8202}, {5603,8213}, {5604,8219}, {5605,8217}, {6461,8210}, {8192,8195}
X(8212) lies on these lines: {4,493}, {5,8222}, {381,8220}, {1598,8194}, {1699,8188}, {3091,6462}, {5587,8214}, {5603,8210}, {6201,8218}, {6202,8216}, {6461,8213}, {8196,8201}, {8203,8208}
X(8213) lies on these lines: {4,494}, {5,8223}, {381,8221}, {1598,8195}, {1699,8189}, {3091,6463}, {5587,8215}, {5603,8211}, {6201,8219}, {6202,8217}, {6461,8212}, {8196,8202}, {8203,8209}
X(8214) lies on these lines: {1,8222}, {8,6462}, {10,493}, {517,8220}, {519,8210}, {3679,8188}, {5587,8212}, {5688,8218}, {5689,8216}, {6461,8215}, {8193,8194}, {8197,8201}, {8204,8208}
X(8215) lies on these lines: {1,8223}, {8,6463}, {10,494}, {517,8221}, {519,8211}, {3679,8189}, {5587,8213}, {5688,8219}, {5689,8217}, {6461,8214}, {8193,8195}, {8197,8202}, {8204,8209}
X(8216) lies on these lines: {6,493}, {1271,2896}, {5589,8188}, {5591,8222}, {5595,8194}, {5605,8210}, {5689,8214}, {6202,8212}, {6215,8220}, {6461,8217}, {8198,8201}, {8205,8208}
X(8217) lies on these lines: {6,494}, {1271,6463}, {5589,8189}, {5591,8223}, {5595,8195}, {5605,8211}, {5689,8215}, {6202,8213}, {6215,8221}, {6461,8216}, {8198,8202}, {8205,8209}
X(8218) lies on these lines: {6,493}, {1270,6462}, {5588,8188}, {5590,8222}, {5594,8194}, {5604,8210}, {5688,8214}, {6201,8212}, {6214,8220}, {6461,8219}, {8199,8201}, {8206,8208}
X(8219) lies on these lines: {6,494}, {1270,2896}, {5588,8189}, {5590,8223}, {5594,8195}, {5604,8211}, {5688,8215}, {6201,8213}, {6214,8221}, {6461,8218}, {8199,8202}, {8206,8209}
X(8220) lies on these lines: {3,8194}, {4,6462}, {5,493}, {381,8212}, {517,8214}, {952,8210}, {1181,6461}, {5587,8188}, {6214,8218}, {6215,8216}, {8200,8201}, {8207,8208}
X(8221) lies on these lines: {3,8195}, {4,6463}, {5,494}, {381,8213}, {517,8215}, {952,8211}, {1181,6461}, {5587,8189}, {6214,8219}, {6215,8217}, {8200,8202}, {8207,8209}
X(8222) lies on these lines: {1,8214}, {2,493}, {3,8194}, {5,8212}, {8,8210}, {69,1589}, {326,1267}, {394,3926}, {487,6465}, {491,6805}, {494,6339}, {1698,8188}, {3785,5407}, {5408,6337}, {5491,6504}, {5590,8218}, {5591,8216}, {5599,8201}, {5600,8208}
X(8222) = {X(394),X(3926)}-harmonic conjugate of X(8223)
X(8223) lies on these lines: {1,8215}, {2,494}, {3,8195}, {5,8213}, {8,8211}, {69,1590}, {326,3084}, {394,3926}, {488,6466}, {492,6806}, {493,6339}, {1698,8189}, {3785,5406}, {5409,6337}, {5490,6504}, {5590,8219}, {5591,8217}, {5599,8202}, {5600,8209}
X(8223) = {X(394),X(3926)}-harmonic conjugate of X(8222)
X(8224) lies on these lines: {2,8228}, {3,142}, {4,8230}, {40,8234}, {55,8243}, {56,8239}, {165,8231}, {1721,6204}, {4220,8246}, {7580,8233}, {7676,8237}, {8075,8247}, {8076,8248}
X(8224) = {X(3),X(7596)}-harmonic conjugate of X(8225)
X(8225) lies on these lines: {1,372}, {2,8230}, {3,142}, {4,8228}, {21,7595}, {55,5405}, {56,482}, {238,371}, {405,8233}, {3576,8234}, {4649,6420}, {6135,7090}, {7588,8248}, {7677,8237}, {8077,8247}
X(8225) = midpoint of X(i),X(j) for these (i,j): (1,6213)
X(8225) = {X(3),X(7596)}-harmonic conjugate of X(8224)
As a point on the Euler line, X(8226) has Shinagawa coefficients: (2R + r, 4R + r)
X(8226) lies on these lines: {2,3}, {9,1699}, {10,7957}, {11,118}, {12,950}, {63,5805}, {72,946}, {165,3826}, {200,3419}, {329,5817}, {355,3870}, {495,3488}, {496,3487}, {497,954}, {516,3683}, {517,3690}, {583,1713}, {952,3957}, {971,5249}, {1125,7958}, {1260,3434}, {1329,7989}, {1490,8227}, {1503,1746}, {1708,1836}, {1709,5880}, {1736,6354}, {1750,3816}, {1861,6708}, {3219,5762}, {3338,4355}, {3574,5777}, {3586,7951}, {3876,5763}, {3929,5735}, {4666,5886}, {4930,5734}, {5226,5809}, {5231,5715}, {5248,6253}, {5274,7678}, {5281,7679}, {5436,5691}, {5439,6245}, {5511,5513}, {5779,5905}, {5806,6734}, {8079,8085}, {8080,8086}, {8228,8233}
X(8226) = midpoint of X(i),X(j) for these (i,j): (4,1006), (3925,7965), (6839,6912)
X(8226) = reflection of X(6881) in X(5)
X(8226) = complement of X(7411)
X(8226) = orthocentroidal-circle-inverse of X(7580)
X(8227) lies on these lines: {1,5}, {2,40}, {3,1699}, {4,1125}, {8,5056}, {9,6832}, {10,3090}, {19,7537}, {20,5550}, {30,7987}, {35,6911}, {36,3560}, {46,6862}, {55,6918}, {56,6913}, {57,499}, {63,6884}, {72,5231}, {78,5178}, {84,5249}, {140,165}, {142,6847}, {226,3086}, {354,5777}, {381,1385}, {388,6939}, {392,7686}, {405,5715}, {411,5284}, {442,7681}, {474,2077}, {497,6864}, {498,1697}, {515,3091}, {516,631}, {517,1656}, {519,5071}, {547,3656}, {551,944}, {580,748}, {581,3720}, {590,1702}, {615,1703}, {908,6886}, {936,2886}, {942,5693}, {950,6843}, {958,5087}, {960,5705}, {988,3944}, {993,6920}, {997,6829}, {999,5290}, {1001,3149}, {1012,3838}, {1056,3947}, {1071,3742}, {1158,3306}, {1203,5707}, {1210,3485}, {1420,1478}, {1449,5816}, {1453,5713}, {1479,3601}, {1482,3679}, {1490,8226}, {1519,5437}, {1537,6667}, {1538,3824}, {1572,7746}, {1621,6796}, {1706,6983}, {1737,3340}, {1770,6892}, {1836,5433}, {1902,5094}, {2257,5747}, {2550,6700}, {2646,3586}, {3085,6964}, {3295,7743}, {3338,3582}, {3359,6958}, {3525,6361}, {3526,3579}, {3533,5493}, {3542,7713}, {3544,3636}, {3577,6933}, {3583,3612}, {3585,6929}, {3617,5734}, {3622,5068}, {3628,7991}, {3632,5079}, {3634,4301}, {3653,3845}, {3655,5066}, {3813,6765}, {3814,6975}, {3816,6831}, {3822,6941}, {3825,6830}, {3832,5731}, {3877,7504}, {3894,6583}, {3901,5694}, {3911,4295}, {3925,6769}, {4187,7680}, {4292,7288}, {4298,5714}, {4299,6930}, {4302,6885}, {4304,5225}, {4311,5229}, {4466,6173}, {4512,7483}, {4652,5057}, {4668,5844}, {4679,5812}, {5010,6924}, {5082,6745}, {5119,6959}, {5248,6905}, {5253,5450}, {5274,5703}, {5316,6766}, {5438,6854}, {5439,6001}, {5535,6852}, {5542,5817}, {5709,6861}, {5805,6675}, {5880,6691}, {5887,5902}, {6256,6957}, {6261,6828}, {6282,7956}, {6856,7682}, {6860,7971}, {6887,7308}, {6914,7280}, {6946,7704}, {6981,7962}, {7529,8185}, {7675,7678}, {7688,8167}, {8081,8085}, {8082,8086}, {8228,8234}, {8229,8235}
X(8227) = midpoint of X(i),X(j) for these (i,j): (3091,3616), (3617,5734)
X(8227) = reflection of X(1698 in X(1655)
X(8227) = X(3091)-of-Fuhrmann-triangle
X(8227) = X(3)-of-cross-triangle of these triangles: Aquila and anti-Aquila
X(8227) = homothetic center of Ae (aka K798e) triangle and cross-triangle of 2nd Fuhrmann and Ae triangles
X(8228) lies on these lines: {2,8224}, {4,8225}, {5,7596}, {11,8243}, {12,8239}, {1699,8231}, {7678,8237}, {8085,8247}, {8086,8248}, {8227,8234}
X(8228) = {X(5),X(7596)}-harmonic conjugate of X(8230)
Let A', B', C' be the feet of the internal bisectors and A", B", C" the feet of the external bisectors. The perpendicular bisectors of AA', BB', CC' bound a triangle DEF called the first Sharygin triangle. The perpendicular bisectors of AA", BB", CC" bound a triangle D'E'F' called the second Sharygin triangle. Reference
Trilinears for the A-vertex of the 1st Sharygin triangle: bc - a2 : ab + c2 : ac + b2
Trilinears for the A-vertex of the 2nd Sharygin triangle: bc - a2 : ab - c2 : ac - b2
(César Lozada, October 22, 2015)
As a point on the Euler line, X(8229) has Shinagawa coefficients (2r(R + r), -E - F)
X(8229) lies on these lines: {1,7683}, {2,3}, {11,1284}, {12,8240}, {114,1281}, {132,243}, {147,2651}, {230,3285}, {511,3909}, {515,3011}, {517,3006}, {614,6261}, {846,1699}, {896,2792}, {946,2292}, {1150,1352}, {2254,3667}, {2783,4442}, {3705,3869}, {3815,4286}, {3817,4425}, {5086,7081}, {5480,5718}, {7678,8238}, {7988,8245}, {8085,8249}, {8086,8250}, {8227,8235}, {8228,8246}
X(8229) = inverse of X(3109) in orthoptic circle of Steiner inellipse
X(8230) lies on these lines: {2,8225}, {4,8224}, {5,7596}, {11,8239}, {12,8243}, {442,8233}, {1698,6212}, {5587,8234}, {7679,8237}, {8087,8247}, {8088,8248}
X(8230) = {X(5),X(7596)}-harmonic conjugate of X(8228)
X(8231) lies on these lines: {1,372}, {2,1766}, {3,8234}, {9,7595}, {40,7596}, {57,8243}, {165,8224}, {258,8248}, {572,3084}, {846,8246}, {1445,8237}, {1697,8239}, {1698,6212}, {1699,8228}, {1743,7347}, {2285,5393}, {5816,6347}, {8078,8247}
X(8232) lies on these lines: {1,5809}, {2,7}, {4,390}, {12,480}, {37,347}, {72,5686}, {77,5308}, {85,344}, {119,6843}, {281,342}, {346,1441}, {348,4687}, {388,452}, {405,3600}, {442,7679}, {498,4312}, {516,3085}, {518,3485}, {651,3945}, {950,8236}, {971,6847}, {1056,6913}, {1125,4321}, {1440,1903}, {1490,5703}, {1750,4326}, {1864,3475}, {3008,4328}, {3086,5542}, {3434,7674}, {3487,5045}, {3579,5714}, {3668,3731}, {4199,8238}, {4308,5436}, {4648,6180}, {5175,5853}, {5222,7190}, {5274,7678}, {5281,7580}, {5729,6832}, {5736,5776}, {5762,6825}, {5779,6824}, {5805,6848}, {5841,6987}, {5843,6862}, {5927,7671}, {6147,6887}, {6907,8164}, {8233,8237}
X(8233) lies on these lines: {4,1123}, {9,7595}, {226,1465}, {405,8225}, {442,8230}, {950,8239}, {1490,8234}, {1750,8244}, {4199,8246}, {7580,8224}, {8079,8247}, {8080,8248}, {8226,8228}, {8232,8237}
X(8234) lies on these lines: {1,7595}, {3,8231}, {4,990}, {40,8224}, {84,2067}, {1490,8233}, {3220,6213}, {3576,8225}, {5587,8230}, {7675,8237}, {8081,8247}, {8082,8248}, {8227,8228}, {8235,8246}
X(8235) lies on these lines: {1,256}, {3,846}, {4,4425}, {21,84}, {40,612}, {1385,3073}, {1490,4199}, {1724,7609}, {3009,4300}, {3430,3743}, {5051,5587}, {5205,6684}, {7675,8238}, {8081,8249}, {8082,8250}, {8227,8229}, {8234,8246}
X(8236) lies on these lines: {1,7}, {2,3158}, {8,344}, {9,145}, {11,7679}, {12,7678}, {37,5838}, {55,5435}, {56,7676}, {65,7673}, {142,3622}, {144,3243}, {329,3957}, {376,5049}, {405,6764}, {497,3748}, {518,1992}, {519,5686}, {673,5308}, {938,3295}, {950,8232}, {952,954}, {971,7967}, {1058,5703}, {1156,1317}, {1279,5222}, {1445,1697}, {1482,5759}, {1483,5779}, {1621,5273}, {2550,3616}, {3057,5572}, {3058,3475}, {3244,5223}, {3476,8162}, {3729,4779}, {3826,5550}, {3973,4924}, {4419,4864}, {5129,6765}, {5274,7988}, {5698,5852}, {5728,5766}, {8237,8239}, {8238,8240}
X(8236) = reflection of X(i) in X(j) for these (i,j): (4349, 4021)
X(8236) = anticomplement of X(38200)
X(8237) lies on these lines: {7,1659}, {390,7596}, {1445,8231}, {2346,7595}, {4326,8244}, {7675,8234}, {7676,8224}, {7677,8225}, {7678,8228}, {7679,8230}, {8232,8233}, {8236,8239}, {8238,8246}
X(8238) lies on these lines: {7,21}, {256,2346}, {846,1445}, {1423,1621}, {2292,7672}, {4199,8232}, {4220,7676}, {4326,8245}, {5051,7679}, {7675,8235}, {7678,8229}, {8236,8240}, {8237,8246}
X(8239) lies on these lines: {1,7595}, {8,7090}, {12,8228}, {55,5405}, {56,8224}, {950,8233}, {1697,8231}, {8236,8237}, {8240,8246}, {8241,8247}, {8242,8248}
X(8240) lies on these lines: {1,256}, {8,21}, {11,5051}, {12,8229}, {43,3601}, {56,4220}, {846,1697}, {855,2292}, {950,3741}, {1104,1193}, {1107,2269}, {1212,2347}, {1281,3023}, {2268,4426}, {2330,5247}, {8236,8238}, {8239,8246}, {8241,8249}, {8242,8250}
X(8241) lies on these lines: {1,167}, {8,188}, {11,8087}, {12,8085}, {55,8077}, {56,8075}, {236,259}, {950,8079}, {1317,8103}, {1697,8078}, {3057,8093}, {7962,8101}, {8239,8247}, {8240,8249}
Let (Oa) be the circle tangent to the incircle and sides CA and AB, such that its center Oa lies between A and X(1). Define (Ob) and (Oc) cyclically. Then X(8241) is the radical center of (Oa), (Ob), (Oc). (Randy Hutson, October 27, 2015)
X(8241) = reflection of X(i) in X(j) for these (i,j): (8,2090), (174,1)
X(8241) = {X(1),X(8422)}-harmonic conjugate of X(8242)
Let (O'a) be the circle tangent to the incircle and sides CA and AB, such that its center O'a lies on the opposite side of X(1) from A. Define (O'b) and (O'c) cyclically. Then X(8242) is the radical center of (O'a), (O'b), (O'c). (Randy Hutson, October 27, 2015)
X(8242) lies on these lines: {1,167}, {8,7028}, {11,8088}, {12,8086}, {55,7588}, {56,8076}, {145,8125}, {179,483}, {188,6732}, {236,3616}, {258,1697}, {400,3082}, {505,1488}, {950,8080}, {1317,8104}, {1482,8129}, {3057,8094}, {3303,7589}, {3304,7587}, {3622,8126}, {5558,7707}, {7962,8102}, {8239,8248}, {8240,8250}
X(8242) = reflection of X(2089) in X(1)
X(8242) = {X(1),X(8422)}-harmonic conjugate of X(8241)
X(8243) lies on these lines: {1,7595}, {7,1659}, {11,8228}, {12,8230}, {55,8224}, {56,482}, {57,8231}, {174,8248}, {176,3622}, {226,1465}, {1284,8246}, {2089,8247}
X(8244) lies on these lines: {1,7595}, {165,8224}, {1699,1721}, {1750,8233}, {3062,7133}, {4326,8237}, {7987,8225}, {7988,8228}, {7989,8230}, {8089,8247}, {8090,8248}, {8245,8246}
X(8245) lies on these lines: {1,256}, {21,3062}, {165,846}, {182,7609}, {572,2112}, {1281,3729}, {1580,1743}, {1654,2784}, {1699,4425}, {2292,7991}, {3098,7611}, {4326,8238}, {8089,8249}, {8090,8250}
X(8246) lies on these lines: {21,7595}, {256,7133}, {846,8231}, {1284,8243}, {8234,8235}, {8237,8238}, {8239,8240}, {8247,8249}, {8248,8250}
X(8247) lies on these lines: {1,8248}, {177,7133}, {2089,8243}, {7596,8091}, {8075,8224}, {8077,8225}, {8078,8231}, {8079,8233}, {8081,8234}, {8085,8228}, {8087,8230}, {8089,8244}, {8239,8241}, {8246,8249}
X(8248) lies on these lines: {1,8247}, {174,8243}, {258,8231}, {7588,8225}, {7596,8092}, {8076,8224}, {8080,8233}, {8082,8234}, {8086,8228}, {8088,8230}, {8090,8244}, {8239,8242}, {8246,8250}
X(8249) lies on these lines: {1,8250}, {21,177}, {846,8078}, {1284,2089}, {2292,8093}, {4199,8079}, {4220,8075}, {5051,8087}, {8081,8235}, {8085,8229}, {8089,8245}, {8240,8241}, {8246,8247}
X(8250) lies on these lines: {1,8249}, {21,7588}, {174,1284}, {258,846}, {2292,8094}, {4199,8080}, {4220,8076}, {5051,8088}, {8082,8235}, {8086,8229}, {8090,8245}, {8240,8242}, {8246,8248}
X(8251) lies on these lines: {1,3}, {2,6197}, {4,3101}, {5,19}, {68,71}, {1158,1503}, {1753,6907}, {1763,5777}, {1766,1901}, {1842,6929}, {1869,6917}, {2550,6643}, {3611,5562}, {5250,7515}, {5886,7561}, {6913,7713}, {7723,7724}
X(8252) lies on these lines: {2,6}, {3,3367}, {4,6410}, {5,1152}, {30,6412}, {53,3536}, {140,486}, {371,3526}, {372,1656}, {381,6396}, {485,3594}, {494,2963}, {498,3297}, {499,3298}, {547,6438}, {549,6411}, {631,3071}, {632,3592}, {641,6119}, {1506,6423}, {1578,3549}, {1579,3548}, {1587,5067}, {1588,3317}, {1698,7968}, {2045,5339}, {2046,5340}, {3070,3090}, {3093,7505}, {3312,5070}, {3535,6748}, {3545,6434}, {3624,7969}, {3830,6452}, {3843,6456}, {3851,6450}, {5054,6200}, {5055,6398}, {5056,6430}, {5071,6469}, {5073,6497}, {5079,6454}, {5094,5413}, {5254,7376}, {5356,6203}, {6421,7746}, {6424,7749}, {6432,7583}, {6470,7582}, {7300,7348}, {7375,7745}
X(8252) = crosspoint of X(2) and X(3317)
X(8252) = crosssum of X(6) and X(3312)
X(8252) = {X(2),X(6)}-harmonic conjugate of X(8253)
X(8253) lies on these lines: {2,6}, {3,3366}, {4,6409}, {5,1151}, {30,6411}, {53,3535}, {140,485}, {371,1656}, {372,3526}, {381,6200}, {486,3592}, {493,2963}, {498,3298}, {499,3297}, {547,6437}, {549,6412}, {631,3070}, {632,3594}, {642,6118}, {1506,6424}, {1578,3548}, {1579,3549}, {1587,3316}, {1588,5067}, {1698,7969}, {2045,5340}, {2046,5339}, {3071,3090}, {3092,7505}, {3311,5070}, {3536,6748}, {3545,6433}, {3624,7968}, {3830,6451}, {3843,6455}, {3851,6449}, {5054,6396}, {5055,6221}, {5056,6429}, {5071,6468}, {5073,6496}, {5079,6453}, {5094,5412}, {5254,7375}, {5356,6204}, {6422,7746}, {6423,7749}, {6431,7584}, {6471,7581}, {7300,7347}, {7376,7745}
X(8253) = crosspoint of X(2) and X(3316)
X(8253) = crosssum of X(6) and X(3311)
X(8253) = {X(2),X(6)}-harmonic conjugate of X(8252)
Midpoints associated with pedal and antipedal triangles
This section was contributed by Peter Moses, October 17, 2015.
In TCCT, pp. 187-188, it is noted that if ABC is a triangle and P is a point not on a sideline of ABC, then the pedal triangle of P is homothetic to the antipedal triangle of the isogonal conjugate, P', of P, and also that the product of the areas of the two triangles is the square of the area of ABC. Moreover, the vertices of the pedal triangles of P and P' are concyclic, and the center of the circle is the midpoint of P and P'. The following table shows examples of P, P', and their midpoint.
| P | P' | midpoint |
|---|---|---|
| X(1) | X(1) | X(1) |
| X(2) | X(6) | X(597) |
| X(3) | X(4) | X(5) |
| X(5) | X(54) | X(8254) |
| X(7) | X(55) | X(8255) |
| X(8) | X(56) | X(8256) |
| X(9) | X(57) | X(8257) |
| X(10) | X(58) | X(8258) |
| X(13) | X(15) | X(396) |
| X(14) | X(16) | X(397) |
| X(17) | X(61) | X(8259) |
| X(18) | X(62) | X(8260) |
| X(20) | X(64) | X(5894) |
| X(21) | X(65) | X(8261) |
| X(23) | X(67) | X(8262) |
| X(25) | X(69) | X(8263) |
| X(36) | X(80) | X(1737) |
| X(40) | X(84) | X(1158) |
Continuing, let P = p : q : r (barycentrics)and let C(P) denote the circle defined above. Then
area(pedal triangle of P) = 2(a^2 q r + b^2 r p + c^2 p q) S^3 / (a^2 b^2 c^2 (p + q + r)^2)
area(pedal triangle of P') = 2 p q r (p + q + r) S^3 / (a^2 q r + b^2 r p + c^2 p q)^2
(radius of C(P)) = sqrt[(-a^2+b^2+c^2) q r+b^2 r^2+c^2 q^2) ((-b^2+c^2+a^2) r p+c^2 p^2+a^2 r^2) ((-c^2+a^2+b^2) p q+a^2 q^2+b^2 p^2)/(4 (p+q+r)^2 (a^2 q r + b^2 r p + c^2 p q)^2]
X(8254) lies on the bianticevian conic of X(2) and X(6) and also on these lines: {2,195}, {5,49}, {30,3574}, {137,5501}, {140,389}, {468,6152}, {539,547}, {549,7691}, {1209,1493}, {1656,2888}, {2914,6143}, {3589,5965}, {5432,6286}, {5433,7356}, {5943,6153}
X(8254) = midpoint of X(i) and X(j) for these {i,j}: {5, 54}, {1209, 1493}
X(8254) = reflection of X(i) and X(j) for these (i,j): (140, 6689), (1209, 3628)
X(8255) lies on these lines: {1,528}, {7,55}, {9,6690}, {11,7671}, {142,2886}, {442,5696}, {495,2801}, {498,5729}, {517,5542}, {527,4640}, {954,8069}, {971,7680}, {1737,3826}, {3035,8257}, {3085,5220}, {3243,5855}, {5698,5703}, {5805,5842}, {8256,8261}
X(8255) = midpoint of X(7) and X(55)
X(8255) = reflection of X(i) and X(j) for these (i,j): (9,6690), (2886,142)
X(8256) lies on these lines: {1,1145}, {2,2098}, {5,10}, {8,56}, {46,529}, {55,5554}, {65,6735}, {78,5855}, {144,1654}, {214,1483}, {318,1846}, {355,1158}, {495,3754}, {496,2802}, {518,4848}, {528,1837}, {958,5657}, {1146,3501}, {1210,3880}, {1388,6921}, {1728,3419}, {1737,3813}, {1776,5086}, {1828,1861}, {2099,5552}, {2841,3042}, {3057,3816}, {3123,4642}, {3486,4421}, {3632,5438}, {3698,3826}, {3919,6147}, {3922,5249}, {4187,5697}, {4640,5795}, {4861,5433}, {5175,5825}, {5176,7354}, {5687,8069}, {7173,7705}, {8255,8261}
X(8256) = midpoint of X(i) and X(j) for these {i,j}: {8, 56}, {4848, 6736}
X(8256) = reflection of X(i) in X(j) for these (i,j): (1,6691), (1329,10)
X(8256) = complement of X(2098)
X(8256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,946,5123), (10,3878,3820), (10,5836,2886), (10,5837,3740), (1706,3679,5794)
X(8257) lies on these lines: {1,3939}, {2,7}, {5,1158}, {6,6510}, {40,5804}, {46,5084}, {100,7671}, {443,1728}, {474,5729}, {516,3359}, {517,1001}, {518,997}, {528,5722}, {920,6856}, {971,6911}, {1006,6282}, {1125,5761}, {1737,2550}, {2096,5817}, {2269,7614}, {2801,5720}, {3007,5826}, {3035,8255}, {3243,4511}, {3262,4384}, {3358,6826}, {4000,7961}, {5044,5220}, {5422,6505}, {5440,5728}, {5572,6600}, {5732,6905}, {5735,6963}, {5759,6947}, {5805,6882}
X(8257) = midpoint of X(9) and X(57)
X(8257) = reflection of X(i) in X(j) for these (i,j): (142,6692), (3452,6666)
X(8257) = {X(474),X(5729)}-harmonic conjugate of X(5784)
X(8258) lies on these lines: {2,1046}, {5,2792}, {8,5429}, {10,58}, {44,1213}, {81,3178}, {191,4425}, {442,4697}, {511,6684}, {516,7683}, {519,3704}, {540,3828}, {579,1761}, {758,942}, {896,5051}, {1330,1698}, {1714,3980}, {3579,4085}, {3831,5294}, {3923,5292}, {4438,5711}
X(8258) = midpoint of X(10) and X(58)
X(8258) = reflection of X(i) and X(j) for these (i,j): (1125,6693), (1125,6693), (3454,3634)
X(8258) = {X(942),X(6679)}-harmonic conjugate of X(1125)
X(8259) lies on these lines: {5,14}, {6,627}, {395,629}, {532,5007}, {597,7807}, {635,6673}
X(8259) = midpoint of X(17) and X(61)
X(8259) = reflection of X(i) in X(j) for these (i,j): (629,6692), (635,6673)
X(8260) lies on these lines: {5,13}, {6,628}, {396,630}, {533,5007}, {597,7807}, {636,6674}
X(8260) = midpoint of X(18) and X(62)
X(8260) = reflection of X(i) in X(j) for these (i,j): (630,6695), (636,6674)
X(8261) lies on these lines: {1,6596}, {5,2771}, {21,65}, {30,7686}, {191,405}, {442,1737}, {517,5428}, {551,6583}, {758,942}, {1837,2475}, {1858,3838}, {3649,5087}, {3754,6797}, {4004,5441}, {5426,5903}, {5728,6598}, {6001,6841}
X(8261) = midpoint of X(21) and X(65)
X(8261) = reflection of X(i) and X(j) for these (i,j): (442,3812), (960,6675)
X(8262) lies on these lines: {5,141}, {23,67}, {468,524}, {542,7575}, {599,1995}, {858,6698}, {1352,3581}, {2854,3580}
X(8262) = midpoint of X(i) and X(j) for these {i,j}: {23, 67}, {1352, 3581}
X(8262) = reflection of X(i) in X(j) for these (i,j): (858,6698), (6593,468)
X(8263) lies on these lines: {5,5181}, {6,6387}, {25,69}, {30,599}, {141,1368}, {511,1596}, {597,5972}, {1370,3620}, {2882,3491}, {3564,6644}
X(8263) = midpoint of X(25) and X(69)
X(8263) = reflection of X(i) in X(j) for these (i,j): (6,6677), (1368,141)
X(8264) lies on the cubic K075 and these lines: {2,308}, {6,194}, {8,704}, {22,385}, {69,706}, {192,700}, {315,710}, {708,6327}, {3186,3511}, {3229,6374}, {3852,5596}
X(8264) = anticomplement of X(1502)
X(8264) = X(32)-Ceva conjugate of X(2)
X(8264) = {X(194),X(2998)}-harmonic conjugate of X(6)
X(8264) = polar conjugate of isogonal conjugate of X(23173)
X(8264) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (31,315), (32,6327), (560,69), (1101,670), (1501,8), (1917,2), (1923,1369), (1924,3448), (1927,5207), (1980,150), (2205,1330)
X(8265) is the center of the inellipse that is the barycentric square of the Lemoine axis. The Brianchon point (perspector) of this inellipse is X(32). (Randy Hutson, October 15, 2018)
X(8265) lies on these lines: {2,308}, {6,694}, {10,704}, {32,206}, {37,700}, {39,698}, {141,706}, {216,230}, {626,710}, {708,2887}, {1180,7875}, {1194,7792}, {3118,4173}, {7664,7806}
X(8265) = isogonal conjugate of X(38830)
X(8265) = complement of X(1502)
X(8265) = X(38)-isoconjugate of X(3115)
X(8265) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,626), (32,8023), (4577,688)
X(8265) = crosssum of X(i) and X(j) for these {i,j}: {6, 76}, {782, 35078}
X(8265) = crosspoint of X(2) and X(32)
X(8265) = complementary conjugate of isogonal conjugate of X(38829)
X(8265) = crossdifference of every pair of points on line X(804)X(5152)
X(8265) = X(i)-isoconjugate of X(j) for these {i,j}: {1, 38830}, {2, 38847}, {38, 3115}, {561, 38826}, {2156, 38842}
X(8265) = trilinear product X(i)*X(j) for these {i,j}: {6, 2085}, {19, 4173}, {31, 20859}, {32, 4118}, {42, 16717}, {82, 3118}, {560, 626}, {1501, 20627}
X(8265) = trilinear quotient X(i)/X(j) for these (i,j): (1, 38830), (6, 38847), (82, 3115), (560, 38826), (626, 561), (1760, 38842), (2085, 2), (3118, 38), (4118, 76), (4173, 63), (16717, 86), (20627, 1502), (20859, 75)
X(8265) = barycentric product X(i)*X(j) for these {i,j}: {1, 2085}, {4, 4173}, {6, 20859}, {31, 4118}, {32, 626}, {37, 16717}, {83, 3118}, {560, 20627}
X(8265) = barycentric quotient X(i)/X(j) for these (i,j): (6, 38830), (22, 38842), (31, 38847), (251, 3115), (626, 1502), (1501, 38826), (2085, 75), (3118, 141), (4118, 561), (4173, 69), (16717, 274), (20627, 1928), (20859, 76)
X(8265) = polar conjugate of isotomic conjugate of X(4173)
X(8265) = polar conjugate of isogonal conjugate of X(23209)
X(8265) = {X(39),X(6375)}-harmonic conjugate of X(3589)
X(8265) = X(i)-complementary conjugate of X(j) for these (i,j): (31,626), (32,2887), (560,141), (1501,10), (1917,2), (1924,125), (1927,5031), (1980,116), (2205,3454)
X(8266) lies on the bianticevian conic of X(2) and X(6) and also on these lines: {2,3613}, {3,6}, {22,157}, {69,160}, {141,237}, {230,7467}, {308,1078}, {385,6636}, {1624,7493}, {2916,3511}, {2925,2926}, {3785,5596}, {6664,8177}
X(8266) = anticomplement of X(3613)
X(8266) = X(i)-Ceva conjugate of X(j) for these (i,j): (308,6), (1078,2)
X(8266) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,160,1634), (216,3313,3001), (216,5188,3313)
X(8266) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,3410), (31,7785), (1078,6327), (1101,1634), (1629,5906), (5012,8)
X(8267) lies on the bianticevian conic of X(2) and X(6) and also on these lines: {2,39}, {6,6664}, {22,7754}, {99,1627}, {193,2393}, {251,7760}, {385,6636}, {698,3051}, {1369,6655}, {1613,4576}, {1975,5359}, {7394,7774}
X(8267) = reflection of X(8024) in X(1194)
X(8267) = anticomplement of X(8024)
X(8267) = X(i)-Ceva conjugate of X(j) for these (i,j): (251,2), (7760,6)
X(8267) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,1180,2), (1194,8024,2), (1196,3266,2)
X(8267) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (31,1369), (82,315), (251,6327), (560,2896), (4630,7192)
Centers of apedal conics
This section was contributed by Clark Kimberling and Peter Moses, October 19, 2015.
Suppose that ABC is a triangle and P is a point not on a sideline of ABC. Let P' denote the isogonal conjugate of P. The vertices of the antipedal triangles of P and P' lie on a conic, here denoted by apedal(P). The following table shows examples of P, P', and the center of apedal(P').
| P | P' | center of apedal(P) |
|---|---|---|
| X(2) | X(6) | X(8268) |
| X(3) | X(4) | X(4577) |
| X(40) | X(80) | X(8269) |
Conjecture: For every ABC and P, the conic apedal(P) is a hyperbola. If you have The Geometer's Sketchpad, you can view an interactive sketch supporting the conjecture.
X(8268) is the center of the conic that passes through the following six points: the vertices of the antipedal triangle of X(2) and the vertices of the antipedal triangle of X(6).
X(8268) lies on no line X(i)X(j) for 1 <= i < j <= 8267.
X(8269) is the center of the conic H that passes through the following six points: the vertices of the antipedal triangle of X(40) and the vertices of the antipedal triangle of X(84). A barycentric equation for H follows:
f(a,b,c) + f(b,c,a) + f(c,a,b) = 0, where f(a,b,c) = bc(b - c)(b + c - a)2(a2 + b2 + c2 - 2bc)2
The conic H passes through X(i) for these i: 9, 57, 8270, 8271. Extensive sampling suggests that H is a hyperbola.
X(8269) lies on these lines: {7,1037}, {692,6183}, {883,934}, {1025,1332}, {1275,1633}, {3939,6516}, {4616,7256}, {7131,8257}
X(8269) = X(i)-cross conjugate of X(j) for these (i,j): (6,1275), (3870,4564), (7177,7045)
X(8269) = X(i)-isoconjugate of X(j) for these (i,j): (497,663), (513,4319), (522,7083), (614,3900), (649,6554), (650,2082), (657,4000), (1459,1863), (1633,2310), (3064,7124), (4041,5324), (4105,7195)
X(8270) lies on the conic at X(8269) and these lines: {1,3}, {2,4318}, {7,3920}, {8,1943}, {10,34}, {31,1708}, {33,516}, {63,109}, {72,221}, {73,3811}, {77,3870}, {81,7672}, {197,1763}, {200,223}, {201,1395}, {210,1456}, {222,518}, {226,612}, {227,5687}, {278,2550}, {388,1448}, {553,4327}, {608,1041}, {614,3911}, {651,3681}, {664,3996}, {956,1455}, {975,3485}, {976,1042}, {990,3474}, {997,1457}, {1376,1465}, {1407,3242}, {1421,5272}, {1458,3938}, {1728,3073}, {1777,7330}, {1870,5657}, {1876,7085}, {1945,2319}, {2000,3434}, {2003,3751}, {3961,5018}, {4348,4848}, {5219,5268}, {5226,5297}, {5435,7191}, {6198,6361}
X(8270) = reflection of X(i) in X(j) for these (i,j): (1,1060), (1763,197)
X(8270) = X(348)-Ceva conjugate of X(9)
X(8270) = crosssum of X(663) and X(7004)
X(8270) = crosspoint of X(664) and X(7012)
X(8270) = barycentric product X(664)*X(2509)
X(8270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,1040), (10,4347,34), (65,1460,57), (200,223,4551), (241,3744,1617), (612,2263,226)
X(8271) lies on the conic at X(8269) and these lines: {1,6}, {19,2809}, {57,1037}, {77,3870}, {78,4684}, {145,4318}, {241,6600}, {269,3174}, {279,7674}, {527,4319}, {948,6601}, {1066,1818}, {1445,3939}, {1448,3189}, {1998,4551}, {2136,7273}, {2191,3008}, {2263,5853}
X(8271) = X(6604)-Ceva conjugate of X(57)
X(8272) lies on the conic at X(4577) and these lines: {2,1031}, {6,6655}, {5596,7929}
X(8272) = anticomplement of X(1031)
X(8272) = X(2896)-Ceva conjugate of X(2)
X(8272) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (163,2528), (1964,1031), (2896,6327)
Perspectors: X(8273)-X(8350)
This section was contributed by César Eliud Lozada, Oct. 21, 2015. Perspectors are given for pairs of triangles Many of these triangles were introduced in connection with specific triangle centers in ETC. The following table identifies the triangles and locations in ETC.
| Triangle(s) | Location |
|---|---|
| Andromeda triangle | X(5573) |
| 1st anti-Brocard triangle | X(5939) |
| Antlia triangle | X(5574) |
| Apus triangle | X(5584) |
| Aquila triangle | X(5586) |
| Ara triangle | X(5594) |
| Aries triangle | X(5596) |
| Auriga triangles | X(5597) |
| Caelum triangle | X(5603) |
| Euler triangles | X(3758) |
| Extouch triangles | X(5927) |
| Garcia-inner/outer triangles | X(5587) |
| Honsberger triangle | X(7676) |
| Hutson-inner/outer triangles | X(363) |
| Hutson-intouch/extouch triangles | X(5731) |
| Lucas(n) Brocard triangles | X(6421) |
| Lucas(n) antipodal triangles | X(6457) |
| Lucas(n) homothetic triangles | X(493) |
| Lucas(n) inner-tangential triangles | X(6394) |
| Lucas(n) reflection triangles | X(6401) |
| Lucas(n) secondary central/tangents triangles | X(6199) |
| McCay triangle | X(7606) |
| Mixtilinear triangles | X(7955) |
| Pamfilos-Zhou triangles | X(7594) |
| Schroeter triangle | X(8286) |
| Sharygin triangles | X(8229) |
| Tangential-midarc triangles | X(8075) |
| Trinh triangle | X(7688) |
X(8273) lies on these lines: {1,3}, {4,4423}, {11,6908}, {12,6865}, {20,1001}, {48,220}, {71,1208}, {84,3683}, {154,7959}, {198,1212}, {218,572}, {405,4297}, {443,6253}, {631,4413}, {954,4298}, {956,6743}, {958,5731}, {990,6051}, {1125,7580}, {1191,4300}, {1253,4322}, {1376,3523}, {1621,3522}, {1699,3824}, {1750,3646}, {2829,6936}, {3091,8167}, {3146,5284}, {3616,7411}, {3711,5534}, {3816,6838}, {3893,7966}, {3913,6764}, {4313,7677}, {4679,6260}, {5082,6067}, {5432,6926}, {5433,6988}, {5842,6897}, {6284,6916}, {6600,6762}, {6690,6890}, {6691,6962}, {6899,7680}, {6987,7354}
X(8274) lies on these lines: {1,71}, {40,500}, {65,3679}, {209,1698}, {1044,2093}, {5691,6254}
X(8275) lies on these lines: {1,631}, {8,4342}, {9,5854}, {10,4345}, {11,3679}, {191,1697}, {390,519}, {495,7982}, {497,4677}, {517,4312}, {1320,5231}, {1698,2098}, {1768,5119}, {1837,4816}, {2269,4898}, {3057,3632}, {3577,5559}, {4293,7991}, {4302,7992}, {4669,5274}, {5586,5903}, {5691,5697}, {6744,7320}
X(8275) = reflection of X(i) in X(j) for these (i,j): (1,1000), (4900,8)
X(8276) lies on these lines: {3,485}, {6,1147}, {24,3068}, {25,371}, {486,5020}, {642,1584}, {1131,2071}, {1151,7387}, {1588,1995}, {1593,6564}, {1598,6561}, {3071,7529}, {3311,7506}, {5899,6407}, {6221,7517}, {6644,7583}, {7393,8253}, {7484,8280}
X(8276) = {X(6),X(6642)}-harmonic conjugate of X(8277)
X(8277) lies on these lines: {3,486}, {6,1147}, {24,3069}, {25,372}, {485,5020}, {641,1583}, {1132,2071}, {1152,7387}, {1587,1995}, {1593,6565}, {1598,6560}, {3070,7529}, {3312,7506}, {5899,6408}, {6398,7517}, {6644,7584}, {7393,8252}, {7484,8281}
X(8277) = {X(6),X(6642)}-harmonic conjugate of X(8276)
X(8278) lies on these lines: {56,197}, {100,4308}
X(8279) = R^2 (J^2 - 5) X(945) + 4 (R^2 - r^2) X(3149)
X(8279) = (J^2 - 5) R^2 X(1) + 12 r R X(2) - 2 [r^2 + s^2 + (J^2 - 7) R^2] X(3)
X(8279) lies on this line:
{78,517}
X(8280) lies on these lines: {2,372}, {6,8281}, {25,6564}, {371,427}, {590,1368}, {1370,6200}, {3070,6676}, {5133,6565}, {5418,7386}, {6396,7499}, {6560,7494}, {6561,7378}, {7484,8276}
X(8281) lies on these lines: {2,371}, {6,8280}, {25,6565}, {372,427}, {615,1368}, {1370,6396}, {3071,6676}, {5133,6564}, {5420,7386}, {6200,7499}, {6560,7378}, {6561,7494}, {7484,8277}
X(8282) lies on these lines: {4,2331}, {10,223}, {387,7682}, {2093,3987}, {3421,5930}
X(8283) lies on these lines: {1,945}, {29,388}, {56,515}, {65,990}, {108,944}, {221,513}, {859,5358}, {1457,4186}, {1479,1846}, {2098,2817}
X(8284) lies on these lines: {3,3582}
X(8285) lies on these lines: {40,170}, {279,291}
Let A'B'C' and A"B"C" be the medial and orthic triangle of ABC, respectively. Let A* = B'C'∩(B"C", and define B* and C* cyclically. The triangle A*B*C* is called the Schroeter triangle of ABC (Catalin Barbu, "Teoreme fundamentale din geometria triunghiului", 2008, pp. 482-, in Romanian.
Barycentrics for the A-vertex of the Schroeter triangle: c2 - b2 : c2 - a2 : a2 - b2. (Peter Moses, October 22, 2015)
X(8286) lies on these lines: {1,6739}, {2,643}, {11,125}, {115,124}, {244,656}, {442,1145}, {523,1365}, {1086,2968}, {1109,2632}, {1577,4939}, {2887,3696}, {3454,3626}, {3935,3936}
X(8286) = complement of X(643)
X(8286) = X(8)-Ceva conjugate of X(523)
X(8286) = X(110)-isoconjugate of X(6011)
X(8286) = {X(11),X(125)}-harmonic conjugate of X(8287)
X(8286) = X(i)-complementary conjugate of X(j) for these (i,j): (7,512), (34,8062), (42,4521), (56,523), (57,4369), (65,513), (181,661), (226,3835), (512,9), (513,960), (523,1329), (608,525), (649,5745), (661,3452), (670,3037), (798,1212), (1015,4858), (1042,522), (1118,520), (1356,1084), (1357,244), (1365,125), (1397,647), (1400,514), (1401,3005), (1402,650), (1416,4458), (1426,521), (1427,4885), (2171,4129), (3120,124), (3122,1146), (3669,3739), (3676,3741), (3709,6554), (3733,4999), (3952,3038), (4017,10), (4077,2887), (4516,5514), (4524,5574), (4557,3039), (4559,4422), (4565,620), (6591,6708), (7143,656), (7178,141), (7180,2), (7216,142), (7250,1), (7316,690), (7337,6587)
X(8287) lies on these lines: {2,662}, {11,125}, {80,6739}, {115,116}, {127,5517}, {142,5949}, {338,1577}, {429,1887}, {513,5954}, {523,4092}, {583,5740}, {651,3013}, {656,2310}, {661,3942}, {857,2245}, {868,4459}, {1213,6666}, {2611,6741}, {5074,5164}
X(8287) = midpoint of X(4092) and X(4934)
X(8287) = complement of X(662)
X(8287) = complementary conjugate of X(4369)
X(8287) = isotomic conjugate of isogonal conjugate of X(20982)
X(8287) = polar conjugate of isogonal conjugate of X(22094)
X(8287) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,523), (1029,514), (2994,525), (3969,7265), (7331,520), (7357,512), (8044,513)
X(8287) = X(i)-isoconjugate of X(j) for these (i,j): {163,6742}, {476,1983}, {1789,7115}, {2149,3615}, {2160,4570}, {4567,6186}
X(8287) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,125,8286), (1577,4858,338)
X(8287) = X(i)-complementary conjugate of X(j) for these (i,j): (1,4369), (2,512), (6,523), (10,3835), (19,8062), (25,525), (32,647), (37,513), (39,3005), (42,514), (65,4885), (82,8060), (110,620), (111,690), (112,5972), (115,125), (125,127), (181,1577), (187,1649), (213,650), (251,826), (308,688), (351,2482), (393,520), (512,2), (513,3739), (514,3741), (520,6389), (523,141), (525,1368), (647,3), (649,1125), (650,960), (661,10), (663,5745), (667,3666), (669,39), (686,131), (690,126), (694,804), (733,5113), (756,4129), (798,37), (810,1214), (850,626), (878,441), (881,3229), (882,325), (1015,244), (1042,7658), (1171,6367), (1334,4521), (1383,3906), (1400,522), (1402,905), (1427,3900), (1438,4458), (1495,5664), (1500,661), (1577,2887), (1637,113), (1648,5099), (1880,521), (1918,6586), (1974,2485), (1976,2799), (1989,526), (2054,2786), (2081,128), (2088,3258), (2165,924), (2207,6587), (2279,4913), (2333,3239), (2350,4151), (2353,3265), (2395,511), (2422,230), (2433,30), (2485,206), (2489,6), (2492,6593), (2501,5), (2623,140), (2643,8287), (2963,1510), (2971,6388), (2998,3221), (3005,6292), (3049,216), (3108,7927), (3120,116), (3121,1015), (3122,1086), (3124,115), (3125,11), (3221,6374), (3228,888), (3269,122), (3271,4858), (3445,2487), (3456,2525), (3569,114), (3572,740), (3669,3742), (3700,1329), (3709,9), (4017,142), (4024,3454), (4041,3452), (4079,1213), (4120,121), (4524,6554), (4557,4422), (4559,3035), (4674,4928), (4705,1211), (5027,5976), (5113,8290), (5466,625), (5638,3414), (5639,3413), (6137,618), (6138,619), (6378,798), (6388,5139), (6531,6130), (6587,2883), (6591,942), (6753,1147), (6791,5512), (7178,2886), (7180,1), (7250,4000), (7252,4999), (8034,6547), (8105,2575), (8106,2574)
X(8288) lies on Lemoine inellipse and these lines: {2,353}, {6,6032}, {115,125}, {338,850}, {1501,1853}, {3016,7603}, {6034,6792}
X(8288) = complement of X(35356)
X(8288) = X(2)-Ceva conjugate of X(17436)
X(8289) lies on these lines: {2,353}, {3,8310}, {6,1916}, {98,5092}, {99,187}, {115,7875}, {542,3314}, {543,3972}, {574,5152}, {671,7804}, {1281,8296}, {3329,5182}, {3620,5984}, {5477,7837}, {5969,7766}, {5978,6777}, {5979,6778}, {6054,7925}, {6199,8304}, {6200,8313}, {6221,8306}, {6395,8305}, {6396,8312}, {6398,8307}, {6433,8308}, {6434,8309}, {6435,8314}, {6436,8315}
X(8289) = {X(8310),X(8311)}-harmonic conjugate of X(3)
X(8290) is the center of hyperbola H = {{A,B,C,PU(1)}}, which is the isogonal conjugate of line PU(1), which is the line X(39)X(512)). Also, H is the isotomic conjugate of line PU(11), which is the line X(141)X(523). Also, H passes through X(83), X(99) and X(880). (Randy Hutson, October 27, 2015
X(8290) lies on cubics K252, K699, and these lines: {2,4048}, {3,147}, {39,83}, {76,8150}, {98,5092}, {114,5999}, {115,6704}, {148,7770}, {238,1281}, {385,732}, {618,5978}, {619,5979}, {620,5152}, {629,5982}, {630,5983}, {754,2482}, {1125,5988}, {1649,7711}, {2076,7779}, {2329,7061}, {3552,6337}, {5017,7837}, {6033,7470}, {6287,8295}, {7786,8178}
X(8290) = midpoint of X(i),X(j) for these (i,j): (83,99)
X(8290) = reflection of X(i) in X(j) for these (i,j): (115,6704), (6292,620)
X(8290) = X(2)-Ceva conjugate of X(385)
X(8290) = complementary conjugate of X(5103)
X(8290) = crosssum of circumcircle intercepts of line PU(1)
X(8290) = isotomic conjugate of X(9477)
X(8290) = 1st-Brocard-to-ABC similarity image of X(83)
X(8290) = barycentric product X(385)*X(7779)
X(8291) lies on these lines: {3,5979}, {182,5980}, {1916,3106}, {5152,8292}, {5463,7840}, {5617,5978}, {5982,5989}, {6294,7781}
X(8292) lies on these lines: {3,5978}, {182,5981}, {1916,3107}, {5152,8291}, {5464,7840}, {5613,5979}, {5983,5989}, {6581,7781}
X(8293) lies on these lines: {114,5152}, {1916,3102}, {5989,6289}
X(8293) = {X(114),X(5152)}-harmonic conjugate of X(8294)
X(8294) lies on these lines: {114,5152}, {1916,3103}, {5989,6290}
X(8294) = {X(114),X(5152)}-harmonic conjugate of X(8293)
X(8295) lies on these lines: {3,1916}, {147,5152}, {6287,8290}
X(8295) = circumtangential-isogonal conjugate of X(35375)
X(8296) lies on these lines: {3,8326}, {6,256}, {846,3247}, {1281,8289}, {3973,8245}, {6199,8320}, {6200,8329}, {6221,8322}, {6395,8321}, {6396,8328}, {6398,8323}, {6433,8324}, {6434,8325}, {6435,8330}, {6436,8331}
X(8296) = {X(8326),X(8327)}-harmonic conjugate of X(3)
X(8297) lies on these lines: {3,8342}, {6,291}, {100,238}, {105,8298}, {256,753}, {984,2246}, {1281,8289}, {6199,8336}, {6200,8345}, {6221,8338}, {6395,8337}, {6396,8344}, {6398,8339}, {6433,8340}, {6434,8341}, {6435,8346}, {6436,8347}
X(8297) = {X(8342),X(8343)}-harmonic conjugate of X(3)
X(8298) lies on these lines: {1,1929}, {6,2108}, {42,81}, {55,846}, {105,8297}, {238,1914}, {244,4038}, {385,740}, {518,2076}, {659,4155}, {678,896}, {758,5184}, {904,3903}, {906,5247}, {940,1054}, {1580,4433}, {1962,3722}, {3158,3550}, {3570,4368}
X(8299) lies on cubic K251 and these lines: {1,39}, {2,11}, {8,4595}, {9,1282}, {21,6626}, {35,6292}, {36,2482}, {42,1386}, {43,3158}, {56,6337}, {101,1083}, {141,8053}, {214,3126}, {238,1914}, {239,4433}, {244,1962}, {350,1281}, {518,672}, {544,993}, {641,8225}, {659,812}, {678,899}, {902,2239}, {1002,1280}, {1025,1362}, {1040,6509}, {1054,5437}, {1086,4436}, {1111,2795}, {1193,4161}, {1279,1575}, {1283,4199}, {1403,5435}, {1458,6168}, {2223,3912}, {3271,3882}, {3286,4966}, {3703,3969}, {3741,4154}, {3771,4192}, {3923,4376}, {3941,4851}, {4030,4651}, {4422,4557}, {4441,5695}
X(8299) = midpoint of X(i),X(j) for these (i,j): (1,1018)
X(8299) = bicentric sum of PU(134)
X(8299) = PU(134)-harmonic conjugate of X(659)
X(8300) lies on cubic K673 and these lines: {1,41}, {3,8348}, {6,291}, {31,43}, {58,2665}, {81,244}, {82,872}, {87,1958}, {238,1914}, {239,1281}, {560,3759}, {662,741}, {678,5524}, {751,753}, {756,1621}, {765,2382}, {1054,2999}, {1203,5213}, {1253,7220}, {1283,6044}, {1757,4712}, {2310,7281}, {3033,3271}, {3311,8336}, {3312,8337}, {4366,4368}, {4443,4471}, {4974,5009}, {5058,8334}, {5062,8335}, {6409,8340}, {6410,8341}, {6425,8338}, {6426,8339}, {6427,8346}, {6428,8347}
X(8300) = isogonal conjugate of X(30663)
X(8300) = crossdifference of every pair of points on line X(876)X(2254) (the perspectrix of Gemini triangles 31 and 33)
X(8300) = perspector of unary cofactor triangles of Gemini triangles 31 and 33
X(8300) = trilinear square of X(239)
X(8300) = {X(8348),X(8349)}-harmonic conjugate of X(3)
X(8301) lies on cubic K323 and these lines: {1,1929}, {2,11}, {6,291}, {56,664}, {75,1281}, {171,1054}, {197,4362}, {238,1575}, {244,940}, {371,8336}, {372,8337}, {518,910}, {519,5144}, {659,918}, {753,4492}, {760,5011}, {958,1146}, {1018,1083}, {1151,8334}, {1152,8335}, {1631,4361}, {1958,3056}, {1961,3750}, {2246,4712}, {2669,3286}, {2876,3033}, {3550,7290}, {3722,5311}, {4363,4381}, {6429,8338}, {6430,8339}, {6431,8346}, {6432,8347}
X(8301) = isogonal conjugate of X(2113)
X(8301) = anticomplement of X(20531)
X(8301) = crosspoint of PU(134)
X(8301) = polar conjugate of isotomic conjugate of X(20742)
X(8302) lies on these lines: {2,99}, {1151,5989}, {1281,8318}, {1916,6421}, {4027,5058}
X(8303) lies on these lines: {2,99}, {1152,5989}, {1281,8319}, {1916,6422}, {4027,5062}
X(8304) lies on these lines: {3,1916}, {98,6222}, {371,5989}, {372,8309}, {1281,8320}, {3311,4027}, {3312,8317}, {6199,8289}, {6221,8316}, {6407,8306}, {6446,8307}, {6447,8312}, {6449,8314}, {6451,8315}, {6453,8308} p
X(8304) = {X(3),X(1916)}-harmonic conjugate of X(8305)X(8305) lies on these lines: {3,1916}, {98,6399}, {371,8308}, {372,5989}, {1281,8321}, {3311,8316}, {3312,4027}, {6395,8289}, {6398,8317}, {6408,8307}, {6445,8306}, {6448,8313}, {6450,8315}, {6452,8314}, {6454,8309} p
X(8305) = {X(3),X(1916)}-harmonic conjugate of X(8304)X(8306) lies on these lines: {3,8307}, {371,8317}, {385,6567}, {1151,1916}, {1281,8322}, {4027,6425}, {5989,6429}, {6221,8289}, {6407,8304}, {6445,8305}, {6453,8316}, {6468,8308}, {6470,8309}, {6472,8310}, {6474,8311}, {6476,8312}, {6478,8313}, {6480,8314}, {6482,8315}
X(8307) lies on these lines: {3,8306}, {372,8316}, {385,6566}, {1152,1916}, {1281,8323}, {4027,6426}, {5989,6430}, {6398,8289}, {6408,8305}, {6446,8304}, {6454,8317}, {6469,8309}, {6471,8308}, {6473,8311}, {6475,8310}, {6477,8313}, {6479,8312}, {6481,8315}, {6483,8314}
X(8308) lies on these lines: {3,8309}, {371,8305}, {1151,5989}, {1281,8324}, {1916,6425}, {4027,6409}, {6411,8317}, {6433,8289}, {6453,8304}, {6468,8306}, {6471,8307}, {6484,8310}, {6486,8311}, {6488,8313}, {6490,8314}, {6492,8315}
X(8309) lies on these lines: {3,8308}, {372,8304}, {1152,5989}, {1281,8325}, {1916,6426}, {4027,6410}, {6412,8316}, {6434,8289}, {6454,8305}, {6469,8307}, {6470,8306}, {6485,8311}, {6487,8310}, {6489,8312}, {6491,8315}, {6493,8314}
X(8310) lies on these lines: {3,8289}, {371,5989}, {1281,8326}, {1916,6417}, {3312,4027}, {6449,8316}, {6456,8317}, {6472,8306}, {6475,8307}, {6484,8308}, {6487,8309}, {6494,8314}, {6496,8313}, {6498,8315}
X(8310) = {X(3),X(8289)}-harmonic conjugate of X(8311)
X(8311) lies on these lines: {3,8289}, {372,5989}, {1281,8327}, {1916,6418}, {3311,4027}, {6450,8317}, {6455,8316}, {6473,8307}, {6474,8306}, {6485,8309}, {6486,8308}, {6495,8315}, {6497,8312}, {6499,8314}
X(8311) = {X(3),X(8289)}-harmonic conjugate of X(8310)
X(8312) lies on these lines: {3,8313}, {1151,5989}, {1281,8328}, {1916,6419}, {3312,4027}, {6199,8314}, {6396,8289}, {6447,8304}, {6476,8306}, {6479,8307}, {6489,8309}, {6497,8311}, {6500,8315}
X(8313) lies on these lines: {3,8312}, {1152,5989}, {1281,8329}, {1916,6420}, {3311,4027}, {6200,8289}, {6395,8315}, {6448,8305}, {6477,8307}, {6478,8306}, {6488,8308}, {6496,8310}, {6501,8314}
X(8314) lies on these lines: {3,8315}, {372,1916}, {1281,8330}, {4027,6427}, {5989,6431}, {6199,8312}, {6425,8316}, {6435,8289}, {6449,8304}, {6452,8305}, {6480,8306}, {6483,8307}, {6490,8308}, {6493,8309}, {6494,8310}, {6499,8311}, {6501,8313}
X(8315) lies on these lines: {3,8314}, {371,1916}, {1281,8331}, {4027,6428}, {5989,6432}, {6395,8313}, {6426,8317}, {6436,8289}, {6450,8305}, {6451,8304}, {6481,8307}, {6482,8306}, {6491,8309}, {6492,8308}, {6495,8311}, {6498,8310}, {6500,8312}
X(8316) lies on these lines: {3,4027}, {371,1916}, {372,8307}, {385,2460}, {1151,5989}, {1281,8332}, {3311,8305}, {6200,8289}, {6221,8304}, {6412,8309}, {6425,8314}, {6449,8310}, {6453,8306}, {6455,8311}
X(8316) = {X(3),X(4027)}-harmonic conjugate of X(8317)
X(8317) lies on these lines: {3,4027}, {371,8306}, {372,1916}, {385,2459}, {1152,5989}, {1281,8333}, {3312,8304}, {6396,8289}, {6398,8305}, {6411,8308}, {6426,8315}, {6450,8311}, {6454,8307}, {6456,8310}
X(8317) = {X(3),X(4027)}-harmonic conjugate of X(8316)
X(8318) lies on these lines: {256,6421}, {574,8319}, {1151,8324}, {1281,8302}, {1580,5058}
X(8319) lies on these lines: {256,6422}, {574,8318}, {1152,8325}, {1281,8303}, {1580,5062}
X(8320) lies on these lines: {3,256}, {371,8326}, {372,8325}, {1281,8304}, {1580,3311}, {3312,8333}, {6199,8296}, {6221,8332}, {6407,8322}, {6446,8323}, {6447,8328}, {6449,8330}, {6451,8331}, {6453,8324}
X(8320) = {X(3),X(256)}-harmonic conjugate of X(8321)
X(8321) lies on these lines: {3,256}, {371,8324}, {372,8327}, {1281,8305}, {1580,3312}, {3311,8332}, {6395,8296}, {6398,8333}, {6408,8323}, {6445,8322}, {6448,8329}, {6450,8331}, {6452,8330}, {6454,8325}
X(8321) = {X(3),X(256)}-harmonic conjugate of X(8320)
X(8322) lies on these lines: {3,8323}, {256,1151}, {371,8333}, {1281,8306}, {1580,6425}, {6221,8296}, {6445,8321}, {6453,8332}, {6468,8324}, {6470,8325}, {6472,8326}, {6474,8327}, {6476,8328}, {6478,8329}, {6480,8330}, {6482,8331}
X(8323) lies on these lines: {3,8322}, {256,1152}, {372,8332}, {1281,8307}, {1580,6426}, {6398,8296}, {6408,8321}, {6446,8320}, {6454,8333}, {6469,8325}, {6471,8324}, {6473,8327}, {6475,8326}, {6477,8329}, {6479,8328}, {6481,8331}, {6483,8330}
X(8324) lies on these lines: {3,8325}, {256,6425}, {371,8321}, {1151,8318}, {1281,8308}, {1580,6409}, {6411,8333}, {6433,8296}, {6453,8320}, {6468,8322}, {6471,8323}, {6484,8326}, {6486,8327}, {6488,8329}, {6490,8330}, {6492,8331}
X(8325) lies on these lines: {3,8324}, {256,6426}, {372,8320}, {1152,8319}, {1281,8309}, {1580,6410}, {6412,8332}, {6434,8296}, {6454,8321}, {6469,8323}, {6470,8322}, {6485,8327}, {6487,8326}, {6489,8328}, {6491,8331}, {6493,8330}
X(8326) lies on these lines: {3,8296}, {256,6417}, {371,8320}, {1281,8310}, {1580,3312}, {6449,8332}, {6456,8333}, {6472,8322}, {6475,8323}, {6484,8324}, {6487,8325}, {6494,8330}, {6496,8329}, {6498,8331}
X(8326) = {X(3),X(8296)}-harmonic conjugate of X(8327)
X(8327) lies on these lines: {3,8296}, {256,6418}, {372,8321}, {1281,8311}, {1580,3311}, {6450,8333}, {6455,8332}, {6473,8323}, {6474,8322}, {6485,8325}, {6486,8324}, {6495,8331}, {6497,8328}, {6499,8330}
X(8327) = {X(3),X(8296)}-harmonic conjugate of X(8326)
X(8328) lies on these lines: {3,8329}, {256,6419}, {1151,8318}, {1281,8312}, {1580,3312}, {6199,8330}, {6396,8296}, {6447,8320}, {6476,8322}, {6479,8323}, {6489,8325}, {6497,8327}, {6500,8331}
X(8329) lies on these lines: {3,8328}, {256,6420}, {1152,8319}, {1281,8313}, {1580,3311}, {6200,8296}, {6395,8331}, {6448,8321}, {6477,8323}, {6478,8322}, {6488,8324}, {6496,8326}, {6501,8330}
X(8330) lies on these lines: {3,8331}, {256,372}, {1281,8314}, {1580,6427}, {6199,8328}, {6425,8332}, {6435,8296}, {6449,8320}, {6452,8321}, {6480,8322}, {6483,8323}, {6490,8324}, {6493,8325}, {6494,8326}, {6499,8327}, {6501,8329}
X(8331) lies on these lines: {3,8330}, {256,371}, {1281,8315}, {1580,6428}, {6395,8329}, {6426,8333}, {6436,8296}, {6450,8321}, {6451,8320}, {6481,8323}, {6482,8322}, {6491,8325}, {6492,8324}, {6495,8327}, {6498,8326}, {6500,8328}
X(8332) lies on these lines: {3,1580}, {256,371}, {372,8323}, {1151,8318}, {1281,8316}, {3311,8321}, {6200,8296}, {6221,8320}, {6412,8325}, {6425,8330}, {6449,8326}, {6453,8322}, {6455,8327}
X(8332) = {X(3),X(1580)}-harmonic conjugate of X(8333)
X(8333) lies on these lines: {3,1580}, {256,372}, {371,8322}, {1152,8319}, {1281,8317}, {3312,8320}, {6396,8296}, {6398,8321}, {6411,8324}, {6426,8331}, {6450,8327}, {6454,8323}, {6456,8326}
X(8333) = {X(3),X(1580)}-harmonic conjugate of X(8332)
X(8334) lies on these lines: {291,6421}, {574,8335}, {1151,8301}, {1281,8302}, {5058,8300}
X(8335) lies on these lines: {291,6422}, {574,8334}, {1152,8301}, {1281,8303}, {5062,8300}
.png){kind=link}
andX(8161).png){kind=link}