| 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}
X(8336) lies on these lines: {3,291}, {371,8301}, {372,8341}, {1281,8304}, {3311,8300}, {3312,8349}, {6199,8297}, {6221,8348}, {6407,8338}, {6446,8339}, {6447,8344}, {6449,8346}, {6451,8347}, {6453,8340}
X(8336) = {X(3),X(291)}-harmonic conjugate of X(8337)
X(8337) lies on these lines: {3,291}, {371,8340}, {372,8301}, {1281,8305}, {3311,8348}, {3312,8300}, {6395,8297}, {6398,8349}, {6408,8339}, {6445,8338}, {6448,8345}, {6450,8347}, {6452,8346}, {6454,8341}
X(8337) = {X(3),X(291)}-harmonic conjugate of X(8336)
X(8338) lies on these lines: {3,8339}, {291,1151}, {371,8349}, {1281,8306}, {6221,8297}, {6407,8336}, {6425,8300}, {6429,8301}, {6445,8337}, {6453,8348}, {6468,8340}, {6470,8341}, {6472,8342}, {6474,8343}, {6476,8344}, {6478,8345}, {6480,8346}, {6482,8347}
X(8339) lies on these lines: {3,8338}, {291,1152}, {372,8348}, {1281,8307}, {6398,8297}, {6408,8337}, {6426,8300}, {6430,8301}, {6446,8336}, {6454,8349}, {6469,8341}, {6471,8340}, {6473,8343}, {6475,8342}, {6477,8345}, {6479,8344}, {6481,8347}, {6483,8346}
X(8340) lies on these lines: {3,8341}, {291,6425}, {371,8337}, {1151,8301}, {1281,8308}, {6409,8300}, {6411,8349}, {6433,8297}, {6453,8336}, {6468,8338}, {6471,8339}, {6484,8342}, {6486,8343}, {6488,8345}, {6490,8346}, {6492,8347}
X(8341) lies on these lines: {3,8340}, {291,6426}, {372,8336}, {1152,8301}, {1281,8309}, {6410,8300}, {6412,8348}, {6434,8297}, {6454,8337}, {6469,8339}, {6470,8338}, {6485,8343}, {6487,8342}, {6489,8344}, {6491,8347}, {6493,8346}
X(8342) lies on these lines: {3,8297}, {291,6417}, {371,8301}, {1281,8310}, {3312,8300}, {6449,8348}, {6456,8349}, {6472,8338}, {6475,8339}, {6484,8340}, {6487,8341}, {6494,8346}, {6496,8345}, {6498,8347}
X(8342) = {X(3),X(8297)}-harmonic conjugate of X(8343)
X(8343) lies on these lines: {3,8297}, {291,6418}, {372,8301}, {1281,8311}, {3311,8300}, {6450,8349}, {6455,8348}, {6473,8339}, {6474,8338}, {6485,8341}, {6486,8340}, {6495,8347}, {6497,8344}, {6499,8346}
X(8343) = {X(3),X(8297)}-harmonic conjugate of X(8342)
X(8344) lies on these lines: {3,8345}, {291,6419}, {1151,8301}, {1281,8312}, {3312,8300}, {6199,8346}, {6396,8297}, {6447,8336}, {6476,8338}, {6479,8339}, {6489,8341}, {6497,8343}, {6500,8347}
X(8345) lies on these lines: {3,8344}, {291,6420}, {1152,8301}, {1281,8313}, {3311,8300}, {6200,8297}, {6395,8347}, {6448,8337}, {6477,8339}, {6478,8338}, {6488,8340}, {6496,8342}, {6501,8346}
X(8346) lies on these lines: {3,8347}, {291,372}, {1281,8314}, {6199,8344}, {6425,8348}, {6427,8300}, {6431,8301}, {6435,8297}, {6449,8336}, {6452,8337}, {6480,8338}, {6483,8339}, {6490,8340}, {6493,8341}, {6494,8342}, {6499,8343}, {6501,8345}
X(8347) lies on these lines: {3,8346}, {291,371}, {1281,8315}, {6395,8345}, {6426,8349}, {6428,8300}, {6432,8301}, {6436,8297}, {6450,8337}, {6451,8336}, {6481,8339}, {6482,8338}, {6491,8341}, {6492,8340}, {6495,8343}, {6498,8342}, {6500,8344}
X(8348) lies on these lines: {3,8300}, {291,371}, {372,8339}, {1151,8301}, {1281,8316}, {3311,8337}, {6200,8297}, {6221,8336}, {6412,8341}, {6425,8346}, {6449,8342}, {6453,8338}, {6455,8343}
X(8348) = {X(3),X(8300)}-harmonic conjugate of X(8349)
X(8349) lies on these lines: {3,8300}, {291,372}, {371,8338}, {1152,8301}, {1281,8317}, {3312,8336}, {6396,8297}, {6398,8337}, {6411,8340}, {6426,8347}, {6450,8343}, {6454,8339}, {6456,8342}
X(8349) = {X(3),X(8300)}-harmonic conjugate of X(8348)
X(8350) lies on these lines: {3,7840}
See Yff Central Circle at MathWorld.
The radius of the Yff central circle is r/[1 + sin(A/2) + sin(B/2) + sin(C/2)] (Peter Moses, October 29, 2015)
X(8351) lies on these lines: (1, 167), (3, 7587), (4, 5935), (8, 8126), (10, 236), (40, 173), (188, 1130), (258, 3333), (495, 8088), (496, 8086), (517, 8130), (942, 8083), (999, 7588), (1125, 7028), (1385, 8129), (1387, 8104), (3295, 8076), (3487, 8080), (3616, 8125), (5045, 8100), (5657, 8128), (6724, 8079), (8132, 8193)
X(8351) = {X(i), X(j)}- harmonic conjugate of X(j) for these {i,j,k}: {1,174,8092}, {1,177,8091}, {7587,7589,3}, {8083,8084,942}
As a point on the Euler line, X(8352) has Shinagawa coefficients ((E+F)2, -9S2).
X(8352) lies on these lines: {2,3}, {69,7620}, {115,3849}, {148,7840}, {183,7615}, {187,5461}, {316,524}, {325,543}, {574,8176}, {597,598}, {625,2482}, {5215,6722}, {5254,7812}, {7745,7827}, {7747,7817}, {7748,7775}, {7801,7825}, {7810,7842}
X(8352) = midpoint of X(i) and X(j) for these {i,j}: {148, 7840}, {316, 671}, {3543, 5999}
X(8352) = reflection of X(i) in X(j) for these (i,j): (187, 5461), (1513, 381), (2482, 625)
X(8352) = crosspoint of X(598) and X(671)
As a point on the Euler line, X(8353) has Shinagawa coefficients ((E+F)2+4S2, -9S2).
X(8353) lies on these lines: {2,3}, {538,7750}, {6390,7898}, {6781,7792}, {7757,7762}, {7789,7910}
X(8353) = midpoint of X(7757) and X(7802)
X(8353) = reflection of X(i) in X(j) for these (i,j): (2,8354), (7762,7757), (8356,7833), (8370,8356)
As a point on the Euler line, X(8354) has Shinagawa coefficients (2(E+F)2+5S2, -9S2).
X(8354) lies on these lines: {2,3}, {538,7767}, {597,5033}, {3589,6781}, {5305,7847}, {6390,7761}, {7750,7757}
X(8354) = midpoint of X(i) and X(j) for these {i,j}: {2,8353}, {7750,7757}, {7833,8356}
X(8354) = reflection of X(i) in X(j) for these (i,j): (2,8358), (8359,8356)
As a point on the Euler line, X(8355) has Shinagawa coefficients (4(E+F)2-9S2, -9S2).
X(8355) lies on these lines: {2,3}, {125,6092}, {141,7617}, {524,625}, {597,7844}, {671,6390}, {3849,6722}, {5305,7775}, {7615,7778}
X(8355) = midpoint of X(i) and X(j) for these {i,j}: {625,5461}, {671,6390}
As a point on the Euler line, X(8356) has Shinagawa coefficients ((E+F)2+2S2, -3S2).
X(8356) is the centroid of the mid-triangle of the circumcevian triangles of PU(1). Randy Hutson, July 31 2018)
X(8356) lies on these lines: {2,3}, {39,754}, {99,141}, {183,2549}, {187,4045}, {194,7767}, {230,7771}, {315,5013}, {316,3815}, {325,574}, {524,3094}, {538,7810}, {543,5976}, {597,1691}, {599,6393}, {620,7853}, {1078,5254}, {1506,7842}, {1975,7800}, {2021,3849}, {2482,7880}, {2896,3933}, {3053,7803}, {3096,7782}, {3314,6390}, {3564,7709}, {3589,3972}, {3785,7738}, {3788,7935}, {3793,7766}, {3926,7879}, {3934,7756}, {5024,7774}, {5206,7834}, {5305,7793}, {5306,7827}, {6292,7816}, {6337,7881}, {6683,7747}, {6781,7804}, {7745,7786}, {7746,7872}, {7748,7815}, {7749,7861}, {7752,7910}, {7763,7784}, {7764,7873}, {7765,7780}, {7769,7911}, {7777,7898}, {7781,7854}, {7796,7936}, {7799,7883}, {7801,7865}, {7813,7848}, {7835,7937}, {7836,7928}, {7849,7863}, {7857,7918}, {7891,7938}, {7906,7929}
X(8356) = midpoint of X(i) and X(j) for these {i,j}: {2,7833}, {7757,7811}, {8353,8370}, {8354,8359}
X(8356) = reflection of X(i) in X(j) for these (i,j): (2,8359), (7833,8354), (8353,7833), (8359,8358), (8370,2)
X(8356) = complement of X(11361)
X(8356) = circumcircle-inverse of X(37903)
X(8356) = {X(2),X(20)}-harmonic conjugate of X(14033)
As a point on the Euler line, X(8357) has Shinagawa coefficients (2(E+F)2+S2, -5S2).
X(8357) lies on these lines: {2,3}, {141,7748}, {148,7928}, {230,7830}, {325,7847}, {524,7765}, {543,7849}, {626,6390}, {698,3631}, {1692,6329}, {2549,3933}, {3589,7747}, {3629,5028}, {3793,5305}, {3815,7825}, {3849,7829}, {4045,7745}, {5254,7751}, {5306,7902}, {6781,7852}, {7738,7776}, {7756,7789}, {7762,7864}, {7792,7802}
X(8357) = midpoint of X(i) and X(j) for these {i,j}: {6655,6656}, {7765,7873}
X(8357) = reflection of X(i) in X(j) for these (i,j): (384,8364), (7819,6656)
X(8357) = complement of X(19687)
X(8357) = anticomplement of X(19697)
X(8357) = pole of Napoleon axis wrt conic {{X(13),X(14),X(15),X(16),X(141)}}
As a point on the Euler line, X(8358) has Shinagawa coefficients (4(E+F)2+7S2, -9S2).
X(8358) lies on these lines: {2,3}, {6390,7831}, {7757,7767}
X(8358) = midpoint of X(i) and X(j) for these {i,j}: {2,8354}, {7757,7767}, {8356,8359}
As a point on the Euler line, X(8359) has Shinagawa coefficients (2(E+F)2+3S2, -3S2).
X(8359) lies on these lines: {2,3}, {6,3793}, {32,597}, {39,524}, {69,5024}, {141,574}, {187,3589}, {230,4045}, {325,7831}, {543,3934}, {598,7802}, {599,3933}, {625,3055}, {671,7847}, {1078,5305}, {1153,7886}, {1384,3618}, {1992,3785}, {2482,6292}, {2896,7840}, {3053,8182}, {3054,7844}, {3096,7870}, {3631,7813}, {3767,7610}, {3788,7622}, {3815,7761}, {3849,6683}, {5008,6329}, {5171,5476}, {5215,7852}, {5254,7815}, {5461,7861}, {5569,7834}, {7617,7872}, {7618,7795}, {7750,7786}, {7762,7904}, {7771,7792}, {7825,8176}
X(8359) = midpoint of X(i) and X(j) for these {i,j}: {2,8356}, {39,7810}, {7750,7812}, {7833,8370}
X(8359) = reflection of X(i) in X(j) for these (i,j): (7767,7810), (8354,8356), (8356,8358), (8370, 8367)
X(8359) = complement X(8370)
X(8359) = anticomplement X(8367)
X(8359) = {X(2),X(3)}-harmonic conjugate of X(8369)
As a point on the Euler line, X(8360) has Shinagawa coefficients (4(E+F)2-3S2, -3S2).
X(8360) lies on these lines: {2,3}, {141,7844}, {230,7810}, {325,7827}, {524,626}, {543,7789}, {597,7775}, {598,7846}, {599,3767}, {625,2030}, {671,7832}, {1992,7776}, {2482,7874}, {3793,7806}, {3815,7913}, {3849,6680}, {3933,7851}, {3934,5461}, {5254,7801}, {5306,7818}, {6390,7790}, {7610,7800}, {7617,7914}, {7745,7852}, {7750,7942}, {7762,7932}, {7767,7828}, {7792,7812}, {7797,7840}, {7808,8176}
X(8360) = midpoint of X(i) and X(j) for these {i,j}: {2,33184}, {626,7817}, {5254,7801}, {5306,7818}, {7841,8369}
X(8360) = reflection of X(i) in X(j) for these (i,j): (2,33213), (5305, 7817), (8368,2), (8369,8365)
X(8360) = complement X(8369)
X(8360) = anticomplement X(8365)
X(8360) = orthocentroidal-circle-inverse of X(8367)
X(8360) = {X(2),X(4)}-harmonic conjugate of X(33237)
X(8360) = {X(2),X(5)}-harmonic conjugate of X(8367)
X(8360) = {X(2),X(20)}-harmonic conjugate of X(33197)
As a point on the Euler line, X(8361) has Shinagawa coefficients (2(E+F)2-3S2, -S2).
X(8361) lies on these lines: {2,3}, {115,7789}, {141,5028}, {230,626}, {315,3793}, {325,5305}, {524,7755}, {620,7861}, {625,6680}, {1506,1692}, {2023,3934}, {3054,7815}, {3055,6683}, {3629,5346}, {3767,3933}, {3788,5254}, {3815,7834}, {5306,7759}, {5309,7888}, {7603,7889}, {7735,7776}, {7749,7853}, {7750,7857}, {7752,7792}, {7762,7806}, {7763,7851}, {7764,7817}, {7769,7919}, {7777,7932}, {7790,7940}, {7797,7925}, {7814,7856}
X(8361) = midpoint of X(i) and X(j) for these {i,j}: {5025,7807}, {7755,7821}
X(8361) = complement X(7807)
X(8361) = {X(2),X(4)}-harmonic conjugate of X(32954)
X(8361) = {X(2),X(5)}-harmonic conjugate of X(7819)
X(8361) = orthocentroidal-circle-inverse of X(32954)
As a point on the Euler line, X(8362) has Shinagawa coefficients ((E+F)2+S2, -S2).
X(8362) lies on these lines: {2,3}, {6,7767}, {32,3589}, {39,141}, {83,7750}, {183,5305}, {187,5103}, {230,7815}, {325,3096}, {524,7772}, {574,7789}, {597,5007}, {599,7758}, {620,7915}, {626,3815}, {1078,7792}, {1506,7853}, {2021,5031}, {2548,7784}, {2896,3329}, {3054,7886}, {3055,7862}, {3618,3785}, {3619,3926}, {3629,5041}, {3630,7890}, {3631,7855}, {3763,5013}, {3788,7914}, {3934,4045}, {5188,5480}, {5306,7780}, {5475,7935}, {6704,7804}, {7736,7776}, {7745,7761}, {7746,7913}, {7749,7852}, {7752,7937}, {7753,7873}, {7759,7865}, {7763,7868}, {7764,7849}, {7769,7944}, {7771,7846}, {7774,7879}, {7777,7938}, {7785,7928}, {7787,7904}, {7793,7875}, {7811,7878}, {7812,7936}, {7838,7848}, {7857,7943}, {7858,7883}, {7921,7929}
X(8362) = midpoint of X(i) and X(j) for these {i,j}: {7770,7791}, {7772,7854}
X(8362) = complement X(7770)
X(8362) = {X(39),X(6292)}-harmonic conjugate of X(141)
X(8362) = {X(2),X(3)}-harmonic conjugate of X(7819)
X(8362) = {X(2),X(20)}-harmonic conjugate of X(16045)
As a point on the Euler line, X(8363) has Shinagawa coefficients (3(E+F)2-2S2, -S2).
X(8363) lies on these lines: {2,3}, {115,7915}, {141,7828}, {230,3096}, {325,7772}, {524,7856}, {597,7858}, {625,7889}, {626,5007}, {3314,5305}, {3589,7752}, {3629,7917}, {3763,6393}, {3767,7868}, {3788,7913}, {3793,7929}, {3815,7859}, {3933,7797}, {4045,7874}, {5254,7832}, {5286,7881}, {5306,7768}, {5309,7869}, {5319,7788}, {5346,7896}, {5355,7895}, {5368,7882}, {5976,6722}, {6292,7886}, {6390,7864}, {6680,7750}, {6704,7603}, {7735,7879}, {7745,7846}, {7746,7914}, {7755,7849}, {7765,7880}, {7767,7806}, {7778,7803}, {7789,7790}, {7794,7817}, {7795,7851}, {7796,7884}, {7801,7902}, {7820,7861}, {7821,7829}, {7822,7844}, {7827,7909}, {7835,7918}, {7836,7923}, {7857,7937}, {7875,7912}, {7897,7920}
X(8363) = midpoint of X(i) and X(j) for these {i,j}: {7856,7922}, {7892,7933}
X(8363) = complement X(7892)
X(8363) = {X(2),X(4)}-harmonic conjugate of X(33217)
X(8363) = orthocentroidal-circle-inverse of X(33217)
As a point on the Euler line, X(8364) has Shinagawa coefficients (4(E+F)2-S2, -S2).
X(8364) lies on these lines: {2,3}, {141,5305}, {230,6292}, {325,7859}, {524,7829}, {597,7759}, {599,5319}, {625,6704}, {626,3589}, {698,3934}, {736,6683}, {2896,3793}, {3096,6179}, {3618,7776}, {3629,7896}, {3631,7805}, {3763,3767}, {3815,7867}, {3933,7803}, {4045,7789}, {5254,7822}, {5306,7854}, {6329,7838}, {6390,7832}, {7745,7853}, {7750,7846}, {7762,7875}
X(8364) = midpoint of X(i) and X(j) for these {i,j}: {384,8357}, {6656,7819}, {7829,7849}
X(8364) = complement X(7819)
X(8364) = pole of van Aubel line wrt conic {{X(13),X(14),X(15),X(16),X(141)}}
X(8364) = {X(2),X(3)}-harmonic conjugate of X(33185)
As a point on the Euler line, X(8365) has Shinagawa coefficients (8(E+F)2-9S2, 3S2).
X(8365) lies on these lines: {2,3}, {524,6680}, {597,3788}, {2482,7852}, {3793,7931}, {5215,6292}, {5305,7801}, {5569,7914}, {5969,6683}, {6390,7827}, {7767,7930}, {7789,7817}, {7792,7870}
X(8365) = midpoint of X(i) and X(j) for these {i,j}: {2,8368}, {5305,7801}, {7789,7817}, {8360,8369}
X(8365) = complement X(8360)
X(8365) = reflection of X(33213) in X(2)
As a point on the Euler line, X(8366) has Shinagawa coefficients (5(E+F)2-6S2, 3S2).
X(8366) lies on these lines: {2,3}, {6,7870}, {524,7881}, {543,7851}, {597,7763}, {598,7899}, {599,7832}, {671,7942}, {1384,7931}, {1975,7817}, {2482,7834}, {3053,7883}, {3849,7867}, {5023,7944}, {5210,7937}, {5215,7815}, {5346,6680}, {5569,6292}, {5969,7786}, {6704,7619}, {7610,7857}, {7775,7874}, {7778,7812}, {7810,7868}, {7827,7835}, {7840,7945}
X(8366) = midpoint of X(5346) and X(7801)
As a point on the Euler line, X(8367) has Shinagawa coefficients (4(E+F)2+3S2, 3S2).
X(8367) lies on these lines: {2,3}, {141,7775}, {524,3934}, {543,6683}, {597,5305}, {598,7750}, {599,2548}, {3055,7820}, {3589,7817}, {3815,7801}, {5461,6704}, {7617,7834}, {7745,7810}, {7767,7812}
X(8367) = midpoint of X(i) and X(j) for these {i,j}: {7745,7810}, {7767,7812}, {8359,8370}
X(8367) = complement of X(8359)
X(8367) = {X(2),X(5)}-harmonic conjugate of X(8360)
As a point on the Euler line, X(8368) has Shinagawa coefficients (4(E+F)2-5S2, 3S2).
X(8368) lies on these lines: {2,3}, {230,7820}, {524,7880}, {538,5305}, {620,3589}, {3314,3793}, {3629,7908}, {5306,7801}, {6390,7757}, {7745,7874}, {7750,7930}, {7762,7945}, {7767,7832}
X(8368) = midpoint of X(i) and X(j) for these {i,j}: {2,8369}, {5306,7801}
X(8368) = reflection of X(i) in X(j) for these (i,j): (2,8365), (8360,2)
X(8368) = complement of X(33184)
X(8368) = anticomplement of X(33213)
X(8368) = {X(2),X(4)}-harmonic conjugate of X(33240)
X(8368) = {X(2),X(20)}-harmonic conjugate of X(33196)
X(8368) = orthocentroidal-circle-inverse of X(33240)
As a point on the Euler line, X(8369) has Shinagawa coefficients (2(E+F)2-3S2, 3S2).
X(8369) lies on these lines: {2,3}, {6,6390}, {32,524}, {39,597}, {69,1384}, {83,5503}, {99,7792}, {141,187}, {230,3734}, {325,3972}, {525,6041}, {538,5306}, {543,5254}, {574,3589}, {598,7752}, {599,3053}, {620,3815}, {626,3849}, {671,7828}, {754,7880}, {1975,5305}, {1992,3926}, {3398,5182}, {3618,5024}, {3629,5008}, {3763,5210}, {3788,7745}, {5007,7863}, {5013,7618}, {5023,7800}, {5206,7822}, {5215,7749}, {5461,7886}, {5569,7815}, {6055,6248}, {6781,7853}, {7622,7808}, {7737,7778}, {7747,7874}, {7750,7832}, {7756,7852}, {7762,7836}, {7782,7846}, {7787,7891}, {7802,7930}, {7823,7945}, {7830,7915}, {7862,8176}
X(8369) = midpoint of X(i) and X(j) for these {i,j}: {2,1003}, {32,7801}, {7816,7817}
X(8369) = reflection of X(i) in X(j) for these (i,j): (2,8368), (3933,7801), (5254,7817), (7801,7789), (7817,6680), (7841,8360), (8360,8365), (33184,2)
X(8369) = complement X(7841)
X(8369) = anticomplement(X(8360)
X(8369) = orthocentroidal-circle-inverse of X(11318)
X(8369) = {X(2),X(3)}-harmonic conjugate of X(8359)
X(8369) = {X(2),X(4)}-harmonic conjugate of X(11318)
X(8369) = {X(2),X(20)}-harmonic conjugate of X(33190)
X(8369) = {X(37172),X(37173)}-harmonic conjugate of X(20)
As a point on the Euler line, X(8370) has Shinagawa coefficients ((E+F)2, 3S2).
X(8370) lies on these lines: {2,3}, {39,543}, {76,524}, {83,597}, {99,3815}, {115,5939}, {141,316}, {148,3329}, {183,7737}, {230,3972}, {315,599}, {325,3734}, {495,4366}, {496,6645}, {538,7753}, {542,6248}, {620,7603}, {625,7820}, {1506,2482}, {1975,2548}, {1992,7754}, {3053,7610}, {3589,7790}, {3631,7850}, {3767,7615}, {3788,8176}, {3849,3934}, {3933,7785}, {5032,5395}, {5206,5569}, {5286,7620}, {5305,7787}, {5461,6680}, {6292,7842}, {6390,7777}, {6683,7756}, {7617,7746}, {7748,7808}, {7752,7789}, {7767,7823}, {7773,7795}, {7794,7843}, {7822,7825}, {7861,7889}
X(8370) = midpoint of X(i) and X(j) for these {i,j}: {2,11361}, {76,7812}, {7747,7810}
X(8370) = reflection of X(i) in X(j) for these (i,j): (7750,7810), (7762,7812), (7810,3934), (7812,7745), (7833,8359), (8353,8356), (8356,2), (8359,8367)
X(8370) = complement X(7833)
X(8370) = anticomplement(X(8359)
X(8370) = orthocentroidal-circle-inverse of X(7841)
X(8370) = sum of vertices of triangle A"B"C" defined at X(5640)
X(8370) = {X(2),X(4)}-harmonic conjugate of X(7841)
X(8370) = {X(2),X(20)}-harmonic conjugate of X(33215)
Let HP be the Hutson-Parry circle (see X(5466)), with center X(8371) and pass-through point X(2). HP passes through the point X(k) for k = 2, 111, 476, 1144, 1171, 1523, 1525, 1527, 1564, 3850, 5373, 5394, 5466, 5626, 5640, 6032, 6792, 7698, 9140, 9159, 11628, 11639, 11640, 13636, 13722, 14846, 14932, 16446, 34320. Also, HP meets the circumcircle at X(111) and X(476), and the Parry circle at X(111) and X(2). Inverse pairs on HP include {115,1648}, {125,868}, and {1637,1640}. The radius of HP is
2/9*|sqrt(9*R^2-2*SW)*S*(SW^2-3*S^2)/((b^2-c^2)*(c^2-a^2)*(a^2-b^2))|.
X(8371) and properties of HP are contributed by César Eliud Lozada, October 31, 2015.
X(8371) lies on the radical axis of the orthocentroidal circle and the McCay circumcircle. (Randy Hutson, November 18, 2015)
The circle HP is tangent to the Euler line at X(2). (Randy Hutson, November 18, 2015)
Also, HP is the Parry circle of the orthocentroidal triangle. (Randy Hutson, December 26, 2015)
X(8371) lies on the cubic K218 and these lines: (2,523), (115,125), (351,2793), (373,512), (381,1499), (669,1995), (2501,5094), (5070,8151), (5544,5652)
X(8371) = midpoint of X(i) and X(j) for these {i,j}: {2,5466}, {1649,8029}
X(8371) = reflection of X(i) in X(j) for these (i,j): (1649,2), (8029,5466)
X(8371) = pole of line X(2)X(6) wrt orthocentroidal circle
X(8371) = crossdifference of every pair of points on line X(110)X(187)
X(8371) = X(351) of orthocentroidal triangle
X(8371) = centroid of degenerate side-triangle of outer and inner Napoleon triangles
Perspectors of pairs of triangles: X(8372)-X(8424)
Central triangles A'B'C' and A''B''C'' are perspective if the lines A'A'', B'B'', C'C'' concur. The point of concurrence is their perspector (or center of perspective). If A'B'C' and A''B''C'' are not only perspective, but homothetic, their perspector is their homothetic center. Centers X(8372)-X(8424) are perspectors, including homothetic centers. Contributed by César Eliud Lozada, November 2, 2015. See Perspectivities in Tables (accessible at the top of this page).
X(8372) lies on the Feuerbach hyperbola and these lines: {1,8111}, {9,164}, {21,8110}, {266,8138}, {2346,8386}
X(8373) lies on these lines: {2,494}, {184,5058}, {353,8375}
X(8374) lies on these lines: {2,493}, {184,5062}, {353,8376}
X(8375) lies on these lines: {3,6}, {353,8373}, {486,3054}, {493,1383}, {3055,5418}, {8289,8302}, {8296,8318}, {8297,8334}
X(8375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,187,8376), (6,1151,574), (8398,8403,3)
X(8376) lies on these lines: {3,6}, {353,8374}, {485,3054}, {494,1383}, {3055,5420}, {8289,8303}, {8296,8319}, {8297,8335}
X(8376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,187,8375), (6,1152,574), (8411,8418,3)
X(8377) lies on these lines: {2,8107}, {4,8109}, {5,8380}, {11,8113}, {12,8390}, {363,1699}, {6732,8086}, {7678,8385}, {7988,8140}, {8085,8133}, {8111,8227}
X(8378) lies on these lines: {2,8108}, {4,8110}, {5,8381}, {11,8114}, {12,8392}, {168,1699}, {7678,8386}, {7707,8379}, {7988, 8140}, {8085,8135}, {8086,8138}, {8112,8227}
X(8379) lies on these lines: {1,8087}, {2,7589}, {4,7587}, {5,8351}, {11,174}, {173,1699}, {177,8085}, {226,8083}, {236,2886}, {496,8092}, {497,8076}, {1210,8094}, {3086,7588}, {3816,7028}, {7590,8227}, {7593,8226}, {7678,8389}, {7707,8378}, {7988,8423}, {8229,8425}
X(8380) lies on these lines: {2,8109}, {4,8107}, {5,8377}, {11,8390}, {12,8113}, {363,1698}, {442,5934}, {5587,8111}, {6732,8088}, {7679,8385}, {7989,8140}, {8087,8133}
X(8381) lies on these lines: {2,8110}, {4,8108}, {5,8378}, {11,8392}, {12,8114}, {168,1698}, {442,5935}, {5587,8112}, {7679,8386}, {7707,8382}, {7989,8140}, {8087,8135}, {8088,8138}
X(8382) lies on these lines: {1,8085}, {2,7587}, {4,7589}, {5,8351}, {12,174}, {173,1698}, {177,8087}, {226,8094}, {236,1329}, {258,5290}, {388,7588}, {442,7593}, {495,8092}, {1210,8083}, {3085,8076}, {5051,8425}, {5587,7590}, {7679,8389}, {7707,8381}, {7989,8423}
X(8383) lies on these lines: {6,3156}, {376,5861}, {3917,8384}
X(8384) lies on these lines: {6,3155}, {376,5860}, {3917,8383}
X(8385) lies on these lines: {7,8113}, {363,1445}, {4326,8140}, {5934,8232}, {6732,8388}, {7675,8111}, {7676,8107}, {7677,8109}, {7678,8377}, {7679,8380}, {8133,8387}, {8236,8390}, {8238,8391}
X(8386) lies on these lines: {7,174}, {168,1445}, {2346,8372}, {4326,8140}, {5935,8232}, {7675,8112}, {7676,8108}, {7677,8110}, {7678,8378}, {7679,8381}, {8135,8387}, {8236,8392}
X(8387) lies on these lines: {1,7670}, {7,1488}, {177,2346}, {188,7022}, {390,8091}, {1156,8103}, {1445,8078}, {4326,8089}, {5728,8099}, {7672,8093}, {7675,8081}, {7676,8075}, {7677,8077}, {7678,8085}, {7679,8087}, {8079,8232}, {8133,8385}, {8135,8386}, {8236,8241}, {8237,8247}, {8238,8249}
X(8388) lies on these lines: {1,7670}, {7,174}, {9,8125}, {142,8126}, {258,1445}, {390,8092}, {1156,8104}, {2346,7589}, {4326,8090}, {5728,8100}, {5759,8129}, {6732,8385}, {7588,7677}, {7672,8094}, {7675,8082}, {7676,8076}, {7678,8086}, {7679,8088}, {8080,8232}, {8236,8242}, {8237,8248}, {8238,8250}
X(8389) lies on these lines: {7,174}, {9,8126}, {142,8125}, {173,1445}, {177,2346}, {390,8351}, {4326,8423}, {5759,8130}, {7587,7677}, {7589,7676}, {7590,7675}, {7593,8232}, {7678,8379}, {7679,8382}, {8238,8425}
X(8390) lies on these lines: {1,8111}, {11,8380}, {12,8377}, {55,8109}, {56,8107}, {188,6732}, {363,1697}, {950,5934}, {8133,8241}, {8236,8385}, {8240,8391}
X(8391) lies on these lines: {21,8109}, {363,846}, {1284,8113}, {6732,8250}, {8111,8235}, {8133,8249}, {8238,8385}, {8240,8390}
X(8392) lies on these lines: {1,8111}, {8,178}, {11,8381}, {12,8378}, {55,8110}, {56,8108}, {168,1697}, {177,7962}, {950,5935}, {8135,8241}, {8138,8242}, {8236,8386}
X(8393) lies on these lines: {1,6462}, {37,493}, {42,6339}, {63,2066}, {326,1267}, {1124,6461}, {3057,8210}
X(8394) lies on these lines: {1,6463}, {37,494}, {42,6339}, {63,5414}, {326,3084}, {1335,6461}, {3057,8211}
X(8395) lies on these lines: {1588,3523}
X(8396) lies on these lines: {3,6}, {637,7866}, {1271,7388}, {1588,5591}, {1702,3641}, {3068,6202}, {5871,6459}, {8302,8304}, {8318,8320}, {8334,8336}
X(8396) = {X(3),X(6421)}-harmonic conjugate of X(8400)
X(8397) lies on these lines: {3,6}, {8302,8306}, {8318,8322}, {8334,8338}
X(8398) lies on these lines: {3,6}, {8302,8310}, {8318,8326}, {8334,8342}
X(8398) = {X(3),X(8375)}-harmonic conjugate of X(8403)
X(8399) lies on these lines: {3,6}, {8302,8314}, {8318,8330}, {8334,8346}
X(8400) lies on these lines: {3,6}, {8302,8305}, {8318,8321}, {8334,8337}
X(8400) = {X(3),X(6421)}-harmonic conjugate of X(8396)
X(8401) lies on these lines: {3,6}, {8302,8307}, {8318,8323}, {8334,8339}
X(8402) lies on these lines: {3,6}, {8302,8309}, {8318,8325}, {8334,8341}
X(8403) lies on these lines: {3,6}, {8302,8311}, {8318,8327}, {8334,8343}
X(8403) = {X(3),X(8375)}-harmonic conjugate of X(8398)
X(8404) lies on these lines: {3,6}, {1131,5395}, {3317,7612}, {8302,8313}, {8318,8329}, {8334,8345}
X(8405) lies on these lines: {3,6}, {8302,8315}, {8318,8331}, {8334,8347}
X(8406) lies on these lines: {3,6}, {5420,6290}, {8302,8317}, {8318,8333}, {8334,8349}
X(8406) = {X(182),X(5013)}-harmonic conjugate of X(8414)
X(8407) lies on these lines: {3,6}, {8303,8304}, {8319,8320}, {8335,8336}
X(8408) lies on these lines: {3,493}, {193,371}, {1151,6465}, {3555,8210}
X(8409) lies on these lines: {3,6}, {8303,8306}, {8319,8322}, {8335,8338}
X(8410) lies on these lines: {3,6}, {8303,8308}, {8319,8324}, {8335,8340}
X(8411) lies on these lines: {3,6}, {8303,8310}, {8319,8326}, {8335,8342}
X(8411) = reflection of X(i) in X(j) for these (i,j): (6424,5165)
X(8412) lies on these lines: {3,6}, {1132,5395}, {3316,7612}, {8303,8312}, {8335,8344}
X(8413) lies on these lines: {3,6}, {8303,8314}, {8319,8330}, {8335,8346}
X(8414) lies on these lines: {3,6}, {5418,6289}, {8303,8316}, {8319,8332}, {8335,8348}
X(8414) = reflection of X(6425) in X(5111)
X(8415) lies on these lines: {1587,3523}
X(8416) lies on these lines: {3,6}, {638,7866}, {1270,7389}, {1587,5590}, {1703,3640}, {3069,6201}, {5870,6460}, {8303,8305}, {8319,8321}, {8335,8337}
X(8417) lies on these lines: {3,6}, {8303,8307}, {8319,8323}, {8335,8339}
X(8418) lies on these lines: {3,6}, {8303,8311}, {8319,8327}, {8335,8343}
X(8419) lies on these lines: {3,6}, {8303,8315}, {8319,8331}, {8335,8347}
X(8420) lies on these lines: {3,494}, {193,372}, {1152,6466}, {3555,8211}
X(8421) lies on these lines: {8,192}, {21,1149}, {846,988}, {986,5084}
X(8422) lies on these lines: {1,167}, {8,7048}, {65,2091}, {100,260}, {145,8094}, {164,1697}, {188,258}, {390,8084}, {944,8096}, {3340,8114}, {7670,8236}
X(8422) = reflection of X(i) in X(j) for these (i,j): (65,5571), (177,1)
X(8422) = X(40)-of-intouch-triangle
X(8422) = {X(8241),X(8242)}-harmonic conjugate of X(1)
X(8423) lies on these lines: {1,167}, {165,173}, {200,8126}, {258,8083}, {1130,6726}, {1750,7593}, {3062,7707}, {4326,8389}, {6769,8130}, {7587,7987}, {7988,8379}, {7989,8382}, {8245,8425}
X(8424) lies on the cubic K132 and these lines: {3,3923}, {6,256}, {7,21}, {9,8245}, {37,171}, {55,192}, {75,1281}, {198,1376}, {199,4418}, {284,6007}, {371,8320}, {372,8321}, {958,1503}, {960,8235}, {993,2792}, {1030,4436}, {1151,8318}, {1152,8319}, {1220,4026}, {1631,4363}, {1716,4426}, {1901,4199}, {2292,5710}, {3511,8053}, {4361,4471}, {4425,4657}, {6429,8322}, {6430,8323}, {6431,8330}, {6432,8331}
X(8424) = perspector of tangential triangle and obverse triangle of X(1)
X(8425) lies on these lines: {21,177}, {173,846}, {174,1284}, {256,7707}, {4199,7593}, {4220,7589}, {5051,8382}, {8229,8379}, {8238,8389}, {8245,8423}
X(8426) lies on the orthocentroidal circle, the Moses-Parry circle, and these lines: {2,2593}, {4,8105}, {111,1345}, {112,1344}, {115,1347}, {187,1113}, {1312,5913}, {1313,5523}, {1346,1560}, {5094,8427}
X(8427) lies on the orthocentroidal circle, the Moses-Parry circle, and these lines: {2,2592}, {4,8106}, {111,1344}, {112,1345}, {115,1346}, {187,1114}, {1313,5913}, {1312,5523}, {1347,1560}, {5094,8426}
X(8428) and X(2079) are the points of intersection of the tangential circle and the Moses-Parry circle, denoted by MP and defined here as the circle tangent to the Parry circle and which has center X(2492). MP passes through X(i) for these i: 6, 111, 112, 115, 187, 1560, 2079, 3569, 5000, 5001, 5523, 5913, 6032, 8105, 8106, 8426, 8427. MP is mentioned briefly in TCCT, p. 228. (Based on notes from Peter Moses, November 2, 2015)
MP is orthogonal to each circle in the coaxal system of the circumcircle and nine-point circle.
MP is orthogonal to circumcircle at X(111) and X(112).
MP is orthogonal to nine-point circle at X(115) and X(1560)
MP is orthogonal to orthocentroidal circle at X(8426) and X(8427).
MP is orthogonal to the tangential circle at X(2079) and X(8428)
MP is orthogonal to the Moses circle at X(115)
MP meets the Hutson-Parry circle at X(111) and X(6032)
MP meets the Brocard circle at X(6) and X(8429)
MP has radius = a^3 b^3 c^3 J^2 / (2 (a^2-b^2) (b^2-c^2) (c^2-a^2) (a^2+b^2+c^2)), where J = |OH|/R
MP has A-power = b^2 c^2 (a^2-b^2-c^2) (2 a^2-b^2-c^2) / (2 (a^2-b^2) (a^2-c^2) (a^2+b^2+c^2))
X(8428) lies on the tangential circle, the cubic K108, and these lines: {3,1560}, {6,1112}, {22,112}, {23,5523}, {24,111}, {25,115}, {186,5913}, {378,6032}, {1609,6103}, {1611,2079}, {2931,3569}
X(8428) = isogonal conjugate of isotomic conjugate of X(34163)
X(8428) = Dao-Moses-Telv-circle-inverse of X(34131)
X(8428) = circumcircle-inverse of X(1560)
X(8428) = X(i)-Ceva conjugate of X(j) for these (i,j): (23,25), (5523,6)
X(8429) lies on the Brocard circle and these lines: {3,3569}, {6,157}, {115,1316}, {5523,7473}, {6032,6232}
X(8429) = circumcircle-inverse of X(3569)
X(8429) = Parry-circle-inverse of X(35901)
X(8429) = X(2871)-vertex conjugate of X(3569)
X(8429) = X(2857) of 1st Brocard triangle
X(8430) lies on the Moses-Parry circle and these lines: {6,512}, {111,647}, {112,250}, {115,523}, {187,2492}, {262,5466}, {325,2799}, {511,3569}, {526,5107}, {1560,2501}, {1637,5913}, {2079,2485}
X(8430) = reflection of X(187) in X(2942)
X(8430) = antipode of X(187) in Moses-Parry circle
X(8430) = X(i)-isoconjugate of X(j) for these {i,j}: {293,4235}, {662,5967}, {896,2966}, {1821,5467}, {1910,5468}
X(30)-Ceva conjugates: X(8431)-X(8533 )
Centers X(8431)-X(8533) are contributed by Peter Moses, November 4, 2015. They lie on the Neuberg cubic, K001 in Bernard Gibert's Cubics in the Triangle Plane. Each of these centers is the perspector of two triangles from a list of 25 triangles found in Table 19.1. That list is reproduced in the following table, in which the isogonal conjugate of a point P is denoted by P*. Triangles T5 and T6 are named in TCCT, p. 178.
| triangle | A-vertex |
|---|---|
| T1 = ABC, the reference triangle | A |
| T2 = excentral triangle | Ia = A-excenter |
| T3 = reflection triangle | A' = reflection of A in sideline BC |
| T4 | AX(3)∩A'X(30) |
| T5 = outer Fermat triangle | Ae |
| T6 = inner Fermat triangle | Ai |
| T7 | Ae* |
| T8 | Ai* |
| T9 = cevian triangle of X(30) | Ao |
| T10 | A1 = A'X(1)∩IaX(3) |
| T11 | A2 = AoX(74)∩BoCo |
| T12 | A3 = AoX(3)∩A'X(74) |
| T13 | A4 = A'X(3)∩B'C' |
| T14 | A5 = AoX(1)∩IaX(74) |
| T15 | A1* |
| T16 | A2* |
| T17 | A3* |
| T18 | A4* |
| T19 | A5* |
| T20 | Ae' |
| T21 | Ai' |
| T22 | Ae'* |
| T23 | Ai'* |
| T24 | A6 |
| T25 | A6* |
Suppose that P and Q are points. The cevian triangle of P is perspective to the anticevian triangle of Q. If the perspector is written as θP(Q) then θP(θP(Q)) = Q, so that the mapping θP is a conjugacy. The term Ceva conjugate appeared as early as 1998 in TCCT (p. 57-58); the term cevian quotient appeared in Hyacinthos in 2000. See Francisco Javier Garcia Capitán, "Some Simple Results on Cevian Quotients", Forum Geometricorum 13 (2013) 227-231: Abstract and access to a pdf
X(8431) is the perspector of the triangle pairs {T1, T24} and {T3, T11} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8431) lies on the Neuberg cubic K001 and these lines: {1, 5680}, {3, 2132}, {4, 8443}, {13, 8458}, {14, 8448}, {15, 8453}, {16, 8463}, {30, 5667}, {74, 8440}, {399, 3484}, {484, 8485}, {616, 5681}, {617, 5682}, {1138, 5676}, {1157, 5683}, {1263, 5670}, {1276, 8459}, {1277, 8449}, {1337, 5678}, {1338, 5679}, {2133, 8517}, {3065, 8432}, {3284, 5668}, {3440, 8516}, {3441, 8515}, {3464, 3465}, {3479, 8532}, {3480, 8533}, {3482, 5671}, {3483, 5677}, {5623, 8174}, {5624, 8175}, {5674, 8441}, {5675, 8442}, {7164, 8527}, {7327, 8512}, {7329, 8503}, {8435, 8509}, {8436, 8508}, {8444, 8468}, {8446, 8465}, {8447, 8466}, {8454, 8476}, {8456, 8473}, {8457, 8474}, {8470, 8478}, {8480, 8504}, {8481, 8525}, {8482, 8526}, {8483, 8523}, {8484, 8524}, {8487, 8510}, {8494, 8511}, {8495, 8513}, {8496, 8514}, {8497, 8519}, {8498, 8520} X(8431) = isogonal conjugate of X(5667)X(8432) is the perspector of the triangle pairs {T2, T16}, {T9, T19} and {T15, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8432) lies on the Neuberg cubic K001 and these lines: {1, 2133}, {3, 7328}, {4, 7327}, {30, 8488}, {74, 7164}, {484, 8486}, {1138, 3466}, {3065, 8431}, {3465, 8493}, {7059, 8455}, {7060, 8445}, {7165, 8487}, {7325, 8452}, {7326, 8462}, {7329, 8439}, {8489, 8501}, {8490, 8502}, {8491, 8499}, {8492, 8500} X(8432) = isogonal conjugate of X(8488) X(8432) = X(30)-Ceva conjugate of X(7164)X(8433) is the perspector of the triangle pair {T2, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8433) lies on the Neuberg cubic K001 and these lines: {1, 1337}, {3, 8482}, {4, 8483}, {13, 1276}, {16, 8444}, {30, 8501}, {74, 8435}, {484, 8495}, {617, 3065}, {1138, 8508}, {1263, 8434}, {2133, 8525}, {3440, 3465}, {3464, 8491}, {3466, 8441}, {3480, 3483}, {3484, 8500}, {5668, 7326}, {5673, 8446}, {5675, 7329}, {5679, 7327}, {5681, 7164}, {7059, 8174}, {7060, 8451}, {7165, 8497}, {7325, 8478}, {7328, 8532}, {8438, 8480}, {8445, 8449}, {8450, 8454}, {8485, 8490}, {8487, 8524}, {8488, 8515}, {8494, 8507}, {8502, 8530} X(8433) = isogonal conjugate of X(8501)X(8434) is the perspector of the triangle pair {T2, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8434) lies on the Neuberg cubic K001 and these lines: {1, 1338}, {3, 8481}, {4, 8484}, {14, 1277}, {15, 8454}, {30, 8502}, {74, 8436}, {484, 8496}, {616, 3065}, {1138, 8509}, {1263, 8433}, {2133, 8526}, {3441, 3465}, {3464, 8492}, {3466, 8442}, {3479, 3483}, {3484, 8499}, {5669, 7325}, {5672, 8456}, {5674, 7329}, {5678, 7327}, {5682, 7164}, {7059, 8461}, {7060, 8175}, {7165, 8498}, {7326, 8470}, {7328, 8533}, {8437, 8480}, {8444, 8460}, {8455, 8459}, {8485, 8489}, {8487, 8523}, {8488, 8516}, {8494, 8506}, {8501, 8528} X(8434) = isogonal conjugate of X(8502)X(8435) is the perspector of the triangle pair {T2, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8435) lies on the Neuberg cubic K001 and these lines: {1, 3479}, {4, 8481}, {15, 7060}, {30, 8499}, {74, 8433}, {484, 3441}, {616, 3466}, {1138, 8506}, {1157, 8502}, {1277, 8471}, {1338, 7165}, {2133, 8523}, {3065, 8437}, {3464, 8489}, {3465, 8529}, {3481, 8484}, {5672, 8462}, {5674, 7164}, {5678, 7328}, {5685, 8492}, {7059, 8460}, {7325, 8172}, {7326, 8469}, {7327, 8519}, {7329, 8521}, {8431, 8509}, {8436, 8439}, {8455, 8464}, {8488, 8513}, {8500, 8528} X(8435) = isogonal conjugate of X(8499)X(8436) is the perspector of the triangle pair {T2, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8436) lies on the Neuberg cubic K001 and these lines: {1, 3480}, {4, 8482}, {16, 7059}, {30, 8500}, {74, 8434}, {484, 3440}, {617, 3466}, {1138, 8507}, {1157, 8501}, {1276, 8479}, {1337, 7165}, {2133, 8524}, {3065, 8438}, {3464, 8490}, {3465, 8531}, {3481, 8483}, {5673, 8452}, {5675, 7164}, {5679, 7328}, {5685, 8491}, {7060, 8450}, {7325, 8477}, {7326, 8173}, {7327, 8520}, {7329, 8522}, {8431, 8508}, {8435, 8439}, {8445, 8472}, {8488, 8514}, {8499, 8530} X(8436) = isogonal conjugate of X(8500)X(8437) is the perspector of the triangle pair {T3, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8437) lies on the Neuberg cubic K001 and these lines: {1, 8483}, {3, 1337}, {4, 8497}, {13, 8174}, {14, 8451}, {16, 8447}, {30, 8495}, {74, 8441}, {617, 1263}, {1138, 5681}, {1276, 8444}, {2133, 8532}, {3065, 8435}, {3440, 3484}, {3465, 8501}, {3480, 3482}, {3483, 8482}, {5667, 8491}, {5668, 8446}, {5675, 8494}, {5679, 8487}, {5683, 8490}, {7327, 8525}, {7329, 8508}, {8434, 8480}, {8445, 8453}, {8450, 8457}, {8456, 8478}, {8486, 8515}, {8496, 8530} X(8437) = isogonal conjugate of X(8495)X(8438) is the perspector of the triangle pair {T3, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8438) lies on the Neuberg cubic K001 and these lines: {1, 8484}, {3, 1338}, {4, 8498}, {13, 8461}, {14, 8175}, {15, 8457}, {30, 8496}, {74, 8442}, {616, 1263}, {1138, 5682}, {1277, 8454}, {2133, 8533}, {3065, 8436}, {3441, 3484}, {3465, 8502}, {3479, 3482}, {3483, 8481}, {5667, 8492}, {5669, 8456}, {5674, 8494}, {5678, 8487}, {5683, 8489}, {7327, 8526}, {7329, 8509}, {8433, 8480}, {8446, 8470}, {8447, 8460}, {8455, 8463}, {8486, 8516}, {8495, 8528} X(8438) = isogonal conjugate of X(8496)X(8439) is the perspector of the triangle pairs {T3, T24} and {T11, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8439) lies on the Neuberg cubic K001 and these lines: {1, 8485}, {3, 3462}, {4, 3463}, {13, 8463}, {14, 8453}, {30, 3484}, {74, 8443}, {399, 3482}, {616, 8441}, {617, 8442}, {1138, 8440}, {1263, 2132}, {1337, 5682}, {1338, 5681}, {3065, 5680}, {3440, 8533}, {3441, 8532}, {3464, 3483}, {5623, 8447}, {5624, 8457}, {5668, 8175}, {5669, 8174}, {5670, 8494}, {5674, 8497}, {5675, 8498}, {5676, 8487}, {5677, 8480}, {5678, 8495}, {5679, 8496}, {7327, 8527}, {7329, 8432}, {8435, 8436}, {8444, 8459}, {8446, 8458}, {8448, 8456}, {8449, 8454}, {8451, 8470}, {8461, 8478}, {8483, 8509}, {8484, 8508}, {8486, 8517}, {8491, 8516}, {8492, 8515}, {8501, 8526}, {8502, 8525} X(8439) = isogonal conjugate of X(3484)X(8440) is the perspector of the triangle pairs {T4, T16} and {T9, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8440) lies on the Neuberg cubic K001 and these lines: {1, 7328}, {4, 2133}, {30, 8493}, {74, 8431}, {484, 8488}, {1138, 8439}, {1157, 8486}, {3440, 8489}, {3441, 8490}, {3466, 7164}, {3481, 8487}, {7165, 7327}, {8445, 8471}, {8452, 8462}, {8455, 8479}, {8491, 8529}, {8492, 8531} X(8440) = isogonal conjugate of X(8493)X(8441) is the perspector of the triangle pair {T4, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8441) lies on the Neuberg cubic K001 and these lines: {4, 3479}, {15, 8471}, {30, 8529}, {74, 8437}, {399, 8489}, {484, 8499}, {616, 8439}, {1138, 8521}, {1157, 3441}, {1338, 3481}, {2133, 8519}, {3466, 8433}, {5674, 8431}, {5684, 8492}, {7164, 8506}, {7165, 8481}, {7328, 8523}, {8172, 8462}, {8452, 8469}, {8455, 8467}, {8460, 8479}, {8493, 8513}, {8528, 8531} X(8441) = isogonal conjugate of X(8529)X(8442) is the perspector of the triangle pair {T4, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8442) lies on the Neuberg cubic K001 and these lines: {4, 3480}, {16, 8479}, {30, 8531}, {74, 8438}, {399, 8490}, {484, 8500}, {617, 8439}, {1138, 8522}, {1157, 3440}, {1337, 3481}, {2133, 8520}, {3466, 8434}, {5675, 8431}, {5684, 8491}, {7164, 8507}, {7165, 8482}, {7328, 8524}, {8173, 8452}, {8445, 8475}, {8450, 8471}, {8462, 8477}, {8493, 8514}, {8529, 8530} X(8442) = isogonal conjugate of X(8531)X(8443) is the perspector of the triangle pairs {T4, T25} and {T13, T16} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8443) lies on the Neuberg cubic K001 and these lines: {4, 8431}, {74, 8439}, {399, 8493}, {484, 7328}, {1138, 3481}, {1157, 2133}, {3440, 8529}, {3441, 8531}, {3479, 8490}, {3480, 8489}, {5684, 8486}, {5685, 8488}, {7164, 7165}, {8452, 8471}, {8462, 8479}, {8499, 8500} X(8443) = X(30)-Ceva conjugate of X(8439)X(8444) is the perspector of the triangle pair {T5, T10} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8444) lies on the Neuberg cubic K001 and these lines: {1, 8172}, {4, 5672}, {13, 8506}, {14, 5685}, {15, 484}, {16, 8433}, {74, 8464}, {399, 7060}, {1157, 1277}, {1276, 8437}, {3065, 8467}, {3441, 8472}, {3464, 8471}, {3466, 5623}, {3479, 5673}, {5624, 8499}, {5669, 7165}, {5671, 7325}, {5674, 7059}, {5677, 8462}, {5684, 8454}, {7164, 8466}, {7326, 8519}, {7328, 8465}, {8173, 8481}, {8431, 8468}, {8434, 8460}, {8439, 8459}, {8449, 8529}, {8452, 8523}, {8455, 8504}, {8456, 8505}, {8469, 8482}, {8475, 8502}, {8476, 8489}, {8479, 8509} X(8444) = X(30)-Ceva conjugate of X(8464)X(8445) is the perspector of the triangle pair {T5, T11} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8445) lies on the Neuberg cubic K001 and these lines: {1, 8468}, {3, 8466}, {4, 8458}, {13, 8513}, {14, 5670}, {15, 2132}, {16, 5678}, {30, 5623}, {74, 8465}, {399, 5669}, {484, 8459}, {616, 5624}, {1157, 8463}, {1276, 8523}, {1277, 5677}, {1338, 8474}, {3441, 8473}, {3464, 5672}, {3465, 8464}, {3479, 8448}, {3484, 8467}, {5667, 8172}, {5668, 5674}, {5671, 8175}, {5673, 8509}, {5675, 8470}, {5676, 8462}, {5679, 8460}, {5681, 8469}, {5682, 8173}, {7060, 8432}, {7325, 8503}, {8174, 8519}, {8433, 8449}, {8436, 8472}, {8437, 8453}, {8440, 8471}, {8442, 8475}, {8454, 8504}, {8455, 8510}, {8456, 8511}, {8457, 8518}, {8461, 8520}, {8476, 8481}, {8479, 8516} X(8445) = isogonal conjugate of X(5623)X(8446) is the perspector of the triangle pair {T5, T12} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8446) lies on the Neuberg cubic K001 and these lines: {1, 8464}, {3, 8467}, {4, 5623}, {13, 8519}, {14, 5671}, {15, 399}, {16, 5674}, {30, 8172}, {74, 8466}, {484, 5672}, {616, 8173}, {617, 8469}, {1157, 5669}, {1276, 8506}, {1277, 5685}, {1338, 8475}, {2132, 8471}, {3441, 8474}, {3466, 8468}, {3479, 5624}, {3481, 8463}, {5668, 8437}, {5670, 8462}, {5673, 8433}, {5675, 8460}, {5677, 7060}, {5678, 8479}, {5684, 8175}, {7059, 8523}, {7165, 8459}, {7325, 8504}, {8174, 8521}, {8431, 8465}, {8438, 8470}, {8439, 8458}, {8448, 8529}, {8452, 8513}, {8454, 8505}, {8455, 8511}, {8456, 8518}, {8461, 8522}, {8472, 8481}, {8473, 8489}, {8476, 8499}, {8477, 8528} X(8446) = isogonal conjugate of X(8172)X(8447) is the perspector of the triangle pair {T5, T13} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8447) lies on the Neuberg cubic K001 and these lines: {4, 8172}, {13, 8521}, {14, 5684}, {15, 1157}, {16, 8437}, {74, 8467}, {399, 8471}, {3441, 8475}, {3466, 8464}, {3479, 8173}, {3480, 8469}, {3481, 5669}, {5623, 8439}, {5624, 8529}, {5671, 8462}, {5672, 7165}, {5674, 8479}, {5685, 7060}, {7059, 8506}, {7325, 8505}, {8431, 8466}, {8438, 8460}, {8452, 8519}, {8455, 8518}, {8472, 8499}, {8474, 8489} X(8447) = X(30)-Ceva conjugate of X(8467)X(8448) is the perspector of the triangle pairs {T5, T16} and {T8, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8448) lies on the Neuberg cubic K001 and these lines: {4, 8455}, {13, 8489}, {14, 8431}, {15, 2133}, {74, 8462}, {1138, 8471}, {1277, 7328}, {3441, 8452}, {3466, 7325}, {3479, 8445}, {5669, 8493}, {5672, 8488}, {7060, 7164}, {7326, 8499}, {8172, 8486}, {8439, 8456}, {8446, 8529}, {8479, 8492} X(8448) = X(30)-Ceva conjugate of X(8462)X(8449) is the perspector of the triangle pair {T5, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8449) lies on the Neuberg cubic K001 and these lines: {1, 8462}, {4, 7325}, {13, 8499}, {14, 3466}, {15, 7164}, {74, 7060}, {484, 8455}, {1276, 8489}, {1277, 8431}, {2133, 5672}, {3065, 8471}, {3441, 7059}, {3479, 7326}, {5623, 8488}, {5669, 7328}, {7165, 8456}, {7327, 8172}, {8433, 8445}, {8439, 8454}, {8444, 8529}, {8452, 8481}, {8459, 8493}, {8464, 8486}, {8479, 8502} X(8449) = X(30)-Ceva conjugate of X(7060)X(8450) is the perspector of the triangle pair {T5, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8450) lies on the Neuberg cubic K001 and these lines: {3, 8460}, {4, 8461}, {13, 8528}, {14, 148}, {15, 1338}, {74, 8470}, {1263, 8469}, {1277, 8481}, {3441, 5669}, {3479, 8175}, {5623, 8492}, {5672, 8502}, {5674, 8456}, {5678, 8455}, {5682, 8462}, {7060, 8436}, {7325, 8509}, {8172, 8496}, {8433, 8454}, {8437, 8457}, {8442, 8471}, {8463, 8489} X(8450) = X(30)-Ceva conjugate of X(8470)X(8451) is the perspector of the triangle pair {T5, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8451) lies on the Neuberg cubic K001 and these lines: {4, 8460}, {14, 8437}, {15, 3479}, {74, 8469}, {616, 8471}, {3441, 8172}, {3481, 8461}, {5623, 8489}, {5669, 8529}, {5672, 8499}, {5674, 8462}, {7060, 8433}, {7325, 8506}, {8439, 8470}, {8455, 8519}, {8456, 8521}, {8467, 8492}, {8479, 8528} X(8451) = X(30)-Ceva conjugate of X(8469)X(8452) is the perspector of the triangle pairs {T5, T24} and {T8, T11} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8452) lies on the Neuberg cubic K001 and these lines: {1, 8459}, {3, 5623}, {4, 8463}, {13, 5678}, {14, 2132}, {15, 5667}, {16, 5682}, {30, 5669}, {74, 8458}, {399, 8175}, {616, 5668}, {617, 8470}, {1138, 8465}, {1263, 8466}, {1276, 8509}, {1277, 3464}, {1338, 5624}, {3065, 8468}, {3441, 8448}, {3465, 5672}, {3479, 8453}, {3482, 8467}, {3483, 8464}, {3484, 8172}, {5670, 8456}, {5671, 8457}, {5673, 8436}, {5674, 8174}, {5675, 8461}, {5676, 8455}, {5677, 8454}, {5680, 7060}, {5681, 8460}, {7059, 8526}, {7325, 8432}, {8173, 8442}, {8440, 8462}, {8441, 8469}, {8443, 8471}, {8444, 8523}, {8446, 8513}, {8447, 8519}, {8449, 8481}, {8472, 8484}, {8473, 8492}, {8474, 8496}, {8475, 8498}, {8476, 8502}, {8478, 8528}, {8479, 8533} X(8452) = isogonal conjugate of X(5669)X(8453) is the perspector of the triangle pair {T5, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8453) lies on the Neuberg cubic K001 and these lines: {4, 8462}, {13, 8529}, {14, 8439}, {15, 8431}, {16, 8489}, {74, 8471}, {1157, 8455}, {2133, 8172}, {3441, 8479}, {3466, 7060}, {3479, 8452}, {3481, 8456}, {5623, 8493}, {5672, 7328}, {7059, 8499}, {7165, 7325}, {8437, 8445}, {8460, 8490}, {8464, 8488}, {8467, 8486} X(8453) = X(30)-Ceva conjugate of X(8471)X(8454) is the perspector of the triangle pair {T6, T10} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8454) lies on the Neuberg cubic K001 and these lines: {1, 8173}, {4, 5673}, {13, 5685}, {14, 8507}, {15, 8434}, {16, 484}, {74, 8472}, {399, 7059}, {1157, 1276}, {1277, 8438}, {3065, 8475}, {3440, 8464}, {3464, 8479}, {3466, 5624}, {3480, 5672}, {5623, 8500}, {5668, 7165}, {5671, 7326}, {5675, 7060}, {5677, 8452}, {5684, 8444}, {7164, 8474}, {7325, 8520}, {7328, 8473}, {8172, 8482}, {8431, 8476}, {8433, 8450}, {8439, 8449}, {8445, 8504}, {8446, 8505}, {8459, 8531}, {8462, 8524}, {8467, 8501}, {8468, 8490}, {8471, 8508}, {8477, 8481} X(8454) = X(30)-Ceva conjugate of X(8472)X(8455) is the perspector of the triangle pair {T6, T11} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8455) lies on the Neuberg cubic K001 and these lines: {1, 8476}, {3, 8474}, {4, 8448}, {13, 5670}, {14, 8514}, {15, 5679}, {16, 2132}, {30, 5624}, {74, 8473}, {399, 5668}, {484, 8449}, {617, 5623}, {1157, 8453}, {1276, 5677}, {1277, 8524}, {1337, 8466}, {3440, 8465}, {3464, 5673}, {3465, 8472}, {3480, 8458}, {3484, 8475}, {5667, 8173}, {5669, 5675}, {5671, 8174}, {5672, 8508}, {5674, 8478}, {5676, 8452}, {5678, 8450}, {5681, 8172}, {5682, 8477}, {7059, 8432}, {7326, 8503}, {8175, 8520}, {8434, 8459}, {8435, 8464}, {8438, 8463}, {8440, 8479}, {8441, 8467}, {8444, 8504}, {8445, 8510}, {8446, 8511}, {8447, 8518}, {8451, 8519}, {8468, 8482}, {8471, 8515} X(8455) = isogonal conjugate of X(5624)X(8456) is the perspector of the triangle pair {T6, T12} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8456) lies on the Neuberg cubic K001 and these lines: {1, 8472}, {3, 8475}, {4, 5624}, {13, 5671}, {14, 8520}, {15, 5675}, {16, 399}, {30, 8173}, {74, 8474}, {484, 5673}, {616, 8477}, {617, 8172}, {1157, 5668}, {1276, 5685}, {1277, 8507}, {1337, 8467}, {2132, 8479}, {3440, 8466}, {3466, 8476}, {3480, 5623}, {3481, 8453}, {5669, 8438}, {5670, 8452}, {5672, 8434}, {5674, 8450}, {5677, 7059}, {5679, 8471}, {5684, 8174}, {7060, 8524}, {7165, 8449}, {7326, 8504}, {8175, 8522}, {8431, 8473}, {8437, 8478}, {8439, 8448}, {8444, 8505}, {8445, 8511}, {8446, 8518}, {8451, 8521}, {8458, 8531}, {8462, 8514}, {8464, 8482}, {8465, 8490}, {8468, 8500}, {8469, 8530} X(8456) = isogonal conjugate of X(8173)X(8457) is the perspector of the triangle pair {T6, T13} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8457) lies on the Neuberg cubic K001 and these lines: {4, 8173}, {13, 5684}, {14, 8522}, {15, 8438}, {16, 1157}, {74, 8475}, {399, 8479}, {3440, 8467}, {3466, 8472}, {3479, 8477}, {3480, 8172}, {3481, 5668}, {5623, 8531}, {5624, 8439}, {5671, 8452}, {5673, 7165}, {5675, 8471}, {5685, 7059}, {7060, 8507}, {7326, 8505}, {8431, 8474}, {8437, 8450}, {8445, 8518}, {8462, 8520}, {8464, 8500}, {8466, 8490} X(8457) = X(30)-Ceva conjugate of X(8475)X(8458) is the perspector of the triangle pairs {T6, T16} and {T7, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8458) lies on the Neuberg cubic K001 and these lines: {4, 8445}, {13, 8431}, {14, 8490}, {16, 2133}, {74, 8452}, {1138, 8479}, {1276, 7328}, {3440, 8462}, {3466, 7326}, {3480, 8455}, {5668, 8493}, {5673, 8488}, {7059, 7164}, {7325, 8500}, {8173, 8486}, {8439, 8446}, {8456, 8531}, {8471, 8491} X(8458) = X(30)-Ceva conjugate of X(8452)X(8459) is the perspector of the triangle pair {T6, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8459) lies on the Neuberg cubic K001 and these lines: {1, 8452}, {4, 7326}, {13, 3466}, {14, 8500}, {16, 7164}, {74, 7059}, {484, 8445}, {1276, 8431}, {1277, 8490}, {2133, 5673}, {3065, 8479}, {3440, 7060}, {3480, 7325}, {5624, 8488}, {5668, 7328}, {7165, 8446}, {7327, 8173}, {8434, 8455}, {8439, 8444}, {8449, 8493}, {8454, 8531}, {8462, 8482}, {8471, 8501}, {8472, 8486} X(8459) = X(30)-Ceva conjugate of X(7059)X(8460) is the perspector of the triangle pair {T6, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8460) lies on the Neuberg cubic K001 and these lines: {3, 8450}, {4, 8451}, {13, 148}, {14, 8530}, {16, 1337}, {74, 8478}, {1263, 8477}, {1276, 8482}, {3440, 5668}, {3480, 8174}, {5624, 8491}, {5673, 8501}, {5675, 8446}, {5679, 8445}, {5681, 8452}, {7059, 8435}, {7326, 8508}, {8173, 8495}, {8434, 8444}, {8438, 8447}, {8441, 8479}, {8453, 8490} X(8460) = X(30)-Ceva conjugate of X(8478)X(8461) is the perspector of the triangle pair {T6, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8461) lies on the Neuberg cubic K001 and these lines: {4, 8450}, {13, 8438}, {16, 3480}, {74, 8477}, {617, 8479}, {3440, 8173}, {3481, 8451}, {5624, 8490}, {5668, 8531}, {5673, 8500}, {5675, 8452}, {7059, 8434}, {7326, 8507}, {8439, 8478}, {8445, 8520}, {8446, 8522}, {8471, 8530}, {8475, 8491} X(8461) = X(30)-Ceva conjugate of X(8477)X(8462) is the perspector of the triangle pairs {T6, T24} and {T7, T11} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8462) lies on the Neuberg cubic K001 and these lines: {1, 8449}, {3, 5624}, {4, 8453}, {13, 2132}, {14, 5679}, {15, 5681}, {16, 5667}, {30, 5668}, {74, 8448}, {399, 8174}, {616, 8478}, {617, 5669}, {1138, 8473}, {1263, 8474}, {1276, 3464}, {1277, 8508}, {1337, 5623}, {3065, 8476}, {3440, 8458}, {3465, 5673}, {3480, 8463}, {3482, 8475}, {3483, 8472}, {3484, 8173}, {5670, 8446}, {5671, 8447}, {5672, 8435}, {5674, 8451}, {5675, 8175}, {5676, 8445}, {5677, 8444}, {5680, 7059}, {5682, 8450}, {7060, 8525}, {7326, 8432}, {8172, 8441}, {8440, 8452}, {8442, 8477}, {8443, 8479}, {8454, 8524}, {8456, 8514}, {8457, 8520}, {8459, 8482}, {8464, 8483}, {8465, 8491}, {8466, 8495}, {8467, 8497}, {8468, 8501}, {8470, 8530}, {8471, 8532} X(8462) = isogonal conjugate of X(5668)X(8463) is the perspector of the triangle pair {T6, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8463) lies on the Neuberg cubic K001 and these lines: {4, 8452}, {13, 8439}, {14, 8531}, {15, 8490}, {16, 8431}, {74, 8479}, {1157, 8445}, {2133, 8173}, {3440, 8471}, {3466, 7059}, {3480, 8462}, {3481, 8446}, {5624, 8493}, {5673, 7328}, {7060, 8500}, {7165, 7326}, {8438, 8455}, {8450, 8489}, {8472, 8488}, {8475, 8486} X(8463) = X(30)-Ceva conjugate of X(8479)X(8464) is the perspector of the triangle pair {T7, T15} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8464) lies on the Neuberg cubic K001 and these lines: {1, 8446}, {3, 7326}, {13, 3065}, {14, 8501}, {16, 7329}, {74, 8444}, {1138, 1276}, {1263, 7059}, {1277, 8491}, {1337, 7325}, {3440, 8454}, {3465, 8445}, {3466, 8447}, {3483, 8452}, {5668, 7327}, {5673, 8487}, {7060, 8495}, {7164, 8174}, {8435, 8455}, {8449, 8486}, {8453, 8488}, {8456, 8482}, {8457, 8500}, {8462, 8483}, {8479, 8480} X(8464) = X(30)-Ceva conjugate of X(8444)X(8465) is the perspector of the triangle pair {T7, T16} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8465) lies on the Neuberg cubic K001 and these lines: {13, 2133}, {16, 8486}, {74, 8445}, {1138, 8452}, {1276, 8488}, {3440, 8455}, {7059, 7327}, {7164, 7326}, {7328, 8444}, {8174, 8493}, {8431, 8446}, {8456, 8490}, {8462, 8491}, {8479, 8487} X(8465) = X(30)-Ceva conjugate of X(8445)X(8466) is the perspector of the triangle pair {T7, T17} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8466) lies on the Neuberg cubic K001 and these lines: {3, 8445}, {13, 1138}, {14, 8491}, {16, 8487}, {74, 8446}, {1263, 8452}, {1276, 7327}, {1337, 8455}, {2133, 8174}, {3065, 7326}, {3440, 8456}, {5668, 8486}, {7059, 7329}, {7164, 8444}, {7325, 8501}, {8431, 8447}, {8457, 8490}, {8462, 8495}, {8479, 8494} X(8466) = X(30)-Ceva conjugate of X(8446)X(8467) is the perspector of the triangle pair {T7, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8467) lies on the Neuberg cubic K001 and these lines: {3, 8446}, {13, 1263}, {14, 8495}, {16, 8494}, {74, 8447}, {1138, 8174}, {1276, 7329}, {1337, 8456}, {3065, 8444}, {3440, 8457}, {3482, 8452}, {3483, 7326}, {3484, 8445}, {5668, 8487}, {7059, 8480}, {7325, 8483}, {8175, 8491}, {8441, 8455}, {8451, 8492}, {8453, 8486}, {8454, 8501}, {8462, 8497} X(8467) = X(30)-Ceva conjugate of X(8447)X(8468) is the perspector of the triangle pair {T7, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8468) lies on the Neuberg cubic K001 and these lines: {1, 8445}, {13, 7164}, {16, 7327}, {74, 7326}, {1138, 7059}, {1276, 2133}, {3065, 8452}, {3440, 7325}, {3466, 8446}, {5668, 8488}, {5673, 8486}, {7060, 8491}, {7328, 8174}, {7329, 8479}, {8431, 8444}, {8454, 8490}, {8455, 8482}, {8456, 8500}, {8462, 8501} X(8468) = X(30)-Ceva conjugate of X(7326)X(8469) is the perspector of the triangle pair {T7, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8469) lies on the Neuberg cubic K001 and these lines: {13, 1337}, {16, 8495}, {74, 8451}, {617, 8446}, {1138, 8478}, {1263, 8450}, {1276, 8501}, {3440, 8174}, {3480, 8447}, {5668, 8491}, {5681, 8445}, {7059, 8483}, {7326, 8435}, {8441, 8452}, {8444, 8482}, {8456, 8530}, {8477, 8494}, {8479, 8497} X(8469) = X(30)-Ceva conjugate of X(8451)X(8470) is the perspector of the triangle pair {T7, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8470) lies on the Neuberg cubic K001 and these lines: {13, 3480}, {16, 3440}, {74, 8450}, {617, 8452}, {1138, 8477}, {1276, 8500}, {1337, 8479}, {5668, 8490}, {5675, 8445}, {7059, 8482}, {7326, 8434}, {8173, 8491}, {8174, 8531}, {8431, 8478}, {8438, 8446}, {8439, 8451}, {8462, 8530} X(8470) = X(30)-Ceva conjugate of X(8450)X(8471) is the perspector of the triangle pair {T7, T24} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8471) lies on the Neuberg cubic K001 and these lines: {3, 3166}, {13, 5667}, {14, 5681}, {15, 8441}, {16, 3484}, {30, 8174}, {74, 8453}, {399, 8447}, {616, 8451}, {617, 8175}, {1138, 8448}, {1263, 5624}, {1276, 3465}, {1277, 8435}, {1337, 5669}, {1338, 8478}, {2132, 8446}, {3065, 8449}, {3440, 8463}, {3464, 8444}, {3482, 8173}, {3483, 5673}, {5623, 8495}, {5672, 8483}, {5675, 8457}, {5679, 8456}, {5680, 7326}, {5683, 8479}, {7059, 8485}, {7325, 8525}, {7329, 8476}, {8172, 8497}, {8440, 8445}, {8442, 8450}, {8443, 8452}, {8454, 8508}, {8455, 8515}, {8458, 8491}, {8459, 8501}, {8461, 8530}, {8462, 8532}, {8472, 8480}, {8473, 8487}, {8474, 8494}, {8477, 8498} X(8471) = isogonal conjugate of X(8174)X(8472) is the perspector of the triangle pair {T8, T15} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8472) lies on the Neuberg cubic K001 and these lines: {1, 8456}, {3, 7325}, {13, 8502}, {14, 3065}, {15, 7329}, {74, 8454}, {1138, 1277}, {1263, 7060}, {1276, 8492}, {1338, 7326}, {3441, 8444}, {3465, 8455}, {3466, 8457}, {3483, 8462}, {5669, 7327}, {5672, 8487}, {7059, 8496}, {7164, 8175}, {8436, 8445}, {8446, 8481}, {8447, 8499}, {8452, 8484}, {8459, 8486}, {8463, 8488}, {8471, 8480} X(8472) = X(30)-Ceva conjugate of X(8454)X(8473) is the perspector of the triangle pair {T8, T16} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8473) lies on the Neuberg cubic K001 and these lines: {14, 2133}, {15, 8486}, {74, 8455}, {1138, 8462}, {1277, 8488}, {3441, 8445}, {7060, 7327}, {7164, 7325}, {7328, 8454}, {8175, 8493}, {8431, 8456}, {8446, 8489}, {8452, 8492}, {8471, 8487} X(8473) = X(30)-Ceva conjugate of X(8455)X(8474) is the perspector of the triangle pair {T8, T17} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8474) lies on the Neuberg cubic K001 and these lines: {3, 8455}, {13, 8492}, {14, 1138}, {15, 8487}, {74, 8456}, {1263, 8462}, {1277, 7327}, {1338, 8445}, {2133, 8175}, {3065, 7325}, {3441, 8446}, {5669, 8486}, {7060, 7329}, {7164, 8454}, {7326, 8502}, {8431, 8457}, {8447, 8489}, {8452, 8496}, {8471, 8494} X(8474) = X(30)-Ceva conjugate of X(8456)X(8475) is the perspector of the triangle pair {T8, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8475) lies on the Neuberg cubic K001 and these lines: {3, 8456}, {13, 8496}, {14, 1263}, {15, 8494}, {74, 8457}, {1138, 8175}, {1277, 7329}, {1338, 8446}, {3065, 8454}, {3441, 8447}, {3482, 8462}, {3483, 7325}, {3484, 8455}, {5669, 8487}, {7060, 8480}, {7326, 8484}, {8174, 8492}, {8442, 8445}, {8444, 8502}, {8452, 8498}, {8461, 8491}, {8463, 8486} X(8475) = X(30)-Ceva conjugate of X(8457)X(8476) is the perspector of the triangle pair {T8, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8476) lies on the Neuberg cubic K001 and these lines: {1, 8455}, {14, 7164}, {15, 7327}, {74, 7325}, {1138, 7060}, {1277, 2133}, {3065, 8462}, {3441, 7326}, {3466, 8456}, {5669, 8488}, {5672, 8486}, {7059, 8492}, {7328, 8175}, {7329, 8471}, {8431, 8454}, {8444, 8489}, {8445, 8481}, {8446, 8499}, {8452, 8502} X(8476) = X(30)-Ceva conjugate of X(7325)X(8477) is the perspector of the triangle pair {T8, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8477) lies on the Neuberg cubic K001 and these lines: {14, 1338}, {15, 8496}, {74, 8461}, {616, 8456}, {1138, 8470}, {1263, 8460}, {1277, 8502}, {3441, 8175}, {3479, 8457}, {5669, 8492}, {5682, 8455}, {7060, 8484}, {7325, 8436}, {8442, 8462}, {8446, 8528}, {8454, 8481}, {8469, 8494}, {8471, 8498} X(8477) = X(30)-Ceva conjugate of X(8461)X(8478) is the perspector of the triangle pair {T8, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8478) lies on the Neuberg cubic K001 and these lines: {14, 3479}, {15, 3441}, {74, 8460}, {616, 8462}, {1138, 8469}, {1277, 8499}, {1338, 8471}, {5669, 8489}, {5674, 8455}, {7060, 8481}, {7325, 8433}, {8172, 8492}, {8175, 8529}, {8431, 8470}, {8437, 8456}, {8439, 8461}, {8452, 8528} X(8478) = X(30)-Ceva conjugate of X(8460)X(8479) is the perspector of the triangle pair {T8, T24} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8479) lies on the Neuberg cubic K001 and these lines: {3, 3165}, {13, 5682}, {14, 5667}, {15, 3484}, {16, 8442}, {30, 8175}, {74, 8463}, {399, 8457}, {616, 8174}, {617, 8461}, {1138, 8458}, {1263, 5623}, {1276, 8436}, {1277, 3465}, {1337, 8470}, {1338, 5668}, {2132, 8456}, {3065, 8459}, {3441, 8453}, {3464, 8454}, {3482, 8172}, {3483, 5672}, {5624, 8496}, {5673, 8484}, {5674, 8447}, {5678, 8446}, {5680, 7325}, {5683, 8471}, {7060, 8485}, {7326, 8526}, {7329, 8468}, {8173, 8498}, {8440, 8455}, {8441, 8460}, {8443, 8462}, {8444, 8509}, {8445, 8516}, {8448, 8492}, {8449, 8502}, {8451, 8528}, {8452, 8533}, {8464, 8480}, {8465, 8487}, {8466, 8494}, {8469, 8497} X(8479) = isogonal conjugate of X(8175)X(8480) is the perspector of the triangle pair {T10, T13} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8480) lies on the Neuberg cubic K001 and these lines: {1, 5684}, {4, 5685}, {74, 8505}, {399, 7165}, {484, 1157}, {3464, 3481}, {3466, 5671}, {3479, 8507}, {3480, 8506}, {5677, 8439}, {7059, 8467}, {7060, 8475}, {7164, 8518}, {7328, 8511}, {8431, 8504}, {8433, 8438}, {8434, 8437}, {8464, 8479}, {8471, 8472}, {8481, 8522}, {8482, 8521}, {8499, 8520}, {8500, 8519}, {8523, 8531}, {8524, 8529} X(8480) = X(30)-Ceva conjugate of X(8505)X(8481) is the perspector of the triangle pair {T10, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8481) lies on the Neuberg cubic K001 and these lines: {1, 617}, {3, 8434}, {4, 8435}, {13, 5673}, {16, 1276}, {30, 8482}, {74, 8508}, {399, 8501}, {484, 1337}, {1138, 8524}, {1157, 8483}, {1263, 8507}, {1277, 8450}, {3065, 5675}, {3440, 3464}, {3465, 3480}, {3466, 5681}, {3483, 8438}, {5624, 7326}, {5667, 8500}, {5668, 7059}, {5677, 8491}, {5679, 7164}, {5680, 8490}, {5685, 8495}, {7060, 8478}, {7165, 8441}, {7327, 8514}, {7328, 8515}, {7329, 8520}, {8173, 8444}, {8431, 8525}, {8445, 8476}, {8446, 8472}, {8449, 8452}, {8454, 8477}, {8480, 8522}, {8485, 8531} X(8481) = isogonal conjugate of X(8482)X(8482) is the perspector of the triangle pair {T10, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8482) lies on the Neuberg cubic K001 and these lines: {1, 616}, {3, 8433}, {4, 8436}, {14, 5672}, {15, 1277}, {30, 8481}, {74, 8509}, {399, 8502}, {484, 1338}, {1138, 8523}, {1157, 8484}, {1263, 8506}, {1276, 8460}, {3065, 5674}, {3441, 3464}, {3465, 3479}, {3466, 5682}, {3483, 8437}, {5623, 7325}, {5667, 8499}, {5669, 7060}, {5677, 8492}, {5678, 7164}, {5680, 8489}, {5685, 8496}, {7059, 8470}, {7165, 8442}, {7327, 8513}, {7328, 8516}, {7329, 8519}, {8172, 8454}, {8431, 8526}, {8444, 8469}, {8455, 8468}, {8456, 8464}, {8459, 8462}, {8480, 8521}, {8485, 8529} X(8482) = isogonal conjugate of X(8481)X(8483) is the perspector of the triangle pair {T10, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8483) lies on the Neuberg cubic K001 and these lines: {1, 8437}, {4, 8433}, {74, 8506}, {399, 8499}, {484, 3479}, {616, 7165}, {1157, 8481}, {3065, 8521}, {3441, 5685}, {3464, 8529}, {3466, 5674}, {3481, 8436}, {5672, 8471}, {5677, 8489}, {5684, 8502}, {7059, 8469}, {7060, 8172}, {7164, 8519}, {7325, 8467}, {7328, 8513}, {8431, 8523}, {8439, 8509}, {8462, 8464}, {8492, 8505} X(8483) = X(30)-Ceva conjugate of X(8506)X(8484) is the perspector of the triangle pair {T10, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8484) lies on the Neuberg cubic K001 and these lines: {1, 8438}, {4, 8434}, {74, 8507}, {399, 8500}, {484, 3480}, {617, 7165}, {1157, 8482}, {3065, 8522}, {3440, 5685}, {3464, 8531}, {3466, 5675}, {3481, 8435}, {5673, 8479}, {5677, 8490}, {5684, 8501}, {7059, 8173}, {7060, 8477}, {7164, 8520}, {7326, 8475}, {7328, 8514}, {8431, 8524}, {8439, 8508}, {8452, 8472}, {8491, 8505} X(8484) = X(30)-Ceva conjugate of X(8507)X(8485) is the perspector of the triangle pairs {T10, T25} and {T13, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8485) lies on the Neuberg cubic K001 and these lines: {1, 8439}, {4, 3466}, {74, 7165}, {399, 7328}, {484, 8431}, {1157, 7164}, {2133, 5685}, {3065, 3481}, {3479, 8500}, {3480, 8499}, {5671, 8488}, {5677, 8493}, {5684, 7327}, {7059, 8471}, {7060, 8479}, {8433, 8490}, {8434, 8489}, {8481, 8531}, {8482, 8529}, {8486, 8505} X(8485) = X(30)-Ceva conjugate of X(7165)X(8486) is the perspector of the triangle pair {T11, T12} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8486) lies on the Neuberg cubic K001 and these lines: {1, 8503}, {3, 8511}, {4, 5676}, {15, 8473}, {16, 8465}, {30, 5670}, {74, 8510}, {399, 2132}, {484, 8432}, {616, 8514}, {617, 8513}, {1157, 8440}, {3464, 5677}, {3465, 8504}, {3466, 8512}, {3484, 8518}, {5623, 5624}, {5667, 5671}, {5668, 8466}, {5669, 8474}, {5672, 8476}, {5673, 8468}, {5674, 5679}, {5675, 5678}, {5680, 5685}, {5681, 8519}, {5682, 8520}, {5684, 8443}, {7165, 8527}, {8172, 8448}, {8173, 8458}, {8437, 8515}, {8438, 8516}, {8439, 8517}, {8449, 8464}, {8453, 8467}, {8459, 8472}, {8463, 8475}, {8485, 8505}, {8506, 8525}, {8507, 8526}, {8508, 8523}, {8509, 8524}, {8521, 8532}, {8522, 8533} X(8486) = isogonal conjugate of X(5670)X(8487) is the perspector of the triangle pair {T11, T13} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8487) lies on the Neuberg cubic K001 and these lines: {1, 8504}, {3, 8518}, {4, 5670}, {15, 8474}, {16, 8466}, {30, 1117}, {74, 8511}, {484, 5677}, {616, 8520}, {617, 8519}, {1157, 2132}, {3464, 5685}, {3465, 8505}, {3466, 8503}, {3479, 8514}, {3480, 8513}, {3481, 8440}, {5623, 8173}, {5624, 8172}, {5667, 5684}, {5668, 8467}, {5669, 8475}, {5672, 8472}, {5673, 8464}, {5674, 5675}, {5676, 8439}, {5678, 8438}, {5679, 8437}, {5681, 8521}, {5682, 8522}, {7165, 8432}, {8431, 8510}, {8433, 8524}, {8434, 8523}, {8465, 8479}, {8471, 8473}, {8506, 8508}, {8507, 8509} X(8487) = isogonal conjugate of X(5671)X(8488) is the perspector of the triangle pair {T11, T14} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8488) lies on the Neuberg cubic K001 and these lines: {1, 5676}, {3, 8503}, {4, 8527}, {30, 8432}, {74, 8512}, {399, 5680}, {484, 8440}, {1276, 8465}, {1277, 8473}, {2132, 3464}, {3065, 8510}, {3465, 5670}, {3466, 8517}, {3483, 8511}, {3484, 8504}, {5623, 8449}, {5624, 8459}, {5667, 5677}, {5668, 8468}, {5669, 8476}, {5671, 8485}, {5672, 8448}, {5673, 8458}, {5674, 8525}, {5675, 8526}, {5678, 8508}, {5679, 8509}, {5681, 8523}, {5682, 8524}, {5683, 8505}, {5685, 8443}, {8433, 8515}, {8434, 8516}, {8435, 8513}, {8436, 8514}, {8453, 8464}, {8463, 8472}, {8506, 8532}, {8507, 8533} X(8488) = isogonal conjugate of X(8432)X(8489) is the perspector of the triangle pair {T11, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8489) lies on the Neuberg cubic K001 and these lines: {1, 8525}, {3, 5679}, {4, 8532}, {13, 8448}, {16, 8453}, {30, 5681}, {74, 8515}, {399, 8441}, {617, 5667}, {1263, 8514}, {1276, 8449}, {1337, 2132}, {3440, 8440}, {3464, 8435}, {3465, 8508}, {3480, 8443}, {3482, 8520}, {3483, 8524}, {3484, 5675}, {5623, 8451}, {5624, 8174}, {5669, 8478}, {5670, 8495}, {5671, 8497}, {5676, 8491}, {5677, 8483}, {5680, 8482}, {5682, 8530}, {5683, 8438}, {8432, 8501}, {8434, 8485}, {8444, 8476}, {8446, 8473}, {8447, 8474}, {8450, 8463} X(8489) = isogonal conjugate of X(5681)X(8490) is the perspector of the triangle pair {T11, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8490) lies on the Neuberg cubic K001 and these lines: {1, 8526}, {3, 5678}, {4, 8533}, {14, 8458}, {15, 8463}, {30, 5682}, {74, 8516}, {399, 8442}, {616, 5667}, {1263, 8513}, {1277, 8459}, {1338, 2132}, {3441, 8440}, {3464, 8436}, {3465, 8509}, {3479, 8443}, {3482, 8519}, {3483, 8523}, {3484, 5674}, {5623, 8175}, {5624, 8461}, {5668, 8470}, {5670, 8496}, {5671, 8498}, {5676, 8492}, {5677, 8484}, {5680, 8481}, {5681, 8528}, {5683, 8437}, {8432, 8502}, {8433, 8485}, {8453, 8460}, {8454, 8468}, {8456, 8465}, {8457, 8466} X(8490) = isogonal conjugate of X(5682)X(8491) is the perspector of the triangle pair {T11, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8491) lies on the Neuberg cubic K001 and these lines: {1, 8523}, {3, 8519}, {4, 5678}, {14, 8466}, {15, 5623}, {30, 5674}, {74, 8513}, {399, 616}, {484, 8509}, {1157, 5682}, {1277, 8464}, {1338, 5671}, {2132, 3479}, {3441, 5670}, {3464, 8433}, {3465, 8506}, {3481, 8533}, {3484, 8521}, {5624, 8460}, {5667, 8437}, {5668, 8469}, {5669, 8172}, {5675, 8528}, {5676, 8489}, {5677, 8481}, {5684, 8442}, {5685, 8436}, {7060, 8468}, {7165, 8526}, {8173, 8470}, {8175, 8467}, {8432, 8499}, {8439, 8516}, {8440, 8529}, {8458, 8471}, {8461, 8475}, {8462, 8465}, {8484, 8505}, {8492, 8511}, {8496, 8518}, {8502, 8504} X(8491) = isogonal conjugate of X(5674)X(8492) is the perspector of the triangle pair {T11, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8492) lies on the Neuberg cubic K001 and these lines: {1, 8524}, {3, 8520}, {4, 5679}, {13, 8474}, {16, 5624}, {30, 5675}, {74, 8514}, {399, 617}, {484, 8508}, {1157, 5681}, {1276, 8472}, {1337, 5671}, {2132, 3480}, {3440, 5670}, {3464, 8434}, {3465, 8507}, {3481, 8532}, {3484, 8522}, {5623, 8450}, {5667, 8438}, {5668, 8173}, {5669, 8477}, {5674, 8530}, {5676, 8490}, {5677, 8482}, {5684, 8441}, {5685, 8435}, {7059, 8476}, {7165, 8525}, {8172, 8478}, {8174, 8475}, {8432, 8500}, {8439, 8515}, {8440, 8531}, {8448, 8479}, {8451, 8467}, {8452, 8473}, {8483, 8505}, {8491, 8511}, {8495, 8518}, {8501, 8504} X(8492) = isogonal conjugate of X(5675)X(8493) is the perspector of the triangle pair {T11, T24} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8493) lies on the Neuberg cubic K001 and these lines: {1, 8527}, {3, 5676}, {30, 8440}, {74, 8517}, {399, 8443}, {616, 8515}, {617, 8516}, {1263, 8510}, {2132, 5667}, {3065, 8512}, {3464, 5680}, {3465, 8432}, {3482, 8511}, {3483, 8503}, {3484, 5670}, {5623, 8453}, {5624, 8463}, {5668, 8458}, {5669, 8448}, {5671, 5683}, {5674, 8532}, {5675, 8533}, {5677, 8485}, {5678, 5681}, {5679, 5682}, {8174, 8465}, {8175, 8473}, {8441, 8513}, {8442, 8514}, {8449, 8459}, {8508, 8526}, {8509, 8525} X(8493) = isogonal conjugate of X(8440)X(8494) is the perspector of the triangle pair {T12, T13} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8494) lies on the Neuberg cubic K001 and these lines: {1, 8505}, {4, 5671}, {15, 8475}, {16, 8467}, {30, 5684}, {74, 8518}, {399, 1157}, {484, 5685}, {616, 8522}, {617, 8521}, {2132, 3481}, {3466, 8504}, {3479, 8520}, {3480, 8519}, {5670, 8439}, {5674, 8438}, {5675, 8437}, {5677, 7165}, {8172, 8173}, {8431, 8511}, {8433, 8507}, {8434, 8506}, {8466, 8479}, {8469, 8477}, {8471, 8474}, {8513, 8531}, {8514, 8529} X(8494) = isogonal conjugate of X(5684)X(8495) is the perspector of the triangle pair {T12, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8495) lies on the Neuberg cubic K001 and these lines: {1, 8506}, {3, 8521}, {4, 5674}, {14, 8467}, {15, 8172}, {16, 8469}, {30, 8437}, {74, 8519}, {399, 3479}, {484, 8433}, {616, 1157}, {1338, 5684}, {2132, 8529}, {3441, 5671}, {3466, 8523}, {3481, 5682}, {5623, 8471}, {5670, 8489}, {5677, 8499}, {5678, 8439}, {5685, 8481}, {7060, 8464}, {7165, 8509}, {8173, 8460}, {8431, 8513}, {8438, 8528}, {8462, 8466}, {8492, 8518}, {8502, 8505} X(8495) = isogonal conjugate of X(8437)X(8496) is the perspector of the triangle pair {T12, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8496) lies on the Neuberg cubic K001 and these lines: {1, 8507}, {3, 8522}, {4, 5675}, {13, 8475}, {15, 8477}, {16, 8173}, {30, 8438}, {74, 8520}, {399, 3480}, {484, 8434}, {617, 1157}, {1337, 5684}, {2132, 8531}, {3440, 5671}, {3466, 8524}, {3481, 5681}, {5624, 8479}, {5670, 8490}, {5677, 8500}, {5679, 8439}, {5685, 8482}, {7059, 8472}, {7165, 8508}, {8172, 8450}, {8431, 8514}, {8437, 8530}, {8452, 8474}, {8491, 8518}, {8501, 8505} X(8496) = isogonal conjugate of X(8438)X(8497) is the perspector of the triangle pair {T13, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8497) lies on the Neuberg cubic K001 and these lines: {4, 8437}, {74, 8521}, {399, 8529}, {616, 3481}, {1157, 3479}, {3441, 5684}, {3466, 8506}, {5671, 8489}, {5674, 8439}, {5685, 8499}, {7165, 8433}, {8172, 8471}, {8431, 8519}, {8462, 8467}, {8469, 8479} X(8497) = X(30)-Ceva conjugate of X(8521)X(8498) is the perspector of the triangle pair {T13, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8498) lies on the Neuberg cubic K001 and these lines: {4, 8438}, {74, 8522}, {399, 8531}, {617, 3481}, {1157, 3480}, {3440, 5684}, {3466, 8507}, {5671, 8490}, {5675, 8439}, {5685, 8500}, {7165, 8434}, {8173, 8479}, {8431, 8520}, {8452, 8475}, {8471, 8477} X(8498) = X(30)-Ceva conjugate of X(8522)X(8499) is the perspector of the triangle pair {T14, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8499) lies on the Neuberg cubic K001 and these lines: {1, 5681}, {3, 8508}, {13, 8449}, {30, 8435}, {74, 8525}, {399, 8483}, {484, 8441}, {617, 3465}, {1263, 8524}, {1276, 5668}, {1277, 8478}, {1337, 3464}, {2132, 8501}, {3065, 5679}, {3440, 5680}, {3466, 8532}, {3480, 8485}, {3482, 8507}, {3483, 5675}, {3484, 8434}, {5624, 8444}, {5667, 8482}, {5672, 8451}, {5673, 8174}, {5677, 8495}, {5685, 8497}, {7059, 8453}, {7164, 8515}, {7326, 8448}, {7329, 8514}, {8432, 8491}, {8436, 8530}, {8443, 8500}, {8446, 8476}, {8447, 8472}, {8480, 8520} X(8499) = isogonal conjugate of X(8435)X(8500) is the perspector of the triangle pair {T14, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8500) lies on the Neuberg cubic K001 and these lines: {1, 5682}, {3, 8509}, {14, 8459}, {30, 8436}, {74, 8526}, {399, 8484}, {484, 8442}, {616, 3465}, {1263, 8523}, {1276, 8470}, {1277, 5669}, {1338, 3464}, {2132, 8502}, {3065, 5678}, {3441, 5680}, {3466, 8533}, {3479, 8485}, {3482, 8506}, {3483, 5674}, {3484, 8433}, {5623, 8454}, {5667, 8481}, {5672, 8175}, {5673, 8461}, {5677, 8496}, {5685, 8498}, {7060, 8463}, {7164, 8516}, {7325, 8458}, {7329, 8513}, {8432, 8492}, {8435, 8528}, {8443, 8499}, {8456, 8468}, {8457, 8464}, {8480, 8519} X(8500) = isogonal conjugate of X(8436)X(8501) is the perspector of the triangle pair {T14, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8501) lies on the Neuberg cubic K001 and these lines: {1, 5674}, {3, 8506}, {4, 8509}, {14, 8464}, {15, 5672}, {30, 8433}, {74, 8523}, {399, 8481}, {484, 616}, {1157, 8436}, {1276, 8469}, {1277, 8172}, {1338, 5685}, {2132, 8499}, {3065, 8519}, {3441, 5677}, {3464, 3479}, {3465, 8437}, {3466, 5678}, {3483, 8521}, {5623, 7060}, {5671, 8502}, {5673, 8460}, {5680, 8529}, {5682, 7165}, {5684, 8484}, {7164, 8513}, {7325, 8466}, {8432, 8489}, {8434, 8528}, {8439, 8526}, {8454, 8467}, {8459, 8471}, {8462, 8468}, {8492, 8504}, {8496, 8505} X(8501) = isogonal conjugate of X(8433)X(8502) is the perspector of the triangle pair {T14, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8502) lies on the Neuberg cubic K001 and these lines: {1, 5675}, {3, 8507}, {4, 8508}, {13, 8472}, {16, 5673}, {30, 8434}, {74, 8524}, {399, 8482}, {484, 617}, {1157, 8435}, {1276, 8173}, {1277, 8477}, {1337, 5685}, {2132, 8500}, {3065, 8520}, {3440, 5677}, {3464, 3480}, {3465, 8438}, {3466, 5679}, {3483, 8522}, {5624, 7059}, {5671, 8501}, {5672, 8450}, {5680, 8531}, {5681, 7165}, {5684, 8483}, {7164, 8514}, {7326, 8474}, {8432, 8490}, {8433, 8530}, {8439, 8525}, {8444, 8475}, {8449, 8479}, {8452, 8476}, {8491, 8504}, {8495, 8505} X(8502) = isogonal conjugate of X(8434)X(8503) is the perspector of the triangle pairs {T15, T16} and {T17, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8503) lies on the Neuberg cubic K001 and these lines: {1, 8486}, {3, 8488}, {74, 7327}, {1138, 7164}, {1263, 7328}, {2133, 3065}, {3466, 8487}, {3483, 8493}, {7325, 8445}, {7326, 8455}, {7329, 8431} X(8503) = X(30)-Ceva conjugate of X(7327)X(8504) is the perspector of the triangle pairs {T15, T17} and {T18, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8504) lies on the Neuberg cubic K001 and these lines: {1, 8487}, {3, 7327}, {74, 7329}, {1138, 3065}, {1263, 7164}, {2133, 3483}, {3465, 8486}, {3466, 8494}, {3482, 7328}, {3484, 8488}, {7325, 8446}, {7326, 8456}, {8431, 8480}, {8444, 8455}, {8445, 8454}, {8491, 8502}, {8492, 8501} X(8504) = X(30)-Ceva conjugate of X(7329)X(8505) is the perspector of the triangle pair {T15, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8505) lies on the Neuberg cubic K001 and these lines: {1, 8494}, {3, 7329}, {74, 8480}, {1138, 3483}, {1263, 3065}, {3465, 8487}, {3482, 7164}, {3484, 7327}, {5683, 8488}, {7325, 8447}, {7326, 8457}, {8444, 8456}, {8446, 8454}, {8483, 8492}, {8484, 8491}, {8485, 8486}, {8495, 8502}, {8496, 8501} X(8505) = X(30)-Ceva conjugate of X(8480)X(8506) is the perspector of the triangle pair {T15, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8506) lies on the Neuberg cubic K001 and these lines: {1, 8495}, {3, 8501}, {13, 8444}, {74, 8483}, {617, 7329}, {1138, 8435}, {1263, 8482}, {1276, 8446}, {1337, 3065}, {3440, 3483}, {3465, 8491}, {3466, 8497}, {3480, 8480}, {3482, 8500}, {5681, 7327}, {7059, 8447}, {7164, 8441}, {7325, 8451}, {7326, 8174}, {8434, 8494}, {8486, 8525}, {8487, 8508}, {8488, 8532} X(8506) = X(30)-Ceva conjugate of X(8483)X(8507) is the perspector of the triangle pair {T15, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8507) lies on the Neuberg cubic K001 and these lines: {1, 8496}, {3, 8502}, {14, 8454}, {74, 8484}, {616, 7329}, {1138, 8436}, {1263, 8481}, {1277, 8456}, {1338, 3065}, {3441, 3483}, {3465, 8492}, {3466, 8498}, {3479, 8480}, {3482, 8499}, {5682, 7327}, {7060, 8457}, {7164, 8442}, {7325, 8175}, {7326, 8461}, {8433, 8494}, {8486, 8526}, {8487, 8509}, {8488, 8533} X(8507) = X(30)-Ceva conjugate of X(8484)X(8508) is the perspector of the triangle pair {T15, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8508) lies on the Neuberg cubic K001 and these lines: {1, 3441}, {3, 8499}, {4, 8502}, {14, 7060}, {15, 7325}, {74, 8481}, {484, 8492}, {616, 7164}, {1138, 8433}, {1277, 8462}, {1338, 3466}, {2133, 8509}, {3065, 3479}, {3465, 8489}, {3483, 8529}, {5672, 8455}, {5674, 7327}, {5678, 8488}, {5682, 7328}, {7165, 8496}, {7326, 8460}, {7329, 8437}, {8431, 8436}, {8439, 8484}, {8454, 8471}, {8486, 8523}, {8487, 8506}, {8493, 8526} X(8508) = X(30)-Ceva conjugate of X(8481)X(8509) is the perspector of the triangle pair {T15, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8509) lies on the Neuberg cubic K001 and these lines: {1, 3440}, {3, 8500}, {4, 8501}, {13, 7059}, {16, 7326}, {74, 8482}, {484, 8491}, {617, 7164}, {1138, 8434}, {1276, 8452}, {1337, 3466}, {2133, 8508}, {3065, 3480}, {3465, 8490}, {3483, 8531}, {5673, 8445}, {5675, 7327}, {5679, 8488}, {5681, 7328}, {7165, 8495}, {7325, 8450}, {7329, 8438}, {8431, 8435}, {8439, 8483}, {8444, 8479}, {8486, 8524}, {8487, 8507}, {8493, 8525} X(8509) = X(30)-Ceva conjugate of X(8482)X(8510) is the perspector of the triangle pair {T16, T17} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8510) lies on the Neuberg cubic K001 and these lines: {74, 8486}, {1138, 2133}, {1263, 8493}, {3065, 8488}, {7164, 7327}, {7328, 7329}, {8431, 8487}, {8445, 8455} X(8510) = X(30)-Ceva conjugate of X(8486)X(8511) is the perspector of the triangle pair {T16, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8511) lies on the Neuberg cubic K001 and these lines: {3, 8486}, {74, 8487}, {1263, 2133}, {3065, 7327}, {3482, 8493}, {3483, 8488}, {7164, 7329}, {7328, 8480}, {8431, 8494}, {8445, 8456}, {8446, 8455}, {8491, 8492} X(8511) = X(30)-Ceva conjugate of X(8487)X(8512) is the perspector of the triangle pair {T16, T19} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8512) lies on the Neuberg cubic K001 and these lines: {74, 8488}, {1138, 7328}, {2133, 7164}, {3065, 8493}, {3466, 8486}, {7327, 8431} X(8512) = X(30)-Ceva conjugate of X(8488)X(8513) is the perspector of the triangle pair {T16, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8513) lies on the Neuberg cubic K001 and these lines: {13, 8445}, {74, 8491}, {617, 8486}, {1138, 3440}, {1263, 8490}, {1337, 2133}, {3480, 8487}, {7164, 8501}, {7327, 8482}, {7328, 8483}, {7329, 8500}, {8431, 8495}, {8435, 8488}, {8441, 8493}, {8446, 8452}, {8494, 8531} X(8513) = X(30)-Ceva conjugate of X(8491)X(8514) is the perspector of the triangle pair {T16, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8514) lies on the Neuberg cubic K001 and these lines: {14, 8455}, {74, 8492}, {616, 8486}, {1138, 3441}, {1263, 8489}, {1338, 2133}, {3479, 8487}, {7164, 8502}, {7327, 8481}, {7328, 8484}, {7329, 8499}, {8431, 8496}, {8436, 8488}, {8442, 8493}, {8456, 8462}, {8494, 8529} X(8514) = X(30)-Ceva conjugate of X(8492)X(8515) is the perspector of the triangle pair {T16, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8515) lies on the Neuberg cubic K001 and these lines: {74, 8489}, {616, 8493}, {1138, 8529}, {2133, 3479}, {3441, 8431}, {7164, 8499}, {7328, 8481}, {8433, 8488}, {8437, 8486}, {8439, 8492}, {8455, 8471} X(8515) = X(30)-Ceva conjugate of X(8489)X(8516) is the perspector of the triangle pair {T16, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8516) lies on the Neuberg cubic K001 and these lines: {74, 8490}, {617, 8493}, {1138, 8531}, {2133, 3480}, {3440, 8431}, {7164, 8500}, {7328, 8482}, {8434, 8488}, {8438, 8486}, {8439, 8491}, {8445, 8479} X(8516) = X(30)-Ceva conjugate of X(8490)X(8517) is the perspector of the triangle pair {T16, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8517) lies on the Neuberg cubic K001 and these lines: {74, 8493}, {2133, 8431}, {3466, 8488}, {7164, 7328}, {8439, 8486} X(8517) = X(30)-Ceva conjugate of X(8493)X(8518) is the perspector of the triangle pair {T17, T18} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8518) lies on the Neuberg cubic K001 and these lines: {3, 8487}, {74, 8494}, {1138, 1263}, {2133, 3482}, {3065, 7329}, {3483, 7327}, {3484, 8486}, {7164, 8480}, {8445, 8457}, {8446, 8456}, {8447, 8455}, {8491, 8496}, {8492, 8495} X(8518) = X(30)-Ceva conjugate of X(8494)X(8519) is the perspector of the triangle pair {T17, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8519) lies on the Neuberg cubic K001 and these lines: {3, 8491}, {13, 8446}, {74, 8495}, {617, 8487}, {1138, 1337}, {1263, 3440}, {2133, 8441}, {3065, 8501}, {3480, 8494}, {3482, 8490}, {5681, 8486}, {7164, 8483}, {7326, 8444}, {7327, 8435}, {7329, 8482}, {8174, 8445}, {8431, 8497}, {8447, 8452}, {8451, 8455}, {8480, 8500} X(8519) = X(30)-Ceva conjugate of X(8495)X(8520) is the perspector of the triangle pair {T17, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8520) lies on the Neuberg cubic K001 and these lines: {3, 8492}, {14, 8456}, {74, 8496}, {616, 8487}, {1138, 1338}, {1263, 3441}, {2133, 8442}, {3065, 8502}, {3479, 8494}, {3482, 8489}, {5682, 8486}, {7164, 8484}, {7325, 8454}, {7327, 8436}, {7329, 8481}, {8175, 8455}, {8431, 8498}, {8445, 8461}, {8457, 8462}, {8480, 8499} X(8520) = X(30)-Ceva conjugate of X(8496)X(8521) is the perspector of the triangle pair {T18, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8521) lies on the Neuberg cubic K001 and these lines: {3, 8495}, {13, 8447}, {74, 8497}, {617, 8494}, {1138, 8441}, {1263, 1337}, {3065, 8483}, {3440, 3482}, {3483, 8501}, {3484, 8491}, {5681, 8487}, {7329, 8435}, {8174, 8446}, {8451, 8456}, {8480, 8482}, {8486, 8532} X(8521) = X(30)-Ceva conjugate of X(8497)X(8522) is the perspector of the triangle pair {T18, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8522) lies on the Neuberg cubic K001 and these lines: {3, 8496}, {14, 8457}, {74, 8498}, {616, 8494}, {1138, 8442}, {1263, 1338}, {3065, 8484}, {3441, 3482}, {3483, 8502}, {3484, 8492}, {5682, 8487}, {7329, 8436}, {8175, 8456}, {8446, 8461}, {8480, 8481}, {8486, 8533} X(8522) = X(30)-Ceva conjugate of X(8498)X(8523) is the perspector of the triangle pair {T19, T20} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8523) lies on the Neuberg cubic K001 and these lines: {1, 8491}, {13, 7326}, {74, 8501}, {617, 7327}, {1138, 8482}, {1263, 8500}, {1276, 8445}, {1337, 7164}, {2133, 8435}, {3065, 3440}, {3466, 8495}, {3480, 7329}, {3483, 8490}, {5681, 8488}, {7059, 8446}, {7328, 8441}, {8431, 8483}, {8434, 8487}, {8444, 8452}, {8480, 8531}, {8486, 8508} X(8523) = X(30)-Ceva conjugate of X(8501)X(8524) is the perspector of the triangle pair {T19, T21} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8524) lies on the Neuberg cubic K001 and these lines: {1, 8492}, {14, 7325}, {74, 8502}, {616, 7327}, {1138, 8481}, {1263, 8499}, {1277, 8455}, {1338, 7164}, {2133, 8436}, {3065, 3441}, {3466, 8496}, {3479, 7329}, {3483, 8489}, {5682, 8488}, {7060, 8456}, {7328, 8442}, {8431, 8484}, {8433, 8487}, {8454, 8462}, {8480, 8529}, {8486, 8509} X(8524) = X(30)-Ceva conjugate of X(8502)X(8525) is the perspector of the triangle pair {T19, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8525) lies on the Neuberg cubic K001 and these lines: {1, 8489}, {74, 8499}, {616, 7328}, {2133, 8433}, {3065, 8529}, {3441, 3466}, {3479, 7164}, {5674, 8488}, {7060, 8462}, {7165, 8492}, {7325, 8471}, {7327, 8437}, {8431, 8481}, {8439, 8502}, {8486, 8506}, {8493, 8509} X(8525) = X(30)-Ceva conjugate of X(8499)X(8526) is the perspector of the triangle pair {T19, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8526) lies on the Neuberg cubic K001 and these lines: {1, 8490}, {74, 8500}, {617, 7328}, {2133, 8434}, {3065, 8531}, {3440, 3466}, {3480, 7164}, {5675, 8488}, {7059, 8452}, {7165, 8491}, {7326, 8479}, {7327, 8438}, {8431, 8482}, {8439, 8501}, {8486, 8507}, {8493, 8508} X(8526) = X(30)-Ceva conjugate of X(8500)X(8527) is the perspector of the triangle pair {T19, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8527) lies on the Neuberg cubic K001 and these lines: {1, 8493}, {4, 8488}, {74, 7328}, {2133, 3466}, {7164, 8431}, {7165, 8486}, {7327, 8439} X(8527) = X(30)-Ceva conjugate of X(7328)X(8528) is the perspector of the triangle pair {T20, T23} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8528) lies on the Neuberg cubic K001 and these lines: {13, 8450}, {74, 8530}, {617, 3440}, {1337, 3480}, {5675, 8491}, {5681, 8490}, {8434, 8501}, {8435, 8500}, {8438, 8495}, {8441, 8531}, {8446, 8477}, {8451, 8479}, {8452, 8478} X(8528) = X(30)-Ceva conjugate of X(8530)X(8529) is the perspector of the triangle pair {T20, T24} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8529) lies on the Neuberg cubic K001 and these lines: {3, 5681}, {13, 8453}, {30, 8441}, {74, 8532}, {399, 8497}, {617, 3484}, {1138, 8515}, {1263, 5679}, {1337, 5667}, {2132, 8495}, {3065, 8525}, {3440, 8443}, {3464, 8483}, {3465, 8435}, {3480, 5683}, {3482, 5675}, {3483, 8508}, {5624, 8447}, {5668, 8174}, {5669, 8451}, {5680, 8501}, {8175, 8478}, {8440, 8491}, {8442, 8530}, {8444, 8449}, {8446, 8448}, {8480, 8524}, {8482, 8485}, {8494, 8514} X(8529) = isogonal conjugate of X(8441)X(8530) is the perspector of the triangle pair {T21, T22} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8530) lies on the Neuberg cubic K001 and these lines: {14, 8460}, {74, 8528}, {616, 3441}, {1338, 3479}, {5674, 8492}, {5682, 8489}, {8433, 8502}, {8436, 8499}, {8437, 8496}, {8442, 8529}, {8456, 8469}, {8461, 8471}, {8462, 8470} X(8530) = X(30)-Ceva conjugate of X(8528)X(8531) is the perspector of the triangle pair {T21, T24} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8531) lies on the Neuberg cubic K001 and these lines: {3, 5682}, {14, 8463}, {30, 8442}, {74, 8533}, {399, 8498}, {616, 3484}, {1138, 8516}, {1263, 5678}, {1338, 5667}, {2132, 8496}, {3065, 8526}, {3441, 8443}, {3464, 8484}, {3465, 8436}, {3479, 5683}, {3482, 5674}, {3483, 8509}, {5623, 8457}, {5668, 8461}, {5669, 8175}, {5680, 8502}, {8174, 8470}, {8440, 8492}, {8441, 8528}, {8454, 8459}, {8456, 8458}, {8480, 8523}, {8481, 8485}, {8494, 8513} X(8531) = isogonal conjugate of X(8442)X(8532) is the perspector of the triangle pair {T22, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8532) lies on the Neuberg cubic K001 and these lines: {4, 8489}, {74, 8529}, {2133, 8437}, {3441, 8439}, {3466, 8499}, {3479, 8431}, {3481, 8492}, {5674, 8493}, {7328, 8433}, {8462, 8471}, {8486, 8521}, {8488, 8506} X(8532) = X(30)-Ceva conjugate of X(8529)X(8533) is the perspector of the triangle pair {T23, T25} in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane; see the preamble just before X(8431).
X(8533) lies on the Neuberg cubic K001 and these lines: {4, 8490}, {74, 8531}, {2133, 8438}, {3440, 8439}, {3466, 8500}, {3480, 8431}, {3481, 8491}, {5675, 8493}, {7328, 8434}, {8452, 8479}, {8486, 8522}, {8488, 8507} X(8533) = X(30)-Ceva conjugate of X(8531)X(8534) lies on the Neuberg cubic K001 and these lines: {1,8512}, {3,8510}, {4,8517}, {30,5676}, {399,8440}, {484,8527}, {3464,8432}, {3465,8503}, {3484,8511}, {5623,8448}, {5624,8458}, {5667,5670}, {5668,8465}, {5669,8473}, {5671,8443}, {5674,8515}, {5675,8516}, {5677,5680}, {5678,5679}, {5681,8513}, {5682,8514}, {5683,8518}, {8449,8468}, {8453,8466}, {8459,8476}, {8463,8474}, {8485,8504}, {8519,8532}, {8520,8533}, {8523,8525}, {8524,8526}
X(8534) = X(74)-cross conjugate of X(2132)X(8535) lies on the Neuberg cubic K001 and these lines: {3,8513}, {4,8516}, {14,8465}, {15,8458}, {30,5678}, {399,5682}, {484,8526}, {616,2132}, {1157,8533}, {1277,8468}, {1338,5670}, {3441,5676}, {3464,8509}, {3465,8523}, {3479,8440}, {3484,8519}, {5623,5669}, {5624,8470}, {5667,5674}, {5671,8442}, {5672,8459}, {5677,8436}, {5679,8528}, {5680,8433}, {5683,8521}, {8172,8463}, {8175,8466}, {8432,8481}, {8437,8443}, {8448,8460}, {8453,8469}, {8461,8474}, {8484,8504}, {8485,8506}, {8489,8517}, {8492,8510}, {8496,8511}, {8498,8518}, {8499,8527}, {8502,8503}
X(8535) = X(74)-cross conjugate of X(616)Barycentrics (2*sqrt(3)*(a^16-6*(b^2-c^2)*a^14+(14*b^4+3*b^2*c^2-20*c^4)*a^12-(14*b^6-10*c^6+3*(19*b^2-21*c^2)*b^2*c^2)*a^10+6*(b^2-c^2)*(14*b^4+9*b^2*c^2-c^4)*c^2*a^8+2*(b^2-c^2)*(7*b^8-5*c^8-(11*b^4+48*b^2*c^2-25*c^4)*b^2*c^2)*a^6-(b^2-c^2)^3*(14*b^6-20*c^6+3*(13*b^2+c^2)*b^2*c^2)*a^4+3*(b^2-c^2)^4*(2*b^6+2*c^6+(5*b^2+9*c^2)*b^2*c^2)*a^2-(b^2+c^2)*(b^2-c^2)^7)*S+a^18-(3*b^2+7*c^2)*a^16-(6*b^4-b^2*c^2-20*c^4)*a^14+2*(21*b^6-9*b^4*c^2-14*c^6)*a^12-(84*b^8-14*c^8-(155*b^4-126*b^2*c^2+39*c^4)*b^2*c^2)*a^10+2*(b^2-c^2)*(42*b^8-7*c^8-(64*b^4+45*b^2*c^2-30*c^4)*b^2*c^2)*a^8-(b^2-c^2)*(42*b^10-28*c^10+(27*b^6+11*c^6-(309*b^2-161*c^2)*b^2*c^2)*b^2*c^2)*a^6+2*(b^2-c^2)^3*(3*b^8-10*c^8+(70*b^4+3*b^2*c^2-30*c^4)*b^2*c^2)*a^4+(3*b^8-7*c^8-(47*b^4+84*b^2*c^2+27*c^4)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^7)*(2*sqrt(3)*(a^16+6*(b^2-c^2)*a^14-(20*b^4-3*b^2*c^2-14*c^4)*a^12+(10*b^6-14*c^6+3*(21*b^2-19*c^2)*b^2*c^2)*a^10+6*(b^2-c^2)*(b^4-9*b^2*c^2-14*c^4)*b^2*a^8+2*(b^2-c^2)*(5*b^8-7*c^8-(25*b^4-48*b^2*c^2-11*c^4)*b^2*c^2)*a^6-(b^2-c^2)^3*(20*b^6-14*c^6-3*(b^2+13*c^2)*b^2*c^2)*a^4+3*(b^2-c^2)^4*(2*b^6+2*c^6+(9*b^2+5*c^2)*b^2*c^2)*a^2+(b^2+c^2)*(b^2-c^2)^7)*S+a^18-(7*b^2+3*c^2)*a^16+(20*b^4+b^2*c^2-6*c^4)*a^14-2*(14*b^6+9*b^2*c^4-21*c^6)*a^12+(14*b^8-84*c^8+(39*b^4-126*b^2*c^2+155*c^4)*b^2*c^2)*a^10+2*(b^2-c^2)*(7*b^8-42*c^8-(30*b^4-45*b^2*c^2-64*c^4)*b^2*c^2)*a^8-(b^2-c^2)*(28*b^10-42*c^10-(11*b^6+27*c^6+(161*b^2-309*c^2)*b^2*c^2)*b^2*c^2)*a^6+2*(b^2-c^2)^3*(10*b^8-3*c^8+(30*b^4-3*b^2*c^2-70*c^4)*b^2*c^2)*a^4-(b^2-c^2)^4*(7*b^8-3*c^8+(27*b^4+84*b^2*c^2+47*c^4)*b^2*c^2)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^7)*(2*sqrt(3)*(-a^2+b^2+c^2)*S+5*a^4-4*(b^2+c^2)*a^2-(b^2-c^2)^2) : :
X(8536) lies on the Neuberg cubic K001 and these lines: {3,8514}, {4,8515}, {13,8473}, {16,8448}, {30,5679}, {399,5681}, {484,8525}, {617,2132}, {1157,8532}, {1276,8476}, {1337,5670}, {3440,5676}, {3464,8508}, {3465,8524}, {3480,8440}, {3484,8520}, {5623,8478}, {5624,5668}, {5667,5675}, {5671,8441}, {5673,8449}, {5677,8435}, {5678,8530}, {5680,8434}, {5683,8522}, {8173,8453}, {8174,8474}, {8432,8482}, {8438,8443}, {8450,8458}, {8451,8466}, {8463,8477}, {8483,8504}, {8485,8507}, {8490,8517}, {8491,8510}, {8495,8511}, {8497,8518}, {8500,8527}, {8501,8503}
X(8536) = X(74)-cross conjugate of X(617)
Centers associated with Ehrmann circles: X(8537)-X(8550 )
This section is contributed by César Eliud Lozada, November 9, 2015.
In Hyacinthos #6098), December 2, 2002, Jean-Pierre Ehrmann defines a circle as follows. Let P be a point in the plane of ABC and not on the lines BC, CA, AB. Let AB the the point of intersection of the circle {{P,B,C}} and the line AB. Define AC symmetrically, and define BC, BA, CA, CB cyclically. These six points of intersection are on a circle if and only if P = X(6). This circle is here named the Ehrmann circle. The circles {{X(6),B,C}}, {{X(6),C,A}}, {{X(6),A,B,}} are the A-Ehrmann circle, B-Ehrmann circle, C-Ehrmann circle, respectively.
The center of the Ehrmann circle is X(576) and the radius, sqrt(1+9*tan(ω)^2)*(R/2). No ETC-center X(i) lies on this circle, for 1 <= i <= 8550.
Let A' be the center of the A-Ehrmann circle, and define B' and C' cyclically. The triangle A'B'C' is here named the 1st Ehrmann triangle. Trilinears are given by
A'= -(a^4-b^4+4*b^2*c^2-c^4)*a : (a^2*(a^2+5*c^2)-(b^2-c^2)*(b^2-2*c^2))*b : (a^2*(a^2+5*b^2)-(c^2-b^2)*(c^2-2*b^2))*c.
The 1st Ehrmann triangle is similar to the circummedial triangle with similitude center X(23). (Randy Hutson, December 2, 2017)
Let A'' = BCBA∩CACB, and define B'' and C'' cyclically. The triangle A''B''C'' is here named the 2nd Ehrmann triangle. Trilinears are given by
A"= a*(a^2-2*b^2-2*c^2)/(2*a^2-b^2-c^2) : b : c
A'B'C' is perspective at X(3) to the following triangles: Ara, 1st Brocard, 1st and 2nd circumperp, 2nd Euler, Fuhrmann, Johnson, McCay, medial, inner Napoleon, outer Napoleon, 1st Neuberg, 2nd Neuberg, tangential, Trinh, inner Vecten, and outer Vecten. A'B'C' is perspective to the circumsymmedial triangle at X(1296) and to the 4th Brocard triangle at X(10870).
A''B''C'' is perspective at X(6) to the following: ABC, 2nd Brocard, circumsymmedial, inner Grebe, outer Grebe, symmedial, and tangential. A''B''C'' is homothetic to the following triangles: circumorthic at X(8537), 2nd Euler at X(8538), extangents at X(8539), intangents at X(8540), orthic triangle at X(8541), and Trinh at X(511). A''B''C'' is perspective to the medial triangle at X(8542).
A'B'C' and A''B''C'' are are orthologic with centers X(6) and X(576).
Following are notes about triangles mentioned above. The center of the Ehrmann circle is X(576). Also, the Ehrmann circle is the incircle of the 2nd Ehrmann triangle if ABC is acute. Otherwise it is an excircle of the 2nd Ehrmann triangle. The 2nd Ehrmann triangle is congruent to the Trinh triangle, and also is the reflection of the Kosnita triangle in X(575). (Randy Hutson, November 18, 2015)
X(8537) lies on these lines: {2,8538}, {4,542}, {6,24}, {68,193}, {186,575}, {511,3520}, {524,1594}, {847,4994}, {1173,3542}, {1351,1593}, {1598,5093}, {1614,2393}, {1843,5097}, {5032,7487}, {6197,8539}, {6198,8540}
X(8538) lies on these lines: {2,8537}, {3,6}, {5,8541}, {68,895}, {69,5449}, {343,5159}, {597,7542}, {1062,8540}, {1209,8542}, {1992,6643}, {3618,6689}, {5622,7689}, {8251,8539}
X(8539) lies on these lines: {6,31}, {19,8541}, {36,4260}, {40,576}, {65,651}, {511,7688}, {524,3925}, {1992,2550}, {4273,4735}, {5010,5138}, {6197,8537}, {8251,8538}
X(8540) lies on these lines: {1,576}, {6,31}, {11,524}, {33,8541}, {35,575}, {36,511}, {182,5010}, {193,5274}, {239,4459}, {390,5032}, {497,1992}, {518,5048}, {542,3583}, {597,5432}, {611,5093}, {613,999}, {1015,5107}, {1062,8538}, {1124,6283}, {1944,4124}, {2316,7077}, {2323,3271}, {3057,4663}, {3589,5326}, {3751,7962}, {5476,7951}, {6198,8537}
X(8541) lies on these lines: {4,542}, {5,8538}, {6,25}, {19,8539}, {24,575}, {32,3455}, {33,8540}, {112,6323}, {182,186}, {193,7378}, {237,5158}, {340,5117}, {378,511}, {403,5476}, {427,524}, {460,6749}, {468,597}, {599,5094}, {1204,1205}, {1351,1597}, {1352,1568}, {1829,4663}, {2387,5028}, {3092,6406}, {3093,6291}, {3148,3284}, {3629,3867}, {5032,6995}, {5986,7766}
X(8541) = reflection of X(184) in X(6)
Let A'B'C' be the 1st Ehrmann triangle. Let Pa be the pole of line B'C' wrt the A-Ehrmann circle, and define Pb and Pc cyclically. The lines A'Pa, B'Pb, C'Pc concur in X(8542). (Randy Hutson, November 18, 2015)
X(8542) lies on the cubic K284 and these lines: {2,895}, {3,2393}, {5,524}, {6,373}, {110,5505}, {113,1352}, {141,5159}, {182,1511}, {184,2930}, {511,4550}, {523,3734}, {575,1147}, {599,5094}, {942,4663}, {1209,8538}, {5092,8547}
X(8542) = midpoint of X(110 and X(5505)
X(8542) = complement of X(5486)
X(8542) = orthocenter-of-1st-Ehrmann-triangle
X(8542) = Dao image of X(6)
X(8543) lies on these lines: {1,651}, {4,390}, {7,21}, {9,1405}, {12,528}, {35,411}, {37,4318}, {55,5226}, {57,5284}, {65,5047}, {100,5219}, {105,7179}, {226,1005}, {354,1776}, {404,5880}, {518,4861}, {527,2975}, {958,4323}, {962,5766}, {1056,6930}, {1386,7269}, {1441,3685}, {1442,1456}, {1445,3339}, {1466,5550}, {1476,3622}, {1836,7411}, {1858,5572}, {2099,5220}, {2476,2550}, {3085,6932}, {3219,5173}, {3486,8236}, {3560,5843}, {3576,8544}, {3671,5259}, {3746,3947}, {4127,5223}, {4295,6986}, {4296,6051}, {4312,7280}, {4423,5435}, {4511,5784}, {5086,5853}, {5253,6173}, {5728,5887}, {5729,6920}, {6600,6871}, {6828,7678}, {6982,8164}
X(8544) lies on these lines: {1,7}, {3,8545}, {9,404}, {44,5781}, {46,2801}, {63,210}, {78,527}, {84,1156}, {142,2478}, {377,5795}, {936,6172}, {971,1445}, {1012,5126}, {1467,3146}, {1750,5435}, {1836,4666}, {1998,2094}, {2136,3868}, {3243,3885}, {3333,7671}, {3474,3870}, {3522,5766}, {3576,8543}, {5450,7677}, {5708,5728}, {5714,6865}, {5880,7354}, {5884,7672}
X(8545) lies on these lines: {1,651}, {2,7}, {3,8544}, {12,5880}, {19,7282}, {20,5766}, {37,77}, {40,5261}, {44,5228}, {45,241}, {46,3947}, {65,3951}, {78,5784}, {84,5703}, {85,190}, {222,5287}, {238,4327}, {269,3731}, {388,5250}, {390,515}, {484,4312}, {516,1478}, {518,2099}, {528,3895}, {653,7079}, {664,4664}, {758,5223}, {912,5728}, {942,5729}, {948,4419}, {954,971}, {984,2263}, {993,4321}, {1001,1319}, {1020,7177}, {1253,1721}, {1255,1422}, {1419,1442}, {1441,3692}, {1449,7269}, {1467,5047}, {1532,5805}, {1697,3146}, {1765,5736}, {2161,5845}, {2346,3062}, {2475,3882}, {2550,6735}, {2951,7676}, {3358,6935}, {3359,8164}, {3487,7330}, {3587,5759}, {3646,5265}, {3665,7131}, {3811,5696}, {3822,7679}, {3973,7274}, {4318,7174}, {4323,6762}, {5709,5714}, {5732,6909}, {5735,6932}, {5762,6907}, {5817,6939}, {5851,8255}
X(8545) = midpoint of X(144) and X(5905)
X(8545) = reflection of X(i) in X(j) for these (i,j): (7,226), (63,9), (7675,954)
X(8546) lies on these lines: {2,2930}, {3,524}, {6,23}, {159,6329}, {182,1511}, {184,6593}, {575,2393}, {597,1995}, {599,7496}, {895,5012}, {1992,7492}, {3431,5085}
X(8546) = midpoint of X(6) and X(8547)\
The bisecting circle of 3 circles is defined at X(7623). The Ehrmann bisecting circle is here defined as the bisecting circle of the A-Ehrmann circle, B-Ehrmann circle, and C-Ehrmann, these being defined just before X(8534). The bisecting circle has squared-radius R^2*|(3*SA+SW)*(3*SB+SW)*(3*SC+SW)*(9*R^2-SW)|/SW^4. No ETC-center X(i) lies on this circle, for 1 ≤ i ≤ 8550.
X(8547) lies on these lines: {3,2854}, {6,23}, {182,2393}, {376,524}, {597,3066}, {599,3448}, {5092,8542}, {6329,7716}
X(8547) = reflection of X(6) in X(8546)
X(8547) = X(3) of antipedal triangle of X(6)
X(8548) is the orthologic center of the 2nd Ehrmann triangle to the following triangles: circumorthic, extangents, intangents, orthic, and tangential
X(8548) lies on these lines: {3,895}, {5,6}, {26,2393}, {30,8549}, {52,8541}, {54,5050}, {69,3548}, {140,5486}, {193,3541}, {394,5159}, {511,7689}, {575,1147}, {912,4663}, {1351,1593}, {1993,5094}, {3167,5422}, {5448,5476}, {5562,8538}, {6237,8539}, {6238,8540}
X(8548) = reflection of X(1147) in X(575)
X(8548) = X(4)-of-2nd-Ehrmann-triangle
X(8549) is the orthologic center of the 2nd Ehrmann triangle to the following triangles: midheight, orthocentroidal, and reflection.
X(8549) lies on these lines: {3,2393}, {4,6}, {24,5622}, {30,8548}, {51,1619}, {64,895}, {66,3564}, {141,3546}, {154,1995}, {155,542}, {159,182}, {161,3796}, {185,8541}, {206,5050}, {394,858}, {511,3357}, {524,6247}, {575,6759}, {576,6000}, {1177,7530}, {1350,7691}, {1660,5020}, {4663,6001}, {6241,8537}, {6254,8539}, {6285,8540}
X(8549) = reflection of X(i) in X(j) for these (i,j): (159,182), (6759,575)
The reciprocal orthologic center of the 2nd Ehrmann and MacBeath triangles is X(4).
Let P be a point on the circumcircle. Let A' be the orthogonal projection of P on the A-altitude, and define B' and C' cyclically. As P traces the circumcircle, the locus of the symmedian point of A'B'C' is an ellipse with center X(8550); see X(5884). (Randy Hutson, November 18, 2015)
Let T be the triangle whose vertices are the orthocenters of the altimedial triangles; then X(8550) = X(6)-of-T. (Randy Hutson, November 18, 2015)
Let T' be the triangle whose vertices are the symmedian points of the altimedial triangles; then X(8550) = orthocenter of T'; see X(5884). (Randy Hutson, November 18, 2015)
X(8550) lies on these lines: {3,524}, {4,6}, {5,542}, {20,1992}, {30,576}, {39,5477}, {54,67}, {69,3523}, {98,3815}, {140,141}, {147,7792}, {154,4232}, {159,3517}, {184,468}, {185,1205}, {193,1350}, {343,5012}, {389,2393}, {395,6770}, {396,6773}, {511,550}, {515,4663}, {516,4743}, {518,5882}, {546,5476}, {578,6247}, {599,631}, {639,5874}, {640,5875}, {1351,1657}, {1352,1656}, {1513,5306}, {1692,7755}, {1843,6746}, {1899,5094}, {1994,5189}, {2784,4672}, {2836,5884}, {3054,7607}, {3146,5032}, {3329,5984}, {3410,7570}, {3520,5621}, {3533,3763}, {3575,8541}, {3618,5056}, {3630,5092}, {3796,6515}, {3818,3850}, {3851,6329}, {5059,5102}, {5073,5093}, {5107,7756}, {5182,7807}, {5304,7710}, {6240,8537}, {6284,8540}
X(8550) = midpoint of X(i),X(j) for these (i,j): (6,6776), (193,1350)
X(8550) = reflection of X(i) in X(j) for these (i,j): (5,575), (141,182), (1352,3589), (3629,1353), (5480,6)
X(8550) = center of the perspeconic of these triangles: orthic and circumorthic
Coefficient Points of Circles: X(8551)-X(8579)
This section is contributed by César Eliud Lozada, November 15, 2015.
Every circle has a trilinear equation of the following form:
(Lα + Mβ + Nγ)⋅(aα + bβ + cγ) + K⋅(aβγ + bγα + cαβ) = 0 (1)
and conversely, if K ≠ 0 then the equation (1) represents a circle.
A circle Λ is a central circle if L : M : N is a triangle center and K is a homogeneous symmetric function of (a,b,c); see TCCT, pp 219-226 and CircleFunction at MathWorld). In such a case, L : M : N is called the coefficient point of the circle Λ. (The coefficients L, M, N, K are homogeneous in a,b,c, so that L : M : N is unambiguously defined. In other words, there is no loss in assuming that K = 1 in equation (1).)
In the following list, the appearance of (Λ, i) means that X(i) is the coefficient point of the circle Λ.
(Adams, 8551), (anticomplementary, 32), (Apollonius, 940), (Bevan, 1), (Brocard, 2), (2nd Brocard, 6), (Conway, 213), (Dao-Moses-Telv, 8552), (1st Droz-Farny, 8553), (2nd Droz-Farny, 6), (Ehrmann, 524), (Euler-Gergonne-Soddy, 8554), (excircles radical circle, 56), (extangents, 8555), (Fuhrmann, 48), (Gallatly, 183), (GEOS, 1946), (Half-Moses, 8556), (hexyl, 8557), (incentral, 191), (incircle, 220), (intangents, 8558), (1st Johnson-Yff, 8559), (2nd Johnson-Yff, 8560), (Kenmotu, 492), (1st Lemoine, 141), (2nd Lemoine, 69), (3rd Lemoine, 8561), (Lester, 8562), (Longuet-Higgins, 1500), (Lucas central, 8563), (Lucas(-1) central, 8564), (Lucas-circles radical circle, 2), (Lucas inner, 2), (Lucas(-1) inner, 2), (Macbeath, 8565), (Mandart, 221), (Mccay-centers, 8566), (midheight, 8567), (mixtilinear, 8568), (Moses, 599), (Moses-Longuet-Higgins, 220), (inner-Napoleon, 15), (outer-Napoleon, 16), (Neuberg-circles radical circle, 3229), (1st Neuberg, 8569), (2nd Neuberg2, 8570), (nine-points-circle, 3), (orthocentroidal, 3), (orthoptic of the Steiner inellipse, 3), (Parry, 690), (polar, 3), (reflection, 195), (sine triple-angle, 8571), (Spieker, 8572), (Stammler, 6), (1st Steiner, 8573), (2nd Steiner, 8574), (Stevanovic, 905), (symmedial, 2896), (tangential, 3), (Taylor, 394), (Van Lamoen, 8575), (inner-Vecten, 8576), (outer-Vecten, 8577), (Yff-contact, 8578), (Yiu, 8579)
X(8551) lies on these lines: {1,6}, {1253,6602}, {2293,8012}
X(8552) lies on these lines: {2,2411}, {3,684}, {5,2797}, {140,6130}, {441,525}, {523,7623}, {526,1511}, {620,2492}, {690,6132}, {924,5926}, {3268,5664}
X(8552) = midpoint of X(i),X(j) for these (i,j): (3,684), (3268,5664)
X(8552) = reflection of X(6130) in X(140)
X(8552) = isogonal conjugate of polar conjugate of X(3268)
X(8552) = crossdifference of every pair of points on line X(25)X(1989)
X(8552) = X(92)-isoconjugate of X(14560)
X(8553) lies on these lines: {3,6}, {22,230}, {24,53}, {25,2934}, {26,2165}, {157,237}, {159,7669}, {161,2351}, {186,393}, {233,7747}, {378,6748}, {590,1599}, {615,1600}, {1583,8253}, {1584,8252}, {1879,7517}, {1989,2079}, {1995,3054}, {2164,7297}, {2178,5172}, {2548,7516}, {3087,3520}, {3553,5010}, {3554,7280}, {3815,7485}, {4644,7279}, {4996,5839}, {6636,7735}, {7506,7749}, {7509,7745}, {7514,7737}
X(8553) = circle-{X(371),X(372),PU(1),PU(39)}-inverse of X(36749)
X(8553) = {X(371),X(372)}-harmonic conjugate of X(36749)
X(8554) lies on this line: {3,6}
X(8555) lies on these lines: {1,4}, {184,1782}, {284,501}, {386,1060}, {580,1214}, {991,1062}, {1829,2360}
X(8556) lies on these lines: {2,6}, {381,7810}, {538,5013}, {1003,1078}, {1656,7854}, {2549,8358}, {3053,3934}, {3526,7794}, {3734,5210}, {3851,7873}, {5020,5201}, {5054,7801}, {5055,7818}, {5070,7821}, {5585,7771}
X(8557) lies on these lines: {1,6}, {19,208}, {31,33}, {40,1834}, {41,2302}, {56,2182}, {57,1020}, {58,7330}, {63,4419}, {84,4252}, {169,1731}, {171,1711}, {198,2264}, {216,2277}, {283,1778}, {579,1766}, {604,2261}, {610,2178}, {672,1732}, {728,3943}, {990,3358}, {1068,1249}, {1072,5286}, {1405,2170}, {1420,7113}, {1423,7289}, {1445,4000}, {1572,4274}, {1901,5715}, {1998,3693}, {1999,3719}, {2082,2183}, {2163,7284}, {2260,2285}, {2278,3576}, {2345,6734}, {3008,8257}, {3011,7735}, {3218,4346}, {4257,7171}, {4644,8545}, {5043,5536}, {5415,7133}, {7118,7129}
X(8558) lies on these lines: {2,7}, {4,1729}, {19,1709}, {30,1146}, {46,7079}, {90,1752}, {101,912}, {103,910}, {109,5089}, {163,2074}, {169,7330}, {220,3927}, {241,2338}, {243,522}, {281,3474}, {580,1212}, {760,958}, {1158,7719}, {1735,1783}, {1759,6554}, {1768,2272}, {1770,1855}, {1802,5904}, {2249,2689}, {2312,3220}, {2328,2361}
X(8559) lies on these lines: {6,8071}, {71,8560}
X(8560) lies on these lines: {6,906}, {71,8559}, {220,1415}
X(8561) lies on this line: {5643,6292}
X(8562) = crossdifference of every pair of points on line X(231)X(1989). This line is the radical axis of the circumcircle and Lester circle; it is also the line through the X(13)-Ceva conjugate of X(14) and the X(14)-Ceva conjugate of X(13), and also the line through the circumcircle inverses of X(13) and X(14). (Randy Hutson, November 18, 2015)
X(8562) lies on these lines: {3,1510}, {140,523}, {526,1511}, {570,647}, {1649,6132}, {3005,5888}, {6140,6592}
X(8562) = crossdifference of every pair of points on line X(231)X(1989); see note above.
X(8563) lies on this line: {2,493}
X(8564) lies on this line: {2,494}
X(8565) lies on this line: {216,6798}
X(8566) lies on these lines: {2,99}, {6,8575}, {353,7998}
X(8567) lies on these lines: {2,5893}, {3,64}, {5,5925}, {6,1204}, {20,1853}, {25,1620}, {74,1181}, {221,5217}, {376,6247}, {378,3567}, {549,5878}, {1155,1854}, {1192,1593}, {1350,7691}, {1503,3522}, {1656,2777}, {2192,5204}, {2883,3523}, {2935,7526}, {3520,7592}, {3576,7973}, {5562,7729}
X(8568) lies on these lines: {2,7}, {6,6745}, {10,4875}, {37,3756}, {200,5839}, {218,6700}, {594,1108}, {966,2297}, {3061,4848}, {3693,3950}
X(8569) lies on these lines: {2,39}, {32,2001}, {237,694}, {1613,3148}
X(8570) lies on these lines: {2,32}, {263,3094}
X(8571) lies on these lines: 5,6}, {115,5449}
X(8572) lies on these lines: {1,4004}, {3,1616}, {6,41}, {9,7963}, {36,1191}, {55,3445}, {106,3295}, {995,4252}, {999,4255}, {1015,4258}, {1086,3616}, {1149,5217}, {1201,3052}, {1279,7987}, {1420,3752}, {3486,3756}, {4256,7373}, {5013,6184}, {5230,5298}, {5312,5563}
X(8573) lies on these lines: {3,6}, {9,8069}, {22,5304}, {24,1249}, {25,393}, {53,1598}, {55,3553}, {56,2262}, {193,3964}, {198,2264}, {230,5020}, {232,1184}, {910,1108}, {1449,8071}, {1583,3068}, {1584,3069}, {1593,3087}, {1597,6748}, {1599,7585}, {1600,7586}, {1990,3517}, {2079,3163}, {2165,7529}, {2257,7742}, {5305,7387}, {6389,6617}, {7484,7736}
X(8574) lies on these lines: {32,512}, {39,647}, {115,6328}, {523,5305}, {525,7789}, {690,2510}, {850,7828}, {2395,3767}, {2489,3199}, {2491,6140}, {2492,2793}
X(8575) lies on these lines: {6,8566}, {574,599}
X(8576) lies on these lines: {2,371}, {3,494}, {6,3155}, {25,6424}, {32,51}, {37,2066}, {184,5058}, {372,589}, {393,5412}, {493,3167}, {588,6419}, {1322,3128}, {1599,3102}, {1600,2460}, {2165,6413}, {2987,5408}, {3071,6457}
X(8576) = isogonal conjugate of X(491)
X(8576) = X(75)-isoconjugate of X(372)
X(8576) = {X(32),X(51)}-harmonic conjugate of X(8577)
X(8577) lies on these lines: {2,372}, {3,493}, {6,3156}, {25,6423}, {32,51}, {37,5414}, {184,5062}, {371,588}, {393,5200}, {494,3167}, {589,6420}, {1321,3127}, {1599,2459}, {1600,3103}, {2165,6414}, {2987,5409}, {3070,6458}
X(8577) = isogonal conjugate of X(492)
X(8577) = X(75)-isoconjugate of X(371)
X(8577) = {X(32),X(51)}-harmonic conjugate of X(8576)
X(8578) lies on these lines: {1,522}, {32,649}, {34,1027}, {513,1104}, {663,1201}, {764,1042}, {3737,5592}
This section is contributed by Clark Kimberling and Peter Moses, November 22, 2015.
Suppose that V is a point outside a circle (U,u). Let (V,v) be the circle with center V that is orthogonal to (U,u), so that v2 = |UV|2 - u2 = power of V with respect to (U,u). Let UV,A be the circle (V,w) obtained from (U,r) = A-excircle and V = incenter, and let LA be the radical axis of UV,A and the A-excircle. Define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The triangle A'B'C' is here named the Atik triangle (Atik being the name of a star). The Atik triangle is perspective to ABC at X(3062).
If you have The Geometer's Sketchpad, you can view X(3062) and the Atik triangle.
The A-vertex of the Atik triangle is given by these barycentrics:
a^2 b-2 a b^2+b^3+a^2 c+3 b^2 c-2 a c^2+3 b c^2+c^3 : -b (a^2-2 a b+b^2+2 a c+2 b c-3 c^2) : -c (a^2+2 a b-3 b^2-2 a c+2 b c+c^2)
Radius-squared of UV,A: w2 = 4arR/sA - r2A
Power of A with respect to UV,A: (-a2 - b2 - c2 - 2ab - 2ac + 6bc)/4
Power of B with respect to UV,A: - (a + b - c )2/4
Power of C with respect to UV,A: - (a + b - c )2/4
Another construction of the Atik triangle, found near the end of the discussion at X(5927), is as follows. Let PA be the polar of the incenter with respect to the A-excircle, and define PB and PC cyclically. Then A' = PB∩PC, B' = PC∩PA, C' = PA∩PB. The triangle A'B'C' is the Atik triangle, and its centroid is X(5927). (Randy Hutson, July 11, 2014)
The inverse-in-incircle triangle defined at X(5571) is homothetic to the Atik triangle. (Randy Hutson, November 24, 2015).
X(8580) lies on these lines: {1, 2}, {3, 5234}, {5, 6769}, {9, 165}, {40, 5044}, {55, 7308}, {56, 3983}, {57, 210}, {63, 5785}, {72, 3339}, {100, 3305}, {140, 5534}, {171, 1743}, {188, 8089}, {236, 8423}, {269, 5232}, {282, 2947}, {329, 4312}, {354, 3711}, {391, 2297}, {443, 5290}, {474, 3361}, {497, 5316}, {518, 5437}, {750, 4722}, {956, 3921}, {958, 5438}, {960, 1706}, {965, 1754}, {1001, 3158}, {1054, 4712}, {1155, 3715}, {1215, 7274}, {1329, 7989}, {1449, 4682}, {1490, 6684}, {1699, 2550}, {1707, 3973}, {1864, 5696}, {2093, 5692}, {2099, 4731}, {2551, 5691}, {2886, 7988}, {2900, 6690}, {3035, 5531}, {3036, 7993}, {3174, 6666}, {3242, 5573}, {3243, 3742}, {3295, 3646}, {3306, 3681}, {3340, 3698}, {3666, 7322}, {3689, 4423}, {3693, 3731}, {3752, 7174}, {3817, 5328}, {3820, 5587}, {3925, 5219}, {3928, 5220}, {3947, 4208}, {3967, 4659}, {4002, 5730}, {4005, 5221}, {4104, 7271}, {4219, 7079}, {4298, 5815}, {4314, 5129}, {4321, 5435}, {4326, 5281}, {4327, 5772}, {4335, 5296}, {4383, 5269}, {4662, 6762}, {4855, 5260}, {5123, 5538}, {5199, 5527}, {5249, 5833}, {5273, 5732}, {5400, 5743}, {5777, 7992}, {5936, 7190}, {7028, 8090}
X(8580) = homothetic center of medial triangle and 3rd antipedal triangle of X(1)
Let A'B'C' be the incircle-antipode of the Mandart-incircle triangle. X(8581) = X(9)-of-A'B'C'. (Randy Hutson, November 30, 2015)
X(8581) lies on these lines: {1, 971}, {6, 4327}, {7, 8}, {9, 56}, {11, 118}, {12, 142}, {37, 1458}, {38, 1427}, {44, 1471}, {55, 5732}, {57, 210}, {72, 4298}, {144, 960}, {222, 3745}, {241, 984}, {269, 7174}, {390, 3476}, {392, 4315}, {480, 1466}, {516, 3057}, {517, 4312}, {527, 5434}, {612, 1407}, {651, 1386}, {942, 5290}, {948, 4310}, {954, 2646}, {999, 5779}, {1001, 1319}, {1056, 6001}, {1434, 3786}, {1445, 5220}, {1465, 4003}, {1478, 5805}, {1617, 3683}, {1697, 2951}, {1788, 3983}, {1898, 3485}, {2099, 3243}, {2263, 3242}, {2264, 5781}, {3086, 5817}, {3303, 4326}, {3333, 5777}, {3340, 5696}, {3361, 5044}, {3555, 3671}, {3660, 5219}, {3740, 5435}, {3742, 5226}, {3751, 5228}, {3812, 5261}, {3947, 5439}, {4292, 5920}, {4293, 5759}, {4355, 5904}, {4654, 5173}, {5433, 6666}, {5843, 5887}
X(8582) lies on these lines: {1, 2}, {5, 1538}, {7, 8165}, {9, 1788}, {11, 3698}, {12, 142}, {40, 5084}, {56, 5795}, {57, 2551}, {65, 3452}, {100, 4314}, {165, 452}, {226, 1329}, {329, 3339}, {388, 5437}, {404, 4297}, {405, 6684}, {442, 5927}, {443, 5587}, {474, 515}, {478, 2122}, {497, 1706}, {516, 2478}, {527, 5221}, {631, 1512}, {908, 3671}, {942, 3820}, {946, 3753}, {950, 1376}, {958, 3911}, {960, 4848}, {986, 4656}, {1212, 8568}, {1466, 5745}, {1519, 6975}, {1699, 6919}, {1837, 4413}, {2899, 3729}, {3062, 5177}, {3306, 3436}, {3333, 3421}, {3359, 6893}, {3486, 5438}, {3660, 5123}, {3816, 5836}, {3817, 4193}, {3825, 3918}, {3841, 6702}, {3947, 5249}, {4197, 7705}, {4315, 5253}, {4512, 5129}, {5047, 5537}, {5128, 5698}, {5218, 5436}, {5234, 5744}, {5691, 6904}, {5804, 6769}
X(8582) = {X(1),X(10)}-harmonic conjugate of X(6736)
X(8583) lies on these lines: {1, 2}, {3, 4512}, {9, 56}, {21, 3062}, {40, 392}, {45, 8572}, {55, 5438}, {57, 960}, {63, 3361}, {65, 5437}, {72, 3333}, {73, 5257}, {142, 3485}, {165, 404}, {210, 3304}, {269, 348}, {329, 4298}, {377, 1699}, {388, 3452}, {394, 1203}, {405, 1490}, {442, 7681}, {443, 946}, {452, 1750}, {515, 5084}, {516, 6904}, {908, 5290}, {958, 1420}, {965, 2257}, {999, 5044}, {1001, 3601}, {1191, 5269}, {1213, 3554}, {1219, 5423}, {1376, 1697}, {1385, 5720}, {1450, 2324}, {1458, 5296}, {1467, 5745}, {1468, 1743}, {1512, 6983}, {1519, 6897}, {1621, 4855}, {1706, 3057}, {1788, 5837}, {2093, 3878}, {2098, 3698}, {2136, 5919}, {2476, 7988}, {2478, 5691}, {2551, 5316}, {2646, 4423}, {2951, 3522}, {2975, 3305}, {3158, 3303}, {3306, 3339}, {3338, 5692}, {3340, 3812}, {3476, 5795}, {3612, 5259}, {3649, 6173}, {3683, 5204}, {3731, 7963}, {3753, 7982}, {3816, 5794}, {3817, 5177}, {3873, 3984}, {3876, 5223}, {3877, 7991}, {3940, 5045}, {3947, 5748}, {3962, 4860}, {4187, 5587}, {4193, 7989}, {4197, 5538}, {4301, 7994}, {4327, 5749}, {4355, 5905}, {4679, 7354}, {5129, 5731}, {5261, 5328}, {5265, 5273}, {5439, 5730}, {5603, 6769}, {5836, 7962}, {6261, 6705}
Let A'B'C' be the orthic triangle. Let A" be the symmedian point of AB'C', and define B" and C" cyclically. A"B"C" is the medial triangle of the reflection triangle of X(6), and is perspective to ABC at X(2). X(8584) = X(3) of A"B"C". (Randy Hutson, November 20, 2015)
X(8584) lies on these lines: {2, 6}, {22, 8546}, {30, 576}, {51, 2854}, {427, 5095}, {428, 8541}, {518, 3898}, {519, 4527}, {542, 1353}, {549, 575}, {648, 6749}, {1351, 3534}, {1384, 7618}, {1503, 3830}, {1569, 5052}, {2482, 5008}, {3058, 8540}, {3363, 7753}, {3564, 5066}, {3758, 4665}, {3759, 7263}, {3818, 3860}, {4363, 4405}, {4395, 4644}, {4478, 5749}, {4670, 4700}, {4677, 5846}, {4722, 4884}, {4745, 5847}, {5007, 7863}, {5024, 8182}, {5028, 8354}, {5041, 7810}, {5077, 7739}, {5111, 8353}, {5222, 7238}, {5254, 7812}, {5305, 7775}, {5839, 7227}, {7576, 8537}, {7751, 8367}, {7757, 8598}, {7759, 8360}, {7760, 8370}, {7762, 7827}, {7772, 8359}, {7817, 7838}
X(8584) = midpoint of X(6) and X(1992)
X(8584) = reflection of X(597) in X(6)
X(8584) = centroid of the six points of intersection of the Ehrmann circle and the sidelines of ABC
X(8584) = X(5) of reflection triangle of X(6)
X(8584) = {X(12158),X(12159)}-harmonic conjugate of X(11159)
Let P be the perspector of the Ehrmann circle. X(8585) is the trilinear pole of the polar of P with respect to the Ehrmann circle. (Randy Hutson, November 20, 2015)
X(8585) lies on these lines: {2, 99}, {6, 373}, {23, 8588}, {25, 5210}, {53, 468}, {110, 7708}, {182, 2502}, {187, 1995}, {352, 5640}, {843, 6787}, {1384, 5020}, {5475, 5913}
X(8585) = crossdifference of every pair of points on line X(351)X(1499)
Let A'B'C' be the anti-McCay triangle. Let A" be the isogonal conjugate of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(8586). (Randy Hutson, September 14, 2016)
X(8586) lies on these lines: {3, 6}, {316, 524}, {323, 2502}, {352, 3124}, {599, 625}, {842, 1971}, {1383, 1915}, {3055, 7608}, {3620, 5031}, {3631, 5103}, {3849, 7798}, {4663, 5184}
X(8586) = perspector of ABC and the reflection of the circumsymmedial tangential triangle in the Lemoine axis
X(8586) = isogonal conjugate of X(8587)
X(8586) = crossdifference of every pair of points on line X(523)X(8584)
X(8586) = inverse-in-circumcircle of X(8588)
X(8586) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(8590)
X(8586) = Schoute-circle-inverse of X(38225)
X(8586) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (15,16,38225), (574,576,6), (1379,1380,8588), (1687,1688,8589)
X(8586) = inverse-in-circle-{X(574),X(576),PU(1)} of X(6)
The anti-McCay triangle, A'B'C', is here introduced as the triangle of which ABC is the McCay triangle. In the following list of pairs (i,j), X(i)-of-A'B'C' = X(j): (2,2), (3,671), (4,8591), (5,2482), (6,8593), (13,8594), (14,8595), (20,8596), (30,543), (98,385), (99,8597), (111,23), (114,325), (115,8598), (351,8599), (381,99), (523,2793), (524,542), (1499,690). It is homothetic to the McCay triangle at X(2), similar to the 4th Brocard triangle, and perspective to ABC, medial triangle, anticomplementary triangle, 1st Brocard triangle, and 1st anti-Brocard triangle. (Randy Hutson, November 20, 2015)
The A-vertex of the anti-McCay triangle has these barycentrics:
5 a^4-2 a^2 b^2+2 b^4-2 a^2 c^2-5 b^2 c^2+2 c^4 : -(2 a^2-3 a b+2 b^2-c^2) (2 a^2+3 a b+2 b^2-c^2) : -(2 a^2-b^2-3 a c+2 c^2) (2 a^2-b^2+3 a c+2 c^2)}
X(8587) lies on the Kiepert hyperbola and these lines: {76, 2482}, {98, 8590}, {187, 671}, {262, 6055}, {351, 5466}, {385, 5503}, {542, 7607}, {575, 7608}, {598, 5461}, {2996, 8596}, {5485, 8591}, {7610, 8289}
X(8587) = isogonal conjugate of X(8586)
X(8587) = X(7607) of anti-McCay triangle
X(8587) = trilinear pole of line X(523)X(8584)
Tangents from P(2) to the Brocard circle touch the Brocard circle in two points, one of which is U(1). Call the other point P*. Tangents from U(2) to the Brocard circle touch the Brocard circle in two points, one of which is P(1). Call the other point U*. X(8588) is the intersection of lines P(2)P* and U(2)U*. (Randy Hutson, November 20, 2015)
X(8588) lies on these lines: {2, 6781}, {3, 6}, {20, 7749}, {23, 8585}, {30, 3054}, {69, 2482}, {111, 7492}, {115, 376}, {193, 7618}, {439, 7800}, {548, 7748}, {549, 3055}, {550, 7746}, {620, 7818}, {631, 7747}, {1506, 3523}, {1569, 6194}, {2241, 7280}, {2242, 5010}, {3522, 7756}, {3524, 7737}, {3528, 3767}, {3552, 7815}, {3619, 7820}, {3620, 7810}, {3630, 6390}, {3631, 7801}, {3734, 7771}, {3785, 7863}, {5054, 7603}, {5077, 5215}, {5569, 8598}, {6337, 7826}, {7622, 7777}, {7750, 7888}, {7751, 7782}, {7781, 7793}, {7802, 7862}, {7807, 7935}, {7811, 7908}, {7825, 7907}, {7830, 7867}, {7833, 7844}, {7835, 7865}, {7857, 7872}, {7869, 7904}, {7891, 7896}, {7913, 8356}
X(8588) = midpoint of X(6200) and X(6396)
X(8588) = inverse-in-circumcircle of X(8586)
X(8588) = inverse-in-1st-Brocard-circle of X(8589)
X(8588) = inverse-in-circle-O(15,16) of X(576)
X(8588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,X(8589)), (15,16,576), (1379,1380,8586)
X(8588) = radical center of Lucas(-2/9 cot ω) circles
X(8588) = intersection of tangents at PU(2) to conic {{X(574),PU(1),PU(2)}}
X(8588) = homothetic center of Trinh triangle and mid-triangle of inner and outer tri-equilateral triangles
X(8589) lies on the circle {{X(6),X(115),X(7603)}} and these lines: {3, 6}, {30, 3055}, {69, 7618}, {111, 7496}, {115, 549}, {140, 7756}, {141, 2482}, {193, 8182}, {353, 7998}, {376, 5475}, {538, 7771}, {548, 7747}, {550, 1506}, {620, 7853}, {625, 7833}, {631, 7748}, {671, 1153}, {1015, 5010}, {1500, 7280}, {1574, 5267}, {2502, 5650}, {2548, 3528}, {2549, 3524}, {3199, 3520}, {3523, 7746}, {3530, 7749}, {3552, 6683}, {3620, 7801}, {3630, 7813}, {3631, 6390}, {3815, 6781}, {3849, 7777}, {3934, 7782}, {6337, 7854}, {7619, 8352}, {7763, 7873}, {7769, 7842}, {7780, 7783}, {7791, 7874}, {7799, 7848}, {7816, 7824}, {7820, 8359}, {7821, 7830}, {7831, 7880}, {7847, 7886}, {7849, 7891}, {7861, 7907}, {7895, 7904}
X(8589) = inverse-in-1st-Brocard-circle of X(8588)
X(8589) = inverse-in-circle-O(15,16) of X(575)
X(8589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):(3,6,X(8588)), (15,16,575)
X(8589) = radical center of Lucas(2/9 cot ω) circles
X(8589) = midpoint of P* and U* as described at X(8588)
X(8589) = reflection of X(39) in [X(39) of 2nd Brocard triangle]
X(8589) = QA-P8 (Midray Homothetic Center) of quadrangle ABCX(2)
X(8590) lies on these lines: {3, 6}, {98, 8587}, {5970, 8600}, {7766, 8350}
X(8590) = center of circle {X(187),X(575),PU(1)}
X(8590) = midpoint of X(187) and X(575)
X(8590) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(8586)
X(8590) = {X(1687),X(1688)}-harmonic conjugate of X(8586)
Let A'B'C' be the anticomplementary triangle of ABC. The Kiepert circumconic of A'B'C' and the Steiner circumconic of A'B'C' intersect in 4 points: A', B', C', and X(8591). (Peter Moses, November 23, 2015)
If you have The Geometer's Sketchpad, you can view X(8591). X(8591) lies on the Steiner circumellipse of the anticomplementary triangle (aka, the permutation ellipse E(X(4440))), the bianticevian conic of X(1) and X(2), and these lines: {1, 2796}, {2, 99}, {3, 7616}, {4, 8724}, {5, 12355}, {6, 11164}, {8, 9881}, {10, 9875}, {20, 542}, {22, 9876}, {23, 2936}, {30, 147}, {63, 29617}, {69, 9830}, {76, 34504}, {98, 10304}, {100, 12326}, {114, 3839}, {145, 9884}, {187, 11054}, {193, 8593}, {194, 1992}, {316, 15301}, {325, 8597}, {376, 2782}, {384, 597}, {385, 8598}, {388, 12350}, {485, 35698}, {486, 35699}, {497, 12351}, {519, 13174}, {524, 9855}, {530, 617}, {531, 616}, {538, 8782}, {547, 38732}, {549, 14651}, {599, 1975}, {627, 5463}, {628, 5464}, {690, 9131}, {1003, 10336}, {1270, 9883}, {1271, 9882}, {1634, 36182}, {1916, 14033}, {2418, 10787}, {2784, 34628}, {2786, 17487}, {2793, 37749}, {2794, 15683}, {2795, 15677}, {2975, 22565}, {2996, 10153}, {3023, 10385}, {3085, 10054}, {3086, 10070}, {3091, 9880}, {3146, 14981}, {3180, 8594}, {3181, 8595}, {3314, 5077}, {3434, 12348}, {3436, 12349}, {3448, 11006}, {3455, 7492}, {3522, 38664}, {3524, 11632}, {3529, 7946}, {3534, 9862}, {3543, 6054}, {3545, 6321}, {3552, 37809}, {3616, 12258}, {3845, 38733}, {3849, 7779}, {3926, 33192}, {4027, 33187}, {4240, 12347}, {4576, 14916}, {4590, 35087}, {5032, 10754}, {5056, 38734}, {5068, 20399}, {5071, 15561}, {5181, 14833}, {5182, 7787}, {5186, 7714}, {5215, 32457}, {5459, 22577}, {5460, 22578}, {5485, 8587}, {5525, 16563}, {5601, 12345}, {5602, 12346}, {5976, 33008}, {5984, 38738}, {5989, 13586}, {6033, 15682}, {6034, 7738}, {6036, 15708}, {6055, 15692}, {6337, 15814}, {6390, 8352}, {6462, 12352}, {6463, 12353}, {6655, 7801}, {6658, 7781}, {6777, 33608}, {6778, 33609}, {7486, 38751}, {7585, 19058}, {7586, 19057}, {7748, 7870}, {7756, 7883}, {7757, 19686}, {7772, 19693}, {7774, 8592}, {7777, 8786}, {7783, 8370}, {7785, 34511}, {7793, 33208}, {7796, 19691}, {7797, 8369}, {7799, 31173}, {7810, 33260}, {7816, 7827}, {7817, 33225}, {7836, 7841}, {7839, 20583}, {7864, 33237}, {7891, 11318}, {7898, 32817}, {7925, 37350}, {8030, 10553}, {8182, 10810}, {8290, 11286}, {8359, 17128}, {8703, 12188}, {8716, 11163}, {8859, 27088}, {8860, 33274}, {8972, 13908}, {9168, 14443}, {9263, 17147}, {9770, 9773}, {9998, 30229}, {10519, 19905}, {10528, 12356}, {10529, 12357}, {10722, 15640}, {11001, 14976}, {11149, 33259}, {11159, 31859}, {11168, 33273}, {11599, 25055}, {11623, 15717}, {11646, 21356}, {11656, 15035}, {11711, 38314}, {12036, 35279}, {12040, 17005}, {12042, 19708}, {13574, 20063}, {13941, 13968}, {14002, 34013}, {14041, 22110}, {14645, 20065}, {15048, 35954}, {15687, 38743}, {15697, 38749}, {15700, 38635}, {15702, 38224}, {15703, 38229}, {15709, 38750}, {15719, 38739}, {15721, 38748}, {15933, 24472}, {16509, 17006}, {17578, 38745}, {18823, 33799}, {20016, 30579}, {20112, 37647}, {25561, 37336}, {26070, 29576}, {26081, 31144}, {26613, 32456}, {31276, 32822}, {32528, 33017}, {32552, 33612}, {32553, 33613}, {32819, 33013}, {32820, 33256}, {32824, 32997}, {32833, 33264}, {32836, 33207}, {33376, 36775}, {33610, 35751}, {33611, 35750}, {34897, 35923}, {37299, 38499}
X(8591) = reflection of X(i) and X(j) for these {i,j}: (2, 99), (148, 2), (193, 8593), (385, 8598), (671, 2482), (3180, 8594), (3181, 8595), (3543, 6054), (8352, 6390), (8596, 671), (8597, 325)
X(8591) = isotomic conjugate of anticomplement of X(39061)
X(8591) = complement of X(8596)
X(8591) = anticomplement of X(671)
X(8591) = anticomplementary conjugate of X(316)
X(8591) = anticomplementary isotomic conjugate of X(524)
X(8591) = antipode of X(2) in bianticevian conic of X(1) and X(2)
X(8591) = inverse-in-Steiner-circumellipse of X(2482)
X(8591) = {X(99),X(671)}-harmonic conjugate of X(2482)
X(8591) = X(524)-Ceva conjugate of X(2)
X(8591) = X(4) of anti-McCay triangle
X(8591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,8596,671), (99,671,2482), (671,2482,2), (671,8596,148)
X(8591) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1,316), (31,524), (48,858), (163,690), (187,8), (524,6327), (604,4442), (896,69), (922,2), (923,671), (1101,5468), (2157,67), (2642,3448), (3292,4329), (5467,7192)
X(8592) lies on these lines: {2, 353}, {99, 3849}, {147, 376}, {385, 8593}, {543, 598}, {551, 5988}, {671, 3329}, {2482, 3314}, {5152, 7622}, {5463, 5978}, {5464, 5979}, {5999, 6054}, {7774, 8591}, {7880, 8290}
X(8593) lies on these lines: {2, 98}, {6, 598}, {69, 2482}, {99, 524}, {148, 5032}, {193, 8591}, {385, 8592}, {543, 1992}, {599, 5026}, {1503, 8352}, {3618, 5461}, {4563, 8030}, {5503, 7774}, {7840, 8289}, {8370, 8550}
X(8593) = reflection of X(671) in X(6)
X(8593) = {X(8594),X(8595)}-harmonic conjugate of X(99)
X(8593) = X(671)-of-anti-Artzt-triangle
X(8593) = 1st-tri-squares-to-anti-Artzt similarity image of X(13640)
X(8593) = orthologic center of these triangles: anti-Artzt to 1st Brocard
X(8594) = reflection of X(8595) in X(8598)
X(8594) = {X(99),X(8593)}-harmonic conjugate of X(8595)
X(8594) = X(13)-antipedal-to-X(14)-antipedal similarity image of X(2)
X(8594) lies on these lines: {2, 14}, {99, 524}, {298, 2482}, {299, 3849}, {396, 671}, {543, 5472}, {3180, 8591}
X(8595) lies on these lines: {2, 13}, {99, 524}, {298, 3849}, {299, 2482}, {395, 671}, {543, 5471}, {3181, 8591}
X(8595) = reflection of X(8594) in X(8598)
X(8595) = {X(99),X(8593)}-harmonic conjugate of X(8594)
X(8595) = X(14)-antipedal-to-X(13)-antipedal similarity image of X(2)
X(8596) lies on these lines: {2, 99}, {8, 2796}, {30, 5984}, {542, 3146}, {599, 6655}, {2782, 3543}, {2996, 8587}, {3839, 6321}, {7779, 8597}
X(8596) =anticomplement of X(8591)
X(8596) =reflection of X(2) in X(148)
X(8596) =antipode of X(2) in conic through X(2), X(8), and the extraversions of X(8)
As a point on the Euler line, X(8597) has Shinagawa coefficients ((E+F)2+3S2, -18S2).
X(8597) lies on the anti-McCay circumcircle and these lines: {2, 3}, {148, 524}, {316, 543}, {325, 8591}, {385, 671}, {598, 3329}, {599, 7898}, {2482, 7925}, {5461, 6781}, {7617, 7771}, {7747, 7827}, {7748, 7812}, {7775, 7783}, {7779, 8596}, {7801, 7885}, {7825, 7870}, {7842, 7883}
X(8597) = anticomplement of X(8598)
X(8597) = reflection of X(385) in X(671)
As a point on the Euler line, X(8598) has Shinagawa coefficients ((E+F)2-6S2, 9S2).
X(8598) lies on these lines: {2, 3}, {99, 524}, {183, 8182}, {187, 543}, {230, 671}, {325, 2482}, {385, 8591}, {530, 6783}, {531, 6782}, {597, 3972}, {598, 3815}, {1285, 5032}, {2396, 8030}, {5210, 7610}, {5215, 5461}, {5475, 7622}, {5569, 8588}, {6390, 7840}, {7603, 7619}, {7618, 7737}, {7750, 7801}, {7756, 7817}, {7757, 8584}, {7782, 7812}, {7789, 7883}, {7802, 7870}, {7810, 7816}
X(8598) =complement of X(8597)
X(8598) =midpoint of X(8594) and X(8595)
X(8599) = lies on these lines: {110, 892}, {351, 523}, {512, 598}, {690, 850}, {1383, 2395}, {1499, 8352}, {5996, 8371}
X(8599) = trilinear pole of line X(115)X(2793)
X(8600) lies on the circumcircle and these lines: {111, 576}, {5970, 8590}
X(8600) = Thomson-isogonal conjugate of X(32479)
X(8600) = Lucas-isogonal conjugate of X(32479)
X(8600) = Λ(perspectrix of ABC and anti-McCay triangle)
X(8600) = Λ(radical axis of circumcircle and McCay circumcircle)
Points associated with orthocevian triangles: X(8601)-X(8614)
This section is contributed by Clark Kimberling and Peter Moses, November 22, 2015.
Let (O,R) be the circumcircle, and let A′B′C′ be the cevian triangle of a point X. Let (OA,rA) be the circle through B′ and C′ and orthogonal to (O,R). That is, (OA,rA) is the circle that passes through the points B' and C' and also their inverses in (O,R). Define (OB,rB) and (OC,rC) cyclically. The triangle T(X) = OAOBOC is here named the orthocevian triangle of X. If P = p : q : r (barycentrics) and T(P) is perspective to ABC, then the perspector is the point whose isogonal conjugate has the following barycentrics:
a2(p2qr + q2r2) - b2(p2qr + p2r2) - c2(p2qr + p2q2) : :
The appearance of (i,j) in the following list means that T(X(i)) is perspective to ABC and that X(j) is the perspector.
(1,2160), (2,25), (3,8612), (4,2165), (6,8601), (7,1), (8,8602), (13,8603), (14,8604), (74,3003), (75,8615), (81,8605), (88,3689), (92,8606), (98,230), (99,523), (100,650), (101,6586), (102,8607), (103,8608), (104,8609), (105,3290), (106,8610), (107,6587), (108,6588), (109,6589), (110,647), (111,3291), 112,2485), (190,773), (253,3532), (280,198), (476,1637), (477,3018), (651,3900), (658,4105), (660,4435), (662,4041), (673,2340), (675,3011), (691,2492), (789,4874), (805,2491), (835,6590), (842,2493), (901,3310), (925,2501), (927),676), (934,6129)
If P is on the Darboux cubic, then the orthocevian triangle of P is perspective to the tangential triangle. The appearance of (i,j) in the next list means that X(i) is on the Darboux cubic, and X(j) is the perspector of T(P) and the tangential triangle: (1,8614), (4,25), (20,1498).
If P is on (O,R), then (perspector of T(P) and ABC) = (cevapoint of X(6) and P*)*, where * denotes isogonal conjugate. For points associated with orthoanticevian triangles, see the preamble to X(8735).
The orthocevian triangle of X(3) is homothetic to the circumorthic triangle at X(54). (Randy Hutson, December 4, 2015)
X(8601) lies on these lines: {39,6234}, {193,732}, {3978,7745}
X(8602) lies on these lines: {9,1158}, {1436,3554}, {2270,2316}, {2343,3197}, {2432,4394}
X(8603) lies on these lines: {5,14}, {6,3132}, {15,1154}, {298,1273}, {2307,2599}, {570,8604}
X(8603) = isogonal conjugate of X(8838)
X(8603) = barycentric product X(15)*X(17)
X(8604) lies on these lines: {5,13}, {6,3131}, {16,1154}, {299,1273}, {570,8603}
X(8604) = isogonal conjugate of X(8836)
X(8604) = barycentric product X(16)*X(18)
X(8605) lies on these lines: {209,3059}, {354,3671}, {2198,8012}, {2352,4512}, {3868,4673}
X(8606) lies on these lines: {1,1786}, {3,7100}, {31,3003}, {35,79}, {55,199}, {939,5217}, {1789,1792}, {1816,3615}, {4183,7110}
X(8606) = isogonal conjugate of X(7282)
X(8607) lies on these lines: {6,41}, {37,216}, {53,1881}, {230,231}, {800,1108}, {1609,5301}, {2182,7117}, {2252,4559}, {2352,3192}, {3553,4261}
X(8607) = isogonal conjugate of X(2988)
X(8607) = complement of X(35516)
X(8607) = X(2)-Ceva conjugate of X(117)
X(8607) = crossdifference of every pair of points on line X(3)X(522)
X(8607) = perspector of hyperbola {{A,B,C,X(4),X(109)}} (circumconic centered at X(117))
X(8607) = intersection of trilinear polars of X(4) and X(109)
X(8607) = X(63)-isoconjugate of X(32706)
X(8607) = center of circumconic that is locus of trilinear poles of lines passing through X(117)
X(8608) lies on these lines: {6,31}, {37,800}, {39,4646}, {53,1860}, {216,1108}, {230,231}, {517,3002}, {1319,7117}, {1458,2272}, {1945,2161}, {1951,5172}, {2197,2264}, {2257,4261}, {3554,8270}
X(8608) = isogonal conjugate of X(2989)
X(8608) = complement of X(35517)
X(8608) = X(2)-Ceva conjugate of X(118)
X(8608) = crossdifference of every pair of points on line X(3)X(514)
X(8608) = perspector of hyperbola {A,B,C,X(4),X(101)} (circumconic centered at X(118))
X(8608) = intersection of trilinear polars of X(4) and X(101)
X(8608) = center of circumconic that is locus of trilinear poles of lines passing through X(118)
X(8608) = X(63)-isoconjugate of X(917)
X(8608) = polar conjugate of isotomic conjugate of X(916)
X(8609) lies on these lines: {1,6}, {2,3262}, {19,1609}, {53,225}, {101,1731}, {216,7561}, {230,231}, {241,1086}, {393,1068}, {517,2245}, {571,5301}, {594,6734}, {909,1319}, {910,2078}, {1055,2173}, {1072,5254}, {1214,3772}, {1284,3827}, {1385,2278}, {1400,1953}, {1404,2265}, {1465,2006}, {1612,6198}, {1778,3193}, {1824,2352}, {1880,2165}, {2170,2183}, {2171,2260}, {2174,2264}, {2223,4516}, {2277,5230}, {2305,3072}, {3693,3943}, {4261,5292}
X(8609) = isogonal conjugate of X(2990)
X(8609) = X(2)-Ceva conjugate of X(119)
X(8609) = crossdifference of every pair of points on line X(3)X(513)
X(8609) = perspector of hyperbola {{A,B,C,X(4),X(100)}} (circumconic centered at X(119))
X(8609) = intersection of trilinear polars of X(4) and X(100)
X(8609) = enter of circumconic that is locus of trilinear poles of lines passing through X(119)
X(8609) = X(63)-isoconjugate of X(915)
X(8609) = polar conjugate of isotomic conjugate of X(912)
X(8610) lies on these lines: {1,4277}, {2,3264}, {6,1201}, {37,39}, {44,1015}, {45,2275}, {106,5053}, {230,231}, {594,3831}, {674,3009}, {1055,3285}, {1149,2183}, {1334,4286}, {1575,3943}, {2092,3723}, {2178,3053}, {2245,3230}, {3247,4261}, {3726,4053}, {3731,5069}, {4484,4517}, {5035,5563}
X(8610) = complement of X(3264)
X(8610) = X(2)-Ceva conjugate of X(121)
X(8610) = crossdifference of every pair of points on line X(3)X(3667)
X(8610) = perspector of hyperbola {{A,B,C,X(4),X(1293)}} (circumconic centered at X(121))
X(8610) = intersection of trilinear polars of X(4) and X(1293)
X(8610) = center of circumconic that is locus of trilinear poles of lines passing through X(121)
X(8611) lies on these lines: {71,822}, {101,2766}, {521,652}, {523,661}, {647,656}, {650,663}, {1459,2522}, {3064,3239}, {3119,6741}
X(8611) = crossdifference of every pair of points on line X(28)X(34) (line is the locus of the P-beth conjugate of P, for P on the Euler line)
X(8612) lies on the Jerabek hyperbola, but on no line X(i)X(j) for 1 <= i <= 8611
X(8612) = isogonal conjugate of X(8613)
As a point on the Euler line, X(8613) has Shinagawa coefficients (S2-F2, -2S2).
X(8613) lies on these lines: {2,3}, {97,324}, {216,275}, {219,7361}, {222,6360}, {577,2052}, {925,1298}, {1993,3164}
X(8613) = reflection of X(i) and X(j) for these (i,j): (472, 465), (473, 466)
X(8613) = isogonal conjugate of X(8612)
X(8613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22,1370,5999), (3152,7538,7513).
X(8614) lies on these lines: {1,399}, {3,6149}, {6,1406}, {12,651}, {55,500}, {56,58}, {60,3028}, {65,267}, {73,1399}, {81,3649}, {109,2594}, {221,2099}, {223,1454}, {1388,1616}, {1718,5885}, {1725,7100}, {1745,5348}, {2361,4303}, {2392,2915}, {3562,6284}, {5217,7078}
X(8615) lies on these lines: {10,4220}, {21,3220}, {25,3924}, {41,2092}, {55,976}, {1193,2194}, {1257,5285}, {1284,2218}, {2204,2354}
X(8615) = isogonal conjugate of X(7270)
X(8615) = X(i)-isoconjugate of X(j) for these {i,j}: {1,7270}, {2,5279}, {6,2064}, {8,4296}, {75,5285}
Gibert circumtangential conjugates: X(8616)-X(8665)
This section is contributed by Peter Moses, November 26, 2015, based on notes from Bernard Gibert, November 2, 2015, in connection with the cubic K024.
The Gibert circumtangential conjugate of a point U = u : v : w (barycentrics) is defined as the point
a^2 (a^2 b^2 c^2 u v - b^2 c^4 u v + a^4 c^2 v^2 - a^2 c^4 v^2 + a^2 b^2 c^2 u w - b^4 c^2 u w - 3 a^2 b^2 c^2 v w + a^4 b^2 w^2 - a^2 b^4 w^2) : :
A related point is the circumtangential-isogonal conjugate of U, defined by
a^2 (a^2 c^2 u v - c^4 u v + a^2 c^2 v^2 - c^4 v^2 + a^2 b^2 u w - b^4 u w + a^4 v w - a^2 b^2 v w - a^2 c^2 v w - 2 b^2 c^2 v w + a^2 b^2 w^2 - b^4 w^2) : :
If U is on the circumcircle, then its Gibert circumtangential conjugate, denoted by M1(U), is on the Lemoine axis, X(187)X(237), and the circumtangential-isogonal conjugate, denoted by M2(U), is on the line at infinity, X(30)X(511). Thus, if U is on X(187)X(237), then M1(U) is on the circumcircle, and if U is on X(30)X(511), then M2(U) is on the circumcircle. For a selection of circumtangential-isogonal conjugates, see X(8666)-X(8714).
The appearance of (i,j) in the following list means that the Gibert circumtangential conjugate of X(i) is X(j):
(2,3), (74,1495), (98,237), (99,669), (100,667), (101,649), (105,2223), (106,902), (108,1946), (109,663), (110,512), (111,187), (112,647), (351,691), (665,919), (699,3229), (727,3009), (729,3231), (739,3230), (741,3747), (759,3724), (785,2978), (805,5027), (825,3250), (827,3005), (842,5191), (843,2502), (890,898), (901,1960), (907,3804), (1055,2291), (1291,6140), (1379,5638), (1380,5639), (1995,8182), (2701,5075), (2702,5029), (2703,5040), (2712,5168), (2715,3569), (4588,4775), (5092,5888), (5106,5970), (5994,6138), (5995,6137), (6287,6636)
Let U* denote the isogonal conjugate of a point U not on a sideline of ABC. The points U*, M1(U), M2(U) are collinear. If U lies on the cubic K024, then M1(U) = U*; if U is on the circumcircle or at infinity, then M2(U) = U*. If U is on the circumcircle or the Lemoine axis, then M1(U) = X(32)/U (barycentric quotient), The barycentric product U*M1(U) lies on the Brocard axis. (Bernard Gibert, November 27, 2015)
X(8616) lies on these lines: {1,21}, {2,902}, {6,3750}, {9,983}, {35,978}, {43,55}, {100,748}, {105,165}, {171,1001}, {200,8300}, {312,4432}, {354,4650}, {405,5255}, {518,7262}, {551,4257}, {750,5284}, {982,1279}, {984,3683}, {985,1961}, {1044,7742}, {1197,2176}, {1201,4189}, {1215,4676}, {1283,6210}, {1486,5329}, {1613,3230}, {1617,4334}, {1698,5192}, {1714,4309}, {1724,3746}, {1740,8053}, {1743,2280}, {1757,3870}, {2205,3294}, {2382,6079}, {3011,3944}, {3158,5524}, {3208,4426}, {3219,3938}, {3246,3752}, {3295,5247}, {3576,5197}, {3679,5278}, {3681,3722}, {3685,4362}, {3731,5276}, {3748,4641}, {3751,3979}, {3757,3923}, {3771,4388}, {3886,4154}, {3916,3976}, {4011,7081}, {4090,4759}, {4414,7191}, {4438,4514}, {5259,5264}
X(8616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4512,846), (2,902,3550), (6,4428,3750), (9,3749,3961), (21,3915,1), (31,1621,1), (55,238,43), (165,5272,1054), (595,5248,1), (1001,3052,171), (1279,4640,982), (3683,3744,984)
X(8617) lies on these lines: {2, 6}, {110, 5033}, {111, 3098}, {187, 6787}, {353, 5092}, {574, 5888}, {2502, 7712}, {3291, 7998}
X(8618) lies on these lines: {3, 142}, {31, 4116}, {32, 560}, {39, 2309}, {160, 3941}, {187, 237}, {766, 1755}, {1634, 6629}, {2876, 3002}, {4215, 5310}
X(8619) lies on these lines: {10, 75}, {187, 237}, {712, 2227}, {2085, 3721}, {3116, 3735}
X(8620) lies on these lines: {2, 37}, {187, 237}, {292, 3218}, {518, 3121}, {726, 2229}, {893, 3920}, {899, 6377}, {2275, 4392}, {2300, 3051}
X(8621) lies on these lines: {6, 75}, {187, 237}, {213, 899}, {753, 795}, {869, 1908}, {2231, 3264}
X(8622) lies on these lines: {2, 31}, {10, 1923}, {42, 1197}, {55, 1613}, {187, 237}, {612, 3508}, {743, 813}, {2236, 3263}
X(8623) lies on these lines: {2, 32}, {3, 695}, {39, 3051}, {112, 420}, {171, 172}, {187, 237}, {385, 3978}, {694, 2076}, {1084, 5201}, {1193, 1197}, {1196, 5188}, {1976, 2458}, {5162, 8569}, {5206, 6030}, {8265, 8266}
X(8623) = isogonal conjugate of X(14970)
X(8623) = cevapoint of X(32) and X(9480)
X(8623) = crosspoint of X(i) and X(j) for these {i,j}: {6, 733}, {385, 1691}, {805, 4590}
X(8623) = crosssum of X(i) and X(j) for these {i,j}: {2, 732}, {694, 1916}, {804, 3124}
X(8623) = crossdifference of every pair of points on line X(2)X(881)
X(8624) lies on these lines: {1, 32}, {3, 2176}, {36, 292}, {38, 2205}, {39, 213}, {55, 1908}, {187, 237}, {238, 2235}, {239, 5291}, {731, 813}, {761, 919}, {980, 3218}, {2240, 3006}, {3774, 5132}
X(8625) lies on these lines: {1, 82}, {101, 4093}, {187, 237}, {213, 3778}, {1631, 2176}, {3725, 5285}
X(8626) lies on these lines: {6, 560}, {31, 36}, {42, 2251}, {187, 237}, {239, 8297}, {244, 5144}, {674, 922}, {717, 795}, {2177, 4262}
X(8627) lies on these lines: {6, 22}, {23, 1691}, {110, 2076}, {187, 237}, {323, 5104}, {703, 783}, {1915, 6636}, {3094, 7492}, {5017, 6800}, {5092, 7467}, {5206, 5651}
X(8628) lies on these lines: {22, 55}, {31, 1501}, {100, 3797}, {187, 237}, {743, 919}, {1401, 1402}, {2200, 5369}
X(8629) lies on these lines: {22, 1602}, {187, 237}, {560, 1501}
X(8630) is the intersection of these three lines: 1) line PU(12); 2) the trilinear polar of the cevapoint of PU(12); 3) the line through [P(12)-Ceva conjugate of U(12)] and [U(12)-Ceva conjugate of P(12)] (Randy Hutson, March 21, 2019)
X(8630) lies on these lines: {187, 237}, {795, 825}, {814, 7255}, {1918, 1919}, {1924, 1980}, {4057, 8053}
X(8630) = crossdifference of every pair of points on line X(2)X(561)
X(8631) lies on these lines: {187, 237}, {514, 1924}, {824, 4560}, {1919, 4367}
X(8632) lies on these lines: {6, 3768}, {100, 101}, {106, 739}, {187, 237}, {513, 1919}, {572, 2827}, {659, 4435}, {661, 830}, {731, 743}, {798, 4057}, {812, 4366}, {1015, 1977}, {1019, 3960}, {1027, 1438}, {1911, 3572}, {2112, 2254}, {2483, 4079}, {2484, 4502}, {3766, 4107}
X(8632) = isogonal conjugate of X(4562)
X(8632) = complement of X(21303)
X(8632) = anticomplement of X(21261)
X(8632) = crossdifference of every pair of points on line X(2)X(38)
X(8632) = polar conjugate of isotomic conjugate of X(22384)
X(8633) lies on these lines: {187, 237}, {661, 1980}, {788, 7252}, {832, 1491}, {1919, 7234}, {3835, 4164}
X(8634) lies on this line: {187, 237}
X(8635) lies on these lines: {187, 237}, {788, 1980}, {798, 1933}, {2254, 3733}
X(8636) lies on these lines: {187, 237}, {810, 1980}
X(8637) lies on these lines: {187, 237}, {522, 1324}, {659, 784}, {810, 838}
X(8637) = crossdifference of every pair of points on line X(2)X(313)
X(8637) = isogonal conjugate of anticomplement of X(39016)
X(8637) = anticomplement of complementary conjugate of X(39016)
X(8638) lies on these lines: {55, 4435}, {187, 237}, {659, 885}, {900, 8053}
X(8638) = anticomplement of complementary conjugate of X(39014)
X(8638) = bicentric difference of PU(95)
X(8638) = PU(95)-harmonic conjugate of X(9449)
X(8639) lies on these lines: {187, 237}, {523, 1325}, {798, 3049}, {4041, 7234}, {4455, 4822}
X(8640) lies on these lines: {31, 1980}, {100, 7035}, {187, 237}, {513, 3510}, {522, 659}, {650, 4455}, {798, 4507}, {875, 1402}, {3185, 4394}, {4057, 4782}, {4367, 4932}, {4378, 7192}
X(8640) = isogonal conjugate of X(18830)
X(8640) = crosspoint of X(i) and X(j) for these {i,j}: {6, 932}, {31, 100}
X(8640) = crosssum of X(i) and X(j) for these (i,j): {2, 4083}, {75, 513}, {321, 2533}, {514, 3840}
X(8640) = crossdifference of every pair of points on line X(2)X(330)
X(8641) lies on these lines: {1, 1938}, {55, 650}, {187, 237}, {513, 2078}, {652, 926}, {657, 4105}, {661, 2520}, {692, 2149}, {693, 1621}, {1001, 4885}, {1021, 3900}, {3239, 4477}, {4428, 4762}
X(8641) = isogonal conjugate of X(4569)
X(8642) lies on these lines: {25, 884}, {55, 4394}, {110, 2704}, {184, 1980}, {187, 237}, {692, 6066}, {1001, 4106}, {1621, 4380}, {1635, 6608}, {3733, 7659}, {4057, 6129}
X(8643) lies on these lines: {1, 4401}, {101, 6065}, {110, 2705}, {187, 237}, {657, 1919}, {659, 4449}, {1019, 4794}, {1459, 4057}, {1635, 3900}, {2441, 3052}, {3667, 4881}, {4041, 6050}, {4162, 4394}, {4367, 4724}, {4462, 4504}, {4730, 4959}, {4782, 4879}
X(8643) = isogonal conjugate of isotomic conjugate of X(3667)
X(8644) lies on the line X(1499)X(8644), which is the perspectrix of ABC and the 1st Parry triangle, as well as the trilinear polar of X(1992), and also the orthic axis of the Thompson triangle. (The 1st Parry triangle is defined in the preamble to X(9123).)
X(8644) lies on these lines: {25, 2489}, {51, 3221}, {110, 2709}, {154, 924}, {187, 237}, {1384, 2444}, {1499, 4786}, {1962, 4139}, {2780, 5926}
X(8644) = isogonal conjugate of X(35179)
X(8644) = crossdifference of every pair of points on line X(2)X(2418)
X(8644) = X(187)-of-1st-Parry-triangle
X(8644) = intersection of Lemoine axes of ABC and 1st Parry triangle
X(8644) = intersection of orthic axes of anti-McCay and anti-Artzt triangles
X(8645) lies on these lines: {55, 1635}, {100, 4763}, {187, 237}, {692, 2874}, {812, 1621}, {884, 6187}, {1001, 4728}, {1769, 4491}, {4928, 5284}
X(8645) = isogonal conjugate of X(35171)
X(8645) = crossdifference of every pair of points on line X(2)X(1111)
X(8646) lies on these lines: {187, 237}, {513, 5078}, {832, 2254}, {1980, 3063}, {4025, 4367}
X(8647) lies on these lines: {31, 1475}, {41, 55}, {56, 1149}, {187, 237}, {672, 1438}, {692, 1404}, {1253, 2347}, {1400, 1486}, {2175, 2293}, {3217, 6600}, {3601, 4511}
X(8647) = isogonal conjugate of X(35160)
X(8647) = crossdifference of every pair of points on line X(2)X(3676)
X(8648) lies on these lines: {3, 2254}, {21, 3716}, {35, 3887}, {36, 3960}, {55, 4895}, {187, 237}, {692, 1415}, {993, 3762}, {1635, 6187}, {3271, 7117}, {3738, 4996}, {4057, 6615}
X(8648) = isogonal conjugate of X(35174)
X(8648) = crossdifference of every pair of points on line X(2)X(2006)
X(8649) lies on these lines: {1, 8297}, {6, 101}, {37, 214}, {110, 5170}, {115, 6739}, {172, 5315}, {187, 237}, {292, 2163}, {574, 6184}, {1149, 2251}, {1201, 5007}, {1252, 2226}, {1500, 4256}, {2087, 2246}, {2112, 5008}, {2176, 4257}, {2241, 3207}
X(8649) = isogonal conjugate of X(35168)
X(8649) = crossdifference of every pair of points on line X(2)X(900)
X(8650) lies on these lines: {187, 237}, {659, 4750}, {2254, 4491}, {4401, 4786}
X(8651) lies on these lines: {25, 6753}, {187, 237}, {690, 3265}, {924, 6132}, {2353, 8574}, {2491, 3221}, {2501, 6562}, {3566, 3798}, {3709, 7234}, {4455, 7180}, {4765, 6050}
X(8651) = isogonal conjugate of X(35136)
X(8651) = crossdifference of every pair of points on line X(2)X(1975)
X(8652) lies on the circumcircle and these lines: {106, 1468}, {835, 4756}, {3573, 6013}
X(8652) = isogonal conjugate of X(4802)
X(8652) = trilinear pole of line X(6)X(35)
X(8652) = Ψ(X(6), X(35))
X(8653) lies on these lines: {187, 237}, {1499, 3798}, {2499, 6589}, {3709, 4524}, {4765, 4843}
X(8654) lies on this line: {187, 237}
X(8655) lies on these lines: {187, 237}, {659, 4802}, {4057, 4724}, {4455, 4813}, {4893, 7234}
X(8656) lies on these lines: {41, 1919}, {56, 4057}, {101, 6017}, {187, 237}, {1635, 4814}, {2516, 4041}, {3667, 7987}, {4394, 4895}, {4401, 4449}
X(8657) lies on these lines: {187, 237}, {1919, 3768}, {3063, 4057}, {3207, 4394}, {4401, 4435}
X(8658) lies on these lines: {101, 6014}, {187, 237}, {1635, 6161}, {3960, 4790}
X(8659) lies on these lines: {6, 6085}, {101, 1293}, {106, 2441}, {187, 237}, {650, 6004}, {690, 4773}, {826, 4976}, {891, 4435}, {1015, 1357}, {1438, 2440}, {3063, 6363}, {3309, 4394}
X(8660) lies on these lines: {187, 237}, {4057, 4943}
X(8661) lies on these lines: {187, 237}, {659, 3218}, {899, 6165}, {900, 4358}, {1646, 8027}
X(8662) lies on these lines: {187, 237}, {659, 4765}, {4057, 4394}
X(8663) lies on these lines: {187, 237}, {661, 4155}, {1962, 4979}, {4983, 4988}
X(8664) lies on these lines: {187, 237}, {684, 1510}, {690, 2528}, {2474, 7651}, {2492, 2514}, {7234, 8034}
X(8664) = isogonal conjugate of X(35137)
X(8664) = crossdifference of every pair of points on line X(2)X(3108)
Barycentrics a^2*(b - c)*(b + c)*(a^2 + 2*b^2 + 2*c^2) : :
X(8665) lies on these lines: {187, 237}, {523, 2528}, {690, 3806}, {2525, 7927}
Circumtangential-isogonal conjugates: X(8666)-X(8714)
This section is contributed by Peter Moses, November 26, 2015, based on notes from Bernard Gibert, November 2, 2015, in connection with the cubic K024.
The circumtangential-isogonal conjugate of U, denoted by M2(U), is defined by
a^2 (a^2 c^2 u v - c^4 u v + a^2 c^2 v^2 - c^4 v^2 + a^2 b^2 u w - b^4 u w + a^4 v w - a^2 b^2 v w - a^2 c^2 v w - 2 b^2 c^2 v w + a^2 b^2 w^2 - b^4 w^2) : :
If U is on the circumcircle, then M2(U), is on the line at infinity, X(30)X(511), so that if U is on X(30)X(511), then M2(U) is on the circumcircle. Related conjugates are M1(U) defined in the preamble to X(8616) and the circumnormal-isogonal conjugate M3(U) defined in the preamble to X(8715).
The appearance of (i,j) in the following list means that the circumtangential-isogonal conjugate of X(i) is X(j):
(2,3098), (3,3), (4,7689), (5,7691), (15,6582), (16,6295), (20,3357), (21,3579), (30,74), (39,6308), (40,5450), (98,511), (99,512), (100,513), (101,514), (102,515), (103,516), (104,517), (105,518), (106,519), (107,520), (108,521), (109,522), (110,523), (111,524), (112,525), (182,6194), (371,6312), (372,6316), (376,4550), (476,526), (477,5663), (527,2291), (528,840), (530,2378), (531,2379), (532,2380), (533,2381), (536,739), (537,2382), (538,729), (539,2383), (541,841), (542,842), (543,843), (545,2384), (574,8182), (674,675), (680,681), (688,689), (690,691), (696,697), (698,699), (700,701), (702,703), (704,705), (706,707), (708,709), (710,711), (712,713), (714,715), (716,717), (718,719), (720,721), (722,723), (724,725), (726,727), (730,731), (732,733), (734,735), (736,737), (740,741), (742,743), (744,745), (746,747), (752,753), (754,755), (758,759), (760,761), (766,767), (768,769), (772,773), (776,777), (778,779), (780,781), (782,783), (784,785), (786,787), (788,789), (790,791), (792,793), (794,795), (796,797), (802,803), (804,805), (806,807), (808,809), (812,813), (814,815), (816,817), (818,819), (824,825), (826,827), (830,831), (832,833), (834,835), (838,839), (891,898), (900,901), (907,3800), (912,915), (916,917), (918,919), (924,925), (926,927), (928,929), (930,1510), (932,4083), (933,6368), (934,3900), (952,953), (971,972), (1113,2574), (1114,2575), (1141,1154), (1292,3309), (1293,3667), (1294,6000), (1295,6001), (1296,1499), (1297,1503), (1308,3887), (1379,3413), (1380,3414), (1381,3307), (1382,3308), (1477,5853), (2222,3738), (2365,2385), (2366,2386), (2367,2387), (2368,2388), (2369,2389), (2370,2390), (2371,2391), (2372,2392), (2373,2393), (2687,2771), (2688,2772), (2689,2773), (2690,2774), (2691,2775), (2692,2776), (2693,2777), (2694,2778), (2695,2779), (2696,2780), (2697,2781), (2698,2782), (2699,2783), (2700,2784), (2701,2785), (2702,2786), (2703,2787), (2704,2788), (2705,2789), (2706,2790), (2707,2791), (2708,2792), (2709,2793), (2710,2794), (2711,2795), (2712,2796), (2713,2797), (2714,2798), (2715,2799), (2716,2800), (2717,2801), (2718,2802), (2719,2803), (2720,2804), (2721,2805), (2722,2806), (2723,2807), (2724,2808), (2725,2809), (2726,2810), (2727,2811), (2728,2812), (2729,2813), (2730,2814), (2731,2815), (2732,2816), (2733,2817), (2734,2818), (2735,2819), (2736,2820), (2737,2821), (2738,2822), (2739,2823), (2740,2824), (2741,2825), (2742,2826), (2743,2827), (2744,2828), (2745,2829), (2746,2830), (2747,2831), (2748,2832), (2749,2833), (2750,2834), (2751,2835), (2752,2836), (2753,2837), (2754,2838), (2755,2839), (2756,2840), (2757,2841), (2758,2842), (2759,2843), (2760,2844), (2761,2845), (2762,2846), (2763,2847), (2764,2848), (2765,2849), (2766,2850), (2767,2851), (2768,2852), (2769,2853), (2770,2854), (2855,2869), (2856,2870), (2857,2871), (2858,2872), (2859,2873), (2860,2874), (2861,2875), (2862,2876), (2863,2877), (2864,2878), (2865,2879), (2866,2880), (2867,2881), (2868,2882), (3221,3222), (3563,3564), (3565,3566), (3849,6323), (4588,4777), (4843,5545), (5965,5966), (5969,5970), (6002,6010), (6003,6011), (6004,6012), (6005,6013), (6006,6014), (6007,6015), (6008,6016), (6009,6017), (6078,6084), (6079,6085), (6080,6086), (6081,6087), (6082,6088), (6083,6089), (6135,6364), (6136,6365), (6182,6183), (6367,6578), (6379,6380), (6550,6551), (7927,7953), (7950,7954), (8058,8059), (8063,8064)
Regarding the list just above, most of the points X(i) are on the circumcircle. Following is a sublist of those (i,j) for which X(i) is not on the circumcircle: (2,3098), (3,3), (4,7689), (5,7691), (15,6582), (16,6295), (20,3357), (21,3579), (39,6308), (40,5450), (182,6194), (371,6312), (372,6316), (376,4550), (574,8182)
X(8666) lies on these lines: {1,21}, {2,5258}, {3,519}, {4,535}, {5,529}, {8,36}, {10,56}, {30,3813}, {35,145}, {37,5042}, {40,104}, {46,3872}, {55,3244}, {57,3754}, {72,1319}, {78,214}, {100,3632}, {106,979}, {182,518}, {377,4317}, {388,3822}, {404,3679}, {405,551}, {442,5434}, {495,4999}, {499,3436}, {515,6985}, {517,5450}, {527,3560}, {528,550}, {536,7781}, {546,3829}, {596,2217}, {936,1476}, {944,6876}, {952,6796}, {958,999}, {995,5247}, {997,1420}, {1001,3636}, {1004,4311}, {1012,4301}, {1015,4426}, {1043,4278}, {1107,2242}, {1201,1724}, {1376,3626}, {1388,4067}, {1444,3875}, {1455,4347}, {1470,4848}, {1478,6871}, {1483,5855}, {1698,5253}, {1759,2170}, {2099,4084}, {2178,3686}, {2275,5291}, {2329,4253}, {2476,5270}, {2646,3555}, {2718,6079}, {2801,6261}, {3057,3916}, {3086,3825}, {3189,7688}, {3218,4861}, {3241,3746}, {3295,3635}, {3338,5883}, {3340,4757}, {3361,3918}, {3421,5193}, {3428,4297}, {3434,4299}, {3501,5030}, {3576,3811}, {3579,3880}, {3582,4193}, {3600,3841}, {3612,3870}, {3616,5251}, {3622,5259}, {3624,5260}, {3625,5204}, {3633,3871}, {3682,4322}, {3820,6691}, {3919,5221}, {3924,3953}, {3927,5289}, {3928,6906}, {3988,5223}, {3997,5021}, {4051,5011}, {4191,4685}, {4254,4856}, {4257,5255}, {4420,4881}, {4511,5904}, {4652,5119}, {4996,7972}, {5080,7741}, {5881,6905}, {6737,7742}, {6765,7987}, {6909,7991}
X(8666) = midpoint of X(3811) and X(6762)
X(8666) = 2nd-isogonal-triangle-of-X(1)-to-ABC similarity image of X(8)
X(8666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,63,3878), (1,191,3877), (1,993,5248), (1,2975,993), (1,3899,5330), (1,5250,3898), (1,6763,3869), (36,5288,8), (56,956,10), (405,3304,551), (499,3436,3814), (958,999,1125), (1001,7373,3636), (3218,4861,5903), (3241,4189,3746), (3244,5267,55), (3576,6762,3811), (3632,7280,100), (3633,5010,3871), (3647,3898,5250), (3871,5303,5010), (3873,3897,1), (5258,5563,2)
X(8667) lies on these lines: {2, 6}, {3, 538}, {22, 7669}, {25, 5201}, {55, 4396}, {56, 4400}, {76, 1003}, {98, 1350}, {99, 5210}, {140, 7758}, {262, 5102}, {381, 754}, {536, 4421}, {543, 3534}, {698, 6194}, {732, 5085}, {1078, 5013}, {1384, 3734}, {1447, 4361}, {1656, 7759}, {1975, 5023}, {2549, 8354}, {2896, 7851}, {3052, 4713}, {3526, 7764}, {3598, 7263}, {3767, 7767}, {3785, 5254}, {3793, 7737}, {3830, 3849}, {3845, 7615}, {3851, 7843}, {4363, 7081}, {4387, 4760}, {4665, 7172}, {5024, 7798}, {5055, 7775}, {5305, 7800}, {5309, 7810}, {5319, 8362}, {5346, 6292}, {6179, 7770}, {7223, 7267}, {7739, 8359}, {7746, 7776}, {7749, 7855}, {7755, 7854}, {7768, 7887}, {7773, 7893}, {7795, 8368}, {7805, 7815}, {7811, 7841}, {7817, 7865}, {7828, 7879}, {7844, 7848}, {7857, 7881}, {7862, 7882}, {7886, 7896}
X(8668) lies on these lines: {3, 3880}, {8, 21}, {35, 2136}, {36, 3680}, {100, 7288}, {104, 5854}, {518, 1158}, {519, 5450}, {997, 3295}, {1376, 3086}, {1737, 5687}, {3158, 3746}, {3174, 5785}, {3524, 4421}, {3811, 5887}, {5537, 6762}, {5853, 6684}
X(8669) lies on these lines: {1, 2}, {3, 726}, {21, 3971}, {65, 4434}, {98, 3430}, {183, 3905}, {230, 4136}, {712, 7780}, {946, 6287}, {3794, 4134}, {4090, 5247}, {4096, 5302}, {4135, 7283}, {4918, 4995}, {4920, 7767}
X(8670) lies on this line: {649, 834}
X(8671) lies on these lines: {35, 291}, {55, 574}, {99, 100}, {644, 667}, {932, 1016}, {1376, 3734}, {2810, 3098}, {3733, 3908}, {3939, 6373}
X(8672) lies on these lines:
{30, 511}, {647, 661}, {656, 4705}, {669, 4724}, {850, 4374}, {1109, 3937}, {1459, 4378}, {2451, 3063}, {2512, 2526}, {2532, 4394}, {2533, 4086}, {2605, 4833}, {2643, 4014}, {3737, 4367}, {4010, 4815}, {4404, 4761}, {4502, 4526}, {7178, 7250}
X(8672) = isogonal conjugate of X(931)
X(8672) = crossdifference of every pair of points on line X(6)X(21)
X(8673) lies on these lines: {3, 2435}, {30, 511}, {68, 879}, {3569, 6753}
X(8673) = isogonal conjugate of X(1289)
X(8673) = isotomic conjugate of polar conjugate of X(2485)
X(8673) = crossdifference of every pair of points on line X(6)X(66)
X(8673) = infinite point of trilinear polar of X(22)
X(8673) = isogonal conjugate of the polar conjugate of X(33294)
X(8673) = Thomson-isogonal conjugate of X(34168)
X(8673) = Lucas-isogonal conjugate of X(34168)
X(8674) lies on these lines: {11, 125}, {30, 511}, {74, 104}, {80, 7727}, {100, 110}, {113, 119}, {146, 153}, {149, 3448}, {245, 2643}, {265, 3657}, {656, 2605}, {895, 2991}, {1112, 1862}, {1317, 3028}, {1320, 7984}, {1769, 4895}, {2254, 3722}, {2642, 5029}, {2677, 5520}, {2948, 4730}, {3031, 3032}, {3035, 5972}, {3045, 3047}, {3737, 8043}, {4036, 7253}, {4491, 6161}, {4705, 4833}, {4922, 6224}, {5642, 6174}, {6667, 6723}, {6699, 6713}
X(8674) = isogonal conjugate of X(1290)
X(8674) = isotomic conjugate of X(35156)
X(8674) = X(2)-Ceva conjugate of X(35090)
X(8674) = crossdifference of every pair of points on line X(6)X(1718)
X(8674) = crosspoint, wrt incentral triangle, of X(11) and X(214)
X(8674) = X(523)-of-inner-Garcia triangle
X(8675) lies on these lines: {6, 647}, {30, 511}, {69, 850}, {74, 841}, {879, 5486}, {2451, 2485}, {2528, 3313}, {7625, 8542}
X(8675) = isogonal conjugate of X(1302)
X(8675) = crossdifference of every pair of points on line X(6)X(30)
X(8676) lies on these lines: {30, 511}, {652, 663}, {654, 1946}, {810, 6589}, {1243, 3657}, {4822, 6608}
X(8676) = isogonal conjugate of X(1305)
X(8676) = crossdifference of every pair of points on line X(6)X(226)
X(8677) lies on these lines: {30, 511}, {109, 692}, {1361, 1769}, {1364, 3270}, {1459, 1946}
X(8677) = isogonal conjugate of X(1309)
X(8677) = crossdifference of every pair of points on line X(6)X(281)
X(8677) = isotomic conjugate of isogonal conjugate of X(23220)
X(8677) = isotomic conjugate of polar conjugate of X(3310)
X(8677) = X(19)-isoconjugate of X(13136)
X(8678) lies on these lines: {30, 511}, {649, 4041}, {650, 667}, {659, 3803}, {661, 663}, {905, 1491}, {1019, 1734}, {1027, 2334}, {1577, 7662}, {2526, 2530}, {4162, 4775}, {4394, 4770}, {4397, 4581}, {4482, 4553}, {4522, 8045}, {4729, 4814}, {4730, 4790}, {4813, 4822}
X(8678) = isogonal conjugate of X(1310)
X(8678) = crossdifference of every pair of points on line X(6)X(63)
X(8679) lies on these lines: {2, 375}, {6, 41}, {30, 511}, {46, 3293}, {51, 354}, {69, 313}, {141, 1329}, {197, 222}, {210, 3917}, {212, 1626}, {603, 2933}, {611, 8069}, {692, 3220}, {910, 1362}, {1086, 1463}, {1155, 3937}, {1279, 3271}, {1376, 3784}, {1401, 3752}, {1478, 5820}, {1757, 3792}, {1818, 4557}, {1827, 1828}, {2098, 3056}, {2330, 4265}, {2979, 3681}, {3000, 3779}, {3006, 3909}, {3060, 3873}, {3589, 6691}, {3717, 4553}, {3740, 3819}, {3742, 5943}, {3781, 5220}, {3848, 6688}, {3961, 7186}, {4260, 4663}, {5480, 7681}
X(8679) = isogonal conjugate of X(1311)
X(8679) = crossdifference of every pair of points on line X(6)X(522)
X(8680) lies on these lines: {1, 5327}, {7, 2294}, {19, 27}, {30, 511}, {37, 226}, {71, 1441}, {144, 3958}, {192, 3151}, {243, 1430}, {307, 1826}, {313, 4019}, {349, 1840}, {388, 2292}, {857, 4466}, {958, 4363}, {984, 1478}, {1375, 7359}, {1486, 8424}, {1944, 5088}, {1962, 3475}, {2650, 3486}, {2667, 4319}, {3729, 5227}, {3739, 5745}, {3822, 3842}, {4095, 4377}, {4643, 5794}, {4647, 4659}
X(8680) = isogonal conjugate of X(2249)
X(8680) = isotomic conjugate of X(35145)
X(8680) = X(2)-Ceva conjugate of X(35075)
X(8680) = crossdifference of every pair of points on line X(6)X(810)
X(8680) = Thomson-isogonal conjugate of X(36516)
X(8681) lies on these lines: {6, 1196}, {30, 511}, {51, 1992}, {68, 5486}, {69, 305}, {155, 576}, {187, 2936}, {193, 1843}, {389, 6193}, {575, 1147}, {597, 6688}, {599, 3819}, {895, 3292}, {1353, 5946}, {1634, 3003}, {1993, 8541}, {5032, 5640}, {5476, 5654}, {5504, 5505}, {5866, 6091}
X(8681) = isogonal conjugate of X(2374)
X(8681) = isotomic conjugate of polar conjugate of X(3291)
X(8681) = crossdifference of every pair of points on line X(6)X(3566)
X(8682) lies on these lines: {30, 511}, {69, 3735}, {81, 239}, {385, 5977}, {1211, 3912}, {1655, 2895}, {1930, 3780}, {3008, 6703}, {3702, 4721}, {4465, 4975}, {4647, 4754}
X(8682) = isogonal conjugate of X(2375)
X(8683) lies on these lines: {3, 2802}, {35, 4674}, {55, 244}, {100, 190}, {109, 6014}, {537, 4421}, {643, 3733}, {1293, 3939}, {1376, 4011}, {2743, 6011}, {2835, 6244}, {4068, 4689}, {4738, 5687}
X(8684) lies on the circumcircle and these lines: {99, 7255}, {100, 4621}, {109, 660}, {983, 2382}
X(8684) = isogonal conjugate of X(3808)
X(8684) = trilinear pole of line X(6)X(983)
X(8684) = Ψ(X(6), X(983))
X(8685) lies on the circumcircle and these lines: {104, 983}, {105, 3924}, {741, 1408}, {813, 1415}, {825, 4559}, {833, 4551}
X(8685) = isogonal conjugate of X(3810)
X(8686) lies on the circumcircle and these lines: {1, 1293}, {36, 2743}, {56, 100}, {57, 6014}, {99, 1014}, {101, 604}, {108, 1398}, {109, 1106}, {110, 1408}, {898, 7677}, {901, 1319}, {934, 7023}, {999, 1292}, {2222, 5193}, {2742, 5126}, {5563, 6011}
X(8686) = isogonal conjugate of X(3880)
X(8686) = cevapoint of X(56) and X(1319)
X(8686) = trilinear pole of line X(6)X(4394)
X(8686) = Ψ(X(6), X(4394))
X(8686) = trilinear product of circumcircle intercepts of line X(1)X(3667)
Let P be a point on line X(2)X(12) other than X(2). Let A'B'C' be the cevian triangle of P. Let A" be the {B,C}-harmonic conjugate of A' (or equivalently, A" = BC/\B'C'), and define B" and C" cyclically. The circumcircles of AB"C", BC"A", CA"B" concur in X(8687). (Randy Hutson, November 30, 2015)
X(8687) lies on the circumcircle and these lines: {59, 2703}, {99, 4552}, {100, 1415}, {102, 2359}, {104, 2298}, {105, 961}, {110, 4559}, {651, 1310}, {759, 1169}, {929, 4581}, {931, 5546}, {1220, 1311}, {2766, 7115}
X(8687) = isogonal conjugate of X(3910)
X(8687) = Ψ(X(I), X(j)) for these (i,j): (2,12, (6,181), (8,6), (76,7)
X(8687) = trilinear pole of line X(6)X(181)
X(8687) = barycentric product of circumcircle intercepts of line X(2)X(12)
X(8688) lies on these lines: {1, 474}, {56, 7419}, {1120, 3621}, {3616, 4854}
X(8689) lies on these lines: {9, 649}, {513, 6687}, {514, 1960}, {522, 659}, {1443, 1447}, {3667, 4782}, {3716, 4106}, {3835, 4448}
X(8690) lies on the circumcircle and these lines: {643, 1293}, {931, 1634}, {3733, 6079}
X(8690) = isogonal conjugate of X(4139)
X(8690) = trilinear pole of line X(6)X(404)
X(8690) = Ψ(X(6), X(404))
X(8691) lies on the circumcircle and these lines: {1, 111}, {36, 2721}, {56, 1366}, {105, 551}, {106, 1386}, {392, 2291}, {662, 691}, {739, 5315}, {741, 995}, {759, 1001}, {2375, 4649}, {2748, 4557}
X(8691) = isogonal conjugate of X(4160)
X(8691) = trilinear pole of line X(6)X(896)
X(8691) = Ψ(X(1), X(524))
X(8691) = Ψ(X(6), X(896))
X(8691) = trilinear product of circumcircle intercepts of line X(1)X(524)
X(8692) lies on these lines: {1, 6}, {31, 8167}, {748, 902}, {1125, 7232}, {2177, 4383}, {3286, 7419}, {3929, 4906}, {4361, 4432}, {6221, 8225}
X(8693) lies on the circumcircle and these lines: {6, 105}, {99, 644}, {100, 2284}, {106, 2279}, {187, 2711}, {323, 2856}, {352, 2721}, {651, 927}, {692, 919}, {934, 4559}, {1018, 6013}, {1292, 2428}, {2177, 2291}, {2725, 5526}
X(8693) = isogonal conjugate of X(4762)
X(8693) = trilinear pole of line X(6)X(2223)
X(8693) = Ψ(X(6), X(2223))
X(8693) = circumcircle-intercept, other than X(105), of the circle {{X(15), X(16), X(105)}}
X(8694) lies on the circumcircle and these lines: {99, 3699}, {100, 4069}, {104, 4866}, {105, 612}, {106, 386}, {109, 4557}, {110, 3939}, {675, 5936}, {739, 4264}, {741, 7077}, {927, 4624}, {934, 4551}, {1026, 1310}, {1331, 4588}
X(8694) = isogonal conjugate of X(4778)
X(8694) = trilinear pole of line X(6)X(1334)
X(8694) = Ψ(X(7), X(10))
X(8694) = Ψ(X(6), X(1334))
X(8695) lies on the circumcircle and these lines: {675, 2320}, {739, 2163}, {789, 4597}, {898, 4604}
X(8695) = isogonal conjugate of X(4844)
X(8696) lies on the circumcircle and this line: {99, 4921}
X(8696) = isogonal conjugate of X(4912)
X(8697) lies on the circumcircle and these lines: {104, 1392}, {106, 3915}, {1331, 6014}
X(8697) = isogonal conjugate of X(4926)
X(8697) = trilinear pole of line X(6)X(5563)
X(8697) = Ψ(X(6), X(5563))
X(8698) lies on the circumcircle and these lines: {106, 3746}, {2718, 5131}
X(8698) = isogonal conjugate of X(4943)
X(8699) lies on the circumcircle and these lines: {104, 7982}, {106, 1616}, {4752, 6574}
X(8699) = isogonal conjugate of X(4962)
X(8700) lies on the circumcircle and these lines: {99, 8025}, {100, 1100}, {101, 2308}, {593, 6578}
X(8700) = isogonal conjugate of X(4971)
X(8701) lies on the circumcircle and these lines: {98, 6539}, {99, 3952}, {103, 1796}, {104, 6986}, {105, 1255}, {106, 1126}, {110, 4557}, {675, 1268}, {741, 1171}, {759, 5293}, {831, 1026}, {835, 3699}, {927, 4608}, {1252, 2702}, {1311, 4102}, {2372, 6538}, {2718, 5529}
X(8701) = reflection of X(28173) in X(3)
X(8701) = isogonal conjugate of X(4977)
X(8701) = circumcircle-antipode of X(28173)
X(8701) = trilinear pole of line X(6)X(595)
X(8701) = Ψ(X(7), X(12))
X(8701) = Ψ(X(6), X(595))
X(8701) = X(11792)-of-excentral-triangle
X(8702) lies on these lines: {8, 4036}, {11, 2677}, {30, 511}, {100, 6584}, {1459, 4814}, {2605, 4041}, {3733, 4730}
X(8702) = isogonal conjugate of X(5606)
X(8702) = crossdifference of every pair of points on line X(6)X(3336)
As a point on the Euler line, X(8703) has Shinagawa coefficients (7, -9).
Let A'B'C' be the anticomplementary triangle. Let A" be the pole of line B'C' wrt the A-power circle, and define B" and C" cyclically. Then X(8703) = X(2)-of-A"B"C". (Randy Hutson, September 14, 2016)
X(8703) lies on these lines: {2, 3}, {35, 5434}, {36, 3058}, {40, 1483}, {165, 952}, {187, 5306}, {397, 5352}, {398, 5351}, {495, 4995}, {496, 5298}, {515, 4745}, {517, 3892}, {519, 3579}, {524, 3098}, {541, 1511}, {597, 5092}, {1350, 1353}, {1587, 6455}, {1588, 6456}, {1770, 4870}, {2549, 5210}, {3053, 7739}, {3068, 6451}, {3069, 6452}, {3576, 3656}, {3582, 6284}, {3584, 7354}, {3587, 3928}, {3653, 5901}, {3815, 6781}, {3917, 5663}, {3929, 7171}, {3933, 7782}, {4297, 4669}, {4304, 5122}, {4316, 5432}, {4324, 5433}, {4654, 5719}, {5023, 5305}, {5206, 5309}, {5447, 5876}, {5655, 6030}, {5731, 5844}, {6390, 7788}, {6407, 7581}, {6408, 7582}, {6409, 7583}, {6410, 7584}, {6411, 6560}, {6412, 6561}, {6445, 7585}, {6446, 7586}, {6449, 6460}, {6450, 6459}, {7750, 7799}, {7789, 7865}, {7830, 7880}
X(8703) = complement of X(3830)
X(8703) = anticomplement of X(5066)
X(8703) = Thomson-isogonal conjugate of X(5643)
X(8703) = {X(2),X(3)}-harmonic conjugate of X(12100)
Let AB, AC, BC, BA, CA, CB be the points on the Dao 6-point circle as defined at X(5569). Triangles BACBAC and CAABBC are perspective at X(2), and X(8704) is the infinite point of their perspectrix. (Randy Hutson, August 17, 2020)
X(8704) lies on these lines: {3, 6322}, {6, 6232}, {30, 511}, {262, 5466}, {691, 6236}, {842, 6325}, {2408, 7757}, {6031, 6194}
X(8704) = isogonal conjugate of X(6233)
X(8705) lies on these lines: {2, 6324}, {4, 6235}, {6, 23}, {30, 511}, {69, 5189}, {125, 8262}, {141, 858}, {182, 7575}, {186, 5085}, {323, 2930}, {373, 468}, {568, 8550}, {597, 5640}, {691, 5104}, {842, 6233}, {1350, 7464}, {1352, 7574}, {1469, 7286}, {1495, 6593}, {2070, 5050}, {3056, 5160}, {3098, 8542}, {3313, 3631}, {5093, 5899}
X(8705) = isogonal conjugate of X(6325)
X(8705) = crossdifference of every pair of points on line X(6)X(3906)
X(8705) = X(524)-of-circumsymmedial-triangle
X(8706) lies on the circumcircle and these lines: {10, 106}, {101, 4169}, {105, 1261}, {109, 3699}, {110, 7256}, {190, 1293}, {664, 6571}, {668, 934}, {901, 3952}, {1324, 2757}, {2743, 4076}
X(8706) = isogonal conjugate of X(6363)
X(8706) = trilinear pole of line X(6)X(145)
X(8706) = Ψ(X(6), X(145))
X(8706) = Λ(trilinear polar of X(3445))
X(8706) = center of rectangular hyperbola that passes through X(1), X(8), and the excenters
X(8707) lies on the circumcircle, the circumconic with center X(3161), and these lines: {100, 646}, {101, 3699}, {104, 1791}, {105, 3757}, {106, 1125}, {108, 6335}, {109, 190}, {110, 645}, {660, 805}, {668, 1310}, {675, 1240}, {739, 2298}, {741, 1961}, {815, 3799}, {901, 4427}, {931, 4557}, {934, 4554}, {1016, 2703}, {1018, 6010}, {4588, 4756}, {4633, 5545}, {6540, 6578}
X(8707) = isogonal conjugate of X(6371)
X(8707) = isotomic conjugate of X(3004)
X(8707) = trilinear pole of line X(6)X(8)
X(8707) = Ψ(X(6), X(8))
X(8707) = center of rectangular hyperbola passing through X(1), X(10) and the excenters
X(8708) lies on the circumcircle and these lines: {99, 4557}, {190, 6013}, {739, 1197}
X(8708) = isogonal conjugate of X(6372)
X(8708) = trilinear pole of line X(6)X(1621)
X(8708) = Ψ(X(6), X(1621))
X(8709) lies on the circumcircle and these lines: {100, 7035}, {101, 1016}, {106, 3226}, {109, 4998}, {110, 4600}, {238, 727}, {659, 4583}, {667, 668}, {739, 4607}, {741, 3510}, {813, 3570}, {889, 4491}, {2382, 3253}
X(8709) = isogonal conjugate of X(6373)
X(8709) = isotomic conjugate of X(3837)
X(8709) = trilinear pole of line X(6)X(190)
X(8709) = Ψ(X(6), X(190))
X(8709) = X(19)-isoconjugate of X(22092)
X(8709) = intersection, other than A, B, and C, of circumcircle and hyperbola {{A,B,C,PU(24),PU(58)}}
X(8710) lies on these lines: {30, 511}, {650, 4546}, {663, 4163}, {3239, 4162}
X(8710) = isogonal conjugate of X(6571)
X(8711) lies on this line: {30,511}
X(8711) = isogonal conjugate of X(6573)
X(8711) = crossdifference of every pair of points on line X(6)X(6664)
X(8712) lies on these lines: {1, 3803}, {30, 511}, {649, 3669}, {650, 4498}, {764, 4834}, {905, 4063}, {1022, 2441}, {2526, 4041}, {4106, 4391}, {4435, 4790}
X(8712) = isogonal conjugate of X(8574)
X(8712) = crossdifference of every pair of points on line X(6)X(200)
X(8713) lies on these lines: {30, 511}, {663, 3676}, {885, 5665}, {3126, 5592}, {4077, 4822}, {4163, 4468}
X(8713) = isogonal conjugate of X(6575)
X(8713) = crossdifference of every pair of points on line X(6)X(8012)
X(8714) lies on these lines: {10, 1734}, {30, 511}, {656, 4985}, {665, 3700}, {693, 4905}, {905, 1125}, {1491, 4129}, {1577, 2254}, {2530, 4010}, {3159, 7265}, {3762, 4041}, {3766, 4467}, {4040, 4560}, {4075, 4522}, {4804, 4978}
X(8714) = isogonal conjugate of X(6577)
Circumnormal-isogonal conjugates: X(8715)-X(8725)
This section is contributed by Peter Moses, November 28, 2015, based on notes from Bernard Gibert, November 28, 2015, as a sequel to the preceding section on circumtangential-isogonal conjugates.
The circumnormal-isogonal conjugate of U, denoted by M3(U), is defined by
a^2 (-2 a^2 b^4 c^2 u^2+2 b^6 c^2 u^2-2 a^2 b^2 c^4 u^2-4 b^4 c^4 u^2+2 b^2 c^6 u^2+a^6 c^2 u v-6 a^4 b^2 c^2 u v+5 a^2 b^4 c^2 u v-3 a^4 c^4 u v+4 a^2 b^2 c^4 u v-b^4 c^4 u v+3 a^2 c^6 u v+2 b^2 c^6 u v-c^8 u v-a^6 c^2 v^2+a^2 b^4 c^2 v^2+3 a^4 c^4 v^2+2 a^2 b^2 c^4 v^2+b^4 c^4 v^2-3 a^2 c^6 v^2-2 b^2 c^6 v^2+c^8 v^2+a^6 b^2 u w-3 a^4 b^4 u w+3 a^2 b^6 u w-b^8 u w-6 a^4 b^2 c^2 u w+4 a^2 b^4 c^2 u w+2 b^6 c^2 u w+5 a^2 b^2 c^4 u w-b^4 c^4 u w+a^8 v w-3 a^6 b^2 v w+3 a^4 b^4 v w-a^2 b^6 v w-3 a^6 c^2 v w+a^2 b^4 c^2 v w+2 b^6 c^2 v w+3 a^4 c^4 v w+a^2 b^2 c^4 v w-4 b^4 c^4 v w-a^2 c^6 v w+2 b^2 c^6 v w-a^6 b^2 w^2+3 a^4 b^4 w^2-3 a^2 b^6 w^2+b^8 w^2+2 a^2 b^4 c^2 w^2-2 b^6 c^2 w^2+a^2 b^2 c^4 w^2+b^4 c^4 w^2) : :
The appearance of (i,j) in the following list means that the circumnormal-isogonal conjugate of X(i) is X(j): (1,6796), (3,3), (4,6759), (20,1147), (30,110), (54,550), (74,523), (98,512), (99,511), (100,517), (101,516), (102,522), (103,514), (104,513), (105,3309), (106,3667), (107,6000), (108,6001), (109,515), (111,1499), (112,1503), (182,376), (476,5663), (477,526), (518,1292), (519,1293), (520,1294), (521,1295), (524,1296), (525,1297), (528,2742), (542,691), (543,2709), (690,842), (740,6010), (741,6002), (758,6011), (759,6003), (804,2698), (805,2782), (840,2826), (843,2793), (900,953), (901,952), (916,1305), (917,8676), (924,1300), (926,2724), (927,2808), (928,2723), (929,2807), (930,1154), (934,971), (935,2781), (972,3900), (1075,3357), (1113,2575), (1114,2574), (1141,1510), (1290,2771), (1304,2777), (1308,2801), (1309,2818), (1379,3414), (1380,3413), (1381,3308), (1382,3307), (1385,3651), (1745,5450), (2222,2800), (2687,8674), (2688,2774), (2689,2779), (2690,2772), (2691,2836), (2692,2842), (2694,2850), (2695,2773), (2696,2854), (2699,2787), (2700,2786), (2701,2792), (2702,2784), (2703,2783), (2704,2795), (2705,2796), (2706,2797), (2707,2798), (2708,2785), (2710,2799), (2711,2788), (2712,2789), (2713,2790), (2714,2791), (2715,2794), (2716,3738), (2717,3887), (2718,2827), (2719,2828), (2720,2829), (2721,2830), (2722,2831), (2725,2820), (2726,2821), (2727,2822), (2728,2823), (2729,2824), (2730,2835), (2731,2841), (2732,2846), (2733,2849), (2734,8677), (2735,2852), (2736,2809), (2737,2810), (2738,2811), (2739,2812), (2740,2813), (2743,2802), (2744,2803), (2745,2804), (2746,2805), (2747,2806), (2751,2814), (2752,2775), (2757,2815), (2758,2776), (2762,2816), (2765,2817), (2766,2778), (2768,2819), (2770,2780), (3098,7709), (3563,3566), (3564,3565), (3849,6233), (5171,6337), (5840,6099), (5951,8702), (6088,6093), (6236,8705), (6323,8704)
Regarding the list just above, most of the points X(i) are on the circumcircle or the line at infinity. Following is a sublist of those (i,j) for which X(i) is not on the circumcircle or line at infinity: (1,6796), (3,3), (4,6759), (20,1147), (54,550), (182,376), (1075,3357), (1385,3651), (1745,5450), (3098,7709), (5171,6337)
If U is on the circumcircle, then M3(U) is the isogonal conjugate of the circumcircle-antipode of U. (Randy Hutson, November 28, 2015)
X(8715) lies on these lines: {1,88}, {2,3746}, {3,519}, {5,528}, {8,35}, {9,4015}, {10,55}, {12,6154}, {20,535}, {21,3679}, {31,3293}, {36,145}, {40,758}, {41,1018}, {42,5264}, {43,595}, {46,3870}, {56,3244}, {57,3881}, {58,3550}, {72,3689}, {78,3878}, {101,3208}, {140,3813}, {149,7741}, {165,6765}, {191,3681}, {198,3950}, {200,1005}, {227,4347}, {228,2901}, {386,5255}, {388,3256}, {411,7991}, {474,551}, {484,3868}, {496,3035}, {497,3825}, {498,3434}, {516,5812}, {517,6796}, {518,3098}, {529,550}, {536,7751}, {572,3169}, {612,3743}, {902,1724}, {910,3991}, {944,2077}, {952,5450}, {956,3625}, {958,3626}, {976,4424}, {997,1697}, {999,3635}, {1001,3634}, {1011,4685}, {1043,4276}, {1125,1376}, {1155,3555}, {1158,2801}, {1203,3240}, {1259,4304}, {1385,3880}, {1466,4315}, {1479,3814}, {1483,5854}, {1486,4078}, {1500,4386}, {1575,2241}, {1621,1698}, {1696,4098}, {1706,3918}, {1759,3930}, {1788,2078}, {2093,4757}, {2136,3576}, {2271,3997}, {2329,4262}, {2476,3584}, {2478,4309}, {2550,3841}, {2805,8143}, {2949,3174}, {2975,3632}, {3057,5440}, {3058,4187}, {3085,3822}, {3086,6681}, {3149,4301}, {3189,5657}, {3216,3915}, {3241,4188}, {3336,3873}, {3337,3889}, {3338,3892}, {3436,4302}, {3454,4660}, {3501,4251}, {3612,3872}, {3617,5251}, {3621,5288}, {3628,3829}, {3633,7280}, {3636,6767}, {3670,3938}, {3680,6940}, {3683,3697}, {3684,3730}, {3717,7295}, {3748,5439}, {3828,4428}, {3924,3987}, {3935,5904}, {3956,4512}, {4006,5282}, {4018,5183}, {4066,5695}, {4075,4557}, {4097,5847}, {4193,4857}, {4294,7080}, {4420,5692}, {4511,5697}, {4646,5266}, {4701,8168}, {4856,5120}, {4917,4973}, {4995,7483}, {5082,5218}, {5223,7676}, {5428,5690}, {5493,7580}, {5853,6684}, {5881,6906}, {6048,8616}, {6905,7982}
X(8715) = midpoint of X(i) and X(j) for these {i,j}: {3, 3913}, {40, 3811}, {1158, 5534}
X(8715) = reflection of X(i) and X(j) for these (i,j): (3813, 140), (8666, 3)
X(8715) = X(2167)-Ceva conugate of X(9)
X(8715) = intouch-to-ABC barycentric image of X(5)
X(8715) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4855,214), (8,35,993), (8,4189,5258), (10,55,5248), (35,5258,4189), (40,3158,3811), (46,3870,3874), (55,5687,10), (78,5119,3878), (100,3871,1), (474,3303,551), (902,3214,1724), (956,5217,5267), (997,1697,3884), (1376,3295,1125), (1479,5552,3814), (3241,4188,5563), (3625,5267,956), (3632,5010,2975), (3895,4855,1), (3913,4421,3), (4189,5258,993)
X(8716) lies on these lines: {2,1975}, {3,538}, {6,99}, {76,8556}, {114,381}, {194,3053}, {315,8353}, {376,524}, {382,7764}, {385,5210}, {547,7615}, {549,7610}, {550,7758}, {599,6393}, {698,5085}, {754,3534}, {1384,7798}, {1657,7759}, {2482,5309}, {2549,6390}, {3734,5024}, {3830,7775}, {3926,7784}, {3933,8354}, {5023,7754}, {5071,7620}, {5073,7843}, {5077,7818}, {7739,8369}, {7756,7776}, {7788,7833}, {7799,7841}, {7800,8358}, {7847,7881}, {7863,7866}, {7884,8366}
X(8716) = reflection of X(i) and X(j) for these (i,j): (3830, 7775), (7610, 7618), (8667, 3)
X(8716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (99,7757,1003), (1003,7757,6), (1975,7783,5013), (2549,6390,7778), (7754,7782,5023)
X(8717) lies on these lines: {3,1495}, {20,54}, {30,182}, {74,7492}, {110,376}, {184,3534}, {206,2777}, {511,8547}, {548,5894}, {550,1147}, {567,1657}, {569,3529}, {1480,3304}, {1498,5447}, {3066,5892}, {3098,5663}, {3303,6580}, {3357,7525}, {6699,7493}
X(8718) lies on these lines: {3,6030}, {4,83}, {20,1147}, {22,6241}, {30,54}, {74,7488}, {110,550}, {156,3534}, {184,3529}, {376,6225}, {382,5012}, {569,3543}, {1204,7556}, {1385,1621}, {1498,2916}, {5093,7592}, {5663,7691}, {5890,7387}, {6000,7512}
X(8719) lies on these lines: {3,3734}, {6,7709}, {20,325}, {30,7618}, {98,5210}, {99,1350}, {262,5013}, {376,599}, {1003,5085}, {1975,6194}, {2782,8667}, {2794,3534}, {3424,3522}, {5102,7757}
X(8720) lies on these lines: {3,726}, {10,4201}, {40,376}, {58,99}, {404,3971}, {946,2796}, {988,1125}, {1201,4427}, {1468,4970}, {2901,4973}, {3840,7283}, {4362,4652}, {4672,4719}, {4920,6390}
X(8721) lies on these lines: {3,66}, {4,39}, {20,99}, {32,6776}, {69,5188}, {154,441}, {237,1899}, {376,7801}, {511,6309}, {542,5171}, {577,5596}, {631,7822}, {1350,3933}, {1513,3767}, {2896,3522}, {3091,7790}, {3146,7785}, {3424,3523}, {3785,5921}, {5056,7919}, {5059,7900}, {5085,7819}, {5207,6337}, {5984,7793}, {5999,7763}
X(8722) lies on these lines: {3,6}, {4,7815}, {5,7935}, {20,1078}, {74,6233}, {83,3523}, {98,376}, {99,6194}, {114,7818}, {147,7811}, {237,5651}, {631,7808}, {1352,7810}, {1513,7761}, {2698,2709}, {3522,6392}, {3785,5921}, {3793,8550}, {5480,8359}, {5999,7771}, {7709,7798}
X(8722) = reflection of X(574) in X(3)
X(8722) = inverse-in-circle-{X(1687),X(1688),PU(1),PU(2)} of X(1384)
X(8722) = {X(1687),X(1688)}-harmonic conjugate of X(1384)
X(8723) lies on these lines: {3,512}, {39,2422}, {54,826}, {99,110}, {182,523}, {525,1147}, {574,3288}, {842,2698}, {1649,5651}, {3049,5028}, {3050,3094}, {3111,5968}, {3566,6759}
X(8723) = pole of Euler line with respect to the Brocard circle
X(8723) = intersection of tangents to Brocard circle at X(3) and X(1316)
X(8723) = Brocard-circle-inverse of X(36177)
X(8723) = antipode of X(182) in circle {{X(3),X(110),X(182),X(1316),X(2698),X(8723)}}
X(8724) lies on these lines: {2,2782}, {3,67}, {4,8591}, {5,671}, {30,99}, {39,6034}, {98,549}, {114,381}, {115,5024}, {147,376}, {148,3545}, {524,2080}, {530,5613}, {531,5617}, {620,5054}, {690,5655}, {946,2796}, {1352,7618}, {1384,5477}, {1569,5309}, {1656,5461}, {2794,3534}, {3091,8596}, {3095,5969}, {3398,5182}, {3524,8289}, {3564,8593}, {6114,6772}, {6115,6775}, {6777,6780}, {6778,6779}
X(8724) = Stammler-circle-inverse of X(32254)
X(8724) = circumperp-conjugate of X(32305)
X(8724) = harmonic center of antipedal circles of X(13) and X(14)
X(8725) lies on these lines: {3,2916}, {20,3095}, {30,83}, {99,550}, {376,2896}, {381,6704}, {382,4045}, {754,3534}, {6033,7470}
Perspectors involving the Ascella triangle: X(8726)-X(8734)
This section is contributed by Clark Kimberling and Peter Moses, December 3, 2015.
Let A' = incircle-inverse of A, and define B' and C' cyclically. Let OA be the circle {{B,C,B',C'}}, and define OB and OC cyclically. The circles OA, OB, OC are orthogonal to the incircle. Let VA be the center of OA, and define VB and VC cyclically. The triangle VAVBVC is here named the Ascella triangle. The Ascella triangle is also the mid-triangle of the excentral triangle and the intouch triangle. Barycentrics are given by
VA = 2a2 : a2 - b2 + c2 - 2ab - 2ac : a2 - c2 + b2 - 2ab - 2ac
X(8726) lies on these lines: {1,3}, {2,1490}, {4,142}, {5,1750}, {9,1071}, {10,5768}, {63,6986}, {77,3345}, {78,3523}, {84,405}, {140,5720}, {169,572}, {200,6684}, {223,4303}, {226,6865}, {392,7971}, {411,3306}, {443,515}, {581,2999}, {614,4300}, {631,936}, {938,7675}, {950,6916}, {962,4666}, {1012,5436}, {1125,6847}, {1158,4512}, {1210,6908}, {1698,6989}, {1699,6851}, {1709,5259}, {2324,3730}, {2951,5805}, {3149,5437}, {3358,7992}, {3560,7171}, {3586,6850}, {3624,6824}, {3911,6988}, {4292,6987}, {4654,5812}, {4853,5882}, {5084,6260}, {5129,6223}, {5219,6922}, {5249,5715}, {5439,7580}, {5587,5787}, {5657,6765}, {5691,6826}, {5705,6889}, {5731,6904}, {5777,7308}, {5791,8580}, {6261,6705}, {6692,6927}, {6841,7988}, {6881,7989}, {6883,7330}
X(8726) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,6282), (1,165,6769), (3,942,40), (3,1385,3601), (3,2095,3579), (3,5709,165), (5249,6836,5715)
X(8727) lies on these lines: {1,5787}, {2,3}, {11,57}, {12,3601}, {35,6253}, {40,5791}, {63,5762}, {72,5763}, {84,5715}, {92,2968}, {142,1538}, {165,3925}, {200,355}, {226,971}, {329,5779}, {495,515}, {496,942}, {516,2886}, {517,4847}, {908,5927}, {952,3870}, {962,5789}, {990,3772}, {1071,6147}, {1088,1565}, {1210,5806}, {1483,3957}, {1484,1537}, {1498,5707}, {1503,5138}, {1750,5219}, {1765,5798}, {2051,4260}, {2095,5770}, {2550,6244}, {3624,7958}, {3813,4301}, {3820,5587}, {3927,5758}, {3928,5735}, {4666,5901}, {5226,5658}, {5231,5709}, {5273,5759}, {5603,5768}, {5714,6223}, {5799,7683}, {5812,7330}, {5843,5905}, {6705,7681}, {7678,8166}
X(8727) = midpoint of X(i) and X(j) for these {i,j}: {4, 1012}, {1709, 1836}
X(8727) = reflection of X(i) and X(j) for these (i,j): (495, 7680), (550, 7508), (6907, 5)
X(8727) = complement of X(7580)
X(8727) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(2,8226,5), (3,381,6826), (3,382,6869), (3,1656,6989), (3,6824,6675), (3,6841,5), (3,6861,140), (4,6830,1532), (4,6831,5), (4,6833,3149), (4,6844,381), (4,6845,6831), (4,6847,3), (4,6956,6848), (11,1699,7956), (11,7965,1699), (20,6828,442), (20,6857,3), (57,1699,5805), (381,6882,5), (411,6888,7483), (442,6828,5), (946,6245,942), (1532,6830,5), (1532,6831,6830), (1699,1709,1836), (3091,4187,5), (3091,6943,4187), (3149,6833,140), (3529,6873,6937), (4219,4224,3), (6824,6851,3), (6835,6890,474), (6836,6837,405), (6848,6956,1656), (6849,6891,6918), (6869,6892,3)
X(8728) lies on these lines: {1,3925}, {2,3}, {7,3927}, {10,141}, {11,3601}, {12,57}, {40,5805}, {72,5249}, {75,3695}, {78,5719}, {226,3824}, {274,3933}, {277,1390}, {387,4648}, {496,1125}, {498,4413}, {551,3813}, {579,1213}, {596,4884}, {940,1714}, {975,3772}, {1329,3634}, {1479,4423}, {1699,3646}, {1706,3254}, {1722,5725}, {1738,3931}, {1770,3683}, {1847,6356}, {2550,3295}, {3008,5717}, {3216,5718}, {3358,6259}, {3454,5743}, {3487,3940}, {3589,5138}, {3649,5692}, {3753,5690}, {3814,5122}, {3912,5295}, {3914,6051}, {3918,8256}, {4002,6735}, {4265,5358}, {4384,5814}, {4847,5045}, {4866,5557}, {5082,6767}, {5251,7354}, {5258,5434}, {5259,6284}, {5261,7679}, {5275,5305}, {5437,5705}, {5439,6734}, {5442,7951}, {5587,5787}, {5708,5815}, {5768,5818}, {5817,6223}, {5886,8583}, {6282,7956}, {6684,7680}
X(8728) = midpoint of X(377 and X(405)
X(8728) = complement of X(405)
X(8728) = X(i)-complementary conjugate of X(j) for these (i,j): (2215,2), (2335,3452)
X(8728) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,6675), (2,377,405), (2,404,7483), (2,442,5), (2,443,3), (2,474,140), (2,2475,5047), (2,2476,4187), (2,4197,442), (2,4208,4), (2,5177,5084), (2,6856,1656), (2,6904,6857), (3,381,6851), (3,1656,6824), (3,6881,5), (5,140,6922), (10,142,942), (12,1698,3820), (20,6991,8226), (57,1698,5791), (72,5249,6147), (404,7483,549), (442,4187,2476), (443,6857,6904), (631,6829,6831), (1125,2886,496), (1125,3841,2886), (1532,3090,5), (1656,6842,5), (2476,4187,5), (2476,6857,6841), (3090,6937,1532), (3533,6874,6963), (3634,3822,1329), (3824,5044,226), (5084,5177,381), (6826,6989,3), (6829,6831,5), (6832,6897,1012), (6850,6887,6913), (6854,6889,3149), (6857,6904,3), (6991,8226,5)
X(8729) lies on these lines: {2,7593}, {3,7587}, {7,8080}, {57,173}, {142,7028}, {236,5745}, {942,8092}, {5744,8126}
X(8730) lies on these lines: {3,5853}, {55,6067}, {57,3174}, {100,7674}, {142,1376}, {496,1001}, {518,5534}, {521,3211}, {942,3913}, {1467,2136}, {5744,7676}
As a point on the Euler line, X(8731) has Shinagawa coefficients ((E+2$bc$)S2-$bcSBSC$, -$bc$S2).
X(8731) lies on these lines: {2,3}, {43,3601}, {57,846}, {142,4425}, {284,2238}, {942,2292}, {993,3771}, {1001,6703}, {2276,5336}, {2886,8053}, {3741,4154}, {5248,6693}, {6707,8167}
X(8732) lies on these lines: {2,7}, {3,390}, {8,1467}, {10,4321}, {56,2550}, {77,5222}, {100,7674}, {241,347}, {269,3008}, {277,279}, {388,3826}, {443,956}, {499,4312}, {516,3086}, {518,1788}, {610,5838}, {631,954}, {653,1119}, {673,1436}, {938,7675}, {942,5657}, {948,1418}, {971,6848}, {1001,1466}, {1210,5732}, {1420,5853}, {1471,4307}, {1512,5768}, {1617,6601}, {2256,4648}, {2346,5281}, {3085,5542}, {3243,4848}, {3601,8236}, {3668,4859}, {3697,5686}, {4331,7613}, {4452,4552}, {5261,7679}, {5308,7190}, {5704,6245}, {5708,5771}, {5728,6908}, {5729,6834}, {5759,6926}, {5762,6891}, {5779,6944}, {5805,6847}, {5817,6964}, {5843,6959}, {6857,8543}, {7678,8166}
X(8732) = X(7)-cross conjugate of X(3174)
X(8732) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7,8232), (7,5435,1445), (57,142,7), (57,3911,5744), (241,4000,347), (390,5265,7677), (1210,5732,5809).
X(8733) lies on these lines: {1,8734}, {2,8079}, {3,8075}, {57,2089}, {188,5745}, {942,8093}, {3601,8241}
X(8733) = {X(2089),X(8078)}-harmonic conjugate of X(8101)
X(8734) lies on these lines: {1,8733}, {2,8080}, {3,7588}, {7,7593}, {57,173}, {142,236}, {942,8083}, {3601,8242}, {5744,8125}, {5745,7028}, {8100,8130}
X(8734) = {X(174),X(258)}-harmonic conjugate of X(8102)
Points associated with orthoanticevian triangles: X(8735)-X(8756)
This section is contributed by Clark Kimberling, Peter Moses, and Randy Hutson, November 9, 2015.
Let (O,R) be the circumcircle, and let A′B′C′ be the anticevian triangle of a point X. Let (OA,rA) be the circle through B′ and C′ and orthogonal to (O,R). That is, (OA,rA) is the circle that passes through the points B' and C' and also their inverses in (O,R). Define (OB,rB) and (OC,rC) cyclically. The triangle T(X) = OAOBOC is here named the orthoanticevian triangle of X. If P = p : q : r (barycentrics) and T(P) is perspective to ABC, then the perspector is barycentric product of P and the orthocenter; i.e.,
p tan A : q tan B : r tan C
The appearance of (i,j) in the following list means that T(X(i)) is perspective to ABC and that X(j) is the perspector, for i≤200:
(1, 19), (2, 4), (3, 6), (4, 393), (5, 53), (6, 25), (7, 278), (8, 281), (9, 33), (10, 1826), (19, 1096), (20, 1249), (21, 1172), (25, 2207), (28, 5317), (30, 1990), (31, 1973), (32, 1974), (37, 1824), (39, 1843), (40, 2331), (41, 2212), (42, 2333), (48, 31), (49, 2965), (51, 3199), (55, 607), (56, 608), (57, 34), (58, 1474), (59, 7115), (60, 2189), (63, 1), (65, 1880), (68, 2165), (69, 2), (71, 42), (72, 37), (73, 1400), (75, 92), (76, 264), (77, 57), (78, 9), (81, 28), (84, 7129), (85, 273), (86, 27), (92, 158), (94, 6344), (95, 275), (97, 54), (98, 6531), (99, 648), (100, 1783), (107, 6529), (110, 112), (122, 1562), (125, 115), (140, 6748), (141, 427), (154, 3172), (155, 1609), (158, 6520), (159, 3162), (171, 7119), (183, 458), (184, 32), (185, 800), (190, 1897), (193, 6353), (194, 3186), (198, 3195), (200, 7079)
This perspector is the polar conjugate of the isotomic conjugate of P. For points associated with orthocevian triangles, see the preamble to X(8601).
The triangle T(X(6)) is identical to the orthocevian triangle of X(2). Also, T(X(6)) is homothetic to the following triangles, with centers of homothety indicated: ABC, X(25); medial, X(6676); anticomplementary, X(7493); Euler, X(6756); Johnson, X(3549); Ara, X(25); 3rd antipedal tiangle of X(4), X(6697). (Randy Hutson, December 4, 2015)
X(8735) lies on these lines: {4, 218}, {11, 5190}, {19, 53}, {30, 1951}, {115, 5521}, {136, 5517}, {281, 3161}, {393, 608}, {650, 6506}, {906, 5840}, {1015, 6591}, {1479, 7124}, {1785, 5089}, {1826, 5151}, {1839, 1865}, {1852, 2204}, {1877, 1886}, {1884, 2201}, {2082, 5254}, {2164, 2165}, {2170, 2969}, {2973, 4904}, {3064, 4534}, {5146, 5523}
X(8735) = barycentric product X(4)*X(11)
X(8735) = pole wrt polar circle of trilinear polar of X(4998) (line X(190)X(644))
X(8735) = X(48)-isoconjugate (polar conjugate) of X(4998)
X(8735) = X(63)-isoconjugate of X(59)
X(8735) = crossdifference of every pair of points on line X(1331)X(1813)
The trilinear polar of X(8736) passes through X(4705). (Randy Hutson, December 11, 2015) X(8736) lies on these lines: {4, 608}, {12, 37}, {19, 53}, {30, 1950}, {181, 1824}, {219, 3436}, {273, 335}, {278, 469}, {281, 5296}, {393, 607}, {407, 1400}, {756, 7140}, {1409, 1901}, {1478, 2286}, {2165, 2178}, {2285, 5254}, {7119, 7120}
X(8736) = barycentric product X(4)*X(12)
X(8736) = pole wrt polar circle of trilinear polar of X(261) (line X(3904)X(3910))
X(8736) = X(48)-isoconjugate (polar conjugate) of X(261)
X(8736) = X(63)-isoconjugate of X(60)
X(8737) lies on these lines: {4, 13}, {18, 93}, {24, 6104}, {25, 1989}, {53, 462}, {225, 2153}, {264, 300}, {393, 3457}, {463, 1990}, {472, 648}, {1300, 5995}, {3443, 7755}
X(8737) = barycentric product X(4)*X(13)
X(8737) = pole wrt polar circle of trilinear polar of X(298) (line X(3268)X(6137))
X(8737) = X(48)-isoconjugate (polar conjugate) of X(298)
X(8737) = X(63)-isoconjugate of X(15)
X(8737) = trilinear pole of line X(462)X(2501)
X(8738) lies on these lines: {4, 14}, {17, 93}, {24, 6105}, {25, 1989}, {53, 463}, {225, 2154}, {264, 301}, {393, 3458}, {462, 1990}, {473, 648}, {1300, 5994}, {3442, 7755}
X(8738) = barycentric product X(4)*X(14)
X(8738) = pole wrt polar circle of trilinear polar of X(299) (line X(3268)X(6138))
X(8738) = X(48)-isoconjugate (polar conjugate) of X(299)
X(8738) = X(63)-isoconjugate of X(16)
X(8738) = trilinear pole of line X(463)X(2501)
The trilinear polar of X(8739) passes through X(6137). (Randy Hutson, December 11, 2015)
X(8739) lies on these lines: {4, 14}, {6, 25}, {13, 403}, {15, 186}, {16, 378}, {17, 7505}, {18, 1594}, {24, 61}, {32, 3439}, {33, 7127}, {112, 2378}, {202, 1870}, {216, 3132}, {235, 397}, {298, 340}, {395, 427}, {396, 468}, {398, 3575}, {462, 6749}, {463, 1990}, {577, 3131}, {2902, 5962}, {3129, 3284}, {3130, 5158}, {3520, 5237}, {5335, 6623}, {6198, 7006}, {6748, 8604}
X(8739) = barycentric product X(4)*X(15)
X(8739) = pole wrt polar circle of trilinear polar of X(300)
X(8739) = X(48)-isoconjugate (polar conjugate) of X(300)
X(8739) = X(63)-isoconjugate of X(13)
The trilinear polar of X(8740) passes through X(6138). (Randy Hutson, December 11, 2015)
X(8740) lies on these lines: {4, 13}, {6, 25}, {14, 403}, {15, 378}, {16, 186}, {17, 1594}, {18, 7505}, {24, 62}, {32, 3438}, {34, 2307}, {112, 2379}, {203, 1870}, {216, 3131}, {235, 398}, {299, 340}, {395, 468}, {396, 427}, {397, 3575}, {462, 1990}, {463, 6749}, {577, 3132}, {2903, 5962}, {3129, 5158}, {3130, 3284}, {3520, 5238}, {5334, 6623}, {6198, 7005}, {6748, 8603}
X(8740) = barycentric product X(4)*X(16)
X(8740) = pole wrt polar circle of trilinear polar of X(301)
X(8740) = X(48)-isoconjugate (polar conjugate) of X(301)
X(8740) = X(63)-isoconjugate of X(14)
X(8741) lies on these lines: {4, 16}, {13, 6344}, {25, 2934}, {53, 462}, {115, 3438}, {264, 299}, {6748, 8604}
X(8741) = barycentric product X(4)*X(17)
X(8741) = pole wrt polar circle of trilinear polar of X(302)
X(8741) = X(48)-isoconjugate (polar conjugate) of X(302)
X(8741) = X(63)-isoconjugate of X(61)
X(8741) = trilinear pole of line X(2501)X(6137)
X(8742) lies on these lines: {4, 16}, {13, 6344}, {25, 2934}, {53, 462}, {115, 3438}, {264, 299}, {6748, 8604}
X(8742) = barycentric product X(4)*X(18)
X(8742) = pole wrt polar circle of trilinear polar of X(303)
X(8742) = X(48)-isoconjugate (polar conjugate) of X(303)
X(8742) = X(63)-isoconjugate of X(62)
X(8742) = trilinear pole of line X(2501)X(6138)
The trilinear polar of X(8743) passes through X(2485). (Randy Hutson, December 11, 2015)
X(8743) lies on these lines: {2, 2138}, {3, 112}, {4, 6}, {8, 1783}, {24, 32}, {25, 251}, {33, 5280}, {34, 5299}, {39, 378}, {76, 648}, {83, 264}, {186, 3053}, {216, 7509}, {218, 3195}, {230, 7505}, {235, 5305}, {297, 1993}, {324, 458}, {340, 7877}, {386, 2332}, {403, 3767}, {406, 5276}, {419, 3168}, {420, 1613}, {451, 5275}, {847, 6531}, {1033, 7395}, {1184, 6353}, {1235, 7770}, {1384, 3515}, {1594, 2548}, {1843, 5039}, {3089, 5304}, {3192, 4251}, {3199, 5007}, {3516, 5024}, {3520, 5013}, {3541, 7736}, {3542, 7735}, {4232, 5354}, {5475, 7547}, {6103, 7746}, {6240, 7737}, {7290, 7719}
X(8743) = isogonal conjugate of X(14376)
X(8743) = polar conjugate of X(18018)
X(8743) = barycentric product X(4)*X(22)
X(8743) = X(63)-isoconjugate of X(66)
X(8743) = Cundy-Parry Phi transform of X(1297)
X(8743) = Cundy-Parry Psi transform of X(1503)
The trilinear polar of X(8744) passes through X(2492). (Randy Hutson, December 11, 2015)
X(8744) lies on these lines: {4, 6}, {24, 1384}, {25, 1383}, {32, 3518}, {112, 186}, {216, 7550}, {297, 323}, {378, 5024}, {419, 2501}, {420, 3231}, {574, 1968}, {2138, 6622}, {3055, 6143}, {3162, 6353}, {3199, 5008}, {5359, 7714}, {6344, 6531}
X(8744) = isogonal conjugate of X(34897)
X(8744) = barycentric product X(4)*X(23)
X(8744) = polar conjugate of X(18019)
X(8744) = X(63)-isoconjugate of X(67)
The trilinear polar of X(8745) passes through X(6753). (Randy Hutson, December 11, 2015)
X(8745) lies on these lines: {4, 6}, {22, 232}, {24, 571}, {112, 1299}, {216, 1968}, {317, 467}, {378, 570}, {403, 2165}, {2052, 5422}, {3162, 7735}, {3172, 8573}
X(8745) = barycentric product X(4)*X(24)
X(8745) = X(63)-isoconjugate of X(68)
X(8745) = polar conjugate of X(20563)
X(8746) lies on these lines: {4, 6}, {24, 2965}, {112, 8553}, {216, 7514}, {230, 3162}, {232, 571}, {570, 1968}, {1609, 3172}
X(8746) = polar conjugate of X(20564)
X(8746) = barycentric product X(4)*X(26)
X(8746) = X(63)-isoconjugate of X(70)
X(8747) lies on these lines: {1, 29}, {4, 6}, {10, 2322}, {19, 2215}, {27, 58}, {28, 56}, {106, 107}, {112, 917}, {162, 1780}, {225, 2299}, {232, 6998}, {269, 1847}, {281, 2336}, {286, 3673}, {297, 1330}, {386, 7513}, {412, 1754}, {447, 1043}, {837, 3436}, {939, 4183}, {1167, 1785}, {1214, 1940}, {1396, 1413}, {1474, 1842}, {1714, 5125}, {1857, 2334}, {1886, 2332}, {1897, 2901}, {2360, 5930}, {3226, 6528}, {4252, 7554}, {5137, 5146}
X(8747) = barycentric product X(4)*X(27)
X(8747) = pole wrt polar circle of trilinear polar of X(306) (line X(525)X(656))
X(8747) = X(48)-isoconjugate (polar conjugate) of X(306)
X(8747) = X(63)-isoconjugate of X(71)
X(8747) = isogonal conjugate of X(3682)
X(8747) = X(1842)-cross conjugate of X(4)
X(8747) = trilinear pole of line X(649)X(7649)
X(8748) lies on these lines: {4, 6}, {9, 318}, {19, 158}, {27, 57}, {28, 1436}, {29, 284}, {55, 281}, {107, 2291}, {216, 7567}, {232, 7413}, {297, 2893}, {333, 1948}, {412, 579}, {673, 823}, {909, 1474}, {1096, 2258}, {1713, 1715}, {1741, 3559}, {1751, 2052}, {1781, 1784}, {1826, 2259}, {1880, 1945}, {2164, 7040}
X(8748) = barycentric product X(4)*X(29)
X(8748) = pole wrt polar circle of trilinear polar of X(307) (line X(525)X(8611))
X(8748) = X(48)-isoconjugate (polar conjugate) of X(307)
X(8748) = X(63)-isoconjugate of X(73)
X(8748) = trilinear pole of line X(663)X(3064)
Let A'B'C' and A"B"C" be the orthocentroidal and anti-orthocentroidal triangles, resp. Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(8749). (Randy Hutson, March 21, 2019)
X(8749) lies on these lines: {2, 648}, {4, 6128}, {6, 74}, {37, 1783}, {111, 232}, {115, 393}, {186, 3003}, {263, 8541}, {403, 1989}, {895, 4230}, {1249, 2165}, {1400, 2159}, {1560, 7736}, {2071, 3284}, {2395, 6531}
X(8749) = isogonal conjugate of X(11064)
X(8749) = barycentric product X(4)*X(74)
X(8749) = pole wrt polar circle of trilinear polar of X(3260) (line X(1637)X(5664))
X(8749) = X(48)-isoconjugate (polar conjugate) of X(3260)
X(8749) = X(63)-isoconjugate of X(30)
X(8749) = cevapoint of X(6) and X(3003)
X(8749) = trilinear pole of line X(25)X(512)
X(8749) = barycentric product of (real or nonreal) circumcircle intercepts of line X(4)X(523)
X(8750) lies on these lines: {4, 595}, {6, 3270}, {19, 2195}, {25, 3052}, {31, 33}, {34, 3915}, {55, 3192}, {58, 6198}, {101, 112}, {108, 109}, {162, 190}, {204, 212}, {213, 2332}, {220, 3172}, {238, 1861}, {292, 1474}, {318, 4676}, {378, 995}, {406, 5264}, {602, 1753}, {653, 7012}, {692, 2498}, {1018, 1783}, {1104, 1902}, {1191, 1593}, {1253, 2331}, {1279, 1876}, {1398, 1616}, {1459, 1461}, {1824, 2161}, {1826, 1918}, {1968, 2176}, {2426, 4557}, {4584, 5379}, {5134, 5523}, {5698, 7952}
X(8750) = barycentric product X(4)*X(101)
X(8750) = pole wrt polar circle of trilinear polar of X(3261) (line X(1111)X(3120))
X(8750) = X(48)-isoconjugate (polar conjugate) of X(3261)
X(8750) = X(63)-isoconjugate of X(514)
X(8750) = isogonal conjugate of X(4025)
X(8750) = trilinear pole of line X(25)X(41)
X(8750) = barycentric product of circumcircle intercepts of line X(4)X(9)
X(8750) = trilinear product of circumcircle intercepts of Stevanovic circle
X(8751) lies on these lines: {4, 218}, {19, 2195}, {28, 105}, {34, 1438}, {281, 6559}, {286, 648}, {608, 1119}, {915, 919}, {1118, 2207}, {2201, 2356}
X(8751) = isogonal conjugate of X(25083)
X(8751) = barycentric product X(4)*X(105)
X(8751) = pole wrt polar circle of trilinear polar of X(3263) (line X(918)X(4437))
X(8751) = X(48)-isoconjugate (polar conjugate) of X(3263)
X(8751) = X(63)-isoconjugate of X(518)
X(8751) = trilinear pole of line X(25)X(884)
X(8752) lies on these lines: {19, 1743}, {25, 3052}, {27, 648}, {106, 112}
X(8752) = barycentric product X(4)*X(106)
X(8752) = pole wrt polar circle of trilinear polar of X(3264) (line X(3762)X(4120))
X(8752) = X(48)-isoconjugate (polar conjugate) of X(3264)
X(8752) = X(63)-isoconjugate of X(519)
X(8752) = isogonal conjugate of X(3977)
X(8752) = trilinear pole of line X(25)X(8643)
X(8753) lies on these lines: {4, 542}, {25, 111}, {186, 691}, {378, 5968}, {683, 1235}, {1426, 7316}, {1597, 2967}, {1783, 1824}, {6524, 6529}
X(8753) = barycentric product X(4)*X(111)
X(8753) = pole wrt polar circle of trilinear polar of X(3266) (line X(690)X(5181))
X(8753) = X(48)-isoconjugate (polar conjugate) of X(3266)
X(8753) = X(63)-isoconjugate of X(524)
X(8753) = isogonal conjugate of X(6390)
X(8753) = trilinear pole of line X(25)X(2489)
The trilinear polar of X(8754) passes through X(8029). (Randy Hutson, December 11, 2015)
X(8754) lies on these lines: {4, 542}, {25, 1989}, {53, 1843}, {115, 2971}, {125, 136}, {135, 137}, {225, 2652}, {254, 1092}, {264, 5117}, {393, 1974}, {460, 1990}, {1084, 2489}, {1576, 3018}, {1826, 6543}, {1865, 2870}, {2501, 6791}, {2872, 6784}, {3563, 6036}, {5140, 5523}
X(8754) = crossdifference of every pair of points on line X(4558)X(8552)
X(8754) = barycentric product X(4)*X(115)
X(8754) = pole wrt polar circle of trilinear polar of X(4590) (line X(99)X(110))
X(8754) = X(48)-isoconjugate (polar conjugate) of X(4590)
X(8754) = orthic-isogonal conjugate of X(2501)
X(8754) = X(4)-Ceva conjugate of X(2501)
X(8754) = X(63)-isoconjugate of X(249)
X(8754) = X(651) of orthic triangle if ABC is acute
X(8754) = excentral-to-ABC functional image of X(651)
X(8754) = trilinear pole, wrt orthic triangle, of Euler line
X(8755) lies on these lines: {1, 281}, {6, 1826}, {19, 208}, {53, 1839}, {112, 2695}, {204, 1857}, {230, 231}, {1108, 1880}, {1783, 2323}, {1856, 3195}, {2257, 5292}, {3975, 6335}
X(8755) = barycentric product X(4)*X(515)
X(8755) = X(63)-isoconjugate of X(102)
X(8755) = polar conjugate of X(34393)
X(8756) lies on these lines: {2, 3007}, {4, 9}, {28, 5258}, {44, 1877}, {48, 5882}, {112, 2758}, {230, 231}, {374, 1828}, {423, 648}, {515, 2173}, {517, 7359}, {534, 857}, {653, 5236}, {1146, 2182}, {1172, 5559}, {1222, 1474}, {1404, 4530}, {1731, 1737}, {1732, 1788}, {1781, 5270}, {1838, 5445}, {2218, 7154}, {2325, 3992}, {2331, 3554}, {2355, 7140}, {3264, 3977}, {3689, 3943}, {5338, 7102}
X(8756) = barycentric product X(4)*X(519)
X(8756) = pole wrt polar circle of trilinear polar of X(903) (line X(2)X(514))
X(8756) = X(48)-isoconjugate (polar conjugate) of X(903)
X(8756) = X(63)-isoconjugate of X(106)
X(8756) = isogonal conjugate of X(1797)
X(8756) = complement of X(3007)
X(8756) = crosspoint of X(4) and X(6336)
X(8756) = trilinear pole of line X(4120)X(4895)
X(8757) lies on these lines: {1,1898}, {3,1745}, {4,651}, {5,222}, {30,7078}, {34,912}, {73,3560}, {81,5714}, {90,1079}, {155,225}, {221,355}, {223,7330}, {255,2635}, {603,6911}, {942,6180}, {971,1062}, {1046,2640}, {1060,5777}, {1068,1069}, {1394,5720}, {1406,1737}, {1498,6259}, {1777,4551}, {1838,3173}, {2003,5707}, {2956,3359}, {4185,7352}, {4303,6883}, {7299,7742}
X(8758) lies on these lines: {1,3}, {11,1465}, {73,774}, {90,1079}, {158,1068}, {225,235}, {227,1837}, {230,231}, {422,1632}, {651,1776}, {912,1725}, {920,3157}, {1148,3147}, {1158,1406}, {1254,2654}, {1376,2000}, {1427,1836}, {1458,7004}, {1464,6001}, {1736,4551}, {1745,1898}, {2310,2635}, {2658,3725}, {3073,3468}, {3562,7098}
X(8758) = isogonal conjugate of X(8759)
X(8758) = crossdifference of every pair of points on line X(3)X(650)
X(8759) lies on the MacBeath circumconic and these lines: {1,1813}, {4,651}, {8,1332}, {9,1331}, {21,4558}, {90,255}, {104,3100}, {110,1172}, {314,4563}, {648,1896}, {885,1814}, {990,7284}, {1041,1830}, {1936,1937}
X(8759) = isogonal conjugate of X(8758)
X(8759) = isotomic conjugate of polar conjugate of X(20624)
X(8759) = trilinear pole of line X(3)X(650)
X(8760) lies on these lines: {3,650}, {4,693}, {5,4885}, {30,511}, {550,8142}
X(8760) = crossdifference of every pair of points on line X(6)X(8758)
X(8761) lies on these lines: {6,7120}, {48,1950}, {219,3362}, {1172,7049}, {1812,7361}, {1949,2178}
X(8761) = isogonal conjugate of X(6360)
X(8761) = X(19)-cross conjugate of X(6)
X(8762) lies on these lines: {3,653}, {4,46}, {65,7531}, {158,1155}, {196,631}, {376,3176}, {1068,1758}, {1895,3579}, {4295,7551}
X(8763) lies on these lines: {1,4}, {65,820}, {520,647}, {1155,2638}, {2646,7138}
X(8763) = isogonal conjugate of X(8764)
X(8763) = crossdifference of every pair of points on line X(4)X(652)
X(8764) lies on these lines: {3,653}, {78,6335}, {107,284}, {158,2638}, {219,1897}, {243,296}, {283,648}, {332,6331}
X(6764) = isogonal conjugate of X(8763)
X(6764) = trilinear pole of line X(4)X(652)
X(8765) lies on these lines: {19,1707}, {33,4641}, {63,162}, {165,1783}, {171,7079}, {208,1935}, {238,1435}, {255,1712}, {896,1096}, {1743,4219}, {1936,7008}, {2331,4640}
X(8766) lies on these lines: {1,19}, {31,6508}, {326,1792}, {336,1966}, {521,656}, {560,774}, {820,1964}, {896,2632}, {1038,7114}, {1040,2188}
X(8766) = isogonal conjugate of X(8767)
X(8766) = crossdifference of every pair of points on line X(19)X(656)
X(8767) lies on these lines: {63,162}, {72,1783}, {108,1214}, {240,293}, {304,811}, {306,1897}, {1096,2184}, {2582,2587}, {2583,2586}
X(8767) = isogonal conjugate of X(8766)
X(8767) = trilinear pole of line X(19)X(656)
X(8768) lies on these lines: {19,656}, {30,511}, {4329,7253}
X(8769) lies on these lines: {1,1958}, {10,2996}, {19,1707}, {37,1376}, {65,3751}, {158,1733}, {225,1738}, {326,2643}, {610,1910}, {759,3565}, {897,1760}, {1740,2186}
X(8769) = isogonal conjugate of X(1707)
X(8769) = X(63)-cross conjugate of X(1)
Let A'B'C' be the tangential triangle and L the orthic axis. Let A'' = L∩B'C', and define B'' and C'' cyclically. Let Ta be the line, other than B'C', through A'' tangent to the circumcircle, and define Tb and Tc cyclically. Let A* = Tb∩Tc, and define B* and C* cyclically. Then A*B*C* is perspective to ABC, and the perspector is X(8770). (Angel Montesdeoca, July 17, 2019)
X(8770) lies on the hyperbola {{A,B,C,X(2),X(6)}} and these lines: {2,1975}, {6,1196}, {22,111}, {25,1611}, {37,1376}, {69,6339}, {154,1976}, {183,2998}, {230,393}, {251,1184}, {263,1613}, {394,2987}, {427,5203}, {493,5409}, {494,5408}, {694,1350}, {941,5275}, {1880,3290}, {2165,6676}, {2248,4252}, {2395,6587}, {3051,3066}, {3228,8667}, {3767,6677}, {5913,7391}, {7398,7745}
X(8770) = isogonal conjugate of X(193)
X(8770) = complement X(19583)
X(8770) = X(25)-vertex conjugate of X(25)
X(8770) = X(3)-cross conjugate of X(6)
X(8771) lies on these lines: {19,662}, {63,610}, {326,2173}, {969,4273}, {1740,2617}, {1973,2128}
X(8772) lies on these lines: {1,21}, {244,1429}, {560,1953}, {661,663}, {756,2329}, {922,2173}, {1910,1933}, {1973,2181}, {2170,2210}, {2171,7122}
X(8772) = isogonal conjugate of X(8773)
X(8772) = crossdifference of every pair of points on line X(63)X(661)
X(8772) = X(2)-Ceva conjugate of X(39069)
X(8772) = perspector of conic {{A,B,C,X(19),X(662)}}
X(8773) lies on these lines: {1,4592}, {10,4561}, {19,662}, {37,1332}, {65,6516}, {91,304}, {158,811}, {225,664}, {326,2643}, {1310,3563}, {1910,1959}, {1930,2962}
X(8773) = isogonal conjugate of X(8772)
X(8773) = isotomic conjugate of X(1733)
X(8773) = trilinear pole of line X(63)X(661)
X(8774) lies on these lines: {30,511}, {63,661}, {226,4369}, {1478,4761}, {5905,7192}
X(8775) lies on these lines: {19,109}, {48,1950}, {601,2294}, {603,1953}, {608,2252}, {1771,1839}, {1777,1826}, {1841,3215}
X(8776) lies on these lines: {1,5179}, {6,41}, {42,3010}, {661,663}, {672,3002}, {800,1195}, {910,1464}, {1064,2280}, {1457,2170}, {2252,3003}, {3100,5746}, {4262,4337}, {4559,8608}
X(8776) = isogonal conjugate of X(8777)
X(8776) = crossdifference of every pair of points on line X(63)X(522)
X(8776) = X(2)-Ceva conjugate of X(39070)
X(8776) = perspector of conic {{A,B,C,X(19),X(109)}}
X(8777) lies on these lines: {2,6506}, {8,1332}, {29,662}, {92,664}, {312,4561}, {333,4592}, {394,1146}, {1944,1952}
X(8777) = isogonal conjugate of X(8776)
X(8777) = trilinear pole of line X(63)X(522)
X(8778) lies on these lines: {3,112}, {4,1384}, {6,1204}, {25,1611}, {32,1593}, {172,7071}, {187,2207}, {232,5023}, {577,1033}, {1249,3522}, {1285,3088}, {1398,1914}, {1562,5925}, {1657,5523}, {1885,7735}, {1950,3209}, {1951,7154}, {2332,4252}, {5094,7745}, {7507,7737}
X(8779) lies on the Simson quartic (Q101) and these lines: {6,25}, {30,1562}, {32,185}, {112,6000}, {125,230}, {132,1503}, {172,1425}, {187,3269}, {217,5007}, {287,385}, {394,1073}, {426,577}, {511,1297}, {520,647}, {800,1501}, {1204,3053}, {1498,3172}, {1899,7735}, {1914,3270}, {2030,5622}, {5304,6776}, {5305,6146}
X(8779) = isogonal conjugate of X(6330)
X(8779) = crosssum of X(2) and X(297)
X(8779) = crosspoint of X(6) and X(248)
X(8779) = crossdifference of every pair of points on line X(4)X(525)
X(8779) = X(2)-Ceva conjugate of X(39071)
X(8779) = perspector of conic {{A,B,C,X(3),X(112)}}
X(8779) = intersection of Simson line of X(112) (line X(132)X(1503)) and trilinear polar of X(112) (line X(6)X(25))
X(8779) = barycentric product X(3)*X(1503) (see http://bernard-gibert.fr/curves/q101.html)
X(8780) lies on these lines: {3,64}, {22,6090}, {25,110}, {49,7529}, {113,3830}, {155,3517}, {156,6642}, {184,373}, {394,1495}, {575,5644}, {1147,1598}, {1384,1613}, {1511,3426}, {1853,5972}, {3564,6353}, {3796,5651}, {6677,6776}, {6800,7484}
X(8781) lies on the Kiepert hyperbola and these lines: {2,2987}, {4,99}, {10,4561}, {69,6036}, {76,2023}, {83,3815}, {94,3266}, {98,325}, {115,2996}, {147,3424}, {183,7607}, {262,5976}, {305,5392}, {598,1003}, {620,2548}, {671,7799}, {1078,3406}, {1916,7925}, {2052,6331}, {3268,5466}, {3407,7777}, {5152,7814}, {5989,6054}, {6321,6390}, {6722,7869}, {7840,8587}
X(8781) = isogonal conjugate of X(1692)
X(8781) = isotomic conjugate of X(230)
X(8781) = antigonal image of X(2996)
X(8781) = Kiepert hyperbola antipode of X(2996)
X(8781) = reflection of X(99) in X(6337)
X(8781) = reflection of X(2996) in X(115)
X(8781) = trilinear pole of line X(69)X(523)
X(8781) = antipode of X(99) in the circumconic that is the isotomic conjugate of the orthic axis (conic {A,B,C,X(99),X(4554),X(4563)})
X(8781) = pole wrt polar circle of trilinear polar of X(460)
X(8781) = X(48)-isoconjugate (polar conjugate) of X(460)
X(8781) = Vu circlecevian point V(X(485),X(486))
Loci associated with selected triangles (1): X(8782)-X(8855)
This preamble and centers X(8782)-X(8855) were contributed by César Eliud Lozada, December 11, 2015.
Let ABC be a triangle and let P1 = p1 : q1 : r1, P2 = p2 : q2 : r2, P3 = p3 : q3 : r3 (barycentrics) be three non-collinear points, none on a sideline of ABC. The locus of a point M such that the cevian triangle of M and the triangle T = P1P2P3 are perspective, with perspector denoted by Z(M), is given by
F(x,y,z) = δ11(r1y-q1z)x2+δ22(p2z-r2x)y2+ δ33(q3x-p3y)z2+(p2q3r1-p3q1r2)xyz = 0 (1)
where δij is the (i, j)-minor of the vertex matrix of the triangle T.
If T and ABC are perspective we have p2q3r1 - p3q1r2 = 0 and, if the equation (1) is non-degenerate, it represents a pivotal cubic with pole Ω(T) and pivot P(T) given by:
Ω(T) = δ22δ33p2p3 : δ33δ11p3q1 : δ11δ22p2r1
P(T) = δ22δ33p3r2 : δ33δ11p3r1 : δ11δ22p2r1
This section deals with the poles and pivots of these cubics and the perspectors Z(M) for most of the centers M lying on them. In the following table, each column contains:
| Triangle T | Locus of M for perspectivity of triangles T and cevian-of-M | Gibert's catalogue | Pole* | Pivot* | ( M,Z(M) )** | Locus of Z(M) |
|---|---|---|---|---|---|---|
| 1st anti-Brocard | Σ[(a^4-b^2*c^2)*x*(b^2*y^2-c^2*z^2)] | -- | 76 | 3978 | (2,8290), (6,4027), (75,1281), (76,5976), (290,98), (325,147), (698,8782), (1502,8783), (1916,1916), (3978,5989), (5989,8784) | K699 |
| anti-McCay | Σ[((3*SA+SW)^2-(3*b*c)^2)*((3*SB+SW)*y^2-(3*SC+SW)*z^2)*x] | -- | 598 | 8785 | (2,8786), (524,8591), (598,8787), (8587,8587) | |
| 1st Brocard | Σ[(a^4-b^2*c^2)*(b^2*y-c^2*z)*x^2] | K322 | 1916 | 694 | (2,3), (76,76), (141,2896), (257,3496), (297,4), (335,1), (384,3492), (385,32), (694,384), (698,194), (1916,39), (2998,3224) | K020 |
| 2nd Brocard | Σ[(a^4-b^4-c^4+b^2*c^2)*(c^2*y-b^2*z)*x^2] | K531 | 3455 | 67 | (6,6), (67,141), (141,8788), (524,69) | K538 |
| 3rd Brocard | Σ[(a^4-b^4-c^4+b^2*c^2)*(c^2*y-b^2*z)*x^2] | K532 | 8789 | 694 | (6,194), (25,4), (32,32), (237,3), (384,8790), (385,76), (694,384), (733,83), (904,6196), (1911,1), (2076,2896), (3051,3499) | K020 |
| 4th Brocard | Σ[(b^2+c^2-a^2)*(a^4-b^4-c^4+b^2*c^2)*(y-z)*x^2] | K533 | 8791 | 8791 | (2,2), (427,8792), (468,25) | K539 |
| circummedial | Σ[a^2*(b^2+c^2)*(y-z)*x^2] | -- | 308 | 308 | (2,2), (76,22), (83,251), (264,25), (308,1799), (1799,8793) | pK(251,1799) |
| circumorthic | Σ[(SB+SC)*(SB*SC+S^2)*SA^3*(SB*y-SC*z)*x^2] | -- | 8794 | 8795 | (4,4), (264,3), (275,54), (317,6193), (2052,24) | |
| 2nd circumperp | Σ[(b+c)*(c*y-b*z)*x^2] | K317 | 81 | 86 | (1,1), (2,3), (7,56), (21,2360), (29,28), (77,1394), (81,58), (86,21) | K318 |
| Euler | Σ[SA*(SB*SC-2*S^2)*(SB*y-SC*z)*x^2] | -- | 8796 | 8797 | (2,5), (4,4), (253,8798), (3091,8799) | |
| 2nd Euler | Σ[SB*SC*(SA^2-S^2)*x*(y^2-z^2)] | -- | 2 | 317 | (2,3), (4,5), (68,68), (69,5562), (317,4), (6193,155), (6504,8800), (6515,52) | K044 |
| 5th Euler | Σ[(SA-2*SW)*SA*(y-z)*x^2] | -- | 8801 | 8801 | (2,2), (4,427) | |
| extangents | Σ[a*(b+c)*((s-c)*y-(s-b)*z)*y*z] | K033 | 37 | 8 | (1,55), (4,19), (8,40), (10,71), (40,3197), (65,65), (72,3198), (3176,8802), (5930,8803) | pK(213,40) |
| 2nd extouch | Σ[(SC*y-SB*z)*y*z] | K007 | 2 | 69 | (2,9), (4,4), (7,226), (8,72), (20,1490), (69,329), (189,1903), (253,2184), (329,8804), (1032,8805), (1034,8806), (5932,8807) | pK(37,329) |
| 3rd extouch | Σ[(SC*y-SB*z)*y*z] | K007 | 2 | 69 | (2,223), (4,4), (7,1439), (8,5930), (20,3182), (69,5932), (189,8808), (253,8809), (329,8807), (1032,8810), (1034,8811), (5932,8812) | |
| 4rd extouch | Σ[(s-a)*(4*R*r+2*r^2+b*c)*(SC*y-SB*z)*x^2] | -- | 8813 | 8814 | (7,65), (69,69), (5933,8815) | |
| 5th extouch | Σ[(s-b)*(s-c)*(b*c+SW)*((s-b)*(SW-a*c)*y-(s-c)*(SW-a*b)*z)*y*z] | -- | 8816 | 8817 | (7,65), (8,5930), (388,388) | |
| Feuerbach | Σ[(b+c)*(2*SA+b*c)*((a+b)^2*(s-b)*y-(a+c)^2*(s-c)*z)*x^2] | -- | 8818 | 3615 | (1,11), (5,8819), (12,12), (3615,5), (6757,523) | |
| outer-Garcia | Σ[(b+c)*(b*y-c*z)*y*z] | K366 | 321 | 75 | (2,1), (8,40), (10,10), (75,8), (307,5930), (318,4), (321,72), (1441,65) | K033 |
| inner-Grebe | Σ[a^2*((b^2-S)*y-(c^2-S)*z)] | -- | 494 | 5491 | (2,5591), (4,1163), (6,6), (494,8820), (1271,--), (5491,1271) | |
| outer-Grebe | Σ[a^2*((b^2+S)*y-(c^2+S)*z)] | -- | 493 | 5490 | (2,5590), (4,1162), (6,6), (493,8821), (1270,--), (5490,1270) | |
| hexyl | Σ[(s^2-a*s+2*R*r-b*c)*((a+c)*c*y^2-(a+b)*b*z^2)*x] | K344 | 81 | 8822 | (7,1), (20,1490), (21,3), (27,4), (63,40), (84,84) | K004 |
| Johnson | Σ[(SB*SC+S^2)*(b^2*SB*y-SC*c^2*z)*y*z] | K674 | 324 | 264 | (2,3), (4,155), (5,5), (264,4), (311,5562), (324,52), (847,68) | K044 |
| Kosnita | Σ[b^2*c^2*(SB*SC+S^2)*x*(y^2-z^2)] | -- | 2 | 311 | (2,3), (4,24), (54,54), (69,0), (311,7488), (1994,52), (2888,2917), (3459,8823), (7488,8824) | K388 |
| Lucas central | Σ[b^2*c^2*(a^2+2*S)*(SC*c^2*y-SB*b^2*z)*x^2] | -- | 8825 | 588 | (3,3), (6,3311), (371,8826), (588,371) | |
| Lucas tangents | Σ[b^2*c^2*(S+SB+SC)*((S+SC)*c^2*y-(S+SB)*b^2*z)*x^2] | 8950 | 493 | (6,3), (371,371), (493,1151), (1151,8912) | ||
| McCay | $((SW+3*SA)^2-(3*b*c)^2)*((SW*SB-3*S^2)*y-(SC*SW-3*S^2)*z)*x^2] | -- | 8587 | 8827 | (2,3), (468,4), (7607,7607), (8587,575) | |
| midheight | Σ[(SC*y-SB*z)*y*z] | K007 | 2 | 69 | (2,6), (4,4), (7,57), (8,1), (20,3), (69,2), (189,282), (253,1073), (329,9), (1032,3344), (1034,3342), (5932,223) | K002 |
| mixtilinear | Σ[b*c*(s-a)*((2*SA-2*SW)*b*c+S^2)*(c*y-b*z)*x^2] | -- | 8828 | 8829 | (1,1), (57,40), (1697,--) | |
| 3rd mixtilinear | Σ[b*c*(s-a)^3*((s-b)*c^2*y-(s-c)*b^2*z)*x^2] | -- | 7366 | 269 | (56,56), (57,1), (269,1420), (1407,1616), (1420,7963) | |
| 4th mixtilinear | Σ[b*c*(s-c)^2*((s-c)*c^2*y-(s-b)*b^2*z)*x^2] | -- | 1253 | 9 | (1,57), (9,165), (55,55), (165,8830), (220,1615), (2066,8831), (4166,8832), (5414,8833) | |
| 5th mixtilinear | Σ[(s-a)*(c*y-b*z)*x^2] | K365 | 57 | 7 | (1,1), (2,8), (7,145), (57,2136), (145,8834) | K201 |
| 6th mixtilinear | Σ[(3*a^2-2*a*b-2*a*c-b^2+2*b*c-c^2)*(c*y^2-b*z^2)*x] | K202 | 1 | 144 | (7,1), (9,165), (144,2951), (366, {165,365}∩{364,3062}), (2951,8835), (3062,3062) | |
| inner-Napoleon | Σ[(t*SA-S)*((t*SB-3*S)*y-(t*SC-3*S)*z)*x^2],t=sqrt(3) | K420a | 14 | 8836 | (2,3), (5,8837), (14,61), (18,18), (302,627), (395,62), (471,4) | K005 |
| outer-Napoleon | Σ[(t*SA+S)*((t*SB+3*S)*y-(t*SC+3*S)*z)*x^2],t=sqrt(3) | K420b | 13 | 8838 | (2,3), (5,8839), (13,62), (17,17), (303,628), (396,61), (470,4) | K005 |
| 1st Neuberg | Σ[(SW^2-S^2+2*SW*SA)*x^2*((SW*SB-S^2)*y-(SW*SC-S^2)*z)] | -- | 3407 | 8840 | (2,3), (25,4), (98,98), (183,6194), (385,511), (5989,147), (5999,8841) | K422 |
| 2nd Neuberg | Σ[(S^2+SW*SA)*((-2*SW*SB+SW^2-S^2)*y^2-(SW^2-2*SW*SC-S^2)*z^2)*x] | -- | 1916 | 8842 | (2,3), (262,262), (325,147), (427,4), (1916,511), (3329,182), (4518,6211), (5999,8840), (7249,1) | K422 |
| orthocentroidal | Σ[(2*SA+b*c)*(2*SA-b*c)*(SB*y-SC*z)*x^2] | K060 | 1989 | 265 | (4,4), (5,195), (13,62), (14,61), (30,3), (79,3336), (80,1), (265,5), (621,628), (622,627), (1117,3471), (1141,54), (5627,3470), (6761,3462) | K005 |
| reflection | Σ[(2*SA+b*c)*(2*SA-b*c)*(SB*y-SC*z)*x^2] | K060 | 1989 | 265 | (4,4), (5,3), (13,15), (14,16), (30,399), (79,1), (80,484), (265,30), (621,616), (622,617), (1117,5671), (1141,1157), (5627,74), (6761,5667) | K001 |
| 1st Sharygin | Σ[(a^2+b*c)*x*(c*y^2-b*z^2)] | K132 | 1 | 894 | (6,1580), (7,1284), (9,8843), (37,846), (75,1281), (86,21), (87,8843), (192,8844), (256,256), (366, {364,8245}∩{366,8424}), (894,8424), (1045,8845), (1654,8846), (8424,8847) | pK(1914,8424) |
| 2nd Sharygin | Σ[(a^2-b*c)*x*(c*y^2-b*z^2)] | K323 | 1 | 329 | (1,8298), (2,8299), (6,8300), (75,1281), (239,8301), (291,291), (366, {365,2108}∩{366,8301}), (518,1282), (673,105), (1575,2108), (2319,8848), (2669,8849), (3212,8850), (3226,8851), (7061,8852), (8301,8853) | pK(1914,8301) |
| inner-squares | Σ[(SA+S)*(SB*y^2-SC*z^2)*x] | K070b | 4 | 1585 | (2,6), (4,371), (485,485), (491,2), (1585,3068), (1659,6204), (3068,8854) | K424a |
| outer-squares | Σ[(SA-S)*(SB*y^2-SC*z^2)*x] | K070a | 4 | 1586 | (2,6), (4,372), (486,486), (492,2), (1586,3069), (3069,8855), (7090,1) | K424b |
| inner-Vecten | Σ[(SA-S)*(SB*y^2-SC*z^2)*x] | K070a | 4 | 1586 | (2,3), (4,371), (486,486), (492,488), (1586,4), (3069,372), (7090,6213) | K006 |
| outer-Vecten | Σ[(SA+S)*(SB*y^2-SC*z^2)*x] | K070b | 4 | 1585 | (2,3), (4,372), (485,485), (491,487), (1585,4), (1659,1), (3068,371) | K006 |
*: The appearance of i in columns 4 and 5 means that X(i) is the pole or pivot of the cubic.
**: The appearance of (i,j) in this column means that X(j) is the perspector of T and the cevian-triangle-of-X(i). Some perspectors having very long coordinates are either not calculated (--) or are represented as intersections of lines.
X(8782) lies on the cubics K128, K699, the bianticevian conic of X(1) and X(2), and these lines: {1,1281}, {2,694}, {6,8290}, {20,2782}, {22,3504}, {32,99}, {63,2319}, {76,148}, {98,3098}, {115,3096}, {147,511}, {385,698}, {538,8591}, {543,7811}, {620,7846}, {627,3104}, {628,3105}, {671,7865}, {726,2959}, {1078,8178}, {3506,8784}, {5149,7787}, {5152,7793}, {5992,6646}, {6033,7900}, {7785,8149}
X(8782) = anticomplementary conjugate of X(5207)
X(8782) = anticomplement of X(1916)
X(8782) = reflection of X(i) in X(j) for these (i,j): (148,76), (194,99), (1916,5976)
X(8782) = perspector of these triangles: 1st anti-Brocard and anticevian of X(194)
X(8782) = antipode of X(194) in bianticevian conic of X(1) and X(2)
X(8782) = anticomplementary isotomic conjugate of X(7779)
X(8782) = 1st-Brocard-to-5th-Brocard similarity image of X(6)
X(8782) = X(99)-of-5th-Brocard-triangle
X(8783) lies on the cubic K699 and these lines: {76,148}, {98,689}, {147,305}, {385,710}, {1502,5989}, {3978,4027}
X(8783) = perspector of these triangles: 1st anti-Brocard and anticevian-of-X(3978)
X(8784) lies on the cubic K699 and these lines: {1580,7061}, {1916,6660}, {3506,8782}
X(8784) = perspector of these triangles: 1st anti-Brocard and anticevian-of-X(4027)
X(8785) lies on these lines: {2,187}
X(8786) lies on these lines: {39,671}, {316,2482}, {5976,7840}, {7777,8591}, {8587,8593}
X(8786) = X(7608) of anti-McCay triangle
X(8787) lies on these lines: {2,8587}, {5,542}, {6,598}, {141,1153}, {187,524}, {194,1992}, {543,8584}, {599,1078}, {3455,8546}, {4027,7840}, {5032,8596}, {7766,8592}
X(8787) = midpoint of X(i),X(j) for these (i,j): (6,8593)
X(8788) lies on the cubic K538 and these lines: {69,184}, {353,3620}, {574,3619}, {3618,6387}
X(8788) = perspector of these triangles: 2nd Brocard and anticevian of X(69)
X(8789) lies on these lines: {6,1916}, {83,3115}, {98,2086}, {110,251}, {699,805}, {1501,8023}, {1922,1967}, {1927,2205}
X(8790) lies on the cubic K020 and these lines: {1,7346}, {3,6374}, {76,695}, {83,3224}, {3403,3495}
X(8790) = perspector of these triangles: 3rd Brocard and anticevian of X(76)
X(8791) lies on the cubic K533 and these lines: {2,339}, {4,1383}, {6,67}, {25,115}, {111,468}, {112,251}, {232,1989}, {308,6331}, {393,2970}, {1400,2157}, {2165,2493}, {2987,3580}
X(8791) = isogonal conjugate of X(22151)
X(8791) = isotomic conjugate of X(37804)
X(8791) = X(23)-isoconjugate of X(63)
X(8791) = polar conjugate of X(316)
X(8791) = pole wrt polar circle of trilinear polar of X(316) (line X(2492)X(7664))
X(8791) = X(48)-isoconjugate (polar conjugate) of X(316)
X(8791) = trilinear pole of line X(512)X(1843)
X(8791) = perspector of ABC and orthoanticevian triangle of X(67)
X(8792) lies on the cubic K539 and these lines: {6,5064}, {25,251}, {3162,5094}
X(8793) lies on these lines: {2,66}, {25,251}, {83,6997}, {1799,7493}
X(8793) = isogonal conjugate of X(39129)
X(8793) = cevapoint of X(206) and X(20993)
X(8793) = trilinear product X(i)*X(j) for these {i,j}: {82, 159}, {251, 18596}, {3162, 34055}
X(8794) lies on these lines: {2,276}, {6,275}, {25,1093}, {54,436}, {263,6524}, {324,4993}, {467,6528}
X(8794) = polar conjugate of X(5562)
Let A'B'C' be the circumorthic triangle, and let A'' = {B,C}-reciprocal conjugate of A, with respect to A'B'C', and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(8795). (Randy Hutson, December 26, 2015) Here, the term "{X,Y}-reciprocal conjugate" designates the reciprocal conjugate that swaps X and Y, where X and Y are not on sidelines of ABC and not collinear with A or B or C. Reciprocal conjugation is introduced in Keith Dean and Floor van Lamoen, 'Geometric Construction of Reciprocal Conjugations," Forum Geometricorm 1 (2001) 115-120. Specifically, if P = p : q : r, U = u : v : w, X = x : y: z are trilinears for points, none on a sideline of ABC, then the {P,U}-reciprocal conjugate of X is the point pu/x : qv/y : rw/z. See "reciprocal conjugate" in Glossary.
X(8795) lies on the Jerabek hyperbola, the conic {{A,B,C,X(264),X(2052)}}, and these lines: {3,95}, {4,6752}, {6,275}, {53,1987}, {68,317}, {97,324}, {265,6528}, {340,3519}, {1093,3527}, {1173,4994}
X(8795) = isogonal conjugate of X(418)
X(8795) = isotomic conjugate of X(5562)
X(8795) = polar conjugate of X(216)
X(8795) = pole wrt polar circle of trilinear polar of X(216)
X(8795) = perspector of ABC and orthoanticevian triangle of X(276)
X(8796) lies on the Kiepert hyperbola and these lines: {2,53}, {4,3527}, {25,7612}, {76,324}, {96,7487}, {98,6995}, {193,6504}, {262,7378}, {275,393}, {1585,3316}, {1586,3317}, {2996,6515}, {3424,7408}, {4232,7607}, {5056,6750}, {5395,5422}, {6747,7608}
X(8796) = isogonal conjugate of X(36748)
X(8796) = cevapoint of X(i) and X(j) for these {i,j}: {6, 3517}, {53, 8887}, {3527, 34818}
X(8796) = polar conjugate of X(631)
X(8796) = X(63)-isoconjugate of X(11402)
X(8796) = isotomic conjugate of isogonal conjugate of X(34818)
Let A'B'C' be the Euler triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2). Let A* be the trilinear pole of line B"C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(8797). (Randy Hutson, December 26, 2015)
X(8797) lies on these lines: {2,53}, {4,95}, {5,69}, {253,5056}, {264,3090}, {286,6969}, {287,3618}, {305,311}, {317,3545}, {1441,6933}, {1494,5071}, {1799,7392}, {2165,4993}, {6337,7404}, {6527,7486}
X(8797) = isotomic conjugate of X(631)
X(8797) = polar conjugate of X(3087)
X(8797) = pole wrt polar circle of trilinear polar of X(3087)
X(8798) lies on the cubics K026, the K096, the K558 and these lines: {3,64}, {122,6247}, {253,264}, {546,6662}, {1301,3518}
X(8798) = isogonal conjugate of X(38808)
X(8798) = crosspoint of X(253) and X(1073)
X(8798) = crosssum of X(154) and X(1249)
X(8798) = trilinear product X(i)*X(j) for these {i,j}: {5, 19614}, {51, 19611}, {216, 2184}
X(8798) = trilinear quotient X(i)/X(j) for these (i,j): (5, 1895), (51, 204), (216, 610), (2184, 275), (19611, 95), (19614, 54)
X(8799) lies on these lines: {4,216}, {52,381}, {324,3832}
X(8800) lies on the cubic K044 and these lines: {3,2165}, {4,155}, {52,53}, {569,5254}, {2055,6321}, {2980,7553}, {3613,7403}, {6337,7404}
X(8800) = isogonal conjugate of X(8883)
X(8800) = perspector of 2nd Euler triangle and tangential triangle of orthic triangle
Let Ab and Ac be the intersections of lines CA and AB, respectlivey, with the parallel to BC through X(4). Define Bc and Ca cyclically, and define and Ba, Cb cyclically. The points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic centered at the midpoint of X(4) and X(1249). The perspector of the conic is X(8801) (Randy Hutson, December 26, 2015)
X(8801) lies on these lines: {4,141}, {264,7378}, {393,427}, {907,1300}, {1179,3541}, {1217,1595}, {1826,1851}, {3087,6531}, {6526,7507}
X(8801) = isogonal conjugate of X(3796)
X(8801) = isotomic conjugate of X(3785)
X(8801) = polar conjugate of X(3618)
X(8801) = pole wrt polar circle of trilinear polar of X(3618) (line X(3800)X(3804))
X(8801) = trilinear pole of line X(826)X(2501)
X(8802) lies on these lines: {19,1857}, {33,64}, {40,1712}, {55,204}, {2192,3213}
X(8803) lies on these lines: {19,208}, {40,3182}, {55,64}, {65,2357}, {71,227}, {109,5897}, {1712,6523}
X(8804) lies on these lines: {1,5746}, {4,9}, {6,950}, {20,610}, {37,226}, {42,4319}, {48,4297}, {72,1903}, {101,1294}, {190,4150}, {198,7580}, {204,1249}, {209,1864}, {219,515}, {253,306}, {282,6282}, {284,4304}, {307,857}, {380,4294}, {391,5175}, {405,5750}, {442,5257}, {452,5749}, {497,2257}, {579,1210}, {672,1713}, {946,5798}, {1449,3488}, {1479,1723}, {1490,2324}, {1741,5709}, {1743,3586}, {1770,1781}, {1953,4301}, {2264,6284}, {2294,3671}, {2893,4416}, {3159,3950}, {3247,3487}, {3330,3990}, {3419,3686}, {3436,3692}, {3610,4082}, {4183,5285}, {5177,5296}
X(8805) lies on these lines: {9,3341}, {72,3176}, {329,1032}, {1490,2324}
X(8806) lies on these lines: {9,1249}, {20,78}, {72,5930}, {2184,3346}, {3694,8804}
X(8807) lies on these lines: {4,65}, {9,223}, {57,5776}, {72,5930}, {226,1439}, {329,1032}, {1425,1868}, {1427,3330}, {1490,3182}, {1498,1712}, {1708,2182}
X(8807) = X(8807) is also the perspector of the 3rd extouch triangle and the cevian-triangle-of-X(329)
X(8807) = perspector of (cross-triangle of ABC and 2nd extouch triangle) and (cross-triangle of ABC and 3rd extouch triangle)
X(8808) lies on the Kiepert hyperbola and these lines: {2,77}, {4,57}, {10,227}, {98,8059}, {226,1439}, {268,5745}, {273,2052}, {280,6734}, {307,321}, {515,7011}, {940,1433}, {1394,7498}, {1413,7532}, {1436,1751}, {3597,7146}, {5125,7020}, {5909,6922}
X(8809) lies on the Feuerbach hyperbola and these lines: {1,64}, {4,3668}, {8,253}, {9,223}, {21,77}, {57,1172}, {84,269}, {104,4341}, {273,1896}, {294,2155}, {885,7661}, {1119,7149}, {1838,6526}, {4866,8282}, {6247,6355}, {7013,7070}
X(8809) = isogonal conjugate of X(7070)
X(8810) lies on these lines: {40,3182}, {196,1439}, {222,3194}, {223,3344}, {329,1032}, {1422,3345}, {1804,1817}, {2184,8808}
X(8811) lies on the Jerabek hyperbola and these lines: {3,223}, {4,7149}, {6,208}, {34,64}, {69,1034}, {71,227}, {72,5930}, {225,1903}
X(8811) = isogonal conjugate of X(13614)
X(8812) lies on these lines: {4,3668}, {223,3344}, {1427,8808}, {1439,8811}, {2131,3182}
X(8813) lies on these lines: {3,1014}, {77,3682}, {279,1214}, {348,3998}
X(8814) lies on the Jerabek hyperbola and these lines: {3,1014}, {6,1396}, {7,72}, {57,71}, {64,3332}, {65,1119}, {73,269}, {479,1439}, {1245,2263}
X(8814) = isogonal conjugate of X(13615)
X(8815) lies on these lines: {69,226}, {1214,2092}, {5929,8807}
X(8816) lies on these lines: {2,1435}, {279,304}, {1037,3600}, {1041,4296}
X(8817) lies on these lines: {2,1462}, {7,3263}, {8,479}, {57,345}, {69,200}, {75,1119}, {171,7084}, {307,1041}, {332,1014}, {333,1396}, {348,8270}, {497,4554}, {518,7055}, {693,3434}, {2550,6063}
X(8817) = isogonal conjugate of X(7083)
X(8817) = isotomic conjugate of X(497)
X(8817) = cevapoint of X(7) and X(8)
X(8817) = trilinear pole of line X(918)X(3669)
X(8818) lies on these lines: {4,584}, {5,583}, {6,13}, {7,8287}, {9,46}, {33,430}, {210,8013}, {312,1230}, {391,6871}, {661,1953}, {1478,6739}, {2174,3585}, {2278,5747}, {2321,4053}, {2478,3615}, {3767,4275}, {4204,6186}, {5153,5254}
X(8818) = isotomic conjugate of X(34016)
X(8818) = SS(A → A') of X(19), where A'B'C' is the incentral triangle
X(8819) lies on these lines: {5,3737}, {11,2605}, {12,523}, {656,900}, {1532,4129}, {3259,8286}
X(8820) lies on these lines: {494,1271}, {5591,6421}
X(8821) lies on these lines: {493,1270}, {5590,6422}
Let A'B'C' be the hexyl triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(63). Let A* be the trilinear pole of line B"C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(8822). (Randy Hutson, December 26, 2015)
Let A1B1C1 be the 1st Conway triangle. Let A' be the cevapoint of B1 and C1, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(8822). (Randy Hutson, December 10, 2016)
X(8822) lies on the cubic K344 and these lines: {2,1901}, {7,21}, {19,27}, {20,64}, {29,307}, {30,2893}, {40,322}, {58,3663}, {81,2255}, {84,309}, {99,102}, {142,4877}, {144,2287}, {190,5279}, {198,329}, {208,342}, {221,347}, {264,412}, {272,903}, {273,3559}, {284,527}, {377,5224}, {662,2327}, {1010,4292}, {1765,6996}, {1778,4000}, {1812,7152}, {2096,4221}, {2303,4419}, {2328,3668}, {3664,4653}, {3786,5784}, {3868,4360}, {3879,4304}, {4021,4658}, {4189,5736}, {4229,5732}, {5046,5740}, {5738,6872}
X(8822) = isogonal conjugate of X(2357)
X(8822) = isotomic conjugate of X(39130)
X(8822) = anticomplement of X(1901)
X(8822) = crossdifference of every pair of points on line X(810)X(3709)
X(8822) = crosspoint, wrt excentral triangle, of X(20) and X(63)
X(8822) = crosspoint, wrt anticomplementary triangle, of X(20) and X(63)
X(8822) = pivot of K344 (from http://bernard-gibert.fr/Exemples/k344.html)
X(8822) = barycentric product of X(86) and X(329)
X(8822) = perspector of 1st Conway triangle and cross-triangle of ABC and 1st Conway triangle
X(8823) lies on the cubic K388 and these lines: {3459,7488},
X(8824) lies on the cubic K388 and these lines: {3,6293}, {24,571}, {3575,6747}
The trilinear polar of X(8825) passes through X(3049). (Randy Hutson, December 26, 2015)
X(8825) lies on this line: {25,588}
X(8825) = X(92)-isoconjugate of X(590)
X(8826) lies on these lines: {1147,1151}
X(8827) lies on these lines: {2,8587}
X(8828) lies on these lines: {6,6612}, {33,1466}, {56,7074}, {57,2324}, {198,1407}, {223,738}, {1435,2331}, {5438,7091}
Let A'B'C' be the mixtilinear triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(57). Let A* be the trilinear pole of line B"C", and define B* and C* cyclcially. The lines AA*, BB*, CC* concur in X(8829). (Randy Hutson, December 26, 2015)
X(8829) lies on these lines: {7,7080}, {40,269}, {57,2324}, {281,1767}, {347,479}, {1119,3663}
X(8830) lies on these lines: {55,1419}, {57,7955}, {165,2124}
X(8831) lies on these lines: {6,1200}, {55,1152}, {57,2067}, {165,2066}, {1124,6244}
X(8832) lies on these lines: {55,365}, {57,367}, {165,364}
X(8833) lies on these lines: {6,1200}, {55,1151}, {57,6502}, {165,5414}, {1335,6244}
X(8834) lies on the cubic K201 and these lines: {1,3161}, {8,3452}, {145,8055}, {190,6049}, {644,1616}, {1265,4345}, {2098,5423}, {3616,6703}, {4308,4488}
X(8834) = reflection of X(i) in X(j) for these (i,j): (8,6552), (6553,1)
X(8835) lies on these lines: {1,971}, {165,220}, {1282,7991}, {7994,8159}
X(8836) lies on the cubic 420a and these lines: {2,14}, {5,49}, {18,94}, {23,7685}, {301,302}, {323,624}, {395,1989}, {6105,6672}
X(8836) = isogonal conjugate of X(8604)
X(8837) lies on the cubic K005 and these lines: {1,7344}, {3,3166}, {4,15}, {51,61}, {54,62}, {216,3463}, {3469,6191}
X(8838) lies on the cubic K420b and these lines: {2,13}, {5,49}, {17,94}, {23,7684}, {300,303}, {323,623}, {396,1989}, {6104,6671}
X(8838) = isogonal conjugate of X(8603)
X(8839) lies on the cubic K005 and these lines: {1,7345}, {3,3165}, {4,16}, {51,62}, {54,61}, {216,3463}, {3469,6192}
X(8840) lies on these lines: {2,1501}, {183,327}, {237,5976}, {385,694}, {669,804}, {1284,3113}, {3978,5989}
X(8841) lies on the cubic K422 and these lines: {1,7350}, {3,3491}, {4,39}, {98,3229}, {512,684}, {3117,6776}
X(8842) lies on these lines: {2,694}, {183,6784}, {262,6234}, {290,325}, {427,6331}, {1934,4518}, {3493,7752}, {5999,8840}
X(8843) lies on these lines: {3,2053}, {21,741}, {87,8424}, {846,2162}, {1580,1914}
X(8843) = perspector of the 1st Sharygin triangle and the cevian triangle of X(87)
X(8844) lies on these lines: {9,43}, {55,192}, {63,3056}, {350,1281}, {522,659}, {712,8618}, {1403,4203}, {1580,1914}, {2236,2309}, {2292,4093}, {3509,4447}
X(8845) lies on these lines: {21,238}, {385,740}, {1045,8424}
X(8846) lies on these lines: {21,6626}, {1580,2238}, {1654,8424}, {4433,8844}
X(8847) lies on these lines: {256,8300}
X(8848) lies on these lines: {3,2053}, {1428,8300}, {2319,8301}
X(8849) lies on these lines: {21,99}, {39,404}, {42,81}, {58,2665}, {86,8299}, {1621,4418}, {2106,2108}, {2669,3286}, {3786,4712}, {4188,7766}
X(8850) lies on these lines: {43,57}, {56,664}, {100,1403}, {350,1281}, {659,3808}, {1423,2108}, {1428,8300}, {1429,8298}, {1432,1929}, {1463,1575}, {2082,2275}
X(8851) lies on these lines: {8,2053}, {21,7257}, {25,1897}, {31,43}, {41,644}, {55,3699}, {56,664}, {643,2194}, {904,3903}, {3253,8299}, {6186,6742}
X(8851) = isogonal conjugate of X(1463)
X(8852) lies on the cubic K432 and these lines: {3,2053}, {21,6626}, {25,1403}, {31,1469}, {41,2276}, {55,846}, {56,7204}, {75,1281}, {105,1284}, {244,6186}, {256,8300}, {291,1580}, {884,4455}, {1631,7241}, {1755,2076}, {2194,3736}, {4471,4492}
X(8852) = isogonal conjugate of X(4645)
X(8852) = perspector of ABC and cross-triangle of 1st and 2nd Sharygin triangles
X(8853) lies on these lines: {105,1929}, {291,1580}, {1281,8847}, {2108,2112}
X(8854) lies on the cubic K424a and these lines: {1,7347}, {2,372}, {5,8281}, {6,1196}, {22,6200}, {23,6453}, {25,371}, {427,6564}, {486,7392}, {590,6676}, {1368,3070}, {1588,7398}, {1611,6423}, {1707,6204}, {1995,6419}, {3068,5413}, {5268,5414}, {5272,6502}, {5418,7494}, {6396,7484}, {6459,7714}, {6560,7386}, {6561,6995}, {6565,6997}, {6677,7583}
X(8854) = {X(6),X(5020)}-harmonic conjugate of X(8855)
X(8855) lies on the cubic K424b and these lines: {1,7348}, {2,371}, {5,8280}, {6,1196}, {22,6396}, {23,6454}, {25,372}, {427,6565}, {485,7392}, {615,6676}, {1368,3071}, {1587,7398}, {1611,6424}, {1707,6203}, {1995,6420}, {2066,5268}, {2067,5272}, {3069,5412}, {5420,7494}, {6200,7484}, {6460,7714}, {6560,6995}, {6561,7386}, {6564,6997}, {6677,7584}
X(8855) = {X(6),X(5020)}-harmonic conjugate of X(8854)
Loci associated with selected triangles (2): X(8856)-X(8950)
This preamble and centers X(8856)-X(8950) were contributed by César Eliud Lozada, December 11, 2015.
Let ABC be a triangle and let P1 = p1 : q1 : r1, P2 = p2 : q2 : r2, P3 = p3 : q3 : r3 (barycentrics) be three non-collinear points, none on a sideline of ABC. The locus of a point M such that the anticevian triangle of M and the triangle T = P1P2P3 are perspective, with perspector denoted by Z(M), is given by
F(x,y,z) = (r3δ32y2-δ23q2z2)x+(δ31r3x2-δ13p1z2)y+(δ21q2x2-δ12p1y2)z+(p2q3r1-p3q1r2)xyz = 0 (1)
where δij is the (i, j)-minor of the vertex matrix of the triangle T.
If T and ABC are perspective we have p2q3r1 - p3q1r2 = 0 and, if the equation (1) is non-degenerate, it represents a pivotal cubic with pole Ω(T) and pivot P(T) given by:
Ω(T) = p1δ12δ13 : q2δ21δ13 : r3δ31δ12
P(T) = δ13δ32 : -δ13δ31 : δ31δ12
This section deals with the poles and pivots of these cubics and the perspectors Z(M) for most of the centers M lying on them. In the following table, each column contains:
| Triangle X | Locus of M for perspectivity of triangles X and anticevian-of-M | Gibert's catalogue | Pole* | Pivot* | ( M,Z(M) )** | Locus of Z(M) |
|---|---|---|---|---|---|---|
| 1st anti-Brocard | Σ[(a^4-b^2*c^2)*(c^2*y-b^2*z)*y*z] | -- | 385 | 6 | (2,147), (6,5989), (83,8856), (194,8782), (239,1281), (287,98), (385,8290), (732,5976), (894,8857), (1916,1916), (3225,8858), (3978,8783), (4027,8784) | K699 |
| anti-McCay | Too long | -- | 8859 | 8860 | (2,8591), (8587,8587) | |
| 1st Brocard | Σ[(a^2-b*c)*(a^2+b*c)*x*(c^2*y^2-b^2*z^2)] | K128 | 6 | 385 | (1,3496), (2,2896), (6,3), (32,3492), (76,76), (98,8861), (385,384), (511,39), (694,3493), (1423,8862), (2319,3494), (3186,8863), (3225,8864), (3229,3491), (3504,3224), (3505,695), (3506,32), (3507,3501), (3508,8865), (3509,1), (3510,8866), (3511,3499), (3512,8867), (7166,{32,3502})∩{194,3497}), (7167,{4,6196}∩{32,3503}), (7168,8868) | K020 |
| 2nd Brocard | Σ[(3*SA-SW)*(SC*c^2*y^2-SB*b^2*z^2)*x] | K534 | 3 | 524 | (6,6), (69,8788), (524,141), (895,8869), (2930,2916) | K538 |
| 3rd Brocard | Σ[(a^2-b*c)*(a^2+b*c)*x*(c^2*y^2-b^2*z^2)] | K128 | 6 | 385 | (1,6196), (2,194), (6,3499), (32,32), (76,8790), (98,8870), (385,384), (511,3491), (694,8871), (1423,3503), (2319,8872), (3186,4), (3225,8873), (3229,39), (3504,{76,3504}∩{695,3224}), (3505,8874), (3506,3492), (3507,8865), (3508,3501), (3509,8866), (3510,1), (3511,3), (3512,8875), (7166,8876), (7167,{3,7346}∩{76,3495}), (7168,{76,7168}∩{3224,3495}) | K020 |
| 4th Brocard | Σ[(3*SA-SW)*(SB*c^2*y^2-SC*b^2*z^2)*x] | K535 | 25 | 468 | (2,2), (25,8792), (111,8877), (468,427), (7665,8878) | K539 |
| circummedial | Σ[((a^2+c^2)*c^2*y^2-(a^2+b^2)*b^2*z^2)*x] | K644 | 251 | 83 | (2,2), (4,8879), (6,22), (83,1799), (251,8793), (1176,251), (1342,8880), (1343,8881) | |
| circumorthic | Σ[SA*(SB*SC+S^2)*(c^2*y-b^2*z)*x^2] | -- | 8882 | 275 | (2,6193), (4,4), (6,24), (54,8883), (275,8884), (1993,3) | |
| 2nd circumperp | Σ[b*c*(b+c)*(c^2*y-b^2*z)*x^2] | K319 | 1333 | 81 | (1,1), (6,3), (57,1394), (58,2360), (81,21), (222,56), (284,58), (1172,8885), (1433,8886), (3194,28) | K318 |
| Euler | Σ[SB*SC*(SB*SC+S^2)*(y-z)*y*z] | K671 | 53 | 2 | (2,3091), (4,4), (5,8799), (53,8887), (216,5), (1249,8888), (2052,{5,2052}∩{1093,3574}) | |
| 2nd Euler | Σ[a^2*SA*(SB*SC+S^2)*((2*R^2-SC)*y-(2*R^2-SB)*z)*y*z] | -- | 216 | 6515 | (6,3), (52,5), (68,68), (343,5562), (6515,4) | |
| 5th Euler | Σ[(b^2+c^2)*(SB*y-SC*z)*y*z] | K517 | 427 | 4 | (2,2), (4,8889), (76,8890), (141,8891), (193,8892), (427,8893), (1843,427) | |
| extangents | Σ[a^3*(b+c)*(c*y-b*z)*y*z] | K362 | 213 | 1 | (1,40), (6,55), (33,19), (37,3198), (42,71), (55,3197), (65,65), (73,8803), (2331,8802) | pK(213,40) |
| 2nd extouch | Σ[a*(b+c)*((s-c)*y-(s-b)*z)*y*z] | K033 | 37 | 8 | (1,9), (4,4), (8,329), (10,72), (40,1490), (65,226), (72,8804), (3176,8894), (5930,8807) | pK(37,329) |
| 3rd extouch | Σ[a*(b+c)*(s-b)*(s-c)*((s-b)*y-(s-c)*z)*y*z] | -- | 1427 | 7 | (1,3182), (4,4), (7,5932), (57,223), (65,5930), (196,{196,1439}∩{223,1249}), (226,8807), (1439,--), (3668,1439) | |
| 4rd extouch | Σ[a*(b+c)*SA*((s-b)*y-(s-c)*z)*y*z] | -- | 1214 | 7 | (7,5933), (63,8895), (65,8815), (69,69), (72,8896), (224,8897), (226,65) | |
| 5th extouch | Σ[a*(b+c)*(s-b)*(s-c)*(b*c+SW)*((s-b)*y-(s-c)*z)*y*z] | -- | 8898 | 7 | (7,8), (33,8899), (65,5930), (226,65), (388,388), (2285,8900) | |
| Feuerbach | Σ[(b^2-c^2)^2*y*z*(c*y-b*z)] | K672 | 115 | 1 | (1,5), (11,8819), (12,12), (523,11), (1109,523), (2588,1313), (2589,1312), (2616,8901), (2618,8902) | pK(115,5) |
| outer-Garcia | Σ[a*(b+c)*y*z*(y-z)] | K345 | 37 | 2 | (1,40), (2,8), (9,1), (10,10), (37,72), (226,5930), (281,3176), (1214,65), (7952,4) | K033 |
| inner-Grebe | Σ[a^2*(y-z)*(b^2+c^2-S)*y*z] | -- | 6421 | 2 | (2,1271), (3,8903), (6,6), (3128,1163), (5591,--), (6421,8820) | |
| outer-Grebe | Σ[a^2*(y-z)*(b^2+c^2+S)*y*z] | -- | 6422 | 2 | (2,1270), (3,8904), (6,6), (3127,1162), (5590,--), (6422,8821) | |
| hexyl | Σ[a^2*(SC*c*y-SB*b*z)*y*z] | K343 | 6 | 63 | (1,3), (9,40), (19,4), (40,1490), (57,1), (63,20), (84,84), (610,1498), (1712,3183), (2184,64) | K004 |
| Johnson | Σ[a^2*SA*(SB*SC+S^2)*(y-z)*y*z] | K612 | 216 | 2 | (2,4), (3,155), (5,5), (6,3), (216,52), (343,8905), (2165,8906) | K044 |
| Kosnita | Σ[a^4*(SA^2-S^2)*((SC^2-3*S^2)*c^2*y-(SB^2-3*S^2)*b^2*z)*y*z] | -- | 571 | 1994 | (6,3), (52,8824), (54,54), (195,2917), (1993,8907), (1994,7488), (2904,24) | K388 |
| Lucas central | Σ[a^4*SA*(SA+4*S)*y*z*(c^2*y-b^2*z)] | -- | 8908 | 6 | (3,3), (6,371), (3167,8909), (3311,8826), (5406,8910) | |
| Lucas tangents | Σ[a^4*SA*(SA+2*S)*y*z*(c^2*y-b^2*z)] | 8911 | 6 | (3,8912), (6,1151), (371,371), (3167,3), (5408,8913) | ||
| McCay | Σ[a^4*SA*(SA+S)*(c^2*y-b^2*z)*y*z] | -- | 6 | 8859 | (1,7609), (2,7616), (6,3), (187,575), (576,--), (671,8914), (7607,7607) | |
| midheight | Σ[(2*SB*SC-S^2)*x*(c^2*y^2-b^2*z^2)] | K004 | 6 | 20 | (1,57), (3,6), (4,4), (20,2), (40,1), (64,1073), (84,282), (1490,9), (1498,3), (2130,3343), (2131,3356), (3182,223), (3183,1249), (3345,3342), (3346,3344), (3347,3352), (3348,3349), (3353,3341), (3354,--), (3355,3350), (3472,3351), (3473,--), (3637,--) | K002 |
| mixtilinear | Σ[a^2*(s-a)*(s*(s-a)-b*c+2*R*r)*((s-b)*c*y-(s-c)*b*z)*y*z] | -- | 198 | 57 | (1,1), (40,--), (57,1697), (1743,8915), (2324,--) | |
| 3rd mixtilinear | Σ[b*c*(s-a)*(c^2*y-b^2*z)*x^2] | -- | 604 | 57 | (1,7963),6,1616), (56,56),57,1420) | |
| 4th mixtilinear | Σ[a^3*(c*y-b*z)*y*z] | -- | 31 | 1 | (1,165), (6,1615), (55,55), (57,8830), (365,--), (1419,57), (2067,8831), (6502,8833), (7133,--) | |
| 5th mixtilinear | Σ[a*(s-a)*(y-z)*y*z] | -- | 9 | 2 | (1,1), (2,145), (8,8834), (9,2136), (3161,8), (8056,0) | K201 |
| 6th mixtilinear | Σ[a*(s-a)*x*(c^2*y^2-b^2*z^2)] | K351 | 6 | 9 | (1,165), (9,2951), (57,1), (165,8835), (364,--), (2124,8916), (2125,8917), (3062,3062), (6212,--), (6213,0) | |
| inner-Napoleon | Σ[(sqrt(3)*a^2-2*S)*x*(c^2*y^2-b^2*z^2)] | K129a | 6 | 395 | (1,6192), (2,627), (6,3), (14,8918), (16,61), (18,18), (62,8837), (395,5), (1653,3468), (6151,{4,617}∩{62,110}), (7088,{4,8436}∩{62,6191}) | K005 |
| outer-Napoleon | Σ[(sqrt(3)*a^2+2*S)*x*(c^2*y^2-b^2*z^2)] | K129b | 6 | 396 | (1,6191), (2,628), (6,3), (13,8919), (15,62), (17,17), (61,8839), (396,5), (1652,3468), (2981,{4,616}∩{61,110}), (7089,{4,8435}∩{61,6192}) | K005 |
| 1st Neuberg | Σ[(a^2-b*c)*(a^2+b*c)*x*(c^2*y^2-b^2*z^2)] | K128 | 6 | 385 | (1,6210), (2,6194), (6,3), (32,182), (76,8920), (98,98), (385,5999), (511,--), (694,6234), (1423,1), (2319,{3,8872}∩{4,3494})), (3186,4), (3225,8921), (3229,511), (3504,{4,3224}∩{511,3504}), (3505,{4,695}∩{511,3505}), (3506,8922), (3507,6211), (3508,{4,8865}∩{511,3508}), (3509,8923), (3510,8924), (3511,8925), (3512,8926), (7166,{3,8876}∩{147,7350}), (7167,8927), (7168,{4,8872}∩{511,7168}) | K422 |
| 2nd Neuberg | Σ[(a^4+2*a^2*(b^2+c^2)+b^2*c^2)*x*(c^2*y^2-b^2*z^2)] | K423 | 6 | 3329 | (1,6211), (2,147), (6,3), (39,511), (83,8928), (182,8922), (262,262), (3329,5999) | K422 |
| orthocentroidal | Σ[(S^2-3*SB*SC)*x*(c^2*y^2-b^2*z^2)] | K001 | 6 | 30 | ((1,3336), (3,195), (4,4), (13,8929), (14,8930), (15,62), (16,61), (30,5), (74,3470), (399,3), (484,1), (616,628), (617,627), (1138,--), (1157,54), (1263,--), (1276,--), (1277,--), (1337,--), (1338,0), (2132,--), (2133,--), (3065,--), (3440,--), (3441,--), (3464,3468), (3465,3460), (3466,--), (3479,--)) (Shortened list) | K005 |
| reflection | Σ[(SB*SC+S^2)*x*(c^2*y^2-b^2*z^2)] | K005 | 6 | 5 | (1,484), (3,399), (4,4), (5,30), (17,8172), (18,8173), (54,1157), (61,16), (62,15), (195,3), (627,617), (628,616), (2120,3484), (2121,{3,2121}∩{174,2298}), (3336,1) | K001 |
| 1st Sharygin | Σ[a^2*(a^2-b*c)*(c^2*y-b^2*z)*y*z] | K673 | 1914 | 6 | (1,846), (6,8424), (43,8931), (81,21), (238,8932), (239,1281), (256,256), (291,8933), (294,{105,1284}∩{294,1580}), (1580,8847), (2068,{364,8245}∩{366,8424}), (2069,--), (2238,8846), (2665,8934), (8300,1580) | pK(1914,8424) |
| 2nd Sharygin | Σ[a^2*(a^2-b*c)*(c^2*y-b^2*z)*y*z] | K673 | 1914 | 6 | (1,1282), (6,8301), (43,2108), (81,8935), (238,8298), (239,1281), (256,8936), (291,291), (294,105), (1580,8300), (2068,--), (2069,--), (2238,8299), (2665,8937), (8300,8853) | pK(1914,8301) |
| inner-squares | Σ[SB*SC*x*(c^2*y^2-b^2*z^2)] | K006 | 6 | 4 | (1,6204), (3,6), (4,3068), (46,8938), (90,{2,914}∩{90,371}), (155,8939), (254,{2,6503}∩{254,371}), (371,8854), (372,371), (485,485), (486,8940), (487,2), (488,6462), (6212,1), (6213,8941) | K424a |
| outer-squares | Σ[SB*SC*x*(c^2*y^2-b^2*z^2)] | K006 | 6 | 4 | (1,6203), (3,6), (4,3069), (46,8942), (90,{2,914}∩{90,372}), (155,8943), (254,0), (371,372), (372,0), (485,8944), (486,486), (487,6463), (488,2), (6212,8945), (6213,1) | K424b |
| inner-Vecten | Σ[(SB+SC-S)*x*(c^2*y^2-b^2*z^2)] | K424b | 6 | 3069 | (1,6213), (2,488), (6,3), (372,371), (486,486), (494,8946), (3069,4), (6203,1), (6463,487), (7348,8947) | K006 |
| outer-Vecten | Σ[(SB+SC+S)*x*(c^2*y^2-b^2*z^2)] | K424a | 6 | 3068 | (1,6212), (2,487), (6,3), (371,372), (485,485), (493,8948), (3068,4), (6204,1), (6462,488), (7347,8949) | K006 |
*: The appearance of i in columns 4 and 5 means that X(i) is the pole or pivot of the cubic.
**: The appearance of (i,j) in this column means that X(j) is the perspector of T and the cevian-triangle-of-X(i). Some perspectors having very long coordinates are either not calculated (--) or are represented as intersections of lines.
X(8856) lies on the cubic K699 and these lines: {83,115}, {1799,5976}, {3112,7061}
X(8857) lies on the cubic K699 and these lines: {894,4027}, {3509,8782}
X(8858) lies on the cubic K699 and these lines: {25,5989}, {32,99}, {184,4563}, {305,3504}, {1402,4554}, {2200,4561}, {2353,5152}
X(8858) = isogonal conjugate of the complement of X(32529)
X(8858) = Neuberg-circles-radical circle-inverse of X(699)e
Let Ab and Ac be the intersections of CA and AB, respectively, with the parallel to BC through X(2). Define Bc and Ca cyclically, and define Ba and Cb cyclically. The points Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse, E, centered at X(2). X(8859) = inverse-in-E of X(6). (Randy Hutson, December 26, 2015)
Let P* and U* be as described at X(8588). Then X(8859) = crossdifference of P* and U*. (Randy Hutson, December 26, 2015)
X(8859) lies on these lines: {2,6}, {32,598}, {39,1153}, {115,8597}, {140,7616}, {148,8598}, {187,671}, {316,5461}, {538,5215}, {543,5152}, {576,7607}, {1078,7817}, {2030,8593}, {2896,8360}, {3767,7833}, {3793,8355}, {5277,7621}, {5309,5569}, {5999,6055}, {6179,7775}, {7618,7783}, {7619,7749}, {7622,7757}, {7746,7812}, {7751,7870}, {7755,7824}, {7780,7883}, {7793,7841}, {7797,8359}, {7801,7857}, {7810,7828}, {7886,7939}
X(8859) = pivot of the cubic-locus-for perspectivity of these triangles: McCay and anticevian-of-P
X(8859) = trilinear pole of perspectrix of ABC and McCay triangle
X(8860) lies on these lines: {2,6}, {3,671}, {115,5569}, {187,7617}, {574,1153}, {598,1384}, {1656,7812}, {1975,2482}, {3734,5215}, {3934,8366}, {5070,6179}, {5077,7771}, {5461,7746}, {7615,8598}, {7810,7887}, {7851,8359}, {8182,8352}
X(8860) = reflection of X(i) in X(j) for these (i,j): (2,3054)
X(8860) = trilinear pole of perspectrix of ABC and anti-McCay triangle
X(8861) lies on the cubic K020 and these lines: {4,32}, {194,287}, {2966,7750}, {3404,3497}
X(8862) lies on the cubic K020 and these lines: {1,4}, {32,1423}, {39,3497}, {695,3503}
X(8863) lies on the cubic K020 and these lines: {4,695}, {19,3497}, {32,3186}
X(8864) lies on the cubic K020 and these lines: {32,99}, {3502,6196}
X(8865) lies on the cubic K020 and these lines: {1,83}, {3,984}, {32,983}, {76,3502}, {194,869}, {695,3501}, {3496,3954}
X(8865) = perspector of these triangles: 3rd Brocard and anticevian of X(3507)
X(8866) lies on the cubic K020 and these lines: {1,695}, {4,3495}, {32,3510}, {76,3497}, {83,3500}, {3224,3502}
X(8866) = perspector of these triangles: 3rd Brocard and anticevian of X(3509)
X(8867) lies on the cubic K020 and these lines: {4,3502}, {32,3497}
X(8868) lies on the cubic K020 and these lines: {1,8871}, {4,8872}, {32,6196}, {39,8876}, {194,3212}, {384,8875}, {3497,8790}, {3499,3502}, {3500,8873}, {3501,8870}
X(8868) = isogonal conjugate of X(8875)
X(8869) lies on the cubic K538 and these lines: {69,125}, {691,1843}, {6091,8588}
X(8870) lies on the cubic K020 and these lines: {{3,76}, {83,3493}, {695,8861}, {3404,3495}, {3501,8868}, {5967,7787}, {7346,8862}
X(8871) lies on the cubic K020 and these lines: {1,8868}, {3,3224}, {76,694}, {83,8864}, {695,3493}, {3494,3495}, {3496,7346}
X(8872) lies on these lines: {1,3224}, {39,7346}, {76,2319}, {695,3494}, {3497,8864}
X(8873) lies on these lines: {76,3224}, {695,8864}, {3494,7346}
X(8874) lies on the cubic K020 and these lines: {76,3505}, {3497,7346}
X(8875) lies on the cubic K020 and these lines: {1,3493}, {3,2053}, {76,3496}, {384,8868}, {695,8867}, {2896,3495}, {3224,8862}, {3492,7346}, {3500,8861}, {3501,8864}
X(8876) lies on these lines: {1,8864}, {76,3494}, {2896,7346}, {3224,3496}, {3493,3500}
X(8877) lies on the cubic K539 and these lines: {25,111}, {691,6636}, {892,8024}, {2514,3108}, {5968,7485}
X(8878) lies on the cubic K539 and these lines: {2,32}, {22,7823}, {25,7665}, {148,8267}, {193,7378}, {194,7391}, {305,7759}, {316,1194}, {385,5133}, {427,7762}, {1031,7779}, {1180,6655}, {1184,7773}, {1196,7843}, {1370,3164}, {3051,5207}, {4159,7737}, {5025,5359}, {5064,7754}, {6392,7409}, {7485,7777}
X(8878) = anticomplement of X(1799)
X(8879) lies on these lines: {2,2138}, {4,251}, {25,393}, {112,1370}, {427,3172}, {1249,7494}, {5523,7500}
X(8880) lies on these lines: {2,1343}, {6,25}, {22,1671}, {251,1342}
X(8880) = {X(25),X(184)}-harmonic conjugate of X(8881)
X(8881) lies on these lines: {2,1342}, {6,25}, {22,1670}, {251,1343}
X(8881) = {X(25),X(184)}-harmonic conjugate of X(8880)
Let P1 and P2 be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. Let Q1 and Q2 be the respective points of tangency. Q1 and Q2 are also the circumcircle intercepts of line X(186)X(523). X(8882) is the barycentric product of Q1 and Q2. See also X(2190). (Randy Hutson, July 20, 2016)
X(8882) lies on these lines: {2,95}, {4,96}, {6,24}, {32,393}, {37,2190}, {50,252}, {53,112}, {111,933}, {186,570}, {216,7488}, {232,251}, {248,8795}, {263,1974}, {276,308}, {1166,3518}, {1400,2148}, {1843,1976}, {2433,2623}, {3147,5063}, {5412,8577}, {5413,8576}
X(8882) = isogonal conjugate of X(343)
X(8882) = isotomic conjugate of X(28706)
X(8882) = polar conjugate of X(311)
X(8882) = trilinear pole of line X(512)X(2623)
X(8882) = X(63)-isoconjugate of X(5)
X(8883) lies on these lines: {3,54}, {4,96}, {50,6146}, {95,315}, {110,3133}, {275,3541}, {427,4994}, {570,1199}, {933,1299}, {1614,3135}
X(8883) = isogonal conjugate of X(8800)
Let A'B'C' be the circumorthic triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(8884). (Randy Hutson, December 26, 2015)
Let {X,Y}-isoconjugate denote the isoconjugation that swaps X and Y, as discussed at X(8795). Let A'B'C' be the circumorthic triangle. Let A* be the {B,C}-isoconjugate, wrt A'B'C', of A, and define B*, C* cyclically. The lines A'A*, B'B*, C'C* concur in X(8884). (Randy Hutson, December 26, 2015)
X(8884) lies on the hyperbola {{A,B,C,X(4),X(93)}} and these lines: {3,95}, {4,54}, {20,97}, {24,96}, {25,1093}, {30,1105}, {32,393}, {52,648}, {53,1970}, {93,186}, {107,1141}, {225,2190}, {228,7412}, {254,8883}, {317,6193}, {324,7488}, {933,1300}, {1166,1179}, {1826,2200}, {3088,8721}, {3089,6526}, {3091,4993}, {3199,6529}, {6530,6756}
X(8884) = isogonal conjugate of X(5562)
X(8884) = polar conjugate of X(343)
X(8884) = reflection of X(i) in X(j) for these (i,j): (4,6750)
X(8884) = cevapoint of X(4) and X(24)
X(8884) = X(3575)-cross conjugate of X(4)
X(8884) = trilinear pole of line X(421)X(2501)
X(8884) = trilinear product of vertices of circumorthic triangle
X(8884) = perspector of circumorthic triangle and cross-triangle of ABC and circumorthic triangle
X(8885) lies on the cubic K318 and these lines: {1,204}, {3,1033}, {21,7149}, {28,56}, {58,84}, {108,5930}, {387,4219}, {648,1792}, {1108,5317}, {1783,3682}, {1858,2906}
X(8886) lies on the cubic K318 and these lines: {1,268}, {3,3341}, {28,1436}, {56,84}, {58,1433}
X(8887) lies on these lines: {4,51}, {5,53}, {133,1906}, {324,5562}, {436,8884}, {1594,6747}, {3091,8796}
X(8887) = polar conjugate of X(37872)
X(8888) lies on these lines: {2,3183}, {4,154}, {5,1249}, {122,631}, {3176,8227}, {3186,6622}, {3462,3545}
As a point on the Euler line, X(8889) has Shinagawa coefficients [2*F, E+F]
X(8889) lies on these lines: {2,3}, {33,5272}, {34,5268}, {69,8541}, {230,3087}, {264,1007}, {275,7612}, {305,1235}, {393,3815}, {612,1870}, {614,6198}, {1119,7179}, {1125,7718}, {1249,7736}, {1398,5261}, {1843,3619}, {1853,6776}, {1876,5226}, {1892,5435}, {3068,8280}, {3069,8281}, {3167,5921}, {3616,5090}, {3634,7713}, {3705,7046}, {3763,3867}, {3917,6403}, {5274,7071}, {6032,8879}, {6666,7717}
X(8889) = polar conjugate of X(5395)
X(8889) = inverse of X(6353) in orthocentroidal circle
X(8889) = perspector of 5th Euler triangle and cross-triangle of ABC and 5th Euler triangle
X(8890) lies on these lines: {76,427}, {305,7764}, {1502,3815}, {6032,8024}
X(8891) lies on these lines: {2,39}, {22,3734}, {141,427}, {251,7804}, {308,1241}, {384,1799}, {626,5133}, {1369,7848}, {1370,7800}, {1627,7780}, {5064,7784}, {5149,5986}, {5169,7849}, {5359,7751}, {6248,7467}, {6636,7816}, {7391,7761}, {7485,7815}, {7499,7789}, {7539,7778}, {7571,7862}, {7768,8878}
X(8891) = isotomic conjugate of X(37876)
X(8891) = complement of X(1180)
X(8892) lies on these lines: {2,3053}, {193,427}, {2996,7378}, {3164,7396}, {3620,8891}, {5139,6995}
X(8893) lies on these lines: {2,1843}, {7786,8889}
X(8894) lies on these lines: {4,1903}, {9,1249}, {72,3176}, {196,226}, {1490,3183}
X(8895) lies on these lines: {63,65}, {69,224}, {1038,6505}
X(8896) lies on these lines: {8,1869}, {10,12}, {19,5739}, {55,4028}, {63,69}, {219,940}, {329,1826}, {440,4047}, {1038,3682}, {1762,4416}, {1848,3869}, {2895,3101}, {5928,8804}
X(8897) lies on these lines: {2,2082}, {3,X(8822)4}, {46,4028}, {57,3169}, {63,69}, {65,224}, {141,2339}, {664,1435}, {940,1100}, {1038,4855}, {1211,3305}, {1400,8815}, {1708,3882}, {1763,3912}, {1999,3212}, {3895,3938}, {6332,6546}
X(8898) lies on these lines: {9,7273}, {37,226}, {56,5336}, {227,2092}, {269,7146}, {279,304}, {1042,2171}, {1254,1400}, {1448,1766}, {1455,5019}, {1465,2277}, {1841,8557}, {1892,4331}, {2285,2286}, {2298,4296}, {2345,7365}
X(8899) lies on these lines: {33,64}, {34,6059}, {281,388}, {607,3213}
X(8900) lies on these lines: {6,19}, {8,1943}, {1455,8192}, {1460,4320}
Let A'B'C' be the Feuerbach triangle. Let La be the tangent to conic {{A',B',C',B,C}} at A', and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(8901). (Randy Hutson, December 26, 2015)
X(8901) lies on these lines: {5,49}, {11,2605}, {96,2986}, {97,1368}, {98,275}, {115,136}, {122,5522}, {125,526}, {235,8884}, {339,2972}, {1595,4994}, {1942,8795}
X(8901) = X(1101)-isoconjugate of X(5)
X(8902) lies on these lines: {11,2618}, {12,1091}, {125,137}, {140,6592}, {338,2972}, {389,546}, {523,8901}, {570,2493}, {2970,7668}, {6036,6676}
X(8903) lies on the Stammler hyperbola and these lines: {3,5591}, {6,6416}, {22,8904}, {25,1163}, {155,1161}, {195,6277}, {399,7725}, {1152,8820}, {1498,6267}
X(8904) lies on the Stammler hyperbola and these lines: {3,5590}, {6,6415}, {22,8903}, {25,1162}, {155,1160}, {195,6276}, {399,7726}, {1151,8821}, {1498,6266}
X(8905) is the point on the Euler central cubic K044 for which the perspector (on the Euler perspector cubic K045) is the isotomic conjugate of X(254). (Randy Hutson, December 26, 2015)
X(8905) lies on the cubic K044 and these lines: {3,68}, {5,8800}, {570,5254}, {4558,8883}, {5562,6751}
X(8905) = X(64) of orthic triangle
X(8905) = orthic isogonal conjugate of X(5562)
X(8905) = X(4)-Ceva conjugate of X(5562)
X(8905) = QA-P17 (Involutary Conjugate of QA-P5) of quadrangle ABCX(4)
X(8906) lies on the cubic K044 and these lines: {3,2165}, {68,5562}, {91,1479}, {847,6816}, {925,3542}, {5392,6643}
X(8907) lies on the cubic K388 and these lines: {3,70}, {22,5562}, {24,52}, {26,110}, {54,6644}, {69,7488}, {182,6467}, {186,6193}, {394,2917}, {1209,7509}
X(8908) lies on these lines: {6,6415}, {32,8825}, {184,418}, {5158,6413}
X(8909) lies on these lines: {3,6413}, {6,1147}, {68,590}, {155,371}, {493,3167}, {1069,2066}, {2067,3157}, {3068,6193}, {3071,5654}, {5409,6503}, {5449,8253}, {6409,7689}
X(8909) = X(371)-Ceva conjugate of X(3)
X(8910) lies on these lines: {3311,5406}, {5408,8909}
X(8911) lies on these lines: {3,6414}, {6,3156}, {24,371}, {48,605}, {96,486}, {184,418}, {372,7592}, {1181,6458}, {1300,6561}, {3364,8837}, {3389,8839}
X(8911) = X(92)-isoconjugate of X(485)
X(8912) lies on these lines: {3,6415}, {371,3167}, {1147,6199}, {6221,8909}
X(8913) lies on these lines: {3,49}, {487,641}
X(8914) lies on these lines: {4,542}
X(8915) lies on these lines: {1,1407}, {3,7963}, {40,1743}, {165,978}, {1046,2941}, {2951,5255}, {2956,5119}
X(8915) = complement of X(35661)
X(8915) = anticomplement of X(35680)
Continuing from the preamble before X(7955), define the 7th mixtilinear triangle as A7B7C7, where A7 = (C''a-A''c)∩(A''b-B''a), and B7 and C7 are defined cyclically. Then A7B7C7 and the excentral triangle are perspective, and X(8916) is the perspector. (Randy Hutson, December 26, 2015)
Trilinears for the A-vertex of the 7th mixtilinear triangle are given by (a-b+c)*(a+b-c)*(a^2-2*(b+c)*a+(b-c)^2)/(2*a*(-a+b+c)) : b^2+(a-c)*(a-2*b+3*c) : c^2+(a-b)*(a-2*c+3*b). (César Lozada, January 8, 2016)
X(8916) lies on these lines: {1,3599}, {57,3062}, {165,2124}
X(8917) lies on these lines: {1,3599}, {55,1419}, {144,200}, {165,220}, {2391,7994}
X(8917) = isogonal conjugate of X(2951)
X(8918) lies on the cubic K005 and these lines: {3,5624}, {4,14}, {17,3470}, {3336,3376}
X(8919) lies on the cubic K005 and these lines: {3,5623}, {4,13}, {18,3470}, {3336,3383}
X(8920) lies on the cubic K422 and these lines: {3,6374}, {4,69}, {98,3222}, {147,305}, {670,1350}, {1502,1503}, {3403,7350}, {3978,6776}
X(8921) lies on these lines: {3,8873}, {4,8864}, {147,8858}, {511,3225}
X(8922) lies on these lines: {1,7351}, {3,3492}, {4,32}, {147,8784}, {511,3506}
X(8923) lies on the cubic K422 and these lines: {1,4}, {3,8866}, {98,3510}, {147,8857}, {182,7350}, {295,511}
X(8924) lies on these lines: {1,3}, {4,3495}, {98,813}, {147,4645}, {182,985}, {262,3402}, {291,511}, {580,1923}, {3403,7350}, {3405,7351}
X(8925) lies on the Stammler hyperbola, the cubic K422 and these lines: {4,3499}, {6,98}, {511,3511}, {1351,6234}, {1740,7350}, {5999,6038}
X(8926) lies on the cubic K422 and these lines: {1,147}, {3,2053}, {4,3502}, {98,7166}, {511,1757}, {7350,8922}
X(8927) lies on the cubic K422 and these lines: {3,7346}, {4,6196}, {98,7168}, {147,7224}, {291,511}, {1740,7350}, {3223,6210}
X(8928) lies on the cubic K422 and these lines: {4,83}, {98,783}, {147,8856}, {1503,4577}, {1799,6194}, {3405,7351}
X(8929) lies on the cubic K005 and these lines: {3,13}, {4,5623}, {54,8919}, {61,8014}, {62,3471}, {3383,3467}, {3459,8839}, {3489,8918}
X(8930) lies on the cubic K005 and these lines: {3,14}, {4,5624}, {54,8918}, {61,3471}, {62,8015}, {3376,3467}, {3459,8837}, {3490,8919}
X(8931) lies on these lines: {1,256}, {3,2053}, {8,1281}, {20,2944}, {41,43}, {165,3501}, {846,1334}, {2329,8424}, {3208,8844}
X(8932) lies on these lines: {9,8245}, {55,846}, {71,1633}, {220,8931}, {238,1284}, {256,294}, {3684,8844}, {4220,8012}
X(8932) = perspector of the 1st Sharygin triangle and the cevian triangle of X(9)
X(8933) lies on these lines: {291,1580}, {1284,5018}, {3509,4447}
X(8934) lies on these lines: {1438,1580}
X(8935) lies on these lines: {21,6626}, {81,244}, {1014,8850}, {1931,2108}
X(8936) lies on these lines: {256,8300}, {1432,1929}
X(8937) lies on these lines: {58,2665}, {893,8299}, {2107,2108}, {2248,8301}
X(8937) = 2nd-Lemoine-circle-inverse of X(34575)
X(8938) lies on the cubic K424a and these lines: {1,485}, {2,914}, {6,6203}, {46,371}, {493,1880}, {1773,2067}, {2066,2961}
Let La be the line tangent to the A-Lucas circle at the antipode of A. Define Lb and Lc cyclically. Let Ta = Lb∩Lc, and define Tb and Tc cyclically. The triangle TaTbTc is homothetic to the tangential triangle, and homothetic center is X(8939). (Randy Hutson, December 26, 2015)
X(8939) lies on the Stammler hyperbola, the cubic K424a and these lines: {2,6503}, {3,485}, {6,493}, {155,371}, {159,3155}, {195,3311}, {399,6221}, {1151,1498}, {1584,8253}, {1599,1609}, {2164,7347}, {2178,6204}, {6219,8904}
X(8940) lies on the cubic K424a and these lines: {2,371}, {1322,6459}, {3053,8944}, {3378,7347}
X(8941) lies on the cubic K424a and these lines: {1,2}, {371,1707}, {493,1711}, {982,3641}, {1376,7969}, {1659,1738}
X(8941) = {X(1),X(43)}-harmonic conjugate of X(8945)
X(8942) lies on the cubic K424b and these lines: {1,486}, {2,914}, {6,6204}, {46,372}, {494,1880}, {1773,6502}, {2961,5414}
Let La be the line tangent to the A-Lucas(-1) circle at the antipode of A. Define Lb and Lc cyclically. Let Ta = Lb∩Lc, and define Tb and Tc cyclically. The triangle TaTbTc is homothetic to the tangential triangle, and homothetic center is X(8943). (Randy Hutson, December 26, 2015)
X(8943) lies on the Stammler hyperbola, the cubic K424b and these lines: {2,6503}, {3,486}, {6,494}, {155,372}, {159,3156}, {195,3312}, {399,6398}, {1152,1498}, {1583,8252}, {1600,1609}, {2164,7348}, {2178,6203}, {6220,8903}
X(8944) lies on the cubic K424b and these lines: {2,372}, {1321,6460}, {3053,8940}, {3377,7348}
X(8945) lies on the cubic K424b and these lines: {1,2}, {372,1707}, {494,1711}, {982,3640}, {1376,7968}
X(8945) = {X(1),X(43)}-harmonic conjugate of X(8941)
Let A'B'C' be the Lucas(-1) tangents triangle. Let A" be the trilinear pole of B'C', and define B" and C" cyclically; A"B"C" is perspective to ABC at X(6). Let A* be the trilinear pole of B"C", and define B* and C* cyclically; A*B*C* is perpsective to ABC at X(494). Let A''' be the trilinear pole of B*C*, and define B''' and C''' cyclically. A'''B'''C''' is perspective to ABC at X(8946). (Randy Hutson, December 26, 2015)
The trilinear polar of X(8946) passes through X(2489). (Randy Hutson, December 26, 2015)
X(8946) lies on the cubics K006 and K171 and these lines: {3,6406}, {4,487}, {25,372}, {155,1351}, {485,8940}, {1307,3563}, {1824,6212}, {2207,5412}
X(8946) = isogonal conjugate of X(487)
X(8946) = {X(1598),X(1843)}-harmonic conjugate of X(8948)
X(8946) = X(3)-cross conjugate of X(8948)
X(8946) = X(3053)-vertex conjugate of X(8948)
X(8947) lies on the cubic K006 and these lines: {1,487}, {4,1716}, {372,1707}, {485,8941}
X(8947) = isogonal conjugate of X(8949)
X(8947) = {X(1721),X(1722)}-harmonic conjugate of X(8949)
Let A'B'C' be the Lucas tangents triangle; let A" be the trilinear pole of B'C', and define B" and C" cyclically; A"B"C" is perspective to ABC at X(6). Let A* be the trilinear pole of B"C", and define B* and C* cyclically; A*B*C* is perpsective to ABC at X(493). Let A''' be the trilinear pole of B*C*; define B''' and C''' cyclically; A'''B'''C''' is perspective to ABC at X(8948). (Randy Hutson, December 26, 2015)
The trilinear polar of X(8948) passes through X(2489). (Randy Hutson, December 26, 2015)
X(8948) lies on the cubics K006 and K171 and these lines: {3,6291}, {4,488}, {25,371}, {46,8947}, {155,1351}, {486,8944}, {1306,3563}, {1824,6213}, {2207,5413}, {5200,6524}
X(8948) = isogonal conjugate of X(488)
X(8948) = {X(1598),X(1843)}-harmonic conjugate of X(8946)
X(8948) = X(3)-cross conjugate of X(8946)
X(8948) = X(3053)-vertex conjugate of X(8946)
X(8949) lies on the cubic K006 and these lines: {1,488}, {4,1716}, {371,1707}, {486,8945}
X(8949) = isogonal conjugate of X(8947)
X(8949) = {X(1721),X(1722)}-harmonic conjugate of X(8947)
X(8950) lies on these lines: {6,8948}, {32,8911}, {493,3167}
(Contributed by Peter Moses, December 10, 2015) As in the construction of X(5423), let A' be the point in which the A-excircle is tangent to the circle OA that passes through vertices B and C, and define OB and OC cyclically. Let UA be the outer Apollonian circle of OA, OB, OC, and let A'' be the touch-point of OA and UA. Define B'' and C'' cyclically. The center of the circumcircle of A''B''C'' is X(8951). The circumcircle of A''B''C'' tangent to the Apollonius circle of the excircles at X(3030). The vertex A'' is given by
A'' = 4 a^2 (a^2-6 a b+b^2-c^2) (a^2-b^2-6 a c+c^2) : (-a-b-c) (a^2-6 a b+b^2-c^2) (a^3-9 a^2 b-a b^2+b^3+3 a^2 c-2 a b c-b^2 c+3 a c^2-b c^2+c^3) : (-a-b-c) (a^2-b^2-6 a c+c^2) (a^3+3 a^2 b+3 a b^2+b^3-9 a^2 c-2 a b c-b^2 c-a c^2-b c^2+c^3) (barycentrics)
X(8951) lies on these lines: {1,210}, {3,3973}, {9,4255}, {44,5438}, {58,936}, {200,3915}, {386,3731}, {582,5720}, {970,3030}, {978,5223}, {1757,3361}, {2176,7323}, {2999,3876}, {3445,6762}, {3677,4005}, {5529,7987}
(Contributed by Peter Moses, December 10, 2015) As in the construction of X(5423), let A' be the point in which the A-excircle is tangent to the circle OA that passes through vertices B and C, and define OB and OC cyclically. Let VA be the inner Apollonian circle of OA, OB, OC, and let A* be the touch-point of OA and VA. Define B'' and C'' cyclically. The center of the circumcircle of A*B*C* is X(8952). The vertex A* is given by
A* = 2 a^2 (a+b-c) (a-b+c) (a^2-6 a b+b^2-c^2) (a^2-b^2-6 a c+c^2) (a^2+2 a b+b^2-2 a c+2 b c+c^2) (a^2-2 a b+b^2+2 a c+2 b c+c^2)
: (a+b-c) (a^2-6 a b+b^2-c^2) (a^2+2 a b+b^2+2 a c-2 b c+c^2) (a^2-2 a b+b^2+2 a c+2 b c+c^2) (a^5-5 a^4 b+4 a^3 b^2+4 a^2 b^3-5 a b^4+b^5-a^4 c+2 a^3 b c-26 a^2 b^2 c+10 a b^3 c-b^4 c-2 a^3 c^2-4 a^2 b c^2+4 a b^2 c^2-2 b^3 c^2+2 a^2 c^3-10 a b c^3+2 b^2 c^3+a c^4+b c^4-c^5)
: (a-b+c) (a^2-b^2-6 a c+c^2) (a^2+2 a b+b^2+2 a c-2 b c+c^2) (a^2+2 a b+b^2-2 a c+2 b c+c^2) (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-5 a^4 c+2 a^3 b c-4 a^2 b^2 c-10 a b^3 c+b^4 c+4 a^3 c^2-26 a^2 b c^2+4 a b^2 c^2+2 b^3 c^2+4 a^2 c^3+10 a b c^3-2 b^2 c^3-5 a c^4-b c^4+c^5) (barycentrics)
X(8952) lies on these lines: {40,1000}, {58,936}
Centers associated with quadsquare triangles: X(8953)-X(8998)
This preamble and centers X(8953)-X(8998) were contributed by César Eliud Lozada, December 21, 2015.
"In every quadrilateral may be inscribed at least one square having a vertex on each of the four sides. If there is more than one such square in a given quadrilateral, there is an infinite number". Reference: Hebbert, C. M., Annals of Mathematics, Second Series, Vol. 16, No. 1/4 (1914 - 1915), pp. 38-42.
Let P be a point in the plane Π of ABC and let ϒA = ACA'BA'CAB be a square inscribed in the quadrilateral ABPC, with AC on line AB, A'B on BP, A'C on PC, and AB on CA. It is easy to prove, using homogeneous coordinates, that ϒA is unique if P does not lie on the conic κA given by the trilinear equation
Sa2u2 + bc(a2 + S)vw + ca(b2 + S)wu + ab(c2 + S)uv = 0
Note that B and C lie on κA. Define κB and κC cyclically. If
P ∈ Π - κA∪κB∪κC (1),
then there exist unique squares ϒA, ϒB, ϒC inscribed in the quadrilaterals ABPC, BCPA, CAPB, respectively. If P = u : v : w (trilinears) satisfies (1), then the vertices of ϒA are as shown here:
AB = (au + bv + cw)uS + (abv + SCu)cw : 0 : bvwS + (bcv + SAw)au
AC = (au + bv + cw)uS + (acw + SBu)bv : cvwS + (bcw + SAv)au : 0
A'B = u(auS + bSBv) : (S + SA)auv + abcuw + (S + SC)cvw : w(auS + bSBv)
A'C = u(auS + cSCw) : v(auS + cSCw) : abcuv + (S + SA)auw + (S + SB)bvw
If P lies on the side BC of ABC then ϒA is the A-inner-inscribed square of ABC, as defined at Mathworld: Inner Inscribed Squares Triangle. The square ϒA is here named here the A-quadsquare triangle of P. The sidelength of ϒA is given by
LA = S*Sqrt(abc(2uvw(aSAu+S(bv + cw)) + bc(v^2 w^2 + u^2 w^2 + u^2 v^2)a))/|(au + cw)(au + bv)S + (avw + buw + cuv)abc|
Let OA, OB, OC be the centers of the squares ϒA, ϒB, ϒC, respectively. The triangle OAOBOC is here named the P-quadsquares triangle. The vertex OA are given by
OA = a(bcvw + 2Su^2) + (S + SB)buv + (S + SC)cuw : abcuw + (S + SA)auv + (S + SC)cvw : abcuv + (S + SA)auw + (S + SB)bvw,
and cyclically for OB and OC.
The P-quadsquares triangle is perspective to ABC for all P satisyfing (1). The perspector is given by
Z(P) = u((S + SB)bv + (S+SC)cw) + abcvw : :
The transformation P → Z(P) maps lines onto conics. For P on the Euler line, the locus of Z(P) is the conic through X(6), X(485) and the vertices of the three inner-inscribed-squares triangle.
The appearance of (i, j) in the following list means that X(j) is the perspector of the X(i)-quadsquares triangle and ABC:
(1, 8953), (2, 3068), (3, 8954), (4, 485), (5, 8955), (6, 8956), (7, 1659), (8, 8957), (9, 8958), (10, 8959), (253, 3535), (485, 8960), (486, 371), (492, 493), (1132, 4), (1328, 6564), (1586, 6), (3069, 8940), (7090, 1)
If P lies on K070a (Shoemaker cubic), then the P-quadsquares triangle is perspective to the medial triangle and the inner-inscribed-squares triangle. In the next list, the appearance of (i,j) means that X(j) is the perspector of the X(i)-quadsquares triangle and the medial triangle: (2, 590)(homothetic), (4, 5), (486, 8961), (492, 8962), (1586, 8963), (3069, 8964), (7090, 8965). For the inner-inscribed-squares triangle: (2, 8966), (4,485), (486, 8967), (492,8968), (1586,8969), (3069, 8970), (7090, 8971).
Next, the appearance of (T, i, j) means that X(j) is the perspector of T and the X(i)-quadsquares triangle:
(anticomplementary, 2, 8972)
(circumorthic,1132, 4)
(2nd Brocard, 1586, 6)
(2nd circumperp, 7090, 1)
(circumsymmedial, 1, 8973)
(circumsymmedial, 1586, 6)
(Euler, 2, 371)
(Euler, 4, 5)
(Euler, 1132, 4)
(excentral, 7090, 1)
(inner Grebe, 2, 8974)
(inner Grebe, 1586, 6)
(outer Grebe, 2, 8975)
(outer Grebe, 1586, 6)
(incentral, 7090, 1)
(Johnson, 2, 8976)
(midheight, 1132, 4)
(mixtilinear, 7090, 1)
(orthic, 1132, 4)
(orthocentroidal, 1132, 4)
(reflection, 1132, 4)
(outer-squares, 7090, 8977)
(symmedial, 1, 8978)
(symmedial, 1586, 6)
(inner Vecten, 4, 4)
(outer Vecten, 4, 485)
X(8953) lies on these lines: {1,371}, {6,8973}, {37,579}, {46,7133}, {65,8965}, {176,1587}, {372,1708}, {485,1659}, {1123,1788}, {1336,3475}, {3487,6352}
X(8954) lies on these lines: {3,6}, {24,6413}, {485,1585}, {486,6809}, {1075,3068}, {6414,7592}, {6564,8887}
X(8955) lies on these lines: {6,17}, {275,485}, {371,6750}, {3068,3462}
X(8956) lies on these lines: {2,3103}, {25,371}, {39,3981}, {51,3102}, {110,588}, {372,1583}, {485,1585}, {1599,2459}, {1915,5058}, {2004,3389}, {2005,3364}, {3068,3186}, {3311,8780}, {5020,6422}
X(8957) lies on these lines: {1,1336}, {4,6204}, {9,1210}, {485,1659}, {497,6212}, {1067,2066}, {1123,1738}, {1737,7090}, {1788,6213}, {2047,3931}, {3673,7389}
X(8958) lies on these lines: {485,1659}
X(8959) lies on these lines: {4,371}, {386,590}, {572,7583}
X(8960) lies on these lines: {2,6420}, {4,371}, {5,6419}, {6,17}, {30,6453}, {140,372}, {381,3592}, {382,6425}, {468,8854}, {486,3590}, {492,6118}, {550,3070}, {631,6454}, {1151,1657}, {1327,3146}, {1328,3855}, {1587,3523}, {1588,5068}, {1991,7751}, {2066,4857}, {2067,5270}, {3069,3316}, {3071,3850}, {3103,8992}, {3311,3851}, {3312,8253}, {3515,8276}, {3522,6560}, {3526,3594}, {3533,5420}, {3591,7486}, {3830,6447}, {5054,6426}, {5055,6427}, {5059,6480}, {5070,6428}, {5073,6221}, {5094,8280}, {5882,8983}, {6418,8252}, {6435,7582}
X(8961) lies on these lines: {3,6}, {5,8963}, {590,8967}, {1147,6414}
X(8962) lies on these lines: {2,39}, {6,493}, {32,1599}, {42,5346}, {51,3102}, {176,7201}, {184,371}, {216,1590}, {232,1585}, {394,6422}, {510,8957}, {570,590}, {574,1600}, {588,1994}, {615,5421}, {674,7112}, {800,7585}, {892,3342}, {1107,6348}, {1134,8340}, {1488,5454}, {1496,3704}, {1504,1993}, {1505,5422}, {1546,4068}, {1575,6347}, {1584,5013}, {1591,5254}, {1592,3815}, {2018,2138}, {2052,6433}, {2275,3083}, {2276,3084}, {2296,8256}, {2462,4353}, {2585,5603}, {2920,8498}, {3103,3917}, {3156,8854}, {3552,4386}, {4272,7936}, {4294,8081}, {4653,7867}, {5012,6945}, {5153,5313}, {5405,7777}, {5406,6423}, {5462,8413}, {5530,8799}, {5724,7037}, {6805,7738}, {6806,7736}
X(8962) = crosspoint of X(2) and X(588)
X(8962) = crosssum of X(6) and X(590)
X(8962) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(2) and X(588)
X(8963) lies on these lines: {2,216}, {3,3093}, {5,8961}, {6,494}, {39,3068}, {53,1592}, {182,6413}, {372,1092}, {566,8253}, {570,590}, {577,1600}, {615,3003}, {642,8969}, {800,3069}, {1180,8974}, {1974,3156}
As a point on the Euler line, X(8964) has Shinagawa coefficients: [-E-3*F+2*S, E+F-2*S]
X(8964) lies on these lines: {2,3}, {6,8940}, {184,6402}, {487,3167}, {5418,8967}
X(8964) = complement of X(32587)
X(8965) lies on these lines: {1,6}, {2,586}, {3,7133}, {55,8276}, {65,8953}, {241,482}, {642,3666}, {946,8233}, {1214,1659}, {1373,1418}, {1377,3083}, {2047,3931}, {3085,6352}, {3086,6351}, {3302,3584}
X(8965) = complement of isotomic conjugate of X(34216)
X(8965) = {X(1),X(9)}-harmonic conjugate of X(1124)
X(8966) lies on these lines: {3,485}, {5,8909}, {6,8968}, {371,6750}, {393,493}, {3155,5593}
X(8967) lies on these lines: {5,6}, {590,8961}, {642,8968}, {3155,8276}, {5418,8964}
X(8968) lies on these lines: {2,372}, {5,578}, {6,8966}, {216,590}, {233,615}, {371,1585}, {394,639}, {642,8967}, {1249,3068}, {1583,8276}, {1586,6564}, {1589,6560}, {1590,5418}, {3155,8989}, {3167,6289}
X(8969) lies on these lines: {4,371}, {216,590}, {578,7583}, {642,8963}, {8276,8989}
X(8970) lies on these lines: {2,372}, {3068,3186}, {3128,6564}, {3155,8276}
X(8971) lies on these lines: {6,1210}, {46,485}, {226,8977}
X(8972) lies on these lines: {2,6}, {4,3590}, {5,1132}, {8,8983}, {20,485}, {30,6445}, {140,6395}, {147,8980}, {148,8997}, {194,8992}, {371,3091}, {376,6451}, {486,7486}, {549,6446}, {631,6398}, {632,6418}, {1151,2671}, {1587,3523}, {1588,5056}, {1656,7582}, {2066,5274}, {2067,5261}, {2888,8995}, {2896,8993}, {3070,3522}, {3071,5068}, {3090,3311}, {3312,3525}, {3317,5070}, {3448,8998}, {3524,6452}, {3529,6449}, {3543,6480}, {3591,6435}, {3617,7969}, {3627,6407}, {3628,6417}, {3723,6352}, {3731,5393}, {3832,6437}, {3839,6561}, {5059,6433}, {5067,7584}, {5410,8889}, {5412,7378}, {6224,8988}, {6412,6460}, {7488,8276}
X(8973) lies on these lines: {1,6396}, {6,8953}
X(8974) lies on these lines: {2,6}, {25,1164}, {371,6202}, {485,3424}, {1161,8981}, {1180,8963}, {3641,8983}, {5305,7375}, {6215,8976}, {6227,8980}, {6258,8987}, {6273,8992}, {6275,8993}, {6277,8995}, {6319,8997}, {6813,8396}
X(8975) lies on these lines: {2,6}, {371,6201}, {485,5870}, {631,8416}, {1160,8981}, {3127,5410}, {6214,8976}, {6226,328}, {6257,8987}, {6262,8988}, {6272,8992}, {6274,8993},6276,8995}, {6320,8997}, {7733,8998}
X(8976) lies on these lines: {2,3312}, {3,485}, {4,3590}, {5,1588}, {6,17}, {20,6455}, {30,6449}, {140,1587}, {265,8998}, {355,8983}, {371,381}, {372,3526}, {376,1131}, {382,1151}, {486,5055}, {546,6447}, {549,6456}, {550,6451}, {615,5070}, {631,6450}, {639,1991}, {1132,3544}, {1152,5054}, {1594,5410}, {1657,6200}, {3069,3628}, {3071,3851}, {3090,6427}, {3523,6452}, {3525,6448}, {3530,6497}, {3534,6409}, {3592,5072}, {3595,7375}, {3627,6519}, {3830,6407}, {3843,6561}, {5056,7582}, {5067,7586}, {5073,6445}, {5076,6453}, {5079,6419}, {5411,7505}, {5420,6395}, {5790,7969}, {6033,8980}, {6214,8975}, {6215,8974}, {6287,8993}, {6288,8995}, {6321,8997}, {6420,8252}
X(8977) lies on these lines: {226,8971}, {3338,6203}
X(8978) lies on these lines: {1,372}, {6,8953}, {176,1588}, {284,501}, {486,1659}, {995,7969}, {6204,6419}
The reciprocal orthologic center of these triangles is X(4)
X(8979) lies on these lines: {4,487}, {1322,3536}
X(8979) = reflection of X(i) in X(j) for these (i,j): (4,8985)
X(8979) = anticomplement of X(8990)
The reciprocal orthologic center of these triangles is X(3)
X(8980) lies on these lines: {6,6036}, {98,3068}, {114,590}, {115,371}, {147,8972}, {485,2794}, {486,6722}, {542,8998}, {620,5418}, {690,8994}, {2460,6781}, {2782,8981}, {6033,8976}, {6221,6321}, {6226,8975}, {6227,8974}, {6721,8253}
The reciprocal orthologic center of these triangles is X(4)
X(8981) lies on these lines: {2,3311}, {3,1587}, {4,3590}, {5,371}, {6,140}, {20,6449}, {26,8276}, {30,485}, {230,1504}, {372,549}, {376,6455}, {381,6459}, {382,6407}, {395,3364}, {396,3389}, {486,3592}, {491,7767}, {495,2067}, {496,2066}, {517,8983}, {524,641}, {546,6425}, {548,6409}, {550,3070}, {615,632}, {631,3312}, {639,7886}, {640,754}, {642,3589}, {971,8987}, {1131,3529}, {1132,5071}, {1152,3530}, {1154,8995}, {1160,8975}, {1161,8974}, {1588,1656}, {1595,5412}, {1657,6445}, {1702,5886}, {2782,8980}, {3069,3526}, {3091,3316}, {3146,6519}, {3147,5411}, {3299,5433}, {3301,5432}, {3522,6451}, {3523,6398}, {3524,6450}, {3525,6427}, {3528,6496}, {3541,5410}, {3564,8909}, {3595,7376}, {3627,6453}, {3815,5058}, {3850,6437}, {3853,6429}, {5054,6418}, {5305,6422}, {5663,8994}, {5690,7969}, {6000,8991}, {6431,8252}
X(8981) = midpoint of X(i),X(j) for these (i,j): (485,1151)
X(8981) = orthologic center of X(2)-quadsquares triangles to these triangles: extangents, Kosnita, orthic, tangential
X(8981) = X(3) of X(2)-quadsquares triangle
The reciprocal orthologic center of these triangles is X(5)
X(8982) lies on these lines: {3,490}, {4,372}, {20,185}, {32,1587}, {99,487}, {230,3070}, {315,488}, {376,5861}, {489,3564}, {631,640}, {1131,7607}, {1152,6813}, {1505,1588}, {2794,5870}, {5871,8721}
X(8982) = reflection of X(i) in X(j) for these (i,j): (4,372), (638,3)
The reciprocal orthologic center of these triangles is X(3)
X(8983) lies on these lines: {1,1336}, {6,1125}, {8,8972}, {10,590}, {226,2067}, {355,8976}, {371,946}, {485,515}, {516,1151}, {517,8981}, {551,7968}, {631,1703}, {730,8992}, {952,8988}, {1377,6700}, {1385,7583}, {1587,3576}, {1588,8227}, {1659,7490}, {1699,6459}, {1702,5603}, {2362,3911}, {3069,3624}, {3070,4297}, {3071,3817}, {3311,5886}, {3316,5818}, {3616,7585}, {3634,8253}, {3641,8974}, {5393,5745}, {5418,6684}, {5550,7586}, {5882,8960}, {6001,8987}, {6460,7987}
X(8983) = orthologic center of these triangles: 2nd circumperp to X(2)-quadsquares
X(8983) = orthologic center of these triangles: X(2)-quadsquares to excentral
X(8983) = orthologic center of these triangles: X(2)-quadsquares to hexyl
X(8983) = orthologic center of these triangles: X(2)-quadsquares to intangents
X(8983) = X(1) of X(2)-quadsquares triangle
The reciprocal orthologic center of these triangles is X(1)
X(8984) lies on these lines: {1,7}, {3,7090}
X(8984) = midpoint of X(i),X(j) for these (i,j): (1,8986), (20,176)
X(8984) = reflection of X(i) in X(j) for these (i,j): (7090,3)
The reciprocal orthologic center of these triangles is X(4)
X(8985) lies on these lines: {4,487}
X(8985) = midpoint of X(i),X(j) for these (i,j): (4,8979)
The reciprocal orthologic center of these triangles is X(1)
X(8986) lies on these lines: {1,7}, {165,7090}
X(8986) = reflection of X(i) in X(j) for these (i,j): (1,8984)
The reciprocal orthologic center of these triangles is X(40)
X(8987) lies on these lines: {6,6705}, {84,3068}, {515,1151}, {590,6260}, {971,8981}, {2829,8988}, {5787,6221}, {6001,8983}, {6257,8975}, {6258,8974}
The reciprocal orthologic center of these triangles is X(3)
X(8988) lies on these lines: {6,6702}, {80,3068}, {214,590}, {485,2800}, {952,8983}, {2829,8987}, {6224,8972}, {6262,8975}
The reciprocal orthologic center of these triangles is X(5)
X(8989) lies on these lines: {3,640}, {22,5409}, {24,372}, {26,206}, {186,8982}, {371,3425}, {638,7488}, {3155,8968}, {6222,7669}, {8276,8969}
X(8989) = midpoint of X(i),X(j) for these (i,j): (3,8996)
The reciprocal orthologic center of these triangles is X(4)
X(8990) lies on these lines: {2,8979}, {3,486}, {5,8985}
X(8990) = Complement of X(8979)
The reciprocal orthologic center of these triangles is X(4)
X(8991) lies on these lines: {6,6696}, {371,6247}, {590,6812}, {1151,1503}, {1853,6459}, {6000,8981}
The reciprocal orthologic center of these triangles is X(3)
X(8992) lies on these lines: {6,3934}, {39,590}, {76,3068}, {194,8972}, {371,6248}, {485,511}, {640,1506}, {730,8983}, {732,8993}, {2782,8980}, {3070,5188}, {3103,8960}, {3311,7697}, {6272,8975}, {6273,8974}, {6683,8253}
The reciprocal orthologic center of these triangles is X(3)
X(8993) lies on these lines: {6,6704}, {83,3068}, {371,6249}, {732,8992}, {2896,8972}, {6287,8976}
The reciprocal orthologic center of these triangles is X(4)
X(8994) lies on these lines: {6,6699}, {113,590}, {125,371}, {265,6221}, {485,2777}, {486,6723}, {542,8997}, {690,8980}, {5418,5972}, {5663,8981}
The reciprocal orthologic center of these triangles is X(4)
X(8995) lies on these lines: {6,6689}, {54,3068}, {371,3574}, {539,8909}, {590,1209}, {1154,8981}, {2888,8972}, {2917,8276}, {6276,8975}, {6277,8974}, {6288,8976}
The reciprocal orthologic center of these triangles is X(5)
X(8996) lies on these lines: {3,640}, {22,638}, {24,8982}, {25,372}, {155,159}, {485,2351}
X(8996) = reflection of X(i) in X(j) for these (i,j): (3,8989)
The reciprocal parallelogic center of these triangles is X(6)
X(8997) lies on these lines: {6,620}, {99,3068}, {114,371}, {115,590}, {148,8972}, {486,6721}, {542,8994}, {641,5058}, {690,8998}, {1151,2794}, {2782,8980}, {5418,6036}, {6033,6221}, {6319,8974}, {6320,8975}, {6321,8976}, {6722,8253}
The reciprocal parallelogic center of these triangles is X(2)
X(8998) lies on these lines: {6,5181}, {110,3068}, {113,371}, {125,590}, {265,8976}, {542,8980}, {690,8997}, {1151,2777}, {1511,7583}, {2931,8276}, {3448,8972}, {5418,6699}, {5663,8981}, {6221,7728}, {6723,8253}, {7733,8975}
Crossdifferences involving X(6): X(8999)-X(9111)
Centers X(8999)-X(9111) were contributed by Peter J. C. Moses, December 28-31, 2015.
Suppose that P = p : q : r and U = u : v : w (barycentrics) are points. It is well known that the trilinear pole of the line PU, given by barycentrics 1/(rv - qw) : 1/(pw - ru) : 1/(qu - sv) , lies on the circumconic with perspector P and on the circumconic with perspector U (e.g., the trilinear pole of X(2)U is the point 1/(v - w) : 1/(w - u) : 1/(u - v), on the Steiner circumellpse.
If P lies on the circumcircle, then P is the trilinear pole of the line X(6)Q, where Q = crossdifference of X(6) and the isogonal conjugate of P; e.g., X(99) = trilinear pole of X(6)X(524), where X(524) = crossdifference of X(6) and X(512). The appearance of (i,j,k) in the following list means that X(i), on the circumcircle, X(j) is the isogonal conjugate of X(i), and X(k) is the crossdifference of (X(6) and X(j), and X(i) is the trilinear pole of X(6)X(k).
{74,30,8675}, {98,511,523}, {99,512,524}, {100,513,518}, {101,514,674}, {105,518,513}, {107,520,1503}, {108,521,3827}, {109,522,8679}, {110,523,511}, {111,524,512}, {112,525,2393}, {476,526,542}, {675,674,514}, {689,688,732}, {691,690,2854}, {699,698,3221}, {733,732,688}, {743,742,788}, {789,788,742}, {835,834,5847}, {842,542,526}, {901,900,2810}, {919,918,2876}, {925,924,3564}, {927,926,5845}, {929,928,5848}, {930,1510,5965}, {1113,2574,2575}, {1114,2575,2574}, {1290,8674,2836}, {1297,1503,520}, {1302,8675,30}, {1311,8679,522}, {2373,2393,525}, {2374,8681,3566}, {2715,2799,2871}, {2726,2810,900}, {2752,2836,8674}, {2770,2854,690}, {2857,2871,2799}, {2862,2876,918}, {3222,3221,698}, {3563,3564,924}, {3565,3566,8681}, {5966,5965,1510}, {5970,5969,888}, {6325,8705,3906}, {8707,6371,5846}
X(8999) lies on these lines: {6, 652}, {30, 511}
X(8999) = isogonal conjugate of X(9056)
X(9000) lies on these lines: {6, 657}, {30, 511}, {69, 3261}
X(9000) = isogonal conjugate of X(9057)
X(9001) lies on these lines: {6, 650}, {30, 511}, {69, 693}, {141, 4885}, {182, 7626}, {1769, 4449}, {3004, 4131}
X(9001) = isogonal conjugate of X(9058)
X(9002) lies on these lines: {6, 649}, {30, 511}, {141, 3835}, {663, 4491}, {1428, 3733}, {1459, 4057}, {3250, 3768}, {4724, 4833}
X(9002) = isogonal conjugate of X(9059)
X(9003) lies on these lines: {6, 1637}, {30, 511}, {69, 3268}, {110, 1302}, {684, 1649}, {879, 5505}, {895, 2986}
X(9003) = isogonal conjugate of X(9060)
X(9004) lies on these lines: {6,354}, {30,511}, {69,3263}, {72,5486}, {141,3740}, {165,7289}, {193,4430}, {210,599}, {241,4557}, {597,3742}, {942,4663}, {1386,5049}, {1486,8271}, {1992,3873}, {2340,3942}, {3242,5919}, {3589,3848}, {3751,5902}
X(9004) = isogonal conjugate of X(9061)
X(9005) lies on this line: {30, 511}
X(9005) = isogonal conjugate of X(9062)
X(9006) lies on these lines: {30, 511}, {669, 881}
X(9006) = isogonal conjugate of X(9063)
X(9007) lies on these lines: {6,2430}, {30,511}, {69,3265}, {2435,5486}
X(9007) = isogonal conjugate of X(9064)
X(9008) lies on these lines: {30, 511}, {1924, 1980}
X(9008) = isogonal conjugate of X(9065)
X(9009) lies on these lines: {6, 669}, {30, 511}, {182, 5926}, {1843, 2501}, {2491, 8644}
X(9009) = isogonal conjugate of X(9066)
X(9010) lies on these lines: {6, 667}, {30, 511}, {663, 3768}, {875, 4663}, {1469, 3669}, {3056, 4162}, {3751, 4063}
X(9010) = isogonal conjugate of X(9067)
X(9011) lies on these lines: {6, 3250}, {30, 511}
X(9011) = isogonal conjugate of X(9068)
X(9012) lies on these lines: {6, 3005}, {30, 511}
X(9012) = isogonal conjugate of X(9069)
X(9013) lies on these lines: {6, 661}, {30, 511}, {69, 7192}, {141, 4369}, {656, 3733}, {1491, 5040}, {3416, 4761}
X(9013) = isogonal conjugate of X(9070)
X(9014) lies on these lines: {6, 1491}, {30, 511}, {141, 4874}
X(9014) = isogonal conjugate of X(9071)
X(9015) lies on these lines: {6, 693}, {30, 511}, {141, 650}, {3589, 4885}
X(9015) = isogonal conjugate of X(9072)
X(9016) lies on these lines: {6, 292}, {30, 511}, {37, 3271}, {75, 3888}, {295, 8301}, {572, 3939}, {2310, 3056}, {3681, 3807}, {3780, 5369}
X(9016) = isogonal conjugate of X(9073)
X(9017) lies on these lines: {30, 511}, {69, 561}, {4443, 4531}
X(9017) = isogonal conjugate of X(9074)
X(9018) lies on these lines: {6, 560}, {30, 511}, {1826, 1843}, {3056, 4336}, {3416, 4710}
X(9018) = isogonal conjugate of X(9075)
X(9019) lies on these lines: {6, 22}, {23, 6593}, {30, 511}, {51, 597}, {52, 8550}, {67, 5189}, {69, 1369}, {141, 427}, {143, 575}, {159, 3167}, {182, 5946}, {237, 3001}, {378, 1350}, {576, 8546}, {599, 2979}, {858, 6698}, {973, 7512}, {1181, 8718}, {1351, 8547}, {1576, 6660}, {3521, 5486}, {3589, 5943}, {3629, 6467}
X(9019) = isogonal conjugate of X(9076)
X(9020) lies on these lines: {6, 38}, {30, 511}, {141, 1215}, {1220, 3868}, {3416, 4692}, {3589, 6682}, {4259, 5904}, {5135, 6763}
X(9020) = isogonal conjugate of X(9077)
X(9021) lies on these lines: {6, 977}, {30, 511}, {72, 141}, {942, 3589}, {1386, 3874}, {3218, 5096}, {3242, 3869}, {3416, 5904}, {3678, 3844}, {3751, 3901}, {3763, 3876}, {3782, 4463}
X(9021) = isogonal conjugate of X(9078)
X(9022) lies on these lines: {6, 321}, {30, 511}, {141, 306}, {3416, 4424}, {4363, 4968}, {4665, 5835}
X(9022) = isogonal conjugate of X(9079)
X(9023) lies on these lines: {6, 351}, {30, 511}, {6784, 6791}
X(9023) = isogonal conjugate of X(9080)
X(9023) = ideal point of PU(62)
X(9024) lies on these lines: {6, 100}, {11, 141}, {30, 511}, {69, 149}, {80, 3416}, {104, 1350}, {119, 5480}, {214, 1386}, {597, 6174}, {751, 3809}, {1083, 3908}, {1086, 3888}, {1317, 1469}, {1320, 3242}, {1843, 1862}, {3032, 4260}, {3035, 3589}, {3271, 4422}, {3629, 3779}, {3751, 5541}, {3799, 4370}, {3844, 6702}, {4265, 4996}
X(9024) = isogonal conjugate of X(9081)
X(9025) lies on these lines: {1, 4503}, {6, 43}, {30, 511}, {44, 4553}, {69, 350}, {141, 3816}, {193, 3779}, {239, 3888}, {995, 1386}, {1266, 4014}, {1469, 3476}, {1742, 3169}, {2183, 4447}, {2223, 3882}, {2228, 3248}, {3262, 4459}, {3271, 3912}, {3589, 6686}, {3688, 4416}, {3812, 5717}, {4562, 7077}
X(9025) = isogonal conjugate of X(9082)
X(9026) lies on these lines: {6, 1201}, {30, 511}, {69, 3264}, {141, 2885}, {210, 5650}, {354, 373}, {1401, 4849}, {1458, 4557}, {1469, 3214}, {3216, 3338}, {3271, 4864}, {3681, 7998}, {3689, 3937}, {3873, 5640}, {4553, 4899}
X(9026) = isogonal conjugate of X(9083)
X(9027) lies on these lines: {6, 373}, {30, 511}, {69, 3266}, {323, 895}, {599, 5650}, {1495, 2930}, {1843, 6144}, {1992, 5640}, {3098, 8547}, {3580, 5181}, {5092, 8546}, {5943, 8584}
X(9027) = isogonal conjugate of X(9084)
X(9028) lies on these lines: {6, 226}, {30, 511}, {48, 307}, {63, 69}, {141, 5745}, {142, 3211}, {150, 1944}, {193, 1839}, {388, 4644}, {651, 5236}, {958, 4643}, {1478, 3751}, {1565, 6510}, {1891, 3868}, {2293, 3938}, {3157, 4667}, {3486, 4419}, {4259, 4292}, {4363, 5794}
X(9028) = isogonal conjugate of X(9085)
X(9028) = isotomic conjugate of polar conjugate of X(3011)
X(9029) lies on these lines: {6, 663}, {30, 511}, {69, 4406}, {657, 1919}, {3242, 4449}, {3751, 4040}, {4394, 4524}, {4663, 4794}
X(9029) = isogonal conjugate of X(9086)
X(9030) lies on these lines: {6, 850}, {30, 511}, {141, 647}, {3050, 3267}
X(9030) = isogonal conjugate of X(9087)
X(9031) lies on these lines: {6, 3239}, {30, 511}, {69, 4025}, {141, 7658}, {652, 5227}
X(9031) = isogonal conjugate of X(9088)
X(9032) lies on these lines: {6, 1960}, {30, 511}, {659, 3751}, {875, 7077}, {2087, 3271}
X(9032) = isogonal conjugate of X(9089)
Let W be the circumconic with center X(1650). One of the asymptotes of W is the Euler line. The other is in the direction of X(9033). Also, X(9033) is the infinity point of the isotomic line of the Euler line. For a sketch, click X(9033). (Angel Montesdeoca, April 19, 2016)
X(9033) lies on these lines: {30, 511}, {67, 2435}, {69, 6333}, {74, 1294}, {107, 110}, {113, 133}, {122, 125}, {247, 8754}, {265, 6334}, {287, 879}, {402, 5972}, {1494, 3268}, {1636, 1637}, {3024, 7158}, {3028, 3324}, {6699, 8552}
X(9033) = isogonal conjugate of X(1304)
X(9033) = isotomic conjugate of X(16077)
X(9033) = crossdifference of every pair of points on line X(6)X(74)
X(9033) = X(2)-Ceva conjugate of X(39008)
X(9033) = perspector of hyperbola {{A,B,C,X(30),X(525)}}
X(9033) = barycentric square root of X(39008)
X(9034) lies on these lines: {30, 511}, {651, 662}, {1814, 2987}, {4131, 4885}, {6506, 8287}
X(9034) = isogonal conjugate of X(9090)
X(9035) lies on these lines: {30, 511}, {69, 3569}, {141, 2492}, {287, 2422}, {648, 670}, {1084, 6388}, {1494, 2433}, {2451, 3267}, {5027, 6131}
X(9035) = isogonal conjugate of X(9091)
X(9036) lies on these lines: {{6, 359}, {30, 511}
X(9036) = isogonal conjugate of X(9092)
X(9037) lies on these lines: {6, 36}, {30, 511}, {44, 3792}, {51, 3742}, {69, 5080}, {141, 3814}, {210, 2979}, {354, 3060}, {375, 3819}, {484, 3751}, {751, 1319}, {1155, 3240}, {1350, 2077}, {1843, 1878}, {1985, 5087}, {3056, 5048}, {3246, 3271}, {3416, 5176}, {3589, 6681}, {3740, 3917}, {3779, 5183}, {3844, 5123}, {3848, 5943}, {4260, 5122}, {5535, 7289}
X(9037) = isogonal conjugate of X(9093)
X(9038) lies on these lines: {6, 748}, {30, 511}, {69, 4651}, {3056, 6144}
X(9038) = isogonal conjugate of X(9094)
X(9039) lies on these lines: {6, 1149}, {30, 511}, {375, 3873}, {899, 4860}
X(9039) = isogonal conjugate of X(9095)
X(9040) lies on these lines: {6, 4367}, {30, 511}, {1019, 3751}, {1469, 7178}, {3242, 4879}, {4981, 5996}
X(9040) = isogonal conjugate of X(9096)
X(9041) lies on these lines: {1, 597}, {2, 1280}, {6, 644}, {8, 599}, {30, 511}, {57, 4952}, {69, 903}, {141, 3679}, {145, 190}, {551, 3589}, {1279, 4899}, {3243, 4929}, {3244, 4432}, {3416, 4677}, {3621, 4440}, {3623, 4473}, {3656, 5480}, {3712, 4141}, {3717, 4864}, {3751, 8584}, {3844, 4745}, {3870, 4884}
X(9041) = isogonal conjugate of X(9097)
X(9042) lies on these lines: {6, 7133}, {30, 511}
X(9042) = isogonal conjugate of X(9098)
X(9043) lies on this line: {30, 511}
X(9043) = isogonal conjugate of X(9099)
X(9044) lies on this line: {30, 511}
X(9044) = isogonal conjugate of X(9100)
X(9045) lies on these lines: {6, 1239}, {30, 511}, {69, 8267}, {141, 1194}
X(9045) = isogonal conjugate of X(9101)
X(9046) lies on this line: {30, 511}
X(9046) = isogonal conjugate of X(9102)
X(9047) lies on these lines: {6, 35}, {30, 511}, {51, 3740}, {210, 3060}, {354, 2979}, {1279, 3792}, {1386, 2646}, {1843, 1900}, {2274, 4787}, {3416, 5086}, {3742, 3917}, {3779, 4663}, {3819, 3848}, {4735, 5145}, {5889, 7957}
X(9047) = isogonal conjugate of X(9103)
X(9048) lies on these lines: {6, 4394}, {30, 511}, {69, 4106}, {193, 4380}
X(9048) = isogonal conjugate of X(9104)
X(9049) lies on these lines: {6, 1334}, {30, 511}, {209, 3938}, {210, 373}, {354, 5650}, {375, 3681}, {3242, 3779}, {3690, 3748}, {3873, 7998}, {4553, 4684}
X(9049) = isogonal conjugate of X(9105)
X(9050) lies on these lines: {30, 511}, {373, 3873}, {1357, 5524}, {3681, 5650}, {4260, 6048}, {4430, 5640}, {4661, 7998}
X(9050) = isogonal conjugate of X(9106)
X(9051) lies on these lines: {6,2431}, {30,511}, {69,4131}, {3657,6391}
X(9051) = isogonal conjugate of X(9107)
X(9052) lies on these lines: {1, 3688}, {6, 595}, {8, 3963}, {30, 511}, {51, 3681}, {55, 5138}, {69, 2891}, {72, 3883}, {141, 2140}, {181, 3961}, {209, 3744}, {210, 5943}, {242, 5185}, {354, 3819}, {970, 3811}, {1362, 5018}, {1428, 2078}, {1469, 3340}, {1621, 3690}, {1697, 3056}, {1757, 3271}, {2979, 4430}, {3030, 5524}, {3060, 4661}, {3169, 6765}, {3242, 4259}, {3555, 3879}, {3740, 6688}, {3873, 3917}, {4553, 4966}
X(9052) = isogonal conjugate of X(9108)
X(9053) lies on these lines: {1, 3589}, {6, 145}, {8, 141}, {9, 4929}, {30, 511}, {38, 4030}, {44, 4899}, {55, 4884}, {69, 3621}, {182, 1483}, {200, 4952}, {239, 4437}, {597, 3241}, {748, 4126}, {1265, 1616}, {1279, 3717}, {1317, 1428}, {1386, 3244}, {1482, 5480}, {2136, 7289}, {2550, 7263}, {3243, 4851}, {3416, 3631}, {3617, 3763}, {3618, 3623}, {3619, 4678}, {3626, 3844}, {3633, 3751}, {3699, 5211}, {3703, 3938}, {3712, 3722}, {3756, 5205}, {3782, 5014}, {3871, 4265}, {3912, 4864}, {3920, 6703}, {4364, 7174}, {4415, 4514}, {4645, 7238}, {5085, 7967}, {5263, 7227}, {5838, 5839}
X(9053) = isogonal conjugate of X(9109)
X(9054) lies on these lines: {6, 1621}, {30, 511}, {141, 3779}, {3056, 3629}
X(9054) = isogonal conjugate of X(9110)
X(9055) lies on these lines: {2, 3807}, {6, 190}, {30, 511}, {37, 3589}, {69, 1278}, {75, 141}, {193, 4788}, {597, 4370}, {599, 903}, {673, 4361}, {984, 4026}, {1015, 4568}, {1386, 3993}, {3125, 4986}, {3242, 4363}, {3263, 3726}, {3618, 4473}, {3619, 4772}, {3620, 4821}, {3629, 3644}, {3630, 4409}, {3631, 4686}, {3722, 4760}, {3744, 4797}, {3763, 4699}, {3797, 3943}, {3938, 4376}, {3952, 4465}, {4681, 6329}, {4799, 5014}
X(9055) = isogonal conjugate of X(9111)
X(9055) = isotomic conjugate of X(35172)
X(9055) = X(2)-Ceva conjugate of X(35126)
X(9056) lies on the circumcircle and these lines: {2, 102}, {23, 2695}, {74, 7413}, {98, 7449}, {99, 7450}, {104, 4224}, {109, 2406}, {112, 7452}, {1292, 7451}, {1296, 7462}, {1311, 1995}, {2756, 5205}
X(9056) = orthoptic-circle-of-Steiner-inellipse-inverse of X(117)
X(9056) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(151)
X(9056) = trilinear pole of line X(6)X(515)
X(9056) = Ψ(X(6), X(515))
X(9056) = Λ(X(6), X(652))
X(9057) lies on the circumcircle and these lines: {2, 103}, {23, 2688}, {25, 917}, {74, 6998}, {98, 7453}, {99, 4243}, {101, 2398}, {104, 4223}, {112, 4241}, {675, 1995}, {691, 7479}, {1292, 7437}, {1296, 4237}, {2750, 3006}
X(9057) = orthoptic-circle-of-Steiner-inellipse-inverse of X(118)
X(9057) = orthoptic-circle-of-Steiner-circumellipse-inverse of (X(152)
X(9057) = trilinear pole of line X(6)X(516)
X(9057) = Ψ(X(6), X(516))
X(9057) = Λ(X(6), X(657))
X(9057) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,X(2),X(658)}}
X(9058) lies on the circumcircle and these lines: {2, 104}, {22, 1295}, {23, 2687}, {25, 915}, {74, 4220}, {98, 4239}, {99, 3658}, {100, 2397}, {103, 7465}, {105, 1995}, {106, 614}, {112, 4246}, {651, 2720}, {691, 7477}, {692, 6099}, {759, 4228}, {858, 2694}, {901, 1633}, {917, 7466}, {1289, 7435}, {1296, 4236}, {1300, 4231}, {1309, 6335}, {2696, 7475}, {2718, 5121}, {2752, 7426}, {2757, 5205}, {3563, 7438}
X(9058) = isogonal conjugate of X(9001)
X(9058) = orthoptic-circle-of-Steiner-inellipse-inverse of X(119)
X(9058) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(153)
X(9058) = trilinear pole of line X(6)X(517)
X(9058) = Ψ(X(6), X(517))
X(9058) = Λ(X(6), X(650))
X(9059) lies on the circumcircle and these lines: {2, 106}, {22, 2370}, {23, 2758}, {109, 3952}, {110, 3699}, {183, 675}, {190, 901}, {739, 5276}, {741, 4518}, {759, 7081}, {953, 3006}, {1293, 2415}, {2703, 3799}, {2718, 5205}, {2729, 5971}, {4588, 4767}, {4756, 8697}, {4781, 6014}, {6551, 6632}
X(9059) = trilinear pole of line X(6)X(519)
X(9059) = Ψ(X(6), X(519))
X(9059) = Λ(X(6), X(649))
X(9059) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(21290)
X(9059) = orthoptic-circle-of-Steiner-inellipse-inverse of X(121)
X(9060) lies on the circumcircle and these lines: {2, 477}, {3, 841}, {22, 2693}, {23, 74}, {98, 7426}, {99, 7471}, {104, 7469}, {110, 8675}, {111, 3003}, {112, 7480}, {468, 1300}, {476, 2410}, {523, 1302}, {842, 1995}, {858, 1294}, {935, 4240}, {1292, 7477}, {1296, 7468}, {2687, 4228}, {2688, 7474}, {2691, 3658}, {2696, 4226}, {2697, 7493}
X(9060 = reflection of X(841) in X(3)
X(9060) = trilinear pole of line X(6)X(5663) (the tangent to Moses-Parry circle at X(6))
X(9060) = Ψ(X(6), X(5663))
X(9060) = Λ(X(6), X(1637))
X(9060) = circumcircle intercept, other than X(74), of circle {{X(2),X(3),X(74)}}
X(9060) = orthoptic-circle-of-Steiner-circumellipse-inverse of X(34193)
X(9060) = orthoptic-circle-of-Steiner-inellipse inverse of X(25641)
X(9061) lies on the circumcircle and these lines: {2, 1292}, {21, 1296}, {23, 2691}, {74, 7423}, {98, 7458}, {99, 4228}, {100, 344}, {101, 3870}, {105, 2402}, {108, 4232}, {109, 1445}, {112, 4233}, {691, 7469}, {1290, 7426}, {1325, 2696}, {4239, 6011}
X(9061) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5511)
X(9061) = trilinear pole of line X(6)X(3309)
X(9061) = Ψ(X(6), X(3309))
X(9061) = Λ(X(6), X(354))
X(9062) lies on the circumcircle and these lines: {2, 703}, {183, 733}
X(9062) = trilinear pole of line X(6)X(702)
X(9062) = Ψ(X(6), X(702))
X(9063) lies on the circumcircle and these lines: {2, 707}, {75, 723}, {76, 737}, {110, 880}, {308, 733}, {670, 805}, {699, 3407}, {715, 3113}, {729, 3114}, {773, 4586}
X(9063) = trilinear pole of line X(6)X(706)
X(9063) = Ψ(X(6), X(706))
X(9063) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {{A,B,C,PU(11)}}
X(9064) lies on the circumcircle and these lines: {2, 133}, {22, 5897}, {23, 2693}, {25, 74}, {98, 4232}, {99, 4240}, {104, 4233}, {107, 2404}, {186, 841}, {468, 477}, {691, 7480}, {1292, 4246}, {1295, 4228}, {1296, 4230}, {1297, 1995}, {1300, 6353}, {2694, 7469}, {2696, 7473}, {2697, 7426}
X(9064) =orthoptic-circle-of-Steiner-inellipse-inverse of X(133)
X(9064) = trilinear pole of line X(6)X(1597)
X(9064) = Ψ(X(6), X(1597))
X(9064) = Λ(X(6), X(2430))
X(9065) lies on the circumcircle and these lines: {1, 705}, {2, 721}, {75, 735}, {76, 747}, {101, 6386}, {110, 4602}
X(9065) = trilinear pole of line X(6)X(561)
X(9065) = Ψ(X(6), X(561))
X(9066) lies on the circumcircle and these lines: {2, 729}, {111, 183}, {251, 6579}, {670, 4108}, {703, 3329}, {843, 5971}, {6380, 7766}
X(9066) = Isotomic conjugate of X(5996)
X(9066) = trilinear pole of line X(6)X(538)
X(9066) = Ψ(X(6), X(538))
X(9066) = Λ(X(6), X(669))
X(9067) lies on the circumcircle and these lines: {2, 739}, {105, 183}, {190, 6016}, {668, 898}, {717, 5276}, {799, 8690}, {2721, 5971}, {3570, 8693}
X(9067) = trilinear pole of line X(6)X(536)
X(9067) = Ψ(X(6), X(536))
X(9067) = Λ(X(6), X(667))
X(9068) lies on the circumcircle and these lines: {2,753}
X(9068) = trilinear pole of line X(6)X(752)
X(9068) = Ψ(X(6), X(752))
X(9068) = Λ(X(6), X(3250))
X(9069) lies on the circumcircle and these lines: {2, 755}, {111, 7792}, {729, 5354}, {1634, 2858}
X(9069) = trilinear pole of line X(6)X(754)
X(9069) = Ψ(X(6), X(754))
X(9069) = Λ(X(6), X(3005))
X(9070) lies on the circumcircle and these lines: {2, 759}, {104, 4220}, {105, 4239}, {106, 7191}, {110, 3909}, {112, 4242}, {915, 4231}, {2222, 4552}, {2372, 7081}, {2718, 5211}, {2758, 5205}
X(9070) = trilinear pole of line X(6)X(758)
X(9070) = Ψ(X(6), X(758))
X(9070) = Λ(X(6), X(661))
X(9071) lies on the circumcircle and these lines: {2,761}
X(9071) = trilinear pole of line X(6)X(760)
X(9071) = Ψ(X(6), X(760))
X(9071) = Λ(X(6), X(1491))
X(9072) lies on the circumcircle and these lines: {2,767}
X(9072) = trilinear pole of line X(6)X(766)
X(9072) = Ψ(X(6), X(766))
X(9072) = Λ(X(6), X(693))
X(9073) lies on the circumcircle and these lines: {2, 813}, {100, 350}, {101, 239}, {109, 1447}, {741, 7192}, {919, 6654}, {7033, 8684}
X(9073) = trilinear pole of line X(6)X(812)
X(9073) = Ψ(X(6), X(812))
X(9073) = Λ(X(6), X(292))
X(9074) lies on the circumcircle and these lines: {2, 815}, {99, 4215}, {101, 4362}
X(9074) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5509)
X(9074) = trilinear pole of line X(6)X(814)
X(9074) = Ψ(X(6), X(814))
X(9074) = Λ(X(69), X(561))
X(9075) lies on the circumcircle and these lines: {2, 825}, {101, 3661}, {109, 7179}, {561, 789}
X(9075) = trilinear pole of line X(6)X(824)
X(9075) = Ψ(X(6), X(824))
X(9075) = Λ(X(6), X(560))
X(9076) lies on the circumcircle and these lines: {2, 827}, {23, 1287}, {67, 110}, {99, 1799}, {112, 251}, {125, 4630}, {550, 1296}, {689, 1502}, {691, 5189}, {813, 2157}, {935, 5938}, {1141, 7418}, {4577, 7664}, {5966, 7417}
X(9076) = trilinear pole of line X(6)X(826)
X(9076) = Ψ(X(6), X(826))
X(9076) = Λ(X(6), X(22))
X(9076) = cevapoint of X(67) and X(3455)
X(9076) = trilinear pole, wrt circummedial triangle, of line X(25)X(183)
X(9077) lies on the circumcircle and these lines: {2, 831}, {101, 3920}
X(9077) = trilinear pole of line X(6)X(830)
X(9077) = Ψ(X(6), X(830))
X(9077) = Λ(X(6), X(38))
X(9078) lies on the circumcircle and these lines: {2, 833}, {101, 976}
X(9078) = trilinear pole of line X(6)X(832)
X(9078) = Ψ(X(6), X(832))
X(9078) = Λ(X(6), X(977))
X(9079) lies on the circumcircle and these lines: {2, 839}, {100, 4261}
X(9079) = trilinear pole of line X(6)X(838)
X(9079) = Ψ(X(6), X(838))
X(9079) = Λ(X(6), X(321))
X(9080) lies on the circumcircle and these lines: {2, 843}, {99, 4108}, {111, 6094}, {183, 2770}, {2709, 5468}
X(9080) = trilinear pole of line X(6)X(543)
X(9080) = Ψ(X(6), X(543))
X(9080) = Ψ(X(352), X(2))
X(9080) = Λ(X(6), X(351))
X(9080) = X(111)-of-circummedial-triangle
X(9081) lies on the circumcircle and these lines: {2, 898}, {99, 5990}, {100, 536}, {101, 899}, {513, 739}, {750, 813}, {1155, 6016}
X(9081) = trilinear pole of line X(6)X(891)
X(9081) = Ψ(X(6), X(891))
X(9081) = Λ(X(6), X(100))
X(9082) lies on the circumcircle and these lines: {2, 932}, {43, 101}, {100, 192}, {109, 1423}, {4220, 6010}
X(9082) = isogonal conjugate of X(9025)
X(9082) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5518)
X(9082) = trilinear pole of line X(6)X(4083)
X(9082) = Ψ(X(6), X(4083))
X(9082) = Λ(X(6), X(43))
X(9083) lies on the circumcircle and these lines: {2, 1293}, {23, 2692}, {74, 7434}, {98, 7448}, {99, 7419}, {101, 145}, {104, 7459}, {106, 2403}, {109, 5435}, {112, 4248}, {404, 1292}, {691, 7478}, {1296, 4234}, {2696, 7481}
X(9083) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5510)
X(9083) = trilinear pole of line X(6)X(3667)
X(9083) = Ψ(X(6), X(3667))
X(9083) = Λ(X(6), X(1201))
X(9084) lies on the circumcircle and these lines: {2, 1296}, {23, 2696}, {74, 7417}, {99, 1995}, {104, 7458}, {110, 1992}, {111, 2408}, {112, 4232}, {691, 7426}, {1292, 4239}, {3565, 7493}, {5182, 6233}
X(9084) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5512)
X(9084) = trilinear pole of line X(6)X(1499)
X(9084) = Ψ(X(6), X(1499))
X(9084) = Λ(X(6), X(373))
X(9085 lies on the circumcircle and these lines: {2, 1305}, {4, 5513}, {19, 100}, {25, 101}, {27, 99}, {109, 579}, {110, 1474}, {468, 2690}, {691, 2073}, {901, 8752}, {913, 6099}, {925, 7474}, {934, 1435}, {1292, 4219}, {1294, 7433}, {1295, 7445}, {1296, 7431}, {1297, 7432}, {2222, 5089}, {3565, 4184}
X(9085) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5190)
X(9085) = inverse-in-polar-circle-of-X(5513)
X(9085) = trilinear pole of line X(6)X(8676)
X(9085) = Ψ(X(6), X(8676))
X(9085) = Λ(X(6), X(226))
X(9085) = X(63)-isoconjugate of X(3011)
X(9086) lies on the circumcircle and these lines: {2, 2291}, {183, 1311}, {1477, 3598}, {1978, 8706}, {2768, 5971}
X(9086) = trilinear pole of line X(6)X(527)
X(9086) = Ψ(X(6), X(527))
X(9086) = Λ(X(6), X(663))
X(9087) lies on the circumcircle and these lines: {2, 2367}
X(9087) = trilinear pole of line X(6)X(2387)
X(9087) = Ψ(X(6), X(2387))
X(9087) = Λ(X(6), X(850))
X(9088) lies on the circumcircle and these lines: {2, 2370}, {25, 106}, {102, 3478}, {104, 614}, {162, 8690}, {468, 2758}, {901, 8750}, {1897, 8706}, {2757, 5121}, {4242, 6012}
X(9088) = trilinear pole of line X(6)X(2390)
X(9088) = Ψ(X(6), X(2390))
X(9088) = Λ(X(6), X(3239))
X(9089) lies on the circumcircle and these lines: {2, 2384}, {101, 6633}, {183, 2726}, {874, 6079}, {1447, 8686}
X(9089) = trilinear pole of line X(6)X(545)
X(9089) = Ψ(X(6), X(545))
X(9089) = Λ(X(6), X(1960))
X(9090) lies on the circumcircle and these lines: {2, 2856}, {99, 4391}, {100, 3700}, {101, 4041}, {105, 230}, {108, 2501}, {109, 661}, {110, 650}, {934, 7178}, {2721, 6792}, {3563, 5089}
X(9090) = inverse-in-Stevanovic-circle of X(110)
X(9090) = trilinear pole of line X(6)X(2870)
X(9090) = Ψ(X(6), X(2870))
X(9091) lies on the circumcircle and these lines: {2, 2868}, {74, 3231}, {98, 3291}, {99, 647}, {107, 2489}, {110, 3049}, {112, 669}, {232, 2374}, {385, 2373}, {689, 4580}, {729, 1495}, {2452, 2770}
X(9091) = orthoptic-circle-of-Steiner-inellipse-inverse of X(99)
X(9091) = trilinear pole of line X(6)X(2882)
X(9091) = Ψ(X(6), X(2882))
X(9091) = inverse-in-Moses-radical-circle of X(99)
X(9092) lies on the circumcircle and these lines: {2, 3067}
X(9092) = trilinear pole of line X(6)[isogonal conjugate of X(3067)]
X(9092) = Ψ(X(6), [isogonal conjugate of X(3067)])
X(9092) = Λ(X(6), X(359))
X(9093) lies on the circumcircle and these lines: {2, 4588}, {100, 4671}, {101, 3679}, {109, 750}, {110, 5235}, {214, 8695}, {901, 4945}, {934, 7223}
X(9093) = trilinear pole of line X(6)X(4777)
X(9093) = Ψ(X(6), X(4777))
X(9093) = Λ(X(6), X(36))
X(9094) lies on the circumcircle and these lines: {2, 6013}, {100, 4687}
X(9094) = trilinear pole of line X(6)X(6005)
X(9094) = Ψ(X(6), X(6005))
X(9094) = Λ(X(6), X(748))
X(9095) lies on the circumcircle and these lines: {2, 6014}, {101, 3241}
X(9095) = trilinear pole of line X(6)X(6006)
X(9095) = Ψ(X(6), X(6006))
X(9095) = Λ(X(6), X(1149))
X(9096) lies on the circumcircle and these lines: {2, 6015}
X(9096) = trilinear pole of line X(6)X(6007)
X(9096) = Ψ(X(6), X(6007))
X(9096) = Λ(X(6), X(4367))
X(9097) lies on the circumcircle and these lines: {1, 2748}, {2, 5516}, {6, 6078}, {100, 1279}, {101, 1149}, {106, 8643}, {551, 2759}, {614, 2743}, {901, 3052}, {902, 1293}, {1201, 2705}, {1386, 2753}, {3669, 8686}
X(9097) = orthoptic-circle-of-Steiner-inellipse-inverse of X(5516)
X(9097) = trilinear pole of line X(6)X(6085)
X(9097) = Ψ(X(6), X(6085))
X(9097) = Λ(X(i),X(j)) for these (i,j): (1,597), (6,644), (8,599)
X(9098) lies on the circumcircle and these lines: {2, 6135}, {100, 1267}, {101, 3083}
X(9098) = trilinear pole of line X(6)X(6364)
X(9098) = Ψ(X(6), X(6364))
X(9098) = Λ(X(6), X(7133))
X(9099) lies on the circumcircle and these lines: {2, 6136}, {100, 5391}, {101, 3084}
X(9099) = trilinear pole of line X(6)X(6365)
X(9099) = Ψ(X(6), X(6365))
X(9100) lies on the circumcircle and these lines: {2, 6323}, {183, 6325}
X(9100) = trilinear pole of line X(6)X(3849)
X(9100) = Ψ(X(6), X(3849))
X(9101) lies on the circumcircle and these lines: {2, 6572}, {99, 1180}
X(9101) = trilinear pole of line X(6)[isogonal conjugate of X(6572)]
X(9101) = Ψ(X(6), [isogonal conjugate of X(6572)])
X(9101) = Λ(X(6), X(1239))
X(9102) lies on the circumcircle and these lines: {2, 6579}, {729, 1799}
X(9102) = trilinear pole of line X(6)[isogonal conjugate of X(6579)]
X(9102) = Ψ(X(6), [isogonal conjugate of X(6579)])
X(9103) lies on the circumcircle and these lines: {2, 8652}, {101, 1698}, {109, 4654}, {110, 5333}
X(9103) = trilinear pole of line X(6)X(4802)
X(9103) = Ψ(X(6), X(4802))
X(9103) = Λ(X(6), X(35))
X(9104) lies on the circumcircle and these lines: {2, 8686}, {106, 5272}, {646, 6079}
X(9104) = trilinear pole of line X(6)X(3880)
X(9104) = Ψ(X(6), X(3880))
X(9104) = Λ(X(6), X(4394))
X(9105) lies on the circumcircle and these lines: {2, 8694}, {101, 3616}, {1434, 5545}
X(9105) = trilinear pole of line X(6)X(4778)
X(9105) = Ψ(X(6), X(4778))
X(9105) = Λ(X(6), X(1334))
X(9106) lies on the circumcircle and these lines: {2, 8698}, {101, 3635}
X(9106) = trilinear pole of line X(6)X(4943)
X(9106) = Ψ(X(6), X(4943))
X(9107) lies on the circumcircle and these lines: {2, 1295}, {23, 2694}, {25, 104}, {74, 4231}, {98, 7438}, {99, 4246}, {103, 7466}, {105, 4232}, {108, 2405}, {112, 7435}, {468, 2687}, {759, 4233}, {915, 6353}, {1292, 4242}, {1294, 4220}, {1296, 4238}, {1297, 4239}, {2696, 7476}, {3565, 3658}
X(9107) = trilinear pole of line X(6)X(6001)
X(9107) = Ψ(X(6), X(6001))
X(9107) = Λ(X(6), X(2431))
X(9108) lies on the circumcircle and these lines: {2, 8701}, {100, 3757}, {101, 1125}, {109, 553}, {110, 8025}, {1509, 6578}
X(9108) = trilinear pole of line X(6)X(4977)
X(9108) = Ψ(X(6), X(4977))
X(9108) = Λ(X(6), X(595))
X(9109) lies on the circumcircle and these lines: {2, 8706}, {100, 3744}, {101, 1201}, {5211, 6079}
X(9109) = trilinear pole of line X(6)X(6363)
X(9109) = Ψ(X(6), X(6363))
X(9109) = Λ(X(6), X(145))
X(9110) lies on the circumcircle and these lines: {2, 8708}, {100, 3739}, {101, 3720}
X(9110) = trilinear pole of line X(6)X(6372)
X(9110) = Ψ(X(6), X(6372))
X(9110) = Λ(X(6), X(1621))
X(9111) lies on the circumcircle and these lines: {2, 8709}, {31, 813}, {100, 1575}, {101, 2210}, {649, 727}, {898, 2242}, {919, 1627}
X(9111) = trilinear pole of line X(6)X(6373)
X(9111) = Ψ(X(6), X(6373))
X(9111) = Λ(X(6), X(190))
Let No = AoBoCo be the outer-Napoleon triangle of ABC. The equilateral triangle T' = A'B'C' inscribed in ABC with least area is homothetic to No with homothetic center X(6), and A' = X(6)Ao∩BC. The center of T' is X(396). The equilateral triangle T" = A"B"C" circumscribed to ABC and having greatest area is homothetic to No with homothetic center X(13), and A'' = reflection of X(13) in Ao. (The vertex A" lies on the major-arc BC of the circle centered at Ao and passing through B. This arc is the locus of a point P satisfying angle(BPC) = π/3.) The center of T" is X(5463). The triangles T' and T" are homothetic with homothetic center X(9112). (César Lozada, January 3, 2016)
X(9112) lies on these lines: {2, 13646}, {4, 31683}, {6, 13}, {15, 5473}, {16, 21156}, {17, 37637}, {18, 7603}, {39, 42990}, {61, 7737}, {62, 3815}, {98, 47864}, {99, 12155}, {112, 11612}, {187, 6779}, {230, 16267}, {303, 41633}, {393, 31687}, {395, 22489}, {396, 5463}, {397, 15048}, {530, 1285}, {574, 41100}, {616, 41408}, {618, 11488}, {620, 9763}, {671, 41621}, {1033, 31685}, {1384, 35751}, {1506, 3411}, {1992, 40671}, {2549, 41107}, {3053, 3412}, {3055, 41944}, {3068, 6302}, {3069, 6306}, {3087, 35714}, {3390, 36762}, {3767, 42992}, {5032, 22574}, {5052, 43538}, {5318, 36961}, {5334, 5478}, {5335, 41022}, {5353, 10062}, {5357, 10078}, {5459, 37641}, {5611, 38730}, {5617, 11542}, {6108, 7736}, {6115, 7735}, {6303, 13760}, {6307, 13640}, {6669, 11489}, {6770, 14482}, {6771, 11486}, {6772, 18907}, {6776, 43954}, {6782, 18582}, {6783, 36776}, {7684, 41019}, {8553, 37848}, {8593, 22573}, {8742, 34321}, {8838, 31898}, {9114, 9117}, {9115, 16644}, {9166, 47858}, {9749, 37665}, {9762, 32553}, {9862, 16941}, {10611, 42098}, {11408, 12142}, {11485, 36772}, {13651, 13876}, {13770, 13929}, {14639, 47862}, {14853, 41045}, {16268, 31415}, {16530, 16966}, {16806, 34374}, {16960, 36763}, {16963, 31489}, {16965, 44526}, {18584, 49908}, {18800, 42036}, {19101, 25186}, {19780, 36782}, {21843, 41943}, {22541, 25185}, {22701, 32465}, {23302, 36770}, {25178, 36773}, {25220, 36783}, {36767, 49905}, {36768, 49862}, {36769, 46453}, {36784, 41630}, {36785, 41632}, {36968, 44541}, {41112, 43448}, {41121, 43620}, {41638, 49603}, {44647, 51727}, {50855, 51201}
X(9112) = reflection of X(i) in X(j) for these {i,j}: {13, 42974}, {36772, 11485}
X(9112) = homothetic center of pedal triangle of X(15) and antipedal triangle of X(13)
X(9112) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5472, 13}, {6, 15484, 9113}, {13, 51200, 41042}, {15, 23006, 5473}, {115, 41672, 9113}, {396, 5463, 36764}, {396, 41745, 5463}, {1992, 40671, 42035}, {5617, 11542, 36771}, {6782, 18582, 36765}, {6782, 47855, 18582}, {12155, 37786, 36775}, {16960, 36766, 36763}, {41620, 47857, 2}, {47861, 47863, 4}, {50719, 50720, 41043}
Let No = AoBoCo be the inner-Napoleon triangle of ABC. The equilateral triangle T' = A'B'C' inscribed in ABC with greatest area is homothetic to No with homothetic center X(6), and A' = X(6)Ao∩BC. The center of T' is X(395). The equilateral triangle T" = A"B"C" circumscribed to ABC and having least area is homothetic to No with homothetic center X(14), and A'' = reflection of X(14) in Ao. The triangles T' and T" are homothetic with homothetic center X(9113); see X(9112). (César Lozada, January 3, 2016 and Peter Moses, January 4, 2016 )
X(9113) lies on these lines: {2, 13645}, {4, 31684}, {6, 13}, {15, 21157}, {16, 5474}, {17, 7603}, {18, 37637}, {39, 42991}, {61, 3815}, {62, 7737}, {98, 47863}, {99, 12154}, {112, 11613}, {187, 6780}, {230, 16268}, {302, 41643}, {393, 31688}, {395, 5464}, {396, 22490}, {398, 15048}, {531, 1285}, {574, 41101}, {617, 41409}, {619, 11489}, {620, 9761}, {671, 41620}, {1033, 31686}, {1384, 36329}, {1506, 3412}, {1992, 40672}, {2549, 41108}, {3053, 3411}, {3055, 41943}, {3068, 6303}, {3069, 6307}, {3087, 35715}, {3767, 42993}, {5032, 22573}, {5052, 43539}, {5321, 36962}, {5334, 41023}, {5335, 5479}, {5353, 10077}, {5357, 10061}, {5460, 37640}, {5613, 11543}, {5615, 38730}, {6109, 7736}, {6114, 7735}, {6302, 13760}, {6306, 13640}, {6670, 11488}, {6773, 14482}, {6774, 11485}, {6775, 18907}, {6776, 43953}, {6783, 18581}, {8553, 37850}, {8593, 22574}, {8741, 34322}, {8836, 31899}, {9115, 9116}, {9117, 16645}, {9166, 47857}, {9750, 37665}, {9760, 32552}, {9862, 16940}, {10612, 42095}, {11409, 12141}, {13651, 13875}, {13770, 13928}, {14639, 47861}, {14853, 41044}, {16267, 31415}, {16529, 16967}, {16807, 34376}, {16962, 31489}, {16964, 44526}, {18584, 49907}, {18800, 42035}, {19101, 25190}, {21843, 41944}, {22541, 25189}, {22702, 32466}, {36967, 44541}, {41113, 43448}, {41122, 43620}, {43274, 44219}, {46453, 47867}, {50858, 51204}
X(9113) = reflection of X(14) in X(42975)
X(9113) = homothetic center of pedal triangle of X(16) and antipedal triangle of X(14)
X(9113) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 5471, 14}, {6, 15484, 9112}, {14, 51203, 41043}, {16, 23013, 5474}, {115, 41672, 9112}, {395, 41746, 5464}, {1992, 40672, 42036}, {6783, 47856, 18581}, {41621, 47858, 2}, {47862, 47864, 4}, {50719, 50720, 41042}
Orthologic centers: X(9114)-X(9117)
Contributed by César Eliud Lozada, January 5, 2016
Let Li and Gc be the triangles T' and T", respectively, defined at X(9112), i.e., Li is the equilateral triangle with least area inscribed in ABC and Gc is the equilateral triangle with greatest area circumscribed to ABC.
Let Gi and Lc be the triangles T' and T", respectively, defined at X(9113), i.e., Gi is the equilateral triangle with greatest area inscribed in ABC and Lc is the equilateral triangle with least area circumscribed to ABC.
The following tables show the orthologic and parallelogic centers among these and the Napoleon triangles.
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X(9114) lies on these lines: {2,5469}, {13,543}, {14,2482}, {99,298}, {148,5459}, {524,6779}, {530,617}, {533,8595}, {542,1350}, {619,671}, {9112,9117}
X(9114) = midpoint of X(i),X(j) for these (i,j): (617,8591)
X(9114) = reflection of X(i) in X(j) for these (i,j): (13,5464), (14,2482), (148,5459), (671,619), (5463,99), (6777,5463)
X(9115) lies on these lines: {2,5472}, {6,5463}, {13,5055}, {14,5615}, {16,542}, {30,6782}, {99,3181}, {114,6108}, {115,395}, {187,524}, {298,754}, {299,620}, {396,532}, {531,6781}, {543,5471}, {616,6772}, {9113,9116}
X(9115) = midpoint of X(i),X(j) for these (i,j): (14,6779), (99,3181)
X(9115) = reflection of X(i) in X(j) for these (i,j): (115,395), (299,620), (9117,187)
X(9115) = {X(2482),X(5477)}-harmonic conjugate of X(9117)
X(9116) lies on these lines: {2,5470}, {13,2482}, {14,543}, {99,299}, {148,5460}, {524,6780}, {531,616}, {532,8594}, {542,1350}, {618,671}, {9113,9115}
X(9116) = midpoint of X(i),X(j) for these (i,j): (616,8591)
X(9116) = reflection of X(i) in X(j) for these (i,j): (13,2482), (14,5463), (148,5460), (671,618), (5464,99), (6778,5464)
X(9117) lies on these lines: {2,5471}, {6,5464}, {13,5611}, {14,5055}, {15,542}, {30,6783}, {99,3180}, {114,6109}, {115,396}, {187,524}, {298,620}, {299,754}, {395,533}, {530,6781}, {543,5472}, {617,6775}, {9112,9114}
X(9117) = midpoint of X(i),X(j) for these (i,j): (13,6780), (99,3180)
X(9117) = reflection of X(i) in X(j) for these (i,j): (115,396), (298,620), (9115,187)
X(9117) = {X(2482),X(5477)}-harmonic conjugate of X(9115)
Loud centers: X(9118)-X(9122)
This preamble and centers X(9118)-X(9122) were contributed by César Eliud Lozada, January 8, 2016.
In 1891, Frank Herbert Loud published the following theorem.
Let P1, P2, P3, Q1, Q2, Q3, R1, R2, R3 be nine distinct points on a cubic Ψ such that:
(1) P1, P2, P3 are collinear on a line, LP;
(2) P1, Q1, R1 are collinear on a line, L1;
(3) P2, Q2, R2 are collinear on a line, L2;
(4) P3, Q3, R3 are collinear on a line, L3.
Define S1, S2, S3 and T1, T2, T3 as follows:
(i) S1 = Q2R3∩Q1R1 and T1 = Q2R3∩Ψ - {Q2, R3}
(ii) S2 = Q3R1∩Q2R2 and T2 = Q3R1∩Ψ - {Q3, R1}
(iii) S3 = Q1R2∩Q3R3 and T3 = Q1R2∩Ψ - {Q1, R2}
Then (a) the points S1, S2 and S3 are collinear (on a line LS ); (b) the points T1, T2 and T3 are collinear (on a line LT ); and
(c) the lines LP, LS and LT concur.
Reference: Loud F.H. "A theorem in plane cubics", Annals of Mathematics, Vol. 6, No. 1 (Jun., 1891), pp. 5-6
The following table show how the 15 point and 6 lines in Loud's theorem are related. All the points except S1, S2, S3, lie on the cubic Ψ.
| Lines | L1 | L2 | L3 |
| LP | P1 | P2 | P3 |
| Q1 | Q2 | Q3 | |
| R1 | R2 | R3 | |
| --- | --- | --- | --- |
| LS | S1 | S2 | S3 |
| LT | T1 | T2 | T3 |
The point of concurrence of the lines LP, LS and LT, as described in the part (c) of the theorem, are represented here as the (P1, Q1, R1)-(P2, Q2, R2)-(P3, Q3, R3)-Loud point of Ψ, using ETC indexes; e.g., X(1249) is represented by 1249. Although the 3rd point in each triad is determined by the first 2 points, the 3rd is included for completeness.
X(9118) lies on these lines: {1,4}, {9,3341}, {1433,5776}, {2184,3176}, {5777,5910}
X(9119) lies on these lines: {1,6}, {3,1741}, {4,1903}, {19,5776}, {57,282}, {63,2062}, {65,281}, {226,6708}, {278,8807}, {284,8558}, {389,916}, {517,8804}, {1146,1845}, {1409,5089}, {1490,2270}, {1708,5120}, {1712,7129}, {1826,5798}, {1841,3330}, {1858,2264}, {1864,5802}, {3197,7719}
X(9120) lies on these lines: {1,3}, {4,282}, {610,1753}, {1422,3182}
Let ABC be a triangle with incenter I = X(1), De Longchamps point L = X(20), and Bevan point Be = X(40). Let DEF = cevian triangle of L. Let A' = ABe∩ID, and define B' and C' cyclically. The triangles ABC and A'B'C' are orthologic. The center of orthology of ABC with respect to A'B'C' is X(3345) and the center of orthology of A'B'C' with respect to ABC is X(9121). (Angel Montesdeoca, October 3, 2023 *)
X(9121) lies on these lines: {1,4}, {10,282}, {40,219}, {84,1214}, {227,2192}, {937,2257}, {1103,7070}, {2939,7991}, {2999,3149}, {3182,7011}, {3190,6769}, {3198,7973}, {3682,6282}, {4341,5768}
As a point on the Euler line, X(9122) has Shinagawa coefficients [F+E+r^2+4*R*r, -F-2*E-r^2-6*R*r]
X(9122) lies on these lines: {2,3}, {40,219}, {77,942}, {284,5706}
X(9122) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6918,7523), (3,7497,405), (3,7562,6883), (20,1817,3), (411,7520,3), (3651,7501,3)
Parry triangles and associated centers: X(9123)-X(9214)
This preamble and centers X(9123)-X(9214) were contributed by Randy Hutson, January 15, 2016.
Let A1 be the intersection, other than X(111), of the Parry circle and the line AX(111), and define B1 and C1 are cyclically. The triangle A1B1C1, here named the 1st Parry triangle, is similar to ABC and orthogonal to ABC, with similitude center X(110). Let A2 be the intersection, other than X(110), of the Parry circle and line AX(110), and define B2 and C2 cyclically. The triangle A2B2C2, here named the 2nd Parry triangle, is similar to ABC and orthogonal to ABC, with similitude center X(111). On the Parry circle, A1B1C1 and A2B2C2 are antipodes. The Euler line of A1B1C1 is the Euler line of A2B2C2, and the Brocard axis of A1B1C1 is the Brocard axis of A2B2C2. The Brocard axis of A1B1C1, and also of A2B2C2, is the Lemoine axis of ABC.
Let A3 be the intersection, other than X(2), of the Parry circle and the A-median, and define B3 and C3 cyclically. The triangle A3B3C3, here named the 3rd Parry triangle, is similar to the 4th Brocard triangle, with similitude center the intersection, other than X(2), of the Parry circle and the orthocentroidal circle. Also, A3B3C3 is the reflection of the circumsymmedial triangle of A2B2C2 in the common Brocard axis of A1B1C1 (which is also the Brocard axis of A2B2C2 and A3B3C3). The triangles A2B2C2 and A3B3C3 are perspective at X(647). Let A'B'C' be the reflection of the circumsymmedial triangle in the Brocard axis. Then A3B3C3 is similar to A'B'C', with similitude center X(111).
The appearance of (i,j) in the following list means that X(i)-of-ABC = X(j)-of A1B1C1: (2,99), (23,691), (15,X(9202)), (16,X(9203)), (23,691), (30,523), (110,110), (111,1296), (351,3), (352,2709), (353,6242), (523,30), (5027,182), (5638,1380), (5639,1379), (8644,187)
The appearance of (i,j) in the following list means that X(i)-of-ABC = X(j)-of A2B2C2: (2,98), (15,2378), (16,2379), (23,842), (30,523), (110,74), (111,111), (351,3), (352,843), (353,6323), (523,30), (647,187), (1637,115), (3569,6), (5638,1379), (5639,1380), (6137,16), (6138,15)
The appearance of (i,j) in the following list means that X(i)-of-ABC = X(j)-of A3B3C3: (23,1296), (111,691), (351,3), (352,110), (511,512), (512,511), (647,187), (3569,6), (6137,16), (6138,15)
A-vertices of the three Parry triangles are given by Peter Moses (January 17, 2016):
A1 = -3 a^4+2 a^2 b^2-b^4+2 a^2 c^2+b^2 c^2-c^4,b^2 (a^2+b^2-2 c^2),c^2 (a^2-2 b^2+c^2)
A2 = (b^2-c^2) (a^4-b^4+b^2 c^2-c^4), (a^2-b^2) b^2 (2 a^2-b^2-c^2),c^2 (c^2-a^2) (2 a^2-b^2-c^2)
A3 = a^2 (a^4-3 a^2 b^2+2 b^4-3 a^2 c^2+b^2 c^2+2 c^4),b^2 c^2 (2 a^2-b^2-c^2),b^2 c^2 (2 a^2-b^2-c^2)
X(9123) lies on these lines: {2,2793}, {98,9215}, {99,110}, {111,2408}, {351,523}, {804,1649}, {888,5640}, {1499,8598}, {2826,9978}, {3667,9810}, {4778,9811}, {5466,9189}, {8371,11176}, {8644,8704}, {9003,9138}, {9158,9213}
X(9123) = isogonal conjugate of X(9124)
X(9123) = reflection of X(2) in X(9125)
X(9123) = crossdifference of every pair of points on line X(574)X(3124)
X(9123) = X(2)-of-1st-Parry-triangle
X(9123) = X(376)-of-2nd-Parry-triangle
X(9124) lies on these lines: {512,9145}, {523,9146}, {2709,6088}, {3908,4705}, {5107,9027}
X(9124) = isogonal conjugate of X(9123)
X(9124) = trilinear pole of line X(574)X(3124)
X(9125) lies on these lines: {2,2793}, {230,231}, {351,690}, {373,888}, {1499,4786}, {2799,9168}, {5027,5651}, {5108,9130}, {5466,9131}, {5648,9003}, {9147,9191}
X(9125) = midpoint of X(2) and X(9123)
X(9125) = tripolar centroid of X(1992)
X(9125) = centroid of (degenerate) side-triangle of ABC and 1st Parry triangle
X(9125) = crossdifference of every pair of points on line X(3)X(111)
X(9125) = intersection of orthic axes of ABC and anti-Artzt triangle
X(9126) lies on these lines: {3,351}, {30,11176}, {182,9023}, {186,9213}, {511,9188}, {512,5926}, {526,1511}, {549,804}, {1499,4786}, {3524,9147}, {5054,9148}, {6088,9175}
X(9126) = midpoint X(3) and X(351)
X(9126) = center of circle {{X(3),X(110),X(111),X(187),X(351)}}
X(9127) lies on circle {{X(6),X(187),X(351),X(9127),X(9128),X(9129)}} and these lines: {2,6}, {351,1499}, {511,9172}, {549,10166}, {2502,8598}
X(9127) = midpoint of X(2) and X(352)
X(9128) lies on circle {{X(6),X(187),X(351),X(9127),X(9128),X(9129)}} and these lines: {6,23}, {30,10166}, {351,5926}, {858,10160}, {7426,9830}
X(9128) = midpoint of X(23) and X(353)
X(9129) lies on the Brocard circle, the Parry circle of the 2nd Brocard triangle, the circle {{X(6),X(187),X(351),X(9127),X(9128),X(9129)}} and these lines: {3, 351}, {6, 110}, {125, 6719}, {126, 5972}, {523, 9179}, {542, 9169}, {543, 1316}, {690, 5108}, {1495, 9181}, {1641, 9144}, {2793, 6795}
X(9129) = midpoint of X(110) and X(111)
X(9129) = radical trace of circumcircle and Parry circle
X(9129) = radical trace of circumcircle and circle {{X(3),X(110),X(111),X(187),X(351)}}
X(9129) = inverse-in-circumcircle of X(351)
X(9129) = inverse-in-Parry-circle of X(3)
X(9129) = inverse-in-circle-{{X(3),X(110),X(111),X(187),X(351)}} of X(9130)
X(9129) = {X(3),X(351)}-harmonic conjugate of X(9130)
X(9129) = X(110)-of-2nd-Brocard-triangle
X(9129) = anticenter of cyclic quadrilateral X(98)X(99)X(110)X(111)
X(9129) = QA-P2 (Euler-Poncelet Point)-of-quadrangle X(98)X(99)X(110)X(111)
X(9130) lies on these lines: {3, 351}, {110, 9184}, {2502, 5210}, {2854, 9215}, {5108, 9125}
X(9130) = midpoint of PU(63)
X(9130) = midpoint of X(9215) and X(9216)
X(9130) = inverse-in-circle-{{X(3),X(110),X(111),X(187),X(351)}} of X(9129)
X(9130) = {X(3),X(351)}-harmonic conjugate of X(9129)
X(9130) = harmonic center of circumcircle and Parry circle; see X(8072)
X(9131) lies on these lines: {2, 9134}, {110, 925}, {111, 2374}, {351, 523}, {514, 9811}, {522, 9810}, {525, 9135}, {690, 8591}, {804, 3268}, {826, 5027}, {2793, 6054}, {2799, 9147}, {2804, 9980}, {3566, 6562}, {3569, 3800}, {5466, 9125}, {6131, 7664}, {7927, 9208}, {8029, 11176}, {9033, 9138}, {9148, 10190}
X(9131) = isogonal conjugate of X(9132)
X(9131) = isotomic conjugate of X(9133)
X(9131) = anticomplement of X(9134)
X(9131) = radical center of {polar circle, outer Vecten circle, inner Vecten circle}
X(9131) = X(4)-of-1st-Parry-triangle
X(9131) = X(20)-of-2nd-Parry-triangle
The trilinear polar of X(9132) passes through X(574). (Randy Hutson, January 15, 2016)
X(9132) lies on these lines: {924, 9145}, {2872, 10425}, {6563, 9133}
X(9132) = isogonal conjugate of X(9131)
X(9132) = vertex conjugate of PU(63)
X(9133) lies on this line: {6563, 9132}
X(9133) = isotomic conjugate of X(9131)
X(9133) = trilinear pole of line X(599)X(626)
X(9134) lies on these lines: {2, 9131}, {125, 136}, {351, 2793}, {523, 7625}, {671, 690}, {804, 1637}, {826, 850}, {1194, 7656}, {2501, 3566}, {2799, 8029}, {10189, 11176}
X(9134) = complement of X(9131)
X(9134) = tripolar centroid of X(6) with respect to the orthic triangle
X(9134) = crosssum of X(3) and X(351)
X(9134) = crosspoint of X(4) and X(892)
X(9134) = crossdifference of every pair of points on line X(1384)X(1501)
X(9134) = centroid of degenerate side-triangle of outer and inner Vecten triangles
X(9135) lies on these lines: {6, 6088}, {23, 9212}, {110, 1296}, {111, 2444}, {154, 5653}, {187, 237}, {525, 9131}, {526, 2930}, {690, 8030}, {804, 5652}, {1499, 8598}, {1976, 2422}, {3309, 9810}, {3800, 9979}, {9138, 9157}, {9148, 11183}, {9485, 11152}
X(9135) = reflection of X(351) in X(5027)
X(9135) = X(6)-of-1st-Parry-triangle
X(9135) = X(1350)-of-2nd-Parry-triangle
X(9135) = X(1350)-of-3rd-Parry-triangle
X(9136) lies on the circumcircle, circle {{X(2),X(6),X(110)}}, and these lines: {2, 6082}, {6, 2709}, {98, 5914}, {99, 9487}, {110, 2030}, {111, 8644}, {187, 1296}, {691, 1384}, {1691, 6233}, {2696, 7735}
X(9136) = trilinear pole of line X(6)X(6088)
X(9136) = Λ(X(115), X(599))
X(9136) = Ψ(X(2), X(2793))
X(9136) = Ψ(X(6), X(9135))
X(9136) = circumcircle intercept, other than X(110), of circle {{X(2),X(6),X(110)}}
X(9136) = Λ(X(6), X(2482))
X(9136) = barycentric product of circumcircle intercepts of line X(2)X(2793)
X(9137) lies on these lines: {2, 9175}, {23, 8644}, {110, 249}, {111, 10102}, {351, 523}, {842, 9215}, {1995, 9178}, {5640, 9171}, {8675, 9138}, {9147, 9158}
X(9137) = X(23)-of-1st-Parry-triangle
X(9137) = crossdifference of every pair of points on line X(574)X(1648)
X(9138) lies on the Parry circle and these lines: {2, 690}, {15, 6138}, {16, 6137}, {23, 512}, {74, 111}, {110, 351}, {352, 647}, {511, 9213}, {523, 9158}, {542, 9147}, {686, 5622}, {804, 9140}, {895, 9023}, {2771, 9980}, {2854, 9156}, {5027, 9999}, {5113, 7711}, {8674, 9978}, {8675, 9137}, {9003, 9123}, {9033, 9131}, {9135, 9157}, {9984, 9998}
X(9138) = reflection of X(110) in X(351)
X(9138) = crossdifference of every pair of points on line X(115)X(2502)
X(9138) = X(74)-of-1st-Parry-triangle
X(9138) = X(110)-of-2nd-Parry-triangle
X(9138) = X(843)-of-3rd-Parry-triangle
X(9138) = Parry-circle-antipode of X(110)
X(9138) = perspector of 1st Parry triangle and 1st Parry triangle of 2nd Parry triangle
X(9138) = trilinear pole, wrt 2nd Parry triangle, of Lemoine axis
X(9139) lies on these lines: {74, 511}, {111, 232}, {250, 5191}, {262, 2452}, {325, 892}, {523, 9154}, {2394, 2408}, {3545, 5627}, {5968, 9717}, {8430, 10630
X(9139) = isogonal conjugate of X(5642)
X(9139) = trilinear pole of line X(74)X(111) (the line through X(74) of ABC and the 1st Parry triangle)
X(9139) = vertex conjugate of X(6) and X(250)
Let A'B'C' be the anti-orthocentroidal triangle. Let A" be the reflection of A in line B'C', and define B" and C" cyclically. Then X(9140) is the centroid of A"B"C"; see X(23) and X(11002). (Randy Hutson, December 10, 2016)
X(9140) lies on these lines: {2, 98}, {4, 541}, {5, 5643}, {6, 6032}, {13, 5916}, {14, 5917}, {22, 5621}, {30, 74}, {67, 524}, {69, 5505}, {111, 1648}, {113, 3545}, {115, 6792}, {141, 5648}, {146, 3839}, {210, 2836}, {290, 850}, {323, 9976}, {381, 5640}, {389, 7565}, {399, 5055}, {511, 10989}, {519, 7984}, {526, 9148}, {528, 10778}, {539, 5504}, {599, 2854}, {671, 690}, {804, 9138}, {1112, 5064}, {1494, 3268}, {1503, 7426}, {1511, 5054}, {1656, 5609}, {1853, 2781}, {2574, 10719}, {2575, 10720}, {2771, 3753}, {2772, 10710}, {2773, 10716}, {2774, 10708}, {2777, 3543}, {2779, 10709}, {2782, 9159}, {2842, 10713}, {2850, 10715}, {2930, 6698}, {3024, 11238}, {3028, 11237}, {3146, 10990}, {3524, 6699}, {3582, 10091}, {3584, 10088}, {3656, 7978}, {3818, 10545}, {3830, 10113}, {3845, 7728}, {5032, 5095}, {5169, 5476}, {5449, 7552}, {5465, 9166}, {5653, 11182}, {6033, 7698}, {6143, 9706}, {8674, 10707}, {9003, 9191}, {9517, 10718}, {9830, 9832}, {11117, 11118}
X(9140) = midpoint of X(2) and X(3448)
X(9140) = reflection of X(i) in X(j) for these (i,j): (2,125), (110,2), (9144,115)
X(9140) = isotomic conjugate of X(9141)
X(9140) = complement of X(9143)
X(9140) = anticomplement of X(5642)
X(9140) = X(2)-Ceva conjugate of X(36904)
X(9140) = X(23) of orthocentroidal triangle
X(9140) = reflection of X(2) in line X(115)X(125)
X(9140) = reflection of X(671) in the Fermat axis
X(9140) = inverse-in-orthocentroidal-circle of X(5640)
X(9140) = inverse-in-{circumcircle, nine-point circle}-inverter of X(6055)
X(9140) = intersection of the tangent to circle {{X(13),X(14),X(15)}} at X(14) and the tangent to circle {{X(13),X(14),X(16)}} at X(13)
X(9140) = intersection of tangents at X(13) and X(14) to the Brocard (second) cubic (K018)
X(9140) = circle-{{X(381),X(599),X(9169),X(36194)}}-inverse of X(2)
X(9140) = Artzt-to-McCay similarity image of X(23)
X(9140) = X(11002)-of-anti-orthocentroidal-triangle
X(9141) lies on these lines: {30, 3268}, {110, 5641}, {316, 5468}, {325, 3233}, {340, 4240}, {476, 1494}, {524, 9979}, {542, 850}, {892, 10557}
X(9141) = isogonal conjugate of X(9142)
X(9141) = isotomic conjugate of X(9140)
X(9142) lies on these lines: {2, 3014}, {3, 2854}, {6, 157}, {74, 526}, {98, 523}, {111, 351}, {187, 2393}, {323, 3001}, {512, 2378}, {524, 9888}, {729, 755}, {843, 6323}, {895, 5467}, {1495, 3003}, {1632, 7668}, {2080, 8705}, {2380, 2381}, {2930, 9155}, {8546, 11171}, {9019, 9301}
X(9142) = reflection of X(9145) in X(3)
X(9142) = isogonal conjugate of X(9141)
X(9142) = X(3)-of-2nd-anti-Parry-triangle
X(9143) lies on these lines: {2, 98}, {4, 5609}, {6, 7693}, {20, 541}, {23, 524}, {30, 146}, {69, 5648}, {74, 10304}, {99, 9184}, {111, 5477}, {113, 3839}, {148, 9144}, {156, 2888}, {265, 3545}, {376, 5663}, {519, 2948}, {526, 9147}, {543, 10554}, {597, 7605}, {599, 6800}, {690, 8591}, {895, 5032}, {1112, 7714}, {1503, 10989}, {1511, 3524}, {1641, 10488}, {1992, 2854}, {2502, 6792}, {2771, 3655}, {2781, 11206}, {2836, 3873}, {3024, 10385}, {3292, 5189}, {3543, 10706}, {3564, 7426}, {5054, 10264}, {5055, 10272}, {5191, 8724}, {5468, 10553}, {5476, 7533}, {6053, 10733}, {7519, 9970}, {8703, 10620}, {10552, 10754}
X(9143) = reflection of X(2) in X(110)
X(9143) = anticomplement of X(9140)
X(9143) = isotomic conjugate of anticomplement of X(36904)
X(9143) = crosspoint of X(399) and X(2930) with respect to both the excentral and tangential triangles
X(9144) lies on the orthocentroidal circle and these lines: {2, 690}, {4, 542}, {74, 6055}, {98, 541}, {99, 5642}, {110, 543}, {113, 6054}, {115, 6792}, {125, 9166}, {146, 11177}, {148, 9143}, {381, 11005}, {526, 6787}, {1641, 9129}, {2782, 5655}, {5648, 5969}, {5663, 6785}
X(9144) = reflection of X(9140) in X(115)
X(9144) = reflection of X(2) in the Fermat axis
X(9144) = reflection of X(671) in line X(115)X(125)
X(9144) = reflection of X(691) in the centroid of its (degenerate) pedal triangle
X(9144) = X(691)-of-orthocentroidal-triangle
X(9144) = reflection of X(2) in X(5465)
X(9144) = Λ(Lemoine axis) wrt orthocentroidal triangle
X(9144) = trilinear pole, wrt orthocentroidal triangle, of line X(2)X(6)
The trilinear polar of X(9145) passes through X(574). (Randy Hutson, January 15, 2016) p
X(9145) lies on these lines: {3, 2854}, {6, 9155}, {99, 523}, {110, 351}, {157, 10607}, {187, 9027}, {512, 9124}, {524, 2080}, {574, 8542}, {924, 9132}, {1296, 6088}, {1326, 9026}, {1350, 2871}, {2709, 6233}, {2930, 5191}, {3733, 5546}, {4557, 4565}, {5063, 5651}, {5486, 9516}, {5610, 5614}, {8675, 9181}
X(9145) = reflection of X(9142) in X(3)
X(9145) = isogonal conjugate of X(8599)
X(9145) = crossdifference of every pair of points on line X(115)X(2793)
X(9145) = X(3)-of-1st-anti-Parry-triangle
X(9145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (110,4558,5467), (110,5467,1576), (1634,4558,1576), (1634,5467,110)
X(9146) lies on these lines: {2, 5503}, {69, 5505}, {99, 110}, {111, 5108}, {126, 6792}, {183, 7998}, {352, 538}, {511, 5971}, {523, 9124}, {524, 10717}, {599, 8288}, {645, 7192}, {670, 850}, {805, 9066}, {1296, 1499}, {1649, 9216}, {2709, 8704}, {3098, 6031}, {3265, 4558}, {3952, 4573}, {4069, 7203}, {6563, 9132}, {7798, 9463}, {8030, 10554}, {9182, 9186}
X(9146) = isotomic conjugate of X(8599)
X(9146) = anticomplement of X(6791)
X(9146) = trilinear pole of line X(574)X(599)
X(9147) lies on the Parry circle and these lines: {2, 351}, {3, 9153}, {23, 385}, {30, 9213}, {98, 111}, {99, 110}, {352, 1499}, {353, 7709}, {376, 2780}, {512, 11229}, {524, 9212}, {526, 9143}, {530, 9163}, {531, 9162}, {542, 9138}, {543, 9156}, {647, 5996}, {686, 6776}, {740, 1635}, {812, 1962}, {850, 8651}, {888, 3228}, {1649, 7711}, {1992, 9023}, {2783, 9980}, {2787, 9978}, {2799, 9131}, {3413, 5639}, {3414, 5638}, {3524, 9126}, {3569, 9862}, {4108, 8644}, {4120, 4368}, {4728, 10180}, {5113, 11182}, {8782, 9999}, {9125, 9191}, {9137, 9158}, {9479, 11123}
X(9147) = reflection of X(2) in X(351)
X(9147) = anticomplement of X(9148)
X(9147) = inverse-in-circumcircle of X(9153)
X(9147) = crossdifference of every pair of points on line X(39)X(373)
X(9147) = X(9150)-Ceva conjugate of X(2)
X(9147) = vertex conjugate of X(2) and X(9149)
X(9147) = X(98)-of-1st-Parry-triangle
X(9147) = X(99)-of-2nd-Parry-triangle
X(9147) = Parry-circle-antipode of X(2)
X(9147) = orthocenter of X(2)X(3)X(9149)
X(9147) = radical center of circumcircle, Artzt circle, anti-Artzt circle
X(9147) = trilinear pole, wrt 2nd Parry triangle, of line X(1499)X(1513)
X(9148) lies on these lines: {2, 351}, {114, 126}, {115, 125}, {325, 523}, {381, 2780}, {526, 9140}, {599, 9023}, {647, 7631}, {740, 4928}, {782, 11205}, {881, 8842}, {888, 6786}, {1916, 5466}, {2799, 8029}, {3835, 8663}, {4155, 4728}, {4379, 9279}, {5054, 9126}, {5652, 6033}, {6088, 10717}, {9131, 10190}, {9135, 11183}, {9185, 9478}, {9479, 9979}
X(9148) = reflection of X(351) in X(2)
X(9148) = isogonal conjugate of X(32717)
X(9148) = isotomic conjugate of X(9150)
X(9148) = complement of X(9147)
X(9148) = crossdifference of every pair of points on line X(32)X(110)
X(9148) = tripolar centroid of X(76)
X(9148) = centroid of de Longchamps-line intercepts with sidelines of ABC
X(9148) = radical center of nine-point circles of ABC, Artzt and anti-Artzt triangles
X(9148) = radical center of de Longchamps circles of ABC, Artzt and anti-Artzt triangles
X(9148) = Hutson-Parry-circle inverse of X(11182)
X(9148) = midpoint of Kiepert hyperbola intercepts of de Longchamps line
X(9148) = {X(13636),X(13722)}-harmonic conjugate of X(11182)
X(9149) lies on these lines: {2, 1634}, {3, 76}, {6, 110}, {22, 7669}, {23, 385}, {25, 648}, {147, 5877}, {542, 7418}, {4230, 6103}, {7485, 8556}, {7492, 8266}
X(9149) = crossdifference of every pair of points on line X(39)X(690)
X(9149) = vertex conjugate of Parry circle antipodes X(2) and X(9147)
X(9149) = perspector of circummedial triangle and circumcevian triangle of X(9147)
X(9149) = orthocenter of X(2)X(3)X(9147)
Let A' = BC∩X(2)X(39), and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X(9150). (Randy Hutson, January 15, 2016)
X(9150) lies on the circumcircle and these lines: {2, 5970}, {99, 669}, {100, 4601}, {101, 4600}, {106, 4615}, {109, 4620}, {110, 4590}, {111, 385}, {670, 4108}, {729, 3231}, {739, 2106}, {805, 5468}, {842, 6035}, {843, 5108}, {2378, 6295}, {2379, 6582}, {2407, 9091}, {6031, 9831}
X(9150) = isogonal conjugate of X(888)
X(9150) = isotomic conjugate of X(9148)
X(9150) = anticomplement of X(9151)
X(9150) = cevapoint of X(2) and X(9147) (Parry circle antipodes)
X(9150) = {circumcircle, nine-point circle}-inverter-inverse of X(9152)
X(9150) = trilinear pole of line X(6)X(99)
X(9150) = Ψ(X(6), X(99))
X(9150) = Λ(X(351), X(865))
X(9150) = Λ(X(670), X(888))
X(9150) = Λ(X(694), X(882))
X(9151) lies on the nine-point circle and these lines: {2, 5970}, {126, 325}, {1648, 2679}
X(9151) = complement of X(9150)
X(9151) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5970)
X(9152) lies on the nine-point circle and these lines: {2, 5970}, {115, 325}, {125, 5031}, {141, 2679}, {626, 5099}
X(9152) = complement of X(5970)
X(9152) = inverse-in-{circumcircle, nine-point circle}-inverter of X(9150)
X(9153) lies on the Parry circle and these lines: {2, 1634}, {3, 9147}, {110, 3111}, {7468, 9158}, {7496, 9828}
X(9153) = inverse-in-circumcircle of X(9147)
X(9153) = intersection, other than X(2), of Parry circle and circle O(2,3)
X(9154) lies on these lines: {30, 98}, {111, 6037}, {125, 5641}, {290, 892}, {523, 9139}, {685, 1990}, {879, 5466}, {2395, 2408}, {2698, 6784}, {5967, 9214}, {5968, 11174}
X(9154) = isogonal conjugate of X(9155)
X(9154) = trilinear pole of line X(98)X(111) "(the line through the Tarry points of ABC and the 1st Parry triangle; also the radical axis of the circumcircle and Artzt circle)
X(9155) lies on these lines: {2, 2782}, {3, 74}, {6, 9145}, {23, 7711}, {39, 373}, {99, 1316}, {111, 5024}, {114, 868}, {187, 3292}, {237, 511}, {323, 2080}, {325, 5112}, {351, 690}, {468, 3266}, {574, 2502}, {684, 2491}, {694, 11175}, {1634, 2854}, {1976, 5085}, {2021, 3231}, {2396, 5976}, {2930, 9142}, {2967, 4230}, {2987, 5093}, {3001, 8705}, {3003, 9027}, {3095, 11002}, {3793, 10552}, {5026, 5967}, {5029, 6184}, {5162, 8627}, {5467, 6593}, {9306, 9734}
X(9155) = isogonal conjugate of X(9154)
X(9155) = crossdifference of every pair of points on line X(98)X(111)
X(9155) = pole of line X(2)X(98) with respect to the Parry circle
X(9155) = pole of Lemoine axis wrt Thomson-Gibert-Moses hyperbola
X(9155) = intersection of tangents to Parry circle at X(2) and X(110)
X(9155) = intersection of tangents to Thomson-Gibert-Moses hyperbola at X(5638) and X(5639)
X(9156) lies on the Parry circle and these lines: {2, 2793}, {23, 8644}, {110, 1296}, {111, 351}, {352, 512}, {353, 5027}, {511, 9212}, {543, 9147}, {804, 10717}, {2805, 9980}, {2830, 9978}, {2854, 9138}, {9023, 10765}, {9208, 9998}
X(9156) = reflection of X(111) in X(351)
X(9156) = crossdifference of every pair of points on line X(2482)X(6791)
X(9156) = X(111)-of-1st-Parry-triangle
X(9156) = X(1296)-of-2nd-Parry-triangle
X(9156) = X(842)-of-3rd-Parry-triangle
X(9156) = Parry-circle-antipode of X(111)
X(9156) = trilinear pole, wrt 1st Parry triangle, of line X(511)X(9135)
X(9156) = perspector of 2nd Parry triangle and 2nd Parry triangle of 1st Parry triangle
X(9157) lies on the Parry circle and these lines: {2, 2794}, {22, 110}, {25, 111}, {51, 251}, {127, 7494}, {132, 6995}, {184, 353}, {351, 2881}, {352, 1495}, {427, 10735}, {2393, 10313}, {2799, 9131}, {2806, 9980}, {2831, 9978}, {2848, 9185}, {6636, 7711}, {6676, 10749}, {6720, 7392}, {7598, 8854}, {7599, 8855}, {7664, 10565}, {8644, 9213}, {9135, 9138}
X(9157) = X(112)-of-1st-Parry-triangle
X(9157) = X(1297)-of-2nd-Parry-triangle
X(9157) = 1st-orthosymmedial-to-ABC similarity image of X(2)
X(9157) = X(2)-of-1st-anti-orthosymmedial-triangle
X(9157) = orthoptic-circle-of-Steiner-inellipse-inverse of X(36519)
X(9158) lies on the Parry circle and these lines: {2, 9159}, {30, 110}, {111, 230}, {352, 3016}, {353, 2549}, {523, 9138}, {3258, 7711}, {7468, 9153}, {9123, 9213}, {9137, 9147}
X(9158) = X(476)-of-1st-Parry-triangle
X(9158) = X(477)-of-2nd-Parry-triangle
X(9158) = intersection of the tangent to circle {{X(13),X(14),X(15)}} at X(15) and the tangent to circle {{X(13),X(14),X(16)}} at X(16)
X(9159) lies on the Hutson-Parry circle and these lines: {2, 9158}, {3, 476}, {30, 5640}, {110, 6795}, {111, 9828}, {262, 10989}, {523, 7998}, {566, 858}, {1316, 7698}, {2088, 2549}, {2782, 9140}
X(9159) = intersection of the tangent to circle {{X(13),X(15),X(16)}} at X(13) and the tangent to circle {{X(14),X(15),X(16)}} at X(14)
X(9160) lies on the circumcircle and these lines: {3, 9161}, {98, 5663}, {99, 526}, {111, 3016}, {249, 10420}, {476, 512}, {477, 511}, {648, 1300}, {842, 3581}, {1291, 9218}
X(9160) = reflection of X(476) in the Brocard axis
X(9160) = reflection of X(99) in line X(3)X(74)
X(9160) = circumcircle- antipode of X(9161)
X(9160) = reflection of X(9161) in X(3)
X(9160) = isogonal conjugate of infinite point of tangent to Brocard circle at X(1316)
X(9160) = Λ(X(98), X(477))
X(9160) = Λ(X(99), X(476))
X(9161) lies on the circumcircle and these lines: {3, 9160}, {50, 2715}, {98, 526}, {99, 5663}, {107, 1112}, {476, 511}, {477, 512}
X(9161) = reflection of X(9160) in X(3)
X(9161) = reflection of X(477) in the Brocard axis
X(9161) = reflection of X(98) in line X(3)X(74)
X(9161) = circumcircle-antipode of X(9160)
X(9161) = Λ(X(2), X(9158))
X(9161) = Λ(X(98), X(476))
X(9161) = Λ(X(99), X(477))
X(9162) lies on the Parry circle and these lines: {15, 351}, {16, 512}, {23, 5608}, {110, 9202}, {111, 2378}, {352, 6137}, {511, 9163}, {531, 9147}
X(9162) = reflection of X(15) in X(351)
X(9162) = X(2378)-of-1st-Parry-triangle
X(9162) = Parry-circle-antipode of X(15)
X(9163) lies on the Parry circle and these lines: {15, 512}, {16, 351}, {23, 5607}, {110, 9203}, {111, 2379}, {352, 6138}, {511, 9162}, {530, 9147}
X(9163) = reflection of X(16) in X(351)
X(9163) = X(2379)-of-1st-Parry-triangle
X(9163) = Parry-circle-antipode of X(16)
X(9164) lies on these lines: {2, 5914}, {523, 2482}, {524, 620}, {599, 5967}, {10415, 10717}
X(9164) = isotomic conjugate of X(9166)
X(9164) = trilinear pole of line X(690)X(8030) (the Fermat axis of the 1st Parry triangle)
X(9164) = anticomplement of X(9165)
X(9165) lies on these lines: {2, 5914}, {523, 5461}, {524, 6722}
X(9165) = complement of X(9164)
X(9165) = X(2)-Ceva conjugate of X(9167)
Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. Let A" be the nine-point center of BCX(14), and define B" and C" cyclically. In A'B'C', let U be the internal angle bisector of angle A'; in A''B''C'', let V be the internal angle bisector of angle A''. Let A* = U∩V, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC, and center of homothety is X(9166). (Randy Hutson, January 15, 2016)
X(9166) lies on these lines: {2, 99}, {4, 6055}, {5, 6054}, {6, 11161}, {13, 5460}, {14, 5459}, {76, 5503}, {83, 7817}, {98, 381}, {114, 5071}, {125, 9144}, {187, 8597}, {230, 8352}, {315, 9740}, {316, 3793}, {325, 8355}, {376, 6036}, {524, 5103}, {530, 5470}, {531, 5469}, {542, 3545}, {547, 8724}, {549, 6321}, {551, 9884}, {597, 8593}, {599, 7934}, {625, 7840}, {1078, 7610}, {1916, 9466}, {2023, 7757}, {2782, 5055}, {2794, 3839}, {3091, 7856}, {3363, 7792}, {3525, 10992}, {3582, 10054}, {3584, 10070}, {3679, 7983}, {3767, 7812}, {3828, 9881}, {3832, 10991}, {3845, 10722}, {3849, 8859}, {3851, 7878}, {5025, 7854}, {5066, 6033}, {5077, 7771}, {5182, 7884}, {5254, 9771}, {5309, 8176}, {5463, 6670}, {5464, 6669}, {5465, 9140}, {5476, 10753}, {5485, 8781}, {5569, 7746}, {6174, 10769}, {6784, 6787}, {7752, 9770}, {7760, 7775}, {7763, 9741}, {7801, 7899}, {7810, 7911}, {7825, 9939}, {7828, 8370}, {7831, 11168}, {7859, 8367}, {7870, 7887}, {7944, 8360}, {8288, 9169}, {8371, 9180}, {8592, 10352}
X(9166) = isotomic conjugate of X(9164)
X(9166) = anticomplement of X(9167)
X(9166) = centroid of X(2)X(13)X(14)
X(9167) = QA-P34 (Euler-Poncelet Point of the Centroid Quadrangle) of quadrangle ABCX(2); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/62-qa-p34.html. (Randy Hutson, January 15, 2016)
X(9167) lies on these lines: {2, 99}, {30, 10256}, {32, 9770}, {114, 549}, {140, 6055}, {381, 6721}, {524, 1692}, {542, 5054}, {547, 9880}, {599, 5477}, {625, 8598}, {631, 6054}, {1003, 8176}, {1153, 7880}, {1506, 8369}, {1569, 9466}, {2794, 3524}, {3455, 7484}, {3619, 11161}, {3628, 10992}, {3767, 9741}, {3788, 5569}, {5309, 11165}, {6036, 8724}, {7610, 7749}, {7753, 11184}, {7794, 7870}, {7799, 8859}, {7807, 9698}, {7818, 8182}, {7833, 7940}, {7874, 8359}, {7889, 8366}, {7934, 11149}, {10303, 11177}
X(9167) = complement of X(9166)
X(9167) = X(2)-Ceva conjugate of X(9165)
X(9168) lies on these lines: {2, 523}, {99, 110}, {376, 1499}, {512, 7998}, {525, 9741}, {543, 9180}, {631, 8151}, {647, 9465}, {669, 7492}, {804, 8592}, {842, 2770}, {850, 11059}, {1316, 2394}, {2793, 6054}, {2799, 9125}, {3533, 10279}, {3906, 7757}, {4108, 8704}, {5463, 9205}, {5464, 9204}, {6563, 7493}, {9189, 9979}
X(9168) = anticomplement of X(8371)
X(9168) = reflection of X(2) in X(1649)
X(9169) lies on the McCay circumcircle and these lines: {2, 6}, {110, 8787}, {111, 9830}, {125, 5461}, {182, 10166}, {381, 1499}, {542, 9129}, {543, 6791}, {598, 843}, {671, 3124}, {2482, 6388}, {2502, 8593}, {5969, 10717}, {8288, 9166}
X(9169) = inverse-in-orthocentroidal circle of X(8371)
X(9169) = X(110)-of-McCay-triangle
X(9169) = Artzt-to-McCay similarity image of X(111)
X(9169) = inverse-in-Hutson-Parry-circle of X(381)
The line X(99)X(524), of which X(9170) is the trilinear pole) is the line through X(99) parallel to the trilinear polar of X(99). (Randy Hutson, January 15, 2016)
X(9170) lies on these lines: {2, 892}, {99, 1649}, {843, 5108}, {2396, 2418}, {4590, 5468}
X(9170) = isogonal conjugate of X(9171)
X(9170) = isotomic conjugate of X(8371)
X(9170) = trilinear pole of line X(99)X(524)
X(9171) lies on these lines: {6, 512}, {351, 865}, {511, 9175}, {523, 597}, {524, 11182}, {882, 9009}, {895, 5653}, {1641, 8371}, {2489, 8541}, {2492, 9023}, {3221, 9971}, {5027, 6088}, {5640, 9137}, {7625, 8542}
X(9171) = isogonal conjugate of X(9170)
X(9171) = crossdifference of every pair of points on line X(99)X(524)
X(9172) lies on these lines: {2, 99}, {5, 10162}, {30, 5512}, {351, 2793}, {373, 597}, {468, 8754}, {511, 9127}, {524, 5914}, {542, 9129}, {1084, 3291}, {1296, 3524}, {1641, 3124}, {2805, 4755}, {3325, 5298}, {3839, 10734}, {3849, 5913}, {4995, 6019}, {5054, 11258}, {5055, 10748}, {6088, 11176}, {6094, 7610}, {8176, 9745}, {10166, 10168}
X(9172) = midpoint of X(2) and X(111)
X(9172) = radical trace of Parry circle and Hutson-Parry circle
X(9172) = common radical trace of the similitude circles of pairs of the McCay circles
X(9172) = radical trace of Parry circles of ABC and Artzt triangle
X(9173) lies on these lines: {6, 1344}, {351, 2793}, {691, 1113}
X(9173) = reflection of X(9174)in X(9175)
X(9173) = internal center of similitude of Parry circle and Hutson-Parry circle
X(9173) = {X(351),X(8371)}-harmonic conjugate of X(9174)
X(9174) lies on these lines: {6, 1345}, {351, 2793}, {691, 1114}
X(9174) = reflection of X(9173) in X(9175)
X(9174) = {X(351),X(8371)}-harmonic conjugate of X(9173)
X(9174) = external center of similitude of Parry circle and Hutson-Parry circle
X(9175) lies on these lines: {2, 9137}, {3, 9178}, {182, 512}, {351, 2793}, {378, 2489}, {511, 9171}, {523, 549}, {542, 11182}, {597, 1499}, {1995, 8644}, {2492, 2780}, {6041, 7418}, {6088, 9126}, {7624, 8675}, {8704, 11171}, {10168, 11183}
X(9175) = midpoint of X(9173) and X(9174)
X(9175) = center of circle {{X(2),X(3),X(6),X(111),X(691)}}
X(9175) = isogonal conjugate, wrt triangle X(2)X(3)X(6), of X(5652)
X(9175) = harmonic center of Parry circle and Hutson-Parry circle
X(9175) = harmonic center of circles {{X(6),X(13),X(16)}} and {{X(6),X(14),X(15)}}
X(9176) lies on these lines: {2, 9177}, {187, 1995}, {351, 2793}, {3291, 5968}
X(9176) = {X(9173),X(9174)}-harmonic conjugate of X(9172)
X(9176) = intersection of tangents to circle {{X(2),X(3),X(6),X(111),X(691)}} at X(2) and X(111)
X(9177) lies on the Brocard circle and these lines: {2, 9176}, {3, 6}, {351, 690}, {2434, 9717}
X(9177) = crossdifference of every pair of points on line X(111)X(523)
X(9177) = intersection of tangents to circle {{X(2),X(3),X(6),X(111),X(691)}} at X(2) and X(691)
X(9178) lies on these lines: {2, 523}, {3, 9175}, {6, 512}, {25, 2489}, {37, 4705}, {42, 4079}, {111, 351}, {263, 9009}, {526, 895}, {597, 5652}, {599, 11182}, {669, 1383}, {671, 804}, {691, 5467}, {694, 882}, {876, 897}, {924, 11216}, {1976, 2422}, {1995, 9137}, {2485, 8770}, {2514, 3108}, {2780, 6096}, {2793, 6094}, {2854, 5653}, {3005, 10562}, {3288, 11166}, {3569, 9023}, {5996, 11163}, {8749, 8753}, {9206, 9207}
X(9178) = reflection of X(3) in X(9175)
X(9178) = isogonal conjugate of X(5468)
X(9178) = antipode of X(3) in circle {{X(2),X(3),X(6),X(111),X(691)}}
X(9178) = trilinear pole of line X(512)X(3124)
X(9178) = radical center of circumcircle and circles O(13,14) and O(15,16)
X(9179) lies on these lines: {3, 8371}, {111, 230}, {523, 9129}, {2854, 7471}, {3258, 6719}
X(9179) = midpoint of X(111) and X(476)
X(9179) = radical trace of circumcircle and Hutson-Parry circle
Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. Let A" be the nine-point center of BCX(14), and define B" and C" cyclically. Let A* = B'C'∩\B"C", and define B* and C* cyclically (so that A*B*C* is the side-triangle of A'B'C' and A"B"C"). The lines AA*, BB*, CC* concur in X(9180). (Randy Hutson, January 15, 2016)
Let Ha be the orthocenter of AX(13)X(14), and define Hb and Hc cyclically. The triangle HaHbHc is perspective to ABC at X(690), and X(9180) is the centroid of HaHbHc. (Randy Hutson, January 15, 2016)
X(9180) lies on the Kiepert hyperbola and these lines: {2, 690}, {4, 2793}, {98, 843}, {99, 1649}, {115, 5466}, {523, 671}, {525, 5503}, {543, 9168}, {598, 804}, {1916, 3906}, {2799, 5485}, {8371, 9166}, {9479, 10302}
X(9180) = isogonal conjugate of X(9181)
X(9180) = isotomic conjugate of X(9182)
X(9180) = orthocenter of X(2)X(13)X(14)
The trilinear polar of X(9181) passes through X(2502). (Randy Hutson, January 15, 2016)
X(9181) lies on these lines: {3, 6}, {30, 5465}, {99, 3906}, {110, 249}, {112, 2709}, {690, 7472}, {1296, 2715}, {1495, 9129}, {1576, 9192}, {4558, 9190}, {5099, 5972}, {5994, 9202}, {5995, 9203}, {8675, 9145}
X(9181) = isogonal conjugate of X(9180)
X(9181) = midpoint of X(110) and X(691)
X(9181) = orthogonal projection of X(110) on its trilinear polar (the Brocard axis)
X(9182) lies on these lines: {2, 6}, {99, 523}, {110, 9080}, {805, 9009}, {2854, 5939}, {4563, 9190}, {9146, 9186}, {9196, 9198}, {9197, 9199}
X(9182) = isotomic conjugate of X(9180)
X(9182) = trilinear pole of line X(543)X(1641)
X(9183) lies on these lines: {115, 523}, {671, 1649}, {690, 5461}, {8371, 9166}
X(9183) = nine-point center of X(2)X(13)X(14)
X(9183) = centroid of A*B*C* as defined at X(9180)
X(9184) lies on the circumcircle and these lines: {98, 9191}, {99, 9143}, {110, 9130}, {111, 526}, {323, 691}, {325, 9080}, {352, 2715}, {476, 524}, {477, 1499}, {1296, 5663}
X(9184) = Λ(X(111), X(230))
X(9184) = perspector of ABC and the reflection of the 1st Parry triangle in X(110)
X(9184) = reflection of X(111) in line X(3)X(74)
X(9185) is the point in which the extended legs X(9201)X(6138) and X(9200)X(6137) of the trapezoid X(9200)X(9201)X(6138)X(6137) meet. Trapezoid X(9200)X(9201)X(6138)X(6137) is similar and orthogonal to trapezoid X(13)X(15)X(14)X(16), with similitude center X(111). Diagonals of trapezoid X(9200)X(9201)X(6138)X(6137) intersect at X(3569), which is X(6) of the 2nd Parry triangle. (Randy Hutson, January 15, 2016)
X(9185) lies on these lines: {2, 690}, {98, 111}, {99, 9216}, {110, 1302}, {351, 523}, {647, 8704}, {804, 8371}, {892, 9080}, {1499, 1513}, {1649, 3268}, {2491, 9465}, {2799, 9125}, {2826, 9980}, {2848, 9157}, {3667, 9811}, {3906, 5113}, {4778, 9810}, {5607, 6108}, {5608, 6109}, {9148, 9478}
X(9185) = isogonal conjugate of X(9186)
X(9185) = isotomic conjugate of X(9187)
X(9185) = crossdifference of every pair of points on line X(574)X(2502)
X(9185) = X(2)-of-2nd-Parry-triangle
X(9185) = X(376)-of-1st-Parry-triangle
X(9185) = center of circle {{X(9140),X(9158),X(9159)}}
X(9185) = pole of Fermat axis wrt {circumcircle, nine-point circle}-inverter
Line X(574)X(2502) is the radical axis of the Brocard and Parry circles. (Randy Hutson, January 15, 2016)
X(9186) lies on these lines: {110, 9044}, {351, 9192}, {691, 9023}, {8675, 9145}, {9146, 9182}
X(9186) = isogonal conjugate of X(9185)
X(9186) = trilinear pole of line X(574)X(2502)
X(9187) lies on these lines: {690, 9080}, {892, 9191}, {9146, 9182}
X(9187) = isogonal conjugate of X(9188)
X(9187) = isotomic conjugate of X(9185)
X(9187) = trilinear pole of line X(543)X(599)
X(9188) lies on these lines: {6, 351}, {182, 2780}, {511, 9126}, {512, 2030}, {524, 11176}, {526, 6593}, {597, 804}, {647, 9044}, {691, 9192}, {1576, 2420}, {1976, 2422}, {2491, 8644}, {3049, 11186}, {5027, 6088}, {5166, 9212}, {6132, 8675}
X(9188) = midpoint of X(6) and X(351)
X(9188) = isogonal conjugate of X(9187)
X(9188) = crossdifference of every pair of points on line X(543)X(599)
X(9188) = center of circle {X(6),X(187),X(351),X(9127),X(9128),X(9129)}
X(9189) lies on these lines: {2, 690}, {230, 231}, {351, 2793}, {1649, 2799}, {5466, 9123}, {5642, 9003}, {9168, 9979}
X(9189) = midpoint of X(9194) and X(9195)
X(9189) = midpoint of X(2) and X(9185)
X(9189) = isogonal conjugate of X(9190)
X(9189) = complement of X(9191)
X(9189) = crossdifference of every pair of points on line X(3)X(2502)
X(9189) = intersection of orthic axes of ABC and Artzt triangle
X(9189) = center of inverse-in-{circumcircle, nine-point circle}-inverter of Fermat axis
X(9190) lies on these lines: {895, 5107}, {4558, 9181}, {4563, 9182}, {9023, 9192}
X(9190) = isogonal conjugate of X(9189)
X(9190) = trilinear pole of line X(3)X(2502) (the polar of X(6) wrt the Parry circle)
X(9191) lies on these lines: {2, 690}, {98, 9184}, {325, 523}, {804, 1649}, {892, 9187}, {1499, 4108}, {2793, 6054}, {2799, 5466}, {8371, 9979}, {9003, 9140}, {9125, 9147}
X(9191) = isogonal conjugate of X(9192)
X(9191) = isotomic conjugate of X(9080)
X(9191) = anticomplement of X(9189)
X(9191) = crossdifference of every pair of points on line X(32)X(2502)
X(9191) = intersection of de Longchamps lines of ABC and Artzt triangle
X(9192) lies on these lines: {110, 9080}, {351, 9186}, {691, 9188}, {1576, 9181}, {1976, 6094}, {9023, 9190}
X(9192) = isogonal conjugate of X(9191)
X(9192) = trilinear pole of line X(32)X(2502)
X(9193) lies on the nine-point circle and this line: {2, 843}
X(9193) = complement of X(9080)
X(9193) = X(111)-of-5th-Euler-triangle
X(9194) lies on these lines: {2, 690}, {396, 523}, {5459, 8371}
X(9194) = reflection of X(9195) in X(9189)
X(9194) = isogonal conjugate of X(9197)
X(9194) = isotomic conjugate of X(9199)
X(9194) = crossdifference of every pair of points on line X(16)X(2502)
X(9194) = circumcenter of X(2)X(13)X(15)
X(9195) lies on these lines: {2, 690}, {395, 523}, {5460, 8371}
X(9195) = reflection of X(9194) in X(9189)
X(9195) = isogonal conjugate of X(9196)
X(9195) = isotomic conjugate of X(9198)
X(9195) = crossdifference of every pair of points on line X(15)X(2502)
X(9195) = circumcenter of X(2)X(14)X(16)
X(9196) lies on this line: {9182, 9198}
X(9196) = isogonal conjugate of X(9195)
X(9196) = trilinear pole of line X(15)X(2502) (the tangent to the Parry circle at X(15))
X(9197) lies on this line: {9182, 9199}
X(9197) = isogonal conjugate of X(9194)
X(9197) = trilinear pole of line X(16)X(2502) (the tangent to the Parry circle at X(16))
X(9198) lies on this line: {9182, 9196}
X(9198) = isotomic conjugate of X(9195)
X(9198) = trilinear pole of line X(298)X(543)
X(9199) lies on this line: {9182, 9197}
X(9199) = isotomic conjugate of X(9194)
X(9199) = trilinear pole of line X(299)X(543)
X(9200) lies on these lines: {2, 9204}, {14, 5466}, {110, 5618}, {111, 5916}, {115, 125}, {395, 523}, {2799, 9205}, {5607, 6108}
X(9200) = reflection in X(1637) of X(9201)
X(9200) = crossdifference of every pair of points on line X(15)X(110)
X(9200) = tripolar centroid of X(13)
X(9200) = X(13)-of-2nd-Parry-triangle
X(9200) = center of circle {{X(13),X(14),X(16)}}
X(9200) = X(5473)-of-2nd-Parry-triangle
X(9200) = inverse-in-Hutson-Parry-circle of X(9201)
X(9200) = {X(13636),X(13722)}-harmonic conjugate of X(9201)
X(9201) lies on these lines: {2, 9205}, {13, 5466}, {110, 5619}, {111, 5917}, {115, 125}, {396, 523}, {2799, 9204}, {5608, 6109}
X(9201) = reflection in X(1637) of X(9200)
X(9201) = crossdifference of every pair of points on line X(16)X(110)
X(9201) = X(14) of 2nd Parry triangle
X(9201) = center of circle {{X(13),X(14),X(15)}}
X(9201) = tripolar centroid of X(14)
X(9201) = X(5474)-of-1st-Parry-triangle
X(9201) = inverse-in-Hutson-Parry-circle of X(9200)
X(9201) = {X(13636),X(13722)}-harmonic conjugate of X(9200)
Let A' be the nine-point center of BCX(14), and define B' and C' cyclically. Then A'B'C' is an equilateral triangle concyclic with X(2) and X(14). The isogonal (and isotomic) conjugate of X(14) with respect to A'B'C' is the isogonal conjugate of X(9202). (Randy Hutson, January 15, 2016) p
Let A15B15C15 and A16B16C16 be the (equilateral) circumcevian triangles of X(15) and X(16), respectively. Let A'16 be the reflection of A16 in line B15C15, and define B'16 and C'16 cyclically. Then triangle A'16B'16C'16 is congruent and homothetic to A15B15C15, with center X(9202), which is also the perspector of A16B16C16 and A'16B'16C'16. (Randy Hutson, January 15, 2016)X(9202) lies on the circumcircle and these lines: {3, 2378}, {15, 111}, {16, 843}, {98, 531}, {110, 9162}, {477, 5474}, {511, 2379}, {512, 9124}, {2381, 5611}, {2770, 5464}, {5467, 5995}, {5994, 9181}, {6138, 9216}
X(9202) = reflection of X(2378) in X(3)
X(9202) = Λ(X(9200),X(6137))
X(9202) = circumcircle-antipode of X(2378)
X(9202) = X(15)-of-1st-anti-Parry-triangle
X(9202) = perspector of ABC and triangle formed by line X(2)X(13) reflected in sides of ABC
X(9202) = isogonal conjugate of outer-Napoleon-isogonal conjugate of X(14)
Let A' be the nine-point center of BCX(13), and define B' and C' cyclically. Then A'B'C' is an equilateral triangle concyclic with X(2) and X(13). The isogonal (and isotomic) conjugate of X(13) with respect to A'B'C' is the isogonal conjugate of X(9203).(Randy Hutson, January 15, 2016)
Let A15B15C15 and A16B16C16 be the (equilateral) circumcevian triangles of X(15) and X(16), respectively. Let A'15 be the reflection of A15 in line B16C16, and define B'15 and C'15 cyclically. The triangle A'15B'15C'15' is congruent and homothetic to A16B16C16, with center X(9203), which is also the perspector of A15B15C15 and A'15B'15C'15. (Randy Hutson, January 15, 2016)
X(9203) lies on the circumcircle and these lines: {3, 2379}, {15, 843}, {16, 111}, {98, 530}, {110, 9163}, {477, 5473}, {511, 2378}, {512, 9124}, {2380, 5615}, {2770, 5463}, {5467, 5994}, {5995, 9181}, {6137, 9216}
X(9203) = reflection of X(2379) of X(3)
X(9203) = circumcircle-antipode of X(2379)
X(9203) = Λ(X(9201),X(6138))
X(9203) = X(16)-of-1st-anti-Parry-triangle
X(9203) = perspector of ABC and triangle formed by line X(2)X(14) reflected in sides of ABC
X(9203) = isogonal conjugate of inner-Napoleon-isogonal conjugate of X(13)
X(9204) lies on these lines: {2, 9200}, {351, 690}, {2799, 9201}, {3268, 6137}, {5464, 9168
X(9204) = crossdifference of every pair of points on line X(111)X(3457)
X(9205) lies on these lines: {2, 9201}, {351, 690}, {2799, 9200}, {3268, 6138}, {5463, 9168}
X(9205) = crossdifference of every pair of points on line X(111)X(3458)
X(9206) lies on these lines: {111, 11081}, {691, 5995}, {9178, 9207}
X(9206) = isogonal conjugate of X(9204)
X(9206) = trilinear pole of line X(111)X(5995) (the radical axis of the circumcircle and circle {{X(6),X(13),X(16),X(111)}})
X(9207) lies on these lines: {111, 11086}, {691, 5994}, {9178, 9206}
X(9207) = isogonal conjugate of X(9205)
X(9207) = trilinear pole of line X(111)X(3458) (the radical axis of the circumcircle and circle {{X(6),X(14),X(15),X(111)}})
X(9208) lies on these lines: {2, 690}, {111, 6323}, {187, 237}, {352, 11215}, {526, 6593}, {694, 882}, {804, 11182}, {826, 9979}, {2491, 3117}, {2492, 9023}, {6004, 9811}, {7927, 9131}, {9156, 9998}, {11176, 11183}
X(9208) = crossdifference of every pair of points on line X(2)X(353)
X(9208) = X(3098)-of-1st-Parry-triangle
X(9208) = X(182)-of-2nd-Parry-triangle
X(9208) = X(182)-of-3rd-Parry-triangle
X(9209) = QA-P32 (Centroid of the Circumcenter Quadrangle) of quadrangle X(13)X(15)X(14)X(16); see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/60-qa-p32.html. (Randy Hutson, January 15, 2016)
X(9209) lies on these lines: {2, 525}, {111, 477}, {115, 3154}, {230, 231}, {1499, 1513}, {1636, 8057}, {2395, 8371}, {3800, 4108}, {5466, 7612}
X(9209) = crossdifference of every pair of points on line X(3)X(1495)
X(9209) = centroid of X(9200)X(9201)X(6138)X(6137)
X(9209) = X(230)-of-2nd-Parry-triangle
X(9209) = pole of van Aubel line wrt {circumcircle, nine-point circle}-inverter
X(9209) = inverse-in-{circumcircle, nine-point circle}-inverter of X(6794)
X(9209) = intersection of orthic axes of ABC and 2nd Parry triangle
X(9209) = PU(4)-harmonic conjugate of X(1990)
X(9210) lies on these lines: {2, 525}, {187, 237}, {420, 2501}, {526, 3049}, {826, 1637}, {1636, 8673}, {2451, 2485}, {8617, 10097}, {9213, 9998}
X(9210) = isotomic conjugate of X(9211)
X(9210) = crossdifference of every pair of points on line X(2)X(1495)
X(9210) = X(1691)-of-2nd-Parry-triangle
X(9210) = X(1691)-of-3rd-Parry-triangle
The line X(30)X(76), of which X(9211) is the trilinear pole, is the line through X(76) perpendicular to the trilinear polar of X(76). Also, this line is tangent to circle {{X(13),X(14),X(76)}} at X(76). (Randy Hutson, January 15, 2016)
X(9211) lies on these lines: {670, 2407}, {4240, 6331}
X(9211) = isotomic conjugate of X(9210)
X(9211) = trilinear pole of line X(30)X(76)
X(9212) lies on the Parry circle and these lines: {2, 1499}, {23, 9135}, {110, 2709}, {111, 512}, {351, 352}, {353, 669}, {511, 9156}, {524, 9147}, {5166, 9188}
X(9212) = reflection of X(352) in X(351)
X(9212) = X(74)-of-3rd-Parry-triangle
X(9212) = X(843)-of-1st-Parry-triangle
X(9212) = X(2709)-of-2nd-Parry-triangle
X(9212) = Parry-circle-antipode of X(352)
X(9213) lies on the Parry circle, the hypernbola {{A,B,C,X(2),X(15),X(16)}} (the isogonal conjugate of the Fermat axis), and on these lines: {2, 523}, {15, 6137}, {16, 6138}, {23, 351}, {30, 9147}, {110, 249}, {111, 647}, {186, 9126}, {323, 526}, {352, 3569}, {353, 3288}, {511, 9138}, {669, 9999}, {804, 10989}, {895, 8675}, {2780, 7464}, {3005, 7711}, {3268, 7799}, {5996, 6054}, {8644, 9157}, {9123, 9158}, {9210, 9998}
X(9213) = reflection of X(23) in X(351)
X(9213) = isogonal conjugate of X(14559)
X(9213) = crossdifference of every pair of points on line X(187)X(1648)
X(9213) = X(111)-of-3rd-Parry-triangle
X(9213) = X(691)-of-2nd-Parry-triangle
X(9213) = X(842)-of-1st-Parry-triangle
X(9213) = Parry-circle-antipode of X(23)
\X(9213) = trilinear pole, wrt 2nd Parry triangle, of line X(74)X(111)
X(9213) = trilinear pole, wrt 3rd Parry triangle, of line X(511)X(3569)
Let A'B'C' be the 2nd Parry triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(9214); if '2nd Parry triangle' is replaced with '1st Parry triangle', the resulting lines concur in X(1992). (Randy Hutson, January 15, 2016)
X(9214) lies on these lines: {2, 523}, {4, 542}, {30, 2407}, {69, 892}, {110, 10555}, {111, 1302}, {376, 477}, {381, 2452}, {1138, 3524}, {1370, 8877}, {1990, 4240}, {3014, 7774}, {3421, 5380}, {3545, 5627}, {5967, 9154}
X(9214) = isogonal conjugate of X(9717)
X(9214) = isotomic conjugate of X(36890)
X(9214) = crosspoint of X(36307) and X(36310)
X(9214) = trilinear pole of line X(30)X(1637), which is the perspectrix of ABC and the 2nd Parry triangle
X(9215) lies on these lines: {3, 74}, {98, 9123}, {111, 351}, {842, 9137}, {2502, 7669}, {2854, 9130}
X(9215) = reflection of X(9216) in X(9130)
X(9215) = crossdifference of every pair of points on line X(1637)X(2482)
X(9215) = homothetic center of ABC and 2nd Parry triangle of 1st Parry triangle
X(9215) = homothetic center of 1st Parry triangle and 2nd anti-Parry triangle
X(9216) lies on these lines: {3, 111}, {99, 9185}, {110, 351}, {1649, 9146}, {2854, 9130}, {6137, 9203}, {6138, 9202}
X(9216) = reflection of X(9215) in X(9130)
X(9216) = crossdifference of every pair of points on line X(115)X(9125)
X(9216) = homothetic center of ABC and 1st Parry triangle of 2nd Parry triangle
X(9216) = homothetic center of 2nd Parry triangle and 1st anti-Parry triangle
Centers related to bicentric pairs: X(9217)-X(9461)
These centers were contributed by César Eliud Lozada - January 17, 2016 and January 19, 2016
The trilinear polar of X(9217) passes through X(351). (Randy Hutson, Februay 10, 2016)
X(9217) lies on the cubic K559 and these lines: {6,9218}, {524,2076}, {2641,9395}, {3566,3568}
X(9217) = isogonal conjugate of X(148)
X(9217) = vertex conjugate of X(3440) and X(3441)
X(9217) = X(92)-isoconjugate of X(22143)
X(9217) = perspector of ABC and the reflection of the tangential triangle in the Brocard axis
X(9217) = X(512)-vertex conjugate of X(512)
X(9218) lies on the cubics K367, the K579, the K629 and these lines: {6,9217}, {36,1326}, {99,826}, {110,249}, {112,805}, {163,2702}, {511,2071}, {525,7472}, {2076,9019}, {2612,2959}, {2701,4575}, {2715,3565}
Let A'B'C' be the circumcevian triangle of X(512). Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(9218). (Randy Hutson, August 19, 2019)
X(9218) = midpoint of X(i),X(j) for these (i,j): (249,691)
X(9218) = reflection of X(i) in X(j) for these (i,j): (110,249)
X(9218) = isogonal conjugate of X(9293)
X(9218) = crosssum of PU(40)
X(9218) = circumcircle-inverse of X(39138)
X(9219) lies on these lines: {1,564}, {19,2290}, {91,2964}, {92,6149}, {2491,9420}, {7951,9222}
X(9219) = trilinear product X(7741)*X(7951)
X(9220) lies on these lines: {4,50}, {5,566}, {6,13}, {25,2934}, {53,403}, {231,7747}, {570,7603}, {2165,2965}, {2493,5169}, {3054,7426}, {3815,5133}, {7394,7735}, {7545,7746}
X(9221) lies on the Kiepert hyperbola and these lines: {2,568}, {3,7578}, {4,566}, {5,94}, {76,1273}, {83,7550}, {96,1199}, {186,275}, {2052,7577}, {2599,7741}
X(9221) = isogonal conjugate of X(567)
X(9221) = trilinear pole of line X(523)X(2081)
Starting with the bicentric pair PU(5), let D = P(5), E = U(5), F = D-Ceva conjugate of E, G = E-Ceva conjugate of D. The X(9222) = DF∩EG. (Randy Hutson, February 10, 2016)
X(9222) lies on these lines: {5,94}, {51,381}, {566,1656}, {7951,9219}
X(9223) lies on these lines: {1,799}
X(9224) lies on these lines: {2,9307}, {6,99}
X(9225) lies on these lines: {2,2056}, {6,373}, {23,352}, {32,1613}, {50,647}, {110,1691}, {111,8586}, {154,5023}, {323,3124}, {394,3981}, {698,4563}, {1495,2076}, {2080,5106}, {3266,5108}, {5116,5650}, {5965,6388}
X(9225) = isogonal conjugate of X(9227)
X(9225) = X(2)-Ceva conjugate of X(39027)
X(9225) = perspector of conic {{A,B,C,X(54),X(1296)}}
X(9226) lies on these lines: {54,1296}
X(9226) = isogonal conjugate of X(9228)
X(9227) lies on these lines: {126,6374}, {194,1992}, {3186,4232}, {3291,8859}
X(9227) = isogonal conjugate of X(9225)
X(9227) = polar conjugate of X(38294)
X(9228) = isogonal conjugate of X(9226)
The trilinear polar of X(9229) meets the line at infinity at X(525). (Randy Hutson, February 10, 2016)
X(9229) lies on the cubic K739 and these lines: {2,3186}, {69,194}, {95,7824}, {239,3778}, {264,5025}, {287,6467}, {305,3314}, {306,3797}, {384,1843}, {385,1194}, {1494,7924}, {1916,8783}, {2998,3981}
X(9229) = isogonal conjugate of X(1915)
X(9229) = isotomic conjugate of X(384)
X(9229) = anticomplement of X(37891)
X(9230) lies on the cubic K354 and these lines: {2,2998}, {4,69}, {6,706}, {39,8790}, {75,291}, {141,308}, {305,7774}, {384,710}, {427,1241}, {668,3688}, {1031,7779}, {1915,6657}, {1916,8783}, {2345,6382}, {3117,8264}, {4000,6383}
X(9230) = isotomic conjugate of X(695)
X(9230) = polar conjugate of isogonal conjugate of X(37894)
X(9231) lies on these lines: {30,511}, {1918,1919}
X(9232) lies on these lines: {1918,1919}
Let A'B'C' and A"B"C" be the 5th Brocard and 5th anti-Brocard triangles, resp. Let A* be the barycentric product B"*C", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(9233). (Randy Hutson, July 11, 2019)
X(9233) lies on these lines: {6,6660}, {32,206}, {115,2980}, {827,6579}, {1084,1974}, {4577,8264}
X(9233) = isogonal conjugate of isotomic conjugate of X(1501)
X(9233) = barycentric square of X(32)
X(9234) lies on these lines: {30,511}, {1923,1924}
X(9235) lies on these lines: {1923,1924}
X(9236) lies on these lines: {1,9239}, {48,9288}, {695,1914}, {1580,9285}, {1927,1932}
X(9236) = isogonal conjugate of X(1925)
X(9237) lies on these lines: {30,511}, {1577,1930}, {3801,4444}, {4107,4467}
X(9238) lies on these lines: {1577,1930}
X(9239) lies on these lines: {1,9236}, {304,9285}, {350,695}, {1925,1934}, {1966,9288}
X(9239) = isogonal conjugate of X(1932)
X(9239) = isotomic conjugate of X(1582)
X(9240) lies on these lines: {30,511}, {73,652}
X(9241) lies on these lines: {73,652}, {828,2174}, {946,2657}
X(9242) lies on these lines: {30,511}, {185,647}
X(9243) lies on these lines: {3,3229}, {185,647}, {216,7542}
X(9244) lies on these lines: {1075,3542}, {1941,1942}
X(9244) = isogonal conjugate of X(1941)
X(9245) lies on these lines: {30,511}, {663,1400}
X(9246) lies on these lines: {663,1400}
X(9247) lies on these lines: {1,163}, {19,2148}, {31,1932}, {32,1397}, {47,1755}, {48,255}, {184,2200}, {560,1917}, {2201,3073}, {3072,7119}
X(9247) = isogonal conjugate of X(1969)
X(9247) = X(92)-isoconjugate of X(75)
X(9247) = intersection of tangents at X(255) and X(1917) to the inellipse centered at X(23993)
X(9248) lies on these lines: {30,511}
X(9249) lies on these lines: {30,511}, {656,1953}
X(9250) lies on these lines: {37,1861}, {656,1953}, {1486,2110}
X(9251) lies on these lines: {10,9290}, {19,2313}, {759,1303}, {1955,2190}, {1956,9252}
X(9251) = isogonal conjugate of X(1954)
X(9251) = polar conjugate of X(9252)
X(9252) lies on these lines: {19,27}, {823,1953}, {1956,9251}, {1958,1969}
X(9252) = polar conjugate of X(9251)
X(9253) lies on these lines: {30,511}, {48,656}
X(9254) lies on these lines: {48,656}
X(9255) lies on these lines: {240,774}, {293,1958}, {610,1707}, {1957,2155}, {5930,9307}
X(9255) = isogonal conjugate of X(1957)
X(9256) lies on these lines: {30,511}, {31,661}, {2887,4369}, {3287,4705}, {4680,4761}, {6327,7192}
X(9257) lies on these lines: {31,661}, {513,9314}, {993,9284}
X(9258) lies on these lines: {610,1707}, {1910,1957}, {1959,6508}, {3198,5360}, {4028,8804}
X(9258) = isogonal conjugate of X(1958)
X(9259) lies on these lines: {1,1929}, {3,9261}, {6,101}, {31,5168}, {36,3230}, {56,292}, {172,1201}, {213,5563}, {214,5110}, {649,9262}, {1054,4919}, {1055,1149}, {1319,3290}, {1438,3207}, {1500,9327}, {1616,3053}, {2087,5540}, {2271,7373}, {2275,9310}, {2295,5253}, {3286,9264}, {3570,9263}, {3726,4511}, {3730,9351}, {4103,6789}, {4128,5147}, {4561,9055}, {5013,6184}, {5301,9412}, {7113,8610}, {7200,9318}
X(9259) = isogonal conjugate of X(6630)
X(9259) = crosspoint of PU(25)
X(9259) = inverse-in-circumcircle of X(9261)
X(9259) = polar conjugate of isotomic conjugate of X(22148)
X(9260) lies on these lines: {2,9269}, {30,511}, {764,3632}, {899,4449}, {1022,4677}, {3241,4448}, {3633,6161}, {3699,6631}, {4147,4871}, {4776,6542}, {6164,6630}
X(9261) lies on these lines: {3,9259}, {1015,1960}, {5029,6377}
X(9261) = inverse-in-circumcircle of X(9259)
X(9262) lies on these lines: {649,9259}, {1635,3722}
X(9262) = isogonal conjugate of X(6631)
X(9263) lies on the permutation ellipse E(X(4440)) and these lines: {1, 1655}, {2, 668}, {6, 18047}, {8, 291}, {75, 7200}, {145, 194}, {148, 149}, {192, 537}, {193, 2810}, {239, 9317}, {335, 2170}, {513, 9267}, {518, 19565}, {519, 17759}, {664, 32029}, {672, 10027}, {812, 4440}, {891, 17154}, {894, 9457}, {956, 16998}, {999, 16997}, {1107, 25303}, {1475, 17752}, {1565, 31129}, {1909, 17448}, {2087, 18061}, {2094, 3210}, {2241, 17692}, {2275, 24524}, {2895, 17946}, {3125, 35957}, {3244, 25264}, {3570, 9259}, {3616, 17793}, {3617, 27318}, {3622, 27269}, {3780, 24522}, {3871, 7783}, {3975, 26113}, {4033, 26076}, {4430, 17486}, {4473, 24508}, {4560, 6630}, {4595, 20331}, {4922, 25048}, {4986, 7208}, {5773, 37683}, {6224, 17148}, {6384, 24528}, {6553, 30695}, {7257, 37128}, {7785, 20060}, {8591, 17147}, {9055, 33946}, {9359, 21100}, {10453, 21223}, {14712, 20067}, {16604, 25280}, {16722, 17169}, {16781, 16916}, {17300, 26140}, {17480, 21216}, {18135, 21219}, {18149, 21893}, {18159, 27918}, {20041, 20109}, {20050, 32005}, {20057, 32095}, {20065, 20076}, {20081, 32035}, {23447, 27033}, {23646, 32454}, {24485, 37129}, {24507, 36798}, {24722, 36222}, {25298, 26048}, {29588, 31036}, {30661, 32913}, {30806, 33891}, {31997, 39028}, {33888, 35956}, {36854, 36858}
X(9263) = reflection of X(i) in X(j) for these (i,j): (2,3227), (8,291), (668,1015)
X(9263) = isogonal conjugate of X(9265)
X(9263) = isotomic conjugate of X(9295)
X(9263) = complement of X(31298)
X(9263) = anticomplement of X(668)
X(9263) = anticomplementary conjugate of X(21301)
X(9263) = crosspoint of PU(27)
X(9263) = polar conjugate of isogonal conjugate of X(22158)
X(9264) lies on these lines: {39,1083}, {187,237}, {238,1015}, {3286,9259}
X(9265) lies on these lines: {6,9266}, {667,1979}, {2664,3230}, {2669,9295}
X(9265) = isogonal conjugate of X(9263)
X(9265) = X(92)-isoconjugate of X(22158)
X(9266) lies on these lines: {6,9265}, {100,667}, {101,1924}, {190,4367}, {238,1149}, {644,813}, {659,6631}, {660,663}, {1083,2975}, {3573,9323}, {4401,6633}
X(9266) = isogonal conjugate of X(9267)
X(9266) = crosssum of PU(27)
X(9267) lies on these lines: {513,9263}, {891,4928}
X(9267) = isogonal conjugate of X(9266)
X(9268) lies on these lines: {1,765}, {6,1252}, {34,7012}, {56,59}, {58,4570}, {86,4600}, {106,6551}, {269,7045}, {677,2424}, {901,4638}, {996,1016}, {1027,3257}, {1110,2163}, {1320,1411}, {1438,2316}, {2284,5548}, {3226,6635}
X(9268) = isogonal conjugate of X(1647)
X(9269) lies on these lines: {1,513}, {2,9260}, {354,4083}, {3315,6164}, {3720,4449}
X(9269) = isogonal conjugate of X(9271)
X(9269) = midpoint of X(i),X(j) for these (i,j): (1022,3251)
X(9270) lies on these lines: {1,6}
X(9271) lies on these lines: {1,9325}, {100,9272}, {3251,4618}
X(9271) = isogonal conjugate of X(9269)
X(9272) lies on these lines: {100,9271}, {1022,1023}
X(9273) lies on these lines: {35,4570}, {3219,4567}
X(9274) lies on these lines: {35,4570}
X(9275) lies on these lines: {1,60}, {3,6}, {21,5692}, {55,5127}, {56,501}, {81,1325}, {163,2280}, {849,1468}, {993,2185}, {994,2363}, {1098,5248}, {1175,3601}, {1408,5221}, {1437,4658}, {2194,4653}
X(9275) = PU(29)-harmonic conjugate of X(1983)
X(9275) = crossdifference of every pair of points on the trilinear polar of X(12) wrt the Feuerbach triangle
X(9276) lies on these lines: {12,2614}, {495,523}
X(9277) lies on these lines: {1,1654}, {6,846}, {87,5256}, {292,2092}, {740,1220}, {1100,6158}, {1126,1757}, {1213,1961}, {1929,1963}
X(9277) = isogonal conjugate of X(1961)
X(9278) lies on the cubics K137, the K323 and these lines: {1,1929}, {6,2640}, {10,115}, {37,2054}, {44,897}, {75,1654}, {267,5540}, {662,1100}, {759,2702}, {910,1910}, {1213,6158}, {1247,3496}, {1575,1581}, {1961,9281}, {2245,2652}
X(9278) = isogonal conjugate of X(1931)
X(9278) = complement of X(20538)
X(9278) = anticomplement of X(20529)
X(9278) = antitomic conjugate of isogonal conjugate of X(38814)
X(9279) lies on these lines: {30,511}, {351,4893}, {661,1962}, {2642,4770}
X(9279) = midpoint of X(i),X(j) for these (i,j): (746,918)
X(9280) lies on these lines: {661,9279}, {2653,3743}
X(9281) lies on these lines: {1,2248}, {6,846}, {111,3920}, {1169,1914}, {1171,1931}, {1961,9278}, {1962,2054}, {4093,9403}
X(9281) = isogonal conjugate of X(1963)
X(9282) lies on the cubic K661 and these lines: {1,6163}, {44,3684}, {238,1319}, {242,1877}, {513,1052}, {519,1757}, {1046,6788}, {1326,5150}, {1447,5121}, {3667,9355}, {3737,9359}
X(9282) = reflection of X(i) in X(j) for these (i,j): (1054,1052)
X(9282) = isogonal conjugate of X(1054)
X(9282) = trilinear pole of line X(1635)X(3722)
X(9283) lies on these lines: {9,1054}, {100,8649}, {244,665}, {1086,4763}
X(9284) lies on these lines: {2,292}, {37,908}, {38,661}, {39,4109}, {893,4388}, {982,3981}, {993,9257}, {2276,4144}, {2653,3874}, {3821,6377}
X(9285) lies on these lines: {63,1740}, {72,695}, {304,9239}, {306,3797}, {1580,9236}, {1581,1965}, {1921,2887}
X(9285) = isogonal conjugate of X(1582)
X(9285) = isotomic conjugate of X(1965)
X(9286) lies on these lines: {30,511}, {798,1964}
X(9287) lies on these lines: {798,1964}
X(9288) lies on these lines: {48,9236}, {63,1740}, {71,695}, {239,3778}, {1582,1967}, {1966,9239}
X(9288) = isogonal conjugate of X(1965)
X(9288) = isotomic conjugate of X(1925)
X(9288) = complement of anticomplementary conjugate of X(17485)
X(9289) lies on the cubic K718 and these lines: {20,185}, {64,1105}, {76,3269}, {184,7783}, {192,1425}, {217,7757}, {249,7782}, {287,1975}, {297,3981}, {330,3270}, {385,1204}, {1562,5025}, {3552,8779}, {6337,6509}
X(9289) = isogonal conjugate of X(1968)
X(9289) = isotomic conjugate of X(9308)
X(9289) = trilinear pole of line X(684)X(8057)
X(9290) lies on the Kiepert hyperbola and these lines: {4,3164}, {10,9251}, {98,185}, {262,3574}, {275,401}, {1972,9291}, {2986,7783}
X(9290) = isogonal conjugate of X(1970)
X(9290) = polar conjugate of X(436
X(9291) lies on these lines: {4,69}, {5,276}, {217,648}, {1105,6394}, {1972,9290}
X(9292) lies on the cubic K041 and these lines: {20,185}, {154,237}, {376,4173}, {512,3767}, {1204,1297}, {1968,1976}, {2211,3172}, {3198,5360}, {5395,5943}, {6525,6620}
X(9292) = isogonal conjugate of X(1975)
X(9292) = trilinear pole of line X(2491)X(3221)
The trilinear polar of X(9293) passes through X(1648). (Randy Hutson, February 10, 2016)
Let A'B'C' be the circumcevian triangle of X(512). Let A" be the crosssum of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(9293). (Randy Hutson, August 19, 2019)
X(9293) lies on the cubic K630 and these lines: {148,523}, {620,690}, {804,5186}, {3566,3568}
X(9293) = isogonal conjugate of X(9218)
X(9293) = isotomic conjugate of X(31998)
X(9293) = X(2)-cross conjugate of X(523)
X(9294) lies on these lines: {30,511}, {350,3835}, {646,4562}
X(9295) lies on these lines: {2,9296}, {513,9263}, {899,9361}, {2669,9265}
X(9295) = isogonal conjugate of X(1979)
X(9295) = isotomic conjugate of X(9263)
X(9295) = X(19)-isoconjugate of X(22158)
X(9296) lies on these lines: {2,9295}, {190,798}, {350,1266}, {513,668}, {522,4583}, {646,4562}, {1978,4374}, {3063,5383}
X(9296) = isogonal conjugate of X(9299)
X(9296) = isotomic conjugate ofX(9267)
X(9296) = complement of X(9295)
X(9296) = crosssum of PU(42)
X(9296) = X(2)-Ceva conjugate of X(668)
X(9296) = center of hyperbola {{A,B,C,PU(41)}}
X(9297) lies on these lines: {30,511}, {890,1977}
X(9298) lies on these lines: {3,1979}, {890,1977}
X(9298) = circumcircle-inverse of X(1979)
X(9299) lies on these lines: {667,1979}
X(9299) = isogonal conjugate of X(9296)
X(9300) lies on these lines: {2,6}, {5,5309}, {25,6749}, {30,39}, {32,549}, {50,251}, {53,5064}, {83,6661}, {115,5066}, {140,5007}, {172,5298}, {232,428}, {262,1503}, {376,5013}, {381,2548}, {383,397}, {398,1080}, {427,1990}, {519,4095}, {542,2023}, {546,7765}, {547,1506}, {570,7667}, {574,8703}, {754,8359}, {800,7734}, {1194,2493}, {1368,5158}, {1447,7277}, {1572,3654}, {1656,5319}, {1914,4995}, {1989,3108}, {2275,5434}, {2276,3058}, {2549,3830}, {3053,3524}, {3087,7714}, {3284,6676}, {3534,5024}, {3543,7738}, {3545,5286}, {3582,5280}, {3584,5299}, {3628,7755}, {3767,5055}, {3845,5475}, {3849,8354}, {3933,7808}, {4969,7081}, {5332,5432}, {5355,7603}, {5433,7296}, {6034,6054}, {6390,7804}, {6656,7809}, {6683,7767}, {6704,7895}, {7750,7921}, {7752,7884}, {7757,8370}, {7759,7865}, {7762,7786}, {7764,7819}, {7785,7924}, {7807,7878}, {7812,8356}, {7814,8363}, {7821,8364}, {7829,8361}, {7843,8357}
X(9300) = midpoint of X(i),X(j) for these (i,j): (39,7753), (7757,8370), (7762,7811), (7812,8356)
X(9300) = reflection of X(i) in X(j) for these (i,j): (7745,7753)
X(9300) = complement of X(37671)
X(9300) = midpoint of PU(43)
X(9300) = centroid of pedal triangle of X(39)
X(9300) = centroid of pedal feet of PU(1)
Let P and U be the intersections of the circumcircles of the reflection triangles of PU(1). Then X(9301) = {P,U}-harmonic conjugate of X(6). (Randy Hutson, July 31 2018)
X(9301) lies on these lines: {3,6}, {5,2896}, {30,148}, {183,316}, {237,323}, {376,7766}, {399,3511}, {517,3099}, {549,3329}, {625,5055}, {754,6033}, {827,842}, {1656,3096}, {3526,7846}, {3830,3849}, {5070,7914}, {5103,7800}, {5148,6767}, {5191,6660}, {5194,7373}, {5207,7767}, {5354,7467}, {7813,8724}
X(9301) = isogonal conjugate of X(9302)
X(9301) = circumcircle-inverse of X(5092)
X(9301) = Stammler circle-inverse of X(6)
X(9301) = reflection of X(i) in X(j) for these (i,j): (3,2080)
X(9301) = intersection of Lemoine axes of antipedal triangles of PU(1)
X(9302) lies on the Kiepert hyperbola and these lines: {2,7711}, {83,542}, {262,5309}, {598,3818}, {3524,5989}
X(9302) = isogonal conjugate of X(9301)
X(9303) lies on these lines: {1,6}, {3749,7084}
X(9304) lies on these lines: {6,31}
X(9305) lies on these lines: {4,120}, {20,958}, {55,85}, {170,2329}, {1376,6554}, {3061,5527}
For P on the nine-point circle, let E(P) be the isogonal conjugate of the trilinear pole of the tangent at P. The locus of E(P) is an ellipse, E, with center X(9306). Let F = X(11) (the Feuerbach point), and let A'B'C' be the Feuerbach triangle. Let (O) denote the circumcircle. Then E∩(O) = {F, A', B', C'}. Also, E(X(11)) = X(101), E(X(125)) = X(1625), and E(X(115)) = X(1634). (Randy Hutson, February 10, 2016)
Let (O*) be circle {{X(371),X(372),PU(1),PU(39)}}, and let P be the perspector of (O*). Then X(9306) is the trilinear pole of the polar of P with respect to (O*). (Randy Hutson, February 10, 2016)
The trilinear polar of X(9306) passes through X(2451). (Randy Hutson, February 10, 2016)
Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the medial triangle at X(9306). Also, X(9306) = X(1899)-of-A'B'C'. (Randy Hutson, March 14, 2018)
X(9306) lies on these lines: {2,98}, {3,64}, {4,801}, {5,578}, {6,1196}, {9,3955}, {22,1495}, {23,2979}, {24,5562}, {25,394}, {26,1216}, {32,1613}, {49,569}, {51,576}, {52,7506}, {54,3090}, {57,7193}, {61,2005}, {62,2004}, {69,1974}, {76,419}, {81,4223}, {140,156}, {141,206}, {155,389}, {171,2175}, {237,5171}, {238,1397}, {242,1943}, {249,6787}, {264,436}, {323,3060}, {343,468}, {373,5422}, {405,1437}, {427,3818}, {450,2052}, {458,6248}, {524,8263}, {567,5055}, {574,2936}, {575,6688}, {577,1971}, {631,1614}, {648,3168}, {692,1376}, {846,5197}, {940,2194}, {1011,1790}, {1176,3619}, {1209,6639}, {1304,2706}, {1368,1503}, {1428,5272}, {1501,3231}, {1504,8956}, {1511,4550}, {1634,5063}, {1935,7335}, {1936,6056}, {1944,7009}, {1994,5640}, {2330,5268}, {2909,3788}, {3051,5039}, {3066,5097}, {3155,5409}, {3156,5408}, {3202,3934}, {3203,7808}, {3220,3784}, {3492,7795}, {3528,8718}, {3564,6677}, {3574,7544}, {3734,4074}, {3763,5157}, {3767,8970}, {3781,5285}, {3787,5017}, {3796,5092}, {3981,5028}, {4245,5398}, {4846,6053}, {5504,7687}, {5650,6800}, {5707,7535}, {5788,7532}, {5876,7689}, {5965,6515}, {6030,7712}, {6414,8963}, {6593,8542}, {6636,7998}, {7512,7999}, {8703,8717}
X(9306) = midpoint of X(i),X(j) for these (i,j): (25,394)
X(9306) = isogonal conjugate of X(9307)
X(9306) = complement of X(1899)
X(9306) = Brocard circle-inverse of X(5972)
X(9306) = X(25) of 1st Brocard triangle
X(9306) = 1st-Brocard-isogonal conjugate of X(2549)
X(9306) = 1st-Brocard-isotomic conjugate of X(3981)
X(9306) = crossdifference of every pair of points on line X(2450)X(3566)
X(9306) = inverse-in-Thomson-Gibert-Moses-hyperbola of X(184)
X(9306) = {X(2),X(110)}-harmonic conjugate of X(184)
Let A' be the radical center of the polar circle and the B- and C-power circles. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(9307). (Randy Hutson, February 10, 2016)
X(9307) lies on the cubic K675 and these lines: {6,1632}, {20,185}, {98,9308}, {232,800}, {235,6530}, {250,2452}, {325,1368}, {1084,9224}, {1975,6391}, {2790,5186}, {4028,8804}, {5930,9255}
X(9307) = reflection of X(i) in X(j) for these (i,j): (3186,800)
X(9307) = isogonal conjugate of X(9306)
X(9307) = isotomic conjugate of X(1975)
X(9307) = polar conjugate of X(9308)
X(9307) = trilinear pole of line X(2450)X(3566)
X(9308) lies on the cubic K675 and these lines: {2,253}, {3,3164}, {4,193}, {6,264}, {25,385}, {53,317}, {64,1105}, {69,297}, {76,683}, {92,1999}, {98,9307}, {112,1003}, {141,1990}, {157,523}, {183,232}, {192,1013}, {194,1593}, {219,1948}, {222,1947}, {239,273}, {250,2453}, {311,8745}, {318,894}, {324,1993}, {330,1398}, {384,3172}, {394,801}, {427,7774}, {436,3167}, {450,6090}, {472,3181}, {473,3180}, {653,3209}, {1235,7770}, {1352,6530}, {1785,4416}, {1968,1975}, {1992,3087}, {3168,5020}, {3199,7751}, {3515,7793}, {3516,7783}, {3552,8778}, {3629,6748}, {5064,7837}, {5094,7777}, {5523,7841}, {6331,6374}, {7507,7785}
X(9308) = isotomic conjugate of X(9289)
X(9308) = polar conjugate of X(9307)
X(9308) = reflection of X(i) in X(j) for these (i,j): (317,53)
X(9309) lies on these lines: {6,1633}, {7,3271}, {43,165}, {59,2175}, {105,6180}, {144,145}, {346,9025}, {513,4000}, {651,1037}, {1122,3598}, {1201,1419}, {1828,1876}, {1977,9082}, {3286,4225}, {3688,6172}
X(9309) = isogonal conjugate of X(1376)
X(9309) = trilinear pole of line X(665)X(4083)
X(9310) lies on these lines: {1,41}, {2,1429}, {3,1055}, {6,1201}, {9,604}, {10,4390}, {12,6506}, {31,172}, {32,3230}, {36,3730}, {37,48}, {39,8649}, {42,8770}, {45,2267}, {55,1615}, {56,220}, {57,6602}, {65,1802}, {71,2178}, {78,3930}, {85,4564}, {100,3208}, {142,7225}, {144,7175}, {145,3684}, {192,1958}, {198,2256}, {213,1468}, {218,999}, {219,1400}, {281,7120}, {284,3247}, {404,644}, {572,3731}, {584,3723}, {595,609}, {603,4559}, {612,2187}, {662,4664}, {663,2440}, {728,5438}, {748,4426}, {750,2295}, {902,3053}, {910,3057}, {956,3691}, {960,5282}, {976,1973}, {993,3294}, {995,5280}, {1066,8776}, {1100,3204}, {1212,1319}, {1376,4513}, {1405,2323}, {1412,3929}, {1466,7368}, {1496,1951}, {1617,8012}, {1759,3878}, {1813,8545}, {2171,2289}, {2174,4289}, {2177,5168}, {2202,7952}, {2241,2251}, {2260,2911}, {2275,9259}, {2285,2324}, {3290,3924}, {3303,4258}, {3476,6554}, {3496,3877}, {3509,3869}, {3673,9317}, {3746,4262}, {3991,5440}, {4056,5074}, {4253,5526}, {4520,4640}, {5011,5697}
X(9310) = isogonal conjugate of X(9311)
X(9310) = {X(37),X(48)}-harmonic conjugate of X(2268)
X(9310) = crosspoint of PU(93)
X(9310) = crossdifference of every pair of points on line X(2254)X(3667)
X(9311) is the trilinear pole of line X(2254)X(3667), which is the radical axis of incircle and excircles radical circle, and also the polar of X(1) wrt {circumcircle, nine-point circle}-inverter. (Randy Hutson, February 10, 2016)
Line X(2254)X(3667) is also the line of the (degenerate) cross-triangle of Gemini triangles 7 and 8. (Randy Hutson, November 30, 2018)
Let DEF be the intouch triangle of triangle ABC. Let L be the line through D parallel to CA, and let CA = L∩AB, and define AB and BC cyclically. Let L' be the line through D parallel to AB, and let BA = L'∩CA, and define CB and AC cyclically. Let A'B'C' be the triangle having sidelines BACA, CBAB, ACBC. Then A'B'C' is perspective to ABC, and the perspector is X(9311). Click here for a sketch showing X(9311). (Angel Montesdeoca, March 20, 2016.)
X(9311) lies on these lines: {1,3732}, {41,4564}, {75,4051}, {85,2170}, {144,145}, {241,2275}, {514,3673}, {664,2082}, {673,9312}, {1222,3680}, {3061,3452}, {3928,9315}, {3959,7233}
X(9311) = isogonal conjugate of X(9310)
X(9311) = isotomic conjugate of X(3729)
X(9312) lies on these lines: {1,85}, {2,3160}, {7,145}, {8,279}, {9,3177}, {10,348}, {40,5088}, {43,7196}, {46,7183}, {57,239}, {63,3188}, {65,7223}, {69,3668}, {75,269}, {77,1441}, {78,1446}, {150,5881}, {169,514}, {190,4936}, {200,1088}, {226,2996}, {241,4384}, {347,4357}, {349,3761}, {355,1565}, {388,3674}, {404,934}, {519,6604}, {673,9311}, {738,1706}, {894,1419}, {948,3912}, {1025,3501}, {1231,1448}, {1418,4361}, {1420,1447}, {1424,3503}, {1434,3339}, {1909,6063}, {1996,6745}, {2082,9317}, {2099,4059}, {2263,3886}, {2898,3452}, {3476,7195}, {3598,4308}, {3623,5543}, {3665,5252}, {3687,7365}, {3729,4513}, {4363,6610}, {4552,8545}, {4554,6376}, {4566,7177}, {4872,5691}
X(9312) = crosssum of PU(93)
X(9312) = trilinear pole of line X(4885)X(6168)
X(9313) lies on these lines: {30,511}, {31,649}, {2483,4834}, {2887,3835}, {4455,6586}
X(9314) lies on these lines: {31,649}, {513,9257}
X(9315) lies on these lines: {43,165}, {57,6169}, {614,649}, {1438,9316}, {2176,2223}, {2319,3729}, {3928,9311}
X(9315) = isogonal conjugate of X(3729)
X(9316) lies on these lines: {1,1106}, {2,9364}, {3,1042}, {6,1200}, {7,171}, {31,57}, {35,4306}, {38,8270}, {40,1496}, {42,222}, {43,651}, {46,255}, {47,3336}, {55,1407}, {56,1149}, {58,3339}, {65,603}, {73,1406}, {145,9363}, {165,269}, {196,1430}, {212,1155}, {221,1193}, {226,750}, {238,5435}, {241,4640}, {411,1044}, {595,3361}, {601,942}, {604,1403}, {653,1957}, {748,3911}, {774,1158}, {896,1708}, {902,1617}, {982,4318}, {986,4296}, {1038,2292}, {1088,7045}, {1096,1767}, {1210,1777}, {1214,4414}, {1376,6168}, {1395,1876}, {1399,1451}, {1400,8770}, {1414,6043}, {1438,9315}, {1441,3980}, {1445,1707}, {1454,3215}, {1456,3752}, {1457,1470}, {1497,3338}, {1709,2310}, {1754,3668}, {1771,4292}, {1788,1935}, {1936,3474}, {2177,3256}, {2199,2285}, {2209,7175}, {2260,8775}, {3000,7580}, {3075,4295}, {3214,9370}, {3295,4322}, {3550,4334}, {3600,5255}, {3670,4347}, {4298,5264}, {4327,5269}, {5128,7273}, {5268,8545}, {7121,7132}, {7677,8616}
X(9317) lies on these lines: {1,2140}, {6,7200}, {41,85}, {57,4566}, {75,4390}, {80,116}, {101,1111}, {142,6224}, {239,9263}, {277,944}, {379,7146}, {514,5540}, {664,673}, {672,5088}, {952,4904}, {1054,2789}, {1055,1447}, {1083,2795}, {1358,5845}, {1475,7176}, {2082,9312}, {2246,3732}, {2646,6706}, {3212,4209}, {3476,4000}, {3583,5074}, {3673,9310}, {4482,4986}
X(9317) = isogonal conjugate of X(9322)
X(9318) lies on these lines: {1,514}, {2,7}, {11,5845}, {38,9414}, {41,3673}, {75,3570}, {80,544}, {85,4564}, {101,1111}, {106,7208}, {320,4766}, {536,3689}, {545,6174}, {673,2246}, {742,4396}, {1055,5088}, {1647,4644}, {2170,3732}, {2280,6654}, {3257,3758}, {3938,4336}, {4251,7264}, {4363,4413}, {4376,4713}, {4403,8649}, {4414,4419}, {7200,9259}, {7278,9327}
X(9318) = isogonal conjugate of X(9319)
X(9318) = anticomplement of X(24318)
X(9318) = X(2)-Ceva conjugate of X(39047)
X(9318) = perspector of conic {{A,B,C,X(664),X(673)}}
X(9319) lies on these lines: {1,1024}, {6,2283}, {9,1026}, {55,2284}, {101,2195}, {664,673}
X(9319) = isogonal conjugate of X(9318)
X(9320) lies on these lines: {30,511}, {101,667}, {218,884}, {649,2340}, {663,672}, {1282,4063}, {1362,3669}, {1734,3783}, {3022,4162}
X(9321) lies on these lines: {44,4794}, {101,7193}, {106,292}, {214,6184}, {663,672}, {665,1155}, {910,4401}, {1015,1279}
X(9322) lies on these lines: {1053,4040}
X(9322) = isogonal conjugate of X(9317)
X(9323) lies on these lines: {100,2701}, {101,4040}, {514,1110}, {663,4564}, {692,4367}, {1053,2170}, {1308,4905}, {1308,4905}, {3573,9266},
X(9324) lies on these lines: {1,88}, {43,4274}, {44,9325}, {45,846}, {165,2246}, {171,1051}, {899,2108}, {1155,1282}, {1283,5217}, {2629,2939}, {2636,2947}, {2944,3579}, {3246,8301}, {3752,9337}, {5010,6187}
X(9324) = isogonal conjugate of X(9325)
X(9325) lies on these lines: {1,9271}, {44,9324}, {519,4480}
X(9325) = isogonal conjugate of X(9324)
X(9326) lies on these lines: {1,9271}, {44,88}, {320,4945}, {758,4792}, {903,4395}
X(9327) lies on these lines: {1,41}, {37,5053}, {39,106}, {58,3230}, {220,7373}, {284,3723}, {551,2329}, {572,3247}, {595,2242}, {999,3730}, {1018,5253}, {1055,3746}, {1149,5280}, {1334,5030}, {1500,9259}, {1759,3890}, {3207,6767}, {3303,4262}, {3304,4253}, {3496,3898}, {3509,3884}, {3624,4390}, {3635,3684}, {3754,4919}, {5045,6603}, {5074,7247}, {7278,9318}
X(9327) = isogonal conjugate of X(9328)
X(9328) lies on these lines: {518,3635}, {673,9329}
X(9328) = isogonal conjugate of X(9327)
X(9329) lies on these lines: {673,9328}, {3635,4743}
X(9330) lies on these lines: {2,38}, {9,5297}, {10,4671}, {37,3240}, {44,9347}, {45,100}, {81,3715}, {3219,5268}, {3305,3920}, {3688,5640}, {3720,4661}, {4363,4756}, {5281,7069}, {7191,7308}
X(9331) lies on these lines: {1,39}, {35,5023}, {37,3679}, {55,609}, {101,2177}, {192,3761}, {668,4664}, {941,3950}, {1107,3633}, {1571,3337}, {1573,4677}, {2176,5312}, {2242,5010}, {3230,5313}, {3295,5280}, {3303,5299}, {3632,5283}, {3746,7031}, {4099,4385}, {4390,4653}, {4515,6051}
X(9332) lies on these lines: {42,81}, {238,9345}, {1001,4038}
X(9332) = isogonal conjugate of X(9333)
X(9333) lies on these lines: {291,9334}
X(9333) = isogonal conjugate of X(9332)
X(9334) lies on these lines: {291,9333}
X(9335) lies on these lines: {2,38}, {8,4694}, {55,88}, {57,7292}, {89,2308}, {145,4695}, {354,3240}, {899,4430}, {1376,3315}, {1393,5265}, {1401,5640}, {1739,3241}, {3218,5272}, {3306,5269}, {3616,4424}, {3617,3976}, {3670,5550}, {3677,5297}, {3681,3999}, {3742,4850}, {3752,4883}, {3848,4003}, {3870,8056}, {3873,4849}, {3920,5437}, {4135,4871}
X(9336) lies on these lines: {1,39}, {36,5023}, {41,106}, {56,1384}, {218,3445}, {330,3760}, {609,5563}, {999,5299}, {1019,7153}, {1149,4253}, {1572,3337}, {1574,4677}, {1575,3633}, {2241,7280}, {3061,4694}, {3227,6376}, {3304,5280}, {3673,7208}
X(9337) lies on these lines: {42,81}, {238,9350}, {1054,4906}, {1376,8616}, {3550,4383}, {3749,8056}, {3750,4421}, {3752,9324}
X(9337) = isogonal conjugate of X(9338)
X(9338) lies on these lines: {291,9339}
X(9338) = isogonal conjugate of X(9337)
X(9339) lies on these lines: {291,9338}, {726,3633}
X(9340) lies on these lines: {31,57}, {38,4650}, {58,484}, {100,4722}, {171,756}, {678,3550}, {750,1707}, {846,1255}, {902,3748}, {1155,2308}, {1193,5122}, {1254,1399}, {1468,5119}, {1743,9350}, {1962,4640}, {2310,5348}, {3011,3982}, {4512,9345}, {4995,7277}
X(9341) lies on these lines: {32,56}, {35,172}, {36,5007}, {39,609}, {595,8649}, {1017,4257}, {1384,2241}, {1506,7294}, {2242,3053}, {2275,5008}, {3052,9351}, {4258,9346}, {4299,5309}, {5204,7772}, {5260,5277}, {5433,7753}, {7354,7755}
X(9342) lies on these lines: {1,3968}, {2,11}, {8,7373}, {9,9352}, {10,5253}, {21,3634}, {36,3828}, {38,88}, {81,899}, {404,993}, {474,2975}, {756,1054}, {1706,3890}, {2999,9347}, {3218,3740}, {3242,9335}, {3304,4678}, {3306,3681}, {3315,3961}, {3337,4015}, {3624,3871}, {3689,3848}, {3711,4430}, {3742,3935}, {3752,5297}, {3814,6175}, {3841,7504}, {3873,5437}, {4002,4861}, {4359,5205}, {4420,5439}, {4661,4860}, {4850,5268}, {5333,6685}
X(9342) = isogonal conjugate of X(9343)
X(9343) lies on these lines: {105,9344}, {518,3635}
X(9343) = isogonal conjugate of X(9342)
X(9344) lies on these lines: {105,9343}
X(9345) lies on these lines: {1,88}, {2,3775}, {31,940}, {38,5287}, {42,4413}, {57,1962}, {81,748}, {238,9332}, {354,5311}, {612,3243}, {756,5223}, {1125,1150}, {1468,5251}, {1961,3873}, {2308,4423}, {3120,4675}, {3305,4722}, {3723,4003}, {4512,9340}
X(9346) lies on these lines: {6,101}, {32,1468}, {42,574}, {58,2241}, {99,4393}, {213,9351}, {940,1573}, {993,1100}, {1475,7772}, {1500,5021}, {2267,5042}, {4258,9341}
X(9347) lies on these lines: {1,88}, {2,1386}, {6,5297}, {31,1961}, {44,9330}, {81,612}, {171,4414}, {908,4349}, {940,3242}, {968,1255}, {1100,3240}, {1621,5269}, {1962,3550}, {2999,9342}, {3666,9352}, {3758,3952}, {3989,4650}, {4307,5057}, {4427,4664}
X(9347) = isogonal conjugate of X(9348)
X(9348) lies on these lines: {88,9349}
X(9348) = isogonal conjugate of X(9347)
X(9349) lies on these lines: {44,5275}, {88,9348}
X(9350) lies on these lines: {1,3968}, {2,2177}, {31,899}, {41,1574}, {42,4413}, {43,81}, {100,748}, {106,4677}, {200,244}, {238,9337}, {404,6048}, {474,3214}, {756,8580}, {936,4642}, {997,4695}, {1054,3681}, {1647,4863}, {1698,4653}, {1743,9340}, {1757,9352}, {3030,3917}, {3722,5272}, {3740,4414}, {3873,5524}, {3938,4906}
X(9351) lies on these lines: {3,8649}, {32,3230}, {37,5114}, {58,2176}, {101,2241}, {213,9346}, {220,1015}, {574,1334}, {1055,5206}, {1201,7772}, {1500,4255}, {1574,4513}, {3052,9341}, {3730,9259}
X(9352) lies on these lines: {2,1155}, {9,9342}, {31,1054}, {40,3890}, {43,4722}, {46,404}, {56,8668}, {57,100}, {63,5785}, {65,4188}, {88,614}, {101,6205}, {165,1621}, {171,4850}, {244,3550}, {484,3877}, {750,1961}, {899,4650}, {993,5131}, {1158,6915}, {1376,3218}, {1739,4257}, {1757,9350}, {1770,4193}, {1788,4190}, {2099,4881}, {3052,7292}, {3219,4413}, {3315,3749}, {3336,3868}, {3337,3889}, {3338,3871}, {3339,4855}, {3434,5435}, {3524,3579}, {3585,7705}, {3666,9347}, {3679,4973}, {3689,4430}, {3696,5372}, {3753,5122}, {3754,3897}, {3812,4189}, {3885,5563}, {3957,4421}, {4293,5176}, {4295,6921}, {4511,4930}, {4652,5260}, {5010,5883}, {5284,5437}
X(9352) = isogonal conjugate of X(9353)
X(9353) lies on these lines: {1280,9354},3625,4133}
X(9353) = isogonal conjugate of X(9352)
X(9354) lies on these lines: {518,1351}, {1280,9353}
X(9355) lies on these lines: {1,651}, {3,7609}, {4,1046}, {8,2943}, {9,1742}, {43,1709}, {57,4014}, {84,978}, {90,1745}, {165,2246}, {171,5927}, {238,971}, {294,1721}, {513,2957}, {516,1757}, {572,2112}, {573,2938}, {650,9357}, {984,5779}, {1044,1728}, {1053,3309}, {1054,1768}, {1086,5851}, {1633,2265}, {1707,1750}, {1722,7992}, {1734,2958}, {1736,5018}, {1758,1776}, {1858,2647}, {1898,1935}, {2808,3271}, {2820,5540}, {2951,3973}, {2956,9372}, {3065,6127}, {3667,9282}, {3817,4896}, {4866,8915}, {5293,5777}, {5524,5537}, {5526,5527}, {5529,6909}, {5974,6045}, {7262,7580}
X(9355) = isogonal conjugate of X(9357)
X(9355) = trilinear pole, wrt excentral triangle, of line X(1)X(3)
X(9355) = X(648)-of-excentral-triangle
X(9355) = excentral isogonal conjugate of X(649)
X(9355) = excentral isotomic conjugate of X(3309)
X(9356) lies on these lines: {44,513}, {241,9358}, {1758,8609}, {6603,9357}, {6610,7045}
X(9357) lies on these lines: {1,9358}, {650,9355}, {1054,7658}, {4480,6745}, {6603,9356}
X(9357) = isogonal conjugate of X(9355)
X(9358) lies on these lines: {1,9357}, {100,4105}, {241,9356}, {650,651}, {653,2501}, {5435,6654}
X(9359) lies on the conic TCX(1054) and these lines: {1, 190}, {6, 1045}, {9, 87}, {10, 26076}, {31, 4579}, {40, 979}, {43, 4274}, {44, 2664}, {56, 1050}, {58, 5150}, {142, 7240}, {191, 17954}, {192, 23524}, {291, 3271}, {645, 741}, {649, 9361}, {651, 36276}, {659, 1054}, {672, 3510}, {846, 24492}, {894, 21352}, {978, 28283}, {1052, 21173}, {1086, 24722}, {1449, 17475}, {1580, 5053}, {1740, 1743}, {1979, 21893}, {2108, 3882}, {2235, 39252}, {2309, 2663}, {2629, 16575}, {2640, 5539}, {3122, 24482}, {3123, 4499}, {3226, 3875}, {3685, 23579}, {3729, 18194}, {3731, 24661}, {3737, 9282}, {4164, 23394}, {4253, 6196}, {4440, 27846}, {4465, 26102}, {4489, 4700}, {5272, 26273}, {7032, 17350}, {7184, 17353}, {9263, 21100}, {10436, 39044}, {15601, 21214}, {16482, 16726}, {17296, 25572}, {17319, 23532}, {17351, 18170}, {18754, 33863}, {19950, 23774}, {21371, 27663}, {25570, 37681}
X(9359) = isogonal conjugate of X(9361)
X(9359) = trilinear pole, wrt excentral triangle, of Nagel line
X(9360) lies on these lines: {44,513}, {239,6631}, {536,7035}, {2664,3230}, {3009,3689}, {3290,5205}
X(9361) lies on these lines: {1,9362}, {649,9359}, {899,9295}, {2664,3230}
X(9361) = isogonal conjugate of X(9359)
X(9362) lies on these lines: {1,9361}, {100,669}, {190,649}, {239,6631}, {650,4562}, {660,3699}, {668,4369}, {2664,4434}
X(9363) lies on these lines: {1,84}, {8,1106}, {21,4322}, {31,4308}, {34,3976}, {56,978}, {58,4315}, {104,1066}, {145,9316}, {238,1420}, {354,2647}, {603,3476}, {651,1201}, {999,8757}, {1319,1935}, {1458,2975}, {1468,3600}, {1496,5731}, {2943,3057}, {3304,6180}, {3889,4332}, {4306,8666}
X(9363) = isogonal conjugate of X(9368)
X(9363) = crosspoint of PU(92)
X(9364) lies on these lines: {1,3}, {2,9316}, {7,750}, {8,1106}, {31,5435}, {43,222}, {100,1458}, {109,238}, {221,978}, {244,4318}, {320,765}, {404,1042}, {603,1788}, {651,899}, {653,1430}, {902,7677}, {979,1413}, {1044,3149}, {1054,1465}, {1376,1407}, {1394,1722}, {1708,4650}, {1736,1768}, {1772,4351}, {2263,3306}, {2647,3812}, {2669,4573}, {3871,4322}, {3915,5265}, {4413,6180}, {6048,9370}
X(9364) = isogonal conjugate of X(9365)
X(9364) = perspector of conic {{A,B,C,PU(92)}}
X(9364) = X(2)-Ceva conjugate of X(39049)
X(9365) lies on the Feuerbach hyperbola and these lines: {1,2810}, {7,244}, {8,2310}, {84,978}, {104,238}, {646,6736}, {651,1201}, {899,1156}, {983,7083}, {984,1000}, {1266,2481}, {2195,5377}, {3062,5400}
X(9365) = isogonal conjugate of X(9364)
X(9366) lies on these lines: {30,511}, {650,3057}, {4885,5836}
X(9367) lies on these lines: {3,9374}, {37,216}, {220,1575}, {650,3057}, {958,8770}, {1212,3039}, {2275,6554}, {3295,6181}
X(9368) lies on these lines: {40,978}, {2324,3169}
X(9368) = isogonal conjugate of X(9363)
X(9369) lies on these lines: {1,979}, {3,4737}, {8,20}, {10,1054}, {30,5100}, {56,341}, {145,3685}, {190,1222}, {404,4723}, {519,7283}, {529,7270}, {956,4385}, {958,3757}, {1050,1201}, {1089,5288}, {1265,3476}, {2975,4696}, {3177,3729}, {3263,7176}, {3436,3705}, {3992,5563}, {4308,5423}, {4692,5258}, {4899,6737}
X(9369) = crosssum of PU(92)
X(9370) lies on these lines: {1,1864}, {3,4551}, {6,388}, {8,221}, {10,222}, {12,940}, {20,7074}, {34,518}, {40,2956}, {43,1466}, {55,1935}, {56,978}, {63,227}, {65,3751}, {73,958}, {78,1455}, {81,5261}, {84,1103}, {109,5687}, {200,1394}, {201,5220}, {210,1038}, {219,5930}, {355,3157}, {394,3436}, {478,3713}, {515,7078}, {517,8757}, {603,1376}, {607,5781}, {611,5252}, {748,4322}, {899,1106}, {965,2286}, {1191,3476}, {1407,1788}, {1413,7080}, {1478,3173}, {1617,1724}, {1745,3428}, {1943,4385}, {2003,5711}, {2099,2647}, {2122,6736}, {3214,9316}, {3681,4296}, {4513,4559}, {6048,9364}
X(9370) = isogonal conjugate of X(9375)
X(9371) lies on these lines: {1,3}, {11,1738}, {20,227}, {33,1376}, {37,5218}, {43,1864}, {44,1776}, {63,7074}, {78,1854}, {84,1103}, {100,3100}, {109,2739}, {212,4640}, {280,341}, {390,4850}, {497,3752}, {516,1465}, {518,7004}, {522,650}, {750,4336}, {899,2310}, {971,4551}, {1074,7680}, {1158,7078}, {1253,4414}, {1427,3474}, {1455,6909}, {2654,3812}, {3486,4646}, {3703,6736}, {3740,7069}
X(9371) = isogonal conjugate of X(9372)
X(9371) = X(1)-line conjugate of X(56)
X(9372) lies on the Feuerbach hyperbola and these lines: {1,1461}, {7,4617}, {8,221}, {9,109}, {21,4565}, {84,1106}, {294,2182}, {314,4573}, {608,7003}, {885,1462}, {1320,4318}, {2263,3577}, {2956,9355}
X(9372) = isogonal conjugate of X(9371)
X(9371) = X(2)-Ceva conjugate of X(39050)
X(9371) = perspector of hyperbola {{A,B,C,X(8),X(651)}}
X(9373) lies on these lines: {30,511}, {56,650}, {693,3436}, {998,2424}, {1329,4885}
X(9374) lies on these lines: {3,9367}, {56,650}, {404,7123}, {1385,6181}
X(9375) lies on these lines: {978,1044}
X(9375) = isogonal conjugate of X(9370)
X(9376) lies on these lines: {40,1256}, {56,280}, {84,1728}, {189,1788}, {271,1376}, {979,1413}, {1433,1771}, {1804,7157}, {2192,7038}
X(9377) lies on these lines: {564,2167}, {1109,2190}
X(9378) lies on these lines: {54,1879}, {115,8882}
X(9379) lies on these lines: {110,1209}, {264,1147}
X(9380) lies on these lines: {3,6}, {5189,6103}
X(9380) = isogonal conjugate of X(9381)
X(9381) lies on the Kiepert hyperbola and these lines: {4,7730}, {264,7578}, {275,338}
X(9381) = isogonal conjugate of X(9380)
X(9383) lies on these lines: {30,511}, {2599,2600}
X(9384) lies on these lines: {30,511}, {692,4556}, {2245,2605}
X(9385) lies on these lines: {30,511}, {2610,2611}
X(9386) lies on these lines: {30,511}, {2290,2618}
X(9387) lies on these lines: {2621,2625}
X(9387) = isogonal conjugate of X(2619)
X(9388) lies on these lines: {30,511}, {1109,2624}
X(9389) lies on these lines: {2619,2627}
X(9389) = isogonal conjugate of X(2625)
X(9390) lies on these lines: {1,2633}, {656,2629}
X(9390) = isogonal conjugate of X(2629)
X(9391) lies on these lines: {30,511}, {2631,2632}
X(9392) lies on these lines: {656,2629}
X(9392) = isogonal conjugate of X(2633)
X(9393) lies on these lines: {44,513}
X(9394) lies on these lines: {1,2639}, {652,2636}
X(9394) = isogonal conjugate of X(2636)
X(9395) lies on these lines: {1,2644}, {661,2640}, {2641,9217}
X(9395) = isogonal conjugate of X(2640)
X(9396) lies on these lines: {661,2640}
X(9396) = isogonal conjugate of X(2644)
X(9397) lies on these lines: {30,511}, {650,2646}, {693,5086}
X(9398) lies on these lines: {1,3144}, {3,1046}}, {407,2647}, {409,2648}, {1807,5293}
X(9398) = isogonal conjugate of X(2647)
X(9399) lies on the Jerabek hyperbola and these lines: {3,1046}, {72,3178}, {2647,2652}
X(9399) = isogonal conjugate of X(409)
X(9400) lies on these lines: {30,511}, {649,2666}, {2978,4380}, {3835,4507}, {4063,4455}
X(9401) lies on these lines: {2,1045}, {1258,3747}, {2665,2668}
X(9401) = isogonal conjugate of X(2663)
X(9402) lies on these lines: {30,511}, {798,2667}, {3063,5027}, {3709,4093}
X(9403) lies on these lines: {2,1045}, {2107,2663}, {4093,9281}
X(9403) = isogonal conjugate of X(2668)
X(9404) lies on these lines: {6,647}, {9,1021}, {44,513}, {55,4524}, {210,4477}, {850,5278}, {3219,4467}, {3709,7252}
X(9404) = PU(86)-harmonic conjugate of X(1464)
X(9404) = crossdifference of every pair of points on line X(1)X(30)
X(9404) = crosssum of Kiepert-hyperbola intercepts of Gergonne line
X(9405) lies on these lines: {30,3464}, {56,2606}, {57,80}, {74,1099}, {542,1354}, {1425,2595}, {2607,3028}
X(9406) lies on these lines: {31,1932}, {163,1755}, {667,788}, {1725,2247}, {2172,4020}, {2312,3708}
X(9406) = isogonal conjugate of X(33805)
X(9407) lies on these lines: {32,682}, {50,237}, {206,5063}, {669,688}, {1495,3284}, {3003,5191}
X(9407) = isogonal conjugate of isotomic conjugate of X(1495)
X(9407) = crossdifference of every pair of points on line X(76)X(2394)
X(9408) lies on the Brocard inellipse and these lines: {6,74}, {32,3124}, {248,3531}, {1511,2420}, {6103,7687}
X(9408) = isogonal conjugate of X(31621)
X(9408) = crosspoint of X(6) and X(1495)
X(9408) = crosssum of X(2) and X(1494)
X(9408) = PU(87)-harmonic conjugate of X(9409)
X(9408) = barycentric square of X(2173)
X(9408) = crossdifference of every pair of points on line X(1494)X(3268)
X(9409) is the center of this circle: V(X(74)) = {{15,16,74,112}}; see the preamble to X(6137). (Randy Hutson, March 26, 2016)
X(9409) lies on these lines: {3,684}, {4,6130}, {20,2797}, {74,526}, {112,1576}, {187,237}, {248,878}, {690,5489}, {3184,9033}, {5667,6086}
X(9409) = reflection of X(i) in X(j) for these (i,j): (4,6130) , (684,3)
X(9409) = isogonal conjugate of X(16077)
X(9409) = anticomplement of complementary conjugate of X(39008)
X(9409) = bicentric sum of PU(109)
X(9409) = PU(87)-harmonic conjugate of X(9408)
X(9409) = PU(109)-harmonic conjugate of X(3269)
X(9409) = crossdifference of every pair of points on line X(2)X(648)
X(9410) lies on the cubic K472 and these lines: {30,340}, {1531,7809}
X(9410) = complement of isogonal conjugate of X(9412)
X(9410) = complement of isotomic conjugate of X(39358)
X(9410) = X(2)-Ceva conjugate of X(1494)
X(9411) lies on these lines: {30,511}
X(9412) lies on these lines: {6,74}, {399,2420}, {1990,5667}, {3053,5191}, {5301,9259}
X(9412) = isogonal conjugate of isotomic conjugate of X(39358)
X(9412) = isogonal conjugate of anticomplement of X(9410)
X(9413) lies on these lines: {1,7095}, {43,57}, {56,7170}, {98,9414}, {982,2606}, {3027,3571}
X(9414) lies on these lines: {1,256}, {9,3287}, {38,9318}, {55,3571}, {98,9413}, {984,2607}, {3023,7170}
X(9414) = isogonal conjugate of X(9415)
X(9415) lies on these lines: {1,446}
X(9415) = isogonal conjugate of X(9414)
X(9416) lies on these lines: {98,1756}, {1821,7235}
X(9417) lies on these lines: {1,19}, {31,3402}, {32,7104}, {101,3508}, {560,1917}, {662,1966}, {667,788}, {1933,1967}, {1958,3403}, {1959,3405}, {1964,2085}, {2312,3404}
X(9418) lies on these lines: {6,25}, {32,3202}, {110,385}, {183,9306}, {237,3289}, {290,419}, {325,8840}, {669,688}, {1501,9233}, {1625,5167}, {3051,8265}, {3203,5007}, {3292,5201}, {3329,5012}, {3917,8266}
X(9418) = crossdifference of every pair of points on line X(76)X(525)
X(9419) lies on these lines: {6,98}, {32,8789}, {39,185}, {51,1196}, {232,511}, {690,1569}, {1625,2782}, {2421,5976}, {2491, 9420}, {2781,3094}
X(9419) = reflection of X(3269) in X(39)
X(9419) = Brocard-inellipse-antipode of X(3269)
X(9419) = PU(89)-harmonic conjugate of X(9420)
X(9419) = barycentric square of X(1755)
X(9420) lies on these lines: {98,804}, {187,237}, {2491,9419}
X(9420) = isogonal conjugate of anticomplement of X(39009)
X(9420) = center of circle {{X(15),X(16),X(98)}} (or V(X(98)))
X(9420) = PU(89)-harmonic conjugate of X(9419)
X(9421) lies on these lines: {99,4094}, {874,4418}, {1018,5539}, {3903,4128}
X(9421) = isogonal conjugate of X(3571)
X(9422) lies on these lines: {30,511}, {291,4705}, {668,2533}, {2238,4367}
X(9423) lies on these lines: {292,4589}, {740,9264}, {899,5029}, {1015,4974}, {1575,9321}, {2238,4367}
X(9424) lies on these lines: {1019,5539}, {9421,9425}
X(9424) = isogonal conjugate of X(7170)
X(9425) lies on these lines: {1,2142}, {99,813}, {799,2533}, {9421,9424}
X(9426) lies on these lines: {6,3221}, {110,4590}, {206,924}, {512,1691}, {523,5027}, {669,688}, {888,2451}, {1974,2422}
X(9426) = midpoint of X(i),X(j) for these (i,j): (669,3049)
X(9426) = isogonal conjugate of X(4609)
X(9426) = crosssum of Kiepert hyperbola intercepts of de Longchamps line
X(9427) lies on the Brocard inellipse and these lines: {6,99}, {32,8789}, {76,3224}, {115,2086}, {213,2107}, {887,1084}, {1186,3511}, {2031,3289}, {2032,2211}, {3051,5008}, {3124,5027}
X(9427) = trilinear pole, wrt symmedial triangle, of line X(2)X(6)
X(9427) = barycentric square of X(798)
X(9427) = PU(91)-harmonic conjugate of X(887)
X(9428) lies on these lines: {512,670}
X(9429) lies on these lines: {30,511}, {694,5027}, {881,2491}, {887,1084}
X(9430) lies on these lines: {3,3224}, {887,1084}
X(9430) = circumcircle-inverse of X(9431)
X(9431) lies on these lines: {3,3224}, {6,99}, {32,3499}, {148,2086}, {1384,1613}, {3225,3360}
X(9431) = circumcircle-inverse of X(9430)
X(9431) = trilinear pole wrt, tangential triangle, of line X(2)X(6)
X(9432) lies on the cubic K040 and these lines: {1,2810}, {40,979}, {56,3248}, {87,1054}, {190,1222}, {269,1357}, {292,2183}, {518,1120}, {996,2802}, {998,2841}, {1027,6085}, {1220,5836}, {2279,4274}, {2297,3030}, {5205,9025}, {5378,7077}, {9363,114}
X(9432) = isogonal conjugate of X(5205)
X(9433) lies on these lines: {30,511}, {649,1201}, {3216,4507}
X(9434) lies on these lines: {649,1201}
X(9435) lies on these lines: {40,978}, {9363,9432}
X(9435) = isogonal conjugate of X(9369)
Let La be the line through A parallel to the Gergonne line, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to ABC, with ratio 3. Let A"B"C" be the reflection of A'B'C' in the Gergonne line. The triangle A"B"C" is homothetic to ABC, and its centroid is X(9436); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, February 10, 2016)
X(9436) lies on the cubics K296, the K623 and these lines: {1,348}, {2,7}, {8,279}, {10,85}, {12,4059}, {36,6516}, {65,760}, {69,200}, {75,1088}, {77,3870}, {78,4350}, {86,2328}, {103,516}, {109,2862}, {116,5179}, {141,1418}, {145,3160}, {150,515}, {152,1541}, {171,3664}, {193,1419}, {238,1416}, {241,3693}, {291,1738}, {320,765}, {326,4341}, {331,1838}, {347,3875}, {350,4554}, {517,1565}, {518,1362}, {519,664}, {522,693}, {524,6610}, {666,3008}, {738,6762}, {910,5845}, {934,2751}, {948,4384}, {971,1536}, {982,3663}, {1010,1434}, {1111,1737}, {1210,3673}, {1231,1930}, {1319,7181}, {1414,6629}, {1427,3687}, {1441,4967}, {1442,3957}, {1443,3935}, {1446,6734}, {1458,4684}, {1770,4056}, {1788,7195}, {1847,5125}, {1996,5231}, {2263,3883}, {2792,5991}, {3212,4848}, {3263,3717}, {3622,5543}, {3649,4955}, {3672,3677}, {3732,8074}, {3741,7196}, {3945,5269}, {4104,7271}, {4292,4911}, {4347,7210}, {4416,6180}, {5018,5847}, {5252,7223}, {5285,7411}, {5853,9451}
X(9436) = midpoint of X(i),X(j) for these (i,j): (150,5088)
X(9436) = reflection of X(i) in X(j) for these (i,j): (152,1541), (664,1323), (666,3008), (3732,8074), (5179,116)
X(9436) = isogonal conjugate of X(2195)
X(9436) = isotomic conjugate of X(14942)
X(9436) = X(2)-Ceva conjugate of X(36905)
X(9436) = inverse-in-Steiner-circumellipse of X(7)
X(9436) = X(2)-Hirst inverse of X(7)
X(9436) = perspector of hyperbola {A,B,C,PU(47)}
X(9436) = crossdifference of every pair of points on line X(41)X(663)
X(9437) lies on these lines: {30,511}, {41,663}
X(9438) lies on these lines: {41,663}
X(9439) lies on these lines: {1,3732}, {101,7084}, {663,2082}, {1201,1419}, {2176,2223}, {2195,9310}, {2340,3158}, {2356,3192}
X(9439) = isogonal conjugate of X(9312)
X(9440) lies on these lines: {1,6}, {3,4334}, {7,1253}, {55,1742}, {105,2347}, {142,3939}, {165,7271}, {171,3664}, {212,3475}, {651,2293}, {943,1066}, {1496,5703}, {2223,7175}, {2646,9363}, {2647,3057}, {3000,7676}, {3295,8757}, {3303,9370}, {4307,5255}, {4319,8545}, {5218,9364}, {5281,9316}, {8012,9445}
X(9440) = isogonal conjugate of X(9445)
X(9440) = crosspoint of PU(104)
X(9441) lies on these lines: {1,3}, {6,1742}, {7,1253}, {9,1721}, {20,5247}, {31,5222}, {42,7411}, {43,170}, {44,9355}, {58,4229}, {100,2340}, {105,8647}, {109,2724}, {212,948}, {220,1376}, {238,516}, {279,9316}, {294,672}, {390,1471}, {527,3939}, {582,3073}, {595,5493}, {602,6361}, {650,1734}, {651,3000}, {750,5308}, {971,1757}, {984,990}, {991,4649}, {1044,7078}, {1053,2958}, {1170,9445}, {1212,3496}, {1323,7045}, {1445,4319}, {1468,3522}, {1633,2183}, {1709,7262}, {1743,2951}, {1770,3074}, {2108,2115}, {2239,6999}, {2293,7676}, {2361,5723}, {2711,6011}, {2807,3792}, {3062,3973}, {3751,5732}, {4641,5918}, {6603,9356}, {6684,6998}
X(9441) = reflection of X(i) in X(j) for these (i,j): (9355,44)
X(9441) = isogonal conjugate of X(9442)
X(9441) = perspector of conic {{A,B,C,PU(104)}}
X(9441) = intersection of the trilinear polars of PU(104) (the 1st and 2nd bicentrics of the Gergonne line)
X(9442) lies on the Feuerbach hyperbola, the cubic K623 and these lines: {1,1362}, {7,2310}, {8,3177}, {9,1742}, {294,672}, {651,2293}, {885,2254}, {1156,3000}, {2481,20115}, {2795,6598}, {3732,8285}
X(9442) = isogonal conjugate of X(9441)
X(9442) = isotomic conjugate of X(33677)
X(9443) lies on these lines: {30,511}, {354,650}, {657,4449}, {693,3681}, {3676,4524}, {3740,4885}
X(9444) lies on these lines: {37,3911}, {354,650}, {942,9367}, {999,6181}, {7373,9374}
X(9445) lies on these lines: {57,1742}, {105,1200}, {291,5728}, {1170,9441}, {8012,9440}
X(9445) = isogonal conjugate of X(9440)
X(9446) lies on these lines: {2,7}, {55,1088}, {85,1376}, {279,5281}, {354,658}, {390,2898}, {497,1996}, {2223,7176}, {3158,9312}, {3475,7056}, {7196,8299}
X(9446) = crosssum of PU(104)
X(9447) lies on these lines: {6,1626}, {9,4123}, {32,1397}, {41,212}, {163,609}, {560,1501}, {643,7075}, {1918,1974}, {1933,2209}, {2149,9315}
X(9448) lies on these lines: {41,4531}, {1501,1917}, {2172,8628}, {9247,9455}
X(9449) lies on these lines: {6,7}, {41,904}, {213,3010}, {919,3449}, {1197,1200}
X(9449) = isogonal conjugate of isotomic conjugate of X(16588)
X(9449) = PU(95)-harmonic conjugate of X(8638)
X(9449) = polar conjugate of isotomic conjugate of X(22368)
X(9450) lies on these lines: {6,3433}, {919,3449}
X(9451) lies on the cubic K294 and these lines: {1,39}, {8,9317}, {9,105}, {57,100}, {171,3979}, {244,612}, {518,910}, {982,1054}, {1416,3939}, {1961,3742}, {3169,8271}, {3315,5287}, {3722,5269}, {3751,8300}, {3757,5745}, {4318,6168}, {5853,9436}
X(9451) = reflection of X(i) in X(j) for these (i,j): (1282,8301), (4919,1)
X(9452) lies on these lines: {57,9453}, {238,1282}, {239,5853}, {1279,1429}
X(9452) = isogonal conjugate of X(9451)
X(9453) lies on these lines: {57,9452}, {105,238}, {241,1279}, {294,518}
X(9454) lies on these lines: {6,41}, {32,560}, {101,238}, {184,20126}, {213,1964}, {609,5156}, {662,2669}, {667,788}, {672,1818}, {673,1429}, {1001,9310}, {1174,1412}, {1334,8053}, {1911,2210}, {2205,3051}, {2225,3724}, {2329,5263}, {2352,5364}, {3736,5280}, {4251,4649}
X(9454) = isogonal conjugate of X(18031)
X(9454) = crossdifference of every pair of points on line X(75)X(522)
X(9455) lies on these lines: {31,184}, {105,1428}, {110,2106}, {560,1501}, {669,688}, {692,1914}, {1755,8628}, {1918,3051}, {2330,5276}, {9247,9448}
X(9456) lies on the conic {{A,B,C,X(101),X(163)}} and these lines: {6,101}, {31,692}, {44,5548}, {81,88}, {163,1333}, {292,9268}, {604,1415}, {608,8752}, {739,901}, {903,4586}, {909,2423}, {1022,2224}, {1100,1320}, {1404,1457}, {1407,1461}, {1449,2214}, {1462,6610}, {1797,2221}, {2087,2161}, {2162,2278}, {2251,7113}, {3227,5376}, {4792,5114}
X(9456) = isogonal conjugate of X(4358)
X(9456) = perspector of ABC and unary cofactor triangle of Gemini triangle 14
X(9456) = polar conjugate of isotomic conjugate of X(36058)
X(9456) = trilinear product X(6)*X(106)
X(9456) = barycentric product X(1)*X(106)
X(9456) = X(63)-isoconjugate of X(38462)
X(9456) = barycentric product of circumcircle intercepts of line X(1)X(513)
X(9457) lies on these lines: {1,3952}, {8,1054}, {106,4738}, {145,1046}, {190,1120}, {484,519}, {537,1320}, {894,9263}, {1317,9041}, {4361,9317}
X(9458) lies on these lines: {1,2}, {9,649}, {37,9360}, {75,7035}, {80,121}, {88,537}, {100,4432}, {106,4738}, {244,3699}, {312,9350}, {750,3570}, {867,1329}, {1054,3952}, {1215,9342}, {2607,4418}, {4152,9041}, {4358,4693}, {4363,4413}, {4422,6174}, {4781,9324}
X(9458) = midpoint of X(i),X(j) for these (i,j): (88,4767)
X(9458) = anticomplement of X(25377)
X(9459) lies on these lines: {31,7113}, {32,560}, {669,688}, {692,2210}, {760,2244}, {902,3285}, {922,2223}, {2177,4289}, {2245,8626}, {5035,7122}, {5168,8610}
X(9460) lies on these lines: {320,519}, {545,6631}, {3936,4945}
X(9460) = X(2)-Ceva conjugate of X(903)
X(9461) lies on these lines: {30,511}, {1017,1960}
Let A', B', C' be the reflections of X(39) in A, B, C, resp. X(9466) is the centroid of A'B'C'. (Randy Hutson, November 30, 2018)
X(9466) = midpoint of X(2) and X(76)
X(9466) = reflection of X(39) in X(2)
X(9466) = complement of X(7757)
X(9466) = SS(a → bc) of X(597) (barycentric substitution)
X(9467) lies on the cubics K252, K570 and these lines: {3,3493}, {98,385}, {238,1581}, {512,5149}, {694,2076}
X(9467) = isogonal conjugate of X(9469)
X(9467) = complement of X(34214)
X(9467) = circumcircle-inverse of X(34132)
X(9467) = X(2)-Ceva conjugate of X(9468)
X(9467) = complementary conjugate of complement of X(5989)
X(9467) = center of hyperbola {{A,B,C,X(805),X(34238)}}
X(9468) lies on the cubics K222, K252, K482, K532, the curve Q053 and these lines: {2,3114}, {3,3224}, {6,694}, {32,8789}, {39,83}, {111,1645}, {171,292}, {187,729}, {213,904}, {232,419}, {385,3225}, {881,887}, {882,2422}, {1691,20201}, {1918,1927}
X(9468) = isogonal conjugate of X(3978)
X(9468) = complement of isotomic conjugate of X(34214)
X(9468) = barycentric product of circumcircle intercepts of line PU(1)
X(9468) = X(2)-Ceva conjugate of X(9467)
X(9468) = perspector of hyperbola {{A,B,C,X(805),X(34238)}}
X(9469) lies on the cubics K354, K699 and these lines: {147,511}, {3978,5989}, {4027,8784}
X(9469) = isogonal conjugate of X(9467)
X(9469) = isogonal conjugate of complement of X(34214)
X(9470) lies on the cubic K251 and these lines: {2,2113}, {238,292}, {239,335}, {291,1757}, {334,1966}, {752,7245}, {875,6165}, {1001,3252}, {1447,1463}, {3685,4562}
X(9470) = isogonal conjugate of X(9472)
X(9470) = complement of X(2113)
X(9470) = complementary conjugate of X(20531)
X(9470) = X(2)-Ceva conjugate of X(292)
X(9470) = center of hyperbola {{A,B,C,X(660),X(813)}}
X(9471) lies on these lines: {3,8301}, {659,812}, {3802,8298}
X(9471) = circumcircle-inverse of X(8301)
X(9472) lies on the cubic K135 and these lines: {239,8301}, {672,1282}, {3802,8298}, {6651,8299}, {8849,8935}
X(9472) = isogonal conjugate of X(9470)
X(9473) lies on these lines: {147,325}, {297,385}
X(9473) = isotomic conjugate of X(147)
X(9473) = anticomplement of X(36899)
X(9473) = cyclocevian conjugate of X(290)
X(9473) = trilinear pole of Lemoine axis of 1st orthosymmedial triangle
X(9474) lies on these lines: {20,8861}, {237,694}, {248,2076}, {511,1297}, {2030,2065}
X(9475) lies on these lines: {3,112}, {6,157}, {39,185}, {74,5024}, {187,9408}, {232,237}, {426,3162}, {648,6394}, {684,2491}, {1193,7117}, {1249,3164}, {1316,6531}, {1560,1650}, {5013,8567}
X(9475) = isogonal conjugate of X(9476)
X(9475) = crossdifference of every pair of points on line X(98)X(1297)
X(9475) = X(2)-Ceva conjugate of X(39073)
X(9475) = perspector of hyperbola {{A,B,C,X(511),X(1503),X(2715)}}
X(9476) lies on these lines: {287,297}, {325,441}, {401,1297}, {2395,2419}
X(9476) = isogonal conjugate of X(9475)
X(9476) = isotomic conjugate of X(15595)
X(9476) = trilinear pole of line X(98)X(1297)
X(9476) = polar conjugate of X(132)
X(9477) lies on these lines: {732,1916}
X(9477) = isotomic conjugate of X(8290)
X(9478) lies on these lines: {2,4048}, {5,83}, {99,8362}, {115,3934}, {141,1916}, {148,7876}, {183,2896}{308,8783}, {325,732}, {385,5103}, {754,5461}, {1513,6036}, {3589,4027}, {3618,5984}, {3836,5988}, {5152,7819}, {5939,6704}, {6108,6670}, {6109,6669}, {6308,7750}, {7697,7790}, {7746,8150}, {7822,8178}, {9148,9185}
X(9478) = midpoint of X(115) and X(6292)
X(9478) = complement of X(8290)
X(9478) = midpoint of PU(136)
X(9479) lies on these lines: {30,511}, {99,827}, {351,3268}, {2394,9302}, {2485,2524}, {2492,2507}
X(9479) = isotomic conjugate of isogonal conjugate of X(5113)
X(9480) lies on these lines: {6,3506}, {32,733}
X(9481) lies on these lines: {2,32}, {39,827}, {1176,5116}, {4577,7783}
X(9481) = isogonal conjugate of X(9484)
X(9482) lies on these lines: {6,22}, {688,3005}, {732,8782}, {826,6781}
X(9482) = isogonal conjugate of X(9483)
X(9482) = X(2)-Ceva conjugate of X(39074)
X(9482) = perspector of hyperbola {{A,B,C,X(39),X(827)}}
X(9483) lies on these lines: {141,4577}, {689,8024}
X(9483) = isogonal conjugate of X(9482)
X(9484) lies on these lines: {6,6655}, {755,7828}
X(9484) = isogonal conjugate of X(9481)
X(9485) lies on these lines: {2,2793}, {23,385}, {351,5466}, {690,8591}, {804,8592}, {1649,8786}, {2408,6088}
X(9485) = reflection of X(i) in X(j) for these (i,j): (2,9123), (5466,351)
X(9485) = PU(138)-harmonic conjugate of X(2)
X(9486) lies on these lines: {3,111}, {6,9145}, {110,1384}, {187,237}, {352,2080}, {574,3124}, {2482,6791}, {6792,8724}, {7664,8369}
X(9486) = isogonal conjugate of X(9487)
X(9486) = X(2)-Ceva conjugate of X(39075)
X(9486) = perspector of hyperbola {{A,B,C,X(6),X(2709),X(9124)}}
X(9486) = pole of line X(111)X(352) wrt Parry circle
X(9487) lies on the Steiner circumellipse and these lines: {99,9136}, {671,1499}, {892,1992}
X(9487) = isogonal conjugate of X(9486)
X(9487) = trilinear pole of line X(2)X(2793)
X(9488) lies on these lines: {6,538}, {729,3329}, {3231,5008}
X(9489) lies on these lines: {32,9426}, {39,3221}, {187,237}
X(9489) = midpoint of X(i),X(j) for these (i,j): (669,887)
X(9490) lies on these lines: {5,2086}, {6,194}, {39,9427}, {83,3225}, {3051,3229}, {3231,6179}, {7783,9431}
X(9490) = isogonal conjugate of isotomic conjugate of X(6375)
X(9490) = polar conjugate of isotomic conjugate of X(23221)
X(9490) = eigencenter of the cevian triangle of X(670)
X(9490) = eigencenter of the anticevian triangle of X(669)
X(9490) = {X,Y}-harmonic conjugate of X(1), where X = SS(a → bc) of X(11), and Y = SS(a → bc) of X(12) (trilinear substitution)
X(9491) lies on these lines: {187,237}, {804,3267}, {2524,3221}, {3049,9006}
X(9491) = midpoint of X(i),X(j) for these (i,j): (887,9494)
X(9491) = reflection of X(i) in X(j) for these (i,j): (669,9494)
X(9491) = crossdifference of every pair of points on line X(2)X(2998)
X(9492) lies on these lines: {2,39}, {3224,9493}
X(9493) lies on these lines: {2,2998}, {3224,9492}, {3225,3978}
X(9494) lies on these lines: {6,9005}, {32,881}, {83,9498}, {99,9428}, {187,237}, {688,2531}, {783,827}, {804,5152}, {9006,9426}
X(9494) = midpoint of X(i),X(j) for these (i,j): (669,9491)
X(9494) = reflection of X(i) in X(j) for these (i,j): (887,9491)
X(9494) = PU(141)-harmonic conjugate of X(83)
X(9494) = perspector wrt the tangential triangle of the bianticevian conic of X(2) and X(6)
X(9494) = bicentric difference of PU(159)
X(9494) = PU(159)-harmonic conjugate of X(76)
X(9495) lies on these lines: {6,76}, {689,7783}
X(9496) lies on these lines: {2,308}, {688,3005}
X(9496) = isogonal conjugate of X(9497)
X(9496) = X(2)-Ceva conjugate of X(39076)
X(9497) lies on these lines: {689,3051}
X(9497) = isogonal conjugate of X(9496)
X(9498) lies on these lines: {30,511}, {83,9494}
, (9,9499)
X(9499) lies on the cubic K294 and these lines: {1,9453}, {9,9470}, {238,241}, {242,5236}, {516,3685}, {518,910}
X(9499) = isogonal conjugate of X(1282)
X(9500) lies on these lines: {7,6654}, {55,3252}, {105,2113}, {518,910}, {672,2112}, {1458,1914}, {1876,2201}, {8424,8934}
X(9501) lies on these lines: {105,238}, {294,1757}, {518,677}
X(9501) = isogonal conjugate of X(9504)
X(9502) lies on these lines: {1,41}, {9,77}, {37,2293}, {73,1212}, {201,220}, {218,3157}, {241,672}, {500,5777}, {664,6559}, {665,1642}, {1214,8012}, {2340,2356}, {3160,3177}, {3252,9309}, {3731,9355}, {5526,6126}, {6602,8270}, {8299,9504}
X(9502) = isogonal conjugate of X(9503)
X(9502) = X(2)-Ceva conjugate of X(39077)
X(9502) = perspector of hyperbola {{A,B,C,X(516),X(518),X(2736),X(36086)}}
X(9503) lies on these lines: {103,9441}, {241,294}, {514,6185}, {518,677}, {666,2338}, {1886,5236}, {2400,2402}, {6654,9311}
X(9503) = isogonal conjugate of X(9502)
X(9504) lies on these lines: {238,910}, {676,812}, {2114,8301}, {8299,9502}
X(9504) = isogonal conjugate of X(9501)
X(9505) lies on these lines: {238,1931}, {291,1757}, {335,740}, {518,9278}, {1284,5018}
X(9505) = isogonal conjugate of X(8298)
X(9506) lies on the cubic K135 and these lines: {239,9278}, {291,1757}, {292,3747}, {672,2054}, {741,1326}, {4584,6157}
X(9506) = isogonal conjugate of X(6651)
X(9507) lies on these lines: {1,1929}, {37,291}, {81,105}, {86,1281}, {100,4682}, {244,1962}, {518,2238}, {676,918}, {846,982}, {891,9269}, {896,3246}, {1100,8300}
X(9507) = midpoint of X(i),X(j) for these (i,j): (1,3125)
X(9508) lies on these lines: {1,4730}, {2,4010}, {8,4922}, {10,2787}, {44,513}, {88,897}, {100,110}, {244,2611}, {522,4874}, {523,2487}, {667,1734}, {812,3837}, {834,4507}, {876,9278}, {891,3960}, {900,3035}, {905,4083}, {911,1910}, {918,2977}, {1019,4705}, {1054,2640}, {1960,3887}, {2108,4455}, {2496,4926}, {2530,4063}, {2533,4560}, {2775,3579}, {3005,4132}, {3309,6050}, {3733,7234}, {3777,4498}, {4041,4367}, {4088,4750}, {4122,4467}, {4155,20241}, {4160,4770}, {4164,8301}, {4401,6004}, {4453,4802}, {4622,5380}, {4728,4810}, {4729,4879}, {4777,4789}, {4824,7192}
X(9508) = midpoint of X(i),X(j) for these (i,j): (1,4730), (8,4922), (649,1491), (659,2254), (661,4784), (667,1734), (1019,4705), (2530,4063), (2533,4560), (3777,4498), (4041,4367), (4122,4467), (4369,4913), (4729,4879), (4824,7192)
X(9508) = reflection of X(i) in X(j) for these (i,j): (4782,4394)
X(9508) = isogonal conjugate of X(37135)
X(9508) = complement of X(4010)
X(9508) = cevapoint of X(21196) and X(25381)
X(9508) = crosspoint of X(i) and X(j) for these {i,j}: {1, 38135}, {2, 4589}, {100, 291}
X(9508) = crosssum of X(i) and X(j) for these {i,j}: {1, 9508}, {6, 4455}, {238, 513}
X(9508) = crossdifference of every pair of points on line X(1)X(1929)
X(9509) lies on the Stammler hyperbola and these lines: {1,1929}, {6,662}, {37,2640}, {43,2248}, {148,6543}, {1030,2110}, {1740,2305}, {1910,3207}, {1931,2238}, {2134,6626}
X(9509) = isogonal conjugate of X(9510)
X(9510) lies on these lines: {1757,2959}, {1931,2238}, {4037,6542}
X(9510) = isogonal conjugate of X(9509)
X(9510) = isotomic conjugate of anticomplement of X(37128)
X(9511) lies on these lines: {44,513}, {55,1638}, {57,926}, {100,658}, {165,6139}, {918,1376}, {1054,2821}, {1639,4413}, {4025,4477}, {4458,8058}
X(9511) = X(25)-of-2nd-Sharygin-triangle
X(9511) = 1st-Sharygin-to-2nd-Sharygin similarity image of X(13097)
X(9511) = excentral-to-2nd-Sharygin similarity image of X(57)
X(9511) = parallelogic center of these triangles: 2nd Sharygin to mixtilinear
X(9512) lies on these lines: {6,3613}, {98,648}, {338,1316}, {419,9407}, {523,3447}, {1141,7731}, {1632,5191}, {2782,4558}, {2794,8754}
X(9513) lies on the Jerabek hyperbola, the cubic K166 and these lines: {2,879}, {3,2421}, {6,4230}, {69,2396}, {110,248}, {125,290}, {182,5649}, {265,2782}, {5968,6787}
X(9513) = reflection of X(i) in X(j) for these (i,j): (290,125)
X(9513) = isogonal conjugate of X(1316)
X(9513) = antigonal conjugate of X(290)
X(9513) = antipode of X(290) in Jerabek hyperbola
X(9513) = antitomic conjugate of X(110); see the preamble just before X(14941)
X(9513) = cevapoint of S1 and S2 on the Brocard (third) cubic (K019)
X(9513) = trilinear pole of line X(511)X(647), which is the Lemoine axis of the 2nd Parry triangle
X(9514) lies on these lines: {99,2525}, {647,2966}
X(9515) lies on these lines: {6,2936}, {32,524}, {187,1501}, {251,305}, {427,5475}
X(9515) = isogonal conjugate of X(7790)
X(9515) = X(75)-isoconjugate of X(9645)
X(9515) lies on these lines: {6,3266}, {32,524}, {141,468}, {523,3734}, {1235,2207}, {1918,4062}, {2393,7816}, {2422,9030}, {2854,4048}, {4590,7835}, {5026,8546}, {5486,9145}, {7754,9045}
X(9516) = isogonal conjugate of X(9465)
X(9516) = isotomic conjugate of X(7790)
X(9516) = cevapoint of X(6) and X(599)
X(9516) = X(7820)-crossconjugate of X(2)
X(9516) = trilinear pole of line X(669) X(690)
Midpoints on the infinity line: X(9517)-X(9532)
These centers were contributed by Peter Moses, February 11, 2016.
Suppose that P = p : q : r and U = u : v : w (barycentrics) are distinct points on the line L at infinity; that is, p + q + r = 0 and u + v + w = 0. As in the definition of orthopoint (copied below from Glossary), P may be regarded as a "direction" in the plane of the reference triangle ABC, and all the (parallel) lines in this direction meet in P, and likewise for U. Let L(P) be any line that meets L in P, and let L(U) be any line that meets L in U. Let W1 and W2 be the angle bisectors of the angles between L(P) and L(U) at L(P)∩L(U). The midpoints of P and U are here defined as W1∩L and W2∩L. Since L(P) and L(U) are perpendicular lines, the two midpoints are a pair of orthopoints, defined in the Glossary of ETC.
The appearance of {i, j}, {h,k} in the following list means that X(i) and X(j) are points on L and X(h) and X(k) are their midpoints.
{30,542}, {3414,3413}; {30,2771}, {3308,3307}; {30,5663}, {2575,2574}; {511,542}, {2575,2574}; {511,2782}, {3414,3413}; {511,2783}, {3308,3307}; {512,690}, {2574,2575}; {512,804}, {3413,3414}; {512,2787}, {3307,3308}; {513,900}, {3307,3308}; {513,2787}, {3413,3414}; {513,8674}, {2574,2575}; {514,2774}, {2574,2575}; {514,2786}, {3413,3414}; {514,3887}, {3307,3308}; {515,2779}, {2575,2574}; {515,2792}, {3414,3413}; {515,2800}, {3308,3307}; {516,2772}, {2575,2574}; {516,2784}, {3414,3413}; {516,2801}, {3308,3307}; {517,952}, {3308,3307}; {517,2771}, {2575,2574}; {517,2783}, {3414,3413}; {518,528}, {3307,3308}; {518,2795}, {3413,3414}; {518,2836}, {2574,2575}; {519,2796}, {3413,3414}; {519,2802}, {3307,3308}; {519,2842}, {2574,2575}; {520,2797}, {3413,3414}; {520,2803}, {3307,3308}; {520,9033}, {2574,2575}; {521,2798}, {3413,3414}; {521,2804}, {3307,3308}; {521,2850}, {2574,2575}; {522,2773}, {2574,2575}; {522,2785}, {3413,3414}; {522,3738}, {3307,3308}; {523,526}, {2574,2575}; {523,690}, {3413,3414}; {523,8674}, {3307,3308}; {524,543}, {3413,3414}; {524,2805}, {3307,3308}; {524,2854}, {2574,2575}; {525,2799}, {3413,3414}; {525,2806}, {3307,3308}; {526,804}, {690,542}; {526,900}, {8674,2771}; {526,926}, {2774,2772}; {526,2881}, {9517,2781}; {526,6084}, {2775,2836}; {526,6085}, {2842,2776}; {526,6086}, {9033,2777}; {526,6087}, {2850,2778}; {526,6088}, {2854,2780}; {526,8677}, {2773,2779}; {530,531}, {3414,3413}; {536,9024}, {3307,3308}; {538,5969}, {3413,3414}; {804,900}, {2787,2783}; {804,926}, {2786,2784}; {804,2881}, {2799,2794}; {804,6084}, {2788,2795}; {804,6085}, {2789,2796}; {804,6086}, {2797,2790}; {804,6087}, {2798,2791}; {804,6088}, {2793,543}; {804,8677}, {2785,2792}; {826,9479}, {3413,3414}; {900,926}, {3887,2801}; {900,2881}, {2806,2831}; {900,6084}, {2826,528}; {900,6085}, {2802,2827}; {900,6086}, {2803,2828}; {900,6087}, {2804,2829}; {900,6088}, {2830,2805}; {900,8677}, {3738,2800}; {912,5840}, {3308,3307}; {926,2881}, {9518,2825}; {926,6084}, {2820,2809}; {926,6085}, {2810,2821}; {926,6086}, {2811,2822}; {926,6087}, {2812,2823}; {926,6088}, {2813,2824}; {926,8677}, {928,2807}; {952,2782}, {2783,2787}; {952,2808}, {2801,3887}; {952,2818}, {2800,3738}; {952,5663}, {2771,8674}; {1499,2780}, {2575,2574}; {1499,2793}, {3414,3413}; {1499,2830}, {3308,3307}; {1503,2781}, {2575,2574}; {1503,2794}, {3414,3413}; {1503,2831}, {3308,3307}; {2775,3309}, {2575,2574}; {2776,3667}, {2575,2574}; {2777,6000}, {2575,2574}; {2778,6001}, {2575,2574}; {2782,2808}, {2784,2786}; {2782,2818}, {2792,2785}; {2782,5663}, {542,690}; {2788,3309}, {3414,3413}; {2789,3667}, {3414,3413}; {2790,6000}, {3414,3413}; {2791,6001}, {3414,3413}; {2808,2818}, {2807,928}; {2808,5663}, {2772,2774}; {2818,5663}, {2779,2773}; {2826,3309}, {3308,3307}; {2827,3667}, {3308,3307}; {2828,6000}, {3308,3307}; {2829,6001}, {3308,3307}; {2881,6084}, {2838,9523}; {2881,6085}, {2844,9527}; {2881,6086}, {2848,9530}; {2881,8677}, {2853,9532}; {3880,5854}, {3307,3308}; {3900,6366}, {3307,3308}; {6084,6085}, {2832,9519}; {6084,6086}, {2833,9520}; {6084,6087}, {2834,9521}; {6084,6088}, {2837,9522}; {6084,8677}, {2835,2814}; {6085,6086}, {2839,9524}; {6085,6087}, {2840,9525}; {6085,6088}, {2843,9526}; {6085,8677}, {2841,2815}; {6086,6087}, {2845,9528}; {6086,6088}, {2847,9529}; {6086,8677}, {2846,2816}; {6087,6088}, {2851,9531}; {6087,8677}, {2849,2817}; {6088,8677}, {2852,2819}; {8675,9003}, {2574,2575}
Let D(a,b,c) = [b2r2 + c2q2 + (b2 + c2 - a2)qr]1/2[b2w2 + c2v2 + (b2 + c2 - a2)vw]1/2.
Then the two midpoints are f(a,b,c) : f(b,c,a) : f(c,a,b) and g(a,b,c) : g(b,c,a) : g(c,a,b), where
f(a,b,c) = u[b2r2 + c2q2 + (b2 + c2 - a2)qr] + pD(a,b,c)
g(a,b,c) = u[b2r2 + c2q2 + (b2 + c2 - a2)qr] - pD(a,b,c)
Let P' = isogonal conjugate of P, and U' = isogonal conjugate of U. Then P' and U' lie on the circumcircle of ABC. Let M be the midpoint of P' and U', and assume that M is not the circumcenter, O. Then the line OM intersects the circumcircle in two points, given by the combos M + (|OM|/R - 1)*O and M + ( - |OM|/R - 1)*O. The isogonal conjugates of these two points are the midpoints of P and U.
X(9517) lies on these lines: {3, 684}, {5, 6130}, {30, 511}, {74, 1297}, {110, 112}, {113, 132}, {125, 127}, {265, 879}, {2492, 6593}, {3024, 6020}, {3028, 3320}, {3167, 5653}, {5972, 6720}, {9135, 9138}
X(9517) = isogonal conjugate of X(935)
X(9517) = isotomic conjugate of polar conjugate of X(2492)
X(9517) = X(19)-isoconjugate of X(17708)
X(9518) lies on these lines: {30, 511}, {101, 112}, {103, 1297}, {116, 127}, {118, 132}, {1362, 3320}, {3022, 6020}, {6710, 6720}
X(9519) lies on these lines: {30, 511}, {105, 165}, {106, 1292}, {120, 3817}, {121, 5511}, {354, 1357}, {1358, 3663}, {1766, 3973}, {3038, 3740}, {3057, 4014}, {7991, 9355}
X(9520) lies on these lines: {30, 511}, {105, 1294}, {107, 1292}, {120, 133}, {122, 5511}, {1358, 7158}, {3021, 3324}
X(9521) lies on these lines: {3, 676}, {30, 511}, {105, 1295}, {108, 1292}, {123, 5511}, {355, 4528}, {962, 3904}, {1358, 3318}, {1359, 3021}
X(9522) lies on these lines: {30, 511}, {105, 1296}, {111, 1292}, {120, 5512}, {126, 5511}, {1358, 4854}, {2941, 5540}, {3021, 3325}
X(9523) lies on these lines: {30, 511}, {105, 1297}, {112, 1292}, {120, 132}, {127, 5511}, {1358, 6020}, {3021, 3320}
X(9524) lies on these lines: {30, 511}, {106, 1294}, {107, 1293}, {121, 133}, {122, 5510}, {1357, 7158}, {3324, 6018}
X(9525) lies on these lines: {4, 4768}, {30, 511}, {40, 1769}, {106, 1295}, {108, 1293}, {123, 5510}, {1357, 3318}, {1359, 6018}
X(9526) lies on these lines: {30, 511}, {106, 1296}, {111, 1293}, {121, 5512}, {126, 5510}, {1054, 2938}, {1357, 4890}, {3325, 6018}
X(9527) lies on these lines: {30, 511}, {106, 1297}, {112, 1293}, {121, 132}, {127, 5510}, {1357, 6020}, {3320, 6018}
X(9528) lies on these lines: {21, 107}, {30, 511}, {108, 1294}, {122, 442}, {123, 133}, {1359, 7158}, {3318, 3324}, {6675, 6716}
X(9529) lies on these lines: {30, 511}, {107, 1296}, {111, 1294}, {122, 5512}, {126, 133}, {376, 1637}, {3268, 3543}, {3324, 6019}, {3325, 7158}
X(9530) lies on these lines: {2, 107}, {20, 648}, {30, 511}, {112, 376}, {127, 133}, {253, 317}, {549, 6720}, {3058, 3320}, {3324, 5434}
X(9531) lies on these lines: {30, 511}, {108, 1296}, {111, 1295}, {123, 5512}, {1359, 6019}, {3318, 3325}
X(9532) lies on these lines: {30, 511}, {102, 112}, {109, 1297}, {117, 127}, {124, 132}, {151, 2893}, {1361, 6020}, {1364, 3320}, {6711, 6720}
X(9533) lies on the cubic K577 and these lines: {1,3599}, {2,658}, {7,1699}, {55,934}, {57,279}, {165,3160}, {222,4617}, {883,6555}, {1122,3598}, {1439,3945}, {4454,4569}, {4635,8033}
X(9533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,479,279), (658,7056,2)
X(9533) = X(i)-Ceva conjugate of X(j) for these {i,j}: {7, 279}, {658, 7658}
X(9533) = X(1419)-cross conjugate of X(3160)
X(9533) = cross point of X(7) and X(3160)
X(9533) = X(220)-isoconjugate of X(3062)
X(9533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,479,279), (658,7056,2)
Centers of similitude: X(9534)-X(9715)
This preamble and centers X(9534)-X(9715) were contributed by César Eliud Lozada, February 27, 2016.
| Circle Φ1 | The appearance of (Φ2, i, j) in the following lists means that the insimilcenter and the exsimilcenter of the circles Φ1 and Φ2 are X(i) and X(j), respectively. |
|---|---|
| anticomplementary | (Apollonius, 9534, 9535), (Bevan, 10, 516), (extangents, 9536, 9537), (Half-Moses, 7736, 7738), (hexyl, 946, 515), (intangents, 9538, 9539), (1st Johnson-Yff, 5229, 388), (2nd Johnson-Yff, 497, 5225), (Lucas circles radical circle, 9540, 9541), (Lucas inner, 9542, 9543), (Moses, 2548, 2549), (sine triple-angle, 9544, 9545), (Spieker, 2550, 2551), (Stammler, 5, 30), (1st Steiner, 3832, 20), (tangential, 23, 7488) |
| Apollonius | (Bevan, 1695, 43), (Half-Moses, 9546, 9547), (hexyl, 9548, 9549), (intangents, 9550, 9551), (1st Johnson-Yff, 9552, 9553), (2nd Johnson-Yff, 9554, 9555), (Lucas circles radical circle, 9556, 9557), (Lucas inner, 9558, 9559), (Moses, 9560, 9561), (sine triple-angle, 9562, 9563), (Spieker, 9564, 9565), (Stammler, 9566, 9567), (1st Steiner, 9568, 9569) (tangential, 9570, 9571) |
| Bevan | (extangents, 9572, 9573), (Half-Moses, 9574, 9575), (hexyl, 3, 517), (intangents, 9576, 9577), (1st Johnson-Yff, 9578, 9579), (2nd Johnson-Yff, 9580, 9581), (Lucas circles radical circle, 9582, 9583), (Lucas inner, 9584, 9585), (Moses, 1571, 1572), (sine triple-angle, 9586, 9587), (Spieker, 9, 1706), (Stammler, 3579, 517), (1st Steiner, 9588, 9589), (tangential, 9590, 9591) |
| Half-Moses | (hexyl, 9592, 9593), (intangents, 9594, 9595), (1st Johnson-Yff, 9596, 9597), (2nd Johnson-Yff, 9598, 9599), (Lucas circles radical circle, 9600, 6424), (Lucas inner, 9601, 9602), (Moses, 39, 39), (sine triple-angle, 9603, 9604), (Spieker, 1107, 1575), (Stammler, 5024, 9605), (1st Steiner, 9606, 9607), (tangential, 9608, 9609) |
| hexyl | (intangents, 9610, 9611), (1st Johnson-Yff, 9612, 9613), (2nd Johnson-Yff, 3586, 9614), (Lucas circles radical circle, 9615, 9616), (Lucas inner, 9617, 9618), (Moses, 9619, 9620), (sine triple-angle, 9621, 9622), (Spieker, 9623, 936), (Stammler, 1385, 517), (1st Steiner, 9624, 5881), (tangential, 9625, 9626) |
| intangents | (1st Johnson-Yff, 9627, 9628), (2nd Johnson-Yff, 9629, 9630), (Lucas circles radical circle, 9631, 9632), (Lucas inner, 9633, 9634), (Moses, 9635, 9636), (sine triple-angle, 9637, 9638), (Spieker, 9639, 9640), (Stammler, 9641, 9642), (1st Steiner, 9643, 9644), (tangential, 55, 9645) |
| 1st Johnson-Yff | (2nd Johnson-Yff, 1, 4), (Lucas circles radical circle, 9646, 9647), (Lucas inner, 9648, 9649), (Moses, 9650, 9651), (sine triple-angle, 9652, 9653), (Spieker, 377, 3436), (Stammler, 9654, 9655), (1st Steiner, 9656, 9657), (tangential, 9658, 9659) |
| 2nd Johnson-Yff | (Lucas circles radical circle, 9660, 9661), (Lucas inner, 9662, 9663), (Moses, 9664, 9665), (sine triple-angle, 9666, 9667), (Spieker, 2478, 3434), (Stammler, 9668, 9669), (1st Steiner, 9670, 9671), (tangential, 9672, 9673) |
| Lucas circles radical circle | (Lucas inner, 6480, 6453), (Moses, 9674, 9675), (sine triple-angle, 9676, 9677), (Spieker, 9678, 9679), (Stammler, 6449, 6221), (1st Steiner, 9680, 9681), (tangential, 9682, 9683) |
| Lucas inner | (Moses, 9684, 9685), (sine triple-angle, 9686, 9687), (Spieker, 9688, 9689), (Stammler, 9690, 9691), (1st Steiner, 9692, 9693), (tangential, 9694, 9695) |
| Moses | (sine triple-angle, 9696, 9697), (Spieker, 1573, 1574), (Stammler, 5013, 6), (1st Steiner, 9698, 7765), (tangential, 9699, 9700) |
| sine triple-angle | (Spieker, 9701, 9702), (Stammler, 9703, 9704), (1st Steiner, 9705, 9706), (tangential, 9707, 1993) |
| Spieker | (Stammler, 9708, 9709), (1st Steiner, 9710, 9711), (tangential, 9712, 9713) |
| Stammler | (1st Steiner, 3526, 382), (tangential, 2070, 2937) |
| 1st Steiner | (tangential, 9714, 9715) |
X(9534) lies on these lines: {1,2}, {3,333}, {4,970}, {5,5233}, {6,1010}, {9,7283}, {12,4023}, {20,391}, {21,5278}, {56,4042}, {69,274}, {72,75}, {148,3029}, {149,3032}, {150,3033}, {181,388}, {193,4340}, {194,1654}, {210,4385}, {213,2345}, {312,5044}, {341,3697}, {346,3294}, {377,1330}, {392,4673}, {404,1150}, {405,1043}, {442,4417}, {497,1682}, {515,9548}, {516,1695}, {579,3686}, {594,2176}, {958,5132}, {960,3696}, {966,2092}, {1008,2238}, {1038,1943}, {1046,3980}, {1107,4261}, {1265,3596}, {1587,1686}, {1588,1685}, {1683,2547}, {1684,2546}, {1693,2543}, {1694,2542}, {1724,4195}, {1834,5743}, {2019,2545}, {2020,2544}, {2051,3091}, {2322,7513}, {2476,5741}, {3031,3448}, {3295,3996}, {3681,4968}, {3714,3740}, {3868,4359}, {3890,3902}, {3936,4197}, {3966,5015}, {4101,5249}, {4188,5361}, {4189,4276}, {4255,5737}, {4292,4416}, {4352,5232}, {4647,5692}, {4662,4737}, {4720,5047}, {4886,5814}, {5156,5247}, {7738,9546}
X(9534) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,970,9535), (10, 386, 2), (10, 6685, 1698), (377, 5739, 1330), (3679, 6048, 10), (4886, 7270, 5814), (5044, 5295, 312)
X(9535) lies on these lines: {2,573}, {4,970}, {10,962}, {20,386}, {43,516}, {146,3031}, {147,3029}, {152,3033}, {153,3032}, {165,6685}, {181,497}, {194,6999}, {223,5088}, {312,517}, {329,3687}, {333,2050}, {355,4886}, {388,1682}, {946,9548}, {980,5712}, {1211,7377}, {1587,1685}, {1588,1686}, {1683,2546}, {1684,2547}, {1693,2542}, {1694,2543}, {1754,4279}, {2019,2544}, {2020,2545}, {4295,5530}, {4383,6996}, {7738,9547}
X(9535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,970,9534), (573,2051,2)
X(9536) lies on these lines: {2,19}, {4,8141}, {20,6197}, {23,55}, {40,3146}, {1726,5011}, {1993,3197}, {2550,7391}, {3060,3611}, {3091,8251}, {5341,5712}
X(9536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,8141,9537), (19,3101,2)
X(9537) lies on these lines: {2,6197}, {4,8141}, {8,20}, {19,3091}, {46,347}, {55,7488}, {517,7520}, {2071,5584}, {2093,4296}, {3611,5889}
X(9537) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,8141,9536), (40,3101,20), (6197,8251,2)
X(9538) lies on these lines: {1,7}, {2,1062}, {4,8144}, {22,3295}, {33,3091}, {34,3543}, {55,7488}, {56,2071}, {280,4511}, {323,1069}, {496,858}, {1040,3523}, {1058,1370}, {1060,3522}, {1068,6895}, {1479,3153}, {1870,3146}, {3270,5889}, {6840,7952}, {7071,7503}, {7191,7396}
X(9538) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3100,20), (1,4354,4294), (4,8144,9539), (1062,6198,2)
X(9539) lies on these lines: {1,3146}, {2,33}, {4,8144}, {20,1060}, {22,7071}, {23,55}, {346,4123}, {390,7500}, {497,7391}, {1062,3091}, {1717,4295}, {1870,3543}, {1993,2192}, {3060,3270}, {3085,4354}, {3219,7070}, {3920,4319}, {4296,5059}, {6895,7952}, {7191,7221}
X(9539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,8144,9538), (33,3100,2)
X(9540) lies on these lines: {2,371}, {3,1587}, {4,590}, {5,6221}, {6,631}, {20,485}, {22,8276}, {30,6449}, {40,8983}, {74,8998}, {98,8997}, {99,8980}, {110,8994}, {140,3069}, {372,3523}, {376,3070}, {381,6407}, {382,6445}, {488,7793}, {491,3785}, {546,6519}, {548,6451}, {549,3312}, {550,6455}, {615,3525}, {640,3595}, {641,1270}, {1124,7288}, {1125,1702}, {1132,7486}, {1152,3524}, {1328,6478}, {1335,5218}, {1490,8987}, {1498,8991}, {1579,7494}, {2066,3086}, {2067,3085}, {3071,3090}, {3088,5412}, {3091,6453}, {3092,6353}, {3103,6194}, {3146,6564}, {3522,6560}, {3526,6199}, {3528,6411}, {3530,6398}, {3533,8252}, {3543,6484}, {3545,6429}, {3628,6447}, {3832,6480}, {3855,6468}, {5054,6417}, {5056,6565}, {5059,6486}, {5067,6437}, {5407,6805}, {5420,6419}, {5657,7969}, {5870,6811}, {6202,6813}, {6422,7735}, {6424,7736}, {6496,8703}, {7396,8280}, {7691,8995}
X(9540) = midpoint of X(i),X(j) for these (i,j): (6449,8976)
X(9540) = homothetic center of X(2)-quadsquares triangle and cevian triangle of X(3)
X(9540) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 371, 1588), (3, 3068, 1587), (3, 7583, 6460), (3, 8981, 3068), (4,1151,9541), (5, 6221, 6459), (20, 8972, 485), (140, 3311, 3069), (371, 5418, 2), (485, 6200, 20), (590, 1151, 4), (615, 3592, 7582), (3068, 6460, 7583), (3070, 6409, 376), (3071, 8253, 3090), (3523, 7585, 372), (3524, 7581, 1152), (3525, 7582, 615), (3526, 6199, 7584), (6425, 8253, 3071), (6460, 7583, 1587)
X(9541) lies on these lines: {2,1328}, {3,1588}, {4,590}, {5,6449}, {6,376}, {20,371}, {30,3068}, {140,6455}, {372,3522}, {381,6445}, {382,6407}, {485,3146}, {486,3523}, {487,2896}, {548,3312}, {549,6451}, {550,3311}, {615,3524}, {631,3071}, {1131,8960}, {1152,3528}, {1327,6476}, {1657,7583}, {1702,4297}, {2066,4293}, {2067,4294}, {3070,3529}, {3091,5418}, {3530,6496}, {3534,6199}, {3543,6480}, {3544,6488}, {3545,6433}, {3592,7581}, {3627,6519}, {3832,6484}, {5056,6486}, {5407,6806}, {5870,8721}, {6361,7969}, {6396,7586}, {6398,8703}, {6424,7738}
X(9541) = reflection of X(i) in X(j) for these (i,j): (3068,6221)
X(9541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 6459, 1588), (4,1151,9540), (20, 371, 1587), (20, 7585, 6560), (371, 6560, 7585), (382, 6407, 8981), (550, 3311, 6460), (615, 6411, 3524), (3071, 6409, 631), (3528, 7582, 1152), (6200, 6561, 2), (6560, 7585, 1587)
X(9542) lies on these lines: {2,6221}, {4,6407}, {20,1151}, {371,3523}, {376,6445}, {590,3839}, {615,6425}, {1656,6472}, {3069,6437}, {3091,6453}, {3146,6519}, {3522,6449}, {3524,6199}, {3526,6474}, {3543,6480}, {5056,6429}, {5418,7486}, {6200,7585}, {6447,7582}, {6455,7581}
X(9542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6407,9543), (8972,9541,3543)
X(9543) lies on these lines: {2,489}, {4,6407}, {20,6221}, {369,2681}, {485,3146}, {631,6445}, {1657,6472}, {3068,5059}, {3523,6449}, {3528,6199}, {3534,6474}, {3543,8981}, {3832,6480}, {5068,6561}, {6409,7586}, {6425,7585}, {6437,6460}, {6447,7581}, {6455,7582}, {6468,8972}, {6482,6565}
X(9543) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6407,9542)
X(9544) lies on these lines: {2,98}, {4,49}, {20,1147}, {22,323}, {23,154}, {25,1994}, {54,3091}, {146,3043}, {148,3044}, {149,3045}, {150,3046}, {155,7488}, {193,206}, {194,3202}, {215,497}, {388,2477}, {394,6636}, {399,3431}, {567,3545}, {569,5056}, {578,3832}, {1092,3522}, {1154,7556}, {1176,3620}, {1199,7506}, {1437,4189}, {1495,3060}, {1501,2056}, {1660,7500}, {1995,8780}, {2888,3549}, {2979,3292}, {3098,6030}, {3146,6759}, {3153,5654}, {3168,4240}, {3203,7787}, {6090,7485}, {7392,7605}, {7766,9418}
X(9544) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,49,9545), (22, 3167, 323), (49, 156, 4), (110, 184, 2), (110, 3047, 3448), (110, 5012, 9306), (154, 1993, 23), (184, 9306, 5012), (394, 6800, 6636), (1147, 1614, 20), (1501, 2056, 9463), (5012, 9306, 2)
X(9545) lies on these lines: {2,54}, {3,323}, {4,49}, {20,184}, {24,1994}, {110,578}, {146,3047}, {147,3044}, {152,3046}, {153,3045}, {182,3620}, {195,1658}, {215,388}, {497,2477}, {567,3090}, {568,1493}, {1092,3523}, {1181,2071}, {1199,6644}, {1614,3146}, {1993,7488}, {3043,3448}, {3167,7503}, {3543,6759}, {5056,9306}, {5944,6243}
X(9545) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6193,2888), (4,49,9544), (54,1147,2), (110,578,3091), (1092,5012,3523), (5944,6243,7556)
X(9546) lies on these lines: {3,6}, {10,5254}, {181,2276}, {1682,2275}, {2023,3029}, {2051,3815}, {2269,3778}, {7736,9535}, {7738,9534}
X(9546) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,970,9547), (39,9560,970)
X(9547) lies on these lines: {3,6}, {10,3815}, {181,2275}, {1682,2276}, {2051,5254}, {7738,9535}
X(9547) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,970,9546), (39,9561,970)
X(9548) lies on these lines: {1,181}, {3,43}, {4,9}, {57,5530}, {165,6048}, {386,3576}, {515,9534}, {517,1695}, {602,4279}, {631,6685}, {946,9535}, {986,1423}, {987,3550}, {1698,1764}, {2051,8227}, {3032,6264}, {4260,8726}, {5725,5755}
X(9548) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,970,9549), (10,573,40)
X(9549) lies on these lines: {1,181}, {3,1695}, {10,3090}, {40,386}, {43,517}, {515,9535}, {573,1449}, {946,9534}, {1764,5313}, {2051,5587}, {3032,6326}, {3340,5530}, {4260,6282}, {5657,6685}
X(9549) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,970,9548)
X(9550) lies on these lines: {10,33}, {386,3100}, {573,6198}, {970,8144}, {1062,2051}
X(9550) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (970,8144,9551)
X(9551) lies on these lines: {10,1062}, {33,2051}, {386,6198}, {573,3100}, {970,8144}
X(9551) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (970,8144,9550)
X(9552) lies on these lines: {4,1682}, {10,56}, {12,386}, {181,388}, {355,5530}, {573,7354}, {970,1478}, {1010,1397}, {3687,5794}, {5229,9535}
X(9552) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388,9534,181), (970,1478,9553)
X(9553) lies on these lines: {4,181}, {10,1836}, {12,573}, {56,2051}, {386,7354}, {388,1682}, {970,1478}, {4274,5230}, {5229,9534}
X(9553) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388,9535,1682), (970,1478,9552)
X(9554) lies on these lines: {1,9553}, {4,1682}, {10,4679}, {11,573}, {55,2051}, {181,497}, {386,6284}, {970,1479}, {3586,9549}, {3687,5695}, {5225,9534}
X(9554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1682,9552), (497,9535,181), (970,1479, 9555)
X(9555) lies on these lines: {1,9552}, {4,181}, {8,314}, {10,55}, {11,386}, {29,2175}, {80,987}, {150,388}, {355,5711}, {497,1682}, {573,6284}, {970,1479}, {1460,5786}, {3586,9548}, {4769,7270}, {5225,9535}, {5530,5722}
X(9555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,181,9553), (497,9534,1682), (970,1479,9554)
X(9556) lies on these lines: {3,6}, {10,6561}, {2051,5418}
X(9557) lies on these lines: {3,6}, {10,5418}, {2051,6561}
X(9558) lies on these lines: {3,6}, {43,9584}, {1695,9585}, {9534,9543}, {9535,9542}, {9548,9618}, {9549,9617}, {9550,9634}, {9551,9633}, {9552,9649}, {9553,9648}, {9554,9663}, {9555,9662}, {9562,9687}, {9563,9686}, {9564,9689}, {9565,9688}, {9568,9693}, {9569,9692}, {9570,9695}, {9571,9694}
X(9558) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (970,6407,9559)
X(9559) lies on these lines: {3,6}, {43,9585}, {1695,9584}, {9534,9542}, {9535,9543}, {9548,9617}, {9549,9618}, {9550,9633}, {9551,9634}, {9552,9648}, {9553,9649}, {9554,9662}, {9555,9663}, {9562,9686}, {9563,9687}, {9564,9688}, {9565,9689}, {9568,9692}, {9569,9693}
X(9559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (970,6407,9558)
X(9560) lies on these lines: {3,6}, {9,1247}, {10,115}, {43,1571}, {181,1500}, {1015,1682}, {1334,3124}, {1506,2051}, {1572,1695}, {2238,3496}, {2276,2653}, {2548,9535}, {2549,9534}
X(9560) = isogonal conjugate of isotomic conjugate of X(34528)
X(9560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 970,9561), (970,9546,39)
X(9561) lies on these lines: {3,6}, {10,1506}, {43,1572}, {115,2051}, {181,1015}, {1500,1682}, {1569,3029}, {1571,1695}, {2548,9534}, {2549,9535}
X(9561) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 970,9560), (970,9547,39)
X(9562) lies on these lines: {10,54}, {49,970}, {110,2051}, {181,215}, {184,199}, {386,1147}, {1682,2477}, {3031,3043}
X(9562) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,970,9563)
X(9563) lies on these lines: {10,110}, {49,970}, {54,2051}, {181,2477}, {184,386}, {215,501}, {573,1147}, {3029,3044}, {3031,3047}, {3032,3045}, {3033,3046}, {3203,4279}
X(9563) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,970,9562)
X(9564) lies on these lines: {2,181}, {5,10}, {8,1682}, {9,43}, {72,5530}, {171,4274}, {210,3687}, {377,9553}, {386,958}, {573,1376}, {936,9548}, {1107,9547}, {1377,1686}, {1378,1685}, {1573,9561}, {1574,9560}, {1575,9546}, {1680,1684}, {1681,1683}, {1695,1706}, {2013,2020}, {2014,2019}, {2478,9555}, {2550,9535}, {2551,9534}, {3030,3038}, {3032,3036}, {3033,3041}, {3034,3039}, {3037,5213}, {3434,9554}, {3436,9552}, {3596,7064}, {3688,7081}, {4260,5745}
X(9564) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,970,9565), (10,2051,2886)
X(9565) lies on these lines: {1,2092}, {2,1682}, {5,10}, {8,181}, {9,1695}, {43,1706}, {65,3687}, {377,9552}, {386,1376}, {404,5061}, {573,958}, {936,9549}, {1107,9546}, {1377,1685}, {1378,1686}, {1573,9560}, {1574,9561}, {1575,9547}, {1680,1683}, {1681,1684}, {2013,2019}, {2014,2020}, {2478,9554}, {2550,9534}, {2551,9535}, {3032,3035}, {3042,5810}, {3271,4195}, {3434,9555}, {3436,9553}, {3597,5603}, {3753,5530}, {4260,5847}, {4274,5247}, {5794,5814}
X(9565) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,970,9564), (10,2051,1329)
X(9566) lies on these lines: {3,6}, {5,9535}, {10,381}, {30,9534}, {40,6048}, {43,3579}, {181,3295}, {517,1695}, {999,1682}, {1385,9549}, {1656,2051}, {3687,3927}
X(9566) = midpoint of X(i),X(j) for these (i,j): (1695,9548)
X(9566) = {X(3),X(970)}-harmonic conjugate of X(9567)
X(9567) lies on these lines: {3,6}, {5,5233}, {10,1482}, {30,9535}, {43,517}, {181,999}, {381,2051}, {399,3031}, {1385,9548}, {1682,3295}, {1695,3579}, {3687,3940}, {4792,8148}, {6048,7982}
X(9567) = midpoint of X(i),X(j) for these (i,j): (43,9549)
X(9567) = {X(3),X(970)}-harmonic conjugate of X(9566)
X(9568) lies on these lines: {5,10}, {20,391}, {382,9566}, {386,631}, {3244,6176}, {3526,9567}, {3832,9535}, {4848,5718}, {5396,6684}, {5881,9548}, {7765,9560}
X(9568) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,970,9569), (5,9569,2051), (10,970,2051), (10,9569,5)
X(9569) lies on these lines: {5,10}, {20,386}, {382,9567}, {515,5754}, {573,631}, {962,5400}, {3526,9566}, {3671,5718}, {3832,9534}, {4192,5493}, {4297,5396}, {5881,9549}, {7765,9561}
X(9569) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,970,9568), (5,9568,10), (970,2051,10), (2051,9568,5)
X(9570) lies on these lines: {3,10}, {22,573}, {23,9535}, {24,386}, {25,2051}, {26,970}, {55,9551}, {1993,9562}, {2070,9567}, {2931,3031}, {2937,9566}, {7488,9534}
X(9570) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,970,9571)
X(9571) lies on these lines: {3,2051}, {10,25}, {22,386}, {23,9534}, {24,573}, {26,970}, {55,9550}, {1993,p230}, {2070,9566}, {2937,9567}, {7488,9535}
X(9571) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,970,9570)
X(9572) lies on these lines: {1,6197}, {10,9537}, {19,1699}, {30,40}, {165,3101}, {516,9536}, {1698,8251}
X(9572) = reflection of X(i) in X(j) for these (i,j): (1,7501)
X(9572) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40,8141,9573)
X(9573) lies on these lines: {1,3101}, {10,9536}, {19,1698}, {30,40}, {46,2257}, {165,6197}, {516,9537}, {1699,8251}, {3624,7561}
X(9573) = reflection of X(i) in X(j) for these (i,j): (1,7520)
X(9573) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40,8141,9572)
X(9574) lies on these lines: {1,5013}, {6,165}, {9,1575}, {10,7738}, {37,5437}, {39,40}, {43,9546}, {57,2276}, {516,7736}, {517,5024}, {574,3576}, {988,3501}, {1107,1706}, {1449,3550}, {1500,3333}, {1695,9547}, {1697,2275}, {1698,5254}, {1699,3815}, {1702,6421}, {1703,6422}, {2549,5587}, {3097,3099}, {5286,6684}
X(9574) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,40,9575), (39,1571,40)
X(9575) lies on these lines: {1,6}, {10,7736}, {32,3576}, {39,40}, {43,9547}, {57,893}, {165,5013}, {169,995}, {172,1420}, {230,3624}, {232,7713}, {516,7738}, {946,5286}, {988,3496}, {1015,3333}, {1125,7735}, {1193,2082}, {1575,1706}, {1695,9546}, {1697,2276}, {1698,3815}, {1699,5254}, {1702,6422}, {1703,6421}, {1914,3601}, {1959,5256}, {2270,2277}, {2548,5587}, {2999,7146}, {3612,7031}, {3674,4000}, {3677,3721}, {3767,8227}, {4386,5438}, {5305,5886}, {5691,7745}
X(9575) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1743,2329), (39,40,9574), (39,1572,40)
X(9576) lies on these lines: {1,7}, {10,9539}, {33,451}, {40,8144}, {43,9550}, {55,9573}, {165,6198}, {191,7070}, {1040,3624}, {1062,1699}, {1695,9551}, {1837,5160}, {2960,3601}
X(9576) = reflection of X(i) in X(j) for these (i,j): (1,9538)
X(9577) lies on these lines: {1,4}, {10,9538}, {40,8144}, {43,9551}, {55,9572}, {165,3100}, {516,9539}, {1062,1698}, {1695,9550}, {3295,8185}
X(9577) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,33,1699), (40,8144,9576)
X(9578) lies on these lines: {1,5}, {2,1420}, {4,1697}, {7,3617}, {8,226}, {9,3436}, {10,57}, {40,1478}, {43,9552}, {46,5270}, {55,5691}, {56,1698}, {65,3679}, {85,668}, {145,5226}, {165,7354}, {200,5794}, {329,5837}, {377,1706}, {405,2078}, {498,3576}, {515,3085}, {516,5229}, {519,3485}, {631,4311}, {942,5790}, {944,6956}, {946,7962}, {956,5705}, {1056,1210}, {1125,3476}, {1319,3624}, {1329,8583}, {1441,4696}, {1454,6763}, {1467,8728}, {1695,9553}, {1699,3057}, {1737,3333}, {1836,7991}, {1935,5264}, {2003,5711}, {2093,5690}, {2099,3632}, {2136,3434}, {2171,4007}, {2475,3882}, {2550,6736}, {2551,7308}, {2647,3961}, {2886,4853}, {3245,4338}, {3256,5687}, {3295,3586}, {3361,5434}, {3419,6765}, {3475,6738}, {3526,5126}, {3584,3612}, {3585,5119}, {3600,3911}, {3621,4323}, {3626,3671}, {3634,4315}, {3649,4668}, {3680,6871}, {3696,7201}, {3698,8581}, {3826,4321}, {3851,7743}, {3854,7320}, {3870,5086}, {4292,5128}, {4293,6684}, {4297,5218}, {4355,5221}, {4385,6358}, {5080,5250}, {5123,5193}, {5175,5853}, {5249,5554}, {5316,8165}, {5432,7987}, {5438,5552}, {5880,8256}, {6734,6762}, {6982,7982}, {7319,8236}
X(9578) = reflection of X(i) in X(j) for these (i,j): (3485,3947), (3601,3085)
X(9578) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12,5219), (1,355,5727), (1,5726,12), (1,7951,8227), (1,7989,11), (7,3617,4848), (8,226,3340), (8,5261,226), (10,388,57), (10,4298,1788), (12,5252,1), (65,5290,4654), (355,495,1), (355,5534,5881), (377,6735,1706), (388,1788,4298), (1056,5818,1210), (1788,4298,57), (3634,4315,7288), (3679,5290,65), (4292,5657,5128), (5252,5726,5219), (5711,9370,2003)
X(9579) lies on these lines: {1,30}, {3,5219}, {4,57}, {7,950}, {8,527}, {9,377}, {10,3474}, {11,3361}, {12,165}, {20,226}, {33,1448}, {34,990}, {35,4333}, {36,3560}, {40,1478}, {43,9553}, {46,3585}, {55,5290}, {56,1699}, {63,2475}, {65,971}, {142,452}, {225,1394}, {354,4355}, {355,2093}, {376,5714}, {382,942}, {386,2635}, {388,516}, {443,7308}, {497,4298}, {515,3340}, {529,4853}, {553,938}, {610,1901}, {908,4190}, {912,4338}, {920,7701}, {946,1420}, {962,7962}, {1012,5715}, {1155,1698}, {1419,3332}, {1445,6894}, {1454,1768}, {1467,5805}, {1473,4214}, {1479,3333}, {1656,5122}, {1695,9552}, {1706,3436}, {1708,6839}, {1754,1935}, {1777,3072}, {1837,3339}, {1839,2257}, {1885,1892}, {2003,5706}, {2476,4652}, {2478,5437}, {2654,4306}, {3091,3911}, {3220,4185}, {3306,5046}, {3338,3583}, {3434,6762}, {3452,6904}, {3475,4314}, {3476,4301}, {3485,4297}, {3486,3671}, {3487,3529}, {3522,5226}, {3576,4299}, {3577,5553}, {3612,4316}, {3616,5556}, {3624,3824}, {3627,5722}, {3663,5716}, {3679,3927}, {3729,7270}, {3817,7288}, {3830,5708}, {3832,5435}, {3839,5704}, {3916,5705}, {3925,5234}, {3928,6734}, {3947,5218}, {4311,5603}, {4313,5059}, {4331,7273}, {4858,5342}, {4901,5300}, {5119,5270}, {5177,5745}, {5249,5436}, {5252,5762}, {5433,7988}, {5555,7091}, {5709,6923}, {5812,6282}, {6173,8544}, {6692,6919}, {6705,6844}
X(9579) = reflection of X(i) in X(j) for these (i,j): (40,6850), (1697,388), (3340,4295), (3486,3671), (7330,6917)
X(9579) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,4292,57), (7,3146,950), (10,3474,5128), (20,226,3601), (40,1478,9578), (46,3585,5587), (65,5691,5727), (382,942,3586), (908,4190,5438), (946,4293,1420), (1478,1770,40), (1836,7354,1), (3474,5229,10), (3487,3529,4304), (4312,5691,65), (5249,6872,5436), (6917,7330,5587)
X(9580) lies on these lines: {1,30}, {3,7743}, {4,1697}, {9,3434}, {10,5225}, {11,165}, {20,1420}, {35,6911}, {40,1479}, {43,9554}, {46,4857}, {55,1538}, {57,497}, {63,149}, {200,528}, {226,390}, {329,5853}, {354,4312}, {515,7962}, {517,1864}, {908,3158}, {946,3601}, {950,962}, {1040,1421}, {1058,4292}, {1210,5128}, {1695,9555}, {1706,2478}, {1770,3333}, {1788,5493}, {1837,7991}, {1898,5904}, {2078,7580}, {2093,5722}, {2136,3436}, {2481,7243}, {2550,7308}, {2886,4512}, {3057,5691}, {3243,5905}, {3303,5290}, {3476,4342}, {3485,4314}, {3486,4301}, {3529,4311}, {3534,5126}, {3576,4302}, {3583,5119}, {3612,4330}, {3624,5217}, {3679,3715}, {3681,7673}, {3729,4514}, {3749,3944}, {3817,5218}, {3838,4428}, {3870,5057}, {3886,4388}, {3895,5080}, {3911,5274}, {3914,7290}, {3929,4847}, {4304,5603}, {4308,5059}, {4326,8255}, {4333,5563}, {4421,5087}, {4640,5231}, {4666,6173}, {4678,7319}, {4679,8580}, {4863,5223}, {4883,4888}, {4901,5014}, {5175,5837}, {5284,7676}, {5432,7988}, {5697,5881}
X(9580) = reflection of X(i) in X(j) for these (i,j): (40,6827), (57,497), (2093,5722), (3476,4342), (5727,3586)
X(9580) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1836,4654), (4,1697,9578), (55,1699,5219), (946,4294,3601), (950,962,3340), (1836,3058,1), (3583,5119,5587), (4847,5698,3929)
Let
(I) = incircle;
T = a point on (I);
t = line tangent to (I) at T;
T1 and T2 = points of intersection of t and the nine-point circle;
t1 = line through T1 tangent to (I);
t2 = line through T2 tangent to (I).
The locus of t1∩t2 as T varies on (I) is a conic, Γ, with center X(50443).
The center of the reciprocal polar of Γ wrt (I) is X(9581).
The center of the reciprocal polar of (I) wrt Γ is X(50444).
If you have Geogebra, you can view X(9581)
(Angel Montesdeoca, June 8, 2022)
X(9581) lies on these lines: {1,5}, {2,950}, {3,3586}, {4,57}, {7,3832}, {8,3452}, {9,2478}, {10,497}, {20,3911}, {33,5142}, {35,6883}, {36,6985}, {40,1479}, {43,9555}, {46,3583}, {55,1698}, {56,5691}, {63,5046}, {65,1699}, {78,4193}, {142,5177}, {145,5748}, {165,6284}, {191,7082}, {200,1329}, {226,938}, {318,4858}, {354,5290}, {377,5437}, {381,942}, {386,2654}, {390,6666}, {405,5705}, {452,5745}, {498,6887}, {499,3576}, {515,1420}, {516,1788}, {553,3839}, {631,4304}, {908,5187}, {920,5535}, {936,3419}, {944,6969}, {946,3340}, {958,5231}, {962,4848}, {997,3825}, {1058,5818}, {1125,3486}, {1394,1877}, {1445,6895}, {1449,5747}, {1453,5292}, {1467,1750}, {1478,3333}, {1482,7743}, {1490,1532}, {1519,7971}, {1657,5122}, {1695,9554}, {1706,3434}, {1708,6840}, {1714,7070}, {1724,1936}, {1728,5709}, {1738,4907}, {1771,3073}, {1826,2257}, {1834,2999}, {1836,3339}, {1838,1857}, {1858,5902}, {1895,7541}, {2098,3632}, {2136,6735}, {2364,3615}, {2475,3306}, {2550,8582}, {2551,4847}, {2635,4306}, {2646,3624}, {2899,3717}, {3057,3679}, {3075,7524}, {3090,3488}, {3146,5435}, {3158,5552}, {3189,6745}, {3220,4186}, {3338,3585}, {3361,7354}, {3436,6762}, {3475,3947}, {3485,3817}, {3487,3545}, {3621,4345}, {3626,4342}, {3633,5048}, {3634,4314}, {3701,4901}, {3811,3814}, {3813,4853}, {3816,5794}, {3826,4326}, {3843,5708}, {3850,6147}, {3855,5714}, {3871,7705}, {3913,5123}, {4197,7675}, {4294,6684}, {4297,7288}, {4298,5229}, {4308,7319}, {4312,5221}, {4355,4860}, {4857,5119}, {4863,4882}, {5056,5703}, {5068,5226}, {5230,7290}, {5249,6871}, {5433,7987}, {5665,6870}, {5729,5735}, {5768,6260}, {5853,7080}, {6282,6922}, {6692,6904}, {6702,8715}, {6907,8726}, {6929,7330}
X(9581) = (1,5,5219), (1,80,5881), (1,1837,5727), (1,5587,9578), (1,7741,8227), (1,7989,12), (2,950,3601), (4,57,9579), (4,1210,57), (5,5722,1), (8,6919,3452), (10,497,1697), (10,5084,7308), (11,1837,1), (20,5704,3911), (40,1479,9580), (80,5533,6264), (355,496,1), (938,3091,226), (1479,1737,40), (2478,6734,9), (3419,4187,936), (3634,4314,5218), (3816,5794,8583), (3817,6738,3485), (3947,6744,3475), (5804,6844,946)
X(9582) lies on these lines: {1,6200}, {3,1702}, {10,9541}, {40,1151}, {43,9556}, {165,371}, {516,9540}, {517,6449}, {1385,6455}, {1695,9557}, {1698,6561}, {1699,5418}, {3576,6409}, {3579,6221}, {6361,8983}, {6411,7968}, {6424,9574}, {6459,6684}
X(9582) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (165,371,1703)
X(9583) lies on these lines: {1,371}, {3,1703}, {4,8983}, {6,3576}, {10,9540}, {40,1151}, {43,9557}, {165,6200}, {355,8981}, {372,7987}, {485,5691}, {486,3624}, {515,3068}, {516,9541}, {517,6221}, {590,5587}, {946,6459}, {1124,1420}, {1125,1588}, {1335,3601}, {1377,5438}, {1385,3311}, {1587,4297}, {1695,9556}, {1698,5418}, {1699,6561}, {3071,8227}, {3301,3612}, {3579,6449}, {3592,7968}, {5731,7585}, {6424,9575}, {6425,7982}, {6453,7991}, {6565,7988}, {8185,8276}
X(9583) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,371,1702), (40,1151,9582)
X(9584) lies on these lines: {1,1151}, {10,9543}, {40,6407}, {43,9558}, {165,6221}, {516,9542}, {1695,9559}
X(9585) lies on these lines: {1,6221}, {10,9542}, {40,6407}, {43,9559}, {165,1151}, {371,7987}, {516,9543}, {1482,6472}, {1695,9558}, {6459,7988}, {7989,9540}, {7991,9583}
X(9586) lies on these lines: {1,1147}, {10,9545}, {40,49}, {43,9562}, {54,1698}, {57,215}, {110,1699}, {165,184}, {516,9544}, {578,7989}, {1092,7987}, {1695,20230}, {1697,2477}, {1768,3045}, {2948,3043}, {7988,9306}
X(9587) lies on these lines: {1,184}, {10,9544}, {40,49}, {54,1699}, {57,2477}, {110,1698}, {156,5587}, {165,1147}, {215,1697}, {516,9545}, {569,7988}, {1282,3046}, {1437,5010}, {1614,5691}, {1695,9562}, {2187,2964}, {2948,3047}, {3045,5541}, {3097,3202}, {3624,5012}
X(9587) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40,49,9586)
X(9588) lies on these lines: {1,631}, {2,4301}, {3,3679}, {5,40}, {8,7987}, {10,20}, {12,4312}, {43,9568}, {46,5290}, {140,3654}, {191,3359}, {200,224}, {355,548}, {382,3579}, {405,5537}, {484,4338}, {495,4355}, {498,2093}, {515,3528}, {516,3832}, {517,3526}, {519,3523}, {632,3656}, {936,6962}, {944,4668}, {946,5067}, {962,3634}, {971,3983}, {991,3214}, {1125,5734}, {1155,9578}, {1385,3633}, {1571,7765}, {1695,9569}, {1737,4309}, {1907,7713}, {3085,3339}, {3091,3828}, {3340,5432}, {3524,4677}, {3530,3576}, {3582,6967}, {3584,6889}, {3586,4330}, {3617,4297}, {3626,5731}, {3855,6361}, {4197,5735}, {4292,5726}, {4325,6955}, {4857,6947}, {5070,8227}, {5119,5445}, {5217,5727}, {5223,7080}, {5270,6897}, {5281,6738}, {5433,7962}, {5705,6943}, {5744,6736}, {5775,6743}, {6282,6892}, {6986,8715}
X(9588) = {X(10),X(20)}-harmonic conjugate of X(37714)
X(9588) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,165,5691), (12,5128,4312), (40,1698,1699), (140,3654,7982), (962,3634,7988), (3576,5690,3632), (3828,5493,3091), (4848,5218,1), (5657,6684,1)
X(9589) lies on these lines: {1,7}, {2,5493}, {4,3679}, {5,40}, {10,3832}, {11,5128}, {30,7982}, {43,9569}, {65,9580}, {78,5180}, {165,631}, {329,4882}, {354,5586}, {355,3853}, {382,517}, {484,6943}, {497,3339}, {515,3633}, {519,3146}, {529,3680}, {546,3654}, {548,3576}, {550,3656}, {551,3522}, {1158,5536}, {1479,2093}, {1572,7765}, {1695,9568}, {1697,1836}, {1709,6763}, {1750,5758}, {1906,7713}, {3057,9579}, {3062,8001}, {3149,5537}, {3241,5059}, {3303,4654}, {3340,6284}, {3361,3474}, {3526,3579}, {3528,5603}, {3529,5882}, {3530,5886}, {3543,4677}, {3582,6890}, {3746,7580}, {3813,3928}, {3817,7486}, {3828,5068}, {3843,5587}, {3855,5657}, {3861,5690}, {3901,6001}, {4197,5250}, {4848,5225}, {4857,6836}, {5045,5918}, {5067,6684}, {5082,5223}, {5234,5698}, {5259,5584}, {5270,6925}, {5288,8158}, {5541,5812}, {5692,7957}, {5715,6937}, {5762,6766}, {5842,7971}, {5850,6764}, {6048,9535}, {6282,6885}, {7354,7962}
X(9589) = anticomplement of X(5493)
X(9589) = reflection of X(i) in X(j) for these (i,j): (1,962), (20,4301), (3529,5882), (3632,5691), (4677,3543), (5881,382), (6361,946), (7991,4)
X(9589) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4312,4355), (4,7991,3679), (5,40,9588), (5,9588,1698), (20,962,4301), (20,4301,1), (40,1699,1698), (165,946,3624), (382,5881,5691), (390,3671,1), (1697,1836,5290), (1699,9588,5), (3600,4342,1)
X(9590) lies on these lines: {1,24}, {3,1698}, {10,7488}, {22,165}, {23,516}, {25,1699}, {26,40}, {36,1455}, {43,9570}, {55,9572}, {186,515}, {355,1658}, {517,2070}, {910,2919}, {912,2931}, {946,3518}, {952,7575}, {1478,7501}, {1495,2807}, {1695,9571}, {1768,3220}, {1781,5089}, {1993,9586}, {1995,7988}, {2937,3579}, {3576,6644}, {3624,6642}, {5657,7556}, {6684,7512}, {7503,7989}, {7506,8227}
X(9590) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,8185,5691)
X(9591) lies on these lines: {1,22}, {3,1699}, {10,23}, {24,165}, {25,1698}, {26,40}, {35,37}, {36,2920}, {43,9571}, {55,9573}, {191,5285}, {516,7488}, {517,2937}, {946,7512}, {1125,6636}, {1203,5347}, {1386,2916}, {1695,9570}, {1993,9587}, {2070,3579}, {2271,5280}, {3101,4354}, {3220,6763}, {3337,7293}, {3518,6684}, {3616,7492}, {3679,8185}, {4302,7520}, {5587,7517}, {5691,7387}, {5886,7525}, {6361,7556}, {7509,7988}
X(9591) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,40,9590), (8185,8193,3679)
X(9592) lies on these lines: {1,39}, {3,9575}, {6,3576}, {32,7987}, {40,5013}, {115,7988}, {165,574}, {380,5110}, {515,7736}, {517,5024}, {936,1107}, {946,7738}, {988,3061}, {1125,5286}, {1506,7989}, {1571,7991}, {1573,8580}, {1699,2549}, {2548,5691}, {3553,5069}, {3554,4261}, {3612,5299}, {3624,3767}, {3815,5587}, {5254,8227}, {5283,8583}
X(9592) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (574,1572,165)
X(9593) lies on these lines: {1,39}, {3,9574}, {6,40}, {9,986}, {10,5286}, {32,165}, {46,5280}, {115,7989}, {213,2999}, {220,3752}, {515,7738}, {517,9575}, {574,7987}, {610,2273}, {614,1334}, {728,3677}, {936,1575}, {946,7736}, {988,2329}, {1254,2285}, {1385,5024}, {1449,5255}, {1506,7988}, {1570,5184}, {1572,7772}, {1574,8580}, {1698,3767}, {1699,2548}, {1743,3496}, {2082,4642}, {2242,3361}, {2271,9441}, {2277,2324}, {3242,4515}, {3553,8607}, {3679,7739}, {3815,8227}, {5119,5299}, {5254,5587}, {5319,9588}, {6684,7735}
X(9593) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,39,9592), (32,1571,165)
X(9594) lies on these lines: {6,3100}, {33,3815}, {39,8144}, {230,1040}, {1062,5254}, {5013,6198}, {7736,9539}, {7738,9538}
X(9594) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,8144,9595)
X(9595) lies on these lines: {1,7745}, {6,6198}, {33,5254}, {39,8144}, {1062,3815}, {3100,5013}, {7736,9538}, {7738,9539}
X(9595) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,8144,9594)
X(9596) lies on these lines: {1,2548}, {2,172}, {4,2276}, {6,12}, {32,498}, {35,7737}, {37,2478}, {39,1478}, {44,966}, {55,7745}, {56,3815}, {193,4400}, {218,442}, {377,1575}, {388,2275}, {499,1506}, {574,4299}, {1107,3436}, {1212,2551}, {1329,5275}, {1479,1500}, {1571,1770}, {1909,7774}, {1914,3085}, {2241,7753}, {2273,5747}, {2549,3585}, {3053,5432}, {3584,7031}, {3761,7758}, {3767,5280}, {3772,5222}, {4095,4865}, {4302,7747}, {4386,5552}, {4857,9331}, {5013,7354}, {5229,7738}, {5332,8164}, {6645,7777}, {7296,7735}
X(9596) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388,7736,2275), (1500,5475,1479), (1506,2242,499), (5280,7951,3767)
X(9597) lies on these lines: {1,2549}, {4,2275}, {6,7354}, {12,5013}, {20,1914}, {32,4299}, {36,3767}, {39,1478}, {56,5254}, {115,499}, {172,4293}, {230,5204}, {330,6655}, {377,1107}, {388,2276}, {498,574}, {609,4325}, {1015,1479}, {1572,1770}, {1575,3436}, {1909,7791}, {2241,4302}, {2242,4317}, {2345,5484}, {2548,3585}, {3761,7800}, {3785,4400}, {4190,4386}, {4316,7031}, {4396,6392}, {4857,9336}, {5229,7736}, {5299,7737}, {5346,9341}, {6645,7864}
X(9597) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,1478,9596), (388,7738,2276), (1015,7748,1479), (2241,7756,4302), (4293,5286,172)
X(9598) lies on these lines: {1,2549}, {4,2276}, {6,6284}, {11,5013}, {20,172}, {32,4302}, {35,3767}, {37,377}, {39,1479}, {55,5254}, {115,498}, {192,6655}, {230,5217}, {350,7791}, {386,5134}, {497,2275}, {499,574}, {609,4324}, {1107,3434}, {1478,1500}, {1571,1737}, {1575,2478}, {1914,4294}, {2241,4309}, {2242,4299}, {2548,3583}, {3586,9593}, {3760,7800}, {3785,4396}, {4330,5319}, {4366,7864}, {4400,6392}, {4426,6872}, {5225,7736}, {5270,9331}, {5280,7737}, {5299,7739}
X(9598) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2549,9597), (4,2276,9596), (497,7738,2275), (1500,7748,1478), (2242,7756,4299), (4294,5286,1914)
X(9599) lies on these lines: {1,2548}, {2,1914}, {4,2275}, {6,11}, {32,499}, {36,7737}, {39,1479}, {55,3815}, {56,7745}, {172,3086}, {193,4396}, {350,7774}, {497,2276}, {498,1506}, {574,4302}, {609,3582}, {1015,1478}, {1107,2478}, {1572,1737}, {1575,3434}, {2242,7753}, {2549,3583}, {3053,5433}, {3586,9592}, {3760,7758}, {3767,5299}, {3816,5275}, {4299,7747}, {4366,7777}, {5013,6284}, {5222,7261}, {5225,7738}, {5270,9336}
X(9599) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2548,9596), (4,2275,9597), (39,1479,9598), (497,7736,2276), (1015,5475,1478), (1506,2241,498), (5299,7741,3767)
The exsimilcenter of these circles is X(6424)
X(9600) lies on these lines: {3,6}, {115,8253}, {491,8356}, {493,1600}, {590,2549}, {1571,7969}, {3815,6561}, {5254,5418}, {5407,8962}, {7736,9541}, {7738,9540}
X(9600) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6422,6423), (3,8407,1151), (6,6221,8375), (6,6396,8376), (39,1151,6424), (371,5013,6421), (5024,6221,6)
X(9601) lies on these lines: {3,6}, {7736,9543}, {7738,9542}
X(9602) lies on these lines: {3,6}, {7736,9542}, {7738,9543}
X(9603) lies on these lines: {6,1147}, {39,49}, {54,3815}, {110,5254}, {156,2549}, {184,5013}, {215,2275}, {567,1506}, {1092,3053}, {2023,3044}, {2276,2477}, {3094,3202}, {3203,5038}, {7736,9545}, {7738,9544}
X(9604) lies on these lines: {6,25}, {39,49}, {54,5254}, {110,3815}, {115,567}, {156,2548}, {215,2276}, {230,5012}, {1147,5013}, {1614,7745}, {2275,2477}, {2965,3051}, {3796,8553}, {7736,9544}, {7738,9545}
X(9604) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,49,9603)
The insimilcenter of these circles is X(5024).
Let P'1 and U'1 be the circle-(X(3),|OK|)-inverses of PU(1). Then X(9605) = P(1)U'1∩U(1)P'1. (Randy Hutson, March 29, 2020)
X(9605) lies on these lines: {1,728}, {2,3933}, {3,6}, {5,5286}, {25,1180}, {30,7738}, {55,5299}, {56,5280}, {69,8362}, {83,1975}, {99,7878}, {112,3516}, {115,3851}, {140,7735}, {141,7758}, {169,3752}, {183,7760}, {193,7767}, {194,3329}, {218,1193}, {220,995}, {230,3526}, {232,1598}, {325,7803}, {378,3172}, {381,2548}, {382,2549}, {393,1595}, {474,5276}, {517,9575}, {524,7800}, {538,7808}, {597,7789}, {599,6292}, {609,5204}, {625,7902}, {631,5304}, {980,4383}, {988,1743}, {999,2275}, {1003,7783}, {1007,8361}, {1015,7373}, {1078,7894}, {1191,3730}, {1194,5020}, {1249,3088}, {1385,9592}, {1449,5266}, {1500,6767}, {1506,5055}, {1593,8743}, {1597,2207}, {1656,3767}, {1657,7737}, {1992,3785}, {2276,3295}, {2896,7837}, {3087,6756}, {3096,7788}, {3445,9327}, {3520,8778}, {3579,9574}, {3589,7795}, {3618,3926}, {3763,7794}, {3788,7829}, {3830,7748}, {3843,5475}, {3934,7798}, {4045,7759}, {5054,5306}, {5070,5355}, {5073,7747}, {5077,7812}, {5217,7031}, {5346,7749}, {5359,7484}, {5480,8721}, {5523,7507}, {5938,9407}, {6144,7826}, {6337,8369}, {6655,7921}, {6656,7774}, {6683,7751}, {7592,9475}, {7761,7838}, {7762,7791}, {7763,7792}, {7764,7778}, {7766,7824}, {7769,7856}, {7773,7790}, {7775,7861}, {7777,7797}, {7779,7876}, {7781,7804}, {7785,7841}, {7796,7859}, {7799,7846}, {7801,7889}, {7805,7815}, {7809,7918}, {7813,7822}, {7814,7919}, {7816,8716}, {7817,7862}, {7821,7913}, {7831,7877}, {7836,7875}, {7840,7938}, {7843,7872}, {7845,7935}, {7849,7916}, {7852,7888}, {7853,7903}, {7854,7890}, {7865,7882}, {7871,7944}, {7883,7949}, {7884,7899}, {7895,7914}, {7897,7948}, {7900,7924}, {7908,7915}, {7909,7943}, {7911,7926}, {7912,7923}, {7917,7937}, {7925,7932}, {7928,7946}, {7933,7941}
X(9605) = isogonal conjugate of X(18841)
X(9605) = crossdifference of every pair of points on line X(523)X(3804)
X(9605) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(5085)
X(9605) = radical center of Lucas(4 cot ω) circles
X(9605) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7839,7754), (2,7906,7881), (3,39,5024), (6,39,3), (6,570,8573), (6,1350,5039), (6,1504,6417), (6,1505,6418), (6,3053,5007), (6,4255,4251), (6,4261,4254), (6,5013,32), (6,5022,58), (6,5024,1384), (6,5028,5093), (6,5069,5120), (6,6421,3312), (6,6422,3311), (32,39,5013), (32,5013,3), (32,5041,6), (32,7772,5041), (39,5007,574), (39,5041,32), (39,7772,6), (83,7757,1975), (194,3329,7770), (325,7803,7866), (371,372,5085), (574,3053,3), (574,5007,3053), (1689,1690,5188), (2021,5111,9301), (2548,5254,381), (2548,7739,5254), (2549,7745,382), (3096,7905,7788), (3311,3312,5050), (3618,3926,7819), (3767,3815,1656), (4045,7759,7784), (4284,5105,6), (5254,9300,2548), (5286,7736,5), (6292,7855,599), (6656,7774,7776), (7739,9300,381), (7760,7786,183), (7764,7834,7778), (7777,7797,7887), (7779,7876,7879), (7783,7787,1003), (7785,7864,7841), (7790,7858,7773), (7796,7859,7868), (7805,7815,8667)
X(9606) lies on these lines: {3,9300}, {5,39}, {6,631}, {20,5013}, {24,6749}, {32,3530}, {140,5306}, {141,7786}, {230,3526}, {232,1907}, {325,7876}, {382,2548}, {548,574}, {549,5007}, {550,7753}, {597,7807}, {632,7755}, {1078,3629}, {1656,7739}, {1990,3541}, {2549,3843}, {3054,5305}, {3055,3767}, {3589,7763}, {3628,5309}, {3630,7905}, {3832,7738}, {3853,5475}, {3861,7748}, {3933,6683}, {5067,5286}, {5881,9592}, {6179,8584}, {6390,7808}, {6656,7814}, {7759,8359}, {7764,7849}, {7775,8357}, {7777,7933}, {7858,8356}, {7888,8364}
X(9606) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,3815,5254), (140,7772,5306), (3526,5319,230), (3526,9605,5319), (5013,7736,7745), (8259,8260,597)
X(9607) lies on these lines: {3,5306}, {4,9300}, {5,39}, {6,20}, {30,7772}, {32,548}, {53,1907}, {61,5474}, {62,5473}, {140,5309}, {141,194}, {230,631}, {232,1906}, {325,7864}, {382,2549}, {384,597}, {397,6770}, {398,6773}, {524,7791}, {538,8362}, {549,7755}, {550,5007}, {574,3530}, {858,1180}, {1593,1990}, {1975,3589}, {2548,3843}, {2896,3630}, {3053,3528}, {3054,3526}, {3055,5067}, {3575,6749}, {3627,7753}, {3629,7750}, {3832,7736}, {3853,7748}, {3861,5475}, {3933,4045}, {4325,5280}, {4330,5299}, {5023,5304}, {5041,7756}, {5881,9593}, {6390,7834}, {6656,7757}, {7751,8359}, {7759,8357}, {7760,8356}, {7762,7847}, {7767,7798}, {7781,7819}, {7783,7792}, {7789,7803}, {7790,7814}, {7799,8363}, {7801,8364}, {7807,7827}, {7829,8369}, {7833,8584}, {7888,8360}, {7902,8361}
X(9607) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,39,9606), (5,7765,5254), (5,9606,3815), (39,5254,3815), (39,7765,5), (2549,9605,7745), (5013,5286,230), (5254,9606,5), (7750,7839,3629)
X(9608) lies on these lines: {3,2548}, {6,24}, {22,5013}, {23,7738}, {25,5254}, {26,39}, {32,6644}, {55,9595}, {230,6642}, {631,8553}, {1506,7514}, {1993,9603}, {2070,9605}, {2549,7517}, {2937,5024}, {3055,7393}, {3518,5286}, {3767,7506}, {5475,7526}, {7488,7736}, {7530,7748}
X(9609) lies on these lines: {3,230}, {6,22}, {23,7736}, {24,5013}, {25,160}, {26,39}, {55,9594}, {115,7514}, {232,5063}, {571,1194}, {574,6644}, {1609,5306}, {1971,3094}, {1993,9604}, {2070,5024}, {2548,7517}, {2937,9605}, {2965,5359}, {3054,7484}, {3055,5020}, {3553,5345}, {3554,7298}, {5304,7492}, {5305,7525}, {5475,7530}, {6636,7735}, {7387,7745}, {7488,7738}, {7516,7746}
X(9609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,39,9608), (6636,7735,8553)
X(9610) lies on these lines: {1,30}, {3,9577}, {33,8227}, {40,3100}, {515,9538}, {517,9576}, {946,9539}, {1062,5587}, {1697,4354}, {1753,3576}, {2910,6326}
X(9611) lies on these lines: {1,30}, {3,9576}, {33,5587}, {40,6198}, {515,9539}, {517,9577}, {946,9538}, {1062,8227}, {2002,3100}, {3601,4354}
X(9611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1717,9579), (1,8144,9610)
X(9612) lies on these lines: {1,4}, {2,4292}, {3,5219}, {5,57}, {7,1210}, {8,4054}, {9,46}, {10,329}, {11,3333}, {12,40}, {20,5226}, {30,3601}, {35,7580}, {36,405}, {56,6913}, {63,2476}, {65,5587}, {72,3679}, {78,2475}, {80,5665}, {84,6831}, {90,7363}, {142,5084}, {144,5833}, {158,342}, {165,498}, {171,1777}, {235,1892}, {273,7563}, {355,3340}, {377,908}, {380,1839}, {381,942}, {443,3452}, {452,1125}, {484,4338}, {495,1697}, {499,3361}, {516,3085}, {517,9578}, {519,5175}, {546,5722}, {553,3545}, {610,5747}, {938,3832}, {943,5561}, {954,3746}, {958,3838}, {962,5261}, {1005,5248}, {1006,7280}, {1158,5924}, {1420,5886}, {1445,6991}, {1453,3772}, {1466,6918}, {1467,6893}, {1708,3336}, {1714,1743}, {1717,9576}, {1724,7522}, {1737,3339}, {1754,3074}, {1768,8068}, {1837,3649}, {1876,7507}, {2003,5707}, {2096,6705}, {2099,5881}, {3086,3817}, {3090,3911}, {3146,4304}, {3218,5141}, {3295,9580}, {3305,4197}, {3306,4193}, {3337,6990}, {3338,4355}, {3419,3632}, {3434,6765}, {3474,6684}, {3526,5122}, {3543,4313}, {3544,4031}, {3576,7354}, {3616,4311}, {3627,5719}, {3646,4679}, {3651,4333}, {3670,4862}, {3731,8804}, {3851,5708}, {3855,3982}, {3876,6175}, {3929,5791}, {4187,5437}, {4299,6987}, {4325,6936}, {4848,5818}, {5057,5250}, {5068,5704}, {5119,9589}, {5252,7982}, {5259,7742}, {5436,5443}, {5439,6173}, {5542,5809}, {5709,6842}, {5720,6917}, {5726,5758}, {5732,6836}, {5735,6932}, {5745,6856}, {5748,6700}, {5902,5927}, {5905,6734}, {6282,6850}, {6361,8164}, {6598,7700}, {6827,8726}, {7179,7385}, {7308,8728}, {7373,7743}
X(9612) = midpoint of X(i),X(j) for these (i,j): (3485,5229)
X(9612) = reflection of X(i) in X(j) for these (i,j): (1,3485), (3085,3947)
X(9612) = homothetic center of 2nd extouch triangle and reflection triangle of X(1)
X(9612) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,3586), (1,3585,5691), (4,226,1), (4,1490,5691), (4,3487,950), (4,5714,226), (4,5715,1699), (7,3091,1210), (9,442,1698), (10,4295,2093), (12,1836,40), (46,79,4312), (46,7951,1698), (63,2476,5705), (79,7951,46), (225,5713,1), (226,950,3487), (329,5177,10), (377,908,936), (381,942,9581), (388,946,1), (498,1770,165), (546,6147,5722), (950,3487,1), (1698,4312,46), (1699,5290,1), (2096,6956,6705), (3146,5703,4304), (3339,7989,1737), (3361,7988,499), (3817,4298,3086), (4654,9581,942), (5219,9579,3), (5707,8757,2003), (5748,6904,6700), (5812,6907,40), (5905,6871,6734)
X(9613) lies on these lines: {1,4}, {2,4311}, {3,9578}, {5,1420}, {8,2093}, {10,4293}, {12,3576}, {30,1697}, {36,474}, {40,5252}, {46,529}, {56,5587}, {57,355}, {65,5881}, {79,3633}, {80,3338}, {165,4299}, {382,9580}, {443,5795}, {495,3601}, {498,5726}, {499,6964}, {517,9579}, {518,3632}, {936,3436}, {942,5727}, {952,3340}, {999,9581}, {1125,6919}, {1210,3600}, {1319,8227}, {1385,5219}, {1467,6826}, {1656,5126}, {1737,3361}, {1770,7991}, {1777,5255}, {1836,7982}, {1837,3333}, {2078,3560}, {2975,5705}, {3085,4297}, {3086,4315}, {3091,4308}, {3419,6762}, {3624,4187}, {3843,7743}, {3911,5818}, {4190,6735}, {4325,6955}, {4355,5902}, {4911,9312}, {5128,5690}, {5231,8666}, {5251,7742}, {5261,5731}, {5697,9589}, {6940,7280}
X(9613) = reflection of X(i) in X(j) for these (i,j): (1,388)
X(9613) = complementary conjugate of X(38991)
X(9613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1478,9612), (1,3585,1699), (1,5270,5290), (1,5691,3586), (8,4292,2093), (226,944,1), (946,3476,1), (950,1056,1), (1737,4317,3361), (1837,5434,3333), (3476,5229,946), (3485,5882,1), (5252,7354,40), (5726,7987,498)
The insimilcenter of these circles is X(3586)
X(9614) lies on these lines: {1,4}, {3,7743}, {5,1697}, {10,6919}, {11,40}, {30,1420}, {35,474}, {46,9589}, {55,6918}, {57,496}, {78,149}, {79,4355}, {80,3632}, {90,3254}, {165,499}, {355,7962}, {381,9578}, {498,6964}, {516,3086}, {517,9581}, {551,4305}, {908,6765}, {920,5536}, {936,3434}, {960,3679}, {962,1210}, {999,9579}, {1062,1421}, {1125,4294}, {1467,6851}, {1482,5727}, {1537,7971}, {1657,5126}, {1698,1706}, {1737,7991}, {1768,5533}, {1770,3361}, {1836,3333}, {1837,7982}, {2078,6985}, {2098,5881}, {3057,5587}, {3085,3817}, {3146,4311}, {3295,5219}, {3296,3982}, {3337,4338}, {3338,4312}, {3340,5722}, {3452,5082}, {3543,4308}, {3576,6284}, {3601,5886}, {3616,4304}, {3646,3925}, {3911,6361}, {3913,5087}, {3953,4862}, {4302,7987}, {4330,6955}, {4654,5045}, {5010,6940}, {5187,6735}, {5250,5705}, {6963,9588}
X(9614) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,9613), (1,1479,3586), (1,1699,9612), (1,3583,5691), (226,1058,1), (497,946,1), (950,5603,1), (962,1210,2093), (962,5274,1210), (1706,4187,1698), (5119,7741,1698)
X(9615) lies on these lines: {1,1151}, {3,1703}, {6,7987}, {20,8983}, {40,6200}, {165,6409}, {371,3576}, {515,9540}, {517,6449}, {590,5691}, {946,9541}, {1125,6459}, {1385,1702}, {1420,2066}, {1482,6445}, {2067,3601}, {3068,4297}, {3071,3624}, {3579,6455}, {3622,9543}, {5418,5587}, {6424,9592}, {6425,7968}, {6561,8227}, {7989,8253}
X(9615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1385,6221,1702)
X(9616) lies on these lines: {1,1151}, {3,1702}, {6,165}, {10,6459}, {40,371}, {57,2066}, {145,9543}, {515,9541}, {516,3068}, {517,6221}, {590,1699}, {946,9540}, {962,8983}, {975,6212}, {1385,6449}, {1482,6407}, {1571,5058}, {1588,6684}, {1697,2067}, {1698,3071}, {1703,3311}, {2362,5128}, {3297,3361}, {3576,6200}, {4319,6204}, {5418,8227}, {5587,6561}, {6409,7968}, {6424,9593}, {6425,7969}, {6429,9585}, {6453,7982}, {7988,8253}
X(9616) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1151,9615), (1702,9582,3), (3311,3579,1703)
X(9617) lies on these lines: {1,6407}, {3,9585}, {40,1151}, {165,6445}, {515,9542}, {517,9584}, {946,9543}, {1702,6429}
X(9618) lies on these lines: {1,6407}, {3,9584}, {40,6221}, {515,9543}, {517,9585}, {946,9542}, {1151,1702}, {1703,6437}, {7982,9616}
X(9619) lies on these lines: {1,39}, {3,1572}, {6,1385}, {32,3576}, {40,574}, {115,8227}, {187,7987}, {355,3815}, {515,2548}, {517,1571}, {551,7739}, {936,1573}, {944,7736}, {946,2549}, {988,3735}, {997,1107}, {1125,3767}, {1420,2242}, {1482,5024}, {1506,5587}, {1699,7748}, {1914,3612}, {2092,3554}, {2241,3601}, {3616,5286}, {3624,7746}, {3653,5306}, {3655,9300}, {3751,5034}, {4297,7737}, {5058,9583}, {5114,8557}, {5254,5886}, {5475,5691}, {5603,7738}, {6421,7969}, {6422,7968}, {7603,7989}, {7982,9574}
X(9619) = reflection of X(i) in X(j) for these (i,j): (1571,5013)
X(9619) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9592,39), (3576,9575,32)
X(9620) lies on these lines: {1,39}, {3,1571}, {6,517}, {8,5286}, {9,3735}, {10,3767}, {19,2273}, {32,40}, {38,4390}, {41,4642}, {42,3010}, {46,172}, {57,2242}, {63,5291}, {71,5336}, {115,5587}, {165,187}, {169,3959}, {355,5254}, {484,609}, {515,2549}, {516,7737}, {519,7739}, {574,3576}, {614,3230}, {712,3729}, {760,3751}, {936,1574}, {944,7738}, {946,2548}, {986,2329}, {997,1575}, {1334,3924}, {1385,5013}, {1482,9605}, {1506,8227}, {1697,2241}, {1698,7746}, {1699,5475}, {1702,5058}, {1703,5062}, {1902,2207}, {1914,5119}, {2092,3553}, {2271,4646}, {3053,3579}, {3654,5306}, {3656,9300}, {3679,5309}, {3752,6603}, {3753,5275}, {3815,5886}, {5007,7991}, {5299,5697}, {5305,5690}, {5603,7736}, {5657,7735}, {5691,7748}, {5881,7765}, {6421,7968}, {6422,7969}, {7603,7988}, {7772,7982}
X(9620) = reflection of X(i) in X(j) for these (i,j): (1572,6)
X(9620) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,39,9619), (1,9593,39), (3576,9574,574)
X(9621) lies on these lines: {1,49}, {3,9587}, {40,1147}, {54,8227}, {110,5587}, {156,5691}, {184,3576}, {515,9544}, {517,9586}, {567,7988}, {946,9545}, {3045,6264}
X(9622) lies on these lines: {1,49}, {3,9586}, {40,184}, {54,5587}, {110,8227}, {156,1699}, {515,9545}, {517,9587}, {567,7989}, {946,9544}, {1147,3576}, {3045,6326}
X(9622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,49,9621)
The exsimilcenter of these circles is X(936)
X(9623) lies on these lines: {1,2}, {3,1706}, {4,5795}, {9,374}, {40,958}, {46,5258}, {56,3698}, {57,956}, {63,2093}, {72,3340}, {142,1056}, {165,993}, {210,2099}, {214,7993}, {226,3421}, {269,996}, {355,1490}, {377,9613}, {392,7308}, {405,1697}, {442,9578}, {443,1467}, {474,1420}, {515,2550}, {518,5785}, {529,5880}, {758,5223}, {937,1220}, {942,6762}, {944,8726}, {946,2551}, {950,5082}, {960,7982}, {966,2324}, {998,2297}, {999,5437}, {1000,6666}, {1001,3880}, {1107,9593}, {1319,4413}, {1329,8227}, {1376,3576}, {1385,5438}, {1442,5936}, {1449,5783}, {1453,5710}, {1482,5044}, {1532,2886}, {1573,9620}, {1574,9619}, {1575,9592}, {1621,3895}, {1699,6957}, {1721,4660}, {2136,3295}, {2346,4900}, {2478,9614}, {2975,9352}, {3035,6264}, {3036,6326}, {3303,3893}, {3305,3877}, {3333,3812}, {3338,5288}, {3339,3754}, {3361,3918}, {3419,5727}, {3434,3586}, {3436,9612}, {3452,5603}, {3488,5853}, {3601,5687}, {3678,4866}, {3680,7160}, {3692,3731}, {3697,5730}, {3740,5289}, {3814,7988}, {3820,5886}, {3822,5726}, {3885,5047}, {3897,4855}, {3916,5128}, {3922,5221}, {3925,5252}, {4311,6904}, {4318,5772}, {4423,5919}, {4512,5119}, {5234,6912}, {5250,5260}, {5273,7994}, {5657,5745}, {5690,5791}, {5691,6925}, {5720,5790}, {5777,7971}, {5794,5881}, {5806,8158}, {5818,6969}, {6945,7989}, {6966,9588}
X(9623) = midpoint of X(i),X(j) for these (i,j): (1,4915)
X(9623) = reflection of X(i) in X(j) for these (i,j): (1056,142)
X(9623) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8,6765), (1,10,936), (1,1698,8583), (1,3679,200), (1,4882,3811), (1,8580,997), (10,997,8580), (956,3753,57), (958,5836,40), (997,8580,936), (1319,4731,4413), (2136,5436,3295), (3241,4666,1), (3626,3811,4882), (5119,5251,4512), (7308,7962,392)
The exsimilcenter of these circles is X(5881)
X(9624) lies on these lines: {1,5}, {2,5734}, {3,9589}, {4,551}, {8,7486}, {10,5067}, {20,946}, {36,4338}, {40,631}, {57,6892}, {140,3656}, {165,3530}, {354,5693}, {382,1385}, {498,7962}, {499,3340}, {515,3622}, {516,3528}, {517,3526}, {519,3090}, {546,3655}, {547,4677}, {548,7987}, {550,3653}, {632,3654}, {944,3636}, {1001,5735}, {1319,9612}, {1388,9613}, {1420,4317}, {1482,1698}, {1656,3679}, {1697,6970}, {1776,3333}, {2093,5433}, {2646,9614}, {3091,5882}, {3241,5056}, {3244,5818}, {3303,6918}, {3304,4870}, {3524,5493}, {3560,4654}, {3582,6862}, {3584,6959}, {3601,4309}, {3612,4330}, {3633,5790}, {3680,6983}, {3746,6911}, {3843,5691}, {3894,5694}, {3901,6583}, {4315,5714}, {4325,9579}, {4857,6917}, {5231,5730}, {5250,5535}, {5270,6929}, {5289,5705}, {5315,5707}, {5319,9575}, {5436,6936}, {5550,6684}, {5733,7290}, {5761,7308}, {6173,6906}, {6920,8666}, {6946,8715}, {7765,9619}
X(9624) = midpoint of X(i),X(j) for these (i,j): (1,7989)
X(9624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,5881), (1,5443,5219), (1,5886,8227), (1,7741,5727), (1,7988,355), (1,8227,5587), (5,5881,5587), (140,3656,7991), (631,4301,40), (631,5603,4301), (946,3616,3576), (1125,4301,631), (1125,5603,40), (3624,9588,3526), (3636,3817,944), (5881,8227,5), (5886,5901,1)
X(9625) lies on these lines: {1,26}, {3,1699}, {10,3518}, {22,3576}, {23,515}, {24,40}, {25,5587}, {36,1421}, {55,9611}, {165,6644}, {186,516}, {517,2070}, {946,7488}, {1125,7512}, {1385,2937}, {1698,7506}, {1993,9621}, {3517,8193}, {5603,7556}, {5691,7517}, {5881,8185}, {5886,7502}, {7514,7988}
X(9626) lies on these lines: {1,26}, {3,1698}, {10,7512}, {22,40}, {23,946}, {24,3576}, {25,8227}, {55,9610}, {186,4297}, {355,7502}, {515,7488}, {517,2937}, {944,7556}, {1125,3518}, {1385,2070}, {1699,7517}, {1993,9622}, {3624,7506}, {5690,7555}, {6636,6684}, {6644,7987}, {7514,7989}
X(9626) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,26,9625), (3,8185,5587)
X(9627) lies on these lines: {1,3}, {4,9629}, {12,403}, {30,4354}, {388,9538}, {550,4351}, {774,2361}, {1478,8144}, {1870,6284}, {3085,7505}, {3100,7354}, {4337,7100}, {5229,9539}
X(9627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1062,56), (1478,8144,9628)
X(9628) lies on these lines: {1,382}, {4,9630}, {12,3100}, {33,56}, {35,2937}, {55,7387}, {388,9539}, {495,4354}, {517,1717}, {971,8614}, {990,5221}, {1478,8144}, {3746,8143}, {5229,9538}, {6198,7354}, {6285,6293}
X(9628) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1478,8144,9627)
X(9629) lies on these lines: {1,382}, {4,9627}, {5,4354}, {11,858}, {19,25}, {30,4351}, {390,7519}, {497,7391}, {942,1717}, {990,4860}, {1479,8144}, {2310,2361}, {3326,5497}, {3586,9611}, {4123,4387}, {4162,4775}, {5225,9538}, {6198,6240}, {7070,7082}
X(9629) = homothetic center of X(1)- and X(4)-Ehrmann triangles
X(9629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,5160,3100), (1479,8144,9630)
X(9630) lies on these lines: {1,3}, {4,9628}, {11,1594}, {33,7507}, {34,7221}, {497,9538}, {614,5094}, {858,7191}, {1399,7004}, {1479,8144}, {1870,6240}, {3085,7558}, {3100,6284}, {3586,9610}, {3920,7495}, {4857,7574}
X(9630) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1062,55), (1479,8144,9629)
X(9631) lies on these lines: {33,5418}, {371,3100}, {486,1040}, {1062,6561}, {1151,8144}, {2067,4354}, {6198,6200}, {6424,9594}
X(9631) = {X(1151),X(8144)}-harmonic conjugate of X(9632)
X(9632) lies on these lines: {1,485}, {33,6561}, {371,6198}, {1060,6560}, {1062,5418}, {1151,8144}, {1870,6564}, {3100,6200}, {3295,8276}, {3920,8854}, {6424,9595}, {7191,8280}
X(9632) = {X(1151),X(8144)}-harmonic conjugate of X(9631)
X(9633) lies on these lines: {1151,6198}, {1870,9541}, {3100,6221}, {6407,8144}, {6453,9631}
X(9633) = {X(6407),X(8144)}-harmonic conjugate of X(9634)
X(9634) lies on these lines: {34,8972}, {1038,7585}, {1151,3100}, {1870,8981}, {2067,3920}, {3068,4296}, {4318,8983}, {6198,6221}, {6407,8144}, {6453,9632}, {6480,9631}
X(9634) = {X(6407),X(8144)}-harmonic conjugate of X(9633)
X(9635) lies on these lines: {1,7756}, {32,3100}, {33,1506}, {39,8144}, {115,1062}, {574,6198}, {1040,7749}, {1571,9577}, {1572,9576}, {2241,4354}, {2548,9539}, {2549,9538}
X(9635) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,8144,9636), (8144,9594,39)
X(9636) lies on these lines: {1,7747}, {32,6198}, {33,115}, {39,8144}, {574,3100}, {1060,6781}, {1062,1506}, {1571,9576}, {1572,9577}, {2548,9538}, {2549,9539}
X(9636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,8144,9635), (8144,9595,39)
X(9637) lies on these lines: {33,110}, {35,5889}, {49,8144}, {54,1062}, {55,1993}, {184,3100}, {193,2330}, {212,4184}, {378,3157}, {651,4219}, {1040,5012}, {1147,6198}, {3167,7071}, {3219,6056}, {3520,7352}, {3580,5432}, {5218,6515}
X(9637) = {X(49),X(8144)}-harmonic conjugate of X(9638)
X(9638) lies on these lines: {1,1614}, {22,1069}, {33,54}, {36,6241}, {49,8144}, {74,7280}, {110,1062}, {184,6198}, {1147,3100}, {1870,6759}, {3520,6285}, {6238,7488}
X(9638) = {X(49),X(8144)}-harmonic conjugate of X(9637)
X(9639) lies on these lines: {1,529}, {9,9577}, {10,8144}, {33,2886}, {936,9610}, {958,6198}, {1040,3035}, {1062,1329}, {1107,9595}, {1376,3100}, {1573,9636}, {1574,9635}, {1575,9594}, {1706,9576}, {2550,9539}, {2551,9538}, {3434,9629}, {3436,9627}, {3920,4428}, {3932,4123}, {4354,5687}
X(9639) = {X(10),X(8144)}-harmonic conjugate of X(9640)
X(9640) lies on these lines: {1,528}, {9,9576}, {10,8144}, {33,1329}, {377,9627}, {405,4354}, {936,9611}, {958,3100}, {1040,4999}, {1062,2886}, {1107,9594}, {1376,6198}, {1573,9635}, {1574,9636}, {1575,9595}, {1706,9577}, {2478,9629}, {2550,9538}, {2551,9539}, {3434,9630}, {3436,9628}, {3704,4123}
X(9640) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,8144,9639)
X(9641) lies on these lines: {1,1657}, {3,3100}, {5,9539}, {6,9635}, {30,9538}, {33,1656}, {34,382}, {55,2937}, {381,1062}, {517,9576}, {1040,3526}, {1385,9611}, {1479,5160}, {1870,5073}, {3270,6243}, {3295,4354}, {3579,9577}, {5013,9636}, {5024,9595}, {6221,9631}, {6449,9632}
X(9641) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,8144,9642), (3100,8144,3)
X(9642) lies on these lines: {1,382}, {3,3100}, {5,9538}, {6,9636}, {30,9539}, {33,381}, {34,5076}, {55,2070}, {399,3157}, {517,9577}, {568,3270}, {1040,5054}, {1060,3534}, {1062,1656}, {1385,9610}, {1482,2817}, {1870,3830}, {3295,7517}, {3579,9576}, {4302,5160}, {5013,9635}, {5024,9594}, {6221,9632}, {6449,9631}
X(9642) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,8144,9641)
X(9643) lies on these lines: {1,7}, {5,33}, {22,3746}, {34,382}, {78,3469}, {548,1060}, {612,7493}, {614,858}, {631,1040}, {1038,3528}, {3526,9642}, {3832,9539}, {5881,9610}, {7765,9635}
X(9643) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4302,4348), (1,4354,4319), (5,8144,9644), (1062,8144,33), (3100,9538,1)
X(9644) lies on these lines: {1,382}, {5,33}, {20,1060}, {34,3853}, {496,7221}, {631,3100}, {1040,3526}, {1717,4338}, {3746,7517}, {3832,9538}, {5881,9611}, {7765,9636}
X(9644) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,8144,9643), (5,9643,1062), (33,8144,1062), (33,9643,5)
The insimilcenter of these circles is X(55)
X(9645) lies on these lines: {1,7387}, {3,33}, {4,9660}, {22,6198}, {23,9538}, {24,3100}, {25,1062}, {26,55}, {30,56}, {108,6851}, {511,1069}, {999,4320}, {1040,6642}, {1498,7352}, {1658,5217}, {1993,9638}, {2070,9641}, {2192,6238}, {2937,9642}, {3157,6759}, {3295,5310}, {7071,9644}, {7488,9539}
X(9645) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,8144,55)
X(9646) lies on these lines: {1,590}, {2,1124}, {4,9660}, {5,2066}, {6,498}, {12,371}, {35,3070}, {55,485}, {56,5418}, {140,6502}, {372,5432}, {388,9540}, {495,2067}, {499,3297}, {615,3299}, {750,3076}, {1058,3316}, {1151,1478}, {1255,1336}, {1335,3068}, {1377,5552}, {1587,5218}, {1702,5219}, {3071,7951}, {3295,8976}, {3301,3584}, {3614,6565}, {4299,6409}, {5217,6560}, {5229,9541}, {5414,7583}, {6200,7354}, {6284,6564}, {6424,9596}, {8965,8969}
X(9646) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1151,1478,9647)
X(9647) lies on these lines: {1,9660}, {4,9661}, {6,4299}, {12,6200}, {20,1335}, {30,2067}, {36,3071}, {56,6561}, {371,7354}, {388,9541}, {486,5204}, {498,6409}, {550,5414}, {590,3585}, {615,7280}, {1124,4293}, {1151,1478}, {1377,4190}, {1770,7969}, {3297,4317}, {3298,4302}, {3299,4325}, {3301,4316}, {5229,9540}, {5433,6565}, {6424,9597}
X(9647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1151,1478,9646)
X(9648) lies on these lines: {11,9540}, {12,6221}, {371,5432}, {388,9542}, {1151,7354}, {1478,6407}, {1588,5326}, {3614,6459}, {4299,6445}, {5229,9543}, {6284,8981}, {6453,9646}, {6480,9647}
X(9648) = {X(1478),X(6407)}-harmonic conjugate of X(9649)
X(9649) lies on these lines: {12,1151}, {388,9543}, {498,6445}, {1478,6407}, {3071,7294}, {5229,9542}, {5432,6449}, {5433,6459}, {6221,7354}, {6284,9541}, {6453,9647}, {6480,9646}, {6561,7173}
X(9649) = {X(1478),X(6407)}-harmonic conjugate of X(9648
X(9650) lies on these lines: {1,5475}, {4,1500}, {5,2242}, {12,32}, {39,1478}, {55,7747}, {56,1506}, {172,7746}, {187,498}, {330,7858}, {377,1574}, {388,1015}, {495,2241}, {499,7603}, {574,7354}, {1571,9579}, {1572,9578}, {1573,3436}, {1909,7759}, {2275,5270}, {2276,3585}, {2476,5291}, {2549,5229}, {3085,7737}, {3761,7855}, {3822,4426}, {5080,5283}, {5206,5432}, {5217,6781}, {5280,5309}, {5346,7296}, {6645,7752}
X(9650) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,1478,9651), (172,7951,7746), (388,2548,1015), (495,7745,2241), (1478,9596,39), (2276,3585,7748)
X(9651) lies on these lines: {1,7748}, {4,1015}, {12,574}, {30,2241}, {32,7354}, {36,7746}, {39,1478}, {55,7756}, {56,115}, {172,5309}, {187,4299}, {316,330}, {377,1573}, {388,1500}, {535,4426}, {609,5346}, {1571,9578}, {1572,9579}, {1574,3436}, {1834,9346}, {1909,7761}, {2242,5254}, {2275,3585}, {2276,5270}, {2548,5229}, {3761,7854}, {3767,4293}, {4056,7200}, {5204,7749}, {6645,7790}, {7735,9341}
X(9651) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,1478,9650), (388,2549,1500), (1478,9597,39), (2275,3585,5475)
X(9652) lies on these lines: {1,156}, {4,215}, {12,184}, {49,1478}, {55,1614}, {56,110}, {388,2477}, {569,3614}, {1147,7354}, {5229,9545}, {5433,9306}, {5651,7294}, {6284,6759}, {7085,7144}
X(9652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,1478,9653), (388,9544,2477)
X(9653) lies on these lines: {4,2477}, {11,578}, {12,1147}, {49,1478}, {54,56}, {156,3585}, {184,7354}, {215,388}, {499,567}, {569,5433}, {1092,5432}, {3614,9306}, {5012,5204}, {5229,9544}
X(9653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,1478,9652), (388,9545,215)
X(9654) lies on these lines: {1,381}, {3,12}, {4,390}, {5,388}, {6,9650}, {7,5818}, {8,5714}, {10,527}, {11,3851}, {20,8164}, {30,3085}, {35,1657}, {36,3526}, {40,5726}, {55,382}, {56,1656}, {65,5790}, {75,5827}, {119,6918}, {140,4293}, {153,6828}, {226,355}, {405,5080}, {442,3436}, {443,3820}, {496,1056}, {497,546}, {499,3614}, {515,3947}, {517,9578}, {550,5218}, {942,5290}, {944,5226}, {952,3485}, {956,2476}, {958,3822}, {1058,3832}, {1159,3649}, {1330,5774}, {1385,5219}, {1388,5443}, {1398,1594}, {1479,3843}, {1482,5252}, {1737,5708}, {1870,7507}, {1909,7776}, {2067,8976}, {2475,5687}, {2551,8728}, {3090,3600}, {3297,6565}, {3298,6564}, {3303,3583}, {3304,5072}, {3333,7989}, {3421,5177}, {3454,5793}, {3476,5901}, {3486,5719}, {3529,5281}, {3534,3584}, {3579,9579}, {3617,6175}, {3624,5126}, {3627,4294}, {3628,7288}, {3679,3962}, {3746,5076}, {3830,6284}, {3845,5225}, {3855,5274}, {3920,5064}, {3940,5794}, {4295,5690}, {4302,5073}, {4317,5070}, {4338,5183}, {5013,9651}, {5024,9597}, {5045,9581}, {5054,5204}, {5067,5265}, {5079,5563}, {5339,7005}, {5340,7006}, {5707,9370}, {5711,8757}, {5722,6744}, {6221,9646}, {6244,6850}, {6256,7680}, {6449,9647}, {6645,7887}
X(9654) = midpoint of X(i),X(j) for these (i,j): (3085,5229)
X(9654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,9669), (4,495,3295), (4,5261,495), (5,388,999), (12,1478,3), (12,7354,498), (55,3585,382), (56,7951,1656), (498,1478,7354), (498,7354,3), (499,3614,5055), (1056,3091,496), (3614,5434,499), (3843,6767,1479), (3851,7373,11), (4299,5432,3), (5219,9613,1385), (5270,7951,56), (5290,5587,942)
X(9655) lies on these lines: {1,382}, {3,12}, {4,496}, {5,4293}, {6,9651}, {11,3843}, {20,495}, {30,388}, {35,3534}, {36,1656}, {46,5790}, {55,1657}, {56,381}, {79,2099}, {80,5221}, {202,5339}, {203,5340}, {355,4292}, {376,5261}, {474,5080}, {497,3627}, {499,3851}, {515,3671}, {517,9579}, {535,958}, {546,3086}, {548,5218}, {550,3085}, {942,4355}, {944,4323}, {952,4295}, {956,2475}, {1056,3146}, {1058,3543}, {1385,9612}, {1387,4308}, {1479,3830}, {1482,1836}, {1698,5122}, {1770,5252}, {1837,5708}, {3304,3583}, {3474,5690}, {3486,6147}, {3522,8164}, {3526,4325}, {3545,5265}, {3579,9578}, {3586,5045}, {3614,5070}, {3820,6904}, {3853,5225}, {3927,5794}, {4056,7223}, {4298,5722}, {4305,5719}, {4311,5886}, {4312,5881}, {4316,5217}, {4911,7185}, {5013,9650}, {5024,9596}, {5054,7280}, {5055,5433}, {5073,6284}, {5126,8227}, {5556,5734}, {5587,5789}, {5603,6049}, {5714,5731}, {6449,9646}, {6645,7841}, {7489,7742}, {7965,8171}
X(9655) = reflection of X(i) in X(j) for these (i,j): (3295,388), (3486,6147), (3927,5794)
X(9655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9628,9642), (3,1478,9654), (4,3600,496), (12,4299,3), (12,7354,4299), (56,3585,381), (496,3600,999), (1478,4299,12), (1478,7354,3), (1479,5434,7373), (3830,7373,1479), (4293,5229,5), (4325,7951,5204), (5073,6767,6284), (5204,7951,3526)
X(9656) lies on these lines: {1,3843}, {4,3058}, {5,56}, {11,3855}, {12,20}, {36,5070}, {55,382}, {79,5790}, {381,3304}, {388,3832}, {495,3853}, {496,3856}, {498,548}, {529,6871}, {631,5326}, {1388,9613}, {1479,3861}, {1657,3584}, {2099,5881}, {3091,5434}, {3526,4325}, {3528,5432}, {3529,4995}, {3530,4299}, {3582,5072}, {3600,7173}, {3614,4293}, {3746,3830}, {3851,5563}, {4197,5080}, {4301,5252}, {5056,5298}, {5221,5587}, {5433,7486}, {7765,9650}
X(9656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (381,5270,3304), (495,3853,4309), (3526,4325,5204), (3526,9655,4325), (3585,9654,55), (4325,7951,3526), (7951,9655,5204)
X(9657) lies on these lines: {1,382}, {3,3584}, {4,3304}, {5,56}, {11,3600}, {12,631}, {20,55}, {30,3303}, {36,3526}, {65,5881}, {79,1482}, {80,5708}, {354,5691}, {355,5221}, {377,529}, {381,5563}, {405,535}, {495,548}, {496,3861}, {498,3530}, {517,4338}, {944,3649}, {958,4197}, {999,3585}, {1056,6284}, {1155,9578}, {1319,9612}, {1466,6885}, {1479,3853}, {1657,3746}, {1836,2098}, {1837,4298}, {2646,5290}, {3057,9579}, {3058,3146}, {3085,3528}, {3086,3855}, {3090,5298}, {3295,4330}, {3336,5790}, {3436,4413}, {3476,5734}, {3522,4995}, {3582,3851}, {3583,7373}, {3614,7288}, {3830,4857}, {4345,5556}, {4355,5727}, {4911,7223}, {5067,5433}, {5070,7951}, {5261,5432}, {5735,8581}, {6580,7706}, {6645,7933}, {7765,9651}
X(9657) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5434,3304), (5,1478,9656), (5,4317,56), (12,4293,5204), (56,9656,5), (388,7354,55), (495,4299,5217), (1478,4317,5), (1837,4298,4860), (3600,5229,11)
X(9658) lies on these lines: {1,7517}, {3,3585}, {12,22}, {23,388}, {24,7354}, {25,34}, {26,1478}, {36,7506}, {55,7387}, {65,8185}, {154,2477}, {1454,3220}, {1479,7530}, {1993,9652}, {1995,5433}, {2070,9655}, {2937,9654}, {3295,5899}, {3518,4293}, {3614,7509}, {4299,6644}, {5204,6642}, {5229,7488}, {5329,7299}
X(9659) lies on these lines: {1,3}, {11,7503}, {12,24}, {22,7354}, {23,5229}, {26,1478}, {155,2477}, {186,3085}, {378,6284}, {388,7488}, {495,1658}, {498,6644}, {499,7514}, {1479,7526}, {1993,9653}, {1995,3614}, {2070,9654}, {2937,9655}, {3520,4294}, {3585,7517}, {4293,7512}, {5225,7527}, {5433,7509}, {7506,7951}
X(9659) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,1478,9658)
X(9660) lies on these lines: {1,9647}, {4,9646}, {6,4302}, {11,6200}, {20,1124}, {30,2066}, {35,3071}, {55,6561}, {371,6284}, {486,5217}, {497,9541}, {499,6409}, {550,6502}, {590,3583}, {615,5010}, {1151,1479}, {1335,4294}, {1378,6872}, {3297,4299}, {3298,4309}, {3299,4324}, {3301,4330}, {3586,9616}, {5225,9540}, {5432,6565}, {6424,9598}
X(9660) = {X(1151),X(1479)}-harmonic conjugate of X(9661)X(9661) lies on these lines: {1,590}, {2,1335}, {4,9647}, {5,2067}, {6,499}, {11,371}, {36,3070}, {55,5418}, {56,485}, {88,1123}, {140,5414}, {372,5433}, {496,2066}, {497,9540}, {498,3298}, {615,3301}, {642,3666}, {748,3077}, {999,8976}, {1056,3316}, {1124,3068}, {1151,1479}, {1210,8983}, {1587,7288}, {1737,7969}, {3071,7741}, {3299,3582}, {3586,9615}, {4302,6409}, {5204,6560}, {5225,9541}, {6200,6284}, {6424,9599}, {6502,7583}, {6564,7354}, {6565,7173}
X(9661) = {X(1151),X(1479)}-harmonic conjugate of X(9660)X(9662) lies on these lines: {1,9649}, {4,9648}, {11,1151}, {497,9543}, {499,6445}, {1479,6407}, {3071,5326}, {3586,9618}, {3614,6561}, {5225,9542}, {5432,6459}, {5433,6449}, {6221,6284}, {6453,9660}, {6480,9661}, {7354,9541}
X(9662) = {X(1479),X(6407)}-harmonic conjugate of X(9663)
X(9663) lies on these lines: {1,9648}, {4,9649}, {11,6221}, {12,9540}, {371,5433}, {497,9542}, {1151,6284}, {1479,6407}, {1588,7294}, {3586,9617}, {4302,6445}, {5225,9543}, {6453,9661}, {6459,7173}, {6480,9660}, {7354,8981}
X(9663) = {X(1479),X(6407)}-harmonic conjugate of X(9662)
X(9664) lies on these lines: {1,7748}, {4,1500}, {6,5134}, {11,574}, {30,2242}, {32,6284}, {35,7746}, {39,1479}, {55,115}, {56,7756}, {187,4302}, {192,316}, {350,7761}, {497,1015}, {1571,9581}, {1572,9580}, {1573,3434}, {1574,2478}, {1914,5309}, {2241,5254}, {2275,4857}, {2276,3583}, {2548,5225}, {3586,9620}, {3760,7854}, {3767,4294}, {4037,4680}, {4366,7790}, {5217,7749}, {5346,7031}
X(9664) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7748,9651), (4,1500,9650), (497,2549,1015), (1479,9598,39), (2276,3583,5475)
X(9665) lies on these lines: {1,5475}, {4,1015}, {5,2241}, {11,32}, {39,1479}, {55,1506}, {56,7747}, {187,499}, {192,7858}, {350,7759}, {496,2242}, {497,1500}, {498,7603}, {574,6284}, {1571,9580}, {1572,9581}, {1573,2478}, {1574,3434}, {1914,7741}, {2275,3583}, {2276,4857}, {2549,5225}, {3086,7737}, {3586,9619}, {3760,7855}, {3825,4386}, {4366,7752}, {5204,6781}, {5206,5433}, {5299,5309}, {5332,5346}
X(9665) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5475,9650), (4,1015,9651), (39,1479,9664), (496,7745,2242), (497,2548,1500), (1479,9599,39), (1914,7741,7746), (2275,3583,7748)
X(9666) lies on these lines: {1,9653}, {4,215}, {11,1147}, {12,578}, {49,1479}, {54,55}, {156,3583}, {184,6284}, {497,2477}, {498,567}, {569,5432}, {1092,5433}, {3586,9622}, {5012,5217}, {5225,9544}, {7173,9306}
X(9666) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,215,9652), (497,9545,2477)
X(9667) lies on these lines: {1,156}, {4,2477}, {11,184}, {49,1479}, {55,110}, {56,1614}, {215,497}, {569,7173}, {1147,6284}, {1473,3025}, {3586,9621}, {5225,9545}, {5326,5651}, {5432,9306}, {6759,7354}
X(9667) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,156,9652), (4,2477,9653), (49,1479,9666), (497,9544,215)
X(9668) lies on these lines: {1,382}, {3,11}, {4,390}, {5,4294}, {6,5134}, {8,4756}, {12,3843}, {20,496}, {30,497}, {35,1656}, {36,3534}, {55,381}, {56,1657}, {149,956}, {376,5274}, {388,3627}, {498,3851}, {515,4342}, {516,5722}, {517,1864}, {546,3085}, {548,7288}, {550,3086}, {942,4312}, {944,1537}, {1056,3543}, {1058,3146}, {1385,9614}, {1387,5731}, {1478,3058}, {1770,5708}, {2098,7972}, {3303,3585}, {3526,4330}, {3545,5281}, {3576,7743}, {3579,9581}, {3790,5015}, {3814,4421}, {3822,4428}, {3839,8164}, {3853,5229}, {4304,5886}, {4305,5901}, {4324,5204}, {4366,7841}, {4387,4680}, {5010,5054}, {5013,9665}, {5024,9599}, {5045,9579}, {5046,5687}, {5055,5432}, {5070,7173}, {5073,7354}, {5119,5790}, {5339,7006}, {5340,7005}, {5694,5697}, {5881,8275}, {6221,9660}, {6244,6827}, {6449,9661}
X(9668) = midpoint of X(i),X(j) for these (i,j): (3586,9580)
X(9668) = reflection of X(i) in X(j) for these (i,j): (999,497), (6244,6827)
X(9668) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,382,9655), (1,9629,9642), (4,390,495), (4,3295,9654), (11,4302,3), (11,6284,4302), (55,3583,381), (390,495,3295), (1478,3058,6767), (1479,4302,11), (1479,6284,3), (3830,6767,1478), (4294,5225,5), (4330,7741,5217), (5073,7373,7354), (5217,7741,3526)
X(9669) lies on these lines: {1,381}, {3,11}, {4,496}, {5,497}, {6,9665}, {8,4767}, {12,3851}, {30,3086}, {35,3526}, {36,1657}, {55,1656}, {56,382}, {79,4860}, {80,2098}, {140,4294}, {149,4193}, {202,5340}, {203,5339}, {350,7776}, {388,546}, {390,3090}, {495,1058}, {498,3058}, {517,9581}, {550,7288}, {942,1699}, {944,1387}, {946,5722}, {950,5886}, {954,6990}, {956,5046}, {1056,3832}, {1376,3825}, {1385,3586}, {1478,3843}, {1482,1837}, {1490,1538}, {1594,7071}, {1836,5708}, {1898,5570}, {3057,5790}, {3297,6564}, {3298,6565}, {3303,5072}, {3304,3585}, {3434,4187}, {3486,5901}, {3529,5265}, {3534,3582}, {3579,9580}, {3627,4293}, {3628,5218}, {3746,5079}, {3811,5087}, {3814,3913}, {3820,5082}, {3830,7354}, {3841,8167}, {3845,5229}, {3855,5261}, {3871,5154}, {4299,5073}, {4309,5070}, {4366,7887}, {4673,5827}, {5013,9664}, {5022,5134}, {5024,9598}, {5045,9612}, {5049,5290}, {5054,5217}, {5064,7191}, {5067,5281}, {5068,8164}, {5076,5563}, {5704,6361}, {6198,7507}, {6221,9661}, {6244,6922}, {6449,9660}
X(9669) = midpoint of X(i),X(j) for these (i,j): (3086,5225)
X(9669) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,9654), (3,1479,9668), (4,496,999), (4,999,9655), (4,5274,496), (5,497,3295), (11,1479,3), (11,6284,499), (55,7741,1656), (56,3583,382), (149,4193,5687), (498,7173,5055), (499,1479,6284), (499,6284,3), (1058,3091,495), (3058,7173,498), (3843,7373,1478), (3851,6767,12), (4302,5433,3), (4857,7741,55), (5082,6919,3820)
X(9670) lies on these lines: {1,382}, {3,3582}, {4,3058}, {5,55}, {11,631}, {12,390}, {20,56}, {30,3304}, {35,3526}, {65,9580}, {149,958}, {381,3746}, {388,8162}, {495,3861}, {496,548}, {499,3530}, {516,5221}, {528,2478}, {942,4338}, {950,2099}, {999,4325}, {1001,4197}, {1058,7354}, {1478,3853}, {1480,7706}, {1657,5563}, {1770,4860}, {2476,4428}, {2646,9614}, {3057,3586}, {3085,3855}, {3086,3528}, {3090,4995}, {3146,5434}, {3295,3583}, {3486,5734}, {3522,5298}, {3584,3851}, {3585,6767}, {3612,7743}, {3748,9612}, {3813,6872}, {3829,6910}, {3830,5270}, {3913,5046}, {4193,4421}, {4366,7933}, {4387,5015}, {5067,5432}, {5070,7741}, {5218,7173}, {5274,5433}, {5691,5919}, {7765,9664}
X(9670) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,382,9657), (4,3058,3303), (5,4309,55), (11,4294,5217), (390,5225,12), (496,4302,5204), (497,6284,56), (1479,4309,5)
X(9671) lies on these lines: {1,3843}, {4,3304}, {5,55}, {11,20}, {12,3855}, {35,5070}, {56,382}, {381,3303}, {390,3614}, {495,3856}, {496,3853}, {497,3832}, {499,548}, {528,5187}, {631,6284}, {1388,7743}, {1478,3861}, {1657,3582}, {1837,4301}, {2098,5881}, {3058,3091}, {3526,4330}, {3528,5433}, {3529,5298}, {3530,4302}, {3584,5072}, {3586,9624}, {3746,3851}, {3829,6872}, {3830,5563}, {4197,4423}, {4294,5067}, {4338,5221}, {4421,5154}, {4428,5141}, {4995,5056}, {5274,7354}, {5432,7486}, {7765,9665}, {8162,9654}
X(9671) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3843,9656), (5,1479,9670), (5,9670,55), (381,4857,3303), (496,3853,4317), (3526,4330,5217), (3526,9668,4330), (3583,9669,56), (4330,7741,3526), (7741,9668,5217)
X(9672) lies on these lines: {1,3}, {4,9658}, {11,24}, {12,7503}, {22,6284}, {23,5225}, {26,1479}, {155,215}, {186,3086}, {378,7354}, {496,1658}, {497,7488}, {498,7514}, {499,6644}, {1478,7526}, {1993,9666}, {1995,7173}, {2070,9669}, {2937,9668}, {3520,4293}, {3583,7517}, {3586,9626}, {4294,7512}, {5229,7527}, {5432,7509}, {7506,7741}
X(9672) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,9659), (55,56,9630)
X(9673) lies on these lines: {1,7517}, {3,3583}, {4,9659}, {11,22}, {19,25}, {23,497}, {24,6284}, {26,1479}, {35,7506}, {56,7387}, {154,215}, {999,5899}, {1478,7530}, {1993,9667}, {1995,5432}, {2070,9668}, {2937,9669}, {3057,8185}, {3518,4294}, {3586,9625}, {4302,6644}, {5217,6642}, {5225,7488}, {5285,7082}, {5348,7295}, {7173,7509}
X(9673) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7517,9658), (26,1479,9672)
X(9674) lies on these lines: {3,6}, {115,5418}, {485,7756}, {491,7830}, {590,7748}, {1506,6561}, {1571,9583}, {1572,9582}, {2548,9541}, {2549,9540}
X(9674) = {X(39),X(1151)}-harmonic conjugate of X(9675)
X(9675) lies on these lines: {3,6}, {115,6561}, {485,7747}, {486,7749}, {491,754}, {492,620}, {590,5475}, {1506,5418}, {1571,9582}, {1572,9583}, {2066,2242}, {2067,2241}, {2548,9540}, {2549,9541}, {3068,7737}, {3071,7746}, {3767,6459}, {6560,6781}, {7603,8253}, {7745,8981}
X(9675) = {X(39),X(1151)}-harmonic conjugate of X(9674)
X(9676) lies on these lines: {49,1151}, {54,5418}, {110,6561}, {184,6200}, {371,1147}, {372,1092}, {567,8253}, {6424,9603}, {6565,9306}
X(9676) = {X(49),X(1151)}-harmonic conjugate of X(9677)
X(9677) lies on these lines: {6,7517}, {49,1151}, {54,6561}, {110,5418}, {156,590}, {160,8825}, {184,371}, {485,1614}, {486,5012}, {569,6565}, {1147,6200}, {6424,9604}, {6564,6759}
X(9677) = {X(49),X(1151)}-harmonic conjugate of X(9676)
X(9678) lies on these lines: {3,1377}, {6,993}, {9,9583}, {10,1151}, {21,1335}, {371,958}, {486,4999}, {936,9615}, {956,2066}, {1107,6424}, {1124,2975}, {1152,5267}, {1329,5418}, {1376,6200}, {1573,9675}, {1574,9674}, {1575,9600}, {1706,9582}, {2362,3916}, {2550,9541}, {2551,9540}, {2886,6561}, {3297,8666}, {3298,5248}, {3434,9660}, {3436,9646}, {3814,8253}, {4426,6422}, {5407,6347}
X(9678) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,1151,9679), (371,958,1378)
X(9679) lies on these lines: {3,1378}, {9,9582}, {10,1151}, {100,1335}, {371,1376}, {377,9646}, {404,1124}, {474,2066}, {486,3035}, {936,9616}, {958,6200}, {993,6409}, {1107,9600}, {1329,6561}, {1573,9674}, {1574,9675}, {1575,6424}, {1702,5438}, {1706,9583}, {2067,5687}, {2478,9660}, {2550,9540}, {2551,9541}, {2886,5418}, {3298,8715}, {3436,9647}, {4386,6422}, {5267,6411}, {5407,6348}
X(9679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,1151,9678), (371,1376,1377)
X(9680) lies on these lines: {2,6453}, {5,1151}, {6,3530}, {17,2041}, {18,2042}, {20,485}, {140,6425}, {371,631}, {376,8960}, {382,590}, {486,3526}, {548,6409}, {549,3592}, {1124,9648}, {1327,1657}, {1328,3090}, {1335,9663}, {1588,9542}, {1656,6519}, {3068,3528}, {3070,6455}, {3071,5070}, {3523,6419}, {3524,6420}, {3627,6488}, {3832,6484}, {3843,6445}, {3853,6433}, {5054,6447}, {5056,6482}, {5067,6459}, {5881,9615}, {6411,7583}, {6424,9606}, {6437,7584}, {6486,6564}, {6565,7486}, {7765,9674}
X(9680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1151,9681), (5418,9681,5)
X(9681) lies on these lines: {4,6453}, {5,1151}, {6,548}, {13,2041}, {14,2042}, {20,371}, {30,6425}, {372,3528}, {376,6419}, {381,6519}, {382,485}, {486,631}, {550,3592}, {590,3843}, {615,6455}, {632,6488}, {1327,3146}, {1328,1656}, {1657,6447}, {2066,4317}, {2067,4309}, {3071,3526}, {3522,6420}, {3530,5420}, {3545,6482}, {3594,8703}, {3832,6480}, {3853,6429}, {3859,6439}, {3861,6468}, {5067,6484}, {5070,6445}, {5881,9616}, {6231,6278}, {6411,7584}, {6424,9607}, {6437,7583}, {6476,8972}, {7765,9675}
X(9681) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1151,9680), (5,9680,5418), (9540,9543,6480)
X(9682) lies on these lines: {3,485}, {6,1511}, {22,6200}, {23,9541}, {24,371}, {25,6561}, {26,1151}, {55,9632}, {186,3068}, {378,6564}, {486,6642}, {1599,8968}, {1658,8981}, {1993,9676}, {1995,6565}, {2070,6221}, {2937,6449}, {3071,7506}, {3518,6459}, {6424,9608}, {7485,8280}, {7488,9540}, {7514,8253}
X(9682) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,8276,485). (26,1151,9683)
X(9683) lies on these lines: {3,486}, {22,371}, {23,9540}, {24,6200}, {25,5418}, {26,1151}, {55,9631}, {485,7387}, {590,7517}, {639,1599}, {1588,6636}, {1993,9677}, {2070,6449}, {2937,6221}, {5899,8976}, {6409,6644}, {6424,9609}, {6459,7512}, {6565,7509}, {7488,9541}
X(9683) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,1151,9682)
X(9684) lies on these lines: {3,6}, {1571,9585}, {1572,9584}, {2548,9543}, {2549,9542}
X(9684) = {X(39),X(6407)}-harmonic conjugate of X(9685)
X(9685) lies on these lines: {3,6}, {1571,9584}, {1572,9585}, {2548,9542}, {2549,9543}
X(9685) = {X(39),X(6407)}-harmonic conjugate of X(9684)
X(9686) lies on these lines: {49,6407}, {184,1151}, {371,1092}, {578,9540}, {1147,6221}, {3071,5651}, {6453,9676}, {6459,9306}, {6480,9677}, {6759,9541}
X(9686) = {X(49),X(6407)}-harmonic conjugate of X(9687)
X(9687) lies on these lines: {49,6407}, {184,6221}, {569,6459}, {1092,6449}, {1147,1151}, {6453,9677}, {6480,9676}, {6560,8717}, {6759,8981}
X(9687) = {X(49),X(6407)}-harmonic conjugate of X(9686)
X(9688) lies on these lines: {9,9585}, {10,6407}, {377,9649}, {936,9617}, {958,6221}, {1107,9602}, {1151,1376}, {1573,9685}, {1574,9684}, {1575,9601}, {1706,9584}, {2067,4428}, {2478,9663}, {2550,9543}, {2551,9542}, {3434,9662}, {5267,6199}, {6453,9678}, {6480,9679}
X(9689) lies on these lines: {9,9584}, {10,6407}, {377,9648}, {936,9618}, {958,1151}, {993,6445}, {1107,9601}, {1376,6221}, {1573,9684}, {1574,9685}, {1575,9602}, {1706,9585}, {2478,9662}, {2550,9542}, {2551,9543}, {3434,9663}, {3436,9649}, {5289,9616}, {6453,9679}, {6480,9678}
X(9690) lies on these lines: {3,6}, {5,9543}, {30,9542}, {382,8972}, {517,9584}, {1385,9618}, {3579,9585}, {3830,9541}, {3843,9540}, {5070,6459}, {5073,8981}
X(9690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,371,6500), (3,6474,371)
X(9691) lies on these lines: 3,6}, {5,9542}, {30,9543}, {517,9585}, {1385,9617}, {3579,9584}, {3830,8981}, {3851,9540}, {5055,6459}, {5073,9541}, {5076,8972}, {8148,9616}, {8976,9681}
X(9692) lies on these lines: {2,6453}, {4,6519}, {5,6407}, {20,1151}, {382,8972}, {486,6476}, {548,6445}, {631,6221}, {1132,5070}, {1587,6484}, {3316,3861}, {3523,6425}, {3524,6447}, {3526,9691}, {3528,6449}, {3530,7586}, {3590,3627}, {3832,9540}, {5059,6482}, {5420,6478}, {5734,9616}, {5881,9617}, {6433,6460}, {6459,6468}, {7765,9684}
X(9692) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3832,9543,9681), (6480,9540,9543), (9540,9681,3832)
X(9693) lies on these lines: {2,6519}, {4,6453}, {5,6407}, {20,6221}, {371,3528}, {376,6425}, {382,9691}, {485,6476}, {548,7581}, {615,631}, {3069,6484}, {3316,3843}, {3522,6447}, {3526,9690}, {3530,6445}, {3832,9542}, {3853,8972}, {3855,9540}, {5067,6459}, {5881,9618}, {6429,9541}, {6449,7582}, {6478,6560}, {7765,9685}
X(9694) lies on these lines: {3,7581}, {22,1151}, {23,9543}, {24,6221}, {26,6407}, {55,9634}, {378,8981}, {1593,8972}, {1993,9686}, {1995,6459}, {2070,9691}, {2937,9690}, {6453,9682}, {7488,9542}, {7503,9540}, {8276,9541}
X(9695) lies on these lines: {3,7582}, {22,6221}, {23,9542}, {24,1151}, {26,6407}, {55,9633}, {378,9541}, {1993,9687}, {2070,9690}, {2937,9691}, {3316,5198}, {6445,6644}, {6453,9683}, {6459,7509}, {6480,9682}, {7488,9543}
X(9696) lies on these lines: {6,3200}, {32,1147}, {39,49}, {54,1506}, {110,115}, {112,3043}, {156,7748}, {184,574}, {215,1015}, {567,7603}, {1092,5206}, {1500,2477}, {1571,9587}, {1572,9586}, {1614,7756}, {2548,9545}, {2549,9544}
X(9696) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,9603,39)
X(9697) lies on these lines: {6,3205}, {32,184}, {39,49}, {54,115}, {110,1506}, {156,5475}, {215,1500}, {574,1147}, {1015,2477}, {1569,3044}, {1571,9586}, {1572,9587}, {1614,7747}, {2548,9544}, {2549,9545}, {5012,7749}
X(9697) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,49,9696), (49,9604,39)
The exsimilcenter of these circles is X(7765)
X(9698) lies on these lines: {2,3108}, {3,7753}, {5,39}, {6,3411}, {20,574}, {32,631}, {61,6774}, {62,6771}, {83,620}, {140,5007}, {183,7890}, {187,3530}, {230,5041}, {233,5421}, {325,6292}, {382,5013}, {384,2482}, {395,629}, {396,630}, {548,6781}, {626,7777}, {632,5306}, {1007,7867}, {1078,7838}, {1571,9589}, {1572,9588}, {1656,5309}, {1907,3199}, {2549,3832}, {3055,5305}, {3090,7739}, {3329,6680}, {3528,7737}, {3548,5158}, {3589,7874}, {3767,5067}, {3788,7889}, {3843,5024}, {3855,7738}, {3934,7813}, {4045,7752}, {4309,9599}, {4317,9596}, {5038,5477}, {5070,5355}, {5286,7486}, {5881,9619}, {6103,6143}, {6704,7832}, {6722,7797}, {7608,9302}, {7759,7810}, {7763,7808}, {7770,7863}, {7771,7921}, {7774,7826}, {7775,7791}, {7785,7830}, {7800,7903}, {7803,7862}, {7807,9167}, {7821,8362}, {7831,7941}, {7843,8356}, {7859,7925}, {7873,8359}, {7875,7940}, {7878,7907}, {7904,7926}
X(9698) = midpoint of X(i),X(j) for these (i,j): (7824,7858)
X(9698) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7764,7794), (2,7772,7755), (5,39,7765), (5,7765,115), (5,9606,39), (39,1506,115), (39,3815,1506), (39,7603,5254), (140,9300,5007), (230,5041,5368), (325,6683,6292), (574,2548,7747), (1506,7765,5), (3329,7769,6680), (3815,9606,5), (5013,5475,7756), (7746,9605,5355), (7763,7808,7820), (7774,7815,7826), (7777,7786,626), (7777,7876,7814), (7786,7814,7876), (7814,7876,626)
X(9699) lies on these lines: {3,1506}, {6,2070}, {22,574}, {23,2549}, {24,32}, {25,115}, {26,39}, {55,9636}, {186,7737}, {187,6644}, {1384,2079}, {1571,9591}, {1572,9590}, {1658,7745}, {1993,9696}, {2548,7488}, {2937,5013}, {3517,7755}, {3518,3767}, {3815,7502}, {6243,9603}, {6642,7749}, {7387,7756}, {7506,7746}, {7514,7603}, {7517,7748}, {7556,7736}
X(9699) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,9608,39)
X(9700) lies on these lines: {3,115}, {6,2937}, {22,32}, {23,2548}, {24,574}, {25,1506}, {26,39}, {55,9635}, {230,7525}, {1571,9590}, {1572,9591}, {1993,9697}, {2070,5013}, {2549,7488}, {3172,8428}, {3767,7512}, {5254,7502}, {5305,7555}, {5475,7517}, {6243,9604}, {7387,7747}, {7556,7738}
X(9700) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,39,9699), (26,9609,39)
X(9701) lies on these lines: {2,2477}, {8,215}, {9,9587}, {10,49}, {54,2886}, {110,1329}, {184,958}, {377,9653}, {936,9621}, {1107,9604}, {1147,1376}, {1573,9697}, {1574,9696}, {1575,9603}, {1706,9586}, {2478,9667}, {2550,9545}, {2551,9544}, {3036,3045}, {3041,3046}, {3434,9666}, {3436,9652}, {4999,5012}
X(9702) lies on these lines: {2,215}, {8,2477}, {9,9586}, {10,49}, {54,1329}, {110,2886}, {184,1376}, {567,3814}, {936,9622}, {958,1147}, {1107,9603}, {1573,9696}, {1574,9697}, {1575,9604}, {1706,9587}, {2550,9544}, {2551,9545}, {3035,3045}, {3434,9667}, {3436,9653}
X(9702) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,49,9701)
X(9703) lies on these lines: {3,49}, {5,9545}, {6,3200}, {24,195}, {30,9544}, {54,1656}, {110,381}, {154,5899}, {156,382}, {215,999}, {323,7502}, {378,399}, {517,9586}, {567,5055}, {569,5070}, {578,3851}, {1385,9622}, {1493,3567}, {1511,5890}, {1614,1657}, {1993,2070}, {2477,3295}, {3579,9587}, {5012,5054}, {5013,9697}, {5024,9604}, {5073,6759}, {6193,6639}, {6221,9676}, {6449,9677}
X(9703) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,1147,3), (567,9306,5055), (3200,3201,6)
X(9704) lies on these lines: {3,49}, {5,9544}, {6,3205}, {26,195}, {30,9545}, {54,156}, {110,1656}, {206,5093}, {215,3295}, {323,7525}, {382,1614}, {399,3047}, {517,9587}, {567,3851}, {569,5055}, {578,3843}, {999,2477}, {1385,9621}, {1493,3060}, {1657,8718}, {1993,2937}, {3526,5012}, {3579,9586}, {3830,6759}, {5013,9696}, {5024,9603}, {5070,9306}, {5889,5944}, {6221,9677}, {6449,9676}, {6640,6776}
X(9704) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,49,9703), (49,184,3), (54,156,381), (3205,3206,6)
X(9705) lies on these lines: {3,7666}, {5,49}, {20,1147}, {156,382}, {184,631}, {542,6143}, {569,7486}, {578,3855}, {1092,3528}, {3205,3412}, {3206,3411}, {3292,7512}, {3526,5012}, {3832,9545}, {5067,9306}, {5881,9621}, {5944,7691}, {7765,9696}
X(9705) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,110,54), (1147,9544,1614)
X(9706) lies on these lines: {5,49}, {20,184}, {156,3843}, {195,5944}, {382,1614}, {569,5067}, {578,3832}, {631,1147}, {1493,2070}, {3200,3411}, {3201,3412}, {3431,7689}, {3526,9703}, {5881,9622}, {6143,9140}, {7486,9306}, {7765,9697}
X(9706) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,49,9705), (5,9705,110), (49,54,110), (54,9705,5)
The exsimilcenter of these circles is X(1993)
X(9707) lies on these lines: {3,74}, {4,154}, {6,3518}, {22,1147}, {23,9545}, {24,184}, {25,54}, {26,49}, {55,9638}, {70,7568}, {155,7488}, {186,1181}, {378,6759}, {394,7512}, {569,1995}, {578,1495}, {631,3796}, {1498,3520}, {1853,6143}, {2070,9704}, {2917,6242}, {2931,3047}, {2937,9703}, {3060,9706}, {3147,6776}, {3167,9705}, {3515,5890}, {3517,3567}, {3525,5085}, {5012,6642}, {5422,7506}, {6146,7505}, {6193,7493}, {7395,8780}, {7509,9306}
X(9707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6090,7999), (24,184,7592), (26,49,1993), (156,5944,3), (7488,9544,155)
X(9708) lies on these lines: {1,210}, {2,495}, {3,10}, {5,2551}, {6,1573}, {8,405}, {9,374}, {21,3617}, {30,2550}, {36,4413}, {40,5234}, {46,3698}, {55,3679}, {56,1698}, {63,3753}, {65,3927}, {78,3697}, {145,5047}, {218,3691}, {281,7497}, {333,5774}, {377,9655}, {381,2886}, {388,8728}, {392,3305}, {442,3436}, {452,5082}, {474,2975}, {496,5084}, {519,1001}, {529,3826}, {551,8167}, {758,1159}, {859,5235}, {936,1385}, {946,8158}, {952,6883}, {954,5686}, {960,1482}, {997,3740}, {1012,5273}, {1058,5129}, {1107,9605}, {1125,7373}, {1155,4731}, {1220,2049}, {1329,1656}, {1377,3312}, {1378,3311}, {1384,4386}, {1478,3925}, {1574,5013}, {1575,5024}, {1597,1861}, {1617,5252}, {1706,3579}, {1724,5710}, {2093,3929}, {2099,3715}, {2478,9669}, {2646,3983}, {3035,5054}, {3085,6675}, {3090,8165}, {3149,5818}, {3241,5284}, {3294,4513}, {3303,3632}, {3304,3624}, {3428,5587}, {3434,9668}, {3452,5886}, {3487,5815}, {3526,4999}, {3560,5690}, {3576,8580}, {3626,3913}, {3634,8666}, {3654,5325}, {3683,5119}, {3746,4668}, {3757,4737}, {3811,4662}, {3812,5708}, {3814,5055}, {3824,5290}, {3871,4678}, {3876,5730}, {3878,8148}, {3916,4002}, {3921,5440}, {3951,4018}, {3984,4533}, {4185,7140}, {4414,4695}, {4421,4745}, {4428,4669}, {4691,8715}, {4847,5722}, {5045,6762}, {5221,6763}, {5247,5711}, {5275,5291}, {5302,5836}, {5436,6765}, {5552,7483}, {5584,5691}, {5705,6918}, {5779,6001}, {6221,9678}, {6449,9679}, {6857,7080}, {6939,7956}
X(9708) = complement of X(1056)
X(9708) = midpoint of X(i),X(j) for these (i,j): (8,3488), (9,9623)
X(9708) = reflection of X(i) in X(j) for these (i,j): (6767,1001)
X(9708) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,210,3940), (2,956,999), (2,3421,495), (8,405,3295), (8,5260,405), (10,958,3), (10,993,1376), (10,5795,355), (21,3617,5687), (442,3436,9654), (958,1376,993), (993,1376,3), (1012,5657,6244), (1698,5258,56), (2099,3715,5692), (3624,5288,3304), (3626,5248,3913), (3632,5259,3303), (3679,5251,55)
X(9709) lies on these lines: {1,3689}, {2,496}, {3,10}, {4,3820}, {5,2550}, {6,1574}, {8,474}, {9,3579}, {30,2551}, {40,5044}, {43,5711}, {46,210}, {55,1698}, {56,3679}, {63,3697}, {65,3940}, {78,3753}, {100,405}, {171,6048}, {191,3715}, {200,942}, {377,9654}, {381,1329}, {404,956}, {442,1260}, {443,495}, {451,7071}, {480,3824}, {498,3925}, {517,936}, {518,5708}, {519,7373}, {750,3214}, {940,3293}, {975,4646}, {997,1482}, {1001,3634}, {1012,5818}, {1107,5024}, {1125,3913}, {1155,3983}, {1158,5779}, {1159,3754}, {1193,9350}, {1377,3311}, {1378,3312}, {1384,4426}, {1385,5438}, {1466,9578}, {1573,5013}, {1575,9605}, {1598,1861}, {1656,2886}, {1722,5266}, {2478,9668}, {2646,4731}, {3035,3526}, {3085,8728}, {3149,5657}, {3216,5710}, {3244,8168}, {3303,3624}, {3304,3632}, {3306,3555}, {3333,4882}, {3340,4930}, {3359,5777}, {3421,6904}, {3434,4187}, {3436,9655}, {3616,9342}, {3654,5837}, {3711,5221}, {3746,4423}, {3811,3812}, {3814,3851}, {3826,6600}, {3828,4421}, {3870,5439}, {3916,3921}, {3931,5268}, {3951,4533}, {3984,4018}, {4002,5440}, {4015,5220}, {4208,8164}, {4309,6154}, {4383,5264}, {4668,5563}, {4691,8666}, {4999,5054}, {5045,5437}, {5204,5258}, {5217,5251}, {5218,6675}, {5289,8148}, {5445,7742}, {5537,7989}, {5584,9588}, {5690,6911}, {5722,8582}, {5774,9534}, {5780,5887}, {5806,6769}, {5827,7270}, {5886,6700}, {6221,9679}, {6449,9678}, {6964,7956}, {8163,8275}
X(9709) = complement of X(1058)
X(9709) = midpoint of X(i),X(j) for these (i,j): (936,1706), (3333,4882)
X(9709) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5082,496), (2,5687,3295), (3,10,9708), (8,474,999), (10,1376,3), (10,5794,5790), (10,6684,5791), (40,8580,5044), (46,210,3927), (404,3617,956), (443,7080,495), (997,5836,1482), (1125,3913,6767), (3434,4187,9669), (3634,8715,1001), (3711,5221,5904), (5437,6765,5045), (5438,9623,1385)
X(9710) lies on these lines: {1,3826}, {2,3303}, {5,10}, {8,3475}, {9,9589}, {12,3617}, {20,958}, {141,9049}, {226,4662}, {377,529}, {382,9708}, {405,528}, {442,3679}, {495,3626}, {496,3634}, {516,5302}, {519,8728}, {548,993}, {631,1376}, {908,3983}, {936,9624}, {956,4317}, {1001,5082}, {1058,8167}, {1107,9607}, {1573,7765}, {1574,9698}, {1575,9606}, {1698,3816}, {1861,1907}, {2478,9671}, {2551,3832}, {3035,3526}, {3036,6937}, {3058,5047}, {3214,5718}, {3434,9670}, {3436,9656}, {3649,3681}, {3654,6841}, {3674,4967}, {3698,6734}, {3812,4847}, {3822,4691}, {3829,4187}, {4101,4113}, {4295,5220}, {4323,7679}, {4325,5258}, {4330,5251}, {4413,6691}, {4421,6857}, {5260,6284}, {5289,5734}, {5325,5493}, {5552,6668}, {5657,6845}, {5687,6690}, {5794,5881}, {6675,8715}, {7991,8226}
X(9710) = complement of X(3303)
X(9710) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,946,3740), (10,2886,1329), (3626,3841,495)
X(9711) lies on these lines: {2,3304}, {5,10}, {7,12}, {8,1997}, {9,9588}, {11,3617}, {20,1376}, {104,631}, {141,2885}, {377,9656}, {382,9709}, {452,4421}, {474,529}, {495,3634}, {496,3626}, {518,8582}, {528,2478}, {908,3698}, {936,5881}, {956,6691}, {993,3530}, {1001,7080}, {1058,8168}, {1107,9606}, {1158,3652}, {1210,4662}, {1573,9698}, {1574,7765}, {1575,9607}, {1698,3338}, {1706,9589}, {1737,3697}, {1834,6048}, {1861,1906}, {2550,3832}, {2899,5695}, {3264,3701}, {3434,9671}, {3436,4413}, {3526,4999}, {3679,3680}, {3825,4691}, {3828,8728}, {3829,4193}, {3913,5084}, {3983,6734}, {4189,6174}, {4309,5687}, {4428,5129}, {5260,5432}, {5302,6684}, {5316,6736}, {5552,6690}, {5554,5855}, {5794,8580}, {5818,6845}
X(9711) = complement of X(3304)
X(9711) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,10,9710), (5,9710,2886), (10,960,8256), (10,1329,2886), (10,3452,5836), (10,3820,1329), (1329,9710,5), (3679,4187,3813)
X(9712) lies on these lines: {3,2886}, {9,9591}, {10,26}, {22,958}, {23,2551}, {24,1376}, {25,1329}, {55,9640}, {377,9659}, {936,9625}, {1107,9609}, {1573,9700}, {1574,9699}, {1575,9608}, {1706,9590}, {1993,9701}, {2070,9709}, {2478,9673}, {2550,7488}, {2937,9708}, {3035,6642}, {3434,9672}, {3436,9658}
X(9713) lies on these lines: {3,119}, {9,9590}, {10,26}, {22,1376}, {23,2550}, {24,958}, {25,2886}, {55,9639}, {377,9658}, {936,9626}, {993,6644}, {1107,9608}, {1573,9699}, {1574,9700}, {1575,9609}, {1706,9591}, {1993,9702}, {2070,9708}, {2478,9672}, {2551,7488}, {2937,9709}, {3434,9673}, {3436,9659}, {3814,7514}, {3820,7502}, {4999,6642}, {5794,8185}, {8193,8256}
X(9713) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,26,9712)
X(9714) = As a point on the Euler line X(9714) has Shinagawa coefficients [-E-6*F, 5*E+6*F]
X(9714) lies on these lines: {2,3}, {49,206}, {52,154}, {55,9644}, {155,1495}, {567,3527}, {1993,9705}, {3060,9706}, {3167,6243}, {3567,6800}, {3796,5462}, {5881,8185}, {6090,6101}, {7765,9699}
X(9714) = reflection of X(i) in X(j) for these (i,j): (3,3515)
X(9714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,25,7529)
X(9715) lies on these lines: {2,3}, {54,1351}, {55,9643}, {154,5562}, {159,2917}, {206,1092}, {389,3796}, {1216,6090}, {1993,9706}, {3167,9705}, {3567,5050}, {3964,7796}, {5217,9628}, {5594,6278}, {5595,6281}, {5881,9626}, {5889,6800}, {7071,9644}, {7765,9700}
X(9715) = isogonal conjugate of X(38442)
X(9715) = reflection of X(i) in X(j) for these (i,j): (3516,3), (7507,3549)
X(9715) = homothetic center of orthocevian triangle of X(2) and cevian triangle of X(3)
Contributed by Peter Moses, March 5, 2016.
X(9716) lies on the cubic K759, the Thomson-Gibert-Moses hyperbola, and these lines: {2,575}, {3,323}, {4,5609}, {23,154}, {49,7556}, {110,576}, {155,7527}, {182,5888}, {184,6030}, {194,3906}, {373,5645}, {392,3897}, {394,7496}, {511,7712}, {1493,2888}, {1992,5648}, {1994,1995}, {2904,3518}, {3091,5654}, {3146,5656}, {3557,5638}, {3558,5639}, {5544,6090}, {5643,5651}, {7555,9704}, {7575,9703}
X(9716) = anticomplement of X(38397)
X(9716) = isogonal conjugate (with respect to ABC) of X(1657)-of-Thomson-triangle
Contributed by Peter Moses, March 5, 2016.
X(9717) lies on the cubics K297, K626, K759 and these lines: {3,74}, {6,647}, {25,842}, {183,1494}, {1316,2394}, {1495,5502}, {1649,5967}, {2434,9177}, {3292,5467}, {5055,5627}, {5468,6390}, {5968,9139}
X(9717) = isogonal conjugate of X(9214)
X(9717) = X(9139)-Ceva conjugate of X(74)
X(9717) = X(i)-isoconjugate of X(j) for these (i,j): {1,9214}, {30,897}, {671,2173}, {895,1784}, {923,3260}, {1099,9139}
X(9717) = crossdifference of every pair of centers on X(30)X(1637)
.
X(9717) = crosssum of X(i) and X(j) for (i,j): {1637,2682}, {30,5642}
X(9717) = crosspoint of X(74) and X(9139)
X(9717) = barycentric product X(i)*X(j) for these (i,j): {74,524}, {187,1494}, {896,2349}, {2394,5467}, {2433,5468}, {2482,9139}, {6390,8749}
X(9717) = perspector of ABC and unary cofactor of 2nd Parry triangle
Centers related to Dou circle: X(9718)-X(9721)
This preamble and centers X(9718)-X(9721) were contributed by César Eliud Lozada, March 10, 2016.
The Dou circle is defined in MathWorld as the circle cutting the sidelines of the reference triangle ABC at A', A", B', B", C', C" so that angle(A'AA") = angle(B'BB") = angle(C'CC") = 90º. The center of the Dou circle is X(155).
Let f(a,b,c) = (a^2)[SA*(2*R^2-SA) and g(a,b,c) = bc(S^2+SB*SC-2*(SB+SC)*R^2). A trilinear equation for the Dou circle is then
f(a,b,c)x^2 + f(b,c,a)y^2 + f(c,a,b)z^2 + g(a,b,c)yz + g(b,c,a)zx + g(c,a,b)xy = 0,
and the radius-squared is (4*R^6-S^2*(5*R^2-SW))/(4*R^2-SW)^2.
Let S = {circumcircle, nine-points-circle, orthocentroidal circle, orthoptic circle of the Steiner inellipse, polar circle (for obtuse ABC), tangential circle, Taylor circle}, and suppose that U and V are in S. The radical circle of {U, V, Dou circle} is here named the Dow radical circle of U and V, denoted by Dou(U,V). The center of Dou(U,V) is X(2501), and the radius-squared of Dou(U,V) is the power of X(2501) with repect to U and V. Moreover, Dou(U,V) is orthogonal to the Dou circle and every circle in S. The Stevanovic circle is orthogonal to the Apollonius circle, Bevan circle, excircles-radical circle, as well as every circle in S except the Taylor circle. The radical axis of the Stevanovic and each Dou-radical circle is the Euler line, and their radical trace is X(468). If ABC is acute then Dou(U,V) passes through X(5000) and X(5001).
X(9718) lies on these lines: {513,676}, {924,2501}
X(9719) lies on these lines: {11,244}, {2501,3566}, {2520,3309}
X(9720) lies on the cubic K418 and these lines: {3,49}, {924,2501}
X(9721) lies on these lines: {5,6}, {131,187}, {2501,3566}
Let P be the perspector of the Dou circle. Then X(9722) is the trilinear pole of the polar of P with respect to the Dou circle. (Randy Hutson, March 14, 2016)
X(9722) lies on these lines: {2,6503}, {4,1609}, {5,6}, {30,8553}, {32,7403}, {39,233}, {115,131} et al
X(9722) = complement of X(9723)
Let La be the line tangent to the A-Lucas circle at the antipode of A. Define Lb and Lc cyclically. Let Ta = Lb∩Lc, and define Tb and Tc cyclically. Triangle TaTbTc is here introduced as the Lucas antipodal tangents triangle. Let La' be the line tangent to the A-Lucas(-1) circle at the antipode of A. Define Lb' and Lc' cyclically. Let Ta' = Lb'∩Lc', and define Tb' and Tc' cyclically. Triangle Ta'Tb'Tc' is here introduced as the Lucas(-1) antipodal tangents triangle. The triangles TaTbTc and Ta'Tb'Tc' are homothetic, and their center of homothety is X(9723). See also X(8939) and X(8943). (Randy Hutson, March 14, 2016)
X(9723) lies on these lines: {2,6503}, {3,69}, {6,2987}, {24,317}, {25,1007}, {76,95}, {99,264} et al
X(9723) = isotomic conjugate of X(847)
X(9723) = anticomplement of X(9722)
X(9723) = X(25)-isoconjugate of X(91)
X(9723) = exsimilicenter of Lucas antipodal circle and Lucas(-1) antipodal circle; the insimilicenter is X(3)
Let A'B'C' be the incentral triangle of ABC, and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a conic with center X(9724). (César Lozada, March 17, 2016) The conic is here named the 1st Lozada conic.
X(9724) lies on these lines: {284,501}, {3982,6610}
Let A'B'C' be the Lucas antipodal triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a conic with center X(9725). César Lozada, March 17, 2016. The conic is here named the 2nd Lozada conic.
X(9725) lies on these lines: {394,493}, {1599,1609}
Let A'B'C' be the Lucas(-1) antipodal triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a conic with center X(9726). César Lozada, March 17, 2016. The conic is here named the 3rd Lozada conic.
X(9726) lies on these lines: {394,494}, {1600,1609}
Let A'B'C' be the Lucas central triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a conic with center X(9727). César Lozada, March 17, 2016. The conic is here named the 4th Lozada conic.
Let A'B'C' be the Lucas(-1) central triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a conic with center X(9728). César Lozada, March 17, 2016. The conic is here named the 5th Lozada conic.
X(9729) = (8*R^2-SW)*X(3)+SW*X(6)
Let A'B'C' be the midheight triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a circle with center X(9729) and radius-squared (s^4+(r^2+4*R*r+8*R^2)^2+2*(r^2-8*R^2-4*R*r)*s^2)/(64*R^2). César Lozada, March 17, 2016. The circle is here named the 1st Lozada circle.
X(9729) lies on these lines: {2,185}, {3,6}, {4,5943}, {5,2883}, {20,51}, {30,5462}, {143,548}, {373,3091}, {376,3567}, {517,6738}, {549,1216}, {550,5446}, {631,3819}, {632,5876}, {858,3574}, {916,5044}, {974,5972}, {1092,7592}, {1125,2807}, {1154,3530}, {1181,9306}, {1204,7503}, {1352,6803}, {1498,5020}, {1660,6644}, {1715,4192}, {1899,6815}, {2808,5777}, {3060,3522}, {3066,5198}, {3090,6241}, {3146,5640}, {3515,3796}, {3523,3917}, {3526,5891}, {3589,6696}, {3628,5663}, {3818,7401}, {6642,6759}, {6689,6699}, {7514,7689}, {8550,8681}
X(9729) = midpoint of X(i),X(j) for these (i,j): (3,389), (143,548), (185,5907), (550,5446), (974,5972), (1216,6102), (3819,5890)
X(9729) = reflection of X(i) in X(j) for these (i,j): (5447,3530), (6688,5892)
X(9729) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,185,5907), (3,9730,389), (371,372,5065), (1192,5085,3)
X(9729) = complement of X(5907)
X(9729) = X(8)-of-submedial-triangle if ABC is acute
X(9730) = (6*R^2-SW)*X(3)+SW*X(6)
Let A'B'C' be the orthocentroidal triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a circle with center X(9730) and radius-squared ((r^2+4*R*r+6*R^2)^2+s^4-2*(6*R^2-r^2+4*R*r)*s^2)/(36*R^2). César Lozada, March 17, 2016. The circle is here named the 2nd Lozada circle.
Let P be a point on the circumcircle. Let Pa be the orthogonal projection of P on the A-altitude, and define Pb and Pc cyclically. The locus of the centroid of PaPbPc as P varies is an ellipse centered at X(9730). See also X(185). (Randy Hutson, March 25, 2016)
Let Ga, Gb, Gc be the centroids of the A-, B-, and C-altimedial triangles. Then X(9730) is the orthocenter of GaGbGc. (Randy Hutson, March 25, 2016)
Let Ha, Hb, Hc be the orthocenters of the A-, B-, and C-altimedial triangles. Then X(9730) is the centroid of HaHbHc. (Randy Hutson, March 25, 2016)
X(9730) lies on these lines: {2,5654}, {3,6}, {4,4846}, {5,113}, {20,3567}, {24,6800}, {30,51}, {54,5504}, {68,6815}, {74,7527}, {140,5562}, {143,550}, {155,6090}, {184,6644}, {186,5012}, {376,3060}, {378,5422}, {381,1853}, {517,3058}, {520,5664}, {546,1514}, {549,1154}, {597,2781}, {631,1216}, {1147,7592}, {1181,6642}, {1204,7526}, {1209,7399}, {1498,7529}, {1656,5907}, {1986,6699}, {2079,9604}, {2779,5883}, {2807,5886}, {2842,5884}, {2854,8550}, {2979,3524}, {3091,6241}, {3523,5447}, {3530,6101}, {3628,5876}, {3819,5054}, {5055,6688}, {5498,8254}, {6247,7403}, {6723,7723}, {6759,7506}, {6785,7827}, {7503,7689}
X(9730) = midpoint of X(i),X(j) for these (i,j): (2,5890), (3,568), (376,3060)
X(9730) = reflection of X(i) in X(j) for these (i,j): (2,5892), (51,5946), (52,568), (381,5943), (568,389), (3917,549), (5891,2)
X(9730) = anticomplement of X(10170)
X(9730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,389,52), (371,372,5063), (389,9729,3), (5890,5892,5891)
Let A'B'C' be the 1st Ehrmann triangle of ABC and let BA and CA be the orthogonal projections of B' and C' on BC, respectively. Build AB, CB, AC and BC similarly. These last six points lie on a ellipse with center X(9731). César Lozada, March 17, 2016. The ellipse is here named the Ehrmann-Lozada ellipse.
X(9731) lies on these lines: {6,598}, {524,3934}, {597,5008}, {5039,7606}
X(9732) = (SW-S)*X(3)-SW*X(6)
Let A'B'C' be the inner-Gebe triangle of ABC and let Ab and Ac be the intersections of B'C' with AC and AB, respectively. Build Ba, Bc, Ca and Cb similarly. These last six points lie on a circle with center X(9732) and radius-squared R^2*(2*S^2-2*SW*S+SW^2)/S^2. (César Lozada, March 19, 2016) The circle is here named the 3rd Lozada circle.
X(9732) lies on these lines: {3,6}, {4,487}, {20,6462}, {25,5409}, {30,1991}, {51,1584}, {98,6222}, {325,637}, {394,3155}, {638,7750}, {639,7778}, {640,7784}, {1306,3563}, {1583,3917}, {1599,2979}, {1600,3060}, {3156,5407}
X(9732) = midpoint of X(3) and X(1161)
X(9732) = reflection of X(9733) in X(3)
X(9733) = (SW+S)*X(3)-SW*X(6)
Let A'B'C' be the outer-Gebe triangle of ABC and let Ab and Ac be the intersections of B'C' with AC and AB, respectively. Build Ba, Bc, Ca and Cb similarly. These last six points lie on a circle with center X(9733) and radius-squared R^2*(2*S^2+2*SW*S+SW^2)/S^2. César Lozada, March 19, 2016. The circle is here named the 4th Lozada circle.
X(9733) lies on these lines: {3,6}, {4,488}, {20,6463}, {25,5408}, {51,1583}, {98,6399}, {325,638}, {394,3156}, {637,7750}, {639,7784}, {640,7778}, {1307,3563}, {1584,3917}, {1599,3060}, {1600,2979}, {3155,5406}
X(9733) = midpoint of X(3) and X(1160)
X(9733) = reflection of X(9732) in X(3)
X(9734) = (6*S^2-SW^2)*X(3)+SW^2*X(6)
Let A'B'C' be the McCay triangle of ABC defined in X(7606). The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9734) and radius-squared R^2*(36*S^4-11*SW^2*S^2+SW^4)/(36*S^4). César Lozada, March 19, 2016. The circle is here named the 5th Lozada circle.
X(9734) lies on these lines: {3,6}, {30,7622}, {140,7844}, {542,7618}, {631,7790}, {2549,6036}, {9155,9306}
X(9734) = {X(9735),X(9736)}-harmonic conjugate of X(182)
X(9735) = (SW-2*sqrt(3)*S)*X(3)-SW*X(6)
Let A'B'C' be the inner-Napoleon triangle of ABC. The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9735) and radius-squared R^2*(13*S^2-4*SW*sqrt(3)*S+SW^2)/(12*S^2). César Lozada, March 19, 2016. The circle is here named the 6th Lozada circle.
X(9735) lies on these lines: {3,6}, {531,7622}, {617,5617}, {2782,6295}, {3131,9306}, {6671,7844}, {6771,6772}
X(9735) = reflection of X(9736) in X(3)
X(9735) = {X(182),X(9734)}-harmonic conjugate of X(9736)
X(9736) = (SW+2*sqrt(3)*S)*X(3)-SW*X(6)
Let A'B'C' be the outer-Napoleon triangle of ABC. The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9736) and radius-squared R^2*(13*S^2+4*SW*sqrt(3)*S+SW^2)/(12*S^2). César Lozada, March 19, 2016. The circle is here named the 7th Lozada circle.
X(9736) lies on these lines: {3,6}, {530,7622}, {616,5613}, {2782,6582}, {3132,9306}, {6672,7844}, {6774,6775}
X(9736) = reflection of X(9735) in X(3)
X(9736) = {X(182),X(9734)}-harmonic conjugate of X(9735)
The function p = (r + 4R)/s is the Tucker parameter for X(9737); see the preamble to X(13323.)
Let A'B'C' be the 1st Neuberg triangle of ABC. The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9737) and radius-squared R^2*(4*S^4-3*SW^2*S^2+SW^4)/(4*S^4). César Lozada, March 19, 2016. The circle is here named the 8th Lozada circle. A similar construction using the 2nd Neuberg triangle leads to a circle of center X(5171) and radius squared R^2*(SW^2+S^2)*(SW^2+4*S^2)/(4*S^4). César Lozada, March 19, 2016. The circle is here named the 9th Lozada circle.
X(9737) lies on these lines: {3,6},{4,99},{5,3734},{20,7785},{30,7775},{98,194},{140,7834},{147,9873},{262,384},{376,7812},{542,9888},{631,7803},{1078,12251},{1147,2909},{1352,3926},{1656,7874},{1843,9723},{1968,2967},{1975,6248},{2782,7781},{2794,7764},{3097,12197},{3148,9306},{3522,12122},{3526,7852},{3552,10352},{3767,6036},{3818,6390},{4558,8541},{5476,8369},{5965,7758},{5999,7783},{6461,11394},{7626,8760},{7709,12203},{7751,10104},{7782,11676},{7798,12042},{7801,11178},{7808,11272},{7836,10356},{7906,9863},{7908,9996},{8671,11248},{10607,12167},{11165,11645}
X(9737) = reflection of X(i) in X(j) for these (i,j): (5171,3), (7751,10104)
X(9737) = circumcircle-inverse of X(35383)
X(9737) = Schoute-circle-inverse of X(5033)
X(9737) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(1692)
X(9737) = harmonic center of Gallatly circle and circle {X(371),X(372),PU(1),PU(39)}
X(9737) = exsimilicenter of circumcircle and circle {X(4),X(194),X(3557),X(3558)}; the insimilicenter is X(32)
X(9737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,39,182), (3,1351,3053), (3,2080,5206), (3,3095,32), (3,9821,8722), (3,10983,6), (4,7763,114), (15,16,5033), (32,3095,576), (262,384,10358), (371,372,1692), (3001,5063,11511), (3557,3558,1351), (5171,9734,3), (5999,7783,11257), (9732,9733,1351), (9738,9739,182)
X(9738) = (SW-2*S)*X(3)-SW*X(6)
Let A'B'C' be the inner-Vecten triangle of ABC. The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9738) and radius-squared R^2*(SW^2-4*S*SW+5*S^2)/(4*S^2). César Lozada, March 19, 2016. The circle is here named the 10th Lozada circle.
X(9738) lies on these lines: {3,6}, {5,642}, {25,5407}, {51,1600}, {114,489}, {485,8997}, {487,1352}, {637,7763}, {639,3788}, {640,7761}, {1583,3819}, {1584,5943}, {1599,3917}, {3155,5409}
X(9738) = midpoint of X(i),X(j) for these (i,j): (3,9732), (1161,9733)
X(9738) = reflection of X(9739) in X(3)
X(9738) = circumtangential-isogonal conjugate of X(33370)
X(9738) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(5058)
X(9738) = {X(371),X(372)}-harmonic conjugate of X(5058)
X(9738) = {X(182),X(9737)}-harmonic conjugate of X(9739)
X(9739) = (SW+2*S)*X(3)-SW*X(6)
Let A'B'C' be the outer-Vecten triangle of ABC. The parallel through B' to AC cuts BC in Ba and the parallel through C' to AB cuts BC in Ca. Build Ab, Cb, Ac and Bc similarly. These last six points lie on a circle with center X(9739) and radius-squared R^2*(SW^2+4*S*SW+5*S^2)/(4*S^2). César Lozada, March 19, 2016. The circle is here named the 11th Lozada circle.
X(9739) lies on these lines: {3,6}, {5,641}, {25,5406}, {51,1599}, {114,490}, {488,1352}, {638,7763}, {639,7761}, {640,3788}, {1583,5943}, {1584,3819}, {1600,3917}, {3156,5408}, {8577,8962}
X(9739) = midpoint of X(i),X(j) for these (i,j): (3,9733), (1160,9732)
X(9739) = reflection of X(9738) in X(3)
X(9739) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5062)
X(9739) = circumtangential-isogonal conjugate of X(33371)
X(9739) = {X(371),X(372)}-harmonic conjugate of X(5062)
X(9739) = {X(182),X(9737)}-harmonic conjugate of X(9738)
Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(2) wrt Pa, and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Then X(9740) = X(20) of A'B'C'. (Randy Hutson, March 25, 2016)
X(9740) lies on the cubic K764 and these lines: {2,6}, {20,543}, {30,3424}, {315,9166}, {390,4396}, {538,6194}, {754,3839}, {3091,7843}, {3523,5569}, {3543,3849}, {3600,4400}, {3785,7847}, {5056,7775}, {5286,7810}, {5503,6055}, {6392,7833}, {7486,7759}
X(9740) = reflection of X(i) in X(j) for these (i,j): (3543, 7620), (5503, 6055), (5569, 7780), (9741,3)
X(9740) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (183,1992,2), (599,7735,2), (1007,8860,2)
X(9741) lies on the cubic K764 and these lines: {2,2418}, {4,543}, {99,1285}, {193,8598}, {376,524}, {525,9168}, {538,3524}, {551,3923}, {631,7610}, {671,1007}, {1003,5032}, {2482,7735}, {3545,7620}, {3767,9167}, {5071,7615}, {5463,6782}, {5464,6783}, {6337,7857}, {7738,7801}, {7763,9166}, {7774,8591}
X(9741) = isogonal conjugate of X(39236)
X(9741) = crosssum of X(6) and X(33979)
X(9741) = crossdifference of every pair of points on line X(8644)X(14328)
X(9741) = reflection of X(i) in X(j) for these (i,j): (5485,2), (9740,3)
X(9741) = Thomson-triangle-isogonal-conjugate of X(1350)
X(9741) = X(4)-of-antipedal-triangle-of-X(2)
Artzt triangle: X(9742)-X(9775)
This preamble and centers X(9742)-X(9775) were contributed by César Eliud Lozada, March 25, 2016.
The A-Artzt parabola of a triangle ABC is the parabola tangent at B and C to the sidelines AB and AC, respectively. See William Gallatly, The Modern Geometry of the Triangle, 2nd ed. (London: Hodgson, 1913), p. 42.
The A-Artzt parabola has trilinear equation a2x2 - 4bcyvz = 0. Its axis is the line a(b2 - c2)x + b(a2 - b2 - 3c2)y - (a2 - 3b2 - c2)cz = 0, and its directrix is the line (a2 - b2 - c2)ax + 2bc2y + 2b2cz = 0.
The B- and C-Artzt parabolas are defined cyclically. The foci of the three parabolas are the vertices of the 2nd Brocard triangle.
The trilinear poles of the directrices of the Artzt parabolas lie on the cubics K675 and K676.
The triangle A'B'C' bounded by the directrices of the Artzt parabolas is here named the Artzt triangle. Coordinates for A' follow:
A' = -(3a4 + (b2 - c2)2)/(2a) : (b2(c2 + 2a2) + (a2 - c2)c2)/b : (c2(b2 + 2a2) + (a2 - b2)b2)/c (trilinears)
A' = SB*SC-(SB+SC)^2 : S^2+SC*SW : S^2+SB*SW (barycentrics)
Moreover, |B'C'| =SW*sqrt(3*SA+SW)/(3*S), and area(A'B'C') = SW^2/(6*S).
The appearance of (i,j) in the following list means that X(i)-of-A'B'C' = X(j)-of-ABC.
(2, 2), (3, 7610), (4,9770), (5, 9771), (6, 381), (20, 9740), (69, 376), (99, 98), (110, 111), (115, 114), (125, 126), (141, 549), (148, 147), (182, 7617), (193, 3543), (597, 5), (598, 262), (599, 3), (620, 6036), (671, 6054), (1316, 9169), (1352, 7618), (1992, 4), (2482, 6055), (2549, 1352), (3363, 3815), (3534, 8667), (3589, 547), (3618, 5071), (3734, 182), (5032, 3839), (5077, 599), (5104, 2080), (5112, 9127), (5182, 9166), (5468, 7417), (5476, 8176), (5485, 7710), (5640, 6032), (5642, 9172), (5972, 6719), (6189, 6039), (6190, 6040), (6722, 6721), (6772, 5617), (6775, 5613), (6776, 7620), (7426, 5913), (7472, 5912), (7840, 5999), (8352, 325), (8584, 3845), (8593, 671), (8597, 7840), (8599, 5996), (9123, 9185), (9125, 9189)
The appearance of (T,i) in the following list means that A'B'C' is perspective to T and that X(i) is the perspector.
(ABC, 262)
(1st anti-Brocard, 9772)
(anticomplementary, 9742)
(anti-McCay, 9773)
(1st Brocard, 9743)
(4th Brocard, 9744)
(1st circumperp, 9746)
(circumsymmedial, 98)
(1st Ehrmann, 9775)
(medial,7710)
(Macbeath, 9747)
(McCay, 9774)
(midheight, 9748)
(inner Napoleon, 9749)
(outer Napoleon, 9750)
(1st Neuberg, 9751)
(2nd Neuberg, 262)
(orthic, 9752)
(orthocentroidal, 9753)
(reflection, 9754)
(symmedial, 9755)
(tangential, 9756)
(inner Vecten, 9757)
(outer Vecten, 9758)
The Artzt triangle A'B'C' is directly similar to the circumsymmedial triangle and inversely similar to the 4th Brocard triangle, with center of inverse similitude X(9745). Also, A'B'C' is perspective to the 1st and 2nd Sharygin triangles, as well as all the Lucas(±1) triangles, except these two: the antipodal and homothetic Lucas(±1) triangles.
The locus of P such that the cevian triangle of P and A'B'C' are perspective is the cubic K677, and the locus of perspector is the cubic pK(X7735,X9756).
The locus of P such that the anticevian triangle of P and A'B'C' are perspective is the circumcubic pK(X7735,X6), which has barycentric equation ∑[ (SW SA+S^2) (SB y^2-SC z^2) x ]=0; this cubic passes through A,B,C, the vertices of the symmedial triangle and X(i) for these i: 2, 4, 6, 194, 262, 3168, 6776, 7735, 9307.
The locus of P such that the circumcevian triangle of P and A'B'C' are perspective is the quartic with trilinear equation ∑[ a*SA*(b*(SB*SW+S^2)*v-(SC*SW+S^2)*c*w)*u^3 ]=0; this quartic passes through A,B,C and X(i) for these i: 6, 262, 523, 1344, 1345, 5000, 5001.
The appearance of (T,i,j) in the following list means A'B'C' and T are orthologic with centers X(i) and X(j).
(ABC, 2, 2)
(anticomplementary, 2, 2)
(1st Brocard, 6054, 599)
(4th Brocard, 9759, 2)
(circummedial, 381, 2)
(Euler, 2, 381)
(inner Grebe, 2, 5861)
(outer Grebe, 2, 5860)
(Johnson, 2, 381)
(medial, 2, 2)
(inner Napoleon, 9760, 9761)
(outer Napoleon, 9762, 9763)
(1st Neuberg, 9764, 8667)
(2nd Neuberg, 9765, 9766)
(inner Vecten, 9767, 591)
(outer Vecten, 9768, 1991)
The Artzt and the 4th Brocard triangles are parallelogic with centers X(9769) and X(4).
An alternate construction for the Artzt triangle, A'B'C' follows. Let Pa be the parabola with focus A and directrix BC. Let La be the polar of X(3) wrt Pa, and define Lb and Lc cyclically. Then A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Also, the Artzt triangle is the 1st anti-Brocard triangle of the McCay triangle. (Randy Hutson, April 9, 2016)
X(9742) lies on these lines: {2,3167}, {20,325}, {114,193}, {147,3424}, {194,262}, {1007,5921}, {2896,3523}, {3543,6054}, {3926,7694}, {5056,7736}, {6289,6462}, {6290,6463}
X(9742) = anticomplement of X(7612)
X(9743) lies on these lines: {262,1513}, {316,7694}, {599,6054}, {1352,7710}, {6194,9742}
Let A'B'C' be the Artzt triangle. Let A" be the reflection of A in B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(9744). (Randy Hutson, December 10, 2016)
X(9744) lies on these lines: {2,98}, {3,315}, {4,39}, {5,7803}, {6,1513}, {20,7785}, {140,7868}, {183,3564}, {230,8550}, {371,6811}, {372,6813}, {376,3849}, {511,7774}, {574,2794}, {631,3788}, {754,8722}, {858,6795}, {1007,5207}, {1503,3815}, {1614,2909}, {1692,7735}, {2896,3523}, {3090,7834}, {3424,7608}, {3491,9729}, {3522,7900}, {3525,7874}, {4846,9513}, {5050,7792}, {5067,7852}, {5085,7778}, {5188,7759}, {5480,9300}, {5999,7777}, {6194,7779}, {6747,7378}, {7607,7612}, {7758,8149}, {8982,9739}
X(9744) = reflection of X(i) in X(j) for these (i,j): (4,5475)
X(9744) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,147,1352), (2,6776,98), (4,7709,2549), (4,7736,262), (114,182,2), (2548,8721,4), (2549,7694,4), (7710,7736,4)
X(9745) lies on these lines: {2,6}, {22,6781}, {111,381}, {378,1560}, {858,2549}, {1003,7664}, {1995,5475}, {5094,8426}, {5476,6791}, {7426,7737}, {7603,8585}, {8176,9172}
X(9745) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (111,6032,381)
X(9746) lies on these lines: {1,1447}, {2,165}, {3,9305}, {10,3424}, {40,6998}, {183,3886}, {262,3402}, {519,9740}, {740,3158}, {1281,3729}, {1376,3185}, {2784,3679}, {3598,5542}, {3667,4375}, {3755,7735}, {4307,5435}, {4312,7179}, {5745,7710}
X(9747) lies on these lines: {20,76}, {262,9307}, {2052,6353}
X(9748) lies on these lines: {2,51}, {4,3172}, {6,7710}, {20,7864}, {132,3087}, {385,3091}, {3523,7875}, {3545,9740}, {3832,6249}, {5068,7900}, {5480,7735}, {5921,7766}, {6530,6995}, {6811,8416}, {6813,8396}, {7738,8719}, {7774,9742}
X(9749) lies on these lines: {3,114}, {4,6115}, {13,262}, {15,1513}, {16,9744}, {18,98}, {147,627}, {616,9742}, {618,7710}, {1503,5617}, {2782,6294}, {3389,6811}, {3390,6813}, {3642,9743}, {5463,6054}, {5473,5979}, {5999,8291}, {6108,6770}, {6776,6782}
X(9749) = reflection of X(9750) in X(114)
X(9750) lies on these lines: {3,114}, {4,6114}, {14,262}, {15,9744}, {16,1513}, {17,98}, {147,628}, {617,9742}, {619,7710}, {1503,5613}, {2782,6581}, {3364,6811}, {3365,6813}, {3643,9743}, {5464,6054}, {5474,5978}, {5999,8292}, {6109,6773}, {6776,6783}
X(9750) = reflection of X(9749) in X(114)
X(9751) lies on these lines: {2,6030}, {3,83}, {4,6704}, {5,8725}, {20,6249}, {98,5092}, {140,6287}, {182,6194}, {325,3530}, {549,6054}, {631,6292}, {732,5085}, {754,3524}, {1385,7977}, {1587,8993}, {2896,3523}, {5206,7736}, {6274,6316}, {6275,6312}, {7709,7781}
X(9751) = X(83)-Gibert-Moses centroid; see the preamble just before X(21153)X(9752) lies on these lines: {2,51}, {4,230}, {5,3785}, {32,7694}, {98,3424}, {114,193}, {1007,1351}, {1513,6776}, {2896,5056}, {3091,7793}, {3543,6055}, {3545,7610}, {5254,8719}, {5286,7709}, {5304,9744}, {6353,6525}, {6459,6813}, {6460,6811}, {7486,7938}, {7755,8721}
X(9752) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9748,262), (1513,7735,6776)
X(9753) = X(4)+2*X(32)
X(9753) lies on these lines: {2,51}, {3,7792}, {4,32}, {5,183}, {6,1513}, {20,7797}, {25,3425}, {61,9749}, {62,9750}, {114,576}, {147,7766}, {230,5017}, {325,1351}, {371,6813}, {372,6811}, {385,1352}, {626,3090}, {631,6680}, {754,3545}, {760,5603}, {1503,5306}, {1992,6054}, {3095,5976}, {4045,8722}, {4769,5818}, {5028,7736}, {5067,7867}, {5071,7818}, {5171,7791}, {5188,7834}, {5304,6776}, {5319,8721}, {5939,6321}, {5999,7806}, {7374,8982}, {7709,7739}
X(9753) = X(2)-of-X(4)X(371)X(372)
X(9753) = centroid of {X(4),X(371),X(372)}
X(9753) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7735,98), (6,1513,9744), (114,576,7774), (9748,9752,262)
X(9754) lies on these lines: {2,51}, {4,5206}, {98,5033}, {230,8550}, {376,5461}, {631,4045}, {3090,7815}, {3424,7607}, {3545,8182}, {3767,7709}, {5056,7898}, {5067,6292}, {6250,6811}, {6251,6813}, {6722,8722}
X(9754) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9752,262), (262,9752,9753), (7612,7710,98)
X(9755) lies on these lines: {2,3167}, {3,194}, {4,3172}, {6,98}, {25,3168}, {140,7881}, {147,7806}, {182,183}, {230,8550}, {263,6038}, {523,878}, {631,3933}, {732,5085}, {1003,2782}, {1351,5999}, {1352,7792}, {1353,7774}, {1503,5306}, {1513,6776}, {1656,7875}, {2794,5309}, {3053,8719}, {3524,9740}, {3684,9746}, {3767,7694}, {5965,7788}, {6054,8787}, {6811,7583}, {6813,7584}
X(9755) = perspector of ABC and cross-triangle of ABC and Artzt triangle
X(9755) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98,262,9756), (230,8550,9744), (3424,5304,9748), (3424,9748,4), (5999,7766,1351), (6776,7735,1513), (6776,9752,7710), (7710,7735,9752), (7710,9752,1513)
Let A'B'C' be the Artzt triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(9756). (Randy Hutson, April 9, 2016)
X(9756) lies on the cubic K677 and these lines: {2,154}, {3,3734}, {4,230}, {5,7694}, {6,98}, {30,7610}, {140,8721}, {183,1350}, {264,9747}, {371,6222}, {372,6399}, {381,2794}, {511,8667}, {1007,5921}, {1352,7778}, {1376,3185}, {1513,9754}, {1656,6287}, {1661,5020}, {2352,5307}, {2782,8716}, {3425,7669}, {3815,6776}, {3818,6036}, {5013,7709}, {5480,7735}, {5984,7777}, {5989,8295}, {6312,9733}, {6316,9732}, {7736,8550}
X(9756) = midpoint of X(i),X(j) for these (i,j): {3424,7710}
X(9756) = reflection of X(i) in X(j) for these (i,j): (7694,5), (8719,3)
X(9756) = complement of X(7710)
X(9756) = perspector of Artzt triangle and cross-triangle of ABC and Artzt triangle
X(9756) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3424,7710), (4,7612,9752), (98,262,9755), (183,5999,1350), (262,9755,6), (7612,9752,230), (98,262,9756)
X(9757) lies on these lines: {3,7694}, {262,485}, {371,9752}, {486,7612}, {488,9742}, {641,7710}, {6250,7000}, {6289,9756}
X(9758) lies on these lines: {3,7694}, {262,486}, {372,9752}, {485,7612}, {487,9742}, {642,7710}, {1991,3564}, {6251,7374}, {6290,9756}
X(9759) lies on these lines: {2,98}, {74,9100}, {111,381}, {376,7664}, {3543,7665}, {5655,5913}
X(9759) = X(111)-of-Artzt-triangle
X(9759) = anti-Artzt-to-Artzt similarity image of X(110)
X(9760) lies on these lines: {2,14}, {30,9750}, {114,381}, {383,530}, {524,5613}, {542,9749}, {671,6115}, {1992,6783}, {3815,6775}, {5979,9116}, {5982,8592}, {6782,8593}
X(9760) = reflection of X(9762) in X(114)
X(9761) lies on these lines: {2,6}, {3,531}, {13,8176}, {14,543}, {15,7622}, {16,3849}, {62,7775}, {381,530}, {532,5055}, {533,5054}, {538,3106}, {542,9749}, {620,9113}, {2482,5471}, {3411,7759}, {3643,5460}, {4851,5242}, {5077,6775}, {5464,5569}, {5475,9115}
X(9761) = midpoint of X(5858) and X(9763)
X(9761) = anticomplement of X(33475)
X(9761) = reflection of X(i) in X(j) for these (i,j): (5859,9763), (9763,2)
X(9762) lies on these lines: {2,13}, {30,9749}, {114,381}, {524,5617}, {531,1080}, {542,9750}, {671,6114}, {1992,6782}, {3815,6772}, {5476,9761}, {5978,9114}, {5983,8592}, {6783,8593}
X(9762) = reflection of X(9760) in X(114)
X(9763) lies on these lines: {2,6}, {3,530}, {13,543}, {14,8176}, {15,3849}, {16,7622}, {61,7775}, {381,531}, {532,5054}, {533,5055}, {538,3107}, {542,9750}, {620,9112}, {2482,5472}, {3412,7759}, {3642,5459}, {4851,5243}, {5077,6772}, {5463,5569}, {5475,9117}, {5476,9760}
X(9763) = midpoint of X(i),X(j) for these {i,j}: {5859,9761}
X(9763) = reflection of X(i) in X(j) for these (i,j): (5858,9761), (9761,2)
X(9763) = anticomplement of X(33474)
X(9764) lies on these lines: {2,39}, {262,698}, {385,5033}, {511,7710}, {732,5085}, {5182,5976}, {5969,6054}, {5999,7781}
X(9764) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6309,8149,76)
X(9765) lies on these lines: {2,32}, {262,732}, {381,9764}, {8290,8781}
X(9765) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2548,6292,83)
X(9766) lies on the cubic K395 and these lines: {2,6}, {3,754}, {5,7758}, {25,1634}, {30,8716}, {39,7776}, {83,7871}, {99,7926}, {114,1351}, {194,7773}, {262,732}, {315,5013}, {381,538}, {382,7781}, {543,3830}, {574,7845}, {620,1384}, {625,7798}, {626,9605}, {1003,7799}, {1078,7949}, {1350,9744}, {1506,7855}, {1656,7751}, {1975,7785}, {2548,3933}, {3053,7762}, {3526,7780}, {3534,3849}, {3564,9756}, {3705,4363}, {3788,7838}, {3926,7745}, {3934,7916}, {4361,7179}, {5007,7888}, {5024,7761}, {5041,7867}, {5066,7615}, {5102,9753}, {5319,8361}, {5475,7813}, {5969,6054}, {6390,7737}, {6683,7896}, {7618,8703}, {7746,7890}, {7752,7754}, {7753,7801}, {7757,7809}, {7760,7814}, {7769,7877}, {7770,7796}, {7772,7821}, {7783,7900}, {7786,7879}, {7787,7947}, {7804,7908}, {7805,7862}, {7808,7895}, {7815,7882}, {7824,7946}, {7836,7921}, {7839,7851}, {7854,9698}, {7878,7909}, {7894,7899}
X(9766) = reflection of X(i) in X(j) for these (i,j): (381,7775), (8667,2)
X(9766) = anticomplement of X(13468)
X(9766) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1992,5306), (2,7788,599), (2,7840,7788), (2,8667,7610), (6,325,7778), (39,7776,7784), (39,7903,7776), (83,7871,7881), (193,1007,230), (194,7941,7773), (325,7774,6), (491,5861,1991), (591,1991,6), (3329,7897,7868), (5860,5861,193), (7752,7905,7754), (7757,7809,7841), (7759,7764,3), (7760,7814,7887), (7762,7763,3053), (7772,7821,7866), (7777,7779,183), (7781,7843,382), (7785,7906,1975), (7786,7917,7879), (7796,7858,7770), (7799,7812,1003), (7839,7912,7851)
X(9767) lies on these lines: {2,371}, {381,9768}, {1007,6565}, {1328,8781}, {6281,6813}, {6290,9766}, {7388,7909}
X(9767) = X(1328)-of-Artzt-triangle
X(9768) lies on these lines: {2,372}, {381,9767}, {1007,6564}, {1327,8781}, {1991,3564}, {6278,6811}, {6289,9766}, {7389,7909}
X(9768) = X(1327)-of-Artzt-triangle
X(9769) lies on these lines: {67,230}, {74,98}, {110,183}, {111,6325}, {125,7735}, {385,895}, {542,7710}, {2781,9756}, {2854,8667}, {5095,7736}, {5622,9755}, {7610,9759}
X(9770) lies on these lines: {2,6}, {4,543}, {30,7710}, {32,9167}, {262,538}, {263,6786}, {376,3849}, {381,7620}, {511,9743}, {530,9749}, {531,9750}, {598,7799}, {631,5569}, {754,3524}, {2482,7737}, {2548,7801}, {3090,7758}, {3529,7843}, {3533,7780}, {3543,8716}, {3839,9742}, {3926,8370}, {5067,7751}, {5071,7617}, {5461,7798}, {6337,7785}, {7738,7841}, {7752,9166}, {7763,7812}, {7776,8359}, {7810,7903}, {7814,7827}, {7833,7941}, {7858,7870}, {8360,9605}
X(9770) = midpoint of X(i),X(j) for these (i,j): (4,9741), (5503,6054), (5569,7759)
X(9770) = reflection of X(i) in X(j) for these (i,j): (376,7618), (5485,7615), (7615,8176), (7620,381), (8182,7622), (9740,7610)
X(9770) = anticomplement of X(7610)
X(9770) = complement of X(9740)
X(9770) = X(147)-of-McCay-triangle
X(9770) = anti-Artzt-to-Artzt similarity image of X(1992)
X(9770) = X(1992)-of-Artzt-of-Artzt-triangle
X(9770) = orthoptic-circle-of-Steiner-inellipe-inverse of X(37745)
X(9770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1992,7735), (2,7774,1992), (2,7840,69), (2,9740,7610), (597,7778,2), (599,3815,2), (1007,1992,2), (1007,7774,7735), (3545,5485,7615), (7615,8176,3545), (7622,8182,3524), (7777,7840,2)
X(9771) lies on these lines: {2,6}, {5,543}, {30,7622}, {140,5569}, {381,7618}, {547,7617}, {549,3849}, {754,1153}, {1506,8369}, {2482,3363}, {3090,9741}, {3788,8367}, {5054,8182}, {5055,7615}, {5071,7620}, {5215,7753}, {5254,9166}, {5503,7608}, {6055,8550}, {7769,8370}, {7808,8365}, {7827,9606}, {7862,8360}, {7935,8359}
X(9771) = midpoint of X(i),X(j) for these (i,j): (381,7618), (5569,7775), (7610,9770), (7620,8716), (7622,8176)
X(9771) = reflection of X(i) in X(j) for these (i,j): (549,7619), (5569,140), (7617,547)
X(9771) = complement of X(7610)
X(9771) = X(114)-of-McCay-triangle
X(9771) = anti-Artzt-to-Artzt similarity image of X(597)
X(9771) = centroid of BaCaCbAbAcBc used in construction of 5th Lozada circle; see X(9734)
X(9771) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1007,599), (2,3815,597), (2,7774,8860), (2,9770,7610), (2482,7603,3363), (3054,7777,3629)
X(9772) lies on these lines: {2,2782}, {69,147}, {76,9743}, {98,5092}, {99,5999}, {114,262}, {511,6054}, {542,8592}, {3663,5988}, {4027,9755}, {5939,8350}, {5969,9770}, {5978,9749}, {5979,9750}, {5980,8292}, {5981,8291}, {5989,8295}, {6248,7783}, {8782,9742}
X(9772) = midpoint of X(i),X(j) for these {i,j}: {147,6194}
X(9772) = reflection of X(i) in X(j) for these (i,j): (262,114), (1916,262), (6194,5976)
X(9772) = isotomic conjugate of isogonal conjugate of X(33876)
These triangles are inversely similar with center of inverse similitude X(2).
X(9773) lies on these lines: {8289,8786}, {8591,9770}, {8594,9762}, {8595,9760}, {8724,9742}
These triangles are inversely similar with center of inverse similitude X(2).
X(9774) lies on these lines: {2,1495}, {3,6054}, {30,262}, {98,6233}, {325,8703}, {376,3849}, {524,9764}, {542,8592}, {1503,9743}, {3098,7840}, {3524,7710}, {5054,9751}, {6055,7607}, {6776,9740}, {7470,7775}
X(9774) = X(76)-of-Artzt-triangle
X(9774) = X(1916)-of-McCay-triangle
X(9775) lies on these lines: {2,98}, {30,1296}, {99,7417}, {111,2782}, {126,2794}, {511,5971}, {2698,9066}, {3564,5913}
X(9775) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6054,9759)
Conway triangle: X(9776)-X(9812)
This preamble and centers X(9776)-X(9812) were contributed by César Eliud Lozada, April 2, 2016.
Let ABC be a a triangle with opposite sidelengths a,b,c, respectively. Let AB be the point P on line AB such that |BP| = b and B lies between A and P. Define BC and CA cyclically. Likewise, let AC be the point Q on line AC such that |CP| = c and C lies between A and P. Define BA and CB cyclically. It is known that these six points lie on the Conway circle, with center X(1) and radius-squared r2 + s2, where r = inradius and s = semiperimeter of ABC.
Let A' = BABC∩CACB, and define B' and C' cyclically. The triangle A'B'C' is here named the 2nd Conway triangle of ABC. Let A''B''C'' be the (1st) Conway triangle, introduced at X(7411), and let A*B*C* be the intouch triangle; A' = reflection of A'' in A*, and likewise for B' and C', as noted by Peter Moses, April 2, 2016.
area(A'B'C') = 8Rs
|B'C'| = sqrt(8*S*R*a/((s-b)*(s-c)))
A' = - (a + b + c) : a + b - c : a - b + c (barycentric coordinates)
The vertices A', B', C' lie on these cubics: K007 (Lucas cubic), K028, K461, K651.
Another construction of AB and AC follows. Let pa be the parabola tangent to BC at the A-cevian-trace-of-X(75) and also tangent to the sidelines AB and AC. AB and AC are the touchpoints of pa with AB and AC.
A'B'C' is perspective to ABC with perspector X(7); it is also perspective with perspector X(8) to these triangle: anticomplementary, Fuhrmann and outer-Garcia.
A'B'C' and Fuhrmann triangles are inversely similar, with X(9782) as center of inverse similitude.
The appearance of (T,i,j) in the following list means that A'B'C' and T are homothetic with X(i) as homothetic center and X(j) as and endo-homothetic center:
(Ascella, 9776, 9777)The appearance of (T,i,j) in the following list means that A'B'C' and T are orthologic with centers X(i) and X(j):
(ABC, 962, 1)The appearance of (T,i,j) in the following list means that A'B'C' and T are parallelogic with centers X(i) and X(j):
(Fuhrmann, 9809, 4)
(1st Parry, 962, 9810)
(2nd Parry, 962, 9811)
(2nd Sharygin, 8, 2254)
The appearance of (i,j) in the following list means that X(i)-of-2nd-Conway-triangle = X(j)-of-ABC:
(1,9807), (2,9812), (3,962), (4,8), (5,4), (6,7), (25,329), (53,69), (65,7057), (110,9809), (115,150), (186,5180), (187,5195), (230,4872), (235,3436), (317,9801), (403,5080), (427,3434), (428,3681), (468,5057), (546,355), (1843,144), (1859,556), (1990,320), (2501,693), (5254,6604), (6116,622), (6525,8055), (6530,4645), (6748,75), (7745,85)
The locus of P such that the cevian triangle of P and A'B'C' are perspective is the circumcubic pK(X274,X314), with barycentric equation
∑[a*(b+c)*((s-b)*y-(s-c)*z)*x^2] = 0.
This cubic passes through X(7), X(69), X(75), X(86), X(286), X(309), X(314), X(8822) and the vertices of the cevian triangle of X(314). The locus of the perspector is the Lucas cubic, K007.The locus of P such that the anticevian triangle of P and A'B'C' are perspective is the cubic K034, and the locus of the perspectors is the Lucas cubic, K007.
The 2nd Conway triangle is also the extraversion triangle of X(8), and the anticomplement of the excentral triangle. (Randy Hutson, April 9, 2016)
X(9776) = 6*R*X(2)+(4*R+r)*X(7)
X(9776) lies on these lines: {1,6904}, {2,7}, {3,962}, {4,5439}, {8,443}, {20,8726}, {69,4886}, {81,277}, {85,189}, {86,1817}, {92,1119}, {175,3083}, {176,3084}, {200,5542}, {284,8025}, {306,4869}, {354,2550}, {377,938}, {388,3812}, {390,4666}, {391,4001}, {404,5703}, {452,4292}, {474,1260}, {497,3742}, {612,4310}, {614,4307}, {631,5758}, {940,4000}, {948,1407}, {1001,3474}, {1125,4295}, {1210,5177}, {1376,3475}, {1466,3485}, {1467,3600}, {1473,4223}, {1519,6847}, {2095,5657}, {2096,6913}, {2476,5704}, {2999,3664}, {3090,5811}, {3187,4402}, {3601,3622}, {3650,6675}, {3666,4648}, {3671,8583}, {3672,5287}, {3752,4277}, {3824,6856}, {3889,6764}, {3914,7613}, {3925,4860}, {3945,5256}, {4187,5714}, {4190,4313}, {4208,6734}, {4344,7191}, {4383,4644}, {4423,5698}, {4656,4862}, {5045,5082}, {5122,5180}, {5290,8582}, {5550,6857}, {5708,5815}, {5743,7232}, {5768,6826}, {5770,6881}, {5804,6850}, {5886,6935}, {6705,8227}, {6919,9612}
X(9776) = anticomplement of X(7308)
X(9776) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7,329), (2,57,5744), (2,144,3305), (2,226,5748), (2,3218,5273), (57,142,2), (226,5437,2), (377,938,5175), (443,942,8), (1056,3753,8), (3306,5249,2), (3742,5880,497), (3752,4675,5712), (5219,6692,2), (5437,6173,226), (9612,20011,6919)
X(9777) = 2*SW*X(6)+(6*R^2-SW)*X(25)
X(9777) lies on these lines: {2,1351}, {3,143}, {4,3527}, {5,6515}, {6,25}, {22,5050}, {52,7395}, {54,3517}, {193,7392}, {237,9605}, {343,7539}, {373,5102}, {389,1593}, {393,6755}, {394,576}, {428,6776}, {493,3102}, {494,3103}, {511,7484}, {569,9715}, {575,3796}, {578,3515}, {1112,5622}, {1181,5198}, {1184,5052}, {1597,5890}, {1598,7592}, {1899,5064}, {1993,5020}, {1994,1995}, {2261,2355}, {2979,5644}, {3066,5097}, {3087,6524}, {3155,3312}, {3156,3311}, {3564,6997}, {3618,7499}, {5200,7582}, {6243,7393}, {6391,7398}, {6423,8576}, {6424,8577}, {7506,9703}
X(9777) = isogonal conjugate of X(36948)
X(9777) = polar conjugate of isotomic conjugate of X(36751)
X(9777) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,51,25), (576,5943,394), (1899,5480,5064), (1993,5020,6090), (1993,5640,5020), (1994,1995,3167), (3060,5422,3), (5020,5093,1993), (5093,5640,6090)
X(9778) = (4*R+r)*X(7)-4*(R+r)*X(55) = X(8)+2*X(20)
Let A' be the reflection of A in the A-excenter, and define B' and C' cyclically. Then X(9778) is the centroid of A'B'C'. (Randy Hutson, April 9, 2016)
The endo-homothetic center of these triangles is X(3060)
X(9778) lies on these lines: {1,3522}, {2,165}, {3,962}, {4,2355}, {7,55}, {8,20}, {10,3146}, {21,5584}, {30,5657}, {31,5222}, {35,4295}, {42,1742}, {46,938}, {57,390}, {65,4313}, {81,4229}, {100,329}, {144,200}, {145,4297}, {170,2340}, {190,5423}, {197,1633}, {210,6172}, {226,5281}, {347,7070}, {354,8236}, {355,3529}, {376,517}, {382,5818}, {484,4302}, {497,1155}, {518,5918}, {527,3158}, {548,1482}, {550,944}, {612,1721}, {651,7074}, {672,5838}, {901,2723}, {946,3523}, {950,5128}, {952,3534}, {968,5308}, {971,3681}, {986,4339}, {990,3920}, {1040,4318}, {1088,3599}, {1125,9589}, {1293,1311}, {1319,4345}, {1376,5698}, {1385,3528}, {1479,5704}, {1490,4420}, {1657,5690}, {1697,3600}, {1698,3832}, {1708,5809}, {1709,3219}, {1766,3161}, {1770,3085}, {1788,6284}, {1836,5218}, {2077,5180}, {2093,4304}, {2098,6049}, {2550,4640}, {2646,4323}, {2801,4661}, {2807,2979}, {2938,4418}, {2941,4427}, {3052,4000}, {3057,4308}, {3091,6684}, {3100,8270}, {3160,7056}, {3336,4309}, {3339,4314}, {3359,6987}, {3419,5775}, {3428,6909}, {3434,5744}, {3485,5217}, {3524,5886}, {3543,5587}, {3617,5059}, {3622,4301}, {3634,5068}, {3651,5758}, {3666,4344}, {3667,6546}, {3672,5269}, {3729,7172}, {3749,4310}, {3826,7965}, {3868,7957}, {3870,5732}, {3911,5274}, {3916,5082}, {3928,5853}, {3929,5686}, {3951,7992}, {4293,5119}, {4329,5285}, {4511,6282}, {4689,5712}, {4781,6327}, {5057,5748}, {5080,6925}, {5178,5787}, {5205,8055}, {5250,6904},
X(9778) = reflection of X(i) in X(j) for these (i,j): (2,165), (962,5603), (3241,5731), (3543,5587), (5603,3), (5731,376)
X(9778) = anticomplement of X(1699)
X(9778) = midpoint of X(i),X(j) for these {i,j}: {5603,6361}
X(9778) = X(2)-of-polar-triangle-of AC-incircle
X(9778) = perspector of Hutson extouch triangle and cross-triangle of ABC and Hutson extouch triangle
X(9778) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,962,3616), (20,40,8), (35,4295,5703), (46,4294,938), (55,3474,7), (497,1155,5435), (946,3523,5550), (1621,9776,3616), (1836,5218,5226), (2550,4640,5273), (3617,5059,5691), (3911,9580,5274), (4297,7991,145), (4301,7987,3622), (5732,7994,3870), (5759,7580,329), (6244,7580,100)
X(9779) = (4*R+r)*X(7)-4*(R-2*r)*X(11)
The endo-homothetic center of these triangles is X(5640).
X(9779) lies on these lines: {1,3832}, {2,165}, {4,1385}, {5,962}, {7,11}, {8,908}, {10,5068}, {12,9785}, {20,5550}, {40,5056}, {119,9802}, {144,5231}, {145,3854}, {149,5660}, {226,5274}, {329,5817}, {355,3855}, {381,952}, {390,5219}, {496,5714}, {497,3748}, {515,3839}, {517,3545}, {546,944}, {938,6870}, {990,7292}, {1056,7743}, {1125,3146}, {1479,5703}, {1482,3850}, {1483,3856}, {1519,6844}, {1656,6361}, {1750,4666}, {1788,7173}, {1836,5435}, {1837,4323}, {2550,5087}, {2807,5640}, {3161,5510}, {3240,5400}, {3434,5748}, {3487,9669}, {3522,3624}, {3543,3576}, {3579,5067}, {3617,4301}, {3622,5691}, {3634,9589}, {3816,7965}, {3829,5852}, {3843,5901}, {3848,5918}, {3851,5818}, {3869,5806}, {3873,5927}, {4197,7958}, {4292,5556}, {4295,5704}, {4305,5443}, {4308,5229}, {4313,5225}, {4342,5726}, {4345,5252}, {4423,7411}, {5057,5744}, {5059,7987}, {5066,5790}, {5072,5690}, {5080,6957}, {5121,7613}, {5222,7384}, {5265,9579}, {5273,5832}, {5281,9580}, {5284,7580}, {5758,6990}, {5770,6841}, {5804,6866}, {6244,9342}, {6684,7486}, {6828,7681}, {6845,9782}, {6945,7680}, {8085,9793}, {8086,9795}, {8228,9789}, {8229,9791}, {8377,9783}, {8378,9787}, {8727,9776}
X(9779) = midpoint of X(i),X(j) for these {i,j}: {1699,7988}
X(9779) = reflection of X(i) in X(j) for these (i,j): (2,7988), (7988,3817)
X(9779) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5886,5731), (20,8227,5550), (946,3091,8), (1699,3817,2), (2550,5087,5328), (4295,7741,5704), (4301,7989,3617), (5731,5886,3616)
X(9780) = 2*X(1)-9*X(2) = (4*R+r)*X(7)-4*(2*r+R)*X(12)
X(9780) lies on these lines: {1,2}, {3,5260}, {4,2355}, {5,962}, {7,12}, {9,5128}, {11,9710}, {20,5587}, {21,1376}, {40,3091}, {44,966}, {45,1213}, {46,3219}, {55,5047}, {57,5261}, {63,4208}, {65,3740}, {69,3844}, {72,5775}, {75,3701}, {80,4305}, {89,1224}, {100,405}, {119,5811}, {140,944}, {142,5686}, {149,6702}, {165,3146}, {192,3842}, {210,3812}, {281,5125}, {329,442}, {341,4003}, {344,4026}, {346,5257}, {354,4662}, {355,631}, {377,1155}, {381,6361}, {388,5435}, {390,6666}, {391,5750}, {404,958}, {443,3436}, {468,7718}, {474,2975}, {515,3523}, {516,3832}, {517,3090}, {518,3619}, {726,4772}, {748,5255}, {750,5247}, {756,986}, {942,3681}, {946,5056}, {950,5281}, {952,3526}, {956,5253}, {960,3698}, {984,4699}, {993,4188}, {1001,3871}, {1010,5235}, {1220,5737}, {1320,6667}, {1329,2476}, {1385,3525}, {1388,7294}, {1392,7317}, {1478,5445}, {1482,3628}, {1512,6847}, {1574,5283}, {1621,5687}, {1656,5603}, {1697,5274}, {1699,5068}, {1706,5250}, {1829,8889}, {1837,4313}, {1861,4194}, {1902,6622}, {1995,8193}, {2049,5278}, {2292,4903}, {2475,3647}, {2478,2550}, {2886,4193}, {3035,6224}, {3144,7102}, {3245,3814}, {3295,5284}, {3303,8167}, {3339,3947}, {3416,3618}, {3421,5828}, {3428,6915}, {3434,5084}, {3474,3648}, {3476,5433}, {3486,5432}, {3522,5691}, {3533,7967}, {3600,3911}, {3620,3751}, {3654,5071}, {3670,7226}, {3672,3790}, {3678,5902}, {3696,4687}, {3714,4734}, {3739,5772}, {3742,3889}, {3753,3869}, {3754,5692}, {3817,7991}, {3822,5905}, {3833,4430}, {3841,4295}, {3854,5493}, {3873,3921}, {3874,3956}, {3877,4731}, {3878,3968}, {3901,4134}, {3913,4423}, {3916,9352}, {3918,5903}, {3953,9335}, {4015,5883}, {4189,5251}, {4298,5726}, {4301,7988}, {4308,5252}, {4346,4357}, {4359,4385}, {4363,4748}, {4402,4657}, {4419,4708}, {4445,6707}, {4460,5564}, {4470,4643}, {4472,4644}, {4478,4916}, {4488,7229}, {4647,4671}, {4652,5234}, {4690,4798}, {4848,5219}, {5067,5734}, {5070,5901}, {5086,6857}, {5090,6353}, {5126,5176}, {5174,7498}, {5178,5722}, {5192,5263}, {5259,8715}, {5325,9579}, {5592,6544}, {5708,5815}, {5745,6904}, {5758,6829}, {5766,6594}, {5768,6989}, {5794,6910}, {5804,6887}, {5880,6172}, {6636,8185}, {6683,7976}, {6721,7970}, {6722,7983}, {6723,7984}, {6991,7680}, {7378,7713}, {7485,9798}, {7486,8227}, {8087,9793}, {8088,9795}, {8230,9789}, {8380,9783}, {8381,9787}
X(9780) = midpoint of X(i),X(j) for these {i,j}: {3622,4678}, {7989,9588}
X(9780) = reflection of X(i) in X(j) for these (i,j): (8,4678), (3622,3624), (3832,7989)
X(9780) = isotomic conjugate of X(28626)
X(9780) = complement of X(3622)
X(9780) = anticomplement of X(3624)
X(9780) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,5550), (1,10,3617), (1,1698,3634), (1,3617,8), (1,3625,145), (1,3626,3621), (1,3634,2), (1,3679,3625), (1,5550,3616), (2,8,3616), (2,10,8), (2,145,1125), (2,3617,1), (2,3622,3624), (2,3661,5308), (2,4678,3622), (5,962,9779), (5,5657,962), (8,3616,3241), (8,5550,1), (10,551,4691), (10,1125,3679), (10,1698,2), (10,3178,8013), (10,3244,4745), (10,3624,4678), (10,3634,1), (10,3828,1698), (10,8582,6734), (12,1788,7), (12,3826,4197), (65,3740,3876), (140,5790,944), (145,3679,8), (210,3812,3868), (355,631,5731), (377,2551,5080), (405,9709,100), (443,5791,5744), (474,9708,2975), (551,3632,3623), (551,4691,3632), (612,1722,5262), (942,3697,3681), (958,4413,404), (1125,3625,1), (1125,3679,145), (1213,2345,5296), (1329,3925,2476), (1656,5690,5603), (1706,7308,5250), (1737,3085,938), (1837,5218,4313), (2345,5296,3161), (2975,9342,474), (3244,4745,4668), (3617,3621,3626), (3617,3634,5550), (3621,3626,8), (3623,4691,8), (3636,4669,3633), (3753,5044,3869), (3826,9711,12), (3911,9578,3600), (4015,5883,5904), (5044,6856,5748), (5252,7288,4308), (5587,6684,20), (6734,7080,8)
X(9781) = X(4)+6*X(51) = 2*(6*R^2-SW)*X(25)-(5*R^2-2*SW)*X(54)
X(9781) lies on these lines: {2,5446}, {3,5640}, {4,51}, {5,3060}, {6,1173}, {20,5462}, {23,569}, {25,54}, {52,3091}, {68,7394}, {143,381}, {156,7545}, {373,3525}, {378,1192}, {382,5946}, {511,3090}, {546,568}, {578,3518}, {631,5943}, {973,7547}, {1112,7507}, {1154,3851}, {1199,6759}, {1216,5056}, {1594,5480}, {1598,7592}, {1656,2979}, {1993,7529}, {2888,7533}, {3146,9730}, {3522,5892}, {3529,9729}, {3533,6688}, {3542,6403}, {3545,5562}, {3574,6242}, {3580,7403}, {3628,7998}, {3843,6102}, {3855,5907}, {3917,5067}, {5012,7517}, {5055,6101}, {5068,5891}, {5169,5449}, {5422,7387}, {5476,7552}, {5752,6920}
X(9781) = reflection of X(i) in X(j) for these (i,j): (7999,3090)
X(9781) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,51,3567), (4,389,6241), (4,3567,5890), (143,381,5889), (389,6241,5890), (1173,1614,6), (1598,9777,7592), (3567,6241,389)
X(9782) = (4*R+r)*(R-2*r)*X(7)-4*R*(R+2*r)*X(12)
X(9782) lies on these lines: {2,191}, {3,962}, {7,12}, {8,2891}, {79,3833}, {355,5885}, {942,5178}, {1125,5131}, {1158,3306}, {2475,5883}, {3189,3241}, {3648,5047}, {3701,7321}, {3812,5080}, {3951,5586}, {4295,5443}, {5249,5535}, {5704,8070}, {5811,6829}, {5880,7671}, {6256,6839}, {6845,9779}, {7989,7997}
X(9782) = anticomplement of X(5506)
X(9783) lies on these lines: {2,363}, {7,8113}, {20,8111}, {329,5934}, {516,8140}, {3616,8109}
X(9784) lies on these lines: {511,9788}
X(9785) = 8*R*X(1)-(4*R+r)*X(7)
X(9785) lies on these lines: {1,7}, {2,1697}, {4,7320}, {8,210}, {10,5274}, {11,9710}, {12,9779}, {40,5435}, {55,404}, {56,9778}, {72,5809}, {144,6762}, {145,329}, {149,5175}, {165,5265}, {220,5838}, {355,1000}, {388,5919}, {392,5082}, {405,5766}, {496,5657}, {517,938}, {518,20016}, {519,5815}, {529,2098}, {946,5226}, {952,5811}, {999,6361}, {1125,5281}, {1191,5222}, {1219,3729}, {1420,3522}, {1482,3488}, {1616,4000}, {1699,5261}, {2136,3452}, {2269,5296}, {3085,6953}, {3086,5119}, {3090,7743}, {3091,9614}, {3146,9580}, {3149,3295}, {3189,5289}, {3303,3485}, {3304,3474}, {3361,5493}, {3434,3890}, {3476,6284}, {3487,6767}, {3528,5126}, {3543,9613}, {3601,3622}, {3617,9581}, {3621,5727}, {3623,5905}, {3625,8275}, {3649,8162}, {3680,5795}, {3832,9578}, {3895,5328}, {3913,8169}, {5048,5180}, {5129,9623}, {5218,5550}, {5225,5252}, {5250,5273}, {5584,7677}, {5759,8158}, {5818,9669}, {6260,7966}, {6735,6919}, {6736,8165}, {7673,7957}, {7676,8273}, {7679,7958}
X(9785) = reflection of X(i) in X(j) for these (i,j): (8,2551), (938,1058), (3600,1)
X(9785) = anticomplement of X(1706)
X(9785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,20,4308), (1,390,4313), (1,962,7), (1,1742,4322), (1,4294,5731), (1,4309,4305), (1,5731,6049), (1,9589,4298), (8,8055,341), (175,176,269), (390,4342,4345), (496,5657,5704), (497,3057,8), (950,7962,145), (2098,3058,3486), (2098,3486,3241), (2478,3885,8), (3295,5603,5703), (4313,4345,1), (4323,8236,1)
X(9786) = 2*(SW-4*R^2)*X(3)-SW*X(6)
X(9786) lies on these lines: {3,6}, {4,64}, {24,154}, {25,185}, {51,1204}, {74,9781}, {125,7507}, {141,6803}, {155,2929}, {169,5777}, {184,3515}, {186,7592}, {343,6815}, {378,3567}, {394,5889}, {1112,2935}, {1490,2270}, {1503,7487}, {1596,5878}, {1597,3357}, {1598,6000}, {1715,7580}, {1730,3149}, {1753,1864}, {1854,1905}, {1885,5925}, {1899,3575}, {2883,3089}, {3066,3091}, {3088,5480}, {3516,9777}, {3517,6759}, {3525,5646}, {3527,3532}, {3574,5094}, {3796,7488}, {5020,5907}, {5449,7706}, {5462,7689}, {5706,7412}, {5892,7393}, {5893,6623}, {5946,7526}, {7485,7691}
X(9786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,389,6), (6,1192,3), (6,5023,1970)
X(9787) lies on these lines: {2,168}, {7,174}, {8,8372}, {20,8112}, {329,5935}, {516,8140}, {3616,8110}, {8108,9778}, {8378,9779}, {8381,9780}
X(9788) lies on these lines: {511,9784}
X(9789) lies on these lines: {2,1766}, {7,1659}, {8,637}, {20,8234}, {329,8233}, {516,8244}, {962,7596}, {3616,8225}, {8224,9778}, {8228,9779}, {8230,9780}, {8239,9785}
X(9790) lies on these lines: {51,577}
X(9791) = (2*s^2+SW)*X(8)+2*(2*s^2-SW)*X(192)
X(9791) lies on these lines: {1,6646}, {2,846}, {7,21}, {8,192}, {20,8235}, {37,4645}, {45,1213}, {78,4335}, {190,4026}, {329,4199}, {333,4854}, {516,8245}, {524,3241}, {962,20008}, {1330,3743}, {1423,5250}, {1469,3877}, {1962,4683}, {3416,4664}, {3685,4357}, {3775,4693}, {3786,6007}, {3993,6542}, {4220,9778}, {4356,4416}, {4364,5263}, {4442,5235}, {4656,7081}, {4657,4676}, {4687,5880}, {5224,5695}, {8229,9779}, {8240,9785}, {8246,9789}, {8731,9776}
X(9791) = reflection of X(i) in X(j) for these (i,j): (8,1654)
X(9791) = anticomplement of X(24342)
X(9791) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (846,4425,2), (4418,6536,2)
Let A'B'C' be the circumorthic triangle. Let A" be the {B',C'}-reciprocal conjugate of A', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(54). The lines A'A", B'B", C'C" concur in X(9792). (Randy Hutson, April 9, 2016)
X(9792) lies on these lines: {4,6752}, {6,24}, {51,107}, {95,511}, {97,3060}, {389,8884}, {3168,8794}, {4993,5640}, {4994,9781}
X(9792) = circumorthic isotomic conjugate of X(185)
X(9793) lies on these lines: {1,9795}, {2,8078}, {4,8099}, {20,8081}, {329,8079}, {516,8089}, {962,8091}, {3616,8077}, {8075,9778}, {8085,9779}, {8087,9780}, {8241,9785}, {8247,9789}, {8249,9791}, {8733,9776}
X(9793) = {X(1),X(9807)}-harmonic conjugate of X(9795)
Let A'B'C' be the Artzt triangle. Let A" be the reflection of A in B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(9794). (Randy Hutson, July 20, 2016)
X(9794) lies on these lines: {51,8121}, {52,8123}, {185,8122}, {389,9796}, {3060,8117}, {3567,8119}, {5890,8120}, {8124,9730}
X(9795) lies on these lines: {1,9793}, {4,8100}, {8,8094}, {20,8082}, {329,8080}, {516,8090}, {962,8092}, {8076,9778}, {8086,9779}, {8088,9780}, {8242,9785}, {8248,9789}, {8250,9791}
X(9795) = {X(1),X(9807)}-harmonic conjugate of X(9793)
X(9796) lies on these lines: {51,8122}, {52,8124}, {185,8121}, {389,9794}, {3060,8118}, {3567,8120}, {5890,8119}, {8123,9730}
X(9797) = 2*(8*R+r)*X(1)-3*(4*R+r)*X(2)
X(9797) lies on these lines: {1,2}, {20,6766}, {279,3875}, {390,6762}, {518,9785}, {944,8158}, {958,8236}, {962,971}, {1058,5815}, {1219,3886}, {3158,5265}, {3160,4460}, {3189,4308}, {3303,5273}, {3600,5853}, {3813,5226}, {3871,8273}, {3885,7957}, {3913,5435}, {5558,9776}, {7982,9799}
X(9797) = reflection of X(i) in X(j) for these (i,j): (8, 938), (5815, 1058)
X(9797) = anticomplement of X(4882)
X(9797) = X(5558)-anticomplementary conjugate of X(3436)
X(9797) = {X(1),X(6764)}-harmonic conjugate of X(8)
X(9798) = 2*R^2*X(1)-(6*R^2-SW)*X(25) = SW*X(3)-4*R^2*X(10) = R^2*X(8)+(3*R^2-SW)*X(22)z
X(9798) lies on these lines: {1,25}, {3,10}, {8,22}, {23,145}, {24,944}, {26,952}, {28,388}, {36,1722}, {40,3220}, {46,1473}, {48,2333}, {55,8190}, {56,998}, {58,1460}, {104,1603}, {159,518}, {169,198}, {219,4456}, {497,4222}, {517,3556}, {595,7083}, {602,2183}, {607,2172}, {610,7719}, {859,5358}, {946,1598}, {956,2915}, {1072,1842}, {1125,5020}, {1385,6642}, {1478,4185}, {1479,4186}, {1482,7517}, {1486,3295}, {1498,2807}, {1593,5691}, {1604,1622}, {1617,5930}, {1633,6361}, {1698,7484}, {1699,5198}, {1995,3616}, {2098,9673}, {2099,9658}, {3085,4224}, {3157,8679}, {3515,9590}, {3517,5882}, {3518,7967}, {3585,4214}, {3617,6636}, {3632,9591}, {3640,5594}, {3641,5595}, {3703,5687}, {4678,7492}, {5247,5329}, {5255,7295}, {5587,7395}, {5818,7509}, {5881,9626}, {5886,7529}, {5899,8148}, {7485,9780}
X(9798) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8185,25), (8,22,8193), (25,8192,1), (8190,8191,55)
X(9799) = 2*(r+2*R)*X(4)-(4*R+r)*X(7) = (2*R+r)*X(8)-2*R*X(20)
X(9799) lies on these lines: {2,1490}, {3,5273}, {4,7}, {5,5658}, {8,20}, {10,5732}, {21,3427}, {77,9121}, {224,6890}, {329,6836}, {355,6916}, {387,990}, {411,5744}, {516,7992}, {517,6764}, {912,5758}, {944,1012}, {962,3868}, {1210,1750}, {1259,6909}, {1498,3562}, {1709,4294}, {1765,3730}, {2551,5784}, {2947,4303}, {3091,5249}, {3149,5435}, {3339,4292}, {3487,8727}, {3523,6705}, {3927,5759}, {4208,5587}, {5084,5927}, {5175,6925}, {5226,6831}, {5328,6922}, {5550,6884}, {5703,6847}, {5704,6848}, {5720,6926}, {5748,6943}, {5770,6985}, {5777,6865}, {5781,6554}, {5811,6827}, {5905,6895}, {6256,6839}, {6835,9776}, {6987,7330}, {7982,9797}, {8095,9793}, {8096,9795}
X(9799) = reflection of X(i) in X(j) for these (i,j): (4,5787), (20,84), (5758,6851)
X(9799) = anticomplement of X(1490)
X(9799) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5768,938), (944,1012,4313)
X(9800) lies on these lines: {4,3753}, {8,144}, {20,5250}, {65,5809}, {758,6764}, {938,1479}, {962,3868}, {1750,7080}, {3522,4512}
X(9801) = ((4*R+r)^2-s^2)*X(8)-4*R*(4*R+r)*X(144)
X(9801) lies on these lines: {2,1721}, {8,144}, {20,2128}, {726,6764}, {990,3616}, {1766,3161}, {2951,3912}, {2961,3218}, {4907,9436}
X(9801) = anticomplement of X(1721)
X(9802) = 3*X(8)-4*X(80)
X(9802) lies on these lines: {1,9782}, {2,5541}, {7,528}, {8,80}, {11,9710}, {20,6264}, {100,474}, {104,9778}, {119,9779}, {319,3902}, {382,952}, {516,7993}, {519,5180}, {1000,3434}, {1058,6797}, {1484,5657}, {2796,9457}, {2800,9799}, {3880,5080}, {3885,5252}, {3895,5219}, {4295,7972}, {4301,5531}, {4304,4861}, {5533,5704}
X(9802) = reflection of X(i) in X(j) for these (i,j): (8,149), (20,6264), (5531,4301), (6224,1320)
X(9802) = anticomplement of X(5541)
X(9802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1320,6224,3241)
X(9802) = antipode of X(8) in conic through X(7), X(8), and the extraversions of X(8)
X(9803) = 4*R*X(3)-(3*R+2*r)*X(8) = (4*R+r)*X(7)-4*(R+r)*X(80)
X(9803) lies on these lines: {1,6888}, {2,6326}, {3,8}, {4,2771}, {7,80}, {10,5531}, {11,938}, {20,1768}, {119,6829}, {145,6264}, {149,151}, {355,5885}, {515,3218}, {517,9802}, {519,5538}, {758,6840}, {912,5080}, {1156,5809}, {1320,3427}, {1484,5603}, {2320,7967}, {2475,5884}, {2829,9799}, {3336,4293}, {3648,7491}, {3649,7548}, {5046,5693}, {5226,8068}, {5400,6788}, {5660,6702}, {5694,6902}, {5734,6845}, {5854,6764}
X(9803) = reflection of X(i) in X(j) for these (i,j): (20,1768), (145,6264), (153,80), (962,149), (5531,10), (6224,104)
X(9803) = anticomplement of X(6326)
X(9803) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,5768,5731), (104,6224,5731), (355,5885,6901), (5885,6901,9782)
X(9804) lies on these lines: {7,3555}, {8,3305}, {516,8001}, {938,4002}, {5920,9785}
X(9805) lies on these lines: {1,289}, {10,8380}, {40,8107}, {65,8113}, {72,5934}, {946,8377}, {2292,8391}, {3057,8390}, {6732,8094}, {7672,8385}, {7991,8140}, {8093,8133}
X(9805) = X(4)-of-inner-Hutson-triangle
X(9805) = reflection of X(9806) in X(7991)
X(9806) = X(4)-of-outer-Hutson-triangle
X(9806) = reflection of X(9805) in X(7991)
X(9806) lies on these lines: {1,168}, {8,8372}, {10,8381}, {40,8108}, {65,8114}, {72,5935}, {946,8378}, {3057,8392}, {7672,8386}, {7991,8140}, {8093,8135}, {8094,8138}
X(9807) lies on these lines: {1,9793}, {2,164}, {7,177}, {167,516}, {962,7057}, {2089,8390}, {8422,9785}
X(9807) = anticomplement of X(164)
X(9807) = X(1)-of-2nd-Conway-triangle
X(9807) = {X(9793),X(9795)}-harmonic conjugate of X(1)
X(9808) lies on these lines: {1,372}, {8,637}, {10,8230}, {40,8224}, {65,8243}, {72,8233}, {517,7596}, {946,8228}, {2292,8246}, {3057,8239}, {7672,8237}, {7991,8244}, {8093,8247}, {8094,8248}
X(9808) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8231,8225), (40,8234,8224)
X(9808) = X(4)-of-2nd-Pamfilos-Zhou-triangle
X(9809) = (4*R+r)*X(7)-4*R*X(11)
X(9809) lies on these lines: {2,1768}, {4,2771}, {5,9782}, {7,11}, {8,153}, {20,6326}, {79,6894}, {80,4295}, {100,329}, {104,3560}, {119,5811}, {149,152}, {382,952}, {411,3648}, {515,5180}, {516,3935}, {971,5057}, {1145,5815}, {1317,9785}, {1836,7672}, {2950,5552}, {3120,9355}, {3652,6853}, {4301,7993}, {5080,6001}, {5758,5840}, {6888,7701}, {8103,9793}, {8104,9795}
X(9809) = reflection of X(i) in X(j) for these (i,j): (8,153), (20,6326), (7993,4301)
X(9809) = anticomplement of X(1768)
X(9809) = X(110)-of-2nd-Conway-triangle
X(9810) lies on these lines: {110,6011}, {351,513}, {522,9131}, {1491,5040}, {2786,3268}, {3309,9135}, {3667,9123}, {3960,4813}, {4778,9185}, {5027,6004}
X(9810) = X(1)-of-1st-Parry-triangle
X(9810) = reflection of X(9811) in X(351)
X(9811) lies on these lines: {351,513}, {514,9131}, {846,2610}, {3309,3569}, {3667,9185}, {3960,5168}, {4724,5075}, {4778,9123}, {6004,9208}
X(9811) = X(1)-of-2nd-Parry-triangle
X(9811) = reflection of X(9810) in X(351)
X(9812) = 4*X(4)-X(8)
X(9812) lies on these lines: {1,3146}, {2,165}, {3,5284}, {4,8}, {5,6361}, {7,354}, {10,3832}, {11,3474}, {20,946}, {30,5603}, {33,4318}, {40,3091}, {46,5704}, {55,5226}, {57,5274}, {65,5225}, {79,5558}, {100,5748}, {144,4847}, {145,4301}, {149,152}, {153,9802}, {226,390}, {376,5886}, {381,5657}, {382,944}, {388,5919}, {499,5131}, {515,3241}, {546,5818}, {614,1721}, {748,9441}, {938,1479}, {952,3830}, {971,3873}, {990,7191}, {1001,7411}, {1058,5049}, {1125,3522}, {1376,5328}, {1385,3529}, {1482,3627}, {1537,6224}, {1621,7580}, {1657,5901}, {1697,5261}, {1698,5068}, {1706,8165}, {1709,3218}, {1742,3720}, {1750,3870}, {1770,3086}, {1864,7672}, {1996,3599}, {2550,3740}, {2807,3060}, {2886,5273}, {3057,5229}, {3058,3475}, {3090,3579}, {3295,5714}, {3428,6912}, {3476,4345}, {3485,4313}, {3486,4323}, {3488,9668}, {3523,8227}, {3600,9579}, {3617,7991}, {3622,4297}, {3649,9670}, {3656,7967}, {3667,6545}, {3677,4346}, {3742,5918}, {3839,5587}, {3843,5690}, {3845,5790}, {3848,5880}, {3854,7989}, {3876,7957}, {3914,5222}, {4229,5333}, {4292,9614}, {4294,5703}, {4308,7354}, {4420,6769}, {4666,5732}, {5047,5584}, {5056,6684}, {5076,8148}, {5177,5250}, {5205,6557}, {5219,5281}, {5272,7613}, {5308,6999}, {5744,8727}, {5759,8226}, {5805,9776}, {6932,7680}, {6943,7681}, {8055,9519}, {8166,9352}, {8972,9616}
X(9812) = reflection of X(i) in X(j) for these (i,j): (2,1699), (20,3576), (165,3817), (376,5886), (3576,946), (3681,5927), (5657,381), (5731,5603), (5790,3845), (5918,3742), (7967,3656)
X(9812) = anticomplement of X(165)
X(9812) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1699,9779), (4,962,8), (7,2898,479), (11,3474,5435), (20,946,3616), (40,3091,9780), (165,1699,3817), (165,3817,2), (226,9580,390), (329,3434,8), (497,1836,7), (1479,4295,938), (2886,5698,5273), (3058,3475,8236), (3434,5057,329), (3485,6284,4313), (3622,5059,4297), (3869,5175,8), (4301,5691,145), (5082,5815,8), (9778,9779,2)
Submedial triangle and related centers: X(9813)-X(9827)
This preamble and centers X(9813)-X(9827) were contributed by César Eliud Lozada, April 15, 2016.
Let ABC be a triangle, and suppose that C' is a point on side AB and that B' is a point on side AC. Let r(B'C') be the rectangle whose vertices are B', C', and the orthogonal projections of B' and C' onto side BC. Let RA be the rectangle r(B'C') of maximal area, which is obtained by taking A'B'C' to be the medial triangle of ABC. Let OA be the center RA, and define OB and OC cyclically. The central triangle OAOBOC is here named the submedial triangle; c.f. the Calabi triangle (https://en.wikipedia.org/wiki/Calabi_triangle).
The A-vertex of OAOBOC is given by
OA = 2*a*b*c : (3*a^2+b^2-c^2)*c : (3*a^2-b^2+c^2)*b (trilinears)
The vertices of the submedial triangle lie on the cubic K281, and
X(6688) = X(2)-of-OAOBOC
X(3628) = X(3)-of-OAOBOC
X(5462) = X(4)-of-OAOBOC
The circumcircle of OAOBOC passes through these points: X(6667), X(6721), X(6722), X(6723), and the X(5) is the radical center of the circumcircles of the rectangles RA, RB, RC.
The appearance of (T,i) in the following list means that OAOBOC and T are perspective with perspector X(i), and an asterisk (*) indicates that the triangles are homothetic:
(circumorthic*, 3090)
(2nd Ehrmann*, 9813)
(Euler, 9815)
(2nd Euler*, 1656)
(3rd Euler, 5)
(4th Euler, 5)
(extangents*, 9816)
(intangents*, 9817)
(Kosnita*, 6642)
(medial, 6)
(midheight, 6)
(orthic, 2)
(inner-squares, 6)
(outer-squares, 6)
(tangential*, 5020)
(Trinh*, 9818)
The appearance of (T,i,j) in the following list means that OAOBOC and T are orthologic with centers X(i) and X(j):
(ABC, 5, 3)
(anticomplementary, 5, 4)
(Aquila, 5, 40)
(Ara, 5, 7387)
(Aries, 9820, 7387)
(5th Brocard, 5, 9821)
(circumorthic, 5462, 5889)
(1st Ehrmann, 9822, 576)
(2nd Ehrmann, 5462, 8548)
(Euler, 5, 5)
(2nd Euler, 5462, 5562)
(extangents, 5462, 6237)
(outer Garcia, 5, 355)
(inner Grebe, 5, 1161)
(outer Grebe, 5, 1160)
(intangents, 5462, 6238)
(Johnson, 5, 4)
(Kosnita, 5462, 1147)
(Lucas antipodal, 642, 3)
(Lucas central, 9823, 3)
(Lucas homothetic, 5, not computed)
(Lucas(-1) antipodal, 641, 3)
(Lucas(-1) central, 9824, 3)
(Lucas(-1) homothetic, 5, not computed)
(Macbeath, 9825, 4)
(medial, 5, 5)
(midheight, 9729, 389)
(5th mixtilinear [also called Caelum], 5, 1482)
(orthic, 5462, 52)
(orthocentroidal, p314, 568)
(reflection, p315, 6243)
(tangential, 5462, 155)
(Trinh, 5462, 7689)
The triangle OAOBOC is parallelogic to the 1st and 2nd Parry triangles with centers X(5) and X(351) in both cases.
The locus of P such that OAOBOC and the cevian-triangle-of P are perspective is the circumcubic given by
∑[SA*(2*SA+SW)*((2*SB+SC)*y-(2*SC+SB)*z)*x^2] + 2*(SB-SC)*(SC-SA)*(SA-SB)*x*y*z = 0 (barycentrics).
The curve passes through these points: X(2), X(4), X(5395).
The locus of P such that OAOBOC and the anticevian-triangle-of P are perspective is the circumcubic given by
∑[a^2*SA*((SA+3*SB-SC)*y-(SA+3*SC-SB)*z)*y*z] + 2*(SB-SC)*(SC-SA)*(SA-SB)*x*y*z = 0 (barycentrics)
The curve passes through these points: X(3), X(6), X(69), X(193), X(1351), X(2996).
X(9813) lies on these lines: {2,8541}, {5,524}, {6,1196}, {22,1843}, {182,2393}, {575,6642}, {597,6677}, {895,5642}, {1351,5891}, {1656,8538}, {1992,7392}, {3090,8537}, {3098,9019}, {5462,8548}, {8549,9729}
X(9814) lies on these lines: {1,6610}, {7,1699}, {9,8169}, {55,2951}, {165,8545}, {527,1478}, {3816,6173}, {7962,8581}, {7987,8544}
X(9815) lies on these lines: {2,3574}, {3,3589}, {4,5943}, {5,9786}, {51,6815}, {68,5946}, {125,146}, {182,7487}, {185,6997}, {235,3066}, {373,6816}, {381,5878}, {389,1352}, {511,6803}, {546,4846}, {1899,7544}, {3090,7699}, {3146,5643}, {5907,7392}, {6688,6804}, {6928,7686}, {7528,9730}
X(9816) lies on these lines: {1,7535}, {2,19}, {5,40}, {9,1730}, {33,4223}, {55,5020}, {57,1723}, {65,1722}, {71,3305}, {169,226}, {373,3611}, {940,2264}, {1001,3198}, {1040,1859}, {2082,5712}, {2294,5256}, {5227,5271}, {5249,7289}, {5462,6237}, {6254,9729}, {8539,9813}
X(9817) lies on these lines: {1,5}, {2,33}, {4,1038}, {9,1936}, {34,3091}, {35,6642}, {55,5020}, {57,1736}, {63,7069}, {77,2635}, {78,2654}, {200,3706}, {212,3305}, {222,5927}, {225,6835}, {373,3270}, {381,1060}, {390,5297}, {497,612}, {499,7404}, {750,2310}, {899,4336}, {908,2000}, {936,7532}, {940,1864}, {950,975}, {990,3911}, {1062,1656}, {1068,6896}, {1074,6854}, {1155,1721}, {1479,7401}, {1699,8270}, {1707,5348}, {1785,6826}, {1838,6849}, {1854,3812}, {1870,3545}, {1877,6957}, {3055,9594}, {3075,7330}, {3090,6198}, {3306,7004}, {3554,9599}, {3601,7535}, {3628,8144}, {3740,7074}, {3832,4296}, {3920,5274}, {4318,9779}, {4319,5218}, {4320,5229}, {4413,9371}, {5010,6644}, {5067,9643}, {5070,9644}, {5432,6677}, {5462,6238}, {6285,9729}, {6864,7952}, {7070,7308}, {7280,7526}, {7298,9673}, {7393,9645}, {7486,9538}, {8540,9813}
X(9817) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,33,1040), (5348,7082,1707)
As a point on the Euler line, X(9818) has Shinagawa coefficients (E+2*F, E-2*F)
X(9818) lies on these lines: {2,3}, {6,4550}, {36,9817}, {114,2936}, {143,3527}, {155,578}, {159,3818}, {182,6000}, {311,3964}, {394,5891}, {511,9813}, {541,5621}, {568,9777}, {569,1181}, {1154,1351}, {1176,3426}, {1609,7737}, {2931,7687}, {2935,6699}, {3357,9729}, {3589,4846}, {4549,5480}, {5050,5622}, {5422,5890}, {5462,7689}, {7688,9816}
X(9818) = midpoint of X(3) and X(1597)
X(9818) = {X(3),X(4)}-harmonic conjugate of X(7387)
X(9818) = homothetic center of Ara triangle and Ehrmann mid-triangle
X(9818) = homothetic center of anti-Ascella triangle and Ehrmann side-triangle
X(9819) lies on these lines: {1,3}, {2,4342}, {8,4082}, {9,3880}, {10,5274}, {145,4314}, {200,3877}, {388,9589}, {390,519}, {392,8580}, {497,3679}, {515,1000}, {516,9814}, {551,4345}, {758,4326}, {936,3884}, {944,7990}, {946,8164}, {950,3632}, {958,3680}, {960,2136}, {962,5290}, {1056,4312}, {1282,4845}, {1682,6048}, {1699,5726}, {1837,4668}, {2269,3731}, {2391,4419}, {2802,9623}, {2943,8915}, {3058,4677}, {3086,9588}, {3158,5289}, {3244,4313}, {3486,3633}, {3600,5493}, {3878,6765}, {3885,4853}, {3890,8583}, {4050,7323}, {4315,9778}, {5252,9580}, {5315,7074}, {5836,8167}, {6001,7966}, {7989,9614}
X(9819) = reflection of X(i) in X(j) for these (i,j): (4312,1056), (4900,4915), (4915,9)
X(9819) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,40,3361), (1,5119,165), (1,7991,3339), (40,7982,8158), (55,3057,7962), (55,7962,1), (57,5919,1), (960,2136,4882), (3877,3895,200), (3885,5250,4853), (4345,5281,551), (4853,5250,5234)
X(9820) lies on these lines: {2,155}, {3,4549}, {5,578}, {30,5448}, {49,2072}, {52,468}, {68,1656}, {110,1594}, {113,1885}, {140,9729}, {156,1503}, {343,6639}, {389,5972}, {394,3549}, {450,3462}, {486,8909}, {498,1069}, {499,3157}, {539,547}, {549,7689}, {575,3564}, {597,8548}, {858,1614}, {912,1125}, {1181,3548}, {1216,6676}, {1495,7553}, {1993,7505}, {3090,6193}, {3574,5642}, {5432,6238}, {5433,7352}, {5462,6677}, {5562,7542}, {5651,7405}, {5663,6696}, {6642,9815}, {7495,7999}
X(9820) = midpoint of X(5) and X(1147)
X(9820) = reflection of X(5449) in X(3628)
X(9820) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (49,2072,6146), (1656,3167,68)
Let X be a point on the 2nd Brocard circle. The locus of X(20) of triangle XPU(1) as X varies is a circle with center X(9821). This circle is the reflection of the 2nd Brocard circle in X(5188). See also X(3095). (Randy Hutson, July 20, 2016)
X(9821) lies on these lines: {3,6}, {4,2896}, {5,3096}, {20,2782}, {30,76}, {40,3099}, {140,262}, {194,376}, {237,2979}, {315,5976}, {381,3934}, {382,6248}, {385,7470}, {524,6309}, {538,3534}, {542,7826}, {549,7786}, {732,8725}, {754,8149}, {1656,7914}, {1916,7793}, {3522,7709}, {3818,7854}, {3830,9466}, {3926,8724}, {5054,6683}, {6054,7917}, {6308,8177}, {7467,9465}, {7757,8703}
X(9821) = X(3)-of-the-5th Brocard triangle
X(9821) = 2nd-Brocard-circle-inverse-of-X(3098)
X(9821) = reflection of X(i) in X(j) for these (i,j): (3,5188), (382,6248), (3094,3098), (3095,3), (3830,9466), (6033,5976), (7697,6194), (7757,8703)
X(9821) = circumperp conjugate of X(35375)
X(9821) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,9301,32), (32,3098,3)
X(9822) lies on these lines: {2,1843}, {3,7716}, {5,141}, {6,1196}, {51,69}, {66,3818}, {159,182}, {193,5640}, {373,3618}, {389,1352}, {1176,1495}, {1368,3867}, {1503,9729}, {1634,5421}, {1974,1995}, {2386,7804}, {2393,3589}, {2854,6329}, {3056,9817}, {3060,3620}, {3090,6403}, {3098,9818}, {3313,3763}, {3564,5462}, {3619,3917}, {3779,9816}, {4260,7535}, {5092,6644}, {5140,8370}
X(9822) = midpoint of X(389),X(1352)
X(9822) = X(7)-of-submedial-triangle if ABC is acute
X(9822) = {X(9823),X(9824)}-harmonic conjugate of X(5)
X(9822) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (373,6467,3618), (3313,3763,3819)
X(9823) lies on these lines: {2,6291}, {5,141}, {485,8681}, {1151,5020}, {3090,6239}, {6283,9817}
X(9823) = X(176)-of-submedial-triangle if ABC is acute
X(9823) = orthic-to-submedial similarity image of X(6291)
X(9823) = {X(5),X(9822)}-harmonic conjugate of X(9824)
X(9824) lies on these lines: {2,6406}, {5,141}, {486,8681}, {1152,5020}, {3090,6400}, {6404,9816}, {6405,9817}, {7692,9818}
X(9824) = {X(5),X(9822)}-harmonic conjugate of X(9823)
X(9824) = X(175)-of-submedial-triangle if ABC is acute
X(9824) = orthic-to-submedial similarity image of X(6406)
As a point on the Euler line, X(9825) has Shinagawa coefficients (E-F, E+3*F)
X(9825) lies on these lines: {2,3}, {6,9815}, {389,3564}, {1352,9786}, {1353,6193}, {1503,9729}, {3818,6247}, {6284,9817}, {6696,6697}, {8550,9813}
X(9825) = midpoint of X(3) and X(6756)
X(9825) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,381,3088), (3,7401,5), (3,7528,1595), (4,631,7396), (4,5020,5), (5,6642,6677), (5,6644,140), (24,7399,6676), (25,6815,6823)
Let A'B'C' be the orthic triangle. Let La, Lb, Lc be the orthic axes of AB'C', BC'A', CA'B', resp. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. Triangle A"B"C" is inversely similar to ABC, with similitude center X(6). X(9826) = X(5) of A"B"C". (Randy Hutson, July 20, 2016)
X(9826) lies on these lines: {2,1986}, {3,1112}, {5,113}, {6,1511}, {74,9818}, {110,6642}, {389,5972}, {399,5020}, {542,9822}, {1539,4846}, {1656,7723}, {2777,9729}, {2781,3589}, {2935,7526}, {3090,7722}, {3448,7401}, {5654,6102}, {5943,7687}, {7724,9816}, {7727,9817}
X(9826) = midpoint of X(i),X(j) for these {i,j}: {3,1112}, {113,974}, {389,5972}
X(9827) lies on these lines: {2,6152}, {5,51}, {6,1493}, {54,5422}, {68,5946}, {195,5020}, {539,5462}, {2888,7401}, {2917,6644}, {3090,6242}, {3519,6515}, {3589,6153}, {4846,6145}, {5943,9820}, {5965,9822}, {6255,9816}, {6286,9817}, {6677,8254}, {7691,9818}, {7730,7999}, {9825,9826}
X(9827) = midpoint of X(i),X(j) for these {i,j}: {973,1209}, {6153,6689}
Contributed by Peter Moses, April 16, 2016.
X(9828) lies on the cubics K795 and K796 and these lines: {2,351}, {3,76}, {110,5026}, {111,9159}, {187,5912}, {542,6786}, {690,3111}, {2491,5661}, {7496,9153}
X(9828) = {X(98),X(99)}-harmonic conjugate of X(9149)
X(9828) = inverse-in-circumcircle of X(9149)
X(9828) = inverse-in-orthoptic-circle-of-Steiner-inellipe of X(9148)
Contributed by Peter Moses, April 16, 2016.
X(9829) lies on the cubic K796 and these lines: {2,187}, {110,599}, {111,7610}, {183,6322}, {1641,7998}, {4108,8704}, {5971,7622}, {7600,8860}, {9080,9831}
X(9829) = midpoint of X(2) and X(6031)
X(9829) = reflection of X(6032) in X(2)
X(9829) = X(i)-vertex conjugate of X(j) for these {i,j}: {804, 9149}, {9149, 804}
X(9829) = X(5661)-line conjugate of X(2491)
X(9829) = X(2)-of-circummedial-triangle
Contributed by Peter Moses, April 16, 2016.
X(9830) lies on the cubic K796 and these lines: {2,353}, {6,598}, {30,511}, {69,8591}, {98,6233}, {99,599}, {111,9169}, {114,9771}, {115,597}, {141,2482}, {147,9770}, {148,1992}, {182,7606}, {193,8596}, {1153,5092}, {1352,7618}, {1691,8859}, {3589,5461}, {3818,8176}, {5182,7884}, {5465,6593}, {5477,8584}, {5503,9766}, {5613,9762}, {5617,9760}, {5980,9761}, {5981,9763}, {5984,9740}, {6776,7620}, {7426,9128}, {9140,9832}
Contributed by Peter Moses, April 16, 2016.
X(9831) lies on the circumcircle and these lines: {98,8704}, {99,3849}, {110,5104}, {476,6032}, {511,6233}, {512,6323}, {574,691}, {2709,3098}, {6031,9150}, {9080,9829}
X(9831) = X(99)-of-circummedial-triangle
X(9831) = trilinear pole of line X(6)X(9208)
X(9831) = trilinear pole, wrt circummedial triangle, of line X(183)X(6322)
X(9831) = Λ(X(2), X(353))
X(9831) = Λ(X(182),X(7606)); the line X(182)X(7606) is the Brocard axis of the McCay triangle
X(9831) = 3rd-Parry-to-circumsymmedial similarity image of X(353)
X(9831) = X(805)-of-circumsymmedial-triangle
Contributed by Peter Moses, April 16, 2016.
X(9832) lies on the Brocard circle of the circummedial triangle, the cubic K796, and these lines: {2,3}, {98,6795}, {111,9159}, {183,523}, {385,2452}, {691,7771}, {805,2770}, {5099,7761}, {5967,6800}, {9140,9830}
X(9832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,23,1316), (2,7417,1995)
Orthologic centers: X(9833)-X(10000)
Centers X(9833)-X(10000) were contributed by César Eliud Lozada - April 28, 2016.
X(9833) = 2*(R^2-SW)*X(3)+SW*X(66) = (R^2-SW)*X(4)-(5*R^2-2*SW)*X(54)
The reciprocal orthologic center of these triangles is X(68).
X(9833) lies on these lines: {3,66}, {4,54}, {5,154}, {6,6756}, {20,2979}, {24,1899}, {25,6146}, {26,68}, {30,155}, {52,2393}, {64,550}, {125,3147}, {133,6616}, {140,1853}, {156,5654}, {182,7401}, {206,569}, {343,9715}, {376,3357}, {382,2883}, {389,1843}, {511,5596}, {1092,1370}, {1154,6293}, {1181,3575}, {1495,3542}, {1594,9707}, {1619,7387}, {1971,3767}, {2777,3529}, {2917,2918}, {3146,5656}, {3534,5894}, {3796,7399}, {3818,7404}, {3830,5893}, {4299,7355}, {4302,6285}, {4549,5876}, {5012,7544}, {5706,7511}, {5786,7510}, {6449,8991}, {6643,9306}, {7576,7592}, {8549,9815}, {8567,8703}
X(9833) = orthologic center of anti-McCay triangle to anticomplementary triangle, with reciprocal center X(6193)
X(9833) = midpoint of X(i),X(j) for these {i,j}: {3529,6225}
X(9833) = reflection of X(i) in X(j) for these (i,j): (4,6759), (64,550), (68,26), (382,2883), (1352,159), (5878,1498)
X(9833) = X(72)-of-tangential-triangle if ABC is acute
X(9833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6776,7487,389)
The reciprocal orthologic center of these triangles is X(4).
X(9834) lies on these lines: {55,515}, {5691,8186}
X(9834) = reflection of X(i) in X(j) for these (i,j): (9835,55)
X(9834) = X(4)-of-1st-Auriga-triangle
The reciprocal orthologic center of these triangles is X(4).
X(9835) lies on these lines: {55,515}, {5691,8187}
X(9835) = reflection of X(i) in X(j) for these (i,j): (9834,55)
X(9835) = X(4)-of-2nd-Auriga-triangle
The reciprocal orthologic center of these triangles is X(1).
X(9836) lies on these lines: {1,8111}, {3,8107}, {4,5934}, {5,8377}, {84,177}, {259,5935}, {266,6732}, {390,8385}
X(9836) = reflection of X(9837) in X(1)
X(9836) = X(3) of inner Hutson triangle
X(9836) = {X(8107),X(8109)}-harmonic conjugate of X(3)
X(9836) = orthologic center of inner Hutson tirnalge to anticomplementary triangle, with reciprocal center X(8)
The reciprocal orthologic center of these triangles is X(1).
X(9837) lies on these lines: {1,8111}, {3,8108}, {4,5935}, {5,8378}, {40,168}, {174,515}, {177,3577}, {188,6326}, {236,5587}, {355,8130}, {390,8386}, {517,9806}
X(9837) = reflection of X(9836) in X(1)
X(9837) = X(3) of outer Hutson triangle
X(9837) = {X(8108),X(8110)}-harmonic conjugate of X(3)
X(9837) = orthologic center of outer Hutson triiangle to anticomplementary triangle, with reciprocal center X(8)
The reciprocal orthologic center of these triangles is X(4).
X(9838) lies on these lines: {3,8194}, {4,493}, {20,6462}, {40,8214}, {944,8210}, {5691,8188}, {5870,8218}, {5871,8216}, {6461,9839}
X(9838) = X(4)-of-Lucas-homothetic-triangle
X(9838) = orthologic center of triangles Lucas homothetic to anticomplementary, with reciprocal center X(20)
The reciprocal orthologic center of these triangles is X(4).
X(9839) lies on these lines: {3,8195}, {4,494}, {20,6463}, {40,8215}, {944,8211}, {5691,8189}, {5870,8219}, {5871,8217}, {6461,9838}
X(9839) = X(4)-of-Lucas(-1)-homothetic-triangle
X(9839) = orthologic center of triangles Lucas(-1) homothetic to anticomplementary, with reciprocal center X(20)
As a point on the Euler line, X(9840) has Shinagawa coefficients: (2*r^2+SW+4*R*r, -2*r^2-SW). The reciprocal orthologic center of these triangles is X(1).
X(9840) lies on these lines: {1,256}, {2,3}, {40,846}, {182,1724}, {390,8238}, {500,1064}, {517,2292}, {540,8666}, {542,5258}, {573,5283}, {946,4425}, {956,3564}, {958,1503}, {962,9791}, {971,9852}, {1281,2782}, {1580,5247}, {1834,4267}, {1935,3955}, {2783,4647}, {2784,8846}, {2975,4388}, {3185,5794}, {3430,4653}, {4375,6002}, {5288,5965}, {6003,6161}, {7596,8246}, {8091,8249}, {8092,8250}
X(9840) = orthologic center of triangles 1st Sharygin to anticomplementary, with reciprocal center X(8)
X(9840) = reflection of X(i) in X(j) for these (i,j): (500,1385)
X(9841) = 8*R*X(3)-(4*R+r)*X(9)
The reciprocal orthologic center of these triangles is X(1).
X(9841) lies on these lines: {1,1407}, {2,9842}, {3,9}, {4,5437}, {20,57}, {40,376}, {56,5918}, {63,3522}, {90,7285}, {165,3916}, {474,1750}, {515,1706}, {516,1058}, {548,3587}, {550,5709}, {553,962}, {603,7070}, {946,6173}, {960,7992}, {988,1742}, {1012,5436}, {1040,1394}, {1106,4319}, {1697,5731}, {1709,7987}, {2257,4252}, {2951,3361}, {3146,3306}, {3243,6769}, {3523,7308}, {3555,7994}, {3576,5248}, {3601,6909}, {3746,7284}, {3881,7982}, {4220,9852}, {4512,8273}, {4652,7411}, {5219,6890}, {5493,6766}, {5587,6897}, {5658,6700}, {6244,6765}, {6260,6926}, {6705,6908}, {6836,9579}, {6925,9581}, {7580,9844}, {7676,9846}, {8075,9853}, {8076,9854}, {8108,9849}, {9778,9797}
X(9841) = anticomplement of X(9842)
X(9841) = midpoint of X(i),X(j) for these {i,j}: {20,938}, {40,9845}, {4882,9851}
X(9841) = reflection of X(i) in X(j) for these (i,j): (4,9843), (936,3)
X(9841) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,84,9), (3,1490,5438), (3,7171,84), (40,944,2136), (165,9851,4882), (1012,8726,5436)
X(9842) = 8*R*X(5)-(4*R+r)*X(142)
The reciprocal orthologic center of these triangles is X(1).
X(9842) lies on these lines: {2,9841}, {4,936}, {5,142}, {12,9848}, {20,5316}, {84,6692}, {226,938}, {355,381}, {496,3817}, {515,6893}, {516,9709}, {527,5811}, {908,3832}, {950,6957}, {1210,5927}, {1490,6939}, {1699,4882}, {1750,5084}, {3841,6842}, {3911,6953}, {5068,5249}, {5705,5817}, {5745,6848}, {5777,7682}, {6666,6908}, {6705,6944}, {7678,9846}, {7988,9851}, {8085,9853}, {8086,9854}, {8226,9844}, {8227,9845}, {8229,9852}, {8378,9849}, {9779,9797}
X(9842) = complement of X(9841)
X(9842) = midpoint of X(i),X(j) for these {i,j}: {4,936}
X(9842) = reflection of X(i) in X(j) for these (i,j): (9843,5)
X(9842) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (84,6964,6692)
X(9843) = (2*R+r)*X(1)+3*(2*R-r)*X(2)
The reciprocal orthologic center of these triangles is X(1).
X(9843) lies on these lines: {1,2}, {3,6692}, {4,5437}, {5,142}, {11,9848}, {57,5084}, {72,5316}, {84,6939}, {226,4187}, {392,4848}, {404,4304}, {405,1466}, {442,9844}, {443,9581}, {474,950}, {515,6918}, {516,6865}, {527,5708}, {631,5436}, {942,3452}, {946,3812}, {999,5795}, {1001,6684}, {1058,1706}, {1229,4066}, {1329,3742}, {1413,7532}, {1490,6964}, {2478,3306}, {2551,3333}, {3149,4297}, {3488,5438}, {3576,6927}, {3586,6904}, {3670,4656}, {3671,5883}, {3754,4301}, {3817,3825}, {3820,5045}, {4193,5249}, {4311,5253}, {4679,5221}, {5051,9852}, {5129,5435}, {5587,9845}, {5709,8257}, {5714,6173}, {5791,6666}, {5804,6282}, {5853,9709}, {6705,6913}, {6848,8726}, {6919,9612}, {6956,8227}, {7679,9846}, {7989,9851}, {8087,9853}, {8088,9854}, {8381,9849}
X(9843) = complement of X(936)
X(9843) = midpoint of X(i),X(j) for these {i,j}: {4,9841}, {936,938}, {1058,1706}, {2551,3333}
X(9843) = reflection of X(i) in X(j) for these (i,j): (9842,5)
X(9843) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,6700), (1,1698,7080), (1,8582,10), (2,938,936), (2,1210,10), (2,5704,5705), (2,5705,3634), (1698,4847,10), (2478,3306,4292), (3812,3816,946), (4187,5439,226), (6919,9776,9612)
X(9844) = r*(8*R+r)*X(4)-(2*R+r)*(4*R+r)*X(7)
The reciprocal orthologic center of these triangles is X(1).
X(9844) lies on these lines: {1,5927}, {2,9859}, {4,7}, {9,3913}, {40,5729}, {55,3697}, {72,519}, {210,4314}, {329,9797}, {392,3486}, {405,936}, {442,9843}, {452,3876}, {497,3555}, {1058,5811}, {1490,9845}, {1750,9851}, {1837,3753}, {1858,4018}, {2310,3931}, {3488,5777}, {4199,9852}, {4313,5044}, {5045,5274}, {5173,5225}, {5261,7671}, {5290,5572}, {5439,9581}, {5935,9849}, {6744,8581}, {7580,9841}, {8079,9853}, {8080,9854}, {8226,9842}, {8232,9846}
X(9844) = complement of X(9859)
X(9844) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (950,1864,72)
The reciprocal orthologic center of these triangles is X(1).
X(9845) lies on these lines: {1,971}, {3,4882}, {20,6766}, {40,376}, {84,4313}, {515,938}, {936,958}, {1158,7966}, {1385,3646}, {1490,9844}, {1750,3304}, {2951,8158}, {3655,7330}, {5434,5691}, {5436,6920}, {5587,9843}, {5768,5881}, {6738,7091}, {7675,9846}, {8081,9853}, {8082,9854}, {8112,9849}, {8227,9842}, {8235,9852}
X(9845) = midpoint of X(i),X(j) for these {i,j}: {1,9851}, {20,9797}
X(9845) = reflection of X(i) in X(j) for these (i,j): (40,9841), (4882,3)
X(9845) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4297,6762,40)
The reciprocal orthologic center of these triangles is X(1).
X(9846) lies on these lines: {7,9797}, {390,944}, {519,7672}, {936,7677}, {1445,4882}, {3059,4308}, {3241,8581}, {4326,9851}, {7676,9841}, {7678,9842}, {7679,9843}, {8232,9844}, {8236,9848}, {8238,9852}, {8386,9849}, {8387,9853}, {8388,9854}
The reciprocal orthologic center of these triangles is X(1).
X(9847) lies on these lines: {363,4882}, {519,9805}, {936,8109}, {971,9836}, {5934,9844}, {6732,9854}, {8107,9841}, {8111,9845}, {8133,9853}, {8140,9849}, {8377,9842}, {8380,9843}, {8385,9846}, {8390,9848}, {8391,9852}, {9783,9797}
The reciprocal orthologic center of these triangles is X(1).
X(9848) lies on these lines: {1,971}, {11,9843}, {12,9842}, {55,936}, {56,5918}, {65,497}, {72,519}, {210,1697}, {354,3671}, {390,960}, {392,4314}, {518,9785}, {1058,6001}, {1191,4319}, {1837,5082}, {1898,3486}, {3555,4342}, {3698,9581}, {3812,5274}, {3893,5727}, {4301,5728}, {4313,5784}, {4323,7671}, {8236,9846}, {8240,9852}, {8241,9853}, {8242,9854}, {8392,9849}
X(9848) = reflection of X(i) in X(j) for these (i,j): (65,938), (9850,1)
The reciprocal orthologic center of these triangles is X(1).
X(9849) lies on these lines: {168,4882}, {519,9806}, {936,8110}, {971,9837}, {5935,9844}, {8108,9841}, {8112,9845}, {8135,9853}, {8138,9854}, {8140,9847}, {8378,9842}, {8381,9843}, {8386,9846}, {8392,9848}, {9787,9797}
X(9849) = reflection of X(i) in X(j) for these (i,j): (9847,9851)
The reciprocal orthologic center of these triangles is X(1).
X(9850) lies on these lines: {1,971}, {7,9797}, {11,9842}, {12,9843}, {20,3057}, {37,4322}, {55,9841}, {56,210}, {57,4882}, {65,519}, {72,4315}, {200,7091}, {226,9844}, {227,4003}, {354,388}, {381,5045}, {405,1319}, {443,3698}, {518,3600}, {942,5881}, {960,4308}, {1284,9852}, {1420,5234}, {1466,3689}, {1697,5918}, {1864,3304}, {3059,4321}, {3242,4320}, {3333,6918}, {3740,5265}, {3742,5261}, {3911,3983}, {4293,7957}, {4314,5919}, {4332,4864}, {4662,5435}, {5302,7677}, {8114,9849}
X(9850) = midpoint of X(i),X(j) for these {i,j}: {7,9846}
X(9850) = reflection of X(i) in X(j) for these (i,j): (9848,1)
X(9850) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3555,4298,65)
The reciprocal orthologic center of these triangles is X(1).
X(9851) lies on these lines: {1,971}, {3,4866}, {20,519}, {165,3916}, {515,3339}, {516,9797}, {936,993}, {938,4298}, {944,7990}, {1750,9844}, {2951,6762}, {3149,3361}, {4297,5223}, {4326,9846}, {7988,9842}, {7989,9843}, {8089,9853}, {8090,9854}, {8245,9852}
X(9851) = reflection of X(i) in X(j) for these (i,j): (1,9845), (4882,9841), (5691,938)
X(9851) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (944,7992,9819), (944,9819,7990), (4882,9841,165)
X(9852) = 2*r*(3*R+2*r)*X(21)-(3*r^2+4*R*r-s^2)*X(936)
The reciprocal orthologic center of these triangles is X(1).
X(9852) lies on these lines: {21,936}, {519,2292}, {846,4882}, {971,9840}, {1284,9850}, {4220,9841}, {5051,9843}, {8229,9842}, {8235,9845}, {8238,9846}, {8240,9848}, {8245,9851}, {8249,9853}, {8250,9854}, {9791,9797}
The reciprocal orthologic center of these triangles is X(1).
X(9853) lies on these lines: {1,9854}, {519,8093}, {936,8077}, {971,8091}, {2089,9850}, {4882,8078}, {8075,9841}, {8079,9844}, {8081,9845}, {8085,9842}, {8087,9843}, {8089,9851}, {8133,9847}, {8135,9849}, {8241,9848}, {8249,9852}, {8387,9846}, {9793,9797}
The reciprocal orthologic center of these triangles is X(1).
X(9854) lies on these lines: {1,9853}, {174,9850}, {258,4882}, {519,8094}, {936,7588}, {971,8092}, {6732,9847}, {8076,9841}, {8080,9844}, {8082,9845}, {8086,9842}, {8088,9843}, {8090,9851}, {8138,9849}, {8242,9848}, {8250,9852}, {8388,9846}, {9795,9797}
X(9854) = reflection of X(i) in X(j) for these (i,j): (9853,1)
X(9855) = (9*S^2+SW^2)*X(2)-12*S^2*X(3)
The reciprocal orthologic center of these triangles is X(671)
X(9855) lies on the cubic K751 and these lines: {2,3}, {99,3849}, {187,671}, {316,2482}, {385,543}, {511,8593}, {524,8591}, {530,8594}, {531,8595}, {574,598}, {1499,9485}, {4316,4366}, {4324,6645}, {5104,9830}, {5210,8860}, {6054,9773}, {7617,8588}, {7618,7777}, {7756,7827}, {7775,7782}, {7783,7812}, {7801,7802}, {7816,7883}, {7839,8584}, {7870,7885}, {8586,8787}
X(9855) = reflection of X(i) in X(j) for these (i,j): (2,8598), (316,2482), (671,187), (3543,1513), (5999,376), (7840,99), (8586,8787), (8597,2)
X(9855) = anticomplement of X(8352)
X(9855) = orthologic center of triangles anti-McCay to anticomplementary, with reciprocal center X(8591)
X(9855) = McCay-to-anti-McCay similarity image of X(3)
X(9855) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5077,7924), (187,671,8859)
The reciprocal orthologic center of these triangles is X(1).
X(9856) lies on these lines: {1,971}, {3,4512}, {4,8}, {5,1538}, {10,9842}, {20,392}, {40,5044}, {56,1709}, {65,1699}, {84,999}, {210,7991}, {495,6260}, {496,942}, {497,5787}, {518,4301}, {936,6244}, {990,1191}, {1058,9799}, {1210,7956}, {1361,1887}, {1490,3295}, {1537,2771}, {1697,1750}, {1858,5173}, {1864,3340}, {1898,2099}, {2800,6797}, {3057,5691}, {3333,7992}, {3698,7989}, {3812,3817}, {3940,6769}, {4295,5805}, {4300,6051}, {5126,5450}, {5223,6766}, {5231,5789}, {5692,7957}, {5704,8166}, {5779,8158}, {5918,7987}
X(9856) = midpoint of X(i),X(j) for these {i,j}: {72,962}, {3057,5691}, {3062,8581}, {7957,9589}
X(9856) = reflection of X(i) in X(j) for these (i,j): (40,5044), (65,5806), (942,946)
X(9856) = anticomplement of X(31787)
X(9856) = X(3)-of-Atik-triangle
X(9856) = orthologic center of triangles Atik to anticomplementary, with reciprocal center X(8)
X(9856) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (65,1699,5806), (5692,9589,7957)
X(9857) = (S^2-3*SW^2)*X(10)-(S^2-SW^2)*X(32)
X(9857) lies on these lines: {1,3096}, {8,2896}, {10,32}, {40,20041}, {355,9821}, {515,3098}, {517,20164}, {519,7865}, {730,3094}, {1125,7914}, {1698,7846}, {3099,3679}, {3735,4660}, {5587,9993}, {5657,9862}, {5688,9995}, {5689,9881}, {5790,9301}, {9878,9881}, {9923,9928}
X(9857) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,2896,9941)
X(9858) = r*(r+8*R)*X(3)-(r+2*R)*(r+4*R)*X(9)
The reciprocal orthologic center of these triangles is X(1).
X(9858) lies on these lines: {2,9844}, {3,9}, {57,4882}, {142,3813}, {443,938}, {517,6885}, {519,942}, {997,9856}, {1706,5708}, {2550,5045}, {3059,3361}, {3601,9848}, {3868,6904}, {3878,5493}, {5558,9776}, {8726,9845}, {8727,9842}, {8728,9843}, {8731,9852}, {8732,9846}, {8733,9853}, {8734,9854}
X(9858) = complement of X(20012)
X(9858) = midpoint of X(i),X(j) for these {i,j}: {4882,9850}
X(9858) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5438,5784,5044)
X(9859) = 3*r*X(20)-2*(r+2*R)*X(72)
The reciprocal orthologic center of these triangles is X(1).
X(9859) lies on these lines: {2,9844}, {7,9797}, {20,72}, {21,936}, {63,4882}, {377,938}, {519,3868}, {3434,3889}, {3873,4355}, {3877,4294}, {3983,5273}, {4197,9843}, {4297,5696}, {4313,5784}, {4652,7411}, {5436,7675}, {5732,9851}, {6762,8544}
X(9859) = reflection of X(i) in X(j) for these (i,j): (9797,9850), (9844,9858)
X(9859) = anticomplement of X(9844)
X(9859) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9844,9858,2)
The reciprocal orthologic center of these triangles is X(5999).
X(9860) lies on the Bevan circle and these lines: {1,98}, {8,1281}, {10,147}, {40,2782}, {57,3023}, {99,165}, {114,1698}, {115,1572}, {148,516}, {515,3099}, {542,2948}, {1054,2789}, {1478,2792}, {1569,1571}, {1695,3029}, {1697,3027}, {1768,2787}, {2023,9575}, {2783,5541}, {2788,5540}, {2794,5691}, {3044,9586}, {3212,7061}, {3624,6036}, {5587,6033}, {5588,6226}, {5589,6227}
X(9860) = midpoint of X(i),X(j) for these {i,j}: {8,5984}
X(9860) = reflection of X(i) in X(j) for these (i,j): (1,98), (147,10)
X(9860) = X(3023)-of-tangential-triangle-of-excentral-triangle
X(9861) = (6*R^2-SW)*X(25)-2*R^2*X(98)
The reciprocal orthologic center of these triangles is X(5999).
X(9861) lies on these lines: {3,114}, {22,147}, {23,5984}, {25,98}, {115,1598}, {161,542}, {1597,3455}, {1624,5020}, {2080,2386}, {2782,7387}, {5594,6226}, {5595,6227}, {7970,8192}, {8185,9860}
X(9862) = (S^2-3*SW^2)*X(4)-4*(S^2-SW^2)*X(32)
The reciprocal orthologic center of these triangles is X(5999).
X(9862) lies on these lines: {2,5191}, {3,147}, {4,32}, {15,6770}, {16,6773}, {20,2782}, {24,9861}, {30,148}, {40,2784}, {69,74}, {114,631}, {315,5152}, {377,5985}, {378,3438}, {381,7806}, {515,3099}, {549,7931}, {620,3524}, {754,8178}, {804,9409}, {1370,5986}, {1503,2076}, {1742,2792}, {2790,3186}, {3023,4293}, {3027,4294}, {3090,6036}, {3094,3269}, {3146,6321}, {3448,4226}, {3525,7914}, {3533,6721}, {3534,8591}, {3545,6055}, {3569,9147}, {3785,5976}, {3818,3972}, {4027,7791}, {5071,6722}, {5092,7831}, {5149,7800}, {5989,7470}, {6034,9408}, {7967,7970}
X(9862) = midpoint of X(i),X(j) for these {i,j}: {20,5984}
X(9862) = reflection of X(i) in X(j) for these (i,j): (4,98), (147,3), (3146,6321), (8591,3534), (8782,9821)
X(9862) = anticomplement of X(6033)
X(9862) = X(98)-of-5th-Brocard-triangle
X(9862) = 1st-Brocard-to-5th-Brocard similarity image of X(3)
X(9863) = (S^2+(16*R^2-3*SW)*SW)*X(20)-4*SW*(4*R^2-SW)*X(64)
The reciprocal orthologic center of these triangles is X(147).
X(9863) lies on these lines: {3,147}, {4,385}, {5,7787}, {20,64}, {76,2794}, {98,5025}, {114,7907}, {182,7876}, {194,3564}, {262,7921}, {315,5999}, {384,1352}, {511,7893}, {542,7833}, {1513,7793}, {3091,7735}, {3095,7837}, {3186,3575}, {5188,7811}, {5611,5873}, {5615,5872}, {5877,7527}, {5984,6655}, {6194,7767}, {6776,7791}, {7773,9756}, {7797,9755}, {7824,9744}, {7906,9737}
X(9863) = X(4) of the 6th Brocard triangle
X(9863) = reflection of X(i) in X(j) for these (i,j): (20,7750), (7823,4)
The reciprocal orthologic center of these triangles is X(5999).
X(9864) lies on these lines: {1,114}, {8,147}, {10,98}, {40,2794}, {80,256}, {99,515}, {115,5587}, {355,2782}, {517,6033}, {519,6054}, {542,2948}, {620,3576}, {946,7983}, {1018,6211}, {1698,6036}, {1837,3027}, {2792,3732}, {3023,5252}, {3617,5984}, {3624,6721}, {5657,9862}, {5688,6226}, {5689,6227}, {8193,9861}, {8997,9583}
X(9864) = midpoint of X(i),X(j) for these {i,j}: {8,147}
X(9864) = reflection of X(i) in X(j) for these (i,j): (1,114), (98,10), (7983,946)
X(9864) = X(1)-of-X(511)-Fuhrmann triangle (the reflection of ABC in X(114))
X(9865) = 3*(S^2-3*SW^2)*X(2)+8*SW^2*X(39)
The reciprocal orthologic center of these triangles is X(98).
X(9865) lies on these lines: {2,39}, {15,8291}, {16,8292}, {98,8295}, {99,736}, {147,511}, {183,5116}, {325,698}, {385,732}, {524,8592}, {726,5988}, {1513,9772}, {2023,7925}, {2782,5999}, {3094,3314}, {3095,7906}, {3705,3865}, {3906,5996}, {5092,8350}, {5969,7840}, {6194,6776}, {7893,9821}
X(9865) = midpoint of X(i),X(j) for these {i,j}: {7779,8782}
X(9865) = reflection of X(i) in X(j) for these (i,j): (385,5976), (1916,325)
X(9865) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39,3934,7859), (39,7915,7786), (76,6309,194), (194,7836,39), (385,8290,1691)
The reciprocal orthologic center of these triangles is X(147).
X(9866) lies on the cubic K460 and these lines: {2,32}, {316,9865}, {325,8290}, {385,5103}, {732,1916}, {6033,9772}, {8350,9744}
X(9866) = reflection of X(i) in X(j) for these (i,j): (385,9478), (8290,325)
X(9866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (83,7944,6704), (2896,7785,83)
The reciprocal orthologic center of these triangles is X(6231).
X(9867) lies on these lines: {2,371},, {316,9868}
The reciprocal orthologic center of these triangles is X(6230).
X(9868) lies on these lines: {2,372}, {316,9867}
X(9869) = SW*(SW^2+S^2)*X(3)+(9*R^2*S^2-S^2*SW-SW^3)*X(111)
The reciprocal orthologic center of these triangles is X(9870).
X(9869) lies on these lines: {3,111}, {511,5971}, {2780,9759}, {2854,8667}
X(9869) = 1st-tri-squares-to-Artzt similarity image of X(13641)
The reciprocal orthologic center of these triangles is X(9869).
X(9870) lies on these lines: {2,2418}, {23,385}, {111,538}, {352,5969}, {698,5108}, {3734,9465}, {3972,5354}, {6031,8667}
X(9870) = reflection of X(i) in X(j) for these (i,j): (5971,111)
X(9870) = 1st-tri-squares-to-4th-anti-Brocard similarity image of X(13641)
The reciprocal orthologic center of these triangles is X(1296).
X(9871) lies on these lines: {3,8617}, {1499,9870}
X(9871) = X(74)-of-4th-anti-Brocard-triangle
X(9872) = (3*S^2*SW+SW^3+27*R^2*S^2)*X(6)-54*R^2*S^2*X(373)
The reciprocal orthologic center of these triangles is X(1296).
X(9872) lies on these lines: {6,373}, {111,524}, {187,2930}, {352,2854}, {512,5104}, {574,599}
X(9873) = (S^2-3*SW^2)*X(4)-2*(S^2-SW^2)*X(32)
The reciprocal orthologic center of these triangles is X(4).
X(9873) lies on these lines: {3,3096}, {4,32}, {5,7846}, {20,1352}, {30,76}, {40,20025}, {147,9737}, {194,542}, {376,7795}, {378,2353}, {381,7828}, {382,9301}, {384,3818}, {511,7893}, {515,9941}, {549,7930}, {631,7914}, {944,20165}, {1503,3094}, {2777,9984}, {3099,5691}, {3104,9982}, {3105,9981}, {5092,7876}, {5152,6033}, {5870,9995}, {5871,9994}, {5999,7885}, {6054,7763}, {7470,7761}, {8359,9774}
X(9873) = X(4)-of-5th-Brocard-triangle
X(9873) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,9996,3096), (4,32,9993), (4,9862,32), (20,2896,3098), (9988,9989,7811)
The reciprocal orthologic center of these triangles is X(40).
X(9874) lies on these lines: {1,7674}, {2,7160}, {7,3555}, {8,6835}, {144,962}, {145,8000}, {153,5175}, {405,5766}, {3832,5815}, {4295,9814}
X(9874) = anticomplement of X(7160)
X(9874) = reflection of X(i) in X(j) for these (i,j): (145,8000)
X(9874) = X(176)-of-2nd-Ehrmann-triangle if ABC is acute
X(9874) = orthic-to-2nd-Ehrmann similarity image of X(6291)
The reciprocal orthologic center of these triangles is X(9855).
X(9875) lies on these lines: {1,671}, {8,2796}, {10,8591}, {30,9860}, {148,519}, {542,5691}, {543,3679}, {1698,2482}, {3624,5461}, {3751,9830}, {5587,8724}
X(9875) = midpoint of X(i),X(j) for these {i,j}: {8,8596}
X(9875) = reflection of X(i) in X(j) for these (i,j): (1,671), (8591,10)
X(9875) = X(175)-of-2nd-Ehrmann-triangle if ABC is acute
X(9875) = orthic-to-2nd-Ehrmann similarity image of X(6406)
The reciprocal orthologic center of these triangles is X(9855).
X(9876) lies on these lines: {3,67}, {22,8591}, {23,8596}, {25,671}, {30,9861}, {159,9830}, {5020,5461}, {8185,9875}
X(9876) = circumcircle-inverse of X(32257)
X(9876) = Stammler-circle-inverse of X(32272)
X(9876) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2936,3455,3)
The reciprocal orthologic center of these triangles is X(385).
X(9877) lies on these lines: {2,99}, {98,6233}, {114,9770}, {147,9740}, {262,5503}, {385,9773}, {524,1513}, {530,9750}, {531,9749}, {538,9772}, {542,7710}, {690,9759}, {2782,9743}, {5939,8860}, {7684,9762}, {7685,9760}, {8592,8859}
X(9877) = midpoint of X(i),X(j) for these {i,j}: {147,9740}
X(9877) = reflection of X(i) in X(j) for these (i,j): (98,7610), (671,7615), (9770,114)
X(9877) = X(98)-of-Artzt-triangle
X(9877) = X(99)-of-Artzt-of-Artzt-triangle
X(9877) = anti-Artzt-to-Artzt similarity image of X(99)
The reciprocal orthologic center of these triangles is X(9855).
X(9878) lies on these lines: {2,4159}, {30,148}, {32,671}, {99,7865}, {194,542}, {543,7811}, {599,1975}, {2076,9855}, {2482,3096}, {3094,9830}, {3099,9875}, {5461,7846}, {5989,7924}, {6034,7787}, {7809,8178}
X(9878) = reflection of X(i) in X(j) for these (i,j): (8591,7833), (8782,7811)
The reciprocal orthologic center of these triangles is X(99).
X(9879) lies on the anit-McCay circumcircle and these lines: {23,9871}, {385,512}, {511,8597}, {6310,7824}, {6787,7840}
X(9879) = reflection of X(i) in X(j) for these (i,j): (7840,6787)
X(9879) = X(6323)-of-anti-McCay-triangle
X(9879) = circumsymmedial-to-anti-McCay similarity image of X(99)
X(9879) = McCay-to-anti-McCay similarity image of X(12525)
The reciprocal orthologic center of these triangles is X(9855).
X(9880) lies on these lines: {2,9734}, {3,5461}, {4,542}, {5,2482}, {30,115}, {98,3543}, {99,3545}, {114,381}, {148,3839}, {262,9773}, {376,6036}, {511,8352}, {530,5479}, {531,5478}, {547,9167}, {620,5055}, {1352,7620}, {1598,9876}, {1699,9875}, {2782,3845}, {2794,3830}, {3091,8591}, {3455,7530}, {3832,8596}, {5054,6722}, {5071,6721}, {5480,9830}, {5969,6248}
X(9880) = midpoint of X(i),X(j) for these {i,j}: {4,671}, {98,3543}, {148,6054}, {381,6321}
X(9880) = reflection of X(i) in X(j) for these (i,j): (3,5461), (114,381), (376,6036), (2482,5), (6055,115)
X(9880) = polar-circle-inverse of X(32234)
X(9880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115,31709,31710, (148,3839,6054), (376,9166,6036)
The reciprocal orthologic center of these triangles is X(9855).
X(9881) lies on these lines: {1,2482}, {8,8591}, {10,190}, {30,9864}, {40,542}, {99,519}, {291,4424}, {484,1018}, {517,8724}, {524,5184}, {543,3679}, {551,7983}, {1698,5461}, {2782,3654}, {3416,9830}, {3617,8596}, {3828,9166}, {4781,6542}, {5587,9880}, {5847,8593}, {8193,9876}
X(9881) = midpoint of X(i),X(j) for these {i,j}: {8,8591}
X(9881) = reflection of X(i) in X(j) for these (i,j): (1,2482), (671,10), (7983,551)
The reciprocal orthologic center of these triangles is X(9855).
X(9882) lies on these lines: {6,598}, {30,6227}, {530,6271}, {531,6270}, {542,5871}, {543,5861}, {1271,8591}, {2482,5591}, {5589,9875}, {5595,9876}, {5689,9881}, {5969,6273}, {6202,9880}, {6215,8724}
X(9882) = reflection of X(i) in X(j) for these (i,j): (6319,5861), (9883,671)
The reciprocal orthologic center of these triangles is X(9855).
X(9883) lies on these lines: {6,598}, {30,6226}, {530,6269}, {531,6268}, {542,5870}, {543,5860}, {1270,8591}, {2482,5590}, {5588,9875}, {5594,9876}, {5688,9881}, {5969,6272}, {6201,9880}, {6214,8724}
X(9883) = reflection of X(i) in X(j) for these (i,j): (6320,5860), (9882,671)
The reciprocal orthologic center of these triangles is X(9855).
X(9884) lies on these lines: {1,671}, {8,2482}, {30,7970}, {98,3655}, {99,519}, {145,8591}, {518,8593}, {530,7974}, {531,7975}, {542,944}, {543,3241}, {551,9166}, {952,8724}, {2796,3244}, {3242,9830}, {3616,5461}, {3623,8596}, {5603,9880}, {5604,9883}, {5605,9882}, {5969,7976}, {8192,9876}
X(9884) = midpoint of X(i),X(j) for these {i,j}: {145,8591}
X(9884) = reflection of X(i) in X(j) for these (i,j): (8,2482), (98,3655), (671,1), (7983,3241)
The reciprocal orthologic center of these triangles is X(8594).
X(9885) lies on these lines: {2,99}, {3,530}, {15,524}, {30,9749}, {187,8595}, {303,8589}, {531,5617}, {542,9735}, {616,8182}, {618,9761}, {1992,9115}, {2936,3131}, {3106,5969}, {3849,5979}, {5464,9830}, {5569,5980}, {8593,9117}
X(9885) = reflection of X(i) in X(j) for these (i,j): (9761,618), (9886,2482)
X(9885) = anticomplement of X(33477)
X(9885) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7618,9886)
The reciprocal orthologic center of these triangles is X(8595).
X(9886) lies on these lines: {2,99}, {3,531}, {16,524}, {30,9750}, {187,8594}, {302,8589}, {530,5613}, {542,9736}, {617,8182}, {619,9763}, {1992,9117}, {2936,3132}, {3107,5969}, {3849,5978}, {5463,9830}, {5569,5981}, {8593,9115}
X(9886) = reflection of X(i) in X(j) for these (i,j): (9763,619), (9885,2482)
X(9886) = anticomplement of X(33476)
X(9886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7618,9885)
The reciprocal orthologic center of these triangles is X(9888).
X(9887) lies on these lines: {2,39}, {1916,8352}, {3906,9485}, {5969,9855}
X(9887) = reflection of X(i) in X(j) for these (i,j): (9865,7757)
The reciprocal orthologic center of these triangles is X(9887).
X(9888) lies on these lines: {2,99}, {3,5969}, {39,5182}, {98,538}, {524,9142}, {542,9737}, {1916,2021}, {1976,5118}, {2782,8716}, {2936,3148}, {3098,8182}, {3849,5503}, {5024,5026}, {5152,7757}, {6034,8369}, {6082,6323}, {6287,8724}
X(9888) = reflection of X(i) in X(j) for these (i,j): (9877,7622)
X(9889) = (9*S^4-14*S^2*SW^2-7*SW^4)*X(2)-12*(S^4-SW^4)*X(32)
The reciprocal orthologic center of these triangles is X(9890).
X(9889) lies on these lines: {2,32}
The reciprocal orthologic center of these triangles is X(9889).
X(9890) lies on these lines: {2,99}, {147,754}, {237,2936}, {524,9301}, {542,5171}, {3095,5969}, {5569,9302}, {8182,9830}
X(9890) = reflection of X(i) in X(j) for these (i,j): (7615,9877), (9888,2482)
X(9890) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2482,9888,7618)
X(9891) = (3*S^2-4*S*SW-SW^2)*SW*X(2)-6*(S^2-SW^2)*S*X(371)
The reciprocal orthologic center of these triangles is X(9892).
X(9891) lies on these lines: {2,371}
X(9892) = (S+SW)*X(2)+(S-SW)*X(99)
The reciprocal orthologic center of these triangles is X(9891).
X(9892) lies on these lines: {2,99}, {542,9738}, {2936,3155}, {3102,5969}, {6289,8724}
X(9892) = reflection of X(i) in X(j) for these (i,j): (9894,2482)
X(9892) = complement of X(33433)
X(9892) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9888,9894), (99,9890,9894), (671,7618,9894)
X(9893) = (3*S^2+4*S*SW-SW^2)*SW*X(2)+6*(S^2-SW^2)*S*X(372)
The reciprocal orthologic center of these triangles is X(9894).
X(9893) lies on these lines: {2,372}, {5162,9891}
X(9894) = (S-SW)*X(2)+(S+SW)*X(99)
The reciprocal orthologic center of these triangles is X(9893).
X(9894) lies on these lines: {2,99}, {542,9739}, {2936,3156}, {3103,5969}, {6290,8724}
X(9894) = reflection of X(i) in X(j) for these (i,j): (9892,2482)
X(9894) = complement of X(33432)
X(9894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9888,9892), (99,9890,9892), (671,7618,9892)
The reciprocal orthologic center of these triangles is X(3).
Let Wa be the inverter of the circumcircle and A-excircle, and define Wb, Wc cyclically. X(9895) is the radical center of Wa, Wb, Wc. (Randy Hutson, July 20, 2016)
X(9895) lies on these lines: {1,7535}, {3,19}, {5,10}, {58,1731}, {65,1714}, {72,5271}, {197,6642}, {389,916}, {392,7532}, {405,1824}, {442,1829}, {612,3295}, {942,6678}, {1697,9817}, {1872,6913}, {1902,8226}, {1953,3682}, {2262,5752}, {2355,2915}, {2475,5146}, {3101,7523}, {3702,5082}, {9052,9822}
X(9895) = midpoint of X(i),X(j) for these {i,j}: {3,1871}
X(9895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9816,7535)
The reciprocal orthologic center of these triangles is X(9833).
X(9896) lies on these lines: {1,68}, {10,6193}, {155,5587}, {355,3564}, {539,3679}, {912,4338}, {1069,9581}, {1147,1698}, {3157,9578}, {3586,6238}, {3624,5449}, {5534,8270}, {5654,7989}, {6237,8274}, {7352,9613}
X(9896) = reflection of X(i) in X(j) for these (i,j): (1,68), (6193,10)
X(9897) = (R+6*r)*X(1)-8*r*X(5)
The reciprocal orthologic center of these triangles is X(3).
X(9897) lies on these lines: {1,5}, {8,191}, {10,6224}, {100,993}, {104,6796}, {149,519}, {150,4089}, {153,3585}, {214,1698}, {484,515}, {528,4677}, {1145,4668}, {1320,3633}, {1862,5155}, {2800,5691}, {2801,4312}, {2802,3632}, {2829,7992}, {3336,4293}, {3586,8275}, {3621,9802}, {3624,6702}, {4325,4848}, {4342,4857}, {4530,5540}, {5586,9613}, {5588,6262}, {5589,6263}, {5694,5697}, {5840,7991}, {5902,6797}
X(9897) = midpoint of X(i),X(j) for these {i,j}: {3621,9802}
X(9897) = reflection of X(i) in X(j) for these (i,j): (1,80), (3633,1320), (5541,8), (6224,10), (6326,355), (7972,11)
X(9897) = isogonal conjugate of X(32899)
X(9897) = anticomplement of X(33337)
X(9897) = inverse-in-circumcircle-of-reflection-triangle-of-X(1) of X(11)
X(9897) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,7972,1), (80,7972,11)
The reciprocal orthologic center of these triangles is X(40).
X(9898) lies on these lines: {1,5920}, {10,9874}, {388,4312}, {497,4866}, {950,3632}, {1697,5223}, {2093,5586}, {3057,3715}, {5119,7992}
X(9898) = reflection of X(i) in X(j) for these (i,j): (1,7160), (9874,10)
The reciprocal orthologic center of these triangles is X(4).
X(9899) lies on these lines: {1,64}, {10,6225}, {30,9896}, {40,2939}, {57,6285}, {165,1498}, {1697,7355}, {1698,2883}, {1699,6247}, {2192,3361}, {3357,3576}, {3624,6696}, {5587,5878}, {5588,6266}, {5589,6267}, {5656,6684}, {5904,6001}
X(9899) = reflection of X(i) in X(j) for these (i,j): (1,64), (6225,10)
The reciprocal orthologic center of these triangles is X(3).
X(9900) lies on these lines: {1,14}, {10,617}, {165,5474}, {515,6773}, {530,9875}, {531,3679}, {542,3751}, {619,1698}, {1699,5479}, {3576,6774}, {3624,6670}, {5587,5613}, {5588,6269}, {5589,6271}
X(9900) = reflection of X(i) in X(j) for these (i,j): (1,14), (617,10)
The reciprocal orthologic center of these triangles is X(3).
X(9901) lies on these lines: {1,13}, {10,616}, {165,5473}, {515,6770}, {530,3679}, {531,9875}, {542,3751}, {618,1698}, {1699,5478}, {3576,6771}, {3624,6669}, {5587,5617}, {5588,6268}, {5589,6270}
X(9901) = reflection of X(i) in X(j) for these (i,j): (1,13), (616,10)
The reciprocal orthologic center of these triangles is X(3).
X(9902) lies on these lines: {1,76}, {8,726}, {10,194}, {39,1698}, {40,2782}, {43,3765}, {148,1330}, {262,7989}, {313,1740}, {511,5691}, {538,3679}, {698,3416}, {732,3751}, {1699,6248}, {3095,5587}, {3550,4039}, {3624,3934}, {4297,6194}, {5588,6272}, {5589,6273}, {5969,9875}, {6684,7709}, {7697,8227}
X(9902) = reflection of X(i) in X(j) for these (i,j): (1,76), (194,10)
X(9902) = X(76)-of-Aquila-triangle
X(9902) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,194,3097)
The reciprocal orthologic center of these triangles is X(3).
X(9903) lies on these lines: {1,83}, {8,1757}, {10,2896}, {355,9860}, {732,3751}, {754,3679}, {1572,3632}, {1698,6292}, {1699,6249}, {3624,6704}, {5587,6287}, {5588,6274}, {5589,6275}, {7987,9751}
X(9903) = reflection of X(i) in X(j) for these (i,j): (1,83), (2896,10)
X(9903) = X(83)-of-Aquila-triangle
The reciprocal orthologic center of these triangles is X(4).
X(9904) lies on the Bevan circle and these lines: {1,74}, {10,146}, {40,2940}, {46,7727}, {110,165}, {113,1698}, {125,1699}, {399,3579}, {516,3448}, {541,3679}, {690,9860}, {1054,2776}, {1282,2772}, {1386,5621}, {1695,3031}, {1697,3028}, {1717,1854}, {2100,2575}, {2101,2574}, {2771,5541}, {2775,5540}, {2777,5691}, {2781,3751}, {2931,9591}, {3043,9587}, {3047,9586}, {3624,6699}, {5587,7728}, {5588,7726}, {5589,7725}
X(9904) = reflection of X(i) in X(j) for these (i,j): (1,74), (146,10), (399,3579), (2948,40)
X(9904) = antipode of X(2948) in Bevan circle
X(9904) = X(3023)-of-tangential triangle-of-excentral-triangle
The reciprocal orthologic center of these triangles is X(4).
X(9905) lies on these lines: {1,54}, {3,5904}, {10,2888}, {40,1154}, {46,7356}, {165,7691}, {195,517}, {355,2948}, {539,3679}, {1209,1698}, {1493,7982}, {1699,3574}, {2917,9590}, {3624,6689}, {5119,6286}, {5587,6288}, {5588,6276}, {5589,6277}, {5886,8254}, {8274,9572}
X(9905) = reflection of X(i) in X(j) for these (i,j): (1,54), (2888,10)
The reciprocal orthologic center of these triangles is X(3).
X(9906) lies on these lines: {1,486}, {10,487}, {176,5261}, {355,3564}, {642,1698}, {1699,6251}, {3624,6119}, {5587,6290}, {5588,6280}, {5589,6281}
X(9906) = reflection of X(i) in X(j) for these (i,j): (1,486), (487,10)
X(9906) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355,3751,9907)
The reciprocal orthologic center of these triangles is X(3).
X(9907) lies on these lines: {1,485}, {10,488}, {175,5261}, {355,3564}, {641,1698}, {1699,6250}, {3624,6118}, {5587,6289}, {5588,6278}, {5589,6279}
X(9907) = reflection of X(i) in X(j) for these (i,j): (1,485), (488,10)
X(9907) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (355,3751,9906)
X(9908) = (SW^2+2*R^2*(R^2-2*SW))*X(3)+2*(7*R^2-2*SW)*R^2*X(49)
The reciprocal orthologic center of these triangles is X(9833).
X(9908) lies on these lines: {3,49}, {22,6193}, {25,68}, {26,159}, {143,8548}, {912,3556}, {1352,6642}, {1619,7387}, {5020,5449}, {5654,7395}, {7393,9820}, {8185,9896}
X(9908) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1147,7689,5447)
X(9909) = 4*R^2*X(2)-SW*X(3)
The reciprocal orthologic center of these triangles is X(2)
As a point on the Euler line, X(9909) has Shinagawa coefficients (-E-3*F, 3*E+3*F).
X(9909) lies on these lines: {2,3}, {51,3796}, {55,7298}, {56,5345}, {154,511}, {159,524}, {161,542}, {184,1351}, {187,1611}, {197,4421}, {394,1495}, {519,9798}, {534,1486}, {539,9908}, {541,2931}, {543,9876}, {569,3527}, {612,7302}, {614,5370}, {999,5322}, {1151,8854}, {1152,8855}, {1184,1384}, {1196,3053}, {1350,9306}, {1460,7295}, {1661,9530}, {1994,7712}, {2979,6090}, {3060,5093}, {3220,3928}, {3241,8192}, {3295,5310}, {3679,8185}, {3920,8144}, {3929,5285}, {5012,9777}, {5023,8770}, {5085,5943}, {5204,5272}, {5217,5268}, {5306,8573}, {5329,7083}, {5594,5860}, {5595,5861}, {5640,5644}, {7753,9700}, {8266,8556}, {8276,9683}, {9300,9609}
X(9909) = reflection of X(i) in X(j) for these (i,j): (3167,154)
X(9909) = circumcircle-inverse-of X(5159)
X(9909) = Thomson-isogonal conjugate of X(32605)
X(9909) = X(2)-of-3rd-antipedal-triangle-of-X(3)
X(9909) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,428,381), (3,25,5020), (3,7517,1598), (3,9714,3517), (4,9715,3), (20,3515,3), (20,6353,1368), (22,23,25), (22,25,3), (22,1995,6636), (22,7485,7492), (25,3515,6353), (25,7484,1995), (25,9715,6676), (26,7387,3), (51,3796,5050), (394,1495,8780), (8185,9591,8193)
X(9910) = (6*R^2-SW)*X(25)-2*R^2*X(84)
The reciprocal orthologic center of these triangles is X(40).
X(9910) lies on these lines: {3,3452}, {25,84}, {971,7387}, {5020,6705}, {5594,6257}, {5595,6258}, {6001,9798}, {7971,8192}, {7992,8185}
X(9911) = (6*R^2-SW)*X(25)-2*R^2*X(40)
The reciprocal orthologic center of these triangles is X(4).
X(9911) lies on these lines: {3,142}, {4,8193}, {10,1598}, {22,962}, {24,6361}, {25,40}, {161,7973}, {515,9910}, {517,3556}, {580,7083}, {999,4347}, {1699,7395}, {3515,9625}, {3517,5493}, {3579,6642}, {5020,6684}, {5198,5587}, {5690,7530}, {7484,8227}, {7503,9812}, {7982,8192}, {7991,8185}, {9589,9591}
X(9911) = reflection of X(i) in X(j) for these (i,j): (9798,7387)
X(9912) = (6*R^2-SW)*X(25)-2*R^2*X(80)
The reciprocal orthologic center of these triangles is X(3).
X(9912) lies on these lines: {3,214}, {22,6224}, {25,80}, {26,952}, {100,8193}, {104,1602}, {2771,3556}, {2829,9910}, {5020,6702}, {5594,6262}, {5595,6263}, {5840,9911}, {7972,8192}, {8185,9897}
X(9913) = (6*R^2-SW)*X(25)-2*R^2*X(104)
The reciprocal orthologic center of these triangles is X(40).
X(9913) lies on these lines: {3,119}, {11,1598}, {22,153}, {25,104}, {515,9912}, {952,7387}, {1484,7530}, {1768,8185}, {2787,9861}, {2800,9798}, {2802,9911}, {5020,6713}
X(9914) = (6*R^2-SW)*X(25)-2*R^2*X(64)
The reciprocal orthologic center of these triangles is X(4).
X(9914) lies on these lines: {3,1661}, {20,1619}, {22,6225}, {25,64}, {30,9908}, {40,197}, {84,7169}, {159,1350}, {1593,3574}, {1598,6247}, {1853,5198}, {3357,6642}, {5020,6696}, {5594,6266}, {5595,6267}, {5646,8567}, {6000,7387}, {6001,9911}, {7973,8192}, {8185,9899}
The reciprocal orthologic center of these triangles is X(3).
X(9915) lies on these lines: {3,619}, {14,25}, {22,617}, {24,6773}, {159,542}, {530,9876}, {531,9909}, {1598,5479}, {3129,3439}, {5020,6670}, {5594,6269}, {5595,6271}, {6642,6774}, {7974,8192}, {8185,9900}
The reciprocal orthologic center of these triangles is X(3).
X(9916) lies on these lines: {3,618}, {13,25}, {22,616}, {24,6770}, {159,542}, {530,9909}, {531,9876}, {1598,5478}, {3130,3438}, {5020,6669}, {5594,6268}, {5595,6270}, {6642,6771}, {7975,8192}, {8185,9901}
X(9917) = (6*R^2-SW)*X(25)-2*R^2*X(76)
The reciprocal orthologic center of these triangles is X(3).
X(9917) lies on these lines: {3,6}, {22,194}, {25,76}, {159,732}, {237,3926}, {262,7395}, {538,9909}, {730,9798}, {1598,6248}, {2353,3504}, {2782,7387}, {3167,3202}, {3934,5020}, {5286,7467}, {5594,6272}, {5595,6273}, {5969,9876}, {7484,7786}, {7529,7697}, {7976,8192}, {8185,9902}
X(9917) = X(76)-of-Ara-triangle
X(9918) = (6*R^2-SW)*X(25)-2*R^2*X(83)
The reciprocal orthologic center of these triangles is X(3).
X(9918) lies on these lines: {3,2916}, {22,2896}, {25,83}, {26,9861}, {159,732}, {754,9909}, {1598,6249}, {5020,6704}, {5594,6274}, {5595,6275}, {7977,8192}, {8185,9903}
X(9918) = X(83)-of-Ara-triangle
X(9919) = (6*R^2-SW)*X(25)-2*R^2*X(74)
The reciprocal orthologic center of these triangles is X(4).
X(9919) lies on these lines: {3,113}, {22,146}, {25,74}, {125,1598}, {159,399}, {541,2931}, {690,9861}, {1177,5050}, {1498,5898}, {1539,9818}, {2778,3556}, {2937,5878}, {3581,5899}, {5020,6699}, {5594,7726}, {5595,7725}, {5621,7687}, {5663,7387}, {7517,9914}, {7978,8192}, {8185,9904}
X(9920) = (6*R^2-SW)*X(25)-2*R^2*X(54)
The reciprocal orthologic center of these triangles is X(4).
X(9920) lies on these lines: {3,161}, {22,2888}, {25,54}, {154,9704}, {159,195}, {539,9908}, {1154,7387}, {1482,3556}, {1498,5898}, {1598,3574}, {2937,9833}, {5020,6689}, {5544,6642}, {5594,6276}, {5595,6277}, {5899,6243}, {7592,7730}, {7979,8192}, {8185,9905}
X(9920) = reflection of X(i) in X(j) for these (i,j): (3,2917)
X(9921) = (6*R^2-SW)*X(25)-2*R^2*X(486)
The reciprocal orthologic center of these triangles is X(3).
X(9921) lies on these lines: {3,640}, {22,487}, {25,486}, {26,159}, {1598,6251}, {3155,8276}, {5020,6119}, {5594,6280}, {5595,6281}, {7980,8192}, {8185,9906}
X(9921) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,159,9922)
X(9922) = (6*R^2-SW)*X(25)-2*R^2*X(485)
The reciprocal orthologic center of these triangles is X(3).
X(9922) lies on these lines: {3,639}, {22,488}, {25,485}, {26,159}, {1598,6250}, {3156,8277}, {5020,6118}, {5594,6278}, {5595,6279}, {7981,8192}, {8185,9907}, {8996,9909}
X(9922) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (26,159,9921)
X(9923) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(68)
The reciprocal orthologic center of these triangles is X(9833).
X(9923) lies on these lines: {32,68}, {539,7811}, {1147,3096}, {2896,6193}, {3094,3564}, {3099,9896}, {5449,7846}
X(9924) = "(8*R^2-SW)*X(6)-2*(6*R^2-SW)*X(25)
The reciprocal orthologic center of these triangles is X(9925).
X(9924) lies on the cubic K075 and these lines: {6,25}, {20,64}, {66,599}, {141,1853}, {221,1469}, {511,1498}, {524,5596}, {1181,6403}, {1351,6759}, {1619,8681}, {1854,3057}, {2192,3056}, {3197,3779}, {3564,9833}, {3589,8547}, {3763,5646}, {5085,8549}, {6391,9909}, {6776,9786}
X(9924) = reflection of X(i) in X(j) for these (i,j): (6,159), (64,1350), (1351,6759)
X(9924) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,159,154), (25,6467,6)
X(9925) = (R^2-SW)*X(3)+3*R^2*X(69)
The reciprocal orthologic center of these triangles is X(9924).
X(9925) lies on these lines: {3,69}, {6,1493}, {26,524}, {155,2930}, {193,3518}, {539,7514}, {575,1147}, {599,7516}, {1992,7506}, {1994,1995}, {3292,8538}, {5050,9545}, {7555,9908}
X(9925) = reflection of X(i) in X(j) for these (i,j): (8548,1147)
X(9926) = R^2*(6*R^2-SW)*X(68)+(2*R^2-SW)*(9*R^2-2*SW)*X(895)
The reciprocal orthologic center of these triangles is X(7387).
X(9926) lies on these lines: {6,1147}, {68,895}, {155,8541}, {3628,8542}, {5621,7689}, {6193,8537}, {7514,8548}, {9813,9820}
X(9927) = (3*R^2-SW)*X(4)+R^2*X(52)
The reciprocal orthologic center of these triangles is X(9833).
X(9927) lies on these lines: {3,125}, {4,52}, {5,578}, {30,3357}, {113,2904}, {155,195}, {389,7706}, {541,5895}, {542,8548}, {546,576}, {912,7686}, {1092,2072}, {1209,7503}, {1598,9908}, {1656,7666}, {1699,9896}, {1993,7547}, {3091,5654}, {3167,3851}, {3448,6241}, {3580,6240}, {3583,6238}, {3585,7352}, {5447,6643}, {5462,9815}, {5892,6815}, {7746,8571}
X(9927) = midpoint of X(i),X(j) for these {i,j}: {4,68}
X(9927) = reflection of X(i) in X(j) for these (i,j): (3,5449), (155,5448), (1147,5)
X(9927) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (155,381,5448)
X(9927) = X(6261)-of-orthic-triangle if ABC is acute
X(9927) = X(4)-of-Ehrmann-vertex-triangle
The reciprocal orthologic center of these triangles is X(9833).
X(9928) lies on these lines: {1,1147}, {3,63}, {8,6193}, {10,68}, {40,6237}, {46,4551}, {65,921}, {155,517}, {165,7689}, {539,3679}, {946,5654}, {1069,3057}, {1482,3167}, {1698,5449}, {1699,5448}, {2182,3211}, {2948,5691}, {3416,3564}, {5119,6238}, {5446,7713}, {5587,9927}, {5886,9820}, {7969,8909}, {8193,9908}
X(9928) = midpoint of X(i),X(j) for these {i,j}: {8,6193}
X(9928) = reflection of X(i) in X(j) for these (i,j): (1,1147), (68,10)
X(9929) = 2*(2*R^2-SW)*(S-2*SW)*X(5)+SW*(S+8*R^2-4*SW)*X(6)
The reciprocal orthologic center of these triangles is X(9833).
X(9929) lies on these lines: {5,6}, {30,6267}, {52,1163}, {1147,5591}, {5589,9896}, {5595,9908}, {6202,9927}
X(9930) = 2*(2*R^2-SW)*(S+2*SW)*X(5)+SW*(4*SW+S-8*R^2)*X(6)
The reciprocal orthologic center of these triangles is X(9833).
X(9930) lies on these lines: {5,6}, {1147,5590}, {6201,9927}
X(9930) = reflection of X(i) in X(j) for these (i,j): (9929,68)
X(9931) = (8*R^2+r^2-s^2-SW)*X(33)-(SW+r^2-s^2)*X(155)
The reciprocal orthologic center of these triangles is X(7387).
X(9931) lies on these lines: {33,155}, {68,1062}, {3564,8144}, {6193,6198}, {8540,9926}, {9645,9908}, {9817,9820}
X(9932) = (3*R^2-SW)*X(3)+R^2*X(68)
The reciprocal orthologic center of these triangles is X(7387).
X(9932) lies on these lines: {3,68}, {22,6241}, {24,110}, {26,6759}, {35,9931}, {182,8548}, {186,6193}, {389,1147}, {575,9926}, {1658,3564}, {5449,7514}, {5654,7506}, {5878,7387}, {6642,9815}, {7526,9927}, {7689,8717}, {8909,9682}
X(9932) = reflection of X(i) in X(j) for these (i,j): (9926,575)
X(9932) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (155,2931,24)
The reciprocal orthologic center of these triangles is X(9833).
X(9933) lies on these lines: {1,68}, {8,1147}, {30,7973}, {145,6193}, {155,952}, {355,5654}, {519,9928}, {539,3241}, {912,3057}, {962,7978}, {1483,3242}, {3616,5449}, {5446,7718}, {5603,9927}, {5604,9930}, {5605,9929}, {5731,7689}, {5790,9820}, {8192,9908}
X(9933) = midpoint of X(i),X(j) for these {i,j}: {145,6193}
X(9933) = reflection of X(i) in X(j) for these (i,j): (8,1147), (68,1)
X(9934) = R^2*X(20)-(5*R^2-SW)*X(110)
The reciprocal orthologic center of these triangles is X(5654).
X(9934) lies on these lines: {20,110}, {25,974}, {26,1498}, {30,5504}, {74,186}, {113,206}, {125,3542}, {154,1511}, {156,5895}, {159,399}, {265,1177}, {542,5596}, {1112,1181}, {1974,5622}, {2883,7728}, {3047,3146}, {3546,5972}
X(9934) = midpoint of X(i),X(j) for these {i,j}: {399,9919}
X(9934) = reflection of X(i) in X(j) for these (i,j): (110,6759), (2935,1511), (7728,2883)
X(9934) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (154,2935,1511)
X(9935) = 6*R^2*X(51)-(2*R^2+SW)*X(54)
The reciprocal orthologic center of these triangles is X(9936).
X(9935) lies on these lines: {20,2888}, {51,54}, {154,1493}, {159,195}, {539,9833}, {1154,1498}, {1209,6697}, {1503,3519}, {2917,6644}, {5596,5965}, {6152,7592}
X(9936) = 2*(2*R^2+SW)*X(5)-3*SW*X(6)
The reciprocal orthologic center of these triangles is X(9935).
X(9936) lies on the cubic K120 and these lines: {4,539}, {5,6}, {20,6193}, {193,5446}, {524,7387}, {631,1147}, {912,3057}, {1154,6293}, {1216,6776}, {3167,3526}, {3292,3548}, {3528,7689}, {3832,9927}, {3855,5448}, {4317,7352}, {5067,5449}, {5070,9820}, {5965,6759}, {7393,8550}, {9644,9931}, {9715,9908}
X(9936) = reflection of X(i) in X(j) for these (i,j): (68,155)
X(9936) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (68,155,5654)
X(9937) = (2*R^2-SW)*X(3)+2*R^2*X(68)
The reciprocal orthologic center of these triangles is X(7387).
X(9937) lies on the Stammler hyperbola and these lines: {3,68}, {6,1147}, {24,6193}, {25,52}, {26,159}, {55,9931}, {161,1498}, {195,973}, {399,7517}, {539,2917}, {912,9798}, {1209,7395}, {1609,3133}, {2918,9715}, {2931,3515}, {3580,8907}, {5020,9820}, {5449,7393}, {5654,7529}, {5663,9914}, {8903,9929}, {9714,9936}, {9818,9927}
X(9937) = reflection of X(i) in X(j) for these (i,j): (3,9932), (9908,26)
X(9938) = (5*R^2-SW)*X(3)-R^2*X(68)
The reciprocal orthologic center of these triangles is X(7387).
X(9938) lies on these lines: {3,68}, {36,9931}, {155,378}, {511,9926}, {1147,4550}, {3520,6193}, {5504,7723}, {5888,7509}, {6644,9927}, {9818,9820}
X(9938) = reflection of X(i) in X(j) for these (i,j): (9932,3)
X(9939) = (3*S^2-5*SW^2)*X(2)-4*(S^2-SW^2)*X(32)
The reciprocal orthologic center of these triangles is X(6054).
X(9939) lies on these lines: {2,32}, {3,7840}, {20,542}, {30,9863}, {69,5104}, {76,3849}, {187,7850}, {194,524}, {384,599}, {385,7841}, {543,7802}, {597,7876}, {598,3934}, {671,7751}, {1384,7931}, {1975,9855}, {1992,7791}, {2482,7796}, {3053,7939}, {3314,8369}, {3552,7768}, {3793,7806}, {3933,8598}, {3972,7848}, {5007,7936}, {5008,7937}, {5023,7947}, {5077,7754}, {5171,6054}, {5206,7917}, {5215,7940}, {5569,7769}, {5969,9878}, {6179,7817}, {7610,7773}, {7622,7903}, {7761,7766}, {7762,7904}, {7763,8182}, {7767,7823}, {7771,7845}, {7780,7860}, {7782,7882}, {7784,7932}, {7805,7910}, {7825,9166}, {7830,7877}, {7837,8356}, {7885,8859}
X(9939) = anticomplement of X(7812)
X(9939) = midpoint of X(i),X(j) for these {i,j}: {7833,7893}
X(9939) = reflection of X(i) in X(j) for these (i,j): (2,7811), (194,7833), (7762,8359), (7812,7810), (7823,8370), (7833,7750), (7837,8356), (8370,7767)
X(9939) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,7883,7938), (2,7900,7775), (2,7929,7883), (32,7883,2), (32,7929,7938), (187,7850,7897), (315,7793,7912), (1078,7775,2), (3053,7939,7945), (6179,7873,7933), (7750,7893,194), (7810,7812,2), (7811,7812,7810)
The reciprocal orthologic center of these triangles is X(10).
Let Oa be the circle tangent to the incircle and passing through B and C, and define Ob and Oc cyclically. Let A* be the intersection, other than A, of circles Ob and Oc, and define B* and C* cyclically. Triangle A*B*C* is perspective to ABC at X(57), and X(9940) = X(3)-of-A*B*C*. See also X(479). (Randy Hutson, July 20, 2016)
X(9940) lies on these lines: {1,3}, {2,1071}, {4,5439}, {5,142}, {7,5810}, {30,5806}, {72,631}, {84,6913}, {140,912}, {226,6922}, {244,4300}, {355,443}, {452,2096}, {515,3812}, {518,5771}, {581,3752}, {938,6916}, {944,3753}, {946,3742}, {960,5884}, {1001,1158}, {1006,3916}, {1125,6001}, {1210,6907}, {1439,5909}, {1465,4303}, {1490,5437}, {1768,5259}, {1829,7501}, {1858,5433}, {1871,7490}, {1876,7412}, {2771,5972}, {2801,3634}, {3090,5927}, {3149,3306}, {3218,6986}, {3419,6897}, {3487,6926}, {3523,3868}, {3555,5657}, {3616,6935}, {3698,5881}, {4297,5883}, {4675,5713}, {5249,6831}, {5435,6988}, {5440,6940}, {5534,9709}, {5658,6964}, {5705,5784}, {5715,6173}, {5722,6850}, {5728,6908}, {5770,5791}, {5787,6826}, {5805,6851}, {5836,5882}, {5886,6847}, {5887,6857}, {6864,9799}, {8099,8733}, {8582,8728}
X(9940) = midpoint of X(i),X(j) for these {i,j}: {3,942}, {960,5884}, {4297,7686}, {5836,5882}
X(9940) = reflection of X(i) in X(j) for these (i,j): (5044,140)
X(9940) = complement of X(5777)
X(9940) = X(5)-of-Ascella-triangle
X(9940) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1482,6282), (3,2095,40), (3,5708,5709), (3,5709,3579), (7,6865,5812), (57,8726,3), (443,5768,355), (1490,5437,6918), (4297,5883,7686), (5770,6989,5791)
The reciprocal orthologic center of these triangles is X(3).
X(9941) lies on these lines: {1,32}, {8,2896}, {10,3096}, {40,3098}, {515,9873}, {517,9821}, {518,3094}, {519,7811}, {944,9862}, {984,3730}, {986,3874}, {1125,7846}, {1482,9301}, {1698,7914}, {2076,3242}, {3679,7865}
X(9941) = X(1)-of-5th-Brocard-triangle
X(9941) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3099,32), (8,2896,9857)
The reciprocal orthologic center of these triangles is X(72).
X(9942) lies on these lines: {3,960}, {5,142}, {84,405}, {515,942}, {518,5534}, {1040,1498}, {1729,9119}, {1837,5768}, {1864,6848}, {3812,6826}, {5572,5805}, {5658,6834}, {5730,6282}, {5744,6962}, {5745,5777}, {5784,6908}, {5787,6256}, {6675,6705}, {6835,9776}, {8095,8733}, {8096,8734}
The reciprocal orthologic center of these triangles is X(65).
X(9943) lies on these lines: {1,1407}, {3,960}, {4,3812}, {8,9859}, {9,7992}, {10,971}, {20,65}, {30,7686}, {40,518}, {46,7580}, {63,5584}, {72,165}, {73,9371}, {84,958}, {103,831}, {221,1040}, {241,774}, {354,962}, {392,7987}, {405,1709}, {411,1155}, {412,1859}, {464,6254}, {515,5836}, {516,942}, {517,550}, {912,3579}, {944,3880}, {946,3742}, {986,1742}, {990,5711}, {991,3931}, {1001,8726}, {1038,1854}, {1044,1427}, {1046,9441}, {1064,4719}, {1125,9856}, {1254,3000}, {1329,6260}, {1376,1490}, {1385,8717}, {1538,3825}, {1698,5927}, {1699,5439}, {1768,3916}, {1788,1864}, {1836,6836}, {1837,6925}, {1938,8142}, {2550,9799}, {2646,6909}, {2951,3339}, {3057,5731}, {3522,3869}, {3555,7991}, {3666,4300}, {3683,6986}, {3697,9588}, {3740,5777}, {3753,5691}, {3811,6244}, {3838,6831}, {3848,8227}, {3868,7957}, {4301,5045}, {4662,5657}, {4711,5690}, {4915,9851}, {4999,6705}, {5087,6922}, {5250,8273}, {5289,7971}, {5302,7330}, {5794,6916}, {5806,5883}, {9776,9800}
X(9943) = midpoint of X(i),X(j) for these {i,j}: {20,65}, {2951,5728}, {3555,7991}, {3868,7957}, {3874,5493}
X(9943) = reflection of X(i) in X(j) for these (i,j): (4,3812), (946,9940), (960,3), (4301,5045), (5777,6684), (9856,1125)
X(9943) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1158,4640), (65,5918,20), (946,9940,3742), (3868,9778,7957), (5777,6684,3740)
The reciprocal orthologic center of these triangles is X(65).
X(9944) lies on these lines: {3,37}, {57,1721}, {516,942}, {971,4260}, {1445,1827}, {9776,9801}
X(9945) = 5*r*X(3)-(2*R+r)*X(8)
The reciprocal orthologic center of these triangles is X(8).
X(9945) lies on these lines: {1,6154}, {3,8}, {5,4855}, {11,3601}, {28,1862}, {30,908}, {57,1317}, {63,8703}, {72,548}, {78,550}, {80,6174}, {119,8727}, {142,214}, {144,376}, {149,443}, {329,3534}, {519,5122}, {549,3419}, {942,2802}, {2800,9942}, {2951,6282}, {3032,4260}, {3035,3634}, {3036,4745}, {3522,3927}, {3526,5175}, {3530,6734}, {3576,6067}, {3830,5748}, {4190,6147}, {4304,5316}, {4305,9709}, {5126,5853}, {5763,6934}, {6264,8726}, {9776,9802}
X(9945) = midpoint of X(i),X(j) for these {i,j}: {1,6154}, {1145,6224}, {1317,5541}
X(9945) = reflection of X(i) in X(j) for these (i,j): (1387,214)
X(9945) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (100,6224,1145)
X(9946) = (R+r)*X(3)-(2*R+r)*X(214)
The reciprocal orthologic center of these triangles is X(3869).
X(9946) lies on these lines: {3,214}, {57,6326}, {80,6826}, {100,5709}, {116,119}, {153,5768}, {355,5883}, {517,9945}, {912,3911}, {942,952}, {1768,8726}, {2095,8730}, {2771,5972}, {2829,9942}, {3306,5720}, {3628,5777}, {5047,7330}, {5660,5770}, {9776,9803}
The reciprocal orthologic center of these triangles is X(10).
X(9947) lies on these lines: {3,5234}, {4,8}, {5,3947}, {10,971}, {20,3697}, {40,3062}, {65,9656}, {84,9709}, {165,3983}, {210,5691}, {354,7989}, {515,5044}, {516,4662}, {518,5806}, {942,5290}, {944,5129}, {1385,5720}, {1490,9708}, {1698,5789}, {1864,9578}, {2551,5787}, {2801,3812}, {3091,3555}, {3579,7330}, {3740,4297}, {3889,5068}, {5049,8227}, {5261,5728}, {5534,6913}, {5817,9844}, {5918,9588}, {6918,7091}, {7956,9842}, {8582,8728}
X(9947) = midpoint of X(i),X(j) for these {i,j}: {8,9856}, {355,5777}
X(9947) = reflection of X(i) in X(j) for these (i,j): (5045,5)
X(9947) = X(5)-of-Atik-triangle
X(9947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,5927,9856)
X(9948) = (4*R+3*r)*X(8)-(4*R-r)*X(20)
The reciprocal orthologic center of these triangles is X(72).
X(9948) lies on these lines: {4,3062}, {8,20}, {10,971}, {442,5927}, {496,942}, {516,5787}, {950,1709}, {1490,6684}, {1698,5658}, {1750,1788}, {1768,7098}, {2800,4018}, {3295,5882}, {3427,7091}, {3474,4848}, {4297,4640}, {6244,6743}
X(9948) = midpoint of X(i),X(j) for these {i,j}: {4,7992}, {40,9799}
X(9948) = reflection of X(i) in X(j) for these (i,j): (1490,6684)
X(9948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3671,8727,946)
The reciprocal orthologic center of these triangles is X(65).
X(9949) lies on these lines: {8,144}, {10,5927}, {84,4315}, {496,942}, {944,4314}, {1709,4297}, {2476,8582}, {3868,4301}, {4298,7992}, {5690,9947}
The reciprocal orthologic center of these triangles is X(65).
X(9950) lies on these lines: {8,144}, {165,3161}, {346,2951}, {982,3663}, {990,8583}, {1721,8580}, {1742,3950}, {3685,4297}, {4082,9778}, {5051,8582}
X(9950) = midpoint of X(i),X(j) for these {i,j}: {3729,9801}
X(9951) = (2*R-r)*X(8)-(4*R-r)*X(80)
The reciprocal orthologic center of these triangles is X(8).
X(9951) lies on these lines: {8,80}, {11,3698}, {100,8583}, {952,9856}, {1145,3921}, {1317,8581}, {1320,2801}, {2800,4018}, {3681,4900}, {5541,8580}
X(9952) = (4*R-r)*X(3)-(2*R+3*r)*X(8)
The reciprocal orthologic center of these triangles is X(3869).
X(9952) lies on these lines: {3,8}, {11,3340}, {80,2093}, {517,9951}, {1537,6844}, {1768,5128}, {2771,9947}, {2800,6797}, {2801,3036}, {2829,9948}, {3577,8727}, {6326,8580}
X(9952) = midpoint of X(i),X(j) for these {i,j}: {1145,9803}
The reciprocal orthologic center of these triangles is X(3555).
X(9953) lies on these lines: {8,3305}, {72,9951}, {3062,8001}
X(9954) = r*(4*R-r)*X(4)-(8*R^2-r^2)*X(8)
The reciprocal orthologic center of these triangles is X(1).
X(9954) lies on these lines: {4,8}, {57,210}, {200,971}, {518,3452}, {936,7091}, {942,3820}, {956,3305}, {999,5044}, {3059,3062}, {3359,3927}, {3740,6692}, {3921,5775}, {4002,5828}, {4847,7956}
X(9954) = orthologic center of triangles Atik to 2nd mixtilinear, , with reciprocal center X(1)
X(9954) = midpoint of X(i),X(j) for these {i,j}: {72,3421}
X(9954) = reflection of X(i) in X(j) for these (i,j): (942,3820), (999,5044)
X(9954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (210,8581,8580)
The reciprocal orthologic center of these triangles is X(10).
X(9955) is the centroid of the maltitude quadrangle of quadrangle ABCX(1), which consists of X(946) and the extraversions of X(10). (Randy Hutson, July 20, 2016)
X(9955) lies on these lines: {1,381}, {2,3579}, {3,1699}, {4,1385}, {5,10}, {8,3545}, {11,113}, {30,1125}, {40,1656}, {65,7741}, {72,6990}, {79,3582}, {125,5950}, {140,516}, {145,355}, {165,3526}, {226,496}, {376,5550}, {382,3576}, {392,2476}, {403,1829}, {497,6849}, {499,1836}, {500,3720}, {515,546}, {519,5066}, {547,3634}, {551,3845}, {582,748}, {631,9812}, {912,5448}, {944,3832}, {952,3635}, {962,3090}, {999,9612}, {1001,6985}, {1058,5226}, {1155,5442}, {1319,3585}, {1386,3818}, {1420,9655}, {1482,3632}, {1483,3857}, {1519,6831}, {1594,1902}, {1657,7987}, {1698,5055}, {1702,8976}, {1737,7173}, {1770,5122}, {2646,3583}, {2807,5462}, {3057,7951}, {3218,3652}, {3295,5219}, {3337,7701}, {3434,6896}, {3485,5722}, {3487,5274}, {3525,9778}, {3543,3653}, {3574,5777}, {3587,3646}, {3601,9668}, {3622,3655}, {3627,4297}, {3628,6684}, {3648,3916}, {3651,5284}, {3654,5071}, {3666,8143}, {3670,5492}, {3679,8148}, {3753,4193}, {3812,3825}, {3834,5482}, {3843,5691}, {3858,5882}, {3869,6873}, {3877,5141}, {3922,6971}, {3927,5231}, {3962,5694}, {4004,6830}, {4018,5887}, {4324,5444}, {4663,5476}, {4668,5072}, {4678,5068}, {4701,5844}, {5056,5657}, {5070,9589}, {5074,6706}, {5079,7991}, {5082,5748}, {5126,7354}, {5183,5445}, {5290,7373}, {5439,6845}, {5440,6900}, {5586,5708}, {5698,5805}, {5715,6913}, {5728,7678}, {5812,6846}, {5885,6001}, {6564,7968}, {6565,7969}, {6855,8166}, {6881,7958}, {7393,9911}, {8085,8099}, {8086,8100}, {8164,9785}, {8727,9940}
X(9955) = midpoint of X(i),X(j) for these {i,j}: {4,1385}, {5,946}, {546,5901}, {551,3845}, {1386,3818}, {3627,4297}, {4301,5690}
X(9955) = reflection of X(i) in X(j) for these (i,j): (6684,3628)
X(9955) = complement of X(3579)
X(9955) = X(5)-of-3rd-Euler-triangle
X(9955) = homothetic center of anti-Aquila triangle and Ehrmann mid-triangle
X(9955) = homothetic center of 2nd Fuhrmann triangle and cross-triangle of 2nd Fuhrmann and Ae (aka K798e) triangles
X(9955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,5886,1385), (40,7988,1656), (226,496,5045), (946,3817,5), (946,7681,5806), (1125,3838,3824), (1482,3851,5587), (1519,6831,9856), (1699,8227,3), (1770,5433,5122), (3091,5603,355), (3583,5443,2646), (5072,5790,7989), (5219,9614,3295), (7982,7989,5790)
The reciprocal orthologic center of these triangles is X(10).
X(9956) lies on these lines: {1,1656}, {2,355}, {3,1698}, {4,2355}, {5,10}, {8,3090}, {12,942}, {30,3828}, {35,7489}, {40,381}, {57,9654}, {65,5694}, {72,6829}, {80,2646}, {100,6920}, {113,5952}, {116,6706}, {119,125}, {140,515}, {145,7486}, {165,382}, {392,4193}, {403,1902}, {495,1210}, {498,1837}, {499,5252}, {511,3844}, {516,546}, {518,6583}, {519,547}, {549,4297}, {551,1483}, {912,3812}, {938,8164}, {952,1125}, {958,6911}, {962,3545}, {971,3826}, {993,6924}, {999,9578}, {1056,5704}, {1155,3585}, {1376,3560}, {1482,3679}, {1512,6831}, {1532,9856}, {1538,4002}, {1594,1829}, {1697,9669}, {1699,3851}, {1871,5142}, {2476,3753}, {2550,6893}, {2551,5791}, {2800,3918}, {2975,6946}, {3057,7741}, {3085,5722}, {3091,5657}, {3167,9896}, {3295,9581}, {3333,5726}, {3419,5552}, {3434,6898}, {3436,6854}, {3525,5731}, {3526,3576}, {3542,5090}, {3544,9779}, {3616,5067}, {3617,5056}, {3624,5070}, {3626,5844}, {3632,9624}, {3652,6175}, {3656,5071}, {3697,6991}, {3698,6980}, {3754,3838}, {3832,6361}, {3841,6001}, {3843,9588}, {3855,9812}, {3858,5493}, {3869,6874}, {3877,5154}, {3916,5080}, {3925,6842}, {3947,6147}, {4325,5442}, {5054,7987}, {5072,7991}, {5079,7982}, {5086,5440}, {5122,7354}, {5126,5433}, {5174,7551}, {5219,5780}, {5260,6905}, {5285,7562}, {5290,5708}, {5499,9943}, {5550,7967}, {5554,6933}, {5705,6918}, {5719,6738}, {5728,7679}, {5787,6989}, {5794,6862}, {5812,6843}, {5927,6937}, {6197,7559}, {6797,8068}, {6913,9709}, {6940,9342}, {7393,9798}, {7529,8193}, {8087,8099}, {8088,8100}, {8582,8728}
X(9956) = midpoint of X(i),X(j) for these {i,j}: {4,3579}, {5,10}, {65,5694}, {355,1385}, {946,5690}, {9940,9947}
X(9956) = reflection of X(i) in X(j) for these (i,j): (140,3634), (1125,3628), (5885,3812), (9955,5)
X(9956) = complement of X(1385)
X(9956) = X(5)-of-4th-Euler-triangle
X(9956) = homothetic center of Fuhrmann triangle and cross-triangle of Fuhrmann and Ai (aka K798i) triangles
X(9956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,355,1385), (2,5818,355), (5,5690,946), (8,3090,5886), (10,946,5690), (10,1329,5044), (10,3814,960), (12,1737,942), (40,7989,381), (495,1210,5045), (1482,5055,8227), (1656,5790,1), (1698,5587,3), (3057,7741,7743), (3585,5445,1155), (3617,5056,5603), (3679,8227,1482), (3812,3822,3824)
X(9957) = (2*R-r)*X(1)+r*X(3)
The reciprocal orthologic center of these triangles is X(10).
X(9957) lies on these lines: {1,3}, {2,3885}, {4,7320}, {5,7743}, {8,392}, {10,496}, {11,9956}, {12,9955}, {37,4266}, {72,145}, {78,4917}, {210,3632}, {221,1480}, {355,497}, {376,4308}, {381,9578}, {382,9580}, {390,944}, {495,946}, {515,9856}, {518,3244}, {519,960}, {550,4311}, {551,3812}, {758,3635}, {912,1483}, {936,2136}, {943,1320}, {950,952}, {956,5250}, {962,1056}, {995,4646}, {997,3913}, {1005,3957}, {1064,5399}, {1100,5053}, {1125,1387}, {1149,4642}, {1210,5690}, {1317,2771}, {1365,6018}, {1389,2346}, {1479,5252}, {1490,7966}, {1497,5398}, {1621,4861}, {1699,9654}, {1770,5434}, {1788,3654}, {1829,4222}, {1870,1902}, {2262,3247}, {2292,2611}, {3061,3991}, {3085,5886}, {3241,3555}, {3294,4875}, {3476,4294}, {3485,3656}, {3486,5887}, {3600,6361}, {3616,3753}, {3621,3876}, {3622,5439}, {3623,3868}, {3624,3698}, {3625,4662}, {3626,3740}, {3633,5692}, {3636,3742}, {3655,4305}, {3679,3893}, {3680,7160}, {3683,5258}, {3811,5289}, {3820,6736}, {3825,5123}, {3870,5730}, {3871,5440}, {3873,4018}, {3892,4084}, {3895,5687}, {3899,3962}, {3921,4678}, {3927,6762}, {3940,6765}, {3956,4746}, {3983,4668}, {4015,4701}, {4059,7278}, {4187,6735}, {4251,6603}, {4314,5882}, {4345,5703}, {4640,8666}, {4673,5295}, {4719,4868}, {4853,9708}, {5011,9327}, {5195,7247}, {5274,5818}, {5587,9669}, {5603,5806}, {5691,9668}, {5728,5766}, {5790,9581}, {5881,9947}, {7992,9845}, {8099,8241}, {8100,8242}, {8583,9709}
X(9957) = midpoint of X(i),X(j) for these {i,j}: {1,3057}, {65,5697}, {72,145}, {3244,3878}, {3555,3869}
X(9957) = reflection of X(i) in X(j) for these (i,j): (8,5044), (65,5045), (942,1), (960,3884), (3625,4662), (3754,3636), (4701,4015), (5836,1125), (5881,9947), (6797,1387)
X(9957) = X(20)-of-incircle-circles-triangle
X(9957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,35,1319), (1,40,999), (1,46,3304), (1,55,1385), (1,57,7373), (1,65,5045), (1,942,5049), (1,1697,3), (1,3612,1388), (1,3746,2646), (1,5119,56), (1,5697,65), (1,7962,1482), (1,7991,3333), (1,9819,40), (8,392,5044), (8,1058,5722), (8,3890,392), (55,1388,3612), (56,3579,5122), (56,5119,3579), (65,3057,5697), (65,5045,942), (145,3877,72), (1000,1058,8), (1388,3612,1385), (1482,6767,1), (3241,3869,3555), (3636,3754,3742), (4015,4701,4711), (9578,9614,381), (9580,9613,382)
The reciprocal orthologic center of these triangles is X(500).
X(9958) lies on these lines: {4,5278}, {5,182}, {10,30}, {381,1714}, {500,612}, {6000,9895}
X(9958) = X(4)-of-Ayme-triangle
X(9958) = Ayme-isogonal conjugate of X(11259)
The reciprocal orthologic center of these triangles is X(10).
X(9959) lies on these lines: {3,846}, {4,9791}, {5,4425}, {10,2783}, {21,104}, {40,8245}, {256,3931}, {511,3743}, {517,2292}, {942,1284}, {1281,6998}, {3579,4220}, {4199,5777}, {5051,9956}, {5728,8238}, {5884,6176}, {8099,8249}, {8100,8250}, {8229,9955}, {8240,9957}, {8731,9940}
X(9959) = midpoint of X(i),X(j) for these {i,j}: {3,5492}, {2292,9840}
X(9959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (846,8235,3)
X(9960) = (4*R^2+4*R*r+r^2)*X(4)-(4*R^2+5*R*r+r^2)*X(7)
The reciprocal orthologic center of these triangles is X(72).
X(9960) lies on these lines: {2,9942}, {4,7}, {20,3869}, {21,84}, {63,411}, {224,6909}, {515,3868}, {912,6869}, {1158,7411}, {1498,3100}, {2476,6260}, {3874,5735}, {5273,5777}, {5658,6825}, {5732,7992}, {5784,9943}, {5927,6856}, {6855,9940}
The reciprocal orthologic center of these triangles is X(65).
X(9961) lies on these lines: {2,9943}, {7,9800}, {8,971}, {20,3869}, {21,1709}, {40,3681}, {63,7992}, {64,3101}, {65,3146}, {72,9778}, {84,2975}, {100,1490}, {165,3876}, {221,3100}, {376,5887}, {411,1158}, {516,3868}, {517,3529}, {774,1044}, {912,6361}, {942,9812}, {960,3522}, {1742,2292}, {1788,1898}, {1836,6895}, {1854,4296}, {1858,3474}, {2801,7991}, {3149,9352}, {3151,6254}, {3219,5584}, {3434,9799}, {3543,7686}, {3616,9856}, {3812,3832}, {3874,9589}, {3877,4297}, {3889,4301}, {3890,5731}, {4329,6225}, {4420,6244}, {5057,6836}, {5086,6925}, {5250,5732}, {5284,8726}, {5439,9779}, {5493,5904}, {5552,5658}, {5880,6894}, {5927,9780}, {6734,9948}, {6737,9859}
X(9961) = reflection of X(i) in X(j) for these (i,j): (3146,65), (3869,20), (5904,5493), (9589,3874)
X(9961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (960,5918,3522)
The reciprocal orthologic center of these triangles is X(65).
X(9962) lies on these lines: {2,9944}, {7,9801}, {20,192}, {21,990}, {63,1721}, {516,3868}, {1766,7411}, {5732,7996}
X(9963) = 8*(r+2*R)*X(10)-5*(3*R+2*r)*X(21)
The reciprocal orthologic center of these triangles is X(8).
X(9963) lies on these lines: {2,9945}, {7,528}, {8,6154}, {10,21}, {11,4197}, {20,952}, {63,4677}, {104,7411}, {149,377}, {319,4720}, {404,5722}, {1862,4198}, {2475,5719}, {2800,9960}, {2802,3633}, {3036,5273}, {3146,5763}, {3871,5252}, {4292,7972}, {5528,7675}, {5732,7993}
X(9963) = reflection of X(i) in X(j) for these (i,j): (8,6154), (1320,6224), (9802,1317)
X(9963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1317,9802,1320), (6224,9802,1317)
X(9964) = (4*R^2+9*R*r+2*r^2)*X(7)-2*(2*R^2+3*R*r+r^2)*X(80)
The reciprocal orthologic center of these triangles is X(3869).
X(9964) lies on these lines: {2,9946}, {7,80}, {20,2800}, {21,104}, {63,4996}, {517,9963}, {912,3218}, {942,7548}, {952,3868}, {2829,9960}, {3878,5731}, {5046,5768}, {5777,7504}, {9799,9809}
X(9965) = 3*(r+2*R)*X(2)-2*(4*R+r)*X(7)
The reciprocal orthologic center of these triangles is X(1).
X(9965) lies on these lines: {2,7}, {4,2095}, {8,2093}, {20,145}, {21,999}, {27,4373}, {31,4310}, {38,4307}, {46,7080}, {69,2097}, {72,6904}, {81,2255}, {84,962}, {193,3210}, {200,5850}, {222,347}, {279,394}, {320,345}, {321,4454}, {354,5698}, {377,3421}, {390,3873}, {391,4359}, {443,3927}, {452,942}, {518,3474}, {758,4293}, {934,6612}, {940,4419}, {1259,4188}, {1376,5852}, {1443,6505}, {1478,4880}, {1788,5123}, {2551,5221}, {3101,7289}, {3146,9799}, {3187,4452}, {3241,4304}, {3475,4640}, {3487,3916}, {3522,6282}, {3600,3869}, {3623,4313}, {3666,4644}, {3681,5784}, {3820,4197}, {3832,7682}, {3870,5732}, {3874,4294}, {3894,4302}, {3901,4299}, {3935,8544}, {3957,7675}, {4000,4641}, {4312,4847}, {4512,5542}, {4652,5703}, {5057,5274}, {5084,5708}, {5175,9579}, {5234,5586}, {5735,9812}, {5737,7228}, {5770,6844}, {6147,6857}, {6244,7411}, {7291,9536}, {7992,9800}
X(9965) = reflection of X(i) in X(j) for these (i,j): (2,2094), (4,2095), (8,2093), (20,2096), (69,2097), (329,57)
X(9965) = complement of X(20214)
X(9965) = anticomplement of X(329)
X(9965) = orthologic center of triangles Conway to 2nd mixtilinear, with reciprocal center X(1)
X(9965) = isotomic conjugate of isogonal conjugate of X(37519)
X(9965) = polar conjugate of isogonal conjugate of X(23072)
X(9965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,63,2), (7,5273,5249), (9,553,9776), (9,9776,2), (20,3868,145), (57,329,2), (63,5249,5273), (226,3928,5744), (226,5744,2), (329,2094,57), (908,5435,2), (3218,5905,2), (3911,5748,2)
The reciprocal orthologic center of these triangles is X(6).
X(9966) lies on these lines: {3,67}, {23,8593}, {99,6093}, {182,2502}, {8546,9830}
X(9966) = reflection of X(13233) in X(8546)
X(9966) = X(1379)-of-1st-Ehrmann-triangle
X(9967) = (2*R^2-SW)*X(3)-(4*R^2-SW)*X(6)
The reciprocal orthologic center of these triangles is X(576).
X(9967) lies on these lines: {2,6403}, {3,6}, {5,1843}, {26,1974}, {51,6676}, {68,69}, {97,2987}, {125,343}, {141,1209}, {161,9306}, {542,7723}, {632,9827}, {1060,1469}, {1062,3056}, {1154,1353}, {1205,5663}, {1352,2393}, {1656,9822}, {2967,7467}, {2979,6515}, {3060,7494}, {3546,5447}, {3547,5446}, {3564,4173}, {3589,7542}, {3618,5462}, {3620,7999}, {3779,8251}, {3867,7403}, {4549,6776}, {5480,9019}, {7514,8541}, {7529,7716}
X(9967) = complement of X(6403)
X(9967) = midpoint of X(i),X(j) for these {i,j}: {5562,6467}
X(9967) = reflection of X(i) in X(j) for these (i,j): (52,6), (69,1216), (1843,5)
X(9968) = 2*X(3)-3*X(206)
The reciprocal orthologic center of these triangles is X(9969).
X(9968) lies on these lines: {3,206}, {66,3091}, {511,9925}, {575,6000}, {576,1353}, {1498,2393}, {1660,3292}, {2781,5609}, {2883,8542}, {3090,6697}, {3146,5596}, {5878,8538}
Let A'B'C' be the midheight triangle. Let AB, AC be the reflections of A' in CA, AB, resp. Define BC, BA, CA, CB cyclically. Let A" = BCBA∩CACB, and define B" and C" cyclically. Triangle A"B"C" is homothetic to the orthic triangle at X(25). The lines A'A", B'B", C'C" concur in X(9969). (Randy Hutson, March 29, 2020)
The reciprocal orthologic center of these triangles is X(9968).
In the plane of a triangle ABC, let
Γ = circumcircle of ABC
TaTbTc = tangential triangle of ABC
DEF = cevian triangle of X(1176)
Ab = the point, other than B, of intersection of circles Γ and (BDT)
Ac = the point, other than C, of intersection of circles Γ and (CDT)
La = AbAc, and define Lb and Lc cyclically
A' = Lb∩Lc, and define B' and C' cyclically
Then A'B'C' is perspective to ABC, and the perspector is X(9969). (Angel Montesdeoca, February 10, 2021)
X(9969) lies on these lines: {2,3313}, {4,66}, {5,141}, {6,25}, {22,5157}, {23,1176}, {26,182}, {32,157}, {39,160}, {52,1352}, {69,3060}, {143,3564}, {193,9027}, {237,570}, {263,2165}, {389,1503}, {427,6697}, {571,3148}, {576,9925}, {1350,7395}, {1351,7529}, {1691,2934}, {2781,7687}, {2979,3619}, {3098,7514}, {3456,5007}, {3542,6403}, {3549,9967}, {3567,6776}, {3589,5943}, {3618,5640}, {3629,8681}, {3763,3917}, {3852,7745}, {5050,9714}, {5085,9715}, {5092,5892}, {5198,9968}, {5596,6995}, {5965,6153}, {6329,8705}, {6664,9045}, {7715,8550}
X(9969) = midpoint of X(i),X(j) for these {i,j}: {6,1843}, {52,1352}
X(9969) = reflection of X(i) in X(j) for these (i,j): (141,9822), (182,5462)
X(9969) = complement of X(3313)
X(9969) = complement of X(6) wrt orthic triangle
X(9969) = X(142)-of-orthic-triangle if ABC is acute
X(9969) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,25,206), (6,7716,159), (51,1843,6)
X(9970) = 2*X(5)-X(67)
The reciprocal orthologic center of these triangles is X(9971).
X(9970) lies on the cubics K298, K305 and these lines: {3,1177}, {4,542}, {5,67}, {6,5663}, {23,110}, {74,182}, {113,1352}, {146,6776}, {155,2930}, {185,575}, {265,5480}, {399,1351}, {524,5655}, {613,3028}, {1092,7556}, {1350,1511}, {1503,7728}, {1656,6698}, {5169,5476}, {5621,7526}, {5642,7493}, {5878,8538}, {7519,9143}
X(9970) = midpoint of X(i),X(j) for these {i,j}: {146,6776}, {399,1351}
X(9970) = reflection of X(i) in X(j) for these (i,j): (3,6593), (67,5), (74,182), (265,5480), (895,576), (1350,1511), (1352,113), (2930,5609), (9140,5476)
X(9970) = complement of X(32247)
X(9971) = 2*SW*X(4)+(9*R^2-2*SW)*X(67)
The reciprocal orthologic center of these triangles is X(9970).
X(9971) lies on these lines: {2,9019}, {4,67}, {6,25}, {32,7669}, {50,3148}, {69,7394}, {141,2979}, {157,2965}, {182,2070}, {195,576}, {237,566}, {263,1989}, {381,511}, {403,5480}, {524,3060}, {542,568}, {597,5640}, {1154,1352}, {1350,9818}, {1503,5890}, {1634,3095}, {1992,2854}, {2916,5157}, {3221,9171}, {3313,3763}, {3567,8550}, {5085,5892}, {5169,8262}, {5648,5654}
X(9971) = midpoint of X(i),X(j) for these {i,j}: {51,1843}
X(9971) = reflection of X(i) in X(j) for these (i,j): (6,51), (51,9969), (2979,141), (3313,3819), (3819,9822)
X(9971) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1843,9969,6), (3313,9822,3763)
The reciprocal orthologic center of these triangles is X(9973).
X(9972) lies on these lines: {6,1493}, {54,575}, {193,576}, {511,6242}, {524,3519}, {1173,5097}, {1209,8538}, {2393,9935}, {3541,8541}, {5643,5972}, {6193,9815}, {6288,9970}, {7577,8537}, {8254,8263}, {9306,9716}
X(9973) = (5*R^2-SW)*X(6)-(6*R^2-SW)*X(25)
The reciprocal orthologic center of these triangles is X(9972).
X(9973) lies on these lines: {6,25}, {32,5938}, {50,157}, {66,67}, {69,1369}, {141,858}, {160,566}, {193,2854}, {338,3186}, {382,511}, {571,7669}, {599,3313}, {1503,6240}, {2979,3631}, {2980,9512}, {3056,9629}, {3060,3629}, {5102,5446}, {5157,9813}, {5640,6329}, {5965,6243}, {6144,8681}
X(9973) = reflection of X(i) in X(j) for these (i,j): (6,1843), (6467,9969)
X(9973) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,1843,9971), (206,8541,6), (1843,6467,9969), (6467,9969,6)
X(9974) = S*X(3)-(3*S+SW)*X(6)
The reciprocal orthologic center of these triangles is X(3).
X(9974) lies on these lines: {3,6}, {485,524}, {597,5420}, {1124,6283}, {1335,7362}, {1587,1992}, {3093,6291}, {6239,8537}, {6252,8539}, {6560,8550}, {9813,9823}
X(9974) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3312,8376), (6,1152,575)
X(9975) = S*X(3)-(3*S-SW)*X(6)
The reciprocal orthologic center of these triangles is X(3).
X(9975) lies on these lines: {3,6}, {486,524}, {597,5418}, {1124,7353}, {1335,8540}, {1588,1992}, {3092,6406}, {6400,8537}, {6404,8539}, {6561,8550}, {9813,9824}
X(9975) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3311,8375), (6,576,9974), (6,1151,575)
The reciprocal orthologic center of these triangles is X(568).
X(9976) lies on these lines: {6,13}, {67,5965}, {74,511}, {110,373}, {182,1511}, {323,9140}, {576,5663}, {1986,8541}, {2771,4663}, {2777,8549}, {2930,5050}, {5092,5622}, {5097,9970}, {5504,5505}, {7722,8537}, {7723,8538}, {7724,8539}, {7727,8540}, {9813,9826}
X(9976) = Second Lemoine circle-inverse-of-X(6034)
X(9976) = reflection of X(i) in X(j) for these (i,j): (110,575), (9970,5097)
The reciprocal orthologic center of these triangles is X(6243).
X(9977) lies on these lines: {6,17}, {54,575}, {511,7691}, {539,8548}, {542,6288}, {576,1154}, {597,8254}, {1493,8542}, {2393,6153}, {3292,5643}, {3574,5476}, {5097,5562}, {6152,8541}, {6242,8537}, {6255,8539}, {6286,8540}, {8550,9976}, {9813,9827}
X(9977) = midpoint of X(i),X(j) for these {i,j}: {54,9972}
X(9977) = reflection of X(i) in X(j) for these (i,j): (54,575)
The reciprocal orthologic center of these triangles is X(1).
X(9978) lies on the Parry circle and these lines: {2,2783}, {21,104}, {37,100}, {214,5168}, {351,900}, {352,3230}, {404,8143}, {2787,9147}, {2802,3743}, {2826,9123}, {2827,9810}, {2830,9156}, {2831,9157}, {3035,3712}, {3738,9811}
X(9978) = X(100)-of-1st-Parry-triangle
X(9978) = X(104)-of-2nd-Parry-triangle
X(9978) = Parry circle antipode of X(9980)
The reciprocal orthologic center of these triangles is X(9131).
The trilinear polar of X(9979) passes through X(5099). (Randy Hutson, July 20, 2016)
X(9979) lies on these lines: {2,1637}, {20,9529}, {107,110}, {111,2373}, {297,525}, {338,3124}, {351,523}, {476,2966}, {514,9810}, {522,9811}, {524,9141}, {671,690}, {804,8029}, {826,9208}, {1180,2507}, {1992,9003}, {2492,7664}, {2804,9978}, {2986,2987}, {3800,9135}, {5027,7927}, {5113,7950}, {6563,6587}, {8371,9191}, {9148,9479}, {9168,9189}
X(9979) = reflection of X(i) in X(j) for these (i,j): (2,1637), (3268,2), (9123,9185), (9131,351), (9168,9189), (9191,8371)
X(9979) = isotomic conjugate of X(17708)
X(9979) = polar conjugate of X(935)
X(9979) = pole wrt polar circle of trilinear polar of X(935) (line X(6)X(67), the radical axis of orthocentroidal and Dao-Moses-Telv circles)
X(9979) = X(20) of 1st Parry triangle
X(9979) = X(4) of 2nd Parry triangle
X(9979) = crossdifference of every pair of points on line X(574)X(3455)
X(9979) = barycentric product X(316)*X(523)
X(9979) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (351,9131,9123), (2592,2593,850), (9131,9185,351)
The reciprocal orthologic center of these triangles is X(1).
X(9980) lies on the Parry circle and these lines: {2,2787}, {23,667}, {100,110}, {104,111}, {214,5029}, {351,900}, {2771,9138}, {2783,9147}, {2804,9131}, {2805,9156}, {2806,9157}, {2826,9185}, {2827,9811}, {3738,9810}
X(9980) = reflection of X(i) in X(j) for these (i,j): (9978,351)
X(9980) = Parry circle-antipode-of X(9978)
X(9980) = X(104)-of-1st-Parry-triangle
X(9980) = X(100)-of-2nd-Parry-triangle
The reciprocal orthologic center of these triangles is X(3).
X(9981) lies on these lines: {14,32}, {15,5613}, {16,6773}, {530,9878}, {531,7811}, {542,1569}, {617,2896}, {619,3096}, {3098,5474}, {3099,9900}, {3105,9873}, {5464,7865}, {6670,7846}
X(9981) = X(14)-of-5th-Brocard-triangle
The reciprocal orthologic center of these triangles is X(3).
X(9982) lies on these lines: {13,32}, {15,6770}, {16,5617}, {530,7811}, {531,9878}, {542,1569}, {616,2896}, {618,3096}, {3098,5473}, {3099,9901}, {3104,9873}, {5463,7865}, {6669,7846}
X(9982) = X(13)-of-5th-Brocard-triangle
X(9983) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(76)
The reciprocal orthologic center of these triangles is X(3).
X(9983) lies on these lines: {3,9865}, {20,2782}, {32,76}, {39,3096}, {69,194}, {262,7941}, {315,1916}, {511,7893}, {538,7811}, {698,7750}, {730,9941}, {1078,8149}, {1975,2076}, {2021,7891}, {2023,7912}, {3095,7779}, {3099,9902}, {3934,7806}, {5969,9878}, {5976,7793}, {6309,7783}, {7757,7865}, {7786,7888}
X(9983) = X(76)-of-5th-Brocard-triangle
X(9983) = 5th-Brocard-isogonal conjugate of X(32)
X(9983) = 5th-Brocard-isotomic conjugate of X(3094)
X(9983) = perspector of the 5th and 6th Brocard triangles
X(9983) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,2896,3094)
X(9984) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(74)
The reciprocal orthologic center of these triangles is X(4).
X(9984) lies on these lines: {32,74}, {110,3098}, {113,3096}, {146,2896}, {541,7811}, {542,8782}, {690,9862}, {3094,9419}, {3099,9904}, {5663,9821}, {6699,7846}
X(9985) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(54)
The reciprocal orthologic center of these triangles is X(4).
X(9985) lies on these lines: {32,54}, {195,9301}, {539,7811}, {1154,9821}, {1204,3098}, {1209,3096}, {2888,2896}, {3099,9905}, {6689,7846}
X(9986) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(486)
The reciprocal orthologic center of these triangles is X(3).
X(9986) lies on these lines: {32,486}, {316,6316}, {487,2896}, {642,3096}, {3094,3564}, {3099,9906}, {3103,9863}, {6119,7846}
X(9987) = 2*(S^2-SW^2)*X(32)-(S^2-3*SW^2)*X(485)
The reciprocal orthologic center of these triangles is X(3).
X(9987) lies on these lines: {32,485}, {316,6312}, {488,2896}, {641,3096}, {3094,3564}, {3099,9907}, {3102,9863}, {6118,7846}
The reciprocal orthologic center of these triangles is X(5617).
X(9988) lies on these lines: {3,5978}, {16,621}, {20,616}, {30,76}, {194,533}, {315,5979}, {383,5171}, {384,3642}, {530,9939}, {531,3104}, {532,7893}, {617,2896}, {754,6294}, {5025,6109}, {5980,9862}, {5981,7761}, {6108,7793}, {9982,9983}
X(9988) = reflection of X(9989) in X(7750)
X(9988) = {X(7811),X(9873)}-harmonic conjugate of X(9989)
The reciprocal orthologic center of these triangles is X(5613).
X(9989) lies on these lines: {3,5979}, {15,622}, {20,617}, {30,76}, {194,532}, {315,5978}, {384,3643}, {530,3105}, {531,9939}, {533,7893}, {616,2896}, {754,6581}, {1080,5171}, {5025,6108}, {5980,7761}, {5981,9862}, {6109,7793}, {9981,9983}
X(9989) = reflection of X(i) in X(j) for these (i,j): (9988,7750)
X(9989) = {X(7811),X(9873)}-harmonic conjugate of X(9988)
The reciprocal orthologic center of these triangles is X(147).
X(9990) lies on these lines: {3,9866}, {83,7761}, {141,384}, {315,8290}, {316,8150}, {732,7893}, {754,7757}, {3972,6292}, {6704,7937}, {7793,9478}, {7802,9983}
X(9990) = reflection of X(i) in X(j) for these (i,j): (2896,7750), (7823,83)
X(9991) = 4*S^2*X(3)-(3*S^2-SW^2)*X(9867)
The reciprocal orthologic center of these triangles is X(6231).
X(9991) lies on these lines: {3,9867}, {487,2896}, {7802,9992}
X(9992) = 4*S^2*X(3)-(3*S^2-SW^2)*X(9868)
The reciprocal orthologic center of these triangles is X(6230).
X(9992) lies on these lines: {3,9868}, {384,6228}, {488,2896}, {7802,9991}
X(9993) = (S^2-3*SW^2)*X(4)+2*(S^2-SW^2)*X(32)
X(9993) lies on these lines: {2,3098}, {3,7846}, {4,32}, {5,3096}, {15,383}, {16,1080}, {30,3972}, {114,8782}, {125,9984}, {147,576}, {183,316}, {262,1513}, {376,4045}, {385,3818}, {511,3314}, {542,7766}, {597,9774}, {946,9941}, {1348,6039}, {1349,6040}, {1699,3099}, {2679,6785}, {2896,3091}, {3090,7914}, {3104,6115}, {3105,6114}, {3329,5476}, {3545,7865}, {3574,9985}, {5092,7875}, {5418,6813}, {5420,6811}, {5478,9982}, {5479,9981}, {5976,7752}, {6033,7812}, {6054,7774}, {6248,9983}, {6250,9987}, {6251,9986}, {6747,6995}, {6776,9748}, {7470,7834}, {9878,9880}, {9923,9927}
X(9993) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,32,9873), (4,9753,98), (5,9821,3096), (1513,5480,262)
X(9994) = S^2*(S-2*SW)*X(3)+(S^2-S*SW-SW^2)*SW*X(6)
X(9994) lies on these lines: {3,6}, {1271,2896}, {3096,5591}, {3099,5589}, {3641,9941}, {5861,7811}, {5871,9873}, {6202,9993}, {6227,9862}, {6270,9982}, {6271,9981}, {6273,9983}, {6277,9985}, {6279,9987}, {6281,9986}, {6319,8782}, {7725,9984}, {9878,9882}
X(9994) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,3103,182), (32,3094,9995)
X(9995) = S^2*(S+2*SW)*X(3)-(S^2+S*SW-SW^2)*SW*X(6)
X(9995) lies on these lines: {3,6}, {1270,2896}, {3096,5590}, {3099,5588}, {3640,9941}, {5860,7811}, {5870,9873}, {6201,9993}, {6226,9862}, {6268,9982}, {6269,9981}, {6272,9983}, {6276,9985}, {6278,9987}, {6280,9986}, {6320,8782}, {7726,9984}, {9878,9883}
X(9995) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,3094,9994), (372,3102,182)
X(9996) = (S^2-3*SW^2)*X(5)-(S^2-SW^2)*X(32)
X(9996) lies on these lines: {2,5191}, {3,3096}, {4,2896}, {5,32}, {15,5613}, {16,5617}, {30,141}, {98,7919}, {140,7914}, {155,9923}, {157,7514}, {183,316}, {262,7926}, {355,9941}, {511,7848}, {542,4045}, {754,8177}, {1352,2549}, {1656,7846}, {3095,7779}, {3099,5587}, {5066,7617}, {5167,5891}, {6214,9995}, {6215,9994}, {6288,9985}, {6289,9987}, {6290,9986}, {6321,8782}, {7470,7928}, {7728,9984}, {7908,9737}, {8724,9878}
X(9996) = midpoint of X(i),X(j) for these {i,j}: {3818,7761}
X(9996) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,2896,9821), (381,9301,9993), (3096,9873,3), (7811,9993,9301)
X(9997) = (S^2-3*SW^2)*X(1)-(S^2-SW^2)*X(32)
X(9997) lies on these lines: {1,32}, {8,3096}, {10,7914}, {145,2896}, {517,3098}, {519,7865}, {944,9873}, {952,9996}, {1482,9821}, {3094,3242}, {3241,7811}, {3616,7846}, {5603,9993}, {5604,9995}, {5605,9994}, {7967,7970}, {7974,9981}, {7975,9982}, {7976,9983}, {7978,9984}, {7979,9985}, {7980,9986}, {7981,9987}, {7983,8782}, {9878,9884}, {9923,9933}
X(9997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9941,32)
X(9998) lies on the Parry circle and these lines: {2,694}, {6,7711}, {23,2076}, {32,110}, {111,3098}, {352,9301}, {353,9486}, {1627,3506}, {2021,8569}, {2502,9999}, {3117,8570}, {3569,9147}, {9138,9984}, {9156,9208}, {9210,9213}
X(9998) = X(729)-of-2nd-Parry-triangle
X(9998) = inverse of X(2023) in the orthoptic circle of the Steiner inellipse
X(9999) lies on the Parry circle, the cubics K728, K730 and these lines: {2,5191}, {3,7711}, {23,9301}, {32,111}, {110,3098}, {352,2076}, {353,3094}, {669,9213}, {2502,9998}, {5027,9138}, {6031,7664}, {8782,9147}
X(9999) = circumcircle-inverse of X(7711)
Let A'B'C' be the 1st Brocard triangle and A"B"C" the first anti-Brocard triangle. X(10000) is the radical center of circumcircles of A"B'C', B"C'A', C"A'B'. (Randy Hutson, July 20, 2016)
X(10000) lies on these lines: {2,4159}, {3,3096}, {32,76}, {98,7697}, {99,737}, {574,8290}, {599,1003}, {761,6012}, {1352,9862}, {2896,3552}, {3314,5162}, {3643,9982}, {5989,7790}, {5999,7934}, {6228,9987}, {6229,9986}, {7470,7910}, {7824,7914}
X(10000) = perspector of 1st Brocard triangle and cross-triangle of ABC and 1st-Brocard-of-1st-Brocard triangle
| PART 1: | Introduction and Centers X(1) - X(1000) | PART 2: | Centers X(1001) - X(3000) | PART 3: | Centers X(3001) - X(5000) |
| PART 4: | Centers X(5001) - X(7000) | PART 5: | Centers X(7001) - X(10000) | PART 6: | Centers X(10001) - X(12000) |
| PART 7: | Centers X(12001) - X(14000) | PART 8: | Centers X(14001) - X(16000) | PART 9: | Centers X(16001) - X(18000) |
| PART 10: | Centers X(18001) - X(20000) | PART 11: | Centers X(20001) - X(22000) | PART 12: | Centers X(22001) - X(24000) |
| PART 13: | Centers X(24001) - X(26000) | PART 14: | Centers X(26001) - X(28000) | PART 15: | Centers X(28001) - X(30000) |
| PART 16: | Centers X(30001) - X(32000) | PART 17: | Centers X(32001) - X(34000) | PART 18: | Centers X(34001) - X(36000) |
| PART 19: | Centers X(36001) - X(38000) | PART 20: | Centers X(38001) - X(40000) | PART 21: | Centers X(40001) - X(42000) |
| PART 22: | Centers X(42001) - X(44000) | PART 23: | Centers X(44001) - X(46000) | PART 24: | Centers X(46001) - X(48000) |
| PART 25: | Centers X(48001) - X(50000) | PART 26: | Centers X(50001) - X(52000) | PART 27: | Centers X(52001) - X(54000) |
| PART 28: | Centers X(54001) - X(56000) | PART 29: | Centers X(56001) - X(58000) | PART 30: | Centers X(58001) - X(60000) |
| PART 31: | Centers X(60001) - X(62000) | PART 32: | Centers X(62001) - X(64000) | PART 33: | Centers X(64001) - X(66000) |
| PART 34: | Centers X(66001) - X(68000) | PART 35: | Centers X(68001) - X(70000) | PART 36: | Centers X(70001) - X(72000) |
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