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This is PART 5: Centers X(7001) - 10000

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) -


leftri  Centers associated with extra-triangles: X(7001) - X(7373)  rightri

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.

underbar

X(7001) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(173)

Trilinears    cot(A/4) : :
Trilinears    cot(A/2) + csc(A/2) : :
Barycentrics    cos(A/2)[1 + cos(A/2)] : :

X(7001) lies on the cubic K351 and these lines: {9,173}, {57,557}, {164,6212}, {258,3082}, {505,6213}

X(7001) = isogonal conjugate of X(7010)
X(7001) = {X(9),X(188)}-harmonic conjugate of X(7010)


X(7002) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(179)

Barycentrics    a sec4(B/4+ C/4)

X(7002) lies on this line: {174,5435}


X(7003) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(196)

Barycentrics    (1 + sec A)/(1 + cos A - cos B - cos C) : :

X(7003) lies on the Feuerbach hyperbola and these lines: {1,281}, {4,1903}, {7,92}, {19,84}, {21,268}, {104,1436}, {278,459}, {393,1146}, {1119,4858}, {1148,3296}, {1172,2192}, {1826,3577}, {2257,3341}

X(7003) = isogonal conjugate of X(7011)
X(7003) = X(189)-Ceva conjugate of X(4)
X(7003) = X(i)-cross conjugate of X(j) for these (i,j): (19,281), (1857,4), (1903,282), (2192,280)
X(7003) = cevapoint of X(1146) & X(3064)
X(7003) = pole wrt polar circle of trilinear polar of X(347)
X(7003) = X(48)-isoconjugate (polar conjugate) of X(347)


X(7004) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(201)

Barycentrics    a(b - c)2(b + c - a)(b2 + c2 - a2) : :

X(7004) lies on the de Longchamps ellipse and these lines: {1,104}, {3,201}, {4,1393}, {7,1937}, {11,244}, {33,57}, {34,84}, {36,1725}, {38,55}, {56,774}, {58,2906}, {63,212}, {73,1071}, {78,4571}, {80,1772}, {125,656}, {225,6245}, {238,1776}, {240,243}, {255,1062}, {390,4392}, {497,982}, {515,1735}, {602,920}, {650,3119}, {654,2170}, {756,5432}, {896,2361}, {942,2654}, {950,3670}, {971,1465}, {984,5218}, {986,3486}, {1193,1858}, {1357,2821}, {1364,3270}, {1407,2192}, {1457,6001}, {1736,3911}, {1864,3752}, {1936,3100}, {2272,5089}, {2292,2646}, {2632,2638}, {2801,4551}, {3024,3025}, {3075,6198}, {3326,6075}

X(7004) = isogonal conjugate of X(7012)
X(7004) = reflection of X(2635) in X(1465)
X(7004) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1459), (3,656), (57,650), (63,652), (77,905), (78,521), (84,513), (103,2254), (270,3737), (273,514), (1565,3942), (3497,3287), (3718,6332), (4858,2170)
X(7004) = X(i)-line conjugate of X(j) for these (i,j): (33,108), (2801,4551)
X(7004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1768,109), (63,1040,212), (244,2310,11), (971,1465,2635), (3100,3218,1936), (3270,3937,1364)
X(7004) = crosssum of X(i) and X(j) for these (i,j): (1,109), (33,1783), (34,108), (55,4559), (100,3869), (101,212), (201,4551), (2324,3939)
X(7004) = crossdifference of every pair of centers on the line X(101)X(108)
X(7004) = crosspoint of X(i) andX(j) for these (i,j): (1,522), (7,4560), (63,4025), (77,905), (78,521), (270,3737), (273,514), (513,2217), (1422,3676), (3718,6332)
X(7004) = X(92)-isoconjugate of X(2149)


X(7005) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(202)

Barycentrics    a2(b2 + c2 - a2 + 4bc + 2*31/2S) : :

X(7005) lies on these lines: {1,61}, {3,202}, {6,595}, {11,17}, {12,14}, {13,1479}, {15,56}, {16,35}, {18,498}, {36,5238}, {55,62}, {215,3205}, {396,496}, {398,495}, {1124,3365}, {1276,5018}, {1335,3364}, {2066,3389}, {2477,3201}, {3390,5414}, {3884,5239}, {5010,5351}, {5204,5352}, {5217,5237}, {5261,5334}

X(7005) = X(1095)-cross conjugate of X(16)
X(7005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,61,203), (6,3295,7006)


X(7006) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(203)

Barycentrics    a2(b2 + c2 - a2 + 4bc - 2*31/2S)

X(7006) lies on these lines: {1,62}, {3,203}, {6,595}, {11,18}, {12,13}, {14,1479}, {15,35}, {16,56}, {17,498}, {36,5237}, {55,61}, {215,3206}, {395,496}, {397,495}, {484,2306}, {1124,3390}, {1277,5018}, {1335,3389}, {2066,3364}, {2477,3200}, {3365,5414}, {3884,5240}, {5010,5352}, {5204,5351}, {5217,5238}, {5261,5335}

X(7006) = X(1094)-cross conjugate of X(15)
X(7006) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,62,202), (6,3295,7005)


X(7007) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(207)

Barycentrics    a(1 + sec A)/(1 + sec A - sec B - sec C) : :

X(7007) lies on these lines: {1,196}, {19,2192}, {33,6525}, {34,64}, {55,204}, {963,1455}, {2188,3213}

X(7007) = X(1096)-cross conjugate of X(33)
X(7007) = {X(1),X(3183)}-harmonic conjugate of X(207)


X(7008) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(208)

Barycentrics    a(1 + sec A)/(1 + cos A - cos B - cos C) : :

X(7008) lies on these lines: {4,57}, {6,33}, {11,1435}, {19,1857}, {25,1436}, {108,1750}, {189,5809}, {210,2343}, {271,333}, {280,452}, {282,284}, {909,2208}, {1096,2310}, {1709,1767}, {1728,1753}, {1859,2358}

X(7008) = isogonal conjugate of X(7013)
X(7008) = X(84)-Ceva conjugate of X(19)
X(7008) = X(i)-cross conjugate of X(j) for these (i,j): (25,33), (2357,2192), (3119,3064)
X(7008) = {X(4),X(1712)}-harmonic conjugate of X(208)
X(7008) = crosssum of X(57) & X(3182)
X(7008) = crosspoint of X(9) & X(3347)


X(7009) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(242)

Barycentrics    (a2 + bc)/(b2 + c2 - a2) : :

X(7009) lies on these lines: {1,4}, {10,3144}, {19,2319}, {25,92}, {27,295}, {28,1791}, {29,1867}, {98,108}, {171,4032}, {281,6353}, {286,1221}, {318,4185}, {321,4206}, {407,5174}, {412,1902}, {419,1215}, {428,2969}, {444,3963}, {511,1943}, {894,3955}, {1441,4220}, {1460,4008}, {1503,6354}, {1736,1746}, {1826,4213}, {1861,3741}, {1894,5081}, {1957,2212}, {4186,5342}, {5088,6063}, {5090,5125}, {5285,6358}

X(7009) = isogonal conjugate of X(7015)
X(7009) = isotomic conjugate of X(7019)
X(7009) = X(172)-cross conjugate of X(894)
X(7009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,92,242), (27,1897,1824), (33,5307,4), (225,1891,4)
X(7009) = cevapoint of X(4367) & X(4459)
X(7009) = trilinear pole of the line X(2533)X(3287)
X(7009) = pole wrt polar circle of trilinear polar of X(257) (line X(522)X(1491))
X(7009) = X(48)-isoconjugate (polar conjugate) of X(257)
X(7009) = crosspoint of polar conjugates of PU(10)


X(7010) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(258)

Trilinears    tan(A/4) : :
Trilinears    cot(A/2) - csc(A/2) : :
Barycentrics    [1 - cos(A/2)] cos(A/2) : :

X(7010) lies on these lines: {9,173}, {57,558}, {164,3645}, {258,483}, {505,6212}

X(7010) = isogonal conjugate of X(7001)
X(7010) = {X(9),X(188)}-harmonic conjugate of X(7001)


X(7011) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(268)

Barycentrics    (cos A)(1 - cos A)(1 + cos A - cos B - cos C) : :
Barycentrics    a2SA[S2 - 2abcsa]/sa2a : :

X(7011) lies on these lines: {1,3}, {7,6349}, {48,222}, {63,268}, {109,154}, {196,347}, {198,223}, {219,1073}, {221,2360}, {345,6516}, {394,1813}, {577,1407}, {653,6360}, {856,956}, {859,1396}, {1412,2193}, {1415,4548}, {1427,2178}, {1435,1465}, {1461,6609}, {1604,1763}, {1661,3556}, {1708,5120}, {1767,3213}, {5435,6350}

X(7011) = isogonal conjugate of X(7003)
X(7011) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,222), (347,221), (1804,3), (1817,223)
X(7011) = X(198)-cross conjugate of X(3)
X(7011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56,2352,1617), (63,6617,268), (198,6611,223)
X(7011) = crosssum of X(1146) and X(3064)
X(7011) = crosspoint of X(1262) and X(1813)
X(7011) = X(92)-isoconjugate of X(2192)


X(7012) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(270)

Barycentrics    a/[(b - c)2(b + c - a)(b2 + c2 - a2)] : :

X(7012) lies on these lines: {59,517}, {108,901}, {109,522}, {162,655}, {250,270}, {516,1785}, {521,651}, {765,1861}, {1041,5377}, {1252,5089}, {1262,3100}, {2398,2405}, {4564,4570}

X(7012) = isogonal conjugate of X(7004)
X(7012) = reflection of X(5081) in X(1861)
X(7012) = X(5379)-Ceva conjugate of X(59)
X(7012) = X(i)-cross conjugate of X(j) for these (i,j): (1,1897), (4,162), (9,651), (19,653), (33,1783), (34,108), (40,100), (169,658), (201,4551), (212,101), (573,662), (1766,190), (1830,4), (1845,4242), (2149,4564), (6210,1492), (6211,660)
X(7012) = cevapoint of X(i) and X(j) for these (i,j): (1,109), (33,1783), (34,108), (55,4559), (100,3869), (101,212), (201,4551), (2324,3939)
X(7012) = trilinear pole of the line through X(101) & X(108)
X(7012) = pole wrt polar circle of trilinear polar of X(4858)
X(7012) = X(48)-isoconjugate (polar conjugate) of X(4858)


X(7013) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(271)

Barycentrics    (cot A)(1 - cos A)(1 + cos A - cos B - cos C) : :

X(7013) lies on these lines: {2,7}, {3,77}, {20,3182}, {36,4341}, {40,347}, {46,3668}, {69,271}, {208,342}, {223,1817}, {255,269}, {273,412}, {283,1014}, {326,6516}, {610,651}, {934,6282}, {1020,1766}, {1068,1119}, {1442,3601}, {1813,2289}

X(7013) = isogonal conjugate of X(7008)
X(7013) = isotomic conjugate of X(7020)
X(7013) = X(69)-Ceva conjugate of X(77)
X(7013) = X(40)-cross conjugate of X(63)
X(7013) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1439,77), (57,579,1445)
X(7013) = cevapoint of X(57) and X(3182)


X(7014) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(289)

Barycentrics    .(1 - cos A/2) cos2(A/2) : :

X(7014) lies on the line {259,5414}

X(7014) = isogonal conjugate of X(557)
X(7014) = crosspoint of X(483) & X(558)


X(7015) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(295)

Barycentrics    a2(b2 + c2 - a2)/(a2 + bc) : :

X(7015) lies on these lines: {1,256}, {29,242}, {77,3784}, {78,3781}, {284,893}, {314,4594}, {517,1065}, {904,1036}, {1409,2359}, {3491,3496}, {3688,3961}, {3903,4451}

X(7015) = isogonal conjugate of X(7009)
X(7015) = X(257)-Ceva conjugate of X(893)
X(7015) = crosssum of X(4367) and X(4459)
X(7015) = crossdifference of every pair of centers on the line X(2533)X(3287)


X(7016) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(296)

Barycentrics    a2(b + c - a)(b2 + c2 - a2)/[(bc - a2)(b2 + c2 - a2) + 2b2c2] : :

X(7016) lies on these lines: {43,46}, {73,1942}, {243,1858}, {296,1935}

X(7016) = isogonal conjugate of X(1940)
X(7016) = X(1936)-cross conjugate of X(296)
X(7016) = crosssum of X(i) and X(j) for these (i,j): (46,1047), (1148,3144)
X(7016) = crosspoint of X(90) and X(1248)
X(7016) = cevapoint of PU(16)
X(7016) = perspector of ABC and the vertex-triangle of the 1st and 2nd bicentrics of the orthic triangle


X(7017) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(331)

Barycentrics    b2c2(b + c - a)/(b2 + c2 - a2) : :
Barycentrics    : (csc 2A)(cot A/2) : :
Barycentrics    (sec A)/(1 - cos A) : :
Barycentrics    (sec A)(csc^2 A/2) : :

X(7017) lies on these lines: {2,6335}, {8,1857}, {76,331}, {92,264}, {158,4385}, {281,345}, {318,341}, {321,2052}, {324,4671}, {1784,4692}, {1897,3192}, {1947,3729}

X(7017) = isotomic conjugate of X(222)
X(7017) = X(1969)-Ceva conjugate of X(264)
X(7017) = X(i)-cross conjugate of X(j) for these (i,j): (8,3596), (318,264), (321,312), (4391,6335)
X(7017) = {X(76),X(1969)}-harmonic conjugate of X(331)
X(7017) = cevapoint of X(8) and X(281)
X(7017) = trilinear pole of the line X(2804)X(4397)
X(7017) = pole wrt polar circle of trilinear polar of X(56) (line X(649)X(854))
X(7017) = X(48)-isoconjugate (polar conjugate) of X(56)


X(7018) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(334)

Barycentrics    bc/(a2 + bc) : :

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(7018) = isotomic conjugate of X(171)
X(7018) = X(i)-cross conjugate of X(j) for these (i,j): (1921,334), (3847,2)
X(7018) = {X(1920),X(2887)}-harmonic conjugate of X(334)
X(7018) = perspector of the inconic with center X(3847)
X(7018) = cevapoint of X(i) and X(j) for these (i,j): (2,4388), (257,4451), (693,3120)
X(7018) = trilinear pole of the line through X(824) & X(4391)


X(7019) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(337)

Barycentrics    (b2 + c2 - a2)/(a2 + bc) : :

X(7019) lies on these lines: {75,325}, {239,257}

X(7019) = isotomic conjugate of X(7009)


X(7020) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(342)

Barycentrics    (1 + cos A)/[(sin 2A)(1 + cos A - cos B - cos C)] : :

X(7020) lies on the cubic pK(X2052,X264) (the polar conjugate of the Thomson cubic) and these lines: {2,280}, {4,189}, {29,282}, {84,412}, {85,264}, {92,946}, {271,333}, {1440,4200}, {2994,5081}

X(7020) = isogonal conjugate of X(7114)
X(7020) = isotomic conjugate of X(7013)
X(7020) = X(309)-Ceva conjugate of X(92)
X(7020) = X(i)-cross conjugate of X(j) for these (i,j): (4,318), (1856,281)
X(7020) = cevapoint of X(1) and X(1753)
X(7020) = pole wrt polar circle of trilinear polar of X(223)
X(7020) = X(48)-isoconjugate (polar conjugate) of X(223)


X(7021) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(359)

Barycentrics    a2 / (B + C) : :

X(7021) = isogonal conjugate of X(1115)
X(7021) = 1st Saragossa point of X(359)


X(7022) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(400)

Barycentrics    sin(A) / [1 - sin(A/2)]2 : :

X(7022) lies on the line {7,177}

X(7022) = trilinear square of X(2089)


X(7023) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(480)

Barycentrics    a2 / (b + c - a)3 : :

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(7024) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(506)

Barycentrics    a[sin(A/2)]-2/3 : :


X(7025) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(507)

Barycentrics    a[sin(A/2)]-1/2 : :


X(7026) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(554)

Barycentrics    1 / [1 + 31/2*tan(A/2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(555)

Barycentrics    cot(A/2)csc(A/2) : :

X(7027) lies on these lines: {178,312}, {188,556}, {346,5430}

X(7027) = isotomic conjugate of X(7371)


X(7028) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(558)

Barycentrics    1 - sin(A/2) : :

X(7028) lies on these lines: {1,188}, {2,174}, {9,258}, {173,5437}, {289,1488}

X(7028) = X(9)-cross conjugate of X(188)
X(7028) = {X(2),X(174)}-harmonic conjugate of X(236)


X(7029) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(585)

Barycentrics    1 / [S(1/b + 1/c - 1/a) + a - b - c] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(586)

Barycentrics    1 / [S(1/b + 1/c - 1/a) + b + c - a] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(609)

Barycentrics    a2(2a2 - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(869)

Barycentrics    a3(b2 - bc + c2) : :

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)


X(7033) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(870)

Barycentrics    bc / (b2 - bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(871)

Barycentrics    1 / [a3(b2 - bc + c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(873)

Barycentrics    bc / (b - c)2 : :

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)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1028)

Barycentrics    a / (B + C)2 : :

X(7036) = isogonal conjugate of X(7044)


X(7037) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1035)

Barycentrics    a2*sa / [s*sa*SB*SC - a*sb*sc*SA*(b + c)] : :
Barycentrics    a2(b + c - a) / [a6 - a4(b + c) 2 + {(b + c)2 - a2}(b2 - c2)2 - 2a(b + c){a4 + (b - c)2(b2 + c2)} + 4a3(b3 + c3)] : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1044)

Barycentrics    a / [(1 - cos(A))(cos(B) + cos(C)) - cos(A) + cos(B)cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1049)

Barycentrics    a(B + C) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1068)

Barycentrics    tan(A) / [cos(B) + cos(C) - cos(A)] : :

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(7041) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1077)

Barycentrics    a / (B + C) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1079)

Barycentrics    a / [cos(B) + cos(C) - cos(A)]2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1081)

Barycentrics    1 / [1 - 31/2.tan(A/2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1085)

Barycentrics    a.(B + C)2 : :

X(7044) = isogonal conjugate of X(7036)


X(7045) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1098)

Trilinears    1/(cos B - cos C)^2 : :
Barycentrics    a / [(b - c)(b + c - a)]2 : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1119)

Trilinears    (1 + sec A)/(1 - cos A) : :
Barycentrics    tan A cot2 A/2 : :
Barycentrics    tan(A)[1 + cos(A)] / [1 - cos(A)] : :
Barycentrics    (b + c - a)2 / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1133)

Barycentrics    a / [1 + 31/2cot(A/3)] : :

X(7047) lies on the line: {1,1133}


X(7048) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1143)

Barycentrics    tan[(B+C)/4] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1148)

Barycentrics    tan(A) / [sec(B) + sec(C) - sec(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1191)

Barycentrics    a2 / [(a + b + c)2 - 4bc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1250)

Barycentrics    a2[1 - 31/2.tan(A/2)] : :

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(7052) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1251)

Barycentrics    a2 / [1 + 2cos(A+π/3)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1260)

Trilinears    (1 - cos A)/(1 + sec A) : : Barycentrics    cot(A)[1 - cos(A)]2 : :
Barycentrics    a2(b2 + c2 - a2) / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1262)

Barycentrics    a2(b + c - a)2 / (b + c)2 : :

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(7055) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1264)

Barycentrics    cot(A) / [1 + sec(A)] : :
Barycentrics    (b2 + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1265)

Barycentrics    cot A tan2 A/2 : :
Barycentrics    (b2 + c2 - a2) / (b + c - a)2 : :

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)


X(7057) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1274)

Barycentrics    cot[(B+C)/4] : :
Barycentrics    cos B/2 + cos C/2 - cos A/2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1275)

Barycentrics    (b + c - a)2 / (b + c)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1276)

Barycentrics    a / [31/2(1 + cosA - cosB - cosC) + sinB + sinC - sinA] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1277)

Barycentrics    a / [31/2(1 + cosA - cosB - cosC) + sinA - sinB - sinC] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1281)

Barycentrics    (a2 + bc) / (b3 + c3 - a3 - abc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1355)

Barycentrics    a4(b + c - a)(b4 + c4 - a2b2 - a2c2)2 : :

X(7062) lies on the Mandart inellipse and these lines: {11,1211}, {3688,4092}, {4548,6056}


X(7063) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1356)

Barycentrics    a4(b + c - a)(b2 - c2)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1357)

Barycentrics    a2(b + c - a)(b + c)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1363)

Barycentrics    a4(b + c - a)(b2 - c2)2(b2 + c2 - a2)4 : :

X(7065) lies on the Mandart inellipse and the line {219, 6862}


X(7066) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1364)

Trilinears    (cos^2 A)(1 + cos(B - C)) : :
Trilinears    cos2A cos2(B/2 - C/2) : :
Barycentrics    a2(b + c)2(b2 + c2 - a2)2 / (b + c - a) : :

See ADGEOM #2046 by Tran Quang Hung, 12/16/2014.

X(7066) lies on these lines: {3,1794}, {10,12}, {40,1745}, {55,581}, {56,219}, {71,73}, {78,296}, {185,212}, {201,1425}, {255,1364}, {329,1118}, {389,3074}, {511,1935}, {517,1838}, {603,3917}, {1038,3781}, {1361,3869}, {1469,5227}, {1490,6254}, {1859,5777}, {1936,5907}, {2218,3271}

X(7066) = X(72)-Ceva conjugate of X(201)
X(7066) = X(2632)-cross conjugate of X(520)
X(7066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (255,5562,1364), (1425,3690,201)
X(7066) = crosspoint of X(72) and X(3682)
X(7066) = trilinear square of X(7591)


X(7067) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1366)

Barycentrics    (b + c - a)(2a2 - b2 - c2)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1367)

Barycentrics    (b + c - a)(b2 - c2)2(b2 + c2 - a2)2 : :

X(7068) lies on the Mandart inellipse and these lines: {8,6062}, {219,6867}, {345,6056}, {3695, 6866}, {3703,6862}


X(7069) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1393)

Barycentrics    a[a2(b2 + c2) - (b2 - c2)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1394)

Barycentrics    a[2a2(b2 + c2) + (b2 - c2)2 - 3a4] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1398)

Barycentrics    a2(b + c - a)2 / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1406)

Barycentrics    sin(A)[1 + cos(A)] / [cos(B) + cos(C) - cos(A)] : :

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(7073) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1411)

Barycentrics    a(b + c - a) / (b2 + bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1413)

Barycentrics    sin(A)[1 + cos(A)].[1 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1424)

Barycentrics    a(b + c - a)(a2b2 + a2c2 - b2c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1430)

Barycentrics    [tan(A) + sin(A)][cos2(A) + cos(B)cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1431)

Barycentrics    a2(b + c - a) / (a2 - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1433)

Barycentrics    sin(2A)[1 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1435)

Barycentrics    a(b + c - a)2 / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1440)

Barycentrics    [cot(A) + csc(A)][1 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1447)

Barycentrics    (b + c - a)(a2 + bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1454)

Barycentrics    sin(A)[1 + cos(A)][1 - 2sin(B)sin(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1460)

Barycentrics    a2(b + c - a)[a2 + (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1472)

Barycentrics    sin(A)[1 + cos(A)] / [1 - cos(B)cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1473)

Barycentrics    sin(2A)[1 + cos(A)][1 + cos(B)cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1481)

Barycentrics    sin(A)[2 + cos(A)][2 + cos(A) - cos(B) - cos(C)] : :

X(7086) lies on these lines: {3,4322}, {55,5398}, {1407,3579}, {1480,5119}


X(7087) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1631)

Barycentrics    a2 / (b3 + c3 - a3) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1652)

Barycentrics    a / [31/2{1 - cos(A) + cos(B) + cos(C)} + sin(A) - sin(B) - sin(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1653)

Barycentrics    a / [31/2{1 - cos(A) + cos(B) + cos(C)} + sin(B) + sin(C) - sin(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1659)

Trilinears    1/(1 + sin A - cos A) : :
Barycentrics    1 / [1 + csc(A) - cot(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1697)

Barycentrics    sin(A) / [3 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1711)

Barycentrics    a / [a.SA2 -(b + c)(a2.sb.sc + s.sa.(b - c)2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1716)

Barycentrics    a / [a(b + c)(a2 + (b - c)2) - bc(b2 + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1726)

Barycentrics    a / [a3(b2 + c2 - a2) - a2(b3 + c3) + (b2 - c2)(b3 - c3)] : :

X(7094) lies on these lines: {264,1726}, {573,1759}, {1631,3185}

X(7094) = isogonal conjugate of X(1726)
X(7094) = X(184)-cross conjugate of X(1)


X(7095) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1756)

Barycentrics    a / [a4(b + c) + a3(b2 + c2) - a2(b3 + c3) - a(b4 + c4) + bc(b + c)(b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1759)

Barycentrics    a / (b3 + c3 - a3) : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1763)

Barycentrics    a / [a5 + a2(b + c)(a2 - 2bc) - a(b2 - c2)2 - (b - c)(b4 - c4)] : :

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) = X(25)-cross conjugate of X(1)
X(7097) = trilinear pole of the line through X(2522) and X(3900)


X(7098) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1776)

Barycentrics    a[a4 - a2(2b2 + 3bc + 2c2) + (b + c)(b3 + c3)] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1802)

Barycentrics    cos(A)[1 - cos(A)]2 : :
Barycentrics    a3.SA / sa2 : :

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)


X(7100) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1807)

Barycentrics    sin(A) / [2 + sec(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1847)

Barycentrics    sec(A)csc2(A)[1 + cos(A)]2 : :
Barycentrics    sec(A)cot2(A/2) : :
Barycentrics    bc(b + c - a)2 / (b2 + c2 - a2) : :
Barycentrics    (1 + sec A)/(1 - cos A) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1851)

Barycentrics    [a2 + (b + c)2] / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1863)

Barycentrics    [a2 + (b + c)2] / [(b + c - a)2(b2 + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1922)

Barycentrics    a4 / (a2 + bc) : :

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(7105) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1937)

Trilinears    1/(cos^2 A + cos B cos C) : :
Barycentrics    a(b + c - a) / [a2(b2 + c2 - a2) - bc(a + b + c)(b + c - a)] : :
Barycentrics    a.sa / (a2.SA - 2bcs.sa) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1945)

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

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1949)

Barycentrics    sin2(A)cos(A) / [cos2(A) + cos(B)cos(C)] : :
Barycentrics    a3.sa.SA / (a2.SA - 2bc.s.sa) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1952)

Barycentrics    1 / [cos2(A) + cos(B)cos(C)] : :
Barycentrics    (sec A)/(sec^2 A + sec B sec C) : :
Barycentrics    sa / (a2.SA - 2bc.s.sa) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(1977)

Barycentrics    a4(b + c)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2006)

Barycentrics    (b + c - a) / (b2 + bc + c2 - a2) : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2062)

Barycentrics    a2.sa.SA.[a8 - a6(2b2 - 3bc + 2c2) + 2a2(b - c)2(5bcSA + b4 + b2c2 + c4) - (b2 - c2)2[(b - c)(b3 - c3) + 4b2c2]] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2064)

Barycentrics    bc[a2(a2 - bc) - (b - c)(b3 - c3)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2174)

Barycentrics    a3(a2 - b2 + bc - c2) : :

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) = 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(1807)


X(7114) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2188)

Barycentrics    cot(A)[1 - cos(A)][1 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2189)

Trilinears    (tan A)/[1 - cos(B - C)] : :
Trilinears: tan A sin2(B/2 - C/2) : :
Barycentrics    a2 / [(b - c)2(b + c - a)(b2 + c2 - a2)] : :

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) = 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(7116) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2196)

Barycentrics    a3(b2 + c2 - a2) / (a2 + bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2197)

Trilinears    (sin 2A)[1 - cos(B - C)] : :
Trilinears    sin 2A sin2(B/2 - C/2) : :
Barycentrics    (sin 2A)(sin B - sin C)(cos B - cos C) : :
Barycentrics    a2(b - c)2(b + c - a)(b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2199)

Barycentrics    sin2(A)[1 + cos(A)] / [1 + cos(A) - cos(B) - cos(C)] : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2201)

Barycentrics    a(a2 + bc) / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2202)

Barycentrics    a(a2SA - 2bc.s.sa) / (sa.SA) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2209)

Barycentrics    a3 / (ab + ac - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2210)

Barycentrics    a3(a2 + bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2221)

Barycentrics    a2 / [a2 + (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2286)

Barycentrics    sin2(A)cos(A)[1 - cos(B)cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2289)

Barycentrics    cos2(A)[1 - cos(A)] : :
Barycentrics    a3(b2 + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2306)

Barycentrics    a / [1 - 31/2tan(A/2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2307)

Barycentrics    sin(A)[1 + cos(A) - 31/2sin(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2326)

Trilinears    1/[(b - c)^2(b + c - a)^2(b^2 + c^2 - a^2)] : :
Barycentrics    [sec(A) - 1] / [1 - cos(B - C)] : :

X(7128) lies on these lines: {59,517}, {162,1624}, {514,653}, {905,934}, {908,4564}, {1262,1465}, {1845,2717}, {2405,2406}

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), (40,101), (48,109), (57,1461)
X(7128) = trilinear pole of the line through X(108) and X(109)


X(7129) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2331)

Barycentrics    sin(A)tan(A) / [1 + cos(A) - cos(B) - cos(C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2337)

Barycentrics    [1 - cos(A)] / [1 - 2sin(B)sin(C)] : :

X(7130) = cevapoint of X(56) and X(2178)


X(7131) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2339)

Barycentrics    1/(1 - cos B cos C) : :
Barycentrics    a / [(b + c - a)(a2 + (b - c)2)] : :
Barycentrics    a / [sa(Sω - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2344)

Barycentrics    a / [(b + c - a)(b2 - bc + c2)] : :

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)


X(7133) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2362)

Trilinears    1/[1 + tan(A/2)] : :
Trilinears    (tan A)(1 - cot A/2) : :
Barycentrics    sin2(A) / [1 + sin(A) - cos(A)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2597)

Barycentrics    sin(A) / [cos(A-B)cos(A-C) + cos2(B-C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2603)

Barycentrics    sin(A) / [sec(A-B)sec(A-C) + sec2(B-C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2608)

Barycentrics    sin(A) / [sin(A-B)sin(A-C) + sin2(B-C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2614)

Barycentrics    sin(A) / [csc(A-B)csc(A-C) + csc2(B-C)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2638)

Barycentrics    a3(b + c)2(b2 + c2 - a2)2 / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2908)

Barycentrics    a3 / [a3(b2 + c2 - a2) - a2(b3 + c3) + (b2 - c2)(b3 - c3)] : :

X(7139) lies on the line {4,2908}


X(7140) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2969)

Barycentrics    (b + c)2 / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(2973)

Barycentrics    (b + c)2 / [a2(b2 + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3020)

Barycentrics    (b + c)2(b2 + bc + c2)2 / (b + c - a) : :

X(7142) lies on the line {594,3027}


X(7143) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3022)

Barycentrics    a2(b + c)2 / (b + c - a) 3 : :

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)


X(7144) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3025)

Barycentrics    a2(b + c)2(b2 + bc + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3033)

Barycentrics    a2[(b - c)2(a2 + bc)2 - (b3 + c3)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3061)

Barycentrics    a(b2 + bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3119)

Barycentrics    a(b + c)2 / (b + c - a)3 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3123)

Barycentrics    a(b + c)2 / (ab + ac - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3176)

Barycentrics    1 / [SA{s.sa.SB.SC - a(b + c)sb.sc.SA}] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3179)

Barycentrics    sin(A) / tan(A/2 - π/6) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3195)

Barycentrics    a2 / [SA{a.s.sa - (b + c)sb.sc}] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3197)

Barycentrics    a2 / [s.sa.SB.SC - a(b + c)sb.sc.SA] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3208)

Barycentrics    a / [(b + c - a)(ab + ac - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3209)

Barycentrics    a.sa / [SA{a.s.sa - (b + c)sb.sc}] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3212)

Barycentrics    (b + c - a) / (ab + ac - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3213)

Barycentrics    a(b + c - a)[3a4 - 2a2(b2 + c2) - (b2 - c2)2] / (b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3318)

Barycentrics    (b + c)2 / [(b + c - a){[b + c][(b - c)2 + a(b + c - a)] - a3}2] : :

X(7157) lies on these lines: {56,7003}, {1256,1411}


X(7158) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3324)

Barycentrics    (b - c)2(b + c - a)[a6 - a4(2b2 + 3bc + 2c2) + a2(b + c)2(b2 + c2) + bc(b2 - c2)2]2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3327)

Barycentrics    (b + c)2[a6 - 2a4(b2 + c2) + a2(b + c)(b3 + c3) - a2b2c2 - bc(b2 - c2)2]2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3333)

Barycentrics    a / [(a + b + c)(a - b + c)(a + b - c) + 4abc] : :

X(7160) lies on the Feuerbach hyperbola and these lines: {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) = isogonal conjugate of X(3333)
X(7160) = X(3303)-cross conjugate of X(1)
X(7160) = cevapoint of X(55) and X(2256)


X(7161) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3337)

Barycentrics    a / [(a + b + c)(a - b + c)(a + b - c) + abc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3338)

Barycentrics    a / [(a + b + c)(a - b + c)(a + b - c) + 2abc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3422)

Barycentrics    a2 / [a4 - 2a2bc - (b2 - c2)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3464)

Barycentrics    a / [(b + c)(a - b + c)(a + b - c)(b2 - bc + c2 - a2){a2(b2 + c2 - a2) - (a2 - b2 + c2)(a2 + b2 - c2)} - a{a6(b2 + c2 - a2) - a4b2c2 - (b2 - c2)2((b2 + c2 - a2)(3a2 - 2b2 - 2c2) - b2c2)}] : :

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(7165) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3483)

Barycentrics    a[a(b2 - bc + c2 - a2) - (b + c)(a - b + c)(a + b - c)] / [(a2 - b2 + c2)(a2 + b2 - c2)(b2 - bc + c2 - a2) + a(b + c)(a - b + c)(a + b - c)(b2 + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3507)

Barycentrics    a / [a(b + c)(a2 + b2 + c2) - (a2 + bc)(b2 + bc + c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3508)

Barycentrics    a / [(b + c)(b2c2 + c2a2 + a2b2) - a(a2 + bc)(b2 + bc + c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3510)

Barycentrics    a / [a3(b3 + c3) - b2c2(a2 + bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3556)

Barycentrics    a2 / [(b + c)(b2 + c2 - a2){a2 + (b - c)2} - a(a2 - b2 + c2)(a2 + b2 - c2)] : :

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(63),X(1619)}-harmonic conjugate of X(3556)
X(7169) = cevapoint of X(i) and X(j) for (i,j) = (667,3270)


X(7170) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3571)

Barycentrics    a[a4bc - a2(b4 + c4 + bc(b - c)2) + b3c3] : :

X(7170) lies on these lines: {1,39}, {99,3571}, {512,5539}, {3903,4128}

X(7170) = crosssum of PU(90)


X(7171) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3587)

Barycentrics    a[a6 - a4 (3b - c)(3c - b) + a2{3(b2 - c2)2 - 8bc(b2 - bc + c2)} - (b - c)2(b + c)4] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3598)

Barycentrics    (b + c - a)[3a2 + (b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3614)

Barycentrics    sin(A)[1 - 3cos(B-C)] : :

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), (11,3614,1), (35,3628,5326), (1479,1656,5432), (5217,5225,6284)


X(7174) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3677)

Barycentrics    a[a2 +3b2 + 2bc + 3c2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3684)

Barycentrics    a(a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3685)

Barycentrics    (a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3692)

Barycentrics    a(b2 + c2 - a2) / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3700)

Barycentrics    (b2 - c2) / (b + c - a) : :

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(7179) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3705)

Barycentrics    (b2 + bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3709)

Barycentrics    a2(b2 - c2) / (b + c - a) : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3712)

Barycentrics    (2a2 - b2 - c2) / (b + c - a) : :

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)


X(7182) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3718)

Barycentrics    bc(b2 + c2 - a2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3719)

Barycentrics    a(b2 + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3783)

Barycentrics    a(a2 + bc)(b2 - bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3790)

Barycentrics    (b2 - bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3792)

Barycentrics    a2(b2 + bc + c2 - a2)(b2 - bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3797)

Barycentrics    (a2 + bc)(b2 - bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3802)

Barycentrics    a(a2 + bc)2(b2 - bc + c2) : :

X(7188) lies on these lines: {1,7153}, {171,7176}, {256,7145}


X(7189) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3809)

Barycentrics    a(2a2 - bc)(b2 - bc + c2) : :

X(7189) lies on these lines: {1,3790}, {3589,3809}


X(7190) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3872)

Barycentrics    a(b2 + 4bc + c2 - a2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3920)

Barycentrics    a(a2 + b2 - bc + c2) : :

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) = {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(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(7192) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3952)

Barycentrics    (b - c) / (b + c) : :

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)

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) = anticomplement of X(661)
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(7193) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3955)

Barycentrics    a2(a2 - bc)(b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3961)

Barycentrics    a / (a2 + b2 + c2 + bc - ab - ac) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3974)

Barycentrics    [a2 + (b - c)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(3975)

Barycentrics    bc(a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4012)

Barycentrics    [a2 + (b + c)2] / (b + c - a)3 : :

X(7197) lies on these lines: {7,354}, {34,269}, {279,961}, {738,3668}


X(7198) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4030)

Barycentrics    (2a2 + b2 + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4033)

Barycentrics    bc(b - c) / (b + c) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4037)

Barycentrics    (b - c)2(a2 + bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4051)

Barycentrics    a(b2 + 3bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4053)

Barycentrics    a(b - c)2(b2 + bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4069)

Barycentrics    a(b - c) / [(b + c)(b + c - a)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4073)

Barycentrics    a(b2 + bc + c2) / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4087)

Barycentrics    b2c2(a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4089)

Barycentrics    (b + c)2(b2 + bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4094)

Barycentrics    a(b - c)2(a2 + bc)2 : :

X(7207) lies on these lines: {31,1414}, {99,4094}, {4128,4367}


X(7208) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4099)

Barycentrics    (b - c)2(2a2 + bc) : :

X(7208) lies on these lines: {75,3227}, {194,4099}, {244,514}, {330,1930}, {538,4975}, {604,2224}, {1015,1111}, {1086,2087}


X(7209) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4110)

Barycentrics    bc / [(b + c - a)(ab + ac - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4123)

Barycentrics    a(b4 + c4 - a4) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4124)

Barycentrics    (b + c)2(a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4140)

Barycentrics    (b2 - c2)(a2 - bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4149)

Barycentrics    a / [(b + c - a)(b3 + c3 - a3)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4157)

Barycentrics    (b4 + c4 - 2a4) / (b + c - a) : :

X(7214) lies on these lines: {7,1397}, {241,514}

X(7214) = X(2244)-cross conjugate of X(754)


X(7215) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4158)

Barycentrics    a2(b - c)2(b2 + c2 - a2)3 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4171)

Barycentrics    a(b2 - c2) / (b + c - a) 2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4178)

Barycentrics    (b4 + c4) / (b + c - a) : :

X(7217) lies on these lines: {7,1397}, {1365,6063}, {4121,4178}

X(7217) = X(4118)-cross conjugate of X(626)


X(7218) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4321)

Barycentrics    a(a + b - c) / [(b + c - a)3 + 8abc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4329)

Barycentrics    1 / [a4(a + b + c) - 2a2bc(b + c) - a(b2 - c2)2 - (b - c)(b4 - c4)] : :

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) = cyclocevian conjugate of X(2994)
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(7220) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4334)

Barycentrics    a(b + c - a) / [a2(b + c) - a(b2 + c2) + bc(a + b + c)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4348)

Barycentrics    a(b + c - a)[3a4 - 4(b - c)(b3 - c3) + (b2 - c2)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4371)

Barycentrics    b2 - 6bc + c2 - 3a2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4387)

Barycentrics    (a2 + 2bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4388)

Barycentrics    1 / (b3 + c3 - a3 + abc) : :

X(7224) lies on these lines: {63,2896}, {69,3974}, {77,612}, {561,4645}, {1965,4388}

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(7225) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4390)

Barycentrics    1 / a(a2 - 2bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4392)

Barycentrics    a(2b2 + bc + 2c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4395)

Barycentrics    2(a2 + bc) + (b + c)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4399)

Barycentrics    2(a2 + bc) - (b - c)2 : :

X(7228) lies on these lines: {1,6552}, {6,4395}, {7,141}, {8,3630}, {10,5852}, {37,545}, {69,4478}, {75,524}, {86,4440}, {142,4422}, {145,6691}, {320,594}, {519,4726}, {527,3739}, {536,3664}, {597,4000}, {894,1086}, {1100,1266}, {1213,6646}, {1449,4795}, {2321,4896}, {3218,5356}, {3629,4361}, {3663,4670}, {3686,4715}, {3729,4675}, {3758,6329}, {3782,6703}, {3879,4686}, {4357,4472}, {4364,6707}, {4373,4747}, {4405,5839}, {4416,4688}, {4454,4648}, {4657,4862}, {4659,4851}, {4667,4852}, {4698,4912}, {4887,5750}, {5743,5905}

X(7228) = midpoint of X(3879)X(4686)
X(7228) = reflection of X(i) in X(j) for these (i,j): (4399,75), (3686,4739)
X(7228) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,4363,141), (69,4665,4478), (75,524,4399), (320,594,3631), (894,1086,3589), (3879,4686,4971), (4361,4644,3629), (4659,4888,4851), (4715,4739,3686)


X(7229) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4402)

Barycentrics    3a2 + b2 + 6bc + c2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4403)

Barycentrics    (b + c)2(a2 - 2bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4405)

Barycentrics    b2 - 8bc + c2 - 4a2 : :

X(7231) lies on these lines: {7,141}, {8,6691}, {75,3629}, {319,3630}, {597,894}, {1268,6646}, {4021,4670}, {4357,4499}, {4399,4644}, {4419,6707}, {4464,4686}, {4659,4898}, {4667,4726}, {4912,5257}

X(7231) = {X(75),X(3629)}-harmonic conjugate of X(4405)


X(7232) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4445)

Barycentrics    2(b2 - bc + c2) - a2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4451)

Barycentrics    1 / [(b + c - a)(a2 - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4455)

Barycentrics    a2(b2 - c2)(a2 + bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4459)

Barycentrics    (b + c)2(a2 - bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4471)

Barycentrics    a2 / (b3 + c3 - a3 + 2abc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4475)

Barycentrics    a(b + c)(b3 + c3) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4478)

Barycentrics    3b2 - 4bc + 3c2 - 2a2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4481)

Barycentrics    a(b3 + c3) / (b - c) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4489)

Barycentrics    a(a2 + bc)(b2 - 3bc + c2) : :

X(7240) lies on these lines: {7,87}, {193,4489}, {1045,4667}, {1054,2347}, {1740,4644}, {3598,5272}


X(7241) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4492)

Barycentrics    a / (2a2 - bc) : :

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) = 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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4493)

Barycentrics    a[a2(b4 + c4) - b3c3] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4494)

Barycentrics    bc(a2 - 2bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4495)

Barycentrics    bc(a2 + bc)(a2 - 2bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4496)

Barycentrics    (a2 + 2bc) / (a2 - bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4497)

Barycentrics    a2 / (a3 + 2abc - b3 - c3) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4514)

Barycentrics    (a2 + b2 + bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4517)

Barycentrics    a2(b2 - bc + c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4518)

Barycentrics    1 / [(b + c - a)(a2 + bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4524)

Barycentrics    a2(b2 - c2) / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4548)

Barycentrics    a4(b4 + c4 - a4) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4559)

Barycentrics    a2(b - c)(b + c - a) / (b + c) : :

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(7253) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4566)

Barycentrics    (b - c)(b + c - a)2 / (b + c) : :

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(7254) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4574)

Barycentrics    a2(b - c)(b2 + c2 - a2) / (b + c) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4613)

Barycentrics    (b - c) / (b3 + c3) : :

X(7255) lies on these lines: {824,4560}, {3113,4367}

X(7255) = cevapoint of X(514) and X(667)


X(7256) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4616)

Barycentrics    (b + c - a)2 / (b2 - c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4625)

Barycentrics    bc(b + c - a) / (b2 - c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4635)

Barycentrics    bc(b + c - a)2 / (b2 - c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4637)

Barycentrics    a(b + c - a)2 / (b2 - c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4639)

Barycentrics    bc / [(b2 - c2)(a2 + bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4645)

Barycentrics    1 / (b3 + c3 - a3 - abc) : :

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(7262) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4650)

Barycentrics    a(b2 + bc + c2 - 2a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4665)

Barycentrics    b2 - 4bc + c2 : :

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) = {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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4692)

Barycentrics    bc[2a2 + (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4707)

Barycentrics    (b2 - c2)(b2 + bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4736)

Barycentrics    a(b - c)2(b2 + bc + c2 - a2)2 : :

X(7266) lies on these lines: {1,1414}, {99,4736}, {1019,3708}, {1565,4897}


X(7267) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4760)

Barycentrics    (a2 + bc)(b2 + c2 - 2a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4779)

Barycentrics    (7a2 + b2 + 6bc + c2) / (b + c - a) : :

X(7268) lies on these lines: {1,7}, {145,1434}, {956,1014}


X(7269) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4861)

Barycentrics    a(b2 + 3bc + c2 - a2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4872)

Barycentrics    b4 + c4 - a4 + bc(b2 + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4882)

Barycentrics    a(a2 - b2 + 6bc - c2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4894)

Barycentrics    b4 + c4 - a4 - bc(2a2 + b2 + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4907)

Barycentrics    a[a2 + 3(b + c)2] / (b +c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4915)

Barycentrics    a(b2 + 10bc + c2 - a2) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4941)

Barycentrics    a(b2 + 3bc + c2) / (ab + ac - bc) : :

X(7275) lies on these lines: {8,291}, {10,6384}, {87,2334}


X(7276) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4965)

Barycentrics    (b + c)2(2a2 + bc) / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4969)

Barycentrics    4a2 - (b - c)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4975)

Barycentrics    bc[4a2 - (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(4996)

Barycentrics    a2(b2 + bc + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5010)

Barycentrics    a2[2(b2 + c2 - a2) - bc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5018)

Barycentrics    a(b + c - a) / (b3 + c3 - a3 - abc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5081)

Barycentrics    (b2 + bc + c2 - a2) / [(b + c - a)(b2 + c2 - a2] : :

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) = {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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5088)

Barycentrics    (a2 - bc)(b2 + c2 - a2) - 2b2c2 : :

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(7284) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5119)

Barycentrics    a / [(b + c)(a - b + c)(a + b - c) - a(b2 + 4bc + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5128)

Barycentrics    a / [3(a + b + c)(a - b + c)(a + b - c) - 8abc] : :

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)


X(7286) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5160)

Barycentrics    a[2(b4 + c4 - a4 - a2bc) + bc(b - c)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5216)

Barycentrics    a2(b + c)(2b2 - 3bc + 2c2) / (b - c) : :

X(7287) lies on these lines: {1,295}, {512,1018}


X(7288) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5218)

Barycentrics    [3a2 - (b + c)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5227)

Barycentrics    a(b2 + c2 - a2)[a2 + (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5269)

Barycentrics    a[3a2 + (b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5279)

Barycentrics    a[b4 + c4 - a4 - bc(b2 + c2 - a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5297)

Barycentrics    a(a2 + b2 - 3bc + c2) : :

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) = 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(7293) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5314)

Barycentrics    a2(b2 + c2 - a2)(a2 + b2 - bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5326)

Barycentrics    [3(b + c)2 - 4a2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5329)

Barycentrics    a2 (b4 + c4 - a4 + 2a2bc) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5332)

Barycentrics    a2(2a2 + b2 + bc + c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5341)

Barycentrics    a[(a2 - b2 + c2)(a2 + b2 - c2) - a2bc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5345)

Barycentrics    a2[2(b4 + c4 - a4) + bc(a2 + b2 + c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5348)

Barycentrics    a[a4 + (bc - a2)(b + c)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5356)

Barycentrics    a[(a2 - b2 + c2)(a2 + b2 - c2) - 3a2bc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5363)

Barycentrics    a2(b4 + c4 - a4 + 3a2bc - b2c2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5370)

Barycentrics    a2[3(b4 + c4 - a4) + bc(a2 + b2 + c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5378)

Barycentrics    a / [(b + c)2(a2 + bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5383)

Barycentrics    b2c2 / (b + c)2 - a2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5384)

Barycentrics    a / [(b + c)(b3 + c3)] : :

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(7306) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5387)

Barycentrics    1 / [(b + c)2(a2 + b2 + 3bc + c2)] : :


X(7307) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5388)

Barycentrics    1 / [a2(b + c)(b3 + c3)] : :

X(7307) lies on the line {310,2206}

X(7307) = cevapoint of X(81) and X(310)


X(7308) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5437)

Barycentrics    a(b2 + 6bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5456)

Barycentrics    sin[2(B+C)/3] : :

X(7309) lies on these lines: {3273,3603}, {3277,3607}

X(7309) = cevapoint of X(3277) and X(3604)


X(7310) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5457)

Barycentrics    sin(A) / [cos(A) - cos((B+C)/3)] : :

X(7310) lies on the line {1134,5632}

X(7310) = isogonal conjugate of X(6124)


X(7311) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5458)

Barycentrics    sin(A) / [4cos(A) - sec((B+C)/3)] : :


X(7312) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5524)

Barycentrics    a / [a2 + b2 + c2 - 3(ab + ac - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5525)

Barycentrics    a / [(b + c - a)(a2 + b2 + c2) - 3abc] : :

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)


X(7314) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5532)

Barycentrics    (b + c)4 / (b + c - a)3 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5539)

Barycentrics    a / [a2bc(b2 + c2 - a2) + a(b + c)(b - c)2(a2 + bc) - b3c3] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5547)

Barycentrics    a2 / [(b + c - a)(b2 + c2 - 2a2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5551)

Barycentrics    1 / [3(b2 + c2 - a2) - 8bc] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5552)

Barycentrics    1 / [(b + c - a){(a + b + c)(a - b + c)(a + b - c) - 2abc}] : :
Barycentrics    1 / (S - 2Rsa) : :

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)


X(7319) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5556)

Trilinears    (csc2A)/(2 sec2(A/2) - csc2(A/2)) : :
Barycentrics    1 / [3(b2 + c2 - a2) - 2bc] : :

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(7320) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5558)

Trilinears    (csc2A)/(2 sec2(A/2) + csc2(A/2)) : :
Trilinears    1/(3 - cos A) : :
Barycentrics    1 / (b2 - 6bc + c2 - a2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5564)

Barycentrics    a2 - b2 + 3bc - c2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5574)

Barycentrics    a[a2 + 3(b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5575)

Barycentrics    a(b + c - a)[a2 + 3(b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5577)

Barycentrics    a2(b + c)2(b2 + 4bc + c2 - a2)2 / (b + c - a) : :

X(7324) lies on these lines: {181,7144}, {756,7066 }


X(7325) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5673)

Barycentrics    (pending)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5672)

Barycentrics    (pending)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5677)

Barycentrics    (pending)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5680)

Barycentrics    (pending)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5685)

Barycentrics    (pending)

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5709)

Barycentrics    a[a6 - a4 (3b2 - 2bc + 3c2) + a2(3b4 + 2b2c2 + 3c4) - (b + c)4(b - c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5906)

Barycentrics    1 / [(b + c - a){a6 + a3(b + c)(a2 - 2b2 + 3bc - 2c2) + (b - c)2 [(a + b + c)(b3 + c3) - a2(a2 + b2 + 3bc + c2)]}] : :

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)


X(7332) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(5949)

Barycentrics    (b - c)2 / [a(a2 - b2 + 3bc - c2) + (b + c)(a - b + c)(a + b - c)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6022)

Barycentrics    a2(b - c)2[a2(2b2 + 3bc + 2c2) - b2c2]2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6026)

Barycentrics    (b + c)2[a4(b2 - bc + c2) - b3c3] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6056)

Trilinears    (sin2A)(1 + cos 2A)/(1 + cos A) : :
Barycentrics    (sin2A)(sin 2A)/(1 + sec A) : :
Barycentrics    sin22A tan A/2 : :
Barycentrics    a4(b2 + c2 - a2)2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6058)

Barycentrics    (b + c - a)(b - c)4 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6059)

Barycentrics    a2 / [(b + c - a)(b2 + c2 - a2)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6060)

Barycentrics    [3a4 - 2a2(b2 + c2) - (b2 - c2)2]2 / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6061)

Barycentrics    a2 / [(b - c)2(b + c - a) 3] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6064)

Barycentrics    1 / [(b + c - a)(b2 - c2)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6065)

Barycentrics    a2 / [(b + c - a)(b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6066)

Barycentrics    a4 / [(b + c - a)(b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6126)

Barycentrics    a2(b2 + bc + c2 - a2) / [a3 - a(b2 + bc + c2) + (b + c)(a - b + c)(a + b - c)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6191)

Barycentrics    a / [a.s.sa - (b + c)sb.sc + 31/2S.sa] : :

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(7345) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6192)

Barycentrics    a / [a.s.sa - (b + c)sb.sc - 31/2S.sa] : :

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(7346) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6196)

Barycentrics    a / [a3(b3 + c3) + b2c2(a2 - bc)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6203)

Barycentrics    a / (2s.sb.sc + S.sa) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6204)

Barycentrics    a / (2s.sb.sc - S.sa) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6205)

Barycentrics    a / [a3 - 2a(b2 + c2) + (b + c)(2a2 - b2 + 3bc - c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6210)

Barycentrics    a / [(b + c){a4 + bc(b - c)2} + a(b2 - bc + c2){a2 - (b + c)(a + b + c)}] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6211)

Barycentrics    a / [a5 + a2(b + c - a)(b2 + bc + c2) - abc(b + c)2 - (b2 - c2)(b3 - c3)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6238)

Barycentrics    a2(b2 + c2 - a2)[a4(b2 + bc + c2) - 2a2(b4 - b2c2 + c4) + (b2 - c2)2(b2 - bc + c2)] : :

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(1425),X(5562)}-harmonic conjugate of X(1060)


X(7353) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6283)

Barycentrics    a2[a2(b2 + bc + c2) - (b - c)(b3 - c3) - 4bcS] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6284)

Barycentrics    2a4 - (b - c)2[a2 + (b + c)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6285)

Barycentrics    a2[a6(b2 + bc + c2) + a2(b2 - c2)2(3b2 - bc + 3c2) - a4(b + c)2(3b2 - 5bc + 3c2) - (b4 - c4)2 + bc(b + c)2(b - c)4] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6286)

Barycentrics    a2[a6(b2 + bc + c2) + a2{3(b6 + c6) - bc(b4 + b2c2 + c4)} - a4{3(b4 + c4) + bc(b + c)2} - (b + c)(b - c)2(b5 + c5)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6327)

Barycentrics    1 / (b3 + c3 - a3) : :

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) = 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(7358) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6355)

Barycentrics    (b - c)2(b + c - a)2(b2 + c2 - a2)[a(a + b + c)(b + c - a) - (b + c)(a - b + c)(a + b - c)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6357)

Barycentrics    (b + c - a)[2a4 - a2(b2 + c2) - (b2 - c2)2] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6359)

Barycentrics    (b + c - a)2[a4 - a2(b2 - bc + c2) - bc(b - c)2] : :

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)


X(7361) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6360)

Barycentrics    1/(sec B + sec C - sec A) : :
Barycentrics    1 / [a5(b + c) - 2a3(b3 + c3) + a(b - c)(b4 - c4) + bc(a2 - b2 + c2)(a2 + b2 - c2)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6405)

Barycentrics    a2[sb.sc.(b2 + bc + c2) + bcS] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6506)

Barycentrics    (b + c)2 / [(b + c - a)2{(b + c)(a - b + c)(a + b - c) - a(b2 + c2 - a2)}] : :

X(7363) lies on these lines: {2,6512}, {63,6506}, {222,2165}, {223,2006}, {225,431}, {1069,5713}, {6504,6511}


X(7364) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6518)

Barycentrics    cos(A)[cos2(A) + cos(B)cos(C)] : :
Barycentrics    a(b2 + c2 - a2)[a4 - a2(b2 + bc + c2) + bc(b + c)2] / (b + c - a) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6554)

Barycentrics    [a2 + (b + c)2] / (b + c - a)2 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6602)

Barycentrics    a3 / (b + c - a)3 : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6611)

Barycentrics    a2(b + c - a)2 / [(b + c)(a - b + c)(a + b - c) - a(a + b + c)(b +c - a)] : :

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(7368) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6612)

Barycentrics    a2(b + c - a)2[(b + c)(a - b + c)(a + b - c) - a(a + b + c).(b +c - a)] : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6652)

Barycentrics    (a2 + bc)3 : :

X(7369) lies on the line {894,7211}


X(7370) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6726)

Barycentrics    sin3(A/2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6731)

Trilinears    csc A' cot A' : : , where A'B'C' is the excentral triangle
Trilinears:    |IAJA| : : , where IAIBIC is the mixtilinear incentral triangle, and JAJBJC is the mixtilinear excentral triangle (Randy Hutson, April 11, 2015))
Barycentrics    sin(A/2)tan(A/2) : :

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) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6758)

Barycentrics    (b - c) / [(b + c){a(a2 - b2 + 3bc - c2) + (b + c)(a - b + c)(a + b - c)}] : :

X(7372) lies on the line {4560,7265}

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(7373) = PERSPECTOR OF ABC AND THE EXTRA-TRIANGLE OF X(6767)

Barycentrics    a2(a2 - b2 + 8bc - c2) : :

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)


X(7374) =  (EULER LINE)∩X(98)X(1131)

Barycentrics    (pending)

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)


X(7375) =  (EULER LINE)∩X(76)X(3316)

Barycentrics    (pending)

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}


X(7376) =  (EULER LINE)∩X(76)X(3317)

Barycentrics    (pending)

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}


X(7377) =  (EULER LINE)∩X(57)X(4911)

Barycentrics    (pending)

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(7378) =  (EULER LINE)∩X(33)X(5274)

Trilinears    2 sec A + csc A tan ω : :
Trilinears: csc A + 2 sec A cot ω : :
Barycentrics: 2 tan A + tan ω : :
Barycentrics: 1 + 2 tan A cot ω : :
Barycentrics: (a2 + 3b2 + 3c2)/(b2 + c2 - a2) : :

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)


X(7379) =  (EULER LINE)∩X(1)X(147)

Barycentrics    (pending)

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}


X(7380) =  (EULER LINE)∩X(10)X(262)

Barycentrics    (pending)

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}


X(7381) =  (EULER LINE)∩X(69)X(1230)

Barycentrics    (pending)

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}


X(7382) =  (EULER LINE)∩X(1478)X(5256)

Barycentrics    (pending)

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}


X(7383) =  (EULER LINE)∩X(95)X(315)

Barycentrics    (pending)

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}


X(7384) =  (EULER LINE)∩X(239)X(946)

Barycentrics    (pending)

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}


X(7385) =  (EULER LINE)∩X(86)X(5480)

Barycentrics    (pending)

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(7386) =  (EULER LINE)∩X(68)X(5447)

Barycentrics: tan B + tan C - tan ω : :
Barycentrics: (b2 + c2 - a2)(a2 + b2 + c2 + 2bc)(a2 + b2 + c2 - 2bc) : :

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)


X(7387) =  (EULER LINE)∩X(6)X(5446)

Barycentrics    (pending)

As a point on the Euler line, X(7387) has Shinagawa coefficients (E - $a2$, $a2$).

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) = 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(7388) =  (EULER LINE)∩X(6)X(638)

Barycentrics    (pending)

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


X(7389) =  (EULER LINE)∩X(6)X(637)

Barycentrics    (pending)

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}


X(7390) =  (EULER LINE)∩X(10)X(3424)

Barycentrics    (pending)

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}


X(7391) =  (EULER LINE)∩X(69)X(1369)

Barycentrics    b^2(c^4 + a^4 - b^4) + c^2(a^4 + b^4 - c^4) - a^2(b^4 + c^4 - a^4) : :
Barycentrics    sin 2B + sin 2C - sin 2A - tan ω : :

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) = anticomplement of X(22)
X(7391) = inverse-in-anticomplementary-circle of X(23)


X(7392) =  (EULER LINE)∩X(51)X(69)

Barycentrics    (pending)

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) = {X(2),X(4)}-harmonic conjugate of X(7386)


X(7393) =  (EULER LINE)∩X(6)X(1216)

Barycentrics &nbsnbsp;  (pending)

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)


X(7394) =  (EULER LINE)∩X(51)X(3818)

Barycentrics    (pending)

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(7395) =  (EULER LINE)∩X(6)X(5562)

Barycentrics    (pending)

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


X(7396) =  (EULER LINE)∩X(69)X(1853)

Barycentrics    tan A - tan B - tan C + 2 tan ω : :

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


X(7397) =  (EULER LINE)∩X(40)X(3008)

Barycentrics    (pending)

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}


X(7398) =  (EULER LINE)∩X(51)X(193)

Barycentrics    (pending)

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}


X(7399) =  (EULER LINE)∩X(66)X(1498)

Barycentrics    (pending)

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}


X(7400) =  (EULER LINE)∩X(69)X(1181)

Barycentrics    (pending)

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}


X(7401) =  (EULER LINE)∩X(6)X(6193)

Barycentrics    (pending)

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) = {X(2),X(4)}-harmonic conjugate of X(6643)


X(7402) =  (EULER LINE)∩X(355)X(5222)

Barycentrics    (pending)

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}


X(7403) =  (EULER LINE)∩X(52)X(5480)

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

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}


X(7404) =  (EULER LINE)∩X(182)X(5596)

Barycentrics    (pending)

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}


X(7405) =  (EULER LINE)∩X(39)X(233)

Barycentrics    (pending)

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}


X(7406) =  (EULER LINE)∩X(239)X(962)

Barycentrics    (pending)

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}


X(7407) =  (EULER LINE)∩X(511)X(5232)

Barycentrics    (pending)

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}


X(7408) =  (EULER LINE)∩X(53)X(5304)

Barycentrics    (5a^2 + 3b^2 + 3c^2)/(b^2 + c^2 - a^2) : :

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}


X(7409) =  (EULER LINE)∩X(193)X(3867)

Barycentrics    (3a^2 + 5b^2 + 5c^2)/(b^2 + c^2 - a^2) : :

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}


X(7410) =  (EULER LINE)∩X(230)X(387)

Barycentrics    (pending)

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}


X(7411) =  (EULER LINE)∩X(7)X(55)

Trilinears    a^5 - 2a^4(b + c) - a^3bc + a^2(b + c)(2b^2 - bc + 2c^2) - a(b - c)^2(b^2 + bc + c^2) + bc(b - c)^2(b + c) : :

As a point on the Euler line, X(7411) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(1)X(102)

Barycentrics    (pending)

As a point on the Euler line, X(7412) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(98)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7413) has Shinagawa coefficients (pending).

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) = Euler line intercept, other than X(29), of circle {X(29),PU(4)}


X(7414) =  (EULER LINE)∩X(1)X(1835)

Barycentrics    (pending)

As a point on the Euler line, X(7414) has Shinagawa coefficients (pending).

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(7413) = Euler line intercept, other than X(29), of circle {X(29),PU(4)}


X(7415) =  (EULER LINE)∩X(99)X(102)

Barycentrics    (pending)

As a point on the Euler line, X(7415) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(55)X(103)

Barycentrics    (pending)

As a point on the Euler line, X(7416) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(98)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7417) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(74)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7418) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(106)X(110)

Barycentrics    (pending)

As a point on the Euler line, X(7419) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(101)X(102)

Barycentrics    (pending)

As a point on the Euler line, X(7420) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(74)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7421) has Shinagawa coefficients (pending).

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(7422) =  (EULER LINE)∩X(74)X(98)

Barycentrics    (pending)

As a point on the Euler line, X(7422) has Shinagawa coefficients (pending).

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(7423) =  (EULER LINE)∩X(104)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7423) has Shinagawa coefficients (pending).

X(7423) lies on these lines:
{2, 3}, {98, 105}, {104, 111}, {842, 2752}, {915, 2373}, {1295, 2374}, {2687, 2770}


X(7424) =  (EULER LINE)∩X(102)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7424) has Shinagawa coefficients (pending).

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(7425) =  (EULER LINE)∩X(98)X(104)

Barycentrics    (pending)

As a point on the Euler line, X(7425) has Shinagawa coefficients (pending).

X(7425) lies on these lines:
{2, 3}, {74, 105}, {98, 104}, {477, 2752}, {842, 2687}, {915, 1297}, {1295, 3563}


X(7426) =  (EULER LINE)∩X(111)X(476)

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

As a point on the Euler line, X(7426) has Shinagawa coefficients (pending).

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) = 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(7427) =  (EULER LINE)∩X(104)X(105)

Barycentrics    (pending)

As a point on the Euler line, X(7427) has Shinagawa coefficients (pending).

X(7427) lies on these lines:
{2, 3}, {98, 759}, {104, 105}, {511, 4585}, {517, 3573}, {2687, 2752}, {2716, 2726}


X(7428) =  (EULER LINE)∩X(56)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7428) has Shinagawa coefficients (pending).

X(7428) lies on these lines:
{2, 3}, {56, 106}, {160, 1486}, {956, 1222}, {2689, 2758}, {3060, 5754}, {3185, 3612}


X(7429) =  (EULER LINE)∩X(74)X(104)

Barycentrics    (pending)

As a point on the Euler line, X(7429) has Shinagawa coefficients (pending).

X(7429) lies on these lines: {2, 3}, {74, 104}, {477, 2687}, {841, 2752}, {915, 1294}, {1295, 1300}


X(7430) =  (EULER LINE)∩X(74)X(101)

Barycentrics    (pending)

As a point on the Euler line, X(7430) has Shinagawa coefficients (pending).

X(7430) lies on these lines: {2, 3}, {71, 74}, {477, 2690}, {573, 5890}, {1300, 1305}, {2178, 2335}


X(7431) =  (EULER LINE)∩X(103)X(112)

Barycentrics    (pending)

As a point on the Euler line, X(7431) has Shinagawa coefficients (pending).

X(7431) lies on these lines: {2, 3}, {55, 1396}, {58, 103}, {99, 917}, {935, 2688}, {1172, 3286}


X(7432) =  (EULER LINE)∩X(103)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7432) has Shinagawa coefficients (pending).

X(7432) lies on these lines: {2, 3}, {98, 675}, {103, 111}, {917, 2373}, {2688, 2770}


X(7433) =  (EULER LINE)∩X(98)X(103)

Barycentrics    (pending)

As a point on the Euler line, X(7433) has Shinagawa coefficients (pending).

X(7433) lies on these lines: {2, 3}, {74, 675}, {98, 103}, {842, 2688}, {917, 1297}


X(7434) =  (EULER LINE)∩X(98)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7434) has Shinagawa coefficients (pending).

X(7434) lies on these lines: {2, 3}, {98, 106}, {842, 2758}, {2370, 3563}, {3430, 6789}


X(7435) =  (EULER LINE)∩X(107)X(108)

Barycentrics    (pending)

As a point on the Euler line, X(7435) has Shinagawa coefficients (pending).

X(7435) lies on these lines: {2, 3}, {100, 1301}, {107, 108}, {162, 4565}, {1304, 2766}


X(7436) =  (EULER LINE)∩X(102)X(112)

Barycentrics    (pending)

As a point on the Euler line, X(7436) has Shinagawa coefficients (pending).

X(7436) lies on these lines: {2, 3}, {102, 112}, {935, 2695}, {1172, 3428}, {3194, 4267}


X(7437) =  (EULER LINE)∩X(100)X(101)

Barycentrics    (pending)

As a point on the Euler line, X(7437) has Shinagawa coefficients (pending).

X(7437) lies on these lines: {2, 3}, {100, 101}, {108, 1305}, {885, 2283}, {1290, 2690}


X(7438) =  (EULER LINE)∩X(108)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7438) has Shinagawa coefficients (pending).

X(7438) lies on these lines: {2, 3}, {100, 2374}, {108, 111}, {1824, 5297}, {2766, 2770}


X(7439) =  (EULER LINE)∩X(102)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7439) has Shinagawa coefficients (pending).

X(7439) lies on these lines: {2, 3}, {98, 1311}, {102, 111}, {2695, 2770}


X(7440) =  (EULER LINE)∩X(74)X(103)

Barycentrics    (pending)

As a point on the Euler line, X(7440) has Shinagawa coefficients (pending).

X(7440) lies on these lines: {2, 3}, {74, 103}, {477, 2688}, {917, 1294}


X(7441) =  (EULER LINE)∩X(98)X(102)

Barycentrics    (pending)

As a point on the Euler line, X(7441) has Shinagawa coefficients (pending).

X(7441) lies on these lines: {2, 3}, {74, 1311}, {98, 102}, {842, 2695}


X(7442) =  (EULER LINE)∩X(103)X(104)

Barycentrics    (pending)

As a point on the Euler line, X(7442) has Shinagawa coefficients (pending).

X(7442) lies on these lines: {2, 3}, {103, 104}, {917, 1295}, {2687, 2688}


X(7443) =  (EULER LINE)∩X(102)X(105)

Barycentrics    (pending)

As a point on the Euler line, X(7443) has Shinagawa coefficients (pending).

X(7443) lies on these lines: {2, 3}, {102, 105}, {104, 1311}, {2695, 2752}


X(7444) =  (EULER LINE)∩X(74)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7444) has Shinagawa coefficients (pending).

X(7444) lies on these lines: {2, 3}, {74, 106}, {477, 2758}, {1300, 2370}


X(7445) =  (EULER LINE)∩X(103)X(105)

Barycentrics    (pending)

As a point on the Euler line, X(7445) has Shinagawa coefficients (pending).

X(7445) lies on these lines: {2, 3}, {103, 105}, {104, 675}, {2688, 2752}


X(7446) =  (EULER LINE)∩X(103)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7446) has Shinagawa coefficients (pending).

X(7446) lies on these lines: {2, 3}, {103, 106}, {917, 2370}, {2688, 2758}


X(7447) =  (EULER LINE)∩X(104)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7447) has Shinagawa coefficients (pending).

X(7447) lies on these lines: {2, 3}, {104, 106}, {915, 2370}, {2687, 2758}


X(7448) =  (EULER LINE)∩X(106)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7448) has Shinagawa coefficients (pending).

X(7448) lies on these lines: {2, 3}, {106, 111}, {2370, 2374}, {2758, 2770}


X(7449) =  (EULER LINE)∩X(109)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7449) has Shinagawa coefficients (pending).

X(7449) lies on these lines: {2, 3}, {109, 111}, {228, 5297}, {2689, 2770}


X(7450) =  (EULER LINE)∩X(109)X(110)

Barycentrics    (pending)

As a point on the Euler line, X(7450) has Shinagawa coefficients (pending).

X(7450) lies on these lines: {2, 3}, {109, 110}, {476, 2689}, {1624, 1633}


X(7451) =  (EULER LINE)∩X(100)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7451) has Shinagawa coefficients (pending).

X(7451) lies on these lines: {2, 3}, {100, 109}, {1290, 2689}, {3871, 5399}


X(7452) =  (EULER LINE)∩X(107)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7452) has Shinagawa coefficients (pending).

X(7452) lies on these lines: {2, 3}, {107, 109}, {110, 1897}, {1304, 2689}


X(7453) =  (EULER LINE)∩X(101)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7453) has Shinagawa coefficients (pending).

X(7453) lies on these lines: {2, 3}, {42, 101}, {1305, 2374}, {2690, 2770}


X(7454) =  (EULER LINE)∩X(74)X(102)

Barycentrics    (pending)

As a point on the Euler line, X(7454) has Shinagawa coefficients (pending).

X(7454) lies on these lines: {2, 3}, {74, 102}, {477, 2695}


X(7455) =  (EULER LINE)∩X(102)X(103)

Barycentrics    (pending)

As a point on the Euler line, X(7455) has Shinagawa coefficients (pending).

X(7455) lies on these lines: {2, 3}, {102, 103}, {2688, 2695}


X(7456) =  (EULER LINE)∩X(102)X(104)

Barycentrics    (pending)

As a point on the Euler line, X(7456) has Shinagawa coefficients (pending).

X(7456) lies on these lines: {2, 3}, {102, 104}, {2687, 2695}


X(7457) =  (EULER LINE)∩X(102)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7457) has Shinagawa coefficients (pending).

X(7457) lies on these lines: {2, 3}, {102, 106}, {2695, 2758}


X(7458) =  (EULER LINE)∩X(105)X(111)

Barycentrics    (pending)

As a point on the Euler line, X(7458) has Shinagawa coefficients (pending).

X(7458) lies on these lines: {2, 3}, {105, 111}, {2752, 2770}


X(7459) =  (EULER LINE)∩X(105)X(106)

Barycentrics    (pending)

As a point on the Euler line, X(7459) has Shinagawa coefficients (pending).

X(7459) lies on these lines: {2, 3}, {105, 106}, {2752, 2758}


X(7460) =  (EULER LINE)∩X(101)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7460) has Shinagawa coefficients (pending).

X(7460) lies on these lines: {2, 3}, {101, 109}, {2689, 2690}


X(7461) =  (EULER LINE)∩X(108)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7461) has Shinagawa coefficients (pending).

X(7461) lies on these lines: {2, 3}, {108, 109}, {2689, 2766}


X(7462) =  (EULER LINE)∩X(99)X(109)

Barycentrics    (pending)

As a point on the Euler line, X(7462) has Shinagawa coefficients (pending).

X(7462) lies on these lines: {2, 3}, {99, 109}, {691, 2689}


X(7463) =  (EULER LINE)∩X(109)X(112)

Barycentrics    (pending)

As a point on the Euler line, X(7463) has Shinagawa coefficients (pending).

X(7463) lies on these lines: {2, 3}, {109, 112}, {935, 2689}


X(7464) =  (EULER LINE)∩X(74)X(511)

Barycentrics    a^2[a^8 - 2a^6(b^2 + c^2) + 11a^4b^2c^2 + 2a^2(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^2(b^4 + 5b^2c^2 + c^4)] : :

As a point on the Euler line, X(7464) has Shinagawa coefficients (-2 (E - 2 F), 5 E - 4 F).

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(4) in the de Longchamps line
X(7464) = Thomson-isogonal conjugate of X(5648)
X(7464) = inverse-in-circumcircle of X(376)


X(7465) =  (EULER LINE)∩X(75)X(100)

Barycentrics    (pending)

As a point on the Euler line, X(7465) has Shinagawa coefficients (pending).

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(7466) =  (EULER LINE)∩X(19)X(100)

Barycentrics    (pending)

As a point on the Euler line, X(7466) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(98)X(689)

Barycentrics    (pending)

As a point on the Euler line, X(7467) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(99)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7468) has Shinagawa coefficients (pending).

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(7469) =  (EULER LINE)∩X(105)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7469) has Shinagawa coefficients (pending).

X(7469) lies on these lines: {2, 3}, {105, 476}, {110,518}, {1302,2687}}


X(7470)&nbnbsp;=  (EULER LINE)∩X(74)X(689)

Barycentrics    (pending)

As a point on the Euler line, X(7470) has Shinagawa coefficients (pending).

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) =  (EULER LINE)∩X(110)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7471) has Shinagawa coefficients (pending).

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) = Euler line intercept of axis of Kiepert parabola


X(7472) =  (EULER LINE)∩X(99)X(523)

Barycentrics    (pending)

As a point on the Euler line, X(7472) has Shinagawa coefficients (pending).

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(7473) =  (EULER LINE)∩X(110)X(525)

Barycentrics    (2a^6 - b^6 - c^6 - 2a^4b^2 - 2a^4c^2 + a^2b^4 + a^2c^4 + b^4c^2 + b^2c^4)/[(b^2 - c^2)(b^2 + c^2 - a^2)] : :

As a point on the Euler line, X(7473) has Shinagawa coefficients (pending).

X(7473) lies on these lines:
{2, 3}, {99, 1304}, {107, 691}, {110, 525}, {112, 476}, {250, 523}, {933, 1287}, {1552, 1553}

X(7473) = orthogonal projection of X(648) on its trilinear polar
X(7473) = trilinear pole of line X(542)X(6103)


X(7474) =  (EULER LINE)∩X(86)X(110)

Barycentrics    (pending)

As a point on the Euler line, X(7474) has Shinagawa coefficients (pending).

X(7474) lies on these lines:
{2, 3}, {58, 3011}, {81, 3475}, {86, 110}, {103, 1302}, {333, 3681}, {498, 5358}, {1043, 3006}


X(7475) =  (EULER LINE)∩X(100)X(691)

Barycentrics    (pending)

As a point on the Euler line, X(7475) has Shinagawa coefficients (pending).

X(7475) lies on these lines:
{2, 3}, {99, 693}, {100, 691}, {110, 2691}, {476, 1292}, {523, 4436}, {2766, 3565}


X(7476) =  (EULER LINE)∩X(100)X(935)

Barycentrics    (pending)

As a point on the Euler line, X(7476) has Shinagawa coefficients (pending).

X(7476) lies on these lines:
{2, 3}, {99, 2766}, {100, 935}, {107, 2691}, {108, 691}, {112, 1290}, {1292, 1304}


X(7477) =  (EULER LINE)∩X(100)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7477) has Shinagawa coefficients (pending).

X(7477) lies on these lines:
{2, 3}, {100, 476}, {110, 513}, {925, 2766}, {1302, 2691}, {3233, 3733}


X(7478) =  (EULER LINE)∩X(106)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7478) has Shinagawa coefficients (pending).

X(7478) lies on these lines:
{2, 3}, {106, 476}, {110, 519}, {229, 5434}, {759, 3582}, {1304, 2370}


X(7479) =  (EULER LINE)∩X(101)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7479) has Shinagawa coefficients (pending).

X(7479) lies on these lines:
{2, 3}, {101, 476}, {110, 514}, {1304, 1305}, {3233, 7192}


X(7480) =  (EULER LINE)∩X(107)X(476)

Barycentrics    (pending)

As a point on the Euler line, X(7480) has Shinagawa coefficients (pending).

X(7480) lies on these lines:
{2, 3}, {107, 476}, {110, 250}, {523, 1624}, {935, 1302}


X(7481) =  (EULER LINE)∩X(106)X(691)

Barycentrics    (pending)

As a point on the Euler line, X(7481) has Shinagawa coefficients (pending).

X(7481) lies on these lines:
{2, 3}, {99, 2758}, {106, 691}, {935, 2370}, {1403, 7286}


X(7482) =  (EULER LINE)∩X(99)X(935)

Barycentrics    (pending)

As a point on the Euler line, X(7482) has Shinagawa coefficients (pending).

X(7482) lies on these lines:
{2, 3}, {99, 935}, {107, 2696}, {112, 250}, {1296, 1304}


X(7483) =  {X(4),X(24)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7483) has Shinagawa coefficients (pending).

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) =  {X(2),X(3)}-harmonic conjugate of X(25)

Barycentrics    (pending)

As a point on the Euler line, X(7484) has Shinagawa coefficients (pending).

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(7485) =  {X(2),X(3)}-harmonic conjugate of X(22)

Barycentrics    (pending)

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(7486) =  {X(2),X(5)}-harmonic conjugate of X(20)

Barycentrics    (pending)

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(7487) =  {X(2),X(21)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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(7488) =  {X(3),X(24)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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) = anticomplement of X(1594)
X(7488) = inverse-in-circumcircle of X(3153)
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) =  {X(5),X(21)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7489) has Shinagawa coefficients (pending).

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) =  {X(2),X(27)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7490) has Shinagawa coefficients (pending).

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) =  {X(4),X(21)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7491) has Shinagawa coefficients (pending).

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}


X(7492) =  {X(3),X(23)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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(7493) =  {X(2),X(23)}-harmonic conjugate of X(4)

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

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) = anticomplement of X(5094)
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


X(7494) =  {X(2),X(22)}-harmonic conjugate of X(4)

Barycentrics    tan B + tan C + tan ω : :
Barycentrics    (b^2 + c^2 - a^2)(3a^4 - b^4 - c^4 + 2a^2b^2 + 2a^2c^2 + 2b^2c^2) : :

As a point on the Euler line, X(7494) has Shinagawa coefficients (-E - 2 F, Sω).

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)


X(7495) =  {X(2),X(23)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7495) has Shinagawa coefficients (-3 (E + 2 F), 2 Sω).

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(7496) =  {X(2),X(3)}-harmonic conjugate of X(23)

Barycentrics    (pending)

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) =  {X(4),X(28)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7497) has Shinagawa coefficients (pending).

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) =  {X(2),X(29)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7498) has Shinagawa coefficients (pending).

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}


X(7499) =  {X(2),X(22)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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}


X(7500) =  {X(4),X(22)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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) =  {X(3),X(28)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7501) has Shinagawa coefficients (pending).

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}


X(7502) =  {X(3),X(26)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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(7503) =  {X(3),X(4)}-harmonic conjugate of X(22)

Barycentrics    (pending)

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) =  {X(2),X(5)}-harmonic conjugate of X(21)

Barycentrics    (pending)

As a point on the Euler line, X(7504) has Shinagawa coefficients (pending).

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(7505) =  {X(5),X(24)}-harmonic conjugate of X(4)

Barycentrics    (tan A)(- cos 2A + cos 2B + cos 2C) : :

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)


X(7506) =  {X(2),X(26)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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)


X(7507) =  {X(4),X(5)}-harmonic conjugate of X(25)

Barycentrics    (pending)

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) =  {X(3),X(21)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7508) has Shinagawa coefficients (pending).

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}


X(7509) =  {X(2),X(3)}-harmonic conjugate of X(24)

Barycentrics    (pending)

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) =  {X(4),X(29)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7510) has Shinagawa coefficients (pending).

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) =  {X(4),X(28)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7511) has Shinagawa coefficients (pending).

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}


X(7512) =  {X(3),X(22)}-harmonic conjugate of X(4)

Barycentrics    (pending)

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) =  {X(3),X(4)}-harmonic conjugate of X(27)

Barycentrics    (pending)

As a point on the Euler line, X(7513) has Shinagawa coefficients (pending).

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}


X(7514) =  {X(3),X(5)}-harmonic conjugate of X(26)

Barycentrics    (pending)

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) =  {X(2),X(29)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7515) has Shinagawa coefficients (pending).

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)


X(7516) =  {X(2),X(3)}-harmonic conjugate of X(26)

Barycentrics    (pending)

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}


X(7517) =  {X(4),X(26)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,26,3), (5,22,3)


X(7518) =  {X(4),X(29)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7518) has Shinagawa coefficients (pending).

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(7517) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,26,3), (5,22,3)


X(7519) =  {X(4),X(23)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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) =  {X(3),X(28)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7520) has Shinagawa coefficients (pending).

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) =  {X(2),X(28)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7521) has Shinagawa coefficients (pending).

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) =  {X(2),X(27)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7522) has Shinagawa coefficients (pending).

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) =  {X(2),X(3)}-harmonic conjugate of X(28)

Barycentrics    (pending)

As a point on the Euler line, X(7523) has Shinagawa coefficients (pending).

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) =  {X(4),X(29)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7524) has Shinagawa coefficients (pending).

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}


X(7525) =  {X(3),X(22)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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}


X(7526) =  {X(3),X(4)}-harmonic conjugate of X(26)

Barycentrics    (pending)

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}


X(7527) =  {X(3),X(4)}-harmonic conjugate of X(23)

Barycentrics    (pending)

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}


X(7528) =  {X(5),X(26)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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(7529) =  {X(5),X(25)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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}


X(7530) =  {X(4),X(23)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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) =  {X(3),X(29)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7531) has Shinagawa coefficients (pending).

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) =  {X(2),X(29)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7532) has Shinagawa coefficients (pending).

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(7533) =  {X(5),X(23)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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) =  {X(4),X(27)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7534) has Shinagawa coefficients (pending).

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) =  {X(2),X(28)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7535) has Shinagawa coefficients (pending).

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(7536) =  {X(2),X(27)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7536) has Shinagawa coefficients (pending).

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) =  {X(5),X(28)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7537) has Shinagawa coefficients (pending).

X(7537) lies on these lines:
{2, 3}, {278, 499}, {946, 6197}, {1068, 2006}, {1210, 1870}, {1852, 7173}, {1861, 6700}, {3086, 3176}


X(7538) =  {X(3),X(29)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7538) has Shinagawa coefficients (pending).

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)


X(7539) =  {X(2),X(5)}-harmonic conjugate of X(25)

Barycentrics    (pending)

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) =  {X(4),X(23)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7540) has Shinagawa coefficients (pending).

X(7540) lies on these lines: {2, 3}, {52, 542}, {187, 1879}, {524, 6243}, {567, 5480}, {568, 1503}, {569, 5476}


X(7541) =  {X(4),X(5)}-harmonic conjugate of X(29)

Barycentrics    (pending)

As a point on the Euler line, X(7541) has Shinagawa coefficients (pending).

X(7541) lies on these lines: {2, 3}, {92, 5587}, {653, 1737}, {908, 5081}, {946, 5174}, {1807, 1870}, {2907, 3574}


X(7542) =  {X(2),X(24)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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) =  {X(5),X(27)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7543) has Shinagawa coefficients (pending).

X(7543) lies on these lines: {2, 3}, {19, 3462}, {278, 1148}, {1838, 1940}, {1871, 5439}, {5174, 5440}


X(7544) =  {X(5),X(24)}-harmonic conjugate of X(2)

Barycentrics    (pending)

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(7545) =  {X(5),X(23)}-harmonic conjugate of X(3)

Barycentrics    (pending)

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) =  {X(4),X(27)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7546) has Shinagawa coefficients (pending).

X(7546) lies on these lines: {2, 3}, {52, 916}, {53, 58}, {386, 6748}, {1865, 5398}, {6147, 7282}


X(7547) =  {X(4),X(5)}-harmonic conjugate of X(24)

Barycentrics    (pending)

As a point on the Euler line, X(7547) has Shinagawa coefficients (pending).

X(7547) lies on these lines: {2, 3}, {1853, 6241}, {1986, 3567}, {2548, 5523}, {2904, 3574}, {5876, 6746}


X(7548) =  {X(4),X(5)}-harmonic conjugate of X(21)

Barycentrics    (pending)

As a point on the Euler line, X(7548) has Shinagawa coefficients (pending).

X(7548) lies on these lines: {2, 3}, {355, 1389}, {946, 5086}, {1699, 3878}, {3817, 5443}, {3869, 5587}


X(7549) =  {X(3),X(5)}-harmonic conjugate of X(28)

Barycentrics    (pending)

As a point on the Euler line, X(7549) has Shinagawa coefficients (pending).

X(7549) lies on these lines: {2, 3}, {81, 5562}, {216, 5317}, {946, 5285}, {1871, 3101}, {5758, 7085}


X(7550) =  {X(3),X(5)}-harmonic conjugate of X(23)

Barycentrics    (pending)

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) =  {X(5),X(29)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7551) has Shinagawa coefficients (pending).

X(7551) lies on these lines: {1, 3462}, {2, 3}, {92, 5886}, {281, 1953}, {1148, 3485}


X(7552) =  {X(5),X(23)}-harmonic conjugate of X(4)

Barycentrics    (pending)

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}


X(7553) =  {X(4),X(22)}-harmonic conjugate of X(5)

Barycentrics    (pending)

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) =  {X(3),X(27)}-harmonic conjugate of X(4)

Barycentrics    (pending)

As a point on the Euler line, X(7554) has Shinagawa coefficients (pending).

X(7554) lies on these lines: {2, 3}, {19, 1075}, {54, 1246}, {92, 3916}, {278, 603}


X(7555) =  {X(3),X(23)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7555) has Shinagawa coefficients (pending).

X(7555) lies on these lines: {2, 3}, {143, 575}, {2918, 2930}, {3292, 6101}, {5562, 5609}


X(7556) =  {X(3),X(23)}-harmonic conjugate of X(4)

Barycentrics    (pending)

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) =  {X(5),X(28)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7557) has Shinagawa coefficients (pending).

X(7557) lies on these lines: {2, 3}, {226, 5262}, {975, 3586}, {5279, 6734}


X(7558) =  {X(5),X(22)}-harmonic conjugate of X(4)

Barycentrics    (pending)

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) =  {X(4),X(5)}-harmonic conjugate of X(28)

Barycentrics    (pending)

As a point on the Euler line, X(7559) has Shinagawa coefficients (pending).

X(7559) lies on these lines: {2, 3}, {1852, 3614}, {1891, 3817}, {2906, 3574}


X(7560) =  {X(3),X(27)}-harmonic conjugate of X(2)

Barycentrics    (pending)

As a point on the Euler line, X(7560) has Shinagawa coefficients (pending).

X(7560) lies on these lines: {2, 3}, {278, 1950}, {3101, 3164}, {4652, 5271}


X(7561) =  {X(2),X(28)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7561) has Shinagawa coefficients (pending).

X(7561) lies on these lines: {2, 3}, {343, 1437}, {975, 1062}, {1214, 5433}


X(7562) =  {X(5),X(28)}-harmonic conjugate of X(3)

Barycentrics    (pending)

As a point on the Euler line, X(7562) has Shinagawa coefficients (pending).

X(7562) lies on these lines: {2, 3}, {942, 2003}, {2262, 2323}


X(7563) =  {X(4),X(5)}-harmonic conjugate of X(27)

Barycentrics    (pending)

As a point on the Euler line, X(7563) has Shinagawa coefficients (pending).

X(7563) lies on these lines: {2, 3}, {1210, 7282}, {2905, 3574}


X(7564) =  {X(4),X(5)}-harmonic conjugate of X(26)

Barycentrics    (pending)

As a point on the Euler line, X(7564) has Shinagawa coefficients (pending).

X(7564) lies on these lines: {2, 3}, {1993, 6288}, {3818, 5448}


X(7565) =  {X(4),X(5)}-harmonic conjugate of X(23)

Barycentrics    (pending)

As a point on the Euler line, X(7565) has Shinagawa coefficients (pending).

X(7565) lies on these lines: {2, 3}, {524, 2888}, {542, 3574}


X(7566) =  {X(4),X(5)}-harmonic conjugate of X(22)

Barycentrics    (pending)

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) =  {X(3),X(5)}-harmonic conjugate of X(29)

Barycentrics    (pending)

As a point on the Euler line, X(7567) has Shinagawa coefficients (pending).

X(7567) lies on these lines: {2, 3}, {226, 3075}, {333, 5562}


X(7568) =  {X(2),X(26)}-harmonic conjugate of X(5)

Barycentrics    (pending)

As a point on the Euler line, X(7568) has Shinagawa coefficients (pending).

X(7568) lies on these lines: {2, 3}, {511, 6689}, {524, 1493}


X(7569) =  {X(2),X(5)}-harmonic conjugate of X(24)

Barycentrics    (pending)

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) =  {X(2),X(5)}-harmonic conjugate of X(23)

Barycentrics    (pending)

As a point on the Euler line, X(7570) has Shinagawa coefficients (pending).

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)


X(7571) =  {X(2),X(5)}-harmonic conjugate of X(22)

Barycentrics    (pending)

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) =  {X(2),X(3)}-harmonic conjugate of X(29)

Barycentrics    (pending)

As a point on the Euler line, X(7572) has Shinagawa coefficients (pending).

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) =  {X(2),X(3)}-harmonic conjugate of X(27)

Barycentrics    (pending)

As a point on the Euler line, X(7573) has Shinagawa coefficients (pending).

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) =  REFLECTION OF X(3) IN THE DE LONGCHAMPS LINE

Barycentrics    3 R^2 [2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2] - a^6 - a^4(b^2 + c^2) + a^2(b^4 + b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2) : :
X(7574) = 4 X[140] - 3 X[186] = 4 X[468] - 5 X[1656] = 4 X[468] - 3 X[2070] = 5 X[1656] - 3 X[2070] = 2 X[550] - 3 X[2071] = 5 X[1656] - 6 X[2072] = 2 X[468] - 3 X[2072] = X[4] - 3 X[3153] = 6 X[403] - 7 X[3851] = 7 X[3526] - 8 X[5159] = 3 X[3153] + X[5189] = 7 X[3851] - 3 X[5899] = 3 X[5055] - 2 X[7426]

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) = {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)


X(7575) =  REFLECTION OF X(5) IN THE ORTHIC AXIS

Barycentrics    3 R^2 [2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2] - 2 a^2 (a^4 - b^4 - c^4 + b^2c^2) : :
X(7575) = X[3] - 3 X[186] = X[23] + 3 X[186] = 3 X[403] - 2 X[546] = X[23] - 3 X[2070] = X[3] + 3 X[2070] = 5 X[3] - 3 X[2071] = 5 X[186] - X[2071] = 5 X[2070] + X[2071] = 5 X[23] + 3 X[2071] = 7 X[3090] - 3 X[3153] = 3 X[2072] - 4 X[3628] = 5 X[632] - 4 X[5159] = 5 X[631] - X[5189] = 5 X[23] - 3 X[5899] = 5 X[2070] - X[5899] = 5 X[186] + X[5899] = 5 X[3] + 3 X[5899] = 9 X[2071] - 5 X[7464] = 3 X[3] - X[7464] = 9 X[186] - X[7464] = 3 X[23] + X[7464] = 9 X[2070] + X[7464] = 9 X[5899] + 5 X[7464]

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) = 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) = complement of X(7574)
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)


X(7576) =  KOSNITA-TO-ORTHIC SIMILARITY IMAGE OF X(2)

Barycentrics    (pending)

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) = {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)


X(7577) =  INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(186)

Barycentrics    tan A + cot D/2 : : , where cot D/2 = 2S/(a^2 + b^2 + c^2 - 6R^2)
Barycentrics    (b^2 + c^2 - 3 R^2)/(b^2 + c^2 - a^2) : :
X(7577) = 7X(3090) - X(7556)

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) =  ISOGONAL CONJUGATE OF X(566)

Trilinears    1/(sin A + cos A cot D/2) : :, where cot D/2 = 2S/(a^2 + b^2 + c^2 - 6R^2)
Barycentrics    1/(b^2 + c^2 - 3 R^2) : :

X(7578) = trilinear pole of line X(523)X(5926)
X(7578) = X(48)-isoconjugate (polar conjugate) of X(7577)
X(7578) = pole wrt polar circle of trilinear polar of X(7577)

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(7579) =  MIDPOINT OF X(1346) AND X(1347)

Barycentrics    2 a^8 b^2-4 a^6 b^4+4 a^2 b^8-2 b^10+2 a^8 c^2-3 a^6 b^2 c^2-a^4 b^4 c^2-4 a^2 b^6 c^2+6 b^8 c^2-4 a^6 c^4-a^4 b^2 c^4-4 b^6 c^4-4 a^2 b^2 c^6-4 b^4 c^6+4 a^2 c^8+6 b^2 c^8-2 c^10 : : (Peter Moses, June 21, 2015)

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)


X(7580) =  LOZADA-EULER POINT

Trilinears    = a^5-2*(b+c)*a^4+2*(b^3+c^3)*a^2-(b+c)*(b-c)^2*((b+c)*a-2*b*c) : :
X(7580) = -3R*X(2) + (4R + r)*X(3) = 3 X[165] - X[1709] = 3 X[1699] - 4 X[3838] = 3 X[165] - 2 X[4640] = 3 X[1012] - 4 X[6914] = 3 X[3] - 2 X[6914] = 3 X[376] - X[6938] = 7 X[3523] - 5 X[6974] = 5 X[1012] - 8 X[7508] = 5 X[6914] - 6 X[7508] = 5 X[3] - 4 X[7508]

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(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)



leftri  Endo-homothetic centers: X(7581) - X(7588)  rightri

This preamble and centers X(7581)-X(7586) were contributed by César 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)

underbar

X(7581) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER GREBE AND EULER

Trilinears    cos A - 4 sin A - cos(B - C) : :
X(7581) = S*X(4) + 4Sω*X(6)

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)


X(7582) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER GREBE AND EULER

Trilinears    cos A + 4 sin A - cos(B - C) : :
X(7582) = -S*X(4) + 4Sω*X(6)

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)


X(7583) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER GREBE AND JOHNSON

Trilinears    2 sin A + cos(B - C) : :
X(7583) = 2S*X(5) + Sω*X(6)

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)


X(7584) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER GREBE AND JOHNSON

Trilinears    2 sin A - cos(B - C) : :
X(7584) = 2S*X(5) - Sω*X(6)

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)


X(7585) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER GREBE AND MEDIAL

Trilinears    2 sin A + sin B sin C : :
Barycentrics    2a2 + S : :
X(7585) = 3S*X(2) + 4Sω*X(6)

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)


X(7586) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER GREBE AND MEDIAL

Trilinears    2 sin A - sin B sin C : :
Barycentrics    2a2 - S : :
X(7586) = -3S*X(2) + 4Sω*X(6)

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)


X(7587) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND EXTANGENTS

Trilinears    cos(B/2 - C/2) + sin(A/2) + 2 sin2(A/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)


X(7588) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND INTANGENTS

Trilinears    cos(B/2 - C/2) + sin(A/2) - 2 sin2(A/2) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: YFF-CENTRAL AND 1st CIRCUMPERP

Trilinears    (y-z)*(x^3+y*z*(y+z))*sin(A/2)+y*(z^3-x^3)*sin(B/2)+z*(x^3-y^3)*sin(C/2) : : , where x = cos(A/2), y = cos(B/2), z = cos(C/2)

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: YFF-CENTRAL AND HEXYL

Trilinears    F(a,b,c,A,B,C) + G(a,b,c,A,B,C) - G(a,c,b,A,C,B) : : , where
F(a,b,c,A,B,C) = 2*(b-c)*(s-a)*(2*s*a^2-(b+c)^2*a-(b-c)*(b^2-c^2))*cos(B/2)*cos(C/2)+4*(b-c)*S*(2*cos(A/2)*a*(s-a)+S)*s/a
G(a,b,c,A,B,C) = (-2*(a^3*(2*s-3*b)-a*(b+c)*(-2*b^2+2*c*(s+b))-(b^2-c^2)*(b-c)*b)*(s-c)*cos(A/2)+8*S*s*(s-b)*b)*cos(B/2)

X(7590) lies on these lines:
{1, 167}, {3, 173}, {40, 7589}, {236, 936}, {258, 942}, {1490, 7593}, {3333, 7588}, {3576, 7587}


X(7591) =  EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(4)

Trilinears    cos A cos(B/2 - C/2) : :

X(7591) lies on this line: {259,260}

X(7591) = trilinear square root of X(7066)


X(7592) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH AND 2nd CIRCUMPERP

Trilinears    (2*cos(A)^2-3)* cos(A)+(2*cos(A)^2-1)*cos(B-C) : :
Barycentrics    a^2[a^8 - 4a^6(b^2 + c^2) + 2a^4(3b^4 + b^2c^2 + 3c^4) - 4a^2(b^2 - c^2)^2(b^2 + c^2) + (b^2 - c^2)^2(b^4 + c^4)] : :

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) = {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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH AND YFF CENTRAL

Barycentrics    F(a,b,c,A,B,C) + G(a,b,c,A,B,C) - G(a,c,b,A,C,B) : :, where
F(a,b,c,A,B,C) = (4*b^2-4*c^2)*b*c*sin(A/2)-(b-c)*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c))
G(a,b,c,A,B,C) = (2*a^3-2*a^2*b+2*(c^2-b^2)*a+2*(b^2-c^2)*(-2*c+b))*a*sin(B/2)

X(7593) lies on these lines:
{4, 5935}, {9, 173}, {174, 226}, {405, 7587}, {1490, 7590}, {7580, 7589}


X(7594) =  PERSPECTOR OF ABC AND 1st PAMFILOS-ZHOU TRIANGLE

Trilinears    SA/(a2S + bcSω) : :

The Pamfilos-Zhou A-rectangle RA = AABAAAC is the rectangle of maximal area such that A is a vertex of RA, B lies on the line AAAC, and C lies on the line AAAB. Trilinears for the vertices, found by César Lozada, are as follows:

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) =  PERSPECTOR OF ABC AND 2nd PAMFILOS-ZHOU TRIANGLE

Trilinears    1/[a2S + aS(SA - S)] : :

The Pamfilos-Zhou A-rectangle RA = AABAAAC is the rectangle of maximal area such that A is a vertex of RA, B lies on the line AAAC, and C lies on the line AAAB. 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 X(7594).

X(7595) lies on the Feuerbach hyperbola and these lines: {1, 7596}, {8, 637}


X(7596) =  2nd-PAMFILOS-ZHOU-TRIANGLE-TO-ABC ORTHOLOGY CENTER

Trilinears    (((b+c)*a^3+(b-c)^2*(a^2-2*(b+c)*s))*s-S*(2*a^3-(b+c)*(a^2+(b-c)^2)))/a : :

X(7596) lies on these lines: {1, 7595}, {3, 142}, {4, 1123}, {482, 1565}

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


leftri  Touchpoints of pairs of circles: X(7597) - X(7602)  rightri

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.

underbar

X(7597) =  X(11) OF HEXYL TRIANGLE

Trilinears    a*((a*s+b*c)*u - b*(s-c)*v - c*(s-b)*w -(2*r*(R+r)-(s-b)*(s-c))*b*c/r^2) : : where u = csc(A/2), v = csc(B/2), w = csc(C/2)

X(7597) is the touchpoint of the circumcircle-of-ABC and the incircle-of-hexyl-triangle. See Tran Quang Hung, Randy Hutson, and César Lozada, ADGEOM 2511 (May 2015)

X(7597) lies on the circumcircle and these lines: {3,164}, {109,260}

X(7597) = reflection of X(3659) in X(3)
X(7597) = X(11)-of-hexyl-triangle
X(7597) = X(119)-of-excentral-triangle
X(7597) = X(104)-of-1st-circumperp-triangle
X(7597) = X(100)-of-2nd-circumperp-triangle
X(7597) = circumcircle-antipode of X(3659)
X(7597) = Λ(X(1), X(167))
X(7597) = Λ(X(10), X(2090))
X(7597) = Λ(X(40), X(164))
X(7597) = Λ(X(178), X(946))


X(7598) =  X(11) OF LUCAS CENTRAL TRIANGLE

Trilinears    a[SA(S2ω - 3S2) + S(SB -SC)2] : :

X(7598) is the touchpoint of the nine-point-circle-of-Lucas-central-triangle and the circumcircle-of--Lucas-tangents-circle. X(7598) is the first known center on the Lucas circles radical circle. See X(7599) and César Lozada, ADGEOM 2520 (May 9, 2015)

X(7598) lies on the Parry circle, the Lucas radical circle, and these lines:
{3, 3124}, {23, 2460}, {110, 371}, {111, 6200}, {372, 7602}, {493, 1976}, {2502, 6221}, {2987, 5408}, {6453, 7601}

X(7598) = reflection of X(7599) in X(7600)
X(7598) = X(11) of Lucas central triangle
X(7598) = {X(3),X(3124)}-harmonic conjugate of X(7599)
X(7598) = trilinear pole, wrt the Lucas tangents triangle, of the Brocard axis


X(7599) =  X(11) OF LUCAS(-1) CENTRAL TRIANGLE

Trilinears    a[SA(S2ω - 3S2) - S(SB - SC)2] : :

X(7599) is the touchpoint of the nine-point-circle-of-Lucas(-1)-central triangle and the circumcircle-of-Lucas(-1)-tangents-triangle. See X(7598). The points of intersection of the Parry circle and the polar of X(3124) are X(7598) and X(7599). (César Lozada, May 10, 2015)

X(7599) lies on the Parry circle, the Lucas(-1) radical circle, and these lines:
{3,3124}, {23,2459}, {110,372}, {111,6396}, {371,7601}, {494,1976}, {2502,6398}, {2987,5409}, {6454,7602}

X(7599) = reflection of X(7598) in X(7600)
X(7599) = X(11) of Lucas(-1) central triangle
X(7599) = {X(3),X(3124)}-harmonic conjugate of X(7598)
X(7599) = trilinear pole, wrt the Lucas(-1) tangents triangle, of the Brocard axis


X(7600) =  MIDPOINT OF X(7598) AND X(7599)

Trilinears    a*(3*(18*R^2*S^2-SW*(S^2+SW^2))*SA^2+(-9*S^2*(4*SW*R^2+S^2)+SW^2*(8*S^2+SW^2))*SA+(4*S^2-SW^2)*(18*R^2*S^2-SW*(S^2+SW^2)))

X(7600) lies on these lines: {3,3124}, {351,2872}, {(2482,6388}

X(7600) = midpoint of X(7598) and X(7599)
X(7600) = circumcircle-inverse of X(3124)


X(7601) =  X(11) OF LUCAS INNER TANGENTIAL TRIANGLE

Trilinears    a[4SA(-3S2 + S2ω) + (S2A + 2SBSC + 2S2ω - 7S2)S] : :

X(7601) is the touchpoint of the circumcircle-of-Lucas-inner-triangle and the nine-point-circle-of-Lucas-inner-tangential triangle.

X(7601) lies on the Parry circle and these lines:
{2, 6568}, {3, 7602}, {23, 6567}, {110, 1151}, {111, 6221}, {371, 7599}, {2502, 6468}, {3124, 6425}, {6453, 7598}

X(7601) = inverse-in-Lucas-circles-radical-circle of X(110)
X(7601) = X(11) of Lucas inner tangential triangle
X(7601) = trilinear pole, wrt the Lucas inner triangle, of the Brocard axis


X(7602) =  X(11) OF LUCAS(-1) INNER TANGENTAIL TRIANGLE

Trilinears    a[4SA(-3S2 + S2ω) + (S2A - 2SBSC + 2S2ω - 7S2)S] : :

X(7602) is the touchpoint of the circumcircle-of-Lucas(-1)-inner triangle and the nine-point-circle-of-Lucas(-1)-inner-tangential-triangle.

X(7602) lies on the Parry circle and these lines:
{2, 6569}, {3, 7601}, {23, 6566}, {110, 1152}, {111, 6398}, {372, 7598}, {2502, 6469}, {3124, 6426}, {6454, 7599}

X(7602) = X(11) of Lucas(-1) inner tangential triangle
X(7602) = inverse-in-Lucas(-1)-circles-radical-circle of X(110)
X(7602) = trilinear pole, wrt the Lucas(-1) inner triangle, of the Brocard axis


X(7603) =  X(2)-HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER

Trilinears    bc[3a2(b2 + c2) - 2(b2 - c2)2] : :

Suppose that P is a point in the plane of a triangle ABC. Let NA be the nine-point center of PBC, and define NB and NC cyclically. Let N1 be the nine-point center of NNBNC, and define N2 and N3 cyclically. The triangle N1N2N3 is homothetic to ABC. Let H(P) denote the center of homothety, here named the P-Hatzipolakis-Lozada homothetic center. If P = u : v : w, (trilinears), then

H(P) = (a^3*b*c*v*w + c*((2*b^2+c^2)*a^2-(b^2-c^2)^2)*u*w + b*((2*c^2+b^2)*a^2-(b^2-c^2)^2)*u*v - a*(a^4-3*(b^2+c^2)*a^2+2*(b^2-c^2)^2)*u^2)/(a*u) : :

The appearance of (i,j) in the following list indicates that H(X(i)) = X(j): (1,5443), (2,7603), (3,6143), (4,5), (5,7604), (6,7605), (523,476), (1138,3258), (3459,5).

See Antreas Hatzipolakis and César Lozada, Hyacinthos #23244, May 18 2015.

X(7603) is the QA-P7 center (QA-Nine-point Center Homothetic Center) of quadrangle ABCX(2); see Chris van Tienhoven, Quadrangle Objects. (In general, the P-Hatzipolakis-Lozada homothetic center is the QA-P7 center of quadrangle ABCP.)

X(7603) lies on these lines:
{2,187}, {5,39}, {6,5055}, {30,3055}, {32,1656}, {216,2072}, {230,547}, {232,7577}, {233,800}, {373,1648}, {381,574}, {395,5459}, {396,5460}, {549,6781}, {566,7579}, {597,5477}, {1078,7843}, {1285,5067}, {1573,3814}, {1594,3199}, {2482,3363}, {2548,3090}, {2549,3545}, {3053,5070}, {3314,3934}, {3526,5206}, {3767,5041}, {3851,5013}, {5025,6683}, {5071,5309}, {5107,5476}, {5154,5283}

X(7603) = complement of X(7771)


X(7604) =  X(5)-HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER

Trilinears    sec(B - C) [7/2 - cos 2A + 2 cos A cos(B - C) + 3 cos(2B - 2C)] : :

X(7604) is the QA-P7 center (QA-Nine-point Center Homothetic Center) of quadrangle ABCX(5); see X(7603).

X(7604) lies on these lines:
{5,49}, {252,1656}, {1157,3628}


X(7605) =  X(6)-HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER

Trilinears    (11 - 4 cos 2A) cos(B - C) + 2 cos A cos(2B - 2C) + 6 cos A - 3 cos 3A : :

X(7605) is the QA-P7 center (QA-Nine-point Center Homothetic Center) of quadrangle ABCX(6); see X(7603).

X(7605) lies on these lines:
{2,51}, {5,399}, {23,2916}, {125,5643}, {182,7533}, {1112,6143}, {3410,5422}, {3580,7570}, {5056,5645}, {5480,7496}


X(7606) =  X(182)-OF-McCAY TRIANGLE

Barycentrics    5a^6 + 2b^6 + 2c^6 - 6a^4(b^2 + c^2) - 18a^2b^2c^2 - 3b^4c^2 - 3b^2c^4 : :

The McCay triangle, here denoted by M, is the triangle whose vertices are the centers of the McCay circles. Along with properties of M given at X(7606)-X(7615) are the following: M is similar to the 4th Brocard triangle with X(111) as center of similitude. The circumcircle of M is orthogonal to the Parry circle. The inverse-in-circumcircle-of-M of X(2) is X(111).

The appearance of (i,j) in the following list indicates that X(i)-of-M = X(j):
(2,2), (6,182), (23,111), (30,543), (69,1352), (98,7610), (99,381), (115,549), (148,376), (182,7606), (193,6776), (230,6036), (298,5617), (299,5613), (325,114), (385,98), (395,6774), (396,6771), (468,6719), (523,2793), (524,542), (530,531), (531,530), (542,524), (543,30), (620,547), (671,3), (690,1499), (858,126), (1499,690), (2482,5), (2782,3849), (2793,523), (3180,6770), (3181,6773), (3413,3414), (3414,3413), (3849,2782), (5077,3734), (5461,140)

Let LA be the radical axis of the A-McCay circle and the A-Neuberg circle, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Then A'B'C' is the reflection of ABC in X(6). Let TA be the radical trace of the A-McCay circle and the A-Neuberg circle, and define TB and TC cyclically. The triangle TATBTC is perspective to ABC at X(2996). See ADGEOM #2541, Randy Hutson, May 19, 2015.

The A-vertex of the McCay triangle is given by barycentrics -a^2 (a^2+b^2+c^2) : 2 a^4-2 a^2 b^2+2 b^4-3 a^2 c^2-3 b^2 c^2+c^4 : 2 a^4-3 a^2 b^2+b^4-2 a^2 c^2-3 b^2 c^2+2 c^4; moreover, (power of A in A-McCay circle) = (b2 + c2 - a2)/3, (power of B in A-McCay circle) = a2/3, (power of C in A-McCay circle) = a2/3. (Peter Moses, May 23, 2015)

X(7606) lies on these lines: {2,6}, {182,7617}, {5969,7622}, {511,1153}, {598,1691}, {5476,5569}

X(7606) = midpoint of X(5476) and X(5569)


X(7607) =  PERSPECTOR OF ABC AND McCAY TRIANGLE

Trilinears    1/(3 sin A - cos A cot ω) : :
Trilinears    1/(cos A - 3 sin A tan ω) : :
Barycentrics    1/(a^4 + 2b^4 + 2c^4 - 3a^2b^2 - 3a^2c^2 - 2b^2c^2) : :

The McCay triangle MAMBMC, defined at X(7606), is perspective to ABC, and the perspector is X(7607). See ADGEOM #2541, Randy Hutson, May 19, 2015.

X(7607) lies on the Kiepert hyperbola and these lines: {2,575}, {3,671}, {4,187}, {5,598}, {6,7608}, {76,140}, {83,1656}, {230,262}, {275,5094}, {468,2052}, {631,1153}, {647,5466}, {1327,6811}, {1328,6813}, {1916,6036}, {2996,3523}, {3533,3788}, {5056,5395}, {5392,7495}, {5503,7610}

X(7607) = isogonal conjugate of X(576)
X(7607) = trilinear product of the vertices of the McCay triangle
X(7607) = X(3054)-cross conjugate of X(2)
X(7607) = X(25)-vertex conjugate of X(262)
X(7607) = perspector of ABC and 1st Brocard triangle of Artzt triangle


X(7608) =  ISOGONAL CONJUGATE OF X(575)

Trilinears    1/(3 sin A + cos A cot ω) : :
Trilinears    1/(cos A + 3 sin A tan ω) : :

Let MAMBMC be the McCay triangle, defined at X(7606). Let A' be the reflection of MA in line BC, and define B' and C' cyclically. Then the lines AA', BB', CC" concur in X(7608). Also, X(7608) is the trilinear product A'*B'*C'. See X(7606) and ADGEOM #2541, Randy Hutson, May 19, 2015.

X(7608) lies on the Kiepert hyperbola and these lines: {2,576}, {3,598}, {4,574}, {5,671}, {6,7607}, {76,1656}, {83,140}, {98,3815}, {275,468}, {1327,6813}, {1328,6811}, {2052,5094}, {2996,5056}, {3090,5485}, {3523,5395}, {3533,6680}

X(7608) = isogonal conjugate of X(575)
X(7608) = complement of X(7616)
X(7608) = X(3055)-cross conjugate of X(2)
X(7608) = perspector of ABC and the medial triangle of the McCay triangle


X(7609) =  PERSPECTOR OF McCAY TRIANGLE AND EXCENTRAL TRIANGLE

Barycentrics    (pending)

See X(7606) and ADGEOM #2541, Randy Hutson, May 19, 2015.

X(7609) lies on these lines:
{1,576}, {4,9}, {238,1385}, {984,1482}, {1423,3338}, {1756,3336}, {2792,7385}, {4672,6998}, {7613, 7614}

X(7609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,6210,6211), (576,7611,1), (6191,6192,3496)


X(7610) =  X(98)-OF-McCAY-TRIANGLE

Barycentrics    (pending)
X(7610) = 4 X[1153] - 3 X[5054] = 3 X[3524] + X[5485]

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)
X(7610) = antipode of X(381) in the circumcircle of the McCay triangle
X(7610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,183,599), (2,1992,3815)
X(7610) = X(3)-of-Artzt-triangle
X(7610) = anti-Artzt-to-Artzt similarity image of X(599)


X(7611) =  1st HUTSON-McCAY POINT

Barycentrics    (pending)
X(7611) = MA/RA + MB/RB + MA/RC, where RA = radius of A-McCay circle, and RB and RC are defined cyclically

Let MAMBMC be the McCay triangle, defined at X(7606). Let A' be the insimilicenter of the B- and C- McCay circles, and define B' and C' cyclically. The triangles A'B'C' and MAMBMC are perspective, and X(7611) is their perspector. The excimilicenters of pairs of McCay circles lie on the sidelines of the excentral triangle.

See X(7606) and ADGEOM #2541, Randy Hutson, May 19, 2015.

X(7611) lies on these lines: {1,576}, {2,2783}, {37,517}, {1001,2801}, {1284,5902}

X(7611) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7609,576), (2,183,599), (2,1992,3815)


X(7612) =  2nd HUTSON-McCAY POINT

Trilinears    1/(a - R cot ω cos A) : :
Trilinears: 1/(2 sin A - cos A cot ω) : :
Trilinears: 1/(cos A - 2 sin A tan ω) : :
Barycentrics: 1/[a^4 - 4a^2(b^2 + c^2) + 3b^4 - 2b^2c^2 + 3c^4] : :
Barycentrics    (tan A)/(2 tan A - cot ω) : :
X(7612) = 2 X[3] + X[2996] = 5 X[631] - 2 X[6337]

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)
X(7612) = isotomic conjugate of X(1007)
X(7612) = X(6776)-cross conjugate of X(4)
X(7612) = X(i)-vertex conjugate of X(j) for these (i,j): (4,25), (3424,3425)
X(7612) = perspector of ABC and medial triangle of Artzt triangle


X(7613) =  1st LOZADA-McCAY POINT

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

Let OM be the circle tangent to all three McCay circles. The center of OM is X(7613), and the radius-squared is r2(S2ω - 3S2)/(3s2 - 4Sω)2).

See ADGEOM #2542, César Lozada, May 20, 2015.

If you have The Geometer's Sketchpad, you can view X(7613) and X(7614).

X(7613) lies on these lines:
{2,846}, {7,1738}, {10,4862}, {277,5805}, {346,3836}, {391,4655}, {516,4859}, {726,4373}, {740,4869}, {984,4346}, {986,4208}, {1086,2550}, {1386,4000}, {1698,7229}, {3008,4312}, {3315,3434}, {3755,6173}, {3826,4419}, {3846,4748}, {4402,5847}, {4887,5223}, {4902,5850}, {7609,7614}

X(7613) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1086,2550,4310), (4000,5880,4307)


X(7614) =  2nd LOZADA-McCAY POINT

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

Continuing from X(7613), let QA be the touchpoint of OM and the A-McCay circle, and define QB and QC cyclically. The triangle QAQBQC is perspective to the excentral triangle of ABC, and the perspector is X(7614).

See ADGEOM #2542, César Lozada, May 20, 2015.

If you have The Geometer's Sketchpad, you can view X(7613) and X(7614).

X(7614) lies on these lines: {9,4454}, {373,4414}, {1445,2347}, {7609,7613}


X(7615) =  REFLECTION OF X(376) IN X(5569)

Barycentrics    a^4 + 2*(b^2+c^2)*a^2 - 5*b^4 + 14*b^2*c^2 - 5*c^4 : :
X(7615) = 4 X[1153] - 3 X[3524] = 3 X[3545] + X[5485]

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)
X(7615) = anticomplement of X(7622)
X(7615) = X(376)-of-McCay-triangle
X(7615) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,671,2549), (3734,5461,2)


X(7616) =  PERSPECTOR OF McCAY TRIANGLE AND ANTICOMPLEMENTARY TRIANGLE

Barycentrics    (pending)

X(7616) lies on the Steiner rectangular hyperbola, the anticomplement of the Kiepert hyperbola, and these lines: {2,576}, {5,2896}, {20,1078}, {147,183}, {194,631}, {626,7486}

X(7616) = anticomplement of X(7608)
X(7616) = X(575)-anticomplementary conjugate of X(8)


X(7617) =  X(3)-OF-McCAY-TRIANGLE

Barycentrics    a^4-4 a^2 b^2+4 b^4-4 a^2 c^2-10 b^2 c^2+4 c^4 : :

Peter Moses (May 24, 2015) contributes the following theorem and examples. Suppose that P is a point on the Euler line of the McCay triangle, and let Q be the complement (wrt ABC) of P. Then P = X(2)X(99)∩QX(524). Specifically, if X = a2SA + kSBSC : : is a point on the Euler line, then the point P = X-of-McCay-triangle is given by

P = 2 (a^4-4 a^2 b^2+4 b^4-4 a^2 c^2-10 b^2 c^2+4 c^4) + k(7 a^4-10 a^2 b^2+b^4-10 a^2 c^2+2 b^2 c^2+c^4) : :

If k = -1, then X = X(20) and P = X(7620);
if k = -1/2, then X = X(376) and P = X(7615);
if k = = - 2abc/(a^3-a^2 b-a b^2+b^3-a^2 c-b^2 c-a c^2-b c^2+c^3), then X = X(21) and P = X(7621).
See X(7617)-X(7622).

X(7617) lies on these lines:
{2,99}, {3,1153}, {5,524}, {30,5569}, {32,598}, {182,7606}, {230,3363}, {381,2080}, {538,5055}, {599,625}, {754,3545}, {1003,5215}, {2548,5032}, {3090,5485}

X(7617) = midpoint of X(7618) and X(7620)
X(7617) = reflection of X(7618) in X(7619)
X(7617) = complement of X(7618)
X(7617) = anticomplement of X(7619)
X(7617) = center of McCay circumcircle
X(7617) = harmonic center of nine-point circle and Ehrmann circle
X(7617) = X(182)-of-Artzt-triangle
X(7617) = anti-Artzt-to-Artzt similarity image of X(3734)


X(7618) =  X(4)-OF-McCAY-TRIANGLE

Barycentrics    7 a^4-10 a^2 b^2+b^4-10 a^2 c^2+2 b^2 c^2+c^4 : :

See X(7617).

X(7618) lies on these lines:
{2,99}, {3,524}, {32,5032}, {187,1992}, {376,3849}, {538,3524}, {549,7610}, {597,5024}, {598,2548}, {599,6390}, {631,1153}, {2709,6093}, {3523,7616}, {3793,5585}, {5215,5309}

X(7618) = reflection of X(i) in X(j) for these (i,j): (7617,7619), (7620,7617)
X(7618) = anticomplement of X(7617)


X(7619) =  X(5)-OF-McCAY-TRIANGLE

Barycentrics    8 a^4-14 a^2 b^2+5 b^4-14 a^2 c^2-8 b^2 c^2+5 c^4 : :
X(7619) = (1 + 20 sin2ω)*X(2) + (-1 + 4 sin2ω)*X(99)

See X(7617).

Let G be the centroid of a triangle ABC, and
Oa = circumcenter of GBC, and define Ob and Oc cyclically
Na = nine-point center of GObOc, and define Nb and Nc cyclically
L = Euler line of NaNbNc
L' = Euler line of McCay triangle
Then X(7619) = L∩L'. See Angel Montesdeoca, X(7619) and Hyacinthos #24690.

X(7619) lies on these lines: {2,99}, {140,524}, {549,3849}, {598,1506}, {754,5054}, {3533,5485}

X(7619) = midpoint of X(7617) and X(7618)
X(7619) = complement of X(7617)


X(7620) =  X(20)-OF-McCAY-TRIANGLE

Barycentrics    5 a^4-2 a^2 b^2-7 b^4-2 a^2 c^2+22 b^2 c^2-7 c^4 : :

See X(7617).

Let GA be the antipode of X(2) in the A-McCay circle, and define GB and GC cyclically; then GAGBGC is homothetic to the McCay triangle at X(2). Let LA be the line tangent to the A-McCay circle at GA, and define LB and LC cyclically. Let A" = LB∩LC, and define B" and C" cyclically. Then A"B"C" is the antipedal triangle of X(2) with respect to GAGBGC, and A"B"C" is inversely similar to ABC and homothetic to the 1st Brocard triangle. Also, X(7620) = X(2)-of-A"B"C" = X(376)-of GAGBGC. (Randy Hutson, May 27, 2015)

X(7620) lies on these lines: {2,99}, {4,524}, {376,7610}, {538,3839}, {598,2996}, {1153,3523}, {3543,3849}

X(7620) = reflection of X(7618) in X(7617)


X(7621) =  X(21)-OF-McCAY-TRIANGLE

Barycentrics    a^4-4 a^2 b^2+4 b^4-9 a^2 b c-9 a b^2 c-4 a^2 c^2-9 a b c^2-10 b^2 c^2+4 c^4 : :

See X(7617).

X(7621) lies on these lines: {2,99}, {404,1153}, {442,524}, {1698,4363}, {3849,6175}


X(7622) =  X(381)-OF-McCAY-TRIANGLE

Barycentrics    5 a^4-8 a^2 b^2+2 b^4-8 a^2 c^2-2 b^2 c^2+2 c^4 : :
X(7622) = 2 X[1153] - 3 X[5054] = 3 X[5054] - X[7610] = 2 X[7615] - 3 X[7617] = X[7617] + 2 X[7618] = X[7615] + 3 X[7618] = X[7615] - 6 X[7619] = X[7617] - 4 X[7619] = X[7618] + 2 X[7619] = 5 X[7615] - 3 X[7620] = 5 X[7617] - 2 X[7620] = 5 X[2] - X[7620] = 10 X[7619] - X[7620] = 5 X[7618] + X[7620]

See 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(7622) = complement of X(7615)
X(7622) = reflection of X(i) in X(j) for these (i,j): (2,7619), (5569,549), (7610,1153), (7617,2)
X(7622) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2482,3734), (2,2549,5461), (5054,7610,1153), (7618,7619,7617)


X(7623) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, INCIRCLE, NINE-POINT

Barycentrics    (b-c) (2 a^7+a^6 b-8 a^5 b^2-a^4 b^3+10 a^3 b^4-a^2 b^5-4 a b^6+b^7+a^6 c-8 a^5 b c+13 a^4 b^2 c-13 a^2 b^4 c+8 a b^5 c-b^6 c-8 a^5 c^2+13 a^4 b c^2-8 a^3 b^2 c^2+6 a^2 b^3 c^2+4 a b^4 c^2-3 b^5 c^2-a^4 c^3+6 a^2 b^2 c^3-16 a b^3 c^3+3 b^4 c^3+10 a^3 c^4-13 a^2 b c^4+4 a b^2 c^4+3 b^3 c^4-a^2 c^5+8 a b c^5-3 b^2 c^5-4 a c^6-b c^6+c^7) : :

If U, V, W are circles, there exists a unique circle B = B(U,V,W) which bisects each of the circles U, V, W. Let P, Q, R be the centers of U,V,W, respectively, let O be the circumcenter of PQR, and let O' be the radical center of U, V, W. Then B = reflection of O' in O. For B = X(7623), the radical center O' is X(676). (Peter Moses, May 26, 2015). For properties of bisecting circles, see P. H. Daus's article in the American Mathematical Monthly 47 (1940) 519-529: Bisecting Circles.

Examples:

X(40) = center of B(Soddy circles)
X(382) = center of B(power circles)
X(550) = center of B(Stammler circles)
X(6361) = center of B(Longuet-Higgins circles)
X(5493) = center of B(excentral circles)
X(7615) = center of B(McCay circles)
X(7623) = center of B(circumcircle, incircle, nine-point circle)
X(7624) = center of B(circumcircle, Brocard, nine-point circle)
X(7625) = center of B(circumcircle, Brocard, orthocentroidal circle)
X(7626) = center of B(circumcircle, nine-point circle, Apollonius circle)
X(7627) = center of B(circumcircle, nine-point circle, Bevan circle)
X(8547) = center of B(A-Ehrmann circle, B-Ehrmann circle, C-Ehrmann circle)

X(7623) lies on this line: {523,7624}


X(7624) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, BROCARD, NINE-POINT

Barycentrics    a^2 (b-c) (b+c) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+2 a^4 b^2 c^2-10 b^4 c^4+2 a^2 c^6-c^8) : :

The radical center O' for the three circles is X(647). See X(7623).

X(7624) lies on these lines: {2,647}, {512,7625}, {523,7623}, {525,2492}


X(7625) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES CIRCUMCIRCLE, BROCARD, ORTHOCENTROIDAL

Barycentrics    a^2 (b-c) (b+c) (a^8-6 a^6 b^2+4 a^4 b^4+6 a^2 b^6-5 b^8-6 a^6 c^2+4 a^4 b^2 c^2-2 a^2 b^4 c^2+4 a^4 c^4-2 a^2 b^2 c^4-26 b^4 c^4+6 a^2 c^6-5 c^8) : :

The radical center O' for the three circles is X(647). See X(7623). See X(7623).

X(7625) lies on this line: {512,7624}, {523,7624}


X(7626) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, APOLLONIUS}

Barycentrics    a (b-c) (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^2 b^2 c-2 a b^3 c-b^4 c-2 a^3 c^2+2 a^2 b c^2+2 a^2 c^3-2 a b c^3+a c^4-b c^4-c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7626) lies on these lines: {39, 650}, {512, 7635}, {513, 7640}, {523, 7623}, {3788, 4885}


X(7627) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, BEVAN}

Barycentrics    a (b-c) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-2 a^4 b c+4 a^2 b^3 c-2 b^5 c-3 a^4 c^2+8 a^2 b^2 c^2-12 a b^3 c^2+b^4 c^2+4 a^2 b c^3-12 a b^2 c^3+4 b^3 c^3+3 a^2 c^4+b^2 c^4-2 b c^5-c^6) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7627) lies on these lines: {37, 650}, {513, 7629}, {523, 7623}, {4977, 7645}


X(7628) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, ORTHOCENTROIDAL, FUHRMANN

Barycentrics    (a - b - c)*(b - c)*(a^3 + a^2*b + a*b^2 + b^3 + a^2*c + 6*a*b*c - 5*b^2*c + a*c^2 - 5*b*c^2 + c^3) : :

The radical center O' for the three circles is X(7649). See X(7623).

X(7628) lies on these lines: {522, 3717}, {523, 7625}, {3667, 3679}


X(7629) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, FUHRMANN, BEVAN

Barycentrics    a*(a - b - c)*(b - c)*(a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 2*a^3*b*c + 2*a*b^3*c - b^4*c + 2*a*b^2*c^2 - 2*a^2*c^3 + 2*a*b*c^3 - b*c^4 + c^5) : :

The radical center O' for the three circles is X(656). See X(7623).

X(7629) lies on these lines: {1, 656}, {513, 7627}, {522, 3717}, {5552, 7253}


X(7630) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, 1st LEMOINE

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

The radical center O' for the three circles is X(2485). See X(7623).

X(7630) lies on these lines: {39, 2485}, {182, 520}, {512, 7637}, {523, 7623}, {525, 7648}, {2489, 2799}


X(7631) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, 2nd LEMOINE

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

The radical center O' for the three circles is X(2489). See X(7623).

X(7631) lies on these lines: {39, 2489}, {182, 924}, {512, 7633}, {523, 7623}, {690, 2485}, {3566, 7648}


X(7632) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: APOLLONIUS, BEVAN, SPIEKER RADICAL

Barycentrics    a*(b - c)*(a^5 + a^4*b - 4*a^3*b^2 + 3*a*b^4 - b^5 + a^4*c - 8*a^3*b*c + 10*a^2*b^2*c - 6*a*b^3*c - b^4*c - 4*a^3*c^2 + 10*a^2*b*c^2 - 6*a*b*c^3 + 3*a*c^4 - b*c^4 - c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7632) lies on these lines: {513, 7636}, {514, 7634}, {6371, 7635}


X(7633) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, 2nd LEMOINE, ORTHOCENTROIDAL

Barycentrics    a^2*(b - c)*(b + c)*(a^8 - 6*a^6*b^2 + 4*a^4*b^4 + 6*a^2*b^6 - 5*b^8 - 6*a^6*c^2 + 4*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 4*a^4*c^4 + 10*a^2*b^2*c^4 - 14*b^4*c^4 + 6*a^2*c^6 - 5*c^8) : :

The radical center O' for the three circles is X(2489). See X(7623).

X(7633) lies on these lines: {512, 7631}, {523, 7625}, {526, 2485}


X(7634) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, BEVAN, SPIEKER RADICAL

Barycentrics    a*(b - c)*(a^5 + a^4*b - 4*a^3*b^2 + 3*a*b^4 - b^5 + a^4*c - 6*a^3*b*c + 10*a^2*b^2*c - 6*a*b^3*c + b^4*c - 4*a^3*c^2 + 10*a^2*b*c^2 - 6*a*b^2*c^2 - 6*a*b*c^3 + 3*a*c^4 + b*c^4 - c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7634) lies on these lines: {103, 2716}, {513, 7627}, {514, 7632}, {522, 7636}, {650, 3730}


X(7635) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, APOLLONIUS, BEVAN

Barycentrics    a*(b - c)*(a^5 - a^4*b - 4*a^3*b^2 + a*b^4 - b^5 - a^4*c - 4*a^3*b*c + 12*a^2*b^2*c - 4*a*b^3*c - b^4*c - 4*a^3*c^2 + 12*a^2*b*c^2 - 4*a*b^2*c^2 - 4*a*b*c^3 + a*c^4 - b*c^4 - c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7635) lies on these lines: {512, 7626}, {513, 7627}, {6371, 7632}


X(7636) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, APOLLONIUS, SPIEKER RADICAL

Barycentrics    a*(b - c)*(a^5 + a^4*b - 4*a^3*b^2 + 3*a*b^4 - b^5 + a^4*c - 4*a^3*b*c + 6*a^2*b^2*c - 2*a*b^3*c - b^4*c - 4*a^3*c^2 + 6*a^2*b*c^2 - 4*a*b^2*c^2 - 2*a*b*c^3 + 3*a*c^4 - b*c^4 - c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7636) lies on these lines: {512, 7626}, {513, 7632}, {522, 7634}


X(7637) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, 1st LEMOINE, ORTHOCENTROIDAL

Barycentrics    a^2*(b - c)*(b + c)*(a^8 - 6*a^6*b^2 + 4*a^4*b^4 + 6*a^2*b^6 - 5*b^8 - 6*a^6*c^2 - 2*a^4*b^2*c^2 + 10*a^2*b^4*c^2 - 6*b^6*c^2 + 4*a^4*c^4 + 10*a^2*b^2*c^4 - 14*b^4*c^4 + 6*a^2*c^6 - 6*b^2*c^6 - 5*c^8) : :

The radical center O' for the three circles is X(2485). See X(7623).

X(7637) lies on these lines: {512, 7630}, {523, 7625}


X(7638) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, GALLATLY

Barycentrics    a^2*(b - c)*(b + c)*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 + a^8*c^2 - 4*a^6*b^2*c^2 + 5*a^4*b^4*c^2 - 3*a^6*c^4 + 5*a^4*b^2*c^4 - 2*a^2*b^4*c^4 - 2*b^6*c^4 + 3*a^4*c^6 - 2*b^4*c^6 - a^2*c^8) : :

The radical center O' for the three circles is X(2491). See X(7623).

X(7638) lies on these lines: {39, 690}, {523, 7623}


X(7639) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, ORTHOCENTROIDAL, BEVAN

Barycentrics    a*(b - c)*(a^6 - 7*a^4*b^2 + 11*a^2*b^4 - 5*b^6 - 6*a^4*b*c + 12*a^2*b^3*c - 6*b^5*c - 7*a^4*c^2 + 18*a^2*b^2*c^2 - 36*a*b^3*c^2 + 5*b^4*c^2 + 12*a^2*b*c^3 - 36*a*b^2*c^3 + 12*b^3*c^3 + 11*a^2*c^4 + 5*b^2*c^4 - 6*b*c^5 - 5*c^6) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7639) lies on these lines: {513, 7627}, {523, 7625}


X(7640) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: NINE-POINT, ORTHOCENTROIDAL, APOLLONIUS

Barycentrics    a*(b - c)*(5*a^6 + a^4*b^2 - 17*a^2*b^4 + 11*b^6 + 6*a^4*b*c - 6*a^2*b^3*c + 6*a*b^4*c + 6*b^5*c + a^4*c^2 + 12*a^2*b^2*c^2 + 6*a*b^3*c^2 - 5*b^4*c^2 - 6*a^2*b*c^3 + 6*a*b^2*c^3 - 17*a^2*c^4 + 6*a*b*c^4 - 5*b^2*c^4 + 6*b*c^5 + 11*c^6) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7640) lies on these lines: {513, 7626}


X(7641) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, BROCARD, BEVAN

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

The radical center O' for the three circles is X(649). See X(7623).

X(7641) lies on these lines: {512, 7624}, {513, 7627}


X(7642) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, ORTHOCENTROIDAL, APOLLONIUS

Barycentrics    a*(b - c)*(b + c)*(a^7 - 7*a^5*b^2 + 11*a^3*b^4 - 5*a*b^6 - 3*a^5*b*c - 3*a^4*b^2*c - 3*a*b^5*c - 3*b^6*c - 7*a^5*c^2 - 3*a^4*b*c^2 + 6*a^3*b^2*c^2 - 7*a*b^4*c^2 - 3*b^5*c^2 + 11*a^3*c^4 - 7*a*b^2*c^4 - 3*a*b*c^5 - 3*b^2*c^5 - 5*a*c^6 - 3*b*c^6) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7642) lies on these lines: {512, 7626}, {523, 7625}


X(7643) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, 2nd LEMOINE, BEVAN

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

The radical center O' for the three circles is X(2484). See X(7623).

X(7643) lies on these lines: {512, 7631}, {513, 7627}


X(7644) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, SPIEKER, BEVAN

Barycentrics    a*(b - c)*(3*a^5 + 3*a^4*b - 14*a^3*b^2 + 2*a^2*b^3 + 11*a*b^4 - 5*b^5 + 3*a^4*c - 24*a^3*b*c + 42*a^2*b^2*c - 24*a*b^3*c + 3*b^4*c - 14*a^3*c^2 + 42*a^2*b*c^2 - 22*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 - 24*a*b*c^3 + 2*b^2*c^3 + 11*a*c^4 + 3*b*c^4 - 5*c^5) : :

The radical center O' for the three circles is X(2516). See X(7623).

X(7644) lies on this line: {513, 7627}


X(7645) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: NINE-POINT, BEVAN, SPIEKER RADICAL

Barycentrics    (b - c)*(a^6 + 3*a^5*b - 4*a^4*b^2 - 4*a^3*b^3 + 3*a^2*b^4 + a*b^5 + 3*a^5*c - 8*a^4*b*c + 10*a^3*b^2*c - 6*a^2*b^3*c - a*b^4*c + 2*b^5*c - 4*a^4*c^2 + 10*a^3*b*c^2 - 4*a^3*c^3 - 6*a^2*b*c^3 - 4*b^3*c^3 + 3*a^2*c^4 - a*b*c^4 + a*c^5 + 2*b*c^5) : :

The radical center O' for the three circles is X(650). See X(7623).

X(7645) lies on these lines: {514, 7632}, {4977, 7627}


X(7646) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, FUHRMANN, CONWAY

Barycentrics    (a - b - c)*(b - c)*(2*a^5*b - 4*a^3*b^3 + 2*a*b^5 + 2*a^5*c - 5*a^4*b*c + 2*a^3*b^2*c + 4*a^2*b^3*c - 4*a*b^4*c + b^5*c + 2*a^3*b*c^2 - 10*a^2*b^2*c^2 + 2*a*b^3*c^2 - 4*a^3*c^3 + 4*a^2*b*c^3 + 2*a*b^2*c^3 - 2*b^3*c^3 - 4*a*b*c^4 + 2*a*c^5 + b*c^5) : :

The radical center O' for the three circles is X(7650). See X(7623).

X(7646) lies on this line: {522, 3717}


X(7647) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, FUHRMANN

Barycentrics    (a - b - c)*(b - c)*(a^6 - a^4*b^2 - a^2*b^4 + b^6 + 2*a^4*b*c - 4*a^3*b^2*c + 4*a*b^4*c - 2*b^5*c - a^4*c^2 - 4*a^3*b*c^2 + 8*a^2*b^2*c^2 - 4*a*b^3*c^2 - b^4*c^2 - 4*a*b^2*c^3 + 4*b^3*c^3 - a^2*c^4 + 4*a*b*c^4 - b^2*c^4 - 2*b*c^5 + c^6) : :

The radical center O' for the three circles is X(7649). See X(7623).

X(7647) lies on these lines: {522, 3717}, {523, 7623}


X(7648) =  CENTER OF BISECTING CIRCLE OF 3 CIRCLES: NINE-POINT, 1st LEMOINE, 2nd LEMOINE

Barycentrics    a^2*(b - c)*(b + c)*(a^10 - a^8*b^2 - 2*a^6*b^4 + 2*a^4*b^6 + a^2*b^8 - b^10 - a^8*c^2 - 8*a^6*b^2*c^2 + 8*a^4*b^4*c^2 + b^8*c^2 - 2*a^6*c^4 + 8*a^4*b^2*c^4 + 2*a^2*b^4*c^4 - 4*b^6*c^4 + 2*a^4*c^6 - 4*b^4*c^6 + a^2*c^8 + b^2*c^8 - c^10) : :

The radical center O' for the three circles is X(2506). See X(7623).

X(7648) lies on these lines: {39, 2506}, {182, 512}, {525, 7630}, {3566, 7631}


X(7649) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, ORTHOCENTROIDAL FUHRMANN

Barycentrics    (b - c)*(-a^2 + b^2 - c^2)*(a^2 + b^2 - c^2) : :

For the corresponding bisecting circle, see X(7628). X(7649) is also the center of the radical center of these 3 circles: circumcircle, nine-point, Fuhrmann; see X(7647) for the corresponding center of the bisecting circle.

X(7649) lies on these lines:
{4, 2457}, {19, 1024}, {25, 4057}, {27, 4786}, {108, 2222}, {112, 2690}, {230, 231}, {240, 522}, {242, 514}, {244, 1090}, {513, 1835}, {653, 7012}, {885, 1041}, {900, 7655}, {1474, 1919}, {1851, 6545}, {2969, 3756}, {3239, 4064}, {3261, 4025}

X(7649) = isogonal conjugate of X(1331)
X(7649) = X(2)-Ceva conjugate of X(5190)
X(7649) = X(2969)-cross conjugate of X(4)
X(7649) = crossdifference of every pair of points on line X(3)X(48)
X(7649) = inverse-in-polar-circle of X(6788)
X(7649) = X(48)-isoconjugate (polar conjugate) of X(190)
X(7649) = X(63)-isoconjugate of X(101)
X(7649) = pole wrt polar circle of trilinear polar of X(190) (i.e., the Nagel line)
X(7649) = perspector of hyperbola {{A,B,C,X(4),X(27)|} (the circumconic centered at X(5190))
X(7649) = intersection of trilinear polars of X(4) and X(27)
X(7649) = PU(4)-harmonic conjugate of X(3011)
X(7649) = trilinear pole of line X(2170)X(2969) (the line through the polar conjugates of PU(24))


X(7650) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, FUHRMANN, CONWAY

Barycentrics    b*(b - c)*c*(-a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2) : :

For the corresponding bisecting circle, see X(7646).

X(7650) lies on these lines:
{240, 522}, {320, 350}, {514, 4815}, {523, 4391}, {3667, 4823}, {4036, 4397}, {4086, 4791}, {4462, 4802}, {4778, 4978}, {4801, 4977}


X(7651) =  RADICAL CENTER OF 3 CIRCLES: 1st LEMOINE, 2nd LEMOINE, ORTHOCENTROIDAL

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

X(7651) lies on these lines: {6, 512}, {525, 2492}, {647, 826}, {690, 2489}, {2485, 3906}


X(7652) =  RADICAL CENTER OF 3 CIRCLES: BROCARD, 2nd LEMOINE, ORTHOCENTROIDAL

Barycentrics    a^2*(b - c)*(b + c)*(a^6 + a^4*b^2 - a^2*b^4 - b^6 + a^4*c^2 - 6*a^2*b^2*c^2 + 5*b^4*c^2 - a^2*c^4 + 5*b^2*c^4 - c^6) : :

X(7652) lies on these lines: {39, 647}, {512, 1570}, {525, 2485}, {690, 2489}, {850, 5286}


X(7653) =  RADICAL CENTER OF 3 CIRCLES: SPIEKER, BEVAN, CONWAY

Barycentrics    (b - c)*(5*a^2 + a*b + a*c + 4*b*c) : :

X(7653) lies on these lines: {513, 3716}, {514, 2487}, {3798, 4777}, {4379, 4790}, {4977, 7658}


X(7654) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, APOLLONIUS, FUHRMANN

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

X(7654) lies on these lines: {240, 522}, {512, 650}, {784, 7178}, {3900, 6589}


X(7655) =  RADICAL CENTER OF 3 CIRCLES: APOLLONIUS, FUHRMANN, SPIEKER RADICAL

Barycentrics    a*(b - c)*(a^3 - 3*a^2*b - a*b^2 + 3*b^3 - 3*a^2*c - 2*a*b*c + b^2*c - a*c^2 + b*c^2 + 3*c^3) : :

X(7655) is also the radical center of these 3 circles: nine-point, Apollonius, Fuhrmann

X(7655) lies on these lines:
{44, 513}, {521, 3669}, {522, 7178}, {900, 7649}, {905, 6003}, {926, 7250}, {2457, 4926}, {3900, 4017}, {4162, 6129}, {4885, 7253}


X(7656) =  RADICAL CENTER OF 3 CIRCLES: NINE-POINT, 2nd LEMOINE, GALLATLY

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

X(7656) lies on these lines: {512, 2025}, {523, 2524}, {804, 2023}, {2489, 2506}


X(7657) =  RADICAL CENTER OF 3 CIRCLES: SPIEKER, FUHRMANN, BEVAN

Barycentrics    (b - c)*(b + c)*(-3*a^2 + 2*a*b + 5*b^2 + 2*a*c - 6*b*c + 5*c^2) : :

X(7657) lies on these lines: {514, 2487}, {523, 656}


X(7658) =  RADICAL CENTER OF 3 CIRCLES: INCIRCLE, BEVAN, SPIKER RADICAL

Barycentrics    (b - c)( -3a2 + b2 + c2 + 2ab + 2ac - 2bc) : :

X(7658) lies on these lines:
{1, 4105}, {2, 2400}, {57, 652}, {241, 514}, {513, 2473}, {522, 676}, {693, 4765}, {918, 4521}, {1734, 6608}, {2254, 3667}, {2499, 6005}, {2516, 6084}, {3752, 6589}, {4453, 4468}, {4724, 4932}, {4928, 4962}, {4940, 6006}, {4977, 7653}

X(7658) = complement of X(3239)


X(7659) =  RADICAL CENTER OF 3 CIRCLES: APOLLONIUS, CONWAY, SPIEKER RADICAL

Barycentrics    a(b - c)(a2 - b2 - c2 + 4ab + 4ac + 2bc) : :

X(7659) is also the radical center of these 3 circles: nine-point, Apollonius, Conway.

X(7659) lies on these lines:
{44, 513}, {512, 3669}, {522, 4897}, {900, 7662}, {905, 6005}, {1019, 3309}, {2505, 4943}, {3667, 4369}, {3716, 6006}, {4162, 4367}, {4507, 6363}, {4778, 4913}


X(7660) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, BROCARD, SPIEKER

Barycentrics    a^2 (b-c) (5 a^2 b+2 a b^2-3 b^3+5 a^2 c-4 a b c-b^2 c+2 a c^2-b c^2-3 c^3) : :

X(7660) lies on these lines: {187, 237}, {522, 2490}, {2532, 4777}, {6363, 6589}


X(7661) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, INCIRCLE, FUHRMANN

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

X(7661) lies on these lines: {240, 522}, {513, 676}, {514, 6129}, {1459, 4040}, {4025, 4811}


X(7662) =  RADICAL CENTER OF 3 CIRCLES: CIRCUMCIRCLE, NINE-POINT, CONWAY

Barycentrics    (b - c)(a3 + ab2 + ac2 + 2abc + 2b2c + 2bc2) : :

X(7662) lies on these lines:
{230, 231}, {320, 350}, {514, 3716}, {522, 3798}, {649, 4804}, {659, 4762}, {784, 905}, {824, 4458}, {830, 4823}, {900, 7659}, {1491, 4885}, {2254, 4379}, {2526, 3837}, {2533, 3900}, {4139, 4507}, {4160, 4791}, {4777, 4789}, {4810, 6008}


X(7663) =  RADICAL CENTER OF 3 CIRCLES: NINE-POINT, 2nd LEMOINE, MOSES

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

X(7663) lies on these lines: {115, 804}, {2489, 2506}, {6036, 6132}


X(7664) =  X(2)X(99)∩X(69)X(110)

Barycentrics    (b4 + c4 - a4 - b2c2)(b2 + c2 - 2a2) : :

Let LA be the radical axis of the circumcircle and the A-McCay circle, and define LB and LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. The triangle A'B'C' is homothetic to ABC at X(111) and homothetic to the medial triangle at X(7664). (Randy Hutson, May 27, 2015)

X(7664) lies on these lines:
{2, 99}, {23, 316}, {69, 110}, {114, 7417}, {141, 2502}, {311, 7495}, {317, 4232}, {325, 3233}, {339, 6676}, {468, 3266}, {1560, 4235}, {1648, 5026}, {1649, 3268}, {2770, 4590}, {3124, 3589}, {4576, 5972}, {5182, 6792}, {5468, 5642}}


X(7665) =  X(2)X(99)∩X(110)X(193)

Barycentrics    3a^6 + b^6 + c^6 - 2a^4(b^2 + c^2) - a^2(4b^4 - 7b^2c^2 + 4c^4) : :

Continuing from X(7664), the triangle A'B'C' is homothetic to the anticomplementary triangle at X(7665). (Randy Hutson, May 27, 2015)

X(7665) lies on these lines:
{2, 99}, {69, 2502}, {110, 193}, {147, 7417}, {351, 6131}, {385, 7426}, {865, 3511}, {3124, 3618}, {3164, 7493}, {5139, 6995}, {5182, 6791}


X(7666) =  GIUGIUC CENTER OF SIMILITUDE

Trilinears    (4 + 6 cos 2A) cos(B - C) - 16 cos A - 9 cos 3A : : (César Lozada)
Barycentrics    a^2 (9 a^8 - 24 a^6 (b^2 + c^2) + a^4 (18 b^4 + 37 b^2 c^2 + 18 c^4) - 15 a^2 b^2 c^2 (b^2 + c^2) - (b^2 - c^2)^2 (3 b^4 + 4 b^2 c^2 + 3 c^4)) : : (Angel Montesdeoca)

Let OA = reflection of X(3) in line AX(4), let HA = reflection of X(4) in OA, let MA = midpoint of segment OAHA, and define MB and MC cyclically. Then MAMBMC is similar to ABC, and the center of similitude is X(7666). See Hyacinthos 23263 and 23277.

X(7666) lies on these lines:
{140,2888}, {154,1657}, {195,1511}, {399,3357}, {2904,3515}, {3576,5694}


X(7667) =  1st HATZIPOLAKIS-MOSES-EULER CENTROID

Barycentrics    2 a^6+a^4 b^2-2 a^2 b^4-b^6+a^4 c^2-4 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6 : :
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)
X(7667) = (X(i),X(j))-harmonic conjugate of X(k) for these (i,j,k): (3,427,7499), (3,1370,427), (20,7386,25), (22,1368,468), (548,6676,6636), (550,1368,22), (631,7378,7539), (858,6636,6676), (1657,5020,7500), (3522,7396,7494), (7391,7485,5), (7396,7494,5094)


X(7668) =  POLE OF X(115)X(125) WITH RESPECT TO THE NINE-POINT CIRCLE

Trilinears    [(2 cos A - cos 3A + cos(B - C)] sin2(B - C) : :
Barycentrics    (b2 - c2)2(a4 - a2b2 - a2c2 - b2c2) : :

Dao Thanh Oai posed the following conjecture. Let P be a point on the Neuberg cubic. Let PA be the reflection of P in line BC, and define PB and PC cyclically. It is known that the lines APA, BPB, CPC concur. Let Q(P) be the point of concurrence. Then the following 4 points lie on a circle: X(13), X(14), P, Q(P).

César Lozada found that as P traces the Neuberg cubic, the center of the circle {{X(13), X(14), P, Q(P)}} traces the line X(115)X(125) (which, notably, is parallel to the lines X(74)X(98), X(99)X(110), X(113)X(114), and is perpendicular to X(2)X(98) at X(125) and perpendicular to X(6)X(13) at X(115)). X(7668) is the pole of X(115)X(125) with respect to the nine-pont circle of ABC. See ADGEOM 2546.

Let T be the tangential triangle of the medial triangle, and let T' be the tangential triangle of the orthic triangle. Let S be the side-triangle of T and T'. Let V be the vertex triangle of T and T'. Then S and V are perspective, and their perspector is X(7668). (Randy Hutson, June 4, 2015)

X(7668) lies on these lines:
(2,1634), (5,542), (6,3613), (11,3141), (98,1576), (115,804), (125,526), (338,523)

X(7668) = complement of X(1634)
X(7668) = complementary conjugate of X(3005)
X(7668) = crossdifference of every pair of points on line X(1625)X(1634)


X(7669) =  POLE OF X(115)X(125) WITH RESPECT TO THE CIRCUMCIRCLE

Trilinears    (cos(B-C)+2*cos(A)*cos(2*(B-C))-2*cos(A)+cos(3*A))*sin(A)^2 : :

Continuing from X(7668), the pole of X(115)X(125) with respect to the circumcircle is X(7669). See ADGEOM 2546.

For a sketch, click X(3447)andX(7669). (Angel Montesdeoca, April 22, 2016)

X(7669) lies on these lines:
(3,67), (6,157), (25,1989), (50,2393), (98,338) et al

X(7669) = reflection of X(6) in X(1976)
X(7669) = trilinear pole, with respect to the tangential triangle, of the Euler line


X(7670) =  X(1)-OF-HONSBERGER TRIANGLE

Barycentrics    a (a+b-c) (a-b+c) (b c (-a+b+c) Sin[A/2]+c (a^2+c (-b+c)-a (b+2 c)) Sin[B/2]+b (a^2+b (b-c)-a (2 b+c)) Sin[C/2]) : :

Let DEF be the cevian triangle of X(7); i.e., DEF = intouch triangle. Let LA be the line through X(7) parallel to EF, and let AB = AB∩LA and AC = AC∩LA. Define BC and CA cyclically, and define BA and CB cyclically. (The six points AB, BC, CA, AC, BA, CB lie on the Adams circle.) Let HA = ABCB∩ACBC, and define HB and HC cyclically. The triangle Let HAHBHC is here named the Honsberger triangle, after the triangle labeled XYZ on page 98 of Ross Honsberger's Episodes in Nineteenth and Twentieth Century Euclidean Geometry (Mathematical Association of America, 1995). Peter Moses found barycentrics for the A-vertex:

HA = a(a - b + c)(a + b - c) : (a + b - c)[(b - c)2 - a(b + c)] : (a - b + c)[(b - c)2 - a(b + c)]

The appearance of (i,j) in the following list indicates that X(i)-of-HAHBHC = X(j):

(3,390), (5,5728), (6,7), (26,5759), (30,517), (110,1156), (154,6172), (156,5779), (159,144), (182,1), (206,9), (511,516), (512,514), (520,3667), (523,513), (525,3309), (526,900), (542,2801), (575,5542), (576,4312), (597,354), (690,3887), (804,926), (924,522), (1154,30), (1176,2346), (1177,100), (1386,177), (1503,518), (1510,523), (1576,673), (1974,1445), (1976,651), (2030,1323), (2393,527), (2574,3308), (2575,3307), (2777,2802), (2781,528), (2782,2808), (2790,2810), (2794,2809), (2797,2821), (2799,2820), (2848,2832), (2854,5851), (2871,5845), (2881,6084), (3564,971), (3566,3900), (3589,5572), (5026,3022), (5476,5902), (5480,65), (5663,952), (6000,519), (6086,6085), (6368,6003), (6593,11), (6759,5223), (7514,3488)

The Honsberger triangle is perspective to the following triangles, with perspectors as indicated: anticomplementary triangle, X(7674); hexyl, X(7675); 1st circumperp, X(7676), 2nd circumperp, X(7677); 3rd Euler, X(7678), 4th Euler, X(7679). Indeed, the Honsberger triangle is homothetic to each of those triangles except the anticomplementary; it is also homothetic to the intouch triangle at X(7) and the excentral triangle at X(1445).

The vertices of the Honsberger triangle lie on the circumconic centered at X(9). (Randy Hutson, December 10, 2016)

X(7670) lies on these lines: {7,177}, {164,1445}, {167,4326}, {174,6732}

X(7670) = reflection of X(7) in X(177)


X(7671) =  X(2)-OF-HONSBERGER TRIANGLE

Barycentrics    a (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+a^2 b c-3 a b^2 c+b^3 c-3 a^2 c^2-3 a b c^2+3 a c^3+b c^3-c^4) : :
X(7671) = X[7] - 4 X[5572] = 4 X[1125] - X[5696] = X[3868] + 2 X[5698] = X[390] + 2 X[5728] = 5 X[3616] - 2 X[5784]

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)
X(7671) = reflection of X(i) in X(j) for these (i,j): (3059,3740), (354,5572), (3681,9), (7,354)


X(7672) =  X(4)-OF-HONSBERGER TRIANGLE

Barycentrics    a (a+b-c) (a-b+c) (a^2 b-2 a b^2+b^3+a^2 c-3 a b c-2 a c^2+c^3) : :
X(7672) = 2 X[3243] - 3 X[3873] = 4 X[1001] - 3 X[3877] = 2 X[72] - 3 X[5686] = 3 X[5817] - 2 X[5887] = 2 X[5542] - 3 X[5902]

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)
X(7672) = {X(2263), X(3751}-harmonic conjugate of X(651)


X(7673) =  X(20)-OF-HONSBERGER TRIANGLE

Barycentrics    a (a^4 b-2 a^3 b^2+2 a b^4-b^5+a^4 c-9 a^3 b c+7 a^2 b^2 c-3 a b^3 c+4 b^4 c-2 a^3 c^2+7 a^2 b c^2+2 a b^2 c^2-3 b^3 c^2-3 a b c^3-3 b^2 c^3+2 a c^4+4 b c^4-c^5) : :
X(7673) = 2 X[2550] - 3 X[3877] = 4 X[142] - 5 X[3890] = 3 X[390] - 2 X[5728]

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)


X(7674) =  PERSPECTOR OF HONSBERGER TRIANGLE AND ANTICOMPLEMENTARY TRIANGLE

Barycentrics    (a-b-c) (3 a^4-6 a^3 b+4 a^2 b^2-2 a b^3+b^4-6 a^3 c+2 a b^2 c-4 b^3 c+4 a^2 c^2+2 a b c^2+6 b^2 c^2-2 a c^3-4 b c^3+c^4) : :
X(7674) = 2 X[142] - 3 X[3158] = 3 X[2] - 4 X[6600]

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(7874) = reflection of X(i) and X(j) for these (i,j): (7, 3174), (2550, 3913), (6601, 6600)
X(7674) = anticomplement of X(6601)
X(7674) = X(6604)-Ceva conjugate of X(2)
X(7674) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (109,3309), (218,329), (269,6601), (1445,69), (1617,8), (2149,644), (3870,3436), (4350,3434), (6604,6327)
X(7874) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390,5686,452), (6600,6601,2)


X(7675) =  PERSPECTOR OF HONSBERGER TRIANGLE AND HEXYL TRIANGLE

Barycentrics    a (a-b-c) (a^4-2 a^3 b+2 a b^3-b^4-2 a^3 c-6 a^2 b c-2 a b^2 c+2 b^3 c-2 a b c^2-2 b^2 c^2+2 a c^3+2 b c^3-c^4) : :
X(7675) = X[7] + 2 X[4304] = X[4302] + 2 X[5542]

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(7875) = {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).


X(7676) =  PERSPECTOR OF HONSBERGER TRIANGLE AND 1st CIRCUMPERP TRIANGLE

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

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)


X(7677) =  PERSPECTOR OF HONSBERGER TRIANGLE AND 2nd CIRCUMPERP TRIANGLE

Barycentrics    a (a+b-c) (a-b+c) (a^3-2 a^2 b+a b^2-2 a^2 c-a b c+b^2 c+a c^2+b c^2) : :
X(7677) = X[1156] + 8 X[5126]

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)


X(7678) =  PERSPECTOR OF HONSBERGER TRIANGLE AND 3nd EULER TRIANGLE

Barycentrics    a^3 b^2-3 a^2 b^3+3 a b^4-b^5-3 a^3 b c+3 a^2 b^2 c-3 a b^3 c+3 b^4 c+a^3 c^2+3 a^2 b c^2-2 b^3 c^2-3 a^2 c^3-3 a b c^3-2 b^2 c^3+3 a c^4+3 b c^4-c^5 : :

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)


X(7679) =  PERSPECTOR OF HONSBERGER TRIANGLE AND 4th EULER TRIANGLE

Barycentrics    (a+b-c) (a-b+c) (a^2 b^2-2 a b^3+b^4+3 a^2 b c-5 a b^2 c+a^2 c^2-5 a b c^2-2 b^2 c^2-2 a c^3+c^4) : :

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)


X(7680) =  HATZIPOLAKIS-MOSES IMAGE OF X(7)

Barycentrics    a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7-3 a^4 b^2 c+2 a^2 b^4 c+b^6 c+a^5 c^2-3 a^4 b c^2+4 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3-4 a^2 b^2 c^3-3 b^4 c^3-2 a^3 c^4+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
X(7680) = 5 X[3091] - X[3434] = 3 X[1699] + X[5119] = X[3419] - 3 X[5587] = X[2099] - 3 X[5603]

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

The set of points HM(P(k)) as k goes through all appropriate functions of a,b,c (including all nonzero homogeneous symmetric functions of (a,b,c)), is the Jerabek hyperbola of the Euler line, a right hyperbola that passes through X(i) for these i: 4, 5, 125, 1352, 2574, 2575, 3574, 5480, 5777, 6247. The hyperbola is discussed further at X(7687). See Hyacinthos 23282, June 4, 2015

X(7680) lies on these lines: {1, 6831}, {2, 3428}, {3, 6690}, {4, 12}, {5, 10}, {8, 6828}, {11, 2099}, {40, 442}, {42, 5721}, {56, 6833}, {71, 5798}, {84, 5290}, {100, 6839}, {104, 5434}, {116, 117}, {119, 381}, {197, 7497}, {200, 3419}, {226, 6001}, {227, 1838}, {235, 1824}, {281, 5514}, {355, 3811}, {388, 6847}, {495, 515}, {496, 6738}, {497, 6844}, {498, 3149}, {516, 3822}, {674, 5480}, {944, 6845}, {958, 6824}, {962, 2476}, {1001, 6827}, {1012, 1478}, {1064, 5718}, {1072, 3666}, {1125, 6922}, {1210, 5173}, {1376, 6826}, {1482, 3813}, {1532, 1699}, {1621, 6840}, {2550, 6843}, {2551, 6846}, {2975, 6888}, {3035, 6911}, {3086, 6956}, {3091, 3434}, {3359, 5880}, {3436, 6837}, {3475, 5768}, {3577, 3679}, {3614, 6941}, {3616, 6943}, {3656, 3829}, {3816, 5886}, {3826, 5805}, {3913, 6866}, {3925, 5657}, {3947, 6260}, {4293, 6935}, {4298, 6705}, {4413, 6854}, {4423, 6947}, {4999, 6862}, {5080, 6912}, {5172, 6906}, {5204, 6977}, {5217, 6934}, {5230, 5706}, {5253, 6972}, {5260, 6884}, {5432, 6905}, {5433, 6952}, {5441, 5691}, {5552, 6835}, {5584, 6889}, {5711, 5713}, {5818, 6990}, {5841, 6914}, {5884, 6147}, {6361, 6937}, {6668, 6863}, {6691, 6958}

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)


X(7681) =  HATZIPOLAKIS-MOSES IMAGE OF X(8)

Barycentrics    a^5 b^2-a^4 b^3-2 a^3 b^4+2 a^2 b^5+a b^6-b^7+a^4 b^2 c+4 a^3 b^3 c-2 a^2 b^4 c-4 a b^5 c+b^6 c+a^5 c^2+a^4 b c^2-4 a^3 b^2 c^2-a b^4 c^2+3 b^5 c^2-a^4 c^3+4 a^3 b c^3+8 a b^3 c^3-3 b^4 c^3-2 a^3 c^4-2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a^2 c^5-4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
X(7681) = X[46] + 3 X[1699] = 5 X[3091] - X[3436] = X[2098] - 3 X[5603]

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)


X(7682) =  HATZIPOLAKIS-MOSES IMAGE OF X(9)

Barycentrics    a^6 b-2 a^5 b^2-a^4 b^3+4 a^3 b^4-a^2 b^5-2 a b^6+b^7+a^6 c-4 a^5 b c+a^4 b^2 c-4 a^3 b^3 c-a^2 b^4 c+8 a b^5 c-b^6 c-2 a^5 c^2+a^4 b c^2+2 a^2 b^3 c^2+2 a b^4 c^2-3 b^5 c^2-a^4 c^3-4 a^3 b c^3+2 a^2 b^2 c^3-16 a b^3 c^3+3 b^4 c^3+4 a^3 c^4-a^2 b c^4+2 a b^2 c^4+3 b^3 c^4-a^2 c^5+8 a b c^5-3 b^2 c^5-2 a c^6-b c^6+c^7 : :
X(7682) = 3 X[1699] + X[2093] = 3 X[381] + X[2095] = 3 X[57] - X[2096] = 3 X[4] + X[2096] = X[329] - 5 X[3091] = X[2094] + 3 X[3839] = X[3421] - 3 X[5587]

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)


X(7683) =  HATZIPOLAKIS-MOSES IMAGE OF X(10)

Barycentrics    2 a^5 b^2+a^4 b^3-2 a^3 b^4-b^7+a^4 b^2 c+a^3 b^3 c-a^2 b^4 c-a b^5 c+2 a^5 c^2+a^4 b c^2+2 a^3 b^2 c^2+a^2 b^3 c^2+2 b^5 c^2+a^4 c^3+a^3 b c^3+a^2 b^2 c^3+2 a b^3 c^3-b^4 c^3-2 a^3 c^4-a^2 b c^4-b^3 c^4-a b c^5+2 b^2 c^5-c^7 : :
X(7683) = X[1046] + 3 X[1699] = X[1330] - 5 X[3091] = 3 X[5429] + X[5691]

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)


X(7684) =  HATZIPOLAKIS-MOSES IMAGE OF X(13)

Barycentrics    2 a^2 S (Sqrt[3] (-a^2+b^2+c^2)+2 S)+(a^2+b^2-c^2) (a^2-b^2+c^2) (a^2+b^2+c^2+2 Sqrt[3] S) : :
X(7684) = X[621] - 5 X[3091] = 3 X[381] + X[5611].

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}
X(7684) = reflection of X(i) in X(j) for these (i,j): (3,6671), (623,5)
X(7684) = inverse-in-nine-point-circle of X(7685)
X(7684) = X(15)-of-Euler-triangle
X(7684) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,5480,7685), (2039,2040,7685)


X(7685) =  HATZIPOLAKIS-MOSES IMAGE OF X(14)

Barycentrics    2 a^2 (Sqrt[3] (-a^2+b^2+c^2)-2 S) S-(a^2+b^2-c^2) (a^2-b^2+c^2) (a^2+b^2+c^2-2 Sqrt[3] S) : :
X(7685) = X[622] - 5 X[3091] = 3 X[381] + X[5615].

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}
X(7685) = reflection of X(i) in X(j) for these (i,j): (14,383), (624,5)
X(7685) = inverse-in-nine-point-circle of X(7684)
X(7685) = X(16)-of-Euler triangle
X(7685) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,5480,7684), (2039,2040,7684)


X(7686) =  HATZIPOLAKIS-MOSES IMAGE OF X(21)

Barycentrics    (pending)
X(7686) = 3 X[354] - X[944] = 2 X[1385] - 3 X[3742] = X[40] - 3 X[3753] = 5 X[3091] - X[3869] = 3 X[3817] - X[3878] = 3 X[3576] - 5 X[5439] = X[72] - 3 X[5587] = X[3057] - 3 X[5603] = 5 X[3698] - 3 X[5657] = X[3885] - 5 X[5734] = 2 X[4662] - 3 X[5790] = 3 X[210] - 5 X[5818] = X[4297] - 3 X[5883] = 3 X[381] - X[5887] = X[1071] - 3 X[5902] = X[5691] + 3 X[5902] = 3 X[1699] + X[5903] = X[3529] - 3 X[5918] = X[5693] - 3 X[5927] = X[4018] + 3 X[5927] = 7 X[3922] - X[6361]

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)


X(7687) =  CENTER OF JERABEK HYPERBOLA OF EULER TRIANGLE

Barycentrics    2 a^10-2 a^8 b^2-3 a^6 b^4+a^4 b^6+5 a^2 b^8-3 b^10-2 a^8 c^2+8 a^6 b^2 c^2-a^4 b^4 c^2-14 a^2 b^6 c^2+9 b^8 c^2-3 a^6 c^4-a^4 b^2 c^4+18 a^2 b^4 c^4-6 b^6 c^4+a^4 c^6-14 a^2 b^2 c^6-6 b^4 c^6+5 a^2 c^8+9 b^2 c^8-3 c^10 : :
X(7687) = 3 X[4] + X[74] = X[74] - 3 X[125] = X[113] - 3 X[381] = X[265] + 3 X[381] = 3 X[113] - X[399] = 9 X[381] - X[399] = 3 X[265] + X[399] = 3 X[5] - X[1511] = 3 X[51] - X[1986] = X[110] - 5 X[3091] = 5 X[399] - 9 X[5655] = 5 X[113] - 3 X[5655] = 5 X[381] - X[5655] = 5 X[265] + 3 X[5655] = 2 X[1511] - 3 X[5972] = 6 X[5655] - 5 X[6053] = 2 X[399] - 3 X[6053] = 6 X[381] - X[6053] = 2 X[265] + X[6053]

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}
X(7687) = reflection of X(i) in X(j) for these (u,j): (3,6723), (5972,5), (6053,113)
X(7687) = inverse-in-Kiepert-hyperbola of X(3163)
X(7687) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (13,14,3163), (265,381,113)


X(7688) =  PERSPECTOR OF TRINH TRIANGLE AND EXTANGENTS TRIANGLE

Trilinears    2 + 3 cos A + cos 2A + 2 sin(3A/2) cos(B/2 - C/2) - 4 sin A/2 : :
X(7688) = R*X(1) - (6R + 2r)*X(3) = 4X(3579) + X(5425)

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)


X(7689) =  ORTHOLOGIC CENTER OF TRINH TRIANGLE AND EXTANGENTS TRIANGLE

Trilinears    (cos A)[2 + cos 2A - 4 cos A cos(B - C)] : :

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.

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(7689) = reflection of X(i) in X(j) for these (i,j): (4,5449), (1147,3), (6759,1658)
X(7689) = anticomplement of X(5448)
X(7689) = X(4)-of -Trinh-triangle


X(7690) =  ORTHOLOGIC CENTER OF TRINH TRIANGLE AND LUCAS CENTRAL TRIANGLE

Trilinears    a(S2 - 4SSA - 3SωSA) : :

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)


X(7691) =  ORTHOLOGIC CENTER OF TRINH TRIANGLE AND REFLECTION TRIANGLE

Trilinears    cos 3A - (3 + 4 cos 2A) cos(B - C) : :

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(7692) =  ORTHOLOGIC CENTER OF TRINH TRIANGLE AND LUCAS(-1) CENTRAL TRIANGLE

Trilinears    a(S2 + 4SSA - 3SωSA) : :

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)



leftri  Similar triangles and centers of similitude: X(7693) - X(7709)  rightri

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

underbar

X(7693) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: ANTICOMPLEMENTARY AND ORTHOCENTROIDAL

Trilinears    (-8*cos(2*A)+13)*cos(B-C)+2*cos(A)*cos(2*(B-C))+2*cos(A)+cos(3*A) : :

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)
X(7693) = X(5888)-of-anticomplementary triangle
X(7693) = X(5888)-of-orthocentroidal triangle


X(7694) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND EULER

Trilinears    (6*cos(2*A)-6*cos(4*A)+4)*cos(B-C)+(-6*cos(A)-2*cos(3*A))*cos(2*(B-C)) +(-6*cos(2*A)+2)*cos(3*(B-C))+3*cos(5*A)+24*cos(A)-19*cos(3*A) : :

X(7694) lies on these lines:
{2,2794}, {4,39}, {30,7618}, {83,3091}, {115,6776}, {381,597}, {542,7615}, {1348,3414}, {1349,3413}, {1352,6033), {1499,4846}


X(7695) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND INNER GREBE

Trilinears    (-8*(2*a^6+(b^2+c^2)*a^4+2*(b^2+c^2)^2*a^2+3*(b^2+c^2)*b^2*c^2)*Delta +4*a^8+7*(b^2+c^2)*a^6+(19*b^2*c^2+6*c^4+6*b^4)*a^4 -(b^2+c^2)*(b^4-20*b^2*c^2+c^4)*a^2+b^2*c^2*(7*c^4+7*b^4+18*b^2*c^2))/a : :

X(7695) lies on these lines: {3734,6319}, {3818,5591}


X(7696) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND OUTER GREBE

Trilinears    (8*(2*a^6+(b^2+c^2)*a^4+2*(b^2+c^2)^2*a^2+3*b^2*c^2*(b^2+c^2))*Delta +4*a^8+7*(b^2+c^2)*a^6+(19*b^2*c^2+6*c^4+6*b^4)*a^4 -(b^2+c^2)*(b^4-20*b^2*c^2+c^4)*a^2+b^2*c^2*(7*c^4+7*b^4+18*b^2*c^2))/a : :

X(7696) lies on these lines: {3734,6320}, {3818,5590}


X(7697) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND JOHNSON

Trilinears    22*cos(B-C)+(12*cos(A)-2*cos(3*A))*cos(2*(B-C)) +2*cos(3*(B-C))-cos(3*A)+15*cos(A) : :

X(7697) lies on these lines:
{2,2782}, {3,3734}, {4,2896}, {5,76}, {39,1656}, {115,3094}, {183,2080}, {194,3090}, {381,511}, {382,5188}, {538,5055}, {1352,6033}, {1569,6721}, {3642,5613}, {3643,5617}, {5070,6683}, {5969,7615}, {5976,6321}

X(7697) = midpoint of X(i) and X(j) for these {i,j}: {4,6194}, {76,262}
X(7697) = reflection of X(i) in X(j) for these (i,j): (262,5), (3095,262)
X(7697) = X(262)-of-Johnson-triangle
X(7697) = X(262)-of-Brocard-triangle


X(7698) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND ORTHOCENTROIDAL

Trilinears    (a^10+2*(b^2+c^2)*a^8-2*(2*b^4-b^2*c^2+2*c^4)*a^6 +(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^4 +b^2*c^2*(7*b^4-9*b^2*c^2+7*c^4)*a^2 +2*(b^4-c^4)*(b^2-c^2)*b^2*c^2)/a : :

X(7698) lies on these lines: {2,3098}, {83,5466}


X(7699) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: EULER AND ORTHOCENTROIDAL

Trilinears    (6*cos(2*A)+3)*cos(B-C)-4*cos(A)*cos(2*(B-C))-5*cos(A) : :

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(7700) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 2nd EXTOUCH AND FUHRMANN

Trilinears    (a^7-2*(b+c)*a^6-2*b*c*a^5+(b+c)*(3*c^2-4*b*c+3*b^2)*a^4 +(-3*b^2-3*c^2+2*b*c)*(b+c)^2*a^3+2*b*c*(b+c)*(c^2-4*b*c+b^2)*a^2 +2*(b^2-c^2)^2*a*(b^2+3*b*c+c^2)-(b^2-c^2)^3*(b-c))/a : :

X(7700) lies on these lines: {4,3877}, {9,4333}, {3419,3625}


X(7701) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: FUHRMANN AND HEXYL

Trilinears    a^6-3*(b^2-b*c+c^2)*a^4+b*c*(b+c)*a^3-(-3*c^4+b*c^3 -3*b^4+b^3*c)*a^2-(b^2-c^2)*(b-c)*a*b*c-(b^2-c^2)^2*(b+c)^2 : :

Let A' be the reflection of the A-excenter in A, and define B' and C' cyclically. Then X(7701) = X(3)-of-A'B'C'. (Randy Hutson, June 10, 2015)

X(7701) lies on these lines:
{1,399}, {4,5535}, {5,1768}, {9,2173}, {21,84}, {30,40}, {46,1749}, {57,79}, {63,2894}, {381,3336}, {549,5506}, {758,6762}, {1012,5693}, {1158,2475}, {1697,5441}, {1698,5499}, {1717,6149}, {1727,3585}, {1776,4292}, {2077,5777}, {3333,3649}, {3358,6675}, {3467,5131}, {3579,3983}, {4466,6173}, {5428,7171}, {5437,6701}, {5538,5694}, {5884,6912}, {6326,6906}

X(7701) = reflection of X(i) in X(j) for these (i,j): (40,191), (79,6841), (191,3652), (3651,3647)
X(7701) = X(79)-of-tangential triangle-of-excentral triangle


X(7702) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: FUHRMANN AND INTOUCH

Trilinears    (a^5-(b^2+c^2)*a^3+(c^2-b^2)*(b-c)*a^2+(b^2-c^2)^2*(b+c))/(a*(a-b-c)) : :

X(7702) lies on these lines:
{1,5840}, {4,5553}, {12,5880}, {56,946}, {57,79}, {65,68}, {224,3649}, {225,1406}, {603,3120}, {1155,5812}, {1319,4299}, {1388,4297}, {1399,3772}, {1837,5884}, {1887,1892}, {2099,3244}, {4652,5433}, {5057,7288}, {5221,5729}

X(7702) = midpoint of X(4) and X(5553)
X(7702) = X(2904)-of-intouch-triangle
X(7702) = X(2904)-of-Fuhrmann-triangle


X(7703) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: MEDIAL AND ORTHOCENTROIDAL

Trilinears    (2*cos(2*A)-3)*cos(B-C)-cos(A)*(4*cos(2*(B-C))-3) : :

X(7703) lies on these lines:
{2,1495}, {6,6032}, {66,3618}, {74,381}, {110,5094}, {113,7577}, {125,5169}, {141,858}, {264,850}, {378,2931}, {427,3060}, {625,6787}, {1346,2575}, {1347,2574}, {1853,5012}, {3545,4846}, {3763,5888}, {5663,7579}


X(7704) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 3rd EULER AND FUHRMANN

Trilinears    (10*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)-4*sin(A/2)*cos(3*(B-C)/2) +(-2*cos(A)+3)*cos(B-C)+5*cos(A)-6 : :

X(7704) lies on these lines:
{4,1319}, {5,1145}, {145,355}, {517,5154}, {946,1737}, {962,6978}, {1699,5450}, {3890,6980}, {5805,6847}


X(7705) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 4th EULER AND FUHRMANN

Trilinears    (-a+b+c)*((b+c)*a^2-(b^2-b*c+c^2)*a-2*(b^2-c^2)*(b-c))/a : :

X(7705) lies on these lines:
{1,6702}, {2,355}, {5,1537}, {8,1392}, {10,3877}, {11,3885}, {21,1698}, {404,5450}, {517,5154}, {1210,3889}, {1329,3876}, {1512,6943}, {1737,3868}, {2478,2550}, {3090,5554}, {3189,5552}, {3679,5330}, {3753,5141}, {3814,3869}, {3825,3890}, {5177,5817}, {5187,5657}, {5250,5506}

X(7705) = midpoint of X(8) and X(1392)


X(7706) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: JOHNSON AND ORTHOCENTROIDAL

Trilinears    (6*cos(2*A)+2)*cos(B-C)-2*cos(A)*cos(2*(B-C))-7*cos(A)-cos(3*A) : :

X(7706) lies on these lines:
{2,4549}, {3,3574}, {4,4846}, {5,4550}, {30,182}, {74,5169}, {113,1995}, {125,381}, {146,7533}, {373,1531}, {1204,5576}, {1216,6815}, {2088,5475}, {2929,5448}, {3426,3521}, {3448,5890}, {3543,5645}, {3818,5663}, {5094,6699}, {5972,6644}

X(7706) = midpoint of X(4) and X(4846)
X(7706) = reflection of X(4550) in X(5)
X(7706) = complement of X(4549)


X(7707) =  HOMOTHETIC CENTER OF THESE TRIANGLES: YFF CENTRAL AND OUTER HUTSON

Trilinears    cos(B/2) + cos(C/2) : :
Trilinears    b(csc B/2) + c(csc C/2) : :

X(7707) lies on the Feuerbach hyperbola and these lines:
{1,168}, {7,174}, {8,178} et al

X(7707) = perspector of 1st tangential mid-arc conic


X(7708) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND McCAY

Trilinears    a (2 a^4-2 a^2 b^2+2 b^4-2 a^2 c^2-11 b^2 c^2+2 c^4) : :

X(7708) lies on these lines:
{2,6}, {32,5643}, {111,182}, {187,5640}, {373,2030}, {1383,1691}, {3288,5466}

X(7708) = X(111),X(182)}-harmonic conjugate of X(353)


X(7709) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 1st BROCARD AND REFLECTION-IN-X(3)-OF-ABC

Trilinears    (-12*cos(3*A)*cos(A)+13)*cos(B-C)-2*cos(3*A)*cos(2*(B-C))-cos(3*(B-C))+15*cos(A)-13*cos(3*A) : :
Barycentrics    (3*S^2+SW^2)*SA^2-(S^2+SW^2)*SW*SA+S^2*(S^2-SW^2) : :
Barycentrics    3 a^6 b^2-4 a^4 b^4+a^2 b^6+3 a^6 c^2-7 a^4 b^2 c^2-a^2 b^4 c^2-b^6 c^2-4 a^4 c^4-a^2 b^2 c^4+2 b^4 c^4+a^2 c^6-b^2 c^6 : :
X(7709) = X[4] - 4 X[39] = 2 X[3] + X[194] = 2 X[76] - 5 X[631] = X[98] + 2 X[1569] = X[20] + 2 X[3095] = 11 X[3525] - 8 X[3934] = 7 X[3528] - 4 X[5188] = 7 X[3090] - 4 X[6248] = 13 X[5067] - 16 X[6683] = 2 X[3094] + X[6776]

X(7709) lies on the cubic K048 these lines:
{2,2782}, {3,194}, {4,39}, {20,3095}, {76,631}, {98,574}, {99,182}, {376,511}, {515,3097}, {538,3524}, {575,3972}, {698,5085}, {726,3576}, {730,5657}, {879,3431}, {1003,5050}, {1285,5052}, {1340,3413}, {1341,3414}, {3090,6248}, {3094,3269}, {3102,6459}, {3103,6460}, {3106,6773}, {3107,6770}, {3398,3552}, {3399,3424}, {3525,3934}, {3528,5188}, {5067,6683}, {6795,7464}

X(7709) = midpoint of X(194) and X(6194)
X(7709) = reflection of X(i) in X(j) for these (i,j): (4,262), (262, 39), (6194,3)
X(7709) = {X(2544),X(2545)}-harmonic conjugate of X(2549)
X(7709) = X(6194)-of-Brocard triangle
X(7709) = X(6194)-of-reflection-in-X(3)-of-ABC


X(7710) =  ENDO-SIMILARITY IMAGE OF THESE TRIANGLES: 1st BROCARD AND EULER

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

X(7710) is the center of the circumconic described by Francisco Javier Garcia Capitan (Hyacinthos 21466, January 30, 2013), in response to a problem by Antreas P. Hatzipolakis, Conics centered at O, January 29, 2013)

Let A'B'C' be the cevian triangle of X(1), let A"B"C" be the medial triangle, let A*B*C* be the circumcevian triangle of X(2) with respect to A"B"C", and let DEF be the triangle bounded by the perpendiculars to lines A"A*, B"B*, C"C* at A*,B*,C*, respectively. The lines A"D, B"E, C"F concur in X(7710). See Hyacinthos 21665 (A. Hatzipolakis and R. Hutson, March 5, 2013)

Another construction for DEF follows: Let PA be the parabola with focus A and directrix BC. Let LA be the polar of X(3) with respect to PA. Define LB and LC cyclically. Then D = LB∩LC, E = LC∩LA, F = LA∩LB. Note that the lines AD, BE, CF concur in X(262). (Randy Hutson, June 15, 2015)

X(7710)-of-1st-Brocard-triangle = X(7710)-of-Euler-triangle = X(7694)-of-ABC. The term "similarity image" is defined in the preamble to X(6724). The term "endo-similarity image" is introduced here to match "endo-homothetic" defined in the preamble to X(7581); the term "similarity image" is defined in the preamble to X(6724). (Peter Moses, June 11, 2015)

If U and V are similar triangles with similitude center S, then the endo-similarity center of U and V is the U-to-ABC functional image of S (which is also the V-to-ABC functional image of S). (Randy Hutson, June 11, 2015)

X(7710) lies on these lines:
{2,154}, {4,39}, {20,325}, {22,6503}, {69,147}, {98,5033}, {132,1249}, {183,5921}, {376,2482}, {497,3666}, {631,6292}, {1007,5999}, {1499,5664}, {1513,6776}, {5870,6811}, {5871,6813}, {6459,7374}, {6460,7000}, {6509,7386}

X(7710) = reflection of X(4) in X(7694)
X(7710) = complement of X(3424)


X(7711) =  ENDO-SIMILARITY IMAGE OF THESE TRIANGLES: 1st BROCARD AND ORTHOCENTROIDAL

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

X(7711)-of-1st-Brocard-triangle = X(7711)-of-orthocentroidal-triangle = X(7698)-of-ABC.

X(7711) lies on the Parry circle and these lines:
{39,111}, {110,5092}, {352,1691}, {353,5116}, {5971,7664}


X(7712) =  ENDO-SIMILARITY IMAGE OF THESE TRIANGLES: MEDIAL AND ORTHOCENTROIDAL

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

X(7712)-of-medial-triangle = X(7712)-of-orthocentroidal-triangle = X(7703)-of-ABC.

X(7712) lies on these lines:
{2,1495}, {6,23}, {20,5654}, {22,323}, {25,5644}, {30,3431}, et al



leftri  Hatzipolakis-Lozada Homothetic Centers X(7713) - X(7718)  rightri

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.

underbar

X(7713) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(1))

Trilinears    (a2 + b2 +c2 + 2ab + 2ca)/(a2 - b2 - c2) : :

See the preamble to X(7713). Let A' be the perspector of the A-mixtilinear incircle, and define B' and C' cyclically. (The lines AA', BB', CC' concur in X(57).) Let LA be the polar of A' with respect to the A-mixtilinear incircle, and define LB and LC cyclically. Let A'' = LB∩LC, B''' = LC∩LA, C'' = LA∩LB. The lines AA'', BB'', CC'' concur in X(7713); see also X(7719). (Randy Hutson, June 16, 2015)

X(7713) lies on these lines:
{1,25}, {4,9}, {8,6995}, {24,3576}, {27,4384}, {28,34}, {33,1697}, {46,1707), (56,3420}, {63,4198}, {518,7716}, {519,7714}, {952,7715} et al

X(7713) = {X(19),X(2333)}-harmonic conjugate of X(7719)


X(7714) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(2))

Barycentrics    (5a2 + b2 +c2)/(a2 - b2 - c2) : :
X(7714) = (8R2 + Sω)*X(2) - 2Sω*X(3)

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}


X(7715) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(5))

Trilinears    (cos 2B + cos 2C - 8 sin 2A) sec A : :
X(7715) = 3(4R2 + Sω)*X(2) - (4R2 - 5Sω)*X(3)

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)


X(7716) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(6))

Trilinears    a*(a^2+3*b^2+3*c^2)/(a^2-b^2-c^2) : :

See the preamble to X(7713).

X(7716) lies on these lines:
{2,3867}, {4,141}, {6,25}, {19,3059}, {24,5085}, {64,66}, {69,6995}, {518,7713}, {524,7714}, {3564,7715}, {5845,7717}, {5846,7718} et al


X(7717) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(7))

Trilinears    (3*a^4-2*a^2*b*c-(4*(b+c))*a^3+(b^2+c^2)*(b-c)^2)/((a^2-b^2-c^2)*a) : :

See the preamble to X(7713).

X(7717) lies on these lines:
{4,9}, {7,25}, {142,6353}, {144,6995}, {278,2212}, {390,1829}, {428,6172}, {518,7718}, {527,7714}, {5843,7715}, {5845,7716}


X(7718) =  HATZIPOLAKIS-LOZADA HOMOTHETIC CENTER HL(X(8))

Trilinears    (3*a^3+(b+c)*a^2-(b^2+c^2)*a+(b+c)*(b^2+c^2))/(a*(a^2-b^2-c^2)) : :

See the preamble to X(7713).

X(7718) lies on these lines:
{1,4}, {2,5090}, {8,25}, {10,6353}, {19,3189}, {20,1902}, {24,5657}, {28,1043}, {518,7717}, {519,7713}, {5844,7715}, {5846,7716}


X(7719) =  {X(19), X(2333)}-HARMONIC CONJUGATE OF X(7713)

Trilinears    (a2 + b2 + c2 - 2ab - 2ca)/(a2 - b2 - c2) : :
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)



leftri  Centers related to the orthocentroidal triangle: X(7720) - X(7733)  rightri

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).


underbar

X(7720) =  PERSPECTOR OF THESE TRIANGLES: ORTHOCENTROIDAL AND INNER GREBE

Barycentrics    a^8-2*(7*b^2+7*c^2-4*S)*a^6-4*(3*b^2*c^2-2*(b^2+c^2)*S)*a^4+2*(b^2-c^2)^2*a^2*(7*b^2+7*c^2-6*S)-(b^2-c^2)^2*((b^2-c^2)^2+4*(b^2+c^2)*S) : :

See the preamble to X(7720).

X(7720) lies on these lines: {4,7725}, {381,1161}, {5871, 5890}, {6202,7699}


X(7721) =  PERSPECTOR OF THESE TRIANGLES: ORTHOCENTROIDAL AND OUTER GREBE

Barycentrics    a^8-2*(7*b^2+7*c^2+4*S)*a^6-4*(3*b^2*c^2+2*(b^2+c^2)*S)*a^4+2*(b^2-c^2)^2*a^2*(7*b^2+7*c^2+6*S)-(b^2-c^2)^2*((b^2-c^2)^2-4*(b^2+c^2)*S) : :

See the preamble to X(7720).

X(7721) lies on these lines: {4,7726}, {5870, 5890}, {6201,7699}


X(7722) =  CIRCUMORTHIC-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (16*cos(A)^3*cos(B-C)-12*cos(A)^2+1)*sec(A) : :

See the preamble to X(7720).

X(7722) lies on these lines:
{2,7723}, {3,3043}, {4,94}, {24,399}, {54,74}, {110,186}, {113,5449}, {125,5890}, {542,6403}, {2781,6776}, {3567,7687}, {6128,7727}

X(7722) = midpoint of X(6241) and X(7731)
X(7722) = reflection of X(i) in X(j) for these (i,j): (4,1986), (74,185), (265,6102)
X(7722) = anticomplement of X(7723)
X(7722) = circumcevian antigonal image of X(4) (antigonal image, wrt cirumorthic triangle, of X(4))


X(7723) =  2nd-EULER-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    cos(A)*(-(6*cos(A)+cos(3*A))*cos(B-C)+(cos(2*A)+3/2)*cos(2*(B-C))+2*cos(2*A)+2) : :

See the preamble to X(7720).

Let A'B'C' be the reflection of ABC in X(3) (ie, the circumcevian triangle of X(3)). Let A" be the reflection of A' in line BC, and define B" and C" cyclically. (A"B"C" is the X(3)-Fuhrmann triangle.) Let A* be the reflection of A in B'C', and define B* and C* cyclically. Triangle A*B*C* is inversely similar to ABC, with similitude center X(265), and A*B*C* is perspective to ABC at X(74). A*B*C* is congruent to A''B''C'' and also homothetic to A"B"C", with center of homothety X(7723). Also, X(7723) = X(2072)-of-A"B"C" = X(2072)-of-A*B*C*. (Randy Hutson, June 19, 2015)

X(7723) lies on these lines:
{2,7722}, {3,74}, {5,1986}, {52,7687}, {68,265}, {113,1209}, {125,1568}, {185,6699}, {381,1112}, {974,5654}, {1026,7727}, {1352,2781}, {3830,6403}, {5891,5972}, {5946,7699}

X(7723) = reflection of X(i) in X(j) for these (i,j): (52,7687), (113,5907), (185,6699), (1987,5)
X(7723) = complement of X(7722)


X(7724) =  EXTANGENTS-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    -12*sin(A/2)*cos(A)*cos((B-C)/2)+2*(2*sin(A/2)+sin(5*A/2))*cos(3*(B-C)/2)+(4*cos(A)+2*cos(2*A)+2*cos(3*A)+4)*cos(B-C)+cos(3*A)-4*cos(A)-2*cos(2*A)-1 : :

See the preamble to X(7720).

X(7724) lies on these lines:
{19,1986}, {40,2940}, {55,399}, {65,79}, {71,74}, {542,3779}, {2777,6254}, {6253,6255}

X(7724) = reflection of X(7727) in X(399)


X(7725) =  INNER-GREBE-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (-6*cos(2*A)-6*cos(4*A)+2*sin(2*A)-sin(4*A)+12)*cos(B-C)+(12*cos(A)+6*cos(3*A)-3*sin(A)-2*sin(3*A))*cos(2*(B-C))-42*cos(A)+24*cos(A)^3+sin(3*A)*(cos(2*A)+2) : :

See the preamble to X(7720).

X(7725) lies on these lines:
{4,7720}, {6,74}, {113,5591}, {125,6202}, {146,1271}, {541,5861}, {542,6319}, {1161,5663}, {2777,5871}, {6215,7728}, {6218,7687}

X(7725) = reflection of X(i) in X(j) for these (i,j): (7726,74), (7732,1161)


X(7726) =  OUTER-GREBE-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (-6*cos(2*A)-6*cos(4*A)-2*sin(2*A)+sin(4*A)+12)*cos(B-C)+(12*cos(A)+6*cos(3*A)+3*sin(A)+2*sin(3*A))*cos(2*(B-C))-42*cos(A)+24*cos(A)^3-sin(3*A)*(cos(2*A)+2) : :

See the preamble to X(7720).

X(7726) lies on these lines:
{4,7721}, {6,74}, {113,5590}, {125,6201}, {146,1270}, {541,5860}, {542,6320}, {690,6226}, {1160,5663}, {2777,5870}, {6214,7728}, {6217,7687}

X(7726) = reflection of X(i) in X(j) for these (i,j): (7725,74), (7733,1160)


X(7727) =  INTANGENTS-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (2*cos(A)+2*cos(2*A)+3)*cos(B-C)-4*cos(A)-cos(2*A)-3/2 : :

See the preamble to X(7720).

X(7727) lies on these lines:
{33,1986}, {35,110}, {36,74}, {55,399}, {146,1478}, {265,3583}, {542,3056}, {1062,7723}, {1479,3448}, {1511,5010}, {2293,2772}, {2771,3057}, {2948,5119}, {3019,5902}, {3028,5663}, {3585,7728}, {6198,7722}

X(7727) = reflection of X(7724) in X(399)


X(7728) =  JOHNSON-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (6*cos(2*A)+8)*cos(B-C)-2*cos(A)*cos(2B-2C)-10*cos(A)-cos(3*A) : :

See the preamble to X(7720).

X(7728) lies on the cubic K530 and these lines:
{3,113}, {4,94}, {5,74}, {20,1511}, {30,110}, {67,3818}, {125,381}, {155,382}, {156,3043}, {542,1351}, {690,6033}, {1352,2781}, {1479,3028}, {1514,3581}, {1531,7574}, {1656,6699}, {2778,5887}, {2931,7517}, {3146,5609}, {3534,5642}, {3585,7727}, {3843,7687}, {5055,6723}, {5073,6053}, {6214,7726}, {6215,7725}

X(7728) = midpoint of X(i) and X(j) for these {i,j}: {4,146}, {382,399}, {2935,5895}
X(7728) = reflection of X(i) in X(j) for these (i,j): (3,113), (4,1539), (20,1511), (67,3818), (74,5), (265,4), (3534,5642), (7574,1531)
X(7728) = isogonal conjugate of X(4)-vertex conjugate of X(30)
X(7728) = X(3) of X(30)-Fuhrmann triangle


X(7729) =  MIDHEIGHT-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (22*cos(2*A)+2*cos(4*A)+24)*cos(B-C)+(-6*cos(A)-2*cos(3*A))*cos(2B-2C)-33*cos(A)-7*cos(3*A) : :

See the preamble to X(7720).

X(7729) lies on these lines:
{4,974}, {6,64}, {52,5925}, {195,2935}, {381,1853}, {389,5895}, {568,2777}, {1498,6642}, {1992,2781}, {4846,6145}, {5654,5663}, {5889,5894}


X(7730) =  ORTHOCENTROIDAL-TO-REFLECTION ORTHOLOGIC CENTER

Trilinears    (-2*cos(2*A)+2*cos(4*A)+3)*cos(B-C)-2*cos(3*A)*cos(2B-2C))-cos(3*A) : :

See the preamble to X(7720).

X(7730) lies on these lines:
{4,7731}, {6,24}, {52,2888}, {110,143}, {381,1154}, {7526,7691}


X(7731) =  REFLECTION-TO-ORTHOCENTROIDAL ORTHOLOGIC CENTER

Trilinears    (6*cos(2*A)+2*cos(4*A)+1)*cos(B-C)+(-4*cos(A)-2*cos(3*A))*cos(2B-2C)-3*cos(3*A) : :

See the preamble to X(7720).

X(7731) lies on these lines:
{4,7730}, {6,74}, {26,110}, {52,3448}, {125,3567}, {184,2914}, {265,3060}, {382,5663}, {399,1154}, {1112,7507}, {1511,2979}, {5640,7579}

X(7731) = reflection of X(i) in X(j) for these (i,j): (74,1986), (3448,52), (6241,7722)


X(7732) =  INNER-GREBE-TO-ORTHOCENTROIDAL PARALLELOGIC CENTER

Trilinears    a*((36*R^2-10*Sω+S)*S^2-SA*(3*SA-2*Sω)*(2*Sω-S)) : :

See the preamble to X(7720).

X(7732) lies on these lines:
{6,110}, {113,6202}, {125,5591}, {265,6215}, {542,5861}, {690,6319}, {1161,5663}, {1271,3448}, {2948,5589}

X(7732) = reflection of X(i) in X(j) for these (i,j): (7725,1161), (7733,110)


X(7733) =  OUTER-GREBE-TO-ORTHOCENTROIDAL PARALLELOGIC CENTER

Trilinears    a*((36*R^2-10*Sω-S)*S^2-SA*(3*SA-2*Sω)*(2*Sω+S)) : :

See the preamble to X(7720).

X(7733) lies on these lines:
{6,110}, {113,6201}, {125,5590}, {265,6214}, {542,5860}, {690,6320}, {1160,5663}, {1270,3448}, {2948,5588}

X(7733) = reflection of X(i) in X(j) for these (i,j): (7726,1160), (7732,110)


X(7734) =  2nd HATZIPOLAKIS-MOSES-EULER CENTROID

Barycentrics    2 a^6+a^4 b^2-2 a^2 b^4-b^6+a^4 c^2-16 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6 : :
X(7734) = (|OH|2 + 7 R2)*X(2) + 2Sω*X(3) = 5 X[2] - X[428] = 7 X[3526] - X[6756] = 3 X[2] + X[7667] = 3 X[428] + 5 X[7667]

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) =  X(2)X(6)∩X(4)X(32)

Barycentrics    S2 - a2Sω : :
Barycentrics    3a4 + (b2 - c2)2 : :
Barycentrics    cot B + cot C - tan ω : :
X(7735) = 3S2*X(2) - 2SωX(6).

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) = anticomplement of X(7778)
X(7735) = X(2)-Ceva-conjugate of X(7710)
X(7735) = crosspoint of X(2) and X(3424)
X(7735) = crosssum of X(i) and X(j) for these {i,j}: {6,1350}, {183,1975}
X(7735) = crossdifference of every pair of points on X(512)X(684)
X(7735) = X(i)-complementary conjugate of X(j) for these (i,j): (31,7710), (3424,2887)
X(7735) = {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): (2,6,7736), (2,193,325), (2,385,69), (2,5304,6), (3,5286,7738), (3,5305,5286), (4,1285,7737), (6,230,2), (6,1613,3289), (6,5306,5304), (32,115,7737), (32,3767,4), (32,7737,1285), (39,5346,5319), (115,7737,4), (187,2549,376), (187,5309,2549), (230,5304,7736), (230,5306,6), (395,396,599), (574,5355,7739), (590,615,3763), (597,7610,2), (800,1196,232), (1249,6353,232), (1513,6776,7710), (3053,5254,20), (3068,3069,69), (3767,7737,115)


X(7736) =  X(2)X(6)∩X(4)X(39)

Barycentrics    S2 + a2Sω : :
Barycentrics    a4 + 4a2(b2 + c2) - (b2 - c2)2 : :
Barycentrics    cot B + cot C + tan ω : :
X(7736) = 3S2*X(2) + 2SωX(6).

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)
X(7736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,7735), (2,193,183), (2,3314,3619), (2,3329,3618), (2,5304,230), (4,39,7738), (6,230,5304), (6,3815,2), (39,2548,4), (39,5475,2549), (230,5304,7735), (574,7737,376), (1007,3618,2), (1285,3524,187), (1506,3767,3090), (2544,2545,262), (2548,2549,5475), (2549,5475,4), (3068,3069,3618), (3815,5306,3055)


X(7737) =  X(4)X(32)∩X(6)X(30)

Barycentrics    3a4 - (b2 - c2)2 : :
X(7737) = 5*X(3618) - 4*X(4045) = 4*X(6) - 3*X(7739) = 2*X(2549) - 3*X(7739)

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(7737) = barycentric product X(98)*X(5112)
X(7737) = anticomplement of X(7761)
X(7737) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,32,3767), (4,1285,7735), (4,7735,115), (6,2549,7739), (32,115,7735), (115,7735,3767), (187,5475,2), (316,3972,2), (376,7736,574), (381,1384,230), (574,6781,376), (1285,7735,32), (1506,5206,631), (5008,5309,5304)


X(7738) =  X(4)X(39)∩X(6)X(20)

Barycentrics    a4 - 4a2(b2 + c2) - (b2 - c2)2 : :

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) =  X(2)X(39)∩X(6)X(30)

Barycentrics    a4 + 4a2(b2 + c2) + (b2 - c2)2 : :
X(7739) = 4*X(6) - X(7737) = 2*X(2549) + X(7737)

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) =  MIDPOINT OF X(3) AND X(5502)

Trilinears    (1-4*cos(A)^2)*((4*cos(2*A)+5)*cos(B-C)-cos(A)*cos(2*(B-C))-7*cos(A)-cos(3*A)) : :
Barycentrics    a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (2 a^10-2 a^8 b^2-5 a^6 b^4+7 a^4 b^6-a^2 b^8-b^10-2 a^8 c^2+12 a^6 b^2 c^2-7 a^4 b^4 c^2-6 a^2 b^6 c^2+3 b^8 c^2-5 a^6 c^4-7 a^4 b^2 c^4+14 a^2 b^4 c^4-2 b^6 c^4+7 a^4 c^6-6 a^2 b^2 c^6-2 b^4 c^6-a^2 c^8+3 b^2 c^8-c^10) : :
Barycentrics    ((-a^2+b^2+c^2)^2-b^2*c^2)*(2*a^10-2*(b^2+c^2)*a^8-(5*b^4-12*b^2*c^2+5*c^4)*a^6+7*(b^4-c^4)*(b^2-c^2)*a^4-(b^2-c^2)^2*(b^4+8*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3)*a^2 : :

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)
X(7740) = crossdifference of every pair of centers on the line X(1989)X(6587)
X(7740) = barycentric product X(323) X(2777)


X(7741) =  CENTER OF NINE-POINT-CIRCLE-INVERSE OF INCIRCLE

Trilinears    1 - 2 cos(B - C) : :
Barycentrics    a2(b2 + c2 - bc) - (b2 - c2)2 : :
X(7741) = R*X(1) - 6r*X(2) + 2r*X(3) = R*X(1) - 4r*X(5) = -6r*X(2) + (R + 2r)*X(35) = -2r*X(4) + (R - 2r)*X(36)

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

X(7741) lies on these lines:
{1,5}, {2,35}, {3,3583}, {4,36}, {8,3814}, {10,3877}, {30,5433}, {33,1594), (34,403}, {46,1699}, {55,1656}, {56,381}, {57,79}, {72,5087}, {115,2275), {125,7727}, {140,5010}, {165,6922}, {172,5475}, {226,6990}, {230,7031}, {312,7206}, (325,3760}, {382,4316}, {388,3545}, {390,7486}, {427,5272}, {428,5345}, {442,3586}, {485,3299}, {486,3301}, {515,6941}, {516,6943}, {517,6971}, {546,7354}, {547,3058}, {549,7294}, {567,2477}, {614,5133}, {631,4302}, {908,5904}, {946,1737}, {950,6829}, {962,3245}, {978,3142}, {993,5046}, {999,3851}, {1056,3544}, {1058,5071}, {1062,2072}, {1087,1090}, {1089,3705}, {1111,7185}, {1125,2476}, {1203,5292}, {1210,3671}, {1329,3679}, {1385,6980}, {1393,1725}, {1428,3818}, {1447,4056}, {1500,7603}, {1506,2276}, {1532,5691}, {1621,7504}, {1698,1706}, {1724,1985}, {1727,1836}, {1728,5715}, {1770,3911}, {1834,5313}, {2077,6958}, {2098,5790}, {2478,5251}, {2548,5280}, {2611,6757}, {2957,3460}, {2964,5348}, {3303,5079}, {3304,5072}, {3338,4355}, {3434,6931}, {3436,5288}, {3485,5425}, {3486,6874}, {3526,4330}, {3574,7356}, {3576,6842}, {3601,6881}, {3616,3822}, {3628,5432}, {3632,3813}, {3670,3944}, {3767,5299}, {3832,4293}, {3838,5439}, {3839,5265}, {3843,4325}, {3845,5298}, {3855,4317}, {4188,6681}, {4197,4304}, {4295,5704}, {4297,6932}, {4305,5550}, {4309,5067}, {4894,7081}, {5066,5434}, {5169,7292}, {5187,5258}, {5230,5315}, {5322,7394}, {5370,7519}, {5448,7352}, {5536,5812}, {5537,6978}, {5570,5777}, {5692,6734}, {6149,7299}, {6198,7577}, {6256,6968}, {6502,6564}, {6684,6963}, {7179,7264}, {7298,7499}

X(7741) = reflection of X(7280) in X(5433)
X(7741) = {X(1),X(5)}-harmonic conjugate of X(7951)
X(7741) = homothetic center of 3rd Euler triangle and reflection triangle of X(1)
X(7741) = homothetic center of 2nd isogonal triangle of X(1) and Euler triangle; see X(36)
X(7741) = homothetic center of outer Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)


X(7742) =  CENTER OF CIRCUMCIRCLE-INVERSE OF INCIRCLE

Trilinears    cos(A)*(cos(A)+2*cos((B-C)/2)*sin(A/2)+1)-1 : :
X(7742) = R2*X(1) - (2Rr + r2)*X(3)

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(7742) = {X(55),X(56)}-harmonic conjugate of X(942)


X(7743) =  CENTER OF INCIRCLE-INVERSE OF ORTHOCENTROIDAL CIRCLE

Trilinears    3 - cos B - cos C - cos(B - C) : :
X(7743) = (2R - r)*X(1) - 6r*X(2) + 3r*X(3)

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) =  CENTER OF ORTHOCENTROIDAL-CIRCLE-INVERSE OF INCIRCLE

Trilinears    (-4*cos(A)-12)*cos(B-C)+(6*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+4*sin(A/2)*cos(3*(B-C)/2)+6*cos(A)+3*cos(2*A)+5 : :
X(7744) = (9R2 - 2Sω)*X(1) + 6r(r - 2R)*X(2) - 3r(r -2R)*X(3)

X(7744) lies on these lines: {1,381}, {5094,5121}

X(7744) = {X(2463),X(2464)}-harmonic conjugate of X(7743)



leftri  The form k1a4 + k2a2(b2 + c2) + k3b2c2 + k4(b4 + c4)  rightri

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.


underbar

X(7745) =  X(4)X(6)∩X(5)X(32)

Barycentrics    -2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7745) = (pending)

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(7745) = crosssum of X(3) and X(39)
X(7745) = crosspoint of X(4) and X(83)
X(7745) = midpoint of the orthocenters of the pedal triangles of the 1st and 2nd Brocard points


X(7746) =  X(2)X(39)∩X(5)X(32)

Barycentrics    a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7746) = (pending)

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) = crosssum of X(3557) and X(3558)
X(7746) = complement of X(7763)
X(7746) = {X(6),X(1656)}-harmonic conjugate of X(1506)


X(7747) =  X(4)X(32)∩X(5)X(187)

Barycentrics    -2*a^4 + b^4 - 2*b^2*c^2 + c^4 : :
X(7747) = (pending)

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(7747) = (pending)


X(7748) =  X(3)X(115)∩X(4)X(39)

Barycentrics    -a^4 + a^2*b^2 + b^4 + a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7748) = (pending)

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(7748) = (pending)


X(7749) =  X(2)X(32)∩X(3)X(115)

Barycentrics    2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4 : :
Barycentrics    a4 - 4S2 : :
X(7749) = (pending)

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) =  X(20)X(64)∩X(30)X(76)

Barycentrics    -2*a^4 + a^2*b^2 + b^4 + a^2*c^2 + c^4 : :
X(7750) = (pending)

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(7750) = crosspoint, wrt excentral triangle, of X(20) and X(2896)
X(7750) = crosspoint, wrt anticomplementary triangle, of X(20) and X(2896)


X(7751) =  X(32)X(76)∩X(39)X(183)

Barycentrics    -a^4 + 2*b^2*c^2 : :
X(7751) = (pending)

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(7751) = (pending)


X(7752) =  X(2)X(32)∩X(4)X(99)

Barycentrics    -(a^2*b^2) + b^4 - a^2*c^2 - b^2*c^2 + c^4 : :
Barycentrics    a^4 - b^4 - c^4 + 4S^2 : :
X(7752) = (pending)

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(7752) = complement of X(7793)
X(7752) = anticomplement of X(7749)
X(7752) = {X(4),X(1007)}-harmonic conjugate of X(7763)


X(7753) =  X(2)X(32)∩X(6)X(13)

Barycentrics    -2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7753) = (pending)

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-triangle
X(7753) = X(39)-of-orthocentroidal-triangle
X(7753) = complement of X(7811)
X(7753) = {X(6),X(381)}-harmonic conjugate of X(5309)


X(7754) =  X(3)X(194)∩X(4)X(193)

Barycentrics    -a^4 - a^2*b^2 - a^2*c^2 + 2*b^2*c^2 : :
X(7754) = (pending)

X(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) = (pending)


X(7755) =  X(4)X(32)∩X(6)X(17)

Barycentrics    2*a^4 + b^4 - 2*b^2*c^2 + c^4 : :
X(7755) = (pending)

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(7755) = (pending)


X(7756) =  X(20)X(32)∩X(30)X(39)

Barycentrics    -2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7756) = (pending)

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 triangle


X(7757) =  X(2)X(39)∩X(6)X(99)

Barycentrics    -2*a^2*b^2 - 2*a^2*c^2 + b^2*c^2 : :
X(7757) = X(2) + X(194) = X(2) - 2X(39)

X(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(7757) = centroid of X(194)PU(1)
X(7757) = SS(a->bc) of X(599) (barycentric substitution)
X(7757) = isotomic conjugate of X(9462)
X(7757) = anticomplement of X(9466)


X(7758) =  X(32)X(193)∩X(39)X(69)

Barycentrics    -a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7758) = (pending)

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(7758) = (pending)


X(7759) =  X(32)X(325)∩X(39)X(315)

Barycentrics    -a^4 - a^2*b^2 + b^4 - a^2*c^2 + c^4 : :
X(7759) = (pending)

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(7759) = (pending)


X(7760) =  X(6)X(76)∩X(32)X(99)

Barycentrics    -a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2 : :
X(7760) = (pending)

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) =  X(2)X(187)∩X(3)X(114)

Barycentrics    -a^4 + a^2*b^2 + b^4 + a^2*c^2 + c^4 : :
X(7761) = (pending)

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(7761) = anticomplement of X(7804)


X(7762) =  X(4)X(193)∩X(30)X(194)

Barycentrics    -2*a^4 - a^2*b^2 + b^4 - a^2*c^2 + c^4 : :
X(7762) = (pending)

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(7762) = (pending)


X(7763) =  X(2)X(39)∩X(4)X(99)

Barycentrics    a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4 : :
X(7763) = (pending)

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(7763) = anticomplement of X(7746)
X(7763) = {X(4),X(1007)}-harmonic conjugate of X(7752)


X(7764) =  X(32)X(620)∩X(39)X(325)

Barycentrics    -2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4 : :
X(7764) = (pending)

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(7764) = (pending)


X(7765) =  X(5)X(39)∩X(20)X(32)

Barycentrics    2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4 : :
X(7765) = (pending)

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(7765) = (pending)


X(7766) =  X(2)X(6)∩X(32)X(99)

Barycentrics    -2*a^4 - a^2*b^2 - a^2*c^2 + b^2*c^2 : :
X(7766) = (pending)

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(7766) = {X(6),X(183)}-harmonic conjugate of X(3329)


X(7767) =  X(3)X(69)∩X(30)X(76)

Barycentrics    (a^2 - b^2 - c^2)*(2*a^2 + b^2 + c^2) : :
X(7767) = (pending)

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) = (pending)


X(7768) =  X(2)X(5007)∩X(4)X(69)

Barycentrics    -a^4 + b^4 + b^2*c^2 + c^4 : :
X(7768) = (pending)

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)
X(7768) = anticomplement of X(5007)
X(7768) = cevapoint of X(69) and X(1369)
X(7768) = midpoint of X(633) and X(634)


X(7769) =  X(2)X(39)∩X(5)X(99)

Barycentrics    a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4 : :
Barycentrics    3 - cot2A : :
Barycentrics    csc2A sin(A + π/6) sin(A - π/6) : :
X(7769) = (pending)

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)


X(7770) =  X(2)X(3)∩X(6)X(76)

Barycentrics    a^4+a^2 b^2+a^2 c^2+2 b^2 c^2 : :
X(7770) = (pending)

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(7770) = anticomplement of X(8362)


X(7771) =  X(2)X(187)∩X(3)X(76)

Barycentrics    -2*a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2 : :
X(7771) = (pending)

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)


X(7772) =  X(2)X(3108)∩X(3)X(6)

Barycentrics    a^2*(a^2 + 2*b^2 + 2*c^2) : :
Trilinears    3 sin A + cos A tan ω : :
Trilinears    cos A + 3 sin A cot ω : :
Trilinears    2 sin(A + ω) + sin(A - ω) : :
Trilinears    2 cos(A + 2ω) - cos(A - 2ω) - cos A : :
X(7772) = (pending)

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) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5007), 61,62,182), (371,372,5092)
X(7772) = {X(3),X(6)}-harmonic conjugate of X(5007)
X(7772) = inverse-in-1st-Brocard-circle of X(5007)
X(7772) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(5092)
X(7772) = radical center of Lucas(6 cot ω) circles
X(7772) = X(6)-of X(6)PU(1)


X(7773) =  X(3)X(316)∩X(5)X(183)

Barycentrics    -a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7773) = (pending)

X(7773) lies on these lines:
{2,3053}, {3,316}, {4,325}, {5,183}, {6,5025}, {30,7763}, {32,625} and others

X(7773) = (pending)


X(7774) =  X(2)X(6)∩X(32)X(620)

Barycentrics    -a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4 : :
Barycentrics    cot B + cot C - cot A - tan ω : :
X(7774) = (pending)

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) = crossdifference of every pair of points on the radical axis of the Brocard circle and 1st and 2nd Kenmotu circles
X(7774) = {X(2),X(193)}-harmonic conjugate of X(385)
X(7774) = anticomplement of X(183)


X(7775) =  X(2)X(32)∩X(5)X(524)

Barycentrics    -a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7775) = (pending)

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(7775) = (pending)


X(7776) =  X(5)X(69)∩X(76)X(381)

Barycentrics    -a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*c^4 : :
X(7776) = (pending)

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(7776) = (pending)


X(7777) =  X(2)X(6)∩X(11)X(192)

Barycentrics    -2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7777) = (pending)

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) = (pending)


X(7778) =  X(2)X(6)∩X(381)X(625)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*c^4 : :
X(7778) = (pending)

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) =  X(2)X(6)∩X(99)X(754)

Barycentrics    -a^4 - a^2*b^2 + b^4 - a^2*c^2 + b^2*c^2 + c^4 : :
X(7779) = (pending)

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) =  X(5)X(754)∩X(32)X(183)

Barycentrics    -2*a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2 : :
X(7780) = (pending)

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(7780) = (pending)


X(7781) =  X(3)X(538)∩X(32)X(99)

Barycentrics    a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2 : :
X(7781) = (pending)

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(7781) = (pending)


X(7782) =  X(3)X(76)∩X(20)X(316)

Barycentrics    2*a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^2*c^2 : :
X(7782) = (pending)

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(7782) = (pending)


X(7783) =  X(3)X(194)∩X(39)X(83)

Barycentrics    a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^2*c^2 : :
X(7783) = (pending)

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(7783) = (pending)


X(7784) =  X(3)X(114)∩X(6)X(315)

Barycentrics    -a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*c^4 : :
X(7784) = (pending)

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(7784) = (pending)


X(7785) =  X(2)X(32)∩X(4)X(147)

Barycentrics    -a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4 : :
X(7785) = (pending)

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(7785) = anticomplement of X(1078)


X(7786) =  X(2)X(39)∩X(3)X(83)

Barycentrics    2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2 : :
X(7786) = (pending)

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(7786) = center of circle that is locus of crosssums of antipodes on the Moses circle


X(7787) =  X(2)X(32)∩X(6)X(194)

Barycentrics    2*a^4 + a^2*b^2 + a^2*c^2 + b^2*c^2 : :
X(7787) = (pending)

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(7787) = anticomplement of X(3096)
X(7787) = homothetic center of 5th anti-Brocard triangle and anticomplementary triangle
X(7787) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 5th anti-Brocard triangle
X(7787) = homothetic center of 5th anti-Brocard triangle and cross-triangle of ABC and 5th anti-Brocard triangle


X(7788) =  X(2)X(6)∩X(30)X(315)

Barycentrics    -a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7788) = (pending)

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) =  X(2)X(1975)∩X(3)X(66)

Barycentrics    2*a^4 - a^2*b^2 + b^4 - a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7789) = (pending)

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) =  X(2)X(99)∩X(4)X(83)

Barycentrics    a^2*b^2 + b^4 + a^2*c^2 - b^2*c^2 + c^4 : :
X(7790) = (pending)

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)
X(7790) = anticomplement of X(7820)


X(7791) =  X(2)X(3)∩X(39)X(315)

Barycentrics    -a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4 : :
X(7791) = (pending)

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) = anticomplement of X(7770)
X(7791) = {X(194),X(2896)}-harmonic conjugate of X(69)


X(7792) =  X(2)X(6)∩X(5)X(83)

Barycentrics    2*a^4 + a^2*b^2 + b^4 + a^2*c^2 + c^4 : :
X(7792) = (pending)

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) =  X(2)X(32)∩X(20)X(98)

Barycentrics    -2*a^4 + a^2*b^2 + a^2*c^2 + b^2*c^2 : :
Barycentrics    3a^4 - b^4 - c^4 - 4S^2 : :
X(7793) = (pending)

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(7793) = anticomplement of X(7752)


X(7794) =  X(2)X(3108)∩X(3)X(67)

Barycentrics    (b^2 + c^2)^2 : :
X(7794) = (pending)

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(7794) = (pending)


X(7795) =  X(2)X(39)∩X(3)X(66)

Barycentrics    a^4 + b^4 + 2*b^2*c^2 + c^4 : :
X(7795) = (pending)

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(7795) = anticomplement of X(7834)


X(7796) =  X(5)X(76)∩X(20)X(99)

Barycentrics    -(a^2*b^2) + b^4 - a^2*c^2 + b^2*c^2 + c^4 : :
X(7796) = (pending)

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) =  X(2)X(39)∩X(5)X(147)

Barycentrics    a^4 + a^2*b^2 + b^4 + a^2*c^2 - b^2*c^2 + c^4 : :
X(7797) = (pending)

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) =  X(32)X(99)∩X(39)X(183)

Barycentrics    -a^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2 : :
X(7798) = (pending)

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(7798) = (pending)


X(7799) =  X(2)X(39)∩X(30)X(99)

Barycentrics    (a^2 - b^2 - b*c - c^2)*(a^2 - b^2 + b*c - c^2) : :
X(7799) = (pending)

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) = isotomic conjugate of X(1989)
X(7799) = trilinear pole of line X(526)X(3268)


X(7800) =  X(2)X(32)∩X(3)X(66)

Barycentrics    -a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7800) = (pending)

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) =  X(2)X(39)∩X(3)X(67)

Barycentrics    a^4 - a^2*b^2 + b^4 - a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7801) = (pending)

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) =  X(20)X(99)∩X(30)X(76)

Barycentrics    -2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - b^2*c^2 + c^4 : :
X(7802) = (pending)

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(7802) = (pending)


X(7803) =  X(2)X(39)∩X(4)X(83)

Barycentrics    a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4 : :
X(7803) = (pending)

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) =  X(2)X(187)∩X(39)X(83)

Barycentrics    2*a^4 + a^2*b^2 + a^2*c^2 + 2*b^2*c^2 : :
X(7804) = (pending)

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) =  X(32)X(538)∩X(39)X(385)

Barycentrics    -2*a^4 - a^2*b^2 - a^2*c^2 + 2*b^2*c^2 : :
X(7805) = (pending)

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(7805) = (pending)


X(7806) =  X(2)X(6)∩X(32)X(316)

Barycentrics    2*a^4 + b^4 - b^2*c^2 + c^4 : :
X(7806) = (pending)

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)


X(7807) =  X(2)X(3)∩X(32)X(325)

Barycentrics    2 a^4-a^2 b^2+b^4-a^2 c^2+c^4 : :
X(7807) = (pending)

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) = complement of X(5025)
X(7807) = anticomplement of X(8361)


X(7808) =  X(2)X(32)∩X(5)X(182)

Barycentrics    a^4 + 2*a^2*b^2 + 2*a^2*c^2 + 2*b^2*c^2 : :
X(7808) = (pending)

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) =  X(2)X(32)∩X(30)X(99)

Barycentrics    -a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7809) = (pending)

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(7809) = (pending)


X(7810) =  X(2)X(32)∩X(3)X(67)

Barycentrics    -2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7810) = (pending)

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)


X(7811) =  X(2)X(32)∩X(30)X(76)

Barycentrics    -2*a^4 + a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 + c^4 : :
X(7811) = (pending)

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(7811) = {X(9988),X(9989)}-harmonic conjugate of X(9873)


X(7812) =  X(2)X(32)∩X(6)X(316)

Barycentrics    -2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4 : :
X(7812) = (pending)

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) =  X(32)X(193)∩X(39)X(141)

Barycentrics    (2*a^2 - b^2 - c^2)*(b^2 + c^2) : :
X(7813) = (pending)

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(7813) = (pending)


X(7814) =  X(5)X(76)∩X(20)X(316)

Barycentrics    -2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7814) = (pending)

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(7814) = (pending)


X(7815) =  X(2)X(32)∩X(39)X(183)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*a^2*c^2 + 2*b^2*c^2 : :
X(7815) = (pending)

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) =  X(4)X(625)∩X(39)X(83)

Barycentrics    2*a^4 - a^2*b^2 - a^2*c^2 + 2*b^2*c^2 : :
X(7816) = (pending)

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(7816) = (pending)


X(7817) =  X(2)X(39)∩X(6)X(625)

Barycentrics    2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7817) = (pending)

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)


X(7818) =  X(2)X(32)∩X(69)X(115)

Barycentrics    -a^4 + 2*b^4 + 2*c^4 : :
X(7818) = (pending)

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) = (pending)


X(7819) =  X(2)X(3)∩X(32)X(141)

Barycentrics    2*a^4 + a^2*b^2 + b^4 + a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7819) = (pending)

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) = complement of X(6656)
X(7819) = anticomplement of X(8364)


X(7820) =  X(2)X(99)∩X(32)X(69)

Barycentrics    2*a^4 + b^4 + 2*b^2*c^2 + c^4 : :
X(7820) = (pending)

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) =  X(39)X(325)∩X(76)X(625)

Barycentrics    -(a^2*b^2) + 2*b^4 - a^2*c^2 + 2*c^4 : :
X(7821) = (pending)

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) =  X(2)X(39)∩X(32)X(141)

Barycentrics    a^4 + a^2*b^2 + b^4 + a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7822) = (pending)

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) =  X(30)X(194)∩X(32)X(316)

Barycentrics    -2*a^4 + b^4 - b^2*c^2 + c^4 : :
X(7823) = (pending)

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}

X(7823) = (pending)


X(7824) =  X(2)X(3)∩X(83)X(187)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2 : :
X(7824) = (pending)

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) = (pending)


X(7825) =  X(3)X(625)∩X(20)X(620)

Barycentrics    -a^4 + 2*b^4 - 2*b^2*c^2 + 2*c^4 : :
X(7825) = (pending)

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(7825) = (pending)


X(7826) =  X(32)X(69)∩X(39)X(524)

Barycentrics    -2*a^4 + b^4 + 2*b^2*c^2 + c^4 : :
X(7826) = (pending)

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(7826) = (pending)


X(7827) =  X(2)X(39)∩X(6)X(316)

Barycentrics    a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7827) = (pending)

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(7827) = (pending)


X(7828) =  X(2)X(39)∩X(5)X(83)

Barycentrics    a^4 + b^4 - b^2*c^2 + c^4 : :
X(7828) = (pending)

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(7828) = anticomplement of X(7874)


X(7829) =  X(2)X(3108)∩X(39)X(620)

Barycentrics    2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4 : :
X(7829) = (pending)

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(7829) = (pending)


X(7830) =  X(2)X(5206)∩X(3)X(114)

Barycentrics    -2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + c^4 : :
X(7830) = (pending)

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(7830) = (pending)


X(7831) =  X(2)X(187)∩X(99)X(141)

Barycentrics    -a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7831) = (pending)

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(7831) = (pending)


X(7832) =  X(2)X(39)∩X(98)X(140)

Barycentrics    a^4 + b^4 + b^2*c^2 + c^4 : :
X(7832) = (pending)

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)
X(7832) = anticomplement of X(7852)


X(7833) =  X(2)X(3)∩X(39)X(3849)

Barycentrics    -2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7833) = (pending)

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) = inverse-in-2nd-Brocard-circle of X(23)
X(7833) = centroid of 6th Brocard triangle


X(7834) =  X(2)X(39)∩X(5)X(182)

Barycentrics    a^4 + a^2*b^2 + b^4 + a^2*c^2 + c^4 : :
X(7834) = (pending)

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(7834) = anticomplement of X(7915)


X(7835) =  X(2)X(99)∩X(76)X(230)

Barycentrics    2*a^4 - a^2*b^2 + b^4 - a^2*c^2 + b^2*c^2 + c^4 : :
X(7835) = (pending)

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(7835) = (pending)


X(7836) =  X(2)X(39)∩X(3)X(147)

Barycentrics    a^4 - a^2*b^2 + b^4 - a^2*c^2 + b^2*c^2 + c^4 : :

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) =  X(2)X(6)∩X(30)X(194)

Barycentrics    -2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 : :

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) = {X(3180),X(3181)}-harmonic conjugate of X(6)


X(7838) =  X(6)X(626)∩X(32)X(620)

Barycentrics    -2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + c^4 : :

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(7838) = (pending)


X(7839) =  X(6)X(194)∩X(39)X(385)

Barycentrics    -a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^2*c^2 : :
X(7839) = (pending)

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(7839) = (pending)


X(7840) =  X(2)X(6)∩X(30)X(147)

Barycentrics    -a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7840) = (pending)

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) = McCay-to-Artzt similarity image of X(99)


X(7841) =  X(2)X(3)∩X(6)X(316)

Barycentrics    -a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7841) = (pending)

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) = (pending)


X(7842) =  X(3)X(625)∩X(30)X(626)

Barycentrics    -2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7842) = (pending)

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(7842) = (pending)


X(7843) =  X(32)X(625)∩X(39)X(316)

Barycentrics    -2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7843) = (pending)

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(7843) = (pending)


X(7844) =  X(2)X(99)∩X(6)X(625)

Barycentrics    a^4 + 2*b^4 - 2*b^2*c^2 + 2*c^4 : :
X(7844) = (pending)

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(7844) = (pending)


X(7845) =  X(39)X(315)∩X(115)X(524)

Barycentrics    -2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*c^4 : :
X(7845) = (pending)

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(7845) = (pending)


X(7846) =  X(2)X(32)∩X(76)X(5305)

Barycentrics    2*a^4 + a^2*b^2 + b^4 + a^2*c^2 + b^2*c^2 + c^4 : :
X(7846) = (pending)

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) =  X(30)X(83)∩X(39)X(316)

Barycentrics    -a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7847) = (pending)

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(7847) = (pending)


X(7848) =  X(2)X(5008)∩X(30)X(3631)

Barycentrics    -2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7848) = (pending)

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(7848) = (pending)


X(7849) =  X(2)X(5007)∩X(5)X(141)

Barycentrics    a^2*b^2 + 2*b^4 + a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7849) = (pending)

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(7849) = midpoint of X(635) and X(636)


X(7850) =  X(2)X(5008)∩X(4)X(69)

Barycentrics    -2*a^4 + 2*b^4 + b^2*c^2 + 2*c^4 : :
X(7850) = (pending)

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) =  X(2)X(1975)∩X(83)X(381)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7851) = (pending)

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(7851) = (pending)


X(7852) =  X(2)X(39)∩X(83)X(625)

Barycentrics    2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*c^4 : :
X(7852) = (pending)

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) =  X(2)X(187)∩X(39)X(325)

Barycentrics    a^2*b^2 + 2*b^4 + a^2*c^2 + 2*c^4 : :
X(7853) = (pending)

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(7853) = (pending)


X(7854) =  X(3)X(67)∩X(39)X(69)

Barycentrics    -a^4 + a^2*b^2 + b^4 + a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7854) = (pending)

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(7854) = (pending)


X(7855) =  X(32)X(524)∩X(39)X(69)

Barycentrics    -a^4 - a^2*b^2 + b^4 - a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7855) = (pending)

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(7855) = (pending)


X(7856) =  X(2)X(3108)∩X(6)X(5031)

Barycentrics    2*a^4 + a^2*b^2 + b^4 + a^2*c^2 - b^2*c^2 + c^4 : :
X(7856) = (pending)

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(7856) = (pending)


X(7857) =  X(2)X(32)∩X(76)X(230)

Barycentrics    2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4 : :
X(7857) = (pending)

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(7857) = (pending)


X(7858) =  X(2)X(5007)∩X(6)X(5031)

Barycentrics    -a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7858) = (pending)

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(7858) = (pending)


X(7859) =  X(2)X(39)∩X(4)X(5092)

Barycentrics    (a^2 + b^2 - b*c + c^2)*(a^2 + b^2 + b*c + c^2) : :
X(7859) = (pending)

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) = (pending)


X(7860) =  X(4)X(69)∩X(99)X(1657)

Barycentrics    -2*a^4 + 2*b^4 - b^2*c^2 + 2*c^4) : :
X(7860) = (pending)

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(7860) = (pending)


X(7861) =  X(5)X(4045)∩X(39)X(625)

Barycentrics    a^2*b^2 + 2*b^4 + a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7861) = (pending)

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(7861) = (pending)


X(7862) =  X(2)X(32)∩X(3)X(625)

Barycentrics    a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7862) = (pending)

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(7862) = (pending)


X(7863) =  X(3)X(67)∩X(32)X(193)

Barycentrics    2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7863) = (pending)

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(7863) = (pending)


X(7864) =  X(2)X(1975)∩X(39)X(625)

Barycentrics    2*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7864) = (pending)

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(7864) = (pending)


X(7865) =  X(2)X(32)∩X(30)X(141)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7865) = (pending)

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}

X(7865) = (pending)


X(7866) =  X(2)X(3)∩X(6)X(626)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*c^4 : :
X(7866) = (pending)

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) = (pending)


X(7867) =  X(2)X(32)∩X(141)X(5028)

Barycentrics    a^4 + 2*b^4 + 2*c^4 : :
X(7867) = (pending)

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(7867) = (pending)


X(7868) =  X(2)X(6)∩X(3)X(3096)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7868) = (pending)

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) = (pending)


X(7869) =  X(2)X(3108)∩X(4)X(626)

Barycentrics    a^4 + 2*b^4 + 2*b^2*c^2 + 2*c^4 : :
X(7869) = (pending)

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(7869) = (pending)


X(7870) =  X(2)X(39)∩X(3)X(6054)

Barycentrics    2*a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7870) = (pending)

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(7870) = (pending)


X(7871) =  X(2)X(5041)∩X(5)X(76)

Barycentrics    -2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7871) = (pending)

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(7871) = (pending)


X(7872) =  X(4)X(4045)∩X(20)X(6680)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7872) = (pending)

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(7872) = (pending)


X(7873) =  X(39)X(315)∩X(187)X(626)

Barycentrics    -2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*c^4 : :
X(7873) = (pending)

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(7873) = (pending)


X(7874) =  X(2)X(39)∩X(49)X(182)

Barycentrics    2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*c^4 : :
X(7874) = (pending)

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) =  X(2)X(6)∩X(76)X(5355)

Barycentrics    2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7875) = (pending)

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) = (pending)


X(7876) =  X(2)X(3)∩X(6)X(2896)

Barycentrics    2*a^2*b^2 + b^4 + 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7876) = (pending)

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) = (pending)


X(7877) =  X(69)X(83)∩X(76)X(524)

Barycentrics    -2*a^4 - a^2*b^2 + b^4 - a^2*c^2 + b^2*c^2 + c^4 : :
X(7877) = (pending)

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(7877) = (pending)


X(7878) =  X(2)X(5007)∩X(6)X(76)

Barycentrics    2*a^4 + 2*a^2*b^2 + 2*a^2*c^2 + b^2*c^2 : :
X(7878) = (pending)

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(7878) = (pending)


X(7879) =  X(3)X(147)∩X(4)X(3620)

Barycentrics    -a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7879) = (pending)

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(7879) = (pending)


X(7880) =  X(2)X(39)∩X(30)X(626)

Barycentrics    2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7880) = (pending)

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(7880) = X(32) of X(2)-Brocard triangle


X(7881) =  X(2)X(3933)∩X(3)X(147)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7881) = (pending)

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(7881) = (pending)


X(7882) =  X(5)X(3630)∩X(39)X(2896)

Barycentrics    -2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7882) = (pending)

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(7882) = (pending)


X(7883) =  X(2)X(32)∩X(3)X(6054)

Barycentrics    -a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7883) = (pending)

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(7883) = (pending)


X(7884) =  X(2)X(39)∩X(83)X(381)

Barycentrics    2 a^4 + 2 a^2*b^2 + 2*b^4 + 2 a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7884) = (pending)

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(7884) = (pending)


X(7885) =  X(2)X(3053)∩X(4)X(3314)

Barycentrics    -a^4 + 2*b^4 - b^2*c^2 + 2*c^4 : :
X(7885) = (pending)

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(7885) = (pending)


X(7886) =  X(2)X(39)∩X(5)X(2794)

Barycentrics    2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7886) = (pending)

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)


X(7887) =  X(2)X(3)∩X(32)X(625)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7887) = (pending)

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) = (pending)


X(7888) =  X(2)X(3108)∩X(32)X(325)

Barycentrics    a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + 2*c^4 : :
X(7888) = (pending)

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(7888) = (pending)


X(7889) =  X(2)X(32)∩X(39)X(698)

Barycentrics    2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7889) = (pending)

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(7889) = (pending)


X(7890) =  X(32)X(193)∩X(39)X(524)

Barycentrics    -2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4 : :
X(7890) = (pending)

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(7890) = (pending)


X(7891) =  X(2)X(1975)∩X(3)X(147)

Barycentrics    2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7891) = (pending)

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}

X(7891) = (pending)


X(7892) =  X(2)X(3)∩X(32)X(3314)

Barycentrics    2*a^4 + b^4 + b^2*c^2 + c^4 : :
X(7892) = (pending)

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) = (pending)


X(7893) =  X(6)X(2896)∩X(69)X(384)

Barycentrics    -2*a^4 + b^4 + b^2*c^2 + c^4 : :
X(7893) = (pending)

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(7893) = (pending)


X(7894) =  X(2)X(5041)∩X(6)X(76)

Barycentrics    -2*a^4 - 2*a^2*b^2 - 2*a^2*c^2 + b^2*c^2 : :
X(7894) = (pending)

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(7894) = (pending)


X(7895) =  X(2)X(5041)∩X(76)X(625)

Barycentrics    -(a^2*b^2) + 2*b^4 - a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7895) = (pending)

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(7895) = (pending)


X(7896) =  X(5)X(3631)∩X(32)X(3314)

Barycentrics    -a^4 + 2*b^4 + 2*b^2*c^2 + 2*c^4 : :
X(7896) = (pending)

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(7896) = (pending)


X(7897) =  X(2)X(6)∩X(76)X(625)

Barycentrics    -(a^2*b^2) + 2*b^4 - a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7897) = (pending)

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) =  X(2)X(187)∩X(69)X(148)

Barycentrics    -2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7898) = (pending)

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(7898) = (pending)


X(7899) =  X(2)X(32)∩X(76)X(2023)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7899) = (pending)

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(7899) = (pending)


X(7900) =  X(2)X(32)∩X(194)X(316)

Barycentrics    -2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7900) = (pending)

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)


X(7901) =  X(2)X(3)∩X(83)X(625)

Barycentrics    a^4 + 2*b^4 - b^2*c^2 + 2*c^4 : :
X(7901) = (pending)

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) = (pending)


X(7902) =  X(32)X(6655)∩X(315)X(5355)

Barycentrics    a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - 2*b^2*c^2 + 2*c^4 : :
X(7902) = (pending)

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(7902) = (pending)


X(7903) =  X(32)X(325)∩X(315)X(574)

Barycentrics    -a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + 2*c^4 : :
X(7903) = (pending)

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(7903) = (pending)


X(7904) =  X(3)X(147)∩X(76)X(543)

Barycentrics    -2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7904) = (pending)

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(7904) = (pending)


X(7905) =  X(2)X(5041)∩X(39)X(2896)

Barycentrics    -a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7905) = (pending)

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(7905) = (pending)


X(7906) =  X(2)X(3933)∩X(194)X(325)

Barycentrics    -2*a^2*b^2 + b^4 - 2*a^2*c^2 + b^2*c^2 + c^4 : :
X(7906) = (pending)

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}

X(7906) = (pending)


X(7907) =  X(2)X(3)∩X(76)X(620)

Barycentrics    2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7907) = (pending)

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) = (pending)


X(7908) =  X(2)X(5355)∩X(69)X(620)

Barycentrics    a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7908) = (pending)

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(7908) = (pending)


X(7909) =  X(2)X(3108)∩X(99)X(626)

Barycentrics    a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7909) = (pending)

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(7909) = (pending)


X(7910) =  X(2)X(3096)∩X(76)X(148)

Barycentrics    -2*a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7910) = (pending)

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(7910) = (pending)


X(7911) =  X(2)X(5206)∩X(4)X(3096)

Barycentrics    -a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7911) = (pending)

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(7911) = (pending)


X(7912) =  X(2)X(32)∩X(20)X(114)

Barycentrics    -(a^2*b^2) + 2*b^4 - a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7912) = (pending)

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(7912) = (pending)


X(7913) =  X(2)X(99)∩X(32)X(6656)

Barycentrics    a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 2*c^4 : :
X(7913) = (pending)

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(7913) = (pending)


X(7914) =  X(2)X(32)∩X(5)X(3098)

Barycentrics    a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7914) = (pending)

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(7914) = (pending)


X(7915) =  X(2)X(39)∩X(140)X(1503)

Barycentrics    2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7915) = (pending)

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) =  X(2)X(5368)∩X(69)X(5034)

Barycentrics    -a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + 2*c^4 : :
X(7916) = (pending)

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(7916) = (pending)


X(7917) =  X(20)X(99)∩X(69)X(576)

Barycentrics    -a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7917) = (pending)

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(7917) = (pending)


X(7918) =  X(4)X(5092)∩X(76)X(141)

Barycentrics    2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7918) = (pending)

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(7918) = (pending)


X(7919) =  X(2)X(99)∩X(83)X(3407)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7919) = (pending)

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(7919) = (pending)


X(7920) =  X(2)X(3933)∩X(6)X(5025)

Barycentrics    2*a^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7920) = (pending)

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(7920) = (pending)


X(7921) =  X(6)X(5025)∩X(39)X(3849)

Barycentrics    -2*a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 - b^2*c^2 + c^4 : :
X(7921) = (pending)

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(7921) = (pending)


X(7922) =  X(2)X(5007)∩X(76)X(115)

Barycentrics    2*b^4 + b^2*c^2 + 2*c^4 : :
X(7922) = (pending)

X(7922) lies on these lines:
{2, 5007}, {3, 6054}, {76, 115}, {315, 3972}, {325, 3096}, {2896, 3788}, {3090, 5476}, {3620, 5107}

X(7922) = (pending)


X(7923) =  X(2)X(1975)∩X(115)X(6704)

Barycentrics    a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7923) = (pending)

X(7923) lies on these lines:
{2, 1975}, {115, 6704}, {385, 2896}, {1656, 7709}, {2548, 3329}, {3096, 5309}, {3314, 5286}, {5068, 7710}

X(7923) = (pending)


X(7924) =  X(2)X(3)∩X(141)X(148)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7924) = (pending)

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) = anticomplement of X(6661)


X(7925) =  X(2)X(6)∩X(99)X(625)

Barycentrics    a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7925) = (pending)

X(7925) lies on these lines:
{2, 6}, {99, 625}, {114, 5999}, {140, 2896}, {148, 6390}, {316, 620}, {384, 3788}, {2549, 5025}

X(7925) = (pending)


X(7926) =  X(2)X(5008)∩X(76)X(5475)

Barycentrics    -2*a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7926) = (pending)

X(7926) lies on these lines:
{2, 5008}, {76, 5475}, {230, 6179}, {315, 7736}, {316, 2549}, {325, 3972}, {598, 3734}, {5025, 5355}

X(7926) = (pending)


X(7927) =  X(30)X(511)∩X(661)X(4808)

Barycentrics    (b - c)*(b + c)*(2*a^2 + b^2 + c^2) : :

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) =  X(2)X(3053)∩X(141)X(6655)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7928) = (pending)

X(7928) lies on these lines:
{2, 3053}, {141, 6655}, {315, 3329}, {316, 6292}, {384, 3096}, {385, 2896}, {3314, 3926}

X(7928) = (pending)


X(7929) =  X(2)X(32)∩X(69)X(698)

Barycentrics    -2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7929) = (pending)

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) =  X(2)X(39)∩X(98)X(3526)

Barycentrics    2*a^4 + 2*b^4 + b^2*c^2 + 2*c^4 : :
X(7930) = (pending)

X(7930) lies on these lines:
{2, 39}, {98, 3526}, {315, 1285}, {626, 3972}, {1352, 3525}, {2030, 3619}, {3314, 6179}

X(7930) = (pending)


X(7931) =  X(2)X(6)∩X(316)X(384)

Barycentrics    a^4 + 2*b^4 + b^2*c^2 + 2*c^4 : :
X(7931) = (pending)

X(7931) lies on these lines:
{2, 6}, {316, 384}, {3096, 3788}, {3186, 5094}, {3818, 5999}, {5092, 6054}, {6390, 6656}

X(7931) = (pending)


X(7932) =  X(2)X(39)∩X(147)X(3090)

Barycentrics    2*a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7932) = (pending)

X(7932) lies on these lines:
{2, 39}, {147, 3090}, {626, 5368}, {2896, 7735}, {3314, 5305}, {3552, 6680}, {5056, 6776}

X(7932) = (pending)


X(7933) =  X(2)X(3)∩X(115)X(3096)

Barycentrics    a^2*b^2 + 2*b^4 + a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7933) = (pending)

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) = (pending)


X(7934) =  X(2)X(187)∩X(76)X(115)

Barycentrics    2*b^4 - b^2*c^2 + 2*c^4 : :
X(7934) = (pending)

X(7934) lies on these lines:
{2, 187}, {5, 3096}, {76, 115}, {315, 6179}, {3545, 3619}, {3788, 6655}, {3815, 6656}

X(7934) = (pending)


X(7935) =  X(2)X(5206)∩X(4)X(6292)

Barycentrics    -a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + 2*c^4 : :
X(7935) = (pending)

X(7935) lies on these lines:
{2, 5206}, {4, 6292}, {32, 6656}, {315, 4045}, {382, 3763}, {574, 626}, {3096, 3734}

X(7935) = (pending)


X(7936) =  X(3)X(6054)∩X(76)X(148)

Barycentrics    -2*a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7936) = (pending)

X(7936) lies on these lines:
{3, 6054}, {76, 148}, {315, 7736}, {3096, 3972}, {5025, 5461}, {5306, 6179}

X(7936) = (pending)


X(7937) =  X(2)X(187)∩X(76)X(141)

Barycentrics    2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7937) = (pending)

X(7937) lies on these lines:
{2, 187}, {76, 141}, {315, 3618}, {2896, 6179}, {3314, 4045}, {5025, 6292}

X(7937) = (pending)


X(7938) =  X(2)X(32)∩X(5)X(6194)

Barycentrics    a^2*b^2 + 2*b^4 + a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7938) = (pending)

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) =  X(69)X(5025)∩X(315)X(384)

Barycentrics    -a^4 + 2*b^4 + b^2*c^2 + 2*c^4 : :
X(7939) = (pending)

X(7939) lies on these lines: {69, 5025}, {315, 384}, {325, 2896}, {385, 626}, {3096, 3329}, {3933, 6655}

X(7939) = (pending)


X(7940) =  X(2)X(39)∩X(140)X(3096)

Barycentrics    2*a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7940) = (pending)

X(7940) lies on these lines: {2, 39}, {140, 3096}, {325, 6179}, {620, 5025}, {625, 3552}, {3533, 3619}

X(7940) = (pending)


X(7941) =  X(325)X(384)∩X(626)X(3329)

Barycentrics    -a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 - b^2*c^2 + 2*c^4 : :
X(7941) = (pending)

X(7941) lies on these lines: {325, 384}, {626, 3329}, {2548, 3314}, {2896, 3815}, {5025, 5286}, {6390, 6658}

X(7941) = (pending)


X(7942) =  X(2)X(39)∩X(98)X(1656)

Barycentrics    2*a^4 + 2*b^4 - b^2*c^2 + 2*c^4 : :
X(7942) = (pending)

X(7942) lies on these lines: {2, 39}, {98, 1656}, {230, 3096}, {626, 6179}, {1352, 5067}, {3972, 5025}

X(7942) = (pending)


X(7943) =  X(2)X(39)∩X(2030)X(5207)

Barycentrics    2*a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7943) = (pending)

X(7943) lies on these lines: {2, 39}, {2030, 5207}, {3090, 3818}, {3096, 6179}, {3972, 6656}

X(7943) = (pending)


X(7944) =  X(2)X(32)∩X(99)X(6656)

Barycentrics    a^4 + a^2*b^2 + 2*b^4 + a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7944) = (pending)

X(7944) lies on these lines: {2, 32}, {99, 6656}, {140, 6033}, {1916, 3934}, {5103, 5104}

X(7944) = (pending)


X(7945) =  X(2)X(39)∩X(147)X(631)

Barycentrics    2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7945) = (pending)

X(7945) lies on these lines: {2, 39}, {147, 631}, {620, 3096}, {626, 3552}, {2030, 3620}

X(7945) = (pending)


X(7946) =  X(2)X(5007)∩X(194)X(315)

Barycentrics    -2*a^4 - a^2*b^2 + 2*b^4 - a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7946) = (pending)

X(7946) lies on these lines: {2, 5007}, {194, 315}, {524, 5025}, {626, 5368}, {754, 3552}

X(7946) = anticomplement of X(6179)


X(7947) =  X(2)X(3933)∩X(325)X(384)

Barycentrics    a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7947) = (pending)

X(7947) lies on these lines: {2, 3933}, {325, 384}, {385, 3788}, {3926, 5025}, {6390, 6655}

X(7947) = (pending)


X(7948) =  X(2)X(3)∩X(385)X(3096)

Barycentrics    a^4 + 2*a^2*b^2 + 2*b^4 + 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7948) = (pending)

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(7948) = (pending)


X(7949) =  X(76)X(5475)∩X(315)X(7738)

Barycentrics    -2*a^4 - 2*a^2*b^2 + 2*b^4 - 2*a^2*c^2 + b^2*c^2 + 2*c^4 : :
X(7949) = (pending)

X(7949) lies on these lines: {76, 5475}, {315, 7738}, {325, 6179}

X(7949) = (pending)


X(7950) =  X(30)X(511)∩X(4770)X(4808)

Barycentrics    (b - c)*(b + c)*(a^2 + 2*b^2 + 2*c^2) : :

X(7950) lies on these lines: {30, 511}, {4770, 4808}

X(7950) = isogonal conjugate of X(7954)


X(7951) =  X(1)X(5)∩X(2)X(36)

Trilinears    1 + 2 cos(B - C) : :

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(7951) = X(5397)-cevaconjugate of X(1)
X(7951) = homothetic center of 4th Euler triangle and reflection triangle of X(1)
X(7951) = homothetic center of 2nd isogonal triangle of X(1) and medial triangle; see X(36)
X(7951) = homothetic center of ABC and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(7951) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,7741), (1,5587,80), (2,1478,36), (2,5080,993), (4,498,35), (4,5218,4302), (5,12,1), (5,119,5587), (5,495,11), (5,496,7173), (10,908,5692), (11,12,495), (11,495,1), (12,3614,5), (46,1698,5445), (55,381,3583), (79,5445,46), (140,7354,7280), (226,1737,5902), (382,5217,4324), (388,499,5563), (388,3090,499), (442,1329,1698), (498,4302,5218), (631,5229,4299), (1479,3085,3746), (1532,7680,1699), (3085,3091,1479), (3583,3584,55), (3616,5154,3825), (3814,3822,2), (3838,5123,3753), (4302,5218,35), (5056,5261,3086), (5219,5587,1), (5252,5886,1), (5587,6326,355)


X(7952) =  CENTER OF INVERSE-IN-POLAR-CIRCLE OF INCIRCLE

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) : :
Barycentrics    (tan A)(cos B + cos C - cos A - 1) : :

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(7952) = complement of X(280)
X(7952) = X(i)-cevapoint of X(j) for these {i,j}: {1,1720}, {198,3195}
X(7952) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,281), (318,4), (342,196), (1309,6087), (1895,3176)
X(7952) = X(2)-crosspoint of X(347)
X(7952) = X(i)-cross conjugate of X(j) for these (i,j): (198,329), (208,4), (227,40), (2331,196)
X(7952) = X(i)-crosssum of X(j) for these {i,j}: {6,2192}, {1364,1459}
X(7952) = X(i)-complementary conjugate of X(j) for these (i,j): (31,281), (40,1329), (56,946), (198,3452), (208,5), (221,10), (223,141), (227,3454), (347,2887), (604,57), (1106,3086), (1397,1108), (1402,1901), (2187,9), (2199,2), (2360,960), (3209,226), (6129,124), (6611,142), (7013,1368), (7114,3)
X(7952) = X(i)-isoconjugate of X(j) for these (i,j): (1,1433), (3,84), (7,2188), (48,189), (56,271), (57,268), (63,1436), (69,2208), (73,285), (77,2192), (78,1413), (184,309), (212,1440), (219,1422), (222,282), (280,603), (326,7151), (348,7118), (394,7129), (1256,7078), (1444,2357), (1790,1903), (1804,7008), (2358,6514), (3692,6612), (7003,7125), (7020,7335), (7154,7183), (7177,7367)
X(7952) = polar conjugate of X(189)
X(7952) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1785,4), (4,1068,278), (33,225,4), (37,393,281), (158,3085,281), (1123,1336,281)
X(7952) = pole wrt polar circle of trilinear polar of X(189) (the line X(522)X(905))
X(7952) = X(48)-isoconjugate (polar conjugate) of X(189)


X(7953) =  ISOGONAL CONJUGATE OF X(7927)

Barycentrics    a2/[(b2 - c2)(2a2 + b2 + c2)] : :

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)
X(7953) = center of bianticevian conic of X(1) and X(38)


X(7954) =  ISOGONAL CONJUGATE OF X(7950)

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)


leftri  Centers associated with mixtilinear triangles: X(7955) - X(7999)  rightri

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)

Next (T,i) means that A5B5C5 is endo-homothetic to T with endo-homethetic center X(i): (anticomplementary,8), (Euler,7967), (inner Grebe, 7968), (outer Grebe, 7969), (Johnson, 952), (medial, 145)

Next, (T,i,j) means that A5B5C5 is orthologic to T and the orthologic centers are X(i) and X(j): (ABC,944,4), (anticomplementary,944,20, (1st Brocard,7970), (circumorthic,1482,4), (1st circumperp,1,3), (2nd circumperp,1,3), (Euler,944,4), (excentral,1,40), (extangents,1482,40), (extouch,7971,40), (Fuhrmann,7972,3), (inner Grebe,944,5871), (outer Grebe,944,5870), (hexyl,1,1), (intangents,1482,1), (intouch,1,1), (Johnson,944,3), (medial,944,3), (midheight,7973,4), (inner Napoleon,7974,3), (outer Napleon,7975,3), (Neuberg,7976,3), (2nd Neuberg,7977,3), (orthic,1482,4), (orthocentroidal,7978,4), (reflection 7979,4), (tangential,1482,3), (inner Vecten,7980,3), (outer Vecten,7981,3), (2nd Euler,1482,3), (3rd Euler,1,5), (4th Euler,1,5), (2nd extouch,1,4), (3rd extouch,7982,4)

A5B5C5 is parallelogic to the 1st Brocard triangle with centers X(7983) and X(6), and parallelogic to the orthocentroidal triangle with centers X(7984) and X(2).

A5B5C5 is inversely similar to the 1st Brocard triangle with center X(7985) of similitude, and also inversely similar to the orthocentroidal triangle with center X(7986) of similitude.


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.

In the next list, the appearance of (T,i) means that A6B6C6 is perspective to T and the perspector is X(i), and (T,i)* means that A6B6C6 is homothetic to T: (ABC,3062), (1st circumperp,165)*, (2nd circumperp,7987)*, (excentral,165)*, (hexyl,1)*, (intangents, 1), (intouch,1)*, (3rd Euler,7988)*, (4th Euler,7989)*, (2nd extouch,1750)*, (4th mixtilinear,165), (5th mixtilinear,7990)

In the next list, the appearance of (T,i) means that A6B6C6 is orthologic to T and the orthologic centers are X(i) and X(j): (ABC,1,1), (anticomplementary,1,8), (1st circumperp,7991,40), (2nd circumperp,7991,1), (Euler,1,946), (excentral,7991,1), (extouch,7992,72), (Fuhrmann,7993,8), (inner Grebe,1,3641), (outer Grebe,1,3640), (hexyl,7991,40), (incentral,40,1), (intouch,7991,65), (Johnson,1,355), (medial,1,10), (midarc,167,1), (mixtilinear, 7994,1), (2nd mixtilinear,7994,1), (5th mixtilinear,1,1), (3rd Euler,7991,946), (4th Euler,7991,10), (2nd extouch,7991,72), (4th extouch,7995,65), (5th extouch,7996,65)

A6B6C6 is parallelogic to the Fuhrmann triangle with centers X(5531) and X(4).

A6B6C6 is inversely similar to the Fuhrmann triangle with center X(7997) of similitude.

In the next list, the appearance of (T,i) means that A6B6C6 is endo-homothetic to T with endo-homethetic center X(i): (1st circumperp,2), (2nd circumperp,631), (excentral,2), (hexyl,3), (intouch,3), (3rd Euler,7998), (4th Euler,7999), (2nd extouch,394)

7th mixtilinear triangle: (See X(8916)

underbar

X(7955) =  CENTER OF CONIC THROUGH TOUCHPOINTS OF MIXTILINEAR EXCIRCLES

Trilinears    (2*sin(A/2)*(6-19*sin(A/2)^2+4*sin(A/2)^4)*cos((B-C)/2)-2*sin(A/2)*(­2+sin(A/2)^2)*cos(3*(B-C)/2) +(cos(A)^2-8*cos(A)-9)*cos(B-C)+cos(A)*(cos(A)^2+15-4*cos(A))+4)*tan(A/2)
Trilinears    (4*R+r)*(2*(4*R+r)*R*r-(4*R-r)*b*c)/(s-a)+s*(4*R-r)^2

X(7955) lies on these lines: (220,2124), (223,3160)


X(7956) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 3rd EULER AND MIXTILINEAR

Barycentrics    (b^2+c^2)*a^4+2*(c^2-b^2)*(b-c)*a^3+6*b*c*(b-c)^2*a^2+2*(b^2-c^2)*(b­c)*a*(b^2-4*b*c+c^2)-(b^2-c^2)^2*(b-c)^2 ::
X(7956) lies on these lines:
(2,6244), (4,496), (5,10), (11,57), (118,2810), (119,3577), (226,1538), (329,5817), (495,1532), (516,3816), (527,3829), (956,6957), (962,4187), (1484,3845), (2093,7741), (2095,5789), (3091,3421), (3295,6848), (3359,6922), (4915,5587), (5193,7354), (5687,6953), (5704,6831)

X(7956) = complement of X(6244)
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(7957) =  PERSPECTOR OF THESE TRIANGLES: 2nd MIXTILINEAR AND EXTANGENTS

Trilinears    2*sin(A)*cos(A/2)*cos((B-C)/2)+(-cos(A)+1)*cos(B-C)-3*cos(A)-1 ::
X(7957) = (3*R + r)*X(1) - (4R + r)*X(3)
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) =  PERSPECTOR OF THESE TRIANGLES: 2nd MIXTILINEAR AND FEUERBACH

Trilinears    3*sin(A/2)*cos((B-C)/2)+sin(A/2)*cos(3*(B-C)/2)+(cos(A)+3)*cos(B­C)+cos(A)-1 ::

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd MIXTILINEAR AND MIDHEIGHT

Trilinears    8*(3*sin(A/2)-sin(3*A/2))*cos((B-C)/2)+2*(4*cos(A)+cos(2*A)+3)*cos(B­C)-9*cos(A)+cos(3*A)-8 ::
X(7959) = (4*R^2-s^2)*X(1)+(4*R*r+r^2)*X(84)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd MIXTILINEAR AND 1st MIXTILINEAR

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

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st MIXTILINEAR AND 2nd MIXTILINEAR

Barycentrics    a^5-3*(b+c)*a^4+4*b*c*a^3+4*(b+c)*(b^2-3*b*c+c^2)*a^2­(b^2+6*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3::

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON INTOUCH AND 1st MIXTILINEAR

Trilinears    : (-a+b+c)*(a^2-2*a*(b+c)-3*(b-c)^2) : :
X(7962) = (2R - 3r)*X(1) + 2r*X(3)

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) =  PERSPECTOR OF THESE TRIANGLES: 3rd. MIXTILINEAR AND EXCENTRAL

Trilinears    5*a^3-5*a^2*(b+c)-(9*b^2-14*b*c+9*c^2)*a+(b+c)^3 ::

X(7963) lies on these lines:
(1,474), (36,2956), (56,1743), (106,6765), (165,2943), (1046,3361), (3616,4859)


X(7964) =  PERSPECTOR OF THESE TRIANGLES: 4th MIXTILINEAR AND EXTANGENTS

Trilinears    2*a^5-3*(b+c)*a^4-2*(b+c)^2*a^3+4*(b+c)*(b^2+c^2)*a^2-(b^2-c^2)*(b-c)^3
X(7964) = (-3R - r)*X(1) + (8R + 3r)*X(3)

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) =  PERSPECTOR OF THESE TRIANGLES: 4th MIXTILINEAR AND FEUERBACH

Trilinears    4*sin(A/2)*cos((B-C)/2)^3+(cos(A)+3)*cos(B-C)-3*cos(A)-1 ::

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) =  PERSPECTOR OF THESE TRIANGLES: 5th MIXTILINEAR AND HEXYL

Trilinears    16*sin(A/2)*cos((B-C)/2)+(cos(A)-3)*cos(B-C)+9*cos(A)+1/2*cos(2*A)­15/2 : :
X(7966) = (6*R-r)*X(1)-6*R*X(2)+(4*R+2*r)*X(3)

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th. MIXTILINEAR AND EULER

Trilinears    5*cos(A)+4*cos(B)+4*cos(C)-cos(B-C)-4 : :
X(7967) = 4*X(1)-3*X(2)+2*X(3)

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(7968) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND INNER GREBE

Trilinears    a^2+(b+c)*a-2*S ::
X(7968) = S*X(1) - SωX(6)

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND OUTER GREBE

Trilinears    a^2+(b+c)*a+2*S : :
X(7969) = S*X(1) + SωX(6)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 1st BROCARD

Barycentrics    a^8-2*(b+c)*a^7+(-3*b^2+4*b*c-3*c^2)*a^6+4*(b^3+c^3)*a^5 +(3*b^4-4*b^3*c+3*b^2*c^2-4*b*c^3+3*c^4)*a^4-2*(b+c)*(2*b^4­2*b^3*c+b^2*c^2-2*b*c^3+2*c^4)*a^3 +(-b^4+2*b^3*c+4*b^2*c^2+2*b*c^3-c^4)*(b-c)^2*a^2+2*(b^2-c^2)*(b­c)*(b^4-b^2*c^2+c^4)*a +(b^2-c^2)^2*b^2*c^2

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)


X(7971) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND EXTOUCH

Trilinears    (-24*sin(A/2)+8*sin(3*A/2))*cos((B-C)/2)+(-6*cos(A)+2)*cos(B-C)­6*cos(A)+cos(2*A)+9
X(7971) = (2R + 3r)*X(1) - 6R*X(2) + (4R - 2r)*X(3)

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) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND FUHRMANN

Trilinears    4*cos(B)*cos(C)-4*cos(A)-6*cos(B)-6*cos(C)+7
X(7972) = (R -6r)*X(1) + 6r*X(2) - 2r*X(3)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND MIDHEIGHT

Trilinears    4*(3*sin(A/2)-2*sin(3*A/2)+sin(5*A/2))*cos((B-C)/2)­8*sin(A/2)*cos(A)*cos(3*(B-C)/2)+2*(4*cos(A)+cos(2*A)+3)*cos(B-C)+cos(3*A)­9*cos(A)-4*cos(2*A)-4

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND INNER NAPOLEON

Trilinears    -sqrt(3)*(4*sin(A)*cos(A/2)*cos((B-C)/2)-4*cos(A)*cos(B-C)+cos(2*A)-5) +(2*cos(A/2)+2*sin(3*B/2+3*C/2))*cos((B-C)/2)-8*sin(A)*cos(B-C)­7*sin(2*A)+4*cos(A/2)*cos(3*(B-C)/2) Reflection of: (14/1)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND OUTER NAPOLEON

Trilinears    sqrt(3)*(4*sin(A)*cos(A/2)*cos((B-C)/2)-4*cos(A)*cos(B-C)+cos(2*A)-5) +(2*cos(A/2)+2*sin(3*B/2+3*C/2))*cos((B-C)/2)-8*sin(A)*cos(B-C)­7*sin(2*A)+4*cos(A/2)*cos(3*(B-C)/2)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 1st NEUBERG

Barycentrics    2*(b^2+c^2)*a^3-b^2*c^2*(-a+b+c)

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(7977) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 2nd NEUBERG

Barycentrics    a^5-(b+c)*a^4+5*(b^2+c^2)*a^3­(b+c)*(b^2+c^2)*a^2+(2*b^2+c^2)*(2*c^2+b^2)*a-b^2*c^2*(b+c)

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(7978) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND ORTHOCENTROIDAL

Trilinears    4*(4*sin(A/2)-2*sin(3*A/2)+sin(5*A/2))*cos((B-C)/2)-8*sin(A/2)*cos(A)*cos(3*(B­C)/2)+(8*cos(A)+1)*cos(B-C) +cos(3*A)-4*cos(2*A)-6

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th. MIXTILINEAR AND REFLECTION

Trilinears    4*(-2*sin(3*A/2)+sin(5*A/2))*cos((B-C)/2)-8*sin(A/2)*cos(A)*cos(3*(B­C)/2)+(8*cos(A)-4*cos(2*A)-3)*cos(B-C) +cos(3*A)-4*cos(2*A)+2

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th. MIXTILINEAR AND INNER VECTEN

Trilinears    (-2*cos(A/2)+2*sin(A/2)+2*sin(3*A/2))*cos((B-C)/2)-2*cos(A/2)*cos(3*(B-C)/2)+(­4*cos(A)+6*sin(A))*cos(B-C) +cos(2*A)+4*sin(2*A)-5

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND OUTER VECTEN

Trilinears    (2*cos(A/2)+2*sin(A/2)+2*sin(3*A/2))*cos((B-C)/2)+2*sin(B/2+C/2)*cos(3*(B­C)/2)+(-4*cos(A)-6*sin(A))*cos(B-C) +cos(2*A)-4*sin(2*A)-5

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 3rd EXTOUCH

Trilinears    a^3-3*a^2*(b+c)-(b^2-6*b*c+c^2)*a+3*(b^2-c^2)*(b-c)
X(7982) = 3X(1) - 2X(3)

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) =  PARALLELOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND 1st BROCARD

Barycentrics    a^5-(b+c)*a^4­(b^2+c^2)*a^3+(b+c)*(b^2+c^2)*a^2+(2*c^4+2*b^4-3*b^2*c^2)*a­b^2*c^2*(b+c)

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(7983) = midpoint of X(145) and X(148)
X(7983) = reflection of X(i) in X(j) for these (i,j): (8,115), (99,1)


X(7984) =  PARALLELOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND ORTHOCENTROIDAL

Trilinears    a^6-(b+c)*a^5-(b^2+c^2)*a^4+(b+c)*(b^2+c^2)*a^3+(c^2­2*b^2)*(b^2-2*c^2)*a^2-b^2*c^2*(b+c)*a +(b^2-c^2)^2*(2*c^2+2*b^2)

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) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 5th MIXTILINEAR AND 1st BROCARD

Barycentrics    3*(b^2+c^2)*a^3+(-b^2+c^2)*(b-c)*a^2+(-2*b^4+b^2*c^2­2*c^4)*a+b*c*(b+c)*(2*c^2-b*c+2*b^2)

X(7985) lies on this line: (726,3241)


X(7986) =  CENTER OF SIMILITUDE OF THESE TRIANGLES: 5th. MIXTILINEAR AND ORTHOCENTROIDAL

Trilinears    (10*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+4*cos(A)*cos(B-C)-cos(2*A)-1

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)


X(7987) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 2nd CIRCUMPERP

Trilinears    5*a^3-(b+c)*a^2+(2*b*c-5*c^2-5*b^2)*a+(b^2-c^2)*(b-c)
Trilinears    5 cos A + cos B + cos C - 1 : : (César Lozada, ADGEOM #209, 6/19/2013)
X(7987) = X(1) + 4X(3)

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)

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(7988) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 3rd EULER

Barycentrics    a^3-2*(b+c)*a^2-3*(b-c)^2*a+4*(b^2-c^2)*(b-c) Trilinears: r/R+4*cos(B-C)
X(7988) = X(1) + 12X(2) - 4X(3) = X(1) + 8X(5)

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 4th EULER

Barycentrics    a^4-(b+c)*a^3+(3*c^2+2*b*c+3*b^2)*a^2+(b^2-c^2)*(b-c)*a­4*(b^2-c^2)^2 Trilinears: r/R-4*cos(B-C)
X(7989) = X(1) + 12X(2) + 4X(3) = X(1) - 8X(5)

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) =  PERSPECTOR OF THESE TRIANGLES: 5th. MIXTILINEAR AND 6th. MIXTILINEAR

Trilinears    (54*sin(A/2)-2*sin(3*A/2))*cos((B-C)/2)+(3*cos(A)-11)*cos(B­C)+33*cos(A)+(cos(2*A)-51)/2
X(7990) = (5R - 3r/4)*X(1) - 6R*X(2) + (4R + r)*X(3)

X(7990) lies on these lines:
(1,5806), (100,4853), (145,4297), (165,2136), (390,3062), (952,4866), (2951,3243), (3339,5882)


X(7991) =  ORHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 1st CIRCUMPERP

Trilinears    a^3+3*(b+c)*a^2-(b^2+6*b*c+c^2)*a-3*(b^2-c^2)*(b-c)
X(7991) = -3X(1) + 4X(3)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND EXTOUCH

Trilinears    (12*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)+(6*cos(A)+2)*cos(B-C)­6*cos(A)+cos(2*A)-3
X(7992) = (R + 3r/4)*X(1) - 3R*X(2) + (2R - r)*X(3)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND FUHRMANN

Trilinears    (-30*sin(A/2)+4*sin(3*A/2))*cos((B-C)/2)+(-2*cos(A)+6)*cos(B-C)­15*cos(A)+cos(2*A)+14
X(7993) = (9R - 2r)*X(1) - 12R*X(2) + 4R*X(3) = (9R - 2r)*X(1) - 8R*X(5)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 1st MIXTILINEAR

Trilinears    a^5-3*(b+c)*a^4+2*(b^2+8*b*c+c^2)*a^3+2*(b^2-6*b*c+c^2)*(b+c)*a^2­3*(b^2-c^2)^2*a+(b^2-c^2)*(b-c)^3
X(7994) = (-6R - r)*X(1) + 8R*X(3)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 4th EXTOUCH

Trilinears    (20*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)+(6*cos(A)+2)*cos(B-C)­2*cos(A)+cos(2*A)-7
X(7995) = (-2R - 3r)*X(1) + 12R*X(2) + (-8R + 4r)*X(3)

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 5th EXTOUCH

Trilinears    a^4+2*(b+c)*a^3-2*(b^2-3*b*c+c^2)*a^2+2*(b+c)*(b^2-4*b*c+c^2)*a­(3*b^2+8*b*c+3*c^2)*(b-c)^2

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) =  CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: 6th. MIXTILINEAR AND FUHRMANN

Trilinears    (2*sin(A/2)-4*sin(3*A/2))*cos((B-C)/2)+(6*cos(A)+10)*cos(B-C)­7*cos(A)+cos(2*A)+2

X(7997) lies on these lines:
(165,7701), (355,3627), (1006,1490), (3339,7702), (5538,5777), (5557,7741)


X(7998) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 3rd EULER

Trilinears    (-(b^2+c^2)*a^2+b^4+3*b^2*c^2+c^4)*a

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(7999) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 4th EULER

Trilinears    (2*cos(2*A)+3)*cos(B-C)+3*cos(A) : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 5th MIXTILINEAR AND HUTSON EXTOUCH

Trilinears    8*(5*sin(A/2)+sin(3*A/2))*cos((B-C)/2)+2*(-3*cos(A)+1)*cos(B-C)+10*cos(A)+cos(2*A)-7 : :
X(8000) = (6R + r)*X(1) - 2R*X(5920)

X(8000) lies on these lines: (pending)


X(8001) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND HUTSON EXTOUCH

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: (pending)



leftri  Centers associated with the Stammler triangle: X(8002) -X(8011)  rightri

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).


underbar

X(8002) =  HOMOTHETIC CENTER OF THESE TRIANGLES: STAMMLER AND 1st MORLEY

Trilinears    cos A - 2 cos(A/3) - 4 cos(B/3) cos(C/3) : :

X(8002) lies on these lines: (3,356), (357,3280), (358,8003), (1135,8004), (3605,8005)


X(8003) =  HOMOTHETIC CENTER OF THESE TRIANGLES: STAMMLER AND 2nd MORLEY

Trilinears    cos A + 2 cos(A/3 + π/3) - 4 cos(B/3 + π/3) cos(C/3 + π/3) : :

X(8003) lies on these lines: (3,3276), (358,8002), (1137,8004), (3606,8006)


X(8004) =  HOMOTHETIC CENTER OF THESE TRIANGLES: STAMMLER AND 3rd MORLEY

Trilinears    cos A + 2 cos(A/3 - π/3) - 4 cos(B/3 - π/3) cos(C/3 - π/3) : :

X(8004) lies on these lines: (3,3277), (1135,8002), (1137,8003), (3607,8007)


X(8005) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: STAMMLER TO 1st MORLEY-ADJUNCT

Trilinears    af(A,B,C) + bg(A,B,C) + cg(A,C,B) : : , where
f(A,B,C) = (cos A)(-2 cos(B/3) cos(C/3) + cos A + 2 cos(A/3))
g(A,B,C) = (cos A)(cos B + 2 cos(A/3)*cos(C/3) - 2 cos(B/3)) - (cos B)(4 cos(B/3) cos(C/3) - 4 cos(A/3))

X(8005) lies on these lines: (3,3276), (3605,8002)

X(8005) = reflection of X(3) in X(3281)


X(8006) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: STAMMLER TO 2nd MORLEY-ADJUNCT

Trilinears    af(A,B,C) + bg(A,B,C) + cg(A,C,B) : : , where
f(A,B,C) = (cos(A)-2*cos((A+π/3)-2*cos((B+π)/3)*cos((C+π)/3))*cos(A)
g(A,B,C) = cos(A)*(cos(B)+2*cos((B+π)/3)+2*cos((A+π)/3)*cos((C+π)/3))-(4*cos((A+π)/3)+4*cos((B+π)/3)*cos((C+π)/3))*cos(B)

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)


X(8007) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: STAMMLER TO 3rd MORLEY-ADJUNCT

Trilinears    af(A,B,C) + bg(A,B,C) + cg(A,C,B) : : , where
f(A,B,C) = (cos(A)-2*cos((A-π/3)-2*cos((B-π)/3)*cos((C-π)/3))*cos(A)
g(A,B,C) = cos(A)*(cos(B)+2*cos((B-π)/3)+2*cos((A-π)/3)*cos((C-π)/3))-(4*cos((A-π)/3)+4*cos((B-π)/3)*cos((C-π)/3))*cos(B)

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) =  PARALLELOGIC CENTER OF THESE TRIANGLES: STAMMLER TO INNER NAPOLEON

Trilinears    -2*(2*cos(2*A + π/3) + 1)*cos(B - C) + 8*(cos(2*A + 2*π/3) + 1)*cos((B - C)/3) + 16*cos(A + π/3)*cos(2*(B - C)/3) + 4*cos(5*(B - C)/3) - 5*cos(A) - sqrt(3)*cos(A + π/2) - 2*cos(3*A + 2*π/3) : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: STAMMLER TO INNER NAPOLEON

Trilinears    -au cos B cos C + (bv + cw)cos A : : , where u : v : w = X(8008)

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) =  PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO STAMMLER

Trilinears    -(2*(2*q^2-1)*(16*q^4-16*q^2+1)*cos(A)-(80*q^4-120*q^2+43)*cos(A+Pi/3) +(48*q^4-72*q^2+25)*cos(A+2*Pi/3)-(8*(q^2-1)*q-sqrt(3)*(4*q^2-1)*cos(A+Pi/2))*(4*q^2-3)^2)/(4*q^2-3)^2 : : , where q = cos(B/3 - C/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(8009) = reflection of X(8011) in X(2)


X(8011) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-NAPOLEON TO STAMMLER

Trilinears    (-au/2 + bv + cw)/a : : , where u : v: w = X(8010)

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)



leftri  Danneels points: X(8012) -X(8042)  rightri

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)


underbar

X(8012) =  DANNEELS POINT OF X(9)

Barycentrics    a^2 (a-b-c)^2 (a b-b^2+a c+2 b c-c^2) : :

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) =  DANNEELS POINT OF X(10)

Barycentrics    (b+c)^2 (2 a+b+c) : :

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) =  DANNEELS POINT OF X(13)

Barycentrics    (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4+2 Sqrt(3) a^2 S)^2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4-2 Sqrt(3) b^2 S-2 Sqrt(3) c^2 S) : :

X(8014) lies on these lines: {2, 13}, {51, 512}, {2153, 7126}

X(8014) = X(13)-Ceva conjugate of X(396)


X(8015) =  DANNEELS POINT OF X(14)

Barycentrics    (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-2 Sqrt(3) a^2 S)^2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4+2 Sqrt(3) b^2 S+2 Sqrt(3) c^2 S) : :

X(8015) lies on these lines: {2, 14}, {51, 512}

X(8015) = X(14)-Ceva conjugate of X(395)


X(8016) =  DANNEELS POINT OF X(15)

Barycentrics    a^4 (Sqrt(3) a^2-Sqrt(3) b^2-Sqrt(3) c^2-2 S)^2 (Sqrt(3) a^2 b^2-Sqrt(3) b^4+Sqrt(3) a^2 c^2+2 Sqrt(3) b^2 c^2-Sqrt(3) c^4+2 b^2 S+2 c^2 S) : :

X(8016) lies on this line: {2, 14}


X(8017) =  DANNEELS POINT OF X(16)

Barycentrics    a^4 (Sqrt(3) a^2-Sqrt(3) b^2-Sqrt(3) c^2+2 S)^2 (Sqrt(3) a^2 b^2-Sqrt(3) b^4+Sqrt(3) a^2 c^2+2 Sqrt(3) b^2 c^2-Sqrt(3) c^4-2 b^2 S-2 c^2 S) : :

X(8017) lies on this line: {2, 13}


X(8018) =  DANNEELS POINT OF X(17)

Barycentrics    (a^2+2 Sqrt(3) S) (a^2+b^2-c^2+2 Sqrt(3) S) (a^2-b^2+c^2+2 Sqrt(3) S) : :

X(8018) lies on these lines: {2, 17}, {7603, 8019}


X(8019) =  DANNEELS POINT OF X(18)

Barycentrics    (a^2-2 Sqrt(3) S) (a^2+b^2-c^2-2 Sqrt(3) S) (a^2-b^2+c^2-2 Sqrt(3) S) : :

X(8019) lies on these lines: {2, 18}, {7603, 8018}


X(8020) =  DANNEELS POINT OF X(19)

Barycentrics    a^2 (b+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2+b^2-2 b c+c^2) : :

X(8020) lies on these lines: {2, 19}, {756, 862}, {1851, 2082}, {2201, 5276}, {2489, 8034}

X(8020) = X(332)-isoconjugate of X(7131)


X(8021) =  DANNEELS POINT OF X(21)

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

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) =  DANNEELS POINT OF X(31)

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

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) =  DANNEELS POINT OF X(32)

Barycentrics    a8 (b4+c4) : :

X(8023) lies on these lines: {2, 32}, {710, 8039}


X(8024) =  DANNEELS POINT OF X(76)

Barycentrics    b2c2(b2 + c2) : :
Barycentrics    |AP(1)|^2 + |AU(1)|^2 : :

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) =  DANNEELS POINT OF X(86)

Barycentrics    (a + b)(a + c)(2a + b + c) : :

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) =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) =  DANNEELS POINT OF X(192)

Barycentrics    bc(bc - ab - ac)2 : :

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) =  DANNEELS POINT OF X(513)

Barycentrics    (ab - ac)3 : :
X(8027) = 5X(649) - 2X(4507) = X(2978) + 2X(4790)

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) = 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) =  DANNEELS POINT OF X(519)

Barycentrics    (2a - b - c)3 : :
X(8028) = 2X(2) - 3X(1644) = 4X(2) - 3X(1647)

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) =  DANNEELS POINT OF X(523)

Barycentrics    (b2 - c2)3 : :
X(8029) = 4X(2) - 3X(1649) = X(669) - 4X(2501) = 4X(850) - X(2528) = X(1649) - 4X(5466) = X(2) - 3X(5466)

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) = 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(8030) =  DANNEELS POINT OF X(524)

Barycentrics    (2a2 - b2 - c2)3 : :
X(8030) = 2 X(2) - 3 X(1641) = 4 X(2) - 3 X(1648) = X(1648) - 4 X(5468) = X(2) - 3 X(5468)

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) =  DANNEELS POINT OF X(536)

Barycentrics    (ab + ac - 2bc)3 : :
X(8031) = 4 X(2) - 3 X(1646)

X(8031) lies on this line: {2,37}

X(8031) = barycentric cube of X(536)


X(8032) =  DANNEELS POINT OF X(752)

Barycentrics    (2a3 - b3 - c3)3 : :

X(8032) lies on this line: {2,31}

X(8032) = barycentric cube of X(752)


X(8033) =  DANNEELS POINT OF X(799)

Barycentrics    bc(a + b)(a + c) (a2 + bc) : :

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) =  DANNEELS POINT OF X(876)

Barycentrics    a2(b + c)(b - c)3 : :

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) =  DANNEELS POINT OF X(485)

Barycentrics    (a^2+2 S) (a^2+b^2-c^2+2 S) (a^2-b^2+c^2+2 S) : :

X(8035) lies on these lines: {2, 372}, {5475, 8036}

X(8035) = X(485)-Ceva conjugate of X(590)


X(8036) =  DANNEELS POINT OF X(486)

Barycentrics    (a^2-2 S) (a^2+b^2-c^2-2 S) (a^2-b^2+c^2-2 S) : :

X(8036) lies on these lines: {2, 371}, {5475, 8035}

X(8036) =X(486)-Ceva conjugate of X(615)


X(8037) =  DANNEELS POINT OF X(1131)

Barycentrics    (a^2+S) (a^2+b^2-c^2+S) (a^2-b^2+c^2+S) : :

X(8037) lies on this line: {2, 490}

X(8037) =X(1131)-Ceva conjugate of X(3068)


X(8038) =  DANNEELS POINT OF X(1132)

Barycentrics    (a^2-S) (a^2+b^2-c^2-S) (a^2-b^2+c^2-S) : :

X(8038) lies on this line: {2,489}

X(8038) = X(1132)-Ceva conjugate of X(3069)


X(8039) =  DANNEELS POINT OF X(1502)

Barycentrics    b4c4(b4 + c4) : :

X(8039) lies on these lines: {2, 308}, {710, 8023}

X(8039) = X(1502)-Ceva conjugate of X(626)


X(8040) =  DANNEELS POINT OF X(4427)

Barycentrics    (b + c)(2a + b + c)2 : :

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)


X(8041) =  DANNEELS POINT OF X(4576)

Barycentrics    a2(b2 + c2)2 : :

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) =  DANNEELS POINT OF X(7192)

Barycentrics    a(a + b)(a + c)(b - c)3 : :

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)


X(8043) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(1)

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

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) 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) = 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) =  ISOGONAL CONJUGATE OF X(199)

Barycentrics    1/[b4 + c4 - a4 + (b2 + c2 - a2)(bc + ca + ab)] : :

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(8045) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(75)

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

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) =  CYCLOCEVIAN CONJUGATE OF X(80)

Barycentrics    1/(a^3+a^2 b-a b^2-b^3+a^2 c-5 a b c+3 b^2 c-a c^2+3 b c^2-c^3) : :

X(8046) lies on these lines: {320,4358}, {484,519}, {1443,3911}

X(8046) = isogonal conjugate of X(3196)
X(8046) = trilinear pole of line X(900)X(1387)


X(8047) =  CYCLOCEVIAN CONJUGATE OF X(668)

Barycentrics    1/(a^3-a^2 b+a b^2-b^3-a^2 c-a b c+b^2 c+a c^2+b c^2-c^3) : :

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) =  CYCLOCEVIAN CONJUGATE OF X(4373)

Barycentrics    1/(a^4-b^4+2 a^2 b c-2 a b^2 c-2 a b c^2+2 b^2 c^2-c^4) : :

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) =  CYCLOCEVIAN CONJUGATE OF X(6625)

Barycentrics    1/(a^2 b-a b^2+a^2 c-b^2 c-a c^2-b c^2) : :

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(8050) =  CYCLOCEVIAN CONJUGATE OF X(6630)

Barycentrics    1/((b-c) (a^2+a b+a c-b c)) : :

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) = trilinear pole of line X(37)X(39)
X(8050) = anticomplement of X(8054)


X(8051) =  CYCLOCEVIAN CONJUGATE OF X(7319)

Barycentrics    1/((-a+b+c) (a^2+2 a b+b^2+2 a c-6 b c+c^2)) : :

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) =  ISOTOMIC CONJUGATE OF X(8045)

Barycentrics    1/[(b - c)(a^3 + b^3 + c^3 + abc + 2b^2 c + 2bc^2)] : :

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(8053) =  ISOGONAL CONJUGATE OF X(8049)

Trilinears    (b - c)(a^2 + ab + ac - bc) : :

X(8053) lies on these lines:
{1,3286}, {3,142}, {6,31}, {9,4557}, {21,5263}, {56,7225} et al

X(8053) = X(3613)-of-excentral triangle
X(8053) = crossdifference of every pair of points on line X(514)X(6586)
X(8053) = perspector, wrt excentral triangle, of the circumcircle


X(8054) =  COMPLEMENT OF X(8050)

Barycentrics    a^2/((b-c) (a^2+a b+a c-b c)) : :

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) = 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))


X(8055) =  ISOTOMIC CONJUGATE OF X(8051)

Barycentrics    (b + c - a)(a^2 + b^2 + c^2 + 2ab + 2ac - 6bc) : :

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 (pending)

X(8055) = X(7)-Ceva conjugate of X(8)
X(8055) = anticomplement of X(8056)


X(8056) =  ISOGONAL CONJUGATE OF X(1743)

Trilinears    1/(3a - b - c) : :

X(8056) lies on these lines:
{1,474}, {2,2415}, {8,6553}, {10,1219}, {57,1122}, {63,88} et al

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(8057) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(20)

Trilinears    bz(ax + by - cz) - cy(ax + cz - by) : : , where x : y : z = X(235)
Barycentrics    (cos A)(cos^2 B - cos^2 C)(cos A - cos B cos C) : :

X(8057) lies on these lines:
{3,2416}, {30,511}, {69,2419} et al

X(8057) = isogonal conjugate of X(1301)
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) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(329)

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

X(8058) lies on these lines: {8,2399}, {30,511}, {3064,3239} et al (pending)

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) =  ISOGONAL CONJUGATE OF X(8058)

Trilinears    a/[(b + c - a)(b - c)(a^3 - b^3 - c^3 + a^2 b + a^2 c - a b^2 - a c^2 - 2 a b c + b^2 c + b c^2) : :

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): (2,77), (4,57), (6,603), (8,20)
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) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(330)

Barycentrics    b(c^4 - a^4) + c(a^4 - b^4) : :

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) =  ANTICOMPLEMENT OF X(8060)

Trilinears    b4 - c4 : c4 - a4 : a4 - b4

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)


X(8062) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(2994)

Barycentrics    sin B (tan C - tan A) + sin C (tan A - tan B) : :
Barycentrics    (b - c)[a^4 - a^2(b^2 + bc + c^2) + bc(b + c)^2] : :

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) = complement of X(656)
X(8062) = crossdifference of every pair of points on the line X(2176)X(2178)


X(8063) =  FARHANGI-CYCLOCEVIAN IMAGE OF X(5932)

Barycentrics    (cos B cot B/2)/(1 + cos B - cos C - cos A) - (cos C cot C/2)/(1 + cos C - cos A - cos B) : :

X(8063) lies on these lines: {30,511} et al
(pending)

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) =  ISOGONAL CONJUGATE OF X(8063)

Trilinears    a/[(cos B cot B/2)/(1 + cos B - cos C - cos A) - (cos C cot C/2)/(1 + cos C - cos A - cos B)] : :

X(8064) lies on the circumcircle and these lines: {107,7152} et al (pending)

X(8064) = trilinear pole of line X(6)X(2188)


X(8065) =  6th MORLEY-KIRIKAMI POINT

Trilinears    sin(A/3)(sin(2B/3) + sin(2C/3)) + sin(B/3)sin(C/3) : :

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)


X(8066) =  7th MORLEY-KIRIKAMI POINT

Barycentrics    Sin[A] (Cos[B/3-Pi/6] Cos[C/3-Pi/6]-Cos[A/3-Pi/6] (Cos[2 B/3+Pi/6]+Cos[2 C/3+Pi/6])) : :

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)


X(8067) =  8th MORLEY-KIRIKAMI POINT

Barycentrics    Sin[A] (Cos[B/3+Pi/6] Cos[C/3+Pi/6] -Cos[A/3+Pi/6] (Cos[2 B/3-Pi/6]+Cos[2 C/3-Pi/6])) : :

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)


X(8068) =  MIDPOINT OF X(11) AND X(12)

Barycentrics    a^5(b^2 + c^2) - a^4(b + c)(b^2 + c^2) - 2a^3(b^4 + c^4 - b^3c - bc^3 - b^2c^2) + 2a^2(b - c)^2(b + c)(b^2 + bc + c^2) + a(b - c)^4(b + c)^2 - (b - c)^4(b + c)^3 : :
X(8068) = R2*X(1) - 4r2*X(5)

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) =  MIDPOINT OF X(55) AND X(56)

Trilinears    1 + (cos A)(1 - cos A - 2 sin(A/2) cos(B/2 - C/2)) : :
X(8069) = R2*X(1) - r2*X(3)

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) =  {X(1),X(5)}-HARMONIC CONJUGATE OF X(8068)

Trilinears    (pending)
X(8070) = R2*X(1) + 4r2*X(5)

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) =  {X(1),X(3)}-HARMONIC CONJUGATE OF X(8069)

Trilinears    1 - (cos A)(1 - cos A - 2 sin(A/2) cos(B/2 - C/2)) : :
X(8071) = R2*X(1) + r2*X(3)

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


X(8072) =  1st HARMONIC TRACE OF THE EXCIRCLES

Barycentrics    (2 a^4-a^3 b+a^2 b^2-3 a b^3+b^4-a^3 c-2 a^2 b c+3 a b^2 c+a^2 c^2+3 a b c^2-2 b^2 c^2-3 a c^3+c^4) + 2 (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) Sqrt[s^2-4 r R-4 R^2] : :

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) = {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) =  2nd HARMONIC TRACE OF THE EXCIRCLES

Barycentrics    (2 a^4-a^3 b+a^2 b^2-3 a b^3+b^4-a^3 c-2 a^2 b c+3 a b^2 c+a^2 c^2+3 a b c^2-2 b^2 c^2-3 a c^3+c^4) - 2 (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) Sqrt[s^2-4 r R-4 R^2] : :

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) = 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) =  MIDPOINT OF X(8072) AND X(8073)

Barycentrics    2 a^4-a^3 b+a^2 b^2-3 a b^3+b^4-a^3 c-2 a^2 b c+3 a b^2 c+a^2 c^2+3 a b c^2-2 b^2 c^2-3 a c^3+c^4 : :
X(8074) = 3 X[5011] + X[5134] = X[5134] - 3 X[5179] = X[5134] - 6 X[5199] = X[5011] + 2 X[5199]

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)


leftri  Points associated with tangential-midarc triangles: X(8075) -X(8104)  rightri

This section was contributed by César 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

The next table lists triangles orthologic to the 1st and 2nd tangential-midarc triangles, with pairs of othologic centers labeled "1st" for the orthologic centers of the triangle and the 1st tangential mid-arc triangle, and likewise for "2nd".

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) : : .

underbar

X(8075) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 1st CIRCUMPERP

Trilinears    -(s-a)^2*a*sin(A/2)+(a-c)*(s-b)^2*sin(B/2)+(a-b)*(s-c)^2*sin(C/2)-S^2/(4*s) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 1st CIRCUMPERP

Trilinears    -(s-a)^2*a*sin(A/2)+(a-c)*(s-b)^2*sin(B/2)+(a-b)*(s-c)^2*sin(C/2)+S^2/(4*s) : :

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(8077) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 2nd CIRCUMPERP

Trilinears    -a*sin(A/2)+(a+c)*sin(B/2)+(a+b)*sin(C/2)-s : :

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)


X(8078) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND EXCENTRAL

Trilinears    -sin(A/2)+sin(B/2)+sin(C/2)-1 : :
Trilinears    (a*(cos(A/2)+cos(B/2)+cos(C/2))-s*cos(A/2))/cos(A/2) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 2nd EXTOUCH

Trilinears    -(b+c)*b*c*sin(A/2)+SC*c*sin(B/2)+SB*b*sin(C/2)-S*(r+2*R) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 2nd EXTOUCH

Trilinears    -(b+c)*b*c*sin(A/2)+SC*c*sin(B/2)+SB*b*sin(C/2)+S*(r+2*R) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND HEXYL

Trilinears    (s-a)*r*(r+2*R)*sin(A/2)+(s-b)*(2*R*r+c^2-c*s+r^2)*sin(B/2)+(s-c)*(2*R*r+b^2­b*s+r^2)*sin(C/2)+S^2/(4*s) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND HEXYL

Trilinears    (s-a)*r*(r+2*R)*sin(A/2)+(s-b)*(2*R*r+c^2-c*s+r^2)*sin(B/2)+(s-c)*(2*R*r+b^2­b*s+r^2)*sin(C/2)-S^2/(4*s) : :

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) =  PERSPECTOR OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND MIDARC

Trilinears    cos(B/2)^2 + cos(B/2)*cos(C/2) + cos(C/2)^2 : :

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) =  PERSPECTOR OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND MIDARC

Trilinears    cos(B/2)^2*(cos(A/2)+cos(B/2))-cos(B/2)*cos(C/2)*(cos(A/2)+2*cos(B/2)+2*cos(C/2))+cos(C/2)^2*(cos(A/2)+cos(C/2)) : :

X(8084) lies on these lines:
(164, 8076), (167, 258), (5571, 8083), (8077, 8082), (8078, 8090), (8091, 8100), (8092, 8093)


X(8085) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 3rd EULER

Barycentrics    (b-c)^2*(s-a)^2*sin(A/2)-(c*a+b^2-c^2)*(s-b)^2*sin(B/2)-(a*b-b^2+c^2)*(s­c)^2*sin(C/2)+a*S^2/(4*s) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 3rd EULER

Barycentrics    (b-c)^2*(s-a)^2*sin(A/2)-(c*a+b^2-c^2)*(s-b)^2*sin(B/2)-(a*b-b^2+c^2)*(s­c)^2*sin(C/2)-a*S^2/(4*s) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 4th EULER

Barycentrics    (b+c)^2*sin(A/2)+(c*a-b^2+c^2)*sin(B/2)+(a*b+b^2-c^2)*sin(C/2)+s*a : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 4th EULER

Barycentrics    (b+c)^2*sin(A/2)+(c*a-b^2+c^2)*sin(B/2)+(a*b+b^2-c^2)*sin(C/2)-s*a : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 6th MIXTILINEAR

Trilinears    (s-a)*r*(4*R+r)*sin(A/2)-(s-b)*(-4*R*r-2*c^2+2*c*s-r^2)*sin(B/2)-(s-c)*(-4*R*r­2*b^2+2*b*s-r^2)*sin(C/2)+ S^2/(2*s) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 6th MIXTILINEAR

Trilinears    (s-a)*r*(4*R+r)*sin(A/2)-(s-b)*(-4*R*r-2*c^2+2*c*s-r^2)*sin(B/2)-(s-c)*(-4*R*r­2*b^2+2*b*s-r^2)*sin(C/2)-S^2/(2*s) : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO ABC

Trilinears    -(s-a)*a*b*c*sin(A/2)+(s-b)*SC*c*sin(B/2)+(s-c)*SB*b*sin(C/2)-S^2/2 : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO ABC

Barycentrics    a - (a + b + c)*sin(A/2) : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO EXCENTRAL

Trilinears    (b+c)*sin(A/2)-b*sin(B/2)-c*sin(C/2)+s : :

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) = 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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO EXCENTRAL

Trilinears    (b+c)*sin(A/2)-b*sin(B/2)-c*sin(C/2)-s : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO EXTOUCH

Trilinears    -(s-a)*SA*r*((b+c)*(r+2*R)-4*R*s)*sin(A/2)+(s-b)*SB*(2*R*r+c^2­c*s+r^2)*b*sin(B/2)+(s-c)*SC*(2*R*r+b^2-b*s+r^2)*c*sin(C/2)+S^4/(8*s^2) : :

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(8096) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO EXTOUCH

Trilinears    -(s-a)*SA*r*((b+c)*(r+2*R)-4*R*s)*sin(A/2)+(s-b)*SB*(2*R*r+c^2­c*s+r^2)*b*sin(B/2)+(s-c)*SC*(2*R*r+b^2-b*s+r^2)*c*sin(C/2)-S^4/(8*s^2) : :

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(8097) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO FUHRMANN

Trilinears    -b*c*((3*b-2*s)*(2*cos(C)-1)*a*sin(B/2)+(3*c-2*s)*(2*cos(B)-1)*a*sin(C/2)-(3*a-2*s)^2*sin(A/2))+4*(R-2*r)*r*s^2 : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO FUHRMANN

Trilinears    -b*c*((3*b-2*s)*(2*cos(C)-1)*a*sin(B/2)+(3*c-2*s)*(2*cos(B)-1)*a*sin(C/2)-(3*a-2*s)^2*sin(A/2))-4*(R-2*r)*r*s^2 : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO INCENTRAL

Trilinears    -b*c*((3*b-2*s)*(2*cos(C)-1)*a*sin(B/2)+(3*c-2*s)*(2*cos(B)-1)*a*sin(C/2)-(3*a-2*s)^2*sin(A/2))-4*(R-2*r)*r*s^2 : :
Trilinears    (s-a)*(a*(a*b+a*c+b*c)+2*b*c*s-b^3-c^3)*sin(A/2)+(s-b)*(a*(a*b-a*c-b*c)-2*b*c*(s-c)-b^3+c^3)*sin(B/2)+(s-c)*(a*(-a*b+a*c-b*c)-2*b*c*(s-b)-c^3+b^3)*sin(C/2)+2*S^2 : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO INCENTRAL

Trilinears    (s-a)*(a*(a*b+a*c+b*c)+2*b*c*s-b^3-c^3)*sin(A/2)+(s-b)*(a*(a*b-a*c-b*c)-2*b*c*(s­c)-b^3+c^3)*sin(B/2)+(s-c)*(a*(-a*b+a*c-b*c)-2*b*c*(s-b)-c^3+b^3)*sin(C/2)-2*S^2 : :
Trilinears    (s-a)*(a*(a*b+a*c+b*c)+2*b*c*s-b^3-c^3)*sin(A/2)+(s-b)*(a*(a*b-a*c-b*c)-2*b*c*(s-c)-b^3+c^3)*sin(B/2)+(s-c)*(a*(-a*b+a*c-b*c)-2*b*c*(s-b)-c^3+b^3)*sin(C/2)-2*S^2 : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO MIXTILINEAR

Trilinears    2*a*b*c*sin(A/2)+(a^2-2*a*b-b^2+2*b*c-c^2)*c*sin(B/2)+(a^2-2*a*c-b^2+2*b*c­c^2)*b*sin(C/2)+2*S*(2*R-r) : :

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) =  ORTHOLOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO MIXTILINEAR

Trilinears    2*a*b*c*sin(A/2)+(a^2-2*a*b-b^2+2*b*c-c^2)*c*sin(B/2)+(a^2-2*a*c-b^2+2*b*c­c^2)*b*sin(C/2)-2*S*(2*R-r) : :

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) =  PARALLELOGIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC TO FUHRMANN

Trilinears    -(b-c)^2*(s-a)^2*b*c*sin(A/2)+(a-c)*((a-c)^2+b*(a-b))*(s-b)^2*c*sin(B/2)+(a-b)*((a­b)^2+c*(a-c))*(s-c)^2*b*sin(C/2)-S^2*r*(R-2*r) : :

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) =  PARALLELOGIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC TO FUHRMANN

Trilinears    -(b-c)^2*(s-a)^2*b*c*sin(A/2)+(a-c)*((a-c)^2+b*(a-b))*(s-b)^2*c*sin(B/2)+(a-b)*((a­b)^2+c*(a-c))*(s-c)^2*b*sin(C/2)+S^2*r*(R-2*r) : :

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)


X(8105) =  X(2)-CEVA CONJUGATE OF X(1313)

Barycentrics    (b^2-c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4 + a^2 (a^2-b^2-c^2) J) : : , where J = |OH|/R (as at X(1113)

The trilinear polar of X(8105) meets the line at infinity at X(512).

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)


X(8106) =  X(2)-CEVA CONJUGATE OF X(1312)

Barycentrics    (b^2-c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4 - a^2 (a^2-b^2-c^2) J) : : , where J = |OH|/R (as at X(1113)

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND 1st CIRCUMPERP

Trilinears    -(s-a)*a*sin(A/2)+(a-c)*(s-b)*sin(B/2)+(a-b)*(s-c)*sin(C/2)+(r*(4*R+r)-2*s*(-s+a)- 2*b*c) : :

X(8107) lies on these lines:
(3, 8109), (40, 8111), (55, 8113), (165, 166), (167, 7987), (259, 503), (5934, 7580), (6732, 8076)


X(8108) =  HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND 1st CIRCUMPERP

Trilinears    -(s-a)*a*sin(A/2)+(a-c)*(s-b)*sin(B/2)+(a-b)*(s-c)*sin(C/2)-(r*(4*R+r)-2*s*(-s+a)- 2*b*c) : :

X(8108) lies on these lines:
(3, 8110), (40, 8112), (55, 8114), (164, 6726), (165, 166), (5935, 7580), (7589, 7707)


X(8109) =  HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND 2nd CIRCUMPERP

Trilinears    a*(s-b)*(s-c)*sin(A/2) -(a+c)*(s-c)*(s-a)*sin(B/2)-(a+b)*(s-a)*(s-b)*sin(C/2) +(4*R*r-2*a*s-2*b*c+3*r^2+2*s^2)*s : :

X(8109) lies on these lines:
(1, 289), (3, 8107), (56, 8113), (3576, 8111), (6732, 7588), (7987, 8110)


X(8110) =  HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND 2nd CIRCUMPERP

Trilinears    a*(s-b)*(s-c)*sin(A/2) -(a+c)*(s-c)*(s-a)*sin(B/2)-(a+b)*(s-a)*(s-b)*sin(C/2)- (4*R*r-2*a*s-2*b*c+3*r^2+2*s^2)*s : :

X(8110) lies on these lines:
(1, 168), (3, 8108), (56, 8114), (405, 5935), (3576, 8112), (7987, 8109)


X(8111) =  HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND HEXYL

Trilinears    r*(r+2*R)*sin(A/2)+(2*R*r+r^2+c^2-c*s)*sin(B/2)+(2*R*r+r^2+b^2-b*s)*sin(C/2)- (S/(2*s))^2 : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND HEXYL

Trilinears    r*(r+2*R)*sin(A/2)+(2*R*r+r^2+c^2-c*s)*sin(B/2)+(2*R*r+r^2+b^2-b*s)*sin(C/2) +(S/(2*s))^2 : :

X(8112) lies on these lines:
(1, 8111), (3, 168), (40, 8108), (84, 7590), (164, 517), (1490, 5935), (3576, 8110)


X(8113) =  HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND INTOUCH

Trilinears    (s-c)*c*sin(B/2)+(s-b)*b*sin(C/2)-r*(4*R+r) : :

X(8113) lies on these lines:
(1, 8111), (2, 178), (55, 8107), (56, 8109), (57, 363), (174, 6732), (226, 5934)


X(8114) =  HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND INTOUCH

Trilinears    (s-c)*c*sin(B/2)+(s-b)*b*sin(C/2)+r*(4*R+r) : :

X(8114) lies on these lines:
(1, 8111), (7, 174), (55, 8108), (56, 8110), (57, 168), (188, 3870), (226, 5935), (289, 1488)


X(8115) =  ISOGONAL CONJUGATE OF X(8105)

Barycentrics    2 - (1-J) a^2 SA / (SB SC) : : , where J = |OH|/R

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(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) =  ISOGONAL CONJUGATE OF X(8106)

Barycentrics    2 - (1+J) a^2 SA / (SB SC) : : , where J = |OH|/R

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(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}


leftri  Endo-homothetic centers: X(8117) -X(8140)  rightri

This section was contributed by César Lozada, September 7, 2015.


underbar

X(8117) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 1st CIRCUMPERP

Barycentrics    SB*SC*(1-sin(A/2))-S^2/2 : :

X(8117) lies on these lines:
(2, 8121), (3, 8119), (4, 8123), (20, 8118), (30, 8120), (376, 8124), (3146, 8122)


X(8118) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 1st CIRCUMPERP

Barycentrics    SB*SC*(1+sin(A/2))-S^2/2 : :

X(8118) lies on these lines:
(2, 8122), (3, 8120), (4, 8124), (20, 8117), (30, 8119), (376, 8123), (3146, 8121)


X(8119) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND 2nd CIRCUMPERP

Barycentrics    SB*SC*(1-2*sin(A/2)) : :

X(8119) lies on these lines:
(2, 8123), (3, 8117), (4, 5934), (20, 8124), (30, 8118)


X(8120) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND 2nd CIRCUMPERP

Barycentrics    SB*SC*(1+2*sin(A/2)) : :

X(8120) lies on these lines:
(2, 8124), (3, 8118), (4, 5934), (20, 8123), (30, 8117)


X(8121) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND EXCENTRAL

Barycentrics    SB*SC*(1-sin(A/2)) : :

X(8121) lies on these lines:
(2, 8117), (4, 5934), (5, 8123), (30, 8124), (3146, 8118)


X(8122) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND EXCENTRAL

Barycentrics    SB*SC*(1+sin(A/2)) : :

X(8122) lies on these lines:
(2, 8118), (4, 5934), (5, 8124), (30, 8123), (3146, 8117)


X(8123) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND HEXYL

Barycentrics    a^2*SA+2*SB*SC*sin(A/2) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND HEXYL

Barycentrics    a^2*SA-2*SB*SC*sin(A/2) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND 1st CIRCUMPERP

Barycentrics    1-2*sin(A/2) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND 1st CIRCUMPERP

Barycentrics    1+2*sin(A/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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND 2nd CIRCUMPERP

Barycentrics    2*S^2*(1-sin(A/2))-SB*SC : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND 2nd CIRCUMPERP

Barycentrics    2*S^2*(1+sin(A/2))-SB*SC : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND HEXYL

Barycentrics    a^2*SA-2*S^2*sin(A/2) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND HEXYL

Barycentrics    a^2*SA+2*S^2*sin(A/2) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND INTOUCH

Barycentrics    a^2*(SW*SA-(b*c)^2*sin(A/2)) : :

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER HUTSON AND INTOUCH

Barycentrics    a^2*(SW*SA+(b*c)^2*sin(A/2)) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND INNER HUTSON

Trilinears    F(a,A) + G(a,b,c,A,B,C)+ G(a,c,b,A,C,B) : : , where F(a,A) = 2*(s-a)^2*a*sin(A/2)+S^2/s and G(a,b,c,A,B,C) = (2*(s-c)*(a-b)*c*sin(A/2)-(b-c)^2*(s-a)*sin(C/2)+(s-b)*((b-2*c)*(a-c)-b^2))*sin(B/2)

X(8133) lies on these lines:
(1, 6724), (2, 178), (4, 5934), (177, 266), (363, 8078), (8077, 8109), (8081, 8111), (8089, 8135)


X(8134) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND INNER HUTSON

Barycentrics    a^2*SA*(1-sin(A/2)) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND OUTER HUTSON

Trilinears    F(a,b,c,A) + G(a,b,c,A,B,C) + G(a,c,b,A,C,B) : : , where F(a,b,c,A) = (s-a)*((b-c)^2-(b+c)*a)*sin(A/2)-S^2/s and G(a,b,c,A,B,C) = sin(B/2)*(2*(s-c)*(a-b)*c*sin(A/2)+2*(a-c)*(s-b)^2-(b-c)^2*(s-a)* sin(C/2))

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st TANGENTIAL-MIDARC AND OUTER HUTSON

Barycentrics    a^2*SA+(S^2+SB*SC)*sin(A/2) : :

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)


X(8137) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND INNER HUTSON

Barycentrics    a^2*SA-(S^2+SB*SC)*sin(A/2) : :

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) =  HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND OUTER HUTSON

Trilinears    F(a,A) + G(a,b,c,A,B,C) + G(a,c,b,A,C,B) : : , where F(a,A) = 2*(s-a)^2*a*sin(A/2)-S^2/s G(a,b,c,A,B,C) = sin(B/2)*(-2*(s-c)*(a-b)*c*sin(A/2)+(s-b)*(a*(b-2*c)-(b+2*c)*(b-c))+(b-c)^2*(s-a)*sin(C/2))

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) =  ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd TANGENTIAL-MIDARC AND OUTER HUTSON

Barycentrics    a^2*SA*(1+sin(A/2)) : :

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)


X(8140) =  HOMOTHETIC CENTER OF THESE TRIANGLES: INNER HUTSON AND OUTER HUTSON

Barycentrics    r*(4*R+r)*sin(A/2)-(2*c*(s-c)-(4*R+r)*r)*sin(B/2)-(2*b*(s-b)-(4*R+r)*r)*sin(C/2) : :

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)


leftri  Centers of central circles: X(8141) -X(8155)  rightri

This section was contributed by César 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(8144)
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(8351)
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(8352)
Yff contact X(5592)
Yiu X(8154)


underbar

X(8141) =  CENTER OF THE EXTANGENTS CIRCLE

Trilinears    a^9+(b+c)*a^8-2*(b^2+c^2)*a^7-2*(b+c)*(b^2+c^2)*a^6-b*c*(b^2+c^2)*a^5+b*c*(b+c)*(b^2+c^2)*a^4+2*(b^4+c^4-b*c*(b^2-b*c+c^2))*(b+c)^2*a^3 +2*(b+c)*(b^4+c^4+b*c*(b^2+b*c+c^2))*(b-c)^2*a^2-(b^3+c^3)*(b^2-c^2)^2*(b+c)*a-(b^3-c^3)*(b^2-c^2)^3 : :

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) =  CENTER OF THE GEOS CIRCLE

Barycentrics    (b-c)*(7*a^5*(a-b-c)-6*(b^2-b*c+c^2)*a^4+6*(b+c)*(b^2+c^2)*a^3-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^2+(b^2-c^2)^2*((b+c)*a-2*b*c)) : :

See GEOS Circle at MathWorld.

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) =  CENTER OF THE INCENTRAL CIRCLE

Trilinears    (b+c)*a^5+(b^2+c^2)*a^4-(b+c)*(2*c^2+b*c+2*b^2)*a^3+(-2*b^2*c^2-2*b^3*c-2*c^4-2*b*c^3-2*b^4)*a^2+(b+c)*(c^2+3*b*c+b^2)*(b-c)^2*a+(b-c)^2*(b+c)^4 : :

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) =  CENTER OF THE INTANGENTS CIRCLE

Trilinears    a^6-(b+c)^2*a^4+(b^3*c+2*b^2*c^2-b^4+b*c^3-c^4)*a^2+(b^2+b*c+c^2)*(b^2-c^2)^2 : :

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)


X(8145) =  CENTER OF THE 3rd LEMOINE CIRCLE

Barycentrics    6*a^10+11*(b^2+c^2)*a^8-2*(10*b^4+10*c^4+31*b^2*c^2)*a^6-3*(b^2+c^2)*(4*b^4+45*b^2*c^2+4*c^4)*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(7*b^4+26*b^2*c^2+7*c^4)*a^2+(b^2-c^2)^2*(b^2+c^2)^3 : :
Barycentrics    2*(27*R^2*S^2+11*SW^3)*S^2+(-SW^3+(27*R^2-9*SW)*S^2)*SA^2-(-SW^3+(27*R^2+9*SW)*S^2)*SA*SW : :
X(8145) = [-9*S^2*(-3*r^2+6*s^2+10*SW)+108*s^2*(SW-s^2)^2+272*SW^3]*X(5)+72*SW*(3*S^2-SW^2)*X(6054)
X(8145) = 18 SW (3 S^2 + SW^2) X[2] - (27 SA SB SC + SW (9 S^2 + 4 SW^2)) X[5] + 36 SW^3 X[6]
X(8145) = 9 (a^2+b^2+c^2) (a^4+b^4+c^4-4 b^2 c^2-4 c^2 a^2-4 a^2 b^2) X[2]+2 (-a^2+2 b^2+2 c^2) (2 a^2-b^2+2 c^2) (2 a^2+2 b^2-c^2) X[5]-9 (a^2+b^2+c^2)^3 X[6]

See Third Lemoine Circle at MathWorld.

X(8145) lies on this line: {5,6054}


X(8146) =  CENTER OF THE MACBEATH CIRCLE

Trilinears    cos(2*A)*cos(3*(B-C)) -cos(A)*(-1+8*cos(A)^2)*cos(2*(B-C)) -(8*cos(A)^2+1-24*cos(A)^4)*cos(B-C)-cos(A)*(5-12*cos(A)^2+16*cos(A)^4) : :

See MacBeath Circle at MathWorld.

X(8146) lies on these lines: (4,2917), (5,5961)


X(8147) =  CENTER OF THE MIXTILINEAR CIRCLE

Trilinears    (a^6+(-7*c^2+4*b*c-7*b^2)*a^4+8*(b+c)*(b^2+c^2)*a^3 +(b^2-10*b*c+c^2)*(3*c^2+3*b^2-2*b*c)*a^2-8*(b+c)*(b-c)^4*a+(3*b^2+8*b*c+3*c^2)*(b-c)^4)*a : :

X(8147) lies on these lines: (6,6244), (991,999), (4648,7956)


X(8148) =  CENTER OF THE MOSES-LONGUET-HIGGINS CIRCLE

Trilinears    a^3-4*(b+c)*a^2-(b^2-8*b*c+c^2)*a+4*(b^2-c^2)*(b-c) : :
X(8148) = 4*X(1)-3*X(3)

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) =  CENTER OF THE 1st NEUBERG CIRCLE

Barycentrics    ((b^2+c^2)^2-b^2*c^2)*a^4+b^2*c^2*((b^2+c^2)*a^2-b^4-c^4) : :

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) = {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) =  CENTER OF THE 2nd NEUBERG CIRCLE

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

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(8151) =  CENTER OF THE 2nd STEINER CIRCLE

Barycentrics    (a^8-4*(b^2+c^2)*a^6+(6*b^4+b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2+(b^4+c^4)*(b^2-c^2)^2)*(b^2-c^2) : :

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) =  CENTER OF THE SYMMEDIAL CIRCLE

Trilinears    ((b^2+c^2)*a^6-6*a^4*b^2*c^2-(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^2+b^2*c^2*((b^2-c^2)^2-4*b^2*c^2)+b^8+c^8)*a : :

X(8152) lies on these lines: (39,6234), (185,575)


X(8153) =  CENTER OF THE EULER-GERGONNE-SODDY CIRCLE

Barycentrics    14*a^9-6*(b+c)*a^8-19*(b^2+c^2)*a^7-(b+c)*(3*b^2-20*b*c+3*c^2)*a^6-3*(b^4-10*b^2*c^2+c^4)*a^5+(b+c)*(21*b^4-36*b^3*c+22*b^2*c^2-36*b*c^3+21*c^4)*a^4+7*(b^2-c^2)^2*a^3*(b^2+c^2)-(b+c)*(9*b^4+6*b^3*c+26*b^2*c^2+6*b*c^3+9*c^4)*(b-c)^2*a^2+(b^2-c^2)^4*a-(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^3*(b-c) : :
X(8153) = X(857) - 5X(3522)
X(8153) = (-(3*(r+2*R))*(r+4*R)+6*s^2)*X(2)+((6*(r+2*R))*(r+4*R)-11*s^2)*X(3)

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) =  CENTER OF THE YIU CIRCLE

Trilinears    (16*cos(A)^4-24*cos(A)^2+7)*cos(A)+(4*cos(A)^2-1)*cos(B-C)-2*cos(A)*cos(2*(B-C)) : :
Barycentrics   
a^2 (a^14-5 a^12 b^2+10 a^10 b^4-10 a^8 b^6+5 a^6 b^8-a^4 b^10-5 a^12 c^2+14 a^10 b^2 c^2-13 a^8 b^4 c^2+4 a^6 b^6 c^2+a^4 b^8 c^2-2 a^2 b^10 c^2+b^12 c^2+10 a^10 c^4-13 a^8 b^2 c^4+3 a^6 b^4 c^4+3 a^2 b^8 c^4-3 b^10 c^4-10 a^8 c^6+4 a^6 b^2 c^6-2 a^2 b^6 c^6+2 b^8 c^6+5 a^6 c^8+a^4 b^2 c^8+3 a^2 b^4 c^8+2 b^6 c^8-a^4 c^10-2 a^2 b^2 c^10-3 b^4 c^10+b^2 c^12) : :

See Yiu Circle T MathWorld

X(8154) lies on this line: {1157,8157}


X(8155) =  CENTER OF THE LUCAS CENTRAL CIRCLE

Trilinears    a*(5*(a^8-b^8-c^8)-14*(b^2+c^2)*a^6+4*((b^2-c^2)^2-4*b^2*c^2)*a^4+2*(b^2+c^2)*(5*b^4+22*b^2*c^2+5*c^4)*a^2-2*(2*b^4-13*b^2*c^2+2*c^4)*b^2*c^2+4*(-7*(b^2+c^2)*a^4+4*(2*b^4+3*b^2*c^2+2*c^4)*a^2-(b^2+c^2)*(b^4-9*b^2*c^2+c^4))*S) : :

See Lucas Central Circle at MathWorld.


X(8156) =  CENTER OF THE LUCAS(-1) CENTRAL CIRCLE

Trilinears    a*(5*(a^8-b^8-c^8)-14*(b^2+c^2)*a^6+4*((b^2-c^2)^2-4*b^2*c^2)*a^4+2*(b^2+c^2)*(5*b^4+22*b^2*c^2+5*c^4)*a^2-2*(2*b^4-13*b^2*c^2+2*c^4)*b^2*c^2+4*(-7*(b^2+c^2)*a^4+4*(2*b^4+3*b^2*c^2+2*c^4)*a^2-(b^2+c^2)*(b^4-9*b^2*c^2+c^4))*(-S)) : :

Contributed by Randy Hutson, September 26, 2015.


X(8157) =  PARRY-POHOATA POINT

Barycentrics    a^2(a^20 +
a^18 (-6 (b^2 + c^2)) +
a^16 (2 (7 b^4 + 12 b^2 c^2 + 7 c^4)) +
a^14 (-2 (b^2 + c^2) (7 b^4 + 10 b^2 c^2 + 7 c^4)) +
a^12 (b^2 c^2 (18 b^4 + 25 b^2 c^2 + 18 c^4)) +
a^10 (b^2 + c^2) (14 b^8 - 15 b^6 c^2 + 8 b^4 c^4 - 15 b^2 c^6 + 14 c^8) +
a^8 (-14 b^12 + b^10 c^2 + 5 b^8 c^4 - 2 b^6 c^6 + 5 b^4 c^8 + b^2 c^10 - 14 c^12) +
a^6 (b^2 - c^2)^2 (b^2 + c^2) (6 b^8 + 2 b^6 c^2 + 5 b^4 c^4 + 2 b^2 c^6 + 6 c^8) +
a^4 (-(b^2 - c^2)^2 (b^12 - 2 b^10 c^2 - b^8 c^4 - 6 b^6 c^6 - b^4 c^8 - 2 b^2 c^10 + c^12)) +
a^2 (-b^2 c^2 (b^2 - c^2)^4 (b^2 + c^2) (3 b^4 + b^2 c^2 + 3 c^4))
+ b^2 c^2 (b^2 - c^2)^6 (b^2 + c^2)^2) : :
X(8157) = X(4) + (J2 - 2)*X(933)

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}


X(8158) =  CENTER OF THE APOLLONIAN CIRCLE OF THE EXTERNAL MIXTILINEAR CIRCLES

Barycentrics    a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c-8 a^2 b^2 c-2 a b^3 c+9 b^4 c-2 a^3 c^2-8 a^2 b c^2+18 a b^2 c^2-8 b^3 c^2+2 a^2 c^3-2 a b c^3-8 b^2 c^3+a c^4+9 b c^4-c^5) : :
X(8158) = 3X(5603) - 2X(5763) = 4R*X(1) - (r + 4R)*X(3)

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).

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) = 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)


X(8159) =  X(3)-OF-EXTERNAL-MIXTILINEAR-TRIANGLE (aka 2nd MIXTILINEAR TRIANGLE)

Barycentrics    a^2 (a^8-2 a^7 b-6 a^6 b^2+22 a^5 b^3-20 a^4 b^4-6 a^3 b^5+22 a^2 b^6-14 a b^7+3 b^8-2 a^7 c+10 a^6 b c-2 a^5 b^2 c-54 a^4 b^3 c+106 a^3 b^4 c-82 a^2 b^5 c+26 a b^6 c-2 b^7 c-6 a^6 c^2-2 a^5 b c^2+52 a^4 b^2 c^2-68 a^3 b^3 c^2+2 a^2 b^4 c^2+38 a b^5 c^2-16 b^6 c^2+22 a^5 c^3-54 a^4 b c^3-68 a^3 b^2 c^3+116 a^2 b^3 c^3-50 a b^4 c^3+34 b^5 c^3-20 a^4 c^4+106 a^3 b c^4+2 a^2 b^2 c^4-50 a b^3 c^4-38 b^4 c^4-6 a^3 c^5-82 a^2 b c^5+38 a b^2 c^5+34 b^3 c^5+22 a^2 c^6+26 a b c^6-16 b^2 c^6-14 a c^7-2 b c^7+3 c^8) : :
X(8159) = 4R*X(1) - (r + 4R)*X(8147)
X(8159) = (s2/[(4 + 4R)2 - 1]*X(220) + X(6244)

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}


X(8160) =  CENTER OF THE OUTER MONTESDEOCA-LEMOINE CIRCLE

Barycentrics    a^2(a^6(b^2+c^2)+2a^4(b^2+c^2)^2-a^2(4b^6+7b^4c^2+7b^2c^4+4c^6)+b^8-3b^6c^2-2b^4c^4-3b^2c^6+c^8 + 4S^3(2a^2-b^2-c^2) csc ω) : :
Barycentrics    a^2 (SA + S Cot[θ]) : : , where θ = ArcTan[-(2 + Cos[w]) Csc[w]]
Barycentrics    a^2 (SA - (S Sin[ω]) / (2 + Cos[ω])) : :
X(8160) = 3 X[3] - X[1671] = 3 X[1670] + X[1671] = 2 X[1671] - 3 X[8161] = 2 X[1670] + X[8161] = (1 + 2 Sec[ω]) X[3] - X[6]

Let ABC be a triangle, and let A'B'C' be the cevian triangle of X(6). Let U be the line through A' parallel to AB, and let V be the line through A' parallel to AC. Let U' = U∩AC and V' = V∩AB. The 4 points B, C, U', V' lie on a circle, (O)A. Define (O)B and (O)C cyclically. Let M be the circle tangent to (O)A, (O)B, (O)C that encompasses them; call M the outer Montesdeoca-Lemoine circle. Let M' be the circle tangent to (O)A, (O)B, (O)C that is encompassed by each of them; call M' the inner Montesdeoca-Lemoine circle. Then X(8160) is the center of M, and X(8161) is the center of M'. The contact points of M with (O)A, (O)B, (O)C are the vertices of a triangle perspective to ABC, with perspector X(1343). Likewise, the contact points of M' with (O)A, (O)B, (O)C are the vertices of a triangle perspective to ABC, with perspector X(1342). (Based on notes from Angel Montesdeoca, October 2, 2015)

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) =  CENTER OF THE INNER MONTESDEOCA-LEMOINE CIRCLE

Barycentrics    a^2(a^6(b^2+c^2)+2a^4(b^2+c^2)^2-a^2(4b^6+7b^4c^2+7b^2c^4+4c^6)+b^8-3b^6c^2-2b^4c^4-3b^2c^6+c^8 - 4S^3(2a^2-b^2-c^2) csc ω) : :
Barycentrics    a^2 (SA + S Cot[θ]) : : , where θ = ArcTan[(2 - Cos[w]) Csc[w]]
Barycentrics    a^2 (SA + (S Sin[ω]) / (2 - Cos[ω])) : :
X(8161) = 3 X[3] - X[1670] = X[1670] + 3 X[1671] = 2 X[1670] - 3 X[8160] = 2 X[1671] + X[8160] = (1 - 2 Sec[ω]) X[3] - X[6]

See X(8160).

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)



leftri  Similicenters of pairs of circles: X(8162)-X(8171)  rightri

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)



underbar



X(8162) =  INSIMILICENTER OF THESE CIRCLES: AIMC AND INCIRCLE

Barycentrics    a^2 (a^2-b^2-c^2-14 b c) : :
Barycentrics    Sin[A] (Cos[A] + 7) : :
X(8162) = 7R*(X(1) + r*X(3)

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) =  INSIMILICENTER OF THESE CIRCLES: AEMC AND INCIRCLE

Barycentrics    a^2 (a-b-c) (a^4-2 a^2 b^2+b^4+8 a^2 b c-8 a b^2 c-16 b^3 c-2 a^2 c^2-8 a b c^2+30 b^2 c^2-16 b c^3+c^4) : :
X(8163) = (7r - 4R)R*X(1) - r(r + 4R)*X(3)

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) =  INSIMILICENTER OF THESE CIRCLES: AIMC AND NINE-POINT CIRCLE

Barycentrics    a^4-4 a^2 b^2+3 b^4-8 a^2 b c-4 a^2 c^2-6 b^2 c^2+3 c^4 : :
X(8164) = r*X(4) - (8r + 4R)*X(12)

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) =  INSIMILICENTER OF THESE CIRCLES: AEMC AND NINE-POINT CIRCLE

Barycentrics    (a-b-c) (a^3+a^2 b+3 a b^2+3 b^3+a^2 c+2 a b c-3 b^2 c+3 a c^2-3 b c^2+3 c^3) : :
X(8165) = (3r + 12R)*X(2) - (8r + 2R)*X(12)

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) =  EXSIMILICENTER OF THESE CIRCLES: AEMC AND NINE-POINT CIRCLE

Barycentrics    a^6-2 a^5 b-3 a^4 b^2+8 a^3 b^3-a^2 b^4-6 a b^5+3 b^6-2 a^5 c+6 a^4 b c-8 a^3 b^2 c-16 a^2 b^3 c+26 a b^4 c-6 b^5 c-3 a^4 c^2-8 a^3 b c^2+34 a^2 b^2 c^2-20 a b^3 c^2-3 b^4 c^2+8 a^3 c^3-16 a^2 b c^3-20 a b^2 c^3+12 b^3 c^3-a^2 c^4+26 a b c^4-3 b^2 c^4-6 a c^5-6 b c^5+3 c^6 : :
X(8166) = (r + 4R)*X(4) - (8r - 4R)*X(11)

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) =  INSIMILICENTER OF THESE CIRCLES: AIMC AND SPIEKER

Barycentrics    a (a^2-a b-a c-6 b c) : :
X(8167) = (3r + 9R)*X(2) + (R - 2r)*X(11)

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) =  EXSIMILICENTER OF THESE CIRCLES: AIMC AND SPIEKER

Barycentrics    a (a-b-c) (a^2+a b+a c-6 b c) : :
X(8168) = 7R*X(8) - (2r + 3R)*X(21) = 2X(999) - 3X(1376) = 3X(5289) - 2X(7962) = 3X(200) - X(7962)

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) =  INSIMILICENTER OF THESE CIRCLES: AEMC AND SPIEKER

Barycentrics    a (a-b-c) (a^4-a^3 b-a^2 b^2+a b^3-a^3 c+4 a^2 b c-5 a b^2 c-6 b^3 c-a^2 c^2-5 a b c^2+12 b^2 c^2+a c^3-6 b c^3) : :

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) =  EXSIMILICENTER OF THESE CIRCLES: AEMC AND SPIEKER

Barycentrics    a (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4-a b^5-a^5 c+2 a^4 b c-14 a^3 b^2 c+4 a^2 b^3 c+15 a b^4 c-6 b^5 c-2 a^4 c^2-14 a^3 b c^2+30 a^2 b^2 c^2-30 a b^3 c^2+2 a^3 c^3+4 a^2 b c^3-30 a b^2 c^3+12 b^3 c^3+a^2 c^4+15 a b c^4-a c^5-6 b c^5) : :

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) =  EXSIMILICENTER OF THESE CIRCLES: AIMC AND AEMC

Barycentrics    a^2 (3 a^4-6 a^3 b+6 a b^3-3 b^4-6 a^3 c+10 a^2 b c-10 a b^2 c+6 b^3 c-10 a b c^2-6 b^2 c^2+6 a c^3+6 b c^3-3 c^4) : :
X(8171) = 8R2*X(1) - r(3r + 4R)*X(3)

X(8171) lies on these lines: {1,3}, {4428,5542}, {5284,5748}


X(8172) =  1st GIBERT-NEUBERG PERSPECTOR

Barycentrics    Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (3 a^10-4 a^8 b^2-5 a^6 b^4+9 a^4 b^6-2 a^2 b^8-b^10-4 a^8 c^2-15 a^6 b^2 c^2+6 a^4 b^4 c^2+7 a^2 b^6 c^2+3 b^8 c^2-5 a^6 c^4+6 a^4 b^2 c^4-10 a^2 b^4 c^4-2 b^6 c^4+9 a^4 c^6+7 a^2 b^2 c^6-2 b^4 c^6-2 a^2 c^8+3 b^2 c^8-c^10) + 2 (a^12-13 a^10 b^2+29 a^8 b^4-18 a^6 b^6-5 a^4 b^8+7 a^2 b^10-b^12-13 a^10 c^2+33 a^8 b^2 c^2+12 a^6 b^4 c^2-26 a^4 b^6 c^2-12 a^2 b^8 c^2+6 b^10 c^2+29 a^8 c^4+12 a^6 b^2 c^4+8 a^4 b^4 c^4+5 a^2 b^6 c^4-15 b^8 c^4-18 a^6 c^6-26 a^4 b^2 c^6+5 a^2 b^4 c^6+20 b^6 c^6-5 a^4 c^8-12 a^2 b^2 c^8-15 b^4 c^8+7 a^2 c^10+6 b^2 c^10-c^12) S : :

X(8172) is the perspector of triangles 3 and 7 in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane.

X(8172) lies on the Neuberg cubic K001, the cubic K061a, and these lines: {3,13}, {14,1337}, {16,1263}, {74,8174}, {1138,5668}, {1276,3065}, {3440,8175}, {3465,7326}, {3483,7059}, {5673,7329}


X(8173) =  2nd GIBERT-NEUBERG PERSPECTOR

Barycentrics    Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (3 a^10-4 a^8 b^2-5 a^6 b^4+9 a^4 b^6-2 a^2 b^8-b^10-4 a^8 c^2-15 a^6 b^2 c^2+6 a^4 b^4 c^2+7 a^2 b^6 c^2+3 b^8 c^2-5 a^6 c^4+6 a^4 b^2 c^4-10 a^2 b^4 c^4-2 b^6 c^4+9 a^4 c^6+7 a^2 b^2 c^6-2 b^4 c^6-2 a^2 c^8+3 b^2 c^8-c^10) - 2 (a^12-13 a^10 b^2+29 a^8 b^4-18 a^6 b^6-5 a^4 b^8+7 a^2 b^10-b^12-13 a^10 c^2+33 a^8 b^2 c^2+12 a^6 b^4 c^2-26 a^4 b^6 c^2-12 a^2 b^8 c^2+6 b^10 c^2+29 a^8 c^4+12 a^6 b^2 c^4+8 a^4 b^4 c^4+5 a^2 b^6 c^4-15 b^8 c^4-18 a^6 c^6-26 a^4 b^2 c^6+5 a^2 b^4 c^6+20 b^6 c^6-5 a^4 c^8-12 a^2 b^2 c^8-15 b^4 c^8+7 a^2 c^10+6 b^2 c^10-c^12) S : :

X(8173) is the perspector of triangles 3 and 8 in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane.

X(8173) lies on the Neuberg cubic K001, the cubic K061b, and these lines: {3,14}, {13,1338}, {15,1263}, {74,8175}, {1138,5669}, {1277,3065}, {3441,8174}, {3465,7325}, {3483,7060}, {5672,7329}


X(8174) =  3rd GIBERT-NEUBERG PERSPECTOR

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(8174) is the perspector of triangles 4 and 5 in Table 19-1 of Bernard Gibert's Cubics in the Triangle Plane.

X(8174) lies on the Neuberg cubic K001 and these lines: {4,15}, {14,1157}, {16,3479}, {74,8172}, {484,3375}, {1277,7165}, {3441,8173}, {3466,5672}, {3481,8175}, {5685,7325}


X(8175) =  4th GIBERT-NEUBERG PERSPECTOR

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}



leftri  Centers associated with Van Lamoen circles: X(8176)-X(8182)  rightri

This section was contributed by César 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 medial-van Lamoen circle is ρ1 = 2-1/2(S2ω + 9S2)/[1296S4(3SA + Sω)(3SB + Sω)(3SC + Sω)]

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



underbar



X(8176) =  CENTER OF THE ANTICOMPLEMENTARY-VAN LAMOEN CIRCLE

Barycentrics    a^4+5*(b^2+c^2)*a^2-5*b^4+8*b^2*c^2-5*c^4 : :

X(8176) lies on these lines: (2,187), (3,7619), (4,7618), (5,524), (6,5461), (30,7622), (114,381), (262,538), (597,7844), (671,7777), (754,5055), (1506,7841), (2476,7621), (2548,7817), (3090,7843), (3091,7620), (3363,3734), (3544,7758), (3767,5032), (3850,7781), (3851,7764), (5056,7780), (7746,7812), (7752,7801), (7773,7810)

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)


X(8177) =  CENTER OF THE 1st BROCARD-VAN LAMOEN CIRCLE

Barycentrics: a^6-2*a^2*b^2*c^2-b^2*c^2*(b^2+c^2) : :

X(8177) lies on these lines:
(2,6), (3,698), (15,6581), (16,6294), (76,1691), (157,6660), (160,3511), (182,732), (194,5116), (206,3506), (308,3114), (315,5103), (511,7780), (538,5092), (1078,3094), (1428,4400), (1799,3981), (2076,7793), (2330,4396), (3098,5969), (5026,5033), (5031,7746), (6034,7811)

X(8177) = midpoint of X(182) and X(7751)


X(8178) =  CENTER OF THE 1st ANTI-BROCARD-VAN LAMOEN CIRCLE

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

X(8178) lies on these lines:
(2,99), (3,8150), (32,1916), (39,5989), (98,7751), (147,7764), (736,5999), (2023,5149), (2782,7781), (3098,5969), (3818,6298), (4027,7772), (5939,7798), (5976,7815), (5980,6294), (5981,6581), (5984,7758), (6033,7775)

X(8178) = midpoint of X(5984) and X(7758)
X(8178) = reflection of X(i) in X(j) for these (i,j): (147,7764), (7751,98)


X(8179) =  CENTER OF THE 2nd NEUBERG-VAN LAMOEN CIRCLE

Barycentrics: 2*(9*S^4+16*S^2*SW^2-SW^4)*S^2-SA*(3*S^2-SW^2)*((3*S^2-SW^2)*SA+(S^2+SW^2)*SW) : :

X(8179) lies on these lines: (262,2080), (511,1153), (574,2023), (576,7815), (2782,7617), (7608,7899)


X(8180) =  CENTER OF THE OUTER-VECTEN-VAN LAMOEN CIRCLE

Barycentrics: (SA*(S+SW)+S*(5*S+3*SW))*(SB+SC+2*S) : :

If you have The Geometer's Sketchpad, you can view X(8180).

X(8180) lies on these lines: (5,6118), (485,490), (590,641)


X(8181) =  CENTER OF THE McCAY-VAN LAMOEN CIRCLE

Barycentrics: 120 a^16-718 a^14 b^2+1955 a^12 b^4-3173 a^10 b^6+3360 a^8 b^8-2374 a^6 b^10+1090 a^4 b^12-296 a^2 b^14+36 b^16-718 a^14 c^2+2414 a^12 b^2 c^2-2977 a^10 b^4 c^2+718 a^8 b^6 c^2+2183 a^6 b^8 c^2-2731 a^4 b^10 c^2+1388 a^2 b^12 c^2-282 b^14 c^2+1955 a^12 c^4-2977 a^10 b^2 c^4+1487 a^8 b^4 c^4-658 a^6 b^6 c^4+2180 a^4 b^8 c^4-2386 a^2 b^10 c^4+966 b^12 c^4-3173 a^10 c^6+718 a^8 b^2 c^6-658 a^6 b^4 c^6-1012 a^4 b^6 c^6+1299 a^2 b^8 c^6-1905 b^10 c^6+3360 a^8 c^8+2183 a^6 b^2 c^8+2180 a^4 b^4 c^8+1299 a^2 b^6 c^8+2370 b^8 c^8-2374 a^6 c^10-2731 a^4 b^2 c^10-2386 a^2 b^4 c^10-1905 b^6 c^10+1090 a^4 c^12+1388 a^2 b^2 c^12+966 b^4 c^12-296 a^2 c^14-282 b^2 c^14+36 c^16 : :

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(8182) = EXSIMILCENTER OF THESE CIRCLES: MEDIAL-VAN LAMOEN AND ANTICOMPLEMENTARY-VAN LAMOEN

Barycentrics: 11*a^4-(b^2+c^2)*(b^2+8*a^2+c^2) : :

The insimilcenter of both circles is X(2). Their radical trace is X(625) .

Let A' be the circumcenter of BCX(2) and define B' and C' cyclically. Then X(8182) = X(3)-of-A'B'C'. (Randy Hutson, October 13, 2015)

X(8182) lies on these lines:
(2,187), (3,524), (4,7617), (20,7620), (30,7610), (39,5032), (69,2482), (98,376), (230,5077), (377,7621), (439,7854), (538,6194), (574,1992), (597,1384), (599,5210), (631,7619), (754,3524), (843,6082), (3522,7780), (3528,7751), (3767,7833), (3785,7801), (5023,7800), (5206,7795), (5585,6390)

X(8182) = midpoint of X(20) and X(7620)
X(8182) = reflection of X(i) in X(j) for these (i,j): (2,5569), (4,7617), (7615,7610), (7618,3), (7775,7619), (8176,1153)
X(8182) = anticomplement of X(8176)
X(8182) = {X(1153),X(8176)}-harmonic conjugate of X(2)


X(8183) = MONTESDEOCA DEGENERATE CONICS POINT

Barycentrics: a - a1/3b1/3c1/3 : b - a1/3b1/3c1/3 : c - a1/3b1/3c1/3

(Contributed by Angel Montesceoca, Ocftober 9, 2015) Let ABC be a triangle and k a real number. Let BA on line AB and CA on line AC be points such that BACA is parallel to BC at distance |kr(A)|, where r(A) is the inradius of triangle ABACA. Points CB, AB, AC, BC are defined cyclically. The six points BA, CA, CB, AB, AC, BC lie on a conic, with barycentric equation

0 = cyclic sum of kbc(a + b + c)x2 - a(a2 + b2 + c2 + 2bc + 2ca + 2ab + bck2)yz

The 4 degenerate real conics are given by these values of k: -(a + b + c)/a, -(a + b + c)/b, -(a + b + c)/c, and -(a + b + c)a-1/3b-1/3c-1/3. The 4 singular points of degenerate conics (i.e., points of intersection of the pairs of lines comprising each degenerate conic) are X(8183) and

bc - a2 : ba - bc : ca - cb
ab - ac : ac - b2 : cb - ca
ac - ab : bc - ba : ba - c2

These last three points are collinear on the trilinear polar of X(86).

X(8183) lies on these lines: (pending)

X(8183) = perspector of conic {{A,B,C,PU(124)}}


X(8184) =  CENTER OF THE INNER-VECTEN-VAN LAMOEN CIRCLE

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}



leftri  Centers associated with homothetic pairs of triangles: X(8185)-X(8253)  rightri

This section was contributed by César Lozada, October 13, 2015.

These homothetic centers are related to named triangles defined in ETC. The table lists such centers.

Triangle UThe 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).



underbar



X(8185) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND ARA

Trilinears    a*(a^5+(b+c)*a^4-(b^4+c^4)*a-(b^2-c^2)^2*(b+c)) : :
X(8185) = R^2*X(1)+(SW-6*R^2)*X(25) = R^2*X(8)+(-2*SW+9*R^2)*X(23) = R^2*X(8)+OH^2*X(23)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 1st AURIGA

Trilinears    (a+b+c)*(a-b-c)*a+2*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND 2nd AURIGA

Trilinears    (a+b+c)*(a-b-c)*a-2*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND LUCAS HOMOTHETIC

Trilinears    2*(a^5+(b+c)*a^4-2*(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2+(b^2-c^2)^2*a+(b+c)*((b^2-c^2)^2-4*b^2*c^2))*a +(a^4-6*a^2*(b^2+c^2)-8*(b+c)*(b^2+c^2)*a+(b^2+c^2)^2)*S : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND LUCAS(-1) HOMOTHETIC

Trilinears    2*(a^5+(b+c)*a^4-2*(b^2+c^2)*a^3-2*(b+c)*(b^2+c^2)*a^2+(b^2-c^2)^2*a+(b+c)*((b^2-c^2)^2-4*b^2*c^2))*a -(a^4-6*a^2*(b^2+c^2)-8*(b+c)*(b^2+c^2)*a+(b^2+c^2)^2)*S : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 1st AURIGA

Trilinears    a*(a^2*b^2*c^2*(a+b+c)*(a-b-c)-(a^5+(b+c)*a^4-(b^2+c^2)^2*a-(b^2-c^2)^2*(b+c))*S*D), : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2nd AURIGA

Trilinears    a*(a^2*b^2*c^2*(a+b+c)*(a-b-c)+(a^5+(b+c)*a^4-(b^2+c^2)^2*a-(b^2-c^2)^2*(b+c))*S*D), : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND CAELUM

Trilinears    a*(a^5+(b+c)*a^4-(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)^2*(b+c)) : :
X(8192) = 4*R^2*X(1)-6*R^2*X(2)+SW*X(3) = -4*R^2*X(1)+(6*R^2-SW)*X(25) = -SW*X(3)+2*R^2*X(8)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND OUTER-GARCIA

Trilinears    a*(a^5+(b+c)*a^4-(b^2-c^2)^2*a-(b+c)*(b^2+c^2)^2) : :
X(8193) = 2*R^2*X(1)-SW*X(3)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND LUCAS HOMOTHETIC

Trilinears    a*(b^2+c^2-a^2)*(4*a^2*b^2*c^2*(3*a^2+b^2+c^2)+ (a^6+3*(b^2+c^2)*a^4+(3*b^2+c^2)*(b^2+3*c^2)*a^2+(b^2+c^2)*(b^2-c^2)^2)*S) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND LUCAS(-1) HOMOTHETIC

Trilinears    a*(b^2+c^2-a^2)*(4*a^2*b^2*c^2*(3*a^2+b^2+c^2)-(a^6+3*(b^2+c^2)*a^4+(3*b^2+c^2)*(b^2+3*c^2)*a^2+(b^2+c^2)*(b^2-c^2)^2)*S) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND EULER

Barycentrics    S^2*(-a+b+c)*a^2-((b+c)*a^3+a^2*(b-c)^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*S*D : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-GARCIA

Trilinears    (b+c-a)*(a^3+(b+c)*a^2+4*S*D)/a : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND INNER-GREBE

Trilinears    s*(s-a)*(-2*SW+S)*a-(S+(b+c)*a-b^2-c^2)*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND OUTER-GREBE

Trilinears    s*(s-a)*(-2*SW-S)*a-(-S+(b+c)*a-b^2-c^2)*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND JOHNSON

Barycentrics    S^2*(a-b-c)*a^2-(a^4-(b+c)*a^3+2*a^2*b*c+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*S*D, : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND LUCAS HOMOTHETIC

Trilinears    F(a,b,c,S)+(G(a,b,c,S)+H(a,b,c))*S*D : : , where
D=sqrt(R*r+4*R^2)
F(a,b,c,S) = (a+b+c)*(a-b-c)*(8*a^2*b^2*c^2+(a^2+b^2+c^2)^2*S)*a
G(a,b,c,S) = 4*(a^4-2*(b^2+c^2)*a^2-4*(b+c)*(b^2+c^2)*a+(b^2+c^2)^2)*S
H(a,b,c) =4*(a-b-c)*(a^4+2*(b+c)*a^3+4*a^2*b*c-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2+4*b^2*c^2)*a

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st AURIGA AND LUCAS(-1) HOMOTHETIC

Trilinears    F(a,b,c,-S)+(G(a,b,c,-S)+H(a,b,c))*S*D : : , where D, F(a,b,c,S), G(a,b,c,S) and H(a,b,c) are given at X(8201).

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND EULER

Barycentrics    S^2*(-a+b+c)*a^2+((b+c)*a^3+a^2*(b-c)^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*S*D : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-GARCIA

Trilinears    (b+c-a)*(a^3+(b+c)*a^2-4*S*D)/a : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND INNER-GREBE

Trilinears    s*(s-a)*(-2*SW+S)*a+(S+(b+c)*a-b^2-c^2)*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND OUTER-GREBE

Trilinears    s*(s-a)*(-2*SW-S)*a+(-S+(b+c)*a-b^2-c^2)*S*D : : , where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND JOHNSON

Barycentrics    S^2*(a-b-c)*a^2+(a^4-(b+c)*a^3+2*a^2*b*c+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)*S*D, : : where D=sqrt(R*r+4*R^2)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND LUCAS HOMOTHETIC

Trilinears    F(a,b,c,S)-(G(a,b,c,S)+H(a,b,c))*S*D : : , where D, F(a,b,c,S), G(a,b,c,S) and H(a,b,c) are given at X(8201).

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd AURIGA AND LUCAS(-1) HOMOTHETIC

Trilinears    F(a,b,c,-S)-(G(a,b,c,-S)+H(a,b,c))*S*D : : , where D, F(a,b,c,S), G(a,b,c,S) and H(a,b,c) are given at X(8201).

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: CAELUM AND LUCAS HOMOTHETIC

Trilinears    (a^2*(a+b+c)*(a^2-2*b^2-2*c^2)+(b^4+10*b^2*c^2+c^4)*a+(b+c)*((b^2-c^2)^2-4*b^2*c^2))*a + (2*a^4-2*(b^2+c^2)*(-b^2+2*a*b+2*c*a-c^2))*S : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: CAELUM AND LUCAS(-1) HOMOTHETIC

Trilinears    (a^2*(a+b+c)*(a^2-2*b^2-2*c^2)+(b^4+10*b^2*c^2+c^4)*a+(b+c)*((b^2-c^2)^2-4*b^2*c^2))*a - (2*a^4-2*(b^2+c^2)*(-b^2+2*a*b+2*c*a-c^2))*S : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND LUCAS HOMOTHETIC

Barycentrics    S*(3*SA^2-8*R^2*SA+4*S^2)*a^2+SW^2*(2*S^2-SW*SA)-(S^2-SW^2)*SA^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND LUCAS(-1) HOMOTHETIC

Barycentrics    -S*(3*SA^2-8*R^2*SA+4*S^2)*a^2+SW^2*(2*S^2-SW*SA)-(S^2-SW^2)*SA^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND LUCAS HOMOTHETIC

Barycentrics    (b+c-a)*F(a,b,c)+4*G(a,b,c)*S : : , where
F(a,b,c) = 5*(b+c)*a^7+(5*b^2+22*b*c+5*c^2)*a^6-(b+c)*(7*b^2-26*b*c+7*c^2)*a^5-(7*b^2+26*b*c+7*c^2)*(b-c)^2*a^4+(b+c)*(3*b^4+3*c^4-2*(10*b^2-11*b*c+10*c^2)*b*c)*a^3+(b^2+c^2)*(3*b^2+4*b*c+3*c^2)*(b-c)^2*a^2-(b+c)*(b-c)^2*(b^2+c^2)^2*a-(b^4-c^4)^2
G(a,b,c) = a^6*(a+b+c)-(3*b^2+3*c^2+2*b*c)*a^5+(b+c)*(2*b^2-b*c+2*c^2)*a^4+(3*b^4+3*c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a^3+(b^2+c^2)*(b+c)^3*a^2-(b+c)*(b^2+c^2)^2*(a*b+b*c+c*a)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND LUCAS(-1) HOMOTHETIC

Barycentrics    (b+c-a)*F(a,b,c)-4*G(a,b,c)*S : : , where F(a,b,c) and G(a,b,c) are given in X(8214)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND LUCAS HOMOTHETIC

Trilinears    (a^6-(b^2+c^2)*(a^2*(2*a^2-b^2-c^2)+4*b^2*c^2)-4*(b^4+c^4)*S)*a : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND LUCAS(-1) HOMOTHETIC

Trilinears    a*(a^6-3*(b^2+c^2)^2*a^2+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)-2*(a^4-2*a^2*(b^2+c^2)-b^4-6*b^2*c^2-c^4)*S) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND LUCAS HOMOTHETIC

Trilinears    a*(a^6-3*(b^2+c^2)^2*a^2+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)+2*(a^4-2*a^2*(b^2+c^2)-b^4-6*b^2*c^2-c^4)*S) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND LUCAS(-1) HOMOTHETIC

Trilinears    (a^6-(b^2+c^2)*(a^2*(2*a^2-b^2-c^2)+4*b^2*c^2)+4*(b^4+c^4)*S)*a : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND LUCAS HOMOTHETIC

Barycentrics    (b^2+c^2-a^2)*(F(a,b,c)+2*G(a,b,c)*S) : : , where
F(a,b,c) = 2*a^10+(b^2+c^2)*a^8-4*(11*b^2*c^2+3*c^4+3*b^4)*a^6+2*(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4)*a^4-2*(b^2-c^2)^2*a^2*(3*b^4+4*b^2*c^2+3*c^4)+(b^2+c^2)*(b^2-c^2)^4
G(a,b,c) = 2*a^8-11*(b^2+c^2)*a^6+(3*b^4-22*b^2*c^2+3*c^4)*a^4-(b^2+c^2)*((b^2-c^2)^2-4*b^2*c^2)*a^2-(b^4-c^4)^2

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND LUCAS(-1) HOMOTHETIC

Barycentrics    (b^2+c^2-a^2)*(F(a,b,c)-2*G(a,b,c)*S) : : , where F(a,b,c) and G(a,b,c) are given in X(8220)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC AND MEDIAL

Trilinears    (a^2+S)*(b^2+c^2-a^2)^2/a : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC AND MEDIAL

Trilinears    (a^2-S)*(b^2+c^2-a^2)^2/a : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES 1st CIRCUMPERP AND 2nd PAMFILOS-ZHOU

Trilinears    s*(2*a^3-(b+c)*a^2-(b+c)*(b-c)^2)*b*c+S*(a^4-(b+c)*a^3+a^2*(b^2+c^2)-(b+c)*(b^2-b*c+c^2)*a+b*c*(b-c)^2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES 2nd CIRCUMPERP AND 2nd PAMFILOS-ZHOU

Trilinears    a^5+2*a^3*b*c-a^2*b*(b+c)*c-(b^2+c^2)^2*a-b*c*(b+c)*(b-c)^2-2*(2*a+b+c)*S*b*c : :

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)


X(8226) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND 2nd EXTOUCH

Trilinears    ((b-c)^2*a^4-2*(b^3+c^3)*a^3+2*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^2*(b-c)^2)/a : :
X(8226)=3*(3*R+r)*X(2)-(4*R+r)*X(3)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND HEXYL

Trilinears    (a^4-(b+c)*a^3-(3*b^2-2*b*c+3*c^2)*a^2+(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2)/a : :
X(8227)=X(1)+4*X(5) = 6*X(2)-X(40)

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(8228) = HOMOTHETIC CENTER OF THESE TRIANGLES 3rd EULER AND 2nd PAMFILOS-ZHOU

Barycentrics    s*((b+c)*a^2-(b-c)^2*(-2*a+3*b+3*c))*a*b*c +S*((b-c)^2*a^3-(b+c)*(b^2-3*b*c+c^2)*a^2+(b^3-c^3)*(b-c)*a-(b^4-c^4)*(b-c)) : :

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)


X(8229) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd EULER AND 1st SHARYGIN

Barycentrics    (b+c)*a^3*(a^2-2*b^2+3*b*c-2*c^2)+(b-c)^2*a^4+b*c*(b-c)^2*a^2+(b^3+c^3)*(b-c)^2*(a-b-c) : :
X(8229)= 3*(3*r^2+s^2-3*SW)*X(2)+4*SW*X(3)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4th EULER AND 2nd PAMFILOS-ZHOU

Barycentrics    (b+c)^2*a^4-a^2*b*c*(a*b-4*b*c+c*a)-b*c*(b+c)*((b-c)^2-2*S)*a-(b^2+c^2)*(b^2-c^2)^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: EXCENTRAL AND 2nd PAMFILOS-ZHOU

Trilinears    a^2*(a^2+2*b*c)-2*b*c*((b+c)*a+2*S)-(b^2-c^2)^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH AND HONSBERGER

Trilinears    (a^3-3*(b+c)*a^2+3*(b+c)^2*a-(b+c)*(b-c)^2)/(a*(a-b-c)) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EXTOUCH AND 2nd PAMFILOS-ZHOU

Barycentrics    3*(b+c)*a^4-4*a^3*b*c-2*(b+c)*(b-c)^2*a^2-(b-c)^2*(b+c)^3-2*(b+c)*(a-b+c)*(a+b-c)*S : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HEXYL AND 2nd PAMFILOS-ZHOU

Trilinears    2*s*(a^2+b^2-c^2)*(a^2-b^2+c^2)*b*c+S*(a^5+(-b-c)*a^4-2*a^3*b*c-(b^2+c^2)*(b-c)^2*a+(b-c)^2*(b+c)^3) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HEXYL AND 1st SHARYGIN

Trilinears    a^6+(b+c)*a^5-2*(b+c)*a^3*b*c+(b^2+3*b*c+c^2)*a^4-(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^2-(b+c)*(b^4+c^4-2*b*c*(b^2-3*b*c+c^2))*a+(b^2+b*c+c^2)*(b^2-c^2)^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND HUTSON-INTOUCH

Trilinears    (5*a^3-7*(b+c)*a^2+(b-c)^2*(3*a-b-c))/a : :

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(8237) = HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND 2nd PAMFILOS-ZHOU

Barycentrics    (2*((b+c)*a^2-2*(b^2+b*c+c^2)*a+(b+c)*(b-c)^2)*S-(a-b-c)*(a^4+(b+c)*a^3-(b^2+4*b*c+c^2)*a^2-(b+c)*(b-c)^2*a-2*b*c*(b-c)^2))/(-a+b+c) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HONSBERGER AND 1st SHARYGIN

Trilinears    ((b+c)*a^4+(b+c)*b^2*c^2-(2*b^2+2*c^2+3*b*c)*a^3+(b+2*c)*(2*b+c)*a*b*c+(b+c)*(b^2-3*b*c+c^2)*a^2)/(a-b-c) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON-INTOUCH AND 2nd PAMFILOS-ZHOU

Barycentrics    (-a+b+c)*(4*a^5-(b+c)*a^4-2*(b^2-4*b*c+c^2)*a^3-2*(b^2+c^2)*(b-c)^2*a+(b+c)*(b-c)^4+((2*c+2*b)*a^2+4*a*(b-c)^2+2*(b+c)*(b-c)^2)*S) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON-INTOUCH AND 1st SHARYGIN

Trilinears    (-a+b+c)*((b+c)*a^4+2*(b^2+c^2)*a^3+(b+c)*(b^2+b*c+c^2)*a^2+2*b*c*(b^2+c^2)*a-b*c*(b+c)*(b-c)^2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON-INTOUCH AND TANGENTIAL-MIDARC

Trilinears    4*(s-a)*(a*b*c*sin(A/2)+(s-c)^2*b*sin(C/2)+(s-b)^2*c*sin(B/2))+S^2 : :

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)


X(8242) = HOMOTHETIC CENTER OF THESE TRIANGLES: HUTSON-INTOUCH AND 2nd TANGENTIAL-MIDARC

Barycentrics    -2a + (a + b + c)*sin(A/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: INTOUCH AND 2nd PAMFILOS-ZHOU

Trilinears    ((b+c)*((b-c)^2-a^2+2*S)+4*a*b*c)/(a*(-a+b+c)) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 2nd PAMFILOS-ZHOU

Trilinears    S*(a^4-2*(b+c)*a^3+2*(b^2-b*c+c^2)*a^2-2*(b+c)*(b-c)^2*a+(b^2+4*b*c+c^2)*(b-c)^2) + 2*s*(a^3+(b-c)^2*(a-2*b-2*c))*b*c : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 6th MIXTILINEAR AND 1st SHARYGIN

Trilinears    a^4-3*a^2*b*c+(b+c)*a^3-(b+c)*(3*b^2-4*b*c+3*c^2)*a+(b^2+3*b*c+c^2)*(b-c)^2 : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND 1st SHARYGIN

Trilinears    (b+c)*a^6-b*c*(b-c)^2*a^3-(b+c)*(c^4-b*c*(2*b-c)*(b-2*c)+b^4)*a^2-b*c*(c^4+2*b*c*(b^2-b*c+c^2)+b^4)*a+b^2*c^2*(b+c)*(b-c)^2-2*(a*(a+b+c)*(b+c)+b^3+c^3)*S*b*c : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND TANGENTIAL-MIDARC

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+S^2*(2*R+SW/s) : : , where
F(a,b,c) = ((b+c)*(-2*(c-s)*(b-s)+S)+2*a*b*c)*b*c
G(a,b,c) = -c*((b-c)*S*b+2*(-b+s)*(a^2*(-c+s)+c*(b-c)*(s-a)))

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd PAMFILOS-ZHOU AND 2nd TANGENTIAL-MIDARC

Trilinears    F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)-S^2*(2*R+SW/s) : : , where F(a,b,c) and G(a,b,c) are given in X(8247).

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st SHARYGIN AND TANGENTIAL-MIDARC

Trilinears    a^3+b^3+c^3+a*b*c+2*(b+c)*(a^2-b*c)*sin(A/2)-2*(a+c)*(a*b+c^2)*sin(B/2)-2*(a+b)*(a*c+b^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1st SHARYGIN AND 2nd TANGENTIAL-MIDARC

Trilinears    -( a^3+b^3+c^3+a*b*c)+2*(b+c)*(a^2-b*c)*sin(A/2)-2*(a+c)*(a*b+c^2)*sin(B/2)-2*(a+b)*(a*c+b^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER AND EXTANGENTS

Trilinears    cos(A)*(-2*cos((B-C)/2)*sin(3*A/2)+2*cos(3*(B-C)/2)*sin(A/2)+cos(2*A)+(-2*cos(A)-2)*cos(B-C)-1) : :

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) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND INNER-GREBE

Trilinears    sin A - 4 sin B sin C : :
Barycentrics    a2 - 4S : :
X(8252) = 6*S*X(2)-SW*X(6)

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) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTICOMPLEMENTARY AND OUTER-GREBE

Trilinears    sin A + 4 sin B sin C : :
Barycentrics    a2 + 4S : :
X(8253) = 6*S*X(2)+SW*X(6)

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)



leftri  Midpoints associated with pedal and antipedal triangles   rightri

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]



underbar



X(8254) = MIDPOINT OF X(5) AND X(54)

Barycentrics    2 a^10-7 a^8 b^2+8 a^6 b^4-2 a^4 b^6-2 a^2 b^8+b^10-7 a^8 c^2+6 a^6 b^2 c^2+5 a^4 b^4 c^2-a^2 b^6 c^2-3 b^8 c^2+8 a^6 c^4+5 a^4 b^2 c^4+6 a^2 b^4 c^4+2 b^6 c^4-2 a^4 c^6-a^2 b^2 c^6+2 b^4 c^6-2 a^2 c^8-3 b^2 c^8+c^10 : :
X(8254) = 3 X[2] + X[195] = 5 X[1656] - X[2888] = X[1493] + 2 X[3628] = 3 X[5943] - X[6153] = 3 X[5] - X[6288] = 3 X[54] + X[6288] = 3 X[549] - X[7691]

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) = MIDPOINT OF X(7) AND X(55)

Barycentrics    2 a^4 b-5 a^3 b^2+3 a^2 b^3+a b^4-b^5+2 a^4 c-3 a^2 b^2 c-2 a b^3 c+3 b^4 c-5 a^3 c^2-3 a^2 b c^2+2 a b^2 c^2-2 b^3 c^2+3 a^2 c^3-2 a b c^3-2 b^2 c^3+a c^4+3 b c^4-c^5 : :

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) = MIDPOINT OF X(8) AND X(56)

Barycentrics    2 a^3 b-a^2 b^2-2 a b^3+b^4+2 a^3 c-4 a^2 b c+4 a b^2 c-a^2 c^2+4 a b c^2-2 b^2 c^2-2 a c^3+c^4 : :
X(8256) = X[3436] - 5 X[3617] = X[46] + 3 X[3679] = r X[5] + (R - 2 r) X[10]

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) = MIDPOINT OF X(9) AND X(57)

Barycentrics    a (a^4-2 a^3 b+2 a b^3-b^4-2 a^3 c+4 a^2 b c-4 a b^2 c+2 b^3 c-4 a b c^2-2 b^2 c^2+2 a c^3+2 b c^3-c^4) : : X(8257) = X[2096] + 3 X[5817] = 3(R - r) X[2] + (r + R) X[7]

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) = MIDPOINT OF X(10) AND X(58)

Barycentrics    2 a^4+3 a^3 b+b^4+3 a^3 c+2 a^2 b c+a b^2 c+b^3 c+a b c^2+b c^3+c^4 : :
X(8258) = 3 X[2] + X[1046] = X[1330] - 5 X[1698] = X[8] + 3 X[5429]

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) = MIDPOINT OF X(17) AND X(61)

Barycentrics    7 a^2 b^2-b^4+7 a^2 c^2+2 b^2 c^2-c^4+2 Sqrt[3] (4 a^2+b^2+c^2) S : :

X(8259) lies on these lines: {5,14}, {6,627}, {395,629}, {532,5007}, {597,7807}, {635,6673} (pending)

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) = MIDPOINT OF X(18) AND X(62)

Barycentrics    7 a^2 b^2-b^4+7 a^2 c^2+2 b^2 c^2-c^4-2 Sqrt[3] (4 a^2+b^2+c^2) S : :

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) = MIDPOINT OF X(21) AND X(65)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-a^3 b^2 c-2 a^2 b^3 c+2 b^5 c-a^4 c^2-a^3 b c^2-6 a^2 b^2 c^2-3 a b^3 c^2+b^4 c^2-2a^3 c^3-2 a^2 b c^3-3 a b^2 c^3-4 b^3 c^3+2 a^2 c^4+b^2 c^4+a c^5+2 b c^5-c^6) : :
X(8261) = 5 X[4004] + X[5441] = X[191] + 3 X[5902] = 3 X[5426] + X[5903]

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) = MIDPOINT OF X(23) AND X(67)

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

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) = MIDPOINT OF X(25) AND X(69)

Barycentrics    3 a^6 b^2-a^4 b^4-3 a^2 b^6+b^8+3 a^6 c^2-6 a^4 b^2 c^2+7 a^2 b^4 c^2-a^4 c^4+7 a^2 b^2 c^4-2 b^4 c^4-3 a^2 c^6+c^8 : :

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) = ANTICOMPLEMENT OF X(1502)

Barycentrics    b-4 + c-4 - a-4 : :

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) = 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) = COMPLEMENT OF X(1502)

Barycentrics    b-4 + c-4 : :

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) = 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(6) and X(76)
X(8265) = crosspoint of X(2) and X(32)
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) = ANTICOMPLEMENT OF X(3613)

Barycentrics    a^2 (a^4 b^2-a^2 b^4+a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4) : :
X(8266) = S*X(3) - R2Sin(2ω)*X(6)

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) = ANTICOMPLEMENT OF X(8024)

Barycentrics    a^4 b^2+a^2 b^4+a^4 c^2-b^4 c^2+a^2 c^4-b^2 c^4 : :
X(8267) = 3X(2) - 4X(1194)

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)



leftri  Midpoints associated with pedal and antipedal triangles   rightri

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.



underbar



X(8268) = CENTER OF THIS CONIC: APEDAL(X(2))

Barycentrics    (a-b) (a+b) (a-c) (a+c) (a^4+10 a^2 b^2+b^4+3 a^2 c^2+3 b^2 c^2) (a^4+3 a^2 b^2+10 a^2 c^2+3 b^2 c^2+c^4) (3 a^10-15 a^8 b^2+18 a^6 b^4+126 a^4 b^6-5 a^2 b^8+b^10-15 a^8 c^2+36 a^6 b^2 c^2-78 a^4 b^4 c^2+236 a^2 b^6 c^2-51 b^8 c^2+18 a^6 c^4-78 a^4 b^2 c^4-846 a^2 b^4 c^4+50 b^6 c^4+126 a^4 c^6+236 a^2 b^2 c^6+50 b^4 c^6-5 a^2 c^8-51 b^2 c^8+c^10) : :

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) = CENTER OF THIS CONIC: APEDAL(X(40))

Barycentrics    a/((b + c - a)2(b - c) (a2 + b2 + c2 - 2bc) : :

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) = REFLECTION OF X(1) IN X(1060)

Barycentrics    a (a+b-c) (a-b+c) (a^3-a^2 b+a b^2-b^3-a^2 c+2 a b c-b^2 c+a c^2-b c^2-c^3) : :
X(8270) = [(r + 2 R)^2 - s^2]*X(1) - 2rR(X(40)

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) = X(6604)-CEVA CONJUGATE OF X(57)

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

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) = ANTICOMPLEMENT OF X(1031)

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

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) <



leftri  Perspectors: X(8273) - X(8350)  rightri

This section was contributed by César 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 triangleX(5573)
1st anti-Brocard triangleX(5939)
Antlia triangleX(5574)
Apus triangleX(5584)
Aquila triangleX(5586)
Ara triangleX(5594)
Aries triangleX(5596)
Auriga trianglesX(5597)
Caelum triangleX(5603)
Euler trianglesX(3758)
Extouch trianglesX(5927)
Garcia-inner/outer trianglesX(5587)
Honsberger triangleX(7676)
Hutson-inner/outer trianglesX(363)
Hutson-intouch/extouch trianglesX(5731)
Lucas(n) Brocard trianglesX(6421)
Lucas(n) antipodal trianglesX(6457)
Lucas(n) homothetic trianglesX(493)
Lucas(n) inner-tangential trianglesX(6394)
Lucas(n) reflection trianglesX(6401)
Lucas(n) secondary central/tangents trianglesX(6199)
McCay triangleX(7606)
Mixtilinear trianglesX(7955)
Pamfilos-Zhou trianglesX(7594)
Schroeter triangleX(8286)
Sharygin trianglesX(8229)
Tangential-midarc trianglesX(8075)
Trinh triangleX(7688)



underbar



X(8273) = PERSPECTOR OF THESE TRIANGLES: APUS AND 2nd MIXTILINEAR

Trilinears    (a^5-(b+c)*a^4-2*(b^2+4*b*c+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+((b^2-c^2)^2+8*b*c*(b^2+c^2))*a-(b^2-c^2)^2*(b+c))*a : :
X(8273) = R*X(1)+(4*R+r)*X(3)

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) = PERSPECTOR OF THESE TRIANGLES: AQUILA AND EXTANGENTS

Trilinears
(4*b^2+7*b*c+4*c^2)*a^4+(b+c)*(4*b^2+11*b*c+4*c^2)*a^3-(4*b^2-11*b*c+4*c^2)*(b+c)^2*a^2-2*(b^2-c^2)^2*b*c-(b+c)*(4*b^4+4*c^4+3*b*c*(b-c)^2)*a : :

X(8274) lies on these lines:
{1,71}, {40,500}, {65,3679}, {209,1698}, {1044,2093}, {5691,6254}


X(8275) = PERSPECTOR OF THESE TRIANGLES: AQUILA AND HUTSON-INTOUCH

Barycentrics    (-a+b+c)*(3*a^3-6*(b+c)*a^2-7*(b-c)^2*a+2*(b+c)*(b-c)^2) : :
X(8275) = (4*R-9*r)*X(1)+10*r*X(631)

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) = PERSPECTOR OF THESE TRIANGLES: ARA AND LUCAS CENTRAL

Trilinears    (a^8+4*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^2-c^2)^4-8*S*a^2*b^2*c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: ARA AND LUCAS(-1) CENTRAL

Trilinears    (a^8+4*a^4*b^2*c^2-2*(b^2+c^2)*a^6+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^2-c^2)^4+8*S*a^2*b^2*c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: ARA AND 3RD MIXTILINEAR

Trilinears
(a^6-(b-c)^2*a^4+4*(b+c)*b*c*a^3-(b^4+c^4-2*b*c*(2*b^2-11*b*c+2*c^2))*a^2+4*(b+c)^3*b*c*a+(b^2-c^2)^2*(b+c)^2)*a/(-a+b+c) : :

X(8278) lies on these lines:
{56,197}, {100,4308}


X(8279) = PERSPECTOR OF THESE TRIANGLES: 2nd EULER AND INNER-GARCIA

Trilinears    1-(cos(A)^2-cos(A)+1)*cos(A)-(-cos(A)^2-cos(A)+1)*cos(B-C)-2*sin(A/2)*(1-3*sin(A/2)^2+4*sin(A/2)^4)*cos((B-C)/2)+2*sin(A/2)^3*cos(3*(B-C)/2) : :
Barycentrics    a (a^9-a^8 b-2 a^7 b^2+4 a^6 b^3-6 a^4 b^5+2 a^3 b^6+4 a^2 b^7-a b^8-b^9-a^8 c+2 a^7 b c-6 a^5 b^3 c+6 a^4 b^4 c+6 a^3 b^5 c-8 a^2 b^6 c-2 a b^7 c+3 b^8 c-2 a^7 c^2+8 a^5 b^2 c^2-4 a^4 b^3 c^2-10 a^3 b^4 c^2+4 a^2 b^5 c^2+4 a b^6 c^2+4 a^6 c^3-6 a^5 b c^3-4 a^4 b^2 c^3+12 a^3 b^3 c^3+2 a b^5 c^3-8 b^6 c^3+6 a^4 b c^4-10 a^3 b^2 c^4-6 a b^4 c^4+6 b^5 c^4-6 a^4 c^5+6 a^3 b c^5+4 a^2 b^2 c^5+2 a b^3 c^5+6 b^4 c^5+2 a^3 c^6-8 a^2 b c^6+4 a b^2 c^6-8 b^3 c^6+4 a^2 c^7-2 a b c^7-a c^8+3 b c^8-c^9) : :

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) = PERSPECTOR OF THESE TRIANGLES: 5TH EULER AND INNER-SQUARES

Barycentrics    (8*R^2-3*SW)*S^2-SW*(SB*SC+(SB+SC)*S) : :

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) = PERSPECTOR OF THESE TRIANGLES: 5TH EULER AND OUTER-SQUARES

Barycentrics    (8*R^2-3*SW)*S^2-SW*(SB*SC-(SB+SC)*S) : :

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) = PERSPECTOR OF THESE TRIANGLES: 3RD EXTOUCH AND OUTER-GARCIA

Trilinears
a^9+5*(b+c)*a^8+4*(2*c+b)*(c+2*b)*a^7+20*b*c*(b+c)*a^6-2*(7*b^2-8*b*c+7*c^2)*(b+c)^2*a^5-2*(b+c)*(7*b^4+7*c^4+2*(3*b^2-5*b*c+3*c^2)*b*c)*a^4-4*(b^2-c^2)^2*b*c*a^3+4*(2*b^2-b*c+2*c^2)*(b-c)^2*(b+c)^3*a^2+(5*b^2+6*b*c+5*c^2)*(b+c)^2*(b-c)^4*a+(b^2-6*b*c+c^2)*(b-c)^2*(b+c)^5 : :

X(8282) lies on these lines:
{4,2331}, {10,223}, {387,7682}, {2093,3987}, {3421,5930}


X(8283) = PERSPECTOR OF THESE TRIANGLES: 2nd CIRCUMPERP AND 5TH EXTOUCH

Trilinears
(a^7-(b^2+4*b*c+c^2)*a^5+6*b*c*(b+c)*a^4-(b^4+6*b^2*c^2+c^4)*a^3-4*b*c*(b+c)*(b-c)^2*a^2+(b^2-c^2)^2*(b^2+4*b*c+c^2)*a-2*b*c*(b^2-c^2)^2*(b+c))/(-a+b+c) : :

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) = PERSPECTOR OF THESE TRIANGLES: INNER-GARCIA AND TRINH

Trilinears    4*(16*p^4-16*p^2+5)*p*q^3+(64*p^6-144*p^4+100*p^2-27)*p*q+(-64*p^6+48*p^4-6)*q^2-(64*p^6-48*p^4+9)*(p^2-1) : : , where p=sin(A/2) and q=cos((B-C)/2)
Barycentrics    a^2 (a^11-a^10 b-3 a^9 b^2+3 a^8 b^3+2 a^7 b^4-2 a^6 b^5+2 a^5 b^6-2 a^4 b^7-3 a^3 b^8+3 a^2 b^9+a b^10-b^11-a^10 c+3 a^9 b c+2 a^8 b^2 c-6 a^7 b^3 c-a^6 b^4 c+a^4 b^6 c+6 a^3 b^7 c-2 a^2 b^8 c-3 a b^9 c+b^10 c-3 a^9 c^2+2 a^8 b c^2+8 a^7 b^2 c^2-6 a^6 b^3 c^2-6 a^5 b^4 c^2+6 a^4 b^5 c^2-2 a^2 b^7 c^2+a b^8 c^2+3 a^8 c^3-6 a^7 b c^3-6 a^6 b^2 c^3+17 a^5 b^3 c^3+2 a^4 b^4 c^3-9 a^3 b^5 c^3+a^2 b^6 c^3-2 a b^7 c^3+2 a^7 c^4-a^6 b c^4-6 a^5 b^2 c^4+2 a^4 b^3 c^4-2 a b^6 c^4+5 b^7 c^4-2 a^6 c^5+6 a^4 b^2 c^5-9 a^3 b^3 c^5+10 a b^5 c^5-5 b^6 c^5+2 a^5 c^6+a^4 b c^6+a^2 b^3 c^6-2 a b^4 c^6-5 b^5 c^6-2 a^4 c^7+6 a^3 b c^7-2 a^2 b^2 c^7-2 a b^3 c^7+5 b^4 c^7-3 a^3 c^8-2 a^2 b c^8+a b^2 c^8+3 a^2 c^9-3 a b c^9+a c^10+b c^10-c^11) : :

X(8284) lies on these lines:
{3,3582}


X(8285) = PERSPECTOR OF THESE TRIANGLES: 2nd MIXTILINEAR AND 2nd SHARYGIN

Trilinears
a^6*b*c-(b+c)*(b^2+3*b*c+c^2)*a^5+(4*b^4+4*c^4+3*(2*b^2-b*c+2*c^2)*b*c)*a^4+(b-c)^4*b^2*c^2-2*(b+c)*(3*b^4+3*c^4-(b+c)^2*b*c)*a^3+(4*b^4+4*c^4+3*(3*b^2+4*b*c+3*c^2)*b*c)*(b-c)^2*a^2-(b^3+c^3)*(b^2-c^2)^2*a : :

X(8285) lies on these lines:
{40,170}, {279,291}


X(8286) = PERSPECTOR OF THESE TRIANGLES: SCHROETER AND EXTOUCH

Barycentrics    (b+c)*(b-c)^2*(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+b^3+c^3) : :

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) = PERSPECTOR OF THESE TRIANGLES: SCHROETER AND INTOUCH

Barycentrics    (a^2-b^2-b*c-c^2)*(b+c)*(b-c)^2 : :

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) = 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) = PERSPECTOR OF THESE TRIANGLES: SCHROETER AND LEMOINE

Barycentrics    (b^2-c^2)^2*(a^2-2*b^2-2*c^2) : :

X(8288) lies on Lemoine inellipse and these lines:
{2,353}, {6,6032}, {115,125}, {338,850}, {1501,1853}, {3016,7603}, {6034,6792}


X(8289) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND CIRCUMSYMMEDIAL

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

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND MEDIAL

Barycentrics    (a^2*(a^2+b^2+c^2)-(b^2+c^2)^2+b^2*c^2)*(a^4-b^2*c^2) : :
Barycentrics    (sin B)/(sin B - 2 cos B sin 2ω) + (sin C)/(sin C - 2 cos C sin 2ω) : :

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(8291) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND INNER-NAPOLEON

Barycentrics
(9*S^2+SW^2)*SA^2*SW^2-(3*S^2-2*SW^2)*(S^2+SW^2)*S^2+(3*S^4-6*S^2*SW^2-SW^4)*SA*SW-3^(1/2)*((S^2+SW^2)*S^2+(-3*S^2+5*SW^2)*SA^2+4*(S-SW)*(S+SW)*SA*SW)*SW*S : :

X(8291) lies on these lines:
{3,5979}, {182,5980}, {1916,3106}, {5152,8292}, {5463,7840}, {5617,5978}, {5982,5989}, {6294,7781}


X(8292) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND OUTER-NAPOLEON

Barycentrics
-4*(9*S^2+SW^2)*SA^2*SW^2+4*(3*S^2-2*SW^2)*(S^2+SW^2)*S^2-4*(3*S^4-6*S^2*SW^2-SW^4)*SA*SW-4*3^(1/2)*((S^2+SW^2)*S^2+(-3*S^2+5*SW^2)*SA^2+4*(-S-SW)*(-S+SW)*SA*SW)*SW*S : :

X(8292) lies on these lines:
{3,5978}, {182,5981}, {1916,3107}, {5152,8291}, {5464,7840}, {5613,5979}, {5983,5989}, {6581,7781}


X(8293) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND INNER-VECTEN

Barycentrics    (S^2+SW^2)*(2*S^2+SA*SW-SW^2)*(SA^2-SB*SC)-SW*((S^2+SW^2)*S^2-(3*S^2-5*SW^2)*SA^2+4*(S^2-SW^2)*SA*SW)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND OUTER-VECTEN

Barycentrics    (S^2+SW^2)*(2*S^2+SA*SW-SW^2)*(SA^2-SB*SC)-SW*((S^2+SW^2)*S^2-(3*S^2-5*SW^2)*SA^2+4*(S^2-SW^2)*SA*SW)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND 1st NEUBERG

Barycentrics    2*SA*SW*(S^2-SW^2)*((7*S^2-SW^2)*SA*SW+(S^2-SW^2)^2-4*S^2*SW^2)-(S^2+SW^2)*((S^2+2*SW^2)^2-5*SW^4)*S^2 : :

X(8295) lies on these lines:
{3,1916}, {147,5152}, {6287,8290}


X(8296) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND CIRCUMSYMMEDIAL

Trilinears    2*a^4-b*c*a^2-(b^3+c^3)*a-3*b^2*c^2 : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND CIRCUMSYMMEDIAL

Trilinears    2*a^4-3*b*c*a^2-(b^3+c^3)*a+3*b^2*c^2 : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND INCENTRAL

Trilinears    (a^2-b*c)*(a^2-b^2-b*c-c^2+(b+c)*a) : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND MEDIAL

Trilinears    (a^2-b*c)*((b+c)*a-b^2-c^2) : :
X(8299) = (-2*R*s^2+R*SW)*X(1)+3*R*SW*X(2)+(-2*r*s^2+SW*r)*X(3)

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND SYMMEDIAL

Trilinears    (a^2-b*c)^2 : :

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) = {X(8348),X(8349)}-harmonic conjugate of X(3)


X(8301) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND TANGENTIAL

Trilinears    a^4-b*c*a^2-(b^3+c^3)*a+2*b^2*c^2 : :

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) = crosspoint of PU(134)


X(8302) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS BROCARD

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

X(8302) lies on these lines:
{2,99}, {1151,5989}, {1281,8318}, {1916,6421}, {4027,5058}


X(8303) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) BROCARD

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

X(8303) lies on these lines:
{2,99}, {1152,5989}, {1281,8319}, {1916,6422}, {4027,5062}


X(8304) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS CENTRAL

Barycentrics    SW*SA*(-3*S^2+SW^2)*a^2-(-c^4+a^2*b^2)*(-b^4+a^2*c^2)*S : :

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} X(8304) = {X(3),X(1916)}-harmonic conjugate of X(8305)


X(8305) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) CENTRAL

Barycentrics    SW*SA*(-3*S^2+SW^2)*a^2+(-c^4+a^2*b^2)*(-b^4+a^2*c^2)*S : :

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} X(8305) = {X(3),X(1916)}-harmonic conjugate of X(8304)


X(8306) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS INNER

Barycentrics    8*SW*SA*(-3*S^2+SW^2)*a^2+(2*a^8-7*a^4*b^2*c^2+(3*(b^2+c^2))*(b^4-b^2*c^2+c^4)*a^2-b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) INNER

Barycentrics    8*SW*SA*(-3*S^2+SW^2)*a^2-(2*a^8-7*a^4*b^2*c^2+(3*(b^2+c^2))*(b^4-b^2*c^2+c^4)*a^2-b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS INNER-TANGENTIAL

Barycentrics    8*SW*SA*(-3*S^2+SW^2)*a^2+(3*a^8-7*b^2*c^2*a^4+(b^6+c^6)*a^2+2*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) INNER-TANGENTIAL

Barycentrics    8*SW*SA*(-3*S^2+SW^2)*a^2-(3*a^8-7*b^2*c^2*a^4+(b^6+c^6)*a^2+2*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS SECONDARY CENTRAL

Barycentrics    SW*SA*(3*S^2-SW^2)*a^2-(2*a^8-3*b^2*c^2*a^4-(b^6+c^6)*a^2+3*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) SECONDARY CENTRAL

Barycentrics    SW*SA*(3*S^2-SW^2)*a^2+(2*a^8-3*b^2*c^2*a^4-(b^6+c^6)*a^2+3*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS 1st SECONDARY TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2-(3*a^8-4*a^4*b^2*c^2-(2*(b^6+c^6))*a^2+5*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) 1st SECONDARY TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2+(3*a^8-4*a^4*b^2*c^2-(2*(b^6+c^6))*a^2+5*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS 2nd SECONDARY TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2+(a^8+4*a^4*b^2*c^2-(6*(b^6+c^6))*a^2+7*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) 2nd SECONDARY TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2-(a^8+4*a^4*b^2*c^2-(6*(b^6+c^6))*a^2+7*b^4*c^4)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2-(a^4-b^2*c^2)^2*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND LUCAS(-1) TANGENTS

Barycentrics    2*SW*SA*(3*S^2-SW^2)*a^2+(a^4-b^2*c^2)^2*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS BROCARD

Trilinears    (a^2*(a^2-2*b*c)-2*a*(b^3+c^3)-3*b^2*c^2)*S-(b^2-b*c+c^2)*a^4+(b^3+c^3)*a^3+b^2*c^2*(a^2+b^2+c^2) : :

X(8318) lies on these lines:
{256,6421}, {574,8319}, {1151,8324}, {1281,8302}, {1580,5058}


X(8319) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) BROCARD

Trilinears    -(a^2*(a^2-2*b*c)-2*a*(b^3+c^3)-3*b^2*c^2)*S-(b^2-b*c+c^2)*a^4+(b^3+c^3)*a^3+b^2*c^2*(a^2+b^2+c^2) : :

X(8319) lies on these lines:
{256,6422}, {574,8318}, {1152,8325}, {1281,8303}, {1580,5062}


X(8320) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS CENTRAL

Trilinears    a*(b^2+c^2-a^2)*(a^3+b*c*a+b^3+c^3)+4*(b*a+c^2)*(c*a+b^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) CENTRAL

Trilinears    a*(b^2+c^2-a^2)*(a^3+b*c*a+b^3+c^3)-4*(b*a+c^2)*(c*a+b^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS INNER

Trilinears    2*a*(b^2+c^2-a^2)*(a^3+b*c*a+b^3+c^3)+(2*a^4+3*b*c*a^2+3*a*(b^3+c^3)+b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) INNER

Trilinears    2*a*(b^2+c^2-a^2)*(a^3+b*c*a+b^3+c^3)-(2*a^4+3*b*c*a^2+3*a*(b^3+c^3)+b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS INNER TANGENTIAL

Trilinears    -2*(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a+(3*a^4+b*c*a^2+(b^3+c^3)*a-2*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) INNER TANGENTIAL

Trilinears    -2*(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a-(3*a^4+b*c*a^2+(b^3+c^3)*a-2*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS SECONDARY CENTRAL

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a+4*(2*a^4-b*c*a^2-(b^3+c^3)*a-3*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) SECONDARY CENTRAL

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a-4*(2*a^4-b*c*a^2-(b^3+c^3)*a-3*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS 1st SECONDARY TANGENTS

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a+2*(3*a^4-2*b*c*a^2-2*(b^3+c^3)*a-5*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) 1st SECONDARY TANGENTS

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a-2*(3*a^4-2*b*c*a^2-2*(b^3+c^3)*a-5*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS 2nd SECONDARY TANGENTS

Trilinears    (a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a+2*(a^4-6*b*c*a^2-6*(b^3+c^3)*a-7*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) 2nd SECONDARY TANGENTS

Trilinears    (a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a-2*(a^4-6*b*c*a^2-6*(b^3+c^3)*a-7*b^2*c^2)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS TANGENTS

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a+2*(a^2-b*c)*(a^2+b*c)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st SHARYGIN AND LUCAS(-1) TANGENTS

Trilinears    -(a^2-b^2-c^2)*(a^3+b*c*a+b^3+c^3)*a-2*(a^2-b*c)*(a^2+b*c)*S : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS BROCARD

Trilinears    (b^2+b*c+c^2)*a^4-(b^3+c^3)*a^3-b*c*(2*b^2-b*c+2*c^2)*a^2+b^2*c^2*(b^2+c^2)-(a^4-2*(b^3+c^3)*a+3*b^2*c^2)*S : :

X(8334) lies on these lines:
{291,6421}, {574,8335}, {1151,8301}, {1281,8302}, {5058,8300}


X(8335) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) BROCARD

Trilinears    (b^2+b*c+c^2)*a^4-(b^3+c^3)*a^3-b*c*(2*b^2-b*c+2*c^2)*a^2+b^2*c^2*(b^2+c^2)+(a^4-2*(b^3+c^3)*a+3*b^2*c^2)*S : :

X(8335) lies on these lines:
{291,6422}, {574,8334}, {1152,8301}, {1281,8303}, {5062,8300}


X(8336) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS CENTRAL

Trilinears    4*(a*b-c^2)*(a*c-b^2)*S+(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) CENTRAL

Trilinears    -4*(a*b-c^2)*(a*c-b^2)*S+(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS INNER

Trilinears    (2*a^4-7*b*c*a^2+3*(b^3+c^3)*a-b^2*c^2)*S-2*(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) INNER

Trilinears    -(2*a^4-7*b*c*a^2+3*(b^3+c^3)*a-b^2*c^2)*S-2*(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS INNER TANGENTIAL

Trilinears    (3*a^4-7*a^2*b*c+(b^3+c^3)*a+2*b^2*c^2)*S-2*(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) INNER TANGENTIAL

Trilinears    -(3*a^4-7*a^2*b*c+(b^3+c^3)*a+2*b^2*c^2)*S-2*(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS SECONDARY CENTRAL

Trilinears    4*(2*a^4-3*a^2*b*c+3*b^2*c^2-(b^3+c^3)*a)*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) SECONDARY CENTRAL

Trilinears    -4*(2*a^4-3*a^2*b*c+3*b^2*c^2-(b^3+c^3)*a)*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS 1st SECONDARY TANGENTS

Trilinears    2*(3*a^4-4*a^2*b*c-2*(b^3+c^3)*a+5*b^2*c^2)*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) 1st SECONDARY TANGENTS

Trilinears    -2*(3*a^4-4*a^2*b*c-2*(b^3+c^3)*a+5*b^2*c^2)*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS 2nd SECONDARY TANGENTS

Trilinears    2*(a^4+4*a^2*b*c-6*(b^3+c^3)*a+7*b^2*c^2)*S+(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) 2nd SECONDARY TANGENTS

Trilinears    -2*(a^4+4*a^2*b*c-6*(b^3+c^3)*a+7*b^2*c^2)*S+(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS TANGENTS

Trilinears    2*(a^2-b*c)^2*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 2nd SHARYGIN AND LUCAS(-1) TANGENTS

Trilinears    -2*(a^2-b*c)^2*S-(a+b+c)*(a^2-(b+c)*a+b^2-b*c+c^2)*(a^2-b^2-c^2)*a : :

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) = PERSPECTOR OF THESE TRIANGLES: 1st ANTI-BROCARD AND MCCAY

Barycentrics    2*(15*S^2-SW^2)*SW^2*SA^2+2*(3*S^4-12*S^2*SW^2+SW^4)*SW*SA-(3*S^2+SW^2)*(S^2+SW^2)*S^2 : :
X(8350) = 4*SW^2*(15*S^2-SW^2)*X(3)+3*(3*S^2-SW^2)*(S^2+SW^2)*X(7840)
X(8350) = 3 (3 S^2 - 5 SW^2) (S^2 + SW^2) X[2] + 4 SW^2 (15 S^2 - SW^2) X[3] + 12 SW^2 (S^2 + SW^2) X[6]

X(8350) lies on these lines:
{3,7840}


X(8351) = CENTER OF THE YFF CENTRAL CIRCLE

Barycentrics    a + (a + b + c) sin(A/2) : :

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}


X(8352) = MIDPOINT OF X(148) AND X(7840)

Barycentrics    4 a^4-a^2 b^2-5 b^4-a^2 c^2+8 b^2 c^2-5 c^4 : :

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)


X(8353) = MIDPOINT OF X(7757) AND X(7802)

Barycentrics    6 a^4-5 a^2 b^2-3 b^4-5 a^2 c^2+4 b^2 c^2-3 c^4 : :

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): (7757,7762), (7833,8356), (8354,2), (8356,8370)


X(8354) = MIDPOINT OF X(2) AND X(8353)

Barycentrics    6 a^4-7 a^2 b^2-3 b^4-7 a^2 c^2+2 b^2 c^2-3 c^4 : :

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): (8356,8359), (8358,2)


X(8355) = MIDPOINT OF X(625) AND X(5461)

Barycentrics    2 a^4-5 a^2 b^2+11 b^4-5 a^2 c^2-14 b^2 c^2+11 c^4 : :

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}


X(8356) = MIDPOINT OF X(2) AND X(7833)

Barycentrics    2 a^4-3 a^2 b^2-b^4-3 a^2 c^2-c^4 : :

As a point on the Euler line, X(8356) has Shinagawa coefficients ((E+F)2+2S2, -3S2).

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): (2,8370},(7833,8353), (8354,7833), (8358,8359), (8359,2)


X(8357) = MIDPOINT OF X(6655) AND X(6656)

Barycentrics    2 a^4-3 a^2 b^2-3 b^4-3 a^2 c^2+2 b^2 c^2-3 c^4 : :

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): (6656,7819), (8364,384)


X(8358) = MIDPOINT OF X(2) AND X(8354)

Barycentrics    6 a^4-11 a^2 b^2-3 b^4-11 a^2 c^2-2 b^2 c^2-3 c^4 : :

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}


X(8359) = MIDPOINT OF X(2) AND X(8356)

Barycentrics    2 a^4-5 a^2 b^2-b^4-5 a^2 c^2-2 b^2 c^2-c^4 : :

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): (7810,7767), (8356,8354), (8358,8356), (8367,8370)
X(8359) = complement X(8370)
X(8359) = anticomplement X(8367)


X(8360) = MIDPOINT OF X(626) AND X(7817)

Barycentrics    2 a^4+a^2 b^2+5 b^4+a^2 c^2-2 b^2 c^2+5 c^4 : :

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}: {626,7817}, {5254,7801}, {5306,7818}, {7841,8369}
X(8360) = reflection of X(i) in X(j) for these (i,j): (2,8368), (7817,5305), (8365,8369)
X(8360) = complement X(8369)
X(8360) = anticomplement X(8365)


X(8361) = MIDPOINT OF X(5025) AND X(7807)

Barycentrics    2 a^4-a^2 b^2+3 b^4-a^2 c^2-2 b^2 c^2+3 c^4 : :

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) = reflection of X(i) in X(j) for these (i,j): (5025,7807), (7755,7821)
X(8361) = complement X(7807)


X(8362) = MIDPOINT OF X(7770) AND X(7791)

Barycentrics    (b^2+c^2) (3 a^2+b^2+c^2) : :

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(8363) = MIDPOINT OF X(7856) AND X(7922)

Barycentrics    2 a^4+a^2 b^2+3 b^4+a^2 c^2+3 c^4 : :

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(8364) = MIDPOINT OF X(384) AND X(8357)

Barycentrics    2 a^4+3 a^2 b^2+3 b^4+3 a^2 c^2+2 b^2 c^2+3 c^4 : :

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(8365) = MIDPOINT OF X(2) AND X(8368)

Barycentrics    10 a^4-a^2 b^2+7 b^4-a^2 c^2+2 b^2 c^2+7 c^4 : :

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(8366) = MIDPOINT OF X(5346) AND X(7801)

Barycentrics    7 a^4-a^2 b^2+4 b^4-a^2 c^2+2 b^2 c^2+4 c^4 : :

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)


X(8367) = MIDPOINT OF X(7745) AND X(7810)

Barycentrics    2 a^4+7 a^2 b^2-b^4+7 a^2 c^2+10 b^2 c^2-c^4 : :

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(8368) = MIDPOINT OF X(2) AND X(8369)

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

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): (2,8360), (8365,2)


X(8369) = MIDPOINT OF X(2) AND X(1003)

Barycentrics    4 a^4-a^2 b^2+b^4-a^2 c^2+2 b^2 c^2+c^4 : :

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): (6680,7817), (7789,7801), (7801,3933), (7817,5254), (8360,7841), (8365,8360), (8368,2)
X(8369) = complement X(7841)
X(8369) = anticomplement(X(8360)


X(8370) = MIDPOINT OF X(76) AND X(7812)

Barycentrics    2 a^4+a^2 b^2-b^4+a^2 c^2+4 b^2 c^2-c^4 : :

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}: {76,7812}, {7747,7810}
X(8370) = reflection of X(i) in X(j) for these (i,j): (2,8356), (3934,7810), (7745,7812), (7810,7750), (7812,7762), (8356,8353), (8359,7833), (8367,8359)
X(8370) = complement X(7833)
X(8370) = anticomplement(X(8359)
X(8370) = inverse-in-orthocentroidal-circle of X(7841)
X(8370) = sum of vertices of triangle A"B"C" defined at X(5640)


X(8371) = MIDPOINT OF X(2) AND X(5466)

Barycentrics    (2*a^4-2*(b^2+c^2)*a^2-c^4+4*b^2*c^2-b^4)*(b^2-c^2) : :
Barycentrics   (3*SA^2+6*SB*SC-SW^2)*(SB-SC) : :

Let HP be the Hutson-Parry circle (see X(5466)), with center X(8371) and pass-through point X(2). HP passes through these 10 points: X(2), X(13), X(14), X(111), X(476), X(5466), X(5640), X(6032), X(6792), X(7698). 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 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



leftri  Perspectors of pairs of triangles: X(8372) - X(8424)  rightri

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 Lozada, November 2, 2015. See Perspectivities in Tables (accessible at the top of this page).

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X(8372) = PERSPECTOR OF THESE TRIANGLES: ABC AND OUTER-HUTSON

Trilinears    (b-c)/((sin(B/2)-sin(C/2))*a-sin(A/2)*(b-c)) : :

X(8372) lies on the Feuerbach hyperbola and these lines:
{1,8111}, {9,164}, {21,8110}, {266,8138}, {2346,8386}


X(8373) = PERSPECTOR OF THESE TRIANGLES: 2nd BROCARD AND LUCAS BROCARD

Trilinears    a*(a^6-(b^2+c^2)*a^4-5*(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)+4*(2*(b^2+c^2)*a^2+b^4+c^4)*S) : :

X(8373) lies on these lines:
{2,494}, {184,5058}, {353,8375}


X(8374) = PERSPECTOR OF THESE TRIANGLES: 2nd BROCARD AND LUCAS(-1) BROCARD

Trilinears    a*(a^6-(b^2+c^2)*a^4-5*(b^2+c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)-4*(2*(b^2+c^2)*a^2+b^4+c^4)*S) : :

X(8374) lies on these lines:
{2,493}, {184,5062}, {353,8376}


X(8375) = PERSPECTOR OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND LUCAS BROCARD

Trilinears    (2*a^2-b^2-c^2-3*S)*a : :
X(8375) = 3*S^2*X(3)+SW*(3*S-SW)*X(6)

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) = PERSPECTOR OF THESE TRIANGLES: CIRCUMSYMMEDIAL AND LUCAS(-1) BROCARD

Trilinears    (2*a^2-b^2-c^2+3*S)*a : :
X(8376) = 3*S^2*X(3)-SW*(3*S+SW)*X(6)

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD EULER AND INNER-HUTSON

Barycentrics
a^3-2*(b+c)*a^2+(b-c)^2*(-3*a+4*b+4*c)+2*(b-c)^2*(a-b-c)*sin(A/2)+2*(a-b+c)*(a*c+b^2-c^2)*sin(B/2)+2*(a+b-c)*(a*b-b^2+c^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD EULER AND OUTER-HUTSON

Barycentrics
a^3-2*(b+c)*a^2+(b-c)^2*(-3*a+4*b+4*c)-2*(b-c)^2*(a-b-c)*sin(A/2)-2*(a-b+c)*(a*c+b^2-c^2)*sin(B/2)-2*(a+b-c)*(a*b-b^2+c^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD EULER AND YFF-CENTRAL

Barycentrics
(a+b-c)*(a-b+c)*(a-b-c)*a-2*(b-c)^2*(a-b-c)^2*sin(A/2)-2*(a-b+c)*(a^2*c-(b-c)*(b-2*c)*a+(b+c)*(b-c)^2)*sin(B/2)-2*(a+b-c)*(a^2*b-(2*b-c)*(b-c)*a+(b+c)*(b-c)^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4TH EULER AND INNER-HUTSON

Barycentrics
a^4-(b+c)*a*(a+b-c)*(a-b+c)+(3*b^2+2*b*c+3*c^2)*a^2-4*(b^2-c^2)^2-2*(b+c)^2*(a+b-c)*(a-b+c)*sin(A/2)+2*(a-b-c)*(a+b-c)*(a*c-b^2+c^2)*sin(B/2)+2*(a-b-c)*(a-b+c)*(a*b+b^2-c^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4TH EULER AND OUTER-HUTSON

Barycentrics
a^4-(b+c)*a*(a+b-c)*(a-b+c)+(3*b^2+2*b*c+3*c^2)*a^2-4*(b^2-c^2)^2+2*(b+c)^2*(a+b-c)*(a-b+c)*sin(A/2)-2*(a-b-c)*(a+b-c)*(a*c-b^2+c^2)*sin(B/2)-2*(a-b-c)*(a-b+c)*(a*b+b^2-c^2)*sin(C/2) : :

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) = HOMOTHETIC CENTER OF THESE TRIANGLES: 4TH EULER AND YFF-CENTRAL

Barycentrics    2*(a+b+c)*a*b*c*sin(A/2)+(b+c)^2*(a+b-c)*(a-b+c) : :

X(8382) lies on these lines:
{1,8085}, {2,7587}, {4,7589}, {5,8351}, {1