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This is PART 4: Centers X(5001) -

Introduction and Centers X(1) - X(1000): PART 1
Centers X(1001) - X(3000): PART 2
Centers X(3001) - X(5000): PART 3


X(5001) = INVERSE-IN-CIRCUMCIRCLE OF X(5000)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S2 - SBSC - k2(S2 - 3SBSC),
                     where k2 = (3SASBSC - S2Sω - 2(SASBSCS2Sω)1/2)/(9SASBSC - S2Sω)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = SSBSC(3SA - Sω) + (S2 - 3SBSC)(SBSBSBSω)1/2     (Peter Moses, December 8, 2014)
X(5001) = (1 - k2)X(3) + k2X(4)
X(5001) = 3k2X(2) + (1 - 3k2)X(3)

Click here for a 3-dimensional representation of X(5001).

X(5001) lies on this line {2,3}:


X(5002) = 1st WALSMITH-MOSES POINT

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S2 - SBSC + k2(S2 - 3SBSC),
                     where k2 = (3SASBSC - S2Sω - 4(SASBSCS2Sω)1/2)/(-9SASBSC + S2Sω)
X(5002) = -3k2X(2) + (1 + 3k2)X(3)
X(5002) = (1 + k2)X(3) - k2X(4)

X(5002) is the point whose tripolar distances in the plane of triangle ABC are proportional to (a, b, c); the actual tripolar distances in case ABC is acute are ka, kb, kc.

If the reference triangle ABC is obtuse, then the barycentrics of X(5002) are a nonreal complex numbers. Nevertheless, for all triangles ABC, the midpoint of X(5002) and X(5003) is X(858).

A method for converting from homogeneous tripolar coordinates (henceforth simply "tripolars") to homogeneous barycentrics, found by Peter Moses (March, 2012), depends on finding the point of intersection of the radical axes of radical circles centered at A, B, C. Write the tripolars for a point U as u : v : w, and let

da = b2 + c2 - a2, db = c2 + a2 - c2, dc = a2 + b2 - c2.

Then barycentrics x : y : z for U are given by

x = a2da + k2[dcv2 + dbw2 - 2a2u2]
y = b2db + k2[daw2 + dcu2 - 2b2v2]
z = c2dc + k2[dbu2 + dav2 - 2c2w2],

where k2 has two values (as in the quadratic formula): (-f - g)/h or (-f + g)/h, where

f = - a2u2da - b2v2db - c2w2dc
g = 2S[(-au + bv + cw)(au - bv + cw)(au + bv - cw)(au + bv + cw)]1/2
h = 2[a2(u2 - v2)(u2 - w2) + b2(v2 - w2)(v2 - u2) + c2(w2 - u2)(w2 - v2)]

The meaning of k can be stated thus: starting with tripolars u : v : w, the actual tripolar distances are ku, kv, kw. That is, |UA| = ku, |UB| = kv, |UC| = kw.

X(5002) = inverse-in-circumcircle of X(5003)

X(5002) lies on this line: {2,3}


X(5003) = INVERSE-IN-CIRCUMCIRCLE OF X(5002)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S2 - SBSC - k2(S2 - 3SBSC),
                     where k2 = (3SASBSC - S2Sω - 4(SASBSCS2Sω)1/2)/(-9SASBSC + S2Sω)

The four points, X(i) for i=5000, 5001, 5002, 5003, all lie on the Euler line of triangle ABC, and all are nonreal complex-valued if ABC is obtuse.

If the reference triangle ABC is obtuse, then the barycentrics of X(5002) are a nonreal complex numbers. Nevertheless, for all triangles ABC, the midpoint of X(5002) and X(5003) is X(858).

X(5003) = inverse-in-circumcircle of X(5002)

X(5003) lies on this line: {2,3}


X(5004) = 2nd WALSMITH-MOSES POINT

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2S(S2 + (SA)2 - 4SBSC + abc(S2 - 3SBSC)(2Sω)1/2

X(5004) is the point in the plane of triangle ABC whose tripolar distances are proportional to ((b2 + c2)1/2, (c2 + a2)1/2, (a2 + b2)1/2). Like X(5000) and X(5002), the point X(5004) lies on the Euler line; unlike X(5000) and X(5002), this point is real-valued when ABC is obtuse.

X(5004) is the inverse-in-circumcircle of X(5005). The midpoint of X(5004) and X(5005) is X(23). Of the two points, X(5004) is the one inside the circumcircle. (Peter Moses, March 7, 2012)

If you have The Geometer's Sketchpad, you can view X(5004) and X(5005).

X(5004) lies on this line: {2,3}


X(5005) = INVERSE-IN-CIRCUMCIRCLE OF X(5004)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2S(S2 + (SA)2 - 4SBSC - abc(S2 - 3SBSC)(2Sω)1/2

X(5005) is the inverse-in-circumcircle of X(5004). The midpoint of X(504) and X(5005) is X(23). (Peter Moses, March 7, 2012)

X(5005) lies on this line: {2,3}


X(5006) = INVERSE-IN-CIRCUMCIRCLE OF X(1333)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(A,B,C) = a2(a + b)(a + c)(a3 + abc - b2c - bc2)

X(5006) lies on these lines:
{3, 6}, {60, 213}, {99, 712}, {104, 2715}, {110, 3230}, {112, 2699}, {163, 1914}, {172, 849}, {249, 1931}, {691, 739}, {713, 805}, {1325, 3125}


X(5007) = INVERSE-IN-MOSES-CIRCLE OF X(1691)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(A,B,C) = a2(2a2 + b2 + c2)

X(5007) lies on these lines:
{3, 6}, {23, 251}, {44, 3678}, {83, 385}, {115, 546}, {172, 1015}, {194, 3972}, {211, 4173}, {230, 1506}, {232, 3518}, {248, 1173}, {384, 538}, {395, 635}, {396, 636}, {609, 2275}, {632, 3815}, {1078, 3329}, {1100, 3881}, {1193, 2251}, {1196, 1995}, {1500, 1914}, {1573, 4426}, {1574, 4386}, {2223, 2308}, {2241, 3303}, {2242, 3304}, {2243, 3670}, {2548, 3090}, {2549, 3529}, {3051, 3229}, {3091, 3767}, {3629, 3933}


X(5008) = INVERSE-IN-MOSES-CIRCLE OF X(2030)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(A,B,C) = a2(4a2 + b2c + c2)

X(5008) lies on these lines:
{3, 6}, {111, 251}, {115, 3845}, {230, 547}, {538, 3972}, {609, 1015}, {1285, 2549}, {1506, 3054}, {3589, 3793}, {3767, 3832}


X(5009) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4283)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b)(a + c)(a2 - bc)

X(5009) lies on these lines:
{1, 82}, {3, 6}, {21, 976}, {60, 1178}, {81, 982}, {110, 2382}, {238, 2210}, {333, 4438}, {740, 1580}, {741, 919}, {757, 763}, {765, 1110}


X(5010) = INVERSE-IN-CIRCUMCIRCLE OF X(3245)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b2 + 2c2 - a2 + bc)

X(5010) lies on these lines:
{1, 3}, {2, 3583}, {4, 4324}, {9, 1030}, {10, 4189}, {11, 549}, {12, 550}, {20, 498}, {31, 4256}, {33, 186}, {34, 3520}, {42, 4257}, {43, 4184}, {78, 191}, {90, 3467}, {99, 3761}, {100, 993}, {187, 609}, {203, 1250}, {214, 3877}, {376, 1478}, {386, 2308}, {388, 3528}, {404, 3624}, {495, 4995}, {497, 3524}, {499, 3523}, {574, 1914}, {601, 2964}, {631, 1479}, {672, 4262}, {678, 1623}, {750, 4653}, {902, 995}, {956, 4421}, {975, 1719}, {1006, 3586}, {1054, 4218}, {1078, 3760}, {1125, 4188}, {1151, 3301}, {1152, 3299}, {1203, 4255}, {1737, 4304}, {2163, 2177}, {2267, 2316}, {2278, 2364}, {2330, 3098}, {2975, 3632}, {3085, 3522}, {3086, 4309}, {3218, 3894}, {3614, 3627}, {3633, 3871}, {3647, 3876}, {3651, 4333}, {3751, 4265}, {3811, 4652}, {3872, 4996}, {3873, 4973}, {3899, 4511}


X(5011) = INVERSE-IN-BEVAN CIRCLE OF X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a3 + abc - b2c - bc2)

X(5011) lies on these lines:
{1, 1055}, {4, 9}, {8, 1759}, {20, 1729}, {30, 1146}, {32, 3959}, {36, 2170}, {46, 2082}, {57, 1323}, {65, 2301}, {101, 517}, {113, 1566}, {116, 4872}, {163, 1325}, {191, 3691}, {239, 514}, {284, 501}, {484, 672}, {519, 3509}, {572, 2262}, {758, 3684}, {759, 2702}, {995, 1572}, {1155, 1308}, {1212, 3579}, {1404, 3339}, {1475, 3336}, {1482, 3207}, {1652, 3638}, {1653, 3639}, {1730, 3101}, {1731, 2245}, {1761, 3686}, {1781, 2269}, {1845, 2202}, {1914, 3125}, {1951, 1983}, {2173, 2323}, {2246, 3245}, {2249, 2690}, {2328, 2355}, {3735, 4386}, {3871, 3970}, {3916, 4875}, {4165, 4680}, {4316, 4530}


X(5012) = INVERSE-IN-1st-BROCARD CIRCLE OF X(3448)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - a2b2 - a2c2 - b2c2)

X(5012) lies on these lines:
{2, 98}, {3, 54}, {4, 569}, {5, 1614}, {6, 22}, {20, 578}, {23, 51}, {26, 3567}, {30, 567}, {49, 140}, {52, 1199}, {60, 386}, {143, 2937}, {154, 1995}, {156, 1656}, {206, 3618}, {237, 3398}, {249, 3111}, {323, 3917}, {404, 1437}, {511, 1994}, {572, 4184}, {580, 4225}, {620, 3044}, {631, 1147}, {692, 1621}, {1078, 3203}, {1092, 3523}, {1194, 1692}, {1627, 1691}, {1790, 4210}, {2056, 3231}, {2206, 4279}, {2330, 3920}, {3035, 3045}, {3218, 3955}, {3292, 3819}


X(5013) = INVERSE-IN-1st-BROCARD CIRCLE OF X(3053)

Trilinears       sin A + 2 cos A tan ω : sin B + 2 cos B tan ω : sin C + 2 cos C tan ω
Trilinears       2 cos A + sin A cot ω : 2 cos B + sin B cot ω : 2 cos C + sin C tan ω
Trilinears       2 cos A sin ω + sin A cos ω : 2 cos B sin ω + sin B cos ω : 2 cos C sin ω + sin C cos ω
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3b2 + 3c2 - a2)

Let U be the circle centered at X(371) and passing through X(1151), and let U' the circle centered at X(372) and passing through X(1152); then X(5013) is the insimilicenter of U and U'. Let V be the circle centered at X(1151) and passing through X(371), and let V' be the circle centered at X(371) and passing through X(1151); then X(5013) is the insimilicenter of V and V'. Let W be the circle having diameter X(371)X(372), and let W' be the circle having diameter X(1151)X(1152); Then X(5013) is the exsimilicenter of W and W'. (Randy Hutson, September 5, 2014)

X(5013) lies on these lines:
{2, 1975}, {3, 6}, {4, 3815}, {5, 2549}, {30, 2548}, {37, 988}, {53, 3088}, {55, 2275}, {56, 2276}, {83, 1003}, {99, 2023}, {115, 1656}, {140, 3767}, {141, 3926}, {154, 3148}, {183, 194}, {230, 631}, {232, 1593}, {378, 2207}, {381, 1506}, {517, 1571}, {524, 3785}, {599, 3933}, {958, 1575}, {999, 1500}, {1015, 3295}, {1107, 1376}, {1180, 1184}, {1181, 3269}, {1194, 1611}, {1572, 3579}, {1597, 3199}, {1968, 3516}, {3054, 3525}, {3055, 3090}, {3329, 3552}, {3788, 4045}

X(5013) = isogonal conjugate of X(5395)
X(5013) = radical center of Lucas(cot ω) circles
X(5013) = {(X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3053), (3,39,6), (371,372,5050)


X(5014) = INVERSE-IN-FUHRMANN CIRCLE OF X(4696)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 + a2b + a2c - ab2 - ac2

X(5014) lies on these lines:
{1, 4202}, {2, 1279}, {4, 8}, {10, 748}, {38, 4660}, {42, 4865}, {55, 3006}, {63, 4450}, {75, 1369}, {100, 3705}, {149, 312}, {319, 4441}, {320, 4430}, {497, 4358}, {519, 3891}, {528, 3703}, {740, 4137}, {902, 4438}, {1150, 1754}, {1479, 3701}, {2280, 4071}, {2550, 4359}, {2886, 4030}, {2887, 3938}, {3058, 3932}, {3416, 4863}, {3632, 4442}, {3696, 4914}, {3722, 3771}, {3870, 3936}, {3873, 4645}, {3886, 3969}, {3935, 4417}, {3966, 4651}


X(5015) = INVERSE-IN-FUHRMANN CIRCLE OF X(4385)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + a2bc + b3c + bc3

X(5015) lies on these lines:
{1, 977}, {3, 3705}, {4, 8}, {10, 82}, {12, 4030}, {21, 3006}, {69, 3673}, {75, 315}, {76, 319}, {149, 3702}, {312, 1479}, {320, 3874}, {333, 1780}, {345, 4294}, {442, 3757}, {518, 1330}, {528, 3704}, {752, 1046}, {942, 4645}, {986, 4660}, {1089, 3583}, {1930, 4872}, {2475, 4968}, {3178, 3750}, {3496, 4136}, {3585, 4692}, {3684, 4109}, {3685, 3695}, {3811, 4417}, {4153, 4251}


X(5016) = INVERSE-IN-FUHRMANN CIRCLE OF X(321)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + b3c + bc3 + ab2c + + abc2

X(5016) lies on these lines:
{1, 3454}, {2, 1104}, {4, 8}, {10, 31}, {40, 4450}, {41, 4109}, {44, 391}, {75, 2475}, {141, 4950}, {145, 4514}, {149, 4673}, {306, 950}, {377, 4359}, {518, 4812}, {519, 4101}, {958, 3006}, {1330, 3868}, {1478, 4968}, {1479, 3702}, {1834, 3187}, {1837, 3056}, {2478, 4358}, {2887, 3924}, {2975, 3705}, {3496, 4165}, {3586, 3969}, {4201, 4850}, {4642, 4660}


X(5017) = INVERSE-IN-CIRCUMCIRCLE OF X(2021)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2( b4 + c4 - a4 - 2a2b2 - 2a2c2)

X(5017) lies on these lines:
{3, 6}, {22, 3051}, {24, 2211}, {25, 694}, {69, 384}, {141, 315}, {154, 2056}, {159, 3499}, {172, 3056}, {193, 3552}, {251, 2979}, {263, 3148}, {352, 1383}, {394, 1915}, {524, 1003}, {599, 754}, {626, 3763}, {732, 1975}, {760, 3242}, {1184, 3981}, {1460, 2162}, {1469, 1914}, {1501, 1993}, {1627, 3060}, {1843, 1968}, {1995, 3231}


X(5018) = INVERSE-IN-INCIRCLE OF X(4298)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b - c)(a - b + c)(b3 + c3 - a3 - abc)

X(5018) lies on these lines:
{1, 7}, {34, 87}, {43, 223}, {46, 3468}, {57, 985}, {59, 484}, {65, 4649}, {109, 1758}, {171, 1427}, {226, 1961}, {238, 241}, {296, 3466}, {514, 4581}, {651, 1757}, {664, 740}, {741, 927}, {846, 1214}, {934, 2700}, {982, 1407}, {1020, 1756}, {1046, 1409}, {1054, 1465}, {1386, 1418}, {1404, 3339}, {1419, 3751}, {1735, 2958}


X(5019) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(2092)

Trilinears        a2(as + bc) : b2(bs + ca) : c2(cs + ab)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a2 + ab + ac + 2bc)

X(5019) lies on these lines:
{2, 261}, {3, 6}, {9, 172}, {31, 184}, {36, 2277}, {37, 993}, {48, 213}, {56, 478}, {87, 1716}, {609, 1743}, {672, 2273}, {941, 4189}, {980, 1444}, {992, 1724}, {1100, 2241}, {1172, 1968}, {1449, 1914}, {1468, 2268}, {1572, 3554}, {1631, 4749}, {1761, 3735}, {2298, 2975}, {3169, 3550}, {3686, 4386}


X(5020) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(1368)

Trilinears        a2(aR - bc) : b2(bR - ca) : c2(cR - ab)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - 6b2c2)

X(5020) lies on these lines:
{2, 3}, {6, 1196}, {32, 1611}, {51, 394}, {111, 907}, {115, 2936}, {154, 182}, {159, 3589}, {184, 373}, {197, 1001}, {238, 1460}, {262, 801}, {612, 3295}, {614, 999}, {1007, 3964}, {1070, 3011}, {1184, 3291}, {1350, 3819}, {1376, 1486}, {1473, 3306}, {1495, 3796}, {1619, 1853}, {3556, 3812}


X(5021) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(2271)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2 - 2ab - 2ac - 2bc)

X(5021) lies on these lines:
{2, 967}, {3, 6}, {25, 2350}, {31, 1475}, {56, 213}, {172, 218}, {220, 2242}, {474, 2238}, {604, 2200}, {672, 1468}, {750, 3691}, {956, 2295}, {999, 2176}, {1015, 1191}, {1046, 3061}, {1106, 1400}, {1571, 4646}, {1834, 2549}, {2241, 3052}, {3230, 3304}, {3290, 3338}, {3496, 4650}


X(5022) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4258)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3b2 + 3c2 - a2 - 2ab - 2ac - 2bc)

X(5022) lies on these lines:
{2, 1434}, {3, 6}, {9, 3361}, {36, 218}, {37, 3333}, {55, 1475}, {56, 220}, {57, 1212}, {604, 1802}, {999, 3730}, {1011, 2350}, {1015, 1616}, {1146, 1788}, {1155, 2082}, {1191, 2275}, {1334, 3304}, {1732, 2182}, {2332, 3516}, {3230, 3445}, {3691, 4413}


X(5023) = INVERSE-IN-CIRCUMCIRCLE OF X(1570)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3b2 + 3c2 - 5a2)

X(5023) lies on these lines:
{3, 6}, {20, 230}, {22, 1611}, {26, 2079}, {64, 1971}, {69, 439}, {115, 1657}, {160, 682}, {183, 3552}, {186, 2207}, {248, 3532}, {548, 2549}, {549, 2548}, {550, 3767}, {599, 3785}, {1003, 1078}, {1968, 3515}, {3054, 3091}, {3523, 3815}


X(5024) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1384)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(5b2 + 5c2 - a2)

Let A'B'C' be the circumcevian triangle of the symmedian point (Lemoine point), X(6). The sidelines BC, CA, AB meet the sidelines of B'C', C'A', A'B' in 9 points, of which 6 do not lie on the trilinear polar of K; barycentrics for the 6 points are 0 : b2 : 2c2, 0 : 2b2 : c2, 2a2 : 0 : c2, a2 : 0 : 2c2, a2 : 2b2 : 0, 2a2 : b2 : 0. The 6 points lie on a conic with center X(5024) and equation

2(b4c4x2 + c4a4y2 + a4b4z2) -5a2b2c2(a2yz + b2zx + c2xy) = 0.


Moreover, the center of the conic tangent to the 6 lines BC, CA, AB, B'C', C'A', A'B' is X(39), and an equation for this conic is

b4c4x2 + c4a4y2 + a4b4z2 -2a2b2c2(a2yz + b2zx + c2xy) = 0.


(From Angel Montesdeoca, March 28, 2013)

X(5024) lies on these lines:
{2, 2418}, {3, 6}, {22, 1383}, {232, 1597}, {353, 3148}, {381, 2549}, {382, 2548}, {988, 3731}, {999, 2276}, {1003, 3329}, {1506, 3851}, {1656, 3055}, {1992, 3793}, {2275, 3295}, {3054, 3526}, {3172, 3520}, {3331, 3426}, {3619, 3926}, {3620, 3933}


X(5025) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(384)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - b2c2

X(5025) lies on these lines:
{2, 3}, {32, 316}, {39, 625}, {76, 115}, {83, 3407}, {99, 3788}, {148, 1975}, {183, 2896}, {194, 325}, {315, 385}, {623, 3104}, {624, 3105}, {1348, 2559}, {1349, 2558}, {1479, 4366}, {1506, 4045}, {2548, 3329}, {3096, 3934}


X(5026) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3734)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a4 - b2c2)(b2 + c2 - 2a2)

X(5026) lies on these lines:
{2, 353}, {6, 99}, {114, 1503}, {115, 3589}, {141, 542}, {148, 3618}, {182, 2782}, {187, 524}, {385, 732}, {538, 2030}, {543, 597}, {698, 1569}, {804, 4107}, {1428, 3027}, {2330, 3023}, {2796, 3946}, {2854, 3111}


X(5027) = INVERSE-IN-PARRY-CIRCLE OF X(669)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)(a4 - b2c2)

X(5027) lies on these lines:
{6, 888}, {99, 110}, {111, 729}, {182, 2793}, {187, 237}, {688, 3050}, {707, 737}, {804, 4107}, {808, 3267}, {882, 2422}, {1511, 2780}, {1580, 4367}, {1976, 2395}, {2492, 2872}, {2799, 3506}, {3049, 3221}, {4155, 4435}

X(5027) = inverse-in-Parry-circle of X(669)
X(5027) = inverse-in-2nd-Lemoine-circle of X(2456)


X(5028) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1692)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + 2b4 + 2c4 - a2b2 - a2c2)

X(5028) lies on these lines:
{2, 2987}, {3, 6}, {69, 626}, {115, 1352}, {193, 315}, {394, 1196}, {611, 1500}, {613, 1015}, {754, 1992}, {760, 3751}, {1180, 1994}, {1184, 3787}, {1194, 1993}, {1469, 2242}, {2241, 3056}, {2549, 2794}


X(5029) = INVERSE-IN-PARRY-CIRCLE OF X(649)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(b2 + c2 - a2 - ab - ac + bc)

X(5029) lies on these lines:
{2, 4107}, {101, 110}, {106, 111}, {187, 237}, {245, 3708}, {661, 4367}, {798, 2605}, {1015, 3124}, {1635, 3722}, {2054, 3572}, {2183, 2609}, {3723, 4145}, {3733, 4079}, {3960, 4813}, {4024, 4560}, {4160, 4893}


X(5030) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4262)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b2 + 2c2 - a2 - ab - ac - bc)

X(5030) lies on these lines:
{3, 6}, {35, 1475}, {36, 101}, {56, 3730}, {106, 292}, {484, 2170}, {595, 2275}, {661, 1019}, {1155, 1308}, {2285, 3361}, {2332, 3520}, {2350, 4184}, {3247, 3333}, {3509, 4973}


X(5031) = INVERSE-IN-NINE-POINT-CIRCLE OF X(626)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a2b2c2

X(5031) lies on these lines:
{2, 1501}, {5, 141}, {114, 1503}, {115, 698}, {316, 2076}, {325, 732}, {524, 1570}, {1352, 2456}, {1506, 1692}, {2024, 3815}, {3788, 3818}


X(5032) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(352)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - 11a2

X(5032) lies on these lines:
{2, 6}, {20, 576}, {145, 4663}, {376, 1351}, {381, 1353}, {542, 3839}, {575, 3523}, {598, 2996}, {1570, 3849}, {3241, 3751}, {3545, 3564}


X(5033) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(1570)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3a4 - a2b2 - a2c2 - 4b2c2)

X(5033) lies on these lines:
{3, 6}, {69, 620}, {184, 3231}, {206, 3016}, {729, 3565}, {1078, 3620}, {1196, 3796}, {1428, 2241}, {2242, 2330}, {3618, 4045}


X(5034) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(187)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - 3a2b2 - 3a2c2)

X(5034) lies on these lines:
{3, 6}, {83, 2996}, {193, 1078}, {611, 1015}, {613, 1500}, {1352, 1506}, {1428, 2242}, {2241, 2330}, {3564, 3815}, {3618, 3767}


X(5035) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4277)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a2 + ab + ac + 3bc)

X(5035) lies on these lines:
{3, 6}, {31, 692}, {37, 2975}, {44, 172}, {45, 2242}, {593, 662}, {1405, 1415}, {1468, 2267}, {1914, 4689}


X(5036) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4287)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b3 + 2c3 - 2a2b - 2a2c + abc)

X(5036) lies on these lines:
{3, 6}, {9, 484}, {45, 71}, {966, 2475}, {1213, 2476}, {1400, 2099}, {2209, 4484}, {3196, 3197}


X(5037) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4284)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2a3 + b3 + c3 + a2b + a2c - abc)

X(5037) lies on these lines:
{3, 6}, {9, 976}, {595, 2911}, {609, 2260}, {995, 2174}, {1449, 3509}, {1914, 2273}, {2251, 2277}


X(5038) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(2076)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - 2a2b2 - 2a2c2 - 3b2c2)

X(5038) lies on these lines:
{2, 2056}, {3, 6}, {83, 597}, {98, 3815}, {524, 1078}, {542, 1506}, {2023, 3329}


X(5039) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(2030)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + 3a2b2 + 3a2c2 + 2b2c2)

X(5039) lies on these lines:
{3, 6}, {69, 83}, {184, 251}, {206, 3203}, {609, 1428}, {732, 3734}, {1078, 3618}


X(5040) = INVERSE-IN-PARRY-CIRCLE OF X(667)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(a3 + abc - b2c - bc2)

X(5040) lies on these lines:
{2, 4164}, {31, 4455}, {100, 110}, {111, 739}, {187, 237}, {650, 1980}, {1977, 3124}


X(5041) = INVERSE-IN-MOSES-CIRCLE OF X(2076)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2a2 + 3b2 + 3c2)

X(5041) lies on these lines:
{3, 6}, {83, 538}, {115, 3850}, {547, 1506}, {597, 3933}, {2548, 3545}, {3329, 3934}


X(5042) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4263)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a2 + ab + ac + 4bc)

X(5042) lies on these lines:
{3, 6}, {9, 2242}, {172, 1743}, {213, 604}, {594, 996}, {1449, 2241}, {4497, 4749}


X(5043) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4289)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b3 + 2c3 - 2a2b - 2a2c - 3abc)

X(5043) lies on these lines:
{3, 6}, {9, 3337}, {31, 4484}, {45, 672}, {1334, 2260}, {2503, 4383}


X(5044) = INVERSE-IN-SPIEKER-CIRCLE OF X(3814)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 2abc + 3b2c + 3bc2)

X(5044) = r*X(3) - (r + 4R)(X(9)
X(5044) = r*X(5) - (r + 2R)*X(10)
X(5044) = X(1) + 3X(210)
X(5044) = 3X(2) + x(72)     (Peter Moses, April 3, 2012)

X(5044) lies on these lines:
{1, 210}, {2, 72}, {3, 9}, {5, 10}, {6, 975}, {8, 392}, {35, 3683}, {37, 386}, {43, 3931}, {44, 58}, {45, 4255}, {46, 4413}, {56, 3715}, {57, 3927}, {63, 474}, {65, 1698}, {78, 405}, {140, 912}, {191, 1155}, {200, 3295}, {201, 1465}, {226, 3824}, {281, 1871}, {329, 443}, {354, 3624}, {355, 2551}, {404, 3219}, {442, 908}, {496, 4847}, {500, 1818}, {518, 1125}, {519, 4015}, {536, 3159}, {581, 1212}, {631, 1071}, {748, 976}, {756, 1193}, {758, 3634}, {762, 3230}, {899, 2292}, {958, 997}, {966, 3781}, {978, 984}, {1001, 3811}, {1018, 4520}, {1089, 4009}, {1203, 3745}, {1376, 3579}, {1479, 4679}, {1621, 4420}, {1864, 3601}, {2140, 3739}, {2478, 3419}, {2771, 3035}, {2802, 4540}, {3057, 3679}, {3216, 3666}, {3290, 3954}, {3294, 3693}, {3303, 3711}, {3306, 3951}, {3555, 3616}, {3617, 3877}, {3625, 3898}, {3626, 3880}, {3636, 4547}, {3687, 3695}, {3689, 3746}, {3702, 4651}, {3742, 3874}, {3753, 3869}, {3754, 3828}, {3827, 3844}, {3833, 4127}, {3838, 3841}, {3848, 3988}, {3873, 4539}, {3885, 4678}, {3889, 4661}, {3893, 4668}, {3899, 4731}, {3952, 4968}, {4113, 4975}, {4158, 4187}


X(5045) = INVERSE-IN-INCIRCLE OF X(484)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c - 6abc - b2c - bc2)

X(5045) lies on these lines:
{1, 3}, {2, 3555}, {5, 3947}, {7, 1058}, {8, 4002}, {10, 3742}, {30, 4298}, {37, 4253}, {58, 1279}, {72, 3616}, {145, 3753}, {210, 3624}, {226, 496}, {355, 938}, {392, 3622}, {404, 3957}, {405, 4666}, {474, 3870}, {495, 1210}, {500, 1458}, {518, 1125}, {519, 3812}, {536, 596}, {550, 4314}, {551, 960}, {582, 1471}, {758, 3636}, {936, 3243}, {946, 971}, {975, 3242}, {1100, 2174}, {1149, 2650}, {1387, 2771}, {1621, 3916}, {1770, 3058}, {2886, 3824}, {2891, 4886}, {2901, 4891}, {3086, 3475}, {3306, 4917}, {3488, 3600}, {3626, 3833}, {3632, 3698}, {3634, 3848}, {3635, 3754}, {3655, 4308}, {3876, 4430}, {3877, 4018}, {3894, 3962}, {3898, 4084}, {3968, 4701}, {4533, 4661}, {4668, 4731}


X(5046) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(2475)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + a2bc + ab2c + abc2 - 2b2c2

X(5046) lies on these lines:
{2, 3}, {8, 80}, {10, 3583}, {11, 2975}, {12, 1621}, {35, 3814}, {36, 3825}, {78, 3586}, {100, 1329}, {145, 497}, {153, 944}, {191, 3467}, {324, 1896}, {355, 3877}, {388, 1388}, {519, 4857}, {908, 950}, {1058, 3623}, {1125, 3585}, {1210, 3218}, {1478, 3616}, {1749, 3648}, {1837, 3869}, {1842, 3101}, {1877, 4296}, {1994, 3193}, {2551, 3434}, {3419, 3876}, {3421, 3621}, {3924, 3944}, {4297, 4881}, {4514, 4696}


X(5047) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4197)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 - 3abc - 3b2c - 3bc2)

X(5047) lies on these lines:
{1, 748}, {2, 3}, {8, 344}, {9, 3868}, {10, 1621}, {35, 3634}, {81, 1724}, {100, 1698}, {373, 970}, {908, 1125}, {942, 3219}, {956, 3622}, {958, 3304}, {968, 1722}, {993, 3624}, {1320, 3890}, {3074, 3562}, {3214, 3750}, {3216, 4653}, {3295, 3617}, {3336, 3647}, {3583, 3841}, {3683, 3812}, {3697, 3935}, {3701, 3757}, {3740, 4420}, {3748, 4662}, {3889, 4666}, {3915, 4279}


X(5048) = INVERSE-IN-INCIRCLE OF X(3057)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(3b2 + 3c2 - 2a2 + ab + ac - 6bc)

X(5048) = (R - 3r)*X(1) + r*X(3)

Let I be the incenter of a triangle ABC, let NA be the nine-point circle of the triangle IBC, and define NB and NC cyclically. Let RA be the reflection of NA in the line AI, and define RB and RC cyclically. Then X(5048) is the radical center of the circles RA, RB, RC.      (Angel Montesdeoca, Hyacinthos #22502, July 7, 2014.)

X(5048) = reflection of X(1319) in X(1)
X(5048) = inverse-in-incircle of X(3057)

X(5048) lies on these lines:
{1, 3}, {8, 1392}, {11, 519}, {33, 1878}, {78, 3893}, {145, 1837}, {210, 3872}, {495, 4870}, {497, 3241}, {513, 4162}, {515, 1317}, {535, 3058}, {950, 3635}, {960, 4861}, {1318, 1320}, {1387, 1737}, {1391, 1870}, {1478, 3656}, {1836, 3476}, {2170, 2348}, {2269, 3723}, {3021, 3328}, {3318, 3319}, {3486, 3623}, {3655, 4302}, {3683, 3877}, {3693, 4919}, {3711, 4915}


X(5049) = INVERSE-IN-INCIRCLE OF X(3245)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2b + a2c + 10abc - b3 - b2c - bc2 - c3)

X(5049) lies on these lines:
{1, 3}, {2, 3921}, {10, 3848}, {72, 3622}, {101, 1100}, {374, 1449}, {392, 3873}, {496, 3817}, {518, 551}, {519, 3742}, {956, 4666}, {960, 3636}, {962, 3296}, {1125, 3740}, {1387, 2801}, {3241, 3753}, {3243, 3940}, {3244, 3812}, {3555, 3616}, {3621, 4002}, {3633, 3698}, {3828, 4711}, {3885, 4004}, {3890, 4018}, {4677, 4731}


X(5050) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(2080)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3a4 + b4 + c4 - 4a2b2 - 4a2c2 - 6b2c2)

X(5050) lies on these lines:
{2, 3167}, {3, 6}, {5, 3618}, {51, 3796}, {69, 140}, {141, 3526}, {184, 373}, {193, 631}, {381, 597}, {549, 1992}, {611, 999}, {613, 2330}, {632, 3619}, {895, 1511}, {1176, 3527}, {1352, 1656}, {1385, 3751}, {1386, 1482}, {1495, 3066}, {1598, 1974}, {1843, 3517}, {3525, 3620}, {3818, 3851}


X(5051) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(964)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a3 + b3 + c3 + a2b + a2c + ab2 + ac2 + abc)

X(5051) lies on these lines:
{1, 3454}, {2, 3}, {8, 1211}, {10, 321}, {12, 1284}, {45, 1213}, {75, 1228}, {81, 1330}, {115, 1281}, {225, 1441}, {846, 1698}, {984, 4812}, {1046, 4683}, {1193, 3847}, {1230, 4385}, {1962, 3178}, {2298, 4645}, {2901, 3969}, {3017, 3578}, {3214, 4085}, {3695, 3995}, {3704, 4854}


X(5052) = INVERSE-IN-MOSES-CIRCLE OF X(2021)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3a2b2 + 3a2c2 + 2b2c2 - b4 - c4)

X(5052) lies on these lines:
{2, 3787}, {3, 6}, {51, 1196}, {69, 2548}, {76, 193}, {141, 1506}, {251, 1994}, {263, 3117}, {373, 3231}, {538, 1992}, {611, 2241}, {613, 2242}, {726, 4856}, {732, 3629}, {1015, 1469}, {1194, 3060}, {1353, 2782}, {1500, 3056}, {1572, 3751}, {1843, 2211}, {1974, 3202}


X(5053) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4266)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3 - ab2 - ac2 + 3abc - b2c - bc2)

X(5053) lies on these lines:
{1, 2267}, {2, 1412}, {3, 6}, {9, 604}, {36, 909}, {44, 101}, {48, 1732}, {59, 672}, {241, 1461}, {527, 1429}, {602, 947}, {661, 3737}, {1174, 2364}, {1449, 1697}, {1474, 4222}, {1630, 1723}, {1724, 2360}, {1731, 2182}, {1766, 3554}, {2223, 3939}, {3684, 4700}


X(5054) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(547)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5a4 + 2b4 + 2c4 - 7a2b2 - 7a2c2 - 4b2c2

X(5054) lies on these lines:
{2, 3}, {10, 3655}, {46, 4870}, {55, 3582}, {56, 3584}, {182, 599}, {355, 3828}, {499, 3058}, {519, 3653}, {538, 1153}, {551, 1482}, {568, 3917}, {597, 1351}, {1125, 3656}, {1384, 3815}, {1385, 3679}, {2549, 3054}, {3017, 4255}, {3295, 4995}, {3579, 3624}


X(5055) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(549)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + 4b4 + 4c4 - 5a2b2 - 5a2c2 - 8b2c2

X(5055) lies on these lines:
{2, 3}, {10, 3656}, {355, 551}, {498, 3058}, {499, 3614}, {515, 3653}, {517, 4731}, {539, 3167}, {597, 1352}, {599, 1351}, {946, 3654}, {999, 3582}, {1125, 3655}, {1159, 1737}, {1479, 4995}, {1482, 3679}, {2549, 3055}, {3295, 3584}


X(5056) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3523)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + 5b4 + 5c4 - 6a2b2 - 6a2c2 - 10b2c2

X(5056) lies on these lines:
{2, 3}, {233, 393}, {355, 3622}, {371, 1132}, {372, 1131}, {388, 3614}, {390, 498}, {485, 3591}, {486, 3590}, {499, 3600}, {637, 3595}, {638, 3593}, {962, 1698}, {1482, 4678}, {1699, 3634}, {3311, 3316}, {3312, 3317}


X(5057) = INVERSE-IN-POLAR-CIRCLE OF X(1824)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 + abc - b2c - bc2

X(5057) lies on these lines:
{1, 535}, {2, 1155}, {4, 8}, {7, 3660}, {10, 3245}, {11, 1776}, {21, 36}, {30, 4511}, {31, 3944}, {46, 4193}, {63, 1699}, {100, 516}, {114, 1281}, {149, 518}, {190, 3006}, {214, 4316}, {226, 1005}, {238, 3120}, {239, 4442}, {243, 3326}, {320, 350}, {388, 3890}, {404, 1770}, {411, 2077}, {484, 1698}, {497, 3873}, {524, 4956}, {527, 1156}, {528, 3935}, {758, 3583}, {901, 1311}, {946, 2975}, {960, 2475}, {1319, 3485}, {1330, 3702}, {1478, 3877}, {1479, 3868}, {1839, 2287}, {2325, 4071}, {2478, 4295}, {2886, 3219}, {2895, 3706}, {3058, 3957}, {3306, 4312}, {3336, 3825}, {3416, 4671}, {3486, 3623}, {3582, 4973}, {3585, 3878}, {3648, 3916}, {3685, 3936}, {3717, 4756}, {3741, 4683}, {3847, 4418}, {3874, 4857}, {3883, 4054}, {3920, 4415}, {4062, 4693}, {4358, 4645}, {4432, 4892}, {4654, 4666}, {4661, 4863}, {4713, 4799}


X(5058) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1505)

Trilinears        a2(aR - bc) : b2(bR - ca) : c2(cR - ab)   César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 - [(b + c - a)(c + a - b)(a + b - c)(a + b + c)]1/2)

X(5058) lies on these lines:
{3, 6}, {115, 3071}, {172, 3299}, {251, 588}, {492, 3788}, {590, 1506}, {615, 642}, {1015, 2067}, {1124, 2242}, {1335, 2241}, {1500, 2066}, {1588, 3767}, {1914, 3301}, {2548, 3068}

X(5058) = {X(3),X(6)}-harmonic conjugate of X(5062)


X(5059) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3854)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5b4 + 5c4 - 11a4 + 6a2b2 + 6a2c2 - 10b2c2

X(5059) lies on these lines:
{2, 3}, {40, 4678}, {145, 516}, {323, 1498}, {515, 3621}, {962, 3623}, {1131, 1151}, {1132, 1152}, {3085, 4324}, {3086, 4316}, {3622, 4297}, {4299, 4857}


X(5060) = INVERSE-IN-CIRCUMCIRCLE OF X(284)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b)(a + c)(a3 + b3 + c3 - 2a2b - 2a2c + abc)

X(5060) lies on these lines:
{3, 6}, {19, 1247}, {21, 3496}, {36, 163}, {102, 2715}, {110, 1055}, {112, 2708}, {162, 2202}, {691, 2291}, {759, 2702}, {1951, 2249}, {3735, 4653}


X(5061) = INVERSE-IN-CIRCUMCIRCLE OF X(1402)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b - c)(a - b + c)(a3 + abc - b2c - bc2)

X(5061) lies on these lines:
{1, 3}, {2, 1397}, {12, 1408}, {59, 4600}, {81, 181}, {98, 2720}, {108, 2699}, {109, 1284}, {604, 750}, {741, 2222}, {899, 1404}, {1428, 3911}


X(5062) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1504)

Trilinears        a2(aR + bc) : b2(bR + ca) : c2(cR + ab)    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + [(b + c - a)(c + a - b)(a + b - c)(a + b + c)]1/2)

X(5062) lies on these lines:
{3, 6}, {115, 3070}, {172, 3301}, {251, 589}, {491, 3788}, {590, 641}, {615, 1506}, {1124, 2241}, {1335, 2242}, {1587, 3767}, {1914, 3299}, {2548, 3069}

X(5062) = {X(3),X(6)}-harmonic conjugate of X(5058)


X(5063) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3003)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(a4 + b4 + c4 - 2a2b2 - 2a2c2 + 4b2c2)

X(5063) lies on these lines:
{2, 2986}, {3, 6}, {53, 1885}, {160, 1974}, {184, 1576}, {231, 3767}, {468, 3815}, {1968, 1990}, {2393, 3148}, {2549, 3018}, {3087, 3542}


X(5064) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(428)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - b2 + c2)(a2 + b2 - c2)(a2 + 2b2 + 2c2)

X(5064) lies on these lines:
{2, 3}, {51, 1853}, {115, 1184}, {394, 3818}, {524, 3867}, {553, 1892}, {599, 1843}, {1498, 3574}, {1829, 3679}, {1876, 4654}


X(5065) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(800)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a2 + 2b2 + 2c2)

X(5065) lies on these lines:
{2, 801}, {3, 6}, {393, 2549}, {1249, 1968}, {1950, 2082}, {1951, 2285}, {2241, 3554}, {2242, 3553}, {2548, 3087}, {3618, 4558}


X(5066) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3534)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 7b4 + 7c4 - 2a4 - 5a2b2 - 5a2c2 - 14b2c2

X(5066) lies on these lines:
{2, 3}, {517, 3956}, {597, 3818}, {946, 4669}, {952, 3817}, {1699, 3654}, {3583, 4995}, {3584, 3614}, {3656, 4677}


X(5067) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3525)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 + 5b4 + 5c4 - 8a2b2 - 8a2c2 - 10b2c2

X(5067) lies on these lines:
{2, 3}, {6, 3316}, {373, 3567}, {498, 1058}, {499, 1056}, {944, 3624}, {3614, 4293}, {3634, 4301}


X(5068) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3522)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 7b4 + 7c4 - a4 - 6a2b2 - 6a2c2 - 14b2c2

X(5068) lies on these lines:
{2, 3}, {8, 3817}, {355, 3623}, {497, 3614}, {946, 3617}, {1131, 3069}, {1132, 3068}, {3085, 4857}


X(5069) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(2220)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(ab2 + ac2 - abc + b3 + b2c + bc2 + c3)

X(5069) lies on these lines:
{2, 3770}, {3, 6}, {37, 2275}, {42, 3941}, {44, 2277}, {749, 1100}, {980, 3589}, {3056, 4735}, {3061, 4016}


X(5070) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(632)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 + 4b4 + 4c4 - 7a2b2 - 7a2c2 - 8b2c2

X(5070) lies on these lines:
{2, 3}, {17, 3411}, {18, 3412}, {373, 1216}, {1351, 3763}, {1482, 1698}, {2548, 3054}, {3055, 3767}


X(5071) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3524)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + 7b4 + 7c4 - 8a2b2 - 8a2c2 - 14b2c2

X(5071) lies on these lines:
{2, 3}, {388, 3582}, {497, 3584}, {542, 3618}, {1587, 3317}, {1588, 3316}, {3086, 3614}, {3817, 3828}


X(5072) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(548)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 6b4 + 6c4 - a4 - 5a2b2 - 5a2c2 - 12b2c2

X(5072) lies on these lines:
{2, 3}, {355, 3635}, {946, 4691}, {1351, 3630}, {1482, 3625}, {3295, 3614}


X(5073) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3858)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4b4 + 4c4 - 7a4 + 3a2b2 + 3a2c2 - 8b2c2

X(5073) lies on these lines:
{2, 3}, {516, 4701}, {999, 4857}, {2996, 3793}, {3357, 3581}, {3426, 3519}


X(5074) = INVERSE-IN-CIRCUMCIRCLE OF X(1631)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3b + a3c - 2a2bc - b4 + b3c + bc3 - c4

X(5074) lies on these lines:
{3, 142}, {101, 4872}, {116, 517}, {226, 1323}, {304, 4153}, {514, 661}


X(5075) = INVERSE-IN-PARRY-CIRCLE OF X(663)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(a3 + b3 + c3 - 2a2b - 2a2c + abc)

X(5075) = center of this circle: V(X(846)) = {{15,16,846,1054,1283,5197}}; see the preamble to X(6137).

X(5075) lies on these lines:
{109, 110}, {111, 2291}, {187, 237}, {659, 1769}, {846, 2786}, {4414, 4750}


X(5076) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3861)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 6b4 + 6c4 - 7a4 + a2b2 + a2c2 - 12b2c2

X(5076) lies on these lines:
{2, 3}, {355, 4746}, {517, 4816}, {3303, 3585}, {3304, 3583}


X(5077) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3363)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4b4 + 4c4 - 5a4 + 5a2b2 + 5a2c2 - 4b2c2

X(5077) lies on these lines:
{2, 3}, {6, 3849}, {183, 671}, {524, 2549}, {543, 599}


X(5078) = INVERSE-IN-CIRCUMCIRCLE OF X(3666)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - 3a2bc + ab2c + abc2)

X(5078) lies on these lines:
{1, 3}, {22, 3052}, {197, 4383}, {595, 2915}, {1979, 2076}


X(5079) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3530)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + 6b4 + 6c4 - 7a2b2 - 7a2c2 - 12b2c2

X(5079) lies on these lines:
{2, 3}, {355, 3636}, {999, 3614}, {1351, 3631}, {1482, 3626}


X(5080) = INVERSE-IN-POLAR-CIRCLE OF X(1829)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - a2bc + ab2c + abc2 - 2b2c2

X(5080) lies on these lines:
{2, 36}, {4, 8}, {5, 2975}, {10, 191}, {11, 529}, {12, 21}, {20, 2077}, {30, 100}, {56, 4193}, {79, 3754}, {80, 758}, {119, 4996}, {145, 1479}, {149, 519}, {153, 515}, {316, 668}, {377, 1155}, {381, 956}, {388, 1319}, {404, 1329}, {452, 2078}, {495, 1621}, {497, 3241}, {498, 4189}, {513, 2517}, {666, 671}, {946, 4861}, {958, 2476}, {1168, 4080}, {1330, 2392}, {1698, 4652}, {1699, 3872}, {1737, 3218}, {1793, 2222}, {1837, 3868}, {1877, 4318}, {3244, 4857}, {3245, 3617}, {3586, 3870}, {3614, 4999}, {4188, 4299}


X(5081) = INVERSE-IN-FUHRMANN-CIRCLE OF X(318)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + b2 - c2)(a2 - b2 + c2)(b2 + c2 - a2 - bc)

X(5081) lies on these lines:
{4, 8}, {10, 275}, {25, 3705}, {27, 3687}, {29, 270}, {30, 2968}, {33, 3872}, {34, 78}, {36, 4242}, {69, 273}, {75, 317}, {102, 515}, {162, 447}, {186, 4996}, {200, 4680}, {225, 4101}, {239, 297}, {242, 1884}, {264, 319}, {280, 3146}, {281, 391}, {320, 340}, {458, 3661}, {518, 1875}, {519, 1785}, {521, 1948}, {758, 1845}, {765, 1861}, {860, 1870}, {1043, 3559}, {1325, 2766}, {1841, 3965}, {1852, 3704}, {1876, 4645}, {1990, 4969}, {2202, 3684}, {2322, 3686}, {2345, 3087}, {4853, 4894}


X(5082) = INVERSE-IN-FUHRMANN-CIRCLE OF X(3421)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + 4a2bc - 4ab2c - 4abc2 - 2b2c2

X(5082) lies on these lines:
{1, 142}, {2, 496}, {4, 8}, {7, 3555}, {10, 497}, {20, 956}, {40, 4847}, {65, 4863}, {69, 2891}, {100, 631}, {145, 377}, {149, 1145}, {200, 946}, {376, 2975}, {388, 519}, {390, 405}, {515, 4853}, {518, 4295}, {528, 958}, {938, 3753}, {944, 3872}, {966, 3294}, {1000, 3885}, {1210, 1706}, {1376, 3086}, {1478, 3632}, {1479, 2551}, {1699, 4882}, {2475, 3621}, {2886, 3085}, {3296, 3889}, {3303, 3925}, {3485, 3811}, {3487, 3870}, {3583, 4668}, {3585, 4677}, {3983, 4679}, {4421, 4999}


X(5083) = INVERSE-IN-INCIRCLE OF X(109)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b - c)(a - b + c)(b3 + c3 + a2b + a2c - 2ab2 - 2ac2)

X(5083) lies on these lines:
{1, 104}, {7, 149}, {11, 118}, {56, 214}, {57, 100}, {65, 1317}, {73, 3953}, {80, 388}, {119, 1210}, {153, 938}, {244, 4551}, {518, 3035}, {528, 553}, {651, 1421}, {758, 1319}, {942, 952}, {950, 2829}, {1071, 1537}, {1145, 3555}, {1320, 3340}, {1387, 2771}, {1388, 3878}, {1420, 3868}, {1457, 4694}, {1465, 3999}, {1466, 2932}, {1787, 3333}, {1862, 1876}, {2078, 3218}, {2099, 3892}, {2835, 3937}, {3036, 3812}, {3256, 3957}, {3738, 4458}


X(5084) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(443)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + 4a2bc + 4ab2c + 4abc2 - 2b2c2

X(5084) lies on these lines:
{1, 2551}, {2, 3}, {8, 392}, {9, 1210}, {10, 497}, {12, 4423}, {65, 4679}, {72, 938}, {78, 3488}, {145, 3940}, {226, 1467}, {329, 942}, {387, 4383}, {388, 1125}, {908, 3487}, {936, 950}, {958, 3086}, {962, 3753}, {966, 4266}, {997, 3486}, {1001, 1329}, {1056, 3436}, {1376, 4294}, {1478, 3624}, {1479, 1698}, {2078, 3814}, {2899, 4385}, {3295, 3820}, {3670, 4419}, {3812, 4295}, {3983, 4863}


X(5085) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1350)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - 3a4 + 2a2b2 + 2a2c2 + 6b2c2)

X(5085) lies on these lines:
{2, 154}, {3, 6}, {4, 3589}, {20, 3618}, {23, 3066}, {25, 373}, {26, 2916}, {35, 613}, {36, 611}, {40, 1386}, {55, 1428}, {56, 2330}, {64, 1176}, {69, 3523}, {140, 1352}, {141, 631}, {206, 1498}, {376, 597}, {518, 3576}, {524, 3524}, {549, 599}, {1177, 2935}, {1385, 3242}, {1407, 3955}, {1511, 2930}, {1593, 1974}, {1656, 3818}, {1843, 3515}, {3167, 3819}, {4220, 4383}


X(5086) = INVERSE-IN-FUHRMANN-CIRCLE OF X(3869)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + a3b + a3c - a2bc - ab3 - ac3 - 2b2c2

X(5086) lies on these lines:
{1, 2476}, {2, 1837}, {4, 8}, {5, 4511}, {10, 21}, {65, 2475}, {79, 4084}, {145, 3485}, {149, 3057}, {153, 2894}, {224, 4197}, {388, 3873}, {404, 1737}, {411, 515}, {497, 3890}, {758, 3585}, {950, 1621}, {952, 4861}, {997, 4193}, {1441, 2893}, {1478, 3868}, {1479, 3877}, {1698, 4855}, {1788, 4190}, {1826, 2287}, {3583, 3878}, {3621, 4863}, {3884, 4857}, {4325, 4973}


X(5087) = INVERSE-IN-NINE-POINT-CIRCLE OF X(2886)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b + a2c + ab2 + ac2 - 4abc - 2b3 + 2b2c + 2bc2 - 2c3

X(5087) lies on these lines:
{2, 1155}, {5, 10}, {11, 518}, {36, 405}, {65, 4193}, {120, 3259}, {145, 1837}, {149, 3689}, {226, 3660}, {381, 997}, {388, 1319}, {429, 1878}, {513, 3716}, {516, 1538}, {535, 1125}, {942, 3825}, {1001, 2078}, {1376, 1699}, {1647, 3999}, {1698, 3245}, {2077, 3149}, {3006, 4009}, {3011, 3246}, {3705, 3967}, {3752, 3944}, {3812, 4187}, {3829, 4847}, {4442, 4706}


X(5088) = INVERSE-IN-INCIRCLE OF X(3664)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - a2b2 - a2c2 + a2bc - b3c - bc3 + 2b2c2

X(5088) lies on these lines:
{1, 7}, {3, 85}, {4, 348}, {28, 242}, {30, 1565}, {36, 1111}, {46, 3212}, {56, 3673}, {75, 956}, {104, 927}, {150, 515}, {169, 3177}, {187, 4403}, {239, 514}, {273, 1804}, {304, 1975}, {411, 1446}, {517, 664}, {675, 1308}, {910, 3732}, {934, 2723}, {942, 1434}, {1366, 3328}, {2369, 2736}, {2646, 4059}, {3665, 4911}


X(5089) = INVERSE-IN-STEVANOVIC-CIRCLE OF X(468)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 - c2)(a2 - b2 + c2)(b2 + c2 - ab - ac)

X(5089) lies on these lines:
{1, 607}, {2, 92}, {9, 608}, {19, 25}, {28, 1390}, {34, 1212}, {111, 2766}, {112, 2074}, {225, 1855}, {230, 231}, {427, 1826}, {428, 1839}, {614, 1108}, {653, 1447}, {672, 1876}, {976, 1973}, {1172, 2346}, {1334, 1902}, {1729, 1771}, {1783, 1870}, {1829, 2333}, {1861, 3693}, {2299, 3745}, {2322, 3757}, {2340, 2356}


X(5090) = INVERSE-IN-FUHRMANN-CIRCLE OF X(1829)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 + b3 + c3 + b2c + bc2)

X(5090) lies on these lines:
{1, 427}, {4, 8}, {10, 25}, {19, 594}, {27, 3661}, {33, 429}, {34, 1883}, {40, 3575}, {65, 66}, {80, 1039}, {239, 469}, {388, 1876}, {428, 3679}, {468, 1698}, {515, 1593}, {607, 1826}, {944, 3088}, {952, 1595}, {1385, 3541}, {1843, 3416}, {1861, 1891}, {2204, 4426}, {3516, 4297}


X(5091) = INVERSE-IN-CIRCUMCIRCLE OF X(2223)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 - a3b - a3c + a2bc + b3c + bc3 - 2b2c2)

X(5091) lies on these lines:
{1, 3}, {2, 1083}, {6, 513}, {7, 59}, {81, 3110}, {516, 1428}, {651, 4014}, {692, 1086}, {760, 3218}, {840, 1002}, {1023, 4413}, {1026, 1376}, {1290, 2711}, {1397, 3474}, {1404, 3000}, {1572, 2087}, {1618, 2175}, {1633, 3271}, {2330, 3663}, {2720, 2724}, {3735, 4414}, {4440, 4579}


X(5092) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3098)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - 2a4 + a2b2 + a2c2 + 4b2c2)

X(5092) lies on these lines:
{2, 1495}, {3, 6}, {23, 373}, {30, 3589}, {35, 1428}, {36, 2330}, {69, 3431}, {74, 827}, {140, 1503}, {141, 542}, {184, 3819}, {186, 1843}, {206, 4550}, {323, 3917}, {376, 3618}, {378, 1974}, {631, 1352}, {1386, 3579}, {2070, 2916}, {3523, 3620}, {3530, 3564}, {3934, 4048}}}


X(5093) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(187)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3a4 + 5b4 + 5c4 - 8a2b2 - 8a2c2 - 6b2c2)

X(5093) lies on these lines:
{3, 6}, {4, 1353}, {5, 193}, {25, 1994}, {30, 5032}, {49, 1974}, {51, 3167}, {69, 1656}, {143, 3517}, {155, 3527}, {373, 394}, {381, 1992}, {399, 895}, {518, 4930}, {1352, 3629}, {1482, 3751}, {1503, 3830}, {1993, 5020}, {3066, 3292}, {3526, 3618}, {3620, 3628}


X(5094) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(468)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2b2 + 2c2 - a2)(a2 + b2 - c2)(a2 - b2 + c2)

X(5094) lies on these lines:
{2, 3}, {6, 67}, {12, 1398}, {53, 3055}, {126, 136}, {183, 340}, {184, 1853}, {232, 566}, {264, 2970}, {281, 2969}, {1235, 3266}, {1351, 3580}, {1506, 2207}, {1698, 1829}, {1843, 3763}, {1892, 3911}, {1990, 3815}, {2453, 3258}, {2548, 3172}


X(5095) = INVERSE-IN-POLAR-CIRCLE OF X(671)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - 2a2)2(a2 + b2 - c2)(a2 - b2 + c2)

X(5095) lies on these lines:
{4, 542}, {6, 67}, {25, 2930}, {110, 193}, {113, 3564}, {114, 2407}, {184, 1177}, {185, 1205}, {468, 524}, {511, 1986}, {868, 3163}, {1112, 1843}, {1829, 2836}, {1839, 2969}, {1858, 3270}, {1899, 2892}, {2452, 2794}


X(5096) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4265)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - a2bc - ab2c - abc2 + 2 b2c2)

X(5096) lies on these lines:
{3, 6}, {21, 3589}, {22, 4383}, {35, 1386}, {36, 518}, {44, 3220}, {56, 976}, {69, 4188}, {141, 404}, {474, 3763}, {656, 3733}, {674, 1428}, {1155, 3827}, {2915, 2916}, {3618, 4189}


X(5097) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(2076)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2a4 + 3b4 + 3c4 - 5a2b2 - 5a2c2 + 4b2c2)

X(5097) lies on these lines:
{3, 6}, {5, 3629}, {51, 110}, {323, 373}, {524, 547}, {542, 1353}, {1352, 1992}, {1503, 3853}, {3533, 3618}, {3543, 5032}, {3564, 3850}, {3628, 3631}, {3818, 3832}


X(5098) = INVERSE-IN-PARRY-CIRCLE OF X(665)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b - c)(b4 + c4 - ab3 - ac3 + 2a2b2 + ab2c - abc2)

X(5098) lies on these lines:
{110, 919}, {111, 840}, {187, 237}, {244, 661}, {513, 3290}, {518, 650}, {523, 3726}, {672, 3709}, {1638, 4776}, {1914, 2605}, {3509, 3737}, {3700, 4358}, {3797, 4467}


X(5099) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(2453)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2(b2 + c2 - 2a2)(b4 + c4 - a4 - b2c2)

Let ABC be a triangle with orthic triangle DEF, and let L be a line in the plane of ABC. Let A' be the reflection of A in L, and define B' and C' cyclically. The circumcircles of DB'C', EC'A', FA'C' concur.    Antreas Hatzipolakis, Anolpolis #816, September 2013.

If L is the Euler line of ABC, then the circumcircles concur in A(5099).      Seiichi Kirikami, September 25, 2013.

X(5099) lies on these lines:
{2, 691}, {4, 842}, {23, 316}, {30, 114}, {113, 511}, {115, 523}, {125, 512}, {126, 625}, {132, 403}, {187, 468}, {381, 2453}, {690, 2682}, {868, 1649}


X(5100) = INVERSE-IN-FUHRMANN-CIRCLE OF X(4737)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + 3a2bc - 2ab2c - 2abc2 + b3c + bc3

X(5100) lies on these lines:
{1, 3836}, {4, 8}, {10, 4514}, {149, 3701}, {319, 1269}, {341, 1479}, {3006, 3871}, {3496, 4119}, {3555, 4645}, {3625, 4792}, {3632, 4680}, {3679, 4894}, {3992, 4857}


X(5101) = INVERSE-IN-FUHRMANN-CIRCLE OF X(1828)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a2 + b2 + c2 - ab - ac)

X(5101) lies on these lines:
{1, 1883}, {4, 8}, {10, 4186}, {11, 33}, {19, 428}, {25, 1376}, {429, 1717}, {1709, 1726}, {1753, 3575}, {1830, 1836}, {1837, 1853}, {1864, 1899}, {1891, 4214}


X(5102) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(2030)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3a4 + 7b4 + 7c4 - 10a2b2 - 10a2c2 - 6b2c2)

X(5102) lies on these lines:
{3, 6}, {4, 3629}, {154, 3060}, {193, 3832}, {323, 3066}, {524, 3545}, {547, 599}, {1352, 3850}, {1503, 1992}, {3090, 3631}, {3533, 3589}, {3564, 3845}


X(5103) = INVERSE-IN-NINE-POINT-CIRCLE OF X(3934)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a4b2 - a4c2 + a2b4 + a2c4 - 2a2b2c2

X(5103) lies on these lines:
{2, 2076}, {4, 4048}, {5, 141}, {6, 5025}, {83, 316}, {115, 732}, {325, 698}, {597, 1692}, {1503, 2456}, {1570, 3629}


X(5104) = INVERSE-IN-CIRCUMCIRCLE OF X(574)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b4 + 2c4 - a4 - 2a2b2 - 2a2c2 + b2c2)

X(5104) lies on these lines:
{3, 6}, {22, 2056}, {23, 352}, {99, 524}, {111, 694}, {141, 316}, {599, 3734}, {625, 3763}, {1915, 2979}, {1971, 2781}


X(5105) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4264)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 + a2b + a2c + 2ab2 + 2ac2 + abc + 2b2c + 2bc2)

X(5105) lies on these lines:
{1, 2321}, {3, 6}, {9, 1193}, {37, 995}, {42, 1449}, {43, 3686}, {966, 3216}, {1201, 3247}, {2276, 2300}, {3214, 4034}


X(5106) = INVERSE-IN-PARRY-CIRCLE OF X(3231)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4b2 + a4c2 - 2a2b4 - 2a2c4 + b4c2 + b2c4)

X(5106) lies on these lines:
{2, 99}, {6, 694}, {32, 110}, {39, 373}, {187, 237}, {1384, 1613}, {1976, 5033}, {2021, 3291}, {3051, 5008}


X(5107) = INVERSE-IN-MOSES-CIRCLE OF X(574)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2a4 + 5b4 + 5c4 - 5a2b2 - 5a2c2 - 2b2c2)

X(5107) lies on these lines:
{3, 6}, {69, 625}, {111, 323}, {115, 524}, {193, 316}, {352, 3291}, {843, 3565}, {1992, 2549}, {2502, 3292}


X(5108) = INVERSE-IN-CIRCUMCIRCLE OF X(669)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - 2a4b2 - 2a4c2 + 5a2b2c2 - b4c2 - b2c4

X(5108) lies on these lines:
{2, 6}, {3, 669}, {99, 2502}, {110, 5026}, {126, 542}, {805, 2770}, {1078, 2142}, {1316, 3734}, {3124, 4563}


X(5109) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4290)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 + a2b + a2c + 2ab2 + 2ac2 + 2b2c + 2bc2)

X(5109) lies on these lines:
{1, 3943}, {3, 6}, {37, 1201}, {42, 678}, {44, 1193}, {45, 995}, {1100, 2295}, {1404, 2594}, {3293, 4969}


X(5110) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(2305)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 - a3 + 2ab2 + 2ac2 + abc + 2b2c + 2bc2)

X(5110) lies on these lines:
{3, 6}, {21, 992}, {35, 2300}, {48, 2276}, {55, 1964}, {141, 332}, {171, 1100}, {2268, 2277}


X(5111) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(182)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + 2b4 + 2c4 - 2a2b2 - 2a2c2 - b2c2)

X(5111) lies on these lines:
{3, 6}, {316, 3629}, {323, 3124}, {385, 2023}, {694, 2987}, {1915, 3060}, {1993, 2056}


X(5112) = INVERSE-IN-POLAR-CIRCLE OF X(458)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b4 + c4 - a2b2 - a2c2)(b4 + c4 - 3a4 - 2b2c2)

X(5112) lies on these lines:
{2, 3}, {373, 4045}, {523, 3569}, {754, 3292}, {1495, 2794}, {1555, 2777}, {2782, 3580}


X(5113) = INVERSE-IN-PARRY-CIRCLE OF X(3005)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)(b4 + c4 - a4 - a2b2 - a2c2 + b2c2)

X(5113) lies on these lines:
{110, 827}, {111, 755}, {187, 237}, {620, 690}, {826, 4142}, {888, 2492}, {2485, 3221}


X(5114) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4274)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3 - a2b - a2c - 2ab2 - 2ac2 + 2b2c + 2bc2)

X(5114) lies on these lines:
{3, 6}, {42, 1397}, {44, 993}, {213, 2267}, {2175, 2309}, {2268, 2300}, {2276, 2323}


X(5115) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4272)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3 + 2a2b + 2a2c + ab2 + ac2 + 4abc + b2c + bc2)

X(5115) lies on these lines:
{2, 757}, {3, 6}, {31, 1100}, {37, 1468}, {172, 3204}, {560, 2308}, {1107, 2214}


X(5116) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(2076)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 + a2b2 + a2c2 + 3b2c2)

X(5116) lies on these lines:
{2, 4048}, {3, 6}, {384, 3589}, {732, 1078}, {2056, 3819}, {2211, 3520}, {3552, 3618}


X(5117) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(419)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(b2 + c2 - bc)(b2 + c2 + bc)

X(5117) lies on these lines:
{2, 3}, {141, 3186}, {275, 3406}, {2052, 3399}, {2887, 3061}, {3096, 3819}


X(5118) = INVERSE-IN-BROCARD-CIRCLE OF X(3111)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 - b2)(a2 - c2)(a2b2 +a2c2 + 2b2c2)

X(5118) lies on these lines:
{3, 6}, {99, 110}, {512, 2421}, {691, 805}, {1316, 3734}


X(5119) = INVERSE-IN-BEVAN-CIRCLE OF X(3245)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a3 - a2b - a2c + ab2 + ac2 + 4abc - b2c - bc2)

X(5119) = (R - r)X(1) + 2r*X(3)
X(5119) = r(r + 4R)*X(9) - R(2r - R)X(80)
X(5119) = 2rR*X(8) + (R2 - 2rR - r2)*X(90)

Let A'B'C' be the orthic triangle of ABC. Let LA be the antiorthic axis of AB'C', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. Then A''B''C'' is inversely similar to ABC, with similitude center X(9), and X(5119) = A''B''C''-to-ABC similarity image of X(1). (Randy Hutson, December 4, 2014)

X(5119) lies on these lines:
{1, 3}, {8, 90}, {9, 80}, {10, 1479}, {63, 519}, {71, 1723}, {72, 2900}, {78, 3878}, {100, 997}, {145, 4305}, {169, 1334}, {190, 4737}, {191, 2136}, {376, 1000}, {388, 1770}, {392, 1376}, {404, 3890}, {495, 1836}, {497, 1737}, {498, 946}, {515, 1709}, {516, 1478}, {549, 1387}, {551, 3306}, {674, 3751}, {748, 4695}, {758, 3870}, {846, 855}, {920, 3486}, {944, 1158}, {950, 1728}, {956, 3880}, {962, 3085}, {993, 2802}, {1001, 3753}, {1056, 3474}, {1058, 1788}, {1150, 3902}, {1253, 1718}, {1317, 3655}, {1449, 4268}, {1532, 1699}, {1571, 2275}, {1572, 2276}, {1698, 1706}, {1702, 3301}, {1703, 3299}, {1708, 3488}, {1717, 1773}, {1722, 3987}, {1742, 2807}, {1745, 2943}, {1763, 3465}, {1766, 2269}, {1824, 4186}, {2082, 3730}, {2270, 3731}, {2975, 3885}, {3058, 3654}, {3158, 3899}, {3208, 3496}, {3218, 3241}, {3243, 3894}, {3656, 4995}, {3689, 3940}, {3729, 4692}, {3811, 3869}, {3820, 4679}, {3915, 4642}, {3929, 4677}, {4067, 4917}, {4189, 4861}, {4384, 4714}

X(5119) = reflection of X(1) in X(55)
X(5119) = {X(1),X(40)}-harmonic conjugate of X(46)


X(5120) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4254)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 - a3 - a2b - a2c + ab2 + ac2 - 4abc + b2c + bc2)

X(5120) lies on these lines:
{2, 1014}, {3, 6}, {9, 56}, {36, 198}, {37, 999}, {44, 2178}, {46, 2262}, {48, 218}, {55, 1449}, {219, 604}, {268, 1741}, {391, 404}, {474, 966}, {517, 3554}, {602, 1622}, {859, 1778}, {956, 2345}, {1055, 3217}, {1100, 3295}, {1108, 1766}, {1172, 1593}, {1260, 2352}, {1376, 3686}, {1385, 3553}, {1402, 4047}, {1420, 2324}, {1436, 2270}, {1444, 3618}, {1445, 1804}, {1475, 2268}, {1486, 4497}, {1604, 2183}, {1723, 2182}, {1728, 1903}, {1732, 2261}, {2256, 3730}, {2257, 3428}, {2260, 2267}, {2343, 3451}, {3247, 3304}


X(5121) = INVERSE-IN-SPIEKER-RADICAL-CIRCLE OF X(43)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b + a2c + 2ab2 + 2ac2 - 6abc + b2c + bc2 - b3 - c3

X(5121) lies on these lines:
{1, 2}, {11, 1738}, {88, 5057}, {105, 2743}, {109, 238}, {244, 908}, {516, 1054}, {518, 3756}, {982, 3452}, {988, 5084}, {1086, 5087}, {1279, 3035}, {1362, 3660}, {2254, 3667}, {3752, 3816}

X(5121) = complement of X(5205)
X(5121) = inverse-in-{circumcircle, nine-point circle}-inverter of X(1)
X(5121) = radical trace of incircle and excircles-radical circle


X(5122) = INVERSE-IN-CIRCUMCIRCLE OF X(3295)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - 4a3 - a2b - a2c + 4ab2 + 4ac2 - 2abc - b2c - bc2)

X(5122) lies on these lines:
{1, 3}, {7, 3524}, {28, 1878}, {30, 3911}, {72, 4188}, {140, 4292}, {186, 1876}, {226, 549}, {404, 3219}, {474, 3305}, {513, 4401}, {518, 4973}, {535, 3828}, {548, 950}, {550, 1210}, {582, 603}, {910, 5030}, {938, 3528}, {1439, 3431}, {3476, 3654}, {3530, 3982}, {3534, 3586}, {3752, 4257}, {3897, 4004}, {3928, 3940}, {4742, 4781}


X(5123) = INVERSE-IN-SPIEKER-CIRCLE OF X(960)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2b3 + 2c3 - a2b - a2c + ab2 + ac2 - 2b2c - 2bc2)

X(5123) lies on these lines:
{2, 1319}, {5, 10}, {8, 1392}, {9, 484}, {11, 3880}, {12, 3812}, {36, 474}, {377, 1155}, {495, 3742}, {513, 3823}, {515, 3035}, {518, 1737}, {519, 1387}, {529, 3911}, {535, 3828}, {1012, 1376}, {1878, 1883}, {2476, 3698}, {3057, 4193}, {3586, 4421}, {3634, 4999}, {3753, 3838}, {3992, 4858}, {4711, 4847}

X(5123) = complement of X(1319)


X(5124) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1030)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 - a3 - a2b - a2c + ab2 + ac2 - abc + b2c + bc2)

X(5124) lies on these lines:
{2, 1029}, {3, 6}, {35, 1100}, {36, 37}, {45, 1696}, {55, 4497}, {56, 2171}, {141, 1444}, {165, 3554}, {198, 2265}, {404, 1213}, {594, 2975}, {672, 2174}, {966, 4188}, {992, 4225}, {1006, 1901}, {1078, 3770}, {1172, 3520}, {1449, 5010}, {2238, 4210}, {3252, 3446}, {3815, 4220}


X(5125) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(29)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(b3 + c3 - a2b - a2c - abc)

X(5125) lies on these lines:
{2, 3}, {6, 2907}, {8, 278}, {10, 92}, {19, 4429}, {34, 78}, {46, 1748}, {75, 225}, {158, 1737}, {162, 1724}, {208, 1445}, {243, 1837}, {608, 4645}, {653, 1118}, {960, 1888}, {1068, 1897}, {1096, 1722}, {1210, 1785}, {1848, 1869}, {1859, 3812}, {1871, 3753}


X(5126) = INVERSE-IN-CIRCUMCIRCLE OF X(999)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a3 + b3 + c3 - a2b - a2c - 4ab2 - 4ac2 + 6abc - b2c - bc2)

X(5126) lies on these lines:
{1, 3}, {5, 4311}, {44, 101}, {104, 971}, {106, 1279}, {214, 518}, {495, 4315}, {496, 4297}, {513, 1960}, {516, 1387}, {535, 1125}, {631, 4308}, {934, 953}, {952, 3911}, {1055, 2246}, {1483, 4848}, {1538, 2829}, {1878, 4222}, {3474, 3656}, {3634, 4999}, {3935, 4881}


X(5127) = INVERSE-IN-CIRCUMCIRCLE OF X(501)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b)(a + c)(a3 + b3 + c3 - a2b - a2c - ab2 - ac2 - abc + b2c + bc2)

X(5127) lies on these lines:
{1, 21}, {3, 501}, {5, 580}, {10, 1098}, {35, 60}, {36, 110}, {44, 2341}, {71, 2150}, {162, 1785}, {163, 672}, {229, 3336}, {249, 1101}, {409, 3754}, {484, 1325}, {517, 759}, {519, 643}, {656, 3737}, {1323, 1414}, {1437, 4278}, {2194, 4276}, {3286, 3446}

X(5127) = isogonal conjugate of X(5620)


X(5128) = INVERSE-IN-BEVAN-CIRCLE OF X(1319)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b3 + 3c3 - 3a3 - 3a2b - 3a2c + 3ab2 + 3ac2 + 2abc - 3b2c - 3bc2)

X(5128) lies on these lines:
{1, 3}, {8, 3928}, {10, 3474}, {12, 4312}, {20, 4848}, {44, 2270}, {63, 1706}, {80, 4333}, {227, 1419}, {516, 1788}, {728, 3509}, {962, 3911}, {1044, 4551}, {1698, 1836}, {3000, 3214}, {3085, 4654}, {3158, 3868}, {3218, 3621}, {3243, 3871}, {3812, 4512}, {4430, 4917}


X(5129) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4208)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 + 3a4 + 2a2b2 + 2a2c2 + 8a2bc + 8ab2c + 8abc2 - 2b2c2

X(5129) lies on these lines:
{2, 3}, {8, 3305}, {9, 938}, {10, 390}, {144, 942}, {145, 392}, {388, 4423}, {519, 4866}, {908, 3616}, {936, 4313}, {1001, 2551}, {1125, 3600}, {1330, 4869}, {1698, 4294}, {1788, 3683}, {2899, 3757}, {3189, 3740}, {3241, 3984}, {3485, 4679}, {3624, 4293}


X(5130) = INVERSE-IN-FUHRMANN-CIRCLE OF X(1824)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 + b3 + c3 + 2abc + b2c + bc2)

X(5130) lies on these lines:
{1, 429}, {4, 8}, {6, 1826}, {10, 4185}, {12, 34}, {25, 958}, {29, 2203}, {33, 1904}, {65, 1899}, {388, 1426}, {407, 1211}, {431, 2886}, {469, 1999}, {996, 4186}, {1861, 4214}, {1869, 1889}, {2333, 3691}, {2355, 4198}, {2975, 4231}


X(5131) = INVERSE-IN-CIRCUMCIRCLE OF X(3746)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - 3a3 - a2b - a2c + 3ab2 + 3ac2 - abc - b2c - bc2)

X(5131) lies on these lines:
{1, 3}, {10, 4325}, {21, 3833}, {79, 140}, {100, 4973}, {191, 404}, {516, 3582}, {1054, 1325}, {1210, 4324}, {1698, 4652}, {1737, 4316}, {1749, 1768}, {1770, 3817}, {3530, 3649}, {3583, 3911}, {3624, 4338}, {3740, 3916}, {3814, 4197}, {3901, 4855}


X(5132) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3286)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3b + a3c - ab3 - ac3 - 2ab2c - 2abc2 - b3c - bc3)

X(5132) lies on these lines:
{2, 11}, {3, 6}, {35, 238}, {36, 4649}, {81, 4210}, {86, 404}, {228, 3666}, {940, 4191}, {984, 4557}, {1009, 3589}, {1011, 4383}, {1193, 1918}, {1386, 2223}, {1818, 2269}, {2209, 2274}, {2703, 2711}, {3923, 4436}, {4245, 4653}


X(5133) = INVERSE-IN-NINE-POINT-CIRCLE OF X(23)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a4b2 - a4c2 - 2a2b2c2 - b4c2 - b2c4

X(5133) lies on these lines:
{2, 3}, {12, 3920}, {51, 3580}, {114, 137}, {115, 1194}, {141, 2979}, {184, 3818}, {230, 1627}, {311, 325}, {316, 1799}, {343, 3060}, {1176, 3589}, {1352, 1993}, {1503, 5012}, {1989, 3108}, {1994, 3410}, {4074, 5031}

X(5133) = inverse-in-orthocentroidal-circle of X(22)


X(5134) = INVERSE-IN-POLAR-CIRCLE OF X(1839)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - a3b - a3c + a2b2 + a2c2 + a2bc - 2b2c2

X(5134) lies on these lines:
{4, 9}, {11, 5030}, {30, 101}, {190, 316}, {220, 382}, {514, 4024}, {672, 3583}, {995, 2549}, {1055, 4316}, {1334, 3585}, {1475, 4857}, {1479, 4253}, {1657, 3207}, {2372, 2702}, {2475, 3294}, {4262, 4302}


X(5135) = INVERSE-IN-BROCARD-CIRCLE OF X(4259)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - a2b2 - a2c2 - a2bc - ab2c - abc2 - 2b2c2)

X(5135) lies on these lines:
{1, 692}, {2, 2194}, {3, 6}, {35, 674}, {60, 404}, {65, 82}, {81, 5012}, {184, 940}, {377, 3618}, {442, 3589}, {518, 2330}, {673, 1492}, {1001, 2175}, {1974, 4185}, {3612, 3751}


X(5136) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(860)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 - ab2 - ac2 - b2c - bc2)

X(5136) lies on these lines:
{2, 3}, {6, 281}, {10, 212}, {33, 997}, {47, 1724}, {92, 1870}, {225, 1125}, {264, 811}, {318, 4511}, {392, 1824}, {1068, 3616}, {1395, 1877}, {1825, 3878}, {1826, 2267}


X(5137) = INVERSE-IN-CIRCUMCIRCLE OF X(2352)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a5 - a3b2 - a3c2 + a3bc + b4c + bc4 - b3c2 - b2c3)

X(5137) lies on these lines:
{1, 3}, {11, 1428}, {184, 3772}, {513, 1430}, {692, 3011}, {917, 2720}, {1284, 2361}, {1397, 1836}, {1404, 2635}, {1408, 4292}, {1548, 2829}, {3025, 3320}, {3782, 3955}

X(5137) = crossdifference of every pair of points on the line X(72)X(650)


X(5138) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4260)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - a2b2 - a2c2 - 2a2bc - 2ab2c - 2abc2 - 2b2c2)

X(5138) lies on these lines:
{1, 2175}, {3, 6}, {28, 1974}, {35, 3779}, {57, 985}, {69, 261}, {81, 184}, {206, 942}, {443, 3618}, {518, 993}, {611, 2810}, {940, 2194}, {2330, 3601}


X(5139) = INVERSE-IN-POLAR-CIRCLE OF X(99)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2(b2 + c2 - 3a2)(a2 + b2 - c2)(a2 - b2 + c2)

X(5139) is the center of the hyperbola {A,B,C,X(4),X(25)}, which meets the circumcircle at X(3573) (and A,B,C) and tangents to the line X(4)X(69) at X(4) and tangent to the line X(6)(25) at X(25). Moreover, X(5139) is the perspector of the circumconic centered at X(2489). (Randy Hutson, November 22, 2014)

X(5139) lies on the nine-point circle and these lines:
{2, 2374}, {4, 99}, {25, 1560}, {113, 1596}, {115, 2971}, {120, 429}, {122, 868}, {123, 3140}, {126, 427}, {127, 3143}, {131, 381}, {132, 235}

X(5139) = midpoint of X(4) and X(3563)
X(5139) = complement of X(3565)
X(5139) = X(2)-Ceva conjugate of X(2489)


X(5140) = INVERSE-IN-POLAR-CIRCLE OF X(69)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 - c2)(a2 - b2 + c2)(a2b2 + a2c2 + b4 - 4b2c2 + c4)

X(5140) lies on these lines:
{4, 69}, {25, 187}, {115, 2386}, {133, 2679}, {232, 2971}, {427, 625}, {428, 3849}, {460, 512}, {1598, 2080}, {1692, 2207}, {1974, 2030}, {2021, 3199}


X(5141) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4189)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b4 + 2c4 - 2a2b2 - 2a2c2 - a2bc - ab2c - abc2 - 4b2c2

X(5141) lies on these lines:
{2, 3}, {8, 4867}, {10, 3899}, {11, 3622}, {12, 145}, {149, 3085}, {495, 3623}, {1125, 2320}, {2886, 3614}, {3245, 3814}, {3616, 3822}, {3624, 4881}


X(5142) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(28)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(ab2 + ac2 + abc + b3 + b2c + bc2 + c3)

X(5142) lies on these lines:
{2, 3}, {6, 2906}, {10, 1848}, {12, 278}, {19, 1698}, {34, 975}, {264, 1969}, {281, 1329}, {1125, 1891}, {1172, 1714}, {1228, 1235}, {1826, 1838}


X(5143) = INVERSE-IN-CIRCUMCIRCLE OF X(171)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3b + a3c - a2bc - ab3 - ac3 + ab2c + abc2 - b2c2)

X(5143) lies on these lines:
{1, 3}, {31, 5012}, {43, 3185}, {98, 2222}, {100, 740}, {109, 2699}, {172, 1908}, {513, 3510}, {741, 901}, {1756, 4551}, {4225, 4642}, {4276, 4868}


X(5144) = INVERSE-IN-CIRCUMCIRCLE OF X(1001)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a4 - 2ab3 - 2ac3 + ab2c + abc2 - b3c - bc3 + 2b2c2)

X(5144) lies on these lines:
{1, 1055}, {3, 142}, {36, 105}, {56, 1323}, {100, 2725}, {187, 1279}, {238, 5030}, {514, 659}, {910, 2809}, {3361, 5018}, {4251, 4649}, {4471, 4667}

X(5144) = X(187)-of-2nd-circumperp triangle


X(5145) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4279)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2b2 + a2c2 + ab3 + ac3 + ab2c + abc2 + b3c + bc3 + b2c2)

X(5145) lies on these lines:
{1, 87}, {3, 6}, {10, 1740}, {35, 2209}, {42, 3097}, {76, 86}, {81, 4203}, {238, 993}, {869, 1757}, {984, 1964}, {1911, 3864}, {3051, 4476}


X(5146) = INVERSE-IN-POLAR-CIRCLE OF X(72)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 + b3 + c3 - abc - b2c - bc2)

X(5146) lies on these lines:
{4, 8}, {19, 484}, {28, 36}, {133, 3259}, {225, 2078}, {242, 860}, {278, 1319}, {1168, 1877}, {1869, 3245}, {1870, 1884}, {2077, 4219}


X(5147) = INVERSE-IN-PARRY-CIRCLE OF X(3747)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(a4 - a2b2 - a2c2 - ab3 - ac3 + ab2c + abc2 + b2c2)

X(5147) lies on these lines:
{2, 4154}, {31, 110}, {42, 2054}, {100, 4094}, {111, 2177}, {187, 237}, {662, 3571}, {1402, 2107}, {1911, 4117}, {1962, 3722}, {1976, 2187}


X(5148) = INVERSE-IN-INCIRCLE OF X(3056)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b4 + c4 + a2b2 + a2c2 - 4a2bc + 2b3c + 2bc3 - 4b2c2)

X(5148) lies on these lines:
{1, 256}, {11, 625}, {55, 187}, {316, 497}, {512, 4162}, {538, 3023}, {1500, 2021}, {1914, 2031}, {2030, 2330}, {2080, 3295}, {3058, 3849}

X(5148) = X(187)-of-Mandart-incircle triangle
X(5148) = homothetic center of the intangents triangle and the reflection of the extangents triangle in X(187)


X(5149) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4048)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a8 - a4b2c2 - a2b4c2 - a2b2c4 + b6c2 + b2c6

Let A'B'C' be the 1st Brocard triangle. The radical center of the circumcircles of AB'C', BC'A', CA'B' concur in X(5149). Let A'' be the A'B'C'-isogonal-conjugate of A, and define B'' and C'' cyclically; the lines A'A'', B'B'', C'C'' concur in X(5149). (Randy Hutson, November 22, 2014)

X(5149) lies on these lines:
{2, 4159}, {3, 114}, {39, 83}, {76, 4027}, {98, 3934}, {182, 2782}, {538, 1692}, {736, 1691}, {754, 2076}, {1003, 2482}, {1569, 1975}

X(5149) = X(1691)-of-1st-Brocard triangle
X(5149) = 1st-Brocard-triangle-isogonal-conjugate of X(76)


X(5150) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3923)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a5 - a3b2 - a3c2 + a3bc - 2ab2c2 + b3c2 + b2c3)

X(5150) lies on these lines:
{1, 4579}, {9, 48}, {31, 43}, {182, 2783}, {184, 4011}, {386, 987}, {528, 597}, {692, 4432}, {726, 1428}, {2787, 4164}, {3840, 3955}

X(5150) = X(36)-of-1st-Brocard trangle


X(5151) = INVERSE-IN-POLAR-CIRCLE OF X(1320)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(a2 + b2 - c2)(a2 - b2 + c2)(ab + ac + b2 + c2 - 4bc)

X(5151) lies on these lines:
{4, 145}, {11, 1883}, {25, 2932}, {100, 2899}, {900, 1846}, {1145, 4723}, {1317, 1877}, {1811, 1997}, {1828, 2802}, {1878, 3880}


X(5152) = INVERSE-IN-CIRCUMCIRCLE OF X(76)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a8 - a6b2 - a6c2 + a4b4 + a4c4 - a2b6 - a2c6 + b4c4

X(5152) lies on these lines:
{2, 4159}, {3, 76}, {32, 1916}, {39, 4027}, {83, 2023}, {115, 384}, {148, 3552}, {316, 2794}, {671, 1003}, {2854, 4590}


X(5153) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4275)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 + a2b + a2c + 2ab2 + 2ac2 + 2abc + 2b2c + 2bc2)

X(5153) lies on these lines:
{1, 594}, {3, 6}, {37, 992}, {42, 1100}, {604, 2594}, {1009, 1386}, {1201, 3723}, {1213, 3216}, {2260, 3588}, {2309, 4749}


X(5154) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4275)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b4 + 2c4 - 2a2b2 - 2a2c2 + a2bc + ab2c + abc2 - 4b2c2

X(5154) lies on these lines:
{2, 3}, {8, 3814}, {11, 145}, {12, 3622}, {496, 3623}, {519, 1392}, {1329, 3617}, {3614, 3816}, {3616, 3825}


X(5155) = INVERSE-IN-FUHRMANN-CIRCLE OF X(1900)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 + b3 + c3 + 4abc + b2c + bc2)

X(5155) lies on these lines:
{1, 1904}, {4, 8}, {10, 4214}, {25, 993}, {34, 429}, {608, 1826}, {1875, 1892}, {1891, 4186}, {3897, 4194}


X(5156) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3736)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3b + a3c + a2b2 + a2c2 + 2a2bc + ab2c + abc2 + b2c2)

X(5156) lies on these lines:
{1, 1918}, {2, 31}, {3, 6}, {36, 2274}, {593, 5012}, {595, 1001}, {1010, 1724}, {1468, 2209}, {1740, 2228}


X(5157) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3313)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a6 - a2b4 - a2c4 - a2b2c2 - 2b4c2 - 2b2c4)

X(5157) lies on these lines:
{2, 66}, {3, 6}, {69, 5012}, {110, 3619}, {141, 184}, {159, 3796}, {427, 1974}, {1370, 3618}, {3575, 3867}


X(5158) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(3284)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)(2b4 + 2c4 - a4 - a2b2 - a2c2 - 4b2c2)

X(5158) = X(92)-isoconjugate of X(3431)

X(5158) lies on these lines:
{2, 648}, {3, 6}, {5, 1990}, {53, 546}, {232, 1995}, {233, 1249}, {393, 3091}, {441, 597}, {3087, 3146}


X(5159) = INVERSE-IN-NINE-POINT-CIRCLE OF X(1368)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)(3b4 + 3c4 - 2a4 + a2b2 + a2c2 - 6b2c2)

X(5159) = (inverse-in-de-Longchamps-circle of X(22)) = (radical trace of nine-point circle and de-Longchamps circle) = (radical trace of polar circle and de-Longchamps circle) = (reflection of X(23) in de Longchamps line)    (Randy Hutson, August-September, 2013)

X(5159) lies on these lines:
{2, 3}, {125, 3292}, {216, 3055}, {230, 3284}, {339, 3266}, {523, 4885}, {577, 3054}, {1007, 2452}

X(5159) = complement of X(468)
X(5159) = inverse-in-{circumcircle, nine-point circle}-inverter of X(20)
X(5159) = inverse-in-complement-of-polar-circle of X(2)


X(5160) = INVERSE-IN-INCIRCLE OF X(3058)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2b4 + 2c4 - 2a4 + 2a2bc - b3c - bc3 - 2b2c2)

X(5160) lies on these lines:
{1, 30}, {11, 858}, {12, 4354}, {23, 55}, {33, 468}, {403, 3614}, {511, 3024}, {523, 4724}

X(5160) = X(23)-of-Mandart-incircle triangle
X(5160) = homothetic center of intangents triangle and reflection of extangents triangle in X(23)


X(5161) = INVERSE-IN-CIRCUMCIRCLE OF X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + a3b + a3c + a2bc - ab2c - abc2 - b3c - bc3)

X(5161) lies on these lines:
{3, 31}, {81, 849}, {560, 4414}, {649, 834}, {896, 2210}, {902, 1818}, {2206, 3666}, {3218, 5009}


X(5162) = INVERSE-IN-CIRCUMCIRCLE OF X(3094)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b6 + c6 - a6 - a2b4 - a2c4 - a2b2c2 + b4c2 + b2c4)

X(5162) s the point of intersection of the Lemoine axes of the circumcevian triangles of the 1st and 2nd Brocard points. (Randy Hutson, November 22, 2014)

X(5162) lies on these lines:
{3, 6}, {99, 736}, {315, 3552}, {316, 384}, {737, 805}, {754, 2482}, {1003, 3849}, {2387, 3455}


X(5163) = INVERSE-IN-PARRY-CIRCLE OF X(3230)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3b + a3c - 2a2bc - 2ab3 - 2ac3 + ab2c + abc2 + b3c + bc3)

X(5163) lies on these lines:
{6, 3121}, {37, 100}, {110, 739}, {187, 237}, {574, 4414}, {1977, 2300}, {2092, 3030}


X(5164) = INVERSE-IN-CIRCUMCIRCLE OF X(1030)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(b4 + c4 - a3b - a3c - a2b2 - a2c2 + ab3 + ac3)

X(5164) lies on these lines:
{3, 6}, {115, 517}, {502, 594}, {512, 661}, {730, 3029}, {1500, 2653}, {3124, 3230}

X(5164) = crossdfference of every pair of points on the line X(81)X(523)


X(5165) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(4273)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 - 2a2b - 2a2c - ab2 - ac2 - 3abc - b2c - bc2)

X(5165) lies on these lines:
{2, 44}, {3, 6}, {37, 3868}, {45, 3927}, {603, 1405}, {1201, 2260}, {2308, 3764}


X(5166) = INVERSE-IN-2nd-LEMOINE-CIRCLE OF X(1992)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a6 + b6 + c6 - 3a4b2 - 3a4c2 - 3a2b4 - 3a2c4 + 9a2b2c2)

X(5166) lies on these lines:
{2, 6}, {111, 2393}, {112, 843}, {729, 2696}, {895, 3291}, {1499, 3049}


X(5167) = INVERSE-IN-POLAR-CIRCLE OF X(264)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4b4 + a4c4 - a2b6 - a2c6 + b6c2 + b2c6 - 2b4c4)

X(5167) lies on these lines:
{4, 69}, {113, 2679}, {115, 2387}, {187, 237}, {206, 1691}, {217, 1692}


X(5168) = INVERSE-IN-PARRY-CIRCLE OF X(902)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3 - 2a3 + a2b + a2c + ab2 + ac2 - 2b2c - 2bc2)

X(5168) lies on these lines:
{6, 2054}, {42, 101}, {58, 106}, {187, 237}, {1015, 2308}, {1017, 3124}


X(5169) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(23)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a4b2 - a4c2 - a2b2c2 - b4c2 - b2c4

X(5169) lies on these lines:
{2, 3}, {6, 3448}, {94, 262}, {110, 3818}, {323, 1352}, {1993, 3410}


X(5170) = INVERSE-IN-CIRCUMCIRCLE OF X(3285)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b)(a + c)(2a3 + b3 + c3 - 2a2b - 2a2c + 2abc - b2c - bc2)

X(5170) lies on these lines:
{3, 6}, {31, 512}, {163, 1015}, {249, 593}, {691, 2384}, {953, 2715}


X(5171) = INVERSE-IN-CIRCUMCIRCLE OF X(2456)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a6 - 4a4b2 - 4a4c2 + 3a2b4 + 3a2c4 + 2b4c2 + 2b2c4)

X(5171) = center of the circumcircle-inverse of the 1st Lemoine circle. (Randy Hutson, November 22, 2014)

Let H be the hyperbola of these five points: X(182), PU(1), PU(2). One vertex of H is X(182); the other is X(5171). (Randy Hutson, November 22, 2014)

X(5171) lies on these lines:
{3, 6}, {4, 1078}, {20, 98}, {83, 631}, {114, 315}, {1352, 3785}

X(5171) = {X(I),X(J)-harmonic conjugate of X(k) for these (I,J,K): (371, 372, 5052), (1687,1688,3053)


X(5172) = INVERSE-IN-CIRCUMCIRCLE OF X(65)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b - c)(a - b + c)(a3 + b3 + c3 - a2b - a2c - ab2 - ac2 - abc + b2c + bc2)

X(5172) lies on these lines:
{1, 3}, {12, 21}, {58, 2594}, {59, 3286}, {73, 1399}, {74, 2720}, {108, 186}, {109, 1464}, {181, 4276}, {187, 1415}, {198, 1609}, {388, 4189}, {404, 2886}, {405, 3814}, {474, 3841}, {513, 1946}, {674, 1428}, {759, 859}, {902, 1457}, {906, 3002}, {1030, 1400}, {1055, 2272}, {1317, 4996}, {1333, 2197}, {1408, 4278}, {1437, 2477}, {1458, 3446}, {1469, 4265}, {1725, 1807}, {1727, 2771}, {2161, 2173}, {2932, 3911}, {3434, 4188}


X(5173) = INVERSE-IN-CIRCUMCIRCLE OF X(2078)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b - c)(a - b + c)(a2b + a2c - 2ab2 - 2ac2 - 2abc + b3 - b2c - bc2 + c3)

X(5173) lies on these lines:
{1, 3}, {7, 3434}, {42, 1465}, {72, 3485}, {81, 4318}, {105, 2982}, {222, 2263}, {226, 518}, {278, 1002}, {388, 3419}, {528, 553}, {672, 2171}, {910, 1630}, {971, 1836}, {1001, 1708}, {1071, 4295}, {1202, 2170}, {1360, 3024}, {1362, 1365}, {1445, 4666}, {1456, 2003}, {1468, 4332}, {1699, 1864}, {1838, 1887}, {2900, 3243}, {3600, 3889}, {3671, 3874}, {3742, 3911}, {3812, 4848}, {3869, 4323}, {3881, 4298}, {3892, 4315}


X(5174) = INVERSE-IN-FUHRMANN-CIRCLE OF X(92)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a3 + b3 + c3 - a2b - a2c - ab2 - ac2 - abc)

X(5174) lies on these lines:
{4, 8}, {10, 29}, {19, 3692}, {27, 306}, {28, 100}, {34, 3870}, {40, 1748}, {80, 1896}, {145, 278}, {162, 2907}, {225, 1897}, {240, 4642}, {270, 447}, {281, 3617}, {286, 319}, {317, 322}, {412, 515}, {427, 3757}, {518, 1888}, {519, 1838}, {528, 1852}, {653, 4848}, {958, 1013}, {1214, 3152}, {1441, 2475}, {1826, 2322}, {1844, 3754}, {1848, 4514}, {2349, 2816}, {2975, 4219}


X(5175) = INVERSE-IN-FUHRMANN-CIRCLE OF X(329)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 + 2a3b + 2a3c - 2ab3 - 2ac3 - 2ab2c - 2ac2 - 6b2c2

X(5175) lies on these lines:
{2, 950}, {4, 8}, {7, 2475}, {9, 3617}, {10, 452}, {12, 3189}, {20, 4652}, {63, 3146}, {78, 3091}, {100, 405}, {145, 226}, {377, 938}, {442, 496}, {546, 3940}, {908, 3832}, {958, 1005}, {1490, 3872}, {1750, 4853}, {1837, 2550}, {2000, 4296}, {2094, 4292}, {2551, 3983}, {2886, 3486}, {3241, 3487}, {3476, 3813}, {3529, 3916}, {3627, 3927}, {4084, 4295}


X(5176) = INVERSE-IN-POLAR-CIRCLE OF X(1828)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + a3b + a3c - 3a2bc - ab3 - ac3 + 2ab2c + 2abc2 - 2b2c2

X(5176) lies on these lines:
{1, 3814}, {2, 1319}, {4, 8}, {5, 4861}, {10, 36}, {30, 1145}, {63, 484}, {80, 519}, {100, 515}, {145, 1837}, {149, 3880}, {498, 3897}, {513, 4397}, {529, 3036}, {758, 1109}, {901, 2370}, {952, 4511}, {1155, 3617}, {1479, 3885}, {2478, 3890}, {2802, 3583}, {2995, 3596}, {3035, 4881}, {3245, 3626}, {3586, 3895}

X(5176) = anticomplement of X(1319)
X(5176) = excircle-radical-circle-inverse of X(573)


X(5177) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(452)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - a4 - 2a2b2 - 2a2c2 - 4a2bc - 4ab2c - 4abc2 - 6b2c2

X(5177) lies on these lines:
{2, 3}, {8, 226}, {10, 329}, {12, 480}, {72, 3617}, {145, 3419}, {200, 3947}, {225, 347}, {253, 318}, {388, 2886}, {950, 3616}, {954, 3871}, {966, 1901}, {1125, 3586}, {1655, 2996}, {1698, 1770}, {1864, 3812}, {2551, 3925}, {2893, 3945}, {3011, 4339}, {3085, 3822}, {3485, 3838}, {3488, 3622}, {3614, 4413}


X(5178) = INVERSE-IN-FUHRMANN-CIRCLE OF X(3681)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 + a3b + a3c + a2bc - ab3 - ac3 - 2ab2c - 2abc2 - 2b2c2

X(5178) lies on these lines:
{1, 3841}, {2, 3189}, {4, 8}, {5, 4420}, {10, 1621}, {12, 3935}, {80, 3626}, {145, 3475}, {149, 960}, {377, 3873}, {390, 1837}, {518, 2475}, {1043, 3006}, {1479, 3876}, {1483, 4861}, {1699, 3984}, {1834, 3920}, {2476, 3811}, {2975, 4297}, {3583, 3678}, {3647, 4330}, {4361, 4950}


X(5179) = INVERSE-IN-POLAR-CIRCLE OF X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3b2 + a3c - a2b2 - a2c2 + ab3 + ac3 - ab2c - abc2 - b4 - c4 + 2b2c2

X(5179) lies on these lines:
{4, 9}, {5, 1212}, {30, 910}, {37, 495}, {80, 294}, {101, 515}, {119, 1566}, {218, 1837}, {220, 355}, {514, 661}, {517, 1146}, {519, 4919}, {672, 1737}, {948, 1323}, {950, 4251}, {1210, 4253}, {1479, 2082}, {3732, 4872}, {3911, 5030}, {4262, 4304}

Let U be the radical circle of the excircles. Then X(5179) is the U-inverse of X(573). (Randy Hutson, November 22, 2014)


X(5180) = INVERSE-IN-POLAR-CIRCLE OF X(1900)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - 2a3b - 2a3c + 3a2bc + 2ab3 + 2ac3 - ab2c - abc2 - 2b2c2

X(5180) lies on these lines:
{2, 484}, {4, 8}, {7, 1319}, {36, 3616}, {79, 3884}, {149, 758}, {320, 4742}, {513, 4801}, {516, 4511}, {529, 1320}, {535, 3241}, {1537, 4996}, {1727, 3218}, {1836, 3877}, {2475, 3878}, {2975, 3648}, {3245, 3814}, {4084, 4857}, {4301, 4861}


X(5181) = INVERSE-IN-CIRCUMCIRCLE OF X(2936)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - 2a2)(b6 + c6 - a4b2 - a4c2 + 2a2b2c2 - b4c2 - b2c4)

X(5181) lies on these lines:
{2, 895}, {3, 67}, {20, 1632}, {69, 110}, {113, 511}, {125, 126}, {468, 524}, {684, 1649}, {858, 2393}, {960, 2836}, {1176, 3047}, {1205, 3917}, {1350, 2777}, {1352, 4550}, {1511, 3564}, {2781, 2883}, {3448, 3620}

X(5181) = complement of X(895)


X(5182) = INVERSE-IN-1st-LEMOINE-CIRCLE OF X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a6 - 2a4b2 - 2a4c2 + a2b4 + a2c4 + 3a2b2c2 + b4c2 + b2c4

X(5182) lies on these lines:
{2, 98}, {6, 99}, {30, 2456}, {32, 1992}, {69, 620}, {83, 597}, {115, 3618}, {249, 524}, {384, 575}, {385, 2030}, {538, 1692}, {543, 5034}, {576, 3552}, {599, 1078}, {754, 2458}, {5032, 5039}


X(5183) = INVERSE-IN-BEVAN-CIRCLE OF X(1697)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b3 + 3c3 - 2a3 - 3a2b - 3a2c + 2ab2 + 2ac2 + 4abc - 3b2c - 3bc2)

X(5183) lies on these lines:
{1, 3}, {9, 4731}, {19, 1878}, {44, 4695}, {405, 3922}, {513, 4041}, {535, 4669}, {758, 3689}, {1478, 3654}, {1706, 3983}, {2308, 4642}, {2348, 5011}, {3218, 3880}, {3650, 4691}, {3683, 3753}, {3814, 3828}


X(5184) = INVERSE-IN-BEVAN-CIRCLE OF X(1697)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 - 2a3b - 2a3c + ab3 + ac3 + ab2c + abc2 - b2c2)

X(5184) lies on these lines:
{1, 187}, {10, 316}, {40, 511}, {238, 5011}, {291, 484}, {512, 659}, {517, 2080}, {625, 1698}, {761, 2702}, {986, 1326}, {1386, 1691}, {1572, 2021}, {2076, 3242}, {3679, 3849}, {4649, 4868}


X(5185) = INVERSE-IN-POLAR-CIRCLE OF X(150)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 - c2)(a2 - b2 + c2)(a2b2 + a2c2 - 2ab3 - 2ac3 + b4 + c4)

X(5185) lies on these lines:
{4, 150}, {25, 101}, {33, 181}, {34, 1362}, {103, 1593}, {116, 427}, {118, 235}, {428, 544}, {1112, 2774}, {1827, 1845}, {1829, 2809}, {1830, 2821}, {1843, 2810}, {1862, 3887}


X(5186) = INVERSE-IN-POLAR-CIRCLE OF X(148)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a4b2 + a4c2 - 4a2b2c2 + b4c2 + b242)

X(5186) lies on these lines:
{4, 147}, {25, 99}, {33, 3023}, {34, 3027}, {98, 1593}, {114, 235}, {115, 427}, {428, 543}, {468, 620}, {690, 1112}, {1569, 3199}, {1862, 2787}, {1885, 2794}, {1974, 5026}


X(5187) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4190)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - a4 - 2a2b2 - 2a2c2 + 2a2bc + 2ab2c + 2abc2 - 6b2c2

X(5187) lies on these lines:
{2, 3}, {11, 3436}, {145, 1837}, {960, 3617}, {1001, 3614}, {1320, 3621}, {1329, 3434}, {1478, 3825}, {1479, 3814}, {1728, 3218}, {2899, 3006}, {3476, 3622}, {3947, 4666}


X(5188) = INVERSE-IN-MOSES-CIRCLE OF X(2025)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b6 + c6 - 3a4b2 - 3a4c2 + 2a2b4 + 2a2c4 + 3b4c2 + 3b4c2)

Let T1 be the antipedal triangle of the 1st Brocard point, and let T2 be the antipedal triangles of 2nd Brocard point (these points comprising the bicentric pair PU(1)). Then X(5188) is the point in which the Brocard axis meets the line of the circumcenters of T1 and T2. Also, X(5188) is the radical trace of the circumcircles of T1 and T2, as well as the insimilicenter of those circles and the midpoint of their centers. (Randy Hutson, November 22, 2014)

X(5188) lies on these lines:
{3, 6}, {4, 3934}, {20, 76}, {194, 3522}, {237, 3917}, {262, 631}, {376, 538}, {550, 2782}, {626, 1513}, {730, 4297}, {827, 1297}, {1092, 3202}, {3117, 3787}

X(5188) = inverse-in-2nd-Brocard-crcle of X1350)
X(5188) = (39)-of-circumcevian triangle of X(511)
X(5188) = {X(371),X(372)}-harmonic-conjugate of X(5039).


X(5189) = INVERSE-IN-POLAR-CIRCLE OF X(428)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a6 - a4b2 - a4c2 + a2b4 + a2c4 + a2b2c2 - b4c2 - b2c4

X(5189) lies on these lines:
{2, 3}, {98, 1291}, {149, 4442}, {316, 3266}, {323, 1503}, {511, 3448}, {523, 2528}, {842, 930}, {933, 2697}, {2393, 2892}, {2453, 3314}, {2979, 3410}

X(5189) = anticomplement of X(23)
X(5189) = inverse-in-anticomplementary-circle of X(2)
X(5189) = inverse-in-deLongchamps-circle of X(22)
X(5189) = inverse-in-{circumcircle, nine-point circle}-inverter of X(140)
X(5189) = reflection of X(23) in the deLongchamps line


X(5190) = INVERSE-IN-POLAR-CIRCLE OF X(101)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)2(a2 + b2 - c2)(a2 - b2 + c2)(b3 + c3 - a2b - a2c - abc)

X(5190) is the center of the hyperbola {A,B,C,X(4),X(27)}, which meets the circumcircle at X(917) and is tangent to the line X(4)X(9) at X(4), and tangent to the line X(27)X(86) at X(27). (Randy Hutson, November 22, 2014)

X(5190) lies on the nine-point circle and these lines:
{2, 1305}, {4, 101}, {19, 117}, {113, 1839}, {116, 2973}, {119, 1826}, {120, 1855}, {121, 281}, {122, 3138}, {125, 1146}, {132, 1842}, {1560, 1860}

X(5190) = midpoint of X(4) and X(917)
X(5190) = complement of X(1305)


X(5191) = INVERSE-IN-PARRY-CIRCLE OF X(1495)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b6 + c6 - 2a6 + 2a4b2 + 2a4c2 - a2b4 - a2c4 - b4c2 - b2c4)

X(5191) lies on these lines:
{3, 74}, {6, 157}, {23, 2080}, {25, 111}, {32, 3124}, {51, 5008}, {98, 1316}, {184, 574}, {187, 237}, {868, 2794}, {2782, 4226}, {3098, 3506}

X(5191) = pole of the line X(23)X(110) with respect to the Parry circle.


X(5192) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(4202)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + a3b + a3c + a2b2 + a2c2 + ab3 + ac3 + b3c + bc3 + 2b2c2

X(5192) lies on these lines:
{1, 996}, {2, 3}, {8, 1191}, {10, 748}, {31, 3831}, {614, 4968}, {1089, 3891}, {1150, 1724}, {1220, 2899}, {1468, 3840}, {1479, 4972}, {2292, 4011}


X(5193) = INVERSE-IN-CIRCUMCIRCLE OF X(1420)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b - c)(a - b + c)(a3 + b3 + c3 - a2b - a2c - ab2 - ac2 + 5abc - 2b2c - 2bc2)

X(5193) lies on these lines:
{1, 3}, {104, 1519}, {106, 1457}, {108, 1877}, {109, 1149}, {388, 3814}, {995, 2003}, {1398, 1878}, {1404, 2316}, {1421, 1455}, {1428, 2810}, {2975, 3452}


X(5194) = INVERSE-IN-INCIRCLE OF X(1469)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b - c)(a - b + c)(a2b2 + a2c2 + 4a2bc + b4 - 2b3c - 4b2c2 - 2bc3 + c4)

X(5194) lies on these lines:
{1, 256}, {12, 625}, {56, 187}, {172, 2031}, {316, 388}, {512, 3669}, {538, 3027}, {999, 2080}, {1015, 2021}, {1357, 1429}, {1428, 2030}


X(5195) = INVERSE-IN-INCIRCLE OF X(4021)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - a3b - a3c + 3a2bc + ab3 + ac3 - ab2c - abc2 - b3c - bc3

X(5195) lies on these lines:
{1, 7}, {2, 5011}, {30, 664}, {72, 319}, {74, 927}, {150, 517}, {514, 4024}, {534, 1944}, {1479, 3212}, {3057, 4911}


X(5196) = INVERSE-IN-CIRCUMCIRCLE OF X(4184)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b)(a + c)(b4 + c4 - a4 - a3b - a3c + a2b2 + a2c2 + a2bc - 2b2c2)

X(5196) lies on these lines:
{2, 3}, {60, 1770}, {99, 3006}, {103, 476}, {110, 516}, {523, 4467}, {593, 3914}, {675, 691}, {759, 4316}, {1326, 3120}


X(5197) = INVERSE-IN-1st-BROCARD-CIRCLE OF X(1054)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - a2b2 - a2c2 - a2bc - ab2c - abc2 - b3c - bc3 + b2c2)

X(5197) lies on these lines:
{31, 36}, {43, 3955}, {57, 985}, {81, 1325}, {110, 4414}, {182, 1054}, {986, 1437}, {991, 1283}, {1326, 1790}, {2194, 4650}


X(5198) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(1907)

Trilinears       3 sec A - cos A : 3 sec B - cos B : 3 sec C - cos C
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 - c2)(a2 - b2 + c2)(a4 + b4 + c4 - 2a2b2 - 2a2c2 - 10b2c2)

X(5198) lies on these lines:
{2, 3}, {33, 3303}, {34, 3304}, {51, 1498}, {53, 1033}, {159, 3574}, {1173, 3527}, {1753, 2355}, {2207, 5007}


X(5199) = INVERSE-IN-SPIEKER-CIRCLE OF X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(3b3 + 3c3 - 2a3 + a2b + a2c - 2ab2 - 2ac2 + 4abc - 3b2c - 3bc2)

X(5199) lies on these lines:
{2, 1323}, {4, 9}, {121, 1566}, {220, 3626}, {514, 4521}, {519, 1146}, {1212, 3634}, {2297, 5018}


X(5200) = INVERSE-IN-ORTHOCENTROIDAL-CIRCLE OF X(3127)

Barycentrics   (S + a2)/SA : (S + b2)/SB : (S + c2)/SC
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(2a2 +[(b + c - a)(c + a - b)(a + b - c)(a + b + c)]1/2)

X(5200) lies on these lines:
{2, 3}, {6, 1162}, {51, 1588}, {154, 3070}, {184, 1587}, {615, 1165}, {1164, 3087}


X(5201) = INVERSE-IN-CIRCUMCIRCLE OF X(3111)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4b2 + a4c2 - a2b4 - a2c4 + 2a2b2c2 - b4c2 - b2c4)

X(5201) lies on these lines:
{3, 6}, {23, 385}, {160, 193}, {183, 1995}, {237, 524}, {340, 4230}, {2930, 3511}


X(5202) = INVERSE-IN-PARRY-CIRCLE OF X(3724)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)(a5 - a3b2 - a3c2 + ab2c2 - b4c + b3c2 + b2c3 - bc4)

X(5202) lies on these lines:
{1, 60}, {31, 4128}, {187, 237}, {213, 3124}, {692, 2643}, {1400, 2054}


X(5203) = INVERSE-IN-POLAR-CIRCLE OF X(193)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(a2 + b2 - 3c2)(a2 - 3b2 + c2)(2a2 - b2 - c2)

X(5203) lies on these lines:
{4, 193}, {30, 3565}, {126, 468}, {2501, 3566}

X(5203) = antigonal image of X(468)


X(5204) = INVERSE-IN-CIRCUMCIRCLE OF X(5048)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[s(b + c - a) - (a - b + c)(a + b - c)]
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(3b2 + 3c2 - 3a2 - 2bc)
X(5204) = R*X(1) - 3r*X(3)
X(5204) = (2r - R)*X(11) + r*X(20)

X(5204) lies on these lines:
{1, 3}, {2, 3614}, {5, 4299}, {11, 20}, {12, 631}, {21, 4423}, {30, 499}, {33, 3516}, {34, 3515}, {41, 5022}, {44, 198}, {45, 1696}, {100, 3621}, {140, 1478}, {145, 4421}, {172, 5013}, {212, 1106}, {215, 1092}, {218, 5030}, {220, 1055}, {376, 3086}, {377, 4999}, {382, 4316}, {388, 3523}, {404, 958}, {474, 993}, {495, 3530}, {496, 548}, {497, 3522}, {498, 549}, {518, 4855}, {550, 1479}, {602, 1399}, {603, 2361}, {611, 5092}, {613, 3098}, {672, 3207}, {859, 4278}, {896, 1473}, {899, 4191}, {936, 3715}, {956, 3626}, {960, 4652}, {997, 3916}, {1001, 4189}, {1125, 1836}, {1152, 2067}, {1193, 4252}, {1201, 3052}, {1259, 4996}, {1350, 1428}, {1376, 2975}, {1436, 2173}, {1443, 1804}, {1450, 4300}, {1468, 4255}, {1469, 5085}, {1475, 4258}, {1656, 3585}, {1657, 3583}, {1837, 3911}, {1914, 5023}, {2071, 5160}, {2275, 3053}, {2886, 4190}, {3035, 3436}, {3085, 3524}, {3240, 4210}, {3286, 4225}, {3474, 3616}, {3526, 4325}, {3528, 4294}, {3534, 3582}, {3622, 4428}, {3624, 3824}, {3869, 4881}, {3928, 3962}, {5087, 5121}

X(5204) = {X(55),X(56}-harmonic conjugate of X(3304)


X(5205) = INVERSE-IN-SPIEKER-CIRCLE OF X(3687)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - a2b - a2c + 3abc - b2c - bc2

X(5205) lies on these lines:
{1, 2}, {20, 2899}, {36, 3992}, {56, 341}, {75, 4413}, {100, 2726}, {125, 3936}, {171, 4672}, {190, 1155}, {238, 4434}, {295, 660}, {312, 1376}, {319, 4023}, {333, 3740}, {404, 3701}, {474, 4385}, {496, 5100}, {497, 1997}, {518, 3699}, {645, 2651}, {649, 3239}, {675, 2748}, {726, 1054}, {750, 894}, {851, 3948}, {908, 4645}, {999, 4737}, {1156, 4607}, {1265, 1788}, {1311, 2743}, {1447, 3263}, {3035, 3932}, {3218, 3952}, {3452, 4388}, {3550, 4011}, {3717, 3911}, {3769, 4383}, {3816, 4514}, {3975, 4447}, {4187, 5015}, {4997, 5087}

X(5205) = complement of X(5211)
X(5205) = anticomplement of X(5121)
X(5205) = inverse-in-{circumcircle, nine-point circle}-inverter of X(10)


X(5206) = INVERSE-IN-CIRCUMCIRCLE OF X(5111)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b2 + 2c2 - 3a2)

Let H be the ellipse of these five points: X(574), PU(1), PU(2). One vertex of H is X(574); the other is X(5206). (Randy Hutson, November 22, 2014)

X(5206) lies on these lines:
{3, 6}, {20, 115}, {22, 3291}, {35, 2242}, {36, 2241}, {172, 5010}, {186, 1968}, {230, 550}, {315, 620}, {376, 3767}, {439, 3785}, {546, 3054}, {631, 1506}, {1003, 3934}, {1078, 3552}, {1658, 2493}, {1971, 3357}, {2079, 2937}, {2482, 3926}, {2548, 3523}, {2549, 3522}, {3199, 3515}, {3530, 3815}, {3787, 3796}, {5087, 5121}

X(5206) = {X(371),X(372)}-harmonic conjugate of X(5097)


X(5207) = INVERSE-IN-ANTICOMPLEMENTARY-CIRCLE OF X(315)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b6 + c6 - a6 - a2b2c2

X(5207) lies on these lines:
{2, 1501}, {4, 69}, {6, 5025}, {141, 384}, {147, 325}, {148, 698}, {187, 3619}, {193, 5111}, {334, 1966}, {512, 3267}, {625, 1692}, {626, 2458}, {732, 1916}, {1570, 1992}, {1965, 4388}, {2080, 3785}, {3620, 5104}, {4576, 5189}, {5087, 5121}

X(5207) = anticomplement of X(1691)
X(5207) = crosspoint of X(147) and X(2896) with respect to the excentral triangle
X(5207) = crosspoint of X(147) and X(2896) with respect to the anticomplementary triangle


X(5208) = INVERSE-IN-CONWAY-CIRCLE OF X(2651)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b)(a + c)(b3 + c3 - ab2 - ac2 - abc)

X(5208) lies on these lines:
{1, 21}, {2, 3786}, {7, 310}, {27, 295}, {65, 1043}, {86, 354}, {228, 4225}, {284, 3509}, {333, 518}, {942, 1010}, {982, 3736}, {1412, 5083}, {3218, 4184}, {5087, 5121}


X(5209) = INVERSE-IN-CONWAY-CIRCLE OF X(314)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a + b)(a + c)(a3 + abc - b2c - bc2)

X(5209) lies on these lines:
{1, 75}, {36, 99}, {80, 334}, {313, 757}, {670, 4495}, {730, 741}, {811, 1785}, {1019, 1577}, {1323, 4625}, {1509, 1909}, {1931, 3948}, {1963, 3963}, {4039, 4600}, {5087, 5121}


X(5210) = INVERSE-IN-CIRCUMCIRCLE OF X(5107)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(5b2 + 5c2 - 7a2)

X(5210) lies on these lines:
{3, 6}, {4, 3054}, {22, 111}, {115, 3534}, {154, 5191}, {230, 376}, {439, 3620}, {548, 3767}, {631, 3055}, {2548, 3530}, {3524, 3815}, {3630, 3926}, {3631, 3785}, {5087, 5121}


X(5211) = INVERSE-IN-CONWAY-CIRCLE OF X(1999)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 - 2ab2 - 2ac2 + 3abc

X(5211) lies on these lines:
{1, 2}, {110, 2726}, {244, 4645}, {320, 3999}, {497, 3210}, {675, 2705}, {982, 4388}, {1330, 3953}, {3315, 3936}, {3667, 4025}, {3752, 4514}, {4440, 5057}, {5087, 5121}

X(5211) = anticomplement of X(5205)


X(5212) = INVERSE-IN-SPIEKER-RADICAL-CIRCLE OF X(2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - 5a2b - 5a2c + 6abc + b2c + bc2

X(5212) lies on these lines:
{1, 2}, {518, 3030}, {661, 3667}, {908, 4442}, {1155, 4831}, {1266, 4706}, {1738, 4892}, {3684, 4700}, {3879, 4413}, {4023, 4357}, {4656, 4734}, {5087, 5121}


X(5213) = INVERSE-IN-SPIEKER-RADICAL-CIRCLE OF X(115)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)2(a2 - ab + ac - b2)(a2 + ab - ac - c2)

Let (N) be the nine-point circle of a triangle ABC. Let (IA) be the A-excircle of ABC, and define (IB) and (IC) cyclically. Let A' be the midpoint of side BC, and define B' and C' cyclically. Let (KA) be the circle, other than (N), that passes through B' and C' and touches (IA), and define (KB) and (KC) cyclically. Let (K) be the circle externally tangent to (KA), (KB), (KC), and let (L) be the circle externally tangent to (IA), (IB), (IC). Then X(5213) = (K)∩(L). (Tran Quang Hung, July 16, 2014)

If you have The Geometer's Sketchpad, you can view X(5213).

X(5213) lies on the Apollonius circle and these lines:
{10, 115}, {181, 1356}, {214, 1015}, {386, 741}, {573, 759}, {1018, 3124}, {1575, 5164}, {2238, 5011}, {5087, 5121}


X(5214) = INVERSE-IN-CONWAY-CIRCLE OF X(3109)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b)(a + c)(b - c)(a2 + 2b2 + 2c2 - ab - ac + 4bc)

X(5214) lies on these lines:
{1, 523}, {513, 4960}, {522, 1019}, {3733, 4777}, {4151, 4581}, {4802, 4833}, {4840, 4926}, {5087, 5121}

X(5213) = pole of the Euler line with respect to the Conway circle


X(5215) = INVERSE-IN-VAN-LAMOEN-CIRCLE OF X(598)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 10a4 + 4b4 + 4c4 - 7a2b2 - 7a2c2 - 4b2c2

X(5215) = centroid of {X(13), X(14), X(15), X(16), X(5463), X(5464)} (Randy Hutson, November 22, 2014)

X(5215) lies on these lines:
{2, 187}, {230, 2482}, {511, 5054}, {524, 1692}, {597, 5107}, {599, 2030}, {5087, 5121}


X(5216) = INVERSE-IN-CONWAY-CIRCLE OF X(3110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a + b)(a + c)(b - c)(2b2 + 2c2 + 3bc)

X(5216) lies on these lines:
{1, 512}, {513, 4960}, {834, 3737}, {1734, 4481}, {2978, 4040}, {3733, 4834}, {5087, 5121}

X(5216) = pole of the Brocard axis with respect to the Conway circle


X(5217) = INTERSECTION OF LINES X(1)X(3) AND X(12)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3b2 + 3c2 - 3a2 + 2bc)

X(5217) = R*X(1) + 3r*X(3)
X(5217) = (2r + R)*X(12) + r*X(20)

X(5217) lies on these lines:
{1, 3}, {4, 3614}, {5, 4302}, {8, 4421}, {11, 631}, {12, 20}, {21, 1376}, {30, 498}, {31, 4255}, {33, 3515}, {34, 3516}, {42, 4252}, {45, 198}, {73, 3532}, {78, 4640}, {100, 958}, {140, 1479}, {172, 5023}, {186, 5160}, {191, 3940}, {212, 1399}, {218, 4262}, {376, 3085}, {382, 4324}, {388, 3522}, {404, 1001}, {405, 3634}, {474, 4423}, {480, 1259}, {495, 548}, {496, 3530}, {497, 3523}, {499, 549}, {518, 4652}, {550, 1478}, {601, 2361}, {603, 1253}, {611, 3098}, {613, 5092}, {672, 4258}, {899, 1011}, {902, 1191}, {936, 3683}, {956, 3625}, {960, 4855}, {991, 2594}, {993, 3626}, {1092, 2477}, {1152, 2066}, {1193, 3052}, {1334, 2272}, {1350, 2330}, {1468, 2334}, {1500, 5206}, {1621, 4188}, {1656, 3583}, {1657, 3585}, {1788, 4313}, {1837, 4304}, {1898, 5044}, {1914, 5013}, {2276, 3053}, {2280, 5022}, {2478, 3035}, {2975, 3621}, {3056, 5085}, {3058, 3086}, {3240, 4184}, {3434, 4999}, {3474, 3649}, {3526, 4330}, {3528, 4293}, {3534, 3584}, {3616, 4428}, {3811, 3916}, {3890, 4881}, {3911, 4314}, {3929, 4005}


X(5218) = INTERSECTION OF LINES X(2)X(11) AND X(4)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 3a2 - 2bc)

X(5218) = 9r*X(2) - 2(2r - R)*X(11)
X(5218) = 2(2r + R)*X(12) + r*X(20)

X(5218) lies on these lines:
{1, 631}, {2, 11}, {3, 388}, {4, 35}, {5, 4294}, {7, 1155}, {8, 2320}, {9, 1776}, {10, 3486}, {12, 20}, {21, 2551}, {36, 1056}, {40, 3485}, {46, 3487}, {56, 3523}, {57, 3475}, {69, 2330}, {140, 3086}, {165, 226}, {171, 212}, {197, 4224}, {243, 281}, {329, 4640}, {344, 5205}, {345, 3790}, {346, 3712}, {355, 4305}, {376, 1478}, {391, 4023}, {452, 1329}, {496, 3526}, {499, 1058}, {549, 999}, {601, 3074}, {612, 1040}, {650, 885}, {750, 1253}, {899, 2293}, {944, 3612}, {950, 1698}, {966, 2268}, {991, 4551}, {993, 3421}, {1125, 1697}, {1479, 3090}, {1737, 3488}, {1742, 2635}, {1837, 4313}, {1858, 3876}, {1864, 3740}, {2066, 3069}, {2098, 3622}, {2999, 4989}, {3011, 4000}, {3056, 3618}, {3057, 3616}, {3158, 4847}, {3161, 4009}, {3296, 3337}, {3436, 4189}, {3452, 4512}, {3476, 3576}, {3528, 4299}, {3529, 3585}, {3545, 3583}, {3579, 4295}, {3600, 5204}, {3614, 3832}, {3634, 4314}, {3671, 5128}, {3855, 4330}, {3913, 4999}, {4309, 5067}, {4402, 4706}, {4414, 4419}


X(5219) = INTERSECTION OF LINES X(1)X(5) AND X(2)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2b + 2c - a)/(b + c - a)

X(5219) lies on these lines:
{1, 5}, {2, 7}, {4, 3601}, {10, 3340}, {34, 975}, {37, 1465}, {40, 498}, {55, 1538}, {56, 3624}, {65, 1698}, {78, 2476}, {85, 4554}, {109, 750}, {165, 1836}, {191, 1454}, {200, 2886}, {208, 451}, {210, 5173}, {278, 1826}, {319, 4417}, {381, 3586}, {388, 1125}, {442, 936}, {468, 1892}, {497, 3817}, {499, 3333}, {551, 3476}, {631, 4292}, {938, 5056}, {940, 2003}, {942, 1656}, {946, 1697}, {948, 1323}, {950, 3091}, {991, 2635}, {997, 3822}, {1000, 1512}, {1001, 2078}, {1155, 4312}, {1210, 3090}, {1376, 3256}, {1419, 4648}, {1441, 4358}, {1466, 3824}, {1478, 3576}, {1617, 4423}, {1788, 3634}, {1876, 5094}, {2099, 3679}, {2475, 4855}, {2999, 3553}, {3006, 4901}, {3158, 3434}, {3339, 3649}, {3488, 3545}, {3584, 5119}, {3585, 3612}, {3617, 4323}, {3832, 4313}, {4032, 4687}, {4054, 4659}, {4295, 5128}, {4671, 4873}, {5054, 5122}

X(5219) = isogonal conjugate of X(2364)
X(5219) = {X(2),X(7)}-harmonic conjugate of X(3911)


X(5220) = INTERSECTION OF LINES X(1)X(6) AND X(7)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b2 + 2c2 - a2 - ab - ac + 2bc)

X(5220) lies on these lines:
{1, 6}, {2, 3715}, {3, 2801}, {7, 12}, {8, 190}, {10, 527}, {38, 4383}, {40, 4662}, {46, 3697}, {55, 1776}, {56, 3876}, {57, 3740}, {63, 210}, {65, 3951}, {69, 3932}, {78, 4005}, {100, 3711}, {142, 3634}, {144, 1654}, {183, 4518}, {191, 4436}, {200, 3929}, {319, 3790}, {321, 4042}, {329, 2886}, {344, 4966}, {354, 3305}, {355, 382}, {390, 3621}, {480, 1259}, {612, 4641}, {672, 3789}, {726, 4361}, {756, 940}, {758, 1159}, {971, 1158}, {993, 3940}, {997, 5126}, {1150, 3952}, {1621, 4661}, {1706, 4866}, {1890, 5130}, {2246, 4712}, {2646, 3984}, {3052, 3961}, {3218, 4413}, {3245, 3679}, {3286, 3786}, {3416, 3717}, {3625, 4133}, {3632, 4693}, {3683, 3870}, {3696, 3729}, {3773, 4445}, {3873, 4423}, {3883, 4899}, {3916, 4533}, {4078, 4851}, {4671, 4756}

X(5220) = X(67) of Fuhrman triangle


X(5221) = INTERSECTION OF LINES X(1)X(3) AND X(7)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2b + 2c + a)/(b + c - a)

X(5221) lies on these lines:
{1, 3}, {2, 3649}, {6, 1406}, {7, 12}, {10, 553}, {11, 4295}, {34, 2355}, {44, 2285}, {45, 1400}, {63, 3812}, {72, 4413}, {79, 381}, {88, 959}, {89, 961}, {208, 1827}, {221, 1393}, {226, 3634}, {227, 1418}, {244, 1191}, {386, 1464}, {388, 3617}, {405, 3647}, {474, 758}, {936, 3962}, {938, 3474}, {952, 4317}, {956, 3754}, {958, 3218}, {960, 3306}, {997, 4018}, {1046, 4383}, {1210, 1836}, {1254, 1407}, {1317, 4308}, {1376, 3868}, {1399, 1451}, {1417, 4792}, {1434, 3212}, {1452, 1876}, {1469, 3214}, {1475, 2272}, {1698, 3715}, {1722, 4641}, {1835, 4185}, {1837, 4292}, {2334, 4646}, {2594, 4306}, {2650, 4255}, {3125, 5021}, {3600, 3621}, {3624, 4870}, {3626, 4031}, {3671, 3911}, {3740, 3951}, {3873, 3913}, {3901, 3940}, {3924, 4252}, {3947, 3982}


X(5222) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 + (b - c)2

X(5222) lies on these lines:
{1, 2}, {6, 7}, {9, 3672}, {20, 1453}, {27, 3194}, {41, 1429}, {44, 4419}, {57, 279}, {69, 3759}, {75, 3618}, {81, 277}, {142, 1449}, {144, 1743}, {192, 3161}, {193, 3662}, {218, 329}, {220, 4383}, {223, 4350}, {241, 2275}, {273, 1249}, {278, 607}, {319, 3619}, {320, 1992}, {344, 4360}, {346, 3875}, {347, 1445}, {390, 3755}, {391, 4357}, {527, 4346}, {594, 4371}, {597, 4363}, {599, 4969}, {857, 1834}, {966, 4657}, {1100, 4648}, {1104, 4313}, {1203, 4295}, {1212, 3666}, {1266, 4454}, {1386, 2550}, {1423, 2347}, {1442, 3554}, {1468, 4209}, {1738, 4307}, {2345, 3589}, {3664, 4859}, {3729, 4452}, {3731, 4021}, {3739, 4798}, {3751, 4310}, {3879, 4869}, {4460, 4852}, {4470, 4688}


X(5223) = INTERSECTION OF LINES X(1)X(6) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3b2 + 3c2 - a2 - 2ab - 2ac + 2bc)

X(5223) lies on these lines:
{1, 6}, {3, 480}, {7, 10}, {8, 144}, {38, 2999}, {40, 971}, {55, 3929}, {56, 4005}, {57, 210}, {63, 100}, {69, 3717}, {142, 1698}, {190, 3886}, {191, 3174}, {329, 1699}, {344, 4684}, {354, 3715}, {390, 519}, {443, 4355}, {474, 4533}, {517, 4915}, {527, 1478}, {528, 4677}, {537, 673}, {668, 3403}, {936, 1445}, {991, 2340}, {997, 4134}, {1155, 3711}, {1156, 2802}, {1376, 3928}, {1706, 4662}, {1707, 3961}, {1738, 4862}, {2184, 2947}, {2809, 4752}, {2810, 3781}, {2975, 3984}, {3008, 4310}, {3158, 4640}, {3219, 3870}, {3305, 3873}, {3333, 5044}, {3340, 3962}, {3416, 4901}, {3576, 3940}, {3677, 4383}, {3696, 4659}, {3755, 4419}, {3869, 4853}, {3925, 4654}, {4420, 4652}, {4430, 4666}


X(5224) = INTERSECTION OF LINES X(2)X(6) AND X(7)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 + bc + ca + ab

X(5224) lies on these lines:
{1, 319}, {2, 6}, {3, 3437}, {7, 12}, {8, 4026}, {9, 1760}, {10, 75}, {37, 3661}, {38, 4446}, {85, 307}, {142, 4751}, {190, 2345}, {192, 594}, {239, 4657}, {261, 1078}, {274, 4283}, {286, 5125}, {320, 1698}, {326, 936}, {334, 1218}, {404, 1444}, {405, 2893}, {638, 2047}, {894, 4643}, {1086, 4699}, {1100, 4690}, {1125, 3879}, {1278, 4665}, {1330, 2049}, {1975, 4201}, {2321, 4664}, {3305, 4872}, {3616, 4966}, {3617, 3672}, {3625, 4464}, {3626, 4021}, {3634, 3664}, {3644, 4431}, {3662, 3739}, {3679, 3875}, {3686, 3759}, {3728, 4443}, {3758, 4416}, {3779, 3789}, {3786, 4259}, {3912, 4687}, {3943, 4704}, {4441, 4972}, {4472, 4741}


X(5225) = INTERSECTION OF LINES X(1)X(4) AND X(11)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 + 4a2bc - 6b2c2

X(5225) lies on these lines:
{1, 4}, {5, 4294}, {8, 3967}, {11, 20}, {12, 390}, {30, 3086}, {35, 3090}, {36, 3529}, {55, 3091}, {56, 3146}, {100, 5187}, {149, 3436}, {376, 499}, {381, 3085}, {382, 496}, {452, 2886}, {495, 3843}, {498, 3545}, {516, 1788}, {546, 3295}, {631, 4302}, {908, 3189}, {938, 1836}, {960, 5175}, {962, 1837}, {999, 3627}, {1001, 5177}, {1210, 3474}, {1452, 2961}, {1898, 3868}, {2478, 2550}, {2551, 3434}, {3058, 3839}, {3153, 5160}, {3421, 3625}, {3525, 5010}, {3528, 4324}, {3601, 3817}, {3616, 3838}, {3626, 5082}, {3634, 5084}, {3855, 4309}, {3925, 5129}, {3974, 5015}, {4208, 4423}, {4330, 5067}


X(5226) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3b + 3c - a)/(b + c - a)

X(5226) lies on these lines:
{1, 3091}, {2, 7}, {4, 4313}, {5, 938}, {8, 12}, {11, 3475}, {56, 5047}, {65, 3740}, {78, 5177}, {223, 1442}, {278, 469}, {312, 1441}, {381, 3488}, {388, 1319}, {390, 1699}, {479, 1996}, {484, 498}, {495, 1532}, {497, 3748}, {612, 4318}, {631, 5122}, {651, 940}, {857, 948}, {936, 4208}, {942, 3090}, {950, 3832}, {962, 3085}, {975, 4296}, {1000, 3656}, {1125, 3600}, {1210, 5056}, {1456, 4682}, {1698, 3671}, {1788, 3649}, {2550, 3838}, {2900, 5175}, {3146, 3601}, {3241, 4870}, {3339, 3634}, {3340, 3617}, {3523, 4292}, {3543, 4304}, {3585, 4305}, {3586, 3839}, {3624, 4298}, {3681, 5173}


X(5227) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2)(a2 + b2 + c2 + 2bc)

Let H be the homothety that maps the 2nd extouch triangle onto the excentral triangle; then X(5227) = H(X(6)). (Randy Hutson, November 22, 2014)

X(5227) lies on these lines:
{1, 6}, {3, 3694}, {8, 19}, {40, 1503}, {48, 78}, {57, 141}, {63, 69}, {84, 1350}, {144, 4329}, {159, 197}, {169, 3686}, {193, 3219}, {198, 3965}, {210, 965}, {281, 3421}, {284, 3811}, {319, 1760}, {329, 1848}, {388, 2285}, {515, 1766}, {524, 3929}, {599, 3928}, {612, 2303}, {988, 4261}, {1038, 2286}, {1474, 2287}, {1781, 3679}, {1792, 4288}, {1826, 3436}, {1839, 3434}, {1953, 3872}, {2171, 4390}, {2182, 3713}, {2268, 3930}, {3169, 3496}, {3218, 3620}, {3305, 3618}, {3306, 3619}, {3927, 4047}, {3951, 3958}


X(5228) = INTERSECTION OF LINES X(1)X(3) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - ab - ac - 2bc)/(b + c - a)

X(5228) lies on these lines:
{1, 3}, {2, 220}, {6, 7}, {9, 4328}, {37, 1445}, {63, 1212}, {75, 3713}, {77, 1100}, {81, 279}, {85, 239}, {142, 219}, {175, 3297}, {176, 3298}, {218, 226}, {222, 553}, {269, 1449}, {277, 2982}, {307, 4657}, {481, 1124}, {482, 1335}, {518, 4327}, {664, 4393}, {965, 3739}, {1001, 1471}, {1119, 1172}, {1231, 4359}, {1323, 4031}, {1373, 3301}, {1374, 3299}, {1376, 2340}, {1386, 2263}, {1427, 4350}, {1441, 4361}, {1616, 4323}, {2256, 4648}, {3668, 3946}, {3912, 4513}, {4334, 4649}

X(5228) = crossdifference of every pair of points on the line X(650)X(926)


X(5229) = INTERSECTION OF LINES X(1)X(4) AND X(12)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 - 4a2bc - 6b2c2

X(5229) lies on these lines:
{1, 4}, {2, 3614}, {5, 4293}, {7, 1837}, {8, 1836}, {10, 3474}, {11, 3600}, {12, 20}, {30, 3085}, {35, 3529}, {36, 3090}, {55, 3146}, {56, 3091}, {144, 1654}, {355, 4295}, {376, 498}, {377, 1155}, {381, 3086}, {382, 495}, {443, 3634}, {496, 3843}, {499, 3545}, {518, 5175}, {546, 999}, {631, 4299}, {958, 5177}, {1420, 3817}, {1788, 4292}, {3295, 3627}, {3421, 3626}, {3434, 3621}, {3528, 4316}, {3601, 3947}, {3625, 5082}, {3855, 4317}, {4312, 4848}, {4325, 5067}


X(5230) = INTERSECTION OF LINES X(1)X(2) AND X(4)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + b4 + c4 + 2a3b + 2a3c - 2b2c2

X(5230) lies on these lines:
{1, 2}, {4, 31}, {6, 12}, {11, 1191}, {19, 208}, {40, 3914}, {55, 1834}, {58, 1478}, {65, 3772}, {171, 377}, {213, 3767}, {227, 1108}, {235, 3195}, {238, 2478}, {278, 1254}, {318, 4008}, {388, 1468}, {443, 750}, {497, 3915}, {595, 1479}, {748, 5084}, {902, 4294}, {959, 2006}, {1068, 1148}, {1104, 1837}, {1329, 4383}, {1460, 4185}, {1788, 4000}, {2650, 3487}, {3120, 4295}, {4257, 4299}, {4307, 5177}, {4339, 5175}


X(5231) = INTERSECTION OF LINES X(1)X(2) AND X(9)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(2b2 + 2c2 - a2 - ab - ac - 4bc)

X(5231) lies on these lines:
{1, 2}, {9, 11}, {36, 1004}, {57, 2886}, {63, 1699}, {75, 4554}, {165, 3434}, {244, 4859}, {329, 3817}, {377, 3361}, {442, 3333}, {497, 4512}, {993, 1005}, {1260, 4423}, {1376, 2078}, {1697, 3813}, {1836, 3928}, {2550, 3911}, {3120, 4862}, {3158, 4863}, {3218, 4312}, {3419, 3576}, {3601, 4999}, {3677, 3772}, {3693, 4519}, {3829, 3929}, {3838, 4654}, {4297, 5175}, {4298, 5177}, {4855, 5178}


X(5232) = INTERSECTION OF LINES X(2)X(6) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b2 + 3c2 - a2 + 2ab + 2ac + 2bc

X(5232) lies on these lines:
{2, 6}, {7, 10}, {8, 3672}, {37, 4748}, {75, 3617}, {77, 936}, {144, 2345}, {145, 319}, {279, 307}, {320, 3823}, {346, 3661}, {390, 3775}, {452, 2893}, {474, 1014}, {594, 4419}, {997, 1442}, {1122, 3983}, {1444, 4188}, {1698, 3664}, {3616, 3879}, {3621, 4360}, {3632, 4021}, {3663, 3679}, {3723, 4916}, {3946, 4034}, {4364, 4445}, {4389, 4452}, {4657, 4690}, {4708, 4851}


X(5233) = INTERSECTION OF LINES X(2)X(6) AND X(8)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 + ab + ac - bc)

X(5233) lies on these lines:
{2, 6}, {8, 11}, {9, 4070}, {43, 3847}, {75, 908}, {200, 4514}, {210, 3705}, {312, 2321}, {320, 3306}, {345, 3161}, {474, 1330}, {497, 3996}, {899, 4429}, {997, 998}, {1043, 2478}, {1054, 4655}, {1376, 4388}, {3210, 4415}, {3242, 5211}, {3416, 5205}, {3685, 4679}, {3696, 5087}, {3755, 5212}, {3790, 4009}, {3807, 4671}, {3911, 4416}, {4389, 4850}, {4413, 4645}, {4734, 4854}


X(5234) = INTERSECTION OF LINES X(1)X(6) AND X(10)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(3a2 + b2 + c2 + 4ab + 4ac + 6bc)

X(5234) lies on these lines:
{1, 6}, {2, 3361}, {8, 4314}, {10, 20}, {21, 200}, {55, 4882}, {63, 3339}, {65, 3929}, {142, 4355}, {144, 3671}, {191, 2093}, {210, 3601}, {443, 1478}, {452, 4847}, {936, 993}, {1697, 3683}, {1706, 4640}, {2646, 3715}, {2975, 3305}, {3158, 4662}, {3452, 3624}, {3576, 5044}, {3698, 5128}, {3812, 3928}, {3885, 4853}, {5123, 5131}


X(5235) = INTERSECTION OF LINES X(2)X(6) AND X(10)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2b + 2c - a)/(b + c)

X(5235) lies on these lines:
{2, 6}, {10, 21}, {27, 281}, {28, 5130}, {45, 4671}, {58, 750}, {63, 1781}, {88, 274}, {314, 4358}, {899, 3736}, {958, 4225}, {1014, 3911}, {1043, 3617}, {1155, 3846}, {1255, 1999}, {1376, 4184}, {2177, 3679}, {3218, 3739}, {3286, 4413}, {3624, 4658}, {3681, 5208}, {3712, 4733}, {3757, 4981}, {3977, 4967}, {4384, 4850}, {4396, 4708}


X(5236) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - ab - ac)/[(b + c - a)(b2 + c2 - a2)]

X(5236) lies on these lines:
{1, 4}, {2, 1435}, {7, 19}, {27, 1803}, {28, 4298}, {85, 92}, {108, 2725}, {142, 281}, {241, 5089}, {273, 1826}, {514, 3064}, {518, 1861}, {908, 4564}, {958, 1398}, {1430, 3011}, {1456, 1503}, {1783, 3008}, {1890, 1892}, {2331, 4000}, {3947, 5142}


X(5237) = INTERSECTION OF LINES X(3)X(6) AND X(14)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(5b2 + 5c2 - 5a2 - (12)1/2S)

X(5237) lies on these lines:
{3, 6}, {13, 140}, {14, 20}, {17, 631}, {18, 30}, {35, 202}, {203, 5204}, {395, 550}, {396, 3530}, {397, 549}, {398, 548}, {530, 630}, {532, 628}, {616, 636}, {619, 634}, {627, 3642}, {1092, 3201}, {2306, 5131}


X(5238) = INTERSECTION OF LINES X(3)X(6) AND X(13)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(5b2 + 5c2 - 5a2 + (12)1/2S)

X(5238) lies on these lines:
{3, 6}, {13, 20}, {14, 140}, {17, 30}, {18, 631}, {35, 203}, {202, 5204}, {395, 3530}, {396, 550}, {397, 548}, {398, 549}, {531, 629}, {533, 627}, {617, 635}, {618, 633}, {628, 3643}, {1092, 3200}, {2307, 5010}


X(5239) = INTERSECTION OF LINES X(1)X(6) AND X(10)X(17)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2 + 2bc - (12)1/2S)

X(5239) lies on these lines:
{1, 6}, {2, 559}, {3, 1277}, {8, 1251}, {10, 17}, {16, 214}, {56, 1653}, {61, 3878}, {63, 1082}, {65, 1652}, {142, 3638}, {203, 758}, {471, 1833}, {517, 1276}, {527, 3639}, {2307, 3869}


X(5240) = INTERSECTION OF LINES X(1)X(6) AND X(10)X(18)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + 2bc + (12)1/2S)

X(5240) lies on these lines:
{1, 6}, {2, 1082}, {3, 1276}, {10, 18}, {15, 214}, {21, 1251}, {36, 3179}, {56, 1652}, {62, 3878}, {63, 559}, {65, 1653}, {142, 3639}, {202, 758}, {470, 1832}, {517, 1277}, {527, 3638}


X(5241) = INTERSECTION OF LINES X(2)X(6) AND X(10)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 + a2 + ac2 + 6abc + b2c + bc2

X(5241) lies on these lines:
{1, 4023}, {2, 6}, {10, 11}, {354, 4104}, {594, 4358}, {899, 4026}, {908, 3739}, {3216, 4205}, {3306, 4643}, {3775, 4871}, {3846, 5087}, {3847, 3925}, {4054, 4688}, {4239, 5096}, {4359, 4415}, {4364, 4850}, {4665, 4671}


X(5242) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(18)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 - ab - ac - (12)1/2S)

X(5242) lies on these lines:
{2, 7}, {10, 18}, {302, 3912}, {303, 4416}, {395, 1100}, {946, 1277}


X(5243) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(17)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 - ab - ac + (12)1/2S)

X(5243) lies on these lines:
{2, 7}, {10, 17}, {302, 4416}, {303, 3912}, {396, 1100}, {946, 1276}


X(5244) = INTERSECTION OF LINES X(6)X(7) AND X(10)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(2a2 + b2 + c2 + ab + ac)/(b + c - a)

X(5244) lies on these lines:
{6, 7}, {10, 12}, {57, 1759}, {241, 3674}, {1386, 1890}, {2295, 4415}


X(5245) = INTERSECTION OF LINES X(8)X(9) AND X(10)X(17)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + ab + ac + (12)1/2S)

X(5245) lies on these lines:
{8, 9}, {10, 17}, {515, 1277}, {1652, 4848}


X(5246) = INTERSECTION OF LINES X(8)X(9) AND X(10)X(18)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + ab + ac - (12)1/2S)

X(5246) lies on these lines:
{8, 9}, {10, 18}, {515, 1276}, {1653, 4848}


X(5247) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + a2b + a2c + abc - b2c - bc2)

X(5247) lies on these lines:
{1, 6}, {2, 1468}, {3, 43}, {4, 1957}, {8, 31}, {10, 58}, {21, 42}, {28, 291}, {32, 3684}, {35, 3293}, {36, 3216}, {40, 1707}, {46, 4650}, {56, 978}, {57, 1722}, {63, 986}, {65, 1046}, {71, 1778}, {100, 3214}, {109, 4848}, {145, 3915}, {162, 2907}, {172, 2238}, {191, 4424}, {212, 3486}, {227, 1758}, {239, 384}, {256, 1245}, {341, 3769}, {355, 3072}, {386, 993}, {388, 1451}, {404, 899}, {484, 3987}, {515, 580}, {517, 3073}, {519, 595}, {602, 944}, {603, 1788}, {614, 3976}, {651, 1042}, {744, 4647}, {748, 3616}, {846, 3931}, {896, 4642}, {902, 3871}, {938, 1496}, {959, 1405}, {961, 1400}, {976, 3681}, {988, 2999}, {1009, 3783}, {1043, 1918}, {1126, 4653}, {1183, 2347}, {1193, 2975}, {1253, 4313}, {1330, 2887}, {1376, 4252}, {1430, 5125}, {1445, 4320}, {1471, 3600}, {1478, 1714}, {1572, 4051}, {1610, 2183}, {1737, 3075}, {1738, 4292}, {1739, 3336}, {1777, 2093}, {1837, 1936}, {1891, 2299}, {1914, 3780}, {2239, 4201}, {2292, 3219}, {2650, 4722}, {3008, 4298}, {3052, 3913}, {3240, 4189}, {3436, 5230}, {3647, 4868}, {3686, 4264}, {3720, 5047}, {3868, 3924}, {4234, 4685}, {4355, 4859}, {4362, 4385}, {4640, 4646}, {4673, 4676}


X(5248) = INTERSECTION OF LINES X(1)X(21) AND X(2)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 - 2abc - b2c - bc2)

X(5248) lies on these lines:
{1, 21}, {2, 35}, {3, 142}, {4, 3822}, {8, 3746}, {9, 943}, {10, 55}, {32, 37}, {36, 3616}, {40, 1006}, {41, 3294}, {42, 1724}, {56, 551}, {72, 3683}, {86, 4278}, {100, 1698}, {101, 2304}, {140, 3816}, {165, 3833}, {198, 3986}, {200, 4015}, {214, 3612}, {238, 386}, {354, 3916}, {377, 4302}, {388, 535}, {392, 2646}, {404, 3624}, {411, 1699}, {452, 3085}, {474, 4423}, {496, 4999}, {498, 2478}, {515, 3560}, {519, 958}, {581, 3073}, {631, 2077}, {748, 3216}, {759, 931}, {936, 4326}, {942, 4640}, {956, 3244}, {976, 2210}, {978, 4256}, {997, 3601}, {999, 3636}, {1012, 4297}, {1013, 1838}, {1100, 4047}, {1104, 3931}, {1107, 2241}, {1214, 4347}, {1259, 4847}, {1376, 3634}, {1500, 4426}, {1617, 4298}, {1697, 2802}, {1706, 3968}, {1748, 1844}, {1777, 4303}, {1788, 3256}, {1792, 3886}, {2177, 3293}, {2293, 3682}, {2346, 5223}, {2476, 3583}, {2901, 4362}, {2922, 3145}, {3006, 4894}, {3149, 3817}, {3158, 3956}, {3246, 4719}, {3338, 4652}, {3428, 4301}, {3434, 4309}, {3454, 3771}, {3555, 3748}, {3579, 3812}, {3626, 3913}, {3670, 4414}, {3679, 3871}, {3689, 3697}, {3828, 4421}, {3924, 4424}, {4004, 5183}, {4197, 4330}, {5084, 5218}


X(5249) = INTERSECTION OF LINES X(2)X(7) AND X(21)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a2b - a2c - 2abc - b2c - bc2

X(5249) lies on these lines:
{1, 224}, {2, 7}, {5, 1071}, {8, 4208}, {10, 3681}, {11, 3742}, {12, 3812}, {20, 946}, {21, 36}, {27, 86}, {37, 3782}, {42, 1738}, {55, 1004}, {75, 306}, {77, 278}, {78, 443}, {81, 3664}, {85, 92}, {171, 3011}, {210, 3826}, {239, 2890}, {312, 1269}, {320, 333}, {321, 1930}, {354, 2886}, {379, 2140}, {442, 942}, {474, 1259}, {495, 3753}, {497, 4666}, {516, 1621}, {518, 3925}, {528, 3748}, {551, 4304}, {554, 5239}, {914, 1441}, {938, 5177}, {940, 3772}, {948, 4350}, {950, 2475}, {960, 3649}, {1001, 1836}, {1012, 1519}, {1056, 3872}, {1081, 5240}, {1086, 3666}, {1210, 2476}, {1211, 3739}, {1215, 3836}, {1659, 3084}, {1737, 3822}, {1838, 4303}, {1959, 3674}, {2550, 3475}, {2895, 3686}, {2975, 4298}, {2999, 4859}, {3075, 3561}, {3120, 3720}, {3187, 3879}, {3220, 4228}, {3601, 4190}, {3622, 4313}, {3671, 3869}, {3687, 3936}, {3706, 4966}, {3741, 5208}, {3757, 4645}, {3771, 3980}, {3814, 3833}, {3841, 3874}, {3847, 4892}, {3848, 5087}, {3873, 4847}, {3890, 4301}, {3897, 4311}, {3969, 4431}, {4312, 4512}

X(5249) = isogonal conjugate of X(2259)
X(5249) = complement of X(3319)


X(5250) = INTERSECTION OF LINES X(1)X(21) AND X(2)X(40)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + b2 + c2 + 2ab + 2ac)

X(5250) lies on these lines:
{1, 21}, {2, 40}, {3, 392}, {8, 9}, {10, 1479}, {19, 29}, {35, 997}, {46, 1125}, {55, 78}, {56, 4640}, {57, 3616}, {65, 1001}, {72, 3295}, {77, 221}, {100, 936}, {145, 3219}, {165, 404}, {169, 3294}, {200, 3871}, {210, 3913}, {220, 4520}, {333, 4673}, {377, 516}, {380, 2287}, {405, 517}, {443, 3587}, {474, 3579}, {484, 3624}, {518, 3303}, {551, 3338}, {614, 986}, {631, 3359}, {748, 1722}, {908, 3085}, {942, 4666}, {958, 3057}, {964, 1766}, {976, 3749}, {988, 1201}, {999, 3916}, {1005, 1490}, {1039, 2212}, {1158, 3576}, {1191, 3666}, {1220, 4676}, {1329, 4679}, {1698, 4193}, {1699, 2476}, {1708, 3340}, {1709, 4297}, {2255, 2256}, {2334, 4663}, {2944, 4203}, {3158, 4420}, {3218, 3333}, {3241, 3929}, {3555, 3927}, {3586, 5086}, {3601, 4511}, {3652, 3655}, {3678, 4917}, {3679, 5178}, {3704, 3966}, {3714, 4387}, {3715, 4662}, {3729, 4968}, {3742, 5221}, {3746, 3811}, {3748, 3962}, {3812, 4423}, {3885, 4853}, {4255, 4689}, {4329, 4357}, {4383, 4646}


X(5251) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 - abc - 2b2c - 2bc2)

X(5251) lies on these lines:
{1, 6}, {2, 36}, {3, 1698}, {8, 3746}, {10, 21}, {28, 1224}, {30, 3925}, {40, 3560}, {42, 4653}, {55, 3679}, {56, 3624}, {63, 4880}, {65, 191}, {71, 4877}, {119, 140}, {165, 1012}, {261, 5209}, {404, 3634}, {442, 3585}, {443, 4299}, {452, 1479}, {484, 3753}, {498, 2551}, {499, 5084}, {515, 1006}, {517, 3683}, {519, 1621}, {748, 995}, {750, 4257}, {758, 3219}, {846, 4424}, {899, 4256}, {908, 1125}, {936, 3612}, {997, 3305}, {999, 4423}, {1308, 2752}, {1334, 4752}, {1376, 5010}, {1573, 1914}, {1699, 3428}, {2099, 3899}, {2475, 3841}, {2550, 4302}, {2646, 5044}, {2886, 3583}, {3086, 5129}, {3295, 3632}, {3303, 3633}, {3336, 3812}, {3579, 3698}, {3582, 3816}, {3626, 3871}, {3647, 3754}, {3691, 4251}, {3715, 3940}, {3757, 4692}, {3826, 4316}, {3833, 4973}, {3844, 4265}, {3884, 4861}, {3901, 3927}, {3913, 4668}, {4015, 4420}, {4187, 4999}, {4223, 5144}, {4309, 5082}, {4428, 4677}, {4512, 5119}


X(5252) = INTERSECTION OF LINES X(1)X(5) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 + c2 - ab - ac + 2bc)/(b + c - a)

X(5252) lies on these lines:
{1, 5}, {2, 1319}, {4, 1000}, {7, 8}, {10, 56}, {30, 5119}, {34, 1883}, {55, 515}, {57, 3679}, {63, 529}, {145, 3485}, {210, 3421}, {225, 5130}, {226, 519}, {354, 1056}, {392, 4679}, {443, 3698}, {484, 3654}, {498, 1385}, {517, 1478}, {528, 3895}, {553, 4669}, {594, 2285}, {899, 1450}, {944, 2646}, {946, 2098}, {950, 954}, {960, 3436}, {962, 5229}, {993, 5172}, {999, 1737}, {1010, 1408}, {1125, 1388}, {1155, 4293}, {1210, 3304}, {1376, 1470}, {1415, 4386}, {1420, 1698}, {1788, 3600}, {1826, 2256}, {1877, 5101}, {2475, 3909}, {2476, 4861}, {2886, 3872}, {3036, 3306}, {3058, 3586}, {3241, 4870}, {3244, 3947}, {3339, 4668}, {3340, 3632}, {3434, 3880}, {3474, 5183}, {3488, 3748}, {3579, 4299}, {3584, 3655}, {3621, 5178}, {3625, 3671}, {3626, 4031}, {3877, 5080}, {3890, 5046}, {3893, 5082}, {4297, 5217}, {4311, 5204}, {4415, 5155}, {4654, 4677}


X(5253) = INTERSECTION OF LINES X(1)X(88) AND X(2)X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 + 3abc + b2c + bc2)

X(5253) lies on these lines:
{1, 88}, {2, 12}, {3, 962}, {5, 104}, {8, 474}, {11, 2475}, {21, 36}, {35, 551}, {40, 3890}, {46, 3877}, {55, 3622}, {57, 3869}, {63, 3361}, {78, 3333}, {81, 1193}, {85, 934}, {86, 4225}, {145, 1376}, {171, 1201}, {191, 4973}, {377, 3086}, {411, 3576}, {484, 3884}, {497, 4190}, {499, 2476}, {758, 3337}, {908, 4298}, {936, 3681}, {942, 4511}, {960, 3218}, {976, 3976}, {978, 1468}, {993, 3624}, {997, 3338}, {1001, 4189}, {1004, 4313}, {1014, 4357}, {1104, 4239}, {1210, 5086}, {1290, 3109}, {1319, 3812}, {1470, 3485}, {1476, 5176}, {1478, 4193}, {2260, 2287}, {2306, 5240}, {2478, 4293}, {2646, 3742}, {3294, 5030}, {3336, 3878}, {3428, 3523}, {3555, 4420}, {3585, 3825}, {3601, 4666}, {3617, 4413}, {3623, 3913}, {3636, 3746}, {3753, 4861}, {3811, 3889}, {3816, 5046}, {4187, 5080}, {4696, 5205}, {5187, 5229}


X(5254) = INTERSECTION OF LINES X(4)X(6) AND X(30)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2 + a2(b2 + c2)

Let U be the circle obtained as the inverse-in-the-polar-circle of the 2nd Lemoine circle. The center of U is X(5254). (Randy Hutson, November 22, 2014)

X(5254) lies on these lines:
{2, 1975}, {3, 230}, {4, 6}, {5, 39}, {11, 2275}, {12, 2276}, {20, 3053}, {30, 32}, {76, 141}, {83, 597}, {140, 574}, {148, 384}, {184, 460}, {185, 1562}, {187, 550}, {194, 325}, {232, 235}, {290, 695}, {297, 3981}, {315, 524}, {316, 3629}, {338, 1235}, {376, 5023}, {381, 2548}, {395, 616}, {396, 617}, {427, 1194}, {489, 3068}, {490, 3069}, {495, 1500}, {496, 1015}, {538, 626}, {548, 5206}, {594, 4385}, {595, 5134}, {726, 4136}, {1086, 3673}, {1105, 1970}, {1107, 2886}, {1146, 3959}, {1180, 5133}, {1184, 1370}, {1196, 1368}, {1329, 1575}, {1353, 1570}, {1384, 1657}, {1574, 3820}, {1596, 3199}, {1656, 3055}, {1885, 1968}, {3061, 3944}, {3522, 5210}, {3564, 5028}, {3589, 4048}, {3627, 5007}, {3721, 3782}, {3845, 5041}, {3934, 4045}, {4173, 5167}

X(5254) = complement of X(1975)
X(5254) = {X(39),X(115)}-harmonic conjugate of X(5)


X(5255) = INTERSECTION OF LINES X(1)X(3) AND X(8)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + a2b + a2c - abc + b2c + bc2)

X(5255) lies on these lines:
{1, 3}, {2, 3915}, {4, 983}, {6, 979}, {8, 31}, {9, 989}, {10, 82}, {21, 902}, {32, 2329}, {37, 3496}, {42, 3871}, {44, 4662}, {58, 519}, {72, 3961}, {100, 1193}, {145, 1468}, {213, 3684}, {341, 4676}, {355, 3073}, {404, 1201}, {518, 1046}, {528, 1834}, {582, 3654}, {601, 944}, {603, 3476}, {643, 2363}, {750, 3616}, {752, 1330}, {958, 3052}, {976, 3869}, {978, 1191}, {1106, 4308}, {1203, 3293}, {1253, 4344}, {1254, 4318}, {1279, 3812}, {1386, 4646}, {1572, 3061}, {1706, 1722}, {1724, 3679}, {1743, 3713}, {1914, 2295}, {2176, 4386}, {2269, 2298}, {2292, 3920}, {2321, 4264}, {2650, 3722}, {2901, 4693}, {3434, 5230}, {3743, 5184}, {3769, 4673}, {3868, 3938}, {3923, 4385}, {3973, 4866}, {3997, 4251}, {4255, 4421}, {4418, 4968}, {4649, 5145}


X(5256) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(63)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c)2 - 2bc
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(a2 + b2 + c2 + 2ab + 2ac)

X(5256) lies on these lines:
{1, 2}, {6, 63}, {7, 223}, {21, 1453}, {27, 34}, {33, 469}, {37, 3305}, {38, 3751}, {55, 1386}, {56, 4719}, {57, 77}, {58, 4652}, {92, 2331}, {193, 4001}, {204, 1013}, {226, 3946}, {238, 968}, {312, 4360}, {321, 3875}, {329, 3672}, {333, 3759}, {345, 3618}, {380, 3101}, {440, 1062}, {464, 1040}, {553, 4667}, {748, 1962}, {894, 3210}, {908, 3553}, {940, 1100}, {982, 4649}, {988, 1468}, {1211, 4272}, {1214, 1445}, {1230, 3760}, {1376, 3745}, {1427, 4350}, {1707, 2308}, {1743, 3219}, {1763, 2172}, {2177, 3749}, {2352, 5132}, {3052, 4689}, {3247, 3930}, {3434, 3755}, {3677, 3873}, {3886, 3896}, {3923, 4970}, {3966, 4026}, {3993, 4011}, {4021, 4656}, {4085, 4865}, {4255, 4855}, {4270, 4357}, {4285, 4643}, {4413, 4682}, {4868, 5119}, {4886, 5224}

X(5256) = {X(1),X(2)}-harmonic conjugate of X(5287)


X(5257) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(3a + b + c)

X(5257) lies on these lines:
{1, 966}, {2, 7}, {6, 1125}, {8, 3247}, {10, 37}, {19, 406}, {45, 3634}, {71, 3294}, {75, 4044}, {86, 4416}, {141, 4698}, {145, 4034}, {192, 4967}, {198, 405}, {225, 281}, {228, 4204}, {238, 4264}, {391, 1449}, {392, 2262}, {461, 4512}, {551, 1100}, {573, 946}, {756, 3778}, {860, 1826}, {978, 5105}, {993, 2178}, {1001, 4254}, {1010, 4877}, {1211, 4035}, {1266, 4699}, {1654, 3879}, {1698, 1738}, {1743, 3624}, {2171, 4848}, {2238, 4104}, {3008, 4657}, {3244, 3723}, {3617, 4007}, {3622, 4982}, {3632, 4545}, {3636, 4856}, {3663, 3739}, {3664, 4643}, {3671, 4047}, {3679, 4060}, {3912, 4687}, {3946, 4384}, {3949, 3970}, {3965, 4847}, {3985, 4656}, {4021, 4361}, {4061, 4771}, {4260, 5044}, {4389, 4751}, {4431, 4664}, {4648, 4748}, {4665, 4681}, {4668, 4898}


X(5258) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 + abc - 2b2c - 2bc2)

X(5258) lies on these lines:
{1, 6}, {3, 3679}, {8, 35}, {10, 36}, {21, 519}, {55, 3632}, {56, 1698}, {65, 4880}, {100, 3626}, {101, 3691}, {172, 1573}, {191, 517}, {200, 3612}, {210, 1385}, {214, 4015}, {442, 529}, {443, 4317}, {484, 3916}, {498, 3421}, {499, 2551}, {515, 3651}, {528, 4330}, {535, 2475}, {551, 5047}, {961, 1224}, {999, 3624}, {1005, 4847}, {1319, 5044}, {1388, 3715}, {1444, 4967}, {1478, 5177}, {1482, 3899}, {1621, 3244}, {2099, 3927}, {2550, 4299}, {2802, 3647}, {2886, 3585}, {3214, 4256}, {3218, 3754}, {3219, 3878}, {3295, 3633}, {3336, 3753}, {3337, 3812}, {3560, 3929}, {3582, 4187}, {3625, 3871}, {3678, 4511}, {3681, 3897}, {3730, 4390}, {3813, 4857}, {3820, 5193}, {3913, 4677}, {3918, 4973}, {3956, 4881}, {4302, 5082}, {4668, 5010}, {4853, 5119}


X(5259) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 - 3abc - 2b2c - 2bc2)

X(5259) lies on these lines:
{1, 6}, {2, 35}, {3, 1699}, {10, 1621}, {12, 2078}, {21, 36}, {28, 1839}, {46, 4512}, {55, 1698}, {56, 4355}, {58, 3720}, {100, 3634}, {105, 1224}, {140, 2077}, {142, 1770}, {191, 942}, {386, 748}, {411, 3817}, {442, 3583}, {443, 4302}, {452, 1478}, {474, 5010}, {484, 3812}, {498, 5084}, {551, 2975}, {846, 3670}, {946, 1006}, {993, 3616}, {1089, 3757}, {1193, 4653}, {1259, 5231}, {1283, 5051}, {1329, 3584}, {1838, 4183}, {2260, 4877}, {2308, 4658}, {2550, 4309}, {2886, 4857}, {3085, 5129}, {3218, 3647}, {3219, 3874}, {3245, 3754}, {3293, 3750}, {3295, 3679}, {3303, 3632}, {3305, 3811}, {3336, 4640}, {3337, 3742}, {3560, 3576}, {3582, 4999}, {3685, 4647}, {3822, 5046}, {3848, 5131}, {3894, 3927}, {3898, 4861}, {3935, 4015}, {4068, 4716}


X(5260) = INTERSECTION OF LINES X(2)X(12) AND X(10)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 - abc - 3b2c - 3bc2)

X(5260) lies on these lines:
{1, 748}, {2, 12}, {8, 405}, {9, 1405}, {10, 21}, {36, 3634}, {55, 3617}, {63, 3339}, {65, 3219}, {104, 140}, {145, 1001}, {191, 3754}, {355, 1006}, {392, 4861}, {404, 993}, {442, 5080}, {452, 3434}, {484, 3647}, {502, 1224}, {644, 3294}, {846, 4642}, {950, 5178}, {956, 3616}, {984, 3924}, {997, 3897}, {1043, 4651}, {1104, 3920}, {1320, 3884}, {1376, 4189}, {1478, 4197}, {1722, 4850}, {1757, 2650}, {1891, 4233}, {2078, 5176}, {2475, 3925}, {2646, 3740}, {2886, 5046}, {3091, 3428}, {3218, 3812}, {3293, 4653}, {3303, 3621}, {3337, 3833}, {3579, 4002}, {3585, 3841}, {3622, 4423}, {3626, 3746}, {3679, 3871}, {3697, 4420}, {3698, 4640}, {3757, 4696}, {3872, 3890}, {3913, 4678}, {3935, 4662}, {4183, 5174}, {4188, 4413}, {4511, 5044}


X(5261) = INTERSECTION OF LINES X(2)X(12) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + 3b2 + 3c2 + 6bc)/(b + c - a)

X(5261) lies on these lines:
{1, 3091}, {2, 12}, {4, 390}, {5, 1056}, {7, 10}, {8, 226}, {11, 5068}, {20, 35}, {34, 3920}, {55, 3146}, {65, 3617}, {85, 341}, {145, 3485}, {192, 2996}, {355, 3487}, {381, 1058}, {387, 1126}, {391, 1405}, {442, 3421}, {452, 2078}, {496, 3545}, {497, 3832}, {498, 3523}, {519, 4323}, {612, 4296}, {976, 2647}, {984, 1254}, {986, 4346}, {999, 3090}, {1125, 4308}, {1219, 3705}, {1393, 4392}, {1441, 4385}, {1469, 3620}, {1479, 3839}, {1617, 5047}, {1698, 4298}, {1722, 4327}, {1837, 3475}, {2099, 3621}, {3086, 5056}, {3303, 5225}, {3304, 3614}, {3361, 3634}, {3476, 3622}, {3522, 5218}, {3543, 3585}, {3584, 4299}, {3616, 5219}, {3624, 4315}, {3649, 4678}, {3671, 3679}, {3704, 4461}, {3870, 5175}, {3961, 4332}, {4654, 4848}


X(5262) = INTERSECTION OF LINES X(1)X(2) AND X(7)X(34)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + b3 + c3 + a2b + a2c + ab2 + ac2 + abc)

X(5262) lies on these lines:
{1, 2}, {3, 4850}, {6, 977}, {7, 34}, {21, 1104}, {28, 60}, {31, 986}, {37, 5047}, {57, 4296}, {58, 3218}, {63, 1453}, {65, 82}, {75, 964}, {77, 1467}, {238, 2292}, {257, 1178}, {312, 5192}, {350, 1228}, {377, 4000}, {404, 3752}, {452, 3672}, {595, 4424}, {758, 1203}, {950, 3100}, {982, 1468}, {990, 3146}, {1010, 4359}, {1040, 4313}, {1046, 2308}, {1062, 3488}, {1100, 2303}, {1191, 3877}, {1220, 4968}, {1245, 4388}, {1325, 2363}, {1442, 3212}, {1449, 2082}, {1621, 3931}, {1724, 3219}, {1743, 3951}, {2476, 3772}, {2646, 4719}, {3210, 4195}, {3315, 5045}, {3337, 4351}, {3339, 4347}, {3744, 3871}, {3745, 3812}, {3746, 4868}, {3876, 4383}, {3891, 4385}, {4972, 5015}, {5090, 5142}


X(5263) = INTERSECTION OF LINES X(1)X(75) AND X(2)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + ab2 + ac2 + abc + b2c + bc2

X(5263) lies on these lines:
{1, 75}, {2, 11}, {6, 8}, {9, 4676}, {10, 82}, {19, 29}, {31, 333}, {37, 3685}, {38, 4418}, {42, 3996}, {69, 4307}, {85, 2263}, {87, 1222}, {141, 4645}, {171, 3741}, {190, 984}, {239, 1386}, {312, 612}, {321, 3920}, {516, 4357}, {518, 894}, {519, 4649}, {752, 3775}, {958, 4195}, {982, 3980}, {993, 4234}, {1008, 5224}, {1125, 1738}, {1211, 4388}, {1215, 3961}, {1266, 4353}, {1279, 3739}, {1441, 4318}, {1757, 4672}, {1861, 5174}, {1999, 3706}, {2049, 3295}, {2607, 3878}, {2975, 3286}, {3219, 4981}, {3241, 4499}, {3242, 4363}, {3246, 3846}, {3416, 3661}, {3616, 4000}, {3664, 4684}, {3744, 3757}, {3751, 3758}, {3842, 4432}, {3879, 4349}, {3993, 4693}, {4709, 4716}, {4732, 4974}

X(5263) = anticomplement of X(4026)


X(5264) = INTERSECTION OF LINES X(1)X(3) AND X(10)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + a2b + a2c + b2c + bc2)

X(5264) lies on these lines:
{1, 3}, {2, 595}, {6, 3293}, {8, 58}, {10, 31}, {32, 2295}, {37, 1759}, {41, 3997}, {43, 1203}, {44, 3697}, {79, 983}, {80, 987}, {81, 3871}, {82, 4429}, {90, 989}, {100, 386}, {109, 388}, {191, 984}, {213, 4386}, {238, 1698}, {404, 995}, {405, 3052}, {474, 1191}, {515, 601}, {519, 1468}, {573, 2298}, {594, 4275}, {609, 2329}, {748, 3634}, {750, 1125}, {758, 976}, {956, 4252}, {1046, 3961}, {1089, 3923}, {1104, 3753}, {1106, 4315}, {1210, 1497}, {1253, 4349}, {1254, 4347}, {1376, 3216}, {1451, 4848}, {1453, 1706}, {1478, 1777}, {1714, 2550}, {2308, 3214}, {2345, 4264}, {2975, 4257}, {3085, 4307}, {3754, 3924}, {3874, 3938}, {4362, 4647}, {4450, 5051}


X(5265) = INTERSECTION OF LINES X(2)X(12) AND X(20)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - 5a2 + 2bc)/(b + c - a)

X(5265) lies on these lines:
{1, 3523}, {2, 12}, {3, 390}, {7, 1125}, {8, 1420}, {10, 4308}, {11, 3146}, {20, 36}, {34, 4232}, {43, 4322}, {57, 3616}, {65, 3622}, {108, 4200}, {140, 1056}, {145, 1319}, {193, 1428}, {201, 4392}, {238, 1106}, {279, 1447}, {348, 3598}, {376, 496}, {391, 604}, {404, 1617}, {439, 4366}, {495, 3525}, {497, 3522}, {499, 3091}, {551, 3339}, {614, 4296}, {631, 999}, {938, 3576}, {944, 5126}, {956, 1476}, {978, 1458}, {988, 3672}, {993, 5129}, {1388, 3623}, {1445, 3333}, {1466, 1621}, {1470, 4189}, {1471, 3945}, {1478, 5056}, {1698, 4315}, {3241, 4848}, {3295, 3524}, {3304, 5218}, {3476, 3617}, {3543, 3582}, {3624, 4298}, {3660, 3868}, {5059, 5225}, {5068, 5229}


X(5266) = INTERSECTION OF LINES X(1)X(3) AND X(32)X(37)

Trilinears        arSA - SSA : brSA - SSB : crSA - SSC    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a3 + b3 + c3 + a2b +a2c + b2c + bc2)

X(5266) lies on these lines:
{1, 3}, {2, 5015}, {4, 4339}, {6, 3694}, {10, 1104}, {21, 3920}, {31, 72}, {32, 37}, {38, 3916}, {39, 1100}, {42, 1009}, {44, 3678}, {58, 518}, {187, 3723}, {200, 1453}, {210, 1724}, {238, 5044}, {386, 1386}, {387, 3189}, {392, 1472}, {405, 612}, {442, 3011}, {474, 614}, {519, 3704}, {595, 960}, {601, 1071}, {902, 2292}, {943, 2298}, {975, 1001}, {983, 987}, {997, 1191}, {1010, 3757}, {1125, 1279}, {1384, 3247}, {1427, 4347}, {1468, 3555}, {1707, 3927}, {1770, 3782}, {1785, 1852}, {2204, 5089}, {3242, 4252}, {3293, 3689}, {3419, 5230}, {3475, 4340}, {3487, 4307}, {3753, 3924}, {3831, 4434}, {3879, 3933}, {3881, 4864}, {4195, 4385}, {4256, 4719}


X(5267) = INTERSECTION OF LINES X(3)X(10) AND X(21)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a3 - 2ab2 - 2ac2 - b2c - bc2)

X(5267) lies on these lines:
{1, 89}, {2, 3585}, {3, 10}, {8, 5010}, {12, 535}, {21, 36}, {30, 4999}, {35, 519}, {46, 3919}, {55, 3244}, {56, 551}, {58, 2185}, {63, 3612}, {78, 4134}, {100, 3626}, {140, 3814}, {149, 4330}, {187, 1107}, {191, 4511}, {214, 960}, {404, 3634}, {405, 5204}, {501, 1098}, {549, 1329}, {550, 2886}, {574, 4426}, {758, 2646}, {942, 4973}, {956, 3625}, {1011, 3840}, {1030, 3686}, {1055, 3294}, {1155, 3754}, {1319, 3884}, {1385, 3878}, {1444, 3664}, {1621, 3636}, {1698, 4188}, {1861, 3520}, {2178, 3986}, {2475, 4316}, {2550, 3528}, {2551, 3524}, {3035, 3530}, {3560, 3817}, {3635, 3746}, {3741, 4184}, {3812, 5122}, {3927, 4525}, {3940, 4537}, {4386, 5206}


X(5268) = INTERSECTION OF LINES X(1)X(2) AND X(25)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + 4bc)

X(5268) lies on these lines:
{1, 2}, {6, 3740}, {9, 171}, {12, 1038}, {22, 5010}, {25, 35}, {31, 3305}, {33, 5218}, {37, 1376}, {38, 3306}, {45, 4640}, {55, 5020}, {57, 984}, {63, 750}, {69, 4104}, {87, 2297}, {100, 968}, {165, 846}, {181, 3781}, {210, 940}, {230, 3553}, {305, 3761}, {345, 4078}, {427, 5155}, {474, 988}, {1001, 3749}, {1196, 2276}, {1215, 3718}, {1370, 3585}, {1447, 4328}, {1448, 3947}, {1469, 3819}, {1742, 1750}, {2263, 5226}, {2650, 3984}, {3158, 3750}, {3242, 3742}, {3247, 3290}, {3550, 4512}, {3554, 3815}, {3666, 4413}, {3715, 4641}, {3729, 3971}, {3744, 4423}, {3745, 4383}, {3772, 3826}, {3929, 4650}, {3966, 5241}, {3967, 4363}, {4339, 5129}

X(5258) = {X(1),X(2)}-harmonic conjugate of X(5272)


X(5269) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a2 + b2 + c2 + 2bc)

X(5269) lies on these lines:
{1, 3}, {2, 3883}, {6, 200}, {9, 31}, {10, 1453}, {33, 1395}, {37, 3052}, {38, 3928}, {42, 1449}, {63, 3920}, {81, 3870}, {84, 601}, {181, 3056}, {197, 2270}, {204, 281}, {210, 1743}, {226, 3424}, {380, 3198}, {388, 1394}, {553, 4310}, {595, 975}, {611, 2003}, {614, 750}, {869, 2258}, {902, 968}, {950, 4339}, {984, 1707}, {985, 1961}, {987, 989}, {1001, 4682}, {1254, 4348}, {1376, 1386}, {1397, 2330}, {1407, 4321}, {1706, 4695}, {1999, 3886}, {2303, 2328}, {2318, 3997}, {3243, 3938}, {3474, 3663}, {3475, 3664}, {3632, 4046}, {3683, 3731}, {3715, 3973}, {3751, 3961}, {3782, 4312}, {3791, 4457}, {3923, 4135}, {4418, 4659}, {4641, 5223}


X(5270) = INTERSECTION OF LINES X(1)X(4) AND X(12)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - 3a2bc - 2b2c2

X(5270) lies on these lines:
{1, 4}, {2, 4317}, {3, 3584}, {5, 3582}, {8, 3901}, {10, 3218}, {11, 3850}, {12, 36}, {21, 535}, {30, 3746}, {35, 495}, {55, 1657}, {56, 1656}, {65, 2962}, {79, 517}, {80, 942}, {149, 3635}, {377, 3679}, {381, 3304}, {382, 3303}, {442, 529}, {484, 4292}, {496, 3858}, {498, 3523}, {499, 3600}, {519, 2475}, {548, 4995}, {551, 5046}, {952, 3649}, {999, 3851}, {1125, 5080}, {1698, 3436}, {1737, 3337}, {1935, 2964}, {2550, 4668}, {2975, 3822}, {3058, 3627}, {3085, 3522}, {3086, 5068}, {3146, 4309}, {3295, 5073}, {3434, 3633}, {3754, 5176}, {3874, 5086}, {3884, 5057}, {3920, 5189}, {3947, 4311}, {4302, 5059}


X(5271) = INTERSECTION OF LINES X(1)X(2) AND X(19)X(27)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - ab2 - ac2 - 2abc - 2b2c - 2bc2

X(5271) lies on these lines:
{1, 2}, {7, 4001}, {9, 321}, {19, 27}, {45, 3175}, {55, 3696}, {57, 1150}, {77, 1943}, {226, 3686}, {278, 307}, {312, 3305}, {322, 3306}, {329, 391}, {344, 3610}, {379, 4968}, {440, 3419}, {469, 5174}, {518, 4042}, {740, 968}, {850, 1021}, {940, 3739}, {964, 1453}, {1001, 3706}, {1211, 3772}, {1376, 2352}, {1621, 3886}, {1707, 4418}, {1746, 1766}, {1790, 1958}, {1817, 2975}, {2886, 3966}, {3219, 3729}, {3416, 3925}, {3434, 3883}, {3487, 4101}, {3578, 4654}, {3666, 4361}, {3715, 3967}, {3731, 3995}, {3782, 4643}, {3846, 4682}, {3875, 5235}, {3891, 4981}, {3929, 4659}, {3936, 4034}, {3969, 4007}, {4363, 4641}, {4417, 4886}


X(5272) = INTERSECTION OF LINES X(1)X(2) AND X(57)X(238)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - 4bc)

X(5272) lies on these lines:
{1, 2}, {6, 3742}, {9, 982}, {11, 1040}, {25, 36}, {31, 3306}, {38, 3305}, {56, 5020}, {57, 238}, {63, 244}, {87, 269}, {105, 165}, {142, 1716}, {230, 3554}, {305, 3760}, {354, 3751}, {405, 988}, {497, 1738}, {968, 4850}, {984, 3677}, {990, 3817}, {1001, 3752}, {1191, 3812}, {1196, 2275}, {1279, 1376}, {1370, 3583}, {1386, 3848}, {1435, 1957}, {1449, 4038}, {1699, 1721}, {1724, 3338}, {1739, 5119}, {3052, 3246}, {3056, 3819}, {3242, 3740}, {3271, 3784}, {3315, 3681}, {3361, 4223}, {3553, 3815}, {3666, 4423}, {3729, 4011}, {3744, 4413}, {3772, 3816}, {3782, 4679}, {3895, 4695}, {4327, 5226}, {4641, 4860}

X(5272) = {X(1),X(2)}-harmonic conjugate of X(5268)


X(5273) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 3a2 - 2ab - 2ac - 2bc)

X(5273) lies on these lines:
{2, 7}, {8, 21}, {10, 20}, {27, 281}, {31, 4344}, {81, 219}, {189, 268}, {191, 4295}, {210, 5218}, {220, 940}, {261, 1264}, {312, 3161}, {348, 479}, {354, 960}, {377, 1155}, {390, 4512}, {391, 3687}, {405, 938}, {443, 3916}, {497, 3683}, {631, 1071}, {910, 966}, {936, 3523}, {1002, 5208}, {1200, 3691}, {1210, 5129}, {1212, 3666}, {1214, 3160}, {1329, 4197}, {1479, 2894}, {1617, 2975}, {1698, 4208}, {1707, 4307}, {1764, 3730}, {2096, 3820}, {2550, 4640}, {3187, 4460}, {3210, 4402}, {3241, 3748}, {3474, 3925}, {3487, 3927}, {3679, 4304}, {3711, 4995}, {3772, 4419}, {3869, 4323}, {3877, 4345}, {4860, 4999}

X(5273) = {X(2),X(63)}-harmonic conjugate of X(7)


X(5274) = INTERSECTION OF LINES X(2)X(11) AND X(20)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 + 3b2 + 3c2 - 6bc)

X(5274) lies on these lines:
{1, 3091}, {2, 11}, {4, 496}, {5, 1058}, {7, 1699}, {8, 3452}, {12, 5068}, {20, 36}, {56, 3146}, {145, 1837}, {150, 4845}, {279, 2898}, {330, 2996}, {346, 3705}, {381, 1056}, {388, 3832}, {495, 3545}, {499, 3523}, {519, 4345}, {614, 3100}, {938, 946}, {950, 3616}, {962, 1210}, {982, 2310}, {1125, 4208}, {1478, 3839}, {1788, 5183}, {1864, 3873}, {2098, 3621}, {2551, 3813}, {2900, 4511}, {3056, 3620}, {3057, 3617}, {3085, 5056}, {3090, 3295}, {3304, 5229}, {3486, 3622}, {3543, 3583}, {3582, 4302}, {3598, 4872}, {3624, 4314}, {3679, 4342}, {3741, 5232}, {3944, 4310}, {3945, 4038}, {4187, 5082}


X(5275) = INTERSECTION OF LINES X(2)X(6) AND X(19)X(25)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + ab2 + ac2 + 2abc + 2b2c + 2bc2)

X(5275) lies on these lines:
{1, 2271}, {2, 6}, {9, 171}, {19, 25}, {21, 3053}, {22, 1030}, {32, 405}, {39, 474}, {45, 2243}, {56, 1107}, {169, 975}, {172, 958}, {220, 2295}, {305, 3770}, {392, 1572}, {404, 5013}, {406, 2207}, {442, 3767}, {614, 1100}, {672, 750}, {956, 1573}, {984, 3509}, {1001, 1914}, {1194, 4261}, {1196, 2092}, {1376, 2276}, {1447, 5228}, {1449, 4038}, {1468, 3691}, {1575, 4413}, {1610, 3207}, {1655, 1975}, {2235, 5205}, {2280, 3720}, {2548, 4187}, {3242, 3726}, {3247, 3750}, {3263, 4363}, {3291, 4277}, {3550, 3731}, {3923, 3985}, {4189, 5023}, {4223, 4258}, {4254, 5020}, {4262, 4653}, {4655, 4987}


X(5276) = INTERSECTION OF LINES X(2)X(6) AND X(9)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + ab2 + ac2 + abc + b2c + bc2)

X(5276) lies on these lines:
{1, 41}, {2, 6}, {9, 31}, {21, 32}, {25, 941}, {37, 82}, {38, 3509}, {39, 404}, {42, 3684}, {100, 743}, {171, 672}, {172, 1107}, {284, 4224}, {384, 1655}, {573, 1754}, {584, 4228}, {595, 3294}, {609, 993}, {614, 1449}, {894, 3263}, {910, 3666}, {984, 985}, {1100, 3290}, {1180, 4261}, {1194, 2092}, {1196, 2670}, {1206, 3757}, {1333, 1627}, {1500, 3871}, {1572, 3877}, {1778, 4275}, {1922, 4518}, {2207, 4194}, {2292, 3496}, {2348, 3745}, {2476, 3767}, {2548, 4193}, {2651, 4274}, {3053, 4189}, {3598, 5228}, {3930, 3961}, {4188, 5013}, {4209, 4352}, {4239, 4277}, {4424, 5011}, {5007, 5047}


X(5277) = INTERSECTION OF LINES X(2)X(32) AND X(35)X(37)

Trilinears        a3r + bcS : b3r + caS : c3r + abS>    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + abc + b2c + bc2)

X(5277) lies on these lines:
{1, 1929}, {2, 32}, {6, 474}, {8, 2242}, {9, 2305}, {10, 172}, {12, 1415}, {21, 187}, {35, 37}, {36, 1107}, {39, 404}, {41, 750}, {58, 2238}, {99, 1655}, {100, 1500}, {101, 2295}, {112, 451}, {115, 2475}, {171, 213}, {199, 612}, {230, 442}, {274, 385}, {377, 3767}, {391, 5042}, {405, 3053}, {406, 1968}, {468, 2204}, {574, 4188}, {609, 1698}, {762, 2248}, {763, 1654}, {846, 2135}, {940, 2271}, {966, 5019}, {992, 4264}, {1125, 1914}, {1213, 1333}, {1573, 2975}, {2092, 2303}, {2160, 4016}, {2241, 3616}, {2549, 4190}, {3291, 4239}, {3509, 3954}, {3727, 5011}, {4189, 5206}


X(5278) = INTERSECTION OF LINES X(2)X(6) AND X(10)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - ab2 - ac2 - 2abc - b2c - bc2

X(5278) lies on these lines:
{1, 4981}, {2, 6}, {8, 405}, {9, 321}, {10, 31}, {37, 3187}, {45, 3995}, {55, 4651}, {63, 169}, {75, 3219}, {100, 1011}, {142, 4001}, {226, 1405}, {306, 2280}, {317, 445}, {573, 1746}, {748, 3741}, {756, 4362}, {896, 3980}, {956, 4245}, {968, 3896}, {984, 3891}, {1001, 4042}, {1125, 4101}, {1212, 3998}, {1229, 3719}, {1330, 4197}, {1441, 1708}, {1714, 5051}, {2177, 4685}, {2205, 4426}, {2476, 2651}, {2550, 4450}, {3006, 3966}, {3011, 4104}, {3120, 4703}, {3305, 4358}, {3681, 3757}, {3683, 3696}, {3691, 3765}, {3715, 3952}, {3729, 4980}, {3739, 4641}, {3791, 3842}, {3883, 5014}, {5081, 5136}


X(5279) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 - a2bc + b3c + bc3)

X(5279) lies on these lines:
{2, 7}, {6, 977}, {8, 19}, {10, 1781}, {20, 346}, {21, 37}, {27, 321}, {28, 72}, {40, 3692}, {48, 4511}, {69, 1760}, {71, 1761}, {75, 379}, {78, 610}, {100, 3694}, {101, 2327}, {169, 391}, {198, 1259}, {219, 608}, {272, 335}, {281, 3436}, {306, 2897}, {377, 2345}, {380, 3870}, {518, 2264}, {573, 1759}, {604, 3061}, {910, 3965}, {965, 3876}, {975, 3731}, {1172, 4463}, {1330, 4456}, {1442, 1959}, {1723, 4310}, {1817, 3998}, {1826, 5080}, {1953, 4861}, {2092, 2240}, {2171, 2329}, {2173, 3949}, {2174, 4053}, {2256, 3877}, {2269, 3496}, {2354, 4388}, {3950, 4304}


X(5280) = INTERSECTION OF LINES X(1)X(6) AND X(32)X(35)

Trilinears        SR + aSω : SR + bSω : SR + cSω    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 + c2 + bc)

X(5280) lies on these lines:
{1, 6}, {3, 609}, {31, 3730}, {32, 35}, {36, 39}, {41, 386}, {42, 251}, {48, 5105}, {58, 672}, {71, 4264}, {81, 3912}, {83, 350}, {101, 1193}, {304, 3758}, {595, 1334}, {651, 3674}, {894, 1930}, {986, 1759}, {1015, 5041}, {1126, 1438}, {1174, 2299}, {1197, 3507}, {1384, 5217}, {1448, 2285}, {1468, 4253}, {1500, 1914}, {1890, 3755}, {1922, 3864}, {1973, 4270}, {2174, 5153}, {2242, 2275}, {2260, 4284}, {2503, 2653}, {3053, 5010}, {3056, 5039}, {3293, 3684}, {3496, 4424}, {3509, 3670}, {3685, 4099}, {3710, 3997}, {3744, 3991}, {3934, 4396}, {3961, 4006}, {4642, 5011}, {5024, 5204}


X(5281) = INTERSECTION OF LINES X(2)X(11) AND X(20)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 5a2 - 2bc)

X(5281) lies on these lines:
{1, 3523}, {2, 11}, {3, 1056}, {7, 165}, {8, 3158}, {9, 1200}, {10, 4313}, {12, 3146}, {20, 35}, {33, 4232}, {43, 2293}, {140, 1058}, {144, 4640}, {145, 2646}, {171, 1253}, {193, 2330}, {376, 495}, {388, 3522}, {391, 2268}, {496, 3525}, {498, 3091}, {516, 5226}, {551, 4345}, {612, 3100}, {631, 3295}, {999, 3524}, {1040, 3920}, {1155, 3475}, {1447, 3672}, {1479, 5056}, {1697, 3616}, {1698, 4314}, {1961, 4336}, {3057, 3622}, {3086, 3746}, {3486, 3617}, {3487, 3579}, {3543, 3584}, {3550, 4307}, {3614, 3854}, {3712, 3974}, {4293, 5010}, {5059, 5229}, {5068, 5225}


X(5282) = INTERSECTION OF LINES X(2)X(7) AND X(6)X(38)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a3 + b2c + bc2)

X(5282) lies on these lines:
{2, 7}, {6, 38}, {8, 3496}, {10, 1759}, {31, 37}, {32, 976}, {41, 72}, {44, 4003}, {45, 896}, {55, 3930}, {66, 71}, {141, 4376}, {169, 3691}, {191, 3730}, {198, 199}, {201, 220}, {210, 910}, {218, 3927}, {517, 4390}, {518, 2280}, {748, 3290}, {956, 2170}, {984, 985}, {997, 1055}, {1212, 1451}, {1395, 5089}, {1707, 1961}, {1709, 1766}, {1761, 2345}, {1914, 3938}, {2235, 3116}, {2239, 2276}, {2243, 4386}, {2246, 4712}, {2269, 5227}, {2329, 3869}, {2911, 3958}, {2975, 3061}, {3679, 5011}, {3681, 3684}, {3693, 4640}, {3721, 3924}, {4119, 5014}, {4136, 5016}


X(5283) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(39)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(ab2 + ac2 + abc + b2c + bc2)

X(5283) lies on these lines:
{1, 6}, {2, 39}, {8, 1500}, {10, 2276}, {21, 32}, {35, 4386}, {42, 3691}, {115, 2476}, {172, 993}, {187, 4189}, {232, 406}, {377, 2549}, {386, 2238}, {391, 941}, {404, 574}, {474, 5013}, {612, 1011}, {756, 869}, {846, 3496}, {940, 5021}, {966, 2092}, {968, 2082}, {986, 3125}, {992, 5105}, {1015, 3616}, {1125, 2275}, {1213, 4261}, {1475, 3720}, {1506, 4193}, {1575, 1698}, {1621, 2241}, {2242, 2975}, {2268, 2304}, {2292, 3735}, {2295, 3730}, {2303, 5019}, {2478, 2548}, {3199, 4194}, {3666, 4384}, {3815, 4187}, {3959, 4424}, {4185, 5089}, {4251, 4653}, {4264, 4877}

X(5283) = isotomic conjugate of X(1218)
X(5283) = {X(1),X(9)}-harmonic conjugate of X(213)


X(5284) = INTERSECTION OF LINES X(2)X(11) AND X(21)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - ab - ac - 3bc)

X(5284) lies on these lines:
{1, 748}, {2, 11}, {9, 3873}, {21, 36}, {37, 3108}, {38, 3315}, {44, 4883}, {81, 238}, {210, 3957}, {244, 846}, {329, 405}, {354, 3219}, {404, 3624}, {484, 3833}, {496, 943}, {899, 3750}, {958, 3622}, {968, 4850}, {1155, 3848}, {1279, 3920}, {1320, 3898}, {1479, 4197}, {1617, 5226}, {1698, 3871}, {1848, 4233}, {2895, 4966}, {3218, 3683}, {3246, 3745}, {3303, 3617}, {3306, 4512}, {3337, 3647}, {3436, 5129}, {3634, 3746}, {3685, 4359}, {3715, 4661}, {3740, 3748}, {3741, 5235}, {3757, 4358}, {3812, 5183}, {3841, 4857}, {3936, 4204}, {4228, 4872}, {4418, 4432}, {4430, 5220}


X(5285) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(25)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - a2bc + b3c + bc3)

X(5285) lies on these lines:
{1, 3}, {9, 25}, {10, 28}, {22, 63}, {23, 3219}, {31, 579}, {33, 1766}, {42, 284}, {48, 3190}, {71, 1474}, {72, 2915}, {73, 3430}, {100, 306}, {101, 2318}, {109, 1297}, {154, 219}, {159, 197}, {181, 2330}, {184, 2323}, {198, 1260}, {199, 228}, {209, 2194}, {212, 573}, {222, 1350}, {226, 4220}, {291, 1283}, {511, 2003}, {516, 1848}, {951, 1042}, {1376, 3844}, {1397, 3056}, {1473, 3928}, {1486, 4512}, {1495, 3690}, {1631, 3185}, {1995, 3305}, {2187, 2289}, {2222, 2747}, {2299, 4456}, {2360, 3682}, {3098, 3784}, {3752, 5096}, {4221, 4304}


X(5286) = INTERSECTION OF LINES X(2)X(39) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 + c2)2 - 4b2c2

X(5286) lies on these lines:
{2, 39}, {4, 6}, {20, 32}, {83, 2996}, {115, 147}, {140, 5024}, {148, 4027}, {172, 4293}, {187, 3522}, {193, 315}, {230, 631}, {232, 3089}, {316, 1570}, {376, 3053}, {385, 3785}, {390, 2241}, {487, 3068}, {488, 3069}, {550, 1384}, {574, 3523}, {578, 1217}, {609, 4299}, {672, 5230}, {962, 1572}, {1212, 3772}, {1285, 3529}, {1506, 5056}, {1851, 2082}, {1885, 3172}, {1914, 4294}, {2242, 3600}, {2275, 3086}, {2276, 3085}, {2345, 4385}, {3054, 3533}, {3090, 3815}, {3096, 3620}, {3146, 5007}, {3528, 5023}, {3673, 4000}, {3832, 5041}, {4644, 4911}, {5008, 5059}


X(5287) = INTERSECTION OF LINES X(1)X(2) AND X(9)X(81)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c)2 + 2bc
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + 2ab + 2ac + 4bc)

X(5287) lies on these lines:
{1, 2}, {6, 3305}, {9, 81}, {27, 33}, {34, 469}, {37, 63}, {45, 4641}, {46, 3743}, {55, 4682}, {57, 1255}, {77, 226}, {86, 312}, {171, 968}, {223, 1442}, {329, 3945}, {333, 4687}, {440, 1060}, {464, 1038}, {750, 1962}, {756, 3751}, {984, 4038}, {1001, 3745}, {1100, 4383}, {1211, 4851}, {1230, 3761}, {1386, 4423}, {1453, 5047}, {1790, 2268}, {1817, 3601}, {2334, 4662}, {3175, 4363}, {3219, 3731}, {3242, 4883}, {3306, 3666}, {3664, 4656}, {3715, 4663}, {3723, 3752}, {3729, 3995}, {3737, 4789}, {3782, 4675}, {3875, 4359}, {3980, 3993}

X(5287) = {X(1),X(2)}-harmonic conjugate of X(5256)


X(5288) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - ab2 - ac2 + 3abc - 2b2c - 2bc2)

X(5288) lies on these lines:
{1, 6}, {3, 3632}, {8, 36}, {21, 3244}, {35, 519}, {46, 4853}, {55, 3633}, {56, 3679}, {100, 3625}, {145, 993}, {191, 3057}, {214, 4420}, {404, 3626}, {499, 3421}, {528, 4324}, {529, 3585}, {758, 4861}, {999, 1698}, {1329, 3582}, {1376, 4668}, {1388, 3940}, {1621, 3635}, {1759, 4051}, {2098, 3899}, {2099, 3901}, {2178, 4034}, {2550, 4317}, {3219, 3884}, {3304, 3624}, {3337, 3753}, {3579, 3893}, {3583, 3813}, {3584, 4999}, {3636, 5047}, {3872, 4880}, {3880, 3916}, {3913, 5010}, {4253, 4390}, {4278, 4720}, {4299, 5082}, {4816, 5204}


X(5289) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2b2 + 2c2 - a2 + ab + ac - 2bc)

X(5289) lies on these lines:
{1, 6}, {2, 2099}, {3, 214}, {8, 11}, {10, 1482}, {21, 2320}, {36, 3899}, {55, 3877}, {56, 3218}, {63, 1319}, {65, 3306}, {78, 3057}, {145, 2551}, {200, 3880}, {210, 3872}, {329, 529}, {517, 997}, {519, 3452}, {527, 4315}, {551, 4930}, {758, 999}, {965, 1953}, {1388, 2975}, {1389, 3090}, {2390, 3784}, {3207, 3496}, {3295, 3884}, {3303, 3890}, {3304, 3868}, {3338, 4018}, {3340, 3812}, {3445, 3976}, {3576, 4640}, {3616, 4999}, {3679, 5123}, {3680, 4882}, {3876, 4861}, {3885, 4420}, {4421, 5119}, {4662, 4853}, {4711, 4915}


X(5290) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + 2b2 + 2c2 + ab + ac + 4bc)/(b + c - a)

X(5290) lies on these lines:
{1, 4}, {2, 3361}, {5, 3333}, {7, 10}, {8, 3671}, {12, 57}, {40, 495}, {56, 3624}, {65, 3679}, {79, 5119}, {85, 1930}, {142, 2551}, {165, 3085}, {200, 377}, {381, 5045}, {551, 4308}, {553, 1788}, {612, 1448}, {975, 4320}, {986, 4862}, {1074, 1103}, {1125, 3600}, {1388, 4870}, {1435, 5142}, {1697, 1836}, {1722, 4859}, {1773, 1781}, {2099, 3633}, {2475, 3870}, {2476, 5231}, {2550, 4882}, {3146, 4314}, {3244, 4323}, {3340, 3632}, {3616, 4315}, {3704, 4659}, {3920, 4347}, {3982, 4848}, {4666, 5046}, {4847, 5177}


X(5291) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + abc - b2c - bc2)

X(5291) lies on these lines:
{1, 6}, {2, 2242}, {8, 32}, {10, 172}, {21, 1500}, {31, 4390}, {36, 1575}, {39, 2975}, {58, 2295}, {100, 187}, {101, 2238}, {111, 898}, {115, 5080}, {145, 2241}, {232, 1783}, {385, 668}, {404, 1574}, {519, 1914}, {594, 1333}, {609, 3679}, {650, 667}, {759, 813}, {899, 1055}, {993, 2276}, {1016, 1252}, {1150, 3661}, {1571, 4652}, {1572, 3872}, {1759, 3959}, {2239, 5091}, {2243, 5011}, {2251, 3684}, {2345, 5019}, {2703, 5164}, {3125, 3509}, {3436, 3767}, {3734, 4441}, {3780, 4251}, {4112, 4362}


X(5292) = INTERSECTION OF LINES X(1)X(2) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + b4 + c4 + 2a3b + 2a3c + 2a2bc - 2b2c2

X(5292) lies on these lines:
{1, 2}, {3, 1834}, {4, 58}, {5, 6}, {20, 4257}, {30, 4252}, {31, 1479}, {46, 1076}, {57, 225}, {69, 3454}, {81, 2476}, {140, 4255}, {230, 2271}, {283, 1724}, {345, 2901}, {442, 940}, {496, 1191}, {497, 595}, {579, 1766}, {631, 4256}, {902, 4309}, {942, 3772}, {959, 994}, {967, 1889}, {1046, 3944}, {1068, 4000}, {1150, 5051}, {1468, 1478}, {1719, 3336}, {2163, 4325}, {3072, 5156}, {3192, 3542}, {3193, 4193}, {3769, 5015}, {3824, 4675}, {3927, 4415}, {4187, 4383}, {4340, 5177}


X(5293) = INTERSECTION OF LINES X(1)X(2) AND X(9)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 + b3 + c3 + abc + 2b2c + 2bc2)

X(5293) lies on these lines:
{1, 2}, {3, 984}, {9, 32}, {12, 2647}, {21, 756}, {31, 3876}, {35, 228}, {37, 1247}, {38, 404}, {58, 1757}, {72, 171}, {100, 2292}, {201, 1758}, {238, 5044}, {474, 982}, {750, 3868}, {872, 4281}, {943, 2648}, {970, 3688}, {986, 1376}, {1010, 1215}, {1054, 3670}, {1104, 3740}, {1220, 3699}, {1468, 3681}, {1490, 1742}, {2303, 3949}, {3242, 3976}, {3496, 4386}, {3509, 3954}, {3731, 4262}, {3847, 5015}, {3927, 4650}, {4005, 4641}, {4096, 4234}, {4252, 5220}, {4267, 4557}, {4332, 5226}


X(5294) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a3 + b3 + c3 + a2b + a2c + b2c + bc2

X(5294) lies on these lines:
{1, 3710}, {2, 7}, {6, 306}, {8, 1453}, {10, 31}, {38, 1125}, {44, 1211}, {81, 3912}, {141, 4001}, {228, 1009}, {345, 3618}, {474, 1473}, {516, 4972}, {519, 3969}, {896, 3634}, {1210, 5192}, {1215, 3011}, {1386, 3703}, {1698, 1707}, {1730, 4456}, {1738, 4418}, {1861, 2299}, {1890, 4429}, {2221, 4383}, {2321, 3187}, {2325, 3995}, {2887, 4672}, {3008, 4359}, {3586, 4217}, {3589, 3666}, {3683, 4026}, {3717, 3920}, {3745, 3932}, {3772, 4054}, {3773, 3791}, {3836, 4697}, {3914, 3923}, {4202, 4292}


X(5295) = INTERSECTION OF LINES X(4)X(8) AND X(10)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a3 - ab2 - ac2 - 2abc - 2b2c - 2bc2)

X(5295) lies on these lines:
{1, 2049}, {4, 8}, {5, 3687}, {10, 37}, {12, 4046}, {65, 4647}, {75, 942}, {200, 3191}, {210, 1089}, {306, 442}, {312, 5044}, {319, 1330}, {341, 4043}, {387, 2345}, {392, 3702}, {728, 3294}, {964, 3187}, {1010, 1999}, {1150, 3916}, {1479, 3966}, {2292, 4365}, {3159, 3626}, {3175, 3679}, {3295, 3886}, {3555, 4968}, {3617, 3995}, {3678, 3967}, {3697, 3701}, {3698, 4714}, {3729, 3927}, {3876, 4671}, {3878, 4717}, {3952, 4533}, {3983, 3992}, {4015, 4125}, {4054, 4101}, {4658, 4670}, {4894, 4914}


X(5296) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - a2 + 4ab + 4ac + 2bc

X(5296) lies on these lines:
{1, 391}, {2, 7}, {6, 3616}, {8, 37}, {10, 346}, {19, 4194}, {21, 198}, {45, 1213}, {69, 4687}, {141, 4748}, {145, 3247}, {200, 4343}, {344, 5224}, {573, 962}, {958, 1696}, {1125, 1743}, {1449, 3622}, {1621, 4254}, {2262, 3877}, {2297, 4334}, {2321, 3617}, {3621, 4034}, {3624, 3973}, {3625, 4898}, {3626, 4098}, {3672, 4384}, {3679, 3950}, {3739, 4419}, {3912, 5232}, {3945, 4416}, {4000, 4364}, {4007, 4029}, {4072, 4691}, {4363, 4488}, {4461, 4967}, {4643, 4648}, {4755, 4851}


X(5297) = INTERSECTION OF LINES X(1)X(2) AND X(37)X(100)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + 3bc)

X(5297) lies on these lines:
{1, 2}, {12, 858}, {22, 5217}, {23, 35}, {33, 4232}, {37, 100}, {45, 2243}, {55, 1995}, {81, 210}, {86, 3699}, {88, 1390}, {110, 2330}, {171, 756}, {741, 4518}, {750, 984}, {894, 3952}, {940, 3681}, {1010, 3701}, {1370, 5229}, {1442, 4551}, {1500, 3291}, {1870, 5094}, {1909, 3266}, {2895, 4104}, {3100, 5218}, {3306, 4392}, {3579, 4220}, {3585, 5189}, {3614, 5133}, {3740, 3745}, {3842, 4434}, {3971, 4418}, {4096, 4697}, {4318, 5219}, {4413, 4850}, {4670, 4767}, {4995, 5160}


X(5298) = INTERSECTION OF LINES X(2)X(12) AND X(11)X(30)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [4a2 - (b + c)2](b + c - a)
X(5298) = R*X(1) - r*X(2) - r*X(3)   (Peter Moses, April 2, 2013)

X(5298) lies on these lines:
{1, 549}, {2, 12}, {3, 3058}, {11, 30}, {46, 3656}, {55, 3524}, {65, 551}, {140, 3584}, {214, 519}, {376, 3086}, {381, 499}, {484, 1387}, {524, 1428}, {528, 5172}, {546, 4325}, {547, 3614}, {548, 4857}, {553, 1125}, {597, 1469}, {631, 3304}, {999, 5054}, {1358, 1447}, {1388, 1788}, {1420, 3679}, {1478, 5055}, {1479, 3534}, {1656, 4317}, {1737, 5126}, {2482, 3027}, {3303, 3523}, {3361, 4654}, {3530, 3746}, {3545, 4293}, {3585, 5066}, {3616, 5221}, {3813, 4188}, {3830, 4299}


X(5299) = INTERSECTION OF LINES X(1)X(6) AND X(32)X(36)

Trilinears        SR - aSω : SR - bSω : SR - cSω    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 + c2 - bc)

X(5299) lies on these lines:
{1, 6}, {31, 4253}, {32, 36}, {35, 39}, {41, 995}, {42, 3108}, {48, 5037}, {56, 609}, {58, 163}, {71, 4284}, {83, 1909}, {101, 1201}, {169, 614}, {172, 1015}, {239, 1930}, {304, 3759}, {386, 2280}, {572, 4300}, {595, 672}, {604, 2172}, {982, 1759}, {1193, 4251}, {1384, 5204}, {1429, 2003}, {1432, 2224}, {1469, 5039}, {1500, 5041}, {2241, 2276}, {2260, 4264}, {3216, 3684}, {3496, 3670}, {3509, 3953}, {3730, 3915}, {3934, 4400}, {5010, 5013}, {5024, 5217}


X(5300) = INTERSECTION OF LINES X(7)X(8) AND X(10)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - ab2c - abc2 + b3c + bc3

X(5300) lies on these lines:
{1, 4202}, {2, 5015}, {3, 3006}, {4, 3701}, {7, 8}, {10, 31}, {41, 4071}, {145, 5100}, {306, 379}, {315, 3263}, {341, 5080}, {404, 3705}, {516, 3710}, {540, 1046}, {612, 5051}, {976, 2887}, {1125, 4894}, {1193, 4865}, {1330, 3681}, {1478, 4696}, {1479, 4358}, {1839, 3610}, {2177, 3178}, {2292, 4660}, {2475, 4385}, {3434, 3702}, {3436, 4723}, {3616, 4514}, {3757, 4197}, {3811, 3936}, {3876, 4388}, {3902, 5082}, {4193, 5205}, {4198, 5174}, {4200, 5081}, {4417, 4420}


X(5301) = INTERSECTION OF LINES X(6)X(31) AND X(32)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a3 + a2b + a2c - b2c - bc2)

X(5301) lies on these lines:
{1, 1333}, {6, 31}, {9, 2220}, {19, 2204}, {32, 37}, {35, 4261}, {44, 3694}, {48, 3285}, {53, 1852}, {56, 1950}, {213, 584}, {284, 595}, {560, 3747}, {577, 1108}, {594, 4426}, {609, 3247}, {906, 1723}, {1030, 2277}, {1100, 2241}, {1172, 1612}, {1213, 4386}, {1449, 5035}, {1474, 2352}, {1621, 2303}, {1839, 3011}, {1841, 1968}, {2174, 2176}, {2178, 3053}, {2242, 3723}, {2251, 3204}, {2275, 5124}, {2278, 2300}, {3730, 5037}, {3749, 5227}, {4026, 4660}


X(5302) = INTERSECTION OF LINES X(1)X(6) AND X(10)X(30)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2a2 + b2 + c2 + 3ab + 3ac + 4bc)

X(5302) lies on these lines:
{1, 6}, {3, 3740}, {8, 3683}, {10, 30}, {21, 210}, {35, 3697}, {55, 4662}, {56, 3305}, {58, 4682}, {63, 3812}, {65, 3219}, {78, 3715}, {100, 3983}, {191, 3753}, {333, 3714}, {354, 5047}, {375, 970}, {377, 1155}, {484, 4002}, {846, 4646}, {993, 5044}, {1329, 3634}, {1698, 3916}, {2646, 3876}, {3158, 4866}, {3214, 4689}, {3338, 3848}, {3452, 4999}, {3617, 5086}, {3694, 4877}, {3826, 4292}, {3913, 4512}, {4383, 4719}, {4390, 4520}, {4413, 4652}


X(5303) = INTERSECTION OF LINES X(3)X(8) AND X(21)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a3 - 3ab2 - 3ac2 + abc - b2c - bc2)

X(5303) lies on these lines:
{1, 4757}, {2, 3614}, {3, 8}, {21, 36}, {35, 3244}, {46, 3897}, {55, 3623}, {56, 1621}, {140, 5080}, {145, 5217}, {191, 214}, {320, 1444}, {404, 993}, {958, 4188}, {960, 4881}, {1030, 4969}, {1420, 3890}, {1476, 2078}, {2475, 4999}, {2476, 4299}, {2646, 3218}, {3434, 3522}, {3436, 3523}, {3576, 3869}, {3579, 4861}, {3601, 3873}, {3612, 3868}, {3621, 4421}, {3633, 3871}, {3681, 4855}, {3754, 5131}, {3822, 4325}, {3916, 4511}, {4297, 5086}


X(5304) = INTERSECTION OF LINES X(2)X(6) AND X(20)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5a4 + b4 + c4 + 2a2b2 + 2a2c2 - 2b2c2

X(5304) lies on these lines:
{2, 6}, {4, 3172}, {20, 32}, {25, 1249}, {30, 1285}, {39, 3523}, {98, 5039}, {111, 3163}, {115, 3839}, {172, 3600}, {216, 1180}, {232, 4232}, {251, 393}, {376, 1384}, {387, 4251}, {390, 1914}, {577, 1627}, {609, 4293}, {800, 1194}, {910, 3598}, {1202, 2257}, {1447, 5222}, {2243, 4346}, {2548, 5056}, {2996, 3407}, {3053, 3522}, {3091, 3767}, {3509, 4310}, {3524, 5024}, {3543, 5008}, {3553, 3920}, {4220, 4254}


X(5305) = INTERSECTION OF LINES X(5)X(6) AND X(30)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a4 + b4 + c4 + a2b2 + a2c2 - 2b2c2

X(5305) lies on these lines:
{2, 3933}, {4, 3172}, {5, 6}, {20, 1384}, {30, 32}, {39, 140}, {112, 1885}, {115, 546}, {169, 3772}, {187, 548}, {218, 5230}, {251, 428}, {385, 2896}, {393, 1598}, {524, 626}, {547, 1506}, {549, 5013}, {550, 2549}, {574, 3530}, {631, 5024}, {732, 3589}, {1104, 5179}, {1184, 1368}, {1249, 3089}, {1285, 3146}, {1596, 2207}, {1759, 3782}, {1834, 4251}, {1901, 4264}, {1990, 3199}, {3628, 3815}, {3853, 5008}


X(5306) = INTERSECTION OF LINES X(2)X(6) AND X(30)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a4 + b4 + c4 + a2b2 + a2c2 - 2b2c2

X(5306) lies on these lines:
{2, 6}, {5, 5007}, {25, 1990}, {30, 32}, {39, 549}, {50, 1627}, {51, 2871}, {53, 428}, {114, 1353}, {115, 3845}, {251, 1989}, {376, 3053}, {381, 3767}, {383, 398}, {397, 1080}, {519, 4136}, {566, 1180}, {1084, 1196}, {1194, 3003}, {1368, 3284}, {1384, 2549}, {1572, 3656}, {1914, 3058}, {2023, 5052}, {2031, 3849}, {2243, 3782}, {2276, 4995}, {2548, 5055}, {3017, 4251}, {3524, 5013}, {3705, 4969}


X(5307) = INTERSECTION OF LINES X(1)X(4) AND X(19)X(27)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + ab + ac + 2bc)/(b2 + c2 - a2)

X(5307) lies on these lines:
{1, 4}, {2, 1826}, {8, 1869}, {19, 27}, {28, 993}, {56, 1882}, {193, 1839}, {273, 1435}, {312, 1840}, {321, 5227}, {407, 1211}, {518, 1824}, {535, 5146}, {912, 1871}, {958, 1867}, {960, 1868}, {1465, 2050}, {1503, 1836}, {1708, 1746}, {1723, 1751}, {1842, 4198}, {1851, 1890}, {1861, 4196}, {1865, 3772}, {1880, 3666}, {1894, 5155}, {1957, 2299}, {2250, 2282}, {2333, 4384}, {2501, 4897}, {3822, 5142}


X(5308) = INTERSECTION OF LINES X(1)X(2) AND X(7)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - a2 - 4ab - 4ac - 2bc

X(5308) lies on these lines:
{1, 2}, {7, 37}, {9, 3945}, {45, 4644}, {57, 1334}, {69, 4687}, {81, 218}, {86, 344}, {142, 3247}, {144, 3664}, {220, 940}, {226, 279}, {277, 1255}, {354, 4517}, {379, 4313}, {391, 3879}, {599, 4748}, {857, 948}, {894, 3161}, {966, 4690}, {1001, 4344}, {2295, 5228}, {2345, 4472}, {3950, 4461}, {4021, 4859}, {4029, 4659}, {4357, 4869}, {4360, 4402}, {4361, 4460}, {4413, 4433}, {4643, 4755}


X(5309) = INTERSECTION OF LINES X(2)X(39) AND X(6)X(13)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + b4 + c4 + a2b2 + a2c2 - 2b2c2

X(5309) lies on these lines:
{2, 39}, {4, 5007}, {6, 13}, {30, 32}, {148, 3972}, {183, 4045}, {187, 376}, {230, 549}, {395, 3643}, {396, 3642}, {519, 4153}, {524, 5028}, {543, 1003}, {547, 3815}, {597, 5034}, {671, 3407}, {1506, 5055}, {1570, 1992}, {1596, 1990}, {2241, 3058}, {2275, 3582}, {2276, 3584}, {2452, 5099}, {2548, 3545}, {3053, 3534}, {3162, 5064}, {3543, 5008}, {5013, 5054}

X(5309) = X(32)-of-4th-Brocard-triangle
X(5309 = X(32)-of orthocentroidal-triangle
X(5309 = inverse-in-Kiepert-hyperbola of X(3818)
X(5309 = {X(13),X(14)}-harmonic conjugate of X(3818)


X(5310) = INTERSECTION OF LINES X(1)X(22) AND X(2)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 + a2bc + b3c + bc3)

X(5310) lies on these lines:
{1, 22}, {2, 35}, {3, 614}, {12, 428}, {19, 25}, {23, 3743}, {31, 579}, {38, 3220}, {42, 251}, {51, 2330}, {56, 4348}, {184, 3056}, {199, 2223}, {350, 1799}, {354, 4265}, {613, 3796}, {674, 2194}, {858, 4330}, {1030, 3290}, {1194, 1914}, {1281, 1283}, {1370, 4302}, {1631, 2352}, {2920, 3057}, {2922, 3670}, {3011, 4220}, {3583, 5133}, {4228, 4276}


X(5311) = INTERSECTION OF LINES X(1)X(2) AND X(31)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + ab + ac + 2bc)

X(5311) lies on these lines:
{1, 2}, {6, 756}, {9, 2308}, {31, 37}, {33, 1839}, {38, 940}, {55, 199}, {63, 3989}, {81, 984}, {171, 4414}, {192, 4418}, {197, 1953}, {210, 1100}, {748, 1386}, {750, 3666}, {902, 968}, {985, 1255}, {1460, 2171}, {2177, 3723}, {2206, 2303}, {3681, 4649}, {3791, 3842}, {3873, 4038}, {3923, 3995}, {4349, 4656}, {4722, 5220}


X(5312) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2ab + 2ac + 2b2 + 2c2 + 3bc)

X(5312) lies on these lines:
{1, 2}, {6, 35}, {9, 4272}, {36, 4255}, {55, 1203}, {57, 2594}, {58, 5010}, {73, 3339}, {165, 581}, {595, 2177}, {749, 3736}, {750, 4658}, {986, 3901}, {999, 2334}, {1126, 1468}, {1449, 5153}, {1743, 4270}, {1745, 4312}, {3555, 4719}, {3670, 3894}, {3743, 3876}, {3869, 4868}, {3874, 4850}, {3916, 4663}, {4023, 4205}


X(5313) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2ab + 2ac + 2b2 + 2c2 + bc)

X(5313) lies on these lines:
{1, 2}, {3, 1203}, {6, 36}, {9, 5153}, {31, 4256}, {35, 3052}, {57, 1464}, {72, 4719}, {73, 3361}, {165, 1064}, {748, 4653}, {751, 3736}, {758, 4850}, {982, 3894}, {1191, 3746}, {1420, 2594}, {1449, 4272}, {1453, 3612}, {1470, 2003}, {1743, 5105}, {2308, 4257}, {3670, 3901}, {3792, 4277}, {3877, 4868}, {3899, 4424}


X(5314) = INTERSECTION OF LINES X(3)X(63) AND X(31)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)(a2 + b2 + c2 + bc)

X(5314) lies on these lines:
{3, 63}, {9, 22}, {25, 3305}, {31, 35}, {36, 38}, {55, 1386}, {71, 1176}, {100, 3687}, {184, 3781}, {209, 5135}, {219, 3796}, {284, 672}, {378, 3587}, {908, 4220}, {1707, 5010}, {1790, 1818}, {2003, 2979}, {2172, 3730}, {2221, 4255}, {2323, 5012}, {2915, 5044}, {3219, 3220}, {3666, 5096}, {3917, 3955}, {4265, 4641}


X(5315) = INTERSECTION OF LINES X(1)X(6) AND X(31)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 + c2 + 2ab + 2ac - bc)

X(5315) lies on these lines:
{1, 6}, {31, 36}, {35, 595}, {40, 1480}, {56, 2163}, {58, 106}, {65, 1421}, {81, 551}, {109, 1450}, {221, 3361}, {386, 2177}, {484, 3752}, {651, 4315}, {982, 4880}, {1017, 5007}, {1046, 3953}, {1149, 2308}, {1319, 2003}, {1834, 4857}, {1999, 4975}, {2382, 2703}, {2999, 5119}, {3052, 5010}, {3679, 4383}, {3792, 4749}


X(5316) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a2b - a2c + 8abc - b2c - bc2

X(5316) lies on these lines:
{2, 7}, {10, 11}, {312, 4431}, {516, 4413}, {519, 3711}, {899, 3755}, {936, 950}, {946, 1698}, {956, 1125}, {960, 4848}, {984, 5121}, {1000, 3679}, {1150, 3707}, {1210, 5044}, {2321, 4358}, {3601, 5129}, {3698, 4301}, {3740, 3816}, {3752, 4656}, {3817, 3925}, {3826, 5087}, {3840, 4104}, {3883, 5205}, {3912, 5233}

X(5316) = complement of X(3306)


X(5317) = INTERSECTION OF LINES X(4)X(6) AND X(19)X(31)

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

X(5317) lies on these lines:
{4, 6}, {19, 31}, {27, 2221}, {28, 1104}, {29, 2303}, {34, 604}, {37, 4183}, {81, 286}, {107, 739}, {112, 915}, {158, 2214}, {162, 1778}, {232, 4220}, {240, 1761}, {608, 1118}, {648, 2991}, {1119, 1396}, {1430, 2260}, {1880, 2204}, {1896, 2298}, {2287, 5016}, {2322, 2345}, {2331, 2332}, {4219, 4261}

X(5317) = isogonal conjugate of X(3998)


X(5318) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(16)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 - 6b2c2 - (12)1/2a2S

X(5318) lies on these lines:
{4, 6}, {5, 16}, {12, 1250}, {13, 15}, {14, 3845}, {17, 550}, {18, 3850}, {61, 3627}, {62, 546}, {141, 622}, {230, 1080}, {381, 395}, {383, 3815}, {463, 1495}, {524, 621}, {530, 623}, {590, 2043}, {615, 2044}, {633, 3630}, {634, 3631}, {1546, 3003}, {3411, 3856}, {3628, 5237}

X(5318) = crosssum of X(3) and X(15)
X(5318) = crosspoint of X(4) and X(13)


X(5319) = INTERSECTION OF LINES X(5)X(6) AND X(20)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 + b4 + c4 +2a2b2 + 2a2c2 - 2b2c2

X(5319) lies on these lines:
{2, 3108}, {4, 5007}, {5, 6}, {20, 32}, {39, 631}, {115, 3832}, {172, 4317}, {187, 3528}, {193, 626}, {230, 3526}, {548, 3053}, {609, 4325}, {1249, 3199}, {1572, 4301}, {1598, 1990}, {1906, 2207}, {1914, 4309}, {3530, 5013}, {3547, 5158}, {3618, 3934}, {3785, 4045}, {3815, 5070}, {5041, 5067}


X(5320) = INTERSECTION OF LINES X(6)X(25) AND X(31)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a3 - ab2 - ac2 - 2abc - 2b2c - 2bc2)

X(5320) lies on these lines:
{2, 5138}, {4, 1175}, {6, 25}, {22, 4260}, {31, 32}, {42, 2175}, {55, 584}, {65, 2355}, {81, 4223}, {182, 4220}, {198, 4275}, {199, 579}, {284, 1011}, {386, 3145}, {1200, 2357}, {1395, 1409}, {1397, 1400}, {1751, 3136}, {1824, 2264}, {2174, 2352}, {2206, 5019}, {2328, 4251}, {4383, 5135}

X(5320) = crosssum of X(2) and X(377)


X(5321) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(15)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 - 6b2c2 + (12)1/2a2S

X(5321) lies on these lines:
{4, 6}, {5, 15}, {13, 3845}, {14, 16}, {17, 3850}, {18, 550}, {61, 546}, {62, 3627}, {141, 621}, {230, 383}, {381, 396}, {462, 1495}, {524, 622}, {531, 624}, {590, 2044}, {615, 2043}, {633, 3631}, {634, 3630}, {1080, 3815}, {1545, 3003}, {3412, 3856}, {3628, 5238}

X(5321) = crosssum of X(3) and X(16)
X(5321) = crosspoint of X(4) and X(14)


X(5322) = INTERSECTION OF LINES X(1)X(22) AND X(2)X(26)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - a2bc - b3c - bc3)

X(5322) lies on these lines:
{1, 22}, {2, 36}, {3, 612}, {11, 428}, {25, 34}, {31, 3220}, {35, 3920}, {51, 1428}, {104, 4231}, {172, 1194}, {184, 1469}, {210, 5096}, {611, 3796}, {858, 4325}, {1370, 4299}, {1460, 1473}, {1626, 2352}, {1799, 1909}, {3011, 4224}, {3585, 5133}, {3745, 4265}, {4640, 5078}


X(5323) = INTERSECTION OF LINES X(7)X(21) AND X(28)X(34)

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

X(5323) lies on these lines:
{1, 1412}, {3, 4340}, {7, 21}, {28, 34}, {65, 81}, {73, 3736}, {229, 4228}, {333, 1788}, {388, 1010}, {404, 4417}, {894, 1791}, {1038, 2285}, {1043, 3476}, {1325, 5221}, {1350, 2213}, {1400, 1778}, {1420, 4653}, {1466, 1817}, {1470, 4225}, {1848, 4292}, {3340, 4658}, {4224, 4252}


X(5324) = INTERSECTION OF LINES X(8)X(21) AND X(28)X(34)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 - 2bc)(b + c - a)/(b + c)

X(5324) lies on these lines:
{6, 4224}, {8, 21}, {27, 3423}, {28, 34}, {81, 105}, {165, 4221}, {479, 1014}, {672, 1778}, {759, 3256}, {859, 1617}, {910, 1333}, {940, 4223}, {1040, 2082}, {1183, 2646}, {1350, 4383}, {1437, 3660}, {1473, 1851}, {1633, 3914}, {1812, 3794}, {1817, 3286}, {2287, 2348}, {3060, 4259}


X(5325) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(30)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b2 + c2 - 4a2 - 3ab - 3ac - 2bc)

X(5325) lies on these lines:
{2, 7}, {10, 30}, {210, 4995}, {306, 3578}, {333, 2321}, {345, 3686}, {519, 958}, {549, 5044}, {551, 960}, {846, 3755}, {936, 3524}, {971, 3740}, {1125, 3927}, {1999, 4029}, {2551, 3585}, {3058, 3683}, {3679, 5234}, {3687, 3707}, {3712, 4061}, {4035, 4416}, {4042, 4923}, {4641, 4667}


X(5326) = INTERSECTION OF LINES X(2)X(11) AND X(12)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(3b2 + 3c2 - 4a2 - 6bc)

X(5326) lies on these lines:
{1, 632}, {2, 11}, {3, 3614}, {5, 5010}, {12, 36}, {35, 3628}, {56, 3525}, {498, 999}, {547, 3583}, {1125, 5048}, {1478, 5054}, {1479, 5070}, {1914, 3055}, {2276, 3054}, {2646, 3634}, {3057, 3918}, {3085, 3533}, {3090, 5217}, {3530, 3585}, {3850, 4324}, {4302, 5055}


X(5327) = INTERSECTION OF LINES X(4)X(6) AND X(7)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a4 + b4 + c4 - 2a3b - 2a3c - 2b2c2)/(b + c)

X(5327) lies on these lines:
{4, 6}, {7, 21}, {27, 1836}, {28, 3556}, {29, 65}, {58, 946}, {81, 497}, {226, 2328}, {238, 1780}, {284, 516}, {333, 2651}, {411, 5132}, {960, 1010}, {990, 3736}, {1430, 1848}, {1817, 3474}, {1858, 2905}, {2287, 2550}, {2303, 4307}, {2360, 4292}, {5057, 5137}


X(5328) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(3b2 + 3c2 - a2 + 2ab + 2ac - 6bc)

X(5328) lies on these lines:
{2, 7}, {8, 11}, {10, 5056}, {153, 214}, {497, 3689}, {936, 3091}, {938, 4187}, {960, 3698}, {1997, 4417}, {2478, 4313}, {2550, 5087}, {2551, 3616}, {3061, 3119}, {3090, 5044}, {3436, 4308}, {4310, 5121}, {4671, 4858}, {4679, 5218}, {5175, 5187}


X(5329) = INTERSECTION OF LINES X(1)X(3) AND X(22)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - 2a2bc)

X(5329) lies on these lines:
{1, 3}, {22, 31}, {24, 602}, {25, 238}, {43, 197}, {159, 1740}, {181, 182}, {199, 985}, {394, 3792}, {511, 1397}, {748, 1995}, {1376, 5096}, {1469, 3955}, {1473, 4650}, {1626, 3286}, {1707, 3220}, {1790, 3736}, {2076, 2162}, {2178, 3509}


X(5330) = INTERSECTION OF LINES X(1)X(21) AND X(8)X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 - ab - ac - 2b2 - 2c2 + 3bc)

X(5330) lies on these lines:
{1, 21}, {2, 1482}, {8, 11}, {78, 2136}, {145, 1058}, {392, 5047}, {404, 517}, {452, 3623}, {644, 3061}, {952, 5046}, {960, 4861}, {1788, 2099}, {3057, 3871}, {3244, 4867}, {3579, 4881}, {3621, 3940}, {3872, 3876}, {3880, 4420}, {4673, 4720}


X(5331) = INTERSECTION OF LINES X(6)X(21) AND X(27)X(34)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b + c)(a2 + ab + ac + 2bc)]

X(5331) lies on these lines:
{1, 333}, {6, 21}, {27, 34}, {29, 3192}, {42, 1043}, {56, 81}, {58, 2185}, {86, 1193}, {87, 3736}, {106, 931}, {269, 1434}, {270, 1474}, {284, 2363}, {386, 1010}, {958, 2334}, {1126, 4653}, {2215, 4269}, {2279, 3601}


X(5332) = INTERSECTION OF LINES X(6)X(31) AND X(32)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2a2 + b2 + c2 - bc)

X(5332) lies on these lines:
{1, 5007}, {6, 31}, {32, 36}, {39, 5010}, {44, 3681}, {172, 999}, {238, 3789}, {239, 4376}, {609, 1015}, {893, 2364}, {982, 2243}, {995, 2251}, {1040, 3284}, {1100, 3873}, {1403, 1404}, {2220, 2277}, {2300, 5037}, {3703, 4969}


X(5333) = INTERSECTION OF LINES X(2)X(6) AND X(21)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + 2b + 2c)/(b + c)

X(5333) lies on these lines:
{1, 4720}, {2, 6}, {21, 36}, {58, 748}, {142, 1817}, {226, 1014}, {274, 321}, {314, 4359}, {1001, 4184}, {1010, 3616}, {1043, 3622}, {1412, 5219}, {1698, 4658}, {3219, 4670}, {3286, 4423}, {3720, 3736}, {3786, 3873}, {4654, 4877}


X(5334) = INTERSECTION OF LINES X(2)X(14) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 - 6b2c2 + (48)1/2a2S

X(5334) lies on these lines:
{2, 14}, {4, 6}, {13, 3839}, {16, 20}, {17, 5068}, {18, 3523}, {61, 3091}, {62, 3146}, {193, 622}, {376, 395}, {396, 3545}, {633, 3620}, {1131, 3367}, {1132, 3366}, {1250, 4294}, {2043, 3069}, {2044, 3068}


X(5335) = INTERSECTION OF LINES X(2)X(13) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b4 + 3c4 - 3a4 - 6b2c2 - (48)1/2a2S

X(5335) lies on these lines:
{2, 13}, {4, 6}, {14, 3839}, {15, 20}, {17, 3523}, {18, 5068}, {61, 3146}, {62, 3091}, {193, 621}, {376, 396}, {395, 3545}, {634, 3620}, {1131, 3392}, {1132, 3391}, {1250, 3085}, {2043, 3068}, {2044, 3069}


X(5336) = INTERSECTION OF LINES X(1)X(6) AND X(19)X(32)

Trilinears        a3s - SBSC : b3s - SCSA : c3s - SASB    César Lozada (9/07/2013)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 + b4 + c4 + 2a3b + 2a3c - 2b2c2)

X(5336) lies on these lines:
{1, 6}, {19, 32}, {25, 1096}, {31, 2171}, {46, 2305}, {609, 1781}, {800, 2331}, {992, 997}, {1184, 5089}, {1400, 3924}, {1572, 1953}, {1731, 5037}, {1826, 3767}, {2285, 5019}, {2321, 4362}, {3290, 5020}, {3612, 5110}


X(5337) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 + a3b + a3c + a2bc + b3c + bc3)

X(5337) lies on these lines:
{1, 3}, {2, 32}, {6, 3882}, {39, 81}, {58, 1009}, {63, 3954}, {69, 5019}, {141, 1333}, {172, 3912}, {193, 5042}, {524, 5035}, {1150, 3661}, {2220, 3589}, {3793, 5241}, {4044, 4396}, {4220, 5188}, {4384, 4386}


X(5338) = INTERSECTION OF LINES X(19)X(25) AND X(28)X(34)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(3a + b + c)/(b2 + c2 - a2)

X(5338) lies on these lines:
{2, 1890}, {4, 165}, {19, 25}, {28, 34}, {51, 2261}, {154, 2262}, {204, 1841}, {212, 2270}, {354, 1829}, {461, 4512}, {607, 1190}, {1155, 1878}, {1474, 2280}, {1598, 1753}, {1839, 4207}, {1871, 3517}


X(5339) = INTERSECTION OF LINES X(3)X(14) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - 2b2c2 + 31/2a2S

X(5339) lies on these lines:
{3, 14}, {4, 6}, {13, 3843}, {15, 1656}, {16, 1657}, {17, 3851}, {20, 395}, {61, 381}, {62, 382}, {154, 462}, {396, 3091}, {599, 633}, {621, 3763}, {3526, 5238}, {3534, 5237}


X(5340) = INTERSECTION OF LINES X(3)X(13) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b4 + c4 - a4 - 2b2c2 - 31/2a2S

X(5340) lies on these lines:
{3, 13}, {4, 6}, {14, 3843}, {15, 1657}, {16, 1656}, {18, 3851}, {20, 396}, {61, 382}, {62, 381}, {154, 463}, {395, 3091}, {599, 634}, {622, 3763}, {3526, 5237}, {3534, 5238}


X(5341) = INTERSECTION OF LINES X(6)X(19) AND X(35)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 - a2bc - 2b2c2)

X(5341) lies on these lines:
{6, 19}, {9, 484}, {35, 37}, {45, 1766}, {50, 1950}, {583, 1731}, {759, 1333}, {910, 3256}, {1400, 1989}, {1719, 4640}, {1723, 5043}, {1760, 4363}, {2171, 2173}, {2178, 5172}, {4271, 5011}


X(5342) = INTERSECTION OF LINES X(4)X(8) AND X(29)X(34)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a + b + c)(b2 + c2 - a2)

X(5342) lies on these lines:
{4, 8}, {27, 4384}, {29, 34}, {75, 1890}, {242, 4185}, {278, 4194}, {281, 4200}, {391, 4047}, {452, 1441}, {461, 3616}, {469, 3912}, {1039, 2481}, {1904, 2969}, {4101, 4673}


X(5343) = INTERSECTION OF LINES X(4)X(6) AND X(14)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5b4 + 5c4 - 5a4 - 10b2c2 + (48)1/2a2S

X(5343) lies on these lines:
{2, 5238}, {4, 6}, {14, 20}, {15, 5056}, {16, 5059}, {17, 3091}, {18, 3522}, {61, 3832}, {62, 3543}, {395, 3529}, {396, 3855}, {1131, 3364}, {1132, 3365}


X(5344) = INTERSECTION OF LINES X(4)X(6) AND X(13)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5b4 + 5c4 - 5a4 - 10b2c2 - (48)1/2a2S

X(5344) lies on these lines:
{2, 5237}, {4, 6}, {13, 20}, {15, 5059}, {16, 5056}, {17, 3522}, {18, 3091}, {61, 3543}, {62, 3832}, {395, 3855}, {396, 3529}, {1131, 3389}, {1132, 3390}


X(5345) = INTERSECTION OF LINES X(1)X(22) AND X(25)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2b4 + 2c4 - 2a4 - a2bc - b3c - bc3)

X(5345) lies on these lines:
{1, 22}, {2, 3585}, {23, 614}, {25, 36}, {609, 1194}, {612, 5010}, {846, 3415}, {988, 2915}, {990, 1719}, {1370, 4316}, {1707, 3220}, {1799, 3761}, {5020, 5204}


X(5346) = INTERSECTION OF LINES X(6)X(17) AND X(30)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a4 + b4 + c4 + a2b2 + a2c2 - 2b2c2

X(5346) lies on these lines:
{2, 5041}, {4, 5008}, {6, 17}, {30, 32}, {39, 631}, {115, 3843}, {187, 3522}, {230, 632}, {385, 3096}, {1186, 2086}, {2548, 5071}, {3091, 3767}


X(5347) = INTERSECTION OF LINES X(1)X(3) AND X(6)X(22)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - a2bc - ab2c - abc2)

X(5347) lies on these lines:
{1, 3}, {2, 5096}, {6, 22}, {25, 4383}, {81, 4265}, {184, 4259}, {199, 5132}, {386, 2915}, {1626, 4497}, {2194, 4260}, {3220, 4641}, {4184, 5124}


X(5348) = INTERSECTION OF LINES X(1)X(3) AND X(11)X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + bc - b2)(a2 + bc - c2)

X(5348) lies on these lines:
{1, 3}, {2, 2361}, {4, 1399}, {5, 47}, {11, 31}, {12, 255}, {58, 1837}, {109, 1836}, {181, 1364}, {212, 750}, {394, 1376}, {1253, 4995}


X(5349) = INTERSECTION OF LINES X(4)X(6) AND X(18)X(30)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5b4 + 5c4 - 5a4 - 10b2c2 + (12)1/2a2S

X(5349) lies on these lines:
{4, 6}, {5, 5238}, {13, 3861}, {14, 3627}, {15, 3850}, {17, 3858}, {18, 30}, {61, 3845}, {62, 3853}, {382, 395}, {396, 546}


X(5350) = INTERSECTION OF LINES X(4)X(6) AND X(17)X(30)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 5b4 + 5c4 - 5a4 - 10b2c2 - (12)1/2a2S

X(5350) lies on these lines:
{4, 6}, {5, 5237}, {13, 3627}, {14, 3861}, {16, 3850}, {17, 30}, {18, 3858}, {61, 3853}, {62, 3845}, {382, 396}, {395, 546}


X(5351) = INTERSECTION OF LINES X(3)X(6) AND X(18)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(7b2 + 7c2 - 7a2 - (12)1/2a2S)

X(5351) lies on these lines:
{3, 6}, {13, 631}, {14, 550}, {17, 549}, {18, 20}, {202, 5217}, {395, 548}, {397, 3530}, {622, 630}, {1092, 3206}, {3411, 3528}


X(5352) = INTERSECTION OF LINES X(3)X(6) AND X(17)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(7b2 + 7c2 - 7a2 + (12)1/2a2S)

X(5352) lies on these lines:
{3, 6}, {13, 550}, {14, 631}, {17, 20}, {18, 549}, {203, 5217}, {396, 548}, {398, 3530}, {621, 629}, {1092, 3205}, {3412, 3528}


X(5353) = INTERSECTION OF LINES X(1)X(6) AND X(15)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2S - 31/2bc)

X(5353) lies on these lines:
{1, 6}, {15, 35}, {16, 36}, {42, 2981}, {61, 3746}, {395, 3582}, {396, 3584}, {398, 4857}, {651, 3639}, {1082, 2003}, {1094, 3170}


X(5354) = INTERSECTION OF LINES X(2)X(6) AND X(23)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + b4 + c4 + 2a2b2 + 2a2c2 - b2c2)

X(5354) lies on these lines:
{2, 6}, {22, 1384}, {23, 32}, {25, 1383}, {111, 251}, {187, 1194}, {574, 1180}, {1915, 2502}, {2030, 5012}, {3291, 5007}, {3767, 5169}


X(5355) = INTERSECTION OF LINES X(6)X(13) AND X(20)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a4 + b4 + c4 + 2a2b2 + 2a2c2 - 2b2c2

X(5355) lies on these lines:
{5, 5041}, {6, 13}, {20, 32}, {30, 5008}, {39, 140}, {385, 4045}, {543, 3972}, {574, 3524}, {1506, 3090}, {2548, 5068}, {3627, 5007}


X(5356) = INTERSECTION OF LINES X(6)X(19) AND X(36)X(37)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 - 3a2bc - 2b2c2)

X(5356) lies on these lines:
{1, 4287}, {6, 19}, {9, 3336}, {36, 37}, {44, 1781}, {46, 5036}, {1385, 1766}, {1400, 2963}, {1950, 2965}, {2160, 2183}, {2161, 2260}


X(5357) = INTERSECTION OF LINES X(1)X(6) AND X(15)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(2S + 31/2bc)

X(5357) lies on these lines:
{1, 6}, {15, 36}, {16, 35}, {62, 1250}, {395, 3584}, {396, 3582}, {397, 4857}, {559, 2003}, {651, 3638}, {1095, 3171}


X(5358) = INTERSECTION OF LINES X(1)X(6) AND X(15)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 + 2a2bc - 2b2c2)/(b + c)

X(5358) lies on these lines:
{1, 4228}, {10, 21}, {22, 1714}, {28, 34}, {169, 284}, {386, 4224}, {1210, 4233}, {1817, 3008}, {1842, 3220}, {4269, 4456}


X(5359) = INTERSECTION OF LINES X(2)X(6) AND X(22)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + b4 + c4 + 2a2b2 + 2a2c2)

X(5359) lies on these lines:
{2, 6}, {3, 1180}, {4, 3162}, {22, 32}, {25, 251}, {51, 5039}, {169, 614}, {1196, 1995}, {3767, 5133}


X(5360) = INTERSECTION OF LINES X(4)X(8) AND X(31)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c)(b4 + c4 - a2b2 - a2c2)

X(5360) lies on these lines:
{4, 8}, {31, 32}, {37, 263}, {42, 4531}, {100, 2698}, {237, 1755}, {511, 1959}, {512, 661}, {674, 4053}


X(5361) = INTERSECTION OF LINES X(2)X(6) AND X(8)X(35)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a3 - 2ab2 - 2ac2 - abc - 2b2c - 2bc2

X(5361) lies on these lines:
{2, 6}, {8, 35}, {63, 4659}, {100, 4042}, {956, 4216}, {3219, 4671}, {3679, 4257}, {3757, 4430}, {3769, 4981}


X(5362) = INTERSECTION OF LINES X(2)X(6) AND X(15)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[2aS - 31/2(abc + b2c + bc2)]

X(5362) lies on these lines:
{2, 6}, {15, 21}, {16, 404}, {37, 2981}, {61, 5047}, {100, 1250}, {470, 1172}, {2323, 5243}


X(5363) = INTERSECTION OF LINES X(2)X(6) AND X(15)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a4 - 3a2bc - b2c2)

X(5363) lies on these lines:
{1, 3}, {23, 31}, {181, 575}, {238, 1995}, {576, 1397}, {1283, 3941}, {1395, 3518}, {1740, 2930}


X(5364) = INTERSECTION OF LINES X(2)X(7) AND X(31)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b3 + c3 - ab2 - ac2 - abc)

X(5364) lies on these lines:
{2, 7}, {31, 32}, {198, 1755}, {292, 1613}, {846, 3730}, {968, 1334}, {1707, 2664}, {4020, 5021}


X(5365) = INTERSECTION OF LINES X(4)X(6) AND X(18)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 7b4 + 7c4 - 7a4 - 14b2c2 + (48)1/2a2S

X(5365) lies on these lines:
{4, 6}, {14, 3146}, {15, 5068}, {17, 3854}, {18, 20}, {61, 3839}, {3412, 3832}


X(5366) = INTERSECTION OF LINES X(4)X(6) AND X(17)X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 7b4 + 7c4 - 7a4 - 14b2c2 - (48)1/2a2S

X(5366) lies on these lines:
{4, 6}, {13, 3146}, {16, 5068}, {17, 20}, {18, 3854}, {62, 3839}, {3411, 3832}


X(5367) = INTERSECTION OF LINES X(2)X(6) AND X(16)X(21)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[2aS + 31/2(abc + b2c + bc2)]

X(5367) lies on these lines:
{2, 6}, {15, 404}, {16, 21}, {62, 5047}, {471, 1172}, {1250, 1621}, {2323, 5242}


X(5368) = INTERSECTION OF LINES X(6)X(17) AND X(20)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a4 + b4 + c4 + 2a2b2 +2a2c2 - 2b2c2

X(5368) lies on these lines:
{6, 17}, {20, 32}, {39, 549}, {115, 546}, {230, 5041}, {3545, 3767}


X(5369) = INTERSECTION OF LINES X(7)X(8) AND X(31)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b4 + c4 - ab3 - ac3 - ab2c - abc2)

X(5369) lies on these lines:
{7, 8}, {31, 32}, {674, 3721}, {1046, 1282}, {1193, 4531}, {2292, 3688}


X(5370) = INTERSECTION OF LINES X(1)X(22) AND X(23)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(3b4 + 3c4 - 3a4 - a2bc - b3c - bc3)

X(5370) lies on these lines:
{1, 22}, {23, 36}, {25, 5204}, {612, 5217}, {858, 4316}, {896, 3220}


X(5371) = INTERSECTION OF LINES X(6)X(22) AND X(31)X(32)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(a3 - abc - b2c - bc2)

X(5371) lies on these lines:
{6, 22}, {31, 32}, {81, 1915}, {584, 2276}, {2194, 3051}, {2277, 4275}


X(5372) = INTERSECTION OF LINES X(2)X(6) AND X(8)X(36)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a3 - 2ab2 - 2ac2 + abc - 2b2c - 2bc2

X(5372) lies on these lines:
{2, 6}, {3, 4720}, {8, 36}, {63, 4671}, {1330, 5141}, {4362, 4392}


X(5373) = EQUIAREALITY CENTER

Trilinears   x : y : z = f(A,B,C) : f(B,C,A) : f(C,A,B) where (x,y,z) is the solution of the following system:
(y2 + z2)cos A + 2yz = sin A
(z2 + x2)cos B + 2zx = sin B
(x2 + y2)cos C + 2xy = sin C
(There is a unique solution with real x,y,z if the reference triangle ABC is acute.)

For any point X inside an acute triangle ABC, let A' B' C' denote the pedal triangle of X. Then X(5373) is the point X for which the quadrilaterals AC'XB', BA'XC', CB'XA' all have the same area.

X(5373) is discussed in the following articles:

Apoloniusz Tyszka, "Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 181-185: Tyszka article.

Jean Pierre Ehrmann, "Constructive solution of a generalization of Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 187-190: Ehrmann article.

Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.

X(5373) is the incenter of the Thomson triangle, as proved in Thomson Triangle.


X(5374) = TRILINEAR SQUARE ROOT OF X(63)

Trilinears   (cot A)1/2 : (cot B)1/2 : (cot C)1/2

For any point P on segment BC of an acute triangle ABC, let Q be the point on AB nearest to P and let R be the point on AC nearest to P. Let A' be the choice of P for which area(A'QB) = area(A'RC). Define B' and C' cyclically. Then the lines AA', BB', CC' concur in X(5374).

X(5374) is introduced in

Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.


X(5375) = INTERSECTION OF LINES X(100)X(650) and X(101)X(661)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(b3 + c3 - a3 - b2a - c2a + a2b + a2c + abc)

X(5375) is the center of the circumconic with perspector X(100). This conic passes through the bicentric pairs P(26), U(26), and P(33), U(33) (Randy Hutson, 9/10/2012) and is a hyperbola (Peter Moses, 10/10/2012). It is introduced here as the Hutson-Moses hyperbola, discussed in the preamble to X(5376).

X(5375) lies on these lines:
{44, 3290}, {100, 650}, {101, 661}, {190, 4467}, {644, 3239}, {651, 3676}, {666, 693}, {901, 4394}, {908, 3008}, {2323, 4700}




leftri Points on the Hutson-Moses hyperbola: X(5376) - X(5389) rightri


The Hutson-Moses hyperbola, introduced at X(5375), is given by the following barycentric equation:

a(a - b)(a - c)yz + b(b - c)(b - a)zx + c(c - a)(c - b)xy = 0.


The hyperbola has perspector X(100), center X(5375), meets the circumcircle in X(898) and the Steiner circumellipse in X(666), and is the isogonal conjugate of the line X(244)X(665). If X = x : y : z (barycentrics) is a point on the circumcircle, then the point

H(X) = x/(a(b - c)) : y/(b(c - a)) : z/(c(a - b))


is on the Hutson-Moses hyperbola. Examples are shown in the following table:

X H(X)
X(99) X(4601)
X(100) X(1016)
X(101) X(765)
X(105) X(666)
X(106) X(3257)
X(109) X(4564)
X(110) X(4567)
X(739) X(898)
X(741) X(4584)
X(934) X(1275)
X(901) X(5376)
X(919) X(5377)
X(813) X(5378)
X(112) X(5379)
X(111) X(5380)
X(898) X(5381)
X(1293) X(5382)
X(932) X(5383)
X(825) X(5384)
X(4588) X(5385)
X(753) X(5386)
X(2748) X(5387)
X(789) X(5388)
X(755) X(5389)

The acute angle Ψ between the asymptotes of the circumhyperbola with perspector X = x : y : z is given by

tan(Ψ) = S*T/(x*SA + y*SB + z*SC),


where T = (x2 + y2 + z2 - 2(yz + zx + xy))1/2, and the eccentricity e is then given by e = sec(Ψ/2). (Peter Moses, 10/11/12). For the Hutson-Moses hyperbola, (x, y, z) = a2/(b2 - c2), b2/(c2 - a2), c2/(a2 - b2).

underbar

X(5376) = H(X(901))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(a - 2b + c)(a + b - 2c)

X(5376) lies on the Hutson-Moses hyperbola and these lines:
{1, 765}, {2, 1016}, {57, 4564}, {81, 4567}, {89, 1252}, {100, 3251}, {105, 1320}, {106, 291}, {274, 4601}, {279, 1275}, {666, 4555}, {898, 901}, {1022, 1023}, {1929, 4674}, {2006, 4997}, {2397, 2401}, {4584, 4622}

X(5376) = trilinear product of PU(28)


X(5377) = H(X(919))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(a2 + b2 - ac - bc)(a2 + c2 - ab - cb)

X(5377) is the trilinear pole of the line X(100)X(650), this line being tangent at X(100) to the conic {A, B, C, X(2), X(100), PU*(46)}, where PU*(46) are the isogonal conjugates of the bicentric pair PU(46). (Randy Hutson, September 29, 2014)

X(5377) lies on the Hutson-Moses hyperbola and these lines:
{1, 1053}, {7, 59}, {8, 1016}, {9, 765}, {21, 4567}, {100, 3126}, {105, 1320}, {294, 1642}, {314, 4601}, {666, 885}, {673, 3254}, {898, 919}, {927, 2742}, {1027, 3257}, {1438, 4876}, {4998, 5218}

X(5377) = isogonal conjugate of X(3675)


X(5378) = H(X(813))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(b2 - ac)(c2 - ab)

X(5378) is the trilinear pole of the line X(100)X(649), this line being tangent at X(100) to the hyperbola {A, B, C, X(81), X(100), PU(8)}. (Randy Hutson, September 29, 2014)

X(5378) lies on the Hutson-Moses hyperbola and these lines:
{1, 1016}, {6, 765}, {56, 4564}, {58, 4567}, {86, 4601}, {87, 4076}, {106, 291}, {269, 1275}, {660, 876}, {666, 1026}, {813, 898}, {1411, 4518}, {1438, 4876}


X(5379) = H(X(112))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b)(a + c)(a - b)2(a - c)2(a2 - b2 + c2)(a2 + b2 - c2)

X(5379) lies on the Hutson-Moses hyperbola and these lines:
{59, 5080}, {100, 1304}, {110, 1309}, {112, 898}, {162, 3257}, {250, 2074}, {422, 4601}, {648, 666}, {685, 692}, {2397, 2409}, {4564, 4570}


X(5380) = H(X(111))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(a2 - 2b2 + c2)(a2 + b2 - 2c2)

X(5379) lies on the Hutson-Moses hyperbola and these lines:
{100, 691}, {111, 898}, {291, 4584}, {666, 671}, {668, 892}, {765, 1018}, {897, 1757}, {1016, 3952}, {1275, 4566}, {2397, 2408}, {4551, 4564}


X(5381) = H(X(898))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b)2(a - c)2(2ab - ac - bc)(2ac - ab - bc)

X(5381) lies on the Hutson-Moses hyperbola and these lines:
{6, 1016}, {31, 765}, {81, 4601}, {604, 4564}, {666, 889}, {1275, 1407}, {1333, 4567}, {3257, 3570}


X(5382) = H(X(1293))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(a - 3b + c)(a + b - 3c)

X(5382) lies on the Hutson-Moses hyperbola and these lines:
{644, 3669}, {765, 1279}, {898, 1293}, {1016, 3008}, {1332, 3257}, {2397, 2415}


X(5383) = H(X(932))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b)2(a - c)2(bc + ab - ac)(bc - ab + ac)

X(5383) lies on the Hutson-Moses hyperbola and these lines:
{87, 4076}, {190, 1919}, {645, 4584}, {898, 932}, {3257, 4598}, {3287, 4583}


X(5384) = H(X(825))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(a2 + ab + b2)(a2 + ac + c2)

X(5384) lies on the Hutson-Moses hyperbola and these lines:
{110, 4584}, {666, 4586}, {825, 898}, {1016, 1110}, {1492, 3257}, {4570, 4601}


X(5385) = H(X(4588))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)2(a - c)2(2a - b + 2c)(2a + 2b - c)

X(5385) lies on the Hutson-Moses hyperbola and these lines:
{89, 1252}, {100, 4825}, {101, 3257}, {666, 4597}, {898, 4588}


X(5386) = H(X(753))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(a3 - 2b3 + c3)(a3 + b3 - 2c3)

X(5386) lies on the Hutson-Moses hyperbola and these lines: {753, 898}, {765, 3799}, {1016, 3807}


X(5387) = H(X(2748))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b)2(a - c)2(a2 + b2 + c2 - 3ab)(a2 + b2 + c2 - 3ac)

X(5387) lies on the Hutson-Moses hyperbola and these lines: {898, 2748}, {1016, 3759}


X(5388) = H(X(789))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(a - b)2(a - c)2(a2 + ab + b2)(a2 + ac + c2)

X(5388) lies on the Hutson-Moses hyperbola and these lines: {789, 898}, {799, 4584}


X(5389) = H(X(755))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a - b)(a - c)(a4 - 2b4 + c4)(a4 + b4 - 2c4)

X(5389) lies on the Hutson-Moses hyperbola and these lines: {755, 898}, {4553, 4567}


X(5390) = EULER-MORLEY-ZHAO POINT

Trilinears       T(A,B,C) : T(B,C,A) : T(C,A,B), where T(A,B,C) = cos(B - C) - cos(B + C) - cos(B/3 + C/3) + cos(5B/3 + 5C/3) - sin(C - B/3 - π/6) - sin(B - C/3 - π/6) + sin(B + 5C/3 - π/6) + sin(C + 5B/3 - π/6)  (Chris van Tienhoven, April 7, 2013)

Barycentrics   a*f(A/3, B/3, C/3) : b*f(B/3, C/3, A/3) : c*f(C/3, A/3, B/3), where f(x,y,z) is defined using the abbreviations [m,n] for sin(x + 2my + 2nz) + sin(x + 2ny + 2mz) and [m] for [m,m]/2, as follows:

f(x,y,z) = [-3,-1] + 5[-2,-1] + [-2,0] - 5[-1,1] - 3[-1,2] - [-1,3] + [0,1] + 2[0,2] - [0,3] - 2[1,3] - 2[2,3] + [-3] - 2[-2] + 3[-1] + 3[0] + 3[1] + 3[2] - 2[3] - [4]   (Barry Wolk)

Let DEF be the classical Morley triangle. The Euler lines of the three triangles AEF, BFD, CDE to concur in X(5390), as discovered by Zhao Yong of Anhui, China, October 2, 2012. For a construction and derivation of barycentric coordinates by Shi Yong, see Problem 20 at Unsolved Problems and Rewards. For further developments, including the development of trilinear and barycentric coordinates as shown above, type X(5390) into Search at Hyacinthos.

If you have The Geometer's Sketchpad, you can view X(5390).

X(5390) lies on this line: {357, 1136}


X(5391) = ISOTOMIC CONJUGATE OF X(1336)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(1 - sin A)


X(5392) = INTERSECTION OF LINES X(4)X(52) AND X(22)X(98)

Barycentrics   sec 2A : sec 2B : sec 2C

X(5392) lies on the Euler perspective cubic (K045), corresponding to the point X(68) on the Euler central cubic (K044). (Randy Hutson, November 22, 2014)

X(5392) lies on the Kiepert hyperbola and these lines: {2, 311}, {3, 96}, {4, 52}, {10, 91}, {22, 98}, {226, 914}, {262, 5133}, {264, 275}, {338, 394}, {467, 2052}

X(5392) = isogonal conjugate of X(571)
X(5392) = isotomic conjugate of X(1993)


X(5393) = CENTER OF THE PAACHE-MYAKISHEV ELLIPSE

Barycentrics   2 + cot(B/2) + cot(C/2) : 2 + cot(C/2) + cot(A/2) : 2 + cot(A/2) + cot B/2)
Barycentrics   a + 2r : b + 2r : c + 2r
X(5393) = s*X(1) + 3r*X(2)   (Peter Moses, January 2, 2013)

Let W(BA) and W(CA) be the two congruent circles, within triangle ABC, each tangent to the other and to sideline BC of triangle ABC, with W(BA) also tangent to sideline AB and W(CA) also tangent to sideline AC; cf. the Paache configuration at X(1123). Let BA and CA be the touchpoints of these circles with sideline BC. Define the points CB, AC cyclically and define the points AB, BC cyclically. The six points lie on an ellipse having center X(5393) and equation

d(2 + d)x2 + e(2 + e)y2 + f(2 + f)z2 - 2(2 + e + f + ef)yz - 2(2 + f + d + fd)zx - 2(2 + d + e + de)xy = 0,

where d = cot(A/2), e = cot(B/2), f = cot(C/2). Let X = X(5393). Then |GX|/|IX| = s/(3r), where G = centroid, I = incenter, r = inradius, and s = semiperimeter. (Alexei Myakishev, December 25, 2012).

An associated conic, the Paache-Myakishev-Moses conic, is introduced at X(5405). This conic results from the two congruent circles that do not lie within triangle ABC.

If you have The Geometer's Sketchpad, you can view

X(5393), including the ellipse. You can also view the configuration for pairs of circles used in the constructions of X(5393) and X(5405): Pairs of Circles.

X(5393) lies on these lines: {1, 2 }, {9, 3068}, {37, 590}, {57, 482}, {81, 3300}, {175, 5226}, {226, 481}, {491, 4357}, {492, 3879}, {515, 2048}, {615, 1100}, {642, 3666}, {940, 1335}, {1124, 4383}, {1255, 3302}, {1267, 3875}, {1449, 3069}, {1585, 1785}, {1991, 4643}

X(5393) = {X(1),X(2)}-harmonic conjugate of X(5405)


X(5394) = CONGRUENT INCIRCLES POINT

Barycentrics   (unknown)

X(5394) is the point X for which the three triangles AXB, BXC, CXA have congruent incircles. The existence of this point is proved by Noam Elkies in Mathematics Magazine 60 (1987) 117. His proof applies to a much wider range of functions (with the inradius replaced by the area, semiperimeter, etc., or any positive combination thereof).

Following is a copy-and-run Mathematica program that computes actual trilinear distances (1.7916..., 1.7057..., 1.6328...) of X(5394) for the triangle given by (a,b,c) = (6,9,13).

(1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)] {x/a, y/b, z/c} /. #1 /.
NSolve[{x + y + z == 1, (a + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2])/
x == (b + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
y == (c + Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
z /. #1}, {x, y, z}, WorkingPrecision -> 40][[1]] &)[Thread[{a, b, c} -> {6, 9,13}]] (* Program by Peter Moses, October 23, 2012. *)

X(5394) lies on no line X(i)X(j) for 1 <= i < j <= 5393.


X(5395) = ISOTOMIC CONJUGATE OF X(3620)

Trilinears       1/(sin A + 2 cos A tan ω) : 1/(sin B + 2 cos B tan ω) : 1/(sin C + 2 cos C tan ω)
Trilinears       1/(2 cos A + sin A cot ω) : 1/(2 cos B + sin B cot ω) : 1/(2 cos C + sin C tan ω)
Trilinears       1/(2 cos A sin ω + sin A cos ω) : 1/(2 cos B sin ω + sin B cos ω) : 1/(2 cos C sin ω + sin C cos ω)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(3b2 + 3c2 - a2)

Let X, Y, Z be the points defined by Dominik Burek as at X(1217). If the initial point P is the centroid, then the perspector of the triangles XYZ and ABC is X(5395). (Peter Moses, June 9, 2012)

X(5395) lies on these lines:
{2, 3053}, {4, 5050}, {6, 2996}, {20, 262}, {76, 193}, {83, 5033}, {98, 3091}, {439, 3815}, {458, 459}, {620, 2548}, {671, 5286}, {3146, 3329}, {3424, 3832}

X5395) = isogonal conjugate of X(5013)


X(5396) = INTERSECTION OF LINES X(1)X(5) AND X(3)X(6)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos(B - A) + cos(C - A)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)(cos A + cos B + cos C) + (sin A)(sin A + sin B + sin C)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = R*(cos A)(cos A + cos B + cos C) + s*sin A
X(5396) = (b + c)(c + a)(a + b)*X(1) - 2abc*X(5)   (Peter Moses, January 2, 2013)

X(5396)-X(5400) were submitted with trilinears by Randy Hutson, December 12, 2012.

X(5396) = {X(3),X(6)}-harmonic conjugate of X(5398)

X(5396) lies on these lines:
{1, 5}, {3, 6}, {35, 2361}, {40, 5312}, {42, 517}, {51, 859}, {54, 60}, {73, 942}, {140, 3216}, {515, 2051}, {912, 3666}, {1066, 5045}, {1155, 4337}, {1193, 1385}, {1450, 5126}, {1871, 1880}, {2800, 4868}, {3060, 4216}, {3190, 3940}, {3576, 5313},{3579, 4300}, {3682, 5044}}

X(5396) = isogonal conjugate of X(5397)
X(5396) = crossdifference of every pair of points on line X(523)X(654)
X(5396) = {X(371),X(372)}-harmonic conjugate of X(2278)


X(5397) = ISOGONAL CONJUGATE OF X(5396)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 + cos(B - A)) + cos(C - A))
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/[(cos A)(cos A + cos B + cos C) + (sin A)(sin A + sin B + sin C)]
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 1/[R*(cos A)(cos A + cos B + cos C) + s*sin A]

X(5397) lies on the hyperbola that passes through the points A, B, C, X(1), X(36), as well as the Kiepert hyperbola. X(5397) is the trilinear pole of the line X(523)X(654). (Randy Hutson, Dec. 31, 2012)

X(5397) lies on the Kiepert hyperbola and these lines:
{4, 2278}, {5, 60}, {10, 2323}, {12,54}, {36, 226}, {59, 495}, {94, 3615}, {275, 860}, {321, 4511}, {1443, 1446}, {2051, 4276}, {2052, 5136}, {2618, 3737}

X(5397) = isogonal conjugate of X(5396)


X(5398) = {X(3), X(6)}-HARMONIC CONJUGATE OF X(5396)

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)(cos A + cos B + cos C) - (sin A)(sin A + sin B + sin C)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = R*(cos A)(cos A + cos B + cos C) - s*sin A
X(5398) = 2r(r + R)*X(3) + (r2 + 4rR - s2)*X(6)    (Peter Moses, January 2, 2013)

X(5398) lies on these lines:
{1, 2361}, {3, 6}, {4, 162}, {5, 1724}, {30, 1754}, {31, 517}, {36, 2003}, {46, 1399}, {47, 65}, {56, 215}, {81, 1006}, {184, 859}, {255, 942}, {283, 405}, {355, 3072}, {595, 1482}, {601, 3579}, {602, 1385}, {603, 1465}, {912, 4641}, {1060, 1708}, {1064, 2308}, {1411, 2964}, {1496, 5045}, {1718, 3336}, {1737, 5348}, {1780, 3560}, {2979, 4218}, {4216, 5012}

X(5398) = {X(371),X(372)}-harmonic conjugate of X(2245)


X(5399) = INTERSECTION OF LINES X(1)X(5) AND X(54)X(59)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - cos(A - B) - cos(A - C)
X(5399) = (4R2 - 2rR - r2 - s2)*X(1) + 4rR*X(5)    (Peter Moses, January 2, 2013)

X(5399) = {X(1), X(2594)}-harmonic conjugate of X(5396)

X(5399) lies on these lines:
{1, 5}, {3, 947}, {42, 942}, {54, 59}, {55, 500}, {73, 517}, {386, 999}, {581, 3295}, {1048, 2607}, {1060, 3811}, {1870, 5174}, {3333, 5312}, {3579, 4303}, {4322, 5126}}


X(5400) = INTERSECTION OF LINES X(1)X(5) AND X(2)X(991)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos(A - B) + cos(A - C) - 3 cos(B - C)
X(5400) = (a + b)(a + c)(b + c)*X(1) - 8abc*X(5)    (Peter Moses, January 2, 2013)

X(5400) = trilinear pole, with respect to the excentral triangle, of the Brocard axis (Randy Hutson, Dec. 31, 2012)

X(5400) lies on these lines:
{1, 5}, {2, 991}, {4, 3216}, {42, 3817}, {43, 1699}, {118, 2999}, {165, 2108}, {200, 5014}, {244, 2801}, {386, 3091}, {500, 3628}, {516, 899}, {581, 3090}, {946, 3293}, {970, 3030}, {1054, 1768}, {1465, 1736}, {1724, 3149}, {1754, 4383}, {2635, 3911}, {2800, 4674}, {3214, 4301}, {3634, 4300}


X(5401) = SEC(A + π/5) POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A + π/5)

X(5401) lies on the Kiepert hyperbola and these lines:
{2, 3379}, {3, 3382}, {4, 3380}, {5, 3368}, {6, 3381}, {1139, 3395}, {1140, 3393}, {3370, 3394}, {3396, 3397}

X(5401) = isogonal conjugate of X(3393)


X(5402) = CSC(A + π/5) POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A + π/5)

X(5402) lies on the Kiepert hyperbola and these lines:
{2, 3380}, {3, 3381}, {4, 3379}, {5, 3368}, {6, 3382}, {1139, 3396}, {1140, 3394}, {3370, 3393}, {3395, 3397}

X(5402) = isogonal conjugate of X(3394)


X(5403) = SEC(A - ω/2) POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - ω/2)

X(5403) lies on the Kiepert hyperbola and these lines:
{2, 1670}, {3, 1676}, {4, 1671}, {5, 141}, {6, 1677}, {11, 3238}, {12, 3237}, {83, 1342}, {98, 1343}, {485, 1690}, {486, 1689}, {1348, 1664}, {1349, 1665}, {2009, 3102}, {2010, 3103}

X(5403) = isogonal conjugate of X(1342)


X(5404) = CSC(A - ω/2) POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - ω/2)

Let B* be a point such that (angle CBB*) = ω/2, and let C* be a point such that (angle CBC*) = ω/2. Let A' = (line BB*)∩(line CC*), and define B' and C' cyclically. The lines AA', BB', CC' concur in X(5404). (Randy Hutson, September 5, 2014)

X(5404) lies on the Kiepert hyperbola and these lines:
{2, 1671}, {3, 1677}, {4, 1670}, {5, 141}, {6, 1676}, {11, 3237}, {12, 3238}, {83, 1343}, {98, 1342}, {485, 1689}, {486, 1690}, {1348, 1665},{1349, 1664},{2009, 3103},{2010, 3102}

X(5404) = isogonal conjugate of X(1343)


X(5405) = CENTER OF THE PAACHE-MYAKISHEV-MOSES CONIC

Barycentrics   2 - cot(B/2) - cot(C/2) : 2 - cot(C/2) - cot(A/2) : 2 - cot(A/2) - cot B/2)
Barycentrics   a - 2r : b - 2r : c - 2r
X(5405) = s*X(1) - 3r*X(2)   (Peter Moses, January 2, 2013)

For the construction of this conic, see X(5393), where the associated Paache-Myakishev ellipse is introduced.

If you have The Geometer's Sketchpad, you can view X(5405), including the conic.

X(5405) lies on these lines: {1, 2}, {9, 3069}, {37, 615}, {57, 481}, {81, 3299}, {176, 5226}, {226, 482}, {491, 3879}, {492, 4357}, {590, 1100}, {591, 4643}, {641, 3666}, {940, 1124}, {946, 2048}, {1255, 3300}, {1335, 4383}, {1449, 3068}, {1586, 1785}, {1659, 5219}

X(5405) = {X(1),X(2)}-harmonic conjugate of X(5393)


X(5406) = 1st LUCAS POLAR PERSPECTOR

Trilinears        (cos A)(2 + cot A) : (cos B)(2 + cot B) : (cos C)(2 + cot C)

Let A'B'C' be the Lucas central triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5406), which is the {X(3), X(394)}-harmonic conjugate of X(5407). (Randy Hutson, February 9, 2013)

X(5406) lies on these lines: {3, 49}, {6, 588}


X(5407) = 2nd LUCAS POLAR PERSPECTOR

Trilinears        (cos A)(2 - cot A) : (cos B)(2 - cot B) : (cos C)(2 - cot C)

Let A'B'C' be the Lucas(-1) central triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5407), which is the {X(3), X(394)}-harmonic conjugate of X(5406). (Randy Hutson, February 9, 2013)

X(5407) lies on these lines: {3, 49}, {6, 589}


X(5408) = 3rd LUCAS POLAR PERSPECTOR

Trilinears        (cos A)(1 + cot A) : (cos B)(1 + cot B) : (cos C)(1 + cot C)

Let A'B'C' be the Lucas tangents triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5408). The same is true if A'B'C' is the Lucas(2) central triangle. X(5408) = {X(3), X(394)}-harmonic conjugate of X(5409). (Randy Hutson, February 9, 2013)

X(5408) lies on the conic {A, B, C, X(69), X(97)} and these lines: {3, 49}, {6, 493}

X(5408) = X(2)-Ceva conjugate of X(5409)


X(5409) = 4th LUCAS POLAR PERSPECTOR

Trilinears        (cos A)(1 - cot A) : (cos B)(1 - cot B) : (cos C)(1 - cot C)

Let A'B'C' be the Lucas(-1) tangents triangle. Let LA be the trilinear polar of A', and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5409). The same is true if A'B'C' is the Lucas(-2) central triangle. X(5409) = {X(3), X(394)}-harmonic conjugate of X(5408). (Randy Hutson, February 9, 2013)

X(5409) lies on the conic {A, B, C, X(69), X(97)} and these lines: {3, 49}, {6, 494}

X(5409) = X(2)-Ceva conjugate of X(5408)


X(5410) = 5th LUCAS POLAR PERSPECTOR

Trilinears        (sin A)(2 - tan A) : (sin B)(2 - tan B) : (sin C)(2 - tan C)

Let A' be the perspector of the A-Lucas circle, and define B' and C' cyclically. (The lines AA', BB', CC' concur in X(1151).) Let LA be the polar of A' with respect to the A-Lucas circle, and define LB and LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. The lines AA*, BB*, CC* concur in X(5410). X(5410) = {X(6), X(25)}-harmonic conjugate of X(5411). (Randy Hutson, February 10, 2013)

X(5410) lies on these lines: {6, 25}, {4, 1131}


X(5411) = 6th LUCAS POLAR PERSPECTOR

Trilinears        (sin A)(2 + tan A) : (sin B)(2 + tan B) : (sin C)(2 + tan C)

Let A' be the perspector of the A-Lucas(-1) circle, and define B' and C' cyclically. (The lines AA', BB', CC' concur in X(1152).) Let LA be the polar of A' with respect to the A-Lucas(-1) circle, and define LB and LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. The lines AA*, BB*, CC* concur in X(5411). X(5411) = {X(6), X(25)}-harmonic conjugate of X(5410). (Randy Hutson, February 10, 2013)

X(5411) lies on these lines: {6, 25}, {4, 1132}


X(5412) = 1st KENMOTU HOMOTHETIC CENTER

Trilinears        (sin A)(1 - tan A) : (sin B)(1 - tan B) : (sin C)(1 - tan C)

Let U, V, W be the congruent squares described at X(371). Let LA be the extended diagonal of U that does not contain X(371), and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is homothetic to the orthic triangle, and the center of homothety is X(5412). Also, A'B'C' is homothetic to the tangential triangle at X(6), to the intangents triangle at X(2066), and to the extangents triangle at X(5415). X(5412) = {X(6), X(25)}-harmonic conjugate of X(5413). (Randy Hutson, February 9, 2013)

X(5412) lies on the conic {A, B, C, X(4), X(24)} and the line {6, 25}

X(5412) = X(393)-Ceva conjugate of X(5413)
X(5412) = X(571)-cross conjugate of X(5413)


X(5413) = 2nd KENMOTU HOMOTHETIC CENTER

Trilinears        (sin A)(1 + tan A) : (sin B)(1 + tan B) : (sin C)(1 + tan C)

Let U', V', W' be the congruent squares as described at X(371), but with two vertices each on the extended sides of triangle ABC, and having common vertex X(372). Let LA be the extended diagonal of U' that does not contain X(372), and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The triangle A'B'C' is homothetic to the orthic triangle, and the center of homothety is X(5413). Also, A'B'C' is homothetic to the tangential triangle at X(6), to the intangents triangle at X(5414), and to the extangents triangle at X(5416). X(5413) = {X(6), X(25)}-harmonic conjugate of X(5412). (Randy Hutson, February 9, 2013)

X(5413) lies on the conic {A, B, C, X(4), X(24)} and the line {6, 25}

X(5413) = X(393)-Ceva conjugate of X(5412)
X(5413) = X(571)-cross conjugate of X(5412)


X(5414) = 3rd KENMOTU HOMOTHETIC CENTER

Trilinears        1 - sin A + cos A : 1 - sin B + cos B : 1 - sin C + cos C

The A'B'C' defined at X(5413) is homothetic to the intangents triangle, and the center of homothety is X(5414). Also, X(5414) = {X(6), X(55)}-harmonic conjugate of X(2066) and X(5414) = {X(3), X(1335)}-harmonic conjugate of X(2067). (Randy Hutson, February 9, 2013)

X(5414) lies on the conic {A, B, C, X(1), X(3)} and these lines {1, 372}, {3, 1335}, {6, 31}

X(5414) = X(9)-Ceva conjugate of X(2066)
X(5414) = cevapoint of X(48) and X(606)


X(5415) = 4th KENMOTU HOMOTHETIC CENTER

Trilinears        a(2R + r + s - a) : b(2R + r + s - b) : c(2R + r + s - c)   (César Lozada, April 7, 2013; Hyacinthos #21900)
Trilinears        (sin A)(2R sin A - 2R - r - s) : (sin B)(2R sin B - 2R - r - s) : (sin C)(2R sin C - 2R - r - s)   (César Lozada, April 7, 2013)

The A'B'C' defined at X(5412) is homothetic to the extangents triangle, and the center of homothety is X(5415). Also, X(5415) = {X(6), X(55)}-harmonic conjugate of X(5416). (Randy Hutson, February 9, 2013)

X(5415) lies on the line {6, 31}


X(5416) = 5th KENMOTU HOMOTHETIC CENTER

Trilinears        a(2R + r - s + a) : b(2R + r - s + b) : c(2R + r - s + c)   (César Lozada, April 7, 2013)
Trilinears        (sin A)(2R sin A + 2R + r - s) : (sin B)(2R sin B + 2R + r - s) : (sin C)(2R sin C + 2R + r - s)   (César Lozada, April 7, 2013)

The A'B'C' defined at X(5413) is homothetic to the extangents triangle, and the center of homothety is X(5416). Also, X(5416) = {X(6), X(55)}-harmonic conjugate of X(5415). (Randy Hutson, February 9, 2013)

X(5416) lies on the line {6, 31}


X(5417) = PERSPECTOR OF 1st KENMOTU CIRCLE

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A + sin A + 2 sin B sin C)

The 1st Kenmotu circle is defined at MathWorld. Let A' be the pole of line BC with respect to the 1st Kenmotu circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(5417). (Randy Hutson, February 9, 2013)

X(5417) lies on the conic {A, B, C, X(2), X(1173)} and the line {5419, 5421}

X(5417) = isogonal conjugate of X(5418)


X(5418) = ISOGONAL CONJUGATE OF X(5417)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + sin A + 2 sin B sin C

X(5418) = {X(6),X(140}-harmonic conjugate of X(5420). (Randy Hutson, February 9, 2013)

X(5418) lies on lines {2, 486}, {3, 485}, {6, 140}

X(5418) = isogonal conjugate of X(5417)


X(5419) = PERSPECTOR OF 2nd KENMOTU CIRCLE

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A - sin A + 2 sin B sin C)

The 2nd Kenmotu circle has center X(372) and passes through the six contact points of the congruent squares in the construction of the 2nd Kenmotu point. Let Let A' be the pole of line BC with respect to the 2nd Kenmotu circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(5419). (Randy Hutson, February 9, 2013)

X(5419) lies on the conic {A, B, C, X(2), X(1173)} and the line {5417, 5421}

X(5419) = isogonal conjugate of X(5420)


X(5420) = ISOGONAL CONJUGATE OF X(5419)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A - sin A + 2 sin B sin C

X(5420) = {X(6),X(140}-harmonic conjugate of X(5418). (Randy Hutson, February 9, 2013)

X(5420) lies on lines {2, 485}, {3, 486}, {6, 140}

X(5420) = isogonal conjugate of X(5418)


X(5421) = INTERSECTION OF LINES X(3)X(6) and X(5417)X(5419)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)(1 + 2 sin2B + 2 sin2C)

X(5421) is the center of the bicevian conic of X(61) and X(62) and lies on these lines:
{2, 1232}, {3, 6}, {51, 160}, {53, 1907}, {232, 428}, {1194, 3815}, {1879, 5254}, {2165, 5286}, {3055, 3291}, {5417, 5419}

X(5421) = complement of X(1232)
X(5421) = {X(6),X(39)}-harmonic conjugate of X(570)


X(5422) = INTERSECTION OF LINES X(493)X(589) AND X(494)X(588)

Barycentrics   a2 + 2R2 : b2 + 2R2 : c2 + 2R2

X(5422) is the point of intersection of the following pairs of lines:
(1) the line joining the center of the 1st Kenmotu circle and the perspector of the 2nd Kenmotu circle, these two points being X(371) and X(5419);
(2) the line joining the center of the 2nd Kenmotu circle and the perspector of the 1st, these being X(372) and X(5417). See X(5446).
Also, X(5422) = {X(2), X(6)}-harmonic conjugate of X(1993).   (Randy Hutson, April 8, 2013)

X(5422) lies on these lines:
{2, 6}, {3, 143}, {22, 51}, {23, 3796}, {24, 569}, {25, 5012}, {83, 5392}, {110, 5020}, {154, 3066}, {155, 1199}, {184, 575}, {195, 5070}, {324, 458}, {371, 1600}, {372, 1599}, {493, 589}, {494, 588}, {576, 3917}, {613, 3920}, {1181, 3091}, {1194, 5034}, {1351, 2979}, {1498, 3832}, {1583, 3312}, {1584, 3311}, {1853, 5169}, {1899, 5133}, {2003, 3306}, {2323, 3305}, {3083, 3301}, {3084, 3299}, {3148, 3398}, {3193, 5084}, {3819, 5097}, {3981, 5038}


X(5423) = ISOTOMIC CONJUGATE OF X(479)

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

X(5423) = isotomic conjugate of X(479)

X(5423) lies on these lines:
{2, 3677}, {7, 3263}, {8, 210}, {55, 1261}, {200, 346}, {280, 2057}, {329, 2835}, {345, 3699}, {756, 5296}, {1260, 4578}, {2325, 3158}, {2550, 3967}, {3021, 4387}, {3452, 4901}, {3474,4488},{3596, 4441}, {3705, 5328}


X(5424) = HATZIPOLAKIS-EULER-SCHIFFLER POINT

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a3 + 2b3 + c3 - 2a2b - 2ab2 - 2ac2 - 2bc2 - a2c - b2c - abc)(2a3 + b3 + 2c3 - 2a2c - 2ac2 - 2ab2 - 2b2c - a2b - bc2 - abc)
X(5424) = 6(r + R)*X(21) + (2r - R)*X(4867)
X(5424) = R*X(79) + 4(2r + R)*X(2646)   (Peter Moses, February 8, 2013)

Let I be the incenter and L the Euler line of triangle ABC. Let LA be the Euler line of IBC, and define LB and LC cyclically. (The four Euler lines concur in the Schiffler point, X(21).) Let OA be the circumcenter of IBC, and define OB and OC cyclically.

Continuing, let AB, AC be the orthogonal projections of OA on LB and LC, respecitively, and define BC, BA and CA, CB cyclically. Let A' be the circumcenter of OAABAC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(5424), and the circumcenter X(5428) of A'B'C' lies on L.   (Antreas Hatzipolakis, February 8, 2013)

A summary of Hyacinthos discussions of centers X(5424)-X(5429) is presented at Euler Lines, Circumcircles.

X(5424) lies on the Feuerbach hyperbola and these lines:
{21, 4867}, {79, 2646}, {80, 3584}, {758, 2320}, {1389, 3746}


X(5425) = ISOGONAL CONJUGATE OF X(5424)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(2a3 + 2b3 + c3 - 2a2b - 2ab2 - 2ac2 - 2bc2 - a2c - b2c - abc)(2a3 + b3 + 2c3 - 2a2c - 2ac2 - 2ab2 - 2b2c - a2b - bc2 - abc)]
X(5425) = (4r + 5R)*X(1) - 2r*X(3)
X(5425) = (2r + 3R)*X(21) + 2R*X(4084)   (Peter Moses, February 8, 2013)

X(5425) lies on these lines:
{1,3},{2,4867},{8,3841},{21,4084},{79,3671},{80,226},{81,759},{100,3919},{191,4018},{515,3982},{519,5249},{758,3219},{956,3894},{958,3901},{993,4880},{1001,3899},{1100,5341},{1203,3924},{1411,2003},{2802,3957},{3585,3649},{3636,5330},{3868,5258},{3869,5259},{3874,5288},{3881,4861},{3918,4420},{4067,5260}


X(5426) = HATZIPOLAKIS-EXCENTRAL PERSPECTOR

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a3 + b3 + c3 - a2b - a2c - 3ab2 - 3ac2 - 3b2c - 3bc2 - 3abc
X(5426) = X(1) + 2*X(21)   (Peter Moses, February 8, 2013)

The triangle A'B'C' of circumcenters at X(5424) is perspective to the excentral triangle, and the perspector is X(5426).   (Peter Moses, February 8, 2013)

X(5426) lies on these lines:
{1,21},{30,1699},{35,3753},{36,3742},{100,3968},{210,5251},{214,5284},{442,3586},{484,3919},{1125,2475},{1420,3649},{1698,1837},{2320,3065},{2646,5259},{3158,3679},{3219,4525},{3336,4189},{3337,5267},{3616,4299},{3636,3648},{3683,4867},{3746,3880},{3956,5260},{4316,5249},{4539,5302},{4677,4933}


X(5427) = HATZIPOLAKIS-INTOUCH PERSPECTOR

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)(a - b + c)(2a4 - 2a3b - 2a3c - 2a2b2 - 2a2c2 - 2a2bc + 2ab3 + 2ac3 + b3c + bc3 + 2b2c2)
X(5427) = R(r + 4R)*X(7) - r(4r + 7R)*X(21)   (Peter Moses, February 8, 2013)

The triangle A'B'C' of circumcenters at X(5424) is perspective to the intouch triangle, and the perspector is X(5427).   (Peter Moses, February 8, 2013)

X(5427) lies on these lines:
{7,21},{11,30},{12,5251},{100,5172},{191,1420},{392,3647},{758,1319},{1317,2078},{1411,1758},{2771,5126},{3651,5204},{4189,5221}


X(5428) = HATZIPOLAKIS-EULER CIRCUMCENTER

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a^6 - 2a5b - 2a5c - 4a4b2 - 4a4c2 + 4a3b3 + 4a3c3 + 2a2b4 + 2a2c4 + a2b3c + a2bc3 + 4a2b2c2 + 2ab4c + 2abc4 - 2ab5 - 2ac5 - b5c - bc5 + 2b3c3
X(5428) = 3R*X(2) + (4r + 3R)*X(3)   (Peter Moses, February 8, 2013)

X(5428) is the circumcenter of the triangle A'B'C' defined at X(5424). X(5428) lies on the Euler line of ABC.    (Antreas Hatzipolakis, February 8, 2013)

X(5428) lies on these lines:
{2,3},{36,3649},{191,3576},{214,960},{758,1385},{952,5258},{1837,5010},{3579,3754},{3650,5303}


X(5429) = HATZIPOLAKIS-BROCARD-EXCENTRAL PERSPECTOR

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a3 + b3 + c3 + 2a2b + 2a2c + 3abc
X(5429) = (r2 - 3s2)*X(1) + 4r(2r + 3R)*X(21)   (Peter Moses, February 8, 2013)

In the construction at X(5424), if L is taken to be the Brocard axis instead of the Euler line, then the resulting triangle A'B'C' of circumcenters is not perspective to ABC. However, it is perspective to the excentral triangle, and X(5429) is the perspector, and its center lies on the line L. The triangle is also perspective to the intouch, hexyl, Yff, and 1st and 2nd cirumperp triangles.    (Peter Moses, February 8, 2013)

For more, see Four Concurrent Lines, Circumcircles.

X(5429) lies on these lines:
{1,21},{36,199},{171,3753},{210,5247},{511,3576},{740,4234},{976,4661},{978,1453},{986,4252},{1104,3742},{1125,1330},{1193,4881},{1247,2363},{1757,4134},{1961,5251},{2308,4511},{2938,4221},{3454,3624},{3880,5255}


X(5430) = CENTER OF THE 1st GRIGORIEV CONIC

Barycentrics   (1 + csc A/2)(csc B/2 + csc C/2) - cot2A/2

Let LA be the line parallel to line BC and tangent to the circumcircle of triangle ABC on the negative side of BC (the region that does not contain A) , and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Let AB be the reflection of A in line A'B', and let AC be the reflection of A in line A'C'. Let C1 be the touchpoint of the incircle of triangle ABAB and line AB, and let B2 be the touchpoint of the incircle of ACAC and line AC. Define A1, B1 and C2, A2 cyclically. Then |AB2| = |CB1|, |BA1| = |CA2|, |AC1| = |BC2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic. A barycentric equation for this 1st Grigoriev conic follows:

x2 + y2 + z2 - 2 csc(A/2) yz - 2 csc(B/2) zx - 2 csc(C/2) xy = 0


(Communicated on behalf of Dmitry Grigoriev, Moscow, by Alexei Myakishev, March 28, 2013)

The perspector of the 1st Grigoriev conic is X(188).   (Randy Hutson, March 30, 2013)

If you have The Geometer's Sketchpad, you can view X(5430)

X(5430) lies on these lines: {8, 188}, {236, 3161}


X(5431) = CENTER OF THE 2nd GRIGORIEV CONIC

Barycentrics   (1 + sec A/2)(sec B/2 + sec C/2) - tan2A/2

Let LA be the line parallel to line BC and tangent to the circumcircle of triangle ABC on the positive side of BC (the region that includes A), and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. Let AB be the reflection of A in line A'B', and let AC be the reflection of A in line A'C'. Let C1 be the touchpoint of the incircle of triangle ABAB and line AB, and let B2 be the touchpoint of the incircle of ACAC and line AC. Define A1, B1 and C2, A2 cyclically. Then |AB2| = |CB1|, |BA1| = |CA2|, |AC1| = |BC2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic. A barycentric equation for this 2nd Grigoriev conic follows:

x2 + y2 + z2 - 2 sec(A/2) yz - 2 sec(B/2) zx - 2 sec(C/2) xy = 0


The perspector of the 2nd Grigoriev conic is X(5451).   (Randy Hutson, April 8, 2013)
See also X(5452).

If you have The Geometer's Sketchpad, you can view X(5431)

X(5431) lies on this line: {178, 5451}


X(5432) = INTERSECTION OF LINES X(2)X(11) AND X(3)X(12)

Barycentrics   (b + c - a)(b2 + c2 - 2a2 - 2bc)
X(5432) = R*X(1) + 3r*X(2) + r*X(3)   (Peter Moses, April 2, 2013)

The coefficients in the combo shown just above are (R, 3r, r); for comparison, the coefficients for X(5433), X(11), and X(12) are, respectively, (R, -3r, -r), (R, -3r, r), and (R, 3r, -r).

X(5432) lies on these lines:
{1,140},{2,11},{3,12},{4,3614},{5,35},{8,4999},{9,3255},{10,2646},{21,1329},{30,5010},{33,468},{36,495},{46,3649},{56,631},{141,2330},{165,1836},{171,2361},{212,750},{215,5012},{226,1155},{230,2276},{312,3712},{333,4023},{354,3911},{355,3612},{381,4302},{388,3523},{496,632},{499,3295},{550,3585},{551,5048},{615,2066},{620,3023},{846,2607},{908,4640},{950,3634},{999,5054},{1006,5172},{1040,5268},{1058,3533},{1125,3057},{1213,2268},{1399,3074},{1479,1656},{1697,3624},{1698,1837},{1852,5142},{1858,5044},{1914,3815},{2098,3616},{2320,3036},{3011,3752},{3056,3589},{3086,3303},{3090,4294},{3158,4863},{3452,3683},{3474,5226},{3475,4860},{3487,5221},{3522,5229},{3524,4293},{3576,5252},{3627,4324},{3689,4847},{3699,4126},{3705,4030},{3715,5273},{3813,3871},{3820,5251},{3967,3977},{4187,5248},{4255,5230},{4309,5070},{4414,4415},{4512,4679},{4870,5183},{5056,5225}


X(5433) = INTERSECTION OF LINES X(2)X(12) AND X(3)X(11)

Barycentrics   (a - b + c)(a + b - c)(b2 + c2 + 2a2 - 2bc)
X(5433) = R*X(1) - 3r*X(2) - r*X(3)   (Peter Moses, April 2, 2013)

The coefficients in the combo shown just above are (R, -3r, r); for comparison, the coefficients for X(5432), X(11), and X(12) are, respectively, (R, 3r, r), (R, -3r, r), and (R, 3r, -r).

X(5433) lies on these lines:
{1,140},{2,12},{3,11},{4,5204},{5,36},{8,1317},{10,1319},{21,3816},{34,468},{35,496},{55,631},{57,191},{65,392},{100,3813},{141,1428},{172,3815},{201,244},{230,2275},{238,1399},{348,1358},{381,4299},{395,2307},{404,2886},{405,1470},{474,3925},{495,632},{497,3523},{498,999},{550,3583},{551,4848},{602,5348},{603,748},{604,1213},{615,2067},{620,3027},{946,1155},{993,4187},{1038,5272},{1056,3533},{1210,2646},{1385,1737},{1420,1698},{1447,3665},{1454,3306},{1466,4423},{1469,3589},{1478,1656},{1770,5122},{1788,2099},{1837,3576},{2361,3075},{2477,5012},{2594,3216},{3085,3304},{3090,4293},{3295,4995},{3303,5218},{3361,5219},{3485,5221},{3487,4860},{3522,5225},{3524,4294},{3530,5010},{3627,4316},{3660,5044},{3671,4870},{3678,5083},{3820,5193},{3825,5267},{4301,5183},{4317,5070},{4881,5086},{5046,5303},{5056,5229}


X(5434) = INTERSECTION OF LINES X(1)X(30) AND X(2)X(12)

Barycentrics   (a - b + c)(a + b - c)(b2 + c2 + 2a2 + 2bc)
X(5433) = R*X(1) + r*X(2) - r*X(3)   (Peter Moses, April 2, 2013)

X(5434) lies on these lines:
{1, 30}, {2, 12}, {3, 4317}, {4, 3304}, {5, 3582}, {7, 528}, {8, 5221}, {11, 381}, {20, 3303}, {34, 428}, {36, 495}, {46, 3654}, {55, 376}, {57, 3679}, {65, 519}, {172, 5306}, {226, 535}, {354, 515}, {355, 3338}, {496, 3585}, {497, 3543}, {498, 5054}, {499, 3614}, {524, 1469}, {537, 4032}, {541, 3024}, {542, 3023}, {543, 3027}, {544, 1362}, {550, 3746}, {597, 1428}, {752, 1463}, {956, 3925}, {1388, 3485}, {1398, 5064}, {1420, 5290}, {1479, 3830}, {1657, 4309}, {1837, 3333}, {2098, 4295}, {2242, 5309}, {2475, 3813}, {2646, 4311}, {3057, 4292}, {3085, 3524}, {3086, 3545}, {3295, 3534}, {3339, 4677}, {3340, 4355}, {3421, 4413}, {3627, 4857}, {3748, 4304}, {3816, 5080}, {3828, 3911}, {3839, 5229}, {3849, 5194}, {3913, 4190}, {4669, 4848}

X(5434) = reflection of X(3058) in X(1)


X(5435) = 2(r + 2R)*X(1) - 3r*X(2) - 4r*X(3)

Barycentrics   (3a - b - c)(a - b + c)(a + b - c)
X(5435) = 2(r + 2R)*X(1) - 3r*X(2) - 4r*X(3)   (Peter Moses, April 3, 2013)

X(5435) lies on these lines:
{1,3523},{2,7},{3,938},{8,56},{10,3361},{11,3474},{20,1210},{21,1466},{43,1458},{46,962},{65,3616},{77,2999},{78,1467},{88,278},{100,1617},{140,3487},{145,1420},{165,390},{171,1471},{175,5405},{176,5393},{190,1997},{208,4200},{223,1443},{241,2275},{279,3008},{333,1014},{354,5218},{376,5122},{452,4652},{479,658},{497,1155},{498,3337},{499,3336},{516,5274},{517,4345},{604,3684},{614,4318},{631,942},{651,1407},{673,2898},{950,3522},{978,1042},{1000,3654},{1038,5262},{1058,3579},{1106,5247},{1125,3339},{1214,4850},{1319,3241},{1442,5256},{1532,2096},{1698,4298},{1707,5121},{1722,4320},{1737,4293},{1999,4460},{2099,5298},{2263,5272},{2295,5228},{3052,3756},{3085,3338},{3091,4292},{3210,4552},{3216,4306},{3340,3622},{3475,4860},{3485,5221},{3486,5204},{3624,3671},{3634,5290},{3660,3873},{3679,4315},{3817,4312},{3916,5084},{3947,4355},{4032,4699},{4190,5175},{4302,5131},{4327,5268},{4430,5083}

X(5435) = {X(8),X(56)}-harmonic conjugate of X(4308)


X(5436) = (r + 2R)*X(1) + 6R*X(2) + 2r*X(3)

Barycentrics   a(3a3 + b3 + c3 - a2b - a2c - 3ab2 -3ac2 - 6abc -5b2c - 5bc2)
X(5436) = (r + 2R)*X(1) + 6R*X(2) + 2r*X(3)   (Peter Moses, April 3, 2013)

X(5436) lies on these lines:
{1,6},{2,950},{4,1125},{10,3158},{20,142},{21,57},{34,4183},{40,1006},{55,1706},{65,4512},{78,5047},{84,3560},{165,3812},{200,3983},{226,452},{329,3622},{382,3824},{442,3586},{443,4304},{551,3487},{936,2900},{942,3928},{943,3680},{968,3924},{993,3333},{1005,5253},{1043,4384},{1260,3303},{1385,1490},{1451,2328},{1621,1697},{1698,3419},{1708,3340},{2136,3295},{2478,5219},{2550,4314},{2646,4423},{2975,4666},{3306,4189},{3339,4640},{3361,3742},{3452,5129},{3811,4015},{3868,3929},{3870,5260},{4678,4917}


X(5437) = (r + 2R)*X(1) + 6R*X(2) - 2r*X(3)

Barycentrics   a(b2 + c2 - a2 - 6bc)
X(5437) = (r + 2R)*X(1) + 6R*X(2) - 2r*X(3)   (Peter Moses, April 3, 2013)

X(5437) lies on these lines:
{1,474},{2,7},{5,84},{10,1056},{40,631},{46,3624},{85,738},{88,4606},{100,4666},{165,1001},{171,5272},{200,354},{210,4860},{236,258},{244,612},{281,1435},{312,4659},{392,2093},{404,3601},{443,1210},{549,3587},{614,750},{936,942},{940,1449},{958,3361},{960,3339},{982,5268},{1155,4423},{1329,5290},{1420,5253},{1656,3824},{1697,3616},{1698,3338},{1699,3816},{1709,3838},{2098,3922},{2551,4298},{3208,5308},{3220,5020},{3247,3666},{3304,3698},{3576,3833},{3740,5223},{3763,5227},{3772,4859},{3925,5231},{3980,4871},{4035,4869},{4292,5084},{4415,4862},{4652,5047},{4850,5287},{5128,5250}


X(5438) = (r + 2R)*X(1) - 6R*X(2) + 2r*X(3)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c)2(b + c - a) - 4a(b2 + c2 - a2)
Barycentrics   a(3a3 + b3 + c3 - a2b - a2c - 3ab2 - 3ac2 + 2abc + 3b2c + 3bc2)
X(5438) = (r + 2R)*X(1) - 6R*X(2) + 2r*X(3)   (Peter Moses, April 3, 2013)

X(5438) lies on these lines:
{1,474},{2,950},{3,9},{8,1420},{10,631},{20,3452},{40,997},{56,200},{57,78},{63,4188},{72,3928},{80,1698},{100,1697},{165,960},{210,5204},{377,5219},{386,1449},{452,5316},{480,4321},{518,3361},{908,4190},{975,3247},{976,3677},{988,5293},{1058,1125},{1193,5269},{1260,1466},{1319,4853},{1453,3216},{1496,3939},{1743,4252},{2270,3430},{2551,4297},{2646,4413},{2886,3624},{3146,5328},{3218,3984},{3243,3333},{3304,3689},{3305,4189},{3340,4511},{3421,4311},{3586,4187},{3617,4881},{3623,4917},{3740,5234},{3869,5128},{3870,5253},{3876,3929},{3927,5122},{4304,5084},{4512,5217},{5096,5227}


X(5439) = (r + 2R)*X(1) + 3R*X(2) - r*X(3)

Barycentrics   a(b3 + c3 - a2b - a2c - 4abc - 3b2c - 3bc2)
X(5439) = (r + 2R)*X(1) + 3R*X(2) - r*X(3)   (Peter Moses, April 3, 2013)

X(5439) lies on these lines:
{1,474},{2,72},{3,3306},{5,1071},{7,5084},{8,4002},{10,354},{37,3670},{46,1001},{57,405},{65,392},{142,442},{145,5049},{210,3634},{226,4187},{388,3660},{406,1876},{443,938},{517,631},{518,1698},{519,3698},{551,3057},{750,5266},{912,1656},{956,3333},{958,3338},{960,3624},{971,3091},{1155,5248},{1214,1393},{1279,5264},{1385,5253},{1426,5136},{1621,3579},{1788,5173},{2476,3824},{3218,5047},{3244,3918},{3295,4666},{3305,3927},{3336,4640},{3337,5251},{3617,3889},{3625,3968},{3626,3892},{3635,3893},{3636,3922},{3720,3931},{3740,4533},{3811,4413},{3828,3983},{3873,3921},{3884,3919},{3894,4539},{3897,5126},{4189,5122},{4359,5295},{4423,5221}


X(5440) = (r + 2R)*X(1) - 3R*X(2) + r*X(3)

Barycentrics   a(b + c - 2a)(b2 + c2 - a2)
X(5440) = (r + 2R)*X(1) - 3R*X(2) + r*X(3)   (Peter Moses, April 3, 2013)

X(5440) lies on these lines:
{1,474},{2,3419},{3,63},{8,631},{10,2646},{21,5044},{30,908},{35,960},{36,518},{44,2251},{46,4018},{48,3694},{55,392},{56,3555},{80,5123},{100,517},{101,2751},{109,2756},{200,956},{210,993},{214,519},{329,376},{404,942},{405,936},{484,4867},{521,656},{551,3748},{572,3965},{758,1155},{950,4187},{958,3612},{995,3744},{999,3870},{1055,3930},{1104,3216},{1125,3925},{1149,3722},{1193,5266},{1375,3912},{1386,5313},{1437,1792},{1455,4551},{1737,3035},{2077,2932},{2551,4305},{2802,5048},{2975,4420},{3086,3189},{3090,5175},{3218,5122},{3452,4304},{3579,3869},{3583,5087},{3616,5082},{3617,3897},{3666,4256},{3678,5267},{3740,5251},{3868,4188},{3876,4189},{3935,4881},{3957,5049},{4257,4641},{4313,5084},{4421,5119},{4539,5220},{4640,5010},{4662,5258},{4694,4864},{4702,4975},{4880,5131},{5045,5253}

X(5440) = crossdifference of every pair of points on the line X(19)X(4394)


X(5441) = (2r + 3R)*X(1) - 6r*X(2) + 6r*X(3)

Barycentrics   b4 + c4 - 3a4 + a3b + a3c + 2a2b2 + 2a2c2 + a2bc - ab3 - ac3 + ab2c + abc2 - 2b2c2
X(5441) = (2r + 3R)*X(1) - 6r*X(2) + 6r*X(3)   (Peter Moses, April 3, 2013)

X(5441) lies on these lines:
{1, 30}, {8, 3647}, {10, 21}, {36, 950}, {65, 4324}, {145, 758}, {191, 2136}, {354, 4325}, {442, 3586}, {515, 3746}, {517, 4330}, {548, 5131}, {550, 3336}, {942, 4316}, {944, 4309}, {952, 3065}, {1385, 4857}, {1478, 4313}, {1479, 2475}, {1837, 5010}, {2646, 3583}, {2771, 3057}, {3486, 4302}, {3488, 4299}, {3534, 5221}, {3633, 3650}


X(5442) = (2r + 3R)*X(1) - 6r*X(2) - 6r*X(3)

Barycentrics   a4 + b4 + c4 + a3b + a3c - 4a2b2 - 4a2c2 - ab3 - ac3 + ab2c + abc2 - 2b2c2
X(5442) = (2r + 3R)*X(1) - 6r*X(2) - 6r*X(3)   (Peter Moses, April 3, 2013)

X(5442) lies on these lines:
{1, 549}, {2, 79}, {5, 5131}, {10, 36}, {35, 3911}, {46, 3624}, {140, 3336}, {1145, 3632}, {3579, 3582}, {3585, 5122}, {3616, 3884}, {5054, 5221}


X(5443) = (2r + R)*X(1) + 6r*X(2) - 2r*X(3)

Barycentrics   a4 + b4 + c4 - a3b - a3c - 2a2b2 - 2a2c2 + a2bc + ab3 + ac3 - ab2c - abc2 - 2b2c2
X(5443) = (2r + R)*X(1) + 6r*X(2) - 2r*X(3)   (Peter Moses, April 3, 2013)

X(5443) is the QA-P7 center of quadrangle ABCX(1); see QA-Nine-point Center Homothetic Center.

X(5443) lies on these lines:
{1, 5}, {2, 3754}, {21, 36}, {35, 946}, {46, 3624}, {140, 484}, {191, 4999}, {214, 2475}, {451, 1845}, {499, 3485}, {908, 5258}, {942, 3582}, {1319, 5270}, {1385, 3585}, {1478, 3616}, {1479, 4313}, {1656, 2099}, {1699, 3612}, {1749, 3337}, {2646, 3583}, {3057, 3584}

X(5443) = {X(1),X(5)}-harmonic conjugate of X(80)


X(5444) = (2r + R)*X(1) + 6r*X(2) + 2r*X(3)

Barycentrics   3a4 + b4 + c4 - a3b - a3c - 4a2b2 - 4a2c2 + a2bc + ab3 + ac3 - ab2c - abc2 - 2b2c2
X(5444) = (2r + R)*X(1) + 6r*X(2) + 2r*X(3)   (Peter Moses, April 3, 2013)

X(5444) lies on these lines:
{1, 140}, {2, 80}, {35, 404}, {36, 226}, {442, 3586}, {484, 549}, {498, 3476}, {499, 3488}, {952, 5326}, {1319, 3584}, {1387, 4995}, {2099, 5054}, {3616, 3754}, {3653, 5252}, {3822, 4881}, {4870, 5122}


X(5445) = (2r + R)*X(1) - 6r*X(2) - 2r*X(3)

Barycentrics   a4 + b4 + c4 + a3b + a3c - 2a2b2 - 2a2c2 + a2bc - ab3 - ac3 + ab2c + abc2 - 2b2c2
X(5445) = (2r + R)*X(1) - 6r*X(2) - 2r*X(3)   (Peter Moses, April 3, 2013)

X(5445) lies on these lines:
{1, 140}, {2, 3754}, {3, 80}, {5, 484}, {8, 214}, {9, 46}, {10, 36}, {12, 3336}, {35, 950}, {201, 1772}, {495, 3337}, {498, 1788}, {942, 3584}, {946, 3245}, {1125, 5330}, {1155, 3585}, {1210, 3746}, {1837, 5010}, {2099, 3526}, {3057, 3582}, {3579, 3583}, {3828, 4292}, {3916, 5123}, {4325, 5122}


X(5446) =  INTERSECTION OF LINES X(371)X(5417) AND X(372)X(5419)

Trilinears        2R2cos A - a2cos(B - C) : 2R2cos B - b2cos(C - A) : 2R2cos C - c2cos(A - B)   (Randy Hutson, April 2013)
Trilinears        cos(2A) cos(B - C) - 2 cos B cos C : cos(2B) cos(C - A) - 2 cos C cos A : cos(2C) cos(A - B) - 2 cos A cos B   (César Lozada, April 10, 2013; Hyacinthos #21922)

Continuing the discussion at X(5422), the point X(5446) lies on the following pairs of lines:
(1) the line joining the center of the 1st Kenmotu circle and its perspector, these two points being X(371) and X(5417);
(2) the line joining the center of the 2nd Kenmotu circle and its perspector, these two points being X(372) and X(5419).
Also, X(5446) is the complement of X(3) with respect to the orthic triangle.   (Randy Hutson, April 8, 2013)

X(5446) = midpoint of X(4) and X(52)
X(5446) = reflection of X(i) in X(j) for these (i,j): (389, 143), (1216,5)
X(5446) = anticomplement of X(5447)

X(5446) lies on these lines:
{2, 5447},{3, 51}, {4, 52}, {5, 1216}, {22, 569}, {23, 54}, {25,1147}, {143,389}, {155,1351},{371, 5417}, {372, 5419}


X(5447) =  COMPLEMENT OF X(5446)

Trilinears        (cos A)(3 - cos (2B) - cos(2C)) : (cos B)(3 - cos (2C) - cos(2A)) : (cos C)(3 - cos (2A) - cos(2B))   (César Lozada, April 10, 2013; Hyacinthos #21922)

Let A'B'C' be the tangential triangle of triangle ABC. Let OA be the circle with center A' that is tangent to line BC. Define OB and OC cyclically. Then X(5447) is the radical center of the three circles.   (Randy Hutson, April 8, 2013)

X(5447) = complement of X(5446)
X(5447) = midpoint of X(3) and X(1216)

X(5447) lies on these lines: {2, 5446}, {3, 49}, {5,3819}, {51,3526}, {52,631}, {140,143}


X(5448) =  1st HATZIPOLAKIS-MOSES POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),where f(A,B,C) = (cos A)(2 + 2 cos(2B) + 2 cos(2C) + cos(2B - 2C))   (César Lozada, April 15, 2013; Hyacinthos #21954)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b2 + c2 - a2)(b8 + c8 + 2a6b2 + 2a6c2 - 3a4b4 - 3a4c4 + 4a2b2c2 - 4b6c2 - 4b2c6 + 6b4c4)
X(5448) = X(3) + 2X(4) + X(155) = 3X(2) + 2X(3) - X(68)

Let A'B'C' be the pedal triangle of the orthocenter, X(4), and let A"B"C" be the circumcevian triangle of X(4) with respect to A'B'C'. Let RA be the radical axis of the circles (B", |B'C"|) and (C",|C'B"|), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5448). For figures, see Concurrent Radical Axes.   (Antreas Hatzipolakis, April 10, 2013)

X(5448) = midpoint of X(4) and X(1147)

X(5448) lies on these lines:
{3,1568},{4,110},{5,389},{52,403},{68,1173},{155,195},{185,2072},{541,3357},{1533,5073},{1614,3153},{3167,3843},{3546,4846},{3564,3850}


X(5449) =  2nd HATZIPOLAKIS-MOSES POINT

Trilinears        cos A cos(2B - 2C) : cos B cos(2C - 2A) : cos C cos(2A - 2B)   (César Lozada, April 14, 2013; Hyacinthos #21951)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)(b8 + c8 + a4b4 + a4c4 - 2a2b6 - 2a2c6 + 2a2b4c2 + 2a2b2c4 - 4b6c2 - 4b2c6 + 6b4c4)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = SA[a4(S2 - SA2) - 8S2SBSC]
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = b4SB(S2 - SB2) + c4SC(S2 - SC2)
Barycentrics   sin 4B + sin 4C : sin 4C + sin 4A : sin 4A + sin 4B    (Randy Hutson, August 26, 2014)
X(5449) = 3X(2) + X(68) = 3X(2) - X(1147)

Let A'B'C' be the pedal triangle of the circumcenter, X(3), and let A"B"C" be the circumcevian triangle of X(3) with respect to A'B'C'. Let RA be the radical axis of the circles (B', |B'C"|) and (C',|C'B"|), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5449). The midpoint of X(5448) and X(5449) is X(5). For figures, see Concurrent Radical Axes.   (Antreas Hatzipolakis, April 10, 2013)

Let D = X(68); then X(5449) is the centroid of ABCD. (Randy Hutson, August 25, 2014)

X(5449) = midpoint of X(68) and X(1147) X(5449) = complement of X(1147)

X(5449) lies on these lines:
{2,54},{3,125},{5,389},{52,1594},{136,847},{155,1656},{156,542},{343,1216},{568,3574},{575,3564},{912,3812},{1614,3448},{1899,3549},{3167,5070}


X(5450) =  3rd HATZIPOLAKIS-MOSES POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a6 - a5b - a5c - 2a4b2 - 2a4c2 + 4a4bc + 2a3b3 + 2a3c3 - a3b2c - a3bc2 + a2b4 + a2c4 - 3a2b3c - 3a2bc3 + 4a2b2c2 + 2ab4c + 2abc4 - ab3c2 - ab2c3 - ab5 - ac5 - b5c - bc5 + 2b3c3)
X(5450) = R*X(1) + (2r - R)X(104)
X(5450) = (r - R)*X(3) + R*X(10)
X(5450) = (2r + 3R)*X(21) + R*X(84)
X(5450) = 4r*X(3) + R*X(8) - R*X(20)

Let A'B'C' be the circumcevian triangle of X(1). Let RA be the radical axis of the circles (B, |BC'|) and (C,|CB'|), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5450). For figures, see Concurrent Radical Axes. See also X(1147).    (Antreas Hatzipolakis, April 10, 2013)

X(5450) = midpoint of X(1) and X(1158)

X(5450) lies on these lines:
{1,104},{3,10},{4,36},{5,2829},{8,2077},{21,84},{30,3829},{35,944},{40,2975},{48,1765},{56,946},{318,1309},{411,5303},{631,5251},{995,3073},{999,3671},{1006,1490},{1071,2646},{1125,3560},{1210,1470},{1385,5248},{1457,1777},{1482,4084},{2096,3485},{3072,4257},{3149,5204},{4231,5345}


X(5451) =  PERSPECTOR OF 2nd GRIGORIEV CONIC

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sec(B/2) sec(C/2) + sec(A/2))    (Paul Yiu, Hacinthos #21935, April 12, 2013)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/(cos(B/2) cos(C/2) + cos(A/2))    (Paul Yiu, Hacinthos #21935, April 12, 2013)

The 1st and 2nd Grigoriev conics are presented at X(5430) and X(5431). Their perspectors are X(188) and X(5451), respectively,    (Randy Hutson, April 2013)

X(5451) lies on this line: {178, 5431}


X(5452) = CENTER OF THE PRIVALOV CONIC

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos2(A/2)[-sin A cos2(A/2) + sin B cos2(B/2) + sin C cos2(C/2)]   (Randy Hutson, April 19, 2013)

Barycentrics   a2(b + c - a)[a3 - a2b - a2c + ab2 + ac2 - (b + c)(b - c)2]

Let A'B'C' be the anticomplementary triangle of triangle ABC. Let A'' be the reflection of A' in the perpendicular bisector of segtment BC, and define B'' and C'' cyclically. Let A1 be the touchpoint of the incircle of A''BC and line BC, and let A2 be the touchpoint of the incircle of A'BC and line BC. Define the points B1, B2, C1, C2 cyclically. Then |AC2| = |BC1|, |BA1| = |CA2|, |CB1| = |AB2|, so that by Carnot's theorem, the six points A1, A2, B1, B2, C1, C2 lie on a conic, named in honor of Alexander Privalov. A barycentric equation for the Privalov conic follows:

x2 + y2 + z2 + f(a,b,c)yz + f(b,c,a)zx + f(c,a,b)xy = 0, where f(a,b,c) = 2[(b - c)2 + a2]/[(b - c)2 - a2],


or, equivalently, by

x2 + y2 + z2 - g(A,B,C)yz - g(B,C,A)zx - g(C,A,B)xy = 0, where g(A,B,C) = tan(B/2) tan(C/2) [cot2(B/2) + cot2(C/2)].


(Communicated by Dmitry Grigoriev, April 15, 2013.)

The Privalov conic is the bicevian conic of X(7) and X(8) - that is, the conic through the vertices of the intouch and extouch triangles. Its center X(5452) is also the center of the conic through A, B, C, X(101), X(294), X(651), X(666), which is the isogonal conjugate of the Gergonne line. This circumconic is the locus of trilinear poles of lines passing through X(55). Also, X(5452) = crossdifference of every pair of points on the polar of X(6) with respect to the incircle. See also X(5545).    (Randy Hutson, April 19, 2013)

If you have The Geometer's Sketchpad, you can view X(5452)

X(5452) lies on these lines:
{2, 1814}, {6, 354}, {9, 1040}, {33, 210}, {55, 2195}, {218, 226}, {219, 3686}, {294, 497}, {650, 1376}, {2238, 2911}

X(5452) = X(2)-Ceva conjugate of X(55)


X(5453) =  CENTER OF HATZIPOLAKIS CIRCLE

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 + 2 cos A + cos(B - C) + 4 sin(3A/2) cos(B/2 - C/2)   (César Lozada, April 17, 2013)

Let A'B'C' be the cevian triangle of I (the incenter, X(1)). Let NA be the nine-point center of triangle IB'C', and define NB and NC cyclically. The points I, NA, NB, NC are concyclic, and their circle, described by Antreas Hatzipolakis, April 17, 2013.

X(5453) lies on these lines:
{1,30}, {3,81}, {5,581}, {21,323}, {58,5428}, {140,3216}, {186,2906}, {386,549}, {511,1385}, {550,991}, {1154,2646}, {2771,3743}

X(5453) = midpoint of X(1) and X(500)


X(5454) =  1st MORLEY-KIRIKAMI POINT

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((-f(2A/3) + f(4A/3) + f(2B/3) - f(4B/3))(h(C/3 + π/6)(2f(C/3)g(A/3) + g(B/3)) + (2 + f(C/3)g(A/3)g(B/3))g(A/3 + 2π/3) + (f(2A/3) - f(4A/3) - f(2C/3) + f(4C)/3)(h(B/3 + π/6)(2f(B/3)g(A/3) + g(C/3)) + (2 + f(B/3)g(A/3)g(C/3))g(A/3 + 2π/3)), where f = cos, g = sec, h = csc    (Peter Moses, April 26, 2013)

Let DEF be the 1st Morley triangle of triangle ABC. The Newton lines of the quadrilaterals AEDF, BFED, CDFE concur in X(5454).   (Seiichi Kirikami, April 26, 2013)

If you have The Geometer's Sketchpad, you can view X(5454).

X(5454) lies on this line: {356, 1134}


X(5455) =  2nd MORLEY-KIRIKAMI POINT

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (4 sin A)g(A,B,C) + (sin B)g(B,C,A) + (sin C)g(C,A,B), where g(A,B,C) = cos(A/3) + 2 cos(B/3) cos(C/3)    (Peter Moses, April 26, 2013)

Let DEF be the 1st Morley triangle of triangle ABC. Let LA be the line of the centroid of AEF and the centroid of BCD, and define LB and LC cyclically. The lines LA, LB, LC concur in X(5455).   (Seiichi Kirikami, April 26, 2013)

If you have The Geometer's Sketchpad, you can view X(5455).

X(5455) lies on this line: {2, 356}


X(5456) =  3rd MORLEY-KIRIKAMI POINT

Barycentrics   sin(2A/3) : sin(2B/3) : sin(2C/3)    (Peter Moses, May 14, 2013)

Let DEF be the 1st Morley triangle of triangle ABC, and let D1E1F1 be the 1st Morley adjunct triangle (defined at MathWorld). Let D2 = DD1∩BC, and define E2 and F2 cyclically. The lines AD2, BE2, CF2 concur in X(5456).    (Seiichi Kirikami, April 26, 2013)

Let DEF be the 1st Morley triangle. Let D' be the trilinear pole of line EF, and define E', F' cyclically. Let D" be the trilinear pole of line E'F', and define E", F" cyclically. The lines AD", BE", CF" concur in X(5456). (Randy Hutson, September 29, 2014)

If you have The Geometer's Sketchpad, you can view X(5456).

X(5456) lies on this line: {356, 3605}


X(5457) =  4th MORLEY-KIRIKAMI POINT

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A + cos(A/3))    (Peter Moses, April 26, 2013)
Trilinears   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A/3)/sin(4A/3)    (Randy Hutson, August 9, 2014)

Let DEF be the 1st Morley triangle of triangle ABC, and let D3 be the reflection of D in line BC, and define E3 and F3 cyclically. The lines AD3, BE3, CF3 concur in X(5457).    (Seiichi Kirikami, April 26, 2013)

X(5457) is the Hofstadter -1/3 point; see X(359). (Randy Hutson, August 0, 2014)

If you have The Geometer's Sketchpad, you can view X(5457) and X(5458).

X(5457) lies on these lines: (pending)

X(5457) = isogonal conjugate of X(6123)


X(5458) =  5th MORLEY-KIRIKAMI POINT

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(4 cos A + sec(A/3))    (Peter Moses, April 26, 2013)
Trilinears   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A/3)/sin(5A/3)    (Randy Hutson, August 9, 2014)

Let DEF be the 1st Morley triangle of triangle ABC, and let D1E1F1 be the 1st Morley adjunct triangle (defined at MathWorld). Let D4 = be the reflection of D1 in line BC, and define E4 and F4 cyclically. The lines AD4, BE4, CF4 concur in X(5458).    (Seiichi Kirikami, April 26, 2013)

X(5458) is the Hofstadter -2/3 point; see X(359). (Randy Hutson, August 0, 2014)

X(5458) lies on these lines: (pending)


X(5459) =  MIDPOINT OF X(2) AND X(13)

Trilinears   f(a,b,c,A,B,C) : f(b,c,a,B,C,A) :f(c,a,b,C,A,B), where f(a,b,c,A,B,C) = 4abc csc(A + π/3) + b2c csc(B + π/3) + bc2csc(C + π/3)    (Randy Hutson, April 25, 2013)

X(5459) is the center of the equilateral triangle formed by the nine-point centers of the triangles BCF, CAF, ABF, where F is the Fermat point, X(13). Also, X(5459) is the center of the circle through X(2), X(13), and the previously mentioned nine-point centers, and X(5459) = {X(5),X(597)}-harmonic conjugate of X(5460).    (Randy Hutson, April 25, 2013)

X(5459) lies on these lines: {2, 13}, {5, 542}, {17, 671}, {115, 396}, {524, 623}, {543,619}, {599,635}, {630, 2482}

X(5459) = complement of X(5463)


X(5460) =  MIDPOINT OF X(2) AND X(14)

Trilinears   f(a,b,c,A,B,C) : f(b,c,a,B,C,A) :f(c,a,b,C,A,B), where f(a,b,c,A,B,C) = 4abc csc(A - π/3) + b2c csc(B - π/3) + bc2csc(C - π/3)]    (Randy Hutson, April 25, 2013)

X(5460) is the center of the equilateral triangle formed by the nine-point centers of the triangles BCF', CAF', ABF', where F' = X(14). Also, X(5460) is the center of the circle through X(2), X(14), and the previously mentioned nine-point centers.   (Randy Hutson, April 25, 2013)

X(5460) lies on these lines: {2, 14}, {5, 542}, {18,671}, {115, 395}, {125, 5465}, {524,624}, {543,618}, {599,636}, {629,2482}

X(5460) = complement of X(5464)


X(5461) =  MIDPOINT OF X(5459) AND X(5460)

Trilinears   f(a,b,c) : f(b,c,a) :f(c,a,b), where f(a,b,c) = bc[4(b2 - c2)2 + (a2 - b2)2 + (a2 - c2 )2]    (Randy Hutson, April 25, 2013)

X(5461) is the center of the rectangle having vertices X(2), X(115), X(125), and X(5465). (Paul Yiu, Anopolis #170, April 25, 2013)

X(5461) lies on these lines: {2, 99}, {5, 542}, {98, 3545}, {114, 5055}, {125, 5465}, {230, 3849}, {381, 2794}, {524, 625}, {538, 2023}, {547, 2782}, {599, 626}, {1153, 3054}, {1992, 3767}, {1995, 3455}, {2796, 3634}

X(5461) = midpoint of X(I) and X(J) for these (I,J): (2,115), (125,5465)
X(5461) = complement of X(2482)


X(5462) =  INTERSECTION OF LINES X(2)X(52) AND X(3)X(51)

Trilinears   2R2cos A + a2cos(B - C) : 2R2cos B + b2cos(C - A) : 2R2cos C + a2cos(A - B)

X(5462) = (X(I),X(J))-harmonic conjugate of X(K) for these (I,J,K): (2,52,1216), (3,51,5446), (24,5422,569).   (Randy Hutson, April 24, 2013)

X(5462) lies on these lines: {2, 52}, {3, 51}, {4, 4846}, {5, 389}, {6, 1147}, {24, 569}, {140, 143}, {155, 5020}, {185, 381}, {195, 3292}

X(5462) = midpoint of X(I) and X(J) for these (I,J): (5,389), (140,143)
X(5462) = complement of X(1216)


X(5463) =  REFLECTION OF X(13) IN X(2)

Barycentrics    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A sec(A - π/6) - 2 sin B sec(B - π/6) - 2 sin C sec(C - π/6)    (Peter Moses, May 3, 2013)

Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463)}; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014). X(5463) is the center of the equilateral antipedal triangle of X(13), and X(5463) = (X(3), X(599))-harmonic conjugate of X(5464).    (Randy Hutson, May 2, 2013)

The circle {{X(2), X(110), X(5463), X(5464)}} has center X(1649) and passes through X(2770). It is tangent to the Euler line at X(2) and is the reflection in the Euler line of the circle {{X(2), X(13), X(14), X(111), X(476)}}.    (Randy Hutson, August 26, 2014)

X(5463) = Thomson-isogonal-conjugate-of-X(15) (Peter Moses, August 16, 2014)

X(5463) lies on these lines: {2, 13}, {3, 67}, {14, 543}, {15, 524}, {18, 671}, {61, 1992}, {62, 597}, {99, 298}

X(5463) = reflection of X(5464) in X(2482)
X(5463) = anticomplement of X(5459)


X(5464) =  REFLECTION OF X(14) IN X(2)

Barycentrics    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A sec(A + π/6) - 2 sin B sec(B + π/6) - 2 sin C sec(C + π/6)    (Peter Moses, May 3, 2013)

Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463):; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014).

X(5464) is the center of the equilateral antipedal triangle of X(14), and X(5464) = (X(3), X(599))-harmonic conjugate of X(5463).    (Randy Hutson, May 2, 2013)

X(5464) = Thomson-isogonal-conjugate-of-X(16) (Peter Moses, August 16, 2014)

X(5464) lies on these lines: {2, 14}, {3, 67}, {13, 543}, {16, 524}, {17, 671}, {61, 597}, {62, 1992}, {99, 299}

X(5464) = reflection of X(5463) in X(2482)
X(5464) = anticomplement of X(5460)


X(5465) =  ORTHOGONAL PROJECTION OF X(2) ON THE FERMAT AXIS

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a10 - 4a8(b2 + c2) + 4a6(2b4 + 2c4 - b2c2) - a4(b2 + c2)(7b4 + 7c4 - 10b2c2) - a2(b8 + c8 - 14b6c2 - 14b2c6 + 24b4c4) + (b4 - c4)2(b2 + c2)(b2 - 2c2)(2b2 - c2)    (Paul Yiu, Anopolis #170, April 25, 2013)

X(5465) is the point, other than X(2), of intersection of the circle defined at X(5459) and the circle defined at X(5460).    (Randy Hutson, May 2, 2013)

X(5465) is the fourth vertex of a rectangle determined by three vertices X(2), X(115), and X(125); the center of this rectangle is X(5461). (Paul Yiu, Anopolis #170, April 25, 2013)

X(5465) lies on these lines: {2, 690}, {6, 13}, {110, 671}, {125, 5461}, {543, 1316}, {2780, 3111}

X(5465) = midpoint of X(110) and X(671)
X(5465) = reflection of X(125) and X(5461)


X(5466) =  TRILINEAR POLE OF LINE X(115)X(523)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 - c2)/(b2 + c2 - 2a2)

Let L be the line tangent at X(13) to the circle defined at X(5459), and let L' be the line tangent to X(14) to the circle defined at X(5460). Then X(5466) = L∩L'. Also, on the circle passing through X(2), X(13), X(14), X(111), and X(476), the antipode of X(2) is X(5466).    (Randy Hutson, May 3 2013)

X(5466) lies on the Kiepert hyperbola and these lines:
{2,523},{4,1499},{10,4024},{76,850},{98,111},{321,4036},{476,691},{512,598},{671,690},{685,4240},{868,2394},{895,2986}


X(5467) =  ISOGONAL CONJUGATE OF X(5466)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - 2a2)/(b2 - c2)

X(5467) = {X(1576),X(4558)}-harmonic conjugate of X(1634)   (Peter Moses, May 7, 2013)

X(5467) lies on these lines: {3,6},{110,351},{112,1296},{250,4230},{523,2407},{2709,2715},{2794,3014},{2854,5191},{4436,4612}

X(5467) = midpoint of X(2407) and X(4226)
X(5467) = crossdifference of every pair of points on line X(115)X(523)
X(5467) = X(111)-isoconjugate of X(1577)


X(5468) =  ISOTOMIC CONJUGATE OF X(5466)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 + c2 - 2a2)/(b2 - c2)

X(5468) = {X(110),X(4563)}-harmonic conjugate of X(4576)   (Peter Moses, May 7, 2013)

X(5468) lies on these lines: {2,6},{99,110},{877,4240},{2418,2434},{2715,2858},{3266,3292}

X(5468) = crossdifference of every pair of points on line X(512)X(3124)


X(5469) =  {X(14),X(115)}-HARMONIC CONJUGATE OF X(13)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = csc(A + π/3)[a csc(A - π/3) + b csc(B - π/3) + c csc(C - π/3)] + 2 csc(A - π/3)[a csc(A + π/3) + b csc(B + π/3) + c csc(C + π/3)]
X(5469) = X(13) + 2*X(14)

Let A'B'C' be the antipedal triangle of X(13), let A'' be the nine-point center of the triangle BCX(14), and define B'' and C'' cyclically. Then X(5469) is the homothetic center of A'B'C' and A''B''C''.   (Randy Hutson, May 3, 2013)

X(5469) = reflection of X(5470) in X(115)   (Randy Hutson, May 3, 2013)
X(5469) = {X(14),X(115)}-harmonic conjugate of X(13)   (Peter Moses, May 7, 2013)

X(5469) lies on these lines: {6,13}, {18,671}, {148,618}, {98,5478}


X(5470) =  {X(13),X(115)}-HARMONIC CONJUGATE OF X(14)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = csc(A - π/3)[a csc(A + π/3) + b csc(B + π/3) + c csc(C + π/3)] + 2 csc(A + π/3)[a csc(A - π/3) + b csc(B - π/3) + c csc(C - π/3)]
X(5470) = 2*X(13) + X(14)

Let A'B'C' be the antipedal triangle of X(14), let A'' be the nine-point center of the triangle BCX(13), and define B'' and C'' cyclically. Then X(5470) is the homothetic center of A'B'C' and A''B''C''.   (Randy Hutson, May 3, 2013)

X(5470) = reflection of X(5469) in X(115)   (Randy Hutson, May 3, 2013)
X(5470) = {X(13),X(115)}-harmonic conjugate of X(14)   (Peter Moses, May 7, 2013)

X(5470) lies on these lines: {6,13}, {17,671}, {148,619}, {98,5479}


X(5471) =  {X(6),X(14)}-HARMONIC CONJUGATE OF X(115)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4S2 + a2(2a2 - b2 - c2) - (12)1/2a2S
X(5471) = (pending)

Let A'B'C' be the pedal triangle of X(15), let A'' be the nine-point center of the triangle BCX(14), and define B'' and C'' cyclically. Then X(5471) is the homothetic center of A'B'C' and A''B''C''.   (Peter Moses, May 7, 2013)

X(5471) = isogonal conjugate (and isotomic conjugate) of X(16) with respect to the pedal triangle of X(16). Also, X(5471) = {X(I), X(J)}-harmonic conjugate of X(K) for these (I,J,K): (6,5474,5472), (115,5477,5472). (Randy Hutson, May 7, 2013)

X(5471) lies on these lines:
{6,13},{39,398},{61,1506},{187,395},{233,2903},{302,620},{1569,3106},{2549,5334}


X(5472) =  {X(6),X(13)}-HARMONIC CONJUGATE OF X(115)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4S2 + a2(2a2 - b2 - c2) + (12)1/2a2S
X(5472) = (pending)

Let A'B'C' be the pedal triangle of X(16), let A'' be the nine-point center of the triangle BCX(13), and define B'' and C'' cyclically. Then X(5472) is the homothetic center of A'B'C' and A''B''C''.   (Peter Moses, May 7, 2013)

X(5472) = isogonal conjugate (and isotomic conjugate) of X(15) with respect to the pedal triangle of X(15). Also, X(5472) = {X(I), X(J)}-harmonic conjugate of X(K) for these (I,J,K): (6,5475,5471), (115,5477,5471). (Randy Hutson, May 7, 2013)

X(5472) lies on these lines:
{6,13},{39,397},{62,1506},{187,396},{233,2902},{303,620},{1569,3107},{2549,5335}


X(5473) =  INTERSECTION OF LINES X(3)X(13) AND X(4)X(618)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 31/2a2(a4 - 3b4 - 3c4 + 2a2b2 + 2a2c2 - 2b2c2) + 4(area ABC)(7a4 - 2b4 - 2c4 - 5a2b2 - 5a2c2 + 4b2c2)    (Peter Moses, May 12, 2013)

X(5473) is the isogonal conjugate (and isotomic conjugate) of X(13) with respect to the antipedal triangle of X(13). X(5473) = {X(1350), X(3534)}-harmonic conjugate of X(5474).   (Randy Hutson, May 7, 2013)

X(5473) lies on these lines: {2,5478},{3,13},{4,618},{16,2549},{20,616},{376,530},{542,1350}

X(5473) = anticomplement of X(5478)


X(5474) =  INTERSECTION OF LINES X(3)X(14) AND X(4)X(619)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 31/2a2(a4 - 3b4 - 3c4 + 2a2b2 + 2a2c2 - 2b2c2) - 4(area ABC)(7a4 - 2b4 - 2c4 - 5a2b2 - 5a2c2 + 4b2c2)    (Peter Moses, May 12, 2013)

X(5474) is the isogonal conjugate (and isotomic conjugate) of X(14) with respect to the antipedal triangle of X(14).   (Randy Hutson, May 7, 2013)

X(5474) = {X(1350), X(3534)}-harmonic conjugate of X(5473).

X(5474) lies on these lines: {2,5479},{3,14},{4,619},{15,2549},{20,617},{376,531},{542,1350}

X(5474) = anticomplement of X(5479)


X(5475) =  INTERSECTION OF LINES X(2)X(187) AND X(4)X(39)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2) - (b2 - c2)2    (Peter Moses, May 12, 2013)
X(5475) = cot2ω*X(6) + 3*X(381)

X(5475) is the {X(I),X(J)}-harmonic conjugate of X(K) for these (I,J,K): (5472,5471,6), (13,14,5476), and X(5475) is the inverse-in-Kiepert-hyperbola of X(5476).    (Randy Hutson, May 7, 2013)

X(5475) is the {X(I),X(J)}-harmonic conjugate of X(K) for these (I,J,K): (4,2548,39), (6,115,5309), (6,381,115), (3767,5007,5346) (Peter Moses, May 12, 2013)

X(5475) lies on these lines:
{2,187},{3,1506},{4,39},{5,32},{6,13},{11,2242},{12,2241},{30,574},{51,5167},{83,3407},{140,5206},{183,754},{233,1609},{315,3934},{325,3734},{382,5013},{384,3788},{485,5058},{486,5062},{524,3363},{546,5254},{547,3054},{549,3055},{620,1003},{1015,1478},{1285,5071},{1316,5099},{1352,5052},{1384,5055},{1479,1500},{1503,5034},{1504,3071},{1505,3070},{1594,1968},{1596,5065},{1656,3053},{2275,3585},{2276,3583},{2458,5103},{3091,3767},{3526,5023},{3545,5008},{3589,5033},{3814,4386},{3830,5024},{3832,5041},{3850,5305},{3855,5319},{4193,5277},{5046,5283},{5054,5210},{5066,5306}

X(5475) = reflection of X(574) in X(3815)


X(5476) =  MIDPOINT OF X(6) AND X(381)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 + b6 + c6 - 4a4b2 - 4a4c2 + 2a2b4 + 2a2c4 - 6a2b2c2 - b4c2 - b2c4    (Peter Moses, May 12, 2013)

Let A' be the intersection, other than X(4), of the A-altitude and the orthocentroidal circle, and define B' and C' cyclically. The triangle A'B'C', introduced here as the orthocentroidal triangle, is inversely similar to ABC, with center X(6) of similitude. If, in that definition, X(4) replaced by X(2) and A-altitude by A-median, the resulting triangle is the 4th Brocard triangle. Regarding a point X as a function of a triangle, X(A'B'C') - that is, X of A'B'C' - is the reflection of X(ABC) in the centroid of the pedal triangle of X. X(5476) = X(182) of the orthocentroidal triangle, and X(5476) = X(182) of the 4th Brocard triangle. Also, X(5476) = inverse-in-Kiepert-hyperbola of X(5475), and X(5476) = {X(13),X(14)}-harmonic conjugate of X(5475).    (Randy Hutson, May 7, 2013)

The vertices of the orthocentroidal triangle are given by Peter Moses (June 17, 2014): A' = a2 : a2 + b2 - c2 : a2 - b2 + c2
B' = b2 - c2 + a2 : b2 : b2 + c2 - a2
C' = c2 + a2 - b2 : c2 - a2 + b2 : c2

The orthocentroidal triangle is the circumsymmedial triangle of the 4th Brocard triangle. (Randy Hutson, November 22, 2014)

X(5476) lies on these lines:
{2,51},{4,575},{5,524},{6,13},{30,182},{69,5071},{141,547},{376,3618},{549,3098},{599,1351},{1350,5054},{1352,1992},{1469,3582},{1503,3845},{3056,3584},{3091,5032},{3534,5085},{3564,5066},{3830,5050},{5039,5306}


X(5477) =  REFLECTION OF X(115) IN X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2(a2 - b2 - c2) + 2a2[(a2 - b2)2 + (a2 - c2)2]

X(5477) = {X(5472), X(5471)}-harmonic conjugate of X(115)    (Randy Hutson, May 7, 2013)

X(5477) lies on these lines:
{6,13},{30,5107},{69,620},{98,5034},{99,193},{114,230},{147,5304},{187,524},{511,1569},{543,1992},{575,1506},{671,5032},{690,5095},{1353,2782},{1503,1570},{2458,4027},{2549,2794},{2796,4856}


X(5478) =  MIDPOINT OF X(4) AND X(13)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 31/2(3a4b2 + 3a4c2 - 2a2b4 - 2a2c4 + 4a2b2c2 + b4c2 + b2c4 - b6 - c6) + 4(area ABC)(4a4 - 5b4 - 5c4 + a2b2 + a2c2 + 10b2c2)    (Peter Moses, May 12, 2013)

X(5478) = X(13) of Euler triangle, and X(5478) = {X(3845),X(5480)}-harmonic conjugate of X(5479)    (Randy Hutson, May 8, 2013)

X(5478) lies on these lines:
{2,5473},{4,13},{5,618},{98,5469},{107,473},{115,5318},{381,530},{542,1353},{616,3091},{624,3734}}

X(5478) = midpoint of X(4) and X(13)
X(5478) = complement of X(5473)


X(5479) =  MIDPOINT OF X(4) AND X(14)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 31/2(3a4b2 + 3a4c2 - 2a2b4 - 2a2c4 + 4a2b2c2 + b4c2 + b2c4 - b6 - c6) - 4(area ABC)(4a4 - 5b4 - 5c4 + a2b2 + a2c2 + 10b2c2)    (Peter Moses, May 12, 2013)

X(5479) = X(14) of Euler triangle, and X(5478) = {X(3845),X(5480)}-harmonic conjugate of X(5478)    (Randy Hutson, May 8, 2013)

X(5479) lies on these lines:
{2,5474},{4,14},{5,619},{98,5470},{107,472},{115,5321},{381,531},{542,1353},{617,3091},{623,3734}

X(5479) = midpoint of X(4) and X(14)
X(5479) = complement of X(5474)


X(5480) =  MIDPOINT OF X(4) AND X(6)

Trilinears   sin A tan ω + 2 cos B cos C : sin B tan ω + 2 cos C cos A : sin C tan ω + 2 cos A cos B
Trilinears   cos B cos(C - ω) + cos C cos(B - ω) : cos C cos(A - ω) + cos A cos(C - ω) : cos A cos(B - ω) + cos B cos(A - ω)

Let O' denote the orthosymmedial circle, introduced here as the circle having segment X(4)X(6) as diameter, so that X(5480) is the center of O'. Note that O'∩(Euler line) = {X(4), X(1316)} and O'∩(Brocard circle) = {X(6), X(1316)}; X(5480) = X(6) of Euler triangle; X(5480) = {X(5478),X(5479)}-harmonic conjugate of X(3845); X(5480) = inverse-in-Jerabek-hyperbola of X(51).    (Randy Hutson, May 8, 2013)

The orthosymmedial circle is the inverse-in-polar-circle of the line X(297)X(525).    (Randy Hutson, August 26, 2014)

X(5480) lies on these lines:
{2,1350},{3,3589},{4,6},{5,141},{11,1469},{12,3056},{20,3618},{30,182},{51,125},{66,3527},{69,3091},{98,5306},{113,2854},{115,5052},{118,2810},{140,3098},{159,1598},{184,428},{185,1907},{193,3832},{206,578},{230,5017},{235,1843},{262,1513},{343,3060},{381,524},{382,5050},{383,396},{389,1595},{395,1080},{515,1386},{516,4085},{518,946},{542,1353},{546,576},{550,5092},{575,3627},{599,3545},{611,1479},{613,1478},{698,3095},{1596,2393},{1699,3751},{1848,1864},{1861,2262},{1890,2182},{1899,5064},{1974,3575},{1992,3839},{2051,4260},{3054,5104},{3090,3763},{3580,5169},{3619,5056},{3620,5068},{3630,3850},{3631,3851},{3843,5093},{5039,5305}

X(5480) = isogonal conjugate of X(5481)
X(5480) = complement of X(1350)
X(5480) = crosspoint of X(4) and X(262)
X(5480) = crosssum of X(3) and X(182)


X(5481) =  ISOGONAL CONJUGATE OF X(5480)

Trilinears   1/(sin A tan ω + 2 cos B cos C) : 1/(sin B tan ω + 2 cos C cos A) : 1/(sin C tan ω + 2 cos A cos B)
Trilinears   1/(cos B cos(C - ω) + cos C cos(B - ω)) : 1/(cos C cos(A - ω) + cos A cos(C - ω)) : 1/(cos A cos(B - ω) + cos B cos(A - ω))

X(5481) lies on the hyperbola {A,B,C,X(2),X(3)} and these lines: {2,1629},{216,1297},{394,5012},{1078,3523}

X(5481) = isogonal conjugate of X(5480)
X(5481) = cevapoint of X(3) and X(182)


X(5482) =  1st HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5b2 + a5c2 - 2a5b2c2 + a3b3 + a3c3 + a3b2c + a3bc2 + a2b4 + a2c4 - 3a2b3c - 3a2bc3 - ab5 - ac5 - ab4c - abc4 - bc(b2 - c2)2    (Angel Montesdeoca, May 13, 2013)
X(5482) = 3*X(549) - X(970)
X(5482) = (R - 2r)*X(140) - R*X(143)

Let ABC be a triangle and let A'B'C' be the cevian triangle of the incenter, X(1). Let R be the radical center of the circles (A', |A'B|, {B',|B'C|), (C', |C'A|), and let S be the radical center of the circles (A',|A'C|), (B',|B'A|), (C',|C'B|). X(5482) is the midpoint of the segment RS.    (Antreas Hatzipolakis, May 4, 2013)

X(5482) is the {X(3),X(1764)}-harmonic conjugate of X(3579)   (Peter Moses, May 13, 2013)

For the construction and generalizations, see Hechos Geométricos en el Triángulo.

X(5482) lies on these lines: {1,3}, {140,143}, {549,970}


X(5483) =  CENTER OF HUTSON ELLIPSE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a5 + b5 + c5 + a4b + a4c - 2a3b2 - 2a3c2 - 2a2b3 - 2a2c3 - 3a2b2c - 3a2bc2 + ab4 + ac4 - 2ab3c -2abc3 - 3ab2c2 - b3c2 - b2c3)    (Peter Moses, May 17, 2013)
X(5483) = 4(r + R)*X(226) - (2r + R)*X(1029)

Let A'B'C' be the cevian triangle of the incenter. Let AB = (reflection of A' in BB'), and define BC and CA cyclically. Let AC = (reflection of A' in CC'), and define BA and CB cyclically. The ellipse passing through the points AB, AC, BC, BA, CA, CB is here introduced as the Hutson Ellipse, and X(5483) is its center. (Antreas Hatzipolakis, May 17, 2013)

X(5483) lies on these lines: {1,5180}, {81,593}, {226,1029}


X(5484) =  INTERSECTION OF LINES X(2)X(12) and X(8)X(38)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - b4 - c4 - a3b - a3c - 2a2b2 - 2a2c2 + a2bc - ab3 - ac3 - 3a2bc - 3abc2 - b3c - bc3    (Peter Moses, May 18, 2013)
X(5484) = 3X(2) - 2*X(1220)

Let A'B'C' be the cevian triangle of point X. Let AB = (reflection of A' in BB'), and define BC and CA cyclically. Let AC = (reflection of A' in CC'), and define BA and CB cyclically. Let HA = (orthocenter of A'ABAC), and define HB and HC cyclically. The orthocentric triangle of X is here introduced as the (central) triangle HAHBHC.   (Antreas Hatzipolakis, May 17, 2013)

For X = X(1), the orthocentric triangle HAHBHC is perspective to the anticomplementary triangle, and X(5484) is the perspector.    (Peter Moses, May 17, 2013)

Also, HAHBHC is perspective to ABC at X(10).    (Randy Hutson, May 18, 2013)

X(5484) is the crosspoint of X(1) and X(8) with respect to the extraversion triangle of X(8).    (Randy Hutson, August 26, 2014)

X(5484) lies on these lines:
{1,1330},{2,12},{8,38},{10,1054},{69,145},{519,2891},{1469,3869},{1626,4189},{3662,4327}

X(5484) = anticomplement of X(1220)


X(5485) =  KIRIKAMI-EULER IMAGE OF THE CENTROID

Barycentrics   1/(5a2 - b2 - c2) : 1/(5b2 - c2 - a2) : 1/(5c2 - a2 - b2)    (Seichii Kirikami, May 21, 2013)

Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The Euler lines of the triangles AHBHC, BHCHA, CHAHB concur in the Kirikami-Euler image of P. Let P be given by barycentrics p : q : r. Then Q is given by

Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/[(a2 - b2 - c2)p2 + (a2 - b2 + c2)pq + (a2 + b2 - c2)pr + 2a2qr]

If P = X(2), then Q = X(5485).   (Seichii Kirikami, May 20, 2013)

The Kirikami-Euler image K(P) of a point P is related to the mapping H(P) called "pedal antipodal perspector", defined in Hyacinthos #20403 and #20405, November 2011, by Randy Hutson, with general coordinates given in #20404 by Francisco Javier. X(5485) = H(X(I)) for I = 6 and I = 187. In general, K(P) = H(P') = H(P*), where P' denotes the isogonal conjugate of P, and P* = (inverse-in-circumcircle of P'); for example, K(X(1)) = X(8), K(X(3)) = X(68), and K(X(6)) = X(5486).    (Randy Hutson, May 22, 2013)

If you have The Geometer's Sketchpad, you can view X(5485).

X(5485) lies on the Kiepert hyperbola and these lines:
{2,2418},{4,524},{10,4419},{30,3424},{69,671},{98,376},{262,538},{525,5466},{598,1992},{631,1153},{5032,5395}}

X(5485) = isogonal conjugate of X(1384)
X(5485) = isotomic conjugate of X(1992)


X(5486) =  KIRIKAMI-EULER IMAGE OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(b4 + c4 - a4 - 4b2c2)

Using the notation at X(5485) for Kirikami-Euler image and pedal antipodal perspector, X(5486) = K(X(6)) = H(X(2)) = H(X(23)). Also, X(5485) is the trilinear pole of the line X(647)X(690).   (Randy Hutson, May 22, 2013)

X(5486) lies on the Jerabek hyperbola and these lines:
{2,895},{3,524},{4,2393},{6,468},{67,1899},{69,3266},{71,4062},{184,1177},{193,1176},{248,5063},{265,1352},{511,4846},{523,2549},{1173,3542},{1503,3426},{3531,5480}

X(5486) = isogonal conjugate of X(1995)


X(5487) =  KIRIKAMI-EULER IMAGE OF X(17)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(7a2 - b2 - c2 + (12)1/2S)

X(5487) lies on these lines: {13,633},{14,627}


X(5488) =  KIRIKAMI-EULER IMAGE OF X(18)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(7a2 - b2 - c2 - (12)1/2S)

X(5488) lies on these lines: {13,628},{14,634}


X(5489) =  KIRIKAMI-EULER IMAGE OF X(125)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)3(b2 + c2 - a2)2

X(5489) lies on these lines: {3,525},{4,523},{39,647},{669,2353},{826,3574},{3265,3926}

X(5489) = crossdifference of every pair of pints on line X(23)X(232)


X(5490) =  KIRIKAMI-EULER IMAGE OF X(485)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(S + a2)

X(5490) lies on these lines:
{2,493},{4,488},{10,5391},{69,485},{83,3069},{98,637},{486,641},{491,3316},{1131,1270},{1132,3593},{1271,3590}


X(5491) =  KIRIKAMI-EULER IMAGE OF X(486)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(S - a2)

X(5491) lies on these lines:
{2,494},{4,487},{10,1267},{69,486},{83,3068},{98,638},{485,642},{492,3317},{1131,3595},{1132,1271},{1270,3591}


X(5492) =  ORTHOCENTER OF ORTHOCENTRIC TRIANGLE OF X(1)

Barycentrics   a(a4b2 + a4c2 - 2a3b2c - 2a3bc2 - 2a2b4 -2a2c4 - a2b3c - a2bc3 + 2ab4c + 2abc4 - 2ab3c2 - 2ab2c3 + b6 + c6 + b5c + bc5 - b4c2 - b2c4 - 2b3c3)
X(5492) = X(500) - 2*X(3743)
X(5492) = 3*X(1962) - 2*X(5453)

Let T be the orthocentric triangle HAHBHC of X(1), as defined at X(5484). X(5492) is the orthocenter of T, and T is perspective to the Fuhrmann triangle with perspector X(1), and T is perspective to the anticomplementary triangle, with perspector X(5484).    (Peter Moses, May 17, 2013)

T is similar to the incentral triangle, with center of similitude I, the incenter. Let A''B''C'' be the antipedal triangle of X(1) with repect to the incentral triangle. X(5492) is the nine-point center of A''B''C''. (The triangle A''B''C'' is also the triangle formed by the lines LA, LB, LC, where LA is the polar of A with respect to the circle BCI, and LB and LC are defined cyclically.)    (Randy Hutson, May 18, 2013)

X(5492) lies on these lines:
{1,399},{3,846},{5,3120},{30,2292},{58,3652},{355,2783},{381,986},{500,3743},{1725,3649},{1772,3614},{1962,5453}

X(5492) = reflection of X(3743) in X(500)


X(5493) =  CENTER OF CIRCLE BISECTING THE EXCIRCLES

Barycentrics   b4 + c4 - 4a4 - 3a3b - 3a3c + 3a2b2 + 3a2c2 + 6a2bc + 3ab3 + 3ac3 - 3ab2c - 3abc2 - 2b2c2
X(5493) = 3*X(1) + 3*X(2) - 8*X(3)
X(5493) = 3*X(1) - 6*X(3) + X(4)
X(5493) = 2*X(4) - 3*X(10)

X(5493) is the center of the circle Y that bisects each of the three excircles of ABC. Let J be the radius of Y; then 4J2 = r2 + 16rR + 64R2 - 7s2.    (Paul Yiu, Francisco Javier, AdvPlGeom, May 17, 2013)

X(5493) lies on these lines:
{1,3522},{3,551},{4,9},{8,5059},{20,519},{30,4669},{55,3671},{56,4342},{65,4314},{140,946},{144,4882},{165,962},{355,5073},{382,3654},{390,3339},{484,1210},{497,5128},{515,1657},{517,550},{527,3913},{553,3303},{1656,3817},{1697,3474},{1698,5068},{1699,3634},{1770,5270},{1836,3947},{2093,4294},{3057,4315},{3091,3828},{3146,3679},{3428,5267},{3543,4745},{3663,5255},{3931,4349},{4229,4658},{4292,5119},{4848,5183}

X(5493) = reflection of X(I) in X(J) for these (I,J): (10,40), (946,3579), (962,1125), (3244,4297), (3543,4745),(4301,3)


X(5494) =  2nd HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a9 - a8(b + c) - a7( b - c)2 + a6(2b3 - b2c - bc2 + 2c3) - a5(3b4 + b3c - 7b2c2 + bc3 + 3c4) + 4a4bc(b - c)2(b + c) + a3(b2 - c2)2(5b2 - 4bc + 5c2) - a2(b - c)2(2b5 + 5b4c + b3c2 + b2c3 + 5bc4 + 2c5) - a(b2 - c2)2(2b4 - 3b3c + 5b2c2 - 3bc3 + 2c4) + (b - c)4(b + c)3(b2 + c2)]    (Angel Montesdeoca, May 25, 2013)
X(5494) = (2r + R)*X(110) - 4(r + R)X(1385)
X(5494) = 2R*X(65) + (2r + R)*X(74)

Let ABC be a triangle and let A'B'C' be the cevian triangle of the incenter, X(1). Let AB be the reflection of A' in line BB', and define BC and CA cyclically. Let AC be the reflection of A' in line CC', and define BA and CB cyclically. Let L be the Euler line of ABC, let LA be the Euler line of AABAC, and define LB and LC cyclically. Let MA be the reflection of LA in AA', and define MB and MC cyclically. The lines MA, MB, MC concur in X(5494). Moreover, the four Euler lines L, LA, LB, LC are parallel, concurring in X(30).    (Antreas Hatzipolakis, May 25, 2013)

For the construction and discussion, see

Hechos Geométricos en el Triángulo.

X(5494) lies on these lines: {1,2779},{21,104},{36,1725},{65,74},{125,860}


X(5495) =  3rd HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[a7(b + c) - a6(b2 + c2) - a5(3b3 + 2b2c + 2bc2 + 3c3) + a4(3b4 - b3c + 4b2c2 - bc3 + 3c4) + a3(3b5 + b4c + 2b3c2 + 2b2c3 + bc4 + 3c5) - a2(3b6 - 2b5c - 2bc5 + 3c6) - a(b7 - b4c3 - b3c4 + c7) + (b2 - c2)2(b4 - b3c - bc3 - b2c2 + c4)]    (Angel Montesdeoca, May 28, 2013)

Let ABC be a triangle and let A'B'C' be the cevian triangle of the incenter, X(1). Let LA be the line through A' perpendicular to line AA', and define LB add LC cyclically. Let

UA = reflection of LA in AA'
UB = reflection of LA in BB'
UC = reflection of LA in CC'

VA = reflection of LB in AA'
VB = reflection of LB in BB'
VC = reflection of LB in CC'

WA = reflection of LC in AA'
WB = reflection of LC in BB'
WC = reflection of LC in CC'

TA = triangle formed by the lines in UA, UB, UC
TB = triangle formed by the lines in VA, VB, VC
TC = triangle formed by the lines in WA, WB, WC

OA = circumcenter of TA, OB = circumcenter of TA, OC = circumcenter of TA, O = X(3) = circumcenter of ABC. The points O, OA, OB, OC are concyclic. The center of their circle is X(5495).    (Antreas Hatzipolakis, May 28, 2013)

For the construction and discussion, see Concyclic Circumcenters.

X(5495) lies on these lines: (pending)


X(5496) =  4th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)[a5 - 2a3(b2 + c2) - a2bc(b+c) + a(b4 - b2c2 + c4) + bc(b + c)(b - c)2    (Angel Montesdeoca, May 29, 2013)

Let ABC be a triangle and let A'B'C' be the cevian triangle of the incenter, X(1). Let LA be the line through A' perpendicular to line AA', and define LB add LC cyclically. Using the notation at X(5495), let MA be the line parallel to UA through B', and define MB and MC cyclically. Let A'' = MB∩MC, and define B'' and C'' cyclically. Let OA = circumcenter of A''B'C', and define OB and OC cyclically. Then the points X(1), OA, OB, OC are concyclic, and the center of their circle is X(5496).    (Antreas Hatzipolakis, May 29, 2013)

For a discussion, see Concurrent Circles.

X(5496) lies on these lines: (pending)


X(5497) =  5th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a7 - a6(b + c) - a5(b + c)2 + a4(2b3 + b2c + bc2 + 2c3) - a2(b4 - b3c - 3b2c2 - bc3 + c4) + abc(b2 + c2)(b2 - c2)2    (Angel Montesdeoca, May 29, 2013)

Let ABC be a triangle and let A'B'C' be the cevian triangle of the incenter, X(1). The circles OA, OB, OC defined at X(5496) concur in X(5497).    (Antreas Hatzipolakis, May 29, 2013)

For a discussion, see Hechos Geométricos en el Triángulo.

X(5497) lies on these lines: (pending)


X(5498) =  6th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a10 - 5a8(b2 + c2) + 2a6(b4 + 5b2c2 + c4) + a4(4b6 - 5b4c2 - 5b2c4 + 4c6) - a2(b2 - c2)2(4b4 + 5b2c2 + 4c4) + (b2 - c2)sup>4(b2 + c2)    (Angel Montesdeoca, May 30, 2013)

Let ABC be a triangle, let NA be the nine-point center of the triangle BCO, where O = X(3), and define NB and NC cyclically. The nine-point center of the triangle NANBNC is X(5498), which lies on the Euler line of ABC.   (Antreas Hatzipolakis, May 30, 2013)

X(5498) lies on these lines: (2,3}, (more pending)


X(5499) =  7th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5(b2 + 4bc + c2) - a4(b3 + b2c - bc2 + c3) + a3(2b4 + 3b3c + 3bc3 + 2c4) + 2a2(b5 - b3c2 - b2c3 + c5) + a(b2 - c2)2(b2 - bc + c2) - (b - c)4(b + c)3    (Angel Montesdeoca, May 30, 2013)

Let IA be the A-excenter of a triangle ABC and let NA be the nine-point center of IABC. Define NB and NC cyclically. The circumcenter of NANBNC is X(5499), which lies on the Euler line of ABC.    (Antreas Hatzipolakis, May 30, 2013)

Let A'B'C' be the Feuerbach triangle, and let A'' be the reflection of X(11) in line B'C'; define B'' and C'' cyclically. Then A'', B'', C'' are collinear, and their line, X(12)X(79) is here named the Feuerbach line. X(5499) is the point of intersection of the Feuerbach line and the Euler line.   (Randy Hutson, August 26, 2014)

X(5499) lies on these lines: (2,3}, (more pending)


X(5500) =  8th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =
2a22
- 15a20(b2 + c2)
+ 6a18(8b4 + 13b2c2 + 8c4)
- a16(81b6 + 52b4c2 + 152b2c4 + 81c6)
+ a14(64b8 + 111b6c2 + 128b4c4 + 111b2c6 + 64c8)
+ a^12(14b10 + 29b8c2 + 36b6c4 + 36b4c6 + 29b2c8 + 14c10)
- a10(84b^12 + 67b10c2 + 56b8c4 + 48b6c6 + 56b4c8 + 67b2c10 + 84c12)
+ a8(82b14 - 23b12c2 - 31b10c4 - 19b8c6 - 19b6c8 - 31b4c10 - 23b2c^12+ 82c14)
- a6(b2 - c2)2(34b12 + 11b10c2 - 30b8c4 - 35b6c6 - 30b4c8 + 11b2c10 + 34c12)
+ a4(b2- c2)4(b10 - 2b8c2 - 22b6c4 - 22b4c6 - 2b2c8+ c10)
+ a2(b2 - c2)6(4b8 + 5b6c2 + 8b4c4 + 5b2c6 + 4c8)
- (b2 - c2)8(b6 + b4c2 + b2c4 + c6)    (Angel Montesdeoca, May 30, 2013)

Let A'B'C' be the antipedal triangle of the nine-point center, N = X(5) of a triangle ABC. Let NA be the nine-point center of NB'C', and define NB and NC cyclically. The nine-point center of NANBNC is X(5500), which lies on the Euler line of ABC.    (Antreas Hatzipolakis, May 30, 2013)

X(5500) lies on these lines: (2,3}, (more pending)


X(5501) =  9th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =
- 2a16
+ 2a14(b2 + c2)
- a12(13b4 + 18b2c2 + 13c4)
- a10(b6 + b4c2 + b2c4 + c6)
+ a8(25b8 + 10b6c2 + 8b4c4 + 10b2c6 + 25c8)
+ a6(-33b10 + 31b8c2 + 11b6c4 + 11b4c6 + 31b2c8 - 33c10)
+ a4(b2 - c2)2(21b8 - 20b6c2 - 25b4c4 - 20b2c6 + 21c8)
- a2(b2 - c2)4(7b6 -13b4c2 - 13b2c4 + 7c6)
+ (b2 - c2)6(b2 - 4b4c2 + c4)    (Angel Montesdeoca, June 2, 2013)

Let N be a the nine-point center of triangle ABC. Let NA be the nine-point center of NBC, and define NB and NC cyclically. The circumcenter of NANBNC is X(5501), which lies on the Euler line of ABC.    (Antreas Hatzipolakis, June 2, 2013)

See For a discussion, see Hechos Geométricos en el Triángulo.

X(5501) lies on these lines: (2,3}, (more pending)


X(5502) =  10th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 - b2)[a2 - c2)(a6 - a4(b2 + c2) + a2(a2 -b2)(a2 - c2) + 3(b2 - c2)2(b2 + c2)]    (Angel Montesdeoca, June 3, 2013)

Let L be the Euler line of a triangle ABC. Let LA be the reflection of L in line BC, and define LB and LC cyclically. Let A' = L∩BC, and define B' and C' cyclically. The circles whose diameters are the segments AA', BB', CC' are coaxial. Let D be their coaxial axis (the line X(4)X(74)); let DA be the reflection of D in line BC, and define DB and DC cyclically. Let HA = LB∩DC, and define HB and HC cyclically. Let MA = LC∩DB, and define MB and MC cyclically. The triangles HAHBHC and MAMBMC are perspective, and their perspector is X(5502).    (Antreas Hatzipolakis, June 3, 2013)

See For a discussion, see Hechos Geométricos en el Triángulo.

X(5502) lies on these lines: {3,64}, {110, 351}


X(5503) =  KIRIKAMI CONCURRENT CIRCLES IMAGE OF THE CENTROID

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(4a4 + b4 + c4 - 4b2c2 - a2b2 - a2c2)    (Seichii Kirikami, June 2, 2013)

Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. Let OA be the circle of the points A, HB, HC, and define OB and OC cyclically. The circles OA, OB, OC concur in a point Q, the Kirikami concurrent circles image of P. Let P be given by barycentrics p : q : r. Then Q given by

Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/[2a4qr(p + q)(p + r) + b4pr(p + q)(q + r - p) + c4pq(p + r)(q + r - p) - b2c2p(p2q + pq2 + pq2 + pr2 + 2q2r + 2qr2) + a2b2pr(p + q)(p - 3q + r) + a2c2pq(p + r)(p + q - 3r)]

If P = X(2), then Q = X(5503).   (Seichii Kirikami, June 2, 2013)

If P is on the circumcircle, then Q(P) = P. This follows from the fact that the denominators of g(a,b,c,p,q,r) and g(b,c,a,q,r,p} are polynomial multiples of a2qr + b2rp+c2pq.    (Seichii Kirikami, July 27, 2013)

If you have The Geometer's Sketchpad, you can view X(5503).

X(5503) lies on these lines: {4,543},{98,524},{99,598},{115,5485},{325,671},{542,3424},{2799,5466},{3407,5182}

X(5503) = reflection of X(5485) in X(115)


X(5504) =  KIRIKAMI CONCURRENT CIRCLES IMAGE OF X(3)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos A)/(1 + cos 2B + cos 2C)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b2 + c2 - a2)/[a4(b2 + c2) + (b2 - c2)2(b2 + c2) - 2a2(b4 - b2c2 + c4)]    (Seichii Kirikami, June 2, 2013)

See X(5503) for the definition of Kirikami concurrent circles image Q of a point P and X(5509) for an occurrence of X(5504) as a point of concurrence given by the Hatzipolakis-Moses Theorem.

If P = X(3), then Q = X(5504).    (Seichii Kirikami, June 2, 2013)

Let A'B'C' be the tangential triangle. Let LA be the line through A' parallel to the Euler line, and define LB and LC cyclically. Let RA be the reflection of LA in BC, and define RB and RC cyclically. The lines RA, RB, RC concur in X(5504); see X(399). (Randy Hutson, August 17, 2014)

Continuing, let A'' be the reflection of A' in line BC, and define B'' and C'' cyclically. The circumcircles of AB''C'', BC''A'', CA''B'' concur in X(5504). Morevoer, X(5504) is the antigonal image of X(68), the trilinear pole of line X(577)X(647), and the X(92)-isoconjugate of X(3003). (Randy Hutson, August 17, 2014)

X(5504) lies on the Jerabek hyperbola and these lines:
{3,974},{4,110},{6,1511},{20,3047},{49,3521},{64,155},{66,542},{67,3564},{68,125},{70,3448},{74,323},{182,5486},{184,4846},{265,2072},{290,1236},{399,3167},{511,1177},{1069,3024},{1986,1993},{2850,3657},{3028,3157},{3431,5012}

X(5504) = reflection of X(I) in X(J) for these (I,J): (110,1147), (68,125), (2931, 1511)
X(5504) = isogonal conjugate of X(403)


X(5505) =  KIRIKAMI CONCURRENT CIRCLES IMAGE OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/[4a6 - a4(b2 + c2) + (b2 - c2)2(b2 + c2) - 2a2(2b4 - 3b2c2 + 2c4)]    (Seichii Kirikami, June 2, 2013)

If P = X(6), then Q = X(5505).   (Seichii Kirikami, June 2, 2013)

X(5505) lies on these lines:
{3,2854},{72,3908},{74,2393},{125,5486},{265,524},{323,895},{542,4846},{1177,1495},{2781,3426}

X(5505) = reflection of X(5486) in X(125)


X(5506) =  WOLK PERSPECTOR

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - b3 - c3 + a2b + a2c - ab2 - ac2 - 5abc - 5b2c - 5bc2)
X(5506) = R*X(1) + 2(r + 7R)*X(3305)
X(5506) = 12R*X(2) + (2r + 3R)*X(191)
X(5506) = 12R*X(5) + (2r + 3R)*X(40)

Let I be the incenter of a triangle ABC. Let NA be the nine-point center of IBC, and define NB and NC cyclically. The triangle NANBNC is X(5501)is both similar to and perspective to the excentral triangle of ABC. The perspector is X(5506).    (Barry Wolk, June 1, 2013)

X(5506) lies on these lines:
{1,748},{2,191},{5,40},{9,583},{10,149},{140,1768},{405,5426},{411,2951},{484,3634},{1006,1490},{1045,3216},{1385,5251},{2136,3679},{2950,5316},{3219,3337},{3647,5131},{3740,3746},{5044,5259}


X(5507) = 5th HATZIPOLAKIS-YIU POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(bc - 2S)[abc(b + c - a) + 2(b2 + c2 - a2)S]

Let OA be the circle with center A and radius R, the circumradius of triangle ABC. Let BA be the point where OA meets line AB nearest to B. Define CB and AC cyclically. Let CA be the point where OA meets line AC nearest to C. Define AB and BC cyclically. X(5507) is the radical center of the circles ABACA, BCBAB, CACBC. If "nearest to" is changed to "farthest from" in the construction, the resulting point is X(600). See also X(600). (Peter Moses, June 19, 2013)

If you have The Geometer's Sketchpad, you can view X(5507).

X(5507) lies on this line: {600, 4640}


X(5508) =  KIRIKAMI CONCURRENT CIRCLES IMAGE (2nd KIND) OF X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a6 - a4(b2 + c2) + a3(b3 + c3) - a(b5 + c5) + b3c3]    (Seichii Kirikami, July 2, 2013)

Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The circumcircles of HABC, HBCA, HCAB concur in a point Q, the Kirikami concurrent circles image (2nd kind) of P; see X(5503). Let P be given by barycentrics p : q : r. Then Q is given by

Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = p/[(a2 - b2 - c2)p2 + a2qr + (a2 - b2)pq + (a2 - c2)pr].

If P = X(31), then Q = X(5508).   (Seichii Kirikami, July 2, 2013)

The barycentrics for Q show that "concurrent circles image (2nd kind)" is the same as "antigonal image".    (Randy Hutson, July 15, 2013)

If you have The Geometer's Sketchpad, you can view X(5508).

X(5508) lies on these lines: {31, 5509}, {815, 2887}


X(5509) =  KIRIKAMI SIX CIRCLES IMAGE OF X(31)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(a3 - b2c - bc2)(a2(b2 + c2 + bc) - b4 - c4 - b3c - bc3)    (Seichii Kirikami, July 2, 2013)

Let P be a point in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. The nine-point circles of the six triangles HABC, HBCA, HCAB, AHBHC, BHCHA, CHAHB concur in a point, Q = Q(P), the Kirikami six circles image of P. Let P be given by barycentrics p : q : r. Then Q(P) is given by

Q = h(a,b,c,p,q,r) : h(b,c,a,q,r,p) : h(c,a,b,r,p,q), where h(a,b,c,p,q,r) = p[(a2 - b2 + c2)q - (a2 + b2 - c2)r][(pr + qr)b2 - (pq + rq)c2].

The point Q lies on the nine-point circle of ABC. If P = X(31), then Q = X(5509). If P = X(1), then Q = X(11); if P = X(2), then Q = X(115); if P = X(3), then Q = X(125).    (Seichii Kirikami, July 2, 2013)

Q maps each right circumhyperbola onto its center. Special cases: Q maps the Feuerbach hyperbola onto X(11), the Kiepert hyperbola onto X(115), and the Jerabek hyperbola onto X(125).    (Peter Moses, July 7, 2013)

The Kirikami six circles image, Q(P), of a point P is also the point of concurrence of the nine-point circles of BCP, CAP, ABP (these being the same as the nine-point circles of BCHA, CAHB, ABHC). Also, Q(P) is the center of the rectangular hyperbola passing through P, and Q(P) lies on the cevian circle of P.    (Randy Hutson, July 15, 2013)

The Kirikami six circles image of P is also the QA-P2 center (Euler-Poncelet Point) of the quadrangle ABCP; see Encyclopedia of Quadri-Figures.

If you have The Geometer's Sketchpad, you can view X(5509).

The Kirikami six circles configuration led to a conjecture by Antreas Hatzipolakis (July 5, 2013), proved by Peter Moses, and stated here as the Hatzipolakis-Moses Theorem: Suppose that P and P* are an isogonal conjugate pair of points in the plane of triangle ABC. Let HA be the orthocenter of triangle PBC, and define HB and HC cyclically. Let H'A be the orthocenter of triangle P*BC, and define H'B and H'C cyclically. Then circumcircles of HAHBHC concur and the circumcircles of H'AH'BH'C concur.

The known proof of the theorem depends on a Mathematica program that runs for several minutes. Barycentrics for most choices of P are too long to be included here. An exception is P = X(3), for which P* = X(4) and the two points of concurrence are H(3) = X(265) and H(4) = X(5504).

Related links:
http://tech.groups.yahoo.com/group/Hyacinthos/message/21992

X(5509) lies on these lines: {2,185}, {31,5508}, {115,3271}


X(5510) =  KIRIKAMI SIX CIRCLES IMAGE OF X(106)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a - b - c)(b - c)2(a2b + a2c - 3abc - b3 + 2b2c + 2bc2 - c3)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5510) is the homothetic center of the cyclic quadrilateral ABCX(106) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5510) is the anticenter of ABCX(106)    (Randy Hutson, July 15, 2013)

X(5510) = X(106)-of-Euler-triangle (Randy Hutson, August 17, 2014)

X(5510) = reflection of X(121) in X(5)
X(5510) = midpoint of X(4) and X(106)
X(5510) = complement of X(1293)

X(5510) lies on these lines:
{2, 1293}, {4, 106}, {5, 121}, {11, 1357}, {113, 2842}, {114, 2796}, {115, 2789}, {116, 2821}, {117, 2841}, {118, 2810}, {119, 946}, {120, 3817}, {124, 2815}, {125, 2776}, {132, 2844}, {133, 2839}, {1054, 1699}, {2051, 3030}, {2886, 3038}, {3667, 3756}


X(5511) =  KIRIKAMI SIX CIRCLES IMAGE OF X(105)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a2 + b2 + c2 - 2ab - 2ac)(a3 - a2b - a2c + ab2 + ac2 - b3 + b2c + bc2 - c3)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5511) is the homothetic center of the cyclic quadrilateral ABCX(105) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5511) is the anticenter of ABCX(105)    (Randy Hutson, July 15, 2013)

X(5511) = X(105)-of-Euler-triangle (Randy Hutson, August 17, 2014)

X(5511) = reflection of X(120) in X(5)
X(5511) = midpoint of X(4) and X(105)
X(5511) = complement of X(1292)

X(5511) lies on these lines:
{2, 1292}, {4, 105}, {5, 120}, {11, 1111}, {12, 3021}, {113, 2836}, {114, 2795}, {115, 2788}, {116, 2820}, {117, 2835}, {118, 946}, {119, 381}, {124, 2814}, {125, 2775}, {132, 2838}, {133, 2833}, {1596, 2834}, {2051, 3034}, {2886, 3039}, {3309, 4904}


X(5512) =  KIRIKAMI SIX CIRCLES IMAGE OF X(111)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2(b2 + c2 -5a2)(b4 + c4 - a4 - 4b2c2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5512) is the homothetic center of the cyclic quadrilateral ABCX(111) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5512) is the anticenter of ABCX(111)    (Randy Hutson, July 15, 2013)

X(5512) = X(111)-of-Euler-triangle (Randy Hutson, August 17, 2014)

X(5512) = reflection of X(126) in X(5)
X(5512) = midpoint of X(4) and X(111)
X(5512) = complement of X(1296)

X(5512) lies on these lines:
{2, 1296}, {4, 111}, {5, 126}, {11, 2830}, {54, 3048}, {113, 2854}, {114, 381}, {115, 2793}, {116, 2824}, {117, 2852}, {118, 2813}, {119, 2805}, {124, 2819}, {125, 2780}, {132, 1596}, {133, 2847}, {1499, 2686}


X(5513) =  KIRIKAMI SIX CIRCLES IMAGE OF X(190)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b3 + c3 - ab2 - ac2)(b3 + c3 + 2a3 - a2b - a2c - b2c - bc2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5513) is the perspector of the circumconic centered at X(3011)    (Randy Hutson, July 15, 2013)

Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5513) is the inverse-in-O* of X(101). (Randy Hutson, August 17, 2014)

X(5513) = complement of X(675)
X(5513) = X(2)-Ceva conjugate of X(3011)

X(5513) lies on these lines:
{2, 101}, {9, 124}, {11, 37}, {115, 3136}, {118, 4120}, {125, 1213}, {127, 440}, {427, 5190}, {430, 5139}, {3259, 4370}, {3690, 5509}


X(5514) =  KIRIKAMI SIX CIRCLES IMAGE OF X(40)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(b + c - a)2(b3 + c3 - a3 - a2b - a2c + ab2 + ac2 + 2abc - b2c - bc2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5514) is the point of intersection, other than X(11), of the nine-point circle and the Mandart circle.    (Randy Hutson, July 15, 2013)

X(5514) is the center of the hyperbola {A,B,C,X(4),X(40)}, and X(5514) = X(972)-of-Euler-triangle. (Randy Hutson, August 17, 2014)

X(5514) = midpoint of X(4) and X(972)
X(5514) = complement of X(934)

X(5514) lies on these lines:
{2, 934}, {4, 972}, {9, 119}, {10, 118}, {11, 1146}, {12, 208}, {117, 374}, {120, 1329}, {3814, 5199}


X(5515) =  KIRIKAMI SIX CIRCLES IMAGE OF X(75)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a2 + b2 + c2 + ab + ac + bc)(a2 + b2 + c2 + 2bc)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5515) lies on these lines: {2, 835}, {116, 244}, {117, 5230}, {118, 2999}, {121, 1054}, {124, 3120}, {125, 1086}, {127, 2968}


X(5516) =  KIRIKAMI SIX CIRCLES IMAGE OF X(145)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b + c - 3a)(b - c)2(b2 + c2 + ab + ac - 4bc)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5516) lies on these lines: {120, 5121}, {121, 519}, {1647, 3259}, {3667, 3756}

X(5516) = complement of X(6079)


X(5517) =  KIRIKAMI SIX CIRCLES IMAGE OF X(81)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a2 + b2 + c2 + 2bc)(a3 - b3 - c3 + a2b + a2c - ab2 - ac2 - 2abc - b2c - bc2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5517) lies on these lines: {2, 1310}, {11, 3125}, {120, 1698}, {123, 1146}


X(5518) =  KIRIKAMI SIX CIRCLES IMAGE OF X(291)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(ab + ac - bc)(a2b + a2c - ab2 - ac2 - abc + b2c + bc2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5518) is the touchpoint, other than X(11), of the line through X(1086) tangent to the nine-point circle.    (Randy Hutson, July 15, 2013)

X(5518) = complement of X(932)

X(5518) lies on these lines: {2, 932}, {12, 85}, {121, 3822}


X(5519) =  KIRIKAMI SIX CIRCLES IMAGE OF X(218)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a2 + b2 + c2 - 2ab - 2ac)(b2 + c2 - ab - ac)(2a2 + b2 + c2 - ab - ac - 2bc)   (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5519) lies on these lines: {120, 518}, {1566, 3323}, {3309, 4904}

X(5519) = anticomplement of X(6078)


X(5520) =  KIRIKAMI SIX CIRCLES IMAGE OF X(267)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a3 + b3 + c3 - a2b - a2c - ab2 - ac2 - abc + b2c + bc2)(a4 - b4 - c4 + a2bc - ab2c - abc2 + 2b2c2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5520) is the touchpoint, other than X(11), of the line through X(867) tangent to the nine-point circle. Also, X(5520) is the reflection of X(11) in the Euler line.    (Randy Hutson, July 15, 2013)

Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5520) is the inverse-in-O* of X(2752). Also, X(5520) = inverse-in-polar-circle of X(2766). (Randy Hutson, August 17, 2014)

X(5520) = complement of X(1290)

X(5520) lies on these lines:
{2, 1290}, {4, 2687}, {11, 523}, {12, 2222}, {30, 119}, {113, 517}, {115, 650}, {116, 4369}, {120, 858}, {125, 513}, {1325, 5080}, {1560, 5089}, {1985, 2453}, {2074, 5172}, {3139, 3258}, {3140, 5099}


X(5521) =  KIRIKAMI SIX CIRCLES IMAGE OF X(19)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2(a2 - b2 + c2)(a2 + b2 - c2)(a3 + b3 + c3 - a2b - a2c - ab2 - a2c - 2abc + b2c + bc2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5521) = inverse-in-polar-circle of X(100).    (Randy Hutson, July 15, 2013)

X(5521) is the center of the hyperbola {A,B,C,X(4),X(19)}, and X(5521) = X(915)-of-Euler-triangle. (Randy Hutson, August 17, 2014)

X(5521) = midpoint of X(4) and X(915)

X(5521) lies on these lines:
{4, 100}, {11, 2969}, {113, 1829}, {117, 1828}, {118, 1824}, {120, 427}, {121, 1883}, {122, 3139}, {123, 867}, {127, 3140}, {403, 5146}, {431, 1842}, {1560, 1841}


X(5522) =  KIRIKAMI SIX CIRCLES IMAGE OF X(95)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)2(a4 + b4 + c4 - 2a2b2 - 2a2c2 - 4b2c2)(3a4 + b4 + c4 - 4a2b2 - 4a2c2 - 2b2c2)    (Peter Moses, July 10, 2013)

Kirikami six circles images are introduced at X(5509).

X(5522) lies on these lines: {113, 3091}, {132, 5064}, {133, 1906}, {2970, 5139}


X(5523) =  ORTHOASSOCIATE (BUREK CONCURRENT CIRCLES IMAGE) OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 - c2)(a2 - b2 + c2)(b6 + c6 - a4b2 - a4c2 + 2a2b2c2 - b4c2 - b2c4)    (Peter Moses, June 15, 2013)

Let P be a point in the plane of a triangle ABC with orthocenter H. Let DEF be the orthic triangle of ABC. The circumcircles of APD, BPE, CPF concur in two points, P and Q. The point Q = Q(P) is the orthoassociate, or Burek concurrent circles image, of P. Examples: Q(X(1)) = X(1785), Q(X(2)) = X(468), Q(X(3)) = X(403), Q(X(4)) = X(4), Q(X(5)) = X(186).    (Dominic Burek, July 15, 2013)

The mapping Q is included as an orthoassociate of P in Bernard Gibert's paper, "Orthocorrespondence and Orthopivotal Cubics," Forum Geometricorum 3 (2003) 1-27. If P is given by barycentrics p : q : r, then Q is given by

Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = SBSC[(pq + pr)SA - q2SB - r2SC].

Properties reported by Randy Hutson, July 15, 2013:
X(5523) = reflection of X(112) in the orthic axis
X(5523) = inverse-in-polar-circle of X(6)
X(5523) = radical trace of the polar circle and the orthosymmedial circle
X(5523) = pole with respect to the polar circle of the line X(6)X(525)
X(5523) = X(48)-isoconjugate of X(2373)

If you have The Geometer's Sketchpad, you can view X(5523) and X(5523) generalized. The latter has a movable point P.

X(5523) lies on these lines:
{4,6},{24,3767},{30,112},{39,1594},{111,468},{115,232},{186,230},{297,525},{316,648},{378,2549},{382,3172},{427,1180},{459,5485},{858,1560},{1300,2715},{1783,5080},{3575,5305},{5024,5094}


X(5524) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + b2 + c2 + 3bc - 3ab -3ac)    (Peter Moses, June 16, 2013)

Let P be a point in the plane of a triangle ABC with orthocenter H. Let DEF be the excentral triangle of ABC. The circumcircles of APD, BPE, CPF concur in two points, P and Q. The point Q = Q(P) is the Gibert-Burek-Moses concurrent circles image of P. The points X(1), P, Q(P) are collinear, and circumcenters of APD, BPE, CPF are collinear. Let L denote the line of the circumcenters; then Q is the reflection of P in L. Examples: Q(X(3)) = X(484), Q(X(4)) = X(3465), Q(X(15)) = X(1276), Q(X(16)) = X(1277), Q(X(20)) = X(5018).    (Peter Moses, June 16, 2013)

The appearance of (I,J) in the following list means that Q(X(I)) = X(J):
(36, 40), (46, 2077), (74, 3464), (100,1054), (105,1282), (109, 1768), (165, 1155), (759, 2948), (1381, 2449), (1382, 2448)    (Randy Hutson, July 19, 2013)

Let O denote the imaginary circle with center X(1) and squared radius -4rR. Then Q(P) is the O-inverse of P; see Bernard Gibert's "Antiorthocorrespondents of Circumconics," Forum Geometricorum 3 (2003) 231-249. Accordingly, if U is an arbitrary circle, then Q(U) is a circle; here "circle" includes lines, regarded as circles of infinite radius. Examples: Q(circumcircle) = Bevan circle; Q(Euler line) is a circle with center X(3737); Q(antiorthic axis) is a circle with center X(3476); If P is a point on the circumcircle, then Q(P) is the Brisse transform of P with respect to the tangential triangle of the excentral triangle of ABC. (This paragraph is based on notes received from Bernard Gibert and Randy Hutson, July 17-19, 2013.)

Let P be given by barycentrics p : q : r. Then Q is given by

Q = g(a,b,c,p,q,r) : g(b,c,a,q,r,p) : g(c,a,b,r,p,q), where g(a,b,c,p,q,r) = a[bcp2 - caq2 - abr2 + (a - b - c)(aqr - bpr - cpq)]    (Peter Moses, June 16, 2013)

If you have The Geometer's Sketchpad, you can view X(5524) and X(5524) generalized. The latter has a movable point P.

X(5524) lies on these lines:
(pending)


X(5525) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - b3 - c3 - a2b - a2c + ab2 + ac2 + 3abc - b2c - bc2)    (Peter Moses, June 16, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524).

X(5525) lies on these lines:
{1,6},{35,3991},{36,3693},{46,728},{101,2752},{111,2748},{169,3632},{191,1334},{346,4293},{484,1018},{644,758},{1759,3208},{1781,2321},{2082,3633},{3065,4876},{3218,3912},{3309,4790},{3336,3501},{3950,4304}


X(5526) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a2 + b2 + c2 - 2ab - 2ac + bc)    (Peter Moses, June 16, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524).

X(5526) lies on these lines:
{1,6},{35,41},{36,101},{71,2301},{80,294},{115,2238},{187,1017},{484,910},{517,2348},{519,644},{573,3217},{645,5209},{650,1734},{651,1323},{739,2748},{902,1110},{908,3008},{1018,3684},{1334,3746},{1783,1785},{1795,2338},{2246,3245},{2291,2742},{2503,5164},{3509,4880},{3632,4513},{3997,5276},{5219,5228}


X(5527) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(7)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - 3a5(b + c) + a4(3b2 + 3c2 + 7bc) - 2a3(b + c)(b2 + c2) + 3a2(b2 + c2)(b - c)2 - a(b + c)(b - c)2(3b2 + 3c2 - 2bc) + (b - c)4(b2 + c2 + 3bc)    (César Lozada, August 15, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524).

X(5527) = X(316)-of-excentral triangle; also, X(5527) is the excentral isotomic conjugate of X(5528).    (Randy Hutson, July 18, 2013)

X(5527) = reflection of X(5536) in X(1308)

X(5527) lies on these lines: {1, 7}, {165, 5011}, {514,4105}, {1053, 2958}, {1308, 5536}, {1699,5074}


X(5528) =  REFLECTION OF X(9) IN X(100)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - 4a3(b + c) + a2(6b2 + 6c2 + bc) - a(4b3 + 4c3 - b2c - bc2) + (b -c)2(b2 + c2 + 4bc)    (Randy Hutson, July 18, 2013)

X(5528) is the antipode of X(9) in the rectangular hyperbola that passes through X(1), X(9), and the 3 excenters. Also, X(5528) is X(67)-of-the-excentral triangle, the excentral isogonal conjugate of X(5536), and the excentral isotomic conjugate of X(5527).    (Randy Hutson, July 18, 2013)

X(5528) lies on these lines: {1, 528}, {9, 100}, {11, 4329}, {2951, 5531}

X(5528) = reflection of X(I) in X(J) for these (I,J):
{1,528}, {9,100}, {11,4326}, {35,5506}, {40,2801}, {142,149}, {191,4436}, {518,3245}, {527,3935}, {971,2950}, {1317,4321}, {2136,3868}, {2802,3243}, {2949,3579}, {2951,5531}, {3020,3340}, {3646,5248}


X(5529) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(10)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - 2a2(b + c) - a(2b2 + 2c2 - bc) + (b + c)(b2 + c2 + bc)    (Randy Hutson, July 18, 2013)

X(5529) is the inverse-in-excircles-radical-circle of X(5530).   (Randy Hutson, July 18, 2013)

X(5529) lies on these lines:
{1,2}, {9,5110}, {36,1757}, {238,5440}, {404,1046}, {609,1743}, {758,1054}, {846,4256}, {982,3940}, {1326,5150}, {1739,4867}, {2948,5131}, {3667,4040}, {5400,5538}


X(5530) =  HUTSON RADICAL CIRCLE POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c) + a2(3b2 + 3c2 + 4bc) + a(b + c)(b2 + c2) - (b2 - c2)2    (César Lozada, August 15, 2013)

Let A' be the inverse-in-excircles-radical-circle of A, and define B' and C' cyclically. Let IA be the inverse-in-excircles-radical-circle of the A-excenter, and define IB and IC cyclically. The lines A'IA, B'IB, C'IC concur in X(5530).   (Randy Hutson, July 18, 2013)

X(5530) is the inverse-in-excircles-radical-circle of X(5529).   (Randy Hutson, July 18, 2013)

X(5530) lies on these lines:
{1,2}, {5,3931}, {12,3666}, {36,961}, {37,1329}, {46,573}, {65,970}, {171,580}, {181,942}, {226,986}, {388,988}, {429,1785}, {442,1738}, {517,1682}, {908,2292}, {968,2478}, {1686,2362}, {1695,2093}, {1838,1880}, {2051,4424}, {2476,3914}, {2886,4646}, {3596,4078}, {3614,4854}, {3663,3947}, {3743,3814}, {4339,5281}


X(5531) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(11)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5 - 3a4(b + c) + a3(2b + c)(b + 2c) + 2a2(b + c)(b2 + c2 - 3bc) - a(b - c)2(3b2 + 3c2 + 5bc) + (b + c)(b2 - c2)2

Four of the circles that are tangent to two of the sidelines BC, CA, AB pass through X(11), namely, the incircle and 3 others. The centers of those 3 are collinear. (See Barry Wolk's Hyacinthos messages #21431, #21433, etc., January 2013). Let A'B'C' be the triangle formed by the radical axes of these circles and the corresponding mixtilinear excircle. A'B'C' is homothetic to the hexyl triangle, and the center of homothety is X(5531). Moreover, X(5531) is the Fuhrmann-triangle-to-excentral triangle similarity image of X(40). Further, in the definition of X(5495), if A'B'C' is the excentral triangle, then the circumcircles of TA, TB, TC concur in X(5531). Also, X(5531) is the inverse of X(1) in the circumcircle of OA, OB, OC.    (Randy Hutson, July 18, 2013)

X(5531) lies on these lines:
{1,5}, {3,3711}, {40,2771}, {63,100}, {101,3119}, {104,4866}, {149,1699}, {153,3811}, {214,936}, {484,912}, {515,5538}, {516,3935}, {518,5536}, {528,1750}, {971,3689}, {1145,4882}, {1156,4326}, {1490,2800}, {1709,3158}, {2951,5528}, {3062,3174}, {3817,3957}, {4297,4420}

X(5531) = reflection of X(I) in X(J) for these (I,J): (1768,100), (2951,5528), (5537,3689)


X(5532) =  WOLK-FEUERBACH POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)4(b + c - a)3

X(5532) is the point of intersection, other than X(11), of the three collinear circles described at X(5531).    (Barry Wolk, Hyacinthos #21433, January 18, 2013)

X(5532) lies on these lines: {11,514}, {516,5183}, {1111,3323}, {1146,3022}, {2310,4041}, {3689,5199}, {4081,4163}


X(5533) =  INVERSE-IN-INCIRCLE OF X(5)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b + c - a)(b2 + c2 - 4bc) - 2a3(b4 + c4 - 3b3c - 3bc3 + 5b2c2) + 2a2(b + c)(b - c)2(b2 + c2 - bc) + a(b + c)2(b - c)4 - (b - c)(b2 - c2)3    (César Lozada, August 15, 2013)

X(5533) is the Gibert-Burek-Moses concurrent circles image of X(5534).    (Randy Hutson, July 18, 2013)

X(5533) = inverse-in-incircle of X(5), and X(5533) = {X(11),X(1317)}-harmonic conjugate of X(5).

X(5533) lies on these lines:
{1,5}, {100,499}, {104,1479}, {149,3086}, {528,3582}, {1145,3813}, {1647,1772}, {1737,2802}, {2829,3583}, {3036,4187}


X(5534) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(5533)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - 2a5(b +c) - a4(b + c)2 + 4a3(b + c)(b2 + c2) - a2(b4 + c4 + 6b2c2) - 2a(b + c)(b2 - c2)2 + (b + c)2(b2 - c2)2       (César Lozada, August 15, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). In the definition of X(5495), if A'B'C' is the excentral triangle, then X(5534) is the center of the circumcircle of OA, OB, OC. In this case, unlike that of X(5495), the circle does not also pass through O. Also, X(5534) = X(5)-of-3rd-antipedal-triangle-of-X(1).    (Randy Hutson, July 18, 2013)

X(5534) lies on these lines:
{1,5}, {3,200}, {4,3870}, {20,3935}, {40,912}, {78,944}, {84,3158}, {104,4855}, {515,3811}, {517,1490}, {936,1385}, {971,3174}, {1062,1103}, {1158,2801}, {1728,2078}, {1998,3149}, {2057,5440}, {3072,3751}, {3073,3749}, {3090,4666}, {3091,3957}, {3576,5258}


X(5535) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(35)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - a4(3b2 + 3c2 + bc) + a3bc(b + c) + a2(3b4 + 3c4 - b3c - bc3) - abc(b + c)(b - c)2 - (b + c)2(b - c)4    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5535) = inverse-in-Bevan-circle-of- X(3) = X(2070)-of-excentral-triangle = X(36)-of-tangential-triangle-of-excentral triangle.    (Randy Hutson, July 18, 2013)

X(5535) = midpoint of X(484) and X(5536)
X(5535) = reflection of X(I) in X(J) for these (I,J): (40,484), (104,4973), (2077,1155), (5180,946), (5538,3)
X(5535) = inverse-in-Bevan-circle of X(3)

X(5535) lies on these lines:
{1,3}, {5,191}, {9,3814}, {30,1768}, {63,5080}, {104,4973}, {442,2949}, {515,3218}, {535,3928}, {546,3652}, {912,4880}, {946,5180}, {1727,3583}, {2272,5011}, {3628,5506}


X(5536) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(55)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a5 - a4(b + c) - a3(2b2 + 2c2 - bc) + 2a2(b3 + c3) + a(b - c)2(b2 + c2 - bc) - (b + c)(b - c)4    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5536) = inverse-in-Bevan-circle-of-X(165) = X(23)-of-excentral-triangle = X(1155)-of-tangential-triangle-of-excentral-triangle = excentral isogonal conjugate of X(5528).    (Randy Hutson, July 18, 2013)

X(5536) = reflection of X(I) in X(J) for these (I,J): (484,5535), (1768,3218), (5527,1308), (5537,1155), (5538,36)
X(5536) = inverse-in-Bevan-circle of X(165)

X(5536) lies on these lines:
{1,3}, {9,5087}, {63,1699}, {103,1290}, {110,2717}, {149,516}, {191,946}, {411,3874}, {518,5531}, {672,2957}, {910,2323}, {1308,5527}, {1421,2361}, {1709,3928}, {1757,5400}, {2949,5506}, {3219,3817}


X(5537) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(57)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4 - 2a3(b + c) + 7a2bc + 2a(b + c)(b2 + c2 - 3bc) - (b - c)2(b2 + c2 + 3bc)]    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5537) is the radical trace of each pair of the 1st, 2nd, and 3rd antipedal circles of X(1); also, X(5537) = X(23)-of-1st-circumperp-triangle = X(858)-of-excentral-triangle.    (Randy Hutson, July 18, 2013)

X(5537) = reflection of X(I) in X(J) for these (I,J): (36,2077), (3245,40), (5526,2742), (5531,3689), (5536,1155)
X(5537) = inverse-in-circumcircle of X(165)

X(5537) lies on these lines: {1,3}, {20,535}, {100,516}, {103,677}, {105,2743}, {200,1709}, {404,4301}, {411,5493}, {480,3062}, {518,1768}, {840,1293}, {971,3689}, {972,2222}, {991,2177}, {1012,3679}, {1260,1750}, {1376,1699}, {1618,2272}, {2291,2742}, {2800,4867}, {2801,3935}, {3091,3814}, {3146,5080}, {3871,4297}, {5288,5450}


X(5538) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(65)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - 2a5(b + c) - a4(b2 + c2 - 7bc) + a3(b + c)(4b2 + 4c2 - 5bc) - a2(b2 + c2 - bc)(b2 + c2 + 6bc) - a(b + c)(b - c)2(2b2 + 2c2 - bc) + (b + c)2(b - c)4    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5538) = X(2070)-of-hexyl-triangle.    (Randy Hutson, July 18, 2013)

X(5538) = reflection of X(I) in X(J) for these (I,J): (484,2077), (5535,3), (5536,36)
X(5538) = inverse-in-hexyl-circle of X(3)

X(5538) lies on these lines:
{1,3}, {78,5080}, {200,5176}, {515,5531}, {516,4511}, {758,1768}, {936,3814}, {997,1699}, {1006,5426}, {5400,5529}


X(5539) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(99)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4bc - a3(b + c)(b - c)2 - a2bc(b2 + c2) - abc(b + c)(b - c)2 + b3c3    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5539) = X(1356)-of-tangential-triangle-of-excentral-triangle.    (Randy Hutson, July 18, 2013)

X(5539) = reflection of X(1) in X(741)

X(5539) lies on the Bevan circle and these lines: {1,99}, {9,3037}, {43,5213}, {57,1356}, {484,3510}, {1015,3571}, {1045,5541}, {1046,1282}, {1716,3464}, {1740,2948}, {2640,5540)


X(5540) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(101)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - a2(b + c) + a(b2 + c2 - bc) - (b + c)(b - c)2    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5540) = X(112)-of-excentral-triangle = X(1358)-of-tangential-triangle-of-excentral-triangle = excentral-isogonal-conjugate-of-X(3309) = trilinear-pole-with-respect-to-excentral-triangle-of-the-line-X(2)X(7)    (Randy Hutson, July 18, 2013)

X(5540) is the point of concurrence of the reflections of the line X(1)X(6) in the sides of the excentral triangle.    (Randy Hutson, August 14, 2013)

X(5540) = reflection of X(I) in X(J) for these (I,J): (1,105), (5526,2348)

X(5540) lies on the Bevan circle and these lines:
{1,41}, {6,1718}, {9,80}, {19,1743}, {35,1212}, {36,910}, {37,3196}, {43,3034}, {44,3245}, {57,1358}, {115,2503}, {120,1698}, {165,1292}, {190,4986}, {191,2795}, {484,672}, {517,2348}, {519,5525}, {579,3464}, {610,909}, {614,5354}, {644,2802}, {654,1768}, {657,2957}, {673,1111}, {952,4534}, {1023,4919}, {1053,4040}, {1054,1635}, {1475,3337}, {1697,3021}, {1699,5511}, {1713,2833}, {1723,2270}, {1724,2838}, {1731,2183}, {1766,3973}, {2173,5053}, {2238,5164}, {2448,2591}, {2449,2590}, {2640,5539}, {2814,5400}, {3336,4253}, {3583,5179}, {4875,5258}, {5030,5131}


X(5541) =  GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(106)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + a2(b + c) - a(b2 + c2 + 5bc) - (b + c)(b2 + c2 - 4bc)    (Randy Hutson, July 18, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). X(5541) = Bevan-circle-antipode-of-X(1768) = X(74)-of-excentral-triangle = {X(100), X(1320)}-harmonic-conjugate-of-X(214) = X(1317)-of-tangential-triangle-of-excentral-triangle. Also, X(5541) is the antipode of X(1) in the rectangular hyperbola that passes through X(1), X(9) and the 3 excenters, and X(5541) is the inverse of X(214) in the circumconic centered at X(1).    (Randy Hutson, July 18, 2013)

Let A'B'C' be the excentral triangle. Let EA be the Euler line of BCA'. Let LA be the line through A' parallel to EA, and define LB and LC cyclically. The lines LA, LB, LC concur in X(5541).    (Randy Hutson, August 14, 2013)

X(5541) = reflection of X(I) in X(J) for these (I,J): (1,100), (80,1145), (149,10), (1320,214), (1768,40), (4867,3689), (4880,5183)

X(5541) lies on the Bevan circle and these lines: {1,88}, {8,191}, {9,80}, {10,149}, {11,1697}, {36,2932}, {40,550}, {43,3032}, {46,2136}, {55,5426}, {57,1317}, {63,4677}, {104,165}, {119,1699}, {145,3336}, {153,516}, {190,4738}, {200,3899}, {484,519}, {515,2950}, {517,3689}, {518,3245}, {984,2805}, {1045,5539}, {1050,3216}, {1282,3887}, {1490,2800}, {1706,3035}, {1759,4050}, {2093,3174}, {2246,4752}, {2448,3307}, {2449,3308}, {2801,2951}, {2948,4730}, {3219,4669}, {3244,3337}, {3339,5083}, {3464,4707}, {3579,3893}, {3654,4863}, {3813,5445}, {3919,3957}, {3968,5284}, {4880,5183}, {5011,5525}


X(5542) =  MIDPOINT OF X(1) AND X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2(b + c) - 2a(b - c)2 - (b + c)(b - c)2    (Randy Hutson, July 18, 2013)

X(5542) = X(182)-of-intouch-triangle = {X(175), X(176)}-harmonic conjugate of X(5543).    (Randy Hutson, July 18, 2013)

X(5542) = midpoint of X(I) and X(J) for these (I,J): (1,7), (390,4312), (962,2951), (2550,3243), (3059,3555), (4295,4326)
X(5542) = reflection of X(I) on X(J) for these (I,J): (9,1125), (10,142)
X(5542) = complement of X(5223)

X lies on line X(I),X(J) for these (I,J):
{1,7}, {2,5223}, {6,4989}, {9,1125}, {10,141}, {11,118}, {35,2346}, {55,553}, {56,954}, {57,3475}, {75,4684}, {144,3616}, {320,3883}, {474,480}, {497,4654}, {519,1056}, {527,551}, {537,4078}, {673,4649}, {726,3950}, {938,5290}, {946,971}, {1086,3755}, {1155,4031}, {1210,3947}, {1386,4667}, {1445,3338}, {1836,3982}, {1870,1890}, {2321,4966}, {3008,3751}, {3059,3555}, {3242,4675}, {3244,4780}, {3295,5493}, {3452,3742}, {3649,4890}, {3720,4656}, {3748,4114}, {3782,4883}, {3790,3912}, {3873,4847}, {3911,4860}, {4061,4359}


X(5543) =  {X(175),X(176)}-HARMONIC CONJUGATE OF X(5542)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(5a2 + b2 + c2 - 6ab - 6ac - 2bc)    (Peter Moses, August 13, 2013)

X(5543) = {X(I),X(J)}-harmonic conjugate of X(K) for these (I,J,K): (1,7,3160), (1,4350,1442), (175,176,5542), (3945,4328,7), (5228,5308,5435).    (Randy Hutson, July 18, 2013 and Peter Moses, August 13, 2013)

X(5543) lies on these lines: {1,7},{85,3241},{354,3599},{1441,4460},{2295,5228},{3340,3598},{3772,5222}


X(5544) =  HIRIART-URRUTY MINIMIZER

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 - 4a2b2 - 4a2c2 + 3b4 + 3c4 - 26b2c2)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = a2(7S2 + 5S2A + SBSC)    (Peter Moses, August 30, 2013)

Let X be a point in the plane of a triangle ABC, and let A'B'C' be the pedal triangle of X. The sum |AX|2 + |BX|2 + |CX|2 + |A'X|2 + |B'X|2 + |C'X|2 is minimized by X = X(5544).    (Jean-Baptiste Hiriart-Urruty; Toulouse, France; August 30, 2013)

The minimal value is (4S4 - 12PT + 15S2T2)/(20S2T - 18P), where P = SASBSC and T = SA + SB + SC.   (Peter Moses, August 30, 2013)

X(5544) is the only point whose polar conic in the Thomson cubic (K002) is a circle. (Bernard Gibert, June 22, 2014)

If you have The Geometer's Sketchpad, you can view X(5544).

X(5544) lies on these lines:
{2,1351}, {3,373}, {110,5050}, {125,5055}, {154,182}, {354,3751}, {392,1482}, {3124,5024}, {3526,3527}

X(5542) = midpoint of X(3) and X(3531)


X(5545) =  ISOGONAL CONJUGATE OF X(4843)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b + c - a)(b + c + 3a)(b2 - c2)]

Suppose that P is a point in the plane of triangle ABC. Let A'B'C' be the anticevian triangle of P and let A''B''C'' be the 1st circumperp triangle. The locus of P for which the lines A'A'', B'B'', C'C'' concur is the union of the line X(1)X(6) and the conic U = {A, B, C, X(66), X(101), X(294), X(651)}; i.e., the isogonal conjugate of the Gergonne line. The conic U has center X5452) and is given by the trilinear equation

a(b + c - a)yz + b(c + a - b)zx + c(a + b - c)xy = 0.

For X on X(1)X(6)∪U, let F(X) be the point of concurrence. Then if X is on X(1)X(6), the image F(X) is on the line X(1)X(3); a pair (I,J) in the following list indicates that F(X(I)) = X(J): (1,165), (6,3), (9,40), (37,3579), (44,517), 281,55), 1713,1715), (1723,46), (1724,1754), (1743,1), (2323,2077), (5247,171), 5526,5537). On the other hand, if X is on U, the image F(X) is on the circumcircle; a pair (I,J) in the following list indicates that F(X(I)) = X(J): (101,109), (110,5543), (111,5543), (294,105), (644,100), (645,99), (651,934), (666,927), (1783,108), (2311,741), (2316,106), (4627,5545).   (César Lozada; August 29, 2013)

Suppose that P is on X(1)X(6). If P = p : q : r (trilinears), then F(P) = a(b + c - a)/[(b - c)p] : b(c + a - b)/[(c - a)q] : c(a + b - c)/[(a - b)r];
If P = p : q : r (barycentrics), then F(P) = a3(b + c - a)/[(b - c)p] : b3(c + a - b)/[(c - a)q] : c3(a + b - c)/[(a - b)r]
Suppose that P is on U. If P = p : q : r (trilinears), then F(P) = (b + c - a)p : (c + a - b)q : (a + b - c)r];
If P = p : q : r (barycentrics), then F(P) = then F(P) = (b + c - a)p : (c + a - b)q : (a + b - c)r].    (Peter Moses; September 2, 2013)

X(5545) = trilinear pole of the line X(6)X(1412)

X(5545) lies on the circumcircle and these lines:
{100,1414}, {101,4565}, {105,5323}, {835,4624}


X(5546) =  X(100)X(112)∩X(101)X(110)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 - b2)(a2 - c2)

X(5546) = F(X(110)), where F is the mapping defined at X(5545).    (Peter Moses; September 2, 2013)

X(5546) lies on these lines:
{9,1793},{21,294},{41,60},{58,1810},{99,666},{100,112},{101,110},{283,2338},{284,2316},{345,4548},{593,609},{643,644},{645,4612},{648,4552},{651,662},{672,5060},{910,1325},{1018,1021},{1333,1811},{1576,4557},{1625,2427},{1809,2193},{1951,4511},{1983,2610},{2251,5006},{2328,4845},{3732,4237},{3939,4587},{4556,4627},{5127,5526}


X(5547) =  POINT ARNEB

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(a2 + b2 - 2c2)(a2 - 2b2 + c2)

X(5547) = F(X(111)), where F is the mapping defined at X(5545).    (Peter Moses; September 2, 2013)

X(5547) lies on these lines:
{8,645},{42,101},{65,651},{210,644},{666,671},{1334,3939},{1783,1824},{2334,4627}


X(5548) =  POINT ASCELLUS AUSTRALIS

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)/[(b - c)(b + c - 2a)]

X(5548) = F(X(44)), where F is the mapping defined at X(5545).    (Peter Moses; September 3, 2013)

X(5548) lies on these lines:
{101,649},{106,5526},{294,1320},{644,650},{645,4560},{651,3257},{663,3939},{666,4555},{1318,2316},{1783,5375},{2340,4845},{2423,2427},{2429,2441},{4591,4627}


X(5549) =  POINT ASCELLUS BOREALIS

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)/[(b - c)(2b + 2c - a)]

X(5549) = F(X(45)), where F is the mapping defined at X(5545).    (Peter Moses; September 3, 2013)

X(5549) lies on these lines:
{21,2341},{41,2316},{101,4588},{294,2320},{651,4604},{666,4597},{1783,4242},{4558,4627}


X(5550) =  GARCIA POINT G(1/4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c) + (1/4)(a - b - c)
Barycentrics   5a +3b + 3c : 3a + 5b + 3c + 3a + 3b + 5c

Let O(A,t) be the circle with center A and radius t*r, where r is the inradius of triangle ABC. Define O(B,t) and O(C,t) cyclically. Of the two parallel lines tangent to O(B,t) and O(C,t), let TA be the closer to A, and define TB and TC cyclically. Let D = TB∩TC, and define E and F cyclically. Let A' be the midpoint of segment BC, and define B' and C' cyclically. Let A'' be the touchpoint of TA and the incircle of DEF. Then AA', BB', CC' concur and AA", BB", CC" concur.    (Emmanuel José Garcia; September 11, 2013)

The triangle DEF has incenter X(1) and is similar to ABC with dilation factor 1-t. Let G(t) = AA'∩BB'∩ CC' and GF(t) = AA''∩BB''∩ CC''. The point G(t) lies on the line X(1)X(2) and has barycentric coordinates given by

G(t) = a + b + c + (b + c - a)t : a + b + c + (c + a - b)t : a + b + c + (a + b - c)t

and satisfies |X(1)G(t)|/|X(2)G(T)| = 3(1 - t)/(2t).    (Peter Moses; September 12, 2013)

The point GF(t) lies on the Feuerbach hyperbola (the isogonal conjugate of the line X(1)X(3)) and has barycentric coordinates given (Peter Moses; September 12, 2013) by

GF(t) = f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)t]

In the following list, the appearance of {n,t} indicates that X(n) = G(t):

{1,1}, {2,0}, {10,-1}, {78,1+(2 R)/r}, {145,2}, {200,1+(4 R)/r}, {498,R/(2 r+R)}, {499,-(R/(2 r-R))}, {551,3/5}, {936,(r+2 R)/(r-2 R)}, {938,1+r/(2 R)}, {997,(r+R)/(r-R)}, {1125,1/3}, {1210,-((r+2 R)/(r-2 R))}, {1698,-(1/3)}, {1737,-((r+R)/(r-R))}, {3085,R/(r+R)}, {3086,-(R/(r-R))}, {3241,3/2}, {3244,5/3}, {3582,-((3 R)/(4 r-3 R))}, {3584,(3 R)/(4 r+3 R)}, {3616,1/2},{3617,-2}, {3621,4}, {3622,2/3}, {3623,4/3}, {3624,1/5}, {3625,7}, {3626,-5}, {3632,5}, {3633,7/3}, {3634,-(1/5)}, {3635,7/5}, {3636,5/7}, {3679,-3}, {3811,(r+3 R)/(r+R)}, {3828,-(3/7)}, {3870,(r+4 R)/(r+2 R)}, {3872,1-(2 R)/r}, {3935,(r+4 R)/(r+R)}, {3957,(r+4 R)/(r+3 R)}, {4420,1+(3 R)/r}, {4511,(r+R)/r}, {4666,(r+4 R)/(r+6 R)}, {4668,-7}, {4678,-4},v{4691,-(7/3)}, {4847,-1-(4 R)/r}, {4853,1-(4 R)/r}, {4861,1-R/r}, {4882,1+(8 R)/r}, {4915,1-(8 R)/r}, {5231,-((r+4 R)/(3 r))    (Peter Moses; September 14, 2013)

In the next list, the appearance of {n,t} indicates that X(n) = GF(t):

{1,1}, {7,0}, {8,2}, {9,(r+4 R)/(r+2 R)}, {21,(r+2 R)/(r+R)}, {79,-1}, {80,3}, {84,1+(2 R)/r}, {90,(r+3 R)/(r+R)}, {104,(r+R)/r}, {943,(r+3 R)/(r+2 R)}, {1000,3/2}, {1156,(r+4 R)/(r+R)}, {1320,(r-2 R)/(r-R)}, {1389,1-R/r}, {1392,(2 (r-R))/(2 r-R)}, {1476,r/(r-R)}, {2320,(2 (r+R))/(2 r+R)}, {2346,(r+4 R)/(r+3 R)}, {3062,1+(4 R)/r}, {3065,(2 r+5 R)/(2 r+R)}, {3254,-((r+4 R)/(r-2 R))}, {3255,(r+4 R)/(3 r+2 R)}, {3296,1/2}, {3427,(2 (r+R))/r}, {3467,(2 r+7 R)/(2 r+3 R)}, {3577,1-(2 R)/r}, {3680,(r-4 R)/(r-2 R)}, {4866,(r+8 R)/(r+4 R)}, {4900,(r-8 R)/(r-4 R)}, {5424,(4 r+7 R)/(4 r+5 R)}    (Peter Moses; September 14, 2013)

X(5550) lies on these lines:
{1,2},{3,5284},{11,4197},{12,4308},{21,4423},{44,5296},{56,5047},{63,3646},{65,3848},{210,3889},{226,5265},{354,3876},{355,5067},{377,5225},{404,1001},{405,5253},{474,1621},{515,5056},{517,3525},{631,962},{632,1482},{756,3976},{944,1656},{946,3523},{952,5070},{958,5328},{999,5260},{1155,5180},{1385,3090},{1386,3619},{1420,5261},{1479,5444},{1699,3522},{1788,4323},{2098,5326},{2476,3816},{2478,5229},{3091,3576},{3146,3817},{3219,3338},{3246,4645},{3305,3333},{3485,5221},{3600,5219},{3601,5274},{3614,4193},{3618,4663},{3653,5071},{3678,4430},{3681,5045},{3697,5049},{3698,3885},{3742,3868},{3753,3890},{3812,3877},{3822,5154},{3825,5141},{3832,4297},{3869,5439},{3871,4413},{3873,4539},{3874,4532},{3881,4661},{3993,4772},{4188,5248},{4189,5259},{4295,5443},{4419,4798},{4747,4758},{4860,4999},{5080,5084},{5128,5250},{5177,5436}


X(5551) =  GARCIA-FEUERBACH POINT GF(1/4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(3b2 + 3c2 - 3a2 + 8bc)

Garcia-Feuerbach points are defined at X(5550).

X(5551) lies on these lines: {1,4114}, {8,4004}, {943,5204}


X(5552) =  GARCIA POINT GF(R/r)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c) + (R/r)(a - b - c)

Garcia points are defined at X(5550).

X(5552) lies on these lines:
{1,2},{3,3436},{4,100},{5,3434},{9,1195},{11,3913},{12,377},{20,2077},{21,2551},{40,908},{55,1329},{56,3035},{140,956},{149,5154},{318,406},{329,3359},{345,3701},{355,5440},{388,404},{405,3820},{442,1260},{452,5281},{474,495},{475,5081},{480,3826},{497,3871},{515,4855},{529,5204},{631,2975},{944,5176},{958,5432},{962,1519},{1056,5253},{1145,1482},{1213,3713},{1331,1771},{1478,4190},{1479,3814},{1621,5084},{1706,5219},{1788,3868},{1837,5123},{1877,4200},{2476,2550},{2899,4194},{3090,5082},{3256,5177},{3295,4187},{3303,3816},{3452,5250},{3524,5303},{4188,4293},{4294,5046},{5193,5265}


X(5553) =  GARCIA-FEUERBACH POINT GF(R/r)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)R/r]

Garcia-Feuerbach points are defined at X(5550).

X(5553) lies on these lines:
{8,912},{9,2252},{21,2096},{84,1519},{90,499},{944,1320},{962,1392},{1389,4295}


X(5554) =  GARCIA POINT G(r/R)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c) + (r/R)(a - b - c)

Garcia points are defined at X(5550).

X(5554) lies on these lines:
{1,2}, {20,3359}, {63,4848}, {65,3436}, {100,3486}, {119,2476}, {355,377}, {388,5176}, {404,944}, {474,952}, {515,4190}, {517,2478}, {529,5221}, {631,3897}, {908,3340}, {946,5187}, {962,5046}, {1058,3885}, {1145,3295}, {1220,2994}, {1329,2099}, {1470,1788}, {1478,3754}, {1482,4187}, {1519,3091}, {1837,3434}, {2077,4189}, {2098,3816}, {2550,5086}, {2551,3869}, {3256,5273}, {3421,3868}, {3476,5253}, {3488,3871}, {3812,5252}, {3877,5084}, {4295,5080}, {4308,5193}


X(5555) =  GARCIA-FEUERBACH POINT GF(r/R)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)r/R]

Garcia-Feuerbach points are defined at X(5550).

X(5555) lies on these lines:
{21,1470}, {90,1210}, {388,1320}, {497,1476}, {943,5281}, {1039,1877}, {1392,4323}, {3434,3680}


X(5556) =  GARCIA-FEUERBACH POINT GF(-2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) + 2(b2 + c2 - a2)]

Garcia-Feuerbach points are defined at X(5550).

X(5556) = orthocenter of triangle X(1)X(4)X(8) (Randy Hutson, November 22, 2014 )

X(5556) lies on these lines:
{1,3146}, {7,5225}, {8,1836}, {9,5128}, {21,4423}, {79,938}, {80,4295}, {962,1000}, {1156,5221}, {3474,3614}, {3617,4866}, {3621,4900}, {3832,4312}, {5217,5226}


X(5557) =  GARCIA-FEUERBACH POINT GF(1/3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)(1/3)]

Garcia-Feuerbach points are defined at X(5550).

X(5557) = isogonal conjugate of X(3746)

X(5557) lies on these lines:
{1, 550}, {8, 2891}, {9, 583}, {21, 551}, {35, 2346}, {36, 943}, {79, 354}, {80, 942}, {90, 3333}, {140, 3337}, {256, 3953}, {553, 3746}, {1320, 3635}, {1385, 5424}, {1389, 5425}, {1476, 3671}, {1656, 4860}, {3065, 3649}, {3467, 5443}, {3487, 5444}


X(5558) =  GARCIA-FEUERBACH POINT GF(2/3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)(2/3)]

Garcia-Feuerbach points are defined at X(5550).

X(5558) = isogonal conjugate of X(3303)

X(5558) lies on these lines:
{1, 3522}, {2, 4866}, {4, 5045}, {8, 354}, {9, 1475}, {21, 3304}, {56, 2346}, {80, 938}, {145, 4900}, {942, 1000}, {943, 999}, {962, 3296}, {1156, 3485}, {1476, 4323}, {3062, 5542}, {3241, 3680}, {3333, 3523}, {3854, 5290}, {4298, 5059}


X(5559) =  GARCIA-FEUERBACH POINT GF(5/3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - (b2 + c2 - a2)(5/3)]

Garcia-Feuerbach points are defined at X(5550).

X(5559) lies on these lines:
{1, 140}, {2, 1392}, {8, 3884}, {9, 3632}, {10, 1320}, {21, 519}, {35, 104}, {36, 1476}, {79, 517}, {80, 3057}, {84, 5119}, {90, 1697}, {145, 2320}, {314, 3264}, {518, 3255}, {952, 3065}, {1389, 5443}, {1656, 2098}, {3254, 4553}, {3679, 3680}, {4668, 4900}, {4677, 4866}


X(5560) =  GARCIA-FEUERBACH POINT GF(5)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) - 5(b2 + c2 - a2)]

Garcia-Feuerbach points are defined at X(5550).

X(5560) lies on these lines:
{1, 381}, {7, 3585}, {8, 3583}, {21, 1698}, {40, 3467}, {46, 3065}, {79, 1837}, {84, 3336}, {90, 484}, {943, 3586}, {1000, 1479}, {1125, 2320}, {1320, 3633}, {1389, 1699}, {1392, 3244}, {1478, 3296}


X(5561) =  GARCIA-FEUERBACH POINT GF(-3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(a + b + c)(b + c - a) + 3(b2 + c2 - a2)]

Garcia-Feuerbach points are defined at X(5550).

Let A'B'C' be the cevian triangle of X(1) with respect to the incentral triangle. Let A'' be the reflection of A' in BC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5561). (Randy Hutson, August 17, 2014)

X(5561) = isogonal conjugate of X(5010)

X(5561) lies on these lines:
{1, 382}, {7, 3583}, {8, 3585}, {9, 484}, {21, 3624}, {30, 5424}, {46, 3467}, {57, 3065}, {80, 1836}, {84, 3337}, {90, 3336}, {104, 1699}, {551, 2320}, {1000, 1478}, {1156, 4312}, {1392, 3635}, {1479, 3296}, {1770, 3634}, {3830, 5425}, {5010, 5219}


X(5562) =  REFLECTION OF X(52) IN X(5)

Trilinears        cos2A cos(B - C) : cos2B cos(C - A) : cos2C cos(A - B)
Trilinears        (cos A)(cos 2B + cos 2C) : (cos B)(cos 2C + cos 2A) : (cos C)(cos 2A + cos 2B)
Barycentrics   (cot A)(csc 2B + csc 2C) : (cot B)(csc 2C + csc 2A) : (cot C)(csc 2A + csc 2B)
Barycentrics   (sin 2A)(cos 2B + cos 2C) : (sin 2B)(cos2C + cos 2A) : (sin 2C)(cos 2A + cos 2B)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2)2[a2b2 + a2c2 - (b2 - c2)2]
X(5562) = 4X(5) - 3*X(51)

Let A' be the point, other than A, in which the line parallel to BC meets the circumcircle of ABC, and define B' and C' cyclically. Let PA be the point in which the line through A' perpendicular to BC meets BC, and define PB and PC cyclically; the points PA, PB, PC are collinear, forming the so-called Simson line of A'. The Simson lines of A', B', C' concur in X(5562).      (Dao Thanh Oai, October 2, 2013)

X(5562) = isotomic conjugate of isogonal conjugate of X(418), and A'B'C' is homothetic to the orthic triangle of ABC from X(2) with ratio -2.      (Peter Moses, October 4, 2013)

Let A'B'C' be the cevian triangle of X(3). Let A''B''C'' be the reflection of A'B'C' in X(3). Let A*B*C be the tangential triangle, with respect to A'B'C', of the circumconic of A'B'C' centered at X(3) (that is, the bicevian conic of X(3) and X(394)). The lines A''B*, B''B*, C''C* concur in X(5562). (Randy Hutson, August 17, 2014)

The following items were contributed by Randy Hutson, August 17, 2014:
X(5562) = X(20)-of-orthic-triangle
X(5562)-of-excentral-triangle = X(20)
X(5562)-of-hexyl-triangle = X(4)
X(5562)-of-intouch-triangle = X(950)
X(5562) is the QA-P5 center (Isotomic Center) of the quadrangle ABCX(4); see Isotomic Center
. X(5562) is the QA-P37 center of quadrangle ABCX(4); see QA-P37. (For certain special cases of quadrangles, such as orthocentric systems, some QA points coincide.)

The tangents at A, B, C to the Euler central cubic (K044) concur in X(5562), which lies on the Euler central cubic. (Randy Hutson, November 22, 2014)

If you have The Geometer's Sketchpad, you can view X(5562).

X(5562) lies on the hyperbola {A,B,C,X(4),X(51)} and these lines:
{2,389},{3,49},{4,69},{5,51},{20,2979},{26,1495},{39,3289},{40,2807},{99,1298},{146,2889},{159,1350},{195,567},{216,217},{255,1364},{265,3519},{373,568},{381,5446},{399,2918},{417,2972},{520,5489},{542,1205},{575,1199},{578,1993},{631,3819},{916,1071},{970,1812},{1060,1425},{1062,3270},{1503,3313},{2055,3463},{2072,5449},{2781,2883},{2818,3869},{2888,3153},{3060,3091},{3090,3567},{3564,4173},{3719,4158}

X(5562) = reflection of X(I) in X(J) for these (I,J): (185,3), (52,5), (3,1216), (1843,1352)
X(5562) = complement of X(5889)
X(5562) = anticomplement of X(389)
X(5662) = X(343)-Ceva conjugate of X(216)
X(5662) = crosspoint of X(3) and X(68)
X(5662) = crosssum of X(4) and X(24)
X(5662) = crossdifference of every pair of points on the line X(421)X(2501)


X(5563) =  ISOGONAL CONJUGATE OF X(5559)

Trilinears        3 - 2 cos A : 3 - 2 cos B : 3 - 2 cos C
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 - a2 - 3bc)
X(5563) = 3R*X(1) - 2r*X(3)

X(5563) lies on these lines:
{1,3},{2,5258},{4,4317},{5,3582},{10,5253},{11,546},{12,3628},{21,551},{23,5322},{30,4325},{58,106},{61,5357},{62,5353},{73,1173},{79,104},{80,1210},{100,3244},{101,1475},{140,3584},{172,1015},{191,392},{202,2307},{214,3881},{226,5443},{229,759},{376,4309},{388,499},{404,519},{474,3679},{495,632},{496,3583},{497,3529},{498,1056},{529,4187},{535,5046},{550,3058},{575,1428},{576,1469},{595,1149},{614,1995},{908,1125},{956,1698},{958,3624},{993,3616},{995,1203},{997,3984},{1014,3663},{1054,3987},{1055,4251},{1058,4302},{1066,1450},{1106,1497},{1108,1781},{1124,3592},{1250,5237},{1283,1623},{1290,2718},{1334,5030},{1335,3594},{1376,3632},{1398,5198},{1449,2178},{1478,3086},{1479,3146},{1621,3636},{1696,3973},{1804,4328},{1866,1870},{2067,3299},{2163,3445},{2242,2275},{3085,5265},{3218,3878},{3241,4188},{3530,4995},{3560,4654},{3622,5248},{3635,3871},{3723,5124},{3731,5120},{3754,4861},{3825,5080},{3868,4867},{3869,4880},{3874,4511},{3884,4973},{3892,4881},{3911,5445},{3915,4257},{4225,4658},{4253,5526},{4297,5441},{5302,5506}


X(5564) =  ISOTOMIC CONJUGATE OF X(5557)

Trilinears        a2(3 + 2 cos A) : b23 - 2 cos B : c23 - 2 cos C
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2 + 3bc
X(5564) = r(r + 4R)*X(7) + 5s2*X(8)

X(5564) lies on these lines:
{2,3723},{7,8},{10,4360},{86,519},{190,3686},{239,594},{314,3264},{316,5015},{321,4886},{326,4853},{333,3977},{350,4651},{527,4545},{536,1654},{668,1269},{872,4489},{894,3629},{966,4664},{1086,4478},{1125,1268},{1213,4971},{1266,4746},{1267,3595},{1278,4643},{2345,3759},{2895,4980},{3593,5391},{3619,4402},{3625,3879},{3626,4357},{3661,3763},{3662,4445},{3663,4669},{3664,4701},{3672,4678},{3679,3875},{3729,4034},{3757,4046},{3912,4060},{3975,4043},{4007,4384},{4021,4691},{4389,4668},{4419,4764},{4440,4726},{4675,4772},{4686,4690},{4698,4727},{4699,4851},{4741,4821}


X(5565) =  OUTER DOMINICAN IMAGE OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a4b4 + a4c4 - a2b6 - a2c6 + a2b4c2 + a2b2c4 + 4a2bc(b2 + c2)S - 2b4c4]

Let X be a point in the plane of a triangle ABC. Let A* = AX∩BC, and define B* and C* cyclically. Let O(BA*) be the circle having diameter BA* and O(A*C) the circle having diameter A*C. There are two lines tangent to the circles O(BA*) and O(A*C). Let UA be the inner one (i.e., closer to A) and VA the outer. Define UB and UC cyclically and VB and VC cyclically. Let A' = VB∩VC, and define B' and C' cyclically. The lines AA', BB', CC' concur in the outer Dominican image of X, denoted by D(X). Let A'' = UB∩UC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in the inner Dominican image of X, denoted by E(X).      (Emmanuel José García, September 28, 2013)

Suppose that X = x : y : z (barycentrics). Let

f(a,b,c,x,y,z) = a2[a2y2z2 - x2(2(y + z)(yz)1/2S + y2SB + z2SC)] and
g(a,b,c,x,y,z) = a2[a2y2z2 - x2(2(y + z)(yz)1/2(- S) + y2SB + z2SC)].

Then D(X) = f(a,b,c,x,y,z) : f(b,c,a,y,z,x) : f(c,a,b,z,x,y) and E(X) = g(a,b,c,x,y,z) : g(b,c,a,y,z,x) : g(c,a,b,z,x,y).

Note that D(X) and E(X) lie in the real plane of ABC if and only if X lies inside ABC; equivalently, yz >0, zx > 0, xy > 0.      (Peter Moses, September 30, 2013)

If the construction is modified by using the A-internal tangent and the B- and C- external tangents, the resulting triangle is perspective to ABC, and likewise for 5 other perspectivities, for a total of 8 perspectors, of which only two (D(X) and E(X)) are central if X is central. The 8 perspectors are given by barycentrics

a2[a2y2z2 - x2(2(y + z)(yz)1/2S*i + y2SB + z2SC )] : b2[b2z2x2 - y2(2(z + x)(zx)1/2S*j + z2SB + x2SC )] : c2[c2x2y2 - z2(2(x + y)(xy)1/2S*k + x2SB + y2SC)],

where (i,j,k) ranges through 8 3-tuples listed here as additive-inverse pairs: (-1,-1,-1) & (1,1,1), (-1,-1,1) & (1,1,-1), (-1,1,-1) & (1,-1,1), (-1,1,1) & (1,-1,-1). Each pair determines a line, and the four lines concur in the point having 1st barycentric

a2t/(t2 - w2), where t = x2(y2SB + z2SC - a2y2z2, w = 2x2(y + z)(yz)1/2S.

The 4 lines determined by pairs differing only in the first coordinate, such as (-1,1,1) & (1,1,1), concur in A; those 4 differing only in the 2nd coordinate concur in B, and those 4 differing only in the 3rd coordinate concur in C.      (Peter Moses, October 1, 2013)

X(5565) lies on these lines: (pending)


X(5566) =  INNER DOMINICAN IMAGE OF X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a4b4 + a4c4 - a2b6 - a2c6 + a2b4c2 + a2b2c4 - 4a2bc(b2 + c2)S - 2b4c4]

For definitions and discussion, see X(5565).

X(5566) lies on these lines: (pending)


X(5567) =  OUTER DOMINICAN IMAGE OF X(76)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/[2a6 - b6 - c6 - a2b4 - a2c4 + b4c2 + b2c4 - 4bc(b2 + c2)S]

For definitions and discussion, see X(5565).

X(5567) lies on these lines: (pending)


X(5568) =  INNER DOMINICAN IMAGE OF X(76)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/[2a6 - b6 - c6 - a2b4 - a2c4 + b4c2 + b2c4 + 4bc(b2 + c2)S]

For definitions and discussion, see X(5565).

X(5568) lies on these lines: (pending)


X(5569) =  CENTER OF THE DAO 6-POINT CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 7a4 + b4 + c4 - 7a2b2 - 7a2c2 - 4b2c2

Let ABC be a triangle, and let AB be the center of the circle through A and tangent to the B-median at X(2), and define BC and CA cyclically. Let AC be the center of the circle through A and tangent to the C-median at X(2), and define BA and CB cyclically. The points AB, BA, BC, CB, CA, AC lie on a circle, of which X(5569) is the center.      (Dao Thanh Oai, Nov. 3, 2013)

The following properties were communicated by Peter Moses, November 4, 2013. Let Δ = area of ABC, r = radius of the Dao 6-point circle, and ω = Brocard angle of ABC. Let fa = 2b2 + 2c2 - a2, and define fb and fc cyclically. Then

r = [fafbfc(b2c2 + c2a2 + a2b2)]1/2/(144Δ)2

|ABBA| = |BCCB| = |CAAC| = [fafbfc]1/2/(36Δ)

Let X = X(5569). Then angle(ABXBA) = angle(BCXCB) = angle(CAXAC) = Tan-1[(a2 + b2 + c2)/(4Δ)]

angle(ABBAX) = angle(BCCBX) = angle(CAACX) = π/2 - ω

If you have The Geometer's Sketchpad, you can view X(5569).

X(5569) lies on these lines: {2,187}, {3,543}, {182,524}, {183,2482}, {538,3524}, {599,620}, {754, 5054}, {3406, 5503}, {5077,5461}

X(5569) = reflection of X(2) in X(1153)


X(5570) =  INVERSE-IN-INCIRCLE OF X(3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a(a5b + a5c - a4b2 - a4c2 - 2a3b3 - 2a3c3 + 2a2b4 + 2a2c4 - 2a2b3c - 2a2bc3 + 4a2b2c2 + ab5 + ac5 - ab4c - abc4 - b6 - c6 + 2b5c + 2bc5 + b4c2 + b2c4 - 4b3c3)
X(5570) = (r2 + 2rR - R2)*X(1) - r2*X(3)   (Peter Moses, November 9, 2013)

X(5570) = X(2072)-of-intouch-triangle. (Randy Hutson, July 18, 2014)

X(5570) lies on these lines:
{1,3}, {11,912}, {72,499}, {496,1858}, {498,5439}, {515,5083}, {518,1737}, {938,5080}, {971,3583}, {1066,1393}, {1071,1479}, {1210,3814}, {1785,1876}, {2771,5533}, {3086,3868}, {3873,5176}


X(5571) =  X(1) OF INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[(b + c - a)(ab + ac - (b - c)2)sin(A/2) + b(c + a - b)2sin(B/2) + c(a + b - c)2Sin(C/2)]

Let ABC be a triangle. Let A' = (inverse-in-incircle) of A, and define B' and C' cyclically. The triangle with vertices A,' B, C' is introduced here as the inverse-in-incircle triangle. A'B'C' is a central triangle of type 2. The following properties were communicated by Peter Moses, November 9, 2013:

Barycentric coordinates for vertices:
A' = (b - c)2 - ab - ac : b(a - b - c) : c(a - b - c)
B' = a(b - c - a) : (c - a)2 - bc - ba : c(b - c - a)
C' = a(c - a - b) : b(c - a - b) : (a - b)2 : a(c - a - b).

|B'C'|2 = (a - b - c)2[a2 - (b - c)2]/(16bc)
area(A'B'C')/area(ABC) = (b + c - a)(c + a - b)(a + b - c)/(16abc)

X(354) = centroid of A'B'C'
X(942) = circumcenter of A'B'C'
X(1) = orthocenter of A'B'C'
X(5045) = nine-point center of A'B'C'

The following triangles are perspective to A'B'C', with perspector X(1): reflection of T in X(3), excentral, incentral, mid-arc, circum-mid-arc, mixtilinear, 1st circumperp. Also, A'B'C' is perspective to other central triangles, with perspectors as shown:

medial, X(142)
intouch, X(354)
hexyl, X(3333)
2nd circumperp, X(57)

X(5571) = X(10)-of-intouch-triangle. (Randy Hutson, July 18, 2014)

X(5571) lies on these lines:
{1,164}, {65,209}, {177,354}


X(5572) =  X(6) OF INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a(a3b + a3c - 3a2b2 - 3a2c2 + 3ab3 + 3ac3 - 3ab2c - 3abc2 - b4 - c4 + 2b3c + 2bc3 - 2b2c2)

See X(5571) for the inverse-in-incircle triangle.

X(5572) = X(7) - 3X(354)   (Peter Moses, November 9, 2013)

X(5572) = X(141)-of-intouch-triangle. Let A' be the inverse-in-incircle of the A-excenter, and define B' and C' cyclically. Then X(5572) = X(9)-of-A'B'C'. (Randy Hutson, July 18, 2014)

X(5572) lies on these lines:
{1,6}, {2,3059}, {7,354}, {55,1445}, {57,4326}, {65,390}, {105,2264}, {142,2886}, {144,3873}, {241,2293}, {480,3870}, {516,942}, {938,2550}, {946,971}, {982,4335}, {1210,3826}, {1376,3174}, {1387,2801}

X(5572) = complement of X(3059)


X(5573) =  PERSPECTOR OF MEDIAL AND ANDROMEDA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 3b2 + 3c2 - 6bc)

Let A' be the center of the inverse-in-incircle of the A-excircle, and define B' and C' cyclically. The triangle with vertices A,' B, C' is introduced here as the Andromeda triangle. A'B'C' is a central triangle of type 2. The following properties were communicated by Peter Moses, November 9, 2013:

Barycentric coordinates for vertices:
A' = a[a2 + 3(b - c)2] : b[3a2 + (b - c)2] : c[3a2 + (b - c)2]
B' = a[3b2 + (c - a)2] : b[b2 + 3(c - a)2] : c[3b2 + (c - a)2]
C' = a[3c2 + (a - b)2] : b[3c2 + (a - b)2] : c[c2 + 3(a - b)2]

The triangle A'B'C' is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, circum-mid-crc, mixtilinear, 2nd circumperp. Also, A'B'C' is perspective to the intouch triangle at X(4907).

X(5573) lies on these lines:
{1,474}, {2,3677}, {9,982}, {31,57}, {43,3243}, {165,1279}, {223,3660}, {238,3928}, {354,2999}, {748,3929}, {988,5436}, {1054,3749}, {1086,1699}, {1104,3361}, {1191,3339}, {1201,3340}, {1261,3872}, {1420,3924}, {1453,3338}, {1722,3976}, {2276,3247}, {2886,4859}, {3305,4392}, {3306,5269}, {3315,3870}, {3452,4310}, {3756,3772}, {3915,5128}, {3999,4383}, {4003,4423}, {4666,4850}, {4907,5274}

X(5573) = complement of X(5423)


X(5574) =  PERSPECTOR OF MEDIAL AND ANTLIA TRIANGLES

Barycentrics   a(b + c - a)3(a2 + 3b2 + 3c2 - 6bc)

Let A' be the center of the inverse-in-A-excircle of the incircle, and define B' and C' cyclically. The triangle with vertices A,' B, C' is introduced here as the Antlia triangle. A'B'C' is a central triangle of type 2. The following properties were communicated by Peter Moses, November 10, 2013:

Barycentric coordinates for vertices:
A' = a[a2 + 3(b - c)2] : - b[3a2 + (b - c)2] : - c[3a2 + (b - c)2]
B' = - a[3b2 + (c - a)2] : b[b2 + 3(c - a)2] : - c[3b2 + (c - a)2]
C' = - a[3c2 + (a - b)2] : - b[3c2 + (a - b)2] : c[c2 + 3(a - b)2]

The triangle A'B'C' is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, circum-mid-crc, mixtilinear, 2nd circumperp.

X(5574) lies on these lines:
{2,479}, {9,165}, {200,3119}, {2391,3452}, {3041,5223}, {3817,5199}


X(5575) =  PERSPECTOR OF INTOUCH AND ANTLIA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a2 - (b - c)2][a2 + 3(b - c)2]

See X(5574) for the Antlia triangle.

X(5575) lies on these lines:
{7,346}, {57,1122}, {269,604}, {1463,5223}, {1469,3339}, {3062,4014}


X(5576) =  CENTER OF NINE-POINT-CIRCLE-INVERSE OF CIRCUMCIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b10 + c10 - a8(b2 + c2) + 2a6(b4 + c4 + b2c2) + 2a4b2c2(b2 + c2) - 2a2(b8 + c8 - 4b4c4) - 3b8c2 - 3b2c8 + 2b6c4 + 2b4c6
X(5576) = 3(-5 + J2)*X(2) + (7 - J2)*X(3), where J = |OH|/R. (See X(1113) for J = J(a,b,c).)    (Peter Moses, November 10, 2013)

X(5576) = inverse-in-orthocentroidal-circle of X(26) (Randy Hutson, July 18, 2014)

X(5576) lies on these lines:
{2,3},{51,5449},{125,5462},{143,3580},{195,3564},{511,1209},{524,3519},{570,1506},{1199,3448}


X(5577) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(106)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b - c)2[b2 + c2 - a2 - 4bc]2

The inverter of circles (U,u) and (V,v) is introduced here as the circle (W,w) such that (V,v) is the inverse-in-(W,w) of (U,u), where W is the insimilicenter of (U,u) and (V,v).

Peter Moses (Nov. 12, 2013) found representations for W and w, as follows. The center W of (W,w) is the combo uV + vU; that is, barycentrics for W are given by u(vA, vB, vC) + v(uA, uB, uC), where (uA, uB, uC) are normalized barycentrics for U, and (vA, vB, vC) are normalized barycentrics for V. The radius of (W,w) is w = sqrt[uv(1 - (|UV|/(u + v))2], so that the inverter is real if and only if u + v >= |UV|. Moses also gave properties for the case that (U,u) = (O,R) = circumcircle and (V,v) = (I,r) = incircle, for which the inverter is given by (W,w) = (X(55), (r/(r + R))sqrt(rR + 4R2)). The power of A with respect to (W,w) is

- abc(b + c - a)2/D, where D = 2(a3 + b3 + c2 - a2b - a2c - ab2 - ac2 - b2c - bc2); likewise, (power of B) = - abc(a - b + c)2/D and (power of C) = - abc(a + b - c)2/D.

The alternate inverter of circles (U,u) and (V,v) is introduced here as the circle (W',w') such that (V,v) is the inverse-in-(W,w) of (U,u), where W' is the exsimilicenter of (U,u) and (V,v). The center W' is the combo uV - vU, and the radius w' of W' is given by sqrt[uv(- 1 + (|UV|/(u - v))2], so that the alternate inverter is real if and only if |u - v| <= |UV|. (Peter Moses, September 3, 2014)

The appearance of (I,J) in the following list means that X(I) is on the circumcircle, X(J) is on the incircle, and each is the inverse-in-(W,w) of the other: (98, 5578), (99, 5579), (100, 3021), (101, 5580), (103, 1364), (105, 11), (106, 5577), (108, 1360), (109, 1362), (840, 3025), (934, 3321), (939, 5582), (972, 3318), (1381, 2447), (1382, 2446), (1477, 1357), (2222, 3322), (2291, 3022), (2384, 5583), (2717, 3326).

The barycentrics for X(5577) are of the form g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(b + c - a)(b - c)2[b2 + c2 - a2 - kbc]2, where k is a homogeneous of degree zero and symmetric in (a,b,c); every such point lies on the incircle. (Peter Moses, August 28, 2014)

If you have The Geometer's Sketchpad, you can view Inverter.

X(5577) lies on the incircle and these lines: {55,106}, {57,1361}, {244,1364}, {354,1317}, {1086,3326}, {1362,4860}, {3025,3271}, {3319,3660}


X(5578) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(98)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)2(a5 + a4b + a4c - a3b2 - a3c2 - a3bc - a2b3 - a2c3 - ab3c - abc3 - 2ab2c2 + b4c + bc4 - b3c2 - b2c3)2

Inverters are dicussed at X(5577).

X(5578) lies on the incircle and these lines: {55,98}, {354,1355}


X(5579) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(99)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a3b + a3c + a2b2 + a2c2 - 2a2bc - ab2c - abc2 + b3c - 2b2c2 + bc3)2

Inverters are discussed at X(5577).

X(5579) lies on the incircle and these lines: {11,4357}, {55,99}, {354,1356}, {1357,3664}, {1358,3666}, {1365,3663}


X(5580) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(101)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b3 + c3 + a2b + a2c - 2ab2 - 2ac2 + 2abc - b2c - bc2)2

Inverters are discussed at X(5577).

X(5580) lies on the incircle and these lines: {11,142}, {55,101}, {354,1358}, {1357,4860}, {1364,3056}, {1365,4890}


X(5581) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(739)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b - c)2(2a2b + 2a2c - 2ab2 - 2ac2 + abc + b2c + bc2)2

Inverters are discussed at X(5577).

X(5581) lies on the incircle and this line: {55,739}


X(5582) =  INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(2384)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c - a)(b - c)2(b3 + c3 - a3 + 5a2b + 5a2c - 5ab2 - 5ac2 - abc)

Inverters are discussed at X(5577).

X(5582) lies on the incircle and this line: {55, 2384}


X(5583) =  CENTER OF INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF EULER LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)(4a5 + b5 + c5 - a4b - a4c - 2a3b2 - 2a3c2 + 4a2b2c + 4a2bc2 - 2ab4 - 2ac4 + 4ab2c2 - 3b4c - 3bc4 + 2b3c2 + 2b2c3)

Inverters are discussed at X(5577).

X(5583) lies on these lines: (pending)


X(5584) =  PERSPECTOR OF EXTANGENTS AND APUS TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b5 + c5 - a5 + a4b + a4c + 2a3b2 + 2a3c2 + 4a3bc - 2a2b3 - 2a2c3 + 2a2b2c + 2a2bc2 - ab4 - ac4 - 4ab3c - 4abc3 - 6ab2c2 - 3b4c - 3bc4 + 2b3c2 + 2b2c3)

Let A' be the insimilicenter of the circumcircle and A-excircle, and define B' and C' cyclically. The triangle with vertices A,' B, C' is introduced here as the Apus triangle. A'B'C' is a central triangle of type 2. The following properties were communicated by Peter Moses, November 12, 2013:

Barycentric coordinates for vertices:
A' = a2(a - b + c)(a + b - c) : b2(b - c - a)(a + b + c) : c2(c - b - a)(a + b + c)
B' = a2(a - c - b)(a + b + c) : b2(b - c + a)(b + c - a) : c2(c - a - b)(a + b + c)
C' = a2(a - b - c)(a + b + c) : b2(b - a - c)(a + b + c) : c2(c - a + b)(c + a - b)

The triangle A'B'C' is perspective to ABC at X(55), the excentral and hexyl triangles at X(3), the incentral triangle at X(56), the tangential triangle at X(198), the Feuerbach triangle at X(4), and the Apollonius triangle at X(573).

The Apus triangle is the extraversion triangle of X(56). (Randy Hutson, July 18, 2014)

X(5584) lies on these lines:
{1,3},{4,3925},{6,4300},{19,1212},{20,958},{64,71},{72,480},{104,3528},{201,1854},{210,1490},{212,221},{218,573},{227,1035},{380,5120},{405,516},{411,1376},{946,4423},{954,3671},{956,4297},{962,1001},{1042,1253},{1151,5415},{1152,5416},{1204,3611},{1350,3779},{1407,1496},{1742,5247},{1753,1859},{1802,3207},{1804,3160},{2266,4258},{2951,5234},{2975,3522},{3146,5260},{3149,4413},{5248,5493}


X(5585) =  CENTER OF AQUARIUS CONIC

Barycentrics   a2(11b2 + 11c2 - 13a2) : b2(11c2 + 11a2 - 13b2) : c2(11a2 + 11b2 - 13c2)

Let A'B'C' be the tangential triangle, so that A' is the center of the circle OA through B and C that is orthogonal to the circumcircle (whence OA is self-inverse with respect to the circumcircle). Define OB and OC cyclically. Let O(A,B) be the circle which is the inverse-in-OA of OB; define O(B,C) and O(C,A) cyclically. Let O(A,C) be the circle which is the inverse-in-OA of OC; define O(B,A) and O(C,B) cyclically. The centers of these six circles lie on a conic, introduced here as the Aquarius conic, of which X(5585) is the center. The following properties were found by Peter Moses (Nov. 18, 2013).

The centers of the 6 circles are given by the following barycentrics:

- a2 : b2 : 3c2,       3a2 : - b2 : c2        a2 : 3b2 : - c2;
- a2 : 3b2 : c2,       a2 : - b2 : 3c2        3a2 : b2 : - c2

The radius of O(A,B) is abc/(-a2 + b2 + 3c2); the remaining 5 radii are found by cyclical and bicentric modifications. The Aquarius conic has equation

b4c4x2 + c4a4y2 + a4b4z2 + 11a2b2c2(a2yz + b2zx + c2xy) = 0

The major axis of the Aquarius conic is the Brocard axis, and the perspector is X(6).      (Randy Hutson, November 30, 2013)

X(5585) lies on these lines: {3,6}, {20,3054}, {439,3619}, {3055,3523}


X(5586) =  PERSPECTOR OF AQUILA AND INTOUCH TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b + c)(a + b - c)(3a + b + c)(a + 2b + 2c)

Let A' = reflection of the incenter in A, and define B' and C' cyclically. The triangle A'B'C' is introduced here as the Aquila triangle. The following properties were found by Peter Moses (Nov. 18, 2013).

A'' = a + 2b + 2c : - b - c      B'' = b + 2c + 2a      C'' = c + 2a + 2b;      area(A''B''C'') = 4*area(ABC).

The Aquila triangle is perspective to the following triangles with perspector X(1): ABC, excentral, incentral, mid arc, circum-mid-arc, 2nd circumperp, and mixtilinear. The Aquila triangle is perspective to other triangles with perspectors as listed here: medial, X(1698); anticomplementary, X(10); intouch, X(5586); Euler, X(1699); hexyl, X(1768); tangential 1st circumperp, X(35); tangential 2nd circumperp, X(36); Carnot, X(5587); outer Grebe, X(5588); inner Grebe, (X5589).

X(5586) lies on these lines:
{1,376},{7,10},{57,191},{65,3632},{145,4298},{388,4114},{942,4312},{986,4888},{1046,4859},{1317,3340},{1537,1768},{1698,3715},{1788,3982},{3361,3616},{3485,4031},{3600,3635},{3633,5434}


X(5587) =  PERSPECTOR OF AQUILA AND CARNOT TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b4 + 2c4 - a4 + a3b + a3c - a2b2 - a2c2 - 2a2bc - ab3 - ac3 + ab2c + abc2 - 4b2c2

The Aquila triangle is introduced at X(5586). Not only is X(5587) the perspector of the Aquila and Carnot triangles, but also, X(5587) is also the perspector of the Euler triangle and the outer Garcia triangle, defined as follows. Let TATBTC be the extouch triangle of a triangle ABC, and let LA be the line perpendicular to line BC at TA. Of the two points on LA at distance r from TA, let A' be the one farther from A and let A'' be the closer. Define B', C' and B'', C'' cyclically. We call A'B'C' the outer Garcia triangle and A''B''C'' the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014). In subsequent postings, Paul Yiu reports that A'B'C' is oppositely congruent to ABC at X(10), the Euler lines of the four triangles AB'C', BC'A', CA'B', ABC concur in X(2475), and the circles (BCA'), (CAB'), (ABC') concur in X(80). Peter Moses reports that

A' = -a : a + c : a + b,     B' = b + c : -b : b + a,     C' = c + b : c + a : -cj,

and Randy Hutson, that the inner Garcia triangle is the anticomplement of (orthic triangle of Fuhrmann triangle), that X(5587) = X(2)-of-Fuhrmann triangle, and X(5587) = {X(355), X(5)}-harmonic conjugate of X(1).

Let MA be the midpoint of segment BC and A' the reflection of X(1) in MA; define B' and C' cyclically. Then A'B'C' is the outer Garcia triangle. See Emmanuel José Garcia, A Note on Reflections.

Triangle A'B'C' is perspective to the cevian triangle of the points on the cubic (K366) and to the anticevian triangles of points on the cubic (K345), as well at the following triangles, with perspectors:

medial, X(1)
excentral, extouch, extangents, X(40)
anticomplementary, Fuhrmann, X(8)
circumcircle midard, 2nd circumperp, X(100)
tangential 1st circumperp, X(5587)
tangential 2nd circumperp, X(956)
Euler, X(5587)
outer Grebe, X(5688)
inner Grebe, X(5689)

The appearance of (I,J) is the following list means that (X(I) of A'B'C') = X(J): (1,8), (2,3679), (3,355), (4,40), (5, 5690), (6,3416), (7,5223), (8,1), (9,2550), (10,10), (11,1145), (20,5691). (Peter Moses, June 21, 2014)

X(5587) lies on these lines:
{1,5},{2,515},{3,1698},{4,9},{8,908},{30,165},{35,3560},{46,3585},{55,3586},{57,1478},{63,5080},{78,5086},{84,377},{145,5068},{149,3895},{153,3306},{200,3419},{210,381},{235,5090},{262,730},{265,2948},{282,1549},{376,3828},{382,3579},{388,1210},{404,5450},{411,5260},{442,1490},{498,3601},{499,1420},{519,3545},{547,3655},{551,5071},{631,3634},{912,4654},{936,1329},{938,5261},{942,5290},{944,1125},{950,3085},{956,5231},{958,3149},{962,3617},{997,3814},{1000,4342},{1012,1376},{1071,3812},{1158,2475},{1350,3844},{1352,3751},{1385,1656},{1453,5230},{1479,1697},{1482,3632},{1532,2886},{1537,3036},{1572,5475},{1702,3071},{1703,3070},{1709,3359},{1724,3072},{1750,3925},{1770,5128},{1771,1935},{1785,1857},{1788,4292},{1834,2910},{1836,2093},{1853,3753},{2364,5397},{2782,3097},{3073,5264},{3338,5270},{3416,5480},{3421,4847},{3487,3947},{3583,5119},{3616,5056},{3626,3855},{3633,5072},{3646,5084},{3654,3845},{3656,4677},{3850,4668},{3854,4678},{3872,5176},{3911,4293},{3949,4007},{4295,4848},{4304,5218},{5046,5250},{5087,5289}

X(5587) = midpoint of X(1699) and X(3679)
X(5587) = reflection of X(I) in X(J) for these (I,J): (1699, 381), (3576,2)
X(5587) = crossdifference of every pair of points on the line X(654)X(1459)


X(5588) =  PERSPECTOR OF AQUILA AND OUTER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - b2 - c2 + 2ab + 2ac - S)

The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5588) lies on these lines: {1,6},{10,1270},{40,1160},{1374,1738},{1698,5590}


X(5589) =  PERSPECTOR OF AQUILA AND INNER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - b2 - c2 + 2ab + 2ac + S)

The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5589) lies on these lines: {1,6},{10,1271},{40,1161},{1373,1738},{1698,5591}


X(5590) =  PERSPECTOR OF MEDIAL AND OUTER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 + S

The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5590) = {X(2), X(141)}-harmonic conjugate of X(5591)

X(5590) lies on these lines:
{2,6}, {3,5594}, {4,639}, {5,1160}, {8,5604}, {10,3640}, {76,5490}, {626,638}, {631,641}, {640,3090}, {642,3525}, {1162,1165}, {1267,3662}, {1698,5588}, {3535,5413}, {3661,5391}


X(5591) =  PERSPECTOR OF MEDIAL AND INNER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2 + c2 - S

The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5591) = {X(2), X(141)}-harmonic conjugate of X(5590)

X(5591) lies on these lines:
{2,6}, {3,5595}, {4,640}, {5,1161}, {8,5605}, {10,3641}, {76,5491}, {626,637}, {631,642},{639,3090},{641,3525}, {1163,1164}, {1267,3661}, {1698,5589}, {3536,5412}, {3662,5391}


X(5592) =  CIRCUMCENTER OF CEVIAN TRIANGLE OF X(190)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c) (3a3 + b3 + c3 - 2a2b - 2a2c - abc)

If you have The Geometer's Sketchpad, you can view X(5592).

X(5592) lies on these lines:
{1,514}, {20,3667}, {513,960}, {659,2785}, {661,5051}, {764,4778}, {1960,4458}, {2789,3762}, {2899,3239}


X(5593) =  CENTER OF YIU CONIC OF THE TANGENTIAL TRIANGLE (IF ABC IS ACUTE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2SA(SA - SB)(SA - SC)(g(a,b,c) - g(a,c,b)), where g(a,b,c) = [b2SBSB(S2 - SCSC)(4S4 - (S2 + SASC)(3S2 + SBSB - 2SASC))]

The Yiu conic is presented at X(478); it passes through the 6 of the nine touch-points of the sidelines of a triangle and the excircles of the triangle. When the triangle is the tangential, the conic has center X(5593).

Let u(a,b,c) = 4a2b4c4 and v(a,b,c) = a2(a8 + b8 + c8 - 2a6b2 - 2a6c2 + 2a4b4 + 2a4c4 - 2a2b6 - 2a2c6 + 6a2b4c2 + 6a2b2c4 - 4b6c2 - 4b6c2 + 6b4c4).

The Yiu conic of the tangential triangle of a triangle ABC is given by

u(a,b,c)x + u(b,c,a)y + u(c,a,b) z + v(a,b,c)yz + v(b,c,a)zx + v(c,a,b)xy = 0. (Peter Moses, Nov. 19, 2013)

If you have The Geometer's Sketchpad, you can view X(5593).

X(5593) lies on these lines:
{4,157}, {184,216}


X(5594) =  PERSPECTOR OF ARA AND OUTER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[a6 - b6 - c6 + a4b2 + a4c2 - a2b4- a2c4 - 2a2b2c2 + b4c2 + b2c4 + (a2 - b2 + c2)(a2 + b2 - c2)S]

Let A'B'C' be the tangential triangle of triangle ABC. Let A" be the center of the A'-excircle of A'B'C', unless this is also the circumcircle of ABC, in which case let A" be the incenter of A'B'C'. Define B", C" cyclically. The triangle A''B''C'' is introduced here as the Ara triangle, which appears in the sketch at X(5593). The vertices of the Ara triangle are given by Peter Moses (Nov. 19, 2013):

A'' = - a2(a2 + b2 + c2) : b2(a2 + b2 - c2) : c2(a2 - b2 + c2)
B'' = a2(b2 - c2 + a2) : - b2(b2 + c2 + a2) : c2(b2 + c2 - a2)
C'' = a2(c2 + a2 - b2) : b2(c2 - a2 + b2) : - c2(c2 + a2 + b2)

The Ara triangle is perspective to triangles as listed here with perspectors: ABC, X(25); anticomplementary, X(22); Euler, X(1598); tangential 1st circumperp, X(197).      (Peter Moses, Nov. 19, 2013)

The Ara triangle is homothetic to triangle ABC. If ABC is acute then the Ara triangle is the excentral triangle of the tangential triangle. (Randy Hutson, August 17, 2014)

X(5594) lies on these lines: (pending)


X(5595) =  PERSPECTOR OF ARA AND INNER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[a6 - b6 - c6 + a4b2 + a4c2 - a2b4- a2c4 - 2a2b2c2 + b4c2 + b2c4 - (a2 - b2 + c2)(a2 + b2 - c2)S]

The Ara triangle is introduced at X(5594).

X(5595) lies on these lines: (pending)


X(5596) =  PERSPECTOR OF ARIES AND ANTICOMPLEMENTARY TRIANGLES (IF ABC IS ACUTE)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b8 + c8 - 3a8 + 2a4b4 + 2a4c4 - 2b4c4

Let A'B'C' be the tangential triangle of an acute triangle ABC. Let A'' be the touchpoint of the A-excircle of A'B'C' and the line B'C'; define B'' and C'' cyclically. The triangle A''B''C'' is introduced here as the Aries triangle, with vertices given by Peter Moses (Nov. 21, 2013):

A'' = a4 + b4 + c4 - 2b2c2 : 2b2(c2 - b2) : 2c2(b2 - c2)
B'' = 2a2(c2 - a2) : b4 + c4 + a4 - 2c2a2 : 2c2(a2 - c2)
C'' = 2a2(b2 - a2) : 2b2(a2 - b2) : c4 + a4 + b4 - 2a2b2

The Aries triangle is perspective to the tangential triangle, with perspector X(1498).

If ABC is acute, the Aries triangle is the extouch triangle of the tangential triangle. (Randy Hutson, July 18, 2014)

If you have The Geometer's Sketchpad, you can view X(5596).

X(5596) lies on these lines: {2,66}, {4,6}, {20,3313}, {22,69}, {110,2892}


X(5597) =  PERSPECTOR OF ABC AND 1st AURIGA TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - a2(b + c)2 - 4aS(rR + 4R2)1/2

Let U be the inverter of the circumcircle and incircle, as described at X(5577). There are two triangles that circumscribe U and are homothetic to triangle ABC, one of which has A-vertex on the same side of line BC as A. This triangle, A'B'C', is introduced here as the 1st Auriga triangle, and the other, as the 2nd Auriga triangle. Each is the reflection of the other in the center, X(55), of the inverter. The six points A', B', C', A'', B'', C'' lie on a conic introduced here as the Auriga conic. Let D = (rR + 4R2)1/2; barycentrics for the six points and conic were found by Peter Moses (Nov. 21, 2013):

A' = a4 - a2(b + c)2 - 4(b + c)SD : b4 - b2(c + a)2 + 4bSD : c4 - c2(a + b)2 + 4cSD

A'' = a4 - a2(b + c)2 + 4(b + c)SD : b4 - b2(c + a)2 - 4bSD : c4 - c2(a + b)2 - 4cSD

where B', C', B'', C'' are determined cyclically.

The Auriga conic is given by {cyclic sum[g(a,b,c)x2 + h(a,b,c)yz} = 0, where

g(a,b,c) = bc(b + c - a)(b5 + c5 + 3a3bc + a2b3 + a2c3 - a2b2c - a2bc2 + ab3c + abc3 - 2ab2c2 + 3b4c + 3bc4 - 2b3c2 - 2b2c3)

h(a,b,c) = a[a7 - 2a6(b + c) - a5(b2 + c2 - 4a5bc) + 4a4(b3 + c3)
- a3(b4 + c4 + 4b3c + 4bc3 - 8b2c2)
- 2a2(b5 + c5 - b4c - bc4 + 5b3c2 + 5b2c3)
+ a(b6 + c6 + b4c2 + b2c4 - 4b3c3)
+ 2(b5c2 + b2c5 - b4c3 - b3c4)

The two Auriga triangles are perspective with perpsector X(55), which is the center of the Auriga conic.

X(5597) = {X(1), X(55)}-harmonic conjugate of X(5598)

If you have The Geometer's Sketchpad, you can view X(5597).

X(5597) lies on these lines: {1,3},{2,5599},{8,5600},{145,5602}


X(5598) =  PERSPECTOR OF ABC AND 2nd AURIGA TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - a2(b + c)2 - 4aS(r + 4R)1/2

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5598) = {X(1), X(55)}-harmonic conjugate of X(5597)

If you have The Geometer's Sketchpad, you can view X(5598).

X(5598) lies on these lines: {1,3},{2,5600},{8,5599},{145,5601}


X(5599) =  PERSPECTOR OF MEDIAL AND 1st AURIGA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - a2(b + c)2 - 4(b + c)S(r + 4R)1/2

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5599) = {X(10), X(55)}-harmonic conjugate of X(5600)

X(5599) lies on these lines: {2,5597},{8,5598},{10,55},{3617,5602}


X(5600) =  PERSPECTOR OF MEDIAL AND 2nd AURIGA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - a2(b + c)2] + 4(b + c)S(r + 4R)1/2

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5600) = {X(10), X(55)}-harmonic conjugate of X(5599)

X(5600) lies on these lines: {2,5598},{8,5597},{10,55},{3617,5601}


X(5601) =  PERSPECTOR OF ANTICOMPLEMENTARY AND 1st AURIGA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[a2(a + b + c) - 2S(r + 4R)1/2]

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5601) = {X(8), X(55)}-harmonic conjugate of X(5602)

X(5601) lies on these lines: {{2,5597},{8,21},{145,5598},{3617,5600}


X(5602) =  PERSPECTOR OF ANTICOMPLEMENTARY AND 2nd AURIGA TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[a2(a + b + c) + 2S(r + 4R)1/2]

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5602) = {X(8), X(55)}-harmonic conjugate of X(5601)

X(5602) lies on these lines: {2,5598},{8,21},{145,5597},{3617,5599}


X(5603) =  PERSPECTOR OF EULER AND CAELUM TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + b4 + c4 - 2a3b - 2a3c - 2a2b2 - 2a2c2 + 4a2bc + 2ab3 + 2ac3 - 2ab2c -2abc2 - 2b2c2

The reflection of triangle ABC in the incenter, here called the Caelum triangle, is perspective to the medial triangle with perspector X(8), to the anticomplementary and intouch triangles with perspector X(145), and to the Euler triangle at X(5603). See also X(5604) and X(5605). The vertices of the Caelum triangle were found by Peter Moses (Nov. 21, 2013):

A' = a - b - c : 2b : 2c       B' = 2a : b - c - a : 2c       C' = 2a : 2b : c - a - b.

Let A' be the orthocenter of triangle BCX(7), and define B' and C' cyclically. Then X(5603) is the centroid of A'B'C'. (Randy Hutson, November 22, 2014)

X(5603) lies on these lines:
{1,4},{2,392},{3,962},{5,8},{7,104},{10,3090},{11,2099},{12,2098},{20,1385},{29,945},{36,3474},{40,631},{56,4295},{65,3086},{78,5082},{79,4317},{84,3296},{86,4221},{119,1320},{140,5550},{145,355},{165,3524},{281,1953},{329,956},{376,516},{381,952},{495,1532},{496,938},{498,5443},{499,1788},{519,3545},{546,1483},{908,3421},{912,3873},{929,953},{971,5049},{995,4000},{997,2550},{1000,1512},{1001,1006},{1060,4318},{1065,3478},{1071,5045},{1158,3338},{1210,3340},{1279,3332},{1312,2102},{1313,2103},{1319,1836},{1420,4292},{1468,3073},{1476,5553},{1698,5067},{1829,3089},{1872,4200},{1902,3088},{2093,3911},{2476,5330},{2646,4294},{2792,5429},{2801,3892},{2829,5434},{2886,5289},{2975,3560},{3057,3085},{3072,3915},{3149,3295},{3242,5480},{3244,3855},{3304,3649},{3306,3359},{3333,3671},{3434,4511},{3436,4861},{3523,3579},{3525,3624},{3529,3636},{3543,3655},{3617,5056},{3621,5068},{3623,3832},{3679,5071},{3820,5328},{4193,5554},{5048,5252},{5119,5218},{5450,5563}

X(5603) = midpoint of X(1) and X(1699)
X(5603) = relection of X(I) in X(J) for these (I,J): (4,1699), (376,3576), (1699,946), (3576,551)


X(5604) =  PERSPECTOR OF CAELUM AND OUTER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 2b2 + 2c2 - ab - ac + 2S)

The Caelum triangle is defined at X(5603).

X(5604) = {X(1), X(3242)}-harmonic conjugate of X(5605)

X(5604) lies on these lines: {1,6},{8,5590},{145,1270},{1160,1482}


X(5605) =  PERSPECTOR OF CAELUM AND INNER GREBE TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 + 2b2 + 2c2 - ab - ac - 2S)

The Caelum triangle is defined at X(5603).

X(5605) = {X(1), X(3242)}-harmonic conjugate of X(5604)

X(5605) lies on these lines: {1,6},{8,5591},{145,1271},{1161,1482}


X(5606) =  HATZIPOLAKIS CIRCUMCIRCLE POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b - c)(a3 + b3 + c3 - a2b -a2c - ab2 - ac2 - abc + 3b2c + 3bc2)]

Let I be the incenter and A' the nine-point center of triangle IBC. Define B' and C' cyclically. The circles AB'C', BC'A', CA'B' concur in X(5606).      (Antreas Hatzipolakis, June 2, 2013: see Concurrent Circumcircles)

Let L be the Euler line of the incentral triangle of ABC, so that L is the line X(500)X(1962). Let LA be the reflection of L in line BC, and define LB and LC cyclically. Let A'' = LB∩LC, B'' = LC∩LA, C'' = LA∩LB. The lines AA'', BB'', CC'' concur in X(5606). (Randy Hutson, July 18, 2014)

X(5606) lies on the circumcircle and these lines: {74,1385}, {229,759}, {2372,5253}

X(5606) = anticomplement of X(5952)
X(5606) = cevapoint of X(513) and X(3337)
X(5606) = trilinear pole of line X(6)X(3336)


X(5607) =  CENTER OF 1st POHOATA-DAO-MOSES CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)[31/2(a8 - 3a6b2 - 3a6c2 + 3a4b4 + 3a4c4 - a4b2c2 - a2b6 - a2c6 + 2b4c4) + 2S(a6 - 2b6 - 2c6 - 4a4b2 - 4a4c2 + 5a2b4 + 5a2c4 - 3a2b2c2 + 2b4c2 + 2b2c4)]

A construction of the 1st Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617).     (Dao Thanh Oai and Peter Moses, Nov., 2013)

X(5607) lies on this line: {526, 5608}


X(5608) =  CENTER OF 2nd POHOATA-DAO-MOSES CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)[31/2(a8 - 3a6b2 - 3a6c2 + 3a4b4 + 3a4c4 - a4b2c2 - a2b6 - a2c6 + 2b4c4) - 2S(a6 - 2b6 - 2c6 - 4a4b2 - 4a4c2 + 5a2b4 + 5a2c4 - 3a2b2c2 + 2b4c2 + 2b2c4)]

A construction of the 2nd Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617).     (Dao Thanh Oai and Peter Moses, Nov., 2013)

X(5608) lies on this line: {526, 5607}


X(5609) =  RADICAL TRACE OF 1st AND 2nd POHOATA-DAO-MOSES CIRCLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a8 + b8 + c8 - 7a6b2 - 7a6c2 + 9a4b4 + 9a4c4 + 4a4b2c2 - 5a2b6 - 5a2c6 + 3b6c2 + 3b2c6 - 8b4c4)
X(5009) = 5X(3) - 3X(74)

See X(5607) and X(5608).      (Dao Thanh Oai and Peter Moses, Nov., 2013)

Let A'B'C' be the Euler triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in line BC, and define M' and N' cyclically. The lines L', M', N' concur in X(5609); c.f., X(113), X(399), and X(1511). Let NA be the reflection of X(5) in the perpendicular bisector of BC, and define NB and NC cyclically. Then X(5609) = X(23)-of-NANBNC. (Randy Hutson, July 18, 2014)

X(5609) lies on these lines: {3,74},{5,542},{23,1154},{30,3292},{113,137},{125,3628},{526,5607}


X(5610) =  INTERSECTION OF LINES X(13)X(531) AND X(15)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending))

X(5610) lies on these lines: {13,531}, {15,110},{511,2379}

X(5610) = reflection of X(2378) in X(15)


X(5611) =  INTERSECTION OF LINES X(3)X(6) AND X(5)X(303)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[S2 + 31/2SSA - SASA + SBSC      (Wimalasiri Perera, December 18, 2013)

X(5611) lies on these lines:
{3,6}, {5,303}, {110,3129}, {147,1080}, {381,531}, {623,1656}, {1993,3131}, {3060,3132}, {5464,5476}

X(5611) = reflection of X(I) in X(J) for these (I,J): (3,15), (621,5)


X(5612) =  INTERSECTION OF LINES X(5)X(14) AND X(15)X(399)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending))

X(5612) lies on these lines:
{3,3201}, {5,14}, {15,399}, {16,323}, {62,195}, {3166,5238}

X(5612) = X(13)-Ceva conjugate of X(16)
X(5612) = trilinear product X(16)*X(1749)


X(5613) =  INTERSECTION OF LINES X(2)X(98) AND X(5)X(14)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2[SASA + SBSC) + 6SBSBASC - (12)1/2S3      (Wimalasiri Perera, December 15, 2013)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2S3 - 31/2[S2SA + (SA + SB + SC)SBSC]     (Peter Moses, June 20, 2014)

Lengths of segments: |X(5613)X(13)| = |X(5617)X(14)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier Garcia Capitan), ADGEOM #1256, April 23, 2014)

The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco Javier Garcia Capitan), ADGEOM #1259, April 23, 2014)

The Euler line is tangent to the circle {{X(3), X(5613), X(5617)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

Suppose that P is a point. The P-Fuhrmann triangle is the triangle A''B''C'', where A'' is the reflection in line BC of the A-vertex of the cercumcevian triangle of P, and B'' and C'' are defined cyclically (so that taking P to be X(1), X(4), and X(6) yields the classical Fuhrmann triangle, the single-point triangle X(4), and the 4th Brocard triangle, respectively. The X(16)-Fuhrmann triangle is equilateral, and X(5613) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5617). (Randy Hutson, July 7, 2014)

X(5613) lies on these lines:
{2,98}, {3,619}, {4,617}, {5,14}, {13,2782}, {30,5464}, {99,622}, {299,383}, {303,1080}, {381,531}, {395,3564}, {576,3180}, {5055,5460}

X(5613) = midpoint of X(I) and X(J) for these (I,J): (4,617), (299,383)
X(5613) = reflection of X(I) in X(J) for these (I,J): (14,5), (3,619), (5617,114)


X(5614) =  INTERSECTION OF LINES X(14)X(530) AND X(16)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending))

X(5614) lies on these lines: {14,530}, {16,110},{511,2378}

X(5614) = reflection of X(2379) in X(16)


X(5615) =  INTERSECTION OF LINES X(3)X(6) AND X(5)X(302)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[S2 - 31/2SSA - SASA + SBSC      (Wimalasiri Perera, December 18, 2013)

X(5615) lies on these lines:
{3,6}, {5,302}, {110,3130}, {147,383}, {381,530}, {624,1656}, {1993,3132}, {3060,3131}, {5463,5476}

X(5615) = reflection of X(I) in X(J) for these (I,J): (3,16), (622,5)


X(5616) =  INTERSECTION OF LINES X(5)X(13) AND X(16)X(399)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending))

X(5616) lies on these lines:
{3,3200}, {5,13}, {15,399}, {15,323}, {16,399}, {61,195},{3165,5237}

X(5616) = X(14)-Ceva conjugate of X(15)
X(5616) = trilinear product X(15)*X(1749)


X(5617) =  INTERSECTION OF LINES X(2)X(98) AND X(5)X(13)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2[SASA + SBSC) + 6SBSBASC + (12)1/2S3      (Wimalasiri Perera, December 16, 2013)

Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 2S3 + 31/2[S2SA + (SA + SB + SC)SBSC]     (Peter Moses, June 20, 2014)

Lengths of segments: |X(5617)X(14)| = |X(5613)X(13)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier Garcia Capitan), ADGEOM #1256, April 23, 2014)

The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco Javier Garcia Capitan), ADGEOM #1259, April 23, 2014)

The Euler line is tangent to the circle {{X(3), X(5617), X(5661)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

The X(15)-Fuhrmann triangle, defined at X(5613), is equilateral, and X(5617) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5613). (Randy Hutson, July 7, 2014)

X(5617) lies on these lines:
{2,98},{3,618},{4,616},{5,13},{14,2782},{30,5463},{99,621},{298,511},{302,383},{381,530},{396,3564},{576,3181},{5055,5459}

X(5617) = midpoint of X(I) and X(J) for these (I,J): (4,616), (298,1080)
X(5617) = reflection of X(I) in X(J) for these (I,J): (13,5), (3,618), (5613,114)


X(5618) =  1st MONTESDEOCA EQUILATERAL TRIANGLES POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b2 - c2)(271/2b2c2SA + S(S2 + 9SASA))]

Let AP, BP, CP be the cevians of a point P in the plane of a triangle ABC. The A-positive Montesdeoca equilateral triangle is constructed as follows: let LA be the line through A perpendicular to CP; let UA be the 30-degree rotation of LA, where the angle BAC, for present purposes, defines the positive direction of rotation, and angle CAB, the negative (used in X(5619)). Let A1 = UA∩CP, let VA be the - 60 degree rotation of CP about A1, let AB = VA∩BP, let AC be the - 60 degree rotation of segment AAB about A. Then AABAC is an equilateral triangle. Define BBCBA and CCACB cyclically. These are the positive Montesdeoca equilateral triangles. X(5618) is the unique choice of P on the circumcircle of ABC for which the lines ABAC, BCBA, CBCA concur. For arbitrary P, the centers of the three equilateral triangles are collinear with P; denote their line by L(P). If P is on the circumcircle of ABC, then L(P) passes through X(110).    (Angel Montesdeoca, November 3, 2013)

For details, see Hechos Geométricos en el Triángulo.

If you have The Geometer's Sketchpad, you can view Montesdeoca Equilateral Triangles.

X(5618) lies on the circumcircle and these lines: {13,74}, {115,2378}, {1989,2380}


X(5619) =  2nd MONTESDEOCA EQUILATERAL TRIANGLES POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b2 - c2)(271/2b2c2SA - S(S2 + 9SASA))]

The negative Montesdeoca equilateral triangles for a point P are constructed as follows: in the construction of the positive Montesdeoca equilateral triangles atX(5618), replace the rotation angles (30, -60, -60) by (-30, 60, 60). Barycentrics for X(5619) are obtained from those of X(5618) by replacing S by - S.    (Peter Moses, November 8, 2013)

If you have The Geometer's Sketchpad, you can view Montesdeoca Equilateral Triangles.

X(5619) lies on the circumcircle and these lines: {14,74}, {115,2379}, {1989,2381}


X(5620) =  ISOGONAL CONJUGATE OF X(5127)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)[a6 - a4(b2 + c2) - a2(b4 + c4 - 3b2c2) - 2abc(b + c)(b - c)2 + (b + c)2(b - c)4]
X(5620) = R*X(65) - (2r + R)*X(1365)

Let A'B'C' be the excentral triangle of ABC. Let NA be the nine-point center of A'BC, and let OA be the circumcircle of NABC. Define OB and OC cyclically. The circles OA, OB, OC concur in X(5620).      (Angel Montesdoca, Anapolis #1120, November 2013: see Concurrent Circumcircles)

Let A'B'C' be the incentral triangle and let A'' be the point such that triangle A''BC is similar to A'B'C' and A'' is on the same side of line BC as A. Define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5620); see also X(502). (Randy Hutson, July 7, 2014)

X(5620) lies on these lines:
{1,149},{10,1109},{36,759},{37,115},{65,1365},{162,1838},{267,3336},{897,1738},{1054,1247},{1737,2166},{2218,2915}

X(5620) = isogonal conjugate of X(5127)
X(5620) = X(2245)-cross conjugate of X(226)
X(5620) = X(I)-isoconjugate of X(J) for these (I,J): (3,2074), (21,5172)
X(5620) = trilinear pole of line X(661)X(2294)
X(5620) = trilinear product of X(523) and X(1290)
X(5620) = barycentric product of X(1290) and X(1577)


X(5621) =  1st KIRIKAMI-EULER-LEMOINE POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b2(SC(SA - SB)(-2SASB + SASC + SBSC) + a2c2SB(SA - SC)(-2SASC + SASB + SBSC)
+ (a2 + b2 + c2)(-2SASB + SASC + SBSC)(-2SASC + SASB + SBSC)
X(5621) = X(6) + 2X(74)

Let L be the 1st Lemoine circle of a triangle ABC. Let AB be the point nearer A where line AB meets L, and define BC and CA cyclically. Let AC be the point nearer A where line AC meets L, and define BA and CB cyclically. The Euler lines of the triangles AABAC, BBCBA, CCACB concur in X(5621).      (Seiichi Kirikami, November 7, 2013)

See Concurrent Euler lines.

X(5621) lies on these lines:
{3, 67}, {6, 74}, {25, 125}, {64, 1177}, {110, 3796}, {146, 3589}, {165, 2836}, {186, 1503}, {246, 1976}, {343, 3448}, {399, 5092}, {524, 2071}, {895, 3532}, {1204, 1205}, {1597, 2777}, {2453, 2790}, {2916, 2931}, {3516, 5095}


X(5622) =  2nd KIRIKAMI-EULER-LEMOINE POINT

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a2b2(SC(SA - SB)(-2SASB + SASC + SBSC) + 2a2c2SB(SA - SC)(-2SASC + SASB + SBSC)
+ (a2 + b2 + c2)(-2SASB + SASC + SBSC)(-2SASC + SASB + SBSC)
X(5622) = 2X(6) + X(74)

Continuing from the configuration in X(5621), let K denote the symmedian point (Lemoine point, X(6)) of ABC. The Euler lines of the triangles KABAC, KBCBA, KCACB concur in X(5622).       (Seiichi Kirikami, November 7, 2013)

See Concurrent Euler lines.

X(5622) lies on these lines:
{2, 98}, {3, 895}, {4, 1177}, {6, 74}, {54, 67}, {69, 5504}, {113, 3618}, {185, 575}, {186, 2393}, {217, 5038}, {265, 1176}, {389, 1205}, {403, 1503}, {511, 2071}, {576, 1204}, {578, 5095}, {631, 5181}, {1316, 2790}, {2854, 5085}, {2892, 3541}, {3431, 5505}


X(5623) =  REFLECTION OF X(13) IN X(5618)

Barycentrics    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = HA (HAVA - HBVB - HCVC), where HA = 31/2a2S + S2 + 3SBSC and VA = (S2 - 3SASB)(S2 - 3SASC)       (Peter Moses, June 20, 2014)

Referring to the construction of X(5618), the lines ABAC, BCBA, CACB concur in X(5623).    (Peter Moses, December 4, 2013)

X(5623) lies on the Neuberg cubic and these lines: {13,74}, {14,3440}, {16,1138}, {3065,3383}

X(5623) = reflection of X(13) in X(5618)
X(5623) = X(30)-Ceva conjugate of X(13)


X(5624) =  REFLECTION OF X(14) IN X(5619)

Barycentrics    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = KA (KAVA - KBVB - KCVC), where KA = - 31/2a2S + S2 + 3SBSC and VA = (S2 - 3SASB)(S2 - 3SASC)       (Peter Moses, June 20, 2014)

Referring to the construction of X(5619), the lines ABAC, BCBA, CACB concur in X(5624).    (Peter Moses, December 4, 2013)

X(5624) lies on the Neuberg cubic and these lines: {14,74}, {13,3441}, {15,1138}, {3065,3376}

X(5624) = reflection of X(14) in X(5619)
X(5624) = X(30)-Ceva conjugate of X(14)


X(5625) =  MIDPOINT OF X(1) AND X(86)

Barycentrics    (2a + b + c)(a2 + 2ab + 2ac + bc) : (2b + c + a)(b2 + 2bc + 2ba + ca) : (2c + a + b)(c2 + 2ca + 2cb + ab)
X(5625) = X(1654) - 5*X(3616)

Suppose that P is a point in the plane of a triangle ABC. Let LA be the line through P parallel to BC, and let BA = LA∩AB and CA = LA∩CA. Define AB and CB cyclically, and define AC and BC cyclically. Let UA be the line of the midpoints of segments ABAC and BCCB, and define UB and UC cyclically. The lines UA, UB, UC concur in a point Q = Q(P). If P is given by barycentrics p : q : r, then Q = g(p, q, r) : g(q, r, p) : g(r, p, q), where g(p,q,r) = (2p + q + r)(p2 + 2pq + 2pr + qr). If P = X(1), then Q = X(5625).    (Seiichi Kirikami, December 8, 2013)

X(5625) lies on these lines:
{1,75},{10,4478},{519,4733},{524,551},{726,3723},{1100,1125},{1255,4756},{1279,3636},{1654,3616},{1961,3699},{1962,4427},{2796,4353},{3244,4923},{3624,3759},{3842,4649},{3945,4655},{3993,4670}

X(5625) = midpoint of X(1) and X(86)
X(5625) = reflection of X(1213) in X(1125)
X(5625) = trilinear product of X(I) and X(J) for these (I,J): (1125,4649), (4427,4784) X(5625) = barycentric product of X(4359) and X(4649)


X(5626) =  CENTER OF ELECTROSTATIC POTENTIAL

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b), where [(gaga - 1)(a2 - (bgb - gcc)2)]1/2, where

ga = coth(aλ/(a + b + c)),   gb = coth(bλ/(a + b + c)),   gc = coth(cλ/(a + b + c)),    where λ is the unique positive solution of the equation

[(u2 - a2) (a2 - (v - w)2)]1/2 + [(v2 - b2) (b2 - (w - u)2)]1/2 + [(w2 - c2) (c2 - (u - v)2)]1/2 = [2(b2c2 + c2a2 + a2b2) - a4 - b4 - c4)]1/2,

where u = aga,   v = bgb,   w = cgc

X(5626) is the point of maximal electrostatic potential inside a triangle ABC having a homogeneous surface charge distribution.    (Hrvoje Abraham and Vjekoslav Kovac, December 11, 2013)

Download From electrostatic potentials to yet another triangle center.

Here is a Mathematica program which gives λ = 4.6547... for the (user-chosen) triangle ABC as testTriangle = {6,9,13), followed by the normalized barycentric coordinates and then normalized trilinear coordinates for X(5626).

\[Lambda] =.; testTriangle = {6, 9, 13}; {u, v, w} = Map[# Coth[# \[Lambda]/(a + b + c)] &, {a, b, c}]; {lhs, area} = {Sqrt[(u^2 - a^2) (a^2 - (v - w)^2)] + Sqrt[(v^2 - b^2) (b^2 - (w - u)^2)] + Sqrt[(w^2 - c^2) (c^2 - (u - v)^2)], Sqrt[(-a + b + c) (a + b - c) (a - b + c) (a + b + c)]/4} /. Thread[{a, b, c} -> testTriangle]; \[Lambda] = \[Lambda] /. FindRoot[lhs == 4 area, {\[Lambda], 1}, WorkingPrecision -> 50]; {\[Lambda], #, 2 area #/testTriangle} &[#/Apply[Plus, #] &[ Sqrt[(u^2 - a^2) (a^2 - (v - w)^2)] /. Map[Thread[{a, b, c} -> #] &, NestList[RotateLeft, testTriangle, 2]]]] (* Peter Moses, December 20, 2013 *)

If you have The Geometer's Sketchpad (version 5.05 or later), you can view X(5626).


X(5627) =  YIU REFLECTION POINT

Trilinears    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A csc 3A)/(cos A - 2 cos B cos C)    (Randy Hutson, Jan. 8, 2014)
Barycentrics    g(a,b,c) : g(b,c,a): g(c,a,b), where g(a,b,c) = 1/[(a2SA - 2SBSC)(S2 - 3SASA)]
Barycentrics    h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = 1/[3a6(b2 + c2) - 6a4(b4 + c4) + 3a2(b6 + c6) - 2b8 + 3b6c2 - 6b4c4 + 3b2c6 - 2c8]    (Randy Hutson, Jan. 8, 2014)
X(5627) = 2X(265) + X(476)
X(5627) = 4X(125) - X(477)    (Peter Moses, January 2, 2014)

Paul Yiu introduced this point on New Year's Day, January 1, 2014. He noted that X(74) is the unique point whose reflections in the sidelines of triangle ABC are collinear and perspective to ABC. The perspector is X(5627). The line of the reflections is perpendicular to the Euler line at X(4), and the rectangular circumhyperbola through X(5627), here called the Yiu hyperbola, YH, has asymptotes parallel and perpendicular to the Euler line. The center of YH is X(3258), and the perspector of YH is X(1637); YH meets the circumcircle in X(477), which is the reflection of X(74) in the Euler line.    (Paul Yiu, ADGEOM, "An easy new year puzzle", January 1, 2014)

The line tangent to YH at X(5627) is parallel to the line X(74)X(1138). The axes of YH are the Wallace-Simson lines of X(74) and X(110). The Steiner circumellipse meets YH in four points: A, B, C, and X(5641). The isogonal conjugate of YH is the line X(3)X(74). X(5627) is the cevapoint of the 1st and 2nd Fermat Points.    (Peter Moses, January 2, 2014)

X(5627) is the perspector of ABC and the reflection of the Euler triangle in the Euler line.    (Randy Hutson, Jan. 8, 2014)

Let A'B'C' be the tangential triangle of the Kiepert hyperbola. Let A'' be the intersection, other than X(3258) of the nine-point circle and the line A'X(3258); define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(5627).    (Randy Hutson, Jan. 8, 2014)

X(5627) lies on these lines:
{5,1117},{30,74},{125,477},{328,1494},{403,1989},{1138,3258},{1141,1304}

X(5627) = reflection of X(1138) in X(3258)
X(5627) = isogonal conjugate of X(1511)
X(5627) = isotomic conjugate of X(6148)
X(5627) = cevapoint of X(I) and X(J) for these (I,J): (13,14), (74,3470)
X(5627) = crossconjugate of X(I) and X(J) for these (I,J): (4,1141), (115,2394), (523, 476)
X(5627) = isoconjugate of X(I) and X(J) for these (I,J): (1,1511), (323,2173), (1101,3258), (2407,2624)
X(5627) = trilinear pole of line X(1637) X(1989)
X(5627) = trilinear product X(I)*X(J) for these (74,94), (476,2394), (1494,1989), (2166,2349)


X(5628) =  1st MORLEY - VAN TIENHOVEN POINT

Trilinears    cos(A/3) sec(2A/3) : cos(B/3) sec(2B/3) : cos(C/3) sec(2C/3)

X(5628) and several other triangle centers are perspectors of each pair of the following triangles in the plane of a triangle ABC:
T1 = 1st Morley triangle; MathWorld: First Morley triangle
T2 = 2nd Morley triangle; MathWorld: Second Morely triangle, etc.
T3 = 3rd Morley triangle>
T4 = 1st adjunct Morley triangle; MathWorld: First adjunct Morley triangle, etc.)
T5 = 2nd adjunct Morley triangle
T6 = 3rd adjunct Morley triangle
T7 = 1st p-Morley triangle (defined below)
T8 = 2nd p-Morley triangle
T9 = 3rd p-Morley triangle
T10 = 1st p-adjunct Morley triangle (defined below)
T11 = 2nd p-adjunct Morley triangle
T12 = 3rd p-adjunct Morley triangle

The p-Morley triangles T7, T8, T9 have as vertices the points of intersection of pairs of perpendiculars to trisectors at corresponding vertices that form T1, T2, T3, respectively. For example, T7 is formed as follows from the 1st Morley triangle A'B'C': let LB be the line perpendicular to BA' at B, let LC be the line perpendicular to CA' at C; then the A-vertex of T7 is LB∩LC, and the B-vertex and C-vertex are defined cyclically. Similarly, the p-adjunct Morley triangles T10, T11, T12 are defined from T4, T5, T6.

X(5628) = 1st Morley-van Tienhoven point = perspector of ABC and T7
X(5629) = 2nd Morley-van Tienhoven point = perspector of ABC and T10
X(5630) = 3rd Morley-van Tienhoven point = perspector of ABC and T8
X(5631) = 4th Morley-van Tienhoven point = perspector of ABC and T11
X(5632) = 5th Morley-van Tienhoven point = perspector of ABC and T9
X(5633) = 6th Morley-van Tienhoven point = perspector of ABC and T12
X(356) = 7th Morley-van Tienhoven point = perspector of each pair of T1, T4, T7
X(3276) = 8th Morley-van Tienhoven point = perspector of each pair of T2, T5, T8
X(3277) = 9th Morley-van Tienhoven point = perspector of each pair of T3, T6, T9
X(5634) = 10th Morley-van Tienhoven point = perspector of T7 and T10
X(5635) = 11th Morley-van Tienhoven point = perspector of T8 and T11
X(5636) = 12th Morley-van Tienhoven point = perspector of T9 and T12
X(5637) = 13th Morley-van Tienhoven point

X(5628)-X(5633) were found in connection with Chris van Tienhoven's rotations of Morley trisectors and subsequent collaborations with Bernard Gibert, including is the cubic K587 in Gibert's catalog of cubics: Morley - van Tienhoven cubic.

X(5628) lies on these lines: {357,5457},{3272,3274},{3602,3606}

X(5628) = isogonal conjugate of X(5629)


X(5629) =  2nd MORLEY - VAN TIENHOVEN POINT

Trilinears    cos(2A/3) sec(A/3) : cos(2B/3) sec(B/3) : cos(2/3) sec(C/3)

X(5629) = perspector of ABC and T10; see X(5628)

X(5629) lies on these lines: {356,357},{3273,3281},{3274,3606}

X(5629) = isogonal conjugate of X(5628)


X(5630) =  3rd MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(A - 2π)/3] sec[(2A - 4π)/3] : cos[(B - 2π)/3] sec[(2B - 4π)/3] : cos[(C - 2π)/3] sec[(2C - 4π)/3]

X(5630) = perspector of ABC and T8; see X(5628)

X(5630) lies on these lines: {3603,3607},{3272,3275}

X(5630) = isogonal conjugate of X(5631)


X(5631) =  4th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(2A - 4π)/3] sec[((A - 2π)/3] : cos[(2B - 4π)/3] sec[((B - 2π)/3] : cos[(2C - 4π)/3] sec[((C - 2π)/3]

X(5631) = perspector of ABC and T11; see X(5628)

X(5631) lies on these lines: {1136,1137},{3274,3283},{3275,3607}

X(5631) = isogonal conjugate of X(5630)


X(5632) =  5th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(A - 4π)/3] sec[(2A - 8π)/3] : cos[(B - 4π)/3] sec[(2B - 8π)/3] : cos[(C - 4π)/3] sec[(2C - 8π)/3]

X(5632) = perspector of ABC and T9; see X(5628)

X(5632) lies on these lines: {356,3272}, {3604,3605}

X(5632) = isogonal conjugate of X(5633)


X(5633) =  6th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(2A - 8π)/3] sec[((A - 4π)/3] : cos[(2B - 8π)/3] sec[((B - 4π)/3] : cos[(2C - 8π)/3] sec[((C - 4π)/3]

X(5633) = perspector of ABC and T12; see X(5628)

X(5633) lies on these lines: {356,1134}, {3273,3605}, {3275,3279}

X(5633) = isogonal conjugate of X(5632)


X(5634) =  10th MORLEY - VAN TIENHOVEN POINT

Trilinears    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(2A/3) cos(B/3) cos(C/3) - cos(A/3) cos(2B/3) cos(2C/3)

X(5634) = perspector of the triangles T7 and T10 listed at X(5628).

X(5634) lies on these lines: {356,3605}, {3276,3606}


X(5635) =  11th MORLEY - VAN TIENHOVEN POINT

Trilinears    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos[(2(A - 2π)/3] cos[(B - 2π)/3] cos[(C - 2π)/3] - cos[(A - 2π)/3] cos[(2B - 4π)/3] cos[(2C - 4π)/3]

X(5635) = perspector of the triangles T8 and T11 listed at X(5628).

X(5635) lies on these lines: {3276,3606}, {3277,3607}


X(5636) =  12th MORLEY - VAN TIENHOVEN POINT

Trilinears    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos[(2A - 8π)/3] cos[(B - 4π)/3] cos[C - 4π)/3] - cos[(A - 4π)/3] cos[(2B - 8π)/3] cos[(2C - 8π)/3]

X(5636) = perspector of the triangles T9 and T12 as listed at X(5628).

X(5636) lies on these lines: {356,3605}, {3277,3607}


X(5637) =  13th MORLEY - VAN TIENHOVEN POINT

Trilinears    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin(B/3 + C/3) sin(B/3 - B/3) [cos(A/3) - 2 cos(B/3)cos(C/3)]

Using the notation at X(5628), let
L1 = perspectrix of each pair of the triangles ABC, T1, T4
L2 = perspectrix of each pair of the triangles ABC, T2, T5
L3 = perspectrix of each pair of the triangles ABC, T3, T6
The lines L1, L2, L3 concur in X(5637).    (Chris van Tienhoven, January 3, 2014)

X(5637) lies on this line: {396,523}


X(5638) =  INSIMILICENTER OF CIRCLES X(14)X(15)X(16) AND X(13)X(15)X(16)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc B)(e cos C + cos(C - ω)] - (csc C)(e cos B + cos(B - ω)]
Barycentrics    g(A,B,C) : g(B,C,A ) : g(C,A,B), where g(A,B,C) = sin(A - ω)/[e cos A + cos(A + ω]
Barycentrics    h(A,B,C) : h(B,C,A) : hf(C,A,B), where h(A,B,C) = (S2B - S2C)[(S + 31/2SA)p - (S - 31/2SA)q], where p = (Sω - 31/2S)1/2 and q = (Sω + 31/2S)1/2]    César Lozada (ADGEOM #1341, June 22, 2014)

The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5638) is the farther from X(2); also, X(5638) = intersection farther from X(2) of the Parry circle and Lemoine axis. X(5638) = (F1, F2)-harmonic conjugate of X(1341), where F1 and F2 are the foci of the Steiner inellipse. (Randy Hutson, June 16, 2014)

X(5638) lies on the Parry circle, the Thomson-Gibert-Moses hyperbola, and these lines: {2,1341}, {110,1379}, {111,1380}, {187,237}, {353,1340}

X(5638) = reflection of X(5639) in X(351)
X(5638) = X(110)-Ceva conjugate of X(5639)
X(5638) = X(3124)-cross conjugate of X(5639)
X(5638) = X(6)-vertex conjugate of X(5639)
X(5638) = inverse-in-circumcircle of X(6141)
X(5638) = antipode-in-Parry-circle of X(5639)
X(5638) = trilinear pole of the line X(512)X(2028)
X(5638) = crossdifference of every pair of points on the line X(2)X(1340)


X(5639) =  EXSIMILICENTER OF CIRCLES X(14)X(15)X(16) AND X(13)X(15)X(16)

Barycentrics    Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc B)(e cos C - cos(C - ω)] - (csc C)(e cos B - cos(B - ω)]
Barycentrics    g(A,B,C) : g(B,C,A ) : g(C,A,B), where g(A,B,C) = sin(A - ω)/[e cos A - cos(A + ω]
Barycentrics    h(A,B,C) : h(B,C,A) : hf(C,A,B), where h(A,B,C) = (S2B - S2C)[(S + 31/2SA)p + (S - 31/2SA)q], where p = (Sω - 31/2S)1/2 and q = (Sω + 31/2S)1/2]    César Lozada (ADGEOM #1341, June 22, 2014)

The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5638) is the closer to X(2); also, X(5639) = the intersection closer to X(2) of the Parry circle and Lemoine axis. X(5639) = perspector of the hyperbola (A,B,C,X(6), F1, F2, where F1 and F2 are the foci of the Steiner inellipse; also X(5639) = intersection of the trilinear polars of X(6), F1, and F2. (Randy Hutson, June 16, 2014)

X(5639) lies on the Parry circle, the Thomson-Gibert-Moses hyperbola, and these lines: {2,1340}, {110,1380}, {111,1379}, {187,237}, {353,1341}

X(5639) = reflection of X(5638) in X(351)
X(5639) = X(110)-Ceva conjugate of X(5638)
X(5639) = X(3124)-cross conjugate of X(5638)
X(5639) = X(6)-vertex conjugate of X(5638)
X(5639) = inverse-in-circumcircle of X(6142)
X(5639) = antipode-in-Parry-circle of X(5638)
X(5639) = trilinear pole of the line X(512)X(2029)
X(5639) = crossdifference of every pair of points on the line X(2)X(1341)


X(5640) =  CENTROID OF ORTHOCENTROIDAL TRIANGLE

Barycentrics    a2(b4 + c4 + a2b2 + a2c2 + 3b2c2)/(b2 - c2)2

The orthocentroidal triangle is introduced at X(5476). For another construction of X(5640), let A'B'C' be the orthic triangle, let A'' be the centroid of AB'C', and define B'' and C'' cyclically. Then X(5640) is the centroid of A''B''C''. Also, X(5640) is the trilinear pole of the polar, with respect to the Moses circle, of the perspector of the Moses circle. (Randy Hutson, June 16, 2014)

Vertices of the central triangle A"B"C" are given by barycentrics as follows:

A'' = b4 + c4 - a2b2 - a2c2 - 4b2c2 : b2(a2 - b2 - c2) : c2(a2 - b2 - c2) (Peter Moses, June 20, 2014)

A''B''C'' is perspective to ABC, the tangential triangle and the second Brocard triangle, all with perspector X(6), and perspective to the Euler triangle at X(125). The Euler line of A''B''C'' passes througth X(I) for these I: 6,110,111,895,1995,2493,2502,2503,2854,2930,3066,3124 (Peter Moses, June 20, 2014)

X(5640) = isogonal conjugate of X(2) with respect to the pedal triangle of X(2)

X(5640) lies on these lines:
{2,51},{4,4846},{5,568},{6,110},{22,5085},{23,182},{25,5012},{52,3090},{125,5169},{143,1656},{185,3832},{323,576},{324,3168},{375,3681},{381,5663},{389,3091},{394,5102},{476,1316},{512,598},{569,3518},{575,1495},{631,5446},{858,5480},{1112,5094},{1154,5055},{1180,3981},{1216,5067},{1383,2030},{1843,4232},{1993,5020},{3056,5297},{3111,3972},{3291,5052},{3292,5097},{3448,3818},{3533,5447},{5039,5354}

X(5640) = midpoint of X(51) and X(373)
X(5640) = reflection of X(2) in X(373)
X(5640) = anticomplement of X(5650)
X(5640) = crossdifference of every pair of points on line X(690)X(3288)


X(5641) =  ISOTOMIC CONJUGATE OF X(542)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[2a sec(A + ω) - b sec(B + ω) - c sec(C + ω)]

The Steiner circumellipse meets the Yiu Hyperbola, defined at X(5627), in four points: A, B, C, and X(5641).    (Peter Moses, January 2, 2014)

X(5641) lies on these lines: {2,2966},{30,99},{69,892},{290,850},{297,340},{523,1494},{525,671},{670,3260}

X(5641) = reflection of X(2966) in X(2)
X(5641) = isotomic conjugate of X(542)
X(5641) = cevapoint of X(2) and X(542)
X(5641) = X(542)-cross conjugate of X(2)
X(5641) = isoconjugate of X(I) and X(J) for these (I,J): (6,2247), (163,1640)
X(5641) = trilinear pole of line X(2)X(1637)
X(5641) = trilinear product X(75)*X(842)
X(5641) = barycentric product X(76)*X(842)


X(5642) =  CENTER OF THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2a2 - b2 - c2)(2a4 - a2b2 - a2c2 + 2b2c2 - b4 - c4)

The point X(5542) minimizes a certain sum involving the pedal triangle A'B'C' of a triangle ABC. During May, 2014, Peter Moses generalized the sum to include a parameter t, as follows:

|XA|2 + |XB|2 + |XC|2 + t(|XA'|2 + |XB'|2 + |XC'|2),

and found the solution to the extremal problem to be the point H(t) = f(A,B,C) : f(B,C,A) : f(C,A,B), where

f(A,B,C) = a2[(t2 + 3t + 3)S2 + (2t + 3) S2A + tSBSC].

Let u = 3abc[-3abc + (9a2b2c2 - 8S2S2W)1/2/(4S2S2W) and v = 3abc[-3abc - (9a2b2c2 - 8S2S2W)1/2/(4S2S2W)

Moses found that if t > u, then H(t) minimizes the sum, and if t < v, then H(t) maximizes the sum, and if t > -1, then H(t) minimizes the sum. The extreme value in all cases is

[(6t + 6)SASBSCSW - (2t + 3)(t + 2)S2S2W - t(t + 1)2S4]/[(9t + 9)SASBSC - (2t + 3)(t + 3)S2SW].

The locus of H(t) as t ranges through the extended real number line is a rectangular hyperbola, here named the Thomson-Gibert-Moses hyperbola, which passes through the triangle centers X(I) for I = 2, 3, 6, 110, 154, 354, 392, 1201, 2574, 2575, 3167, 5544, 5638, 5639, 5643, 5644, 5645, 5646, 5648, 5652-5656. The axes of this hyperbola are parallel to the Simson-Wallace lines of X(1113) and X(1114), these being the points of intersection of the Euler line and the circumcircle. See X(5643). (based on notes from Peter Moses, June 7, 2014)

X(5642) is the radical trace of the Parry circles of ABC and the 1st Brocard triangle, and also the centroid of the (degenerate) pedal triangle of X(110). The Thomson-Gibert-Moses hyperbola intersects the circumcircle in X(110) and the vertices of the Thomson triangle (see Thomson Triangle. The 4 points of intersection form an X(74)-centric system; i.e., each is X(74) of the triangle of the other three. Moreover, the Thomson-Gibert-Moses hyperbola is the Thomson isogonal conjugate (i.e., isogonal-conjugate-with-respect-to-the-Thomson-triangle) of the Euler line. In general, the Thomson isogonal conjugate of a point P is the centroid of the antipedal triangle of the isogonal conjugate of P; consequently, the Thomson-Gibert-Moses hyperbola is the locus of the centroid of the antipedal triangle of a point P that traverses the Jerebek hyperbola. Indeed, the Thomson-Gibert-Moses hyperbola is the Jerabek hyperbola of the Thomson triangle, as noted at Thomson Triangle (Randy Hutson, June 16-17, 2014)

Let O = X(3) and suppose that P is a point other than O. Let OP be the circle with segment PO as diameter. Let A' be the point of intersection, other than O, of OP and the perpendicular bisector of segment BC, and define B' and C' cyclically. Triangle A'B'C' is called the P-Brocard triangle, and X(5642) is X(23)-of-the-X(2)-Brocard triangle. (Randy Hutson, June 16-17, 2014)

The Thomson-Gibert-Moses hyperbola is the image of the Euler line under a mapping T discussed in connection with the third Deaux cubic (K609). (Bernard Gibert, June 22, 2014)

Let O* be the circle with segment X(13)X(14) as diameter (and center X(115). Let P be the perspector of O*. Then X(5642) is the trilinear pole of the polar of P with respect to O*. See X(3292) for a similar property involving the segment X(15)X(16). (Randy Hutson, July 18, 2014)

X(5642) lies on these lines:
{2,98}, {3,541}, {30,113}, {74,3524}, {115,2502}, {126,5026}, {265,5055}, {351,690}, {373,597}, {376,2777}, {399,5054}, {468,524}, {543,1316}, {620,5108}, {1636,1637}, {1648,5477}, {1995,5476}, {2781,3917}, {2836,3742}, {3024,4995}, {3028,5298}, {3849,5112}

X(5642) = midpoint of X(2) and X(110)
X(5642) = reflection of X(125) in X(2)
X(5642) = crossdifference of every pair of points on the line X(74)X(111)
X(5642) = X(3524) line-conjugate of X(74)


X(5643) =  H(2) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2(13S2 + 7S2A + 2SBSC)    (Peter Moses, June 9, 2014)

X(5643) is the only point whose polar conic in the Napoleon cubic (K005) is a circle. (Bernard Gibert, June 22, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642). Points H(t) on this hyperbola include the following:

t H(t)
-3 X(3167)
-2 X(110)
-3/2 X(154)
-1/2 X(5646)
-1 X(3)
0 X(2)
1 X(5544)
2 X(5643)
3 X(5644)
4 X(5645)
infinity X(6)
3R/r X(354)
6rR/(r2 - 2rR - s2 X(354)

Let A' be the centroid of the A-altimedial triangle, and define B' and C' cyclically; then X(5643) is the center of similitude of ABC and A'B'C'. (Randy Hutson, July 7, 2014)

X(5643) lies on these lines:
{2,576},{83,5466},{110,373},{111,5038},{154,1995},{354,4663},{392,5047},{597,895},{632,1173},{3090,5449},{3167,5422},{3580,3628}


X(5644) =  H(3) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2(7S2 + 3S2A + SBSC)    (Peter Moses, June 9, 2014)

X(5644) is the Thomson isogonal conjugate of X(3522); see X(5642).

X(5644) lies on these lines: {2,5093},{3,51},{110,5020},{154,5050},{343,5070},{373,3167},{394,5544},{1351,3819},{1899,3851}


X(5645) =  H(4) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2(31S2 + 11S2A + 4SBSC)    (Peter Moses, June 9, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642).

X(5645) lies on these lines: {2,5097},{154,3066},{323,5544},{2889,3533},{3448,3545}


X(5646) =  H(-1/2) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a2(7S2 + 8S2A - 2SBSC)    (Peter Moses, June 9, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642). X(5646) = (X(6) of the Thomson triangle); see Thomson Triangle (Randy Hutson, June 16, 2014)

X(5646) lies on these lines: {2,1350},{40,392},{64,631},{110,5085},{182,3167},{354,612},{511,5544},{1201,2177},{1351,3819}


X(5647) =  HATZIPOLAKIS-EULER POINT

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A cos(B - C))(4 cos4A + (-1 + 4 cos2(B - C)) cos2A + cos(B - C)(4 cos A + cos(B - C))    (César Lozada, May 18, 2014)

Let A'B'C' be the cevian triangle of the circumcenter. Let

LAB = reflection of AA' in AB, LAC = reflection of AA' in AC; HAB = orthogonal projection of A' on LAB, HAC = orthogonal projection of A' on LAC, and define LBC, LCA, LBA, LCB, and HBC, HCA, HBA, HCB cyclically,

M11 = midpoint(HAB, HAC),      M12 = midpoint(HBC, HBA),      M13 = midpoint(HCA, HCB)
M21 = midpoint(HBA, HCA),      M22 = midpoint(HCB, HAB),      M23 = midpoint(HAC, HBC)
M31 = midpoint(HBC, HCB),      M32 = midpoint(HCA, HAC),      M33 = midpoint(HAB, HBA).

Antreas Hatzipolakis (Anopolis , May 17, 2014) asked if the Euler lines of the triangles M11M12M13, M21M22M23, M31M32M33 concur, and César Lozada established that they do. The point of concurrence is X(5647).

X(5647) lies on these lines: {5,51}, {154,157}


X(5648) =  ANTIPODE OF X(6) IN THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8+3 a^6 b^2-2 a^4 b^4-3 a^2 b^6+b^8+3 a^6 c^2-11 a^4 b^2 c^2+7 a^2 b^4 c^2-2 a^4 c^4+7 a^2 b^2 c^4-2 b^4 c^4-3 a^2 c^6+c^8

X(5648) is the radical center of the the circumcircle, the circle {{X(13), X(15), X(5463), X(5464)}} and the circle {{X(14), X(16), X(5463), X(5464)}}. (Randy Hutson, August 26, 2014)

X(5648) lies on these lines:
{2,2854},{3,67},{6,5642},{110,524},{125,5646},{141,5888},{392,2836},{511,5655},{523,5653},{526,5652},{541,1350},{543,2453},{597,895},{2781,5656}

X(5648) = crossdifference of every pair of points on the line X(2492)X(2780)


X(5649) =  ISOGONAL CONJUGATE OF X(1640)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b2 - c2)(2a6 - b6 - c6 - 2a4b2 - 2a4c2 + a2b4 + a2c4 + b4c2 + b2c4)]
Trilinears   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cot A)/[sin 3B csc 2B tan C cos(C + ω) - sin 3C csc 2C tan B cos(B + ω)]

X(5649) is the trilinear pole of the line X(23)X(110); at X(110), this line is tangent to the Thomson-Gibert-Moses hyperbola and parallel to the trilinear polar of X(110). (Randy Hutson, June 16, 2014)

X(5649) lies on the hyperbola {A, B, C, X(2), X(110)} and this line: {250, 4230}

X(5649) = isogonal conjugate of X(1640)


X(5650) =  REFLECTION OF X(373) IN X(2)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(2), and let V be the tangent at X(3); then X(5650) = U∩V. Also, X(5650) is the centroid of the X(2)-Brocard triangle; see X(5642). (Randy Hutson, June 16, 2014)

X(5650) = pole of the Euler line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)

X(5650) lies on these lines: {2,51}, {3,1495}, {6,5646}, {110,5092}, {549,5642}

X(5650) = reflection of X(373) in X(2)
X(5650) = complement of X(5640)


X(5651) =  {X(2), X(110)}-HARMONIC CONJUGATE OF X(182)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(2), and let V be the tangent at X(6); then X(5651) = U∩V. Also, X(5650) is the inverse-in-Thomson-Gibert-Moses hyperbola of X(182). (Randy Hutson, June 16, 2014)

X(5651) = pole of the Brocard axis line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)

X(5651) lies on these lines: {2,98}, {3,1495}, {6,373}

X(5651) = crossdifference of every pair of points on the line X(1499)X(3569)


X(5652) =  X(4) OF TRIANGLE X(2)X(3)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5652) = antipode of X(5653) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)

X(5652) lies on the circle {{X(3), X(6), X(110), X(353), X(843)}}. (Randy Hutson, November 22, 2014)

X(5652) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2,3111}, {3,669}, {6,523}, {83,5466}, {99,110}

X(5652) = reflection of X(5653) in X(5642)


X(5653) =  X(4) OF TRIANGLE X(3)X(6)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5653) = antipode of X(5652) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)

X(5653) lies on the Thomson-Gibert-Moses hyperbola, the circle {{X(2), X(3), X(6), X(111), X(691)}}, and these lines: {2,690}, {3,351}, {6,526}, {110,249}

X(5653) is denoted QA-P9 (QA-Miquel Center) of the quadrangle X(13)X(14)X(15)X(16); see Chris van Tienhoven's site.

X(5653) = crossdifference of every pair of points on the line X(542)X(1648)
X(5653) = reflection of X(5652) in X(5642)


X(5654) =  INTERSECTION OF LINES X(4)X(110) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 - b2 - c2)(a8 - b8 - c8 - a6b2 - a6c2 + 4a4b4 + 4a4c4 - 4a4b2c2 + 4b6c2 - 6b4c4 + 4b2c6)

X(5654) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(22). (Peter Moses, June 18, 2014)

X(5654) lies on these lines:
{3,4549}, {4,110}, {5,6}, {11,3157}, {12,1069}, {30,154}, {52,3542}, {140,5646}, {184,1568}, {185,3548}, {354,912}, {381,3167}, {382,1514}, {403,1993}, {539,3545}, {1216,3547}, {1656,5544}, {1899,2072}, {3089,5446}, {3090,5449}, {3549,5562}, {5055,5644}, {5056,5645}


X(5655) =  INTERSECTION OF LINES X(6)X(13) AND X(30)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5655) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(23). (Peter Moses, June 18, 2014)
X(5655) = antipode of X(3) in Thomson-Gibert-Moses hyperbola

Let A'B'C' be the orthocentroidal triangle. Let L be the lines through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and L' cyclically. The lines L',M',N' concur in X(5655); c.f. X(i) for i = 74, 113, 265, 399, 1147, 1511, 5504, 5609. (Randy Hutson, August 26, 2014 *)

X(5655) lies on these lines:
{3,541}, {4,5609}, {5,5643}, {6,13}, {30,110}, {74,549}, {125,5055}, {146,376}, {154,2777}, {354,2771}, {1539,3543}, {3167,3830}, {3448,3545}, {5054,5646}


X(5656) =  INTERSECTION OF LINES X(4)X(6) AND X(20)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5656) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(25). (Peter Moses, June 18, 2014)

X(5656) lies on these lines:
{4,6}, {5,5544}, {20,110}, {30,3167}, {64,631}, {154,376}, {185,3089}, {193,1533}, {221,1058}, {354,5603}, {378,1619}, {381,5644}, {1056,2192}, {1853,3545}, {3091,5643}, {3357,3523}, {3832,5645}


X(5657) =  INTERSECTION OF LINES X(3)X(8) AND X(4)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + b4 + c4 + 2a3b + 2a3c - a2b2 - a2c2 - 4a2bc -2ab3 -2ac3 + 2ab2c + 2abc2 - 2b2c2

X(5657) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(55). (Peter Moses, June 18, 2014)

X(5657) = {X(10),X(40)}-harmonic conjugate of X(4); X(5657) = centroid of antipedal triangle of X(7). Let A* be the parabola with focus A and directrix BC, and let A** be the polar of X(1) with respect to A*. Define B** and C** cyclically, and let A' = B**∩C**, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1000), and the centroid of A'B'C' is X(5657). (Randy Hutson, July 7, 2014)

X(5657) lies on these lines:
{1,631},{2,392},{3,8},{4,9},{5,962},{7,495},{12,4295},{20,355},{21,5554},{35,3486},{36,3476},{43,1064},{46,388},{55,1006},{57,1056},{63,2096},{65,3085},{80,4302},{140,1482},{145,1385},{165,376},{191,2950},{201,1148},{226,2093},{333,4221},{387,1108},{484,1478},{497,1737},{498,3485},{499,5445},{519,3158},{549,3241},{580,5264},{581,3293},{601,5247},{602,5255},{912,3681},{938,3295},{946,1698},{993,2077},{999,5435},{1012,5273},{1058,1210},{1072,1738},{1083,2726},{1125,3525},{1155,4293},{1376,3428},{1387,4345},{1483,3530},{1519,3452},{1532,3820},{1537,5328},{1699,3545},{1714,3987},{1770,5229},{1829,3088},{1836,5183},{1837,4294},{1872,4194},{1902,3089},{2098,5433},{2099,5432},{3035,5289},{3057,3086},{3214,4300},{3240,5396},{3522,4678},{3526,5550},{3528,3626},{3533,3624},{3560,5260},{3634,4301},{3651,5584},{3817,3828},{3869,5552},{4292,5128},{4305,5217},{4642,5230},{5084,5250},{5251,5537},{5258,5450},{5442,5559}

X(5657) = midpoint of X(165) ande X(3679)
X(5657) = reflection of X(I) in X(J) for these (I,J): (376, 165), (3817, 3828)
X(5657) = anticomplement of X(5886)
X(5657) = crossdifference of every pair of points on the line X(1459)X(3310)


X(5658) =  INTERSECTION OF LINES X(1)X(4) AND X(2)X(971)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5658) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(56). (Peter Moses, June 18, 2014)

X(5658) is the centroid of the antipedal triangle of X(8). Let AAABAC be the orthic triangle of the A-extouch triangle, let BABBBC be the orthic triangle of the B-extouch triangle, and let CACBCC be the orthic triangle of the C-extouch triangle. Let A' be the centroid of AABACA, let B' be the centroid of ABBBCB, and let C' be the centroid of ACBCCC. Then triangle A'B'C' is homothetic to ABC with center of homothety the {X(2),X(9)}-harmonic conjugate of X(57), and X(5658) = X(84)-of-A'B'C'. (Randy Hutson, July 7, 2014)

X(5658) lies on these lines:
{1,4},{2,971},{9,2272},{20,5440},{84,631},{100,329},{516,3158},{1538,5274},{1709,5218},{3305,3358}


X(5659) =  INTERSECTION OF LINES X(1)X(140) AND X(9)X(1699)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5659) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(35). (Peter Moses, June 18, 2014)

X(5659) lies on these lines: {1,140},{9,1699},{100,4847},{515,3651},{3925,5536}


X(5660) =  INTERSECTION OF LINES X(1)X(5) AND X(100)X(516)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5660) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(36). (Peter Moses, June 18, 2014)

Let A'B'C' be the orthic triangle. Let A* be the antiorthic axis of triangle AB'C', and define B* and C* cyclically. Let A'' = B*∩C*, and define B'' and C'' cyclically. Then X(5660) = X(165)-of-A''B''C''. (Randy Hutson, July 7, 2014)

X(5660) lies on these lines:
{1,5},{2,2801},{9,1768},{100,516},{104,5251},{153,214},{528,1699},{1512,4867},{1537,5541},{1538,3689},{1639,2826}


X(5661) =  MINIMIZER ON BROCARD AXIS OF x2 + y2 + z2

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a6b4 + a6c4 - 2a4b6 - 2a4c6 + a2b8 + a2c8 - a2b6c2 - a2b2c6 + 2a2b4c4 + b8c2 + b2c8 - b6c4 - b4c6)

Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given (Peter Moses, June 23, 2014) by

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)

X(5661) is the minimizer T when P = X(3) and Q = X(6), so that PU is the Brocard axis. Other examples follow:

If PU = Lemoine axis (P = X(187), Q = X(237)), then T = X(3229)
If PU = orthic axis (P = X(230), Q = X(223)), then T = X(230)
If PU = anti-orthic axis (P = X(44), Q = X(513)), then T = X(1575)
If PU = De Longchamps line (P = X(325), Q = X(523)), then T = X(325)
If PU = Gergonne line (P = X(241), Q = X(514)), then T = X(3008)
If P = X(1) and Q = X(3), then T = X(5662)
If P = X(1) and Q = X(6), then T = X(5701)

X(5661) lies on these lines: {2,647}, {3,6}, {3672,4235}

X(5661) = crossdifference of every pair of points on the line X(237)X(523)
X(5661) = crosssum of X(6) and X(5091)


X(5662) =  MINIMIZER ON LINE X(1)X(3) OF x2 + y2 + z2

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a5b2 + a5c2 - 2a4b2c - 2a4bc2 - 2a3b4 - 2a3c4 + 3a3b3c + 3a3bc3 + a2b4c + a2bc4 - a2b3c2 -a2b2c3 + ab6 + ac6 - 3ab5c - 3abc5 + 3ab4c2 + 3ab2c4 - 2ab3c3 + b6c + bc6 - b5c2 - b2c5)

Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given (Peter Moses, June 23, 2014) by

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)

X(5662) is the minimizer T when P = X(1) and Q = X(3).

X(5662) lies on these lines: {1,3}, {2,905}, {63,2427}


X(5663) =  ISOGONAL CONJUGATE OF X(477)

Trilinears   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(5663) is X(30)-of-T for these triangles T: orthocentroidal, X(2)-Brocard, and X(4)-Brocard; see X(5642). (Randy Hutson, July 7, 2014)

X(5663) lies on the line at infinity and these lines:
{1,3024},{3,74},{4,94},{5,113},{23,3581},{26,1498},{30,511},{40,2940},{49,3043},{51,3845},{52,3627},{67,1352},{64,155},{182,4550},{389,546},{500,3746},{548,1216},{895,1351},{1147,3357},{1350,2930},{1353,5095},{1597,5093},{1625,3269},{2088,2493},{3567,3843},{3850,5462}

X(5663) = isogonal conjugate of X(477)
X(5663) = X(2693)-Ceva conjugate of X(3)
X(5663) = X(477)-anticomplementary conjugate of X(8)
X(5663) = complementary conjugate of X(10)
X(5663) = crossdifference of X(6) and X(1637)
X(5663) = crosssum of X(I) and X(J) for these {I,J}: {55, 3013}, {523,3154}
X(5663) = X(3)-vertex conjugate of X(526)
X(5663) = trilinear product of X(2410) and X(2624)
X(5663) = barycentric product of X(I) and X(J) for these {I,J}: {526,2410}, {2437,3268}


X(5664) =  CENTER OF CIRCLE {X(3), X(5613), X(5617)}

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 - c2)(2a4 - b4 - c4 - a2b2 - a2c2 + 2b2c2)(b2 + c2 - a2 - bc)(b2 + c2 - a2 + bc) (Peter Moses, June 20, 2014)

Let U be the circle (X(5617), R) and V the circle (X(5613),R), so that U passes through X(14) and V passes through X(13), and let Γ be the circumcircle of ABC. Then X(5664) is the radical center of the circles U, V, and Γ. (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

X(5664) is the isotomic conjugate of the trilinear pole of line X(30)X(74). (Randy Hutson, July 7, 2014)

X(5664) lies on these lines: (pending)


X(5665) =  ISOGONAL CONJUGATE OF X(3601)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(s - a)((s2 - SA)]

Let A'B'C' be the intouch triangle of a triangle ABC. Let (BA') be the circle having diameter BA', and likewise for (A'C); define circles (CB') and (AC') cyclically, and define circles (B'A) and (C'B) cyclically. Let U be the point, other than A', in which (BA') meets the incircle, and define V and W cyclically. Let U' be the point, other than A', in which (A'C) meets the incircle, and define V' and W' cyclically. Let A''B''C'' be the triangle formed by the lines UU', VV', WW'. Let A''' be the point, other than A, in which the circles (AB') and (AC') meet, and define B''' and C''' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is X(5665). Moreover, A''B''C'' is perspective to A''B''C'', and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)

X(5665) lies on the Feuerbach hyperbola and these lines:
(1, 1427), (4, 3671), (7, 950), (8, 226), (9, 65), (21, 57), (34, 1172), (40, 943), (72, 4866), (79, 3586), (84, 942), (85, 314), (104, 3333), (388, 3243), (405, 3339), (728, 2171), (946, 3427), (1000, 3487), (1420, 2320), (1490, 3577), (1697, 2346), (1728, 3467), (1896, 5342), (2099, 2900), (2263, 2298), (2335, 3247), (3296, 3488), (3419, 5290), (3600, 5558), (3612, 5424), (4332, 5269), (4355, 5557)

X(5665) = isogonal conjugate of X(3601)


X(5666) =  INTERSECTION OF LINES X(1)X(1427) AND X(1054)X(3339)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [s(s2 + 4r2 + 2bc) - (b + c)(s2 + 2r2)]/(s - a)

Let A'B'C' be the intouch triangle of a triangle ABC. Let (BA') be the circle having diameter BA', and likewise for (A'C); define circles (CB') and (AC') cyclically, and define circles (B'A) and (C'B) cyclically. Let U be the point, other than A', in which (BA') meets the incircle, and define V and W cyclically. Let U' be the point, other than A', in which (A'C) meets the incircle, and define V' and W' cyclically. Let A''B''C'' be the triangle formed by the lines UU', VV', WW'. Let A''' be the point, other than A, in which the circles (AB') and (AC') meet, and define B''' and C''' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is X(5665). Moreover, A''B''C'' is perspective to A''B''C'', and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)

X(5666) lies on these lines: (1, 1427), (1054, 3339), (1707, 3361)


X(5667) =  X(30)-CEVA CONJUGATE OF X(4)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

As X(30)-Ceva conjugates of points on the Neuberg cubic (K001), the points X(5667) - X(5685) are also on the Neuberg cubic. Specifically, if P is a point on the Neuberg cubic, then the points X(74), P, and the X(30)-Ceva conjugate of P are collinear, since X(74) is the isopivot (or secondary pivot) of the cubic. See Table 9: Points on the Neuberg Cubic.

X(5667) inverse-in-circumconic-centered-at-X(4) of X(133); also, X(5667) is the antipode of X(4) in the bianticevian conic of X(1) and X(4). (Randy Hutson, July 7, 2014)

X(5667) lies on the Neuberg cubic and these lines:
{1,2816},{3,3462},{4,74},{19,2822},{20,1075},{112,376},{122,631},{146,4240},{399,2133},{1138,1157},{1148,4302},{1263,3481},{2790,3186},{3087,3269},{3183,3529},{3324,4293},{3440,3479},{3441,3480}

X(5667) = reflection of X(I() in X(J) for these (I,J): (4,107), (1294,3184)


X(5668) =  X(30)-CEVA CONJUGATE OF X(15)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5668) lies on the Neuberg cubic and these lines: {3,3166},{4,14},{13,3479},{15,74},{16,1495},{61,185},{1277,3466},{2133,5623}

X(5668) = reflection of X(5669) in X(3284)


X(5669) =  X(30)-CEVA CONJUGATE OF X(16)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5669) lies on the Neuberg cubic and these lines: {3,3165},{4,13},{14,3480},{15,1495},{16,74},{62,185},{1276,3466},{2133,5624}

X(5669) = reflection of X(5668) in X(3284)


X(5670) =  X(30)-CEVA CONJUGATE OF X(1138)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5670) lies on the Neuberg cubic and these lines: {3,2133},{74,1138}


X(5671) =  X(30)-CEVA CONJUGATE OF X(1263)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5671) is the tangential of X(3065) on the Neuberg cubic.

X(5671) lies on the Neuberg cubic, the Lester circle, and these lines: {3,1138},{30,1117},{74,1263},{2133,3484}>

X(5671) = X(1117)-Ceva conjugate of X(3471)


X(5672) =  X(30)-CEVA CONJUGATE OF X(1276)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5672) is the Gibert-Burek-Moses-concurrent-circles image of X(13). (Randy Hutson, July 7, 2014)

X(5672) lies on the Neuberg cubic and these lines: {1,13},{16,3065},{74,1276},{1277,3440},{1138,5673},{2940,2953}


X(5673) =  X(30)-CEVA CONJUGATE OF X(1277)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5673) is the Gibert-Burek-Moses-concurrent-circles image of X(14). (Randy Hutson, July 7, 2014)

X(5673) lies on the Neuberg cubic and these lines: {{1,14},{15,3065},{74,1277},{1276,3441},{1138,5672},{2940,2952}


X(5674) =  X(30)-CEVA CONJUGATE OF X(1337)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5674) is the tangential of X(13) on the Neuberg cubic.

X(5674) lies on the Neuberg cubic and these lines: {3,3440},{13,2981},{74,1337},{617,1138},{1263,3480}


X(5675) =  X(30)-CEVA CONJUGATE OF X(1338)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5675) is the tangential of X(14) on the Neuberg cubic.

X(5675) lies on the Neuberg cubic and these lines: {3,3441},{74,1338},{616,1138},{1263,3479}


X(5676) =  X(30)-CEVA CONJUGATE OF X(2133)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5676) lies on the Neuberg cubic and this line: {74,2133}


X(5677) =  X(30)-CEVA CONJUGATE OF X(3065)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5677) lies on the Neuberg cubic and these lines: {1,1138},{74,3065},{1263,3466},{2133,3465}


X(5678) =  X(30)-CEVA CONJUGATE OF X(3440)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5678) lies on the Neuberg cubic and these lines: {74,3440},{617,2133},{1138,3480}


X(5679) =  X(30)-CEVA CONJUGATE OF X(3441)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5679) lies on the Neuberg cubic and these lines: {74,3441},{616,2133},{1138,3479}


X(5680) =  X(30)-CEVA CONJUGATE OF X(3466)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5680) lies on the Neuberg cubic and these lines: {74,3466},{484,2133}


X(5681) =  X(30)-CEVA CONJUGATE OF X(3479)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5681) lies on the Neuberg cubic and these lines: {4,3441},{74,3479}


X(5682) =  X(30)-CEVA CONJUGATE OF X(3480)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5682) lies on the Neuberg cubic and these lines: {4,3440},{74,3480}


X(5683) =  X(30)-CEVA CONJUGATE OF X(3481)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5683) lies on the Neuberg cubic and these lines: {4,3463},{74,3481}


X(5684) =  X(30)-CEVA CONJUGATE OF X(3482)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5684) lies on the Neuberg cubic and these lines: {3,1263},{74,3482},{1138,3484},{3065,3483}


X(5685) =  X(30)-CEVA CONJUGATE OF X(3483)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5685) lies on the Neuberg cubic and these lines: {1,1263},{3,3065},{74,3483},{1138,3465},{3466,3482}


X(5686) =  INTERSECTION OF LINES X(7)X(10) and X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 + 3b3 + 3c3 + 5a2b + 5a2c - ab2 - ac2 + 2abc + 5b2c + 5bc2

X(5686) lies on these lines:
{1, 4924}, {2, 210}, {7, 10}, {8, 9}, {144, 1654}, {145, 1001}, {200, 5273}, {405, 2346}, {480, 958}, {497, 3715}, {516, 3543}, {936, 5265}, {971, 5657}, {984, 3672}, {1145, 1156}, {1445, 3600}, {1698, 5542}, {1738, 4346}, {1743, 4344}, {1757, 4307}, {1788, 3983}, {3059, 4662}, {3158, 5325}, {3189, 5302}, {3243, 3616}, {3485, 4005}, {3696, 4461}, {3711, 5218}, {3751, 3945}, {3974, 4042}, {4313, 5234}, {4321, 5435}, {4326, 4882}, {4384, 4899}, {4678, 5086}, {4847, 5274}, {5231, 5328}


X(5687) =  PERSPECTOR OF OUTER GARCIA TRIANGLE AND TANGENTIAL 1ST CIRCUMPERP TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a2 - ab2 - ac2 - 2abc + 2b2c + 2bc2)

The outer Garcia triangle is defined at X(5587). The definition is re-stated here. Let TATBTC be the extouch triangle of a triangle ABC, and let LA be the line perpendicular to line BC at TA. Of the two points on LA at distance r from TA, let A' be the one farther from A and let A'' be the closer. Define B', C' and B'', C'' cyclically. Then A'B'C' is the outer Garcia triangle, and A''B''C'' the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014).

X(5687) = X(56)-of-inner-Garcia-triangle (Randy Hutson, July 18, 2014)

X(5687) lies on these lines: (pending)


X(5688) =  PERSPECTOR OF OUTER GARCIA TRIANGLE AND OUTER GREBE TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 + b2c + bc2 + (b + c)S

The outer Garcia triangle is defined at X(5587).

X(5688) lies on these lines: (pending)


X(5689) =  PERSPECTOR OF OUTER GARCIA TRIANGLE AND INNER GREBE TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - a3 + b2c + bc2 - (b + c)S

The outer Garcia triangle is defined at X(5587).

X(5689) lies on these lines: (pending)


X(5690) =  NINE-POINT CENTER OF OUTER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The outer Garcia triangle is defined at X(5587).

X(5690) lies on these lines:
{1,140},{2,1482},{3,8},{4,3617},{5,10},{20,4678},{30,40},{46,5252},{65,495},{80,3467} (others pending)


X(5691) =  DE LONGCHAMPS POINT OF OUTER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The outer Garcia triangle is defined at X(5587).

X(5691) lies on these lines:
{1,4},{2,4297},{3,1698},{5,3576},{8,144},{10,20},{11,1420},{12,3601},{30,40},{35,1012},{36,3149},{46,80},{57,1837},{63,5086},{65,971},{78,5080},{79,3577} (others pending)


X(5692) =  CENTROID OF INNER GARCIA TRIANGLE

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = abc + (b + c)(b2 + c2 - a2)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b3 + c3 - a2b - a2c + abc + b2c + bc2)

The inner Garcia triangle A''B''C'' is defined at X(5587). Another construction of A'' is as the reflection of X(1) in the perpendicular bisector of side BC, so that |OA''| = |X(1)X(3)|, and A'', B'', C'', and X(1) are on a circle with center X(3). (Paul Yiu, ADGEOM #1214, April 3, 2014)

A'' = a2 : c2 - b2 + ac : b2 - c2 + ab
B'' = c2 - a2 + bc : b2 : a2 - c2 + ba
C'' = b2 - a2 + cb : a2 - b2 + ca : c2
(Peter Moses, April 4, 2014)

The appearance of (I,J) in the following list means that (X(I) of A''B''C') = X(J): (1,8),(3,3),(10,3878),(11,72),(21,191),(35,2975),(36,100),(40,944),(55,956),(63,4302),(78,1479),(80,3869),(100,1),(104,40),(214,10),(238,190),(662,2607),(663,3762),(667,659),(976,4894),(1001,5220),(1125,3678),(1145,3057),(1149,4738),(1193,1089),(1319,1145),(1320,3632),(1325,2948),(1376,5289),(1459,4768),(1734,3904),(1768,20),(1818,4858),(2077,104),(2932,56),(3032,2901),(3035,960),(3065,3648),(3220,1633),(3286,4436),(4057,4491),(4367,4730),(4511,80),(4855,499),(4996,35),(5150,3923),(5313,4671),(5440,11),(5531,962),(5541,145). (Peter Moses, April 3, 2014)

X(5692) lies on these lines: (pending)

X(5692) = reflection of X(1) in X(392)
X(5692) = anticomplement of X(5883)


X(5693) =  ORTHOCENTER OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b6 + c6 - a5b - a5c + a4b2 + a4c2 + a4bc + 2a3b3 + 2a3c3 - a3b2c - a3bc2 - 2a2b4 - 2a2c4 - a2b3c - a2bc3 + 2a2b2c2 - ab5 - ac5 + 2ab4c + 2abc4 - ab3c2 - ab2c3 - b2c4 - b2c4)

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5693) is the incenter of the X(3)-Fuhrmann triangle, defined at X(5613).

X(5693) lies on these lines: (pending)

X(5693) = anticomplement of X(5884)


X(5694) =  NINE-POINT CENTER OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b6 + c6 - a5b - a5c + a4b2 + a4c2 + 2a4bc + 2a3b3 + 2a3c3 - a3b2c - a3bc2 - 2a2b4 - 2a2c4 - 2a2b3c - 2a2bc3 + 2a2b2c2 - ab5 - ac5 + 2ab4c + 2abc4 - ab3c2 - ab2c3 - b2c4 - b2c4)

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

Let A* be the reflection of X(5) in the perpendicular bisector of segment BC, and define B* and C* cyclically. Triangle A*B*C* is inversely similar to ABC, with similitude center X(3); also, A*B*C* is perspective to ABC, with perspector X(3519), and X(5694) is the incenter of A*B*C*.

X(5694) lies on these lines: (pending)

X(5694) = anticomplement of X(5885)


X(5695) =  SYMMEDIAN POINT OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - a2b - a2c + 2b2c + 2bc2

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5695) lies on these lines: (pending)


X(5696) =  X(7) OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b5 + c5 - a4b - a4c + 2a3b2 + 2a3c2 - a3bc + 2a2b2c + 2a2bc2 - ab3c - abc3 -2ab4 -2ac4 + b4c + bc4 - 2b3c2 - 2b2c3 - 2ab2c2)

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5696) lies on these lines: (pending)


X(5697) =  X(8) OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b3 + c3 - a2b - a2c + 3abc - b2c - bc2)

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5697) = {X(1),X(40)}-harmonic conjugate of X(36)

X(5697) lies on these lines: (pending)


X(5698) =  X(9) OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3 + c3 - 3a3 + a2b + a2c + ab2 + ac2 + 2abc - b2c - bc2

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5698) lies on these lines: (pending)

X(5698) = anticomplement of X(5880)


X(5699) =  X(15) OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2(a + b + c)(a2 - b2 - c2) - 2(31/2)(a3 - a2b - a2c + 2b2c + 2bc2)S

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5699) lies on these lines: (pending)


X(5700) =  X(16) OF INNER GARCIA TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2(a + b + c)(a2 - b2 - c2) + 2(31/2)(a3 - a2b - a2c + 2b2c + 2bc2)S

The inner Garcia triangle A''B''C'' is defined at X(5587); see also X(5692).

X(5700) lies on these lines: (pending)


X(5701) =  MINIMIZER ON LINE X(1)X(6) OF x2 + y2 + z2

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3b2 + a3c2 - 2a2b3 - 2a2c3 + ab4 + ac4 - ab3c - abc3 + 2a2b2 + b4c + bc4 - b3c2 - b2c3)

Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x2 + y2 + z2 is given by

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)     (Peter Moses, June 23, 2014)

X(5701) is the minimizer T when P = X(1) and Q = X(6). See also X(5661) and X(5662).

X(5701) lies on these lines: {1,6},{2,650},{1252,1621},{3693,4702},{5284,5375}


X(5702) =  CENTER OF MONTESDEOCA CONIC

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = SBSC(5a2SA - SBSC)

Let ABC be a triangle, let PA be the polar of A with respect to the circle with diameter BC, and define PB and PC cyclically. Let AB = PA∩AB and AC = PA∩AC, and define BC, CA, BA, and CB cyclically. The six points AB, AC, BC, BA, CA, CB lie on, and define, the Montesdeoca conic. (Angel Montesdeoca, June 23, 2014)

A barycentric equation for the Montesdeoca conic is found from AB = SC : 0 : 2SA and AC = SB : 2SA : 0 to be as follows:

2(S2Ax2 + S2By2 + S2Cz2) - 5(SBSCyz + SCSAzx + SASBxy) = 0      (Peter Moses, June 23, 2014)

The Montesdeoca conic is the anticevian-intersection conic when P = X(4); this conic is defined by Francisco J. Garcia Capitán (The Anticevian Intersection Conic and Hyacinthos #20547 (December 19, 2011). Also, the perspector of the Montesdeoca conic is X(4).

X(5702) lies on these lines:
{4,6},{297,5032},{340,1992},{376,3284},{468,5304},{578,3183},{631,5158},{3163,3545}


X(5703) =  INTERSECTION OF LINES X(1)X(2) AND X(3)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 4*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5703) lies on these lines:
{1, 2}, {3, 7}, {4, 4313}, {5, 3488}, {12, 3486}, {20, 226}, {21, 329}, {35, 4295}, {40, 5281}, {55, 411}, {56, 3475}, {57, 3523}, {65, 5218}, {72, 5273}, {86, 939}, {142, 5438}, {165, 3671}, {307, 3945}, {388, 2646}, {390, 946}, {443, 5440}, {452, 908}, {495, 944}, {515, 5261}, {517, 4323}, {631, 942}, {940, 3562}, {950, 3091}, {988, 4310}, {1056, 1385}, {1445, 3333}, {1446, 3160}, {1478, 4305}, {1699, 4314}, {1788, 5432}, {2287, 5296}, {2476, 5175}, {2886, 3189}, {3146, 4304}, {3149, 3295}, {3361, 5542}, {3452, 5129}, {3474, 3649}, {3522, 4292}, {3576, 3600}, {3586, 3832}, {3612, 4293}, {4000, 4255}, {4252, 4644}, {4297, 5290}, {4344, 5266}, {4855, 5249}, {5084, 5328}


X(5704) =  INTERSECTION OF LINES X(1)X(2) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b - 4*a^2*b^2 - 2*a*b^3 + 3*b^4 + 2*a^3*c + 4*a^2*b*c + 2*a*b^2*c - 4*a^2*c^2 + 2*a*b*c^2 - 6*b^2*c^2 - 2*a*c^3 + 3*c^4

X(5704) lies on these lines:
{1, 2}, {4, 5435}, {5, 7}, {11, 962}, {20, 3911}, {40, 5274}, {57, 3091}, {72, 5328}, {88, 5125}, {90, 5556}, {104, 3149}, {140, 3488}, {226, 5056}, {307, 4346}, {329, 4193}, {355, 4308}, {404, 5175}, {411, 5204}, {496, 5657}, {515, 5265}, {631, 4313}, {942, 3090}, {950, 3523}, {1155, 5225}, {1158, 1445}, {1656, 3487}, {1728, 3218}, {3035, 3189}, {3306, 5177}, {3333, 5261}, {3339, 3817}, {3486, 5433}, {3522, 3586}, {3529, 5122}, {3562, 4383}, {3600, 5587}, {3614, 4860}, {3832, 4292}, {4208, 5437}, {4306, 5400}, {5084, 5273}


X(5705) =  INTERSECTION OF LINES X(1)X(2) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b - 3*a^2*b^2 - a*b^3 + 2*b^4 + a^3*c - 2*a^2*b*c - 3*a*b^2*c - 3*a^2*c^2 - 3*a*b*c^2 - 4*b^2*c^2 - a*c^3 + 2*c^4

X(5705) lies on these lines:
{1, 2}, {5, 9}, {21, 3586}, {40, 2886}, {57, 442}, {63, 2476}, {72, 5219}, {140, 5438}, {283, 5235}, {411, 993}, {443, 3911}, {958, 3149}, {965, 2323}, {1445, 3841}, {1479, 4512}, {1656, 5044}, {2475, 4652}, {3090, 3452}, {3091, 5273}, {3219, 5141}, {3305, 4193}, {3306, 4197}, {3419, 3601}, {3545, 5325}, {3576, 4999}, {3646, 3816}, {3814, 5536}, {3822, 5290}, {4208, 5435}, {4292, 5177}, {4304, 5175}


X(5706) =  INTERSECTION OF LINES X(1)X(3) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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 - 2*a^3*b^2*c + a*b^4*c + 2*b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5706) lies on these lines:
{1, 3}, {4, 6}, {5, 1714}, {7, 3562}, {10, 219}, {20, 81}, {28, 154}, {33, 1712}, {51, 4186}, {58, 1012}, {64, 4219}, {98, 3597}, {184, 4185}, {221, 278}, {222, 4292}, {377, 394}, {386, 3149}, {405, 580}, {429, 1899}, {602, 1001}, {774, 4336}, {990, 1071}, {991, 4658}, {1191, 5603}, {1203, 1699}, {1260, 3191}, {1376, 3682}, {1451, 2654}, {1478, 3173}, {1612, 3052}, {1753, 2285}, {1765, 2257}, {1780, 3560}, {1836, 1838}, {1853, 5142}, {1993, 2475}, {2192, 2982}, {2256, 5657}, {3695, 4513}, {3713, 5295}, {4259, 5562}, {5046, 5422}


X(5707) =  INTERSECTION OF LINES X(1)X(3) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + 2*b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5707) lies on these lines:
{1, 3}, {2, 3193}, {4, 81}, {5, 6}, {7, 1068}, {58, 3560}, {222, 225}, {226, 3157}, {283, 405}, {394, 442}, {581, 4658}, {602, 3720}, {965, 2323}, {1069, 1210}, {1216, 4259}, {1437, 4185}, {1480, 4301}, {1656, 4383}, {1993, 2476}, {1994, 5141}, {3149, 5396}, {3487, 3562}, {4193, 5422}


X(5708) =  INTERSECTION OF LINES X(1)X(3) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b - a*b^2 - 2*b^3 + 2*a^2*c + 4*a*b*c + 2*b^2*c - a*c^2 + 2*b*c^2 - 2*c^3)

X(5708) lies on these lines:
{1, 3}, {2, 3927}, {5, 7}, {28, 89}, {30, 938}, {45, 579}, {63, 5439}, {72, 3306}, {140, 3487}, {142, 3634}, {191, 4423}, {222, 1393}, {226, 1656}, {355, 4298}, {381, 553}, {382, 4031}, {405, 3218}, {443, 3617}, {474, 3868}, {495, 1788}, {496, 4295}, {499, 3649}, {548, 4313}, {550, 3488}, {950, 1657}, {952, 3600}, {1086, 5292}, {1376, 3874}, {1435, 1871}, {1439, 3527}, {1483, 4308}, {1598, 1876}, {3526, 3911}, {3586, 5073}, {3624, 4880}, {3628, 5226}, {3872, 4004}, {3881, 3913}, {3982, 5079}, {4084, 5289}, {4114, 5072}, {4306, 5396}, {4321, 5534}, {4355, 5587}, {4654, 5055}, {5044, 5437}, {5070, 5219}, {5445, 5557}


X(5709) =  INTERSECTION OF LINES X(1)X(3) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 2*a^4*b*c + 2*b^5*c - 3*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - 4*b^3*c^3 + 3*a^2*c^4 + b^2*c^4 + 2*b*c^5 - c^6)

X(5709) = X(26)-of-excentral-triangle.

X(5709) lies on these lines:
{1, 3}, {4, 63}, {5, 9}, {20, 3218}, {28, 283}, {30, 84}, {34, 255}, {72, 3149}, {90, 3583}, {140, 5437}, {155, 610}, {191, 1699}, {212, 1393}, {223, 3157}, {225, 1217}, {381, 3929}, {411, 3868}, {443, 5657}, {516, 1158}, {518, 5534}, {578, 3955}, {579, 1766}, {602, 614}, {631, 3306}, {912, 1490}, {920, 1479}, {1012, 3916}, {1068, 1119}, {1069, 3345}, {1070, 4331}, {1071, 1998}, {1072, 5230}, {1093, 1948}, {1210, 1708}, {1254, 1496}, {1352, 5227}, {1453, 5398}, {1512, 3436}, {1707, 3073}, {1776, 5225}, {1817, 3193}, {2000, 4219}, {3090, 3305}, {3091, 3219}, {5250, 5603}


X(5710) =  INTERSECTION OF LINES X(1)X(3) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c + 2*b^2*c + a*c^2 + 2*b*c^2)

X(5710) lies on these lines:
{1, 3}, {2, 1191}, {6, 8}, {10, 3966}, {21, 3052}, {31, 958}, {37, 5250}, {42, 3913}, {58, 956}, {81, 145}, {87, 2334}, {100, 4255}, {197, 1036}, {218, 3997}, {220, 5276}, {221, 388}, {387, 5082}, {392, 975}, {405, 595}, {474, 995}, {608, 1891}, {611, 5252}, {612, 960}, {614, 3812}, {750, 1201}, {962, 5244}, {978, 4413}, {1001, 1918}, {1056, 4340}, {1100, 3895}, {1193, 1376}, {1203, 3679}, {1406, 5434}, {1407, 3600}, {1449, 2136}, {1616, 3616}, {1698, 5315}, {1706, 2999}, {1722, 3698}, {1834, 3434}, {1999, 4673}, {2176, 5275}, {2256, 2303}, {2650, 3938}, {2886, 5230}, {2975, 4252}, {3242, 3868}, {3486, 4339}, {3782, 4295}, {3869, 3920}, {4363, 4968}, {4646, 5256}


X(5711) =  INTERSECTION OF LINES X(1)X(3) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c + 2*a*b*c + 2*b^2*c + a*c^2 + 2*b*c^2)

X(5711) lies on these lines:
{1, 3}, {4, 608}, {6, 10}, {8, 81}, {31, 405}, {58, 958}, {72, 612}, {213, 5275}, {218, 5276}, {219, 2303}, {220, 3997}, {221, 226}, {222, 388}, {281, 3194}, {341, 3758}, {386, 1376}, {387, 2550}, {406, 3195}, {442, 5230}, {474, 750}, {495, 611}, {551, 1616}, {595, 1001}, {601, 1012}, {614, 5439}, {651, 5261}, {894, 4385}, {938, 4344}, {946, 2050}, {956, 1468}, {960, 975}, {976, 2650}, {984, 1046}, {993, 4252}, {1064, 3149}, {1065, 1433}, {1100, 4646}, {1107, 5021}, {1125, 1191}, {1203, 1698}, {1386, 3812}, {1407, 4298}, {1449, 1706}, {1714, 3925}, {1740, 4649}, {2271, 4386}, {2292, 5311}, {2295, 3695}, {2305, 3743}, {2886, 5292}, {3052, 5248}, {3216, 4413}, {3242, 3874}, {3488, 4339}, {3562, 3945}, {3624, 5315}, {3720, 3915}, {3736, 3913}, {3868, 3920}, {3876, 5297}, {3940, 5293}, {4356, 5493}, {4662, 4663}, {4868, 5110}, {5044, 5268}, {5250, 5287}


X(5712) =  INTERSECTION OF LINES X(1)X(4) AND X(2)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3 + 3*a^2*b + a*b^2 - b^3 + 3*a^2*c + 2*a*b*c + b^2*c + a*c^2 + b*c^2 - c^3

X(5712) lies on these lines:
{1, 4}, {2, 6}, {3, 4340}, {7, 941}, {37, 329}, {42, 2550}, {55, 4307}, {57, 573}, {63, 4644}, {142, 2999}, {171, 212}, {306, 2345}, {312, 1909}, {345, 894}, {354, 1469}, {386, 443}, {387, 442}, {406, 3194}, {553, 4888}, {580, 631}, {908, 5287}, {1100, 3772}, {1104, 3616}, {1125, 1453}, {1212, 5308}, {1215, 3974}, {1730, 4266}, {1788, 5530}, {1834, 5177}, {3247, 4656}, {3296, 3953}, {3622, 5484}, {3663, 4654}, {3672, 3782}, {3677, 5542}, {3744, 4344}, {3752, 4277}, {3931, 4295}, {3982, 4862}, {4000, 5249}, {4349, 5269}, {4641, 5273}, {4658, 5292}


X(5713) =  INTERSECTION OF LINES X(1)X(4) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - a^5*b^2 - 3*a^4*b^3 - a^3*b^4 + 3*a^2*b^5 + a*b^6 - b^7 + a^6*c - 3*a^4*b^2*c - 2*a^3*b^3*c + a^2*b^4*c + 2*a*b^5*c + b^6*c - a^5*c^2 - 3*a^4*b*c^2 - 2*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 - 3*a^4*c^3 - 2*a^3*b*c^3 - 4*a^2*b^2*c^3 - 4*a*b^3*c^3 - 3*b^4*c^3 - a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 + 3*a^2*c^5 + 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5713) lies on these lines: {1, 4}, {2, 283}, {5, 6}, {212, 498}, {499, 1451}, {1899, 3142}, {2299, 3542}, {2476, 3193}


X(5714) =  INTERSECTION OF LINES X(1)X(4) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b + 2*a^2*b^2 - 2*a*b^3 - 3*b^4 + 2*a^3*c + 4*a^2*b*c + 2*a*b^2*c + 2*a^2*c^2 + 2*a*b*c^2 + 6*b^2*c^2 - 2*a*c^3 - 3*c^4

X(5714) lies on these lines:
{1, 4}, {2, 3824}, {3, 5226}, {5, 7}, {9, 3634}, {12, 4295}, {40, 3947}, {45, 1901}, {57, 3090}, {72, 3617}, {79, 498}, {329, 442}, {381, 938}, {382, 4313}, {405, 5253}, {443, 908}, {452, 5126}, {495, 962}, {517, 5261}, {553, 5071}, {631, 4292}, {942, 3091}, {943, 5556}, {952, 4323}, {1000, 4301}, {1006, 5204}, {1210, 3545}, {1656, 5435}, {1770, 5218}, {1836, 3085}, {1892, 3089}, {2345, 3454}, {3333, 3817}, {3419, 3621}, {3529, 3601}, {3614, 5221}, {3651, 5217}, {3671, 5587}, {3911, 5067}, {4208, 5044}, {4317, 5443}, {4330, 5561}, {4644, 5292}, {5045, 5274}, {5084, 5249}


X(5715) =  INTERSECTION OF LINES X(1)X(4) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - 4*a^2*b^5 - a*b^6 + 2*b^7 + 2*a^5*b*c + 2*a^4*b^2*c - 2*a*b^5*c - 2*b^6*c + a^5*c^2 + 2*a^4*b*c^2 - 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + a*b^4*c^2 - 6*b^5*c^2 + 2*a^4*c^3 + 4*a^2*b^2*c^3 + 4*a*b^3*c^3 + 6*b^4*c^3 + a^3*c^4 + a*b^2*c^4 + 6*b^3*c^4 - 4*a^2*c^5 - 2*a*b*c^5 - 6*b^2*c^5 - a*c^6 - 2*b*c^6 + 2*c^7

X(5715) lies on these lines:
{1, 4}, {3, 3824}, {5, 9}, {40, 442}, {72, 5587}, {79, 1709}, {329, 3091}, {355, 3577}, {962, 5177}, {1006, 3624}, {1071, 4654}, {1158, 4312}, {3149, 5219}


X(5716) =  INTERSECTION OF LINES X(1)X(4) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*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*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5716) lies on these lines:
{1, 4}, {2, 1104}, {6, 8}, {10, 1453}, {20, 3666}, {29, 2303}, {37, 452}, {42, 3189}, {55, 4339}, {56, 4220}, {65, 3056}, {85, 3945}, {145, 321}, {171, 1451}, {212, 5255}, {345, 4195}, {377, 4000}, {387, 3419}, {405, 1612}, {580, 5264}, {612, 2551}, {938, 940}, {942, 3784}, {975, 5084}, {986, 3474}, {1036, 1610}, {1427, 3600}, {1697, 1766}, {1834, 5175}, {1837, 3745}, {1841, 4198}, {3085, 5266}, {3146, 3672}, {3436, 3920}, {3616, 5051}, {3677, 4298}, {3772, 5177}, {3868, 4644}, {3931, 4294}, {4190, 4850}, {5218, 5530}


X(5717) =  INTERSECTION OF LINES X(1)X(4) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 - 3*a^3*b - 3*a^2*b^2 - a*b^3 + b^4 - 3*a^3*c - 6*a^2*b*c - 3*a*b^2*c - 3*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4

X(5717) lies on these lines:
{1, 4}, {2, 1453}, {6, 10}, {12, 3745}, {40, 2269}, {57, 4340}, {171, 580}, {204, 406}, {212, 5264}, {306, 964}, {377, 5256}, {387, 1449}, {443, 2999}, {511, 942}, {516, 3931}, {519, 5295}, {553, 3670}, {937, 2551}, {938, 3945}, {940, 1210}, {975, 3452}, {1010, 3687}, {1100, 1834}, {1104, 1125}, {1329, 4682}, {1330, 4357}, {1427, 4298}, {1451, 3911}, {1842, 2294}, {2047, 5405}, {2303, 3194}, {2334, 4863}, {2478, 5287}, {3085, 5269}, {3666, 4292}, {4208, 5222}, {5129, 5308}, {5249, 5262}


X(5718) =  INTERSECTION OF LINES X(1)X(5) AND X(2)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a2(b + c) + a(b2 + c2) - (b - c)2(b + c)

X(5718) lies on these lines:
{1, 5}, {2, 6}, {10, 4023}, {37, 908}, {42, 2886}, {43, 3925}, {55, 4192}, {57, 4888}, {65, 970}, {140, 5398}, {171, 2361}, {226, 1465}, {312, 3963}, {313, 4358}, {377, 4255}, {386, 442}, {469, 1865}, {516, 4689}, {528, 2177}, {536, 4054}, {631, 4340}, {750, 3035}, {851, 5132}, {899, 3826}, {986, 3649}, {1086, 4850}, {1215, 3703}, {1386, 3011}, {1468, 4999}, {1834, 2476}, {1848, 1880}, {3058, 3750}, {3306, 4675}, {3550, 4995}, {3664, 3911}, {3687, 4967}, {3706, 4028}, {3712, 3923}, {3720, 3816}, {3752, 5249}, {3772, 5256}, {3821, 4892}, {3838, 3914}, {3943, 4671}, {3944, 4854}, {3999, 5542}, {4009, 4078}, {4030, 4865}, {4031, 4896}, {4090, 4126}, {4220, 5347}, {4307, 5218}, {4654, 4902}, {5308, 5328}

X(5718) = complement of X(1150)


X(5719) =  INTERSECTION OF LINES X(1)X(5) AND X(3)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 2*a^3*b - 3*a^2*b^2 + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 3*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5719) lies on these lines:
{1, 5}, {2, 3940}, {3, 7}, {30, 226}, {35, 3649}, {37, 3002}, {57, 549}, {73, 5453}, {140, 942}, {381, 3488}, {382, 4313}, {484, 4995}, {518, 1125}, {546, 950}, {548, 4292}, {550, 3601}, {551, 3452}, {553, 5122}, {938, 1656}, {956, 3616}, {999, 3475}, {1000, 1482}, {1086, 4256}, {1159, 5657}, {1210, 3628}, {1385, 4315}, {3295, 3485}, {3296, 5265}, {3530, 4031}, {3579, 3671}, {3584, 5425}, {3586, 3845}, {3622, 5084}, {3748, 4870}, {3874, 4999}, {4415, 4653}, {5054, 5435}, {5249, 5440}, {5298, 5444}


X(5720) =  INTERSECTION OF LINES X(1)X(5) AND X(3)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c + 2*a^4*b*c - 4*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 - 4*a^2*b*c^3 - 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6)

X(5720) lies on these lines:
{1, 5}, {3, 9}, {4, 78}, {8, 1512}, {30, 1750}, {57, 912}, {72, 3149}, {200, 517}, {210, 3428}, {223, 1060}, {381, 2900}, {386, 3553}, {411, 3876}, {474, 1071}, {515, 997}, {581, 975}, {612, 1064}, {944, 5084}, {946, 3811}, {962, 4420}, {1006, 3305}, {1012, 5440}, {1038, 1745}, {1040, 3465}, {1217, 1826}, {1376, 3359}, {1482, 3577}, {1519, 3434}, {1532, 3419}, {1709, 2077}, {1743, 5398}, {3560, 3601}, {3576, 5251}, {3870, 5603}


X(5721) =  INTERSECTION OF LINES X(1)X(5) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6*b + a^5*b^2 + 3*a^4*b^3 - 2*a^3*b^4 + a*b^6 - b^7 - 2*a^6*c + a^4*b^2*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 + 3*a^4*c^3 - 3*b^4*c^3 - 2*a^3*c^4 - a*b^2*c^4 - 3*b^3*c^4 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5721) lies on these lines:
{1, 5}, {3, 1714}, {4, 6}, {30, 1754}, {40, 1723}, {51, 1894}, {65, 1243}, {184, 1884}, {209, 517}, {442, 581}, {518, 1072}, {912, 3782}, {1064, 2886}, {1108, 1512}, {1210, 1465}, {1214, 1737}, {1329, 3682}, {1785, 1864}, {2361, 3073}, {3149, 5292}, {3428, 4192}


X(5722) =  INTERSECTION OF LINES X(1)X(5) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^4 + a^3*b - a*b^3 + b^4 + a^3*c + 2*a^2*b*c + a*b^2*c + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4

X(5722) = X(25)-of-Fuhrmann-triangle; X(5722) = inverse-in-Feuerbach-hyperbola of X(5252); X(5722) = {X(1),X(80)}-harmonic conjugate of X(5252). (Randy Hutson, July 7, 2014)

X(5722) lies on these lines:
{1, 5}, {2, 3419}, {3, 950}, {4, 7}, {6, 5179}, {8, 392}, {10, 1001}, {30, 57}, {55, 1737}, {65, 1479}, {72, 2478}, {78, 4187}, {79, 5561}, {81, 5155}, {90, 3652}, {140, 3601}, {200, 3820}, {222, 1877}, {224, 442}, {226, 381}, {354, 1478}, {376, 5122}, {377, 5439}, {382, 4031}, {388, 5045}, {390, 5657}, {405, 1259}, {443, 5175}, {497, 517}, {499, 2646}, {515, 999}, {519, 3452}, {553, 3830}, {631, 4313}, {912, 1864}, {943, 5047}, {997, 3816}, {1056, 5049}, {1062, 1834}, {1104, 5292}, {1145, 3895}, {1155, 4302}, {1319, 3655}, {1329, 3811}, {1385, 3086}, {1770, 5221}, {1788, 3579}, {1836, 3583}, {1936, 5398}, {2099, 3656}, {3058, 3654}, {3091, 3487}, {3241, 5176}, {3434, 3753}, {3436, 3555}, {3545, 5226}, {3582, 3653}, {3612, 5433}, {3616, 5086}, {3679, 4863}, {3824, 5177}, {3845, 4654}, {3868, 5046}, {3873, 5080}, {4295, 5225}, {5090, 5142}, {5274, 5603}, {5427, 5441}


X(5723) =  INTERSECTION OF LINES X(1)X(5) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)*(a - b + c)*(2*a^3 - 2*a^2*b + a*b^2 - b^3 - 2*a^2*c + b^2*c + a*c^2 + b*c^2 - c^3)

X(5723) lies on these lines:
{1, 5}, {2, 664}, {6, 7}, {57, 1358}, {88, 279}, {226, 544}, {241, 514}, {278, 1783}, {1419, 4859}, {1441, 3589}, {1456, 1738}, {4422, 4552}

X(5723) = crossdifference of every pair of points on the line X(55)X(654)


X(5724) =  INTERSECTION OF LINES X(1)X(5) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 - a^2*b^2 - 2*a*b^3 + b^4 - 4*a^2*b*c - a^2*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5724) lies on these lines:
{1, 5}, {6, 8}, {10, 4434}, {30, 4424}, {38, 529}, {65, 511}, {388, 4310}, {515, 3666}, {519, 1215}, {982, 5434}, {1146, 5276}, {1478, 3782}, {1834, 5086}, {1880, 1891}, {2361, 5255}, {2646, 5530}, {3679, 5269}, {3920, 5176}, {4304, 4689}, {4415, 5080}, {5264, 5398}


X(5725) =  INTERSECTION OF LINES X(1)X(5) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b + 2*a^2*b^2 + a*b^3 - b^4 + a^3*c + 4*a^2*b*c + a*b^2*c + 2*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4

X(5725) lies on these lines:
{1, 5}, {3, 5530}, {4, 941}, {6, 10}, {42, 3419}, {171, 5398}, {388, 1465}, {515, 2050}, {940, 1737}, {942, 1469}, {975, 998}, {1478, 3666}, {1783, 5276}, {1788, 4340}, {1836, 4424}, {2361, 5264}, {3085, 5266}, {3772, 3822}, {3820, 5268}, {4205, 5336}, {4302, 4689}, {4307, 5657}, {4682, 5123}


X(5726) =  INTERSECTION OF LINES X(1)X(5) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)*(a - b + c)*(a^2 - a*b + 4*b^2 - a*c + 8*b*c + 4*c^2)

X(5726) lies on these lines:
{1, 5}, {2, 4315}, {7, 10}, {8, 3947}, {165, 1478}, {226, 3679}, {388, 1698}, {519, 5226}, {946, 1000}, {1788, 4031}, {2099, 4677}, {2476, 4853}, {2886, 4915}, {3085, 4304}, {3340, 4668}, {3436, 5234}, {3485, 3632}, {3600, 3634}, {3617, 3671}, {3625, 4323}, {3731, 5179}, {3828, 5435}, {4312, 5657}, {4512, 5080}, {5123, 5437}


X(5727) =  INTERSECTION OF LINES X(1)X(5) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^3 - a*b^2 + 2*b^3 + 2*a*b*c - 2*b^2*c - a*c^2 - 2*b*c^2 + 2*c^3)

X(5727) lies on these lines:
{1, 5}, {4, 3340}, {8, 9}, {10, 3486}, {20, 4848}, {30, 2093}, {40, 920}, {46, 4316}, {55, 3679}, {57, 515}, {65, 971}, {145, 908}, {388, 5542}, {497, 519}, {517, 1864}, {944, 1210}, {1012, 3256}, {1249, 1826}, {1478, 4654}, {1698, 2646}, {1699, 2099}, {1706, 5554}, {1737, 3576}, {1788, 4297}, {2098, 3633}, {3057, 3632}, {3058, 4677}, {3241, 5274}, {3617, 4313}, {3626, 4314}, {3671, 5229}, {3832, 4323}, {3870, 5176}, {4301, 5225}, {4304, 5657}, {4863, 4915}


X(5728) =  INTERSECTION OF LINES X(1)X(6) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 + 2*a*c^4 + b*c^4 - c^5)

X(5728) = {X(1),X(9)}-harmonic conjugate of X(954)

X(5728) lies on these lines:
{1, 6}, {2, 955}, {3, 1445}, {4, 7}, {10, 3059}, {11, 118}, {40, 4326}, {55, 1708}, {65, 516}, {81, 162}, {142, 442}, {144, 452}, {241, 991}, {329, 3873}, {390, 517}, {480, 3811}, {497, 5173}, {774, 3931}, {943, 2346}, {986, 4335}, {990, 5228}, {1005, 3218}, {1012, 3358}, {1156, 2771}, {1260, 3870}, {1376, 2900}, {1490, 3333}, {1730, 3198}, {1737, 3826}, {1890, 1905}, {1898, 3649}, {2550, 3419}, {2951, 3339}, {3062, 5665}, {3085, 3697}, {3487, 5045}, {3586, 4312}


X(5729) =  INTERSECTION OF LINES X(1)X(6) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 - 4*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 - 2*b^5 - 4*a^3*c^2 - 6*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 + 3*a*c^4 - 2*c^5)

X(5729) lies on these lines:
{1, 6}, {4, 653}, {5, 7}, {56, 2801}, {144, 2478}, {226, 4860}, {516, 1837}, {527, 1210}, {938, 3927}, {971, 1445}, {1155, 1708}, {1260, 3935}, {3245, 3586}, {5435, 5658}


X(5730) =  INTERSECTION OF LINES X(1)X(6) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 - 2*a^2*b - a*b^2 + 2*b^3 - 2*a^2*c + 2*a*b*c - a*c^2 + 2*c^3)

X(5730) lies on these lines:
{1, 6}, {2, 4930}, {3, 3417}, {5, 8}, {10, 2099}, {35, 3899}, {40, 5440}, {55, 3878}, {56, 758}, {57, 4018}, {63, 1385}, {65, 474}, {78, 517}, {145, 1058}, {214, 5204}, {329, 944}, {355, 908}, {381, 5086}, {382, 5057}, {442, 3485}, {519, 1837}, {527, 4311}, {936, 3340}, {946, 3419}, {952, 3436}, {957, 1257}, {999, 3868}, {1319, 3962}, {1320, 3621}, {1388, 4067}, {1457, 3682}, {1759, 3207}, {2093, 5438}, {2271, 3727}, {2800, 2932}, {2975, 3927}, {3057, 3811}, {3219, 3897}, {3244, 4679}, {3295, 3877}, {3303, 3884}, {3304, 3874}, {3445, 4694}, {3576, 3916}, {3579, 4855}, {3612, 4640}, {3624, 5425}, {3626, 3711}, {3632, 5087}, {3681, 4861}, {3754, 4413}, {3820, 5554}, {3872, 3984}, {3885, 3935}, {3901, 5563}, {4084, 5221}, {4255, 4424}


X(5731) =  INTERSECTION OF LINES X(1)X(7) AND X(3)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -5*a^4 + 2*a^3*b + 4*a^2*b^2 - 2*a*b^3 + b^4 + 2*a^3*c - 4*a^2*b*c + 2*a*b^2*c + 4*a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5731) = {X(1),X(20)}-harmonic conjugate of X(962). Let A' be the antipode of the A-extouch point in the A-excircle, and define B' and C' cyclically, and let A'' be the antipode of the A-intouch point in the incircle, and define B'' and C'' cyclically. Then X(5731) is the centroid of {A',B',C',A'',B'',C''}. (Randy Hutson, July 7, 2014)

The triangle A'B'C' is here named the Hutson-extouch triangle, and A''B''C'', the Hutson-intouch triangle - not to be confused with the outer and inner Hutson triangles defined at X(363). Hutson established (July 10, 2014) that A'B'C' and A''B''C'' are orthologic with orthology center X(3555); also that A''B''C'' and A'B'C' are orthologic with orthology center X(5920). X(5731) is the midpoint of the centroids of A'B'C' and A''B''C''; see X(5918) and X(5919).

Peter Moses (July 15, 2014) gives barycentrics for Hutson-extouch triangle,
-4a2 : (a + b + c)(a + b - c) : (a + b + c)(a - b + c)
(a + b + c)(b - c + a) : -4b2 : (a + b + c)(b + c - a)
(a + b + c)(c + a - b) : (a + b + c)(c - a + b) : -4c2

and for the Hutson-intouch triangle,
4a2 : (-a + b + c)(a - b + c) : (-a + b + c)(a + b - c)
(-b + c + a)(b + c - a) : 4b2 : (-b + c + a)(b - c + a)
(-c + a + b)(c - a + b) : (-c + a + b)(c + a - b) : 4c2

X(5731) lies on these lines:
{1, 7}, {2, 515}, {3, 8}, {4, 1385}, {5, 5550}, {10, 3523}, {21, 3427}, {30, 5603}, {36, 5435}, {40, 145}, {55, 3476}, {56, 411}, {84, 5250}, {153, 214}, {165, 519}, {329, 4511}, {355, 631}, {376, 517}, {377, 3897}, {388, 2646}, {392, 971}, {452, 1490}, {497, 1319}, {548, 1483}, {550, 1482}, {551, 1699}, {840, 2737}, {946, 3146}, {950, 1420}, {963, 1043}, {993, 5273}, {999, 3488}, {1012, 1621}, {1071, 3869}, {1125, 3091}, {1210, 5265}, {1478, 5226}, {1602, 1610}, {1788, 5204}, {2099, 3474}, {3085, 3612}, {3149, 5253}, {3189, 5584}, {3421, 5440}, {3475, 5434}, {3528, 3579}, {3545, 3653}, {3586, 5274}, {3624, 5056}, {3635, 5493}, {3817, 3839}, {4188, 5554}, {4189, 5450}, {4420, 5534}, {5218, 5252}

X(5731) = anticomplement of X(5587)


X(5732) =  INTERSECTION OF LINES X(1)X(7) AND X(3)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + 4*a*b^3*c + b^4*c + 2*a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 + 4*a*b*c^3 - 2*b^2*c^3 - 3*a*c^4 + b*c^4 + c^5)

X(5732) = (X(6) of hexyl triangle) = (X(69) of 2nd circumperp triangle) = (X(1352) of excentral triangle) = (isogonal conjugate of X(3)-vertex conjugate of X(57) = isotomic conjugate with respect to X(1)-circumcevian triangle) of X(1) (Randy Hutson, July 7, 2014)

X(5732) lies on these lines:
{1, 7}, {2, 1750}, {3, 9}, {4, 142}, {21, 3062}, {40, 518}, {63, 100}, {78, 144}, {223, 1040}, {376, 527}, {411, 1445}, {464, 2947}, {515, 2550}, {517, 3243}, {912, 3587}, {950, 1467}, {952, 5528}, {954, 3601}, {1001, 1012}, {1699, 5249}, {1709, 4512}, {1818, 2324}, {1998, 3218}, {2808, 3781}, {2900, 3928}, {3059, 5584}, {3333, 5572}, {3452, 5658}, {3579, 5534}, {3826, 5587}, {3868, 3895}

X(5732) = midpoint of X(I) and X(J) for these (I,J): (1,2951), (7,20)
X(5732) = reflection of X(I) in X(J) for these (I,J): (4,142), (9,3)


X(5733) =  INTERSECTION OF LINES X(1)X(7) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^6 + 5*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 - 2*a*b^5 + b^6 + 2*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + 5*a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + 4*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 3*a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6

X(5733) lies on these lines: {1, 7}, {4, 4658}, {5, 6}, {225, 1419}, {631, 4648}, {3193, 4197}


X(5734) =  INTERSECTION OF LINES X(1)X(7) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 - 6*a^3*c + 12*a^2*b*c - 6*a*b^2*c - 4*a^2*c^2 - 6*a*b*c^2 - 2*b^2*c^2 + 6*a*c^3 + c^4

X(5734) lies on these lines:
{1, 7}, {4, 1392}, {5, 8}, {40, 3622}, {145, 946}, {165, 3636}, {329, 4861}, {355, 3855}, {382, 944}, {388, 5048}, {411, 3303}, {515, 3623}, {517, 631}, {519, 3091}, {551, 3523}, {938, 2099}, {952, 3843}, {1385, 3528}, {1388, 3474}, {1483, 3853}, {1699, 3244}, {2093, 5265}, {2098, 3485}, {3262, 4673}, {3525, 3654}, {3526, 5550}, {3529, 3655}, {3621, 5587}, {3632, 3817}, {3679, 5056}, {3878, 5273}, {4197, 5330}


X(5735) =  INTERSECTION OF LINES X(1)X(7) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^6 - 3*a^5*b - 4*a^4*b^2 + 2*a^3*b^3 + 3*a^2*b^4 + a*b^5 - 2*b^6 - 3*a^5*c + 2*a^3*b^2*c - 4*a^2*b^3*c + a*b^4*c + 4*b^5*c - 4*a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + 2*b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 - 2*a*b^2*c^3 - 8*b^3*c^3 + 3*a^2*c^4 + a*b*c^4 + 2*b^2*c^4 + a*c^5 + 4*b*c^5 - 2*c^6

X(5735) lies on these lines:
{1, 7}, {4, 527}, {5, 9}, {63, 1699}, {84, 3254}, {142, 631}, {144, 3832}, {165, 5249}, {382, 971}, {1004, 5537}, {1750, 1998}, {2801, 3868}, {3436, 5223}, {3817, 5273}, {5220, 5587}


X(5736) =  INTERSECTION OF LINES X(2)X(6) AND X(3)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 2*a^3*b^2 + a*b^4 - 2*a^3*b*c - 3*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 3*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - b^2*c^3 + a*c^4 + b*c^4

X(5736) lies on these lines:
{1, 1441}, {2, 6}, {3, 7}, {77, 1446}, {226, 2268}, {255, 307}, {273, 1442}, {284, 379}, {1253, 1754}, {1958, 5249}, {2476, 2893}, {3485, 4329}


X(5737) =  INTERSECTION OF LINES X(2)X(6) AND X(3)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3 - a^2*b - 2*a*b^2 - a^2*c - 2*a*b*c - 2*b^2*c - 2*a*c^2 - 2*b*c^2

X(5737) lies on these lines:
{2, 6}, {3, 10}, {9, 1764}, {42, 4042}, {45, 312}, {57, 3739}, {58, 2049}, {63, 4363}, {226, 4643}, {306, 4445}, {345, 594}, {405, 4267}, {573, 2050}, {968, 3706}, {980, 1107}, {1001, 3741}, {1010, 4252}, {1215, 5220}, {1698, 5247}, {2345, 5273}, {3052, 5263}, {3242, 3757}, {3666, 4361}, {3771, 3775}, {3772, 4357}, {4205, 5292}, {4426, 5337}


X(5738) =  INTERSECTION OF LINES X(2)X(6) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(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 - 4*a^3*b*c - 4*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 4*a^2*b*c^2 - 2*a*b^2*c^2 + 2*a^2*c^3 + a*c^4 + b*c^4 - c^5

X(5738) lies on these lines:
{1, 307}, {2, 6}, {4, 7}, {65, 4329}, {77, 581}, {78, 3879}, {322, 5554}, {377, 2893}, {411, 1014}, {573, 1445}, {1060, 1442}, {1210, 3664}


X(5739) =  INTERSECTION OF LINES X(2)X(6) AND X(4)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(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

Let A'B'C' be the extangents triangle, and let AB be the touch point of the A-excircle and the line A'B', and define BC and CA cyclically. Let AC be the touch point of the A-excircle and the line A'C', and define BA and CB cyclically. Let A'' = BCBA∩CACB, and define B'' and C'' cyclically. The A''B''C'' is homothetic to ABC and to the outer and inner Grebe triangles at X(6), to the medial triangle at X(1211), and to the anticomplementary triangle at X(5739). (Randy Hutson, July 7, 2014)

X(5739) lies on these lines:
{1, 4101}, {2, 6}, {4, 8}, {7, 4359}, {9, 306}, {57, 4001}, {63, 573}, {75, 4886}, {78, 581}, {209, 2550}, {210, 3416}, {223, 307}, {226, 3686}, {312, 319}, {345, 3219}, {346, 3969}, {377, 1330}, {387, 5051}, {388, 959}, {516, 4061}, {518, 3966}, {519, 4656}, {612, 4104}, {740, 4703}, {968, 4028}, {1376, 4023}, {1479, 4044}, {1714, 3454}, {1743, 5294}, {1836, 3696}, {2478, 3948}, {2886, 4042}, {3305, 3912}, {3666, 4277}, {3703, 5220}, {3707, 4035}, {3715, 3932}, {3782, 4361}, {3870, 3883}, {3879, 5287}, {3929, 3977}, {3952, 3974}, {3965, 3998}, {4034, 4054}, {4270, 4357}, {4384, 5249}, {4423, 4966}, {4660, 4685}, {4666, 4684}

X(5739) = anticomplement of X(940)


X(5740) =  INTERSECTION OF LINES X(2)X(6) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4*b - 2*a^2*b^3 + b^5 + a^4*c + 2*a^3*b*c + a^2*b^2*c + a^2*b*c^2 - b^3*c^2 - 2*a^2*c^3 - b^2*c^3 + c^5

X(5740) lies on these lines:
{2, 6}, {5, 7}, {269, 5400}, {273, 2973}, {307, 1210}, {404, 2893}, {579, 857}, {1441, 1737}, {1442, 5396}, {1788, 4329}


X(5741) =  INTERSECTION OF LINES X(2)X(6) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^2*b + b^3 - a^2*c + 2*a*b*c + c^3

X(5741) lies on these lines:
{2, 6}, {5, 8}, {42, 3847}, {43, 4972}, {78, 5016}, {100, 4192}, {149, 3996}, {200, 5014}, {210, 3006}, {226, 4359}, {306, 3452}, {312, 3969}, {321, 908}, {329, 4488}, {386, 5051}, {404, 1330}, {748, 3771}, {899, 2887}, {970, 3869}, {1043, 5046}, {1210, 4101}, {2886, 4023}, {3216, 3454}, {3681, 3705}, {3703, 3952}, {3706, 5087}, {3817, 4061}, {3909, 3917}, {3911, 4001}, {3935, 4514}, {3944, 4442}, {4035, 5316}, {4054, 4980}, {4104, 4981}, {4414, 4703}, {4420, 5015}, {4511, 5396}, {5219, 5271}


X(5742) =  INTERSECTION OF LINES X(2)X(6) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4*b - a^3*b^2 - 3*a^2*b^3 + a*b^4 + b^5 + 2*a^4*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 6*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 + b*c^4 + c^5

X(5742) lies on these lines:
{2, 6}, {5, 9}, {48, 4999}, {71, 2886}, {307, 3739}, {442, 579}, {936, 5396}, {970, 2262}, {1210, 5257}, {1698, 1723}, {1839, 4640}, {1865, 5125}, {1901, 2476}


X(5743) =  INTERSECTION OF LINES X(2)X(6) AND X(5)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 + bc) + b3 + c3 + b2c + bc2

X(5743) lies on these lines:
{2, 6}, {5, 10}, {37, 3687}, {42, 4023}, {43, 4026}, {45, 345}, {57, 4643}, {75, 4415}, {226, 3739}, {312, 594}, {321, 3264}, {329, 4363}, {386, 4205}, {518, 4104}, {536, 4656}, {612, 3966}, {756, 3703}, {984, 4884}, {997, 5396}, {1376, 4192}, {1999, 4886}, {2161, 2339}, {2887, 3826}, {2999, 4657}, {3035, 5150}, {3416, 5268}, {3666, 4364}, {3741, 3816}, {3752, 4357}, {3772, 4384}, {3775, 3840}, {3782, 4359}, {3838, 3846}, {3980, 4703}, {4199, 5132}, {4239, 5347}

X(5743) = complement of X(940)


X(5744) =  INTERSECTION OF LINES X(2)X(7) AND X(3)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^3 + a^2*b + 3*a*b^2 - b^3 + a^2*c - 2*a*b*c + b^2*c + 3*a*c^2 + b*c^2 - c^3

X(5744) = {X(2),X(63)}-harmonic conjugate of X(329)

X(5744) lies on these lines:
{2, 7}, {3, 8}, {4, 3916}, {10, 4293}, {20, 4652}, {21, 938}, {72, 631}, {78, 3523}, {88, 277}, {92, 4359}, {140, 3927}, {145, 3601}, {165, 4847}, {189, 333}, {191, 499}, {346, 3977}, {348, 658}, {376, 3419}, {391, 610}, {392, 942}, {443, 3436}, {452, 1210}, {497, 4640}, {516, 5231}, {518, 5218}, {549, 3940}, {940, 2256}, {958, 1466}, {1000, 2320}, {1108, 3666}, {1155, 2550}, {1467, 5265}, {1473, 4220}, {2000, 3100}, {2095, 5603}, {2886, 3474}, {3011, 4310}, {3035, 5220}, {3161, 4358}, {3189, 5217}, {3427, 3428}, {3485, 4999}, {3524, 5440}, {3579, 5082}, {3752, 5069}, {3870, 5281}, {4054, 4454}, {4189, 4313}, {4292, 5177}, {4305, 5267}, {4384, 5088}, {4850, 5222}, {5122, 5176}

X(5744) = anticomplement of X(5219)


X(5745) =  INTERSECTION OF LINES X(2)X(7) AND X(3)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^2 + a*b - b^2 + a*c + 2*b*c - c^2)

X(5745) = {X(2),X(63)}-harmonic conjugate of X(226)

X(5745) lies on these lines:
{2, 7}, {3, 10}, {8, 3158}, {11, 3683}, {21, 950}, {38, 3011}, {39, 1212}, {55, 4847}, {69, 4035}, {71, 1764}, {81, 2323}, {114, 124}, {140, 912}, {165, 2550}, {200, 5218}, {210, 5432}, {219, 940}, {261, 284}, {281, 5307}, {306, 1150}, {312, 2325}, {321, 3977}, {345, 2321}, {377, 4652}, {405, 1210}, {442, 3916}, {443, 1478}, {497, 4512}, {516, 2886}, {535, 3828}, {551, 4930}, {610, 966}, {631, 936}, {758, 942}, {938, 5436}, {1146, 2482}, {1155, 3925}, {1329, 3634}, {1737, 5251}, {1817, 5235}, {1861, 4219}, {1936, 2328}, {2329, 3912}, {2801, 3035}, {2999, 5105}, {3036, 4745}, {3220, 4220}, {3419, 4304}, {3523, 5438}, {3663, 3772}, {3666, 3946}, {3689, 4995}, {3705, 3883}, {3706, 3712}, {3741, 4154}, {3914, 4414}, {3936, 4001}, {4042, 4061}, {4138, 4655}, {4224, 5285}, {4359, 4858}, {4416, 4417}, {5247, 5530}, {5537, 5659}

X(5745) = complement of X(226)


X(5746) =  INTERSECTION OF LINES X(2)X(7) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 3*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 + b^5 - 3*a^4*c - 4*a^3*b*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5746) lies on these lines:
{2, 7}, {4, 6}, {19, 4295}, {20, 284}, {37, 3487}, {48, 4293}, {65, 281}, {71, 3085}, {72, 2345}, {193, 2893}, {219, 388}, {377, 2287}, {380, 516}, {386, 990}, {391, 5177}, {405, 5120}, {442, 966}, {443, 965}, {452, 5053}, {573, 1715}, {610, 4292}, {946, 2257}, {950, 1449}, {1056, 2256}, {1100, 3488}, {1108, 5603}, {1713, 4253}, {1714, 1743}, {1836, 2264}, {2260, 3086}, {2303, 4340}, {4264, 5304}


X(5747) =  INTERSECTION OF LINES X(2)X(7) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - a^4*b - 2*a^3*b^2 + a*b^4 + b^5 - a^4*c - 2*a^3*b*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5747) lies on these lines:
{1, 1826}, {2, 7}, {3, 1901}, {4, 284}, {5, 6}, {12, 219}, {48, 1478}, {71, 498}, {281, 3485}, {377, 2327}, {380, 1699}, {386, 3553}, {387, 5587}, {442, 965}, {495, 2256}, {499, 2260}, {594, 3940}, {2287, 2476}, {2303, 5142}, {2549, 5110}, {5053, 5084}, {5105, 5286}


X(5748) =  INTERSECTION OF LINES X(2)X(7) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^3 + 3*a^2*b + a*b^2 - 3*b^3 + 3*a^2*c - 6*a*b*c + 3*b^2*c + a*c^2 + 3*b*c^2 - 3*c^3

X(5748) lies on these lines:
{2, 7}, {4, 5440}, {5, 8}, {72, 3090}, {78, 3091}, {92, 4358}, {189, 1997}, {200, 3817}, {312, 3262}, {497, 5087}, {936, 5177}, {938, 4193}, {962, 1519}, {1056, 3436}, {1329, 3485}, {2975, 5550}, {3006, 5423}, {3035, 3474}, {3146, 4855}, {3190, 5400}, {3241, 5176}, {3419, 3545}, {3475, 3816}, {3487, 4187}, {3525, 3916}, {3628, 3927}, {3753, 3869}, {3870, 5274}, {4313, 5046}, {4323, 5554}


X(5749) =  INTERSECTION OF LINES X(2)X(7) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 + (b + c)2

X(5749) lies on these lines:
{1, 346}, {2, 7}, {6, 8}, {10, 391}, {19, 4200}, {37, 2275}, {44, 966}, {45, 5550}, {69, 3758}, {75, 3618}, {86, 344}, {100, 4254}, {141, 4644}, {145, 1449}, {193, 3661}, {198, 404}, {218, 1010}, {281, 608}, {318, 1249}, {319, 1992}, {320, 3619}, {335, 4473}, {377, 2182}, {597, 4361}, {604, 2329}, {612, 5423}, {644, 2256}, {941, 2276}, {962, 1766}, {1100, 3241}, {1125, 3731}, {1698, 3973}, {1901, 5051}, {2092, 2229}, {2171, 3061}, {2264, 2550}, {2268, 4195}, {2269, 3501}, {2322, 3194}, {2325, 3247}, {2975, 5120}, {3240, 4270}, {3553, 4511}, {3554, 4861}, {3589, 4000}, {3617, 3686}, {3621, 4007}, {3623, 4873}, {3624, 3986}, {3629, 4445}, {3632, 4058}, {3635, 4072}, {3636, 4098}, {3663, 4454}, {3664, 4747}, {3672, 3729}, {3739, 4470}, {3745, 3974}, {3875, 4461}, {3912, 3945}, {3946, 4452}, {4034, 4678}, {4371, 4665}, {4393, 4460}, {4416, 5232}, {4419, 4488}, {4648, 4670}, {4698, 4798}, {5042, 5291}


X(5750) =  INTERSECTION OF LINES X(2)X(7) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^2 + a*b + b^2 + a*c + 2*b*c + c^2

X(5750) = {X(6),X(10)}-harmonic conjugate of X(3686)

X(5750) lies on these lines:
{1, 2321}, {2, 7}, {6, 10}, {8, 1449}, {19, 475}, {34, 281}, {37, 39}, {43, 4270}, {44, 1213}, {45, 3986}, {69, 4667}, {75, 3946}, {86, 3912}, {141, 3664}, {145, 4007}, {171, 4264}, {198, 474}, {213, 992}, {218, 965}, {239, 4967}, {284, 1010}, {346, 3247}, {380, 2550}, {442, 2182}, {443, 610}, {478, 2122}, {515, 572}, {516, 4026}, {519, 594}, {536, 4021}, {551, 3950}, {742, 3008}, {946, 1766}, {950, 964}, {958, 5120}, {966, 1698}, {997, 3553}, {1172, 1861}, {1203, 1224}, {1220, 5053}, {1376, 4254}, {1450, 2324}, {1574, 4263}, {1575, 2092}, {1826, 2267}, {1901, 4205}, {2262, 3753}, {2264, 3925}, {2295, 2300}, {2298, 4071}, {2303, 5280}, {2663, 3783}, {3161, 5550}, {3244, 4058}, {3617, 4034}, {3618, 4384}, {3622, 4873}, {3624, 3731}, {3626, 4545}, {3629, 4690}, {3636, 3723}, {3661, 3879}, {3663, 4363}, {3672, 4659}, {3713, 4847}, {3758, 4416}, {3763, 4675}, {4000, 4470}, {4360, 4431}, {4395, 4739}, {4422, 4698}, {4426, 5019}, {4478, 4725}, {4665, 4852}, {4691, 4969}, {5124, 5267}


X(5751) =  INTERSECTION OF LINES X(3)X(6) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 - b^7 + a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - 2*a^2*b^4*c + a*b^5*c + b^6*c + a^5*c^2 + a^4*b*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + b^5*c^2 - a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 - 2*a*b^3*c^3 - b^4*c^3 - 2*a^3*c^4 - 2*a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 + a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7)

X(5751) lies on these lines: {1, 916}, {3, 6}, {4, 7}, {55, 1779}, {81, 4219}, {1817, 3060}


X(5752) =  INTERSECTION OF LINES X(3)X(6) AND X(4)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 + a^2*c^3 - a*b*c^3 - a*c^4 - b*c^4 - c^5)

X(5752) = circumcenter of the triangle A''B''C'' be as defined at X(5739); also X(5752) = {X(371)X(372)}-harmonic conjugate of X(1333). Let Γ be the circle of the points X(371), X(372), PU(1), PU(39); then X(5752) is the inverse-in-Γ of X(1333). (Randy Hutson, July 7, 2014)

X(5752) lies on these lines:
{3, 6}, {4, 8}, {5, 1211}, {21, 3060}, {24, 2203}, {40, 209}, {51, 405}, {184, 2915}, {404, 2979}, {474, 3917}, {631, 5482}, {674, 3811}, {916, 1490}, {966, 3781}, {978, 3792}, {1006, 3567}, {1437, 1993}, {2183, 3682}, {3056, 5266}, {3149, 5562}, {3240, 3579}, {3560, 5446}, {5047, 5640}


X(5753) =  INTERSECTION OF LINES X(3)X(6) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^6*b + a^5*b^2 - 4*a^4*b^3 - 2*a^3*b^4 + 5*a^2*b^5 + a*b^6 - 2*b^7 + a^6*c + 2*a^5*b*c - a^4*b^2*c - 4*a^3*b^3*c - a^2*b^4*c + 2*a*b^5*c + b^6*c + a^5*c^2 - a^4*b*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 - 4*a^4*c^3 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 - 2*b^4*c^3 - 2*a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 - 2*b^3*c^4 + 5*a^2*c^5 + 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - 2*c^7)

X(5753) lies on these lines: {3, 6}, {5, 7}, {57, 5400}, {916, 2260}, {942, 1736}


X(5754) =  INTERSECTION OF LINES X(3)X(6) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^4*b - a^3*b^2 - 3*a^2*b^3 + a*b^4 + 2*b^5 + a^4*c - 2*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c + b^4*c - a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 3*a^2*c^3 + 2*a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4 + 2*c^5)

X(5754) lies on these lines: {3, 6}, {5, 8}, {355, 2051}, {517, 3293}, {3240, 4192}


X(5755) =  INTERSECTION OF LINES X(3)X(6) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(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 - 2*a^4*b*c - a^3*b^2*c + a^2*b^3*c + b^5*c - a^4*c^2 - a^3*b*c^2 + a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 - 2*b^3*c^3 + 2*a^2*c^4 + b^2*c^4 + a*c^5 + b*c^5 - c^6)

X(5755) lies on these lines:
{3, 6}, {5, 9}, {30, 1765}, {40, 1723}, {57, 4888}, {71, 517}, {198, 3211}, {672, 4192}, {942, 1400}, {1385, 2260}, {1766, 2161}, {1781, 5535}, {2197, 5399}, {2361, 5285}, {3973, 5400}


X(5756) =  INTERSECTION OF LINES X(3)X(6) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^3*b + 3*a^2*b^2 - a*b^3 - 3*b^4 + a^3*c + 5*a^2*b*c + 3*a*b^2*c - b^3*c + 3*a^2*c^2 + 3*a*b*c^2 - a*c^3 - b*c^3 - 3*c^4)

X(5756) lies on these lines: {3, 6}, {7, 10}


X(5757) =  INTERSECTION OF LINES X(3)X(7) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^9 - 2*a^8*b - 2*a^7*b^2 + 4*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - 2*a^3*b^6 + a*b^8 - 2*a^8*c - 2*a^7*b*c + 3*a^6*b^2*c + 4*a^5*b^3*c + a^4*b^4*c - 2*a^3*b^5*c - 3*a^2*b^6*c + b^8*c - 2*a^7*c^2 + 3*a^6*b*c^2 + 4*a^5*b^2*c^2 + a^4*b^3*c^2 + 2*a^3*b^4*c^2 - 3*a^2*b^5*c^2 - 4*a*b^6*c^2 - b^7*c^2 + 4*a^6*c^3 + 4*a^5*b*c^3 + a^4*b^2*c^3 + 4*a^3*b^3*c^3 + 6*a^2*b^4*c^3 - 3*b^6*c^3 + 2*a^5*c^4 + a^4*b*c^4 + 2*a^3*b^2*c^4 + 6*a^2*b^3*c^4 + 6*a*b^4*c^4 + 3*b^5*c^4 - 2*a^4*c^5 - 2*a^3*b*c^5 - 3*a^2*b^2*c^5 + 3*b^4*c^5 - 2*a^3*c^6 - 3*a^2*b*c^6 - 4*a*b^2*c^6 - 3*b^3*c^6 - b^2*c^7 + a*c^8 + b*c^8

X(5757) lies on these lines: {3, 7}, {4, 6}, {212, 226}


X(5758) =  INTERSECTION OF LINES X(3)X(7) AND X(4)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - a^6*b + 3*a^5*b^2 + 3*a^4*b^3 - 3*a^3*b^4 - 3*a^2*b^5 + a*b^6 + b^7 - a^6*c + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - b^6*c + 3*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 3*b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5758) lies on these lines:
{3, 7}, {4, 8}, {9, 946}, {40, 226}, {84, 527}, {405, 5603}, {442, 5657}, {499, 5536}, {516, 1490}, {1006, 3616}, {1260, 3149}, {1482, 3488}, {1708, 3086}, {3428, 3485}, {3649, 5584}, {4299, 5538}


X(5759) =  INTERSECTION OF LINES X(3)X(7) AND X(4)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^6 - 4*a^5*b - 3*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 - b^6 - 4*a^5*c - 2*a^4*b*c + 4*a^3*b^2*c + 2*b^5*c - 3*a^4*c^2 + 4*a^3*b*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 4*a^3*c^3 - 4*b^3*c^3 + a^2*c^4 + b^2*c^4 + 2*b*c^5 - c^6

X(5759) = isogonal conjugate of the X(3)-vertex conjugate of X(55)

X(5759) lies on these lines:
{3, 7}, {4, 9}, {20, 72}, {37, 3332}, {63, 3358}, {100, 329}, {142, 631}, {165, 226}, {212, 278}, {376, 527}, {390, 517}, {405, 962}, {497, 1708}, {515, 5223}, {518, 944}, {990, 4419}, {991, 4644}, {1001, 1006}, {1253, 4331}, {1490, 2951}, {2318, 2947}, {2724, 2742}, {3576, 5542}, {4295, 5584}, {4301, 5436}


X(5760) =  INTERSECTION OF LINES X(3)X(7) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^9 + a^8*b + 3*a^7*b^2 - 2*a^6*b^3 - 4*a^5*b^4 + a^4*b^5 + 3*a^3*b^6 - a*b^8 + a^8*c + 2*a^7*b*c - 4*a^5*b^3*c - 4*a^4*b^4*c + 2*a^3*b^5*c + 4*a^2*b^6*c - b^8*c + 3*a^7*c^2 - 4*a^5*b^2*c^2 - 3*a^4*b^3*c^2 - 3*a^3*b^4*c^2 + 2*a^2*b^5*c^2 + 4*a*b^6*c^2 + b^7*c^2 - 2*a^6*c^3 - 4*a^5*b*c^3 - 3*a^4*b^2*c^3 - 4*a^3*b^3*c^3 - 6*a^2*b^4*c^3 + 3*b^6*c^3 - 4*a^5*c^4 - 4*a^4*b*c^4 - 3*a^3*b^2*c^4 - 6*a^2*b^3*c^4 - 6*a*b^4*c^4 - 3*b^5*c^4 + a^4*c^5 + 2*a^3*b*c^5 + 2*a^2*b^2*c^5 - 3*b^4*c^5 + 3*a^3*c^6 + 4*a^2*b*c^6 + 4*a*b^2*c^6 + 3*b^3*c^6 + b^2*c^7 - a*c^8 - b*c^8

X(5760) lies on these lines: {3, 7}, {5, 6}


X(5761) =  INTERSECTION OF LINES X(3)X(7) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 - 3*a^6*b - a^5*b^2 + 7*a^4*b^3 - a^3*b^4 - 5*a^2*b^5 + a*b^6 + b^7 - 3*a^6*c + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + 3*a^2*b^4*c - 2*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 7*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 3*b^4*c^3 - a^3*c^4 + 3*a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 5*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5761) lies on these lines:
{3, 7}, {5, 8}, {140, 2095}, {329, 3560}, {382, 5658}, {517, 3085}, {946, 3811}, {1385, 3475}, {3359, 3671}


X(5762) =  INTERSECTION OF LINES X(3)X(7) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 2*a^5*b + 3*a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + b^6 + 2*a^5*c - 2*a^3*b^2*c + 2*a^2*b^3*c - 2*b^5*c + 3*a^4*c^2 - 2*a^3*b*c^2 - b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 2*b*c^5 + c^6

X(5762) lies on these lines:
{3, 7}, {4, 144}, {5, 9}, {30, 511}, {40, 495}, {140, 142}, {144, 2894}, {165, 4654}, {355, 5223}, {390, 1482}, {495, 4312}, {1385, 5542}, {1483, 3243}, {1484, 3254}, {1699, 3929}, {1754, 3782}, {3332, 4419}, {3817, 5325}, {4654, 4995}


X(5763) =  INTERSECTION OF LINES X(3)X(7) AND X(5)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6*b + a^5*b^2 + 5*a^4*b^3 - 2*a^3*b^4 - 4*a^2*b^5 + a*b^6 + b^7 - 2*a^6*c + 8*a^5*b*c + a^4*b^2*c - 6*a^3*b^3*c + 2*a^2*b^4*c - 2*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 5*a^4*c^3 - 6*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*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 - 4*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5763) lies on these lines:
{3, 7}, {4, 3940}, {5, 10}, {30, 1490}, {140, 5437}, {165, 3649}, {1058, 1482}, {5433, 5536}


X(5764) =  INTERSECTION OF LINES X(3)X(7) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + a^5*b + 2*a^4*b^2 - a*b^5 + a^5*c + 2*a^4*b*c + 3*a^3*b^2*c + 3*a^2*b^3*c - b^5*c + 2*a^4*c^2 + 3*a^3*b*c^2 + 6*a^2*b^2*c^2 + a*b^3*c^2 + 3*a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 - a*c^5 - b*c^5

X(5764) lies on these lines: {1, 4552}, {3, 7}, {6, 8}, {3006, 5294}, {3085, 4307}


X(5765) =  INTERSECTION OF LINES X(3)X(7) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^6 - 4*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + 2*a*b^5 - 4*a^4*b*c - 10*a^3*b^2*c - 6*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - 4*a^4*c^2 - 10*a^3*b*c^2 - 14*a^2*b^2*c^2 - 4*a*b^3*c^2 - 2*a^3*c^3 - 6*a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + 2*a*b*c^4 + 2*a*c^5 + 2*b*c^5

X(5765) lies on these lines: {3, 7}, {6, 10}


X(5766) =  INTERSECTION OF LINES X(3)X(7) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(5*a^5 - 3*a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + a*b^4 + b^5 - 3*a^4*c - 8*a^3*b*c - 2*a^2*b^2*c - 3*b^4*c - 6*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 + a*c^4 - 3*b*c^4 + c^5)

X(5766) lies on these lines:
{3, 7}, {8, 9}, {55, 329}, {72, 4313}, {226, 5281}, {516, 3085}, {527, 3601}, {528, 5175}, {2801, 4305}, {3486, 5220}, {3811, 4326}


X(5767) =  INTERSECTION OF LINES X(3)X(8) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - 2*a^6*b + a^5*b^2 + 2*a^4*b^3 - a^3*b^4 + a*b^6 - 2*a^6*c + a^4*b^2*c + b^6*c + a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + b^5*c^2 + 2*a^4*c^3 - 2*b^4*c^3 - a^3*c^4 - a*b^2*c^4 - 2*b^3*c^4 + b^2*c^5 + a*c^6 + b*c^6

X(5767) lies on these lines:
{3, 8}, {4, 6}, {10, 48}, {184, 5136}, {515, 1754}, {517, 3187}, {860, 1899}, {912, 4463}, {940, 1056}


X(5768) =  INTERSECTION OF LINES X(3)X(8) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 3*a^6*b - a^5*b^2 - 5*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 - 3*a*b^6 + b^7 + 3*a^6*c + 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - 3*a^2*b^4*c + 2*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 5*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 5*a^3*c^4 - 3*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5768) lies on these lines:
{1, 3427}, {3, 8}, {4, 7}, {20, 3218}, {30, 2094}, {57, 515}, {84, 950}, {142, 5587}, {329, 912}, {355, 443}, {390, 3358}, {601, 4339}, {1006, 5273}, {1012, 3488}, {1072, 4310}, {1158, 4294}, {1181, 3562}, {1210, 1467}, {1519, 5274}, {1532, 5658}, {1768, 4302}, {4305, 5450}


X(5769) =  INTERSECTION OF LINES X(3)X(8) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 2*a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - a*b^6 + a^6*c - a^4*b^2*c + a^2*b^4*c - b^6*c - 2*a^5*c^2 - a^4*b*c^2 + a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 - a^4*c^3 + a^2*b^2*c^3 + 2*b^4*c^3 + 2*a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + 2*b^3*c^4 - b^2*c^5 - a*c^6 - b*c^6

X(5769) lies on these lines: {3, 8}, {5, 6}, {495, 940}


X(5770) =  INTERSECTION OF LINES X(3)X(8) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 5*a^5*b^2 - a^4*b^3 + 7*a^3*b^4 - a^2*b^5 - 3*a*b^6 + b^7 + a^6*c + 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - a^2*b^4*c + 2*a*b^5*c - b^6*c - 5*a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*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 - 4*a*b^3*c^3 + 3*b^4*c^3 + 7*a^3*c^4 - a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 - a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5770) lies on these lines:
{2, 912}, {3, 8}, {4, 3218}, {5, 7}, {57, 1478}, {355, 1788}, {381, 2094}, {516, 1158}, {938, 3560}, {942, 3086}, {3359, 4847}


X(5771) =  INTERSECTION OF LINES X(3)X(8) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 7*a^5*b^2 + a^4*b^3 + 8*a^3*b^4 - 2*a^2*b^5 - 3*a*b^6 + b^7 - 4*a^5*b*c + a^4*b^2*c + 2*a^3*b^3*c + 2*a*b^5*c - b^6*c - 7*a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 + a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 8*a^3*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 - 2*a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5771) lies on these lines: {2, 2095}, {3, 8}, {5, 9}, {57, 495}, {140, 942}, {484, 5659}, {1532, 3219}, {3628, 5316}, {3925, 5535}


X(5772) =  INTERSECTION OF LINES X(6)X(8) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(a^2 + 4*a*b - b^2 + 4*a*c + 2*b*c - c^2)

X(5772) lies on these lines: {2, 3677}, {6, 8}, {7, 10}, {894, 3617}, {1215, 5226}, {1698, 4310}, {3679, 4307}, {3755, 4461}, {3932, 5308}, {4645, 4715}


X(5773) =  INTERSECTION OF LINES X(3)X(8) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + a^4*b + a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c - 2*a*b^3*c + b^4*c + a^3*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4

X(5773) lies on these lines: {2, 101}, {3, 8}, {6, 7}, {57, 4566}, {239, 514}, {1055, 3911}, {1647, 5168}, {2398, 2809}


X(5774) =  INTERSECTION OF LINES X(3)X(8) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b - a^2*b^2 - 2*a*b^3 + 2*a^3*c - 2*a^2*b*c - 2*a*b^2*c - 2*b^3*c - a^2*c^2 - 2*a*b*c^2 - 4*b^2*c^2 - 2*a*c^3 - 2*b*c^3

X(5774) lies on these lines: {3, 8}, {6, 10}, {40, 5295}, {69, 495}, {171, 3679}, {381, 4388}, {517, 2050}, {996, 3626}, {1010, 3617}, {1737, 3966}, {3706, 5119}, {3715, 3992}, {3753, 5271}, {3927, 4385}


X(5775) =  INTERSECTION OF LINES X(3)X(8) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 6*a^3*b - 4*a^2*b^2 - 6*a*b^3 + 3*b^4 + 6*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 4*a^2*c^2 - 2*a*b*c^2 - 6*b^2*c^2 - 6*a*c^3 + 3*c^4

X(5775) lies on these lines:
{3, 8}, {7, 10}, {144, 5587}, {519, 5281}, {758, 5226}, {938, 1001}, {1788, 4413}, {2094, 3421}, {3218, 3617}, {3679, 4293}, {5251, 5273}


X(5776) =  INTERSECTION OF LINES X(3)X(9) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c - 2*a^4*b^2*c + 2*a^2*b^4*c + 2*b^6*c - a^5*c^2 - 2*a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 4*b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 - 4*b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 + a*c^6 + 2*b*c^6)

The B-excircle meets the sidelines of ABC in 3 points, and likewise for the C-excircle. The 6 points lie on a conic, denoted by A*. Let A' be the center of A*, and define B' and C' cyclically. Then X(5776) is the perspector of A'B'C' and the 2nd extouch triangle (defined at X(5927). (Randy Hutson, July 7, 2014)

X(5776) lies on these lines:
{3, 9}, {4, 6}, {20, 2287}, {40, 4047}, {72, 1766}, {154, 4183}, {219, 515}, {222, 226}, {281, 3197}, {284, 1012}, {405, 572}, {579, 3149}, {944, 2256}, {1713, 5120}, {1715, 2270}, {1743, 1750}


X(5777) =  INTERSECTION OF LINES X(3)X(9) AND X(4)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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 + 2*a^2*b^3*c - a*b^4*c - 2*b^5*c - a^4*c^2 + b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 + 2*a^2*c^4 - a*b*c^4 + b^2*c^4 + a*c^5 - 2*b*c^5 - c^6)

Let P be a point in the plane of a triangle ABC and let A'B'C' be the cevian triangle of P. Let HA be the orthocenter of triangle AB'C', and define HB and HC cyclically. Then the points A', B', C', HA, HB, HC lie on a conic. (Dominik Burek (ADGEOM #424, Aug 2, 2013)

The conic {A', B', C', HA, HB, HC} is here named the Burek-cevian conic of P. If P = X(8), the conic has center X(5777). More generally, if P is on the Lucas cubic, then the triangles A'B'C' and HAHBHC are homothetic, and HAHBHC is perspective to ABC at a point on the Darboux cubic. (ADGEOM #431, August 3, 2013, and related postings)

X(5777) = (X(5) of 2nd extouch triangle); see X(5776). Also, X(5777) lies on the Burek-Hutson central cubic, K645.

X(5777) lies on these lines:
{1, 1864}, {2, 1071}, {3, 9}, {4, 8}, {5, 226}, {12, 1858}, {20, 3876}, {37, 581}, {40, 210}, {44, 580}, {55, 1898}, {56, 1728}, {63, 3149}, {65, 5587}, {78, 1012}, {119, 125}, {201, 2635}, {342, 1148}, {389, 916}, {392, 452}, {405, 1385}, {411, 3219}, {499, 3660}, {515, 960}, {516, 3678}, {518, 946}, {756, 4300}, {943, 1156}, {950, 952}, {1125, 2801}, {1158, 1376}, {1159, 5665}, {1214, 1745}, {1260, 2057}, {2800, 3036}, {3057, 3586}, {3073, 5266}, {3074, 3465}, {3090, 5439}, {3091, 3868}, {3295, 5534}, {3487, 5045}, {3555, 5603}, {3651, 3652}, {3670, 5400}, {3697, 5657}, {3715, 5584}, {3746, 5531}, {3753, 5177}, {3817, 3874}

X(5777) = midpoint of X(4) and X(72)
X(5777) = reflection of X(942) in X(5)
X(5777) = complement of X(1071)


X(5778) =  INTERSECTION OF LINES X(3)X(9) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c + 2*a^5*b*c - 2*a^3*b^3*c + 2*b^6*c - a^5*c^2 + 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*b^4*c^3 - a^3*c^4 - a*b^2*c^4 - 4*b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 + a*c^6 + 2*b*c^6)

X(5778) lies on these lines:
{3, 9}, {4, 2287}, {5, 6}, {219, 355}, {284, 3560}, {940, 2003}, {952, 2256}, {1012, 2327}, {3713, 3940}


X(5779) =  INTERSECTION OF LINES X(3)X(9) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 - 4*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 - 2*b^5 + 2*a^3*b*c + 2*a^2*b^2*c - 2*a*b^3*c - 2*b^4*c - 4*a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + 4*b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 + 4*b^2*c^3 + 3*a*c^4 - 2*b*c^4 - 2*c^5)

X(5779) lies on these lines:
{3, 9}, {4, 144}, {5, 7}, {40, 3062}, {44, 990}, {45, 991}, {55, 5531}, {119, 3826}, {142, 1656}, {165, 3715}, {210, 1709}, {355, 382}, {381, 527}, {390, 952}, {517, 4915}, {518, 1351}, {954, 3560}, {1001, 2801}, {1012, 3940}, {1538, 5231}, {1750, 3929}, {1768, 4413}, {2951, 3579}, {3711, 5537}, {4312, 5587}, {4326, 5534}, {5273, 5658}


X(5780) =  INTERSECTION OF LINES X(3)X(9) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^5*b + 6*a^3*b^3 - 3*a^2*b^4 - 3*a*b^5 + 2*b^6 - 3*a^5*c + 6*a^4*b*c - 10*a^2*b^3*c + 3*a*b^4*c + 4*b^5*c + 2*a^2*b^2*c^2 - 2*b^4*c^2 + 6*a^3*c^3 - 10*a^2*b*c^3 - 8*b^3*c^3 - 3*a^2*c^4 + 3*a*b*c^4 - 2*b^2*c^4 - 3*a*c^5 + 4*b*c^5 + 2*c^6)

X(5780) lies on these lines: {3, 9}, {5, 8}, {72, 2095}, {355, 3452}, {952, 5084}, {1210, 1656}, {3149, 3876}


X(5781) =  INTERSECTION OF LINES X(3)X(9) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5 - 3*a^5*c - 2*a^4*b*c + 2*a^3*b^2*c + a*b^4*c + 2*b^5*c + 2*a^4*c^2 + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + 2*a^3*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 - 3*a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5781) lies on these lines:
{3, 9}, {6, 7}, {19, 518}, {20, 220}, {21, 3207}, {48, 1001}, {63, 910}, {101, 1012}, {144, 2287}, {169, 1071}, {218, 4292}, {219, 516}, {284, 954}, {390, 2256}, {1376, 2272}, {1503, 2550}, {1615, 5273}, {2173, 5220}, {2257, 4321}, {3059, 5227}, {3713, 5279}} SEARCH: -0.33541830108619423321


X(5782) =  INTERSECTION OF LINES X(3)X(9) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 4*a^2*b*c - a*b^2*c + 2*b^3*c - a^2*c^2 - a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3)

X(5782) lies on these lines:
{3, 9}, {6, 8}, {44, 4386}, {346, 2256}, {391, 2255}, {940, 4670}, {956, 5053}, {958, 2267}, {1211, 3330}, {1376, 2183}, {1404, 4390}, {1743, 5264}, {2057, 3965}, {2221, 4383}, {2235, 5205}, {2257, 2297}, {2550, 5480}, {4363, 5228}


X(5783) =  INTERSECTION OF LINES X(3)X(9) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 2*a^2*b*c + a*b^2*c + 2*b^3*c - a^2*c^2 + a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3)

X(5783) lies on these lines:
{1, 3713}, {3, 9}, {6, 10}, {37, 997}, {45, 5110}, {72, 2285}, {171, 1743}, {210, 1460}, {218, 1010}, {219, 1065}, {332, 344}, {405, 2268}, {474, 1400}, {475, 608}, {478, 1211}, {572, 958}, {573, 1376}, {604, 956}, {651, 5232}, {960, 1766}, {2050, 3452}, {2256, 2321}


X(5784) =  INTERSECTION OF LINES X(3)X(9) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c - b^4*c - 2*a^3*c^2 + 4*a*b^2*c^2 + 2*b^3*c^2 + 2*b^2*c^3 + 2*a*c^4 - b*c^4 - c^5)

X(5784) = X(7) of the X(1)-Brocard triangle (see X(5642)).

X(5784) lies on these lines:
{2, 1864}, {3, 9}, {7, 8}, {10, 1071}, {19, 1350}, {20, 960}, {21, 662}, {37, 1818}, {46, 5223}, {63, 210}, {72, 527}, {141, 1861}, {142, 442}, {144, 4190}, {219, 990}, {224, 1001}, {354, 2886}, {390, 3890}, {392, 4304}, {480, 2057}, {511, 2262}, {516, 3878}, {528, 3057}, {997, 1012}, {1824, 3917}, {2261, 5085}, {2348, 3423}, {3660, 5231}, {3740, 5273}, {3812, 4208}, {3881, 5542}


X(5785) =  INTERSECTION OF LINES X(3)X(9) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5 + a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + 4*a*b^3*c - 3*b^4*c - 6*a^3*c^2 + 2*a^2*b*c^2 + 14*a*b^2*c^2 + 6*b^3*c^2 + 2*a^2*c^3 + 4*a*b*c^3 + 6*b^2*c^3 + 5*a*c^4 - 3*b*c^4 - 3*c^5)

X(5785) lies on these lines: {3, 9}, {7, 10}, {20, 3062}, {144, 4292}, {377, 4312}


X(5786) =  INTERSECTION OF LINES X(3)X(10) AND X(4)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + 3*a^6*b - 2*a^4*b^3 + a^3*b^4 - a^2*b^5 - 2*a*b^6 + 3*a^6*c + 2*a^5*b*c - a^2*b^4*c - 2*a*b^5*c - 2*b^6*c - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 2*b^5*c^2 - 2*a^4*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 4*b^4*c^3 + a^3*c^4 - a^2*b*c^4 + 2*a*b^2*c^4 + 4*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 2*b^2*c^5 - 2*a*c^6 - 2*b*c^6

X(5786) lies on these lines:
{3, 10}, {4, 6}, {20, 333}, {29, 154}, {40, 1765}, {64, 412}, {65, 5307}, {84, 1715}, {243, 1854}, {386, 2050}, {388, 940}, {405, 1746}, {407, 1899}, {572, 2049}, {965, 2551}, {1012, 4267}, {1754, 5247}, {1766, 5295}, {1837, 1891}, {1853, 5125}, {3714, 5227}


X(5787) =  INTERSECTION OF LINES X(3)X(10) AND X(4)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 2*a^6*b - 3*a^4*b^3 + 3*a^3*b^4 - 2*a*b^6 + b^7 + 2*a^6*c + 2*a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - 2*a^2*b^4*c - b^6*c + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 3*b^4*c^3 + 3*a^3*c^4 - 2*a^2*b*c^4 + 2*a*b^2*c^4 + 3*b^3*c^4 - 3*b^2*c^5 - 2*a*c^6 - b*c^6 + c^7

X(5787) lies on these lines:
{3, 10}, {4, 7}, {5, 1490}, {20, 3419}, {30, 84}, {40, 3358}, {57, 1837}, {382, 2095}, {962, 4018}, {990, 1834}, {1467, 1750}, {1699, 3649}, {3091, 5658}, {3601, 5252}, {4219, 5090}


X(5788) =  INTERSECTION OF LINES X(3)X(10) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 2*a^5*b^2 + 3*a^3*b^4 - a^2*b^5 - 2*a*b^6 + a^6*c + 2*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - 2*b^6*c - 2*a^5*c^2 + 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 2*a*b^4*c^2 - 2*b^5*c^2 + 2*a^3*b*c^3 + 4*a^2*b^2*c^3 + 4*a*b^3*c^3 + 4*b^4*c^3 + 3*a^3*c^4 + a^2*b*c^4 + 2*a*b^2*c^4 + 4*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 2*b^2*c^5 - 2*a*c^6 - 2*b*c^6

X(5788) lies on these lines:
{3, 10}, {4, 333}, {5, 6}, {12, 940}, {63, 1867}, {394, 3142}, {970, 2050}, {1150, 3436}, {3560, 4267}, {5247, 5587}


X(5789) =  INTERSECTION OF LINES X(3)X(10) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 6*a^5*b^2 + 9*a^3*b^4 - 3*a^2*b^5 - 4*a*b^6 + 2*b^7 + a^6*c + 2*a^5*b*c + 2*a^4*b^2*c - 2*a^3*b^3*c - a^2*b^4*c - 2*b^6*c - 6*a^5*c^2 + 2*a^4*b*c^2 + 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 4*a*b^4*c^2 - 6*b^5*c^2 - 2*a^3*b*c^3 + 4*a^2*b^2*c^3 + 6*b^4*c^3 + 9*a^3*c^4 - a^2*b*c^4 + 4*a*b^2*c^4 + 6*b^3*c^4 - 3*a^2*c^5 - 6*b^2*c^5 - 4*a*c^6 - 2*b*c^6 + 2*c^7

X(5789) lies on these lines: {3, 10}, {5, 7}, {153, 443}, {381, 3928}


X(5790) =  INTERSECTION OF LINES X(3)X(10) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + a^2*b^2 + 2*a*b^3 - 2*b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c + a^2*c^2 - 2*a*b*c^2 + 4*b^2*c^2 + 2*a*c^3 - 2*c^4

X(5790) lies on these lines:
{1, 1656}, {2, 952}, {3, 10}, {4, 3617}, {5, 8}, {30, 5657}, {40, 382}, {55, 80}, {119, 2886}, {140, 944}, {145, 3090}, {165, 3534}, {210, 381}, {226, 1159}, {442, 5554}, {495, 3475}, {516, 3654}, {519, 5055}, {546, 962}, {547, 3241}, {912, 3753}, {946, 3626}, {956, 5176}, {997, 5123}, {999, 1737}, {1000, 5274}, {1071, 4002}, {1125, 5070}, {1145, 3434}, {1260, 3419}, {1351, 3416}, {1385, 1698}, {1483, 3616}, {1598, 5090}, {1657, 3579}, {1837, 3295}, {2095, 3421}, {2801, 3968}, {3091, 4678}, {3428, 5659}, {3436, 3927}, {3560, 5086}, {3576, 5054}, {3621, 5056}, {3622, 5067}, {3632, 5079}, {3655, 3828}, {3656, 3817}, {3814, 5289}, {3843, 4691}, {4668, 5072}, {5154, 5330}, {5204, 5445}, {5221, 5270}


X(5791) =  INTERSECTION OF LINES X(3)X(10) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b - 2*a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c - 3*a*b^2*c - 2*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4

X(5791) lies on these lines:
{2, 72}, {3, 10}, {4, 5273}, {5, 9}, {7, 3824}, {12, 57}, {21, 3419}, {28, 5130}, {37, 5292}, {46, 3925}, {63, 442}, {140, 936}, {142, 3634}, {191, 1836}, {210, 498}, {226, 3927}, {345, 5295}, {377, 3916}, {381, 5325}, {405, 1259}, {443, 3436}, {496, 5231}, {549, 5438}, {965, 3211}, {997, 4999}, {1479, 3683}, {1656, 2095}, {1706, 3587}, {1714, 3666}, {1837, 5251}, {2476, 3219}, {2550, 3579}, {3218, 4197}, {3295, 4847}, {3305, 4187}, {3416, 5138}, {3454, 4643}, {3601, 3679}, {3656, 3878}, {3697, 5552}, {3746, 4863}, {5067, 5328}, {5234, 5587}, {5252, 5258}


X(5792) =  INTERSECTION OF LINES X(3)X(10) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^5 - a^4*b - a^3*b^2 + a^2*b^3 - 2*a*b^4 - a^4*c + a^2*b^2*c + 2*a*b^3*c - 2*b^4*c - a^3*c^2 + a^2*b*c^2 + 2*b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + 2*b^2*c^3 - 2*a*c^4 - 2*b*c^4

X(5792) lies on these lines: {2, 3207}, {3, 10}, {6, 7}, {19, 4361}, {610, 3739}, {910, 4384}, {2182, 4363}


X(5793) =  INTERSECTION OF LINES X(3)X(10) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^2*b^2 + 2*a*b^3 + 4*a^2*b*c + 2*a*b^2*c + 2*b^3*c + a^2*c^2 + 2*a*b*c^2 + 4*b^2*c^2 + 2*a*c^3 + 2*b*c^3

X(5793) lies on these lines:
{1, 3714}, {3, 10}, {6, 8}, {65, 4363}, {141, 388}, {333, 3617}, {1211, 3436}, {3052, 4195}, {3679, 5247}, {4720, 5331}, {5078, 5176}


X(5794) =  INTERSECTION OF LINES X(3)X(10) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - a^3*b + a*b^3 - b^4 - a^3*c + 2*a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4

X(5794) lies on these lines:
{1, 442}, {2, 1837}, {3, 10}, {4, 960}, {5, 997}, {7, 8}, {12, 78}, {20, 4640}, {46, 529}, {72, 1478}, {80, 1698}, {145, 3475}, {149, 3890}, {210, 3436}, {224, 3925}, {329, 5229}, {392, 1479}, {407, 1211}, {443, 3812}, {474, 1737}, {495, 3811}, {497, 5175}, {498, 5440}, {528, 1697}, {594, 5227}, {936, 1329}, {938, 3742}, {946, 5289}, {950, 1001}, {965, 1826}, {1155, 3617}, {1159, 3625}, {1220, 5135}, {1265, 3967}, {1420, 5231}, {1610, 4220}, {1836, 2475}, {1861, 1891}, {2182, 2345}, {2476, 4511}, {2551, 3740}, {3057, 3434}, {3091, 5087}, {3421, 4662}, {3485, 3838}, {3576, 4999}, {3698, 5554}, {3876, 5080}, {3880, 5082}, {3916, 4299}, {3966, 5016}, {4255, 5530}, {4314, 4428}, {4679, 5046}, {4855, 5432}


X(5795) =  INTERSECTION OF LINES X(3)X(10) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^3 + a^2*b + b^3 + a^2*c + 4*a*b*c - b^2*c - b*c^2 + c^3)

X(5795) lies on these lines:
{1, 2551}, {2, 1420}, {3, 10}, {8, 9}, {20, 1706}, {63, 4848}, {65, 527}, {142, 388}, {145, 3984}, {200, 3486}, {226, 3436}, {281, 1891}, {329, 3340}, {474, 4311}, {495, 1125}, {497, 4853}, {519, 960}, {529, 3812}, {535, 3918}, {936, 944}, {952, 5044}, {956, 1210}, {997, 5534}, {1005, 5086}, {1212, 1573}, {1220, 5053}, {1385, 3820}, {1716, 3755}, {1737, 5258}, {1837, 4847}, {2078, 5176}, {2098, 4679}, {2324, 4270}, {2478, 3872}, {2550, 2951}, {2784, 3041}, {2975, 3911}, {3036, 4691}, {3058, 3893}, {3158, 4313}, {3189, 4882}, {3244, 5289}, {3586, 5082}, {3600, 5437}, {3617, 5273}, {3622, 5328}, {3626, 5302}, {3634, 4999}, {3679, 5234}, {3753, 4292}, {3913, 4314}, {4858, 4968}


X(5796) =  INTERSECTION OF LINES X(4)X(6) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8*b + 2*a^7*b^2 - 4*a^6*b^3 - 4*a^5*b^4 + 4*a^4*b^5 + 2*a^3*b^6 - b^9 + a^8*c + 2*a^7*b*c - a^6*b^2*c - 4*a^5*b^3*c - a^4*b^4*c + 2*a^3*b^5*c + a^2*b^6*c + 2*a^7*c^2 - a^6*b*c^2 - 3*a^4*b^3*c^2 - 2*a^3*b^4*c^2 + a^2*b^5*c^2 + 3*b^7*c^2 - 4*a^6*c^3 - 4*a^5*b*c^3 - 3*a^4*b^2*c^3 - 4*a^3*b^3*c^3 - 2*a^2*b^4*c^3 + b^6*c^3 - 4*a^5*c^4 - a^4*b*c^4 - 2*a^3*b^2*c^4 - 2*a^2*b^3*c^4 - 3*b^5*c^4 + 4*a^4*c^5 + 2*a^3*b*c^5 + a^2*b^2*c^5 - 3*b^4*c^5 + 2*a^3*c^6 + a^2*b*c^6 + b^3*c^6 + 3*b^2*c^7 - c^9

X(5796) lies on these lines: {4, 6}, {5, 7}, {226, 1736}, {3100, 5396}


X(5797) =  INTERSECTION OF LINES X(4)X(6) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6*b - 2*a^5*b^2 - 3*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 + b^7 + a^6*c - 2*a^5*b*c - 2*a^4*b^2*c + a^2*b^4*c + 2*a*b^5*c - 2*a^5*c^2 - 2*a^4*b*c^2 - 4*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - 2*b^5*c^2 - 3*a^4*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + b^4*c^3 + 2*a^3*c^4 + a^2*b*c^4 + b^3*c^4 + a^2*c^5 + 2*a*b*c^5 - 2*b^2*c^5 + c^7

X(5797) lies on these lines: {4, 6}, {5, 8}, {10, 1953}, {355, 3187}


X(5798) =  INTERSECTION OF LINES X(4)X(6) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7*b + a^6*b^2 + 4*a^5*b^3 - a^4*b^4 - 2*a^3*b^5 - a^2*b^6 + b^8 - 2*a^7*c + 4*a^6*b*c + 2*a^5*b^2*c + 2*a^3*b^4*c - 4*a^2*b^5*c - 2*a*b^6*c + a^6*c^2 + 2*a^5*b*c^2 + 2*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*a*b^5*c^2 - 4*b^6*c^2 + 4*a^5*c^3 + 8*a^2*b^3*c^3 + 4*a*b^4*c^3 - a^4*c^4 + 2*a^3*b*c^4 + a^2*b^2*c^4 + 4*a*b^3*c^4 + 6*b^4*c^4 - 2*a^3*c^5 - 4*a^2*b*c^5 - 2*a*b^2*c^5 - a^2*c^6 - 2*a*b*c^6 - 4*b^2*c^6 + c^8

X(5798) lies on these lines: {4, 6}, {5, 9}, {226, 1465}, {442, 573}, {1243, 1903}, {1490, 5396}, {1699, 1723}


X(5799) =  INTERSECTION OF LINES X(4)X(6) AND X(5)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^5*b^2 - 3*a^4*b^3 + 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 + b^7 - 4*a^5*b*c - 3*a^4*b^2*c + 2*a^2*b^4*c + 4*a*b^5*c + b^6*c - 3*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 - b^5*c^2 - 3*a^4*c^3 - 4*a^2*b^2*c^3 - 8*a*b^3*c^3 - b^4*c^3 + 2*a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 + 4*a*b*c^5 - b^2*c^5 + a*c^6 + b*c^6 + c^7

X(5799) lies on these lines: {4, 6}, {5, 10}, {40, 4026}, {51, 1904}, {65, 1848}, {154, 4198}, {573, 4205}, {2050, 5292}


X(5800) =  INTERSECTION OF LINES X(4)X(6) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^4*b^2 - a^2*b^4 - b^6 + 4*a^4*b*c + 4*a^3*b^2*c + a^4*c^2 + 4*a^3*b*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6

X(5800) lies on these lines:
{4, 6}, {7, 8}, {10, 5227}, {28, 159}, {55, 464}, {81, 1370}, {141, 443}, {193, 2475}, {376, 4265}, {497, 1386}, {515, 990}, {631, 5096}, {958, 4026}, {1056, 3242}, {1352, 4260}, {1478, 3751}, {1861, 2285}, {1890, 2082}, {2263, 5236}, {2478, 3618}, {3100, 3486}, {3101, 3474}, {3187, 3434}, {3475, 3920}, {3589, 5084}, {3827, 4295}, {3914, 5307}, {4663, 5229}


X(5801) =  INTERSECTION OF LINES X(4)X(6) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + 2*a^5*b + 7*a^4*b^2 - 5*a^2*b^4 - 2*a*b^5 - 3*b^6 + 2*a^5*c + 18*a^4*b*c + 16*a^3*b^2*c - 2*a*b^4*c - 2*b^5*c + 7*a^4*c^2 + 16*a^3*b*c^2 + 10*a^2*b^2*c^2 + 4*a*b^3*c^2 + 3*b^4*c^2 + 4*a*b^2*c^3 + 4*b^3*c^3 - 5*a^2*c^4 - 2*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 - 3*c^6

X(5801) lies on these lines: {4, 6}, {7, 10}


X(5802) =  INTERSECTION OF LINES X(4)X(6) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^4 + 2*a^3*b + 2*a*b^3 + b^4 + 2*a^3*c - 2*a*b^2*c - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4)

X(5802) lies on these lines: {2, 272}, {4, 6}, {8, 9}, {10, 380}, {20, 579}, {37, 3488}, {48, 3086}, {71, 4294}, {219, 497}, {226, 1449}, {281, 1837}, {329, 3187}, {405, 966}, {515, 2257}, {573, 1713}, {610, 1210}, {944, 1108}, {965, 5084}, {1058, 2256}, {1100, 3487}, {1213, 4258}, {1708, 3101}, {1743, 3586}, {1839, 4295}, {2260, 4293}, {2287, 2478}, {2345, 3419}, {3189, 3694}, {4207, 5320}, {5037, 5304}, {5257, 5436}


X(5803) =  INTERSECTION OF LINES X(4)X(7) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9 - a^8*c - 2*a^7*b*c + 4*a^5*b^3*c + 4*a^4*b^4*c - 2*a^3*b^5*c - 4*a^2*b^6*c + b^8*c - 2*a^7*c^2 + 4*a^5*b^2*c^2 + 2*a^4*b^3*c^2 + 2*a^3*b^4*c^2 - 4*a^2*b^5*c^2 - 4*a*b^6*c^2 + 2*b^7*c^2 + 2*a^6*c^3 + 4*a^5*b*c^3 + 2*a^4*b^2*c^3 + 4*a^3*b^3*c^3 + 6*a^2*b^4*c^3 - 2*b^6*c^3 + 2*a^5*c^4 + 4*a^4*b*c^4 + 2*a^3*b^2*c^4 + 6*a^2*b^3*c^4 + 6*a*b^4*c^4 - 2*a^4*c^5 - 2*a^3*b*c^5 - 4*a^2*b^2*c^5 - 2*a^3*c^6 - 4*a^2*b*c^6 - 4*a*b^2*c^6 - 2*b^3*c^6 + 2*a^2*c^7 + 2*b^2*c^7 + a*c^8 + b*c^8 - c^9

X(5803) lies on these lines: {4, 7}, {5, 6}


X(5804) =  INTERSECTION OF LINES X(4)X(7) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 3*a^6*b - a^5*b^2 - 5*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 - 3*a*b^6 + b^7 + 3*a^6*c - 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - 3*a^2*b^4*c + 10*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 5*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 20*a*b^3*c^3 + 3*b^4*c^3 + 5*a^3*c^4 - 3*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + a^2*c^5 + 10*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5804) lies on these lines: {4, 7}, {5, 8}, {517, 5084}, {946, 3340}, {1532, 3487}, {3149, 3488}


X(5805) =  INTERSECTION OF LINES X(4)X(7) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c - 2*a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6

X(5805) is the Gergonne point of the Euler triangle.

X(5805) lies on these lines:
{3, 142}, {4, 7}, {5, 9}, {11, 57}, {79, 3062}, {144, 3091}, {355, 518}, {381, 527}, {390, 5603}, {392, 443}, {515, 5542}, {517, 2550}, {528, 3656}, {952, 3243}, {954, 3149}, {990, 1086}, {991, 4675}, {1750, 4654}, {3332, 4000}, {4301, 5289}, {5223, 5587}

X(5805) = midpoint of X(4) and X(7)


X(5806) =  INTERSECTION OF LINES X(4)X(7) AND X(5)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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 - 4*a^4*b*c - 2*a^2*b^3*c - a*b^4*c + 6*b^5*c - a^4*c^2 + b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 12*b^3*c^3 + 2*a^2*c^4 - a*b*c^4 + b^2*c^4 + a*c^5 + 6*b*c^5 - c^6)

X(5806) lies on these lines:
{3, 5436}, {4, 7}, {5, 10}, {20, 5439}, {65, 1699}, {72, 3091}, {515, 5045}, {516, 3812}, {546, 912}, {944, 5049}, {962, 3753}, {1465, 2654}, {1482, 3577}, {1709, 5221}, {1837, 5173}, {1902, 5142}, {3057, 5219}, {3585, 5570}, {3742, 4297}, {3832, 3868}, {3876, 5068}, {3940, 4882}


X(5807) =  INTERSECTION OF LINES X(4)X(7) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - 2*a^5*b + a^4*b^2 - a^2*b^4 + 2*a*b^5 - b^6 - 2*a^5*c - 6*a^4*b*c - 6*a^3*b^2*c - 2*a^2*b^3*c + a^4*c^2 - 6*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - a^2*c^4 + b^2*c^4 + 2*a*c^5 - c^6

X(5807) lies on these lines: {4, 7}, {6, 8}, {390, 1766}, {452, 5279}, {950, 2285}, {4200, 5262}, {4220, 5435}, {5269, 5294}


X(5808) =  INTERSECTION OF LINES X(4)X(7) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c - 6*a^4*b*c - 8*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c - a^4*c^2 - 8*a^3*b*c^2 - 6*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 - c^6

X(5808) lies on these lines: {4, 7}, {6, 10}


X(5809) =  INTERSECTION OF LINES X(4)X(7) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 8*a^3*b*c - 2*a^2*b^2*c - 3*b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + 6*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 - 3*a*c^4 - 3*b*c^4 + c^5)

X(5809) lies on these lines:
{4, 7}, {8, 9}, {10, 4326}, {20, 1445}, {142, 5177}, {226, 5274}, {329, 497}, {388, 5572}, {405, 4313}, {480, 3189}, {516, 2093}, {952, 954}, {1001, 3486}, {1728, 4294}, {1750, 4321}, {1837, 2550}, {2551, 3059}, {3100, 5222}, {3755, 4907}


X(5810) =  INTERSECTION OF LINES X(4)X(8) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - a^5*b^2 - a^4*b^3 + a^3*b^4 + a^2*b^5 - a*b^6 - b^7 + a^6*c - a^4*b^2*c + 2*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - b^6*c - a^5*c^2 - a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 + a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + b^3*c^4 + a^2*c^5 - 2*a*b*c^5 + b^2*c^5 - a*c^6 - b*c^6 - c^7

X(5810) lies on these lines: {2, 1437}, {3, 1211}, {4, 8}, {5, 6}, {442, 1899}, {2203, 3542}, {3410, 5141}


X(5811) =  INTERSECTION OF LINES X(4)X(8) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - a^6*b + 3*a^5*b^2 + 3*a^4*b^3 - 3*a^3*b^4 - 3*a^2*b^5 + a*b^6 + b^7 - a^6*c - 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + a^2*b^4*c + 6*a*b^5*c - b^6*c + 3*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 12*a*b^3*c^3 + 3*b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 + 6*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5811) lies on these lines:
{3, 5658}, {4, 8}, {5, 7}, {9, 1158}, {84, 3452}, {104, 405}, {226, 3086}, {390, 5534}, {474, 2096}, {912, 938}, {997, 1490}, {1071, 5084}, {1483, 3488}, {1737, 3339}, {4295, 4848}, {4309, 5531}


X(5812) =  INTERSECTION OF LINES X(4)X(8) AND X(5)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 2*a^5*b^2 + a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + b^7 + 2*a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - b^6*c + 2*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - 3*b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 3*b^4*c^3 - a^3*c^4 + 3*b^3*c^4 - 2*a^2*c^5 - 3*b^2*c^5 - b*c^6 + c^7

X(5812) = X(26)-of-2nd-extouch-triangle

X(5812) lies on these lines:
{3, 226}, {4, 8}, {5, 9}, {11, 1728}, {12, 40}, {30, 1490}, {68, 1903}, {79, 165}, {222, 1076}, {452, 5603}, {580, 3772}, {908, 1259}, {946, 958}, {950, 1482}, {1006, 5253}, {1210, 2095}, {1385, 3487}, {1479, 1864}, {1766, 1901}, {5177, 5657}


X(5813) =  INTERSECTION OF LINES X(4)X(8) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 + a^4*b - a*b^4 - b^5 + a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + b^4*c + 2*a^2*b*c^2 + 2*a*b^2*c^2 - a*c^4 + b*c^4 - c^5

X(5813) lies on these lines: {2, 169}, {4, 8}, {6, 7}, {9, 4329}, {226, 2082}, {307, 2270}, {857, 1211}, {3616, 4223}, {3661, 5179}


X(5814) =  INTERSECTION OF LINES X(4)X(8) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^4 - a^3*b + a*b^3 + b^4 - a^3*c + 3*a*b^2*c + 2*b^3*c + 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + 2*b*c^3 + c^4

The triangle A''B''C'' defined at X(5739) is homothetic to the outer Garcia triangle at X(5814).

X(5814) lies on these lines:
{1, 1211}, {3, 3687}, {4, 8}, {6, 10}, {9, 3695}, {69, 942}, {75, 1330}, {78, 5396}, {209, 4680}, {306, 405}, {442, 5271}, {1125, 4035}, {1479, 3706}, {1836, 4647}, {2895, 3868}, {3187, 5051}, {3295, 3883}, {3454, 3772}, {3927, 4416}, {4651, 5300}, {4863, 4894}, {5244, 5290}


X(5815) =  INTERSECTION OF LINES X(4)X(8) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^4 - 2*a^3*b + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c + 6*a*b^2*c + 6*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5815) lies on these lines:
{1, 5129}, {2, 3333}, {4, 8}, {7, 10}, {20, 200}, {40, 144}, {69, 341}, {210, 388}, {443, 3697}, {452, 3870}, {516, 4882}, {518, 938}, {527, 1706}, {936, 3600}, {944, 3940}, {956, 3616}, {997, 4308}, {1056, 5044}, {1722, 4310}, {2550, 4662}, {3085, 5273}, {3086, 5328}, {3091, 4847}, {3555, 5084}, {3679, 4295}, {3811, 4313}, {3927, 5657}, {3961, 4339}, {4005, 5252}, {4301, 4915}, {4419, 4646}, {4863, 5225}, {5056, 5231}

X(5815) = anticomplement of X(3333)


X(5816) =  INTERSECTION OF LINES X(4)X(9) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^5 + a^4*b - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 2*a^3*b*c + b^4*c - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5816) lies on these lines:
{2, 572}, {3, 1213}, {4, 9}, {5, 6}, {37, 355}, {80, 941}, {119, 5517}, {377, 1765}, {391, 3091}, {498, 2268}, {499, 604}, {515, 5257}, {581, 975}, {946, 3686}, {1400, 1478}, {1474, 3542}, {1479, 2269}, {1737, 2285}, {1899, 3136}, {2328, 4207}, {3419, 3965}, {3553, 4270}


X(5817) =  INTERSECTION OF LINES X(4)X(9) AND X(5)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - 5*a^4*b^2 + 4*a^3*b^3 + 3*a^2*b^4 - 4*a*b^5 + b^6 + 2*a^4*b*c + 4*a^3*b^2*c - 4*a*b^4*c - 2*b^5*c - 5*a^4*c^2 + 4*a^3*b*c^2 - 6*a^2*b^2*c^2 + 8*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 + 8*a*b^2*c^3 + 4*b^3*c^3 + 3*a^2*c^4 - 4*a*b*c^4 - b^2*c^4 - 4*a*c^5 - 2*b*c^5 + c^6

X(5817) lies on these lines:
{2, 971}, {4, 9}, {5, 7}, {44, 3332}, {119, 1156}, {142, 3090}, {144, 3091}, {355, 390}, {443, 3358}, {518, 5603}, {527, 3545}, {944, 1001}, {946, 5223}, {948, 1736}, {952, 954}, {1698, 3062}, {1788, 4312}, {3487, 5045}


X(5818) =  INTERSECTION OF LINES X(4)X(9) AND X(5)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + 2*a^2*b^2 + 2*a*b^3 - 3*b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c + 2*a^2*c^2 - 2*a*b*c^2 + 6*b^2*c^2 + 2*a*c^3 - 3*c^4

X(5818) lies on these lines:
{1, 3090}, {2, 355}, {3, 5260}, {4, 9}, {5, 8}, {12, 3487}, {46, 5229}, {80, 498}, {100, 3560}, {104, 474}, {119, 2476}, {145, 5056}, {165, 3529}, {377, 2096}, {381, 962}, {388, 1737}, {495, 938}, {499, 3476}, {515, 631}, {517, 3091}, {519, 5071}, {547, 1483}, {942, 5261}, {946, 3545}, {952, 1656}, {986, 4947}, {1056, 1210}, {1071, 4208}, {1125, 5067}, {1478, 1788}, {1699, 3855}, {1837, 3085}, {2099, 3614}, {2475, 3652}, {3086, 5252}, {3089, 5090}, {3146, 3579}, {3241, 5055}, {3474, 3585}, {3524, 3828}, {3525, 3576}, {3628, 5550}, {3654, 3839}, {3753, 5177}, {3877, 5187}, {3925, 5658}, {4299, 5445}, {4301, 4691}, {4305, 5432}, {4330, 5560}, {4678, 5068}, {5086, 5552}, {5119, 5225}


X(5819) =  INTERSECTION OF LINES X(4)X(9) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^4 + 2*a^3*b - 2*a^2*b^2 + 2*a*b^3 + b^4 + 2*a^3*c - 2*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5819) lies on these lines:
{2, 910}, {4, 9}, {6, 7}, {20, 1212}, {37, 390}, {41, 3485}, {75, 144}, {101, 5603}, {142, 610}, {198, 1001}, {218, 4295}, {220, 962}, {348, 4209}, {388, 2082}, {518, 2262}, {672, 3474}, {857, 1213}, {954, 4254}, {1449, 5542}, {1478, 5540}, {1738, 1743}, {1836, 2348}, {2170, 3476}, {2280, 3475}, {3207, 3616}, {3487, 4251}, {3686, 5223}, {3772, 5304}, {5263, 5296}


X(5820) =  INTERSECTION OF LINES X(5)X(6) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^4*b^2 + a^2*b^4 - b^6 + 2*a^4*b*c + 2*a^3*b^2*c - a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6

X(5820) lies on these lines: {5, 6}, {7, 8}, {141, 474}, {542, 5138}, {940, 1899}, {1012, 1503}


X(5821) =  INTERSECTION OF LINES X(5)X(6) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - 2*a*b^5 - 3*b^6 + 8*a^4*b*c + 10*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + a^4*c^2 + 10*a^3*b*c^2 + 10*a^2*b^2*c^2 + 4*a*b^3*c^2 + 3*b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 + a^2*c^4 - 2*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 - 3*c^6

X(5821) lies on these lines: {5, 6}, {7, 10}


X(5822) =  INTERSECTION OF LINES X(5)X(6) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^4 - 2*a^2*b^2 + 2*a*b^3 + b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4)

X(5822) lies on these lines: {5, 6}, {8, 9}, {284, 966}, {499, 2317}, {1737, 2261}, {1743, 1826}


X(5823) =  INTERSECTION OF LINES X(5)X(7) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^5*b - 3*a^4*b^2 + a^2*b^4 - a*b^5 + b^6 + a^5*c + 4*a^4*b*c + 3*a^3*b^2*c - a^2*b^3*c + b^5*c - 3*a^4*c^2 + 3*a^3*b*c^2 - 4*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 - a^2*b*c^3 + a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - b^2*c^4 - a*c^5 + b*c^5 + c^6

X(5823) lies on these lines: {5, 7}, {6, 8}, {1736, 3086}


X(5824) =  INTERSECTION OF LINES X(5)X(7) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^6 - 2*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - 2*b^6 - 2*a^5*c - 8*a^4*b*c - 6*a^3*b^2*c + 2*a^2*b^3*c - 2*b^5*c + 2*a^4*c^2 - 6*a^3*b*c^2 + 2*a^2*b^2*c^2 + 2*b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 + a^2*c^4 + 2*b^2*c^4 - 2*b*c^5 - 2*c^6

X(5824) lies on these lines: {5, 7}, {6, 10}, {982, 1736}


X(5825) =  INTERSECTION OF LINES X(5)X(7) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^5 + 3*a^4*b - 10*a^3*b^2 - 2*a^2*b^3 + 7*a*b^4 - b^5 + 3*a^4*c + 8*a^3*b*c + 2*a^2*b^2*c + 3*b^4*c - 10*a^3*c^2 + 2*a^2*b*c^2 - 14*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*b^2*c^3 + 7*a*c^4 + 3*b*c^4 - c^5)

X(5825) lies on these lines: {5, 7}, {8, 9}, {11, 329}, {72, 4345}, {1728, 4293}, {1737, 4312}, {1864, 3740}, {3086, 5542}


X(5826) =  INTERSECTION OF LINES X(5)X(8) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 2*a^4*b - a^3*b^2 + a^2*b^3 + b^5 - 2*a^4*c + 4*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c - 2*b^4*c - a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - 2*b*c^4 + c^5

X(5826) lies on these lines: {5, 8}, {6, 7}


X(5827) =  INTERSECTION OF LINES X(5)X(8) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^2*b^2 - 2*b^4 + 2*a^2*b*c - 4*a*b^2*c - 2*b^3*c + a^2*c^2 - 4*a*b*c^2 - 2*b*c^3 - 2*c^4

X(5827) lies on these lines: {5, 8}, {6, 10}, {355, 2050}, {2551, 3695}, {5295, 5587}


X(5828) =  INTERSECTION OF LINES X(5)X(8) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + 4*a^2*b^2 + 2*a*b^3 - 5*b^4 - 2*a^3*c + 12*a^2*b*c - 10*a*b^2*c + 4*a^2*c^2 - 10*a*b*c^2 + 10*b^2*c^2 + 2*a*c^3 - 5*c^4

X(5828) lies on these lines: {5, 8}, {7, 10}, {341, 3262}, {4853, 5056}


X(5829) =  INTERSECTION OF LINES X(5)X(9) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7 + 2*a^6*b + a^5*b^2 + a^4*b^3 - 4*a^2*b^5 + a*b^6 + b^7 + 2*a^6*c - a^4*b^2*c + 4*a^3*b^3*c - 4*a*b^5*c - b^6*c + a^5*c^2 - a^4*b*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^3*b*c^3 + 4*a^2*b^2*c^3 + 8*a*b^3*c^3 + 3*b^4*c^3 - a*b^2*c^4 + 3*b^3*c^4 - 4*a^2*c^5 - 4*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5829) lies on these lines: {5, 9}, {6, 7}, {142, 1375}, {511, 2262}, {528, 1953}, {910, 5249}, {1723, 4312}


X(5830) =  INTERSECTION OF LINES X(5)X(9) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^5 - 2*a^4*b - a^3*b^2 + a^2*b^3 - a*b^4 + b^5 - 2*a^4*c + 4*a^3*b*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - a*c^4 + b*c^4 + c^5

X(5830) lies on these lines: {5, 9}, {6, 8}


X(5831) =  INTERSECTION OF LINES X(5)X(9) AND X(6)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - a^3*b^2 - a^2*b^3 + b^5 + 2*a^3*b*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 4*a*b^2*c^2 - 2*b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 + b*c^4 + c^5

X(5831) lies on these lines: {5, 9}, {6, 10}, {442, 2285}, {475, 1880}, {498, 3965}, {1766, 2886}, {2268, 3419}


X(5832) =  INTERSECTION OF LINES X(5)X(9) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c + 2*a^4*b*c - 4*a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 4*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6

X(5832) lies on these lines:
{5, 9}, {7, 8}, {63, 1836}, {142, 474}, {495, 1706}, {516, 993}, {611, 1738}, {956, 4292}, {1376, 5249}, {1387, 3254}, {1818, 4675}, {1861, 4363}


X(5833) =  INTERSECTION OF LINES X(5)X(9) AND X(7)X(10)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - 2*a^3*b^3 + a^2*b^4 + 3*a*b^5 - 2*b^6 - a^5*c + 8*a^4*b*c - 2*a^3*b^2*c - 12*a^2*b^3*c + 3*a*b^4*c + 4*b^5*c - 2*a^3*b*c^2 - 10*a^2*b^2*c^2 - 6*a*b^3*c^2 + 2*b^4*c^2 - 2*a^3*c^3 - 12*a^2*b*c^3 - 6*a*b^2*c^3 - 8*b^3*c^3 + a^2*c^4 + 3*a*b*c^4 + 2*b^2*c^4 + 3*a*c^5 + 4*b*c^5 - 2*c^6

X(5833) lies on these lines: {5, 9}, {7, 10}, {63, 4312}, {142, 936}, {515, 2550}, {1699, 5273}


X(5834) =  INTERSECTION OF LINES X(5)X(10) AND X(6)X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4*b - a^3*b^2 + a^2*b^3 + a*b^4 + b^5 - 2*a^4*c + 4*a^3*b*c - 3*a^2*b^2*c + 2*a*b^3*c - b^4*c - a^3*c^2 - 3*a^2*b*c^2 - 6*a*b^2*c^2 + a^2*c^3 + 2*a*b*c^3 + a*c^4 - b*c^4 + c^5

X(5834) lies on these lines: {5, 10}, {6, 7}, {19, 3589}, {141, 2262}, {597, 2182}, {1375, 5437}, {2183, 4364}, {2270, 4657}, {3674, 3752}


X(5835) =  INTERSECTION OF LINES X(5)X(10) AND X(6)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^3*b + a^2*b^2 + b^4 + 2*a^3*c + 4*a*b^2*c + 2*b^3*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 + 2*b*c^3 + c^4

X(5835) lies on these lines:
{1, 3704}, {5, 10}, {6, 8}, {65, 141}, {388, 4363}, {1211, 3869}, {1213, 3959}, {2292, 4364}, {2975, 5078}


X(5836) =  INTERSECTION OF LINES X(5)X(10) AND X(7)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^2*b - b^3 + a^2*c - 2*a*b*c + 3*b^2*c + 3*b*c^2 - c^3)

X(5836) lies on these lines:
{1, 474}, {2, 3057}, {5, 10}, {7, 8}, {12, 3838}, {37, 3208}, {40, 958}, {46, 956}, {56, 3872}, {57, 4853}, {72, 3679}, {78, 2099}, {92, 1888}, {100, 2646}, {145, 354}, {191, 3245}, {200, 3340}, {210, 3617}, {318, 1875}, {341, 3967}, {392, 1698}, {404, 1319}, {405, 5119}, {409, 643}, {442, 1145}, {484, 3916}, {519, 942}, {528, 950}, {529, 4292}, {672, 4875}, {758, 3626}, {891, 4925}, {910, 2329}, {936, 5289}, {962, 2551}, {993, 3579}, {997, 1482}, {1001, 1697}, {1104, 5255}, {1125, 1387}, {1155, 2975}, {1191, 1722}, {1193, 4695}, {1210, 3813}, {1212, 3501}, {1616, 5272}, {1829, 1861}, {1836, 3436}, {1837, 3434}, {1858, 5086}, {1859, 5174}, {1864, 5175}, {1887, 5081}, {2098, 4413}, {2171, 3965}, {2262, 2345}, {2292, 2643}, {2475, 5176}, {2800, 3036}, {2817, 3040}, {3244, 5045}, {3246, 3915}, {3303, 3895}, {3304, 3306}, {3336, 5288}, {3339, 4915}, {3421, 4295}, {3555, 3632}, {3601, 4421}, {3616, 3848}, {3621, 3873}, {3625, 3874}, {3634, 3884}, {3635, 5049}, {3636, 3833}, {3666, 4642}, {3670, 4674}, {3678, 4691}, {3681, 3962}, {3683, 5260}, {3711, 3984}, {3744, 3924}, {3746, 5541}, {3876, 3983}, {3877, 4731}, {3894, 4816}, {3899, 3921}, {3900, 4142}, {4015, 4745}, {4018, 4668}, {4071, 4167}, {4084, 4669}, {4428, 5436}, {4746, 4757}, {4847, 4848}

X(5836) = midpoint of X(8) and X(65)
X(5836) = complement of X(3057)


X(5837) =  INTERSECTION OF LINES X(5)X(10) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^2*b + 2*a*b^2 - b^3 + 3*a^2*c + b^2*c + 2*a*c^2 + b*c^2 - c^3)

X(5837) lies on these lines:
{2, 3340}, {5, 10}, {8, 9}, {65, 142}, {145, 5273}, {226, 3869}, {388, 527}, {392, 1210}, {443, 2093}, {519, 958}, {551, 4999}, {936, 5657}, {1125, 5289}, {1145, 3697}, {1212, 1500}, {1479, 2551}, {2078, 2975}, {3057, 4847}, {3212, 4357}, {3305, 5554}, {3486, 4512}, {3600, 3928}, {3625, 5302}, {3632, 5234}, {3813, 4342}, {3916, 4311}, {4297, 4640}


X(5838) =  INTERSECTION OF LINES X(6)X(7) AND X(8)X(9)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(5*a^3 - a^2*b + 3*a*b^2 + b^3 - a^2*c - 6*a*b*c - b^2*c + 3*a*c^2 - b*c^2 + c^3)

X(5838) lies on these lines:
{6, 7}, {8, 9}, {41, 3616}, {144, 239}, {169, 938}, {218, 962}, {497, 2348}, {516, 1743}, {910, 5435}, {1001, 5296}, {1212, 4313}, {1445, 2270}, {2170, 3241}, {2264, 2550}


X(5839) =  INTERSECTION OF LINES X(6)X(8) AND X(7)X(524)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a2 - (b + c)2

X(5839) lies on these lines:
{1,966},{2,319},{6,8},{7,524},{9,519},{10,1449},{37,145},{44,346},{48,3684},{55,4819},{69,239},{71,3169},{72,5802},{75,193},{78,3554},{141,5222},{144,536},{219,1067},{281,2323},{318,3087},{345,1914},{348,4372},{387,5814},{393,5081},{518,2262},{572,5657},{573,944},{597,4478},{604,1788},{740,3958},{894,1992},{938,965},{956,4254},{982,4771},{1043,1778},{1086,4402},{1108,3965},{1213,3616},{1400,3476},{1654,4393},{1698,4982},{1740,4489},{1743,2321},{1901,5175},{1953,4051},{2082,5227},{2269,3486},{2325,3973},{3061,3949},{3161,3943},{3187,5739},{3240,5153},{3241,5296},{3244,3247},{3293,5105},{3419,5746},{3553,3872},{3589,4445},{3618,3661},{3623,3723},{3625,4007},{3629,4363},{3630,4395},{3633,3707},{3635,3986},{3672,4643},{3679,5750},{3696,4307},{3739,3945},{3758,5564},{3770,4441},{3875,4416},{3879,4384},{4029,4898},{4047,4294},{4058,4701},{4060,4677},{4061,5269},{4454,4686},{4470,4967},{4545,4668},{4657,4690},{5015,5286},{5120,5687},{5271,5712},{5603,5816},{5703,5742}


X(5840) =  INTERSECTION OF LINE X(3)X(11) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7 + 2*a^6*b + 3*a^5*b^2 - 3*a^4*b^3 - a*b^6 + b^7 + 2*a^6*c - 4*a^5*b*c - a^4*b^2*c + 2*a^3*b^3*c + 2*a*b^5*c - b^6*c + 3*a^5*c^2 - a^4*b*c^2 + a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 + 2*a^3*b*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + a*b^2*c^4 + 3*b^3*c^4 + 2*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7

X(5840) lies on these lines:
{3, 11}, {4, 100}, {5, 3035}, {20, 104}, {30, 511}, {36, 5533}, {40, 80}, {140, 3825}, {153, 3146}, {214, 946}, {224, 1537}, {355, 1145}, {550, 1484}, {944, 1320}, {1156, 5759}, {1317, 1482}, {1385, 1387}, {1614, 3045}, {1768, 5709}, {1862, 1872}, {2077, 3583}, {2932, 3149}, {3036, 5690}, {3254, 5732}, {3359, 3586}, {4292, 5083}, {5541, 5691}


X(5841) =  INTERSECTION OF LINE X(3)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 2*a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + a*b^6 - b^7 - 2*a^6*c + 4*a^5*b*c - 3*a^4*b^2*c - 2*a^3*b^3*c + 4*a^2*b^4*c - 2*a*b^5*c + b^6*c - 3*a^5*c^2 - 3*a^4*b*c^2 + 8*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 2*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*a*b^3*c^3 - 3*b^4*c^3 + 4*a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 - 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5841) lies on these lines:
{3, 12}, {4, 2975}, {5, 993}, {30, 511}, {40, 4333}, {63, 355}, {80, 5535}, {119, 4996}, {140, 3822}, {226, 1385}, {1872, 1885}, {2077, 4316}, {4305, 5761}, {5691, 5709}


X(5842) =  INTERSECTION OF LINE X(4)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 2*a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + a*b^6 - b^7 - 2*a^6*c + a^4*b^2*c + b^6*c - 3*a^5*c^2 + a^4*b*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 3*b^4*c^3 - a*b^2*c^4 - 3*b^3*c^4 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5842) lies on these lines:
{3, 2886}, {4, 12}, {5, 5248}, {11, 5172}, {20, 2894}, {30, 511}, {31, 5721}, {40, 1726}, {140, 3841}, {550, 5450}, {944, 2099}, {946, 4314}, {1006, 3925}, {1012, 4302}, {1071, 1770}, {1072, 3744}, {1158, 5787}, {1479, 3149}, {1532, 3583}, {1824, 3575}, {1834, 3072}, {3058, 5603}, {3189, 5758}, {3474, 5768}, {3811, 5812}, {4292, 5173}, {4512, 5587}, {5119, 5691}


X(5843) =  INTERSECTION OF LINE X(5)X(7) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 7*a^4*b^2 - 2*a^3*b^3 - 6*a^2*b^4 + 2*a*b^5 + b^6 - 4*a^4*b*c - 2*a^3*b^2*c + 6*a^2*b^3*c + 2*a*b^4*c - 2*b^5*c + 7*a^4*c^2 - 2*a^3*b*c^2 - 4*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 + 6*a^2*b*c^3 - 4*a*b^2*c^3 + 4*b^3*c^3 - 6*a^2*c^4 + 2*a*b*c^4 - b^2*c^4 + 2*a*c^5 - 2*b*c^5 + c^6

X(5843) lies on these lines:
{3, 144}, {5, 7}, {9, 140}, {30, 511}, {84, 5763}, {142, 3628}, {355, 4312}, {390, 1483}, {546, 5805}, {548, 5732}, {550, 5759}, {1156, 1484}, {2096, 3940}, {2951, 5534}, {3339, 5587}, {3853, 5735}, {3927, 5657}, {5223, 5690}, {5730, 5731}


X(5844) =  INTERSECTION OF LINE X(5)X(8) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 4*a^3*b - a^2*b^2 + 4*a*b^3 - b^4 - 4*a^3*c + 8*a^2*b*c - 4*a*b^2*c - a^2*c^2 - 4*a*b*c^2 + 2*b^2*c^2 + 4*a*c^3 - c^4

X(5844) lies on these lines:
{1, 140}, {3, 145}, {4, 3621}, {5, 8}, {10, 3628}, {30, 511}, {36, 1317}, {40, 548}, {119, 4867}, {165, 3655}, {355, 546}, {495, 2099}, {496, 2098}, {547, 3679}, {549, 3241}, {550, 944}, {631, 3623}, {632, 3616}, {946, 3625}, {962, 3627}, {1000, 2346}, {1056, 1159}, {1145, 4511}, {1320, 1484}, {1385, 3244}, {1387, 1737}, {1656, 3617}, {2136, 5709}, {3036, 3814}, {3090, 4678}, {3488, 5729}, {3526, 3622}, {3576, 3654}, {3656, 4677}, {3680, 5763}, {3820, 5289}, {3861, 4301}, {4187, 5330}, {4534, 5526}, {5535, 5541}


X(5845) =  INTERSECTION OF LINE X(6)X(7) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 2*a^3*b + a^2*b^2 - b^4 - 2*a^3*c + 2*b^3*c + a^2*c^2 - 2*b^2*c^2 + 2*b*c^3 - c^4

X(5845) lies on these lines:
{6, 7}, {9, 141}, {30, 511}, {41, 3665}, {69, 144}, {101, 1565}, {142, 3589}, {150, 1146}, {193, 4440}, {348, 3207}, {390, 3242}, {597, 4795}, {599, 4370}, {903, 1992}, {1001, 4364}, {1350, 5759}, {1352, 5779}, {1386, 4667}, {2550, 4363}, {3416, 4901}, {3618, 4747}, {3620, 4473}, {3751, 4312}, {3763, 4748}, {3826, 4472}, {4904, 5540}, {5480, 5805}

X(5845) = crossdifference of every pair of points on line X(6)X(926)


X(5846) =  INTERSECTION OF LINE X(6)X(8) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = {-2*a^3 - a*b^2 + b^3 + b^2*c - a*c^2 + b*c^2 + c^3

X(5846) lies on these lines:
{1, 141}, {6, 8}, {10, 1386}, {30, 511}, {31, 3703}, {37, 3883}, {42, 4030}, {44, 3717}, {63, 4884}, {69, 145}, {100, 5078}, {182, 5690}, {193, 3621}, {238, 3932}, {306, 3744}, {345, 3052}, {355, 5480}, {595, 3695}, {597, 3679}, {599, 3241}, {612, 3966}, {902, 3712}, {944, 1350}, {1086, 4645}, {1125, 3844}, {1211, 3920}, {1279, 3912}, {1352, 1482}, {1697, 5227}, {1738, 4395}, {1743, 4901}, {1834, 5015}, {1999, 4514}, {2550, 4361}, {2886, 4362}, {2975, 4265}, {3008, 3823}, {3035, 4434}, {3187, 5014}, {3244, 3631}, {3616, 3763}, {3617, 3618}, {3619, 3622}, {3620, 3623}, {3625, 4663}, {3629, 3632}, {3630, 3633}, {3685, 3943}, {3696, 4399}, {3704, 5255}, {3705, 3769}, {3722, 4062}, {3745, 4914}, {3755, 4852}, {3756, 5211}, {3782, 3891}, {3790, 4676}, {3811, 5396}, {3867, 5090}, {4153, 5305}, {4307, 4363}, {4349, 4670}, {4366, 4437}, {4388, 4415}, {4684, 4864}, {5082, 5800}, {5085, 5657}, {5241, 5297}


X(5847) =  INTERSECTION OF LINE X(6)X(10) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^3 - a^2*b + b^3 - a^2*c + b^2*c + b*c^2 + c^3

X(5847) lies on these lines:
{1, 69}, {6, 10}, {8, 193}, {9, 4078}, {30, 511}, {31, 306}, {38, 4001}, {40, 3169}, {44, 3932}, {55, 4028}, {72, 3688}, {141, 1125}, {171, 3687}, {226, 4362}, {238, 3912}, {239, 1738}, {319, 5263}, {322, 4008}, {345, 1707}, {355, 1351}, {551, 599}, {581, 3811}, {597, 3828}, {612, 4104}, {613, 1210}, {896, 3977}, {902, 4062}, {940, 3966}, {946, 1352}, {950, 3056}, {976, 4101}, {984, 4416}, {1001, 4851}, {1100, 4026}, {1211, 3745}, {1279, 4966}, {1350, 4297}, {1353, 5690}, {1428, 3911}, {1698, 3618}, {1733, 3262}, {1757, 3717}, {1992, 3679}, {1999, 4388}, {2308, 5294}, {2321, 3923}, {2887, 3791}, {2895, 3920}, {3008, 3836}, {3011, 3936}, {3187, 3914}, {3242, 3244}, {3510, 3783}, {3578, 4981}, {3589, 3634}, {3616, 3620}, {3619, 3624}, {3626, 3629}, {3630, 3635}, {3631, 3636}, {3663, 4655}, {3703, 4641}, {3722, 4938}, {3755, 4660}, {3759, 4429}, {3769, 4417}, {3771, 4035}, {3772, 4138}, {3773, 4672}, {3821, 3946}, {3896, 4450}, {4085, 4856}, {4133, 5695}, {4265, 5267}, {4432, 4437}, {4656, 4703}, {4682, 5743}, {4769, 5028}, {4847, 4865}, {4909, 5625}, {5093, 5790}, {5138, 5745}


X(5848) =  INTERSECTION OF LINE X(6)X(11) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + 2*a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + 2*a^4*c - 4*a^3*b*c + a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - a^2*c^3 + 2*a*b*c^3 - a*c^4 - b*c^4 + c^5

X(5848) lies on these lines:
{6, 11}, {30, 511}, {69, 100}, {80, 3751}, {119, 1352}, {141, 3035}, {149, 193}, {611, 5820}, {1145, 3416}, {1317, 3242}, {1353, 1484}, {1386, 1387}, {3013, 3140}


X(5849) =  INTERSECTION OF LINE X(6)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 - b^6 - 2*a^3*b^2*c + 2*a^2*b^3*c - 3*a^4*c^2 - 2*a^3*b*c^2 + b^4*c^2 + 2*a^2*b*c^3 + 2*a^2*c^4 + b^2*c^4 - c^6

X(5849) lies on these lines: {6, 12}, {30, 511}, {69, 2975}, {141, 4999}


X(5850) =  INTERSECTION OF LINE X(7)X(10) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^3 - 3*a^2*b + 4*a*b^2 + b^3 - 3*a^2*c - b^2*c + 4*a*c^2 - b*c^2 + c^3

X(5850) lies on these lines:
{1, 144}, {6, 4353}, {7, 10}, {8, 4312}, {9, 1125}, {30, 511}, {72, 4298}, {142, 3634}, {190, 4684}, {210, 553}, {320, 3717}, {390, 3244}, {946, 5779}, {954, 993}, {962, 3062}, {984, 3664}, {997, 4321}, {1001, 3636}, {1738, 4887}, {1743, 4310}, {1757, 3008}, {2325, 4966}, {2550, 3626}, {2951, 5493}, {3243, 3635}, {3475, 3929}, {3663, 3751}, {3685, 4480}, {3811, 5732}, {3817, 5817}, {3874, 5728}, {3881, 5572}, {3925, 3982}, {3946, 4663}, {4021, 4649}, {4031, 4413}, {4297, 5759}, {4349, 4644}, {4356, 4419}, {4645, 4899}, {4860, 5316}, {5715, 5811}


X(5851) =  INTERSECTION OF LINE X(7)X(11) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + 2*a^4*b + 5*a^3*b^2 - 7*a^2*b^3 + a*b^4 + b^5 + 2*a^4*c - 12*a^3*b*c + 7*a^2*b^2*c + 6*a*b^3*c - 3*b^4*c + 5*a^3*c^2 + 7*a^2*b*c^2 - 14*a*b^2*c^2 + 2*b^3*c^2 - 7*a^2*c^3 + 6*a*b*c^3 + 2*b^2*c^3 + a*c^4 - 3*b*c^4 + c^5

X(5851) lies on these lines:
{7, 11}, {9, 1768}, {30, 511}, {80, 4312}, {100, 144}, {104, 1001}, {119, 3826}, {153, 2550}, {390, 1317}, {1145, 5223}, {1387, 5542}, {2951, 5528}, {3062, 3254}, {5083, 5572}, {5220, 5657}, {5289, 5698}


X(5852) =  INTERSECTION OF LINE X(7)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^3 + 2*a^2*b - 3*a*b^2 - b^3 + 2*a^2*c + b^2*c - 3*a*c^2 + b*c^2 - c^3

X(5852) lies on these lines:
{7, 12}, {9, 583}, {30, 511}, {144, 1001}, {190, 4966}, {320, 3932}, {329, 3816}, {345, 3632}, {553, 3740}, {1086, 1757}, {2550, 4678}, {3035, 3218}, {3058, 4430}, {3244, 3772}, {3631, 3773}, {3650, 3746}, {3663, 4663}, {4480, 4684}


X(5853) =  INTERSECTION OF LINE X(8)X(9) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)*(2*a^2 - a*b + b^2 - a*c - 2*b*c + c^2)

X(5853) lies on these lines:
{1, 142}, {2, 3158}, {7, 145}, {8, 9}, {10, 1001}, {11, 3689}, {30, 511}, {40, 5768}, {55, 4847}, {100, 2078}, {144, 3621}, {149, 908}, {190, 4899}, {200, 497}, {210, 3058}, {226, 2900}, {238, 3939}, {306, 5014}, {480, 1837}, {553, 3873}, {673, 3912}, {936, 1058}, {938, 1706}, {944, 5732}, {946, 3811}, {954, 3419}, {956, 4304}, {958, 4314}, {1000, 5785}, {1043, 4483}, {1086, 4864}, {1125, 3813}, {1210, 5687}, {1279, 3008}, {1320, 3254}, {1445, 4848}, {1449, 4344}, {1482, 5805}, {2346, 5178}, {2348, 3021}, {2551, 4882}, {2885, 3622}, {3011, 3722}, {3057, 3059}, {3242, 3663}, {3244, 4780}, {3421, 3586}, {3445, 4678}, {3476, 4321}, {3486, 4326}, {3555, 4292}, {3625, 4133}, {3626, 3773}, {3632, 5223}, {3633, 4312}, {3635, 4743}, {3687, 3996}, {3706, 4030}, {3711, 4679}, {3748, 3925}, {3878, 4523}, {3914, 3938}, {3932, 4702}, {3957, 5249}, {3966, 4061}, {3985, 4541}, {4001, 4450}, {4028, 4865}, {4046, 4914}, {4082, 4387}, {4307, 4667}, {4342, 5289}, {4358, 4939}, {4512, 5325}, {4527, 4701}, {4535, 4746}, {4538, 4662}, {4645, 4684}, {5218, 5231}, {5263, 5750}, {5572, 5836}

X(5853) = isogonal conjugate of X(1477)


X(5854) =  INTERSECTION OF LINE X(8)X(11) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^3 - 2*a^2*b - 3*a*b^2 + b^3 - 2*a^2*c + 8*a*b*c - b^2*c - 3*a*c^2 - b*c^2 + c^3)

X(5854) lies on these lines:
{1, 1145}, {8, 11}, {10, 1387}, {30, 511}, {46, 2136}, {56, 100}, {80, 3632}, {119, 1482}, {149, 3436}, {214, 3244}, {644, 3039}, {1000, 1001}, {1146, 4919}, {1828, 1862}, {1846, 1897}, {3254, 4900}, {3829, 5790}, {3871, 4996}, {4738, 4939}, {4861, 4999}


X(5855) =  INTERSECTION OF LINE X(8)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 + 4*a^3*b + a^2*b^2 - 4*a*b^3 + b^4 + 4*a^3*c - 4*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - 4*a*c^3 + c^4

X(5855) lies on these lines:
{1, 4999}, {8, 12}, {30, 511}, {55, 145}, {1317, 4996}, {1329, 5730}, {1482, 3813}, {2161, 4969}, {3035, 4511}, {3036, 4867}, {3039, 5526}, {3340, 5794}, {3419, 3632}, {3428, 3913}, {3434, 3621}, {3633, 5119}, {3816, 5289}, {3829, 5603}, {4930, 5790}, {5173, 5836}


X(5856) =  INTERSECTION OF LINE X(9)X(11) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^4 - 2*a^3*b - a^2*b^2 + b^4 - 2*a^3*c + 4*a^2*b*c - 4*b^3*c - a^2*c^2 + 6*b^2*c^2 - 4*b*c^3 + c^4)

X(5856) lies on these lines:
{7, 100}, {9, 11}, {30, 511}, {80, 5223}, {104, 5759}, {119, 5805}, {142, 3035}, {144, 149}, {214, 5542}, {390, 1320}, {956, 5698}, {1001, 1387}, {1086, 3939}, {1145, 2550}, {1317, 3243}, {3174, 5528}, {4312, 5541}


X(5857) =  INTERSECTION OF LINE X(9)X(12) AND THE INFINITY LINE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 2*a^5*b + 3*a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + b^6 + 2*a^5*c + 2*a^3*b^2*c - 2*a^2*b^3*c - 2*b^5*c + 3*a^4*c^2 + 2*a^3*b*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 + 4*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 2*b*c^5 + c^6

X(5857) lies on these lines: {7, 2975}, {9, 12}, {30, 511}, {142, 4999}, {219, 4331}, {954, 5698}, {956, 4295}, {4312, 5832}, {5248, 5719}


X(5858) =  CENTROID OF TRIANGLE T(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 + 2(3-1/2)S

Peter Moses (June 27, 2014) constructed families of triangles T(k,n) and U(k,n) as follows. On side BC of a triangle ABC, erect a regular n-sided polygon, externally. Starting at B and going around the polygon until reaching C, label the vertices v(0), v(1), ..., v(n-1), so that line v(0)v(n-1) = BC, and if n is even, then the lines L(A,k,n) = v(k)v(n-k-1), for k = 1, ..., (n-2)/n, are parallel to BC. If n is odd, let L(A,k,n), for k = (n-1)/2, be the line through v((n-1)/2)) that is parallel to BC. Define lines L(B,k,n) and L(C,k,n) cyclically. Then for each k from 1 to floor((n-1)/2), the lines L(A,k,n), L(B,k,n), L(C,k,n) form a triangle T(k,n) homothetic to ABC. T(1,4) and U(1,4) are the outer and inner Grebe triangles, respectively. Triangle centers defined from T(k,n) include the following, given by 1st barycentrics:

centroid of T(k,n): -2a2 + b2 + c2 + S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of T(k,n): a2[2S2A - 2SBSC + SSA csc(kπ/n) csc(π/n+kπ/n) sin(π/n)]
orthocenter of T(k,n): a2SBSC - SA(S2B + S2C) + SSBSC[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of T(k,n): 2SA(a2SA - S2B - S2C) + S(SBSC + S2)[cot(kπ/n)) - cot(kπ/n+π/n)]

If the polygons are erected internally instead of externally, the resulting triangles are denoted by U(k,n), with triangle centers given by changing S to - S in the above first barycentrics:
centroid of U(k,n): -2a2 + b2 + c2 - S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of U(k,n): a2[2S2A - 2SBSC - SSA csc(kπ/n) csc(π/n+kπ/n) sin(π/n)]
orthocenter of U(k,n): a2SBSC - SA(S2B + S2C) - SSBSC[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of U(k,n): 2SA(a2SA - S2B - S2C) - S(SBSC + S2)[cot(kπ/n)) - cot(kπ/n+π/n)]

Note that T(1,6) = T(1,3) and U(1,6) = U(1,3).

X(5858) lies on these lines: {2, 6}, {3, 533}, {381, 532}, {530, 3830}, {531, 3534}, {538, 3104}, {1351, 5617}

X(5858) = reflection of X(5859) in X(2)


X(5859) =  CENTROID OF TRIANGLE U(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 - 2(3-1/2)S

The triangle U(1,3) is defined at X(5858).

X(5859) lies on these lines: {2, 6}, {3, 532}, {381, 533}, {530, 3534}, {531, 3830}, {538, 3105}, {1351, 5613}

X(5859) = reflection of X(5858) in X(2)


X(5860) =  CENTROID OF TRIANGLE T(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 + S

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle..

X(5860) lies on these lines: {2, 6}, {30, 1160}, {519, 3640}, {637, 754}, {1328, 5485}, {3241, 5604}, {3679, 5588}

X(5860) = reflection of X(5861) in X(2)


X(5861) =  CENTROID OF TRIANGLE U(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 - S

The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle.

X(5861) lies on these lines: {2, 6}, {30, 1161}, {519, 3641}, {638, 754}, {1327, 5485}, {3241, 5605}, {3679, 5589}

X(5861) = reflection of X(5860) in X(2)


X(5862) =  CENTROID OF TRIANGLE T(2,6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 + 31/2S

The triangle T(2,6) is defined at X(5858).

X(5862) lies on these lines: {2, 6}, {4, 532}, {376, 533}

X(5862) = reflection of X(5863) in X(2)


X(5863) =  CENTROID OF TRIANGLE U(2,6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a2 + b2 + c2 - 31/2S

The triangle U(2,6) is defined at X(5858).

X(5863) lies on these lines: {2, 6}, {4, 533}, {376, 532}

X(5863) = reflection of X(5862) in X(2)


X(5864) =  CIRCUMCENTER OF TRIANGLE T(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[2S2A - 2SBSC + SSA(4/3)1/2]

The triangle T(1,3) is defined at X(5858).

X(5864) lies on these lines: {3, 6}, {4, 298}, {20, 3181}, {383, 634}, {394, 3130}, {627, 1080}

X(5864) = reflection of X(5865) in X(3)


X(5865) =  CIRCUMCENTER OF TRIANGLE U(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[2S2A - 2SBSC - SSA(4/3)1/2]

The triangle U(1,3) is defined at X(5858).

X(5865) lies on these lines: {3, 6}, {4, 299}, {20, 3180}, {383, 628}, {394, 3129}, {633, 1080}

X(5865) = reflection of X(5864) in X(3)


X(5866) =  INVERSE-IN-CIRCUMCIRCLE OF X(69)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =

X(5866) lies on these lines: {3,69},{25,5203},{99,186},{187,4558},{325,2071},{378,1007},{669,3265},{2373,3266}


X(5867) =  INVERSE-IN-CIRCUMCIRCLE OF X(81)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) =

X(5867) lies on these lines: {3,81},{31,501},{669,2106}


X(5868) =  ORTHOCENTER OF TRIANGLE T(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2SBSC - SA(S2B + S2C) + SSBSC

The triangle U(1,3) is defined at X(5858).

X(5868) lies on these lines:
{3, 618}, {4, 6}, {20, 298}, {64, 2993}, {154, 470}, {463, 1899}, {471, 1853}, {633, 1350}, {3146, 3181}

X(5868) = reflection of X(5869) in X(4)


X(5869) =  ORTHOCENTER OF TRIANGLE U(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2SBSC - SA(S2B + S2C) - SSBSC

The triangle U(1,3) is defined at X(5858).

X(5869) = reflection of X(5868) in X(4)

X(5869) lies on these lines:
{3, 619}, {4, 6}, {20, 299}, {64, 2992}, {154, 471}, {462, 1899}, {470, 1853}, {634, 1350}, {3146, 3180}


X(5870) =  ORTHOCENTER OF TRIANGLE T(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2SBSC - SA(S2B + S2C) - SSBSC
Barycentrics   g(a,b,c) : g(b,c,a) : g(a,b,c), where g(a,b,c) = 2a6 - a4(b2 + c2 - S) - (b2 - c2)2(b2 + c2 + S) (César Lozada, Dec. 20, 2013)

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle. X(5870) is the orthologic center of T(1,4) and ABC. (César Lozada, ADGEOM #978, Dec. 20, 2013)

X(5870) lies on these lines:
{3, 5590}, {4, 6}, {20, 488}, {30, 1160}, {40, 5688}, {147, 487}, {154, 3535}, {184, 3127}, {185, 1162}, {486, 3424}, {489, 3926}, {515, 3640}, {944, 5604}, {1853, 3536}, {1899, 5200}, {5588, 5691}

X(5870) = reflection of X(5871) in X(4)


X(5871) =  ORTHOCENTER OF TRIANGLE U(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2SBSC - SA(S2B + S2C) - SSBSC
Barycentrics   g(a,b,c) : g(b,c,a) : g(a,b,c), where g(a,b,c) = 2a6 - a4(b2 + c2 + S) - (b2 - c2)2(b2 + c2 - S) (César Lozada, Dec. 20, 2013)

The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle. X(5871) is the orthologic center of U(1,4) and ABC. (César Lozada, ADGEOM #978, Dec. 20, 2013)

X(5871) lies on these lines:
{3, 5591}, {4, 6}, {20, 487}, {30, 1161}, {40, 5689}, {147, 488}, {154, 3536}, {184, 3128}, {185, 1163}, {485, 3424}, {490, 3926}, {515, 3641}, {944, 5605}, {1853, 3535}, {5589, 5691}

X(5871) = reflection of X(5870) in X(4)


X(5872) =  NINE-POINT CENTER OF TRIANGLE T(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2SA(a2SA - S2B - S2C) + S(SBSC + 3-1/2S2)

The triangle T(1,3) is defined at X(5858).

X(5872) lies on these lines: {3, 298}, {4, 3181}, {5, 6}, {18, 5613}, {61, 5617}, {182, 635}, {2782, 3104}

X(5872) = reflection of X(5873) in X(5).


X(5873) =  NINE-POINT CENTER OF TRIANGLE U(1,3)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2SA(a2SA - S2B - S2C) - S(SBSC + 3-1/2S2)

The triangle U(1,3) is defined at X(5858).

X(5873) lies on these lines: {3, 299}, {4, 3180}, {5, 6}, {17, 5617}, {62, 5613}, {182, 636}, {2782, 3105}

X(5873) = reflection of X(5872) in X(5).


X(5874) =  NINE-POINT CENTER OF TRIANGLE T(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2SA(a2SA - S2B - S2C) + S(SBSC + S2)

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle.

X(5874) lies on these lines: {3, 1270}, {5, 6}, {26, 5594}, {30, 1160}, {140, 5590}, {355, 5588}, {952, 3640}, {1483, 5604}, {5688, 5690}

X(5874) = reflection of X(5875) in X(5)


X(5875) =  NINE-POINT CENTER OF TRIANGLE U(1,4)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2SA(a2SA - S2B - S2C) - S(SBSC + S2)

The triangle U(1,4) is defined at X(5858), is also the inner Grebe triangle.

X(5875) lies on these lines: {3, 1271}, {5, 6}, {26, 5595}, {30, 1161}, {140, 5591}, {355, 5589}, {952, 3641}, {1483, 5605}, {5689, 5690}

X(5875) = reflection of X(5874) in X(5)


X(5876) =  INTERSECTION OF LINES X(3)X(74) AND X(4)X(93)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5876) is the orthocenter of the triangle A*B*C* defined at X(5694), and X(5876) is the nine-point center of the X(3)-Fuhrmann triangle; see X(5613). (Randy Hutson, July 7, 2014)

X(5876) lies on these lines:
{3,74},{4,93},{5,389},{30,5562},{51,3850},{52,546},{140,185},{143,381},{511,3627},{548,3917},{550,1216},{568,3091},{578,1493},{1498,2918},{1657,2979},{2779,5694},{2807,5690},{3060,3843},{3518,3581},{3567,3851},{3845,5446},{5072,5640}


X(5877) =  INTERSECTION OF LINES X(4)X(523) AND X(5)X(6)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5877) is the similitude center of these equilateral triangles: X(15)-Fuhrmann and X(16)-Fuhrmann. (Randy Hutson, July 7, 2014)

X(5877) lies on these lines: {4,523},{5,6},{1899,3134}


X(5878) =  INTERSECTION OF LINES X(4)X(51) AND X(5)X(64)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S2A(20R2 - 3Sω ) + SA(3S2ω - 20R2Sω + S2) + S2(16R2 - 3Sω)

X(5878) = (X(3) of X(20)-Fuhrmann triangle). (Randy Hutson, July 7, 2014)

X(5878) is the orthologic center of the Carnot (aka Johnson) and half-altitude (midheight) triangles. (César Lozada, Perspective-Orthologic-Parallelogic.pdf, ADGEOM #978, December 20, 2013)

X(5878) lies on these lines:
{2,3357},{3,1661},{4,51},{5,64},{20,110},{30,155},{66,3521},{68,5663},{113,3548},{154,550},{382,1351},{546,1853},{1181,1885},{1204,3542},{3292,3529}

X(5878) = isogonal conjugate of X(5879)
X(5878) = anticomplement of X(3357)


X(5879) =  X(4)-VERTEX CONJUGATE OF X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let A'B'C' be the half-altitude triangle of ABC. Let LA be the reflection of line B'C' in line BC, and define LB and LC cyclically. Let A'' = LB∩LC, B'' = LC∩LA, C'' = LA∩LB. The lines AA'', BB'', CC'' concur in X(5879). (Randy Hutson, July 18, 2014)

X(5879) lies on this line: {1093,1294}

X(5879) = isogonal conjugate of X(5878)


X(5880) =  X(6) OF FUHRMANN TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - b3 - c3 + 2abc + b2c + bc2

X(5880) lies on these lines:
{1,528},{2,1155},{3,142},{4,3812},{5,1158},{6,1738},{7,8},{9,46},{10,527},{11,3306},{19,5829},{40,5735},{55,1004},{57,2886},{63,3925},{78,3649},{171,3772},{200,4654},{226,1260},{241,4331},{329,3740},{354,3434},{355,2801},{390,2646},{405,1770},{443,960},{495,5856},{497,3742},{519,1159},{553,4847},{612,3782},{673,1492},{740,4851},{750,3120},{894,4429},{908,4413},{940,3914},{958,4292},{966,3846},{1056,3880},{1373,3641},{1374,3640},{1386,4000},{1445,1454},{1478,3753},{1479,5439},{1633,4223},{1699,3816},{1706,5290},{1714,5165},{1737,5729},{1788,5177},{1837,2475},{1861,1892},{1890,4185},{2182,5819},{2345,3844},{2887,3980},{3035,5219},{3058,4666},{3243,3633},{3244,4780},{3255,5560},{3333,3813},{3419,5696},{3436,3698},{3579,3824},{3617,4741},{3662,5263},{3664,3755},{3729,3932},{3836,3923},{3841,5791},{3872,5434},{3873,4863},{3886,4966},{3912,5695},{3922,5554},{3946,4349},{3966,4359},{4001,4042},{4082,4942},{4361,5847},{4415,5268},{4644,4663},{4691,5850},{4854,5287},{5218,5766},{5223,5852},{5587,5851}

X(5880) = complement of X(5698)


X(5881) =  DARBOUX IMAGE OF X(40)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3 a^4-3 a^3 b-a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3a/(2s) - SBSC/S2 - 1
X(5881) = 3X(1) - 4X(5)      (barycentrics and combo, Peter Moses, July 14, 2014)

Let A' be the reflection of X(40) in A and let A'' be the reflection of X(40) in line BC. Define B', C', B'', and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5881). Also, X(5881) = (X(20) of the Fuhrmann triangle). (Randy Hutson, July 7, 2014)

More generally, suppose that X is a point in the plane of triangle ABC, and let A' be the reflection of X in A and A'' be the reflection of X in line BC. Define B', C', B'', and C'' cyclically. The triangles A'B'C' and A''B''C'' are perspective if X lies on the Darboux cubic. The perspector is here called the Darboux image of X. The appearance of (I,J) in the following list means that X(J) is the Darboux image of the point X(I) on the Darboux cubic: (1,4312), (3,381), (4,4), (20,5921), (40,5881), (64,5922), (80,5923), (1490,5924), (1498,5925).

If X is on the Darboux cubic and P is the perspector of ABC and the pedal triangle of X, then the Darboux image of X is the reflection of X in P. (Randy Hutson, July 18, 2014)

X(5881) lies on these lines:
{1,5},{3,3679},{4,519},{8,20},{10,631},{30,4677},{46,4325},{57,4317},{78,5176},{100,5450},{140,3655},{145,946},{165,548},{376,4669},{382,517},{516,3625},{518,5735},{546,3656},{550,3654},{551,3090},{573,4034},{632,3653},{912,4338},{962,3621},{996,5767},{1012,3913},{1071,5836},{1125,5067},{1210,3476},{1385,1698},{1420,1737},{1478,3340},{1482,1699},{1490,3419},{1532,3813},{1697,4309},{1750,4863},{1766,4007},{1788,4311},{1907,5090},{2077,5687},{3057,3586},{3091,3241},{3244,3855},{3245,4333},{3247,5816},{3524,4745},{3525,3828},{3528,3626},{3529,5493},{3560,3746},{3617,5731},{3624,5070},{3635,3817},{3853,5844},{3872,5086},{3899,5694},{4293,4848},{4299,5128},{4330,5119},{4654,5270},{4882,5787}

X(5881) = reflection of X(I) in X(J) for these (I,J); (1,355), (40,8)
X(5881) = anticomplement of X(5882)


X(5882) =  COMPLEMENT OF X(5881)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4 a^4-3 a^3 b-3 a^2 b^2+3 a b^3-b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-3 a^2 c^2-3 a b c^2+2 b^2 c^2+3 a c^3-c^4
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3a/(2s) - SBSC/S2
X(5882) = 3X(1) - X(4)      (barycentrics and combo, Peter Moses, July 14, 2014)

X(5882) = X(40) of X(1)-Brocard triangle

X(5882) lies on these lines:
{1,4},{3,519},{5,551},{8,3523},{10,140},{20,3241},{30,4301},{35,104},{36,4848},{40,145},{55,5450},{65,4311},{84,4313},{119,3825},{165,3633},{355,1125},{382,3656},{516,1482},{517,550},{549,4669},{553,4317},{572,2321},{631,3679},{912,3878},{942,4315},{962,3623},{993,5837},{997,5534},{1006,5258},{1012,3303},{1071,1317},{1158,1697},{1210,1319},{1388,1837},{1389,5425},{1698,3533},{2077,3871},{2099,4292},{2360,4248},{2801,3884},{2829,4342},{2894,4861},{3086,5727},{3146,5734},{3149,3304},{3333,4308},{3340,4293},{3524,4677},{3526,3653},{3577,5558},{3579,5844},{3601,5768},{3616,5056},{3622,5068},{3624,5818},{3625,5690},{3632,5657},{3634,5790},{3636,3851},{3671,5842},{3817,3850},{3877,5693},{4745,5054},{5316,5531}


X(5883) =  X(51) OF FUHRMANN TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(a2 - b2 - c2 + 3bc) + 2abc

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5883) = centroid of T. Let T' be the triangle whose vertices are the centroids of the altimedial triangles; then X(5883) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5883) lies on these lines:
{1,88},{2,758},{4,3255},{5,2771},{8,3881},{10,141},{21,3336},{42,1739},{46,5248},{51,2392},{57,993},{58,409},{65,392},{72,3634},{79,5046},{191,5047},{210,3828},{226,3814},{354,519},{373,2842},{405,3647},{484,1621},{513,4049},{517,549},{597,2836},{956,4860},{958,5708},{960,4084},{986,3743},{997,5437},{1006,5535},{1159,5289},{1445,3339},{1698,3678},{1737,3822},{1835,5136},{1844,5125},{1963,4658},{2650,3216},{2690,2699},{2801,5587},{2975,3337},{3035,5719},{3057,3636},{3090,5693},{3218,5251},{3219,4880},{3244,5045},{3290,3997},{3555,3626},{3616,3884},{3622,5697},{3624,3869},{3628,5694},{3632,3889},{3635,3922},{3649,4187},{3660,4315},{3679,3873},{3681,3894},{3720,4424},{3740,4134},{3848,4744},{3876,3901},{3880,5049},{3887,4809},{4002,4691},{4067,5044},{4511,5425},{4666,5119},{4675,5725},{4731,4745},{5083,5252},{5131,5426}

X(5883) = complement of X(5692)


X(5884) =  X(52) OF FUHRMANN TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let P be a point on the circumcircle. Let A' be the orthogonal projection of P on the A-altitude, and define B' and C' cyclically. As P traces the circumcircle, the locus of the incenter of A'B'C' is an ellipse with center X(5884). (Antreas Hatzipolkis, Hyancinthos #20792, February 6, 2012, and subsequent postings)

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5884) = orthocenter of T. Let T' be the triangle whose vertices are the orthocenters of the altimedial triangles; then X(5884) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5884) lies on these lines:
{1,104},{2,5693},{3,758},{4,79},{5,2771},{10,912},{40,3868},{52,2392},{65,515},{73,1735},{117,1425},{140,5694},{165,3901},{185,2779},{191,1006},{355,2801},{411,5535},{496,942},{517,550},{572,1761},{580,1046},{581,986},{631,5692},{944,3474},{1064,3670},{1210,1858},{1385,3878},{1482,3881},{1490,3339},{1656,3833},{1765,2294},{2096,3486},{2695,2719},{3149,5221},{3359,3811},{3576,3869},{3812,5777},{3918,5790},{4295,5768}

X(5884) = complement of X(5693)


X(5885) =  X(143) OF FUHRMANN TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5885) is the nine-point center of the Fuhrmann triangle of the orthic triangle of the Fuhrmann triangle. (Randy Hutson, July 7, 2014)

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5885) = X(5)-of-T. Let T' be the triangle whose vertices are the nine-point centers of the altimedial triangles; then X(5885) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5885) lies on these lines:
{1,3},{2,5694},{5,2771},{140,758},{143,2392},{575,2836},{912,3812},{952,3754},{1656,5693},{1772,2594},{3526,5692},{3628,3833},{3874,5690},{3881,5844}

X(5885) = complement of X(5694)


X(5886) =  X(381) OF FUHRMANN TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let A' be the nine-point center of the triangle IBC, where I = X(1), and define B' and C' cyclically. The triangle A'B'C' is homothetic to the Fuhrmann triangle at X(1), and X(5886) is the centroid of A'B'C'.

X(5886) lies on these lines:
{{1,5},{2,392},{3,142},{4,1385},{8,3090},{10,1482},{30,1699},{36,1836},{40,140},{46,5433},{56,3560},{65,499},{79,5427},{84,3255},{145,5056},{165,549},{226,999},{230,1572},{238,5398},{354,912},{381,515},{382,4297},{405,5812},{475,1872},{498,3057},{519,5055},{546,5691},{547,3679},{631,962},{908,956},{942,3086},{944,3091},{960,5791},{995,3772},{997,2886},{1000,4345},{1006,5284},{1012,1519},{1056,5226},{1058,5703},{1064,3720},{1100,5816},{1104,5713},{1108,5747},{1319,1478},{1352,1386},{1389,5330},{1479,2646},{1537,3306},{1594,5090},{1698,3628},{1737,2099},{1770,5204},{1829,3542},{1902,3541},{2095,5745},{3241,5071},{3244,5079},{3333,5843},{3338,3649},{3359,5437},{3417,3615},{3419,4511},{3421,5748},{3428,4423},{3434,5440},{3474,5122},{3475,5049},{3487,5045},{3488,5274},{3526,4301},{3600,5714},{3634,5070},{3636,3851},{3646,5763},{3671,5708},{3811,3813},{3868,5694},{3897,5046},{3940,4847},{4221,5333},{4293,5126},{4305,5225},{4323,5704},{4679,5251},{5010,5444},{5044,5761},{5067,5734},{5119,5432},{5436,5715},{5542,5779}

X(5886) = complement of X(5657)


X(5887) =  X(119) OF INNER GARCIA TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5887) = (X(4) of the X(1)-Brocard triangle).

X(5887) lies on these lines:
{1,90},{3,960},{4,8},{5,65},{10,119},{19,5778},{21,104},{40,5692},{56,920},{210,5690},{221,1060},{411,3579},{515,3878},{518,1351},{758,946},{936,3359},{942,3086},{944,3877},{952,1898},{971,5698},{1012,5730},{1062,1854},{1064,2292},{1352,3827},{1656,3812},{1697,5534},{2476,3753},{2745,2766},{2801,3884},{3817,4084},{3868,5603},{3876,5657},{3899,5691},{3931,5396},{4047,5755},{4067,4301},{5790,5836}


X(5888) =  INTERSECTION OF LINES X(2)X(3098) AND X(110)X(5092)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5888) is the similitude center of ABC and the X(2)-Brocard triangle.

X(5888) lies on these lines:
{2,3098},{74,549},{110,5092},{140,3581},{141,5648},{323,3819},{354,3920},{392,404},{511,5643},{631,5654},{1201,3746},{1995,5646},{2979,5644},{3167,5012},{3357,3523},{3524,4550},{5113,5653},{5544,5640}


X(5889) =  ORTHOCENTER OF CIRCUMORTHIC TRIANGLE

Trilinears     f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[S2A - S2 - 2SA(4R2 - Sω)]
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cot A)(csc 2B + csc 2C) - (cot B)(csc 2C + csc 2A) - (cot(C)(csc 2A + csc 2B)
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (sin 2A)(cos 2B + cos 2C) - (sin 2B)(cos 2C + cos 2A) - (sin 2C)(cos 2A + cos 2B)

X(5889) is the orthologic center of the circumorthic and orthic triangles. (César Lozada, Perspective-Orthologic-Parallelogic.pdf, ADGEOM #978, December 20, 2013)

Let O =- X(3) and let A' be the isogonal conjugate of A with respect to OBC, and define B' and C' cyclically. Let A'' be the isogonal conjugate of A' with respect to OB'C', and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5889). (Randy Hutson, July 7, 2014)

X(5889) lies on these lines:
{2,389},{3,54},{4,52},{5,568},{20,185},{22,1181},{24,110},{26,1614},{49,1658},{51,3091},{64,895},{143,381},{156,2070},{186,1147},{323,1092},{382,5663},{411,5752},{569,1199},{578,1994},{631,1216},{962,2807},{1204,2071},{1351,1593},{3090,5462},{3167,3515},{3523,3917},{3524,5447},{3564,3575}

X(5889) = reflection of X(i) in X(j) for these (i,j): (3,6102), (4,52)
X(5889) = anticomplement of X(5562)


X(5890) =  CENTROID OF CIRCUMORTHIC TRIANGLE

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [a^3 cos(B - C) - b^3 cos(C - A) - c^3 cos(A - B)]sec A + 2 a^2 cos(B - C) (b sec B + c sec C)
Trilinears   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[a^6 (b^2 + c^2) - a^4(3b^4 - b^2c^2 + 3c^4) + 3a^2(b^6 - b^4c^2 - b^2c^4 + c^6) - b^8 + b^6c^2 + b^2c^6 - c^8]

X(5890) = (X(2) of circumorthic triangle) = (X(4) of orthocentroidal triangle). (Randy Hutson, July 7, 2014)

X(5890) lies on these lines:
{2,5654},{3,54},{4,51},{6,74},{20,52},{24,154},{30,568},{64,1173},{143,382},{184,186},{373,5071},{376,511},{381,5640},{477,2452},{578,1199},{631,3819},{1141,1303},{1216,3523},{1994,2071},{2807,5603},{3091,5462},{3146,5446},{3524,3917},{3651,5752}}

X(5890) = reflection of X(I) in X(J) for these (I,J): (4,51), (2979,3)
X(5890) = anticomplement of X(5891)


X(5891) =  REFLECTION OF X(51) IN X(5)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos(B - C) [a^3 cos(B - C) + b(2b^2 - a^2) cos(C - A) + c(2c^2 - a^2) cos(A - B)]
Trilinears   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[a^2(b^2 + c^2) - (b^2 - c^2)^2][a^4 + b^4 + c^4 - 2a^2 (b^2 + c^2) + 4b^2c^2]

X(5891) = (X(376) of orthic triangle) = (X(4) of X(2)-Brocard triangle); also, (X(5891) of hexyl triangle) = X(2) and (X(5891) of excentral triangle) = X(376). (Randy Hutson, July 7, 2014)

X(5891) lies on these lines:
{2,5654},{3,64},{4,1216},{5,51},{20,5447},{30,3917},{113,127},{128,130},{140,185},{155,569},{216,1625},{373,547},{378,4550},{381,511},{389,1656},{399,5092},{549,5642},{568,5055},{1352,2393},{3060,3545},{3090,5462},{3091,5446},{3313,3818},{3567,5056},{5071,5640}

X(5891) = reflection of X(51) in X(5)
X(5891) = complement of X(5890)
X(5891) = anticomplement of X(5892)


X(5892) =  MIDPOINT OF X(3) AND X(51)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2[a^6 (b^2 + c^2) - a^4 (3b^4 - 4b^2c^2 + 3c^4) + 3a^2 (b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^4]

X(5892) = (X(376) of polar triangle of complement of polar circle) = {X(52),X(631)}-harmonic conjugate of X(5447) (Randy Hutson, July 7, 2014)

X(5892) lies on these lines:
{2,5654},{3,51},{5,2883},{52,631},{140,389},{143,3530},{182,2393},{185,1656},{373,381},{376,5640},{511,549},{512,1116},{547,5663},{568,3917},{2779,3833},{2781,3589},{3060,3524},{3523,3567},{3526,5562}

X(5892) = midpoint of X(I) and X(J) for these {I,J}: {3,51}, {389,3819}
X(5892) = reflection of X(I) in X(J) for these (I,J): (1216,3819), (3819,140)
X(5892) = complement of X(5891)


X(5893) =  CENTER OF HALF-ALTITUDE CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a10 + 3b10 + 3c10 - a8b2 - a8c2 + 12a6b4 + 12a6c4 + 16a6b2c2 - 10a4b6 - 10a4c6 + 10a4b4c2 + 10a4b2c4 - 2a2b8 - 2a2c8 + 16a2b6c2 + 16a2b2c6 + 28a2b4c4 - 9b8c2 - 9b2c8 + 6b6c4 + 6b4c6

The half-altitude circle is the circumcircle of the half-altitude triangle. Its radius is
[R/(16SASBSC)][2(b2c2J2 + S2 - S2A)(c2a2J2 + S2 - S2B)(a2b2J2 + S2 - S2C)]1/2, where J = |OH|/R (as at X(1113).

X(5893) lies on these lines:
{2,5894},{4,6},{5,3357},{30,5448},{64,3091},{140,2777},{154,3146},{221,5225},{546,5462},{1853,3832},{2192,5229}

X(5893) = midpoint of X(4) and X(2883)
X(5893) = complement of X(5894)


X(5894) =  ANTICOMPLEMENT OF X(5893)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a^10 - b^10 - c^10 - 5a^8(b^2 + c^2) - 8a^6(b^4 + c^4 - 3b^2c^2) + 14a^4(b^6 + c^6 - b^4c^2 - b^2c^4) - 4a^2(b^8 +c^8 - 6b^4c^4 + 2b^6c^2 + 2b^2c^6) + 3b^8c^2 + 3b^2c^8 - 2b^6c^4 - 2b^4c^6
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin B)(sec^2 B - sec A sec B sec C)[2 sec C sec A - sec B (sec^2 C + sec^2 A)] + (sin C)(sec^2 C - sec A sec B sec C)[2 sec A sec B - sec C (sec^2 A + sec^2 B)]

X(5894) is the center of the pedal circle of X(20) and of X(64), and the center of the cevian circle of X(69) and of X(253); X(5864) is also (X(64) of X(4)-Brocard triangle. (Randy Hutson, July 7, 2014)

X(5894) lies on these lines:
{2,5893},{3,1661},{4,1192},{5,1539},{20,64},{30,3357},{154,3522},{185,1205},{376,1498},{550,1216},{1204,1885},{1593,5480},{1853,3146},{1854,3474},{2935,3520},{3528,5656},{4219,5799}

X(5894) = midpoint of X(20) and X(64)
X(5894) = complement of X(5895)
X(5894) = anticomplement of X(5893)


X(5895) =  ANTICOMPLEMENT OF X(5984)

Trilinears    : f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec^2 A - sec A sec B sec C)[2 sec B sec C - sec A (sec^2 B + sec^2 C)]
Trilinears     g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B)/(cos C - cos A cos B) + (cos C)/(cos B - cos A cos C)
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)(a^6 + 2b^6 + 2c^6 - 3a^2b^4 - 3a^2c^4 + 6a^2b^2c^2 - 2b^4c^2 - 2b^2c^4)

Let H be the hyperbola {A,B,C,X(4),X(20)}. Let L(X) denote the line tangent to H at a point X on H. Then X(5895) is the point of intersection of L(X(4)) and L(X(20)). (Randy Hutson, July 7, 2014)

X(5895) lies on these lines:
{2,5893},{3,113},{4,64},{6,1885},{20,154},{25,2929},{30,155},{52,382},{193,1503},{235,1192},{381,3357},{468,1620},{1181,2904},{1514,3542},{1562,3172},{1593,3574},{1836,1854},{2778,5693},{2906,5706},{2907,5786},{3529,5656}

X(5895) = anticomplement of X(5984)

X(5895) = crosssum of X(3) and X(64)
X(5895) = crosspoint of X(4) and X(20)


X(5896) =  Λ(X(20), X(154))

Trilinears    : f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a/{(tan B + tan C - tan A)[(2a^2 - b^2 - c^2) tan A + (b^2 - c^2)(tan B - tan C)]}
Trilinears     g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/{(cos A - cos B cos C)[b(a^2 - b^2)(cos B - cos C cos A) - c(c^2 - a^2)(cos C - cos A cos B)]}

Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5896) = Λ(X(3), X(5893) = Λ(X(20), X(154)). (Randy Hutson, July 7, 2014)

X(5896) lies on the circumcircle and these lines: {64,110},{99,253},{107,3146},{1301,3515}}


X(5897) =  Λ(X(4), X(64))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5897) = Λ(X(3), X(1661) = Λ(X(4), X(64) = Λ(X(5), X(5893)) = Λ(X(20), X(394) = Λ(X(146), X(2071). X(5897) is the point of intersection, other than A,B,C, of the circumcircle and the hyperbola {A,B,C,X(3),X(20)}; also, X(5897) is the antipode of X(1301) on the circumcircle. (Randy Hutson, July 7, 2014)

X(5897) lies on the circumcircle and these lines:
{3,1301},{20,107},{110,1498},{112,1033},{376,1289},{393,3344},{1302,1370},{1304,2071}

X(5897) = reflection of X(1301) in X(3)


X(5898) =  REFLECTION OF X(195) IN X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let H be the Stammler hyperbola, and let T be the tangential triangle. X(5898) is the antipode of X(195) in H. The conic H is a rectangular hyperbola passing through X(I) for these I: 1,3,6,155,159,195,399,1498, 2916,2917,2918,2929,2930,2931,2935,2948,3511, the excenters and the vertices of T; the center of H is X(110). H is the isogonal conjugate of the Euler line with respect to T, and H is also the isogonal conjugate of the line X(30)X(40) with respect to the excentral triangle. Also, H is the locus of P for which the P-Brocard triangle is perspective to ABC; see X(5642). X(5898) is the isogonal conjugate of X(5899) with respect to T. (Randy Hutson, July 7, 2014)

X(5898) lies on these lines:
{3,2888},{6,3200},{25,2914},{110,143},{399,1154},{539,2931},{542,2916},{2918,3519}


X(5899) =  INVERSE-IN-CIRCUMCIRCLE OF X(140)

Trilinears     f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B sin 2C sin(A - B) (1 + 2 sin^2 A + 2 sin^2 C) + sin C sin 2B sin(A - C) (1 + 2 sin^2 A + 2 sin^2 B)

X(5899) = X(3) - 4X(23)
X(5899) = 12X(2) + (J2 - 12)X(3) where J = |OH|/R
X(5899) = 4R2X(2) + (3|OG|2 - 4R2)X(3)      (Peter Moses, July 11, 2014)

X(5899) is the isogonal conjugate of X(5898) with respect to the tangential triangle, and X(5899) is the pole with respect to the circumcircle of the line X(140)X(523). (Randy Hutson, July 7, 2014)

X(5899) lies on these lines: {2,3},{195,1614},{399,1154},{1533,2931},{2918,3574}

X(5899) = isogonal conjugate of X(5900)
X(5899) = crossdifference of every pair of points on the line X(647)X(5421)


X(5900) =  ISOGONAL CONJUGATE OF X(5899)

Trilinears     f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sin B sin 2C sin(A - B) (1 + 2 sin^2 A + 2 sin^2 C) + sin C sin 2B sin(A - C) (1 + 2 sin^2 A + 2 sin^2 B)]

X(5900) is the trilinear pole of the line X(647)X(5421), and X(5900) is the the antipode-in-Jerabek-hyperbola of X(1173). Also, X(5900) is the antigonal image of X(1173). (Randy Hutson, July 7, 2014)

X(5900) lies on these lines: {125,1173},{146,3521},{2889,3448}

X(5900) = isogonal conjugate of X(5899)
X(5900) = reflection of X(1173) in X(125)


X(5901) =  COMPLEMENT OF X(5690)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

As at X(5886), let A' be the nine-point center of the triangle IBC, where I = X(1), and define B' and C' cyclically; then X(5901) = (X(5) of A'B'C'), as well as the complement of X(5) with respect to A'B'C'. Let A* be the circle with center A and diameter b + c, and define B* and C* cyclically; then X(5901) is the radical center of A*, B*, C*. Let A'' be the nine-point center of triangle IBC, and define B'' and C'' cyclically; then I, A'', B'', C'' comprise an orthocentric system whose common nine-point circle has center X(5901). (Hyacinthos #21518, February 10, 2013, and following posts by Antreas Hatzipolakis and Randy Hutson)

X(5901) lies on these lines:
{1,5},{2,1482},{3,962},{4,3622},{8,1656},{10,3628},{30,551},{40,549},{104,5606},{140,517},{145,3090},{381,944},{382,5731},{392,5771},{476,953},{498,2098},{499,2099},{515,546},{516,548},{519,547},{550,3576},{632,3624},{912,5045},{999,3485},{1001,5762},{1064,5453},{1159,4323},{1191,5707},{1386,3564},{1388,1478},{1699,3627},{3241,5055},{3336,5298},{3487,5811},{3526,5550},{3530,3579},{3617,5067},{3623,5056},{3649,5563},{3655,3845},{3817,3850},{3874,5694},{3878,4999},{4292,5126},{4308,5714},{4511,5178},{5049,5777},{5180,5303},{5432,5697},{5436,5812},{5542,5843}

X(5901) = midpoint of X(1) and X(5)


X(5902) =  INCENTER OF ORTHOCENTROIDAL TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = abc + (b + c)(a - b + c)(a + b - c)

X(5902) = (X(1) of orthocentroidal triangle) = (X(381) of intouch triangle) = {X(1),X(65)}-harmonic conjugate of X(5903). (Randy Hutson, July 7, 2014)

X(5902) lies on these lines:
{1,3},{2,758},{4,79},{5,3649},{6,1718},{7,80},{8,2891},{10,3681},{20,5441},{43,1739},{47,1451},{58,3924},{61,2306},{63,4880},{72,1698},{81,1325},{90,5665},{145,3881},{191,405},{226,1737},{244,995},{355,5270},{374,1743},{381,2771},{386,2650},{392,3742},{498,1788},{499,3485},{515,553},{518,599},{519,3873},{551,3877},{579,2294},{584,2160},{614,5315},{631,5442},{912,4654},{938,1479},{944,4317},{950,1770},{952,5434},{960,3624},{985,2224},{993,3218},{994,4850},{997,3306},{1002,2809},{1012,1768},{1046,1724},{1051,2939},{1068,1825},{1071,5586},{1125,3869},{1210,3671},{1254,4306},{1464,5396},{1656,5694},{1717,2955},{1725,1779},{1790,4658},{1836,3583},{1837,3585},{1876,1905},{2280,5011},{2362,3301},{2392,3060},{2800,5603},{2802,3241},{2842,5640},{3244,3889},{3296,5559},{3419,5696},{3474,3488},{3475,5657},{3476,5083},{3486,4299},{3501,3970},{3555,3632},{3586,4312},{3616,3878},{3617,3918},{3622,3884},{3634,3876},{3635,3885},{3636,3890},{3752,5313},{3792,4675},{3828,4134},{3922,4668},{3940,4413},{3962,5044},{3968,4661},{3980,5208},{4002,4662},{4116,4128},{4414,4653},{4645,4680},{5432,5719},{5435,5444}}

X(5902) = midpoint of X(65) and X(354)
X(5902) = reflection of X(1) in X(354)


X(5903) =  REFLECTION OF X(1) IN X(65)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - abc + (b + c)(a - b + c)(a + b - c) (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)

X(5903) = {X(1),X(40)}-harmonic conjugate of X(35); X(5903) = {X(1),X(65)}- harmonic conjugate of X(5902). Let A' be the isogonal conjugate of A with respect to triangle IBC, where I = X(1), and define B' and C' cyclically. Let A'' be the isogonal conjugate of A' with respect to IB'C', and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5903). X(5903) is the incenter of the triangle denoted by A''B''C'' at X(5905) and the triangle of the same notation at X(5906). (Randy Hutson, July 14, 2014)

Let PA be the reflection of X(1) in line BC, and define PB and PC cyclically; then X(5903) is the isogonal conjugate of X(1) with respect to PAPBPC. (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)

X(5903) lies on these lines:
{1,3},{2,3754},{4,80},{7,5559},{8,79},{10,908},{12,5690},{37,5036},{41,5011},{43,3987},{63,5258},{72,3679},{78,4867},{90,3577},{145,2802},{150,4056},{169,5526},{181,3944},{191,958},{203,2306},{213,3959},{214,4188},{218,5540},{219,1781},{267,2948},{355,1836},{386,4642},{392,3624},{474,5289},{495,3649},{498,3485},{499,1788},{515,1770},{518,3632},{519,3868},{551,3890},{573,2171},{579,1953},{595,3924},{631,5444},{764,4083},{912,4338},{944,3474},{946,1737},{960,1698},{962,1479},{978,1739},{984,1756},{1000,5557},{1012,1727},{1046,1710},{1068,1835},{1100,4287},{1111,3212},{1122,4902},{1125,3877},{1148,1784},{1210,4301},{1376,5730},{1393,1772},{1411,2964},{1464,5399},{1572,5299},{1717,1854},{1724,3460},{1743,2262},{1759,2329},{1797,4792},{1829,4214},{1837,3583},{1838,1869},{1858,5727},{1871,1888},{1872,1875},{1902,1905},{2170,4253},{2176,3125},{2295,3735},{2362,3299},{3179,5239},{3208,3970},{3216,4674},{3218,4861},{3241,3881},{3244,3873},{3476,4317},{3486,4302},{3488,4309},{3555,3633},{3582,3656},{3584,3654},{3616,3884},{3617,3678},{3622,3898},{3623,3892},{3626,3681},{3635,3889},{3698,5044},{3724,5496},{3740,4002},{3751,3827},{3833,5550},{3872,4880},{3897,5267},{3913,5541},{3962,4668},{4127,4678},{4134,4691},{4153,4165},{4304,5493},{4513,5525},{4646,5312},{4857,5722},{5046,5180},{5250,5259},{5252,5270},{5253,5330},{5435,5734},{5694,5790}

X(5903) = reflection of X(I) in X(J) for these (I,J): (1,65), (5904,8)
X(5903) = anticomplement of X(3878)


X(5904) =  REFLECTION OF X(1) IN X(72)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5904) is the incenter of the triangle A*B*C* described at X(5905) and also the incenter of the triangle A*B*C* described at X(5906). Also, X(5904) = {(X(1),X(9)}-harmonic conjugate of X(5259), and X(5904) = {(X(1),X(72)}-harmonic conjugate of X(5692). (Randy Hutson, July 7, 2014)

X(5904) lies on these lines:
{1,6},{2,3678},{8,79},{10,3681},{20,2801},{35,63},{36,78},{38,386},{40,912},{43,3670},{46,200},{55,191},{56,3940},{58,976},{65,3679},{69,1930},{80,3436},{144,4294},{145,3878},{165,1071},{210,942},{281,1844},{329,1479},{354,3624},{382,517},{474,3337},{484,5687},{519,3869},{527,1770},{551,3889},{579,3949},{595,3938},{651,4347},{936,3338},{978,3953},{982,3216},{986,3293},{997,3984},{1046,3961},{1066,2318},{1125,3873},{1158,5537},{1282,2939},{1376,3336},{1482,5694},{1697,1858},{1699,5777},{1756,4073},{1759,3684},{2093,4882},{2340,4303},{2771,5541},{2774,4088},{2802,3621},{3057,3633},{3059,4312},{3149,5536},{3189,4302},{3218,4420},{3219,5248},{3241,3884},{3244,3877},{3419,3585},{3501,4006},{3579,3689},{3616,3881},{3617,3754},{3622,3892},{3623,3898},{3626,4084},{3635,3890},{3666,5312},{3697,3812},{3711,5221},{3730,3930},{3735,3780},{3740,4533},{3742,4539},{3746,3870},{3753,4662},{3831,4090},{3833,4547},{3875,4523},{3916,5010},{3919,4691},{4018,4668},{4188,4973},{4251,5282},{4292,5850},{4309,5698},{4388,4894},{4413,5708},{4423,5506},{4641,5266},{4658,5311},{4678,4757},{5270,5794},{5445,5552}

X(5904) = reflection of X(I) in X(J) for these (I,J): (1,72), (5903,8)
X(5904) = anticomplement of X(3874)


X(5905) =  ANTICOMPLEMENT OF X(63)

Barycentrics   cos B + cos C - cos A : cos C + cos A - cos B : cos A + cos B - cos C

Let A'B'C' be the orthic triangle, and let LA be the reflection of line B'C' in the internal bisector of angle A, and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The triangle A''B''C'' is homothetic to ABC at X(57), homothetic to the medial triangle at X(908), and to the anticomplementary triangle at X(5905). Let MA be the reflection of the line B'C' in the external bisector of angle A, and define MB and MC cyclically. Let A* = MB∩MC, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC at X(9), homothetic to the medial triangle at X(5249), and to the anticomplementary triangle at X(5905). (Randy Hutson, July 7, 2014)

X(5905) lies on these lines:
{2,7},{4,912},{6,3782},{8,79},{10,3951},{20,5758},{21,3487},{46,5552},{65,3436},{69,321},{72,377},{75,4886},{78,4190},{81,4644},{92,1947},{100,3474},{145,515},{149,152},{192,3151},{193,1839},{200,4312},{239,5813},{278,651},{281,445},{306,3729},{312,320},{345,3936},{355,4018},{388,3869},{390,3957},{442,3927},{443,3876},{481,3084},{482,3083},{497,3873},{516,3870},{518,1836},{529,2099},{535,3241},{537,4865},{554,5240},{938,5046},{940,4415},{942,2478},{944,5841},{958,3649},{993,3616},{1004,1260},{1046,5230},{1056,3877},{1058,3889},{1068,3157},{1071,5812},{1081,5239},{1086,4383},{1210,5187},{1211,4363},{1215,4655},{1329,5221},{1331,1754},{1351,2969},{1479,3874},{1532,2095},{1621,3475},{1707,3011},{1750,1998},{1770,3811},{1797,4080},{1851,3060},{2476,5714},{2550,3681},{2886,5852},{2975,3485},{2999,4862},{3091,5811},{3175,4851},{3210,4440},{3583,3894},{3585,3901},{3617,5815},{3663,5256},{3664,4656},{3715,3826},{3742,4679},{3751,3914},{3772,4641},{3816,4860},{3920,4307},{3925,5220},{3962,5794},{3970,3995},{4001,4054},{4187,5708},{4189,5703},{4293,4511},{4387,4966},{4416,5271},{4419,5712},{4438,4892},{4463,5800},{4666,5542},{4847,5850},{5086,5229},{5154,5704},{5289,5434}

X(5905) = isogonal conjugate of X(2164)
X(5905) = isotomic conjugate of X(2994)
X(5905) = anticomplement of X(63)


X(5906) =  ANTICOMPLEMENT OF X(255)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B cos2B + sin C cos2C - sin A cos2A

Let A'B'C' be the circumorthic triangle, and let LA be the reflection of line B'C' in the internal bisector of angle A, and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The triangle A''B''C'' is homothetic to ABC at X(3075) and homothetic to the anticomplementary triangle at X(5906). Let MA be the reflection of the line B'C' in the external bisector of angle A, and define MB and MC cyclically. Let A* = MB∩MC, and define B* and C* cyclically. The triangle A*B*C*' is homothetic to ABC at X(3074) and to homothetic to the anticomplementary triangle at X(5906). (Randy Hutson, July 7, 2014)

X(5906) lies on these lines:
{2,255},{8,79},{69,349},{78,1448},{651,5125},{860,3157},{962,2817},{1259,3936},{1788,2406},{3868,5081}


X(5907) =  COMPLEMENT OF X(185)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan B)(cos2C + cos2A) + (tan C)(cos2A + cos2B)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2B)[1 - cos B cos(C - A)] + (sin 2C)[1 - cos C cos(A - B)]
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b) , where h(a,b,c) = (b^2 + c^2 - a^2)[2a^8 - 3a^6(b^2 + c^2) - a^4(b^4 - 10b^2c^2 + c^4) + 3a^2(b^2 - c^2)^2(b^2 + c^2) - (b^2 - c^2)^4

Let A'B'C' be the half-altitude triangle. Let A'' be the trilinear pole, with respect to A'B'C', of the line BC, and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5). Let A* be the trilinear pole, with respect to A'B'C', of the line B''C'', and define B* and C* cyclically. The lines A'A*, B'B*, C'C* concur in X(5907). Also, X(5907) is the center of the conic described at X(5777) for P = X(69). In this case, triangle HAHBHC is perspective to ABC at X(64). (Randy Hutson, July 7, 2014)

X(5907) lies on the Burek-Hutson central cubic (K645) and these lines:
{2,185},{3,64},{4,69},{5,389},{10,2807},{20,3917},{30,1216},{40,3781},{51,3091},{52,381},{84,3784},{113,1209},{114,130},{140,5663},{141,2883},{143,3850},{155,578},{182,1181},{235,343},{373,5056},{378,1092},{394,1593},{546,1154},{550,5447},{568,3851},{916,942},{970,3149},{1071,2808},{1147,4550},{1204,5651},{1364,1935},{1568,1594},{2979,3146},{3060,3832},{3523,5650},{3545,3567},{3574,5133},{4260,5706},{5068,5640},{5777,5908}

X(5907) = reflection of X(389) in X(5)
X(5907) = complement of X(185)
X(5907) = X(4)-of-X(5)-Brocard-triangle


X(5908) =  INTERSECTION OF LINES X(1)X(3) AND X(4)X(189)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5908) is the center of the conic described at X(5777) for P = X(189). In this case, triangle HAHBHC is perspective to ABC at X(40). (Randy Hutson, July 7, 2014)

X(5908) lies on the Burek-Hutson central cubic (K645) and these lines: {1,3},{4,189},{5,5909},{222,1753},{282,2262},{971,1872},{1364,1887},{1535,5174},{5777,5907}

X(5908) = reflection of X(5909) in X(5)


X(5909) =  INTERSECTION OF LINES X(3)X(223) AND X(4)X(8)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5909) is the center of the conic described at X(5777) for P = X(329). In this case, triangle HAHBHC is perspective to ABC at X(3345). (Randy Hutson, July 7, 2014)

X(5909) lies on the Burek-Hutson central cubic (K645) and these lines: {3,223},{4,8},{5,5908},{389,942},{960,2817},{2262,5715},{2270,5709}

X(5909) = reflection of X(5908) in X(5)


X(5910) =  INTERSECTION OF LINES X(3)X(64) AND X(4)X(1032)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5910) is the center of the conic described at X(5777) for P = X(1032). In this case, triangle HAHBHC is perspective to ABC at X(1498). (Randy Hutson, July 7, 2014)

X(5910) lies on the Burek-Hutson central cubic (K645) and these lines: {3,64},{4,1032},{3079,5562}


X(5911) =  INTERSECTION OF LINES X(3)X(9) AND X(4)X(1034)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5911) is the center of the conic described at X(5777) for P = X(1034). In this case, triangle HAHBHC is perspective to ABC at X(1490). (Randy Hutson, July 7, 2014)

X(5911) lies on the Burek-Hutson central cubic (K645) and these lines: {3,9},{4,1034}


X(5912) =  EULER-PONCELET POINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a^10 - 4a^8(b^2 + c^2) - 3a^6(b^4 - 6b^2c^2 + c^4) + 2a^4(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - a^2(b^8 + c^8 - 11b^6c^2 - 11b^2c^6 + 18b^4c^4) - 3b^2c^2(b^2 - c^2)^2(b^2 + c^2)

X(5912) is the point QA-P2 center of quadrangle X(13)X(14)X(15)X(16); see Euler-Poncelet Point.

X(5912) is the point common to the nine-points circles of these 4 triangles: X(14)X(15)X(16), X(13)X(15)X(16), X(13)X(14)X(16), X(13)X(14)X(15); also, X(5912) is the center of the rectangular hyperbola that passes through the points X(13), X(14), X(15), X(16). (Randy Hutson, July 7, 2014)

Let O(13,15) be the circle with segment X(13)X(15) as diameter (and center X(396)), and let O(14,16) be the circle with segment X(14)X(16) as diameter (and center X(395)); then X(5912) is the radical trace of O(13,15) and O(14,16). (Randy Hutson, August 17, 2014)

X(5912) lies on these lines: {2,6},{98,843},{111,523}

X(5912) = reflection of X(I) in X(J) for these (I,J): (5913,230), (111,5914)


X(5913) =  GERGONNE-STEINER POINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3 a^4 b^2 + 2 a^2 b^4 - b^6 + 3 a^4 c^2 - 10 a^2 b^2 c^2 + b^4 c^2 + 2 a^2 c^4 + b^2 c^4 - c^6

X(5913) is the QA-P3 center of the quadrangle X(13)X(14)X(15)X(16); see Gergonne-Steiner Point.

X(5913) = inverse-in-{circumcircle, nine-point circle}-inverter of X(6); see X(5577) for the definition of inverter.

The {circumcircle, nine-point circle}-inverter is the orthopic circle of the Steiner inscribed ellipse; its center is X(2), its radius is [(a2 + b2 + c2)/18]1/2, and the powers of A,B,C with respect to this circle are (-a2 + b2 + c2)/6, (a2 - b2 + c2)/6, (a2 + b2 - c2)/6. (Peter Moses, July 16, 2014)T

X(5913) lies on these lines:
{2,6},{23,2079},{30,111},{112,468},{115,858},{403,1560},{843,1302},{1499,1513},{2030,5642}

X(5913) = reflection of X(5912) in X(230)
X(5913) = isogonal conjugate of X(6096)
X(5913) = complement of X(5971)


X(5914) =  PARABOLA AXES CROSSPOINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4 a^10 - 8 a^8 b^2 - 3 a^6 b^4 + 4 a^4 b^6 - 5 a^2 b^8 - 8 a^8 c^2 + 30 a^6 b^2 c^2 - 12 a^4 b^4 c^2 + 25 a^2 b^6 c^2 - 3 b^8 c^2 - 3 a^6 c^4 - 12 a^4 b^2 c^4 - 36 a^2 b^4 c^4 + 3 b^6 c^4 + 4 a^4 c^6 + 25 a^2 b^2 c^6 + 3 b^4 c^6 - 5 a^2 c^8 - 3 b^2 c^8

X(5914) is the QA-P6 center of the quadrangle X(13)X(14)X(15)X(16); see Parabola Axes Crosspoint.

X(5914) lies on these lines: {30,115},{111,523}

X(5914) = midpoint of X(111) and X(5912)


X(5915) =  INSCRIBED SQUARE AXES CROSSPOINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -4 a^12 + 8 a^10 b^2 - 7 a^8 b^4 + 11 a^6 b^6 - 13 a^4 b^8 + 5 a^2 b^10 + 8 a^10 c^2 - 10 a^8 b^2 c^2 - 3 a^6 b^4 c^2 + 17 a^4 b^6 c^2 - 7 a^2 b^8 c^2 + 3 b^10 c^2 - 7 a^8 c^4 - 3 a^6 b^2 c^4 - 12 a^4 b^4 c^4 + 2 a^2 b^6 c^4 - 12 b^8 c^4 + 11 a^6 c^6 + 17 a^4 b^2 c^6 + 2 a^2 b^4 c^6 + 18 b^6 c^6 - 13 a^4 c^8 - 7 a^2 b^2 c^8 - 12 b^4 c^8 + 5 a^2 c^10 + 3 b^2 c^10

X(5915) is the QA-P23 center of the quadrangle X(13)X(14)X(15)X(16); see Inscribed Square Axes Crosspoint

X(5915) is the centroid of the trapezoid X(2378)X(5916)X(2379)X(5917), which is similar to and orthogonal to the trapezoid X(13)X(15)X(14)X(16), with similitude center X(111); see X(5916). (Randy Hutson, July 7, 2014)

X(5915) lies on these lines: {30,115},{98,843},{111,477}


X(5916) =  INTERSECTION OF LINES X(14)X(530) AND X(98)X(2379)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5916) = isogonal conjugate of X(15) with respect to the triangle X(13)X(14)X(16)
X(5917) = isogonal conjugate of X(16) with respect to the triangle X(13)X(14)X(15)
X(2378) = isogonal conjugate of X(13) with respect to the triangle X(14)X(15)X(16)
X(2379) = isogonal conjugate of X(14) with respect to the triangle X(13)X(15)X(16); see X(5915).

X(5916) lies on these lines: {14,530},{98,2379},{523,2378}


X(5917) =  INTERSECTION OF LINES X(13)X(531) AND X(98)X(2378)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5917) = isogonal conjugate of X(16) with respect to the triangle X(13)X(14)X(15); see X(5915).

X(5917) lies on these lines: {13,531},{98,2378},{523,2379}


X(5918) =  CENTROID OF HUTSON-EXTOUCH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).

X(5918) lies on these lines:

X(5918) = reflection of X(I) in X(J) for these (I,J): (210,165), (5919,5731)


X(5919) =  CENTROID OF HUTSON-INTOUCH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The Hutson-intouch and Hutson-extouch triangles are defined at X(5731).

X(5919) lies on these lines: (pending)

X(5919) = reflection of X(I) in X(J) for these (I,J): (354,1), (5918,5731)


X(5920) =  ORTHOLOGY CENTER OF THE HUTSON-EXTOUCH AND HUTSON-INTOUCH TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).

X(5920) lies on these lines:


X(5921) =  DARBOUX IMAGE OF X(20)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Darboux images are discussed at X(5881).

X(5921) lies on these lines:
{2,98},{3,3620},{4,193},{6,3091},{20,64},{68,3089},{141,3523},{153,5848},{381,1353},{511,3146},{524,3543},{546,5093},{611,5261},{613,5274},{962,5847},{1992,3839},{2888,5596},{3090,5050},{3580,4232},{3618,5056},{3619,5085},{3818,3832}

X(5921) = reflection of X(20) in X(69)


X(5922) =  DARBOUX IMAGE OF X(64)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Darboux images are discussed at X(5881).

X(5922) lies on these lines: {20,64},{122,1073},{154,459}

X(5922) = reflection of X(64) in X(253)


X(5923) =  DARBOUX IMAGE OF X(84)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Darboux images are discussed at X(5881).

X(5923) lies on these lines: {8,20},{282,5514},{1256,1837}

X(5923) = reflection of X(84) in X(189)


X(5924) =  DARBOUX IMAGE OF X(1490)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Darboux images are discussed at X(5881).

X(5924) lies on these lines:
{4,2093},{9,119},{20,78},{57,5715},{84,5812},{226,2096},{2095,5735},{2800,3586}

X(5924) = reflection of X(1490) in X(329)


X(5925) =  DARBOUX IMAGE OF X(1498)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Darboux images are discussed at X(5881).

X(5925) lies on these lines:
{3,113},{4,1192},{20,394},{30,64},{154,550},{221,4302},{376,2883},{382,1853},{599,2892},{1503,3529},{1514,3147},{1620,3542},{1770,1854},{2192,4299},{3146,3580}

X(5925) = reflection of X(1498) in X(20)


X(5926) =  CENTER OF INVERSE-IN-CIRCUMCIRCLE OF LINE X(2)X(6))

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 - c2)(2a6 - 4a4b2 - 4a4c2 + 2a2b4 + 2a2c4 + a2b2c2 - b4c2 - b2c4)

The inverse-in-circumcircle of a line is a circle; if the line does not pass through X(3), then the radius of the circle is finite. The appearance if (I,J) in the following list means that X(I) is on the line X(2)X(6) and that X(J) is the inverse of X(I): (2,23), (6,187), (69, 5866), (81,5867), (86,5937), (141,5938), (183, 5939), (193,5940), (230,5941), (352,353), (524,3), (5108,669).

X(5926) lies on these lines: {3,669},{24,2501}

X(5926) = midpoint of X(3) and X(669)


X(5927) =  CENTROID OF 2nd EXTOUCH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

The classical extouch triangle is regarded as the 1st extouch triangle. The 2nd, 3rd, 4th, and 5th extouch triangles are defined by Randy Hutson (July 11, 2014) as follows. Let AA, AB, AC be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define BB, BC, BA and CC, CA, CB cyclically.

Let A1 = AA, B1 = BB, C1 = CC; 1st extouch triangle = A1B1C1

Let A2 = BCBA∩CACB, and define B2 and C2 cyclically; 2nd extouch triangle = A2B2C2

Let A3 = CAAC∩ABBA, and define B3 and C3 cyclically; 3rd extouch triangle = A3B3C3

Let A4 = BCCA∩BACB, and define B4 and C4 cyclically; 4th extouch triangle = A4B4C4

Let D1 = BBBC∩CCCA, and define D2 and D3 cyclically. Let E1 = BBBA∩CCCB, and define E2 and E3 cyclically. Define A5 = D2E2∩D3E3, and define B5 and C5 cyclically; 5th extouch triangle = A5B5C5

Barycentric coordinates for A-vertices of the the five triangles:
A1 = 0 : a - b + c : a + b - c
A2 = 2a(b + c) : - a2 - b2 + c2 : - a2 + b2 - c2
A3 = 2a(b + c)(a - b + c)(a + b - c) : (a + b + c)(a - b - c)(a2 + b2 - c2) : (a + b + c)(a - b - c)(a2 - b2 + c2)
A4 = 2a(b + c)(a + b + c) : (a - b - c)(a2 - b2 + c2) : (a - b - c)(a2 + b2 - c2)
A5 = 2a(b + c)(a - b + c)(a + b - c) : (a - b - c)(a + b - c)[b2 + (a + c)2] : (a - b - c)(a - b + c)[c2 + (a + b)2]

A2B2C2 is perspective to ABC and A3B3C3 at X(4).
A2B2C2 is homothetic to the excentral triangle at X(9).
A2B2C2 is homothetic to the intouch triangle at X(226).
A2B2C2 is perspective to the extouch triangle and (extraversion triangle of X(65)) at X(72).
A2B2C2 is perspective to the anticevian triangle of X(8) at X(329).
A2B2C2 is homothetic to the hexyl triangle at X(1490).
A2B2C2 is perspective to the Feuerbach triangle at X(442).
A2B2C2 is perspective to A4B4C4 at X(5928).
A2B2C2 is homothetic to the inner Hutson triangle at X(5934).
A2B2C2 is homothetic to the outer Hutson triangle at X(5935).
A2B2C2 is homothetic to the 2nd circumperp triangle at X(405).
A2B2C2 is homothetic to the inverse-in-incircle triangle (see X(5571) at X(5728).
A2B2C2 is homothetic to the Hutson-intouch triangle at X(950).
A2B2C2 is homothetic to the Hutson-extouch triangle at X(442).

In the following list, the appearance of (I,J) means that (X(I) of the 2nd extouch triangle) = X(J):
(3,4), (4,72), (5,5777), (6,9), (25,329), (26,5812), (54, 442), (184,226), (185,950), (195,3651), (578, 10), (647,1635), (1181,1)

A3B3C3 is perspective to the intouch triangle and (extraversion triangle of X(65)) at X(1439).
A3B3C3 is perspective to the extouch triangle and A5B5C5 at X(5930).
A3B3C3 is perspective to A4B4C4 at X(5929).
A3B3C3 is perspective to the anticevian triangle of X(7) at X(5932).

A4B4C4 is perspective to ABC at X(69).
A4B4C4 is perspective to the intouch triangle and A5B5C5 at X(65).
A4B4C4 is perspective to the anticevian triangle of X(7) at X(5933).

A5B5C5 is perspective to ABC at X(388).
A5B5C5 is perspective to the anticevian triangle of X(7) at X(8).

X(5927) is also the centroid of the triangle formed by the polars of the incenter with respect to the excircles. (Randy Hutson, July 11, 2014)

X(5927) lies on these lines: (pending)


X(5928) =  PERSPECTOR OF 2nd AND 4th EXTOUCH TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5928) is also the perspector of the 2nd (and 4th) extouch triangle and the polar triangle of the Yiu conic; see X(5927).

X(5928) lies on these lines: (pending)


X(5929) =  PERSPECTOR OF 3rd AND 4th EXTOUCH TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

See X(5927).

X(5929) lies on these lines: (pending)


X(5930) =  PERSPECTOR OF 1st, 3rd AND 5th EXTOUCH TRIANGLES

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)(b + c)/(b + c - a)

See X(5927).

X(5930) lies on these lines: (pending)

X(5930) = isotomic conjugate of X(5731)
X(5930) = X(8)-Ceva conjugate of X(65)


X(5931) =  ISOTOMIC CONJUGATE OF X(5730)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(c + a)(a + b)/(3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)

X(5931) is the trilinear pole of the perspectrix of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). (Randy Hutson, July 11, 2014) See X(5927).

X(5931) lies on these lines: (pending)


X(5932) =  PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND ANTICEVIAN TRIANGLE OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sec^2(A/2) [- a^2 cos B cos C sec^2(A/2) + b^2 cos C cos A sec^2(B/2) + c^2 cos A cos B sec^2(C/2)]
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [a^6 - 2a^5(b + c) - a^4(b + c)^2 + 4a^3(b^3 + c^3) - a^2(b^2 - c^2)^2 - 2a(b - c)^2(b + c)(b^2 + c^2) + (b - c)^2(b + c)^4]/(b + c -a)

X(5932) is the perspector of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). Also, X(5932) is the perspector of ABC and the pedal triangle of X(3182), as well as the perspector of ABC and the triangle obtained by reflecting the pedal triangle of X(223) in X(223). (Randy Hutson, July 11, 2014)

X(5931) lies on the Lucas cubic and these lines: (pending)

X(5932) = isotomic conjugate of X(1034)
X(5932) = anticomplement of X(282)
X(5932) = X(69)-Ceva conjugate of X(7)


X(5933) =  PERSPECTOR OF 4th EXTOUCH TRIANGLE AND ANTICEVIAN TRIANGLE OF X(7)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5932) lies on these lines: (pending)


X(5934) =  PERSPECTOR OF 2nd EXTOUCH TRIANGLE AND INNER HUTSON TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5934) = X(222)-of-2nd-extouch-triangle. (Inner Hutson triangle is defined at X(363).)

X(5934) lies on these lines: (pending)


X(5935) =  PERSPECTOR OF 2nd EXTOUCH TRIANGLE AND OUTER HUTSON TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5935) = X(219)-of-2nd-extouch-triangle. (Outer Hutson triangle is defined at X(363).)

X(5935) lies on these lines: (pending)


X(5936) =  ISOTOMIC CONJUGATE OF X(3616)

Barycentrics   1/(3a + b + c) : 1/(a + 3b + cf) : 1/(a + b + 3c)

X(5936) = trilinear pole of the line X(514)X(1635) (which is the Lemoine axis of the 2nd extouch triangle).

X(5936) lies on these lines: (pending)


X(5937) =  INVERSE-IN-CIRCUMCIRCLE OF X(86)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5937) lies on these lines: {3,86},{669,4367}


X(5938) =  INVERSE-IN-CIRCUMCIRCLE OF X(141)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5938) lies on these lines: {3,66},{25,5523},{353,3148},{525,669},{755,2715}


X(5939) =  INVERSE-IN-CIRCUMCIRCLE OF X(183)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5939) = X(187)-of-circummedial triangle. Let A'B'C' be the triangle of which ABC is the 1st Brocard triangle; here called the 1st anti-Brocard triangle, with circumcircle called the anti-Brocard circle; then X(5939) = X(187)-of-A'B'C', and X(5939) = inverse-in-anti-Brocard-circle of X(99). (Randy Hutson, July 18, 2014)

Barycentrics for the vertices of the 1st anti-Brocard triangle are as follows (Peter Moses, August 21, 2014):

A' = a4 - b2c2 : c4 - a2b2 : b4 - a2c2
B' = c4 - b2a2 : b4 - c2a2 : a4 - b2c2
C' = b4 - c2a2 : a4 - c2b2 : c4 - a2b2

For more properties of A'B'C', see X(5976).

X(5939) lies on these lines:
{2,353},{3,76},{147,1007},{187,543},{325,542},{385,5104},{669,804},{671,3972},{2023,3329}


X(5940) =  INVERSE-IN-CIRCUMCIRCLE OF X(193)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

X(5940) lies on these lines: {3,193},{353,5585}


X(5941) =  INVERSE-IN-CIRCUMCIRCLE OF X(230)

Barycentrics   f(a,b,c) : f(b,c,a) : f(a,b,c), where f(a,b,c) = (pending)

X(5941) lies on these lines: {3,230},{25,669}


X(5942) =  ANTICOMPLEMENT OF X(77)

Barycentrics   f(a,b,c) : f(b,c,a) : f(a,b,c), where f(a,b,c) = b/(1 + sec B) + c/(1 + sec C) - a/(1 + sec A)
Barycentrics   g(A,B,C) : g(B,C,A) : g(A,B,C), where g(A,B,C) = (1 - cos B) cot B + (1 - cos C) cot C - (1 - cos A) cot A
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a4 + b4 + c4 - 2a3(b + c) -2a(b - c)2(b + c) + 2bc(b2 + c2 - 3bc) + 2a2(b2 + c2 - bc) (Randy Hutson, July 18, 2014)

X(5942) lies on these lines:
{2,77},{7,4858},{8,144},{63,3686},{69,1229},{92,1947},{281,651},{329,2893},{894,5554},{1654,3152},{3416,3436}


X(5943) =  CENTROID OF HALF-ALTITUDE TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b4 + c4 - a2b2 - a2c2 - 4b2c2)
X(5943) = 2X(5) + X(389)      (barycentrics and combo, Peter Moses, July 15, 2014)

The locus of the centroid of the pedal triangle of P as P varies around the nine-point circle is an ellipse with center X(5943). Also, X(5943) is the centroid of the pedal triangle of X(5), as well as the centroid of the 6 points of intersection of the nine-point circle and the sidelines of ABC. (Randy Hutson, July 18, 2014). The ellipse is here named the Hutson centroidal ellipse.

If you have The Geometer's Sketchpad, you can view X(5943), with the Hutson centroidal ellipse.

X(5943) lies on these lines:
{2,51},{5,389},{6,1196},{22,5092},{23,5643},{25,182},{30,5892}

X(5943) = midpoint of X(I) and X(J) for these (I,J): (2,51), (5, 5946)
X(5943) = complement of X(3917)
X(5943) = {X(51),X(373)}-isoconjugate of X(2)


X(5944) =  CENTER OF HUNG CIRCLE

Trilinears     f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[S2 + 3S2A - 2SA(3Sω - 5R2)]

Let d be a line tangent to the nine-point circle of a triangle ABC. Let DA be the reflection of d in line BC, and define DB and DC cyclically. Let XYZ be the triangle formed by the lines DA, DB, DC. The envelope of the circumcircle of the variable triangle XYZ is a circle. (Tran Quang Hung, ADGEOM #1387, July 9, 2014). The circle is here named the Hung circle.

The diameter of the Hung circle is the segment X(3)X(1614), so that X(5944) is the midpoint of this segment. The radius of the circle is (R/2)(3 + k)/(1 + k), where k = 2(cos 2A + cos 2B + cos 2C). The lines AX, BY, CZ concur in a point Q = Q(P) on the circumcircle of ABC. The appearance of (i,j) in the following list means that X(j) = Q(X(i)): (11,953), (113,477), (114,2698), (115,2698), (116,2724), (117,2734), (118,2724), (119,953), (124,2734), (125,477), (1312,74), (1313,74), (2039,98), (2040,98). (César Lozada, ADGEOM #1388 et al, July 9, 2014)

The triangles XYZ form a family of similar triangles, and Q is the incenter of XYZ. If P = p : q : r is a point on the nine-point circle, then

Q = Q(P) = a2/[p(v + w) - u(q + r)] : b2/[q(w + u) - v(r + p)] : c2/[r(u + v) - w(p + q)],

where u : v : w = X(5). (Peter Moses, August 1, 2014)

If ABC is acute, then Q is the incenter of XYZ, and XYZ has the orientation opposite that of ABC. Let J = |OH|/R and σ = area(ABC). Maximal area(XYZ) = σ(J - 1)/(J + 1) occurs with P = X(1312) and minimal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313). If ABC is not acute, then Q is an excenter of XYZ, and XYZ has the same orientation as ABC. In this case, maximal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313) and minimal area(XYZ) = 0, which occurs when angle X(5)-to-P-to-X(4) is a right angle and XYZ is a one of the points of intersection of the circumcircle and the Hung circle. Also, there is a local maximum when P = X(1312), and in this case, area(XYZ) = σ(J - 1)/(J + 1). (Peter Moses, August 9, 2014)

Let A' be the reflection of X(5) in line BC, and define B' and C' cyclically. Let A'' be the circumcenter of triangle BCA', and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5944). (Randy Hutson, August 17, 2014)

If you have The Geometer's Sketchpad 5, you can view X(5944).

X(5944) lies on these lines:
{3,74},{24,5946},{49,1154},{52,1493},{54,143},{184,1658},{546,1495},{567,3518},{3146,3431}


X(5945) =  CENTER OF HOFSTADTER 0-ELLIPSE

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [(sin2A)/A][(sin2B)/B + (sin2C)/C - (sin2A)/A]

The family of Hofstadter ellipses are introduced at X(359) and further described at MathWorld. The ellipses are indexed as E(r) for 0 <= r <= 1, and E(1 - r) = E(r). Thus, the Hofstadter 0-ellipse and the Hofstadter 1-ellipse are identical. (Submitted by Valery Nemychnikova, Moscow Chemical Lyceum, July 28, 2014.)

X(5945) = X(2)-Ceva conjugate of X(359)


X(5946) =  NINE-POINT CENTER OF ORTHOCENTROIDAL TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^2 b^4 c^2+3 b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4-4 b^4 c^4+3 a^2 c^6+3 b^2 c^6-c^8)
X(5946) = X(3) + 2X(143) = X(5) + 2X(389)      (Peter Moses, August 9, 2014)

X(5946) lies on these lines:
{2,568},{3,143},{4,3521},{5,389},{6,1511},{24,5944},{26,3796},{30,51},{49,1199},{52,140},{185,546},{186,567},{373,547},{378,1112},{381,5640},{511,549},{550,5446},{569,973},{632,1216},{970,5428},{974,1539},{1147,1493},{1656,5889},{1995,5609},{2070,5012},{2781,5476},{2979,5054},{3628,5562}

X(5946) = midpoint of X(2) and X(568)
X(5946) = reflection of X(5) in X(5943)


X(5947) =  CENTROID OF FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3 a^7 b^2+3 a^6 b^3-9 a^5 b^4-9 a^4 b^5+9 a^3 b^6+9 a^2 b^7-3 a b^8-3 b^9+2 a^7 b c+5 a^6 b^2 c-6 a^5 b^3 c-17 a^4 b^4 c+2 a^3 b^5 c+15 a^2 b^6 c+2 a b^7 c-3 b^8 c+3 a^7 c^2+5 a^6 b c^2-2 a^5 b^2 c^2-14 a^4 b^3 c^2-18 a^3 b^4 c^2-2 a^2 b^5 c^2+16 a b^6 c^2+12 b^7 c^2+3 a^6 c^3-6 a^5 b c^3-14 a^4 b^2 c^3-26 a^3 b^3 c^3-22 a^2 b^4 c^3-2 a b^5 c^3+12 b^6 c^3-9 a^5 c^4-17 a^4 b c^4-18 a^3 b^2 c^4-22 a^2 b^3 c^4-26 a b^4 c^4-18 b^5 c^4-9 a^4 c^5+2 a^3 b c^5-2 a^2 b^2 c^5-2 a b^3 c^5-18 b^4 c^5+9 a^3 c^6+15 a^2 b c^6+16 a b^2 c^6+12 b^3 c^6+9 a^2 c^7+2 a b c^7+12 b^2 c^7-3 a c^8-3 b c^8-3 c^9      (Peter Moses, August 10, 2014)

X(5947) is the Feuerbach-isogonal conjugate of X(5949); i.e., the isogonal-conjugate-with-respect-to-Feuerbach-triangle of X(5949).

X(5947) lies on these lines: {5,5948}, {3614,5949}


X(5948) =  ORTHOCENTER OF FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^11 b^2+a^10 b^3-5 a^9 b^4-5 a^8 b^5+10 a^7 b^6+10 a^6 b^7-10 a^5 b^8-10 a^4 b^9+5 a^3 b^10+5 a^2 b^11-a b^12-b^13+2 a^11 b c+3 a^10 b^2 c-8 a^9 b^3 c-13 a^8 b^4 c+12 a^7 b^5 c+22 a^6 b^6 c-8 a^5 b^7 c-18 a^4 b^8 c+2 a^3 b^9 c+7 a^2 b^10 c-b^12 c+a^11 c^2+3 a^10 b c^2-2 a^9 b^2 c^2-14 a^8 b^3 c^2-5 a^7 b^4 c^2+17 a^6 b^5 c^2+18 a^5 b^6 c^2+2 a^4 b^7 c^2-18 a^3 b^8 c^2-14 a^2 b^9 c^2+6 a b^10 c^2+6 b^11 c^2+a^10 c^3-8 a^9 b c^3-14 a^8 b^2 c^3-6 a^7 b^3 c^3+11 a^6 b^4 c^3+22 a^5 b^5 c^3+18 a^4 b^6 c^3-8 a^3 b^7 c^3-22 a^2 b^8 c^3+6 b^10 c^3-5 a^9 c^4-13 a^8 b c^4-5 a^7 b^2 c^4+11 a^6 b^3 c^4+16 a^5 b^4 c^4+8 a^4 b^5 c^4+13 a^3 b^6 c^4+5 a^2 b^7 c^4-15 a b^8 c^4-15 b^9 c^4-5 a^8 c^5+12 a^7 b c^5+17 a^6 b^2 c^5+22 a^5 b^3 c^5+8 a^4 b^4 c^5+12 a^3 b^5 c^5+19 a^2 b^6 c^5-15 b^8 c^5+10 a^7 c^6+22 a^6 b c^6+18 a^5 b^2 c^6+18 a^4 b^3 c^6+13 a^3 b^4 c^6+19 a^2 b^5 c^6+20 a b^6 c^6+20 b^7 c^6+10 a^6 c^7-8 a^5 b c^7+2 a^4 b^2 c^7-8 a^3 b^3 c^7+5 a^2 b^4 c^7+20 b^6 c^7-10 a^5 c^8-18 a^4 b c^8-18 a^3 b^2 c^8-22 a^2 b^3 c^8-15 a b^4 c^8-15 b^5 c^8-10 a^4 c^9+2 a^3 b c^9-14 a^2 b^2 c^9-15 b^4 c^9+5 a^3 c^10+7 a^2 b c^10+6 a b^2 c^10+6 b^3 c^10+5 a^2 c^11+6 b^2 c^11-a c^12-b c^12-c^13      (Peter Moses, August 10, 2014)

X(5948) is the Feuerbach-isogonal conjugate of X(5) and also the anticomplement of X(5) with respect to the Feuerbach triangle. (Randy Hutson, August 5, 2014)

X(5948) lies on these lines: {5,5947}, {12,79}, {119,5953}


X(5949) =  SYMMEDIAN POINT OF FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)^2 (a^3+a^2 b-a b^2-b^3+a^2 c+3 a b c+b^2 c-a c^2+b c^2-c^3)      (Peter Moses, August 10, 2014)

X(5949) is the Feuerbach-isogonal conjugate of X(5947).

X(5949) lies on these lines:
{2,1029},{5,572},{6,2476},{9,46},{12,594},{37,115},{338,1441},{1030,2475},{1834,5725},{2908,3136},{3841,4047}


X(5950) =  X(74) OF FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (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+4 a^4 b c+a^3 b^2 c-2 a^2 b^3 c-2 a b^4 c-2 b^5 c-a^4 c^2+a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2-2 a^3 c^3-2 a^2 b c^3+a b^2 c^3+4 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6) (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-2 a^5 b c+a^3 b^3 c+a b^5 c-b^6 c+2 a^5 c^2+4 a^3 b^2 c^2+a^2 b^3 c^2-2 a b^4 c^2-3 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+3 b^4 c^3-4 a^3 c^4- 2 a b^2 c^4+3 b^3 c^4-a^2 c^5+a b c^5-3 b^2 c^5+2 a c^6-b c^6+c^7)      (Peter Moses, August 10, 2014)

X(5950) lies on the nine-point circle and these lines: {2,5951},{4,5606},{5,5952},{11,79}

X(5950) = reflection of X(5952) in X(5)
X(5950) = complement of X(5951)


X(5951) =  CEVAPOINT OF X(35) AND X(484)

Trilinears       1/(2E2 + 2F2 - 4EF + 2DE + 2DF + 2D - E - F - 1), where D = cos A, E = cos B, F= cos C         (Randy Hutson, August 17, 2014)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^6+2 a^5 b-a^4 b^2-4 a^3 b^3-a^2 b^4+2 a b^5+b^6-a^5 c+2 a^4 b c-a^3 b^2 c-a^2 b^3 c+2 a b^4 c-b^5 c-2 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2+2 a b^3 c^2-2 b^4 c^2+2 a^3 c^3-a^2 b c^3-a b^2 c^3+2 b^3 c^3+a^2 c^4-4 a b c^4+b^2 c^4-a c^5-b c^5) (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4-a b^5+2 a^5 c+2 a^4 b c+2 a^3 b^2 c-a^2 b^3 c-4 a b^4 c-b^5 c-a^4 c^2-a^3 b c^2+2 a^2 b^2 c^2-a b^3 c^2+b^4 c^2-4 a^3 c^3-a^2 b c^3+2 a b^2 c^3+2 b^3 c^3-a^2 c^4+2 a b c^4-2 b^2 c^4+2 a c^5-b c^5+c^6)     (Peter Moses, August 10, 2014)

X(5951) lies on the circumcircle and these lines: {2,5950},{3,5606},{4,5952},{100,3648},{110,3579}

X(5951) = reflection of X(5606) in X(3)
X(5951) = anticomplement of X(5950)
X(5951) = cevapoint of X(35) and X(484)


X(5952) =  X(110) OF FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)^2(a^3 + b^3 + c^3 - a^2b - a^2c - ab^2 - ac^2 - abc + 3b^2c + 3bc^2)(a^4 - b^4 - c^4 - b^4 - c^4 + 2a^3b + 2a^3c + a^2bc + 2a^3c + 2a^3c + 2a^3c + a^2bc + a^2bc - 2ab^3 - 2ac^3 - 2ac^3 - 2ac^3 - 2ac^3 - 3ab^2c - 3abc^2 + 2b^2c^2)      (Randy Hutson, August 5, 2014; confirmed by Peter Moses, August 10, 2014)

X(5952) lies on the nine-point circle and these lines: {2,5606},{4,5951},{5,5950},{119,3652}

X(5952) = reflection of X(5950) in X(5)
X(5950) = complement of X(5606)


X(5953) =  FEUERBACH ISOGONAL CONJUGATE OF X(12)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 b^2+2 a^7 b^3-2 a^6 b^4-6 a^5 b^5+6 a^3 b^7+2 a^2 b^8-2 a b^9-b^10-2 a^8 b c+2 a^7 b^2 c+6 a^6 b^3 c-8 a^5 b^4 c-12 a^4 b^5 c+6 a^3 b^6 c+10 a^2 b^7 c-2 b^9 c+a^8 c^2+2 a^7 b c^2+8 a^6 b^2 c^2-2 a^5 b^3 c^2-19 a^4 b^4 c^2-10 a^3 b^5 c^2+7 a^2 b^6 c^2+10 a b^7 c^2+3 b^8 c^2+2 a^7 c^3+6 a^6 b c^3-2 a^5 b^2 c^3-18 a^4 b^3 c^3-22 a^3 b^4 c^3-10 a^2 b^5 c^3+6 a b^6 c^3+8 b^7 c^3-2 a^6 c^4-8 a^5 b c^4-19 a^4 b^2 c^4-22 a^3 b^3 c^4-18 a^2 b^4 c^4-14 a b^5 c^4-2 b^6 c^4-6 a^5 c^5-12 a^4 b c^5-10 a^3 b^2 c^5-10 a^2 b^3 c^5-14 a b^4 c^5-12 b^5 c^5+6 a^3 b c^6+7 a^2 b^2 c^6+6 a b^3 c^6-2 b^4 c^6+6 a^3 c^7+10 a^2 b c^7+10 a b^2 c^7+8 b^3 c^7+2 a^2 c^8+3 b^2 c^8-2 a c^9-2 b c^9-c^10      (Peter Moses, August 10, 2014)

Let A'B'C' be the Feuerbach triangle. Let A'' be the isogonal conjugate of A with respect to A'B'C', and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5953). (Randy Hutson, August 5, 2014)

X(5953) lies on these lines: (pending)


X(5954) =  3rd HUTSON-FEUERBACH POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c)^2 (-a^6+a^5 b+a^4 b^2-a^2 b^4-a b^5+b^6+a^5 c-2 a^2 b^3 c+b^5 c+a^4 c^2-3 a^2 b^2 c^2+4 a b^3 c^2-b^4 c^2-2 a^2 b c^3+4 a b^2 c^3-2 b^3 c^3-a^2 c^4-b^2 c^4-a c^5+b c^5+c^6)      (Peter Moses, August 10, 2014)

Let A'B'C' be the Feuerbach triangle, L the line through A and X(11), and A'' = L∩B'C'; and define B'' and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(5954). See also X(3614) and X(3615). (Randy Hutson, August 5, 2014)

Continuing, let A* be the trilinear pole, with respect to A'B'C', of line BC, and define B* and C* cyclically. Let A** be the trilinear pole, with respect to A'B'C', of line B*C*, and define B** and C** cyclically. The lines A'A**, B'B**, C'C** concur in X(5954). (Randy Hutson, August 5, 2014)

X(5954) lies on these lines: {5,2607},{11,523},{12,59}


X(5955) =  CENTER OF INNER HUNG CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4+a^3 b+a b^3+b^4+a^3 c+2 a^2 b c+5 a b^2 c+2 b^3 c+5 a b c^2+2 b^2 c^2+a c^3+2 b c^3+c^4
X(5955) = r2*X(3) + (r2 - s2)X(10)      (barycentrics and combo, Peter Moses, August 9, 2014)

In the configuration for X(5213), the circle internally tangent to the circles (KA), (KB), (KC) is here named the inner Hung circle.

If you have The Geometer's Sketchpad, you can view X(5955).

X(5955) lies on these lines:
{2,3702},{3,10},{46,1211},{171,5814},{899,1245},{975,3704},{997,5835},{1010,5725},{1574,2092},{1698,3712},{2049,5530},{3454,5880},{3687,5711},{3695,5268},{3696,5292},{3927,4104},{3966,5264}


X(5956) =  CENTER OF OUTER HUNG CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^4 b^2+3 a^3 b^3+3 a^2 b^4+a b^5+3 a^3 b^2 c+3 a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+3 a^3 b c^2-2 a^2 b^2 c^2-8 a b^3 c^2-4 b^4 c^2+3 a^3 c^3+3 a^2 b c^3-8 a b^2 c^3-6 b^3 c^3+3 a^2 c^4-a b c^4-4 b^2 c^4+a c^5-b c^5)      (Peter Moses, August 9, 2014)

In the configuration for X(5213), the circle externally tangent to the circles (KA), (KB), (KC) is here named the outer Hung circle; see X(5955) and X(5213).

If you have The Geometer's Sketchpad, you can view X(5955), which includes X(5956).

X(5956) lies on these lines:
{2,3702},{171,1203},{386,1100},{404,593},{970,5213},{1575,5044}


X(5957) =  FEUERBACH ISOGONAL CONJUGATE OF X(11)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-b+c) (2 a^4-a^3 b-3 a^2 b^2+a b^3+b^4-a^3 c-4 a^2 b c-3 a^2 c^2-2 b^2 c^2+a c^3+c^4)      (Peter Moses, August 10, 2014)

For I = 5957, 5958, 5959, 5948, 5949, 5953, X(I) is the isogonal conjugate of a point with respect to the Feuerbach triangle, called a Feuerbach isogonal conjugate.

Follwing is the first barycentric of the Feuerbach isogonal conjugate of an arbitrary point p : q : r. This lengthy function of a,b,c,p,q,r (degree 11 in a,b,c and degree 2 in p,q,r) can be cut-and-pasted into Mathematica. (Peter Moses, August 10, 2014)

(a^9 b^2+a^8 b^3-4 a^7 b^4-4 a^6 b^5+6 a^5 b^6+6 a^4 b^7-4 a^3 b^8-4 a^2 b^9+a b^10+b^11+2 a^9 b c+3 a^8 b^2 c-6 a^7 b^3 c-12 a^6 b^4 c+4 a^5 b^5 c+16 a^4 b^6 c+2 a^3 b^7 c-8 a^2 b^8 c-2 a b^9 c+b^10 c+a^9 c^2+3 a^8 b c^2-8 a^7 b^2 c^2-16 a^6 b^3 c^2+3 a^5 b^4 c^2+15 a^4 b^5 c^2+9 a^3 b^6 c^2+a^2 b^7 c^2-5 a b^8 c^2-3 b^9 c^2+a^8 c^3-6 a^7 b c^3-16 a^6 b^2 c^3+2 a^5 b^3 c^3+23 a^4 b^4 c^3+14 a^3 b^5 c^3+5 a^2 b^6 c^3-3 b^8 c^3-4 a^7 c^4-12 a^6 b c^4+3 a^5 b^2 c^4+23 a^4 b^3 c^4+18 a^3 b^4 c^4+6 a^2 b^5 c^4+4 a b^6 c^4+2 b^7 c^4-4 a^6 c^5+4 a^5 b c^5+15 a^4 b^2 c^5+14 a^3 b^3 c^5+6 a^2 b^4 c^5+4 a b^5 c^5+2 b^6 c^5+6 a^5 c^6+16 a^4 b c^6+9 a^3 b^2 c^6+5 a^2 b^3 c^6+4 a b^4 c^6+2 b^5 c^6+6 a^4 c^7+2 a^3 b c^7+a^2 b^2 c^7+2 b^4 c^7-4 a^3 c^8-8 a^2 b c^8-5 a b^2 c^8-3 b^3 c^8-4 a^2 c^9-2 a b c^9-3 b^2 c^9+a c^10+b c^10+c^11) p^2+2 c (2 a^8 b^2+a^7 b^3-6 a^6 b^4-3 a^5 b^5+6 a^4 b^6+3 a^3 b^7-2 a^2 b^8-a b^9-2 a^8 b c+a^7 b^2 c+a^6 b^3 c-8 a^5 b^4 c-2 a^4 b^5 c+8 a^3 b^6 c+4 a^2 b^7 c-a b^8 c-b^9 c+a^7 b c^2+5 a^6 b^2 c^2-3 a^5 b^3 c^2-12 a^4 b^4 c^2-a^3 b^5 c^2+8 a^2 b^6 c^2+3 a b^7 c^2-b^8 c^2+a^7 c^3+7 a^6 b c^3+3 a^5 b^2 c^3-9 a^4 b^3 c^3-11 a^3 b^4 c^3-a^2 b^5 c^3+6 a b^6 c^3+4 b^7 c^3+a^6 c^4-2 a^5 b c^4-9 a^4 b^2 c^4-11 a^3 b^3 c^4-7 a^2 b^4 c^4-a b^5 c^4+4 b^6 c^4-3 a^5 c^5-11 a^4 b c^5-10 a^3 b^2 c^5-10 a^2 b^3 c^5-10 a b^4 c^5-6 b^5 c^5-3 a^4 c^6-a^3 b c^6-2 a^2 b^2 c^6-3 a b^3 c^6-6 b^4 c^6+3 a^3 c^7+7 a^2 b c^7+6 a b^2 c^7+4 b^3 c^7+3 a^2 c^8+2 a b c^8+4 b^2 c^8-a c^9-b c^9-c^10) p q-(b-c) (a+b+c) (a^8 b-4 a^6 b^3+6 a^4 b^5-4 a^2 b^7+b^9-a^8 c+4 a^4 b^4 c-4 a^2 b^6 c+b^8 c+2 a^6 b c^2-4 a^5 b^2 c^2-a^4 b^3 c^2+6 a^3 b^4 c^2+3 a^2 b^5 c^2-2 a b^6 c^2-4 b^7 c^2+2 a^6 c^3-4 a^5 b c^3+3 a^4 b^2 c^3+10 a^3 b^3 c^3+5 a^2 b^4 c^3-2 a b^5 c^3-4 b^6 c^3+4 a^4 b c^4+6 a^3 b^2 c^4+5 a^2 b^3 c^4+4 a b^4 c^4+6 b^5 c^4+2 a^3 b c^5+a^2 b^2 c^5+4 a b^3 c^5+6 b^4 c^5-4 a^2 b c^6-2 a b^2 c^6-4 b^3 c^6-2 a^2 c^7-2 a b c^7-4 b^2 c^7+b c^8+c^9) q^2-2 b (-a^7 b^3-a^6 b^4+3 a^5 b^5+3 a^4 b^6-3 a^3 b^7-3 a^2 b^8+a b^9+b^10+2 a^8 b c-a^7 b^2 c-7 a^6 b^3 c+2 a^5 b^4 c+11 a^4 b^5 c+a^3 b^6 c-7 a^2 b^7 c-2 a b^8 c+b^9 c-2 a^8 c^2-a^7 b c^2-5 a^6 b^2 c^2-3 a^5 b^3 c^2+9 a^4 b^4 c^2+10 a^3 b^5 c^2+2 a^2 b^6 c^2-6 a b^7 c^2-4 b^8 c^2-a^7 c^3-a^6 b c^3+3 a^5 b^2 c^3+9 a^4 b^3 c^3+11 a^3 b^4 c^3+10 a^2 b^5 c^3+3 a b^6 c^3-4 b^7 c^3+6 a^6 c^4+8 a^5 b c^4+12 a^4 b^2 c^4+11 a^3 b^3 c^4+7 a^2 b^4 c^4+10 a b^5 c^4+6 b^6 c^4+3 a^5 c^5+2 a^4 b c^5+a^3 b^2 c^5+a^2 b^3 c^5+a b^4 c^5+6 b^5 c^5-6 a^4 c^6-8 a^3 b c^6-8 a^2 b^2 c^6-6 a b^3 c^6-4 b^4 c^6-3 a^3 c^7-4 a^2 b c^7-3 a b^2 c^7-4 b^3 c^7+2 a^2 c^8+a b c^8+b^2 c^8+a c^9+b c^9) p r+2 a (a-b) (a-c) (b-c)^2 (a+b+c)^2 (a^4-2 a^2 b^2+b^4-3 a^2 b c+a b^2 c-2 a^2 c^2+a b c^2-2 b^2 c^2+c^4) q r-(b-c) (a+b+c) (a^8 b-2 a^6 b^3+2 a^2 b^7-b^9-a^8 c-2 a^6 b^2 c+4 a^5 b^3 c-4 a^4 b^4 c-2 a^3 b^5 c+4 a^2 b^6 c+2 a b^7 c-b^8 c+4 a^5 b^2 c^2-3 a^4 b^3 c^2-6 a^3 b^4 c^2-a^2 b^5 c^2+2 a b^6 c^2+4 b^7 c^2+4 a^6 c^3+a^4 b^2 c^3-10 a^3 b^3 c^3-5 a^2 b^4 c^3-4 a b^5 c^3+4 b^6 c^3-4 a^4 b c^4-6 a^3 b^2 c^4-5 a^2 b^3 c^4-4 a b^4 c^4-6 b^5 c^4-6 a^4 c^5-3 a^2 b^2 c^5+2 a b^3 c^5-6 b^4 c^5+4 a^2 b c^6+2 a b^2 c^6+4 b^3 c^6+4 a^2 c^7+4 b^2 c^7-b c^8-c^9) r^2

X(5957) lies on the line at infinity and these lines: (pending)


X(5958) =  FEUERBACH ISOGONAL CONJUGATE OF X(115)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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^3 b c-4 a^2 b^2 c-a b^3 c+b^4 c-2 a^3 c^2-4 a^2 b c^2-3 a b^2 c^2-b^3 c^2-2 a^2 c^3-a b c^3-b^2 c^3+a c^4+b c^4+c^5)      (Peter Moses, August 10, 2014)

X(5958) lies on the line at infinity and these lines: (pending)


X(5959) =  FEUERBACH ISOGONAL CONJUGATE OF X(125)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (-a^7+2 a^5 b^2-a^3 b^4+2 a^5 b c-3 a^3 b^3 c-a^2 b^4 c+a b^5 c+b^6 c+2 a^5 c^2-5 a^3 b^2 c^2-3 a^2 b^3 c^2+b^5 c^2-3 a^3 b c^3-3 a^2 b^2 c^3-2 a b^3 c^3-2 b^4 c^3-a^3 c^4-a^2 b c^4-2 b^3 c^4+a b c^5+b^2 c^5+b c^6)      (Peter Moses, August 10, 2014)

X(5959) lies on the line at infinity and these lines: (pending)


X(5960) =  X(1) OF THE FEUERBACH TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)(-(b-c)^2 (a+b+c) Sqrt[(a^3-a^2 b-a b^2+b^3+a^2 c-3 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c-3 a b c-b^2 c-a c^2+b c^2+c^3)]+ (a+b) (a-b+c) (b+c) Sqrt[(-a^3-a^2 b+a b^2+b^3-a^2 c-3 a b c-b^2 c+a c^2-b c^2+c^3) (a^3+a^2 b-a b^2-b^3-a^2 c-3 a b c-b^2 c-a c^2+b c^2+c^3)]+(a+b-c) (a+c) (b+c) Sqrt[(a^3+a^2 b-a b^2-b^3+a^2 c+3 a b c+b^2 c-a c^2+b c^2-c^3) (-a^3+a^2 b+a b^2-b^3-a^2 c+3 a b c-b^2 c+a c^2+b c^2+c^3)])      (Peter Moses, August 10, 2014)


X(5961) =  INVERSE-IN-CIRCUMCIRCLE OF X(265)

Trilinears      f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 4A)/sin(3A)

X(5961) is the Hofstadter 4 point and the antigonal image of X(5964), in accord with the following conjecture and corollary: If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point; thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1, then the antigonal image of the Hofstadter r point is the Hofstadter -r point. (Randy Hutson, August 10, 2014)

The Hofstadter r point is defined at X(359), where further examples are given.

Let A'B'C' be the Kosnita triangle. The circumcircles of A'BC, B'CA, C'AB concur in X(5961). (Randy Hutson, August 10, 2014)

X(5961) lies on these lines: {3,125},{24,136},{94,96},{186,476},{1989,2079}

X(5961) = isogonal conjugate of X(5962)


X(5962) =  ANTIGONAL IMAGE OF X(186)

Trilinears      f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)/sin(4A)

X(5962) is the Hofstadter -3 point; see the conjecture at X(5961).

X(5962) lies on these lines:
{4,52},{30,925},{96,275},{128,186},{136,539},{250,403},{378,2351},{485,1322},{486,1321},{2165,3087}

X(5962) = isogonal conjugate of X(5961)
X(5962) = inverse-in-circumcircle of X(5963)
X(5962) = inverse-in-polar-circle of X(52)


X(5963) =  HOFSTADTER 5 POINT

Trilinears      f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 5A)/sin(4A)

X(5963) is the antigonal image of the Hofstadter -5 point; see the conjecture at X(5961).

X(5963) lies on these lines: (pending)

X(5963) = isogonal conjugate of X(5964)
X(5963) = inverse-in-circumcircle of X(5962)


X(5964) =  HOFSTADTER -4 POINT

Trilinears      f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 4A)/sin(5A)

X(5964) is the antigonal image of the Hofstadter 4 point, X(5961), and the inverse-in-circumcircle of the Hofstadter 6 point; see the conjecture at X(5961).

X(5963) lies on these lines: (pending)

X(5963) = isogonal conjugate of X(5963)


X(5965) =  NAPOLEON INFINITY POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a6 - b6 - c6 - a4b2 - a4c2 + 3a2b4 + 3a2c4 + b4c2 + b2c4

The Napoleon line is the line X(6)X(17), which passes through the Napoleon points, X(17) and X(18). X(5965) is the point in which the Napoleon line meets the line X(30)X(511) at infinity.

X(5965) lies on these lines:
{5,3629},{6,17},{30,511},{54,69},{114,385},{115,5111},{125,323},{140,3631},{141,575},{193,576},{381,5102},{599,5050},{1204,3098},{1351,3818},{1691,5477},{1992,5071},{2914,5095},{2930,5898},{3180,5617},{3181,5613},{3292,3580},{3630,5092},{3858,5480},{5093,5476}

X(5965) = isogonal conjugate of X(5966)


X(5966) =  ISOGONAL CONJUGATE OF NAPOLEON INFINITY POINT

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/(2a6 - b6 - c6 - a4b2 - a4c2 + 3a2b4 + 3a2c4 + b4c2 + b2c4)

Let O* denote the inverter of the circumcircle and nine-point circle, as defined at X(5577). X(5966) is the inverse-in-O* of X(137). (Randy Hutson, August 13, 2014)

X(5966) lies on the circumcircle and these lines:
{2,137},{5,99},{23,1291},{25,933},{51,110},{112,3199},{691,2070},{1141,2413}

X(5966) = isogonal conjugate of X(5965)


X(5967) =  INTERSECTION OF LINES X(2)X(98) AND X(6)X(523)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - 2a2)/(b4 + c4 - a2b2 - 2a2c2)

X(5967) lies on these lines:
{2,98},{4,685},{6,523},{23,5968},{69,4590},{248,5063},{263,2698},{468,1648},{511,4226},{524,5467},{868,1503},{1641,2434},{1992,2966},{2715,2770},{3266,3292}

X(5967) = isogonal conjugate of X(5968)


X(5968) =  INTERSECTION OF LINES X(2)X(523) AND X(6)X(110)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4 - a2b2 - 2a2c2) /(b2 + c2 - 2a2)

Let P denote the antipode of X(23) in the Parry circle. Then X(5968) is the vertex conjugate of X(23) and P, and X(5968) is the perspctor of the circumcevian triangle of X(23) and the circumcevian triangle of P. (Randy Hutson, August 13, 2014)

X(5968) lies on these lines:
{2,523},{3,691},{6,110},{22,3447},{23,5967},{25,250},{183,892},{232,4230},{262,381},{264,2970},{325,868},{511,2421},{956,5380},{2065,5050},{3613,5169}

X(5968) = isogonal conjugate of X(5967)
X(5968) = trilinear pole of the line X(511)X(3569)


X(5969) =  INTERSECTION OF LINES X(2)X(694) AND X(6)X(99)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4b2 + a4c2 - 2a2b4 - 2a2c4 + b4c2 + b2c4

X(5969) lies on these lines:
{2,694},{6,99},{30,511},{39,597},{69,148},{76,338},{98,1350},{111,5108},{114,5480},{115,141},{194,1992},{262,5503},{385,5104},{620,3589},{1469,3027},{1569,5052},{1843,5186},{2076,5152},{2502,5468},{3023,3056},{3029,4260},{3104,5463},{3105,5464},{3629,5477},{3934,5461},{4048,5028}

X(5969) = isogonal conjugate of X(5970)


X(5970) =  INTERSECTION OF LINES X(6)X(805) AND X(99)X(187)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/(a4b2 + a4c2 - 2a2b4 - 2a2c4 + b4c2 + b2c4)

X(5970) is the singular focus of the cubic K688. (Bernard Gibert, August 17, 2014)

The following were contributed by Randy Hutson, August 17, 2014:
X(5970) = Ψ(X(2), X(351))
X(5970) = Λ(X(76), X(338))
X(5970) = inverse-in-O(15,16) of X(99), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(5970) = inverse in O* of X(110), where O* is the circle {{X(1687, X(1688), PU(1), PU(2)}}
X(5970) is the point, other than X(99), in which the circumcircle meets the circle {{X(3), X(6), X(99)}}

X(5970) lies on the circumcircle and these lines:
{6,805},{32,691},{99,187},{110,1691},{111,669},{182,2709},{512,729},{1296,2080},{4027,4590}

X(5970) = isogonal conjugate of X(5969)


X(5971) =  INVERSE IN {CIRCUMCIRCLE, NINE-POINT CIRCLE}-INVERTER OF X(141)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a6 - a2b4 - a2c4 + 5a2b2c2 - 2b4c2 - 2b2c4

See X(5577) for the definition of inverter.

X(5971) = X(111)-of-1st-anti-Brocard-triangle. (Randy Hutson, August 20, 2014)

X(5971) lies on these lines: {2,6},{23,99},{111,538},{126,754},{675,2759},{1003,1383},{2071,2373}

X(5971) = anticomplement of X(5913)


X(5972) =  INVERSE IN {CIRCUMCIRCLE, NINE-POINT CIRCLE}-INVERTER OF X(147)

Barycentrics; f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a6 + b6 + c6 - 2a4b2 - 2a4c2 - a2b4 - a2c4 + 4a2b2c2 - b4c2 - b2c4
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a2 + c2 - b2)(c2 - a2)2 + (a2 + b2 - c2)(a2 - b2)2
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (sin 2B)sin2(A - C) + (sin 2C)sin2(A - B)
X(5972) = 3X(2) + X(110)      (barycentrics g, h and combo, Randy Hutson, August 20, 2014)

Let (A') be the circle centered at A and tangent to the Euler line, and define (B') and (C') cyclically; then X(5972) is the radical center of (A'), (B'), (C'). Let D = X(110), let Q be the cyclic quadrilateral ABCD, of which the centroid is X(5972). Let Q' be the quadrilateral formed by the centroids of the triangles ABC, BCD, CAD, ABD; then Q and Q' are homothetic at X(5972), which is the {X(2),X(110)}-harmonic conjugate of X(125) and also the inverse-in-Thomson-Gibert-Moses-hyperbola of X(125). Also, X(5972) = X(125)-of-medial-triangle = X(468)-of-1st-Brocard-triangle. (Randy Hutson, August 20, 2014)

The X(2)-Ceva conjugate of P, as P traces the Brocard axis, is a hyperbola, H, with center X(5972); H is the Jerabek hyperbola of the medial triangle, and it passes through X(i) for i = 3,5,6,113, 141, 206, 942, 960, 1147, 1209, 2883. (Randy Hutson, August 20, 2014)

X(5972) lies on these lines:
{2,98},{3,113},{5,1511},{6,5181},{67,3763},{69,5095},{74,631},{140,5663},{146,3523},{186,1568},{247,1316},{265,1656},{399,3526},{468,511},{541,549},{550,1539},{578,5504},{620,690},{632,5609},{858,1495},{895,3618},{1365,2607},{1503,5159},{1986,5562},{2482,5465},{2606,4092},{2781,3819},{2836,3848},{2854,3589},{2948,3624},{3024,5432},{3028,5433},{3066,5476},{3154,3233},{3292,3580},{3818,5094},{5054,5646},{5544,5648}

X(5972) = midpoint of X(110) and X(125)
X(5972) = complement of X(125)
X(5972) = inverse-in-{circumcircle, nine-point circle}-inverter of X(147)
X(5972) = X(186)-of-X(5)-Brocard-triangle


X(5973) =  PERSPECTOR OF THE HUNG-FEUERBACH TRIANGLE AND THE FEUERBACH TRIANGLE

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)^2 (-a^14 b-3 a^13 b^2-2 a^12 b^3+2 a^11 b^4+5 a^10 b^5+7 a^9 b^6+4 a^8 b^7-4 a^7 b^8-7 a^6 b^9-5 a^5 b^10-2 a^4 b^11+2 a^3 b^12+3 a^2 b^13+a b^14-a^14 c-2 a^13 b c-8 a^12 b^2 c-16 a^11 b^3 c-8 a^10 b^4 c+4 a^9 b^5 c+9 a^8 b^6 c+16 a^7 b^7 c+11 a^6 b^8 c-2 a^5 b^9 c-2 a^4 b^10 c-2 a^2 b^12 c+b^14 c-3 a^13 c^2-8 a^12 b c^2-8 a^11 b^2 c^2-14 a^10 b^3 c^2-8 a^9 b^4 c^2+25 a^8 b^5 c^2+20 a^7 b^6 c^2-8 a^6 b^7 c^2+a^5 b^8 c^2+6 a^4 b^9 c^2-4 a^3 b^10 c^2-2 a^2 b^11 c^2+2 a b^12 c^2+b^13 c^2-2 a^12 c^3-16 a^11 b c^3-14 a^10 b^2 c^3+2 a^9 b^3 c^3-2 a^8 b^4 c^3+4 a^7 b^5 c^3+36 a^6 b^6 c^3+16 a^5 b^7 c^3-18 a^4 b^8 c^3-4 a^3 b^9 c^3+2 a^2 b^10 c^3-2 a b^11 c^3-2 b^12 c^3+2 a^11 c^4-8 a^10 b c^4-8 a^9 b^2 c^4-2 a^8 b^3 c^4+28 a^7 b^4 c^4-16 a^5 b^6 c^4+20 a^4 b^7 c^4+2 a^3 b^8 c^4-8 a^2 b^9 c^4-8 a b^10 c^4-2 b^11 c^4+5 a^10 c^5+4 a^9 b c^5+25 a^8 b^2 c^5+4 a^7 b^3 c^5+28 a^5 b^5 c^5-4 a^4 b^6 c^5-12 a^3 b^7 c^5-a^2 b^8 c^5-b^10 c^5+7 a^9 c^6+9 a^8 b c^6+20 a^7 b^2 c^6+36 a^6 b^3 c^6-16 a^5 b^4 c^6-4 a^4 b^5 c^6+32 a^3 b^6 c^6+8 a^2 b^7 c^6+5 a b^8 c^6-b^9 c^6+4 a^8 c^7+16 a^7 b c^7-8 a^6 b^2 c^7+16 a^5 b^3 c^7+20 a^4 b^4 c^7-12 a^3 b^5 c^7+8 a^2 b^6 c^7+4 a b^7 c^7+4 b^8 c^7-4 a^7 c^8+11 a^6 b c^8+a^5 b^2 c^8-18 a^4 b^3 c^8+2 a^3 b^4 c^8-a^2 b^5 c^8+5 a b^6 c^8+4 b^7 c^8-7 a^6 c^9-2 a^5 b c^9+6 a^4 b^2 c^9-4 a^3 b^3 c^9-8 a^2 b^4 c^9-b^6 c^9-5 a^5 c^10-2 a^4 b c^10-4 a^3 b^2 c^10+2 a^2 b^3 c^10-8 a b^4 c^10-b^5 c^10-2 a^4 c^11-2 a^2 b^2 c^11-2 a b^3 c^11-2 b^4 c^11+2 a^3 c^12-2 a^2 b c^12+2 a b^2 c^12-2 b^3 c^12+3 a^2 c^13+b^2 c^13+a c^14+b c^14)    (Peter Moses, August 19, 2014)

Let FAFBFC be the Feuerbach triangle of a triangle ABC with excircles (IA), (IB), (IC). Let (KA) be the circle, other than the nine-point circle, which passes through FB and FC and is tangent to (IA), and let D be the touchpoint. Define (KB, (KC) and E, F cyclically. Let TA be the line through D tangent to (IA), and define TB and TC cyclically. The lines TA, TB, TC form a triangle [here named the Hung-Feuerbach triangle] that is perspective to FAFBFC, and X(5973) is the perspector. (Tran Quang Hung, ADGEOM, August 19, 2014) See also X(5974) and X(5975).


X(5974) =  CENTER OF THE HUNG-FEUERBACH CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^6+2 a^5 b+2 a^4 b^2+2 a^3 b^3-2 a b^5-b^6+2 a^5 c+6 a^4 b c+8 a^3 b^2 c-6 a b^4 c-2 b^5 c+2 a^4 c^2+8 a^3 b c^2+a^2 b^2 c^2-8 a b^3 c^2-4 b^4 c^2+2 a^3 c^3-8 a b^2 c^3-6 b^3 c^3-6 a b c^4-4 b^2 c^4-2 a c^5-2 b c^5-c^6)    (Peter Moses, August 19, 2014)

Continuing from X(5973), the Hung-Feuerbach circle is here defined as the circle (K) tangent to each of the following 6 circles: (KA, (KB, (KC), (IA), (IB), (IC), so that (K) is also tangent to the Apollonius circle.    (Tran Quang Hung and Peter Moses, ADGEOM, August 19, 2014)

X(5974) lies on these lines:
{72,171},{191,1045},{846,3931},{970,5975},{1054,5956},{1490,2629},{2959,5687},{3579,5524}}


X(5975) =  TOUCHPOINT HUNG-FEUERBACH CIRCLE AND APOLLONIUS CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b2 + c2 + ab + ac)2(a3 + b3+ c3 + abc - 2bc2)(a3 - b3 + c3 + abc - 2b2c)    (Peter Moses, August 19, 2014)

The Hung-Feuerbach circle is defined at X(5974).

X(5975) is the inverse-in-Speiker-radical-circle of X(5993). (Peter Moses, August 22, 2014)

If you have The Geometer's Sketchpad, you can view X(5975).

X(5975) lies on the Apollonius circle and this line: {970,5974}


X(5976) =  X(39)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a4 - b2c2)(b4 + c4 - a2b2 - a2c2)     (Peter Moses, August 21, 2014)

Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin B csc(B - 2ω) + sin C csc(C - 2ω)     (Randy Hutson, August 22, 2014)

The 1st anti-Brocard triangle is defined at X(5939). Suppose that P = p : q : r (barycentrics) is a point. Then P-of-1st-anti-Brocard-triangle is the point

T(P) = (a4 - b2c2)p + (c4 - a2b2)q + (b4 - a2c2)r : (b4 - c2a2)q + (a4 - b2c2)r + (c4 - b2a2)p : (c4 - a2b2)r + (b4 - c2a2)p + (a4 - c2b2)q

The inverse mapping T*(P), satisfying T*(T(P)) = T(T*(P)) = P, is the given by

T*(P) = a2p + c2q + b2r : c2p + b2q + a2r : b2p + a2q + c2r

The formulas for T and T*, and the following list, were contributed by Peter Moses, August 21, 2014. The appearance of (i,j) in the list means that X(i)-of-1st-anti-Brocard-triangle = X(j); i.e., X(j) = T(X(i)):

(1,1281),(2,2),(3,98),(4,147),(5,114),(6,99),(30,542),(69,148),(76,1916),(99,385),(110,23),(111,5971),(114,1513),(115,325),(125,858),(126,5913),(141,115),(182,3),(184,22),(187,5939),(287,401),(351,4108),(384,4027),(511,2782),(512,804),(513,2787),(514,2786),(515,2792),(516,2784),(517,2783),(518,2795),(519,2796),(520,2797),(521,2798),(522,2785),(523,690),(524,543),(525,2799),(530,531),(531,530),(538,5969),(542,30),(543,524),(574,183),(597,2482),(599,671),(620,230),(690,523),(694,3978),(804,512),(846,3757),(1054,5205),(1083,105),(1316,110),(1352,4),(1499,2793),(1503,2794),(1640,3268),(1691,5152),(1899,1370),(2549,69),(2782,511),(2783,517),(2784,516),(2785,522),(2786,514),(2787,513),(2788,3309),(2789,3667),(2792,515),(2793,1499),(2794,1503),(2795,518),(2796,519),(2797,520),(2798,521),(2799,525),(3094,76),(3120,3006),(3124,3266),(3125,3263),(3309,2788),(3413,3413),(3414,3414),(3448,5189),(3569,850),(3589,620),(3642,13),(3643,14),(3667,2789),(3734,6),(3735,75),(3821,10),(3923,1),(3934,2023),(3944,3705),(3980,612),(3981,305),(4011,614),(4045,141),(4048,32),(4074,1194),(4107,649),(4112,31),(4154,3747),(4159,1501),(4164,667),(4172,560),(4418,3920),(5026,187),(5027,669),(5028,1975),(5091,100),(5108,111),(5116,1078),(5149,1691),(5150,36),(5613,383),(5617,1080),(5622,2071),(5651,1995),(5921,3146),(5967,4226),(5969,538),(5972,468)

Let H be the hyperbola {A,B,C,PU(37)}, which is the isogonal conjugate of the line PU(39), this being the line X(32)X(512); also H is the isotomic conjugate of the line PU(45), which is the line X(6)X(523). This hyperbola passes through the points X(76), X(99), X(877), and X(2396). The center of H is X(5976). Let RA be the radical axis of the nine-point circle of triangle BCP(1) and the nine-point circle of BCU(1), and define RB and RC cyclically. The lines RA, RB, RC concur in X(5976). (Randy Hutson, August 22, 2014; see Hyacinthos #21938-21940, 4/12/2013)

Let L be the Simson line of X(99), and let L' the line normal to the circumcircle at X(99) X(5976). Then X(5976) = L∩L'. (L = X(114)X(325) and L' = X(3)X(76).) (Randy Hutson, December 4, 2014)

X(5976) lies on these lines:
{2,694},{3,76},{32,5149},{39,620},{69,147},{114,325},{115,3934},{187,736},{230,698},{315,6033},{350,1281},{385,732},{538,1569},{618,6109},{619,6108},{641,3102},{642,3103},{877,2967},{1125,5977},{1649,3268},{1909,3023},{2491,2799},{2794,5188},{4357,5988}

X(5976) = midpoint of X(76) and X(88)
X(5976) = reflection of X(I) in X(J) for these (I,J): (39,620), (115,3934), (1916,2023)
X(5976) = inverse-in-circumcircle of X(5989)
X(5976) = complement of X(1916)
X(5976) = anticomplement of X(2023)
X(5976) = X(385)-daleth conjugate of X(732)
X(5976) = X(I)-Ceva conjugate of X(J) for these (I,J): (2,325), (76,732), (99,804)
X(5976) = X(I)-Iso conjugate of X(J) for these (I,J): {98,1967}, {290,1927}, {694,1910}, {733,3404}, {1581,1976}
X(5976) = X(I)-Complementary conjugate of X(J) for these (I,J): (1,5031), (31,325), (163,804), (172,3836), (385,2887), (560,3229), (1580,141), (1691,10), (1914,3847), (1933,2), (1966,626), (2210,4357), (4164,116)
X(5976) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1916,2023), (98,99,5989), (98,5989,5939), (99,183,5939), (99,1078,5152), (183,5989,98)


X(5977) =  X(37)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 b-a^3 b^3+a^5 c-a^2 b^3 c+a b^4 c+b^5 c-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+a b c^4+b c^5

See X(5976).

X(5977) lies on these lines: (pending)


X(5978) =  X(13)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2+b^2+c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)-2 Sqrt[3] (a^2 b^2-b^4+a^2 c^2-c^4) S

See X(5976).

X(5978) lies on these lines: (pending)

X(5978) = anticomplement of X(6109)


X(5979) =  X(14)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2+b^2+c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)+2 Sqrt[3] (a^2 b^2-b^4+a^2 c^2-c^4) S

See X(5976).

X(5979) lies on these lines: (pending)

X(5979) = anticomplement of X(6108)


X(5980) =  X(15)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = Sqrt[3] a^2 (a^2-b^2-c^2) (a^2+b^2+c^2)+2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) S

See X(5976).

X(5980) lies on these lines: (pending)

X(5980) = inverse-in-circumcircle of X(5981)
X(5980) = anticomplement of X(6115)


X(5981) =  X(16)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = Sqrt[3] a^2 (a^2-b^2-c^2) (a^2+b^2+c^2)-2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) S

See X(5976).

X(5981) lies on these lines: (pending)

X(5981) = inverse-in-circumcircle of X(5980)
X(5981) = anticomplement of X(6114)


X(5982) =  X(17)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 - b^2c^2) sec(A - π/3) + (c^4 - a^2b^2) sec(B - π/3) + (b^4 - c^2a^2) sec(C - π/3)

See X(5976).

X(5982) lies on these lines: {2, 5469}, {5, 99}, {17, 1916}, {114, 383}, {147, 627}


X(5983) =  X(18)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 - b^2c^2) sec(A + π/3) + (c^4 - a^2b^2) sec(B + π/3) + (b^4 - c^2a^2) sec(C + π/3)

See X(5976).

X(5983) lies on these lines: {2, 5470}, {5, 99}, {18, 1916}, {114, 1080}, {147, 628}


X(5984) =  X(20)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a^8 - a^6(b^2 + c^2) - 5a^4b^2c^2 - a^2(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^2(b^4 + 3b^2c^2 + c^4)

See X(5976).

X(5984) lies on these lines:
{2, 98}, {8, 1281}, {20, 2782}, {99, 3522}, {115, 3832}, {148, 2794}, {193, 1916}, {385, 1503}, {390, 3027}, {2792, 5905}, {3023, 3600}, {3926, 5152}

X(5984) = anticomplement of X(147)


X(5985) =  X(21)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 - a^6(b^2 + bc + c^2) - a^5bc(b + c) + a^4(b^4 + b^3c - b^2c^2 + bc^3 + c^4) + a^3bc(b + c)(b^2 + c^2) - a^2(b^6 + b^5c - b^4c^2 - b^3c^3 - b^2c^4 + bc^5 + c^6) - abc(b + c)(b^4 - b^2c^2 + c^4) - b^2c^2(b^2 - c^2)^2

See X(5976).

X(5985) lies on these lines:
{2, 98}, {21, 2782}, {99, 4189}, {115, 5046}, {274, 5152}, {385, 518}, {1281, 1283}, {1621, 3027}, {2475, 2794}, {2975, 3023}


X(5986) =  X(22)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^10 - a^4b^2c^2(a^2 + b^2 + c^2) + a^2(b^6c^2 + b^2c^6 + 2b^4c^4 - b^8 - c^8) + b^2c^2(b^2 - c^2)^2(b^2 + c^2)

See X(5976).

X(5986) lies on these lines: {2, 98}, {22, 2782}, {99, 1799}, {305, 5152}, {385, 2393}


X(5987) =  X(23)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^10 - 2a^4b^2c^2(b^2 + c^2) + a^2(2b^6c^2 + 2b^2c^6 + b^4c^4 - b^8 - c^8) - b^2c^2(b^2 - c^2)^2(b^2 + c^2)

See X(5976).

X(5987) lies on these lines:
{2, 98}, {23, 2782}, {67, 3314}, {183, 2930}, {385, 2854}, {2794, 5189}, {3266, 5152}, {5939, 5971}

X(5987) = inverse-in-1st-anti-Brocard-circle of X(2)


X(5988) =  X(10)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b5 + c5 + a4b + a4c - a3b2 - a3c2 - a2b2c - a2bc2 + ab4 + ac4 - b3c2 - b2c3

See X(5976).

X5988) is the anticenter of the four points of intersection of the incircle and the Steiner inellipse; see X(5997) and X(5998). (Randy Hutson, August 22, 2014)

X(5988) lies on these lines:
{1, 147}, {2, 846}, {10, 257}, {11, 114}, {98, 109}, {99, 1010}, {115, 120}, {183, 4655}, {325, 740}, {542, 3745}, {804, 3837}, {908, 2239}, {1575, 1738}, {2254, 2786}, {2793, 4927}, {3699, 4104}, {3920, 5483}

X5988) = complement of X(1281)


X(5989) =  X(32)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a8 - a4b2c2 - a2b6 - a2c6 + 2b4c4

See X(5976).

X(5989) lies on these lines:
{2, 4048}, {3, 76}, {6, 1916}, {75, 1281}, {114, 3818}, {115, 5149}, {147, 325}, {148, 384}, {385, 698}, {538, 5162}, {543, 1003}, {2023, 5026}, {5017, 5969}

X(5989) = inverse-in-circumcircle of X(5976)


X(5990) =  X(100)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 - a^6(b + c) + a^5bc + a^3(b^4 - b^3c - b^2c^2 - bc^3 + c^4) - a^2(b + c)(b^4 - 2b^3c + b^2c^2 - 2bc^3 + c^4) + abc(b^4 - b^3c - b^2c^2 - bc^3 + c^4) - b^2c^2(b - c)^2(b + c)

See X(5976).

X(5990) lies on these lines: {2, 1083}, {98, 901}, {115, 2240}, {385, 513}, {1281, 5205}, {4613, 5143}


X(5991) =  X(101)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 - a^7(b + c) + a^6bc - a^4b^2c^2 + a^3(b + c)(b^2 - bc + c^2)^2 - a^2(b^6 - b^5c + b^3c^3 - bc^5 + c^6) + ab^2c^2(b - c)^2(b + c) - b^3c^3(b - c)^2

See X(5976).

X(5991) lies on these lines: {36, 1111}, {98, 927}, {99, 859}, {385, 514}


X(5992) =  X(8)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - b^5 - c^5 - a^4 b - a^4 c + a^3 b^2 + a^3 c^2 - a^2 b^3 - a^2 c^3 + a^2 b^2c + a^2 b c^2 - a b^4 - a c^4 - a b^2 c^2 + b^4 c + b^3 c^2 + b^2 c^3 + b c^4

b See X(5976).

X(5992) lies on these lines:
{2, 846}, {98, 901}, {145, 2784}, {147, 149}, {148, 1655}, {2403, 2789}, {2792, 5905}, {3314, 5695}, {4440, 4459}

X(5992) = anticomplement of X(1281)


X(5993) =  INTERSECTION OF LINES X(10)X(5975) AND X(11)X(2643)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c)^2 (a^3+b^3+a b c+2 b^2 c+2 b c^2+c^3) (-a^4-a^3 b+a b^3+b^4-a^3 c-a^2 b c+3 a b^2 c+b^3 c+3 a b c^2+a c^3+b c^3+c^4) (Peter Moses, August 22, 2014)

See X(5975) and X(5976).

X(5993) lies on the nine-point circle and these lines: {10,5975},{11,2643},{3259,5954}


X(5994) =  TRILINEAR POLE OF LINE X(6)X(3458)

Trilinears        a/[b csc(B - π/3) - c csc(C - π/3)] : b/[c csc(C - π/3) - a csc(A - π/3)] : c/[a csc(A - π/3) - b csc(B - π/3)]

Let P be a point of the line X(2)X(14), other than X(2). Let A'B'C' be the cevian triangle of P. Let A'' by the {B,C}-harmonic conjugate of A' (i.e., A'' = BC∩B'C'), and define B'' and C'' cyclically. The circumcircles of AB''C'', BC''A'', CA''B'' concur in X(5994). Let L,M,N be the lines obtained by reflecting the line X(4)X(14) in the sidelines of ABC; then L,M,N concur in X(5994). Also, X(5994) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(14),X(15)}}. Further, X(5994) is the trilinear pole of the line X(6)X(3458), and X(5994) = Ψ(X(i), X(j)) for these (i,j): (2,14), (4,14), (6,3458), (16,6), (76,14). (Randy Hutson, August 22, 2014)

X(5994) lies on the circumcircle and these lines:
{6,2379},{14,98},{15,842},{16,74},{111,3458},{187,2378},{301,2367},{512,1576},{759,2154},{1300,6110}

X(5994) = midpoint of X(16) and X(5669)
X(5994) = X(i)-isoconjugate of X(j) for these (i,j): {16,1577},{299,661},{300,2624},{471,656},{850,2152},{2153,3268}


X(5995) =  TRILINEAR POLE OF LINE X(6)X(3457)

Trilinears        a/[b csc(B + π/3) - c csc(C + π/3)] : b/[c csc(C + π/3) - a csc(A + π/3)] : c/[a csc(A + π/3) - b csc(B + π/3)]

Let P be a point of the line X(2)X(13), other than X(2). Let A'B'C' be the cevian triangle of P. Let A'' by the {B,C}-harmonic conjugate of A' (i.e., A'' = BC∩B'C'), and define B'' and C'' cyclically. The circumcircles of AB''C'', BC''A'', CA''B'' concur in X(5995). Let L,M,N be the lines obtained by reflecting the line X(4)X(13) in the sidelines of ABC; then L,M,N concur in X(5995). Also, X(5995) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(13),X(16)}}. Further, X(5995) is the trilinear pole of the line X(6)X(3457), and X(5995) = Ψ(X(i), X(j)) for these (i,j): (2,13), (4,13), (6,3457), (15,6), (76,13). (Randy Hutson, August 22, 2014)

X(5995) lies on the circumcircle and these lines:
{6,2378},{13,98},{15,74},{16,842},{111,3457},{187,2379},{300,2367},{512,1576},{759,2153},{1300,6111}

X(5995) = midpoint of X(15) and X(5668)
X(5994) = X(i)-isoconjugate of X(j) for these (i,j): {15,1577},{298,661},{301,2624},{470,656},{850,2151},{2154,3268}


X(5996) =  INTERSECTION OF LINES X(2)X(512) AND X(325)X(523)

Barycentrics    b2c2(4a2 - 3b2 - 3c2) : c2a2(4b2 - 3c2 - 3a2) : a2b2(4c2 - 3a2 - 3b2)

Let L be the de Longchamps line of ABC, and let L' be the de Longchamps line of the 1st anti-Brocard triangle. Then X(5996) = L∩L'. Let O' be the inverter of the circumcircle and the nine-point circle; then X(5996) is the pole of the Brocard axis with respect to O'. (Randy Hutson, August 22, 2014)

X(5996) lies on these lines: (pending)


X(5997) =  1st HUTSON-WOLK POINT

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

X(5997) and X(5998) are described by Randy Hutson (Hyacinthos #21045) and Barry Wolk (Hyacinthos #21047, June 1, 2012). One of the 4 points of intersection of the incircle and Steiner inellipse is a triangle center, X(5997), and the other 3 are the vertices of a central triangle; of those 3, let A' be the one farthest from A, and define B' and C' cyclically. The triangle A'B'C' is perspective to ABC at X(5998). Barycentrics for A', B', C' follow:

A' = a - [(a - b + c)(a + b - c)]1/2 : b + [(b - c + a)(b + c - a)]1/2 : c + [(c - a + b)(c + a - b)]1/2
B' = a + [(a - b + c)(a + b - c)]1/2 : b - [(b - c + a)(b + c - a)]1/2 : c + [(c - a + b)(c + a - b)]1/2
C' = a + [(a - b + c)(a + b - c)]1/2 : b + [(b - c + a)(b + c - a)]1/2 : c - [(c - a + b)(c + a - b)]1/2

X(5997) is the {X(1),X(508)}-harmonic conjugate of X(5998). (Randy Hutson, August 22, 2014)

X(5997) lies on the incircle, the Steiner inellipse, and these lines: (pending)


X(5998) =  2nd HUTSON-WOLK POINT

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

Continuing from X(5997), the lines AA', BB', CC' concur in X(5998), and X(5998) is the {X(1),X(508)}-harmonic conjugate of X(5997). (Randy Hutson, August 22, 2014)

X(5998) lies on these lines: (pending)


X(5999) =  X(98)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a8 + a4(b4 + c4 + b2c2) - 2a2(b6 + c6 ) - b2c2(b2 - c2)2

X(5999) = {X(6039),X(6040)}-harmonic conjugate of X(3) (Randy Hutson, August 22, 2014)

X(5999) lies on these lines: (Euler line and others, pending)

X(5999) = reflection of X(385) in X(98)
X(5999) = anticomplement of X(1513)




leftri Isogonal Conjugates With Respect To Various Triangles rightri

Many triangle centers are constructible as isogonal conjugates with respect to a central triangle T other than ABC. See, for example, the preamble to X(2883). Other examples are X(5642), for which T is the Thomson triangle, and X(5957), for which T is the Feuerbach triangle. In the latter case, a long formula for barycentric coordinates is given for the Feuerbach isogonal conjugate of an arbitrary point p : q : r is given. Peter Moses (August 11, 2014) developed a much longer formula for the isogonal conjugate of a point p : q : r with respect to a triangle having vertices d1 : d2 : d3, e1 : e2 : e3, f1 : f2 : f3. The total number of characters in the formula is (pending). Although too long to be stated here, the formula served as a basis for finding triangle centers X(i) for i = 6000-6030.

For well-established central triangles T, (e.g., Thomson, Feuerbach, intouch, extouch), the phrase "isogonal conjugate with respect to T can be shortened to "T isogonal conjugate" or "T-I C", as in many of the next triangle centers.

It T is a triangle, then T-isogonal conjugation carries points on the circumcircle of T to the line at infinity. For example, taking T = Brocard triangle, the appearance of (X(i),X(j)) in the following list means that T-I C of X(i) is X(j): (3,2782), (6,804), (1083,2795), (1316,690), (5091,2787), (5108,543).

In each of the next examples (i,j), the point X(i) lies on the incircle, and X(j), the intouch-IC-of-X(i), lies on the line at infinity: (1317,517), (1356,6002), (1357,3667), (1358,3309), (1360,971), (1361,5151), (1362,516), (1364,522), (1365,6003), (2446, 3307), (2447,3308), (3020,6004), (3021,518), (3022,514), (3023,512), (3024,523), (3025,900), (3026,6005), (3027,511), (3028,30), (3318,521), (3319,2800), (3320,1503), (3322,2801), (3323,2820), (3325,1499), (3326,3738), (3327,1510), (3328,3887), (5577,6006), (5579,6007), (5580,520), (5581,6008), (5582,6009)

underbar

X(6000) =  ISOGONAL CONJUGATE OF X(1294)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+4 a^4 b^2 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+6 b^4 c^4+3 a^2 c^6-2 b^2 c^6-c^8)

X(6000) lies on these lines:
{2,5656},{3,64},{4,51},{5,2883},{6,1597},{20,2979},{24,1204},{30,511},{40,2939},{52,382},{65,1844},{66,3818},{74,186},{98,2713},{99,2706},{100,2744},{101,2738},{102,2762},{103,2727},{104,2719},{109,2732},{110,2071},{113,2072},{125,403},{133,1515},{143,3853},{146,1531},{159,3098},{184,378},{187,1971},{206,4550},{221,500},{232,3269},{373,3545},{376,3917},{381,1853},{399,2935},{546,5462},{548,5447},{550,1216},{568,3830},{578,1181},{858,1568},{933,3484},{974,1514},{999,2192},{1092,5879},{1192,3517},{1292,2749},{1293,2755},{1295,2761},{1296,2763},{1297,2764},{1533,3580},{1562,5523},{1614,3520},{1657,5925},{1870,3270},{3060,3543},{3146,5889},{3284,5668},{3524,5650},{3581,5899},{3587,3781},{3627,5446},{3839,5640},{3845,5946}

X(6000) = isogonal conjugate of X(1294)
X(6000) = crosssum of X(4) and X(74)
X(6000) = orthic-IC-of-X(133) = medial-IC-of-X(133)
X(6000) = intouch-IC-of-X(3324)
X(6000) = Euler-IC-of-X(122)
X(6000) = 1st-circumperp-IC-of-X(1294) = circumtangential-IC-of-X(1294)
X(6000) = circumcircle-midarc-IC-of-X(107) = circumorthic-IC-of-X(107) = 2nd-circumperp-IC-of-X(107)
X(6000) = circumnormal-IC-of-X(107) = Thomson-IC-of-X(107)


X(6001) =  ISOGONAL CONJUGATE OF X(1295)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 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+2 a^3 b^2 c-3 a b^4 c-a^4 c^2+2 a^3 b c^2-4 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^3 c^3+2 a b^2 c^3+2 a^2 c^4-3 a b c^4+b^2 c^4+a c^5-c^6)

X(6001) lies on these lines:
{1,84},{3,960},{4,65},{5,3812},{7,3427},{9,3197},{10,5777},{11,1519},{19,5776},{20,3869},{30,511},{36,1727},{40,64},{46,3149},{63,3428},{74,2766},{98,2714},{99,2707},{100,2745},{101,2739},{102,2765},{103,2728},{104,1319},{109,2733},{110,2694},{119,5123},{153,5176},{154,392},{165,5692},{185,1829},{207,1712},{210,5657},{227,1745},{281,1903},{354,5603},{355,5836},{376,5918},{496,942},{497,5768},{573,4047},{581,3931},{774,1042},{944,3057},{962,3868},{1001,3358},{1006,3683},{1064,3666},{1072,3782},{1104,3073},{1292,2750},{1293,2756},{1294,2761},{1296,2767},{1376,3359},{1385,5248},{1430,5706},{1456,1870},{1465,1735},{1532,1737},{1537,5570},{1699,5902},{1739,5400},{1750,2093},{1853,3753},{1871,5786},{2077,2932},{2096,4293},{2194,4227},{2292,4300},{2551,5811},{2935,2948},{3062,3577},{3357,3579},{3698,5818},{3742,5886},{3817,5883},{3874,4301},{3877,5731},{3878,4297},{3889,5734},{3913,5534},{3914,5721},{4018,5895},{4067,5493},{4314,5882},{4662,5690},{4867,5538},{4880,5536},{5691,5903},{5787,5878},{5806,5893}

X(6001) = crosssum of X(4) and X(104)
X(6001) = midarc-of-X(3318)
X(6001) = intouch-IC-of-X(1359)
X(6001) = Euler-IC-of-X(123)
X(6001) = 1st-circumperp-IC-of-X(1295) = circumtangential-IC-of-X(1295)
X(6001) = circumcircle-midarc-IC-of-X(108) = circumorthic-IC-of-X(108) = 2nd-circumperp-IC-of-X(108)
X(6001) = circumnormal-IC-of-X(108) = Thomson-IC-of-X(108)


X(6002) =  INTOUCH-ISOGONAL CONJUGATE OF X(1356)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)*(-a^3 - a^2*b - a^2*c - a*b*c + b^2*c + b*c^2)

X(6002) lies on these lines:
{1, 4170}, {4, 4444}, {8, 4729}, {30, 511}, {649, 4391}, {661, 4560}, {667, 3716}, {905, 3835}, {1019, 1577}, {2533, 4784}, {3669, 4106}, {3762, 4063}, {4010, 4367}, {4380, 4462}, {4382, 4801}, {4705, 4913}, {4879, 4922}

X(6002) = isogonal conjugate of X(6010)


X(6003) =  INTOUCH-ISOGONAL CONJUGATE OF X(1365)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3)

X(6003) lies on these lines:
{1, 4017}, {3, 3733}, {8, 4404}, {30, 511}, {573, 798}, {656, 3737}, {661, 1021}, {1459, 3960}, {1532, 4129}, {4581, 4761}

X(6003) = isogonal conjugate of X(6011)


X(6004) =  INTOUCH-ISOGONAL CONJUGATE OF X(3020)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(b - c)*(a^2 - a*b + b^2 - a*c + c^2)

X(6004) lies on these lines:
{1, 3777}, {30, 511}, {659, 1734}, {663, 1201}, {667, 2254}, {764, 4449}, {905, 1960}, {1016, 3888}, {1491, 3216}, {2473, 2488}, {3214, 4705}, {4367, 4905}, {4498, 4730}

X(6004) = isogonal conjugate of X(6012)


X(6005) =  INTOUCH-ISOGONAL CONJUGATE OF X(3026)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(b - c)*(2*a*b + 2*a*c + b*c)

X(6005) lies on these lines:
{30, 511}, {649, 2664}, {659, 4834}, {661, 1734}, {663, 1019}, {665, 3803}, {667, 4784}, {693, 4170}, {1491, 4983}, {2254, 4822}, {2533, 4791}, {2978, 4932}, {3250, 4979}, {4010, 4823}, {4063, 4724}, {4147, 4807}, {4367, 4775}, {4378, 4879}, {4391, 4761}, {4489, 4813}, {4490, 4730}

X(6005) = isogonal conjugate of X(6013)


X(6006) =  INTOUCH-ISOGONAL CONJUGATE OF X(5577)

Barycentrics   (b - c)(5a - b - c) : (c - a)(5b - c - a) : (a - b)(5c - a - b)

X(6006) lies on these lines:
{7, 3676}, {9, 649}, {30, 511}, {142, 3835}, {144, 4468}, {1769, 4905}, {2254, 4776}, {3239, 4790}, {4401, 4491}, {4406, 4828}, {4462, 4768}

X(6006) = isogonal conjugate of X(6014)


X(6007) =  INTOUCH-ISOGONAL CONJUGATE OF X(5579)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^2*b^2 - a*b^3 + a*b^2*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3)

X(6007) lies on these lines:
{7, 310}, {9, 43}, {30, 511}, {39, 4443}, {86, 4890}, {142, 3840}, {192, 3688}, {239, 3271}, {320, 4014}, {995, 1001}, {1463, 4684}, {1469, 3886}, {1654, 4111}, {1959, 4516}, {2234, 3122}, {2245, 4436}, {3056, 3875}, {3729, 3779}, {3792, 4693}, {3882, 4433}, {3923, 4260}, {3943, 4553}, {4259, 5695}

X(6007) = isogonal conjugate of X(6015)


X(6008) =  INTOUCH-ISOGONAL CONJUGATE OF X(5581)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)*(-3*a^2 - a*b - a*c + 2*b*c)

X(6008) lies on these lines:
{7, 3669}, {9, 4063}, {30, 511}, {144, 4462}, {390, 4162}, {649, 4106}, {650, 4380}, {667, 1001}, {693, 4790}, {1638, 4786}, {3768, 4498}, {3803, 4170}, {3835, 4394}, {4312, 4905}, {4375, 4782}, {4382, 4979}

X(6008) = isogonal conjugate of X(6016)


X(6009) =  INTOUCH-ISOGONAL CONJUGATE OF X(5582)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)*(4*a^2 - a*b + b^2 - a*c - 4*b*c + c^2)

X(6009) lies on these lines:
{30, 511}, {88, 673}, {659, 1001}, {676, 4830}, {1086, 2087}, {1635, 4927}, {2550, 4925}, {3826, 3837}, {4453, 4773}

X(6009) = isogonal conjugate of X(6017)


X(6010) =  ISOGONAL CONJUGATE OF X(6002)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c)*(-a^3 - a^2*b - a^2*c - a*b*c + b^2*c + b*c^2))

X(6010) lies on the circumcircle and these lines:
{3, 741}, {40, 98}, {55, 1356}, {99, 3882}, {103, 3430}, {165, 5539}, {573, 759}, {813, 4574}, {1376, 3037}, {2077, 2699}

X(6010) = isogonal conjugate of X(6002)


X(6011) =  ISOGONAL CONJUGATE OF X(6003)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/((b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3)

X(6011) lies on the circumcircle and these lines:
{3, 759}, {40, 74}, {55, 1365}, {102, 3430}, {104, 3651}, {105, 1283}, {106, 1385}, {107, 4242}, {112, 1983}, {354, 1477}, {573, 2249}, {741, 991}, {840, 5536}, {2077, 2687}

X(6011) = isogonal conjugate of X(6003)
X(6011) = reflection of X(759) in X(3)
X(6011) = circumcircle-antipode of X(759)


X(6012) =  ISOGONAL CONJUGATE OF X(6004)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/((b - c)*(a^2 - a*b + b^2 - a*c + c^2))

X(6012) lies on the circumcircle and these lines: {55, 3020}, {105, 404}, {759, 4234}

X(6012) = isogonal conjugate of X(6004)


X(6013) =  ISOGONAL CONJUGATE OF X(6005)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/((b - c)*(2*a*b + 2*a*c + b*c)

X(6013) lies on the circumcircle and these lines: {55, 3026}, {101, 4436}, {105, 5248}, {741, 4658}

X(6013) = isogonal conjugate of X(6005)


X(6014) =  ISOGONAL CONJUGATE OF X(6006)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((5*a - b - c)*(-b + c)

X(6014) lies on the circumcircle and these lines: {55, 106}, {104, 165}, {672, 2384}, {739, 2280}, {901, 3939}, {953, 5537}

X(6014) = isogonal conjugate of X(6006)


X(6015) =  ISOGONAL CONJUGATE OF X(6007)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a^2*b^2 - a*b^3 + a*b^2*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3)

X(6015) lies on the circumcircle and these lines:
{55, 99}, {57, 4128}, {100, 894}, {101, 171}, {109, 1918}, {110, 2175}, {165, 5539}, {934, 1402}, {1155, 2703}, {1308, 5143}

X(6015) = isogonal conjugate of X(6007)


X(6016) =  ISOGONAL CONJUGATE OF X(6008)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c)*(-3*a^2 - a*b - a*c + 2*b*c)

X(6016) lies on the circumcircle and these lines: {55, 739}, {644, 898}, {672, 2382}, {727, 2280}

X(6016) = isogonal conjugate of X(6008)


X(6017) =  ISOGONAL CONJUGATE OF X(6009)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c)*(4*a^2 - a*b + b^2 - a*c - 4*b*c + c^2))

X(6017) lies on the circumcircle and these lines: {44, 105}, {55, 2384}, {106, 672}, {901, 2284}, {1252, 4588}, {2280, 2382}

X(6017) = isogonal conjugate of X(6009)


X(6018) =  INTOUCH-ISOGONAL CONJUGATE OF X(519)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a b+b^2+a c-4 b c+c^2)^2

Let A' be the point of intersection, other than X(11), of the Mandart inellipse and the incircle that is farthest from A; define B' and C' cyclically. A'B'C' is here named the Mandart-incircle triangle. It is homothetic to ABC at X(55), to the medial triangle at X(11), to the Euler triangle at X(12), and to the anticomplementary triangle at X(497). X(6018) = X(106)-of-Mandart-incircle triangle. (Randy Hutson, August 26, 2014)

X(6018) lies on the incircle and these lines:

X(6018) = reflection of X(1357) in X(1)
X(6018) = X(1294)-of-intouch-triangle


X(6019) =  INTOUCH-ISOGONAL CONJUGATE OF X(524)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b+c)^2 (a^2+b^2-3 b c+c^2)^2

X(6019) lies on the incircle and these lines:

X(6019) = reflection of X(3325) in X(1)
X(6019) = X(111) of Mandart-incrcle triangle (see X(6018))


X(6020) =  INTOUCH-ISOGONAL CONJUGATE OF X(525)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c)^2 (a^4-b^4+a^2 b c-b^3 c-b c^3-c^4)^2

X(6020) lies on the incircle and these lines:

X(6020) = reflection of X(3320) in X(1)
X(6020) = X(112) of Mandart-incrcle triangle (see X(6018))


X(6021) =  INTOUCH-ISOGONAL CONJUGATE OF X(536)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (2 a b^2-2 a b c-b^2 c+2 a c^2-b c^2)^2

X(6021) lies on the incircle and these lines:


X(6022) =  INTOUCH-ISOGONAL CONJUGATE OF X(538)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b+c)^2 (2 a^2 b^2-3 a^2 b c+2 a^2 c^2-b^2 c^2)^2

X(6022) lies on the incircle and these lines:


X(6023) =  INTOUCH-ISOGONAL CONJUGATE OF X(542)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a+b-c) (a-b+c) (b+c)^2 (a^4-b^4-2 a^2 b c+b^3 c+b^2 c^2+b c^3-c^4)^2

X(6023) lies on the incircle and these lines:

X(6023) = reflection of X(6027) in X(1)


X(6024) =  INTOUCH-ISOGONAL CONJUGATE OF X(545)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^2 b-4 a b^2+b^3+a^2 c+4 a b c-4 a c^2+c^3)^2

X(6024) lies on the incircle and these lines:


X(6025) =  INTOUCH-ISOGONAL CONJUGATE OF X(674)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (a^3 b-a^2 b^2+a^3 c+b^3 c-a^2 c^2-2 b^2 c^2+b c^3)^2

X(6025) lies on the incircle and these lines:


X(6026) =  INTOUCH-ISOGONAL CONJUGATE OF X(688)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (b-c)^2 (a^4 b^2+a^4 b c+a^4 c^2+b^3 c^3)^2

X(6026) lies on the incircle and these lines:


X(6027) =  INTOUCH-ISOGONAL CONJUGATE OF X(690)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c)^2 (a^4-b^4+2 a^2 b c-b^3 c+b^2 c^2-b c^3-c^4)^2

X(6027) lies on the incircle and these lines:

X(6027) = reflection of X(6023) in X(1)


X(6028) =  INTOUCH-ISOGONAL CONJUGATE OF X(696)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^4 b^2+a b^5-2 a^4 b c-b^5 c+a^4 c^2+a c^5-b c^5)^2

X(6028) lies on the incircle and these lines:


X(6029) =  INTOUCH-ISOGONAL CONJUGATE OF X(712)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^3 b^2+a b^4-2 a^3 b c-b^4 c+a^3 c^2+a c^4-b c^4)^2

X(6029) lies on the incircle and these lines:


X(6030) =  THOMSON-ISOGONAL CONJUGATE OF X(5)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (3 a^4-a^2 b^2-2 b^4-a^2 c^2-b^2 c^2-2 c^4)
X(6030) = 2X(6) - 5X(1176)
X(6030) = X(6) + 5X(2916)
X(6030) = X(1176) + 2X(2916)
X(6030) = (a2 + b2 + c2)*X(6) - 5(J2 - 3)R2*X(22)
(barycentrics and combos, Peter Moses, August 16, 2014)

X(6030) lies on the Thomson-Gibert-Moses hyperbola; see X(5642). X(6030) = {X(22),X(3796)}-harmonic conjugate of X(3060; also, X(6030) = {X(3060),X(3796)}-harmonic conjugate of X(5012). (Peter Moses, August 16, 2014)

X(6030) lies on these lines:
{6,22},{23,5643},{25,5544},{36,595},{110,3917},{182,5645},{376,5654},{392,4881},{550,3521},{1495,5888},{2937,5946},{2979,3167},{5640,5644}


X(6031) =  X(353)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - b^2c^2)[a^6 - a^2(4b^4 + b^2c^2 + 4c^4) - 2b^2c^2(b^2 + c^2)]

X(6031) lies on these lines:

X(6031) = anticomplement of X(6032)
X(6031) = circumcevian-isogonal conjugate of X(2)


X(6032) =  CENTROID OF 4th BROCARD TRIANGLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - b^2c^2)[6a^4(b^2 + c^2) + 2a^2(b^4 + b^2c^2 + c^4) - (b^2 + c^2)(b^4 - 5b^2c^2 + c^4)]

X(6032) lies on these lines:

X(6032) = complement of X(6031)
X(6032) = inverse-in-orthocentroidal circle of X(111)
X(6032) = X(353)-of-orthocentroidal triangle


X(6033) =  MIDPOINT OF X(4) AND X(147)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3(S2 - S2ω)S2A - Sω(5S2 -3S2ω)SA + 2S2(2S2 - S2ω)

X(6033) = orthologic center of Carnot (Johnson) triangle and 1st Brocard triangle. (César Lozada, August 7, 2013)

Let D = X(99), let HA = X(4)-of-BCD, and define HB, HC, HD cyclically. Then HAHBHCHD is a cyclical quadrilateral, of which the circumcenter is X(6033). (Randy Hutson, August 22, 2014)

X(6033) lies on these lines: (pending)

X(6033) = midpoint of X(4) and X(147)
X(6033) = reflection of X(3) in X(114)
X(6033) = X(3)-of-X(511)-Fuhrmann-triangle
X(6033) = inverse-in-Kiepert-hyperbola of X(6034)
X(6033) = {X(13),X(14)}-harmonic conjugate of X(6034)


X(6034) =  CENTROID OF X(6)X(13)X(14)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (pending)
X(6034) = X(6) + X(13) + X(14) = X(6) + 2X(115)

X(6034) lies on these lines: (pending)

X(6034) = inverse-in-Kiepert-hyperbola of X(6033)
X(6034) = {X(13),X(14)}-harmonic conjugate of X(6033)


X(6035) =  ISOTOMIC CONJUGATE OF X(1640)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b2 - c2)(2a6 - b6 - c6 - 2a4b2 - 2a4c2 + a2b4 + a2c4 + b4c2 + b2c4)]

X(6035) lies on the hyperbola {A, B, C, PU(37)} and these lines: (pending)

X(6035) = isogonal conjugate of X(6041)
X(6035) = isotomic conjugate of X(1640)
X(6035) = trilinear pole of line X(30)X(99)


X(6036) =  MIDPOINT OF X(98) AND X(114)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a sec(A + ω) + b sec(B + ω) + c sec(C + ω)
X(6036) = 3X(2) + X(98)

Let D = X(98), let GA = X(2)-of-BCD, and define GB, GC, GD cyclically. Then GAGBGCGD is a cyclic quadrilateral which is homothetic to the cyclic quadrilateral ABCD, and X(6036) is the homothetic center, as well as the centroid of ABCD and the anticenter of the cyclic quadrilateral whose vertices are X(115) and the vertices of the medial triangle. Let (OA) be the circle centered at the A-vertex of the 1st Brocard triangle and tangent to line BC; define (OB) and (OC) cyclically; then X(6036) is the radical center of (OA), (OB), (OC). Also, X(6036) lies on the axis of the parabola {A, B, C, X(511), X(805}. (Randy Hutson, August 22, 2014)

X(6036) is the QA-P36 center of the quardrangle ABCX(2). See Complement of QA-P30 wrt the QA-Diagonal Triangle.

X(6036) lies on these lines: (pending)

X(6036) = midpoint of X(98) and X(114)
X(6036) = complement of X(114)
X(6036) = X(140)-of-1st-anti-Brocard-triangle


X(6037) =  TRILINEAR POLE OF LINE X(6)X(98)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b sec(C + ω) - c sec(B + ω)]

X(6037) lies on the circumcircle and these lines: (pending)

X(6037) = inverse-in-1st-anti-Brocard-circle of X(6038)
X(6037) = Ψ(X(6),X(98)
X(6037) = Λ(X(290),X(879)
X(6037) = Λ(X(684),X(2491)


X(6038) =  INVERSE-IN-1st-ANTI-BROCARD-CIRCLE OF X(6037)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (pending)

Let A' be the inverse-in-1st-anti-Brocard-circle of A, and define B' and C' cyclically; then X(6038) is the centroid of A'B'C'.

X(6038) lies on these lines: (pending)


X(6039) =  THOMSON-ISOGONAL CONJUGATE OF X(5638)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S4 - a2SASω(-2S2 + S2A + S2B + S2C)1/2 - (S2 - 2SBSC)S2ω
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)[cos B cos C + 3 sin B sin C - (cos A)(3 + 4e*cos ω + 4 cos 2ω)], where e = (1 - 4 sin2ω)1/2
X(6039) = 3(3S2 + S2A + S2B + S2C)*X(2) - 2Sω[Sω + (-S2 + S2A + S2B + S2C)1/2]*X(3)    (barycentrics and combo, Peter Moses, August 24, 2014)

Let O' be the circle with center X(2) that passes through the points X(98) and X(842); then X(6039) and X(6040) are the points of intersection of O' and the Euler line. Also, ({X(6039),X(6040}-harmonic conjugate of X(4)) = X(1513), and ({X(6039),X(6040}-harmonic conjugate of X(383)) = X(1080). (Peter Moses, Augu