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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)

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).

The conversion from homogeneous tripolar coordinates (henceforth simply "tripolars") to homogeneous barycentrics can be carried out by 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)

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

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(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-INCIRCLE 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)(a4c - 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)

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)   C. 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)    C. 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) = a4(a4 + b4 + c4 - 2a2b2 - 2a2c2 + 6b2c2)

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 - 6a4 - 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) 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 + 4b2c)

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)

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(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(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(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(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(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) lies on 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(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(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(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

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


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(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}


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(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)

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(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(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) lies on 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(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 - b4c2)

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(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)

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

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(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(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(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)

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(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(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(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)

X(5213) lies on 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(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) 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(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(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(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)

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(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(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)

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(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)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(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(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(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    C. 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(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(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(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>    C. 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    C. 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(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)

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(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    C. 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(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(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(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(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(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(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(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) = 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    C. 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(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(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) 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(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) 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(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 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(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)

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}


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(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(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(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(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(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)

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(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(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(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(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 this line: {6, 25}


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 this line: {6, 25}


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(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(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(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) = complement of X(1232)

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(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(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(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)

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(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) 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(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) = 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)
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)

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)

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)

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)

If you have The Geometer's Sketchpad, you can view X(5457) and X(5458).

X(5457) lies on these lines: (pending)


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)

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) 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(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(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}, {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)

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)

X(5463) lies on these lines: {2, 13}, {3, 67}, {14, 543}, {15, 524}, {18, 671}, {61, 1992}, {62, 597}, {99, 298}

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)

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) lies on these lines: {2, 14}, {3, 67}, {13, 543}, {16, 524}, {17, 671}, {61, 597}, {62, 1992}, {99, 299}

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}, {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(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(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)

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)

Trilinears   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b2 - c2)2(a2 - b2 - c2) + 2a2bc[(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)

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) 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(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)

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)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(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)

X(5504) lies on 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(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 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(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 is also the point indexed as QA-P2, named the Euler-Poncelet Point, in Chris van Tienhoven's 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) = 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) = 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) = 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)

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) = 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(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(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)

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) = 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 point which is homologous with respect to the excentral triangle as X(40) is to the Fuhrmann triangle. 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)

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)

If you have The Geometer's Sketchpad, you can view X(5544).

X(5542) 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 circumconic having perspector X(55). The conic, denoted by U, is given by the trilinear equation

a(b + c - a)yz + b(c + a - b)zx + c(a + b - c)xy = 0.

For X on X(1)X(6)∪U, let F(X) be the point of concurrence. Then if X is on X(1)X(6), the image F(X) is on the line X(1)X(3); a pair (I,J) in the following list indicates that F(X(I)) = X(J): (1,165), (6,3), (9,40), (37,3579), (44,517), 281,55), 1713,1715), (1723,46), (1724,1754), (1743,1), (2323,2077), (5247,171), 5526,5537). On the other hand, if X is on U, the image F(X) is on the circumcircle; a pair (I,J) in the following list indicates that F(X(I)) = X(J): (101,109), (110,5543), (111,5543), (294,105), (644,100), (645,99), (651,934), (666,927), (1783,108), (2311,741), (2316,106), (4627,5545).   (César Lozada; August 29, 2013)

Suppose that P is on X(1)X(6). If P = p : q : r (trilinears), then F(P) = a(b + c - a)/[(b - c)p] : b(c + a - b)/[(c - a)q] : c(a + b - c)/[(a - b)r];
If P = p : q : r (barycentrics), then F(P) = a3(b + c - a)/[(b - c)p] : b3(c + a - b)/[(c - a)q] : c3(a + b - c)/[(a - b)r]
Suppose that P is on U. If P = p : q : r (trilinears), then F(P) = (b + c - a)p : (c + a - b)q : (a + b - c)r];
If P = p : q : r (barycentrics), then F(P) = then F(P) = (b + c - a)p : (c + a - b)q : (a + b - c)r].    (Peter Moses; September 2, 2013)

X(5545) lies on the circumcircle and these lines:
{100,1414}, {101,4565}, {105,5323}, {835,4624}


X(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) 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).

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)

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 homotheitic to the orthic triangle of ABC from X(2) with ratio -2.      (Peter Moses, October 4, 2013)

If you have The Geometer's Sketchpad, you can view X(5562).

X(5562) = reflection of X(I) in X(J) for these (I,J): (185,3), (52,5), (3,1216), (1843,1352)
X(5562) = anticomplement of X(389)

X(5562) lies on 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(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 S 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 + zCS )]
g(a,b,c,x,y,z) = a2[a2y2z2 - x2(2(y + z)(yz)1/2(- S) + y2SB + zCS )].

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 + zCS )]
: b2[b2z2x2 - y2(2(z + x)(zx)1/2S*j + z2SB + xCS )]
: c2[c2x2y2 - z2(2(x + y)(xy)1/2S*k + x2SB + yCS )],

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, and define BC and CA cyclically. Let AC be the center of the circle through A and tangent to the C-median, 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) 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) 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) 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(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(b + c - a)3[a2 - 3(b - c)2]

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(5574) =  PERSPECTOR OF MEDIAL AND ANTLIA TRIANGLES

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

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) 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-OI-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[a2 - (b - 2c)2]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).

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]. 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 = X(55) and w = (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 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).

If you have The Geometer's Sketchpad, you can view Inverter.

X(5577) lies on the incircle and these lines: (pending)


X(5578) =  INVERSE-IN-OI-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: (pending)


X(5579) =  INVERSE-IN-OI-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: (pending)


X(5580) =  INVERSE-IN-OI-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: (pending)


X(5581) =  INVERSE-IN-OI-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 these lines: (pending)


X(5582) =  INVERSE-IN-OI-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 these lines: (pending)


X(5583) =  CENTER OF INVERSE-IN-OI-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 U be the circumcircle, VA the A-excircle, and WA the inverter of U and VA. Let A' be the center of the inverter of U and WA, 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).

X(5584) lies on these lines: (pending)


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 + c2yz) = 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: (pending)


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: (pending)


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).

X(5587) lies on these lines: (pending)


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: (pending)


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: (pending)


X(5590) =  PERSPECTOR OF ORTHIC 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) lies on these lines: (pending)


X(5591) =  PERSPECTOR OF ORTHIC 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) lies on these lines:
(pending)


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

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 circle, other than the circumcircle, that is tangent to the lines B'C', C'A', A'B'. Define B'' and 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)

X(5594) lies on these lines:
{2,6}, {4,639}, {5,1160}, {10,3640}, {76,5490}, {626,638}, {631,641}, {640,3090}, {642,3525}, {1162,1165}, {1267,3662}, {3535,5413}, {3661,5391}


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:
{2,6}, {4,640}, {5,1161}, {10,3641}, {76,5491}, {626,637}, {631,642}, {639,3090}, {641,3525}, {1163,1164}, {1267,3661}, {3536,5412}, {3662,5391}


X(5596) =  PERSPECTOR OF ARIES AND ANTICOMPLEMENTARY TRIANGLES

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 triangle ABC. Let A'' be the touchpoint of the A-excentral triangle 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 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(r + 4R)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.

If you have The Geometer's Sketchpad, you can view X(5597).

X(5597) lies on these lines: (pending)


X(5598) =  PERSPECTOR OF TRIANGLE 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).

If you have The Geometer's Sketchpad, you can view X(5598).

X(5598) lies on these lines: (pending)


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) lies on these lines: (pending)


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) lies on these lines: (pending)


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) lies on these lines: (pending)


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) lies on these lines: (pending)


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.

X(5603) lies on these lines: (pending)


X(5604) =  PERSPECTOR OF EULER 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) lies on these lines: (pending)


X(5605) =  PERSPECTOR OF EULER 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) lies on these lines: (pending)


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)

X(5606) lies on the circumcircle and these lines: {74,1385}, {229,759}, {2372,5253}

X(5606) = cevapoint of X(513) and X(3337)
X(5606) = trilinear pole of line X(6)X(3336)


X(5607) =  CENTER OF 1st DAO-MOSES CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)[31/2(a8 - 3a6b2 - 3a6c2 + 3a4b4 + 3a4c4 - a4b2c2 - a2b6 - a2c6 + 2b4c4) + 2S(a6 - 2b6 - 2c6 - 4a4b2 - 4a4c2 + 5a2b4 + 5a2c4 - 3a2b2c2 + 2b4c2 - 2b2c4)]

The 1st Dao-Moses 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 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) =  CENTER OF 2nd DAO-MOSES CIRCLE

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)[31/2(a8 - 3a6b2 - 3a6c2 + 3a4b4 + 3a4c4 - a4b2c2 - a2b6 - a2c6 + 2b4c4) - 2S(a6 - 2b6 - 2c6 - 4a4b2 - 4a4c2 + 5a2b4 + 5a2c4 - 3a2b2c2 + 2b4c2 - 2b2c4)]

The 1st Dao-Moses 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 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(5609) =  RADICAL TRACE OF 1st AND 2nd 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)

X(5609) lies on these lines: {3,74},{5,542},{23,1154},{30,3292},{113,137},{125,3628}


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)

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)


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)

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)


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)

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)>BR> 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    (pending)

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 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    (pending)

Referring to the construction of X(5619), the lines ABAC, BCBA, CACB concur in X(5623).    (Peter Moses, December 4, 2013)

X(5624) lies on 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 + ab + ac + bc)
X(5625) = X(1654) - 5*X(3616)

Suppose that P is a point in the plane of a triangle ABC. Let L be the line through P parallel to BC, and let BA = L∩AB and CA = L∩BC. Define CB and AC cyclically, and define AB and BC cyclically. Let UA be the line of the midpoint of segment ABAC, and define UB and UC cyclically. The lines UA, UB, UC concur in 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 X5640). 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) = 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 vertices of T7, T8, T9 are the points of intersection of pairs of perpendiculars at the corresponding trisectors that form T1, T2, T3, respectively. The vertices of T10, T11, T12 are the points of intersection of pairs of perpendiculars at the corresponding trisectors that form T4, T5, T6, respectively.

X(5628) = perspector of ABC and T7
X(5629) = perspector of ABC and T10
X(5630) = perspector of ABC and T8
X(5631) = perspector of ABC and T11
X(5632) = perspector of ABC and T9
X(5633) = perspector of ABC and T12
See also X(5634)-X(5640).

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(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(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(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(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(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(5634) =  7th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos(A/3) + 2 cos(B/3)cos(C/3) : cos(B/3) + 2 cos(C/3)cos(A/3) : cos(C/3) + 2 cos(A/3)cos(B/3)

X(5634) = perspector of each pair of these triangles: T1, T4, T7, as listed at X(5628).


X(5635) =  8th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(A - 2π)/3] + 2 cos[(B - 2π)/3]cos[(C - 2π)/3] : cos[(B - 2π)/3] + 2 cos[(C - 2π)/3]cos[(A - 2π)/3] : cos[(C - 2π)/3] + 2 cos[(A - 2π)/3]cos[(B - 2π)/3]

X(5635) = perspector of each pair of these triangles: T2, T5, T8, as listed at X(5628).


X(5636) =  9th MORLEY - VAN TIENHOVEN POINT

Trilinears    cos[(A - 4π)/3] + 2 cos[(B - 4π)/3]cos[(C - 4π)/3] : cos[(B - 4π)/3] + 2 cos[(C - 4π)/3]cos[(A - 4π)/3] : cos[(C - 4π)/3] + 2 cos[(A - 4π)/3]cos[(B - 4π)/3]

X(5636) = perspector of each pair of these triangles: T3, T6, T9, as listed at X(5628).


X(5637) =  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(5637) = perspector of the triangles T7 and T10 listed at X(5628).


X(5638) =  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(5638) = perspector of the triangles T8 and T11 listed at X(5628).


X(5639) =  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(5639) = perspector of the triangles T9 and T12 as listed at X(5628).


X(5640) =  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(5640).    (Christ van Tienhoven, January 3, 2014)


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(5626), in four points: A, B, C, and X5641).    (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)


This is the end of PART 4.

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