ENCYCLOPEDIA OF TRIANGLE CENTERS Part4

Encyclopedia of Triangle Centers - ETC

This is PART 4: Centers X(5001) - X(7000)

PART 1:	Introduction and Centers X(1) - X(1000)	PART 2:	Centers X(1001) - X(3000)	PART 3:	Centers X(3001) - X(5000)
PART 4:	Centers X(5001) - X(7000)	PART 5:	Centers X(7001) - X(10000)	PART 6:	Centers X(10001) - X(12000)
PART 7:	Centers X(12001) - X(14000)	PART 8:	Centers X(14001) - X(16000)	PART 9:	Centers X(16001) - X(18000)
PART 10:	Centers X(18001) - X(20000)	PART 11:	Centers X(20001) - X(22000)	PART 12:	Centers X(22001) - X(24000)
PART 13:	Centers X(24001) - X(26000)	PART 14:	Centers X(26001) - X(28000)	PART 15:	Centers X(28001) - X(30000)
PART 16:	Centers X(30001) - X(32000)	PART 17:	Centers X(32001) - X(34000)	PART 18:	Centers X(34001) - X(36000)
PART 19:	Centers X(36001) - X(38000)	PART 20:	Centers X(38001) - X(40000)	PART 21:	Centers X(40001) - X(42000)
PART 22:	Centers X(42001) - X(44000)	PART 23:	Centers X(44001) - X(46000)	PART 24:	Centers X(46001) - X(48000)
PART 25:	Centers X(48001) - X(50000)	PART 26:	Centers X(50001) - X(52000)	PART 27:	Centers X(52001) - X(54000)
PART 28:	Centers X(54001) - X(56000)	PART 29:	Centers X(56001) - X(58000)	PART 30:	Centers X(58001) - X(60000)
PART 31:	Centers X(60001) - X(62000)	PART 32:	Centers X(62001) - X(64000)	PART 33:	Centers X(64001) - X(66000)
PART 34:	Centers X(66001) - X(68000)	PART 35:	Centers X(68001) - X(70000)	PART 36:	Centers X(70001) - X(72000)

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

Barycentrics    (SB+SC)*(S*SB*SC + SA*sqrt(SA*SB*SC*SW)) : :
Barycentrics    S² - S_BS_C - k²(S² - 3S_BS_C) : : ,
              where k² = (3S_AS_BS_C - S²S_ω - 2(S_AS_BS_CS²S_ω)^1/2)/(9S_AS_BS_C - S²S_ω)
Barycentrics   SS_BS_C(3S_A - S_ω) + (S² - 3S_BS_C)(S_AS_BS_CS_ω)^1/2 : :      (Peter Moses, December 8, 2014)
Tripolars    Sqrt[SA] : :
X(5001) = (1 - k²)X(3) + k²X(4)
X(5001) = 3k²X(2) + (1 - 3k²)X(3)

As a point on the Euler line, X(5001) has Shinagawa coefficients (1 -k², -1 + 3k²).

X(5000) and X(5001) are the antiorthocorrespondents of X(6); i.e. they share the same orthocorrespondent, X(6). See Bernard Gibert, Table 55: X(5000, X(5001) and related curves.

For a construction of X(5000) and X(5001) see AdGeom 5185.

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

X(5000) lies on 8th Grozdev-Dekov-Parry circle, Dao-Moses-Telv circle, Moses radical circle, Stevanovic circle, Walsmith rectangular hyperbola; cubics K018, K270, K336, K337, K570, K608, K828, K829, K1091, K1092, K1129, K1133a, K1133b; curves Q019, Q021, Q024, Q026, Q037, Q049, Q054, Q098, Q115, Q116, Q117, Q118, Q144, Q146, Q147 and this line: {2,3}

X(5001) = isogonal conjugate of X(32619)
X(5001) = isotomic conjugate of X(42812)
X(5001) = complement of X(5003)
X(5001) = circumcircle-inverse of X(5000)
X(5001) = nine-point-circle-circumcircle-inverse of X(5000)
X(5001) = orthocentrodal-circle-inverse of X(5000)
X(5001) = orthoptic circle of Steiner inellipse-inverse of X(5000)
X(5001) = polar-circle-inverse of X(5000)
X(5001) = tangential-circle-inverse of X(5000)
X(5001) = MacBeath-inconc-inverse of X(5000)
X(5001) = Yff-hypergbola-inverse of X(5000)
X(5001) = Walsmith-rectangular-hyperbola-inverse of X(5000)
X(5001) = polar conjugate of X(41195)
X(5001) = complementary conjugate of the complement of X(34135)
X(5001) = antigonal conjugate of X(42810)
X(5001) = orthoassociate of X(5000)

X(5002) = 1st WALSMITH-MOSES POINT

Barycentrics    S² - S_BS_C + k²(S² - 3S_BS_C) : : ,
             where k² = (3S_AS_BS_C - S²S_ω - 4(S_AS_BS_CS²S_ω^)1/2)/(-9S_AS_BS_C + S²S_ω)
Tripolars    a : b : c
X(5002) = -3k²X(2) + (1 + 3k²)X(3)
X(5002) = (1 + k²)X(3) - k²X(4)

As a point on the Euler line, X(5002) has Shinagawa coefficients (1 +k², -1 - 3k²).

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

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

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

d_a = b² + c² - a², d_b = c² + a² - c², d_c = a² + b² - c².

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

x = a²d_a + k²[d_cv² + d_bw² - 2a²u²]
y = b²d_b + k²[d_aw² + d_cu² - 2b²v²]
z = c²d_c + k²[d_bu² + d_av² - 2c²w²],

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

f = - a²u²d_a - b²v²d_b - c²w²d_c
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) lies on this line: {2,3}

X(5002) = isogonal conjugate of X(34136)
X(5002) = antigonal conjugate of X(34240)
X(5002) = circumcircle-inverse of X(5003)

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

Barycentrics    S² - S_BS_C - k²(S² - 3S_BS_C) : : ,
               where k² = (3S_AS_BS_C - S²S_ω - 4(S_AS_BS_CS²S_ω^)1/2)/(-9S_AS_BS_C + S²S_ω)
Tripolars    a : b : c

As a point on the Euler line, X(5003) has Shinagawa coefficients (1 - k², -1 - 3k²).

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) lies on this line: {2,3}

X(5003) = isogonal conjugate of X(34135)
X(5003) = antigonal conjugate of X(34239)
X(5003) = circumcircle-inverse of X(5002)

X(5004) = 2nd WALSMITH-MOSES POINT

Barycentrics a²S[S² + (S_A)² - 4S_BS_C] + abc(S² - 3S_BS_C)(2S_ω)^1/2 : :
Tripolars Sqrt[b^2 + c^2] : :

As a point on the Euler line, X(5004) has Shinagawa coefficients ((E + 4F)S + abc(2S_ω)^1/2, -4(E + F)S - 3abc(2S_ω)^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 circumcircle-inverse of X(5005). The midpoint of X(5004) and X(5005) is X(23). Of the two points, X(5004) is the one outside the circumcircle. (Peter Moses, March 7, 2012)

X(5004) and X(5005) are the two points whose pedal antipodal perspectors (defined at Hyacinthos #20403 and #20405) are both X(6). (Randy Hutson, Febrary 20, 2015)

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

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

X(5004) = isogonal conjugate of X(34221)

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

Barycentrics a²S[S² + (S_A)² - 4S_BS_C] - abc(S² - 3S_BS_C)(2S_ω)^1/2 : :
Tripolars Sqrt[b^2 + c^2] : :

As a point on the Euler line, X(5005) has Shinagawa coefficients ((E + 4F)S - abc(2S_ω)^1/2, -4(E + F)S + 3abc(2S_ω)^1/2).

X(5005) is the circumcircle-inverse 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(5005) = isogonal conjugate of X(34222)

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)

Trilinears       3 sin A - cos A tan ω : :
Trilinears       cos A - 3 sin A cot ω : :
Trilinears       2 cos(A - 2ω) - cos(A + 2ω) - cos A : :
Barycentrics   a²(2a² + b² + c²): :

Let T be the symmedial triangle, and let H be the bicevian conic of X(2) and X(6). Let V be the triangle formed by the lines tangent to H at the vertices of T. Then T and V are perspective, and their perspector is X(5007). (Randy Hutson, February 20, 2015)

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(5007) = midpoint of X(61) and X(62)
X(5007) = isogonal conjugate of X(10159)
X(5007) = radical center of Lucas(-6 cot ω) circles
X(5007) = {X(371),X(372)}-harmonic conjugate of X(3098)
X(5007) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(3098)
X(5007) = radical trace of Brocard circle and circle {{X(4),X(194),X(3557),X(3558)}}
X(5007) = inverse-in-1st-Brocard-circle of X(7772)
X(5007) = X(39)-of-5th-anti-Brocard-triangle
X(5007) = X(187)-of-X(6)PU(1)
X(5007) = polar conjugate of isotomic conjugate of X(22352)
X(5007) = {X(32),X(39)}-harmonic conjugate of X(187)

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

Barycentrics a²(4a² + b² + 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(5008) = complement of X(7850)
X(5008) = crossdifference of every pair of points on line X(523)X(7840)

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(5009) = isogonal conjugate of the isotomic conjugate of X(33295)
X(5009) = isogonal conjugate of the polar conjugate of X(31905)

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

Trilinears 2a(b^2 + c^2 - a^2) + abc : :
Barycentrics 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}, {2, 5074}, {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(5011) = midpoint of X(1276) and X(1277)
X(5011) = reflection of X(101) in antiorthic axis
X(5011) = complement of X(5195)
X(5011) = anticomplement of X(5074)
X(5011) = X(187)-of-excentral-triangle

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) = is the pole of Brocard axis wrt the ellipse {{X(3), X(6), X(24), X(60), X(143), X(1511), X(1986)} (Randy Hutson, February 20, 2015)

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(5012) = isogonal conjugate of X(3613)

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

Trilinears     sin A + 2 cos A tan ω : :
Trilinears     2 cos A + sin A cot ω : :
Trilinears     2 cos A sin ω + sin A cos ω :
Trilinears    a + 4R cos A tan ω : :
Barycentrics    a²(3b² + 3c² - a²) : :

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

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

X(5013) = isogonal conjugate of X(5395)
X(5013) = radical center of Lucas(cot ω) circles
X(5013) = {(X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,3053), (3,39,6), (371,372,5050), (8406,8414,182)
X(5013) = insimilicenter of circumcircle and (1/2)-Moses circle

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(5014) = anticomplement of X(3744)

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(5016) = anticomplement of X(37539)

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

Trilinears sin A - cos A sin 2ω : :
Barycentrics 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(5017) = reflection of X(6) in X(32)

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    a³(a² + ab + ac + 2bc) : :
Barycentrics    a^2 (a^2 + 4 R r) : :

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(5019) = isogonal conjugate of X(34258)
X(5019) = crosspoint of X(i) and X(j) for these {i,j}: {2, 20029}, {56, 2221}, {1252, 34074}
X(5019) = crosssum of X(i) and X(j) for these (i,j): {2, 5739}, {6, 11337}, {8, 2345}, {386, 573}, {1086, 4801}
X(5019) = crossdifference of every pair of points on line X(523)X(4391)
X(5019) = {X(371),X(372)}-harmonic conjugate of X(970)

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)

As a point on the Euler line, X(5020) has Shinagawa coefficients (E - F,E + F).

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(5020) = circumcircle-inverse of X(37897)
X(5020) = homothetic center of medial triangle and 3rd antipedal triangle of X(3)
X(5020) = trilinear pole of polar of X(3527) wrt 2nd Lemoine circle
X(5020) = {X(8854),X(8855)}-harmonic conjugate of X(6)
X(5020) = {X(2),X(3)}-harmonic conjugate of X(16419)

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

Trilinears cos A - sin A (csc A + csc B + csc C) : : Barycentrics 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(5021) = isogonal conjugate of X(32022)
X(5021) = crosssum of X(2) and X(391)

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

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

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

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

Trilinears    3 sin(A + ω) - 5 sin(A - ω) : :
Trilinears    4 cos A - sin A cot ω : :
Trilinears    sin A - 4 cos A tan ω : :
Barycentrics    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(5023) = midpoint of X(1151) and X(1152)
X(5023) = isogonal conjugate of X(38259)
X(5023) = center of inverse-in-circumcircle-of-Moses-circle
X(5023) = center of inverse-in-circumcircle-of-2nd-Lemoine-circle
X(5023) = {X(371),X(372)}-harmonic conjugate of X(5093) X(5023) = radical center of Lucas(-1/2 cot ω) circles

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

Trilinears    5 sin(A + ω) - sin(A - ω) : :
Trilinears    3 cos A + 2 sin A cot ω : :
Trilinears    2 sin A + 3 cos A tan ω : :
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)

The conic described above is an ellipse with major axis X(512)X(5024). (Randy Hutson, February 20, 2015)

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(5024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1384), (3,39,9605), (6,574,3)
X(5024) = radical center of Lucas(4/3 cot ω) circles
X(5024) = insimilicenter of (1/2)-Moses and Stammler circles; the exsimilicenter is X(9605)

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

As a point on the Euler line, X(5025) has Shinagawa coefficients ((E + F)² - S², -2S²).

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(5025) = midpoint of X(7807) and X(33229)
X(5025) = reflection of X(3552) in X(7807)
X(5025) = reflection of X(33246) in X(2)
X(5025) = complement of X(3552)
X(5025) = anticomplement of X(7807)
X(5025) = {X(2),X(3)}-harmonic conjugate of X(7907)
X(5025) = {X(2),X(4)}-harmonic conjugate of X(384)
X(5025) = {X(2),X(5)}-harmonic conjugate of X(16921)
X(5025) = {X(2),X(20)}-harmonic conjugate of X(16925)

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(5026) = X(187)-of-1st-Brocard-triangle
X(5026) = 1st-Brocard-isogonal conjugate of X(599)

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

Barycentrics a²(b² - c²)(a⁴ - b²c²) : :

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

X(5027) = isogonal conjugate of X(18829)
X(5027) = cevapoint of X(2491) and X(9429)
X(5027) = crosspoint of X(i) and X(j) for these {i,j}: {6, 805}, {25, 685}, {99, 3225}, {110, 1976}, {385, 880}, {733, 4577}, {2966, 14382}
X(5027) = crosssum of X(i) and X(j) for these {i,j}: {2, 804}, {69, 684}, {325, 523}, {512, 3229}, {694, 881}, {732, 3005}, {812, 6682}, {1634, 2421}, {3569, 14251}
X(5027) = crossdifference of every pair of points on line X(2)X(694)
X(5027) = trilinear pole of line X(2086)X(2679)
X(5027) = inverse-in-Parry-circle of X(669)
X(5027) = inverse-in-2nd-Lemoine-circle of X(2456)
X(5027) = radical center of circumcircle, Brocard circle, Brocard circle of 1st Brocard triangle
X(5027) = radical center of Brocard circles of ABC, 1st Brocard triangle, 1st anti-Brocard triangle
X(5027) = (Lemoine axis of ABC)∩(Lemoine axis of the 1st Brocard triangle)
X(5027) = X(182)-of-1st-Parry-triangle
X(5027) = inverse-in-Parry-isodynamic-circle of X(3231); see X(2)
X(5027) = X(669)-of-1st-Brocard-triangle
X(5027) = 1st-Brocard-isogonal conjugate of X(30229)
X(5027) = intersection of perspectrix of ABC and 1st Brocard triangle (line X(804)X(4107)) and perspectrix of ABC and 1st anti-Brocard triangle (line X(187)X(237))
X(5027) = trilinear product X(i)*X(j) for these {i,j}: {31, 804}, {42, 4164}, {171, 4455}, {213, 4107}, {238, 7234}, {385, 798}, {419, 810}, {512, 1580}, {523, 1933}, {560, 14295}, {661, 1691}, {662, 2086}, {667, 4039}, {669, 1966}, {875, 4154}, {880, 4117}, {923, 11183}, {1577, 14602}, {1918, 14296}, {1924, 3978}, {1926, 9426}, {2210, 2533}, {2236, 18105}, {2295, 8632}, {3573, 4128}, {3747, 4367}, {4010, 7122}

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

Trilinears 2 sin(A + 2ω) + sin(A - 2ω) - sin A : :
Barycentrics a²(a⁴ + 2b⁴ + 2c⁴ - a²b² - a²c²) : :

Let P1′ and U1′ be the 2nd-Lemoine-circle-inverses of P(1) and U(1), resp. Then X(5028) = P(1)U1′∩U(1)P1′. (Randy Hutson, January 17, 2020)

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(5028) = reflection of X(32) in X(6)

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(5029) = isogonal conjugate of X(35148)
X(5029) = complement of anticomplementary conjugate of X(39368)
X(5029) = crossdifference of every pair of points on line X(2)X(846)
X(5029) = inverse-in-Parry-isodynamic-circle of X(902); see X(2)

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 b^6 + c^2 + 2 a^2 b^2 c^2 : :
X(5031) = X[141] + 2 X[625], X[2076] - 5 X[3763], X[316] + 5 X[3763], 7 X[3619] - X[5104], 2 X[3631] + X[5107], 3 X[2] + X[5207], X[2458] - 5 X[7867], 5 X[3620] + X[8586], 5 X[7925] + X[11646], 5 X[7925] - X[12215], 3 X[599] + X[15514].

X(5031) lies on these lines: {2,1501}, {5,141}, {6,7752}, {69,5111}, {114,1503}, {115,698}, {125,9152}, {182,7862}, {187,6292}, {316,2076}, {325,732}, {524,1570}, {542,13196}, {599,15514}, {1352,2456}, {1506,1692}, {2021,8362}, {2024,3815}, {2080,7800}, {2458,7867}, {3094,5025}, {3098,7825}, {3619,5104}, {3620,8586}, {3631,5107}, {3788,3818}, {4074,5133}, {5017,7773}, {5039,7775}, {5116,7769}, {5162,7822}, {5969,6393}, {6656,10007}, {6680,16385}, {7746,8177}, {7777,13331}, {7778,10516}, {7785,12212}, {7789,18860}, {7810,10631}, {7815,14693}, {7821,14994}, {7832,24273}, {7842,14810}, {7925,11646}, {14603,18896}, {14712,16898}, {17047,20255}

X(5031) = crosspoint of X(2) and X(18896)
X(5031) = crosssum of X(6) and X(14602)
X(5031) = nine-point-circle-inverse of X(626)
X(5031) = complement of the isogonal conjugate of X(1916)
X(5031) = complement of the isotomic conjugate of X(18896)
X(5031) = X(i)-complementary conjugate of X(j) for these (i,j): {1, 5976}, {2, 19563}, {256, 17755}, {257, 17793}, {694, 37}, {733, 16600}, {805, 14838}, {882, 16592}, {1577, 2679}, {1581, 2}, {1916, 10}, {1927, 8265}, {1934, 141}, {1956, 14382}, {1967, 39}, {7018, 20333}, {9468, 16584}, {14970, 1215}, {17980, 16583}, {18829, 4369}, {18896, 2887}
X(5031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5207, 1691), (5, 141, 24256), (626, 24206, 141), (2039, 2040, 626), (3788, 3818, 4048), (5403, 5404, 18806)

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(5033) = {X(1687),X(1688)}-harmonic conjugate of X(1351)

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

Barycentrics a²(a⁴ - 3a²b² - 3a²c² - 4b²c²) : :

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(5034) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5052), (1687,1688,5050)
X(5034) = inverse-in-1st-Brocard-circle of X(5052)
X(5034) = pole of Lemoine axis wrt 1st Lemoine circle

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)

Trilinears    2 sin A - sin(A - 2ω) : :
Trilinears    a - R sin(A - 2ω) : :
Barycentrics    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(5038) = {X(1687),X(1688)}-harmonic conjugate of X(575)
X(5038) = Brocard axis intercept, other than X(2080), of the circle {{X(2080),PU(1)}}
X(5038) = harmonic center of 2nd Lemoine circle and Ehrmann circle

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(5039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,5188), (1687,1688,5085)

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

Barycentrics 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(5040) = isogonal conjugate of X(35147)
X(5040) = crossdifference of every pair of points on line X(2)X(3125)
X(5040) = inverse-in-Parry-isodynamic-circle of X(3230); see X(2)

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

Trilinears    3 sin(A + ω) + 2 sin(A - ω) : :
Trilinears    cos A + 5 sin A cot ω : :
Trilinears    5 sin A + cos A tan ω : :
Barycentrics    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(5041) = center of inverse-in-Moses-circle-of-circumcircle
X(5041) = radical center of Lucas(10 cot ω) circles

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)

Trilinears        R cos A - s cot A/2 : :
Trilinears        4s^2(b + c - a) - a(b^2 + c^2 - a^2) : :
Barycentrics   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(5044) = complement of X(942)
X(5044) = centroid of {A,B,C,X(72)}

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)

Let A′ be the inverse-in-incircle of the A-excenter, and define B′, C′ cyclically. The triangle A′B′C′ is homothetic to ABC, and its circumcenter is X(5045). (Randy Hutson, February 16, 2015)

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(5045) = complement of X(34790)
X(5045) = X(140)-of-intouch-triangle
X(5045) = X(5) of inverse-in-incircle-triangle

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

As a point on the Euler line, X(5046) has Shinagawa coefficients (abc$a$, - 4S²), and also (R, -2r).

X(5046) lies on these lines: {1, 5080}, {2, 3}, {8, 80}, {10, 3583}, {11, 2975}, {12, 1621}, {35, 3814}, {36, 3825}, {65, 5057}, {78, 3586}, {79, 5883}, {100, 1329}, {115, 5985}, {145, 497}, {153, 944}, {191, 3467}, {210, 5178}, {312, 5016}, {324, 1896}, {341, 5014}, {355, 3877}, {388, 1388}, {519, 4857}, {535, 5563}, {551, 5270}, {908, 950}, {938, 5905}, {952, 5330}, {960, 5086}, {962, 5554}, {1043, 5741}, {1058, 3623}, {1125, 3585}, {1210, 3218}, {1478, 3616}, {1724, 5127}, {1749, 3648}, {1837, 3869}, {1842, 3101}, {1877, 4296}, {1994, 3193}, {2551, 3434}, {2646, 5087}, {2886, 5260}, {3057, 5176}, {3419, 3876}, {3421, 3621}, {3614, 6690}, {3701, 5015}, {3816, 5253}, {3822, 5259}, {3868, 5722}, {3890, 5252}, {3897, 5886}, {3924, 3944}, {4294, 5552}, {4297, 4881}, {4313, 5748}, {4514, 4696}, {4666, 5290}, {4678, 5082}, {4679, 5794}, {4723, 5100}, {5180, 5903}, {5250, 5587}, {5283, 5475}, {5303, 5433}, {5318, 5367}, {5321, 5362}, {5422, 5706}, {5445, 6702}, {5731, 6256}

X(5046) = complement of X(37256)

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)

As a point on the Euler line, X(5047) has Shinagawa coefficients (3abc$a$ + 2S², -2S²).

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(5047) = anticomplement of X(17529)

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

Barycentrics a(b + c - a)(3b² + 3c² - 2a² + ab + ac - 6bc) : :
X(5048) = (R - 3r)*X(1) + r*X(3)

Let I be the incenter of a triangle ABC, let N_A be the nine-point circle of the triangle IBC, and define N_B and N_C cyclically. Let R_A be the reflection of N_A in the line AI, and define R_B and R_C cyclically. Then X(5048) is the radical center of the circles R_A, R_B, R_C. See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos #22502, July 7, 2014. For another construction see Antreas Hatzipolakis and Peter Moses, Hyacinthos #28076.

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(5048) = reflection of X(1319) in X(1)
X(5048) = inverse-in-incircle of X(3057)
X(5048) = X(36) of Mandart-incircle triangle
X(5048) = homothetic center of intangents triangle and reflection of extangents triangle in X(36)

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)

Let A′ be the midpoint of X(1) and A-intouch point. Define B′, C′ cyclically. The centroid of A′B′C′ is X(5049). (Randy Hutson, February 16, 2015)

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(5049) = X(2)-of-incircle-circles-triangle

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

Trilinears        a + R cos A cot ω : b + R cos B cot ω : c + R cos C cot ω
Trilinears        2 sin A + cos A cot ω : 2 cos B + cos B cot ω : 2 cos C + cos C cot ω
Trilinears        cos A + 2 sin A tan ω : cos B + 2 sin B tan ω : cos C + 2 sin C tan ω
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) = X(3) + 2X(6)

Let A′B′C′ be the reflection of ABC in X(6). Let A_B = BC∩A′C′ and define B_C and C_A cyclically. Let A_C = BC∩A′B′, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_C, B_A, C_B lie on the 2nd Lemoine circle. Let A″ be the point of intersection of the tangents to the 2nd Lemoine circle at B_A and C_A, and define B″, C″ cyclically. The centroid of triangle A″B″C″ is X(5050). X(5050) and X(5085) trisect the segment X(3)X(6). Also, X(5050) is the exsimilicenter of the circle with center X(371) and pass-through point X(1151) and the circle with center X(372) and pass-through point X(1152). (Randy Hutson, January 29, 2015)

Continuing, the triangles AA_BA_C, BB_CB_A, CC_AC_B are pairwise similar and and each inversely similar to ABC. Let S_A be the similitude center of BB_CB_A and CC_AC_B, and define S_B and S_C cyclically. Then the triangle S_AS_BS_C is perspective to ABC at X(6) and homothetic to the circumsymmedial triangle at X(6). Moreover, X(5050) = X(3)-of-S_AS_BS_C. (Randy Hutson, October 13, 2015)

X(5050): Let T be a triangle inscribed in the circumcircle and circumscribing the orthic inconic. As T varies, its centroid traces a circle centered at X(5050) with segment X(2)X(14912) as diameter. (Randy Hutson, August 29, 2018)

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(5050) = reflection of X(5085) in X(182)
X(5050) = radical center of the Lucas(4 tan ω) circles
X(5050) = center of inverse-in-1st-Lemoine-circle-of-circumcircle
X(5050) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1351), (371,372,5013), (1687,1688,5034)

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)

As a point on the Euler line, X(5051) has Shinagawa coefficients (2(E + F)² + 2$bc$(E + F) + abc$a$, -2S²).

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(5052) = reflection of X(39) in X(6)
X(5052) = inverse-in-1st-Brocard-circle of X(5034)
X(5052) = bicentric sum of PU(191)
X(5052) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,5034), (371,372,5171)

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 5a⁴ + 2b⁴ + 2c⁴ - 7a²b² - 7a²c² - 4b²c² : :
X(5054) = 4*X(381) - X(382)

As a point on the Euler line, X(5054) has Shinagawa coefficients (7, -3).

Let T be a triangle inscribed in the circumcircle and circumscribing the Steiner inellipse. As T varies, its centroid traces a circle centered at X(5054) with segment X(2)X(3524) as diameter. (Randy Hutson, August 29, 2018)

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(5054) = complement of X(3545)
X(5054) = anticomplement of X(15699)
X(5054) = circumcircle-inverse of X(37967)
X(5054) = trisector nearest X(2) of segment X(2)X(3)
X(5054) = centroid of X(3)PU(116)
X(5054) = centroid of X(3)PU(117)
X(5054) = centroid of X(3)PU(118
) X(5054) = centroid of X(3)PU(119)

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) = 4 X(2) - X(3) = 5 X(3) + 4 X(4)

As a point on the Euler line, X(5055) has Shinagawa coefficients (5, 3).

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(5055) = homothetic center of Johnson triangle and mid-triangle of Euler and anti-Euler triangles
X(5055) = inverse-in-circle-O(PU(5)) of X(20)
X(5055) = homothetic center of X(140)-altimedial and X(3)-anti-altimedial triangles
X(5055) = {X(2),X(3)}-harmonic conjugate of X(15694)
X(5055) = {X(2043),X(2044)}-harmonic conjugate of X(546)
X(5055) = trisector nearest X(5) of segment X(2)X(5)
X(5055) = trisector nearest X(2) of segment X(2)X(381)
X(5055) = homothetic center of Ehrmann side-triangle and submedial triangle
X(5055) = X(3839)-of-Ehrmann-side-triangle
X(5055) = X(3839)-of-submedial-triangle

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

As a point on the Euler line, X(5056) has Shinagawa coefficients (3, 2).

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(5057) = anticomplement of X(1155)
X(5057) = intersection of antiorthic axes of 1st and 2nd Ehrmann circumscribing triangles
X(5057) = intersection of antiorthic axes of anticevian triangles of PU(4)

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

Trilinears    a²(aR - bc) : :    César Lozada (9/07/2013)
Trilinears    cos A cos ω + 2 sin A sin ω : :
Trilinears    2 sin A sin ω - sin(A - ω) : :
Trilinears    cos A + 2 sin A tan ω : :
Trilinears    2 sin A + cos A cot ω : :

Barycentrics    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(6),X(32)}-harmonic conjugate of X(5062)
X(5058) = perspector of symmedial triangle and Lucas Brocard triangle
X(5058) = radical center of Lucas(4 tan ω) circles
X(5058) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9738)
X(5058) = {X(371),X(372)}-harmonic conjugate of X(9738)

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

As a point on the Euler line, X(5059) has Shinagawa coefficients (3, -8).

Let J be the conic {{X(13),X(14),X(15),X(16),X(20)}}. Let U be the line tangent to J at X(15), and let V be the line tangent to J at X(16). Then X(5059) = U∩V. (Randy Hutson, February 16, 2015)

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(5059) = anticomplement of X(3146)
X(5059) = pole of Brocard axis wrt conic {{X(13),X(14),X(15),X(16),X(20)}}
X(5059) = polar conjugate of isogonal conjugate of X(33636)

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) : :    César Lozada (9/07/2013)
Trilinears    2 sin A sin ω + sin(A - ω) : :
Trilinears    cos A - (2 + cot ω) sin A : :
Barycentrics    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(6),X(32)}-harmonic conjugate of X(5058)
X(5062) = perspector of symmedial triangle and Lucas(-1) Brocard triangle
X(5062) = radical center of Lucas(-4 - 2 cot ω) circles
X(5062) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9739)
X(5062) = {X(371),X(372)}-harmonic conjugate of X(9739)

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

Barycentrics a⁴(a⁴ + b⁴ + c⁴ - 2a²b² -2a²c² + 4b²c²) : :

Let A′B′C′ be the Trinh triangle. Let A″ be the barycentric product B′*C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5063). (Randy Hutson, October 13, 2015)

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(5063) = isogonal conjugate of X(34289)
X(5063) = crosssum of X(6) and X(6644)
X(5063) = crossdifference of every pair of points on line X(523)X(11799)
X(5063) = Schoute-circle-inverse of X(37470)
X(5063) = {X(15),X(16)}-harmonic conjugate of X(37470)

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

Trilinears    3 sec A + csc A tan ω : :
Trilinears    csc A + 3 sec A cot ω : :
Barycentrics   (a² - b² + c²)(a² + b² - c²)(a² + 2b² + 2c²) : :
Barycentrics    (sec A)[2 sin(A + ω) + sin(A - ω)] : :
Barycentrics    3 tan A + tan ω : :
Barycentrics    3 tan A cot ω + 1 : :

As a point on the Euler line, X(5064) has Shinagawa coefficients (F, 3E + 3F).

Let W be the orthocentroidal circle, L the line tangent to W at the P(4)-Ceva conjugate of U(4) and M the line tangent to W at the U(4)-Ceva conjugate of P(4). Then X(5064) = L∩M. (Randy Hutson, February 16, 2015)

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(5064) = pole wrt orthocentroidal circle of line X(428)X(523)
X(5064) = {X(2),X(4)}-harmonic conjugate of X(428)
X(5064) = homothetic center of orthic triangle and reflection of tangential triangle in X(2)

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

Barycentrics a^4 (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 + 6 b^2 c^2) : :

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(5065) = isogonal conjugate of X(37874)
X(5065) = crosspoint of X(2) and X(14457)
X(5065) = crosssum of X(i) and X(j) for these {i,j}: {2, 11433}, {6, 17928}, {1146, 2517}
X(5065) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(9729)
X(5065) = {X(371),X(372)}-harmonic conjugate of X(9729)

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) = 7 X(2) - 3 X(3) = X(2) - 3 X(5) = X(5) + X(381) = X(2) + X(3) + 2 X(4) = 2 X(5) - X(547) = X(5) - 2 X(11737)

As a point on the Euler line, X(5066) has Shinagawa coefficients (5, 9).

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(5066) = reflection of X(i) in X(j) for these (i,j): (5,11737), (547,5)
X(5066) = complement of X(8703)
X(5066) = {X(2),X(3)}-harmonic conjugate of X(15713)
X(5066) = {X(2),X(4)}-harmonic conjugate of X(3534)
X(5066) = {X(2),X(5)}-harmonic conjugate of X(10109)
X(5066) = midpoint of X(5) and X(381)

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

Trilinears       4 cos A + 5 cos B cos C : :
Trilinears       5 sec A + 4 sec B sec C : :
Barycentrics   3a⁴ + 5b⁴ + 5c⁴ - 8a²b² - 8a²c² - 10b²c² : :
Barycentrics    cot B cot C + 4 : :

As a point on the Euler line, X(5067) has Shinagawa coefficients (4, 1).

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

X(5067) = {X(3316),X(3317)}-harmonic conjugate of X(6)

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

Barycentrics 7b⁴ + 7c⁴ - a⁴ - 6a²b² - 6a²c² - 14b²c² : :
Trilinears 4 cos B cos C + 3 sin B sin C : :

As a point on the Euler line, X(5068) has Shinagawa coefficients (3, 4).

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

X(5068) = {X(3544),X(3545)}-harmonic conjugate of X(5)

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
Barycentrics cot B cot C + 7 : :

As a point on the Euler line, X(5070) has Shinagawa coefficients (7, 1).

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

X(5070) = homothetic center of X(140)-altimedial and X(2)-anti-altimedial triangles

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) = 7 X(2) - 2 X(3)

As a point on the Euler line, X(5071) has Shinagawa coefficients (4, 3).

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

X(5071) = centroid of cross triangle of Euler and anti-Euler triangles
X(5071) = {X(2),X(3)}-harmonic conjugate of X(15709)
X(5071) = {X(3090),X(3545)}-harmonic conjugate of X(2)

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

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

As a point on the Euler line, X(5072) has Shinagawa coefficients (5, 7).

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

X(5072) = {X(3090),X(3091)}-harmonic conjugate of X(546)

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

Barycentrics 4b⁴ + 4c⁴ - 7a⁴ + 3a²b² + 3a²c² - 8b²c²
X(5073) = 3*X(381) - 5*X(382)

As a point on the Euler line, X(5073) has Shinagawa coefficients (3, -11).

Let O_A be the circle centered at A with radius 2*sqrt(b² + c²), and define O_B and O_C cyclically. Then X(5073) is the radical center of O_A, O_B, O_C. (Randy Hutson, February 16, 2015)

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

X(5073) = circumcircle-inverse of X(37968)
X(5073) = {X(381),X(382)}-harmonic conjugate of X(3627)

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

Barycentrics a³b + a³c - 2a²bc - b⁴ + b³c + bc³ - c⁴ : :

X(5074) lies on these lines: {2, 5011}, {3, 142}, {101, 4872}, {116, 517}, {226, 1323}, {304, 4153}, {514, 661}

X(5074) = isotomic conjugate of X(37213)
X(5074) = complement of X(5011)

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

Barycentrics a²(b - c)(a³ + b³ + c³ - 2a²b - 2a²c + abc) : :

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

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

X(5075) = isogonal conjugate of X(35154)
X(5075) = crossdifference of every pair of points on line X(2)X(9317)
X(5075) = inverse-in-Parry-isodynamic-circle of X(1055); see X(2)

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

Barycentrics 6b⁴ + 6c⁴ - 7a⁴ + a²b² + a²c² - 12b²c² : :
X(5076) = 3*X(381) + 2*X(382)

As a point on the Euler line, X(5076) has Shinagawa coefficients (1, -13).

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

X(5076) = {X(381),X(382)}-harmonic conjugate of X(1657)

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

As a point on the Euler line, X(5077) has Shinagawa coefficients (2(E + F)² + 3S², -9S²).

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

X(5077) = Artzt-to-anti-Artzt similarity image of X(3)

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

As a point on the Euler line, X(5079) has Shinagawa coefficients (7, 5).

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

X(5079) = complement of X(10299)
X(5079) = homothetic center of X(5)-altimedial and X(20)-anti-altimedial triangles
X(5079) = {X(2),X(4)}-harmonic conjugate of X(3530)
X(5079) = {X(2),X(5)}-harmonic conjugate of X(3851)

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(5080) = isogonal conjugate of X(34442)
X(5080) = isotomic conjugate of isogonal conjugate of X(20989)
X(5080) = polar conjugate of isogonal conjugate of X(22123)
X(5080) = anticomplement of X(36)
X(5080) = inverse-in-anticomplementary-circle of X(8)

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(5081) = anticomplement of isogonal conjugate of X(36121)

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) is the center of the rectangular hyperbola, H, that is the locus of perspectors of the pedal triangle of X(1) (the intouch triangle) and A′B′C′, where AA′/AX(1) = BB′/BX(1) = CC′/CX(1). Also, H is the Jerabek hyperbola of the intouch triangle, and H passes through X(1), X(7), X(65), X(145), X(224), X(1071), X(1317), X(1537), X(3174), X(3649), X(5586), and the vertices of the intouch triangle. (Randy Hutson, February 16, 2015)

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(5083) = X(125)-of-intouch-triangle
X(5083) = midpoint of X(i) and X(j) for these {i,j}: {65,1317}, {1071, 1537}

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

As a point on the Euler line, X(5084) has Shinagawa coefficients (abc$a$, -S²).

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)

Trilinears        a + 4R cos A cot ω : b + 4R cos B cot ω : c + 4R cos C cot ω
Trilinears        sin A + 2 cos A cot ω : cos B + 2 cos B cot ω : cos C + 2 cos C cot ω
Trilinears        2 cos A + sin A tan ω : 2 cos B + sin B tan ω : 2 cos C + sin C tan ω
Trilinears        2 cos A cos ω + sin A sin ω : 2 cos B cos ω + sin B sin ω : 2 cos C cos ω + sin C sin ω 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) = 2X(3) + X(6)

X(5085) and X(5050) trisect the segment X(3)X(6). Also, X(5085) is the exsimilicenter of the circle with center X(1151) and pass-through point X(371) and the circle with center X(1152) and pass-through point X(372). (Randy Hutson, January 29, 2015)

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(5085) = reflection of X(5050) in X(182)
X(5085) = radical center of the Lucas(tan ω) circles
X(5085) = centroid of the triangle X(4)X(6)X(20)
X(5085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,1350), (1687,1688,5039)

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(5087) = complement of X(1155)

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)

Trilinears       b + c - a sec A : :
Trilinears       (b² + c² - ab - ac)/(b² + c² - a²) : :
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(5089) = isogonal conjugate of X(1814)
X(5089) = PU(4)-harmonic conjugate of X(650)
X(5089) = pole wrt polar circle of trilinear polar of X(2481) (the line X(2)X(650))
X(5089) = X(48)-isoconjugate (polar conjugate) of X(2481)

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) is the point of intersection, other than X(3), of the Brocard circle and the line X(1)X(3). (Randy Hutson, February 16, 2015)

X(5091) lies on the Brocard circle and 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(5091) = orthogonal projection of X(6) on line X(1)X(3)
X(5091) = X(100)-of-1st-Brocard-triangle
X(5091) = 1st-Brocard-isogonal conjugate of X(2787)
X(5091) = 1st-Brocard-isotomic conjugate of X(24290)

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

Trilinears       sin A + 3 cos A cot ω : :
Trilinears       3 cos A + sin A tan ω : :
Trilinears       2 sin(A - 2ω) - sin(A + 2ω) - sin A : :
Barycentrics   a²(b⁴ + c⁴ - 2a⁴ + a²b² + a²c² + 4b²c²) : :
X(5092) = 3X(3) + X(6) = 2X(3) + X(575)

X(5092) is the centroid of the quadrilateral {{X(3),X(4),X(6),X(20)}}. (Randy Hutson, Febrary 16, 2015)

In the plane of a triangle ABC, let
O = X(3) = circumcenter
K = X(6) = symmedian point
AbAc = line through K parallel to BC, and define BcBa and CaCb cyclically
U = circumcircle of A, Ab, Ac
V = circumcircle of B, C, Cb, Bc
{A1, A2} = U∩V, and define {B1, B2} and {C1, C2} cyclically.

The six points A1, A2, B2, B2, C2, C3 lie on a circle, here named the Dao-symmedial circle, with center X(5092). See Dao-symmedial circle, in which X(5092) is labeled as M. (Dao Thanh Oai, February 11, 2017)

Let u be the radius of the Dao-symmedial circle. Then

u² = R²/2 + |OK|²/16
u² = 3R²(3 - (S/S_ω)²) /16
u² = 3R²(3 - tan²ω) /16
(César Lozada, February 13, 2017)

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(5092) = midpoint of X(3) and X(182)
X(5092) = reflection of X(575) in X(182)
X(5092) = {X(3),X(6)}-harmonic conjugate of X(3098)
X(5092) = intersection of Brocard axes of ABC and X(2)-Brocard triangle
X(5092) = {X(15),X(16)}-harmonic conjugate of X(39)
X(5092) = X(575)-Gibert-Moses centroid; see the preamble just before X(21153)

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

Trilinears 2a - R cos A cot ω : :
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(5093) = isogonal conjugate of isotomic conjugate of X(34803)
X(5093) = centroid of polar triangle of 2nd Lemoine circle
X(5093) = center of inverse-in-2nd-Lemoine-circle-of-circumcircle
X(5093) = {X(371),X(372)}-harmonic conjugate of X(5023)
X(5093) = pole of Lemoine axis wrt 2nd Lemoine circle

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

Trilinears       a² sec A + 4bc : :
Trilinears       sec A + 3 csc A tan ω : :
Trilinears       3 csc A + sec A cot ω : :
Barycentrics   (2b² + 2c² - a²)(a² + b² - c²)(a² - b² + c²) : :

As a point on the Euler line, X(5094) has Shinagawa coefficients (3F, E + F).

Let L_A be the polar of X(3) wrt the A-power circle, and define L_B and L_C cyclically. Let A′ = L_B∩L_C, B′ = L_C∩L_A, C′ = L_A∩B_C. The triangle A′B′C′ is homothetic to ABC, and the center of homothety is X(5094). (Randy Hutson, February 16, 2015)

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(5094) = complement of X(7493)
X(5094) = circumcircle-inverse of X(37969)
X(5094) = intersection of tangents to orthocentroidal circle at PU(4)
X(5094) = pole of orthic axis wrt orthocentroidal circle
X(5094) = pole wrt polar circle of trilinear polar of X(598) (the line X(351)X(523))
X(5094) = X(48)-isoconjugate (polar conjugate) of X(598)
X(5094) = harmonic center of polar circle and {circumcircle, nine-point circle}-inverter
X(5094) = homothetic center of orthic triangle and X(2)-Ehrmann triangle; see X(25)
X(5094) = Euler line intercept, other than X(381), of circle {X(381),PU(4)}
X(5094) = homothetic center of the AOA and AAOA triangles

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

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

X(5095) lies on the orthic inconic and 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(5095) = reflection of X(125) in X(6)
X(5095) = isogonal conjugate of X(15398)
X(5095) = complement of X(32244)
X(5095) = anticomplement of X(32257)
X(5095) = orthic-isogonal conjugate of X(468)
X(5095) = orthic-syngonal conjugate of X(6)
X(5095) = X(1156)-of-orthic-triangle if ABC is acute
X(5095) = crossdifference of every pair of points on the line tangent to the MacBeath circumconic at X(895)

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 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(5097) = midpoint of X(6) and X(576)
X(5097) = reflection of X(575) in X(6)
X(5097) = {X(371),X(372)}-harmonic conjugate of X(5206)

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 X(5099). Seiichi Kirikami, September 25, 2013.

X(5099) is the center of hyperbola H = {{A,B,C,X(4),X(23)}}. H is the isogonal conjugate of line X(3)X(67) and the isotomic conjugate of line X(67)X(69). Also, H passes through X(316), X(842), X(1383). Moreover, H intersects the circumcircle at X(842) and is tangent to line X(4)X(67) at X(4). (Randy Hutson, January 29, 2015)

As a line L varies through the set of all lines that pass through X(2492), the locus of the trilinear pole of L is a circumconic, and its center is X(5099). (Randy Hutson, January 29, 2014)

Let A′B′C′ be the orthic triangle. Let La be the Fermat axis of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5099), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)

X(5099) lies on the nine-point circle, the 2nd Steiner circle, the cevian circle of X(23), and 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(5099) = midpoint of X(i) and X(j) for these {i,j}: {4,842}, {23,316}
X(5099) = complement of X(691)
X(5099) = crosssum of circumcircle intercepts of line X(3)X(67)
X(5099) = perspector of the circumconic centered at X(2492)
X(5099) = X(2)-Ceva conjugate of X(2492)
X(5099) = inverse-in-polar-circle of X(935)
X(5099) = inverse-in-{circumcircle, nine-point circle}-inverter of X(2770)
X(5099) = reflection of X(115) in Euler line
X(5099) = Λ(X(115), X(125)) with respect to the orthic triangle
X(5099) = orthopole of line X(3)X(67)
X(5099) = Kirikami-six-circles image of X(23)

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(5102) = centroid of X(4)X(6)X(193)

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)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos(A + ω) tan ω - cos ω sin A + 3 sinω cos A
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 3 cos(A + ω) tan ω - sin(A - ω) + 2 cos A sin ω
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 6 cos A - (3 tan ω + cot ω) sin A
Barycentrics   k(a,b,c) : k(b,c,a) : k(c,a,b), where k(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(5104) = reflection of X(6) in X(187)
X(5104) = crossdifference of every pair of points on line X(523)X(597)
X(5104) = {X(23),X(352)}-harmonic conjugate of X(2502)
X(5104) = reflection of X(6) in the Lemoine axis

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 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(5106) = inverse-in-Parry-isodynamic-circle of X(669); see X(2)

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

Barycentrics 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(5107) = reflection of X(187) in X(6)
X(5107) = inverse-in-Ehrmann-circle of X(574)
X(5107) = {X(33517), X(33518)}-harmonic conjugate of X(13449)

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

Let U denote the Brocard circle. X(5108) is the intersection, other than X(6), of line X(2)X(6) and U. Also, X(5108) is the intersection, other than X(3), of U and the circle with segment X(2)X(3) as diameter, and X(5108) is the intersection, other than X(2), of line X(2)X(6) and the circle with segment X(2)X(3) as diameter. Also, X(5108) is the intersection, other than X(1316), of U and the circle {{X(2), X(110), X(1316)}}, which is the Parry circle of the 1st Brocard triangle. Let V denote the circle {{X(2), X(110), X(2770), X(5463), X(5464)}}. Then X(5108) = inverse-in-V of X(3). (Randy Hutson, January 29, 2015)

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

X(5108) = anticomplement of X(32525)
X(5108) = X(111)-of-1st-Brocard-triangle
X(5108) = X(111)-of-X(2)-Brocard-triangle
X(5108) = 1st-Brocard-isogonal conjugate of X(543)
X(5108) = intersection, other than X(2), of the Hutson-Parry circles of the inner and outer Vecten triangles
X(5108) = orthogonal projection of X(3) on line X(2)X(6)
X(5108) = intersection of lines X(2)X(6) of antipedal triangles of PU(1)

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 a²(a⁴ + 2b⁴ + 2c⁴ - 2a²b² - 2a²c² - b²c²) : :

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

X(5111) = radical trace of 2nd Lemoine circle and Ehrmann circle

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)

As a point on the Euler line, X(5112) has Shinagawa coefficients (3(E + F)F + S², -(E + F)² - 3S²).

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

X(5112) = nine-point-circle-inverse of X(37988)

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(5113) = crossdifference of PU(147)
X(5113) = isogonal conjugate of isotomic conjugate of X(9479)
X(5113) = X(2)-Ceva conjugate of X(39079)
X(5113) = perspector of hyperbola {{A,B,C,X(6),X(1031),X(2076),X(34214),X(35511)}}

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)

Trilinears       2 cos A + sin A csc 2ω : 2 cos B + sin B csc 2ω : 2 cos C + sin C csc 2ω
Trilinears       sin A + 2 cos A sin 2ω : sin B + 2 cos B sin 2ω : sin C + 2 cos C sin 2ω
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(5116) = radical center of the Lucas(csc 2ω) circles
X(5116) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,2076), (1340,1341,32)

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)

As a point on the Euler line, X(5117) has Shinagawa coefficients (2(E + F)F, (E + F)² - S²).

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

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

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

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

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

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

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

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

For a construction of X(5119) see Peter Moses, Euclid 962

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

X(5119) = reflection of X(1) in X(55)
X(5119) = {X(1),X(40)}-harmonic conjugate of X(46)
X(5119) = X(22)-of-reflection-triangle-of X(1)
X(5119) = {X(1),X(3)}-harmonic conjugate of X(37618)

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

Trilinears a - s cos A : :
Barycentrics 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(5120) = {X(3),X(6)}-harmonic conjugate of X(4254)

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

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

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

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

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

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

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

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

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

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

X(5123) = complement of X(1319)

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

Trilinears a - 2 (a + b + c) cos A : :
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(5124) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37509)
X(5124) = {X(371),X(372)}-harmonic conjugate of X(37509)

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)

As a point on the Euler line, X(5125) has Shinagawa coefficients ($a$F, -$aS_A$ - 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(5125) = complement of X(7538)
X(5125) = anticomplement of X(7515)
X(5125) = pole wrt polar circle of trilinear polar of X(1751) (the line X(523)X(663))
X(5125) = X(48)-isoconjugate (polar conjugate) of X(1751)

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(5126) = midpoint of X(38013) and X(38014)
X(5126) = X(23)-of-incircle-circles-triangle

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(5127) = Conway-circle-inverse of X(35637)

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

As a point on the Euler line, X(5129) has Shinagawa coefficients (2abc$a$ + S², -2S²).

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(5129) = anticomplement of X(17582)

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

As a point on the Euler line, X(5133) has Shinagawa coefficients (E + 2F, 2E + 2F).

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

X(5133) = inverse-in-orthocentroidal-circle of X(22)
X(5133) = homothetic center of polar triangle of nine-point circle and orthoanticevian triangle of X(2)

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)

As a point on the Euler line, X(5136) has Shinagawa coefficients ($a$F, $aS_A$).

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

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

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

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

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

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

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

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

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

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

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

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

X(5139) = midpoint of X(4) and X(3563)
X(5139) = complement of X(3565)
X(5139) = X(2)-Ceva conjugate of X(2489)
X(5139) = crosssum of circumcircle intercepts of line X(3)X(69)
X(5139) = orthopole of line X(3)X(69)
X(5139) = Kirikami-six-circles image of X(25)

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

As a point on the Euler line, X(5141) has Shinagawa coefficients (abc$a$ + 4S², 4S²).

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(5141) = complement of X(17548)

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)

As a point on the Euler line, X(5142) has Shinagawa coefficients ($a$F, $a$(E + F) + abc).

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(5142) = nine-point-circle-inverse of X(37989)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

X(5149) = X(1691)-of-1st-Brocard triangle
X(5149) = 1st-Brocard-triangle-isogonal-conjugate of X(76)
X(5149) = center of the perspeconic of these triangles: 1st Brocard and 6th anti-Brocard

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

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

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

X(5150) = X(36)-of-1st-Brocard trangle
X(5150) = inverse of X(32115) in the 1st Lemoine circle

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(5151) = polar conjugate of isogonal conjugate of X(20972)
X(5151) = polar conjugate of isotomic conjugate of X(16594)

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(5152) = complement of X(32528)

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

As a point on the Euler line, X(5154) has Shinagawa coefficients (abc$a$ - 4S², -4S²).

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(5154) = complement of X(37307)
X(5154) = anticomplement of X(17566)

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)

Let SaSbSc be the Ehrmann side-triangle. Let A′ be the barycentric product Sb*Sc, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5158). (Randy Hutson, June 27, 2018)

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(5158) = isogonal conjugate of isotomic conjugate of X(37638)
X(5158) = isotomic conjugate of polar conjugate of X(34417)
X(5158) = X(92)-isoconjugate of X(3431)

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)

As a point on the Euler line, X(5159) has Shinagawa coefficients (E - 5F, -E - F).

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) = midpoint of X(468) and X(858)
X(5159) = complement of X(468)
X(5159) = inverse-in-{circumcircle, nine-point circle}-inverter of X(20)
X(5159) = inverse-in-complement-of-polar-circle of X(2)

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

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

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

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

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

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

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

X(5161) = isogonal conjugate of X(37842)
X(5161) = crossdifference of every pair of points on line X(10)X(6590)

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

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

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

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

X(5162) = reflection of X(32) in X(187)

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

Barycentrics 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(5163) = isogonal conjugate of X(35155)
X(5163) = crossdifference of every pair of points on line X(2)X(2787)
X(5163) = inverse-in-Parry-isodynamic-circle of X(667); see X(2)

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

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

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

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

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

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

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

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

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

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

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

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

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

X(5167) = anticomplement of X(35060)

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

Barycentrics 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(5168) = isogonal conjugate of X(35153)
X(5168) = crossdifference of every pair of points on line X(2)X(2786)
X(5168) = inverse-in-Parry-isodynamic-circle of X(649); see X(2)

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

As a point on the Euler line, X(5169) has Shinagawa coefficients (E + 4F, 4E + 4F).

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

X(5169) = complement of X(7492)
X(5169) = anticomplement of X(7495)
X(5169) = harmonic center of nine-point circle and {circumcircle, nine-point circle}-inverter

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 a²(a⁶ - 4a⁴b² - 4a⁴c² + 3a²b⁴ + 3a²c⁴ + 2b⁴c² + 2b²c⁴) : :
Barycentrics (sin A)[cos(A - arccot(p))] : : , where p = sin(2ω)/(2 + cos(2ω))
X(5171) = X(3) + ((cot ω)/p)X(6) = 3 X[3576] - X[6210] = X[6] + 2 X[39] = X[1691] - 4 X[2024] = 4 X[39] - X[3094] = 2 X[6] + X[3094] = 2 X[182] + X[3095] = X[76] - 4 X[3589] = X[194] + 5 X[3618] = X[1916] + 2 X[5026] = 5 X[6] - 2 X[5052] = 5 X[39] + X[5052] = 5 X[3094] + 4 X[5052] = 4 X[2021] - X[5104] = 4 X[2025] - X[5111] = 5 X[3763] - 8 X[6683] = 2 X[597] + X[7757] = 2 X[141] - 5 X[7786] = 4 X[5092] - X[9821] = X[69] - 4 X[10007] = 2 X[5480] + X[11257] = X[1352] - 4 X[11272] = 4 X[2023] - X[11646] = 2 X[1386] + X[12782]

The function p = sin(2ω)/(2 + cos(2ω)) is the Tucker parameter for X(5171); see the preamble to X(13323.)

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

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

X(5171) lies on these lines: {2,732}, {3,6}, {69,10007}, {76,3589}, {83,4048}, {141,7786}, {147,2023}, {194,3618}, {262,1503}, {373,1194}, {597,698}, {694,9155}, {1180,3981}, {1352,11272}, {1386,12782}, {1428,12837}, {1613,5650}, {1915,6800}, {1916,5026}, {2056,6090}, {2330,12836}, {2782,6034}, {3051,7998}, {3108,5012}, {3299,12840}, {3301,12841}, {3329,10334}, {3763,6683}, {5031,7777}, {5103,7790}, {5309,7697}, {5480,9607}, {5965,11261}, {6309,7819}, {7760,8177}, {7829,8149}, {7875,9865}, {10347,12216}

X(5171) = inverse-in-Brocard-circle of X(12212)
X(5171) = inverse-in-second-Brocard-circle of X(7772)
X(5171) = circumperp conjugate of X(35383)
X(5171) = {X(i),X(j)-harmonic conjugate of X(k) for these (i,j,k): (3,6,12212), (6,39,3094), (6,2076,5039), (6,5013,5017), (6,5024,5104), (6,5116,32), (39,2021,5024), (39,7772,3095), (182,7772,6), (371,372,12054), (574,5039,2076), (1670,1671,7772), (1689,1690,3098), (3106,3107,3095), (12055,12212,3)

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(5172) = {X(1),X(3)}-harmonic conjugate of X(37564)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

As a point on the Euler line, X(5177) has Shinagawa coefficients (abc$a$ + S², 2S²).

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(5177) = complement of X(17576)
X(5177) = anticomplement of X(6857)

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(5178) = anticomplement of X(37080)

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

Barycentrics a^3*b - a^2*b^2 + a*b^3 - b^4 + a^3*c - a*b^2*c - a^2*c^2 - a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4 : :
X(5179) = X[5011] - 4 X[5199], X[5134] + 4 X[5199], X[5134] + 2 X[8074]

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

X(5179) lies on these lines: {1, 8776}, {2, 5088}, {4, 9}, {5, 1212}, {6, 5722}, {8, 21073}, {12, 16601}, {20, 27541}, {21, 27068}, {30, 910}, {37, 495}, {41, 10572}, {44, 12019}, {72, 22018}, {76, 7112}, {80, 294}, {85, 17671}, {92, 21062}, {101, 515}, {116, 9436}, {119, 1566}, {142, 27471}, {150, 10025}, {218, 1837}, {220, 355}, {225, 25087}, {346, 21074}, {379, 17729}, {514, 661}, {517, 1146}, {519, 4919}, {527, 10708}, {579, 24005}, {644, 5176}, {672, 1737}, {942, 21049}, {946, 24045}, {948, 1323}, {950, 4251}, {952, 6603}, {997, 24247}, {1018, 6735}, {1055, 21578}, {1104, 5305}, {1210, 4253}, {1329, 25066}, {1334, 10039}, {1370, 15487}, {1446, 17758}, {1479, 2082}, {1738, 16611}, {1856, 7070}, {2170, 30384}, {2182, 7359}, {2325, 21066}, {3061, 21616}, {3177, 17181}, {3207, 18481}, {3208, 10915}, {3294, 24987}, {3436, 17742}, {3560, 32561}, {3583, 5540}, {3661, 5195}, {3686, 21065}, {3691, 21029}, {3693, 17757}, {3731, 5726}, {3732, 4872}, {3752, 15048}, {3767, 16968}, {3814, 24036}, {3911, 5030}, {3991, 12607}, {4193, 26690}, {4223, 5144}, {4262, 4304}, {4431, 27492}, {4875, 24390}, {5046, 26793}, {5254, 16583}, {5283, 19754}, {5285, 36010}, {5530, 25092}, {5532, 20683}, {5546, 7424}, {5690, 21872}, {5720, 18328}, {5886, 34522}, {6506, 6882}, {6656, 25994}, {6684, 24047}, {6734, 16552}, {6913, 15288}, {7187, 33837}, {7377, 30854}, {7687, 18327}, {9581, 16572}, {10712, 31160}, {10916, 21384}, {11681, 25082}, {12047, 17451}, {13161, 16600}, {13407, 21808}, {14873, 16589}, {15612, 20623}, {16549, 24982}, {17170, 30694}, {17353, 24279}, {17753, 26531}, {17866, 25002}, {18750, 21621}, {20235, 20236}, {20367, 26001}, {21070, 21078}, {23649, 28096}, {26563, 33839}, {27250, 27254}, {28133, 35270}, {28827, 36698}, {29594, 31142}, {29962, 29967}, {30031, 30063}, {32706, 35182}, {35068, 35122}

X(5179) = midpoint of X(i) and X(j) for these {i,j}: {150, 10025}, {1146, 17747}, {3732, 4872}, {5011, 5134}
X(5179) = reflection of X(i) in X(j) for these {i,j}: {5011, 8074}, {8074, 5199}, {9436, 116}
X(5179) = reflection of X(5011) in the Gergonne line
X(5179) = isotomic conjugate of X(37214)
X(5179) = complement of X(5088)
X(5179) = Spieker-radical-circle-inverse of X(573)
X(5179) = polar-circle-inverse of X(19)
X(5179) = circumcircle-of anticomplementary-triangle-inverse of X(11677)
X(5179) = X(i)-Ceva conjugate of X(j) for these (i,j): {1981, 2811}, {8777, 1}
X(5179) = crosssum of X(6) and X(26884)
X(5179) = crossdifference of every pair of points on line {31, 1459}
X(5179) = barycentric product X(i)*X(j) for these {i,j}: {10, 14956}, {8777, 20623}
X(5179) = barycentric quotient X(14956)/X(86)
X(5179) = intersection of Gergonne lines of 1st and 2nd Ehrmann circumscribing triangles
X(5179) = intersection of Gergonne lines of anticevian triangles of PU(4)
X(5179) = X(i)-complementary conjugate of X(j) for these (i,j): {213, 35075}, {296, 34822}, {1937, 2886}, {1945, 142}, {1949, 17073}, {1952, 17046}, {2249, 3739}, {35145, 21240}
X(5179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 6554, 169}, {85, 17671, 34847}, {672, 21044, 1737}, {5046, 26793, 33950}, {5254, 16583, 23537}, {8804, 20262, 573}

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

X(5181) = reflection of X(6) in X(5972)
X(5181) = complement of X(895)
X(5181) = antipode of X(6) in the bicevian conic of X(2) and X(110)

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(5182) = X(2)-of-6th-anti-Brocard-triangle
X(5182) = inverse-in-Thomson-Gibert-Moses-hyperbola of X(35279)
X(5182) = {X(2),X(110)}-harmonic conjugate of X(35279)

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(5183) = inverse-in-circumcircle of X(5217)

X(5184) = INVERSE-IN-BEVAN-CIRCLE OF X(3751)

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 + b²c²)

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

As a point on the Euler line, X(5187) has Shinagawa coefficients (abc$a$ - 2S², -4S²).

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

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

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

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

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

X(5188) = reflection of X(39) in X(3)
X(5188) = inverse-in-2nd-Brocard-crcle of X1350)
X(5188) = (39)-of-circumcevian triangle of X(511)
X(5188) = Brocard-circle-inverse of X(37479)
X(5188) = {X(3),X(6)}-harmonic conjugate of X(37479)
X(5188) = {X(371),X(372)}-harmonic-conjugate of X(5039).

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

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

As a point on the Euler line, X(5189) has Shinagawa coefficients (3E, -8E - 8F).

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) = isogonal conjugate of X(34437)
X(5189) = anticomplement of X(23)
X(5189) = inverse-in-anticomplementary-circle of X(2)
X(5189) = inverse-in-deLongchamps-circle of X(22)
X(5189) = inverse-in-{circumcircle, nine-point circle}-inverter of X(140)
X(5189) = reflection of X(23) in the deLongchamps line
X(5189) = isotomic conjugate of isogonal conjugate of X(19596)
X(5189) = polar conjugate of isogonal conjugate of X(22121)
X(5189) = antigonal conjugate of X(38946)
X(5189) = nine-point-circle-inverse of X(37990)

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

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

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

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

X(5190) = midpoint of X(4) and X(917)
X(5190) = complement of X(1305)
X(5190) = X(2)-Ceva conjugate of X(7649)
X(5190) = crosssum of circumcircle intercepts of line X(3)X(48)
X(5190) = orthopole of line X(3)X(48)
X(5190) = Kirikami-six-circles image of X(92)

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

Barycentrics a²(b⁶ + c⁶ - 2a⁶ + 2a⁴b² + 2a⁴c² - a²b⁴ - a²c⁴ - b⁴c² - b²c⁴) : :
Barycentrics a^2 (2 a^6 - 2 a^4 (b^2 + c^2) + a^2 (b^4 + c^4) - (b^2 - c^2)^2 (b^2 + c^2)) : :

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

X(5191) = isogonal conjugate of X(5641)
X(5191) = pole of the line X(23)X(110) with respect to the Parry circle
X(5191) = inverse-in-Parry-isodynamic-circle of X(647); see X(2)

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

As a point on the Euler line, X(5192) has Shinagawa coefficients (2(E + F)² + 2$bc$(E + F) - abc$a$, 2S²).

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(5195) = anticomplement of X(5011)

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)

As a point on the Euler line, X(5196) has Shinagawa coefficients (E + 4F + 2$bc$, -4E - 4F - 6$bc$).

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

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

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

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

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

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

As a point on the Euler line, X(5198) has Shinagawa coefficients (F, -3E - F).

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

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

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

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

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

Barycentrics (S + a²)/S_A : (S + b²)/S_B : (S + c²)/S_C
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)

As a point on the Euler line, X(5200) has Shinagawa coefficients (F, -E - F - S).

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

X(5200) = pole wrt polar circle of trilinear polar of X(5490)
X(5200) = X(48)-isoconjugate (polar conjugate) of X(5490)

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

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

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

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

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

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

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

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

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

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

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

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

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

X(5204) = isogonal conjugate of X(7319)
X(5204) = polar conjugate of isotomic conjugate of X(23140)
X(5204) = {X(55),X(56}-harmonic conjugate of X(3304)
X(5204) = {X(3),X(56)}-harmonic conjugate of X(55)

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

Barycentrics a³ - a²b - a²c + 3abc - b²c - bc² : :

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

X(5205) = complement of X(5211)
X(5205) = anticomplement of X(5121)
X(5205) = inverse-in-{circumcircle, nine-point circle}-inverter of X(10)
X(5205) = crossdifference of PU(92)
X(5205) = crossdifference of every pair of points on line X(649)X(1201)
X(5205) = X(2)-Ceva conjugate of X(39059)
X(5205) = perspector of conic {{A,B,C,PU(59)}}

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

Trilinears    2 sin(A + ω) - 3 sin(A - ω) : :
Trilinears    5 cos A - sin A cot ω : :
Trilinears    sin A - 5 cos A tan ω : :
Barycentrics    a²(2b² + 2c² - 3a²) : :

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

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

X(5206) = isogonal conjugate of polar conjugate of X(37453)
X(5206) = {X(371),X(372)}-harmonic conjugate of X(5097)
X(5206) = radical center of Lucas(-2/5 cot ω) circles

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

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

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

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

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

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

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

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

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

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

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

Trilinears    5 sin(A + ω) - 7 sin(A - ω) : :
Trilinears    6 cos A - sin A cot ω : :
Trilinears    sin A - 6 cos A tan ω : :
Barycentrics   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(5210) = radical center of Lucas(-1/3 cot ω) circles
X(5210) = harmonic center of circumcircle and circle O(15,16)

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

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

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

X(5211) = anticomplement of X(5205)
X(5211) = isotomic conjugate of anticomplement of X(39059)

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(5212) = complement of X(38473)

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

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

Let (N) be the nine-point circle of a triangle ABC. Let (I_A) be the A-excircle of ABC, and define (I_B) and (I_C) cyclically. Let A′ be the midpoint of side BC, and define B′ and C′ cyclically. Let (K_A) be the circle, other than (N), that passes through B′ and C′ and touches (I_A), and define (K_B) and (K_C) cyclically. Let (K) be the circle externally tangent to (K_A), (K_B), (K_C), and let (L) be the circle externally tangent to (I_A), (I_B), (I_C). Then X(5213) = (K)∩(L). (Tran Quang Hung, July 16, 2014)

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

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

X(5213) = complement of X(38477)
X(5213) = similitude center of Apollonius triangle and polar triangle of excircles radical circle

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

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

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

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

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

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

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

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

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

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

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

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

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

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(3b² + 3c² - 3a² + 2bc)
X(5217) = R*X(1) + 3r*X(3)
X(5217) = (2r + R)*X(12) + r*X(20)

Let A″B″C″ be as at X(12512). Then A″B″C″ is homothetic to the intouch triangle at X(5217). (Randy Hutson, January 17, 2020)

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(5217) = inverse-in-circumcircle of X(5183)
X(5217) = {X(3),X(55)}-harmonic conjugate of X(56)

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

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

X(5219) = isogonal conjugate of X(2364)
X(5219) = isotomic conjugate of X(30608)
X(5219) = {X(2),X(7)}-harmonic conjugate of X(3911)
X(5219) = {X(2),X(57)}-harmonic conjugate of X(31231)
X(5219) = endo-homothetic center of the AOA and AAOA triangles

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) = X(1) - 3*X(9)

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

X(5220) = X(67) of Fuhrman triangle
X(5220) = perspector of Fuhrmann and outer Johnson triangles

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(5221) = {X(1),X(5708)}-harmonic conjugate of X(4860)
X(5221) = {X(13388),X(13389)}-harmonic conjugate of X(37595)

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(5222) = complement of X(29616)
X(5222) = anticomplement of X(17284)
X(5222) = {X(1),X(2)}-harmonic conjugate of X(5308)

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) = X(1) - 2*X(9)

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(5223) = reflection of X(1) in X(9)
X(5223) = outer-Garcia-to-ABC similarity image of X(7)

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(5224) = complement of X(17379)
X(5224) = anticomplement of X(17398)
X(5224) = {X(2),X(6)}-harmonic conjugate of X(17381)
X(5224) = {X(2),X(69)}-harmonic conjugate of X(86)
X(5224) = {X(2),X(141)}-harmonic conjugate of X(17234)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

X(5233) = complement of X(37684)

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(5235) = isotomic conjugate of X(30588)
X(5235) = complement of X(37635)
X(5235) = trilinear pole of line X(4693)X(4775) (the perspectrix of ABC and Gemini triangle 28)

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(5237) = Schoutte-circle-inverse of X(34755)

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(5238) = Schoutte circle inverse of X(34754)

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(5239) = complement of X(36928)

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(5240) = isogonal conjugate of X(33655)
X(5240) = complement of X(36929)

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(5241) = complement of X(37633)

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(5248) = excentral-to-2nd-circumperp similarity image of X(12514)

X(5249) = INTERSECTION OF LINES X(2)X(7) AND X(21)X(36)

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

X(5249) lies on these lines: {1, 224}, {2, 7}, {5, 1071}, {8, 4208}, {10, 3681}, {11, 3742}, {12, 3812}, {20, 946}, {21, 36}, {27, 86}, {37, 3782}, {42, 1738}, {55, 1004}, {75, 306}, {77, 278}, {78, 443}, {81, 3664}, {85, 92}, {171, 3011}, {210, 3826}, {239, 2890}, {312, 1269}, {320, 333}, {321, 1930}, {354, 2886}, {379, 2140}, {442, 942}, {474, 1259}, {495, 3753}, {497, 4666}, {516, 1621}, {518, 3925}, {528, 3748}, {551, 4304}, {554, 5239}, {914, 1441}, {938, 5177}, {940, 3772}, {948, 4350}, {950, 2475}, {960, 3649}, {1001, 1836}, {1012, 1519}, {1056, 3872}, {1081, 5240}, {1086, 3666}, {1210, 2476}, {1211, 3739}, {1215, 3836}, {1659, 3084}, {1737, 3822}, {1838, 4303}, {1959, 3674}, {2550, 3475}, {2895, 3686}, {2975, 4298}, {2999, 4859}, {3075, 3561}, {3120, 3720}, {3187, 3879}, {3220, 4228}, {3601, 4190}, {3622, 4313}, {3671, 3869}, {3687, 3936}, {3706, 4966}, {3741, 5208}, {3757, 4645}, {3771, 3980}, {3814, 3833}, {3841, 3874}, {3847, 4892}, {3848, 5087}, {3873, 4847}, {3890, 4301}, {3897, 4311}, {3969, 4431}, {4312, 4512}

X(5249) = isogonal conjugate of X(2259)
X(5249) = complement of X(3219)

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)

In the plane of a triangle ABC, let
I = X(1) = incenter;
DEF = intouch triangle;
(O) = circumcircle;
A' = the point, other than A, where the circle {{A,I,D}} meets (O), and define B' and C' cyclically;
Oa = center of the circle {{B',E,F,C'}}, and define Ob and Oc cyclically;
Oaa = A-extraversion of Oa, and define Obb and Occ cyclically;
T = affine transfomation that maps ABC onto OaObOc;
TT = affine transformation that maps ABC onto OaaObbOcc.
Then X(4350) = finite fixed point of T, and X(5250) = finite fixed point of TT. (Angel Montesdeoca, December 29, 2023)

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(5252) = outer-Johnson-to-ABC similarity image of X(1)

X(5253) = INTERSECTION OF LINES X(1)X(88) AND X(2)X(12)

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

X(5253) lies on these lines: {1, 88}, {2, 12}, {3, 962}, {5, 104}, {8, 474}, {11, 2475}, {21, 36}, {35, 551}, {40, 3890}, {46, 3877}, {55, 3622}, {57, 3869}, {63, 3361}, {78, 3333}, {81, 1193}, {85, 934}, {86, 4225}, {145, 1376}, {171, 1201}, {191, 4973}, {377, 3086}, {411, 3576}, {484, 3884}, {497, 4190}, {499, 2476}, {758, 3337}, {908, 4298}, {936, 3681}, {942, 4511}, {960, 3218}, {976, 3976}, {978, 1468}, {993, 3624}, {997, 3338}, {1001, 4189}, {1004, 4313}, {1014, 4357}, {1104, 4239}, {1210, 5086}, {1290, 3109}, {1319, 3812}, {1470, 3485}, {1476, 5176}, {1478, 4193}, {2260, 2287}, {2306, 5240}, {2478, 4293}, {2646, 3742}, {3294, 5030}, {3336, 3878}, {3428, 3523}, {3555, 4420}, {3585, 3825}, {3601, 4666}, {3617, 4413}, {3623, 3913}, {3636, 3746}, {3753, 4861}, {3811, 3889}, {3816, 5046}, {4187, 5080}, {4696, 5205}, {5187, 5229}

X(5254) = INTERSECTION OF LINES X(4)X(6) AND X(30)X(32)

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

Let U be the circle obtained as the inverse-in-the-polar-circle of the 2nd Lemoine circle. The center of U is X(5254). (Randy Hutson, November 22, 2014)

X(5254) lies on these lines: {2, 1975}, {3, 230}, {4, 6}, {5, 39}, {11, 2275}, {12, 2276}, {20, 3053}, {30, 32}, {76, 141}, {83, 597}, {140, 574}, {148, 384}, {184, 460}, {185, 1562}, {187, 550}, {194, 325}, {232, 235}, {290, 695}, {297, 3981}, {315, 524}, {316, 3629}, {338, 1235}, {376, 5023}, {381, 2548}, {395, 616}, {396, 617}, {427, 1194}, {489, 3068}, {490, 3069}, {495, 1500}, {496, 1015}, {538, 626}, {548, 5206}, {594, 4385}, {595, 5134}, {726, 4136}, {1086, 3673}, {1105, 1970}, {1107, 2886}, {1146, 3959}, {1180, 5133}, {1184, 1370}, {1196, 1368}, {1329, 1575}, {1353, 1570}, {1384, 1657}, {1574, 3820}, {1596, 3199}, {1656, 3055}, {1885, 1968}, {3061, 3944}, {3522, 5210}, {3564, 5028}, {3589, 4048}, {3627, 5007}, {3721, 3782}, {3845, 5041}, {3934, 4045}, {4173, 5167}

X(5254) = midpoint of X(3070) and X(3071)
X(5254) = complement of X(1975)
X(5254) = anticomplement of X(7789)
X(5254) = exsimilicenter of nine-point and (1/2)-Moses circles; the insimilicenter is X(3815)
X(5254) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,39,3815), (39,115,5)

X(5255) = INTERSECTION OF LINES X(1)X(3) AND X(8)X(31)

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

X(5255) lies on these lines: {1, 3}, {2, 3915}, {4, 983}, {6, 979}, {8, 31}, {9, 989}, {10, 82}, {21, 902}, {32, 2329}, {37, 3496}, {42, 3871}, {44, 4662}, {58, 519}, {72, 3961}, {100, 1193}, {145, 1468}, {213, 3684}, {341, 4676}, {355, 3073}, {404, 1201}, {518, 1046}, {528, 1834}, {582, 3654}, {601, 944}, {603, 3476}, {643, 2363}, {750, 3616}, {752, 1330}, {958, 3052}, {976, 3869}, {978, 1191}, {1106, 4308}, {1203, 3293}, {1253, 4344}, {1254, 4318}, {1279, 3812}, {1386, 4646}, {1572, 3061}, {1706, 1722}, {1724, 3679}, {1743, 3713}, {1914, 2295}, {2176, 4386}, {2269, 2298}, {2292, 3920}, {2321, 4264}, {2650, 3722}, {2901, 4693}, {3434, 5230}, {3743, 5184}, {3769, 4673}, {3868, 3938}, {3923, 4385}, {3973, 4866}, {3997, 4251}, {4255, 4421}, {4418, 4968}, {4649, 5145}

X(5255) = {X(1),X(3)}-harmonic conjugate of X(37617)

X(5256) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(63)

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

X(5256) lies on these lines: {1, 2}, {6, 63}, {7, 223}, {21, 1453}, {27, 34}, {33, 469}, {37, 3305}, {38, 3751}, {55, 1386}, {56, 4719}, {57, 77}, {58, 4652}, {92, 2331}, {193, 4001}, {204, 1013}, {226, 3946}, {238, 968}, {312, 4360}, {321, 3875}, {329, 3672}, {333, 3759}, {345, 3618}, {380, 3101}, {440, 1062}, {464, 1040}, {553, 4667}, {748, 1962}, {894, 3210}, {908, 3553}, {940, 1100}, {982, 4649}, {988, 1468}, {1211, 4272}, {1214, 1445}, {1230, 3760}, {1376, 3745}, {1427, 4350}, {1707, 2308}, {1743, 3219}, {1763, 2172}, {2177, 3749}, {2352, 5132}, {3052, 4689}, {3247, 3930}, {3434, 3755}, {3677, 3873}, {3886, 3896}, {3923, 4970}, {3966, 4026}, {3993, 4011}, {4021, 4656}, {4085, 4865}, {4255, 4855}, {4270, 4357}, {4285, 4643}, {4413, 4682}, {4868, 5119}, {4886, 5224}

X(5256) = {X(1),X(2)}-harmonic conjugate of X(5287)

X(5257) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(37)

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

X(5257) lies on these lines: {1, 966}, {2, 7}, {6, 1125}, {8, 3247}, {10, 37}, {19, 406}, {45, 3634}, {71, 3294}, {75, 4044}, {86, 4416}, {141, 4698}, {145, 4034}, {192, 4967}, {198, 405}, {225, 281}, {228, 4204}, {238, 4264}, {391, 1449}, {392, 2262}, {461, 4512}, {551, 1100}, {573, 946}, {756, 3778}, {860, 1826}, {978, 5105}, {993, 2178}, {1001, 4254}, {1010, 4877}, {1211, 4035}, {1266, 4699}, {1654, 3879}, {1698, 1738}, {1743, 3624}, {2171, 4848}, {2238, 4104}, {3008, 4657}, {3244, 3723}, {3617, 4007}, {3622, 4982}, {3632, 4545}, {3636, 4856}, {3663, 3739}, {3664, 4643}, {3671, 4047}, {3679, 4060}, {3912, 4687}, {3946, 4384}, {3949, 3970}, {3965, 4847}, {3985, 4656}, {4021, 4361}, {4061, 4771}, {4260, 5044}, {4389, 4751}, {4431, 4664}, {4648, 4748}, {4665, 4681}, {4668, 4898}

X(5258) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(35)

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

X(5258) lies on these lines: {1, 6}, {3, 3679}, {8, 35}, {10, 36}, {21, 519}, {55, 3632}, {56, 1698}, {65, 4880}, {100, 3626}, {101, 3691}, {172, 1573}, {191, 517}, {200, 3612}, {210, 1385}, {214, 4015}, {442, 529}, {443, 4317}, {484, 3916}, {498, 3421}, {499, 2551}, {515, 3651}, {528, 4330}, {535, 2475}, {551, 5047}, {961, 1224}, {999, 3624}, {1005, 4847}, {1319, 5044}, {1388, 3715}, {1444, 4967}, {1478, 5177}, {1482, 3899}, {1621, 3244}, {2099, 3927}, {2550, 4299}, {2802, 3647}, {2886, 3585}, {3214, 4256}, {3218, 3754}, {3219, 3878}, {3295, 3633}, {3336, 3753}, {3337, 3812}, {3560, 3929}, {3582, 4187}, {3625, 3871}, {3678, 4511}, {3681, 3897}, {3730, 4390}, {3813, 4857}, {3820, 5193}, {3913, 4677}, {3918, 4973}, {3956, 4881}, {4302, 5082}, {4668, 5010}, {4853, 5119}

X(5259) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(35)

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

X(5259) lies on these lines: {1, 6}, {2, 35}, {3, 1699}, {10, 1621}, {12, 2078}, {21, 36}, {28, 1839}, {46, 4512}, {55, 1698}, {56, 4355}, {58, 3720}, {100, 3634}, {105, 1224}, {140, 2077}, {142, 1770}, {191, 942}, {386, 748}, {411, 3817}, {442, 3583}, {443, 4302}, {452, 1478}, {474, 5010}, {484, 3812}, {498, 5084}, {551, 2975}, {846, 3670}, {946, 1006}, {993, 3616}, {1089, 3757}, {1193, 4653}, {1259, 5231}, {1283, 5051}, {1329, 3584}, {1838, 4183}, {2260, 4877}, {2308, 4658}, {2550, 4309}, {2886, 4857}, {3085, 5129}, {3218, 3647}, {3219, 3874}, {3245, 3754}, {3293, 3750}, {3295, 3679}, {3303, 3632}, {3305, 3811}, {3336, 4640}, {3337, 3742}, {3560, 3576}, {3582, 4999}, {3685, 4647}, {3822, 5046}, {3848, 5131}, {3894, 3927}, {3898, 4861}, {3935, 4015}, {4068, 4716}

X(5260) = INTERSECTION OF LINES X(2)X(12) AND X(10)X(21)

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

X(5260) lies on these lines: {1, 748}, {2, 12}, {8, 405}, {9, 1405}, {10, 21}, {36, 3634}, {55, 3617}, {63, 3339}, {65, 3219}, {104, 140}, {145, 1001}, {191, 3754}, {355, 1006}, {392, 4861}, {404, 993}, {442, 5080}, {452, 3434}, {484, 3647}, {502, 1224}, {644, 3294}, {846, 4642}, {950, 5178}, {956, 3616}, {984, 3924}, {997, 3897}, {1043, 4651}, {1104, 3920}, {1320, 3884}, {1376, 4189}, {1478, 4197}, {1722, 4850}, {1757, 2650}, {1891, 4233}, {2078, 5176}, {2475, 3925}, {2646, 3740}, {2886, 5046}, {3091, 3428}, {3218, 3812}, {3293, 4653}, {3303, 3621}, {3337, 3833}, {3579, 4002}, {3585, 3841}, {3622, 4423}, {3626, 3746}, {3679, 3871}, {3697, 4420}, {3698, 4640}, {3757, 4696}, {3872, 3890}, {3913, 4678}, {3935, 4662}, {4183, 5174}, {4188, 4413}, {4511, 5044}

X(5261) = INTERSECTION OF LINES X(2)X(12) AND X(7)X(10)

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

X(5261) lies on these lines: {1, 3091}, {2, 12}, {4, 390}, {5, 1056}, {7, 10}, {8, 226}, {11, 5068}, {20, 35}, {34, 3920}, {55, 3146}, {65, 3617}, {85, 341}, {145, 3485}, {192, 2996}, {355, 3487}, {381, 1058}, {387, 1126}, {391, 1405}, {442, 3421}, {452, 2078}, {496, 3545}, {497, 3832}, {498, 3523}, {519, 4323}, {612, 4296}, {976, 2647}, {984, 1254}, {986, 4346}, {999, 3090}, {1125, 4308}, {1219, 3705}, {1393, 4392}, {1441, 4385}, {1469, 3620}, {1479, 3839}, {1617, 5047}, {1698, 4298}, {1722, 4327}, {1837, 3475}, {2099, 3621}, {3086, 5056}, {3303, 5225}, {3304, 3614}, {3361, 3634}, {3476, 3622}, {3522, 5218}, {3543, 3585}, {3584, 4299}, {3616, 5219}, {3624, 4315}, {3649, 4678}, {3671, 3679}, {3704, 4461}, {3870, 5175}, {3961, 4332}, {4654, 4848}

X(5262) = INTERSECTION OF LINES X(1)X(2) AND X(7)X(34)

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

X(5262) lies on these lines: {1, 2}, {3, 4850}, {6, 977}, {7, 34}, {21, 1104}, {28, 60}, {31, 986}, {37, 5047}, {57, 4296}, {58, 3218}, {63, 1453}, {65, 82}, {75, 964}, {77, 1467}, {238, 2292}, {257, 1178}, {312, 5192}, {350, 1228}, {377, 4000}, {404, 3752}, {452, 3672}, {595, 4424}, {758, 1203}, {950, 3100}, {982, 1468}, {990, 3146}, {1010, 4359}, {1040, 4313}, {1046, 2308}, {1062, 3488}, {1100, 2303}, {1191, 3877}, {1220, 4968}, {1245, 4388}, {1325, 2363}, {1442, 3212}, {1449, 2082}, {1621, 3931}, {1724, 3219}, {1743, 3951}, {2476, 3772}, {2646, 4719}, {3210, 4195}, {3315, 5045}, {3337, 4351}, {3339, 4347}, {3744, 3871}, {3745, 3812}, {3746, 4868}, {3876, 4383}, {3891, 4385}, {4972, 5015}, {5090, 5142}

X(5263) = INTERSECTION OF LINES X(1)X(75) AND X(2)X(11)

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

X(5263) lies on these lines: {1, 75}, {2, 11}, {6, 8}, {9, 4676}, {10, 82}, {19, 29}, {31, 333}, {37, 3685}, {38, 4418}, {42, 3996}, {69, 4307}, {85, 2263}, {87, 1222}, {141, 4645}, {171, 3741}, {190, 984}, {239, 1386}, {312, 612}, {321, 3920}, {516, 4357}, {518, 894}, {519, 4649}, {752, 3775}, {958, 4195}, {982, 3980}, {993, 4234}, {1008, 5224}, {1125, 1738}, {1211, 4388}, {1215, 3961}, {1266, 4353}, {1279, 3739}, {1441, 4318}, {1757, 4672}, {1861, 5174}, {1999, 3706}, {2049, 3295}, {2607, 3878}, {2975, 3286}, {3219, 4981}, {3241, 4499}, {3242, 4363}, {3246, 3846}, {3416, 3661}, {3616, 4000}, {3664, 4684}, {3744, 3757}, {3751, 3758}, {3842, 4432}, {3879, 4349}, {3993, 4693}, {4709, 4716}, {4732, 4974}

X(5263) = anticomplement of X(4026)

X(5264) = INTERSECTION OF LINES X(1)X(3) AND X(10)X(31)

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

X(5264) lies on these lines: {1, 3}, {2, 595}, {6, 3293}, {8, 58}, {10, 31}, {32, 2295}, {37, 1759}, {41, 3997}, {43, 1203}, {44, 3697}, {79, 983}, {80, 987}, {81, 3871}, {82, 4429}, {90, 989}, {100, 386}, {109, 388}, {191, 984}, {213, 4386}, {238, 1698}, {404, 995}, {405, 3052}, {474, 1191}, {515, 601}, {519, 1468}, {573, 2298}, {594, 4275}, {609, 2329}, {748, 3634}, {750, 1125}, {758, 976}, {956, 4252}, {1046, 3961}, {1089, 3923}, {1104, 3753}, {1106, 4315}, {1210, 1497}, {1253, 4349}, {1254, 4347}, {1376, 3216}, {1451, 4848}, {1453, 1706}, {1478, 1777}, {1714, 2550}, {2308, 3214}, {2345, 4264}, {2975, 4257}, {3085, 4307}, {3754, 3924}, {3874, 3938}, {4362, 4647}, {4450, 5051}

X(5265) = INTERSECTION OF LINES X(2)X(12) AND X(20)X(36)

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

X(5265) lies on these lines: {1, 3523}, {2, 12}, {3, 390}, {7, 1125}, {8, 1420}, {10, 4308}, {11, 3146}, {20, 36}, {34, 4232}, {43, 4322}, {57, 3616}, {65, 3622}, {108, 4200}, {140, 1056}, {145, 1319}, {193, 1428}, {201, 4392}, {238, 1106}, {279, 1447}, {348, 3598}, {376, 496}, {391, 604}, {404, 1617}, {439, 4366}, {495, 3525}, {497, 3522}, {499, 3091}, {551, 3339}, {614, 4296}, {631, 999}, {938, 3576}, {944, 5126}, {956, 1476}, {978, 1458}, {988, 3672}, {993, 5129}, {1388, 3623}, {1445, 3333}, {1466, 1621}, {1470, 4189}, {1471, 3945}, {1478, 5056}, {1698, 4315}, {3241, 4848}, {3295, 3524}, {3304, 5218}, {3476, 3617}, {3543, 3582}, {3624, 4298}, {3660, 3868}, {5059, 5225}, {5068, 5229}

X(5266) = INTERSECTION OF LINES X(1)X(3) AND X(32)X(37)

Trilinears arS_A - SS_A : brS_A - SS_B : crS_A - SS_C César Lozada (9/07/2013)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a³ + b³ + c³ + a²b +a²c + b²c + bc²)

X(5266) lies on these lines: {1, 3}, {2, 5015}, {4, 4339}, {6, 3694}, {10, 1104}, {21, 3920}, {31, 72}, {32, 37}, {38, 3916}, {39, 1100}, {42, 1009}, {44, 3678}, {58, 518}, {187, 3723}, {200, 1453}, {210, 1724}, {238, 5044}, {386, 1386}, {387, 3189}, {392, 1472}, {405, 612}, {442, 3011}, {474, 614}, {519, 3704}, {595, 960}, {601, 1071}, {902, 2292}, {943, 2298}, {975, 1001}, {983, 987}, {997, 1191}, {1010, 3757}, {1125, 1279}, {1384, 3247}, {1427, 4347}, {1468, 3555}, {1707, 3927}, {1770, 3782}, {1785, 1852}, {2204, 5089}, {3242, 4252}, {3293, 3689}, {3419, 5230}, {3475, 4340}, {3487, 4307}, {3753, 3924}, {3831, 4434}, {3879, 3933}, {3881, 4864}, {4195, 4385}, {4256, 4719}

X(5266) = {X(1),X(3)}-harmonic conjugate of X(37592)

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(5268) = orthoptic-circle-of-Steiner-inellipse-inverse of X(38471)
X(5268) = {X(1),X(2)}-harmonic conjugate of X(5272)

X(5269) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(31)

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

X(5269) lies on these lines: {1, 3}, {2, 3883}, {6, 200}, {9, 31}, {10, 1453}, {33, 1395}, {37, 3052}, {38, 3928}, {42, 1449}, {63, 3920}, {81, 3870}, {84, 601}, {181, 3056}, {197, 2270}, {204, 281}, {210, 1743}, {226, 3424}, {380, 3198}, {388, 1394}, {553, 4310}, {595, 975}, {611, 2003}, {614, 750}, {869, 2258}, {902, 968}, {950, 4339}, {984, 1707}, {985, 1961}, {987, 989}, {1001, 4682}, {1254, 4348}, {1376, 1386}, {1397, 2330}, {1407, 4321}, {1706, 4695}, {1999, 3886}, {2303, 2328}, {2318, 3997}, {3243, 3938}, {3474, 3663}, {3475, 3664}, {3632, 4046}, {3683, 3731}, {3715, 3973}, {3751, 3961}, {3782, 4312}, {3791, 4457}, {3923, 4135}, {4418, 4659}, {4641, 5223}

X(5270) = INTERSECTION OF LINES X(1)X(4) AND X(12)X(36)

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

X(5270) lies on these lines: {1, 4}, {2, 4317}, {3, 3584}, {5, 3582}, {8, 3901}, {10, 3218}, {11, 3850}, {12, 36}, {21, 535}, {30, 3746}, {35, 495}, {55, 1657}, {56, 1656}, {65, 2962}, {79, 517}, {80, 942}, {149, 3635}, {377, 3679}, {381, 3304}, {382, 3303}, {442, 529}, {484, 4292}, {496, 3858}, {498, 3523}, {499, 3600}, {519, 2475}, {548, 4995}, {551, 5046}, {952, 3649}, {999, 3851}, {1125, 5080}, {1698, 3436}, {1737, 3337}, {1935, 2964}, {2550, 4668}, {2975, 3822}, {3058, 3627}, {3085, 3522}, {3086, 5068}, {3146, 4309}, {3295, 5073}, {3434, 3633}, {3754, 5176}, {3874, 5086}, {3884, 5057}, {3920, 5189}, {3947, 4311}, {4302, 5059}

X(5270) = {X(1),X(4)}-harmonic conjugate of X(4857)

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(5271) = isogonal conjugate of X(2215)

X(5272) = INTERSECTION OF LINES X(1)X(2) AND X(57)X(238)

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

X(5272) lies on these lines: {1, 2}, {6, 3742}, {9, 982}, {11, 1040}, {25, 36}, {31, 3306}, {38, 3305}, {56, 5020}, {57, 238}, {63, 244}, {87, 269}, {105, 165}, {142, 1716}, {230, 3554}, {305, 3760}, {354, 3751}, {405, 988}, {497, 1738}, {968, 4850}, {984, 3677}, {990, 3817}, {1001, 3752}, {1191, 3812}, {1196, 2275}, {1279, 1376}, {1370, 3583}, {1386, 3848}, {1435, 1957}, {1449, 4038}, {1699, 1721}, {1724, 3338}, {1739, 5119}, {3052, 3246}, {3056, 3819}, {3242, 3740}, {3271, 3784}, {3315, 3681}, {3361, 4223}, {3553, 3815}, {3666, 4423}, {3729, 4011}, {3744, 4413}, {3772, 3816}, {3782, 4679}, {3895, 4695}, {4327, 5226}, {4641, 4860}

X(5272) = {X(1),X(2)}-harmonic conjugate of X(5268)

X(5273) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(21)

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

X(5273) lies on these lines: {2, 7}, {8, 21}, {10, 20}, {27, 281}, {31, 4344}, {81, 219}, {189, 268}, {191, 4295}, {210, 5218}, {220, 940}, {261, 1264}, {312, 3161}, {348, 479}, {354, 960}, {377, 1155}, {390, 4512}, {391, 3687}, {405, 938}, {443, 3916}, {497, 3683}, {631, 1071}, {910, 966}, {936, 3523}, {1002, 5208}, {1200, 3691}, {1210, 5129}, {1212, 3666}, {1214, 3160}, {1329, 4197}, {1479, 2894}, {1617, 2975}, {1698, 4208}, {1707, 4307}, {1764, 3730}, {2096, 3820}, {2550, 4640}, {3187, 4460}, {3210, 4402}, {3241, 3748}, {3474, 3925}, {3487, 3927}, {3679, 4304}, {3711, 4995}, {3772, 4419}, {3869, 4323}, {3877, 4345}, {4860, 4999}

X(5273) = {X(2),X(63)}-harmonic conjugate of X(7)
X(5273) = complement of isotomic conjugate of X(30711)

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(5275) = {X(2),X(385)}-harmonic conjugate of X(16992)

X(5276) = INTERSECTION OF LINES X(2)X(6) AND X(9)X(31)

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

X(5276) lies on these lines: {1, 41}, {2, 6}, {9, 31}, {21, 32}, {25, 941}, {37, 82}, {38, 3509}, {39, 404}, {42, 3684}, {100, 743}, {171, 672}, {172, 1107}, {284, 4224}, {384, 1655}, {573, 1754}, {584, 4228}, {595, 3294}, {609, 993}, {614, 1449}, {894, 3263}, {910, 3666}, {984, 985}, {1100, 3290}, {1180, 4261}, {1194, 2092}, {1196, 2670}, {1206, 3757}, {1333, 1627}, {1500, 3871}, {1572, 3877}, {1778, 4275}, {1922, 4518}, {2207, 4194}, {2292, 3496}, {2348, 3745}, {2476, 3767}, {2548, 4193}, {2651, 4274}, {3053, 4189}, {3598, 5228}, {3930, 3961}, {4188, 5013}, {4209, 4352}, {4239, 4277}, {4424, 5011}, {5007, 5047}

X(5277) = INTERSECTION OF LINES X(2)X(32) AND X(35)X(37)

Trilinears a³r + bcS : b³r + caS : c³r + abS> César Lozada (9/07/2013)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a³ + abc + b²c + bc²)

X(5277) lies on these lines: {1, 1929}, {2, 32}, {6, 474}, {8, 2242}, {9, 2305}, {10, 172}, {12, 1415}, {21, 187}, {35, 37}, {36, 1107}, {39, 404}, {41, 750}, {58, 2238}, {99, 1655}, {100, 1500}, {101, 2295}, {112, 451}, {115, 2475}, {171, 213}, {199, 612}, {230, 442}, {274, 385}, {377, 3767}, {391, 5042}, {405, 3053}, {406, 1968}, {468, 2204}, {574, 4188}, {609, 1698}, {762, 2248}, {763, 1654}, {846, 2135}, {940, 2271}, {966, 5019}, {992, 4264}, {1125, 1914}, {1213, 1333}, {1573, 2975}, {2092, 2303}, {2160, 4016}, {2241, 3616}, {2549, 4190}, {3291, 4239}, {3509, 3954}, {3727, 5011}, {4189, 5206}

X(5278) = INTERSECTION OF LINES X(2)X(6) AND X(10)X(31)

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

X(5278) lies on these lines: {1, 4981}, {2, 6}, {8, 405}, {9, 321}, {10, 31}, {37, 3187}, {45, 3995}, {55, 4651}, {63, 169}, {75, 3219}, {100, 1011}, {142, 4001}, {226, 1405}, {306, 2280}, {317, 445}, {573, 1746}, {748, 3741}, {756, 4362}, {896, 3980}, {956, 4245}, {968, 3896}, {984, 3891}, {1001, 4042}, {1125, 4101}, {1212, 3998}, {1229, 3719}, {1330, 4197}, {1441, 1708}, {1714, 5051}, {2177, 4685}, {2205, 4426}, {2476, 2651}, {2550, 4450}, {3006, 3966}, {3011, 4104}, {3120, 4703}, {3305, 4358}, {3681, 3757}, {3683, 3696}, {3691, 3765}, {3715, 3952}, {3729, 4980}, {3739, 4641}, {3791, 3842}, {3883, 5014}, {5081, 5136}

X(5279) = INTERSECTION OF LINES X(2)X(7) AND X(8)X(19)

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

X(5279) lies on these lines: {2, 7}, {6, 977}, {8, 19}, {10, 1781}, {20, 346}, {21, 37}, {27, 321}, {28, 72}, {40, 3692}, {48, 4511}, {69, 1760}, {71, 1761}, {75, 379}, {78, 610}, {100, 3694}, {101, 2327}, {169, 391}, {198, 1259}, {219, 608}, {272, 335}, {281, 3436}, {306, 2897}, {377, 2345}, {380, 3870}, {518, 2264}, {573, 1759}, {604, 3061}, {910, 3965}, {965, 3876}, {975, 3731}, {1172, 4463}, {1330, 4456}, {1442, 1959}, {1723, 4310}, {1817, 3998}, {1826, 5080}, {1953, 4861}, {2092, 2240}, {2171, 2329}, {2173, 3949}, {2174, 4053}, {2256, 3877}, {2269, 3496}, {2354, 4388}, {3950, 4304}

X(5280) = INTERSECTION OF LINES X(1)X(6) AND X(32)X(35)

Trilinears SR + aS_ω : SR + bS_ω : SR + cS_ω César Lozada (9/07/2013)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(a² + b² + c² + bc)

X(5280) lies on these lines: {1, 6}, {3, 609}, {31, 3730}, {32, 35}, {36, 39}, {41, 386}, {42, 251}, {48, 5105}, {58, 672}, {71, 4264}, {81, 3912}, {83, 350}, {101, 1193}, {304, 3758}, {595, 1334}, {651, 3674}, {894, 1930}, {986, 1759}, {1015, 5041}, {1126, 1438}, {1174, 2299}, {1197, 3507}, {1384, 5217}, {1448, 2285}, {1468, 4253}, {1500, 1914}, {1890, 3755}, {1922, 3864}, {1973, 4270}, {2174, 5153}, {2242, 2275}, {2260, 4284}, {2503, 2653}, {3053, 5010}, {3056, 5039}, {3293, 3684}, {3496, 4424}, {3509, 3670}, {3685, 4099}, {3710, 3997}, {3744, 3991}, {3934, 4396}, {3961, 4006}, {4642, 5011}, {5024, 5204}

X(5281) = INTERSECTION OF LINES X(2)X(11) AND X(20)X(35)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b² + c² - 5a² - 2bc)

X(5281) lies on these lines: {1, 3523}, {2, 11}, {3, 1056}, {7, 165}, {8, 3158}, {9, 1200}, {10, 4313}, {12, 3146}, {20, 35}, {33, 4232}, {43, 2293}, {140, 1058}, {144, 4640}, {145, 2646}, {171, 1253}, {193, 2330}, {376, 495}, {388, 3522}, {391, 2268}, {496, 3525}, {498, 3091}, {516, 5226}, {551, 4345}, {612, 3100}, {631, 3295}, {999, 3524}, {1040, 3920}, {1155, 3475}, {1447, 3672}, {1479, 5056}, {1697, 3616}, {1698, 4314}, {1961, 4336}, {3057, 3622}, {3086, 3746}, {3486, 3617}, {3487, 3579}, {3543, 3584}, {3550, 4307}, {3614, 3854}, {3712, 3974}, {4293, 5010}, {5059, 5229}, {5068, 5225}

X(5282) = INTERSECTION OF LINES X(2)X(7) AND X(6)X(38)

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

X(5282) lies on these lines: {2, 7}, {6, 38}, {8, 3496}, {10, 1759}, {31, 37}, {32, 976}, {41, 72}, {44, 4003}, {45, 896}, {55, 3930}, {66, 71}, {141, 4376}, {169, 3691}, {191, 3730}, {198, 199}, {201, 220}, {210, 910}, {218, 3927}, {517, 4390}, {518, 2280}, {748, 3290}, {956, 2170}, {984, 985}, {997, 1055}, {1212, 1451}, {1395, 5089}, {1707, 1961}, {1709, 1766}, {1761, 2345}, {1914, 3938}, {2235, 3116}, {2239, 2276}, {2243, 4386}, {2246, 4712}, {2269, 5227}, {2329, 3869}, {2911, 3958}, {2975, 3061}, {3679, 5011}, {3681, 3684}, {3693, 4640}, {3721, 3924}, {4119, 5014}, {4136, 5016}

X(5283) = INTERSECTION OF LINES X(1)X(6) AND X(2)X(39)

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

X(5283) lies on these lines: {1, 6}, {2, 39}, {8, 1500}, {10, 2276}, {21, 32}, {35, 4386}, {42, 3691}, {115, 2476}, {172, 993}, {187, 4189}, {232, 406}, {377, 2549}, {386, 2238}, {391, 941}, {404, 574}, {474, 5013}, {612, 1011}, {756, 869}, {846, 3496}, {940, 5021}, {966, 2092}, {968, 2082}, {986, 3125}, {992, 5105}, {1015, 3616}, {1125, 2275}, {1213, 4261}, {1475, 3720}, {1506, 4193}, {1575, 1698}, {1621, 2241}, {2242, 2975}, {2268, 2304}, {2292, 3735}, {2295, 3730}, {2303, 5019}, {2478, 2548}, {3199, 4194}, {3666, 4384}, {3815, 4187}, {3959, 4424}, {4185, 5089}, {4251, 4653}, {4264, 4877}

X(5283) = isotomic conjugate of X(1218)
X(5283) = complement of X(34284)
X(5283) = {X(1),X(9)}-harmonic conjugate of X(213)

X(5284) = INTERSECTION OF LINES X(2)X(11) AND X(21)X(36)

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

X(5284) lies on these lines: {1, 748}, {2, 11}, {9, 3873}, {21, 36}, {37, 3108}, {38, 3315}, {44, 4883}, {81, 238}, {210, 3957}, {244, 846}, {329, 405}, {354, 3219}, {404, 3624}, {484, 3833}, {496, 943}, {899, 3750}, {958, 3622}, {968, 4850}, {1155, 3848}, {1279, 3920}, {1320, 3898}, {1479, 4197}, {1617, 5226}, {1698, 3871}, {1848, 4233}, {2895, 4966}, {3218, 3683}, {3246, 3745}, {3303, 3617}, {3306, 4512}, {3337, 3647}, {3436, 5129}, {3634, 3746}, {3685, 4359}, {3715, 4661}, {3740, 3748}, {3741, 5235}, {3757, 4358}, {3812, 5183}, {3841, 4857}, {3936, 4204}, {4228, 4872}, {4418, 4432}, {4430, 5220}

X(5285) = INTERSECTION OF LINES X(1)X(3) AND X(9)X(25)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(b⁴ + c⁴ - a⁴ - a²bc + b³c + bc³)

X(5285) lies on these lines: {1, 3}, {9, 25}, {10, 28}, {22, 63}, {23, 3219}, {31, 579}, {33, 1766}, {42, 284}, {48, 3190}, {71, 1474}, {72, 2915}, {73, 3430}, {100, 306}, {101, 2318}, {109, 1297}, {154, 219}, {159, 197}, {181, 2330}, {184, 2323}, {198, 1260}, {199, 228}, {209, 2194}, {212, 573}, {222, 1350}, {226, 4220}, {291, 1283}, {511, 2003}, {516, 1848}, {951, 1042}, {1376, 3844}, {1397, 3056}, {1473, 3928}, {1486, 4512}, {1495, 3690}, {1631, 3185}, {1995, 3305}, {2187, 2289}, {2222, 2747}, {2299, 4456}, {2360, 3682}, {3098, 3784}, {3752, 5096}, {4221, 4304}

X(5286) = INTERSECTION OF LINES X(2)X(39) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a² + b² + c²)² - 4b²c²

X(5286) lies on these lines: {2, 39}, {4, 6}, {20, 32}, {83, 2996}, {115, 147}, {140, 5024}, {148, 4027}, {172, 4293}, {187, 3522}, {193, 315}, {230, 631}, {232, 3089}, {316, 1570}, {376, 3053}, {385, 3785}, {390, 2241}, {487, 3068}, {488, 3069}, {550, 1384}, {574, 3523}, {578, 1217}, {609, 4299}, {672, 5230}, {962, 1572}, {1212, 3772}, {1285, 3529}, {1506, 5056}, {1851, 2082}, {1885, 3172}, {1914, 4294}, {2242, 3600}, {2275, 3086}, {2276, 3085}, {2345, 4385}, {3054, 3533}, {3090, 3815}, {3096, 3620}, {3146, 5007}, {3528, 5023}, {3673, 4000}, {3832, 5041}, {4644, 4911}, {5008, 5059}

X(5286) = anticomplement of X(7795)
X(5286) = barycentric product X(2345)*X(4000)

X(5287) = INTERSECTION OF LINES X(1)X(2) AND X(9)X(81)

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

X(5287) lies on these lines: {1, 2}, {6, 3305}, {9, 81}, {27, 33}, {34, 469}, {37, 63}, {45, 4641}, {46, 3743}, {55, 4682}, {57, 1255}, {77, 226}, {86, 312}, {171, 968}, {223, 1442}, {329, 3945}, {333, 4687}, {440, 1060}, {464, 1038}, {750, 1962}, {756, 3751}, {984, 4038}, {1001, 3745}, {1100, 4383}, {1211, 4851}, {1230, 3761}, {1386, 4423}, {1453, 5047}, {1790, 2268}, {1817, 3601}, {2334, 4662}, {3175, 4363}, {3219, 3731}, {3242, 4883}, {3306, 3666}, {3664, 4656}, {3715, 4663}, {3723, 3752}, {3729, 3995}, {3737, 4789}, {3782, 4675}, {3875, 4359}, {3980, 3993}

X(5287) = {X(1),X(2)}-harmonic conjugate of X(5256)

X(5288) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(36)

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

X(5288) lies on these lines: {1, 6}, {3, 3632}, {8, 36}, {21, 3244}, {35, 519}, {46, 4853}, {55, 3633}, {56, 3679}, {100, 3625}, {145, 993}, {191, 3057}, {214, 4420}, {404, 3626}, {499, 3421}, {528, 4324}, {529, 3585}, {758, 4861}, {999, 1698}, {1329, 3582}, {1376, 4668}, {1388, 3940}, {1621, 3635}, {1759, 4051}, {2098, 3899}, {2099, 3901}, {2178, 4034}, {2550, 4317}, {3219, 3884}, {3304, 3624}, {3337, 3753}, {3579, 3893}, {3583, 3813}, {3584, 4999}, {3636, 5047}, {3872, 4880}, {3880, 3916}, {3913, 5010}, {4253, 4390}, {4278, 4720}, {4299, 5082}, {4816, 5204}

X(5289) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(11)

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

X(5289) lies on these lines: {1, 6}, {2, 2099}, {3, 214}, {8, 11}, {10, 1482}, {21, 2320}, {36, 3899}, {55, 3877}, {56, 3218}, {63, 1319}, {65, 3306}, {78, 3057}, {145, 2551}, {200, 3880}, {210, 3872}, {329, 529}, {517, 997}, {519, 3452}, {527, 4315}, {551, 4930}, {758, 999}, {965, 1953}, {1388, 2975}, {1389, 3090}, {2390, 3784}, {3207, 3496}, {3295, 3884}, {3303, 3890}, {3304, 3868}, {3338, 4018}, {3340, 3812}, {3445, 3976}, {3576, 4640}, {3616, 4999}, {3679, 5123}, {3680, 4882}, {3876, 4861}, {3885, 4420}, {4421, 5119}, {4662, 4853}, {4711, 4915}

X(5290) = INTERSECTION OF LINES X(1)X(4) AND X(7)X(10)

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

X(5290) lies on these lines: {1, 4}, {2, 3361}, {5, 3333}, {7, 10}, {8, 3671}, {12, 57}, {40, 495}, {56, 3624}, {65, 3679}, {79, 5119}, {85, 1930}, {142, 2551}, {165, 3085}, {200, 377}, {381, 5045}, {551, 4308}, {553, 1788}, {612, 1448}, {975, 4320}, {986, 4862}, {1074, 1103}, {1125, 3600}, {1388, 4870}, {1435, 5142}, {1697, 1836}, {1722, 4859}, {1773, 1781}, {2099, 3633}, {2475, 3870}, {2476, 5231}, {2550, 4882}, {3146, 4314}, {3244, 4323}, {3340, 3632}, {3616, 4315}, {3704, 4659}, {3920, 4347}, {3982, 4848}, {4666, 5046}, {4847, 5177}

X(5291) = INTERSECTION OF LINES X(1)X(6) AND X(8)X(32)

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(5294) = complement of X(17184)

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(5297) = isogonal conjugate of X(34916)

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 - aS_ω : SR - bS_ω : SR - cS_ω César Lozada (9/07/2013)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(a² + b² + c² - bc)

X(5299) lies on these lines: {1, 6}, {31, 4253}, {32, 36}, {35, 39}, {41, 995}, {42, 3108}, {48, 5037}, {56, 609}, {58, 163}, {71, 4284}, {83, 1909}, {101, 1201}, {169, 614}, {172, 1015}, {239, 1930}, {304, 3759}, {386, 2280}, {572, 4300}, {595, 672}, {604, 2172}, {982, 1759}, {1193, 4251}, {1384, 5204}, {1429, 2003}, {1432, 2224}, {1469, 5039}, {1500, 5041}, {2241, 2276}, {2260, 4264}, {3216, 3684}, {3496, 3670}, {3509, 3953}, {3730, 3915}, {3934, 4400}, {5010, 5013}, {5024, 5217}

X(5300) = INTERSECTION OF LINES X(7)X(8) AND X(10)X(31)

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

X(5300) lies on these lines: {1, 4202}, {2, 5015}, {3, 3006}, {4, 3701}, {7, 8}, {10, 31}, {41, 4071}, {145, 5100}, {306, 379}, {315, 3263}, {341, 5080}, {404, 3705}, {516, 3710}, {540, 1046}, {612, 5051}, {976, 2887}, {1125, 4894}, {1193, 4865}, {1330, 3681}, {1478, 4696}, {1479, 4358}, {1839, 3610}, {2177, 3178}, {2292, 4660}, {2475, 4385}, {3434, 3702}, {3436, 4723}, {3616, 4514}, {3757, 4197}, {3811, 3936}, {3876, 4388}, {3902, 5082}, {4193, 5205}, {4198, 5174}, {4200, 5081}, {4417, 4420}

X(5301) = INTERSECTION OF LINES X(6)X(31) AND X(32)X(37)

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

X(5301) lies on these lines: {1, 1333}, {6, 31}, {9, 2220}, {19, 2204}, {32, 37}, {35, 4261}, {44, 3694}, {48, 3285}, {53, 1852}, {56, 1950}, {213, 584}, {284, 595}, {560, 3747}, {577, 1108}, {594, 4426}, {609, 3247}, {906, 1723}, {1030, 2277}, {1100, 2241}, {1172, 1612}, {1213, 4386}, {1449, 5035}, {1474, 2352}, {1621, 2303}, {1839, 3011}, {1841, 1968}, {2174, 2176}, {2178, 3053}, {2242, 3723}, {2251, 3204}, {2275, 5124}, {2278, 2300}, {3730, 5037}, {3749, 5227}, {4026, 4660}

X(5302) = INTERSECTION OF LINES X(1)X(6) AND X(10)X(30)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)(2a² + b² + c² + 3ab + 3ac + 4bc)

X(5302) lies on these lines: {1, 6}, {3, 3740}, {8, 3683}, {10, 30}, {21, 210}, {35, 3697}, {55, 4662}, {56, 3305}, {58, 4682}, {63, 3812}, {65, 3219}, {78, 3715}, {100, 3983}, {191, 3753}, {333, 3714}, {354, 5047}, {375, 970}, {377, 1155}, {484, 4002}, {846, 4646}, {993, 5044}, {1329, 3634}, {1698, 3916}, {2646, 3876}, {3158, 4866}, {3214, 4689}, {3338, 3848}, {3452, 4999}, {3617, 5086}, {3694, 4877}, {3826, 4292}, {3913, 4512}, {4383, 4719}, {4390, 4520}, {4413, 4652}

X(5303) = INTERSECTION OF LINES X(3)X(8) AND X(21)X(36)

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

X(5303) lies on these lines: {1, 4757}, {2, 3614}, {3, 8}, {21, 36}, {35, 3244}, {46, 3897}, {55, 3623}, {56, 1621}, {140, 5080}, {145, 5217}, {191, 214}, {320, 1444}, {404, 993}, {958, 4188}, {960, 4881}, {1030, 4969}, {1420, 3890}, {1476, 2078}, {2475, 4999}, {2476, 4299}, {2646, 3218}, {3434, 3522}, {3436, 3523}, {3576, 3869}, {3579, 4861}, {3601, 3873}, {3612, 3868}, {3621, 4421}, {3633, 3871}, {3681, 4855}, {3754, 5131}, {3822, 4325}, {3916, 4511}, {4297, 5086}

X(5304) = INTERSECTION OF LINES X(2)X(6) AND X(20)X(32)

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

X(5304) lies on these lines: {2, 6}, {4, 3172}, {20, 32}, {25, 1249}, {30, 1285}, {39, 3523}, {98, 5039}, {111, 3163}, {115, 3839}, {172, 3600}, {216, 1180}, {232, 4232}, {251, 393}, {376, 1384}, {387, 4251}, {390, 1914}, {577, 1627}, {609, 4293}, {800, 1194}, {910, 3598}, {1202, 2257}, {1447, 5222}, {2243, 4346}, {2548, 5056}, {2996, 3407}, {3053, 3522}, {3091, 3767}, {3509, 4310}, {3524, 5024}, {3543, 5008}, {3553, 3920}, {4220, 4254}

X(5304) = midpoint of X(37640) and X(37641)
X(5304) = {X(2),X(6)}-harmonic conjugate of X(37665)
X(5304) = {X(3068),X(3069)}-harmonic conjugate of X(141)

X(5305) = INTERSECTION OF LINES X(5)X(6) AND X(30)X(32)

Barycentrics 2 a^4+a^2 b^2+b^4+a^2 c^2-2 b^2 c^2+c^4 : :

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(5305) = midpoint of X(7583) and X(7584)
X(5305) = complement of X(3933)

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(5306) = {X(395),X(396)}-harmonic conjugate of X(141)
X(5306) = complement of X(7788)

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(5308) = anticomplement of X(16832)
X(5308) = {X(1),X(2)}-harmonic conjugate of X(5222)

X(5309) = INTERSECTION OF LINES X(2)X(39) AND X(6)X(13)

Barycentrics a⁴ + b⁴ + c⁴ + a²b² + a²c² - 2b²c² : :

X(5309) lies on these lines: {2, 39}, {4, 5007}, {6, 13}, {30, 32}, {148, 3972}, {183, 4045}, {187, 376}, {230, 549}, {395, 3643}, {396, 3642}, {519, 4153}, {524, 5028}, {543, 1003}, {547, 3815}, {597, 5034}, {671, 3407}, {1506, 5055}, {1570, 1992}, {1596, 1990}, {2241, 3058}, {2275, 3582}, {2276, 3584}, {2452, 5099}, {2548, 3545}, {3053, 3534}, {3162, 5064}, {3543, 5008}, {5013, 5054}

X(5309) = complement of X(32833)
X(5309) = anticomplement of X(7880)
X(5309) = X(32)-of-4th-Brocard-triangle
X(5309 = X(32)-of orthocentroidal-triangle
X(5309 = inverse-in-Kiepert-hyperbola of X(3818)
X(5309) = centroid of reflection triangle of X(32)
X(5309 = {X(13),X(14)}-harmonic conjugate of X(3818)
X(5309) = {X(6),X(381)}-harmonic conjugate of X(7753)

X(5310) = INTERSECTION OF LINES X(1)X(22) AND X(2)X(35)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(b⁴ + c⁴ - a⁴ + a²bc + b³c + bc³)

X(5310) lies on these lines: {1, 22}, {2, 35}, {3, 614}, {12, 428}, {19, 25}, {23, 3743}, {31, 579}, {38, 3220}, {42, 251}, {51, 2330}, {56, 4348}, {184, 3056}, {199, 2223}, {350, 1799}, {354, 4265}, {613, 3796}, {674, 2194}, {858, 4330}, {1030, 3290}, {1194, 1914}, {1281, 1283}, {1370, 4302}, {1631, 2352}, {2920, 3057}, {2922, 3670}, {3011, 4220}, {3583, 5133}, {4228, 4276}

X(5311) = INTERSECTION OF LINES X(1)X(2) AND X(31)X(37)

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

X(5311) lies on these lines: {1, 2}, {6, 756}, {9, 2308}, {31, 37}, {33, 1839}, {38, 940}, {55, 199}, {63, 3989}, {81, 984}, {171, 4414}, {192, 4418}, {197, 1953}, {210, 1100}, {748, 1386}, {750, 3666}, {902, 968}, {985, 1255}, {1460, 2171}, {2177, 3723}, {2206, 2303}, {3681, 4649}, {3791, 3842}, {3873, 4038}, {3923, 3995}, {4349, 4656}, {4722, 5220}

X(5312) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(35)

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

X(5312) lies on these lines: {1, 2}, {6, 35}, {9, 4272}, {36, 4255}, {55, 1203}, {57, 2594}, {58, 5010}, {73, 3339}, {165, 581}, {595, 2177}, {749, 3736}, {750, 4658}, {986, 3901}, {999, 2334}, {1126, 1468}, {1449, 5153}, {1743, 4270}, {1745, 4312}, {3555, 4719}, {3670, 3894}, {3743, 3876}, {3869, 4868}, {3874, 4850}, {3916, 4663}, {4023, 4205}

X(5313) = INTERSECTION OF LINES X(1)X(2) AND X(6)X(36)

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

X(5313) lies on these lines: {1, 2}, {3, 1203}, {6, 36}, {9, 5153}, {31, 4256}, {35, 3052}, {57, 1464}, {72, 4719}, {73, 3361}, {165, 1064}, {748, 4653}, {751, 3736}, {758, 4850}, {982, 3894}, {1191, 3746}, {1420, 2594}, {1449, 4272}, {1453, 3612}, {1470, 2003}, {1743, 5105}, {2308, 4257}, {3670, 3901}, {3792, 4277}, {3877, 4868}, {3899, 4424}

X(5314) = INTERSECTION OF LINES X(3)X(63) AND X(31)X(35)

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

X(5314) lies on these lines: {3, 63}, {9, 22}, {25, 3305}, {31, 35}, {36, 38}, {55, 1386}, {71, 1176}, {100, 3687}, {184, 3781}, {209, 5135}, {219, 3796}, {284, 672}, {378, 3587}, {908, 4220}, {1707, 5010}, {1790, 1818}, {2003, 2979}, {2172, 3730}, {2221, 4255}, {2323, 5012}, {2915, 5044}, {3219, 3220}, {3666, 5096}, {3917, 3955}, {4265, 4641}

X(5315) = INTERSECTION OF LINES X(1)X(6) AND X(31)X(36)

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

X(5315) lies on these lines: {1, 6}, {31, 36}, {35, 595}, {40, 1480}, {56, 2163}, {58, 106}, {65, 1421}, {81, 551}, {109, 1450}, {221, 3361}, {386, 2177}, {484, 3752}, {651, 4315}, {982, 4880}, {1017, 5007}, {1046, 3953}, {1149, 2308}, {1319, 2003}, {1834, 4857}, {1999, 4975}, {2382, 2703}, {2999, 5119}, {3052, 5010}, {3679, 4383}, {3792, 4749}

X(5316) = INTERSECTION OF LINES X(2)X(7) AND X(10)X(11)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b³ + c³ - a²b - a²c + 8abc - b²c - bc²

X(5316) lies on these lines: {2, 7}, {10, 11}, {312, 4431}, {516, 4413}, {519, 3711}, {899, 3755}, {936, 950}, {946, 1698}, {956, 1125}, {960, 4848}, {984, 5121}, {1000, 3679}, {1150, 3707}, {1210, 5044}, {2321, 4358}, {3601, 5129}, {3698, 4301}, {3740, 3816}, {3752, 4656}, {3817, 3925}, {3826, 5087}, {3840, 4104}, {3883, 5205}, {3912, 5233}

X(5316) = complement of X(3306)

X(5317) = INTERSECTION OF LINES X(4)X(6) AND X(19)X(31)

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

X(5317) lies on these lines: {4, 6}, {19, 31}, {27, 2221}, {28, 1104}, {29, 2303}, {34, 604}, {37, 4183}, {81, 286}, {107, 739}, {112, 915}, {158, 2214}, {162, 1778}, {232, 4220}, {240, 1761}, {608, 1118}, {648, 2991}, {1119, 1396}, {1430, 2260}, {1880, 2204}, {1896, 2298}, {2287, 5016}, {2322, 2345}, {2331, 2332}, {4219, 4261}

X(5317) = isogonal conjugate of X(3998)

X(5318) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b⁴ + 3c⁴ - 3a⁴ - 6b²c² - (12)^1/2a²S

X(5318) lies on these lines: {4, 6}, {5, 16}, {12, 1250}, {13, 15}, {14, 3845}, {17, 550}, {18, 3850}, {61, 3627}, {62, 546}, {141, 622}, {230, 1080}, {381, 395}, {383, 3815}, {463, 1495}, {524, 621}, {530, 623}, {590, 2043}, {615, 2044}, {633, 3630}, {634, 3631}, {1546, 3003}, {3411, 3856}, {3628, 5237}

X(5318) = crosssum of X(3) and X(15)
X(5318) = crosspoint of X(4) and X(13)

X(5319) = INTERSECTION OF LINES X(5)X(6) AND X(20)X(32)

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

X(5319) lies on these lines: {2, 3108}, {4, 5007}, {5, 6}, {20, 32}, {39, 631}, {115, 3832}, {172, 4317}, {187, 3528}, {193, 626}, {230, 3526}, {548, 3053}, {609, 4325}, {1249, 3199}, {1572, 4301}, {1598, 1990}, {1906, 2207}, {1914, 4309}, {3530, 5013}, {3547, 5158}, {3618, 3934}, {3785, 4045}, {3815, 5070}, {5041, 5067}

X(5320) = INTERSECTION OF LINES X(6)X(25) AND X(31)X(32)

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

X(5320) lies on these lines: {2, 5138}, {4, 1175}, {6, 25}, {22, 4260}, {31, 32}, {42, 2175}, {55, 584}, {65, 2355}, {81, 4223}, {182, 4220}, {198, 4275}, {199, 579}, {284, 1011}, {386, 3145}, {1200, 2357}, {1395, 1409}, {1397, 1400}, {1751, 3136}, {1824, 2264}, {2174, 2352}, {2206, 5019}, {2328, 4251}, {4383, 5135}

X(5320) = crosssum of X(2) and X(377)

X(5321) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(15)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3b⁴ + 3c⁴ - 3a⁴ - 6b²c² + (12)^1/2a²S

X(5321) lies on these lines: {4, 6}, {5, 15}, {13, 3845}, {14, 16}, {17, 3850}, {18, 550}, {61, 546}, {62, 3627}, {141, 621}, {230, 383}, {381, 396}, {462, 1495}, {524, 622}, {531, 624}, {590, 2044}, {615, 2043}, {633, 3631}, {634, 3630}, {1080, 3815}, {1545, 3003}, {3412, 3856}, {3628, 5238}

X(5321) = crosssum of X(3) and X(16)
X(5321) = crosspoint of X(4) and X(14)

X(5322) = INTERSECTION OF LINES X(1)X(22) AND X(2)X(36)

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(5328) = anticomplement of X(31190)

X(5329) = INTERSECTION OF LINES X(1)X(3) AND X(22)X(31)

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

X(5329) lies on these lines: {1, 3}, {22, 31}, {24, 602}, {25, 238}, {43, 197}, {159, 1740}, {181, 182}, {199, 985}, {394, 3792}, {511, 1397}, {748, 1995}, {1376, 5096}, {1469, 3955}, {1473, 4650}, {1626, 3286}, {1707, 3220}, {1790, 3736}, {2076, 2162}, {2178, 3509}

X(5330) = INTERSECTION OF LINES X(1)X(21) AND X(8)X(11)

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

X(5330) lies on these lines: {1, 21}, {2, 1482}, {8, 11}, {78, 2136}, {145, 1058}, {392, 5047}, {404, 517}, {452, 3623}, {644, 3061}, {952, 5046}, {960, 4861}, {1788, 2099}, {3057, 3871}, {3244, 4867}, {3579, 4881}, {3621, 3940}, {3872, 3876}, {3880, 4420}, {4673, 4720}

X(5331) = INTERSECTION OF LINES X(6)X(21) AND X(27)X(34)

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

X(5331) lies on these lines: {1, 333}, {6, 21}, {27, 34}, {29, 3192}, {42, 1043}, {56, 81}, {58, 2185}, {86, 1193}, {87, 3736}, {106, 931}, {269, 1434}, {270, 1474}, {284, 2363}, {386, 1010}, {958, 2334}, {1126, 4653}, {2215, 4269}, {2279, 3601}

X(5332) = INTERSECTION OF LINES X(6)X(31) AND X(32)X(36)

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

X(5332) lies on these lines: {1, 5007}, {6, 31}, {32, 36}, {39, 5010}, {44, 3681}, {172, 999}, {238, 3789}, {239, 4376}, {609, 1015}, {893, 2364}, {982, 2243}, {995, 2251}, {1040, 3284}, {1100, 3873}, {1403, 1404}, {2220, 2277}, {2300, 5037}, {3703, 4969}

X(5333) = INTERSECTION OF LINES X(2)X(6) AND X(21)X(36)

Barycentrics (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(5333) = trilinear pole of line X(4716)X(4802) (the perspectrix of ABC and Gemini triangle 24)
X(5333) = perspector of Gemini triangle 23 and cross-triangle of ABC and Gemini triangle 23

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(5334) = {X(4),X(6)}-harmonic conjugate of X(5335)

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(5335) = {X(4),X(6)}-harmonic conjugate of X(5334)

X(5336) = INTERSECTION OF LINES X(1)X(6) AND X(19)X(32)

Trilinears a³s - S_BS_C : b³s - S_CS_A : c³s - S_AS_B César Lozada (9/07/2013)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a⁴ + b⁴ + c⁴ + 2a³b + 2a³c - 2b²c²)

X(5336) lies on these lines: {1, 6}, {19, 32}, {25, 1096}, {31, 2171}, {46, 2305}, {609, 1781}, {800, 2331}, {992, 997}, {1184, 5089}, {1400, 3924}, {1572, 1953}, {1731, 5037}, {1826, 3767}, {2285, 5019}, {2321, 4362}, {3290, 5020}, {3612, 5110}

X(5337) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(32)

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

X(5337) lies on these lines: {1, 3}, {2, 32}, {6, 3882}, {39, 81}, {58, 1009}, {63, 3954}, {69, 5019}, {141, 1333}, {172, 3912}, {193, 5042}, {524, 5035}, {1150, 3661}, {2220, 3589}, {3793, 5241}, {4044, 4396}, {4220, 5188}, {4384, 4386}

X(5338) = INTERSECTION OF LINES X(19)X(25) AND X(28)X(34)

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

X(5338) lies on these lines: {2, 1890}, {4, 165}, {19, 25}, {28, 34}, {51, 2261}, {154, 2262}, {204, 1841}, {212, 2270}, {354, 1829}, {461, 4512}, {607, 1190}, {1155, 1878}, {1474, 2280}, {1598, 1753}, {1839, 4207}, {1871, 3517}

X(5339) = INTERSECTION OF LINES X(3)X(14) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b⁴ + c⁴ - a⁴ - 2b²c² + 3^1/2a²S

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(5339) = {X(4),X(6)}-harmonic conjugate of X(5340)

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 - 3^1/2a²S

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(5340) = {X(4),X(6)}-harmonic conjugate of X(5339)

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(5343) = {X(4),X(6)}-harmonic conjugate of X(5344)

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(5344) = {X(4),X(6)}-harmonic conjugate of X(5343)

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(5349) = {X(4),X(6)}-harmonic conjugate of X(5350)

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(5350) = {X(4),X(6)}-harmonic conjugate of X(5349)

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 - 3^1/2bc)

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(5353) = {X(1),X(6)}-harmonic conjugate of X(5357)

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 + 3^1/2bc)

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(5357) = {X(1),X(6)}-harmonic conjugate of X(5353)

X(5358) = INTERSECTION OF LINES X(10)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(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 - 3^1/2(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(5362) = {X(2),X(6)}-harmonic conjugate of X(5367)

X(5363) = INTERSECTION OF LINES X(1)X(3) AND X(23)X(31)

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(5365) = {X(4),X(6)}-harmonic conjugate of X(5366)

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(5366) = {X(4),X(6)}-harmonic conjugate of X(5365)

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 + 3^1/2(abc + b²c + bc²)]

X(5367) lies on these lines: {2, 6}, {15, 404}, {16, 21}, {62, 5047}, {471, 1172}, {1250, 1621}, {2323, 5242}

X(5367) = {X(2),X(6)}-harmonic conjugate of X(5362)

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, where S = 2*area(ABC).
(There is a unique solution with real x,y,z if the reference triangle ABC is acute.)

For any point X inside an acute triangle ABC, let A′ B′ C′ denote the pedal triangle of X. Then X(5373) is the point X for which the quadrilaterals AC′XB′, BA′XC′, CB′XA′ all have the same area.

X(5373) is discussed in the following articles:

Apoloniusz Tyszka, "Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 181-185: Tyszka article.

Jean Pierre Ehrmann, "Constructive solution of a generalization of Steinhaus' problem on partition of a triangle," Forum Geometricorum 7 (2007), 187-190: Ehrmann article.

Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.

X(5373) is the incenter of the Thomson triangle, as proved in Thomson Triangle.

X(5374) = TRILINEAR SQUARE ROOT OF X(63)

Trilinears (cot A)^1/2 : (cot B)^1/2 : (cot C)^1/2

For any point P on segment BC of an acute triangle ABC, let Q be the point on AB nearest to P and let R be the point on AC nearest to P. Let A′ be the choice of P for which area(A′QB) = area(A′RC). Define B′ and C′ cyclically. Then the lines AA′, BB′, CC′ concur in X(5374).

X(5374) is introduced in Mowaffaq Hajja and Panagiotis T. Krasopoulos, "Two more triangle centers," Elemente der Mathematik 66 (2011) 164-174.

X(5375) = X(100)X(650)∩X(101)X(661)

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

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}

X(5375) = complement of X(8047)
X(5375) = X(2)-Ceva conjugate of X(100)
X(5375) = complementary conjugate of complement of X(16686)
X(5375) = crosssum of circumcircle intercepts of line PU(27) (line X(244)X(665))

Points on the Hutson-Moses hyperbola: X(5376)-X(5389)

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*S_A + y*S_B + z*S_C),

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).

X(5376) = H(X(901))

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

X(5376) lies on the Hutson-Moses hyperbola and these lines: {1, 765}, {2, 1016}, {57, 4564}, {81, 4567}, {89, 1252}, {100, 3251}, {105, 1320}, {106, 291}, {274, 4601}, {279, 1275}, {666, 4555}, {898, 901}, {1022, 1023}, {1929, 4674}, {2006, 4997}, {2397, 2401}, {4584, 4622}

X(5376) = isogonal conjugate of X(2087)
X(5376) = trilinear product of PU(28)

X(5377) = H(X(919))

Barycentrics a*(a - b)^2*(a - c)^2*(a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2) : :

X(5377) is the trilinear pole of the line X(100)X(650), this line being tangent at X(100) to the conic {A, B, C, X(2), X(100), PU(112)}, where PU(112) are the isogonal conjugates of the bicentric pair PU(46). (Randy Hutson, September 29, 2014)

X(5377) lies on the Hutson-Moses hyperbola, the Feuerbach circumhyperbola, and these lines: {1, 1053}, {2, 38310}, {4, 6074}, {7, 59}, {8, 1016}, {9, 765}, {21, 4567}, {55, 14947}, {79, 40724}, {80, 14942}, {100, 3126}, {104, 59101}, {105, 1320}, {294, 1642}, {314, 4601}, {497, 31633}, {666, 885}, {673, 3254}, {692, 57018}, {898, 919}, {927, 2742}, {1023, 23893}, {1025, 35355}, {1027, 3257}, {1041, 7012}, {1156, 28071}, {1172, 5379}, {1252, 40779}, {1438, 4876}, {2195, 9365}, {2320, 5385}, {2344, 5384}, {2737, 59133}, {3680, 5382}, {4584, 54353}, {4998, 5218}, {5381, 36798}, {5383, 7155}, {6601, 56850}, {11124, 31628}, {11604, 13576}, {11609, 56853}, {13136, 36802}, {23836, 32735}, {43740, 52456}

X(5377) = reflection of X(100) in X(52884)
X(5377) = isogonal conjugate of X(3675)
X(5377) = symgonal image of X(52884)
X(5377) = isogonal conjugate of the complement of X(53358)
X(57536)-Ceva conjugate of X(1252)
X(5377) = X(i)-isoconjugate of X(j) for these (i,j): {1, 3675}, {11, 1458}, {57, 17435}, {109, 52305}, {241, 2170}, {244, 518}, {291, 38989}, {512, 23829}, {513, 2254}, {514, 665}, {522, 53539}, {649, 918}, {650, 53544}, {663, 43042}, {672, 1086}, {673, 35505}, {764, 1026}, {909, 42770}, {926, 3676}, {1015, 3912}, {1019, 24290}, {1027, 3126}, {1111, 2223}, {1357, 3717}, {1358, 2340}, {1438, 35094}, {1565, 2356}, {1643, 52228}, {1647, 34230}, {1769, 57468}, {1818, 2969}, {1861, 3937}, {1876, 7004}, {2162, 23773}, {2195, 3323}, {2283, 21132}, {2284, 6545}, {2310, 34855}, {2428, 23760}, {3120, 3286}, {3121, 18157}, {3122, 30941}, {3125, 18206}, {3248, 3263}, {3252, 27918}, {3271, 9436}, {3693, 53538}, {3733, 4088}, {3737, 53551}, {3930, 16726}, {3942, 5089}, {4712, 43921}, {4858, 52635}, {5236, 7117}, {7649, 53550}, {8638, 52621}, {9454, 23989}, {14942, 61056}, {16727, 39258}, {17205, 20683}, {18210, 54407}, {21143, 42720}, {22116, 27846}, {23225, 46107}, {24002, 46388}, {36819, 42753}, {43035, 56787}, {43924, 50333}, {43929, 53583}, {52614, 58817}
X(5377) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 3675}, {11, 52305}, {3126, 52304}, {5375, 918}, {5452, 17435}, {6184, 35094}, {23980, 42770}, {33675, 23989}, {39026, 2254}, {39029, 38989}, {39054, 23829}, {39063, 3323}
X(5377) = cevapoint of X(i) and X(j) for these (i,j): {55, 2284}, {100, 518}, {105, 36086}, {294, 52927}, {692, 1914}, {11124, 17435}
X(5377) = trilinear pole of line {100, 650}
X(5377) = barycentric product X(i)*X(j) for these {i,j}: {9, 39293}, {59, 36796}, {100, 666}, {101, 51560}, {105, 1016}, {190, 36086}, {294, 4998}, {518, 57536}, {644, 927}, {646, 32735}, {651, 36802}, {668, 919}, {673, 765}, {692, 36803}, {885, 31615}, {1027, 6632}, {1110, 18031}, {1252, 2481}, {1275, 28071}, {1438, 7035}, {1462, 4076}, {1814, 15742}, {1978, 32666}, {3699, 36146}, {3939, 34085}, {4391, 59101}, {4554, 52927}, {4564, 14942}, {4567, 13576}, {4600, 18785}, {4601, 56853}, {5378, 6654}, {5381, 52902}, {6065, 34018}, {6559, 7045}, {31628, 35313}, {39272, 53337}, {42722, 59021}, {43929, 57950}
X(5377) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 3675}, {43, 23773}, {55, 17435}, {59, 241}, {100, 918}, {101, 2254}, {105, 1086}, {109, 53544}, {241, 3323}, {294, 11}, {517, 42770}, {518, 35094}, {644, 50333}, {650, 52305}, {651, 43042}, {662, 23829}, {666, 693}, {673, 1111}, {692, 665}, {765, 3912}, {885, 40166}, {906, 53550}, {919, 513}, {927, 24002}, {1016, 3263}, {1018, 4088}, {1024, 21132}, {1026, 53583}, {1027, 6545}, {1110, 672}, {1252, 518}, {1262, 34855}, {1415, 53539}, {1416, 53538}, {1438, 244}, {1462, 1358}, {1814, 1565}, {1914, 38989}, {1983, 53555}, {2149, 1458}, {2195, 2170}, {2223, 35505}, {2284, 3126}, {2427, 42758}, {2481, 23989}, {4557, 24290}, {4559, 53551}, {4564, 9436}, {4567, 30941}, {4570, 18206}, {4600, 18157}, {4619, 41353}, {4998, 40704}, {5378, 40217}, {5379, 15149}, {6065, 3693}, {6559, 24026}, {7012, 5236}, {7115, 1876}, {8751, 2969}, {9503, 15634}, {13576, 16732}, {14942, 4858}, {15742, 46108}, {17435, 52304}, {18785, 3120}, {23990, 2223}, {28071, 1146}, {28132, 42455}, {31615, 883}, {32641, 57468}, {32658, 3937}, {32666, 649}, {32735, 3669}, {34085, 52621}, {35333, 16892}, {36057, 3942}, {36086, 514}, {36146, 3676}, {36796, 34387}, {36802, 4391}, {36803, 40495}, {39293, 85}, {41339, 1566}, {41934, 43921}, {43929, 764}, {46163, 2530}, {51560, 3261}, {51987, 42753}, {52635, 61056}, {52902, 52626}, {52927, 650}, {54235, 2973}, {54364, 42754}, {56853, 3125}, {57192, 4925}, {57536, 2481}, {57731, 42720}, {59101, 651}, {59149, 1026}
X(5377) = {X(885),X(35313)}-harmonic conjugate of X(666)

X(5378) = H(X(813))

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

X(5378) is the trilinear pole of the line X(100)X(649), this line being tangent at X(100) to the hyperbola {A, B, C, X(81), X(100), PU(8)}. (Randy Hutson, September 29, 2014)

X(5378) lies on the Hutson-Moses hyperbola and these lines: {1, 1016}, {6, 765}, {56, 4564}, {58, 4567}, {86, 4601}, {87, 4076}, {106, 291}, {269, 1275}, {660, 876}, {666, 1026}, {813, 898}, {1411, 4518}, {1438, 4876}

X(5378) = isogonal conjugate of X(27846)

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(5379) = isogonal conjugate of X(18210)
X(5379) = polar conjugate of X(16732)
X(5379) = X(63)-isoconjugate of X(3125)

X(5380) = H(X(111))

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

X(5380) 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(5380) = isogonal conjugate of X(14419)
X(5380) = cevapoint of X(i) and X(j) for these {i,j}: {513, 7292}, {650, 8540}
X(5380) = crosspoint of X(892) and X(36085)
X(5380) = crosssum of X(351) and X(2642)
X(5380) = trilinear pole of X(37)X(100)

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(5381) = isogonal conjugate of X(1646)

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(5382) = isotomic conjugate of complement of X(25268)

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(5383) = isogonal conjugate of X(6377)
X(5383) = isotomic conjugate of X(21138)

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(5385) = isotomic conjugate of X(4957)

X(5386) = H(X(753))

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

X(5386) lies on the Hutson-Moses hyperbola and these lines: {753, 898}, {765, 3799}, {1016, 3807}

X(5387) = H(X(2748))

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

X(5387) lies on the Hutson-Moses hyperbola and these lines: {898, 2748}, {1016, 3759}

X(5388) = H(X(789))

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

X(5388) lies on the Hutson-Moses hyperbola and these lines: {789, 898}, {799, 4584}

X(5389) = H(X(755))

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

X(5389) lies on the Hutson-Moses hyperbola and these lines: {755, 898}, {4553, 4567}

X(5390) = EULER-MORLEY-ZHAO POINT

Trilinears T(A,B,C) : T(B,C,A) : T(C,A,B), where T(A,B,C) = cos(B - C) - cos(B + C) - cos(B/3 + C/3) + cos(5B/3 + 5C/3) - sin(C - B/3 - π/6) - sin(B - C/3 - π/6) + sin(B + 5C/3 - π/6) + sin(C + 5B/3 - π/6) (Chris van Tienhoven, April 7, 2013)

Barycentrics a*f(A/3, B/3, C/3) : b*f(B/3, C/3, A/3) : c*f(C/3, A/3, B/3), where f(x,y,z) is defined using the abbreviations [m,n] for sin(x + 2my + 2nz) + sin(x + 2ny + 2mz) and [m] for [m,m]/2, as follows:

f(x,y,z) = [-3,-1] + 5[-2,-1] + [-2,0] - 5[-1,1] - 3[-1,2] - [-1,3] + [0,1] + 2[0,2] - [0,3] - 2[1,3] - 2[2,3] + [-3] - 2[-2] + 3[-1] + 3[0] + 3[1] + 3[2] - 2[3] - [4] (Barry Wolk)

Let DEF be the classical Morley triangle. The Euler lines of the three triangles AEF, BFD, CDE to concur in X(5390), as discovered by Zhao Yong of Anhui, China, October 2, 2012. For a construction and derivation of barycentric coordinates by Shi Yong, see Problem 20 at Unsolved Problems and Rewards. For further developments, including the development of trilinear and barycentric coordinates as shown above, type X(5390) into Search at Hyacinthos.

If you have The Geometer's Sketchpad, you can view X(5390).

X(5390) lies on this line: {357, 1136}

X(5391) = ISOTOMIC CONJUGATE OF X(1336)

Barycentrics bc(1 - sin A) : :
Barycentrics b c - S : :

X(5391) lies on these lines: {2,37}, {7,491}, {8,492}, {10,5490}, {69,13388}, {100,9099}, {239,3069}, {274,1123}, {319,1270}

X(5391) = isotomic conjugate of X(1336)
X(5391) = {X(2),X(75)}-harmonic conjugate of X(1267)

X(5392) = INTERSECTION OF LINES X(4)X(52) AND X(22)X(98)

Barycentrics sec 2A : sec 2B : sec 2C

X(5392) lies on the Euler perspective cubic (K045), corresponding to the point X(68) on the Euler central cubic (K044). (Randy Hutson, November 22, 2014)

X(5392) lies on the Kiepert hyperbola and these lines: {2, 311}, {3, 96}, {4, 52}, {10, 91}, {22, 98}, {226, 914}, {262, 5133}, {264, 275}, {338, 394}, {467, 2052}

X(5392) = isogonal conjugate of X(571)
X(5392) = isotomic conjugate of X(1993)
X(5392) = pole wrt polar circle of trilinear polar of X(24)
X(5392) = X(48)-isoconjugate (polar conjugate) of X(24)
X(5392) = Cundy-Parry Phi transform of X(96)
X(5392) = Cundy-Parry Psi transform of X(52)
X(5392) = trilinear pole of line X(523)X(2072) (the line of the degenerate cross-triangle of medial and 2nd Euler triangles)

X(5393) = CENTER OF THE PAACHE-MYAKISHEV ELLIPSE

Trilinears    R + r csc A : :
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(B_A) and W(C_A) be the two congruent circles, within triangle ABC, each tangent to the other and to sideline BC of triangle ABC, with W(B_A) also tangent to sideline AB and W(C_A) also tangent to sideline AC; cf. the Paache configuration at X(1123). Let B_A and C_A be the touchpoints of these circles with sideline BC. Define the points C_B, A_C cyclically and define the points A_B, B_C cyclically. The six points lie on an ellipse having center X(5393) and equation

d(2 + d)x² + e(2 + e)y² + f(2 + f)z² - 2(2 + e + f + ef)yz - 2(2 + f + d + fd)zx - 2(2 + d + e + de)xy = 0,

where d = cot(A/2), e = cot(B/2), f = cot(C/2). Let X = X(5393). Then |GX|/|IX| = s/(3r), where G = centroid, I = incenter, r = inradius, and s = semiperimeter. (Alexei Myakishev, December 25, 2012).

An associated conic, the Paache-Myakishev-Moses conic, is introduced at X(5405). This conic results from the two congruent circles that do not lie within triangle ABC.

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

X(5393), including the ellipse. You can also view the configuration for pairs of circles used in the constructions of X(5393) and X(5405): Pairs of Circles.

X(5393) lies on these lines: {1, 2 }, {9, 3068}, {37, 590}, {57, 482}, {81, 3300}, {175, 5226}, {226, 481}, {491, 4357}, {492, 3879}, {515, 2048}, {615, 1100}, {642, 3666}, {940, 1335}, {1124, 4383}, {1255, 3302}, {1267, 3875}, {1449, 3069}, {1585, 1785}, {1991, 4643}

X(5393) = {X(1),X(2)}-harmonic conjugate of X(5405)

X(5394) = CONGRUENT INCIRCLES POINT

Barycentrics (unknown)

X(5394) is the point X for which the three triangles AXB, BXC, CXA have congruent incircles. The existence of this point is proved by Noam Elkies in Mathematics Magazine 60 (1987) 117. His proof applies to a much wider range of functions (with the inradius replaced by the area, semiperimeter, etc., or any positive combination thereof).

Following is a copy-and-run Mathematica program that computes actual trilinear distances (1.7916..., 1.7057..., 1.6328...) of X(5394) for the triangle given by (a,b,c) = (6,9,13).

(1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)] {x/a, y/b, z/c} /. #1 /.
NSolve[{x + y + z == 1, (a + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2])/
x == (b + Sqrt[b^2 x^2 + (a^2 + b^2 - c^2) x y + a^2 y^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
y == (c + Sqrt[c^2 x^2 + (a^2 - b^2 + c^2) x z + a^2 z^2] +
Sqrt[c^2 y^2 + (-a^2 + b^2 + c^2) y z + b^2 z^2])/
z /. #1}, {x, y, z}, WorkingPrecision → 40][[1]] &)[Thread[{a, b, c} → {6, 9,13}]]
(* Code by Peter Moses, October 23, 2012. *)

Let S = 2 area ABC, s1 = a + b + c, s2 = b c + c a + a c, s3 = a b c. Then the radius of the congruent incircles is the least positive solution of the following 8th-degree polynomial equation:

1 -
8 s1 x / S +
(23 s1^2 + 8 s2) x^2 / S^2 -
2 s1 (13 s1^2 + 24 s2) x^3 / S^3 +
2 (3 s1^4 + 42 s1^2 s2 + 8 s2^2 + 12 s1 s3) x^4 / S^4 -
16 s1 (s1^2 s2 + 4 s2^2 + 6 s1 s3) x^5 / S^5 +
4 s1 (2 s1^5 - 9 s1^3 s2 + 4 s1 s2^2 + 24 s1^2 s3 + 24 s2 s3) x^6 / S^6 +
96 s1 s2 x^7 / S^5 -
288 s1 s3 x^8 / S^6 = 0.

For example, if {a,b,c} = {6,9,13} the equation is as follows:

1-4 Sqrt[7/5] x+(2503 x^2)/280-(2021 x^3)/(40 Sqrt[35])+(269373 x^4)/62720-(140289 x^5)/(22400 Sqrt[35])+(187 x^6)/1960+(747 x^7)/(44800 Sqrt[35])-(3159 x^8)/6272000 = 0,

of which the least positive solution is x = 0.803384896325630173615878150981...

Here is a Mathematica code for computing the radius:

testtriangle = {6, 9, 13};
N[First[Sort[Select[x/.Solve[(1 - 8 s1 x / S + (23 s1^2 + 8 s2) x^2 / S^2 - 2 s1 (13 s1^2 + 24 s2) x^3 / S^3 + 2 (3 s1^4 + 42 s1^2 s2 + 8 s2^2 + 12 s1 s3) x^4 / S^4 - 16 s1 (s1^2 s2 + 4 s2^2 + 6 s1 s3) x^5 / S^5 + 4 s1 (2 s1^5 - 9 s1^3 s2 + 4 s1 s2^2 + 24 s1^2 s3 + 24 s2 s3) x^6 / S^6 + 96 s1 s2 x^7 / S^5 - 288 s1 s3 x^8 / S^6 /.Thread[{s1, s2, s3, S} → {a + b + c, b c + c a + a b, a b c, 1/2 Sqrt[(a + b - c) (a - b + c) (-a + b + c) (a + b + c)]}] /.Thread[{a, b, c} → testtriangle]) == 0, x], #>0&]]], 30]

(Notes and code from Peter Moses, July 24, 2019)

Following is another approach to formulating and computing the congruent incircles radius. Let
k0 = s1^2 (s1^3-4 s1 s2+12 s3)^2
k1 = -s1 (11 s1^5-60 s1^3 s2+80 s1 s2^2+48 s1^2 s3-96 s2 s3)
k2 = 4 (11 s1^4-36 s1^2 s2+16 s2^2+24 s1 s3)
k3 = -16 (5 s1^2-8 s2)
k4 = 64, where
s1 = a + b + c
s2 = b c + c a + a b
s3 = a b c.
Next, let x be the maximal root of k0 + k1 x + k2 x^2 + k3 x^3 + k4 x^4 = 0. Then the desired radius is S / (s1 + 2 Sqrt[x]) where S = 2 * Area ABC.

Here is a Mathematica code for computing the radius:

testtriangle={6,9,13}; N[S/(s1+2 Sqrt[Last[Sort[x/.Solve[ (s1^2 (s1^3-4 s1 s2+12 s3)^2-s1 (11 s1^5-60 s1^3 s2+80 s1 s2^2+48 s1^2 s3-96 s2 s3)x+4 (11 s1^4-36 s1^2 s2+16 s2^2+24 s1 s3) x^2-16 (5 s1^2-8 s2) x^3+64 x^4/.#)==0,x]]]])/.#&[Thread[{s1,s2,s3,S} → {a+b+c,b c+c a+a b,a b c,1/2 Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]}]/.Thread[{a,b,c} → testtriangle]],30]

(Notes and code from Peter Moses, August 13, 2019)

X(5394) lies on no line X(i)X(j) for 1 <= i < j <= 33000.

X(5395) = ISOTOMIC CONJUGATE OF X(3620)

Trilinears       1/(sin A + 2 cos A tan ω) : 1/(sin B + 2 cos B tan ω) : 1/(sin C + 2 cos C tan ω)
Trilinears       1/(2 cos A + sin A cot ω) : 1/(2 cos B + sin B cot ω) : 1/(2 cos C + sin C tan ω)
Trilinears       1/(2 cos A sin ω + sin A cos ω) : 1/(2 cos B sin ω + sin B cos ω) : 1/(2 cos C sin ω + sin C cos ω)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(3b² + 3c² - a²)

Let X, Y, Z be the points defined by Dominik Burek as at X(1217). If the initial point P is the centroid, then the perspector of the triangles XYZ and ABC is X(5395). (Peter Moses, June 9, 2012)

X(5395) lies on these lines: {2, 3053}, {4, 5050}, {6, 2996}, {20, 262}, {76, 193}, {83, 5033}, {98, 3091}, {439, 3815}, {458, 459}, {620, 2548}, {671, 5286}, {3146, 3329}, {3424, 3832}

X5395) = isogonal conjugate of X(5013)
X(5395) = polar conjugate of X(8889)

X(5396) = INTERSECTION OF LINES X(1)X(5) AND X(3)X(6)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos(B - A) + cos(C - A)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)(cos A + cos B + cos C) + (sin A)(sin A + sin B + sin C)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = R*(cos A)(cos A + cos B + cos C) + s*sin A
X(5396) = (b + c)(c + a)(a + b)*X(1) - 2abc*X(5)   (Peter Moses, January 2, 2013)

X(5396)-X(5400) were submitted with trilinears by Randy Hutson, December 12, 2012.

X(5396) = {X(3),X(6)}-harmonic conjugate of X(5398)

X(5396) lies on these lines: {1, 5}, {3, 6}, {35, 2361}, {40, 5312}, {42, 517}, {51, 859}, {54, 60}, {73, 942}, {140, 3216}, {515, 2051}, {912, 3666}, {1066, 5045}, {1155, 4337}, {1193, 1385}, {1450, 5126}, {1871, 1880}, {2800, 4868}, {3060, 4216}, {3190, 3940}, {3576, 5313}, {3579, 4300}, {3682, 5044}}

X(5396) = isogonal conjugate of X(5397)
X(5396) = crossdifference of every pair of points on line X(523)X(654)
X(5396) = {X(371),X(372)}-harmonic conjugate of X(2278)

X(5397) = ISOGONAL CONJUGATE OF X(5396)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 + cos(B - A)) + cos(C - A))
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/[(cos A)(cos A + cos B + cos C) + (sin A)(sin A + sin B + sin C)]
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 1/[R*(cos A)(cos A + cos B + cos C) + s*sin A]

X(5397) lies on the hyperbola that passes through the points A, B, C, X(1), X(36), as well as the Kiepert hyperbola. X(5397) is the trilinear pole of the line X(523)X(654). (Randy Hutson, Dec. 31, 2012)

X(5397) lies on the Kiepert hyperbola and these lines: {4, 2278}, {5, 60}, {10, 2323}, {12,54}, {36, 226}, {59, 495}, {94, 3615}, {275, 860}, {321, 4511}, {1443, 1446}, {2051, 4276}, {2052, 5136}, {2618, 3737}

X(5397) = isogonal conjugate of X(5396)

X(5398) = {X(3), X(6)}-HARMONIC CONJUGATE OF X(5396)

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos A)(cos A + cos B + cos C) - (sin A)(sin A + sin B + sin C)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = R*(cos A)(cos A + cos B + cos C) - s*sin A
X(5398) = 2r(r + R)*X(3) + (r² + 4rR - s²)*X(6)    (Peter Moses, January 2, 2013)

X(5398) lies on these lines: {1, 2361}, {3, 6}, {4, 162}, {5, 1724}, {30, 1754}, {31, 517}, {36, 2003}, {46, 1399}, {47, 65}, {56, 215}, {81, 1006}, {184, 859}, {255, 942}, {283, 405}, {355, 3072}, {595, 1482}, {601, 3579}, {602, 1385}, {603, 1465}, {912, 4641}, {1060, 1708}, {1064, 2308}, {1411, 2964}, {1496, 5045}, {1718, 3336}, {1737, 5348}, {1780, 3560}, {2979, 4218}, {4216, 5012}

X(5398) = {X(371),X(372)}-harmonic conjugate of X(2245)

X(5399) = INTERSECTION OF LINES X(1)X(5) AND X(54)X(59)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - cos(A - B) - cos(A - C)
X(5399) = (4R² - 2rR - r² - s²)*X(1) + 4rR*X(5) (Peter Moses, January 2, 2013)

X(5399) = {X(1), X(2594)}-harmonic conjugate of X(5396)

X(5399) lies on these lines: {1, 5}, {3, 947}, {42, 942}, {54, 59}, {55, 500}, {73, 517}, {386, 999}, {581, 3295}, {1048, 2607}, {1060, 3811}, {1870, 5174}, {3333, 5312}, {3579, 4303}, {4322, 5126}}

X(5400) = X(1)X(5)∩X(2)X(991)

Trilinears 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 the hexyl-excentral ellipse and 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 sec(A + π/5) : :
Barycentrics siin A sec(A + π/5) : :

On the sides of ABC construct external regular pentagons ABC1C2C3, BCA1A2A3, CAB1B2B3. The Aubert lines of [ABA1B3], [BCB1C3], [CAC1A3] concur in X(5401). These lines pass through A, B, C. (Aubert lines are defined at X(45010); see also A construction of X(3381) and X(5401) using Aubert lines). For the case of external pentagons, see X(3381). (Ivan Pavlov, April 15, 2022)

X(5401) lies on the Kiepert hyperbola and these lines: {2, 3379}, {3, 3382}, {4, 3380}, {5, 3368}, {6, 3381}, {1139, 3395}, {1140, 3393}, {3370, 3394}, {3396, 3397}

X(5401) = isogonal conjugate of X(3393)
X(5401) = X(3369)-cross conjugate of X(5402)
X(5401) = X(3)-Dao conjugate of X(3393)
X(5401) = barycentric quotient X(6)/X(3393)
X(5401) = {X(5),X(3368)}-harmonic conjugate of X(5402)

X(5402) = CSC(A + π/5) POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A + π/5)

X(5402) lies on the Kiepert hyperbola and these lines: {2, 3380}, {3, 3381}, {4, 3379}, {5, 3368}, {6, 3382}, {1139, 3396}, {1140, 3394}, {3370, 3393}, {3395, 3397}

X(5402) = isogonal conjugate of X(3394)

X(5403) = SEC(A - ω/2) POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec(A - ω/2)

Let A′ be the apex of the isosceles triangle BA′C constructed outward on BC such that ∠A′BC = ∠A′CB = ω/2. Define B′ and C′ cyclically. Let Ha be the orthocenter of BA′C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(5403). (Randy Hutson, July 20, 2016)

X(5403) lies on the Kiepert hyperbola and these lines: {2, 1670}, {3, 1676}, {4, 1671}, {5, 141}, {6, 1677}, {11, 3238}, {12, 3237}, {83, 1342}, {98, 1343}, {485, 1690}, {486, 1689}, {1348, 1664}, {1349, 1665}, {2009, 3102}, {2010, 3103}

X(5403) = isogonal conjugate of X(1342)
X(5403) = circumcircle-inverse of X(34134)

X(5404) = CSC(A - ω/2) POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(A - ω/2)

Let B* be a point such that (angle CBB*) = ω/2, and let C* be a point such that (angle CBC*) = ω/2. Let A′ = (line BB*)∩(line CC*), and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5404). (Randy Hutson, September 5, 2014)

X(5404) lies on the Kiepert hyperbola and these lines: {2, 1671}, {3, 1677}, {4, 1670}, {5, 141}, {6, 1676}, {11, 3237}, {12, 3238}, {83, 1343}, {98, 1342}, {485, 1689}, {486, 1690}, {1348, 1665}, {1349, 1664}, {2009, 3103}, {2010, 3102}

X(5404) = isogonal conjugate of X(1343)

X(5404) = circumcircle-inverse of X(34133)

X(5405) = CENTER OF THE PAACHE-MYAKISHEV-MOSES CONIC

Trilinears    R - r csc A : :
Barycentrics   2 - cot(B/2) - cot(C/2) : 2 - cot(C/2) - cot(A/2) : 2 - cot(A/2) - cot B/2)
Barycentrics   a - 2r : b - 2r : c - 2r
X(5405) = s*X(1) - 3r*X(2)   (Peter Moses, January 2, 2013)

For the construction of this conic, see X(5393), where the associated Paache-Myakishev ellipse is introduced.

If you have The Geometer's Sketchpad, you can view X(5405), including the conic.

X(5405) lies on these lines: {1, 2}, {9, 3069}, {37, 615}, {57, 481}, {81, 3299}, {176, 5226}, {226, 482}, {491, 3879}, {492, 4357}, {590, 1100}, {591, 4643}, {641, 3666}, {940, 1124}, {946, 2048}, {1255, 3300}, {1335, 4383}, {1449, 3068}, {1586, 1785}, {1659, 5219}

X(5405) = {X(1),X(2)}-harmonic conjugate of X(5393)

X(5406) = 1st LUCAS POLAR PERSPECTOR

Trilinears (cos A)(2 + cot A) : (cos B)(2 + cot B) : (cos C)(2 + cot C)

Let A′B′C′ be the Lucas central triangle. Let L_A be the trilinear polar of A′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, 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: {2, 490}, {3, 49}, {6, 588}, {25, 9739}, {323, 6411}, {343, 488}, {372, 1583}, {493, 5062}, {1151, 1993}, {1584, 6396}, {1591, 6560}, {1592, 5420}, {1600, 6410}, {1994, 3592}, {3155, 9733}, {3311, 8910}, {3594, 5422}, {3785, 8223}, {6423, 8962}

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 L_A be the trilinear polar of A′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, 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: {2, 489}, {3, 49}, {6, 589}, {25, 9738}, {323, 6412}, {343, 487}, {371, 1584}, {494, 5058}, {1152, 1993}, {1583, 6200}, {1591, 5418}, {1592, 6561}, {1599, 6409}, {1994, 3594}, {3156, 9732}, {3592, 5422}, {3785, 8222}, {6347, 9678}, {6348, 9679}, {6805, 9540}, {6806, 9541}, {8962, 9600}

X(5408) = 3rd LUCAS POLAR PERSPECTOR

Trilinears (cos A)(1 + cot A) : (cos B)(1 + cot B) : (cos C)(1 + cot C)
Trilinears cos A - sin A + csc A : :

Let A′B′C′ be the Lucas tangents triangle. Let L_A be the trilinear polar of A′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, 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: {2, 372}, {3, 49}, {6, 493}, {25, 9733}, {63, 3083}, {69, 1590}, {317, 492}, {323, 6200}, {371, 1599}, {486, 6504}, {490, 1586}, {494, 8770}, {511, 3155}, {615, 1592}, {1152, 1584}, {1194, 6421}, {1589, 6458}, {1591, 3070}, {1600, 6396}, {1994, 6419}, {2987, 7598}, {3069, 6806}, {3084, 5414}, {3156, 9306}, {5422, 6420}, {6337, 8222}, {6460, 6805}, {8909, 8910}, {8911, 9723}

X(5408) = X(2)-Ceva conjugate of X(5409)

X(5409) = 4th LUCAS POLAR PERSPECTOR

Trilinears (cos A)(1 - cot A) : (cos B)(1 - cot B) : (cos C)(1 - cot C)
Trilinears cos A + sin A - csc A : :

Let A′B′C′ be the Lucas(-1) tangents triangle. Let L_A be the trilinear polar of A′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, 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: {2, 371}, {3, 49}, {6, 494}, {22, 8989}, {25, 9732}, {63, 2067}, {69, 1589}, {317, 491}, {323, 6396}, {372, 1600}, {485, 6504}, {489, 1585}, {493, 8770}, {511, 3156}, {590, 1591}, {1151, 1583}, {1194, 6422}, {1590, 6457}, {1592, 3071}, {1599, 6200}, {1994, 6420}, {2066, 3083}, {2987, 7599}, {3068, 6805}, {3155, 9306}, {5422, 6419}, {6337, 8223}, {6459, 6806}, {6503, 8909}

X(5409) = X(2)-Ceva conjugate of X(5408)

X(5410) = 5th LUCAS POLAR PERSPECTOR

Trilinears (sin A)(2 - tan A) : (sin B)(2 - tan B) : (sin C)(2 - tan C)

Let A′ be the perspector of the A-Lucas circle, and define B′ and C′ cyclically. (The lines AA′, BB′, CC′ concur in X(1151).) Let L_A be the polar of A′ with respect to the A-Lucas circle, and define L_B and L_C cyclically. Let A* = L_B∩L_C, B* = L_C∩L_A, C* = L_A∩L_B. The lines AA*, BB*, CC* concur in X(5410). X(5410) = {X(6), X(25)}-harmonic conjugate of X(5411). (Randy Hutson, February 10, 2013)

X(5410) lies on these lines: {4, 1131}, {6, 25}, {24, 3312}, {186, 6398}, {235, 1588}, {371, 1593}, {372, 3515}, {378, 6221}, {427, 3068}, {468, 3069}, {485, 7507}, {590, 5094}, {1151, 3516}, {1271, 3536}, {1398, 2067}, {1587, 3575}, {1594, 8976}, {1597, 6199}, {1598, 6417}, {1702, 1902}, {1885, 6459}, {2066, 7071}, {2207, 5058}, {3089, 7582}, {3092, 5198}, {3127, 8975}, {3172, 6424}, {3517, 6418}, {3518, 6428}, {3520, 6449}, {3541, 8981}, {3542, 7584}, {6353, 7586}, {7487, 7581}, {8889, 8972}

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 L_A be the polar of A′ with respect to the A-Lucas(-1) circle, and define L_B and L_C cyclically. Let A* = L_B∩L_C, B* = L_C∩L_A, C* = L_A∩L_B. The lines AA*, BB*, CC* concur in X(5411). X(5411) = {X(6), X(25)}-harmonic conjugate of X(5410). (Randy Hutson, February 10, 2013)

X(5411) lies on these lines: {4, 1132}, {6, 25}, {24, 3311}, {186, 6221}, {235, 1587}, {371, 3515}, {372, 1593}, {378, 6398}, {427, 3069}, {468, 3068}, {486, 7507}, {615, 5094}, {1152, 3516}, {1249, 5200}, {1270, 3535}, {1398, 6502}, {1588, 3575}, {1597, 6395}, {1598, 6418}, {1703, 1902}, {1885, 6460}, {2207, 5062}, {3089, 7581}, {3093, 5198}, {3147, 8981}, {3172, 6423}, {3517, 6417}, {3518, 6427}, {3520, 6450}, {3542, 7583}, {5414, 7071}, {6353, 7585}, {7487, 7582}, {7505, 8976}

X(5412) = 1st KENMOTU HOMOTHETIC CENTER

Trilinears (sin A)(1 - tan A) : (sin B)(1 - tan B) : (sin C)(1 - tan C) : :
Trilinears cos A + sin A - sec A : :

Let U, V, W be the congruent squares described at X(371). Let L_A be the extended diagonal of U that does not contain X(371), and define L_B and L_C cyclically. Let A′ = L_B∩L_C, 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); A′B′C′ is named the 1st Kenmotu diagonals triangle at X(31). (Randy Hutson, February 9, 2013)

X(5412) lies on the conic {A, B, C, X(4), X(24)} and these lines: {3, 3093}, {4, 371}, {6, 25}, {19, 5415}, {24, 372}, {33, 2066}, {34, 2067}, {186, 6396}, {216, 3155}, {235, 3071}, {317, 491}, {378, 6200}, {393, 8576}, {403, 6565}, {427, 590}, {468, 615}, {486, 3542}, {577, 3156}, {605, 2212}, {606, 1395}, {1151, 1593}, {1152, 3515}, {1164, 3087}, {1452, 2362}, {1587, 7487}, {1588, 3089}, {1595, 8981}, {1597, 6221}, {1598, 3092}, {1829, 7969}, {1968, 6406}, {2207, 6424}, {3069, 6353}, {3070, 3575}, {3088, 9540}, {3147, 5420}, {3199, 5058}, {3312, 3517}, {3516, 6409}, {3518, 6420}, {3536, 5591}, {3541, 5418}, {3592, 5198}, {4232, 7586}, {5094, 8253}, {5417, 6419}, {6756, 7583}, {6995, 7585}, {7378, 8972}, {8577, 8882}

X(5412) = isogonal conjugate of X(11091)
X(5412) = cevapoint of X(i) and X(j) for these {i,j}: {25, 6424}, {371, 8855}
X(5412) = X(393)-Ceva conjugate of X(5413)
X(5412) = X(571)-cross conjugate of X(5413)
X(5412) = polar conjugate of X(34392)
X(5412) = {X(4),X(371)}-harmonic conjugate of X(11473)
X(5412) = {X(6), X(25)}-harmonic conjugate of X(5413)
X(5412) = X(6203)-of-orthic-triangle if ABC is acute
X(5412) = trilinear product X(i)*X(j) for these {i,j}: {19, 372}, {31, 1586}, {491, 1973}, {1096, 5409}, {1748, 8577}, {3377, 5413}

X(5413) = 2nd KENMOTU HOMOTHETIC CENTER

Trilinears (sin A)(1 + tan A) : (sin B)(1 + tan B) : (sin C)(1 + tan C) : :
Trilinears cos A - sin A - sec A : :

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 L_A be the extended diagonal of U′ that does not contain X(372), and define L_B and L_C cyclically. Let A′ = L_B∩L_C, 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); A′B′C′ is named the 2nd Kenmotu diagonals triangle at X(31). (Randy Hutson, February 9, 2013)

X(5413) lies on the conic {{A, B, C, X(4), X(24)}} and these lines: {3, 3092}, {4, 372}, {6, 25}, {19, 5416}, {24, 371}, {33, 5414}, {34, 6502}, {186, 6200}, {216, 3156}, {235, 3070}, {317, 492}, {378, 6396}, {393, 5200}, {403, 6564}, {427, 615}, {468, 590}, {485, 3542}, {577, 3155}, {605, 1395}, {606, 2212}, {1151, 3515}, {1152, 1593}, {1165, 3087}, {1587, 3089}, {1588, 7487}, {1597, 6398}, {1598, 3093}, {1829, 7968}, {1968, 6291}, {2207, 6423}, {3068, 6353}, {3071, 3575}, {3147, 5418}, {3199, 5062}, {3311, 3517}, {3516, 6410}, {3518, 6419}, {3535, 5590}, {3541, 5420}, {3594, 5198}, {4232, 7585}, {5094, 8252}, {5419, 6420}, {6756, 7584}, {6995, 7586}, {8576, 8882}

X(5413) = isogonal conjugate of X(11090)
X(5413) = cevapoint of X(i) and X(j) for these {i,j}: {25, 6423}, {372, 8854}
X(5413) = X(393)-Ceva conjugate of X(5412)
X(5413) = X(571)-cross conjugate of X(5412)
X(5413) = polar conjugate of X(34391)
X(5413) = {X(4),X(372)}-harmonic conjugate of X(11472)
X(5413) = {X(6), X(25)}-harmonic conjugate of X(5412)
X(5413) = X(6204)-of-orthic-triangle if ABC is acute
X(5413) = trilinear product X(i)*X(j) for these {i,j}: {19, 371}, {31, 1585}, {158, 8911}, {492, 1973}, {1096, 5408}, {1748, 8576}, {3378, 5412}

X(5414) = 3rd KENMOTU HOMOTHETIC CENTER

Trilinears 1 - sin A + cos A : 1 - sin B + cos B : 1 - sin C + cos C

The A′B′C′ defined at X(5413) is homothetic to the intangents triangle, and the center of homothety is X(5414). Also, X(5414) = {X(6), X(55)}-harmonic conjugate of X(2066) and X(5414) = {X(3), X(1335)}-harmonic conjugate of X(2067). (Randy Hutson, February 9, 2013)

X(5414) lies on the conic {A, B, C, X(1), X(3)} and these lines: {1, 372}, {3, 1335}, {6, 31}, {11, 615}, {12, 3070}, {29, 7090}, {33, 5413}, {35, 371}, {36, 6396}, {37, 8577}, {56, 1152}, {63, 8394}, {140, 9661}, {165, 8833}, {172, 6283}, {259, 7014}, {283, 1805}, {388, 6460}, {390, 7586}, {405, 1377}, {485, 498}, {486, 1479}, {497, 3069}, {499, 5420}, {550, 9647}, {590, 5432}, {601, 3077}, {999, 6398}, {1124, 3295}, {1151, 5217}, {1378, 5687}, {1478, 6560}, {1500, 5062}, {1505, 2241}, {1587, 3085}, {1588, 4294}, {2646, 7969}, {2961, 8942}, {3057, 7968}, {3068, 5218}, {3071, 6284}, {3084, 5408}, {3297, 3303}, {3299, 3746}, {3304, 6426}, {3365, 7006}, {3390, 7005}, {3583, 6565}, {4302, 6561}, {5010, 6200}, {5204, 6410}, {5268, 8854}, {5281, 7585}, {5411, 7071}, {5563, 6454}, {6395, 6767}, {6564, 7951}, {7583, 9646}

X(5414) = isogonal conjugate of X(13390)
X(5414) = X(9)-Ceva conjugate of X(2066)
X(5414) = cevapoint of X(48) and X(606)

X(5415) = 4th KENMOTU HOMOTHETIC CENTER

Trilinears a(2R + r + s - a) : b(2R + r + s - b) : c(2R + r + s - c) (César Lozada, April 7, 2013; Hyacinthos #21900)
Trilinears (sin A)(2R sin A - 2R - r - s) : (sin B)(2R sin B - 2R - r - s) : (sin C)(2R sin C - 2R - r - s) (César Lozada, April 7, 2013)

The A′B′C′ defined at X(5412) is homothetic to the extangents triangle, and the center of homothety is X(5415). Also, X(5415) = {X(6), X(55)}-harmonic conjugate of X(5416). (Randy Hutson, February 9, 2013)

X(5415) lies on these lines: {6, 31}, {19, 5412}, {40, 371}, {65, 2067}, {590, 3925}, {1151, 5584}, {1702, 6769}, {2550, 3068}, {3070, 6253}, {6200, 7688}, {7133, 8557}

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 these lines: {6, 31}, {19, 5413}, {40, 372}, {65, 6414}, {615, 3925}, {1152, 5584}, {1703, 6769}, {2550, 3069}, {3071, 6253}, {3553, 7133}, {6396, 7688}

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 these lines: {371, 5446}, {372, 1599}, {491, 639}, {5412, 6419}, {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) lies on these lines: {2, 371}, {3, 485}, {4, 6200}, {5, 1151}, {6, 140}, {10, 9557}, {15, 2046}, {16, 2045}, {20, 6564}, {25, 9683}, {30, 6409}, {33, 9631}, {54, 9676}, {55, 9661}, {56, 9646}, {69, 641}, {110, 9677}, {115, 9674}, {230, 6422}, {316, 2460}, {372, 631}, {376, 1327}, {381, 6449}, {382, 6455}, {468, 3092}, {491, 1078}, {492, 7769}, {498, 2067}, {499, 2066}, {547, 6429}, {549, 1152}, {550, 6411}, {597, 9975}, {615, 3311}, {620, 8980}, {632, 3592}, {1062, 9632}, {1124, 5433}, {1328, 5055}, {1329, 9678}, {1335, 5432}, {1377, 3035}, {1378, 4999}, {1504, 7749}, {1506, 9675}, {1579, 6676}, {1587, 3523}, {1590, 8968}, {1591, 5407}, {1656, 3071}, {1657, 6451}, {1698, 9583}, {1699, 9582}, {1702, 3624}, {2041, 3389}, {2042, 3364}, {2043, 3391}, {2044, 3366}, {2051, 9556}, {2886, 9679}, {3055, 8375}, {3069, 3525}, {3090, 6453}, {3091, 9541}, {3102, 7786}, {3147, 5413}, {3312, 5054}, {3524, 6460}, {3530, 6410}, {3533, 7582}, {3534, 6496}, {3541, 5412}, {3545, 6484}, {3628, 6425}, {3815, 6424}, {3832, 6486}, {3850, 6433}, {3851, 6445}, {3857, 6488}, {5056, 6480}, {5079, 6519}, {5254, 9600}, {5587, 9615}, {5972, 8994}, {6036, 8997}, {6289, 8414}, {6420, 7585}, {6684, 8983}, {6699, 8998}, {6813, 9993}, {7386, 8280}, {7389, 7790}, {7486, 9542}, {7494, 8854}, {8227, 9616}, {8964, 8967}

X(5418) = isogonal conjugate of X(5417)
X(5418) = {X(6),X(140}-harmonic conjugate of X(5420)

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 these lines: {371, 1600}, {372, 5446}, {492, 640}, {5413, 6420}, {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 these lines: {2, 372}, {3, 486}, {4, 6396}, {5, 1152}, {6, 140}, {15, 2045}, {16, 2046}, {20, 6565}, {30, 6410}, {69, 642}, {230, 6421}, {316, 2459}, {371, 631}, {376, 1328}, {381, 6450}, {382, 6456}, {468, 3093}, {491, 7769}, {492, 1078}, {498, 6502}, {499, 5414}, {547, 6430}, {549, 1151}, {550, 6412}, {590, 3312}, {597, 9974}, {632, 3594}, {1124, 5432}, {1327, 5055}, {1335, 5433}, {1377, 4999}, {1378, 3035}, {1505, 7749}, {1578, 6676}, {1588, 3523}, {1592, 5406}, {1656, 3070}, {1657, 6452}, {1703, 3624}, {2041, 3365}, {2042, 3390}, {2043, 3367}, {2044, 3392}, {3055, 8376}, {3068, 3525}, {3090, 6454}, {3103, 7786}, {3147, 5412}, {3311, 5054}, {3524, 6459}, {3530, 6409}, {3533, 7581}, {3534, 6497}, {3541, 5413}, {3545, 6485}, {3628, 6426}, {3815, 6423}, {3832, 6487}, {3850, 6434}, {3851, 6446}, {3857, 6489}, {5056, 6481}, {5079, 6522}, {6290, 8406}, {6395, 8976}, {6419, 7586}, {6478, 9692}, {6811, 9993}, {7386, 8281}, {7388, 7790}, {7494, 8855}

X(5420) = isogonal conjugate of X(5419)

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) lies on these lines: {2, 1232}, {3, 6}, {51, 160}, {53, 1907}, {232, 428}, {233, 9698}, {615, 8962}, {1180, 6997}, {1194, 3815}, {1634, 9822}, {1879, 5254}, {2165, 5286}, {3055, 3291}, {5417, 5419}, {5422, 9723}

X(5421) = complement of X(1232)
X(5421) = the center of the bicevian conic of X(61) and X(62)
X(5421) = {X(6),X(39)}-harmonic conjugate of X(570)

X(5422) = INTERSECTION OF LINES X(2)X(6) AND X(22)X(51)

Barycentrics a² + 2R² : b² + 2R² : c² + 2R²
Barycentrics 1 + 2 sin²A : :

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}, {5, 7592}, {22, 51}, {23, 3796}, {24, 569}, {25, 5012}, {52, 7509}, {54, 6642}, {83, 5392}, {110, 5020}, {154, 3066}, {155, 1199}, {184, 575}, {195, 5070}, {324, 458}, {371, 1600}, {372, 1599}, {373, 9306}, {378, 9730}, {389, 7503}, {401, 7787}, {493, 589}, {494, 588}, {511, 7485}, {567, 6644}, {568, 7514}, {576, 3917}, {611, 7191}, {613, 3920}, {1181, 3091}, {1194, 5034}, {1249, 6819}, {1351, 2979}, {1498, 3832}, {1503, 7394}, {1505, 8962}, {1583, 3312}, {1584, 3311}, {1591, 7584}, {1592, 7583}, {1614, 7529}, {1853, 5169}, {1899, 5133}, {2003, 3306}, {2052, 8745}, {2323, 3305}, {3083, 3301}, {3084, 3299}, {3148, 3398}, {3167, 5643}, {3193, 5084}, {3410, 7605}, {3592, 5407}, {3594, 5406}, {3819, 5097}, {3981, 5038}, {4193, 5707}, {5046, 5706}, {5085, 6636}, {5093, 7998}, {5395, 8796}, {5408, 6420}, {5409, 6419}, {5421, 9723}, {5449, 7569}, {5480, 7391}, {5651, 6688}, {5889, 7395}, {5890, 9818}, {6146, 7544}, {6243, 7516}, {6505, 8257}, {6776, 6997}, {6805, 7582}, {6806, 7581}, {7387, 9781}, {7506, 9707}

X(5423) = ISOTOMIC CONJUGATE OF X(479)

Trilinears csc^2(A/2) cot^2(A/2) : :
Barycentrics (b + c - a)³ : :

Let A′ be the point in which the A-excircle is tangent to the circle O_A that passes through vertices B and C, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5423); for the incircle version, see X(479). (Peter Moses, December 10, 2015)

A′ = -4 a^2 (a+b-c) (a-b+c) : (a-b+c)^3 (a+b+c) : (a+b-c)^3 (a+b+c) (barycentrics)

(center of circle O_A) = -2 a^2 (5 a^3+3 a^2 b-a b^2+b^3+3 a^2 c+2 a b c-b^2 c-a c^2-b c^2+c^3)
: a^5+3 a^4 b+12 a^3 b^2+4 a^2 b^3-5 a b^4+b^5+3 a^4 c+2 a^3 b c+2 a^2 b^2 c+2 a b^3 c-b^4 c+2 a^3 c^2-4 a^2 b c^2+8 a b^2 c^2-2 b^3 c^2-2 a^2 c^3-2 a b c^3+2 b^2 c^3-3 a c^4+b c^4-c^5
: a^5+3 a^4 b+2 a^3 b^2-2 a^2 b^3-3 a b^4-b^5+3 a^4 c+2 a^3 b c-4 a^2 b^2 c-2 a b^3 c+b^4 c+12 a^3 c^2+2 a^2 b c^2+8 a b^2 c^2+2 b^3 c^2+4 a^2 c^3+2 a b c^3-2 b^2 c^3-5 a c^4-b c^4+c^5

(power of A wrt O_A) = {((a+b+c) (a^2+2 a b+b^2+2 a c-2 b c+c^2))/(8 a)

(radius of O_A) = (3 a^3+5 a^2 b+a b^2-b^3+5 a^2 c-2 a b c+b^2 c+a c^2+b c^2-c^3)/(16 S)

The radical center of the circles O_A, O_B, O_C is X(2297). (Peter Moses, December 10, 2015)

If you have The Geometer's Sketchpad, you can view X(5423).

X(5423) lies on these lines: {2, 3677}, {7, 3263}, {8, 210}, {9, 7172}, {11, 6557}, {55, 1261}, {190, 9778}, {200, 346}, {280, 2057}, {329, 2835}, {345, 3699}, {612, 5749}, {644, 7074}, {756, 5296}, {1219, 8583}, {1260, 4578}, {1357, 8051}, {1863, 7046}, {2098, 8834}, {2325, 3158}, {2550, 3967}, {3006, 5748}, {3021, 4387}, {3452, 4901}, {3474, 4488}, {3596, 4441}, {3705, 5328}, {3710, 7080}, {3711, 6057}, {4076, 6632}, {4308, 9369}, {4847, 10005}, {5205, 5435}, {5273, 7081}, {6552, 6736}

X(5423) = isogonal conjugate of X(7023)
X(5423) = isotomic conjugate of X(479)
X(5423) = anticomplement of X(5573)
X(5423) = X(i)-Ceva conjugate of X(j) for these (i,j): (341,346), (4076,6558)
X(5423) = cevapoint of X(i) and X(j) for these {i,j}: {3022,4130}, {3900,4953}, {4081,4163}
X(5423) = crosspoint of X(i) and X(j) for these {i,j}: {346,6556}, {4076,6558}
X(5423) = crosssum of X(1106) and X(7366)
X(5423) = trilinear pole of the line X(4130)X(4163)
X(5423) = trilinear square of X(6731)
X(5423) = barycentric cube of X(8)
X(5423) = polar conjugate of isotomic conjugate of X(30681)
X(5423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,8055,497), (200,4082,346), (210,3974,8), (341,1265,8), (346,6555,200), (497,4009,8055)
X(5423) = X(i)-cross conjugate of X(j) for these (i,j): (728,346), (3022,4130}, (4012,8), (4081,4163), (5574,2)
X(5423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,8055,497), (200,4082,346), (210,3974,8), (341,1265,8), (346,6555,200), (497,4009,8055)
X(5423) = X(i)-isoconjugate of X(j) for these {i,j}: {1,7023}, {2,7366}, {6,738}, {7,1106}, {31,479}, {34,7053}, {56,269}, {57,1407}, {77,1398}, {222,1435}, {223,6612}, {244,7339}, {278,7099}, {279,604}, {513,6614}, {593,7147}, {603,1119}, {608,7177}, {649,4617}, {667,4626}, {757,7143}, {849,6046}, {1014,1042}, {1088,1397}, {1254,7341}, {1357,7045}, {1395,7056}, {1408,3668}, {1412,1427}, {1414,7250}, {1422,6611}, {1461,3669}, {1472,7197}, {4565,7216}, {4637,7180}

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 L_A be the Euler line of IBC, and define L_B and L_C cyclically. (The four Euler lines concur in the Schiffler point, X(21).) Let O_A be the circumcenter of IBC, and define O_B and O_C cyclically.

Continuing, let A_B, A_C be the orthogonal projections of O_A on L_B and L_C, respecitively, and define B_C, B_A and C_A, C_B cyclically. Let A′ be the circumcenter of O_AA_BA_C, 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: {1, 5427}, {4, 5441}, {9, 5426}, {21, 4867}, {30, 5561}, {79, 2646}, {80, 3584}, {758, 2320}, {1385, 5557}, {1389, 3746}, {3612, 5665}, {5560, 7951}

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}, {1210, 5443}, {1389, 5557}, {1411, 2003}, {1770, 5441}, {1835, 6198}, {2802, 3957}, {3305, 5692}, {3485, 6873}, {3584, 5719}, {3585, 3649}, {3624, 5730}, {3636, 5330}, {3830, 5561}, {3868, 5258}, {3869, 5259}, {3874, 5288}, {3881, 4861}, {3911, 5444}, {3918, 4420}, {4067, 5260}, {4511, 5883}, {4640, 5426}, {5226, 7951}, {5270, 6147}, {5357, 7052}, {5542, 7972}

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}, {9, 5424}, {30, 1699}, {35, 3753}, {36, 3742}, {40, 5428}, {55, 5541}, {57, 5427}, {80, 6690}, {100, 3968}, {210, 5251}, {214, 5284}, {355, 10021}, {405, 5506}, {442, 3586}, {484, 3919}, {1006, 5538}, {1125, 2475}, {1420, 3649}, {1698, 1837}, {1768, 6914}, {2320, 3065}, {2646, 5259}, {3158, 3679}, {3219, 4525}, {3336, 4189}, {3337, 5267}, {3616, 4299}, {3636, 3648}, {3651, 7987}, {3683, 4867}, {3746, 3880}, {3956, 5260}, {4316, 5249}, {4539, 5302}, {4640, 5425}, {4677, 4933}, {5131, 5883}, {5535, 7508}, {5691, 6841}, {5903, 8261}, {6326, 7489}

X(5426) = {X(1),X(21)}-harmonic conjugate of X(191)

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: {1, 5424}, {7, 21}, {11, 30}, {12, 5251}, {57, 5426}, {79, 5886}, {100, 5172}, {191, 1420}, {392, 3647}, {442, 5433}, {758, 1319}, {993, 5434}, {1317, 2078}, {1411, 1758}, {1749, 6265}, {2475, 7288}, {2771, 5126}, {3651, 5204}, {4189, 5221}, {5441, 5722}, {5902, 7508}, {6841, 7354}

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)

As a point on the Euler line, X(5428) has Shinagawa coefficients (4$aS_A$ + abc, -4$aS_A$ + abc).

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: {1, 5424}, {2, 3}, {36, 3649}, {40, 5426}, {58, 5453}, {79, 5444}, {191, 3576}, {214, 960}, {517, 8261}, {758, 1385}, {952, 5258}, {970, 5946}, {1837, 5010}, {3579, 3754}, {3650, 5303}, {3652, 7987}, {5690, 8715}, {7171, 7701}

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)
X(5429) = X(1) + 2 X(58)

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}, {8, 8258}, {36, 199}, {171, 3753}, {210, 5247}, {511, 3576}, {740, 4234}, {976, 4661}, {978, 1453}, {986, 4252}, {995, 7032}, {999, 7083}, {1104, 3742}, {1125, 1330}, {1193, 4881}, {1247, 2363}, {1698, 6693}, {1757, 4134}, {1961, 5251}, {2308, 4511}, {2792, 5603}, {2938, 4221}, {3430, 7987}, {3454, 3624}, {3616, 6536}, {3880, 5255}, {5691, 7683}

X(5429) = {X(1),X(58)}-harmonic conjugate of X(1046)

X(5430) = CENTER OF THE 1st GRIGORIEV CONIC

Barycentrics (1 + csc A/2)(csc B/2 + csc C/2) - cot²A/2

Let L_A 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 L_B and L_C cyclically. Let A′ = L_B∩L_C, and define B′ and C′ cyclically. Let A_B be the reflection of A in line A′B′, and let A_C be the reflection of A in line A′C′. Let C₁ be the touchpoint of the incircle of triangle A_BAB and line AB, and let B₂ be the touchpoint of the incircle of A_CAC 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}, {178, 6557}, {236, 3161}, {346, 7027}

X(5431) = CENTER OF THE 2nd GRIGORIEV CONIC

Barycentrics (1 + sec A/2)(sec B/2 + sec C/2) - tan²A/2

Let L_A 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 L_B and L_C cyclically. Let A′ = L_B∩L_C, and define B′ and C′ cyclically. Let A_B be the reflection of A in line A′B′, and let A_C be the reflection of A in line A′C′. Let C₁ be the touchpoint of the incircle of triangle A_BAB and line AB, and let B₂ be the touchpoint of the incircle of A_CAC 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}, {65, 6684}, {119, 6914}, {141, 2330}, {165, 1836}, {171, 2361}, {210, 5745}, {212, 750}, {215, 5012}, {226, 1155}, {230, 2276}, {312, 3712}, {333, 4023}, {345, 6057}, {354, 3911}, {355, 3612}, {371, 9648}, {372, 9646}, {381, 4302}, {388, 3523}, {396, 7127}, {496, 632}, {499, 3295}, {550, 3585}, {551, 5048}, {569, 9666}, {590, 5414}, {597, 8540}, {601, 7299}, {612, 7499}, {615, 2066}, {620, 3023}, {756, 7004}, {846, 2607}, {908, 4640}, {950, 3634}, {958, 5552}, {999, 5054}, {1006, 5172}, {1040, 5268}, {1058, 3533}, {1062, 7542}, {1092, 9653}, {1124, 5420}, {1125, 3057}, {1213, 2268}, {1317, 6713}, {1335, 5418}, {1361, 6711}, {1362, 6712}, {1364, 6718}, {1399, 3074}, {1479, 1656}, {1500, 7749}, {1697, 3624}, {1698, 1837}, {1788, 5703}, {1852, 5142}, {1858, 5044}, {1914, 3815}, {1995, 9673}, {2077, 6907}, {2098, 3616}, {2099, 5657}, {2320, 3036}, {2476, 6668}, {2829, 6950}, {3011, 3752}, {3022, 6710}, {3024, 5972}, {3027, 6036}, {3028, 6699}, {3053, 9596}, {3056, 3589}, {3086, 3303}, {3090, 4294}, {3158, 4863}, {3301, 8981}, {3304, 7288}, {3305, 7082}, {3318, 6717}, {3321, 7056}, {3336, 6147}, {3337, 5442}, {3340, 9588}, {3428, 6954}, {3452, 3683}, {3474, 5226}, {3475, 4860}, {3486, 9780}, {3487, 5221}, {3522, 5229}, {3524, 4293}, {3528, 9656}, {3530, 7280}, {3576, 5252}, {3580, 9637}, {3627, 4324}, {3628, 7741}, {3689, 4847}, {3699, 4126}, {3703, 7081}, {3705, 4030}, {3715, 5273}, {3761, 6390}, {3813, 3871}, {3820, 5251}, {3967, 3977}, {4187, 5248}, {4255, 5230}, {4305, 5818}, {4309, 5070}, {4316, 8703}, {4414, 4415}, {4512, 4679}, {4855, 5794}, {4870, 5183}, {4998, 6066}, {5055, 9668}, {5056, 5225}, {5067, 9670}, {5119, 5886}, {5206, 9650}, {5260, 9711}, {5261, 9657}, {5297, 7495}, {5332, 9300}, {5441, 10021}, {5584, 6988}, {5697, 5901}, {5698, 5748}, {5719, 5902}, {5840, 6980}, {5842, 6830}, {6018, 6715}, {6019, 6719}, {6020, 6720}, {6067, 6600}, {6198, 10018}, {6238, 9820}, {6253, 6796}, {6286, 8254}, {6449, 9649}, {6459, 9662}, {6565, 9660}, {6677, 9817}, {6696, 7355}, {6716, 7158}, {6883, 8069}, {6905, 7680}, {6918, 7958}, {6926, 8273}, {6949, 7681}, {7486, 9671}, {7509, 9672}, {7987, 9578}, {7988, 9580}, {8144, 10020}, {9306, 9667}

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}, {30, 7280}, {34, 468}, {35, 496}, {46, 5886}, {55, 631}, {57, 191}, {65, 392}, {79, 10021}, {85, 7181}, {100, 3813}, {104, 6949}, {141, 1428}, {172, 3815}, {201, 244}, {210, 6700}, {230, 2275}, {238, 1399}, {261, 7342}, {348, 1358}, {371, 9663}, {372, 9661}, {381, 4299}, {395, 2307}, {404, 2886}, {405, 1470}, {442, 5427}, {474, 3925}, {484, 5442}, {495, 632}, {497, 3523}, {498, 999}, {550, 3583}, {551, 4848}, {569, 9653}, {590, 6502}, {602, 5348}, {603, 748}, {604, 1213}, {614, 7499}, {615, 2067}, {620, 3027}, {946, 1155}, {993, 4187}, {1001, 6910}, {1015, 7749}, {1038, 5272}, {1056, 3533}, {1060, 7542}, {1092, 9666}, {1124, 5418}, {1210, 2646}, {1214, 7561}, {1335, 5420}, {1357, 6715}, {1359, 6717}, {1361, 6718}, {1362, 6710}, {1364, 6711}, {1376, 6921}, {1385, 1737}, {1387, 5697}, {1420, 1698}, {1447, 3665}, {1454, 3306}, {1466, 4423}, {1469, 3589}, {1478, 1656}, {1532, 5450}, {1770, 5122}, {1788, 2099}, {1836, 8227}, {1837, 3576}, {1852, 7501}, {1858, 9940}, {1870, 10018}, {1995, 9658}, {2093, 9624}, {2098, 5657}, {2361, 3075}, {2477, 5012}, {2594, 3216}, {2829, 6941}, {3022, 6712}, {3023, 6036}, {3024, 6699}, {3028, 5972}, {3053, 9599}, {3057, 6684}, {3085, 3304}, {3090, 4293}, {3295, 4995}, {3299, 8981}, {3303, 5218}, {3320, 6720}, {3324, 6716}, {3325, 6719}, {3336, 5443}, {3337, 6147}, {3361, 5219}, {3428, 6891}, {3476, 9780}, {3485, 5221}, {3486, 5704}, {3487, 4860}, {3522, 5225}, {3524, 4294}, {3528, 9671}, {3530, 5010}, {3612, 5722}, {3627, 4316}, {3628, 7951}, {3660, 5044}, {3671, 4870}, {3678, 5083}, {3760, 6390}, {3820, 5193}, {3825, 5267}, {3826, 7677}, {3847, 5046}, {4193, 6667}, {4301, 5183}, {4317, 5070}, {4324, 8703}, {4652, 7702}, {4861, 8256}, {4881, 5086}, {5055, 9655}, {5056, 5229}, {5067, 9657}, {5126, 9956}, {5206, 9665}, {5231, 5438}, {5274, 9670}, {5536, 5763}, {5584, 6926}, {5687, 6174}, {5841, 6971}, {5842, 6942}, {5901, 5903}, {6253, 6905}, {6285, 6696}, {6358, 6533}, {6449, 9662}, {6459, 9649}, {6565, 9647}, {6666, 8581}, {6824, 7958}, {6847, 7965}, {6883, 8071}, {6906, 7681}, {6911, 7742}, {6952, 7680}, {6988, 8273}, {7179, 7198}, {7292, 7495}, {7296, 9300}, {7352, 9820}, {7356, 8254}, {7486, 9656}, {7509, 9659}, {7753, 9341}, {7962, 9588}, {7987, 9581}, {7988, 9579}, {9306, 9652}

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}, {35, 8703}, {36, 495}, {46, 3654}, {55, 376}, {57, 3679}, {65, 519}, {85, 7198}, {104, 7680}, {172, 5306}, {226, 535}, {229, 7478}, {330, 7837}, {354, 515}, {355, 3338}, {390, 8162}, {396, 7051}, {442, 8666}, {496, 3585}, {497, 3543}, {498, 5054}, {499, 3614}, {516, 5919}, {524, 1469}, {527, 8581}, {537, 4032}, {541, 3024}, {542, 3023}, {543, 3027}, {544, 1362}, {547, 7951}, {550, 3746}, {597, 1428}, {752, 1463}, {944, 6253}, {952, 5902}, {956, 3925}, {982, 5724}, {993, 5427}, {1015, 7753}, {1388, 3485}, {1398, 5064}, {1406, 5710}, {1411, 7194}, {1420, 5290}, {1470, 6174}, {1479, 3830}, {1565, 7272}, {1652, 7043}, {1653, 7026}, {1657, 4309}, {1770, 9957}, {1837, 3333}, {1870, 7576}, {2098, 4295}, {2242, 5309}, {2275, 9300}, {2475, 3813}, {2646, 4311}, {2829, 5603}, {2886, 6175}, {3057, 4292}, {3085, 3524}, {3086, 3545}, {3091, 9656}, {3146, 9670}, {3295, 3534}, {3324, 6020}, {3336, 5690}, {3339, 4677}, {3340, 4355}, {3361, 9578}, {3421, 4413}, {3475, 5731}, {3627, 4857}, {3633, 5586}, {3665, 7176}, {3748, 4304}, {3816, 5080}, {3828, 3911}, {3839, 5229}, {3849, 5194}, {3872, 5880}, {3913, 4190}, {4302, 6767}, {4312, 7962}, {4320, 7667}, {4321, 6173}, {4669, 4848}, {4846, 6580}, {5066, 7741}, {5258, 8728}, {5289, 5905}, {5691, 9845}, {5842, 7967}, {7179, 7181}

X(5434) = reflection of X(3058) in X(1)

X(5435) = 2(r + 2R)X(1) - 3rX(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}, {4, 5704}, {8, 56}, {10, 3361}, {11, 3474}, {20, 1210}, {21, 1466}, {31, 9364}, {36, 5731}, {40, 9785}, {43, 1458}, {46, 962}, {55, 7677}, {65, 3616}, {77, 2999}, {78, 1467}, {88, 278}, {100, 1617}, {109, 9083}, {140, 3487}, {145, 1420}, {165, 390}, {171, 1471}, {174, 7002}, {175, 5405}, {176, 5393}, {190, 1997}, {208, 4200}, {223, 1443}, {238, 9316}, {241, 2275}, {269, 8051}, {273, 7490}, {279, 3008}, {333, 1014}, {354, 5218}, {376, 5122}, {388, 9780}, {452, 4652}, {479, 658}, {496, 6361}, {497, 1155}, {498, 3337}, {499, 3336}, {516, 5274}, {517, 4345}, {604, 3684}, {614, 4318}, {631, 942}, {651, 1407}, {673, 2898}, {910, 5838}, {912, 6970}, {940, 5933}, {950, 3522}, {978, 1042}, {999, 5657}, {1000, 3654}, {1038, 5262}, {1058, 3579}, {1071, 6927}, {1106, 5247}, {1125, 3339}, {1214, 4850}, {1319, 3241}, {1403, 8299}, {1442, 5256}, {1532, 2096}, {1656, 5714}, {1698, 4298}, {1707, 5121}, {1722, 4320}, {1728, 6953}, {1737, 4293}, {1750, 8544}, {1836, 9779}, {1876, 6353}, {1892, 8889}, {1999, 4460}, {2099, 5298}, {2263, 5272}, {2295, 5228}, {3052, 3756}, {3085, 3338}, {3091, 4292}, {3144, 7103}, {3146, 9581}, {3149, 9799}, {3210, 4552}, {3216, 4306}, {3333, 6684}, {3340, 3622}, {3434, 9352}, {3475, 4860}, {3485, 5221}, {3486, 5204}, {3526, 6147}, {3624, 3671}, {3634, 5290}, {3660, 3873}, {3673, 7397}, {3679, 4315}, {3740, 8581}, {3816, 5698}, {3817, 4312}, {3828, 5726}, {3832, 9579}, {3868, 6921}, {3913, 9797}, {3916, 5084}, {3947, 4355}, {4032, 4699}, {4190, 5175}, {4208, 5705}, {4220, 5807}, {4302, 5131}, {4305, 7280}, {4307, 9746}, {4321, 5686}, {4327, 5268}, {4384, 7176}, {4423, 8543}, {4430, 5083}, {4488, 6557}, {4662, 9850}, {4899, 6555}, {4911, 7402}, {4998, 6632}, {5022, 6554}, {5054, 5719}, {5056, 9612}, {5126, 7967}, {5129, 9843}, {5205, 5423}, {5287, 7269}, {5439, 6857}, {5444, 5902}, {5556, 7173}, {5658, 5729}, {5687, 6764}, {5691, 7319}, {5709, 6926}, {5734, 5903}, {5758, 6891}, {5768, 6905}, {5770, 6911}, {5804, 6906}, {5809, 7580}, {5811, 6944}, {5825, 5927}, {6223, 6848}, {6350, 7011}, {6654, 9358}, {6734, 6904}, {6738, 7987}, {6964, 7330}, {6988, 9940}, {7191, 8270}, {7678, 7965}

X(5435) = {X(8),X(56)}-harmonic conjugate of X(4308)

X(5436) = (r + 2R)X(1) + 6RX(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}, {3, 5437}, {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}, {515, 6846}, {551, 3487}, {631, 9843}, {936, 2900}, {938, 5745}, {942, 3928}, {943, 3680}, {946, 6987}, {968, 3924}, {976, 7322}, {988, 5573}, {993, 3333}, {997, 3646}, {1005, 5253}, {1012, 8726}, {1043, 4384}, {1210, 6857}, {1260, 3303}, {1385, 1490}, {1451, 2328}, {1621, 1697}, {1698, 3419}, {1708, 3340}, {2136, 3295}, {2478, 5219}, {2550, 4314}, {2646, 4423}, {2647, 7273}, {2654, 7070}, {2975, 4666}, {3306, 4189}, {3339, 4640}, {3361, 3742}, {3452, 5129}, {3523, 6692}, {3671, 5698}, {3811, 4015}, {3822, 6990}, {3825, 6829}, {3868, 3929}, {3870, 5260}, {4292, 6173}, {4301, 5759}, {4308, 8232}, {4428, 5836}, {4659, 7283}, {4678, 4917}, {5177, 5550}, {5218, 8582}, {5249, 6872}, {5257, 5802}, {5443, 9612}, {5587, 6832}, {5691, 8226}, {5705, 5722}, {5715, 5886}, {5812, 5901}, {6765, 9708}, {6920, 9845}, {6936, 9624}, {6988, 7682}, {7330, 7489}, {7580, 7987}, {7675, 9859}

X(5437) = (r + 2R)X(1) + 6RX(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}, {3, 5436}, {4, 9841}, {5, 84}, {10, 1056}, {37, 9574}, {38, 7322}, {40, 631}, {46, 3624}, {65, 8583}, {85, 738}, {88, 4606}, {100, 4666}, {140, 5709}, {165, 1001}, {171, 5272}, {173, 7028}, {200, 354}, {210, 4860}, {236, 258}, {244, 612}, {281, 1435}, {312, 4659}, {377, 9581}, {381, 7171}, {388, 7091}, {392, 2093}, {404, 3601}, {443, 1210}, {518, 8580}, {549, 3587}, {614, 750}, {936, 942}, {940, 1449}, {946, 6926}, {950, 6904}, {958, 3361}, {960, 3339}, {982, 5268}, {997, 5883}, {999, 9623}, {1054, 8299}, {1155, 4423}, {1329, 5290}, {1375, 5834}, {1420, 5253}, {1490, 6918}, {1519, 6833}, {1656, 3824}, {1697, 3616}, {1698, 3338}, {1699, 3816}, {1709, 3838}, {1768, 6667}, {1995, 7293}, {2098, 3922}, {2478, 9579}, {2551, 4298}, {3149, 8726}, {3182, 7532}, {3189, 6744}, {3208, 5308}, {3220, 5020}, {3247, 3666}, {3304, 3698}, {3359, 5886}, {3475, 6745}, {3487, 6700}, {3576, 3833}, {3589, 7289}, {3600, 5795}, {3740, 5223}, {3754, 7982}, {3763, 5227}, {3772, 4859}, {3784, 5943}, {3825, 6845}, {3873, 9342}, {3920, 9335}, {3925, 5231}, {3980, 4871}, {4035, 4869}, {4187, 9612}, {4208, 5704}, {4292, 5084}, {4415, 4862}, {4454, 8055}, {4640, 8167}, {4652, 5047}, {4850, 5287}, {5044, 5708}, {5045, 6765}, {5119, 5444}, {5123, 5726}, {5128, 5250}, {5284, 9352}, {5285, 7484}, {5535, 6681}, {5587, 6854}, {5705, 8728}, {5715, 6922}, {5737, 6706}, {6223, 9842}, {6245, 6864}, {6260, 6964}, {6668, 6763}, {6701, 7701}, {6705, 6846}, {6916, 7682}, {7521, 7713}

X(5438) = (r + 2R)X(1) - 6RX(2) + 2r*X(3)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b + c)²(b + c - a) - 4a(b² + c² - a²)
Barycentrics   a(3a³ + b³ + c³ - a²b - a²c - 3ab² - 3ac² + 2abc + 3b²c + 3bc²)
X(5438) = (r + 2R)*X(1) - 6R*X(2) + 2r*X(3)   (Peter Moses, April 3, 2013)

X(5438) lies on these lines: {1, 474}, {2, 950}, {3, 9}, {4, 6700}, {8, 1420}, {10, 631}, {20, 3452}, {21, 7308}, {40, 997}, {44, 8951}, {55, 8583}, {56, 200}, {57, 78}, {63, 4188}, {72, 3928}, {80, 1698}, {100, 1697}, {140, 5705}, {142, 5703}, {165, 960}, {210, 5204}, {214, 6264}, {226, 6904}, {377, 5219}, {386, 1449}, {388, 6745}, {452, 5316}, {480, 4321}, {515, 6926}, {518, 3361}, {549, 5791}, {728, 9310}, {908, 4190}, {938, 6692}, {958, 7987}, {975, 3247}, {976, 3677}, {978, 7290}, {988, 5293}, {999, 6765}, {1058, 1125}, {1193, 5269}, {1260, 1466}, {1319, 4853}, {1329, 5691}, {1377, 9583}, {1385, 9623}, {1453, 3216}, {1496, 3939}, {1702, 9679}, {1743, 4252}, {1788, 6737}, {1861, 7521}, {2093, 5730}, {2270, 3430}, {2551, 4297}, {2646, 4413}, {2886, 3624}, {3146, 5328}, {3149, 6282}, {3218, 3984}, {3243, 3333}, {3304, 3689}, {3305, 4189}, {3340, 4511}, {3359, 7971}, {3421, 4311}, {3476, 6736}, {3486, 8582}, {3487, 6173}, {3488, 9843}, {3523, 5745}, {3586, 4187}, {3617, 4881}, {3623, 4917}, {3632, 8256}, {3646, 5248}, {3679, 5445}, {3740, 5234}, {3814, 6845}, {3869, 5128}, {3870, 5253}, {3876, 3929}, {3927, 5122}, {4304, 5084}, {4386, 9575}, {4512, 5217}, {4847, 7288}, {5096, 5227}, {5231, 5433}, {5289, 7991}, {5552, 9578}, {5587, 6833}, {5709, 6924}, {5719, 5832}, {5731, 5795}, {5735, 5763}, {6734, 6921}, {6854, 8227}, {7091, 8828}

X(5439) = (r + 2R)X(1) + 3RX(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}, {4, 9776}, {5, 1071}, {7, 5084}, {8, 4002}, {10, 354}, {20, 5806}, {37, 3670}, {46, 1001}, {57, 405}, {63, 5708}, {65, 392}, {142, 442}, {145, 5049}, {210, 3634}, {226, 4187}, {355, 6854}, {377, 5722}, {388, 3660}, {406, 1876}, {443, 938}, {498, 5570}, {517, 631}, {518, 1698}, {519, 3698}, {551, 3057}, {614, 5711}, {750, 5266}, {908, 6147}, {912, 1656}, {956, 3333}, {958, 3338}, {960, 3624}, {971, 3091}, {986, 6051}, {1155, 5248}, {1214, 1393}, {1279, 5264}, {1385, 5253}, {1426, 5136}, {1439, 7532}, {1479, 5880}, {1621, 3579}, {1699, 9943}, {1788, 5173}, {1829, 7521}, {1871, 7543}, {1898, 7173}, {2476, 3824}, {3090, 5777}, {3218, 5047}, {3244, 3918}, {3295, 4666}, {3305, 3927}, {3336, 4640}, {3337, 5251}, {3488, 6904}, {3576, 7686}, {3617, 3889}, {3622, 9957}, {3625, 3968}, {3626, 3892}, {3635, 3893}, {3636, 3922}, {3720, 3931}, {3740, 4533}, {3748, 8715}, {3811, 4413}, {3828, 3983}, {3838, 7741}, {3869, 5550}, {3870, 9709}, {3872, 7373}, {3873, 3921}, {3884, 3919}, {3894, 4539}, {3897, 5126}, {3947, 8581}, {4189, 5122}, {4359, 5295}, {4420, 9342}, {4423, 5221}, {5302, 6763}, {5435, 6857}, {5603, 6926}, {5704, 6856}, {5714, 6919}, {5730, 8583}, {5768, 6864}, {5770, 6887}, {5787, 6835}, {5804, 6916}, {5805, 6836}, {5812, 6947}, {5885, 5887}, {5886, 6833}, {6001, 8227}, {6173, 9612}, {6245, 8226}, {6259, 6957}, {6734, 8728}, {6845, 9955}, {7580, 8726}, {8100, 8126}, {9581, 9844}, {9779, 9961}

X(5440) = (r + 2R)X(1) - 3RX(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}, {4, 5748}, {8, 631}, {10, 2646}, {20, 5658}, {21, 5044}, {30, 908}, {35, 960}, {36, 518}, {40, 5730}, {44, 2251}, {46, 4018}, {48, 3694}, {55, 392}, {56, 3555}, {80, 5123}, {100, 517}, {101, 2751}, {109, 2756}, {140, 6734}, {149, 7743}, {200, 956}, {210, 993}, {214, 519}, {238, 5529}, {306, 7536}, {318, 7531}, {329, 376}, {355, 5552}, {404, 942}, {405, 936}, {443, 5703}, {484, 4867}, {498, 5794}, {515, 6745}, {521, 656}, {551, 3748}, {572, 3965}, {758, 1155}, {914, 6699}, {944, 6926}, {950, 4187}, {952, 6735}, {958, 3612}, {971, 6909}, {995, 3744}, {999, 3870}, {1012, 5720}, {1018, 6603}, {1055, 3930}, {1104, 3216}, {1125, 3925}, {1149, 3722}, {1193, 5266}, {1375, 3912}, {1386, 5313}, {1420, 6765}, {1437, 1792}, {1455, 4551}, {1737, 3035}, {2057, 5534}, {2077, 2932}, {2551, 4305}, {2802, 5048}, {2975, 4420}, {3057, 8715}, {3086, 3189}, {3090, 5175}, {3218, 5122}, {3421, 5731}, {3434, 5886}, {3436, 6899}, {3452, 4304}, {3487, 6904}, {3524, 5744}, {3579, 3869}, {3583, 5087}, {3616, 5082}, {3617, 3897}, {3666, 4256}, {3678, 5267}, {3740, 5251}, {3868, 4188}, {3871, 9957}, {3876, 4189}, {3921, 9708}, {3935, 4881}, {3957, 5049}, {3991, 9310}, {4002, 9709}, {4257, 4641}, {4313, 5084}, {4421, 5119}, {4539, 5220}, {4640, 5010}, {4662, 5258}, {4694, 4864}, {4702, 4975}, {4880, 5131}, {5045, 5253}, {5086, 6852}, {5174, 7543}, {5176, 6224}, {5249, 5719}, {5728, 8257}, {5761, 6885}, {5777, 6906}, {5787, 6890}, {5791, 6910}, {5806, 6915}, {5812, 6934}, {5882, 6736}, {5904, 7280}, {6282, 7580}, {6684, 6737}, {6900, 9955}, {6940, 9940}

X(5440) = isogonal conjugate of X(36125)
X(5440) = isotomic conjugate of isogonal conjugate of X(23202)
X(5440) = isotomic conjugate of polar conjugate of X(44)
X(5440) = X(19)-isoconjugate of X(88)
X(5440) = crossdifference of every pair of points on the line X(19)X(4394)
X(5440) = inner-Garcia-to-ABC similarity image of X(11)

X(5441) = (2r + 3R)X(1) - 6rX(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}, {3, 5442}, {4, 5424}, {8, 3647}, {10, 21}, {11, 5499}, {20, 5902}, {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}, {1125, 6175}, {1385, 4857}, {1478, 4313}, {1479, 2475}, {1697, 7701}, {1770, 5425}, {1837, 5010}, {2646, 3583}, {2771, 3057}, {3486, 4302}, {3488, 4299}, {3534, 5221}, {3601, 6841}, {3633, 3650}, {3884, 6224}, {4004, 8261}, {4297, 5563}, {5427, 5722}, {5432, 10021}, {5691, 7680}, {5692, 6872}

X(5442) = (2r + 3R)X(1) - 6rX(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}, {3, 5441}, {5, 5131}, {10, 36}, {35, 3911}, {40, 6713}, {46, 3624}, {65, 5444}, {80, 7280}, {140, 3336}, {484, 5433}, {499, 6361}, {631, 5902}, {1145, 3632}, {1155, 9955}, {3035, 6763}, {3337, 5432}, {3579, 3582}, {3585, 5122}, {3616, 3884}, {4325, 9956}, {5054, 5221}, {5559, 5657}, {5563, 6684}, {5692, 6921}, {5697, 7288}, {7951, 8728}

X(5443) = (2r + R)X(1) + 6rX(2) - 2r*X(3)

Barycentrics a⁴ + b⁴ + c⁴ - a³b - a³c - 2a²b² - 2a²c² + a²bc + ab³ + ac³ - ab²c - abc² - 2b²c²
X(5443) = (2r + R)*X(1) + 6r*X(2) - 2r*X(3) (Peter Moses, April 3, 2013)

X(5443) is the QA-P7 center of quadrangle ABCX(1); see QA-Nine-point Center Homothetic Center.

X(5443) lies on these lines: {1, 5}, {2, 3754}, {3, 5444}, {10, 7504}, {17, 7052}, {21, 36}, {35, 946}, {46, 3624}, {56, 7489}, {140, 484}, {191, 4999}, {214, 2475}, {226, 5563}, {388, 6965}, {451, 1845}, {497, 6900}, {498, 5603}, {499, 3485}, {908, 5258}, {942, 3582}, {1210, 5425}, {1319, 5270}, {1385, 3585}, {1388, 9654}, {1389, 5559}, {1478, 3616}, {1479, 4313}, {1656, 2099}, {1699, 3612}, {1749, 3337}, {1836, 7280}, {2646, 3583}, {2800, 6952}, {3057, 3584}, {3085, 6979}, {3086, 6884}, {3245, 6684}, {3336, 5433}, {3467, 5557}, {3576, 7491}, {3746, 6915}, {3817, 7548}, {4295, 5550}, {4305, 9779}, {4317, 5714}, {4867, 6734}, {5436, 9612}

X(5443) = {X(1),X(5)}-harmonic conjugate of X(80)

X(5444) = (2r + R)X(1) + 6rX(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}, {3, 5443}, {35, 404}, {36, 226}, {65, 5442}, {79, 5428}, {90, 3646}, {442, 3586}, {484, 549}, {498, 3476}, {499, 3488}, {631, 5903}, {952, 5326}, {1319, 3584}, {1387, 4995}, {1479, 5550}, {2099, 5054}, {3487, 5557}, {3576, 6882}, {3616, 3754}, {3653, 5252}, {3822, 4881}, {3911, 5425}, {4324, 9955}, {4870, 5122}, {5010, 5886}, {5119, 5437}, {5298, 5719}, {5435, 5902}

X(5445) = (2r + R)X(1) - 6rX(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}, {40, 6882}, {201, 1772}, {355, 7280}, {495, 3337}, {498, 1788}, {499, 5657}, {942, 3584}, {946, 3245}, {1125, 5330}, {1155, 3585}, {1210, 3746}, {1478, 9780}, {1727, 6907}, {1749, 5499}, {1837, 5010}, {1838, 8756}, {2099, 3526}, {2800, 6949}, {3057, 3582}, {3579, 3583}, {3679, 5438}, {3813, 5541}, {3828, 4292}, {3911, 5563}, {3916, 5123}, {4299, 5818}, {4325, 5122}, {5046, 6702}, {5119, 9588}, {5131, 7354}, {5183, 9955}, {5204, 5790}, {5288, 6735}, {5552, 5904}, {5557, 5708}, {5660, 5693}, {5901, 7294}, {7742, 9709}

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)
Barycentrics    a^2[a^2b^2c^2 SA - 2 S^2(S^2 + SB SC)] : :

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) lies on these lines: {2, 5447}, {3, 51}, {4, 52}, {5, 141}, {6, 7387}, {20, 3567}, {22, 569}, {23, 54}, {25, 1147}, {26, 578}, {30, 143}, {49, 1495}, {113, 6152}, {131, 134}, {140, 5943}, {155, 1351}, {184, 7517}, {185, 382}, {193, 9936}, {235, 5448}, {343, 7403}, {371, 5417}, {372, 5419}, {373, 3526}, {381, 5562}, {394, 7529}, {427, 5449}, {428, 539}, {546, 1154}, {550, 5946}, {567, 2937}, {576, 2393}, {631, 5640}, {632, 6688}, {970, 6914}, {1092, 7506}, {1112, 3575}, {1209, 5133}, {1350, 7393}, {1593, 7689}, {1614, 1994}, {1656, 3917}, {1885, 6746}, {2979, 3090}, {3089, 5654}, {3091, 5891}, {3098, 7516}, {3146, 5890}, {3292, 7545}, {3543, 6241}, {3547, 9967}, {3560, 5752}, {3574, 10024}, {3627, 6000}, {3628, 3819}, {3845, 5876}, {3853, 5663}, {5056, 7999}, {5067, 7998}, {5070, 5650}, {5093, 6467}, {5102, 9973}, {6146, 7553}, {6193, 6995}, {6676, 6689}, {7713, 9928}, {7718, 9933}

X(5446) = midpoint of X(4) and X(52)
X(5446) = reflection of X(i) in X(j) for these (i,j): (389, 143), (1216,5)
X(5446) = anticomplement of X(5447)

X(5446) = X(946)-of-orthic-triangle if ABC is acute

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 O_A be the circle with center A′ that is tangent to line BC. Define O_B and O_C cyclically. Then X(5447) is the radical center of the three circles. (Randy Hutson, April 8, 2013)

X(5447) lies on these lines: {2, 5446}, {3, 49}, {4, 7998}, {5, 3819}, {20, 5891}, {26, 3098}, {51, 3526}, {52, 631}, {68, 7386}, {140, 143}, {389, 549}, {548, 6000}, {550, 5907}, {569, 7485}, {578, 7516}, {632, 5943}, {858, 1209}, {1154, 3530}, {1350, 6642}, {1368, 5449}, {1498, 8717}, {1656, 5650}, {3060, 3525}, {3523, 9730}, {3524, 5889}, {3533, 5640}, {3546, 9967}, {5054, 6243}, {5448, 6823}, {5651, 7517}, {5654, 7400}, {5876, 8703}, {6643, 9927}, {6689, 7499}

X(5447) = complement of X(5446)
X(5447) = midpoint of X(3) and X(1216)

X(5447) = {X(3),X(49)}-harmonic conjugate of X(22352)

X(5448) = 1st HATZIPOLAKIS-MOSES POINT

Trilinears (cos A)(2 + 2 cos(2B) + 2 cos(2C) + cos(2B - 2C)) (César Lozada, April 15, 2013; Hyacinthos #21954)
Barycentrics (b² + c² - a²)(b⁸ + c⁸ + 2a⁶b² + 2a⁶c² - 3a⁴b⁴ - 3a⁴c⁴ + 4a⁴b²c² - 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 R_A be the radical axis of the circles (B″, |B′C″|) and (C″,|C′B″|), and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(5448). For figures, see Concurrent Radical Axes. (Antreas Hatzipolakis, April 10, 2013)

Let A′B′C′ be the orthic triangle. Let (A) be the circle centered at A′ with radius a/2, and define (B), (C) cyclically. X(5448) is the radical center of circles (A), (B), (C). (Randy Hutson, January 29, 2018)

X(5448) lies on these lines: {2, 7689}, {3, 1568}, {4, 110}, {5, 389}, {30, 5893}, {52, 403}, {68, 1173}, {155, 195}, {185, 2072}, {235, 5446}, {525, 7764}, {541, 3357}, {912, 6583}, {1204, 6640}, {1533, 5073}, {1614, 3153}, {1699, 9928}, {2929, 6642}, {3060, 6242}, {3167, 3843}, {3546, 4846}, {3564, 3850}, {3818, 7564}, {3832, 6193}, {3855, 9936}, {5447, 6823}, {5476, 8548}, {5562, 10024}, {6238, 7951}, {6689, 7503}, {7352, 7741}, {7552, 7691}

X(5448) = midpoint of X(4) and X(1147)
X(5448) = complement of X(7689)

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) = S_A[a⁴(S² - S_A²) - 8S²S_BS_C]
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = b⁴S_B(S² - S_B²) + c⁴S_C(S² - S_C²)
Barycentrics   sin 4B + sin 4C : sin 4C + sin 4A : sin 4A + sin 4B    (Randy Hutson, August 26, 2014)
X(5449) = 3X(2) + X(68) = 3X(2) - X(1147)

Let A′B′C′ be the pedal triangle of the circumcenter, X(3), and let A″B″C″ be the circumcevian triangle of X(3) with respect to A′B′C′. Let R_A be the radical axis of the circles (B′, |B′C″|) and (C′,|C′B″|), and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(5449). The midpoint of X(5448) and X(5449) is X(5). For figures, see Concurrent Radical Axes. (Antreas Hatzipolakis, April 10, 2013)

Let D = X(68); then X(5449) is the centroid of ABCD. (Randy Hutson, August 25, 2014)

Let N_AN_BN_C be the reflection triangle of X(5). Let O_A be the circumcenter of AN_BN_C, and define O_B and O_C cyclically. X(5449) is the orthocenter of O_AO_BO_C. (Randy Hutson, June 7, 2019)

X(5449) lies on these lines: {2, 54}, {3, 125}, {4, 7689}, {5, 389}, {30, 6696}, {51, 5576}, {52, 1594}, {69, 8538}, {113, 7722}, {115, 8571}, {136, 847}, {155, 1656}, {156, 542}, {184, 6639}, {185, 10024}, {343, 1216}, {427, 5446}, {511, 6697}, {568, 3574}, {575, 3564}, {912, 3812}, {1092, 6640}, {1368, 5447}, {1614, 3448}, {1698, 9928}, {1853, 7387}, {1899, 3549}, {2072, 5562}, {3090, 5643}, {3091, 7693}, {3167, 5070}, {3616, 9933}, {3624, 9896}, {5020, 9908}, {5067, 9936}, {5169, 9781}, {5422, 7569}, {5889, 7577}, {5892, 7399}, {6146, 7542}, {6238, 7741}, {7352, 7951}, {7393, 9937}, {7514, 9932}, {7552, 9140}, {7706, 9786}, {7846, 9923}, {8253, 8909}

X(5449) = midpoint of X(68) and X(1147)
X(5449) = complement of X(1147)
X(5449) = X(6796)-of-orthic-triangle if ABC is acute

X(5450) = 3rd HATZIPOLAKIS-MOSES POINT

Barycentrics 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 R_A be the radical axis of the circles (B, |BC′|) and (C,|CB′|), and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(5450). For figures, see Concurrent Radical Axes. See also X(1147). (Antreas Hatzipolakis, April 10, 2013)

Let M be the isogonal conjugate of the trilinear polar of X(57) with respect to the circumcevian triangle of X(1). Then M is a conic, and its center is X(5450). (Angel Montesdeoca, August 9, 2019)

X(5450) lies on these lines: {1, 104}, {2, 6256}, {3, 10}, {4, 36}, {5, 2829}, {8, 2077}, {21, 84}, {30, 3829}, {35, 944}, {40, 2975}, {48, 1765}, {55, 5882}, {56, 946}, {100, 5881}, {318, 1309}, {388, 6935}, {404, 5587}, {405, 6260}, {411, 5303}, {498, 6977}, {517, 8666}, {519, 8668}, {550, 5842}, {631, 5251}, {950, 8071}, {952, 8715}, {995, 3073}, {997, 7330}, {999, 3671}, {1006, 1490}, {1071, 2646}, {1125, 3560}, {1210, 1470}, {1385, 5248}, {1457, 1777}, {1478, 6833}, {1479, 6938}, {1482, 4084}, {1532, 5433}, {1621, 7971}, {1698, 6940}, {1699, 7704}, {2096, 3485}, {3072, 4257}, {3149, 5204}, {3436, 6966}, {3522, 7688}, {3585, 6830}, {3624, 6920}, {3746, 7967}, {3814, 6958}, {3822, 6862}, {3825, 6929}, {3885, 6264}, {4189, 5731}, {4231, 5345}, {4293, 6847}, {4305, 5768}, {4325, 6845}, {4511, 5693}, {4999, 6907}, {5080, 6972}, {5126, 9856}, {5229, 6956}, {5253, 6912}, {5258, 5657}, {5288, 5537}, {5538, 6763}, {5563, 5603}, {5691, 6905}, {6259, 7489}, {6681, 6959}, {6831, 7354}, {6946, 7989}, {6952, 7951}, {7677, 8544}

X(5450) = midpoint of X(1) and X(1158)

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 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)]. (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. 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)

As a line L varies through the set of all lines that pass through X(55), the locus of the trilinear pole of L is a circumconic, and its center is X(5452). (Randy Hutson, April 19, 2013)

Let L be a line tangent to the incircle. Let P and U be the circumcircle intercepts of L. Let X be the crossdifference of P and U. As L varies, X traces the circumellipse centered at X(5452). This ellipse shares a major axis with, and is homothetic to, the Privalov conic. Therefore, the Privalov conic is an ellipse for every triangle ABC. (Randy Hutson, July 20, 2016)

Let A′B′C′ be the orthic triangle. Let La be the Gergonne line of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. Triangle A″B″C″ is inversely similar to ABC, with similicenter X(5452). (Randy Hutson, July 31 2018)

Let P be a point on the circumcircle. Let A′B′C′ be the anticevian triangle of P. A′B′C′ is perspective to the unary cofactor triangle of the intangents triangle for all P. As P varies, the perspector traces the circumconic centered at X(5452). (Randy Hutson, July 11, 2019)

For another conic with center X(5452), see X(175) and X(46417.

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}, {212, 8012}, {218, 226}, {219, 3686}, {294, 497}, {650, 1376}, {651, 7056}, {666, 6063}, {1212, 7124}, {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 N_A be the nine-point center of triangle IB′C′, and define N_B and N_C cyclically. The points I, N_A, N_B, N_C 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}, {55, 6097}, {58, 5428}, {73, 5719}, {140, 3216}, {155, 6914}, {186, 2906}, {386, 549}, {511, 1385}, {550, 991}, {1064, 5901}, {1154, 2646}, {1442, 6356}, {1834, 5499}, {1962, 5492}, {2771, 3743}, {3945, 6869}

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)
Trilinears (4*a*g(A)+3*b*g(B)+3*c*g(C))*f(A)+(b*f(B)+c*f(C))*g(A) : :,
where f(A)=cos(A/3)+2*cos(B/3)*cos(C/3) and g(A)= sec((A+2*π)/3) (César Lozada, March 3, 2022)

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 these lines: {356, 1134}, {357, 3280}

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 L_A be the line of the centroid of AEF and the centroid of BCD, and define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in X(5455). (Seiichi Kirikami, April 26, 2013)

If you have The Geometer's Sketchpad, you can view X(5455).

X(5455) lies on this line: {2, 356}

X(5456) = 3rd MORLEY-KIRIKAMI POINT

Barycentrics sin(2A/3) : sin(2B/3) : sin(2C/3) (Peter Moses, May 14, 2013)

Let DEF be the 1st Morley triangle of triangle ABC, and let D₁E₁F₁ be the 1st Morley adjunct triangle (defined at MathWorld). Let D₂ = DD₁∩BC, and define E₂ and F₂ cyclically. The lines AD₂, BE₂, CF₂ concur in X(5456). (Seiichi Kirikami, April 26, 2013)

Let DEF be the 1st Morley triangle. Let D' be the trilinear pole of line EF, and define E', F' cyclically. Let D" be the trilinear pole of line E'F', and define E", F" cyclically. The lines AD", BE", CF" concur in X(5456). (Randy Hutson, September 29, 2014)

If you have The Geometer's Sketchpad, you can view X(5456).

X(5456) lies on these lines: {356, 3605}, {3274, 3602}

X(5457) = 4th MORLEY-KIRIKAMI POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(cos A + cos(A/3)) (Peter Moses, April 26, 2013)
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A/3)/sin(4A/3) (Randy Hutson, August 9, 2014)

Let DEF be the 1st Morley triangle of triangle ABC, and let D₃ be the reflection of D in line BC, and define E₃ and F₃ cyclically. The lines AD₃, BE₃, CF₃ concur in X(5457). (Seiichi Kirikami, April 26, 2013)

X(5457) is the Hofstadter -1/3 point; see X(359). (Randy Hutson, August 9, 2014)

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

X(5457) lies on this line: {357, 5628}

X(5457) = isogonal conjugate of X(6123)

X(5458) = 5th MORLEY-KIRIKAMI POINT

Trilinears 1/(4 cos A + sec(A/3)) : : (Peter Moses, April 26, 2013)
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2A/3)/sin(5A/3) (Randy Hutson, August 9, 2014)

Let DEF be the 1st Morley triangle of triangle ABC, and let D₁E₁F₁ be the 1st Morley adjunct triangle (defined at MathWorld). Let D₄ be the reflection of D₁ in line BC, and define E₄ and F₄ cyclically. The lines AD₄, BE₄, CF₄ concur in X(5458). (Seiichi Kirikami, April 26, 2013)

X(5458) is the Hofstadter -2/3 point; see X(359). (Randy Hutson, August 9, 2014)

D₄ is the BCD-isogonal conjugate of A, and cyclically for E₄ and F₄(Randy Hutson, January 29, 2015

X(5458) lies on these lines: {}

X(5459) = MIDPOINT OF X(2) AND X(13)

Barycentrics 4a csc(A + π/3) + b csc(B + π/3) + c csc(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)

The circle just defined is here given the name 1st Hutson circle. The 2nd Hutson circle and Hutson-Parry circles are defined at X(5460) and X(5466); see also X(8371). (CK, October 31, 2015)

X(5459) lies on these lines: {2, 13}, {5, 542}, {14, 9166}, {17, 671}, {30, 5478}, {61, 5469}, {115, 396}, {148, 9114}, {395, 5472}, {524, 623}, {543, 619}, {599, 635}, {630, 2482}, {3524, 5473}, {3545, 6770}, {3642, 9763}, {5055, 5617}, {5318, 6671}, {5476, 7685}, {8371, 9194}

X(5459) = reflection of X(5460) in X(5461)
X(5459) = complement of X(5463)
X(5459) = radical center of the polar circles of triangles BCX(13), CAX(13), ABX(13)
X(5459) = radical trace of 1st Hutson and Hutson-Parry circles

X(5460) = MIDPOINT OF X(2) AND X(14)

Barycentrics 4a csc(A - π/3) + b csc(B - π/3) + c csc(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)

The circle just described is here given the name 2nd Hutson circle. The 1st Hutson and Hutson-Parry circles are defined at X(5459) and X(5466); see also X(8371). (CK, October 31, 2015)

X(5460) lies on these lines: {2, 14}, {5, 542}, {13, 9166}, {18, 671}, {30, 5479}, {62, 5470}, {115, 395}, {148, 9116}, {396, 5471}, {524, 624}, {543, 618}, {599, 636}, {629, 2482}, {3524, 5474}, {3545, 6773}, {3643, 9761}, {5055, 5613}, {5321, 6672}, {5476, 7684}, {8371, 9195}

X(5460) = reflection of X(5459) in X(5461)
X(5460) = complement of X(5464)
X(5460) = {X(5),X(597)}-harmonic conjugate of X(5459)
X(5460) = radical center of the polar circles of triangles BCX(14), CAX(14), ABX(14)
X(5460) = radical trace of 2nd Hutson and Hutson-Parry circles

X(5461) = MIDPOINT OF X(5459) AND X(5460)

Barycentrics 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}, {3, 9880}, {5, 542}, {6, 8176}, {30, 6036}, {98, 3545}, {114, 5055}, {125, 5465}, {187, 8352}, {230, 3849}, {316, 8859}, {376, 9754}, {381, 2794}, {523, 9165}, {524, 625}, {530, 6670}, {531, 6669}, {538, 2023}, {547, 2782}, {598, 7806}, {599, 626}, {690, 9183}, {754, 9478}, {1153, 3054}, {1506, 7827}, {1656, 8724}, {1698, 9881}, {1992, 3767}, {1995, 3455}, {2796, 3634}, {3090, 7902}, {3363, 7804}, {3589, 9830}, {3616, 9884}, {3618, 8593}, {3624, 9875}, {3934, 5969}, {5020, 9876}, {5025, 7810}, {5054, 6321}, {5071, 6054}, {5077, 5569}, {5182, 7808}, {5215, 8598}, {5463, 5470}, {5464, 5469}, {6680, 8370}, {6704, 8367}, {6781, 8597}, {7610, 7761}, {7746, 7830}, {7749, 7833}, {7755, 7812}, {7798, 9770}, {7801, 7887}, {7846, 9878}, {7861, 8359}, {7886, 8369}

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) = 3X(2) + X(52)

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}, {20, 9781}, {24, 569}, {26, 182}, {30, 9729}, {54, 6153}, {68, 7401}, {110, 1199}, {125, 5576}, {140, 143}, {155, 5020}, {184, 7506}, {185, 381}, {195, 3292}, {343, 7405}, {373, 568}, {468, 6746}, {539, 9827}, {546, 5893}, {570, 3133}, {575, 2393}, {578, 6644}, {631, 3060}, {632, 3819}, {973, 6689}, {1112, 6699}, {1154, 3628}, {1181, 3066}, {1209, 3580}, {1843, 3517}, {1899, 7528}, {1995, 7592}, {2070, 2918}, {2072, 3574}, {2807, 9955}, {2979, 3525}, {3090, 5889}, {3091, 5890}, {3518, 5012}, {3526, 3917}, {3533, 7998}, {3564, 9822}, {3618, 9967}, {3796, 9714}, {3832, 6241}, {3850, 5663}, {5092, 7525}, {5643, 7550}, {5752, 6883}, {6237, 9816}, {6238, 9817}, {6677, 9820}, {7689, 9786}, {8548, 9813}, {9815, 9927}

X(5462) = midpoint of X(i) and X(j) for these (i,j): (5,389), (140,143)
X(5462) = complement of X(1216)
X(5462) = centroid of {A,B,C,X(52)}

X(5463) = REFLECTION OF X(13) IN X(2)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A sec(A - π/6) - 2 sin B sec(B - π/6) - 2 sin C sec(C - π/6) (Peter Moses, May 3, 2013)

Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463)}; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014). X(5463) is the center of the equilateral antipedal triangle of X(13), and X(5463) = (X(3), X(599))-harmonic conjugate of X(5464). (Randy Hutson, May 2, 2013)

The circle {{X(2), X(110), X(5463), X(5464)}} has center X(1649) and passes through X(2770). It is tangent to the Euler line at X(2) and is the reflection in the Euler line of the circle {{X(2), X(13), X(14), X(111), X(476)}}. (Randy Hutson, August 26, 2014)

Let N_AN_BN_C and N′_AN′_BN′_C be the inner and outer Napoleon triangles, resp. Let A′ be the reflection of N_A in line N′_BN′_C, and define B′ and C′ cyclically. Triangle A′B′C′ is equilateral, the reflection of N′_AN′_BN′_C in X(618), and X(5463) is its center. (Randy Hutson, January 17, 2020)

X(5463) lies on these lines: {2, 13}, {3, 67}, {6, 9115}, {14, 543}, {15, 524}, {18, 671}, {30, 5473}, {61, 1992}, {62, 597}, {99, 298}, {299, 7771}, {395, 6772}, {396, 9112}, {519, 7975}, {620, 6778}, {627, 8591}, {2770, 9203}, {2796, 5699}, {3104, 5969}, {3106, 7757}, {3524, 6770}, {3545, 5478}, {5054, 6771}, {5461, 5470}, {5476, 5615}, {5569, 9763}, {5978, 8592}, {6054, 9749}, {6670, 9166}, {6782, 9741}, {7840, 8291}, {7865, 9982}, {9168, 9205}, {9830, 9886}

X(5463) = midpoint of X(2) and X(616)
X(5463) = reflection of X(2) in X(618)
X(5463) = reflection of X(5464) in X(2482)
X(5463) = anticomplement of X(5459)
X(5463) = Thomson-isogonal-conjugate-of-X(15)

X(5464) = REFLECTION OF X(14) IN X(2)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A sec(A + π/6) - 2 sin B sec(B + π/6) - 2 sin C sec(C + π/6) (Peter Moses, May 3, 2013)

Each of the following sets of 4 points are concyclic: {X(13), X(15), X(5463), X(5464)}, {X(14), X(16), X(5463), X(5464)}, {X(2), X(110), X(5463), X(5464)}, {X(3), X(15), X(110), X(5464)}, {X(3), X(16), X(110), X(5463)}. Moreover, the line X(13)X(5463) is tangent to both of the circles of {X(13), X(14), X(15)} and {X(14), X(15), X(5463):; likewise, the line X(14)X(5464) is tangent to both of the circles of {X(13), X(14), X(16)} and {X(13), X(16), X(5464). (Dao Thanh Oai, ADGEOM #1237, April 7, 2014).

X(5464) is the center of the equilateral antipedal triangle of X(14), and X(5464) = (X(3), X(599))-harmonic conjugate of X(5463). (Randy Hutson, May 2, 2013)

Let N_AN_BN_C and N′_AN′_BN′_C be the inner and outer Napoleon triangles, resp. Let A″ be the reflection of N′_A in line N_BN_C, and define B″ and C″ cyclically. Triangle A″B″C″ is equilateral, the reflection of N_AN_BN_C in X(619), and X(5464) is its center. (Randy Hutson, January 17, 2020)

X(5464) lies on these lines: {2, 14}, {3, 67}, {6, 9117}, {13, 543}, {16, 524}, {17, 671}, {30, 5474}, {61, 597}, {62, 1992}, {99, 299}, {298, 7771}, {395, 9113}, {396, 6775}, {519, 7974}, {620, 6777}, {628, 8591}, {2770, 9202}, {2796, 5700}, {3105, 5969}, {3107, 7757}, {3524, 6773}, {3545, 5479}, {5054, 6774}, {5461, 5469}, {5476, 5611}, {5569, 9761}, {5979, 8592}, {6054, 9750}, {6669, 9166}, {6783, 9741}, {7840, 8292}, {7865, 9981}, {9168, 9204}, {9830, 9885}

X(5464) = midpoint of X(2) and X(617)
X(5464) = reflection of X(2) in X(619)
X(5464) = reflection of X(5463) in X(2482)
X(5464) = anticomplement of X(5460)
X(5464) = Thomson-isogonal-conjugate-of-X(16)

X(5465) = ORTHOGONAL PROJECTION OF X(2) ON THE FERMAT AXIS

Barycentrics 2*a^10-4*(b^2+c^2)*a^8+4*(2*b^4-b^2*c^2+2*c^4)*a^6-(b^2+c^2)*(7*b^4-10*b^2*c^2+7*c^4)*a^4-(b^8+c^8-2*b^2*c^2*(7*b^4-12*b^2*c^2+7*c^4))*a^2+(b^4-c^4)*(b^2-c^2)*(b^2-2*c^2)*(2*b^2-c^2) : : (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 the circle O(2,6) (with diameter X(2)X(6)) and these lines: {2, 690}, {6, 13}, {30, 9181}, {110, 671}, {125, 5461}, {541, 6055}, {543, 1316}, {2482, 5972}, {2780, 3111}, {6593, 9830}, {9140, 9166}

X(5465) = midpoint of X(110) and X(671)
X(5465) = reflection of X(125) and X(5461)
X(5465) = complement of X(11006)

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)

The circle just described is here given the name Hutson-Parry circle. (CK, October 31, 2015). The circle is mentioned briefly in TCCT, page 228.

In the plane of a triangle ABC, let
Γ = circumcircle;
D = line tangent to $Gamma; at A;
Ta = D∩BC;
Oab = center of the circle through Ta that is tangent to Γ at B;
Oac = center of the circle through Ta that is tangent to Γ at C;
Ab = orthogonal projection of Oab onto AC;
Ac = orthogonal projection of Oac onto AB;
A' = AbAc∩BC, and define B' and C' cyclically.
The lines AA', BB', CC' concur in X(5466). See X(5466) (Angel Montesdeoca, August 21, 2022)

X(5466) lies on the Kiepert hyperbola and these lines: {2, 523}, {4, 1499}, {10, 4024}, {13, 9201}, {14, 9200}, {76, 850}, {83, 5643}, {98, 111}, {115, 9180}, {262, 5996}, {321, 4036}, {351, 8587}, {476, 691}, {512, 598}, {525, 5485}, {647, 7607}, {671, 690}, {685, 4240}, {868, 2394}, {879, 9154}, {892, 5468}, {895, 2986}, {1916, 9148}, {2799, 5503}, {3268, 8781}, {3288, 7708}, {5067, 8151}, {7612, 9209}, {9123, 9189}, {9125, 9131}

X(5466) = isogonal conjugate of X(5467)
X(5466) = isotomic conjugate of X(5468)
X(5466) = pole wrt polar circle of trilinear polar of X(4235)
X(5466) = X(48)-isoconjugate (polar conjugate) of X(4235)
X(5466) = barycentric product X(111)*X(850)

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}, {23, 5968}, {110, 351}, {112, 1296}, {250, 4230}, {523, 2407}, {524, 5967}, {691, 9178}, {895, 9142}, {1649, 5468}, {1989, 6321}, {2709, 2715}, {2794, 3014}, {2854, 5191}, {3292, 9717}, {4436, 4612}, {5994, 9203}, {5995, 9202}, {6036, 6128}, {6593, 9155}

X(5467) = midpoint of X(2407) and X(4226)
X(5467) = crossdifference of every pair of points on line X(115)X(523)
X(5467) = X(111)-isoconjugate of X(1577)

X(5468) = ISOTOMIC CONJUGATE OF X(5466)

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

X(5468) is the unique point on line X(2)X(6) whose trilinear polar is parallel to line X(2)X(6). (Randy Hutson, March 21, 2019)

X(5468) lies on these lines: {2, 6}, {99, 110}, {126, 5477}, {316, 9141}, {511, 7417}, {691, 6082}, {805, 9150}, {877, 4240}, {892, 5466}, {1316, 6090}, {1649, 5467}, {2418, 2434}, {2502, 5969}, {2709, 9080}, {2715, 2858}, {3111, 7771}, {3266, 3292}, {4590, 9170}, {4615, 6548}, {5642, 7664}, {6390, 9717}

X(5468) = isogonal conjugate of X(9178)
X(5468) = crossdifference of every pair of points on line X(512)X(3124)
X(5468) = trilinear pole of line X(187)X(524)
X(5468) = {X(110),X(4563)}-harmonic conjugate of X(4576)

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: {2, 5982}, {6, 13}, {18, 671}, {61, 5459}, {98, 5479}, {99, 6670}, {148, 618}, {395, 6779}, {531, 9166}, {617, 6669}, {5461, 5464}, {5473, 6321}, {5474, 6036}, {5478, 6773}

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: {2, 5983}, {6, 13}, {17, 671}, {62, 5460}, {98, 5478}, {99, 6669}, {148, 619}, {396, 6780}, {530, 9166}, {616, 6670}, {5461, 5463}, {5473, 6036}, {5474, 6321}, {5479, 6770}

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

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)

Let U be the pedal circle of X(14) (and of X(16). Then X(5471) is the U-inverse of X(187). (Randy Hutson, January 29,2105)

X(5471) lies on these lines: {2, 9117}, {5, 6783}, {6, 13}, {15, 6774}, {16, 6781}, {18, 7749}, {39, 398}, {61, 1506}, {62, 7747}, {187, 395}, {230, 6114}, {233, 2903}, {302, 620}, {396, 5460}, {543, 8595}, {590, 6303}, {615, 6307}, {1569, 3106}, {2482, 9761}, {2549, 5334}, {2782, 6782}, {3815, 6109}, {5318, 5479}, {5339, 7748}, {6775, 7737}

X(5471) = isogonal conjugate (and isotomic conjugate) of X(16) w.r.t to the pedal triangle of X(16)
X(5471) = X(14)-antipedal-to-X(16)-pedal similarity image of X(14)
X(5471) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5474,5472), (115,5477,5472)
X(5471) = X(14)-antipedal-to-X(16)-pedal similarity image of X(14)

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

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)

Let V be the pedal circle of X(13) (and of X(15). Then X(5472) is the V-inverse of X(187). (Randy Hutson, January 29,2105)

X(5472) lies on these lines: {2, 9115}, {5, 6782}, {6, 13}, {15, 6781}, {16, 6771}, {17, 7749}, {39, 397}, {61, 7747}, {62, 1506}, {187, 396}, {230, 6115}, {233, 2902}, {303, 620}, {395, 5459}, {543, 8594}, {590, 6302}, {615, 6306}, {1569, 3107}, {2482, 9763}, {2549, 5335}, {2782, 6783}, {3815, 6108}, {5321, 5478}, {5340, 7748}, {6772, 7737}

X(5472) = isogonal conjugate (and isotomic conjugate) of X(15) w.r.t. the pedal triangle of X(15)
X(5472) = X(13)-antipedal-to-X(15)-pedal similarity image of X(13)
X(5472) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5475,5471), (115,5477,5471)
X(5472) = X(13)-antipedal-to-X(15)-pedal similarity image of X(13)

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) = 3^1/2a²(a⁴ - 3b⁴ - 3c⁴ + 2a²b² + 2a²c² - 2b²c²) + 4(area ABC)(7a⁴ - 2b⁴ - 2c⁴ - 5a²b² - 5a²c² + 4b²c²) (Peter Moses, May 12, 2013)

X(5473) lies on these lines: {2, 5478}, {3, 13}, {4, 618}, {15, 9112}, {16, 2549}, {20, 616}, {30, 5463}, {62, 9607}, {165, 9901}, {376, 530}, {477, 9203}, {511, 6779}, {517, 7975}, {542, 1350}, {631, 6669}, {3098, 9982}, {3524, 5459}, {5469, 6321}, {5470, 6036}, {5979, 9749}

X(5473) = reflection of X(13) in X(3)
X(5473) = anticomplement of X(5478)
X(5473) = isogonal conjugate (and isotomic conjugate) of X(13) wrt the antipedal triangle of X(13)
X(5473) = {X(1350), X(3534)}-harmonic conjugate of X(5474)

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) = 3^1/2a²(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) lies on these lines: {2, 5479}, {3, 14}, {4, 619}, {15, 2549}, {16, 9113}, {20, 617}, {30, 5464}, {61, 9607}, {165, 9900}, {376, 531}, {477, 9202}, {511, 6780}, {517, 7974}, {542, 1350}, {631, 6670}, {3098, 9981}, {3524, 5460}, {5469, 6036}, {5470, 6321}, {5978, 9750}

X(5474) = reflection of X(14) in X(3)
X(5474) = anticomplement of X(5479)
X(5474) = {X(1350), X(3534)}-harmonic conjugate of X(5473)

X(5475) = INTERSECTION OF LINES X(2)X(187) AND X(4)X(39)

Barycentrics a⁴ + a²(b² + c²) - (b² - c²)² (Peter Moses, May 12, 2013)
Barycentrics (SA + SW) (SB + SC) + 4 SB SC : :
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): (5471,5472,6), (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: {1, 9650}, {2, 187}, {3, 1506}, {4, 39}, {5, 32}, {6, 13}, {11, 2242}, {12, 2241}, {30, 574}, {51, 5167}, {69, 7845}, {76, 7759}, {83, 3407}, {99, 7777}, {112, 7577}, {140, 5206}, {141, 7818}, {148, 7757}, {156, 9697}, {172, 7741}, {183, 754}, {194, 7858}, {233, 1609}, {315, 3934}, {325, 3734}, {376, 8589}, {382, 5013}, {384, 3788}, {385, 7812}, {427, 9515}, {485, 5058}, {486, 5062}, {524, 3363}, {538, 7774}, {546, 5254}, {547, 3054}, {549, 3055}, {566, 7574}, {590, 9675}, {620, 1003}, {626, 7770}, {1015, 1478}, {1078, 7823}, {1194, 7394}, {1196, 6997}, {1285, 5071}, {1316, 5099}, {1352, 5052}, {1384, 5055}, {1479, 1500}, {1503, 5034}, {1504, 3071}, {1505, 3070}, {1569, 6321}, {1572, 5587}, {1594, 1968}, {1596, 5065}, {1656, 3053}, {1699, 9620}, {1914, 7951}, {1975, 7764}, {1992, 7615}, {1995, 9745}, {2088, 7706}, {2207, 7507}, {2275, 3585}, {2276, 3583}, {2458, 5103}, {2896, 7860}, {3066, 6388}, {3087, 6623}, {3091, 3767}, {3096, 7885}, {3314, 7809}, {3329, 7790}, {3526, 5023}, {3545, 5008}, {3552, 7769}, {3589, 5033}, {3814, 4386}, {3830, 5024}, {3832, 5041}, {3839, 7739}, {3843, 7765}, {3845, 9300}, {3850, 5305}, {3851, 6287}, {3853, 9606}, {3855, 5319}, {3861, 9607}, {3933, 7903}, {4045, 7841}, {4193, 5277}, {5028, 5480}, {5046, 5283}, {5054, 5210}, {5066, 5306}, {5640, 6787}, {5691, 9619}, {5913, 8585}, {6292, 7784}, {6655, 7786}, {6656, 7808}, {6658, 7782}, {6680, 7887}, {6683, 7791}, {7517, 9700}, {7526, 9608}, {7530, 9609}, {7533, 9465}, {7547, 8743}, {7622, 8598}, {7750, 7815}, {7751, 7762}, {7754, 7838}, {7760, 7921}, {7763, 7816}, {7768, 7900}, {7776, 7794}, {7778, 7820}, {7787, 7828}, {7789, 7888}, {7792, 7844}, {7795, 7821}, {7796, 7941}, {7797, 7878}, {7800, 7873}, {7802, 7824}, {7803, 7861}, {7807, 7862}, {7813, 9766}, {7814, 7836}, {7819, 7867}, {7829, 7851}, {7832, 7912}, {7846, 7901}, {7859, 7933}, {7866, 7889}, {7876, 7911}, {7892, 7899}, {7935, 8362}, {8035, 8036}, {9115, 9761}, {9117, 9763}

X(5475) = reflection of X(574) in X(3815)

X(5476) = MIDPOINT OF X(6) AND X(381)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁶ + b⁶ + c⁶ - 4a⁴b² - 4a⁴c² + 2a²b⁴ + 2a²c⁴ - 6a²b²c² - b⁴c² - b²c⁴ (Peter Moses, May 12, 2013)

Let A′ be the intersection, other than X(4), of the A-altitude and the orthocentroidal circle, and define B′ and C′ cyclically. The triangle A′B′C′, introduced here as the orthocentroidal triangle, is inversely similar to ABC, with center X(6) of similitude. If, in that definition, X(4) replaced by X(2) and A-altitude by A-median, the resulting triangle is the 4th Brocard triangle. Regarding a point X as a function of a triangle, X(A′B′C′) - that is, X of A′B′C′ - is the reflection of X(ABC) in the centroid of the pedal triangle of X. X(5476) = X(182) of the orthocentroidal triangle, and X(5476) = X(182) of the 4th Brocard triangle. Also, X(5476) = inverse-in-Kiepert-hyperbola of X(5475), and X(5476) = {X(13),X(14)}-harmonic conjugate of X(5475). (Randy Hutson, May 7, 2013)

The vertices of the orthocentroidal triangle are given by Peter Moses (June 17, 2014): A′ = a² : a² + b² - c² : a² - b² + c²
B′ = b² - c² + a² : b² : b² + c² - a²
C′ = c² + a² - b² : c² - a² + b² : c²

The orthocentroidal triangle is the circumsymmedial triangle of the 4th Brocard triangle. (Randy Hutson, November 22, 2014)

Another construction of the orthocentroidal triangle: Let NaNbNc and Na'Nb'Nc' be the inner and outer Napoleon triangles, resp. Then A′ = NbNc'∩NcNb', and B′ and C′ are constructed cyclically; i.e, A′B′C′ is the cross-triangle of the inner and outer Napoleon triangles. Also, A′ is the center of inverse similitude of ABC and the A-altimedial triangle, and cyclically for B′ and C′. (Randy Hutson, December 10, 2016)

X(5476) lies on these lines: {2, 51}, {4, 575}, {5, 524}, {6, 13}, {30, 182}, {67, 7579}, {69, 5071}, {141, 547}, {376, 3618}, {403, 8541}, {546, 8550}, {549, 3098}, {569, 7540}, {599, 1351}, {1350, 5054}, {1352, 1992}, {1469, 3582}, {1503, 3845}, {1974, 7576}, {1995, 5642}, {2030, 7737}, {2781, 5946}, {3056, 3584}, {3066, 5972}, {3090, 7922}, {3091, 5032}, {3095, 7801}, {3329, 9993}, {3534, 5085}, {3564, 5066}, {3574, 9977}, {3830, 5050}, {3839, 6776}, {4663, 9955}, {5038, 7748}, {5039, 5306}, {5093, 5965}, {5107, 7603}, {5169, 9140}, {5171, 8359}, {5448, 8548}, {5459, 7685}, {5460, 7684}, {5463, 5615}, {5464, 5611}, {5569, 7606}, {5654, 8681}, {6032, 6792}, {6791, 9745}, {7533, 9143}, {7552, 9781}, {7951, 8540}, {8369, 9737}, {8538, 10024}, {9760, 9763}, {9761, 9762}, {9771, 10011}

X(5476) = centroid of reflection triangle of X(182)

X(5477) = REFLECTION OF X(115) IN X(6)

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

X(5477) lies on these lines: {6, 13}, {30, 5107}, {39, 8550}, {69, 620}, {98, 5034}, {99, 193}, {110, 6388}, {111, 9143}, {114, 230}, {126, 5468}, {147, 5304}, {187, 524}, {511, 1569}, {543, 1992}, {575, 1506}, {576, 7747}, {597, 7603}, {599, 9167}, {671, 5032}, {690, 5095}, {1353, 2782}, {1384, 8724}, {1503, 1570}, {1648, 5642}, {1691, 5965}, {2458, 4027}, {2502, 6791}, {2549, 2794}, {2796, 4856}, {3618, 6722}, {3629, 5969}, {3815, 6055}, {5038, 9698}, {5050, 6036}, {5093, 6321}, {6054, 7735}, {7837, 8289}, {8584, 9830}

X(5477) = {X(5471), X(5472)}-harmonic conjugate of X(115) (Randy Hutson, May 7, 2013)

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) = 3^1/2(3a⁴b² + 3a⁴c² - 2a²b⁴ - 2a²c⁴ + 4a²b²c² + b⁴c² + b²c⁴ - b⁶ - c⁶) + 4(area ABC)(4a⁴ - 5b⁴ - 5c⁴ + a²b² + a²c² + 10b²c²) (Peter Moses, May 12, 2013)

New pair of equilateral triangles (Vu Thanh Tung, November 25, 2019):

Let A₀B₀C₀ be the orthic triangle of ABC. Let A₁₅ = X(15)-of-A₀B₀C₀, and define B₁₅ and C₁₅ cyclically, so that . A₁₅B₁₅C₁₅ is the 3rd isodynamic-Dao equilateral triangle of ABC.

Let A′₁₅ be the point, other than A, in which the line AA₁₅ meets the circle {{A,B₀,C₀}}, and define B′₁₅ and C′₁₅ cyclically. The triangle A′₁₅B′₁₅C′₁₅ is here named the Vu-Dao-X(15)-isodynamic equilateral triangle.

If X(15) is replaced by X(16) in the above construction, the resulting triangle, A′₁₆B′₁₆C′₁₆, is here named the Vu-Dao-X(16)-isodynamic equilateral triangle. The centers of the two equilateral triangles are X(5478) and X(5479), respectively, and the diameters of the circumcircles of the two triangles are the segments X(4)X(13) and X(4)X(14), respectively.

Barycentrics and properties (Peter Moses, November 27, 2019):

A′₁₅ = -2*(a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) + 2*Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) + 2*S) : :

A′₁₆ = -2*(a^2*(a^4 + a^2*b^2 - 2*b^4 + a^2*c^2 + 4*b^2*c^2 - 2*c^4) + 2*Sqrt[3]*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 + b^2 - c^2) + 2*S), (a^2 - b^2 - c^2)*(Sqrt[3]*a^2 + 2*S)*(Sqrt[3]*(a^2 - b^2 + c^2) - 2*S) : :

With perspector X(13), the triangle A′₁₅B′₁₅C′₁₅ is perspective to the following triangles: ABC, outer Fermat (TCCT p178), 1st isodynamic-Dao equilateral (see X(16802)), 3st isodynamic-Dao equilateral (see X(31683)), 1st half-diamonds triangle (see X(33338)), and 2nd Lemoine-Dao equilateral triangle (see X(16940)).

X(5478) lies on these lines: {2, 5473}, {3, 6669}, {4, 13}, {5, 618}, {30, 5459}, {98, 5470}, {107, 473}, {115, 5318}, {381, 530}, {383, 6115}, {531, 9880}, {542, 1353}, {616, 3091}, {624, 3734}, {1080, 6108}, {1598, 9916}, {1699, 9901}, {2043, 6306}, {2044, 6302}, {3545, 5463}, {5321, 5472}, {5334, 9112}, {5469, 6773}, {5603, 7975}, {5613, 6321}, {6201, 6268}, {6202, 6270}, {9982, 9993}

X(5478) = midpoint of X(4) and X(13)
X(5478) = complement of X(5473)
X(5478) = X(13)-of-Euler-triangle
X(5478) = center of circle {{A′₁₅, B′₁₅, C′₁₅, X(4), X(13), X(16179)}}
X(5478) = {X(3845),X(5480)}-harmonic conjugate of X(5479)

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) = 3^1/2(3a⁴b² + 3a⁴c² - 2a²b⁴ - 2a²c⁴ + 4a²b²c² + b⁴c² + b²c⁴ - b⁶ - c⁶) - 4(area ABC)(4a⁴ - 5b⁴ - 5c⁴ + a²b² + a²c² + 10b²c²) (Peter Moses, May 12, 2013)

See X(5478).

X(5479) lies on these lines: {2, 5474}, {3, 6670}, {4, 14}, {5, 619}, {30, 5460}, {98, 5469}, {107, 472}, {115, 5321}, {381, 531}, {383, 6109}, {530, 9880}, {542, 1353}, {617, 3091}, {623, 3734}, {1080, 6114}, {1598, 9915}, {1699, 9900}, {2043, 6303}, {2044, 6307}, {3545, 5464}, {5318, 5471}, {5335, 9113}, {5470, 6770}, {5603, 7974}, {5617, 6321}, {6201, 6269}, {6202, 6271}, {9981, 9993}

X(5479) = midpoint of X(4) and X(14)
X(5479) = complement of X(5474)
X(5479) = X(14)-of-Euler-triangle
X(5479) = {X(3845),X(5480)}-harmonic conjugate of X(5478)

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

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); The orthosymmedial circle is the inverse-in-polar-circle of the line X(297)X(525). (Randy Hutson, August 26, 2014)

X(5480) lies on these lines: {2, 1350}, {3, 3589}, {4, 6}, {5, 141}, {11, 1469}, {12, 3056}, {20, 3618}, {30, 182}, {51, 125}, {52, 7403}, {66, 3527}, {69, 3091}, {86, 7385}, {98, 5306}, {113, 2854}, {114, 5969}, {115, 5052}, {118, 2810}, {119, 9024}, {140, 3098}, {154, 6995}, {159, 1598}, {184, 428}, {185, 1907}, {193, 3832}, {206, 578}, {230, 5017}, {235, 1843}, {262, 1513}, {265, 9970}, {323, 7533}, {343, 3060}, {355, 5846}, {381, 524}, {382, 5050}, {383, 396}, {389, 1595}, {394, 6997}, {395, 1080}, {403, 6403}, {515, 1386}, {516, 4085}, {518, 946}, {542, 1353}, {546, 576}, {550, 5092}, {567, 7540}, {569, 7553}, {575, 3627}, {590, 6813}, {599, 3545}, {611, 1479}, {613, 1478}, {615, 6811}, {674, 7680}, {698, 3095}, {732, 6248}, {858, 5640}, {966, 7407}, {1213, 7380}, {1368, 5943}, {1428, 7354}, {1482, 9053}, {1593, 5894}, {1594, 9781}, {1596, 2393}, {1692, 7747}, {1699, 3751}, {1848, 1864}, {1853, 7378}, {1861, 2262}, {1890, 2182}, {1899, 5064}, {1906, 6467}, {1974, 3575}, {1992, 3839}, {1993, 7394}, {2051, 4260}, {2330, 6284}, {2550, 5782}, {2552, 8115}, {2553, 8116}, {3054, 5104}, {3068, 7000}, {3069, 7374}, {3088, 6696}, {3089, 7716}, {3090, 3763}, {3146, 7864}, {3242, 5603}, {3313, 7399}, {3416, 5587}, {3531, 5486}, {3580, 5169}, {3619, 5056}, {3620, 5068}, {3630, 3850}, {3631, 3851}, {3656, 9041}, {3796, 7500}, {3827, 7686}, {3843, 5093}, {3858, 5965}, {4026, 6210}, {4259, 6831}, {4265, 6905}, {4549, 9818}, {5028, 5475}, {5034, 7748}, {5039, 5305}, {5096, 6906}, {5188, 8362}, {5422, 7391}, {5718, 8229}, {5805, 5845}, {5999, 7792}, {6393, 7752}, {6747, 6755}, {6800, 7519}, {7496, 7605}, {7681, 8679}, {7735, 9748}, {8359, 8722}, {8721, 9605}, {9019, 9967}, {9300, 9744}, {9830, 9880}

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(5480) = inverse-in-Jerabek-hyperbola of X(51)
X(5480) = Johnson-to-Ehrmann-mid similarity image of X(6)
X(5480) = X(1001)-of-orthic-triangle if ABC is acute
X(5480) = {X(34221),X(34222)}-harmonic conjugate of X(51)

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}, {1073, 7484}, {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}, {631, 5752}, {916, 6705}, {3819, 6675}, {3834, 9955}, {3917, 7483}{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 A_B = (reflection of A′ in BB′), and define B_C and C_A cyclically. Let A_C = (reflection of A′ in CC′), and define B_A and C_B cyclically. The ellipse passing through the points A_B, A_C, B_C, B_A, C_A, C_B 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}, {3920, 5988}

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 A_B = (reflection of A′ in BB′), and define B_C and C_A cyclically. Let A_C = (reflection of A′ in CC′), and define B_A and C_B cyclically. Let H_A = (orthocenter of A′A_BA_C), and define H_B and H_C cyclically. The orthocentric triangle of X is here introduced as the (central) triangle H_AH_BH_C. (Antreas Hatzipolakis, May 17, 2013)

For X = X(1), the orthocentric triangle H_AH_BH_C is perspective to the anticomplementary triangle, and X(5484) is the perspector. (Peter Moses, May 17, 2013)

Also, H_AH_BH_C is perspective to ABC at X(10). (Randy Hutson, May 18, 2013)

X(5484) is the crosspoint of X(1) and X(8) with respect to the extraversion triangle of X(8). (Randy Hutson, August 26, 2014)

X(5484) lies on these lines: {1, 1330}, {2, 12}, {8, 38}, {10, 1054}, {69, 145}, {519, 2891}, {1469, 3869}, {1626, 4189}, {2345, 9597}, {3622, 5712}, {3662, 4327}, {5263, 7354}

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 H_A be the orthocenter of triangle PBC, and define H_B and H_C cyclically. The Euler lines of the triangles AH_BH_C, BH_CH_A, CH_AH_B 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}, {115, 5503}, {262, 538}, {459, 5523}, {525, 5466}, {598, 1992}, {631, 1153}, {1327, 5861}, {1328, 5860}, {2799, 9180}, {2996, 7841}, {3090, 7608}, {3524, 7610}, {3533, 7619}, {3590, 7389}, {3591, 7388}, {3855, 7775}, {5032, 5395}, {8587, 8591}, {8781, 9166}

X(5485) = isogonal conjugate of X(1384)
X(5485) = isotomic conjugate of X(1992)
X(5485) = pole wrt polar circle of trilinear polar of X(4232)
X(5485) = X(48)-isoconjugate (polar conjugate) of X(4232)

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}, {66, 6467}, {67, 1899}, {68, 8681}, {69, 3266}, {71, 4062}, {72, 9004}, {74, 6776}, {125, 5505}, {140, 8548}, {141, 6391}, {182, 5504}, {184, 1177}, {193, 1176}, {248, 5063}, {265, 1352}, {511, 4846}, {523, 2549}, {879, 8675}, {1173, 3542}, {1503, 3426}, {1992, 7493}, {2435, 9007}, {3521, 9019}, {3531, 5480}, {9145, 9516}

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}, {3620, 5488}

X(5487) = isotomic conjugate of anticomplement of X(34540)

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}, {3620, 5487}

X(5488) = isotomic conjugate of anticomplement of X(34541)

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}, {140, 5664}, {520, 5562}, {669, 2353}, {684, 6334}, {690, 9409}, {826, 3574}, {868, 6070}, {2409, 5502}, {3265, 3926}

X(5489) = crossdifference of every pair of pints on line X(23)X(232)

X(5490) = KIRIKAMI-EULER IMAGE OF X(485)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(S + a²)

X(5490) lies on these lines: {2, 493}, {4, 488}, {10, 5391}, {69, 485}, {76, 5590}, {83, 3069}, {98, 637}, {99, 6568}, {141, 5491}, {486, 641}, {487, 7612}, {491, 3316}, {1131, 1270}, {1132, 3593}, {1271, 3590}, {6504, 8223}

X(5490) = pole wrt polar circle of trilinear polar of X(5200)
X(5490) = X(48)-isoconjugate (polar conjugate) of X(5200)

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}, {76, 5591}, {83, 3068}, {98, 638}, {99, 6569}, {141, 5490}, {485, 642}, {488, 7612}, {492, 3317}, {1131, 3595}, {1132, 1271}, {1270, 3591}, {6504, 8222}

X(5490) = pole wrt polar circle of trilinear polar of X(5200)
X(5490) = X(48)-isoconjugate (polar conjugate) of X(5200)

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 H_AH_BH_C 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 L_A, L_B, L_C, where L_A is the polar of A with respect to the circle BCI, and L_B and L_C 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}, {405, 7986}, {500, 3743}, {774, 6147}, {1725, 3649}, {1772, 3614}, {1962, 5453}, {3670, 9955}, {3724, 6097}

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}, {2, 9589}, {3, 551}, {4, 9}, {8, 5059}, {20, 519}, {30, 4669}, {55, 3671}, {56, 4342}, {65, 4314}, {140, 946}, {144, 4882}, {165, 962}, {355, 5073}, {376, 7982}, {382, 3654}, {390, 3339}, {411, 5537}, {484, 1210}, {497, 5128}, {515, 1657}, {517, 550}, {527, 3913}, {553, 3303}, {595, 9441}, {758, 7957}, {1656, 3817}, {1697, 3474}, {1698, 5068}, {1699, 3634}, {1770, 5270}, {1788, 9580}, {1836, 3947}, {2093, 4294}, {2951, 5850}, {3057, 4315}, {3091, 3828}, {3146, 3679}, {3295, 5542}, {3361, 9785}, {3428, 5267}, {3516, 8193}, {3517, 9911}, {3524, 9624}, {3529, 5881}, {3533, 8227}, {3543, 4745}, {3555, 5918}, {3600, 7320}, {3626, 5691}, {3635, 5731}, {3636, 7987}, {3663, 5255}, {3746, 7411}, {3811, 7994}, {3854, 9780}, {3858, 9956}, {3878, 9858}, {3931, 4349}, {3962, 6154}, {4015, 5927}, {4067, 6001}, {4192, 9569}, {4229, 4658}, {4292, 5119}, {4304, 5903}, {4311, 5697}, {4356, 5711}, {4525, 5693}, {4848, 5183}, {4866, 6172}, {5248, 5584}, {5325, 9710}, {5904, 9961}, {6766, 9841}, {7580, 8715}

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(5493) = complement of X(9589)

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 A_B be the reflection of A′ in line BB′, and define B_C and C_A cyclically. Let A_C be the reflection of A′ in line CC′, and define B_A and C_B cyclically. Let L be the Euler line of ABC, let L_A be the Euler line of AA_BA_C, and define L_B and L_C cyclically. Let M_A be the reflection of L_A in AA′, and define M_B and M_C cyclically. The lines M_A, M_B, M_C concur in X(5494). Moreover, the four Euler lines L, L_A, L_B, L_C 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 L_A be the line through A′ perpendicular to line AA′, and define L_B add L_C cyclically. Let

U_A = reflection of L_A in AA′
U_B = reflection of L_A in BB′
U_C = reflection of L_A in CC′

V_A = reflection of L_B in AA′
V_B = reflection of L_B in BB′
V_C = reflection of L_B in CC′

W_A = reflection of L_C in AA′
W_B = reflection of L_C in BB′
W_C = reflection of L_C in CC′

T_A = triangle formed by the lines in U_A, U_B, U_C
T_B = triangle formed by the lines in V_A, V_B, V_C
T_C = triangle formed by the lines in W_A, W_B, W_C

O_A = circumcenter of T_A, O_B = circumcenter of T_A, O_C = circumcenter of T_A, O = X(3) = circumcenter of ABC. The points O, O_A, O_B, O_C 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: {1, 30}, {3, 143}, {511, 6097}, {6102, 7416}

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 L_A be the line through A′ perpendicular to line AA′, and define L_B add L_C cyclically. Using the notation at X(5495), let M_AB be the line parallel to U_B through B′, and let M_AC be the line parallel to U_C through C′. Let A″ = M_AB∩M_AC, and define B″ and C″ cyclically. Let O_A = circumcenter of A″B′C′, and define O_B and O_C cyclically. Then the points X(1), O_A, O_B, O_C 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: {1, 21}, {3191, 3678}, {3682, 3841}, {3724, 5903}, {4511, 4647}, {6757, 7073}

X(5497) = 5th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics a[a^6 - a^5(b + c) - a^4(b + c)^2 + a^3(2b^3 + b^2c + bc^2 + 2c^3) - a^2(b^4 - b^3c - 3b^2c^2 - bc^3 + c^4) - a(b^5 + b^4c + bc^4 + c^5) + (b^2 - c^2)^2(b^2 + c^2)] : :

Let ABC be a triangle and let A′B′C′ be the cevian triangle of the incenter, X(1). The circles O_A, O_B, O_C 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: {1, 149}, {37, 101}, {1283, 2292}, {1331, 1780}, {3326, 9629}

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 N_A be the nine-point center of the triangle BCO, where O = X(3), and define N_B and N_C cyclically. The nine-point center of the triangle N_AN_BN_C is X(5498), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)

X(5498) lies on this line: {2,3}

X(5499) = 7th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics (b^2+4*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(2*b^4+2*c^4+3*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

Let I_A be the A-excenter of a triangle ABC and let N_A be the nine-point center of I_ABC. Define N_B and N_C cyclically. The circumcenter of N_AN_BN_C is X(5499), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)

Let A′B′C′ be the Feuerbach triangle, and let A″ be the reflection of X(11) in line B′C′; define B″ and C″ cyclically. Then A″, B″, C″ are collinear, and their line, X(12)X(79) is here named the Feuerbach line. X(5499) is the point of intersection of the Feuerbach line and the Euler line. (Randy Hutson, August 26, 2014)

X(5499) lies on these lines: {2, 3}, {10, 2771}, {11, 5441}, {12, 79}, {119, 3652}, {495, 3649}, {758, 5690}, {1329, 3647}, {1385, 1484}, {1698, 7701}, {1749, 5445}, {1834, 5453}, {3579, 3822}, {3884, 6701}, {9943, 9956}

X(5500) = 8th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics 2*a^22-15*(b^2+c^2)*a^20+6*(8*b^4+13*b^2*c^2+8*c^4)*a^18-(b^2+c^2)*(81*b^4+71*b^2*c^2+81*c^4)*a^16+(b^4+c^4)*(64*b^4+111*b^2*c^2+64*c^4)*a^14+(b^2+c^2)*(14*b^8+14*c^8+3*(5*b^4+7*b^2*c^2+5*c^4)*b^2*c^2)*a^12-(84*b^12+84*c^12+(67*b^8+67*c^8+8*(7*b^4+6*b^2*c^2+7*c^4)*b^2*c^2)*b^2*c^2)*a^10+(b^2+c^2)*(82*b^12+82*c^12-(105*b^8+105*c^8-(74*b^4-93*b^2*c^2+74*c^4)*b^2*c^2)*b^2*c^2)*a^8-(b^2-c^2)^2*(34*b^12+34*c^12+(11*b^8+11*c^8-5*(6*b^4+7*b^2*c^2+6*c^4)*b^2*c^2)*b^2*c^2)*a^6+(b^4-c^4)*(b^2-c^2)^3*(b^8+c^8-(3*b^4+19*b^2*c^2+3*c^4)*b^2*c^2)*a^4+(b^2-c^2)^6*(b^4+c^4)*(4*b^4+5*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^8*(b^2+c^2)*(b^4+c^4) : : (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 N_A be the nine-point center of NB′C′, and define N_B and N_C cyclically. The nine-point center of N_AN_BN_C is X(5500), which lies on the Euler line of ABC. (Antreas Hatzipolakis, May 30, 2013)

X(5500) lies on this line: (2,3}

X(5500) = reflection of X(5) in X(10286)

X(5501) = 9th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics 2*a^16-9*(b^2+c^2)*a^14+(13*b^4+18*b^2*c^2+13*c^4)*a^12+(b^2+c^2)*(b^4+c^4)*a^10-(25*b^8+25*c^8+2*b^2*c^2*(5*b^4+4*b^2*c^2+5*c^4))*a^8+(b^2+c^2)*(33*b^8+33*c^8-b^2*c^2*(64*b^4-53*b^2*c^2+64*c^4))*a^6-(b^2-c^2)^2*(21*b^8+21*c^8-5*b^2*c^2*(4*b^4+5*b^2*c^2+4*c^4))*a^4+(b^4-c^4)*(b^2-c^2)^3*(7*b^4-20*b^2*c^2+7*c^4)*a^2-(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^6 : :

As a point on the Euler line, X(5501) has Shinagawa coefficients (18E² + 128EF + 192F² + 64S², -54E² - 128EF - 64F² + 320S²).

Let N be a the nine-point center of triangle ABC. Let N_A be the nine-point center of NBC, and define N_B and N_C cyclically. The circumcenter of N_AN_BN_C 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: X(5501) lies on these lines: {2, 3}, {137, 8254}, {1154, 31879}, {6150, 23280}, {6592, 31376}, {10095, 20327}, {13372, 23281}, {13856, 15425}, {14140, 15307}, {20414, 32551}, {21230, 24573}, {24385, 36842}, {25150, 34598}, {25340, 32423}, {34597, 34768}

X(5501) = midpoint of X(i) and X(j) for these {i, j}: {3, 20120}, {4, 14142}, {5, 10285}, {10095, 20327}, {10205, 28237}, {20030, 36837}, {20414, 32551}
X(5501) = reflection of X(i) in X(j) for these (i, j): (2, 25403), (3, 15327), (4, 25404), (5, 15957), (546, 19940), (10126, 3628), (10289, 13469), (13856, 15425)
X(5501) = complement of X(10205)
X(5501) = anticomplement of the anticomplement of X(12056)
X(5501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (3628, 27090, 140), (10289, 13469, 547)

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 L_A be the reflection of L in line BC, and define L_B and L_C 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 D_A be the reflection of D in line BC, and define D_B and D_C cyclically. Let H_A = L_B∩D_C, and define H_B and H_C cyclically. Let M_A = L_C∩D_B, and define M_B and M_C cyclically. The triangles H_AH_BH_C and M_AM_BM_C are perspective, and their perspector is X(5502). (Antreas Hatzipolakis, June 3, 2013)

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 1/(4a⁴ + b⁴ + c⁴ - 4b²c² - a²b² - a²c²) (Seichii Kirikami, June 2, 2013)

Let P be a point in the plane of triangle ABC. Let H_A be the orthocenter of triangle PBC, and define H_B and H_C cyclically. Let O_A be the circle of the points A, H_B, H_C, and define O_B and O_C cyclically. The circles O_A, O_B, O_C 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/(2 a^4 q r(p+q) (p+r)+a^2 p (c^2 q (p+q-3 r) (p+r)+b^2 (p+q) r (p-3 q+r))-p (b^4 (p+q) (p-q-r) r-c^4 q (p+r) (-p+q+r)+b^2 c^2 (p^2 (q+r)+2 q r (q+r)+p (q^2+r^2)))) .

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(5503) = isotomic conjugate of X(22329)

X(5504) = KIRIKAMI CONCURRENT CIRCLES IMAGE OF X(3)

Trilinears (cos A)/(1 + cos 2B + cos 2C) : :
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(b² + c² - a²)/[a⁴(b² + c²) + (b² - c²)²(b² + c²) - 2a²(b⁴ - b²c² + c⁴)] (Seichii Kirikami, June 2, 2013)

See X(5503) for the definition of Kirikami concurrent circles image Q of a point P and X(5509) for an occurrence of X(5504) as a point of concurrence given by the Hatzipolakis-Moses Theorem.

If P = X(3), then Q = X(5504). (Seichii Kirikami, June 2, 2013)

Let A′B′C′ be the tangential triangle. Let L_A be the line through A′ parallel to the Euler line, and define L_B and L_C cyclically. Let R_A be the reflection of L_A in BC, and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(5504); see X(399). (Randy Hutson, August 17, 2014)

Continuing, let A″ be the reflection of A′ in line BC, and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5504). Morevoer, X(5504) is the antigonal image of X(68), the trilinear pole of line X(577)X(647), and the X(92)-isoconjugate of X(3003). (Randy Hutson, August 17, 2014)

X(5504) lies on the Jerabek hyperbola and these lines: {3,974}, {4,110}, {6,1511}, {20,3047}, {49,3521}, {64,155}, {66,542}, {67,3564}, {68,125}, {70,3448}, {74,323}, {182,5486}, {184,4846}, {265,2072}, {290,1236}, {399,3167}, {511,1177}, {1069,3024}, {1986,1993}, {2850,3657}, {3028,3157}, {3431,5012}

X(5504) = reflection of X(i) in X(j) for these (i,j): (110,1147), (68,125), (2931, 1511)
X(5504) = isogonal conjugate of X(403)

X(5505) = KIRIKAMI CONCURRENT CIRCLES IMAGE OF X(6)

Barycentrics 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 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 N_A be the nine-point center of IBC, and define N_B and N_C cyclically. The triangle N_AN_BN_C 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(5506) = X(1173)-of-excentral-triangle

X(5507) = 5th HATZIPOLAKIS-YIU POINT

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

Let O_A be the circle with center A and radius R, the circumradius of triangle ABC. Let B_A be the point where O_A meets line AB nearest to B. Define C_B and A_C cyclically. Let C_A be the point where O_A meets line AC nearest to C. Define A_B and B_C cyclically. X(5507) is the radical center of the circles AB_AC_A, BC_BA_B, CA_CB_C. If "nearest to" is changed to "farthest from" in the construction, the resulting point is X(600). See also X(600). (Peter Moses, June 19, 2013)

If you have The Geometer's Sketchpad, you can view X(5507).

X(5507) lies on this line: {600, 4640}

X(5508) = KIRIKAMI CONCURRENT CIRCLES IMAGE (2nd KIND) OF X(31)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[a⁶ - a⁴(b² + c²) + a³(b³ + c³) - a(b⁵ + c⁵) + b³c³] (Seichii Kirikami, July 2, 2013)

Let P be a point in the plane of triangle ABC. Let H_A be the orthocenter of triangle PBC, and define H_B and H_C cyclically. The circumcircles of H_ABC, H_BCA, H_CAB 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)

The Kirikami six circles image of P is simply the center of the rectangular hyperbola {{A, B, C, X(4), P}}.

Let P be a point in the plane of triangle ABC. Let H_A be the orthocenter of triangle PBC, and define H_B and H_C cyclically. The nine-point circles of the six triangles H_ABC, H_BCA, H_CAB, AH_BH_C, BH_CH_A, CH_AH_B 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 BCH_A, CAH_B, ABH_C). Also, Q(P) is the center of the rectangular circumhyperbola passing through P, and Q(P) lies on the cevian circle of P. (Randy Hutson, July 15, 2013)

The Kirikami six-circles-image of P is also the QA-P2 center (Euler-Poncelet Point) of the quadrangle ABCP; see Encyclopedia of Quadri-Figures.

If you have The Geometer's Sketchpad, you can view X(5509).

The Kirikami six circles configuration led to a conjecture by Antreas Hatzipolakis (July 5, 2013), proved by Peter Moses, and stated here as the Hatzipolakis-Moses Theorem: Suppose that P and P* are an isogonal conjugate pair of points in the plane of triangle ABC. Let H_A be the orthocenter of triangle PBC, and define H_B and H_C cyclically. Let H'_A be the orthocenter of triangle P*BC, and define H'_B and H'_C cyclically. Then circumcircles of H_AH_BH_C 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).

See Hyacinthos #21992.

Let P′ be the isogonal conjugate of P. Then the Kirikami-six-circles image of P is the orthopole of line X(3)P′, which is also the crosssum of the circumcircle intercepts of line X(3)P′. (Randy Hutson, March 29, 2020)

X(5509) lies on these lines: {2,185}, {31,5508}, {115,3271}

X(5509) = crosssum of circumcircle intercepts of line X(3)X(75)
X(5509) = orthopole of line X(3)X(75)
X(5509) = center of hyperbola {{A,B,C,X(4),X(31)}}

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)

Let A′B′C′ be the orthic triangle. Let L_A be the Nagel line of AB′C′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5510). Also, X(5510) = X(3)-of-A″B″C″. See X(8054). (Randy Hutson, August 12, 2015)

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(5510) = midpoint of X(4) and X(106)
X(5510) = reflection of X(121) in X(5)
X(5510) = complement of X(1293)
X(5510) = crosssum of circumcircle intercepts of line X(3)X(519)
X(5510) = orthopole of line X(3)X(519)
X(5510) = center of hyperbola {{A,B,C,X(4),X(106)}}
X(5510) = X(106)-of-Euler-triangle
X(5510) = polar-circle-inverse of X(32704)

X(5511) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(105)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)²(a² + b² + c² - 2ab - 2ac)(a³ - a²b - a²c + ab² + ac² - b³ + b²c + bc² - c³) (Peter Moses, July 10, 2013)

Kirikami six-circles-images are introduced at X(5509).

X(5511) is the homothetic center of the cyclic quadrilateral ABCX(105) and the congruent quadrilateral formed by the orthocenters of vertices taken 3 at a time; also, X(5511) is the anticenter of ABCX(105) (Randy Hutson, July 15, 2013)

X(5511) = X(105)-of-Euler-triangle (Randy Hutson, August 17, 2014)

X(5511) = reflection of X(120) in X(5)
X(5511) = midpoint of X(4) and X(105)
X(5511) = complement of X(1292)

X(5511) lies on these lines: {2, 1292}, {4, 105}, {5, 120}, {11, 1111}, {12, 3021}, {113, 2836}, {114, 2795}, {115, 2788}, {116, 2820}, {117, 2835}, {118, 946}, {119, 381}, {124, 2814}, {125, 2775}, {132, 2838}, {133, 2833}, {1596, 2834}, {2051, 3034}, {2886, 3039}, {3309, 4904}

X(5511) = crosssum of circumcircle intercepts of line X(3)X(518)
X(5511) = orthopole of line X(3)X(518)
X(5511) = center of hyperbola {{A,B,C,X(4),X(105)}}

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)

Let A′B′C′ be the orthic triangle. Let La be line X(2)X(6) of triangle AB′C′, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5512), which is X(3)-of-A″B″C″. (Randy Hutson, July 31 2018)

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(5512) = X(111)-of-Euler-triangle
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) = X(14074)-of-orthic-triangle if ABC is acute
X(5512) = crosssum of circumcircle intercepts of line X(3)X(524)
X(5512) = orthopole of line X(3)X(524)
X(5512) = center of hyperbola {{A,B,C,X(4),X(111)}}

X(5513) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(190)

Barycentrics (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).

Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5513) is the inverse-in-O* of X(101). (Randy Hutson, August 17, 2014)

X(5513) lies on the nine-point circle, the Yff contact circle, and 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(5513) = complement of X(675)
X(5513) = X(2)-Ceva conjugate of X(3011)
X(5513) = crosssum of circumcircle intercepts of line X(3)X(649)
X(5513) = orthopole of line X(3)X(649)
X(5513) = center of hyperbola {{A,B,C,X(4),X(190)}}
X(5513) = perspector of the circumconic centered at X(3011)

X(5514) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(40)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)²(b + c - a)²(b³ + c³ - a³ - a²b - a²c + ab² + ac² + 2abc - b²c - bc²) (Peter Moses, July 10, 2013)

Kirikami six-circles-images are introduced at X(5509).

X(5514) is the point of intersection, other than X(11), of the nine-point circle and the Mandart circle. (Randy Hutson, July 15, 2013)

X(5514) is the center of the hyperbola {A,B,C,X(4),X(40)}, and X(5514) = X(972)-of-Euler-triangle. (Randy Hutson, August 17, 2014)

X(5514) = midpoint of X(4) and X(972)
X(5514) = complement of X(934)

X(5514) lies on these lines: {2, 934}, {4, 972}, {9, 119}, {10, 118}, {11, 1146}, {12, 208}, {117, 374}, {120, 1329}, {3814, 5199}

X(5514) = Spieker-radical-circle-inverse of X(34457)
X(5514) = crosssum of the circumcircle intercepts of line X(3)X(9)
X(5514) = orthopole of line X(3)X(9)
X(5514) = point of concurrence of cevian circles of vertices of anticevian triangle of X(8)

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(5515) = X(2)-Ceva conjugate of X(6590)
X(5515) = polar-circle-inverse of X(32691)
X(5515) = crosssum of circumcircle intercepts of line X(3)X(31)
X(5515) = orthopole of line X(3)X(31)
X(5515) = center of hyperbola {{A,B,C,X(4),X(75)}}

X(5516) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(145)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)(b + c - 3a)(b - c)²(b² + c² + ab + ac - 4bc) (Peter Moses, July 10, 2013)

Kirikami six-circles-images are introduced at X(5509).

X(5516) lies on these lines: {120, 5121}, {121, 519}, {1647, 3259}, {3667, 3756}

X(5516) = complement of X(6079)
X(5516) = crosspoint of X(519) and X(3667)
X(5516) = crosssum of X(106) and X(1293)
X(5516) = orthopole of line X(3)X(106)
X(5516) = center of hyperbola {{A,B,C,X(4),X(145)}}

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(5517) = crosssum of circumcircle intercepts of line X(3)X(37)
X(5517) = orthopole of line X(3)X(37)
X(5517) = center of hyperbola {{A,B,C,X(4),X(81)}}

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) lies on these lines: {2, 932}, {12, 85}, {121, 3822}

X(5518) = complement of X(932)

X(5518) = crosssum of circumcircle intercepts of line X(3)X(238)
X(5518) = orthopole of line X(3)X(238)
X(5518) = center of hyperbola {{A,B,C,X(4),X(291)}}

X(5519) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(218)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)²(a² + b² + c² - 2ab - 2ac)(b² + c² - ab - ac)(2a² + b² + c² - ab - ac - 2bc) (Peter Moses, July 10, 2013)

Kirikami six-circles-images are introduced at X(5509).

X(5519) lies on these lines: {120, 518}, {1566, 3323}, {3309, 4904}

X(5519) = complement of X(6078)
X(5519) = crosspoint of X(518) and X(3309)
X(5519) = crosssum of X(105) and X(1292)
X(5519) = orthopole of line X(3)X(105)
X(5519) = center of hyperbola {{A,B,C,X(4),X(218)}}

X(5520) = KIRIKAMI SIX-CIRCLES-IMAGE OF X(267)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)²(a³ + b³ + c³ - a²b - a²c - ab² - ac² - abc + b²c + bc²)(a⁴ - b⁴ - c⁴ + a²bc - ab²c - abc² + 2b²c²) (Peter Moses, July 10, 2013)

Kirikami six-circles-images are introduced at X(5509).

X(5520) is the touchpoint, other than X(11), of the line through X(867) tangent to the nine-point circle. Also, X(5520) is the reflection of X(11) in the Euler line. (Randy Hutson, July 15, 2013)

Let O* be the inverter (defined at X(5577)) of the circumcircle and nine-point circle. X(5520) is the inverse-in-O* of X(2752). Also, X(5520) = inverse-in-polar-circle of X(2766). (Randy Hutson, August 17, 2014)

X(5520) 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(5520) = complement of X(1290)
X(5520) = inverse-in-Stevanovic-circle of X(115)
X(5520) = crosssum of circumcircle intercepts of line X(3)X(191)
X(5520) = orthopole of line X(3)X(191)
X(5520) = center of hyperbola {{A,B,C,X(4),X(267)}}

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) 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(5521) = midpoint of X(4) and X(915)
X(5521) = X(2)-Ceva conjugate of X(6591)
X(5521) = crosssum of circumcircle intercepts of line X(3)X(63)
X(5521) = inverse-in-polar-circle of X(100)
X(5521) = center of the hyperbola {A,B,C,X(4),X(19)}
X(5521) = X(915)-of-Euler-triangle
X(5521) = orthopole of line X(3)X(63)

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(5522) = crosssum of circumcircle intercepts of line X(3)X(51)
X(5522) = orthopole of line X(3)X(51)
X(5522) = center of hyperbola {{A,B,C,X(4),X(95)}}

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) = S_BS_C[(pq + pr)S_A - q²S_B - r²S_C].

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},

X(5523) = reflection of X(112) in the orthic axis
X(5523) = isogonal conjugate of complement of X(34163)
X(5523) = anticomplementary-circle-inverse of X(36851)
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)
X(5523) = {5024,5094}
X(5523) = inverse-in-circle-O(PU(4)) of X(111)

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: X(5524) lies on these lines: {1, 2}, {44, 3684}, {100, 896}, {111, 2748}, {171, 4663}, {210, 846}, {238, 3689}, {518, 1054}, {740, 3699}, {984, 3711}, {1051, 3745}, {1155, 1282}, {3667, 4724}, {3740, 3750}, {3956, 4653}, {3994, 4767}, {4009, 4693}, {4551, 5018}, {4557, 5143}

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(5526) = isogonal conjugate of X(34578)
X(5526) = crosspoint of X(4845) and X(10482)
X(5526) = crosssum of X(i) and X(j) for these {i,j}: {1086, 1638}, {1323, 10481}
X(5526) = crossdifference of every pair of points on line X(354)X(513)
X(5526) = trilinear pole of line X(8645)X(22108)
X(5526) = Conway-circle-inverse of X(35892)

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) lies on these lines: {1, 7}, {165, 5011}, {514,4105}, {1053, 2958}, {1308, 5536}, {1699,5074}

X(5527) = reflection of X(5536) in X(1308)
X(5527) = excentral-isogonal conjugate of X(34925)

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(5528) = Bevan-circle-inverse of X(34925)

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(5529) = Conway-circle-inverse of X(35633)

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 I_A be the inverse-in-excircles-radical-circle of the A-excenter, and define I_B and I_C cyclically. The lines A′I_A, B′I_B, C′I_C concur in X(5530). (Randy Hutson, July 18, 2013)

X(5530) is the inverse-in-excircles-radical-circle of X(5529). (Randy Hutson, July 18, 2013)

X(5530) lies on these lines: {1,2}, {5,3931}, {12,3666}, {36,961}, {37,1329}, {46,573}, {65,970}, {171,580}, {181,942}, {226,986}, {388,988}, {429,1785}, {442,1738}, {517,1682}, {908,2292}, {968,2478}, {1686,2362}, {1695,2093}, {1838,1880}, {2051,4424}, {2476,3914}, {2886,4646}, {3596,4078}, {3614,4854}, {3663,3947}, {3743,3814}, {4339,5281}

X(5531) = GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(11)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁵ - 3a⁴(b + c) + a³(2b + c)(b + 2c) + 2a²(b + c)(b² + c² - 3bc) - a(b - c)²(3b² + 3c² + 5bc) + (b + c)(b² - c²)²

Four of the circles that are tangent to two of the sidelines BC, CA, AB pass through X(11), namely, the incircle and 3 others. The centers of those 3 are collinear. (See Barry Wolk's Hyacinthos messages #21431, #21433, etc., January 2013). Let A′B′C′ be the triangle formed by the radical axes of these circles and the corresponding mixtilinear excircle. A′B′C′ is homothetic to the hexyl triangle, and the center of homothety is X(5531). Moreover, X(5531) is the Fuhrmann-triangle-to-excentral triangle similarity image of X(40). Further, in the definition of X(5495), if A′B′C′ is the excentral triangle, then the circumcircles of T_A, T_B, T_C concur in X(5531). Also, X(5531) is the inverse of X(1) in the circumcircle of O_A, O_B, O_C. (Randy Hutson, July 18, 2013)

X(5531) lies on these lines: {1,5}, {3,3711}, {40,2771}, {63,100}, {101,3119}, {104,4866}, {149,1699}, {153,3811}, {214,936}, {484,912}, {515,5538}, {516,3935}, {518,5536}, {528,1750}, {971,3689}, {1145,4882}, {1156,4326}, {1490,2800}, {1709,3158}, {2951,5528}, {3062,3174}, {3817,3957}, {4297,4420}

X(5531) = reflection of X(i) in X(j) for these (i,j): (1768,100), (2951,5528), (5537,3689)

X(5532) = WOLK-FEUERBACH POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)⁴(b + c - a)³

X(5532) is the point of intersection, other than X(11), of the three collinear circles described at X(5531). (Barry Wolk, Hyacinthos #21433, January 18, 2013)

X(5532) lies on these lines: {11,514}, {516,5183}, {1111,3323}, {1146,3022}, {2310,4041}, {3689,5199}, {4081,4163}

X(5533) = INVERSE-IN-INCIRCLE OF X(5)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁴(b + c - a)(b² + c² - 4bc) - 2a³(b⁴ + c⁴ - 3b³c - 3bc³ + 5b²c²) + 2a²(b + c)(b - c)²(b² + c² - bc) + a(b + c)²(b - c)⁴ - (b - c)(b² - c²)³ (César Lozada, August 15, 2013)

X(5533) is the Gibert-Burek-Moses concurrent circles image of X(5534). (Randy Hutson, July 18, 2013)

X(5533) = inverse-in-incircle of X(5), and X(5533) = {X(11),X(1317)}-harmonic conjugate of X(5).

X(5533) lies on these lines: {1,5}, {100,499}, {104,1479}, {149,3086}, {528,3582}, {1145,3813}, {1647,1772}, {1737,2802}, {2829,3583}, {3036,4187}

X(5534) = GIBERT-BUREK-MOSES CONCURRENT CIRCLES IMAGE OF X(5533)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁶ - 2a⁵(b +c) - a⁴(b + c)² + 4a³(b + c)(b² + c²) - a²(b⁴ + c⁴ + 6b²c²) - 2a(b + c)(b² - c²)² + (b + c)²(b² - c²)² (César Lozada, August 15, 2013)

Gibert-Burek-Moses concurrent circles images are introduced at X(5524). In the definition of X(5495), if A′B′C′ is the excentral triangle, then X(5534) is the center of the circumcircle of O_A, O_B, O_C. 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) 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(5537) = reflection of X(i) in X(j) for these (i,j): (36,2077), (3245,40), (5526,2742), (5531,3689), (5536,1155)
X(5537) = circumcircle-inverse of X(165)

X(5537) = Conway-circle-inverse of X(35645)

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 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) 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)

The 1st and 2nd Montesdeoca bisector triangles are inversely similar to the excentral triangle. Let s₁ and s₂ be the similarity mappings. Then there is a unique point X such that s₁(X) = s₂(X), and X = X(5540). See HGT2017 and AdvGeom3769

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(5540) = reflection of X(i) in X(j) for these (i,j): (1,105), (5526,2348)
X(5540) = X(112)-of-excentral-triangle
X(5540) = Stevanovic-circle-inverse of X(34464)
X(5540) = X(1358)-of-tangential-triangle-of-excentral-triangle
X(5540) = excentral isogonal conjugate of X(3309)
X(5540) = trilinear-pole-with-respect-to-excentral-triangle-of-the-line-X(2)X(7)

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 E_A be the Euler line of BCA′. Let L_A be the line through A′ parallel to E_A, and define L_B and L_C cyclically. The lines L_A, L_B, L_C 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(5541) = {X(10),X(149)}-harmonic conjugate of X(37718)

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) lies on these lines: {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(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(5542) = harmonic center of of inner and outer Soddy circles
X(5542) = X(6)-of-incircle-circles-triangle

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   a^2*(a^4 - 4*a^2*b^2 + 3*b^4 - 4*a^2*c^2 - 26*b^2*c^2 + 3*c^4) : :
Barycentrics   a^2*(7*S^2 + 5*SA^2 + SB*SC) : : (Peter Moses, August 30, 2013)
Barycentrics   Sin[A]^2*(2 + 6*(Cot[A] + Cot[B])*(Cot[A] + Cot[C]) - Cot[A]*Cot[w]) : : (Peter Moses, March 3, 2024)
Barycentrics   Sin[A]^2*(5*Cot[A]^2 + 8*Cot[B]*Cot[C] + 7*Cot[A]*(Cot[B] + Cot[C])) : : (Peter Moses, March 5, 2024)
X(5544) = 6 X[2] - X[40912], X[3] + 2 X[52163], X[5646] - 3 X[59777]

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 = S_AS_BS_C and T = S_A + S_B + S_C. (Peter Moses, August 30, 2013)

X(5544) is the only point whose polar conic in the Thomson cubic. (K002) is a circle. (Bernard Gibert, June 22, 2014)

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

The following notes pertaining to minimal sums were contributed by Peter Moses, March 5, 2024.

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 + |A'X|^2 + |BX|^2 + |B'X|^2 + |CX|^2 + |C'X|^2

is minimized by X(5544). More generally, the sum

|AX|^2 + k*|A'X|^2 + |BX|^2 +k*|B'X|^2 + |CX|^2 + k*|C'X|^2

is minimized by the following point:

X = a^2*(a^4*k^2 - 2*a^2*(b^2 + c^2)*k*(1 + k) + (b^4 + c^4)*k*(2 + k) - 2*b^2*c^2*(6 + 6*k + k^2)) : :

or, equivalentley,

X = = Sin[A]^2*((3 + 2*k)*Cot[A]^2 + (1 + k)*(3 + k)*Cot[B]*Cot[C] + (3 + k*(3 + k))*Cot[A]*(Cot[B] + Cot[C])) : :

The point X lies on the Thomson-Gibert-Moses hyperbola, and its minimum value is

m= (-16*k*(1 + k)^2*S^4 + 96*(1 + k)*SA*SB*SC*SW - 16*(2 + k)*(3 + 2*k)*S^2*SW^2)/(144*(1 + k)*SA*SB*SC - 16*(3 + k)*(3 + 2*k)*S^2*SW).

There are 3 values of k for which m = 0; they are the roots of this equation:

16*k*(1 + k)^2*S^4 - 96*(1 + k)*SA*SB*SC*SW + 16*(2 + k)*(3 + 2*k)*S^2*SW^2 = 0.

The corresponding points X are intersections of the following curves: Thomson-Gibert-Moses hyperbola, Kiepert circumhyperbola of the anticomplementary triangle, and the cubics K210 and K511. Although barycentrics for the three points are unamenable, their barycentric product, trilinear product, and barycentric sum are given at X(62175), X(62176), and X(62177).

The point X is the Thomson-isogonal conjugate of 6*a^2*(a^2 - b^2 - c^2) + (5*a^4 - 4*a^2*b^2 - b^4 - 4*a^2*c^2 + 2*b^2*c^2 - c^4)*k : : = k*X[2] - 2*(1 + k)*X[3] (on the Euler line).

The appearance of (k,X(n)) in the following list means that X = X(n):

(-6,55038), (-4,9716), (-3,3167), (-2,110), (-5/3,55157), (-9/7,61771), (-14/9,55156), (-3/2,154), (-4/3,7712), (-6/5,6030), (-1,3), (-6/7,61772), (-3/4,61773), (-2/3,5888), (-1/2,5646), (-1/3,61774), (0,2), (1/2,14924), (1,5544), (2,5643), (3,5644), (4,5645), (6,61775)

X(5544) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2, 1351}, {3, 373}, {5, 5656}, {6, 16187}, {23, 55682}, {25, 6030}, {51, 55584}, {110, 5050}, {125, 5055}, {140, 21970}, {154, 182}, {323, 5645}, {354, 3751}, {381, 11820}, {382, 35253}, {392, 1482}, {394, 5644}, {474, 48921}, {511, 5646}, {567, 12309}, {576, 12045}, {1350, 5943}, {1495, 55697}, {1656, 5654}, {1995, 7712}, {3030, 37502}, {3090, 32605}, {3124, 5024}, {3167, 5651}, {3357, 11479}, {3426, 40280}, {3526, 3527}, {3545, 15431}, {3589, 23326}, {3618, 6391}, {3628, 11432}, {3819, 55722}, {3843, 44300}, {5054, 32269}, {5070, 12316}, {5079, 45303}, {5085, 32237}, {5097, 10219}, {5422, 55038}, {5640, 5888}, {5643, 11482}, {5648, 5972}, {5650, 44456}, {5652, 8371}, {5653, 11637}, {5892, 11472}, {5921, 45298}, {6090, 9716}, {6244, 16058}, {6388, 31489}, {6642, 9920}, {6723, 15131}, {6800, 30734}, {7392, 39884}, {7393, 32205}, {7395, 11465}, {7484, 11451}, {7485, 55648}, {7496, 55639}, {7529, 13339}, {7998, 55724}, {8547, 40670}, {8717, 18535}, {8780, 11003}, {9171, 34291}, {9306, 55711}, {9818, 12041}, {9909, 53094}, {10545, 55678}, {10546, 55156}, {10620, 41670}, {11002, 55580}, {11328, 34099}, {11433, 61545}, {11477, 15082}, {11484, 15805}, {11579, 40917}, {12099, 38396}, {12167, 52290}, {12174, 15022}, {12310, 15040}, {13361, 14826}, {13364, 58764}, {14810, 17810}, {15107, 55643}, {15448, 38064}, {15693, 20192}, {15703, 58891}, {16042, 26864}, {17508, 31860}, {17809, 55709}, {18928, 40330}, {19124, 21313}, {19347, 19360}, {20190, 41424}, {20998, 55165}, {25555, 59767}, {33586, 55616}, {33750, 37910}, {35259, 55157}, {35283, 39899}, {37493, 55857}, {37672, 55714}, {38110, 40132}, {40916, 55610}, {41462, 55602}, {43650, 55692}, {45311, 51941}, {45578, 55579}, {45579, 55577}, {46219, 61644}, {55587, 58470}

X(5544) = midpoint of X(3) and X(3531)
X(5544) = reflection of X(i) in X(j) for these {i,j}: {3531, 52163}, {18489, 5}, {40912, 44833}
X(5544) = complement of X(44833)
X(5544) = Thomson-isogonal conjugate of X(10304)
X(5544)-Dao conjugate of X(44833)
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {373, 22112, 3066}, {3066, 22112, 3}, {5651, 10601, 53091}, {5651, 53091, 3167}, {6090, 15018, 53092}, {6688, 17825, 5020}
X(5544) = pole of line {3524, 5032} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5544) = pole of line {5093, 5544} with respect to the Thomson-Gibert-Moses hyperbola
X(5544) = pole of line {5646, 54439} with respect to the Parry circle

X(5545) = ISOGONAL CONJUGATE OF X(4843)

Trilinears a/[(b + c - a)(b + c + 3a)(b² - c²)] : :

Suppose that P is a point in the plane of triangle ABC. Let A′B′C′ be the anticevian triangle of P and let A″B″C″ be the 1st circumperp triangle. The locus of P for which the lines A′A″, B′B″, C′C″ concur is the union of the line X(1)X(6) and the conic U = {A, B, C, X(66), X(101), X(294), X(651)}; i.e., the isogonal conjugate of the Gergonne line. The conic U has center X5452) and is given by the trilinear equation

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

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

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

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

X(5545) = trilinear pole of the line X(6)X(1412)
X(5545) = Ψ(X(6), X(1412))
X(5544) = Thomson-isogonal conjugate of X(10304)

X(5546) = X(100)X(112)∩X(101)X(110)

Trilinears 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) is the perspector of the anticevian triangle of X(110) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, July 31 2018)

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(5546) = isogonal conjugate of X(7178)
X(5546) = X(2)-Ceva conjugate of-X(34961)
X(5546) = X(19)-isoconjugate of X(17094)

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) is the perspector of the anticevian triangle of X(111) and the unary cofactor triangle of the intangents triangle. (Randy Hutson, July 31 2018)

X(5547) lies on these lines: {8,645}, {42,101}, {65,651}, {210,644}, {666,671}, {1334,3939}, {1783,1824}, {2334,4627}

X(5547) = isogonal conjugate of X(7181)

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) 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(5548) = isogonal conjugate of X(30725)
X(5548) = F(X(44)), where F is the mapping defined at X(5545) (Peter Moses; September 3, 2013)

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)

Trilinears    r + 3 R sin B sin C : :
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
X(5550) = 2 X(1) + 9 X(2)

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 T_A be the closer to A, and define T_B and T_C cyclically. Let D = T_B∩T_C, 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 T_A and the incircle of DEF. Then AA′, BB′, CC′ concur and AA″, BB″, CC″ concur. (Emmanuel JosÃ© 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)}, {361,1/2}, {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(5550) = {X(1),X(2)}-harmonic conjugate of X(9780)

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 G(R/r)

Barycentrics (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(5553) = perspector of ABC and mid-triangle of hexyl triangle and reflection triangle of X(1)

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(5556) = isotomic conjugate of X(32099)
X(5556) = orthocenter of triangle X(1)X(4)X(8) (Randy Hutson, November 22, 2014 )

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) 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(5557) = isogonal conjugate of X(3746)
X(5557) = perspector of ABC and mid-triangle of 2nd circumperp and 1st Conway triangles

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) 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(5558) = isogonal conjugate of X(3303)
X(5558) = isotomic conjugate of X(32087)

X(5559) = GARCIA-FEUERBACH POINT GF(5/3)

Barycentrics 1/[(a + b + c)(b + c - a) - (b² + c² - a²)(5/3)] : :

Garcia-Feuerbach points are defined at X(5550).

Let Pa be the isotomic conjugate of X(1222) with respect to AX1)X(8), and define Pb and Pc cyclically. Then ABC and PaPbPc are perspective at X(5559). (Angel Montesdeoca, September 23, 2018)

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

Garcia-Feuerbach points are defined at X(5550).

Let A′B′C′ be the cevian triangle of X(1) with respect to the incentral triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5561). (Randy Hutson, August 17, 2014)

X(5561) 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(5561) = isogonal conjugate of X(5010)
X(5561) = isotomic conjugate of X(17360)
X(5561) = perspector of the circle centered at X(1) with radius 2r
X(5561) = X(15110)-of-excentral-triangle

X(5562) = REFLECTION OF X(52) IN X(5)

Trilinears        cos²A cos(B - C) : cos²B cos(C - A) : cos²C cos(A - B)
Trilinears        (cos A)(cos 2B + cos 2C) : (cos B)(cos 2C + cos 2A) : (cos C)(cos 2A + cos 2B)
Barycentrics   (cot A)(csc 2B + csc 2C) : (cot B)(csc 2C + csc 2A) : (cot C)(csc 2A + csc 2B)
Barycentrics   (sin 2A)(cos 2B + cos 2C) : (sin 2B)(cos2C + cos 2A) : (sin 2C)(cos 2A + cos 2B)
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(b² + c² - a²)²[a²b² + a²c² - (b² - c²)²]
Barycentrics    tan B tan C - cot B cot C : :

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 P_A be the point in which the line through A′ perpendicular to BC meets BC, and define P_B and P_C cyclically; the points P_A, P_B, P_C are collinear, forming the so-called Simson line of A′. The Simson lines of A′, B′, C′ concur in X(5562). (Dao Thanh Oai, October 2, 2013)

X(5562) = isotomic conjugate of isogonal conjugate of X(418), and A′B′C′ is homothetic to the orthic triangle of ABC from X(2) with ratio -2. (Peter Moses, October 4, 2013)

Let A′B′C′ be the cevian triangle of X(3). Let A″B″C″ be the reflection of A′B′C′ in X(3). Let A*B*C be the tangential triangle, with respect to A′B′C′, of the circumconic of A′B′C′ centered at X(3) (that is, the bicevian conic of X(3) and X(394)). The lines A″B*, B″B*, C″C* concur in X(5562). (Randy Hutson, August 17, 2014)

The following items were contributed by Randy Hutson, August 17, 2014:
X(5562) = X(20)-of-orthic-triangle
X(5562)-of-excentral-triangle = X(20)
X(5562)-of-hexyl-triangle = X(4)
X(5562)-of-intouch-triangle = X(950)
X(5562) is the QA-P5 center (Isotomic Center) of the quadrangle ABCX(4); see Isotomic Center
. X(5562) is the QA-P37 center of quadrangle ABCX(4); see QA-P37. (For certain special cases of quadrangles, such as orthocentric systems, some QA points coincide.)

The tangents at A, B, C to the Euler central cubic (K044) concur in X(5562), which lies on the Euler central cubic. (Randy Hutson, November 22, 2014)

Let L be the line X(2)X(6). Let U = isogonal conjugate of polar conjugate of L, and let V = polar conjugate of isogonal conjugate of L. Then X(5562) = U∩V. (U = X(3)X(49) and V = X(4)X(69).) (Randuy Hutson, February 16, 2015)

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

X(5562) lies on the hyperbola {A,B,C,X(4),X(51)} and these lines: {2,389}, {3,49}, {4,69}, {5,51}, {20,2979}, {26,1495}, {39,3289}, {40,2807}, {99,1298}, {146,2889}, {159,1350}, {195,567}, {216,217}, {255,1364}, {265,3519}, {373,568}, {381,5446}, {399,2918}, {417,2972}, {520,5489}, {542,1205}, {575,1199}, {578,1993}, {631,3819}, {916,1071}, {970,1812}, {1060,1425}, {1062,3270}, {1503,3313}, {2055,3463}, {2072,5449}, {2781,2883}, {2818,3869}, {2888,3153}, {3060,3091}, {3090,3567}, {3564,4173}, {3719,4158}

X(5562) = reflection of X(i) in X(j) for these (i,j): (185,3), (52,5), (3,1216), (1843,1352)
X(5562) = isogonal conjugate of X(8884)
X(5562) = isotomic conjugate of X(8795)
X(5562) = complement of X(5889)
X(5562) = anticomplement of X(389)
X(5662) = X(343)-Ceva conjugate of X(216)
X(5662) = crosspoint of X(3) and X(68)
X(5662) = crosssum of X(4) and X(24)
X(5662) = crossdifference of every pair of points on the line X(421)X(2501)
X(5662) = orthocenter-of-2nd-Euler-triangle
X(5562) = pole wrt polar circle of trilinear polar of X(8794)
X(5562) = X(48)-isoconjugate (polar conjugate) of X(8794)

X(5563) = ISOGONAL CONJUGATE OF X(5559)

Trilinears 3 - 2 cos A : 3 - 2 cos B : 3 - 2 cos C : :
Barycentrics 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(5563) = {X(1),X(36)}-harmonic conjugate of X(35)

X(5564) = ISOTOMIC CONJUGATE OF X(5557)

Trilinears a²(3 + 2 cos A) : b²(3 - 2 cos B) : c²(3 - 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(5564) = anticomplement of X(3723)

X(5565) = OUTER DOMINICAN IMAGE OF X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a⁴b⁴ + a⁴c⁴ - a²b⁶ - a²c⁶ + a²b⁴c² + a²b²c⁴ + 4a²bc(b² + c²)S - 2b⁴c⁴]

Let X be a point in the plane of a triangle ABC. Let A* = AX∩BC, and define B* and C* cyclically. Let O(BA*) be the circle having diameter BA* and O(A*C) the circle having diameter A*C. There are two lines tangent to the circles O(BA*) and O(A*C). Let U_A be the inner one (i.e., closer to A) and V_A the outer. Define U_B and U_C cyclically and V_B and V_C cyclically. Let A′ = V_B∩V_C, 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″ = U_B∩U_C, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in the inner Dominican image of X, denoted by E(X). (Emmanuel JosÃ© GarcÃa, 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 + y²S_B + z²S_C)] and
g(a,b,c,x,y,z) = a²[a²y²z² - x²(2(y + z)(yz)^1/2(- S) + y²S_B + z²S_C)].

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 + y²S_B + z²S_C )] : b²[b²z²x² - y²(2(z + x)(zx)^1/2S*j + z²S_B + x²S_C )] : c²[c²x²y² - z²(2(x + y)(xy)^1/2S*k + x²S_B + y²S_C)],

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(y²S_B + z²S_C - 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: {}

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: {}

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: {}

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: {}

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 A_B be the center of the circle through A and tangent to the B-median at X(2), and define B_C and C_A cyclically. Let A_C be the center of the circle through A and tangent to the C-median at X(2), and define B_A and C_B cyclically. The points A_B, B_A, B_C, C_B, C_A, A_C 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 f_a = 2b² + 2c² - a², and define f_b and f_c cyclically. Then

r = [f_af_bf_c(b²c² + c²a² + a²b²)]^1/2/(144Δ)²

Let X = X(5569). Then angle(A_BXB_A) = angle(B_CXC_B) = angle(C_AXA_C) = Tan^-1[(a² + b² + c²)/(4Δ)]

angle(A_BB_AX) = angle(B_CC_BX) = angle(C_AA_CX) = 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) = midpoint of X(2) and X(8182)
X(5569) = reflection of X(2) in X(1153)
X(5569) = harmonic center of medial-van Lamoen and anticomplementary-van Lamoen circles

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(5570) = inverse-in-de-Longchamps-ellipse of X(3)
X(5570) = {X(2446),X(2447)}-harmonic conjugate of X(3)
X(5570) = X(2072)-of-intouch-triangle. (Randy Hutson, July 18, 2014)

X(5571) = X(1) OF INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics 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, 2nd circumperp, 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(5571) = X(10)-of-intouch-triangle. (Randy Hutson, July 18, 2014)

X(5572) = X(6) OF INVERSE-IN-INCIRCLE TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a(a³b + a³c - 3a²b² - 3a²c² + 3ab³ + 3ac³ - 3ab²c - 3abc² - b⁴ - c⁴ + 2b³c + 2bc³ - 2b²c²)

See X(5571) for the inverse-in-incircle triangle.

X(5572) = X(7) - 3X(354) (Peter Moses, November 9, 2013)

X(5572) = X(141)-of-intouch-triangle. Let A′ be the inverse-in-incircle of the A-excenter, and define B′ and C′ cyclically. Then X(5572) = X(9)-of-A′B′C′. (Randy Hutson, July 18, 2014)

X(5572) lies on these lines: {1,6}, {2,3059}, {7,354}, {55,1445}, {57,4326}, {65,390}, {105,2264}, {142,2886}, {144,3873}, {241,2293}, {480,3870}, {516,942}, {938,2550}, {946,971}, {982,4335}, {1210,3826}, {1376,3174}, {1387,2801}

X(5572) = complement of X(3059)
X(5572) = X(6)-of-inverse-in-incircle triangle

X(5573) = PERSPECTOR OF MEDIAL AND ANDROMEDA TRIANGLES

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

Let A′ be the center of the inverse-in-incircle of the A-excircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Andromeda triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 9, 2013:

Barycentric coordinates for vertices:
A′ = a[a² + 3(b - c)²] : b[3a² + (b - c)²] : c[3a² + (b - c)²]
B′ = a[3b² + (c - a)²] : b[b² + 3(c - a)²] : c[3b² + (c - a)²]
C′ = a[3c² + (a - b)²] : b[3c² + (a - b)²] : c[c² + 3(a - b)²]

The triangle A′B′C′ is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, mixtilinear, 2nd circumperp. Also, A′B′C′ is perspective to the intouch triangle at X(4907).

X(5573) lies on these lines: {1,474}, {2,3677}, {9,982}, {31,57}, {43,3243}, {165,1279}, {223,3660}, {238,3928}, {354,2999}, {748,3929}, {988,5436}, {1054,3749}, {1086,1699}, {1104,3361}, {1191,3339}, {1201,3340}, {1261,3872}, {1420,3924}, {1453,3338}, {1722,3976}, {2276,3247}, {2886,4859}, {3305,4392}, {3306,5269}, {3315,3870}, {3452,4310}, {3756,3772}, {3915,5128}, {3999,4383}, {4003,4423}, {4666,4850}, {4907,5274}

X(5573) = complement of X(5423)

X(5574) = PERSPECTOR OF MEDIAL AND ANTLIA TRIANGLES

Barycentrics a(b + c - a)³(a² + 3b² + 3c² - 6bc)

Let A′ be the center of the inverse-in-A-excircle of the incircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Antlia triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 10, 2013:

Barycentric coordinates for vertices:
A′ = a[a² + 3(b - c)²] : - b[3a² + (b - c)²] : - c[3a² + (b - c)²]
B′ = - a[3b² + (c - a)²] : b[b² + 3(c - a)²] : - c[3b² + (c - a)²]
C′ = - a[3c² + (a - b)²] : - b[3c² + (a - b)²] : c[c² + 3(a - b)²]

The triangle A′B′C′ is perspective to the following triangles, with perspector X(1): ABC, excentral, incentral, mid-arc, 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 - 2b⁴c⁴) - 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)

As a point on the Euler line, X(5576) has Shinagawa coefficients (E + 4F, 3E + 4F).

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(5576) = polar-circle-inverse of X(37932)
X(5576) = center of inverse-in-polar-circle-of-tangential-circle
X(5576) = center of inverse-in-{circumcircle, nine-point circle}-inverter-of-tangential-circle
X(5576) = inverse-in-orthocentroidal-circle of X(26)

X(5577) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(106)

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

The inverter of circles (U,u) and (V,v) is introduced here as the circle (W,w) such that (V,v) is the inverse-in-(W,w) of (U,u), where W is the insimilicenter of (U,u) and (V,v).

Peter Moses (Nov. 12, 2013) found representations for W and w, as follows. The center W of (W,w) is the combo uV + vU; that is, barycentrics for W are given by u(v_A, v_B, v_C) + v(u_A, u_B, u_C), where (u_A, u_B, u_C) are normalized barycentrics for U, and (v_A, v_B, v_C) are normalized barycentrics for V. The radius of (W,w) is w = sqrt[uv(1 - (|UV|/(u + v))²], so that the inverter is real if and only if u + v >= |UV|. Moses also gave properties for the case that (U,u) = (O,R) = circumcircle and (V,v) = (I,r) = incircle, for which the inverter is given by (W,w) = (X(55), (r/(r + R))sqrt(rR + 4R²)). The power of A with respect to (W,w) is

- abc(b + c - a)²/D, where D = 2(a³ + b³ + c² - a²b - a²c - ab² - ac² - b²c - bc²); likewise, (power of B) = - abc(a - b + c)²/D and (power of C) = - abc(a + b - c)²/D.

The alternate inverter of circles (U,u) and (V,v) is introduced here as the circle (W′,w') such that (V,v) is the inverse-in-(W,w) of (U,u), where W′ is the exsimilicenter of (U,u) and (V,v). The center W′ is the combo uV - vU, and the radius w' of W′ is given by sqrt[uv(- 1 + (|UV|/(u - v))²], so that the alternate inverter is real if and only if |u - v| <= |UV|. (Peter Moses, September 3, 2014)

The appearance of (i,j) in the following list means that X(i) is on the circumcircle, X(j) is on the incircle, and each is the inverse-in-(W,w) of the other: (98, 5578), (99, 5579), (100, 3021), (101, 5580), (103, 1364), (105, 11), (106, 5577), (108, 1360), (109, 1362), (840, 3025), (934, 3321), (939, 5582), (972, 3318), (1381, 2447), (1382, 2446), (1477, 1357), (2222, 3322), (2291, 3022), (2384, 5583), (2717, 3326).

The barycentrics for X(5577) are of the form g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(b + c - a)(b - c)²[b² + c² - a² - kbc]², where k is a homogeneous of degree zero and symmetric in (a,b,c); every such point lies on the incircle. (Peter Moses, August 28, 2014)

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

X(5577) lies on the incircle and these lines: {55,106}, {57,1361}, {244,1364}, {354,1317}, {1086,3326}, {1362,4860}, {3025,3271}, {3319,3660}

X(5578) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(98)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)²(a⁵ + a⁴b + a⁴c - a³b² - a³c² - a³bc - a²b³ - a²c³ - ab³c - abc³ - 2ab²c² + b⁴c + bc⁴ - b³c² - b²c³)²

Inverters are dicussed at X(5577).

X(5578) lies on the incircle and these lines: {55,98}, {354,1355}

X(5579) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(99)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a³b + a³c + a²b² + a²c² - 2a²bc - ab²c - abc² + b³c - 2b²c² + bc³)²

Inverters are discussed at X(5577).

X(5579) lies on the incircle and these lines: {11,4357}, {55,99}, {354,1356}, {1357,3664}, {1358,3666}, {1365,3663}

X(5580) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(101)

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

Inverters are discussed at X(5577).

X(5580) lies on the incircle and these lines: {11,142}, {55,101}, {354,1358}, {1357,4860}, {1364,3056}, {1365,4890}

X(5581) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(739)

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

Inverters are discussed at X(5577).

X(5581) lies on the incircle and this line: {55,739}

X(5582) = INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF X(2384)

Barycentrics a^2*(b-c)^2*(a^3-5*(b+c)*a^2+(5*b^2+b*c+5*c^2)*a-b^3-c^3)^2*(-a+b+c) : :

Inverters are discussed at X(5577).

X(5582) lies on the incircle and this line: {55, 2384}

X(5583) = CENTER OF INVERSE-IN-{INCIRCLE, CIRCUMCIRCLE}-INVERTER OF EULER LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(b - c)(4a⁵ + b⁵ + c⁵ - a⁴b - a⁴c - 2a³b² - 2a³c² + 4a²b²c + 4a²bc² - 2ab⁴ - 2ac⁴ + 4ab²c² - 3b⁴c - 3bc⁴ + 2b³c² + 2b²c³)

Inverters are discussed at X(5577).

X(5583) lies on these lines: {55, 523}, {1946, 6362}, {3737, 10383}

X(5584) = PERSPECTOR OF EXTANGENTS AND APUS TRIANGLES

Barycentrics a²(b⁵ + c⁵ - a⁵ + a⁴b + a⁴c + 2a³b² + 2a³c² + 4a³bc - 2a²b³ - 2a²c³ + 2a²b²c + 2a²bc² - ab⁴ - ac⁴ - 4ab³c - 4abc³ - 6ab²c² - 3b⁴c - 3bc⁴ + 2b³c² + 2b²c³)

Let A′ be the insimilicenter of the circumcircle and A-excircle, and define B′ and C′ cyclically. The triangle with vertices A,′ B, C′ is introduced here as the Apus triangle. A′B′C′ is a central triangle of type 2. The following properties were communicated by Peter Moses, November 12, 2013:

Barycentric coordinates for vertices:
A′ = a²(a - b + c)(a + b - c) : b²(b - c - a)(a + b + c) : c²(c - b - a)(a + b + c)
B′ = a²(a - c - b)(a + b + c) : b²(b - c + a)(b + c - a) : c²(c - a - b)(a + b + c)
C′ = a²(a - b - c)(a + b + c) : b²(b - a - c)(a + b + c) : c²(c - a + b)(c + a - b)

The triangle A′B′C′ is perspective to ABC at X(55), the excentral and hexyl triangles at X(3), the incentral triangle at X(56), the tangential triangle at X(198), the Feuerbach triangle at X(4), and the Apollonius triangle at X(573).

The Apus triangle is the extraversion triangle of X(56). (Randy Hutson, July 18, 2014)

Let A′B′C′ be the intouch triangle of the extangents triangle, if ABC is acute. Then A′B′C′ is homothetic to the cevian triangle of X(3) at X(5584). (Randy Hutson, December 2, 2017)

X(5584) lies on these lines: {1,3}, {4,3925}, {6,4300}, {19,1212}, {20,958}, {64,71}, {72,480}, {104,3528}, {201,1854}, {210,1490}, {212,221}, {218,573}, {227,1035}, {380,5120}, {405,516}, {411,1376}, {946,4423}, {954,3671}, {956,4297}, {962,1001}, {1042,1253}, {1151,5415}, {1152,5416}, {1204,3611}, {1350,3779}, {1407,1496}, {1742,5247}, {1753,1859}, {1802,3207}, {1804,3160}, {2266,4258}, {2951,5234}, {2975,3522}, {3146,5260}, {3149,4413}, {5248,5493}

X(5585) = CENTER OF AQUARIUS CONIC

Barycentrics a²(11b² + 11c² - 13a²) : b²(11c² + 11a² - 13b²) : c²(11a² + 11b² - 13c²)

Let A′B′C′ be the tangential triangle, so that A′ is the center of the circle O_A through B and C that is orthogonal to the circumcircle (whence O_A is self-inverse with respect to the circumcircle). Define O_B and O_C cyclically. Let O(A,B) be the circle which is the inverse-in-O_A of O_B; define O(B,C) and O(C,A) cyclically. Let O(A,C) be the circle which is the inverse-in-O_A of O_C; define O(B,A) and O(C,B) cyclically. The centers of these six circles lie on a conic, introduced here as the Aquarius conic, of which X(5585) is the center. The following properties were found by Peter Moses (Nov. 18, 2013).

The centers of the 6 circles are given by the following barycentrics:

- a² : b² : 3c², 3a² : - b² : c² a² : 3b² : - c²;
- a² : 3b² : c², a² : - b² : 3c² 3a² : b² : - c²

The radius of O(A,B) is abc/(-a² + b² + 3c²); the remaining 5 radii are found by cyclical and bicentric modifications. The Aquarius conic has equation

b⁴c⁴x² + c⁴a⁴y² + a⁴b⁴z² + 11a²b²c²(a²yz + b²zx + c²xy) = 0

The major axis of the Aquarius conic is the Brocard axis, and the perspector is X(6). (Randy Hutson, November 30, 2013)

X(5585) lies on these lines: {3,6}, {20,3054}, {439,3619}, {3055,3523}

X(5586) = PERSPECTOR OF AQUILA AND INTOUCH TRIANGLES

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

Let A′ = reflection of the incenter in A, and define B′ and C′ cyclically. The triangle A′B′C′ is introduced here as the Aquila triangle (in TCCT, , p. 173, as T(1,2)). The following properties were found by Peter Moses (Nov. 18, 2013).

A′ = a + 2b + 2c : -a : -a (trilinears)
B′ = -b : b + 2c + 2a : -b
C′ = -c : -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, 2nd circumperp, and mixtilinear. The Aquila triangle is perspective to other triangles with perspectors as listed here: medial, X(1698); anticomplementary, X(10); intouch, X(5586); Euler, X(1699); hexyl, X(1768); tangential 1st circumperp, X(35); tangential 2nd circumperp, X(36); Carnot, X(5587); outer Grebe, X(5588); inner Grebe, (X5589).

X(5586) lies on these lines: {1,376}, {7,10}, {57,191}, {65,3632}, {145,4298}, {388,4114}, {942,4312}, {986,4888}, {1046,4859}, {1317,3340}, {1537,1768}, {1698,3715}, {1788,3982}, {3361,3616}, {3485,4031}, {3600,3635}, {3633,5434}

X(5587) = PERSPECTOR OF AQUILA AND CARNOT TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b⁴ + 2c⁴ - a⁴ + a³b + a³c - a²b² - a²c² - 2a²bc - ab³ - ac³ + ab²c + abc² - 4b²c²

The Aquila triangle is introduced at X(5586). Not only is X(5587) the perspector of the Aquila and Carnot triangles, but also, X(5587) is also the perspector of the Euler triangle and the outer Garcia triangle, defined as follows. Let T_AT_BT_C be the extouch triangle of a triangle ABC, and let L_A be the line perpendicular to line BC at T_A. Of the two points on L_A at distance r from T_A, let A′ be the one farther from A and let A″ be the closer. Define B′, C′ and B″, C″ cyclically. We call A′B′C′ the outer Garcia triangle and A″B″C″ the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014). In subsequent postings, Paul Yiu reports that A′B′C′ is oppositely congruent to ABC at X(10), the Euler lines of the four triangles AB′C′, BC′A′, CA′B′, ABC concur in X(2475), and the circles (BCA′), (CAB′), (ABC′) concur in X(80). Peter Moses reports that

A′ = -a : a + c : a + b, B′ = b + c : -b : b + a, C′ = c + b : c + a : -c

A″ = a² : ca + c² - b² : ab + b² - c², B″ = bc + c² - a² : b² : ab + a² - c² , C″ = bc + b² - a² : ca + a² - b² : c²

Randy Hutson notes that the inner Garcia triangle is the anticomplement of orthic-triangle-of-Fuhrmann-triangle, that X(5587) = X(2)-of-Fuhrmann triangle, and X(5587) = {X(355), X(5)}-harmonic conjugate of X(1).

Let M_A be the midpoint of segment BC and A′ the reflection of X(1) in M_A; define B′ and C′ cyclically. Then A′B′C′ is the outer Garcia triangle. See Emmanuel José García, A Note on Reflections.

Triangle A′B′C′ is perspective to the cevian triangle of the points on the cubic (K366) and to the anticevian triangles of points on the cubic (K345), as well at the following triangles, with perspectors:

medial, X(1)
excentral, extouch, extangents, X(40)
anticomplementary, Fuhrmann, X(8)
circumcircle midard, 2nd circumperp, X(100)
tangential 1st circumperp, X(5587)
tangential 2nd circumperp, X(956)
Euler, X(5587)
outer Grebe, X(5688)
inner Grebe, X(5689)

The appearance of (i,j) is the following list means that (X(i) of A′B′C′) = X(j): (1,8), (2,3679), (3,355), (4,40), (5, 5690), (6,3416), (7,5223), (8,1), (9,2550), (10,10), (11,1145), (20,5691). (Peter Moses, June 21, 2014)

The point A′ is also the orthocenter of BCI_A and cyclically for B′ and C′, where I_A, I_B, I_C are the excenters. (Randy Hutson, October 8, 2019)

X(5587) lies on these lines: {1,5}, {2,515}, {3,1698}, {4,9}, {8,908}, {30,165}, {35,3560}, {46,3585}, {55,3586}, {57,1478}, {63,5080}, {78,5086}, {84,377}, {145,5068}, {149,3895}, {153,3306}, {200,3419}, {210,381}, {235,5090}, {262,730}, {265,2948}, {282,1549}, {376,3828}, {382,3579}, {388,1210}, {404,5450}, {411,5260}, {442,1490}, {498,3601}, {499,1420}, {519,3545}, {547,3655}, {551,5071}, {631,3634}, {912,4654}, {936,1329}, {938,5261}, {942,5290}, {944,1125}, {950,3085}, {956,5231}, {958,3149}, {962,3617}, {997,3814}, {1000,4342}, {1012,1376}, {1071,3812}, {1158,2475}, {1350,3844}, {1352,3751}, {1385,1656}, {1453,5230}, {1479,1697}, {1482,3632}, {1532,2886}, {1537,3036}, {1572,5475}, {1702,3071}, {1703,3070}, {1709,3359}, {1724,3072}, {1750,3925}, {1770,5128}, {1771,1935}, {1785,1857}, {1788,4292}, {1834,2910}, {1836,2093}, {1853,3753}, {2364,5397}, {2782,3097}, {3073,5264}, {3338,5270}, {3416,5480}, {3421,4847}, {3487,3947}, {3583,5119}, {3616,5056}, {3626,3855}, {3633,5072}, {3646,5084}, {3654,3845}, {3656,4677}, {3850,4668}, {3854,4678}, {3872,5176}, {3911,4293}, {3949,4007}, {4295,4848}, {4304,5218}, {5046,5250}, {5087,5289}

X(5587) = midpoint of X(1699) and X(3679)
X(5587) = reflection of X(i) in X(j) for these (i,j): (1699, 381), (3576,2)
X(5587) = crossdifference of every pair of points on the line X(654)X(1459)
X(5587) = homothetic center of hexyl and 4th Euler triangles
X(5587) = centroid of the six touchpoints of the Johnson circles and the outer Johnson triangle

X(5588) = PERSPECTOR OF AQUILA AND OUTER GREBE TRIANGLES

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

The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5588) lies on these lines: {1,6}, {10,1270}, {40,1160}, {1374,1738}, {1698,5590}

X(5589) = PERSPECTOR OF AQUILA AND INNER GREBE TRIANGLES

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

The Aquila triangle is introduced at X(5586). The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5589) lies on these lines: {1,6}, {10,1271}, {40,1161}, {1373,1738}, {1698,5591}

X(5590) = PERSPECTOR OF MEDIAL AND OUTER GREBE TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b² + c² + S

The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5590) = {X(2), X(141)}-harmonic conjugate of X(5591)

X(5590) lies on these lines: {2,6}, {3,5594}, {4,639}, {5,1160}, {8,5604}, {10,3640}, {76,5490}, {626,638}, {631,641}, {640,3090}, {642,3525}, {1162,1165}, {1267,3662}, {1698,5588}, {3535,5413}, {3661,5391}

X(5590) = homothetic center of ABC and cross-triangle of ABC and outer Grebe triangle

X(5591) = PERSPECTOR OF MEDIAL AND INNER GREBE TRIANGLES

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

The outer and inner Grebe triangles are discussed by Darij Grinberg at Math Forum.

X(5591) = {X(2), X(141)}-harmonic conjugate of X(5590)

X(5591) lies on these lines: {2,6}, {3,5595}, {4,640}, {5,1161}, {8,5605}, {10,3641}, {76,5491}, {626,637}, {631,642}, {639,3090}, {641,3525}, {1163,1164}, {1267,3661}, {1698,5589}, {3536,5412}, {3662,5391}

X(5591) = homothetic center of ABC and cross-triangle of ABC and inner Grebe triangle

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(5592) = Yff-contact-isogonal conjugate of X(8)

X(5593) = CENTER OF YIU CONIC OF THE TANGENTIAL TRIANGLE (IF ABC IS ACUTE)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²S_A(S_A - S_B)(S_A - S_C)(g(a,b,c) - g(a,c,b)), where g(a,b,c) = [b²S_BS_B(S² - S_CS_C)(4S⁴ - (S² + S_AS_C)(3S² + S_BS_B - 2S_AS_C))]

The Yiu conic is presented at X(478); it passes through the 6 of the nine touch-points of the sidelines of a triangle and the excircles of the triangle. When the triangle is the tangential, the conic has center X(5593).

Let u(a,b,c) = 4a²b⁴c⁴ and v(a,b,c) = a²(a⁸ + b⁸ + c⁸ - 2a⁶b² - 2a⁶c² + 2a⁴b⁴ + 2a⁴c⁴ - 2a²b⁶ - 2a²c⁶ + 6a²b⁴c² + 6a²b²c⁴ - 4b⁶c² - 4b⁶c² + 6b⁴c⁴).

The Yiu conic of the tangential triangle of a triangle ABC is given by

u(a,b,c)x + u(b,c,a)y + u(c,a,b) z + v(a,b,c)yz + v(b,c,a)zx + v(c,a,b)xy = 0. (Peter Moses, Nov. 19, 2013)

If you have The Geometer's Sketchpad, you can view X(5593).

X(5593) lies on these lines: {4,157}, {184,216}

X(5594) = PERSPECTOR OF ARA AND OUTER GREBE TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²[a⁶ - b⁶ - c⁶ + a⁴b² + a⁴c² - a²b⁴- a²c⁴ - 2a²b²c² + b⁴c² + b²c⁴ + (a² - b² + c²)(a² + b² - c²)S]

Let A′B′C′ be the tangential triangle of triangle ABC. Let A″ be the center of the A′-excircle of A′B′C′, unless this is also the circumcircle of ABC, in which case let A″ be the incenter of A′B′C′. Define B″, C″ cyclically. The triangle A″B″C″ is introduced here as the Ara triangle, which appears in the sketch at X(5593). The vertices of the Ara triangle are given by Peter Moses (Nov. 19, 2013):

A″ = - a²(a² + b² + c²) : b²(a² + b² - c²) : c²(a² - b² + c²)
B″ = a²(b² - c² + a²) : - b²(b² + c² + a²) : c²(b² + c² - a²)
C″ = a²(c² + a² - b²) : b²(c² - a² + b²) : - c²(c² + a² + b²)

The Ara triangle is perspective to triangles as listed here with perspectors: ABC, X(25); anticomplementary, X(22); Euler, X(1598); tangential 1st circumperp, X(197). (Peter Moses, Nov. 19, 2013)

The Ara triangle is homothetic to triangle ABC. If ABC is acute then the Ara triangle is the excentral triangle of the tangential triangle. (Randy Hutson, August 17, 2014)

X(5594) lies on these lines: {6,25} et al

X(5594) = {X(25),X(159)}-harmonic conjugate of X(5595)

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: {6,25} et al

X(5595) = {X(25),X(159)}-harmonic conjugate of X(5594)

X(5596) = PERSPECTOR OF ARIES AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics b⁸ + c⁸ - 3a⁸ +2a⁴b⁴ + 2a⁴c⁴ - 2b⁴c⁴ : :

Let A′B′C′ be the tangential triangle of an acute triangle ABC. Let A″ be the touchpoint of the A-excircle of A′B′C′ and the line B′C′; define B″ and C″ cyclically. The triangle A″B″C″ is introduced here as the Aries triangle, with vertices given by Peter Moses (Nov. 21, 2013).

A″ = a⁴ + b⁴ + c⁴ - 2b²c² : 2b²(c² - b²) : 2c²(b² - c²)
B″ = 2a²(c² - a²) : b⁴ + c⁴ + a⁴ - 2c²a² : 2c²(a² - c²)
C″ = 2a²(b² - a²) : 2b²(a² - b²) : c⁴ + a⁴ + b⁴ - 2a²b²

The Aries triangle is perspective to the tangential triangle, with perspector X(1498).

If ABC is acute, the Aries triangle is the extouch triangle of the tangential triangle. (Randy Hutson, July 18, 2014)

If ABC is acute, then the vertices of the Aries triangle lie on the cubic K075. (César Lozada, October 21, 2015)

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(5596) = isogonal conjugate of X(34427)
X(5596) = isotomic conjugate of isogonal conjugate of X(20993)
X(5596) = complement of X(20079)
X(5596) = anticomplement of X(66)
X(5596) = polar conjugate of isogonal conjugate of X(22135)

X(5597) = PERSPECTOR OF ABC AND 1st AURIGA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁴ - a²(b + c)² + 4aS(rR + 4R²)^1/2

Let U be the inverter of the circumcircle and incircle, as described at X(5577). There are two triangles that circumscribe U and are homothetic to triangle ABC, one of which has A-vertex on the same side of line BC as A. This triangle, A′B′C′, is introduced here as the 1st Auriga triangle, and the other, as the 2nd Auriga triangle. Each is the reflection of the other in the center, X(55), of the inverter. The six points A′, B′, C′, A″, B″, C″ lie on a conic introduced here as the Auriga conic. Let D = (rR + 4R²)^1/2; barycentrics for the six points and conic were found by Peter Moses (Nov. 21, 2013):

A′ = a⁴ - a²(b + c)² - 4(b + c)SD : b⁴ - b²(c + a)² + 4bSD : c⁴ - c²(a + b)² + 4cSD

A″ = a⁴ - a²(b + c)² + 4(b + c)SD : b⁴ - b²(c + a)² - 4bSD : c⁴ - c²(a + b)² - 4cSD

where B′, C′, B″, C″ are determined cyclically.

The Auriga conic is given by {cyclic sum[g(a,b,c)x² + h(a,b,c)yz} = 0, where

g(a,b,c) = bc(b + c - a)(b⁵ + c⁵ + 3a³bc + a²b³ + a²c³ - a²b²c - a²bc² + ab³c + abc³ - 2ab²c² + 3b⁴c + 3bc⁴ - 2b³c² - 2b²c³)

h(a,b,c) = a[a⁷ - 2a⁶(b + c) - a⁵(b² + c² - 4a⁵bc) + 4a⁴(b³ + c³)
- a³(b⁴ + c⁴ + 4b³c + 4bc³ - 8b²c²)
- 2a²(b⁵ + c⁵ - b⁴c - bc⁴ + 5b³c² + 5b²c³)
+ a(b⁶ + c⁶ + b⁴c² + b²c⁴ - 4b³c³)
+ 2(b⁵c² + b²c⁵ - b⁴c³ - b³c⁴)

The two Auriga triangles are perspective with perpsector X(55), which is the center of the Auriga conic.

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

X(5597) lies on these lines: {1,3}, {2,5599}, {8,5600}, {145,5602}

X(5597) = {X(1), X(55)}-harmonic conjugate of X(5598)
X(5597) = exsimilicenter of circumcircles of ABC and 1st Auriga triangle; the insimilicenter is X(11822)

X(5598) = PERSPECTOR OF ABC AND 2nd AURIGA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁴ - a²(b + c)² - 4aS(rR + 4R²)^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: {1,3}, {2,5600}, {8,5599}, {145,5601}

X(5598) = {X(1), X(55)}-harmonic conjugate of X(5597)
X(5598) = insimilicenter of circumcircles of ABC and 2nd Auriga triangle; the exsimilicenter is X(11823)

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 - 4S(b + c)(rR + 4R²)^1/2

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5599) = {X(10), X(55)}-harmonic conjugate of X(5600)

X(5599) lies on these lines: {2,5597}, {8,5598}, {10,55}, {3617,5602}

X(5600) = PERSPECTOR OF MEDIAL AND 2nd AURIGA TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁴ - a²(b + c)² + 4S(b + c)(rR + 4R²)^1/2

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5600) = {X(10), X(55)}-harmonic conjugate of X(5599)

X(5600) lies on these lines: {2,5598}, {8,5597}, {10,55}, {3617,5601}

X(5601) = PERSPECTOR OF ANTICOMPLEMENTARY AND 1st AURIGA TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[a²(a + b + c) - 2S(rR + 4R²)^1/2]

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5601) = {X(8), X(55)}-harmonic conjugate of X(5602)

X(5601) lies on these lines: {{2,5597}, {8,21}, {145,5598}, {3617,5600}

X(5602) = PERSPECTOR OF ANTICOMPLEMENTARY AND 2nd AURIGA TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)[a²(a + b + c) + 2S(rR + 4R²)^1/2]

The 1st and 2nd Auriga triangles are introduced at X(5597).

X(5602) = {X(8), X(55)}-harmonic conjugate of X(5601)

X(5602) lies on these lines: {2,5598}, {8,21}, {145,5597}, {3617,5599}

X(5603) = PERSPECTOR OF EULER AND CAELUM TRIANGLES

Trilinears r + R cos B cos C : :
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
X(5603) = 2 X(1) + X(4)

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 : 2a : 2a B′ = 2b : b - c - a : 2b C′ = 2c : 2c : c - a - b.

Let A′ be the orthocenter of triangle BCX(7), and define B′ and C′ cyclically. Then X(5603) is the centroid of A′B′C′. (Randy Hutson, November 22, 2014)

X(5603) lies on these lines: {1,4}, {2,392}, {3,962}, {5,8}, {7,104}, {10,3090}, {11,2099}, {12,2098}, {20,1385}, {29,945}, {36,3474}, {40,631}, {56,4295}, {65,3086}, {78,5082}, {79,4317}, {84,3296}, {86,4221}, {119,1320}, {140,5550}, {145,355}, {165,3524}, {281,1953}, {329,956}, {376,516}, {381,952}, {495,1532}, {496,938}, {498,5443}, {499,1788}, {519,3545}, {546,1483}, {908,3421}, {912,3873}, {929,953}, {971,5049}, {995,4000}, {997,2550}, {1000,1512}, {1001,1006}, {1060,4318}, {1065,3478}, {1071,5045}, {1158,3338}, {1210,3340}, {1279,3332}, {1312,2102}, {1313,2103}, {1319,1836}, {1420,4292}, {1468,3073}, {1476,5553}, {1698,5067}, {1829,3089}, {1872,4200}, {1902,3088}, {2093,3911}, {2476,5330}, {2646,4294}, {2792,5429}, {2801,3892}, {2829,5434}, {2886,5289}, {2975,3560}, {3057,3085}, {3072,3915}, {3149,3295}, {3242,5480}, {3244,3855}, {3304,3649}, {3306,3359}, {3333,3671}, {3434,4511}, {3436,4861}, {3523,3579}, {3525,3624}, {3529,3636}, {3543,3655}, {3617,5056}, {3621,5068}, {3623,3832}, {3679,5071}, {3820,5328}, {4193,5554}, {5048,5252}, {5119,5218}, {5450,5563}

X(5603) = midpoint of X(1) and X(1699)
X(5603) = relection of X(i) in X(j) for these (i,j): (4,1699), (376,3576), (1699,946), (3576,551)
X(5603) = {X(1),X(4)}-harmonic conjugate of X(944)

X(5604) = PERSPECTOR OF CAELUM AND OUTER GREBE TRIANGLES

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

The Caelum triangle is defined at X(5603).

X(5604) = {X(1), X(3242)}-harmonic conjugate of X(5605)

X(5604) lies on these lines: {1,6}, {8,5590}, {145,1270}, {1160,1482}

X(5605) = PERSPECTOR OF CAELUM AND INNER GREBE TRIANGLES

Barycentrics a(a² + 2b² + 2c² - ab - ac - 2S) : :

The Caelum triangle is defined at X(5603).

X(5605) = {X(1), X(3242)}-harmonic conjugate of X(5604)

X(5605) lies on these lines: {1,6}, {8,5591}, {145,1271}, {1161,1482}

X(5606) = HATZIPOLAKIS CIRCUMCIRCLE POINT

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

Let I be the incenter and A′ the nine-point center of triangle IBC. Define B′ and C′ cyclically. The circles AB′C′, BC′A′, CA′B′ concur in X(5606). (Antreas Hatzipolakis, June 2, 2013: see Concurrent Circumcircles)

Let L be the Euler line of the incentral triangle of ABC, so that L is the line X(500)X(1962). Let L_A be the reflection of L in line BC, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. The lines AA″, BB″, CC″ concur in X(5606). (Randy Hutson, July 18, 2014)

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

X(5606) = isogonal conjugate of X(8702)
X(5606) = anticomplement of X(5952)
X(5606) = cevapoint of X(513) and X(3337)
X(5606) = trilinear pole of line X(6)X(3336)
X(5606) = Ψ(X(6), X(3336))

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

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(b² - c²)[3^1/2(a⁸ - 3a⁶b² - 3a⁶c² + 3a⁴b⁴ + 3a⁴c⁴ - a⁴b²c² - a²b⁶ - a²c⁶ + 2b⁴c⁴) + 2S(a⁶ - 2b⁶ - 2c⁶ - 4a⁴b² - 4a⁴c² + 5a²b⁴ + 5a²c⁴ - 3a²b²c² + 2b⁴c² + 2b²c⁴)]

A construction of the 1st Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617). (Dao Thanh Oai and Peter Moses, Nov., 2013)

X(5607) lies on this line: {526, 5608}

X(5608) = CENTER OF 2nd POHOATA-DAO-MOSES CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(b² - c²)[3^1/2(a⁸ - 3a⁶b² - 3a⁶c² + 3a⁴b⁴ + 3a⁴c⁴ - a⁴b²c² - a²b⁶ - a²c⁶ + 2b⁴c⁴) - 2S(a⁶ - 2b⁶ - 2c⁶ - 4a⁴b² - 4a⁴c² + 5a²b⁴ + 5a²c⁴ - 3a²b²c² + 2b⁴c² + 2b²c⁴)]

A construction of the 2nd Pohoata-Dao-Moses circle is given at X(399), in connection with the work of Cosmin Pohoata on the Parry reflection point. The circle passes through the points X(13), X(16), X(110), X(399), X(1338), X(2381), X(5610), X(5611), X(5612), X(5613), and the 2nd Pohoata-Dao-Moses circle pass through X(14), X(15), X(110), X(399), X(1337), X(2380), X(5614), X(5615), X(5616), X(5617). (Dao Thanh Oai and Peter Moses, Nov., 2013)

X(5608) lies on this line: {526, 5607}

X(5609) = RADICAL TRACE OF 1st AND 2nd POHOATA-DAO-MOSES CIRCLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(2a⁸ + b⁸ + c⁸ - 7a⁶b² - 7a⁶c² + 9a⁴b⁴ + 9a⁴c⁴ + 4a⁴b²c² - 5a²b⁶ - 5a²c⁶ + 3b⁶c² + 3b²c⁶ - 8b⁴c⁴)
> X(5609) = 5X(3) - 3X(74)

See X(5607) and X(5608). (Dao Thanh Oai and Peter Moses, Nov., 2013)

Let A′B′C′ be the Euler triangle. Let L be the line through A′ parallel to the Euler line, and define M and N cyclically. Let L′ be the reflection of L in line BC, and define M′ and N′ cyclically. The lines L′, M′, N′ concur in X(5609); c.f., X(113), X(399), and X(1511). Let N_A be the reflection of X(5) in the perpendicular bisector of BC, and define N_B and N_C cyclically. Then X(5609) = X(23)-of-N_AN_BN_C. (Randy Hutson, July 18, 2014)

X(5609) lies on these lines: {3,74}, {5,542}, {23,1154}, {30,3292}, {113,137}, {125,3628}, {526,5607}

X(5609) = circumcircle- inverse of X(32609)

X(5610) = INTERSECTION OF LINES X(13)X(531) AND X(15)X(110)

Barycentrics a^2*(-2*sqrt(3)*(a^8-4*(b^2+c^2)*a^6+3*(2*b^4-b^2*c^2+2*c^4)*a^4-(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4)*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)^2)*S+a^10-5*(b^2+c^2)*a^8+(10*b^4+31*b^2*c^2+10*c^4)*a^6-(b^2+c^2)*(11*b^4+16*b^2*c^2+11*c^4)*a^4+(7*b^8+7*c^8+10*b^2*c^2*(b^4+c^4))*a^2-(b^4-c^4)*(b^2-c^2)*(2*b^4+7*b^2*c^2+2*c^4)) : :

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 + 3^1/2SS_A - S_AS_A + S_BS_C) (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 a^2*(-2*S+(-a^2+b^2+c^2)*sqrt(3))*(a^6-3*(b^2+c^2)*a^4+(3*b^4-b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)) : :

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 3a²[S_AS_A + S_BS_C) + 6S_BS_BAS_C - (12)^1/2S³: : (Wimalasiri Perera, December 15, 2013)
Barycentrics 2S³ - 3^1/2[S²S_A + (S_A + S_B + S_C)S_BS_C] (Peter Moses, June 20, 2014)

Lengths of segments: |X(5613)X(13)| = |X(5617)X(14)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier García Capitán), ADGEOM #1256, April 23, 2014)

The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco García Capitán), ADGEOM #1259, April 23, 2014)

The Euler line is tangent to the circle {{X(3), X(5613), X(5617)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

Suppose that P is a point. The P-Fuhrmann triangle is the triangle A″B″C″, where A″ is the reflection in line BC of the A-vertex of the cercumcevian triangle of P, and B″ and C″ are defined cyclically (so that taking P to be X(1), X(4), and X(6) yields the classical Fuhrmann triangle, the single-point triangle X(4), and the 4th Brocard triangle, respectively. The X(16)-Fuhrmann triangle is equilateral, and X(5613) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5617). (Randy Hutson, July 7, 2014)

Let BA′C be the equilateral triangle with side BC and A′ on the side of BC that includes A. Let La be the line through A′ parallel to BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″and C″ cyclically. The Euler lines of B′A″C′, C′B″A′, A′C″B′ concur in X(5613). The acute angle between each pair of these Euler lines is π/6. See Dao Thanh Oai, ADGEOM 4030), and Francisco Javier García Capitán, ADGEOM 4031). See also X(5617) and X(14144).

X(5613) lies on these lines: {2,98}, {3,619}, {4,617}, {5,14}, {13,2782}, {30,5464}, {99,622}, {299,383}, {303,1080}, {381,531}, {395,3564}, {576,3180}, {5055,5460}

X(5613) = midpoint of X(i) and X(j) for these (i,j): (4,617), (299,383)
X(5613) = reflection of X(i) in X(j) for these (i,j): (14,5), (3,619), (5617,114)
X(5613) = inner-Napoleon-to-outer-Napoleon similarity image of X(3)
X(5613) = outer-Napoleon-isogonal conjugate of X(16)

X(5614) = INTERSECTION OF LINES X(14)X(530) AND X(16)X(110)

Barycentrics a^2*(2*sqrt(3)*(a^8-4*(b^2+c^2)*a^6+3*(2*b^4-b^2*c^2+2*c^4)*a^4-(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4)*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)^2)*S+a^10-5*(b^2+c^2)*a^8+(10*b^4+31*b^2*c^2+10*c^4)*a^6-(b^2+c^2)*(11*b^4+16*b^2*c^2+11*c^4)*a^4+(7*b^8+7*c^8+10*b^2*c^2*(b^4+c^4))*a^2-(b^4-c^4)*(b^2-c^2)*(2*b^4+7*b^2*c^2+2*c^4)) : :

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 - 3^1/2SS_A - S_AS_A + S_BS_C) (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 a^2*(2*S+(-a^2+b^2+c^2)*sqrt(3))*(a^6-3*(b^2+c^2)*a^4+(3*b^4-b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2)) : :

X(5616) lies on these lines: {3, 3200}, {5, 13}, {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 3a²[S_AS_A + S_BS_C) + 6S_BS_BAS_C + (12)^1/2S³ : : (Wimalasiri Perera, December 16, 2013)
Barycentrics 2S³ + 3^1/2[S²S_A + (S_A + S_B + S_C)S_BS_C] : : (Peter Moses, June 20, 2014)

Lengths of segments: |X(5617)X(14)| = |X(5613)X(13)| = R, the circumradius of triangle ABC. (Dao Thanh Oai, Francisco Javier García Capitán), ADGEOM #1256, April 23, 2014)

The lines X(5617)X(13) and X(5613)X(14) are parallel to the line X(5)X(39). (Francisco Javier García Capitán), ADGEOM #1259, April 23, 2014)

The Euler line is tangent to the circle {{X(3), X(5617), X(5661)}} at X(3). (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

The X(15)-Fuhrmann triangle, defined at X(5613), is equilateral, and X(5617) is its center. X(5613) is also the {X(2),X(1352)}-harmonic conjugate of X(5613). (Randy Hutson, July 7, 2014)

Let BA′C be the equilateral triangle having A′ on the side of BC that does not include C. Let LA be the line through A′ parallel to BC, and define Lb and Lc cyclicalloy. Let A″ = Lb∩Lc, and define B″ and C″ cyclically.The Euler lines of B′A″C′, C′B″A′, A′C″B′ concur in X(5617), and the acute angle between each pair of these lines is π/2. See Dao Thanh Oai, ADGEOM 4030), and Francisco Javier García Capitán, ADGEOM 4031). See also X(5613) and X(14144).

X(5617) lies on these lines: {2,98}, {3,618}, {4,616}, {5,13}, {14,2782}, {30,5463}, {99,621}, {298,511}, {302,383}, {381,530}, {396,3564}, {576,3181}, {5055,5459}

X(5617) = midpoint of X(i) and X(j) for these (i,j): (4,616), (298,1080)
X(5617) = reflection of X(i) in X(j) for these (i,j): (13,5), (3,618), (5613,114)

X(5617) = outer-Napoleon-to-inner-Napoleon similarity image of X(3)
X(5617) = inner-Napoleon-isogonal conjugate of X(15)

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)(27^1/2b²c²S_A + S(S² + 9S_AS_A))]

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 L_A be the line through A perpendicular to CP; let U_A be the 30-degree rotation of L_A, where the angle BAC, for present purposes, defines the positive direction of rotation, and angle CAB, the negative (used in X(5619)). Let A₁ = U_A∩CP, let V_A be the - 60 degree rotation of CP about A₁, let A_B = V_A∩BP, let A_C be the - 60 degree rotation of segment AA_B about A. Then AA_BA_C is an equilateral triangle. Define BB_CB_A and CC_AC_B 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 A_BA_C, B_CB_A, C_BC_A 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(5618) = intersection, other than X(111), of circumcircle and Parry circle of pedal triangle of X(13)

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)(27^1/2b²c²S_A - S(S² + 9S_AS_A))]

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(5619) = intersection, other than X(111), of circumcircle and Parry circle of pedal triangle of X(14)

X(5620) = ISOGONAL CONJUGATE OF X(5127)

Barycentrics (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 N_A be the nine-point center of A′BC, and let O_A be the circumcircle of N_ABC. Define O_B and O_C cyclically. The circles O_A, O_B, O_C concur in X(5620). (Angel Montesdoca, Anapolis #1120, November 2013: see Concurrent Circumcircles)

Let A′B′C′ be the incentral triangle and let A″ be the point such that triangle A″BC is similar to A′B′C′ and A″ is on the same side of line BC as A. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5620); see also X(502). (Randy Hutson, July 7, 2014)

X(5620) lies on these lines: {1,149}, {10,1109}, {36,759}, {37,115}, {65,1365}, {162,1838}, {267,3336}, {897,1738}, {1054,1247}, {1737,2166}, {2218,2915}

X(5620) = isogonal conjugate of X(5127)
X(5620) = X(2245)-cross conjugate of X(226)
X(5620) = incircle-inverse-of X(33593)
X(5620) = X(i)-isoconjugate of X(j) for these (i,j): (3,2074), (21,5172)
X(5620) = trilinear pole of line X(661)X(2294)
X(5620) = trilinear product X(523)*X(1290)
X(5620) = barycentric product X(1290)*X(1577)
X(5620) = X(10214)-of-excentral-triangle

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) = a²b²S_C(S_A - S_B)(-2S_AS_B + S_AS_C + S_BS_C) + a²c²S_B(S_A - S_C)(-2S_AS_C + S_AS_B + S_BS_C)
+ (a² + b² + c²)(-2S_AS_B + S_AS_C + S_BS_C)(-2S_AS_C + S_AS_B + S_BS_C)
X(5621) = X(6) + 2X(74)

Let L be the 1st Lemoine circle of a triangle ABC. Let A_B be the point nearer A where line AB meets L, and define B_C and C_A cyclically. Let A_C be the point nearer A where line AC meets L, and define B_A and C_B cyclically. The Euler lines of the triangles AA_BA_C, BB_CB_A, CC_AC_B 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 2a²b²S_C(S_A - S_B)(-2S_AS_B + S_AS_C + S_BS_C) + 2a²c²S_B(S_A - S_C)(-2S_AS_C + S_AS_B + S_BS_C) + (a² + b² + c²)(-2S_AS_B + S_AS_C + S_BS_C)(-2S_AS_C + S_AS_B + S_BS_C)
Barycentrics a^2(a^2 - b^2 - c^2)[a^8 - a^6(b^2 + c^2) - a^4(b^4 - 3b^2c^2 + c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + 3b^2c^2(b^2 - c^2)^2] : :
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 KA_BA_C, KB_CB_A, KC_AC_B concur in X(5622). (Seiichi Kirikami, November 7, 2013)

See Concurrent Euler lines.

X(5622) lies on these lines: {2, 98}, {3, 895}, {4, 1177}, {6, 74}, {54, 67}, {69, 5504}, {113, 3618}, {185, 575}, {186, 2393}, {217, 5038}, {265, 1176}, {389, 1205}, {403, 1503}, {511, 2071}, {576, 1204}, {578, 5095}, {631, 5181}, {1316, 2790}, {2854, 5085}, {2892, 3541}, {3431, 5505}

X(5623) = REFLECTION OF X(13) IN X(5618)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = H_A (H_AV_A - H_BV_B - H_CV_C), where H_A = 3^1/2a²S + S² + 3S_BS_C and V_A = (S² - 3S_AS_B)(S² - 3S_AS_C) (Peter Moses, June 20, 2014)

Referring to the construction of X(5618), the lines A_BA_C, B_CB_A, C_AC_B concur in X(5623). (Peter Moses, December 4, 2013)

X(5623) lies on the Neuberg cubic and these lines: {13,74}, {14,3440}, {16,1138}, {3065,3383}

X(5623) = reflection of X(13) in X(5618)
X(5623) = X(30)-Ceva conjugate of X(13)

X(5624) = REFLECTION OF X(14) IN X(5619)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = K_A (K_AV_A - K_BV_B - K_CV_C), where K_A = - 3^1/2a²S + S² + 3S_BS_C and V_A = (S² - 3S_AS_B)(S² - 3S_AS_C) (Peter Moses, June 20, 2014)

Referring to the construction of X(5619), the lines A_BA_C, B_CB_A, C_AC_B concur in X(5624). (Peter Moses, December 4, 2013)

X(5624) lies on the Neuberg cubic and these lines: {14,74}, {13,3441}, {15,1138}, {3065,3376}

X(5624) = reflection of X(14) in X(5619)
X(5624) = X(30)-Ceva conjugate of X(14)

X(5625) = MIDPOINT OF X(1) AND X(86)

Barycentrics (2a + b + c)(a² + 2ab + 2ac + bc) : (2b + c + a)(b² + 2bc + 2ba + ca) : (2c + a + b)(c² + 2ca + 2cb + ab)
X(5625) = X(1654) - 5*X(3616)

Suppose that P is a point in the plane of a triangle ABC. Let L_A be the line through P parallel to BC, and let B_A = L_A∩AB and C_A = L_A∩CA. Define A_B and C_B cyclically, and define A_C and B_C cyclically. Let U_A be the line of the midpoints of segments A_BA_C and B_CC_B, and define U_B and U_C cyclically. The lines U_A, U_B, U_C concur in a point Q = Q(P). If P is given by barycentrics p : q : r, then Q = g(p, q, r) : g(q, r, p) : g(r, p, q), where g(p,q,r) = (2p + q + r)(p² + 2pq + 2pr + qr). If P = X(1), then Q = X(5625). (Seiichi Kirikami, December 8, 2013)

X(5625) lies on these lines: {1,75}, {10,4478}, {519,4733}, {524,551}, {726,3723}, {1100,1125}, {1255,4756}, {1279,3636}, {1654,3616}, {1961,3699}, {1962,4427}, {2796,4353}, {3244,4923}, {3624,3759}, {3842,4649}, {3945,4655}, {3993,4670}

X(5625) = midpoint of X(1) and X(86)
X(5625) = reflection of X(1213) in X(1125)
X(5625) = trilinear product X(i)*X(j) for these (i,j): (1125,4649), (4427,4784)
X(5625) = barycentric product X(4359)*X(4649)

X(5626) = CENTER OF ELECTROSTATIC POTENTIAL

Trilinears [(g_ag_a - 1)(a² - (bg_b - gc_c)²)]^1/2 : : , where

g_a = coth(aλ/(a + b + c)), g_b = coth(bλ/(a + b + c)), g_c = 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 = ag_a, v = bg_b, w = cg_c

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 that 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 *)

Geogebra sketch by David Fernández-De la Cruz: X(5626). (November 21, 2021)

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/[(a²S_A - 2S_BS_C)(S² - 3S_AS_A)]
Barycentrics    h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = 1/[3a⁶(b² + c²) - 6a⁴(b⁴ + c⁴) + 3a²(b⁶ + c⁶) - 2b⁸ + 3b⁶c² - 6b⁴c⁴ + 3b²c⁶ - 2c⁸]    (Randy Hutson, Jan. 8, 2014)
X(5627) = 2X(265) + X(476)
X(5627) = 4X(125) - X(477)    (Peter Moses, January 2, 2014)

Paul Yiu introduced this point on New Year's Day, January 1, 2014. He noted that X(74) is the unique point whose reflections in the sidelines of triangle ABC are collinear and perspective to ABC. The perspector is X(5627). The line of the reflections is perpendicular to the Euler line at X(4), and the rectangular circumhyperbola through X(5627), here called the Yiu hyperbola, YH, has asymptotes parallel and perpendicular to the Euler line. The center of YH is X(3258), and the perspector of YH is X(1637); YH meets the circumcircle in X(477), which is the reflection of X(74) in the Euler line. (Paul Yiu, ADGEOM, "An easy new year puzzle", January 1, 2014)

The line tangent to YH at X(5627) is parallel to the line X(74)X(1138). The axes of YH are the Wallace-Simson lines of X(74) and X(110). The Steiner circumellipse meets YH in four points: A, B, C, and X(5641). The isogonal conjugate of YH is the line X(3)X(74). X(5627) is the cevapoint of the 1st and 2nd Fermat Points. (Peter Moses, January 2, 2014)

X(5627) is the perspector of ABC and the reflection of the Euler triangle in the Euler line. (Randy Hutson, Jan. 8, 2014)

Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Let A″ be the intersection, other than X(3258) of the nine-point circle and the line A′X(3258); define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5627). (Randy Hutson, Jan. 8, 2014)

Let P = X(110) and A1A2, B1B2, C1C2 be the diameters of the circumcircle of ABC, parallel to AP, BP, CP, respectively. The Simson-Wallace lines of the points A1 and A2 intersect orthogonally in Pa, on the nine-point circle. The points Pb and Pc are defined similarly. The triangles ABC and PaPbPc are perspective and inversely similar; the perspector is X(5627). (Angel Montesdeoca, October 21, 2018)

X(5627) lies on these lines: {5,1117}, {30,74}, {125,477}, {328,1494}, {403,1989}, {1138,3258}, {1141,1304}

X(5627) = reflection of X(1138) in X(3258)
X(5627) = isogonal conjugate of X(1511)
X(5627) = isotomic conjugate of X(6148)
X(5627) = cevapoint of X(i) and X(j) for these (i,j): (13,14), (74,3470)
X(5627) = crossconjugate of X(i) and X(j) for these (i,j): (4,1141), (115,2394), (523, 476)
X(5627) = isoconjugate of X(i) and X(j) for these (i,j): (1,1511), (63,39176), (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(5627) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(186)

X(5628) = 1st MORLEY - VAN TIENHOVEN POINT

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

X(5628) and several other triangle centers are perspectors of each pair of the following triangles in the plane of a triangle ABC:
T1 = 1st Morley triangle; MathWorld: First Morley triangle
T2 = 2nd Morley triangle; MathWorld: Second Morely triangle, etc.
T3 = 3rd Morley triangle>
T4 = 1st adjunct Morley triangle; MathWorld: First adjunct Morley triangle, etc.)
T5 = 2nd adjunct Morley triangle
T6 = 3rd adjunct Morley triangle
T7 = 1st p-Morley triangle (defined below)
T8 = 2nd p-Morley triangle
T9 = 3rd p-Morley triangle
T10 = 1st p-adjunct Morley triangle (defined below)
T11 = 2nd p-adjunct Morley triangle
T12 = 3rd p-adjunct Morley triangle

The p-Morley triangles T7, T8, T9 have as vertices the points of intersection of pairs of perpendiculars to trisectors at corresponding vertices that form T1, T2, T3, respectively. For example, T7 is formed as follows from the 1st Morley triangle A′B′C′: let L_B be the line perpendicular to BA′ at B, let L_C be the line perpendicular to CA′ at C; then the A-vertex of T7 is L_B∩L_C, and the B-vertex and C-vertex are defined cyclically. Similarly, the p-adjunct Morley triangles T10, T11, T12 are defined from T4, T5, T6.

X(5628) = 1st Morley-van Tienhoven point = perspector of ABC and T7
X(5629) = 2nd Morley-van Tienhoven point = perspector of ABC and T10
X(5630) = 3rd Morley-van Tienhoven point = perspector of ABC and T8
X(5631) = 4th Morley-van Tienhoven point = perspector of ABC and T11
X(5632) = 5th Morley-van Tienhoven point = perspector of ABC and T9
X(5633) = 6th Morley-van Tienhoven point = perspector of ABC and T12
X(356) = 7th Morley-van Tienhoven point = perspector of each pair of T1, T4, T7
X(3276) = 8th Morley-van Tienhoven point = perspector of each pair of T2, T5, T8
X(3277) = 9th Morley-van Tienhoven point = perspector of each pair of T3, T6, T9
X(5634) = 10th Morley-van Tienhoven point = perspector of T7 and T10
X(5635) = 11th Morley-van Tienhoven point = perspector of T8 and T11
X(5636) = 12th Morley-van Tienhoven point = perspector of T9 and T12
X(5637) = 13th Morley-van Tienhoven point

X(5628)-X(5633) were found in connection with Chris van Tienhoven's rotations of Morley trisectors and subsequent collaborations with Bernard Gibert, including is the cubic K587 in Gibert's catalog of cubics: Morley - van Tienhoven cubic.

X(5628) lies on these lines: {357,5457}, {3272,3274}, {3602,3606}

X(5628) = isogonal conjugate of X(5629)

X(5629) = 2nd MORLEY - VAN TIENHOVEN POINT

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

X(5629) = perspector of ABC and T10; see X(5628)

X(5629) lies on these lines: {356,357}, {3273,3281}, {3274,3606}

X(5629) = isogonal conjugate of X(5628)

X(5630) = 3rd MORLEY - VAN TIENHOVEN POINT

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

X(5630) = perspector of ABC and T8; see X(5628)

X(5630) lies on these lines: {3603,3607}, {3272,3275}

X(5630) = isogonal conjugate of X(5631)

X(5631) = 4th MORLEY - VAN TIENHOVEN POINT

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

X(5631) = perspector of ABC and T11; see X(5628)

X(5631) lies on these lines: {1136,1137}, {3274,3283}, {3275,3607}

X(5631) = isogonal conjugate of X(5630)

X(5632) = 5th MORLEY - VAN TIENHOVEN POINT

Trilinears cos[(A - 4π)/3] sec[(2A - 8π)/3] : cos[(B - 4π)/3] sec[(2B - 8π)/3] : cos[(C - 4π)/3] sec[(2C - 8π)/3]

X(5632) = perspector of ABC and T9; see X(5628)

X(5632) lies on these lines: {356,3272}, {3604,3605}

X(5632) = isogonal conjugate of X(5633)

X(5633) = 6th MORLEY - VAN TIENHOVEN POINT

Trilinears cos[(2A - 8π)/3] sec[(A - 4π)/3] : cos[(2B - 8π)/3] sec[(B - 4π)/3] : cos[(2C - 8π)/3] sec[(C - 4π)/3]

X(5633) = perspector of ABC and T12; see X(5628)

X(5633) lies on these lines: {356,1134}, {3273,3605}, {3275,3279}

X(5633) = isogonal conjugate of X(5632)

X(5634) = 10th MORLEY - VAN TIENHOVEN POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(2A/3) cos(B/3) cos(C/3) - cos(A/3) cos(2B/3) cos(2C/3)

X(5634) = perspector of the triangles T7 and T10 listed at X(5628).

X(5634) lies on these lines: {356,3605}, {3276,3606}

X(5635) = 11th MORLEY - VAN TIENHOVEN POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos[(2(A - 2π)/3] cos[(B - 2π)/3] cos[(C - 2π)/3] - cos[(A - 2π)/3] cos[(2B - 4π)/3] cos[(2C - 4π)/3]

X(5635) = perspector of the triangles T8 and T11 listed at X(5628).

X(5635) lies on these lines: {3276,3606}, {3277,3607}

X(5636) = 12th MORLEY - VAN TIENHOVEN POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos[(2A - 8π)/3] cos[(B - 4π)/3] cos[(C - 4π)/3] - cos[(A - 4π)/3] cos[(2B - 8π)/3] cos[(2C - 8π)/3]

X(5636) = perspector of the triangles T9 and T12 as listed at X(5628).

X(5636) lies on these lines: {356,3605}, {3277,3607}

X(5637) = 13th MORLEY - VAN TIENHOVEN POINT

Trilinears sin(B/3 - C/3) : :

Using the notation at X(5628), let
L1 = perspectrix of each pair of the triangles ABC, T1, T4
L2 = perspectrix of each pair of the triangles ABC, T2, T5
L3 = perspectrix of each pair of the triangles ABC, T3, T6
The lines L1, L2, L3 concur in X(5637). (Chris van Tienhoven, January 3, 2014)

X(5637) lies on this line: {396,523}

X(5637) = isogonal conjugate of X(14146)
X(5637) = crossdifference of every pair of points on line X(16)X(358)
X(5637) = perspector of Morley circumconic

X(5638) = INSIMILICENTER OF CIRCLES {{X(14), X(15), X(16)}} AND {{X(13), X(15), X(16)}}

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc B)[e cos C + cos(C - ω)] - (csc C)[e cos B + cos(B - ω)]
Barycentrics    g(A,B,C) : g(B,C,A ) : g(C,A,B), where g(A,B,C) = sin(A - ω)/[e cos A - cos(A + ω)]
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (S²_B - S²_C)[(S + 3^1/2S_A)p - (S - 3^1/2S_A)q], where p = (S_ω - 3^1/2S)^1/2 and q = (S_ω + 3^1/2S)^1/2    César Lozada (ADGEOM #1341, June 22, 2014)
Barycentrics    Sin[A]*((e*Cos[C] + Cos[C - w])*Csc[B] - (e*Cos[B] + Cos[B - w])*Csc[C]) : : (Peter Moses, March 3, 2024)
Barycentrics    a^2*(2*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 4*b^2*c^2 - c^4 - (2*a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : : (Peter Moses, March 3, 2024)

The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5638) is the farther from X(2); also, X(5638) = intersection farther from X(2) of the Parry circle and Lemoine axis. X(5638) = (F₁, F₂)-harmonic conjugate of X(1341), where F₁ and F₂ are the foci of the Steiner inellipse. (Randy Hutson, June 16, 2014)

X(5638) lies on the Steiner major axis, the circumconic {{A,B,C,X(2),X(6)}}, the Parry circle, the Thomson-Gibert-Moses hyperbola, the cubics K889, K890, K891, K1067, the curve Q090, and these lines: {2, 1341}, {3, 6141}, {6, 6142}, {100, 11651}, {110, 1379}, {111, 1380}, {154, 21032}, {187, 237}, {353, 1340}, {739, 11652}, {2028, 3124}, {2029, 39689}, {2395, 13636}, {3228, 6189}, {3414, 9147}, {3557, 9716}, {5643, 14631}, {5648, 46463}, {5652, 52723}, {16081, 57013}, {21001, 21036}, {36213, 39067}

X(5638) = reflection of X(5639) in X(351)
X(5638) = isogonal conjugate of X(6190)
X(5638) = reflection of X(5638) in the Lemoine axis
X(5638) = Parry-isodynamic-circle-inverse of X(5639)
X(5638) = isogonal conjugate of the anticomplement of X(39023)
X(5638) = isogonal conjugate of the complement of X(39366)
X(5638) = isogonal conjugate of the isotomic conjugate of X(3413)
X(5638) = Thomson-isogonal conjugate of X(6039)
X(5638) = psi-transform of X(1341)
X(5638) = X(31)-complementary conjugate of X(39068)
X(5638) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39068}, {6, 2028}, {110, 5639}, {1379, 6}, {6177, 39023}, {6190, 3558}, {41880, 512}, {57013, 13636}
X(5638) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6190}, {63, 57014}, {75, 1379}, {662, 3414}, {799, 5639}, {13722, 24041}, {36085, 52723}
X(5638) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6190}, {206, 1379}, {1084, 3414}, {3005, 13722}, {3162, 57014}, {13722, 850}, {21905, 46463}, {38988, 52723}, {38996, 5639}, {39023, 76}, {39067, 30508}, {39068, 2}
X(5638) = trilinear pole of line {512, 2028}
X(5638) = complement of anticomplementary conjugate of X(39366)
X(5638) = anticomplement of complementary conjugate of X(39023)
X(5638) = X(2)-Ceva conjugate of X(39068)
X(5638) = X(110)-Ceva conjugate of X(5639)
X(5638) = X(3124)-cross conjugate of X(5639)
X(5638) = X(6)-vertex conjugate of X(5639)
X(5638) = inverse-in-circumcircle of X(6141)
X(5638) = antipode-in-Parry-circle of X(5639)
X(5638) = trilinear pole of the line X(512)X(2028)
X(5638) = crossdifference of every pair of points on the line X(2)X(1340)
X(5638) = X(1380)-of-1st-Parry-triangle
X(5638) = X(1379)-of-2nd-Parry-triangle
X(5638) = crossdifference of PU(i) for these i: 117, 119
X(5638) = PU(118)-harmonic conjugate of X(1341)
X(5638) = X(i)-line conjugate of X(j) for these (i,j): {187, 5639}, {353, 1340}, {1341, 2}, {2395, 13636}, {5652, 52723}, {9147, 3414}
X(5638) = barycentric product X(i)*X(j) for these {i,j}: {6, 3413}, {110, 13636}, {111, 52722}, {512, 6189}, {523, 1380}, {647, 57013}, {691, 46462}, {1379, 39023}, {2028, 6190}, {3414, 41880}, {5639, 30509}
X(5638) = pole of line {7668, 39023} with respect to the Kiepert circumhyperbola
X(5638) = pole of line {2028, 3269} with respect to the Jerabek circumhyperbola
X(5638) = pole of line {2028, 3124} with respect to ABCGK
X(5638) = pole of line {39068, 44312} with respect to ABCGGe
X(5638) = pole of line {1015, 2028} with respect to ABCIK
X(5638) = pole of line {670, 6190} with respect to the Steiner-Wallace right hyperbola
X(5638) = pole of line {99, 1379} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5638) = pole of line {190, 6190} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5638) = pole of line {668, 6190} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5638) = pole of line {670, 6190} with respect to the Kiepert circumhyperbola of the anticomplementary triangle
X(5638) = pole of line {194, 3414} with respect to the Steiner circumellipse
X(5638) = pole of line {3167, 5639} with respect to the MacBeath circumconic
X(5638) = pole of line {39, 3414} with respect to the Steiner inellipse
X(5638) = pole of line {6, 5639} with respect to the Brocard inellipse
X(5638) = pole of line {669, 30509} with respect to the Kiepert parabola
X(5638) = pole of line {5638, 9155} with respect to the Thomson-Gibert-Moses hyperbola
X(5638) = pole of line {6, 5639} with respect to the Aquarius conic (see X(5585))
X(5638) = pole of line {6, 5639} with respect to the circumcircle
X(5638) = pole of line {574, 5639} with respect to the Brocard circle
X(5638) = pole of line {511, 5638} with respect to the Parry circle
X(5638) = pole of line {262, 3414} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5638) = pole of line {3414, 44434} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5638) = pole of line {3414, 44460} with respect to the circumcircle of the inner Napoleon triangle
X(5638) = pole of line {3414, 44464} with respect to the circumcircle of the outer Napoleon triangle
X(5638) = pole of line {5639, 6468} with respect to the Lucas inner circle
X(5638) = pole of line {5639, 6221} with respect to the Lucas circles radical circle
X(5638) = pole of line {5639, 50663} with respect to the outer Montesdeoca-Lemoine circle
X(5638) = pole of line {5639, 50662} with respect to the inner Montesdeoca-Lemoine circle
X(5638) = pole of line {511, 5639} with respect to the Parry isodynamic circle
X(5638) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6190}, {25, 57014}, {32, 1379}, {351, 52723}, {512, 3414}, {669, 5639}, {1379, 57576}, {1380, 99}, {2028, 3413}, {3124, 13722}, {3413, 76}, {5639, 30508}, {6189, 670}, {13636, 850}, {21906, 46463}, {41880, 6189}, {46462, 35522}, {52722, 3266}, {57013, 6331}
X(5638) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 2502, 5639}, {647, 3569, 5639}, {649, 5029, 5639}, {663, 5075, 5639}, {665, 5098, 5639}, {667, 5040, 5639}, {669, 5027, 5639}, {890, 42655, 5639}, {902, 5168, 5639}, {1495, 5191, 5639}, {3005, 5113, 5639}, {3230, 5163, 5639}, {3231, 5106, 5639}, {3724, 5202, 5639}, {3747, 5147, 5639}, {6137, 6138, 5639}, {8644, 9135, 5639}, {8651, 42663, 5639}, {9208, 17414, 5639}, {9409, 42654, 5639}, {14899, 35607, 13722}, {15451, 42651, 5639}, {39162, 39163, 1341}, {39202, 39203, 51493}

X(5639) = EXSIMILICENTER OF CIRCLES {{X(14), X(15), X(16)}} AND {{X(13), X(15), X(16)}}

Trilinears    (csc B)[e cos C - cos(C - ω)] + (csc C)[e cos B - cos(B - ω)] : :
Barycentrics    sin(A - ω)/[e cos A + cos(A + ω)] : :
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (S²_B - S²_C)[(S + 3^1/2S_A)p + (S - 3^1/2S_A)q], where p = (S_ω - 3^1/2S)^1/2 and q = (S_ω + 3^1/2S)^1/2    César Lozada (ADGEOM #1341, June 22, 2014)
Barycentrics    Sin[A]*((e*Cos[C] - Cos[C - w])*Csc[B] - (e*Cos[B] - Cos[B - w])*Csc[C]) : : (Peter Moses, March 3, 2024)
Barycentrics    a^2*(2*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 4*b^2*c^2 - c^4 + (2*a^2 - b^2 - c^2)*Sqrt[a^4 - a^2*b^2 + b^4 - a^2*c^2 - b^2*c^2 + c^4]) : : (Peter Moses, March 3, 2024)

The Parry circle and Thomson-Gibert-Moses hyperbola (introduced at X(5642)), intersect in four points: X(2), X(110), X(5638), X(5639). Of the latter two points, X(5639) is the closer to X(2); also, X(5639) = the intersection closer to X(2) of the Parry circle and Lemoine axis. X(5639) = perspector of the hyperbola {{A,B,C,X(6), F₁, F₂}}, where F₁ and F₂ are the foci of the Steiner inellipse; also X(5639) = intersection of the trilinear polars of X(6), F₁, and F₂. (Randy Hutson, June 16, 2014)

X(5639) lies on the Steiner minor axis, the circumconic {{A,B,C,X(2), X(6)}}, the Parry circle, the Thomson-Gibert-Moses hyperbola, K889, K890, K891, K1067, the curve Q090, and these lines: {2, 1340}, {3, 6142}, {6, 6141}, {100, 11652}, {110, 1380}, {111, 1379}, {154, 21036}, {187, 237}, {353, 1341}, {739, 11651}, {2028, 39689}, {2029, 3124}, {2395, 13722}, {3228, 6190}, {3413, 9147}, {3558, 9716}, {5643, 14630}, {5648, 46462}, {5652, 52722}, {16081, 57014}, {21001, 21032}, {36213, 39068}

X(5639) = reflection of X(5638) in X(351)
X(5639) = isogonal conjugate of X(6189)
X(5639) = complement of anticomplementary conjugate of X(39365)
X(5639) = anticomplement of complementary conjugate of X(39022)
X(5639) = X(3124)-cross conjugate of X(5638)
X(5639) = X(6)-vertex conjugate of X(5638)
X(5639) = circumcircle-inverse of X(6142)
X(5639) = Parry-circle-antipode of X(5638)
X(5639) = trilinear pole of the line X(512)X(2029)
X(5639) = crossdifference of every pair of points on the line X(2)X(1341)
X(5639) = X(1379)-of-1st-Parry-triangle
X(5639) = X(1380)-of-2nd-Parry-triangle
X(5639) = crossdifference of PU(i) for these i: 116, 118
X(5639) = perspector of hyperbola {{A,B,C,X(6),PU(118)}}
X(5639) = isogonal conjugate of the anticomplement of X(39022)
X(5639) = isogonal conjugate of the complement of X(39365)
X(5639) = isogonal conjugate of the isotomic conjugate of X(3414)
X(5639) = Thomson-isogonal conjugate of X(6040)
X(5639) = psi-transform of X(1340)
X(5639) = X(31)-complementary conjugate of X(39067)
X(5639) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 39067}, {6, 2029}, {110, 5638}, {1380, 6}, {6178, 39022}, {6189, 3557}, {41881, 512}, {57014, 13722}
X(5639) = X(i)-isoconjugate of X(j) for these (i,j): {1, 6189}, {63, 57013}, {75, 1380}, {662, 3413}, {799, 5638}, {13636, 24041}, {36085, 52722}
X(5639) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 6189}, {206, 1380}, {1084, 3413}, {3005, 13636}, {3162, 57013}, {13636, 850}, {21905, 46462}, {38988, 52722}, {38996, 5638}, {39022, 76}, {39067, 2}, {39068, 30509}
X(5639) = X(i)-line conjugate of X(j) for these (i,j): {187, 5638}, {353, 1341}, {1340, 2}, {2395, 13722}, {5652, 52722}, {9147, 3413}
X(5639) = pole of line {7668, 39022} with respect to the Kiepert circumhyperbola
X(5639) = pole of line {2029, 3269} with respect to the Jerabek circumhyperbola
X(5639) = pole of line {2029, 3124} with respect to ABCGK
X(5639) = pole of line {39067, 44312} with respect to ABCGGe
X(5639) = pole of line {1015, 2029} with respect to ABCIK
X(5639) = pole of line {670, 6189} with respect to the Steiner / Wallace right hyperbola
X(5639) = pole of line {99, 1380} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5639) = pole of line {190, 6189} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5639) = pole of line {668, 6189} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5639) = pole of line {670, 6189} with respect to the Kiepert circumhyperbola of the anticomplementary triangle
X(5639) = pole of line {194, 3413} with respect to the Steiner circumellipse
X(5639) = pole of line {3167, 5638} with respect to the MacBeath circumconic
X(5639) = pole of line {39, 3413} with respect to the Steiner inellipse
X(5639) = pole of line {6, 5638} with respect to the Brocard inellipse
X(5639) = pole of line {669, 30508} with respect to the Kiepert parabola
X(5639) = pole of line {5639, 9155} with respect to the Thomson-Gibert-Moses hyperbola
X(5639) = pole of line {6, 5638} with respect to the Aquarius conic (see X(5585))
X(5639) = pole of line {6, 5638} with respect to the circumcircle
X(5639) = pole of line {574, 5638} with respect to the Brocard circle
X(5639) = pole of line {511, 5639} with respect to the Parry circle
X(5639) = pole of line {262, 3413} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5639) = pole of line {3413, 44434} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5639) = pole of line {3413, 44460} with respect to the circumcircle of the inner Napoleon triangle
X(5639) = pole of line {3413, 44464} with respect to the circumcircle of the outer Napoleon triangle
X(5639) = pole of line {5638, 6468} with respect to the Lucas inner circle
X(5639) = pole of line {5638, 6221} with respect to the Lucas circles radical circle
X(5639) = pole of line {5638, 50663} with respect to the outer Montesdeoca-Lemoine circle
X(5639) = pole of line {5638, 50662} with respect to the inner Montesdeoca-Lemoine circle
X(5639) = pole of line {511, 5638} with respect to the Parry isodynamic circle
X(5639) = barycentric product X(i)*X(j) for these {i,j}: {6, 3414}, {110, 13722}, {111, 52723}, {512, 6190}, {523, 1379}, {647, 57014}, {691, 46463}, {1380, 39022}, {2029, 6189}, {3413, 41881}, {5638, 30508}
X(5639) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 6189}, {25, 57013}, {32, 1380}, {351, 52722}, {512, 3413}, {669, 5638}, {1379, 99}, {1380, 57575}, {2029, 3414}, {3124, 13636}, {3414, 76}, {5638, 30509}, {6190, 670}, {13722, 850}, {21906, 46462}, {41881, 6190}, {46463, 35522}, {52723, 3266}, {57014, 6331}
X(5639) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {187, 2502, 5638}, {647, 3569, 5638}, {649, 5029, 5638}, {663, 5075, 5638}, {665, 5098, 5638}, {667, 5040, 5638}, {669, 5027, 5638}, {890, 42655, 5638}, {902, 5168, 5638}, {1495, 5191, 5638}, {3005, 5113, 5638}, {3230, 5163, 5638}, {3231, 5106, 5638}, {3724, 5202, 5638}, {3747, 5147, 5638}, {6137, 6138, 5638}, {8644, 9135, 5638}, {8651, 42663, 5638}, {9208, 17414, 5638}, {9409, 42654, 5638}, {15451, 42651, 5638}, {35608, 35609, 13636}, {39204, 39205, 51492}

X(5640) = CENTROID OF ORTHOCENTROIDAL TRIANGLE

Trilinears    sin A (sin 2B + sin 2C) + sin B sin C : :
Barycentrics    a^2(a^2b^2 + a^2c^2 + 3b^2c^2 - b^4 - c^4) : :
Barycentrics    (SB + SC) (SA^2 + 3 SA (SB + SC) + 5 SB SC) : :
X(5640) = X(2) + 2*X(51)

The orthocentroidal triangle is introduced at X(5476). For another construction of X(5640), let A′B′C′ be the orthic triangle, let A″ be the centroid of AB′C′, and define B″ and C″ cyclically. Then X(5640) is the centroid of A″B″C″. Also, X(5640) is the trilinear pole of the polar, with respect to the Moses circle, of the perspector of the Moses circle. (Randy Hutson, June 16, 2014)

Vertices of the central triangle A″B″C″ are given by barycentrics as follows:

A″ = b⁴ + c⁴ - a²b² - a²c² - 4b²c² : b²(a² - b² - c²) : c²(a² - b² - c²) (Peter Moses, June 20, 2014)

A″B″C″ is perspective to ABC, the tangential triangle and the second Brocard triangle, all with perspector X(6), and perspective to the Euler triangle at X(125). The Euler line of A″B″C″ passes througth X(i) for these I: 6,110,111,895,1995,2493,2502,2503,2854,2930,3066,3124 (Peter Moses, June 20, 2014)

Let A′B′C′ be the Artzt triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically; then X(5640) = centroid of A″B″C″; see X(6032). (Randy Hutson, December 10, 2016)

Let A′B′C′ be the anti-Artzt triangle. Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically; then X(5640) = centroid of A″B″C″. (Randy Hutson, December 10, 2016)

X(5640) lies on these lines: {2,51}, {4,4846}, {5,568}, {6,110}, {22,5085}, {23,182}, {25,5012}, {52,3090}, {125,5169}, {143,1656}, {185,3832}, {323,576}, {324,3168}, {375,3681}, {381,5663}, {389,3091}, {394,5102}, {476,1316}, {512,598}, {569,3518}, {575,1495}, {631,5446}, {858,5480}, {1112,5094}, {1154,5055}, {1180,3981}, {1216,5067}, {1383,2030}, {1843,4232}, {1993,5020}, {3056,5297}, {3111,3972}, {3291,5052}, {3292,5097}, {3448,3818}, {3533,5447}, {5039,5354}

X(5640) = midpoint of X(51) and X(373)
X(5640) = reflection of X(2) in X(373)
X(5640) = isotomic conjugate of polar conjugate of X(33885)
X(5640) = complement of X(33884)
X(5640) = anticomplement of X(5650)
X(5640) = crossdifference of every pair of points on line X(690)X(3288)
X(5640) = intercept, other than X(110), of line X(6)X(110) and conic {{X(13),X(14),X(15),X(16),X(110)}}
X(5640) = SS(A/3 → A) of X(8065) (symbolic trilinear substitution)
X(5640) = X(2)-of-reflection-triangle of X(2)
X(5640) = isogonal conjugate of X(2) with respect to the pedal triangle of X(2)
X(5640) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(2)
X(5640) = {X(11624),X(11626)}-harmonic conjugate of X(6)
X(5640) = homothetic center of X(2)-altimedial and X(2)-adjunct anti-altimedial triangles
X(5640) = circummedial-to-X(2)-pedal similarity image of X(2)
X(5640) = {X(2),X(51)}-harmonic conjugate of X(3060)

X(5641) = ISOTOMIC CONJUGATE OF X(542)

Barycentrics 1/[2a sec(A + ω) - b sec(B + ω) - c sec(C + ω)] : :
Barycentrics 1/(2 a^6 - 2 a^4 (b^2 + c^2) + a^2 (b^4 + c^4) - (b^2 - c^2)^2 (b^2 + c^2)) : :

The Steiner circumellipse meets the Yiu Hyperbola, defined at X(5627), in four points: A, B, C, and X(5641). (Peter Moses, January 2, 2014)

X(5641) lies on these lines: {2,2966}, {30,99}, {69,892}, {290,850}, {297,340}, {523,1494}, {525,671}, {670,3260}

X(5641) = reflection of X(2966) in X(2)
X(5641) = reflection of X(2) in X(35088)
X(5641) = isogonal conjugate of X(5191)
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) = antipode of X(2) in hyperbola {{A,B,C,X(2),X(325)}}
X(5641) = trilinear pole of line X(2)X(1637)
X(5641) = trilinear product X(75)*X(842)
X(5641) = barycentric product X(76)*X(842)

X(5642) = CENTER OF THOMSON-GIBERT-MOSES HYPERBOLA

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

The point X(5542) minimizes a certain sum involving the pedal triangle A′B′C′ of a triangle ABC. During May, 2014, Peter Moses generalized the sum to include a parameter t, as follows:

|XA|² + |XB|² + |XC|² + t(|XA′|² + |XB′|² + |XC′|²),

and found the solution to the extremal problem to be the point H(t) = f(A,B,C) : f(B,C,A) : f(C,A,B), where

f(A,B,C) = a²[(t² + 3t + 3)S² + (2t + 3) S²_A + tS_BS_C].

Let u = 3abc[-3abc + (9a²b²c² - 8S²S²_W)^1/2/(4S²S²_W) and v = 3abc[-3abc - (9a²b²c² - 8S²S²_W)^1/2/(4S²S²_W)

Moses found that if t > u, then H(t) minimizes the sum, and if t < v, then H(t) maximizes the sum, and if t > -1, then H(t) minimizes the sum. The extreme value in all cases is

[(6t + 6)S_AS_BS_CS_W - (2t + 3)(t + 2)S²S²_W - t(t + 1)²S⁴]/[(9t + 9)S_AS_BS_C - (2t + 3)(t + 3)S²S_W].

The locus of H(t) as t ranges through the extended real number line is a rectangular hyperbola, here named the Thomson-Gibert-Moses hyperbola, which passes through the triangle centers X(i) for I = 2, 3, 6, 110, 154, 354, 392, 1201, 2574, 2575, 3167, 5544, 5638, 5639, 5643, 5644, 5645, 5646, 5648, 5652-5656. The axes of this hyperbola are parallel to the Simson-Wallace lines of X(1113) and X(1114), these being the points of intersection of the Euler line and the circumcircle. See X(5643). (based on notes from Peter Moses, June 7, 2014)

X(5642) is the radical trace of the Parry circles of ABC and the 1st Brocard triangle, and also the centroid of the (degenerate) pedal triangle of X(110). The Thomson-Gibert-Moses hyperbola intersects the circumcircle in X(110) and the vertices of the Thomson triangle (see Thomson Triangle. The 4 points of intersection form an X(74)-centric system; i.e., each is X(74) of the triangle of the other three. Moreover, the Thomson-Gibert-Moses hyperbola is the Thomson isogonal conjugate (i.e., isogonal-conjugate-with-respect-to-the-Thomson-triangle) of the Euler line. In general, the Thomson isogonal conjugate of a point P is the centroid of the antipedal triangle of the isogonal conjugate of P; consequently, the Thomson-Gibert-Moses hyperbola is the locus of the centroid of the antipedal triangle of a point P that traverses the Jerebek hyperbola. Indeed, the Thomson-Gibert-Moses hyperbola is the Jerabek hyperbola of the Thomson triangle, as noted at Thomson Triangle (Randy Hutson, June 16-17, 2014)

Let O = X(3) and suppose that P is a point other than O. Let O_P be the circle with segment PO as diameter. Let A′ be the point of intersection, other than O, of O_P and the perpendicular bisector of segment BC, and define B′ and C′ cyclically. Triangle A′B′C′ is called the P-Brocard triangle, and X(5642) is X(23)-of-the-X(2)-Brocard triangle. (Randy Hutson, June 16-17, 2014)

The Thomson-Gibert-Moses hyperbola is the image of the Euler line under a mapping T discussed in connection with the third Deaux cubic (K609). (Bernard Gibert, June 22, 2014)

Let O* be the circle with segment X(13)X(14) as diameter (and center X(115). Let P be the perspector of O*. Then X(5642) is the trilinear pole of the polar of P with respect to O*. See X(3292) for a similar property involving the segment X(15)X(16). (Randy Hutson, July 18, 2014)

X(5642) lies on these lines: {2,98}, {3,541}, {30,113}, {74,3524}, {115,2502}, {126,5026}, {265,5055}, {351,690}, {373,597}, {376,2777}, {399,5054}, {468,524}, {543,1316}, {620,5108}, {1636,1637}, {1648,5477}, {1995,5476}, {2781,3917}, {2836,3742}, {3024,4995}, {3028,5298}, {3849,5112}

X(5642) = midpoint of X(2) and X(110)
X(5642) = reflection of X(i) in X(j) for these (i,j): (2,5972), (125,2),
X(5642) = crossdifference of every pair of points on the line X(74)X(111)
X(5642) = X(3524)-line-conjugate of X(74)
X(5642) = radical trace of Parry circles of ABC and anti-Artzt triangle

X(5643) = H(2) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a²(13S² + 7S²_A + 2S_BS_C) (Peter Moses, June 9, 2014)

X(5643) is the only point whose polar conic in the Napoleon cubic (K005) is a circle. (Bernard Gibert, June 22, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642). Points H(t) on this hyperbola include the following:

t	H(t)
-3	X(3167)
-2	X(110)
-3/2	X(154)
-1/2	X(5646)
-1	X(3)
0	X(2)
1	X(5544)
2	X(5643)
3	X(5644)
4	X(5645)
infinity	X(6)
3R/r	X(354)
6rR/(r² - 2rR - s²	X(354)

Let A′ be the centroid of the A-altimedial triangle, and define B′ and C′ cyclically; then X(5643) is the center of similitude of ABC and A′B′C′. (Randy Hutson, July 7, 2014)

X(5643) is the only finite fixed point of the affine transformation that maps a triangle ABC onto the pedal triangle of X(5). (Angel Montesdeoca, August 19, 2016)

X(5643) lies on these lines: {2,576}, {83,5466}, {110,373}, {111,5038}, {154,1995}, {354,4663}, {392,5047}, {597,895}, {632,1173}, {3090,5449}, {3167,5422}, {3580,3628}

X(5643) = Thomson-isogonal conjugate of X(8703)

X(5644) = H(3) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a²(7S² + 3S²_A + S_BS_C) (Peter Moses, June 9, 2014)

X(5644) is the Thomson isogonal conjugate of X(3522); see X(5642).

X(5644) lies on these lines: {2,5093}, {3,51}, {110,5020}, {154,5050}, {343,5070}, {373,3167}, {394,5544}, {1351,3819}, {1899,3851}

X(5644) = Thomson-isogonal conjugate of X(3522)

X(5645) = H(4) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a²(31S² + 11S²_A + 4S_BS_C) (Peter Moses, June 9, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642).

X(5645) lies on these lines: {2,5097}, {154,3066}, {323,5544}, {2889,3533}, {3448,3545}

X(5646) = H(-1/2) ON THE THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics a²(7S² + 8S²_A - 2S_BS_C) (Peter Moses, June 9, 2014)

The Thomson-Gibert-Moses hyperbola is defined at X(5642). X(5646) = (X(6) of the Thomson triangle); see Thomson Triangle (Randy Hutson, June 16, 2014)

X(5646) lies on these lines: {2,1350}, {40,392}, {64,631}, {110,5085}, {182,3167}, {354,612}, {511,5544}, {1201,2177}, {1351,3819}

X(5647) = HATZIPOLAKIS-EULER POINT

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A cos(B - C))(4 cos⁴A + (-1 + 4 cos²(B - C)) cos²A + cos(B - C)(4 cos A + cos(B - C)) (César Lozada, May 18, 2014)

Let A′B′C′ be the cevian triangle of the circumcenter. Let

L_AB = reflection of AA′ in AB, L_AC = reflection of AA′ in AC; H_AB = orthogonal projection of A′ on L_AB, H_AC = orthogonal projection of A′ on L_AC, and define L_BC, L_CA, L_BA, L_CB, and H_BC, H_CA, H_BA, H_CB cyclically,

M₁₁ = midpoint(H_AB, H_AC),      M₁₂ = midpoint(H_BC, H_BA),      M₁₃ = midpoint(H_CA, H_CB)
M₂₁ = midpoint(H_BA, H_CA),      M₂₂ = midpoint(H_CB, H_AB),      M₂₃ = midpoint(H_AC, H_BC)
M₃₁ = midpoint(H_BC, H_CB),      M₃₂ = midpoint(H_CA, H_AC),      M₃₃ = midpoint(H_AB, H_BA).

Antreas Hatzipolakis (Anopolis , May 17, 2014) asked if the Euler lines of the triangles M₁₁M₁₂M₁₃, M₂₁M₂₂M₂₃, M₃₁M₃₂M₃₃ concur, and César Lozada established that they do. The point of concurrence is X(5647).

X(5647) lies on these lines: {5,51}, {154,157}

X(5648) = ANTIPODE OF X(6) IN THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8+3 a^6 b^2-2 a^4 b^4-3 a^2 b^6+b^8+3 a^6 c^2-11 a^4 b^2 c^2+7 a^2 b^4 c^2-2 a^4 c^4+7 a^2 b^2 c^4-2 b^4 c^4-3 a^2 c^6+c^8

X(5648) is the radical center of the the circumcircle, the circle {{X(13), X(15), X(5463), X(5464)}} and the circle {{X(14), X(16), X(5463), X(5464)}}. (Randy Hutson, August 26, 2014)

X(5648) lies on these lines: {2,2854}, {3,67}, {6,5642}, {110,524}, {125,5646}, {141,5888}, {392,2836}, {511,5655}, {523,5653}, {526,5652}, {541,1350}, {543,2453}, {597,895}, {2781,5656}

X(5648) = crossdifference of every pair of points on the line X(2492)X(2780)
X(5648) = circumcircle-inverse of X(34013)
X(5648) = Thomson-isogonal conjugate of X(7464)

X(5649) = ISOGONAL CONJUGATE OF X(1640)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b² - c²)(2a⁶ - b⁶ - c⁶ - 2a⁴b² - 2a⁴c² + a²b⁴ + a²c⁴ + b⁴c² + b²c⁴)]
Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cot A)/[sin 3B csc 2B tan C cos(C + ω) - sin 3C csc 2C tan B cos(B + ω)]

X(5649) is the trilinear pole of the line X(23)X(110); at X(110), this line is tangent to the Thomson-Gibert-Moses hyperbola and parallel to the trilinear polar of X(110). (Randy Hutson, June 16, 2014)

X(5649) lies on the hyperbola {A, B, C, X(2), X(110)} and this line: {250, 4230}

X(5649) = isogonal conjugate of X(1640)
X(5649) = isotomic conjugate of X(18312)

X(5650) = REFLECTION OF X(373) IN X(2)

Barycentrics a^2(b^4 + c^4 - a^2b^2 - a^2c^2 + 6b^2c^2) : :

Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(2), and let V be the tangent at X(3); then X(5650) = U∩V. Also, X(5650) is the centroid of the X(2)-Brocard triangle; see X(5642). (Randy Hutson, June 16, 2014)

X(5650) = pole of the Euler line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)

X(5650) lies on these lines: {2,51}, {3,1495}, {6,5646}, {110,5092}, {549,5642}

X(5650) = reflection of X(373) in X(2)
X(5650) = complement of X(5640)
X(5650) = isotomic conjugate of polar conjugate of X(33843)
X(5650) = {X(3),X(5651)}-harmonic conjugate of X(1495)

X(5651) = {X(2), X(110)}-HARMONIC CONJUGATE OF X(182)

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

Let U be the tangent to the Thomson-Gibert-Moses hyperbola at X(3), and let V be the tangent at X(6); then X(5651) = U∩V. Also, X(5651) is the inverse-in-Thomson-Gibert-Moses hyperbola of X(182). (Randy Hutson, June 16, 2014)

X(5651) = pole of the Brocard axis line with respect to the Thomson-Gibert-Moses hyperbola. (Peter Moses, June 17, 2014)

X(5651) lies on these lines: {2,98}, {3,1495}, {6,373}

X(5651) = crossdifference of every pair of points on the line X(1499)X(3569)
X(5651) = {X(1495),X(5650)}-harmonic conjugate of X(3)

X(5652) = X(4) OF TRIANGLE X(2)X(3)X(6)

Barycentrics (b^2 - c^2)[3a^6 - 2a^4(b^2 + c^2) + a^2(b^4 - 3b^2c^2 + c^4) + b^2c^2(b^2 + c^2)] : :

X(5652) = antipode of X(5653) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)

X(5652) lies on the circle {{X(3), X(6), X(110), X(353), X(843)}}. (Randy Hutson, November 22, 2014)

X(5652) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2,3111}, {3,669}, {6,523}, {83,5466}, {99,110}

X(5652) = reflection of X(5653) in X(5642)
X(5652) = Thomson-isogonal conjugate of X(7418)

X(5653) = X(4) OF TRIANGLE X(3)X(6)X(110)

Barycentrics a^2(b^2 - c^2)[a^10 + a^8(b^2 + c^2) - a^6(5b^4 + 8b^2c^2 + 5c^4) + a^4(b^6 + 15b^4c^2 + 15b^2c^4 + c^6) + a^2(4b^8 - 14b^6c^2 - 3b^4c^4 - 14b^2c^6 + 4c^8) - 2b^10 + 4b^8c^2 + b^6c^4 + b^4c^6 + 4b^2c^8 - 2c^10] : :

X(5653) = antipode of X(5652) in the Thomson-Gibert-Moses hyperbola. (Randy Hutson, June 16, 2014)

X(5653) lies on the Thomson-Gibert-Moses hyperbola, the circle {{X(2), X(3), X(6), X(111), X(691)}}, and these lines: {2,690}, {3,351}, {6,526}, {110,249}

X(5653) is denoted QA-P9 (QA-Miquel Center) of the quadrangle X(13)X(14)X(15)X(16); see Chris van Tienhoven's site.

X(5653) = crossdifference of every pair of points on the line X(542)X(1648)
X(5653) = reflection of X(5652) in X(5642)

X(5654) = INTERSECTION OF LINES X(4)X(110) AND X(5)X(6)

Barycentrics (a^2 - b^2 - c^2)[a^8 - 4a^6(b^2 + c^2) + 4a^4(b^4 - b^2c^2 + c^4) - (b^2 - c^2)^4] : :

X(5654) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(22). (Peter Moses, June 18, 2014)

X(5654) lies on these lines: {3,4549}, {4,110}, {5,6}, {11,3157}, {12,1069}, {30,154}, {52,3542}, {140,5646}, {184,1568}, {185,3548}, {354,912}, {381,3167}, {382,1514}, {403,1993}, {539,3545}, {1216,3547}, {1656,5544}, {1899,2072}, {3089,5446}, {3090,5449}, {3549,5562}, {5055,5644}, {5056,5645}

X(5655) = INTERSECTION OF LINES X(6)X(13) AND X(30)X(110)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^10 - 6a^8(b^2 + c^2) + a^6(11b^4 - b^2c^2 + 11c^4) - a^4(7b^6 - 2b^4c^2 - 2b^2c^4 + 7c^6) + 8a^2b^2c^2(b^2 - c^2)^2 + (b^2 - c^2)^4(b^2 + c^2) : :

Let A′B′C′ be the orthocentroidal triangle. Let L be the lines through A′ parallel to the Euler line, and define M and N cyclically. Let L′ be the reflection of L in sideline BC, and define M′ and L′ cyclically. The lines L′,M′,N′ concur in X(5655); c.f. X(i) for i = 74, 113, 265, 399, 1147, 1511, 5504, 5609. (Randy Hutson, August 26, 2014)

X(5655) is the center of the perspeconic of the orthocentroidal and anti-orthocentroidal triangles (see preamble before X(15254)). This conic is a circle. (Randy Hutson, December 2, 2017)

X(5655) lies on these lines: {3,541}, {4,5609}, {5,5643}, {6,13}, {30,110}, {74,549}, {125,5055}, {146,376}, {154,2777}, {354,2771}, {1539,3543}, {3167,3830}, {3448,3545}, {5054,5646}

X(5655) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(23). (Peter Moses, June 18, 2014)
X(5655) = antipode of X(3) in Thomson-Gibert-Moses hyperbola

X(5656) = INTERSECTION OF LINES X(4)X(6) AND X(20)X(110)

Barycentrics a^10 - 7a^8(b^2 + c^2) + 2a^6(7b^4 - 6b^2c^2 + 7c^4) - 10a^4(b^2 - c^2)^2(b^2 + c^2) + a^2(b^2 - c^2)^2(b^4 + 14b^2c^2 + c^4) + (b^2 - c^2)^4(b^2 + c^2) : :

X(5656) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(25). (Peter Moses, June 18, 2014)

X(5656) lies on these lines: {4,6}, {5,5544}, {20,110}, {30,3167}, {64,631}, {154,376}, {185,3089}, {193,1533}, {221,1058}, {354,5603}, {378,1619}, {381,5644}, {1056,2192}, {1853,3545}, {3091,5643}, {3357,3523}, {3832,5645}

X(5657) = INTERSECTION OF LINES X(3)X(8) AND X(4)X(9)

Barycentrics a^4 + 2a^3(b + c) - 2a^2(b + c)^2 - 2a(b - c)^2(b + c) + (b^2 - c^2)^2 : :
X(5657) = 2 X(3) + X(8)

Let A* be the parabola with focus A and directrix BC, and let A** be the polar of X(1) with respect to A*. Define B** and C** cyclically, and let A′ = B**∩C**, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(1000), and the centroid of A′B′C′ is X(5657). (Randy Hutson, July 7, 2014)

X(5657) lies on these lines: {1,631}, {2,392}, {3,8}, {4,9}, {5,962}, {7,495}, {12,4295}, {20,355}, {21,5554}, {35,3486}, {36,3476}, {43,1064}, {46,388}, {55,1006}, {57,1056}, {63,2096}, {65,3085}, {80,4302}, {140,1482}, {145,1385}, {165,376}, {191,2950}, {201,1148}, {226,2093}, {333,4221}, {387,1108}, {484,1478}, {497,1737}, {498,3485}, {499,5445}, {519,3158}, {549,3241}, {580,5264}, {581,3293}, {601,5247}, {602,5255}, {912,3681}, {938,3295}, {946,1698}, {993,2077}, {999,5435}, {1012,5273}, {1058,1210}, {1072,1738}, {1083,2726}, {1125,3525}, {1155,4293}, {1376,3428}, {1387,4345}, {1483,3530}, {1519,3452}, {1532,3820}, {1537,5328}, {1699,3545}, {1714,3987}, {1770,5229}, {1829,3088}, {1836,5183}, {1837,4294}, {1872,4194}, {1902,3089}, {2098,5433}, {2099,5432}, {3035,5289}, {3057,3086}, {3214,4300}, {3240,5396}, {3522,4678}, {3526,5550}, {3528,3626}, {3533,3624}, {3560,5260}, {3634,4301}, {3651,5584}, {3817,3828}, {3869,5552}, {4292,5128}, {4305,5217}, {4642,5230}, {5084,5250}, {5251,5537}, {5258,5450}, {5442,5559}

X(5657) = midpoint of X(165) and X(3679)
X(5657) = reflection of X(i) in X(j) for these (i,j): (376, 165), (3817, 3828)
X(5657) = anticomplement of X(5886)
X(5657) = crossdifference of every pair of points on the line X(1459)X(3310)
X(5657) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(55). (Peter Moses, June 18, 2014) X(5657) = {X(10),X(40)}-harmonic conjugate of X(4)
X(5657) = centroid of antipedal triangle of X(7)

X(5658) = INTERSECTION OF LINES X(1)X(4) AND X(2)X(971)

Barycentrics a^6 - 4a^5(b + c) + a^4(3b^2 + 2bc + 3c^2) + 4a^3(b - c)^2(b + c) - 5a^2(b^2 - c^2)^2 + 8abc(b - c)^2(b + c) + (b - c)^4(b + c)^2 : :

X(5658) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(56). (Peter Moses, June 18, 2014)

X(5658) is the centroid of the antipedal triangle of X(8). Let A_AA_BA_C be the orthic triangle of the A-extouch triangle, let B_AB_BB_C be the orthic triangle of the B-extouch triangle, and let C_AC_BC_C be the orthic triangle of the C-extouch triangle. Let A′ be the centroid of A_AB_AC_A, let B′ be the centroid of A_BB_BC_B, and let C′ be the centroid of A_CB_CC_C. Then triangle A′B′C′ is homothetic to ABC with center of homothety X(7308), and X(5658) = X(84)-of-A′B′C′. (Randy Hutson, July 7, 2014)

X(5658) lies on these lines: {1,4}, {2,971}, {9,2272}, {20,5440}, {84,631}, {100,329}, {516,3158}, {1538,5274}, {1709,5218}, {3305,3358}

X(5659) = INTERSECTION OF LINES X(1)X(140) AND X(9)X(1699)

Barycentrics a^6 - a^5(b + c) - a^4(3b^2 + bc + 3c^2) + 2a^3(b + c)(2b^2 - bc + 2c^2) + a^2(b - c)^2(b^2 - bc + c^2) - a(b - c)^2(b + c)(3b^2 - 2bc + 3c^2) + (b - c)^4(b + c)^2 : :

X(5659) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(35). (Peter Moses, June 18, 2014)

X(5659) lies on these lines: {1,140}, {9,1699}, {100,4847}, {515,3651}, {3925,5536}

X(5660) = INTERSECTION OF LINES X(1)X(5) AND X(100)X(516)

Barycentrics a^6 - 3a^5(b + c) + a^4(b^2 + 7bc + c^2) + 2a^3(b - 2c)(2b - c)(b + c) - 3a^2(b - c)^2(b^2 + 3bc + c^2) - a(b - c)^2(b + c)(b^2 - 6bc + c^2) + (b - c)^4(b + c)^2 : :

X(5660) = isogonal-conjugate-with-respect-to-the-Thomson-triangle of X(36). (Peter Moses, June 18, 2014)

Let A′B′C′ be the orthic triangle. Let A* be the antiorthic axis of triangle AB′C′, and define B* and C* cyclically. Let A″ = B*∩C*, and define B″ and C″ cyclically. Then X(5660) = X(165)-of-A″B″C″. (Randy Hutson, July 7, 2014)

X(5660) lies on these lines: {1,5}, {2,2801}, {9,1768}, {100,516}, {104,5251}, {153,214}, {528,1699}, {1512,4867}, {1537,5541}, {1538,3689}, {1639,2826}

X(5661) = MINIMIZER ON BROCARD AXIS OF x² + y² + z²

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(a⁶b⁴ + a⁶c⁴ - 2a⁴b⁶ - 2a⁴c⁶ + a²b⁸ + a²c⁸ - a²b⁶c² - a²b²c⁶ + 2a²b⁴c⁴ + b⁸c² + b²c⁸ - b⁶c⁴ - b⁴c⁶)

Suppose that P = p : q : r and U = u : v : w (homogeneous barycentric coordinates). Let (x,y,z) be normalized barycentric coordinates for an arbitrary point X. The point T on the line PU that minimizes x² + y² + z² is given (Peter Moses, June 23, 2014) by

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)

X(5661) is the minimizer T when P = X(3) and Q = X(6), so that PU is the Brocard axis. Other examples follow:

If PU = Lemoine axis (P = X(187), Q = X(237)), then T = X(3229)
If PU = orthic axis (P = X(230), Q = X(223)), then T = X(230)
If PU = anti-orthic axis (P = X(44), Q = X(513)), then T = X(1575)
If PU = De Longchamps line (P = X(325), Q = X(523)), then T = X(325)
If PU = Gergonne line (P = X(241), Q = X(514)), then T = X(3008)
If P = X(1) and Q = X(3), then T = X(5662)
If P = X(1) and Q = X(6), then T = X(5701)

X(5661) lies on these lines: {2,647}, {3,6}, {3672,4235}

X(5661) = crossdifference of every pair of points on the line X(237)X(523)
X(5661) = crosssum of X(6) and X(5091)

X(5662) = MINIMIZER ON LINE X(1)X(3) OF x² + y² + z²

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a⁵b² + a⁵c² - 2a⁴b²c - 2a⁴bc² - 2a³b⁴ - 2a³c⁴ + 3a³b³c + 3a³bc³ + a²b⁴c + a²bc⁴ - a²b³c² -a²b²c³ + ab⁶ + ac⁶ - 3ab⁵c - 3abc⁵ + 3ab⁴c² + 3ab²c⁴ - 2ab³c³ + b⁶c + bc⁶ - b⁵c² - b²c⁵)

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv)

X(5662) is the minimizer T when P = X(1) and Q = X(3).

X(5662) lies on these lines: {1,3}, {2,905}, {63,2427}

X(5663) = ISOGONAL CONJUGATE OF X(477)

Trilinears 4 cos A + cos(A - 2B) + cos(A - 2C) - 3 cos(B - C) : :

Let A′B′C′ be the orthocentroidal triangle. Let A″ be the reflection of A in line B′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5663). (Randy Hutson, July 11, 2019)

X(5663) lies on the line at infinity and these lines: {1,3024}, {3,74}, {4,94}, {5,113}, {23,3581}, {26,1498}, {30,511}, {40,2940}, {49,3043}, {51,3845}, {52,3627}, {67,1352}, {64,155}, {182,4550}, {389,546}, {500,3746}, {548,1216}, {895,1351}, {1147,3357}, {1350,2930}, {1353,5095}, {1597,5093}, {1625,3269}, {2088,2493}, {3567,3843}, {3850,5462}

X(5663) = isogonal conjugate of X(477)
X(5663) = X(2693)-Ceva conjugate of X(3)
X(5663) = anticomplementary conjugate of X(34193)
X(5663) = X(477)-anticomplementary conjugate of X(8)
X(5663) = complementary conjugate of X(10)
X(5663) = crossdifference of X(6) and X(1637)
X(5663) = crosssum of X(i) and X(j) for these {I,J}: {55, 3013}, {523,3154}
X(5663) = X(3)-vertex conjugate of X(526)
X(5663) = trilinear product X(2410)*X(2624)
X(5663) = barycentric product X(i)*X(j) for these {I,J}: {526,2410}, {2437,3268}
X(5663) = Thomson-isogonal conjugate of X(476)
X(5663) = Lucas-isogonal conjugate of X(476)
X(5663) = X(952)-of-orthic-triangle if ABC is acute
X(5663) = Cundy-Parry Phi transform of X(14385)
X(5663) = Cundy-Parry Psi transform of X(14254)
X(5663) = X(92)-isoconjugate of X(32663)

X(5664) = CENTER OF CIRCLE {X(3), X(5613), X(5617)}

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b² - c²)(2a⁴ - b⁴ - c⁴ - a²b² - a²c² + 2b²c²)(b² + c² - a² - bc)(b² + c² - a² + bc) (Peter Moses, June 20, 2014)

Let U be the circle (X(5617), R) and V the circle (X(5613),R), so that U passes through X(14) and V passes through X(13), and let Γ be the circumcircle of ABC. Then X(5664) is the radical center of the circles U, V, and Γ. (Dao Thanh Oai, ADGEOM #1256, April 23, 2014)

X(5664) is the isotomic conjugate of the trilinear pole of line X(30)X(74). (Randy Hutson, July 7, 2014)

X(5664) lies on these lines: {2, 525}, {3, 523}, {39, 2485}, {99, 5649}, {113, 114}, {140, 5489}, {323, 2411}, {2407, 2420}, {2482, 2799}, {2848, 3184}

X(5664) = isotomic conjugate of X(39290)
X(5664) = complement of X(2394)
X(5664) = anticomplement of X(14566)

X(5665) = ISOGONAL CONJUGATE OF X(3601)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(s - a)(s² - S_A)]

Let A′B′C′ be the intouch triangle of a triangle ABC. Let (BA′) be the circle having diameter BA′, and likewise for (A′C); define circles (CB′) and (AC′) cyclically, and define circles (B′A) and (C′B) cyclically. Let U be the point, other than A′, in which (BA′) meets the incircle, and define V and W cyclically. Let U′ be the point, other than A′, in which (A′C) meets the incircle, and define V′ and W′ cyclically. Let A″B″C″ be the triangle formed by the lines UU′, VV′, WW′. Let A″' be the point, other than A, in which the circles (AB′) and (AC′) meet, and define B‴ and C‴ cyclically. Then A″B″C″ is perspective to ABC, and the perspector is X(5665). Moreover, A″B″C″ is perspective to A″B″C″, and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)

X(5665) lies on the Feuerbach hyperbola and these lines: (1, 1427), (4, 3671), (7, 950), (8, 226), (9, 65), (21, 57), (34, 1172), (40, 943), (72, 4866), (79, 3586), (84, 942), (85, 314), (104, 3333), (388, 3243), (405, 3339), (728, 2171), (946, 3427), (1000, 3487), (1420, 2320), (1490, 3577), (1697, 2346), (1728, 3467), (1896, 5342), (2099, 2900), (2263, 2298), (2335, 3247), (3296, 3488), (3419, 5290), (3600, 5558), (3612, 5424), (4332, 5269), (4355, 5557)

X(5665) = isogonal conjugate of X(3601)

X(5666) = INTERSECTION OF LINES X(1)X(1427) AND X(1054)X(3339)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [s(s² + 4r² + 2bc) - (b + c)(s² + 2r²)]/(s - a)

Let A′B′C′ be the intouch triangle of a triangle ABC. Let (BA′) be the circle having diameter BA′, and likewise for (A′C); define circles (CB′) and (AC′) cyclically, and define circles (B′A) and (C′B) cyclically. Let U be the point, other than A′, in which (BA′) meets the incircle, and define V and W cyclically. Let U′ be the point, other than A′, in which (A′C) meets the incircle, and define V′ and W′ cyclically. Let A″B″C″ be the triangle formed by the lines UU′, VV′, WW′. Let A‴ be the point, other than A, in which the circles (AB′) and (AC′) meet, and define B‴ and C‴ cyclically. Then A″B″C″ is perspective to ABC, and the perspector is X(5665). Moreover, A″B″C″ is perspective to A″B″C″, and the perspector is X(5666). (César Lozada ADGEOM #1155, March 8, 2014)

X(5666) lies on these lines: (1, 1427), (1054, 3339), (1707, 3361)

X(5667) = X(30)-CEVA CONJUGATE OF X(4)

Barycentrics [a^12 - a^10(b^2 + c^2) + a^8(3b^4 - 5b^2c^2 + 3c^4) - 6a^6(b^2 - c^2)^2(b^2 + c^2) + a^4(b^2 - c^2)^2(3b^4 + 20b^2c^2 + 3c^4) - a^2(b - c)^2(b + c)^2(b^2 + c^2)(b^4 + 6b^2c^2 + c^4) + (b^2 - c^2)^4(b^4 - 5b^2c^2 + c^4)]/(b^2 - c^2) : :

As X(30)-Ceva conjugates of points on the Neuberg cubic (K001), the points X(5667) - X(5685) are also on the Neuberg cubic. Specifically, if P is a point on the Neuberg cubic, then the points X(74), P, and the X(30)-Ceva conjugate of P are collinear, since X(74) is the isopivot (or secondary pivot) of the cubic. See Table 9: Points on the Neuberg Cubic.

X(5667) inverse-in-circumconic-centered-at-X(4) of X(133); also, X(5667) is the antipode of X(4) in the bianticevian conic of X(1) and X(4). (Randy Hutson, July 7, 2014)

X(5667) lies on the Neuberg cubic and these lines: {1,2816}, {3,3462}, {4,74}, {19,2822}, {20,1075}, {112,376}, {122,631}, {146,4240}, {399,2133}, {1138,1157}, {1148,4302}, {1263,3481}, {2790,3186}, {3087,3269}, {3183,3529}, {3324,4293}, {3440,3479}, {3441,3480}

X(5667) = reflection of X(I() in X(j) for these (i,j): (4,107), (1294,3184)
X(5667) = antigonal conjugate of X(34298)

X(5668) = X(30)-CEVA CONJUGATE OF X(15)

Barycentrics a^2*(2*S+(-a^2+b^2+c^2)*sqrt(3))*(-2*(a^10+(b^2+c^2)*a^8-(8*b^4-9*b^2*c^2+8*c^4)*a^6+2*(b^2+c^2)*(4*b^4-7*b^2*c^2+4*c^4)*a^4-(b^2-c^2)^2*(b^4+9*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3)*S+sqrt(3)*(a^2+b^2-c^2)*(a^8-4*(b^2+c^2)*a^6+(6*b^4+b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*(a^2-b^2+c^2)) : :

X(5668) lies on the Neuberg cubic and these lines: {3,3166}, {4,14}, {13,3479}, {15,74}, {16,1495}, {61,185}, {1277,3466}, {2133,5623}, {3284,5669}

X(5668) = reflection of X(5669) in X(3284)
X(5668) = anticomplement of X(33501)
X(5668) = polar-circle-inverse of X(35715)
X(5668) = homothetic center of X(14)- and X(16)-Ehrmann triangles; see X(25)

X(5669) = X(30)-CEVA CONJUGATE OF X(16)

Barycentrics a^2*(-2*S+(-a^2+b^2+c^2)*sqrt(3))*(2*(a^10+(b^2+c^2)*a^8-(8*b^4-9*b^2*c^2+8*c^4)*a^6+2*(b^2+c^2)*(4*b^4-7*b^2*c^2+4*c^4)*a^4-(b^2-c^2)^2*(b^4+9*b^2*c^2+c^4)*a^2-(b^4-c^4)*(b^2-c^2)^3)*S+sqrt(3)*(a^2+b^2-c^2)*(a^8-4*(b^2+c^2)*a^6+(6*b^4+b^2*c^2+6*c^4)*a^4-(b^2+c^2)*(4*b^4-5*b^2*c^2+4*c^4)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*(a^2-b^2+c^2)) : :

X(5669) lies on the Neuberg cubic and these lines: {3,3165}, {4,13}, {14,3480}, {15,1495}, {16,74}, {62,185}, {1276,3466}, {2133,5624}, {3284,5668}

X(5669) = reflection of X(5668) in X(3284)
X(5669) = anticomplement of X(33499)
X(5669) = polar-circle-inverse of X(35714)
X(5669) = homothetic center of X(13)-Ehrmann triangle and X(15) triangle; see X(25)

X(5670) = X(30)-CEVA CONJUGATE OF X(1138)

Barycentrics (a^8+2*(b^2-2*c^2)*a^6-(6*b^4-b^2*c^2-6*c^4)*a^4+(b^2-c^2)*(2*b^4+3*b^2*c^2+4*c^4)*a^2+(b^2-c^2)^4)*(a^8-2*(2*b^2-c^2)*a^6+(6*b^4+b^2*c^2-6*c^4)*a^4-(b^2-c^2)*(4*b^4+3*b^2*c^2+2*c^4)*a^2+(b^2-c^2)^4)*(5*a^24-27*(b^2+c^2)*a^22+3*(10*b^4+53*b^2*c^2+10*c^4)*a^20+(b^2+c^2)*(121*b^4-472*b^2*c^2+121*c^4)*a^18-3*(153*b^8+153*c^8-(81*b^4+329*b^2*c^2+81*c^4)*b^2*c^2)*a^16+3*(b^2+c^2)*(222*b^8+222*c^8-(78*b^4+353*b^2*c^2+78*c^4)*b^2*c^2)*a^14-(420*b^12+420*c^12+(1260*b^8+1260*c^8-(1143*b^4+1223*b^2*c^2+1143*c^4)*b^2*c^2)*b^2*c^2)*a^12-6*(b^2+c^2)*(9*b^12+9*c^12-(261*b^8+261*c^8-(389*b^4-272*b^2*c^2+389*c^4)*b^2*c^2)*b^2*c^2)*a^10+3*(b^2-c^2)^2*(93*b^12+93*c^12-2*(81*b^8+81*c^8+(199*b^4+80*b^2*c^2+199*c^4)*b^2*c^2)*b^2*c^2)*a^8-(b^4-c^4)*(b^2-c^2)*(191*b^12+191*c^12-2*(98*b^8+98*c^8+(232*b^4-307*b^2*c^2+232*c^4)*b^2*c^2)*b^2*c^2)*a^6+3*(b^2-c^2)^4*(18*b^12+18*c^12+(63*b^8+63*c^8+(17*b^4-61*b^2*c^2+17*c^4)*b^2*c^2)*b^2*c^2)*a^4-3*(b^2-c^2)^6*(b^2+c^2)*(b^8+c^8+4*(4*b^4+5*b^2*c^2+4*c^4)*b^2*c^2)*a^2-(b^6-c^6)*(b^2-c^2)^9) : :

X(5670) lies on the Neuberg cubic and these lines: {3,2133}, {74,1138}

X(5671) = X(30)-CEVA CONJUGATE OF X(1263)

Barycentrics (a^6-(b^2+3*c^2)*a^4-(b^4-b^2*c^2-3*c^4)*a^2+(b^2-c^2)^3)*(a^6-(3*b^2+c^2)*a^4+(3*b^4+b^2*c^2-c^4)*a^2-(b^2-c^2)^3)*(4*a^16-17*(b^2+c^2)*a^14+(19*b^4+58*b^2*c^2+19*c^4)*a^12+(b^2+c^2)*(19*b^4-100*b^2*c^2+19*c^4)*a^10-(65*b^8+65*c^8-2*(38*b^4+27*b^2*c^2+38*c^4)*b^2*c^2)*a^8+(b^2+c^2)*(61*b^8+61*c^8-(120*b^4-109*b^2*c^2+120*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(23*b^8+23*c^8+(28*b^4-3*b^2*c^2+28*c^4)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^3*(b^4+16*b^2*c^2+c^4)*a^2+(b^2-c^2)^8) : :

X(5671) is the tangential of X(3065) on the Neuberg cubic.

X(5671) lies on the Neuberg cubic, the Lester circle, and these lines: {3,1138}, {30,1117}, {74,1263}, {2133,3484}>

X(5671) = X(1117)-Ceva conjugate of X(3471)

X(5672) = X(30)-CEVA CONJUGATE OF X(1276)

Barycentrics a*(2*(-a+b+c)*S+(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*sqrt(3))*(-2*(a^10-3*(b+c)*a^9+(b^2-4*b*c+c^2)*a^8+6*(b^3+c^3)*a^7-(8*b^4+8*c^4-b*c*(4*b^2+9*b*c+4*c^2))*a^6+3*(b+c)*(3*b^2-7*b*c+3*c^2)*b*c*a^5+(8*b^4+8*c^4+b*c*(19*b^2+24*b*c+19*c^2))*(b-c)^2*a^4-6*(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(2*b^2+b*c+2*c^2))*a^3-(b^2-c^2)^2*(b^4+c^4+b*c*(2*b^2+9*b*c+2*c^2))*a^2+3*(b^2-c^2)^2*(b+c)*(b^4+c^4-b*c*(b^2-3*b*c+c^2))*a-(b^2-c^2)^3*(b+c)*(b^3-c^3))*S+sqrt(3)*(a^12-(b+c)*a^11-2*(2*b^2-b*c+2*c^2)*a^10+(b+c)*(5*b^2-8*b*c+5*c^2)*a^9+(b^2+b*c+c^2)*(5*b^2-7*b*c+5*c^2)*a^8-(b+c)*(10*b^4+10*c^4-b*c*(23*b^2-27*b*c+23*c^2))*a^7-(3*b^4+3*c^4+b*c*(3*b^2-8*b*c+3*c^2))*b*c*a^6+(b^2-c^2)*(b-c)*(10*b^4+10*c^4-b*c*(b^2-21*b*c+c^2))*a^5-(b^2-c^2)^2*(5*b^4+5*c^4-b*c*(b^2+3*b*c+c^2))*a^4-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(5*b^4+5*c^4+2*b*c*(5*b^2-2*b*c+5*c^2))*b*c)*a^3+(b^2-c^2)*(b+c)^3*(b^3-c^3)*(4*b^2-7*b*c+4*c^2)*a^2+(b^2-c^2)^3*(b-c)*(b^4+c^4+b*c*(3*b^2+b*c+3*c^2))*a-(b^2-c^2)^3*(b+c)^3*(b^3-c^3))) : :

X(5672) is the Gibert-Burek-Moses-concurrent-circles image of X(13). (Randy Hutson, July 7, 2014)

X(5672) lies on the Neuberg cubic and these lines: {1,13}, {16,3065}, {74,1276}, {1277,3440}, {1138,5673}, {2940,2953}

X(5673) = X(30)-CEVA CONJUGATE OF X(1277)

Barycentrics a*(-2*(-a+b+c)*S+(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))*sqrt(3))*(2*(a^10-3*(b+c)*a^9+(b^2-4*b*c+c^2)*a^8+6*(b^3+c^3)*a^7-(8*b^4+8*c^4-b*c*(4*b^2+9*b*c+4*c^2))*a^6+3*(b+c)*(3*b^2-7*b*c+3*c^2)*b*c*a^5+(8*b^4+8*c^4+b*c*(19*b^2+24*b*c+19*c^2))*(b-c)^2*a^4-6*(b^2-c^2)*(b-c)*(b^4+c^4+b*c*(2*b^2+b*c+2*c^2))*a^3-(b^2-c^2)^2*(b^4+c^4+b*c*(2*b^2+9*b*c+2*c^2))*a^2+3*(b^2-c^2)^2*(b+c)*(b^4+c^4-b*c*(b^2-3*b*c+c^2))*a-(b^2-c^2)^3*(b+c)*(b^3-c^3))*S+sqrt(3)*(a^12-(b+c)*a^11-2*(2*b^2-b*c+2*c^2)*a^10+(b+c)*(5*b^2-8*b*c+5*c^2)*a^9+(b^2+b*c+c^2)*(5*b^2-7*b*c+5*c^2)*a^8-(b+c)*(10*b^4+10*c^4-b*c*(23*b^2-27*b*c+23*c^2))*a^7-(3*b^4+3*c^4+b*c*(3*b^2-8*b*c+3*c^2))*b*c*a^6+(b^2-c^2)*(b-c)*(10*b^4+10*c^4-b*c*(b^2-21*b*c+c^2))*a^5-(b^2-c^2)^2*(5*b^4+5*c^4-b*c*(b^2+3*b*c+c^2))*a^4-(b^2-c^2)*(b-c)*(5*b^6+5*c^6+(5*b^4+5*c^4+2*b*c*(5*b^2-2*b*c+5*c^2))*b*c)*a^3+(b^2-c^2)*(b+c)^3*(b^3-c^3)*(4*b^2-7*b*c+4*c^2)*a^2+(b^2-c^2)^3*(b-c)*(b^4+c^4+b*c*(3*b^2+b*c+3*c^2))*a-(b^2-c^2)^3*(b+c)^3*(b^3-c^3))) : :

X(5673) is the Gibert-Burek-Moses-concurrent-circles image of X(14). (Randy Hutson, July 7, 2014)

X(5673) lies on the Neuberg cubic and these lines: {1,14}, {15,3065}, {74,1277}, {1276,3441}, {1138,5672}, {2940,2952}

X(5674) = X(30)-CEVA CONJUGATE OF X(1337)

Barycentrics a^2*(2*(a^6-6*(b^2+c^2)*a^4+(3*b^4-b^2*c^2+3*c^4)*a^2+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4))*S+(a^8-(b^2+c^2)*a^6-3*(b^4+3*b^2*c^2+c^4)*a^4+(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^2-(2*b^4-3*b^2*c^2+2*c^4)*(b^2-c^2)^2)*sqrt(3))*(2*(a^16-3*(b^2+c^2)*a^14+2*(b^4+13*b^2*c^2+c^4)*a^12+(b^2+c^2)*((b^2+c^2)^2-36*b^2*c^2)*a^10+3*(4*b^4+11*b^2*c^2+4*c^4)*b^2*c^2*a^8-(b^2+c^2)*(b^8+c^8+b^2*c^2*(9*b^4-11*b^2*c^2+9*c^4))*a^6-(b^2-c^2)^2*(2*b^8+2*c^8-b^2*c^2*(11*b^4-15*b^2*c^2+11*c^4))*a^4+3*(b^4-c^4)*(b^2-c^2)*(b^8+c^8-b^2*c^2*(5*b^4-11*b^2*c^2+5*c^4))*a^2-(b^2-c^2)^4*(b^8+c^8-b^2*c^2*(7*b^4+24*b^2*c^2+7*c^4)))*S+(a^18-6*(b^2+c^2)*a^16+3*(5*b^4+8*b^2*c^2+5*c^4)*a^14-(b^2+c^2)*(21*b^4+8*b^2*c^2+21*c^4)*a^12+3*(7*b^8+7*c^8-b^2*c^2*(4*b^4-21*b^2*c^2+4*c^4))*a^10-3*(b^2+c^2)*(7*b^8+7*c^8-b^2*c^2*(29*b^4-48*b^2*c^2+29*c^4))*a^8+(21*b^8+21*c^8-2*b^2*c^2*(13*b^4+48*b^2*c^2+13*c^4))*(b^2-c^2)^2*a^6-3*(b^4-c^4)*(b^2-c^2)*(5*b^8+5*c^8-2*b^2*c^2*(2*b^4+7*b^2*c^2+2*c^4))*a^4+3*(b^2-c^2)^4*(2*b^8+2*c^8+b^2*c^2*(8*b^4+7*b^2*c^2+8*c^4))*a^2-(b^2-c^2)^6*(b^2+c^2)*(b^4+7*b^2*c^2+c^4))*sqrt(3)) : :

X(5674) is the tangential of X(13) on the Neuberg cubic.

X(5674) lies on the Neuberg cubic and these lines: {3,3440}, {13,2981}, {74,1337}, {617,1138}, {1263,3480}

X(5675) = X(30)-CEVA CONJUGATE OF X(1338)

Barycentrics a^2*(-2*(a^6-6*(b^2+c^2)*a^4+(3*b^4-b^2*c^2+3*c^4)*a^2+(b^2+c^2)*(2*b^4-b^2*c^2+2*c^4))*S+(a^8-(b^2+c^2)*a^6-3*(b^4+3*b^2*c^2+c^4)*a^4+(b^2+c^2)*(5*b^4-2*b^2*c^2+5*c^4)*a^2-(2*b^4-3*b^2*c^2+2*c^4)*(b^2-c^2)^2)*sqrt(3))*(-2*(a^16-3*(b^2+c^2)*a^14+2*(b^4+13*b^2*c^2+c^4)*a^12+(b^2+c^2)*((b^2+c^2)^2-36*b^2*c^2)*a^10+3*(4*b^4+11*b^2*c^2+4*c^4)*b^2*c^2*a^8-(b^2+c^2)*(b^8+c^8+b^2*c^2*(9*b^4-11*b^2*c^2+9*c^4))*a^6-(b^2-c^2)^2*(2*b^8+2*c^8-b^2*c^2*(11*b^4-15*b^2*c^2+11*c^4))*a^4+3*(b^4-c^4)*(b^2-c^2)*(b^8+c^8-b^2*c^2*(5*b^4-11*b^2*c^2+5*c^4))*a^2-(b^2-c^2)^4*(b^8+c^8-b^2*c^2*(7*b^4+24*b^2*c^2+7*c^4)))*S+(a^18-6*(b^2+c^2)*a^16+3*(5*b^4+8*b^2*c^2+5*c^4)*a^14-(b^2+c^2)*(21*b^4+8*b^2*c^2+21*c^4)*a^12+3*(7*b^8+7*c^8-b^2*c^2*(4*b^4-21*b^2*c^2+4*c^4))*a^10-3*(b^2+c^2)*(7*b^8+7*c^8-b^2*c^2*(29*b^4-48*b^2*c^2+29*c^4))*a^8+(21*b^8+21*c^8-2*b^2*c^2*(13*b^4+48*b^2*c^2+13*c^4))*(b^2-c^2)^2*a^6-3*(b^4-c^4)*(b^2-c^2)*(5*b^8+5*c^8-2*b^2*c^2*(2*b^4+7*b^2*c^2+2*c^4))*a^4+3*(b^2-c^2)^4*(2*b^8+2*c^8+b^2*c^2*(8*b^4+7*b^2*c^2+8*c^4))*a^2-(b^2-c^2)^6*(b^2+c^2)*(b^4+7*b^2*c^2+c^4))*sqrt(3)) : :

X(5675) is the tangential of X(14) on the Neuberg cubic.

X(5675) lies on the Neuberg cubic and these lines: {3,3441}, {74,1338}, {616,1138}, {1263,3479}

X(5676) = X(30)-CEVA CONJUGATE OF X(2133)

Barycentrics (2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^18-2*(3*b^2-c^2)*a^16+(15*b^4+16*b^2*c^2-25*c^4)*a^14-(21*b^6-53*c^6+3*(17*b^2-5*c^2)*b^2*c^2)*a^12+(21*b^8-31*c^8+(20*b^4+99*b^2*c^2-108*c^4)*b^2*c^2)*a^10-(b^2-c^2)*(21*b^8-31*c^8-(25*b^4-72*b^2*c^2-135*c^4)*b^2*c^2)*a^8+(b^2-c^2)*(21*b^10-53*c^10-(15*b^6-55*c^6+2*(39*b^2-59*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^3*(15*b^8-25*c^8+2*(20*b^4-3*b^2*c^2-30*c^4)*b^2*c^2)*a^4+(6*b^8+2*c^8+(16*b^4+33*b^2*c^2+24*c^4)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^4+b^2*c^2+c^4)*(b^2-c^2)^7)*(3*a^36-20*(b^2+c^2)*a^34+2*(11*b^4+84*b^2*c^2+11*c^4)*a^32+2*(b^2+c^2)*(101*b^4-356*b^2*c^2+101*c^4)*a^30-(971*b^8+971*c^8-8*(38*b^4+305*b^2*c^2+38*c^4)*b^2*c^2)*a^28+2*(b^2+c^2)*(1034*b^8+1034*c^8+5*(23*b^4-524*b^2*c^2+23*c^4)*b^2*c^2)*a^26-(2405*b^12+2405*c^12+3*(2550*b^8+2550*c^8-(1413*b^4+4204*b^2*c^2+1413*c^4)*b^2*c^2)*b^2*c^2)*a^24+2*(b^2+c^2)*(611*b^12+611*c^12+5*(1047*b^8+1047*c^8-2*(383*b^4+415*b^2*c^2+383*c^4)*b^2*c^2)*b^2*c^2)*a^22+(715*b^16+715*c^16-(10098*b^12+10098*c^12+(13348*b^8+13348*c^8-(17962*b^4+9669*b^2*c^2+17962*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^20-2*(b^2+c^2)*(1144*b^16+1144*c^16-(4180*b^12+4180*c^12+(4511*b^8+4511*c^8-(12977*b^4-10856*b^2*c^2+12977*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^18+(b^2-c^2)^2*(3575*b^16+3575*c^16-(66*b^12+66*c^12+(8741*b^8+8741*c^8+2*(11609*b^4-1956*b^2*c^2+11609*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^16-2*(b^4-c^4)*(b^2-c^2)*(2405*b^16+2405*c^16-2*(2180*b^12+2180*c^12-(2787*b^8+2787*c^8-(8472*b^4-11243*b^2*c^2+8472*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^14+(b^2-c^2)^2*(5135*b^20+5135*c^20-(6518*b^16+6518*c^16-(2337*b^12+2337*c^12-2*(1678*b^8+1678*c^8+(10631*b^4-22512*b^2*c^2+10631*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^12-2*(b^4-c^4)*(b^2-c^2)^3*(1958*b^16+1958*c^16-(499*b^12+499*c^12+(1190*b^8+1190*c^8-(5985*b^4-11968*b^2*c^2+5985*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^10+(b^2-c^2)^6*(2009*b^16+2009*c^16+2*(3114*b^12+3114*c^12+(2208*b^8+2208*c^8+(3512*b^4+7701*b^2*c^2+3512*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8-2*(b^2-c^2)^6*(b^2+c^2)*(323*b^16+323*c^16+(931*b^12+931*c^12-(439*b^8+439*c^8+(685*b^4-2656*b^2*c^2+685*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^6+(b^2-c^2)^8*(110*b^16+110*c^16+(870*b^12+870*c^12+(1904*b^8+1904*c^8+(1154*b^4-57*b^2*c^2+1154*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-2*(b^2-c^2)^10*(b^2+c^2)*(2*b^12+2*c^12+(54*b^8+54*c^8+(171*b^4+275*b^2*c^2+171*c^4)*b^2*c^2)*b^2*c^2)*a^2-(b^2-c^2)^14*(b^2+b*c+c^2)^2*(b^2-b*c+c^2)^2)*(a^18+2*(b^2-3*c^2)*a^16-(25*b^4-16*b^2*c^2-15*c^4)*a^14+(53*b^6-21*c^6+3*(5*b^2-17*c^2)*b^2*c^2)*a^12-(31*b^8-21*c^8+(108*b^4-99*b^2*c^2-20*c^4)*b^2*c^2)*a^10-(b^2-c^2)*(31*b^8-21*c^8-(135*b^4+72*b^2*c^2-25*c^4)*b^2*c^2)*a^8+(b^2-c^2)*(53*b^10-21*c^10-(55*b^6-15*c^6+2*(59*b^2-39*c^2)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^3*(25*b^8-15*c^8+2*(30*b^4+3*b^2*c^2-20*c^4)*b^2*c^2)*a^4+(2*b^8+6*c^8+(24*b^4+33*b^2*c^2+16*c^4)*b^2*c^2)*(b^2-c^2)^4*a^2+(b^4+b^2*c^2+c^4)*(b^2-c^2)^7) : :

X(5676) lies on the Neuberg cubic and this line: {74,2133}

X(5677) = X(30)-CEVA CONJUGATE OF X(3065)

Barycentrics a*(a^3-(b-c)*a^2-(b^2-b*c+c^2)*a+(b+c)*(b^2-c^2))*(a^3+(b-c)*a^2-(b^2-b*c+c^2)*a-(b+c)*(b^2-c^2))*(a^15-2*(b+c)*a^14-4*(b^2+b*c+c^2)*a^13+(b+c)*(11*b^2-10*b*c+11*c^2)*a^12+3*(b^4+c^4+2*(b+c)^2*b*c)*a^11-3*(b+c)*(8*b^4+8*c^4-(11*b^2-14*b*c+11*c^2)*b*c)*a^10+2*(5*b^6+5*c^6+3*(2*b^4+2*c^4-3*(b+c)^2*b*c)*b*c)*a^9+(b+c)*(25*b^6+25*c^6-3*(11*b^4+11*c^4-6*(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*a^8-(25*b^8+25*c^8+(28*b^6+28*c^6-(23*b^4+23*c^4+3*(10*b^2+3*b*c+10*c^2)*b*c)*b*c)*b*c)*a^7-(b+c)*(10*b^8+10*c^8-(2*b^6+2*c^6-(25*b^4+25*c^4-3*(19*b^2-18*b*c+19*c^2)*b*c)*b*c)*b*c)*a^6+3*(b^2-c^2)^2*(8*b^6+8*c^6+(4*b^4+4*c^4+9*(b+c)^2*b*c)*b*c)*a^5-3*(b^6-c^6)*(b-c)*(b^4+c^4-2*(b^2+4*b*c+c^2)*b*c)*a^4-(b^2-c^2)^2*(11*b^8+11*c^8-(6*b^6+6*c^6-(13*b^4+13*c^4+3*(8*b^2-b*c+8*c^2)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(4*b^6+4*c^6+(5*b^4+5*c^4+(5*b^2-b*c+5*c^2)*b*c)*b*c)*a^2+(b^2-c^2)^4*(b-c)^2*(2*b^2+c^2)*(b^2+2*c^2)*a-(b^2-c^2)^6*(b+c)*(b^2+b*c+c^2)) : :

X(5677) lies on the Neuberg cubic and these lines: {1,1138}, {74,3065}, {1263,3466}, {2133,3465}

X(5678) = X(30)-CEVA CONJUGATE OF X(3440)

Barycentrics a^2*(2*sqrt(3)*(a^2+b^2-c^2)*S+a^4-2*(b^2-2*c^2)*a^2+(b^2-c^2)*(b^2+5*c^2))*(-2*sqrt(3)*(a^16-6*(b^2+c^2)*a^14+(14*b^4+9*b^2*c^2+14*c^4)*a^12-(b^2+c^2)*(14*b^4-11*b^2*c^2+14*c^4)*a^10+36*(b^2-c^2)^2*b^2*c^2*a^8+2*(b^2+c^2)*(7*b^8+7*c^8-(49*b^4-86*b^2*c^2+49*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(14*b^8+14*c^8-(29*b^4+102*b^2*c^2+29*c^4)*b^2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*(2*b^8+2*c^8+(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(b^8+c^8+2*(5*b^4+7*b^2*c^2+5*c^4)*b^2*c^2))*S+a^18-3*(b^2+c^2)*a^16-(6*b^4-59*b^2*c^2+6*c^4)*a^14+2*(b^2+c^2)*(21*b^4-82*b^2*c^2+21*c^4)*a^12-3*(28*b^8+28*c^8+(5*b^4-132*b^2*c^2+5*c^4)*b^2*c^2)*a^10+4*(b^2+c^2)*(21*b^8+21*c^8+4*(8*b^4-29*b^2*c^2+8*c^4)*b^2*c^2)*a^8-(b^2-c^2)^2*(42*b^8+42*c^8+(239*b^4+474*b^2*c^2+239*c^4)*b^2*c^2)*a^6+6*(b^4-c^4)*(b^2-c^2)*(b^4+c^4-2*(b^2-2*b*c-c^2)*b*c)*(b^4+c^4+2*(b^2+2*b*c-c^2)*b*c)*a^4+(3*b^8+3*c^8+(11*b^4+26*b^2*c^2+11*c^4)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^2+c^2)*(b^2-c^2)^8)*(2*sqrt(3)*(a^2-b^2+c^2)*S+a^4+2*(2*b^2-c^2)*a^2-(b^2-c^2)*(5*b^2+c^2)) : :

X(5678) lies on the Neuberg cubic and these lines: {74,3440}, {617,2133}, {1138,3480}

X(5679) = X(30)-CEVA CONJUGATE OF X(3441)

Barycentrics a^2*(-2*sqrt(3)*(a^2+b^2-c^2)*S+a^4-2*(b^2-2*c^2)*a^2+(b^2-c^2)*(b^2+5*c^2))*(2*sqrt(3)*(a^16-6*(b^2+c^2)*a^14+(14*b^4+9*b^2*c^2+14*c^4)*a^12-(b^2+c^2)*(14*b^4-11*b^2*c^2+14*c^4)*a^10+36*(b^2-c^2)^2*b^2*c^2*a^8+2*(b^2+c^2)*(7*b^8+7*c^8-(49*b^4-86*b^2*c^2+49*c^4)*b^2*c^2)*a^6-(b^2-c^2)^2*(14*b^8+14*c^8-(29*b^4+102*b^2*c^2+29*c^4)*b^2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*(2*b^8+2*c^8+(b^2-4*b*c-c^2)*(b^2+4*b*c-c^2)*b^2*c^2)*a^2-(b^2-c^2)^4*(b^8+c^8+2*(5*b^4+7*b^2*c^2+5*c^4)*b^2*c^2))*S+a^18-3*(b^2+c^2)*a^16-(6*b^4-59*b^2*c^2+6*c^4)*a^14+2*(b^2+c^2)*(21*b^4-82*b^2*c^2+21*c^4)*a^12-3*(28*b^8+28*c^8+(5*b^4-132*b^2*c^2+5*c^4)*b^2*c^2)*a^10+4*(b^2+c^2)*(21*b^8+21*c^8+4*(8*b^4-29*b^2*c^2+8*c^4)*b^2*c^2)*a^8-(b^2-c^2)^2*(42*b^8+42*c^8+(239*b^4+474*b^2*c^2+239*c^4)*b^2*c^2)*a^6+6*(b^4-c^4)*(b^2-c^2)*(b^4+c^4-2*(b^2-2*b*c-c^2)*b*c)*(b^4+c^4+2*(b^2+2*b*c-c^2)*b*c)*a^4+(3*b^8+3*c^8+(11*b^4+26*b^2*c^2+11*c^4)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^2+c^2)*(b^2-c^2)^8)*(-2*sqrt(3)*(a^2-b^2+c^2)*S+a^4+2*(2*b^2-c^2)*a^2-(b^2-c^2)*(5*b^2+c^2)) : :

X(5679) lies on the Neuberg cubic and these lines: {74,3441}, {616,2133}, {1138,3479}

X(5680) = X(30)-CEVA CONJUGATE OF X(3466)

Barycentrics a*(a^6+(b-c)*a^5-(b^2-b*c+c^2)*a^4-2*(b^3-c^3)*a^3-(b^2-c^2)^2*a^2+(b^4-c^4)*(b+c)*a+(b^2-c^2)*(b-c)*(b^3+c^3))*(a^6-(b-c)*a^5-(b^2-b*c+c^2)*a^4+2*(b^3-c^3)*a^3-(b^2-c^2)^2*a^2-(b^4-c^4)*(b+c)*a+(b^2-c^2)*(b-c)*(b^3+c^3))*(a^21+2*(b+c)*a^20-4*(b^2+b*c+c^2)*a^19-(b+c)*(11*b^2-10*b*c+11*c^2)*a^18+(3*b^4+3*c^4+10*(b+c)^2*b*c)*a^17+(b+c)*(24*b^4+24*c^4-(41*b^2-58*b*c+41*c^2)*b*c)*a^16+(9*b^6+9*c^6+(10*b^4+10*c^4-(29*b^2+56*b*c+29*c^2)*b*c)*b*c)*a^15-(b+c)*(27*b^6+27*c^6-(49*b^4+49*c^4-4*(27*b^2-38*b*c+27*c^2)*b*c)*b*c)*a^14-(21*b^8+21*c^8+(50*b^6+50*c^6+(3*b^4+3*c^4-(64*b^2+77*b*c+64*c^2)*b*c)*b*c)*b*c)*a^13+(b+c)*(21*b^8+21*c^8+(10*b^6+10*c^6+(65*b^4+65*c^4-3*(63*b^2-64*b*c+63*c^2)*b*c)*b*c)*b*c)*a^12+(21*b^8+21*c^8-2*(4*b^6+4*c^6-3*(4*b^4+4*c^4+(8*b^2-29*b*c+8*c^2)*b*c)*b*c)*b*c)*(b+c)^2*a^11-(b^2-c^2)*(b-c)*(21*b^8+21*c^8+5*(b^2+c^2)*(19*b^4+19*c^4+(27*b^2+8*b*c+27*c^2)*b*c)*b*c)*a^10-(b^2-c^2)^2*(21*b^8+21*c^8-(34*b^6+34*c^6-(23*b^4+23*c^4+3*(48*b^2+29*b*c+48*c^2)*b*c)*b*c)*b*c)*a^9+(b^2-c^2)*(b-c)*(21*b^10+21*c^10+(52*b^8+52*c^8+(44*b^6+44*c^6+(149*b^4+149*c^4+(203*b^2+102*b*c+203*c^2)*b*c)*b*c)*b*c)*b*c)*a^8+(b^2-c^2)^2*(27*b^10+27*c^10-(50*b^8+50*c^8+(21*b^6+21*c^6+2*(6*b^4+6*c^4-(13*b^2+102*b*c+13*c^2)*b*c)*b*c)*b*c)*b*c)*a^7-(b^2-c^2)^2*(b+c)*(9*b^10+9*c^10-(31*b^8+31*c^8-(46*b^6+46*c^6+(34*b^4+34*c^4-(67*b^2-114*b*c+67*c^2)*b*c)*b*c)*b*c)*b*c)*a^6-(b^2-c^2)^4*(24*b^8+24*c^8-(10*b^6+10*c^6-(37*b^4+37*c^4-(104*b^2-7*b*c+104*c^2)*b*c)*b*c)*b*c)*a^5-(b^2-c^2)^4*(b+c)*(3*b^8+3*c^8+(14*b^6+14*c^6-(31*b^4+31*c^4-(55*b^2-64*b*c+55*c^2)*b*c)*b*c)*b*c)*a^4+(b^2-c^2)^4*(b-c)^2*(11*b^8+11*c^8+2*(16*b^6+16*c^6+(40*b^4+40*c^4+(56*b^2+71*b*c+56*c^2)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^5*(b-c)*(4*b^8+4*c^8+(3*b^6+3*c^6+(3*b^4+3*c^4+(15*b^2+4*b*c+15*c^2)*b*c)*b*c)*b*c)*a^2-(b^2-c^2)^6*(b^2+b*c+c^2)^2*(2*b^2+c^2)*(b^2+2*c^2)*a-(b^3-c^3)*(b^2-c^2)^7*(b^2-b*c+c^2)^2) : :

X(5680) lies on the Neuberg cubic and these lines: {74,3466}, {484,2133}

X(5681) = X(30)-CEVA CONJUGATE OF X(3479)

Barycentrics 324*sqrt(3)*((24*a^22-288*(b^2+c^2)*a^20+6*(145*b^4+266*b^2*c^2+145*c^4)*a^18-18*(b^2+c^2)*(51*b^4+70*b^2*c^2+51*c^4)*a^16+6*(45*b^8+45*c^8+2*(65*b^4-b^2*c^2+65*c^4)*b^2*c^2)*a^14-6*(b^2+c^2)*(141*b^8+141*c^8+2*(44*b^4-465*b^2*c^2+44*c^4)*b^2*c^2)*a^12+6*(447*b^12+447*c^12+2*(144*b^8+144*c^8-(417*b^4+419*b^2*c^2+417*c^4)*b^2*c^2)*b^2*c^2)*a^10-6*(b^2+c^2)*(495*b^12+495*c^12-2*(443*b^8+443*c^8-(492*b^4-671*b^2*c^2+492*c^4)*b^2*c^2)*b^2*c^2)*a^8+6*(243*b^12+243*c^12-(184*b^8+184*c^8+(213*b^4+328*b^2*c^2+213*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^2*a^6-6*(b^4-c^4)*(b^2-c^2)*(55*b^12+55*c^12-(182*b^8+182*c^8+(217*b^4-76*b^2*c^2+217*c^4)*b^2*c^2)*b^2*c^2)*a^4+6*(12*b^12+12*c^12+(34*b^8+34*c^8-(187*b^4+222*b^2*c^2+187*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^4*a^2-6*(b^2-c^2)^6*(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*(2*b^4+11*b^2*c^2+2*c^4))*S+sqrt(3)*(12*a^24-52*(b^2+c^2)*a^22+(17*b^4-78*b^2*c^2+17*c^4)*a^20+2*(b^2+c^2)*(97*b^4+410*b^2*c^2+97*c^4)*a^18-4*(42*b^8+42*c^8+(468*b^4+799*b^2*c^2+468*c^4)*b^2*c^2)*a^16-2*(b^2+c^2)*(237*b^8+237*c^8-(933*b^4+806*b^2*c^2+933*c^4)*b^2*c^2)*a^14+(930*b^12+930*c^12-(1560*b^8+1560*c^8+(1329*b^4+454*b^2*c^2+1329*c^4)*b^2*c^2)*b^2*c^2)*a^12-2*(b^2+c^2)*(213*b^12+213*c^12-(1677*b^8+1677*c^8-(2321*b^4-1786*b^2*c^2+2321*c^4)*b^2*c^2)*b^2*c^2)*a^10-(264*b^16+264*c^16+(2496*b^12+2496*c^12-(2863*b^8+2863*c^8-2*(863*b^4-1767*b^2*c^2+863*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+6*(b^4-c^4)*(b^2-c^2)*(53*b^12+53*c^12+(183*b^8+183*c^8-(91*b^4-226*b^2*c^2+91*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^2*(83*b^16+83*c^16+(360*b^12+360*c^12-(67*b^8+67*c^8+4*(319*b^4-234*b^2*c^2+319*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)^3*(4*b^12+4*c^12-(93*b^8+93*c^8-2*(15*b^4+113*b^2*c^2+15*c^4)*b^2*c^2)*b^2*c^2)*a^2+(4*b^12+4*c^12-(48*b^8+48*c^8+(15*b^4-172*b^2*c^2+15*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^6))*((-4*a^6-2*(3*b^2+c^2)*a^4+2*(2*b^2+c^2)*(3*b^2-c^2)*a^2-2*(b^2-c^2)*(b^4-5*b^2*c^2-2*c^4))*S+sqrt(3)*(2*a^8-(5*b^2+7*c^2)*a^6+(3*b^4-3*b^2*c^2+10*c^4)*a^4+(b^2-c^2)*(b^4+10*b^2*c^2+7*c^4)*a^2-(b^2+2*c^2)*(b^2-c^2)^3))*((-4*a^6-2*(b^2+3*c^2)*a^4-2*(b^2-3*c^2)*(b^2+2*c^2)*a^2-2*(b^2-c^2)*(2*b^4+5*b^2*c^2-c^4))*S+sqrt(3)*(2*a^8-(7*b^2+5*c^2)*a^6+(10*b^4-3*b^2*c^2+3*c^4)*a^4-(b^2-c^2)*(7*b^4+10*b^2*c^2+c^4)*a^2+(2*b^2+c^2)*(b^2-c^2)^3)) : :

X(5681) lies on the Neuberg cubic and these lines: {4,3441}, {74,3479}

X(5682) = X(30)-CEVA CONJUGATE OF X(3480)

Barycentrics (-2*sqrt(3)*(4*a^22-48*(b^2+c^2)*a^20+(145*b^4+266*b^2*c^2+145*c^4)*a^18-3*(b^2+c^2)*(51*b^4+70*b^2*c^2+51*c^4)*a^16+(45*b^8+45*c^8+2*(65*b^4-b^2*c^2+65*c^4)*b^2*c^2)*a^14-(b^2+c^2)*(141*b^8+141*c^8+2*(44*b^4-465*b^2*c^2+44*c^4)*b^2*c^2)*a^12+(447*b^12+447*c^12+2*(144*b^8+144*c^8-(417*b^4+419*b^2*c^2+417*c^4)*b^2*c^2)*b^2*c^2)*a^10-(b^2+c^2)*(495*b^12+495*c^12-2*(443*b^8+443*c^8-(492*b^4-671*b^2*c^2+492*c^4)*b^2*c^2)*b^2*c^2)*a^8+(243*b^12+243*c^12-(184*b^8+184*c^8+(213*b^4+328*b^2*c^2+213*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^2*a^6-(b^4-c^4)*(b^2-c^2)*(55*b^12+55*c^12-(182*b^8+182*c^8+(217*b^4-76*b^2*c^2+217*c^4)*b^2*c^2)*b^2*c^2)*a^4+(12*b^12+12*c^12+(34*b^8+34*c^8-(187*b^4+222*b^2*c^2+187*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^4*a^2-(b^2-c^2)^6*(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*(2*b^4+11*b^2*c^2+2*c^4))*S+12*a^24-52*(b^2+c^2)*a^22+(17*b^4-78*b^2*c^2+17*c^4)*a^20+2*(b^2+c^2)*(97*b^4+410*b^2*c^2+97*c^4)*a^18-4*(42*b^8+42*c^8+(468*b^4+799*b^2*c^2+468*c^4)*b^2*c^2)*a^16-2*(b^2+c^2)*(237*b^8+237*c^8-(933*b^4+806*b^2*c^2+933*c^4)*b^2*c^2)*a^14+(930*b^12+930*c^12-(1560*b^8+1560*c^8+(1329*b^4+454*b^2*c^2+1329*c^4)*b^2*c^2)*b^2*c^2)*a^12-2*(b^2+c^2)*(213*b^12+213*c^12-(1677*b^8+1677*c^8-(2321*b^4-1786*b^2*c^2+2321*c^4)*b^2*c^2)*b^2*c^2)*a^10-(264*b^16+264*c^16+(2496*b^12+2496*c^12-(2863*b^8+2863*c^8-2*(863*b^4-1767*b^2*c^2+863*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^8+6*(b^4-c^4)*(b^2-c^2)*(53*b^12+53*c^12+(183*b^8+183*c^8-(91*b^4-226*b^2*c^2+91*c^4)*b^2*c^2)*b^2*c^2)*a^6-(b^2-c^2)^2*(83*b^16+83*c^16+(360*b^12+360*c^12-(67*b^8+67*c^8+4*(319*b^4-234*b^2*c^2+319*c^4)*b^2*c^2)*b^2*c^2)*b^2*c^2)*a^4-2*(b^4-c^4)*(b^2-c^2)^3*(4*b^12+4*c^12-(93*b^8+93*c^8-2*(15*b^4+113*b^2*c^2+15*c^4)*b^2*c^2)*b^2*c^2)*a^2+(4*b^12+4*c^12-(48*b^8+48*c^8+(15*b^4-172*b^2*c^2+15*c^4)*b^2*c^2)*b^2*c^2)*(b^2-c^2)^6)*(2*(2*a^6+(3*b^2+c^2)*a^4-(2*b^2+c^2)*(3*b^2-c^2)*a^2+(b^2-c^2)*(b^4-5*b^2*c^2-2*c^4))*S+sqrt(3)*(2*a^8-(5*b^2+7*c^2)*a^6+(3*b^4-3*b^2*c^2+10*c^4)*a^4+(b^2-c^2)*(b^4+10*b^2*c^2+7*c^4)*a^2-(b^2+2*c^2)*(b^2-c^2)^3))*(2*(2*a^6+(b^2+3*c^2)*a^4+(b^2-3*c^2)*(b^2+2*c^2)*a^2+(b^2-c^2)*(2*b^4+5*b^2*c^2-c^4))*S+sqrt(3)*(2*a^8-(7*b^2+5*c^2)*a^6+(10*b^4-3*b^2*c^2+3*c^4)*a^4-(b^2-c^2)*(7*b^4+10*b^2*c^2+c^4)*a^2+(2*b^2+c^2)*(b^2-c^2)^3)) : :

X(5682) lies on the Neuberg cubic and these lines: {4,3440}, {74,3480}

X(5683) = X(30)-CEVA CONJUGATE OF X(3481)

Barycentrics a^2*(a^12-3*(b^2+c^2)*a^10+(3*b^4+5*b^2*c^2+c^4)*a^8-2*(b^2-c^2)*(b^4+2*b^2*c^2+3*c^4)*a^6+(b^2-c^2)*(3*b^6+9*c^6+(b^2+3*c^2)*b^2*c^2)*a^4-(b^2-c^2)^3*(3*b^4+4*b^2*c^2+5*c^4)*a^2+(b^4+b^2*c^2-c^4)*(b^2-c^2)^4)*(a^12-3*(b^2+c^2)*a^10+(b^4+5*b^2*c^2+3*c^4)*a^8+2*(b^2-c^2)*(3*b^4+2*b^2*c^2+c^4)*a^6-(b^2-c^2)*(9*b^6+3*c^6+(3*b^2+c^2)*b^2*c^2)*a^4+(5*b^4+4*b^2*c^2+3*c^4)*(b^2-c^2)^3*a^2-(b^4-b^2*c^2-c^4)*(b^2-c^2)^4)*(-a^2+b^2+c^2)*(a^24-6*(b^2+c^2)*a^22+(15*b^4+28*b^2*c^2+15*c^4)*a^20-22*(b^4+b^2*c^2+c^4)*(b^2+c^2)*a^18+(27*b^8+27*c^8+(22*b^4+35*b^2*c^2+22*c^4)*b^2*c^2)*a^16-12*(b^2+c^2)*(3*b^8+3*c^8-2*(b^4+c^4)*b^2*c^2)*a^14+2*(b^2-c^2)^2*(21*b^8+21*c^8+2*(35*b^4+36*b^2*c^2+35*c^4)*b^2*c^2)*a^12-4*(b^4-c^4)*(b^2-c^2)*(9*b^8+9*c^8+(23*b^4+10*b^2*c^2+23*c^4)*b^2*c^2)*a^10+(b^2-c^2)^2*(27*b^12+27*c^12+(18*b^8+18*c^8+83*(b^4+c^4)*b^2*c^2)*b^2*c^2)*a^8-2*(b^4-c^4)*(b^2-c^2)^3*(11*b^8+11*c^8-2*(8*b^4-15*b^2*c^2+8*c^4)*b^2*c^2)*a^6+(b^2-c^2)^6*(15*b^8+15*c^8+2*(11*b^4+9*b^2*c^2+11*c^4)*b^2*c^2)*a^4-2*(b^2-c^2)^6*(b^2+c^2)*(3*b^8+3*c^8+(5*b^4-4*b^2*c^2+5*c^4)*b^2*c^2)*a^2+(b^6-c^6)*(b^2-c^2)^7*(b^4+5*b^2*c^2+c^4)) : :

X(5683) lies on the Neuberg cubic and these lines: {4,3463}, {74,3481}

X(5684) = X(30)-CEVA CONJUGATE OF X(3482)

Barycentrics a^2 (SA - S (3 / (Sqrt[3] + 3 Cot[w])) ) : :

X(5684) lies on the Neuberg cubic and these lines: {3,1263}, {74,3482}, {1138,3484}, {3065,3483}

X(5685) = X(30)-CEVA CONJUGATE OF X(3483)

Barycentrics a^2 (SA + S (3 / (Sqrt[3] - 3 Cot[w])) ) : :

X(5685) lies on the Neuberg cubic and these lines: {1,1263}, {3,3065}, {74,3483}, {1138,3465}, {3466,3482}

X(5686) = INTERSECTION OF LINES X(7)X(10) and X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a³ + 3b³ + 3c³ + 5a²b + 5a²c - ab² - ac² + 2abc + 5b²c + 5bc²

X(5686) lies on these lines: {1, 4924}, {2, 210}, {7, 10}, {8, 9}, {144, 1654}, {145, 1001}, {200, 5273}, {405, 2346}, {480, 958}, {497, 3715}, {516, 3543}, {936, 5265}, {971, 5657}, {984, 3672}, {1145, 1156}, {1445, 3600}, {1698, 5542}, {1738, 4346}, {1743, 4344}, {1757, 4307}, {1788, 3983}, {3059, 4662}, {3158, 5325}, {3189, 5302}, {3243, 3616}, {3485, 4005}, {3696, 4461}, {3711, 5218}, {3751, 3945}, {3974, 4042}, {4313, 5234}, {4321, 5435}, {4326, 4882}, {4384, 4899}, {4678, 5086}, {4847, 5274}, {5231, 5328}

X(5686) = anticomplement of X(38053)

X(5687) = PERSPECTOR OF OUTER GARCIA TRIANGLE AND TANGENTIAL 1ST CIRCUMPERP TRIANGLE

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

The outer Garcia triangle is defined at X(5587). The definition is re-stated here. Let T_AT_BT_C be the extouch triangle of a triangle ABC, and let L_A be the line perpendicular to line BC at T_A. Of the two points on L_A at distance r from T_A, let A′ be the one farther from A and let A″ be the closer. Define B′, C′ and B″, C″ cyclically. Then A′B′C′ is the outer Garcia triangle, and A″B″C″ the inner Garcia triangle. The outer triangle is introduced by Emmanuel Garcia in ADGEOM #1205 (April 2, 2014), and the inner by Garcia in ADGEOM #1212 (April 3, 2014).

X(5687) lies on the cubic K844 and these lines: {1, 474}, {2, 496}, {3, 8}, {4, 1260}, {5, 3434}, {6, 3293}, {9, 3697}, {10, 55}, {19, 3694}, {20, 3421}, {21, 3617}, {25, 3695}, {30, 3436}, {31, 3214}, {35, 958}, {36, 3632}, {40, 64}, {43, 5255}, {46, 518}, {56, 519}, {57, 3555}, {63, 3579}, {65, 3689}, {78, 517}, {101, 4513}, {144, 3650}, {145, 404}, {149, 4193}, {165, 3916}, {169, 3693}, {191, 4436}, {197, 2915}, {198, 2321}, {210, 1898}, {214, 1388}, {218, 3501}, {219, 3362}, {220, 1018}, {221, 4551}, {228, 5295}, {346, 4222}, {355, 1012}, {377, 495}, {382, 5080}, {390, 5084}, {392, 936}, {442, 954}, {480, 516}, {497, 4187}, {498, 2886}, {499, 3035}, {528, 1329}, {529, 4299}, {573, 3713}, {595, 4383}, {612, 3931}, {668, 1975}, {859, 1043}, {899, 3915}, {910, 4515}, {942, 3870}, {960, 5119}, {976, 4642}, {986, 3961}, {993, 3626}, {997, 3057}, {1001, 1698}, {1004, 3868}, {1011, 4651}, {1054, 3976}, {1071, 3359}, {1089, 4557}, {1125, 3303}, {1148, 1897}, {1191, 3216}, {1319, 3893}, {1377, 2066}, {1378, 5414}, {1385, 3872}, {1482, 4511}, {1486, 3932}, {1500, 5275}, {1574, 2241}, {1617, 1788}, {1696, 3950}, {1722, 3749}, {1724, 3052}, {1759, 4006}, {1766, 3965}, {1783, 3172}, {2093, 4018}, {2098, 2802}, {2222, 2756}, {2271, 2295}, {2345, 4254}, {2478, 3820}, {2551, 4294}, {2894, 4197}, {3053, 5291}, {3185, 3714}, {3220, 4901}, {3241, 5253}, {3242, 3670}, {3244, 3304}, {3306, 4917}, {3550, 5247}, {3560, 5086}, {3576, 4853}, {3621, 4188}, {3625, 5204}, {3633, 5563}, {3634, 4423}, {3678, 3711}, {3681, 3927}, {3683, 3983}, {3701, 4186}, {3705, 5100}, {3715, 4015}, {3780, 5021}, {3869, 3940}, {3874, 5221}, {3881, 4860}, {3921, 4512}, {3924, 4695}, {3962, 5183}, {4185, 5300}, {4189, 4678}, {4225, 4720}, {4428, 5259}, {4640, 4662}, {4668, 5010}, {4669, 5267}, {4677, 5288}, {5044, 5250}, {5289, 5541}

X(5687) = anticomplement of X(496)
X(5687) = extouch-isogonal conjugate of X(72)
X(5687) = X(56)-of-inner-Garcia-triangle

X(5688) = PERSPECTOR OF OUTER GARCIA TRIANGLE AND OUTER GREBE TRIANGLE

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

The outer Garcia triangle is defined at X(5587).

X(5688) lies on these lines: {1, 5590}, {6, 10}, {8, 175}, {355, 1160}, {519, 5604}, {3679, 5588}

X(5689) = PERSPECTOR OF OUTER GARCIA TRIANGLE AND INNER GREBE TRIANGLE

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

The outer Garcia triangle is defined at X(5587).

X(5689) lies on these lines: {{1, 5591}, {6, 10}, {8, 176}, {355, 1161}, {519, 5605}, {3679, 5589}

X(5690) = NINE-POINT CENTER OF OUTER GARCIA TRIANGLE

Barycentrics 2*(b+c)*a^3-(b^2+4*b*c+c^2)*a^2-2*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(5690) = X[1] - 3 X[26446], 3 X[1] - 7 X[31423], 2 X[1] - 3 X[38028], 3 X[1] - 4 X[51700], 3 X[1] - 2 X[61597], X[1] + 3 X[63143], 2 X[140] - 3 X[26446], 6 X[140] - 7 X[31423], 4 X[140] - 3 X[38028], 3 X[140] - 2 X[51700], 3 X[140] - X[61597], 2 X[140] + 3 X[63143], 9 X[26446] - 7 X[31423], 9 X[26446] - 4 X[51700], 9 X[26446] - 2 X[61597], 14 X[31423] - 9 X[38028], 7 X[31423] - 4 X[51700], 7 X[31423] - 2 X[61597], 7 X[31423] + 9 X[63143], 3 X[34352] - 2 X[45081], 9 X[38028] - 8 X[51700], 9 X[38028] - 4 X[61597], X[38028] + 2 X[63143], 4 X[51700] + 9 X[63143], 2 X[61597] + 9 X[63143], and many others

The outer Garcia triangle is defined at X(5587).

X(5690) lies on these lines: {1, 140}, {2, 1482}, {3, 8}, {4, 3617}, {5, 10}, {6, 38116}, {7, 38121}, {9, 38126}, {11, 5697}, {12, 5903}, {20, 4678}, {21, 11849}, {24, 12135}, {26, 8193}, {30, 40}, {35, 7508}, {36, 10944}, {46, 5252}, {55, 10573}, {56, 12647}, {63, 31775}, {65, 495}, {72, 6735}, {78, 52265}, {80, 3467}, {119, 5692}, {142, 38041}, {145, 631}, {149, 6902}, {153, 12849}, {165, 548}, {182, 5846}, {200, 37700}, {201, 24028}, {210, 5887}, {219, 59594}, {226, 50193}, {281, 7524}, {333, 15952}, {371, 49233}, {372, 49232}, {376, 50798}, {381, 962}, {382, 6361}, {388, 24470}, {390, 59381}, {392, 17527}, {404, 22765}, {405, 5554}, {411, 18524}, {427, 41722}, {443, 2095}, {474, 10680}, {484, 7354}, {496, 1737}, {498, 2099}, {499, 1387}, {511, 49524}, {515, 550}, {516, 3627}, {518, 10915}, {519, 549}, {528, 60912}, {537, 51048}, {542, 50949}, {546, 5587}, {547, 3656}, {551, 11539}, {572, 17362}, {573, 594}, {590, 35641}, {615, 35642}, {632, 1125}, {730, 32448}, {740, 51046}, {758, 5499}, {900, 59898}, {912, 6736}, {936, 45080}, {938, 6767}, {942, 4848}, {950, 10386}, {958, 6914}, {971, 24393}, {993, 26285}, {997, 37828}, {999, 1788}, {1000, 14986}, {1001, 37622}, {1006, 3871}, {1012, 35448}, {1018, 40997}, {1056, 5708}, {1064, 3214}, {1146, 3730}, {1155, 45287}, {1159, 3487}, {1210, 9957}, {1211, 30449}, {1317, 21842}, {1320, 57298}, {1350, 59407}, {1351, 59406}, {1353, 5847}, {1376, 6924}, {1386, 38110}, {1388, 12735}, {1478, 37567}, {1479, 12019}, {1484, 2802}, {1512, 3697}, {1532, 3876}, {1537, 6941}, {1565, 33298}, {1574, 34460}, {1595, 1829}, {1596, 1902}, {1656, 5603}, {1657, 9778}, {1658, 15177}, {1697, 5722}, {1698, 3628}, {1699, 3850}, {1706, 5709}, {1766, 17275}, {1770, 5183}, {1836, 10827}, {1837, 5119}, {1872, 46878}, {1953, 21012}, {2077, 5258}, {2080, 12195}, {2093, 9578}, {2102, 57323}, {2103, 57322}, {2136, 61122}, {2361, 54350}, {2475, 16150}, {2476, 38058}, {2550, 5762}, {2551, 6929}, {2646, 36920}, {2800, 3678}, {2807, 5876}, {2826, 24093}, {2829, 33559}, {3035, 19907}, {3036, 5840}, {3058, 37563}, {3070, 35611}, {3071, 35610}, {3085, 5719}, {3090, 18493}, {3091, 61262}, {3097, 61625}, {3109, 35193}, {3146, 28182}, {3189, 28466}, {3219, 35460}, {3241, 5054}, {3243, 38122}, {3244, 10165}, {3245, 3585}, {3293, 5396}, {3295, 6883}, {3311, 19065}, {3312, 19066}, {3336, 5434}, {3340, 11374}, {3359, 24467}, {3416, 3564}, {3419, 31789}, {3421, 3927}, {3428, 11499}, {3434, 6928}, {3436, 6923}, {3474, 9655}, {3485, 31479}, {3517, 7718}, {3523, 3621}, {3524, 31145}, {3525, 3622}, {3526, 3616}, {3530, 3576}, {3533, 46934}, {3534, 34627}, {3541, 11396}, {3555, 10202}, {3560, 9708}, {3577, 11530}, {3587, 5787}, {3589, 38040}, {3600, 37545}, {3601, 37739}, {3612, 37740}, {3614, 18393}, {3623, 10303}, {3624, 16200}, {3625, 5882}, {3633, 12108}, {3634, 11230}, {3635, 58441}, {3636, 61853}, {3649, 37719}, {3653, 11812}, {3655, 4677}, {3681, 37401}, {3696, 29010}, {3703, 37619}, {3704, 48930}, {3717, 29327}, {3751, 34380}, {3753, 8728}, {3754, 25466}, {3756, 56804}, {3816, 3884}, {3826, 3918}, {3828, 15699}, {3830, 34632}, {3843, 9812}, {3845, 4745}, {3847, 6702}, {3853, 28216}, {3857, 61260}, {3858, 18483}, {3861, 9589}, {3869, 6842}, {3870, 37615}, {3874, 5885}, {3877, 4187}, {3880, 10916}, {3897, 37298}, {3911, 24928}, {3913, 10267}, {3925, 15104}, {3932, 31394}, {3940, 6825}, {4004, 5249}, {4015, 20117}, {4018, 10273}, {4026, 31395}, {4046, 10434}, {4192, 4651}, {4193, 34122}, {4220, 33091}, {4295, 9654}, {4297, 4669}, {4300, 49984}, {4311, 5122}, {4314, 51787}, {4324, 37006}, {4384, 19512}, {4420, 21740}, {4424, 50067}, {4437, 29697}, {4512, 50243}, {4646, 48847}, {4662, 6001}, {4665, 29069}, {4701, 17502}, {4711, 9943}, {4732, 29054}, {4746, 28236}, {4769, 50772}, {4816, 61792}, {4847, 31786}, {4853, 37611}, {4882, 5534}, {4995, 37571}, {4999, 46920}, {5010, 37706}, {5046, 10738}, {5050, 51192}, {5066, 7989}, {5067, 46932}, {5070, 5734}, {5071, 50872}, {5072, 9779}, {5079, 61267}, {5082, 6827}, {5086, 7491}, {5090, 6756}, {5128, 9613}, {5174, 7510}, {5176, 56288}, {5178, 28459}, {5179, 21872}, {5220, 6256}, {5223, 5843}, {5260, 7489}, {5263, 37510}, {5264, 5398}, {5267, 26086}, {5270, 11246}, {5288, 37561}, {5289, 11729}, {5305, 9620}, {5399, 37558}, {5418, 44635}, {5420, 44636}, {5426, 44254}, {5428, 8715}, {5439, 58561}, {5443, 11280}, {5446, 23841}, {5453, 37698}, {5493, 28146}, {5531, 16132}, {5541, 62354}, {5542, 38111}, {5550, 46219}, {5552, 5730}, {5599, 11253}, {5600, 11252}, {5686, 5779}, {5688, 5874}, {5689, 5875}, {5693, 37725}, {5698, 38211}, {5703, 11041}, {5710, 36754}, {5711, 44414}, {5727, 61763}, {5732, 59414}, {5752, 31785}, {5759, 59413}, {5777, 31798}, {5780, 6848}, {5791, 9623}, {5794, 54286}, {5795, 31445}, {5805, 38200}, {5853, 31658}, {5854, 6713}, {5855, 22836}, {5883, 6583}, {5884, 13145}, {5902, 15888}, {5946, 58487}, {6003, 62323}, {6175, 61552}, {6191, 7026}, {6192, 7043}, {6197, 7511}, {6200, 35843}, {6210, 33165}, {6211, 33076}, {6246, 38213}, {6259, 54156}, {6265, 11014}, {6396, 35842}, {6642, 12410}, {6666, 38043}, {6667, 38044}, {6668, 38045}, {6675, 19860}, {6690, 30147}, {6734, 6922}, {6738, 15935}, {6762, 37534}, {6765, 18443}, {6789, 53799}, {6853, 59382}, {6862, 19843}, {6868, 17784}, {6882, 14923}, {6889, 10528}, {6892, 18231}, {6906, 35000}, {6909, 26321}, {6911, 9709}, {6916, 54398}, {6918, 8158}, {6936, 20075}, {6940, 37535}, {6949, 10698}, {6951, 20060}, {6952, 35457}, {6954, 59591}, {6955, 20076}, {6958, 10527}, {6960, 48667}, {6967, 10529}, {6971, 11680}, {6973, 8165}, {6980, 11681}, {6988, 20007}, {6989, 15934}, {6999, 51353}, {7280, 37707}, {7377, 29593}, {7385, 24808}, {7412, 56877}, {7483, 61520}, {7525, 37557}, {7530, 9911}, {7534, 54294}, {7555, 9626}, {7580, 18518}, {7583, 13911}, {7584, 13973}, {7609, 61621}, {7705, 17533}, {7715, 49542}, {7962, 11373}, {7968, 13966}, {7969, 8981}, {7970, 15561}, {7971, 62218}, {7976, 11171}, {7978, 14643}, {7983, 38224}, {7984, 15061}, {7988, 12812}, {8144, 54295}, {8185, 17714}, {8197, 32146}, {8200, 11823}, {8204, 32147}, {8207, 11822}, {8214, 32177}, {8215, 32178}, {8229, 31079}, {8270, 32047}, {8582, 13600}, {8666, 32612}, {8727, 25006}, {9041, 50977}, {9534, 19543}, {9548, 59313}, {9590, 12107}, {9624, 11224}, {9669, 30305}, {9834, 35245}, {9835, 35244}, {9857, 32151}, {9864, 12783}, {9905, 50708}, {9933, 47391}, {9940, 24391}, {9947, 9949}, {10106, 37582}, {10107, 12609}, {10109, 38021}, {10124, 25055}, {10171, 61907}, {10172, 61270}, {10197, 11281}, {10248, 14269}, {10269, 12513}, {10310, 22758}, {10327, 19544}, {10389, 31436}, {10525, 22938}, {10526, 38178}, {10543, 12104}, {10572, 37568}, {10592, 12047}, {10593, 17606}, {10597, 37462}, {10625, 16980}, {10695, 57297}, {10696, 57303}, {10697, 38764}, {10699, 57327}, {10700, 57328}, {10701, 57329}, {10702, 57330}, {10703, 38776}, {10704, 57331}, {10705, 57332}, {10742, 37437}, {10791, 32134}, {10826, 12701}, {10902, 48696}, {10912, 26492}, {10955, 41686}, {10993, 15338}, {11009, 15950}, {11015, 59347}, {11019, 31792}, {11113, 26878}, {11179, 50783}, {11251, 16210}, {11277, 33858}, {11375, 25415}, {11376, 30323}, {11500, 35239}, {11501, 59317}, {11522, 35018}, {11529, 51784}, {11540, 51105}, {11684, 47032}, {11695, 58535}, {11715, 54176}, {11725, 34127}, {11735, 34128}, {11900, 32162}, {11928, 17556}, {11929, 17532}, {12102, 61257}, {12103, 37712}, {12106, 37546}, {12114, 35238}, {12115, 56879}, {12454, 45620}, {12455, 45621}, {12495, 26316}, {12514, 37290}, {12515, 12751}, {12565, 18528}, {12571, 38071}, {12575, 18527}, {12610, 17239}, {12616, 58637}, {12626, 26451}, {12627, 26341}, {12628, 26348}, {12635, 26487}, {12636, 45623}, {12637, 45624}, {12648, 16203}, {12649, 16202}, {12675, 40296}, {12737, 61566}, {12747, 13199}, {12778, 12785}, {12780, 22896}, {12781, 22851}, {12782, 22697}, {12811, 61263}, {12898, 15035}, {13099, 57304}, {13369, 31787}, {13373, 34791}, {13411, 50194}, {13463, 24387}, {13731, 17751}, {13743, 61622}, {13747, 61521}, {13883, 19117}, {13893, 13925}, {13936, 19116}, {13947, 13993}, {13996, 37726}, {14077, 15584}, {14110, 37356}, {14647, 61556}, {14666, 50925}, {14731, 36154}, {14830, 50880}, {14839, 49111}, {14891, 50804}, {14892, 30308}, {14893, 50865}, {14940, 31948}, {15026, 58469}, {15174, 37724}, {15310, 49693}, {15326, 37572}, {15462, 32298}, {15681, 50864}, {15683, 50809}, {15684, 50797}, {15686, 28208}, {15687, 28198}, {15690, 34628}, {15692, 50818}, {15693, 34748}, {15694, 38314}, {15704, 28160}, {15708, 20049}, {15713, 38068}, {15717, 20052}, {15720, 20050}, {15803, 37709}, {15844, 34353}, {15901, 30329}, {15955, 37646}, {15971, 48928}, {15973, 31778}, {16128, 61605}, {16137, 17718}, {16174, 38182}, {16189, 34595}, {16192, 34200}, {16408, 61535}, {16417, 26062}, {16475, 51732}, {16608, 21231}, {16616, 38139}, {17051, 61852}, {17057, 38109}, {17504, 34641}, {17531, 45977}, {17555, 21664}, {17564, 17614}, {17566, 34123}, {17567, 35272}, {17619, 41012}, {18253, 31649}, {18519, 37022}, {18583, 38047}, {18802, 32554}, {18857, 33956}, {18908, 40263}, {19540, 59296}, {19542, 56810}, {19547, 54433}, {19649, 33090}, {19710, 51070}, {19861, 52264}, {19862, 55859}, {19872, 61877}, {19876, 47599}, {19878, 61876}, {19883, 51077}, {20014, 61820}, {20053, 58230}, {20054, 61814}, {20057, 55863}, {20107, 23960}, {20328, 21258}, {20368, 33169}, {20420, 37584}, {20423, 38087}, {20653, 30272}, {21013, 33299}, {21075, 51362}, {21077, 44663}, {21271, 40999}, {21620, 31794}, {21850, 38165}, {22651, 51688}, {22652, 51690}, {22799, 37821}, {22937, 33856}, {24299, 41575}, {25030, 30196}, {25416, 38032}, {25440, 26286}, {25444, 61699}, {25557, 33815}, {26006, 31186}, {26129, 38084}, {26287, 51111}, {26382, 48519}, {26398, 48493}, {26406, 48520}, {26422, 48494}, {26442, 49428}, {26443, 49427}, {26444, 49086}, {26445, 49087}, {26482, 45288}, {26498, 49402}, {26507, 49401}, {26516, 49060}, {26521, 49061}, {26629, 30136}, {26938, 56960}, {27076, 59703}, {27529, 62826}, {27870, 49600}, {28150, 33697}, {28154, 62044}, {28158, 62047}, {28164, 62155}, {28168, 62159}, {28172, 62162}, {28202, 33699}, {28232, 62006}, {28538, 50979}, {28870, 48932}, {28905, 48944}, {29207, 50308}, {29298, 44824}, {29309, 48888}, {29311, 48934}, {29331, 49772}, {29667, 37360}, {30286, 53053}, {30312, 38055}, {30315, 44904}, {30331, 38130}, {30389, 61291}, {30392, 61288}, {30810, 48381}, {31330, 37365}, {31398, 31406}, {31425, 58221}, {31472, 38235}, {31650, 35016}, {31666, 61802}, {31670, 38144}, {31671, 38149}, {31672, 38154}, {31793, 51755}, {31803, 56762}, {31847, 53800}, {31849, 32142}, {31855, 37732}, {31937, 58631}, {32049, 62858}, {32900, 61293}, {33923, 35242}, {34351, 51696}, {34638, 62154}, {34720, 49176}, {34747, 61827}, {34847, 59615}, {35203, 44039}, {35251, 55868}, {35404, 50814}, {36155, 52200}, {37299, 50890}, {37440, 49553}, {37468, 48363}, {37509, 57280}, {37525, 37734}, {37529, 50418}, {37536, 45955}, {37598, 37715}, {37612, 62874}, {37618, 37738}, {37623, 57284}, {37699, 59294}, {37708, 58887}, {37711, 59316}, {37718, 61601}, {37950, 47492}, {38029, 49681}, {38052, 61509}, {38057, 61511}, {38064, 51000}, {38076, 61956}, {38083, 51069}, {38107, 40333}, {38108, 43166}, {38118, 49684}, {38569, 51631}, {39004, 40944}, {39523, 62805}, {41723, 47515}, {42215, 49226}, {42216, 49227}, {42450, 58647}, {43118, 49330}, {43119, 49329}, {43575, 43827}, {43830, 50476}, {44245, 61247}, {44266, 47496}, {44452, 51725}, {44903, 50801}, {45444, 49355}, {45445, 49356}, {45546, 48772}, {45547, 48773}, {45976, 62318}, {46704, 48917}, {46930, 60781}, {46931, 61886}, {47749, 53283}, {48894, 62185}, {49697, 59620}, {50190, 58605}, {50799, 61978}, {50800, 61985}, {50802, 61942}, {50806, 61936}, {50812, 62114}, {50813, 62116}, {50819, 62088}, {50841, 51717}, {50871, 62089}, {51073, 55861}, {51092, 61833}, {51103, 61851}, {51107, 61845}, {51110, 61860}, {51120, 61949}, {52354, 59586}, {55866, 58238}, {58247, 61919}, {59412, 60922}, {59420, 62126}, {61248, 62123}, {61250, 62144}, {61252, 62151}, {61254, 62034}, {61256, 62026}, {61279, 61858}, {61282, 61835}, {61294, 61801}

X(5690) = midpoint of X(i) and X(j) for these {i,j}: {2, 34718}, {3, 8}, {4, 12702}, {10, 11362}, {20, 18525}, {40, 355}, {72, 37562}, {100, 19914}, {376, 50798}, {381, 50810}, {382, 6361}, {549, 50823}, {550, 37705}, {944, 12645}, {1482, 12245}, {3534, 34627}, {3625, 5882}, {3626, 43174}, {3632, 37727}, {3654, 3679}, {3655, 4677}, {3830, 34632}, {3869, 25413}, {3913, 49168}, {3927, 6850}, {4297, 47745}, {5493, 31673}, {5541, 62354}, {5657, 59503}, {5731, 51515}, {5752, 31785}, {5777, 31798}, {5779, 35514}, {5790, 59417}, {5881, 18481}, {6259, 54156}, {7991, 12699}, {9947, 31797}, {10625, 16980}, {10993, 62616}, {11179, 50783}, {11684, 47032}, {12247, 12331}, {12513, 49169}, {12515, 12751}, {12747, 13199}, {12778, 13211}, {13996, 37726}, {14666, 50925}, {14830, 50880}, {15681, 50864}, {15971, 48928}, {16139, 47033}, {18802, 32554}, {20070, 48661}, {26446, 63143}, {31788, 34790}, {32049, 62858}, {34641, 51705}, {46704, 48917}, {48746, 48747}, {48915, 48937}
X(5690) = reflection of X(i) in X(j) for these {i,j}: {1, 140}, {3, 61524}, {4, 18357}, {5, 10}, {145, 61286}, {355, 61510}, {549, 50821}, {550, 3579}, {946, 9956}, {960, 58630}, {962, 40273}, {1385, 6684}, {1482, 5901}, {1483, 1385}, {1484, 12619}, {1537, 61580}, {3244, 15178}, {3579, 43174}, {3627, 18480}, {3655, 12100}, {3656, 547}, {3874, 5885}, {4297, 31663}, {4301, 9955}, {5446, 23841}, {5694, 3678}, {5882, 13624}, {5884, 13145}, {5887, 31835}, {6147, 37438}, {6265, 61562}, {8715, 32157}, {9957, 31838}, {10222, 1125}, {10246, 61614}, {10283, 11231}, {10284, 3884}, {10543, 12104}, {10680, 61530}, {10738, 61553}, {11278, 13464}, {12675, 40296}, {12699, 546}, {12737, 61566}, {13369, 31787}, {13463, 24387}, {13464, 3634}, {13743, 61622}, {15686, 50808}, {15687, 50796}, {15704, 31730}, {16128, 61605}, {18481, 548}, {18525, 61249}, {19907, 3035}, {20117, 4015}, {20330, 3826}, {22758, 61539}, {22791, 5}, {22793, 19925}, {24474, 61541}, {24475, 34339}, {31162, 5066}, {31649, 18253}, {31803, 56762}, {31837, 58643}, {31870, 3918}, {31937, 58631}, {33668, 5499}, {33699, 34648}, {33858, 11277}, {34628, 15690}, {34773, 3}, {34791, 13373}, {37533, 61533}, {38028, 26446}, {38034, 38042}, {38042, 38112}, {38112, 38127}, {38138, 38176}, {40273, 61259}, {41869, 3853}, {42450, 58647}, {44266, 47496}, {48933, 15973}, {50811, 34200}, {50824, 549}, {50865, 14893}, {51709, 3828}, {52200, 36155}, {58535, 11695}, {61148, 4999}, {61245, 47745}, {61283, 10165}, {61295, 5882}, {61597, 51700}, {62036, 31673}, {62041, 33697}, {62154, 34638}
X(5690) = complement of X(1482)
X(5690) = anticomplement of X(5901)
X(5690) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 140, 38028}, {1, 5445, 5433}, {1, 24914, 15325}, {1, 26446, 140}, {2, 1482, 5901}, {2, 12245, 1482}, {3, 956, 32153}, {3, 5657, 61524}, {3, 5687, 32141}, {3, 12331, 11491}, {3, 12645, 944}, {3, 18526, 5731}, {3, 32153, 38602}, {3, 51515, 18526}, {3, 59503, 8}, {4, 3617, 5790}, {4, 5790, 18357}, {4, 20070, 48661}, {4, 59417, 12702}, {5, 10, 38042}, {5, 22791, 38034}, {5, 38112, 10}, {8, 944, 12645}, {8, 5657, 3}, {8, 61524, 34773}, {10, 946, 9956}, {10, 960, 3820}, {10, 3817, 31399}, {10, 3878, 1329}, {10, 4301, 10175}, {10, 5836, 31419}, {10, 5837, 5044}, {10, 10176, 9711}, {10, 21616, 5123}, {12, 5903, 39542}, {20, 4678, 59388}, {20, 59388, 18525}, {35, 41684, 10950}, {40, 3679, 355}, {46, 5252, 18990}, {55, 10573, 37730}, {65, 495, 6147}, {65, 10039, 495}, {80, 11010, 6284}, {140, 61597, 51700}, {145, 631, 10246}, {145, 10246, 61286}, {165, 4668, 5881}, {165, 5881, 18481}, {165, 18481, 548}, {210, 5887, 31835}, {219, 59671, 59594}, {355, 3654, 40}, {355, 3679, 61510}, {355, 16139, 11827}, {381, 962, 40273}, {381, 5818, 61259}, {388, 36279, 24470}, {392, 24982, 17527}, {484, 37710, 7354}, {498, 2099, 37737}, {499, 2098, 1387}, {549, 1483, 1385}, {550, 59400, 37705}, {632, 10283, 1125}, {946, 9956, 5}, {958, 11248, 6914}, {962, 5818, 381}, {962, 53620, 5818}, {997, 37828, 47742}, {999, 1788, 34753}, {1006, 3871, 37621}, {1125, 10222, 10283}, {1125, 11231, 632}, {1376, 11249, 6924}, {1385, 1483, 50824}, {1385, 6684, 549}, {1385, 50821, 6684}, {1482, 34718, 12245}, {1656, 5603, 61272}, {1656, 8148, 5603}, {1697, 5722, 15172}, {1698, 5886, 3628}, {1698, 7982, 5886}, {1699, 61261, 3850}, {1737, 3057, 496}, {1837, 5119, 15171}, {2093, 9578, 57282}, {3090, 18493, 61269}, {3241, 37624, 61281}, {3244, 10165, 15178}, {3244, 15178, 61283}, {3295, 18391, 12433}, {3340, 31434, 11374}, {3359, 57279, 24467}, {3419, 55104, 31789}, {3523, 3621, 7967}, {3526, 10247, 3616}, {3576, 3632, 37727}, {3579, 3626, 37705}, {3616, 10247, 61278}, {3617, 12702, 18357}, {3617, 59417, 4}, {3624, 16200, 61276}, {3625, 10164, 5882}, {3625, 13624, 61295}, {3627, 38138, 18480}, {3632, 9588, 3576}, {3634, 11230, 55856}, {3634, 13464, 11230}, {3656, 19875, 547}, {3753, 24474, 61541}, {3753, 24987, 8728}, {3826, 20330, 38171}, {3828, 51709, 15699}, {3877, 25005, 4187}, {4297, 4669, 47745}, {4297, 31663, 8703}, {4301, 10175, 9955}, {4669, 31663, 61245}, {4677, 7987, 61296}, {4701, 17502, 61297}, {4848, 31397, 942}, {4882, 30503, 5534}, {5303, 34474, 3}, {5493, 38155, 31673}, {5587, 7991, 12699}, {5587, 12699, 546}, {5603, 9780, 1656}, {5686, 35514, 5779}, {5687, 32141, 51525}, {5697, 18395, 11}, {5790, 12702, 4}, {5818, 50810, 962}, {5882, 10164, 13624}, {6361, 59387, 382}, {6734, 51433, 10914}, {6735, 37562, 10942}, {6940, 54391, 37535}, {7987, 61296, 3655}, {8148, 9780, 61272}, {8227, 11531, 3656}, {9588, 37727, 3530}, {9589, 61258, 3861}, {9708, 10306, 3560}, {9709, 22770, 6911}, {9955, 10175, 5}, {10164, 13624, 15712}, {10222, 11231, 1125}, {10738, 59415, 61553}, {10915, 34339, 32213}, {11230, 11278, 13464}, {11362, 38112, 22791}, {11362, 38127, 10}, {11522, 54447, 61268}, {11531, 19875, 8227}, {11545, 15171, 1837}, {12702, 48661, 20070}, {12787, 12788, 3416}, {13911, 35774, 7583}, {13973, 35775, 7584}, {15694, 50805, 38314}, {15908, 21031, 119}, {16189, 34595, 61275}, {17606, 30384, 10593}, {18483, 38140, 3858}, {18525, 59388, 61249}, {19925, 22793, 3845}, {20053, 58230, 61290}, {21271, 40999, 41007}, {22791, 38042, 5}, {24475, 44222, 31657}, {30305, 54361, 9669}, {31423, 61597, 38028}, {34632, 38074, 3830}, {34632, 51068, 38074}, {34718, 38066, 2}, {35610, 35789, 3071}, {35611, 35788, 3070}, {37563, 37702, 3058}, {37724, 59337, 15174}, {37734, 52793, 37525}, {40273, 61259, 381}, {49226, 49602, 42215}, {49227, 49601, 42216}, {50810, 53620, 381}, {50821, 50823, 50824}, {50821, 50824, 50825}, {50821, 50827, 50823}, {50821, 50828, 50826}, {50821, 50830, 50833}, {50821, 51087, 50829}, {50822, 50823, 50821}, {50822, 50827, 50824}, {50823, 50824, 50830}, {50823, 50826, 50831}, {50824, 50825, 50833}, {50825, 50830, 50824}, {50826, 50831, 50828}, {50828, 50831, 50824}, {50829, 51087, 50832}, {50832, 51087, 50824}, {51073, 58240, 61273}, {51111, 54192, 26287}, {51700, 61597, 1}, {54447, 61268, 35018}
X(5690) = pole of line {9957, 10950} with respect to the Feuerbach circumhyperbola
X(5690) = pole of line {10, 10943} with respect to the Feuerbach circumhyperbola of the medial triangle
X(5690) = pole of line {859, 37535} with respect to the Feuerbach circumhyperbola of the tangential triangle
X(5690) = pole of line {572, 2183} with respect to the Kiepert circumhyperbola of the excentral triangle
X(5690) = pole of line {517, 2975} with respect to the Jerabek circumhyperbola of the excentral triangle
X(5690) = pole of line {34589, 35015} with respect to the Mandart parabola
X(5690) = pole of line {900, 48391} with respect to the circumcircle
X(5690) = pole of line {28217, 39540} with respect to the incircle
X(5690) = pole of line {513, 39508} with respect to the nine point circle
X(5690) = pole of line {28217, 39547} with respect to the Conway circle
X(5690) = pole of line {4977, 39534} with respect to the polar circle
X(5690) = pole of line {4784, 39386} with respect to the orthoptic-circle-of-the-Steiner-inellipse
X(5690) = pole of line {18004, 39386} with respect to the orthoptic-circle-of-the-Steiner-circumellipe
X(5690) = pole of line {28217, 44409} with respect to the Suppa Cucoanes circle (see X(53614))

X(5691) = DE LONGCHAMPS POINT OF OUTER GARCIA TRIANGLE

Trilinears r - 4 R cos B cos C : :
Barycentrics 3 a^4- a^3 (b + c) - a^2 (b - c)^2 + a (b - c)^2 (b + c) - 2 (b^2 - c^2)^2 : : (Angel Montesdeoca, January 21, 2015)
X(5691) = X(1) - 2 X(4)

The outer Garcia triangle is defined at X(5587).

Let I = X(1) and O = X(3). Let A″ be the reflection of I in line AO and let I_A be the reflection of A″ in line AI. Define I_B and I_C cyclically. Then ABC and I_AI_B I_C are orthologic triangles, and X(5691) is the ABC-orthology center of I_AI_B I_C. (Angel Montesdeoca, January 21, 2015)

Let Ja, Jb, Jc be the excenters and I the incenter. Let A′ be the centroid of JbJcI, and define B′ and C′ cyclically. A′B′C′ is also the cross-triangle of the excentral and 2nd circumperp triangles. A′B′C′ is homothetic to the 4th Euler triangle at X(5691). (Randy Hutson, July 31 2018)

X(5691) lies on these lines: {1,4}, {2,4297}, {3,1698}, {5,3576}, {8,144}, {10,20}, {11,1420}, {12,3601}, {30,40}, {35,1012}, {36,3149}, {46,80}, {57,1837}, {63,5086}, {65,971}, {78,5080}, {79,3577}, {5692,5777}

X(5691) = reflection of X(20) in X(10)
X(5691) = {X(1),X(4)}-harmonic conjugate of X(1699)

X(5692) = CENTROID OF INNER GARCIA TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = abc + (b + c)(b² + c² - a²)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b³ + c³ - a²b - a²c + abc + b²c + bc²)

The inner Garcia triangle A″B″C″ is defined at X(5587). Another construction of A″ is as the reflection of X(1) in the perpendicular bisector of side BC, so that |OA″| = |X(1)X(3)|, and A″, B″, C″, and X(1) are on a circle with center X(3). (Paul Yiu, ADGEOM #1214, April 3, 2014)

A″ = a² : c² - b² + ac : b² - c² + ab
B″ = c² - a² + bc : b² : a² - c² + ba
C″ = b² - a² + cb : a² - b² + ca : c²
(Peter Moses, April 4, 2014)

The appearance of (i,j) in the following list means that (X(i) of A″B″C′) = X(j): (1,8), (3,3), (10,3878), (11,72), (21,191), (35,2975), (36,100), (40,944), (55,956), (63,4302), (78,1479), (80,3869), (100,1), (104,40), (214,10), (238,190), (662,2607), (663,3762), (667,659), (976,4894), (1001,5220), (1125,3678), (1145,3057), (1149,4738), (1193,1089), (1319,1145), (1320,3632), (1325,2948), (1376,5289), (1459,4768), (1734,3904), (1768,20), (1818,4858), (2077,104), (2932,56), (3032,2901), (3035,960), (3065,3648), (3220,1633), (3286,4436), (4057,4491), (4367,4730), (4511,80), (4855,499), (4996,35), (5150,3923), (5313,4671), (5440,11), (5531,962), (5541,145). (Peter Moses, April 3, 2014)

X(5692) lies on these lines: {1,6}, {2,758}, {3,191}, {8,80}, {10,908}, {119,5690}, {5691,5777}, {2805,5695}, {4302,5696} et al.

X(5692) = reflection of X(1) in X(392)
X(5692) = anticomplement of X(5883)
X(5692) = {X(1),X(72)}-harmonic conjugate of X(5904)
X(5692) = X(1)-of-X(2)-anti-altimedial-triangle
X(5692) = X(2)-of-X(1)-adjunct-anti-altimedial-triangle
X(5692) = X(12022)-of-excentral-triangle
X(5692) = homothetic center of inner Garcia triangle and X(1)-adjunct anti-altimedial triangle

X(5693) = ORTHOCENTER OF INNER GARCIA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b⁶ + c⁶ - a⁵b - a⁵c + a⁴b² + a⁴c² + a⁴bc + 2a³b³ + 2a³c³ - a³b²c - a³bc² - 2a²b⁴ - 2a²c⁴ - a²b³c - a²bc³ + 2a²b²c² - ab⁵ - ac⁵ + 2ab⁴c + 2abc⁴ - ab³c² - ab²c³ - b⁴c² - b²c⁴)

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5693) is the incenter of the X(3)-Fuhrmann triangle, defined at X(5613).

X(5693) lies on these lines: {1,90}, {2,5884}, {3,191}, {4,758}, {5,3649}, {8,153}, {10,6937}, {944,2801}, {952,5697} et al.

X(5693) = anticomplement of X(5884)

X(5694) = NINE-POINT CENTER OF INNER GARCIA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b⁶ + c⁶ - a⁵b - a⁵c + a⁴b² + a⁴c² + 2a⁴bc + 2a³b³ + 2a³c³ - a³b²c - a³bc² - 2a²b⁴ - 2a²c⁴ - 2a²b³c - 2a²bc³ + 2a²b²c² - ab⁵ - ac⁵ + 2ab⁴c + 2abc⁴ - ab³c² - ab²c³ - b⁴c² - b²c⁴)

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

Let A* be the reflection of X(5) in the perpendicular bisector of segment BC, and define B* and C* cyclically. Triangle A*B*C* is inversely similar to ABC, with similitude center X(3); also, A*B*C* is perspective to ABC, with perspector X(3519), and X(5694) is the incenter of A*B*C*.

X(5694) lies on these lines: {1,195}, {2,5885}, {3,191}, {4,8}, {5,758} et al.

X(5694) = anticomplement of X(5885)

X(5695) = SYMMEDIAN POINT OF INNER GARCIA TRIANGLE

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

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5695) lies on these lines: {1,536}, {2,3712}, {3,2783}, {4,3704}, {6,740}, {8,190}, {9,3696}, {10,45}, {2805,5692} et al.

X(5696) = X(7) OF INNER GARCIA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b⁵ + c⁵ - a⁴b - a⁴c + 2a³b² + 2a³c² - a³bc + 2a²b²c + 2a²bc² - ab³c - abc³ -2ab⁴ -2ac⁴ + b⁴c + bc⁴ - 2b³c² - 2b²c³ - 2ab²c²)

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5696) lies on these lines: {1,5784}, {7,2894}, {8,2801}, {9,35}, {528,5697}, {4302,5692} et al.

X(5697) = X(8) OF INNER GARCIA TRIANGLE

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

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5697) lies on these lines: {1,3}, {2,3884}, {4,5559}, {7,7278}, {8,80}, {10,4193}, {11,5690}, {528,5696}, {952,5693} et al.

X(5697) = X(20)-of-reflection-triangle-of-X(1)
X(5697) = {X(1),X(40)}-harmonic conjugate of X(36)
X(5697) = X(1)-of-X(1)-anti-altimedial-triangle
X(5697) = endo-homothetic center of Ehrmann vertex-triangle and anti-Hutson intouch triangle; the homothetic center is X(382)

X(5698) = X(9) OF INNER GARCIA TRIANGLE

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

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5698) lies on these lines: {1,527}, {2,1155}, {3,1633}, {4,9}, {7,21}, {8,190}, {11,5744}, {944,2801}, {4302,5692} et al.

X(5698) = anticomplement of X(5880)

X(5699) = X(15) OF INNER GARCIA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a²(a + b + c)(a² - b² - c²) - 2(3^1/2)(a³ - a²b - a²c + 2b²c + 2bc²)S

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5699) lies on these lines: X(5699) lies on line {3,2783}, {10,13}, {15,740}, {16,3923} et al.

X(5700) = X(16) OF INNER GARCIA TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a²(a + b + c)(a² - b² - c²) + 2(3^1/2)(a³ - a²b - a²c + 2b²c + 2bc²)S

The inner Garcia triangle A″B″C″ is defined at X(5587); see also X(5692).

X(5700) lies on these lines: {3,2783}, {10,14}, {15,3923}, {16,740} et al.

X(5701) = MINIMIZER ON LINE X(1)X(6) OF x² + y² + z²

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a³b² + a³c² - 2a²b³ - 2a²c³ + ab⁴ + ac⁴ - ab³c - abc³ + 2a²b² + b⁴c + bc⁴ - b³c² - b²c³)

T = (qu - pv)(pv + rv - qu - qw) + (ru - pw)(pw + qw - ru - rv) (Peter Moses, June 23, 2014)

X(5701) is the minimizer T when P = X(1) and Q = X(6). See also X(5661) and X(5662).

X(5701) lies on these lines: {1,6}, {2,650}, {1252,1621}, {3693,4702}, {5284,5375}

X(5702) = CENTER OF MONTESDEOCA CONIC

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S_BS_C(5a²S_A - S_BS_C)

Let ABC be a triangle, let P_A be the polar of A with respect to the circle with diameter BC, and define P_B and P_C cyclically. Let A_B = P_A∩AB and A_C = P_A∩AC, and define B_C, C_A, B_A, and C_B cyclically. The six points A_B, A_C, B_C, B_A, C_A, C_B lie on, and define, the Montesdeoca conic. (Angel Montesdeoca, June 23, 2014)

A barycentric equation for the Montesdeoca conic is found from A_B = S_C : 0 : 2S_A and A_C = S_B : 2S_A : 0 to be as follows:

2(S²_Ax² + S²_By² + S²_Cz²) - 5(S_BS_Cyz + S_CS_Azx + S_AS_Bxy) = 0 (Peter Moses, June 23, 2014)

The Montesdeoca conic is the anticevian-intersection conic when P = X(4); this conic is defined by Francisco J. García Capitán (The Anticevian Intersection Conic and Hyacinthos #20547 (December 19, 2011). Also, the perspector of the Montesdeoca conic is X(4).

X(5702) lies on these lines: {4,6}, {297,5032}, {340,1992}, {376,3284}, {468,5304}, {578,3183}, {631,5158}, {3163,3545}

X(5703) = INTERSECTION OF LINES X(1)X(2) AND X(3)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 4*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5703) lies on these lines: {1, 2}, {3, 7}, {4, 4313}, {5, 3488}, {12, 3486}, {20, 226}, {21, 329}, {35, 4295}, {40, 5281}, {55, 411}, {56, 3475}, {57, 3523}, {65, 5218}, {72, 5273}, {86, 939}, {142, 5438}, {165, 3671}, {307, 3945}, {388, 2646}, {390, 946}, {443, 5440}, {452, 908}, {495, 944}, {515, 5261}, {517, 4323}, {631, 942}, {940, 3562}, {950, 3091}, {988, 4310}, {1056, 1385}, {1445, 3333}, {1446, 3160}, {1478, 4305}, {1699, 4314}, {1788, 5432}, {2287, 5296}, {2476, 5175}, {2886, 3189}, {3146, 4304}, {3149, 3295}, {3361, 5542}, {3452, 5129}, {3474, 3649}, {3522, 4292}, {3576, 3600}, {3586, 3832}, {3612, 4293}, {4000, 4255}, {4252, 4644}, {4297, 5290}, {4344, 5266}, {4855, 5249}, {5084, 5328}

X(5704) = INTERSECTION OF LINES X(1)X(2) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b - 4*a^2*b^2 - 2*a*b^3 + 3*b^4 + 2*a^3*c + 4*a^2*b*c + 2*a*b^2*c - 4*a^2*c^2 + 2*a*b*c^2 - 6*b^2*c^2 - 2*a*c^3 + 3*c^4

X(5704) lies on these lines: {1, 2}, {4, 5435}, {5, 7}, {11, 962}, {20, 3911}, {40, 5274}, {57, 3091}, {72, 5328}, {88, 5125}, {90, 5556}, {104, 3149}, {140, 3488}, {226, 5056}, {307, 4346}, {329, 4193}, {355, 4308}, {404, 5175}, {411, 5204}, {496, 5657}, {515, 5265}, {631, 4313}, {942, 3090}, {950, 3523}, {1155, 5225}, {1158, 1445}, {1656, 3487}, {1728, 3218}, {3035, 3189}, {3306, 5177}, {3333, 5261}, {3339, 3817}, {3486, 5433}, {3522, 3586}, {3529, 5122}, {3562, 4383}, {3600, 5587}, {3614, 4860}, {3832, 4292}, {4208, 5437}, {4306, 5400}, {5084, 5273}

X(5705) = INTERSECTION OF LINES X(1)X(2) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b - 3*a^2*b^2 - a*b^3 + 2*b^4 + a^3*c - 2*a^2*b*c - 3*a*b^2*c - 3*a^2*c^2 - 3*a*b*c^2 - 4*b^2*c^2 - a*c^3 + 2*c^4

X(5705) lies on these lines: {1, 2}, {5, 9}, {21, 3586}, {40, 2886}, {57, 442}, {63, 2476}, {72, 5219}, {140, 5438}, {283, 5235}, {411, 993}, {443, 3911}, {958, 3149}, {965, 2323}, {1445, 3841}, {1479, 4512}, {1656, 5044}, {2475, 4652}, {3090, 3452}, {3091, 5273}, {3219, 5141}, {3305, 4193}, {3306, 4197}, {3419, 3601}, {3545, 5325}, {3576, 4999}, {3646, 3816}, {3814, 5536}, {3822, 5290}, {4208, 5435}, {4292, 5177}, {4304, 5175}

X(5706) = INTERSECTION OF LINES X(1)X(3) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - 2*a^4*b*c - 2*a^3*b^2*c + a*b^4*c + 2*b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5706) lies on these lines: {1, 3}, {4, 6}, {5, 1714}, {7, 3562}, {10, 219}, {20, 81}, {28, 154}, {33, 1712}, {51, 4186}, {58, 1012}, {64, 4219}, {98, 3597}, {184, 4185}, {221, 278}, {222, 4292}, {377, 394}, {386, 3149}, {405, 580}, {429, 1899}, {602, 1001}, {774, 4336}, {990, 1071}, {991, 4658}, {1191, 5603}, {1203, 1699}, {1260, 3191}, {1376, 3682}, {1451, 2654}, {1478, 3173}, {1612, 3052}, {1753, 2285}, {1765, 2257}, {1780, 3560}, {1836, 1838}, {1853, 5142}, {1993, 2475}, {2192, 2982}, {2256, 5657}, {3695, 4513}, {3713, 5295}, {4259, 5562}, {5046, 5422}

X(5706) = {X(1),X(40)}-harmonic conjugate of X(37528)

X(5707) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 + a^5*b - 2*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + a*b^5 + a^5*c - 2*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c + 2*b^5*c - 2*a^4*c^2 - 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5707) lies on these lines: {1, 3}, {2, 3193}, {4, 81}, {5, 6}, {7, 1068}, {58, 3560}, {222, 225}, {226, 3157}, {283, 405}, {394, 442}, {581, 4658}, {602, 3720}, {965, 2323}, {1069, 1210}, {1216, 4259}, {1437, 4185}, {1480, 4301}, {1656, 4383}, {1993, 2476}, {1994, 5141}, {3149, 5396}, {3487, 3562}, {4193, 5422}

X(5707) = {X(1),X(57)}-harmonic conjugate of X(37565)

X(5708) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b - a*b^2 - 2*b^3 + 2*a^2*c + 4*a*b*c + 2*b^2*c - a*c^2 + 2*b*c^2 - 2*c^3)

X(5708) lies on these lines: {1, 3}, {2, 3927}, {5, 7}, {28, 89}, {30, 938}, {45, 579}, {63, 5439}, {72, 3306}, {140, 3487}, {142, 3634}, {191, 4423}, {222, 1393}, {226, 1656}, {355, 4298}, {381, 553}, {382, 4031}, {405, 3218}, {443, 3617}, {474, 3868}, {495, 1788}, {496, 4295}, {499, 3649}, {548, 4313}, {550, 3488}, {950, 1657}, {952, 3600}, {1086, 5292}, {1376, 3874}, {1435, 1871}, {1439, 3527}, {1483, 4308}, {1598, 1876}, {3526, 3911}, {3586, 5073}, {3624, 4880}, {3628, 5226}, {3872, 4004}, {3881, 3913}, {3982, 5079}, {4084, 5289}, {4114, 5072}, {4306, 5396}, {4321, 5534}, {4355, 5587}, {4654, 5055}, {5044, 5437}, {5070, 5219}, {5445, 5557}

X(5708) = {X(4860),X(5221)}-harmonic conjugate of X(1)

X(5709) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 2*a^4*b*c + 2*b^5*c - 3*a^4*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - 4*b^3*c^3 + 3*a^2*c^4 + b^2*c^4 + 2*b*c^5 - c^6)

X(5709) = X(26)-of-excentral-triangle.

X(5709) lies on these lines: {1, 3}, {4, 63}, {5, 9}, {20, 3218}, {28, 283}, {30, 84}, {34, 255}, {72, 3149}, {90, 3583}, {140, 5437}, {155, 610}, {191, 1699}, {212, 1393}, {223, 3157}, {225, 1217}, {381, 3929}, {411, 3868}, {443, 5657}, {516, 1158}, {518, 5534}, {578, 3955}, {579, 1766}, {602, 614}, {631, 3306}, {912, 1490}, {920, 1479}, {1012, 3916}, {1068, 1119}, {1069, 3345}, {1070, 4331}, {1071, 1998}, {1072, 5230}, {1093, 1948}, {1210, 1708}, {1254, 1496}, {1352, 5227}, {1453, 5398}, {1512, 3436}, {1707, 3073}, {1776, 5225}, {1817, 3193}, {2000, 4219}, {3090, 3305}, {3091, 3219}, {5250, 5603}

X(5709) = intouch-to-excentral similarity image of X(3)

X(5710) = INTERSECTION OF LINES X(1)X(3) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c + 2*b^2*c + a*c^2 + 2*b*c^2)

X(5710) lies on these lines: {1, 3}, {2, 1191}, {6, 8}, {10, 3966}, {21, 3052}, {31, 958}, {37, 5250}, {42, 3913}, {58, 956}, {81, 145}, {87, 2334}, {100, 4255}, {197, 1036}, {218, 3997}, {220, 5276}, {221, 388}, {387, 5082}, {392, 975}, {405, 595}, {474, 995}, {608, 1891}, {611, 5252}, {612, 960}, {614, 3812}, {750, 1201}, {962, 5244}, {978, 4413}, {1001, 1918}, {1056, 4340}, {1100, 3895}, {1193, 1376}, {1203, 3679}, {1406, 5434}, {1407, 3600}, {1449, 2136}, {1616, 3616}, {1698, 5315}, {1706, 2999}, {1722, 3698}, {1834, 3434}, {1999, 4673}, {2176, 5275}, {2256, 2303}, {2650, 3938}, {2886, 5230}, {2975, 4252}, {3242, 3868}, {3486, 4339}, {3782, 4295}, {3869, 3920}, {4363, 4968}, {4646, 5256}

X(5711) = INTERSECTION OF LINES X(1)X(3) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^3 + 2*a^2*b + a*b^2 + 2*a^2*c + 2*a*b*c + 2*b^2*c + a*c^2 + 2*b*c^2)

X(5711) lies on these lines: {1, 3}, {4, 608}, {6, 10}, {8, 81}, {31, 405}, {58, 958}, {72, 612}, {213, 5275}, {218, 5276}, {219, 2303}, {220, 3997}, {221, 226}, {222, 388}, {281, 3194}, {341, 3758}, {386, 1376}, {387, 2550}, {406, 3195}, {442, 5230}, {474, 750}, {495, 611}, {551, 1616}, {595, 1001}, {601, 1012}, {614, 5439}, {651, 5261}, {894, 4385}, {938, 4344}, {946, 2050}, {956, 1468}, {960, 975}, {976, 2650}, {984, 1046}, {993, 4252}, {1064, 3149}, {1065, 1433}, {1100, 4646}, {1107, 5021}, {1125, 1191}, {1203, 1698}, {1386, 3812}, {1407, 4298}, {1449, 1706}, {1714, 3925}, {1740, 4649}, {2271, 4386}, {2292, 5311}, {2295, 3695}, {2305, 3743}, {2886, 5292}, {3052, 5248}, {3216, 4413}, {3242, 3874}, {3488, 4339}, {3562, 3945}, {3624, 5315}, {3720, 3915}, {3736, 3913}, {3868, 3920}, {3876, 5297}, {3940, 5293}, {4356, 5493}, {4662, 4663}, {4868, 5110}, {5044, 5268}, {5250, 5287}

X(5712) = INTERSECTION OF LINES X(1)X(4) AND X(2)X(6)

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

X(5712) lies on these lines: {1, 4}, {2, 6}, {3, 4340}, {7, 941}, {37, 329}, {42, 2550}, {55, 4307}, {57, 573}, {63, 4644}, {142, 2999}, {171, 212}, {306, 2345}, {312, 1909}, {345, 894}, {354, 1469}, {386, 443}, {387, 442}, {406, 3194}, {553, 4888}, {580, 631}, {908, 5287}, {1100, 3772}, {1104, 3616}, {1125, 1453}, {1212, 5308}, {1215, 3974}, {1730, 4266}, {1788, 5530}, {1834, 5177}, {3247, 4656}, {3296, 3953}, {3622, 5484}, {3663, 4654}, {3672, 3782}, {3677, 5542}, {3744, 4344}, {3752, 4277}, {3931, 4295}, {3982, 4862}, {4000, 5249}, {4349, 5269}, {4641, 5273}, {4658, 5292}

X(5712) = isotomic conjugate of polar conjugate of X(37384)
X(5712) = complement of X(14552)
X(5712) = anticomplement of X(5737)

X(5713) = INTERSECTION OF LINES X(1)X(4) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - a^5*b^2 - 3*a^4*b^3 - a^3*b^4 + 3*a^2*b^5 + a*b^6 - b^7 + a^6*c - 3*a^4*b^2*c - 2*a^3*b^3*c + a^2*b^4*c + 2*a*b^5*c + b^6*c - a^5*c^2 - 3*a^4*b*c^2 - 2*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 - 3*a^4*c^3 - 2*a^3*b*c^3 - 4*a^2*b^2*c^3 - 4*a*b^3*c^3 - 3*b^4*c^3 - a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 + 3*a^2*c^5 + 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5713) lies on these lines: {1, 4}, {2, 283}, {5, 6}, {212, 498}, {499, 1451}, {1899, 3142}, {2299, 3542}, {2476, 3193}

X(5714) = INTERSECTION OF LINES X(1)X(4) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b + 2*a^2*b^2 - 2*a*b^3 - 3*b^4 + 2*a^3*c + 4*a^2*b*c + 2*a*b^2*c + 2*a^2*c^2 + 2*a*b*c^2 + 6*b^2*c^2 - 2*a*c^3 - 3*c^4

X(5714) lies on these lines: {1, 4}, {2, 3824}, {3, 5226}, {5, 7}, {9, 3634}, {12, 4295}, {40, 3947}, {45, 1901}, {57, 3090}, {72, 3617}, {79, 498}, {329, 442}, {381, 938}, {382, 4313}, {405, 5253}, {443, 908}, {452, 5126}, {495, 962}, {517, 5261}, {553, 5071}, {631, 4292}, {942, 3091}, {943, 5556}, {952, 4323}, {1000, 4301}, {1006, 5204}, {1210, 3545}, {1656, 5435}, {1770, 5218}, {1836, 3085}, {1892, 3089}, {2345, 3454}, {3333, 3817}, {3419, 3621}, {3529, 3601}, {3614, 5221}, {3651, 5217}, {3671, 5587}, {3911, 5067}, {4208, 5044}, {4317, 5443}, {4330, 5561}, {4644, 5292}, {5045, 5274}, {5084, 5249}

X(5715) = INTERSECTION OF LINES X(1)X(4) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - 4*a^2*b^5 - a*b^6 + 2*b^7 + 2*a^5*b*c + 2*a^4*b^2*c - 2*a*b^5*c - 2*b^6*c + a^5*c^2 + 2*a^4*b*c^2 - 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + a*b^4*c^2 - 6*b^5*c^2 + 2*a^4*c^3 + 4*a^2*b^2*c^3 + 4*a*b^3*c^3 + 6*b^4*c^3 + a^3*c^4 + a*b^2*c^4 + 6*b^3*c^4 - 4*a^2*c^5 - 2*a*b*c^5 - 6*b^2*c^5 - a*c^6 - 2*b*c^6 + 2*c^7

X(5715) lies on these lines: {1, 4}, {3, 3824}, {5, 9}, {40, 442}, {72, 5587}, {79, 1709}, {329, 3091}, {355, 3577}, {962, 5177}, {1006, 3624}, {1071, 4654}, {1158, 4312}, {3149, 5219}

X(5716) = INTERSECTION OF LINES X(1)X(4) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^4 - 2*a^3*b - 2*a^2*b^2 - 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5716) lies on these lines: {1, 4}, {2, 1104}, {6, 8}, {10, 1453}, {20, 3666}, {29, 2303}, {37, 452}, {42, 3189}, {55, 4339}, {56, 4220}, {65, 3056}, {85, 3945}, {145, 321}, {171, 1451}, {212, 5255}, {345, 4195}, {377, 4000}, {387, 3419}, {405, 1612}, {580, 5264}, {612, 2551}, {938, 940}, {942, 3784}, {975, 5084}, {986, 3474}, {1036, 1610}, {1427, 3600}, {1697, 1766}, {1834, 5175}, {1837, 3745}, {1841, 4198}, {3085, 5266}, {3146, 3672}, {3436, 3920}, {3616, 5051}, {3677, 4298}, {3772, 5177}, {3868, 4644}, {3931, 4294}, {4190, 4850}, {5218, 5530}

X(5717) = INTERSECTION OF LINES X(1)X(4) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 - 3*a^3*b - 3*a^2*b^2 - a*b^3 + b^4 - 3*a^3*c - 6*a^2*b*c - 3*a*b^2*c - 3*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4

X(5717) lies on these lines: {1, 4}, {2, 1453}, {6, 10}, {12, 3745}, {40, 2269}, {57, 4340}, {171, 580}, {204, 406}, {212, 5264}, {306, 964}, {377, 5256}, {387, 1449}, {443, 2999}, {511, 942}, {516, 3931}, {519, 5295}, {553, 3670}, {937, 2551}, {938, 3945}, {940, 1210}, {975, 3452}, {1010, 3687}, {1100, 1834}, {1104, 1125}, {1329, 4682}, {1330, 4357}, {1427, 4298}, {1451, 3911}, {1842, 2294}, {2047, 5405}, {2303, 3194}, {2334, 4863}, {2478, 5287}, {3085, 5269}, {3666, 4292}, {4208, 5222}, {5129, 5308}, {5249, 5262}

X(5718) = INTERSECTION OF LINES X(1)X(5) AND X(2)X(6)

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

X(5718) lies on these lines: {1, 5}, {2, 6}, {10, 4023}, {37, 908}, {42, 2886}, {43, 3925}, {55, 4192}, {57, 4888}, {65, 970}, {140, 5398}, {171, 2361}, {226, 1465}, {312, 3963}, {313, 4358}, {377, 4255}, {386, 442}, {469, 1865}, {516, 4689}, {528, 2177}, {536, 4054}, {631, 4340}, {750, 3035}, {851, 5132}, {899, 3826}, {986, 3649}, {1086, 4850}, {1215, 3703}, {1386, 3011}, {1468, 4999}, {1834, 2476}, {1848, 1880}, {3058, 3750}, {3306, 4675}, {3550, 4995}, {3664, 3911}, {3687, 4967}, {3706, 4028}, {3712, 3923}, {3720, 3816}, {3752, 5249}, {3772, 5256}, {3821, 4892}, {3838, 3914}, {3943, 4671}, {3944, 4854}, {3999, 5542}, {4009, 4078}, {4030, 4865}, {4031, 4896}, {4090, 4126}, {4220, 5347}, {4307, 5218}, {4654, 4902}, {5308, 5328}

X(5718) = complement of X(1150)

X(5719) = INTERSECTION OF LINES X(1)X(5) AND X(3)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 2*a^3*b - 3*a^2*b^2 + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 3*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5719) lies on these lines: {1, 5}, {2, 3940}, {3, 7}, {30, 226}, {35, 3649}, {37, 3002}, {57, 549}, {73, 5453}, {140, 942}, {381, 3488}, {382, 4313}, {484, 4995}, {518, 1125}, {546, 950}, {548, 4292}, {550, 3601}, {551, 3452}, {553, 5122}, {938, 1656}, {956, 3616}, {999, 3475}, {1000, 1482}, {1086, 4256}, {1159, 5657}, {1210, 3628}, {1385, 4315}, {3295, 3485}, {3296, 5265}, {3530, 4031}, {3579, 3671}, {3584, 5425}, {3586, 3845}, {3622, 5084}, {3748, 4870}, {3874, 4999}, {4415, 4653}, {5054, 5435}, {5249, 5440}, {5298, 5444}

X(5719) = {X(1),X(12)}-harmonic conjugate of X(37730)

X(5720) = INTERSECTION OF LINES X(1)X(5) AND X(3)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a^5*c + 2*a^4*b*c - 4*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - b^4*c^2 + 4*a^3*c^3 - 4*a^2*b*c^3 - 4*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 + 2*b*c^5 + c^6)

X(5720) lies on these lines: {1, 5}, {3, 9}, {4, 78}, {8, 1512}, {30, 1750}, {57, 912}, {72, 3149}, {200, 517}, {210, 3428}, {223, 1060}, {381, 2900}, {386, 3553}, {411, 3876}, {474, 1071}, {515, 997}, {581, 975}, {612, 1064}, {944, 5084}, {946, 3811}, {962, 4420}, {1006, 3305}, {1012, 5440}, {1038, 1745}, {1040, 3465}, {1217, 1826}, {1376, 3359}, {1482, 3577}, {1519, 3434}, {1532, 3419}, {1709, 2077}, {1743, 5398}, {3560, 3601}, {3576, 5251}, {3870, 5603}

X(5721) = INTERSECTION OF LINES X(1)X(5) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6*b + a^5*b^2 + 3*a^4*b^3 - 2*a^3*b^4 + a*b^6 - b^7 - 2*a^6*c + a^4*b^2*c + b^6*c + a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 3*b^4*c^3 - 2*a^3*c^4 - a*b^2*c^4 - 3*b^3*c^4 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5721) lies on these lines: {1, 5}, {3, 1714}, {4, 6}, {30, 1754}, {40, 1723}, {51, 1894}, {65, 1243}, {184, 1884}, {209, 517}, {442, 581}, {518, 1072}, {912, 3782}, {1064, 2886}, {1108, 1512}, {1210, 1465}, {1214, 1737}, {1329, 3682}, {1785, 1864}, {2361, 3073}, {3149, 5292}, {3428, 4192}

X(5722) = INTERSECTION OF LINES X(1)X(5) AND X(4)X(7)

Trilinears cos B + cos C - 2 cos B cos C + 1 : :
Barycentrics -a^4 + a^3*b - a*b^3 + b^4 + a^3*c + 2*a^2*b*c + a*b^2*c + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4 : :

X(5722) lies on these lines: {1, 5}, {2, 3419}, {3, 950}, {4, 7}, {6, 5179}, {8, 392}, {10, 1001}, {30, 57}, {55, 1737}, {65, 1479}, {72, 2478}, {78, 4187}, {79, 5561}, {81, 5155}, {90, 3652}, {140, 3601}, {200, 3820}, {222, 1877}, {224, 442}, {226, 381}, {354, 1478}, {376, 5122}, {377, 5439}, {382, 4031}, {388, 5045}, {390, 5657}, {405, 1259}, {443, 5175}, {497, 517}, {499, 2646}, {515, 999}, {519, 3452}, {553, 3830}, {631, 4313}, {912, 1864}, {943, 5047}, {997, 3816}, {1056, 5049}, {1062, 1834}, {1104, 5292}, {1145, 3895}, {1155, 4302}, {1319, 3655}, {1329, 3811}, {1385, 3086}, {1770, 5221}, {1788, 3579}, {1836, 3583}, {1936, 5398}, {2099, 3656}, {3058, 3654}, {3091, 3487}, {3241, 5176}, {3434, 3753}, {3436, 3555}, {3545, 5226}, {3582, 3653}, {3612, 5433}, {3616, 5086}, {3679, 4863}, {3824, 5177}, {3845, 4654}, {3868, 5046}, {3873, 5080}, {4295, 5225}, {5090, 5142}, {5274, 5603}, {5427, 5441}

X(5722) = midpoint of X(i) in X(j) for these {i,j}: {1,5727}, {4,5768}
X(5722) = X(25)-of-Fuhrmann-triangle
X(5722) = inverse-in-Feuerbach-hyperbola of X(5252)
X(5722) = {X(1),X(80)}-harmonic conjugate of X(5252)
X(5722) = {X(1),X(1837)}-harmonic conjugate of X(355)

X(5723) = INTERSECTION OF LINES X(1)X(5) AND X(6)X(7)

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

X(5723) lies on these lines: {1, 5}, {2, 664}, {6, 7}, {57, 1358}, {88, 279}, {226, 544}, {241, 514}, {278, 1783}, {1419, 4859}, {1441, 3589}, {1456, 1738}, {4422, 4552}

X(5723) = crossdifference of every pair of points on the line X(55)X(654)

X(5724) = INTERSECTION OF LINES X(1)X(5) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 - a^2*b^2 - 2*a*b^3 + b^4 - 4*a^2*b*c - a^2*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5724) lies on these lines: {1, 5}, {6, 8}, {10, 4434}, {30, 4424}, {38, 529}, {65, 511}, {388, 4310}, {515, 3666}, {519, 1215}, {982, 5434}, {1146, 5276}, {1478, 3782}, {1834, 5086}, {1880, 1891}, {2361, 5255}, {2646, 5530}, {3679, 5269}, {3920, 5176}, {4304, 4689}, {4415, 5080}, {5264, 5398}

X(5725) = INTERSECTION OF LINES X(1)X(5) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b + 2*a^2*b^2 + a*b^3 - b^4 + a^3*c + 4*a^2*b*c + a*b^2*c + 2*a^2*c^2 + a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4

X(5725) lies on these lines: {1, 5}, {3, 5530}, {4, 941}, {6, 10}, {42, 3419}, {171, 5398}, {388, 1465}, {515, 2050}, {940, 1737}, {942, 1469}, {975, 998}, {1478, 3666}, {1783, 5276}, {1788, 4340}, {1836, 4424}, {2361, 5264}, {3085, 5266}, {3772, 3822}, {3820, 5268}, {4205, 5336}, {4302, 4689}, {4307, 5657}, {4682, 5123}

X(5726) = INTERSECTION OF LINES X(1)X(5) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)*(a - b + c)*(a^2 - a*b + 4*b^2 - a*c + 8*b*c + 4*c^2)

X(5726) lies on these lines: {1, 5}, {2, 4315}, {7, 10}, {8, 3947}, {165, 1478}, {226, 3679}, {388, 1698}, {519, 5226}, {946, 1000}, {1788, 4031}, {2099, 4677}, {2476, 4853}, {2886, 4915}, {3085, 4304}, {3340, 4668}, {3436, 5234}, {3485, 3632}, {3600, 3634}, {3617, 3671}, {3625, 4323}, {3731, 5179}, {3828, 5435}, {4312, 5657}, {4512, 5080}, {5123, 5437}

X(5727) = INTERSECTION OF LINES X(1)X(5) AND X(8)X(9)

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

X(5727) lies on these lines: {1, 5}, {4, 3340}, {8, 9}, {10, 3486}, {20, 4848}, {30, 2093}, {40, 920}, {46, 4316}, {55, 3679}, {57, 515}, {65, 971}, {145, 908}, {388, 5542}, {497, 519}, {517, 1864}, {944, 1210}, {1012, 3256}, {1249, 1826}, {1478, 4654}, {1698, 2646}, {1699, 2099}, {1706, 5554}, {1737, 3576}, {1788, 4297}, {2098, 3633}, {3057, 3632}, {3058, 4677}, {3241, 5274}, {3617, 4313}, {3626, 4314}, {3671, 5229}, {3832, 4323}, {3870, 5176}, {4301, 5225}, {4304, 5657}, {4863, 4915}

X(5727) = reflection of X(1) in X(5722)

X(5728) = INTERSECTION OF LINES X(1)X(6) AND X(4)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 + 2*a*c^4 + b*c^4 - c^5)

X(5728) = {X(1),X(9)}-harmonic conjugate of X(954)

X(5728) lies on these lines: {1, 6}, {2, 955}, {3, 1445}, {4, 7}, {10, 3059}, {11, 118}, {40, 4326}, {55, 1708}, {65, 516}, {81, 162}, {142, 442}, {144, 452}, {241, 991}, {329, 3873}, {390, 517}, {480, 3811}, {497, 5173}, {774, 3931}, {943, 2346}, {986, 4335}, {990, 5228}, {1005, 3218}, {1012, 3358}, {1156, 2771}, {1260, 3870}, {1376, 2900}, {1490, 3333}, {1730, 3198}, {1737, 3826}, {1890, 1905}, {1898, 3649}, {2550, 3419}, {2951, 3339}, {3062, 5665}, {3085, 3697}, {3487, 5045}, {3586, 4312}

X(5729) = INTERSECTION OF LINES X(1)X(6) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 - 4*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 - 2*b^5 - 4*a^3*c^2 - 6*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 + 3*a*c^4 - 2*c^5)

X(5729) lies on these lines: {1, 6}, {4, 653}, {5, 7}, {56, 2801}, {144, 2478}, {226, 4860}, {516, 1837}, {527, 1210}, {938, 3927}, {971, 1445}, {1155, 1708}, {1260, 3935}, {3245, 3586}, {5435, 5658}

X(5730) = INTERSECTION OF LINES X(1)X(6) AND X(5)X(8)

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

X(5730) lies on these lines: {1, 6}, {2, 4930}, {3, 3417}, {5, 8}, {10, 2099}, {35, 3899}, {40, 5440}, {55, 3878}, {56, 758}, {57, 4018}, {63, 1385}, {65, 474}, {78, 517}, {145, 1058}, {214, 5204}, {329, 944}, {355, 908}, {381, 5086}, {382, 5057}, {442, 3485}, {519, 1837}, {527, 4311}, {936, 3340}, {946, 3419}, {952, 3436}, {957, 1257}, {999, 3868}, {1319, 3962}, {1320, 3621}, {1388, 4067}, {1457, 3682}, {1759, 3207}, {2093, 5438}, {2271, 3727}, {2800, 2932}, {2975, 3927}, {3057, 3811}, {3219, 3897}, {3244, 4679}, {3295, 3877}, {3303, 3884}, {3304, 3874}, {3445, 4694}, {3576, 3916}, {3579, 4855}, {3612, 4640}, {3624, 5425}, {3626, 3711}, {3632, 5087}, {3681, 4861}, {3754, 4413}, {3820, 5554}, {3872, 3984}, {3885, 3935}, {3901, 5563}, {4084, 5221}, {4255, 4424}

X(5731) = INTERSECTION OF LINES X(1)X(7) AND X(3)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -5*a^4 + 2*a^3*b + 4*a^2*b^2 - 2*a*b^3 + b^4 + 2*a^3*c - 4*a^2*b*c + 2*a*b^2*c + 4*a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - 2*a*c^3 + c^4

X(5731) = {X(1),X(20)}-harmonic conjugate of X(962). Let A′ be the antipode of the A-extouch point in the A-excircle, and define B′ and C′ cyclically, and let A″ be the antipode of the A-intouch point in the incircle, and define B″ and C″ cyclically. Then X(5731) is the centroid of {A′,B′,C′,A″,B″,C″}. (Randy Hutson, July 7, 2014)

The triangle A′B′C′ is here named the Hutson-extouch triangle, and A″B″C″, the Hutson-intouch triangle - not to be confused with the outer and inner Hutson triangles defined at X(363). Hutson established (July 10, 2014) that A′B′C′ and A″B″C″ are orthologic with orthology center X(3555); also that A″B″C″ and A′B′C′ are orthologic with orthology center X(5920). X(5731) is the midpoint of the centroids of A′B′C′ and A″B″C″; see X(5918) and X(5919).

Peter Moses (July 15, 2014) gives barycentrics for Hutson-extouch triangle,
-4a² : (a + b + c)(a + b - c) : (a + b + c)(a - b + c)
(a + b + c)(b - c + a) : -4b² : (a + b + c)(b + c - a)
(a + b + c)(c + a - b) : (a + b + c)(c - a + b) : -4c²

and for the Hutson-intouch triangle,
4a² : (-a + b + c)(a - b + c) : (-a + b + c)(a + b - c)
(-b + c + a)(b + c - a) : 4b² : (-b + c + a)(b - c + a)
(-c + a + b)(c - a + b) : (-c + a + b)(c + a - b) : 4c²

X(5731) lies on these lines: {1, 7}, {2, 515}, {3, 8}, {4, 1385}, {5, 5550}, {10, 3523}, {21, 3427}, {30, 5603}, {36, 5435}, {40, 145}, {55, 3476}, {56, 411}, {84, 5250}, {153, 214}, {165, 519}, {329, 4511}, {355, 631}, {376, 517}, {377, 3897}, {388, 2646}, {392, 971}, {452, 1490}, {497, 1319}, {548, 1483}, {550, 1482}, {551, 1699}, {840, 2737}, {946, 3146}, {950, 1420}, {963, 1043}, {993, 5273}, {999, 3488}, {1012, 1621}, {1071, 3869}, {1125, 3091}, {1210, 5265}, {1478, 5226}, {1602, 1610}, {1788, 5204}, {2099, 3474}, {3085, 3612}, {3149, 5253}, {3189, 5584}, {3421, 5440}, {3475, 5434}, {3528, 3579}, {3545, 3653}, {3586, 5274}, {3624, 5056}, {3635, 5493}, {3817, 3839}, {4188, 5554}, {4189, 5450}, {4420, 5534}, {5218, 5252}

X(5731) = anticomplement of X(5587)
X(5731) = X(15030)-of-excentral-triangle
X(5731) = endo-homothetic center of Ehrmann side-triangle and 2nd anti-Conway triangle; the homothetic center is X(568)

X(5732) = INTERSECTION OF LINES X(1)X(7) AND X(3)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + 4*a*b^3*c + b^4*c + 2*a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 + 4*a*b*c^3 - 2*b^2*c^3 - 3*a*c^4 + b*c^4 + c^5)

X(5732) = (X(6) of hexyl triangle) = (X(69) of 2nd circumperp triangle) = (X(1352) of excentral triangle) = (isogonal conjugate of X(3)-vertex conjugate of X(57) = isotomic conjugate with respect to X(1)-circumcevian triangle) of X(1) (Randy Hutson, July 7, 2014)

X(5732) lies on these lines: {1, 7}, {2, 1750}, {3, 9}, {4, 142}, {21, 3062}, {40, 518}, {63, 100}, {78, 144}, {223, 1040}, {376, 527}, {411, 1445}, {464, 2947}, {515, 2550}, {517, 3243}, {912, 3587}, {950, 1467}, {952, 5528}, {954, 3601}, {1001, 1012}, {1699, 5249}, {1709, 4512}, {1818, 2324}, {1998, 3218}, {2808, 3781}, {2900, 3928}, {3059, 5584}, {3333, 5572}, {3452, 5658}, {3579, 5534}, {3826, 5587}, {3868, 3895}

X(5732) = midpoint of X(i) and X(j) for these (i,j): (1,2951), (7,20)
X(5732) = reflection of X(i) in X(j) for these (i,j): (4,142), (9,3)
X(5732) = complement of X(36991)
X(5732) = midpoint of Mandart hyperbola intercepts of Soddy line

X(5733) = INTERSECTION OF LINES X(1)X(7) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^6 + 5*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 - 2*a*b^5 + b^6 + 2*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + 5*a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + 4*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 3*a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6

X(5733) lies on these lines: {1, 7}, {4, 4658}, {5, 6}, {225, 1419}, {631, 4648}, {3193, 4197}

X(5734) = INTERSECTION OF LINES X(1)X(7) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^4 - 6*a^3*b - 4*a^2*b^2 + 6*a*b^3 + b^4 - 6*a^3*c + 12*a^2*b*c - 6*a*b^2*c - 4*a^2*c^2 - 6*a*b*c^2 - 2*b^2*c^2 + 6*a*c^3 + c^4

X(5734) lies on these lines: {1, 7}, {4, 1392}, {5, 8}, {40, 3622}, {145, 946}, {165, 3636}, {329, 4861}, {355, 3855}, {382, 944}, {388, 5048}, {411, 3303}, {515, 3623}, {517, 631}, {519, 3091}, {551, 3523}, {938, 2099}, {952, 3843}, {1385, 3528}, {1388, 3474}, {1483, 3853}, {1699, 3244}, {2093, 5265}, {2098, 3485}, {3262, 4673}, {3525, 3654}, {3526, 5550}, {3529, 3655}, {3621, 5587}, {3632, 3817}, {3679, 5056}, {3878, 5273}, {4197, 5330}

X(5735) = INTERSECTION OF LINES X(1)X(7) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^6 - 3*a^5*b - 4*a^4*b^2 + 2*a^3*b^3 + 3*a^2*b^4 + a*b^5 - 2*b^6 - 3*a^5*c + 2*a^3*b^2*c - 4*a^2*b^3*c + a*b^4*c + 4*b^5*c - 4*a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + 2*b^4*c^2 + 2*a^3*c^3 - 4*a^2*b*c^3 - 2*a*b^2*c^3 - 8*b^3*c^3 + 3*a^2*c^4 + a*b*c^4 + 2*b^2*c^4 + a*c^5 + 4*b*c^5 - 2*c^6

X(5735) lies on these lines: {1, 7}, {4, 527}, {5, 9}, {63, 1699}, {84, 3254}, {142, 631}, {144, 3832}, {165, 5249}, {382, 971}, {1004, 5537}, {1750, 1998}, {2801, 3868}, {3436, 5223}, {3817, 5273}, {5220, 5587}

X(5736) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 2*a^3*b^2 + a*b^4 - 2*a^3*b*c - 3*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 3*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - b^2*c^3 + a*c^4 + b*c^4

X(5736) lies on these lines: {1, 1441}, {2, 6}, {3, 7}, {77, 1446}, {226, 2268}, {255, 307}, {273, 1442}, {284, 379}, {1253, 1754}, {1958, 5249}, {2476, 2893}, {3485, 4329}

X(5737) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3 - a^2*b - 2*a*b^2 - a^2*c - 2*a*b*c - 2*b^2*c - 2*a*c^2 - 2*b*c^2

Let A4B4C4 be the 4th Conway triangle. Let A′ be the barycentric product B4*C4, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5737). (Randy Hutson, December 10, 2016)

Let A′B′C′ be the tangential triangle, wrt the excentral triangle, of the excentral-hexyl ellipse. A′B′C′ is homothetic to the polar triangle of the Spieker circle at X(5737). (Randy Hutson, August 19, 2019)

X(5737) lies on these lines: {2, 6}, {3, 10}, {9, 1764}, {42, 4042}, {45, 312}, {57, 3739}, {58, 2049}, {63, 4363}, {226, 4643}, {306, 4445}, {345, 594}, {405, 4267}, {573, 2050}, {968, 3706}, {980, 1107}, {1001, 3741}, {1010, 4252}, {1215, 5220}, {1698, 5247}, {2345, 5273}, {3052, 5263}, {3242, 3757}, {3666, 4361}, {3771, 3775}, {3772, 4357}, {4205, 5292}, {4426, 5337}

X(5737) = complement of X(5712)

X(5738) = INTERSECTION OF LINES X(2)X(6) AND X(4)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 - a^4*c - 4*a^3*b*c - 4*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 4*a^2*b*c^2 - 2*a*b^2*c^2 + 2*a^2*c^3 + a*c^4 + b*c^4 - c^5

X(5738) lies on these lines: {1, 307}, {2, 6}, {4, 7}, {65, 4329}, {77, 581}, {78, 3879}, {322, 5554}, {377, 2893}, {411, 1014}, {573, 1445}, {1060, 1442}, {1210, 3664}

X(5739) = INTERSECTION OF LINES X(2)X(6) AND X(4)X(8)

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

Let A′B′C′ be the extangents triangle, and let A_B be the touch point of the A-excircle and the line A′B′, and define B_C and C_A cyclically. Let A_C be the touch point of the A-excircle and the line A′C′, and define B_A and C_B cyclically. Let A″ = B_CB_A∩C_AC_B, and define B″ and C″ cyclically. The A″B″C″ is homothetic to ABC and to the outer and inner Grebe triangles at X(6), to the medial triangle at X(1211), and to the anticomplementary triangle at X(5739). (Randy Hutson, July 7, 2014)

X(5739) lies on these lines: {1, 4101}, {2, 6}, {4, 8}, {7, 4359}, {9, 306}, {57, 4001}, {63, 573}, {75, 4886}, {78, 581}, {209, 2550}, {210, 3416}, {223, 307}, {226, 3686}, {312, 319}, {345, 3219}, {346, 3969}, {377, 1330}, {387, 5051}, {388, 959}, {516, 4061}, {518, 3966}, {519, 4656}, {612, 4104}, {740, 4703}, {968, 4028}, {1376, 4023}, {1479, 4044}, {1714, 3454}, {1743, 5294}, {1836, 3696}, {2478, 3948}, {2886, 4042}, {3305, 3912}, {3666, 4277}, {3703, 5220}, {3707, 4035}, {3715, 3932}, {3782, 4361}, {3870, 3883}, {3879, 5287}, {3929, 3977}, {3952, 3974}, {3965, 3998}, {4034, 4054}, {4270, 4357}, {4384, 5249}, {4423, 4966}, {4660, 4685}, {4666, 4684}

X(5739) = anticomplement of X(940)
X(5739) = isotomic conjugate of isogonal conjugate of X(36744)

X(5740) = INTERSECTION OF LINES X(2)X(6) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4*b - 2*a^2*b^3 + b^5 + a^4*c + 2*a^3*b*c + a^2*b^2*c + a^2*b*c^2 - b^3*c^2 - 2*a^2*c^3 - b^2*c^3 + c^5

X(5740) lies on these lines: {2, 6}, {5, 7}, {269, 5400}, {273, 2973}, {307, 1210}, {404, 2893}, {579, 857}, {1441, 1737}, {1442, 5396}, {1788, 4329}

X(5741) = INTERSECTION OF LINES X(2)X(6) AND X(5)X(8)

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

X(5741) lies on these lines: {2, 6}, {5, 8}, {42, 3847}, {43, 4972}, {78, 5016}, {100, 4192}, {149, 3996}, {200, 5014}, {210, 3006}, {226, 4359}, {306, 3452}, {312, 3969}, {321, 908}, {329, 4488}, {386, 5051}, {404, 1330}, {748, 3771}, {899, 2887}, {970, 3869}, {1043, 5046}, {1210, 4101}, {2886, 4023}, {3216, 3454}, {3681, 3705}, {3703, 3952}, {3706, 5087}, {3817, 4061}, {3909, 3917}, {3911, 4001}, {3935, 4514}, {3944, 4442}, {4035, 5316}, {4054, 4980}, {4104, 4981}, {4414, 4703}, {4420, 5015}, {4511, 5396}, {5219, 5271}

X(5741) = complement of X(37639)
X(5741) = anticomplement of X(37634)

X(5742) = INTERSECTION OF LINES X(2)X(6) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4*b - a^3*b^2 - 3*a^2*b^3 + a*b^4 + b^5 + 2*a^4*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 6*a*b^2*c^2 - 2*b^3*c^2 - 3*a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5742) lies on these lines: {2, 6}, {5, 9}, {48, 4999}, {71, 2886}, {307, 3739}, {442, 579}, {936, 5396}, {970, 2262}, {1210, 5257}, {1698, 1723}, {1839, 4640}, {1865, 5125}, {1901, 2476}

X(5743) = INTERSECTION OF LINES X(2)X(6) AND X(5)X(10)

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

X(5743) lies on these lines: {2, 6}, {5, 10}, {37, 3687}, {42, 4023}, {43, 4026}, {45, 345}, {57, 4643}, {75, 4415}, {226, 3739}, {312, 594}, {321, 3264}, {329, 4363}, {386, 4205}, {518, 4104}, {536, 4656}, {612, 3966}, {756, 3703}, {984, 4884}, {997, 5396}, {1376, 4192}, {1999, 4886}, {2161, 2339}, {2887, 3826}, {2999, 4657}, {3035, 5150}, {3416, 5268}, {3666, 4364}, {3741, 3816}, {3752, 4357}, {3772, 4384}, {3775, 3840}, {3782, 4359}, {3838, 3846}, {3980, 4703}, {4199, 5132}, {4239, 5347}

X(5743) = complement of X(940)

X(5744) = INTERSECTION OF LINES X(2)X(7) AND X(3)X(8)

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

X(5744) = {X(2),X(63)}-harmonic conjugate of X(329)

X(5744) lies on these lines: {2, 7}, {3, 8}, {4, 3916}, {10, 4293}, {20, 4652}, {21, 938}, {72, 631}, {78, 3523}, {88, 277}, {92, 4359}, {140, 3927}, {145, 3601}, {165, 4847}, {189, 333}, {191, 499}, {346, 3977}, {348, 658}, {376, 3419}, {391, 610}, {392, 942}, {443, 3436}, {452, 1210}, {497, 4640}, {516, 5231}, {518, 5218}, {549, 3940}, {940, 2256}, {958, 1466}, {1000, 2320}, {1108, 3666}, {1155, 2550}, {1467, 5265}, {1473, 4220}, {2000, 3100}, {2095, 5603}, {2886, 3474}, {3011, 4310}, {3035, 5220}, {3161, 4358}, {3189, 5217}, {3427, 3428}, {3485, 4999}, {3524, 5440}, {3579, 5082}, {3752, 5069}, {3870, 5281}, {4054, 4454}, {4189, 4313}, {4292, 5177}, {4305, 5267}, {4384, 5088}, {4850, 5222}, {5122, 5176}

X(5744) = anticomplement of X(5219)

X(5745) = INTERSECTION OF LINES X(2)X(7) AND X(3)X(10)

Barycentrics (a - b - c)*(2*a^2 + a*b - b^2 + a*c + 2*b*c - c^2) : :
X(5745) = 3X(2) + X(63)

X(5745) = {X(2),X(63)}-harmonic conjugate of X(226)

X(5745) lies on these lines: {2, 7}, {3, 10}, {8, 3158}, {11, 3683}, {21, 950}, {38, 3011}, {39, 1212}, {55, 4847}, {69, 4035}, {71, 1764}, {81, 2323}, {114, 124}, {140, 912}, {165, 2550}, {200, 5218}, {210, 5432}, {219, 940}, {261, 284}, {281, 5307}, {306, 1150}, {312, 2325}, {321, 3977}, {345, 2321}, {377, 4652}, {405, 1210}, {442, 3916}, {443, 1478}, {497, 4512}, {516, 2886}, {535, 3828}, {551, 4930}, {610, 966}, {631, 936}, {758, 942}, {938, 5436}, {1146, 2482}, {1155, 3925}, {1329, 3634}, {1737, 5251}, {1817, 5235}, {1861, 4219}, {1936, 2328}, {2329, 3912}, {2801, 3035}, {2999, 5105}, {3036, 4745}, {3220, 4220}, {3419, 4304}, {3523, 5438}, {3663, 3772}, {3666, 3946}, {3689, 4995}, {3705, 3883}, {3706, 3712}, {3741, 4154}, {3914, 4414}, {3936, 4001}, {4042, 4061}, {4138, 4655}, {4224, 5285}, {4359, 4858}, {4416, 4417}, {5247, 5530}, {5537, 5659}

X(5745) = isotomic conjugate of isogonal conjugate of X(21748)
X(5745) = complement of X(226)
X(5745) = complementary conjugate of X(17052)
X(5745) = polar conjugate of isogonal conjugate of X(22361)
X(5745) = centroid of {A,B,C,X(63)}

X(5746) = INTERSECTION OF LINES X(2)X(7) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 3*a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 + b^5 - 3*a^4*c - 4*a^3*b*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5746) lies on these lines: {2, 7}, {4, 6}, {19, 4295}, {20, 284}, {37, 3487}, {48, 4293}, {65, 281}, {71, 3085}, {72, 2345}, {193, 2893}, {219, 388}, {377, 2287}, {380, 516}, {386, 990}, {391, 5177}, {405, 5120}, {442, 966}, {443, 965}, {452, 5053}, {573, 1715}, {610, 4292}, {946, 2257}, {950, 1449}, {1056, 2256}, {1100, 3488}, {1108, 5603}, {1713, 4253}, {1714, 1743}, {1836, 2264}, {2260, 3086}, {2303, 4340}, {4264, 5304}

X(5747) = INTERSECTION OF LINES X(2)X(7) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - a^4*b - 2*a^3*b^2 + a*b^4 + b^5 - a^4*c - 2*a^3*b*c - 2*a^2*b^2*c + b^4*c - 2*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5747) lies on these lines: {1, 1826}, {2, 7}, {3, 1901}, {4, 284}, {5, 6}, {12, 219}, {48, 1478}, {71, 498}, {281, 3485}, {377, 2327}, {380, 1699}, {386, 3553}, {387, 5587}, {442, 965}, {495, 2256}, {499, 2260}, {594, 3940}, {2287, 2476}, {2303, 5142}, {2549, 5110}, {5053, 5084}, {5105, 5286}

X(5748) = INTERSECTION OF LINES X(2)X(7) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^3 + 3*a^2*b + a*b^2 - 3*b^3 + 3*a^2*c - 6*a*b*c + 3*b^2*c + a*c^2 + 3*b*c^2 - 3*c^3

X(5748) lies on these lines: {2, 7}, {4, 5440}, {5, 8}, {72, 3090}, {78, 3091}, {92, 4358}, {189, 1997}, {200, 3817}, {312, 3262}, {497, 5087}, {936, 5177}, {938, 4193}, {962, 1519}, {1056, 3436}, {1329, 3485}, {2975, 5550}, {3006, 5423}, {3035, 3474}, {3146, 4855}, {3190, 5400}, {3241, 5176}, {3419, 3545}, {3475, 3816}, {3487, 4187}, {3525, 3916}, {3628, 3927}, {3753, 3869}, {3870, 5274}, {4313, 5046}, {4323, 5554}

X(5748) = anticomplement of X(31231)

X(5749) = INTERSECTION OF LINES X(2)X(7) AND X(6)X(8)

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

X(5749) lies on these lines: {1, 346}, {2, 7}, {6, 8}, {10, 391}, {19, 4200}, {37, 2275}, {44, 966}, {45, 5550}, {69, 3758}, {75, 3618}, {86, 344}, {100, 4254}, {141, 4644}, {145, 1449}, {193, 3661}, {198, 404}, {218, 1010}, {281, 608}, {318, 1249}, {319, 1992}, {320, 3619}, {335, 4473}, {377, 2182}, {597, 4361}, {604, 2329}, {612, 5423}, {644, 2256}, {941, 2276}, {962, 1766}, {1100, 3241}, {1125, 3731}, {1698, 3973}, {1901, 5051}, {2092, 2229}, {2171, 3061}, {2264, 2550}, {2268, 4195}, {2269, 3501}, {2322, 3194}, {2325, 3247}, {2975, 5120}, {3240, 4270}, {3553, 4511}, {3554, 4861}, {3589, 4000}, {3617, 3686}, {3621, 4007}, {3623, 4873}, {3624, 3986}, {3629, 4445}, {3632, 4058}, {3635, 4072}, {3636, 4098}, {3663, 4454}, {3664, 4747}, {3672, 3729}, {3739, 4470}, {3745, 3974}, {3875, 4461}, {3912, 3945}, {3946, 4452}, {4034, 4678}, {4371, 4665}, {4393, 4460}, {4416, 5232}, {4419, 4488}, {4648, 4670}, {4698, 4798}, {5042, 5291}

X(5749) = anticomplement of X(17306)

X(5750) = INTERSECTION OF LINES X(2)X(7) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^2 + a*b + b^2 + a*c + 2*b*c + c^2

X(5750) = {X(6),X(10)}-harmonic conjugate of X(3686)

X(5750) lies on these lines: {1, 2321}, {2, 7}, {6, 10}, {8, 1449}, {19, 475}, {34, 281}, {37, 39}, {43, 4270}, {44, 1213}, {45, 3986}, {69, 4667}, {75, 3946}, {86, 3912}, {141, 3664}, {145, 4007}, {171, 4264}, {198, 474}, {213, 992}, {218, 965}, {239, 4967}, {284, 1010}, {346, 3247}, {380, 2550}, {442, 2182}, {443, 610}, {478, 2122}, {515, 572}, {516, 4026}, {519, 594}, {536, 4021}, {551, 3950}, {742, 3008}, {946, 1766}, {950, 964}, {958, 5120}, {966, 1698}, {997, 3553}, {1172, 1861}, {1203, 1224}, {1220, 5053}, {1376, 4254}, {1450, 2324}, {1574, 4263}, {1575, 2092}, {1826, 2267}, {1901, 4205}, {2262, 3753}, {2264, 3925}, {2295, 2300}, {2298, 4071}, {2303, 5280}, {2663, 3783}, {3161, 5550}, {3244, 4058}, {3617, 4034}, {3618, 4384}, {3622, 4873}, {3624, 3731}, {3626, 4545}, {3629, 4690}, {3636, 3723}, {3661, 3879}, {3663, 4363}, {3672, 4659}, {3713, 4847}, {3758, 4416}, {3763, 4675}, {4000, 4470}, {4360, 4431}, {4395, 4739}, {4422, 4698}, {4426, 5019}, {4478, 4725}, {4665, 4852}, {4691, 4969}, {5124, 5267}

X(5750) = complement of X(4357)

X(5751) = INTERSECTION OF LINES X(3)X(6) AND X(4)X(7)

Barycentrics a^2( SA + S (2 (r + 2 R) s / ((r + 2 R) (r + 4 R) - s^2)) ) : :
Barycentrics a^2*(a^5*b^2 - a^4*b^3 - 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 - b^7 + a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - 2*a^2*b^4*c + a*b^5*c + b^6*c + a^5*c^2 + a^4*b*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + b^5*c^2 - a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 - 2*a*b^3*c^3 - b^4*c^3 - 2*a^3*c^4 - 2*a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 + a*b*c^5 + b^2*c^5 + a*c^6 + b*c^6 - c^7) : :

X(5751) lies on these lines: {1, 916}, {3, 6}, {4, 7}, {55, 1779}, {81, 4219}, {1817, 3060}

X(5752) = INTERSECTION OF LINES X(3)X(6) AND X(4)X(8)

Barycentrics a^2 (SA + S (S / (r^2 + 2 r R - s^2)) ) : :
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5 + a^3*b*c + a^2*b^2*c - a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 + a^2*c^3 - a*b*c^3 - a*c^4 - b*c^4 - c^5)

X(5752) = circumcenter of the triangle A″B″C″ be as defined at X(5739); also X(5752) = {X(371)X(372)}-harmonic conjugate of X(1333). Let Γ be the circle of the points X(371), X(372), PU(1), PU(39); then X(5752) is the inverse-in-Γ of X(1333). (Randy Hutson, July 7, 2014)

X(5752) lies on these lines: {3, 6}, {4, 8}, {5, 1211}, {21, 3060}, {24, 2203}, {40, 209}, {51, 405}, {184, 2915}, {404, 2979}, {474, 3917}, {631, 5482}, {674, 3811}, {916, 1490}, {966, 3781}, {978, 3792}, {1006, 3567}, {1437, 1993}, {2183, 3682}, {3056, 5266}, {3149, 5562}, {3240, 3579}, {3560, 5446}, {5047, 5640}

X(5752) = anticomplement of X(37536)
X(5752) = anticomplement of anticomplement of X(34466)

X(5753) = INTERSECTION OF LINES X(3)X(6) AND X(5)X(7)

Barycentrics a^2 (SA + S (4 (r + 2 R) s / ((r + 2 R) (r + 4 R) - 3 s^2)) ) : :
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^6*b + a^5*b^2 - 4*a^4*b^3 - 2*a^3*b^4 + 5*a^2*b^5 + a*b^6 - 2*b^7 + a^6*c + 2*a^5*b*c - a^4*b^2*c - 4*a^3*b^3*c - a^2*b^4*c + 2*a*b^5*c + b^6*c + a^5*c^2 - a^4*b*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 - 4*a^4*c^3 - 4*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 - 2*b^4*c^3 - 2*a^3*c^4 - a^2*b*c^4 - a*b^2*c^4 - 2*b^3*c^4 + 5*a^2*c^5 + 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - 2*c^7)

X(5753) lies on these lines: {3, 6}, {5, 7}, {57, 5400}, {916, 2260}, {942, 1736}

X(5754) = INTERSECTION OF LINES X(3)X(6) AND X(5)X(8)

Barycentrics a^2 (SA + S (2 S / (3 r^2 + 2 r R - s^2) ) : :
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^4*b - a^3*b^2 - 3*a^2*b^3 + a*b^4 + 2*b^5 + a^4*c - 2*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c + b^4*c - a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 - b^3*c^2 - 3*a^2*c^3 + 2*a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4 + 2*c^5)

X(5754) lies on these lines: {3, 6}, {5, 8}, {355, 2051}, {517, 3293}, {3240, 4192}

X(5755) = INTERSECTION OF LINES X(3)X(6) AND X(5)X(9)

Barycentrics a^2 (SA - S (r (r + 2 R) / ((r + R) s)) ) : :
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c - 2*a^4*b*c - a^3*b^2*c + a^2*b^3*c + b^5*c - a^4*c^2 - a^3*b*c^2 + a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 + a^2*b*c^3 + a*b^2*c^3 - 2*b^3*c^3 + 2*a^2*c^4 + b^2*c^4 + a*c^5 + b*c^5 - c^6)

X(5755) lies on these lines: {3, 6}, {5, 9}, {30, 1765}, {40, 1723}, {57, 4888}, {71, 517}, {198, 3211}, {672, 4192}, {942, 1400}, {1385, 2260}, {1766, 2161}, {1781, 5535}, {2197, 5399}, {2361, 5285}, {3973, 5400}

X(5756) = INTERSECTION OF LINES X(3)X(6) AND X(7)X(10)

Barycentrics a^2 (SA + S (3 r + 4 R) s / (r^2 + 4 r R - 2 s^2) ) : :
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(a^3*b + 3*a^2*b^2 - a*b^3 - 3*b^4 + a^3*c + 5*a^2*b*c + 3*a*b^2*c - b^3*c + 3*a^2*c^2 + 3*a*b*c^2 - a*c^3 - b*c^3 - 3*c^4)

X(5756) lies on these lines: {3, 6}, {7, 10}

X(5757) = INTERSECTION OF LINES X(3)X(7) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^9 - 2*a^8*b - 2*a^7*b^2 + 4*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - 2*a^3*b^6 + a*b^8 - 2*a^8*c - 2*a^7*b*c + 3*a^6*b^2*c + 4*a^5*b^3*c + a^4*b^4*c - 2*a^3*b^5*c - 3*a^2*b^6*c + b^8*c - 2*a^7*c^2 + 3*a^6*b*c^2 + 4*a^5*b^2*c^2 + a^4*b^3*c^2 + 2*a^3*b^4*c^2 - 3*a^2*b^5*c^2 - 4*a*b^6*c^2 - b^7*c^2 + 4*a^6*c^3 + 4*a^5*b*c^3 + a^4*b^2*c^3 + 4*a^3*b^3*c^3 + 6*a^2*b^4*c^3 - 3*b^6*c^3 + 2*a^5*c^4 + a^4*b*c^4 + 2*a^3*b^2*c^4 + 6*a^2*b^3*c^4 + 6*a*b^4*c^4 + 3*b^5*c^4 - 2*a^4*c^5 - 2*a^3*b*c^5 - 3*a^2*b^2*c^5 + 3*b^4*c^5 - 2*a^3*c^6 - 3*a^2*b*c^6 - 4*a*b^2*c^6 - 3*b^3*c^6 - b^2*c^7 + a*c^8 + b*c^8

X(5757) lies on these lines: {3, 7}, {4, 6}, {212, 226}

X(5758) = INTERSECTION OF LINES X(3)X(7) AND X(4)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - a^6*b + 3*a^5*b^2 + 3*a^4*b^3 - 3*a^3*b^4 - 3*a^2*b^5 + a*b^6 + b^7 - a^6*c + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - b^6*c + 3*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 3*b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5758) lies on these lines: {3, 7}, {4, 8}, {9, 946}, {40, 226}, {84, 527}, {405, 5603}, {442, 5657}, {499, 5536}, {516, 1490}, {1006, 3616}, {1260, 3149}, {1482, 3488}, {1708, 3086}, {3428, 3485}, {3649, 5584}, {4299, 5538}

X(5759) = INTERSECTION OF LINES X(3)X(7) AND X(4)X(9)

Barycentrics 3*a^6 - 4*a^5*b - 3*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 - b^6 - 4*a^5*c - 2*a^4*b*c + 4*a^3*b^2*c + 2*b^5*c - 3*a^4*c^2 + 4*a^3*b*c^2 - 2*a^2*b^2*c^2 + b^4*c^2 + 4*a^3*c^3 - 4*b^3*c^3 + a^2*c^4 + b^2*c^4 + 2*b*c^5 - c^6 : :
X(5759) = X(4) - 2*X(9)

X(5759) lies on these lines: {3, 7}, {4, 9}, {20, 72}, {37, 3332}, {63, 3358}, {100, 329}, {142, 631}, {165, 226}, {212, 278}, {376, 527}, {390, 517}, {405, 962}, {497, 1708}, {515, 5223}, {518, 944}, {990, 4419}, {991, 4644}, {1001, 1006}, {1253, 4331}, {1490, 2951}, {2318, 2947}, {2724, 2742}, {3576, 5542}, {4295, 5584}, {4301, 5436}

X(5759) = reflection of X(7) in X(3)
X(5759) = isogonal conjugate of the X(3)-vertex conjugate of X(55)
X(5759) = {X(4),X(9)}-harmonic conjugate of X(5817)

X(5760) = INTERSECTION OF LINES X(3)X(7) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^9 + a^8*b + 3*a^7*b^2 - 2*a^6*b^3 - 4*a^5*b^4 + a^4*b^5 + 3*a^3*b^6 - a*b^8 + a^8*c + 2*a^7*b*c - 4*a^5*b^3*c - 4*a^4*b^4*c + 2*a^3*b^5*c + 4*a^2*b^6*c - b^8*c + 3*a^7*c^2 - 4*a^5*b^2*c^2 - 3*a^4*b^3*c^2 - 3*a^3*b^4*c^2 + 2*a^2*b^5*c^2 + 4*a*b^6*c^2 + b^7*c^2 - 2*a^6*c^3 - 4*a^5*b*c^3 - 3*a^4*b^2*c^3 - 4*a^3*b^3*c^3 - 6*a^2*b^4*c^3 + 3*b^6*c^3 - 4*a^5*c^4 - 4*a^4*b*c^4 - 3*a^3*b^2*c^4 - 6*a^2*b^3*c^4 - 6*a*b^4*c^4 - 3*b^5*c^4 + a^4*c^5 + 2*a^3*b*c^5 + 2*a^2*b^2*c^5 - 3*b^4*c^5 + 3*a^3*c^6 + 4*a^2*b*c^6 + 4*a*b^2*c^6 + 3*b^3*c^6 + b^2*c^7 - a*c^8 - b*c^8

X(5760) lies on these lines: {3, 7}, {5, 6}

X(5761) = INTERSECTION OF LINES X(3)X(7) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 - 3*a^6*b - a^5*b^2 + 7*a^4*b^3 - a^3*b^4 - 5*a^2*b^5 + a*b^6 + b^7 - 3*a^6*c + 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + 3*a^2*b^4*c - 2*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 7*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 3*b^4*c^3 - a^3*c^4 + 3*a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 5*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5761) lies on these lines: {3, 7}, {5, 8}, {140, 2095}, {329, 3560}, {382, 5658}, {517, 3085}, {946, 3811}, {1385, 3475}, {3359, 3671}

X(5762) = INTERSECTION OF LINES X(3)X(7) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 2*a^5*b + 3*a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + b^6 + 2*a^5*c - 2*a^3*b^2*c + 2*a^2*b^3*c - 2*b^5*c + 3*a^4*c^2 - 2*a^3*b*c^2 - b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 2*b*c^5 + c^6

X(5762) lies on these lines: {3, 7}, {4, 144}, {5, 9}, {30, 511}, {40, 495}, {140, 142}, {144, 2894}, {165, 4654}, {355, 5223}, {390, 1482}, {495, 4312}, {1385, 5542}, {1483, 3243}, {1484, 3254}, {1699, 3929}, {1754, 3782}, {3332, 4419}, {3817, 5325}, {4654, 4995}

X(5763) = INTERSECTION OF LINES X(3)X(7) AND X(5)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6*b + a^5*b^2 + 5*a^4*b^3 - 2*a^3*b^4 - 4*a^2*b^5 + a*b^6 + b^7 - 2*a^6*c + 8*a^5*b*c + a^4*b^2*c - 6*a^3*b^3*c + 2*a^2*b^4*c - 2*a*b^5*c - b^6*c + a^5*c^2 + a^4*b*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 5*a^4*c^3 - 6*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 3*b^4*c^3 - 2*a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 4*a^2*c^5 - 2*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5763) lies on these lines: {3, 7}, {4, 3940}, {5, 10}, {30, 1490}, {140, 5437}, {165, 3649}, {1058, 1482}, {5433, 5536}

X(5764) = INTERSECTION OF LINES X(3)X(7) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + a^5*b + 2*a^4*b^2 - a*b^5 + a^5*c + 2*a^4*b*c + 3*a^3*b^2*c + 3*a^2*b^3*c - b^5*c + 2*a^4*c^2 + 3*a^3*b*c^2 + 6*a^2*b^2*c^2 + a*b^3*c^2 + 3*a^2*b*c^3 + a*b^2*c^3 + 2*b^3*c^3 - a*c^5 - b*c^5

X(5764) lies on these lines: {1, 4552}, {3, 7}, {6, 8}, {3006, 5294}, {3085, 4307}

X(5765) = INTERSECTION OF LINES X(3)X(7) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^6 - 4*a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + 2*a*b^5 - 4*a^4*b*c - 10*a^3*b^2*c - 6*a^2*b^3*c + 2*a*b^4*c + 2*b^5*c - 4*a^4*c^2 - 10*a^3*b*c^2 - 14*a^2*b^2*c^2 - 4*a*b^3*c^2 - 2*a^3*c^3 - 6*a^2*b*c^3 - 4*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + 2*a*b*c^4 + 2*a*c^5 + 2*b*c^5

X(5765) lies on these lines: {3, 7}, {6, 10}

X(5766) = INTERSECTION OF LINES X(3)X(7) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(5*a^5 - 3*a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + a*b^4 + b^5 - 3*a^4*c - 8*a^3*b*c - 2*a^2*b^2*c - 3*b^4*c - 6*a^3*c^2 - 2*a^2*b*c^2 - 2*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 + a*c^4 - 3*b*c^4 + c^5)

X(5766) lies on these lines: {3, 7}, {8, 9}, {55, 329}, {72, 4313}, {226, 5281}, {516, 3085}, {527, 3601}, {528, 5175}, {2801, 4305}, {3486, 5220}, {3811, 4326}

X(5767) = INTERSECTION OF LINES X(3)X(8) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - 2*a^6*b + a^5*b^2 + 2*a^4*b^3 - a^3*b^4 + a*b^6 - 2*a^6*c + a^4*b^2*c + b^6*c + a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + b^5*c^2 + 2*a^4*c^3 - 2*b^4*c^3 - a^3*c^4 - a*b^2*c^4 - 2*b^3*c^4 + b^2*c^5 + a*c^6 + b*c^6

X(5767) lies on these lines: {3, 8}, {4, 6}, {10, 48}, {184, 5136}, {515, 1754}, {517, 3187}, {860, 1899}, {912, 4463}, {940, 1056}

X(5768) = INTERSECTION OF LINES X(3)X(8) AND X(4)X(7)

Barycentrics -a^7 + 3*a^6*b - a^5*b^2 - 5*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 - 3*a*b^6 + b^7 + 3*a^6*c + 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - 3*a^2*b^4*c + 2*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 5*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 5*a^3*c^4 - 3*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7 : :

X(5768) lies on these lines: {1, 3427}, {3, 8}, {4, 7}, {20, 3218}, {30, 2094}, {57, 515}, {84, 950}, {142, 5587}, {329, 912}, {355, 443}, {390, 3358}, {601, 4339}, {1006, 5273}, {1012, 3488}, {1072, 4310}, {1158, 4294}, {1181, 3562}, {1210, 1467}, {1519, 5274}, {1532, 5658}, {1768, 4302}, {4305, 5450}

X(5768) = reflection of X(4) in X(5722)

X(5769) = INTERSECTION OF LINES X(3)X(8) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 2*a^5*b^2 - a^4*b^3 + 2*a^3*b^4 - a*b^6 + a^6*c - a^4*b^2*c + a^2*b^4*c - b^6*c - 2*a^5*c^2 - a^4*b*c^2 + a^2*b^3*c^2 + a*b^4*c^2 - b^5*c^2 - a^4*c^3 + a^2*b^2*c^3 + 2*b^4*c^3 + 2*a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + 2*b^3*c^4 - b^2*c^5 - a*c^6 - b*c^6

X(5769) lies on these lines: {3, 8}, {5, 6}, {495, 940}

X(5770) = INTERSECTION OF LINES X(3)X(8) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 5*a^5*b^2 - a^4*b^3 + 7*a^3*b^4 - a^2*b^5 - 3*a*b^6 + b^7 + a^6*c + 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - a^2*b^4*c + 2*a*b^5*c - b^6*c - 5*a^5*c^2 + a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 7*a^3*c^4 - a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 - a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5770) lies on these lines: {2, 912}, {3, 8}, {4, 3218}, {5, 7}, {57, 1478}, {355, 1788}, {381, 2094}, {516, 1158}, {938, 3560}, {942, 3086}, {3359, 4847}

X(5770) = anticomplement of X(37713)

X(5771) = INTERSECTION OF LINES X(3)X(8) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 7*a^5*b^2 + a^4*b^3 + 8*a^3*b^4 - 2*a^2*b^5 - 3*a*b^6 + b^7 - 4*a^5*b*c + a^4*b^2*c + 2*a^3*b^3*c + 2*a*b^5*c - b^6*c - 7*a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 + a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + 8*a^3*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 - 2*a^2*c^5 + 2*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5771) lies on these lines: {2, 2095}, {3, 8}, {5, 9}, {57, 495}, {140, 942}, {484, 5659}, {1532, 3219}, {3628, 5316}, {3925, 5535}

X(5772) = INTERSECTION OF LINES X(6)X(8) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(a^2 + 4*a*b - b^2 + 4*a*c + 2*b*c - c^2)

X(5772) lies on these lines: {2, 3677}, {6, 8}, {7, 10}, {894, 3617}, {1215, 5226}, {1698, 4310}, {3679, 4307}, {3755, 4461}, {3932, 5308}, {4645, 4715}

X(5773) = INTERSECTION OF LINES X(3)X(8) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + a^4*b + a^3*b^2 - a^2*b^3 + a*b^4 + a^4*c - 2*a*b^3*c + b^4*c + a^3*c^2 + 2*a*b^2*c^2 - b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - b^2*c^3 + a*c^4 + b*c^4

X(5773) lies on these lines: {2, 101}, {3, 8}, {6, 7}, {57, 4566}, {239, 514}, {1055, 3911}, {1647, 5168}, {2398, 2809}

X(5774) = INTERSECTION OF LINES X(3)X(8) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 2*a^3*b - a^2*b^2 - 2*a*b^3 + 2*a^3*c - 2*a^2*b*c - 2*a*b^2*c - 2*b^3*c - a^2*c^2 - 2*a*b*c^2 - 4*b^2*c^2 - 2*a*c^3 - 2*b*c^3

X(5774) lies on these lines: {3, 8}, {6, 10}, {40, 5295}, {69, 495}, {171, 3679}, {381, 4388}, {517, 2050}, {996, 3626}, {1010, 3617}, {1737, 3966}, {3706, 5119}, {3715, 3992}, {3753, 5271}, {3927, 4385}

X(5775) = INTERSECTION OF LINES X(3)X(8) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + 6*a^3*b - 4*a^2*b^2 - 6*a*b^3 + 3*b^4 + 6*a^3*c - 4*a^2*b*c - 2*a*b^2*c - 4*a^2*c^2 - 2*a*b*c^2 - 6*b^2*c^2 - 6*a*c^3 + 3*c^4

X(5775) lies on these lines: {3, 8}, {7, 10}, {144, 5587}, {519, 5281}, {758, 5226}, {938, 1001}, {1788, 4413}, {2094, 3421}, {3218, 3617}, {3679, 4293}, {5251, 5273}

X(5776) = INTERSECTION OF LINES X(3)X(9) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c - 2*a^4*b^2*c + 2*a^2*b^4*c + 2*b^6*c - a^5*c^2 - 2*a^4*b*c^2 + 2*a^3*b^2*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 4*b^4*c^3 - a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 - 4*b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 + a*c^6 + 2*b*c^6)

The B-excircle meets the sidelines of ABC in 3 points, and likewise for the C-excircle. The 6 points lie on a conic, denoted by A*. Let A′ be the center of A*, and define B′ and C′ cyclically. Then X(5776) is the perspector of A′B′C′ and the 2nd extouch triangle (defined at X(5927). (Randy Hutson, July 7, 2014)

A′B′C′ is also the unary cofactor triangle of the intangents triangle, which is also the cevian triangle of X(1743) wrt excentral triangle. (Randy Hutson, August 29, 2018)

X(5776) lies on these lines: {3, 9}, {4, 6}, {20, 2287}, {40, 4047}, {72, 1766}, {154, 4183}, {219, 515}, {222, 226}, {281, 3197}, {284, 1012}, {405, 572}, {579, 3149}, {944, 2256}, {1713, 5120}, {1715, 2270}, {1743, 1750}

X(5776) = pole of Gergonne line wrt excentral-hexyl ellipse

X(5777) = INTERSECTION OF LINES X(3)X(9) AND X(4)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c + 2*a^2*b^3*c - a*b^4*c - 2*b^5*c - a^4*c^2 + b^4*c^2 - 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 + 2*a^2*c^4 - a*b*c^4 + b^2*c^4 + a*c^5 - 2*b*c^5 - c^6)

Let P be a point in the plane of a triangle ABC and let A′B′C′ be the cevian triangle of P. Let H_A be the orthocenter of triangle AB′C′, and define H_B and H_C cyclically. Then the points A′, B′, C′, H_A, H_B, H_C lie on a conic. (Dominik Burek (ADGEOM #424, Aug 2, 2013)

The conic {A′, B′, C′, H_A, H_B, H_C} is here named the Burek-cevian conic of P. If P = X(8), the conic has center X(5777). More generally, if P is on the Lucas cubic, then the triangles A′B′C′ and H_AH_BH_C are homothetic, and H_AH_BH_C is perspective to ABC at a point on the Darboux cubic. (ADGEOM #431, August 3, 2013, and related postings)

X(5777) = (X(5) of 2nd extouch triangle); see X(5776). Also, X(5777) lies on the Burek-Hutson central cubic, K645.

X(5777) lies on these lines: {1, 1864}, {2, 1071}, {3, 9}, {4, 8}, {5, 226}, {12, 1858}, {20, 3876}, {37, 581}, {40, 210}, {44, 580}, {55, 1898}, {56, 1728}, {63, 3149}, {65, 5587}, {78, 1012}, {119, 125}, {201, 2635}, {342, 1148}, {389, 916}, {392, 452}, {405, 1385}, {411, 3219}, {499, 3660}, {515, 960}, {516, 3678}, {518, 946}, {756, 4300}, {943, 1156}, {950, 952}, {1125, 2801}, {1158, 1376}, {1159, 5665}, {1214, 1745}, {1260, 2057}, {2800, 3036}, {3057, 3586}, {3073, 5266}, {3074, 3465}, {3090, 5439}, {3091, 3868}, {3295, 5534}, {3487, 5045}, {3555, 5603}, {3651, 3652}, {3670, 5400}, {3697, 5657}, {3715, 5584}, {3746, 5531}, {3753, 5177}, {3817, 3874}

X(5777) = midpoint of X(4) and X(72)
X(5777) = reflection of X(942) in X(5)
X(5777) = complement of X(1071)

X(5778) = INTERSECTION OF LINES X(3)X(9) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^7 - 2*a^6*b - a^5*b^2 + 4*a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + a*b^6 - 2*a^6*c + 2*a^5*b*c - 2*a^3*b^3*c + 2*b^6*c - a^5*c^2 + 2*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - a*b^4*c^2 + 2*b^5*c^2 + 4*a^4*c^3 - 2*a^3*b*c^3 - 2*a^2*b^2*c^3 - 4*b^4*c^3 - a^3*c^4 - a*b^2*c^4 - 4*b^3*c^4 - 2*a^2*c^5 + 2*b^2*c^5 + a*c^6 + 2*b*c^6)

X(5778) lies on these lines: {3, 9}, {4, 2287}, {5, 6}, {219, 355}, {284, 3560}, {940, 2003}, {952, 2256}, {1012, 2327}, {3713, 3940}

X(5779) = INTERSECTION OF LINES X(3)X(9) AND X(5)X(7)

Barycentrics a*(a^5 - 4*a^3*b^2 + 2*a^2*b^3 + 3*a*b^4 - 2*b^5 + 2*a^3*b*c + 2*a^2*b^2*c - 2*a*b^3*c - 2*b^4*c - 4*a^3*c^2 + 2*a^2*b*c^2 - 2*a*b^2*c^2 + 4*b^3*c^2 + 2*a^2*c^3 - 2*a*b*c^3 + 4*b^2*c^3 + 3*a*c^4 - 2*b*c^4 - 2*c^5) : :

Let T be a triangle inscribed in the circumcircle and circumscribing the Mandart inellipse. As T varies, its orthocenter traces a circle centered at X(5779) with segment X(4)X(144) as diameter. (Randy Hutson, August 29, 2018)

X(5779) lies on these lines: {3, 9}, {4, 144}, {5, 7}, {40, 3062}, {44, 990}, {45, 991}, {55, 5531}, {119, 3826}, {142, 1656}, {165, 3715}, {210, 1709}, {355, 382}, {381, 527}, {390, 952}, {517, 4915}, {518, 1351}, {954, 3560}, {1001, 2801}, {1012, 3940}, {1538, 5231}, {1750, 3929}, {1768, 4413}, {2951, 3579}, {3711, 5537}, {4312, 5587}, {4326, 5534}, {5273, 5658}

X(5779) = midpoint of X(4) and X(144)
X(5779) = complement of X(36996)
X(5779) = Johnson-isogonal conjugate of X(37820)

X(5780) = INTERSECTION OF LINES X(3)X(9) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^5*b + 6*a^3*b^3 - 3*a^2*b^4 - 3*a*b^5 + 2*b^6 - 3*a^5*c + 6*a^4*b*c - 10*a^2*b^3*c + 3*a*b^4*c + 4*b^5*c + 2*a^2*b^2*c^2 - 2*b^4*c^2 + 6*a^3*c^3 - 10*a^2*b*c^3 - 8*b^3*c^3 - 3*a^2*c^4 + 3*a*b*c^4 - 2*b^2*c^4 - 3*a*c^5 + 4*b*c^5 + 2*c^6)

X(5780) lies on these lines: {3, 9}, {5, 8}, {72, 2095}, {355, 3452}, {952, 5084}, {1210, 1656}, {3149, 3876}

X(5781) = INTERSECTION OF LINES X(3)X(9) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5 - 3*a^5*c - 2*a^4*b*c + 2*a^3*b^2*c + a*b^4*c + 2*b^5*c + 2*a^4*c^2 + 2*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + 2*a^3*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 - 3*a^2*c^4 + a*b*c^4 + a*c^5 + 2*b*c^5)

X(5781) lies on these lines: {3, 9}, {6, 7}, {19, 518}, {20, 220}, {21, 3207}, {48, 1001}, {63, 910}, {101, 1012}, {144, 2287}, {169, 1071}, {218, 4292}, {219, 516}, {284, 954}, {390, 2256}, {1376, 2272}, {1503, 2550}, {1615, 5273}, {2173, 5220}, {2257, 4321}, {3059, 5227}, {3713, 5279}} SEARCH: -0.33541830108619423321

X(5782) = INTERSECTION OF LINES X(3)X(9) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 4*a^2*b*c - a*b^2*c + 2*b^3*c - a^2*c^2 - a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3)

X(5782) lies on these lines: {3, 9}, {6, 8}, {44, 4386}, {346, 2256}, {391, 2255}, {940, 4670}, {956, 5053}, {958, 2267}, {1211, 3330}, {1376, 2183}, {1404, 4390}, {1743, 5264}, {2057, 3965}, {2221, 4383}, {2235, 5205}, {2257, 2297}, {2550, 5480}, {4363, 5228}

X(5783) = INTERSECTION OF LINES X(3)X(9) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4 - a^3*b - a^2*b^2 + a*b^3 - a^3*c + 2*a^2*b*c + a*b^2*c + 2*b^3*c - a^2*c^2 + a*b*c^2 + 4*b^2*c^2 + a*c^3 + 2*b*c^3)

X(5783) lies on these lines: {1, 3713}, {3, 9}, {6, 10}, {37, 997}, {45, 5110}, {72, 2285}, {171, 1743}, {210, 1460}, {218, 1010}, {219, 1065}, {332, 344}, {405, 2268}, {474, 1400}, {475, 608}, {478, 1211}, {572, 958}, {573, 1376}, {604, 956}, {651, 5232}, {960, 1766}, {2050, 3452}, {2256, 2321}

X(5784) = INTERSECTION OF LINES X(3)X(9) AND X(7)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4*b - 2*a^3*b^2 + 2*a*b^4 - b^5 + a^4*c - b^4*c - 2*a^3*c^2 + 4*a*b^2*c^2 + 2*b^3*c^2 + 2*b^2*c^3 + 2*a*c^4 - b*c^4 - c^5)

X(5784) = X(7) of the X(1)-Brocard triangle (see X(5642)).

X(5784) lies on these lines: {2, 1864}, {3, 9}, {7, 8}, {10, 1071}, {19, 1350}, {20, 960}, {21, 662}, {37, 1818}, {46, 5223}, {63, 210}, {72, 527}, {141, 1861}, {142, 442}, {144, 4190}, {219, 990}, {224, 1001}, {354, 2886}, {390, 3890}, {392, 4304}, {480, 2057}, {511, 2262}, {516, 3878}, {528, 3057}, {997, 1012}, {1824, 3917}, {2261, 5085}, {2348, 3423}, {3660, 5231}, {3740, 5273}, {3812, 4208}, {3881, 5542}

X(5785) = INTERSECTION OF LINES X(3)X(9) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5 + a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + 4*a*b^3*c - 3*b^4*c - 6*a^3*c^2 + 2*a^2*b*c^2 + 14*a*b^2*c^2 + 6*b^3*c^2 + 2*a^2*c^3 + 4*a*b*c^3 + 6*b^2*c^3 + 5*a*c^4 - 3*b*c^4 - 3*c^5)

X(5785) lies on these lines: {3, 9}, {7, 10}, {20, 3062}, {144, 4292}, {377, 4312}

X(5786) = INTERSECTION OF LINES X(3)X(10) AND X(4)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + 3*a^6*b - 2*a^4*b^3 + a^3*b^4 - a^2*b^5 - 2*a*b^6 + 3*a^6*c + 2*a^5*b*c - a^2*b^4*c - 2*a*b^5*c - 2*b^6*c - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 2*b^5*c^2 - 2*a^4*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + 4*b^4*c^3 + a^3*c^4 - a^2*b*c^4 + 2*a*b^2*c^4 + 4*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 2*b^2*c^5 - 2*a*c^6 - 2*b*c^6

X(5786) lies on these lines: {3, 10}, {4, 6}, {20, 333}, {29, 154}, {40, 1765}, {64, 412}, {65, 5307}, {84, 1715}, {243, 1854}, {386, 2050}, {388, 940}, {405, 1746}, {407, 1899}, {572, 2049}, {965, 2551}, {1012, 4267}, {1754, 5247}, {1766, 5295}, {1837, 1891}, {1853, 5125}, {3714, 5227}

X(5787) = INTERSECTION OF LINES X(3)X(10) AND X(4)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 2*a^6*b - 3*a^4*b^3 + 3*a^3*b^4 - 2*a*b^6 + b^7 + 2*a^6*c + 2*a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - 2*a^2*b^4*c - b^6*c + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 3*b^4*c^3 + 3*a^3*c^4 - 2*a^2*b*c^4 + 2*a*b^2*c^4 + 3*b^3*c^4 - 3*b^2*c^5 - 2*a*c^6 - b*c^6 + c^7

Let A′B′C′ be the excentral triangle. X(5787) is the radical center of the anticomplementary circles of triangles A′BC, B′CA, C′AB. (Randy Hutson, June 27, 2018)

X(5787) lies on these lines: {3, 10}, {4, 7}, {5, 1490}, {20, 3419}, {30, 84}, {40, 3358}, {57, 1837}, {382, 2095}, {962, 4018}, {990, 1834}, {1467, 1750}, {1699, 3649}, {3091, 5658}, {3601, 5252}, {4219, 5090}

X(5788) = INTERSECTION OF LINES X(3)X(10) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 2*a^5*b^2 + 3*a^3*b^4 - a^2*b^5 - 2*a*b^6 + a^6*c + 2*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - 2*b^6*c - 2*a^5*c^2 + 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 2*a*b^4*c^2 - 2*b^5*c^2 + 2*a^3*b*c^3 + 4*a^2*b^2*c^3 + 4*a*b^3*c^3 + 4*b^4*c^3 + 3*a^3*c^4 + a^2*b*c^4 + 2*a*b^2*c^4 + 4*b^3*c^4 - a^2*c^5 - 2*a*b*c^5 - 2*b^2*c^5 - 2*a*c^6 - 2*b*c^6

X(5788) lies on these lines: {3, 10}, {4, 333}, {5, 6}, {12, 940}, {63, 1867}, {394, 3142}, {970, 2050}, {1150, 3436}, {3560, 4267}, {5247, 5587}

X(5789) = INTERSECTION OF LINES X(3)X(10) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - 6*a^5*b^2 + 9*a^3*b^4 - 3*a^2*b^5 - 4*a*b^6 + 2*b^7 + a^6*c + 2*a^5*b*c + 2*a^4*b^2*c - 2*a^3*b^3*c - a^2*b^4*c - 2*b^6*c - 6*a^5*c^2 + 2*a^4*b*c^2 + 2*a^3*b^2*c^2 + 4*a^2*b^3*c^2 + 4*a*b^4*c^2 - 6*b^5*c^2 - 2*a^3*b*c^3 + 4*a^2*b^2*c^3 + 6*b^4*c^3 + 9*a^3*c^4 - a^2*b*c^4 + 4*a*b^2*c^4 + 6*b^3*c^4 - 3*a^2*c^5 - 6*b^2*c^5 - 4*a*c^6 - 2*b*c^6 + 2*c^7

X(5789) lies on these lines: {3, 10}, {5, 7}, {153, 443}, {381, 3928}

X(5790) = INTERSECTION OF LINES X(3)X(10) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + a^2*b^2 + 2*a*b^3 - 2*b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c + a^2*c^2 - 2*a*b*c^2 + 4*b^2*c^2 + 2*a*c^3 - 2*c^4

X(5790) lies on these lines: {1, 1656}, {2, 952}, {3, 10}, {4, 3617}, {5, 8}, {30, 5657}, {40, 382}, {55, 80}, {119, 2886}, {140, 944}, {145, 3090}, {165, 3534}, {210, 381}, {226, 1159}, {442, 5554}, {495, 3475}, {516, 3654}, {519, 5055}, {546, 962}, {547, 3241}, {912, 3753}, {946, 3626}, {956, 5176}, {997, 5123}, {999, 1737}, {1000, 5274}, {1071, 4002}, {1125, 5070}, {1145, 3434}, {1260, 3419}, {1351, 3416}, {1385, 1698}, {1483, 3616}, {1598, 5090}, {1657, 3579}, {1837, 3295}, {2095, 3421}, {2801, 3968}, {3091, 4678}, {3428, 5659}, {3436, 3927}, {3560, 5086}, {3576, 5054}, {3621, 5056}, {3622, 5067}, {3632, 5079}, {3655, 3828}, {3656, 3817}, {3814, 5289}, {3843, 4691}, {4668, 5072}, {5154, 5330}, {5204, 5445}, {5221, 5270}

X(5790) = anticomplement of X(38028)
X(5790) = centroid of X(3)X(4)X(8)
X(5790) = orthocenter of cross-triangle of Fuhrmann and Ai (aka K798i) triangles

X(5791) = INTERSECTION OF LINES X(3)X(10) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^3*b - 2*a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c - 3*a*b^2*c - 2*a^2*c^2 - 3*a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4

X(5791) lies on these lines: {2, 72}, {3, 10}, {4, 5273}, {5, 9}, {7, 3824}, {12, 57}, {21, 3419}, {28, 5130}, {37, 5292}, {46, 3925}, {63, 442}, {140, 936}, {142, 3634}, {191, 1836}, {210, 498}, {226, 3927}, {345, 5295}, {377, 3916}, {381, 5325}, {405, 1259}, {443, 3436}, {496, 5231}, {549, 5438}, {965, 3211}, {997, 4999}, {1479, 3683}, {1656, 2095}, {1706, 3587}, {1714, 3666}, {1837, 5251}, {2476, 3219}, {2550, 3579}, {3218, 4197}, {3295, 4847}, {3305, 4187}, {3416, 5138}, {3454, 4643}, {3601, 3679}, {3656, 3878}, {3697, 5552}, {3746, 4863}, {5067, 5328}, {5234, 5587}, {5252, 5258}

X(5792) = INTERSECTION OF LINES X(3)X(10) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3*a^5 - a^4*b - a^3*b^2 + a^2*b^3 - 2*a*b^4 - a^4*c + a^2*b^2*c + 2*a*b^3*c - 2*b^4*c - a^3*c^2 + a^2*b*c^2 + 2*b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + 2*b^2*c^3 - 2*a*c^4 - 2*b*c^4

X(5792) lies on these lines: {2, 3207}, {3, 10}, {6, 7}, {19, 4361}, {610, 3739}, {910, 4384}, {2182, 4363}

X(5793) = INTERSECTION OF LINES X(3)X(10) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^2*b^2 + 2*a*b^3 + 4*a^2*b*c + 2*a*b^2*c + 2*b^3*c + a^2*c^2 + 2*a*b*c^2 + 4*b^2*c^2 + 2*a*c^3 + 2*b*c^3

X(5793) lies on these lines: {1, 3714}, {3, 10}, {6, 8}, {65, 4363}, {141, 388}, {333, 3617}, {1211, 3436}, {3052, 4195}, {3679, 5247}, {4720, 5331}, {5078, 5176}

X(5794) = INTERSECTION OF LINES X(3)X(10) AND X(7)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - a^3*b + a*b^3 - b^4 - a^3*c + 2*a^2*b*c + a*b^2*c + a*b*c^2 + 2*b^2*c^2 + a*c^3 - c^4

X(5794) lies on these lines: {1, 442}, {2, 1837}, {3, 10}, {4, 960}, {5, 997}, {7, 8}, {12, 78}, {20, 4640}, {46, 529}, {72, 1478}, {80, 1698}, {145, 3475}, {149, 3890}, {210, 3436}, {224, 3925}, {329, 5229}, {392, 1479}, {407, 1211}, {443, 3812}, {474, 1737}, {495, 3811}, {497, 5175}, {498, 5440}, {528, 1697}, {594, 5227}, {936, 1329}, {938, 3742}, {946, 5289}, {950, 1001}, {965, 1826}, {1155, 3617}, {1159, 3625}, {1220, 5135}, {1265, 3967}, {1420, 5231}, {1610, 4220}, {1836, 2475}, {1861, 1891}, {2182, 2345}, {2476, 4511}, {2551, 3740}, {3057, 3434}, {3091, 5087}, {3421, 4662}, {3485, 3838}, {3576, 4999}, {3698, 5554}, {3876, 5080}, {3880, 5082}, {3916, 4299}, {3966, 5016}, {4255, 5530}, {4314, 4428}, {4679, 5046}, {4855, 5432}

X(5795) = INTERSECTION OF LINES X(3)X(10) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^3 + a^2*b + b^3 + a^2*c + 4*a*b*c - b^2*c - b*c^2 + c^3)

X(5795) lies on these lines: {1, 2551}, {2, 1420}, {3, 10}, {8, 9}, {20, 1706}, {63, 4848}, {65, 527}, {142, 388}, {145, 3984}, {200, 3486}, {226, 3436}, {281, 1891}, {329, 3340}, {474, 4311}, {495, 1125}, {497, 4853}, {519, 960}, {529, 3812}, {535, 3918}, {936, 944}, {952, 5044}, {956, 1210}, {997, 5534}, {1005, 5086}, {1212, 1573}, {1220, 5053}, {1385, 3820}, {1716, 3755}, {1737, 5258}, {1837, 4847}, {2078, 5176}, {2098, 4679}, {2324, 4270}, {2478, 3872}, {2550, 2951}, {2784, 3041}, {2975, 3911}, {3036, 4691}, {3058, 3893}, {3158, 4313}, {3189, 4882}, {3244, 5289}, {3586, 5082}, {3600, 5437}, {3617, 5273}, {3622, 5328}, {3626, 5302}, {3634, 4999}, {3679, 5234}, {3753, 4292}, {3913, 4314}, {4858, 4968}

X(5796) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8*b + 2*a^7*b^2 - 4*a^6*b^3 - 4*a^5*b^4 + 4*a^4*b^5 + 2*a^3*b^6 - b^9 + a^8*c + 2*a^7*b*c - a^6*b^2*c - 4*a^5*b^3*c - a^4*b^4*c + 2*a^3*b^5*c + a^2*b^6*c + 2*a^7*c^2 - a^6*b*c^2 - 3*a^4*b^3*c^2 - 2*a^3*b^4*c^2 + a^2*b^5*c^2 + 3*b^7*c^2 - 4*a^6*c^3 - 4*a^5*b*c^3 - 3*a^4*b^2*c^3 - 4*a^3*b^3*c^3 - 2*a^2*b^4*c^3 + b^6*c^3 - 4*a^5*c^4 - a^4*b*c^4 - 2*a^3*b^2*c^4 - 2*a^2*b^3*c^4 - 3*b^5*c^4 + 4*a^4*c^5 + 2*a^3*b*c^5 + a^2*b^2*c^5 - 3*b^4*c^5 + 2*a^3*c^6 + a^2*b*c^6 + b^3*c^6 + 3*b^2*c^7 - c^9

X(5796) lies on these lines: {4, 6}, {5, 7}, {226, 1736}, {3100, 5396}

X(5797) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6*b - 2*a^5*b^2 - 3*a^4*b^3 + 2*a^3*b^4 + a^2*b^5 + b^7 + a^6*c - 2*a^5*b*c - 2*a^4*b^2*c + a^2*b^4*c + 2*a*b^5*c - 2*a^5*c^2 - 2*a^4*b*c^2 - 4*a^3*b^2*c^2 - 2*a^2*b^3*c^2 - 2*b^5*c^2 - 3*a^4*c^3 - 2*a^2*b^2*c^3 - 4*a*b^3*c^3 + b^4*c^3 + 2*a^3*c^4 + a^2*b*c^4 + b^3*c^4 + a^2*c^5 + 2*a*b*c^5 - 2*b^2*c^5 + c^7

X(5797) lies on these lines: {4, 6}, {5, 8}, {10, 1953}, {355, 3187}

X(5798) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7*b + a^6*b^2 + 4*a^5*b^3 - a^4*b^4 - 2*a^3*b^5 - a^2*b^6 + b^8 - 2*a^7*c + 4*a^6*b*c + 2*a^5*b^2*c + 2*a^3*b^4*c - 4*a^2*b^5*c - 2*a*b^6*c + a^6*c^2 + 2*a^5*b*c^2 + 2*a^4*b^2*c^2 + a^2*b^4*c^2 - 2*a*b^5*c^2 - 4*b^6*c^2 + 4*a^5*c^3 + 8*a^2*b^3*c^3 + 4*a*b^4*c^3 - a^4*c^4 + 2*a^3*b*c^4 + a^2*b^2*c^4 + 4*a*b^3*c^4 + 6*b^4*c^4 - 2*a^3*c^5 - 4*a^2*b*c^5 - 2*a*b^2*c^5 - a^2*c^6 - 2*a*b*c^6 - 4*b^2*c^6 + c^8

X(5798) lies on these lines: {4, 6}, {5, 9}, {226, 1465}, {442, 573}, {1243, 1903}, {1490, 5396}, {1699, 1723}

X(5799) = INTERSECTION OF LINES X(4)X(6) AND X(5)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^5*b^2 - 3*a^4*b^3 + 2*a^3*b^4 + 2*a^2*b^5 + a*b^6 + b^7 - 4*a^5*b*c - 3*a^4*b^2*c + 2*a^2*b^4*c + 4*a*b^5*c + b^6*c - 3*a^5*c^2 - 3*a^4*b*c^2 - 4*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 - b^5*c^2 - 3*a^4*c^3 - 4*a^2*b^2*c^3 - 8*a*b^3*c^3 - b^4*c^3 + 2*a^3*c^4 + 2*a^2*b*c^4 - a*b^2*c^4 - b^3*c^4 + 2*a^2*c^5 + 4*a*b*c^5 - b^2*c^5 + a*c^6 + b*c^6 + c^7

X(5799) lies on these lines: {4, 6}, {5, 10}, {40, 4026}, {51, 1904}, {65, 1848}, {154, 4198}, {573, 4205}, {2050, 5292}

X(5800) = INTERSECTION OF LINES X(4)X(6) AND X(7)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^4*b^2 - a^2*b^4 - b^6 + 4*a^4*b*c + 4*a^3*b^2*c + a^4*c^2 + 4*a^3*b*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 - a^2*c^4 + b^2*c^4 - c^6

X(5800) lies on these lines: {4, 6}, {7, 8}, {10, 5227}, {28, 159}, {55, 464}, {81, 1370}, {141, 443}, {193, 2475}, {376, 4265}, {497, 1386}, {515, 990}, {631, 5096}, {958, 4026}, {1056, 3242}, {1352, 4260}, {1478, 3751}, {1861, 2285}, {1890, 2082}, {2263, 5236}, {2478, 3618}, {3100, 3486}, {3101, 3474}, {3187, 3434}, {3475, 3920}, {3589, 5084}, {3827, 4295}, {3914, 5307}, {4663, 5229}

X(5801) = INTERSECTION OF LINES X(4)X(6) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + 2*a^5*b + 7*a^4*b^2 - 5*a^2*b^4 - 2*a*b^5 - 3*b^6 + 2*a^5*c + 18*a^4*b*c + 16*a^3*b^2*c - 2*a*b^4*c - 2*b^5*c + 7*a^4*c^2 + 16*a^3*b*c^2 + 10*a^2*b^2*c^2 + 4*a*b^3*c^2 + 3*b^4*c^2 + 4*a*b^2*c^3 + 4*b^3*c^3 - 5*a^2*c^4 - 2*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 - 3*c^6

X(5801) lies on these lines: {4, 6}, {7, 10}

X(5802) = INTERSECTION OF LINES X(4)X(6) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^4 + 2*a^3*b + 2*a*b^3 + b^4 + 2*a^3*c - 2*a*b^2*c - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4)

X(5802) lies on these lines: {2, 272}, {4, 6}, {8, 9}, {10, 380}, {20, 579}, {37, 3488}, {48, 3086}, {71, 4294}, {219, 497}, {226, 1449}, {281, 1837}, {329, 3187}, {405, 966}, {515, 2257}, {573, 1713}, {610, 1210}, {944, 1108}, {965, 5084}, {1058, 2256}, {1100, 3487}, {1213, 4258}, {1708, 3101}, {1743, 3586}, {1839, 4295}, {2260, 4293}, {2287, 2478}, {2345, 3419}, {3189, 3694}, {4207, 5320}, {5037, 5304}, {5257, 5436}

X(5803) = INTERSECTION OF LINES X(4)X(7) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^9 - a^8*b - 2*a^7*b^2 + 2*a^6*b^3 + 2*a^5*b^4 - 2*a^4*b^5 - 2*a^3*b^6 + 2*a^2*b^7 + a*b^8 - b^9 - a^8*c - 2*a^7*b*c + 4*a^5*b^3*c + 4*a^4*b^4*c - 2*a^3*b^5*c - 4*a^2*b^6*c + b^8*c - 2*a^7*c^2 + 4*a^5*b^2*c^2 + 2*a^4*b^3*c^2 + 2*a^3*b^4*c^2 - 4*a^2*b^5*c^2 - 4*a*b^6*c^2 + 2*b^7*c^2 + 2*a^6*c^3 + 4*a^5*b*c^3 + 2*a^4*b^2*c^3 + 4*a^3*b^3*c^3 + 6*a^2*b^4*c^3 - 2*b^6*c^3 + 2*a^5*c^4 + 4*a^4*b*c^4 + 2*a^3*b^2*c^4 + 6*a^2*b^3*c^4 + 6*a*b^4*c^4 - 2*a^4*c^5 - 2*a^3*b*c^5 - 4*a^2*b^2*c^5 - 2*a^3*c^6 - 4*a^2*b*c^6 - 4*a*b^2*c^6 - 2*b^3*c^6 + 2*a^2*c^7 + 2*b^2*c^7 + a*c^8 + b*c^8 - c^9

X(5803) lies on these lines: {4, 7}, {5, 6}

X(5804) = INTERSECTION OF LINES X(4)X(7) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 3*a^6*b - a^5*b^2 - 5*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 - 3*a*b^6 + b^7 + 3*a^6*c - 6*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c - 3*a^2*b^4*c + 10*a*b^5*c - b^6*c - a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 3*a*b^4*c^2 - 3*b^5*c^2 - 5*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 20*a*b^3*c^3 + 3*b^4*c^3 + 5*a^3*c^4 - 3*a^2*b*c^4 + 3*a*b^2*c^4 + 3*b^3*c^4 + a^2*c^5 + 10*a*b*c^5 - 3*b^2*c^5 - 3*a*c^6 - b*c^6 + c^7

X(5804) lies on these lines: {4, 7}, {5, 8}, {517, 5084}, {946, 3340}, {1532, 3487}, {3149, 3488}

X(5805) = INTERSECTION OF LINES X(4)X(7) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c - 2*a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6

X(5805) is the Gergonne point of the Euler triangle.

X(5805) lies on these lines: {3, 142}, {4, 7}, {5, 9}, {11, 57}, {79, 3062}, {144, 3091}, {355, 518}, {381, 527}, {390, 5603}, {392, 443}, {515, 5542}, {517, 2550}, {528, 3656}, {952, 3243}, {954, 3149}, {990, 1086}, {991, 4675}, {1750, 4654}, {3332, 4000}, {4301, 5289}, {5223, 5587}

X(5805) = midpoint of X(4) and X(7)
X(5805) = Johnson-isogonal conjugate of X(37822)
X(5805) = center of the perspeconic of these triangles: inner and outer Johnson

X(5806) = INTERSECTION OF LINES X(4)X(7) AND X(5)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + a^5*c - 4*a^4*b*c - 2*a^2*b^3*c - a*b^4*c + 6*b^5*c - a^4*c^2 + b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 - 12*b^3*c^3 + 2*a^2*c^4 - a*b*c^4 + b^2*c^4 + a*c^5 + 6*b*c^5 - c^6)

X(5806) lies on these lines: {3, 5436}, {4, 7}, {5, 10}, {20, 5439}, {65, 1699}, {72, 3091}, {515, 5045}, {516, 3812}, {546, 912}, {944, 5049}, {962, 3753}, {1465, 2654}, {1482, 3577}, {1709, 5221}, {1837, 5173}, {1902, 5142}, {3057, 5219}, {3585, 5570}, {3742, 4297}, {3832, 3868}, {3876, 5068}, {3940, 4882}

X(5806) = complement of X(31793)

X(5807) = INTERSECTION OF LINES X(4)X(7) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - 2*a^5*b + a^4*b^2 - a^2*b^4 + 2*a*b^5 - b^6 - 2*a^5*c - 6*a^4*b*c - 6*a^3*b^2*c - 2*a^2*b^3*c + a^4*c^2 - 6*a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 - a^2*c^4 + b^2*c^4 + 2*a*c^5 - c^6

X(5807) lies on these lines: {4, 7}, {6, 8}, {390, 1766}, {452, 5279}, {950, 2285}, {4200, 5262}, {4220, 5435}, {5269, 5294}

X(5808) = INTERSECTION OF LINES X(4)X(7) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c - 6*a^4*b*c - 8*a^3*b^2*c - 2*a^2*b^3*c + a*b^4*c - a^4*c^2 - 8*a^3*b*c^2 - 6*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^2*b*c^3 - 2*a*b^2*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 - c^6

X(5808) lies on these lines: {4, 7}, {6, 10}

X(5809) = INTERSECTION OF LINES X(4)X(7) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 - 3*a^4*c - 8*a^3*b*c - 2*a^2*b^2*c - 3*b^4*c + 2*a^3*c^2 - 2*a^2*b*c^2 + 6*a*b^2*c^2 + 2*b^3*c^2 + 2*a^2*c^3 + 2*b^2*c^3 - 3*a*c^4 - 3*b*c^4 + c^5)

X(5809) lies on these lines: {4, 7}, {8, 9}, {10, 4326}, {20, 1445}, {142, 5177}, {226, 5274}, {329, 497}, {388, 5572}, {405, 4313}, {480, 3189}, {516, 2093}, {952, 954}, {1001, 3486}, {1728, 4294}, {1750, 4321}, {1837, 2550}, {2551, 3059}, {3100, 5222}, {3755, 4907}

X(5810) = INTERSECTION OF LINES X(4)X(8) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 + a^6*b - a^5*b^2 - a^4*b^3 + a^3*b^4 + a^2*b^5 - a*b^6 - b^7 + a^6*c - a^4*b^2*c + 2*a^3*b^3*c + a^2*b^4*c - 2*a*b^5*c - b^6*c - a^5*c^2 - a^4*b*c^2 + 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a*b^4*c^2 + b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 4*a*b^3*c^3 + b^4*c^3 + a^3*c^4 + a^2*b*c^4 + a*b^2*c^4 + b^3*c^4 + a^2*c^5 - 2*a*b*c^5 + b^2*c^5 - a*c^6 - b*c^6 - c^7

X(5810) lies on these lines: {2, 1437}, {3, 1211}, {4, 8}, {5, 6}, {442, 1899}, {2203, 3542}, {3410, 5141}

X(5811) = INTERSECTION OF LINES X(4)X(8) AND X(5)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 - a^6*b + 3*a^5*b^2 + 3*a^4*b^3 - 3*a^3*b^4 - 3*a^2*b^5 + a*b^6 + b^7 - a^6*c - 2*a^5*b*c + a^4*b^2*c - 4*a^3*b^3*c + a^2*b^4*c + 6*a*b^5*c - b^6*c + 3*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 - 4*a^3*b*c^3 + 2*a^2*b^2*c^3 - 12*a*b^3*c^3 + 3*b^4*c^3 - 3*a^3*c^4 + a^2*b*c^4 - a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 + 6*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5811) lies on these lines: {3, 5658}, {4, 8}, {5, 7}, {9, 1158}, {84, 3452}, {104, 405}, {226, 3086}, {390, 5534}, {474, 2096}, {912, 938}, {997, 1490}, {1071, 5084}, {1483, 3488}, {1737, 3339}, {4295, 4848}, {4309, 5531}

X(5811) = anticomplement of X(37534)

X(5812) = INTERSECTION OF LINES X(4)X(8) AND X(5)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^7 + 2*a^5*b^2 + a^4*b^3 - a^3*b^4 - 2*a^2*b^5 + b^7 + 2*a^5*b*c + a^4*b^2*c - 2*a^3*b^3*c - b^6*c + 2*a^5*c^2 + a^4*b*c^2 - 2*a^3*b^2*c^2 + 2*a^2*b^3*c^2 - 3*b^5*c^2 + a^4*c^3 - 2*a^3*b*c^3 + 2*a^2*b^2*c^3 + 3*b^4*c^3 - a^3*c^4 + 3*b^3*c^4 - 2*a^2*c^5 - 3*b^2*c^5 - b*c^6 + c^7

X(5812) lies on these lines: {3, 226}, {4, 8}, {5, 9}, {11, 1728}, {12, 40}, {30, 1490}, {68, 1903}, {79, 165}, {222, 1076}, {452, 5603}, {580, 3772}, {908, 1259}, {946, 958}, {950, 1482}, {1006, 5253}, {1210, 2095}, {1385, 3487}, {1479, 1864}, {1766, 1901}, {5177, 5657}

X(5812) = anticomplement of X(37623)
X(5812) = X(26)-of-2nd-extouch-triangle
X(5812) = X(40)-of-outer-Johnson-triangle
X(5812) = orthologic center of these triangles: outer-Johnson to 3rd extouch

X(5813) = INTERSECTION OF LINES X(4)X(8) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 + a^4*b - a*b^4 - b^5 + a^4*c - 4*a^3*b*c + 2*a^2*b^2*c + b^4*c + 2*a^2*b*c^2 + 2*a*b^2*c^2 - a*c^4 + b*c^4 - c^5

X(5813) lies on these lines: {2, 169}, {4, 8}, {6, 7}, {9, 4329}, {226, 2082}, {307, 2270}, {857, 1211}, {3616, 4223}, {3661, 5179}

X(5814) = INTERSECTION OF LINES X(4)X(8) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^4 - a^3*b + a*b^3 + b^4 - a^3*c + 3*a*b^2*c + 2*b^3*c + 3*a*b*c^2 + 2*b^2*c^2 + a*c^3 + 2*b*c^3 + c^4

The triangle A″B″C″ defined at X(5739) is homothetic to the outer Garcia triangle at X(5814).

X(5814) lies on these lines: {1, 1211}, {3, 3687}, {4, 8}, {6, 10}, {9, 3695}, {69, 942}, {75, 1330}, {78, 5396}, {209, 4680}, {306, 405}, {442, 5271}, {1125, 4035}, {1479, 3706}, {1836, 4647}, {2895, 3868}, {3187, 5051}, {3295, 3883}, {3454, 3772}, {3927, 4416}, {4651, 5300}, {4863, 4894}, {5244, 5290}

X(5814) = anticomplement of X(37594)

X(5815) = INTERSECTION OF LINES X(4)X(8) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^4 - 2*a^3*b + 2*a*b^3 + b^4 - 2*a^3*c - 4*a^2*b*c + 6*a*b^2*c + 6*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5815) lies on these lines: {1, 5129}, {2, 3333}, {4, 8}, {7, 10}, {20, 200}, {40, 144}, {69, 341}, {210, 388}, {443, 3697}, {452, 3870}, {516, 4882}, {518, 938}, {527, 1706}, {936, 3600}, {944, 3940}, {956, 3616}, {997, 4308}, {1056, 5044}, {1722, 4310}, {2550, 4662}, {3085, 5273}, {3086, 5328}, {3091, 4847}, {3555, 5084}, {3679, 4295}, {3811, 4313}, {3927, 5657}, {3961, 4339}, {4005, 5252}, {4301, 4915}, {4419, 4646}, {4863, 5225}, {5056, 5231}

X(5815) = anticomplement of X(3333)
X(5815) = isotomic conjugate of X(30501)

X(5816) = INTERSECTION OF LINES X(4)X(9) AND X(5)X(6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^5 + a^4*b - 2*a^2*b^3 + a*b^4 + b^5 + a^4*c - 2*a^3*b*c + b^4*c - 2*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*b^2*c^3 + a*c^4 + b*c^4 + c^5

X(5816) lies on these lines: {2, 572}, {3, 1213}, {4, 9}, {5, 6}, {37, 355}, {80, 941}, {119, 5517}, {377, 1765}, {391, 3091}, {498, 2268}, {499, 604}, {515, 5257}, {581, 975}, {946, 3686}, {1400, 1478}, {1474, 3542}, {1479, 2269}, {1737, 2285}, {1899, 3136}, {2328, 4207}, {3419, 3965}, {3553, 4270}

X(5817) = INTERSECTION OF LINES X(4)X(9) AND X(5)X(7)

Barycentrics a^6 - 5*a^4*b^2 + 4*a^3*b^3 + 3*a^2*b^4 - 4*a*b^5 + b^6 + 2*a^4*b*c + 4*a^3*b^2*c - 4*a*b^4*c - 2*b^5*c - 5*a^4*c^2 + 4*a^3*b*c^2 - 6*a^2*b^2*c^2 + 8*a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 + 8*a*b^2*c^3 + 4*b^3*c^3 + 3*a^2*c^4 - 4*a*b*c^4 - b^2*c^4 - 4*a*c^5 - 2*b*c^5 + c^6 : :
X(5817) = X(4) + 2*X(9)

X(5817) lies on these lines: {2, 971}, {4, 9}, {5, 7}, {44, 3332}, {119, 1156}, {142, 3090}, {144, 3091}, {355, 390}, {443, 3358}, {518, 5603}, {527, 3545}, {944, 1001}, {946, 5223}, {948, 1736}, {952, 954}, {1698, 3062}, {1788, 4312}, {3487, 5045}

X(5817) = anticomplement of X(38122)
X(5817) = {X(4),X(9)}-harmonic conjugate of X(5759)

X(5818) = INTERSECTION OF LINES X(4)X(9) AND X(5)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + 2*a^2*b^2 + 2*a*b^3 - 3*b^4 - 2*a^3*c + 4*a^2*b*c - 2*a*b^2*c + 2*a^2*c^2 - 2*a*b*c^2 + 6*b^2*c^2 + 2*a*c^3 - 3*c^4

X(5818) lies on these lines: {1, 3090}, {2, 355}, {3, 5260}, {4, 9}, {5, 8}, {12, 3487}, {46, 5229}, {80, 498}, {100, 3560}, {104, 474}, {119, 2476}, {145, 5056}, {165, 3529}, {377, 2096}, {381, 962}, {388, 1737}, {495, 938}, {499, 3476}, {515, 631}, {517, 3091}, {519, 5071}, {547, 1483}, {942, 5261}, {946, 3545}, {952, 1656}, {986, 4947}, {1056, 1210}, {1071, 4208}, {1125, 5067}, {1478, 1788}, {1699, 3855}, {1837, 3085}, {2099, 3614}, {2475, 3652}, {3086, 5252}, {3089, 5090}, {3146, 3579}, {3241, 5055}, {3474, 3585}, {3524, 3828}, {3525, 3576}, {3628, 5550}, {3654, 3839}, {3753, 5177}, {3877, 5187}, {3925, 5658}, {4299, 5445}, {4301, 4691}, {4305, 5432}, {4330, 5560}, {4678, 5068}, {5086, 5552}, {5119, 5225}

X(5818) = incenter of cross triangle of Euler and anti-Euler triangles

X(5819) = INTERSECTION OF LINES X(4)X(9) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -3*a^4 + 2*a^3*b - 2*a^2*b^2 + 2*a*b^3 + b^4 + 2*a^3*c - 2*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4

X(5819) lies on these lines: {2, 910}, {4, 9}, {6, 7}, {20, 1212}, {37, 390}, {41, 3485}, {75, 144}, {101, 5603}, {142, 610}, {198, 1001}, {218, 4295}, {220, 962}, {348, 4209}, {388, 2082}, {518, 2262}, {672, 3474}, {857, 1213}, {954, 4254}, {1449, 5542}, {1478, 5540}, {1738, 1743}, {1836, 2348}, {2170, 3476}, {2280, 3475}, {3207, 3616}, {3487, 4251}, {3686, 5223}, {3772, 5304}, {5263, 5296}

X(5820) = INTERSECTION OF LINES X(5)X(6) AND X(7)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^4*b^2 + a^2*b^4 - b^6 + 2*a^4*b*c + 2*a^3*b^2*c - a^4*c^2 + 2*a^3*b*c^2 + 2*a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6

X(5820) lies on these lines: {5, 6}, {7, 8}, {141, 474}, {542, 5138}, {940, 1899}, {1012, 1503}

X(5821) = INTERSECTION OF LINES X(5)X(6) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - 2*a*b^5 - 3*b^6 + 8*a^4*b*c + 10*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c + a^4*c^2 + 10*a^3*b*c^2 + 10*a^2*b^2*c^2 + 4*a*b^3*c^2 + 3*b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 + a^2*c^4 - 2*a*b*c^4 + 3*b^2*c^4 - 2*a*c^5 - 2*b*c^5 - 3*c^6

X(5821) lies on these lines: {5, 6}, {7, 10}

X(5822) = INTERSECTION OF LINES X(5)X(6) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^4 - 2*a^2*b^2 + 2*a*b^3 + b^4 + 2*a^2*b*c - 2*a*b^2*c - 2*a^2*c^2 - 2*a*b*c^2 - 2*b^2*c^2 + 2*a*c^3 + c^4)

X(5822) lies on these lines: {5, 6}, {8, 9}, {284, 966}, {499, 2317}, {1737, 2261}, {1743, 1826}

X(5823) = INTERSECTION OF LINES X(5)X(7) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 + a^5*b - 3*a^4*b^2 + a^2*b^4 - a*b^5 + b^6 + a^5*c + 4*a^4*b*c + 3*a^3*b^2*c - a^2*b^3*c + b^5*c - 3*a^4*c^2 + 3*a^3*b*c^2 - 4*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 - a^2*b*c^3 + a*b^2*c^3 - 2*b^3*c^3 + a^2*c^4 - b^2*c^4 - a*c^5 + b*c^5 + c^6

X(5823) lies on these lines: {5, 7}, {6, 8}, {1736, 3086}

X(5824) = INTERSECTION OF LINES X(5)X(7) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -a^6 - 2*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - 2*b^6 - 2*a^5*c - 8*a^4*b*c - 6*a^3*b^2*c + 2*a^2*b^3*c - 2*b^5*c + 2*a^4*c^2 - 6*a^3*b*c^2 + 2*a^2*b^2*c^2 + 2*b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*b^3*c^3 + a^2*c^4 + 2*b^2*c^4 - 2*b*c^5 - 2*c^6

X(5824) lies on these lines: {5, 7}, {6, 10}, {982, 1736}

X(5825) = INTERSECTION OF LINES X(5)X(7) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(3*a^5 + 3*a^4*b - 10*a^3*b^2 - 2*a^2*b^3 + 7*a*b^4 - b^5 + 3*a^4*c + 8*a^3*b*c + 2*a^2*b^2*c + 3*b^4*c - 10*a^3*c^2 + 2*a^2*b*c^2 - 14*a*b^2*c^2 - 2*b^3*c^2 - 2*a^2*c^3 - 2*b^2*c^3 + 7*a*c^4 + 3*b*c^4 - c^5)

X(5825) lies on these lines: {5, 7}, {8, 9}, {11, 329}, {72, 4345}, {1728, 4293}, {1737, 4312}, {1864, 3740}, {3086, 5542}

X(5826) = INTERSECTION OF LINES X(5)X(8) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - 2*a^4*b - a^3*b^2 + a^2*b^3 + b^5 - 2*a^4*c + 4*a^3*b*c - 2*a^2*b^2*c + 2*a*b^3*c - 2*b^4*c - a^3*c^2 - 2*a^2*b*c^2 - 4*a*b^2*c^2 + b^3*c^2 + a^2*c^3 + 2*a*b*c^3 + b^2*c^3 - 2*b*c^4 + c^5

X(5826) lies on these lines: {5, 8}, {6, 7}

X(5827) = INTERSECTION OF LINES X(5)X(8) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 + a^2*b^2 - 2*b^4 + 2*a^2*b*c - 4*a*b^2*c - 2*b^3*c + a^2*c^2 - 4*a*b*c^2 - 2*b*c^3 - 2*c^4

X(5827) lies on these lines: {5, 8}, {6, 10}, {355, 2050}, {2551, 3695}, {5295, 5587}

X(5828) = INTERSECTION OF LINES X(5)X(8) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 - 2*a^3*b + 4*a^2*b^2 + 2*a*b^3 - 5*b^4 - 2*a^3*c + 12*a^2*b*c - 10*a*b^2*c + 4*a^2*c^2 - 10*a*b*c^2 + 10*b^2*c^2 + 2*a*c^3 - 5*c^4

X(5828) lies on these lines: {5, 8}, {7, 10}, {341, 3262}, {4853, 5056}

X(5829) = INTERSECTION OF LINES X(5)X(9) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7 + 2*a^6*b + a^5*b^2 + a^4*b^3 - 4*a^2*b^5 + a*b^6 + b^7 + 2*a^6*c - a^4*b^2*c + 4*a^3*b^3*c - 4*a*b^5*c - b^6*c + a^5*c^2 - a^4*b*c^2 + 4*a^2*b^3*c^2 - a*b^4*c^2 - 3*b^5*c^2 + a^4*c^3 + 4*a^3*b*c^3 + 4*a^2*b^2*c^3 + 8*a*b^3*c^3 + 3*b^4*c^3 - a*b^2*c^4 + 3*b^3*c^4 - 4*a^2*c^5 - 4*a*b*c^5 - 3*b^2*c^5 + a*c^6 - b*c^6 + c^7

X(5829) lies on these lines: {5, 9}, {6, 7}, {142, 1375}, {511, 2262}, {528, 1953}, {910, 5249}, {1723, 4312}

X(5830) = INTERSECTION OF LINES X(5)X(9) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^5 - 2*a^4*b - a^3*b^2 + a^2*b^3 - a*b^4 + b^5 - 2*a^4*c + 4*a^3*b*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 2*a*b^2*c^2 - 2*b^3*c^2 + a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 - a*c^4 + b*c^4 + c^5

X(5830) lies on these lines: {5, 9}, {6, 8}

X(5831) = INTERSECTION OF LINES X(5)X(9) AND X(6)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - a^3*b^2 - a^2*b^3 + b^5 + 2*a^3*b*c - 5*a^2*b^2*c - 2*a*b^3*c + b^4*c - a^3*c^2 - 5*a^2*b*c^2 - 4*a*b^2*c^2 - 2*b^3*c^2 - a^2*c^3 - 2*a*b*c^3 - 2*b^2*c^3 + b*c^4 + c^5

X(5831) lies on these lines: {5, 9}, {6, 10}, {442, 2285}, {475, 1880}, {498, 3965}, {1766, 2886}, {2268, 3419}

X(5832) = INTERSECTION OF LINES X(5)X(9) AND X(7)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - a^4*b^2 + a^2*b^4 + a*b^5 - b^6 - a^5*c + 2*a^4*b*c - 4*a^2*b^3*c + a*b^4*c + 2*b^5*c - a^4*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 4*a^2*b*c^3 - 2*a*b^2*c^3 - 4*b^3*c^3 + a^2*c^4 + a*b*c^4 + b^2*c^4 + a*c^5 + 2*b*c^5 - c^6

X(5832) lies on these lines: {5, 9}, {7, 8}, {63, 1836}, {142, 474}, {495, 1706}, {516, 993}, {611, 1738}, {956, 4292}, {1376, 5249}, {1387, 3254}, {1818, 4675}, {1861, 4363}

X(5833) = INTERSECTION OF LINES X(5)X(9) AND X(7)X(10)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6 - a^5*b - 2*a^3*b^3 + a^2*b^4 + 3*a*b^5 - 2*b^6 - a^5*c + 8*a^4*b*c - 2*a^3*b^2*c - 12*a^2*b^3*c + 3*a*b^4*c + 4*b^5*c - 2*a^3*b*c^2 - 10*a^2*b^2*c^2 - 6*a*b^3*c^2 + 2*b^4*c^2 - 2*a^3*c^3 - 12*a^2*b*c^3 - 6*a*b^2*c^3 - 8*b^3*c^3 + a^2*c^4 + 3*a*b*c^4 + 2*b^2*c^4 + 3*a*c^5 + 4*b*c^5 - 2*c^6

X(5833) lies on these lines: {5, 9}, {7, 10}, {63, 4312}, {142, 936}, {515, 2550}, {1699, 5273}

X(5834) = INTERSECTION OF LINES X(5)X(10) AND X(6)X(7)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4*b - a^3*b^2 + a^2*b^3 + a*b^4 + b^5 - 2*a^4*c + 4*a^3*b*c - 3*a^2*b^2*c + 2*a*b^3*c - b^4*c - a^3*c^2 - 3*a^2*b*c^2 - 6*a*b^2*c^2 + a^2*c^3 + 2*a*b*c^3 + a*c^4 - b*c^4 + c^5

X(5834) lies on these lines: {5, 10}, {6, 7}, {19, 3589}, {141, 2262}, {597, 2182}, {1375, 5437}, {2183, 4364}, {2270, 4657}, {3674, 3752}

X(5835) = INTERSECTION OF LINES X(5)X(10) AND X(6)X(8)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^3*b + a^2*b^2 + b^4 + 2*a^3*c + 4*a*b^2*c + 2*b^3*c + a^2*c^2 + 4*a*b*c^2 + 2*b^2*c^2 + 2*b*c^3 + c^4

X(5835) lies on these lines: {1, 3704}, {5, 10}, {6, 8}, {65, 141}, {388, 4363}, {1211, 3869}, {1213, 3959}, {2292, 4364}, {2975, 5078}

X(5835) = complement of X(37614)

X(5836) = INTERSECTION OF LINES X(5)X(10) AND X(7)X(8)

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

A construction of X(5836) is given by Antreas Hatipolakis and Angel Montesdeoca at 24129.

X(5836) lies on these lines: {1, 474}, {2, 3057}, {5, 10}, {7, 8}, {12, 3838}, {37, 3208}, {40, 958}, {46, 956}, {56, 3872}, {57, 4853}, {72, 3679}, {78, 2099}, {92, 1888}, {100, 2646}, {145, 354}, {191, 3245}, {200, 3340}, {210, 3617}, {318, 1875}, {341, 3967}, {392, 1698}, {404, 1319}, {405, 5119}, {409, 643}, {442, 1145}, {484, 3916}, {519, 942}, {528, 950}, {529, 4292}, {672, 4875}, {758, 3626}, {891, 4925}, {910, 2329}, {936, 5289}, {962, 2551}, {993, 3579}, {997, 1482}, {1001, 1697}, {1104, 5255}, {1125, 1387}, {1155, 2975}, {1191, 1722}, {1193, 4695}, {1210, 3813}, {1212, 3501}, {1616, 5272}, {1829, 1861}, {1836, 3436}, {1837, 3434}, {1858, 5086}, {1859, 5174}, {1864, 5175}, {1887, 5081}, {2098, 4413}, {2171, 3965}, {2262, 2345}, {2292, 2643}, {2475, 5176}, {2800, 3036}, {2817, 3040}, {3244, 5045}, {3246, 3915}, {3303, 3895}, {3304, 3306}, {3336, 5288}, {3339, 4915}, {3421, 4295}, {3555, 3632}, {3601, 4421}, {3616, 3848}, {3621, 3873}, {3625, 3874}, {3634, 3884}, {3635, 5049}, {3636, 3833}, {3666, 4642}, {3670, 4674}, {3678, 4691}, {3681, 3962}, {3683, 5260}, {3711, 3984}, {3744, 3924}, {3746, 5541}, {3876, 3983}, {3877, 4731}, {3894, 4816}, {3899, 3921}, {3900, 4142}, {4015, 4745}, {4018, 4668}, {4071, 4167}, {4084, 4669}, {4428, 5436}, {4746, 4757}, {4847, 4848}

X(5836) = midpoint of X(8) and X(65)
X(5836) = complement of X(3057)

X(5837) = INTERSECTION OF LINES X(5)X(10) AND X(8)X(9)

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

X(5837) lies on these lines: {2, 3340}, {5, 10}, {8, 9}, {65, 142}, {145, 5273}, {226, 3869}, {388, 527}, {392, 1210}, {443, 2093}, {519, 958}, {551, 4999}, {936, 5657}, {1125, 5289}, {1145, 3697}, {1212, 1500}, {1479, 2551}, {2078, 2975}, {3057, 4847}, {3212, 4357}, {3305, 5554}, {3486, 4512}, {3600, 3928}, {3625, 5302}, {3632, 5234}, {3813, 4342}, {3916, 4311}, {4297, 4640}

X(5838) = INTERSECTION OF LINES X(6)X(7) AND X(8)X(9)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(5*a^3 - a^2*b + 3*a*b^2 + b^3 - a^2*c - 6*a*b*c - b^2*c + 3*a*c^2 - b*c^2 + c^3)

X(5838) lies on these lines: {6, 7}, {8, 9}, {41, 3616}, {144, 239}, {169, 938}, {218, 962}, {497, 2348}, {516, 1743}, {910, 5435}, {1001, 5296}, {1212, 4313}, {1445, 2270}, {2170, 3241}, {2264, 2550}

X(5839) = INTERSECTION OF LINES X(6)X(8) AND X(7)X(524)

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

X(5839) lies on these lines: {1,966}, {2,319}, {6,8}, {7,524}, {9,519}, {10,1449}, {37,145}, {44,346}, {48,3684}, {55,4819}, {69,239}, {71,3169}, {72,5802}, {75,193}, {78,3554}, {141,5222}, {144,536}, {219,1067}, {281,2323}, {318,3087}, {345,1914}, {348,4372}, {387,5814}, {393,5081}, {518,2262}, {572,5657}, {573,944}, {597,4478}, {604,1788}, {740,3958}, {894,1992}, {938,965}, {956,4254}, {982,4771}, {1043,1778}, {1086,4402}, {1108,3965}, {1213,3616}, {1400,3476}, {1654,4393}, {1698,4982}, {1740,4489}, {1743,2321}, {1901,5175}, {1953,4051}, {2082,5227}, {2269,3486}, {2325,3973}, {3061,3949}, {3161,3943}, {3187,5739}, {3240,5153}, {3241,5296}, {3244,3247}, {3293,5105}, {3419,5746}, {3553,3872}, {3589,4445}, {3618,3661}, {3623,3723}, {3625,4007}, {3629,4363}, {3630,4395}, {3633,3707}, {3635,3986}, {3672,4643}, {3679,5750}, {3696,4307}, {3739,3945}, {3758,5564}, {3770,4441}, {3875,4416}, {3879,4384}, {4029,4898}, {4047,4294}, {4058,4701}, {4060,4677}, {4061,5269}, {4454,4686}, {4470,4967}, {4545,4668}, {4657,4690}, {5015,5286}, {5120,5687}, {5271,5712}, {5603,5816}, {5703,5742}

X(5839) = anticomplement of X(4851)

X(5840) = INTERSECTION OF LINE X(3)X(11) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^7 + 2*a^6*b + 3*a^5*b^2 - 3*a^4*b^3 - a*b^6 + b^7 + 2*a^6*c - 4*a^5*b*c - a^4*b^2*c + 2*a^3*b^3*c + 2*a*b^5*c - b^6*c + 3*a^5*c^2 - a^4*b*c^2 + a*b^4*c^2 - 3*b^5*c^2 - 3*a^4*c^3 + 2*a^3*b*c^3 - 4*a*b^3*c^3 + 3*b^4*c^3 + a*b^2*c^4 + 3*b^3*c^4 + 2*a*b*c^5 - 3*b^2*c^5 - a*c^6 - b*c^6 + c^7

X(5840) lies on these lines: {3, 11}, {4, 100}, {5, 3035}, {20, 104}, {30, 511}, {36, 5533}, {40, 80}, {140, 3825}, {153, 3146}, {214, 946}, {224, 1537}, {355, 1145}, {550, 1484}, {944, 1320}, {1156, 5759}, {1317, 1482}, {1385, 1387}, {1614, 3045}, {1768, 5709}, {1862, 1872}, {2077, 3583}, {2932, 3149}, {3036, 5690}, {3254, 5732}, {3359, 3586}, {4292, 5083}, {5541, 5691}

X(5841) = INTERSECTION OF LINE X(3)X(12) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 2*a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + a*b^6 - b^7 - 2*a^6*c + 4*a^5*b*c - 3*a^4*b^2*c - 2*a^3*b^3*c + 4*a^2*b^4*c - 2*a*b^5*c + b^6*c - 3*a^5*c^2 - 3*a^4*b*c^2 + 8*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 2*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*a*b^3*c^3 - 3*b^4*c^3 + 4*a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 - 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5841) lies on these lines: {3, 12}, {4, 2975}, {5, 993}, {30, 511}, {40, 4333}, {63, 355}, {80, 5535}, {119, 4996}, {140, 3822}, {226, 1385}, {1872, 1885}, {2077, 4316}, {4305, 5761}, {5691, 5709}

X(5842) = INTERSECTION OF LINE X(4)X(12) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^7 - 2*a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + a*b^6 - b^7 - 2*a^6*c + a^4*b^2*c + b^6*c - 3*a^5*c^2 + a^4*b*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 3*a^4*c^3 - 3*b^4*c^3 - a*b^2*c^4 - 3*b^3*c^4 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7

X(5842) lies on these lines: {3, 2886}, {4, 12}, {5, 5248}, {11, 5172}, {20, 2894}, {30, 511}, {31, 5721}, {40, 1726}, {140, 3841}, {550, 5450}, {944, 2099}, {946, 4314}, {1006, 3925}, {1012, 4302}, {1071, 1770}, {1072, 3744}, {1158, 5787}, {1479, 3149}, {1532, 3583}, {1824, 3575}, {1834, 3072}, {3058, 5603}, {3189, 5758}, {3474, 5768}, {3811, 5812}, {4292, 5173}, {4512, 5587}, {5119, 5691}

X(5843) = INTERSECTION OF LINE X(5)X(7) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 7*a^4*b^2 - 2*a^3*b^3 - 6*a^2*b^4 + 2*a*b^5 + b^6 - 4*a^4*b*c - 2*a^3*b^2*c + 6*a^2*b^3*c + 2*a*b^4*c - 2*b^5*c + 7*a^4*c^2 - 2*a^3*b*c^2 - 4*a*b^3*c^2 - b^4*c^2 - 2*a^3*c^3 + 6*a^2*b*c^3 - 4*a*b^2*c^3 + 4*b^3*c^3 - 6*a^2*c^4 + 2*a*b*c^4 - b^2*c^4 + 2*a*c^5 - 2*b*c^5 + c^6

X(5843) lies on these lines: {3, 144}, {5, 7}, {9, 140}, {30, 511}, {84, 5763}, {142, 3628}, {355, 4312}, {390, 1483}, {546, 5805}, {548, 5732}, {550, 5759}, {1156, 1484}, {2096, 3940}, {2951, 5534}, {3339, 5587}, {3853, 5735}, {3927, 5657}, {5223, 5690}, {5730, 5731}

X(5844) = INTERSECTION OF LINE X(5)X(8) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 4*a^3*b - a^2*b^2 + 4*a*b^3 - b^4 - 4*a^3*c + 8*a^2*b*c - 4*a*b^2*c - a^2*c^2 - 4*a*b*c^2 + 2*b^2*c^2 + 4*a*c^3 - c^4

X(5844) lies on these lines: {1, 140}, {3, 145}, {4, 3621}, {5, 8}, {10, 3628}, {30, 511}, {36, 1317}, {40, 548}, {119, 4867}, {165, 3655}, {355, 546}, {495, 2099}, {496, 2098}, {547, 3679}, {549, 3241}, {550, 944}, {631, 3623}, {632, 3616}, {946, 3625}, {962, 3627}, {1000, 2346}, {1056, 1159}, {1145, 4511}, {1320, 1484}, {1385, 3244}, {1387, 1737}, {1656, 3617}, {2136, 5709}, {3036, 3814}, {3090, 4678}, {3488, 5729}, {3526, 3622}, {3576, 3654}, {3656, 4677}, {3680, 5763}, {3820, 5289}, {3861, 4301}, {4187, 5330}, {4534, 5526}, {5535, 5541}

X(5845) = INTERSECTION OF LINE X(6)X(7) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^4 - 2*a^3*b + a^2*b^2 - b^4 - 2*a^3*c + 2*b^3*c + a^2*c^2 - 2*b^2*c^2 + 2*b*c^3 - c^4

X(5845) lies on these lines: {6, 7}, {9, 141}, {30, 511}, {41, 3665}, {69, 144}, {101, 1565}, {142, 3589}, {150, 1146}, {193, 4440}, {348, 3207}, {390, 3242}, {597, 4795}, {599, 4370}, {903, 1992}, {1001, 4364}, {1350, 5759}, {1352, 5779}, {1386, 4667}, {2550, 4363}, {3416, 4901}, {3618, 4747}, {3620, 4473}, {3751, 4312}, {3763, 4748}, {3826, 4472}, {4904, 5540}, {5480, 5805}

X(5845) = isotomic conjugate of X(35158)
X(5845) = X(2)-Ceva conjugate of X(35093)
X(5845) = crossdifference of every pair of points on line X(6)X(926)

X(5846) = INTERSECTION OF LINE X(6)X(8) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = {-2*a^3 - a*b^2 + b^3 + b^2*c - a*c^2 + b*c^2 + c^3

X(5846) lies on these lines: {1, 141}, {6, 8}, {10, 1386}, {30, 511}, {31, 3703}, {37, 3883}, {42, 4030}, {44, 3717}, {63, 4884}, {69, 145}, {100, 5078}, {182, 5690}, {193, 3621}, {238, 3932}, {306, 3744}, {345, 3052}, {355, 5480}, {595, 3695}, {597, 3679}, {599, 3241}, {612, 3966}, {902, 3712}, {944, 1350}, {1086, 4645}, {1125, 3844}, {1211, 3920}, {1279, 3912}, {1352, 1482}, {1697, 5227}, {1738, 4395}, {1743, 4901}, {1834, 5015}, {1999, 4514}, {2550, 4361}, {2886, 4362}, {2975, 4265}, {3008, 3823}, {3035, 4434}, {3187, 5014}, {3244, 3631}, {3616, 3763}, {3617, 3618}, {3619, 3622}, {3620, 3623}, {3625, 4663}, {3629, 3632}, {3630, 3633}, {3685, 3943}, {3696, 4399}, {3704, 5255}, {3705, 3769}, {3722, 4062}, {3745, 4914}, {3755, 4852}, {3756, 5211}, {3782, 3891}, {3790, 4676}, {3811, 5396}, {3867, 5090}, {4153, 5305}, {4307, 4363}, {4349, 4670}, {4366, 4437}, {4388, 4415}, {4684, 4864}, {5082, 5800}, {5085, 5657}, {5241, 5297}

X(5847) = INTERSECTION OF LINE X(6)X(10) AND THE INFINITY LINE

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

X(5847) lies on these lines: {1, 69}, {6, 10}, {8, 193}, {9, 4078}, {30, 511}, {31, 306}, {38, 4001}, {40, 3169}, {44, 3932}, {55, 4028}, {72, 3688}, {141, 1125}, {171, 3687}, {226, 4362}, {238, 3912}, {239, 1738}, {319, 5263}, {322, 4008}, {345, 1707}, {355, 1351}, {551, 599}, {581, 3811}, {597, 3828}, {612, 4104}, {613, 1210}, {896, 3977}, {902, 4062}, {940, 3966}, {946, 1352}, {950, 3056}, {976, 4101}, {984, 4416}, {1001, 4851}, {1100, 4026}, {1211, 3745}, {1279, 4966}, {1350, 4297}, {1353, 5690}, {1428, 3911}, {1698, 3618}, {1733, 3262}, {1757, 3717}, {1992, 3679}, {1999, 4388}, {2308, 5294}, {2321, 3923}, {2887, 3791}, {2895, 3920}, {3008, 3836}, {3011, 3936}, {3187, 3914}, {3242, 3244}, {3510, 3783}, {3578, 4981}, {3589, 3634}, {3616, 3620}, {3619, 3624}, {3626, 3629}, {3630, 3635}, {3631, 3636}, {3663, 4655}, {3703, 4641}, {3722, 4938}, {3755, 4660}, {3759, 4429}, {3769, 4417}, {3771, 4035}, {3772, 4138}, {3773, 4672}, {3821, 3946}, {3896, 4450}, {4085, 4856}, {4133, 5695}, {4265, 5267}, {4432, 4437}, {4656, 4703}, {4682, 5743}, {4769, 5028}, {4847, 4865}, {4909, 5625}, {5093, 5790}, {5138, 5745}

X(5847) = isogonal conjugate of X(28476)
X(5847) = crossdifference of every pair of points on line X(6)X(834)

X(5848) = INTERSECTION OF LINE X(6)X(11) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + 2*a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + b^5 + 2*a^4*c - 4*a^3*b*c + a^2*b^2*c + 2*a*b^3*c - b^4*c + a^3*c^2 + a^2*b*c^2 - 2*a*b^2*c^2 - a^2*c^3 + 2*a*b*c^3 - a*c^4 - b*c^4 + c^5

X(5848) lies on these lines: {6, 11}, {30, 511}, {69, 100}, {80, 3751}, {119, 1352}, {141, 3035}, {149, 193}, {611, 5820}, {1145, 3416}, {1317, 3242}, {1353, 1484}, {1386, 1387}, {3013, 3140}

X(5849) = INTERSECTION OF LINE X(6)X(12) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a^6 - 3*a^4*b^2 + 2*a^2*b^4 - b^6 - 2*a^3*b^2*c + 2*a^2*b^3*c - 3*a^4*c^2 - 2*a^3*b*c^2 + b^4*c^2 + 2*a^2*b*c^3 + 2*a^2*c^4 + b^2*c^4 - c^6

X(5849) lies on these lines: {6, 12}, {30, 511}, {69, 2975}, {141, 4999}

X(5850) = INTERSECTION OF LINE X(7)X(10) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^3 - 3*a^2*b + 4*a*b^2 + b^3 - 3*a^2*c - b^2*c + 4*a*c^2 - b*c^2 + c^3

X(5850) lies on these lines: {1, 144}, {6, 4353}, {7, 10}, {8, 4312}, {9, 1125}, {30, 511}, {72, 4298}, {142, 3634}, {190, 4684}, {210, 553}, {320, 3717}, {390, 3244}, {946, 5779}, {954, 993}, {962, 3062}, {984, 3664}, {997, 4321}, {1001, 3636}, {1738, 4887}, {1743, 4310}, {1757, 3008}, {2325, 4966}, {2550, 3626}, {2951, 5493}, {3243, 3635}, {3475, 3929}, {3663, 3751}, {3685, 4480}, {3811, 5732}, {3817, 5817}, {3874, 5728}, {3881, 5572}, {3925, 3982}, {3946, 4663}, {4021, 4649}, {4031, 4413}, {4297, 5759}, {4349, 4644}, {4356, 4419}, {4645, 4899}, {4860, 5316}, {5715, 5811}

X(5851) = INTERSECTION OF LINE X(7)X(11) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^5 + 2*a^4*b + 5*a^3*b^2 - 7*a^2*b^3 + a*b^4 + b^5 + 2*a^4*c - 12*a^3*b*c + 7*a^2*b^2*c + 6*a*b^3*c - 3*b^4*c + 5*a^3*c^2 + 7*a^2*b*c^2 - 14*a*b^2*c^2 + 2*b^3*c^2 - 7*a^2*c^3 + 6*a*b*c^3 + 2*b^2*c^3 + a*c^4 - 3*b*c^4 + c^5

X(5851) lies on these lines: {7, 11}, {9, 1768}, {30, 511}, {80, 4312}, {100, 144}, {104, 1001}, {119, 3826}, {153, 2550}, {390, 1317}, {1145, 5223}, {1387, 5542}, {2951, 5528}, {3062, 3254}, {5083, 5572}, {5220, 5657}, {5289, 5698}

X(5852) = INTERSECTION OF LINE X(7)X(12) AND THE INFINITY LINE

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

X(5852) lies on these lines: {7, 12}, {9, 583}, {30, 511}, {144, 1001}, {190, 4966}, {320, 3932}, {329, 3816}, {345, 3632}, {553, 3740}, {1086, 1757}, {2550, 4678}, {3035, 3218}, {3058, 4430}, {3244, 3772}, {3631, 3773}, {3650, 3746}, {3663, 4663}, {4480, 4684}

X(5853) = INTERSECTION OF LINE X(8)X(9) AND THE INFINITY LINE

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

X(5853) lies on these lines: {1, 142}, {2, 3158}, {7, 145}, {8, 9}, {10, 1001}, {11, 3689}, {30, 511}, {40, 5768}, {55, 4847}, {100, 2078}, {144, 3621}, {149, 908}, {190, 4899}, {200, 497}, {210, 3058}, {226, 2900}, {238, 3939}, {306, 5014}, {480, 1837}, {553, 3873}, {673, 3912}, {936, 1058}, {938, 1706}, {944, 5732}, {946, 3811}, {954, 3419}, {956, 4304}, {958, 4314}, {1000, 5785}, {1043, 4483}, {1086, 4864}, {1125, 3813}, {1210, 5687}, {1279, 3008}, {1320, 3254}, {1445, 4848}, {1449, 4344}, {1482, 5805}, {2346, 5178}, {2348, 3021}, {2551, 4882}, {2885, 3622}, {3011, 3722}, {3057, 3059}, {3242, 3663}, {3244, 4780}, {3421, 3586}, {3445, 4678}, {3476, 4321}, {3486, 4326}, {3555, 4292}, {3625, 4133}, {3626, 3773}, {3632, 5223}, {3633, 4312}, {3635, 4743}, {3687, 3996}, {3706, 4030}, {3711, 4679}, {3748, 3925}, {3878, 4523}, {3914, 3938}, {3932, 4702}, {3957, 5249}, {3966, 4061}, {3985, 4541}, {4001, 4450}, {4028, 4865}, {4046, 4914}, {4082, 4387}, {4307, 4667}, {4342, 5289}, {4358, 4939}, {4512, 5325}, {4527, 4701}, {4535, 4746}, {4538, 4662}, {4645, 4684}, {5218, 5231}, {5263, 5750}, {5572, 5836}

X(5853) = isogonal conjugate of X(1477)
X(5853) = isotomic conjugate of X(35160)
X(5853) = X(2)-Ceva conjugate of X(35111)

X(5854) = INTERSECTION OF LINE X(8)X(11) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^3 - 2*a^2*b - 3*a*b^2 + b^3 - 2*a^2*c + 8*a*b*c - b^2*c - 3*a*c^2 - b*c^2 + c^3)

X(5854) lies on these lines: {1, 1145}, {8, 11}, {10, 1387}, {30, 511}, {46, 2136}, {56, 100}, {80, 3632}, {119, 1482}, {149, 3436}, {214, 3244}, {644, 3039}, {1000, 1001}, {1146, 4919}, {1828, 1862}, {1846, 1897}, {3254, 4900}, {3829, 5790}, {3871, 4996}, {4738, 4939}, {4861, 4999}

X(5855) = INTERSECTION OF LINE X(8)X(12) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^4 + 4*a^3*b + a^2*b^2 - 4*a*b^3 + b^4 + 4*a^3*c - 4*a^2*b*c + 2*a*b^2*c + a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 - 4*a*c^3 + c^4

X(5855) lies on these lines: {1, 4999}, {8, 12}, {30, 511}, {55, 145}, {1317, 4996}, {1329, 5730}, {1482, 3813}, {2161, 4969}, {3035, 4511}, {3036, 4867}, {3039, 5526}, {3340, 5794}, {3419, 3632}, {3428, 3913}, {3434, 3621}, {3633, 5119}, {3816, 5289}, {3829, 5603}, {4930, 5790}, {5173, 5836}

X(5856) = INTERSECTION OF LINE X(9)X(11) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c)*(2*a^4 - 2*a^3*b - a^2*b^2 + b^4 - 2*a^3*c + 4*a^2*b*c - 4*b^3*c - a^2*c^2 + 6*b^2*c^2 - 4*b*c^3 + c^4)

X(5856) lies on these lines: {7, 100}, {9, 11}, {30, 511}, {80, 5223}, {104, 5759}, {119, 5805}, {142, 3035}, {144, 149}, {214, 5542}, {390, 1320}, {956, 5698}, {1001, 1387}, {1086, 3939}, {1145, 2550}, {1317, 3243}, {3174, 5528}, {4312, 5541}

X(5857) = INTERSECTION OF LINE X(9)X(12) AND THE INFINITY LINE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2*a^6 + 2*a^5*b + 3*a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + b^6 + 2*a^5*c + 2*a^3*b^2*c - 2*a^2*b^3*c - 2*b^5*c + 3*a^4*c^2 + 2*a^3*b*c^2 - b^4*c^2 - 2*a^3*c^3 - 2*a^2*b*c^3 + 4*b^3*c^3 - 2*a^2*c^4 - b^2*c^4 - 2*b*c^5 + c^6

X(5857) lies on these lines: {7, 2975}, {9, 12}, {30, 511}, {142, 4999}, {219, 4331}, {954, 5698}, {956, 4295}, {4312, 5832}, {5248, 5719}

X(5858) = CENTROID OF TRIANGLE T(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a² + b² + c² + 2(3^-1/2)S

Peter Moses (June 27, 2014) constructed families of triangles T(k,n) and U(k,n) as follows. On side BC of a triangle ABC, erect a regular n-sided polygon, externally. Starting at B and going around the polygon until reaching C, label the vertices v(0), v(1), ..., v(n-1), so that line v(0)v(n-1) = BC, and if n is even, then the lines L(A,k,n) = v(k)v(n-k-1), for k = 1, ..., (n-2)/n, are parallel to BC. If n is odd, let L(A,k,n), for k = (n-1)/2, be the line through v((n-1)/2)) that is parallel to BC. Define lines L(B,k,n) and L(C,k,n) cyclically. Then for each k from 1 to floor((n-1)/2), the lines L(A,k,n), L(B,k,n), L(C,k,n) form a triangle T(k,n) homothetic to ABC. T(1,4) and U(1,4) are the outer and inner Grebe triangles, respectively. Triangle centers defined from T(k,n) include the following, given by 1st barycentrics:

centroid of T(k,n): -2a² + b² + c² + S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of T(k,n): a²[2S²_A - 2S_BS_C + SS_A csc(kπ/n) csc(π/n+kπ/n) sin(π/n)]
orthocenter of T(k,n): a²S_BS_C - S_A(S²_B + S²_C) + SS_BS_C[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of T(k,n): 2S_A(a²S_A - S²_B - S²_C) + S(S_BS_C + S²)[cot(kπ/n)) - cot(kπ/n+π/n)]

If the polygons are erected internally instead of externally, the resulting triangles are denoted by U(k,n), with triangle centers given by changing S to - S in the above first barycentrics:
centroid of U(k,n): -2a² + b² + c² - S*[cot(kπ/n)) - cot(kπ/n+π/n)]
circumcenter of U(k,n): a²[2S²_A - 2S_BS_C - SS_A csc(kπ/n) csc(π/n+kπ/n) sin(π/n)]
orthocenter of U(k,n): a²S_BS_C - S_A(S²_B + S²_C) - SS_BS_C[cot(kπ/n)) - cot(kπ/n+π/n)]
nine-point center of U(k,n): 2S_A(a²S_A - S²_B - S²_C) - S(S_BS_C + S²)[cot(kπ/n)) - cot(kπ/n+π/n)]

Note that T(1,6) = T(1,3) and U(1,6) = U(1,3).

X(5858) lies on these lines: {2, 6}, {3, 533}, {381, 532}, {530, 3830}, {531, 3534}, {538, 3104}, {1351, 5617}

X(5858) = reflection of X(5859) in X(2)
X(5858) = anticomplement of X(33458)

X(5859) = CENTROID OF TRIANGLE U(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a² + b² + c² - 2(3^-1/2)S

The triangle U(1,3) is defined at X(5858).

X(5859) lies on these lines: {2, 6}, {3, 532}, {381, 533}, {530, 3534}, {531, 3830}, {538, 3105}, {1351, 5613}

X(5859) = reflection of X(5858) in X(2)
X(5859) = anticomplement of X(33459)

X(5860) = CENTROID OF TRIANGLE T(1,4)

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

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle..

X(5860) lies on these lines: {2, 6}, {30, 1160}, {519, 3640}, {637, 754}, {1328, 5485}, {3241, 5604}, {3679, 5588}

X(5860) = reflection of X(5861) in X(2)

X(5861) = CENTROID OF TRIANGLE U(1,4)

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

The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle.

X(5861) lies on these lines: {2, 6}, {30, 1161}, {519, 3641}, {638, 754}, {1327, 5485}, {3241, 5605}, {3679, 5589}

X(5861) = reflection of X(5860) in X(2)

X(5862) = CENTROID OF TRIANGLE T(2,6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a² + b² + c² + 3^1/2S

The triangle T(2,6) is defined at X(5858).

X(5862) lies on these lines: {2, 6}, {4, 532}, {376, 533}

X(5862) = reflection of X(5863) in X(2)

X(5863) = CENTROID OF TRIANGLE U(2,6)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a² + b² + c² - 3^1/2S

The triangle U(2,6) is defined at X(5858).

X(5863) lies on these lines: {2, 6}, {4, 533}, {376, 532}

X(5863) = reflection of X(5862) in X(2)

X(5864) = CIRCUMCENTER OF TRIANGLE T(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²[2S²_A - 2S_BS_C + SS_A(4/3)^1/2]

The triangle T(1,3) is defined at X(5858).

X(5864) lies on these lines: {3, 6}, {4, 298}, {20, 3181}, {383, 634}, {394, 3130}, {627, 1080}

X(5864) = reflection of X(5865) in X(3)

X(5865) = CIRCUMCENTER OF TRIANGLE U(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²[2S²_A - 2S_BS_C - SS_A(4/3)^1/2]

The triangle U(1,3) is defined at X(5858).

X(5865) lies on these lines: {3, 6}, {4, 299}, {20, 3180}, {383, 628}, {394, 3129}, {633, 1080}

X(5865) = reflection of X(5864) in X(3)

X(5866) = INVERSE-IN-CIRCUMCIRCLE OF X(69)

Barycentrics a²(b² + c² - a²)[a⁶ - a²(b⁴ - 5b²c² + c⁴) - (b² + c²)(a⁴ - b⁴ + 3b²c² - c⁴)] : : (Richard Hilton, March 2, 2015)

X(5866) lies on these lines: {3,69}, {25,5203}, {99,186}, {187,4558}, {325,2071}, {378,1007}, {669,3265}, {2373,3266}

X(5867) = INVERSE-IN-CIRCUMCIRCLE OF X(81)

Barycentrics a²(a + b)(a + c)[b³(2b² + 3bc + 2c²) - a(2b⁴ + 5b³c + 7b²c² + 5bc³ + 2c⁴) - bc(b + c)[3(b² + c²) - a²]] : : (Richard Hilton, March 2, 2015)

X(5867) lies on these lines: {3,81}, {31,501}, {669,2106}

X(5868) = ORTHOCENTER OF TRIANGLE T(1,3)

Barycentrics    a^2 SB SC+(2 S SB SC)/Sqrt[3]-SA (SB^2+SC^2) : :
Barycentrics    3 (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6)+2 Sqrt[3] (a^2+b^2-c^2) (a^2-b^2+c^2) S : :
Barycentrics    a²S_BS_C - S_A(S²_B + S²_C) + SS_BS_C : :

The triangle U(1,3) is defined at X(5858).

X(5868) lies on these lines: {3, 618}, {4, 6}, {20, 298}, {64, 2993}, {154, 470}, {463, 1899}, {471, 1853}, {633, 1350}, {3146, 3181}

X(5868) = reflection of X(5869) in X(4)

X(5869) = ORTHOCENTER OF TRIANGLE U(1,3)

Barycentrics   a^2 SB SC-(2 S SB SC)/Sqrt[3]-SA (SB^2+SC^2) : :
Barycentrics   3 (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6)-2 Sqrt[3] (a^2+b^2-c^2) (a^2-b^2+c^2) S : :
Barycentrics   a²S_BS_C - S_A(S²_B + S²_C) - SS_BS_C : :

The triangle U(1,3) is defined at X(5858).

X(5869) = reflection of X(5868) in X(4)

X(5869) lies on these lines: {3, 619}, {4, 6}, {20, 299}, {64, 2992}, {154, 471}, {462, 1899}, {470, 1853}, {634, 1350}, {3146, 3180}

X(5870) = ORTHOCENTER OF TRIANGLE T(1,4)

Barycentrics   2a⁶ - a⁴(b² + c² - S) - (b² - c²)²(b² + c² + S) (César Lozada, Dec. 20, 2013)
Barycentrics    a^2 SB SC+S SB SC-SA (SB^2+SC^2) : :
Barycentrics    a²S_BS_C - S_A(S²_B + S²_C) + SS_BS_C : :

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle. X(5870) is the outer-Grebe-triangle-orthologic center of the following triangles: ABC, Euler, anticomplementary, and inner Grebe. (César Lozada, ADGEOM #978, Dec. 20, 2013)

X(5870) lies on these lines: {3, 5590}, {4, 6}, {20, 488}, {30, 1160}, {40, 5688}, {147, 487}, {154, 3535}, {184, 3127}, {185, 1162}, {486, 3424}, {489, 3926}, {515, 3640}, {944, 5604}, {1853, 3536}, {1899, 5200}, {5588, 5691}

X(5870) = reflection of X(5871) in X(4)

X(5871) = ORTHOCENTER OF TRIANGLE U(1,4)

Barycentrics a²S_BS_C - S_A(S²_B + S²_C) - SS_BS_C : :
Barycentrics 2a⁶ - a⁴(b² + c² + S) - (b² - c²)²(b² + c² - S) : : (César Lozada, Dec. 20, 2013)

The triangle U(1,4), defined at X(5858), is also the inner Grebe triangle. X(5871) is the inner-Grebe-triangle-orthologic center of the following triangles: ABC, Euler triangle, anticomplementary triangle, and outer Grebe triangle. (César Lozada, ADGEOM #978, Dec. 20, 2013)

X(5871) lies on these lines: {3, 5591}, {4, 6}, {20, 487}, {30, 1161}, {40, 5689}, {147, 488}, {154, 3536}, {184, 3128}, {185, 1163}, {485, 3424}, {490, 3926}, {515, 3641}, {944, 5605}, {1853, 3535}, {5589, 5691}

X(5871) = reflection of X(5870) in X(4)

X(5872) = NINE-POINT CENTER OF TRIANGLE T(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2S_A(a²S_A - S²_B - S²_C) + S(S_BS_C + 3^-1/2S²)

The triangle T(1,3) is defined at X(5858).

X(5872) lies on these lines: {3, 298}, {4, 3181}, {5, 6}, {18, 5613}, {61, 5617}, {182, 635}, {2782, 3104}

X(5872) = reflection of X(5873) in X(5).

X(5873) = NINE-POINT CENTER OF TRIANGLE U(1,3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2S_A(a²S_A - S²_B - S²_C) - S(S_BS_C + 3^-1/2S²)

The triangle U(1,3) is defined at X(5858).

X(5873) lies on these lines: {3, 299}, {4, 3180}, {5, 6}, {17, 5617}, {62, 5613}, {182, 636}, {2782, 3105}

X(5873) = reflection of X(5872) in X(5).

X(5874) = NINE-POINT CENTER OF TRIANGLE T(1,4)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2S_A(a²S_A - S²_B - S²_C) + S(S_BS_C + S²)

The triangle T(1,4), defined at X(5858), is also the outer Grebe triangle.

X(5874) lies on these lines: {3, 1270}, {5, 6}, {26, 5594}, {30, 1160}, {140, 5590}, {355, 5588}, {952, 3640}, {1483, 5604}, {5688, 5690}

X(5874) = reflection of X(5875) in X(5)

X(5875) = NINE-POINT CENTER OF TRIANGLE U(1,4)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2S_A(a²S_A - S²_B - S²_C) - S(S_BS_C + S²)

The triangle U(1,4) is defined at X(5858), is also the inner Grebe triangle.

X(5875) lies on these lines: {3, 1271}, {5, 6}, {26, 5595}, {30, 1161}, {140, 5591}, {355, 5589}, {952, 3641}, {1483, 5605}, {5689, 5690}

X(5875) = reflection of X(5874) in X(5)

X(5876) = INTERSECTION OF LINES X(3)X(74) AND X(4)X(93)

Barycentrics a²[a⁶(b² + c²) - 3a⁴(b⁴ + c⁴) + 3a²(b⁶ + c⁶) - (b² - c²)²(b⁴ + 3b²c² + c⁴)] : : (Richard Hilton, March 2, 2015)

X(5876) is the orthocenter of the triangle A*B*C* defined at X(5694), and X(5876) is the nine-point center of the X(3)-Fuhrmann triangle; see X(5613). (Randy Hutson, July 7, 2014)

X(5876) is the radical center of the orthocentroidal circles of the adjunct anti-altimedial triangles. (Randy Hutson, November 2, 2017)

X(5876) lies on these lines: {3,74}, {4,93}, {5,389}, {30,5562}, {51,3850}, {52,546}, {140,185}, {143,381}, {511,3627}, {548,3917}, {550,1216}, {568,3091}, {578,1493}, {1498,2918}, {1657,2979}, {2779,5694}, {2807,5690}, {3060,3843}, {3518,3581}, {3567,3851}, {3845,5446}, {5072,5640}

X(5876) = complement of X(34783)

X(5877) = INTERSECTION OF LINES X(4)X(523) AND X(5)X(6)

Barycentrics a¹² + 5a⁸b²c² - [4a⁴ + (b² - c²)²](b² - c²)(b⁶ - c⁶) - a⁶(b² + c²)(2a⁴ - 3b⁴ + 7b²c² - 3c⁴) + 3a²(b² - c²)²(b⁶ + c⁶) : : (Richard Hilton, March 2, 2015)

X(5877) is the similitude center of these equilateral triangles: X(15)-Fuhrmann and X(16)-Fuhrmann. (Randy Hutson, July 7, 2014)

X(5877) lies on these lines: {4,523}, {5,6}, {1899,3134}

X(5878) = INTERSECTION OF LINES X(4)X(51) AND X(5)X(64)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S²_A(20R² - 3S_ω ) + S_A(3S²_ω - 20R²S_ω + S²) + S²(16R² - 3S_ω)

X(5878) = (X(3) of X(20)-Fuhrmann triangle). (Randy Hutson, July 7, 2014)

X(5878) is the orthologic center of the Carnot (aka Johnson) and half-altitude (midheight) triangles. (César Lozada, Perspective-Orthologic-Parallelogic.pdf, ADGEOM #978, December 20, 2013)

X(5878) lies on these lines: {2,3357}, {3,1661}, {4,51}, {5,64}, {20,110}, {30,155}, {66,3521}, {68,5663}, {113,3548}, {154,550}, {382,1351}, {546,1853}, {1181,1885}, {1204,3542}, {3292,3529}

X(5878) = isogonal conjugate of X(5879)
X(5878) = anticomplement of X(3357)

X(5879) = X(4)-VERTEX CONJUGATE OF X(20)

Barycentrics a^2*(a^10+(b^2-3*c^2)*a^8-2*(4*b^4-4*b^2*c^2-c^4)*a^6+2*(b^2-c^2)*(4*b^4+8*b^2*c^2-c^4)*a^4-(b^2-c^2)*(b^6-3*c^6+(13*b^2+5*c^2)*b^2*c^2)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^3)*(a^10-(3*b^2-c^2)*a^8+2*(b^4+4*b^2*c^2-4*c^4)*a^6+2*(b^2-c^2)*(b^4-8*b^2*c^2-4*c^4)*a^4-(b^2-c^2)*(3*b^6-c^6-(5*b^2+13*c^2)*b^2*c^2)*a^2+(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^3) : :

Let A′B′C′ be the half-altitude triangle of ABC. Let L_A be the reflection of line B′C′ in line BC, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. The lines AA″, BB″, CC″ concur in X(5879). (Randy Hutson, July 18, 2014)

X(5879) lies on this line: {1093,1294}

X(5879) = isogonal conjugate of X(5878)

X(5880) = X(6) OF FUHRMANN TRIANGLE

Barycentrics a³ - b³ - c³ + 2abc + b²c + bc² : :
X(5880) = 3*X(7) + X(8)

X(5880) lies on these lines: {1,528}, {2,1155}, {3,142}, {4,3812}, {5,1158}, {6,1738}, {7,8}, {9,46}, {10,527}, {11,3306}, {19,5829}, {40,5735}, {55,1004}, {57,2886}, {63,3925}, {78,3649}, {171,3772}, {200,4654}, {226,1260}, {241,4331}, {329,3740}, {354,3434}, {355,2801}, {390,2646}, {405,1770}, {443,960}, {495,5856}, {497,3742}, {519,1159}, {553,4847}, {612,3782}, {673,1492}, {740,4851}, {750,3120}, {894,4429}, {908,4413}, {940,3914}, {958,4292}, {966,3846}, {1056,3880}, {1373,3641}, {1374,3640}, {1386,4000}, {1445,1454}, {1478,3753}, {1479,5439}, {1633,4223}, {1699,3816}, {1706,5290}, {1714,5165}, {1737,5729}, {1788,5177}, {1837,2475}, {1861,1892}, {1890,4185}, {2182,5819}, {2345,3844}, {2887,3980}, {3035,5219}, {3058,4666}, {3243,3633}, {3244,4780}, {3255,5560}, {3333,3813}, {3419,5696}, {3436,3698}, {3579,3824}, {3617,4741}, {3662,5263}, {3664,3755}, {3729,3932}, {3836,3923}, {3841,5791}, {3872,5434}, {3873,4863}, {3886,4966}, {3912,5695}, {3922,5554}, {3946,4349}, {3966,4359}, {4001,4042}, {4082,4942}, {4361,5847}, {4415,5268}, {4644,4663}, {4691,5850}, {4854,5287}, {5218,5766}, {5223,5852}, {5587,5851}

X(5880) = complement of X(5698)

X(5881) = DARBOUX IMAGE OF X(40)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3 a^4-3 a^3 b-a^2 b^2+3 a b^3-2 b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-2 c^4
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3a/(2s) - S_BS_C/S² - 1
X(5881) = 3X(1) - 4X(5)      (barycentrics and combo, Peter Moses, July 14, 2014)

Let A′ be the reflection of X(40) in A and let A″ be the reflection of X(40) in line BC. Define B′, C′, B″, and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5881). Also, X(5881) = (X(20) of the Fuhrmann triangle). (Randy Hutson, July 7, 2014)

More generally, suppose that X is a point in the plane of triangle ABC, and let A′ be the reflection of X in A and A″ be the reflection of X in line BC. Define B′, C′, B″, and C″ cyclically. The triangles A′B′C′ and A″B″C″ are perspective if X lies on the Darboux cubic. The perspector is here called the Darboux image of X. The appearance of (i,j) in the following list means that X(j) is the Darboux image of the point X(i) on the Darboux cubic: (1,4312), (3,381), (4,4), (20,5921), (40,5881), (64,5922), (80,5923), (1490,5924), (1498,5925).

If X is on the Darboux cubic and P is the perspector of ABC and the pedal triangle of X, then the Darboux image of X is the reflection of X in P. (Randy Hutson, July 18, 2014)

X(5881) lies on these lines: {1,5}, {3,3679}, {4,519}, {8,20}, {10,631}, {30,4677}, {46,4325}, {57,4317}, {78,5176}, {100,5450}, {140,3655}, {145,946}, {165,548}, {376,4669}, {382,517}, {516,3625}, {518,5735}, {546,3656}, {550,3654}, {551,3090}, {573,4034}, {632,3653}, {912,4338}, {962,3621}, {996,5767}, {1012,3913}, {1071,5836}, {1125,5067}, {1210,3476}, {1385,1698}, {1420,1737}, {1478,3340}, {1482,1699}, {1490,3419}, {1532,3813}, {1697,4309}, {1750,4863}, {1766,4007}, {1788,4311}, {1907,5090}, {2077,5687}, {3057,3586}, {3091,3241}, {3244,3855}, {3245,4333}, {3247,5816}, {3524,4745}, {3525,3828}, {3528,3626}, {3529,5493}, {3560,3746}, {3617,5731}, {3624,5070}, {3635,3817}, {3853,5844}, {3872,5086}, {3899,5694}, {4293,4848}, {4299,5128}, {4330,5119}, {4654,5270}, {4882,5787}

X(5881) = reflection of X(i) in X(j) for these (i,j); (1,355), (40,8)
X(5881) = anticomplement of X(5882)
X(5881) = exsimilicenter of hexyl and 1st Steiner circles; the insimilicenter is X(9624)
X(5881) = {X(1),X(5)}-harmonic conjugate of X(9624)

X(5882) = COMPLEMENT OF X(5881)

Trilinears    3 r - 2 R cos B cos C : :
Barycentrics    4 a^4-3 a^3 b-3 a^2 b^2+3 a b^3-b^4-3 a^3 c+6 a^2 b c-3 a b^2 c-3 a^2 c^2-3 a b c^2+2 b^2 c^2+3 a c^3-c^4 : :
Barycentrics    3a/(2s) - S_BS_C/S² : :
X(5882) = 3X(1) - X(4)      (barycentrics and combo, Peter Moses, July 14, 2014)

Let A′B′C′ be the Euler triangle. Let L_A be the reflection of line B′C′ in line BC, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. Triangle A″B″C″ is homothetic to ABC at X(4), and X(5882) = X(1)-of-A″B″C″.

X(5882) lies on these lines: {1,4}, {3,519}, {5,551}, {8,3523}, {10,140}, {20,3241}, {30,4301}, {35,104}, {36,4848}, {40,145}, {55,5450}, {65,4311}, {84,4313}, {119,3825}, {165,3633}, {355,1125}, {382,3656}, {516,1482}, {517,550}, {549,4669}, {553,4317}, {572,2321}, {631,3679}, {912,3878}, {942,4315}, {962,3623}, {993,5837}, {997,5534}, {1006,5258}, {1012,3303}, {1071,1317}, {1158,1697}, {1210,1319}, {1388,1837}, {1389,5425}, {1698,3533}, {2077,3871}, {2099,4292}, {2360,4248}, {2801,3884}, {2829,4342}, {2894,4861}, {3086,5727}, {3146,5734}, {3149,3304}, {3333,4308}, {3340,4293}, {3524,4677}, {3526,3653}, {3577,5558}, {3579,5844}, {3601,5768}, {3616,5056}, {3622,5068}, {3624,5818}, {3625,5690}, {3632,5657}, {3634,5790}, {3636,3851}, {3671,5842}, {3817,3850}, {3877,5693}, {4745,5054}, {5316,5531}

X(5882) = X(40) of X(1)-Brocard triangle
X(5882) = {X(1),X(4)}-harmonic conjugate of X(13464)

X(5883) = X(51) OF FUHRMANN TRIANGLE

Trilinears a^2(b + c) + 2abc - (b + c)(b^2 - 3bc + c^2) : :

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5883) = centroid of T. Let T' be the triangle whose vertices are the centroids of the altimedial triangles; then X(5883) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5883) lies on these lines: {1,88}, {2,758}, {4,3255}, {5,2771}, {8,3881}, {10,141}, {21,3336}, {42,1739}, {46,5248}, {51,2392}, {57,993}, {58,409}, {65,392}, {72,3634}, {79,5046}, {191,5047}, {210,3828}, {226,3814}, {354,519}, {373,2842}, {405,3647}, {484,1621}, {513,4049}, {517,549}, {597,2836}, {956,4860}, {958,5708}, {960,4084}, {986,3743}, {997,5437}, {1006,5535}, {1159,5289}, {1445,3339}, {1698,3678}, {1737,3822}, {1835,5136}, {1844,5125}, {1963,4658}, {2650,3216}, {2690,2699}, {2801,5587}, {2975,3337}, {3035,5719}, {3057,3636}, {3090,5693}, {3218,5251}, {3219,4880}, {3244,5045}, {3290,3997}, {3555,3626}, {3616,3884}, {3622,5697}, {3624,3869}, {3628,5694}, {3632,3889}, {3635,3922}, {3649,4187}, {3660,4315}, {3679,3873}, {3681,3894}, {3720,4424}, {3740,4134}, {3848,4744}, {3876,3901}, {3880,5049}, {3887,4809}, {4002,4691}, {4067,5044}, {4511,5425}, {4666,5119}, {4675,5725}, {4731,4745}, {5083,5252}, {5131,5426}

X(5883) = complement of X(5692)
X(5883) = centroid of the six touchpoints of the Odehnal tritangent circles and the sidelines of ABC

X(5884) = X(52) OF FUHRMANN TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a^5(b + c) - a^4(b^2 + c^2) - a^3(b + c)(2b^2 - 3bc + 2c^2) + a^2(b - c)^2(2b^2 + 3bc + 2c^2) + a(b - c)^2(b^3 + c^3) - (b^2 - c^2)^2(b^2 - bc + c^2)] (Randy Hutson, January 29, 2015)

Let P be a point on the circumcircle. Let A′ be the orthogonal projection of P on the A-altitude, and define B′ and C′ cyclically. As P traces the circumcircle, the locus of the incenter of A′B′C′ is an ellipse with center X(5884). (Antreas Hatzipolkis, Hyancinthos #20792, February 6, 2012, and subsequent postings)

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5884) = orthocenter of T. Let T' be the triangle whose vertices are the orthocenters of the altimedial triangles; then X(5884) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5884) lies on these lines: {1,104}, {2,5693}, {3,758}, {4,79}, {5,2771}, {10,912}, {40,3868}, {52,2392}, {65,515}, {73,1735}, {117,1425}, {140,5694}, {165,3901}, {185,2779}, {191,1006}, {355,2801}, {411,5535}, {496,942}, {517,550}, {572,1761}, {580,1046}, {581,986}, {631,5692}, {944,3474}, {1064,3670}, {1210,1858}, {1385,3878}, {1482,3881}, {1490,3339}, {1656,3833}, {1765,2294}, {2096,3486}, {2695,2719}, {3149,5221}, {3359,3811}, {3576,3869}, {3812,5777}, {3918,5790}, {4295,5768}

X(5884) = complement of X(5693)

X(5885) = X(143) OF FUHRMANN TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a^5(b + c) - a^4(b^2 + c^2) - a^3(b + c)(2b^2 - 3bc + 2c^2) + 2a^2(b^4 - b^3c - b^2c^2 - bc^3 + c^4) + a(b - c)^2(b^3 + c^3) - (b - c)^4(b + c)^2] (Randy Hutson, January 29, 2015)

X(5885) is the nine-point center of the Fuhrmann triangle of the orthic triangle of the Fuhrmann triangle. (Randy Hutson, July 7, 2014)

Let T be the orthic triangle of the Fuhrmann triangle (whose vertices are the incenters of the three altimedial triangles); then X(5885) = X(5)-of-T. Let T' be the triangle whose vertices are the nine-point centers of the altimedial triangles; then X(5885) = incenter of T'. (Randy Hutson, July 18, 2014)

X(5885) lies on these lines: {1,3}, {2,5694}, {5,2771}, {140,758}, {143,2392}, {575,2836}, {912,3812}, {952,3754}, {1656,5693}, {1772,2594}, {3526,5692}, {3628,3833}, {3874,5690}, {3881,5844}

X(5885) = complement of X(5694)

X(5886) = X(381) OF FUHRMANN TRIANGLE

Trilinears r + R cos(B - C) : :
Barycentrics a^4 - a^3(b + c) - 2a^2(b^2 - bc + c^2) + a(b - c)^2(b + c) + (b^2 - c^2)^2 : :

Let A′ be the nine-point center of the triangle IBC, where I = X(1), and define B′ and C′ cyclically. The triangle A′B′C′ is homothetic to the Fuhrmann triangle at X(1), and X(5886) is the centroid of A′B′C′.

X(5886) lies on these lines: {{1,5}, {2,392}, {3,142}, {4,1385}, {8,3090}, {10,1482}, {30,1699}, {36,1836}, {40,140}, {46,5433}, {56,3560}, {65,499}, {79,5427}, {84,3255}, {145,5056}, {165,549}, {226,999}, {230,1572}, {238,5398}, {354,912}, {381,515}, {382,4297}, {405,5812}, {475,1872}, {498,3057}, {519,5055}, {546,5691}, {547,3679}, {631,962}, {908,956}, {942,3086}, {944,3091}, {960,5791}, {995,3772}, {997,2886}, {1000,4345}, {1006,5284}, {1012,1519}, {1056,5226}, {1058,5703}, {1064,3720}, {1100,5816}, {1104,5713}, {1108,5747}, {1319,1478}, {1352,1386}, {1389,5330}, {1479,2646}, {1537,3306}, {1594,5090}, {1698,3628}, {1737,2099}, {1770,5204}, {1829,3542}, {1902,3541}, {2095,5745}, {3241,5071}, {3244,5079}, {3333,5843}, {3338,3649}, {3359,5437}, {3417,3615}, {3419,4511}, {3421,5748}, {3428,4423}, {3434,5440}, {3474,5122}, {3475,5049}, {3487,5045}, {3488,5274}, {3526,4301}, {3600,5714}, {3634,5070}, {3636,3851}, {3646,5763}, {3671,5708}, {3811,3813}, {3868,5694}, {3897,5046}, {3940,4847}, {4221,5333}, {4293,5126}, {4305,5225}, {4323,5704}, {4679,5251}, {5010,5444}, {5044,5761}, {5067,5734}, {5119,5432}, {5436,5715}, {5542,5779}

X(5886) = complement of X(5657)
X(5886) = {X(1),X(5)}-harmonic conjugate of X(355)
X(5886) = {X(1),X(80)}-harmonic conjugate of X(37740)
X(5886) = perspector of [cross-triangle of Fuhrmann and Ai (aka K798i) triangles] and [cross-triangle of 2nd Fuhrmann and Ae (aka K798e) triangles]
X(5886) = endo-homothetic center of Ehrmann side-triangle and 3rd anti-Euler triangle; the homothetic center is X(11459)

X(5887) = X(119) OF INNER GARCIA TRIANGLE

Barycentrics a[a(b + c)[a² - (b - c)²]² - a⁴(b + c)² + 2a²[(b² - c²)² + bc(b² + c²)] - (b² + c²)(b² - c²)²] : : (Richard Hilton, March 2, 2015)

X(5887) lies on these lines: {1,90}, {3,960}, {4,8}, {5,65}, {10,119}, {19,5778}, {21,104}, {40,5692}, {56,920}, {210,5690}, {221,1060}, {411,3579}, {515,3878}, {518,1351}, {758,946}, {936,3359}, {942,3086}, {944,3877}, {952,1898}, {971,5698}, {1012,5730}, {1062,1854}, {1064,2292}, {1352,3827}, {1656,3812}, {1697,5534}, {2476,3753}, {2745,2766}, {2801,3884}, {3817,4084}, {3868,5603}, {3876,5657}, {3899,5691}, {3931,5396}, {4047,5755}, {4067,4301}, {5790,5836}

X(5887) = X(4)-of-X(1)-Brocard triangle)
X(5887) = anticomplement of X(34339)

X(5888) = INTERSECTION OF LINES X(2)X(3098) AND X(110)X(5092)

Barycentrics a²[a⁴ + a²(b² + c²) - 2b⁴ - 11b²c² - 2c⁴] : : (Richard Hilton, March 2, 2015)

X(5888) is the similitude center of ABC and the X(2)-Brocard triangle.

X(5888) lies on the Thomson-Gibert-Moses hyperbola and these lines: {2,3098}, {74,549}, {110,5092}, {140,3581}, {141,5648}, {323,3819}, {354,3920}, {392,404}, {511,5643}, {631,5654}, {1201,3746}, {1995,5646}, {2979,5644}, {3167,5012}, {3357,3523}, {3524,4550}, {5113,5653}, {5544,5640}

X(5888) = Thomson-isogonal conjugate of X(549)

X(5889) = ORTHOCENTER OF CIRCUMORTHIC TRIANGLE

Trilinears     f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[S²_A - S² - 2S_A(4R² - S_ω)]
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cot A)(csc 2B + csc 2C) - (cot B)(csc 2C + csc 2A) - (cot(C)(csc 2A + csc 2B)
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (sin 2A)(cos 2B + cos 2C) - (sin 2B)(cos 2C + cos 2A) - (sin 2C)(cos 2A + cos 2B)
Barycentrics    a^2 (a^6 (b^2 + c^2) - a^4 (3 b^4 + b^2 c^2 + 3 c^4) + a^2 (3 b^6 - b^4 c^2 - b^2 c^4 + 3 c^6) - b^8 + b^6 c^2 + b^2 c^6 - c^8) : :

X(5889) is the circumorthic-triangle-orthologic center of these triangles: extangents, intangents, orthic, and tangential. (César Lozada, ADGEOM #978, Dec. 20, 2013)

Let O =- X(3) and let A′ be the isogonal conjugate of A with respect to OBC, and define B′ and C′ cyclically. Let A″ be the isogonal conjugate of A′ with respect to OB′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5889). (Randy Hutson, July 7, 2014)

X(5889) lies on these lines: {2,389}, {3,54}, {4,52}, {5,568}, {20,185}, {22,1181}, {24,110}, {26,1614}, {49,1658}, {51,3091}, {64,895}, {143,381}, {156,2070}, {186,1147}, {323,1092}, {382,5663}, {411,5752}, {569,1199}, {578,1994}, {631,1216}, {962,2807}, {1204,2071}, {1351,1593}, {3090,5462}, {3167,3515}, {3523,3917}, {3524,5447}, {3564,3575}

X(5889) = reflection of X(i) in X(j) for these (i,j): (3,6102), (4,52)
X(5889) = anticomplement of X(5562)
X(5889) = homothetic center of X(4)-altimedial and X(2)-adjunct anti-altimedial triangles

X(5890) = CENTROID OF CIRCUMORTHIC TRIANGLE

Trilinears   sin A (sin 2B + sin 2C) - sin B sin C : :
Trilinears   [a^3 cos(B - C) - b^3 cos(C - A) - c^3 cos(A - B)]sec A + 2 a^2 cos(B - C) (b sec B + c sec C) : :
Trilinears   a[a^6 (b^2 + c^2) - a^4(3b^4 - b^2c^2 + 3c^4) + 3a^2(b^6 - b^4c^2 - b^2c^4 + c^6) - b^8 + b^6c^2 + b^2c^6 - c^8] : :

X(5890) lies on these lines: {2,5654}, {3,54}, {4,51}, {6,74}, {20,52}, {24,154}, {30,568}, {64,1173}, {143,382}, {184,186}, {373,5071}, {376,511}, {381,5640}, {477,2452}, {578,1199}, {631,3819}, {1141,1303}, {1216,3523}, {1994,2071}, {2807,5603}, {3091,5462}, {3146,5446}, {3524,3917}, {3651,5752}}

X(5890) = reflection of X(i) in X(j) for these (i,j): (4,51), (2979,3)
X(5890) = anticomplement of X(5891)
X(5890) = X(2)-of-circumorthic-triangle
X(5890) = X(4)-of-orthocentroidal-triangle
X(5890) = X(20)-of-Lucas-triangle (defined at X(95))
X(5890) = homothetic center of X(4)-altimedial and X(4)-adjunct anti-altimedial triangles

X(5891) = REFLECTION OF X(51) IN X(5)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos(B - C) [a^3 cos(B - C) + b(2b^2 - a^2) cos(C - A) + c(2c^2 - a^2) cos(A - B)]
Trilinears g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a[a^2(b^2 + c^2) - (b^2 - c^2)^2][a^4 + b^4 + c^4 - 2a^2 (b^2 + c^2) + 4b^2c^2]

X(5891) = (X(376) of orthic triangle) = (X(4) of X(2)-Brocard triangle); also, (X(5891) of hexyl triangle) = X(2) and (X(5891) of excentral triangle) = X(376). (Randy Hutson, July 7, 2014)

X(5891) lies on these lines: {2,5654}, {3,64}, {4,1216}, {5,51}, {20,5447}, {30,3917}, {113,127}, {128,130}, {140,185}, {155,569}, {216,1625}, {373,547}, {378,4550}, {381,511}, {389,1656}, {399,5092}, {549,5642}, {568,5055}, {1352,2393}, {3060,3545}, {3090,5462}, {3091,5446}, {3313,3818}, {3567,5056}, {5071,5640}

X(5891) = reflection of X(51) in X(5)
X(5891) = complement of X(5890)
X(5891) = anticomplement of X(5892)

X(5891) = centroid-of-2nd-Euler-triangle

X(5892) = MIDPOINT OF X(3) AND X(51)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2[a^6 (b^2 + c^2) - a^4 (3b^4 - 4b^2c^2 + 3c^4) + 3a^2 (b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^4]

X(5892) = (X(376) of polar triangle of complement of polar circle) = {X(52),X(631)}-harmonic conjugate of X(5447) (Randy Hutson, July 7, 2014)

X(5892) lies on these lines: {2,5654}, {3,51}, {5,2883}, {52,631}, {140,389}, {143,3530}, {182,2393}, {185,1656}, {373,381}, {376,5640}, {511,549}, {512,1116}, {547,5663}, {568,3917}, {2779,3833}, {2781,3589}, {3060,3524}, {3523,3567}, {3526,5562}

X(5892) = midpoint of X(i) and X(j) for these {I,J}: {3,51}, {389,3819}
X(5892) = reflection of X(i) in X(j) for these (i,j): (1216,3819), (3819,140)
X(5892) = complement of X(5891)

X(5893) = CENTER OF HALF-ALTITUDE CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -2a¹⁰ + 3b¹⁰ + 3c¹⁰ - a⁸b² - a⁸c² + 12a⁶b⁴ + 12a⁶c⁴ + 16a⁶b²c² - 10a⁴b⁶ - 10a⁴c⁶ + 10a⁴b⁴c² + 10a⁴b²c⁴ - 2a²b⁸ - 2a²c⁸ + 16a²b⁶c² + 16a²b²c⁶ + 28a²b⁴c⁴ - 9b⁸c² - 9b²c⁸ + 6b⁶c⁴ + 6b⁴c⁶

The half-altitude circle is the circumcircle of the half-altitude triangle. Its radius is
[R/(16S_AS_BS_C)][2(b²c²J² + S² - S²_A)(c²a²J² + S² - S²_B)(a²b²J² + S² - S²_C)]^1/2, where J = |OH|/R (as at X(1113).

X(5893) lies on these lines: {2,5894}, {4,6}, {5,3357}, {30,5448}, {64,3091}, {140,2777}, {154,3146}, {221,5225}, {546,5462}, {1853,3832}, {2192,5229}

X(5893) = midpoint of X(4) and X(2883)
X(5893) = complement of X(5894)
X(5893) = X(11260)-of-orthic-triangle if ABC is acute

X(5894) = ANTICOMPLEMENT OF X(5893)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4a^10 - b^10 - c^10 - 5a^8(b^2 + c^2) - 8a^6(b^4 + c^4 - 3b^2c^2) + 14a^4(b^6 + c^6 - b^4c^2 - b^2c^4) - 4a^2(b^8 +c^8 - 6b^4c^4 + 2b^6c^2 + 2b^2c^6) + 3b^8c^2 + 3b^2c^8 - 2b^6c^4 - 2b^4c^6
Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin B)(sec^2 B - sec A sec B sec C)[2 sec C sec A - sec B (sec^2 C + sec^2 A)] + (sin C)(sec^2 C - sec A sec B sec C)[2 sec A sec B - sec C (sec^2 A + sec^2 B)]

X(5894) is the center of the pedal circle of X(20) and of X(64), and the center of the cevian circle of X(69) and of X(253); X(5864) is also (X(64) of X(4)-Brocard triangle. (Randy Hutson, July 7, 2014)

X(5894) lies on these lines: {2,5893}, {3,1661}, {4,1192}, {5,1539}, {20,64}, {30,3357}, {154,3522}, {185,1205}, {376,1498}, {550,1216}, {1204,1885}, {1593,5480}, {1853,3146}, {1854,3474}, {2935,3520}, {3528,5656}, {4219,5799}

X(5894) = midpoint of X(20) and X(64)
X(5894) = complement of X(5895)
X(5894) = anticomplement of X(5893)

X(5895) = ANTICOMPLEMENT OF X(5894)

Trilinears    : f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec^2 A - sec A sec B sec C)[2 sec B sec C - sec A (sec^2 B + sec^2 C)]
Trilinears     g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (cos B)/(cos C - cos A cos B) + (cos C)/(cos B - cos A cos C)
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)(a^6 + 2b^6 + 2c^6 - 3a^2b^4 - 3a^2c^4 + 6a^2b^2c^2 - 2b^4c^2 - 2b^2c^4)

Let H be the hyperbola {A,B,C,X(4),X(20)}. Let L(X) denote the line tangent to H at a point X on H. Then X(5895) is the point of intersection of L(X(4)) and L(X(20)). (Randy Hutson, July 7, 2014)

Let A′ be the trilinear pole of the perpendicular bisector of BC, and define B′ and C′ cyclically. A′B′C′ is also the anticomplement of the anticomplement of the midheight triangle. X(5895) = X(3)-of-A′B′C′. (Randy Hutson, January 29, 2018)

X(5895) lies on these lines: {2,5893}, {3,113}, {4,64}, {6,1885}, {20,154}, {25,2929}, {30,155}, {52,382}, {193,1503}, {235,1192}, {381,3357}, {468,1620}, {1181,2904}, {1514,3542}, {1562,3172}, {1593,3574}, {1836,1854}, {2778,5693}, {2906,5706}, {2907,5786}, {3529,5656}

X(5895) = isotomic conjugate of X(34410)
X(5895) = anticomplement of X(5894)
X(5895) = crosssum of X(3) and X(64)
X(5895) = crosspoint of X(4) and X(20)

X(5896) = Λ(X(20), X(154))

Trilinears (4+3*cos(2*A)+2*cos(2*B)+3*cos(2*C))*(4+3*cos(2*A)+3*cos(2*B)+2*cos(2*C))*(3*cos(B)-cos(A-B))*(3*cos(C)-cos(A-C)) : :

Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5896) = Λ(X(3), X(5893) = Λ(X(20), X(154)). (Randy Hutson, July 7, 2014)

X(5896) lies on the circumcircle and these lines: {64,110}, {99,253}, {107,3146}, {1301,3515}}

X(5897) = Λ(X(4), X(64))

Barycentrics a^2*(a^10+(2*b^2-3*c^2)*a^8-2*(b^2-c^2)*(5*b^2+c^2)*a^6+2*(b^2-c^2)*(4*b^4+9*b^2*c^2-c^4)*a^4+(b^2-c^2)*(b^6+3*c^6-5*(3*b^2+c^2)*b^2*c^2)*a^2-(2*b^4+5*b^2*c^2+c^4)*(b^2-c^2)^3)*(a^10-(3*b^2-2*c^2)*a^8+2*(b^2-c^2)*(b^2+5*c^2)*a^6+2*(b^2-c^2)*(b^4-9*b^2*c^2-4*c^4)*a^4-(b^2-c^2)*(3*b^6+c^6-5*(b^2+3*c^2)*b^2*c^2)*a^2+(b^4+5*b^2*c^2+2*c^4)*(b^2-c^2)^3) : :

Λ(P, X) is defined (TCCT, p. 80) as the isogonal conjugate of the point in which the line PX meets the infinity line. X(5897) = Λ(X(3), X(1661) = Λ(X(4), X(64) = Λ(X(5), X(5893)) = Λ(X(20), X(394) = Λ(X(146), X(2071). X(5897) is the point of intersection, other than A,B,C, of the circumcircle and the hyperbola {A,B,C,X(3),X(20)}; also, X(5897) is the antipode of X(1301) on the circumcircle. (Randy Hutson, July 7, 2014)

X(5897) lies on the circumcircle and these lines: {3,1301}, {20,107}, {110,1498}, {112,1033}, {376,1289}, {393,3344}, {1302,1370}, {1304,2071}

X(5897) = reflection of X(1301) in X(3)

X(5898) = REFLECTION OF X(195) IN X(110)

Barycentrics a²[a²[a⁸(a⁴ + b⁴ + c⁴) - a⁴(5b⁸ + 20b⁶c² + 17b⁴c⁴ + 20b²c⁶ + 5c⁸) + 3(b² - c²)²(b⁸ - 2b⁶c² - 2b⁴c⁴ - 2b²c⁶ + c⁸)] - (b² + c²)[3a¹² - 5a⁸(b² + c²)² + a⁴(b⁸ - 16b⁶c² + 15b⁴c⁴ - 16b²c⁶ + c⁸) + (b² - c²)6]] : : (Richard Hilton, March 2, 2015)

Let H be the Stammler hyperbola, and let T be the tangential triangle. X(5898) is the antipode of X(195) in H. The conic H is a rectangular hyperbola passing through X(i) for these I: 1,3,6,155,159,195,399,1498, 2916,2917,2918,2929,2930,2931,2935,2948,3511, the excenters and the vertices of T; the center of H is X(110). H is the isogonal conjugate of the Euler line with respect to T, and H is also the isogonal conjugate of the line X(30)X(40) with respect to the excentral triangle. Also, H is the locus of P for which the P-Brocard triangle is perspective to ABC; see X(5642). X(5898) is the isogonal conjugate of X(5899) with respect to T. (Randy Hutson, July 7, 2014)

X(5898) lies on these lines: {3,2888}, {6,3200}, {25,2914}, {110,143}, {399,1154}, {539,2931}, {542,2916}, {2918,3519}

X(5899) = INVERSE-IN-CIRCUMCIRCLE OF X(140)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B sin 2C sin(A - B) (1 + 2 sin^2 A + 2 sin^2 C) + sin C sin 2B sin(A - C) (1 + 2 sin^2 A + 2 sin^2 B)

X(5899) = X(3) - 4X(23)
X(5899) = 12X(2) + (J² - 12)X(3) where J = |OH|/R
X(5899) = 4R²X(2) + (3|OG|² - 4R²)X(3) (Peter Moses, July 11, 2014)

As a point on the Euler line, X(5899) has Shinagawa coefficients (3E + 8F, -11E - 8F).

X(5899) is the isogonal conjugate of X(5898) with respect to the tangential triangle, and X(5899) is the pole with respect to the circumcircle of the line X(140)X(523). (Randy Hutson, July 7, 2014)

X(5899) lies on these lines: {2,3}, {195,1614}, {399,1154}, {1533,2931}, {2918,3574}

X(5899) = isogonal conjugate of X(5900)
X(5899) = crossdifference of every pair of points on the line X(647)X(5421)
X(5899) = {X(3),X(23)}-harmonic conjugate of X(37923)
X(5899) = {X(15154),X(15155)}-harmonic conjugate of X(4)

X(5900) = ISOGONAL CONJUGATE OF X(5899)

Trilinears 1/[sin B sin 2C sin(A - B) (1 + 2 sin^2 A + 2 sin^2 C) + sin C sin 2B sin(A - C) (1 + 2 sin^2 A + 2 sin^2 B)] : :

X(5900) is the trilinear pole of the line X(647)X(5421), and X(5900) is the the antipode-in-Jerabek-hyperbola of X(1173). Also, X(5900) is the antigonal image of X(1173). (Randy Hutson, July 7, 2014)

X(5900) lies on these lines: {125,1173}, {146,3521}, {2889,3448}

X(5900) = isogonal conjugate of X(5899)
X(5900) = reflection of X(1173) in X(125)

X(5901) = COMPLEMENT OF X(5690)

Barycentrics 2a⁴ - a²(3b² - 4bc + 3c²) + (b² - c²)² - 2a(b + c)[a² - (b - c)²] : : (Richard Hilton, March 2, 2015)

As at X(5886), let A′ be the nine-point center of the triangle IBC, where I = X(1), and define B′ and C′ cyclically; then X(5901) = (X(5) of A′B′C′), as well as the complement of X(5) with respect to A′B′C′. Let A* be the circle with center A and diameter b + c, and define B* and C* cyclically; then X(5901) is the radical center of A*, B*, C*. Let A″ be the nine-point center of triangle IBC, and define B″ and C″ cyclically; then I, A″, B″, C″ comprise an orthocentric system whose common nine-point circle has center X(5901). (Hyacinthos #21518, February 10, 2013, and following posts by Antreas Hatzipolakis and Randy Hutson)

X(5901) lies on these lines: {1,5}, {2,1482}, {3,962}, {4,3622}, {8,1656}, {10,3628}, {30,551}, {40,549}, {104,5606}, {140,517}, {145,3090}, {381,944}, {382,5731}, {392,5771}, {476,953}, {498,2098}, {499,2099}, {515,546}, {516,548}, {519,547}, {550,3576}, {632,3624}, {912,5045}, {999,3485}, {1001,5762}, {1064,5453}, {1159,4323}, {1191,5707}, {1386,3564}, {1388,1478}, {1699,3627}, {3241,5055}, {3336,5298}, {3487,5811}, {3526,5550}, {3530,3579}, {3617,5067}, {3623,5056}, {3649,5563}, {3655,3845}, {3817,3850}, {3874,5694}, {3878,4999}, {4292,5126}, {4308,5714}, {4511,5178}, {5049,5777}, {5180,5303}, {5432,5697}, {5436,5812}, {5542,5843}

X(5901) = midpoint of X(1) and X(5)
X(5901) = {X(1),X(11)}-harmonic conjugate of X(37730)
X(5901) = {X(1),X(80)}-harmonic conjugate of X(37734)

X(5902) = INCENTER OF ORTHOCENTROIDAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = abc + (b + c)(a - b + c)(a + b - c)

X(5902) lies on these lines: {1,3}, {2,758}, {4,79}, {5,3649}, {6,1718}, {7,80}, {8,2891}, {10,3681}, {20,5441}, {43,1739}, {47,1451}, {58,3924}, {61,2306}, {63,4880}, {72,1698}, {81,1325}, {90,5665}, {145,3881}, {191,405}, {226,1737}, {244,995}, {355,5270}, {374,1743}, {381,2771}, {386,2650}, {392,3742}, {498,1788}, {499,3485}, {515,553}, {518,599}, {519,3873}, {551,3877}, {579,2294}, {584,2160}, {614,5315}, {631,5442}, {912,4654}, {938,1479}, {944,4317}, {950,1770}, {952,5434}, {960,3624}, {985,2224}, {993,3218}, {994,4850}, {997,3306}, {1002,2809}, {1012,1768}, {1046,1724}, {1051,2939}, {1068,1825}, {1071,5586}, {1125,3869}, {1210,3671}, {1254,4306}, {1464,5396}, {1656,5694}, {1717,2955}, {1725,1779}, {1790,4658}, {1836,3583}, {1837,3585}, {1876,1905}, {2280,5011}, {2362,3301}, {2392,3060}, {2800,5603}, {2802,3241}, {2842,5640}, {3244,3889}, {3296,5559}, {3419,5696}, {3474,3488}, {3475,5657}, {3476,5083}, {3486,4299}, {3501,3970}, {3555,3632}, {3586,4312}, {3616,3878}, {3617,3918}, {3622,3884}, {3634,3876}, {3635,3885}, {3636,3890}, {3752,5313}, {3792,4675}, {3828,4134}, {3922,4668}, {3940,4413}, {3962,5044}, {3968,4661}, {3980,5208}, {4002,4662}, {4116,4128}, {4414,4653}, {4645,4680}, {5432,5719}, {5435,5444}

X(5902) = midpoint of X(65) and X(354)
X(5902) = reflection of X(1) in X(354)
X(5902) = isogonal conjugate of X(15175)
X(5902) = X(1)-of-orthocentroidal-triangle
X(5902) = X(381)-of-intouch-triangle
X(5902) = {X(1),X(65)}-harmonic conjugate of X(5903)
X(5902) = anticomplement of X(10176)
X(5902) = X(2)-of-reflection-triangle-of-X(1)
X(5902) = Cundy-Parry Phi transform of X(35)
X(5902) = Cundy-Parry Psi transform of X(79)
X(5902) = homothetic center of X(1)-altimedial and X(1)-adjunct anti-altimedial triangles
X(5902) = X(7576)-of-excentral-triangle
X(5902) = {X(1),X(3)}-harmonic conjugate of X(37571)
X(5902) = endo-homothetic center of Ehrmann vertex-triangle and tangential triangle; the homothetic center is X(381)

X(5903) = REFLECTION OF X(1) IN X(65)

Trilinears - abc + (b + c)(a - b + c)(a + b - c) : : (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)
Trilinears 2 cos B + 2 cos C - 1 : :

X(5903) = {X(1),X(40)}-harmonic conjugate of X(35); X(5903) = {X(1),X(65)}- harmonic conjugate of X(5902). Let A′ be the isogonal conjugate of A with respect to triangle IBC, where I = X(1), and define B′ and C′ cyclically. Let A″ be the isogonal conjugate of A′ with respect to IB′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5903). X(5903) is the incenter of the triangle denoted by A″B″C″ at X(5905) and the triangle of the same notation at X(5906). (Randy Hutson, July 14, 2014)

Let P_A be the reflection of X(1) in line BC, and define P_B and P_C cyclically; then X(5903) is the isogonal conjugate of X(1) with respect to P_AP_BP_C. (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)

Let Ja be the A-excenter of the A-adjunct anti-altimedial triangle, and define Jb and Jc cyclically. The lines AJa, BJb, CJc concur in X(5903). (Randy Hutson, November 2, 2017)

Let A₄B₄C₄ and A₆B₆C₆ be the Gemini triangles 4 and 6, resp. Let L_A and M_A be the lines through A₄ and A₆, resp., parallel to BC. Define L_B, L_C, M_B, M_C cyclically. Let A′₄ = L_B∩L_C and define B′₄ and C′₄ cyclically. Let A′₆ = M_B∩M_C and define B′₆ and C′₆ cyclically. Triangles A′₄B′₄C′₄ and A′₆B′₆C′₆ are homothetic at X(5903). (Randy Hutson, November 30, 2018)

X(5903) lies on these lines: {1,3}, {2,3754}, {4,80}, {7,5559}, {8,79}, {10,908}, {12,5690}, {37,5036}, {41,5011}, {43,3987}, {63,5258}, {72,3679}, {78,4867}, {90,3577}, {145,2802}, {150,4056}, {169,5526}, {181,3944}, {191,958}, {203,2306}, {213,3959}, {214,4188}, {218,5540}, {219,1781}, {267,2948}, {355,1836}, {386,4642}, {392,3624}, {474,5289}, {495,3649}, {498,3485}, {499,1788}, {515,1770}, {518,3632}, {519,3868}, {551,3890}, {573,2171}, {579,1953}, {595,3924}, {631,5444}, {764,4083}, {912,4338}, {944,3474}, {946,1737}, {960,1698}, {962,1479}, {978,1739}, {984,1756}, {1000,5557}, {1012,1727}, {1046,1710}, {1068,1835}, {1100,4287}, {1111,3212}, {1122,4902}, {1125,3877}, {1148,1784}, {1210,4301}, {1376,5730}, {1393,1772}, {1411,2964}, {1464,5399}, {1572,5299}, {1717,1854}, {1724,3460}, {1743,2262}, {1759,2329}, {1797,4792}, {1829,4214}, {1837,3583}, {1838,1869}, {1858,5727}, {1871,1888}, {1872,1875}, {1902,1905}, {2170,4253}, {2176,3125}, {2295,3735}, {2362,3299}, {3179,5239}, {3208,3970}, {3216,4674}, {3218,4861}, {3241,3881}, {3244,3873}, {3476,4317}, {3486,4302}, {3488,4309}, {3555,3633}, {3582,3656}, {3584,3654}, {3616,3884}, {3617,3678}, {3622,3898}, {3623,3892}, {3626,3681}, {3635,3889}, {3698,5044}, {3724,5496}, {3740,4002}, {3751,3827}, {3833,5550}, {3872,4880}, {3897,5267}, {3913,5541}, {3962,4668}, {4127,4678}, {4134,4691}, {4153,4165}, {4304,5493}, {4513,5525}, {4646,5312}, {4857,5722}, {5046,5180}, {5250,5259}, {5252,5270}, {5253,5330}, {5435,5734}, {5694,5790}

X(5903) = reflection of X(i) in X(j) for these (i,j): (1,65), (5904,8)
X(5903) = isogonal conjugate of X(15446)
X(5903) = anticomplement of X(3878)
X(5903) = X(4)-of-reflection-triangle-of-X(1)
X(5903) = perspector of reflection triangle of X(1) and 2nd isogonal triangle of X(1)
X(5903) = Cundy-Parry Phi transform of X(36)
X(5903) = Cundy-Parry Psi transform of X(80)
X(5903) = X(6240)-of-excentral-triangle
X(5903) = {X(1),X(3)}-harmonic conjugate of X(37525)

X(5904) = REFLECTION OF X(1) IN X(72)

Barycentrics a²bc - a(b + c)(b² + c² - a²) : : (Richard Hilton, March 2, 2015)

X(5904) is the incenter of the triangle A*B*C* described at X(5905) and also the incenter of the triangle A*B*C* described at X(5906). Also, X(5904) = {(X(1),X(9)}-harmonic conjugate of X(5259), and X(5904) = {(X(1),X(72)}-harmonic conjugate of X(5692). (Randy Hutson, July 7, 2014)

X(5904) lies on these lines: {1,6}, {2,3678}, {8,79}, {10,3681}, {20,2801}, {35,63}, {36,78}, {38,386}, {40,912}, {43,3670}, {46,200}, {55,191}, {56,3940}, {58,976}, {65,3679}, {69,1930}, {80,3436}, {144,4294}, {145,3878}, {165,1071}, {210,942}, {281,1844}, {329,1479}, {354,3624}, {382,517}, {474,3337}, {484,5687}, {519,3869}, {527,1770}, {551,3889}, {579,3949}, {595,3938}, {651,4347}, {936,3338}, {978,3953}, {982,3216}, {986,3293}, {997,3984}, {1046,3961}, {1066,2318}, {1125,3873}, {1158,5537}, {1282,2939}, {1376,3336}, {1482,5694}, {1697,1858}, {1699,5777}, {1756,4073}, {1759,3684}, {2093,4882}, {2340,4303}, {2771,5541}, {2774,4088}, {2802,3621}, {3057,3633}, {3059,4312}, {3149,5536}, {3189,4302}, {3218,4420}, {3219,5248}, {3241,3884}, {3244,3877}, {3419,3585}, {3501,4006}, {3579,3689}, {3616,3881}, {3617,3754}, {3622,3892}, {3623,3898}, {3626,4084}, {3635,3890}, {3666,5312}, {3697,3812}, {3711,5221}, {3730,3930}, {3735,3780}, {3740,4533}, {3742,4539}, {3746,3870}, {3753,4662}, {3831,4090}, {3833,4547}, {3875,4523}, {3916,5010}, {3919,4691}, {4018,4668}, {4188,4973}, {4251,5282}, {4292,5850}, {4309,5698}, {4388,4894}, {4413,5708}, {4423,5506}, {4641,5266}, {4658,5311}, {4678,4757}, {5270,5794}, {5445,5552}

X(5904) = reflection of X(i) in X(j) for these (i,j): (1,72), (5903,8)
X(5904) = anticomplement of X(3874)

X(5905) = ANTICOMPLEMENT OF X(63)

Barycentrics cos B + cos C - cos A : cos C + cos A - cos B : cos A + cos B - cos C
Barycentrics Ra - R : Rb - R : Rc - R, where Ra, Rb, Rc are the exradii

Let A′B′C′ be the orthic triangle, and let L_A be the reflection of line B′C′ in the internal bisector of angle A, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC at X(57), homothetic to the medial triangle at X(908), and to the anticomplementary triangle at X(5905). Let M_A be the reflection of the line B′C′ in the external bisector of angle A, and define M_B and M_C cyclically. Let A* = M_B∩M_C, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC at X(9), homothetic to the medial triangle at X(5249), and to the anticomplementary triangle at X(5905). (Randy Hutson, July 7, 2014)

X(5905) lies on these lines: {2,7}, {4,912}, {6,3782}, {8,79}, {10,3951}, {20,5758}, {21,3487}, {46,5552}, {65,3436}, {69,321}, {72,377}, {75,4886}, {78,4190}, {81,4644}, {92,1947}, {100,3474}, {145,515}, {149,152}, {192,3151}, {193,1839}, {200,4312}, {239,5813}, {278,651}, {281,445}, {306,3729}, {312,320}, {345,3936}, {355,4018}, {388,3869}, {390,3957}, {442,3927}, {443,3876}, {481,3084}, {482,3083}, {497,3873}, {516,3870}, {518,1836}, {529,2099}, {535,3241}, {537,4865}, {554,5240}, {938,5046}, {940,4415}, {942,2478}, {944,5841}, {958,3649}, {993,3616}, {1004,1260}, {1046,5230}, {1056,3877}, {1058,3889}, {1068,3157}, {1071,5812}, {1081,5239}, {1086,4383}, {1210,5187}, {1211,4363}, {1215,4655}, {1329,5221}, {1331,1754}, {1351,2969}, {1479,3874}, {1532,2095}, {1621,3475}, {1707,3011}, {1750,1998}, {1770,3811}, {1797,4080}, {1851,3060}, {2476,5714}, {2550,3681}, {2886,5852}, {2975,3485}, {2999,4862}, {3091,5811}, {3175,4851}, {3210,4440}, {3583,3894}, {3585,3901}, {3617,5815}, {3663,5256}, {3664,4656}, {3715,3826}, {3742,4679}, {3751,3914}, {3772,4641}, {3816,4860}, {3920,4307}, {3925,5220}, {3962,5794}, {3970,3995}, {4001,4054}, {4187,5708}, {4189,5703}, {4293,4511}, {4387,4966}, {4416,5271}, {4419,5712}, {4438,4892}, {4463,5800}, {4666,5542}, {4847,5850}, {5086,5229}, {5154,5704}, {5289,5434}

X(5905) = isogonal conjugate of X(2164)
X(5905) = isotomic conjugate of X(2994)
X(5905) = anticomplement of X(63)
X(5905) = pole wrt polar circle of trilinear polar of X(7040)
X(5905) = X(48)-isoconjugate (polar conjugate) of X(7040)

X(5906) = ANTICOMPLEMENT OF X(255)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B cos²B + sin C cos²C - sin A cos²A

Let A′B′C′ be the circumorthic triangle, and let L_A be the reflection of line B′C′ in the internal bisector of angle A, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The triangle A″B″C″ is homothetic to ABC at X(3075) and homothetic to the anticomplementary triangle at X(5906). Let M_A be the reflection of the line B′C′ in the external bisector of angle A, and define M_B and M_C cyclically. Let A* = M_B∩M_C, and define B* and C* cyclically. The triangle A*B*C*' is homothetic to ABC at X(3074) and to homothetic to the anticomplementary triangle at X(5906). (Randy Hutson, July 7, 2014)

X(5906) lies on these lines: {2,255}, {8,79}, {69,349}, {78,1448}, {651,5125}, {860,3157}, {962,2817}, {1259,3936}, {1788,2406}, {3868,5081}

X(5907) = COMPLEMENT OF X(185)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan B)(cos²C + cos²A) + (tan C)(cos²A + cos²B)
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 2B)[1 - cos B cos(C - A)] + (sin 2C)[1 - cos C cos(A - B)]
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b) , where h(a,b,c) = (b^2 + c^2 - a^2)[2a^8 - 3a^6(b^2 + c^2) - a^4(b^4 - 10b^2c^2 + c^4) + 3a^2(b^2 - c^2)^2(b^2 + c^2) - (b^2 - c^2)^4]

Let A′B′C′ be the half-altitude triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of the line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5). Let A* be the trilinear pole, with respect to A′B′C′, of the line B″C″, and define B* and C* cyclically. The lines A′A*, B′B*, C′C* concur in X(5907). Also, X(5907) is the center of the conic described at X(5777) for P = X(69). In this case, triangle H_AH_BH_C is perspective to ABC at X(64). (Randy Hutson, July 7, 2014)

X(5907) lies on the Burek-Hutson central cubic (K645) and these lines: {2,185}, {3,64}, {4,69}, {5,389}, {10,2807}, {20,3917}, {30,1216}, {40,3781}, {51,3091}, {52,381}, {84,3784}, {113,1209}, {114,130}, {140,5663}, {141,2883}, {143,3850}, {155,578}, {182,1181}, {235,343}, {373,5056}, {378,1092}, {394,1593}, {546,1154}, {550,5447}, {568,3851}, {916,942}, {970,3149}, {1071,2808}, {1147,4550}, {1204,5651}, {1364,1935}, {1568,1594}, {2979,3146}, {3060,3832}, {3523,5650}, {3545,3567}, {3574,5133}, {4260,5706}, {5068,5640}, {5777,5908}

X(5907) = reflection of X(389) in X(5)
X(5907) = complement of X(185)
X(5907) = X(4)-of-X(5)-Brocard-triangle
X(5907) = anticomplement of X(9729)

X(5908) = INTERSECTION OF LINES X(1)X(3) AND X(4)X(189)

Barycentrics a[(b + c)[a⁸ - 2a²(b² + c²)[a⁴ - (b² - c²)²] - (b² - c²)4] + 2a[a⁶(b² - bc + c²) - 3a⁴(b - c)(b³ - c³) + a²(b - c)²[3(b + c)(b³ + c³) + 4b²c²] - (b² - c²)²[(b - c)(b³ - c³) + 4b²c²]]] : : (Richard Hilton, March 2, 2015)

X(5908) is the center of the conic described at X(5777) for P = X(189). In this case, triangle H_AH_BH_C is perspective to ABC at X(40). (Randy Hutson, July 7, 2014)

X(5908) lies on the Burek-Hutson central cubic (K645) and these lines: {1,3}, {4,189}, {5,5909}, {222,1753}, {282,2262}, {971,1872}, {1364,1887}, {1535,5174}, {5777,5907}

X(5908) = reflection of X(5909) in X(5)

X(5909) = INTERSECTION OF LINES X(3)X(223) AND X(4)X(8)

Barycentrics a[(b + c)[a⁸ - 2a⁴(b² + c²)(a² - 2bc) + 2a²(b - c)²[(b² - c²)² - 2bc(b² + c²)] - (b² - c²)²(b - c)4] + 2a[a⁶(b² - bc + c²) - a⁴(3b⁴ - b³c - bc³ + 3c⁴) + (b² - c²)²[a²(3b² + bc + 3c²) - (b + c)(b³ + c³)]]] : : (Richard Hilton, March 2, 2015)

X(5909) is the center of the conic described at X(5777) for P = X(329). In this case, triangle H_AH_BH_C is perspective to ABC at X(3345). (Randy Hutson, July 7, 2014)

X(5909) lies on the Burek-Hutson central cubic (K645) and these lines: {3,223}, {4,8}, {5,5908}, {389,942}, {960,2817}, {2262,5715}, {2270,5709}

X(5909) = reflection of X(5908) in X(5)

X(5910) = INTERSECTION OF LINES X(3)X(64) AND X(4)X(1032)

Barycentrics ((b^2+c^2)*a^18-(9*b^4-4*b^2*c^2+9*c^4)*a^16+4*(b^2+c^2)*(9*b^4-16*b^2*c^2+9*c^4)*a^14-4*(b^2-c^2)^2*(21*b^4+34*b^2*c^2+21*c^4)*a^12+2*(b^4-c^4)*(b^2-c^2)*(63*b^4+86*b^2*c^2+63*c^4)*a^10-2*(b^2-c^2)^2*(63*b^8+63*c^8+2*(97*b^4+95*b^2*c^2+97*c^4)*b^2*c^2)*a^8+4*(b^4-c^4)*(b^2-c^2)*(21*b^8+21*c^8+2*(25*b^4+9*b^2*c^2+25*c^4)*b^2*c^2)*a^6-4*(b^2-c^2)^2*(9*b^12+9*c^12+(26*b^8+26*c^8+(39*b^4-20*b^2*c^2+39*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)^3*(9*b^8+9*c^8+10*(2*b^4+7*b^2*c^2+2*c^4)*b^2*c^2)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^8)*a^2 : :

X(5910) is the center of the conic described at X(5777) for P = X(1032). In this case, triangle H_AH_BH_C is perspective to ABC at X(1498). (Randy Hutson, July 7, 2014)

X(5910) lies on the Burek-Hutson central cubic (K645) and these lines: {3,64}, {4,1032}, {3079,5562}

X(5911) = INTERSECTION OF LINES X(3)X(9) AND X(4)X(1034)

Barycentrics a*((b+c)*a^14-2*(2*b^2-b*c+2*c^2)*a^13+(b+c)*(b^2-6*b*c+c^2)*a^12+4*(4*b^4+4*c^4-b*c*(3*b^2-2*b*c+3*c^2))*a^11-(b^2-c^2)*(b-c)*(19*b^2+10*b*c+19*c^2)*a^10-10*(2*b^4+2*c^4+b*c*(b+c)^2)*(b-c)^2*a^9+5*(b^2-c^2)*(b-c)*(9*b^4+9*c^4+2*b*c*(4*b^2+7*b*c+4*c^2))*a^8-8*(5*b^4+5*c^4+2*b*c*(2*b^2+b*c+2*c^2))*(b-c)^2*b*c*a^7-(b^2-c^2)*(b-c)*(45*b^6+45*c^6+(50*b^4+50*c^4+b*c*(67*b^2+60*b*c+67*c^2))*b*c)*a^6+2*(b^2-c^2)^2*(10*b^6+10*c^6+(15*b^4+15*c^4-2*b*c*(b^2-9*b*c+c^2))*b*c)*a^5+(b^2-c^2)^2*(b+c)*(19*b^6+19*c^6-(10*b^4+10*c^4-b*c*(29*b^2-12*b*c+29*c^2))*b*c)*a^4-4*(b^2-c^2)^2*(b+c)^2*(4*b^6+4*c^6-b*c*(5*b^2-2*b*c+5*c^2)*(b-c)^2)*a^3-(b^2-c^2)^4*(b+c)^5*a^2+2*(b^2-c^2)^4*(2*b^6+2*c^6+(b^4+c^4+2*b*c*(5*b^2+3*b*c+5*c^2))*b*c)*a-(b^2-c^2)^7*(b-c)) : :

X(5911) is the center of the conic described at X(5777) for P = X(1034). In this case, triangle H_AH_BH_C is perspective to ABC at X(1490). (Randy Hutson, July 7, 2014)

X(5911) lies on the Burek-Hutson central cubic (K645) and these lines: {3,9}, {4,1034}

X(5912) = EULER-PONCELET POINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a^10 - 4a^8(b^2 + c^2) - 3a^6(b^4 - 6b^2c^2 + c^4) + 2a^4(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - a^2(b^8 + c^8 - 11b^6c^2 - 11b^2c^6 + 18b^4c^4) - 3b^2c^2(b^2 - c^2)^2(b^2 + c^2)

X(5912) is the point QA-P2 center of quadrangle X(13)X(14)X(15)X(16); see Euler-Poncelet Point.

X(5912) is the point common to the nine-points circles of these 4 triangles: X(14)X(15)X(16), X(13)X(15)X(16), X(13)X(14)X(16), X(13)X(14)X(15); also, X(5912) is the center of the rectangular hyperbola that passes through the points X(13), X(14), X(15), X(16). (Randy Hutson, July 7, 2014)

Let O(13,15) be the circle with segment X(13)X(15) as diameter (and center X(396)), and let O(14,16) be the circle with segment X(14)X(16) as diameter (and center X(395)); then X(5912) is the radical trace of O(13,15) and O(14,16). (Randy Hutson, August 17, 2014)

X(5912) lies on these lines: {2,6}, {98,843}, {111,523}

X(5912) = reflection of X(i) in X(j) for these (i,j): (5913,230), (111,5914)

X(5913) = GERGONNE-STEINER POINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3 a^4 b^2 + 2 a^2 b^4 - b^6 + 3 a^4 c^2 - 10 a^2 b^2 c^2 + b^4 c^2 + 2 a^2 c^4 + b^2 c^4 - c^6

X(5913) is the QA-P3 center of the quadrangle X(13)X(14)X(15)X(16); see Gergonne-Steiner Point.

X(5913) = inverse-in-{circumcircle, nine-point circle}-inverter of X(6); see X(5577) for the definition of inverter.

The {circumcircle, nine-point circle}-inverter is the orthopic circle of the Steiner inscribed ellipse; its center is X(2), its radius is [(a² + b² + c²)/18]^1/2, and the powers of A,B,C with respect to this circle are (-a² + b² + c²)/6, (a² - b² + c²)/6, (a² + b² - c²)/6. (Peter Moses, July 16, 2014)T

X(5913) lies on these lines: {2,6}, {23,2079}, {30,111}, {112,468}, {115,858}, {403,1560}, {843,1302}, {1499,1513}, {2030,5642}

X(5913) = reflection of X(5912) in X(230)
X(5913) = isogonal conjugate of X(6096)
X(5913) = complement of X(5971)

X(5914) = PARABOLA AXES CROSSPOINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4 a^10 - 8 a^8 b^2 - 3 a^6 b^4 + 4 a^4 b^6 - 5 a^2 b^8 - 8 a^8 c^2 + 30 a^6 b^2 c^2 - 12 a^4 b^4 c^2 + 25 a^2 b^6 c^2 - 3 b^8 c^2 - 3 a^6 c^4 - 12 a^4 b^2 c^4 - 36 a^2 b^4 c^4 + 3 b^6 c^4 + 4 a^4 c^6 + 25 a^2 b^2 c^6 + 3 b^4 c^6 - 5 a^2 c^8 - 3 b^2 c^8

X(5914) is the QA-P6 center of the quadrangle X(13)X(14)X(15)X(16); see Parabola Axes Crosspoint.

X(5914) lies on these lines: {30,115}, {111,523}

X(5914) = midpoint of X(111) and X(5912)

X(5915) = INSCRIBED SQUARE AXES CROSSPOINT OF QUADRANGLE X(13)X(14)X(15)X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -4 a^12 + 8 a^10 b^2 - 7 a^8 b^4 + 11 a^6 b^6 - 13 a^4 b^8 + 5 a^2 b^10 + 8 a^10 c^2 - 10 a^8 b^2 c^2 - 3 a^6 b^4 c^2 + 17 a^4 b^6 c^2 - 7 a^2 b^8 c^2 + 3 b^10 c^2 - 7 a^8 c^4 - 3 a^6 b^2 c^4 - 12 a^4 b^4 c^4 + 2 a^2 b^6 c^4 - 12 b^8 c^4 + 11 a^6 c^6 + 17 a^4 b^2 c^6 + 2 a^2 b^4 c^6 + 18 b^6 c^6 - 13 a^4 c^8 - 7 a^2 b^2 c^8 - 12 b^4 c^8 + 5 a^2 c^10 + 3 b^2 c^10

X(5915) is the QA-P23 center of the quadrangle X(13)X(14)X(15)X(16); see Inscribed Square Axes Crosspoint

X(5915) is the centroid of the trapezoid X(2378)X(5916)X(2379)X(5917), which is similar to and orthogonal to the trapezoid X(13)X(15)X(14)X(16), with similitude center X(111); see X(5916). (Randy Hutson, July 7, 2014)

X(5915) lies on these lines: {30,115}, {98,843}, {111,477}

X(5916) = INTERSECTION OF LINES X(14)X(530) AND X(98)X(2379)

Barycentrics 2*a^12 - 4*a^10*b^2 + 2*a^8*b^4 - a^6*b^6 + 2*a^4*b^8 - a^2*b^10 - 4*a^10*c^2 + 8*a^8*b^2*c^2 - 3*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - 3*b^10*c^2 + 2*a^8*c^4 - 3*a^6*b^2*c^4 + 6*a^4*b^4*c^4 - a^2*b^6*c^4 + 12*b^8*c^4 - a^6*c^6 - 4*a^4*b^2*c^6 - a^2*b^4*c^6 - 18*b^6*c^6 + 2*a^4*c^8 + 2*a^2*b^2*c^8 + 12*b^4*c^8 - a^2*c^10 - 3*b^2*c^10 + 2*Sqrt[3]*(a^2 - b^2)*(a^2 - c^2)*(2*a^2 - b^2 - c^2)*(b^2 - c^2)^2*S : : (Richard Hilton, March 2, 2015)

X(5916) lies on these lines: {14,530}, {98,2379}, {523,2378}, {690,5917}

X(5916) = isogonal conjugate of X(15) with respect to the triangle X(13)X(14)X(16)
X(5916) = 2nd-Parry-to-ABC similarity image of X(13)

X(5917) = INTERSECTION OF LINES X(13)X(531) AND X(98)X(2378)

Barycentrics 2*a^12 - 4*a^10*b^2 + 2*a^8*b^4 - a^6*b^6 + 2*a^4*b^8 - a^2*b^10 - 4*a^10*c^2 + 8*a^8*b^2*c^2 - 3*a^6*b^4*c^2 - 4*a^4*b^6*c^2 + 2*a^2*b^8*c^2 - 3*b^10*c^2 + 2*a^8*c^4 - 3*a^6*b^2*c^4 + 6*a^4*b^4*c^4 - a^2*b^6*c^4 + 12*b^8*c^4 - a^6*c^6 - 4*a^4*b^2*c^6 - a^2*b^4*c^6 - 18*b^6*c^6 + 2*a^4*c^8 + 2*a^2*b^2*c^8 + 12*b^4*c^8 - a^2*c^10 - 3*b^2*c^10 - 2*Sqrt[3]*(a^2 - b^2)*(a^2 - c^2)*(2*a^2 - b^2 - c^2)*(b^2 - c^2)^2*S : : (Richard Hilton, March 2, 2015)

X(5917) lies on these lines: {13,531}, {98,2378}, {523,2379}, {690,5916}

X(5917) = isogonal conjugate of X(16) with respect to the triangle X(13)X(14)X(15); see X(5915)
X(5917) = 2nd-Parry-to-ABC similarity image of X(14)

X(5918) = CENTROID OF HUTSON-EXTOUCH TRIANGLE

Barycentrics a[(b + c)[a⁴ - 4a²bc - (b² - c²)²] - 2b³(b² - 4bc + c²) + 2a(b - c)²(b² + c²)] : : (Richard Hilton, March 2, 2015)

The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).

X(5918) lies on these lines: {3,1709}, {20,65}, {55,5732}, {57,2951}, {63,3059}, {84,5584}, {165,210}, {170,241}, {354,516}, {376,6001}, {411,1776}, {517,3534}, {548,5887}, {940,1721}, {960,3522}, {990,3745}, {1040,1456}, {1155,1708}, {1407,4319}, {1427,3000}, {1742,3666}, {1750,4413}, {3057,4297}, {3146,3812}, {3555,5493}, {3689,6244}, {3698,5691}, {4640,5784}, {5731,5919}

X(5918) = reflection of X(i) in X(j) for these (i,j): (210,165), (5919,5731)

X(5919) = CENTROID OF HUTSON-INTOUCH TRIANGLE

Barycentrics a[(b + c)[a² - (b - c)²] - 8abc]] : : (Richard Hilton, March 2, 2015)

The Hutson-intouch and Hutson-extouch triangles are defined at X(5731).

X(5919) lies on these lines: {1,3}, {2,3880}, {8,3740}, {10,3893}, {12,3817}, {37,374}, {72,3244}, {145,960}, {210,392}, {226,4342}, {390,3476}, {497,5252}, {515,3058}, {516,5434}, {518,1992}, {551,2802}, {950,5927}, {956,3683}, {997,3689}, {1001,3872}, {1056,1836}, {1058,1837}, {1100,2267}, {1122,3672}, {1125,3698}, {1149,3752}, {1201,4646}, {1317,2801}, {1320,2346}, {1358,3663}, {1376,3895}, {1483,5887}, {1864,3488}, {1898,3486}, {2256,2264}, {3421,4679}, {3555,3635}, {3616,3848}, {3621,4662}, {3622,3812}, {3623,3869}, {3625,3697}, {3632,5044}, {3636,3922}, {3649,4301}, {3696,3902}, {3870,5289}, {3881,4018}, {3921,4669}, {4009,4737}, {4870,5603}, {5731,5918}

X(5919) = reflection of X(i) in X(j) for these (i,j): (354,1), (5918,5731)

X(5920) = ORTHOLOGIC CENTER OF THESE TRIANGLES: HUTSON-INTOUCH TO HUTSON-EXTOUCH

Barycentrics a*(-a+b+c)*((b+c)*a^7-(b+c)^2*a^6-(b+c)*(3*b^2+8*b*c+3*c^2)*a^5+(3*b^4+3*c^4+2*b*c*(b^2+11*b*c+c^2))*a^4+(b+c)*(3*b^4+3*c^4+2*b*c*(8*b^2+5*b*c+8*c^2))*a^3-(3*b^4+3*c^4+2*b*c*(2*b^2-3*b*c+2*c^2))*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^4+c^4+2*b*c*(5*b^2-3*b*c+5*c^2))*a+(b^4-c^4)*(b^2-c^2)*(b-c)^2) : :

The Hutson-extouch and Hutson-intouch triangles are defined at X(5731).

The reciprocal orthologic center is X(3555). (Randy Hutson, November 2, 2017)

X(5920) lies on these lines: {1,7160}, {1697,12658}, {3601,12439}, {9785,9804}, {9953,10866}

X(5921) = DARBOUX IMAGE OF X(20)

Barycentrics 5*a^6-5*(b^2+c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^2-3*(b^4-c^4)*(b^2-c^2) : :

Darboux images are discussed at X(5881).

X(5921) lies on these lines: {2,98}, {3,3620}, {4,193}, {6,3091}, {20,64}, {68,3089}, {141,3523}, {153,5848}, {381,1353}, {511,3146}, {524,3543}, {546,5093}, {611,5261}, {613,5274}, {962,5847}, {1992,3839}, {2888,5596}, {3090,5050}, {3580,4232}, {3618,5056}, {3619,5085}, {3818,3832}

X(5921) = reflection of X(20) in X(69)
X(5921) = X(20)-of-obverse-triangle-of-X(69)

X(5922) = DARBOUX IMAGE OF X(64)

Barycentrics (7*a^10-10*(b^2+c^2)*a^8+2*(b^4+6*b^2*c^2+c^4)*a^6-(b^2-c^2)^4*a^2+2*(b^4-c^4)*(b^2-c^2)^3)*(a^4+2*(b^2-c^2)*a^2-(3*b^2+c^2)*(b^2-c^2))*(a^4-2*(b^2-c^2)*a^2+(b^2-c^2)*(b^2+3*c^2)) : :

Darboux images are discussed at X(5881).

X(5922) lies on these lines: {20,64}, {122,1073}, {154,459}

X(5922) = reflection of X(64) in X(253)

X(5923) = DARBOUX IMAGE OF X(84)

Barycentrics [3a⁷ - a⁵(7b² + 2bc + 2c²) + b³(b - c)²(5b² + 6bc + 5c²) - a(b² - c²)²(b² - 6bc + c²) + 2(b + c)³[a² - (b - c)²]²} / {a[(b + c)² - a²] - (b + c)[a² - (b - c)²]] : : (Richard Hilton, March 2, 2015)

Darboux images are discussed at X(5881).

X(5923) lies on these lines: {8,20}, {282,5514}, {1256,1837}

X(5923) = reflection of X(84) in X(189)

X(5924) = DARBOUX IMAGE OF X(1490)

Barycentrics 3*a^10-3*(b+c)*a^9-2*(5*b^2-2*b*c+5*c^2)*a^8+8*(b^3+c^3)*a^7+2*(7*b^4+2*b^2*c^2+7*c^4)*a^6-2*(b+c)*(3*b^4+3*c^4-2*b*c*(2*b-c)*(b-2*c))*a^5-4*(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)*a^4+8*(b^2-c^2)^2*(b+c)*b*c*a^3+(b^2-c^2)^2*(7*b^4+2*b^2*c^2+7*c^4)*a^2+(b^2-c^2)^3*(b-c)*(b^2-6*b*c+c^2)*a-2*(b^2-c^2)^4*(b-c)^2 : :

Darboux images are discussed at X(5881).

X(5924) lies on these lines: {4,2093}, {9,119}, {20,78}, {57,5715}, {84,5812}, {226,2096}, {2095,5735}, {2800,3586}

X(5924) = reflection of X(1490) in X(329)

X(5925) = DARBOUX IMAGE OF X(1498)

Barycentrics 5*a^10-6*(b^2+c^2)*a^8-2*(5*b^4-14*b^2*c^2+5*c^4)*a^6+16*(b^4-c^4)*(b^2-c^2)*a^4-3*(b^2-c^2)^2*(b^4+6*b^2*c^2+c^4)*a^2-2*(b^4-c^4)*(b^2-c^2)^3 : :

Darboux images are discussed at X(5881).

X(5925) lies on these lines: {3,113}, {4,1192}, {20,394}, {30,64}, {154,550}, {221,4302}, {376,2883}, {382,1853}, {599,2892}, {1503,3529}, {1514,3147}, {1620,3542}, {1770,1854}, {2192,4299}, {3146,3580}

X(5925) = reflection of X(1498) in X(20)

X(5926) = CENTER OF CIRCUMCIRCLE-INVERSE OF LINE X(2)X(6))

Barycentrics a^2*(b^2 - c^2)*(2*a^6 - 4*a^4*b^2 + 2*a^2*b^4 - 4*a^4*c^2 + a^2*b^2*c^2 - b^4*c^2 + 2*a^2*c^4 - b^2*c^4) : :

Barycentrics a²(b² - c²)(2a⁶ - 4a⁴b² - 4a⁴c² + 2a²b⁴ + 2a²c⁴ + a²b²c² - b⁴c² - b²c⁴) : :
X(5926) = 7 X[3523] + X[31299], 7 X[3526] - 5 X[31279], 3 X[5054] - X[31176], X[11616] + 3 X[39214], 3 X[15724] - X[19912], 3 X[39228] - 4 X[39477], X[39228] - 4 X[39481], X[39477] - 3 X[39481]

The circumcircle-inverse of a line is a circle; if the line does not pass through X(3), then the radius of the circle is finite. The appearance if (i,j) in the following list means that X(i) is on the line X(2)X(6) and that X(j) is the inverse of X(i): (2,23), (6,187), (69, 5866), (81,5867), (86,5937), (141,5938), (183, 5939), (193,5940), (230,5941), (352,353), (524,3), (5108,669).

X(5926) lies on these lines: {2, 39511}, {3, 669}, {24, 2501}, {25, 39533}, {32, 39518}, {140, 23301}, {182, 9009}, {351, 9128}, {512, 6132}, {523, 7575}, {525, 34952}, {549, 25423}, {574, 39501}, {924, 8552}, {1658, 8151}, {1995, 39492}, {2780, 8644}, {3523, 31299}, {3526, 31279}, {3800, 39201}, {4897, 39578}, {5054, 31176}, {6563, 7488}, {6642, 14341}, {7669, 14657}, {8651, 11615}, {9494, 30217}, {11186, 37184}, {14809, 37813}, {15724, 19912}, {16235, 32472}, {16678, 39545}, {19357, 30451}, {19543, 25537}, {34513, 39495}, {41357, 44272}

X(5926) = midpoint of X(3) and X(669)
X(5926) = reflection of X(i) in X(j) for these {i,j}: {6563, 32204}, {11615, 8651}, {23301, 140}, {32231, 9126}
X(5926) = anticomplement of X(39511)
X(5926) = crossdifference of every pair of points on line {566, 3291}

X(5927) = CENTROID OF 2nd EXTOUCH TRIANGLE

Barycentrics a{(b + c)[a⁴ + 2a²bc - (b - c)²(b² + 4bc + c²)] - 2a[a²(b² + c²) - (b² - c²)²]} : : (Richard Hilton, March 2, 2015)

The classical extouch triangle is regarded as the 1st extouch triangle. The 2nd, 3rd, 4th, and 5th extouch triangles are defined by Randy Hutson (July 11, 2014) as follows. Let A_A, A_B, A_C be the touchpoints of the A-excircle and the lines BC, CA, AB, respectively, and define B_B, B_C, B_A and C_C, C_A, C_B cyclically.

Let A₁ = A_A, B₁ = B_B, C₁ = C_C; 1st extouch triangle = A₁B₁C₁

Let A₂ = B_CB_A∩C_AC_B, and define B₂ and C₂ cyclically; 2nd extouch triangle = A₂B₂C₂

Let A₃ = C_AA_C∩A_BB_A, and define B₃ and C₃ cyclically; 3rd extouch triangle = A₃B₃C₃

Let A₄ = B_CC_A∩B_AC_B, and define B₄ and C₄ cyclically; 4th extouch triangle = A₄B₄C₄

Let D₁ = B_BB_C∩C_CC_A, and define D₂ and D₃ cyclically. Let E₁ = B_BB_A∩C_CC_B, and define E₂ and E₃ cyclically. Define A₅ = D₂E₂∩D₃E₃, and define B₅ and C₅ cyclically; 5th extouch triangle = A₅B₅C₅

Barycentric coordinates for A-vertices of the the five triangles:
A₁ = 0 : a - b + c : a + b - c
A₂ = 2a(b + c) : - a² - b² + c² : - a² + b² - c²
A₃ = 2a(b + c)(a - b + c)(a + b - c) : (a + b + c)(a - b - c)(a² + b² - c²) : (a + b + c)(a - b - c)(a² - b² + c²)
A₄ = 2a(b + c)(a + b + c) : (a - b - c)(a² - b² + c²) : (a - b - c)(a² + b² - c²)
A₅ = 2a(b + c)(a - b + c)(a + b - c) : (a - b - c)(a + b - c)[b² + (a + c)²] : (a - b - c)(a - b + c)[c² + (a + b)²]

A₂B₂C₂ is perspective to ABC and A₃B₃C₃ at X(4).
A₂B₂C₂ is homothetic to the excentral triangle at X(9).
A₂B₂C₂ is homothetic to the intouch triangle at X(226).
A₂B₂C₂ is perspective to the extouch triangle and (extraversion triangle of X(65)) at X(72).
A₂B₂C₂ is perspective to the anticevian triangle of X(8) at X(329).
A₂B₂C₂ is homothetic to the hexyl triangle at X(1490).
A₂B₂C₂ is perspective to the Feuerbach triangle at X(442).
A₂B₂C₂ is perspective to A₄B₄C₄ at X(5928).
A₂B₂C₂ is homothetic to the inner Hutson triangle at X(5934).
A₂B₂C₂ is homothetic to the outer Hutson triangle at X(5935).
A₂B₂C₂ is homothetic to the 2nd circumperp triangle at X(405).
A₂B₂C₂ is homothetic to the inverse-in-incircle triangle (see X(5571) at X(5728).
A₂B₂C₂ is homothetic to the Hutson-intouch triangle at X(950).
A₂B₂C₂ is perspective to the Hutson-extouch triangle at X(442).

In the following list, the appearance of (i,j) means that (X(i) of the 2nd extouch triangle) = X(j):
(3,4), (4,72), (5,5777), (6,9), (25,329), (26,5812), (54, 442), (184,226), (185,950), (195,3651), (578, 10), (647,1635), (1181,1)

A₃B₃C₃ is perspective to the intouch triangle and (extraversion triangle of X(65)) at X(1439).
A₃B₃C₃ is perspective to the extouch triangle and A₅B₅C₅ at X(5930).
A₃B₃C₃ is perspective to A₄B₄C₄ at X(5929).
A₃B₃C₃ is perspective to the anticevian triangle of X(7) at X(5932).

A₄B₄C₄ is perspective to ABC at X(69).
A₄B₄C₄ is perspective to the intouch triangle and A₅B₅C₅ at X(65).
A₄B₄C₄ is perspective to the anticevian triangle of X(7) at X(5933).

A₅B₅C₅ is perspective to ABC at X(388).
A₅B₅C₅ is perspective to the anticevian triangle of X(7) at X(8).

X(5927) is also the centroid of the triangle formed by the polars of the incenter with respect to the excircles. (Randy Hutson, July 11, 2014) Note added (11/25/2015): for more about this triangle, named the Atik triangle, see the preamble to X(8580).

X(5927) lies on these lines: {2,971}, {3,3305}, {4,8}, {5,1071}, {9,165}, {11,118}, {12,1898}, {20,5044}, {40,3697}, {44,1754}, {51,916}, {57,5729}, {63,5779}, {84,474}, {210,516}, {374,1903}, {377,6259}, {381,912}, {392,515}, {405,1490}, {442,6260}, {443,6223}, {518,1699}, {942,3091}, {946,3555}, {950,5919}, {960,5691}, {990,4383}, {1012,5440}, {1214,2635}, {1385,5284}, {1427,1736}, {1853,3753}, {2478,5787}, {2808,5943}, {3146,3876}, {3149,3916}, {3452,5784}, {3487,5049}, {3752,5400}, {3832,3868}, {3921,5657}, {4002,5818}, {4015,5493}, {4018,5693}, {4187,6245}, {5435,5825}, {5805,5905}

X(5927) = reflection of X(i) in X(j) for these (i,j): (165,3740), (354,3817), (3753,5587)
X(5927) = complement of X(11220)

X(5928) = PERSPECTOR OF 2nd AND 4th EXTOUCH TRIANGLES

Barycentrics a^6+(b+c)*a^5-(b^2+c^2)*a^4+(b^2-c^2)^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^4-c^4)*(b^2-c^2) : :

X(5928) is also the perspector of the 2nd (and 4th) extouch triangle and the polar triangle of the Yiu conic; see X(5927).

X(5928) lies on these lines: {2,2182}, {4,65}, {6,1848}, {9,440}, {33,1503}, {69,189}, {222,226}, {225,5786}, {355,6358}, {1038,1490}, {1753,6247}, {1824,1899}, {1853,1861}, {4329,5739}, {5307,6354}

X(5928) = 2nd-extouch-isogonal conjugate of X(12689)

X(5929) = PERSPECTOR OF 3rd AND 4th EXTOUCH TRIANGLES

Trilinears (b + c)[a^5(b + c) + 2a^4 (b^2 + c^2) - 2a^2(b^4 - b^3c + b^2c^2 - bc^3 + c^4) - a(b - c)^2(b + c)^3 - 2bc(b - c)^2(b^2 + bc + c^2)] : :

See X(5927).

X(5929) lies on these lines: {4,69}, {65,1439}, {77,851}, {284,940}, {1211,3452}, {2898,5932}

X(5929) = perspector of [cross-triangle of ABC and 3rd extouch triangle] and [cross-triangle of ABC and 4th extouch triangle]

X(5930) = PERSPECTOR OF 1st, 3rd AND 5th EXTOUCH TRIANGLES

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)(b + c)/(b + c - a)

See X(5927).

X(5930) lies on these lines: {1,4}, {8,253}, {10,227}, {20,1394}, {40,3182}, {57,387}, {65,1439}, {77,377}, {109,1294}, {208,1763}, {221,516}, {222,4292}, {603,1754}, {610,1249}, {851,1410}, {1042,3914}, {1074,4303}, {1210,1465}, {1400,2082}, {1419,3332}, {1427,1834}, {1455,4297}, {1456,6284}, {1467,4000}, {1612,2078}, {1630,2202}, {1714,3911}, {1935,2328}, {3682,4551}, {3710,4552}, {3987,4848}, {4304,6357}

X(5930) = isotomic conjugate of X(5931)
X(5930) = X(8)-Ceva conjugate of X(65)

X(5931) = ISOTOMIC CONJUGATE OF X(5930)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(c + a)(a + b)/(3a^4 - b^4 - c^4 - 2a^2b^2 - 2a^2c^2 + 2b^2c^2)

X(5931) is the trilinear pole of the perspectrix of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). (Randy Hutson, July 11, 2014) See X(5927).

X(5931) lies on these lines: {20,64}, {75,1895}

X(5931) = isotomic conjugate of X(5930)

X(5932) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND ANTICEVIAN TRIANGLE OF X(7)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sec^2(A/2) [- a^2 cos B cos C sec^2(A/2) + b^2 cos C cos A sec^2(B/2) + c^2 cos A cos B sec^2(C/2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = [a^6 - 2a^5(b + c) - a^4(b + c)^2 + 4a^3(b^3 + c^3) - a^2(b^2 - c^2)^2 - 2a(b - c)^2(b + c)(b^2 + c^2) + (b - c)^2(b + c)^4]/(b + c -a)

X(5932) is the perspector of these three triangles: 3rd extouch, anticevian of X(7), and extraversion of X(7). Also, X(5932) is the perspector of ABC and the pedal triangle of X(3182), as well as the perspector of ABC and the triangle obtained by reflecting the pedal triangle of X(223) in X(223). (Randy Hutson, July 11, 2014)

X(5932) lies on the Lucas cubic and these lines: {2,77}, {4,7}, {8,253}, {20,3182}, {57,5802}, {69,1034}, {151,4329}, {269,1210}, {329,1032}, {411,1804}, {1264,4554}, {1442,5703}, {1443,5704}, {1445,2270}, {2062,6060}, {2898,5929}, {3086,4341}

X(5932) = isotomic conjugate of X(1034)
X(5932) = complement of X(20212)
X(5932) = anticomplement of X(282)
X(5932) = X(69)-Ceva conjugate of X(7)
X(5932) = perspector of 3rd extouch triangle and cross-triangle of ABC and 3rd extouch triangle

X(5933) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND ANTICEVIAN TRIANGLE OF X(7)

Barycentrics [a³ - a(3b² + 4bc + 3c²) - (b + c)(3a² - b² - c²)] / (b + c - a) : : (Richard Hilton, March 2, 2015)

X(5933) lies on these lines: {7,8}, {57,3879} et al

X(5933) = perspector of 4th extouch triangle and cross-triangle of ABC and 4th extouch triangle

X(5934) = HOMOTHEITIC CENTER OF 2nd EXTOUCH TRIANGLE AND INNER HUTSON TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -S*((2*R+r)*(4*r*R+r^2-2*b*c)+4*s*R*(s-a))+b*c*a^2*(b*cos(B)*sin(C/2)^3+c*cos(C)*sin(B/2)^3)-b*c*(2*s-a)*(s-b)*(s-c)*sin(A/2)      (César Lozada, January 15, 2015)
Trilinears       - p(a,b,c) + q(a,b,c,A,B,C) : - p(b,c,a) + q(b,c,a,B,C,A) : - p(c,a,b) + q(c,a,b,C,A,B), where p(a,b,c) = a*(a^2*(a^2+2*b^2+2*c^2)-3*(b^2-c^2)^2)+(b+c)*(-3*a^4+(b-c)^2*(2*a^2+b^2+6*b*c+c^2)) and
q(a,b,c,A,B,C) = -2*b*(-b-c+a)*(a-b+c)*(c^2+a^2-b^2)*sin(C/2)-2*c*(-b-c+a)*(a+b-c)*(a^2+b^2-c^2)*sin(B/2)-4*b*c*(b+c)*(a+b-c)*(a-b+c)*sin(A/2)      (César Lozada, January 15, 2015)

X(5934) = X(222)-of-2nd-extouch-triangle. (Inner Hutson triangle is defined at X(363).)

X(5934) lies on these lines: {9,363}, {503,1750}

X(5935) = HOMOTHEITIC CENTER OF 2nd EXTOUCH TRIANGLE AND OUTER HUTSON TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S*((2*R+r)*(4*r*R+r^2-2*b*c)+4*s*R*(s-a)) +b*c*a^2*(b*cos(B)*sin(C/2)^3+c*cos(C)*sin(B/2)^3)-b*c*(2*s-a)*(s-b)*(s-c)*sin(A/2)      (César Lozada, January 15, 2015)
Trilinears       p(a,b,c) + q(a,b,c,A,B,C) : - p(b,c,a) + q(b,c,a,B,C,A) : - p(c,a,b) + q(c,a,b,C,A,B), where p(a,b,c) = a*(a^2*(a^2+2*b^2+2*c^2)-3*(b^2-c^2)^2)+(b+c)*(-3*a^4+(b-c)^2*(2*a^2+b^2+6*b*c+c^2)) and
q(a,b,c,A,B,C) = -2*b*(-b-c+a)*(a-b+c)*(c^2+a^2-b^2)*sin(C/2)-2*c*(-b-c+a)*(a+b-c)*(a^2+b^2-c^2)*sin(B/2)-4*b*c*(b+c)*(a+b-c)*(a-b+c)*sin(A/2)      (César Lozada, January 15, 2015)

X(5935) = X(219)-of-2nd-extouch-triangle. (The outer Hutson triangle is defined at X(363).)

X(5935) lies on these lines: {9,164}, {503,1750}

X(5936) = ISOTOMIC CONJUGATE OF X(3616)

Barycentrics 1/(3a + b + c) : 1/(a + 3b + c) : 1/(a + b + 3c)

Let A₂₁B₂₁C₂₁ be the Gemini triangle 21. Let L_A be the line through A₂₁ parallel to BC, and define L_B and L_C cyclically. Let A′₂₁ = L_B∩L_C, and define B′₂₁, C′₂₁ cyclically. Triangle A′₂₁B′₂₁C′₂₁ is homothetic to ABC at X(5936). (Randy Hutson, November 30, 2018)

X(5936) lies on these lines: {2,2321}, {7,10}, {8,86}, {27,281}, {75,3701}, {272,4313}, {310,3596}, {335,4699}, {594,5308}, {673,2345}, {903,5224}, {936,306}, {966,6172}, {1002,4111}, {1088,1441}, {1440,3160}, {1698,3672}, {3241,5564}, {3616,4460}, {3679,3945}, {3879,4678}, {4357,4373}, {4360,5550}, {4409,4748}, {4461,5257}, {4472,5839}, {4677,4909}

X(5936) = trilinear pole of the line X(514)X(1635) (which is the Lemoine axis of the 2nd extouch triangle)

X(5937) = INVERSE-IN-CIRCUMCIRCLE OF X(86)

Barycentrics a²{a⁴(b + c) - (a² + bc)[(b³ + c³) - b²c² / (b + c)] + a[a²(b² + c²) - b⁴ + b²c² - c⁴]] : : (Richard Hilton, March 2, 2015)

X(5937) lies on these lines: {3,86}, {669,4367}

X(5938) = INVERSE-IN-CIRCUMCIRCLE OF X(141)

Barycentrics a²[a⁸ - a²b²c²(b² + c² - a²) - (b⁴ - c⁴)²] : : (Richard Hilton, March 2, 2015)

X(5938) lies on these lines: {3,66}, {25,5523}, {353,3148}, {525,669}, {755,2715}

X(5939) = INVERSE-IN-CIRCUMCIRCLE OF X(183)

Barycentrics 2a⁸ + a⁴(b² - c²)² - b²c²(b⁴ - 4b²c² + c⁴) - a²(b² + c²)(a⁴ + 2b⁴ - 3b²c² + 2c⁴) : : (Richard Hilton, March 2, 2015)

X(5939) = X(187)-of-circummedial triangle. Let A′B′C′ be the triangle of which ABC is the 1st Brocard triangle; here called the 1st anti-Brocard triangle, with circumcircle called the anti-Brocard circle; then X(5939) = X(187)-of-A′B′C′, and X(5939) = inverse-in-anti-Brocard-circle of X(99). (Randy Hutson, July 18, 2014)

Barycentrics for the vertices of the 1st anti-Brocard triangle are as follows (Peter Moses, August 21, 2014):

A′ = a⁴ - b²c² : c⁴ - a²b² : b⁴ - a²c²
B′ = c⁴ - b²a² : b⁴ - c²a² : a⁴ - b²c²
C′ = b⁴ - c²a² : a⁴ - c²b² : c⁴ - a²b²

For more properties of A′B′C′, see X(5976).

X(5939) lies on these lines: {2,353}, {3,76}, {147,1007}, {187,543}, {325,542}, {385,5104}, {669,804}, {671,3972}, {2023,3329}

X(5940) = INVERSE-IN-CIRCUMCIRCLE OF X(193)

Barycentrics a^2*(a^8-6*(b^2+c^2)*a^6+43*b^2*c^2*a^4+2*(b^2+c^2)*(3*b^4-17*b^2*c^2+3*c^4)*a^2-(b^4-5*b^2*c^2+c^4)*(b^2+c^2)^2) : :

X(5940) lies on these lines: {3,193}, {353,5585}

X(5941) = INVERSE-IN-CIRCUMCIRCLE OF X(230)

Barycentrics a^2*(a^12-2*(b^2+c^2)*a^10+(b^4+7*b^2*c^2+c^4)*a^8-5*(b^2+c^2)*b^2*c^2*a^6-(b^8+c^8-3*b^2*c^2*(3*b^4-4*b^2*c^2+3*c^4))*a^4+(b^4-c^4)*(b^2-c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^2-(b^2-c^2)^2*(b^8+c^8-2*(b^4-3*b^2*c^2+c^4)*b^2*c^2)) : :

X(5941) lies on these lines: {3,230}, {25,669}

X(5942) = ANTICOMPLEMENT OF X(77)

Barycentrics   f(a,b,c) : f(b,c,a) : f(a,b,c), where f(a,b,c) = b/(1 + sec B) + c/(1 + sec C) - a/(1 + sec A)
Barycentrics   g(A,B,C) : g(B,C,A) : g(A,B,C), where g(A,B,C) = (1 - cos B) cot B + (1 - cos C) cot C - (1 - cos A) cot A
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a⁴ + b⁴ + c⁴ - 2a³(b + c) -2a(b - c)²(b + c) + 2bc(b² + c² - 3bc) + 2a²(b² + c² - bc) (Randy Hutson, July 18, 2014)

X(5942) lies on these lines: {2,77}, {7,4858}, {8,144}, {63,3686}, {69,1229}, {92,1947}, {281,651}, {329,2893}, {894,5554}, {1654,3152}, {3416,3436}

X(5943) = CENTROID OF HALF-ALTITUDE TRIANGLE

Trilinears    bc + ac cos C + ab cos B : :
Barycentrics    a²(b⁴ + c⁴ - a²b² - a²c² - 4b²c²)
X(5943) = 2X(5) + X(389)      (barycentrics and combo, Peter Moses, July 15, 2014)

The locus of the centroid of the pedal triangle of P as P varies around the nine-point circle is an ellipse with center X(5943). Also, X(5943) is the centroid of the pedal triangle of X(5), as well as the centroid of the 6 points of intersection of the nine-point circle and the sidelines of ABC. (Randy Hutson, July 18, 2014). The ellipse is here named the Hutson centroidal ellipse.

Let A′B′C′ be any one of the following nine triangles: medial, 1st Brocard, McCay, 1st Neuberg, 2nd Neuberg, inner Napoleon, outer Napoleon, inner Vecten, outer Vecten; and let Ba and Ca be the orthogonal projections of B′ and C′ on BC, respectively. Define Cb and Ac cyclically, and define Ab and Bc cyclically. X(5943) is the centroid of {Ba, Ca, Cb, Ab, Ac, Bc}. (Randy Hutson, March 25, 2016)

If you have The Geometer's Sketchpad, you can view X(5943), with the Hutson centroidal ellipse.

X(5943) lies on these lines: {2,51}, {5,389}, {6,1196}, {22,5092}, {23,5643}, {25,182}, {30,5892}

X(5943) = midpoint of X(i) and X(j) for these (i,j): (2,51), (5,5946)
X(5943) = complement of X(3917)
X(5943) = {X(51),X(373)}-isoconjugate of X(2)

X(5944) = CENTER OF HUNG CIRCLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[S² + 3S²_A - 2S_A(3S_ω - 5R²)]

Let d be a line tangent to the nine-point circle of a triangle ABC. Let D_A be the reflection of d in line BC, and define D_B and D_C cyclically. Let XYZ be the triangle formed by the lines D_A, D_B, D_C. The envelope of the circumcircle of the variable triangle XYZ is a circle. (Tran Quang Hung, ADGEOM #1387, July 9, 2014). The circle is here named the Hung circle.

The diameter of the Hung circle is the segment X(3)X(1614), so that X(5944) is the midpoint of this segment. The radius of the circle is (R/2)(3 + k)/(1 + k), where k = 2(cos 2A + cos 2B + cos 2C). The lines AX, BY, CZ concur in a point Q = Q(P) on the circumcircle of ABC. The appearance of (i,j) in the following list means that X(j) = Q(X(i)): (11,953), (113,477), (114,2698), (115,2698), (116,2724), (117,2734), (118,2724), (119,953), (124,2734), (125,477), (1312,74), (1313,74), (2039,98), (2040,98). (César Lozada, ADGEOM #1388 et al, July 9, 2014)

The triangles XYZ form a family of similar triangles, and Q is the incenter of XYZ. If P = p : q : r is a point on the nine-point circle, then

Q = Q(P) = a²/[p(v + w) - u(q + r)] : b²/[q(w + u) - v(r + p)] : c²/[r(u + v) - w(p + q)],

where u : v : w = X(5). (Peter Moses, August 1, 2014)

If ABC is acute, then Q is the incenter of XYZ, and XYZ has the orientation opposite that of ABC. Let J = |OH|/R and σ = area(ABC). Maximal area(XYZ) = σ(J - 1)/(J + 1) occurs with P = X(1312) and minimal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313). If ABC is not acute, then Q is an excenter of XYZ, and XYZ has the same orientation as ABC. In this case, maximal area(XYZ) = σ(J + 1)/(J - 1) occurs with P = X(1313) and minimal area(XYZ) = 0, which occurs when angle X(5)-to-P-to-X(4) is a right angle and XYZ is a one of the points of intersection of the circumcircle and the Hung circle. Also, there is a local maximum when P = X(1312), and in this case, area(XYZ) = σ(J - 1)/(J + 1). (Peter Moses, August 9, 2014)

Let A′ be the reflection of X(5) in line BC, and define B′ and C′ cyclically. Let A″ be the circumcenter of triangle BCA′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5944). (Randy Hutson, August 17, 2014)

If you have The Geometer's Sketchpad 5, you can view X(5944).

X(5944) lies on these lines: {3,74}, {24,5946}, {49,1154}, {52,1493}, {54,143}, {184,1658}, {546,1495}, {567,3518}, {3146,3431}

X(5945) = CENTER OF HOFSTADTER 0-ELLIPSE

\Barycentrics [(sin²A)/A][(sin²B)/B + (sin²C)/C - (sin²A)/A] : :

The family of Hofstadter ellipses are introduced at X(359) and further described at MathWorld. The ellipses are indexed as E(r) for 0 <= r <= 1, and E(1 - r) = E(r). Thus, the Hofstadter 0-ellipse and the Hofstadter 1-ellipse are identical. (Submitted by Valery Nemychnikova, Moscow Chemical Lyceum, July 28, 2014.)

X(5945) = X(2)-Ceva conjugate of X(359)

X(5946) = NINE-POINT CENTER OF ORTHOCENTROIDAL TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^2 b^4 c^2+3 b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4-4 b^4 c^4+3 a^2 c^6+3 b^2 c^6-c^8)
X(5946) = X(3) + 2X(143) = X(5) + 2X(389) (Peter Moses, August 9, 2014)

X(5946) lies on these lines: {2,568}, {3,143}, {4,3521}, {5,389}, {6,1511}, {24,5944}, {26,3796}, {30,51}, {49,1199}, {52,140}, {185,546}, {186,567}, {373,547}, {378,1112}, {381,5640}, {511,549}, {550,5446}, {569,973}, {632,1216}, {970,5428}, {974,1539}, {1147,1493}, {1656,5889}, {1995,5609}, {2070,5012}, {2781,5476}, {2979,5054}, {3628,5562}

X(5946) = midpoint of X(2) and X(568)
X(5946) = reflection of X(5) in X(5943)
X(5946) = orthocentroidal-circle-inverse of X(38724)
X(5946) = X(2)-of-reflection-triangle-of-X(5)

X(5947) = CENTROID OF FEUERBACH TRIANGLE

Barycentrics 3 a^7 b^2+3 a^6 b^3-9 a^5 b^4-9 a^4 b^5+9 a^3 b^6+9 a^2 b^7-3 a b^8-3 b^9+2 a^7 b c+5 a^6 b^2 c-6 a^5 b^3 c-17 a^4 b^4 c+2 a^3 b^5 c+15 a^2 b^6 c+2 a b^7 c-3 b^8 c+3 a^7 c^2+5 a^6 b c^2-2 a^5 b^2 c^2-14 a^4 b^3 c^2-18 a^3 b^4 c^2-2 a^2 b^5 c^2+16 a b^6 c^2+12 b^7 c^2+3 a^6 c^3-6 a^5 b c^3-14 a^4 b^2 c^3-26 a^3 b^3 c^3-22 a^2 b^4 c^3-2 a b^5 c^3+12 b^6 c^3-9 a^5 c^4-17 a^4 b c^4-18 a^3 b^2 c^4-22 a^2 b^3 c^4-26 a b^4 c^4-18 b^5 c^4-9 a^4 c^5+2 a^3 b c^5-2 a^2 b^2 c^5-2 a b^3 c^5-18 b^4 c^5+9 a^3 c^6+15 a^2 b c^6+16 a b^2 c^6+12 b^3 c^6+9 a^2 c^7+2 a b c^7+12 b^2 c^7-3 a c^8-3 b c^8-3 c^9 a: : (Peter Moses, August 10, 2014)
Barycentrics (3*b^2+2*b*c+3*c^2)*a^7+(b+c)*(3*b^2+2*b*c+3*c^2)*a^6-(9*b^4+9*c^4+2*b*c*(3*b^2+b*c+3*c^2))*a^5-(b+c)*(9*b^4+9*c^4+2*b*c*(4*b^2+3*b*c+4*c^2))*a^4+(9*b^6+9*c^6+2*(b^4+c^4-b*c*(9*b^2+13*b*c+9*c^2))*b*c)*a^3+(b^2-c^2)*(b-c)*(9*b^4+9*c^4+b*c*(24*b^2+31*b*c+24*c^2))*a^2-(b^2-c^2)^2*(b+c)^2*(3*b^2-8*b*c+3*c^2)*a-3*(b^2-c^2)^4*(b+c) : : (César Lozada, 5, 2022)

X(5947) is the Feuerbach-isogonal conjugate of X(5949); i.e., the hisogonal-conjugate-with-respect-to-Feuerbach-triangle of X(5949).

X(5947) lies on these lines: {3,31750}, {4,31764}, {5,5948}, {10,31756}, {11,10276}, {52,31754}, {119,10277}, {442,10209}, {946,31759}, {952,10281}, {3614,5953}, {5562,31765}, {14866,31761}, {38109,44847}

X(5948) = ORTHOCENTER OF FEUERBACH TRIANGLE

Barycentrics a^11 b^2+a^10 b^3-5 a^9 b^4-5 a^8 b^5+10 a^7 b^6+10 a^6 b^7-10 a^5 b^8-10 a^4 b^9+5 a^3 b^10+5 a^2 b^11-a b^12-b^13+2 a^11 b c+3 a^10 b^2 c-8 a^9 b^3 c-13 a^8 b^4 c+12 a^7 b^5 c+22 a^6 b^6 c-8 a^5 b^7 c-18 a^4 b^8 c+2 a^3 b^9 c+7 a^2 b^10 c-b^12 c+a^11 c^2+3 a^10 b c^2-2 a^9 b^2 c^2-14 a^8 b^3 c^2-5 a^7 b^4 c^2+17 a^6 b^5 c^2+18 a^5 b^6 c^2+2 a^4 b^7 c^2-18 a^3 b^8 c^2-14 a^2 b^9 c^2+6 a b^10 c^2+6 b^11 c^2+a^10 c^3-8 a^9 b c^3-14 a^8 b^2 c^3-6 a^7 b^3 c^3+11 a^6 b^4 c^3+22 a^5 b^5 c^3+18 a^4 b^6 c^3-8 a^3 b^7 c^3-22 a^2 b^8 c^3+6 b^10 c^3-5 a^9 c^4-13 a^8 b c^4-5 a^7 b^2 c^4+11 a^6 b^3 c^4+16 a^5 b^4 c^4+8 a^4 b^5 c^4+13 a^3 b^6 c^4+5 a^2 b^7 c^4-15 a b^8 c^4-15 b^9 c^4-5 a^8 c^5+12 a^7 b c^5+17 a^6 b^2 c^5+22 a^5 b^3 c^5+8 a^4 b^4 c^5+12 a^3 b^5 c^5+19 a^2 b^6 c^5-15 b^8 c^5+10 a^7 c^6+22 a^6 b c^6+18 a^5 b^2 c^6+18 a^4 b^3 c^6+13 a^3 b^4 c^6+19 a^2 b^5 c^6+20 a b^6 c^6+20 b^7 c^6+10 a^6 c^7-8 a^5 b c^7+2 a^4 b^2 c^7-8 a^3 b^3 c^7+5 a^2 b^4 c^7+20 b^6 c^7-10 a^5 c^8-18 a^4 b c^8-18 a^3 b^2 c^8-22 a^2 b^3 c^8-15 a b^4 c^8-15 b^5 c^8-10 a^4 c^9+2 a^3 b c^9-14 a^2 b^2 c^9-15 b^4 c^9+5 a^3 c^10+7 a^2 b c^10+6 a b^2 c^10+6 b^3 c^10+5 a^2 c^11+6 b^2 c^11-a c^12-b c^12-c^13 : : (Peter Moses, August 10, 2014)
Barycentrics (b+c)^2*a^11+(b+c)^3*a^10-(5*b^4+5*c^4+2*b*c*(4*b^2+b*c+4*c^2))*a^9-(5*b^2-2*b*c+5*c^2)*(b+c)^3*a^8+(10*b^6+10*c^6+(12*b^4+12*c^4-b*c*(5*b^2+6*b*c+5*c^2))*b*c)*a^7+(10*b^4+10*c^4-b*c*(8*b^2-11*b*c+8*c^2))*(b+c)^3*a^6-2*(5*b^8+5*c^8+(4*b^6+4*c^6-(9*b^4+9*c^4+b*c*(11*b^2+8*b*c+11*c^2))*b*c)*b*c)*a^5-2*(b^2-c^2)^2*(b+c)*(5*b^4+5*c^4+b*c*(4*b^2+5*b*c+4*c^2))*a^4+(b^2-c^2)^2*(5*b^6+5*c^6+2*(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*b*c)*a^3+(b^2-c^2)^3*(b-c)*(5*b^4+5*c^4+b*c*(12*b^2+13*b*c+12*c^2))*a^2-(b^2-c^2)^6*a-(b^2-c^2)^6*(b+c) : :

X(5948) is the Feuerbach-isogonal conjugate of X(5) and also the anticomplement of X(5) with respect to the Feuerbach triangle. (Randy Hutson, August 5, 2014)

X(5948) lies on these lines: {3,31764}, {4,31750}, {5,5947}, {10,31759}, {11,10277}, {12,79}, {30,10209}, {52,31765}, {119,5953}, {946,31756}, {5164,14132}, {5562,31754}, {10276,23513}, {12506,31761}

X(5949) = SYMMEDIAN POINT OF FEUERBACH TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)^2 (a^3+a^2 b-a b^2-b^3+a^2 c+3 a b c+b^2 c-a c^2+b c^2-c^3) (Peter Moses, August 10, 2014)

X(5949) is the Feuerbach-isogonal conjugate of X(5947).

X(5949) lies on these lines: {2,1029}, {5,572}, {6,2476}, {9,46}, {12,594}, {37,115}, {338,1441}, {1030,2475}, {1834,5725}, {2908,3136}, {3841,4047}

X(5950) = X(74) OF FEUERBACH TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+4 a^4 b c+a^3 b^2 c-2 a^2 b^3 c-2 a b^4 c-2 b^5 c-a^4 c^2+a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2-2 a^3 c^3-2 a^2 b c^3+a b^2 c^3+4 b^3 c^3+2 a^2 c^4-2 a b c^4+b^2 c^4+a c^5-2 b c^5-c^6) (a^6 b+2 a^5 b^2-a^4 b^3-4 a^3 b^4-a^2 b^5+2 a b^6+b^7+a^6 c-2 a^5 b c+a^3 b^3 c+a b^5 c-b^6 c+2 a^5 c^2+4 a^3 b^2 c^2+a^2 b^3 c^2-2 a b^4 c^2-3 b^5 c^2-a^4 c^3+a^3 b c^3+a^2 b^2 c^3-2 a b^3 c^3+3 b^4 c^3-4 a^3 c^4- 2 a b^2 c^4+3 b^3 c^4-a^2 c^5+a b c^5-3 b^2 c^5+2 a c^6-b c^6+c^7) (Peter Moses, August 10, 2014)

X(5950) lies on the nine-point circle and these lines: {2,5951}, {4,5606}, {5,5952}, {11,79}

X(5950) = reflection of X(5952) in X(5)
X(5950) = complement of X(5951)

X(5951) = CEVAPOINT OF X(35) AND X(484)

Trilinears 1/(2E² + 2F² - 4EF + 2DE + 2DF + 2D - E - F - 1), where D = cos A, E = cos B, F= cos C (Randy Hutson, August 17, 2014)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^6+2 a^5 b-a^4 b^2-4 a^3 b^3-a^2 b^4+2 a b^5+b^6-a^5 c+2 a^4 b c-a^3 b^2 c-a^2 b^3 c+2 a b^4 c-b^5 c-2 a^4 c^2+2 a^3 b c^2+2 a^2 b^2 c^2+2 a b^3 c^2-2 b^4 c^2+2 a^3 c^3-a^2 b c^3-a b^2 c^3+2 b^3 c^3+a^2 c^4-4 a b c^4+b^2 c^4-a c^5-b c^5) (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3+a^2 b^4-a b^5+2 a^5 c+2 a^4 b c+2 a^3 b^2 c-a^2 b^3 c-4 a b^4 c-b^5 c-a^4 c^2-a^3 b c^2+2 a^2 b^2 c^2-a b^3 c^2+b^4 c^2-4 a^3 c^3-a^2 b c^3+2 a b^2 c^3+2 b^3 c^3-a^2 c^4+2 a b c^4-2 b^2 c^4+2 a c^5-b c^5+c^6) (Peter Moses, August 10, 2014)

X(5951) lies on the circumcircle and these lines: {2,5950}, {3,5606}, {4,5952}, {100,3648}, {110,3579}

X(5951) = reflection of X(5606) in X(3)
X(5951) = anticomplement of X(5950)
X(5951) = cevapoint of X(35) and X(484)

X(5952) = X(110) OF FEUERBACH TRIANGLE

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

X(5952) lies on the nine-point circle and these lines: {2,5606}, {4,5951}, {5,5950}, {119,3652}

X(5952) = reflection of X(5950) in X(5)
X(5952) = complement of X(5606)

X(5953) = FEUERBACH ISOGONAL CONJUGATE OF X(12)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 b^2+2 a^7 b^3-2 a^6 b^4-6 a^5 b^5+6 a^3 b^7+2 a^2 b^8-2 a b^9-b^10-2 a^8 b c+2 a^7 b^2 c+6 a^6 b^3 c-8 a^5 b^4 c-12 a^4 b^5 c+6 a^3 b^6 c+10 a^2 b^7 c-2 b^9 c+a^8 c^2+2 a^7 b c^2+8 a^6 b^2 c^2-2 a^5 b^3 c^2-19 a^4 b^4 c^2-10 a^3 b^5 c^2+7 a^2 b^6 c^2+10 a b^7 c^2+3 b^8 c^2+2 a^7 c^3+6 a^6 b c^3-2 a^5 b^2 c^3-18 a^4 b^3 c^3-22 a^3 b^4 c^3-10 a^2 b^5 c^3+6 a b^6 c^3+8 b^7 c^3-2 a^6 c^4-8 a^5 b c^4-19 a^4 b^2 c^4-22 a^3 b^3 c^4-18 a^2 b^4 c^4-14 a b^5 c^4-2 b^6 c^4-6 a^5 c^5-12 a^4 b c^5-10 a^3 b^2 c^5-10 a^2 b^3 c^5-14 a b^4 c^5-12 b^5 c^5+6 a^3 b c^6+7 a^2 b^2 c^6+6 a b^3 c^6-2 b^4 c^6+6 a^3 c^7+10 a^2 b c^7+10 a b^2 c^7+8 b^3 c^7+2 a^2 c^8+3 b^2 c^8-2 a c^9-2 b c^9-c^10 (Peter Moses, August 10, 2014)

Let A′B′C′ be the Feuerbach triangle. Let A″ be the isogonal conjugate of A with respect to A′B′C′, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5953). (Randy Hutson, August 5, 2014)

X(5953) lies on these lines: {5,191}, {119,5948}, {3614,5947}

X(5954) = 3rd HUTSON-FEUERBACH POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c)^2 (-a^6+a^5 b+a^4 b^2-a^2 b^4-a b^5+b^6+a^5 c-2 a^2 b^3 c+b^5 c+a^4 c^2-3 a^2 b^2 c^2+4 a b^3 c^2-b^4 c^2-2 a^2 b c^3+4 a b^2 c^3-2 b^3 c^3-a^2 c^4-b^2 c^4-a c^5+b c^5+c^6) (Peter Moses, August 10, 2014)

Let A′B′C′ be the Feuerbach triangle, L the line through A and X(11), and A″ = L∩B′C′; and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(5954). See also X(3614) and X(3615). (Randy Hutson, August 5, 2014)

Continuing, let A* be the trilinear pole, with respect to A′B′C′, of line BC, and define B* and C* cyclically. Let A** be the trilinear pole, with respect to A′B′C′, of line B*C*, and define B** and C** cyclically. The lines A′A**, B′B**, C′C** concur in X(5954). (Randy Hutson, August 5, 2014)

X(5954) lies on these lines: {5,2607}, {11,523}, {12,59}

X(5955) = CENTER OF INNER HUNG CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4+a^3 b+a b^3+b^4+a^3 c+2 a^2 b c+5 a b^2 c+2 b^3 c+5 a b c^2+2 b^2 c^2+a c^3+2 b c^3+c^4
X(5955) = r²*X(3) + (r² - s²)X(10) (barycentrics and combo, Peter Moses, August 9, 2014)

In the configuration for X(5213), the circle internally tangent to the circles (K_A), (K_B), (K_C) is here named the inner Hung circle.

If you have The Geometer's Sketchpad, you can view X(5955).

X(5955) lies on these lines: {2,3702}, {3,10}, {46,1211}, {171,5814}, {899,1245}, {975,3704}, {997,5835}, {1010,5725}, {1574,2092}, {1698,3712}, {2049,5530}, {3454,5880}, {3687,5711}, {3695,5268}, {3696,5292}, {3927,4104}, {3966,5264}

X(5956) = CENTER OF OUTER HUNG CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^4 b^2+3 a^3 b^3+3 a^2 b^4+a b^5+3 a^3 b^2 c+3 a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+3 a^3 b c^2-2 a^2 b^2 c^2-8 a b^3 c^2-4 b^4 c^2+3 a^3 c^3+3 a^2 b c^3-8 a b^2 c^3-6 b^3 c^3+3 a^2 c^4-a b c^4-4 b^2 c^4+a c^5-b c^5) (Peter Moses, August 9, 2014)

In the configuration for X(5213), the circle externally tangent to the circles (K_A), (K_B), (K_C) is here named the outer Hung circle; see X(5955) and X(5213).

If you have The Geometer's Sketchpad, you can view X(5955), which includes X(5956).

X(5956) lies on these lines: {2,3702}, {171,1203}, {386,1100}, {404,593}, {970,5213}, {1575,5044}

X(5957) = FEUERBACH ISOGONAL CONJUGATE OF X(11)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-b+c) (2 a^4-a^3 b-3 a^2 b^2+a b^3+b^4-a^3 c-4 a^2 b c-3 a^2 c^2-2 b^2 c^2+a c^3+c^4) (Peter Moses, August 10, 2014)

For I = 5957, 5958, 5959, 5948, 5949, 5953, X(i) is the isogonal conjugate of a point with respect to the Feuerbach triangle, called a Feuerbach isogonal conjugate.

Follwing is the first barycentric of the Feuerbach isogonal conjugate of an arbitrary point p : q : r. This lengthy function of a,b,c,p,q,r (degree 11 in a,b,c and degree 2 in p,q,r) can be cut-and-pasted into Mathematica. (Peter Moses, August 10, 2014)

(a^9 b^2+a^8 b^3-4 a^7 b^4-4 a^6 b^5+6 a^5 b^6+6 a^4 b^7-4 a^3 b^8-4 a^2 b^9+a b^10+b^11+2 a^9 b c+3 a^8 b^2 c-6 a^7 b^3 c-12 a^6 b^4 c+4 a^5 b^5 c+16 a^4 b^6 c+2 a^3 b^7 c-8 a^2 b^8 c-2 a b^9 c+b^10 c+a^9 c^2+3 a^8 b c^2-8 a^7 b^2 c^2-16 a^6 b^3 c^2+3 a^5 b^4 c^2+15 a^4 b^5 c^2+9 a^3 b^6 c^2+a^2 b^7 c^2-5 a b^8 c^2-3 b^9 c^2+a^8 c^3-6 a^7 b c^3-16 a^6 b^2 c^3+2 a^5 b^3 c^3+23 a^4 b^4 c^3+14 a^3 b^5 c^3+5 a^2 b^6 c^3-3 b^8 c^3-4 a^7 c^4-12 a^6 b c^4+3 a^5 b^2 c^4+23 a^4 b^3 c^4+18 a^3 b^4 c^4+6 a^2 b^5 c^4+4 a b^6 c^4+2 b^7 c^4-4 a^6 c^5+4 a^5 b c^5+15 a^4 b^2 c^5+14 a^3 b^3 c^5+6 a^2 b^4 c^5+4 a b^5 c^5+2 b^6 c^5+6 a^5 c^6+16 a^4 b c^6+9 a^3 b^2 c^6+5 a^2 b^3 c^6+4 a b^4 c^6+2 b^5 c^6+6 a^4 c^7+2 a^3 b c^7+a^2 b^2 c^7+2 b^4 c^7-4 a^3 c^8-8 a^2 b c^8-5 a b^2 c^8-3 b^3 c^8-4 a^2 c^9-2 a b c^9-3 b^2 c^9+a c^10+b c^10+c^11) p^2+2 c (2 a^8 b^2+a^7 b^3-6 a^6 b^4-3 a^5 b^5+6 a^4 b^6+3 a^3 b^7-2 a^2 b^8-a b^9-2 a^8 b c+a^7 b^2 c+a^6 b^3 c-8 a^5 b^4 c-2 a^4 b^5 c+8 a^3 b^6 c+4 a^2 b^7 c-a b^8 c-b^9 c+a^7 b c^2+5 a^6 b^2 c^2-3 a^5 b^3 c^2-12 a^4 b^4 c^2-a^3 b^5 c^2+8 a^2 b^6 c^2+3 a b^7 c^2-b^8 c^2+a^7 c^3+7 a^6 b c^3+3 a^5 b^2 c^3-9 a^4 b^3 c^3-11 a^3 b^4 c^3-a^2 b^5 c^3+6 a b^6 c^3+4 b^7 c^3+a^6 c^4-2 a^5 b c^4-9 a^4 b^2 c^4-11 a^3 b^3 c^4-7 a^2 b^4 c^4-a b^5 c^4+4 b^6 c^4-3 a^5 c^5-11 a^4 b c^5-10 a^3 b^2 c^5-10 a^2 b^3 c^5-10 a b^4 c^5-6 b^5 c^5-3 a^4 c^6-a^3 b c^6-2 a^2 b^2 c^6-3 a b^3 c^6-6 b^4 c^6+3 a^3 c^7+7 a^2 b c^7+6 a b^2 c^7+4 b^3 c^7+3 a^2 c^8+2 a b c^8+4 b^2 c^8-a c^9-b c^9-c^10) p q-(b-c) (a+b+c) (a^8 b-4 a^6 b^3+6 a^4 b^5-4 a^2 b^7+b^9-a^8 c+4 a^4 b^4 c-4 a^2 b^6 c+b^8 c+2 a^6 b c^2-4 a^5 b^2 c^2-a^4 b^3 c^2+6 a^3 b^4 c^2+3 a^2 b^5 c^2-2 a b^6 c^2-4 b^7 c^2+2 a^6 c^3-4 a^5 b c^3+3 a^4 b^2 c^3+10 a^3 b^3 c^3+5 a^2 b^4 c^3-2 a b^5 c^3-4 b^6 c^3+4 a^4 b c^4+6 a^3 b^2 c^4+5 a^2 b^3 c^4+4 a b^4 c^4+6 b^5 c^4+2 a^3 b c^5+a^2 b^2 c^5+4 a b^3 c^5+6 b^4 c^5-4 a^2 b c^6-2 a b^2 c^6-4 b^3 c^6-2 a^2 c^7-2 a b c^7-4 b^2 c^7+b c^8+c^9) q^2-2 b (-a^7 b^3-a^6 b^4+3 a^5 b^5+3 a^4 b^6-3 a^3 b^7-3 a^2 b^8+a b^9+b^10+2 a^8 b c-a^7 b^2 c-7 a^6 b^3 c+2 a^5 b^4 c+11 a^4 b^5 c+a^3 b^6 c-7 a^2 b^7 c-2 a b^8 c+b^9 c-2 a^8 c^2-a^7 b c^2-5 a^6 b^2 c^2-3 a^5 b^3 c^2+9 a^4 b^4 c^2+10 a^3 b^5 c^2+2 a^2 b^6 c^2-6 a b^7 c^2-4 b^8 c^2-a^7 c^3-a^6 b c^3+3 a^5 b^2 c^3+9 a^4 b^3 c^3+11 a^3 b^4 c^3+10 a^2 b^5 c^3+3 a b^6 c^3-4 b^7 c^3+6 a^6 c^4+8 a^5 b c^4+12 a^4 b^2 c^4+11 a^3 b^3 c^4+7 a^2 b^4 c^4+10 a b^5 c^4+6 b^6 c^4+3 a^5 c^5+2 a^4 b c^5+a^3 b^2 c^5+a^2 b^3 c^5+a b^4 c^5+6 b^5 c^5-6 a^4 c^6-8 a^3 b c^6-8 a^2 b^2 c^6-6 a b^3 c^6-4 b^4 c^6-3 a^3 c^7-4 a^2 b c^7-3 a b^2 c^7-4 b^3 c^7+2 a^2 c^8+a b c^8+b^2 c^8+a c^9+b c^9) p r+2 a (a-b) (a-c) (b-c)^2 (a+b+c)^2 (a^4-2 a^2 b^2+b^4-3 a^2 b c+a b^2 c-2 a^2 c^2+a b c^2-2 b^2 c^2+c^4) q r-(b-c) (a+b+c) (a^8 b-2 a^6 b^3+2 a^2 b^7-b^9-a^8 c-2 a^6 b^2 c+4 a^5 b^3 c-4 a^4 b^4 c-2 a^3 b^5 c+4 a^2 b^6 c+2 a b^7 c-b^8 c+4 a^5 b^2 c^2-3 a^4 b^3 c^2-6 a^3 b^4 c^2-a^2 b^5 c^2+2 a b^6 c^2+4 b^7 c^2+4 a^6 c^3+a^4 b^2 c^3-10 a^3 b^3 c^3-5 a^2 b^4 c^3-4 a b^5 c^3+4 b^6 c^3-4 a^4 b c^4-6 a^3 b^2 c^4-5 a^2 b^3 c^4-4 a b^4 c^4-6 b^5 c^4-6 a^4 c^5-3 a^2 b^2 c^5+2 a b^3 c^5-6 b^4 c^5+4 a^2 b c^6+2 a b^2 c^6+4 b^3 c^6+4 a^2 c^7+4 b^2 c^7-b c^8-c^9) r^2

X(5957) lies on these lines: {30, 511}, {901, 20189}, {953, 13597}, {3259, 11792}

X(5958) = FEUERBACH ISOGONAL CONJUGATE OF X(115)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (b-c) (a^5+a^4 b-2 a^3 b^2-2 a^2 b^3+a b^4+b^5+a^4 c-2 a^3 b c-4 a^2 b^2 c-a b^3 c+b^4 c-2 a^3 c^2-4 a^2 b c^2-3 a b^2 c^2-b^3 c^2-2 a^2 c^3-a b c^3-b^2 c^3+a c^4+b c^4+c^5) (Peter Moses, August 10, 2014)

X(5958) lies on these lines: {30, 511}, {2699, 13597}, {2703, 20189}, {11246, 15544}, {11792, 46671}

X(5959) = FEUERBACH ISOGONAL CONJUGATE OF X(125)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (-a^7+2 a^5 b^2-a^3 b^4+2 a^5 b c-3 a^3 b^3 c-a^2 b^4 c+a b^5 c+b^6 c+2 a^5 c^2-5 a^3 b^2 c^2-3 a^2 b^3 c^2+b^5 c^2-3 a^3 b c^3-3 a^2 b^2 c^3-2 a b^3 c^3-2 b^4 c^3-a^3 c^4-a^2 b c^4-2 b^3 c^4+a b c^5+b^2 c^5+b c^6) (Peter Moses, August 10, 2014)

X(5959) lies on tthese lines: {30, 511}, {1290, 20189}, {2605, 11553}, {2687, 13597}, {5520, 11792}

X(5960) = X(1) OF THE FEUERBACH TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)(-(b-c)^2 (a+b+c) Sqrt[(a^3-a^2 b-a b^2+b^3+a^2 c-3 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c-3 a b c-b^2 c-a c^2+b c^2+c^3)]+ (a+b) (a-b+c) (b+c) Sqrt[(-a^3-a^2 b+a b^2+b^3-a^2 c-3 a b c-b^2 c+a c^2-b c^2+c^3) (a^3+a^2 b-a b^2-b^3-a^2 c-3 a b c-b^2 c-a c^2+b c^2+c^3)]+(a+b-c) (a+c) (b+c) Sqrt[(a^3+a^2 b-a b^2-b^3+a^2 c+3 a b c+b^2 c-a c^2+b c^2-c^3) (-a^3+a^2 b+a b^2-b^3-a^2 c+3 a b c-b^2 c+a c^2+b c^2+c^3)]) (Peter Moses, August 10, 2014)

X(5961) = INVERSE-IN-CIRCUMCIRCLE OF X(265)

Trilinears (sin 4A)/sin(3A) : :
Tripolars a^2((a^2 - b^2 - c^2)^2 - b^2 c^2) : :

X(5961) is the Hofstadter 4 point and the antigonal image of X(5964), in accord with the following conjecture and corollary: If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point; thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1, then the antigonal image of the Hofstadter r point is the Hofstadter -r point. (Randy Hutson, August 10, 2014)

The Hofstadter r point is defined at X(359), where further examples are given.

Let A′B′C′ be the Kosnita triangle. The circumcircles of A′BC, B′CA, C′AB concur in X(5961). (Randy Hutson, August 10, 2014)

X(5961) lies on these lines: {3,125}, {24,136}, {94,96}, {186,476}, {1989,2079}

X(5961) = isogonal conjugate of X(5962)
X(5961) = complement of X(39118)
X(5961) = circumcircle-inverse of X(265)

X(5962) = ANTIGONAL IMAGE OF X(186)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 3A)/sin(4A)

X(5962) is the Hofstadter -3 point; see the conjecture at X(5961).

X(5962) lies on these lines: {4,52}, {30,925}, {96,275}, {128,186}, {136,539}, {250,403}, {378,2351}, {485,1322}, {486,1321}, {2165,3087}

X(5962) = isogonal conjugate of X(5961)
X(5962) = inverse-in-circumcircle of X(5963)
X(5962) = inverse-in-polar-circle of X(52)

X(5963) = HOFSTADTER 5 POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 5A)/sin(4A)

X(5963) is the antigonal image of the Hofstadter -5 point; see the conjecture at X(5961).

X(5963) lies on these lines: {3,5962}, {186,847}, {925,1658}

X(5963) = isogonal conjugate of X(5964)
X(5963) = inverse-in-circumcircle of X(5962)

X(5964) = HOFSTADTER -4 POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin 4A)/sin(5A)

X(5964) is the antigonal image of the Hofstadter 4 point, X(5961), and the inverse-in-circumcircle of the Hofstadter 6 point; see the conjecture at X(5961).

X(5964) lies on these lines: {4, 14104}, {9927, 18403}

X(5964) = isogonal conjugate of X(5963)

X(5965) = NAPOLEON INFINITY POINT

Barycentrics 2a^6 - b^6 - c^6 - 4a^4b^2 - 4a^4c^2 + 3a^2b^4 + 3a^2c^4 + b^4c^2 + b^2c^4 : :

The Napoleon line is the line X(6)X(17), which passes through the Napoleon points, X(17) and X(18). X(5965) is the point in which the Napoleon line meets the line X(30)X(511) at infinity.

X(5965) lies on these lines: {5,3629}, {6,17}, {30,511}, {54,69}, {114,385}, {115,5111}, {125,323}, {140,3631}, {141,575}, {193,576}, {381,5102}, {599,5050}, {1204,3098}, {1351,3818}, {1691,5477}, {1992,5071}, {2914,5095}, {2930,5898}, {3180,5617}, {3181,5613}, {3292,3580}, {3630,5092}, {3858,5480}, {5093,5476}

X(5965) = isogonal conjugate of X(5966)

X(5966) = ISOGONAL CONJUGATE OF NAPOLEON INFINITY POINT

Barycentrics a^2/(2a^6 - b^6 - c^6 - 4a^4b^2 - 4a^4c^2 + 3a^2b^4 + 3a^2c^4 + b^4c^2 + b^2c^4) : :

Let O* denote the inverter of the circumcircle and nine-point circle, as defined at X(5577). X(5966) is the inverse-in-O* of X(137). (Randy Hutson, August 13, 2014)

X(5966) lies on the circumcircle, the hyperbola {{A,B,C,X(2),X(61),X(62)}}, and these lines: {2,137}, {5,99}, {23,1291}, {25,933}, {51,110}, {112,3199}, {691,2070}, {1141,2413}

X(5966) = isogonal conjugate of X(5965)
X(5966) = anticomplement of X(31843)
X(5966) = Thomson-isogonal conjugate of X(32478)
X(5966) = Lucas-isogonal conjugate of X(32478)

X(5967) = INTERSECTION OF LINES X(2)X(98) AND X(6)X(523)

Barycentrics (b² + c² - 2a²)/(b⁴ + c⁴ - a²b² - 2a²c²) : :

X(5967) lies on these lines: {2,98}, {4,685}, {6,523}, {23,5968}, {69,4590}, {248,5063}, {263,2698}, {468,1648}, {511,4226}, {524,5467}, {868,1503}, {1641,2434}, {1992,2966}, {2715,2770}, {3266,3292}

X(5967) = isogonal conjugate of X(5968)
X(5967) = perspector of unary cofactor triangles of 2nd and 3rd Parry triangles

X(5968) = INTERSECTION OF LINES X(2)X(523) AND X(6)X(110)

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

Let P denote the antipode of X(23) in the Parry circle. Then X(5968) is the vertex conjugate of X(23) and P, and X(5968) is the perspector of the circumcevian triangle of X(23) and the circumcevian triangle of P. (Randy Hutson, August 13, 2014)

X(5968) lies on these lines: {2,523}, {3,691}, {6,110}, {22,3447}, {23,5967}, {25,250}, {183,892}, {232,4230}, {262,381}, {264,2970}, {325,868}, {511,2421}, {956,5380}, {2065,5050}, {3613,5169}

X(5968) = isogonal conjugate of X(5967)
X(5968) = trilinear pole of the line X(511)X(3569)

X(5969) = INTERSECTION OF LINES X(2)X(694) AND X(6)X(99)

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

X(5969) lies on these lines: {2,694}, {6,99}, {30,511}, {39,597}, {69,148}, {76,338}, {98,1350}, {111,5108}, {114,5480}, {115,141}, {194,1992}, {262,5503}, {385,5104}, {620,3589}, {1469,3027}, {1569,5052}, {1843,5186}, {2076,5152}, {2502,5468}, {3023,3056}, {3029,4260}, {3104,5463}, {3105,5464}, {3629,5477}, {3934,5461}, {4048,5028}

X(5969) = isogonal conjugate of X(5970)
X(5969) = isotomic conjugate of X(35146)
X(5969) = X(2)-Ceva conjugate of X(35077)

X(5970) = INTERSECTION OF LINES X(6)X(805) AND X(99)X(187)

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

X(5970) is the singular focus of the cubic K688. (Bernard Gibert, August 17, 2014)

X(5970) lies on the circumcircle and these lines: {6,805}, {32,691}, {99,187}, {110,1691}, {111,669}, {182,2709}, {512,729}, {1296,2080}, {4027,4590}

X(5970) = isogonal conjugate of X(5969)
X(5970) = Ψ(X(2), X(351))
X(5970) = Λ(X(76), X(338))
X(5970) = barycentric product of circumcircle intercepts of line X(2)X(351)
X(5970) = inverse-in-O(15,16) of X(99), where O(15,16) is the circle having segment X(15)X(16) as diameter
X(5970) = inverse in O* of X(110), where O* is the circle {{X(1687, X(1688), PU(1), PU(2)}}
X(5970) is the point, other than X(99), in which the circumcircle meets the circle {{X(3), X(6), X(99)}}

X(5971) = INVERSE IN {CIRCUMCIRCLE, NINE-POINT CIRCLE}-INVERTER OF X(141)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁶ - a²b⁴ - a²c⁴ + 5a²b²c² - 2b⁴c² - 2b²c⁴

See X(5577) for the definition of inverter.

X(5971) lies on these lines: {2,6}, {23,99}, {111,538}, {126,754}, {675,2759}, {1003,1383}, {2071,2373}

X(5971) = anticomplement of X(5913)
X(5971) = X(111)-of-1st-anti-Brocard-triangle
X(5971) = intersection, other than X(23), of the line X(2)X(6) and circle {{X(2),X(98),X(99)}}

X(5972) = INVERSE IN {CIRCUMCIRCLE, NINE-POINT CIRCLE}-INVERTER OF X(147)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a⁶ + b⁶ + c⁶ - 2a⁴b² - 2a⁴c² - a²b⁴ - a²c⁴ + 4a²b²c² - b⁴c² - b²c⁴
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a² + c² - b²)(c² - a²)² + (a² + b² - c²)(a² - b²)²
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (sin 2B)sin²(A - C) + (sin 2C)sin²(A - B)
X(5972) = 3X(2) + X(110)      (barycentrics g, h and combo, Randy Hutson, August 20, 2014)

Let (A′) be the circle centered at A and tangent to the Euler line, and define (B′) and (C′) cyclically; then X(5972) is the radical center of (A′), (B′), (C′). Let D = X(110), let Q be the cyclic quadrilateral ABCD, of which the centroid is X(5972). Let Q′ be the quadrilateral formed by the centroids of the triangles ABC, BCD, CAD, ABD; then Q and Q′ are homothetic at X(5972), which is the {X(2),X(110)}-harmonic conjugate of X(125) and also the inverse-in-Thomson-Gibert-Moses-hyperbola of X(125). Also, X(5972) = X(125)-of-medial-triangle = X(468)-of-1st-Brocard-triangle. (Randy Hutson, August 20, 2014)

The X(2)-Ceva conjugate of P, as P traces the Brocard axis, is a hyperbola, H, with center X(5972); H is the Jerabek hyperbola of the medial triangle, and it passes through X(i) for i = 3,5,6,113, 141, 206, 942, 960, 1147, 1209, 2883. H is also the bicevian conic of X(2) and X(110). (Randy Hutson, August 20, 2014)

X(5972) lies on these lines: {2,98}, {3,113}, {5,1511}, {6,5181}, {67,3763}, {69,5095}, {74,631}, {140,5663}, {146,3523}, {186,1568}, {247,1316}, {265,1656}, {399,3526}, {468,511}, {541,549}, {550,1539}, {578,5504}, {620,690}, {632,5609}, {858,1495}, {895,3618}, {1365,2607}, {1503,5159}, {1986,5562}, {2482,5465}, {2606,4092}, {2781,3819}, {2836,3848}, {2854,3589}, {2948,3624}, {3024,5432}, {3028,5433}, {3066,5476}, {3154,3233}, {3292,3580}, {3818,5094}, {5054,5646}, {5544,5648}

X(5972) = midpoint of X(i) and X(j) for these {i,j}: {2,5642}, {3,113}, {5,1511}, {6,5181}, {110, 125}
X(5972) = complement of X(125)
X(5972) = inverse-in-{circumcircle, nine-point circle}-inverter of X(147)
X(5972) = X(186)-of-X(5)-Brocard-triangle

X(5973) = PERSPECTOR OF THE HUNG-FEUERBACH TRIANGLE AND THE FEUERBACH TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)^2 (-a^14 b-3 a^13 b^2-2 a^12 b^3+2 a^11 b^4+5 a^10 b^5+7 a^9 b^6+4 a^8 b^7-4 a^7 b^8-7 a^6 b^9-5 a^5 b^10-2 a^4 b^11+2 a^3 b^12+3 a^2 b^13+a b^14-a^14 c-2 a^13 b c-8 a^12 b^2 c-16 a^11 b^3 c-8 a^10 b^4 c+4 a^9 b^5 c+9 a^8 b^6 c+16 a^7 b^7 c+11 a^6 b^8 c-2 a^5 b^9 c-2 a^4 b^10 c-2 a^2 b^12 c+b^14 c-3 a^13 c^2-8 a^12 b c^2-8 a^11 b^2 c^2-14 a^10 b^3 c^2-8 a^9 b^4 c^2+25 a^8 b^5 c^2+20 a^7 b^6 c^2-8 a^6 b^7 c^2+a^5 b^8 c^2+6 a^4 b^9 c^2-4 a^3 b^10 c^2-2 a^2 b^11 c^2+2 a b^12 c^2+b^13 c^2-2 a^12 c^3-16 a^11 b c^3-14 a^10 b^2 c^3+2 a^9 b^3 c^3-2 a^8 b^4 c^3+4 a^7 b^5 c^3+36 a^6 b^6 c^3+16 a^5 b^7 c^3-18 a^4 b^8 c^3-4 a^3 b^9 c^3+2 a^2 b^10 c^3-2 a b^11 c^3-2 b^12 c^3+2 a^11 c^4-8 a^10 b c^4-8 a^9 b^2 c^4-2 a^8 b^3 c^4+28 a^7 b^4 c^4-16 a^5 b^6 c^4+20 a^4 b^7 c^4+2 a^3 b^8 c^4-8 a^2 b^9 c^4-8 a b^10 c^4-2 b^11 c^4+5 a^10 c^5+4 a^9 b c^5+25 a^8 b^2 c^5+4 a^7 b^3 c^5+28 a^5 b^5 c^5-4 a^4 b^6 c^5-12 a^3 b^7 c^5-a^2 b^8 c^5-b^10 c^5+7 a^9 c^6+9 a^8 b c^6+20 a^7 b^2 c^6+36 a^6 b^3 c^6-16 a^5 b^4 c^6-4 a^4 b^5 c^6+32 a^3 b^6 c^6+8 a^2 b^7 c^6+5 a b^8 c^6-b^9 c^6+4 a^8 c^7+16 a^7 b c^7-8 a^6 b^2 c^7+16 a^5 b^3 c^7+20 a^4 b^4 c^7-12 a^3 b^5 c^7+8 a^2 b^6 c^7+4 a b^7 c^7+4 b^8 c^7-4 a^7 c^8+11 a^6 b c^8+a^5 b^2 c^8-18 a^4 b^3 c^8+2 a^3 b^4 c^8-a^2 b^5 c^8+5 a b^6 c^8+4 b^7 c^8-7 a^6 c^9-2 a^5 b c^9+6 a^4 b^2 c^9-4 a^3 b^3 c^9-8 a^2 b^4 c^9-b^6 c^9-5 a^5 c^10-2 a^4 b c^10-4 a^3 b^2 c^10+2 a^2 b^3 c^10-8 a b^4 c^10-b^5 c^10-2 a^4 c^11-2 a^2 b^2 c^11-2 a b^3 c^11-2 b^4 c^11+2 a^3 c^12-2 a^2 b c^12+2 a b^2 c^12-2 b^3 c^12+3 a^2 c^13+b^2 c^13+a c^14+b c^14) (Peter Moses, August 19, 2014)

Let F_AF_BF_C be the Feuerbach triangle of a triangle ABC with excircles (I_A), (I_B), (I_C). Let (K_A) be the circle, other than the nine-point circle, which passes through F_B and F_C and is tangent to (I_A), and let D be the touchpoint. Define (K_B, (K_C) and E, F cyclically. Let T_A be the line through D tangent to (I_A), and define T_B and T_C cyclically. The lines T_A, T_B, T_C form a triangle [here named the Hung-Feuerbach triangle] that is perspective to F_AF_BF_C, and X(5973) is the perspector. (Tran Quang Hung, ADGEOM, August 19, 2014) See also X(5974) and X(5975).

X(5974) = CENTER OF THE HUNG-FEUERBACH CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^6+2 a^5 b+2 a^4 b^2+2 a^3 b^3-2 a b^5-b^6+2 a^5 c+6 a^4 b c+8 a^3 b^2 c-6 a b^4 c-2 b^5 c+2 a^4 c^2+8 a^3 b c^2+a^2 b^2 c^2-8 a b^3 c^2-4 b^4 c^2+2 a^3 c^3-8 a b^2 c^3-6 b^3 c^3-6 a b c^4-4 b^2 c^4-2 a c^5-2 b c^5-c^6) (Peter Moses, August 19, 2014)

Continuing from X(5973), the Hung-Feuerbach circle is here defined as the circle (K) tangent to each of the following 6 circles: (K_A, (K_B, (K_C), (I_A), (I_B), (I_C), so that (K) is also tangent to the Apollonius circle. (Tran Quang Hung and Peter Moses, ADGEOM, August 19, 2014)

X(5974) lies on these lines: {72,171}, {191,1045}, {846,3931}, {970,5975}, {1054,5956}, {1490,2629}, {2959,5687}, {3579,5524}}

X(5975) = TOUCHPOINT OF HUNG-FEUERBACH CIRCLE AND APOLLONIUS CIRCLE

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

The Hung-Feuerbach circle is defined at X(5974).

X(5975) is the inverse-in-Speiker-radical-circle of X(5993). (Peter Moses, August 22, 2014)

If you have The Geometer's Sketchpad, you can view X(5975).

X(5975) lies on the Apollonius circle and this line: {970,5974}

X(5976) = X(39)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a⁴ - b²c²)(b⁴ + c⁴ - a²b² - a²c²) (Peter Moses, August 21, 2014)

Barycentrics g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin B csc(B - 2ω) + sin C csc(C - 2ω) (Randy Hutson, August 22, 2014)

The 1st anti-Brocard triangle is defined at X(5939). Suppose that P = p : q : r (barycentrics) is a point. Then P-of-1st-anti-Brocard-triangle is the point

T(P) = (a⁴ - b²c²)p + (c⁴ - a²b²)q + (b⁴ - a²c²)r : (b⁴ - c²a²)q + (a⁴ - b²c²)r + (c⁴ - b²a²)p : (c⁴ - a²b²)r + (b⁴ - c²a²)p + (a⁴ - c²b²)q

The inverse mapping T*(P), satisfying T*(T(P)) = T(T*(P)) = P, is the given by

T*(P) = a²p + c²q + b²r : c²p + b²q + a²r : b²p + a²q + c²r

The formulas for T and T*, and the following list, were contributed by Peter Moses, August 21, 2014. The appearance of (i,j) in the list means that X(i)-of-1st-anti-Brocard-triangle = X(j); i.e., X(j) = T(X(i)):

(1,1281), (2,2), (3,98), (4,147), (5,114), (6,99), (30,542), (69,148), (76,1916), (99,385), (110,23), (111,5971), (114,1513), (115,325), (125,858), (126,5913), (141,115), (182,3), (184,22), (187,5939), (287,401), (351,4108), (384,4027), (511,2782), (512,804), (513,2787), (514,2786), (515,2792), (516,2784), (517,2783), (518,2795), (519,2796), (520,2797), (521,2798), (522,2785), (523,690), (524,543), (525,2799), (530,531), (531,530), (538,5969), (542,30), (543,524), (574,183), (597,2482), (599,671), (620,230), (690,523), (694,3978), (804,512), (846,3757), (1054,5205), (1083,105), (1316,110), (1352,4), (1499,2793), (1503,2794), (1640,3268), (1691,5152), (1899,1370), (2549,69), (2782,511), (2783,517), (2784,516), (2785,522), (2786,514), (2787,513), (2788,3309), (2789,3667), (2792,515), (2793,1499), (2794,1503), (2795,518), (2796,519), (2797,520), (2798,521), (2799,525), (3094,76), (3120,3006), (3124,3266), (3125,3263), (3309,2788), (3413,3413), (3414,3414), (3448,5189), (3569,850), (3589,620), (3642,13), (3643,14), (3667,2789), (3734,6), (3735,75), (3821,10), (3923,1), (3934,2023), (3944,3705), (3980,612), (3981,305), (4011,614), (4045,141), (4048,32), (4074,1194), (4107,649), (4112,31), (4154,3747), (4159,1501), (4164,667), (4172,560), (4418,3920), (5026,187), (5027,669), (5028,1975), (5091,100), (5108,111), (5116,1078), (5149,1691), (5150,36), (5613,383), (5617,1080), (5622,2071), (5651,1995), (5921,3146), (5967,4226), (5969,538), (5972,468)

Let H be the hyperbola {{A,B,C,PU(37)}}, which is the isogonal conjugate of the line PU(39), this being the line X(32)X(512); also H is the isotomic conjugate of the line PU(45), which is the line X(6)X(523). This hyperbola passes through the points X(76), X(99), X(877), and X(2396). The center of H is X(5976). Let R_A be the radical axis of the nine-point circle of triangle BCP(1) and the nine-point circle of BCU(1), and define R_B and R_C cyclically. The lines R_A, R_B, R_C concur in X(5976). (Randy Hutson, August 22, 2014; see Hyacinthos #21938-21940, 4/12/2013)

Let L be the Simson line of X(99), and let L′ the line normal to the circumcircle at X(99). Then X(5976) = L∩L′. (L = X(114)X(325) and L′ = X(3)X(76).) (Randy Hutson, December 4, 2014)

X(5976) lies on the bicevian conic of X(2) and X(99) and on these lines: {2,694}, {3,76}, {32,5149}, {39,620}, {69,147}, {114,325}, {115,3934}, {187,736}, {230,698}, {315,6033}, {350,1281}, {385,732}, {538,1569}, {618,6109}, {619,6108}, {641,3102}, {642,3103}, {877,2967}, {1125,5977}, {1649,3268}, {1909,3023}, {2491,2799}, {2794,5188}, {4357,5988}

X(5976) = midpoint of X(76) and X(88)
X(5976) = reflection of X(i) in X(j) for these (i,j): (39,620), (115,3934), (1916,2023)
X(5976) = isogonal conjugate of X(34238)
X(5976) = isotomic conjugate of X(36897)
X(5976) = isotomic conjugate of isogonal conjugate of X(36213)
X(5976) = crosspoint of X(2) and X(385)
X(5976) = crosssum of circumcircle intercepts of line PU(39) (line X(32)X(512))
X(5976) = crosssum of X(i) and X(j) for these (i,j): {6, 694}, {98, 8870}, {512, 15630}
X(5976) = crossdifference of every pair of points on line X(882)X(2422)
X(5976) = circumcircle-inverse of X(5989)
X(5976) = complement of X(1916)
X(5976) = anticomplement of X(2023)
X(5976) = midpoint of PU(133)
X(5976) = X(385)-daleth conjugate of X(732)
X(5976) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,325), (76,732), (99,804)
X(5976) = antipode of X(39) in the bicevian conic of X(2) and X(99)
X(5976) = crosssum of circumcircle intercepts of circle {{X(1687),X(1688),PU(1),PU(2)}}
X(5976) = X(i)-isoconjugate of X(j) for these (i,j): {19,15391}, {98,1967}, {290,1927}, {694,1910}, {733,3404}, {1581,1976}
X(5976) = X(i)-complementary conjugate of X(j) for these (i,j): (1,5031), (31,325), (163,804), (172,3836), (385,2887), (560,3229), (1580,141), (1691,10), (1914,3847), (1933,2), (1966,626), (2210,4357), (4164,116)
X(5976) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1916,2023), (98,99,5989), (98,5989,5939), (99,183,5939), (99,1078,5152), (183,5989,98)

X(5977) = X(37)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 b-a^3 b^3+a^5 c-a^2 b^3 c+a b^4 c+b^5 c-a b^3 c^2-a^3 c^3-a^2 b c^3-a b^2 c^3+a b c^4+b c^5

See X(5976).

X(5977) lies on these lines: {2,3125}, {21,99}, {75,2783}, {98,336}, {114,5509}, {325,758}, {1125,5976}, {2787,3766}, {5971,5990}

X(5978) = X(13)-OF-1st-ANTI-BROCARDTRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2+b^2+c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)-2 Sqrt[3] (a^2 b^2-b^4+a^2 c^2-c^4) S

See X(5976).

X(5978) lies on these lines: {2,14}, {13,1916}, {30,99}, {98,5982}, {114,1080}, {147,616}, {298,542}, {299,383}, {302,5092}, {385,533}, {395,1691}, {396,6034}, {3314,3643}

X(5978) = anticomplement of X(6109)

X(5979) = X(14)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2+b^2+c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)+2 Sqrt[3] (a^2 b^2-b^4+a^2 c^2-c^4) S

See X(5976).

X(5979) lies on these lines: {2,13}, {14,1916}, {30,99}, {98,5983}, {114,383}, {147,617}, {298,511}, {299,542}, {303,5092}, {385,532}, {395,6034}, {396,1691}, {3314,3642}

X(5979) = anticomplement of X(6108)

X(5980) = X(15)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics Sqrt[3] a^2 (a^2-b^2-c^2) (a^2+b^2+c^2)+2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) S : :
Tripolars a Sqrt[b^4 + b^2 c^2 + c^4] : :

See X(5976).

X(5980) lies on these lines: {2,13}, {3,76}, {15,385}, {22,1605}, {62,3329}, {147,627}, {262,5615}, {298,542}, {302,383}, {633,5984}, {1799,3438}, {1916,3104}, {5983,6036}

X(5980) = circumcircle-inverse of X(5981)
X(5980) = anticomplement of X(6115)

X(5981) = X(16)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics Sqrt[3] a^2 (a^2-b^2-c^2) (a^2+b^2+c^2)-2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) S : :
Tripolars a Sqrt[b^4 + b^2 c^2 + c^4] : :

See X(5976).

X(5981) lies on these lines: {2,14}, {3,76}, {16,385}, {22,1606}, {61,3329}, {147,628}, {262,5611}, {299,542}, {303,1080}, {634,5984}, {1799,3439}, {1916,3105}, {5982,6036}

X(5981) = circumcircle-inverse of X(5980)
X(5981) = anticomplement of X(6114)

X(5982) = X(17)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 - b^2c^2) sec(A - π/3) + (c^4 - a^2b^2) sec(B - π/3) + (b^4 - c^2a^2) sec(C - π/3)

See X(5976).

X(5982) lies on these lines: {2, 5469}, {5, 99}, {17, 1916}, {114, 383}, {147, 627}

X(5983) = X(18)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4 - b^2c^2) sec(A + π/3) + (c^4 - a^2b^2) sec(B + π/3) + (b^4 - c^2a^2) sec(C + π/3)

See X(5976).

X(5983) lies on these lines: {2, 5470}, {5, 99}, {18, 1916}, {114, 1080}, {147, 628}

X(5984) = X(20)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a^8 - a^6(b^2 + c^2) - 5a^4b^2c^2 - a^2(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^2(b^4 + 3b^2c^2 + c^4)

See X(5976).

X(5984) lies on these lines: {2, 98}, {8, 1281}, {20, 2782}, {99, 3522}, {115, 3832}, {148, 2794}, {193, 1916}, {385, 1503}, {390, 3027}, {2792, 5905}, {3023, 3600}, {3926, 5152}

X(5984) = anticomplement of X(147)

X(5985) = X(21)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 - a^6(b^2 + bc + c^2) - a^5bc(b + c) + a^4(b^4 + b^3c - b^2c^2 + bc^3 + c^4) + a^3bc(b + c)(b^2 + c^2) - a^2(b^6 + b^5c - b^4c^2 - b^3c^3 - b^2c^4 + bc^5 + c^6) - abc(b + c)(b^4 - b^2c^2 + c^4) - b^2c^2(b^2 - c^2)^2

See X(5976).

X(5985) lies on these lines: {2, 98}, {21, 2782}, {99, 4189}, {115, 5046}, {274, 5152}, {385, 518}, {1281, 1283}, {1621, 3027}, {2475, 2794}, {2975, 3023}

X(5986) = X(22)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^10 - a^4b^2c^2(a^2 + b^2 + c^2) + a^2(b^6c^2 + b^2c^6 + 2b^4c^4 - b^8 - c^8) + b^2c^2(b^2 - c^2)^2(b^2 + c^2)

See X(5976).

X(5986) lies on these lines: {2, 98}, {22, 2782}, {99, 1799}, {305, 5152}, {385, 2393}

X(5987) = X(23)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^10 - 2a^4b^2c^2(b^2 + c^2) + a^2(2b^6c^2 + 2b^2c^6 + b^4c^4 - b^8 - c^8) - b^2c^2(b^2 - c^2)^2(b^2 + c^2)

See X(5976).

X(5987) lies on these lines: {2, 98}, {23, 2782}, {67, 3314}, {183, 2930}, {385, 2854}, {2794, 5189}, {3266, 5152}, {5939, 5971}

X(5987) = inverse-in-1st-anti-Brocard-circle of X(2)

X(5988) = X(10)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b⁵ + c⁵ + a⁴b + a⁴c - a³b² - a³c² - a²b²c - a²bc² + ab⁴ + ac⁴ - b³c² - b²c³

See X(5976).

X5988) is the anticenter of the four points of intersection of the incircle and the Steiner inellipse; see X(5997) and X(5998). (Randy Hutson, August 22, 2014)

X(5988) lies on these lines: {1, 147}, {2, 846}, {10, 257}, {11, 114}, {98, 109}, {99, 1010}, {115, 120}, {183, 4655}, {325, 740}, {542, 3745}, {804, 3837}, {908, 2239}, {1575, 1738}, {2254, 2786}, {2793, 4927}, {3699, 4104}, {3920, 5483}

X5988) = complement of X(1281)

X(5989) = X(32)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics a⁸ - a⁴b²c² - a²b⁶ - a²c⁶ + 2b⁴c⁴

See X(5976).

X(5989) lies on these lines: {2, 4048}, {3, 76}, {6, 1916}, {75, 1281}, {114, 3818}, {115, 5149}, {147, 325}, {148, 384}, {385, 698}, {538, 5162}, {543, 1003}, {2023, 5026}, {5017, 5969}

X(5989) = isogonal conjugate of X(34214)
X(5989) = cevapoint of X(i) and X(j) for these (i,j): {147, 8782}, {4027, 8784}
X(5989) = crosssum of X(512) and X(2679)
X(5989) = crossdifference of every pair of points on line X(2491)X(5113)
X(5989) = circumcircle-inverse of X(5976)
X(5989) = crosspoint of PU(133)
X(5989) = perspector of 1st anti-Brocard triangle and cross-triangle of ABC and 1st anti-Brocard triangle

X(5990) = X(100)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^7 - a^6(b + c) + a^5bc + a^3(b^4 - b^3c - b^2c^2 - bc^3 + c^4) - a^2(b + c)(b^4 - 2b^3c + b^2c^2 - 2bc^3 + c^4) + abc(b^4 - b^3c - b^2c^2 - bc^3 + c^4) - b^2c^2(b - c)^2(b + c)

See X(5976).

X(5990) lies on these lines: {2, 1083}, {98, 901}, {115, 2240}, {385, 513}, {1281, 5205}, {4613, 5143}

X(5991) = X(101)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8 - a^7(b + c) + a^6bc - a^4b^2c^2 + a^3(b + c)(b^2 - bc + c^2)^2 - a^2(b^6 - b^5c + b^3c^3 - bc^5 + c^6) + ab^2c^2(b - c)^2(b + c) - b^3c^3(b - c)^2

See X(5976).

X(5991) lies on these lines: {36, 1111}, {98, 927}, {99, 859}, {385, 514}

X(5992) = X(8)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5 - b^5 - c^5 - a^4 b - a^4 c + a^3 b^2 + a^3 c^2 - a^2 b^3 - a^2 c^3 + a^2 b^2c + a^2 b c^2 - a b^4 - a c^4 - a b^2 c^2 + b^4 c + b^3 c^2 + b^2 c^3 + b c^4

b See X(5976).

X(5992) lies on these lines: {2, 846}, {98, 901}, {145, 2784}, {147, 149}, {148, 1655}, {2403, 2789}, {2792, 5905}, {3314, 5695}, {4440, 4459}

X(5992) = anticomplement of X(1281)

X(5993) = INTERSECTION OF LINES X(10)X(5975) AND X(11)X(2643)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c)^2 (a^3+b^3+a b c+2 b^2 c+2 b c^2+c^3) (-a^4-a^3 b+a b^3+b^4-a^3 c-a^2 b c+3 a b^2 c+b^3 c+3 a b c^2+a c^3+b c^3+c^4) (Peter Moses, August 22, 2014)

See X(5975) and X(5976).

X(5993) lies on the nine-point circle and these lines: {10,5975}, {11,2643}, {3259,5954}

X(5993) = complement of X(38470)
X(5893) = X(11260)-of-orthic-triangle if ABC is acute

X(5994) = TRILINEAR POLE OF LINE X(6)X(3129)

Trilinears    a/[b csc(B - π/3) - c csc(C - π/3)] : b/[c csc(C - π/3) - a csc(A - π/3)] : c/[a csc(A - π/3) - b csc(B - π/3)]
Trilinears    sin A csc(B - C) csc(A - π/3) : sin B csc(C - A) csc(B - π/3) : sin C csc(A - B) csc(C - π/3)
Barycentrics   a^2 (a^2-b^2) (a^2-c^2) (Sqrt[3] (a^2+b^2-c^2)-2 S) (Sqrt[3] (a^2-b^2+c^2)-2 S) : :

Let P be a point of the line X(2)X(14), other than X(2). Let A′B′C′ be the cevian triangle of P. Let A″ by the {B,C}-harmonic conjugate of A′ (i.e., A″ = BC∩B′C′), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5994). Let L,M,N be the lines obtained by reflecting the line X(4)X(14) in the sidelines of ABC; then L,M,N concur in X(5994). Also, X(5994) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(14),X(15)}}. Further, X(5994) is the trilinear pole of the line X(6)X(3129), and X(5994) = Ψ(X(i), X(j)) for these (i,j): (2,14), (4,14), (6,3458), (16,6), (76,14). (Randy Hutson, August 22, 2014)

X(5994) lies on the circumcircle and these lines: {6,2379}, {14,98}, {15,842}, {16,74}, {99,17402}, {110,6137}, {111,3458}, {187,2378}, {249,10409}, {301,2367}, {476,20579}, {477,15743}, {512,1576}, {691,9207}, {759,2154}, {805,14184}, {1141,11582}, {1297,14539}, {1300,6110}, {1625,16806}, {2380,11141}, {2381,11060}, {2710,14538}, {5467,9203}, {6772,18776}, {9181,9202}, {10410,17403}, {11083,15544}, {11085,11135}

X(5994) = midpoint of X(16) and X(5669)
X(5994) = reflection of X(5995) in Brocard
X(5994) = Schoutte-circle-inverse of X(2378)
X(5994) = polar conjugate of isotomic conjugate of X(38413)
X(5994) = X(i)-cross conjugate of X(j) for these (i,j): {512, 16460}, {2420, 5995}, {3130, 250}, {6138, 6}, {11135, 23357}, {19781, 249}
X(5994) = X(5995)-Hirst inverse of X(14560)
X(5994) = cevapoint of X(6) and X(6138)
X(5994) = trilinear pole of line {6, 3129}
X(5994) = barycentric product of circumcircle intercepts of line X(2)X(14)
X(5994) = X(i)-isoconjugate of X(j) for these (i,j): {16, 1577}, {75, 6138}, {299, 661}, {300, 2624}, {471, 656}, {850, 2152}, {897, 9205}, {1109, 17403}, {2153, 3268}, {8740, 14208}
X(5994) = barycentric product X(i)X(j) for these {i,j}: {14, 110}, {15, 476}, {99, 3458}, {249, 20579}, {298, 14560}, {301, 1576}, {524, 9207}, {662, 2154}, {1989, 17402}, {4558, 8738}, {5619, 11130}, {5995, 11092}, {8015, 10410}, {8836, 16807}, {9203, 21467}, {11085, 17403}, {16771, 16806}
X(5994) = barycentric quotient X(i)/X(j) for these {i,j}: {14, 850}, {15, 3268}, {32, 6138}, {110, 299}, {112, 471}, {187, 9205}, {476, 300}, {1576, 16}, {2154, 1577}, {3457, 23283}, {3458, 523}, {5995, 11078}, {8738, 14618}, {9207, 671}, {11060, 20578}, {14560, 13}, {16806, 19779}, {17402, 7799}, {17403, 11128}, {20579, 338}, {23347, 6111}, {23357, 17403}

X(5995) = TRILINEAR POLE OF LINE X(6)X(3130)

Trilinears    a/[b csc(B + π/3) - c csc(C + π/3)] : b/[c csc(C + π/3) - a csc(A + π/3)] : c/[a csc(A + π/3) - b csc(B + π/3)]
Trilinears    sin A csc(B - C) csc(A + π/3) : sin B csc(C - A) csc(B + π/3) : sin C csc(A - B) csc(C + π/3)
Barycentrics    a^2 (a^2-b^2) (a^2-c^2) (Sqrt[3] (a^2+b^2-c^2)+2 S) (Sqrt[3] (a^2-b^2+c^2)+2 S) : :

Let P be a point of the line X(2)X(13), other than X(2). Let A′B′C′ be the cevian triangle of P. Let A″ by the {B,C}-harmonic conjugate of A′ (i.e., A″ = BC∩B′C′), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X(5995). Let L,M,N be the lines obtained by reflecting the line X(4)X(13) in the sidelines of ABC; then L,M,N concur in X(5995). Also, X(5995) is the point, other than X(111), in which the circumcircle meets the circle {{X(6),X(13),X(16)}}. Further, X(5995) is the trilinear pole of the line X(6)X(3130), and X(5995) = Ψ(X(i), X(j)) for these (i,j): (2,13), (4,13), (6,3457), (15,6), (76,13). (Randy Hutson, August 22, 2014)

X(5995) lies on the circumcircle and these lines: {6,2378}, {13,98}, {15,74}, {16,842}, {99,17403}, {110,6138}, {111,3457}, {187,2379}, {249,10410}, {300,2367}, {476,20578}, {477,11586}, {512,1576}, {691,9206}, {759,2153}, {805,14183}, {1141,11581}, {1297,14538}, {1300,6111}, {1625,16807}, {2380,11060}, {2381,11142}, {2710,14539}, {5467,9202}, {6775,18777}, {9181,9203}, {10409,17402}, {11080,11136}, {11088,15544}

X(5995) = midpoint of X(15) and X(5668)
X(5995) = reflection of X(5995) in Brocard axis
X(5995) = Schoutte-circle-inverse of X(2379)
X(5995) = X(i)-cross conjugate of X(j) for these (i,j): {512, 16459}, {2420, 5994}, {3129, 250}, {6137, 6}, {11136, 23357}, {19780, 249}
X(5995) = X(5994)-Hirst inverse of X(14560)
X(5995) = cevapoint of X(6) and X(6137)
X(5995) = trilinear pole of line {6, 3130}
X(5995) = polar conjugate of isotomic conjugate of X(38414)
X(5995) = X(i)-isoconjugate of X(j) for these (i,j): {15, 1577}, {75, 6137}, {298, 661}, {301, 2624}, {470, 656}, {850, 2151}, {897, 9204}, {1109, 17402}, {2154, 3268}, {8739, 14208}
X(5995) = barycentric product of circumcircle intercepts of line X(2)X(13)
X(5995) = barycentric product X(i)X(j) for these {i,j}: {13, 110}, {16, 476}, {99, 3457}, {249, 20578}, {299, 14560}, {300, 1576}, {524, 9206}, {662, 2153}, {1989, 17403}, {4558, 8737}, {5618, 11131}, {5994, 11078}, {8014, 10409}, {8838, 16806}, {9202, 21466}, {11080, 17402}, {16770, 16807}
X(5995) = barycentric quotient X(i)/X(j) for these {i,j}: {13, 850}, {16, 3268}, {32, 6137}, {110, 298}, {112, 470}, {187, 9204}, {476, 301}, {1576, 15}, {2153, 1577}, {3457, 523}, {3458, 23284}, {5994, 11092}, {8737, 14618}, {9206, 671}, {11060, 20579}, {14560, 14}, {16807, 19778}, {17402, 11129}, {17403, 7799}, {20578, 338}, {23347, 6110}, {23357, 17402}

X(5996) = INTERSECTION OF LINES X(2)X(512) AND X(325)X(523)

Barycentrics b²c²(4a² - 3b² - 3c²) : :

Let L be the de Longchamps line of ABC, and let L′ be the de Longchamps line of the 1st anti-Brocard triangle. Then X(5996) = L∩L′. Let O′ be the inverter of the circumcircle and the nine-point circle; then X(5996) is the pole of the Brocard axis with respect to O′. (Randy Hutson, August 22, 2014)

X(5996) lies on these lines: {2,512}, {262,5466}, {325,523}, {1180,2524}, {1499,1513}

X(5997) = 1st HUTSON-WOLK POINT

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

X(5997) and X(5998) are described by Randy Hutson (Hyacinthos #21045) and Barry Wolk (Hyacinthos #21047, June 1, 2012). One of the 4 points of intersection of the incircle and Steiner inellipse is a triangle center, X(5997), and the other 3 are the vertices of a central triangle; of those 3, let A′ be the one farthest from A, and define B′ and C′ cyclically. The triangle A′B′C′ is perspective to ABC at X(5998). Barycentrics for A′, B′, C′ follow:

A′ = a - [(a - b + c)(a + b - c)]^1/2 : b + [(b - c + a)(b + c - a)]^1/2 : c + [(c - a + b)(c + a - b)]^1/2
B′ = a + [(a - b + c)(a + b - c)]^1/2 : b - [(b - c + a)(b + c - a)]^1/2 : c + [(c - a + b)(c + a - b)]^1/2
C′ = a + [(a - b + c)(a + b - c)]^1/2 : b + [(b - c + a)(b + c - a)]^1/2 : c - [(c - a + b)(c + a - b)]^1/2

X(5997) is the {X(1),X(508)}-harmonic conjugate of X(5998). (Randy Hutson, August 22, 2014)

X(5997) lies on the incircle, the Steiner inellipse, and this line: {1,508)

X(5998) = 2nd HUTSON-WOLK POINT

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

Continuing from X(5997), the lines AA′, BB′, CC′ concur in X(5998), and X(5998) is the {X(1),X(508)}-harmonic conjugate of X(5997). (Randy Hutson, August 22, 2014)

X(5998) lies on this line: {1,508}

X(5999) = X(98)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a⁸ + a⁴(b⁴ + c⁴ + b²c²) - 2a²(b⁶ + c⁶ ) - b²c²(b² - c²)²

As a point on the Euler line, X(5999) has Shinagawa coefficients ((E + F)² - S², -2(E + F)²).

Let A′B′C′ be the cross-triangle of the 1st and 2nd Neuberg triangles. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(5999). (Randy Hutson, July 31 2018)

X(5999) lies on these lines: {2,3}, {98,385}, {115,5162}, {147,325}, {182,262}, {183,1350}, {230,2076}, {316,2794}, {516,5988}, {517,5990}, {625,5149}, {1078,5188}, {1352,3314}, {1499,5971}, {1691,2023}, {2456,4027}, {3424,5921}, {3564,5984}, {3815,5116}, {4518,6211}, {5663,5987}

X(5999) = reflection of X(385) in X(98)
X(5999) = anticomplement of X(1513)
X(5999) = {X(6039),X(6040)}-harmonic conjugate of X(3)
X(5999) = pole of Brocard axis wrt conic {{X(13),X(14),X(15),X(16),X(98)}}
X(5999) = {X(2),X(3)}-harmonic conjugate of X(37455)

Isogonal Conjugates With Respect To Various Triangles

Many triangle centers are constructible as isogonal conjugates with respect to a central triangle T other than ABC. See, for example, the preamble to X(2883). Other examples are X(5642), for which T is the Thomson triangle, and X(5957), for which T is the Feuerbach triangle. In the latter case, a long formula for barycentric coordinates is given for the Feuerbach isogonal conjugate of an arbitrary point p : q : r is given. Peter Moses (August 11, 2014) developed a much longer formula for the isogonal conjugate of a point p : q : r with respect to a triangle having vertices d1 : d2 : d3, e1 : e2 : e3, f1 : f2 : f3. The total number of characters in the formula is (pending). Although too long to be stated here, the formula served as a basis for finding triangle centers X(i) for i = 6000-6030.

For well-established central triangles T, (e.g., Thomson, Feuerbach, intouch, extouch), the phrase "isogonal conjugate with respect to T can be shortened to "T isogonal conjugate" or "T-I C″, as in many of the next triangle centers.

It T is a triangle, then T-isogonal conjugation carries points on the circumcircle of T to the line at infinity. For example, taking T = Brocard triangle, the appearance of (X(i),X(j)) in the following list means that T-I C of X(i) is X(j): (3,2782), (6,804), (1083,2795), (1316,690), (5091,2787), (5108,543).

In each of the next examples (i,j), the point X(i) lies on the incircle, and X(j), the intouch-IC-of-X(i), lies on the line at infinity: (1317,517), (1356,6002), (1357,3667), (1358,3309), (1360,971), (1361,5151), (1362,516), (1364,522), (1365,6003), (2446, 3307), (2447,3308), (3020,6004), (3021,518), (3022,514), (3023,512), (3024,523), (3025,900), (3026,6005), (3027,511), (3028,30), (3318,521), (3319,2800), (3320,1503), (3322,2801), (3323,2820), (3325,1499), (3326,3738), (3327,1510), (3328,3887), (5577,6006), (5579,6007), (5580,520), (5581,6008), (5582,6009)

X(6000) = ISOGONAL CONJUGATE OF X(1294)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+4 a^4 b^2 c^2-3 a^2 b^4 c^2-2 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4+6 b^4 c^4+3 a^2 c^6-2 b^2 c^6-c^8)

X(6000) lies on these lines: {2,5656}, {3,64}, {4,51}, {5,2883}, {6,1597}, {20,2979}, {24,1204}, {30,511}, {40,2939}, {52,382}, {65,1844}, {66,3818}, {74,186}, {98,2713}, {99,2706}, {100,2744}, {101,2738}, {102,2762}, {103,2727}, {104,2719}, {109,2732}, {110,2071}, {113,2072}, {125,403}, {133,1515}, {143,3853}, {146,1531}, {159,3098}, {184,378}, {187,1971}, {206,4550}, {221,500}, {232,3269}, {373,3545}, {376,3917}, {381,1853}, {399,2935}, {546,5462}, {548,5447}, {550,1216}, {568,3830}, {578,1181}, {858,1568}, {933,3484}, {974,1514}, {999,2192}, {1092,5879}, {1192,3517}, {1292,2749}, {1293,2755}, {1295,2761}, {1296,2763}, {1297,2764}, {1533,3580}, {1562,5523}, {1614,3520}, {1657,5925}, {1870,3270}, {3060,3543}, {3146,5889}, {3284,5668}, {3524,5650}, {3581,5899}, {3587,3781}, {3627,5446}, {3839,5640}, {3845,5946}

X(6000) = isogonal conjugate of X(1294)
X(6000) = crosssum of X(4) and X(74)
X(6000) = orthic-IC-of-X(133) = medial-IC-of-X(133)
X(6000) = intouch-IC-of-X(3324)
X(6000) = Euler-IC-of-X(122)
X(6000) = 1st-circumperp-IC-of-X(1294)
X(6000) = circumtangential-IC-of-X(1294)
X(6000) = circumcircle-midarc-IC-of-X(107)
X(6000) = circumorthic-IC-of-X(107)
X(6000) = 2nd-circumperp-IC-of-X(107)
X(6000) = circumnormal-IC-of-X(107)
X(6000) = Thomson-IC-of-X(107)
X(6000) = Thomson-isogonal conjugate of X(107)
X(6000) = Lucas-isogonal conjugate of X(107)
X(6000) = Cundy-Parry Phi transform of X(14379)
X(6000) = Cundy-Parry Psi transform of X(14249)
X(6000) = intersection of tangents to Moses-Jerabek conic at X(74) and X(1199)

X(6001) = ISOGONAL CONJUGATE OF X(1295)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+2 a^3 b^2 c-3 a b^4 c-a^4 c^2+2 a^3 b c^2-4 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^3 c^3+2 a b^2 c^3+2 a^2 c^4-3 a b c^4+b^2 c^4+a c^5-c^6)

X(6001) lies on these lines: {1,84}, {3,960}, {4,65}, {5,3812}, {7,3427}, {9,3197}, {10,5777}, {11,1519}, {19,5776}, {20,3869}, {30,511}, {36,1727}, {40,64}, {46,3149}, {63,3428}, {74,2766}, {98,2714}, {99,2707}, {100,2745}, {101,2739}, {102,2765}, {103,2728}, {104,1319}, {109,2733}, {110,2694}, {119,5123}, {153,5176}, {154,392}, {165,5692}, {185,1829}, {207,1712}, {210,5657}, {227,1745}, {281,1903}, {354,5603}, {355,5836}, {376,5918}, {496,942}, {497,5768}, {573,4047}, {581,3931}, {774,1042}, {944,3057}, {962,3868}, {1001,3358}, {1006,3683}, {1064,3666}, {1072,3782}, {1104,3073}, {1292,2750}, {1293,2756}, {1294,2761}, {1296,2767}, {1376,3359}, {1385,5248}, {1430,5706}, {1456,1870}, {1465,1735}, {1532,1737}, {1537,5570}, {1699,5902}, {1739,5400}, {1750,2093}, {1853,3753}, {1871,5786}, {2077,2932}, {2096,4293}, {2194,4227}, {2292,4300}, {2551,5811}, {2935,2948}, {3062,3577}, {3357,3579}, {3698,5818}, {3742,5886}, {3817,5883}, {3874,4301}, {3877,5731}, {3878,4297}, {3889,5734}, {3913,5534}, {3914,5721}, {4018,5895}, {4067,5493}, {4314,5882}, {4662,5690}, {4867,5538}, {4880,5536}, {5691,5903}, {5787,5878}, {5806,5893}

X(6001) = crosssum of X(4) and X(104)
X(6001) = midarc-of-X(3318)
X(6001) = intouch-IC-of-X(1359)
X(6001) = Euler-IC-of-X(123)
X(6001) = 1st-circumperp-IC-of-X(1295)
X(6001) = circumtangential-IC-of-X(1295)
X(6001) = circumcircle-midarc-IC-of-X(108)
X(6001) = circumorthic-IC-of-X(108)
X(6001) = 2nd-circumperp-IC-of-X(108)
X(6001) = circumnormal-IC-of-X(108)
X(6001) = Thomson-IC-of-X(108)
X(6001) = Thomson-isogonal conjugate of X(108)
X(6001) = Lucas-isogonal conjugate of X(108)
X(6001) = Cundy-Parry Psi transform of X(14257)

X(6002) = INTOUCH-ISOGONAL CONJUGATE OF X(1356)

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

X(6002) lies on these lines: {1, 4170}, {4, 4444}, {8, 4729}, {30, 511}, {649, 4391}, {661, 4560}, {667, 3716}, {905, 3835}, {1019, 1577}, {2533, 4784}, {3669, 4106}, {3762, 4063}, {4010, 4367}, {4380, 4462}, {4382, 4801}, {4705, 4913}, {4879, 4922}

X(6002) = isogonal conjugate of X(6010)

X(6003) = INTOUCH-ISOGONAL CONJUGATE OF X(1365)

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

X(6003) lies on these lines: {1, 4017}, {3, 3733}, {8, 4404}, {30, 511}, {573, 798}, {656, 3737}, {661, 1021}, {1459, 3960}, {1532, 4129}, {4581, 4761}

X(6003) = isogonal conjugate of X(6011)

X(6004) = INTOUCH-ISOGONAL CONJUGATE OF X(3020)

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

X(6004) lies on these lines: {1, 3777}, {30, 511}, {659, 1734}, {663, 1201}, {667, 2254}, {764, 4449}, {905, 1960}, {1016, 3888}, {1491, 3216}, {2473, 2488}, {3214, 4705}, {4367, 4905}, {4498, 4730}

X(6004) = isogonal conjugate of X(6012)

X(6005) = INTOUCH-ISOGONAL CONJUGATE OF X(3026)

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

X(6005) lies on these lines: {30, 511}, {649, 2664}, {659, 4834}, {661, 1734}, {663, 1019}, {665, 3803}, {667, 4784}, {693, 4170}, {1491, 4983}, {2254, 4822}, {2533, 4791}, {2978, 4932}, {3250, 4979}, {4010, 4823}, {4063, 4724}, {4147, 4807}, {4367, 4775}, {4378, 4879}, {4391, 4761}, {4489, 4813}, {4490, 4730}

X(6005) = isogonal conjugate of X(6013)

X(6006) = INTOUCH-ISOGONAL CONJUGATE OF X(5577)

Barycentrics (b - c)(5a - b - c) : (c - a)(5b - c - a) : (a - b)(5c - a - b)

X(6006) lies on these lines: {7, 3676}, {9, 649}, {30, 511}, {142, 3835}, {144, 4468}, {1769, 4905}, {2254, 4776}, {3239, 4790}, {4401, 4491}, {4406, 4828}, {4462, 4768}

X(6006) = isogonal conjugate of X(6014)

X(6007) = INTOUCH-ISOGONAL CONJUGATE OF X(5579)

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

X(6007) lies on these lines: {7, 310}, {9, 43}, {30, 511}, {39, 4443}, {86, 4890}, {142, 3840}, {192, 3688}, {239, 3271}, {320, 4014}, {995, 1001}, {1463, 4684}, {1469, 3886}, {1654, 4111}, {1959, 4516}, {2234, 3122}, {2245, 4436}, {3056, 3875}, {3729, 3779}, {3792, 4693}, {3882, 4433}, {3923, 4260}, {3943, 4553}, {4259, 5695}

X(6007) = isogonal conjugate of X(6015)

X(6008) = INTOUCH-ISOGONAL CONJUGATE OF X(5581)

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

X(6008) lies on these lines: {7, 3669}, {9, 4063}, {30, 511}, {144, 4462}, {390, 4162}, {649, 4106}, {650, 4380}, {667, 1001}, {693, 4790}, {1638, 4786}, {3768, 4498}, {3803, 4170}, {3835, 4394}, {4312, 4905}, {4375, 4782}, {4382, 4979}

X(6008) = isogonal conjugate of X(6016)

X(6009) = INTOUCH-ISOGONAL CONJUGATE OF X(5582)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)*(4*a^2 - a*b + b^2 - a*c - 4*b*c + c^2)

X(6009) lies on these lines: {30, 511}, {88, 673}, {659, 1001}, {676, 4830}, {1086, 2087}, {1635, 4927}, {2550, 4925}, {3826, 3837}, {4453, 4773}

X(6009) = isogonal conjugate of X(6017)

X(6010) = ISOGONAL CONJUGATE OF X(6002)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c)*(-a^3 - a^2*b - a^2*c - a*b*c + b^2*c + b*c^2))

X(6010) lies on the circumcircle and these lines: {3, 741}, {40, 98}, {55, 1356}, {99, 3882}, {103, 3430}, {165, 5539}, {573, 759}, {813, 4574}, {1376, 3037}, {2077, 2699}

X(6010) = isogonal conjugate of X(6002)

X(6011) = ISOGONAL CONJUGATE OF X(6003)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b - c)*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c - a*b*c - a*c^2 + c^3)]

X(6011) lies on the circumcircle and these lines: {3, 759}, {40, 74}, {55, 1365}, {102, 3430}, {104, 3651}, {105, 1283}, {106, 1385}, {107, 4242}, {112, 1983}, {354, 1477}, {573, 2249}, {741, 991}, {840, 5536}, {2077, 2687}

X(6011) = isogonal conjugate of X(6003)
X(6011) = reflection of X(759) in X(3)
X(6011) = circumcircle-antipode of X(759)

X(6012) = ISOGONAL CONJUGATE OF X(6004)

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

X(6012) lies on the circumcircle and these lines: {55, 3020}, {105, 404}, {759, 4234}

X(6012) = isogonal conjugate of X(6004)

X(6013) = ISOGONAL CONJUGATE OF X(6005)

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

X(6013) lies on the circumcircle and these lines: {55, 3026}, {101, 4436}, {105, 5248}, {741, 4658}

X(6013) = isogonal conjugate of X(6005)

X(6014) = ISOGONAL CONJUGATE OF X(6006)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/[(5*a - b - c)*(-b + c)]

X(6014) lies on the circumcircle and these lines: {55, 106}, {104, 165}, {672, 2384}, {739, 2280}, {901, 3939}, {953, 5537}

X(6014) = isogonal conjugate of X(6006)

X(6015) = ISOGONAL CONJUGATE OF X(6007)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a^2*b^2 - a*b^3 + a*b^2*c + a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3)

X(6015) lies on the circumcircle and these lines: {55, 99}, {57, 4128}, {100, 894}, {101, 171}, {109, 1918}, {110, 2175}, {165, 5539}, {934, 1402}, {1155, 2703}, {1308, 5143}

X(6015) = isogonal conjugate of X(6007)

X(6016) = ISOGONAL CONJUGATE OF X(6008)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/[(b - c)*(-3*a^2 - a*b - a*c + 2*b*c)]

X(6016) lies on the circumcircle and these lines: {55, 739}, {644, 898}, {672, 2382}, {727, 2280}

X(6016) = isogonal conjugate of X(6008)

X(6017) = ISOGONAL CONJUGATE OF X(6009)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c)*(4*a^2 - a*b + b^2 - a*c - 4*b*c + c^2))

X(6017) lies on the circumcircle and these lines: {44, 105}, {55, 2384}, {106, 672}, {901, 2284}, {1252, 4588}, {2280, 2382}

X(6017) = isogonal conjugate of X(6009)

X(6018) = INTOUCH-ISOGONAL CONJUGATE OF X(519)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a b+b^2+a c-4 b c+c^2)^2

Let A′ be the point of intersection, other than X(11), of the Mandart inellipse and the incircle that is farthest from A; define B′ and C′ cyclically. A′B′C′ is here named the Mandart-incircle triangle. It is homothetic to ABC at X(55), to the medial triangle at X(11), to the Euler triangle at X(12), and to the anticomplementary triangle at X(497). X(6018) = X(106)-of-Mandart-incircle triangle. (Randy Hutson, August 26, 2014)

X(6018) lies on the incircle and these lines: {1,1357}, {8,3038}, {10,11}, {12,5510}, {55,106}, {56,1293}, {65,4661}, {354,4701}, {988,3242}, {1054,1697}, {1317,2827}, {1320,3271}, {1356,4890}, {1358,3663}, {1361,2815}, {1362,2821}, {1364,2098}, {2776,3028}, {2789,3027}, {2796,3023}, {2810,3022}, {2832,3021}, {2840,3318}, {2842,3024}, {2843,6019}, {2844,6020}, {2899,5178}, {3025,5048}, {3616,4731}, {3624,3893}, {3626,4004}, {3632,3698}, {3635,4002}, {3679,3962}, {3753,4746}, {3922,4668}, {4005,5836}, {4533,4691}

X(6018) = reflection of X(1357) in X(1)
X(6018) = X(1294)-of-intouch-triangle

X(6019) = INTOUCH-ISOGONAL CONJUGATE OF X(524)

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

X(6019) lies on the incircle and these lines: {1,3325}, {11,126}, {12,5512}, {55,111}, {56,1296}, {215,3048}, {543,3023}, {1317,2830}, {1357,4890}, {1358,4854}, {1361,2819}, {1362,2824}, {1364,2852}, {2780,3028}, {2793,3027}, {2813,3022}, {2837,3021}, {2843,6018}, {2851,3318}, {2854,3024}, {5148,5160}

X(6019) = reflection of X(3325) in X(1)
X(6019) = X(111) of Mandart-incrcle triangle (see X(6018))

X(6020) = INTOUCH-ISOGONAL CONJUGATE OF X(525)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c)^2 (a^4-b^4+a^2 b c-b^3 c-b c^3-c^4)^2

X(6020) lies on the incircle and these lines: {1,3320}, {11,127}, {12,132}, {55,112}, {56,1297}, {1317,2831}, {1362,2825}, {1364,2853}, {2781,3028}, {2794,3027}, {2799,3023}, {2838,3021}, {2844,6018}, {3324,5434}, {5148,6023}

X(6020) = reflection of X(3320) in X(1)
X(6020) = X(112) of Mandart-incrcle triangle (see X(6018))

X(6021) = INTOUCH-ISOGONAL CONJUGATE OF X(536)

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

X(6021) lies on the incircle and these lines: {11,141}, {55,739}, {1357,4003}

X(6022) = INTOUCH-ISOGONAL CONJUGATE OF X(538)

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

X(6022) lies on the incircle and these lines: {55,729}, {3023,3056}

X(6023) = INTOUCH-ISOGONAL CONJUGATE OF X(542)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a+b-c) (a-b+c) (b+c)^2 (a^4-b^4-2 a^2 b c+b^3 c+b^2 c^2+b c^3-c^4)^2

X(6023) lies on the incircle and these lines: {1,6027}, {12,5099}, {30,3023}, {55,842}, {56,691}, {511,3024}, {512,3028}, {523,3027}, {3325,5194}, {5148,6020}

X(6023) = reflection of X(6027) in X(1)

X(6024) = INTOUCH-ISOGONAL CONJUGATE OF X(545)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^2 b-4 a b^2+b^3+a^2 c+4 a b c-4 a c^2+c^3)^2

X(6024) lies on the incircle and these lines:

X(6025) = INTOUCH-ISOGONAL CONJUGATE OF X(674)

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

X(6025) lies on the incircle and these lines: {11,37}, {55,675}, {544,3022}

X(6026) = INTOUCH-ISOGONAL CONJUGATE OF X(688)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (b-c)^2 (a^4 b^2+a^4 b c+a^4 c^2+b^3 c^3)^2

X(6026) lies on the incircle and this line: {55,689}

X(6027) = INTOUCH-ISOGONAL CONJUGATE OF X(690)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c)^2 (a^4-b^4+2 a^2 b c-b^3 c+b^2 c^2-b c^3-c^4)^2

X(6027) lies on the incircle and these lines: {1,6023}, {11,5099}, {30,3027}, {55,691}, {56,842}, {511,3028}, {512,3024}, {523,3023}, {3320,5194}, {5148,5160}

X(6027) = reflection of X(i) in X(j) for these (i,j): (6023,1), (5160,5148)

X(6028) = INTOUCH-ISOGONAL CONJUGATE OF X(696)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^4 b^2+a b^5-2 a^4 b c-b^5 c+a^4 c^2+a c^5-b c^5)^2

X(6028) lies on the incircle and this line: {55,697}

X(6029) = INTOUCH-ISOGONAL CONJUGATE OF X(712)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (a^3 b^2+a b^4-2 a^3 b c-b^4 c+a^3 c^2+a c^4-b c^4)^2

X(6029) lies on the incircle and these lines: {11,626}, {55,713}

X(6030) = THOMSON-ISOGONAL CONJUGATE OF X(5)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (3 a^4-a^2 b^2-2 b^4-a^2 c^2-b^2 c^2-2 c^4)
X(6030) = 2X(6) - 5X(1176)
X(6030) = X(6) + 5X(2916)
X(6030) = X(1176) + 2X(2916)
X(6030) = (a² + b² + c²)*X(6) - 5(J² - 3)R²*X(22)
(barycentrics and combos, Peter Moses, August 16, 2014)

X(6030) lies on the Thomson-Gibert-Moses hyperbola; see X(5642). X(6030) = {X(22),X(3796)}-harmonic conjugate of X(3060; also, X(6030) = {X(3060),X(3796)}-harmonic conjugate of X(5012). (Peter Moses, August 16, 2014)

X(6030) lies on these lines: {6,22}, {23,5643}, {25,5544}, {36,595}, {110,3917}, {182,5645}, {376,5654}, {392,4881}, {550,3521}, {1495,5888}, {2937,5946}, {2979,3167}, {5640,5644}

X(6031) = X(353)-OF-1st-ANTI-BROCARD TRIANGLE

Barycentrics 4a^6 - a^2(4b^4 + b^2c^2 + 4c^4) - 2b^2c^2(b^2 + c^2) : :

X(6031) lies on these lines: {2,187}, {3,5971}, {22,5938}, {23,183}, {69,5648}, {99,5987}, {353,524}, {385,6322}, {2373,6236}

X(6031) = isogonal conjugate of X(30488)
X(6031) = isotomic conjugate of X(34213)
X(6031) = anticomplement of X(6032)
X(6031) = circumcevian-isogonal conjugate of X(2)
X(6031) = X(2)-pedal-to-circummedial similarity image of X(2)

X(6032) = CENTROID OF 4th BROCARD TRIANGLE

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

Let A′B′C′ be the Artzt triangle. Let A″ be the reflection of A in B′C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(9744), and X(6032) = centroid of A″B″C″; see X(5640). (Randy Hutson, December 10, 2016)

X(6032) lies on these lines: {2,187}, {5,5913}, {111,381}, {112,5094}, {115,5169}, {262,5466}, {427,1180}, {566,858}, {1648,5640}, {1995,2079}

X(6032) = complement of X(6031)
X(6032) = inverse-in-orthocentroidal circle of X(111)
X(6032) = X(353)-of-orthocentroidal triangle
X(6032) = anti-Artzt-to-4th-Brocard similarity image of X(2)

X(6033) = MIDPOINT OF X(4) AND X(147)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3(S² - S²_ω)S²_A - S_ω(5S² -3S²_ω)S_A + 2S²(2S² - S²_ω)
Barycentrics a^8 - a^4 (b^4 + b^2 c^2 + c^4) + a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4) : :

X(6033) = Johnson-triangle-orthologic center of the1st Brocard triangle. (César Lozada, August 7, 2013)

Let D = X(99), let H_A = X(4)-of-BCD, and define H_B, H_C, H_D cyclically. Then H_AH_BH_CH_D is a cyclical quadrilateral, of which the circumcenter is X(6033). (Randy Hutson, August 22, 2014)

X(6033) lies on these lines: {2, 5191}, {3, 114}, {4, 147}, {5, 83}, {6, 13}, {30, 99}, {110, 868}, {315, 5976}, {543, 3830}, {623, 5617}, {624, 5613}, {671, 3845}, {946, 2784}, {1478, 3023}, {1479, 3027}, {1503, 2456}, {1513, 2080}, {1656, 6036}, {2023, 2548}, {2476, 5985}, {2482, 3534}, {3091, 5984}, {4027, 5025}, {5133, 5986}, {5169, 5987}

X(6033) = midpoint of X(4) and X(147)
X(6033) = reflection of X(3) in X(114)
X(6033) = complement of X(9862)
X(6033) = circumcircle-inverse of X(34217)
X(6033) = X(3)-of-X(511)-Fuhrmann-triangle
X(6033) = inverse-in-Kiepert-hyperbola of X(6034)
X(6033) = X(35002)-of-orthocentroidal-triangle
X(6033) = {X(13),X(14)}-harmonic conjugate of X(6034)

X(6034) = CENTROID OF X(6)X(13)X(14)

Barycentrics a⁶ + a²(2b⁴ - 3b²c² + 2c⁴) - (b² + c²)[a⁴ - (b² - c²)²] : : (Richard Hilton, March 2, 2015)
X(6034) = X(6) + X(13) + X(14) = X(6) + 2X(115)

X(6034) lies on these lines: {2,694}, {6,13}, {30,1691}, {83,597}, {98,5306}, {99,3589}, {148,3618}, {182,6321}, {230,5104}, {395,5979}, {396,5978}, {524,5103}, {543,5149}, {599,626}, {1350,6036}, {2482,5013}, {5017,6055}

X(6034) = midpoint of X(671) and X(5182)
X(6034) = reflection of X(5182) in X(597)
X(6034) = inverse-in-Kiepert-hyperbola of X(6033)
X(6034) = {X(13),X(14)}-harmonic conjugate of X(6033)

X(6035) = ISOTOMIC CONJUGATE OF X(1640)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b² - c²)(2a⁶ - b⁶ - c⁶ - 2a⁴b² - 2a⁴c² + a²b⁴ + a²c⁴ + b⁴c² + b²c⁴)]

X(6035) lies on the hyperbola {A, B, C, PU(37)} and these lines: {69, 892}, {877, 4240}, {2396, 2407}

X(6035) = isogonal conjugate of X(6041)
X(6035) = isotomic conjugate of X(1640)
X(6035) = trilinear pole of line X(30)X(99)

X(6036) = MIDPOINT OF X(98) AND X(114)

Barycentrics    2a sec(A + ω) + b sec(B + ω) + c sec(C + ω) : :
Barycentrics    2 a^8 - 4 a^6 (b^2 + c^2) + 5 a^4 (b^4 + c^4) - 2 a^2 (2 b^6 - b^4 c^2 - b^2 c^4 + 2 c^6) + (b^2 - c^2)^4 : :
Barycentrics    4 S^4 + (SA - 2 SW) SW S^2 + SB SC SW^2 : :

X(6036) = 3X(2) + X(98)

Let D = X(98), let G_A = X(2)-of-BCD, and define G_B, G_C, G_D cyclically. Then G_AG_BG_CG_D is a cyclic quadrilateral which is homothetic to the cyclic quadrilateral ABCD, and X(6036) is the homothetic center, as well as the centroid of ABCD and the anticenter of the cyclic quadrilateral whose vertices are X(115) and the vertices of the medial triangle. Let (O_A) be the circle centered at the A-vertex of the 1st Brocard triangle and tangent to line BC; define (O_B) and (O_C) cyclically; then X(6036) is the radical center of (O_A), (O_B), (O_C). Also, X(6036) lies on the axis of the parabola {A, B, C, X(511), X(805}. (Randy Hutson, August 22, 2014)

X(6036) is the QA-P36 center of the quardrangle ABCX(2). See Complement of QA-P30 wrt the QA-Diagonal Triangle.

X(6036) lies on these lines: {2, 98}, {3, 115}, {5, 2794}, {30, 5461}, {99, 631}, {140, 620}, {148, 3523}, {230, 511}, {231, 3001}, {325, 5965}, {338, 2974}, {543, 549}, {575, 3815}, {671, 3524}, {1155, 5988}, {1350, 6034}, {1506, 3398}, {1624, 5020}, {1656, 6033}, {2482, 5054}, {2783, 3035}, {2784, 3634}, {3023, 5433}, {3027, 5432}, {3054, 5092}, {5050, 5477}, {5097, 5306}, {5469, 5474}, {5470, 5473}, {5980, 5983}, {5981, 5982}

X(6036) = midpoint of X(98) and X(114)
X(6036) = complement of X(114)
X(6036) = anticomplement of X(6721)
X(6036) = X(140)-of-1st-anti-Brocard-triangle
X(6036) = center of bicevian conic of X(2) and X(98)

X(6037) = TRILINEAR POLE OF LINE X(6)X(98)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b sec(C + ω) - c sec(B + ω)]

Barycentrics 1/((b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 - 2 b^2 c^2) (a^2 b^2 + a^2 c^2 - b^4 - c^4)) :

X(6037) lies on the circumcircle and these lines: {98,237}, {110,2966}, {112,685}, {262,842}, {263,2698}, {327,2857}, {401,1297}, {419,3563}, {805,4226}

X(6037) = inverse-in-1st-anti-Brocard-circle of X(6038)
X(6037) = Ψ(X(6),X(98)
X(6037) = Λ(X(290),X(879)
X(6037) = Λ(X(684),X(2491)

X(6038) = INVERSE-IN-1st-ANTI-BROCARD-CIRCLE OF X(6037)

Barycentrics 3*(b^2+c^2)*a^10-(b^2-c^2)^2*a^8-2*(b^2-c^2)^2*b^4*c^4+(b^4-c^4)*(b^2-c^2)*a^6-4*(b^4-c^4)*(b^2-c^2)*a^2*b^2*c^2-3*(b^8+c^8)*a^4 : :

Let A′ be the inverse-in-1st-anti-Brocard-circle of A, and define B′ and C′ cyclically; then X(6038) is the centroid of A′B′C′.

X(6038) lies on these lines: {2, 154}, {98, 237}, {669, 804}

X(6039) = THOMSON-ISOGONAL CONJUGATE OF X(5638)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S⁴ - a²S_AS_ω(-2S² + S²_A + S²_B + S²_C)^1/2 - (S² - 2S_BS_C)S²_ω
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)[cos B cos C + 3 sin B sin C - (cos A)(3 + 4e*cos ω + 4 cos 2ω)], where e = (1 - 4 sin²ω)^1/2
X(6039) = 3(3S² + S²_A + S²_B + S²_C)*X(2) - 2S_ω[S_ω + (-S² + S²_A + S²_B + S²_C)^1/2]*X(3)    (barycentrics and combo, Peter Moses, August 24, 2014)

Let O′ be the circle with center X(2) that passes through the points X(98) and X(842); then X(6039) and X(6040) are the points of intersection of O′ and the Euler line. Also, ({X(6039),X(6040}-harmonic conjugate of X(4)) = X(1513), and ({X(6039),X(6040}-harmonic conjugate of X(383)) = X(1080). (Peter Moses, August 24, 2014)

X(6039) lies on these lines: {2,3}, {98,1380}, {262,1340}

X(6039) = reflection of X(6040) in X(2)

X(6040) = THOMSON-ISOGONAL CONJUGATE OF X(5639)

Barycentrics   f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = S⁴ + a²S_AS_ω(-2S² + S²_A + S²_B + S²_C)^1/2 - (S² - 2S_BS_C)S²_ω
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)[cos B cos C + 3 sin B sin C - (cos A)(3 - 4e*cos ω + 4 cos 2ω)], where e = (1 - 4 sin²ω)^1/2
X(6040) = 3(3S² + S²_A + S²_B + S²_C)*X(2) + 2S_ω[S_ω + (-S² + S²_A + S²_B + S²_C)^1/2]*X(3)   (barycentrics and combo, Peter Moses, August 24, 2014)

X(6040) lies on these lines: {2,3}, {98,1379}, {262,1341}

X(6040) = reflection of X(6039) in X(2)

X(6041) = TRIPOLAR CENTROID OF X(1976)

Barycentrics f(a,b,c)) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2(b^2 - c^2)(2a^6 - b^6 - c^6 - 2a^4b^2 - 2a^4c^2 + a^2b^4 + a^2c^4 + b^4c^2 + b^2c^4) (Randy Hutson, September 5, 2014)

(See X(1635) for the definition of tripolar centroid.)

X(6041) lies on these lines: {6,647}, {25,2489}, {32,512}, {351,865}, {523,5306}, {2510,2780}, {3049,3051}

X(6041) = isogonal conjugate of X(6035)
X(6041) = crossdifference of every pair of points on the line X(30)X(99)

X(6042) = PERSPECTOR OF MONTESDEOCA-HUNG TRIANGLE AND ABC

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

Let (Ap) be the Apollonius circle, and let (K_A), (K_B), (K_C) be the circles described at X(5973) in association with the Hung-Feuerbach circle at X(5974). Let L_A be the radical axis of (Ap) and (K_A), and define L_B and L_C cyclically. The lines L_A, L_B, (L_C form a triangle T (here named the Montesdeoca-Hung triangle) that is perspective to ABC, and the perspector is X(6042). (Tran Quang Hung, ADGEOM #1506; Angel Montesdeoca, ADGEOM #1525, August 24, 2014)

The triangle T is a central triangle of type 1, perpsective to each of the following triangles: ABC, excentral (see X(6043), Apollonius, incentral, Feuerbach, and extangents. (Peter Moses, August 24, 2014).

The A-vertex of the central triangle T has these barycentrics: g(a,b,c) : b (a+c)^2 (a^2+a b+b c+c^2)^2 : (a+b)^2 c (a^2+b^2+a c+b c)^2, where
g(a,b,c) = -a (2 a^6+4 a^5 b+4 a^4 b^2+4 a^3 b^3+3 a^2 b^4+2 a b^5+b^6+4 a^5 c+12 a^4 b c+16 a^3 b^2 c+12 a^2 b^3 c+6 a b^4 c+2 b^5 c+4 a^4 c^2+16 a^3 b c^2+20 a^2 b^2 c^2+8 a b^3 c^2+b^4 c^2+4 a^3 c^3+12 a^2 b c^3+8 a b^2 c^3+3 a^2 c^4+6 a b c^4+b^2 c^4+2 a c^5+2 b c^5+c^6) (Peter Moses, August 24, 2014)

X(6042) lies on these lines: {1,5974}, {10,2643}, {73,2632}, {191,849}, {238,3743}, {757,1046}, {846,1098}, {960,1193}, {1104,1962}, {1682,5975}, {3708,3954}, {3847,4647}

X(6043) = PERSPECTOR OF MONTESDEOCA-HUNG TRIANGLE AND EXCENTRAL TRIANGLE

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

Continuing from X(6042), the triangle T is perspective to the excentral triangle, and the perspector is X(6043). (Peter Moses, August 24, 2014)

X(6043) lies on these lines: {1,5974}, {8,2363}, {31,643}, {33,162}, {42,2185}, {43,662}, {58,519}, {81,518}, {82,3791}, {100,593}, {171,319}, {181,5975}, {270,1891}, {849,3293}, {1098,5247}, {1931,4640}, {1963,4682}, {2651,4641}, {4203,5035}

X(6044) = TOUCHPOINT OF THE APOLLONIUS CIRCLE AND THE MOSES-HUNG CIRCLE

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

Following constructions at X(5973), Peter Moses (August 24, 2014) constructs a triangle DEF and circle (J), beginning with the orthic triangle H_AH_BH_C and the excircles (I_A), (I_B), (I_C). Let (J_A) be the circle, other than the nine-point circle, that passes through H_B and H_C and is tangent to (I_A). Let D be the touchpoint. Define (J_B) and (J_C) cyclically, and define E and F cyclically. Let (J) be the circle (here named the Moses-Hung circle) tangent to (J_A), (J_B), (J_C), and let (Ap) be the Apollonius circle. The circles (J) and (Ap) are tangent, and their touchpoint is X(6044). The triangle DEF, here named the Moses-Hung triangle, has vertices given by these barycentrics:

D = 2a³ + b³ + c³ + a²b + a²c - b²c - bc² : -(a + b + c)(a + c)²(a + b - c)³ : -(a + b + c)(a + b)²(a - b + c)³
E = -(a + b + c)(b + a)²(b + c - a)³ : 2b³ + c³ + a³ + b²c + b²a - c²a - ca² : -(a + b + c)(b + a)²(b + c - a)³
F = -(a + b + c)(c + b)²(c + a - b)³ : -(a + b + c)(c + a)²(c + a - b)³ : 2c³ + a³ + b³ + c²a + c²b - a²b - ab²

X(6044) lies on the Apollonius circle and these lines: {10,125}, {43,1726}, {181,1365}, {386,759}, {407,1829}, {440,2968}, {573,2249}, {2092,2264}, {2610,5213}

X(6044) = complement of X(38480)
X(6044) = inverse-in-Spieker-radical-circle of X(125)

X(6045) = CENTER OF MOSES-HUNG CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-5 a^5 b^3 c-3 a^4 b^4 c+6 a^3 b^5 c+2 a^2 b^6 c-a b^7 c+b^8 c+a^7 c^2-6 a^5 b^2 c^2-3 a^4 b^3 c^2+3 a^3 b^4 c^2+4 a^2 b^5 c^2+2 a b^6 c^2-b^7 c^2-5 a^5 b c^3-3 a^4 b^2 c^3+2 a^2 b^4 c^3+a b^5 c^3-3 b^6 c^3-3 a^5 c^4-3 a^4 b c^4+3 a^3 b^2 c^4+2 a^2 b^3 c^4-2 a b^4 c^4+3 b^5 c^4+6 a^3 b c^5+4 a^2 b^2 c^5+a b^3 c^5+3 b^4 c^5+3 a^3 c^6+2 a^2 b c^6+2 a b^2 c^6-3 b^3 c^6-a b c^7-b^2 c^7-a c^8+b c^8) (Peter Moses, August 24, 2014)

The Moses-Hung circle is defined at X(6044).

X(6045) lies on these lines: {4,8}, {442,511}, {573,2264}

X(6046) = PERSPECTOR OF ABC AND MOSES-HUNG TRIANGLE

Trilinears tan²(A/2) cos²(B/2 - C/2) : tan²(B/2) cos²(C/2 - A/2) : tan²(C/2) cos²(A/2 - B/2)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)²(a + b - c)³(a - b + c)³ (Peter Moses, August 24, 2014)

The Moses-Hung triangle is defined at X(6044).

X(6046) lies on these lines: {7,3486}, {11,273}, {55,347}, {56,1119}, {65,1439}, {269,1358}, {279,961}, {1427,1880}, {1441,3925}

X(6046) = isogonal conjugate of X(6061)

X(6047) = PERSPECTOR OF EXCENTRAL TRIANGLE AND MOSES-HUNG TRIANGLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c) (-5 a^5 b-3 a^4 b^2+6 a^3 b^3+2 a^2 b^4-a b^5+b^6-5 a^5 c-10 a^4 b c-2 a^3 b^2 c+4 a^2 b^3 c-a b^4 c-2 b^5 c-3 a^4 c^2-2 a^3 b c^2+4 a^2 b^2 c^2+2 a b^3 c^2-b^4 c^2+6 a^3 c^3+4 a^2 b c^3+2 a b^2 c^3+4 b^3 c^3+2 a^2 c^4-a b c^4-b^2 c^4-a c^5-2 b c^5+c^6) (Peter Moses, August 24, 2014)

The Moses-Hung triangle is defined at X(6044).

X(6047) lies on these lines: {10,12}, {40,1723}, {44,1842}, {55,387}, {71,1834}, {1865,1869}

X(6048) = CENTER OF MOSES HULL CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a²b + a²c + ab² + ac² + abc - 3b²c - 3bc²)
X(6048) = (r² + s²)*X(1) - 16rR*X(10) (barycentrics and combo, Peter Moses, August 26, 2014)

As at X(3596), let A′, B′, C′ denote the respective excircles of a triangle ABC. Let A″ be the circle tangent to A′, B′, C′ whose interior includes A′, and define B″ and C″ cyclically. Let (M) be the circle externally tangent to all three of the circles A′, B′, C′; that is, the circular hull of A′, B′, C′, of which the center is X(6048). Also, (M) is tangent to the Apollonius circle at X(3030), and the tangent line through X(3030) passes through X(2254). The radius of (M) is r(r² + s²)/(16rR - r² - s²). (Peter Moses, August 26, 2014) The circle (M) is here named the Moses hull circle.

The Moses hull circle is the inverse of the incircle in the excircles-radical circle. (Randy Hutson, September 5, 2014)

Let (Ia), (Ib), (Ic) be the excircles of ABC, and let (AP) be the classical Apollonius circle (e.g., TCCT, p. 102). Let (Ja) be the circle internally tangent to (Ia) and externally tangent to (Ib) and (Ic), and define (Jb) and Jc) cyclically. Let (J) be the circle internally tangent to (Ja), (Jb), (Jc). Then X(6048) = center of (J). See ADGEOM #1541 and Problema de Apolonio relativo a las circunferencias exinscritas . (Angel Montesdeoca, February 25, 2017)

X(6048) lies on these lines: {1,2}, {46,1757}, {181,3339}, {210,986}, {238,5687}, {573,3973}, {748,3871}, {970,3030}, {984,3697}, {992,3169}, {1376,4252}, {1724,3550}, {1739,5904}, {1743,4274}, {2092,3731}, {2177,5047}, {2238,3501}, {2640,5974}, {3230,4050}, {3596,3760}, {3666,3983}, {3740,4646}, {3752,4662}, {3812,4849}, {3869,4695}, {3876,4642}, {3987,5692}, {4383,5255}, {4659,4721}

X(6048) = isogonal conjugate of X(36602)
X(6048) = X(979)-Ceva conjugate of X(1)
X(6048) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,43,1), (8,899, 978), (8,978,1), (200,1722,1), (1698,3293,1), (3216,3679,1)

X(6049) = INTERSECTION OF LINES X(1)X(7) AND X(145)X(1420)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 3a)²/(b + c - a) : (c + a - 3b)²/(c + a - b) : (a + b - 3c)²/(a + b - c)
X(6049) = 8r*X(1) - (4R + r)*X(7)
X(6049) = 2(3r - 2R)*X(1) + r*X(20)

X(6049) is the perspector of ABC and a triangle denoted by A₂B₂C₂ in Nikolaos Dergiades, Antirhombi.

X(6049) lies on these lines: {1,7}, {8,1319}, {12,1388}, {55,1476}, {56,3241}, {57,3623}, {145,1420}, {388,4870}, {496,944}, {519,5265}, {551,5261}, {651,1616}, {952,5704}, {1058,3655}, {1317,1788}, {3361,3635}, {3621,3911}, {3622,5226}, {4188,5193}, {5252,5550}

X(6049) = isogonal conjugate of X(33963)
X(6049) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,20,4345), (1,3600,4323), (1,4293,5734), (1,4308,7), (145,1420,5435), (175,176,4862), (1388,3476,3616), (3600,4323,7), (4308,4323,3600), (4315,4355,3600)

X(6050) = RADICAL CENTER OF INNER HUNG CIRCLE, OUTER HUNG CIRCLE, AND CIRCUMCIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)(3a² - b² - c² - 2bc) (Peter Moses, August 26, 2014)

The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).

X(6050) lies on these lines: {512,4394}, {513,4401}, {649,4822}, {650,667}, {659,905}, {663,1635}, {1491,3803}, {1960,2516}, {4162,4730}, {4790,4983}

X(6051) = INSIMILICENTER OF INNER HUNG CIRCLE AND OUTER HUNG CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^2 b+2 a b^2+b^3+a^2 c+6 a b c+3 b^2 c+2 a c^2+3 b c^2+c^3) (Peter Moses, August 26, 2014)

The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).

X(6051) lies on these lines: {1,6}, {2,3702}, {3,968}, {10,3706}, {21,593}, {38,5045}, {42,5044}, {55,975}, {58,3683}, {201,5173}, {227,5219}, {241,3671}, {495,1070}, {501,2360}, {595,3745}, {612,3295}, {750,3579}, {846,3916}, {942,2292}, {962,5308}, {986,5439}, {1010,3685}, {1046,4038}, {1125,3666}, {1193,1962}, {1214,3485}, {1621,5266}, {1698,4646}, {1961,5255}, {2334,3715}, {2478,5725}, {2999,3646}, {3120,3824}, {3178,3847}, {3293,3740}, {3624,3752}, {3634,4868}, {3670,3742}, {3739,4647}, {3750,5293}, {3812,4424}, {3871,5297}, {3874,4883}, {3915,5311}, {3989,5049}, {3995,4968}, {3997,4520}, {4187,5530}, {4295,4648}, {4340,5698}, {4673,4687}, {4682,5264}, {4850,5550}, {5250,5287}, {5262,5284}, {5268,5687}

X(6051) = X(4627)-Ceva conjugate of X(650)
X(6051) = radical center of circles (Ka), (Kb), (Kc) described at X(5213)
X(6051) = {X(i), X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,984,3555), (1,1203,1100), (1,5259,1104), (2292,3720,942), (1125,3743, 3666), (5250,5287,5711)

X(6052) = EXSIMILICENTER OF INNER HUNG CIRCLE AND OUTER HUNG CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^7 b^2+4 a^6 b^3+7 a^5 b^4+8 a^4 b^5+7 a^3 b^6+4 a^2 b^7+a b^8-2 a^7 b c-4 a^6 b^2 c+3 a^5 b^3 c+17 a^4 b^4 c+16 a^3 b^5 c-2 a^2 b^6 c-9 a b^7 c-3 b^8 c+a^7 c^2-4 a^6 b c^2-24 a^5 b^2 c^2-29 a^4 b^3 c^2-45 a^3 b^4 c^2-80 a^2 b^5 c^2-60 a b^6 c^2-15 b^7 c^2+4 a^6 c^3+3 a^5 b c^3-29 a^4 b^2 c^3-120 a^3 b^3 c^3-202 a^2 b^4 c^3-139 a b^5 c^3-33 b^6 c^3+7 a^5 c^4+17 a^4 b c^4-45 a^3 b^2 c^4-202 a^2 b^3 c^4-178 a b^4 c^4-45 b^5 c^4+8 a^4 c^5+16 a^3 b c^5-80 a^2 b^2 c^5-139 a b^3 c^5-45 b^4 c^5+7 a^3 c^6-2 a^2 b c^6-60 a b^2 c^6-33 b^3 c^6+4 a^2 c^7-9 a b c^7-15 b^2 c^7+a c^8-3 b c^8) (Peter Moses, August 26, 2014)

The inner Hung circle and outer Hung circle are defined at X(5955) and X(5956).

X(6052) lies on this line: {2,3702}

X(6053) = INTERSECTION OF LINES X(6)X(13) AND X(20)X(110)

Barycentrics 2*a^10-10*(b^2+c^2)*a^8+17*(b^4+c^4)*a^6-(b^2+c^2)*(11*b^4-14*b^2*c^2+11*c^4)*a^4+(b^4+12*b^2*c^2+c^4)*(b^2-c^2)^2*a^2+(b^4-c^4)*(b^2-c^2)^3 : :

Let A′B′C′ be the half-altitude triangle. Let L be the line through A′ parallel to the Euler line, and define M and N cyclically. Let L′ be the reflection of L in sideline BC, and define M′ and N′ cyclically. The lines L′,M′,N′ concur in X(6053); see also X(i) for i = 74, 113, 265, 399, 1147, 1511, 5504, 5609, 5655. (Randy Hutson, August 26, 2014)

X(6053) lies on these lines: {6,13}, {20,110}, {74,3524}, {125,3090}, {133,648}, {140,5663}, {323,1533}, {541,1511}, {1387,2771}, {1539,3627}, {1596,3629}, {1619,2935}, {3448,5068}

X(6054) = REFLECTION OF X(98) IN X(2)

Barycentrics f(A,B,C) : f(B,C,A) : f(C, A, B), where f(A,B,C) = 2 sin B sec(B + ω) + 2 sin C sec(C + ω) - sin A sec(A + ω)

Let P be a point on the Steiner circumellipse. Let A′ be the orthocenter of BCP, and define B′, C′ cyclically. Let Q be the centroid of A′B′C′. The locus of Q as P varies is an ellipse similar and orthogonal to the Steiner circumellipse, and also centered at X(2). When P = X(99), Q = X(6054). See also X(98) and X(24808). (Randy Hutson, October 15, 2018)

X(6054) lies on these lines: {2,98}, {4,543}, {13,6114}, {14,6115}, {30,99}, {115,3545}, {148,3839}, {262,381}, {376,2482}, {378,2936}, {383,530}, {523,1551}, {524,1513}, {531,1080}, {551,2784}, {620,3524}, {3845,6321}, {5071,5461}

X(6054) = midpoint of X(2) and X(147)
X(6054) = reflection of X(2) in X(114)
X(6054) = anticomplement of X(6055)
X(6054) = Thomson isogonal conjugate of X(187)
X(6054) = centroid of X(511)-Fuhrmann triangle
X(6054) = X(381)-of-1st-anti-Brocard-triangle
X(6054) = antipode of X(98) in the circle O′ described at X(6039)
X(6054) = McCay-to-Artzt similarity image of X(3)
X(6054) = X(671)-of-Artzt-triangle
X(6054) = 1st-tri-squares-to-Artzt similarity image of X(13640)
X(6054) = orthologic center of these triangles: Artzt to 1st Brocard

X(6055) = MIDPOINT OF X(2) AND X(98)

Barycentrics 4*sin(A)*sec(A+ω)+sin(B)*sec(B+ω)+sin(C)*sec(C+ω) : :
Barycentrics 4*a^8-6*(b^2+c^2)*a^6+(7*b^4-2*b^2*c^2+7*c^4)*a^4-2*(b^2+c^2)*(3*b^4-5*b^2*c^2+3*c^4)*a^2+(b^4-4*b^2*c^2+c^4)*(b^2-c^2)^2 : :

X(6055) lies on these lines: {2,98}, {3,543}, {30,115}, {99,3524}, {351,2793}, {376,671}, {381,2794}, {541,5465}, {549,2482}, {620,5054}, {2023,5052}, {2783,6174}, {2784,3828}, {3023,5298}, {3027,4995}, {3534,6321}, {3815,5477}, {5017,6034}, {5055,6033}, {5092,5939}

X(6055) = complement of X(6054)
X(6055) = circumcircle-inverse of X(33900)
X(6055) = {X(6108),X(6109)}-harmonic conjugate of X(115)
X(6055) = center of inverse-in-O′-of-line X(115)X(125), where O′ = inverter of circumcircle and nine-point circle
X(6055) = harmonic center of circles O(13,16) and O(14,15)
X(6055) = McCay-to-Artzt similarity image of X(5)

Lozada Perspectors

Suppose that A′B′C′ be the cevian triangle of a point P. Let L be the line through A′ tangent to the incircle, and let A* be the touchpoint; define B* and C* cyclically. Then A*B*C* is perspective to ABC for all P. (César Lozada, May 7, 2014, Anopolis #1466)

If P = p : q : r (trilinears), then the perspector is P* = a(b + c - a)p² : b(a + c - b)q² : c(a + b - c)r².

In the following list, the appearance of (i,j) means that the Lozada perspector of X(i) is X(j):

{1,55}, {2,8}, {4,1857}, {6,2175}, {7,7}, {8,5423}, {9,480}, {11,5532}, {36,215}, {56,1397}, {57,56}, {63,1259}, {65,181}, {69,1264}, {75,3596}, {77,1804}, {81,60}, {86,261}, {88,1318}, {174,1}, {188,200}, {190,4076}, {226,12}, {241,1362}, {259,1253}, {266,31}, {278,1118}, {279,479}, {365,41}, {366,9}, {507,259}, {508,2}, {509,6}, {513,3271}, {514,11}, {519,4152}, {522,4081}, {523,4092}, {555,1088}, {556,341}, {650,3022}, {651,59}, {664,4998}, {900,4542}, {905,1364}, {1143,179}, {1274,400}, {1323,3321}, {1434,552}, {1465,1361}, {1638,3328}, {1659,1123}, {2003,2477}, {3008,3021}, {3218,4996}, {3666,1682}, {3668,6046}, {3669,1357}, {3676,1358}, {3776,3020}, {3911,1317}, {3960,3025}, {4146,75}, {4182,728}, {4369,3023}, {5374,219}, {5723,3322}

Let P = p : q : r (barycentrics). Let A′B′C′ be the cevian triangle of P, and let A*B*C* be the triangle defined in the preamble just above. The points A′, B′, C′, A*, B*, C* lie on a conic. (Vu Thanh Tung, February 15, 2021)

Peter Moses showed (February 16, 2021) that the conic just defined is given by the equation

f(a,b,c,p,q,r,x,y,z) + f(b,c,a,q,r,p,y,z,x) + f(c,a,b,r,p,q,z,x,y) = 0, where

f(a,b,c,p,q,r,x,y,z) = (a-b-c) q r ((a-b-c)^2 p^2+2 (a-b-c) (a-b+c) p q+(a-b+c)^2 q^2+2 (a-b-c) (a+b-c) p r+(a+b-c)^2 r^2) x^2+p ((a-b-c) p+(a-b+c) q+(a+b-c) r) ((a-b-c) (a-b+c) p q+(a-b+c)^2 q^2+(a-b-c) (a+b-c) p r-2 (a+b-c) (a-b+c) q r+(a+b-c)^2 r^2) y z

Continuing, Moses showed (February 17, 2021) gave the following generalization. Let T be the inscribed conic having perspector U = u : v : w. Let L be the line through A′ tangent to T, and let A*, given by

A* = w*(r*v - q*w^2 : q^2*v*w^2 : r^2*v^2*w,

be the touchpoint. Define B* and C* cyclically. Then A*B*C* is perspective to ABC with perspector v*w*p^2 : w*u*q^2 : u*v*r^2, and the points A′, B′, C′, A*, B*, C* lie on a conic, given by

g(a,b,c,p,q,r,u,v,w,x,y,z) + g(b,c,a,q,r,p,v,w,u,y,z,x) + g(c,a,b,r,p,q,u,u,v,z,x,y) = 0, where

g(a,b,c,p,q,r,u,v,w, x,y,z) = q*r*v*w*(r^2*u^2*v^2 - 2*p*r*u*v^2*w + q^2*u^2*w^2 - 2*p*q*u*v*w^2 + p^2*v^2*w^2)*x^2 - p*u*(-(r*u*v) - q*u*w + p*v*w)*(-(r^2*u*v^2) + 2*q*r*u*v*w + p*r*v^2*w - q^2*u*w^2 + p*q*v*w^2)*y*z

X(6056) = LOZADA PERSPECTOR OF X(3)

Trilinears (sin²2A)/(1 - cos A) : (sin²2B)/(1 - cos B) : (sin²2C)/(1 - cos C)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4*(-a + b + c)*(a^2 - b^2 - c^2)^2

X(6056) lies on these lines: {3,1794}, {11,1751}, {48,184}, {55,219}, {56,580}, {72,993}, {154,197}, {255,1092}, {394,1364}, {577,4055}, {578,3074}, {1397,2352}, {1802,2188}, {2330,5227}

X(6056) = isogonal conjugate of isotomic conjugate of X(1259)
X(6056) = isogonal conjugate of polar conjugate of X(219)
X(6056) = isotomic conjugate of polar conjugate of isogonal conjugate of X(331)
X(6056) = X(19)-isoconjugate of X(331)
X(6056) = X(92)-isoconjugate of X(278)
X(6056) = crossdifference of every pair of points on line X(7178)X(17924)

X(6057) = LOZADA PERSPECTOR OF X(10)

Trilinears cot²(A/2) cos²(B/2 - C/2) : cot²(B/2) cos²(C/2 - A/2) : cot²(C/2) cos²(A/2 - B/2)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)^2*(-a + b + c)

In the consturciton of X(5213), let D be the touchpoints of the circles (K_A) and (I_A), and define E and F cyclically. Then triangle DEF is perspective to ABC at X(6057). (Randy Hutson, August 26, 2014)

X(6057) lies on these lines: {8,3058}, {10,3175}, {11,312}, {12,1089}, {42,3943}, {55,346}, {200,4069}, {210,2321}, {306,3967}, {321,3925}, {345,5432}, {442,4066}, {594,756}, {1211,3773}, {2325,3683}, {2886,4671}, {2887,4135}, {2895,4756}, {3021,3886}, {3685,4030}, {3687,4009}, {3701,3704}, {3706,3717}, {3710,3714}, {3711,5423}, {3712,4995}, {3741,4439}, {3952,3969}, {3994,4415}, {3995,4026}, {4096,4535}, {4420,5160}, {4519,4847}, {4527,4685}, {4819,4849}, {4863,4901}, {4903,5233}, {4942,5905}

X(6057) = isotomic conjugate of X(552)
X(6057) = trilinear square of X(6725)

X(6058) = LOZADA PERSPECTOR OF X(12)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)^2*(a - b + c)^2*(b + c)^4*(-a + b + c)

X(6058) lies on these lines: {181,594}, {756,4092}, {1397,2345}

X(6059) = LOZADA PERSPECTOR OF X(19)

Trilinears        cos²(A/2) tan²A : cos²(B/2) tan²B : cos²(C/2) tan²C
Trilinears        a²(1 + cos A)/(1 + cos 2A) :
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2*(-a + b + c)*(a^2 + b^2 - c^2)^2*(a^2 - b^2 + c^2)^2

X(6059) lies on these lines: {25,1096}, {55,281}, {107,6015}, {158,242}, {181,3195}, {204,1460}, {607,2175}, {608,3271}, {1918,2207}

X(6059) = isogonal conjugate of X(7055)
X(6059) = isogonal conjugate of isotomic conjugate of X(1857)

X(6060) = LOZADA PERSPECTOR OF X(20)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = = (-a + b + c)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4)^2

X(6060) lies on these lines: {{8,21}, {20,1394}, {479,3188}, {2062,5932}, {3474,6046}

X(6061) = LOZADA PERSPECTOR OF X(21)

Trilinears cot²(A/2) sec²(B/2 - C/2) : cot²(B/2) sec²(C/2 - A/2) : cot²(C/2) sec²(A/2 - B/2)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²(a + b)²(a + c)²(b + c - a)³

X(6061) lies on these lines: {21,60}, {27,5057}, {63,110}, {72,2074}, {212,5546}, {219,2189}, {405,1175}, {2287,2326}, {2327,2328}

X(6061) = isogonal conjugate of X(6046)

X(6062) = LOZADA PERSPECTOR OF X(30)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4)^2

X(6062) lies on these lines: {11,214}, {55,4092}, {243,3326}, {1099,1354}, {1731,4542}, {3683,4081}

X(6063) = LOZADA PERSPECTOR OF X(85)

Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a^2*(-a + b + c)]
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = tan(A/2) csc²A
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (csc 2A(/(1 + sec A)
Barycentrics   j(A,B,C) : j(B,C,A) : j(C,A,B), where j(A,B,C) = csc A sec²(A/2)
Barycentrics   k(A,B,C) : k(B,C,A) : k(C,A,B), where k(A,B,C) = (csc A(/(1 + cos A)

Let Q denote isotomic conjugate of isogonal conjugate; e.g., the line X(918)X(3261) = Q(Gergonne line), of which X(6063) is the trilinear pole. Also, X(6063) = Brianchon point (perspector) of the inellipse that is Q(incircle), of which the center is X(2886). (Randy Hutson, September 5, 2014)

X(6063) lies on these lines: {2,4554}, {7,310}, {75,1088}, {76,85}, {181,3212}, {222,4573}, {274,278}, {305,561}, {497,2481}, {666,5452}, {873,4625}, {927,2862}, {1111,3944}, {1376,4998}

X(6063) = isogonal conjugate of X(2175)
X(6063) = isotomic conjugate of X(55)
X(6063) = complement of X(21218)
X(6063) = anticomplement of X(16588)
X(6063) = X(75)-cross conjugate of X(76)
X(6063) = pole with respect to the polar circle of the triilinear polar of X(607)
X(6063) = X(48)-isoconjugate of X(607)
X(6063) = trilinear product of the extraversions of X(8)
X(6063) = trilinear product of vertices of Gemini triangle 40

X(6064) = LOZADA PERSPECTOR OF X(99)

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

X(6064) lies on these lines: {99,4897}, {249,1016}, {422,4601}, {645,3700}, {4062,4570}

X(6065) = LOZADA PERSPECTOR OF X(100)

Trilinears cot²(A/2) csc²(B/2 - C/2)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2*(-a + b + c))/(b - c)^2

X(6065) lies on these lines: {8,1016}, {59,518}, {100,2742}, {110,2748}, {644,3900}, {663,3939}, {840,4996}, {902,1110}

X(6065) = isogonal conjugate of X(1358)
X(6065) = perpsector of ABC and the reflection of the intouch triangle in the line X(1)X(6)

X(6066) = LOZADA PERSPECTOR OF X(101)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4*(-a + b + c))/(b - c)^2

X(6066) lies on these lines: {55,1252}, {1110,2149}, {4998,5432}

X(6067) = LOZADA PERSPECTOR OF X(142)

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

X(6067) lies on these lines: {2,480}, {5,5223}, {7,2886}, {8,3826}, {9,11}, {12,518}, {56,2550}, {75,4081}, {142,354}, {390,3813}, {442,5542}, {516,3916}, {528,4996}, {1001,1259}, {1317,3872}, {1329,5686}, {2646,5853}, {3174,4863}

X(6068) = LOZADA PERSPECTOR OF X(527)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a + b + c)*(a^2 + 2*b*c - a*(b + c) - 2*S_A)^2

X(6068) lies on Mandart hyperbola and these lines: {7,3035}, {8,190}, {9,11}, {12,5880}, {72,2801}, {100,144}, {119,5762}, {200,5528}, {214,5850}, {390,5854}, {516,1145}, {518,1317}, {527,1155}, {952,5223}, {2829,5759}, {3036,5686}, {3057,3271}, {4542,5853}, {4996,5852}, {5779,5840}

X(6068) = reflection of X(11) in X(9)
X(6068) = reflection of X(3059) in X(14740)
X(6068) = antipode of X(3059) in the Mandart hyperbola
X(6068) = antipode in Mandart inellipse of X(11)

X(6069) = POINT AZHA

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^22 - 8 a^20 b^2 + 28 a^18 b^4 - 56 a^16 b^6 + 70 a^14 b^8 - 56 a^12 b^10 + 28 a^10 b^12 - 8 a^8 b^14 + a^6 b^16 - 8 a^20 c^2 + 42 a^18 b^2 c^2 - 92 a^16 b^4 c^2 + 106 a^14 b^6 c^2 - 62 a^12 b^8 c^2 + 7 a^10 b^10 c^2 + 13 a^8 b^12 c^2 - 8 a^6 b^14 c^2 + 4 a^4 b^16 c^2 - 3 a^2 b^18 c^2 + b^20 c^2 + 28 a^18 c^4 - 92 a^16 b^2 c^4 + 113 a^14 b^4 c^4 - 62 a^12 b^6 c^4 + 17 a^10 b^8 c^4 - 9 a^8 b^10 c^4 + 5 a^6 b^12 c^4 - 6 a^4 b^14 c^4 + 13 a^2 b^16 c^4 - 7 b^18 c^4 - 56 a^16 c^6 + 106 a^14 b^2 c^6 - 62 a^12 b^4 c^6 + 4 a^10 b^6 c^6 + 4 a^8 b^8 c^6 + 8 a^6 b^10 c^6 - 6 a^4 b^12 c^6 - 18 a^2 b^14 c^6 + 20 b^16 c^6 + 70 a^14 c^8 - 62 a^12 b^2 c^8 + 17 a^10 b^4 c^8 + 4 a^8 b^6 c^8 - 12 a^6 b^8 c^8 + 8 a^4 b^10 c^8 + 3 a^2 b^12 c^8 - 28 b^14 c^8 - 56 a^12 c^10 + 7 a^10 b^2 c^10 - 9 a^8 b^4 c^10 + 8 a^6 b^6 c^10 + 8 a^4 b^8 c^10 + 10 a^2 b^10 c^10 + 14 b^12 c^10 + 28 a^10 c^12 + 13 a^8 b^2 c^12 + 5 a^6 b^4 c^12 - 6 a^4 b^6 c^12 + 3 a^2 b^8 c^12 + 14 b^10 c^12 - 8 a^8 c^14 - 8 a^6 b^2 c^14 - 6 a^4 b^4 c^14 - 18 a^2 b^6 c^14 - 28 b^8 c^14 + a^6 c^16 + 4 a^4 b^2 c^16 + 13 a^2 b^4 c^16 + 20 b^6 c^16 - 3 a^2 b^2 c^18 - 7 b^4 c^18 + b^2 c^20

Let ABC be a triangle. Let L be a line through A, let M be a line through B parallel to L, and let N be a line through C parallel to L. Let L′ be the reflection of L in line BC, let M′ be the reflection of M in CA, and let N′ be the reflection of N in AB. Let A′B′C′ be the triangle formed by the lines L′, M′, N′. As L rotates about A, the Euler line of A′B′C′ rotates about a fixed point, X(6069).

X(6069) is described at http://anthrakitis.blogspot.com/2014/04/euler-lines-of-triagles-bounded-by.html (and misrepresented there as the Parry-Pohoata point, X(8157)).

X(6069) lies on this line: {68,1658}

Points Associated with the Steiner Deltoid

Suppose that ABC is a triangle. Its Steiner deltoid, SD, is the envelope of the Simson lines of ABC, as shown dynamically at Simson line. Suppose the P = p : q : r (barycentrics) is a point, other than A, B, C, on the circumcircle, denoted by (O), of ABC. Peter Moses (September 14, 2014) defines a mapping from (O) to SD; the mapping is here denoted by M and named the Moses-Steiner image, as follows:

M(P) = b⁴c⁴p⁶(b²rS_B - c²qS_C)²(c²q² + b² r² + 2qrS_A)[S_A(c²q² + b²r²) + 2qr(S² + S²_A)].

The Simson line of P is tangent to SD at M(P), and the Simson line intersects SD, in two other points, which are the points of intersection of the Simson line with the circle that has radius R and center the midpoint of P and the orthocenter, H. The SD is given by the equation s(a,b,c,x,y,z) + s(b,c,a,y,z,x) + s(c,a,b,z,x,y) = 0, where

s(a,b,c,x,y,z) = (a^2 + b^2 - c^2)^2 (a^2 - b^2 + c^2)^2 x^4 - 4 (a^2 + b^2 - c^2) (a^6 - 2 a^4 b^2 + a^2 b^4 - 2 a^4 c^2 - 2 a^2 b^2 c^2 - 4 b^4 c^2 + a^2 c^4 + 4 b^2 c^4) y^3 z - 4 (a^2 - b^2 + c^2) (a^6 - 2 a^4 b^2 + a^2 b^4 - 2 a^4 c^2 - 2 a^2 b^2 c^2 + 4 b^4 c^2 + a^2 c^4 - 4 b^2 c^4) z^3 y + 2 (3 a^8 - 6 a^6 b^2 + 2 a^4 b^4 + 2 a^2 b^6 - b^8 - 6 a^6 c^2 - 8 a^4 b^2 c^2 - 2 a^2 b^4 c^2 + 16 b^6 c^2 + 2 a^4 c^4 - 2 a^2 b^2 c^4 - 30 b^4 c^4 + 2 a^2 c^6 + 16 b^2 c^6 - c^8) y^2 z^2 + 4 (a^8 - 9 a^6 b^2 + 15 a^4 b^4 - 7 a^2 b^6 - 9 a^6 c^2 + 48 a^4 b^2 c^2 + 7 a^2 b^4 c^2 + 2 b^6 c^2 + 15 a^4 c^4 + 7 a^2 b^2 c^4 - 4 b^4 c^4 - 7 a^2 c^6 + 2 b^2 c^6) x^2 y z.

Let L₁ be the line of the midpoint of P and H and the circumcircle-antipode of the isogonal conjugate of P. Let W be the point of intersection of the line of P and the isogonal conjugate of P with the line of the circumcircle-antipode of P and the orthopoint of the isogonal conjugate of P. Let L₂ be the line of W and the point of intersection of the anticomplement of P and the circumcircle-antipode of P. Then M(P) = L₁∩L₂. For example, if P = X(110), then L₁ = X(30)X(113) and L₂ = X(146)X(476), and M(P) = X(1553), the only point X(i), for i < 6070, that lies on SD. (Peter Moses, September 14, 2014).

Let P* denote the circumcircle-antipode of P, and let Q= M(P*); then |P*Q| = 2R. Let U(P) be the reflection of X(4) in the midpoint of segment P*Q. Then U(P) lies on the circumcircle, so that the point T = M(U(P)) is on the SD; indeed, the line M(P)Q is tangent to SD at T (Moses, September 20, 2014). The mapping U is the Moses-Steiner ortho-image, and T, the Moses-Steiner tangential-image. Examples:

U(X(74)) = U(X(110)) = X(901)
U(X(98)) = U(X(99)) = X(805)
U(X(100)) = U(X(104)) = X(476)
U(X(101)) = U(X(103)) = X(927)
U(X(102)) = U(X(109)) = X(1309)
U(X(112)) = U(X(1297)) = X(2867)
U(X(105)) = U(X(1292)) = X(6078)
U(X(106)) = U(X(1293)) = X(6079)
U(X(107)) = U(X(1294)) = X(6080)
U(X(108)) = U(X(1295)) = X(6081)
U(X(111)) = U(X(1296)) = X(6082)
U(X(759)) = U(X(6011)) = X(6083)

Another construction of U(P), along with barycentrics, follows (Moses, September 20, 2014). Let L₃ be the line through P perpendicular to the Simson line of P, and let L₄ be the line through P perpendicular to the Simson line of M(P). Then U = L₃∩L₄. Moreover, L₃ and L₄ both meet the nine-point circle in the midpoint, U′, of U and P; thus, a third construction of U(P) is the -2 dilation of U′ in G. Barycentrics for U are u(a,b,c,p,q,r) : u(b,c,a,q,r,p) : u(c,a,b,r,p,q), where

u(a,b,c,p,q,r) = a⁶q⁶r⁶(a²q² - b²p²)(a²r² - c²p²)/[b² + c² - a²)qr) + b²r² + c²q²]²,

and barycentrics for the point T = M(U(P)) are t(a,b,c,p,q,r) : t(b,c,a,q,r,p) : t(c,a,b,r,p,q), where

t(a,b,c,p,q,r) = a^4 (2 b^2 c^2 q r+c^2 q^2 S_A+b^2 r^2 S_A)^2 (6 b^4 c^4 q^2 r^2 (S^2+S_B S_C)+c^6 q^4 (4 S^2 S_A-b^2 (3 S^2-S_B S_C))+b^6 r^4 (4 S^2 S_A-c^2 (3 S^2-S_B S_C))+4 b^2 c^2 q r (b^4 r^2 (S^2-S_B^2)+c^4 q^2 (S^2-S_C^2))))/(c^2 q^2+b^2 r^2+2 q r S_A)^4.

The vertices of the SD form a central triangle A′B′C′, here named the 1st Steiner equilateral triangle, for which Moses (September 14, 2014) gives these trilinears:

cos(B - C) + 3 cos(B/3 - C/3) : cos(C - A) - 3 cos(B/3 + 2C/3) : cos(A - B) - 3 cos(2B/3 + C/3).

The SD touches the nine-point circle in points that form another central triangle A*B*C*, here named the 2nd Steiner equilateral triangle, for which A* has these trilinears:

cos(B - C) - cos(B/3 - C/3) : cos(C - A) + cos(B/3 + 2C/3) : cos(A - B) + cos(2B/3 + C/3).

Properties of these triangles and the SD are given in C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications (Dover, 1963), and H. Dorrie, Great Problems of Elementary Mathematics (Dover, 1963), p. 226. Such properties are as follows: The triangles A′B′C′ and A*B*C* are homothetic at X(5), with ratio -1/3. The circumcircle of A′B′C′ is centered at X(5) with radius 3R/2. The SD is tangent to the sidelines of ABC at the vertices of the pedal triangle of X(20). The triangles ABC, BCH, CAH, and ABH have the same SD.

A construction of A* is given in H. S. M. Coxeter, Introduction to Geometry, 2nd ed., p. 115: Let DEF be the medial triangle and UVW the orthic triangle; on the ninepoint circle, A* is the point of trisection of the arc DU.

The radical axes of the Stammler circles are tangent to the SD. These axes are the Simson lines of the circumtangential triangle.

The triangle A*B*C* is the complement of the triangle A′B′C′, which is homothetic to the circumtangential triangle at X(20) with ratio 3/2. The circumtangential is homothetic to A*B*C* at X(2) with ratio -1/2. The triangle A*B*C* is homothetic to the circumnormal triangle at X(4) with ratio 1/2, and the circumnormal triangle is homothetic to A′B′C′ at X(631) with ratio -3/2. The homothetic centers X(2), X(4), X(20), X(631) all lie on the Euler line.

The Stammler triangle is homothetic to A′B′C′ at X(3526) with ratio -3/4, and also homothetic to A*B*C* at X(381) with ratio 1/4. The homothetic centers X(3526) and X(381) lie on the Euler line.

Let L_A be the line tangent to the SD at A*, and define L_B and L_C cyclically. The lines L_A, L_B, L_C form an equilateral triangle which is the complement of the Stammler triangle, homothetic to A′B′C′ at X(5) with ratio 2/3. Trilinears for the A-vertex are given by Moses as follows:

cos(B - C) + 2 cos(B/3 - C/3) : cos(C - A) - 2 cos(B/3 + 2C/3) : cos(A - B) - 2 cos(2B/3 + C/3).

As a point P traverses the SD, the locus of the orthopoles of N to P on the SD is an ellipse inscribed to SD; its center is X(546) and it passes through X(125) and Z(1539). (Peter Moses, September 14, 2014). A barycentric equation for this Moses-Steiner ellipse is

f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0, where

f(a,b,c,x,y,z) = (a^2-b^2-2 c^2) (a^2-2 b^2-c^2) (4 a^4-3 a^2 b^2-b^4-3 a^2 c^2+2 b^2 c^2-c^4) x^2 + 2 (2 a^8+2 a^6 b^2-9 a^4 b^4+4 a^2 b^6+b^8+2 a^6 c^2+20 a^4 b^2 c^2-4 a^2 b^4 c^2-9 b^6 c^2-9 a^4 c^4-4 a^2 b^2 c^4+16 b^4 c^4+4 a^2 c^6-9 b^2 c^6+c^8) y z

X(6070) = MOSES-STEINER IMAGE OF X(74)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (S_B - S_C)^2 (a^2 S_A^3 + S_A (4 S_A - a^2) S_B S_C + 2 S_B^2 S_C^2 - 3 S_A^2 (S_B^2 + S_C^2))

The Simson line of X(74) is parallel to the orthic axis.

X(6070) lies on the Steiner deltoid and these lines: {4,5627}, {125,523}, {137,2970}, {476,3448}, {868,5489}, {1522,1523}

X(6071) = MOSES-STEINER IMAGE OF X(98)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a^2 (S_B - S_C)^2 (-a^2 S_A^3 + c^2 S_A S_B^2 + (b^2 S_A - 2 S_B^2) S_C^2)

The Simson line of X(98) is perpendicular to the Brocard axis and parallel to the Lemoine axis.

X(6071) lies on the Steiner deltoid and these lines: {115,512}, {148,805}, {1567,2782}

X(6072) = MOSES-STEINER IMAGE OF X(99)

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a^2 (S_A^2 - S_B S_C)^2 (a^2 S_A^3 + 2 S_B^2 S_C^2 - S_A^2 (S_B^2 + S_C^2) - S_A (S_B^3 + S_C^3))

The Simson line of X(99) is parallel to the Brocard axis.

X(6072) lies on these lines: {114,325}, {147,805}, {1567,2782}

X(6073) = MOSES-STEINER IMAGE OF X(100)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2 b - b^3 + a^2 c - 2 a b c + b^2 c + b c^2 - c^3)^2 (2 a^4 - 2 a^3 b - a^2 b^2 + 2 a b^3 - b^4 - 2 a^3 c + 4 a^2 b c - 2 a b^2 c - a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - c^4)

The Simson line of X(100) is parallel to the line X(1)X(3).

X(6073) lies on the Steiner deltoid and these lines: {119,517}, {153,901}

X(6074) = MOSES-STEINER IMAGE OF X(101)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2 a^3 - a^2 b - b^3 - a^2 c + b^2 c + b c^2 - c^3)^2 (a^4 b^2 - 2 a^3 b^3 + 2 a b^5 - b^6 + a^4 c^2 + 2 a^2 b^2 c^2 - 2 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - 2 a b^2 c^3 + 4 b^3 c^3 - b^2 c^4 + 2 a c^5 - c^6)

The Simson line of X(101) is parallel to the Soddy line.

X(6074) lies on the Steiner deltoid and these lines: {4,5377}, {118,516}, {152,927}, {1517,2808}

X(6075) = MOSES-STEINER IMAGE OF X(104)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)^2 (2 a^4 - 2 a^3 b - a^2 b^2 + 2 a b^3 - b^4 - 2 a^3 c + 4 a^2 b c - 2 a b^2 c - a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - c^4)

The Simson line of X(104) is parallel to the antiorthic axis and perpendicular to X(1)X(3).

X(6075) lies on the Steiner deltoid and these lines: {11,513}, {36,851}, {149,901}, {244,1365}, {764,1647}, {1319,3011}, {2841,5533}, {2969,3756}, {3814,4871}, {5176,5205}

X(6076) = MOSES-STEINER IMAGE OF X(111)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (5 a^2 - b^2 - c^2)^2 (b^2 - c^2)^2 (a^6 b^2 + a^4 b^4 - a^2 b^6 - b^8 + a^6 c^2 - 14 a^4 b^2 c^2 + 10 a^2 b^4 c^2 + 7 b^6 c^2 + a^4 c^4 + 10 a^2 b^2 c^4 - 20 b^4 c^4 - a^2 c^6 + 7 b^2 c^6 - c^8)

The Simson line of X(111) is perpendicular to X(2)X(6).

X(6076) lies on the Steiner deltoid and these lines: {1499,2686}, {6077,6092}

X(6076) = reflection of X(6077) in X(6092)
X(6076) = orthojoin of X(2482)

X(6077) = MOSES-STEINER IMAGE OF X(1296)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2 a^2 - b^2 - c^2)^2 (a^6 b^2 + a^4 b^4 - a^2 b^6 - b^8 + a^6 c^2 - 14 a^4 b^2 c^2 + 10 a^2 b^4 c^2 + 7 b^6 c^2 + a^4 c^4 + 10 a^2 b^2 c^4 - 20 b^4 c^4 - a^2 c^6 + 7 b^2 c^6 - c^8)

The Simson line of X(1296) is parallel to X(2)X(6).

X(6077) lies on the Steiner deltoid and these lines: {126,524}, {6076,6092}

X(6078) = MOSES-STEINER ORTHO-IMAGE OF X(105)

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

Let A′ = BC∩X(8)X(9), and define B′ and C′ cyclically. The circumcircles of AB′C′, BC′A′, CA′B′ concur in X(6078). (Randy Hutson, September 29, 2014)

X(6078) lies on the circumcircle and these lines: {2,5519}, {100,4394}, {101,6065}, {103,1810}, {105,518}, {106,5526}, {109,1252}, {644,1292}, {649,1293}, {672,1477}, {759,5525}, {883,927}, {901,2441}, {919,2284}, {934,4564}, {1023,1308}

X(6078) = isogonal conjugate of X(6084)
X(6078) = anticomplement of X(5519)
X(6078) = trilinear pole of the line X(6)X(3939)
X(6078) = intersection of antipedal lines of X(105) and X(1292)
X(6078) = Ψ(X(6), X(3939)

X(6079) = MOSES-STEINER ORTHO-IMAGE OF X(106)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b) (a-c) (a^2-4 a b+b^2+a c+b c) (a^2+a b-4 a c+b c+c^2)

From Randy Hutson (September 28, 2014):
X(6079) = Ψ(X(6), X(644))
X(6079) = Λ(X(244), X(1357)); the line X(244)X(1357) is the trilinear polar of X(3669)
X(6079) = Λ(X(121), X(6085)); the line X(121)X(6085) is tangent to the nine-point circle at X(121)
X(6079) = Λ(X(2254), X(3030)); the line X(2254)X(3030) is tangent to the Apollonius circle at X(3030)

X(6079) lies on the circumcircle and these lines: {2,5516}, {100,4076}, {104,1811}, {105,5205}, {106,519}, {109,765}, {190,6014}, {741,5524}, {934,4998}, {1293,3667}

X(6079) = isogonal conjugate of X(6085)
X(6079) = isotomic conjugate of X(4927)
X(6079) = anticomplement of X(5516)
X(6079) = intersection of antipedal lines of X(106) and X(1293)
X(6079) = trilinear pole of the line X(6)X(644)

X(6080) = MOSES-STEINER ORTHO-IMAGE OF X(107)

Barycentrics a^2*(a^2-c^2)*(c*a^5-b^2*a^4+2*(b^2-c^2)*c*a^3+2*(b^2-c^2)*b^2*a^2-(b^2-c^2)*(3*b^2+c^2)*c*a-(b^2-c^2)^2*b^2)*(c*a^5+b^2*a^4+2*(b^2-c^2)*c*a^3-2*(b^2-c^2)*b^2*a^2-(b^2-c^2)*(3*b^2+c^2)*c*a+(b^2-c^2)^2*b^2)*(a^2-b^2)*(b*a^5-c^2*a^4-2*(b^2-c^2)*b*a^3-2*(b^2-c^2)*c^2*a^2+(b^2-c^2)*(b^2+3*c^2)*b*a-(b^2-c^2)^2*c^2)*(b*a^5+c^2*a^4-2*(b^2-c^2)*b*a^3+2*(b^2-c^2)*c^2*a^2+(b^2-c^2)*(b^2+3*c^2)*b*a+(b^2-c^2)^2*c^2) : :

X(6080) lies on the circumcircle and these lines: {107,520}, {1294,6000}, {2693,3357}

X(6080) = isogonal conjugate of X(6086)
X(6080) = anticomplement of X(35579)
X(6080) = trilinear pole of line X(6)X(35071)
X(6080) = Ψ(X(6), X(35071))
X(6080) = intersection of antipedal lines of X(107) and X(1294)

X(6081) = MOSES-STEINER ORTHO-IMAGE OF X(108)

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

X(6081) lies on the circumcircle and these lines: {84,2716}, {108,521}, {189,2723}, {271,2733}, {280,2734}, {934,4131}, {1295,6001}, {2357,2708}

X(6081) = isogonal conjugate of X(6087)
X(6081) = anticomplement of X(35580)
X(6081) = trilinear pole of the line X(6) X(268)
X(6081) = intersection of antipedal lines of X(108) and X(1295)
X(6081) = Ψ(X(6),X(268))

X(6082) = MOSES-STEINER ORTHO-IMAGE OF X(111)

Barycentrics (a^2-b^2)*(a^2-c^2)*(9*a^2*c^2-(a^2+b^2+c^2)^2)*(9*a^2*b^2-(a^2+b^2+c^2)^2) : :

Let H be the rectangular circumhyperbola that has center X(6092) and has major axis parallel to the line X(2)X(6). Then X(6082) is the point, other than A, B, C, in which H intersects the circumcircle. (Randy Hutson, September 29, 2014)

X(6082) lies on the circumcircle and these lines: {98,5971}, {111,524}, {352,729}, {691,5468}, {1296,1499}, {5108,5970}

X(6082) = reflection of X(6093) in X(3)
X(6082) = isogonal conjugate of X(6088)
X(6082) = anticomplement of X(31654)
X(6082) = trilinear pole of the line X(6)X(2482)
X(6082) = Ψ(X(96), X(2482))
X(6082) = Λ(X(111), X(351))
X(6082) = Collings transform of X(6092)
X(6082) = circumcircle intercept, other than A, B, C, of conic {{A,B,C,PU(138)}}
X(6082) = intersection of antipedal lines of X(111) and X(1296)

X(6083) = MOSES-STEINER ORTHO-IMAGE OF X(759)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²/[(b² - c²)(2a³ + b³ + c³ - ab² - ac² - b²c - bc²)]

X(6083) lies on the circumcircle and these lines: {99,4897}, {105,2651}, {106,5127}, {108,5379}, {109,4570}, {643,6003}, {758,759}

X(6083) = isogonal conjugate of X(6089)
X(6083) = anticomplement of X(35583)
X(6083) = trilinear pole of the line X(6)X(5546)
X(6083) = Ψ(X(6), X(5546))

X(6084) = ISOGONAL CONJUGATE OF X(6078)

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

X(6084) lies on these lines: {2,4927}, {30,511}, {105,659}, {120,2977}, {190,2415}, {673,2402}, {903,2403}, {1086,1358}, {1635,1638}, {1639,4728}, {2487,3676}, {2490,4885}, {2505,4925}, {2527,4369}, {2976,3021}, {3039,4422}, {3700,4382}, {4106,4468}, {4380,4897}, {4458,4830}, {4750,4773}

X(6084) = crossdifference of every pair of points on the line X(6)X(3939)

X(6085) = ISOGONAL CONJUGATE OF X(6079)

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

X(6085) lies on these lines: {30,511}, {106,1960}, {244,1357}, {659,1054}, {665,3768}, {764,1769}, {1293,3939}, {2254,3030}, {2316,2441}, {3888,3952}

X(6085) = crossdifference of every pair of points on the line X(6)X(644)

X(6086) = ISOGONAL CONJUGATE OF X(6080)

Barycentrics a^2[b^2(sec^2 A - sec^2 B)^2 - c^2(sec^2 C - sec^2 A)^2] : :

X(6086) lies on these lines: {30,511}, {107,1624}

X(6086) = crossdifference of every pair of points on line X(6)X(35071)
X(6086) = complementary conjugate of X(35579)

X(6087) = ISOGONAL CONJUGATE OF X(6081)

Trilinears b(cos C cot C/2)/(-1 - cos C + cos A + cos B) - c(cos B cot B/2)/(-1 - cos B + cos C + cos A) : :

X(6087) lies on these lines: {30,511}, {108,676}

X(6087) = isogonal conjugate of X(6081)
X(6087) = complementary conjugate of X(35580)
X(6087) = crossdifference of every pair of points on the line X(6)X(268)

X(6088) = ISOGONAL CONJUGATE OF X(6082)

Barycentrics a^2(b^2 - c^2)[(a^2 + b^2 + c^2)^2 - 9b^2c^2] : :

X(6088) lies on these lines: {30,511}, {111,351}

X(6088) = crossdifference of every pair of points on the line X(6)X(2482)
X(6088) = X(523)-of-4th-anti-Brocard-triangle
X(6088) = X(526)-of-circumsymmedial-triangle

X(6089) = ISOGONAL CONJUGATE OF X(6083)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b² - c²)(2a³ + b³ + c³ - ab² - ac² - b²c - bc²)

X(6089) lies on these lines: {30,511}, {659,1283}, {1365,2611}, {1635,1637}

X(6089) = crossdifference of every pair of points on the line X(6)X(5546)
X(6089) = complementary conjugate of X(35583)

X(6090) = INTERSECTION OF LINES X(3)X(74) AND X(6)X(373)

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(6090) = X(25) + 3*X(394) (Peter Moses, September 27, 2014)

X(6090) occurs in connection with the Steinhaus cubic: (K698)

Let H be the Thomson-Gibert-Moses hyperbola. Let U be the line tangent at X(6) to H, and let V the the line tangent at X(154) to H. Then X(6090) = U∩V. (Randy Hutson, September 29, 2014)

X(6090) lies on these lines: {2,3167}, {3,74}, {6,373}, {25,394}, {51,5102}, {69,468}, {154,3917}, {184,5085}, {323,1351}, {352,1384}, {426,1073}, {520,2433}, {576,3066}, {599,5642}, {1092,1593}, {1316,5468}, {1350,1495}, {1352,5094}, {1975,4563}, {1993,5020}, {2715,2763}, {3515,5562}, {3516,5907}, {3796,3819}, {5108,5967}

X(6090) = crossdifference of every pair of points on the line X(1499)X(1514)
X(6090) = pole with respect to the Thomson-Gibert-Moses hyperbola of the line X(6)X(25)
X6090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (184,5650,5085), (323,1995,1351), (1993,5640,5093), (3292,5651,6), (5020,5093,5640)

X(6091) = ISOGONAL CONJUGATE OF X(5203)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a²S_A(S_A - a²)/(2S_A - a²)
X(6091) = X(2079) + X(3565) (Peter Moses, September 27, 2014)

X(6091) occurs in connection with the Steinhaus cubic: (K698)

X(6091) lies on these lines: {3,895}, {22,111}, {186,691}, {187,5166}, {376,671}

X(6091) = X(111)-Ceva conjugate of X(895)
X(6091) = inverse-in-circumcircle of X(895)

X(6092) = MIDPOINT OF X(4) AND X(6082)

Barycentrics (a^6*b^2 + a^4*b^4 - a^2*b^6 - b^8 + a^6*c^2 - 14*a^4*b^2*c^2 + 10*a^2*b^4*c^2 + 7*b^6*c^2 + a^4*c^4 + 10*a^2*b^2*c^4 - 20*b^4*c^4 - a^2*c^6 + 7*b^2*c^6 - c^8)*(2*a^8 - 6*a^6*b^2 + 19*a^4*b^4 - 8*a^2*b^6 + b^8 - 6*a^6*c^2 - 20*a^4*b^2*c^2 + 4*a^2*b^4*c^2 + 19*a^4*c^4 + 4*a^2*b^2*c^4 - 2*b^4*c^4 - 8*a^2*c^6 + c^8) : :

X(6092) is the center of the rectangular hyperbola {A, B, C, X(4), X(6082)}. This hyperbola is the isogonal conjugate of the line X(3)X(6088); its major axis is parallel to the line X(2)X(6). (Randy Hutson, September 29, 2014)

For a construction, see, Antreas Hatzipolakis and Peter Moses, Hyacinthos 29114.

X(6092) lies on the nine-point circle and these lines: {2, 6093}, {4, 6082}, {5, 31654}, {125, 8355}, {126, 1499}, {524, 5512}, {599, 5099}, {5139, 16183}, {6076, 6077}, {20388, 20389}

X(6092) = midpoint of X(i) and X(j) for these {i,j}: {4, 6082}, {6076, 6077}
X(6092) = reflection of X(31654) in X(5)
X(6092) = complement of X(6093)
X(6092) = nine-point circle antipode of X(31654)

X(6093) = INTERSECTION OF LINES X(3)X(6082) AND X(524)X(1296)

Barycentrics (a^8+(b^2-7*c^2)*a^6-(b^4+10*b^2*c^2-20*c^4)*a^4-(b^6+7*c^6-2*(7*b^2-5*c^2)*b^2*c^2)*a^2-(b^4-c^4)*(b^2+c^2)*c^2)*(a^8-(7*b^2-c^2)*a^6+(20*b^4-10*b^2*c^2-c^4)*a^4-(7*b^6+c^6+2*(5*b^2-7*c^2)*b^2*c^2)*a^2+(b^4-c^4)*(b^2+c^2)*b^2) : :

X(6093) lies on these lines: {2,6092}, {3,6082}, {111,1499}, {524,1296}, {691,1992}

X(6093) = isogonal conjugate of X(33962)
X(6093) = anticomplement of X(6092)
X(6093) = trilinear pole of line X(6)X(9125)
X(6093) = circumcircle-antipode of X(6082)
X(6093) = Λ(X(3),X(111))

X(6094) = 11th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^4+5 a^2 b^2+b^4-4 a^2 c^2-4 b^2 c^2+c^4) (a^4-4 a^2 b^2+b^4+5 a^2 c^2-4 b^2 c^2+c^4)

Suppose that P = p : q : r (barycentrics) and P* are a pair of isogonal conjugate points in the plane of a triangle ABC. Let A′B′C′ be the pedal triangle of P and A″B″C″ the pedal triangle of P*. Let A* be the reflection of A′ in line B″C″, and define B* and C* cyclically. In Hyacinthos (October 3, 2014), Antreas Hatzipolakis asks for the locus of P for which the circumcircles of A*BC, B*CA, C*AB concur. Angel Montesceoca responds that the three circumcircles concur for all choices of P. He further notes that if P is not on the circumcircle and not on the line at infinity, then the point Q of conurrence of the three circles has barycentrics Q(a,b,c,p,q,r) : Q(b,c,a,q,r,p) : Q(c,a,b,r,p,q) given by

Q(a,b,c,p,q,r) = (q + r)/[a⁴qr(p + q)(p + r) - 2a²(q + r)(r + p)(p + q)(c²q + b²r) + p(q + r)(b⁴r(p + q) + c⁴q(q + r) + 2b²c²(q² + r² + pq + qr + rp))]

Writing Q as Q(P), the occurrence of (i,j) in the following list means that Q(X(i)) = X(j): (1,5620), (2,6094), (3, 1263), (8, 6095), (20, 265), (69, 6096), (1138, 1138). In particular, Q(X(2)) = X(6094).

X(6094) lies on these lines: {6,543}, {263,2854}

X(6094) = isogonal conjugate of X(352)
X(6094) = trilinear pole of the line X(373)X(512)

X(6095) = 12th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^3-a^2 b-a b^2+b^3-2 a^2 c+7 a b c-2 b^2 c-2 a c^2-2 b c^2+c^3) (a^3-2 a^2 b-2 a b^2+b^3-a^2 c+7 a b c-2 b^2 c-a c^2-2 b c^2+c^3)

X(6095) = Q(X(8)); see X(6094).

X(6095) lies on these lines: {1,121}, {56,2802}, {106,1739}

X(6096) = 13th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics a^2*(a^6-(2*b^2+c^2)*a^4-(3*b^4-10*b^2*c^2+c^4)*a^2-(3*b^2-c^2)*(b^2+c^2)*c^2)*(a^6-(b^2+2*c^2)*a^4-(b^4-10*b^2*c^2+3*c^4)*a^2+(b^2-3*c^2)*(b^2+c^2)*b^2) :

X(6096) = Q(X(69)); see X(6094).

X(6096) lies on this line: {25,2854}

X(6096) = isogonal conjugate of X(5913)

X(6097) = 14th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^2-b^2-b c-c^2) (a^5 b-2 a^3 b^3+a b^5+a^5 c-a^2 b^3 c-a b^4 c+b^5 c-2 a^3 c^3-a^2 b c^3-2 b^3 c^3-a b c^4+a c^5+b c^5)
X(6097) = (r² + 2rR - R² + s²)*X(3) + R²*X(4) barycentrics, Peter Moses, October 3, 2014; combo, Angel Montesdoca, October 3, 2014

Let ABC be a triangle and A′B′C′ the cevian triangle of X(1). Let O_AB be the circumcenter of ABA′, and define O_BC and O_CA cyclically; let O_AC be the circumcenter of ACA′, and define O_BA and O_CB cyclically. Let O_A be the circumcenter of triangle AO_ABO_AC, and define O_B and O_C cyclically. Hatzipolakis proposed, and Montesdeoca proved, that the Euler lines concur, in X(186), and that the orthocenter of triangle O_AO_BO_C, which is X(6097), lies on the Euler line. See Anthrakitis (October 3, 2014)

X(6097) lies on these lines: {2,3}, {35,500}, {55,5453}, {511,5495}, {3724,5492}

X(6098) = 15th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^2 - b^2 + b c - c^2) (a^2 (b + c) - 2 a b c - b^3 + b^2 c + b c^2 - c^3) / ((b^2 + c^2 - a^2) (a^6 - a^4 (b^2 - b c + c^2) - a^3 b c (b + c) - a^2 (b^4 - b^3 c - 2 b^2 c^2 - b c^3 + c^4) + a b (b - c)^2 c (b + c) + (b - c)^4 (b + c)^2))

Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let O_A be the circumcenter of A′BC, and define O_B and O_C cyclically. The circumcircles of the four triangles A″B″C″, A″BC, AB″C, ABC″ concur in X(6098). See Hyacinthos 22617, October 8, 2014.

X(6098) lies on this line: {119,1845}

X(6099) = 16th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2/((b - c) (a^3 b - a^2 b^2 - a b^3 + b^4 + a^3 c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + c^4))

Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let O_A be the circumcenter of A′BC, and define O_B and O_C cyclically. The circumcircles of the four triangles ABC, AB″C″, A″BC″, A″B″C concur in X(6099). The lines O_AA″, O_BB″, O_CC″ also concur in X(6099). See Hyacinthos 22617, October 8, 2014.

Let T be the triangle whose sidelines are the reflections of the line X(3)X(11) in the sidelines of ABC. Then T is perspective to ABC, and X(6099) is the perpsector. (Randy Hutson, October 16, 2014)

X(6099) lies on the circumcircle and these lines: {59,108}, {100,1618}, {104,912}, {105,2990}, {107,5379}, {517,915}, {917,5057}, {929,3573}, {1290,3657}, {1300,5080}, {1311,3006}, {3326,6056}

X(6099) = intersection of antipedal lines of circumcircle intercepts of line X(3)X(11)
X(6099) = isotomic conjugate of polar conjugate of X(32698)

X(6100) = 17th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^14 (b + c) - 2 a^13 (b + c)^2 - (b - c)^6 (b + c)^5 (b^2 + c^2)^2 + a^12 (-3 b^3 + 5 b^2 c + 5 b c^2 - 3 c^3) + 2 a^11 (b + c)^2 (4 b^2 - 5 b c + 4 c^2) + a^10 (b^5 - 21 b^4 c + b^3 c^2 + b^2 c^3 - 21 b c^4 + c^5) - 10 a^9 (b^2 - c^2)^2 (b^2 - b c + c^2) + a^8 (5 b^7 + 15 b^6 c - 21 b^5 c^2 + 21 b^4 c^3 + 21 b^3 c^4 - 21 b^2 c^5 + 15 b c^6 + 5 c^7) + 2 a^7 b c (-10 b^6 + 7 b^5 c + 4 b^4 c^2 - 18 b^3 c^3 + 4 b^2 c^4 + 7 b c^5 - 10 c^6) + a^6 (-5 b^9 + 15 b^8 c + 4 b^7 c^2 - 32 b^6 c^3 + 22 b^5 c^4 + 22 b^4 c^5 - 32 b^3 c^6 + 4 b^2 c^7 + 15 b c^8 - 5 c^9) + 2 a^5 (b - c)^2 (5 b^8 + 10 b^7 c - b^6 c^2 + 4 b^5 c^3 + 14 b^4 c^4 + 4 b^3 c^5 - b^2 c^6 + 10 b c^7 + 5 c^8) - a^4 (b - c)^2 (b^9 + 23 b^8 c + 16 b^7 c^2 + 28 b^5 c^4 + 28 b^4 c^5 + 16 b^2 c^7 + 23 b c^8 + c^9) - 2 a^3 (b^2 - c^2)^2 (4 b^8 - 7 b^7 c + b^6 c^2 + 5 b^5 c^3 - 12 b^4 c^4 + 5 b^3 c^5 + b^2 c^6 - 7 b c^7 + 4 c^8) + a^2 (b - c)^4 (b + c)^3 (3 b^6 + 8 b^5 c - 4 b^4 c^2 + 18 b^3 c^3 - 4 b^2 c^4 + 8 b c^5 + 3 c^6) + 2 a (b^2 - c^2)^4 (b^6 - 3 b^5 c + 4 b^4 c^2 - 6 b^3 c^3 + 4 b^2 c^4 - 3 b c^5 + c^6))

Let A′B′C′ be the antipedal triangle of the incenter. Let A″ = X(1986)-of-A′BC, and define B″ and C″ cyclically. Let O_A be the circumcenter of A′BC, and define O_B and O_C cyclically. The points A″, B″, C″, X(6098) lie on a circle, of which the center is X(6100). See Hyacinthos 22617, October 8, 2014.

X(6100) lies on this line: {6001, 12515}

X(6101) = 18th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^6 (b^2 + c^2) - a^4 (3b^4 + 4b^2 c^2 + 3c^4) + a^2(3b^6 + 2b^4 c^2 + 2b^2 c^4 + 3c^6) - b^8 + b^6 c^2 + b^2 c^6 - c^8)

Let A_B be the reflection of A in line OB, where O = circumcenter of ABC, and define B_C and C_A cyclically. Let A_C be the reflection of A in line OC, and define B_A and C_B cyclically. Let A′ be the nine-point center of triangle AA_BA_C, and define B′ and C′ cyclically. The circumcircles of the four triangles A′B′C′, A′BC, AB′C, ABC′ concur in X(6101). (Also, the circumcircles of the four triangles ABC, AB′C′, A′BC′, A′B′C concur in X(930)). See Hyacinthos 22624, October 10, 2014.

X(6101) lies on these lines: {2,143}, {3,54}, {4,2889}, {5,141}, {20,5663}, {22,156}, {26,394}, {30,5562}, {49,323}, {51,3628}, {52,140}, {67,68}, {110,2937}, {155,1350}, {185,548}, {389,549}, {546,5891}, {568,631}, {632,3819}, {1092,1511}, {1147,5944}, {1656,3060}, {2392,5694}, {2781,5609}, {3313,3564}, {3526,3567}, {3627,5907}, {5070,5640}

X(6101) = reflection of X(i) in X(j) for these (i,j): (5,1216), (52,140), (185,548), (389,5447), (3627,5907), (5946,3917)
X(6101) = complement of X(6243)
X(6101) = anticomplement of X(143)
X(6101) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,195,5012), (52,140,5946), (52,3917,140), (389,5447,549), (1092,1658,1511), (3819,5462,632)

X(6102) = 19th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-2 a^2 b^4 c^2+b^6 c^2-3 a^4 c^4-2 a^2 b^2 c^4+3 a^2 c^6+b^2 c^6-c^8) : :

Let A_B be the reflection of A in line OB, where O = circumcenter of ABC, and define B_C and C_A cyclically. Let A_C be the reflection of A in line OC, and define B_A and C_B cyclically. Let A′ be the nine-point center of triangle AA_BA_C, and define B′ and C′ cyclically. Then X(6102) is the ABC-orthology center of A′B′C′ on the circumcircle of A′B′C′. (Also, X(1141) is the ABC-orthology center of ABC on the circumcircle of ABC.) See Hyacinthos 22628 and Hechos Geometricos 121014, October 12, 2014.

Let N_A be the reflection of X(5) in the A-altitude, and define N_B and N_C cyclically. Then X(6102) is the orthocenter of N_AN_BN_C. Let A′B′C′ be the reflection triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of the line BC, and define B″ and C″ cyclically. Let A* be the trilinear pole, with respect to A′B′C′, of line B″C″, and define B* and C* cyclically. The lines A′A*, B′B*,C′C* concur in X(6102), and the lines A′A″, B′B″, C′C″ concur in X(382). (Randy Hutson, October 16, 2014)

X(6102) lies on these lines: {3,54}, {4,94}, {5,389}, {24,156}, {26,1181}, {30,52}, {49,186}, {51,546}, {140,5562}, {155,2929}, {184,1658}, {381,3567}, {382,3060}, {511,550}, {549,1216}, {567,1199}, {576,2781}, {632,5892}, {974,1204}, {1147,1511}, {1614,2070}, {1994,3520}, {3530,3917}, {3627,5446}, {3628,5891}, {3851,5640}

X(6102) = midpoint of X(i) and X(j) for these (i,j): (52,185), (3,5889)
X(6102) = reflection of X(i) in X(j) for these (i,j): (4,143), (5,389), (5562,140), (3627,5446), (5876,5), (5907,5462)
X(6102) = X(5)-of-circumorthic-triangle
X(6102) = X(355)-of-orthic-triangle if ABC is acute
X(6102) = X(80)-of-1st-Hyacinth-triangle if ABC is acute
X(6102) = homothetic center of X(4)-altimedial and X(5)-adjunct anti-altimedial triangles
X(6102) = Ehrmann-side-to-orthic similarity image of X(12111)
X(6102) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,568,143), (5,389,5946), (184,1658,5944), (389,5907,5462), (5462,5907,5), (5876,5946,5), (5889,5890,3)

Dao-Moses-Telv Circle and Associated Points

The Dao-Moses-Telv circle is defined as the circle passing through these six points:

F₁ = X(13), the 1st Fermat point
F₂ = X(14), the 2nd Fermat point
F₃ = inverse of F₁ in the circumcircle
F₄ = inverse of F₂ in the circumcircle
F₅ = inverse of F₁ in the nine-point circle
F₆ = inverse of F₂ in the nine-point circle

Dao Thanh Oai noted that these points are concyclic, Telv Cohl gave a proof, and Peter Moses discovered properties described in this preamble, along with centers X(6103)-X(6111). The circle is first noted at ADGEOM 971. See also ADGEOM 971 DB.

The center of the Dao-Moses-Telv circle is X(1637). (Francisco Javier, ADGEOM 1977, November 3, 2014)

The Dao-Moses-Telv circle is orthogonal to the circumcircle, the nine-point circle, and all the other circles in their coaxal family. The A-power of the circle is

(b² + c² - a²)[((b² + c² - a²)² - b²c²]/[6(a² - b²)(a² - c²)]

The circle passes through X(5000) and X(5001), these being the Walsmith point and its inverse in the circumcircle. (Peter Moses, November 6, 2014)

X(6103) = RADICAL CENTER OF THE DAO-MOSES-TELV CIRCLE, CIRCUMCIRCLE, AND NINE-POINT CIRCLE

Barycentrics f(A,B,C) ; f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)[2a sec(A + ω) - b sec(B + ω) - c sec(C + ω]
Barycentrics (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(2*a^6 - 2*a^4*b^2 + a^2*b^4 - b^6 - 2*a^4*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :

The Dao-Moses-Telv circle is defined in the preamble just before X(6103).

X(6103) = exsimilicenter of the circles {{X(98), X(107), X(125), X(132)}} and {{X(6), X(111), X(112), X(115), X(187), X(1560)}}. The trilinear polar of X(6103) passes through X(1640). (Randy Hutson, November 30, 2014)

X(6103) lies on the Dao-Moses-Telv circle, the cubics K483 and K1353, and these lines: {2, 648}, {4, 32}, {5, 52951}, {6, 67}, {23, 52945}, {25, 1989}, {39, 37118}, {53, 10301}, {74, 6794}, {107, 111}, {113, 40354}, {183, 11331}, {187, 5523}, {216, 7495}, {230, 231}, {233, 7570}, {264, 7806}, {297, 22329}, {317, 63048}, {325, 56021}, {339, 6720}, {340, 385}, {378, 5309}, {403, 14581}, {427, 5306}, {542, 52469}, {543, 4235}, {577, 16063}, {620, 41676}, {671, 40890}, {858, 3284}, {1112, 61675}, {1235, 6680}, {1249, 52290}, {1562, 10990}, {1594, 5007}, {1609, 8428}, {1625, 16534}, {1627, 36423}, {1640, 60509}, {1691, 51434}, {1971, 51363}, {1986, 15544}, {2072, 52950}, {2079, 3515}, {2165, 51936}, {2420, 17702}, {2549, 35485}, {2697, 59108}, {2967, 6036}, {3016, 61209}, {3053, 37196}, {3087, 63097}, {3172, 13881}, {3269, 20417}, {3516, 34866}, {3520, 7765}, {3541, 5319}, {3569, 32119}, {3788, 56015}, {3815, 52293}, {4230, 9149}, {5063, 59768}, {5139, 15546}, {5189, 9380}, {5254, 35903}, {5304, 52284}, {5305, 44468}, {5477, 11005}, {5702, 7736}, {6055, 16279}, {6108, 6110}, {6109, 6111}, {6143, 9698}, {6240, 35007}, {6353, 36898}, {6699, 64645}, {6721, 10414}, {6781, 56369}, {7426, 18487}, {7473, 16092}, {7533, 36412}, {7574, 10317}, {7577, 7753}, {7687, 9408}, {7746, 8743}, {7749, 39575}, {7772, 37119}, {7792, 39931}, {7813, 56016}, {7829, 37125}, {7856, 37337}, {8739, 62199}, {8740, 62200}, {8744, 52192}, {8778, 44518}, {8859, 37765}, {8889, 63006}, {9300, 62958}, {9412, 13202}, {9475, 15000}, {10018, 41366}, {10104, 39604}, {10314, 61327}, {10316, 14791}, {10766, 32247}, {11063, 21284}, {11646, 32250}, {11648, 35481}, {12173, 22331}, {13114, 53345}, {13473, 63543}, {13638, 62956}, {13758, 62957}, {13854, 14910}, {14002, 61314}, {14357, 60507}, {14568, 15014}, {14590, 62298}, {14961, 15122}, {15595, 63464}, {16188, 51456}, {17008, 17907}, {17968, 52477}, {17986, 34366}, {18312, 60513}, {22240, 52300}, {23969, 40118}, {25555, 59208}, {27376, 37458}, {28343, 32274}, {30497, 57608}, {30714, 32661}, {30745, 52058}, {33843, 35484}, {34369, 51428}, {34570, 65060}, {35486, 41361}, {36794, 63019}, {37070, 62512}, {37637, 45141}, {37690, 56013}, {37981, 52952}, {37984, 43291}, {38227, 41204}, {38552, 43087}, {39170, 41392}, {39563, 57584}, {40347, 41489}, {40938, 63846}, {41370, 43620}, {41512, 56399}, {43448, 49670}, {43457, 50718}, {44534, 57262}, {45280, 48981}, {46942, 53723}, {47322, 47327}, {48453, 50188}, {51437, 53475}, {52283, 63029}, {52299, 63024}, {52301, 62195}, {57655, 58908}, {59229, 62981}, {60502, 60511}, {61304, 62950}

X(6103) = isogonal conjugate of X(65308)
X(6103) = Dao-Moses-Telv-circle-inverse of X(468)
X(6103) = Moses-radical-circle-inverse of X(47223)
X(6103) = Moses-Parry-circle-inverse of X(232)
X(6103) = polar conjugate of X(5641)
X(6103) = complement of the isotomic conjugate of X(2697)
X(6103) = isogonal conjugate of the isotomic conjugate of X(60502)
X(6103) = polar conjugate of the isotomic conjugate of X(542)
X(6103) = polar conjugate of the isogonal conjugate of X(5191)
X(6103) = X(i)-complementary conjugate of X(j) for these (i,j): {31, 42426}, {2697, 2887}
X(6103) = X(i)-Ceva conjugate of X(j) for these (i,j): {2, 42426}, {648, 60509}, {16080, 52469}, {35907, 1640}, {40118, 25}, {50188, 36201}, {60502, 542}
X(6103) = X(i)-cross conjugate of X(j) for these (i,j): {1640, 35907}, {5191, 542}, {47427, 6}
X(6103) = X(i)-isoconjugate of X(j) for these (i,j): {1, 65308}, {48, 5641}, {63, 842}, {162, 35911}, {293, 46787}, {336, 52199}, {656, 5649}, {662, 35909}, {810, 6035}, {4575, 14223}, {4592, 14998}, {8552, 36096}, {34055, 46157}, {35200, 51228}, {36060, 52094}
X(6103) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 65308}, {125, 35911}, {132, 46787}, {133, 51228}, {136, 14223}, {1084, 35909}, {1249, 5641}, {1560, 52094}, {3162, 842}, {5139, 14998}, {23967, 69}, {35582, 14417}, {38970, 34765}, {39062, 6035}, {40596, 5649}, {42426, 2}, {48317, 50942}, {61503, 57481}
X(6103) = cevapoint of X(6041) and X(51428)
X(6103) = crosspoint of X(2) and X(2697)
X(6103) = crosssum of X(6) and X(2781)
X(6103) = crossdifference of every pair of points on line {3, 684}
X(6103) = pole of the Euler line with respect to the Dao-Moses-Telv circle
X(6103) = pole of the trilinear polar of X(5641) with respect to the polar circle
X(6103) = PU(4)-harmonic conjugate of X(1637)
X(6103) = barycentric product X(i)*X(j) for these {i,j}: {4, 542}, {6, 60502}, {30, 17986}, {92, 2247}, {98, 54380}, {112, 18312}, {186, 43087}, {232, 46786}, {264, 5191}, {297, 34369}, {403, 51456}, {468, 16092}, {511, 52491}, {523, 7473}, {525, 35907}, {648, 1640}, {690, 53155}, {842, 38552}, {935, 55142}, {1503, 47105}, {1990, 51227}, {2501, 14999}, {2697, 42426}, {2781, 58087}, {6041, 6331}, {14223, 60505}, {14273, 50941}, {16188, 40118}, {16230, 34761}, {17983, 45662}, {18020, 51428}, {18808, 64607}, {23968, 44427}, {44145, 61446}, {45321, 65349}, {46106, 48451}, {51405, 60428}, {52469, 54527}, {60508, 60590}, {60512, 60591}
X(6103) = barycentric quotient X(i)/X(j) for these {i,j}: {4, 5641}, {6, 65308}, {25, 842}, {112, 5649}, {232, 46787}, {419, 57452}, {460, 34174}, {468, 52094}, {512, 35909}, {542, 69}, {647, 35911}, {648, 6035}, {1640, 525}, {1843, 46157}, {1990, 51228}, {2211, 52199}, {2247, 63}, {2489, 14998}, {2501, 14223}, {5191, 3}, {6041, 647}, {7473, 99}, {14273, 50942}, {14581, 48453}, {14999, 4563}, {16092, 30786}, {16230, 34765}, {17986, 1494}, {17994, 23350}, {18312, 3267}, {18384, 54554}, {23347, 51263}, {23968, 60053}, {32696, 53691}, {34369, 287}, {34397, 52179}, {34761, 17932}, {34854, 52492}, {35907, 648}, {42671, 40080}, {43087, 328}, {45662, 6390}, {46786, 57799}, {47105, 35140}, {48451, 14919}, {51428, 125}, {51456, 57829}, {52491, 290}, {53155, 892}, {54380, 325}, {58348, 16163}, {60502, 76}, {60505, 14999}, {61446, 43705}
X(6103) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {98, 115, 36204}, {125, 6793, 6}, {230, 1990, 468}, {230, 3018, 10418}, {230, 16306, 3003}, {230, 16310, 3291}, {230, 16318, 232}, {232, 41358, 1990}, {468, 1990, 232}, {468, 16318, 1990}, {468, 47202, 47204}, {468, 56390, 41360}, {1637, 14273, 16230}, {1637, 47200, 10418}, {1990, 16318, 41358}, {2492, 6130, 1637}, {3003, 14580, 232}, {5304, 52284, 62213}, {5702, 62960, 7736}, {6108, 6110, 57593}, {6109, 6111, 57585}, {8105, 8106, 232}, {8737, 8738, 8754}, {8791, 64213, 1560}, {10314, 61327, 62937}, {16303, 47187, 232}, {16317, 16318, 47162}, {16318, 44467, 3018}, {24007, 24008, 468}, {34366, 35907, 42426}, {41200, 41201, 115}, {44467, 47202, 1637}, {44467, 47228, 232}, {47200, 47204, 468}, {52604, 53264, 25}

X(6104) = CIRCUMCIRCLE-INVERSE OF X(13)

Trilinears (csc 3A sin A )(cos 3A + cos(A - π/3) : (csc 3B sin B )(cos 3B + cos(B - π/3) : (csc 3C sin A )(cos 3C + cos(C - π/3)

X(6104) lies on the Dao-Moses-Telv circle and these lines: {3,13}, {14,1606}, {15,1337}, {16,1511}, {54,62}, {61,143}, {110,5616}, {231,1989}, {300,1078}, {1154,5612}

X(6104) = X(532)-crosssum of X(623)
X(6104) = trilinear product X(2166)*X(3201)
X(6104) = barycentric product X(94)*X(3201)

X(6105) = CIRCUMCIRCLE-INVERSE OF X(14)

Trilinears (csc 3A sin A )(cos 3A + cos(A - π/3) : (csc 3B sin B )(cos 3B + cos(B - π/3) : (csc 3C sin A )(cos 3C + cos(C + π/3)
Barycentrics Csc[3*A]*Sin[A]*(Cos[3*A] + Cos[A + Pi/3]) : :

X(6105) lies on the Dao-Moses-Telv circle, K073, K112, K261b, K947, K1133b, the curve Q049, and these lines: {3, 14}, {13, 1605}, {15, 1511}, {16, 1338}, {23, 6108}, {24, 8738}, {30, 6106}, {35, 42680}, {36, 46075}, {54, 61}, {62, 143}, {94, 40104}, {110, 5612}, {186, 6110}, {187, 32906}, {231, 1989}, {265, 51274}, {301, 1078}, {476, 39424}, {533, 14368}, {618, 11092}, {1154, 5616}, {2380, 36515}, {3130, 15442}, {3200, 11127}, {3439, 46059}, {3518, 52670}, {5961, 45778}, {5994, 39555}, {6107, 32460}, {6117, 11587}, {6672, 8836}, {7575, 11549}, {11060, 19780}, {11085, 48366}, {11087, 60070}, {11126, 36981}, {11133, 52221}, {11142, 43967}, {13349, 54473}, {13859, 38431}, {14169, 30440}, {14179, 51203}, {14270, 23284}, {16257, 36759}, {16462, 21159}, {32223, 51267}, {34008, 61371}, {35470, 36970}, {44879, 46926}

X(6105) = reflection of X(i) in X(j) for these {i,j}: {14, 15930}, {5616, 52971}, {18864, 3}
X(6105) = isogonal conjugate of X(11601)
X(6105) = reflection of X(6106) in the orthic axis
X(6105) = circumcircle-inverse of X(14)
X(6105) = antigonal image of X(42618)
X(6105) = isogonal conjugate of the anticomplement of X(33527)
X(6105) = isogonal conjugate of the isotomic conjugate of X(52221)
X(6105) = X(i)-Ceva conjugate of X(j) for these (i,j): {14, 36209}, {1141, 14}
X(6105) = X(i)-isoconjugate of X(j) for these (i,j): {1, 11601}, {18, 51805}, {2166, 37850}, {2962, 6104}, {3384, 36211}
X(6105) = X(i)-Dao conjugate of X(j) for these (i,j): {3, 11601}, {10639, 11078}, {11131, 299}, {11597, 37850}, {19295, 14921}, {33529, 1273}
X(6105) = cevapoint of X(3129) and X(11063)
X(6105) = crosspoint of X(i) and X(j) for these (i,j): {62, 8456}, {2381, 57383}
X(6105) = crosssum of X(i) and X(j) for these (i,j): {18, 8173}, {526, 30468}, {533, 624}
X(6105) = crossdifference of every pair of points on line {8562, 23283}
X(6105) = barycentric product X(i)*X(j) for these {i,j}: {6, 52221}, {14, 11127}, {15, 8836}, {62, 11092}, {94, 3200}, {298, 11088}, {301, 11136}, {303, 11086}, {323, 11582}, {470, 50469}, {472, 50466}, {1994, 11600}, {3458, 11133}, {7799, 16464}, {10633, 40710}, {10677, 16771}, {11145, 36210}, {16770, 36209}, {20412, 65326}, {23284, 52606}, {30529, 37848}
X(6105) = barycentric quotient X(i)/X(j) for these {i,j}: {6, 11601}, {50, 37850}, {62, 11078}, {1994, 52220}, {2965, 6104}, {3200, 323}, {3458, 11082}, {8836, 300}, {10633, 471}, {10677, 19779}, {11086, 18}, {11088, 13}, {11092, 34390}, {11127, 299}, {11134, 36208}, {11136, 16}, {11137, 10678}, {11142, 36211}, {11582, 94}, {11600, 11140}, {16464, 1989}, {20412, 14918}, {33527, 14921}, {34394, 8604}, {36209, 19778}, {36297, 52204}, {50466, 40711}, {50469, 40709}, {52221, 76}, {55223, 23283}, {64464, 44719}
X(6105) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 11141, 14}, {3, 32628, 31940}, {14, 8930, 11138}, {14, 51269, 18}, {15, 11086, 36209}, {2070, 11063, 6104}, {5474, 21157, 44800}, {8173, 16242, 33498}, {11088, 50469, 62}

X(6106) = INVERSE-IN-NINE-POINT-CIRCLE OF X(13)

Barycentrics Sqrt[3]*(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) - 2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 4*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 - 6*b^4*c^4 + 3*a^2*c^6 + 4*b^2*c^6 - c^8)*S : : (Peter Moses, September 23, 2024)
Barycentrics 3^1/2(b² + c² - a²)[a²(b² + c²)[a⁴ - (b² - c²)²] - a⁴(b⁴ + c⁴) + (b² - c²)⁴] + 2S[a²(b² + c²)[a⁴ + 3(b² - c²)²] - a⁴(3b⁴ - 4b²c² + 3c⁴) - (b² - c²)⁴] : : (Richard Hilton, March 4, 2015)

X(6106) lies on the Dao-Moses-Telv circle and these lines: {2, 14368}, {5, 13}, {14, 3480}, {16, 51269}, {30, 6105}, {113, 15610}, {231, 2072}, {403, 6111}, {623, 3580}, {858, 6109}, {6104, 32461}, {6108, 47119}, {6112, 11543}, {6670, 33527}, {8839, 16965}, {15778, 37850}, {18581, 22648}, {18912, 40694}, {22850, 64253}

X(6106) = midpoint of X(5616) and X(11601)
X(6106) = complement of X(14368)
X(6106) = reflection of X(6105) in the Orthic axis
X(6106) = nine-point-circle-inverse of X(13)
X(6106) = complement of the isogonal conjugate of X(14372)
X(6106) = X(14372)-complementary conjugate of X(10)
X(6106) = X(6106)-Dao conjugate of X(14368)
X(6106) = crosssum of X(16) and X(3200)
X(6106) = {X(13),X(18)}-harmonic conjugate of X(40580)

X(6107) = INVERSE-IN-NINE-POINT-CIRCLE OF X(14)

Barycentrics Sqrt[3]*(a^2 - b^2 - c^2)*(a^6*b^2 - a^4*b^4 - a^2*b^6 + b^8 + a^6*c^2 + a^2*b^4*c^2 - 4*b^6*c^2 - a^4*c^4 + a^2*b^2*c^4 + 6*b^4*c^4 - a^2*c^6 - 4*b^2*c^6 + c^8) + 2*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + a^6*c^2 + 4*a^4*b^2*c^2 - 3*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 3*a^2*b^2*c^4 - 6*b^4*c^4 + 3*a^2*c^6 + 4*b^2*c^6 - c^8)*S : : (Peter Moses, September 23, 2024)
Barycentrics 3^1/2(b² + c² - a²)[a²(b² + c²)[a⁴ - (b² - c²)²] - a⁴(b⁴ + c⁴) + (b² - c²)⁴] - 2S[a²(b² + c²)[a⁴ + 3(b² - c²)²] - a⁴(3b⁴ - 4b²c² + 3c⁴) - (b² - c²)⁴] : : (Richard Hilton, March 4, 2015)

X(6107) lies on the Dao-Moses-Telv circle and these lines: {2, 14369}, {5, 14}, {13, 3479}, {15, 51276}, {30, 6104}, {113, 15609}, {231, 2072}, {403, 6110}, {624, 3580}, {858, 6108}, {6105, 32460}, {6109, 47118}, {6113, 11542}, {6669, 33526}, {8837, 16964}, {15802, 37848}, {16267, 41889}, {18582, 22649}, {18912, 40693}, {22894, 64254}

X(6107) = midpoint of X(5612) and X(11600)
X(6107) = complement of X(14369)
X(6107) = reflection of X(6104) in the Orthic axis
X(6107) = nine-point-circle-inverse of X(14)
X(6107) = complement of the isogonal conjugate of X(14373)
X(6107) = X(14373)-complementary conjugate of X(10)
X(6107) = X(6107)-Dao conjugate of X(14369)
X(6107) = crosssum of X(15) and X(3201)
X(6107) = {X(14),X(17)}-harmonic conjugate of X(40581)
midpoint of X(5612) and X(11600)
X(6107) = complement of X(14369)
X(6107) = reflection of X(6104) in the Orthic axis
X(6107) = nine-point-circle-inverse of X(14)
X(6107) = complement of the isogonal conjugate of X(14373)
X(6107) = X(14373)-complementary conjugate of X(10)
X(6107) = X(6107)-Dao conjugate of X(14369)
X(6107) = crosssum of X(15) and X(3201)
X(6107) = {X(14),X(17)}-harmonic conjugate of X(40581)

X(6108) = INVERSE-IN-ORTHOPIC-CIRCLE-OF-STEINER- INELLIPSE OF X(13)

Barycentrics (a^2 + b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) + 2*Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*S : : (Peter Moses, September 23, 2024)
Barycentrics 4a⁸ + a⁴(7b⁴ - 2b²c² + 7c⁴) + (b² - c²)²(b⁴ - 4b²c² + c⁴) - 2a²(b² + c²)[3(a⁴ + b⁴ + c⁴) - 5b²c²] + 2 3^1/23S(b² - c²)²(a² + b² + c²) : : (Richard Hilton, March 4, 2015)
X61083 X[13] + X[6779], 3 X[16] - X[6779], 2 X[16] + X[33517], 2 X[6779] + 3 X[33517], 3 X[22489] - X[50858], X[51485] + 3 X[59378], X[14] - 3 X[22511], X[6782] + 2 X[53430], X[396] - 3 X[53454], X[1080] - 3 X[38227], X[36969] + 3 X[39554], 3 X[5469] - X[36970], 3 X[5470] - X[23005], 3 X[5470] + X[36968], X[5473] - 3 X[21159], X[5615] + 3 X[59383], X[5617] - 3 X[59404], X[6778] + 3 X[16963], 3 X[16963] - X[22998], X[6770] + 3 X[59398], 3 X[36758] - X[51200], X[6777] - 3 X[16268], 2 X[11623] + X[41034], X[14539] - 3 X[21156], X[16001] + 2 X[21402], X[20429] - 3 X[59401], 3 X[22572] - X[36330], 3 X[26613] - X[35931], X[36961] - 3 X[41037], X[36967] - 3 X[39555], X[36994] - 3 X[59393], X[36995] + 3 X[59394], X[51207] - 3 X[59409]

X(6108) lies on the Dao-Moses-Telv circle, the cubics K892 and K1274, and these lines: {2, 13}, {3, 6772}, {4, 16634}, {14, 98}, {23, 6105}, {25, 1605}, {30, 115}, {62, 37464}, {111, 61472}, {114, 9115}, {147, 16530}, {183, 3642}, {325, 532}, {381, 22513}, {385, 533}, {395, 542}, {396, 511}, {397, 20415}, {427, 31687}, {468, 6111}, {524, 6783}, {531, 22329}, {543, 35303}, {619, 5976}, {671, 35932}, {842, 5618}, {858, 6107}, {1078, 11303}, {1080, 5478}, {1302, 2379}, {1368, 40682}, {1513, 41023}, {1691, 62199}, {2076, 62232}, {2782, 44223}, {2794, 41017}, {3054, 42118}, {3055, 63197}, {3815, 5472}, {3849, 31693}, {3934, 37341}, {5025, 9989}, {5031, 44383}, {5104, 62198}, {5152, 35918}, {5306, 47610}, {5318, 10617}, {5461, 31694}, {5469, 36970}, {5470, 23005}, {5473, 21159}, {5607, 6137}, {5615, 42974}, {5617, 10516}, {5916, 11658}, {5999, 22510}, {6034, 62200}, {6036, 13349}, {6054, 6778}, {6103, 6110}, {6106, 47119}, {6670, 9478}, {6680, 37340}, {6770, 9749}, {6775, 11632}, {6777, 11177}, {7610, 11296}, {7735, 10654}, {7736, 9112}, {7784, 11305}, {7793, 9988}, {7828, 11304}, {8280, 31689}, {8281, 31692}, {8859, 23004}, {9752, 44459}, {9753, 33388}, {9760, 37785}, {9763, 22580}, {10033, 54589}, {10352, 12214}, {10616, 36756}, {11298, 15271}, {11486, 41745}, {11549, 62490}, {11580, 34313}, {11623, 36252}, {11645, 53442}, {11646, 62197}, {11705, 44660}, {12829, 52647}, {13873, 13875}, {13926, 13928}, {14137, 41071}, {14539, 21156}, {14693, 52650}, {14830, 22512}, {15246, 37848}, {16001, 21402}, {16092, 61474}, {16644, 59410}, {16965, 35230}, {17830, 31685}, {18581, 41042}, {19781, 42154}, {20252, 43416}, {20426, 41746}, {20429, 59401}, {22496, 43274}, {22572, 36330}, {22576, 23055}, {22712, 43538}, {22796, 22847}, {23234, 25235}, {23698, 44250}, {25154, 37637}, {25166, 63047}, {26244, 37834}, {26613, 35931}, {29012, 53465}, {31707, 47638}, {32461, 47200}, {33498, 48442}, {35021, 42940}, {36318, 49812}, {36344, 49861}, {36363, 49906}, {36383, 49948}, {36755, 53428}, {36764, 42092}, {36765, 42910}, {36961, 41037}, {36967, 39555}, {36994, 59393}, {36995, 59394}, {37640, 51207}, {37689, 42085}, {37786, 51206}, {40694, 41020}, {41151, 42108}, {42035, 63029}, {42086, 62992}, {43228, 44497}, {44382, 51388}, {47857, 61719}, {53015, 54569}

X(6108) = midpoint of X(i) and X(j) for these {i,j}: {13, 16}, {98, 383}, {385, 5978}, {395, 53430}, {671, 35932}, {6778, 22998}, {8595, 51482}, {23005, 36968}, {42943, 53435}
X(6108) = reflection of X(i) in X(j) for these {i,j}: {114, 52263}, {618, 6672}, {624, 6669}, {6109, 230}, {6782, 395}, {9115, 42913}, {31694, 5461}, {31706, 22847}, {31710, 115}, {33517, 13}, {51388, 44382}, {52021, 45880}, {52650, 14693}
X(6108) = complement of X(5979)
X(6108) = reflection of X(6109) in the orthic axis
X(6108) = orthoptic-circle-of-the-Steiner-inellipse inverse of X(13)
X(6108) = circumcircle-of-outer-Napoleon-triangle-inverse of X(10653)
X(6108) = psi-transform of X(10654)
X(6108) = crossdifference of every pair of points on line {6137, 34291}
X(6108) = X(5607)-line conjugate of X(6137)
X(6108) = radical trace of the circles {{X(13), X(14), X(16)}} and {{X(13), X(15), X(16)}}
X(6108) = X(15)-pedal-to-X(16)-pedal similarity image of X(13)
X(6108) = {X(115),X(6055)}-harmonic conjugate of X(6109)
X(6108) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 13, 6115}, {2, 5980, 618}, {2, 51482, 9762}, {13, 22489, 18582}, {13, 23006, 51482}, {13, 41100, 47859}, {13, 41107, 47861}, {13, 46054, 5459}, {13, 47859, 47865}, {115, 6055, 6109}, {395, 44219, 6114}, {5335, 59378, 13}, {5470, 36968, 23005}, {5915, 46980, 6109}, {6103, 57593, 6110}, {6770, 37641, 51200}, {6778, 16963, 22998}, {10614, 61514, 14139}, {47363, 47364, 10653}

X(6109) = INVERSE-IN-ORTHOPIC-CIRCLE-OF-STEINER- INELLIPSE OF X(14)

Barycentrics (a^2 + b^2 + c^2)*(2*a^4 - a^2*b^2 - b^4 - a^2*c^2 + 2*b^2*c^2 - c^4) - 2*Sqrt[3]*(2*a^4 - a^2*b^2 + b^4 - a^2*c^2 - 2*b^2*c^2 + c^4)*S : : (Peter Moses, September 23, 2024)
Barycentrics 4*a^8 - 6*a^6*b^2 + 7*a^4*b^4 - 6*a^2*b^6 + b^8 - 6*a^6*c^2 - 2*a^4*b^2*c^2 + 4*a^2*b^4*c^2 - 6*b^6*c^2 + 7*a^4*c^4 + 4*a^2*b^2*c^4 + 10*b^4*c^4 - 6*a^2*c^6 - 6*b^2*c^6 + c^8 - 2*Sqrt[3]*(b^2 - c^2)^2*(a^2 + b^2 + c^2)*S : : (Richard Hilton, March 4, 2015)
X(6109) = 3 X[14] + X[6780], 3 X[15] - X[6780], 2 X[15] + X[33518], 2 X[6780] + 3 X[33518], 3 X[22490] - X[50855], X[51484] + 3 X[59379], X[13] - 3 X[22510], X[383] - 3 X[38227], X[36970] + 3 X[39555], X[395] - 3 X[53465], X[6783] + 2 X[53442], 3 X[5469] - X[23004], 3 X[5469] + X[36967], 3 X[5470] - X[36969], X[5474] - 3 X[21158], X[5611] + 3 X[59384], X[5613] - 3 X[59403], X[6777] + 3 X[16962], 3 X[16962] - X[22997], X[6773] + 3 X[59397], 3 X[36757] - X[51203], X[6778] - 3 X[16267], 2 X[11623] + X[41035], X[14538] - 3 X[21157], X[16002] + 2 X[21401], X[20428] - 3 X[59402], 3 X[22571] - X[35752], 3 X[26613] - X[35932], X[36962] - 3 X[41036], X[36968] - 3 X[39554], X[36992] - 3 X[59395], X[36993] + 3 X[59396]

X(6109) lies on the Dao-Moses-Telv circle the cubics K892 and K1274, and these lines: {2, 14}, {3, 6775}, {4, 16635}, {13, 98}, {23, 6104}, {25, 1606}, {30, 115}, {61, 37463}, {111, 61474}, {114, 9117}, {147, 16529}, {183, 3643}, {325, 533}, {381, 22512}, {383, 5479}, {385, 532}, {395, 511}, {396, 542}, {398, 20416}, {427, 31688}, {468, 6110}, {524, 6782}, {530, 22329}, {543, 35304}, {618, 5976}, {671, 35931}, {842, 5619}, {858, 6106}, {1078, 11304}, {1302, 2378}, {1368, 40683}, {1513, 41022}, {1691, 62200}, {2076, 62233}, {2782, 52650}, {2794, 41016}, {3054, 42117}, {3055, 63196}, {3815, 5471}, {3849, 31694}, {3934, 37340}, {5025, 9988}, {5031, 44382}, {5104, 62197}, {5152, 35917}, {5306, 47611}, {5321, 10616}, {5461, 31693}, {5469, 23004}, {5470, 36969}, {5474, 21158}, {5608, 6138}, {5611, 42975}, {5613, 10516}, {5917, 11659}, {5999, 22511}, {6034, 62199}, {6036, 13350}, {6054, 6777}, {6103, 6111}, {6107, 47118}, {6669, 9478}, {6680, 37341}, {6772, 11632}, {6773, 9750}, {6778, 11177}, {7610, 11295}, {7735, 10653}, {7736, 9113}, {7784, 11306}, {7793, 9989}, {7828, 11303}, {8280, 31691}, {8281, 31690}, {8859, 23005}, {9752, 44463}, {9753, 33389}, {9761, 22579}, {9762, 37786}, {10033, 54590}, {10352, 12213}, {10617, 36755}, {11297, 15271}, {11485, 41746}, {11537, 62490}, {11580, 34314}, {11623, 36251}, {11645, 53430}, {11646, 62198}, {11706, 44659}, {12829, 52648}, {13873, 13876}, {13926, 13929}, {14136, 41070}, {14538, 21157}, {14693, 44223}, {14830, 22513}, {15246, 37850}, {16002, 21401}, {16092, 61472}, {16964, 35229}, {17830, 31686}, {18582, 41043}, {19780, 42155}, {20253, 43417}, {20425, 41745}, {20428, 59402}, {22495, 43275}, {22571, 35752}, {22575, 23055}, {22712, 43539}, {22797, 22893}, {23234, 25236}, {25156, 63047}, {25164, 37637}, {26244, 37831}, {26613, 35932}, {29012, 53454}, {31708, 47638}, {32460, 47200}, {33500, 48441}, {35021, 42941}, {35731, 35742}, {36319, 49862}, {36320, 49813}, {36362, 49905}, {36382, 49947}, {36756, 53440}, {36764, 36776}, {36962, 41036}, {36968, 39554}, {36992, 59395}, {36993, 59396}, {37641, 51206}, {37689, 42086}, {37785, 51207}, {40693, 41021}, {41151, 42109}, {42036, 63029}, {42085, 62992}, {43229, 44498}, {44250, 47113}, {44383, 51387}, {47864, 61719}, {53015, 54570}

X(6109) = midpoint of X(i) and X(j) for these {i,j}: {14, 15}, {98, 1080}, {385, 5979}, {396, 53442}, {671, 35931}, {6777, 22997}, {8594, 51483}, {23004, 36967}, {42942, 53447}
X(6109) = reflection of X(i) in X(j) for these {i,j}: {114, 52266}, {619, 6671}, {623, 6670}, {6108, 230}, {6783, 396}, {9117, 42912}, {31693, 5461}, {31705, 22893}, {31709, 115}, {33518, 14}, {44223, 14693}, {44250, 47113}, {51387, 44383}, {52022, 45879}
X(6109) = complement of X(5978)
X(6109) = reflection of X(6108) in the Orthic axis
X(6109) = orthoptic-circle-of-the-Steiner-inellipse-inverse of X(14)
X(6109) = circumcircle-of-inner-Napoleon-triangle-inverse of X(10654)
X(6109) = psi-transform of X(10653)
X(6109) = crossdifference of every pair of points on line {6138, 34291}
X(6109) = X(5608)-line conjugate of X(6138)
X(6109) = radical trace of the circles {{X(13), X(14), X(15)}} and {{X(14), X(15), X(16)}}
X(6109) = X(16)-pedal-to-X(15)-pedal similarity image of X(14)
X(6109) = reflection of X(6108) in X(230)
X(6109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 14, 6114}, {2, 5981, 619}, {2, 51483, 9760}, {14, 22490, 18581}, {14, 23013, 51483}, {14, 41101, 47860}, {14, 41108, 47862}, {14, 46053, 5460}, {14, 47860, 47866}, {115, 6055, 6108}, {5334, 59379, 14}, {5469, 36967, 23004}, {5915, 46980, 6108}, {6103, 57585, 6111}, {6773, 37640, 51203}, {6777, 16962, 22997}, {10613, 61513, 14138}, {47361, 47362, 10654}

X(6110) = INVERSE-IN-POLAR-CIRCLE OF X(13)

Trilinears (cos A - 2 cos B cos C)(sec A sin(A + π/3)) : (cos B - 2 cos C cos A)(sec B sin(B + π/3)) : (cos C - 2 cos A cos B)(sec C sin(C + π/3))
Barycentrics f(A,B,C) : f(B,C,A), F(C,A,B), where f(A,B,C) = (S² - 3S_BS_C)S_BS_C(3^1/2S_B + S)

X(6110) lies on the Dao-Moses-Telv circle and these lines: {4,13}, {14,471}, {15,470}, {16,5667}, {30,1990}, {297,531}, {340,533}, {389,398}, {530,648}, {1300,5994}

X(6110) = reflection of X(6111) in X(1990)
X(6110) = trilinear product X(i)*X(j) for these {i,j}: {15,1784}, {470,2173}
X(6110) = isogonal conjugate of X(39377)
X(6110) = crossdifference of every pair of points on line X(14380)X(36296)
X(6110) = polar conjugate of X(36308)
X(6110) = pole wrt polar circle of trilinear polar of X(36308) (line X(13)X(2394))
X(6110) = barycentric product X(i)*X(j) for these {i,j}: {14, 35201}, {30,470}, {298,1990}, {2154, 14920}, {6137, 24001}, {8739, 14206}
X(6110) = X(i)-Ceva conjugate of X(j) for these {i,j}: {4,6111}, {14,6117}

X(6111) = INVERSE-IN-POLAR-CIRCLE OF X(14)

Trilinears (cos A - 2 cos B cos C)(sec A sin(A - π/3)) : (cos B - 2 cos C cos A)(sec B sin(B - π/3)) : (cos C - 2 cos A cos B)(sec C sin(C - π/3))
Barycentrics f(A,B,C) : f(B,C,A), F(C,A,B), where f(A,B,C) = (S² - 3S_BS_C)S_BS_C(3^1/2S_B - S)

X(6111) lies on the Dao-Moses-Telv circle and these lines: {4,14}, {13,470}, {15,5667}, {16,471}, {30,1990}, {297,530}, {340,532}, {389,397}, {531,648}, {1300,5995}.

X(6111) = reflection of X(6110) in X(1990)
X(6111) = isogonal conjugate of X(39378)
X(6111) = crossdifference of every pair of points on line X(14380)X(36297)
X(6111) = trilinear product X(i)*X(j) for these {i,j}: {13, 35201}, {16,1784}, {471,2173}, {2153, 14920}, {6138, 24001}, {8740, 14206}
X(6111) = barycentric product X(i)*X(j) for these {i,j}: {30,471}, {299,1990}
X(6111) = X(i)-Ceva conjugate of X(j) for these {i,j}: {4,6110}, {13,6116}
X(6111) = polar conjugate of X(36311)
X(6111) = pole wrt polar circle of trilinear polar of X(36311) (line X(14)X(2394))

Moses Radical Circle and Associated Points

Summarizing notes from Peter Moses (November 7, 2014), the circle here named the Moses radical circle, MRC, is defined as the radical circle of the circumcircle, nine-point circle, and Brocard circle. The center of MRC is X(647), and, like the Dao-Moses-Telv circle (see X(6103), MRC passes through X(5000) and X(5001). The radical center of {circumcircle, nine-point circle, MRC} is X(232), and the power of vertex A with respect to MRC is

-b²c²(b² + c² - a²)/[2(a² - b²)(a² - c²)]

If U is any circle in the coaxal family of the circumcircle and nine-point circle, then the inverses of X(15) and X(16) in U lie on MRC. Examples of such inverses include X(6112)-X(6117).

If you have The Geometer's Sketchpad, you can view Moses Radical Circle, including X(6112)-X(6117).

X(6112) = INVERSE-IN-NINE-POINT-CIRCLE OF X(15)

Barycentrics 3^1/2[(b² + c²)[a⁸ - 2a⁴b²c² - (b² - c²)⁴] - 2a⁶(b⁴ + b²c² + c⁴) + 2a²(b⁴ - c⁴)²] - 2S[a²(b² + c²)[a⁴ + 3(b² - c²)²] - a⁴(3b⁴ + 4b²c² + 3c²) - (b² - c²)⁴] : : (Richard Hilton, March 4, 2015)

X(6112) lies on the Moses radical circle and these lines: {5,15}, {570,1506}.

X(6113) = INVERSE-IN-NINE-POINT-CIRCLE OF X(16)

Barycentrics 3^1/2[(b² + c²)[a⁸ - 2a⁴b²c² - (b² - c²)⁴] - 2a⁶(b⁴ + b²c² + c⁴) + 2a²(b⁴ - c⁴)²] + 2S[a²(b² + c²)[a⁴ + 3(b² - c²)²] - a⁴(3b⁴ + 4b²c² + 3c²) - (b² - c²)⁴] : : (Richard Hilton, March 4, 2015)

X(6113) lies on the Moses radical circle and these lines: {5,16}, {570,1506}.

X(6114) = INVERSE-IN-ORTHOPIC-CIRCLE-OF-STEINER-INELLIPSE OF X(15)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = : Sqrt[3](a^2b^2 + a^2c^2 - b^4 - c^4 + 2b^2c^2)(a^2 + b^2 + c^2) - 2S(3a^2b^2 + 3a^2c^2 - b^4 - c^4 + 2b^2c^2)

X(6114) lies on the Moses radical circle and these lines: {2,14}, {5,39}, {6,5613}, {13,6054}, {16,383}, {18,98}, {62,147}, {230,5471}, {325,624}, {395,542}, {629,5982}, {636,3104}, {1080,5479}

X(6114) = complement of X(5981)

X(6115) = INVERSE-IN-ORTHOPIC-CIRCLE-OF-STEINER-INELLIPSE OF X(16)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = Sqrt[3](a^2b^2 + a^2c^2 - b^4 - c^4 + 2b^2c^2)(a^2 + b^2 + c^2) + 2S(3a^2b^2 + 3a^2c^2 - b^4 - c^4 + 2b^2c^2)

X(6115) lies on the Moses radical circle and these lines: {2,13}, {5,39}, {6,5617}, {14,6054}, {15,1080}, {17,98}, {61,147}, {230,5472}, {325,623}, {383,5478}, {396,542}, {630,5983}, {635,3105}

X(6115) = complement of X(5980)

X(6116) = INVERSE-IN-POLAR-CIRCLE OF X(15)

Trilinears sin A cos(B - C) sin(A - π/3) csc 2A : sin B cos(C - A) sin(B - π/3) csc 2B : sin C cos(A - B) sin(C - π/3) csc 2C

X(6116) lies on the Moses radical circle and these lines: {4,15}, {5,53}, {13,275}, {16,471}, {62,3462}, {264,623}, {297,624}, {1833,1838}, {2903,5962}

X(6116) =X(13)-Ceva conjugate of X(6111)
X(6116) = X(53)-Hirst inverse of X(6117)
X(6116) = trilinear product X(i)*X(j) for these {(i,j}: {299,2181}, {324,2152}, {471,1953}
X(6116) = barycentric product X(i)*X(j) for these {(i,j}: {5,471}, {16,324}, {53,299}
X(6116) = X(i)-isoconjugate of X(j) for these {i,j}: {14,2169}, {97,2154}
X(6116) = {X(5),X(53)}-harmonic conjugate of X(6117)
X(6116) = X(15)-of-orthic-triangle
X(6116) = nine-point-circle-inverse of X(6117)
X(6116) = polar conjugate of the isotomic conjugate of X(33530)

X(6117) = INVERSE-IN-POLAR-CIRCLE OF X(16)

Trilinears sin A cos(B - C) sin(A + π/3) csc 2A : sin B cos(C - A) sin(B + π/3) csc 2B : sin C cos(A - B) sin(C + π/3) csc 2C

X(6117) lies on the Moses radical circle and these lines: {4,16}, {5,53}, {14,275}, {15,470}, {61,3462}, {264,624}, {297,623}, {1832,1838}, {2902,5962}

X(6117) =X(14)-Ceva conjugate of X(6110)
X(6117) = X(53)-Hirst inverse of X(6116)
X(6117) = trilinear product X(i)*X(j) for these {(i,j}: {298,2181}, {324,2151}, {470,1953}
X(6117) = barycentric product X(i)*X(j) for these {(i,j}: {5,470}, {15,324}, {53,298}
X(6117) = X(i)-isoconjugate of X(j) for these {i,j}: {13,2169}, {97,2153}
X(6117) = {X(5),X(53)}-harmonic conjugate of X(6116)
X(6117) = X(16)-of-orthic-triangle
X(6117) = nine-point-circle-inverse of X(6116)
X(6117) = polar conjugate of the isotomic conjugate of X(33529)

X(6118) = CENTER OF 1st DAO-VECTEN CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2-b^2-c^2) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) + 2 (6 a^4-7 a^2 b^2+7 b^4-7 a^2 c^2-10 b^2 c^2+7 c^4) S
X(6118) = 3X(2) + X(485)

In the plane of a triangle ABC, let A′B′C′ be the outer Vecten triangle, and let M_AM_BM_C be the medial triangle. Let O_A be the circumcenter of triangle B′C′M_A, and define O_B and O_C cyclically. Let O′_A be the point, other than A′, in which the circles A′B′M_C and C′A′M_B meet, and define O′_B and O′_C cyclically. The six points O_A, O_B, O_C, O′_A, O′_B, O′_C lie on a circle, here named the 1st Dao-Vecten circle. If the construction is carried out using the inner Vecten triangle for A′B′C′, the resulting six points line on the 2nd Dao-Vecten circle. (Thanh Oai Dao, November 4, 2014).

The line X(6118)X(6119) passes through X(575), X(3564), X(3589), X(3628), X(5449), and the radical axis of the 1st and 2nd Dao-Vecten circles passes through X(2501), X(3566), X(5203). X(6118) = {X(3589),X(3628)}-harmonic conjugate of X(6119). (Francisco Javier, ADGEOM, November 4-5, 2014).

The 1st Dao-Vecten circle is the nine-point circle of the outer Vecten triangle. Also, let (O″_A) be the circle centered at A′ and tangent to BC at M_A, and define (O″_B) and (O″_C) cyclically. Then X(6118) is the radical center of (O″_A), (O″_B), (O″_C). (Randy Hutson, January 29, 2015)

If you have The Geometer's Sketchpad, you can view X(6118).

X(6118) lies on these lines: {2,372}, {575,3564}, {590,639}, {3316,5590}

X(6118) = midpoint of X(485) and X(641)
X(6118) = {X(2),X(485)}-harmonic conjugate of X(645)
X(6118) = X(5)-of-outer-Vecten-triangle

X(6119) = CENTER OF 2nd DAO-VECTEN CIRCLE

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2-b^2-c^2) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) - 2 (6 a^4-7 a^2 b^2+7 b^4-7 a^2 c^2-10 b^2 c^2+7 c^4) S
X(6119) = 3X(2) + X(486)

See X(6118) for a construction of the 2nd Dao-Vecten circle and some of its properties.

The 2nd Dao-Vecten circle is the nine-point circle of the inner Vecten triangle. Let A″B″C″ be the inner Vecten triangle. Also, let (O*_A) be the circle centered at A″ and tangent to BC at M_A, and define (O*_B) and (O*_C) cyclically. Then X(6119) is the radical center of (O*_A), (O*_B), (O*_C). (Randy Hutson, January 29, 2015)

X(6119) lies on these lines: {2,371}, {575,3564}, {615,640}, {3317,5591}

X(6119) = midpoint of X(486) and X(642)
X(6119) = {X(2),X(486)}-harmonic conjugate of X(642)
X(6119) = X(5)-of-inner-Vecten-triangle

X(6120) = 1st HATZIPOLAKIS-VAN TIENHOVEN POINT

Trilinears cos A - cos(A/3) : cos B - cos(B/3) : cos C - cos(C/3)
Trilinears sin(A/3) sin(2A/3) : sin(B/3) sin(2B/3) : sin(C/3) sin(2C/3)

In the plane of ABC, let A_B be the internal trisector at A that is closer to B, and define B_C and C_A cyclically. Let A_C be the internal trisector of A that is closer to C, and define B_A and C_B cyclically. Let P_A be the Miquel point (QL-P1) of the quadrilateral having vertices B_C, B_A, C_A, C_B, and define P_B and P_C cyclically. The points P_A, P_B, P_C lie on the circumcircle of ABC, and the triangle P_A, P_B, P_C, here named the 1st Hatzipolakis-Van Tienhoven triangle, is perspective to ABC. The perspector is X(6120). There are 3 sets of trisectors, so that there are 2nd and 3rd triangles and 2nd and 3rd points, as found at X(6121) and X(6122). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014: 22717, 22719)

Let L_A be the line through the circumcircle intercepts of the interior trisector of angle A, and define L_B and L_C cyclically. Let A′ = L_B∩L_C, and define B′ and C′ cyclically. Then A′B′C′ is homothetic to ABC at X(6120). (Randy Hutson, September 18, 2014; see ADGEOM 1733

X(6120) lies on these lines: {3,358}, {357,3279}, {1135,6125}, {1136,3335}, {1137,6124}, {3273,6121}, {3274,6122}

X(6120) = {X(3),X(358)}-harmonic conjugate of X(6123)

X(6121) = 2nd HATZIPOLAKIS-VAN TIENHOVEN POINT

Trilinears sin(A/3 + 2π/3) sin(2A/3 + 4π/3) : sin(B/3 + 2π/3) sin(2B/3 + 4π/3) : sin(C/3 + 2π/3) sin(2C/3 + 4π/3)

X(6121) is the perspector of the 2nd Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6120). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014; see 22717, 22719)

X(6121) lies on these lines: {3,1135}, {357,3335}, {358,6125}, {1134,3283}, {1137,6123}, {3273,6120}, {3275,6122}, {3278,3604}, {3334,3603}

X(6121) = {X(3),X(1135)}-harmonic conjugate of X(6124)

X(6122) = 3rd HATZIPOLAKIS-VAN TIENHOVEN POINT

Trilinears sin(A/3 + 4π/3) sin(2A/3 + 2π/3) : sin(B/3 + 4π/3) sin(2B/3 + 2π/3) : sin(C/3 + 4π/3) sin(2C/3 + 2π/3)

X(6122) is the perspector of the 3nd Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6120). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 8-9, 2014; see 22717, 22719)

X(6122) lies on these lines: {3,1137}, {358,6124}, {1134, 3335}, {1135,6123}, {1136,3281}, {3274,6120}, {3275,6121}, {3282,3603}, {3334,3602}

X(6122) = {X(3),X(1137)}-harmonic conjugate of X(6125)

X(6123) = 4th HATZIPOLAKIS-VAN TIENHOVEN POINT

Trilinears        cos A + cos(A/3) : cos B + cos(B/3) : cos C + cos(C/3)
Trilinears        sin(4A/3) csc(A/3) : sin(4B/3) csc(B/3) : sin(4C/3) csc(C/3)
Trilinears        cos(A/3) cos(2A/3) : cos(B/3) cos(2B/3) : cos(C/3) cos(2C/3)

In the plane of ABC, let A_B be the line perpendicular to the internal trisector at A that is closer to B, and define B_C and C_A cyclically. Let A_C be the line perpendicular to the internal trisector of A that is closer to C, and define B_A and C_B cyclically. Let Q_A be the Miquel point (QL-P1) of the quadrilateral having vertices B_C, B_A, C_A, C_B, and define Q_B and Q_C cyclically. The points Q_A, Q_B, Q_C lie on the circumcircle of ABC, and the triangle Q_A, Q_B, Q_C, here named the 4th Hatzipolakis-Van Tienhoven triangle, is perspective to ABC. The perspector is X(6123). There are 3 sets of trisectors, so that there are 5th and 6th triangles and 5th and 6th points, as found at X(6124) and X(6125). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)

X(6123) lies on these lines: {3,358}, {357,3278}, {1135,6122}, {1136,3334}, {1137,6121}, {3273,6124}, {3274,6125}, {3281,3702}, {3335,3604}

X(6123) = isogonal conjugate of X(5457)
X(6123) = Hofstadter 4/3 point
X(6123) = {X(3),X(358)}-harmonic conjugate of X(6120)

X(6124) = 5th HATZIPOLAKIS-VAN TIENHOVEN POINT

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

X(6124) is the perspector of the 5th Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6123). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)

X(6124) lies on these lines: {3,1135}, {357,3334}, {358,6122}, {1134,3282}, {1137,6120}, {3273,6123}, {3275,6125}, {3279,3604}, {3335,3603}

X(6124) = {X(3),X(1135)}-harmonic conjugate of X(6121)

X(6125) = 6th HATZIPOLAKIS-VAN TIENHOVEN POINT

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

X(6125) is the perspector of the 6th Hatzipolakis-van Tienhoven triangle and ABC, as described at X(6123). (Antreas Hatzipolakis and Chris van Tienhoven, Hyacinthos, November 9, 2014; see 22720)

X(6125) lies on these lines: {3,1137}, {358,6121}, {1134,3334}, {1135,6120}, {1136,3280}, {3274,6123}, {3275,6124}, {3283,3603}, {3335, 3602}

X(6125) = {X(3),X(1137)}-harmonic conjugate of X(6122)

X(6126) = X(1)-CEVA CONJUGATE OF X(36)

Barycentrics (a^2 (a^5 + a^4(b+c) - 2a^3(b^2+c^2) - a^2(2b^3-b*c(b+c)+2c^3)+ a(b^4+b^2c^2+c^4) + (b-c)^2(b^3+c^3)) : :
X(6126) = 3R*X(1) + 2r*X(399)
X(6126) = (a + b + c)(a² + b² + c²)*X(6) + 3abc*X(2948)

X(6126) is the point QA-P41 ('Involutary Conjugate of QA-P4') of the quadrangle ABCX(1). X(6126)-of-orthocentroidal-triangle = X(1). These properties and others are presented in Hyacinthos messages 21651 and 22707-22710 by Shapi Topor (real name of Stefan Dominte), A. Montesdeoca, R. Hutson, and A. Hatzipolakis; see 22710)

X(6126) lies on these lines: {1,399}, {6,1718}, {35,73}, {36,1464}, {58,106}, {80,651}, {221,2778}, {244,1203}, {323,758}, {500,3746}, {1411,2003}, {1419,5119}, {1870,2914}, {2293,2772}, {3157,5903}

X(6126) = X(80)-isoconjugate of X(3065)
X(6126) = orthocentroidal-to-ABC similarity image of X(1)
X(6126) = 4th-Brocard-to-circumsymmedial similarity image of X(1)
X(6126) = perspector of anti-orthocentroidal and incentral triangles

X(6127) = X(36)-CEVA CONJUGATE OF X(1)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^5 b-2 a^3 b^3+a b^5+a^5 c-a^4 b c+2 a^2 b^3 c-a b^4 c-b^5 c-a^2 b^2 c^2-2 a^3 c^3+2 a^2 b c^3+2 b^3 c^3-a b c^4+a c^5-b c^5)
X(6127) = (r² - 3R² + s²)X(1) - rR*X(5)

X(6127) lies on these lines: {1,5}, {43,994}, {386,3120}, {583,1743}, {1046,1054}, {1064,3584}, {1193,5270}, {1742,5010}, {1787,3337}, {2635,4316}

X(6128) = RADICAL CENTER OF DAO-MOSES-TELV CIRCLE, NINE-POINT CIRCLE, LESTER CIRCLE

Barycentrics    f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan A)[2a/(cos A - 2 cos B cos C) + b/(cos B - 2 cos C cos A) + c/(cos C - 2 cos A cos B)]
Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin 3B csc²B)(cos B - 2 cos C cos A) + (sin 3C csc²C)(cos C - 2 cos A cos B)
Barycentrics    h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 2 a^8-2 a^6 b^2-3 a^4 b^4+4 a^2 b^6-b^8-2 a^6 c^2+8 a^4 b^2 c^2-4 a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4-6 b^4 c^4+4 a^2 c^6+4 b^2 c^6-c^8

X(6128) and related radical center of three circles are listed here:
X(231) = radical center of circumcircle, nine-point circle, Lester circle
X(1989) = radical center of circumcircle, Dao-Moses-Telv circle, Lester circle
X(6103) = radical center of circumcircle, nine-point circle, Dao-Moses-Telv circle
X(6128) = radical center of Dao-Moses-Telv circle, nine-point circle, Lester circle.
(Antreas Hatzipolakis, November 30, 2014; see 22823)

X(6128) = X(2407)-Ceva conjugate of X(523)
X(6128) = center of the bicevian conic of X(13) and X(14)
X(6128) = complement X(6148)
X(6128) = inverse-in-Kiepert-hyperbola of X(113)
X(6128) = polar conjugate of cevapoint of X(2) and X(113)
X(6128) = polar conjugate of isotomic conjugate of X(6699)
X(6128) = {X(i),X(j)}-harmonic conjugate of X(k), for these (i,j,k): (6,115,3018), (6,1989,3163), (13,14,113), (115,3163,1989), (1989,3,3018)

X(6128) lies on these lines: {2,2986}, {6,13}, {30,3003}, {44,1737}, {112,3087}, {231,2072}, {427,5306}, {526,1637}, {1194,2493}, {1524,5321}, {1525,5318}, {1634,5020}, {5467,6036}

Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Nine-Point Circle

Continuing Peter Moses's discussion in the preamble to X(6112), suppose that P = p : q : r (barycentrics) is a point other than the circumcenter. Let U(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the coaxal system of the circumcircle and nine-point circle. Then X(5000) and X(5001) lie on U, and the center of U is the point M(U) = f(a,b,c) : f(b,c,a) : f(c,a,b) given by

(b² - c²)(a² - b² - c²)p² + (a² + b² - c²)(a² - b² + c²)pq + a²(a² - b² + c²)q² - (a² + b² - c²)(a² - b² + c²)pr - a²(a² + b² - c²)r²

Following is a list of centers M(U) for selected points P and circles U(P):

X(6129) = center of this circle: U(X(1)) = {{1,36,1785,5000,5001,5121}}
X(6130) = center of this circle: U(X(98)) = {{98,107,125,132,5000,5001}}
X(6131) = center of this circle: U(X(99)) = {{99,126,2374,5000,5001,5139}}
X(6132) = center of this circle: U(X(110)) = {{110,114,136,3563,5000,5001}}
X(6133) = center of this circle: U(X(10)) = {{10,242,1324,3814,5000,5001,5205}}
X(6134) = center of this circle: U(X(141)) = {{141,625,5000,5001,5938,5971,6031}}
X(676) = center of this circle: U(X(105)) = {{11,105,108,1360,3513,3514,4667,5000,5001}}
X(2492) = center of this circle: U(X(111)) = {{6,111,112,115,187,1560,2079,3569,5000,5001,5523,5913,6032}}}

X(6129) = CENTER OF CIRCLE U(X(1))

Trilinears cos B cot(C/2) - cos C cot(B/2) : cos C cot(A/2) - cos A cot(C/2) : cos A cot(B/2) - cos B cot(A/2)
Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (b-c) (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3)
X(6129) = 3X(2) - X(4397)

X(6129) is the center of this circle: U(X(1)) = {{1,36,1785,5000,5001,5121}}; see the preamble to X(6129).

X(6129) lies on these lines: {1,521}, {2,4397}, {37,2509}, {106,2716}, {230,231}, {513,663}, {522,905}, {656,3900}, {2191,2424}, {2517,4885}, {3667,3960}

X(6129) = midpoint of X(i) and X(j) for these (i,j): {663,4017}, {1459,1769}
X(6129) = reflection of X(2517) in X(4885)
X(6129) = complement of X(4397)
X(6129) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,5514), (108,208), (522,513), (905,650)
X(6129) = crossdifference of every pair of points on the line X(3)X(9)
X(6129) = {X(37),X(2509)}-harmonic conjugate of X(4130)
X(6129) = perspector of the hyperbola {{A,B,C,X(4),X(40),X(57),X(972)}}, this being the isogonal conjugate of the line X(3)X(9)
X(6129) = bicentric difference of PU(100)
X(6129) = PU(100)-harmonic conjugate of X(2182)

X(6130) = CENTER OF CIRCLE U(X(98))

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B)(sec C sin³A - sec A sin³C) + (sin C)(sec A sin³B - sec B sin³A)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b^2-c^2) (a^8-2 a^6 b^2+a^4 b^4-2 a^6 c^2+a^4 b^2 c^2+b^6 c^2+a^4 c^4-2 b^4 c^4+b^2 c^6)
X(6130) = 3X(2) - X(684)

X(6130) is the center of this circle: U(X(98)) = {{98,107,125,132,5000,5001}}; see the preamble to X(6129).

Let (O) be the circumcircle and (N) the nine-point circle. Let L be the line tangent to (O) at X(98), and let L′ be the line tangent to (O) at X(107); then X(6130) = L∩L′. Let M be the line tangent to (N) at X(125) and let M′ be the line tangent to (N) at X(132). Then X(6130) = M∩M′. (Randy Hutson, December 4, 2014)

X(6130) lies on these lines: {2,684}, {3,2797}, {98,804}, {107,1624}, {125,526}, {132,2881}, {230,231}, {402,5972}, {879,1987}, {2799,6036}

X(6130) = midpoint of X(879) and X(3569)
X(6130) = complement of X(684)
X(6130) = crosspoint of X(98) and X(107)
X(6130) = crosssum of X(511) and X(520)
X(6130) = crossifference of every pair of points on the line X(3)X(1625)
X(6130) = inverse-in-Dao-Moses-Telv-circle of X(647)
X(6130) = X(i)-Ceva conjugate of X(j) for these (i,j): (879,523), (3569,804)
X(6130) = midpoint of Jerabek hyperbola intercepts of orthic axis

X(6131) = CENTER OF CIRCLE U(X(99))

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b^2-c^2) (a^6+a^4 b^2+a^4 c^2-5 a^2 b^2 c^2+b^4 c^2+b^2 c^4)

X(6131) is the center of this circle: U(X(99)) = {{99,126,2374,5000,5001,5139}}; see the preamble to X(6129).

X(6131) lies on these lines: {99,670}, {126,6088}, {141,2872}, {230,231}

X(6132) = CENTER OF CIRCLE U(X(110))

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (b-c) (b+c) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+a^2 b^2 c^2+3 a^2 c^4-c^6)
X(6132) = 3X(351) + X(684)

X(6132) is the center of this circle: U(X(110)) = {{110,114,136,3563,5000,5001}}; see the preamble to X(6129).

X(6132) lies on these lines: {39,2510}, {110,351}, {114,804}, {182,2869}, {230,231}, {512,5926}, {669,1510}

X(6132) = X(2)-Ceva conjugate of X(36472)
X(6132) = X(4226)-Ceva conjugate of X(511)

X(6133) = CENTER OF CIRCLE U(X(10))

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (a^4+a^3 b+a^3 c-a b^2 c+b^3 c-a b c^2+2 b^2 c^2+b c^3)

X(6133) is the center of this circle: U(X(10)) = {{10,242,1324,3814,5000,5001,5205}}; see the preamble to X(6129).

X(6133) lies on these lines: {10,832}, {230,231}, {513,3823}, {659,2517}, {667,4086}, {814,4036}, {3716,4507}

X(6133) = midpoint of X(i) and X(j) for these {i,j}: {659,2517}, {667,4086}

X(6134) = CENTER OF CIRCLE U(X(141))

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b^2-c^2) (2 a^6+2 a^4 b^2+2 a^4 c^2-a^2 b^2 c^2+2 b^4 c^2+2 b^2 c^4)

X(6134) is the center of this circle: U(X(141)) = {{141,625,5000,5001,5938,5971,6031}}; see the preamble to X(6129).

X(6134) lies on this line: {230,231}

X(6135) = 1st HUNG-SODDY TOUCHPOINT

Barycentrics a*(a-b)*(a-c)*(a*b+S)*(a*c+S) : :

The incircle of the antipedal triangle of the inner Soddy point, X(175), of a triangle ABC is tangent to the circumcircle of ABC, and X(6135) is the touchpoint. (Based on a problem by Tran Quang Hung, with contributions by T. Cohl ane P. Moses, Art of Problem Solving: Tangent to circumcircle, October 16, 2014),

X(6135) lies on these lines: {105,1123}, {662,1307}, {4557,6136}

X(6135) = isogonal conjugate of X(6364)

X(6136) = 2st HUNG-SODDY TOUCHPOINT

Barycentrics a*(a-b)*(a-c)*(a*b-S)*(a*c-S) : :

The incircle or one of the excircles of the antipedal triangle of the outer Soddy point, X(176), of a triangle ABC is tangent to the circumcircle of ABC, and X(6136) is the touchpoint. (Based on a problem by Tran Quang Hung, with contributions by T. Cohl and P. Moses, Art of Problem Solving: Tangent to circumcircle, October 16, 2014),

X(6136) lies on these lines: {105,1336}, {662,1306}, {4557,6135}

X(6136) = isogonal conjugate of X(6365)

Centers of Circles Orthogonal to the Coaxal System of the Circumcircle and Brocard Circle

Continuing Peter Moses's discussion in the preambles to X(6112) and X(6129), suppose that P = p : q : r (barycentrics) is a point other than the circumcenter. Let V(P) be the circle that passes through P and its circumcircle-inverse and is orthogonal to the Schoute coaxal system; that is, the coaxal system of the circumcircle and Brocard circle. Then X(15) and X(16) lie on V, and the center of V is the point M(V) = f(a,b,c) : f(b,c,a) : f(c,a,b) given by

a²[c(²(a² + b² - c²)pq - b²(a² - b² + c²)pr + a²c²q² - a²b²r²]

The power of A with respect to V(P) is

b²c⁴p/[a²(b² - c²)r - c²(a² + b²)p]

Following is a list of centers M(V) for selected points P and circles V(P):

X(351) = center of this circle: V(X(2)) = Parry circle = {{2,15,16,23,110,111,352,353,5638,5639}}
X(649) = center of this circle: V(X(1276)) = {{15,16,1276,1277}}
X(663) = center of this circle: V(X(1)) = {{1,15,16,36,3465,4040,5526,5529}}
X(669) = center of this circle: V(X(5004)) = {{15,16,5004,5005,5980,5981}}
X(887) = center of this circle: V(X(99)) = {{15,16,99,729}}
X(890) = center of this circle: V(X(100)) = {{15,16,100,739}}
X(1960) = center of this circle: V(X(101)) = {{15,16,101,106,214}}
X(2488) = center of this circle: V(X(100)) = {{15,16,3513,3514}}
X(3005) = center of this circle: V(X(5002)) = {{15,16,5002,5003}}
X(5075) = center of this circle: V(X(846)) = {{15,16,846,1054,1283,5197}}
X(6137) = center of this circle: V(X(13)) = {{13,15,16,3165,5616,5669}}
X(6138) = center of this circle: V(X(14)) = {{14,15,16,3166,5612,5668}}
X(6139) = center of this circle: V(X(55)) = {{15,16,55,109,654,1155,2291}}
X(6140) = center of this circle: V(X(115)) = {{15,16,115,128,399,1263,1511,2079}}

X(6137) = CENTER OF CIRCLE V(X(13))

Trilinears        sin A sin(B - C) sin(A + π/3) : sin B sin(C - A) sin(B + π/3) : sin C sin(A - B) sin(C + π/3)
Trilinears        a[b csc(B + π/3) - c csc(C + π/3)] : b[c csc(C + π/3) - a csc(A + π/3)] : c[a csc(A + π/3) - b csc(B + π/3)]
Barycentrics    a^2 (b^2-c^2) [3 (a^2 b^2-b^4+a^2 c^2-c^4)-2 Sqrt[3] (2 a^2-b^2-c^2) S] : :

X(6137) is the center of this circle: V(X(13)) = {{13,15,16,3165,5616,5669}}; see the preamble to X(6137).

X(6137) lies on these lines: {110,5994}, {111,2379}, {187,237}, {396,523}, {2151,2605}, {2433,3457}, {5607,6108}

X(6137) = reflection of X(6138) in X(647)
X(6137) = X(5995)-Ceva conjugate of X(6)
X(6137) = crosspoint of X(i) and X(j) for these (i,j): (6,5995), (100,2981)
X(6137) = crossdifference of every pair of points on the line X(2)X(13)
X(6137) = crosssum of X(396) and X(523)
X(6137) = X(512)-Hirst inverse of X(6138)
X(6137) = X(2)-Ceva conjugate of X(38993)
X(6137) = perspector of hyperbola {{A,B,C,X(6),X(14),X(15)}}
X(6137) = trilinear product X(i)*X(j) for these {i,j}: {14,2624}, {15,661}, {298,798}, {470,810}, {523,2151}, {526,2154}
X(6137) = barycentric product X(i)*X(j) for these {i,j}: {14,526}, {15,523}, {298,512}, {470,647}, {1577,2151}, {3268,3458}
X(6137) = X(i)-isoconjugate of X(j) for these (i,j): {13,662}, {75,5995}, {99,2153}, {163,300}, {799,3457}
X(6137) = X(i)-line conjugate of X(j) for these (i,j): (2433,3457), (5607,6108)
X(6137) = X(16)-of-2nd-Parry-triangle
X(6137) = X(16)-of-3rd-Parry-triangle
X(6137) = {X(351),X(3569)}-harmonic conjugate of X(6138)

X(6138) = CENTER OF CIRCLE V(X(14))

Trilinears        sin A sin(B - C) sin(A - π/3) : sin B sin(C - A) sin(B - π/3) : sin C sin(A - B) sin(C - π/3)
Trilinears        a[b csc(B - π/3) - c csc(C - π/3)] : b[c csc(C - π/3) - a csc(A - π/3)] : c[a csc(A - π/3) - b csc(B - π/3)]
Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (b^2-c^2) [3 (a^2 b^2-b^4+a^2 c^2-c^4)+2 Sqrt[3] (2 a^2-b^2-c^2) S]

X(6138) is the center of this circle: V(X(14)) = {{14,15,16,3166,5612,5668}}; see the preamble to X(6137).

X(6138) lies on these lines: {110,5995}, {111,2378}, {187,237}, {395,523}, {2152,2605}, {2433,3458}, {5608,6109}

X(6138) = reflection of X(6137) in X(647)
X(6138) = X(5994)-Ceva conjugate of X(6)
X(6138) = crosspoint of X(6) and X(5994)
X(6138) = crosssum of X(396) and X(523)
X(6138) = crossdifference of every pair of points on the line X(2)X(14)
X(6138) = X(2)-Ceva conjugate of X(38994)
X(6138) = perspector of hyperbola {{A,B,C,X(6),X(13),X(16)}}
X(6138) = X(512)-Hirst inverse of X(6138)
X(6138) = trilinear product X(i)*X(j) for these {i,j}: {13,2624}, {16,661}, {299,798}, {471,810}, {523,2152}, {526,2153}
X(6138) = barycentric product X(i)*X(j) for these {i,j}: {13,526}, {16,523}, {299,512}, {471,647}, {1577,2152}, {3268,3457}
X(6138) = {X(351,X(3569)}-harmonic conjugate of X(6137)
X(6138) = X(i)-isoconjugate of X(j) for these (i,j): {14,662}, {75,5994}, {99,2154}, {163,301}, {799,3458}
X(6138) = X(i)-line conjugate of X(j) for these (i,j): (2433,3458), (5608,6109)
X(6138) = X(15)-of-2nd-Parry-triangle
X(6138) = X(15)-of-3rd-Parry-triangle

X(6139) = CENTER OF CIRCLE V(X(55))

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a-b-c) (b-c) (2 a^2-a b-b^2-a c+2 b c-c^2)
X(6139) = X(663) - 3X(1946)

X(6139) is the center of this circle: V(X(55)) = {{15,16,55,109,654,1155,2291}}; see the preamble to X(6137).

X(6139) lies on these lines: {31,3310}, {55,654}, {109,692}, {187,237}, {650,2520}, {652,4524}, {884,2195}, {918,4640}, {1155,1638}, {1639,3683}

X(6139) = midpoint of X(55) and X(654)
X(6139) = isogonal conjugate of X(35157)
X(6139) = crossdifference of every pair of points on line X(2)X(664)
X(6139) = X(i)-isoconjugate of X(j) for these (i,j): {651,1121}, {664,1156}, {2291,4554}, {4569,4845}
X(6139) = bicentric sum of PU(103)

X(6140) = CENTER OF CIRCLE V(X(115))

Barycentrics a^2 (b^2-c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) : :

X(6140) is the center of this circle: V(X(115)) = {{15,16,115,128,399,1263,1511,2079}}; see the preamble to X(6137).

X(6140) lies on these lines: {187,237}, {690,6132}, {2380,2381}

X(6140) = X(1291)-Ceva conjugate of X(6)
X(6140) = X(75)-isoconjugate of X(1291)

X(6141) = INVERSE-IN-CIRCUMCIRCLE OF X(5638)

Barycentrics a^2*(((b^2+c^2)*a^6-2*(b^2-c^2)^2*a^4+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-b^8-c^8-2*b^2*c^2*(b^4-5*b^2*c^2+c^4))*sqrt(-3*S^2+SW^2)+a^10-(b^2+c^2)*a^8-(6*b^4-7*b^2*c^2+6*c^4)*a^6+2*(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^4-(4*b^8+4*c^8+b^2*c^2*(7*b^4-24*b^2*c^2+7*c^4))*a^2+b^2*c^2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)) : :

X(6141) lies on the Parry circle, the Brocard circle, and these lines: {3,5638}, {6,5639}, {23,1379}, {110,1341}, {111,1340}, {352,1380}, {574,2502}, {2028,3292}

X(6142) = INVERSE-IN-CIRCUMCIRCLE OF X(5639)

Barycentrics a^2*(-((b^2+c^2)*a^6-2*(b^2-c^2)^2*a^4+(b^2+c^2)*(2*b^4-7*b^2*c^2+2*c^4)*a^2-b^8-c^8-2*b^2*c^2*(b^4-5*b^2*c^2+c^4))*sqrt(-3*S^2+SW^2)+a^10-(b^2+c^2)*a^8-(6*b^4-7*b^2*c^2+6*c^4)*a^6+2*(b^2+c^2)*(5*b^4-8*b^2*c^2+5*c^4)*a^4-(4*b^8+4*c^8+b^2*c^2*(7*b^4-24*b^2*c^2+7*c^4))*a^2+b^2*c^2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)): :

X(6142) lies on the Parry circle, the Brocard circle, and these lines: {3,5639}, {6,5638}, {23,1380}, {110,1340}, {111,1341}, {352,1379}, {574,2502}, {2029,3292}

X(6143) = HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+3 a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)
X(6143) = 3(J² - 2)*X(2) + 2 X(3), where J = |OH|/R
X(6143) = 3(J² - 1)*X(2) - X(4)
X(6143) = 2(J² - 1)*X(3) + (J^2 - 2)*X(4)

As a point on the Euler line, X(6143) has Shinagawa coefficients (8F, E).

Let A′B′C′ be the orthic triangle of ABC. Let A′ be the nine-point center of OBC, where O = X(3), and define B′ and C′ cyclically. Let A″ be the orthogonal projection of A′ on the altitude A′, and define B″ and C″ cyclically. The Euler lines of the jfour triangles ABC, A″BC, AB″C, ABC″ concur in X(6143). (Antreas Hatzipolakis and Peter Moses, November 18, 2014; see 22767)

X(6143) lies on these lines: {2,3}, {49,3448}, {54,125}, {93,2970}, {112,1506}, {252,562}

X(6143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3548,631), (5,3520,4), (5,5498,3), (140,1594,186), (186,1594,4), (427,3518,4), (3526,5094,24)

X(6144) = REFLECTION OF X(6) IN X(193)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2b² + 2c² - 5a²
X(6144) = 6X(2) - 7X(6)

Let A′B′C′ be the orthic triangle and A″B″C″ the medial triangle of ABC. Let A*B*C* be the mid-altitude triangle (midpoints of altitudes). Let U be the reflection of line A″A* in A′, and define V and W cyclically. The lines U,V,W concur in X(6144). (Antreas Hatzipolakis and Peter Moses, November 18, 2014; see 22765)

X(6144) lies on these lines: {2,6}, {7,4969}, {45,3879}, {511,1657}, {518,3633}, {548,1350}, {576,5072}, {621,5340}, {622,5339}, {742,4764}, {1351,3818}, {1352,3850}, {1353,5085}, {3242,3635}, {3416,4691}, {3564,3627}, {3625,5847}, {3729,4725}, {3751,4668}, {3758,4445}, {3793,5210}, {3875,4715}, {4363,5564}, {4409,5845}, {4795,4967}

X(6144) = reflection of X(6) in X(193)

X(6145) = ABC-TO-HATZIPOLAKIS-MOSES-TRIANGLE ORTHOLOGY CENTER

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(-a^8+2 a^6 b^2-2 a^2 b^6+b^8+2 a^6 c^2-a^4 b^2 c^2-b^6 c^2-2 a^2 c^6-b^2 c^6+c^8)
X(6145) = (J² - 3)*X(54) + (2J² - 4)*X1594), where J = |OH|/R

Let A′B′C′ be the orthic triangle of ABC. Let A_B be the orthogonal projection of A′ onto line AB, and define B_C and C_A cyclically. Let A_C be the orthogonal projection of A′ onto line AC, and define B_A and C_B cyclically. Let A′_B be the reflection of A in A_B, and let A′_C be the reflection of A in A_C . Let N_A be the nine-point center of the triangle AA′_BA′_C, and define N_B and N_C cyclically. The triangle N_AN_BN_C, here named the Hatzipolakis-Moses triangle, is orthologic to ABC and perpective to ABC. The A-vertex of this triangle is given by

N_A = -a^4 b^2+2 a^2 b^4-b^6-a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6 : b^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) : c^2 (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4)

The ABC-to-N_AN_BN_C center of orthology is X(6145), and the N_AN_BN_C-to-ABC center of orthology is X(6146). The perspector of ABC and N_AN_BN_C is X(54). (Based on notes , Antreas Hatzipolakis and Peter Moses, November 19, 2014; see 22774)

X(6145) lies on the Jerabek hyperbola and these lines: {3,161}, {4,973}, {6,3574}, {52,265}, {54,1594}, {68,1154}, {69,1225}, {235,1177}, {826,2435}, {1176,1503}, {3426,5895}, {3521,6000}

X(6145) = reflection of X(2917) in X(1209)
X(6145) = complement of X(32354)
X(6145) = anticomplement of X(32391)

X(6146) = HATZIPOLAKIS-MOSES-TRIANGLE-TO-ABC ORTHOLOGY CENTER

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2-b^2-c^2) (2 a^8-3 a^6 b^2+a^4 b^4-a^2 b^6+b^8-3 a^6 c^2-2 a^4 b^2 c^2+a^2 b^4 c^2-4 b^6 c^2+a^4 c^4+a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-4 b^2 c^6+c^8)
X(6146) = (4 - J²)*X(54) + (J² - 2)*X1594)
X(6146) = (J² - 5)R²*X(4) + 2S_ω*X(6), where J = |OH|/R

The Hatzipolakis-Moses triangle is defined at X(6145).

X(6146) lies on these lines: {3,68}, {4,6}, {5,156}, {24,161}, {30,52}, {49,2072}, {54,1594}, {125,128}, {154,3542}, {186,2917}, {389,973}, {403,1614}, {427,578}, {539,1216}, {542,5907}, {550,1204}, {567,5576}, {1092,1368}, {1853,3541}, {1885,6000}, {3547,3796}, {3564,4173}

X(6146) = reflection of X(3575) in X(389)
X(6146) = X(72)-of-orthic-triangle if ABC is acute
X(6146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,68,343), (4,2883,1514)

X(6147) = COMPLEMENT OF X(3927)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a^3(b + c) + a^2(b^2 + 4bc + c^2) - 2a(b - c)^2(b + c) - (b^2 - c^2)^2
X(6147) = X(1) + 3X(4654)
X(6147) = X(3295) - 3X(3475)
X(6147) = 3X(3475) + X(4295)
X(6147) = 7X(3624) - 3X(3929)
X(6147) = r*X(3) + (4R + r)(X(7)
X(6147) = (2R - r)X(57) + 2X(140)

Let A′B′C′ be the pedal triangle of the incenter of ABC. Then X(6147) is the radical center of the nine-point circles of the triangles AA′I, BB′I, CC′I. (Antreas Hatzipolakis and Randy Hutson, December 2, 2014; see 22830)

X(6147) lies on these lines: {1,30}, {2,3927}, {3,7}, {5,226}, {8,1159}, {10,3824}, {12,5902}, {57,140}, {65,495}, {72,5249}, {142,5044}, {354,496}, {355,5290}, {381,938}, {382,3488}, {386,1086}, {388,952}, {405,5905}, {442,3868}, {443,3940}, {498,5221}, {499,4860}, {517,3671}, {527,1125}, {546,5722}, {548,3601}, {549,553}, {550,3982}, {632,3911}, {774,5492}, {908,5439}, {946,971}, {950,3627}, {975,4675}, {999,3485}, {1056,1482}, {1156,3296}, {1329,5883}, {1385,4298}, {1387,3304}, {1484,5083}, {1490,5805}, {1595,1876}, {1656,5226}, {1657,4313}, {1871,5236}, {2886,3874}, {3295,3475}, {3333,5843}, {3336,5432}, {3337,5433}, {3340,5844}, {3467,5443}, {3526,5435}, {3576,4355}, {3586,3853}, {3624,3929}, {3628,5219}, {3664,4920}, {3670,5718}, {3678,3826}, {3811,5880}, {3812,3820}, {3813,3881}, {3925,5904}, {4114,5122}, {5055,5704}, {5244,5752}, {5261,5790}, {5270,5425}, {5715,5787}

X(6147) = midpoint of X(3295) and X(4295)
X(6147) = reflection of X(i) in X(j) for these (i,j): (10,3824), (3560,5901)
X(6147) = complement of X(3927)
X(6147) = crosssum of X(55) and X(584)
X(6147) = (2nd extouch triangle)-to-(intouch triangle) similarity image of X(5)

X(6148) = ISOTOMIC CONJUGATE OF X(5627)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - 3 cot²A)(cos A - 2 cos B cos C)
Barycentrics    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc²A)(sin 3A)(sin B sin C - 3 cos B cos C)
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (csc A)(sin²A - 3 cos²A)(sin B sin C - 3 cos B cos C)
Barycentrics    k(A,B,C) : k(B,C,A) : k(C,A,B), where k(A,B,C) = (csc A)[csc(B + π/3)csc(C - π/3) + csc(C + π/3)csc(B - π/3)]
Barycentrics    m(a,b,c) : m(b,c,a) : m(c,a,b), where m(a,b,c) = (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)

X(6148) lies on these lines: {2,2986}, {22,1634}, {30,3260}, {69,74}, {76,328}, {186,340}, {316,1553}, {320,1443}, {325,3233}, {339,1225}, {526,3268}, {2407,3163}, {4226,5181}, {4590,5641}

X(6148) = anticomplement of X(6128)
X(6148) = isotomic conjugate of X(5627)
X(6148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (69,1272,1494), (99,1494,1272), (616,617,74)

X(6149) = X(13)-ISOCONJUGATE OF X(14)

Trilinears        1 - 4 cos²A : 1 - 4 cos²B : 1 - 4 cos²C
Trilinears        1 + 2 cos 2A : 1 + 2 cos 2B : 1 + 2 cos 2C
Trilinears        csc A sin 3A : csc B sin 3B : csc C sin 3C
Trilinears        sin²A - 3cos²A : sin²B - 3cos²B : sin²C - 3cos²C
Trilinears    directed distance of A to Hatzipolakis axis : :
Barycentrics    sin 3A : sin 3B : sin 3C
Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = = a^3 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2)
X(6149) = 3a²b²c²*X(1) - (-a + b + c)(a - b + c)(a + b - c)(a³ + b³ + c³)*X(31)

Let DEF be the circumtangential triangle. Let D' be the trilinear product E*F, and define E' and F' cyclically. Then D', E', F' all lie on the line X(1)X(21). Let UVW be the circumnormal triangle. Let U′ be the trilinear product V*W, and define V′ and W′ cyclically. Then U′, V′, W′ all line on the line X(656)X(1955). Moreover, X(6149) = X(1)X(21)∩X(656)X(1955), and the trilinear polar of X(6149) passes through X(2624). (Randy Hutson, December 12, 2014)

Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the trilinear product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(6149). (Randy Hutson, December 14, 2014)

Let DEF be any equilateral triangle inscribed in the circumcircle of ABC. Let D' be the barycentric product E*F, and define E', F' cyclically. Then D',E',F' all line on a line passing through X(50) (for which trilinears are sin 3A : sin 3B : sin 3C). In the special case that DEF is the circumtangential triangle, the points D',E',F' lie on the Brocard axis, and in case DEF is the circumnormal triangle, the points D',E'F' lie on the line X(50)X(647). (Randy Hutson, December 15, 2014)

X(6149) is a major center, as noted by César Lozada ( Major Centers)

Let E be the inellipse with perspector X(1101); then X(6149) is the intersection of the tangents to E at X(1094) and X(1095). Let A′B′C′ and A″B″C″ be the (equilateral) circumcevian triangles of X(15) and X(16), let A* be the trilinear product A′*A″, and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(6149). (Randy Hutson, January 29, 2015)

Let AA₁A₂, BB₁B₂, CC₁C₂ be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A′ be the trilinear product A₁*A₂, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6149); see also X(50). (Randy Hutson, October 13, 2015)

Let A′B′C′ be the anti-orthocentroidal triangle. Let A″ be the trilinear product B′*C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6149). (Randy Hutson, December 10, 2016)

X(6149) lies on these lines: {1,21}, {35,500}, {36,1464}, {109,484}, {162,1784}, {163,1755}, {171,3584}, {212,5010}, {223,3215}, {238,3582}, {526,2605}, {580,3336}, {656,1955}, {922,1101}, {1094,2151}, {1095,2152}, {1577,2627}, {1718,5535}, {1935,3585}, {1936,3583}, {2159,2315}, {2190,2962}, {3072,5270}, {3073,4857}, {5398,5902}

X(6149) = isogonal conjugate of X(2166)
X(6149) = crossdifference of every pair of points on the line X(661)X(1953)
X(6149) = trilinear product of X(15) and X(16)
X(6149) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,47,2964), (1,1749,1725), (47,255,1), (896,1725,1749), (1822,1823,1)
X(6149) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2166}, {2,1989}, {4,265}, {5,1141}, {6,94}, {13,14}, {25,328}, {30,5627}, {79,80}, {300,3458}, {301,3457}, {476,523}, {847,5961}

X(6150) = CIRCUMCIRCLE-INVERSE OF X(195)

Trlinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4-2*(b^2+c^2)*a^2+b^4+c^4-b^2*c^2)*(2*a^10-7*(b^2+c^2)*a^8+(10*((b^2+c^2)^2-b^2*c^2))*a^6-(b^2+c^2)*(8*c^4-7*b^2*c^2+8*b^4)*a^4+(b^2-c^2)^2*a^2*(4*b^4+3*b^2*c^2+4*c^4)-(b^2+c^2)*(b^2-c^2)^4)
Barycentrics    g(A,B,C) : g(C,A,B) : g(C,A,B), where g(A,B,C) = (1 - 2 cos 2A)[(2 cos A)(-2 + 4 cos²A + cos(2B - 2C)) + cos(B - C)(1 + 2 cos 2A)]
Barycentrics    h(A,B,C) : h(C,A,B) : h(C,A,B), where h(A,B,C) = a(S²_A - 3S²)[S²_A(-2S_ω + 3R²) + S_A(2S²_ω - 4S² - 3S_ωR²) + 2S²R²]

Let N be the nine-point center of ABC, and let A′ be the nine-point center of the triangle NBC. Define B′ and C′ cyclically. Let U be the reflection of the line NA′ in line BC, and define V and W cyclically. The lines U,V,W concur in X(6150). (A. Hatzipolakis, C. Lozada, P. Moses, December 15, 2014; see 22886)

X(6150) is the orthocenter of the pedal triangle of X(1157). (Randy Hutson, December 16, 2014)

X(6150) lies on these lines: {3,54}, {30,137}, {1291,2070}

X(6150) = midpoint of X(i) and X(j) for these (i,j): (3,1157), (1291,2070)
X(6150) = complementary conjugate of complement of X(34418)
X(6150) = X(1291)-Ceva conjugate of X(1510)
X(6150) = circumcircle-inverse of X(195)

X(6151) = ISOGONAL CONJUGATE OF X(395)

Trlinears 1/[cos(B - C) + 2 cos(A + π/3)] : 1/[cos(C - A) + 2 cos(B + π/3)] : 1/[cos(A - B) + 2 cos(C + π/3)]

Let U be the line tangent to the Neuberg cubic (K001) at X(14), and let V be the line tangent to K001 at X(16). Then X(6151) = U∩V. Let U′ be the line tangent to the Napoleon-Feuerbach cubic (K005) at X(18), and let V′ be the line tangent to K005 are X(62). Then X(6151) = U′∩V′; see X(2981). Also, X(6151) = DC(X(14)) (see the preamble to X(2979). (Randy Hutson, January 5, 2014)

X(6151) lies on the hyperbola {{A,B,C,X(2),X(6)}} and these lines: {14,5675}, {16,1338}, {37,5367}, {39,2981}, {42,5357}, {62,110}, {395,1989}

X(6151) = isogonal conjugate of X(395)
X(6151) = cevapoint of X(i) and X(j) for these (i,j): (6,16, (202, 2245)
X(6151) = X(i)-cross conjugate of X(j) for these (i,j): (323,2981), (6138,110)
X(6151) = trilinear pole of the line X(15)X(512)
X(6151) = X(i)-isoconjugate of X(j) for these {i,j}: {1,395}, {63,462}, {533,2153}, {619,2154}
X(6151) = X(i)-vertex conjugate of X(j) for these {i,j}: {1976,2981}, {2981,1976}
X(6151) = perspector of ABC and unary cofactor triangle of inner Napoleon triangle

X(6152) = ORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trlinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = aS_BS_C[(S² - S²_A)(S_ω - R²) + 2S²S_A]
Trlinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (-1 + cos 2B + cos 2C)(-1 + 2 cos 2A) secA
Barycentrics    h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)
X(6152) = 3X(1209) - 2X(1216)
X(6152) = 3X(54) - 5X(3567)
X(6152) = 6X(973) - 5X(3567)

X(6152) is introduced by Randy Hutson, César Lozada, Peter Moses, December 17, 2014; see 22900

X(6152) is the orthologic center of the pedal triangles of X(4) and X(5). This point is one of several found in connection with the following configuration: Let ABC be a triangle, and let T₁, T₂, T₃ be the pedal triangles of X(3), X(4), X(5), respectively. The three triangles are pairwise orthologic. Let P_i,j be the orthologic center of T_i, T_j. Then the orthocenter of T₃ is the midpoint of P_1,3 and P_2,3. (Antreas Hatzipolakis and César Lozada, December 17, 2014; see 22907

See also Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25814.

X(6152) lies on these lines: {4,93}, {6,24}, {25,195}, {49,143}, {52,539}, {70,6145}, {113,5446}, {155,3060}, {403,3574}, {648,1179}, {1112,2914}, {1209,1216}, {1843,5965}, {1986,3575}

X(6152) = reflection of X(i) in X(j) for these (i,j); (54,973), (1493,143), (1209,6153), (2914,1112)
X(6152) = X(4)-Ceva conjugate of X(1594)
X(6152) = X(4)-crosspoint of X(3518)
X(6152) = X(3)-crosssum of X(3519)
X(6152) = X(2216)-isoconjugate of X(3519)
X(6152) = X(79)-of-orthic-triangle if ABC is acute
X(6152) = trilinear product X(1594)*X(2964)
X(6152) = barycentric product X(1594)*X(1994)
X(6152) = X(10122) of tangential triangle, if ABC is acute
X(6152) = Ehrmann-vertex-to-orthic similarity image of X(6288)

X(6153) = MIDPOINT OF X(52) AND X(2888)

Trlinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [cos 3A + 4 sin²A cos(B - C)][1 + 2 cos A cos(B - C)]
X(6153) = 3X(51) - X(195)

X(6153) is the orthocenter of the pedal triangle of X(5); see X(6154). (Antreas Hatzipolakis and César Lozada, December 17, 2014; see 22907

See also Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 25814.

X(6153) lies on these lines: {51,195}, {52,2888}, {54,5462}, {539,973}, {546,1154}, {1209,1216}

X(6153) = midpoint of X(i) and X(j) for these {i,j}: {52,2888}, {1209,6152}
X(6153) = reflection of X(i) in X(j) for these (i,j); (54,5462), (1216,1209)

X(6154) = X(11) - 2X(100)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4 a^3-4 a^2 b+a b^2-b^3-4 a^2 c+2 a b c+b^2 c+a c^2+b c^2-c^3
X(6154) = 6X(2) - 5X(11)
X(6154) = 3X(2) - X(100)
X(6154) = 4X(2) - 5X(5164)
X(6154) = X(11) - 2X(100)

X(6154) lies on these lines: {2,11}, {19,1862}, {40,550}, {65,1317}, {104,3528}, {119,546}, {165,4863}, {214,3636}, {382,5840}, {516,3689}, {519,5183}, {548,5288}, {678,3120}, {900,4088}, {1086,3722}, {1145,3626}, {1155,5853}, {1320,5558}, {1836,3158}, {1842,5151}, {2805,4681}, {2829,3529}, {3059,6068}, {3174,5528}, {3189,5854}, {3629,3779}, {3698,4314}, {3711,5698}, {3893,4297}, {3952,4152}, {3962,5493}, {4819,5847}, {5082,5217}

X(6154) = anticomplement of anticomplement of X(35023)
X(6154) = intangents-to-extangents similarity image of X(11)
X(6154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,100,6174), (100,149,3035), (149,3035,11), (3434,4421,5432)

X(6155) = BICENTRIC SUM OF PU(113)

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

Suppose that X = x : y : z (trilinears) is a triangle center. The points P(X) = y/z : z/x : x/y and U(X) = z/y : x/z : y/x are introduced at Bicentric Pairs as the 1st and 2nd bicentric quotients of X, respectively. The pair is both a bicentric pair and an isogonal conjugate pair. As an example, P(X(2)) and U(X(2)) are the 1st and 2nd Brocard points. The points PU(113) are the bicentric quotients of X(37) = b + c : c + a : a + b, so that PU(113) consists of the following bicentric pair: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). The bicentric sum of two points p : q : r and u : v : w is the point p + u : q + v : r + w. For the bicentric difference of PU(113), see X(4983).

X(6155) lies on these lines: {1,1929}, {6,191}, {37,762}, {39,3121}, {42,3954}, {81,763}, {213,3931}, {1100,3285}, {2238,3743}, {2295,4868}, {4037,4065}

X(6155) = PU(113)-harmonic conjugate of X(4983)

X(6156) = CROSSSUM OF PU(113)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a+b) (a+c) (a^4+2 a^3 b+a^2 b^2+b^4+2 a^3 c+4 a^2 b c+2 a b^2 c+4 b^3 c+a^2 c^2+2 a b c^2+7 b^2 c^2+4 b c^3+c^4)

X(6156) is the crosssum of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)

X(6156) lies on these lines: {37,2134}, {81,1126}, {762,763}

X(6156) = crosspoint of PU(113)

X(6157) = CROSSDIFFERENCE OF PU(113)

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

X(6157) is the crossdifference of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)

X(6157) lies on these lines: {37,81}, {762,763}, {1654,2248}, {1961,1963}, {2134,3293}

X(6157) = isogonal conjugate of X(6158)
X(6157) = crossdifference of every pair of points on the line X(4983)X(6155)
X(6157) = X(2)-Ceva conjugate of X(39064)
X(6157) = perspector of the conic {{A,B,C,PU(113)}}

X(6158) = TRILINEAR POLE OF LINE PU(113)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c) (a^2+a b+b^2-a c-b c-c^2) (a^2-a b-b^2+a c-b c+c^2) (a^2+3 a b+b^2+a c+b c+c^2) (a^2+a b+b^2+3 a c+b c+c^2)

X(6158) is the trilinear pole of the line of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)

X(6158) lies on this line: {1962, 2054}

X(6158) = isogonal conjugate of X(6157)
X(6158) = trilinear pole of the line X(4983)X(6155)

X(6159) = IDEAL POINT OF LINE PU(113)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (b+c) (a^5+4 a^4 b+5 a^3 b^2+4 a^2 b^3+a b^4+4 a^4 c+8 a^3 b c+10 a^2 b^2 c+8 a b^3 c+2 b^4 c+5 a^3 c^2+10 a^2 b c^2+11 a b^2 c^2+4 b^3 c^2+4 a^2 c^3+8 a b c^3+4 b^2 c^3+a c^4+2 b c^4)

X(6159) is the ideal point of the line of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155)

X(6159) lies on these lines: {30,511}, {4983,6155}

X(6159) = crossdifference of every pair of points on the line X(6)X(6157)

X(6160) = MIDPOINT OF THE BICENTRIC PAIR PU(113)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c) (3 a^5 b+8 a^4 b^2+9 a^3 b^3+4 a^2 b^4+a b^5+3 a^5 c+10 a^4 b c+17 a^3 b^2 c+16 a^2 b^3 c+7 a b^4 c+2 b^5 c+8 a^4 c^2+17 a^3 b c^2+20 a^2 b^2 c^2+11 a b^3 c^2+2 b^4 c^2+9 a^3 c^3+16 a^2 b c^3+11 a b^2 c^3+2 b^3 c^3+4 a^2 c^4+7 a b c^4+2 b^2 c^4+a c^5+2 b c^5)

X(6160) is the midpoint of these points: P(X(37)) = (a + c)/(a + b) : (b + a)/(b + c) : (c + b)/(c + a) and U(X(37)) = (a + b)/(a + c) : (b + c)/(b + a) : (c + a)/(c + b). See X(6155).

X(6160) lies on this line: {4983,6155}

X(6161) = BICENTRIC SUM OF PU(114)

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

Suppose that X = x : y : z (trilinears) is a triangle center. The points P(X) = y/z : z/x : x/y and U(X) = z/y : x/z : y/x are introduced at Bicentric Pairs as the 1st and 2nd bicentric quotients of X, respectively. The pair is both a bicentric pair and an isogonal conjugate pair. As an example, P(X(2)) and U(X(2)) are the 1st and 2nd Brocard points. The points PU(114) are the bicentric quotients of X(513) = b - c : c - a : a - b, so that PU(114) consists of the following bicentric pair: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). For the bicentric difference of PU(114), see X(2087).

X(6161) lies on these lines: {1,513}, {3,667}, {10,4448}, {512,2292}, {514,3244}, {522,5592}, {649,1334}, {659,3887}, {661,1643}, {663,1201}, {830,4983}, {832,1245}, {876,6005}, {891,4895}, {1027,2334}, {1388,3669}, {1491,4794}, {1960,2254}, {2098,4162}, {2136,3900}, {3293,4040}, {3667,4297}, {3803,4834}, {4083,5697}

X(6161) = PU(114)-harmonic conjugate of X(2087)

X(6162) = CROSSSUM OF PU(114)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b) (a-c) (a^4-2 a^3 b+a^2 b^2+b^4-2 a^3 c+4 a^2 b c-2 a b^2 c-4 b^3 c+a^2 c^2-2 a b c^2+7 b^2 c^2-4 b c^3+c^4)

See X(6155) and X(6156).

X(6162) lies on these lines: {1,88}, {764,6163}, {5376,6161}

X(6162) = crosspoint of PU(114)

X(6163) = CROSSDIFFERENCE OF PU(114)

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

See X(6155) and X(6156).

Let P′ be the line tangent to the Hutson-Moses hyperbola H = {{A,B,C,PU(26),PU(33)}} at P(33) , and let U′ be the line tangent to H at U(33). Then X(6163) = P′∩U′. Also, the trilinear polar of X(6163) passes through X(1054). (Randy Hutson, January 6, 2015)

X(6163) lies on these lines: {44,3290}, {100,513}, {190,523}, {238,1149}, {244,1052}, {385,4831}, {518,5048}, {651,660}, {758,1757}

X(6163) = isogonal conjugate of X(6164)
X(6163) = X(2)-Ceva conjugate of X(39065)
X(6163) = crosspoint of PU(33)
X(6163) = crosssum of PU(34)
X(6163) = crossdifference of every pair of points on the line X(2087)X(6166)
X(6163) = perspector of the conic {{A,B,C,PU(114)}}

X(6164) = TRILINEAR POLE OF LINE PU(114)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (a^2-3 a b+b^2+a c+b c-c^2) (a^2+a b-b^2-3 a c+b c+c^2)

X(6164) is the trilinear pole of the line of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6158).

X(6158) lies on these lines: {513,1052}, {900,3035}, {1635,3722}

X(6164) = isogonal conjugate of X(6163)
X(6164) = cevapoint of PU(34)
X(6164) = trilinear pole of the line X(2087)X(6166)

X(6165) = IDEAL POINT OF LINE PU(114)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c) (a^3 b-3 a^2 b^2+a b^3+a^3 c+2 a b^2 c-3 a^2 c^2+2 a b c^2-2 b^2 c^2+a c^3)

X(6165) is the ideal point of the line of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6159).

X(6165) lies on these lines: {30,511}, {238,1960}, {659,1757}

X(6165) = crossdifference of every pair of points on the line X(6)X(6163)

X(6166) = MIDPOINT OF THE BICENTRIC PAIR PU(114)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b-c)^2 (a^5-5 a^3 b^2+4 a^2 b^3-a b^4+6 a^2 b^2 c-2 a b^3 c-5 a^3 c^2+6 a^2 b c^2-9 a b^2 c^2+2 b^3 c^2+4 a^2 c^3-2 a b c^3+2 b^2 c^3-a c^4)

X(6166) is the midpoint of these points: P(X(513)) = (a - c)/(a - b) : (b - a)/(b - c) : (c - b)/(c - a) and U(X(513)) = (a - b)/(a - c) : (b - c)/(b - a) : (c - a)/(c - b). See X(6155) and X(6160).

X(6166) lies on these lines: {1,39}, {2087,6161}

X(6167) = CROSSSUM OF PU(115)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a+b-c) (a-b+c) (a^4-2 a^3 b+2 a^2 b^2-2 a b^3+b^4-2 a^3 c+8 a^2 b c-6 a b^2 c+2 a^2 c^2-6 a b c^2+6 b^2 c^2-2 a c^3+c^4)

X(6167) = crosssum of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).

X(6167) lies on these lines: {1,1462}, {9,2124}, {57,145}, {77,2329}, {194,2128}, {664,2082}, {728,738}

X(6167) = crosspoint of PU(115)

X(6168) = CROSSDIFFERENCE OF PU(115)

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

X(6168) = crossdifference of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).

X(6168) lies on these lines: {2,7}, {279,3501}, {348,1334}, {521,2254}, {651,3684}, {728,738}, {1018,1323}, {1419,3158}, {2124,2136}, {2272,5845}, {3160,3208}

X(6168) = isogonal conjugate of X(6169)
X(6168) = crossdifference of every pair of points on the line X(663)X(2082)
X(6168) = X(2)-Ceva conjugate of X(39066)
X(6168) = perspector of the conic {{A,B,C,PU(115)}}

X(6169) = TRILINEAR POLE OF PU(115)

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

X(6169) = trilinear pole of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).

X(6169) lies on these lines: {6,1633}, {1438,3451}

X(6169) = isogonal conjugate of X(6168)
X(6169) = trilinear pole of the line X(663)X(2082)

X(6170) = IDEAL POINT OF PU(115)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (b-c) (2 a^4 b-2 a^3 b^2+2 a^2 b^3-2 a b^4+2 a^4 c-3 a^3 b c+a^2 b^2 c+3 a b^3 c+b^4 c-2 a^3 c^2+a^2 b c^2-2 a b^2 c^2-b^3 c^2+2 c+a^2 c^3+3 a b c^3-b^2 c^3-2 a c^4+b c^4)

X(6170) = ideal point of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).

X(6170) lies on these lines: {30,511}, {663,2082}

X(6170) = crossdifference of every pair of points on the line X(6)X(6168)

X(6171) = MIDPOINT OF PU(115)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a-b-c) (a^6-2 a^5 b-a^4 b^2-4 a^3 b^3+7 a^2 b^4-2 a b^5+b^6-2 a^5 c+6 a^4 b c+2 a^3 b^2 c-10 a^2 b^3 c-4 b^5 c-a^4 c^2+2 a^3 b c^2+6 a^2 b^2 c^2+2 a b^3 c^2+7 b^4 c^2-4 a^3 c^3-10 a^2 b c^3+2 a b^2 c^3-8 b^3 c^3+7 a^2 c^4+7 b^2 c^4-2 a c^5-4 b c^5+c^6)

X(6171) = midpoint of these points: P(X(9)) = (a + c - b)/(a + b - c) : (b + a - c)/(b + c - a) : (c + b - a)/(c + a - b) and U(X(9) = (a + b - c)/(a + c - b) : (b + c - a)/(b + a - c) : (c + a - b)/(c + b - a). See X(6155).

X(6171) lies on this line: {663, 2082}

X(6172) = REFLECTION OF X(7) IN X(2)

Trlinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = -5 a^2+4 a b+b^2+4 a c-2 b c+c^2
X(6172) = 5X(2) - 4X(142)
X(6172) = X(7) - 4X(9)
X(6172) = 2X(9) + X(144)
X(6172) = X(390) + 2X(5223)

X(6172) lies on these lines: {2,7}, {8,190}, {30,5759}, {44,4419}, {45,4644}, {69,3161}, {72,4313}, {75,4488}, {192,4460}, {220,651}, {346,4416}, {376,971}, {381,5762}, {390,519}, {391,3729}, {452,3951}, {480,4421}, {516,3543}, {518,1992}, {536,5838}, {545,673}, {549,5843}, {551,5850}, {599,4370}, {938,3927}, {956,4345}, {958,4323}, {960,4308}, {966,5936}, {984,4344}, {1014,1696}, {1212,5543}, {1442,2324}, {1743,3672}, {2346,4428}, {2550,5080}, {2801,5692}, {3008,4346}, {3091,5735}, {3177,4552}, {3474,3715}, {3545,5805}, {3616,3758}, {3663,3973}, {3678,5696}, {3686,4461}, {3707,4659}, {3731,3945}, {3829,6067}, {3876,5784}, {4384,4454}, {4473,4741}, {4755,4795}, {4866,5493}, {4916,6144}, {4971,5839}, {5258,5734}, {5851,6174}

X(6172) = midpoint of X(2) and X(144)
X(6172) = reflection of X(i) in X(j) for these (i,j): (7,2), (2,9)
X(6172) = anticomplement of X(6173)
X(6172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,144,7), (44,4419,5222), (45,4644,5308), (329,3219,5273), (329,5273,5226), (1992,4664,3241), (4384,4480,4454), (5220,5698,8)

X(6173) = REFLECTION OF X(9) IN X(2)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2+a b-2 b^2+a c+4 b c-2 c^2
X(6173) = X(1) + 2X(5880)
X(6173) = 5X(2) - X(144)
X(6173) = 2X(3) + X(5735)
X(6173) = 2X(7) + X(9)

X(6173) lies on these lines: {1,528}, {2,7}, {3,5735}, {6,4859}, {30,5732}, {36,1001}, {37,4862}, {69,4034}, {75,4007}, {85,1121}, {320,4384}, {376,516}, {381,971}, {518,599}, {519,1056}, {547,5843}, {549,5762}, {597,4795}, {610,5829}, {673,2364}, {903,4664}, {936,6147}, {942,5784}, {958,4355}, {1125,5698}, {1449,3664}, {1470,4870}, {1698,5220}, {1699,3742}, {2321,4869}, {2323,5228}, {2325,4454}, {2801,5587}, {3008,4644}, {3058,4326}, {3062,3255}, {3158,3475}, {3241,5853}, {3247,3663}, {3487,5438}, {3524,5759}, {3620,4967}, {3624,3916}, {3731,4902}, {3812,5290}, {3824,5705}, {3826,5223}, {3828,5850}, {3834,4363}, {3912,4659}, {3945,3946}, {4292,5436}, {4321,5434}, {4346,5308}, {4361,4725}, {4419,4887}, {4445,4739}, {4667,5222}, {4851,4971}, {4860,5231}, {5055,5779}, {5071,5817}, {5856,6174}

X(6173) = midpoint of X(2) and X(7)
X(6173) = reflection of X(i) in X(j) for these (i,j): (9,2), (2,142)
X(6173) = complement of X(6172)
X(6173) = trilinear product X(i)*X(j) for these {i,j}: {2,4860}, {57,5231}
X(6173) = barycentric product X(i)*X(j) for these ({i,j}: {7,5231}, {75,4860}
X(6173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,144,7), (44,4419,5222), (45,4644,5308), (329,3219,5273), (329,5273,5226), (1992,4664,3241), (4384,4480,4454), (5220,5698,8).

X(6174) = REFLECTION OF X(11) IN X(2)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2 a-b-c) (2 a^2-a b-b^2-a c+2 b c-c^2)
X(6174) = 5X(2) - X(149)
X(6174) = X(11) + 2X(100)
X(6174) = 5X(100) + X(149)
X(6174) = X(104) - 3X(3524)

X(6174) lies on these lines: {2,11}, {30,119}, {104,3524}, {165,5660}, {210,2801}, {214,519}, {376,2829}, {381,5840}, {527,1155}, {529,4996}, {549,952}, {551,2802}, {599,5848}, {678,1647}, {900,1635}, {1387,5541}, {1470,5434}, {1768,3929}, {2482,2787}, {2783,6055}, {2805,4755}, {3241,5854}, {3530,5258}, {3711,5744}, {3712,5205}, {3722,3756}, {3820,5010}, {3887,4763}, {5433,5687}, {5856,6173}

X(6174) = midpoint of X(i) and X(j) for these {i,j}: {2,100}, {165,5660)
X(6174) = reflection of X(i) in X(j) for these (i,j): (11,2), (2,3035)
X(6174) = crossdifference of every pair of points on the line X(106)X(665)
X(6174) = trilinear product X(i)*X(j) for these {i,j}: {44,527}, {519,1155}, {1023,1638}, {1055,4358}, {1323,3689}
X(6174) = barycentric product {519,527}, {1055,3264}, {1155,4358}, {1323,2325}
X(6174) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4421,3058), (100,3035,11), (214,1145,1317), (1376,5432,3925)
X(6174) = X(i)-isoconjugate of X(j) for these (i,j): {88,2291}, {106,1156}

X(6175) = REFLECTION OF X(21) IN X(2)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4+a^2 b^2-2 b^4+3 a^2 b c+3 a b^2 c+a^2 c^2+3 a b c^2+4 b^2 c^2-2 c^4
X(6175) = 2X(2) + X(21) = X(8) + 2X(3649) = 2X(10) + X(79) = X(21) - 4X(442)

As a point on the Euler line, X(6175) has Shinagawa coefficients (3abc$a$ + 2S², 6S²).

X(6175) lies on these lines: {2,3}, {8,3649}, {10,79}, {13,5362}, {14,5367}, {81,3017}, {100,3584}, {519,5178}, {528,2346}, {540,4921}, {671,2795}, {758,3679}, {1125,5441}, {1330,3578}, {1441,1494}, {1698,3647}, {2771,3753}, {2886,5434}, {3241,3475}, {3474,3648}, {3582,5253}, {3583,5284}, {3585,3841}, {3656,5330}, {3838,4511}, {3868,4654}, {3925,5080}, {3936,4720}, {5276,5309}

X(6175) = midpoint of X(2) and X(2475)
X(6175) = reflection of X(i) in X(j) for these (i,j): (21,2), (2,442)
X(6175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,21,15671), (2,3545,4193), (2,3839,2478), (2,4188,5054), (2,4190,3524), (2,5141,5055), (4,4197,5047), (377,2476,404), (377,5177,2476), (442,2475,21), (443,3545,2), (474,5055,2), (3585,3841,5260), (3839,4208,2)

X(6176) = CENTER OF OUTER APOLLONIAN CIRCLE OF ODEHNAL TRITANGENT CIRCLES

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a⁵(b + c) + a⁴(b + c)² - a³(b + c)(3b² - 2bc + 3c²) - a²(b⁴ + 4b³c + 4b²c² + 4bc³ + c⁴) + a(b + c)(b⁴ - 2b³c - 2bc³ + c⁴) + 2bc(b² - c²)²
X(6176) = (b + c)(c + a)(a + b)*X(1) + abc*X(970)
X(6176) = [r²(2r + 5R) + s²(2r + R)]*X(1) + R(3r² - s²)*(X(181)
X(6176) = (r² + s² + 2rR)*X(1) + 6rR(X(2) + (r² + s²)*X(3)
X(6176) = 12rR*X(2) + (3r² + s²)*X(3) - (r² - s² + 4rR)*X(6)

Let ABC be an acute triangle, and let L_A be the circle that is externally tangent to the nine-point circle and to the sidelines AB and AC. Define L_B and L_C cyclically. The three circles are named the Odehnal tritangent circles at X(3822). The center of their outer Apollonian circle is X(6176). (Boris Odehnal, "A Triad of Tritangent Circles, Journal for Geometry and Graphics 18 (2014) no. 1, 61-71)

X(6176) lies on these lines: {1,181}, {3,4653}, {5,515}, {405,1437}, {551,2051}, {995,5396}, {1319,5718}, {3576,4192}

X(6177) = CEVAPOINT OF THE REAL FOCI OF THE STEINER INELLIPSE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(e sin A + sin(A - ω), where e is defined at X(1340)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a⁴ + b⁴ + c⁴ - a²b² - a²c² + (b² + c²)(a⁴ + b⁴ + c⁴ - a²b² - a²c² - b²c²) ^1/2

See also X(3557) and X(6178). Contributed by Bernard Gibert, January 4, 2015.

The angle θ for X(6177) as a Kiepert perspector is given by cot(θ) = - cot(ω) - e csc(ω). (Peter Moses, January 6, 2015)

In the plane of a triangle ABC, let
F1 and F2 be the real foci of the Steiner inellipse
A1 = AF1∩BC
A2 = AF2∩BC
Ga = conic tangent to AF1 at A1, to AF2 at A2, and to F1F2
Ta = Ga∩F1F2
Ao = center of Ga, and define Bo and Co cyclically
The lines AAo, BBo, CCo concur in X(14633), and the lines ATa, BTb, CTc concur in X(6177). See X(6177) and X(14633). (Angel Montesdeoca, July 21, 2023)

X(6177) lies on the cubics K009, K704, the Kiepert hyperbola, and these lines: {3,3414}, {4,1380}, {76,1341}, {98,2558}, {140,141}, {262,3558}, {1078,2559}, {3557,6179}

X(6177) = isogonal conjugate of X(3557)
X(6177) = {X(141,3788)}-harmonic conjugate of X(6178)
X(6177) = cevapoint of PU(118)

X(6178) = CEVAPOINT OF THE IMAGINARY FOCI OF THE STEINER INELLIPSE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(e sin A - sin(A - ω), where e is defined at X(1340)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a⁴ + b⁴ + c⁴ - a²b² - a²c² - (b² + c²)(a⁴ + b⁴ + c⁴ - a²b² - a²c² - b²c²) ^1/2

See also X(3558) and X(6177). Contributed by Bernard Gibert, January 4, 2015.

The angle θ for X(6178) as a Kiepert perspector is given by cot(θ) = - cot(ω) + e csc(ω). (Peter Moses, January 6, 2015)

X(6178) lies on the cubics K009, K704, the Kiepert hyperbola, and these lines: {3,3413}, {4,1379}, {76,1340}, {98,2559}, {140,141}, {262,3557}, {1078,2558}, {3558,6179}

X(6178) = isogonal conjugate of X(3558)

X(6178) = {X(141,3788)}-harmonic conjugate of X(6177)

X(6179) = PAPPUS PIVOT

Barycentrics 2a⁴ - b²c² : 2b⁴ - c²a² : 2c⁴ - a²b²

Contributed by Bernard Gibert, January 4, 2015. See K704.

X(6179) lies on these lines: {2,5007}, {6,1078}, {24,648}, {32,76}, {83,183}, {99,3053}, {187,194}, {193,1692}, {305,1627}, {316,3767}, {350,n+52}, {371,6312}, {372,6316}, {382,671}, {538,3552}, {575,631}, {609,1909}, {754,5025}, {1384,1975}, {1724,3570}, {1799,5359}, {3557,6177}, {3558,6178}, {3785,5304}, {3793,5305}, {3934,5008}, {4027,6309}, {4045,5368}

X(6179) = complement of X(7946)
X(6179) = anticomplement of X(7821)
X(6179) = (X(i),X(i))-harmonic conjugate of X(k) for these (i,,j,k): (32,76,3972), (32,385,76)

X(6180) = CROSSSUM OF PU(112)

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

Contributed by Randy Hutson, January 6, 2015.

X(6180) lies on these lines: {1,971}, {2,1407}, {6,7}, {9,241}, {12,1406}, {37,77}, {44,1418}, {45,1443}, {55,1742}, {56,87}, {57,1122}, {63,1427}, {65,3751}, {69,3713}, {72,1448}, {85,894}, {141,5782}, {144,220}, {175,3298}, {176,3297}, {192,664}, {219,527}, {221,388}, {222,226}, {223,3666}, {278,3782}, {307,965}, {347,2256}, {394,5905}, {405,4306}, {481,1335}, {482,1124}, {513,1037}, {518,2263}, {608,5236}, {644,4488}, {954,991}, {958,1042}, {960,4320}, {984,5018}, {1001,1458}, {1004,1331}, {1014,1778}, {1044,5584}, {1146,5942}, {1191,3600}, {1203,4355}, {1212,4350}, {1253,3000}, {1373,3299}, {1374,3301}, {1376,6168}, {1386,4327}, {1441,4363}, {1449,4328}, {1476,3445}, {1615,3599}, {1616,4308}, {1736,5779}, {1804,2178}, {2003,4654}, {3157,5706}, {3242,4318}, {3729,4513}, {4032,4559}, {5290,5711}

X(6180) = crosspoint of PU(46)
X(6180) = crosssum of PU(112)
X(6180) = {X(222),X(226)}-harmonic conjugate of X(940)

X(6181) = MIDPOINT OF PU(112)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)[a^4 - a^3(b + c) - a^2(3b^2 - 4bc + 3c^2) + 3a(b - c)^2(b + c) - 2bc(b - c)^2]

Contributed by Randy Hutson, January 6, 2015.

X(6181) lies on these lines: {3,1939}, {55,650}, {1376,6184}, {3119,4414}

X(6182) = IDEAL POINT OF PU(112)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin B cos C sec²(C/2) - sin C cos B sec²(B/2)
Trilinears g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b + c - a)(b - c)[a^3 - a^2(b + c) + a(b + c)^2 - (b - c)^2(b + c)]

Contributed by Randy Hutson, January 6, 2015.

X(6182) lies on these lines: {30,511}, {55,650}, {657,4041}, {661,4105}, {693,3434}, {1824,4024}, {2520,4524}, {2886,4885}

X(6182) = isogonal conjugate of X(6183)
X(6182) = crossdifference of every pair of points on the line X(6)X(77)

X(6183) = TRILINEAR POLE OF LINE X(6)X(77)

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[sin B cos C sec²(C/2) - sin C cos B sec²(B/2)]
Trilinears g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[(b + c - a)(b - c)[a^3 - a^2(b + c) + a(b + c)^2 - (b - c)^2(b + c)]]

X(6183) is the point of intersection, other than A, B, C, of the circumcircle and the hyperbola {{A, B, C, PU(46)}}. Contributed by Randy Hutson, January 6, 2015.

X(6183) lies on the circumcircle and these lines: {7,105}, {100,883}, {101,1025}, {108,658}, {112,1414}, {651,919}, {664,1292}, {927,1633}, {2724,5088}

X(6183) = isogonal conjugate of X(6182)
X(6183) = trilinear pole of the line X(6)X(77)
X(6183) = Ψ(X(6), X(77))
X(6183) = Λ(X(657), X(4041))
X(6183) = Λ(PU(112))

X(6184) = X(2)-CEVA CONJUGATE OF X(518)

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

Contributed by Randy Hutson, January 6, 2015.

X(6184) lies on the Steiner inellipse and these lines: {1,39}, {2,2481}, {3,101}, {6,3939}, {9,1742}, {10,1146}, {32,218}, {37,142}, {55,1438}, {100,294}, {106,5024}, {115,120}, {119,1566}, {187,1017}, {241,3693}, {528,5701}, {650,6174}, {665,1642}, {672,2223}, {867,5513}, {80,5308}, {1084,2092}, {1376,6181}, {1575,3008}, {2311,3110}

X(6184) = isogonal conjugate of X(6185)
X(6184) = complement of X(2481)
X(6184) = X(2)-Ceva conjugate of X(518)
X(6184) = crossdifference of every pair of points on the line X(105)X(659)
X(6184) = barycentric square of X(518)
X(6184) = complementary conjugate of X(20544)
X(6184) = polar conjugate of isogonal conjugate of X(20776)
X(6184) = center of the hyperbola {{A,B,C,X(2),X(100),PU(112)}}
X(6184) = crosssum of circumcircle intercepts of line PU(46) (line X(6)X(513))

X(6185) = CEVAPOINT OF X(6) AND X(105)

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(b² + c² - ab - ac)²]

Contributed by Randy Hutson, January 6, 2015.

X(6185) lies on these lines: {105,2223}, {238,516}, {239,294}, {241,927}, {1429,1438}, {5091,5222}

X(6185) = isogonal conjugate of X(6184)
X(6185) = isotomic conjugate of X(4437)
X(6185) = cevapoint of X(6) and X(105)
X(6185) = X(6)-cross conjugate of X(105)
X(6185) = polar conjugate of X(34337)
X(6185) = barycentric square of X(673)
X(6185) = trilinear pole of the line X(105)X(659) (tangent to circumcircle at X(105))

X(6186) = ISOGONAL CONJUGATE OF X(319)

Trilinears a²/(1 + 2 cos A) : b²/(1 + 2 cos B) : c²/(1 + 2 cos C)

X(6186) lies on these lines: {21,36}, {31,1495}, {35,1255}, {55,199}, {105,5322}, {614,3415}, {999,1036}, {2194,2260}, {2206,3122}, {2306,3129}

X(6186) = cevapoint of X(667) and X(3122)
X(6186) = crosspoint of X(56) and X(3444)
X(6186) = crosssum of X(i) and X(j) for these {i,j}: {2,3648}, {8,2895}, {3219,4420}, {3678,3969} X(6186) = trilinear pole of X(3063) and X(4834)
X(6186) = trilinear product X(i)*X(j) for these {i,j}: {6,2160}, {31,79}, {1402,3615}
X(6186) = barycentric product X(i)*X(j) for these {i,j}: {1,2160}, {6,79}, {36,1989}
X(6186) = barycentric product of vertices of reflection triangle of X(1)
X(6186) = perspector of ABC and unary cofactor triangle of Gemini triangle 25
X(6186) = X(i)-isoconjugate of X(j) for these (i,j): {1,319}, {2,3219}, {7,4420}, {8,1442}, {35,75}, {76,2174}, {81,3969}, {86,3678}, {100,4467}, {312,2003}, {314,2594}, {332,1825}, {340,1807}, {668,2605}, {1255,3578}, {1268,3647}, {1399,3596}, {2611,4600}

X(6187) = ISOGONAL CONJUGATE OF X(320)

Trilinears a²/(1 - 2 cos A) : b²/(1 - 2 cos B) : c²/(1 - 2 cos C)

X(6187) lies on the cubic K312 and these lines: {10,21}, {23,5143}, {25,2181}, {31,51}, {36,88}, {41,1017}, {42,692}, {45,55}, {56,244}, {105,2006}, {902,2183}, {976,1036}, {1911,5040}, {1960,3310}, {2204,2333}, {3145,4642}

X(6187) = X(i)-Ceva conjugate of X(j) for these (i,j): (759,2161), (1168,6)
X(6187) = X(2251)-cross conjugate of X(6)
X(6187) = X(i)-vertex conjugate of X(j) for these (i,j): (291,1929), (292,649), (1635,2161)
X(6187) = X(55)-beth conjugate of X(692)
X(6187) = cevapoint of X(1960) and X(3271)
X(6187) = crosspoint of X(i) and X(j) for these {i,j}: {106,909}, {1411,2161}
X(6187) = crossdifference of every pair of points on the line X(3904)X(3960)
X(6187) = crosssum of X(i) and X(j) for these {i,j}: {519,908}, {758,3936}, {3218,4511}, {4089,4453} X(6187) = trilinear pole of X(213) and X(3063)
X(6187) = vertex conjugate of PU(34)
X(6187) = trilinear product X(i)*X(j) for these {i,j}: {6,2161}, {25,1807}, {31,80}, {41,2006}, {42,759}, {55,1411}, {655,3063}, {663,2222}, {902,1168}, {1400,2341}, {1989,2174}
X(6187) = barycentric product X(i)*X(j) for these {i,j}: {1,2161}, {6,80}, {9,1411}, {19,1807}, {35,1989}, {37,759}, {44,1168}, {55,2006}, {65,2341}, {650,2222}, {655,663}, {1793,1880}, {2166,2174}
X(6187) = X(i)-isoconjugate of X(j) for these (i,j): {1,320}, {2,3218}, {7,4511}, {8,1443}, {36,75}, {69,1870}, {77,5081}, {81,3936}, {85,2323}, {86,758}, {100,4453}, {106,1227}, {190,3960}, {214,903}, {274,2245}, {310,3724}, {314,1464}, {332,1835}, {349,4282}, {514,4585}, {651,3904}, {654,4554}, {662,4707}, {664,3738}, {765,4089}, {860,1444}, {870,3792}, {1268,4973}, {1509,4053}, {1983,3261}, {2361,6063}, {2610,4610}, {4025,4242}, {4373,4881}

X(6188) = DAO (a,b,c,R) PERSPECTOR

Barycentrics 1 /[a^8 - 4 a^6 (b^2 + c^2) + (b^2 - c^2)^2 (b^4 + 6 b^2 c^2 + c^4) + a^4 (6 b^4 - 3 b^2 c^2 + 6 c^4) + a^2 (-4 b^6 + 3 b^4 c^2 + 3 b^2 c^4 - 4 c^6)]

Let r_A be a positive valued function of a,b,c, and define r_B and r_C cyclically. Let (A) denote the circle with center A and radius r_A, and define (B) and (C) cyclically. Let P be the radical center of (A), (B), (C). Let r_P be a positive valued function of a,b,c, let L_A be the radical axis of (A), (B), (P), and define L_B and L_C cyclically. Let A′ = L_B∩L_C, and define B′ and C′ cyclically. Then the lines AA′, BB′, CC″ concur in a point, D. Moreover, the six points A, B, C, X(4), P, D lie on a hyperbola. (Dao Thanh Oai, ADGEOM 893, November 26, 2013)

The hyperbola is here called the Dao (r_A,r_B,r_C,r_P) hyperbola. The triangle A′B′C′ is the Dao (r_A,r_B,r_C,r_P) triangle, and the perspector D of ABC and A′B′C′ is the Dao (r_A,r_B,r_C,r_P) perspector.

For details and examples, including coordinates and equations, see Angel Montesdeoca's presentation at Hechos Geometricos.

X(6188) lies on these lines: {20, 399}, {1272, 14615}, {14249, 14940}

X(6189) = 1st INTERCEPT OF X(2)X(6) AND STEINER CIRCUMELLIPSE

Trilinears    1/{(csc B)[e cos C - cos(C - ω)] - (csc C)[e cos B - cos(B - ω)]} : :
Barycentrics    (csc A)[e cos A + cos(A + ω)] : :
Barycentrics    2 a^2-b^2-c^2-Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4] : :
Barycentrics    SA^2 - SB*SC + sqrt(-3*S^2 + SW^2)*SA : :

X(6189) is the tripole of the axis through the real foci of the Steiner circumellipse. The barycentric product X(6189)*X(6190) is the Steiner point, X(99). Contributed by Bernard Gibert, January 7, 2015.

X(6189) lies on the curves K106, K242, K482, Q012, Q017, Q053 and these lines: {2,6}, {76,3558}, {99,1380}, {511,6040}, {538,2028}, {542,6039}, {671,3413}, {892,13636}, {1078,14630}, {1340,7771}, {1341,7757}, {2039,14568}, {2543,14929}, {3228,5638}, {3557,6177}, {3933,19659}, {6178,7871}, {7760,14631}

X(6189) = reflection of X(6190) in X(2)
X(6189) = isogonal conjugate of X(5639)
X(6189) = isotomic conjugate of X(3414)
X(6189) = complement of X(39365)
X(6189) = anticomplement of X(39022)
X(6189) = cevapoint of X(i) and X(j) for these (i,j): {2,3414}, {3557,5639}
X(6189) = X(i)-cross conjugate of X(j) for these (i,j): (3414,2), (5639,6177)
X(6189) = crossdifference of every pair of points on the line X(512)X(2029)
X(6189) = trilinear pole of line X(2)X(1341)
X(6189) = X(i)-isoconjugate of X(j) for these (i,j): (1,5639), (31,3414), (661,1379)
X(6189) = trilinear product X(799)*X(5638)
X(6189) = barycentric product X(i)*X(j) for these {i,j}: {76,1380}, {99,3413}, {670,5638}
X(6189) = X(4590)-Ceva conjugate of X(6190)
X(6189) = trilinear pole of PU(i) for these i: 116, 118
X(6189) = {X(6),X(385)}-harmonic conjugate of X(6190)

X(6190) = 2nd INTERCEPT OF X(2)X(6) AND STEINER CIRCUMELLIPSE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/{(csc B)[e cos C + cos(C - ω)] - (csc C)[e cos B + cos(B - ω)]}
Barycentrics   g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (csc A)[e cos A - cos(A + ω)]
Barycentrics   2 a^2-b^2-c^2+Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4] : :
Barycentrics    SA^2 - SB*SC - sqrt(-3*S^2 + SW^2)*SA : :

X(6190) is the tripole of the axis through the imaginary foci of the Steiner circumellipse. The barycentric product X(6189)*X(6190) is the Steiner point, X(99). Contributed by Bernard Gibert, January 7, 2015.

X(6190) lies on the curves K106, K242, K482, Q012, Q017, Q053 and these lines: {2,6}, {76,3557}, {99,1379}, {511,6039}, {538,2029}, {542,6040}, {671,3414}, {892,13722}, {1078,14631}, {1340,7757}, {1341,7771}, {2040,14568}, {2542,14929}, {3228,5639}, {3558,6178}, {3933,19660}, {6177,7871}, {7760,14630}

X(6190) = reflection of X(6189) in X(2)
X(6190) = isogonal conjugate of X(5638)
X(6190) = isotomic conjugate of X(3413)
X(6190) = complement of X(39366)
X(6190) = anticomplement of X(39023)
X(6190) = cevapoint of X(i) and X(j) for these (i,j): {2,3413}, {3557,5638}
X(6190) = X(i)-cross conjugate of X(j) for these (i,j): (3413,2), (5638,6179)
X(6190) = crossdifference of every pair of points on the line X(512)X(2028)
X(6190) = trilinear pole of line X(2)X(1340)
X(6190) = X(i)-isoconjugate of X(j) for these (i,j): (1,5638), (31,3413), (661,1380)
X(6190) = trilinear product X(799)*X(5639)
X(6190) = barycentric product X(i)*X(j) for these {i,j}: {76,1379}, {99,3414}, {670,5639}
X(6190) = X(4590)-Ceva conjugate of X(6189)
X(6190) = trilinear pole of PU(i) for these i: 117, 119
X(6190) = {X(6),X(385)}-harmonic conjugate of X(6189)

X(6191) = PERSPECTOR OF OUTER NAPOLEON TRIANGLE AND EXCENTRAL TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/2 + sin(A + π/6) - sin(B + π/6) - sin(C + π/6) (Richard Hilton, December 6, 2014)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/2 + cos(A - π/3) - cos(B - π/3) - cos(C - π/3) (Richard Hilton, December 6, 2014)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 4 cos(A - π/3) - (r + sqrt(3)*s)/R (César Lozada, December 20, 2014)

X(6191) lies on these lines: {1,61}, {3,5672}, {4,9}, {14,2946}, {18,1653}, {63,627}, {191,2945}, {1385,5240}, {1483,5239}, {1652,3338}, {3383,3467}

X(6191) = anticomplement of X(33428)
X(6191) = X(62)-of-excentral-triangle
X(6191) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6192), (9,1276,1277)

X(6192) = PERSPECTOR OF INNER NAPOLEON TRIANGLE AND EXCENTRAL TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/2 - sin(A - π/6) + sin(B - π/6) + sin(C - π/6) (Richard Hilton, December 6, 2014)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/2 + cos(A + π/3) - cos(B + π/3) - cos(C + π/3) (Richard Hilton, December 6, 2014)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 4 cos(A + π/3) - (r - sqrt(3)*s)/R (César Lozada, December 20, 2014)

X(6192) lies on these lines: {1,62}, {3,5673}, {4,9}, {13,2945}, {17,1652}, {63,628}, {191,2946}, {1385,5239}, {1482,5240}, {1653,3338}, {3376,3467}

X(6192) = anticomplement of X(33429)
X(6192) = X(61)-of-excentral-triangle
X(6192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6191), (9,1277,1276)

X(6193) = PERSPECTOR OF ANTICOMPLEMENTARY TRIANGLE AND CIRCUMORTHIC TRIANGLE

Trilinears       (cos A)[cos(2B - 2C) + 3 cos 2A] : :
Barycentrics   (a^2-b^2-c^2) (3 a^8-6 a^6 b^2+4 a^4 b^4-2 a^2 b^6+b^8-6 a^6 c^2+2 a^2 b^4 c^2-4 b^6 c^2+4 a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-4 b^2 c^6+c^8) : :
Barycentrics   -tan 2A + tan 2B + tan 2C : :
X(6193) = 3 X[2] - 4 X[1147] = 2 X[5] - 3 X[3167] = 7 X[3832] - 8 X[5448] = 9 X[2] - 8 X[5449] = 3 X[68] - 4 X[5449] = 3 X[1147] - 2 X[5449] = 5 X[3091] - 6 X[5654]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let P be a point in the plane of triangle ABC. Let A′B′C′ be the circumcevian triangle of P, and let A″B″C″ be the reflection of A′B′C′ in X(3). Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines A′A*, B′B*, C′C* concur in a point Q, here introduced as the circumcevian antipodal perspector of P. X(6193) is the circumcevian antipodal perspector of X(4). (Randy Hutson, January 29, 2015)

X(6193) lies on these lines: {2, 54}, {3, 69}, {4, 155}, {5, 3167}, {20, 6241}, {30, 6225}, {49, 3549}, {52, 193}, {70, 3448}, {110, 3542}, {146, 3146}, {184, 3547}, {388, 3157}, {497, 1069}, {511, 5596}, {542, 2892}, {578, 1352}, {912, 944}, {1092, 1899}, {2889, 3522}, {3088, 5921}, {3091, 5654}, {3147, 3580}, {3832, 5448}, {4294, 6238}

X(6193) = reflection of X(i) in X(j) for these (i,j): (4, 155), (68, 1147)
X(6193) = isogonal conjugate of X(34428)
X(6193) = X(317)-Ceva conjugate of X(2)
X(6193) = anticomplement of X(68)
X(6193) = anticomplementary conjugate of X(37444)
X(6193) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (24, 8), (47, 20), (158, 68), (162, 924), (317, 6327), (1101, 925), (1748, 69), (1993, 4329)
X(6193) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (68,1147,2), (1092,1899,3546)

X(6194) = PERSPECTOR OF ANTICOMPLEMENTARY TRIANGLE AND 1st NEUBERG TRIANGLE

Trilinears (csc A)[2S_AS_ω(S²_ω - S_AS_ω + S²) + S²(S² - S²_ω)] : :
Barycentrics 3 a^6 b^2-2 a^4 b^4-a^2 b^6+3 a^6 c^2+a^4 b^2 c^2-5 a^2 b^4 c^2+b^6 c^2-2 a^4 c^4-5 a^2 b^2 c^4-2 b^4 c^4-a^2 c^6+b^2 c^6 : :
X(6194) = X[20] + 2 X[76], 4 X[3] - X[194], 5 X[631] - 2 X[3095], 4 X[39] - 7 X[3523], 5 X[3091] - 8 X[3934], X[20] - 4 X[5188], X[76] + 2 X[5188], X[147] - 4 X[5976], X[3146] - 4 X[6248]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6194) lies on these lines: {1, 1447}, {2, 51}, {3, 194}, {4, 2896}, {20, 76}, {39, 3523}, {63, 6211}, {69, 147}, {98, 3098}, {165, 726}, {183, 1350}, {376, 2782}, {631, 3095}, {730, 5731}, {1351, 3329}, {1513, 3314}, {2080, 3407}, {3091, 3934}, {3146, 6248}, {3552, 5171} {6195, {{6, 538}, {574, 3117}

X(6194) = X(i)-Ceva conjugate of X(j) for these (i,j): (183, 2), (1350, 20), (5999, 147)
X(6194) = anticomplement of X(262)
X(6194) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 1352), (182, 8), (183, 6327), (3403, 315)
X(6194) = Thomson-isogonal conjugate of X(32524)
X(6194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,5188,20), (183,1350,5999)
X(6194) = perspector of 1st Neuberg triangle and cross-triangle of ABC and 2nd Neuberg triangle
X(6194) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(33873)

X(6195) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CIRCUMSYMMEDIAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*[(2*b^4+b^2*c^2+2*c^4)*a^4+4*b^2*c^2*(b^2+c^2)*a^2-4*b^4*c^4]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (2 a^4 b^4+a^4 b^2 c^2+4 a^2 b^4 c^2+2 a^4 c^4+4 a^2 b^2 c^4-4 b^4 c^4)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6195) lies on these lines: {6, 538}, {574, 3117}

X(6195) = vertex conjugate of PU(139)

X(6196) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND EXCENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)*(b^2-b*c+c^2)*a^3+b^2*c^2*(a^2-b*c)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3 b^3+a^2 b^2 c^2+a^3 c^3-b^3 c^3)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6196) lies on these lines: {1, 76}, {3, 238}, {39, 256}, {43, 213}, {83, 3494}, {384, 904}, {386, 1045}, {695, 3496}, {1492, 1932}, {2108, 3216}, {2896, 4388}, {3336, 5539}, {3493, 3497}

X(6196) = X(i)-Ceva conjugate of X(j) for these (i,j): (384, 3496), (904, 1)
X(6196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,4116,1), (1909,4161,1), (2230,4161,1909)

X(6197) = PERSPECTOR OF CIRCUMORTHIC TRIANGLE AND EXTANGENTS TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b*c*(r+2*R)-2*S*s)/(b*c-2*s*(s-a))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^5+a^4 b-2 a^3 b^2-2 a^2 b^3+a b^4+b^5+a^4 c-a^3 b c-a^2 b^2 c+a b^3 c-2 a^3 c^2-a^2 b c^2-b^3 c^2-2 a^2 c^3+a b c^3-b^2 c^3+a c^4+c^5)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6197) lies on these lines: {3, 3101}, {4, 9}, {8, 1748}, {24, 55}, {28, 517}, {34, 2093}, {46, 278}, {54, 65}, {196, 1068}, {240, 5255}, {378, 5584}, {389, 3611}, {484, 1838}, {1181, 3197}, {1594, 3925}, {1715, 1782}, {1844, 3746}, {1871, 3579}, {1872, 4222}, {1888, 5183}, {1902, 2355}, {5889, 6237}, {6239, 6252}, {6240, 6253}, {6241, 6254}, {6242, 6255}

X(6197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (19,40,4), (1871,3579,4219)

X(6198) = PERSPECTOR OF CIRCUMORTHIC TRIANGLE AND INTANGENTS TRIANGLE

Trilinears       2 + sec A : 2 + sec B : 2 + sec C
Barycentrics   f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a (a^2+b^2-c^2) (a^2-b^2-b c-c^2) (a^2-b^2+c^2)
Barycentrics   tan A + 2 sin A : tan B + 2 sin B : tan C + 2 sin C    (Randy Hutson, January 29, 2015)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6198) lies on these lines: {1, 4}, {2, 1062}, {3, 3100}, {7, 1063}, {8, 406}, {10, 451}, {11, 1594}, {12, 403}, {19, 3247}, {20, 1060}, {24, 55}, {25, 3295}, {28, 1255}, {29, 1807}, {30, 4296}, {35, 186}, {36, 3520}, {37, 943}, {42, 4213}, {47, 1776}, {56, 378}, {65, 74}, {72, 4183}, {101, 2332}, {112, 172}, {145, 4194}, {162, 2906}, {208, 3340}, {232, 1500}, {235, 495}, {270, 2074}, {281, 3811}, {318, 4511}, {350, 1235}, {376, 1038}, {389, 3270}, {427, 496}, {461, 3870}, {468, 5297}, {475, 3616}, {500, 1442}, {502, 1826}, {580, 1736}, {631, 1040}, {774, 3072}, {860, 5174}, {912, 3562}, {942, 1902}, {984, 2212}, {999, 1593}, {1000, 1039}, {1013, 3868}, {1041, 3296}, {1069, 1993}, {1125, 1861}, {1148, 4336}, {1181, 2192}, {1214, 3651}, {1249, 3553}, {1319, 1887}, {1385, 1872}, {1387, 1862}, {1398, 1597}, {1425, 6000}, {1452, 5119}, {1717, 1770}, {1753, 3576}, {1827, 1871}, {1829, 4222}, {1830, 5563}, {1835, 5425}, {1841, 3723}, {1876, 5045}, {1892, 4318}, {1905, 3057}, {1968, 2242}, {1986, 3024}, {2310, 3073}, {2326, 5279}, {2915, 3101}, {3083, 3536}, {3084, 3535}, {3085, 3542}, {3086, 3541}, {3092, 3298}, {3093, 3297}, {3147, 5218}, {3518, 3746}, {3622, 4200}, {3720, 4212}, {3931, 4231}, {3969, 4420}, {4291, 5248}, {4861, 5081}, {5090, 5142}, {5889, 6238}, {6239, 6283}, {6240, 6284}, {6241, 6285}, {6242, 6286}

X(6198) = X(i)-cevapoint of X(j) for these (i,j): (1, 1717), (1825, 2594)
X(6198) = X(i)-cross conjugate of X(j) for these (i,j): (1844, 4), (2174, 3219), (2594, 35)
X(6198) = crosspoint of X(i) and X(j) for these (i,j): (1, 3469), (1897, 5379)
X(6198) = crosssum of X(1) and X(3468)
X(6198) = crossdifference of every two points on the line X(652)X(2523)
X(6198) = polar conjugate of X(30690)
X(6198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,1870), (1,33,4), (1,3465,73)
X(6198) = X(i)-isoconjugate of X(j) for these (i,j): (3, 79), (36, 265), (63, 2160), (65, 1789), (69, 6186), (73, 3615)

X(6199) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS CENTRAL TRIANGLE

Trilinears       3 cos A + 4 sin A : 3 cos B + 4 sin B : 3 cos C + 4 sin C
Trilinears       a(3S_A + 4S) : b(3S_B + 4S) : c(3S_C + 4S)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (3 a^2-3 b^2-3 c^2-8 S)

See César Lozada, Perspective-Orthologic-Parallelogic.

Let (L_A), (L_B), (L_C) be the Lucas circles. Let (L′_A) be the circle, other than (L_A), tangent to (L_B), (L_C), and the circumcircle. Define (L′_B), (L′_C) cyclically. The circles (L′_A), (L′_B), (L′_C) are here introduced as the Lucas secondary circles, and the triangle formed by their centers as the Lucas secondary central triangle. Let M_A be the line through the touch points of (L_A) and the adjacent Lucas secondary circles ((L′_B) and (L′_C)). Define M_B and M_C cyclically. The 1st Lucas secondary tangents triangle is the triangle having sidelines M_A, M_B, M_C. Let N_A be the line through the touch points of (L′_A) and the adjacent Lucas circles ((L_B) and (L_C)). Define N_B and N_C cyclically. The 2nd Lucas secondary tangents triangle is the triangle having sidelines N_A, N_B, N_C. The 1st and 2nd Lucas secondary tangents triangles are perspective, and their perspector is X(6199). Moreover, the six touch points of the Lucas circles and Lucas secondary circles lie on a circle, here introduced as the Lucas secondary tangents circle, with center X(6409). X(6199) is the inverse-in-Lucas-secondary-tangents-circle of X(187). (Randy Hutson, January 29, 2015)

X(6199) lies on these lines: {3, 6}, {5, 1132}, {323, 1583}, {381, 3068}, {485, 3843}, {486, 5070}, {590, 5055}, {1131, 3853}, {1327, 3830}, {1482, 1702}, {1587, 1657}, {1588, 1656}, {1597, 5410}, {3069, 5054}, {3070, 5073}, {3071, 3851}

X(6199) = radical center of the Lucas(8/3) circles
X(6199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,371,6221), (6,6221,3), (371,3311,3), (371,3592,3311), (1132,3316,5), (1151,3312,3), (3311,6221,6)

X(6200) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS TANGENTS TRIANGLE

Trilinears       3 cos A + sin A : 3 cos B + sin B : 3 cos C + sin C
Trilinears       a(3S_A + S) : b(3S_B + S) : c(3S_C + S)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (3 a^2-3 b^2-3 c^2-2 S)

See César Lozada, Perspective-Orthologic-Parallelogic.

The circumsymmedial, Lucas tangents, and Lucas(4) central triangles are pairwise perspective, and the perspector of each pair is X(6200). Also, X(6200) is the perspector of Lucas central triangle and circumsymmedial tangential triangle. Let A′, B′, C′ be the free vertices of the Kenmotu squares, and let A″, B″, C″ be the reflections of A′, B′, C′ in X(371). The triangle A″B″C″ is homothetic to ABC at X(6200). (see also X(6396)). (Randy Hutson, February 16, 2015)

Let A′B′C′ and A″B″C″ be the outer and inner Vecten triangles, respectively. Let A* be the trilinear pole, wrt A′B′C′, of the line B″C″. Define B* and C* cyclically. The lines A′A*, B′B*, C′C* concur in X(6200). (Randy Hutson, February 16, 2015)

X(6200) is the trilinear pole, with respect to the inner Vecten triangle, of the line X(2501)X(3566), this being the perspectrix of the outer and inner Vecten triangles. (Randy Hutson, February 20, 2015)

X(6200) lies on these lines: {2, 1328}, {3, 6}, {4, 5418}, {20, 485}, {30, 590}, {35, 2067}, {36, 2066}, {76, 6316}, {140, 3071}, {186, 5413}, {323, 5408}, {376, 3068}, {378, 5412}, {486, 631}, {487, 3620}, {489, 639}, {549, 615}, {550, 3070}, {637, 641}, {1124, 5204}, {1335, 5217}, {1495, 3155}, {1583, 5407}, {1587, 3522}, {1588, 3523}, {1599, 5409}, {369, 3524}, {3092, 3515}, {3093, 3516}, {5010, 5414}

X(6200) = isogonal conjugate of X(1327)
X(6200) = crosssum of X(1327) and X(1327)
X(6200) = insimilicenter of circumcircle and Lucas circles radical circle
X(6200) = radical center of Lucas(2/3) circles
X(6200) = X(1)-isoconjugate of X(1327)
X(6200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6396), (3,371,372), (3,1151,371), (3,3311,1152), (3,6221,6), (6,1151,6221), (6,6221,371), (15,16,371), (1588,3523,5420), (3371,3372,3312), (3385,3386,1151)

X(6201) = PERSPECTOR OF EULER TRIANGLE AND OUTER GREBE TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(3*c^2+3*b^2+S)*a^4-(b^2-c^2)^2*(b^2+c^2+2*a^2+S)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^4 b^2-2 a^2 b^4-b^6+3 a^4 c^2+4 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6-(-a^4+b^4-2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6201) lies on these lines: {4, 6}, {5, 1160}, {51, 3127}, {114, 6320}, {115, 6226}, {381, 5860}, {546, 5874}, {638, 1270}, {946, 3640}, {1162, 3574}, {1598, 5594}, {1699, 5588}, {3832, 6250}, {3839, 6251}, {5478, 6268}, {5479, 6269}, {5587, 5688}, {5603, 5604}, {6245, 6257}, {6246, 6262}, {6247, 6266}, {6248, 6272}, {6249, 6274}

X(6201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,5870), (4,1587,5871), (4,5480,6202), (5,1160,5590)

X(6202) = PERSPECTOR OF EULER TRIANGLE AND INNER GREBE TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(3*c^2+3*b^2-S)*a^4-(b^2-c^2)^2*(b^2+c^2+2*a^2-S)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^4 b^2-2 a^2 b^4-b^6+3 a^4 c^2+4 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6+(-a^4+b^4-2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6202) lies on these lines: {4, 6}, {5, 1161}, {51, 3128}, {114, 6319}, {115, 6227}, {381, 5861}, {546, 5875}, {637, 1271}, {946, 3641}, {1163, 3574}, {1598, 5595}, {1699, 5589}, {3832, 6251}, {3839, 6250}, {5478, 6270}, {5479, 6271}, {5587, 5689}, {5603, 5605}, {6245, 6258}, {6246, 6263}, {6247, 6267}, {6248, 6273}, {6249, 6275}

X(6202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,6,5871), (4,1588,5870), (4,5480,6201), (5,1161,5591)

X(6203) = PERSPECTOR OF EXCENTRAL TRIANGLE AND OUTER INSCRIBED SQUARES TRIANGLE

Trilinears       1 - cos A - sin A + cos B + sin B + cos C + sin C : 1 - cos B - sin B + cos C + sin C + cos A + sin A : 1 - cos C - sin C + cos A + sin A + cos B + sin B
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3+(b+c)*a^2-[(b-c)^2 - 2*S]*a-(b+c)*[(b-c)^2+2*S]
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3+a^2 b-a b^2-b^3+a^2 c+2 a b c+b^2 c-a c^2+b c^2-c^3-(-2 a+2 b+2 c) S)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6203) lies on these lines: {1, 372}, {2, 7}, {19, 3069}, {46, 486}, {169, 481}, {175, 2082}, {494, 1880}, {604, 3084}, {1372, 5540}, {1766, 5405}, {2171, 3083}

X(6203) = X(19)-Ceva conjugate of X(6204)
X(6203) = X(5412)-of-excentral-triangle
X(6203) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2285,6204), (9,57,6204), (63,1400,6204), (579,1708,6204), (672,1445,6204), (1423,3509,6204)

X(6204) = PERSPECTOR OF EXCENTRAL TRIANGLE AND INNER INSCRIBED SQUARES TRIANGLE

Trilinears       1 - cos A + sin A + cos B - sin B + cos C - sin C : 1 - cos B + sin B + cos C - sin C + cos A - sin A : 1 - cos C + sin C + cos A - sin A + cos B - sin B
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3+(b+c)*a^2-[(b-c)^2 + 2*S]*a-(b+c)*[(b-c)^2+2*S]
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3+a^2 b-a b^2-b^3+a^2 c+2 a b c+b^2 c-a c^2+b c^2-c^3+(-2 a+2 b+2 c) S)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6204) lies on these lines: {1, 371}, {2, 7}, {19, 1659}, {46, 485}, {169, 482}, {176, 2082}, {493, 1880}, {604, 3083}, {1371, 5540}, {1722, 2362}, {1766, 5393}, {2171, 3084}

X(6204) = X(i)-Ceva conjugate of X(j) for these (i,j): (19, 6203), (1659, 1)
X(6204) = X(5413)-of-excentral-triangle
X(6204) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,2285,6203), (9,57,6203), (63,1400,6203), (579,1708,6203), (672,1445,6203), (1423,3509,6203)

X(6205) = PERSPECTOR OF EXCENTRAL TRIANGLE AND LEMOINE TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3+2*(b+c)*a^2-2*(b^2+c^2)*a-(b+c)*(c^2-3*b*c+b^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3+2 a^2 b-2 a b^2-b^3+2 a^2 c+2 b^2 c-2 a c^2+2 b c^2-c^3)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6205) lies on these lines: {1, 574}, {9, 46}, {43, 5213}, {57, 1018}, {1100, 5264}, {3208, 3337}, {3218, 3661}, {3336, 3501}, {3970, 5221}, {3987, 5021}, {4390, 4973}

X(6205) = X(598)-Ceva conjugate of X(1)

X(6206) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 2nd MORLEY ADJUNCT TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2-Sec[(A-2 Pi)/3]+Sec[(B-2 Pi)/3]+Sec[(C-2 Pi)/3])      (Peter Moses, January 19, 2015)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 2*cos(A/2+Pi/6)*cos(A/6+Pi/6)-cos(B/3-C/3)*(2*cos(A/3+Pi/3)+1)+4*sin(A/6)*cos(A/3+Pi/3)*cos(B/6-C/6)+1/2      (César Lozada, January 19, 2015)
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (2-Sec[A/3+4\[Pi]/3]+Sec[B/3+4\[Pi]/3]+Sec[C/3+4\[Pi]/3]) Sin[A]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6206) lies on the 3rd Morley cubic (K301) and these lines: {1,1136}, {3606,6209)

X(6206) = X(3276)-Ceva conjugate of X(6208)

X(6207) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 3rd MORLEY ADJUNCT TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2-Sec[(A+2 Pi)/3]+Sec[(B+2 Pi)/3]+Sec[(C+2Pi)/3])       (Peter Moses, January 19, 2015)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 2*sin(A/3+Pi/6)*(cos(B/3-C/3)-2*cos(A/6+Pi/6)*cos(B/6-C/6))+cos(2*A/3)+2*sin(B/3+Pi/6)*sin(C/3+Pi/6)-1/2      (César Lozada, January 19, 2015)
Barycentrics   h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (2-Sec[A/3+2\[Pi]/3]+Sec[B/3+2\[Pi]/3]+Sec[C/3+2\[Pi]/3]) Sin[A]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6207) lies on the 2nd Morley cubic (K030) and these lines: {1,1134}, {3607,6209}

X(6207) = X(3277)-Ceva conjugate of X(6209)

X(6208) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 2nd MORLEY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1+2*[cos(A/3+Pi/3)-cos(B/3+Pi/3)-cos(C/3+Pi/3)]
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = Sin[A] (-1+2 Sin[A/3-\[Pi]/6]-2 Sin[B/3-\[Pi]/6]-2 Sin[C/3-\[Pi]/6])

The perspector of the excentral triangle and the classical (i.e. 1st) Morley triangle is X(1507). See César Lozada, Perspective-Orthologic-Parallelogic.

X(6208) lies on these lines: {1, 1137}, {357, 1507}, {1136, 6209}, {3606, 6206}

X(6208) = X(3276)-Ceva conjugate of X(6206)

X(6209) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 3rd MORLEY TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1+2*[cos(A/3-Pi/3)-cos(B/3-Pi/3)-cos(C/3-Pi/3)]
Barycentrics h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = Sin[A] (1+2 Sin[A/3+\[Pi]/6]-2 Sin[B/3+\[Pi]/6]-2 Sin[C/3+\[Pi]/6])

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6209) lies on these lines: {1, 1135}, {1134, 1507}, {1136, 6208}, {3607, 6207}

X(6209) = X(3277)-Ceva conjugate of X(6207)

X(6210) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)*a^4+(b^2-b*c+c^2)*a^3-(b^3+c^3)*a^2-(b^3+c^3)*(b+c)*a+b*c*(b+c)*(b-c)^2
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^4 b+a^3 b^2-a^2 b^3-a b^4+a^4 c-a^3 b c-a b^3 c+b^4 c+a^3 c^2-b^3 c^2-a^2 c^3-a b c^3-b^2 c^3-a c^4+b c^4)
X(6210) = 2 X[991] - 3 X[3576]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6210) lies on these lines: {1, 256}, {3, 238}, {4, 9}, {31, 4220}, {63, 147}, {72, 4073}, {98, 2319}, {165, 2108}, {262, 3402}, {515, 3883}, {517, 984}, {613, 1429}, {740, 3169}, {946, 4357}, {991, 995}, {1001, 1350}, {1245, 1453}, {1351, 4649}, {1400, 4307}, {1699, 1764}, {3208, 3508}, {3333, 3664}, {3430, 5248}, {3755, 4266}, {3882, 3886}, {4026, 5480}, {4050, 4133}, {4199, 4512}, {4269, 5327}

X(6210) = reflection of X(i) in X(j) for these (i,j): (1742, 3), (40, 573)
X(6210) = crossdifference of every two points on the line X(1459)X(3287)
X(6210) = X(264)-of-hexyl-triangle
X(6210) = hexyl-isotomic conjugate of X(1)
X(6210) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1756,1423), (9,40,6211), (1284,3056,1), (6212,6213,3496)
X(6210) = X(21)-beth conjugate of X(1423)

X(6211) = PERSPECTOR OF EXCENTRAL TRIANGLE AND 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^5-(b^2+b*c+c^2)*a^3+(b+c)*(b^2+b*c+c^2)*a^2-b*c*(b+c)^2*a-(b^2- c^2)*(b^3-c^3)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^5-a^3 b^2+a^2 b^3-b^5-a^3 b c+2 a^2 b^2 c-a b^3 c-a^3 c^2+2 a^2 b c^2-2 a b^2 c^2+b^3 c^2+a^2 c^3-a b c^3+b^2 c^3-c^5)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6211) lies on these lines: {1, 182}, {3, 984}, {4, 9}, {46, 1423}, {63, 6194}, {98, 813}, {100, 2708}, {200, 1726}, {210, 1762}, {238, 517}, {484, 1756}, {511, 1757}, {515, 3717}, {756, 4220}, {759, 2742}, {1324, 2077}, {1350, 5220}, {1503, 3932}, {2792, 4645}, {2938, 3579}, {3430, 3678}, {4073, 5687}, {4518, 5999}, {4901, 5881}

X(6211) = X(4518)-Ceva conjugate of X(1)
X(6211) = excentral-isogonal conjugate of X(2108)
X(6211) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9,40,6210), (6212,6213,3501)
X(6211) = X(188)-aleph conjugate of X(2108)

X(6212) = PERSPECTOR OF EXCENTRAL TRIANGLE AND OUTER VECTEN TRIANGLE

Trilinears       sec A + tan A : sec B + tan B : sec C + tan C
Trilinears       (sec A)(1 + sin A) : (sec B)(1 + sin B) : (sec C)(1 + sin C)
Trilinears       1 + cos A + sin A - cos B - sin B - cos C - sin C : 1 + cos B + sin B - cos C - sin C - cos A - sin A : 1 + cos C + sin C - cos A - sin A - cos B - sin B
Trilinears       (cos A)/(1 + sin A) : (cos B)/(1 + sin B) : (cos C)/(1 + sin C)
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3 + (b+c)*a^2 - [(b+c)^2+2*S]*a - (b+c)*[(b-c)^2-2*S]
Barycentrics   (tan A)(1 - sin A) : (tan B)(1 - sin B): (tan C)(1 - sin C)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3-2 (a-b-c) S)

See César Lozada, Perspective-Orthologic-Parallelogic.

It is a well known property of a cyclic quadrilateral that the incenters and excenters of the component triangles form a rectangular grid. Let P_A be the unique point on circumcircle arc BC such that the rectangle formed by the incenter of P_ABC and the A-excenters of ABC, P_ACA, and P_AAB is a square; define P_B and P_C cyclically. Let J_A, J_B, J_C be the excenters of ABC. Let I_A be the incenter of P_ABC, I_B the incenter of AP_BC, Ic the incenter of ABP_C.

Let (AB) = A-excenter of P_ABC and (AC) = A-excenter of P_AAB. Define (BC) and (CA) cyclically, and define (BA) and (CB) cyclically.

Let L_A be the extended diagonal (AB)(AC) of square I_A(AB)J_A(AC), and define L_B and L_C cyclically.

Let A′ = L_B∩L_C, B′ = L_C∩L_A, C′ = L_A∩L_B. The lines AA′, BB′, CC′ concur in X(6212). Also, X(6212) is the orthocenter of A′B′C′. See also X(6213). The trilinear polar of X(6212) passes through X(1459). (Randy Hutson, January 29, 2015)

Let ABC be a triangle with orthocenter H and orthic triangle A'B'C'. Let Ia, Ib, Ic be the incenters of HB'C', HC'A', HA'B'. Lines AIa, BIb, CIc concur in X(6212). (César Lozada - July 9, 2023)

X(6212) lies on these lines: {1, 371}, {4, 9}, {46, 486}, {57, 481}, {58, 605}, {63, 488}, {90, 3377}, {103, 6136}, {222, 1124}, {372, 1707}, {487, 2128}, {517, 1378}, {607, 3093}, {608, 3092}, {1587, 2082}, {1588, 2285}, {1703, 1743}, {1790, 3083}, {2809, 3641}

X(6212) = reflection of X(6213) in X(169)
X(6212) = isogonal conjugate of X(6213)
X(6212) = X(i)-cross conjugate of X(j) for these (i,j): (605, 3083)
X(6212) = X(372)-of-excentral-triangle
X(6212) = excentral-isogonal conjugate of X(32556)
X(6212) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,19,6213), (9,40,6213), (10,1766,6213), (281,1753,6213), (1276,6191,6213), (1277,6192,6213), (3496,6210,6213), (3501,6211,6213)
X(6212) = X(i)-isoconjugate of X(j) for these (i,j): (1, 6213), (3, 1123), (4, 1335), (19, 3084), (25, 5391), (92, 606), (905, 6135), (1659, 5414)

X(6213) = PERSPECTOR OF EXCENTRAL TRIANGLE AND INNER VECTEN TRIANGLE

Trilinars       sec A - tan A : sec B - tan B : sec C - tan C
Trilinears       (sec A)(1 - sin A) : (sec B)(1 - sin B) : (sec C)(1 - sin C)
Trilinears       1 + cos A - sin A - cos B + sin B - cos C + sin C : 1 + cos B - sin B - cos C + sin C - cos A + sin A : 1 + cos C - sin C - cos A + sin A - cos B + sin B
Trilinears       (cos A)/(1 - sin A) : (cos B)/(1 - sin B): (cos C)/(1 - sin C)
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^3 + (b+c)*a^2 - [(b+c)^2-2*S]*a - (b+c)*[(b-c)^2+2*S]
Barycentrics   (tan A)(1 + sin A) : (tan B)(1 + sin B) : (tan C)(1 + sin C)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3+2 (a-b-c) S)

See César Lozada, Perspective-Orthologic-Parallelogic.

It is a well known property of a cyclic quadrilateral that the incenters of the component triangles form a rectangle. Let Q_A be the unique point on circumcircle arc BC such that the rectangle formed by the incenters of ABC, Q_ABC, Q_ACA, Q_AAB is a square. Define Q_B and Q_C cyclically. Let I = X(1), and let

[AA] = incenter of Q_ABC, [AB] = incenter of Q_ACA, [AC] = incenter of Q_AAB,
[BA] = incenter of Q_BCA, [BB] = incenter of Q_BAB, [BC] = incenter of Q_BBC,
[CA] = incenter of Q_CAB, [CB] = incenter of Q_CBC, [CC] = incenter of Q_CCA.

Let M_A be the extended diagonal [AB][AC] of square I[AA][AB][AC]], and define M_B and M_C cyclically. Let A″ = M_B∩M_C, B″ = M_C∩M_A, C″ = M_A∩M_B. The lines AA″, BB″, CC″ concur in X(6213), and the trilinear polar of X(6213) passes through X(1459). See also X(6212). (Randy Hutson, January 29, 2015)

X(6213) is the radical center of the circumcircles of the squares squares I_A(AB)J_A(AC), I_B(BC)J_B(BA), I_C(CA)J_C(CB). (Randy Hutson, April 11, 2015)

Let ABC be a triangle with orthocenter H and orthic triangle A'B'C'. Let Ja, Jb, Jc be the excenters of HB'C', HC'A', HA'B', corresponding to H. Lines AJa, BJb, CJc concur in X(6213). (César Lozada - July 9, 2023)

X(6213) lies on these lines: {1, 372}, {4, 9}, {46, 485}, {57, 482}, {58, 606}, {63, 487}, {90, 3378}, {103, 6135}, {164, 3645}, {222, 1335}, {371, 1707}, {488, 2128}, {517, 1377}, {607, 3092}, {608, 3093}, {1587, 2285}, {1588, 2082}, {1702, 1743}, {1790, 3084}, {2809, 3640}

X(6213) = isogonal conjugate of X(6212)
X(6213) = reflection of X(6212) in X(169)
X(6213) = X(i)-cross conjugate of X(j) for these (i,j): (606, 3084), (2067, 1)
X(6213) = X(371)-of-excentral-triangle
X(6213) = excentral-isogonal conjugate of X(32555)
X(6213) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,19,6212), (9,40,6212), (10,1766,6212), (281,1753,6212), (1276,6191,6212), (1277,6192,6212), (3496,6210,6212), (3501,6211,6212)
X(6213) = X(i)-isoconjugate of X(j) for these (i,j): (1, 6212), (3, 1336), (4, 1124), (19, 3083), (25, 1267), (92, 605), (905, 6136)

X(6214) = PERSPECTOR OF OUTER GREBE TRIANGLE AND JOHNSON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a^6-(b^2+c^2)*a^4+(a^2*(b^2+c^2)-(b^2-c^2)^2)*(b^2+c^2+S)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6214) lies on these lines: {3, 5590}, {4, 1160}, {5, 6}, {355, 3640}, {381, 5860}, {517, 5688}, {637, 3933}, {639, 1503}, {952, 5604}, {3095, 6272}, {5587, 5588}, {5613, 6269}, {5617, 6268}, {5878, 6266}, {6033, 6226}, {6257, 6259}, {6262, 6265}, {6274, 6287}, {6276, 6288}, {6320, 6321}

X(6214) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1270,1160), (5,1352,6215), (5,5874,6), (5,5875,485), (6,6278,5874), (1352,6289,5), (5590,5870,3)

X(6215) = PERSPECTOR OF INNER GREBE TRIANGLE AND JOHNSON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a^6+a^2*(b^2+c^2)*(-a^2+b^2+c^2-S)-(b^2-c^2)^2*(b^2+c^2-S)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6-(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6215) lies on these lines: {3, 5591}, {4, 1161}, {5, 6}, {355, 3641}, {381, 5861}, {517, 5689}, {638, 3933}, {640, 1503}, {952, 5605}, {3095, 6273}, {5587, 5589}, {5613, 6271}, {5617, 6270}, {5878, 6267}, {6033, 6227}, {6258, 6259}, {6263, 6265}, {6275, 6287}, {6277, 6288}, {6319, 6321}

X(6215) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1271,1161), (5,1352,6214), (5,5874,486), (5,5875,6), (6,6281,5875), (1352,6290,5), (5591,5871,3)

X(6216) = PERSPECTOR OF OUTER GREBE TRIANGLE AND LUCAS TANGENTS TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*[(S+2*S_ω)*(S_A*(3*S+S_A+S_ω)+(4*S+S_ω)*S)-12*S^2*R^2]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 ((3 a^6+2 a^4 b^2-9 a^2 b^4+4 b^6+2 a^4 c^2-18 a^2 b^2 c^2-12 b^4 c^2-9 a^2 c^4-12 b^2 c^4+4 c^6)+2 (4 a^4-5 a^2 b^2-3 b^4-5 a^2 c^2-14 b^2 c^2-3 c^4) S)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6216) lies on this line: {371, 5860}

X(6217) = PERSPECTOR OF OUTER GREBE TRIANGLE AND MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a^8-(2*(9*c^2+9*b^2+4*S))*a^6-(4*(3*S*(b^2+c^2)+8*b^2*c^2))*a^4+(b^2-c^2)^2*((4*(b^2+4*a^2+c^2))*S+18*a^2*(b^2+c^2)-(b^2-c^2)^2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-18 a^6 b^2+18 a^2 b^6-b^8-18 a^6 c^2-32 a^4 b^2 c^2-18 a^2 b^4 c^2+4 b^6 c^2-18 a^2 b^2 c^4-6 b^4 c^4+18 a^2 c^6+4 b^2 c^6-c^8-4 (2 a^6+3 a^4 b^2-4 a^2 b^4-b^6+3 a^4 c^2+8 a^2 b^2 c^2+b^4 c^2-4 a^2 c^4+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6217) lies on these lines: {4, 6266}, {5, 1160}, {51, 1162}, {389, 5870}

X(6218) = PERSPECTOR OF INNER GREBE TRIANGLE AND MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[-a^8+2*(-4*S+9*b^2+9*c^2)*a^6+4*(-3*b^2*S-3*c^2*S+8*b^2*c^2)*a^4+2*(b-c)^2*(b+c)^2*(-9*b^2+8*S-9*c^2)*a^2+(b-c)^2*(b+c)^2*(b^4+4*b^2*S-2*b^2*c^2+c^4+4*c^2*S)]

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-18 a^6 b^2+18 a^2 b^6-b^8-18 a^6 c^2-32 a^4 b^2 c^2-18 a^2 b^4 c^2+4 b^6 c^2-18 a^2 b^2 c^4-6 b^4 c^4+18 a^2 c^6+4 b^2 c^6-c^8+4 (2 a^6+3 a^4 b^2-4 a^2 b^4-b^6+3 a^4 c^2+8 a^2 b^2 c^2+b^4 c^2-4 a^2 c^4+b^2 c^4-c^6) S See César Lozada, Perspective-Orthologic-Parallelogic.

X(6218) lies on these lines: {4, 6267}, {5, 1161}, {51, 1163}, {389, 5871}

X(6219) = PERSPECTOR OF OUTER GREBE TRIANGLE AND REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^8-bc[2*(3*b^2+4*S+3*c^2))*a^6+4*c^2*a^4*b^2+2*(b^2-c^2)^2*(3*b^2+2*S+3*c^2)*a^2+(b^2-c^2)^2*(4*S*(b^2+c^2)-(b^2-c^2)^2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-6 a^6 b^2+6 a^2 b^6-b^8-6 a^6 c^2+4 a^4 b^2 c^2-6 a^2 b^4 c^2+4 b^6 c^2-6 a^2 b^2 c^4-6 b^4 c^4+6 a^2 c^6+4 b^2 c^6-c^8-4 (2 a^6-a^2 b^4-b^6+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6219) lies on these lines: {4, 6276}, {382, 1160}, {5870, 6239}

X(6220) = PERSPECTOR OF INNER GREBE TRIANGLE AND REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[a^8-(2*(9*c^2+9*b^2+4*S))*a^6-(4*(3*S*(b^2+c^2)+8*b^2*c^2))*a^4+(b^2-c^2)^2*((4*(b^2+4*a^2+c^2))*S+18*a^2*(b^2+c^2)-(b^2-c^2)^2)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-6 a^6 b^2+6 a^2 b^6-b^8-6 a^6 c^2+4 a^4 b^2 c^2-6 a^2 b^4 c^2+4 b^6 c^2-6 a^2 b^2 c^4-6 b^4 c^4+6 a^2 c^6+4 b^2 c^6-c^8+4 (2 a^6-a^2 b^4-b^6+2 a^2 b^2 c^2+b^4 c^2-a^2 c^4+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6220) lies on these lines: {4, 6277}, {382, 1161}, {5871, 6241}

X(6221) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND LUCAS TANGENTS TRIANGLE

Trilinears       3 cos A + 2 sin A : 3 cos B + 2 sin B : 3 cos C + 2 sin C
Trilinears       a(3S_A + 2S) : b(3S_B + 2S) : c(3S_C + 2S)
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(3*b^2+3*c^2-3*a^2+4*S)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (3 a^2-3 b^2-3 c^2-4 S)
X(6221) = L_A/R_A + L_B/R_B + L_C/R_C, where L_A, L_B, L_C are the centers of the Lucas circles, and R_A, R_B, R_C their radii. (Randy Hutson, January 29, 2015)

See César Lozada, Perspective-Orthologic-Parallelogic.

Let A′B′C′ be the Lucas central triangle. Let A″ be the trilinear pole, wrt A′B′C′, of BC; define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6221). Let A″B″C: be the Lucas tangents triangle. Let A* be the trilinear pole, wrt A″B″C″, of line BC; define B*, C* cyclically. The lines A″A*, B″B*, C″C* concur in X(6221). The following triangles are perspective from X(6221): circumsymmedial triangle, Lucas inner triangle, and Lucas(2) central triangle. (Randy Hutson, January 29, 2015)

X(6221) lies on these lines: {3, 6}, {4, 3590}, {30, 3068}, {140, 1588}, {186, 5411}, {323, 1599}, {378, 5410}, {381, 590}, {382, 485}, {486, 3526}, {487, 3619}, {549, 3069}, {550, 1587}, {615, 5054}, {999, 2066}, {1132, 5067}, {1385, 1702}, {1597, 5412}, {1656, 3071}, {1657, 3070}, {2067, 3295}, {3092, 3517}, {3299, 5204}, {3301, 5217}, {3316, 3832}

X(6221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,371,3311), (3,3311,3312), (3,6199,6), (6,371,6199), (6,1151,6200), (6,6199,3311), (6,6200,3), (15,16,1151), (371,372,3592), (371,1151,3), (371,6200,6), (3071,5418,1656)

X(6221) = radical center of Lucas(4/3) circles
X(6221) = inverse-in-Lucas-radical-circle of X(187)
X(6221) = exsimilicenter of circumcircle and Lucas inner circle
X(6221) = trilinear pole wrt Lucas central triangle of Lemoine axis
X(6221) = trilinear pole wrt Lucas tangents triangle of Lemoine axis
X(6221) = X(6)-of-Lucas-tangents-triangle
X(6221) = X(7)-of-Lucas-central-triangle
X(6221) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,Y(155) (below)), (3,371,3311), (3,3311,3312), (371,372,3592), (371,1151,3)

X(6222) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND 1st NEUBERG TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)[(S-2*S_ω)*[(S_A-S_ω)*(S+S_ω)+S^2]*S_A-S^3*(2*S+S_ω)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 5 a^8-7 a^6 b^2+7 a^4 b^4-5 a^2 b^6-7 a^6 c^2-2 a^4 b^2 c^2+5 a^2 b^4 c^2-4 b^6 c^2+7 a^4 c^4+5 a^2 b^2 c^4+8 b^4 c^4-5 a^2 c^6-4 b^2 c^6+4 a^2 (a^2 b^2-b^4+a^2 c^2-c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6222) lies on these lines: {2, 6289}, {1350, 6312}, {1991, 6279}

X(6223) = ANTICOMPLEMENTARY-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1-10*cos(A)+cos(2*A)-2*sin(A/2)*cos(3*B/2-3*C/2)-2*sin(3*A/2)*cos(B/2-C/2)-cos(2*B)-cos(2*C)+6*cos(B-C)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^7+a^6 b-3 a^5 b^2-3 a^4 b^3+3 a^3 b^4+3 a^2 b^5-a b^6-b^7+a^6 c+10 a^5 b c-a^4 b^2 c-4 a^3 b^3 c-a^2 b^4 c-6 a b^5 c+b^6 c-3 a^5 c^2-a^4 b c^2+2 a^3 b^2 c^2-2 a^2 b^3 c^2+a b^4 c^2+3 b^5 c^2-3 a^4 c^3-4 a^3 b c^3-2 a^2 b^2 c^3+12 a b^3 c^3-3 b^4 c^3+3 a^3 c^4-a^2 b c^4+a b^2 c^4-3 b^3 c^4+3 a^2 c^5-6 a b c^5+3 b^2 c^5-a c^6+b c^6-c^7
X(6223) = 2 X[3] - 3 X[5658] = 3 X[4] - 2 X[5787] = 5 X[3091] - 4 X[6245] = X[5787] - 3 X[6259] = 3 X[2] - 4 X[6260] = 3 X[5731] - 4 X[6261]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6223) lies on these lines: {2, 84}, {3, 5658}, {4, 7}, {8, 6001}, {20, 78}, {30, 5758}, {40, 144}, {145, 515}, {189, 318}, {443, 5927}, {651, 1498}, {1012, 5703}, {1158, 3219}, {1270, 6257}, {1271, 6258}, {1330, 2897}, {1532, 5704}, {1709, 3085}, {1750, 4292}, {2096, 3149}, {2475, 5554}, {2829, 6224}, {2894, 5175}, {3062, 5290}, {3091, 6245}, {4295, 5691}, {5731, 6261}

X(6223) = reflection of X(i) in X(j) for these (i,j): (20,1490), (4,6250)
X(6223) = isogonal conjugate of X(34432)
X(6223) = X(322)-Ceva conjugate of X(2)
X(6223) = anticomplement of X(84)
X(6223) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 962), (3, 280), (9, 189), (40, 8), (100, 4397), (101, 6332), (198, 2), (221, 145), (223, 7), (227, 2475), (322, 6327), (329, 69), (347, 3434), (651, 4131), (1103, 6223), (1167, 84), (1262, 934), (1817, 75), (2187, 192), (2199, 3210), (2324, 329), (2331, 5905), (2360, 1), (3194, 3868), (3195, 193), (6129, 149) X(6223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1071,938), (84,6260,2), (3146,5905,962)

X(6224) = ANTICOMPLEMENTARY-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[3*a^4-(2*(b+c))*a^3-(-3*b*c+2*c^2+2*b^2)*a^2+(b+c)*(-3*b*c+2*c^2+2*b^2)*a-(b^2-c^2)^2]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^4-2 a^3 b-2 a^2 b^2+2 a b^3-b^4-2 a^3 c+3 a^2 b c-a b^2 c-2 a^2 c^2-a b c^2+2 b^2 c^2+2 a c^3-c^4
X(6224) = 3 X[2] - 4 X[214] = 3 X[8] - 4 X[1145] = 3 X[100] - 2 X[1145] = 4 X[1317] - 3 X[3241] = 2 X[1320] - 3 X[3241] = 4 X[11] - 5 X[3616] = 2 X[908] - 3 X[4511] = 2 X[1737] - 3 X[4881] = 4 X[908] - 3 X[5080] = 2 X[104] - 3 X[5731] = 2 X[3036] - 3 X[6174] = 5 X[3091] - 4 X[6246]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6224) lies on these lines: {1, 149}, {2, 80}, {3, 8}, {4, 6265}, {7, 528}, {11, 2476}, {20, 2800}, {30, 5180}, {59, 3319}, {78, 5531}, {144, 2801}, {145, 2802}, {153, 515}, {484, 519}, {535, 4867}, {900, 3904}, {962, 5840}, {1270, 6262}, {1271, 6263}, {1385, 5086}, {1387, 3488}, {1444, 4720}, {1737, 4881}, {1768, 4297}, {1862, 4214}, {2345, 4287}, {2771, 3648}, {2829, 6223}, {2894, 4861}, {3036, 6174}, {3091, 6246}, {3189, 5854}, {3419, 3655}, {3600, 5083}, {3884, 5441}, {3897, 5794}, {4084, 4325}, {4855, 5881}, {5036, 5839}, {5176, 5440}, {5330, 6284}, {5528, 5853}

X(6224) = reflection of X(i) in X(j) for these (i,j): (1320,1317), (149,1), (1523,6326), (1768,4297), (4,6265), (5080,4511), (5176,5440), (6264,5882), (80,214), (8,100)
X(6224) = isogonal conjugate of X(34431)
X(6224) = isotomic conjugate of X(36917)
X(6224) = complement of X(20085)
X(6224) = X(320)-Ceva conjugate of X(2)
X(6224) = X(i)-cevapoint of X(j) for these (i,j): (5541, 6326)
X(6224) = anticomplement of X(80)
X(6224) = X(484)-of-inner-Garcia-triangle
X(6224) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (80,214,2), (100,104,4996), (104,1145,5744), (1317,1320,3241)
X(6224) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 5080), (36, 8), (58, 758), (101, 3762), (106, 80), (109, 3738), (320, 6327), (758, 1330), (1443, 3434), (1464, 2475), (1870, 4), (1983, 514), (2245, 2895), (2323, 329), (2361, 144), (3218, 69), (3724, 1654), (3960, 150), (4282, 63), (4511, 3436), (4973, 2891), (6149, 3648)

X(6225) = ANTICOMPLEMENTARY-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (csc A)]2*(8*R^2-S_ω))*S^2_A+(S^2-2*S_ω*(8*R^2-S_ω))*S_A+2*S^2*(6*R^2-S_ω)]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^10+3 a^8 b^2-14 a^6 b^4+14 a^4 b^6-3 a^2 b^8-b^10+3 a^8 c^2+20 a^6 b^2 c^2-14 a^4 b^4 c^2-12 a^2 b^6 c^2+3 b^8 c^2-14 a^6 c^4-14 a^4 b^2 c^4+30 a^2 b^4 c^4-2 b^6 c^4+14 a^4 c^6-12 a^2 b^2 c^6-2 b^4 c^6-3 a^2 c^8+3 b^2 c^8-c^10
X(6225) = 3 X[2] - 4 X[2883], 5 X[631] - 4 X[3357], 6 X[154] - 5 X[3522], 6 X[1853] - 7 X[3832], 2 X[3] - 3 X[5656], 7 X[3832] - 8 X[5893], 3 X[1853] - 4 X[5893], 5 X[3522] - 4 X[5894], 3 X[154] - 2 X[5894], 3 X[20] - 2 X[5925], 3 X[1498] - X[5925], 5 X[3091] - 4 X[6247]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let U be the line tangent to K007 (the Lucas cubic) at X(4), and let V be the line tangent to K007 at X(20). Then X6225) = U∩V. (Randy Hutson, May 5, 2015)

X(6225) lies on these lines: {2, 64}, {3, 5656}, {4, 51}, {20, 394}, {74, 3147}, {107, 3183}, {154, 3522}, {193, 1503}, {221, 390}, {388, 6285}, {631, 3357}, {962, 3868}, {1192, 4232}, {1270, 6266}, {1271, 6267}, {1595, 3426}, {1853, 3832}, {2192, 3600}, {2777, 3529}, {3091, 6247}

X(6225) = reflection of X(i) in X(j) for these (i,j): (X[3146], X[5895]), (X[4], X[5878]), (X[64], X[2883])
X(6225) = isotomic conjugate of cyclocevian conjugate of X(35510)
X(6225) = anticomplement of X(64)
X(6225) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 3146), (20, 8), (63, 253), (154, 192), (204, 193), (610, 2), (662, 3265), (775, 64), (1097, 6225), (1249, 5905), (1394, 145), (1895, 4), (2287, 2184), (3198, 1654), (5930, 2475)
X(6225): {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64,2883,2), (154,5894,3522), (1853,5893,3832)

X(6226) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 1st-BROCARD TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2*S_A*S_ω^4+(-S^2+S*S_A-2*S_A^2)*S_ω^3-S*S_A*(S_A+S)*S_ω^2+S^3*(- S_A+S)*S_ω-S^4*(S+3*S_A))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^8 b^2-4 a^6 b^4+3 a^4 b^6-2 a^2 b^8+3 a^8 c^2-4 a^6 b^2 c^2+2 a^2 b^6 c^2-b^8 c^2-4 a^6 c^4+b^6 c^4+3 a^4 c^6+2 a^2 b^2 c^6+b^4 c^6-2 a^2 c^8-b^2 c^8+(a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6226) lies on these lines: {6, 98}, {114, 5590}, {115, 6201}, {147, 1270}, {542, 5860}, {1160, 2782}, {2794, 5870}, {5874, 6274}, {6033, 6214}

X(6226) = reflection of X(i) in X(j) for these (i,j): (6227, 98), (6320, 1160)

X(6227) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 1st-BROCARD TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2*S_A*S_ω^4+(-S^2-S*S_A-2*S_A^2)*S_ω^3-S*S_A*(-S_A+S)*S_ω^2+S^3*(S_A+S)*S_ω+S^4*(S-3*S_A))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^8 b^2-4 a^6 b^4+3 a^4 b^6-2 a^2 b^8+3 a^8 c^2-4 a^6 b^2 c^2+2 a^2 b^6 c^2-b^8 c^2-4 a^6 c^4+b^6 c^4+3 a^4 c^6+2 a^2 b^2 c^6+b^4 c^6-2 a^2 c^8-b^2 c^8-(a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6227) lies on these lines: {6, 98}, {114, 5591}, {115, 6202}, {147, 1271}, {542, 5861}, {1161, 2782}, {2794, 5871}, {5875, 6275}, {6033, 6215}

X(6227) = reflection of X(i) in X(j) for these (i,j): (6226, 98), (6319, 1161)

X(6228) = 1st-BROCARD-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (S*(2*S^2-4*S_A*S_ω+3*S_A^2-S_ω^2)-(3*S_A+S_ω)*S^2+S_A*S_ω*(-S_ω+S_A))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^4 b^2-3 a^2 b^4+b^6+2 a^4 c^2-4 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4-b^2 c^4+c^6+2 (a^4-a^2 b^2-b^4-a^2 c^2-c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6228) lies on these lines: {2, 372}, {3, 6230}, {639, 6250}

X(6228) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (485,488,6312)

X(6229) = 1st-BROCARD-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((-3*S+S_ω)*S_A^2-(S-S_ω)*(3*S-S_ω)*S_A-S*(S+S_ω)*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^4 b^2-3 a^2 b^4+b^6+2 a^4 c^2-4 a^2 b^2 c^2-b^4 c^2-3 a^2 c^4-b^2 c^4+c^6-2 (a^4-a^2 b^2-b^4-a^2 c^2-c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6229) lies on these lines: {2, 371}, {3, 6231}, {640, 6251}

X(6229) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (486,487,6316)

X(6230) = OUTER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 1st BROCARD TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2*S^4-S*((3*S_A+S_ω)*S^2+3*S_A*S_ω*(-S_ω+S_A))-S_ω*(S_ω+S_A)*S^2-S_A*S_ω^2*(-S_ω+S_A))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^10-6 a^8 b^2+7 a^6 b^4-5 a^4 b^6+3 a^2 b^8-b^10-6 a^8 c^2+4 a^6 b^2 c^2+a^4 b^4 c^2-2 a^2 b^6 c^2+3 b^8 c^2+7 a^6 c^4+a^4 b^2 c^4-2 a^2 b^4 c^4-2 b^6 c^4-5 a^4 c^6-2 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10-2 (a^2 b^2-b^4+a^2 c^2-c^4) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6230) lies on these lines: {2, 98}, {3, 6228}, {99, 638}, {115, 485}, {1991, 6321}, {2782, 3070}, {6033, 6290}

X(6230) = reflection of X(6231) in X(114)
X(6230) = complement of X(33430)
X(6230) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (98,1352,6231)

X(6231) = INNER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 1st BROCARD TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (S_ω*(3*S-S_ω)*S_A^2+(S^2-S_ω^2)*(3*S-S_ω)*S_A+S^2*(S+S_ω)*(2*SS_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^10-6 a^8 b^2+7 a^6 b^4-5 a^4 b^6+3 a^2 b^8-b^10-6 a^8 c^2+4 a^6 b^2 c^2+a^4 b^4 c^2-2 a^2 b^6 c^2+3 b^8 c^2+7 a^6 c^4+a^4 b^2 c^4-2 a^2 b^4 c^4-2 b^6 c^4-5 a^4 c^6-2 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10+2 (a^2 b^2-b^4+a^2 c^2-c^4) (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6231) lies on these lines: {2, 98}, {3, 6229}, {99, 637}, {115, 486}, {591, 6321}, {2782, 3071}, {6033, 6289}

X(6231) = reflection of X(6230) in X(114)
X(6231) = complement of X(33431)
X(6231) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (98,1352,6230)

X(6232) = 2nd-BROCARD-TRIANGLE-ORTHOLOGIC CENTER OF CIRCUMMEDIAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((9*S^4+9*S_ω*(-S_ω+6*R^2)*S^2-2*S_ω^4)*S_A^2-2*S_ω*(6*S^4+3*S_ω*(-S_ω+9*R^2)*S^2- S_ω^4)*S_A+S^2*(S_ω^2+6*S^2)*(S_ω^2+S^2))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 4 a^12-14 a^10 b^2+8 a^8 b^4+2 a^6 b^6-14 a^10 c^2+17 a^8 b^2 c^2+26 a^6 b^4 c^2-a^4 b^6 c^2-16 a^2 b^8 c^2+4 b^10 c^2+8 a^8 c^4+26 a^6 b^2 c^4+18 a^4 b^4 c^4+16 a^2 b^6 c^4-12 b^8 c^4+2 a^6 c^6-a^4 b^2 c^6+16 a^2 b^4 c^6+16 b^6 c^6-16 a^2 b^2 c^8-12 b^4 c^8+4 b^2 c^10

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6232) lies on the Brocard circle and these lines: {3, 3849}, {182, 6322}

X(6232) = reflection of X(6322) in X(182)
X(6232) = Brocard circle antipode of X(6322)
X(6232) = 5th-Euler-to-2nd-Brocard similarity image of X(12494)
X(6232) = X(6233)-of-1st-Brocard-triangle

X(6233) = CIRCUMMEDIAL-TRIANGLE-ORTHOLOGIC CENTER OF 2nd BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/((a^4+(5*(b^2+c^2))*a^2-2*(-b^2*c^2+c^4+b^4))*(b^2-c^2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a-b) (a+b) (a-c) (a+c) (2 a^4-2 a^2 b^2+2 b^4-5 a^2 c^2-5 b^2 c^2-c^4) (2 a^4-5 a^2 b^2-b^4-2 a^2 c^2-5 b^2 c^2+2 c^4)

See César Lozada, Perspective-Orthologic-Parallelogic.

Let A_B, A_C, B_C, B_A, C_A, C_B be the points on the Dao 6-point circle as defined at X(5569). The triangles B_AC_BA_C and C_AA_BB_C are perspective from X(2), and X(6233) is the isogonal conjugate of the infinite point of their perspectrix. (Randy Hutson, January 29, 2015)

X(6233) lies on the circumcircle and these lines: {3, 6323}, {111, 182}, {542, 6325}, {690, 6236}, {843, 2080}

X(6233) = reflection of X(6323) in X(3)
X(6233) = isogonal conjugate of X(8704)
X(6233) = X(98)-of-circumsymmedial-triangle
X(6233) = Λ(radical axis of orthocentroidal circle and McCay circumcircle)
X(6233) = Λ(polar of X(2) wrt Dao 6-point circle)
X(6233) = 1st-Parry-to-ABC similarity image of X(353)
X(6233) = X(6785)-of-4th-anti-Brocard-triangle

X(6234) = 1st-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF 3rd BROCARD TRIANGLE

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6234) lies on these lines: {3, 3224}, {4, 147}, {98, 3225}, {511, 694}, {805, 2080}

X(6235) = 4th-BROCARD-TRIANGLE-ORTHOLOGIC CENTER OF CIRCUMSYMMEDIAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*(2*S_ω*(S_A^2+S^2)-(S_ω^2+S^2)*S_A)/(2*S_A*S_ω-S_ω^2+S^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (4 a^10 b^4-8 a^8 b^6+8 a^4 b^10-4 a^2 b^12-4 a^10 b^2 c^2-6 a^8 b^4 c^2+28 a^6 b^6 c^2-22 a^4 b^8 c^2-4 a^2 b^10 c^2+8 b^12 c^2+4 a^10 c^4-6 a^8 b^2 c^4-23 a^6 b^4 c^4+42 a^4 b^6 c^4+31 a^2 b^8 c^4-32 b^10 c^4-8 a^8 c^6+28 a^6 b^2 c^6+42 a^4 b^4 c^6-30 a^2 b^6 c^6+24 b^8 c^6-22 a^4 b^2 c^8+31 a^2 b^4 c^8+24 b^6 c^8+8 a^4 c^10-4 a^2 b^2 c^10-32 b^4 c^10-4 a^2 c^12+8 b^2 c^12)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6235) lies on the orthocentroidal circle and this line: {381, 6324}

X(6235) = reflection of X(6324) in X(381)

X(6236) = CIRCUMSYMMEDIAL-TRIANGLE-ORTHOLOGIC CENTER OF 4th-BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/((2*a^6-2*(b^2+c^2)*a^4-(2*b^4-3*b^2*c^2+2*c^4)*a^2+(b^2+c^2)*(2*c^4-7*b^2*c^2+2*b^4))*(b^2-c^2)*a)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a-b) (a+b) (a-c) (a+c) (2 a^6-2 a^4 b^2-2 a^2 b^4+2 b^6-5 a^4 c^2+3 a^2 b^2 c^2-2 b^4 c^2-5 a^2 c^4-2 b^2 c^4+2 c^6) (2 a^6-5 a^4 b^2-5 a^2 b^4+2 b^6-2 a^4 c^2+3 a^2 b^2 c^2-2 b^4 c^2-2 a^2 c^4-2 b^2 c^4+2 c^6)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6236) lies on the circumcircle and these lines: {3, 6325}, {111, 381}, {542, 6323}, {690, 6233}, {842, 3849}, {2373, 6031}

X(6236) = reflection of X(6325) in X(3)
X(6236) = anticomplement of X(34113)

X(6237) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF ORTHIC TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A)*(1+(2*cos(B)+1)*cos(2*B)+(2*cos(C)+1)*cos(2*C)-cos(2*A)*(1+2*cos(B-C)))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^2-b^2-c^2) (a^7 b^2-a^6 b^3-3 a^5 b^4+3 a^4 b^5+3 a^3 b^6-3 a^2 b^7-a b^8+b^9+a^7 b c-3 a^5 b^3 c+3 a^3 b^5 c-a b^7 c+a^7 c^2-2 a^5 b^2 c^2-a^4 b^3 c^2-a^3 b^4 c^2+4 a^2 b^5 c^2+2 a b^6 c^2-3 b^7 c^2-a^6 c^3-3 a^5 b c^3-a^4 b^2 c^3-2 a^3 b^3 c^3-a^2 b^4 c^3+a b^5 c^3-b^6 c^3-3 a^5 c^4-a^3 b^2 c^4-a^2 b^3 c^4-2 a b^4 c^4+3 b^5 c^4+3 a^4 c^5+3 a^3 b c^5+4 a^2 b^2 c^5+a b^3 c^5+3 b^4 c^5+3 a^3 c^6+2 a b^2 c^6-b^3 c^6-3 a^2 c^7-a b c^7-3 b^2 c^7-a c^8+c^9)

X(6237) is also the extangents-triangle-orthologic center of these triangles: intangents and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.

X(6237) lies on these lines: {3, 48}, {19, 52}, {30, 6254}, {55, 155}, {65, 68}, {539, 6255}, {3564, 3779}, {3611, 5562}, {5777, 5816}, {5889, 6197}

X(6237) = reflection of X(6238) in X(155)
X(6237) = orthocenter-of-extangents-triangle

X(6238) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF ORTHIC TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*(S_A^2-(-S_ω+6*R^2)*S_A)-2*S*R*(-S_ω+4*R^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^2-b^2-c^2) (a^4 b^2-2 a^2 b^4+b^6-a^4 b c+b^5 c+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 b^3 c^3-2 a^2 c^4-b^2 c^4+b c^5+c^6)

X(6238) is also the intangents-triangle-orthologic center of these triangles: extangents, tangential, and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.

X(6238) lies on these lines: {3, 1069}, {30, 6285}, {33, 52}, {35, 1147}, {55, 155}, {68, 1479}, {185, 1060}, {498, 5654}, {539, 6286}, {912, 3057}, {916, 2293}, {942, 5733}, {1040, 1216}, {1062, 3270}, {3056, 3564}, {4296, 6241}, {5889, 6198}

X(6238) = reflection of X(6237) in X(155)
X(6238) = orthocenter-of-intangents triangle
X(6238) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (3270,5562,1062)

X(6239) = CIRCUMORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS CENTRAL TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*((2*S-S_ω)*S_A^2+2*S_A*S^2+S^2*(2*S+S_ω))/S_A
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2-3 a^2 b^2 c^2-2 a^2 c^4+c^6-4 b^2 c^2 S)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6239) lies on these lines: {4, 69}, {24, 1151}, {1306, 3563}, {1588, 3567}, {5870, 6219}, {6197, 6252}, {6198, 6283}

X(6239) = reflection of X(4) in X(6291)
X(6239) = isogonal conjugate of X(6401)

X(6240) = CIRCUMORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF MACBEATH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (S_A-3*S_ω+10*R^2)/(a*S_A)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-b^2 c^4+c^6)
X(6240) = 5 X[4] - 6 X[428], 9 X[428] - 5 X[1885], 3 X[4] - 2 X[1885], 3 X[428] - 5 X[3575], X[1885] - 3 X[3575]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let MaMbMc = medial triangle and DEF = circumorthic triangle. Let (Oa) be the circle with center A that passes through E and F (and X(4)). Let E' be the point, other than E, in which (Oa) meets EMa, and let F' be the point, other than F, in which (Oa) meets FMa. Let A′ = EF'∩FE', and define B′ and C′ cyclically. The lines A′DF, B′E, C′F concur in X(6240). (Angel Montesdeoca, September 25, 2020)

X(6240) lies on these lines: {2, 3}, {32, 5523}, {50, 53}, {52, 1986}, {64, 70}, {74, 6145}, {185, 6152}, {930, 1299}, {933, 1300}, {1105, 1179}, {1503, 6241}, {6197, 6253}, {6198, 6284}

X(6240) = reflection of X(4) in X(3575)
X(6240) = circumcircle-inverse of X(37970)
X(6240) = Kosnita-to-orthic similarity image of X(4)
X(6240) = Ehrmann-vertex-to-orthic similarity image of X(3)
X(6240) = X(5903)-of-orthic-triangle if ABC is acute
X(6240) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,4,1594), (4,20,378), (4,24,403), (4,186,5), (4,376,3541), (4,3147,3091), (4,3518,235), (4,3520,427), (25,382,4), (235,3627,4), (427,550,3520), (3089,3543,4)

X(6241) = CIRCUMORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*(3*S_A^2-(2*(-S_ω+10*R^2))*S_A+S^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+3 a^4 b^2 c^2-3 a^2 b^4 c^2-b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4+4 b^4 c^4+3 a^2 c^6-b^2 c^6-c^8)
X(6241) = 5 X[4] - 6 X[51] = 3 X[51] - 5 X[185] = 9 X[51] - 10 X[389] = 3 X[4] - 4 X[389] = 3 X[185] - 2 X[389]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6241) lies on these lines: {3, 74}, {4, 51}, {24, 1192}, {30, 5889}, {52, 3146}, {54, 64}, {70, 4846}, {143, 3830}, {184, 3357}, {186, 1204}, {376, 5562}, {382, 3060}, {403, 2883}, {511, 3529}, {546, 5640}, {550, 2979}, {568, 3627}, {631, 5907}, {944, 2807}, {1147, 2071}, {1154, 1657}, {1173, 3426}, {1216, 3522}, {1503, 6240}, {1594, 6247}, {1986, 5895}, {2772, 5693}, {2781, 5925}, {3523, 5891}, {3528, 3917}, {3534, 6101}, {3542, 5656}, {3543, 5446}, {3832, 5462}, {3843, 5946}, {3855, 5943}, {4296, 6238}, {5056, 5892}, {5870, 6219}, {5871, 6220}, {6197, 6254}, {6198, 6285}

X(6241) = reflection of X(i) in X(j) for these (i,j): (3146,52), (382,6102), (4,185)
X(6241) = crosssum of X(3) and X(382)
X(6241) = X(20)-of-X(3)-Fuhrmann-triangle
X(6241) = X(4)-of-X(4)-anti-altimedial-triangle
X(6241) = X(4)-of-X(4)-adjunct-anti-altimedial-triangle
X(6241) = X(8)-of-circumorthic-triangle if ABC is acute
X(6241) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,185,5890), (4,5890,3567), (64,1181,378), (74,1614,3), (184,3357,3520), (378,1181,54), (382,6102,3060), (1899,5878,4), (12287,12288, 12283)

X(6242) = CIRCUMORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*((7*R^2-4*S_ω)*S_A^2+4*S^2*S_A+3*S^2*R^2)/S_A
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^8 b^2-4 a^6 b^4+6 a^4 b^6-4 a^2 b^8+b^10+a^8 c^2-5 a^6 b^2 c^2+6 a^4 b^4 c^2-a^2 b^6 c^2-b^8 c^2-4 a^6 c^4+6 a^4 b^2 c^4+a^2 b^4 c^4+6 a^4 c^6-a^2 b^2 c^6-4 a^2 c^8-b^2 c^8+c^10)
X(6242) = 3 X[54] - 4 X[389] = 3 X[568] - 2 X[1493] = 2 X[5876] - 3 X[6288]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6242) lies on the circumcircle and these lines: {4, 93}, {24, 195}, {52, 110}, {54, 186}, {539, 5889}, {568, 1493}, {3060, 5448}, {6197, 6255}, {6198, 6286}

X(6242) = reflection of X(4) in X(6152)

X(6243) = REFLECTION-TRIANGLE-ORTHOLOGIC CENTER OF ORTHIC TRIANGLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a*((-S_ω+R^2)*S_A+S^2)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-3 a^4 b^2 c^2+2 b^6 c^2-3 a^4 c^4-2 b^4 c^4+3 a^2 c^6+2 b^2 c^6-c^8)
Barycentrics    h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a^2[a^6(b^2 + c^2) - 3a^4(b^4 + c^4 + b^2c^2) + 3a^2(b^6 + c^6) - (b^2 - c^2)^2(b^4 + c^4)]
X(6243) = 3 X[2] - 4 X[143] = 3 X[3] - 4 X[389] = 3 X[52] - 2 X[389] = 8 X[389] - 9 X[568] = 2 X[3] - 3 X[568] = 4 X[52] - 3 X[568] = 3 X[51] - 2 X[1216]

X(6243) is also the reflection-triangle-orthologic center of these triangles: intangents, extangents, tangential, and circumorthic. See César Lozada, Perspective-Orthologic-Parallelogic.

Let A′B′C′ be the reflection triangle. Let O_A be the circle with center A and pass-through point A′, and define O_B and O_C cyclically. The radical center of the circles O_A, O_B, O_C is X(6243). If O′_A has center A′ and pass-through point A, and cyclically for O′_B, O′_C, the radical center is X(3519). (Randy Hutson, January 29, 2015)

Let A′B′C′ be the reflection triangle. Let A_B, A_C be the orthogonal projections of A′ on CA, AB, resp. Let A″ = C_AA_C∩A_BB_A, and define B″ and C″ cyclically. Triangle A″B″C″ is homothetic to ABC at X(6). The lines A′A″, B′B″, C′C″ concur in X(6243), which is also X(3)-of-A″B″C″. (Randy Hutson, March 29, 2020)

X(6243) lies on these lines: {2, 143}, {3, 6}, {4, 93}, {5, 3060}, {20, 6102}, {23, 156}, {26, 49}, {30, 5889}, {51, 1216}, {70, 265}, {140, 2979}, {184, 195}, {185, 1657}, {323, 3518}, {343, 5576}, {381, 5446}, {550, 5890}, {631, 5946}, {1112, 3542}, {1147, 2070}, {3146, 5663}, {3526, 3917}, {3628, 5640}, {3843, 5907}, {3851, 5891}, {5054, 5447}, {5070, 5943}, {5073, 6000}

X(6243) = reflection of X(i) in X(j) for these (i,j): (1657, 185), (20,6102), (3519,6152), (3,52), (5562,5446), (6101),143)
X(6243) = anticomplement of X(6101)
X(6243) = inverse-in-circle-{{X(371), X(372), PU(1), PU(39)}} of X(2965)
X(6243) = polar-circle-inverse of X(35718)
X(6243) = Brocard-circle-inverse of X(13353)
X(6243) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,13353), (3,52,568), (26,1993,49), (51,1216,1656), (143,6101,2), (195,2937,184), (371,372,2965), (2979,3567,140), (3917,5462,3526), (5446,5562,381)

X(6244) = 1st-CIRCUMPERP-TRIANGLE-ORTHOLOGIC CENTER OF MIXTILINEAR TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(a^4-2*(b+c)*a^3+10*a^2*b*c+2*(b+c)*(b^2+c^2-4*b*c)*a-(4*b*c+b^2+c^2)*(b-c)^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^4-2 a^3 b+2 a b^3-b^4-2 a^3 c+10 a^2 b c-6 a b^2 c-2 b^3 c-6 a b c^2+6 b^2 c^2+2 a c^3-2 b c^3-c^4)
X(6244) = X[57] - 3 X[165]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6244) lies on these lines: {1, 3}, {4, 3820}, {20, 3421}, {100, 329}, {101, 1615}, {102, 6014}, {103, 1293}, {197, 2823}, {200, 971}, {210, 1709}, {218, 1200}, {219, 2272}, {474, 962}, {480, 2951}, {516, 1376}, {527, 4421}, {910, 1766}, {954, 5281}, {1012, 5273}, {1158, 3927}, {1598, 1753}, {1699, 4413}, {2717, 2743}, {3158, 5732}, {3474, 5762}, {3522, 3871}, {3689, 5918}, {3913, 4297}, {3940, 6001}, {4295, 5763}

X(6244) = midpoint of X(i) and X(j) for these {i,j}: {20, 3421}, {40, 6282}
X(6244) = reflection of X(i) in X(j) for these (i,j): (3359, 3579), (4,3820), (999,3)
X(6244) = anticomplement of X(7956)
X(6244) = X(144)-Ceva conjugate of X(220)
X(6244) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (35,5584,3), (40,2077,3428), (55,165,3), (55,1155,1617), (165,5537,55), (210,1709,5779), (2077,3428,3)
X(6244) = X(1368)-of-excentral-triangle
X(6244) = X(78)-gimel conjugate of X(3428)
X(6244) = radical center of mixtilinear excircles (Forum Geometricorum, FG200601

X(6245) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2*cos(A)*(cos(B-C)-1)+cos((B-C)/2)*(sin(A/2)*(1+2*cos(B-C))-sin(3*A/2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6 b-2 a^5 b^2-a^4 b^3+4 a^3 b^4-a^2 b^5-2 a b^6+b^7+a^6 c+4 a^5 b c+a^4 b^2 c-4 a^3 b^3 c-a^2 b^4 c-b^6 c-2 a^5 c^2+a^4 b c^2+2 a^2 b^3 c^2+2 a b^4 c^2-3 b^5 c^2-a^4 c^3-4 a^3 b c^3+2 a^2 b^2 c^3+3 b^4 c^3+4 a^3 c^4-a^2 b c^4+2 a b^2 c^4+3 b^3 c^4-a^2 c^5-3 b^2 c^5-2 a c^6-b c^6+c^7
X(6245) = 7 X[3090] - 3 X[5658] = X[5709] - 3 X[5770] = 5 X[3091] - X[6223] = 3 X[381] - X[6259]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6245) lies on these lines: {1, 3427}, {2, 1490}, {3, 10}, {4, 57}, {5, 142}, {7, 5715}, {8, 6282}, {20, 4652}, {40, 4847}, {104, 4311}, {149, 2950}, {189, 3345}, {226, 1071}, {381, 6259}, {443, 5587}, {496, 942}, {516, 1158}, {527, 5812}, {944, 3601}, {950, 1012}, {990, 5292}, {1125, 6261}, {1155, 6253}, {1466, 1837}, {1467, 3086}, {1479, 1709}, {1699, 5586}, {1737, 4299}, {1768, 1770}, {1855, 2272}, {1872, 2823}, {2800, 4084}, {2829, 6246}, {3090, 5658}, {3091, 6223}, {3149, 3911}, {3452, 5777}, {3579, 5771}, {3664, 5713}, {4187, 5927}, {5705, 5732}, {5708, 5805}, {6201, 6257}, {6202, 6258}

X(6245) = midpoint of X(i) and X(j) for these {i,j}: {149,2950}, {3,5787}, {4,84}
X(6245) = reflection of X(i) in X(j) for these (i,j): (4297,5450), (6260,5), (6261,1125)
X(6245) = crosspoint of X(189) anmd X(273)
X(6245) = crosssum of X(198) and X(212)
X(6245) = complement of X(1490)
X(6245) = X(i)-complementary conjugate of X(j) for these (i,j): (56, 3341), (1034, 1329), (3342, 6260), (3345, 10)
X(6245) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,5789,5791), (946,5884,3671), (3358,5709,1158)
X(6245) = X(521)-gimel conjugate of X(1770)

X(6246) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (4*(cos(3*(B-C)/2)+4*cos((B-C)/2)))*sin(A/2)-6*sin(3*A/2)*cos((B-C)/2)-4*cos(A)*(cos(A)-2)+cos(BC)*(6*cos(A)-7)-3
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^7-3 a^6 b-a^5 b^2+4 a^4 b^3-4 a^3 b^4+a^2 b^5+3 a b^6-2 b^7-3 a^6 c+8 a^5 b c-5 a^4 b^2 c-a^3 b^3 c+6 a^2 b^4 c-7 a b^5 c+2 b^6 c-a^5 c^2-5 a^4 b c^2+10 a^3 b^2 c^2-7 a^2 b^3 c^2-3 a b^4 c^2+6 b^5 c^2+4 a^4 c^3-a^3 b c^3-7 a^2 b^2 c^3+14 a b^3 c^3-6 b^4 c^3-4 a^3 c^4+6 a^2 b c^4-3 a b^2 c^4-6 b^3 c^4+a^2 c^5-7 a b c^5+6 b^2 c^5+3 a c^6+2 b c^6-2 c^7
X(6246) = X[100] - 3 X[5587] = 5 X[3091] - X[6224] = 3 X[381] - X[6265]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let O_A, O_B, O_C be the circles with collinear centers described at X(5531) and Hyacinthos #21433 (Barry Wolk, January 2013). Let A′B′C′ be the triangle formed by the radical axes of these circles and the corresponding excircle. Then X(6246) is the circumcenter of A′B′C′. (Randy Hutson, January 29, 2015)

X(6246) lies on these lines: {4, 80}, {5, 214}, {10, 5840}, {11, 515}, {100, 5587}, {104, 5560}, {355, 2802}, {381, 6265}, {546, 946}, {1320, 5881}, {1387, 5882}, {1478, 5083}, {1837, 5884}, {2801, 5805}, {2829, 6245}, {3091, 6224}, {5768, 6256}, {6201, 6262}, {6202, 6263}

X(6246) = midpoint of X(i) and X(j) for these {i,j}: {104,5691}, {1320,5881}, {4,80}
X(6246) = reflection of X(i) in X(j) for these (i,j): (214,5), (5882,1387)

X(6247) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((S_ω+S_A)*S^2+8*R^2*S_A*(-S_ω+S_A))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8 b^2-4 a^6 b^4+6 a^4 b^6-4 a^2 b^8+b^10+a^8 c^2+8 a^6 b^2 c^2-6 a^4 b^4 c^2-3 b^8 c^2-4 a^6 c^4-6 a^4 b^2 c^4+8 a^2 b^4 c^4+2 b^6 c^4+6 a^4 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10
X(6247) = 3 X[154] - 5 X[631] = X[4] - 3 X[1853] = X[64] + 3 X[1853] = 3 X[5085] - X[5596] = 7 X[3090] - 3 X[5656] = 3 X[381] - X[5878] = 3 X[381] - 2 X[5893], 3 X[4] - X[5895]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6247) lies on these lines: {2, 1498}, {3, 66}, {4, 64}, {5, 2883}, {6, 3088}, {10, 5777}, {11, 6285}, {20, 343}, {30, 3357}, {51, 1907}, {52, 2781}, {74, 6145}, {125, 235}, {154, 631}, {185, 427}, {221, 3085}, {253, 3346}, {381, 5878}, {389, 1595}, {1181, 3541}, {1204, 3575}, {1368, 5907}, {1593, 1899}, {1594, 6241}, {1753, 5928}, {2192, 3086}, {2777, 3627}, {2935, 3448}, {3090, 5656}, {3091, 6225}, {3146, 3580}, {3925, 6254}, {5085, 5596}, {5890, 6293}, {6201, 6266}, {6202, 6267}

X(6247) = midpoint of X(i) and X(j) for these {i,j}: {2935, 3448}, {3146,5925}, {4,64}
X(6247) = reflection of X(i) in X(j) for these (i,j): (2883,5), (5878,5893), (5894,3357)
X(6247) = crosspoint of X(i) and X(j) for these (i,j): (253, 2052)
X(6247) = crosssum of X(154) and X(577)
X(6247) = complement of X(1498)
X(6247) = X(i)-complementary conjugate of X(j) for these (i,j): (19, 3343), (3346, 10)
X(6247) = X(3913)-of-orthic-triangle if ABC is acute
X(6247) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (64,1853,4), (381,5878,5893), (389,1595,5480)

X(6248) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((S_ω^2+2*S^2)*(S^2+S_A^2)-S_ω*(S_ω^2+S^2)*S_A)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6 b^2-a^2 b^6+a^6 c^2+3 a^2 b^4 c^2-2 b^6 c^2+3 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6
X(6248) = X[194] - 3 X[262] = X[194] - 5 X[3091] = 3 X[262] - 5 X[3091] = 3 X[381] - X[3095] = X[3146] + 3 X[6194]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6248) lies on these lines: {3, 3734}, {4, 69}, {5, 39}, {30, 5188}, {83, 575}, {98, 384}, {183, 5171}, {194, 262}, {381, 538}, {671, 3399}, {730, 946}, {732, 5480}, {3102, 6289}, {3103, 6290}, {3146, 6194}, {3564, 5052}, {6201, 6272}, {6202, 6273}, {6287, 6321}

X(6248) = midpoint of X(4) and X(76)
X(6248) = reflection of X(i) in X(j) for these (i,j): (39,5), (3,3934)
X(6248) = X(76)-of-Euler-triangle
X(6248) = midpoint of orthocenters of antipedal triangles of PU(1)
X(6248) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,3091,262), (2009,2010,5254)

X(6249) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 4 a^6 b^2+a^4 b^4-4 a^2 b^6-b^8+4 a^6 c^2+10 a^4 b^2 c^2+6 a^2 b^4 c^2-2 b^6 c^2+a^4 c^4+6 a^2 b^2 c^4+6 b^4 c^4-4 a^2 c^6-2 b^2 c^6-c^8
X(6249) = X[2896] - 5 X[3091] = 3 X[381] - X[6287]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6249) lies on these lines: {4, 83}, {5, 5188}, {115, 546}, {381, 754}, {382, 4045}, {732, 5480}, {2896, 3091}, {6201, 6274}, {6202, 6275}

X(6249) = midpoint of X(4) and X(83)
X(6249) = reflection of X(6292) in X(5)
X(6249) = X(83)-of-Euler-triangle

X(6250) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((2*S_ω+5*S)*S_A^2+(-5*S*S_ω-S^2-2*S_ω^2)*S_A+3*S^2*(2*S+S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^8-3 a^6 b^2-3 a^4 b^4+7 a^2 b^6-3 b^8-3 a^6 c^2-2 a^4 b^2 c^2-7 a^2 b^4 c^2+12 b^6 c^2-3 a^4 c^4-7 a^2 b^2 c^4-18 b^4 c^4+7 a^2 c^6+12 b^2 c^6-3 c^8-2 (3 a^4 b^2-2 a^2 b^4-b^6+3 a^4 c^2+4 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6) S
X(6250) = X[488] - 5 X[3091] = 7 X[3832] - X[6278] = 9 X[3839] + X[6279] = 3 X[381] - X[6289]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6250) lies on these lines: {3, 6118}, {4, 371}, {5, 641}, {115, 3070}, {381, 591}, {488, 3091}, {546, 576}, {639, 6228}, {1131, 5870}, {2043, 6304}, {2044, 6305}, {3832, 6201}, {3839, 6202}

X(6250) = midpoint of X(4) and X(485)
X(6250) = reflection of X(i) in X(j) for these (i,j): (3,6118), (641,5)
X(6250) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (546,5480,6251)

X(6251) = EULER-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = ((2*S_ω-5*S)*S_A^2+(5*S*S_ω-S^2-2*S_ω^2)*S_A-3*S^2*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^8-3 a^6 b^2-3 a^4 b^4+7 a^2 b^6-3 b^8-3 a^6 c^2-2 a^4 b^2 c^2-7 a^2 b^4 c^2+12 b^6 c^2-3 a^4 c^4-7 a^2 b^2 c^4-18 b^4 c^4+7 a^2 c^6+12 b^2 c^6-3 c^8+2 (3 a^4 b^2-2 a^2 b^4-b^6+3 a^4 c^2+4 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6) S
X(6251) = X[487] - 5 X[3091] = 9 X[3839] + X[6280] = 7 X[3832] - X[6281] = 3 X[381] - X[6290]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6251) lies on these lines: {3, 6119}, {4, 372}, {5, 642}, {115, 3071}, {381, 1991}, {487, 3091}, {546, 576}, {640, 6229}, {1132, 5871}, {2043, 6301}, {2044, 6300}, {3832, 6202}, {3839, 6201}

X(6251) = midpoint of X(4) and X(486)
X(6251) = reflection of X(i) in X(j) for these (i,j): (3,6119), (642,5)
X(6251) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (546,5480,6250)

X(6252) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS CENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((b^2+b*c+c^2)*a^5-(b+c)*(b^2-b*c+c^2)*a^4-2*(b^3*c-2*S*b*c+b^4+c^4+c^3*b+2*b^2*c^2)*a^3+2*(b+c)*(b^4-b^3*c+2*b^2*c^2+2*S*b*c-c^3*b+c^4)*a^2-(b+c)^2*(-b^4+b^3*c+c^3*b+4*S*b*c-c^4)*a-(b+c)*(bc)^2*(b^4+b^3*c+c^3*b+4*S*b*c+c^4))*a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) (a b^2-b^3+a b c+a c^2-c^3)+4 a^2 b c (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6252) lies on these lines: {19, 6291}, {40, 511}, {55, 1151}, {65, 175}, {6197, 6239}

X(6252) = reflection of X(6283) in X(1151)

X(6253) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF MACBEATH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2*sin(A/2)*cos(3*(B-C)/2)+2*cos(B-C)-3*cos(A)-cos(2*A)-4*sin(3*A/2)*cos((B-C)/2)-(2*R^2+3*r*R-s^2+r^2)/R^2
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c-2 a^5 b c+a^4 b^2 c+2 a b^5 c+b^6 c-3 a^5 c^2+a^4 b c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-4 a b^3 c^3-3 b^4 c^3-a b^2 c^4-3 b^3 c^4+2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7
X(6253) = 4 X[946] - 3 X[3058] = 2 X[944] - 3 X[5434]}}

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6253) lies on these lines: {1, 5805}, {3, 3925}, {4, 12}, {5, 5259}, {11, 3149}, {19, 3575}, {20, 958}, {30, 40}, {46, 5787}, {65, 515}, {72, 516}, {165, 5791}, {382, 5840}, {411, 2886}, {528, 962}, {910, 1855}, {944, 5434}, {946, 3058}, {971, 1770}, {1155, 6245}, {1490, 1836}, {1503, 3779}, {1750, 5812}, {1867, 3198}, {1885, 5130}, {3070, 5415}, {3071, 5416}, {3072, 5721}, {3146, 3436}, {5221, 5768}, {5762, 5904}, {5786, 6047}, {6197, 6240}

X(6253) = reflection of X(6284) in X(4)
X(6253) = {X(20),X(2550)}-harmonic conjugate of X(5584)
X(6253) = homothetic center of extangents triangle and reflection of intangents triangle in X(4)

X(6254) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((b^2+b*c+c^2)*a^9-(b+c)*(b^2-b*c+c^2)*a^8-4*(b^2-b*c+c^2)*(b+c)^2*a^7+4*(b+c)*(b^2+b*c+c^2)*(bc)^2*a^6+2*(3*c^4-3*c^3*b+2*b^2*c^2- 3*b^3*c+3*b^4)*(b+c)^2*a^5-2*(b+c)*(3*c^3*b+3*b^4+3*c^4+3*b^3*c+2*b^2*c^2)*(b-c)^2*a^4-4*(b^2+c^2)*(b^2+b*c+c^2)*(bc)^2*(b+c)^2*a^3+4*(b^2+c^2)*(b^2-b*c+c^2)*(bc)^ 2*(b+c)^3*a^2+(c^4-c^3*b+4*b^2*c^2-b^3*c+b^4)*(bc)^2*(b+c)^4*a-(c^4+c^3*b+4*b^2*c^2+b^3*c+b^4)*(b+c)^3*(bc)^4)*a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^9 b^2-a^8 b^3-4 a^7 b^4+4 a^6 b^5+6 a^5 b^6-6 a^4 b^7-4 a^3 b^8+4 a^2 b^9+a b^10-b^11+a^9 b c-4 a^7 b^3 c+6 a^5 b^5 c-4 a^3 b^7 c+a b^9 c+a^9 c^2-4 a^6 b^3 c^2-2 a^5 b^4 c^2+8 a^4 b^5 c^2-4 a^2 b^7 c^2+a b^8 c^2-a^8 c^3-4 a^7 b c^3-4 a^6 b^2 c^3-4 a^5 b^3 c^3-2 a^4 b^4 c^3+4 a^3 b^5 c^3+4 a^2 b^6 c^3+4 a b^7 c^3+3 b^8 c^3-4 a^7 c^4-2 a^5 b^2 c^4-2 a^4 b^3 c^4+8 a^3 b^4 c^4-4 a^2 b^5 c^4-2 a b^6 c^4+6 b^7 c^4+4 a^6 c^5+6 a^5 b c^5+8 a^4 b^2 c^5+4 a^3 b^3 c^5-4 a^2 b^4 c^5-10 a b^5 c^5-8 b^6 c^5+6 a^5 c^6+4 a^2 b^3 c^6-2 a b^4 c^6-8 b^5 c^6-6 a^4 c^7-4 a^3 b c^7-4 a^2 b^2 c^7+4 a b^3 c^7+6 b^4 c^7-4 a^3 c^8+a b^2 c^8+3 b^3 c^8+4 a^2 c^9+a b c^9+a c^10-c^11)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6254) lies on these lines: {4, 65}, {19, 185}, {30, 6237}, {40, 2939}, {55, 1498}, {64, 71}, {1503, 3779}, {3925, 6247}, {6197, 6241}

X(6254) = reflection of X(6285) in X(1498)
X(6254) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (64,3197,5584)

X(6255) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((b^2+b*c+c^2)*a^9-(b+c)*(b^2-b*c+c^2)*a^8+2*(-3*b^2*c^2-2*b^4-2*c^4-2*c^3*b-2*b^3*c)*a^7+2*(b+c)*(2*c^4-2*c^3*b+3*b^2*c^2-2*b^3*c+2*b^4)*a^6+(6*b^6+5*b^3*c^3+6*b^5*c+6*c^6+7*b^4*c^2+7*b^2*c^4+6*b*c^5)*a^5-(b+c)*(6*c^6-6*b*c^5+7*b^2*c^4-5*b^3*c^3+7*b^4*c^2-6*b^5*c+6*b^6)*a^4-(4*b^6- 4*b^5*c+4*b^4*c^2-5*b^3*c^3+4*b^2*c^4-4*b*c^5+4*c^6)*(b+c)^2*a^3+(b+c)*(4*c^6+4*b*c^5+4*b^2*c^4+5*b^3*c^3+4*b^4*c^2+4*b^5*c+4*b^6)*(b-c)^2*a^2+(c^4-c^3*b+b^2*c^2-b^3*c+b^4)*(b-c)^2*(b+c)^4*a-(c^4+c^3*b+b^2*c^2+b^3*c+b^4)*(b+c)^3*(b-c)^4)*a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^9 b^2-a^8 b^3-4 a^7 b^4+4 a^6 b^5+6 a^5 b^6-6 a^4 b^7-4 a^3 b^8+4 a^2 b^9+a b^10-b^11+a^9 b c-4 a^7 b^3 c+6 a^5 b^5 c-4 a^3 b^7 c+a b^9 c+a^9 c^2-6 a^7 b^2 c^2+2 a^6 b^3 c^2+7 a^5 b^4 c^2-a^4 b^5 c^2-4 a^2 b^7 c^2-2 a b^8 c^2+3 b^9 c^2-a^8 c^3-4 a^7 b c^3+2 a^6 b^2 c^3+5 a^5 b^3 c^3-2 a^4 b^4 c^3+a^3 b^5 c^3+a^2 b^6 c^3-2 a b^7 c^3-4 a^7 c^4+7 a^5 b^2 c^4-2 a^4 b^3 c^4+2 a^3 b^4 c^4-a^2 b^5 c^4+a b^6 c^4-3 b^7 c^4+4 a^6 c^5+6 a^5 b c^5-a^4 b^2 c^5+a^3 b^3 c^5-a^2 b^4 c^5+2 a b^5 c^5+b^6 c^5+6 a^5 c^6+a^2 b^3 c^6+a b^4 c^6+b^5 c^6-6 a^4 c^7-4 a^3 b c^7-4 a^2 b^2 c^7-2 a b^3 c^7-3 b^4 c^7-4 a^3 c^8-2 a b^2 c^8+4 a^2 c^9+a b c^9+3 b^2 c^9+a c^10-c^11)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6255) lies on these lines: {19, 6152}, {40, 1154}, {54, 71}, {55, 195}, {65, 2962}, {539, 6237}, {3779, 5965}, {6197, 6242}

X(6255) = reflection of X(6286) in X(195)

X(6256) = FUHRMANN-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-a^7+(b+c)*a^6+(c^2-6*b*c+b^2)*a^5-(b+c)*(c^2-4*b*c+b^2)*a^4+(4*b*c+b^2+c^2)*(b-c)^2*a^3- (4*b*c+b^2+c^2)*(b+c)*(b-c)^2*a^2-(c^2-4*b*c+b^2)*(bc)^2*(b+c)^2*a+(b+c)^3*(b-c)^4)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^7-a^6 b-a^5 b^2+a^4 b^3-a^3 b^4+a^2 b^5+a b^6-b^7-a^6 c+6 a^5 b c-3 a^4 b^2 c-2 a^3 b^3 c+3 a^2 b^4 c-4 a b^5 c+b^6 c-a^5 c^2-3 a^4 b c^2+6 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+a^4 c^3-2 a^3 b c^3-4 a^2 b^2 c^3+8 a b^3 c^3-3 b^4 c^3-a^3 c^4+3 a^2 b c^4-a b^2 c^4-3 b^3 c^4+a^2 c^5-4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7
X(6256) = X[84] - 3 X[5587]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6256) lies on these lines: {1, 4}, {2, 5450}, {3, 119}, {8, 153}, {10, 1158}, {12, 1012}, {20, 2077}, {30, 4421}, {40, 3436}, {46, 1512}, {56, 1532}, {80, 5553}, {84, 377}, {104, 499}, {191, 2950}, {355, 5836}, {382, 5842}, {971, 5880}, {1071, 1837}, {1470, 3149}, {1537, 2098}, {1788, 2096}, {2475, 5554}, {2478, 3576}, {3086, 5193}, {3434, 5881}, {3577, 5555}, {5046, 5731}, {5220, 5690}, {5768, 6246}, {5777, 5794}

X(6256) = midpoint of X(i) and X(j) for these (i,j): (1490,5691), (355,6259)
X(6256) = reflection of X(i) and X(j) for these (i,j): (1158,10), (6261, 6260)
X(6256) = anticomplement of X(5450)
X(6256) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,388,946), (4,944,1479), (20,5552,2077), (3585,5691,4)
X(6256) = X(8)-beth conjugate of X(1158)

X(6257) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)*a^7+(4*b*c+c^2+b^2+S)*a^6-(b+c)*(3*c^2-2*b*c+3*b^2)*a^5-3*(bc)^2*(b^2+c^2+S)*a^4+(b+c)*(2*b*c+3*c^2+3*b^2)*(bc)^2*a^3+(b-c)^2*(3*b^4-2*b^3*c-2*b^2*c^2+3*S*b^2-2*c^3*b+2*S*b*c+3*c^4+3*S*c^2)*a^2-(b+c)^3*(b-c)^4*a-(bc)^2*(b+c)^4*(b^2+c^2+S)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^7 b+a^6 b^2-3 a^5 b^3-3 a^4 b^4+3 a^3 b^5+3 a^2 b^6-a b^7-b^8+a^7 c+4 a^6 b c-a^5 b^2 c+6 a^4 b^3 c-a^3 b^4 c-8 a^2 b^5 c+a b^6 c-2 b^7 c+a^6 c^2-a^5 b c^2-6 a^4 b^2 c^2-2 a^3 b^3 c^2+5 a^2 b^4 c^2+3 a b^5 c^2-3 a^5 c^3+6 a^4 b c^3-2 a^3 b^2 c^3-3 a b^4 c^3+2 b^5 c^3-3 a^4 c^4-a^3 b c^4+5 a^2 b^2 c^4-3 a b^3 c^4+2 b^4 c^4+3 a^3 c^5-8 a^2 b c^5+3 a b^2 c^5+2 b^3 c^5+3 a^2 c^6+a b c^6-a c^7-2 b c^7-c^8)+a (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c+2 a b c-b^2 c-a c^2+b c^2+c^3) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6257) lies on these lines: {6, 84}, {971, 1160}, {1270, 6223}, {2829, 6262}, {3640, 6001}, {5590, 6260}, {6201, 6245}, {6214, 6259}

X(6257) = reflection of X(6258) in X(84)

X(6258) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c)*a^7+(b^2+c^2+4*b*c-S)*a^6-(b+c)*(3*c^2-2*b*c+3*b^2)*a^5-3*(b-c)^2*(b^2+c^2-S)*a^4+(b+c)*(3*c^2+2*b*c+3*b^2)*(b-c)^2*a^3+(b-c)^2*((-3*b^2-2*b*c-3*c^2)*S+(3*b^2+4*b*c+3*c^2)*(b-c)^2)*a^2-(b+c)^3*(b-c)^4*a-(b-c)^2*(b+c)^4*(b^2+c^2-S)2-c^2+S)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^7 b+a^6 b^2-3 a^5 b^3-3 a^4 b^4+3 a^3 b^5+3 a^2 b^6-a b^7-b^8+a^7 c+4 a^6 b c-a^5 b^2 c+6 a^4 b^3 c-a^3 b^4 c-8 a^2 b^5 c+a b^6 c-2 b^7 c+a^6 c^2-a^5 b c^2-6 a^4 b^2 c^2-2 a^3 b^3 c^2+5 a^2 b^4 c^2+3 a b^5 c^2-3 a^5 c^3+6 a^4 b c^3-2 a^3 b^2 c^3-3 a b^4 c^3+2 b^5 c^3-3 a^4 c^4-a^3 b c^4+5 a^2 b^2 c^4-3 a b^3 c^4+2 b^4 c^4+3 a^3 c^5-8 a^2 b c^5+3 a b^2 c^5+2 b^3 c^5+3 a^2 c^6+a b c^6-a c^7-2 b c^7-c^8)-a (a^3-a^2 b-a b^2+b^3+a^2 c+2 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-a^2 c+2 a b c-b^2 c-a c^2+b c^2+c^3) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6258) lies on these lines: {6, 84}, {971, 1161}, {1271, 6223}, {2829, 6263}, {3641, 6001}, {5591, 6260}, {6202, 6245}, {6215, 6259}

X(6258) = reflection of X(6257) in X(84)

X(6259) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2*cos(B-C)*(sin(A/2)*cos(B/2-C/2)-1)+cos(A)*(3-cos(A))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^7-2 a^5 b^2-a^4 b^3+a^3 b^4+2 a^2 b^5-b^7+6 a^5 b c-a^4 b^2 c-2 a^3 b^3 c-4 a b^5 c+b^6 c-2 a^5 c^2-a^4 b c^2+2 a^3 b^2 c^2-2 a^2 b^3 c^2+3 b^5 c^2-a^4 c^3-2 a^3 b c^3-2 a^2 b^2 c^3+8 a b^3 c^3-3 b^4 c^3+a^3 c^4-3 b^3 c^4+2 a^2 c^5-4 a b c^5+3 b^2 c^5+b c^6-c^7
X(6259) = X[20] - 3 X[5658] = X[5787] + 2 X[6223] = 3 X[381] - 2 X[6245]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6259) lies on these lines: {3, 3452}, {4, 7}, {5, 84}, {10, 5779}, {12, 1709}, {20, 5440}, {30, 1490}, {40, 6068}, {355, 5836}, {377, 5927}, {381, 6245}, {382, 515}, {516, 3913}, {952, 3680}, {1158, 3652}, {1538, 3086}, {1836, 3340}, {2829, 6261}, {3587, 5924}, {4208, 5817}, {5044, 5811}, {6214, 6257}, {6215, 6258}

X(6259) = midpoint of X(4) and X(6223)
X(6259) = reflection of X(i) in X(j) for these (i,j): (355, 6256), (3,6260), (5787,4), (5,84)
X(6259) = anticomplement of X(34862)
X(6259) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7,5806), (4,1071,5722)

X(6260) = MEDIAL-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos(B)+cos(C)+(1-cos(B-C)))*(cos(B)+cos(C)-(1-cos(B+C)))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) (a^3 b-a^2 b^2-a b^3+b^4+a^3 c+2 a^2 b c+a b^2 c-a^2 c^2+a b c^2-2 b^2 c^2-a c^3+c^4)
X(6260) = X[1490] - 3 X[5658] = X[4] + 3 X[5658] = 3 X[381] - X[5787] = 3 X[2] + X[6223]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6260) lies on these lines: {1, 4}, {2, 84}, {3, 3452}, {5, 142}, {9, 1158}, {10, 5777}, {20, 908}, {40, 329}, {72, 1145}, {165, 5924}, {214, 2829}, {381, 5787}, {405, 5450}, {442, 5927}, {452, 3576}, {496, 1538}, {498, 1709}, {516, 5812}, {527, 5709}, {631, 5316}, {962, 3895}, {1071, 1210}, {1466, 3149}, {1512, 4848}, {1715, 2183}, {1728, 3911}, {1753, 1763}, {1901, 2092}, {3091, 5249}, {3651, 5660}, {4667, 5707}, {5175, 5881}, {5177, 5587}, {5534, 5853}, {5590, 6257}, {5591, 6258}, {5779, 5791}, {5806, 6147}, {5837, 5887}

X(6260) = midpoint of X(i) and X(j) for these {i,j}: (3,6259), (4,1490), (6256,6261), (84,6223)
X(6260) = reflection of X(6245) in X(5)
X(6260) = X(2)-Ceva conjugate of X(1108)
X(6260) = crosspoint of X(i) and X(j) for these (i,j): (2, 322), (329, 342)
X(6260) = crosssum of X(i) and X(j) for these (i,j): (6, 2208), (1436, 2188)
X(6260) = complement of X(84)
X(6260) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 946), (6, 57), (31, 1108), (40, 10), (42, 1901), (55, 281), (56, 3086), (198, 2), (208, 1210), (221, 1), (223, 142), (227, 442), (322, 2887), (329, 141), (347, 2886), (1103, 6260), (1817, 3739), (2187, 37), (2199, 3752), (2324, 3452), (2331, 226), (2334, 4295), (2360, 1125), (3194, 942), (3195, 6), (3209, 3772), (3342, 6245), (6129, 11)
X(6260) = X(84)-isoconjugate of X(1167)

X(6261) = 2nd-CIRCUMPERP-TRIANGLE-ORTHOLOGIC CENTER OF EXTOUCH TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = -3+4*cos((B-C)/2)*(2*sin(A/2)-sin(3*A/2))+2*cos(A)*cos(BC)+2*cos(A)-cos(2*A)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+2 a^4 b c-2 a^3 b^2 c-2 a^2 b^3 c+4 a b^4 c-a^4 c^2-2 a^3 b c^2+6 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+4 a^3 c^3-2 a^2 b c^3-2 a b^2 c^3-a^2 c^4+4 a b c^4-b^2 c^4-2 a c^5+c^6)
X(6261) = X[84] - 3 X[3576] = 3 X[3576] - 2 X[5450] = X[944] + 3 X[5658] = X[5787] - 3 X[5886] = 3 X[5731] + X[6223]

X(6261) is also the 2nd-circumperp-triangle-orthologic center of the extouch triangle. See César Lozada, Perspective-Orthologic-Parallelogic.

X(6261) lies on these lines: {1, 4}, {3, 960}, {9, 1630}, {10, 5720}, {20, 224}, {21, 84}, {36, 920}, {40, 78}, {56, 1071}, {57, 5884}, {63, 5693}, {65, 3149}, {72, 3428}, {90, 104}, {355, 2886}, {517, 3811}, {519, 5534}, {602, 1780}, {758, 5709}, {958, 5777}, {971, 1001}, {990, 3736}, {1012, 2646}, {1062, 2883}, {1125, 6245}, {1319, 1898}, {1532, 1837}, {1709, 3612}, {2077, 4855}, {2476, 5587}, {2720, 2733}, {2829, 6259}, {3086, 5768}, {3632, 5531}, {3740, 5780}, {3870, 3885}, {3872, 5086}, {3901, 5536}, {3957, 5734}, {5396, 5799}, {5698, 5732}, {5731, 6223}, {5787, 5886}

X(6261) = midpoint of X(1) in X(1490)
X(6261) = reflection of X(i) in X(j) for these {i,j}: {1158,3}, (6245,1125), (6256,6260), (84,5450)
X(6261) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3485,946), (40,6326,78), (84,3576,5450), (411,3869,40)
X(6261) = X(21)-beth conjugate of X(34)
X(6261) = X(9927)-of-excentral-triangle

X(6262) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-2*a^6+(b+c)*a^5+(-S-2*b*c+b^2+c^2)*a^4+(b+c)*(b*c+S)*a^3-b*c*(b^2+c^2+S)*a^2-(b+c)*(b-c)^2*(b^2+c^2+S)*a+(bc)^2*(b+c)^2*(b^2+c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-a^5 b-a^4 b^2+a b^5-b^6-a^5 c+2 a^4 b c-a^3 b^2 c+a^2 b^3 c-a b^4 c-a^4 c^2-a^3 b c^2+b^4 c^2+a^2 b c^3-a b c^4+b^2 c^4+a c^5-c^6+(a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6262) lies on these lines: {6, 80}, {100, 5688}, {214, 5590}, {952, 3640}, {1270, 6224}, {2800, 5870}, {2829, 6257}, {6201, 6246}, {6214, 6265}

X(6262) = reflection of X(6263) in X(80)

X(6263) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (2*a^6-(b+c)*a^5-(S-2*b*c+b^2+c^2)*a^4+(b+c)*(-b*c+S)*a^3-b*c*(-b^2-c^2+S)*a^2-(b+c)*(b-c)^2*(-b^2-c^2+S)*a+(bc)^2*(b+c)^2*(-b^2-c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-a^5 b-a^4 b^2+a b^5-b^6-a^5 c+2 a^4 b c-a^3 b^2 c+a^2 b^3 c-a b^4 c-a^4 c^2-a^3 b c^2+b^4 c^2+a^2 b c^3-a b c^4+b^2 c^4+a c^5-c^6-(a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6263) lies on these lines: {6, 80}, {100, 5689}, {214, 5591}, {952, 3641}, {1271, 6224}, {2800, 5871}, {2829, 6258}, {6202, 6246}, {6215, 6265}

X(6263) = reflection of X(6262) in X(80)
X(6263) = X(21)-beth conjugate of X(1411)

X(6264) = HEXYL-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6-2*(b+c)*a^5-(c^2-11*b*c+b^2)*a^4+(b+c)*(4*c^2-13*b*c+4*b^2)*a^3+(-7*c^3*b-7*b^3*c+20*b^2*c^2-b^4-c^4)*a^2-(b+c)*(2*c^2-9*b*c+2*b^2)*(b-c)^2*a+(b^2-4*b*c+c^2)*(bc)^2*(b+c)^2
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+11 a^4 b c-9 a^3 b^2 c-7 a^2 b^3 c+11 a b^4 c-4 b^5 c-a^4 c^2-9 a^3 b c^2+20 a^2 b^2 c^2-9 a b^3 c^2-b^4 c^2+4 a^3 c^3-7 a^2 b c^3-9 a b^2 c^3+8 b^3 c^3-a^2 c^4+11 a b c^4-b^2 c^4-2 a c^5-4 b c^5+c^6)
X(6264) = 2 X[100] - 3 X[3576] = 3 X[1] - X[5531] = 4 X[11] - 3 X[5587] = 3 X[1] - 2 X[6265] = 2 X[5531] - 3 X[6326] = 4 X[6265] - 3 X[6326]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6264) lies on the hexyl circle and these lines: {1, 5}, {3, 5541}, {40, 104}, {84, 1320}, {100, 3576}, {149, 515}, {153, 946}, {214, 5438}, {517, 1768}, {528, 5732}, {936, 3036}, {1145, 4853}, {1389, 3881}, {1482, 2771}, {2077, 3880}, {2098, 5693}, {2801, 3243}, {2894, 4861}, {3885, 5450}, {5538, 5844}

X(6264) = reflection of X(i) in X(j) for these (i,j): (153,946), (355,1484), (40,104), (5531, 6265), (5541,3), (5881,80), (6224,5882), (6326,1)
X(6264) = X(74)-of-hexyl-triangle
X(6264) = hexyl-isogonal conjugate of X(517)
X(6264) = X(7728)-of-excentral-triangle
X(6264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5531,6265), (5531,6265,6326)

X(6265) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = a^6-cos(B-C)+4*sin(3*A/2)*cos(B/2-C/2)+(2*R^2-r*R+r^2-s^2)/R^2
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+5 a^4 b c-3 a^3 b^2 c-4 a^2 b^3 c+5 a b^4 c-b^5 c-a^4 c^2-3 a^3 b c^2+8 a^2 b^2 c^2-3 a b^3 c^2-b^4 c^2+4 a^3 c^3-4 a^2 b c^3-3 a b^2 c^3+2 b^3 c^3-a^2 c^4+5 a b c^4-b^2 c^4-2 a c^5-b c^5+c^6)
X(6265) = X[1768] - 3 X[3576] = 3 X[1] + X[5531] = X[149] - 3 X[5603] = 2 X[11] - 3 X[5886] = 3 X[381] - 2 X[6246] = 3 X[1] - X[6264] = X[5531] - 3 X[6326] = X[6264] + 3 X[6326]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6265) lies on these lines: {1, 5}, {3, 214}, {4, 6224}, {21, 104}, {78, 1145}, {100, 517}, {149, 5603}, {153, 944}, {224, 1537}, {381, 6246}, {528, 3656}, {912, 1319}, {997, 3035}, {999, 5083}, {1001, 2801}, {1259, 5730}, {1320, 1389}, {1482, 2802}, {1749, 5427}, {1768, 3576}, {2095, 4930}, {2783, 3736}, {2829, 6259}, {2975, 5694}, {3654, 6174}, {3811, 5854}, {3869, 4996}, {5253, 5885}, {6214, 6262}, {6215, 6263}

X(6265) = midpoint of X(i) and X(j) for these {i,j}: {153,944), (1,6326), (4,6224), (5531,6264)
X(6265) = reflection of X(i) in X(j) for these (i,j): (104,1385), (1484, 5901), (355,119), (3654,6174), (3,214), (80,5)
X(6265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5531,6264), (6264,6326,5531)
X(6265) = inner-Garcia-to-ABC similarity image of X(355)
X(6265) = X(10113)-of-excentral-triangle

X(6266) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(2*(S+2*S_ω)*S_A^2-2*(-S_ω+8*R^2)*(S+2*S_ω)*S_A+S^2*(8*R^2+S))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 4 a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+2 a^6 b^2 c^2-2 a^2 b^6 c^2-b^8 c^2-2 a^6 c^4+2 b^6 c^4-2 a^2 b^2 c^6+2 b^4 c^6+2 a^2 c^8-b^2 c^8-c^10)+a^2 (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6266) lies on these lines: {4, 6217}, {6, 64}, {1160, 6000}, {1162, 5895}, {1270, 6225}, {2883, 5590}, {5878, 6214}, {6201, 6247}

X(6266) = reflection of X(6267) in X(64)

X(6267) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*((2*(S-2*S_ω))*S_A^2-(2*(-S_ω+8*R^2))*(S-2*S_ω)*S_A-S^2*(8*R^2-S))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 4 a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2+2 a^6 b^2 c^2-2 a^2 b^6 c^2-b^8 c^2-2 a^6 c^4+2 b^6 c^4-2 a^2 b^2 c^6+2 b^4 c^6+2 a^2 c^8-b^2 c^8-c^10)-a^2 (a^4-2 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2-3 c^4) (a^4+2 a^2 b^2-3 b^4-2 a^2 c^2+2 b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6267) lies on these lines: {4, 6218}, {6, 64}, {1161, 6000}, {1163, 5895}, {1271, 6225}, {2883, 5591}, {5878, 6215}, {6202, 6247}

X(6267) = reflection of X(6266) in X(64)

X(6268) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((33*(2*S_ω+S))*S_A^2+(-6+sqrt(3))*((6+sqrt(3))*(2*S_ω+S)*S_ω-11*S^2)*S_A+(11*(4*S+(2+sqrt(3))*S_ω))*S^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)+2 Sqrt[3] a^2 S^2+ (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6268) lies on these lines: {6, 13}, {530, 5860}, {616, 1270}, {618, 5590}, {2041, 6278}, {5478, 6201}, {5617, 6214}

X(6268) = reflection of X(6270) in X(13)

X(6269) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((33*(S+2*S_ω))*S_A^2+(6+sqrt(3))*((-6+sqrt(3))*(S+2*S_ω)*S_ω+11*S^2)*S_A-(11*(-4*S+(-2+sqrt(3))*S_ω))*S^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)+(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) S-2 Sqrt[3] a^2 S^2

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6269) lies on these lines: {6, 13}, {531, 5860}, {617, 1270}, {619, 5590}, {2042, 6278}, {5479, 6201}, {5613, 6214}

X(6269) = reflection of X(6271) in X(14)

X(6270) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((33*(-S+2*S_ω))*S_A^2-(6+sqrt(3))*((-6+sqrt(3))*(S-2*S_ω)*S_ω-11*S^2)*S_A-(11*(4*S+(-2+sqrt(3))*S_ω))*S^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)-(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) S-2 Sqrt[3] a^2 S^2

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6270) lies on these lines: {6, 13}, {530, 5861}, {616, 1271}, {618, 5591}, {2042, 6281}, {5478, 6202}, {5617, 6215}

X(6270) = reflection of X(6268) in X(13)

X(6271) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((33*(S-2*S_ω))*S_A^2+(-6+sqrt(3))*((6+sqrt(3))*(S-2*S_ω)*S_ω+11*S^2)*S_A-(11*(-4*S+(2+sqrt(3))*S_ω))*S^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6)-(a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) S+2 Sqrt[3] a^2 S^2

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6271) lies on these lines: {{6, 13}, {531, 5861}, {617, 1271}, {619, 5591}, {2041, 6281}, {5479, 6202}, {5613, 6215}

X(6271) = reflection of X(6269) in X(14)

X(6272) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-(b^2+c^2)*a^4+b^2*c^2*(b^2+c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^4 c^2-b^4 c^2-b^2 c^4-b^2 c^2 S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6272) lies on these lines: {{6, 76}, {39, 5590}, {194, 1270}, {511, 5870}, {538, 5860}, {637, 6318}, {730, 3640}, {1160, 2782}, {3095, 6214}, {6201, 6248}

X(6272) = reflection of X(6273) in X(76)
X(6272) = X(76)-of-outer-Grebe-triangle

X(6273) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((b^2+c^2)*a^4+b^2*c^2*(-b^2-c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^4 c^2-b^4 c^2-b^2 c^4+b^2 c^2 S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6273) lies on these lines: {6, 76}, {39, 5591}, {194, 1271}, {511, 5871}, {538, 5861}, {638, 6314}, {730, 3641}, {1161, 2782}, {3095, 6215}, {6202, 6248}

X(6273) = reflection of X(6272) in X(76)
X(6273) = X(76)-of-inner-Grebe-triangle

X(6274) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-(b^2+c^2-S)*a^4+(b^2+c^2)*S*a^2+b^2*c^2*(b^2+c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^4 c^2-b^4 c^2-b^2 c^4-(a^2+b^2) (a^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6274) lies on these lines: {6, 76}, {637, 754}, {1270, 2896}, {5590, 6292}, {5874, 6226}, {6201, 6249}, {6214, 6287}

X(6274) = reflection of X(6275) in X(83)
X(6274) = X(83)-of-outer-Grebe-triangle

X(6275) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((b^2+c^2+S)*a^4+(b^2+c^2)*S*a^2+b^2*c^2*(-b^2-c^2+S))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^4 c^2-b^4 c^2-b^2 c^4+(a^2+b^2) (a^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6275) lies on these lines: {6, 76}, {638, 754}, {1271, 2896}, {5591, 6292}, {5875, 6227}, {6202, 6249}, {6215, 6287}

X(6275) = reflection of X(6274) in X(83)
X(6275) = X(83)-of-inner-Grebe-triangle

X(6276) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*((2*S_ω+S)*S_A^2+2*(-S_ω+2*R^2)*(2*S_ω+S)*S_A-S^2*(S-6*S_ω+20*R^2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2-4 a^6 b^2 c^2+3 a^4 b^4 c^2-2 a^2 b^6 c^2+2 b^8 c^2-2 a^6 c^4+3 a^4 b^2 c^4-b^6 c^4-2 a^2 b^2 c^6-b^4 c^6+2 a^2 c^8+2 b^2 c^8-c^10)+a^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6276) lies on these lines: {4, 6219}, {6, 24}, {539, 5860}, {1154, 1160}, {1162, 3574}, {1209, 5590}, {1270, 2888}, {6214, 6288}

X(6276) = reflection of X(6277) in X(54)

X(6277) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*((S-2*S_ω)*S_A^2+2*(-S_ω+2*R^2)*(S-2*S_ω)*S_A+S^2*(-S-6*S_ω+20*R^2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^8 b^2-2 a^6 b^4+2 a^2 b^8-b^10+a^8 c^2-4 a^6 b^2 c^2+3 a^4 b^4 c^2-2 a^2 b^6 c^2+2 b^8 c^2-2 a^6 c^4+3 a^4 b^2 c^4-b^6 c^4-2 a^2 b^2 c^6-b^4 c^6+2 a^2 c^8+2 b^2 c^8-c^10)-a^2 (a^4-2 a^2 b^2+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-2 a^2 c^2-b^2 c^2+c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6277) lies on these lines: {4, 6220}, {6, 24}, {539, 5861}, {1154, 1161}, {1163, 3574}, {1209, 5591}, {1271, 2888}, {6215, 6288}

X(6277) = reflection of X(6276) in X(54)

X(6278) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((2*S_ω+S)*S_A^2+(3*S+2*S_ω)*(S-S_ω)*S_A+S^2*(2*S+S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^6-4 a^4 b^2+3 a^2 b^4-2 b^6-4 a^4 c^2+2 a^2 b^2 c^2+2 b^4 c^2+3 a^2 c^4+2 b^2 c^4-2 c^6+2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S
X(6278) = 4 X[5] - 3 X[485] = X[20] - 3 X[488] = 5 X[631] - 6 X[641] = 13 X[5067] - 12 X[6118] = 7 X[3832] - 6 X[6250] = 8 X[5] - 3 X[6279] = X[6279] - 4 X[6289] = 2 X[5] - 3 X[6289]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6278) lies on these lines: {4, 5860}, {5, 6}, {20, 488}, {382, 1160}, {631, 641}, {2041, 6268}, {2042, 6269}, {3640, 5881}, {3832, 6201}, {5067, 6118}

X(6278) = reflection of X(i) in X(j) for these (i,j): (485,6289), (6279,485)
X(6278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5874,6280), (5874,6214,6)

X(6279) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((-S+2*S_ω)*S_A^2+(S*S_ω+5*S^2-2*S_ω^2)*S_A-S^2*(2*S+S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 5 a^6-8 a^4 b^2+5 a^2 b^4-2 b^6-8 a^4 c^2-2 a^2 b^2 c^2+2 b^4 c^2+5 a^2 c^4+2 b^2 c^4-2 c^6-2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S
X(6279) = 4 X[5] - 5 X[485] = 5 X[488] - 7 X[3523] = 8 X[5] - 5 X[6278] = 6 X[5] - 5 X[6289] = 3 X[6278] - 4 X[6289] = 3 X[485] - 2 X[6289]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6279) lies on these lines: {5, 6}, {376, 5861}, {488, 1271}, {641, 3525}, {1161, 1657}, {1991, 6222}, {3146, 5871}, {3839, 6202}

X(6279) = reflection of X(6278) in X(485)
X(6279) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6,5875,6281)

X(6280) = OUTER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((2*S_ω+S)*S_A^2+(5*S^2-S*S_ω-2*S_ω^2)*S_A+S^2*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 5 a^6-8 a^4 b^2+5 a^2 b^4-2 b^6-8 a^4 c^2-2 a^2 b^2 c^2+2 b^4 c^2+5 a^2 c^4+2 b^2 c^4-2 c^6+2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S
X(6280) = 4 X[5] - 5 X[486] = 5 X[487] - 7 X[3523] = 8 X[5] - 5 X[6281] = 6 X[5] - 5 X[6290] = 3 X[6281] - 4 X[6290] = 3 X[486] - 2 X[6290]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6280) lies on these lines: {5, 6}, {376, 5860}, {487, 1270}, {642, 3525}, {1160, 1657}, {3146, 5870}, {3839, 6201}

X(6280) = reflection of X(6281) in X(486)
X(6280) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6,5874,6278)

X(6281) = INNER-GREBE-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S-2*S_ω)*S_A^2-(S+S_ω)*(3*S-2*S_ω)*S_A+S^2*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^6-4 a^4 b^2+3 a^2 b^4-2 b^6-4 a^4 c^2+2 a^2 b^2 c^2+2 b^4 c^2+3 a^2 c^4+2 b^2 c^4-2 c^6-2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) S
X(6281) = 4 X[5] - 3 X[486], X[20] - 3 X[487], 5 X[631] - 6 X[642], X[5] - 3 X[6280], X[6280] - 4 X[6290], 2 X[5] - 3 X[6290]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6281) lies on these lines: {4, 5861}, {5, 6}, {20, 487}, {382, 1161}, {631, 642}, {2041, 6271}, {2042, 6270}, {3641, 5881}, {3832, 6202}, {5067, 6119}

X(6281) = reflection of X(i) in X(j) for these (i,j): (486,6290), (6280,486)
X(6281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5875,6279), (5875,6215,6)

X(6282) = HEXYL-TRIANGLE-ORTHOLOGIC CENTER OF MIXTILINEAR TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^6-2*(b+c)*a^5-(c^2+b^2-10*b*c)*a^4+4*(b+c)*(b^2-b*c+c^2)*a^3+(-8*b^3*c+2*b^2*c^2-b^4-c^4-8*b*c^3)*a^2-2*(b+c)*(b^2+c^2)*(b-c)^2*a+(b+c)^2*(b-c)^4
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+10 a^4 b c-8 a^2 b^3 c+2 a b^4 c-2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2+4 a^3 c^3-8 a^2 b c^3+4 b^3 c^3-a^2 c^4+2 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6)
X(6282) = 3 X[165] - X[2093] = 3 X[57] - 2 X[2095] = 3 X[3] - X[2095] = 3 X[376] - X[2096] = 3 X[165] - 2 X[3359]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6282) lies on these lines: {1, 3}, {4, 936}, {8, 6245}, {9, 1012}, {20, 78}, {28, 1753}, {30, 1750}, {72, 84}, {101, 610}, {102, 1292}, {142, 5603}, {200, 515}, {284, 4221}, {376, 527}, {377, 5715}, {411, 4855}, {443, 946}, {516, 997}, {519, 5768}, {971, 3940}, {1308, 2745}, {1709, 5692}, {2057, 3436}, {2716, 2742}, {2951, 6326}, {3149, 5438}, {3358, 5223}, {3577, 3753}, {3654, 5771}, {3811, 4297}, {3820, 5587}, {3870, 5731}, {3872, 5744}, {4292, 5758}, {4882, 5787}, {5657, 5745}

X(6282) = midpoint of X(20) and X(329)
X(6282) = reflection of X(i) in X(j) for these (i,j): (1750,5720), (2093,3359), (40,6244), (4, 3452), (57,3)
X(6282) = anticomplement of X(7682)
X(6282) = X(25)-of-hexyl-triangle
X(6282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3587,165), (20,78,1490), (40,2077,165), (40,3576,3428), (165,2093,3359), (165,5538,1)
X(6282) = X(78)-gimel conjugate of X(165)

X(6283) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS CENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (-(b^2-b*c+c^2)*a^2+(c^4+b^4+b*c^3+b^3*c+4*b*S*c))*a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a-b-c) (a+b+c) (b^2-b c+c^2)-4 a^2 b c S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6283) lies on these lines: {1, 256}, {33, 6291}, {55, 1151}, {172, 5414}, {1335, 2330}, {6198, 6239}

X(6283) = reflection of X(6252) in X(1151)

X(6284) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF MACBEATH CENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(2*a^4-(b+c)^2*a^2-(b^2-c^2)^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^4-a^2 b^2-b^4-2 a^2 b c-a^2 c^2+2 b^2 c^2-c^4
X(6284) = 2 X[1] - 3 X[3058] = 3 X[354] - 2 X[4292] = 4 X[1] - 3 X[5434]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6284) lies on these lines: {1, 30}, {2, 5217}, {3, 11}, {4, 12}, {5, 35}, {8, 190}, {10, 3683}, {20, 56}, {21, 2886}, {31, 1834}, {33, 3575}, {34, 1885}, {36, 496}, {37, 1839}, {40, 1728}, {46, 5722}, {52, 517}, {65, 516}, {72, 1898}, {80, 3467}, {100, 1329}, {140, 5010}, {145, 529}, {149, 2975}, {197, 4186}, {201, 2310}, {215, 1614}, {221, 5895}, {226, 4314}, {227, 1877}, {229, 5196}, {329, 3189}, {354, 4292}, {355, 5119}, {376, 3086}, {377, 1001}, {381, 498}, {382, 1478}, {388, 390}, {404, 3816}, {405, 3925}, {411, 5172}, {427, 5310}, {428, 612}, {440, 2218}, {442, 5248}, {443, 4423}, {452, 2550}, {495, 3585}, {515, 3057}, {519, 3962}, {535, 3244}, {601, 5348}, {936, 4679}, {938, 3474}, {942, 1770}, {944, 1317}, {946, 2646}, {952, 5693}, {958, 3434}, {962, 2099}, {964, 4026}, {976, 4415}, {999, 1657}, {1043, 4388}, {1058, 3304}, {1104, 3914}, {1155, 1210}, {1250, 5321}, {1319, 4297}, {1376, 2478}, {1388, 5731}, {1399, 1936}, {1456, 5930}, {1467, 2951}, {1482, 5841}, {1486, 4185}, {1503, 3056}, {1621, 2475}, {1656, 5326}, {1697, 5252}, {1699, 3601}, {1709, 5787}, {1737, 3579}, {1842, 3198}, {1884, 3185}, {1914, 5254}, {2066, 3070}, {2330, 5480}, {2361, 3073}, {2654, 4300}, {2777, 3028}, {2794, 3027}, {3035, 4193}, {3052, 5230}, {3071, 5414}, {3091, 5218}, {3219, 5178}, {3338, 4333}, {3436, 3913}, {3485, 4313}, {3488, 4295}, {3522, 5274}, {3543, 5229}, {3584, 3845}, {3600, 5059}, {3612, 5886}, {3665, 4872}, {3695, 4680}, {3703, 5015}, {3704, 5016}, {3826, 5047}, {3832, 5281}, {3871, 5080}, {3932, 5300}, {4030, 4385}, {4046, 5814}, {4189, 4999}, {4251, 5134}, {4305, 5603}, {4316, 5563}, {4413, 5084}, {4421, 5552}, {4689, 5530}, {4848, 5183}, {5048, 5882}, {5250, 5794}, {5330, 6224}, {5687, 6154}, {6198, 6240}

X(6284) - reflection of X(i) in X(j) for these (i,j): (1770,942), (5434,3058), (6253,4), (65,950)
X(6284) = crosspoint of X(i) and X(j) for these (i,j): (7, 1751)
X(6284) = crosssum of X(55) and X(579)
X(6284) = orthocenter of Mandart-incircle triangle
X(6284) = homothetic center of intangents triangle and reflection of extangents triangle in X(4)
X(6284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,79,6147), (1,1836,3649), (3,11,5433), (3,1479,11), (4,55,12), (4,4294,55), (5,35,5432), (20,497,56), (35,3583,5), (36,4324,550), (36,4857,496), (40,3586,1837), (100,5046,1329), (149,2975,3813), (376,3086,5204), (381,498,3614), (382,3295,1478), (388,390,3303), (390,3146,388), (495,3627,3585), (496,550,36), (938,3474,5221), (944,2098,1317), (946,4304,2646), (962,3486,2099), (999,1657,4299), (1058,3529,4293), (1058,4293,3304), (1478,4309,3295), (1479,4302,3), (1697,5691,5252), (3086,5204,5298), (3583,4330,35), (3585,3746,495), (3614,4995,498), (4324,4857,36), (4848,5493,5183)

X(6285) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2*a*S_A^2-a*(-S_ω+12*R^2)*S_A-S*(-S*a-2*S_ω*R+8*R^3)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8-a^6 b c+a^4 b^3 c+a^2 b^5 c-b^7 c+a^6 c^2+4 a^4 b^2 c^2-3 a^2 b^4 c^2-2 b^6 c^2+a^4 b c^3-2 a^2 b^3 c^3+b^5 c^3-3 a^4 c^4-3 a^2 b^2 c^4+6 b^4 c^4+a^2 b c^5+b^3 c^5+3 a^2 c^6-2 b^2 c^6-b c^7-c^8)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6285) lies on these lines: {1, 6000}, {11, 6247}, {12, 2883}, {30, 6238}, {33, 185}, {34, 3270}, {36, 3357}, {55, 1498}, {56, 64}, {65, 3332}, {84, 1364}, {154, 5217}, {221, 2293}, {388, 6225}, {944, 3057}, {1040, 5907}, {1478, 5878}, {1503, 3056}, {1854, 2099}, {3085, 5656}, {6198, 6241}

X(6285) = reflection of X(6254) in X(1498)
X(6285) = crosspoint of X(i) and X(j) for these (i,j): (84, 3362)
X(6285) = crosssum of X(40) and X(1745)
X(6285) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (64,2192,56)

X(6286) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*S_A^2-2*a*(-S_ω+3*R^2)*S_A-S*(S*a-4*S_ω*R+10*R^3)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8-a^6 b c+a^4 b^3 c+a^2 b^5 c-b^7 c+a^6 c^2-2 a^4 b^2 c^2+b^6 c^2+a^4 b c^3+a^2 b^3 c^3+b^5 c^3-3 a^4 c^4+a^2 b c^5+b^3 c^5+3 a^2 c^6+b^2 c^6-b c^7-c^8)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6286) lies on these lines: {1, 1154}, {33, 6152}, {35, 54}, {55, 195}, {539, 6238}, {1479, 2888}, {2293, 3746}, {3056, 5965}, {3519, 4857}, {3583, 6288}, {6198, 6242}

X(6286) = reflection of X(6255) in X(195)

X(6287) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((-5*S_ω^2+S^2)*S_A^2-S_ω*(-5*S_ω^2+3*S^2)*S_A-6*S^2*S_ω^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8+a^4 b^4-a^2 b^6-b^8+3 a^4 b^2 c^2+4 a^2 b^4 c^2+a^4 c^4+4 a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-c^8
X(6287) = 3 X[381] - 2 X[6249]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6287) lies on these lines: {3, 2916}, {4, 2896}, {5, 83}, {316, 546}, {381, 754}, {732, 1352}, {2080, 6308}, {2782, 3399}, {3851, 5475}, {5613, 6295}, {5617, 6297}, {6214, 6274}, {6215, 6275}, {6248, 6321}, {6289, 6312}, {6290, 6313}

X(6287) = midpoint of X(4) and X(2896)
X(6287) = reflection of X(i) in X(j) for these (i,k): (3,6292), (83,5)
X(6287) = X(83)-of-Johnson-triangle

X(6288) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF REFLECTION TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*S_ω-7*R^2)*S_A^2+(-3*S_ω^2+7*R^2*S_ω+S^2)*S_A-S^2*(8*R^2-3*S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^10-2 a^8 b^2+a^6 b^4-a^4 b^6+2 a^2 b^8-b^10-2 a^8 c^2+3 a^6 b^2 c^2-2 a^4 b^4 c^2-2 a^2 b^6 c^2+3 b^8 c^2+a^6 c^4-2 a^4 b^2 c^4-2 b^6 c^4-a^4 c^6-2 a^2 b^2 c^6-2 b^4 c^6+2 a^2 c^8+3 b^2 c^8-c^10
X(6288) = X[195] - 3 X[381] = 2 X[4] + X[3519] = 3 X[381] - 2 X[3574] = 2 X[5876] + X[6242]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let (Ha) be the hyperbola having diameter X(3)X(4), passing through A, with an asymptote parallel to the internal bisector of angle A. Let A′ be the reflecttion of A in X(5), and let Ta be the tangent to (Ha) at A″. Define Tb and Tc cyclically. The lines Ta, Tb, Tc concur in X(6288). See X(6288). (Angel Montesdeoca, December 9, 2019)

X(6288) lies on these lines: {3, 161}, {4, 93}, {5, 49}, {68, 568}, {155, 195}, {1351, 3818}, {1493, 3091}, {1511, 6143}, {3521, 5663}, {3575, 3581}, {3583, 6286}, {6214, 6276}, {6215, 6277}

X(6288) = midpoint of X(4) and X(2888)
X(6288) = reflection of X(i) in X(j) for these (i,k): (195,3574), (3519,2888), (3,1209), (54,5)
X(6288) = X(54)-of-X(4)-Brocard-triangle
X(6288) = X(54)-of-Johnson-triangle
X(6288) = X(79)-of-Ehrmann-vertex-triangle if ABC is acute
X(6288) = homothetic center of 2nd Euler triangle and cross-triangle of Ehrmann side-and Ehrmann vertex-triangles
X(6288) = perspector of Ehrmann vertex-triangle and Johnson triangle
X(6288) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3410,5876), (195,381,3574)

X(6289) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S+S_ω)*S_A^2+(-S_ω^2-S*S_ω+S^2)*S_A+S^2*(2*S+S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-2 a^4 b^2 c^2-3 a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4-6 b^4 c^4+3 a^2 c^6+4 b^2 c^6-c^8-2 (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6) S
X(6289) = 3 X[381] - 2 X[6250] = 2 X[5] + X[6278] = 6 X[5] - X[6279] = 3 X[485] - X[6279] = 3 X[6278] + X[6279]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6289) lies on these lines: {2, 6222}, {3, 639}, {4, 488}, {5, 6}, {114, 371}, {325, 637}, {381, 591}, {1591, 1899}, {1656, 6118}, {3095, 6311}, {3102, 6248}, {5613, 6303}, {5617, 6302}, {6033, 6231}, {6287, 6312}

X(6289) = midpoint of X(i) and X(j) for these {i,k}: {485,6278}, {4,488}
X(6289) = reflection of X(i) in X(j) for these (i,k): (3,641), (485,5)
X(6289) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1352,6290), (5,6214,1352)

X(6290) = JOHNSON-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S-S_ω)*S_A^2+(S_ω^2-S*S_ω-S^2)*S_A+S^2*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-2 a^4 b^2 c^2-3 a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4-6 b^4 c^4+3 a^2 c^6+4 b^2 c^6-c^8+2 (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6) S
X(6290) = 3 X[381] - 2 X[6251] = 6 X[5] - X[6280] = 3 X[486] - X[6280] = 2 X[5] + X[6281] = X[6280] + 3 X[6281]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6290) lies on these lines: {3, 640}, {4, 487}, {5, 6}, {114, 372}, {325, 638}, {381, 1991}, {1592, 1899}, {1656, 6119}, {3095, 6314}, {3103, 6248}, {5613, 6300}, {5617, 6301}, {6033, 6230}, {6287, 6313}

X(6290) = midpoint of X(i) and X(j) for these {i,k}: {486,6281}, {4,487}
X(6290) = reflection of X(i) in X(j) for these (i,k): (3,642), (486,5)
X(6290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1352,6289), (5,6215,1352)

X(6291) = ORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS CENTRAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(2*S_A^2+S*S_A+S*(S_ω+2*S))/S_A
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (2 b^2 c^2+b^2 S+c^2 S)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6291) lies on these lines: {4, 69}, {19, 6252}, {25, 1151}, {33, 6283}, {51, 3071}, {1968, 5413}, {1974, 3092}

X(6291) = midpoint of X(4) and X(6239)
X(6291) = crosssum of X(3) and X(488)
X(6291) = X(176)-of-orthic-triangle if ABC is acute

X(6292) = MEDIAL-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b^2+c^2)*(b^2+c^2+2*a^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b^2+c^2) (2 a^2+b^2+c^2)
X(6292) = 3 X[2] + X[2896] = 3 X[6296] - Sqrt[3] X[6313] = 3 X[6297] + Sqrt[3] X[6313] = 3 X[6297] - Sqrt[3] X[6317] = 3 X[6296] + Sqrt[3] X[6317]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6292) lies on these lines: {2, 32}, {3, 2916}, {5, 5188}, {39, 141}, {76, 4045}, {114, 140}, {115, 3934}, {187, 5031}, {211, 3917}, {385, 5368}, {524, 5041}, {574, 3619}, {620, 5152}, {1125, 1279}, {1500, 3666}, {1649, 3005}, {3589, 5007}, {5590, 6274}, {5591, 6275}

X(6292) = midpoint of X(i) and X(j) for these {i,j}: (3,6287), (6296, 6297), (6313,6317), (83,2896)
X(6292) = reflection of X(6249) in X(5)
X(6292) = isotomic conjugate of isogonal conjugate of X(11205)
X(6292) = anticomplement of X(6704)
X(6292) = X(i)-Ceva conjugate of X(j) for these (i,j): (2, 3589), (99, 826)
X(6292) = crosspoint of X(i) and X(j) for these (i,j): (2, 141)
X(6292) = crosssum of X(6) and X(251)
X(6292) = complement of X(83)
X(6292) = polar conjugate of isogonal conjugate of X(22078)
X(6292) = X(i)-complementary conjugate of X(j) for these (i,j): (1, 3934), (6, 1215), (19, 5943), (31, 3589), (38, 141), (39, 10), (141, 2887), (163, 826), (560, 1194), (798, 3124), (799, 688), (810, 339), (1401, 142), (1415, 4142), (1634, 4369), (1843, 226), (1923, 39), (1930, 626), (1964, 2), (1967, 732), (1973, 5305), (2084, 115), (2530, 116), (3051, 37), (3404, 511), (3688, 3452), (3954, 3454), (4020, 3), (4553, 3835)
X(6292) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,626,1506), (2,2896,83), (2,3096,626), (83,6308,32)
X(6292) = X(i)-isoconjugate of X(j) for these (i,j): (82, 3108)
X(6292) = trilinear pole, wrt medial triangle, of de Longchamps line

X(6293) = REFLECTION-TRIANGLE-ORTHOLOGIC CENTER OF MIDHEIGHT TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*((-S_ω+2*R^2)*S_A^2-(-S_ω+2*R^2)*(-S_ω+8*R^2)*S_A-S^2*R^2)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^12 b^2-4 a^10 b^4+5 a^8 b^6-5 a^4 b^10+4 a^2 b^12-b^14+a^12 c^2-a^10 b^2 c^2-a^8 b^4 c^2-2 a^6 b^6 c^2+7 a^4 b^8 c^2-5 a^2 b^10 c^2+b^12 c^2-4 a^10 c^4-a^8 b^2 c^4+4 a^6 b^4 c^4-2 a^4 b^6 c^4+3 b^10 c^4+5 a^8 c^6-2 a^6 b^2 c^6-2 a^4 b^4 c^6+2 a^2 b^6 c^6-3 b^8 c^6+7 a^4 b^2 c^8-3 b^6 c^8-5 a^4 c^10-5 a^2 b^2 c^10+3 b^4 c^10+4 a^2 c^12+b^2 c^12-c^14)
X(6293) = 4 X[389] - 3 X[1853] = 3 X[154] - 2 X[5562] = 4 X[973] - 3 X[6145] = 3 X[5890] - 2 X[6247]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6293) lies on these lines: {4, 973}, {6, 64}, {20, 2781}, {52, 382}, {68, 5663}, {154, 5562}, {161, 1498}, {343, 2883}, {389, 1853}, {399, 2917}, {569, 3357}, {1503, 5889}, {5890, 6247}

X(6293) = reflection of X(64) in X(185)
X(6293) = X(12625)-of-orthic-triangle if ABC is acute

X(6294) = INNER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(-S+S_ω*sqrt(3)))*S_A^2+sqrt(3)*(2*S^2+S_ω*(sqrt(3)*SS_ω))*S_ω)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^2 b^4+a^4 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4-2 Sqrt[3] a^2 (b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6294) lies on these lines: {2, 39}, {3, 6295}, {524, 3105}, {698, 3106}, {732, 3107}, {3095, 5617}, {3104, 5463}, {3643, 6297}

X(6294) = anticomplement of X(33483)

X(6295) = 1st NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(S-S_ω*sqrt(3)))*S_A^2+sqrt(3)*(-S+S_ω*sqrt(3))^2*S_A-(2*S^2+S_ω*(-S_ω+sqrt(3)*S))*S)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 3 a^8-8 a^6 b^2+7 a^4 b^4-2 a^2 b^6-8 a^6 c^2+6 a^2 b^4 c^2-2 b^6 c^2+7 a^4 c^4+6 a^2 b^2 c^4+4 b^4 c^4-2 a^2 c^6-2 b^2 c^6+2 Sqrt[3] a^2 (a^4+a^2 b^2-2 b^4+a^2 c^2-2 b^2 c^2-2 c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6295) lies on these lines: {2, 14}, {3, 6294}, {542, 6298}, {3098, 5969}, {5613, 6287}

X(6295) = reflection of X(6299) in X(619)
X(6295) = circumtangential-isogonal conjugate of X(16)
X(6295) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (6303,6307,3642)

X(6296) = OUTER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3*(S_ω*sqrt(3)-S)*S_A^2-(6*(S_ω*sqrt(3)-S))*S_ω*S_A+sqrt(3)*(S_ω^2+2*S^2+3*S*S_ω*sqrt(3))*S_ω)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6+2 a^4 b^2-a^2 b^4-b^6+2 a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4-c^6+2 Sqrt[3] (b^2+c^2) (2 a^2+b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6296) lies on these lines: {2, 32}, {3, 6298}, {732, 3107}, {5613, 6287}

X(6296) = reflection of X(6297) in X(6292)
X(6296) = anticomplement of X(33484)

X(6297) = INNER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*S+3*S_ω*sqrt(3))*S_A^2-(6*(S+S_ω*sqrt(3)))*S_ω*S_A-sqrt(3)*(- S_ω^2-2*S^2+3*S*S_ω*sqrt(3))*S_ω)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6+2 a^4 b^2-a^2 b^4-b^6+2 a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4-c^6-2 Sqrt[3] (b^2+c^2) (2 a^2+b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6297) lies on these lines: {2, 32}, {3, 6299}, {732, 3106}, {3643, 6294}, {5617, 6287}

X(6297) = reflection of X(6296) in X(6292)
X(6297) = anticomplement of X(33485)

X(6298) = 2nd-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(sqrt(3)*S_ω-S))*S_A^2-sqrt(3)*(3*S_ω^2-S^2)*S_A+(2*S^2+S_ω^2+3*sqrt(3)*S*S_ω)*S)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-5 a^6 b^2+8 a^4 b^4-5 a^2 b^6+b^8-5 a^6 c^2+10 a^4 b^2 c^2+a^2 b^4 c^2-2 b^6 c^2+8 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-5 a^2 c^6-2 b^2 c^6+c^8+2 Sqrt[3] (a^6-b^6+2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6298) lies on these lines: {2, 13}, {3, 6296}, {542, 6295}, {3095, 5617}, {3818, 6299}

X(6299) = 2nd-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*S+3*S_ω*sqrt(3))*S_A^2-sqrt(3)*(-S+S_ω*sqrt(3))*(S+S_ω*sqrt(3))*S_A+(-S_ω^2-2*S^2+3*S*S_ω*sqrt(3))*S)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-5 a^6 b^2+8 a^4 b^4-5 a^2 b^6+b^8-5 a^6 c^2+10 a^4 b^2 c^2+a^2 b^4 c^2-2 b^6 c^2+8 a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-5 a^2 c^6-2 b^2 c^6+c^8-2 Sqrt[3] (a^6-b^6+2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6299) lies on these lines: {2, 14}, {3, 6297}, {3095, 5613}, {3818, 6298}

X(6299) = reflection of X(6295) in X(619)

X(6300) = OUTER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(1+sqrt(3)))*S_A^2+(3*(1+sqrt(3)))*(-S_ω+S)*S_A+S*(2*SS_ω)*(-3+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6+Sqrt[3] (-a^4 b^2+2 a^2 b^4-b^6-a^4 c^2+4 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6)+2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+Sqrt[3] (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4)) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6300) lies on these lines: {2, 371}, {3, 6302}, {2044, 6251}, {3564, 6305}, {3642, 6304}, {5613, 6290}

X(6300) = reflection of X(6301) in X(642)
X(6300) = anticomplement of X(33445)
X(6300): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,486,6301)

X(6301) = INNER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(1-sqrt(3)))*S_A^2-(3*(sqrt(3)-1))*(-S_ω+S)*S_A-S*(2*SS_ω)*(3+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6-Sqrt[3] (-a^4 b^2+2 a^2 b^4-b^6-a^4 c^2+4 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6)+2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-Sqrt[3] (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4)) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6301) lies on these lines: {2, 371}, {3, 6303}, {2043, 6251}, {3564, 6304}, {3643, 6305}, {5617, 6290}

X(6301) = reflection of X(6300) in X(642)
X(6301) = anticomplement of X(33447)
X(6301) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,486,6300)

X(6302) = INNER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(1+sqrt(3)))*S_A^2-(3+sqrt(3))*(sqrt(3)*S_ω+S)*S_A+S*(2*SS_ω)*(sqrt(3)-1))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6-Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (b^2+c^2)-2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4+Sqrt[3] (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)) S
X(6302) = 6 X[115] - Sqrt[3] (3 + Sqrt[3]) X[6303]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6302) lies on these lines: {2, 13}, {3, 6300}, {115, 615}, {542, 6307}, {590, 5472}, {2044, 5478}, {5617, 6289}

X(6302) = reflection of X(6306) in X(618)
X(6302) = complement of X(33441)
X(6302) = inverse of X(33440) in the inner-Napoleon circle
X(6302) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13,6306), (616,5463,6306)

X(6303) = INNER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3*(1-sqrt(3))*S_A^2-(-3+sqrt(3))*(S_ω*sqrt(3)-S)*S_A-S*(2*SS_ω)*(1+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6+Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (b^2+c^2)-2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4-Sqrt[3] (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)) S
X(6303) = 6 X[115] - Sqrt[3] (-3 + Sqrt[3]) X[6302]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6303) lies on these lines: {2, 14}, {3, 6301}, {115, 615}, {542, 6306}, {590, 5471}, {2043, 5479}, {5613, 6289}

X(6303) = reflection of X(6307) in X(619)
X(6303) = complement of X(33443)
X(6303) = outer-Napoleon-circle-inverse of X(33442)
X(6303) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,14,6307), (617,5464,6307), (3642,6295,6307)

X(6304) = OUTER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(1-sqrt(3)))*S_A^2+(3*(sqrt(3)-1))*(S_ω+S)*S_AS*(2*S+S_ω)*(3+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6-Sqrt[3] (-a^4 b^2+2 a^2 b^4-b^6-a^4 c^2+4 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6)-2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-Sqrt[3] (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4)) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6304) lies on these lines: {2, 372}, {3, 6306}, {2043, 6250}, {3564, 6301}, {3642, 6300}, {5613, 6289}

X(6304) = reflection of X(6305) in X(641)
X(6304) = anticomplement of X(33444)
X(6304) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,485,6305)

X(6305) = INNER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3*(1+sqrt(3))*S_A^2-(3*(1+sqrt(3)))*(S_ω+S)*S_A+S*(2*S+S_ω)*(-3+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6+Sqrt[3] (-a^4 b^2+2 a^2 b^4-b^6-a^4 c^2+4 a^2 b^2 c^2+b^4 c^2+2 a^2 c^4+b^2 c^4-c^6)-2 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+Sqrt[3] (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4)) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6305) lies on these lines: {2, 372}, {3, 6307}, {2044, 6250}, {3564, 6300}, {3643, 6301}, {5617, 6289}

X(6305) = reflection of X(6304) in X(641)
X(6305) = anticomplement of X(33446)
X(6305) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2,485,6304)

X(6306) = OUTER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((3*(sqrt(3)-1))*S_A^2+(-3+sqrt(3))*(S+sqrt(3)*S_ω)*S_A+S*(2*S+S_ω)*(1+sqrt(3)))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6+Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (b^2+c^2)+2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4-Sqrt[3] (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)) S
X(6306) = 6 X[115] - Sqrt[3] (-3 + Sqrt[3]) X[6307]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6306) lies on these lines: {2, 13}, {3, 6304}, {115, 590}, {542, 6303}, {615, 5472}, {2043, 5478}, {5617, 6290}

X(6306) = reflection of X(6302) in X(618)
X(6306) = complement of X(33440)
X(6306) = inverse of X(33441) in the inner-Napoleon circle
X(6306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,13,6302), (616,5463,6302)

X(6307) = OUTER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF INNER NAPOLEON TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3*(1+sqrt(3))*S_A^2-(3+sqrt(3))*(S_ω*sqrt(3)-S)*S_A+S*(2*S+S_ω)*(sqrt(3)-1))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6-Sqrt[3] (a-b-c) (a+b-c) (a-b+c) (a+b+c) (b^2+c^2)+2 (4 a^4-5 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4+Sqrt[3] (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)) S
X(6307) = 6 X[115] - Sqrt[3] (3 + Sqrt[3]) X[6306]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6307) lies on these lines: {2, 14}, {3, 6305}, {115, 590}, {542, 6302}, {615, 5471}, {2044, 5479}, {5613, 6290}

X(6307) = reflection of X(6303) in X(619)
X(6307) = complement of X(33442)
X(6307) = inverse of X(33443) in the outer-Napoleon circle
X(6307) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,14,6303), (617,5464,6303), (3642,6295,6303)

X(6308) = 1st-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (S_A*(S^2+3*S_ω^2)*(S_A-2*S_ω)-S_ω^2*(S^2-S_ω^2))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8+3 a^6 b^2-a^4 b^4-a^2 b^6+3 a^6 c^2-a^4 b^2 c^2-5 a^2 b^4 c^2-b^6 c^2-a^4 c^4-5 a^2 b^2 c^4-2 b^4 c^4-a^2 c^6-b^2 c^6

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6308) lies on these lines: {2, 32}, {3, 732}, {98, 5188}, {2080, 6287}

X(6308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,6292,83), (6313,6317,2896)

X(6309) = 2nd-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S^2+3*S_ω^2)*S_A^2+S_ω^2*(S^2-S_ω^2))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^4 b^4+3 a^4 b^2 c^2-b^6 c^2+2 a^4 c^4-2 b^4 c^4-b^2 c^6

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6309) lies on these lines: {2, 39}, {3, 732}, {698, 3095}, {2021, 6337}, {3094, 3933}, {4027, 6179}

X(6309) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (194,3926,39), (6314,6318,194)

X(6310) = 1st-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF SYMMEDIAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(4*S^2*S_A^2+(S^2*S_ω-16*S^2*R^2+S_ω^3)*S_A+S^2*(3*S^2-S_ω^2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (a^4 b^4-a^2 b^6+b^6 c^2+a^4 c^4-4 b^4 c^4-a^2 c^6+b^2 c^6)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6310) lies on these lines: {3, 3229}, {4, 69}, {695, 1196}

X(6310) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (76,5167,3491)

X(6311) = 2nd-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*((S^2-S_A^2)*(S_ω-R^2)+2*S^2*S_A)/S_A
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6 b^2-5 a^4 b^4+4 a^2 b^6-b^8+2 a^6 c^2-8 a^4 b^2 c^2+2 b^6 c^2-5 a^4 c^4-2 b^4 c^4+4 a^2 c^6+2 b^2 c^6-c^8-2 (a^6-b^6+2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6311) lies on these lines: {2, 372}, {3, 6313}, {3095, 6289}, {3564, 6316}, {3818, 6315}

X(6311) = reflection of X(6312) in X(641)

X(6312) = 1st-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF OUTER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (S_A*(S+3*S_ω)*(S-S_A+S_ω)+S*(2*S+S_ω)*(S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^8-5 a^6 b^2+4 a^4 b^4-a^2 b^6-5 a^6 c^2-2 a^4 b^2 c^2+5 a^2 b^4 c^2-2 b^6 c^2+4 a^4 c^4+5 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6-2 a^2 (a^4+a^2 b^2-2 b^4+a^2 c^2-2 b^2 c^2-2 c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6312) lies on these lines: {2, 372}, {3, 6314}, {98, 6320}, {371, 6179}, {1350, 6222}, {3098, 6316}, {3564, 6315}, {6287, 6289}

X(6312) = reflection of X(6311) in X(641)
X(6312) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (485,488,6228)

X(6313) = OUTER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (S_A*(S_A-2*S_ω)*(-3*S_ω+S)-S_ω*(S+S_ω)*(2*S+S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6+2 a^4 b^2-a^2 b^4-b^6+2 a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4-c^6+2 (b^2+c^2) (2 a^2+b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6313) lies on these lines: {2, 32}, {3, 6311}, {732, 3103}, {6287, 6290}

X(6313) = reflection of X(6317) in X(6292)
X(6313) = complement of X(33454)
X(6313) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2896,6308,6317)

X(6314) = OUTER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S+3*S_ω)*S_A^2+S_ω*(2*S+S_ω)*(S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^2 b^4+a^4 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4+2 a^2 (b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6314) lies on these lines: {2, 39}, {3, 6312}, {638, 6273}, {698, 3103}, {732, 3102}, {3095, 6290}

X(6314) = reflection of X(6318) in X(39)
X(6314) = complement of X(33452)
X(6314) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (194,6309,6318)

X(6315) = 2nd-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (S_A*(S+3*S_ω)*(S+S_A-S_ω)-S*(2*S-S_ω)*(S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6 b^2-5 a^4 b^4+4 a^2 b^6-b^8+2 a^6 c^2-8 a^4 b^2 c^2+2 b^6 c^2-5 a^4 c^4-2 b^4 c^4+4 a^2 c^6+2 b^2 c^6-c^8+2 (a^6-b^6+2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6315) lies on these lines: {2, 371}, {3, 6317}, {3095, 6290}, {3564, 6312}, {3818, 6311}

X(6315) = reflection of X(6316) in X(642)

X(6316) = 1st-NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF INNER VECTEN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a*(S_A*(S-3*S_ω)*(S+S_A-S_ω)-S*(S_ω+S)*(2*S-S_ω))/(-S_A+S_ω)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^8-5 a^6 b^2+4 a^4 b^4-a^2 b^6-5 a^6 c^2-2 a^4 b^2 c^2+5 a^2 b^4 c^2-2 b^6 c^2+4 a^4 c^4+5 a^2 b^2 c^4+4 b^4 c^4-a^2 c^6-2 b^2 c^6+2 a^2 (a^4+a^2 b^2-2 b^4+a^2 c^2-2 b^2 c^2-2 c^4) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6316) lies on these lines: {2, 371}, {3, 6318}, {76, 6200}, {98, 6319}, {372, 6179}, {3098, 6312}, {3564, 6311}, {6287, 6290}

X(6316) = reflection of X(6315) in X(642)
X(6316) = {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (486,487,6229)

X(6317) = INNER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 2nd NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (S_A*(S_A-2*S_ω)*(S+3*S_ω)+S_ω*(2*S-S_ω)*(S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2 a^6+2 a^4 b^2-a^2 b^4-b^6+2 a^4 c^2-b^4 c^2-a^2 c^4-b^2 c^4-c^6-2 (b^2+c^2) (2 a^2+b^2+c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6317) lies on these lines: {2, 32}, {3, 6315}, {732, 3102}, {6287, 6289}

X(6317) = reflection of X(6313) in X(6292)
X(6317) = complement of X(33455)
X(6317): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (2896,6308,6313)

X(6318) = INNER-VECTEN-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((S-3*S_ω)*S_A^2-S_ω*(S_ω+S)*(2*S-S_ω))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2+a^2 b^4+a^4 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4+(-2 a^2 b^2-2 a^2 c^2) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6318) lies on these lines: {2, 39}, {3, 6316}, {637, 6272}, {698, 3102}, {732, 3103}, {3095, 6289}

X(6318) = reflection of X(6314) in X(39)
X(6318) = complement of X(33453)
X(6318): {X(i),X(j)}-harmonic conjugate of X(k) for this (i,j,k): (194,6309,6314)

X(6319) = INNER-GREBE-TRIANGLE-PARALLELOGIC CENTER OF 1st BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((a^4+b^2*c^2)*(b^2+c^2-S)-(2*(b^4+c^4)-(b^2+c^2)*S)*a^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2-2 a^2 b^4+a^4 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-(a-b) (a+b) (a-c) (a+c) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6319) lies on these lines: {6, 99}, {98, 6316}, {114, 6202}, {115, 5591}, {148, 1271}, {543, 5861}, {1161, 2782}, {6215, 6321}

X(6319) = reflection of X(i) in X(j) for these (i,j): (6227,1161), (6320,99)

X(6320) = OUTER-GREBE-TRIANGLE-PARALLELOGIC CENTER OF 1st BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((a^4+b^2*c^2)*(b^2+c^2+S)-(2*(b^4+c^4)+(b^2+c^2)*S)*a^2)/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^4 b^2-2 a^2 b^4+a^4 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4+(a-b) (a+b) (a-c) (a+c) S

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6320) lies on these lines: {6, 99}, {98, 6312}, {114, 6201}, {115, 5590}, {148, 1270}, {543, 5860}, {1160, 2782}, {6214, 6321}

X(6320) = reflection of X(i) in X(j) for these (i,j): (6226,1160), (6319,99)

X(6321) = JOHNSON-TRIANGLE-PARALLELOGIC CENTER OF 1st BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((9*S^2-S_ω^2)*S_A^2-S_ω*(7*S^2-S_ω^2)*S_A+2*S^2*(4*S^2-S_ω^2))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^8-2 a^6 b^2+a^4 b^4+a^2 b^6-b^8-2 a^6 c^2+3 a^4 b^2 c^2-2 a^2 b^4 c^2+4 b^6 c^2+a^4 c^4-2 a^2 b^2 c^4-6 b^4 c^4+a^2 c^6+4 b^2 c^6-c^8
X(6321) = 3 X[4] - X[147] = X[147] + 3 X[148] = 2 X[114] - 3 X[381] = X[98] - 3 X[671] = 4 X[620] - 5 X[1656] = 2 X[2482] - 3 X[5055] = 3 X[5054] - 4 X[5461] = 3 X[5469] - X[5473] = 3 X[5470] - X[5474] = 3 X[5093] - 2 X[5477] = 3 X[3543] + X[5984] = 2 X[147] - 3 X[6033] = 2 X[148] + X[6033] = 2 X[182] - 3 X[6034] = 3 X[3] - 4 X[6036] = 3 X[115] - 2 X[6036]

See César Lozada, Perspective-Orthologic-Parallelogic.

Let Q be the quadrilateral ABCX(98). Taking the vertices 3 at a time yields four triangles whose orthocenters are the vertices of a cyclic quadrilateral whose circumcenter is X(6321). (Randy Hutson, January 29, 2015)

X(6321) lies on these lines: {3, 115}, {4, 147}, {5, 99}, {13, 5611}, {14, 5615}, {30, 98}, {114, 381}, {156, 3044}, {182, 6034}, {265, 690}, {382, 2794}, {542, 1351}, {591, 6231}, {620, 1656}, {1352, 5969}, {1478, 3027}, {1479, 3023}, {1569, 5475}, {1989, 5467}, {1991, 6230}, {2023, 2549}, {2482, 5055}, {3398, 5254}, {3455, 5899}, {3534, 6055}, {3543, 5984}, {3845, 6054}, {5054, 5461}, {5093, 5477}, {5469, 5473}, {5470, 5474}, {5478, 5613}, {5479, 5617}, {6214, 6320}, {6215, 6319}, {6248, 6287}

X(6321) = midpoint of X(4) and X(148)
X(6321) = reflection of X(i) in X(j) for these (i,j): (3534,6055), (3,115), (5613,5478), (5617,5479), (6033,4), (6054,3845), (99,5)
X(6321) = anticomplement of X(33813)
X(6321) = crosssum of X(2459) and X(2460)

X(6322) = 2nd-BROCARD-TRIANGLE-PARALLELOGIC CENTER OF CIRCUMMEDIAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (4*a^8-(8*b^4+8*c^4-b^2*c^2)*a^4+(2*(b^2+c^2))*a^2*(-4*b^2*c^2+a^4)-4*b^2*c^2*(b^4+c^4-b^2*c^2))/a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 4 a^8+2 a^6 b^2-8 a^4 b^4+2 a^6 c^2+a^4 b^2 c^2-8 a^2 b^4 c^2-4 b^6 c^2-8 a^4 c^4-8 a^2 b^2 c^4+4 b^4 c^4-4 b^2 c^6

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6322) lies on the Brocard circle and these lines: {6, 3849}, {182, 6232}, {385, 6031}

X(6322) = reflection of X(6232) in X(182)
X(6322) = Brocard-circle-antipode of X(6232)
X(6322) = X(6323)-of-1st-Brocard-triangle

X(6323) = CIRCUMMEDIAL-TRIANGLE-PARALLELOGIC CENTER OF 2nd BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(4*a^4-(b^2+c^2)*(a^2)-2*(b^4+c^4-b^2*c^2))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (2 a^4-2 a^2 b^2+2 b^4+a^2 c^2+b^2 c^2-4 c^4) (2 a^4+a^2 b^2-4 b^4-2 a^2 c^2+b^2 c^2+2 c^4)

See César Lozada, Perspective-Orthologic-Parallelogic.

Let A′B′C′ be the circumsymmedial triangle. Let L_A be the reflection of the Brocard axis in line B′C′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. The lines A′A″, B′B″, C′C″ concur in X(6323). Also, X(6323)-of-orthocentroidal-triangle = X(99)-of-4th-Brocard-triangle. (Randy Hutson, January 29, 2015)

X(6323) lies on the circumcircle and these lines: {3, 6233}, {99, 599}, {110, 353}, {542, 6236}, {690, 6325}, {691, 5104}, {1296, 3098}

X(6323) = reflection of X(6233) in X(3)
X(6323) = isogonal conjugate of X(3849)
X(6323) = crosssum of X(3849) and X(3849)
X(6323) = X(99)-of-circumsymmedial-triangle
X(6323) = X(1)-isoconjugate of X(3849)
X(6323) = Λ(Brocard axis of the 3rd pedal triangle of X(6))
X(6323) = 2nd-Parry-to-ABC similarity image of X(353)
X(6323) = X(6787)-of-4th-anti-Brocard-triangle
X(6323) = Schoute-circle-inverse of X(32694)

X(6324) = 4th-BROCARD-TRIANGLE-PARALLELOGIC CENTER OF CIRCUMSYMMEDIAL TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ((4*((b^2+c^2)^2-b^2*c^2))*a^6+6*b^2*c^2*(b^2+c^2)*a^4+(4*b^2*c^6+3*b^4*c^4+4*b^6*c^2-4*c^8-4*b^8)*a^2-(8*(b^4-c^4))*(b^2-c^2)*b^2*c^2)*a
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a^2 (4 a^6 b^4-4 a^2 b^8+4 a^6 b^2 c^2+6 a^4 b^4 c^2+4 a^2 b^6 c^2-8 b^8 c^2+4 a^6 c^4+6 a^4 b^2 c^4+3 a^2 b^4 c^4+8 b^6 c^4+4 a^2 b^2 c^6+8 b^4 c^6-4 a^2 c^8-8 b^2 c^8)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6324) lies on these lines: {353, 1495}, {381, 6235}

X(6324) = reflection of X(6235) in X(381)

X(6325) = CIRCUMSYMMEDIAL-TRIANGLE-PARALLELOGIC CENTER OF 4th BROCARD TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(a*(2*(b^2+c^2)*a^4-2*b^2*c^2*a^2-(b^2+c^2)*(2*c^4-3*b^2*c^2+2*b^4)))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (2 a^6-a^4 b^2-a^2 b^4+2 b^6+2 a^2 b^2 c^2-2 a^2 c^4-2 b^2 c^4) (2 a^6-2 a^2 b^4-a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2-a^2 c^4+2 c^6)

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6325) lies on the circumcircle and these lines: {3, 6236}, {99, 5987}, {110, 599}, {112, 5094}, {542, 6233}, {690, 6323}, {691, 3849}, {1296, 3534}

X(6325) = reflection of X(6236) in X(3)
X(6325) = isogonal conjugate of X(8705)
X(6325) = trilinear pole of the line X(6)X(3906)
X(6325) = Λ(Euler line of 3rd pedal triangle of X(2))
X(6325) = Λ(Euler line of 3rd antipedal triangle of X(6))
X(6325) = trilinear pole of line X(6)X(3906)
X(6325) = Ψ(X(6), X(3906))

X(6326) = HEXYL-TRIANGLE-PARALLELOGIC CENTER OF FUHRMANN TRIANGLE

Trilinears f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a*(b+c)+4*S_ω-20*R^2)*S^2-S_A*(b*(5*a-3*c)*S_B+c*(5*a-3*b)*S_C)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+3 a^4 b c-a^3 b^2 c-3 a^2 b^3 c+3 a b^4 c-a^4 c^2-a^3 b c^2+4 a^2 b^2 c^2-a b^3 c^2-b^4 c^2+4 a^3 c^3-3 a^2 b c^3-a b^2 c^3-a^2 c^4+3 a b c^4-b^2 c^4-2 a c^5+c^6)
X(6326) = 2 X[104] - 3 X[3576] = 4 X[214] - 3 X[3576] = 2 X[80] - 3 X[5587] = 4 X[119] - 3 X[5587] = X[80] - 3 X[5660] = 2 X[119] - 3 X[5660] = 2 X[1484] - 3 X[5886] = 2 X[5531] + X[6264] = X[6264] - 4 X[6265] = X[5531] + 2 X[6265]

See César Lozada, Perspective-Orthologic-Parallelogic.

X(6326) lies on the hexyl circle, the excentral-hexyl ellipse, and these lines:
{1, 5}, {3, 191}, {9, 48}, {30, 5538}, {35, 5887}, {36, 912}, {40, 78}, {63, 4996}, {72, 2949}, {84, 224}, {110, 1793}, {149, 946}, {153, 515}, {200, 1145}, {404, 5884}, {480, 3428}, {516, 5528}, {517, 3689}, {519, 1512}, {528, 1537}, {758, 5535}, {936, 3035}, {1045, 2783}, {1158, 4855}, {1320, 3577}, {1385, 5251}, {1490, 2829}, {1787, 3333}, {2077, 2932}, {2136, 2802}, {2646, 5777}, {2951, 6282}, {3553, 5153}, {3646, 3897}, {4305, 5811}, {5732, 5851}

Let A′B′C′ be the orthic triangle. Let L_A be the antiorthic axis of AB′C′, and define L_B and L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. Then the triangle A″B″C″ is inversely similar to ABC, with similitude center X(9), and X(6326) = X(40)-of-A″B″C″. (Randy Hutson, January 29, 2015)

X(6326) = midpoint of X(i) and X(j) for these {i,j}: {153,6224}, {1,5531}
X(6326) = reflection of X(i) in X(j) for these (i,j): (104, 214), (149,946), (1768,3), (1,6265), (2077,5440), (40,100), (5587,5660), (6264,1), (80,119)
X(6326) = X(i)-Ceva conjugate of X(j) for these (i,j): (153, 2950), (515, 40), (908, 9), (4511, 1), (5080, 191), (6224, 5541)
X(6326) = crossdifference of every two points on the line X(654)X(1769)
X(6326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5720,5587), (1,6127,1718), (3,5694,191), (78,6261,40), (80,119,5587), (80,5660,119), (104,214,3576), (5531,6265,6264)
X(6326) = excentral-isogonal conjugate of X(484)
X(6326) = X(188)-aleph conjugate of X(484)
X(6326) = X(21)-beth conjugate of X(1421)
X(6326) = antipode in excentral-hexyl ellipse of X(1768)
X(6326) = X(110)-of-hexyl triangle
X(6326) = hexyl-isogonal conjugate of X(513)
X(6326) = Fuhrmann-to-excentral similarity image of X(3)
X(6326) = inner-Garcia-to-ABC similarity image of X(4)
X(6326) = X(265)-of-excentral-triangle
X(6326) = intersection, other than vertices of hexyl triangle, of hexyl circle and excentral-hexyl ellipse

X(6327) = X(1)-ANTICOMPLEMENTARY CONJUGATE OF X(192)

Barycentrics b³ + c³ - a³ : c³ + a³ - b³ : a³ + b³ - c³
X(6327) = 3X(2) - 2X(31)
X(6327) = 3X(2) - 4X(2887)
X(6327) = X(31) - 2X(2887)
X(6327) = X(8) - 2X(4680)

Contributed by Peter Moses, January 15, 2015.

X(6327) lies on these lines: {2, 31}, {6, 4972}, {8, 79}, {38, 4655}, {42, 4660}, {55, 3936}, {63, 3006}, {65, 5016}, {69, 674}, {72, 5300}, {100, 4417}, {192, 744}, {194, 734}, {209, 2550}, {306, 516}, {312, 5057}, {315, 766}, {320, 3873}, {321, 1836}, {329, 2835}, {345, 4427}, {518, 5014}, {742, 4799}, {756, 4703}, {896, 4438}, {902, 3771}, {984, 4683}, {1150, 2886}, {1376, 5741}, {1759, 4153}, {2390, 3436}, {2979, 3888}, {3011, 4138}, {3058, 4966}, {3120, 4362}, {3187, 3914}, {3218, 3705}, {3454, 5264}, {3578, 4042}, {3782, 3891}, {3827, 4463}, {3868, 5015}, {3874, 4894}, {3883, 5249}, {3925, 5278}, {3966, 4359}, {3969, 5695}, {4001, 4847}, {4071, 5282}, {4425, 5311}, {4643, 4981}, {5051, 5711}

X(6327) = reflection of X(i) in X(j) for these (i,j): (8,4680), (31,2887), (3187,3914), (3891,3782)
X(6327) = anticomplement of X(31)
X(6327) = X(561)-Ceva conjugate of X(2)
X(6327) = X(3156)-cevapoint of X(193)
X(6327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31,2887,2), (320,4514,3873), (748,3836,2), (750,3847,2), (1836,3416,321), (2550,5739,4651), (3936,4450,55), (3966,5880,4359), (4388,4645,2), (4655,4865,38)
X(6327) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (1, 192), (3, 3164), (4, 193), (6, 194), (7, 145), (8, 144), (9, 3177), (10, 1654), (13, 3180), (14, 3181), (27, 3187), (37, 1655), (57, 3210), (69, 20), (75, 8), (76, 69), (83, 6), (85, 7), (86, 1), (87, 330), (92, 5905), (95, 3), (98, 385), (99, 523), (141, 2896), (183, 6194), (190, 514), (253, 3146), (261, 2975), (264, 4), (274, 75), (275, 1993), (276, 264), (279, 4452), (286, 3868), (287, 401), (290, 511), (298, 616), (299, 617), (300, 621), (301, 622), (302, 627), (303, 628), (304, 4329), (305, 1370), (306, 3151), (307, 3152), (308, 76), (309, 962), (311, 2888), (312, 329), (313, 1330), (314, 3869), (315, 5596), (317, 6193), (318, 5942), (319, 3648), (320, 6224), (321, 2895), (322, 6223), (325, 147), (327, 1352), (328, 3153), (330, 1278), (333, 63), (334, 4645), (348, 347), (349, 2893), (491, 487), (492, 488), (514, 4440), (523, 148), (561, 6327), (598, 1992), (646, 4462), (648, 525), (658, 4025), (662, 4560), (664, 522), (666, 918), (668, 513), (670, 512), (671, 524), (673, 239), (693, 149), (801, 394), (827, 4580), (850, 3448), (886, 888), (889, 891), (892, 690), (903, 519), (1016, 190), (1121, 527), (1219, 4461), (1220, 894), (1221, 1909), (1222, 3729), (1231, 2897), (1232, 2889), (1233, 2890), (1236, 2892), (1255, 3995), (1268, 10), (1269, 2891), (1275, 664), (1434, 3875), (1441, 2475), (1494, 30), (1502, 315), (1509, 4360), (1799, 22), (2373, 23), (2481, 518), (2966, 2799), (2986, 323), (3222, 669), (3224, 2998), (3225, 698), (3226, 726), (3227, 536), (3228, 538), (3260, 146), (3261, 150), (3262, 153), (3596, 3436), (3699, 4468), (4357, 5484), (4373, 3621), (4554, 693), (4555, 900), (4562, 812), (4564, 4552), (4569, 3900), (4573, 4467), (4577, 826), (4586, 824), (4590, 99), (4597, 4777), (4598, 649), (4600, 4427), (4632, 4608), (4997, 908), (4998, 100), (5376, 2397), (5490, 1270), (5491, 1271), (5641, 542), (5936, 3617), (6063, 3434), (6189, 3414), (6190, 3413), (6330, 297), (6331, 850), (6335, 4391)
X(6327) = isogonal conjugate of X(7087)
X(6327) = trilinear product X(i)*X(j) for these (i,j): {2,1759}, {7,4149}}, {75,1631}, {81,4153}
X(6327) = barycentric product X(i)*X(j) for these (i,j): {75,1759}, {76,1631}, {85,4149}, {86,4153}
X(6327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (31,2887,2), (320,4514,3873), (748,3836,2), (750,3847,2), (1836,3416,321), (2550,5739,4651), (3936,4450,55), (3966,5880,4359), (4388,4645,2), (4655,4865,38)

X(6328) = ANTIGONAL CONJUGATE OF X(115)

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b^2-c^2)^2/(a*(a^6-(b^2+c^2)*a^4+(b^4-b^2*c^2+c^4)*a^2-(b^2+c^2)*(b^2-c^2)^2))
Trilinears    g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin(B-C)^2/(cos(A)*(-7+4*cos(B-C)^2+4*cos(A)^2)+cos(B-C))
Barycentrics   h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = (b-c)^2 (b+c)^2 (-a^6+a^4 b^2-a^2 b^4+b^6+a^4 c^2-a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2+a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6)

The A′B′C′ be the orthic triangle, and let A″ be the intersection of the A-altitude and line B′C′. Define B″ and C″ cyclically. The circumcircles of A″BC, AB″C, ABC″ concur in X(6328). (Antreas Hatzipolakis, César Lozada, January 16, 2014, Hyacinthos 23027, 23029)

Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Then X(6328) is the radical center of the circumcircles of A′BC, AB′C, ABC′. (Randy Hutson, January 16, 2014, Hyacinthos 23032)

X(6328) lies on the curves K072, K474, K567 and this line: {3, 3447}

X(6328) = X(i)-cross conjugate of X(j) for these (i,j): (512, 115)
X(6328) = antigonal conjugate of X(115)
X(6328) = circumcircle-inverse of X(3447)
X(6328) = X(1101)-isoconjugate of X(3448) X(6328) = similitude center of tangential and normal triangles of Kiepert hyperbola

X(6329) = COMPLEMENT OF X(3631)

Barycentrics 6a² - b² - c² : - a² +6b² - c² : -a² - b² + 6c²
X(6329) = 3 X[2] + 5 X[6] = 7 X[6] + X[69] = 3 X[69] - 7 X[141] = 9 X[2] - 5 X[141] = 3 X[6] + X[141] = 9 X[6] - X[193] = 3 X[141] + X[193] = 9 X[69] + 7 X[193] = 5 X[182] - X[550] = X[546] + 5 X[575] = X[141] - 9 X[597] = X[2] - 5 X[597] = X[6] + 3 X[597] = 13 X[141] - 9 X[599] = 13 X[2] - 5 X[599] = 13 X[597] - X[599] = 13 X[6] + 3 X[599] = 11 X[6] - 3 X[1992] = 11 X[597] + X[1992] = 11 X[2] + 5 X[1992] = 11 X[141] + 9 X[1992] = 11 X[599] + 13 X[1992] = 5 X[1386] - X[3244] = 3 X[599] - 13 X[3589] = X[69] - 7 X[3589] = 3 X[2] - 5 X[3589] = X[141] - 3 X[3589] = 3 X[597] - X[3589] = X[193] + 9 X[3589] = 3 X[1992] + 11 X[3589] = X[141] - 5 X[3618] = 9 X[597] - 5 X[3618] = 3 X[3589] - 5 X[3618] = 3 X[6] + 5 X[3618] = X[193] + 15 X[3618] = 17 X[3589] - 7 X[3619] = 17 X[6] + 7 X[3619] = 19 X[141] - 15 X[3620] = 19 X[3589] - 5 X[3620] = 19 X[3618] - 3 X[3620] = 19 X[6] + 5 X[3620] = 15 X[1992] - 11 X[3629] = 5 X[193] - 9 X[3629] = 5 X[6] - X[3629] = 3 X[2] + X[3629] = 15 X[597] + X[3629] = 5 X[3589] + X[3629] = 5 X[141] + 3 X[3629] = 5 X[69] + 7 X[3629] = 15 X[599] + 13 X[3629] = 11 X[69] - 7 X[3630] = 11 X[141] - 3 X[3630] = 11 X[3589] - X[3630] = 11 X[6] + X[3630] = 3 X[1992] + X[3630] = 11 X[3629] + 5 X[3630] = 11 X[193] + 9 X[3630] = 15 X[599] - 13 X[3631] = 5 X[3630] - 11 X[3631] = 5 X[69] - 7 X[3631] = 5 X[141] - 3 X[3631] = 15 X[597] - X[3631] = 5 X[3589] - X[3631] = 5 X[6] + X[3631] = 5 X[193] + 9 X[3631] = 15 X[1992] + 11 X[3631] = 11 X[3620] - 19 X[3763] = 11 X[141] - 15 X[3763] = 11 X[3589] - 5 X[3763] = X[3630] - 5 X[3763] = 11 X[3618] - 3 X[3763] = 11 X[6] + 5 X[3763] = 3 X[1992] + 5 X[3763] = 17 X[6] - 9 X[5032] = 17 X[597] + 3 X[5032] = 17 X[3589] + 9 X[5032] = 7 X[3619] + 9 X[5032] = 17 X[2] + 15 X[5032] = X[382] + 15 X[5050] = 5 X[1352] - 13 X[5079] = 7 X[3528] - 15 X[5085] = 5 X[631] + 3 X[5102] = X[382] - 5 X[5480] = 3 X[5050] + X[5480] = 17 X[193] - 9 X[6144] = 17 X[3629] - 5 X[6144] = 17 X[6] - X[6144] = 9 X[5032] - X[6144] = 17 X[3589] + X[6144] = 7 X[3619] + X[6144] = 17 X[141] + 3 X[6144] = 17 X[3631] + 5 X[6144] = 17 X[69] + 7 X[6144] = 17 X[3630] + 11 X[6144]}}

Let P be a point in the plane of a triangle ABC, and let A′B′C′ be the pedal triangle of P. Let A″B″C″ be the pedal triangle of the circumcenter O′ of A′B′C′. Let A* be the reflection of A′ in O′, and define B* and C* cyclically, so that A*B*C* is the antipodal triangle of A′B′C′. Let D be the midpoint of segment A″A*, and define E and F cyclically. If P = X(2), then the circumcenter of DEF is X(6329). See A. Hatzipolakis and R. Hutson, Hyacinthos 22995, January 10, 2015.

X(6329) lies on these lines: {2, 6}, {140, 5097}, {182, 550}, {382, 5050}, {511, 3530}, {518, 3636}, {545, 3946}, {546, 575}, {631, 5102}, {742, 4739}, {894, 4395}, {1100, 4422}, {1352, 5079}, {1386, 3244}, {1743, 4364}, {2261, 5834}, {3528, 5085}, {3626, 5846}, {3628, 5965}, {3759, 4399}, {3793, 5007}, {4363, 4402}, {4371, 4665}, {4478, 4969}, {4795, 4859}, {4796, 5845}, {4873, 4910}, {4889, 4982}

X(6329) = midpoint of X(i) and X(j) for these {i,j}: (140,a5097), (3629,3631), (6,3589)
X(6329) = complement of X(3631)
X(6329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,3629), (2,3629,3631), (6,597,3589), (6,3618,141), (6,3763,1992), (6,6144,5032), (141,597,3618), (141,3618,3589), (1992,3763,3630), (3589,3631,2), (3619,5032,6144)

X(6330) = ISOTOMIC CONJUGATE OF X(441)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[S_A (-a^2 S_B S_C + S_A (S_B^2 + S²_C))]
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-a^4 b^2-a^2 b^4+b^6+a^2 c^4+b^2 c^4-2 c^6) (a^6+a^2 b^4-2 b^6-a^4 c^2+b^4 c^2-a^2 c^4+c^6)
Barycentrics    h(A,B,C) : h(C,A,B), : h(A, B, C), where h(A,B,C) = 1/[sec B cos(B + ω) + sec C cos(C + ω)]

Let H be the circumhyperbola {{A,B,C,X(2),X(69)}}, and let J be the circumconic centered at X(1249); that is, the conic {{A,B,C,X(107),X(648)}}. Then H and J intersect in four points: A, B, C, and X(6330). (Randy Hutson, January 29, 2015)

X(6330) lies on these lines: {2, 107}, {69, 648}, {253, 393}, {287, 297}, {305, 6331}, {306, 1897}, {307, 653}, {438, 5895}, {2419, 2799}

X(6330) = reflection of X(648) in X(1249)
X(6330) = isogonal conjugate of X(8779)
X(6330) = X(i)-cevapoint of X(j) for these (i,j): (2, 297), (868, 2501), (2972, 3569)
X(6330) = X(i)-cross conjugate of X(j) for these (i,j): (2799, 648)
X(6330) = trilinear pole of the line X(4)X(525)
X(6330) = X(i)-isoconjugate of X(j) for these (i,j): (3, 2312), (31, 441), (48, 1503), (822, 2409)

X(6331) = ISOTOMIC CONJUGATE OF X(647)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a^2 (b^2 - c^2) S_A]
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a-b) b^2 (a+b) (a-c) c^2 (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2)
Barycentrics    csc A csc 2A csc(B - C) : csc B csc 2B csc(C - A) : csc C csc 2C csc(C - A)
Barycentrics    bc/[b sin(A - B) - c sin(A - C)] : ca/[c sin(B - C) - a sin(B - A)] : ab/[a sin(C - A) - b sin(C - B)]

X(6331) lies on the circumconic centered at X(1249); that is, the conic {{A,B,C,X(107),X(648)}}. (Randy Hutson, January 29, 2015)

X(6331) lies on these lines: {25, 5989}, {99, 107}, {110, 685}, {112, 689}, {125, 290}, {264, 2970}, {297, 3978}, {305, 6330}, {648, 670}, {653, 799}, {683, 5254}, {687, 4590}, {811, 1897}, {877, 4576}, {2396, 4609}

X(6331) = isogonal conjugate of X(3049)
X(6331) = isotomic conjugate of X(647)
X(6331) = trilinear pole of the line X(4)X(69)
X(6331) = X(i)-cevapoint of X(j) for these (i,j): (2, 850), (4, 2489), (76, 3267), (99, 648), (343, 3265), (427, 2501), (523, 5254), (647, 1899)
X(6331) = X(i)-cross conjugate of X(j) for these (i,j): (99, 670), (523, 683), (2489, 4), (2799, 290), (3267, 76), (4554, 799), (6335, 811)
X(6331) = X(4)-cross conjugate of isogonal conjugate of X(23216)
X(6331) = crosssum of X(i) and X(j) for these (i,j): (647, 2524), (3049, 3049)
X(6331) = X(i)-isoconjugate of X(j) for these (i,j): (1, 3049), (3, 798), (6, 810), (25, 822), (31, 647), (32, 656), (48, 512), (63, 669), (69, 1924), (71, 667), (72, 1919), (73, 3063), (184, 661), (213, 1459), (217, 2616), (228, 649), (255, 2489), (293, 2491), (306, 1980), (513, 2200), (520, 1973), (525, 560), (603, 3709), (652, 1402), (657, 1410), (663, 1409), (878, 1755), (905, 1918), (906, 3122), (1084, 4592), (1176, 2084), (1331, 3121), (1400, 1946), (1437, 4079), (1576, 3708), (1917, 3267), (1923, 4580), (2196, 4455), (2205, 4025), (3124, 4575), (3248, 4574), (4117, 4563)

X(6332) = ISOTOMIC CONJUGATE OF X(653)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b - c) (b - c) S_A
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a-b-c) (b-c) (a^2-b^2-c^2)
Barycentrics    bc(sec B - sec C) : ca(sec C - sec A) : ab(sec A - sec B)
Barycentrics    (a - b)(1 - cos B) - (a - c)(1 - cos C) : :
X(6332) = 2 X[3239] + X[3904]

X(6332) lies on these lines: {2, 2399}, {8, 4163}, {63, 4091}, {100, 2728}, {110, 2769}, {190, 1813}, {441, 525}, {514, 661}, {522, 663}, {644, 4568}, {650, 3910}, {824, 4529}, {918, 3669}, {1459, 4064}, {3716, 3810}, {3907, 4522}, {4088, 4449}, {4561, 4587}

X(6332) = midpoint of X(i) and X(j) for these {i,j}: {1459,4064}, {3904,4391}, {4088,4449}
X(6332) = reflection of X(i) in X(j) for these (i,j): (4025, 905), (4391, 3239), (8,4163)
X(6332) = isogonal conjugate of X(32674)
X(6332) = isotomic conjugate of X(653)
X(6332) = X(i)-Ceva conjugate of X(j) for these (i,j): (69, 2968), (190, 63), (662, 3687), (664, 8), (1332, 306), (4554, 307), (4561, 78)
X(6332) = trilinear pole of line X(2968)X(4082)
X(6332) = polar conjugate of X(36127)
X(6332) = pole wrt polar circle of trilinear polar of X(36127) (line X(19)X(208))
X(6332) = X(i)-cross conjugate of X(j) for these (i,j): (521, 4025), (525, 4391), (652, 522), (2968, 69)
X(6332) = crosspoint of X(i) and X(j) for these (i,j): (190, 312), (304, 4561), (314, 4554), (348, 664), (1332, 1812)
X(6332) = crosssum of X(i) and X(j) for these (i,j): (604, 649), (607, 663), (1402, 3063)
X(6332) = crossdifference of every two points on the line X(25)X(31)
X(6332) = X(i)-complementary conjugate of X(j) for these (i,j): (2217, 116)
X(6332) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (84, 150), (101, 6223), (109, 5932), (112, 1895), (1436, 149), (1903, 3448), (2208, 4440)
X(6332) = X(i)-isoconjugate of X(j) for these (i,j): (4, 1415), (6, 108), (19, 109), (25, 651), (28, 4559), (31, 653), (33, 1461), (34, 101), (56, 1783), (65, 112), (100, 608), (107, 1409), (110, 1880), (162, 1400), (163, 225), (190, 1395), (278, 692), (604, 1897), (607, 934), (644, 1398), (648, 1402), (658, 2212), (664, 1973), (906, 1118), (919, 1876), (1020, 2299), (1096, 1813), (1295, 2443), (1396, 4557), (1397, 6335), (1414, 2333), (1426, 5546), (1435, 3939), (1474, 4551), (1824, 4565), (1974, 4554), (2203, 4552), (2204, 4566)
X(6332) = X(1812)-beth conjugate of X(4091)

X(6333) = ISOTOMIC CONJUGATE OF X(685)

Barycentrics (b-c) (b+c) (-a^2+b^2+c^2) (-a^2 b^2+b^4-a^2 c^2+c^4) : :
Barycentrics (csc A)(sec B sin^3 C - sec C sin^3 B) : :

X(6333) lies on these lines: {99, 110}, {125, 339}, {287, 2419}, {441, 525}, {523, 4143}, {826, 850}, {2799, 3569}

X(6333) = isogonal conjugate of X(32696)
X(6333) = X(i)-Ceva conjugate of X(j) for these (i,j): (877, 325), (2419, 3265)
X(6333) = X(i)-cross conjugate of X(j) for these (i,j): (684, 2799)
X(6333) = crosspoint of X(i) and X(j) for these (i,j): (325, 877)
X(6333) = crosssum of X(i) and X(j) for these (i,j): (669, 2211), (878, 1976)
X(6333) = crossdifference of every two points on the line X(25)X(1501)
X(6333) = X(i)-isoconjugate of X(j) for these (i,j): (19, 2715), (31, 685), (112, 1910), (162, 1976), (1973, 2966)

X(6334) = ISOTOMIC CONJUGATE OF X(687)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S_A (S_B - S_C) (-S_A (S_B - S_C)^2 + a^2 (S_A^2 - S_B S_C))
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b-c) (b+c) (-a^2+b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6)

X(6334) lies on these lines: {3, 690}, {113, 131}, {115, 127}, {441, 525}, {684, 5489}, {2394, 2986}

X(6334) = midpoint of X(i) and X(j) for these {i,j}: {2394,3268}, {684,5489}
X(6334) = isogonal conjugate of X(32708)
X(6334) = X(i)-Ceva conjugate of X(j) for these (i,j): (328, 125), (2394, 525)
X(6334) = crosspoint of X(1494) and X(4563)
X(6334) = crosssum of X(1495) and X(2489)
X(6334) = crossdifference of every two points on the line X(25)X(1576)
X(6334) = X(i)-complementary conjugate of X(j) for these (i,j): (48, 3258), (163, 1511)
X(6334) = X(i)-isoconjugate of X(j) for these (i,j): (19,10420), (31, 687), (163, 1300)

X(6335) = ISOTOMIC CONJUGATE OF X(905)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(b - c)S_A]
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a-b) b (a-c) c (a^2+b^2-c^2) (a^2-b^2+c^2)
Barycentrics    (sec A)/(b - c) : (sec B)/(c - a): (sec C)/(a - b)

Let U be the circumconic centered at X(1249); that is, the conic {{A,B,C,X(107),X(648)}}, and let V be the circumconic centered at X(3161); that is, the conic {{A,B,C,X(811),X(823). Then U and V intersect in four points: A,B,C, and X(6335). (Randy Hutson, January 29, 2015)

X(6335) lies on these lines: {92, 4997}, {100, 107}, {112, 839}, {190, 653}, {243, 5205}, {264, 281}, {278, 1997}, {286, 334}, {341, 1895}, {645, 648}, {646, 2397}, {651, 4391}, {685, 692}, {687, 4567}, {1118, 2899}, {1265, 3176}, {1633, 2517}, {1784, 3992}, {1897, 3699}, {1948, 3912}, {3239, 4605}

X(6335) = isogonal conjugate of X(22383)
X(6335) = isotomic conjugate of X(905)
X(6335) = X(811)-Ceva conjugate of X(1897)
X(6335) = X(i)-cevapoint of X(j) for these (i,j): (2, 4391), (10, 3239), (37, 4036), (100, 1783), (429, 2501), (650, 1837), (693, 3673)
X(6335) = X(i)-cross conjugate of X(j) for these (i,j): (37, 5379), (100, 668), (4397, 75), (4552, 190), (5552, 4998)
X(6335) = X(6591)-cross conjugate of X(4)
X(6335) = crosspoint of X(811) and X(6331)
X(6335) = crosssum of X(810) and X(3049)
X(6335) = trilinear pole of the line X(4)X(8)
X(6335) = polar conjugate of X(513)
X(6335) = pole wrt polar circle of trilinear polar of X(513) (line X(244)X(665))
X(6335) = X(i)-isoconjugate of X(j) for these (i,j): (3, 649), (6, 1459), (19,23224), (25, 4091), (28, 822), (31, 905), (32, 4025), (48, 513), (56, 652), (57, 1946), (58, 647), (63, 667), (69, 1919), (71, 3733), (77, 3063), (81, 810), (86, 3049), (101, 3937), (184, 514), (212, 3669), (222, 663), (228, 1019), (244, 906), (304, 1980), (512, 1790), (520, 1474), (521, 604), (525, 2206), (603, 650), (656, 1333), (659, 2196), (661, 1437), (692, 3942), (798, 1444), (1015, 1331), (1021, 1410), (1332, 3248), (1357, 4587), (1397, 6332), (1409, 3737), (1461, 3270), (1472, 2522), (1576, 4466), (1795, 3310), (1797, 1960), (1803, 2488), (1813, 3271), (1973, 4131), (1977, 4561), (2193, 4017), (3121, 4592), (3122, 4558), (3125, 4575)

X(6336) = ISOTOMIC CONJUGATE OF X(3977)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(2 a - b - c) (a^2 - b^2 - c^2)]
Barycentrics (tan A)/(2a - b - c) : (tan B)/(2b - c - a) : (tan C)/(2c - a - b)

Let U be the hyperbola {{A,B,C,X(4),X(27)}}, and let V be the circumconic centered at X(1249); that is, the conic {{A,B,C,X(107),X(648). Then U and V intersect in four points: A,B,C, and X(6336). (Randy Hutson, January 29, 2015)

X(6336) lies on these lines: {4, 145}, {27, 648}, {88, 278}, {92, 4997}, {106, 107}, {469, 4945}, {687, 4591}, {901, 917}, {1222, 4054}, {1838, 4792}

X(6336) = isogonal conjugate of X(22356)
X(6336) = isotomic conjugate of X(3977)
X(6336) = X(i)-cevapoint of X(j) for these (i,j): (1, 1731), (19, 1870)
X(6336) = X(i)-cross conjugate of X(j) for these (i,j): (106, 903), (1785, 273)
X(6336) = trilinear pole of the line X(4)X(2457)
X(6336) = polar conjugate of X(519)
X(6336) = pole wrt polar circle of trilinear polar of X(519) (line X(900)X(1635))
X(6336) = X(i)-isoconjugate of X(j) for these (i,j): (3, 44), (6, 5440), (31, 3977), (48, 519), (63, 902), (69, 2251), (72, 3285), (78, 1404), (184, 4358), (212, 3911), (219, 1319), (222, 3689), (603, 2325), (678, 1797), (900, 906), (1023, 1459), (1331, 1635), (1332, 1960), (1437, 3943), (1813, 4895), (1877, 2289), (2196, 4432), (4120, 4575), (4558, 4730)

X(6337) = SS(a → S_A) OF X(9)

Barycentrics    S_A(S_A - S_B - S_C) : :
Barycentrics    (a^2-b^2-c^2) (3 a^2-b^2-c^2) : :
Barycentrics    (cot A)(cot B + cot C - cot A) : :

"SS(a → S_A)" denotes barycentric symbolic substitution, similar to trilinear symbolic substitution defined at X(3221). Here, substituting S_A, S_B, S_C for a,b,c in the barycentrics for X(9) yields X(6337). Symbolic substitution carries lines to lines, conics to conics, and cubics to cubics. For example, since X(1), X(2), X(3), X(4), X(6), X(9), X(57), X(1073), X(1249) lie on the Thomson cubic, their images under SS(a →S_A) lie on the cubic given by

x(S²_Cy² - S²_Bz²) + y(S²_Az² - S²_Cx²) + z(S²_Ax² - S²_Ay²) = 0.

The following four circles intersect in two points, and their crossum is X(6337): (1) circumcircle, (2) 2nd Lemoine circle, (3) {{X(371),X(372),PU(1),PU(39)}} (whose center is X(32)), and (4) {{X(4),X(194),X(3557),X(3558)}} (whose center is X(3095). (Randy Hutson, January 29, 2015)

As a line L varies through the set of all lines that pass through X(69), the locus of the trilinear pole of L is a circumconic, and its center is X(6337). (Randy Hutson, January 29, 2015)

Let A′ be the trilinear product of the vertices of the A-adjunct anti-altimedial triangle, and define B′ and C′ cyclically. Triangle A′B′C′ is the anticomplementary triangle of the 1st Brocard triangle, and is perspective to ABC at X(4), and to the medial triangle at X(6337). (Randy Hutson, November 2, 2017)

X(6337) lies on the curves K009, K164, K168, Q052 and these lines: {2, 1975}, {3, 69}, {4, 99}, {6, 6339}, {20, 325}, {32, 1992}, {39, 3618}, {76, 631}, {183, 3523}, {184, 4176}, {193, 439}, {194, 5976}, {315, 376}, {316, 3529}, {485, 642}, {486, 641}, {524, 5023}, {574, 3619}, {620, 3767}, {988, 1125}, {1078, 3524}, {1649, 3265}, {1909, 5218}, {2021, 6309}, {2275, 3666}, {2549, 3788}, {3630, 5585}, {4384, 5745}

X(6337) = midpoint of X(i) and X(j) for these {i,j}: {99,8781}, 487,488}
X(6337) = reflection of X(i) in X(j) for these (i,j): (485,642), (486,641)
X(6337) = isogonal conjugate of X(14248)
X(6337) = isotomic conjugate of X(34208)
X(6337) = complement of X(2996)
X(6337) = anticomplement of X(13881)
X(6337) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,69), (99,3566)
X(6337) = X(3167)-cross conjugate of X(193)
X(6337) = crosspoint of X(2) and X(193)
X(6337) = crosssum of X(512) and X(2971)
X(6337) = X(i)-complementary conjugate of X(j) for these (i,j): (6,4138), (31,69), (163,3566), (193,2887), (1707,141), (3053,10)
X(6337) = X(4592)-beth conjugate of X(56)
X(6337) = X(i)-Hirst inverse of X(j) for these (i,j): (20,235), (69,3564)
X(6337) = X(i)-isoconjugate of X(j) for these (i,j): {923,5203}, {1973,2996}
X(6337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3926,69), (3,3933,3785), (193,439,3053), (3785,3926,3933), (3785,3933,69)
X(6337) = trilinear product X(i)*X(j) for these (i,j): {63,193}, {69,1707}, {75,3167}, {304,3053}, {1332,3798}, {1444,4028}, {3566,4592}
X(6337) = barycentric product X(i)*X(j) for these (i,j): {69,193}, {76,3167}, {304,1707}, {305,3053}, {3266,6091}, {3566,4563}, {3798,4561}
X(6337) = barycentric quotient X(i)/X(j) for these (i,j): {69,2996}, {193,4}, {524,5203}, {1707,19}, {3053,25}, {3167,6}, {3566,2501}, {3787,1843}, {4028,1826}, {4558,3565}, {6091,111}
X(6337) = X(4592)-beth conjugate of X(56)
X(6337) = perspector of the circumconic centered at X(69)
X(6337) = center of the conic {{A,B,C,X(99),X(4554),X(4563)}}, the isotomic conjugate of the orthic axis)
X(6337) = isogonal conjugate of X(12248)
X(6337) = Cundy-Parry Phi transform of X(3564)
X(6337) = Cundy-Parry Psi transform of X(3563)

X(6338) = SS(a → S_A) OF X(3)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S²_A(S²_A - S²_B - S²_C)
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^2-b^2-c^2)^2 (a^4+2 a^2 b^2+b^4+2 a^2 c^2-6 b^2 c^2+c^4)

"SS(a → S_A)" denotes barycentric symbolic substitution; see X(6337).

X(6338) lies on these lines: {2, 6339}, {69, 1368}, {126, 193}, {141, 5490}, {393, 670}

X(6338) = complement of X(6339)
X(6338) = X(2)-Ceva conjugate of X(3926)
X(6338) = X(i)-cross conjugate of X(j) for these (i,j): (3926, 6341)
X(6338) = X(i)-complementary conjugate of X(j) for these (i,j): (31, 3926), (1611, 10), (2128, 1368)
X(6338) = trilinear product X(69)*X(2128)
X(6338) = barycentric product X(304)*X(2128)
X(6338) = barycentric quotient X(i)/X(j) for these (i,j): (63,2129), (1611,2207), (2128,19), (2519,2489) X(6338) = X(25)-isoconjugate of X(2129)

X(6339) = SS(a → S_A) OF X(4)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(- S²_A + S²_B + S²_C) ]
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^4+2 a^2 b^2+b^4-6 a^2 c^2+2 b^2 c^2+c^4) (a^4-6 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2+c^4)
X(6339) = 3X(2) - 4 X(6387)

"SS(a → S_A)" denotes barycentric symbolic substitution; see X(6337). Also, X(6339) = SS(a → S²_A) of X(7).

X(6339) lies on these lines: {2, 6338}, {6, 6337}, {25, 193}, {3926, 6342}

X(6339) = reflection of X(6338) in X(6387)
X(6339) = isogonal conjugate of X(1611)
X(6339) = isotomic conjugate of X(6392)
X(6339) = anticomplement of X(6338)
X(6339) = X(2)-Ceva conjugate of X(6342)
X(6339) = cevapoint of X(520) and X(1084)
X(6339) = crosssum of X(1611) and X(1611)
X(6339) = X(i)-cross conjugate of X(j) for these (i,j): (3296,2), (6391,69)
X(6339) = X(i)-complementary conjugate of X(j) for this (i,j): (31, 6342)
X(6339) = X(i)-anticomplementary conjugate of X(j) for this (i,j): (2129,1370)
X(6339) = X(i)-isoconjugate of X(j) for these (i,j): (1,1611), (25,2128), (31,6392), (162,2519)
X(6339) = {X(6338),X(6387)}-harmonic conjugate of X(2)
X(6339) = trilinear product X(69)*X(2129)
X(6339) = barycentric product X(304)*X(2129)
X(6339) = barycentric quotient X(i)/X(j) for these (i,j): (6,1611), (63,2128), (647,2519), (2129,19)

X(6340) = SS(a → S_A) OF X(57)

Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = S_A/(- S_A + S_B + S_C)
Barycentrics    g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^2+b^2-3 c^2) (a^2-b^2-c^2) (a^2-3 b^2+c^2)
Barycentrics    h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = (csc A)/(sec A - 2 csc A tan ω)
Barycentrics    k(A,B,C) : k(B,C,A) : k(C,A,B), where k(A,B,C) = (csc A)/(2 csc A - sec A cot ω)

"SS(a → S_A)" denotes barycentric symbolic substitution; see X(6337).

X(6340) lies on these lines: {2, 1975}, {25, 5203}, {69, 1368}, {125, 4176}, {253, 325}, {264, 1007}, {1370, 2373}

X(6340) = isogonal conjugate of X(19118)
X(6340) = isotomic conjugate of X(6353)
X(6340) = X(2996)-Ceva conjugate of X(69)
X(6340) = X(125)-cevapoint of X(3265)
X(6340) = X(i)-cross conjugate of X(j) for these (i,j): (115, 3267), (3926, 69)

X(6341) = SS(a → S_A) OF X(1073)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^2-b^2-c^2)^2 (a^4+2 a^2 b^2+b^4+2 a^2 c^2-6 b^2 c^2+c^4) (a^8+4 a^6 b^2+6 a^4 b^4+4 a^2 b^6+b^8-12 a^6 c^2+44 a^4 b^2 c^2-68 a^2 b^4 c^2+4 b^6 c^2-26 a^4 c^4+44 a^2 b^2 c^4+6 b^4 c^4-12 a^2 c^6+4 b^2 c^6+c^8) (a^8-12 a^6 b^2-26 a^4 b^4-12 a^2 b^6+b^8+4 a^6 c^2+44 a^4 b^2 c^2+44 a^2 b^4 c^2+4 b^6 c^2+6 a^4 c^4-68 a^2 b^2 c^4+6 b^4 c^4+4 a^2 c^6+4 b^2 c^6+c^8)

"SS(a → S_A)" denotes barycentric symbolic substitution; see X(6337).

X(6341) lies on this line: {2, 6342}

X(6341) = X(3926)-cross conjugate of X(6338)

X(6342) = SS(a → S_A) OF X(1249)

Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (a^4+2 a^2 b^2+b^4-6 a^2 c^2+2 b^2 c^2+c^4) (a^4-6 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2+c^4) (a^8+4 a^6 b^2+6 a^4 b^4+4 a^2 b^6+b^8+4 a^6 c^2-68 a^4 b^2 c^2+44 a^2 b^4 c^2-12 b^6 c^2+6 a^4 c^4+44 a^2 b^2 c^4-26 b^4 c^4+4 a^2 c^6-12 b^2 c^6+c^8)

"SS(a → S_A)" denotes barycentric symbolic substitution; see X(6337).

X(6342) lies on these lines: {2, 6341}, {3926, 6339}

X(6342) = X(2)-Ceva conjugate of X(6339)
X(6342) = X(i)-complementary conjugate of X(j) for this (i,j): (31, 6339)

X(6343) = HATZIPOLAKIS-LOZADA-EULER REFLECTIONS POINT

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 4 cos A - 4 cos 3A + 2 cos 5A - 4 cos 7A
+ cos(B - C) (7 + 10 cos 2A + 4 cos 4A - 2 cos 6A)
+ 2 cos(2B - 2C) (cos A + 3 cos 3A - 3 cos 5A)
+ 2 cos(3B - 3C) (2 + cos 2A + cos 4A)
+ 2 cos(4B - 4C) (cos A - cos 3A)

Let N the the nine-point center of a triangle ABC, and let N_A the reflection of N in line BC. Define N_B and N_C cyclically. Let N_AB be the reflection of N_A in line NB, and define N_BC and N_CA cyclically. Let N_AC be the reflection of N_A in line NC, and define N_BA and N_CB cyclically. Let L_A be the Euler line of triangle N_AN_ABN_AC, and define L_B and L_C cyclically. Let L′_A be the reflection of L_A in line BC, and define L′_B and L′_C cyclically. The lines L′_A, L′_B, L′_C concur in X(6343). (Antreas Hatzipolakis and César Lozada, January 22, 2015; see Hyacinthos 23050)

X(6343) lies on these lines: {3,2888}, {54,1263}

X(6343) = reflection of X(1263) in X(54)

X(6344) = ABC-ORTHOLOGIC CENTER OF X(3)-HATZIPOLAKIS-LOZADA TRIANGLE

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)[1 - 4 cos²B][1 - 4 cos²C]
Trilinears        tan A csc 3A : tan B csc 3B : tan C csc 3C
Barycentrics    sin A tan A csc 3A : sin B tan B csc 3B : sin C tan C csc 3C
Barycentrics    (sec A)/(4 sin A - 3 csc A) : (sec B)(4 sin B - 3 csc B) : (sec C)(4 sin C = 3 csc C)
Barycentrics    (tan A)/(3 - 4 sin²A) : (tan B)/(3 - 4 sin²B) : (tan C)/(3 - 4 sin²C)

Suppose that P is a point in the plane of a triangle ABC. Let P_A be the reflection of P in line BC, and define Let P_B and Let P_C cyclically. Let P_A_B be the reflection of P_A in P_B, and let P_A_C be the reflection of P_A in P_C. Let N_A be the nine-point center of the triangle P_AP_A_BP_A_C, and define N_B and N_C cyclically. The triangle N_AN_BN_C is the P-Hatzipolakis-Lozada triangle. For P = X(3), the ABC-orthologic center of this triangle is X(6344). See also X(6345) and X(5346). These four orthologic centers, and also X(1), X(4), X(5), X(1113), and X(1114) lie on the excentral-circum-septic curve described at Hyacinthos 23055. (Antreas Hatzipolakis and César Lozada, January 23, 2015)

X(6344) lies on these lines: {4,94}, {5,93}, {107,1141}, {186,476}, {225,2166}, {264,328}, {393,1989}, {648,2914}, {1105,3520}

X(6344) = isogonal conjugate of X(22115)
X(6344) = X(48)-isoconjugate (polar conjugate) of X(323)
X(6344) = X(63)-isoconjugate of X(50)
X(6344) = X(403)-cross conjugate of X(4)
X(6344) = trilinear pole of line X(53)X(2501)
X(6344) = trilinear product of vertices of Ehrmann vertex-triangle

X(6345) = ABC-ORTHOLOGIC CENTER OF X(5)-HATZIPOLAKIS-LOZADA TRIANGLE

Trilinears f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/[3 cos A + cos 3A + cos 5A - cos 7A + cos(B - C) (3 cos 2A - cos 6A + 1/2) + cos(2B - 2C) (3 cos A + 3 cos 3A - cos 5A) + cos(3B - 3C) (- cos 2A + cos 4A + 2) + cos(4B - 4C) (cos A - cos 3A)]

See X(6344) and X(6346). (Antreas Hatzipolakis and César Lozada, January 23, 2015)

X(6345) lies on the circumcircle and these lines: {930, 32744}, {1291, 15345}, {1510, 32749}

X(6346) = X(5)-HATZIPOLAKIS-LOZADA-TRIANGLE ORTHOLOGIC CENTER OF ABC

Trilinears g(A,B,C) : g(B,C,A) : g(C,A,B), where f(A,B,C) = 5 cos A - 4 cos 3A + 3 cos 5A - cos 7A + cos(B - C) (-7 cos 2A - cos 4A + cos 6A + 7) + cos(2B - 2C) (5 cos A - 3 cos 3A + cos 5A) + cos(3B - 3C) (- 3 cos 2A + 5/2) + cos(4B - 4C) (cos A - 2 cos 3A)

See X(6344) and X(6345). (Antreas Hatzipolakis and César Lozada, January 23, 2015)

X(6346) lies on these lines: {546, 11572}, {11801, 20030}

X(6347) = X(1)X(2)∩X(394)X(1377)

Barycentrics 1 + sin B + sin C : 1 + sin C + sin A : 1 + sin A + sin B

X(6347) lies on these lines: {1, 2}, {394, 1377}, {958, 1584}, {993, 1600}, {1329, 1591}, {1376, 1583}, {1512, 2048}, {1585, 1861}, {1592, 2886}

X(6347) = {X(2),X(10)}-harmonic conjugate of X(6348)

X(6348) = X(1)X(2)∩X(394)X(1378)

Barycentrics - 1 + sin B + sin C : - 1 + sin C + sin A : - 1 + sin A + sin B

X(6348) lies on these lines: {1, 2}, {394, 1378}, {958, 1583}, {993, 1599}, {1329, 1592}, {1376, 1584}, {1586, 1861}, {1591, 2886}

X(6348) = {X(2),X(10)}-harmonic conjugate of X(6347)

X(6349) = 1st SODDY HOMOTHETIC CENTER

Barycentrics 1 + sec B + sec C : 1 + sec C + sec A : 1 + sec A + sec B
Barycentrics (a^2 - b^2 - c^2) (a^4 + 2 a^3 (b + c) - 2 a (b - c)^2 (b + c) - (b^2 - c^2)^2) : :

Let E_A be the ellipse that passes through A and has foci B and C. Call E_A the A-Soddy ellipse. Let L_A be the polar of X(3) with respect to E_A. Define L_B and L_B cyclically. Let A′ = L_B∩L_C, B′ = L_C∩L_A, C′ = L_A∩L_B. Then triangle A′B′C′ is homothetic to the medial triangle, with homothethic center X(6349). Also, A′B′C′ is homothetic to ABC, with homthetic center X(281). The orthocenter of A′B′C′ is X(6361). (Randy Hutson, January 30, 2015)

X(6349) lies on these lines: {2, 92}, {3, 962}, {63, 348}, {189, 3160}, {464, 5249}, {1038, 1457}, {1040, 4666}, {1060, 4511}, {1817, 4329}

X(6349) = isotomic conjugate of polar conjugate of X(4295)
X(6349) = {X(2),X(1214)}-harmonic conjugate of X(6350)

X(6350) = 2nd SODDY HOMOTHETIC CENTER

Barycentrics - 1 + sec B + sec C : - 1 + sec C + sec A : - 1 + sec A + sec B
Barycentrics (a^2 - b^2 - c^2) (a^4 - 2 a^3 (b + c) + 2 a (b - c)^2 (b + c) - (b^2 - c^2)^2) : :

Let H_A be the hyperbola that passes through A and has foci B and C. H_A is the A-Soddy hyperbola, as in (Paul Yiu, Introduction to the Geometry of the Triangle, 2002; 12.4 The Soddy hyperbolas, p. 143). Let M_A be the polar of X(3) with respect to H_A. Define M_B and M_C cyclically. Let A″ = M_B∩M_C, B″ = M_C∩M_A, C″ = M_A∩M_B. Triangle A″B″C″ is homothetic to the medial triangle, with homothetic center X(6350). Also, triangle A″B″C″ is homothetic to ABC, with homothetic center X(278). The orthocenter of A′B′C′ is X(944). (Randy Hutson, January 30, 2015)

X(6350) lies on these lines: {2, 92}, {3, 8}, {20, 5174}, {63, 69}, {189, 268}, {329, 440}, {344, 908}, {1040, 3870}, {1748, 3101}

X(6350) = anticomplement of X(37695)
X(6350) = isotomic conjugate of polar conjugate of X(18391)
X(6350) = {X(2),X(1214)}-harmonic conjugate of X(6349)

X(6351) = X(1)X(1123)∩X(2)X(37)

Barycentrics 1 + csc B + csc C : 1 + csc C + csc A : 1 + csc A + csc B
Barycentrics a b + a c + S : :

X(6351) lies on these lines: {1, 1123}, {2, 37}, {9, 3068}, {45, 590}, {498, 1336}, {1270, 4851}, {1271, 4643}, {1659, 6203}, {3247, 5405}, {3879, 5860}, {3912, 5590}, {4357, 5591}, {4416, 5861}

X(6351) = complement of X(32793)
X(6351) = {X(2),X(37)}-harmonic conjugate of X(6352)

X(6352) = X(1)X(1336)∩X(2)X(37)

Barycentrics - 1 + csc B + csc C : - 1 + csc C + csc A : - 1 + csc A + csc B
Barycentrics a b + a c - S : :

X(6352) lies on these lines: {1, 1336}, {2, 37}, {9, 3069}, {45, 615}, {498, 1123}, {1270, 4643}, {1271, 4851}, {1659, 1826}, {3247, 5393}, {3879, 5861}, {3912, 5591}, {4357, 5590}, {4416, 5860}

X(6352) = complement of X(32794)
X(6352) = {X(2),X(37)}-harmonic conjugate of X(6351)

X(6353) = X(393)-CEVA CONJUGATE OF X(4)

Trilinears        sec A - 2 csc A tan ω : sec B - 2 csc B tan ω : sec C - 2 csc C tan ω
Trilinears        2 csc A - sec A cot ω : 2 csc B - sec B cot ω : 2 csc C - sec C cot ω
Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a² - b² - c²)(a² - b² + c²)(a² + b² - c²)
Barycentrics    tan A - 2 tan ω : tan B - 2 tan ω : tan C - 2 tan ω
Barycentrics    2 - tan A cot ω : 2 - tan B cot ω : 2 - tan C cot ω
Barycentrics    (tan A)(tan B tan C + tan A tan C - tan B tan C) : :
X(6353) = 6S_AS_BS_C*X(2) - S²S_ω*X(4)

As a point on the Euler line, X(6353) has Shinagawa coefficients (2F, -E - F).

X(6353) lies on the cubic K163 and these lines: {2,3}, {33,5218}, {34,5272}, {43,2356}, {69,1974}, {98,459}, {107,3563}, {111,1289}, {112,2374}, {171,2212}, {193,3167}, {230,393}, {232,800}, {238,1395}, {242,278}, {317,1007}, {612,6198}, {614,1870}, {966,1474}, {1068,3011}, {1119,1447}, {1172,5275}, {1192,2883}, {1204,6225}, {1299,1302}, {1398,5265}, {1452,3485}, {1495,1899}, {1611,2207}, {1613,2211}, {1620,5894}, {1829,3616}, {1843,3618}, {1848,5338}, {1876,5435}, {1892,5226}, {2203,5739}, {2299,5712}, {3068,5413}, {3069,5412}, {3087,3815}, {5139,6091}

X(6353) = isogonal conjugate of X(6391)
X(6353) = isotomic conjugate of X(6340)
X(6353) = complement of X(7396)
X(6353) = anticomplement of X(30771)
X(6353) = cevapoint of X(i) and X(j) for these {i,j}: {25,1611}, {193,439}
X(6353) = X(193)-cross conjugate of X(4)
X(6353) = X(4)-Hirst inverse of X(460)
X(6353) = inverse-in-polar-circle of X(5159)
X(6353) = pole wrt polar circle of trilinear polar of X(2996) (line X(523)X(4885))
X(6353) = X(48)-isoconjugate (polar conjugate) of X(2996)
X(6353) = X(i)-Ceva conjugate of X(j) for these {i,j}: {393,4}, {4590,112}
X(6353) = X(i)-cross conjugate of X(j) for these {i,j}: {193,4}, {3053,193}
X(6353) = X(i)-isoconjugate of X(j) for these {i,j}: {31,6340}, {48,2996}, {656,3565}
X(6353) = Euler line intercept, other than X(20), of circle {X(20),PU(4)}
X(6353) = perspector of circumcevian triangle of X(25) and (cross-triangle of ABC and circumcevian triangle of X(25))
X(6353) = cyclocevian conjugate of X(4) wrt anticevian triangle of X(4)
X(6353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,20,1368), (2,23,1370), (2,25,4), (2,4232,25), (3,3089,4), (4,186,376), (4,3147,631), (4,3524,378), (4,3525,3541), (4,5067,1594), (20,235,4), (24,3542,4), (25,468,2), (25,5094,428), (27,4207,4), (28,406,4), (140,1598,3088), (235,3515,20), (429,4198,4), (468,4232,4), (475,4222,4), (1585,5200,4), (1598,3088,4), (3091,3575,4), (4185,4194,4), (4186,4200,4)
X(6353) = trilinear product of X(i)*X(j) for these {i,j}: {4,1707}, {19,193}, {28,4028}, {92,3053}, {158,3167}, {162,3566}, {1096,6337}, {1783,3798}
X(6353) = barycentric product X(i)*X(j) for these {i,j}: {4,193}, {27,4028}, {92,1707}, {264,3053}, {393,6337}, {648,3566}, {1897,3798}, {2052,3167}, {4590,5139}
X(6353) = barycentric quotient X(i)/X(j){i/j} for these {i,j}: {2,6340}, {4,2996}, {112,3565}, {193,69}, {439,6337}, {1707,63}, {3053,3}, {3167,394}, {3566,525}, {3787,3917}, {3798,4025}, {4028,306}, {5139,115}, {6337,3926}

X(6354) = (X(1),X(115))-ANSWER TO QUESTION A

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

Question A is copied here from X(554): Suppose that X and Y are triangle centers. Let

Y_A = (Y of the triangle XBC), Y_B = (Y of the triangle XCA), Y_C = (Y of the triangle XAB).

Let A′ = XY_A∩, and define B′ and C′ cyclically. In Clark Kimberling, "Major Centers of Triangles," Amer. Math. Monthly 104 (1997) 431-438, Question A is this: for what choices of X and Y do the lines AA′, BB′, CC′ concur? A solution (X,Y) is called the (X,Y)-answer to Question A. Answers X(6354)-X(6359) were contributed by Peter Moses, January 28, 2015, in conjunction questions raised by Viktor Kitaisky.

See X(554) for a list of major centers that provide answers to Question A. In the following list, the appearance of (i,j) means that X(j) is a non-major center which is the (X(1),X(i))-answer to Question A: (115,6354), (125,12), (135,6355), (136,6356), (155,84), (231,6357), (249,2185), (250,757), (254,347), (338,6358), (339,1089), (401,894), (421,6359), (426,612), (436,144), (454,1256) (Peter Moses, January 28, 2015)

X(6354) lies on these lines: {6,278}, {7,940}, {12,201}, {33,1836}, {37,226}, {51,2969}, {55,4331}, {57,1020}, {65,225}, {92,1146}, {181,1365}, {196,393}, {220,329}, {223,3553}, {241,5249}, {269,4654}, {279,1255}, {321,349}, {335,1088}, {347,5712}, {394,5905}, {524,1943}, {1042,3649}, {1060,1448}, {1068,5706}, {1211,1441}, {1435,2285}, {1659,3070}, {2171,6046}, {4306,6147}, {4332,6284}, {5307,5928}

X(6355) = (X(1),X(135))-ANSWER TO QUESTION A

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

Regarding Question A, see X(6354).

X(6355) lies on these lines: {226,1439}, {269,282}, {1367,6046}, {1433,1440}

X(6356) = (X(1),X(136))-ANSWER TO QUESTION A

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

Regarding Question A, see X(6354).

X(6356) lies on these lines: {2,1119}, {3,7}, {4,347}, {5,273}, {9,1020}, {12,6046}, {37,226}, {72,307}, {75,2968}, {77,1060}, {216,1086}, {269,1038}, {348,1791}, {405,1398}, {441,894}, {442,1441}, {583,1708}, {594,4605}, {664,2893}, {1040,4328}, {1426,4205}, {1442,5453}, {1536,1827}

X(6357) = (X(1),X(231))-ANSWER TO QUESTION A

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a+b-c) (a-b+c) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)

Regarding Question A, see X(6354).

X(6357) lies on these lines: {7,27}, {34,5722}, {56,5358}, {57,2160}, {73,5453}, {109,2688}, {223,5219}, {241,514}, {319,1943}, {323,651}, {942,1831}, {1387,1457}, {3649,4658}, {4304,5930}

X(6357) = isogonal conjugate of X(15627)

X(6358) = (X(1),X(338))-ANSWER TO QUESTION A

Barycentrics    b c (b+c)^2 / (-a+b+c) : :
Barycentrics    1 + cos(B - C) : :
Barycentrics    cos²(B/2 - C/2) : :

Regarding Question A, see X(6354).

X(6358) lies on these lines: {2,2006}, {9,92}, {10,201}, {12,1089}, {57,75}, {65,4647}, {85,4102}, {109,4418}, {171,1733}, {196,2550}, {222,4363}, {226,306}, {278,2345}, {312,5219}, {355,5928}, {553,4980}, {756,1109}, {894,1943}, {1091,1254}, {1111,3782}, {1215,4551}, {1365,6058}, {1708,5271}, {1766,5307}, {2078,3757}, {3175,3991}, {3262,3687}, {3706,5173}, {3719,3729}, {3911,4359}, {3947,4066}, {4008,5269}, {4671,5226}, {4692,5252}

X(6358) = isogonal conjugate of X(2150)
X(6358) = isotomic conjugate of X(2185)
X(6358) = complement of X(18662)
X(6358) = anticomplement of X(16579)
X(6358) = trilinear pole of line X(4036)X(4064)
X(6358) = pole wrt polar circle of trilinear polar of X(270)
X(6358) = X(48)-isoconjugate (polar conjugate) of X(270)

X(6359) = (X(1),X(421))-ANSWER TO QUESTION A

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = → (a^4-a^2 b^2-a^2 b c+b^3 c-a^2 c^2+2 b^2 c^2+b c^3)/(-a+b+c)^2

Regarding Question A, see X(6354).

X(6359) lies on these lines: {7,90}, {77,3362}, {86,658}, {87,269}, {273,1804}, {348,1119}, {894,1020}, {934,1441}, {1014,1446}, {1247,3668}, {1442,4566}

X(6360) = 3rd SODDY HOMOTHETIC CENTER

Barycentrics sec B + sec C - sec A : sec C + sec A - sec B : sec A + sec B - sec C)
Barycentrics a^5(b + c) + a^4bc - 2a^3(b^3 + c^3) + a(b - c)^2(b + c)(b^2 + c^2) - bc(b^2 - c^2)^2 : :

Let A′B′C′ be as at X(6349) and A″B″C″ as at X(6350). Then A′B′C′ and A″B″C″ are homothetic to each other and to the anticomplementary triangle. The homothetic center is X(6360). (Randy Hutson, January 30, 2015)

X(6360) lies on these lines: {2,92}, {8,3152}, {20,145}, {63,1943}, {192,3151}, {322,3998}, {329,4552}, {394,664}, {653,n+31}, {1148,1816}, {1944,6505}, {1952,2994}, {3100,3957}, {3101,3164}, {3177,3219}

X(6360) = isogonal conjugate of X(8761)
X(6360) = anticomplement of X(92)
X(6360) = anticomplementary conjugate of X(21270)
X(6360) = pole wrt polar circle of trilinear polar of X(7049)
X(6360) = X(48)-isoconjugate (polar conjugate) of X(7049)

X(6361) = 4th SODDY HOMOTHETIC CENTER

Trilinears 1 + cos A - cos B - cos C - cos B cos C : 1 + cos B - cos C - cos A - cos C cos A : 1 + cos C - cos A - cos B - cos A cos B
Barycentrics 3 a^4 + 2 a^3 (b + c) - 2 a^2 (b + c)^2 - 2 a (b - c)^2 (b + c) - (b^2 - c^2)^2 : :
X(6361) = 3 X[4] - 4 X[10] = 2 X[10] - 3 X[40] = 3 X[20] - X[145] = 2 X[1] - 3 X[376]

X(6361) is the orthocenter of the triangle A′B′C′ described at X(6349). (Randy Hutson, January 30, 2015)

Let P(k) = M(A,B,C) : M(B,C,A) : M(C,A,B), where

M(A,B,C) = (1 + Cos[A] - Cos [B] - Cos[C] + k Cos[B] Cos [C])

Then P(k) lies on X(4)X(9). The appearance of (k,m) in the following list means that P(k) = X(m) (-2,516), (-1,6361), (-2/3,5493), (0,40), (1,5657), (2,10), (3,5818), (4,5587) ((2r - 2R)/r, 1512), (4R/(2R - r), 1706), (r + 2R)/R, 2550), (2R - r)/R, 2551), (2R - r)/(r + R), 5698), (-r/(2R - r), 5759). (Peter Moses, February 1, 2015)

X(6361) lies on these lines: {1,376}, {2,3579}, {3,962}, {4,9}, {7,3295}, {8,30}, {20,145}, {35,3485}, {46,497}, {55,3487}, {57,1058}, {63,5082}, {65,3488}, {165,631}, {329,5687}, {348,5195}, {355,3146}, {382,5690}, {387,5165}, {388,1770}, {390,942}, {443,3587}, {452,3753}, {484,1479}, {496,5435}, {499,5442}, {515,3529}, {548,5734}, {549,5550}, {550,1482}, {595,4000}, {758,3189}, {950,2093}, {952,1657}, {1006,5584}, {1044,1066}, {1056,1697}, {1062,4318}, {1125,3524}, {1155,3086}, {1210,5128}, {1385,3522}, {1698,3545}, {1699,3090}, {1737,5225}, {1836,3085}, {1837,5183}, {2099,4305}, {3057,4293}, {3058,5221}, {3149,6244}, {3241,3534}, {3340,4304}, {3475,3746}, {3476,4299}, {3486,4302}, {3523,5886}, {3528,3576}, {3543,3617}, {3586,4848}, {3622,3656}, {3623,3655}, {3627,5790}, {3634,5071}, {3635,4297}, {3817,5067}, {3871,5905}, {3877,4190}, {3931,4307}, {3962,6001}, {4127,5693}, {4309,5902}, {4424,5716}, {4533,5777}, {4668,5691}, {4701,5881}, {5057,5552}, {5122,5265}

X(6361) = reflection of X(i) in X(j) for these (i,j): (4,40), (40,5493), (382,5690), (944,20), (962,3), (1482,550), (3146,355), (3241,3534), (3543,3654)
X(6361) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,553,3296), (3,962,5603), (4,40,5657), (4,5657,5818), (55,4295,3487), (65,4294,3488), (165,946,631), (484,1479,1788), (550,1482,5731), (1697,4292,1056), (1770,5119,388), (1836,3085,5714), (4299,5697,3476), (4302,5903,3486)

X(6362) = POINT ASPIDISKE 1

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B - cos C)(2 + cos B + cos C)

X(6362) lies on these lines: {30, 511}, {442, 1577}, {652, 4976}, {676, 905}, {1946, 5583}, {3649, 4804}, {4990, 6332}

X(6362) = crossdifference of every pair of points on line X(6)X(1174)

X(6363) = POINT ASPIDISKE 2

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (-1 + Cos[A])^2*(Cos[B] - Cos[C])*(-2 + Cos[B] + Cos[C])
Barycentrics g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a²(b - c)(b² + c² + ab + ac - 2bc)

X(6363) lies on these lines: {30, 511}, {58, 3733}, {764, 4017}, {1027, 1431}, {1459, 1960}, {2605, 4491}, {3709, 3768}

X(6363) = isogonal conjugate of X(8706)
X(6363) = crossdifference of every pair of points on line X(6)X(145)

X(6364) = POINT ASPIDISKE 3

Barycentrics (1 + sin A)(sin B - sin C) : (1 + sin B)(sin C - sin A) : (1 + sin C)(sin A - sin B)

X(6364) lies on this line: {30, 511} (the infinity line)

X(6364) = isogonal conjugate of X(6135)

X(6365) = POINT ASPIDISKE 4

Barycentrics (1 - sin A)(sin B - sin C) : (1 - sin B)(sin C - sin A) : (1 - sin C)(sin A - sin B)

X(6365) lies on this line: {30, 511} (the infinity line)

X(6365) = isogonal conjugate of X(6136)

X(6366) = POINT ASPIDISKE 5

Barycentrics f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B - cos C)(2 cos A - cos B - cos C)

X(6366) lies on these lines: {1, 676}, {8, 3904}, {11, 1146}, {30, 511}, {100, 658}, {104, 972}, {885, 1320}, {1145, 3126}, {1317, 1360}, {1638, 6174}, {2976, 6161}, {3004, 4477}, {3700, 4474}, {4391, 4990}

X(6366) = isogonal conjugate of X(14733)
X(6366) = isotomic conjugate of X(35157)
X(6366) = X(2)-Ceva conjugate of X(35091)
X(6366) = crossdifference of every pair of points on line X(6)X(109)

X(6367) = POINT ASPIDISKE 6

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

X(6367) lies on these lines: {30, 511}, {4024, 4705}, {4041, 4838}, {4647, 4978}, {4983, 4988}

X(6367) = isogonal conjugate of X(6578)
X(6367) = crossdifference of every pair of points on line X(6)X(593)

X(6368) = POINT ASPIDISKE 7

Barycentrics    cos²2B - cos²2C : :
Barycentrics    sin 2A sin(2B - 2C) : :
Barycentrics    (b^2 + c^2 - a^2)(b^2 - c^2)(b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :
Barycentrics    cot 2B - cot 2C : :

Let U be the anticomplementary triangle, and let V be the triangle is formed by that tangents to the bianticevian conic of X(2) and X(3) at the vertices of U. Then U and V are perspective, and X(6368) is their perspector.

X(6368) lies on these lines: {30, 511}, {201, 656}, {684, 2525}, {879, 1176}, {2081, 2600}, {3267, 6333}

X(6368) = isogonal conjugate of X(933)
X(6368) = isotomic conjugate of polar conjugate of X(12077)
X(6368) = X(2)-Ceva conjugate of X(39019)
X(6368) = barycentric square root of X(39019)
X(6368) = crossdifference of every pair of points on line X(6)X(24)
X(6368) = perspector of hyperbola {{A,B,C,X(2),X(5)}}
X(6368) = X(163)-isoconjugate of X(275)

X(6369) = POINT ASPIDISKE 8

Barycentrics [cos(A - B) - cos(A - C)] cos(B - C) : [cos(B - C) - cos(B - A)] cos(C - A) : [cos(C - A) - cos(C - B)] cos(A - B)
Barycentrics sec(A - B) - sec(A - C) : :

X(6369) lies on this line: {30, 511} (the infinity line)

X(6369) = isogonal conjugate of X(36078)
X(6369) = crossdifference of every pair of points on line X(6)X(2148)

X(6370) = POINT ASPIDISKE 9

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

X(6370) lies on these lines: {30, 511}, {351, 4809}, {643, 4427}, {1109, 2632}, {1637, 1639}, {1769, 2292}, {3268, 4453}, {3569, 4016}, {4036, 4064}, {4647, 4768}, {4707, 4736}

X(6370) = isogonal conjugate of X(36069)
X(6370) = crossdifference of every pair of points on line X(6)X(163)

X(6371) = POINT ASPIDISKE 10

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

X(6371) lies on these lines: {6,2483}, {30,511}, {649,854}, {650,2532}, {656,2530}, {659,3737}, {665,798}, {667,1459}, {1960,2605}, {2499,2978}, {2526,4524}, {3250,3709}, {3271,4475}, {3733,5009}, {3768,4079}, {4010,4985}

X(6371) = crossdifference of every pair of points on line X(6)X(8)
X(6371) = isogonal conjugate of X(8707)
X(6371) = isotomic conjugate of anticomplement of X(39015)
X(6371) = X(2)-Ceva conjugate of X(39015)
X(6371) = perspector of hyperbola {{A,B,C,X(2),X(56)}}
X(6371) = barycentric square root of X(39015)

X(6372) = POINT ASPIDISKE 11

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

X(6372) lies on these lines: {30,511}, {659,1019}, {661,665}, {663,4378}, {667,4724}, {764,4983}, {1491,4905}, {1734,4490}, {1960,4040}, {2254,4705}, {2533,3762}, {3250,4813}, {3271,4403}, {3837,4129}, {4010,4978}, {4063,4784}, {4449,4775}, {4498,4834}

X(6372) = crossdifference of every pair of points on line X(6)X(1621)
X(6372) = isogonal conjugate of X(8708)

X(6373) = POINT ASPIDISKE 12

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

X(6373) lies on these lines: {30,511}, {42,890}, {291,659}, {665,3572}, {668,3888}, {1015,1960}, {2978,4813}, {4507,4790}

X(6373) = crossdifference of every pair of points on line X(6)X(190)
X(6373) = isogonal conjugate of X(8709)
X(6373) = polar conjugate of isotomic conjugate of X(22092)
X(6373) = ideal point of PU(25)

X(6374) = SS(a → bc) OF X(3)

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

"SS(a → bc)" denotes the barycentric symbolic substitution a → bc, b →ca, c → ab, which carries lines to lines, conics to conics, cubics to cubics, etc.; see X(6337).

X(6374) is the center of conic {{A,B,C,X(670),X(689),X(1978)}}. (This conic is the isogonal conjugate of line X(669)X(688, which is the trilinear polar of X(32)), and it is also the isotomic conjugate of the Lemoine axis.

As a line L varies through the set of all lines that pass through X(76), the locus of the trilinear pole of L is a circumconic, and its center is X(6374). As W varies through the set of all conics that pass through the isotomic conjugates of the incenter and the excenters, the locus of the center of W is X(6374). (Randy Hutson, February 16, 2015)

X(6374) lies on these lines: {2,2998}, {6,670}, {69,3978}, {75,982}, {76,141}, {157,5152}, {192,1978}, {264,5117}, {304,1921}, {305,3314}, {308,3763}, {325,1368}, {5286,6338}

X(6374) = isotomic conjugate of X(3224)
X(6374) = complement of X(2998)
X(6374) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,76), (670,3221)
X(6374) = anticomplement of X(6375)
X(6374) = crosspoint of X(2) and X(194)
X(6374) = crosssum of X(6) and X(3224)
X(6374) = perspector of circumconic centered at X(76)
X(6374) = centroid of X(75) and its three extraversions
X(6374) = X(i)-isoconjugate of X(j) for these {i,j}: {31,3224}, {32,3223}, {560,2998}, {1924,3222}, {1973,3504}
X(6374) = X(i)-complementary conjugate of X(j) for these (i,j): (31,76), (194,2887), (662,3221), (1424,2886), (1613,10), (1740,141)

X(6375) = SS(a → bc) OF X(5)

Barycentrics a^2 (a^2 (b^2 - c^2)^2 - b^2 c^2 (b^2 + c^2)) : :
X(6375) = 3X(2) + X(2998)

See X(6374).

X(6375) lies on these lines: {2,2998}, {6,3229}, {39,698}, {141,1084}, {230,800}, {3117,3618}

X(6375) = complement of X(6374)
X(6375) = isotomic conjugate of isogonal conjugate of X(9490)
X(6375) = polar conjugate of isogonal conjugate of X(23221)
X(6375) = centroid of X(37) and its three extraversions
X(6375) = X(4609)-Ceva conjugate of X(512)
X(6375) = X(i)-complementary conjugate of X(j) for these (i,j): (3223,626), (3224,2887)
X(6375) = bicentric sum of PU(154)
X(6375) = midpoint of PU(154)

X(6376) = SS(a → bc) OF X(9)

Barycentrics bc(bc - ab - ac) : ca(ca - bc - ba) : ab(ab - ca - cb)

See X(6374).

X(6376) is the center of the conic W = {{A,B,C,X(668), X(789), X(799), X(811), X(1978), X(4554), X(4593), X(4602)}. W is the isotomic conjugate of the antiorthic axis, as well as the the isogonal conjugate of the line X(667)X(788). (Randy Hutson, February 16, 2015)

The conic W is also the locus of the trilinear pole of a line that varies through the set of all lines through X(75), as well as the locus of the trilinear product of the two Steiner circumellipse intercepts of a line that varies through the set of all the lines through X(2). The trilinear polar of X(6376) passes through X(3835). (Randy Hutson, February 16, 2015)

X(6376) lies on these lines: 1,668}, {2,330}, {8,350}, {9,1966}, {10,75}, {12,85}, {37,2998}, {41,3570}, {69,2551}, {86,979}, {183,958}, {190,3501}, {192,4110}, {194,1575}, {257,312}, {274,1698}, {319,5722}, {325,1329}, {384,4386}, {385,4426}, {538,1574}, {561,756}, {612,1965}, {750,799}, {993,1078}, {1376,1975}, {1500,4033}, {1573,3934}, {1655,2276}, {1930,3992}, {3061,3452}, {3208,4595}, {3507,3905}, {3617,4441}, {3679,3760}, {3820,3933}, {4377,4708}

X(6376) = isotomic conjugate of X(87)
X(6376) = complement of X(330)
X(6376) = anticomplement of X(16604)
X(6376) = complementary conjugate of X(20255)
X(6376) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,75), (668,4083), (1221,321)
X(6376) = X(i)-cross conjugate of X(j) for these (i,j): (3123,3835), (3971,192), (4147,4595)
X(6376) = perspector of circumconic centered at X(75)
X(6376) = polar conjugate of isogonal conjugate of X(22370)
X(6376) = X(i)-complementary conjugate of X(j) for these (i,j): (6,3840), (31,75), (41,3061), (43,141), (101,4083), (192,2887), (1403,142), (1423,2886), (2176,10), (2209,2), (3208,1329), (4083,116)
X(6376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,76,75), (313,5224,75), (984,1921,75), (1698,3761,274), (3264,4389,75), (3596,4357,75), (3661,3948,312)
X(6376) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2162}, {31,87}, {32,330}, {56,2053}, {604,2319}, {667,932}, {1919,4598}, {1977,5383}

X(6377) = SS(a → bc) OF X(11)

Barycentrics a²(b - c)²(bc - ab - ac) : b²(c - a)²(ca - bc - ba) : c²(a - b)²(ab - ca - cb)

See X(6374).

X(6377) lies on these lines: 2,1978}, {39,4850}, {115,5515}, {239,3229}, {244,665}, {292,1054}, {512,4117}, {649,1977}, {1084,1086}, {1386,1908}, {2666,5625}, {3117,5222}, {3226,4598}

X(6377) = isogonal conjugate of X(5383)
X(6377) = isotomic conjugate of isogonal conjugate of X(21762)
X(6377) = polar conjugate of isogonal conjugate of X(22386)
X(6377) = complementary conjugate of X(21262)
X(6377) = complement of X(1978)
X(6377) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3835), (75,512), (192,4083), (330,513), (2998,514), (3248,1015)
X(6377) = X(3123)-cross conjugate of X(1015)
X(6377) = X(i)-complementary conjugate of X(j) for these (i,j): (31,3835), (32,513), (513,626), (560,514), (649,2887), (667,141), (669,1211), (798,3454), (1333,512), (1397,4885), (1501,650), (1918,4129), (1919,10), (1922,3837), (1924,1213), (1977,11), (1980,2), (2205,661), (2206,4369), (3063,1329), (3121,125), (3248,116), (4612,3037)
X(6377) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2162}, {31,87}, {32,330}, {56,2053}, {604,2319}, {667,932}, {1919,4598}, {1977,5383}

X(6378) = SS(a → bc) OF X(12)

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

See X(6374).

X(6378) lies on these lines: 2,1221}, {9,87}, {37,714}, {330,5283}, {594,1084}, {894,3229}, {932,2375}, {2162,4275}, {3117,5749}

X(6378) = complement of X(34086)
X(6378) = X(3123)-cross conjugate of X(1015)
X(6378) = X(i)-isoconjugate of X(j) for these {i,j}: {{43,1509}, {192,757}, {261,1423}, {552,3208}, {763,3971}, {873,2176}, {2185,3212}, {3123,4590}, {4083,4610}

X(6379) = SS(a → bc) OF X(30)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4 b^4-2 a^4 b^2 c^2+a^2 b^4 c^2+a^4 c^4+a^2 b^2 c^4-2 b^4 c^4

See X(6374).

X(6379) lies on these lines: {2,2998}, {30,511}, {670,3229}

X(6379) = isogonal conjugate of X(6380)

X(6380) = ISOGONAL CONJUGATE OF X(6379)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (2 a^4 b^4-a^4 b^2 c^2-a^2 b^4 c^2-a^4 c^4+2 a^2 b^2 c^4-b^4 c^4) (a^4 b^4+a^4 b^2 c^2-2 a^2 b^4 c^2-2 a^4 c^4+a^2 b^2 c^4+b^4 c^4)

See X(6374).

X(6380) lies on the circumcircle and these lines: {6,3222}, {99,1613}

X(6381) = SS(a → bc) OF X(44)

Barycentrics bc(2bc - ab - ac) : ca(2ca - bc - ba) : ab(2ab - ca - cb)
X(6381) = X(75) - 3X(1921)

See X(6374).

X(6381) lies on these lines: {2,3761}, {8,3760}, {10,75}, {37,716}, {85,3947}, {183,993}, {190,4495}, {274,3634}, {325,3814}, {334,4013}, {350,519}, {514,661}, {538,1575}, {545,4506}, {561,3971}, {799,5209}, {1078,5267}, {1107,3934}, {1111,3263}, {1125,1909}, {1237,4075}, {1323,4554}, {1329,3933}, {1500,4681}, {1930,3701}, {2295,4721}, {3008,3975}, {3230,4465}, {3661,4044}, {3679,4441}, {3734,4386}, {3770,5750}, {4364,4377}, {4396,5291}, {4410,4472}, {4419,4494}, {4479,4669}, {4723,4986}, {5088,5205}

X(6381) = midpoint of X(350) and X(668)
X(6381) = isotomic conjugate of X(37129)
X(6381) = X(3994)-cross conjugate of X(536)
X(6381) = X(715)-complementary conjugate of X(3739)
X(6381) = {X(1111),X(3992)}-harmonic conjugate of X(3263)
X(6381) = X(i)-isoconjugate of X(j) for these {i,j}: {6,739}, {32,3227}, {667,898}, {889,1980}, {1919,4607}, {1977,5381}

X(6382) = SS(a → bc) OF X(55)

Barycentrics b²c²(bc - ab - ac) : c²a²(ca - bc - ba) : a²b²(ab -ca- cb)

See X(6374).

X(6382) lies on these lines: {2,1978}, {8,3978}, {55,874}, {75,982}, {76,321}, {312,1921}, {594,1502}, {670,4363}, {1965,3923}, {1966,4362}, {3263,3705}, {6063,6358}

X(6382) = isotomic conjugate of X(2162)
X(6382) = complement of polar conjugate of isogonal conjugate of X(23183)
X(6382) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,76), (1978,3835)
X(6382) = {X(321),X(561)}-harmonic conjugate of X(76)
X(6382) = X(i)-isoconjugate of X(j) for these {i,j}: {31,2162}, {32,87}, {330,560}, {604,2053}, {932,1919}, {1397,2319}, {1980,4598}

X(6383) = SS(a → bc) OF X(56)

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

See X(6374).

X(6383) lies on these lines: {2,1221}, {7,3978}, {75,982}, {76,3662}, {85,1921}, {87,870}, {274,330}, {334,3596}, {670,4361}, {767,932}, {1086,1502}, {1218,3739}, {1278,1978}

X(6383) = isotomic conjugate of X(2176)
X(6383) = X(561)-cross conjugate of X(76)
X(6383) = X(i)-isoconjugate of X(j) for these {i,j}: {6,2209}, {31,2176}, {32,43}, {41,1403}, {192,560}, {1397,3208}, {1423,2175}, {1980,4595}

X(6384) = SS(a → bc) OF X(57)

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

See X(6374).

X(6384) lies on these lines: {2,330}, {7,350}, {57,1966}, {75,982}, {76,3840}, {86,87}, {244,561}, {312,335}, {334,3705}, {614,1965}, {673,2319}, {675,932}, {748,799}, {903,4479}, {940,2162}, {2296,3720}, {4373,4441}

X(6384) = isogonal conjugate of X(2209)
X(6384) = isotomic conjugate of X(43)
X(6384) = complement of polar conjugate of isogonal conjugate of X(23177)
X(6384) = anticomplement of polar conjugate of isogonal conjugate of X(23213)
X(6384) = X(330)-Ceva conjugate of X(75)
X(6384) = X(i)-cross conjugate of X(j) for these {i,j}: (76,75), (3123,514), (3662,85), (3840,2), (3944,92)
X(6384) = {X(982),X(1920)}-harmonic conjugate of X(75)
X(6384) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2209}, {6,2176}, {31,43}, {32,192}, {41,1423}, {55,1403}, {604,3208}, {692,4083}, {1110,3123}, {1919,4595}, {2175,3212}, {2206,3971}

X(6385) = SS(a → bc) OF X(58)

Barycentrics b³c³(a + b)(a + c) : c³a³(b + c)(b + a) : a³b³(c + a)(c + b)

See X(6374).

X(6385) lies on these lines: {2,1218}, {37,1221}, {38,75}, {76,141}, {86,870}, {99,767}, {274,1107}, {313,334}, {314,670}, {1444,4623}, {1978,4043}, {3770,3978}

X(6385) = isogonal conjugate of X(2205)
X(6385) = isotomic conjugate of X(213)
X(6385) = X(i)-cross conjugate of X(j) for these {i,j}: (76,310), (3125,693), (3662,86)
X(6385) = X(i)-isoconjugate of X(j) for these {i,j}: {1,2205}, {6,1918}, {10,1501}, {25,2200}, {31,213}, {32,42}, {37,560}, {41,1402}, {71,1974}, {100,1924}, {101,669}, {184,2333}, {228,1973}, {321,1917}, {688,4628}, {692,798}, {872,1333}, {1018,1980}, {1084,4570}, {1110,3121}, {1334,1397}, {1400,2175}, {1409,2212}, {1500,2206}, {1576,4079}, {1919,4557}, {1922,3747}, {2207,4055}, {4117,4567}

X(6386) = SS(a → bc) OF X(101)

Barycentrics b³c³(a - b)(a - c) : c³a³(b - c)(b - a) : a³b³(c - a)(c - b)
X(6386) = 3X(2) + X(6339)

See X(6374).

X(6386) lies on these lines: {75,3123}, {76,1086}, {99,839}, {100,689}, {313,334}, {668,670}, {1332,4601}, {1502,3596}, {1920,4377}, {1978,3807}, {6331,6335}

X(6386) = midpoint of X(6338) and X(6339)
X(6386) = isogonal conjugate of X(1980)
X(6386) = isotomic conjugate of X(667)
X(6386) = complement of polar conjugate of isogonal conjugate of X(23190)
X(6386) = anticomplement of polar conjugate of isogonal conjugate of X(23227)
X(6386) = X(4602)-Ceva conjugate of X(1978)
X(6386) = X(i)-cross conjugate of X(j) for these {i,j}: (693,76), (4705,321)
X(6386) = trilinear pole of line X(76)X(321)
X(6386) = X(i)-isoconjugate of X(j) for these {i,j}: {1,1980}, {6,1919}, {31,667}, {32,649}, {58,669}, {81,1924}, {101,1977}, {163,3121}, {512,2206}, {513,560}, {514,1501}, {604,3063}, {663,1397}, {692,3248}, {693,1917}, {798,1333}, {810,2203}, {875,2210}, {1019,2205}, {1084,4556}, {1252,3249}, {1356,4636}, {1395,1946}, {1459,1974}, {1474,3049}, {1576,3122}, {1918,3733}, {1927,4164}

X(6387) = SS(a → S_A) OF X(5)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^8-2 a^6 b^2-6 a^4 b^4-2 a^2 b^6+b^8-2 a^6 c^2+16 a^4 b^2 c^2+2 a^2 b^4 c^2-6 a^4 c^4+2 a^2 b^2 c^4-2 b^4 c^4-2 a^2 c^6+c^8

See X(6374).

X(6387) lies on this line: {2,6338}

X(6387) = complement of X(6338)

X(6388) = SS(a → S_A) OF X(11)

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

See X(6374).

X(6388) lies on these lines: {2,2987}, {6,5181}, {110,5477}, {111,3448}, {115,125}, {126,4576}, {136,2501}, {373,1506}, {468,1692}, {1112,1560}, {3066,5475}, {3291,3580}

X(6388) = complement of X(4563)
X(6388) = trilinear pole, wrt medial triangle, of the line X(5)X(6)

X(6389) = SS(a → S_A) OF X(39)

Barycentrics (a^2-b^2-c^2)^2 (a^4+b^4-2 b^2 c^2+c^4) : :
Barycentrics tan^2 B + tan^2 C : :

See X(6374).

X(6389) lies on these lines: {2,216}, {3,66}, {6,441}, {69,248}, {122,126}, {127,131}, {193,3284}, {317,401}, {338,2165}, {426,1899}, {1589,5590}, {1590,5591}, {2968,4361}, {3546,3788}, {3547,3934}, {3618,5158}, {3926,6338}, {4363,6356}

X(6389) = complement of X(393)

X(6390) = SS(a → S_A) OF X(44)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a^2-b^2-c^2) (2 a^2-b^2-c^2)
X(6390) = 3X(99) + X(316) = X(316) - 3X(325) = X(187) - 3X(2482) = 6X(2482) - X(3793)

See X(6374).

X(6390) lies on these lines: {2,2418}, {3,69}, {5,1975}, {30,99}, {32,3629}, {39,698}, {76,140}, {141,574}, {183,549}, {187,524}, {193,1384}, {194,5305}, {230,538}, {315,550}, {381,1007}, {441,525}, {468,3266}, {543,625}, {732,2021}, {1078,3530}, {3035,6382}, {3053,6144}, {3167,4176}, {3734,3815}, {3760,5433}, {3761,5432}, {3763,5013}, {3788,5254}

X(6390) = midpoint of X(99) and X(325)
X(6390) = reflection of X(i) in X(j) for these (i,j): (230,620), (3793,187)
X(6390) = isotomic conjugate of X(17983)
X(6390) = isogonal conjugate of X(8753)
X(6390) = complement of polar conjugate of X(2374)
X(6390) = crossdifference of every pair of points on line X(25)X(2489)

X(6391) = SS(a → S_A) OF X(65)

Trilinears    1/(sec A - 2 csc A tan ω : :
X(6391): Trilinears    1/(2 csc A - sec A cot ω) : :
Barycentrics    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^2 (a^2+b^2-3 c^2) (a^2-b^2-c^2) (a^2-3 b^2+c^2)
X(6391) = 4X(6) - 3X(3167) = 2X(155) - 3X(5093)

See X(6374).

X(6391) lies on these lines: {4,193}, {6,1196}, {54,5050}, {64,511}, {65,3751}, {66,524}, {69,1368}, {74,3565}, {141,5486}, {155,3527}, {159,1177}, {1350,3532}, {1353,6193}, {2213,4260}, {3618,5544}, {5965,6145}

X(6391) = isogonal conjugate of X(6353)
X(6391) = reflection of X(6193) in X(1353)

X(6392) = SS(a → S_A) OF X(69)

Barycentrics f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a^4+2 a^2 b^2+b^4+2 a^2 c^2-6 b^2 c^2+c^4
Barycentrics cot^2 A - cot^2 B - cot^2 C : :
X(6392) = 3X(2) - 4X(3767) = 9X(2) - 8X(3788) = 3X(3767) - 2X(3788) = 4X(3788) - 3X(3926)

See X(6374). X(6392) = SS(a → S²_A) OF X(8)

X(6392) lies on these lines: {2,39}, {4,193}, {20,385}, {69,5254}, {99,439}, {148,2794}, {230,6337}, {384,5304}, {393,6339}, {1657,3793}, {2549,3785}, {3734,5319}, {5032,5395}

X(6392) = reflection of X(3926) in X(3767)
X(6392) = isotomic conjugate of X(6339)
X(6392) = anticomplement of X(3926)
X(6392) = X(393)-Ceva conjugate of X(2)
X(6392) = X(6338)-cross conjugate of X(2)
X(6392) = polar conjugate of cyclocevian conjugate of X(38259)
X(6392) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (19,1370), (25,4329), (158,315), (393,6327), (1096,69), (1395,347), (1973,20), (1974,6360), (2207,8), (6059,329) X(6392) = X(i)-isoconjugate of X(j) for these {i,j}: {3,2129}, {31,6339}
X(6392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (76,5286,2), (193,2996,4), (3767,3926,2)

X(6393) = SS(a → S_A) OF X(238)

Barycentrics (a^2-b^2-c^2) (a^2 b^2-b^4+a^2 c^2-c^4) : :
Barycentrics csc A cot A cos(A + ω) : :

See X(6374).

X(6393) lies on these lines: {2,4176}, {3,69}, {30,5207}, {76,141}, {99,1503}, {114,325}, {249,524}, {305,343}, {315,1350}, {525,3267}, {620,1692}, {858,4576}, {1352,1975}, {3266,3580}, {3788,5028}, {3917,4121}, {5031,5969}

X(6393) = reflection of X(1692) in X(620)
X(6393) = isotomic conjugate of X(6531)
X(6393) = isotomic conjugate of isogonal conjugate of X(36212)

X(6394) = SS(a → S_A) OF X(292)

Barycentrics (a^2-b^2-c^2)^2 (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) : :
Barycentrics cos A cot A sec(A + ω) : :

See X(6374).

X(6394) lies on these lines: {3,76}, {69,248}, {264,3148}, {305,426}, {325,441}, {336,1231}, {418,1799}, {865,2374}, {879,6333}, {1092,3926}, {2366,2715}, {2972,4563}

X(6394) = isogonal conjugate of X(34854)
X(6394) = isotomic conjugate of X(6530)
X(6394) = X(19)-isoconjugate of X(232)
X(6394) = X(92)-isoconjugate of X(2211)
X(6394) = crossdifference of every pair of points on line X(2491)X(17994) (the tangent to the inellipse that is the barycentric square of the orthic axis, at the barycentric square of X(232))

Centers associated with Lucas triangles and Lucas circles: X(6395)-X(6502)

Contributed by Randy Hutson, February 3, 2015

Definitions are given at MathWorld:
Lucas Central Triangle
Lucas Circles
Lucas Central Circle
Lucas Radical Circle
Lucas Inner Triangle
Lucas Inner Circle
Lucas Tangents Triangle

Additional definitions are given in ETC:

Lucas(t) circles, defined at X(371), where t = L/W
Lucas(t) homothetic triangle, defined at X(493)
Lucas(t) reflection triangle, defined at X(6401)
Lucas secondary circles, defined at X(6199)
Lucas secondary central triangle, defined at X(6199)
1st Lucas secondary tangents triangle, defined at X(6199)
2nd Lucas secondary tangents triangle, defined at X(6199)
Lucas secondary tangents circle, defined at X(6199)
Lucas(t) Brocard triangle, defined at X(6421)
Lucas(t) antipodal triangle, defined at X(6457)
Lucas(t) inner tangential triangle = tangential triangle of the Lucas(t) inner triangle
circumsymmedial tangential triangle = tangential triangle of the circumsymmedial triangle

An effort is being made to identify a publication in which Édouard Lucas (1842-1891) introduced the circles that are now known as Lucas circles. They are described in at least two articles:

Antreas P. Hatzipolakis and Paul Yiu, The Lucas Circles of a Triangle, The American Mathematical Monthly 108 (2001) 444-446.

Peter J. C. Moses, Circles and Triangle Centers Associated with the Lucas Circles, Forum Geometricorum 5 (2005) 97-106.

X(6395) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS(-1) CENTRAL TRIANGLE

Trilinears    3 cos A - 4 sin A : :
Trilinears    a(3S_A - 4S) : :
Barycentrics    a^2 (3 a^2-3 b^2-3 c^2+8 S) : : (Richard Hilton, 2/3/2015)

X(6395) lies on these lines: {3,6}, {5,1131}, {323,1584}, {381,3069}, {485,5070}, {486,3843}, {615,5055}, {1132,3853}, {1328,3830}, {1482,1703}, {1587,1656}, {1588,1657}, {1597,5411}, {3068,5054}, {3070,3851}, {3071,5073}

X(6395) = perspector of 1st and 2nd Lucas(-1) secondary tangents triangles
X(6395) = radical center of Lucas(-8/3) circles
X(6395) = inverse-in-Lucas(-1)-secondary-tangents-circle of X(187)
X(6395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6199), (6,1152,6200), (6,6200,3311), (372,3312,3), (372,3594,3312), (1131,3317,5), (1152,3311,3)

X(6396) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS(-1) TANGENTS TRIANGLE

Trilinears    3 cos A - sin A : :
Trilinears   a(3S_A - S) : :
Barycentrics    a^2 (3 a^2-3 b^2-3 c^2+2 S) : : (Richard Hilton, 2/3/2015)

Let A′, B′, C′ be the free vertices of the 2nd Kenmotu squares. Let A″, B″, C″ be the reflections of A′, B′, C′ in X(372). Triangle A″B″C″ is homothetic to ABC at X(6396). The following 4 triangles are pairwise perspective: (1) circumsymmedial, (2) Lucas(-1) tangents, (3) Lucas(-4) central, and (4) 1st Lucas secondary tangents triangles. Their common perspector is X(6396). (Randy Hutson, January 29, 2015)

Let A′B′C′ and A″B″C″ be the outer and inner Vecten triangles, resp. Let A* be the trilinear pole, wrt A′B′C′, of line B″C″. Define B*, C* cyclically. The lines A′A*, B′B*, C′C* concur in X(6396).

X(6396) lies on these lines: {2,1327}, {3,6}, {4,5420}, {20,486}, {30,615}, {35,6502}, {36,5414}, {76,6312}, {140,3070}, {186,5412}, {323,5409}, {376,3069}, {485,631}, {488,3620}, {490,640}, {549,590}, {550,3071}, {638,642}, {1124,5217}, {1335,5204}, {1495,3156}, {1584,5406}, {1587,3523}, {1588,3522}, {1600,5408}, {2066,5010}, {2362,3612}, {3068,3524}, {3092,3516}, {3093,3515}

X(6396) = isogonal conjugate of X(1328)
X(6396) = perspector of Lucas(-1) central triangle and circumsymmedial tangential triangle
X(6396) = exsimilicenter of circumcircle and Lucas(-1) radical circle
X(6396) = radical center of Lucas(-2/3) circles
X(6396) = trilinear pole, wrt outer Vecten triangle, of line X(2501)X(3566) (the perspectrix of the outer and inner Vecten triangles)
X(6396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6200), (3,372,371), (3,1152,372), (3,3312,1151), (6,1151,6199), (6,6200,371), (15,16,372), (187,3098,6200), (372,6200,6), (574,5092,6200), (1587,3523,5418), (3312,6199,6), (3371,3372,1152), (3385,3386,3311)

X(6397) = PERSPECTOR OF INNER GREBE TRIANGLE AND LUCAS(-1) TANGENTS TRIANGLE

Trilinears a*[(2*S_ω-S)*(S_A*(S_A+S_ω-3*S)+(4*S-S_ω)*S)-12*S^2*R^2] : : (Richard Hilton, 2/3/2015)
Barycentrics a^2 ((3 a^6+2 a^4 b^2-9 a^2 b^4+4 b^6+2 a^4 c^2-18 a^2 b^2 c^2-12 b^4 c^2-9 a^2 c^4-12 b^2 c^4+4 c^6)-2 (4 a^4-5 a^2 b^2-3 b^4-5 a^2 c^2-14 b^2 c^2-3 c^4) S) : : (Richard Hilton, 2/3/2015)

X(6397) lies on this line: {372, 5861}

X(6398) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND LUCAS(-1) TANGENTS TRIANGLE

Trilinears    3 cos A - 2 sin A : :
Trilinears    a(3S_A - 2S) : :
Trilinears   a*(3*b^2+3*c^2-3*a^2-4*S) : : (Richard Hilton, 2/3/2015)
Barycentrics    a^2 (3 a^2-3 b^2-3 c^2+4 S) : : (Richard Hilton, 2/3/2015)

Let Γ be the circumcircle and O(A) the A-Lucas(-1) circle. Let K_A = 1 if O(A) is internally tangent to Γ, and let K_A = -1 if O(A) is externally tangent to Γ. Define K_B and K_C cyclically. Let L_A and R_A be the center and radius of O(A), and define L_B and L_C, and also R_B and R_C, cyclically. Then X(6398) = (K_A/R_A)*L_A + (K_B/R_B)*L_B + (K_C/R_C)*L_C. Also, the circumsymmedial triangle, Lucas(-1) inner triangle, and Lucas(-2) central triangle are pairwise perspective, and their common perspector is X(6398). Let A′B′C′ be the Lucas(-1) central triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6398). Let A*B*C* be the Lucas(-1) tangents triangle. Let A** be the trilinear pole, wrt A*B*C*, of line BC. Define B**, C** cyclically. The lines A*A**, B*B**, C*C** concur in X(6398). (Randy Hutson, January 29, 2015)

X(6398) lies on these lines: {3,6}, {4,3591}, {30,3069}, {140,1587}, {186,5410}, {323,1600}, {378,5411}, {381,615}, {382,486}, {485,3526}, {488,3619}, {549,3068}, {550,1588}, {590,5054}, {999,5414}, {1131,5067}, {1385,1703}, {1597,5413}, {1656,3070}, {1657,3071}, {3093,3517}, {3295,6502}, {3299,5217}, {3301,5204}, {3317,3832}

X(6398) = radical center of Lucas(-4/3) circles
X(6398) = X(6) of Lucas(-1) tangents triangle
X(6398) = inverse-in-Lucas(-1)-radical-circle of X(187)
X(6398) = insimilicenter of circumcircle and Lucas(-1) inner circle
X(6398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6221), (3,372,3312), (3,3312,3311), (3,6199,6200), (6,6200,6199), (6,6221,3311), (15,16,1152), (371,372,3594), (372,1152,3), (3070,5420,1656), (3312,6221,6), (6199,6200,6221)
X(6398) = trilinear pole with respect to Lucas(-1) central triangle of Lemoine axis
X(6398) = trilinear pole with respect to Lucas(-1) tangents triangle of Lemoine axis
X(6398) = X(7) of Lucas(-1) central triangle, if all Lucas(-1) circles are externally tangent

X(6399) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND 1st NEUBERG TRIANGLE

Trilinears (csc A){(S+2*S_ω)*[(S_A-S_ω)*(S-S_ω)+S^2]*S_A-S^3*(2*S-S_ω)} : : (Richard Hilton, 2/3/2015)
Barycentrics 5 a^8-7 a^6 b^2+7 a^4 b^4-5 a^2 b^6-7 a^6 c^2-2 a^4 b^2 c^2+5 a^2 b^4 c^2-4 b^6 c^2+7 a^4 c^4+5 a^2 b^2 c^4+8 b^4 c^4-5 a^2 c^6-4 b^2 c^6-4 a^2 (a^2 b^2-b^4+a^2 c^2-c^4) S : : (Richard Hilton, 2/3/2015)

X(6399) lies on these lines: {2,6290}, {511,6222}, {591,6280}, {1350,6316}.

X(6400) = CIRCUMORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS(-1) CENTRAL TRIANGLE

Trilinears a*((-2*S-S_ω)*S_A^2+2*S_A*S^2+S^2*(-2*S+S_ω))/S_A : :
Barycentrics a^2(a^4b^2 + a^4c^2 - 2a^2b^4 - 2a^2c^4 - 3a^2b^2c^2 + b^6 + c^6 + 4b^2c^2S)/(b^2 + c^2 - a^2) : :

X(6400) lies on these lines: {4, 69}, {24, 1152}, {1307, 3563}, {1587, 3567}, {5871, 6220}, {6197, 6404}, {6198, 6405}

X(6400) = isogonal conjugate of X(6402)
X(6400) = {X(4),X(6403)}-harmonic conjugate of X(6239)

X(6401) = PERSPECTOR OF ABC AND LUCAS REFLECTION TRIANGLE

Barycentrics SA/[(2*S-Sω)*SA^2+2*SA*S^2+S^2*(2*S+Sω)] : :
Barycentrics (b^2 + c^2 - a^2)/(a^4b^2 + a^4c^2 - 2a^2b^4 - 2a^2c^4 - 3a^2b^2c^2 + b^6 + c^6 - 4b^2c^2S) : :

Let A′B′C′ be the Lucas(t) central triangle. Let L_A be the reflection of line B′C′ in line BC, and define L_B, L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. The triangle A″B″C″ is introduced here as the Lucas(t) reflection triangle. The Lucas(t) reflection triangle is perspective to ABC, the Lucas(u) central triangle, the Lucas(u) tangents triangle, the Lucas(u) inner triangle, and the Lucas(u) reflection triangle, for all t, u.

X(6401) lies on the conic {{A,B,C,X(3),X(25),X(32),X(98)}} and these lines: {3, 11941}, {6, 11984}, {32, 7583}, {67, 19390}, {262, 14167}, {371, 11977}, {372, 11975}, {488, 42065}, {1151, 11959}, {1152, 11960}, {1916, 22499}, {2782, 13029}, {3564, 10670}, {6402, 11983}, {6417, 11967}, {6418, 11969}, {6419, 11971}, {6420, 11973}, {6421, 11937}, {6422, 11938}, {6425, 11963}, {6426, 11964}, {6776, 39801}

X(6401) = isogonal conjugate of X(6239)

X(6402) = PERSPECTOR OF ABC AND LUCAS(-1) REFLECTION TRIANGLE

Barycentrics SA/[(-2*S-Sω)*SA^2+2*SA*S^2+S^2*(-2*S+Sω)]: :
Barycentrics (b^2 + c^2 - a^2)/(a^4b^2 + a^4c^2 - 2a^2b^4 - 2a^2c^4 - 3a^2b^2c^2 + b^6 + c^6 + 4b^2c^2S) : :

X(6402) lies on the conic {{A,B,C,X(3),X(25),X(32),X(98)}} and these lines: {3, 11943}, {6, 11985}, {32, 7584}, {67, 19391}, {184, 8964}, {262, 14168}, {371, 11976}, {372, 11978}, {487, 42065}, {1151, 11961}, {1152, 11962}, {1916, 22500}, {2782, 13031}, {3564, 10674}, {6401, 11983}, {6417, 11970}, {6418, 11968}, {6419, 11974}, {6420, 11972}, {6421, 11939}, {6422, 11940}, {6425, 11965}, {6426, 11966}, {6776, 39802}

X(6402) = isogonal conjugate of X(6400)

X(6403) = KOSNITA-TO-ORTHIC SIMILARITY IMAGE OF X(6)

Barycentrics a^2(a^4b^2 + a^4c^2 - 2a^2b^4 - 2a^2c^4 - 3a^2b^2c^2 + b^6 + c^6)/(b^2 + c^2 - a^2) : :

X(6403) lies on these lines: {4,69}, {6,24}, {25,110}, {51,6353}, {52,193}, {66,70}, {112,5017}, {141,1594}, {143,3517}, {159,1614}, {182,186}, {232,5052}, {378,1350}, {389,6467}, {403,5480}, {427,2979}, {468,5640}, {568,1353}, {576,1974}, {935,2698}, {1216,3620}, {1469,1870}, {1503,6240}, {1986,2854}, {2393,5890}, {3056,6198}, {3089,5446}, {3098,3520}, {3147,3618}, {3313,3541}, {3515,5050}, {3564,3575}, {3779,6197}, {5965,6242}

X(6403) = reflection of X(i) in X(j) for these (i,j): (193,52) (6467,389) (4,1843)
X(6403) = {X(6239),X(6400)}-harmonic conjugate of X(4)
X(6403) = anticomplement of X(9967)
X(6403) = X(7)-of-circumorthic-triangle if ABC is acute

X(6404) = EXTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS(-1) CENTRAL TRIANGLE

Trilinears ((b^2+b*c+c^2)*a^5-(b+c)*(b^2-b*c+c^2)*a^4+2*(b^3*c-2*S*b*c+b^4+c^4+c^3*b+2*b^2*c^2)*a^3+2*(b+c)*(b^4-b^3*c+2*b^2*c^2-2*S*b*c-c^3*b+c^4)*a^2-(b+c)^2*(-b^4+b^3*c+c^3*b-4*S*b*c-c^4)*a-(b+c)*(bc)^2*(b^4+b^3*c+c^3*b-4*S*b*c+c^4))*a : : (Richard Hilton, 2/3/2015)
Barycentrics a^2 (a-b-c) (a+b-c) (a-b+c) (a+b+c) (a b^2-b^3+a b c+a c^2-c^3)-4 a^2 b c (a^3+a^2 b-a b^2-b^3+a^2 c-2 a b c+b^2 c-a c^2+b c^2-c^3) S : : (Richard Hilton, 2/3/2015)

X(6404) lies on these lines: {19,6406}, {40,511}, {55,1152}, {65,176}, {6197,6400}

X(6404) = {X(40),X(3779)}-harmonic conjugate of X(6252)

X(6405) = INTANGENTS-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS(-1) CENTRAL TRIANGLE

Trilinears (-(b^2-b*c+c^2)*a^2+(c^4+b^4+b*c^3+b^3*c-4*b*S*c))*a : :
Barycentrics a^2 (a-b-c) (a+b+c) (b^2-b c+c^2)+4 a^2 b c S : :

X(6405) lies on these lines: {1,256}, {55,1152}, {172,2066}, {1124,2330}, {6198,6400}

X(6405) = {X(1),X(3056)}-harmonic conjugate of X(6283)

X(6406) = ORTHIC-TRIANGLE-ORTHOLOGIC CENTER OF LUCAS(-1) CENTRAL TRIANGLE

Trilinears a*(2*S_A^2-S*S_A-S*(S_ω-2*S))/S_A : :
Barycentrics a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (2 b^2 c^2-b^2 S-c^2 S) : :

X(6406) lies on these lines: {4,69}, {19,6404}, {25,1152}, {51,3070}, {1968,5412}, {1974,3093}

X(6406) = {X(4),X(1843)}-harmonic conjugate of X(6291)
X(6406) = crosssum of X(3) and X(487)
X(6406) = X(175)-of-orthic-triangle if ABC is acute

X(6407) = CENTER OF LUCAS INNER CIRCLE

Trilinears 7 cos A + 4 sin A : :
Trilinears a(7S_A + 4S) : :

X(6407) lies on these lines: {3,6}, {30,1131}, {140,3317}, {485,5073}, {590,3843}, {1328,5055}, {1588,5054}, {1656,6459}, {1657,3068}, {3071,5070}, {3316,3845}, {3793,6462}, {3851,6561}

X(6407) = perspector of Lucas central triangle and Lucas inner triangle
X(6407) = radical center of Lucas(8/7) circles
X(6407) = {X(3),X(6)}-harmonic conjugate of X(6408)
X(6407) = {X(371),X(1152)}-harmonic conjugate of X(3311)

X(6408) = CENTER OF LUCAS(-1) INNER CIRCLE

Trilinears 7 cos A - 4 sin A : :
Trilinears a(7S_A - 4S) : :

X(6408) lies on these lines: {3,6}, {30,1132}, {140,3316}, {486,5073}, {615,3843}, {1327,5055}, {1587,5054}, {1656,6460}, {1657,3069}, {3070,5070}, {3317,3845}, {3793,6463}, {3851,6560}

X(6408) = perspector of Lucas(-1) central triangle and Lucas(-1) inner triangle
X(6408) = radical center of Lucas(-8/7) circles
X(6408) = {X(3),X(6)}-harmonic conjugate of X(6407)
X(6408) = {X(372),X(1151)}-harmonic conjugate of X(3312)

X(6409) = CENTER OF LUCAS SECONDARY TANGENTS CIRCLE

Trilinears 4 cos A + sin A : :
Trilinears a(4S_A + S) : :

The Lucas secondary tangents circle is introduced at X(6199).

X(6409) lies on these lines: {3,6}, {20,590}, {30,5418}, {485,550}, {486,549}, {2066,5204}, {3516,5412} .

X(6409) = perspector of symmedial triangle and Lucas inner tangential triangle
X(6409) = radical center of Lucas(1/2) circles
X(6409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6410), (3,371,1152), (3,1151,6), (371,1152,6), (1151,1152,371)

X(6410) = CENTER OF LUCAS(-1) SECONDARY TANGENTS CIRCLE

Trilinears 4 cos A - sin A : :
Trilinears a(4S_A - S) : :

X(6410) lies on these lines: {3,6}, {20,615}, {30,5420}, {485,549}, {486,550}, {3516,5413}, {5204,5414} .

X(6410) = perspector of symmedial triangle and Lucas(-1) inner tangential triangle
X(6410) = radical center of Lucas(-1/2) circles
X(6410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6409), (3,372,1151), (3,1152,6), (372,1151,6), (1151,1152,372)

X(6411) = RADICAL CENTER OF LUCAS SECONDARY CIRCLES

Trilinears 6 cos A + sin A : :
Trilinears a(6S_A + S) : :

X(6411) lies on these lines: {3,6}, {323,5406}, {485,548}, {486,3530}, {487,3631}, {488,3630} (at least).

X(6411) = radical center of Lucas(1/3) circles
X(6411) = perspector of Lucas(-1) tangents triangle and Lucas inner tangential triangle
X(6411) = perspector of 2nd Lucas secondary tangents triangle and circumsymmedial tangential triangle
X(6411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6412), (3,1151,1152)

X(6412) = RADICAL CENTER OF LUCAS(-1) SECONDARY CIRCLES

Trilinears 6 cos A - sin A : :
Trilinears a(6S_A - S) : :

X(6412) lies on these lines: {3,6}, {323,5704}, {485,3530}, {486,548}, {487,3630}, {488,3631} .

X(6412) = radical center of Lucas(-1/3) circles
X(6412) = perspector of Lucas tangents triangle and Lucas(-1) inner tangential triangle
X(6412) = perspector of 2nd Lucas(-1) secondary tangents triangle and circumsymmedial tangential triangle
X(6412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6411), (3,1152,1151)

X(6413) = ISOGONAL CONJUGATE OF X(1585)

Trilinears 1/(csc A + sec A) : :

Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(-1) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let P_A be the polar of A′ with respect to the circumcircle. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. The lines AA″, BB″, CC″ concur in X(6413).

X(6413) Lies on the Jerabek hyperbola and lines {4,371}, {6,3156}, {54,372}, {69,5409}, {184,216} .

X(6413) = X(92)-isoconjugate of X(371)
X(6413) = {X(184),X(216)}-harmonic conjugate of X(6414)
X(6413) = perspector of 1st Kenmotu diagonals triangle and cross-triangle of ABC and 1st Kenmotu diagonals triangle

X(6414) = ISOGONAL CONJUGATE OF X(1586)

Trilinears 1/(csc A - sec A) : :

Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let P_A be the polar of A′ with respect to the circumcircle. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. The lines AA″, BB″, CC″ concur in X(6414).

X(6414) Lies on the Jerabek hyperbola and these lines: {4,372}, {6,3155}, {54,371}, {69,5408}, {184,216} .

X(6414) = X(92)-isoconjugate of X(372)
X(6414) = {X(184),X(216)}-harmonic conjugate of X(6413)
X(6414) = perspector of 2nd Kenmotu diagonals triangle and cross-triangle of ABC and 2nd Kenmotu diagonals triangle

X(6415) = ISOGONAL CONJUGATE OF X(3535)

Trilinears 1/(sec A + 2 csc A) : :

Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(-1/2) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let P_A be the polar of A′ with respect to the circumcircle. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. The lines AA″, BB″, CC″ concur in (X(6415).

X(6415) lies on the Jerabek hyperbola and these lines: {4,1131}, {54,3312}, {64,371}, {184,6416}, {1151,3532}

X(6415) = X(92)-isoconjugate of X(1151)

X(6416) = ISOGONAL CONJUGATE OF X(3536)

Trilinears 1/(sec A - 2 csc A) : :

Constructions by Randy Hutson, February 3, 2015:
(1) Let A′ be the intersection of the tangents to the A-Lucas(1/2) circle at the intersections, other than A, with AB and AC. Define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6).
(2) Let P_A be the polar of A′ with respect to the circumcircle. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. The lines AA″, BB″, CC″ concur in X(6416).

X(6416) lies on the Jerabek hyperbola and these lines: {4,1132}, {54,3311}, {64,372}, {184,6415}, {1152,3532}

X(6416) = X(92)-isoconjugate of X(1152)

X(6417) = PERSPECTOR OF ABC AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears cos A + 4 sin A : :
Trilinears a(S_A + 4S) : :

The Lucas secondary central triangle is introduced at X(6199).

X(6417) lies on these lines: {3,6}, {485,3851}, {486,5055}, {1131,3845}, {1132,3850}

X(6417) = isogonal conjugate of X(34089)
X(6417) = radical center of Lucas(8) circles
X(6417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6418), (6,371,3312), (6,3311,3), (371,3312,3), (3311,3312,371)

X(6418) = PERSPECTOR OF ABC AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears cos A - 4 sin A : :
Trilinears a(S_A - 4S) : :

X(6418) lies on these lines: {3,6}, {485,5055}, {486,3851}, {1131,3850}, {1132,3845} .

X(6418) = isogonal conjugate of X(34091)
X(6418) = radical center of Lucas(-8) circles
X(6418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6417), (6,372,3311), (6,3312,3), (372,3311,3), (3311,3312,372)

X(6419) = PERSPECTOR OF ABC AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears cos A + 3 sin A : :
Trilinears a(S_A + 3S) : :

The 1st Lucas secondary tangents triangle is introduced at X(6199).

X(6419) lies on these lines: {3,6}, {4,1327}, {485,1132}, {486,3068}, {1994,5408}, {2066,3301}, {2067,3299}, {3518,5413}, {5198,5410}, {5409,5422}, {5412,5417} .

X(6419) = perspector of Lucas secondary central triangle and circumsymmedial tangential triangle
X(6419) = radical center of Lucas(6) circles
X(6419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6420), (6,371,372), (6,1151,3312), (6,3311,371), (61,62,371)

X(6420) = PERSPECTOR OF ABC AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears cos A - 3 sin A : :
Trilinears a(S_A - 3S) : :

X(6420) lies on these lines: {3,6}, {486,1131}

X(6420) = perspector of Lucas(-1) secondary central triangle and circumsymmedial tangential triangle
X(6420) = radical center of Lucas(-6) circles
X(6420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6419), (6,372,371), (6,1152,3311), (6,3312,372), (61,62,372)

X(6421) = PERSPECTOR OF ABC AND LUCAS BROCARD TRIANGLE

Trilinears    sin A sin ω - sin(A + ω) : :
Trilinears    cos A - sin A (1 - cot ω) : :
Trilinears    a(b^2 + c^2 - S) : :

Let R_A be the radical axis of the A-Lucas(t) circle and Brocard circle, and define R_B and R_C cyclically. Let A′ = R_B∩R_C, B′ = R_C∩R_A, C′ = R_A∩R_B. Triangle A′B′C′ is here introduced as the Lucas(t) Brocard triangle.

X(6421) lies on these lines: {2,494}, {3,6}, {493,589}

X(6421) = radical center of Lucas(2 cot ω - 2) circles
X(6421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6424), (6,39,6422), (8396,8400,3)
X(6421) = trilinear product of vertices of inner Grebe triangle
X(6421) = exsimilicenter of 2nd Lemoine and (1/2)-Moses circles; the insimilicenter is X(6422)

X(6422) = PERSPECTOR OF ABC AND LUCAS(-1) BROCARD TRIANGLE

Trilinears    sin A sin ω + sin(A + ω) : :
Trilinears    cos A + sin A (1 + cot ω) : :
Trilinears    a(b^2 + c^2 + S) : :

X(6422) lies on these lines: {2,493}, {3,6}, {494,588}

X(6422) = radical center of Lucas(2 cot ω + 2) circles
X(6422) = {X(3),X(6)}-harmonic conjugate of X(6243)
X(6422) = {X(6),X(39)}-harmoninc conjugate of X(6421)
X(6422) = trilinear product of vertices of outer Grebe triangle
X(6422) = insimilicenter of 2nd Lemoine and (1/2)-Moses circles; then exsimilicenter is X(6421)

X(6423) = KIRIKAMI-EULER IMAGE OF X(371)

Trilinears    sin A sin ω + sin(A - ω) : :
Trilinears    cos A - sin A (1 + cot ω) : :
Trilinears    a(a^2 + S) : :

Let (O_A) be the circumcircle of the A-1st-Kenmotu square, and define (O_B) and (O_C) cyclically. Let R_A be the radical axis of the circumcircle and (O_A), and define R_B and R_C cyclically. Let A′ = R_B∩R_C, and define B′, C′ cyclically. The lines AA′, BB′, CC′ concur in X(6423). (Randy Hutson, February 3, 2015)

X(6423) lies on these lines: {3,6}, {251,494}

X(6423) = isogonal conjugate of X(5490)
X(6423) = radical center of Lucas(-2 cot ω - 2) circles
X(6423) = {X(3),X(6)}-harmonic conjugate of X(6422)
X(6423) = {X(3),X(32)}-harmonic conjugate of X(6424)
X(6423) = trilinear product of vertices of anti-outer-Grebe triangle

X(6424) = KIRIKAMI-EULER IMAGE OF X(372)

Trilinears    sin A sin ω - sin(A - ω) : :
Trilinears    cos A + sin A (1 - cot ω) : :
Trilinears    a(a^2 - S) : :

Let (O_A) be the circumcircle of the A-2nd-Kenmotu square, and define (O_B) and (O_C) cyclically. Let R_A be the radical axis of the circumcircle and (O_A), and define R_B and R_C cyclically. Let A′ = R_B∩R_C, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(6424) (Randy Hutson, February 3, 2015)

X(6424) lies on these lines: {3,6}, {251,493}

X(6424) = isogonal conjugate of X(5491)
X(6424) = radical center of Lucas(-2 cot ω + 2) circles
X(6424) = {X(3),X(6)}-harmonic conjugate of X(6421)
X(6424) = {X(3),X(32)}-harmonic conjugate of X(6423)
X(6424) = exsimilicenter of (1/2)-Moses and Lucas radical circles; the insimilicenter is X(9600)
X(6424) = {X(39),X(1151)}-harmonic conjugate of X(9600)
X(6424) = trilinear product of vertices of anti-inner-Grebe triangle

X(6425) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND LUCAS INNER TRIANGLE

Trilinears 4 cos A + 3 sin A : :
Trilinears a(4S_A + 3S) : :

Let L_A and R_A be the center and radius of the A-Lucas circle, and define L_B and L_C, and also R_B and R_C, cyclically. Let L* be the center of the Lucas radical circle and R* its radius. Then X(6425) = L_A/R_A + L_B/R_B + L_C/R_C + L*/R* - X(3)/R.

X(6425) lies on these lines: {3,6}, {485,3627}, {486,632}, {1131,3068}

X(6425) = perspector of ABC and Lucas inner tangential triangle
X(6425) = perspector of Lucas tangents triangle and 2nd Lucas secondary tangents triangle
X(6425) = radical center of Lucas(3/2) circles
X(6425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6426), (371,1151,6), (1152,3311,6)

X(6426) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND LUCAS(-1) INNER TRIANGLE

Trilinears 4 cos A - 3 sin A : :
Trilinears a(4S_A - 3S) : :

X(6426) lies on these lines: {3,6}, {485,632}, {486,3627}, {1132,3069}

X(6426) = perspector of ABC and Lucas(-1) inner tangential triangle
X(6426) = perspector of Lucas(-1) tangents triangle and 2nd Lucas(-1) secondary tangents triangle
X(6426) = radical center of Lucas(-3/2) circles
X(6426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6425), (372,1152,6), (1151,3312,6)

X(6427) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears cos A + 6 sin A : :
Trilinears a(S_A + 6S) : :

X(6427) lies on these lines: {3,6}, {485,5072}, {486,5079}, {546,1588}, {632,3069}, {1173,6415}, {1587,3627}, {3068,3628}, {3070,5076}, {3299,3304}, {3301,3303}, {3518,5411}

X(6427) = radical center of Lucas(12) circles
X(6427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6428), (6,3311,3312), (61,62,1151)

X(6428) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears cos A - 6 sin A : :
Trilinears a(S_A - 6S) : :

X(6428) lies on these lines: {3,6}, {485,5079}, {486,5072}, {546,1587}, {632,3068}, {1173,6416}, {1588,3627}, {3069,3628}, {3071,5076}, {3299,3303}, {3301,3304}, {3518,5410}

X(6428) = radical center of Lucas(-12) circles
X(6428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6427), (6,3312,3311), (61,62,1152)

X(6429) = PERSPECTOR OF TANGENTIAL TRIANGLE AND LUCAS INNER TRIANGLE

Trilinears 8 cos A + 5 sin A : :
Trilinears a(8S_A + 5S) : :

Let A′B′C′ be the Lucas tangents triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6429). Also, let A′B′C′ be the Lucas inner triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6429).

X(6429) lies on these lines: {3,6}, {547,5418}, {590,3832}, {3068,5059}, {3071,5067}, {3850,6561}, {5056,6459}

X(6429) = radical center of Lucas(5/4) circles
X(6429) = {X(3),X(6)}-harmonic conjugate of X(6430)

X(6430) = PERSPECTOR OF TANGENTIAL TRIANGLE AND LUCAS(-1) INNER TRIANGLE

Trilinears 8 cos A - 5 sin A : :
Trilinears a(8S_A - 5S) : :

Let A′B′C′ be the Lucas(-1) tangents triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6430). Also, let A′B′C′ be the Lucas(-1) inner triangle, let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6430).

X(6430) lies on these lines: {3,6}, {547,5420}, {615,3832}, {3069,5059}, {3070,5067}, {3850,6560}, {5056,6460}

X(6430) = radical center of Lucas(-5/4) circles
X(6430) = {X(3),X(6)}-harmonic conjugate of X(6429)

X(6431) = PERSPECTOR OF TANGENTIAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 2 cos A + 5 sin A : :
Trilinears a(2S_A + 5S) : :

Let A′B′C′ be the 2nd Lucas secondary tangents triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6431).

X(6431) lies on these lines: {3,6}, {485,3850}, {486,547}, {487,597}, {590,5067}, {615,3533}, {1588,3545}, {3068,5056}, {3070,3543}, {3071,3832}, {5059,6459}

X(6431) = radical center of Lucas(5) circles
X(6431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6432), (6,371,1152), (371,1152,1151)

X(6432) = PERSPECTOR OF TANGENTIAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 2 cos A - 5 sin A : :
Trilinears a(2S_A - 5S) : :

Let A′B′C′ be the 2nd Lucas(-1) secondary tangents triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6432).

X(6432) lies on these lines: {3,6}, {485,547}, {486,3850}, {488,597}, {590,3533}, {615,5067}, {1587,3545}, {3069,5056}, {3070,3832}, {3071,3543}, {5059,6460}

X(6432) = radical center of Lucas(-5) circles
X(6432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6431), (6,372,1151), (372,1151,1152)

X(6433) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS INNER TANGENTIAL TRIANGLE

Trilinears 12 cos A + 5 sin A : :
Trilinears a(12S_A + 5S) : :

X(6433) lies on these lines: {3,6}, {547,6561}, {590,3543}, {3071,3533}, {3850,5418}

X(6433) = radical center of Lucas(5/6) circles
X(6433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6434), (6456,6458,3)

X(6434) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND LUCAS(-1) INNER TANGENTIAL TRIANGLE

Trilinears 12 cos A - 5 sin A : :
Trilinears a(12S_A - 5S) : :

X(6434) lies on these lines: {3,6}, {547,6560}, {615,3543}, {3070,3533}, {3850,5420}

X(6434) = radical center of Lucas(-5/6) circles
X(6434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6433), (6457,6459,3)

X(6435) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 3 cos A + 11 sin A : :
Trilinears a(3S_A + 11S) : :

X(6435) lies on these lines: {3,6}, {1328,6564}, {1588,3854}

X(6435) = radical center of Lucas(22/3) circles
X(6435) = {X(3),X(6)}-harmonic conjugate of X(6436)

X(6436) = PERSPECTOR OF CIRCUMSYMMEDIAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 3 cos A - 11 sin A : :
Trilinears a(3S_A - 11S) : :

X(6436) lies on these lines: {3,6}, {1327,6565}, {1587,3854}

X(6436) = radical center of Lucas(-22/3) circles
X(6436) = {X(3),X(6)}-harmonic conjugate of X(6435)

X(6437) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS TANGENTS TRIANGLE

Trilinears 6 cos A + 5 sin A : :
Trilinears a(6S_A + 5S) : :

X(6437) lies on these lines: {3,6}, {485,3853}, {590,3545}, {1588,3533}, {3068,3543}, {3070,5059}, {3071,5056}, {3832,6459}, {3845,6561}

X(6437) = radical center of Lucas(5/3) circles
X(6437) = {X(3),X(6)}-harmonic conjugate of X(6438)

X(6438) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS(-1) TANGENTS TRIANGLE

Trilinears 6 cos A - 5 sin A : :
Trilinears a(6S_A - 5S) : :

X(6438) lies on these lines: {3,6}, {486,3853}, {615,3545}, {1587,3533}, {3069,3543}, {3070,5056}, {3071,5059}, {3832,6460}, {3845,6560}

X(6438) = radical center of Lucas(-5/3) circles
X(6438) = {X(3),X(6)}-harmonic conjugate of X(6437)

X(6439) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS INNER TRIANGLE

Trilinears 24 cos A + 13 sin A : :
Trilinears a(24S_A + 13S) : :

X(6439) lies on line {3,6}

X(6439) = radical center of Lucas(13/12) circles
X(6439) = {X(3),X(6)}-harmonic conjugate of X(6440)

X(6440) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS(-1) INNER TRIANGLE

Trilinears 24 cos A - 13 sin A : :
Trilinears a(24S_A - 13S) : :

X(6440) lies on line {3,6}

X(6440) = radical center of Lucas(-13/12) circles
X(6440) = {X(3),X(6)}-harmonic conjugate of X(6439)

X(6441) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 6 cos A + 13 sin A : :
Trilinears a(6S_A + 13S) : :

X(6441) lies on these lines: {3,6}, {485,3859}

X(6441) = radical center of Lucas(13/3) circles
X(6441) = {X(3),X(6)}-harmonic conjugate of X(6442)

X(6442) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 6 cos A - 13 sin A : :
Trilinears a(6S_A - 13S) : :

X(6442) lies on these lines: {3,6}, {486,3859}

X(6442) = radical center of Lucas(-13/3) circles
X(6442) = {X(3),X(6)}-harmonic conjugate of X(6441)

X(6443) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS BROCARD TRIANGLE

Trilinears a(5b^2 + 5c^2 - a^2 - 3S) : :

X(6443) lies on line {3,6}

X(6443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,574,1151), (6,5024,6444)

X(6444) = PERSPECTOR OF CIRCUMSYMMEDIAL TANGENTIAL TRIANGLE AND LUCAS(-1) BROCARD TRIANGLE

Trilinears a(5b^2 + 5c^2 - a^2 + 3S) : :

X(6444) lies on line {3,6}

X(6444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,574,1152), (6,5024,6443)

X(6445) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND LUCAS INNER TRIANGLE

Trilinears 9 cos A + 4 sin A : :
Trilinears a(9S_A + 4S) : :

X(6445) lies on these lines: {3,6}, {590,3830}, {3068,3534}, {3316,3853}, {3526,6459}, {3851,5418}, {5055,6561}

X(6445) = perspector of circumsymmedial triangle and Lucas(3) central triangle
X(6445) = radical center of Lucas(8/9) circles
X(6445) = inverse-in-Lucas(-1)-secondary-radical-circle of X(6509)
X(6445) = {X(3),X(6)}-harmonic conjugate of X(6446)

X(6446) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND LUCAS(-1) INNER TRIANGLE

Trilinears 9 cos A - 4 sin A : :
Trilinears a(9S_A - 4S) : :

X(6446) lies on these lines: {3,6}, {615,3830}, {3069,3534}, {3317,3853}, {3526,6460}, {3851,5420}, {5055,6560}

X(6446) = perspector of circumsymmedial triangle and Lucas(-3) central triangle
X(6446) = radical center of Lucas(-8/9) circles
X(6446) = inverse-in-Lucas-secondary-radical-circle of X(6510)
X(6446) = {X(3),X(6)}-harmonic conjugate of X(6445)

X(6447) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 7 cos A + 6 sin A : :
Trilinears a(7S_A + 6S) : :

X(6447) lies on these lines: {3,6}, {382,1327}, {485,5076}, {546,6459}, {590,5072}, {632,1588}, {1132,3090}, {3068,3627}, {3071,5079}, {3091,3316}

X(6447) = radical center of Lucas(12/7) circles
X(6447) = {X(3),X(6)}-harmonic conjugate of X(6448)

X(6448) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 7 cos A - 6 sin A : :
Trilinears a(7S_A - 6S) : :

X(6448) lies on these lines: {3,6}, {382,1328}, {486,5076}, {546,6460}, {615,5072}, {632,1587}, {1131,3090}, {3069,3627}, {3070,5079}, {3091,3317}

X(6448) = radical center of Lucas(-12/7) circles
X(6448) = {X(3),X(6)}-harmonic conjugate of X(6447)

X(6449) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 5 cos A + 2 sin A : :
Barycentrics a^2 (5 S_A + 2 S) : :

Let L_A and R_A be the center and radius of the A-Lucas circle, and define L_B and L_C, and also R_B and R_C, cyclically. Let L* be the center of the Lucas inner circle and R* its radius. Then X(6449) = L_A/R_A + L_B/R_B + L_C/R_C - L*/R* - X(3)/R; see also X(6426). (Randy Hutson, February 3, 2015)

Let A_B = inverse-in-A-Lucas-circle of B, and define B_C and C_A cyclically. Let A_C = inverse-in-A-Lucas-circle of C, and define B_A and C_B cyclically. The six points A_B, B_C, C_A, A_C, B_A, C_B lie on a circle with center X(6449). See X(6519). (Peter Moses, February 11, 2015)

If you have The Geometer's Sketchpad, you can view X(6449).

X(6449) lies on these lines: {3,6}, {140,6459}, {381,5418}, {382,590}, {485,1657}, {486,5054}, {548,1587}, {549,1588}, {550,3068}, {1132,3533}, {1656,6561}, {3069,3530}, {3070,3534}, {3071,3526}, {3316,3543}, {3520,5410}, {5070,6565}, {5073,6564}

X(6449) = perspector of Lucas tangents triangle and Lucas secondary central triangle
X(6449) = radical center of Lucas(4/5) circles
X(6449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6450), (3,371,3312), (371,3312,3311)

X(6450) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 5 cos A - 2 sin A : :
Trilinears a(5S_A - 2S) : :

X(6450) lies on these lines: {3,6}, {140,6460}, {381,5420}, {382,615}, {485,5054}, {486,1657}, {548,1588}, {549,1587}, {550,3069}, {1131,3533}, {1656,6560}, {3068,3530}, {3070,3526}, {3071,3534}, {3317,3543}, {3520,5411}, {5070,6564}

X(6450) = perspector of Lucas(-1) tangents triangle and Lucas(-1) secondary central triangle
X(6450) = radical center of Lucas(-4/5) circles
X(6450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6449), (3,372,3311), (372,3311,3312)

X(6451) = PERSPECTOR OF LUCAS CENTRAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 9 cos A + 2 sin A : :
Trilinears a(9S_A + 2S) : :

X(6451) lies on these lines: {3,6}, {20,3316}, {590,1327}, {631,1132}, {1657,5418}, {3530,6459}, {5054,6561}

X(6451) = radical center of Lucas(4/9) circles
X(6451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6452), (3,1151,3312)

X(6452) = PERSPECTOR OF LUCAS(-1) CENTRAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 9 cos A - 2 sin A : :
Trilinears a(9S_A - 2S) : :

X(6452) lies on these lines: {3,6}, {20,3317}, {615,1328}, {631,1131}, {1657,5420}, {3530,6460}, {5054,6560}

X(6452) = radical center of Lucas(-4/9) circles
X(6452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6451), (3,1152,3311)

X(6453) = PERSPECTOR OF LUCAS TANGENTS TRIANGLE AND LUCAS INNER TRIANGLE

Trilinears 5 cos A + 3 sin A : :
Trilinears a(5S_A + 3S) : :

Combos by Randy Hutson, February 3, 2015
1) Let L_A and R_A be the center and radius of the A-Lucas circle, and define L_B and L_C, and also R_B and R_C, cyclically. Let L* be the center of the Lucas inner circle and R* its radius. Then X(6453) = L_A/R_A + L_B/R_B + L_C/R_C + L*/R*
(2) Let L** and R** be the center and radius of the Lucas radical circle. Then X(6453) = L_A/R_A + L_B/R_B + L_C/R_C + L**/R**.

X(6453) lies on these lines: {3,6}, {485,3146}, {486,3525}

X(6453) = perspector of Lucas central triangle and Lucas inner tangential triangle
X(6453) = exsimilicenter of Lucas radical circle and Lucas inner circle
X(6453) = radical center of Lucas(6/5) circles
X(6453) = {X(3),X(6)}-harmonic conjugate of X(6454)

X(6454) = PERSPECTOR OF LUCAS(-1) TANGENTS TRIANGLE AND LUCAS(-1) INNER TRIANGLE

Trilinears 5 cos A - 3 sin A : :
Trilinears a(5S_A - 3S) : :

X(6454) lies on these lines: {3,6}, {485,3525}, {486,3146}

X(6454) = perspector of Lucas(-1) central triangle and Lucas(-1) inner tangential triangle
X(6454) = exsimilicenter of Lucas(-1) radical circle and Lucas(-1) inner circle
X(6454) = radical center of Lucas(-6/5) circles
X(6454) = {X(3),X(6)}-harmonic conjugate of X(6453)

X(6455) = PERSPECTOR OF LUCAS TANGENTS TRIANGLE AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears 7 cos A + 2 sin A : :
Trilinears a(7S_A + 2S) : :

X(6455) lies on these lines: {3,6}, {382,5418}, {485,3534}, {548,3068}, {549,6459}, {590,1657}, {1588,3530}, {3071,5054}, {3316,5059}, {3526,6561}

X(6455) = radical center of Lucas(4/7) circles
X(6455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6456), (3,1151,3311)

X(6456) = PERSPECTOR OF LUCAS(-1) TANGENTS TRIANGLE AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears 7 cos A - 2 sin A : :
Trilinears a(7S_A - 2S) : :

X(6456) lies on these lines: {3,6}, {382,5420}, {486,3534}, {548,3069}, {549,6460}, {615,1657}, {1587,3530}, {3070,5054}, {3317,5059}, {3526,6560}

X(6456) = radical center of Lucas(-4/7) circles
X(6456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6455), (3,1152,3312)

X(6457) = PERSPECTOR OF LUCAS TANGENTS TRIANGLE AND LUCAS ANTIPODAL TRIANGLE

Trilinears (cos A)(cos^2 B + cos^2 C) + sin A cos^2 A : :

Let A′ be the antipode of A in the A-Lucas(t) circle, and define B′ and C′ cyclically. The triangle A′B′C′ is here introduced as the Lucas(t) antipodal triangle.

X(6457) lies on these lines: {4,371}, {185,577}

X(6457) = {X(185),X(577)}-harmonic conjugate of X(6458)

X(6458) = PERSPECTOR OF LUCAS(-1) TANGENTS TRIANGLE AND LUCAS(-1) ANTIPODAL TRIANGLE

Trilinears (cos A)(cos^2 B + cos^2 C) - sin A cos^2 A : :

X(6458) lies on these lines: {4,372}, {185,577}

X(6458) = {X(185),X(577)}-harmonic conjugate of X(6457)

X(6459) = INTERSECTION OF LINES X(4)X(371) AND X(6)X(20)

Trilinears cos A + sin A - cos B cos C : :

Let A′B′C′ be the Lucas antipodal triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6459).

X(6459) lies on these lines: {2,489}, {3,1588}, {4,371}, {5,6221}, {6,20}, {30,1587}, {193,490}, {372,376}, {1152,3522}, {3070,3146}

X(6459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1588,3069), (4,371,3068), (6,20,6460)

X(6460) = INTERSECTION OF LINES X(4)X(372) AND X(6)X(20)

Trilinears cos A - sin A - cos B cos C : :

Let A′B′C′ be the Lucas(-1) antipodal triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6460).

X(6460) lies on these lines: {2,490}, {3,1587}, {4,372}, {5,6398}, {6,20}, {30,1588}, {193,489}, {371,376}, {1151,3522}, {3071,3146}

X(6460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1587,3068), (4,372,3069). (6,20,6459)

X(6461) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND LUCAS HOMOTHETIC TRIANGLE

Trilinears    a cot^2 A (cot^2 A - cot^2 B - cot^2 C) : :
Trilinears    a*S_A^2(S_A^2 - S_B^2 - S_C^2) : :
Barycentrics    (cos^2 A)(cot^2 A - cot^2 B - cot^2 C) : :

X(6461) lies on these lines: {6,6337}, {39,493}, {63,7124}, {394,3926}, {1181,8220}, {3796,8194}, {8188,8189}, {8201,8202}, {8208,8209}, {8210,8211}, {8212,8213}, {8214,8215}, {8216,8217}, {8218,8219}

X(6461) = perspector of symmedial, Lucas homothetic, and Lucas(-1) homothetic triangles (see X(493) and X(494))
X(6461) = X(6464)-of-Lucas-homothetic-triangle
X(6461) = X(6464)-of -Lucas(-1)-homothetic-triangle
X(6461) = barycentric product X(6)*X(6338)
X(6461) = {X(6462),X(6463)}-harmonic conjugate of X(6339)
X(6461) = insimilicenter of circumcircles of Lucas homothetic and Lucas(-1) homothetic triangles

X(6462) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND LUCAS HOMOTHETIC TRIANGLE

Barycentrics -8*S*a^2+a^4-6*(b^2+c^2)*a^2+(b^2+c^2)^2 : :

Let A′B′C′ be the Lucas homothetic triangle. Let A″ be the cevapoint of B′ and C′, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X(6462).

X(6462) lies on these lines: {2,493}, {6,6337}, {20,6465}, {148,1131}, {193,371}

X(6462) = {X(6339),X(6461)}-harmonic conjugate of X(6463)

X(6463) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND LUCAS(-1) HOMOTHETIC TRIANGLE

Barycentrics 8*S*a^2+a^4-6*(b^2+c^2)*a^2+(b^2+c^2)^2 : :

Let A′B′C′ be the Lucas(-1) homothetic triangle. Let A″ be the cevapoint of B′ and C′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6463).

X(6463) lies on these lines: {2,494}, {6,6337}, {20,6466}, {148,1132}, {193,372}

X(6463) = {X(6339),X(6461)}-harmonic conjugate of X(6462)

X(6464) = LUCAS-HOMOTHETIC-TO-ABC SIMILARITY IMAGE OF X(6461)

Barycentrics 1/(sin^2 A - sin^2 B sin^2 C) : :
Barycentrics a^2/(a^4 - S^2) : :

X(6464) lies on these lines: {39,493}, {141,5490}, {155,1351}, {6392,6515}

X(6464) = Lucas(-1)-homothetic-to-ABC similarity image of X(6461)
X(6464) = endo-homothetic center of Lucas homothetic and Lucas(-1) homothetic triangles

X(6465) = PERSPECTOR OF LUCAS HOMOTHETIC TRIANGLE AND LUCAS ANTIPODAL TRIANGLE

Barycentrics a^2*(-a^2+b^2+c^2)*(4*((b^2+c^2)*a^2+(b^2-c^2)^2)*S+a^6-(b^2+c^2)*a^4-(b^4-10*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

Let A′ be the midpoint of the intersections, other than A, of the A-Lucas circle and AB and AC. Define B′ and C′ cyclically. Let P_A be the polar with respect to the A-Lucas circle of A′. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. Triangle A″B″C″ is homothetic to the Lucas homothetic triangle at X(6465).

X(6465) lies on these lines: {3,6466}, {20,6462}, {25,371}

X(6465) = {X(3),X(6467)}-harmonic conjugate of X(6466)

X(6466) = PERSPECTOR OF LUCAS(-1) HOMOTHETIC TRIANGLE AND LUCAS(-1) ANTIPODAL TRIANGLE

Barycentrics a^2*(-a^2+b^2+c^2)*(-4*((b^2+c^2)*a^2+(b^2-c^2)^2)*S+a^6-(b^2+c^2)*a^4-(b^4-10*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

Let A′ be the midpoint of the intersections, other than A, of the A-Lucas(-1) circle and AB and AC. Define B′ and C′ cyclically. Let P_A be the polar with respect to the A-Lucas(-1) circle of A′. Define P_B and P_C cyclically. Let A″ = P_B∩P_C, B″ = P_C∩P_A, C″ = P_A∩P_B. The triangle A″B″C″ is homothetic to the Lucas(-1) homothetic triangle at X(6466).

X(6466) lies on these lines: {3,6465}, {20,6463}, {25,372}

X(6466) = {X(3),X(6467)}-harmonic conjugate of X(6465)

X(6467) = CROSSSUM OF X(2) AND X(25)

Trilinears    csc B sin C tan C + csc C sin B tan B : :
Trilinears    (cos A)(sin^2 B cot C + sin^2 C cot B) : :
Trilinears    (sin A)(sin^2 B tan B + sin^2 C tan C) : :
Trilinears    a(b^2 + c^2 - a^2)[a^2(b^2 + c^2) + (b^2 - c^2)^2] : :
Trilinears    a[b^2/(a^2 - b^2 + c^2) + c^2/(a^2 + b^2 - c^2)] : :

X(6467) lies on these lines: {3,6465}, {6,25}, {20,185}, {69,305}

X(6467) = isogonal conjugate of isotomic conjugate of X(1368)
X(6467) = isotomic conjugate of X(683)
X(6467) = crosssum of X(2) and X(25)
X(6467) = crosspoint of X(6) and X(69)
X(6467) = crossdifference of every pair of points on line X(525)X(2451)
X(6467) = polar conjugate of isotomic conjugate of X(22401)
X(6467) = {X(6465), X(6466)}-harmonic conjugate of X(3)

X(6468) = X(6)-OF-LUCAS-INNER-TRIANGLE

Trilinears 12 cos A + 7 sin A : :
Trilinears a(12S_A + 7S) : :

Let A′B′C′ be the Lucas inner triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6468).

X(6468) lies on these lines: {3,6}, {590,3839}, {5066,6561}

X(6468) = perspector of Lucas inner triangle and Lucas inner tangential triangle
X(6468) = radical center of Lucas(7/6) circles
X(6468) = inverse-in-Lucas-inner-circle of X(187)
X(6468) = trilinear pole, with respect to Lucas inner triangle, of Lemoine axis
X(6468) = {X(3),X(6)}-harmonic conjugate of X(6469)

X(6469) = X(6)-OF-LUCAS(-1)-INNER-TRIANGLE

Trilinears 12 cos A - 7 sin A : :
Trilinears a(12S_A - 7S) : :

Let A′B′C′ be the Lucas(-1) inner triangle. Let A″ be the trilinear pole, with respect to A′B′C′, of line BC, and define B″ and C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6469).

X(6469) lies on these lines: {3,6}, {615,3839}, {5066,6560}

X(6469) = perspector of Lucas(-1) inner triangle and Lucas(-1) inner tangential triangle
X(6469) = radical center of Lucas(-7/6) circles
X(6469) = inverse-in-Lucas(-1)-inner-circle of X(187)
X(6469) = trilinear pole, with respect to Lucas(-1) inner triangle, of Lemoine axis
X(6469) = {X(3),X(6)}-harmonic conjugate of X(6468)

X(6470) = PERSPECTOR OF LUCAS INNER TRIANGLE AND LUCAS(-1) INNER TANGENTIAL TRIANGLE

Trilinears 4 cos A + 7 sin A : :
Trilinears a(4S_A + 7S) : :

X(6470) lies on these lines: {3,6}, {485,3858}, {1132,3068}, {1588,3316}, {3071,3855}

X(6470) = radical center of Lucas(7/2) circles
X(6470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6471), (371,3312,1151)

X(6471) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND LUCAS INNER TANGENTIAL TRIANGLE

Trilinears 4 cos A - 7 sin A : :
Trilinears a(4S_A - 7S) : :

X(6471) lies on these lines: {3,6}, {486,3858}, {1131,3069}, {1587,3317}, {3070,3855}

X(6471) = radical center of Lucas(-7/2) circles
X(6471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6476), (372,3311,1152)

X(6472) = PERSPECTOR OF LUCAS INNER TRIANGLE AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears 25 cos A + 16 sin A : :
Trilinears a(25S_A + 16S) : :

X(6472) lies on line {3,6}

X(6472) = radical center of Lucas(32/25) circles
X(6472) = {X(3),X(6)}-harmonic conjugate of X(6473)

X(6473) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears 25 cos A - 16 sin A : :
Trilinears a(25S_A - 16S) : :

X(6473) lies on line {3,6}

X(6473) = radical center of Lucas(-32/25) circles
X(6473) = {X(3),X(6)}-harmonic conjugate of X(6472)

X(6474) = PERSPECTOR OF LUCAS INNER TRIANGLE AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears 23 cos A + 16 sin A : :
Trilinears a(23S_A + 16S) : :

X(6474) lies on these lines: {3,6}, {1131,5073}

X(6474) = radical center of Lucas(32/23) circles
X(6474) = {X(3),X(6)}-harmonic conjugate of X(6475)

X(6475) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears 23 cos A - 16 sin A : :
Trilinears a(23S_A - 16S) : :

X(6475) lies on these lines: {3,6}, {1132,5073}

X(6475) = radical center of Lucas(-32/23) circles
X(6475) = {X(3),X(6)}-harmonic conjugate of X(6474)

X(6476) = PERSPECTOR OF LUCAS INNER TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 21 cos A + 13 sin A : :
Trilinears a(21S_A + 13S) : :

X(6476) lies on line {3,6}

X(6476) = radical center of Lucas(26/21) circles
X(6476) = {X(3),X(6)}-harmonic conjugate of X(6477)

X(6477) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 21 cos A - 13 sin A : :
Trilinears a(21S_A - 13S) : :

X(6477) lies on line {3,6}

X(6477) = radical center of Lucas(-26/21) circles
X(6477) = {X(3),X(6)}-harmonic conjugate of X(6476)

X(6478) = PERSPECTOR OF LUCAS INNER TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 19 cos A + 13 sin A : :
Trilinears a(19S_A + 13S) : :

X(6478) lies on these lines: {3,6}, {590,3859}

X(6478) = radical center of Lucas(26/19) circles
X(6478) = {X(3),X(6)}-harmonic conjugate of X(6479)

X(6479) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 19 cos A - 13 sin A : :
Trilinears a(19S_A - 13S) : :

X(6479) lies on these lines: {3,6}, {615,3859}

X(6479) = radical center of Lucas(-26/19) circles
X(6479) = {X(3),X(6)}-harmonic conjugate of X(6478)

X(6480) = PERSPECTOR OF LUCAS INNER TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 9 cos A + 5 sin A : :
Trilinears a(9S_A + 5S) : :

X(6480) lies on these lines: {3,6}, {547,6565}, {590,3845}, {3543,6564}, {3545,6561}, {5056,5418}, {5067,6459}

X(6480) = radical center of Lucas(10/9) circles
X(6480) = insimilicenter of Lucas radical circle and Lucas inner circle
X(6480) = {X(3),X(6)}-harmonic conjugate of X(6481)

X(6481) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 9 cos A - 5 sin A : :
Trilinears a(9S_A - 5S) : :

X(6481) lies on these lines: {3,6}, {547,6564}, {615,3845}, {3545,6560}, {5056,5420}, {5067,6460}

X(6481) = radical center of Lucas(-10/9) circles
X(6481) = insimilicenter of Lucas(-1) radical circle and Lucas(-1) inner circle
X(6481) = {X(3),X(6)}-harmonic conjugate of X(6480)

X(6482) = PERSPECTOR OF LUCAS INNER TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 29 cos A - 15 sin A : :
Trilinears a(29S_A - 15S) : :

X(6482) lies on these lines: {3,6}, {3543,3590}

X(6482) = radical center of Lucas(-30/29) circles
X(6482) = {X(3),X(6)}-harmonic conjugate of X(6483)

X(6483) = PERSPECTOR OF LUCAS(-1) INNER TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 29 cos A + 15 sin A : :
Trilinears a(29S_A + 15S) : :

X(6483) lies on these lines: {3,6}, {3543,3591}

X(6483) = radical center of Lucas(30/29) circles
X(6483) = {X(3),X(6)}-harmonic conjugate of X(6482)

X(6484) = PERSPECTOR OF LUCAS INNER TANGENTIAL TRIANGLE AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears 11 cos A + 5 sin A : :
Trilinears a(11S_A + 5S) : :

X(6484) lies on these lines: {3,6}, {485,5059}, {590,3853}, {3533,6459}, {3545,5418}, {5056,6561}, {5067,6565}

X(6484) = radical center of Lucas(10/11) circles
X(6484) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6485), (3,6433,6486)

X(6485) = PERSPECTOR OF LUCAS(-1) INNER TANGENTIAL TRIANGLE AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears 11 cos A - 5 sin A : :
Trilinears a(11S_A - 5S) : :

X(6485) lies on these lines: {3,6}, {486,5059}, {615,3853}, {3533,6460}, {3545,5420}, {5056,6560}, {5067,6564}

X(6485) = radical center of Lucas(-10/11) circles
X(6485) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6484), (3,6433,6487)

X(6486) = PERSPECTOR OF LUCAS INNER TANGENTIAL TRIANGLE AND LUCAS(-1) SECONDARY CENTRAL TRIANGLE

Trilinears 13 cos A + 5 sin A : :
Trilinears a(13S_A + 5S) :

X(6486) lies on these lines: {3,6}, {3832,5418}, {5067,6561}

X(6486) = radical center of Lucas(10/13) circles
X(6486) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6487), (3,6433,6484)

X(6487) = PERSPECTOR OF LUCAS(-1) INNER TANGENTIAL TRIANGLE AND LUCAS SECONDARY CENTRAL TRIANGLE

Trilinears 13 cos A - 5 sin A : :
Trilinears a(13S_A - 5S) : :

X(6487) lies on these lines: {3,6}, {3832,5420}, {5067,6560}

X(6487) = radical center of Lucas(-10/13) circles
X(6487) ={X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6486), (3,6434,6485)

X(6488) = PERSPECTOR OF LUCAS INNER TANGENTIAL TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 22 cos A + 9 sin A : :
Trilinears a(22S_A + 9S) : :

X(6488) lies on these lines: {3,6}, {1328,3628}, {3857,5418}

X(6488) = radical center of Lucas(9/11) circles
X(6488) = {X(3),X(6)}-harmonic conjugate of X(6489)

X(6489) = PERSPECTOR OF LUCAS(-1) INNER TANGENTIAL TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 22 cos A - 9 sin A : :
Trilinears a(22S_A - 9S) : :

X(6489) lies on these lines: {3,6}, {1327,3628}, {3857,5420}

X(6489) = radical center of Lucas(-9/11) circles
X(6489) = {X(3),X(6)}-harmonic conjugate of X(6488)

X(6490) = PERSPECTOR OF LUCAS INNER TANGENTIAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 30 cos A + 19 sin A : :
Trilinears a(30S_A + 19S) : :

X(6490) lies on these lines:

X(6490) = radical center of Lucas(19/15) circles
X(6490) = {X(3),X(6)}-harmonic conjugate of X(6491)

X(6491) = PERSPECTOR OF LUCAS(-1) INNER TANGENTIAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 30 cos A - 19 sin A : :
Trilinears a(30S_A - 19S) : :

X(6491) lies on line {3,6}

X(6491) = radical center of Lucas(-19/15) circles
X(6491) = {X(3),X(6)}-harmonic conjugate of X(6490)

X(6492) = PERSPECTOR OF LUCAS INNER TANGENTIAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 26 cos A + 19 sin A : :
Trilinears a(26S_A + 19S) : :

X(6492) lies on line {3,6}

X(6492) = radical center of Lucas(19/13) circles
X(6492) = {X(3),X(6)}-harmonic conjugate of X(6493)

X(6493) = PERSPECTOR OF LUCAS(-1) INNER TANGENTIAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 26 cos A - 19 sin A : :
Trilinears a(26S_A - 19S) : :

X(6493) lies on line {3,6}

X(6493) = radical center of Lucas(-19/13) circles
X(6493) = {X(3),X(6)}-harmonic conjugate of X(6492)

X(6494) = PERSPECTOR OF LUCAS SECONDARY CENTRAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 13 cos A + 22 sin A : :
Trilinears a(13S_A + 22) : :

X(6494) lies on line {3,6}

X(6494) = radical center of Lucas(44/13) circles
X(6494) = {X(3),X(6)}-harmonic conjugate of X(6495)

X(6495) = PERSPECTOR OF LUCAS(-1) SECONDARY CENTRAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 13 cos A - 22 sin A : :
Trilinears a(13S_A - 22) : :

X(6495) lies on line {3,6}

X(6495) = radical center of Lucas(-44/13) circles
X(6495) = {X(3),X(6)}-harmonic conjugate of X(6494)

X(6496) = PERSPECTOR OF LUCAS SECONDARY CENTRAL TRIANGLE AND 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears 11 cos A + 2 sin A : :
Trilinears a(11S_A + 2S) : :

X(6496) lies on these lines: {3,6}, {376,1131}, {1328,5054}, {3317,3523}, {3534,5418}

X(6496) = radical center of Lucas(4/11) circles
X(6496) = {X(3),X(6)}-harmonic conjugate of X(6497)
X(6496) = {X(1152),X(3311)}-harmonic conjugate of X(3312)

X(6497) = PERSPECTOR OF LUCAS(-1) SECONDARY CENTRAL TRIANGLE AND 1st LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears 11 cos A - 2 sin A : :
Trilinears a(11S_A - 2S) : :

X(6497) lies on these lines: {3,6}, {376,1132}, {1327,5054}, {3316,3523}, {3534,5420}

X(6497) = radical center of Lucas(-4/11) circles
X(6497) = {X(3),X(6)}-harmonic conjugate of X(6496)
X(6497) = {X(1151),X(3312)}-harmonic conjugate of X(3311)

X(6498) = PERSPECTOR OF LUCAS SECONDARY CENTRAL TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears cos A + 22 sin A : :
Trilinears a(S_A + 22S) : :

X(6498) lies on line {3,6}

X(6498) = radical center of Lucas(44) circles
X(6498) = {X(3),X(6)}-harmonic conjugate of X(6499)

X(6499) = PERSPECTOR OF LUCAS(-1) SECONDARY CENTRAL TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears cos A - 22 sin A : :
Trilinears a(S_A - 22S) : :

X(6499) lies on line {3,6}

X(6499) = radical center of Lucas(-44) circles
X(6499) = {X(3),X(6)}-harmonic conjugate of X(6498)

X(6500) = PERSPECTOR OF 1st LUCAS SECONDARY TANGENTS TRIANGLE AND 2nd LUCAS(-1) SECONDARY TANGENTS TRIANGLE

Trilinears cos A + 8 sin A : :
Trilinears a(S_A + 8S) : :

X(6500) lies on these lines: {3,6}, {381,1132}, {1327,3071}, {1587,3830}, {1588,3843}, {1656,3316}, {3068,5070}

X(6500) = radical center of Lucas(16) circles
X(6500) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6501), (3311,3312,1151)

X(6501) = PERSPECTOR OF 1st LUCAS(-1) SECONDARY TANGENTS TRIANGLE AND 2nd LUCAS SECONDARY TANGENTS TRIANGLE

Trilinears cos A - 8 sin A : :
Trilinears a(S_A - 8S) : :

X(6501) lies on these lines: {3,6}, {381,1131}, {1328,3070}, {1587,3843}, {1588,3830}, {1656,3317}, {3069,5070}

X(6501) = radical center of Lucas(-16) circles
X(6501) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6500), (3311,3312,1152)

X(6502) = POINT CAROLI III

Trilinears    1 + sin A - cos A : :
Trilinears    (cos A)(cos B + cos C - sin A) : :
Trilinears    2(cos A)(cos B + cos C) - sin 2A : :

Points Caroli I and II are inexed as X(2066) and X(2067).

X(6502) lies on these lines: {1,372}, {3,1124}, {6,41}, {12,615}, {35,6396)}, {36,371}, {55,1152}, {63,3083}, {65,5416}, {485,499}, {486,1478}, {603,605}, {1151,5204}, {3295,6398}, {3301,5563}

X(6502) = homothetic center of 2nd Kenmotu diagonals triangle and anti-tangential midarc triangle
X(6502) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,372,5414), (3,1124,2066), (6,56,2067), (48,73,2067)

X(6503) = SS(a → cos A) of X(3)

Barycentrics Cos[A]^2*(Cos[A]^2 - Cos[B]^2 - Cos[C]^2) : :

See X(6374).

X(6503) lies on these lines: {3, 68}, {25, 114}, {39, 493}, {99, 2052}, {155, 454}, {394, 577}, {641, 1583}, {642, 1584}, {1993, 4558}

X(6503) = complement of X(6504)
X(6503) = X(2)-Ceva conjugate of X(394)
X(6503) = X(i)-complementary conjugate of X(j) for these (i,j): (31,394), (48,3548), (920,141), (1609,10), (1973,2165)
X(6503) = X(i)-isoconjugate of X(j) for these (i,j): {19,254}, {393,921}

X(6504) = SS(a → cos A) of X(4)

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

See X(6374).

X(6504) lies on the Kiepert hyperbola and these lines: {4, 155}, {10, 921}, {69, 5392}, {96, 97}, {98, 1370}, {317, 2052}, {459, 3580}, {485, 5409}, {486, 5408}

X(6504) = isogonal conjugate of X(1609)
X(6504) = isotomic conjugate of X(6515)
X(6504) = anticomplement of X(6503)
X(6504) = cyclocevian conjugate of X(3346)
X(6504) = polar conjugate of X(3542)
X(6504) = X(i)-cross conjugate of X(j) for these (i,j): (68,69), (136,850), (394,2)
X(6504) = X(254)-anticomplementary conjugate of X(4329)
X(6504) = X(i)-isoconjugate of X(j) for these {i,j}: {{1,1609}, {6,920}, {19,155}, {48,3542}}

X(6505) = SS(a → cos A) of X(9)

Barycentrics Cos[A]*(Cos[A] - Cos[B] - Cos[C]) : :

See X(6374).

Let L be a line through X(3). Let P and U be the intersections of L and the Steiner circumellipse. Let Q be the trilinear product P*U. The locus of Q as L varies is the circumconic centered at X(6505); its perspector is X(63).

X(6505) lies on these lines: {1, 224}, {2, 914}, {46, 453}, {57, 1813}, {63, 77}, {69, 6349}, {78, 1060}, {92, 664}, {223, 908}, {306, 326}, {329, 3160}, {940, 1100}, {1212, 3305}, {1708, 1993}, {1944, 6360}, {5287, 5949}

X(6505) = complement of X(2994)
X(6505) = X(2)-Ceva conjugate of X(63)
X(6505) = X(i)-complementary conjugate of X(j) for these (i,j): (31,63), (46,141), (604,499), (1406,142), (2178,10), (5905,2887) X(6505) = {X(394),X(1214)}-harmonic conjugate of X(63)
X(6505) = X(i)-isoconjugate of X(j) for these (i,j): {4,2164}, {19,90}, {25,2994}, {393,1069}

X(6506) = SS(a → cos A) of X(11)

Barycentrics (Cos[B] - Cos[C])^2*(-Cos[A] + Cos[B] + Cos[C]) : :

See X(6374).

X(6506) lies on these lines: {5, 169}, {11, 1146}, {101, 119}, {115, 123}, {124, 5517}, {219, 2165}, {220, 1329}, {910, 1532}, {1212, 4187}, {1566, 5511}, {5190, 5521}

X(6506) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,523), (2990,2804), (2994,522)
X(6506) = X(90)-isoconjugate of X(1262)
X(6506) = {X(11),X(5514)}-harmonic conjugate of X(1146)
X(6506) = X(i)-complementary conjugate of X(j) for these (i,j): (19,4885), (25,522), (29,512), (33,513), (34,3900), (281,3835), (522,1368), (607,514), (652,6389), (663,3), (1096,521), (1172,4369), (1395,6129), (1973,905), (2212,650), (2299,523), (2310,123), (2333,1577), (2356,3126), (3063,1214), (3064,141), (3271,2968), (3709,440), (6059,3239)

X(6507) = SS(a → cos A) of X(31)

Barycentrics Cos[A]^3 : :

See X(6374).

X(6507) lies on these lines: {48, 63}, {92, 1958}, {255, 820}, {304, 2167}, {394, 1804}, {1748, 1959}

X(6507) = isogonal conjugate of X(6520)
X(6507) = isotomic conjugate of X(6521)
X(6507) = X(326)-Ceva conjugate of X(255)
X(6507) = X(4100)-cross conjugate of X(255)
X(6507) = barycentric cube of X(63)
X(6507) = X(i)-isoconjugate of X(j) for these {i,j}: {4,393}, {6,1093}, {19,158}, {25,2052}, {92,1096}, {107,2501}, {264,2207}, {278,1857}, {281,1118}, {331,6059}, {1880,1896}

X(6508) = SS(a → cos A) of X(38)

Barycentrics Cos[A]*(Cos[B]^2 + Cos[C]^2) : :

See X(6374).

X(6508) lies on these lines: {1, 204}, {2, 1952}, {38, 2632}, {48, 63}, {73, 960}, {92, 1953}, {336, 1965}, {774, 820}, {1748, 2173}, {1763, 6261}, {1954, 2629}, {2167, 2349}

X(6508) = X(811)-Ceva conjugate of X(656)
X(6508) = polar conjugate of X(821)
X(6508) = X(i)-isoconjugate of X(j) for these {i,j}: {6,1105}, {19,775}, {25,801}, {48,821}

X(6509) = SS(a → cos A) of X(39)

Barycentrics    Cos[A]^2*(Cos[B]^2 + Cos[C]^2) : :
Barycentrics   sec^2 B + sec^2 C : :
Barycentrics   (cos^2 A)[1 - cos A cos(B - C)] : :

See X(6374).

X(6509) lies on these lines: {2, 216}, {3, 64}, {39, 441}, {51, 852}, {99, 801}, {114, 122}, {184, 426}, {185, 417}, {394, 577}, {418, 2972}, {465, 618}, {466, 619}, {828, 3666}, {1214, 5745}, {1993, 3284}

X(6509) = complement of X(2052)
X(6509) = X(99)-Ceva conjugate of X(520)
X(6509) = {X(418),X(2972)}-harmonic conjugate of X(3917)
X(6509) = X(i)-isoconjugate of X(j) for these (i,j): {6,821}, {19,1105}, {393,775}, {801,1096}
X(6509) = X(i)-complementary conjugate of X(j) for these (i,j): (48,5), (163,520), (184,226), (255,141), (326,626), (394,2887), (560,3767), (563,1147), (577,10), (822,125), (1820,5449), (2148,389), (2149,3042), (2289,1329), (3990,3454), (4055,1211), (4100,3), (6056,3452)

X(6510) = SS(a → cos A) of X(44)

Barycentrics Cos[A]*(2*Cos[A] - Cos[B] - Cos[C]) : :

See X(6374).

X(6510) lies on these lines: {1, 5696}, {63, 77}, {142, 1100}, {241, 2323}, {326, 3694}, {521, 656}, {527, 1323}, {664, 1944}, {908, 6357}, {940, 1449}, {995, 1386}, {1060, 3940}, {1419, 2324}, {1439, 2289}, {3812, 4658}, {5733, 5880}

X(6510) = midpoint of X(664) and X(1944)
X(6510) = X(i)-isoconjugate of X(j) for these (i,j): {4,2291}, {19,1156}, {25,1121}, {278,4845}

X(6511) = SS(a → cos A) of X(55)

Barycentrics Cos[A]^2*(Cos[A] - Cos[B] - Cos[C]) : :

See X(6374).

X(6511) lies on these lines: {394, 1804}, {1800, 3157}, {2178, 5905}, {3964, 3998}

X(6511) = X(63)-Ceva conjugate of X(394)
X(6511) = X(i)-isoconjugate of X(j) for these (i,j): {90,393}, {158,2164}, {1096,2994}

X(6512) = SS(a → cos A) of X(56)

Barycentrics Cos[A]^2*(Cos[A] + Cos[B] - Cos[C])*(Cos[A] - Cos[B] + Cos[C]) : :

See X(6374).

X(6512) lies on these lines: {283, 1069}, {333, 2164}

X(6512) = X(i)-isoconjugate of X(j) for these (i,j): {19,1068}, {46,393}, {158,2178}, {1096,5905}, {1880,3559}

X(6513) = SS(a → cos A) of X(57)

Barycentrics Cos[A]*(Cos[A] + Cos[B] - Cos[C])*(Cos[A] - Cos[B] + Cos[C]) : :

See X(6374).

X(6513) lies on these lines: {2, 914}, {21, 90}, {78, 1069}, {280, 4511}, {2164, 2339}

X(6513) = X(2994)-Ceva conjugate of X(63)
X(6513) = X(i)-cross conjugate of X(j) for these (i,j): (394,63), (1062,77)
X(6513) =X(i)-isoconjugate of X(j) for these (i,j): {4,2178}, {6,1068}, {19,46}, {25,5905}, {281,1406}, {393,3157}, {608,5552}, {1400,3559}, {1880,3193}}.

X(6514) = SS(a → cos A) of X(58)

Barycentrics Cos[A]^2*(Cos[A] + Cos[B])*(Cos[A] + Cos[C]) : :

See X(6374).

X(6514) lies on these lines: {48, 63}, {78, 1800}, {110, 2365}, {261, 284}, {271, 1792}, {394, 577}, {1259, 6056}, {1801, 4575}, {1958, 5307}, {2289, 3719}

X(6314) = X(394)-cross conjugate of X(1812)
X(6314) = X(i)-isoconjugate of X(j) for these (i,j): {4,1880}, {12,5317}, {19,225}, {34,1826}, {37,1118}, {65,393}, {108,2501}, {158,1400}, {226,1096}, {273,2333}, {278,1824}, {281,1426}, {1093,1409}, {1402,2052}, {1427,1857}, {1441,2207}, {1446,6059}

X(6515) = SS(a → cos A) of X(69)

Barycentrics -Cos[A]^2 + Cos[B]^2 + Cos[C]^2 : :
Barycentrics 1 - 2 cos A sin B sin C : :

See X(6374).

X(6515) lies on the cubic K621 and these lines: {2, 6}, {4, 52}, {23, 161}, {24, 6193}, {25, 3564}, {51, 1352}, {92, 1947}, {110, 6353}, {155, 3542}, {317, 2052}, {393, 467}, {406, 3193}, {427, 1351}, {459, 2986}, {468, 3167}, {487, 1600}, {488, 1599}, {511, 1370}, {569, 631}, {914, 1708}, {973, 2888}, {1147, 3147}, {1209, 3090}, {1895, 5081}, {2781, 3448}, {3266, 4176}, {6225, 6293}

X(6515) = isotomic conjugate of X(6504)
X(6515) = reflection of X(1370) in X(1899)
X(6515) = anticomplement of X(394)
X(6515) = complement of polar conjugate of X(36612)
X(6515) = X(i)-Ceva conjugate of X(j) for these (i,j): (317,4), (2052,2)
X(6515) = X(1609)-cross conjugate of X(3542)
X(6515) = polar conjugate of X(254)
X(6515) = X(i)-isoconjugate of X(j) for these (i,j): {6,921}, {19,39109}, {48,254}
X(6515) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,193,1993), (6,343,2), (52,68,4), (1993,3580,2), (2994,5905,92)
X(6515) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (4,4329), (19,20}, (25,6360},(34,347}, (92,1370}, (158,69}, (225,2897}, (393,8}, (823,512}, (1096,2}, (1118,7}, (1824,3151}, (1857,329}, (1880,3152}, (1973,3164}, (2052,6327}, (2207,192}, (5317,1}, (6059,3177)

X(6516) = SS(a → cos A) of X(100)

Barycentrics Cos[A]*(Cos[A] - Cos[B])*(Cos[A] - Cos[C]) : :

See X(6374).

X(6516) lies on these lines: {3, 348}, {7, 1470}, {56, 6337}, {69, 1804}, {73, 1808}, {85, 404}, {99, 108}, {100, 658}, {101, 1025}, {104, 150}, {109, 1310}, {190, 2406}, {279, 4188}, {307, 1444}, {336, 1231}, {644, 4564}, {651, 662}, {668, 4998}, {905, 906}, {927, 1292}, {1004, 1996}, {1055, 6168}, {1332, 1813}, {1461, 3882}, {1810, 4855}, {1812, 1949}, {4561, 4571}, {5172, 5866}

X(6516) = isogonal conjugate of X(18344)
X(6516) = X(i)-Ceva conjugate of X(j) for these (i,j): (99,664), (1275,394), (4554,651), (4620,1812), (4998,69)
X(6516) = X(i)-cross conjugate of X(j) for these (i,j): (78,4564), (394,1275), (521,63), (905,348), (1331,1332), (4025,1444), (4131,69)
X(6516) = {X(100),X(934)}-harmonic conjugate of X(664)
X(6516) = trilinear pole of line X(63)X(77)
X(6516) = X(i)-isoconjugate of X(j) for these (i,j): {4,663}, {6,3064}, {19,650}, {25,522}, {27,3709}, {28,4041}, {29,512}, {33,513}, {34,3900}, {92,3063}, {108,2310}, {158,1946}, {162,4516}, {270,4705}, {278,657}, {281,649}, {284,2501}, {318,667}, {333,2489}, {393,652}, {514,607}, {521,1096}, {523,2299}, {608,3239}, {661,1172}, {693,2212}, {810,1896}, {884,1861}, {885,2356}, {1021,1880}, {1024,5089}, {1119,4105}, {1395,4397}, {1396,4171}, {1398,4163}, {1435,4130}, {1459,1857}, {1474,3700}, {1577,2204}, {1783,2170}, {1824,3737}, {1897,3271}, {1973,4391}, {2189,4024}, {2203,4086}, {2207,6332}, {2333,4560}, {2969,3939}, {4017,4183}, {4025,6059}

X(6517) = SS(a → cos A) of X(101)

Barycentrics Cos[A]^2*(Cos[A] - Cos[B])*(Cos[A] - Cos[C]) : :

See X(6374).

X(6517) lies on these lines: {99, 109}, {1813, 4558}

X(6517) = X(i)-isoconjugate of X(j) for these (i,j): {19,3064}, {107,4516}, {158,663}, {393,650}, {512,1896}, {513,1857}, {522,1096}, {693,6059}, {1093,1946}, {1118,3900}, {1172,2501}, {2052,3063}, {2207,4391}, {3700,5317}

X(6518) = SS(a → cos A) of X(238)

Barycentrics Cos[A]*(Cos[A]^2 - Cos[B]*Cos[C]) : :

See X(6374).

X(6518) lies on these lines: {9, 326}, {63, 77}, {69, 2289}, {226, 801}, {261, 284}, {281, 1958}, {304, 2329}, {521, 1946}, {527, 1275}, {991, 993}, {1944, 5088}, {1948, 2202}

X(6518) = X(i)-isoconjugate of X(j) for these (i,j): {4,1945}, {19,1937}, {25,1952}, {158,1949}, {225,2249}, {296,393}

X(6519) = CENTER OF MOSES-LUCAS CIRCLE

Trilinears 11 cos A + 6 sin A : :
Barycentrics a^2 (11 S_A + 6 S) : :

X(6519) lies on these lines: {3,6}, {3628,6459}, {5072,6561}, {5079,5418}

Let (R) be the radical circle of the Lucas circles, and let (L) = inverse in (R) of the circumcircle. Then (L) tangent to each of the 3 Lucas circles. Let A′ be the touchpoint of (L) and the A-Lucas circle, and define B′ and C′ cyclically. Then

A′ = a²(3S + 4S_A) : 2b² (S + 2S_B) : 2c² (S + 2S_C)
B′ = 2a² (S + 2S_A) : b²(3S + 4S_B) : 2c² (S + 2S_C )
C′ = 2a² (S + 2S_A) : 2b² (S + 2S_B) : c² (3S + 4S_C).

Let A′_B = inverse-in-A-Lucas-circle of B′, and define B′_C and C′_A cyclically. Let A′_C = inverse-in-A-Lucas-circle of C′, and define B′_A and C′_B cyclically. The six points A′_B, B′_C, C′_A, A′_C, B′_A, C′_B lie on a circle with center X(6519). The circle is here called the Moses-Lucas Circle. It and the circle constructed from 6 points at X(6449) are members of the Schoute coaxal system; the Schoute circle with center

a²(pS_A + qS) : b²(pS_B + qS) : c²(pS_C + qS) = pS*X(3) + qS_ω*X(6)

has radius R(S(p² - 3q²)^1/2) / (pS + qS_ω). See X(6449) and X(6522). (Peter Moses, February 11, 2015)

A combo for X(6519) is found as follows: Let L_A and R_A be the center and radius of the A-Lucas circle, and define L_B and L_C, and also R_B and R_C, cyclically. Let L* be the center of the Lucas inner circle and R* its radius. Then X(6519) = L_A/R_A + L_B/R_B + L_C/R_C + L*/R* + X(3)/R. (Circle (L) is the Lucas inner circle, and A′B′C′ is the Lucas inner triangle.) (Randy Hutson, February 12, 2015)

If you have The Geometer's Sketchpad, you can view X(6519).

X(6519) lies on these lines: {3,6}, {3628,6459}, {5072,6561}, {5079,5418}

X(6519) = radical center of the Lucas(12/11) circles
X(6519) = inverse-in-Brocard-circle of X(6522)
X(6519) = inverse-in-Lucas-secondary-tangents-circle of X(6566)
X(6519) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6522), (3,371,6427), (3,3592,3312), (3,6199,6420), (3,6221,6447), (3,6407,6453), (3,6417,6426), (3,6419,6448), (3,6420,6450), (3,6426,6452), (3,6427,6398), (3,6447,3311), (3,6453,6221), (3,6522,6497), (6,1151,6484), (6,6496,6497), (371,1151,6445), (371,6200,6410), (371,6398,3311), (371,6411,6418), (371,6445,6455), (371,6451,6495), (371,6455,6398), (1151,6221,6449), (1151,6407,6221), (1151,6429,6200), (1151,6453,3), (1151,6468,371), (1151,6480,6407), (3311,6449,6451), (3311,6451,6456), (3312,6410,6398), (3312,6450,6438), (3592,6200,3), (3592,6489,6), (3594,6410,6454), (3594,6425,371), (3594,6427,6428), (6199,6409,6450), (6221,6445,6398), (6221,6449,3311), (6221,6452,6437), (6221,6455,371), (6221,6484,6497), (6398,6428,3594), (6398,6449,6455), (6409,6420,3), (6410,6468,6429), (6417,6474,6437), (6417,6487,3312), (6419,6448,6428), (6427,6455,3), (6440,6500,3312), (6445,6455,6449), (6445,6468,6221), (6447,6449,3), (6456,6495,6398), (6484,6496,6449), (6496,6522,3)

X(6520) = ISOTOMIC CONJUGATE OF X(1102)

Barycentrics   sin^2 A sec^3 A : :
Barycentrics   sec A tan^2 A : :
Barycentrics   (tan A)/(csc A - sin A) : :
Barycentrics    b*c*S_B^3*S_C^3 : :

X(6520) lies on these lines: {19, 158}, {48, 821}, {92, 1953}, {393, 1880}, {610, 1784}, {1093, 1826}, {1118, 2358}, {1824, 1857}

X(6520) = isotomic conjugate of X(1102)
X(6520) = pole wrt polar circle of trilinear polar of X(326) (the line X(822)X(4131))
X(6520) = polar conjugate of X(326)
X(6520) = X(1096)-cross conjugate of X(158)
X(6520) = X(i)-isoconjugate of X(j) for these (i,j): {2,1092}, {3,394}, {6,3964}, {31,1102}, {32,4176}, {48,326}, {63,255}, {69,577}, {75,4100}, {77,2289}, {97,5562}, {184,3926}, {219,1804}, {222,1259}, {249,2972}, {348,6056}, {417,801}, {520,4558}, {593,4158}, {603,3719}, {822,4592}, {906,4131}, {1331,4091}, {1437,3998}, {1444,3990}, {1576,4143}, {1790,3682}, {3289,6394}

X(6521) = ISOGONAL CONJUGATE OF X(4100)

Barycentrics sec^3 A : :
Barycentrics b^3*c^3*(-a^2 + b^2 - c^2)^3*(a^2 + b^2 - c^2)^3 : :

X(6521) lies on these lines: {92, 1953}, {158, 774}, {823, 1748}

X(6521) = isogonal conjugate of X(4100)
X(6521) = isotomic conjugate of X(6507)
X(6521) = X(1881)-cross conjugate of X(4)
X(6521) = barycentric cube of X(92)
X(6521) = pole wrt polar circle of trilinear polar of X(255) (line X(680)X(822))
X(6521) = polar conjugate of X(255)
X(6521) = X(i)-isoconjugate of X(j) for these (i,j): {1,4100}, {3,577}, {6,1092}, {32,3964}, {48,255}, {97,418}, {184,394}, {222,6056}, {560,1102}, {603,2289}, {822,4575}, {1437,3990}, {1501,4176}, {1790,4055}

X(6522) = CENTER OF MOSES-LUCAS(-1) CIRCLE

Trilinears 11 cos A - 6 sin A : :
Barycentrics a^2 (11 S_A - 6 S) : :

Let (R) be the radical circle of the Lucas(-1) circles, and let (L) = inverse in (R) of the circumcircle. Then (L) tangent to each of the 3 Lucas(-1) circles. Let A′ be the touchpoint of (L) and the A-Lucas(-1) circle, and define B′ and C′ cyclically. Then

A′ = a²(3S - 4S_A) : 2b²(S - 2S_B) : 2c²(S - 2S_C)
B′ = 2a²(S - 2S_A) : b²(3S - 4S_B) : 2c²(S - 2S_C )
C′ = 2a²(S - 2S_A) : 2b²(S - 2S_B) : c²(3S - 4S_C)

Let A′_B = inverse-in-A-Lucas(-1)-circle of B′, and define B′_C and C′_A cyclically. Let A′_C = inverse-in-A-Lucas(-1)-circle of C′, and define B′_A and C′_B cyclically. The six points A′_B, B′_C, C′_A, A′_C, B′_A, C′_B lie on a circle with center X(6522). The circle is here called the Moses-Lucas(-1) Circle. It is a member of the Schoute coaxal system; see X(6519). Note that (L) is the Lucas(-1) inner circle and A′B′C′ is the Lucas(-1) inner triangle. (Randy Hutson, February 12, 2015)

X(6522) lies on these lines: {3,6}, {3628,6460}, {5072,6560}, {5079,5420}

X(6522) = radical center of the Lucas(-12/11) circles
X(6522) = inverse-in-Lucas(-1)-secondary-tangents-circle of X(6567)
X(6522) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,6519), (3,372,6428), (3,3594,3311), (3,6395,6419), (3,6398,6448), (3,6408,6454), (3,6418,6425), (3,6419,6449), (3,6420,6447), (3,6425,6451), (3,6428,6221), (3,6448,3312), (3,6454,6398), (3,6519,6496), (6,1152,6485), (6,6497,6496), (372,1152,6446), (372,6221,3312), (372,6396,6409), (372,6412,6417), (372,6446,6456), (372,6452,6494), (372,6456,6221), (1152,6398,6450), (1152,6408,6398), (1152,6430,6396), (1152,6454,3), (1152,6469,372), (1152,6481,6408), (3311,6409,6221), (3311,6449,6437), (3312,6450,6452), (3312,6452,6455), (3592,6409,6453), (3592,6426,372), (3592,6428,6427), (3594,6396,3), (3594,6488,6), (6221,6427,3592), (6221,6450,6456), (6395,6410,6449), (6398,6446,6221), (6398,6450,3312), (6398,6451,6438), (6398,6456,372), (6398,6485,6496), (6409,6469,6430), (6410,6419,3), (6418,6475,6438), (6418,6486,3311), (6420,6447,6427), (6428,6456,3), (6439,6501,3311), (6446,6456,6450), (6446,6469,6398), (6448,6450,3), (6455,6494,6221), (6485,6497,6450), (6497,6519,3)

X(6523) = SS(a → S_BS_C) of X(3)

Barycentrics S_B^2*S_C^2*(-(S_A^2*S_B^2) - S_A^2*S_C^2 + S_B^2*S_C^2) : :

See X(6374).

X(6523) lies on the cubic pK(X393,X2), this being the polar conjugate of the Lucas cubic; see Hyacinthos #28116. Also, X(6523) is the centoid of the quadrilateral whose vertices are X(4) together with the vertices of the anticevian triangle of X(4). (Randy Hutson, February 16, 2015)

Let A′B′C′ be the medial triangle. Let A″ be the center of the circle which is the inverse-in-polar-circle of line B′C′. Define B″and C″ cyclically. The lines A′A″, B′B″, C′C″ concur in X(6523). (Randy Hutson, July 11, 2019)

X(6523) lies on these lines: {2, 3346}, {4, 64}, {20, 107}, {133, 5878}, {158, 278}, {393, 800}, {1075, 5656}, {1249, 3349}, {1559, 6225}

X(6523) = midpoint of X(4) and X(3183)
X(6523) = complement of X(3346)
X(6523) = X(2)-Ceva conjugate of X(393)
X(6523) = X(i)-complementary conjugate of X(j) for these (i,j): (1,6247), (31,393), (48,1073), (1033,226), (1498,10), (1712,5)
X(6523) = polar conjugate of X(1032)
X(6523) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,6247), (1093,3089,393)
X(6523) = X(i)-isoconjugate of X(j) for these {i,j}: {48,1032}, {255,3346}

X(6524) = SS(a → S_BS_C) of X(31)

Trilinears       sec A tan²A : :
Barycentrics    S_B^3*S_C^3 : :
Barycentrics    tan³A : :
Barycentrics    1/(b^2 + c^2 - a^2)^3 : :

See X(6374).

X(6524) lies on these lines: {4, 51}, {25, 393}, {107, 3563}, {125, 459}, {154, 1990}, {184, 1249}, {1118, 1426}, {1204, 3183}, {1495, 3079}, {1824, 1857}

X(6524) = X(1093)-Ceva conjugate of X(393)
X(6524) = X(2207)-cross conjugate of X(393)
X(6524) = X(i)-isoconjugate of X(j) for these (i,j): {1,3964}, {3,326}, {6,1102}, {31,4176}, {48,3926}, {63,394}, {69,255}, {75,1092}, {76,4100}, {77,1259}, {78,1804}, {163,4143}, {222,3719}, {304,577}, {348,2289}, {520,4592}, {603,1264}, {757,4158}, {822,4563}, {1331,4131}, {1332,4091}, {1444,3682}, {1790,3998}, {3265,4575}

X(6524) = isogonal conjugate of X(3964)
X(6524) = isotomic conjugate of X(4176)
X(6524) = barycentric cube of X(4)
X(6524) = trilinear pole of line X(2489)X(2508)
X(6524) = pole wrt polar circle of trilinear polar of X(3926) (the line X(520)X(3265))
X(6524) = polar conjugate of X(3926)

X(6525) = SS(a → S_BS_C) of X(55)

Trilinears    (tan^2 A)(cos A - cos B cos C) : :
Barycentrics    S_B^2*S_C^2*(-(S_A*S_B) - S_A*S_C + S_B*S_C) : :
Barycentrics   [3a^4 - 2a^2b^2 - 2a^2c^2 - (b^2 - c^2)^2]/(b^2 + c^2 - a^2)^2 : :

See X(6374).

Let DEF be the orthic triangle and D' the midpoint of AD . Let A′ be the reflection of A in D'X(3), Let La be the perpendicular bisector of A′D, and define Lb and Lc cyclically. Let A″ = Lb?Lc, B″= Lc?La, C″ = La?Lb. The lines DA″, EB″, FC″ concur in X(6525). (Angel Montesdeoca, August 21, 2019)

X(6525) lies on these lines: {2, 107}, {4, 64}, {19, 1857}, {25, 393}, {34, 207}, {51, 3087}, {154, 1249}, {1859, 2262}, {1896, 4198}, {2052, 3424}

X(6525) = X(4)-Ceva conjugate of X(393)
X(6525) = X(3172)-cross conjugate of X(1249)
X(6525) = polar conjugate of X(34403)
X(6525) = Danneels point of X(107)
X(6525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,1853), (4,3183,64), (1249,3079,154)
X(6525) = X(i)-isoconjugate of X(j) for these (i,j): {63,1073}, {64,326}, {253,255}, {394,2184}, {2155,3926}

X(6526) = SS(a → S_BS_C) of X(56)

Barycentrics   (tan^2 A)/(tan B + tan C - tan A) : :
Barycentrics   1/[S_A^2(S^2 - 2 S_B S_C)] : :
Barycentrics    S_B^2*S_C^2*(S_A*S_B - S_A*S_C + S_B*S_C)*(-(S_A*S_B) + S_A*S_C + S_B*S_C) : :

See X(6374).

Let A′B′C′ be the tangential triangle of the Kiepert hyperbola. Let A″ be the intersection, other than X(122), of the nine-point circle and the line A′X(122); define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6526). (Randy Hutson, February 16, 2015)

Let A′B′C′ be the reflection of the orthic triangle in X(4). X(6526) is the trilinear product A′*B′*C′. Moreover, the trilinear polar of X(6526) passes through X(2501). (Randy Hutson, February 16, 2015)

X(6526) lies on these lines: {2, 1105}, {4, 64}, {5, 1073}, {107, 3146}, {122, 3346}, {225, 1857}, {235, 393}, {253, 264}, {254, 403}, {1249, 5922}, {1300, 1301}, {2184, 5715}

X(6526) = isogonal conjugate of X(35602)
X(6526) = cevapoint of X(4) and X(6622)
X(6526) = X(459)-Ceva conjugate of X(393)
X(6526) = polar conjugate of X(37669)
X(6526) = perspector of ABC and the 2nd pedal triangle of X(64)
X(6526) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,459,64), ( 4,3183,5895)
X(6526) = X(i)-isoconjugate of X(j) for these (i,j): {63,15905}, {20,255}, {122,1101}, {154,326}, {204,3964}, {394,610}, {1092,1895}, {1102,3172}, {1259,1394}

X(6527) = SS(a → S_BS_C) of X(69)

Barycentrics S_A^2*S_B^2 + S_A^2*S_C^2 - S_B^2*S_C^2 : :
Barycentrics tan^2 A - tan^2 B - tan^2 C : :
X(6527) = 3X(2) - 4X(6389)

See X(6374).

X(6527) lies on these lines: {2, 216}, {20, 64}, {75, 280}, {99, 5897}, {193, 401}, {317, 3146}, {322, 345}, {340, 5059}, {441, 1249}, {1632, 5596}, {2385, 4329}, {3346, 3926}

X(6527) = reflection of X(393) in X(6389)
X(6527) = isotomic conjugate of X(3346)
X(6527) = anticomplement of X(393)
X(6527) = X(3926)-Ceva conjugate of X(2)
X(6527) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20,253,69), (280,347,75), (393,6389,2)
X(6527) = X(i)-isoconjugate of X(j) for these {i,j}: {31,3346}, {1032,1973}, {2155,3344}
X(6527) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (2,5906), (3,5905), (31,6392), (48,193), (63,4), (75,317), (219,5942), (255,2), (283,92), (326,69), (394,8), (577,192), (662,520), (775,2052), (822,148), (1069,2994), (1092,6360), (1101,648), (1102,1370), (1259,329), (1331,4391), (1437,3187), (1790,3868), (1804,7), (1813,521), (1822,2592), (1823,2593), (2167,5889), (2169,1993), (2289,144), (3682,2895), (3719,3436), (3926,6327), (3964,4329), (3990,1654), (3998,1330), (4055,1655), (4091,149), (4100,3164), (4131,150), (4575,525), (4592,850), (6056,3177)

X(6528) = SS(a → S_BS_C) of X(99)

Trilinears    csc^2 2A csc(B - C) : :
Barycentrics   (csc 2A)/(sin 2B - sin 2C) : :
Barycentrics   1/(tan^2 B - tan^2 C) : :
Barycentrics   1/(sec^2 B - sec^2 C) : :
Barycentrics   (sec^2 A)/(b^2 - c^2) : :
Barycentrics    (S_A - S_B)*S_B^2*(S_A + S_B)*(S_A - S_C)*S_C^2*(S_A + S_C) : :

See X(6374).

X(6528) lies on the Steiner circumellipse and these lines: {4, 290}, {5, 276}, {99, 107}, {112, 2966}, {190, 823}, {264, 339}, {393, 3228}, {648, 1625}, {664, 811}, {670, 877}, {671, 2052}, {1896, 2481}, {2207, 3225}

X(6528) = isogonal conjugate of X(39201)
X(6528) = isotomic conjugate of X(520)
X(6528) = anticomplement of X(35071)
X(6528) = anticomplement of isogonal conjugate of X(34538)
X(6528) = center of bianticevian conic of X(2) and X(4)
X(6528) = trilinear pole of line X(2)X(216)
X(6528) = Brianchon point (perspector) of inscribed parabola with focus X(107)
X(6528) = polar conjugate of X(647)
X(6528) = X(i)-cross conjugate of X(j) for these (i,j): (520,2}, (525,276}, (648,6331}, (850,264}, (3060,250}, (5889,249)
X(6528) = X(i)-isoconjugate of X(j) for these (i,j): {3,810}, {6,822}, {31,520}, {48,647}, {63,3049}, {73,1946}, {163,3269}, {184,656}, {213,4091}, {228,1459}, {255,512}, {326,669}, {394,798}, {418,2616}, {513,4055}, {560,3265}, {577,661}, {649,3990}, {652,1409}, {667,3682}, {905,2200}, {1576,2632}, {1636,2159}, {1918,4131}, {1919,3998}, {1924,3926}, {2501,4100}, {4017,6056}

X(6529) = SS(a → S_BS_C) of X(101)

Barycentrics (tan^2 A)/(sin 2B - sin 2C) : :
Barycentrics (S_B^3*(-S_A + S_B)*(S_A - S_C)*S_C^3) : :

See X(6374).

X(6529) lies on these lines: {4, 1562}, {107, 112}, {115, 393}, {127, 6330}, {133, 1249}, {648, 1625}, {1093, 2207}

X(6529) = isotomic conjugate of X(4143)
X(6529) = trilinear pole of line X(25)X(393)
X(6529) = polar conjugate of X(3265)
X(6529) = X(i)-cross conjugate of X(j) for these (i,j): (2501,393), (3049,25)
X(6529) = trilinear product of vertices of circumanticevian triangle of X(4)
X(6529) = barycentric product X(4)*X(107)
X(6529) = X(i)-isoconjugate of X(j) for these (i,j): {31,4143}, {48,3265}, {63,520}, {69,822}, {71,4131}, {72,4091}, {255,525}, {326,647}, {394,656}, {512,1102}, {661,3964}, {662,2972}, {798,4176}, {810,3926}, {850,4100}, {905,3682}, {1019,4158}, {1092,1577}, {1459,3998}, {2632,4558}, {3269,4592}, {3990,4025}

X(6530) = SS(a → S_BS_C) of X(238)

Barycentrics tan A sec A cos(A + ω) : :
Barycentrics S_B^2*S_C^2*(-S_A^2 + S_B*S_C) : :
X(6530) = X(4) + 2X(1990)

See X(6374).

X(6530) lies on these lines: {4, 6}, {5, 264}, {24, 1485}, {25, 3425}, {30, 250}, {107, 468}, {132, 232}, {186, 3447}, {262, 427}, {297, 511}, {317, 1351}, {324, 5133}, {325, 2967}, {403, 523}, {428, 1629}, {429, 1896}, {460, 685}, {467, 3060}, {648, 3564}, {1594, 3613}

X(6530) = isotomic conjugate of X(6394)
X(6530) = polar conjugate of X(287)
X(6530) = perspector of circumconic through the polar conjugates of PU(37)
X(6530) = X(i)-cross conjugate of X(j) for these (i,j): (132,4), (232,297), (2450,325)
X(6530) = pole wrt polar circle of trilinear polar of X(287) (the line PU(37) = X(3)X(525))
X(6530) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (53,5480,4), (132,232,1513)
X(6530) = X(238)-of-orthic-triangle if ABC is acute
X(6530) = radical center of circumcircle and P(39)- and U(39)-Fuhrmann circles (aka -Hagge circles)
X(6530) = X(i)-isoconjugate of X(j) for these (i,j): {3,293}, {31,6394}, {48,287}, {63,248}, {98,255}, {184,336}, {326,1976}, {394,1910}, {577,1821}, {822,2966}, {878,4592}, {879,4575}

X(6531) = SS(a → S_BS_C) of X(292)

Barycentrics    sin A tan A sec(A + ω) : :
Barycentrics    1/[(b^4 + c^4 - a^2b^2 - a^2c^2)(b^2 + c^2 - a^2)] : :
Barycentrics    (S_B*S_C*(S_B^2 - S_A*S_C)*(S_A*S_B - S_C^2)) : :

See X(6374).

X(6531) lies on these lines: {2, 6394}, {4, 32}, {6, 264}, {107, 3124}, {111, 4240}, {213, 1783}, {225, 1910}, {230, 297}, {232, 419}, {275, 3051}, {393, 1974}, {436, 1196}, {450, 3291}, {578, 1217}, {685, 1990}, {1093, 2207}, {1105, 1970}, {1300, 2715}, {1501, 1629}, {1821, 2281}, {1826, 1918}

X(6531) = isogonal conjugate of X(36212)
X(6531) = isotomic conjugate of X(6393)
X(6531) = trilinear pole of line X(25)X(669) (the line through the polar conjugates of PU(37))
X(6531) = pole wrt polar circle of trilinear polar of X(325) (the line X(2799)X(3569))
X(6531) = polar conjugate of X(325)
X(6531) = barycentric product X(4)*X(98)
X(6531) = cevapoint of X(i) and X(j) for these {i,j}: {4, 419}, {6, 230}, {25, 2211}, {607, 862}, {3124, 17994}
X(6531) = X(i)-cross conjugate of X(j) for these (i,j): (460,4), (1976,98), (2086,2489), (2211,25)
X(6531) = X(i)-isoconjugate of X(j) for these (i,j): {3,1959}, {31,6393}, {48,325}, {63,511}, {69,1755}, {75,3289}, {163,6333}, {232,326}, {237,304}, {240,394}, {255,297}, {656,2421}, {662,684}, {810,2396}, {822,877}, {2799,4575}, {3405,3917}, {3569,4592}

X(6532) = SS(a → b + c) of X(5)

Barycentrics a^2*b^2 + a*b^3 - 6*a^2*b*c - 5*a*b^2*c - b^3*c + a^2*c^2 - 5*a*b*c^2 - 2*b^2*c^2 + a*c^3 - b*c^3 : :
X(6532) = 3X(2) + X(596)

See X(6374).

X(6532) lies on these lines: {2, 596}, {10, 244}, {519, 3812}, {726, 4698}, {1125, 3666}

X(6532) = midpoint of X(596) and X(4075)
X(6532) = complement of X(4075)
X(6532) = X(849)-complementary conjugate of X(4075)
X(6532) = {X(2),X(596)}-harmonic conjugate of X(4075)

X(6533) = SS(a → b + c) of X(12)

Barycentrics b*c*(2*a + b + c)^2 : :

See X(6374).

X(6533) lies on these lines: {1, 3996}, {2, 1089}, {8, 3892}, {10, 244}, {75, 3624}, {190, 5506}, {274, 1111}, {333, 3337}, {341, 1698}, {443, 4680}, {596, 756}, {872, 3216}, {1125, 1962}, {3338, 4384}, {3634, 3992}, {3828, 4696}, {3881, 4651}, {4858, 4999}, {5433, 6358}

X(6533) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1125,4359,4647), (1125,4647,4975), (3634,4968,3992)

X(6534) = SS(a → b + c) of X(30)

Barycentrics 3*a^2*b^2 + 3*a*b^3 - 2*a^2*b*c + a*b^2*c - 3*b^3*c + 3*a^2*c^2 + a*b*c^2 - 6*b^2*c^2 + 3*a*c^3 - 3*b*c^3 : :

See X(6374).

X(6534) lies on these lines: {2, 596}, {30, 511}, {551, 3159}

X(6535) = SS(a → b + c) of X(31)

Barycentrics (b + c)^3 : :

See X(6375).

X(6535) lies on these lines: {10, 3995}, {42, 2321}, {321, 2887}, {594, 756}, {968, 4873}, {1211, 3994}, {1215, 3969}, {1962, 3943}, {2345, 5311}, {3836, 4980}, {3896, 4527}, {4058, 4082}, {4368, 4651}, {4439, 4981}

X(6535) = X(594)-Ceva conjugate of X(762)
X(6535) = barycentric cube of X(10)
X(6535) = X(i)-isoconjugate of X(j) for these {i,j}: {6,763}, {58,757}, {60,1014}, {81,593}, {86,849}, {261,1408}, {552,2194}, {873,2206}, {1019,4556}, {1333,1509}, {1412,2185}, {1434,2150}}
X(6535) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (594,6057,756), (1215,3969,4062), (1215,4535,3969)

X(6536) = SS(a → b + c) of X(38)

Barycentrics (b + c)*(2*a^2 + 2*a*b + b^2 + 2*a*c + c^2) : :

See X(6375).

X(6536) lies on these lines: {1, 2895}, {2, 846}, {10, 3995}, {38, 3122}, {42, 4104}, {86, 4683}, {748, 4657}, {756, 4026}, {1211, 1962}, {1213, 4037}, {1284, 4204}, {1473, 4423}, {2292, 4205}, {2486, 3925}, {3616, 5429}, {3624, 3648}, {3706, 4708}, {3720, 4357}, {3724, 4199}, {3842, 4972}, {3914, 5257}

X(6536) = X(4632)-Ceva conjugate of X(4988)
X(6536) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4425,3120), (1211,1962,4062)

X(6537) = SS(a → b + c) of X(39)

Barycentrics (b + c)^2*(2*a^2 + 2*a*b + b^2 + 2*a*c + c^2) : :

See X(6375).

X(6537) lies on these lines: {2, 1171}, {10, 115}, {39, 5743}, {44, 1213}, {594, 4075}, {626, 5224}, {960, 5164}, {966, 3767}, {1211, 3912}, {2238, 5280}, {3124, 3954}, {3841, 4047}

X(6537) = complement of X(1509)
X(6537) = X(i)-complementary conjugate of X(j) for these (i,j): (37,3741), (41,4999), (42,3739), (181,142), (213,1125), (594,2887), (756,141), (762,3454), (798,244), (813,4155), (872,2), (1018,512), (1089,626), (1110,620), (1334,960), (1400,3742), (1402,3946), (1500,10), (1918,3666), (2171,2886), (2333,942), (3949,1368), (4079,11), (4557,4369), (4705,116), (6378,3840)

X(6538) = SS(a → b + c) of X(58)

Barycentrics (b + c)^2*(a + 2*b + c)*(a + b + 2*c) : :
X(6538) = X(1126) + 3X(4102)

See X(6375).

X(6538) lies on these lines: {10, 3995}, {12, 6058}, {80, 3626}, {313, 4066}, {519, 1126}, {594, 4075}, {1125, 1224}, {1268, 3634}, {1826, 4058}, {3773, 3841}

X(6538) = isotomic conjugate of X(30593)
X(6538) = X(i)-isoconjugate of X(j) for these {i,j}: {553,2150}, {593,1100}, {757,2308}, {849,1125}, {4556,4979}

X(6539) = SS(a → b + c) of X(81)

Barycentrics (b + c)*(a + 2*b + c)*(a + b + 2*c) : :

See X(6375).

Let A₁₃B₁₃C₁₃ be the Gemini triangle 13. Let L_A be the line through A₁₃ parallel to BC, and define L_B and L_C cyclically. Let A′₁₃ = L_B∩L_C, and define B′₁₃ and C′₁₃ cyclically. Triangle A′₁₃B′₁₃C′₁₃ is homothetic to ABC at X(6539). (Randy Hutson, November 30, 2018)

X(6539) lies on the Kiepert hyperbola and these lines: {2, 594}, {4, 3617}, {8, 1126}, {10, 3995}, {1029, 1654}, {1211, 4080}, {4033, 4359}, {4444, 4608}

X(6539) = isotomic conjugate of X(8025)
X(6539) = X(i)-cross conjugate of X(j) for these (i,j): (10,1268), (514,4033), (4024,3952)
X(i)-isoconjugate of X(j) for these {i,j}: {58,1100}, {81,2308}, {110,4979}, {163,4977}, {553,2194}, {593,1962}, {849,1213}, {1125,1333}, {1408,3686}, {1412,3683}, {1437,1839}, {1474,3916}, {1576,4978}, {1790,2355}, {2150,3649}, {2203,4001}, {2206,4359}, {4556,4983}
X(6539) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1255,1268,2), (1268,4102,1255)

X(6540) = SS(a → b + c) of X(99)

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

See X(6375).

X(6540) lies on the Steiner circumellipse and these lines: {99, 3952}, {190, 4103}, {645, 4629}, {668, 4756}, {903, 1268}, {1016, 4115}, {1121, 4102}, {1126, 3226}, {1255, 3227}, {1509, 4075}, {4555, 4608}

X(6540) = isotomic conjugate of X(4977)
X(6540) = complement of X(39348)
X(6540) = anticomplement of X(35076)
X(6540) = X(i)-cross conjugate of X(j) for these (i,j): (10,1016), (190,4632), (319,4998), (4608,1268), (4977,2)
X(6540) = X(i)-isoconjugate of X(j) for these (i,j): {6,4979}, {31,4977}, {32,4978}, {58,4983}, {513,2308}, {553,3063}, {604,4976}, {649,1100}, {667,1125}, {849,6367}, {875,4974}, {1106,4990}, {1269,1980}, {1333,4988}, {1397,4985}, {1459,2355}, {1919,4359}, {1962,3733}, {3248,4427}
X(6540) = {X(4596),X(4632)}-harmonic conjugate of X(99)

X(6541) = SS(a → b + c) of X(238)

Barycentrics (a - b)*(a - c)*(a + 2*b + c)*(a + b + 2*c) : :
X(6541) = X(10) + 2X(3943)

See X(6375).

X(6541) lies on the Steiner circumellipse and these lines: {1, 3790}, {10, 37}, {192, 3821}, {226, 4135}, {238, 519}, {306, 3971}, {335, 726}, {346, 3923}, {518, 4439}, {523, 4129}, {536, 3836}, {537, 4966}, {752, 4908}, {756, 3969}, {1089, 3178}, {1215, 6057}, {1268, 3634}, {1386, 3244}, {2325, 5847}, {2784, 6211}, {2796, 4645}, {2887, 3175}, {3701, 4710}, {3712, 4434}, {3844, 4681}, {3936, 3994}, {3952, 4062}, {3995, 4425}, {4015, 4111}, {4028, 4082}, {4037, 4071}, {4387, 4865}, {4422, 4974}, {4432, 5846}, {4669, 4923

X(6541) = midpoint of X(3932) and X(3943)
X(6541) = reflection of X(i) in X(j) for these (i,j): (10,3932), (4974,4422)
X(6541) = X(i)-isoconjugate of X(j) for these (i,j): {58,1929}, {757,2054}, {1019,2702}
X(6541) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10,3950,3993), (10,4072,4133), (10,4133,4709), (37,3773,10), (594,3842,10), (2321,4078,10), (3842,4535,594), (4028,4082,4090)

X(6542) = SS(a → b + c) of X(239)

Barycentrics -a^2 - a*b + b^2 - a*c + b*c + c^2 : :

See X(6375).

X(6542) lies on the cubic K185 and these lines: {1, 2}, {7, 1278}, {37, 319}, {44, 4473}, {57, 4050}, {63, 3208}, {69, 192}, {75, 4675}, {81, 2295}, {86, 594}, {100, 4433}, {141, 4360}, {190, 524}, {193, 346}, {287, 677}, {297, 1897}, {312, 3765}, {314, 3963}, {320, 536}, {321, 1909}, {335, 740}, {344, 5839}, {514, 4024}, {518, 2113}, {599, 4389}, {668, 3948}, {671, 4080}, {752, 4693}, {894, 2321}, {1016, 1252}, {1086, 4971}, {1100, 4889}, {1213, 4478}, {1330, 2901}, {1334, 3219}, {1655, 2895}, {1931, 6157}, {1959, 3930}, {1978, 3978}, {2112, 3684}, {2345, 4916}, {3620, 3672}, {3662, 3875}, {3664, 4431}, {3681, 4517}, {3685, 5847}, {3739, 5564}, {3751, 3790}, {3770, 4043}, {3773, 4649}, {3836, 4716}, {3888, 6007}, {3946, 4464}, {3950, 4416}, {3975, 4358}, {3994, 4938}, {4060, 4967}, {4366, 4437}, {4422, 4969}, {4445, 5224}, {4643, 4664}, {4648, 4699}, {4774, 4789}
X(6542) = 9 X[2] - 8 X[3008] = 3 X[239] - 4 X[3008] = 3 X[2] - 4 X[3912] = 2 X[3008] - 3 X[3912] = 4 X[44] - 5 X[4473] = X[320] + 2 X[4727] = X[4440] + 4 X[4727]

X(6542) = reflection of X(i) in X(j) for these (i,j): (190,3943), (239,3912), (4440,320), (4716,3836), (4969,4422)
X(6542) = isotomic conjugate of X(6650)
X(6542) = complement of X(20016)
X(6542) = anticomplement of X(239)
X(6542) = anticomplementary conjugate of X(20345)
X(6542) = X(335)-Ceva conjugate of X(2)
X(6542) = inverse-in-Steiner-circumellipse of X(10)
X(6542) = X(2)-Hirst inverse of X(10)
X(6542) = trilinear pole of line X(2786)X(9508) (the perspectrix of ABC and Gemini triangle 19)
X(6542) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (291,69), (292,8), (295,4329), (334,315), (335,6327), (694,4388), (741,75), (813,513), (875,4440), (876,150), (1911,2), (1922,192), (2196,20), (2311,3869), (3572,149), (4584,512), (4876,3436), (5378,668)
X(6542) = X(i)-isoconjugate of X(j) for these (i,j): (6,1929), (81,2054), (513,2702)
X(6542) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3661,2), (2,145,4393), (37,319,1654), (69,4419,4741), (192,4741,4419), (239,3912,2), (306,1999,2), (2321,3879,894), (4433,4447,100)

X(6543) = SS(a → b + c) of X(292)

Barycentrics (b + c)^2*(a^2 + a*b + b^2 - a*c - b*c - c^2)*(a^2 - a*b - b^2 + a*c - b*c + c^2) : :

See X(6375).

X(6543) lies on these lines: {10, 115}, {190, 1213}, {313, 338}, {334, 3948}, {1224, 1929}, {1648, 4080}, {2372, 2702}

X(6543) = X(4037)-cross conjugate of X(594)
X(6543) = X(i)-isoconjugate of X(j) for these {i,j}: {58,1931}, {81,1326}, {423,1437}, {593,1757}

X(6544) = SS(a → b - c) of X(11)

Barycentrics (2*a - b - c)^2*(b - c) : :

See X(6375).

The trilinear polar of X(6544) passes through X(4542). (Randy Hutson, February 16, 2015)

X(6544) lies on the Yff parabola, the cubic K219, and these lines: {2, 514}, {9, 649}, {37, 650}, {121, 5513}, {661, 1213}, {900, 1635}, {1023, 4169}, {1566, 3259}, {1647, 2087}, {3161, 3239}, {3251, 4543}, {4088, 4809}, {4728, 6009}, {4750, 4763}

X(6544) = isogonal conjugate of X(4638)
X(6544) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1647), (190,519), (514,900), (1317,4542)
X(6544) = X(4542)-cross conjugate of X(1317)
X(6544) = tripolar centroid of X(519)
X(6544) = X(i)-complementary conjugate of X(j) for these (i,j):} (31,1647), (44,116), (101,3834), (163,4395), (678,3259), (692,519), (902,11), (1023,141), (1110,900), (1252,4928), (1404,4904), (2251,1086), (3689,124), (3939,5123}
X(6544) = X(i)-isoconjugate of X(j) for these {i,j}: {1,4638), (6,4618}, {88,901}, {100,2226}, {101,679}, {106,3257}, {651,1318}, {1417,4582}, {4591,4674}}

X(6545) = SS(a → b - c) of X(31)

Barycentrics (b - c)^3 : :
X(6545) = X[649] - 4 X[3676] = X[693] + 2 X[3776] = 4 X[693] - X[4024] = 8 X[3776] + X[4024] = 4 X[3837] - X[4088] = 2 X[4025] + X[4382] = 5 X[4024] - 8 X[4500] = 5 X[693] - 2 X[4500] = 5 X[3776] + X[4500] = X[4120] - 4 X[4927] = 4 X[4453] - X[4984] = 4 X[3004] - X[4988] = 5 X[3616] - 2 X[5592]

See X(6375).

X(6545) lies on the cubic K656 and these lines: {2, 514}, {57, 649}, {321, 693}, {354, 513}, {561, 3261}, {614, 1027}, {764, 1647}, {812, 4453}, {918, 4120}, {1230, 1577}, {1427, 3669}, {1635, 1638}, {3004, 4988}, {3616, 5592}, {3837, 4088}, {3938, 4449}, {4025, 4382}, {4448, 4977}

X(6545) = reflection of X(i) in X(j) for these (i,j): (1635, 1638), (4120,4728), (4728,4927), (4750, 4453), (4984, 4750)
X(6545) = isotomic conjugate of X(6632)
X(6545) = barycentric cube of X(514)
X(6545) = X(i)-Ceva conjugate of X(j) for these (i,j): {514,1086}, {693,3120}, {3261,1111}, {3676,244}
X(6545) = X(i)-isoconjugate of X(j) for these (i,j): {59,644}, {100,1252}, {101,765}, {190,1110}, {200,4619}, {651,6065}, {692,1016}, {1018,4570}, {1101,4103}, {1262,4578}, {1415,4076}, {2149,3699}, {2284,5377}, {3939,4564}, {4554,6066}, {4557,4567}, {4574,5379}

X(6546) = SS(a → b - c) of X(38)

Barycentrics (b - c)*(2*a^2 - 2*a*b + b^2 - 2*a*c + c^2) : :
X(6546) = X[2254] - 4 X[2977] = 2 X[659] + X[4088] = 4 X[3239] - X[4382] = X[649] + 2 X[4468] = X[8] + 2 X[5592]

See X(6375).

X(6546) lies on these lines: {2, 514}, {8, 5592}, {63, 649}, {210, 513}, {522, 3158}, {523, 1962}, {612, 1027}, {650, 3752}, {659, 4088}, {661, 1211}, {663, 3938}, {812, 4120}, {918, 1635}, {1639, 4728}, {1647, 4124}, {1763, 4498}, {2254, 2977}, {2786, 4984}, {3239, 4382}, {3961, 4040}, {3995, 4024}, {4453, 4763}

X(6546) = reflection of X(i) in X(j) for these (i,j): (4453,4763), (4728,1639), (4750,1635)

X(6547) = SS(a → b - c) of X(39)

Barycentrics (b - c)^2*(2*a^2 - 2*a*b + b^2 - 2*a*c + c^2) : :

See X(6375).

X(6547) lies on these lines: {2, 1016}, {11, 4162}, {115, 4129}, {239, 3936}, {244, 4041}, {514, 1086}, {519, 3836}, {905, 1015}, {3120, 4983}, {3271, 6004}

X(6547) = complement of X(1016)
X(6547) = X(2)-Ceva-conjugate of X(4422)
X(6547) = X(i)-complementary conjugate of X(j) for these {i,j}: {9,3038}, {31,4422}, {41,3039}, {244,141}, {512,4129}, {513,3835}, {604,3035}, {649,513}, {667,514}, {764,116}, {798,661}, {849,620}, {875,812}, {1015,10}, {1019,512}, {1086,2887}, {1111,626}, {1357,142}, {1919,650}, {1977,37}, {2087,121}, {2170,1329}, {3063,4521}, {3121,1213}, {3122,1211}, {3125,3454}, {3248,2}, {3249,1015}, {3271,3452}, {3572,3837}, {3733,4369}, {3942,1368}, {4817,788}

X(6548) = SS(a → b - c) of X(100)

Barycentrics (b - c)*(a + b - 2*c)*(a - 2*b + c) : :
X(6548) = X[1022] - X[2403] = X[1022] + 2 X[4049] = X[2403] + 8 X[4049] = 8 X[3004] + X[4608] = X[4453] + 2 X[4927]

See X(6375).

X(6548) lies on the cubic K015), the conic {{A, B, C, X(2), X(7)}, and these lines: {2, 514}, {7, 3676}, {75, 693}, {86, 4833}, {88, 673}, {106, 675}, {335, 4080}, {900, 903}, {901, 927}, {918, 4945}, {1268, 3004}, {1647, 4089}, {1797, 2989}, {4025, 4373}, {4555, 4618}, {4615, 5468}

X(6548) = isogonal conjugate of X(23344)
X(6548) = isotomic conjugate of X(17780)
X(6548) = barycentric quotient X(106)/X(101)
X(6548) = X(4555)-Ceva conjugate of X(903)
X(6548) = X(i)-cross conjugate of X(j) for these (i,j): (900,514), (1647,2), (4089,7), (4927,693)
X(6548) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (679,150), (2226,149), (4618,69), (4638,8)
X(6548) = X(i)-isoconjugate of X(j) for these (i,j): {6,1023}, {44,101}, {59,4895}, {100,902}, {109,3689}, {163,3943}, {190,2251}, {519,692}, {644,1404}, {678,901}, {765,1960}, {900,1110}, {1017,3257}, {1018,3285}, {1252,1635}, {1319,3939}, {1333,4169}, {1415,2325}, {1576,3992}, {1639,2149}, {1743,2429}, {4570,4730}

X(6549) = SS(a → b - c) of X(101)

Barycentrics (a + b - 2*c)*(b - c)^2*(a - 2*b + c) : :
X(6549) = 3X(903) + X(4555)

See X(6375).

X(6549) lies on these lines: {88, 3008}, {106, 927}, {320, 519}, {334, 4013}, {514, 1086}, {693, 1111}, {1358, 3676}, {1647, 4089}, {3912, 4080}, {4049, 4444}

X(6549) = X(1647)-cross conjugate of X(1086)
X(6549) = X(i)-isoconjugate of X(j) for these (i,j): {44,1252}, {59,3689}, {101,1023}, {163,4169}, X(6549) = {519,1110}, {765,902}, {1016,2251}, {1017,5376}, {1319,6065}, {2149,2325}
X(6549) = barycentric product X(903)*X(1086)

X(6550) = SS(a → b - c) of X(512)

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

See X(6375).

X(6550) lies on these lines: {30, 511}, {36, 659}, {764, 1647}, {1319, 1960}, {2254, 4695}, {3762, 3992}, {3814, 3837}

X(6550) = isogonal conjugate of X(6551)
X(6550) = isotomic conjugate of X(6635)
X(6550) = crossdifference of every pair of points on line X(6)X(1252)
X(6550) = X(i)-Ceva conjugate of X(j) for these (i,j): (513,3259), (900,1647), (2401,1015)

X(6551) = ISOGONAL CONJUGATE OF X(6550)

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

See X(6375).

X(6551) lies on the circumcircle and these lines: {100, 6161}, {105, 1320}, {214, 765}, {840, 4996}, {919, 5548}, {927, 4555}, {929, 4582}, {1016, 1145}, {1017, 1252}, {1110, 2382}, {1308, 3257}, {2757, 4076}

X(6551) = trilinear pole of line X(6)X(1252)
X(6551) = Ψ(X(6), X(1252))
X(6551) = X(i)-cross conjugate of X(j) for these (i,j): (2427,1016), (5548,5376)
X(6551) = X(i)-isoconjugate of X(j) for these (i,j): {244,900}, {513,1647}, {514,2087}, {519,764}, {1015,3762}, {1086,1635}, {1111,1960}, {1357,4768}, {1358,4895}, {3669,4530}

X(6552) = SS(a → -a + b + c) of X(3)

Barycentrics (a - b - c)^2*(a^2 + 2*a*b + b^2 + 2*a*c - 6*b*c + c^2) : :

See X(6375).

X(6552) is the centroid of the quadrilateral whose vertices are X(8) together with the vertices of the anticevian triangle of X(8). (Randy Hutson, February 16, 2015)

X(6552) lies on these lines: {2, 6553}, {8, 3452}, {10, 4310}, {145, 3699}, {279, 668}, {341, 346}, {1211, 3617}, {1219, 4737}, {1706, 4454}, {3086, 4738}, {3672, 6376}, {3913, 4578}

X(6552) = complement of X(6553)
X(6552) = X(i)-complementary conjugate of X(j) for these (i,j): (31,346), (1616,10), (2136,1329), (4452,2887)

X(6553) = SS(a → -a + b + c) of X(4)

Barycentrics (a^2 + 2*a*b + b^2 - 6*a*c + 2*b*c + c^2)*(a^2 - 6*a*b + b^2 + 2*a*c + 2*b*c + c^2) : :

See X(6375).

X(6553) lies on the cubic K201, the conic {{A,B,C,X(1), X(2)}}, and these lines: {1, 3161}, {2, 6552}, {57, 145}, {81, 3623}, {88, 3621}, {279, 4452}, {330, 4461}, {957, 3555}, {1022, 4962}

X(6553) = isogonal conjugate of X(1616)
X(6553) = isotomic conjugate of X(4452)
X(6553) = anticomplement of X(6552)
X(6553) = X(i)-cross conjugate of X(j) for these (i,j): (346,2), (3680,8)
X(6553) = X(2137)-anticomplementary conjugate of X(3436)
X(6553) = X(i)-isoconjugate of X(j) for these (i,j): {1,1616}, {19,23089}, {31,4452}, {56,2136}

X(6554) = SS(a → -a + b + c) of X(39)

Barycentrics tan^2 B/2 + tan^2 C/2 Barycentrics (a - b - c)^2*(a^2 + b^2 - 2*b*c + c^2) : :

See X(6375).

X(6554) lies on these lines: {2, 85}, {4, 9}, {6, 938}, {8, 220}, {20, 910}, {41, 3486}, {101, 944}, {120, 1329}, {219, 1067}, {277, 1111}, {341, 346}, {344, 6376}, {345, 3975}, {379, 5273}, {497, 2082}, {672, 1788}, {857, 1211}, {958, 4223}, {960, 3789}, {1104, 5304}, {1479, 5540}, {1802, 3189}, {1837, 2348}, {1944, 4644}, {2098, 4534}, {2321, 4882}, {2391, 3452}, {3207, 5731}, {3488, 4251}, {3673, 4000}, {4012, 4319}, {4258, 4313}, {5022, 5435}, {5276, 5716}

X(6554) = isotomic conjugate of X(30705)
X(6554) = complement of X(279)
X(6554) = complementary conjugate of X(21258)
X(6554) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,4000), (668,3900)
X(6554) = X(4319)-cross conjugate of X(497)
X(6554) = X(i)-isoconjugate of X(j) for these (i,j): {57,1037}, {222,1041} X(6554) = X(i)-complementary conjugate of X(j) for these (i,j): {9,2886}, {31,4000}, {41,1}, {55,142}, {101,3900}, {200,141}, {213,1834}, {220,10}, {284,3742}, {341,626}, {346,2887}, {480,3452}, {604,5573}, {607,1210}, {657,11}, {663,4904}, {728,1329}, {1174,5572}, {1253,2}, {1334,442}, {1802,3}, {2175,3752}, {2194,3946}, {2212,3772}, {2287,3741}, {2328,3739}, {2332,942}, {3063,3756}, {3692,1368}, {3900,116}, {3939,4885}, {4130,124}, {4171,125}, {4515,3454}, {4578,3835}, {4936,2885}
X(6554) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3177,348), (4,169,5819), (9,281,2345), (169,5179,4), (220,1146,8).

X(6555) = SS(a → -a + b + c) of X(55)

Barycentrics (a - b - c)^2*(3*a - b - c) : :

See X(6375).

X(6555) lies on these lines: {2, 1280}, {8, 3452}, {55, 4152}, {145, 4487}, {200, 346}, {210, 391}, {936, 1219}, {3158, 3161}, {3434, 4767}, {3711, 3974}, {4090, 4307}, {4126, 5218}, {4163, 4543}, {4899, 5435}

X(6555) = X(i)-Ceva conjugate of X(j) for these (i,j): (8,346), (3699,4521)
X(6555) = X(i)-cross conjugate of X(j) for these (i,j): (4162,4578), (4936,3161), (4953,4546)
X(6555) = X(i)-isoconjugate of X(j) for these {i,j}: {269,3445}, {1106,4373}
X(6555) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (200,5423,346)

X(6556) = SS(a → -a + b + c) of X(56)

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

See X(6375).

X(6556) lies on these lines: {2, 1222}, {8, 3452}, {10, 1219}, {20, 1293}, {75, 3617}, {145, 1997}, {253, 3007}, {318, 4723}, {3621, 3699}, {3679, 4052}

X(6556) = X(i)-cross conjugate of X(j) for these (i,j): (11,4397), (4953,3239), (5423,346)
X(6556) = X(i)-isoconjugate of X(j) for these {i,j}: {56,1420}, {145,1106}, {269,3052}, {604,5435}, {1398,4855}, {1407,1743}, {1408,4848}, {1410,4248}, {1461,4394}

X(6557) = SS(a → -a + b + c) of X(57)

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

See X(6375).

Let A₂₇B₂₇C₂₇ be the Gemini triangle 27. Let L_A be the line through A₂₇ parallel to BC, and define L_B and L_C cyclically. Let A′₂₇ = L_B∩L_C, and define B′₂₇ and C′₂₇ cyclically. Triangle A′₂₇B′₂₇C′₂₇ is homothetic to ABC at X(6557). (Randy Hutson, November 30, 2018)

X(6557) lies on these lines: {2, 2415}, {7, 1997}, {8, 3452}, {11, 5423}, {85, 5226}, {92, 4358}, {178, 5430}, {189, 908}, {312, 5328}, {345, 4997}, {1121, 4417}, {1220, 2899}, {1293, 1311}, {4102, 5233}, {4488, 5435}, {4518, 4903}

X(6557) = isotomic conjugate of X(5435)
X(6557) = X(4373)-Ceva conjugate of X(8)
X(6557) = X(i)-cross conjugate of X(j) for these (i,j): (346,8), (1086,4391), (3680,4373), (4534,522)
X(6557) = X(i)-isoconjugate of X(j) for these (i,j): {6,1420}, {31,5435}, {56,1743}, {57,3052}, {109,4394}, {145,604}, {608,4855}, {1106,3161}, {1333,4848}, {1407,3158}, {1408,3950}, {1409,4248}, {1412,4849}, {1415,3667}, {1461,4162}, {2149,3756}, {4565,4729}}

X(6558) = SS(a → -a + b + c) of X(101)

Barycentrics (a - b)(a - c)(a - b - c)^2 : :

See X(6375).

X(6558) lies on these lines: {8, 4534}, {101, 4169}, {190, 646}, {341, 728}, {346, 1146}, {644, 1639}, {1016, 4561}, {1043, 4515}, {3039, 3161}, {4103, 4752}, {4528, 4578}

X(6558) = X(i)-Ceva conjugate of X(j) for these (i,j): (646,3699), (1016,1265), (4076,5423)
X(6558) = X(i)-cross conjugate of X(j) for these (i,j): (657,200), (3239,346), (3900,1043), (4163,341), (4546,8), (4578,3699), (5423,4076}
X(6558) = X(i)-isoconjugate of X(j) for these (i,j): {56,3669}, {244,1461}, {269,649}, {279,667}, {479,3063}, {513,1407}, {514,1106}, {604,3676}, {651,1357}, {658,3248}, {663,738}, {764,1262}, {905,1398}, {934,1015}, {1019,1042}, {1088,1919}, {1358,1415}, {1412,4017}, {1427,3733}, {1435,1459}, {1977,4569}, {3121,4616}, {3122,4637}, {3271,4617}

X(6559) = SS(a → -a + b + c) of X(292)

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

See X(6375).

X(6558) lies on these lines: {8, 220}, {9, 75}, {105, 958}, {341, 728}, {346, 480}, {666, 2338}, {919, 2370}, {927, 2371}, {1222, 1438}, {1462, 5749}, {3119, 3699}

X(6559) = X(i)-isoconjugate of X(j) for these (i,j): {56,241}, {57,1458}, {222,1876}, {269,672}, {279,2223}, {518,1407}, {603,5236}, {665,934}, {738,2340}, {926,4617}, {1106,3912}, {1262,3675}, {1362,1462}, {1427,3286}, {1435,1818}, {1461,2254}, {2283,3669}

X(6560) = PERSPECTOR OF OUTER VECTEN TRIANGLE AND LUCAS(-1) ANTIPODAL TRIANGLE

Trilinears cos A - sin A - 2 cos B cos C : :

Contributed by Randy Hutson, February 9, 2015.

X(6560) lies on these lines: {2,1327}, {3,485}, {4,372}, {5,1152}, {6,30}, {20,371}, {490,3096}, {550,1151}, {1131,3523}

X(6560) = reflection of X(6561) in X(6)

X(6561) = PERSPECTOR OF INNER VECTEN TRIANGLE AND LUCAS ANTIPODAL TRIANGLE

Trilinears cos A + sin A - 2 cos B cos C : :

Contributed by Randy Hutson, February 9, 2015.

X(6561) lies on these lines: {2,1328}, {3,486}, {4,371}, {5,1151}, {6,30}, {20,372}, {489,3096}, {550,1152}, {1132,3523}

X(6561) = reflection of X(6560) in X(6)
X(6561) = homothetic center of inner Vecten and 3rd anti-tri-squares triangles

X(6562) = CROSSDIFFERENCE OF X(493) AND X(494)

Trilinears a(b^2 - c^2)/[(sin^4 A + 1) sin^2 B sin^2 C - sin^2 A (sin^4 B + sin^4 C )] : :

The perspectrix of any pair of {ABC, outer Vecten triangle, Lucas(-1) antipodal triangle} is the trilinear polar of X(3068). The perspectrix of any pair of {ABC, inner Vecten triangle, Lucas antipodal triangle} is the trilinear polar of X(3069). The intersection of these two perspectrices is X(6562). (Randy Hutson, February 9, 2015)

X(6562) lies on these lines: {23,385}, {804,3265}, {3566,6563}

X(6562) = radical center of {nine-point circle, outer Vecten circle, inner Vecten circle}
X(6562) = crossdifference of every pair of points on line X(39)X(493)

X(6563) = ISOTOMIC CONJUGATE OF X(925)

Barycentrics    bc[b cos(A - B) - c cos(A - C)] : :
Barycentrics    (csc^2 A)(sec 2B - sec 2C) : :
Barycentrics    (b^2 - c^2)(a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2) : :

Contributed by Randy Hutson, February 9, 2015.

X(6563) lies on these lines: {2,2501}, {323,401}, {325,523}, {3566,6562}

X(6563) = isogonal conjugate of X(32734)
X(6563) = isotomic conjugate of X(925)
X(6563) = anticomplement of X(2501)
X(6563) = crossdifference of every pair of points on line X(32)X(51)
X(6563) = radical center of {circumcircle, inner Vecten circle, outer Vecten circle}
X(6563) = pole, wrt de Longchamps circle, of line X(2)X(6)
X(6563) = de-Longchamps-circle-inverse of X(38940)
X(6563) = barycentric product X(523)*X(7763)

X(6564) = INTERSECTION OF LINES X(4)X(371) AND X(6)X(13)

Trilinears cos A + sin A + 4 cos B cos C : :
Barycentrics (SA + S)(SB + SC) + 4 SB SC : :

Let A′B′C′ be the outer Vecten triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6564). (Randy Hutson, February 9, 2015)

X(6564) lies on these lines: {2,1327}, {3,3366}, {4,371}, {5,372}, {6,13}, {20,5418}, {30,590}, {382,1151}, {486,1131}, {1152,1656}

X(6564) = {X(6),X(381)}-harmonic conjugate of X(6565)
X(6564) = homothetic center of 1st Kenmotu diagonals triangle and Ehrmann vertex-triangle

X(6565) = INTERSECTION OF LINES X(4)X(372) AND X(6)X(13)

Trilinears cos A - sin A + 4 cos B cos C : :
Barycentrics (SA - S)(SB + SC) + 4 SB SC : :

Let A′B′C′ be the inner Vecten triangle. Let A″ be the trilinear pole, wrt A′B′C′, of line BC. Define B″, C″ cyclically. The lines A′A″, B′B″, C′C′ concur in X(6565). (Randy Hutson, February 9, 2015)

X(6565) lies on these lines: {2,1328}, {3,3367}, {4,372}, {5,371}, {6,13}, {20,5420}, {30,615}, {382,1152}, {485,1132}, {1151,1656}

X(6565) = homothetic center of 2nd Kenmotu diagonals triangle and Ehrmann vertex-triangle
X(6565) = {X(6),X(381)}-harmonic conjugate of X(6564)

X(6566) = INVERSE-IN-CIRCUMCIRCLE OF X(1151)

Trilinears    2 cos(A + ω) - cos ω sin A + 3 cos A sin ω : :
Trilinears    2 cos(A + ω) - sin(A - ω) + 2 cos A sin ω : :
Trilinears    (2 cos ω + 3 sin ω) cos A - (cos ω + 2 sin ω) sin A : :
Barycentrics    a^2 (SA + S (1 / (-2 + 1 / (2 + Cot[w]))) ) : :

X(6566) is the intersection of the following lines: (1) Brocard axis, (2) the trilinear polar of the cevapoint of X(6) and X(1151) (i.e., X(493)), (3) the line through the X(6)-Ceva conjugate of X(1151) and the X(1151)-Ceva conjugate of X(6). (Randy Hutson, February 9, 2015)

X(6566) lies on these lines: {3,6}, {316,3593}, {625,6460}, {638,3788}, {641,3070}

X(6566) = isogonal conjugate of X(6568)
X(6566) = perspector of ABC and the reflection of the Lucas inner triangle in the Lemoine axis
X(6566) = perspector of the Lucas(-1) central triangle and the reflection of the Lucas central triangle in the Lemoine axis
X(6566) = perspector of the Lucas(-1) tangents triangle and the reflection of the Lucas tangents triangle in the Lemoine axis
X(6566) = inverse-in-Lucas(-1)-radical-circle of X(3)
X(6566) = inverse-in-Lucas(-1)-inner-circle of X(1152)
X(6566) = inverse-in-Lucas(-1)-secondary-radical-circle of X(6445)
X(6566) = inverse-in-Lucas-secondary-tangents-circle of X(6519)
X(6566) = {X(3),X(1570)}-harmonic conjguate of X(6567)
X(6566) = {X(1379),X(1380)}-harmonic conjugate of X(1151)
X(6566) = reflection of X(6567) in X(187)
X(6566) = crossdifference of every pair of points on line X(523)X(3068)
X(6566) = radical trace of circumcircle and 4th Lozada circle

X(6567) = INVERSE-IN-CIRCUMCIRCLE OF X(1152)

Trilinears    2 cos(A + ω) + cos ω sin A - 3 cos A sin ω : :
Trilinears    2 cos(A + ω) + sin(A - ω) - 2 cos A sin ω : :
Trilinears    (2 cos ω - 3 sin ω) cos A + (cos ω - 2 sin ω) sin A : :
Barycentrics    a^2 (SA + S (1 / (2 + 1 / (-2 + Cot[w]))) )::

X(6567) is the intersection of the following lines: (1) Brocard axis, (2) the trilinear polar of the cevapoint of X(6) and X(1152) (i.e., X(494)), (3) the line through the X(6)-Ceva conjugate of X(1152) and the X(1152)-Ceva conjugate of X(6). (Randy Hutson, February 9, 2015)

X(6567) lies on these lines: {3,6}, {316,3595}, {625,6459}, {637,3788}, {642,3071}

X(6567) = isogonal conjugate of X(6569)
X(6567) = perspector of ABC and the reflection of the Lucas(-1) inner triangle in the Lemoine axis
X(6567) = perspector of the Lucas central triangle and the reflection of the Lucas(-1) central triangle in the Lemoine axis
X(6567) = perspector of the Lucas tangents triangle and the reflection of the Lucas(-1) tangents triangle in the Lemoine axis
X(6567) = inverse-in-Lucas-radical-circle of X(3)
X(6567) = inverse-in-Lucas-inner-circle of X(1151)
X(6567) = inverse-in-Lucas-secondary-radical-circle of X(6446)
X(6567) = inverse-in-Lucas(-1)-secondary-tangents-circle of X(6522)
X(6567) = {X(3),X(1570)}-harmonic conjguate of X(6566)
X(6567) = {X(1379),X(1380)}-harmonic conjugate of X(1152)
X(6567) = reflection of X(6566) in X(187)
X(6567) = crossdifference of every pair of points on line X(523)X(3069)
X(6567) = radical trace of circumcircle and 3rd Lozada circle

X(6568) = ANTIGONAL IMAGE OF X(1131)

Trilinears 1/[2 cos(A + ω) - cos ω sin A + 3 cos A sin ω] : :
Trilinears 1/[2 cos(A + ω) - sin(A - ω) + 2 cos A sin ω] : :

X(6568) lies on the Kiepert hyperbola and these lines: {99,5490}, {115,1131}

Contributed by Randy Hutson, February 9, 2015.

X(6568) = isogonal conjugate of X(6566)
X(6568) = antipode in Kiepert hyperbola of X(1131)
X(6568) = reflection of X(1131) in X(115)
X(6568) = trilinear pole of line X(523)X(3068)

X(6569) = ANTIGONAL IMAGE OF X(1132)

Trilinears 1/[2 cos(A + ω) + cos ω sin A - 3 cos A sin ω] : :
Trilinears 1/[2 cos(A + ω) + sin(A - ω) - 2 cos A sin ω] : :

Contributed by Randy Hutson, February 9, 2015.

X(6569) lies on the Kiepert hyperbola and these lines: {99,5491}, {115,1132}

X(6569) = isogonal conjugate of X(6567)
X(6569) = antipode in Kiepert hyperbola of X(1132)
X(6569) = reflection of X(1132) in X(115)
X(6569) = trilinear pole of line X(523)X(3069)

Circumeigencenters (CET): X(6570)-X(6577)

The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the circumcevian triangle of a point U = u : v : w (trilinears), the eigencenter is given by

avw/(av² - aw² + buv - cuw) : bwu/(bw² - bu² + cvw - avu) : cuv/(cu² - cv² + awu - bwv)

This point is here named the circum-eigentransform of U, denoted by CET(U). The point CET(U) lies on the circumcircle. Pairs (i,j) such that CET(X(i)) = X(j) include the following:

(1, 101), (2, 99), (3, 112), (4, 6570), (6, 99), (9, 6571), (15, 5618), (16, 5619), (25, 99), (31, 839), (32, 6572), (37, 99), (39, 6573), (42, 99), (55, 934), (56, 6574), (57, 6575), (75, 773), (76, 6576), (111, 99), (187, 99), (237, 99), (241, 934), (251, 99), (263, 99), (308, 99), (351, 99), (393, 99), (493, 99), (494, 99), (512, 99), (513, 101), (514, 6577), (523, 112), (588, 99), (589, 99), (647, 99), (649, 99), (650, 934), (663, 99), (665, 99), (667, 99), (669, 99), (694, 99), (887, 99), (890, 99), (902, 99), (941, 99), (967, 99), (1055, 99), (1169, 99), (1171, 99), (1218, 99), (1239, 99), (1241, 99), (1383, 99), (1400, 99), (1427, 99), (1495, 99), (1880, 99), (1946, 99), (1960, 99), (1976, 99), (1989, 99), (2054, 99), (2165, 99), (2223, 99), (2248, 99), (2350, 99), (2395, 99), (2433, 99), (2488, 99), (2502, 99), (2574, 112), (2575, 112), (2590, 934), (2591, 934), (2963, 99), (2978, 99), (2981, 99), (2987, 99), (2998, 99), (3005, 99), (3009, 99), (3010, 99), (3016, 99), (3108, 99), (3228, 99), (3229, 99), (3230, 99), (3231, 99), (3250, 99), (3288, 99), (3331, 99), (3444, 99), (3457, 99), (3458, 99), (3513, 101), (3514, 101), (3569, 99), (3572, 99), (3724, 99), (3747, 99), (3804, 99), (4775, 99), (4834, 99), (5027, 99), (5029, 99), (5040, 99), (5075, 99), (5098, 99), (5106, 99), (5113, 99), (5147, 99), (5163, 99), (5167, 99), (5168, 99), (5191, 99), (5202, 99), (5638, 99), (5639, 99), (6094, 99), (6096, 99), (6137, 99), (6138, 99), (6139, 99), (6140, 99), (6151, 99), (6339, 99)

Peter Moses observes (January, 2015) that CET(U) and CIR(U), discussed in the preamble to X(2365), are special cases of points defined by 1st trilinear of the form

avw/(jav² + kaw² - kbuv - jcuw).

Taking (j, k) = (1,1) gives CIR(U), and taking (j, k) = (1,-1) gives CET(U). Also, the locus of X for which CET(X) = X(99) is the union of the line X(187)X(237) and the conic {{A,B,C,X(2),X(6)}}.

If U is given by barycentrics u : v : w, then barycentrics for CET(U) are f(a,b,c) : f(b,c,a) : f(c,a,b), where

f(a,b,c) = a²vw/(a²c²v² - a²b²w² + b²c²uv - b²c²uw).

X(6570) = CET(X(4))

Barycentrics a^2*(a - b)*(a + b)*(a - c)*(a + c)*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + 2*a^6*c^2 - a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 3*b^6*c^2 - 4*a^4*c^4 - a^2*b^2*c^4 - 3*b^4*c^4 + 2*a^2*c^6 + b^2*c^6)*(2*a^6*b^2 - 4*a^4*b^4 + 2*a^2*b^6 + a^6*c^2 - a^4*b^2*c^2 - a^2*b^4*c^2 + b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 3*b^4*c^4 + 3*a^2*c^6 + 3*b^2*c^6 - c^8) : :

X(6570) lies on the circumcircle and these lines: {98, 1181}, {107, 1625}

X(6570) = isoconjugate of X(3900) and X(1577)
X(6570) = trilinear pole of the line X(6)X(418)
X(6570) = Ψ(X(6), X(418))

X(6571) = CET(X(9))

Barycentrics a^2*(a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 - 2*a*b + 5*b^2 + 2*a*c - 2*b*c + c^2)*(a^2 + 2*a*b + b^2 - 2*a*c - 2*b*c + 5*c^2) : :

X(6571) lies on the circumcircle.

X(6571) = isogonal conjugate of X(8710)
X(6571) = isoconjugate of X(3900) and X(4308)

X(6572) = CET(X(32))

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

X(6572) lies on the circumcircle and these lines: {111, 1239}, {670, 827}

X(6572) = Ψ(X(6), X(1239))
X(6572) = trilinear pole of the line X(6)X(1239)
X(6572) = X(i)-isoconjugate of X(j) for these (i,j): {798,1180}, {1924,3096}

X(6573) = CET(X(39))

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

X(6573) lies on the circumcircle and these lines: {83, 699}, {111, 1799}, {733, 1078}

X(6573) = isogonal conjugate of X(8711)
X(6573) = X(4576)-cross conjugate of X(4577)
X(6573) = trilinear pole of line X(6)X(6664)
X(6573) = Ψ(X(6), X(6664))

X(6574) = CET(X(56))

Barycentrics a*(a - b)*(a - c)*(a^2 + 2*a*b + b^2 - 2*a*c + 2*b*c + c^2)*(a^2 - 2*a*b + b^2 + 2*a*c + 2*b*c + c^2) : :

X(6574) lies on the circumcircle and these lines: {101, 4578}, {105, 958}, {106, 2297}, {109, 644}, {190, 934}, {1018, 1293}

X(6574) = isogonal conjugate of X(8712)
X(6574) = trilinear pole of the line X(6)X(200)
X(6574) = X(4512)-cross conjugate of X(765)
X(6574) = Ψ(X(6), X(200))
X(6574) = eigencenter of 3rd mixtilinear triangle
X(6574) = X(i)-isoconjugate of X(j) for these {i,j}: {513,2999}, {514,1191}, {649,3672}, {1019,4646}, {1697,3669}, {3733,4656}

X(6575) = CET(X(57))

Barycentrics a^2*(a - b)*(a - c)*(a^3 + a^2*b - 5*a*b^2 + 3*b^3 - a^2*c - 2*a*b*c - 5*b^2*c - a*c^2 + b*c^2 + c^3)*(a^3 - a^2*b - a*b^2 + b^3 + a^2*c - 2*a*b*c + b^2*c - 5*a*c^2 - 5*b*c^2 + 3*c^3) : :

X(6575) lies on the circumcircle and these lines: {103,5584}, {105,3601}

X(6575) = isogonal conjugate of X(8713)
X(6575) = trilinear pole of line X(6)X(8012)
X(6575) = Ψ(X(6), X(8012))

X(6576) = CET(X(76))

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

X(6576) lies on the circumcircle.

X(6577) = CET(X(514))

Barycentrics a^2*(a - b)*(a - c)*(a^2*b + a*b^2 + a^2*c + b^2*c - a*c^2 - b*c^2)*(a^2*b - a*b^2 + a^2*c - b^2*c + a*c^2 + b*c^2) : :

X(6577) lies on the circumcircle and these lines: {105, 595}, {1918, 2368}, {3939, 6013}

X(6577) = isogonal conjugate of X(8714)
X(6577) = X(4040)-cross conjugate of X(58)

X(6578) = CET(X(661))

Barycentrics a^2*(a - b)*(a + b)^2*(a - c)*(a + c)^2*(a + 2*b + c)*(a + b + 2*c) : :

X(6578) lies on the circumcircle and these lines: {100, 4596}, {101, 4556}, {111, 1171}, {249, 2702}, {476, 4608}, {835, 4632}, {1268, 2372}

X(6578) = isogonal conjugate of X(6367)
X(6578) = trilinear pole of line X(6)X(593)
X(6578) = Ψ(X(6), X(593))
X(6578) = X(i)-cross conjugate of X(j) for these (i,j): (58,249), (110,4629), (4184,250))
X(i)-isoconjugate of X(j) for these (i,j): {1,6367}, {10,4983}, {37,4988}, {181,4985}, {430,656}, {512,4647}, {523,1962}, {594,4979}, {661,1213}, {756,4977}, {798,1230}, {1100,4024}, {1125,4705}, {1254,4990}, {1500,4978}, {2171,4976}, {2308,4036}, {2355,4064}, {2501,3958}, {2643,4427}, {3125,4115}, {3649,4041}, {4017,4046}, {4079,4359}

X(6579) = CIR(X(76))

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

X(6579) lies on the circumcircle and these lines: {32, 689}, {83, 3222}

X(6580) = 2nd SHADOW POINT

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

The 1st Shadow Point is X(1480). See Antreas P. Hatzipolakis and Paul Yiu, Pedal Triangles and Their Shadows, Forum Geometricorum 1 (2001) 81-90. (X(6580) is point M′ on page 88; the trilinears have been corrected here.)

Suppose that t is a number (or symmetric function of A,B,C) and let f(A,B,C,k) = (t + cos A)/(k + cos A - cos B - cos C). Let P(j) = f(A,B,C,j) : f(B,C,A,j) : f(C,A,B,j) (trilinears). The appearance of (j,k) in the following list means that P(j) lies on a line X(1)X(k): (-3,5920), (-1,6), (0,90), (1,84), (2,1406), (3,1407). (Peter Moses, February 9, 2015)

X(6580) lies on these lines: {1,1406}, {56,4550}, {999,3426}, {4846,5434}

X(6581) = OUTER-NAPOLEON-TRIANGLE-ORTHOLOGIC CENTER OF 1st NEUBERG TRIANGLE

Barycentrics a^4 b^2+a^2 b^4+a^4 c^2-2 b^4 c^2+a^2 c^4-2 b^2 c^4+2 Sqrt[3] a^2 (b^2+c^2) S : :

Barycentrics for X(6581) are obtained from X(6294) by the transformation S → -S. (Richard Hilton, February 9, 2015)

X(6581) lies on these lines: {2, 39}, {524, 3104}, {698, 3107}, {732, 3106}, {3095, 5613}, {3105, 5464}, {3642, 6296}

X(6581) = reflection of X(6294) in X(39)
X(6581) = anticomplement of X(33482)

X(6582) = 1st NEUBERG-TRIANGLE-ORTHOLOGIC CENTER OF OUTER NAPOLEON TRIANGLE

Barycentrics 3 a^8-8 a^6 b^2+7 a^4 b^4-2 a^2 b^6-8 a^6 c^2+6 a^2 b^4 c^2-2 b^6 c^2+7 a^4 c^4+6 a^2 b^2 c^4+4 b^4 c^4-2 a^2 c^6-2 b^2 c^6-2 Sqrt[3] a^2 (a^4+a^2 b^2-2 b^4+a^2 c^2-2 b^2 c^2-2 c^4) S : :

Barycentrics for X(6582) are obtained from X(6295) by the transformation S → -S. (Richard Hilton, February 9, 2015)

X(6582) lies on these lines: {2, 13}, {542, 6299}, {3098, 5969}, {5617, 6287}

X(6582) = reflection of X(6298) in X(618)
X(6582) = circumtangential-isogonal conjugate of X(15)
X(6582) = {X(6302),X(6306)}-harmonic conjugate of X(3643)

X(6583) = HATZIPOLAKIS-LOZADA NINE-POINT IMAGE OF X(1)

Trilinears 2*cos((B-C)/2)*sin(3*A/2)+2*(1-cos(A))*cos(B-C)-1 : :

X(6583) = (5R + 2r)*X(1) - (R + 2r)*X(3)

In the plane of ABC, let P be a point, and let A′B′C′ be the pedal triangle of P. Let A_B be the reflection of A in B′, and define B_C and C_A cyclically. Let A_C be the reflection of A in C′, and define B_A and C_B cyclically. Let N_A be the nine-point center of AA_BA_C, and define N_B and N_C cyclically. The nine-point center of N_AN_BN_C is here named the Hatzipolakis-Lozada Nine-Point Image of P. See Hyacinthos 23083. (Antreas Hatzipolakis and César Lozada, February 2, 2015)

X(6583) lies on these lines: {1,3}, {5,3874}, {355,3873}, {546,2801}, {758,5901}, {912,5448}, {946,1484}, {952,3881}, {1393,5399}, {1483,3892}, {1656,5904}, {3628,3678}, {3754,5844}, {3868,5694}, {4430,5818}, {5690,5883}

X(6583) = midpoint of X(i) and X(j) for these {i,j}: {5,3874}, {3868,5694}
X(6583) = reflection of X(i) in X(j) for these (i,j): (3678, 3628), (5885,942)

X(6584) = HATZIPOLAKIS-LOZADA-KOSNITA POINT

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

Let I = X(1), and let N_A be the nine-point center of IBC. Define N_B and N_C cyclically. Let K_A = X(54)-of-IBC; i.e,, the Kosnita point of IBC, and define K_B and K_C cyclically. The circumcircles of the four triangles ABC, AK_BK_C, BK_CK_A, AC_AK_B concur in X(6584); their radical center is X(484). Also, K_A = IBC-isogonal conjugate of N_A, and the points K_A, K_B, K_C lie on the line X(1)X(3). See Hyacinthos 23086. (Antreas Hatzipolakis and César Lozada, February 1, 2015)

X(6584) lies on the circumcircle and this line: {1385,2687}

X(6585) = DOLGIREV-HATZIPOLAKIS-LOZADA POINT

Barycentrics a((a-b-c) (a+b-c) (a-b+c) (a+b+c)+(a-b-c) ((a+b-c) (a-b+c) (a+b+c)+4 a b c)Sin[A/2]+(a-b+c) ((a+b-c) (a-b+c) (a+b+c)-4 a b c) Sin[B/2]+(a+b-c) ((a+b-c) (a-b+c) (a+b+c)-4 a b c) Sin[C/2] ) : : (Peter Moses, August 31, 2016)

The construction below for X(6585) is a tangential triangle variation of the construction described by Pavel Dolgirev: Interest circle: open problem. Let A′B′C′ be the Gergonne triangle of ABC; that is, the cevian triangle of X(7)). Draw tangent lines from the vertices A,B,C to the incircle of A′B′C′ and let A₁, A₂, B₁, B₂, C₁, C₂ be the points of intersection of these tangent lines with the sides of the triangle ABC. These 6 points lie on a circle, of which the center is X(6585). See Hyacinthos 23112. (Antreas Hatzipolakis and César Lozada, February 1, 2015)

X(6585) lies on this line: {1,167}, {40,1130}

X(6585) = midpoint of X(8091) and X(8351)

X(6586) = COMPLEMENT OF X(3261)

Trilinears a*(b - c)*(b² + c² - bc + ab + ac) : :

X(6586) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

X(6586) lies on these lines: {2, 2412}, {6, 657}, {37, 522}, {230, 231}, {513, 665}, {649, 4057}, {667, 2483}, {798, 834}, {905, 918}, {1919, 5029}, {2484, 3733}, {2605, 3063}, {4526, 4926}

X(6586) = isotomic conjugate of X(31624)
X(6586) = complement of X(3261)
X(6586) = X(2)-Ceva conjugate of X(116)
X(6586) = perspector of hyperbola {{A,B,C,X(4),X(103)}} (centered at X(116)
X(6586) = polar conjugate of isogonal conjugate of X(22388)
X(6586) = (trilinear polar of X(4))∩(trilinear polar of X(103))
X(6586) = crossdifference of every pair of points on line X(3)X(142)

X(6587) = COMPLEMENT OF X(3265)

Trilinears     bc(b² - c²)[3a⁴ - (b² - c²)² - 2a²(b² + c²)] : :
Barycentrics    (cot B)(sin 2C - sin 2A) + (cot C)(sin 2A - sin 2B) : :
Barycentrics    (cot B)(tan C - tan A) + (cot C)(tan A - tan B) : :

X(6587) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

Let A′B′C′ be the orthic triangle. Let L_A be the orthic axis of AB′C′, and define L_B and L_C cyclically. Let A″ = L_B∩ L_C, B″ = L_C∩ L_A, C″ = L_A∩ L_B. The triangle A″B″C″ is inversely similar to ABC, with similitude center X(6), and X(6587) = (orthic axis of ABC)∩(orthic axis of A″B″C″. Also, the trilinear polar of X(6587) passes through X(1562). (Randy Hutson, February 20, 2015) X(6587):

X(6587) lies on these lines: {2,2419}, {6,2430}, {230,231} et al.

X(6587) = complement of X(3265)
X(6587) = X(2)-Ceva conjugate of X(122)
X(6587) = crossdifference of every pair of points on line X(3)X(64)
X(6587) = radical center of circumcircle, nine-point circle, and half-altitude circle
X(6587) = X(63)-isoconjugate of X(1301)
X(6587) = perspector of hyperbola {{A,B,C,X(4),X(20)}} (centered at X(122))
X(6587) = polar conjugate of isotomic conjugate of X(8057)
X(6587) = polar conjugate of anticomplement of X(39020)
X(6587) = PU(4)-harmonic conjugate of X(16318)
X(6587) = (trilinear polar of X(4))∩(trilinear polar of X(20)) (orthic axis and perspectrix of ABC and half-altitude triangle)

X(6588) = X(2)-CEVA CONJUGATE OF X(123)

Trilinears a[b^2(sec A - sec B) + c^2(sec C - sec A)] : :
Trilinears (b - c)*[a⁴ + 2a²bc - 2abc(b + c) - (b² - c²)²] : :

X(6588) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

As a line L varies through the set of all lines that pass through X(123), the locus of the trilinear pole of L is a circumconic, and its center is X(6588). (Randy Hutson, February 20, 2014)

X(6588) lies on these lines: {6,2431}, {230,231} et al.

X(6588) = complement of X(35518)
X(6588) = X(2)-Ceva conjugate of X(123)
X(6588) = crossdifference of every pair of points on line X(3)X(960)
X(6588) = (trilinear polar of X(4))∩(trilinear polar of X(1295)
X(6588) = perspector of the hyperbola {{A,B,C,X(4),X(1295)}} (centered at X(123))

X(6589) = X(2)-CEVA CONJUGATE OF X(124)

Trilinears a(b - c)[-b³ - abc - c³ + (b + c)a²] : :
Barycentrics (cos C - cos A)(csc^2 B) + (cos A - cos B)(csc^2 C) : :

X(6589) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

As a line L varies through the set of all lines that pass through X(124), the locus of the trilinear pole of L is a circumconic, and its center is X(6589). (Randy Hutson, February 20, 2014)

X(6589) lies on these lines: {230,231}, {649,834} et al.

X(6589) = X(2)-Ceva conjugate of X(124)
X(6589) = crossdifference of every pair of points on line X(3)X(10)
X(6589) = (trilinear polar of X(4))∩(trilinear polar of X(58)
X(6589) = perspector of the hyperbola {{A,B,C,X(4),X(58)}} (centered at X(124))

X(6590) = X(2)-CEVA CONJUGATE OF X(5515)

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

X(6590) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

X(6590) lies on these lines: {230,231} et al.

X(6590) = isotomic conjugate of X(37215)
X(6590) = X(2)-Ceva conjugate of X(5515)
X(6590) = crossdifference of every pair of points on line X(3)X(31)

X(6591) = ISOGONAL CONJUGATE OF X(1332)

Trilinears (b - c)tan A : :
Trilinears (b - c)/(b² + c² - a²) : :

X(6591) is the perspector of a rectangular circumhyperbola referenced in connection with cyclologic triangles and the nine-point center. See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

As a line L varies through the set of all lines that pass through X(5521), the locus of the trilinear pole of L is a circumconic, and its center is X(6591). (Randy Hutson, February 20, 2014)

X(6591) lies on these lines: {19,4394}, {230,231}, {513,2201}, {661,663}, {693,905}

X(6591) = isogonal conjugate of X(1332)
X(6591) = crosspoint of X(4) and X(6335)
X(6591) = X(2)-Ceva conjugate of X(5521)
X(6591) = crossdifference of every pair of points on line X(3)X(63)
X(6591) = crossdifference, with respect to the orthic triangle, of every pair of points on line X(4)X(8)
X(6591) = (trilinear polar of X(4)∩(trilinear polar of X(19))
X(6591) = X(48)-isoconjugate (polar conjugate) of X(668)
X(6591) = X(63)-isoconjugate of X(100)
X(6591) = PU(4)-harmonic conjugate of X(3290)
X(6591) = perspector of hyperbola {{A,B,C,X(4),X(19)}} (centered at X(5521))
X(6591) = trilinear pole of line X(3125)X(3271) (the line through the polar conjugates of PU(41))
X(6591) = pole wrt the polar circle of the trilinear polar of X(668) (the line X(2)X(37))

X(6592) = COMPLEMENT OF X(1263)

Trilinears (2 cos A)[-2 + cos(2B - 2C) + 4 cos²A](1 + 2 cos 2B + 2 cos 2C - 2 cos 2A) + [-1 + 4 cos²A cos(B - C)] : :
Barycentrics (a^6-3*(b^2+c^2)*a^4+(3*b^4-b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2))*(2*a^10-7*(b^2+c^2)*a^8+10*(b^4+b^2*c^2+c^4)*a^6-(b^2+c^2)*(8*b^4 -7*b^2*c^2+8*c^4)*a^4+(4*b^4+3*b^2*c^2+4*c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3) : :

Let U = A′B′C′ be the medial triangle and X = X(5). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6592) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(137) . See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

X(6592) lies on the cubics K038, K067 and these lines: {2,1263}, {3,2888}, {5,930}, {30,128}, {137,3628} et al.

X(6592) = midpoint of X(5) and X(930)
X(6592) = reflection of X(137) in X(3628)
X(6592) = complement of X(1263)
X(6592) = circumcircle-inverse of X(5898)

X(6593) = COMPLEMENT OF X(67)

Trilinears a(a⁴ - b⁴ - c⁴ + b²c²)(2a² - b² - c²) : :

Let U = A′B′C′ be the medial triangle and X = X(6). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6593) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(125) . See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

Let A′B′C′ be the cevian triangle of X(23). Let A″ be the inverse-in-circumcircle of A′, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(6593). Let P and U be the circumcircle intercepts of the perpendicular to the Euler line through X(2); then X(6593) = crosssum of P and U. Also, X(6593) is the QA-P41 center (Involutary Conjugate of QA-P4) of quadrangle ABCX(2). (Randy Hutson, February 20, 2015)

X(6593) lies on the bicevian conic of X(2) and X(110), on the cubics K042, K043, K565, and on these lines: {2,67}, {3,1177}, {5,542}, {6,110}, {74,5085}, {113,1503}, {125,3589}, {141,5972} et al.

X(6593) = midpoint of X(6) and X(110)
X(6593) = reflection of X(i) in X(j) for these (i,j): (125,3589), (141,5972)
X(6593) = complement of X(67)
X(6593) = complementary conjugate of X(858)
X(6593) = perspector of the circumconic centered at X(187)
X(6593) = X(2)-Ceva conjugate of X(187)
X(6593) = inverse-in-MacBeath-circumconic of X(2930)
X(6593) = {X(110),X(895)}-harmonic conjugate of X(2930)
X(6593) = antipode of X(141) in the bicevian conic of X(2) and X(110)

X(6594) = COMPLEMENT OF X(3254)

Trilinears (2a² - b² - c² - ab - ac + 2bc)(a² + b² + c² - 2ab - 2ac + bc) : :

Let U = A′B′C′ be the medial triangle and X = X(9). Let A″ = X-of-AB′C′, B″ = X-of-A′BC′, C″ = X-of-A′B′C, and V = A″B″C″. Then U and V are cyclologic, and X(6594) is the (V,U)-cyclologic center; the (U,V)-cyclologic center is X(11). See Cyclology with medial triangle (Seiichi Kirikami, December 11, 2014, posted by César Lozada.

X(6594) lies on these lines: {1,3939}, {2,3254}, {3,2801}, {9,100}, {10,528}, {11,6666}, {142,3035} et al.

X(6594) = midpoint of X(9) and X(100)
X(6594) = reflection of X(142) in X(3035)
X(6594) = complement of X(3254)

X(6595) = (ABC-1st-SCHIFFLER) CYCLOLOGIC CENTER

Barycentrics a (a^9+a^7 (-6 b^2+7 b c-6 c^2)-(b-c)^4 (b+c)^3 (2 b^2+5 b c+2 c^2)+a^6 (2 b^3-3 b^2 c-3 b c^2+2 c^3)+a (b^2-c^2)^2 (3 b^4-b^3 c+2 b^2 c^2-b c^3+3 c^4)+a^5 (12 b^4-15 b^3 c+13 b^2 c^2-15 b c^3+12 c^4)+a^4 (-6 b^5+3 b^4 c+5 b^3 c^2+5 b^2 c^3+3 b c^4-6 c^5)+a^3 (-10 b^6+9 b^5 c-3 b^4 c^2+5 b^3 c^3-3 b^2 c^4+9 b c^5-10 c^6)+a^2 (6 b^7+3 b^6 c-11 b^5 c^2+b^4 c^3+b^3 c^4-11 b^2 c^5+3 b c^6+6 c^7)) : :

Let I = X(1), the incenter of a triangle ABC. Let L_A be the Euler line of triangle IBC, and define L_B and L_C cyclically. These lines concur in S = X(21), the Schiffler point of ABC. Let C₂ be the point, other than S, of intersection of the line L_B and the circle (C, |CS|). Let B₃ be the point, other than S, of intersection of the line L_C and the circle (B, |BS|). Let O₁ be the circumcenter of SC₂B₃, and define O₂ and O₃ cyclically. The triangle O₁O₂O₃ is here named the 1st Schiffler triangle. The triangles ABC and O₁O₂O₃ are cyclologic, and X(6595) is the (ABC, O₁O₂O₃)-cyclologic center. See X(6596)-X(6599) and Hyacinthos 23098 and Hyacinthos 23101. (S. Kirikami, A. Hatzipolakis and A.Montesdeoca, February 4-5, 2015)

The points O₁, O₂, O₃ lie on the Feuerbach hyperbola. (Randy Hutson, September 14, 2016)

X(6595) lies on the Feuerbach hyperbola and these lines: {104, 6584}, {1389, 2771}

X(6596) = (1st SCHIFFLER, 2nd SCHIFFLER) CYCLOLOGIC CENTER

Barycentrics a (a^9 + 10 a^7 b c - 3 a^8 (b + c) + (b - c)^6 (b + c)^3 - a (b - c)^4 (b + c)^2 (3 b^2 - 2 b c + 3 c^2) + a^6 (8 b^3 - 6 b^2 c - 6 b c^2 + 8 c^3) - a^5 (6 b^4 + 12 b^3 c - 13 b^2 c^2 + 12 b c^3 + 6 c^4) - a^4 (6 b^5 - 18 b^4 c + 7 b^3 c^2 + 7 b^2 c^3 - 18 b c^4 + 6 c^5) - a^2 b c (6 b^5 - 13 b^4 c + 11 b^3 c^2 + 11 b^2 c^3 - 13 b c^4 + 6 c^5) + a^3 (8 b^6 - 6 b^5 c - 9 b^4 c^2 + 26 b^3 c^3 - 9 b^2 c^4 - 6 b c^5 + 8 c^6)) : :

Continuing from X(6595), let W = X(3065). Let C′₂ be the point, other than S, of intersection of the Euler line of the triangle WCA and the circle (C, |CS|). Let B′₃ be the point, other than S, of intersection of the Euler line of the triangle WAB and the circle (B, |BS|). Let O′₁ be the circumcenter of SC₂B₃, and define O′₂ and O′₃ cyclically. The triangle O′₁O′₂O′₃ is here named the 2nd Schiffler triangle.The triangles ABC, O₁O₂O₃, and O′₁O′₂O′₃ and pairwise cyclologic:

X(6595) = (ABC, O₁O₂O₃)-cyclologic center
X(3065) = (O₁O₂O₃, ABC)-cyclologic center
X(1320) = (ABC, O′₁O′₂O′₃)-cyclologic center
X(1) = (O′₁O′₂O′₃, ABC)-cyclologic center
X(6596) = (O₁O₂O₃, O′₁O′₂O′₃)-cyclologic center
X(6597) = (O′₁O′₂O′₃, O₁O₂O₃)-cyclologic center

The points O′₁, O′₂, O′₃ lie on the Feuerbach hyperbola. (Randy Hutson, September 14, 2016)

See Hyacinthos 23101. (S. Kirikami, A. Hatzipolakis and A.Montesdeoca, February 4-5, 2015)

The 2nd Schiffler triangle is also the reflection of X(100) in every point of the excenters-midpoints triangle. (Randy Hutson, July 11, 2019)

X(6596) lies on the Feuerbach hyperbola and these lines: {4, 6326}, {11, 6598}, {84, 2771}, {104, 758}, {1389, 2802}, {1476, 5083}, {3065, 4867}, {3158, 3577}, {3467, 5692}, {5559, 5854}

X(6597) = (2nd SCHIFFLER, 1st SCHIFFLER)-CYCLOLOGIC CENTER

Barycentrics a (a^9 - a^8 (b + c) - (b - c)^4 (b + c)^5 + a^7 (-4 b^2 + 2 b c - 4 c^2) + a (b - c)^2 (b + c)^4 (b^2 + c^2) + 4 a^6 (b^3 + c^3) + a^5 (6 b^4 - 2 b^3 c + 9 b^2 c^2 - 2 b c^3 + 6 c^4) + a^4 (-6 b^5 + 2 b^4 c + b^3 c^2 + b^2 c^3 + 2 b c^4 - 6 c^5) - a^3 (4 b^6 + 2 b^5 c + 5 b^4 c^2 - 2 b^3 c^3 + 5 b^2 c^4 + 2 b c^5 + 4 c^6) + a^2 (4 b^7 - 5 b^5 c^2 - 3 b^4 c^3 - 3 b^3 c^4 - 5 b^2 c^5 + 4 c^7)) : :

See X(6596).

X(6597) lies on the Feuerbach hyperbola and these lines: {4, 5535}, {11, 6599}, {80, 191}, {758, 1389}

X(6598) = (1st SCHIFFLER, 2nd SCHIFFLER)-ORTHOLOGIC CENTER

Barycentrics a^7 + a^5 b c - 2 a^6 (b + c) - (b - c)^4 (b + c)^3 - a^2 b c (b + c)^3 + 2 a (b^2 - c^2)^2 (b^2 + c^2) - a^3 (b + c)^2 (3 b^2 - 5 b c + 3 c^2) + a^4 (3 b^3 + 2 b^2 c + 2 b c^2 + 3 c^3) : :

See X(6596).

X(6598) lies on the Feuerbach hyperbola and these lines: {1, 442}, {4, 758}, {7, 2475}, {9, 1837}, {10, 943}, {11, 6596}, {21, 950}, {30, 84}, {72, 80}, {90, 191}, {104, 3651}, {226, 5086}, {355, 3577}, {519, 1389}, {1172, 2323}, {1896, 5081}, {2346, 5178}, {2997, 5016}, {3649, 5665}, {3680, 4863}, {3880, 5559}, {5436, 5705}

X(6599) = (2nd SCHIFFLER, 1st SCHIFFLER)-ORTHOLOGIC CENTER

Barycentrics a^13 - 2 a^12 (b + c) - (b - c)^8 (b + c)^5 + a^11 (-3 b^2 + 5 b c - 3 c^2) + 2 a (b - c)^6 (b + c)^4 (b^2 - b c + c^2) - a^8 (b + c) (3 b^2 - 2 b c + 3 c^2)^2 + a^10 (7 b^3 + 2 b^2 c + 2 b c^2 + 7 c^3) + a^2 (b - c)^4 (b + c)^3 (3 b^4 - 3 b^3 c - 5 b^2 c^2 - 3 b c^3 + 3 c^4) + a^9 (4 b^4 - 14 b^3 c + 15 b^2 c^2 - 14 b c^3 + 4 c^4) - 3 a^7 (2 b^6 - 6 b^5 c + 6 b^4 c^2 - 7 b^3 c^3 + 6 b^2 c^4 - 6 b c^5 + 2 c^6) - a^4 (b + c)^3 (4 b^6 - 16 b^5 c + 26 b^4 c^2 - 27 b^3 c^3 + 26 b^2 c^4 - 16 b c^5 + 4 c^6) - a^3 (b^2 - c^2)^2 (7 b^6 - 17 b^5 c + 11 b^4 c^2 - 6 b^3 c^3 + 11 b^2 c^4 - 17 b c^5 + 7 c^6) + a^6 (6 b^7 - 4 b^6 c + 7 b^5 c^2 + 4 b^4 c^3 + 4 b^3 c^4 + 7 b^2 c^5 - 4 b c^6 + 6 c^7) + a^5 (9 b^8 - 20 b^7 c + 3 b^6 c^2 + 3 b^5 c^3 + 6 b^4 c^4 + 3 b^3 c^5 + 3 b^2 c^6 - 20 b c^7 + 9 c^8) : :

In addition to the notes at X(6596), Angel Montesdeoca notes in Hyacinthos 23101 that the following 15 points lie on the Feuerbach hyperbola: O₁, O₂, O₃, O′₁, O′₂, O′₃, the cyclologic ceners X(1), X(1320), X(3065), X(6595)-X(6599), and S.

X(6599) lies on the Feuerbach hyperbola and these lines: {5, 12660}, {11, 6597}, {90, 12769}, {100, 12639}, {5046, 12543}, {5443, 12524}, {6265, 12600}, {7161, 12623}, {7491, 12845}, {10950, 12657}

X(6600) = SS(a → 1 + cos A) OF X(3)

Barycentrics (1 + Cos[A])^2*(-1 + 2*Cos[A] + Cos[A]^2 - 2*Cos[B] - Cos[B]^2 - 2*Cos[C] - Cos[C]^2) : :

Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "SS(a → 1 + cos A) of X" is the point f(1 + cos A, 1 + cos B, 1 + cos C) : f(1 + cos B, 1 + cos C, 1 + cos A) : f(1 + cos C, 1 + cos A, 1 + cos B). See X(6337).

Let V = X(9), and let V_A,V_B,V_C be the vertices of the anticevian triangle of V. Then X(6600) is the centroid of {V, V_A, V_B, V_C}. (Randy Hutson, February 20, 2015)

X(6600) lies on these lines: {2, 2346}, {3, 518}, {6, 3939}, {7, 100}, {9, 55}, {10, 1001}, {21, 5686}, {35, 5223}, {56, 3243}, {119, 381}, {142, 1376}, {214, 999}, {218, 4878}, {219, 949}, {281, 1863}, {390, 2478}, {405, 3189}, {442, 954}, {516, 5812}, {521, 3126}, {527, 4421}, {1445, 1617}, {1466, 4321}, {1486, 4557}, {2136, 3303}, {2177, 4343}, {2223, 5120}, {2271, 3774}, {2951, 5537}, {3161, 4578}, {3358, 5534}, {3423, 4712}, {3647, 5220}, {5432, 6067}, {5759, 5857}

X(6600) = complement of X(6601)

X(6601) = SS(a → 1 + cos A) OF X(4)

Barycentrics (1 + 2*Cos[A] + Cos[A]^2 + 2*Cos[B] + Cos[B]^2 - 2*Cos[C] - Cos[C]^2)*(1 + 2*Cos[A] + Cos[A]^2 - 2*Cos[B] - Cos[B]^2 + 2*Cos[C] + Cos[C]^2) : :

See X(6600).

Let A_kB_kC_k be the cevian triangle of X(k) for k = 7, 8, 9, respectively. Let A′ = {A7,A8}-harmonic conjugate of A9, and define B′ and C′ cyclically. Then A′B′C′ is the cevian triangle of X(6601). (Randy Hutson, February 20, 2015)

X(6601) lies on these lines: {1, 142}, {2, 2346}, {4, 518}, {7, 3434}, {8, 1229}, {9, 497}, {11, 480}, {21, 390}, {55, 6067}, {69, 2481}, {80, 3421}, {84, 516}, {90, 5698}, {104, 376}, {144, 149}, {219, 294}, {278, 1041}, {388, 3243}, {517, 3427}, {519, 3577}, {521, 885}, {527, 3062}, {943, 1001}, {1000, 3880}, {1476, 4190}, {1479, 5223}, {2478, 5686}, {2551, 4866}, {2900, 3475}, {3059, 4863}, {3296, 5880}, {3826, 3913}, {4012, 4858}

X(6601) = isogonal conjugate of X(1617)
X(6601) = isotomic conjugate of X(6604)
X(6601) = complement of X(7674)
X(6601) = anticomplement of X(6600)
X(6601) = X(19)-isoconjugate of X(23144)
X(6601) = perspector of ABC and the reflection of the cevian triangle of X(7) in the centroid of {A,B,C,X(7)}

X(6602) = SS(a → 1 + cos A) OF X(31)

Barycentrics (1 + Cos[A])^3 : :

See X(6600).

X(6602) lies on these lines: {9, 1174}, {37, 2266}, {41, 55}, {101, 165}, {200, 3119}, {218, 1202}, {219, 604}, {1212, 3748}, {2171, 2324}, {2187, 3690}, {2329, 5273}, {3996, 6559}

X(6602) = barycentric cube of X(9)

X(6603) = SS(a → 1 + cos A) OF X(44)

Barycentrics (1 + Cos[A])*(2*Cos[A] - Cos[B] - Cos[C]) : :

See X(6600).

X(6603) is the point of intersection of the line X(1)X(9) and the trilinear polar of X(9); c.f. X(1323). (Randy Hutson, April 11, 2015)

X(6603) lies on these lines: {1, 6}, {21, 4520}, {40, 3207}, {41, 3057}, {65, 1802}, {78, 4513}, {101, 517}, {145, 6554}, {169, 1482}, {210, 4390}, {214, 6184}, {241, 1252}, {294, 1320}, {519, 1146}, {527, 1323}, {536, 1944}, {544, 5074}, {644, 3693}, {650, 663}, {672, 1319}, {908, 5723}, {952, 5179}, {1018, 5440}, {1055, 1155}, {1334, 2646}, {1385, 3730}, {1420, 5022}, {1455, 4559}, {1697, 4258}, {1807, 2338}, {2082, 2098}, {2170, 2348}, {2174, 2301}, {2280, 5919}, {2389, 3022}, {3008, 6547}, {3059, 4336}, {3306, 5228}, {3684, 3880}, {4345, 5838}, {4861, 4875}, {5030, 5126}, {5222, 5328}

X(6603) = isogonal conjugate of X(34056)
X(6603) = inverse-in-circumconic-centered-at-X(1) of X(9)

X(6604) = SS(a → 1 + cos A) OF X(69)

Barycentrics (1 + Cos[A])*(2*Cos[A] - Cos[B] - Cos[C]) : :
Barycentrics (a^2 + b^2 + c^2 - 2 a b - 2 a c)/(a - b - c) : :
X(6604) = 3 X(2) - 2 X(220)

See X(6600).

X(6604) lies on these lines: {1, 348}, {2, 220}, {4, 150}, {7, 8}, {56, 6337}, {57, 345}, {145, 279}, {193, 6180}, {218, 4904}, {226, 4384}, {239, 948}, {269, 3879}, {344, 1445}, {347, 4360}, {349, 4441}, {607, 1814}, {651, 1992}, {944, 5088}, {962, 4872}, {1025, 4253}, {1043, 1434}, {1265, 3263}, {1323, 3244}, {1418, 4851}, {1447, 1788}, {1475, 6168}, {1482, 1565}, {1847, 5174}, {2099, 3665}, {2295, 4648}, {2389, 3434}, {3160, 3241}, {3340, 3674}, {3616, 5543}, {3668, 3875}, {3672, 5738}, {3721, 4419}, {3870, 4350}, {3945, 5710}, {4295, 4911}, {4321, 4684}, {4328, 4357}, {5244, 5739}

X(6604) = isotomic conjugate of X(6601)
X(6604) = polar conjugate of isogonal conjugate of X(23144)
X(6604) = {X(7),X(8)}-harmonic conjugate of X(85)

X(6605) = SS(a → 1 + cos A) OF X(81)

Barycentrics (1 + Cos[A])*(2 + Cos[A] + Cos[B])*(2 + Cos[A] + Cos[C]) : :

See X(6600).

X(6605) lies on these lines: {2, 220}, {9, 1174}, {346, 3996}, {644, 4847}, {1212, 3957}, {2287, 3693}, {4651, 6559}

X(6606) = SS(a → 1 + cos A) OF X(99)

Barycentrics (Cos[A] - Cos[B])*(2 + Cos[A] + Cos[B])*(Cos[A] - Cos[C])*(2 + Cos[A] + Cos[C]) : :

See X(6600).

X(6606) lies on these lines: {99, 883}, {100, 4569}, {664, 3939}, {666, 4552}, {668, 4578}, {927, 4557}, {1170, 3227}, {1441, 2346}, {2283, 4616}

X(6607) = SS(a → 1 + cos A) OF X(512)

Barycentrics Barycentrics sec^4(B/2) - sec^4(C/2) : :
Barycentrics (1 + Cos[A])^2*(Cos[B] - Cos[C])*(2 + Cos[B] + Cos[C]) : :

See X(6600).

X(6607) lies on these lines: {30, 511}, {657, 4105}

X(6607) = complementary conjugate of X(38973)

X(6608) = SS(a → 1 + cos A) OF X(661)

Barycentrics (1 + Cos[A])*(Cos[B] - Cos[C])*(2 + Cos[B] + Cos[C]) : :

See X(6600).

X(6608) lies on these lines: {521, 4724}, {522, 693}, {647, 1962}, {649, 6182}, {650, 663}, {656, 676}, {661, 926}, {1021, 2328}, {3939, 5375}

X(6608) = complement of isotomic conjugate of X(35312)

X(6609) = SS(a → 1 - cos A) OF X(3)

Barycentrics (-1 + Cos[A])^2*(-1 - 2*Cos[A] + Cos[A]^2 + 2*Cos[B] - Cos[B]^2 + 2*Cos[C] - Cos[C]^2) : :

Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "SS(a → 1 - cos A) of X" is the point f(1 - cos A, 1 - cos B, 1 - cos C) : f(1 - cos B, 1 - cos C, 1 - cos A) : f(1 - cos C, 1 - cos A, 1 - cos B). See X(6337) and X(6600).

X(6609) lies on these lines: {57, 1422}, {329, 934}, {946, 999}, {1398, 1851}

X(6609) = complement of X(34546)

X(6610) = SS(a → 1 - cos A) OF X(44)

Barycentrics (-1 + Cos[A])*(2*Cos[A] - Cos[B] - Cos[C]) : :

See X(6600).

X(6610) lies on these lines: {6, 57}, {7, 1100}, {37, 77}, {44, 241}, {73, 1242}, {85, 4670}, {227, 1406}, {279, 4644}, {348, 4643}, {349, 4410}, {513, 663}, {518, 5018}, {527, 1323}, {536, 664}, {604, 1122}, {674, 1362}, {738, 3207}, {948, 4675}, {1104, 4306}, {1108, 4341}, {1386, 4334}, {1442, 3723}, {2099, 2263}, {2182, 3942}, {2293, 3748}, {3160, 4419}, {3945, 4872}, {4318, 4864}, {4648, 5226}

X(6610) = isogonal conjugate of isotomic conjugate of X(37780)
X(6610) = crossdifference of every pair of points on line X(9)X(3900)

X(6611) = SS(a → 1 - cos A) OF X(55)

Barycentrics (-1 + Cos[A])^2*(1 + Cos[A] - Cos[B] - Cos[C]) : :

See X(6600).

X(6611) lies on these lines: {2, 934}, {6, 2155}, {25, 34}, {57, 1422}, {198, 223}, {221, 2187}, {222, 1461}, {604, 1407}, {2654, 3304}

X(6612) = SS(a → 1 - cos A) OF X(56)

Barycentrics (-1 + Cos[A])^2*(-1 + Cos[A] + Cos[B] - Cos[C])*(-1 + Cos[A] - Cos[B] + Cos[C]) : :

See X(6600).

X(6612) lies on these lines: {56, 1413}, {57, 1422}, {84, 999}, {951, 1433}, {2137, 5193}

X(6613) = SS(a → 1 - cos A) OF X(99)

Barycentrics (Cos[A] - Cos[B])*(-2 + Cos[A] + Cos[B])*(Cos[A] - Cos[C])*(-2 + Cos[A] + Cos[C]) : :

See X(6600).

X(6613) lies on these lines: {190, 1461}, {668, 934}, {670, 4616}, {903, 3668}, {1476, 2481}, {4555, 4566}

X(6614) = SS(a → 1 - cos A) OF X(101)

Barycentrics (-1 + Cos[A])^2*(Cos[A] - Cos[B])*(Cos[A] - Cos[C]) : :

See X(6600).

X(6614) lies on these lines: {109, 934}, {738, 1106}, {1015, 1407}, {1414, 4616}, {1415, 1461}, {3939, 6516}

X(6614) = isogonal conjugate of X(4163)

X(6615) = SS(a → 1 - cos A) OF X(661)

Barycentrics (-1 + Cos[A])*(Cos[B] - Cos[C])*(-2 + Cos[B] + Cos[C]) : :

See X(6600).

X(6615) lies on these lines: {21, 3737}, {244, 3259}, {256, 885}, {513, 663}, {521, 4895}, {522, 3717}, {523, 2292}, {656, 900}, {661, 3310}, {1734, 4962}, {2254, 3667}, {4171, 4526}

X(6616) = SS(a → tan A) OF X(78)

Barycentrics (tan A)(tan A - tan B - tan C)(tan²A - tan²B - tan²C) : :

As a point on the Euler line, X(6616) has Shinagawa coefficients (2F², -4EF + S²).

Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "SS(a → tan A) of X" is the point f(tan A, tan B, tan C) : f(tan B, tan C, tan A) : f(tan C, tan A, tan B). Symbolic substitutions carry lines to lines, conics to conics, cubics to cubics, etc. In particular, SS(a → tan A) carries the line X(1)X(2) to the line X(2)X(3), the Euler line.

X(6616) lies on these lines: {2, 3}, {107, 3183}, {1498, 6523}, {1503, 6526}, {2883, 6525}

X(6617) = SS(a → tan A) OF X(306)

Barycentrics (Tan[B] + Tan[C])*(-Tan[A]^2 + Tan[B]^2 + Tan[C]^2) : :

As a point on the Euler line, X(6617) has Shinagawa coefficients (2EF - S², S²).

See X(6616).

X(6617) lies on these lines: {2, 3}, {6, 6509}, {63, 268}, {100, 6060}, {394, 1073}, {1032, 3964}, {1033, 6527}, {1619, 1624}, {2063, 4558}

X(6618) = SS(a → tan A) OF X(612)

Barycentrics Tan[A]*(Tan[A]^2 + Tan[B]^2 + 2*Tan[B]*Tan[C] + Tan[C]^2) : :

As a point on the Euler line, X(6618) has Shinagawa coefficients (2F², -2(E + F)F + S²).

See X(6616).

X(6618) lies on these lines: {2, 3}, {6, 6525}, {154, 393}, {184, 1249}, {185, 3183}, {459, 1899}, {1857, 2182}, {2187, 2202}

X(6619) = SS(a → tan A) OF X(614)

Barycentrics Tan[A]*(Tan[A]^2 + Tan[B]^2 - 2*Tan[B]*Tan[C] + Tan[C]^2) : :

As a point on the Euler line, X(6619) has Shinagawa coefficients (2F², 2(E + F)F - S²).

See X(6616).

X(6619) lies on these lines: {2, 3}, {125, 459}, {393, 1853}, {1249, 1899}

X(6620) = SS(a → tan A) OF X(869)

Barycentrics Tan[A]^3*(Tan[B]^2 + Tan[B]*Tan[C] + Tan[C]^2) : :

As a point on the Euler line, X(6620) has Shinagawa coefficients ((E + F)F, -(E + F)² + S²).

See X(6616).

X(6620) lies on these lines: {2, 3}, {154, 5254}, {184, 5286}, {193, 3186}, {393, 1974}, {1843, 3087}, {2207, 6524}

X(6621) = SS(a → tan A) OF X(936)

Barycentrics Tan[A]*(Tan[A]^3 - Tan[A]^2*Tan[B] - Tan[A]*Tan[B]^2 + Tan[B]^3 - Tan[A]^2*Tan[C] + 2*Tan[A]*Tan[B]*Tan[C] + 3*Tan[B]^2*Tan[C] - Tan[A]*Tan[C]^2 + 3*Tan[B]*Tan[C]^2 + Tan[C]^3) : :

As a point on the Euler line, X(6621) has Shinagawa coefficients (4F², -4EF + S²).

See X(6616).

X(6621) lies on these lines: {2, 3}, {154, 6526}, {459, 1498}, {1249, 3349}, {2883, 3183}

X(6622) = SS(a → tan A) OF X(978)

Barycentrics    (tan A)(cos^2 B + cos^2 C - cos^2 A - sec A sec B sec C) : :
Barycentrics    [S_A(S^2 - 2 S_B S_C) - S_B(S^2 - 2 S_C S_A) - S_C(S^2 - 2 S_A S_B)]/S_A : :
Barycentrics    Tan[A]*(Tan[A]^2*Tan[B] + Tan[A]*Tan[B]^2 + Tan[A]^2*Tan[C] - Tan[A]*Tan[B]*Tan[C] - Tan[B]^2*Tan[C] + Tan[A]*Tan[C]^2 - Tan[B]*Tan[C]^2) : :

As a point on the Euler line, X(6622) has Shinagawa coefficients (2F, -E + F).

See X(6616).

X(6622) lies on these lines: {2, 3}, {1068, 3012}, {1192, 5893}, {1249, 3767}, {1876, 5704}, {5319, 5702}, {6523, 6530}

X(6622) = center of inverse-in-polar-circle-of-de-Longchamps-circle

X(6623) = SS(a → tan A) OF X(995)

Barycentrics Tan[A]^2*(Tan[A]*Tan[B] + Tan[B]^2 + Tan[A]*Tan[C] - Tan[B]*Tan[C] + Tan[C]^2) : :

As a point on the Euler line, X(6623) has Shinagawa coefficients (F, -E + 2F).

See X(6616).

X(6623) lies on these lines: {2, 3}, {113, 193}, {115, 393}, {133, 5139}, {253, 339}, {1093, 6526}, {1870, 5274}, {1899, 5656}, {1902, 5818}, {1974, 5622}, {3087, 5475}, {3620, 5891}, {5261, 6198}, {2, 3}, {1068, 3012}, {1192, 5893}, {1249, 3767}, {1876, 5704}, {5319, 5702}, {6523, 6530}

X(6624) = SS(a → tan A) OF X(997)

Barycentrics Tan[A]*(Tan[A]^3 - Tan[A]^2*Tan[B] - Tan[A]*Tan[B]^2 + Tan[B]^3 - Tan[A]^2*Tan[C] + 2*Tan[A]*Tan[B]*Tan[C] + Tan[B]^2*Tan[C] - Tan[A]*Tan[C]^2 + Tan[B]*Tan[C]^2 + Tan[C]^3) : :

As a point on the Euler line, X(6624) has Shinagawa coefficients (2F², -2(2E - F)F + S²).

See X(6616).

X(6624) lies on these lines: {2, 3}, {133, 1249}, {459, 6000}, {1498, 6526}, {2883, 6523}, {3183, 5878}

X(6625) = SS(a → (a + b)(a + c)) OF X(7)

Barycentrics (a^2 + a*b + b^2 + a*c + b*c - c^2)*(a^2 + a*b - b^2 + a*c + b*c + c^2) : :

See X(6337).

X(6625) lies on the Kiepert hyperbola and these lines: {1, 148}, {2, 1931}, {4, 2905}, {10, 894}, {115, 1509}, {321, 1909}, {2895, 6539}, {2996, 3945}, {4567, 5277}

X(6625) = isogonal conjugate of X(18755)
X(6625) = isotomic conjugate of X(1654)
X(6625) = trilinear pole of line X(523)X(2487)
X(6625) = pole wrt polar circle of trilinear polar of X(4213)
X(6625) = X(19)-isoconjugate of X(22139)
X(6625) = X(48)-isoconjugate (polar conjugate) of X(4213)

X(6626) = SS(a → (a + b)(a + c)) OF X(9)

Barycentrics    (a + b)*(a + c)*(a^2 - a*b - b^2 - a*c - b*c - c^2) : :
Barycentrics    (tan A)(cos^2 B + cos^2 C - cos^2 A - sec A sec B sec C) : :
Barycentrics    [S_A(S^2 - 2 S_B S_C) - S_B(S^2 - 2 S_C S_A) - S_C(S^2 - 2 S_A S_B)]/S_A : :

As a line L varies through the set of all lines that pass through X(86), the locus of the trilinear pole of L is a circumconic, and its center is X(6626). (Randy Hutson, February 20, 2014)

See X(6337).

X(6626) lies on these lines: {2, 1931}, {3, 3437}, {10, 99}, {39, 2669}, {58, 86}, {75, 1247}, {239, 257}, {274, 1111}, {502, 1268}, {620, 6537}, {799, 1909}, {966, 6337}, {1414, 1935}, {1434, 1447}, {1975, 5737}

X(6626) = X(2)-Ceva conjugate of X(86)
X(6626) = perspector of circumconic centered at X(86)

X(6627) = SS(a → (a + b)(a + c)) OF X(11)

Barycentrics (b - c)^2*(b + c)^2*(-a^2 + a*b + b^2 + a*c + b*c + c^2) : :

See X(6337).

X(6627) lies on these lines: {2, 4610}, {115, 125}, {2511, 2615}

X(6627) = complement of X(4610)

X(6628) = SS(a → (a + b)(a + c)) OF X(31)

Barycentrics (a + b)^3*(a + c)^3 : :

See X(6337).

X(6628) lies on these lines: {81, 4610}, {86, 4683}, {171, 4600}, {593, 763}, {757, 873}, {3995, 4632}

X(6628) = barycentric cube of X(86)

X(6629) = SS(a → (a + b)(a + c)) OF X(44)

Barycentrics (a + b)*(a + c)*(2*a^2 - b^2 - c^2) : :

See X(6337).

X(6629) lies on these lines: {58, 86}, {69, 4257}, {99, 519}, {110, 2729}, {187, 524}, {193, 4262}, {239, 514}, {325, 540}, {518, 3110}, {593, 4001}, {799, 5209}, {1323, 4573}, {1366, 4831}, {1444, 4276}, {1931, 3912}, {4592, 5127}

X(6630) = SS(a → (a - b)(a - c)) OF X(7)

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

See X(6337).

X(6630) lies on these lines: {239, 3911}, {514, 4440}, {519, 1757}, {1016, 4473}, {3975, 4358}, {4555, 6547}

X(6630) = isogonal conjugate of X(9259)
X(6630) = isotomic conjugate of X(4440)
X(6630) = cevapoint of PU(24)
X(6630) = X(19)-isoconjugate of X(22148)

X(6631) = SS(a → (a - b)(a - c)) OF X(9)

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

See X(6337).

X(6631) lies on these lines: {100, 4367}, {190, 514}, {519, 1738}, {660, 4083}, {664, 3669}, {666, 5382}, {668, 1577}, {898, 4401}, {3936, 6542}

X(6631) = crosssum of PU(25)
X(6631) = X(2)-Ceva conjugate of X(190)
X(6631) = perspector of circumconic centered at X(190)
X(6631) = trilinear pole of line X(1054)X(4440)
X(6631) = center of hyperbola {{A,B,C,PU(24),PU(58)}}
X(6631) = crosspoint of PU(24)

X(6632) = SS(a → (a - b)(a - c)) OF X(31)

Barycentrics (a - b)^3*(a - c)^3 : :

See X(6337).

X(6632) lies on these lines: {2, 1016}, {31, 765}, {42, 5378}, {81, 4600}, {190, 6546}, {4076, 5423}, {4998, 5435}

X(6632) = isotomic conjugate of X(6545)
X(6632) = barycentric cube of X(190)

X(6633) = SS(a → (a - b)(a - c)) OF X(44)

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

See X(6337).

X(6633) lies on these lines: {1, 2}, {190, 514}, {668, 4791}, {4440, 6549}, {4562, 4597}

X(6634) = SS(a → (a - b)(a - c)) OF X(55)

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

See X(6337).

X(6634) lies on these lines: {2, 1016}, {43, 5378}, {57, 4998}, {171, 765}, {2319, 5383}, {4555, 6545}

X(6635) = SS(a → (a - b)(a - c)) OF X(99)

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

X(6635) is also SS(a → (a - b)(a - c)) OF X(101). See X(6337).

X(6635) lies on the Steiner circumellipse and these lines: {99, 6551}, {190, 6546}, {666, 4582}, {889, 901}, {903, 1644}, {1016, 4370}, {3227, 5376}, {3257, 4562}

X(6635) = isotomic conjugate of X(6550)
X(6635) = trilinear pole of line X(2)X(1016)

X(6636) = {X(2),X(22)}-HARMONIC CONJUGATE OF X(23)

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

As a point on the Euler line, X(6636) has Shinagawa coefficients (3E + 4F, -4E - 4F).

X(6636) lies on these lines: {2,3}, {32,1180}, {35,3920}, {36,5310}, {39,251}, {51,5092}, {81,4265}, {97,933}, {98,930}, {99,1799}, {100,3703}, {110,3917}, {141,2916}, {159,3620}, {160,3314}, {182,3060}, {184,323}, {187,1194}, {343,3448}, {511,1994}, {612,5010}, {842,1291}, {1030,5276}, {1176,3313}, {1184,5023}, {1199,6243}, {1216,1614}, {1350,1993}, {1495,3819}, {1501,3094}, {1503,3410}, {1609,5304}, {2076,3051}, {2917,6247}, {3053,5359}, {3218,5285}, {3219,3220}, {3705,4996}, {5085,5422}, {5297,5370}, {5866,6031}

X(6636) = reflection of X(1994) in X(5012)
X(6636) = anticomplement X(5133)
X(6636) = circumcircle-inverse of X(5189)
X(6636) = crosspoint, wrt the excentral triangle, of X(3) and X(2916)
X(6636) = crosspoint, wrt the tangential triangle, of X(3) and X(2916)
X(6636) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,22,23), (2,5189,427), (3,4,37126), (3,22,2), (3,24,3523), (3,26,631), (3,199,4210), (3,376,2071), (3,550,3520), (3,859,4218), (3,2070,549), (3,2915,404), (3,2937,140), (20,3523,3088), (35,5322,3920), (140,2937,3518), (184,2979,323), (184,3098,2979), (187,1194,1627), (550,6240,20), (1113,1114,5189), (1194,1627,5354), (1350,3796,1993), (3131,3132,237), (3220,5314,3219), (4265,5347,81), (5004,5005,5), (5010,5345,612).

X(6637) = HATZIPOLAKIS-MOSES CIRCUMCENTER

Barycentrics 2 a^10+a^9 b-4 a^8 b^2-3 a^7 b^3+a^6 b^4+3 a^5 b^5+a^4 b^6-a^3 b^7+a^2 b^8-b^10+a^9 c-2 a^8 b c-4 a^7 b^2 c+a^6 b^3 c+4 a^5 b^4 c+3 a^4 b^5 c-a^2 b^7 c-a b^8 c-b^9 c-4 a^8 c^2-4 a^7 b c^2+4 a^6 b^2 c^2+a^5 b^3 c^2-a^4 b^4 c^2+2 a^3 b^5 c^2-2 a^2 b^6 c^2+a b^7 c^2+3 b^8 c^2-3 a^7 c^3+a^6 b c^3+a^5 b^2 c^3-6 a^4 b^3 c^3-a^3 b^4 c^3+a^2 b^5 c^3+3 a b^6 c^3+4 b^7 c^3+a^6 c^4+4 a^5 b c^4-a^4 b^2 c^4-a^3 b^3 c^4+2 a^2 b^4 c^4-3 a b^5 c^4-2 b^6 c^4+3 a^5 c^5+3 a^4 b c^5+2 a^3 b^2 c^5+a^2 b^3 c^5-3 a b^4 c^5-6 b^5 c^5+a^4 c^6-2 a^2 b^2 c^6+3 a b^3 c^6-2 b^4 c^6-a^3 c^7-a^2 b c^7+a b^2 c^7+4 b^3 c^7+a^2 c^8-a b c^8+3 b^2 c^8-b c^9-c^10 : :

Let H be the orthocenter of a triangle ABC. Let L_A be the line through A perpendicular to the line of H and the A-excenter, and define L_B and L_C cyclically. Let A′ = L_B∩L_C, B′ = L_C∩L_A, C′ = L_A∩L_B. The circumcenter of A′B′C′ is X(6637). See

See Hyacinthos #23129.

X(6637) lies on this line: X(1)X(4)

X(6638) = (SS: a → sin 2A) OF X(43)

Barycentrics 1 - (sin 2A)(csc 2B + csc 2C) : :
Barycentrics Sin[2*A]*(Sin[2*A]*Sin[2*B] + Sin[2*A]*Sin[2*C] - Sin[2*B]*Sin[2*C]) : :

As a point on the Euler line, X(6638) has Shinagawa coefficients ((E - F)F - S², (E + F)F + S²).

See X(6337). Centers X(6638)-X(6644) exemplify of the fact that the symbolic substitution (SS: a → sin 2A) takes the Nagel line, X(1)X(2), to the Euler line, X(2)X(3).

X(6638) lies on these lines: {2, 3}, {154, 1624}, {216, 5943}, {511, 6509}, {577, 1971}, {1147, 2055}, {2972, 2979}, {3164, 3168}, {3167, 3289}

X(6638) = isotomic conjugate of the polar conjugate of X(32445)
X(6638) = X(92)-isoconjugate of X(1988)
X(6638) = X(577)-Ceva conjugate of X(3)

X(6639) = (SS: a → sin 2A) OF X(498)

Barycentrics Sin[2*A]^4 - 2*Sin[2*A]^2*Sin[2*B]^2 + Sin[2*B]^4 - 2*Sin[2*A]^2*Sin[2*B]*Sin[2*C] - 2*Sin[2*A]^2*Sin[2*C]^2 - 2*Sin[2*B]^2*Sin[2*C]^2 + Sin[2*C]^4 : :
Barycentrics (a^2-b^2-c^2) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2-2 a^4 b^2 c^2-2 a^2 b^4 c^2+4 b^6 c^2-2 a^2 b^2 c^4-6 b^4 c^4+2 a^2 c^6+4 b^2 c^6-c^8) : :

As a point on the Euler line, X(6639) has Shinagawa coefficients (E + 8F, -E).

See X(6338).

X(6639) lies on these lines: {2, 3}, {49, 68}, {184, 5449}, {195, 6515}, {231, 5158}

X(6639) = complement of X(37119)

X(6640) = (SS: a → sin 2A) OF X(499)

Barycentrics Sin[2*A]^4 - 2*Sin[2*A]^2*Sin[2*B]^2 + Sin[2*B]^4 + 2*Sin[2*A]^2*Sin[2*B]*Sin[2*C] - 2*Sin[2*A]^2*Sin[2*C]^2 - 2*Sin[2*B]^2*Sin[2*C]^2 + Sin[2*C]^4 : :
Barycentrics (a^2-b^2-c^2) (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+2 a^4 b^2 c^2-2 a^2 b^4 c^2+4 b^6 c^2-2 a^2 b^2 c^4-6 b^4 c^4+2 a^2 c^6+4 b^2 c^6-c^8) : :

As a point on the Euler line, X(6640) has Shinagawa coefficients (E - 8F, -E).

See X(6338).

X(6640) lies on these lines: {2, 3}, {49, 1899}, {113, 3357}, {125, 1147}, {974, 5654}, {1092, 5449}, {1204, 5448}

X(6640) = complement of X(7505)

X(6641) = (SS: a → sin 2A) OF X(612)

Barycentrics Sin[2*A]*(Sin[2*A]^2 + Sin[2*B]^2 + 2*Sin[2*B]*Sin[2*C] + Sin[2*C]^2) : :

As a point on the Euler line, X(6641) has Shinagawa coefficients (F² - S², (E + F)F - S²).

See X(6338).

X(6641) lies on these lines: {2, 3}, {51, 577}, {97, 3060}, {154, 157}, {160, 161}, {184, 216}, {5651, 6509}

X(6641) = orthocentroidal-circle-inverse of X(34965)

X(6642) = (SS: a → sin 2A) OF X(936)

Barycentrics Sin[2*A]*(Sin[2*A]^3 - Sin[2*A]^2*Sin[2*B] - Sin[2*A]*Sin[2*B]^2 + Sin[2*B]^3 - Sin[2*A]^2*Sin[2*C] + 2*Sin[2*A]*Sin[2*B]*Sin[2*C] + 3*Sin[2*B]^2*Sin[2*C] - Sin[2*A]*Sin[2*C]^2 + 3*Sin[2*B]*Sin[2*C]^2 + Sin[2*C]^3) : :

As a point on the Euler line, X(6642) has Shinagawa coefficients (E - 2F, E + 2F).

See X(6338).

X(6642) lies on these lines: {2, 3}, {6, 1147}, {51, 1092}, {52, 394}, {54, 5422}, {143, 1351}, {154, 5892}, {155, 389}, {159, 182}, {568, 6090}, {578, 5943}, {1217, 6525}, {1350, 5447}, {1993, 3567}, {2883, 4846}, {2929, 5448}, {2931, 5972}, {3167, 5946}, {5562, 5651}

X(6642) = circumcircle-inverse of X(37971)

X(6643) = (SS: a → sin 2A) OF X(938)

Barycentrics Sin[2*A]^4 - 2*Sin[2*A]^3*Sin[2*B] + 2*Sin[2*A]*Sin[2*B]^3 - Sin[2*B]^4 - 2*Sin[2*A]^3*Sin[2*C] - 4*Sin[2*A]^2*Sin[2*B]*Sin[2*C] - 2*Sin[2*A]*Sin[2*B]^2*Sin[2*C] - 2*Sin[2*A]*Sin[2*B]*Sin[2*C]^2 + 2*Sin[2*B]^2*Sin[2*C]^2 + 2*Sin[2*A]*Sin[2*C]^3 - Sin[2*C]^4 : :
Barycentrics (a^2-b^2-c^2) (a^8-2 a^4 b^4+b^8-4 a^4 b^2 c^2-4 b^6 c^2-2 a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : :

As a point on the Euler line, X(6643) has Shinagawa coefficients (E, -E - 2F).

See X(6338).

X(6643) lies on these lines: {2, 3}, {68, 69}, {96, 97}, {216, 2548}, {388, 1060}, {394, 6193}, {497, 1062}, {577, 3767}, {626, 6389}, {1038, 1478}, {1040, 1479}, {1578, 3070}, {1579, 3071}, {1899, 5562}, {3284, 5319}, {5707, 5800}

X(6643) = orthocentroidal-circle-inverse of X(7401)
X(6643) = homothetic center of 3rd pedal triangle of X(3) and 3rd pedal triangle of X(4)
X(6643) = {X(2),X(4)}-harmonic conjugate of X(7401)

X(6644) = (SS: a → sin 2A) OF X(997)

Barycentrics Sin[2*A]*(Sin[2*A]^3 - Sin[2*A]^2*Sin[2*B] - Sin[2*A]*Sin[2*B]^2 + Sin[2*B]^3 - Sin[2*A]^2*Sin[2*C] + 2*Sin[2*A]*Sin[2*B]*Sin[2*C] + Sin[2*B]^2*Sin[2*C] - Sin[2*A]*Sin[2*C]^2 + Sin[2*B]*Sin[2*C]^2 + Sin[2*C]^3) : :
Barycentrics a^2 (a^8-2 a^6 b^2+2 a^2 b^6-b^8-2 a^6 c^2+4 a^4 b^2 c^2-4 a^2 b^4 c^2+2 b^6 c^2-4 a^2 b^2 c^4-2 b^4 c^4+2 a^2 c^6+2 b^2 c^6-c^8) : :

As a point on the Euler line, X(6644) has Shinagawa coefficients (E - 4F, E + 4F).

See X(6338).

X(6644) lies on these lines: {2, 3}, {6, 1511}, {52, 1092}, {110, 5890}, {155, 2929}, {156, 1181}, {157, 1605}, {182, 2393}, {183, 1236}, {389, 1147}, {394, 1154}, {567, 5422}, {568, 1993}, {569, 6146}, {578, 5462}, {1192, 5876}, {1485, 5961}, {1539, 2935}, {2782, 2936}, {3425, 5968}, {3455, 6055}, {5651, 5891}

X(6644) = midpoint of X(3) and X(25)
X(6644) = inverse-in-circumcircle of [reflection of X(3) in the orthic axis]
X(6644) = harmonic center of nine-point circle and tangential circle
X(6644) = tangential isogonal conjugate of X(35237)
X(6644) = X(999)-of-tangential-triangle if ABC is acute
X(6644) = center of circle that is inverse-in-circumcircle of orthic axis (circle with segment X(3)X(25) as diameter)
X(6644) = center of circle {X(3),X(25),PU(4)} (the inverse-in-circumcircle of the orthic axis)

X(6645) = (SS: a → a² + bc) OF X(6)

Barycentrics (a² + bc)² : :

See X(6338).

X(6645) lies on these lines: {1, 335}, {2, 12}, {6, 330}, {55, 3552}, {76, 2242}, {83, 1015}, {99, 1500}, {172, 385}, {192, 1975}, {257, 3509}, {274, 5291}, {287, 1425}, {401, 4296}, {439, 5281}, {458, 1398}, {668, 5277}, {894, 2329}, {904, 3510}, {1003, 3295}, {1043, 6542}, {1478, 5025}, {2241, 3972}, {2275, 3329}, {3023, 4027}

X(6645) = barycentric square of X(894)

X(6646) = (SS: a → a² + bc) OF X(8)

Barycentrics b² + c² - a² + ab + ac - bc : :

See X(6338).

X(6646) lies on these lines: {2, 7}, {6, 4389}, {8, 726}, {37, 320}, {38, 256}, {69, 192}, {72, 4201}, {75, 1654}, {86, 1931}, {141, 190}, {145, 5847}, {193, 3672}, {239, 3663}, {319, 536}, {333, 3782}, {346, 3620}, {391, 4346}, {524, 4360}, {545, 594}, {960, 1463}, {966, 4699}, {982, 4703}, {984, 4645}, {1100, 4715}, {1266, 3686}, {1284, 2975}, {1469, 3869}, {1757, 3821}, {1999, 4001}, {2896, 3954}, {3210, 5739}, {3631, 3943}, {3661, 3729}, {3688, 3888}, {3758, 4657}, {3870, 4335}, {3957, 4343}, {3986, 4896}, {4026, 5852}, {4361, 4398}, {4363, 5224}, {4366, 5845}, {4384, 4862}, {4392, 5211}, {4429, 5220}, {4454, 5232}, {4664, 4851}, {4675, 4687}, {4686, 4690}

X(6646) = complement of X(31300)
X(6646) = anticomplement of X(894)
X(6646) = isotomic conjugate of isogonal conjugate of X(21008)
X(6646) = polar conjugate of isogonal conjugate of X(22161)

X(6647) = (SS: a → a² + bc) OF X(44)

Barycentrics (a^2 + b*c)*(2*a^2 - a*b - b^2 - a*c + 2*b*c - c^2) : :

See X(6338).

X(6647) lies on these lines: {142, 4315}, {527, 1323}, {535, 5074}, {664, 3509}, {894, 2329}, {2533, 3907}, {4051, 4209}

X(6648) = (SS: a → a² + bc) OF X(99)

Barycentrics (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 + b^2 + a*c + b*c)*(a^2 + a*b + b*c + c^2) : :

See X(6338).

X(6648) lies on the Steiner circumellipse and these lines: {99, 4552}, {109, 190}, {478, 3596}, {553, 903}, {645, 4559}, {651, 668}, {664, 1461}, {666, 4581}, {670, 4573}, {961, 3227}, {1121, 1220}, {1462, 2298}, {4569, 4617}

X(6648) = isotomic conjugate of X(3910)
X(6648) = trilinear pole of line X(2)X(12)

X(6649) = (SS: a → a² + bc) OF X(100)

Barycentrics (a - b)*(a - c)*(a + b - c)*(a - b + c)*(a^2 + b*c) : :

See X(6338).

X(6649) lies on these lines: {85, 750}, {100, 658}, {109, 789}, {651, 3570}, {799, 1414}, {4551, 4573}

X(6650) = (SS: a → a² - bc) OF X(7)

Barycentrics (a^2 + a*b + b^2 - a*c - b*c - c^2)*(a^2 - a*b - b^2 + a*c - b*c + c^2) : :

See X(6338).

X(6650) lies on these lines: {2, 846}, {7, 4393}, {27, 2969}, {75, 1654}, {86, 1086}, {148, 3125}, {190, 1213}, {335, 740}, {524, 903}, {675, 2702}, {4359, 6158}, {4750, 6548}

X(6650) = isotomic conjugate of X(6542)
X(6650) = anticomplement of X(6651)
X(6650) = polar conjugate of X(17927)

X(6651) = (SS: a → a² - bc) OF X(9)

Barycentrics (a^2 - b*c)*(a^2 + a*b - b^2 + a*c - b*c - c^2) : :

See X(6338).

X(6651) lies on these lines: {2, 846}, {9, 1654}, {37, 86}, {238, 239}, {385, 3985}, {524, 4370}, {528, 4733}, {537, 5625}, {1213, 4422}, {1331, 1999}, {1757, 6541}, {1966, 3948}, {2786, 5029}, {3570, 4760}, {4375, 6544}

X(6651) = complement of X(6650)
X(6651) = center of hyperbola {{A,B,C,X(789),PU(6)}}
X(6651) = X(2)-Ceva conjugate of X(239)
X(6651) = crosssum of circumcircle intercepts of line PU(8) (line X(42)X(649))

X(6652) = (SS: a → a² - bc) OF X(31)

Barycentrics (a² - b*c)³ : :

See X(6338).

X(6652) lies on these lines: {2, 31}, {105, 335}, {239, 3573}, {1914, 3253}, {2112, 2145}, {4366, 4368}

X(6653) = (SS: a → a² - bc) OF X(69)

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

See X(6338).

X(6653) lies on these lines: {{1,17565}, {2,11}, {8,6655}, {10,6651}, {69,1278}, {148, 668}, {190, 594}, {335, 740}, {903, 4971}, {966, 4473}, {1086, 4360}, {3661, 4660}, {5025, 5687}

X(6653) = anticomplement of X(4366)

X(6654) = (SS: a → a² - bc) OF X(81)

Barycentrics (a^2 - b*c)*(a^2 + b^2 - a*c - b*c)*(a^2 - a*b - b*c + c^2) : :

See X(6338).

X(6654) lies on the hyperbola {{A,B,C,X(81),X(100),PU(8)}} and these lines: {1, 6185}, {2, 11}, {57, 927}, {239, 3573}, {312, 6632}, {350, 3570}, {870, 1438}, {873, 2185}, {1447, 1914}, {1462, 2162}

X(6654) = isogonal conjugate of X(3252)
X(6654) = trilinear pole of line X(238)X(812)

X(6655) = (SS: a → a⁴ + b²c²) OF X(8)

Barycentrics b⁴ + c⁴ - a⁴ + a²b² + a²c² - b²c² : :

Suppose that X = f(a,b,c) : f(b,c,a) : f(c,a,b) (barycentrics). Then "(SS: a → a⁴ + b²c²) of X" is the point

f(a⁴ + b²c², b⁴ + c²a², c⁴ + a²b²) : f(b⁴ + c²a², c⁴ + a²b², a⁴ + b²c²) : f(c⁴ + a²b², a⁴ + b²c², b⁴ + c²a²).

As a point on the Euler line, X(6655) has Shinagawa coefficients ((E + F)² + S², -4S²).

Symbolic substitutions carry lines to lines, conics to conics, cubics to cubics, etc. In particular, SS(a → a⁴ + b²c²) carries the line X(1)X(2) to the line X(2)X(3), the Euler line. See X(6616).

X(6655) lies on these lines: {2, 3}, {8, 6653}, {39, 316}, {69, 698}, {76, 148}, {83, 4045}, {99, 626}, {115, 1078}, {194, 315}, {385, 5254}, {621, 3104}, {622, 3105}, {1975, 3314}, {2542, 2559}, {2543, 2558}, {2794, 4027}, {3094, 5207}, {3096, 3734}, {3849, 5007}, {3917, 6310}, {4366, 6284}, {5103, 5116}, {5309, 6179}, {7762,7839}

X(6655) = reflection of X(2) in X(7924)
X(6655) = reflection of X(19686) in X(2)
X(6655) = complement of X(6658)
X(6655) = anticomplement of X(384)
X(6655) = anticomplementary conjugate of X(37889)
X(6655) = 2nd-Brocard-Circle-inverse of X(37896)
X(6655) = orthocentroidal-circle-inverse of X(16044)
X(6655) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,33259), (2,4,16044), (2,20,3552), (3,5025,2), (148,2896,76), (315,2549,194)

X(6656) = (SS: a → a⁴ + b²c²) OF X(10)

Barycentrics b⁴ + c⁴ + a²b² + a²c² : :

As a point on the Euler line, X(6656) has Shinagawa coefficients ((E + F)², -S²).

See X(6655).

X(6656) lies on these lines: {2, 3}, {6, 315}, {39, 325}, {69, 5286}, {76, 141}, {83, 316}, {115, 3934}, {183, 3767}, {194, 3314}, {230, 1078}, {239, 5015}, {385, 2896}, {491, 6422}, {492, 6421}, {574, 3788}, {625, 1506}, {639, 3102}, {640, 3103}, {695, 3978}, {754, 5007}, {894, 4911}, {1193, 4766}, {1235, 5523}, {1975, 2549}, {3491, 4173}, {3620, 6392}, {3661, 4385}, {3662, 3673}, {3695, 3797}, {5306, 6179}

X(6656) = midpoint of X(2) and X(7924)
X(6656) = reflection of X(6661) in X(2)
X(6656) = complement of X(384)
X(6656) = anticomplement of X(7819)
X(6656) = crosssum of X(32) and X(39)
X(6656) = crosspoint of X(76) and X(83)
X(6656) = pole of line X(5)X(6) wrt conic {{X(13),X(14),X(15),X(16),X(141)}}
X(6656) = orthocentroidal-circle-inverse of X(7770)
X(6656) = complementary conjugate of X(37890)
X(6656) = {X(2),X(3)}-harmonic conjugate of X(7807)
X(6656) = {X(2),X(4)}-harmonic conjugate of X(7770)
X(6656) = {X(2),X(5)}-harmonic conjugate of X(32992)
X(6656) = {X(2),X(20)}-harmonic conjugate of X(14001)
X(6656) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5025,5), (3,5,1513), (39,626,325), (76,3096,141), (115,6292,3934), (141,3094,6393), (141,5254,76), (194,3314,3933), (626,4045,39)

X(6657) = (SS: a → a⁴ + b²c²) OF X(42)

Barycentrics (a⁴ + b²c²)²*(b⁴ + c⁴ + a²b² + a²c²) : :

As a point on the Euler line, X(6657) has Shinagawa coefficients ((E + F)⁶ + (E + 3F)(E - F)S⁴ - 2(E + F)⁴S², [3(E + F)⁴ - 2(E + F)(E - F)S² - S⁴]S²).

See X(6655).

X(6657) lies on this line: {2,3}

X(6658) = (SS: a → a⁴ + b²c²) OF X(145)

Barycentrics -3*a^4 + a^2*b^2 + b^4 + a^2*c^2 - 3*b^2*c^2 + c^4 : :
X(6658) = 3X(2) - 4X(384)

As a point on the Euler line, X(6658) has Shinagawa coefficients ((E + F)² - 3S²,8S²).

See X(6655).

X(6658) lies on these lines: {2, 3}, {32, 148}, {193, 698}, {2896, 3734}, {3924, 6650}, {6284, 6645}

X(6658) = reflection of X(2) in X(19686)
X(6658) = complement of X(19691)
X(6658) = anticomplement of X(6655)
X(6658) = orthocentroidal-circle-inverse of X(32993)
X(6658) = {X(2),X(4)}-harmonic conjugate of X(32993)
X(6658) = {X(2),X(20)}-harmonic conjugate of X(33260)
X(6658) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3552,2), (382,1003,5025), (439,3832,2)

X(6659) = (SS: a → a⁴ + b²c²) OF X(200)

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

As a point on the Euler line, X(6659) has Shinagawa coefficients ((E + F)⁶ - (E² + 2EF - 31F²)S⁴ + (E + F)⁴S² - S⁶, -[6(E + F)⁴ - 4(E + F)(E - 7F)S² - 10S⁴)]S²).

See X(6655).

X(6659) lies on this line: {2,3}

X(6660) = (SS: a → a⁴ + b²c²) OF X(239)

Barycentrics a²(a⁶ - b⁶ - c⁶ + a²b²c²) : :

As a point on the Euler line, X(6660) has Shinagawa coefficients ((E + F)(E + 3F) - S², -3(E + F)² + S²).

See X(6655).

X(6660) lies on these lines: {2, 3}, {32, 3981}, {39, 1915}, {51, 3398}, {184, 3095}, {256, 1582}, {291, 1580}, {511, 3506}, {525, 669}, {694, 2076}, {1283, 2223}, {1605, 6582}, {1606, 6295}, {1631, 4363}, {2353, 3504}, {3229, 5162}, {3978, 5989}

X(6660) = anticomplement of X(21536)
X(6660) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (22,3148,3), (2479,2480,384)
X(6660) = tangential-isogonal conjugate of X(3511)

X(6661) = (SS: a → a⁴ + b²c²) OF X(551)

Barycentrics 4*a^4 + a^2*b^2 + b^4 + a^2*c^2 + 4*b^2*c^2 + c^4 : :

As a point on the Euler line, X(6661) has Shinagawa coefficients (3(E + F)² - 2S², 3S²).

See X(6655).

X(6661) lies on these lines: {2, 3}, {76, 5306}, {99, 3589}, {141, 3972}, {597, 698}, {1285, 3620}, {3329, 6390}, {3734, 5309}

X(6661) = midpoint of X(2) and X(384)
X(6661) = reflection of X(2) in X(7819)
X(6661) = reflection of X(6656) in X(2)
X(6661) = complement of X(7924)
X(6661) = {X(2),X(4)}-harmonic conjugate of X(33219)
X(6661) = orthocentroidal circle inverse of X(33219)

X(6662) = ISOGONAL CONJUGATE OF X(1614)

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

Let G' be the centroid of the quadrilateral {A,B,C,X(3)}, and let U be the reflection of the cevian triangle of X(3) in G'. Then ABC and U are perspective, and X(6662) is their perspector. The trilinear polar of X(6662) passes through X(1637). (Randy Hutson, February 24, 2015)

X(6662) lies on these lines: {2,6663}, {30,5562}, {140,216}

X(6662) = anticomplement of X(6663)
X(6662) = isotomic conjugate of anticomplement of X(36412)

X(6663) = COMPLEMENT OF X(6662)

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

Let P and Q be the points on the circumcircle whose Steiner lines are tangent to the circumcircle. Then X(6663) = crosssum of P and Q. Let A′B′C′ be the anticevian triangle of X(5). Then X(6663) is the centroid of {A′, B′, C′, X(5)}. (Randy Hutson, February 24, 2015)

X(6663) lies on these lines: {2,6662}, {5,324}, {30,143}

X(6663) = complement of X(6662)

X(6664) = ISOGONAL CONJUGATE OF X(1627)

Barycentrics 1/(a^4 + a^2b^2 + a^2c^2 - b^2c^2) : :

Let G' be the centroid of the quadrilateral {A,B,C,X(6)}, and let U be the reflection of the cevian triangle of X(6) in G'. Then ABC and U are perspective, and X(6664) is their perspector. The trilinear polar of X(6664) passes through X(3005). (Randy Hutson, February 24, 2015)

X(6665) lies on these lines: {2,6665}, {39,698}, {428,524}, {755,6573}

X(6664) = isotomic conjugate of X(7760)
X(6664) = anticomplement of X(6665)

X(6665) = COMPLEMENT OF X(6664)

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

Let A′B′C′ be the anticevian triangle of X(141). Then X(6665) is the centroid of {A′,B′,C′,X(141). (Randy Hutson, February 24, 2015)

X(6665) lies on these lines: {2,6664}, {6,4576}, {698,3934}, {3589,4074}

X(6665) = complement of X(6664)

Centroids of Quadrilaterals {A,B,C,X}: X(6666)-X(6723)

Suppose that X is a point in the plane of a triangle ABC. The centroid, G,′ of {A,B,C,X} is the complement of the complement of X, hence also the midpoint of X and the complement of X, as well as the Kosnita(X,X(2)) point. If X = x : y : z (barycentrics), then G' = 2x + y + z : x + 2y : z : x + y + 2z.

Randy Hutson, February 24, 2015, notes that properties described at X(6600) and X(6601) hold generally, as follows. Let G″ be the complement of X, and let A′B′C′ be the anticevian triangle of G″. The reflection of the cevian triangle of X in G' is perspective to ABC, and the perspector is the anticomplement of the centroid of {A′,B′,C′,G″}. This perspector is also the isogonal conjugate of the TCC-perspector of the isogonal conjugate of G″. (The TCC-perspector of a point is defined in the preamble to X(1601).)

Example 1. X = X(1)
X(596) = perspector of ABC and reflection of cevian triangle of X(1) in centroid of {A,B,C,X(1)}
X(596) = anticomplement of X(4075)
X(596) = isogonal conjugate of TCC-perspector of X(58)
X(4075) = complement of X(596) = centroid of {A′,B′,C′,X(10)), where A′B′C′ = anticevian triangle of X(10)

Example 2. X = X(4)
X(68) = perspector of ABC and reflection of cevian triangle of X(4) in centroid of {A,B,C,X(4)}
X(68) = anticomplement of X(1147)
X(68) = isogonal conjugate of TCC-perspector of X(58)
X(1147) = complement of X(68) = centroid of {A′,B′,C′,X(3)), where A′B′C′ = anticevian triangle of X(3)

Example 8. X = X(8)
X(4) = perspector of ABC and reflection of cevian triangle of X(8) in centroid of {A,B,C,X(8)}
X(4) = anticomplement of X(3)
X(4) = isogonal conjugate of TCC-perspector of X(1)
X(3) = complement of X(4) = centroid of {A′,B′,C′,X(1)), where A′B′C′ = anticevian triangle of X(1)

The appearance of (i,j) in the following list means that the centroid of {A,B,C,X(i)} is X(j):
(1,1125), (2,2), (3,140), (4,5), (5,3628), (6,3589), (7,142), (8,10), (9,6666), (10,3634), (11,6667), (12,6668), (13,6669), (14,6690), (15,6671), (16,6672), (17,6673), (18,6674), (20,3), (21, 6675), (22,6676), (23,468), (25,6677), (27,6678), (31,6679), (32,6680), (36,6681), (37,4698), (38,6682), (39,6683), (40,6684), (42,6685), (43,6686), (44,6687), (51,6688), (52,5462), (54,6689), (55,6690), (56,6691), (57,6692), (58,6693), (61,6694), (62,6695), (63,5745), (64,6696), (65,3812), (66,6697), (67,6698), (68,5449), (69,141), (72,5044), (74,6699), (75,3739), (76,3934), (78,6700), (79,6701), (80,6702), (81,6703), (83,2704), (84,6705), (85,6706), (86,6707), (92,6708), (95,6709), (98,6036), (99,620), (100,3035), (101,6710), (102,6711), (103,6712), (104,6713), (105,6714), (106,6715), (107,6716), (108,6717), (109,6718), (110,5972), (111,6719), (112,6720), (114,6721), (115,6722), (125,6723)

X(6666) = CENTROID OF {A,B,C,X(9)}

Barycentrics 2*a^2 - 3*a*b + b^2 - 3*a*c - 2*b*c + c^2 : :
X(6666) = 9 X[2] - X[7] = 3 X[2] + X[9] = X[7] + 3 X[9] = X[7] - 3 X[142] = 5 X[9] - X[144] = 15 X[2] + X[144] = 5 X[142] + X[144] = 5 X[7] + 3 X[144] = 5 X[1698] - X[2550] = X[3243] - 5 X[3616] = 7 X[3624] + X[5223] = 3 X[3740] + X[5572] = X[3243] + 3 X[5686] = 5 X[3616] + 3 X[5686] = 5 X[631] - X[5732] = 13 X[5067] - X[5735] = 7 X[3090] + X[5759] = 7 X[3526] + X[5779] = 5 X[1656] - X[5805] = 5 X[631] + 3 X[5817] = X[5732] + 3 X[5817] = 7 X[144] - 15 X[6172] = 7 X[9] - 3 X[6172] = 7 X[2] + X[6172] = 7 X[142] + 3 X[6172] = 7 X[7] + 9 X[6172] = 5 X[7] - 9 X[6173] = 5 X[142] - 3 X[6173] = 5 X[2] - X[6173] = 5 X[9] + 3 X[6173] = X[144] + 3 X[6173] = 5 X[6172] + 7 X[6173]

X(6666) lies on these lines: {1, 4878}, {2, 7}, {5, 516}, {10, 1001}, {11, 6594}, {37, 3008}, {44, 3664}, {45, 3663}, {69, 3707}, {75, 2325}, {140, 971}, {239, 4464}, {319, 3686}, {344, 2321}, {405, 4304}, {480, 4423}, {518, 1125}, {528, 3828}, {551, 3940}, {631, 5732}, {673, 1268}, {899, 4343}, {950, 5047}, {954, 1210}, {958, 4315}, {1212, 1323}, {1449, 5308}, {1479, 1698}, {1656, 5805}, {1743, 4648}, {1890, 5142}, {2346, 5284}, {2551, 5726}, {3090, 5759}, {3161, 4659}, {3243, 3616}, {3247, 5222}, {3358, 6260}, {3526, 5779}, {3601, 5809}, {3624, 5223}, {3628, 5762}, {3731, 4000}, {3739, 4422}, {3740, 5572}, {3875, 4029}, {3879, 4700}, {3950, 4361}, {3973, 4644}, {3986, 4657}, {4298, 5302}, {4326, 5218}, {4395, 4681}, {4419, 4859}, {4708, 5845}, {5067, 5735}, {5220, 5542}, {5252, 5795}

X(6666) = midpoint of X(i) and X(j) for these {i,j}: {9,142}, {10,100}, {11,6594}, {3358, 6260}, {3950,4361}, {5220,5542}
X(6666) = reflection of X(3826) in X(3634)
X(6666) = isotomic conjugate of X(32015)
X(6666) = complement of X(142)
X(6666) = {X(2),X(9)}-harmonic conjugate of X(142)

X(6667) = CENTROID OF {A,B,C,X(11)}

Barycentrics 2*a^3 - 2*a^2*b - 3*a*b^2 + 3*b^3 - 2*a^2*c + 8*a*b*c - 3*b^2*c - 3*a*c^2 - 3*b*c^2 + 3*c^3 : :
X(6667) = {3 X[2] + X[11] = 9 X[2] - X[100] = 3 X[11] + X[100] = 5 X[11] - X[149] = 15 X[2] + X[149] = 5 X[100] + 3 X[149] = X[119] - 5 X[1656] = X[1145] - 5 X[1698] = X[100] - 3 X[3035] = X[149] + 5 X[3035] = X[104] + 7 X[3090] = X[1317] - 5 X[3616] = X[80] + 7 X[3624] = 3 X[3742] - X[5083] = X[5080] + 3 X[5298] = 7 X[100] - 3 X[6154] = 7 X[3035] - X[6154] = 7 X[11] + X[6154] = 7 X[149] + 5 X[6154] = 5 X[100] - 9 X[6174] = 5 X[3035] - 3 X[6174] = 5 X[2] - X[6174] = 5 X[11] + 3 X[6174] = X[149] + 3 X[6174]

X(6667) lies on these lines: {1, 3036}, {2, 11}, {5, 2829}, {10, 1387}, {37, 3055}, {80, 3624}, {104, 3090}, {119, 1656}, {140, 3825}, {142, 5851}, {499, 956}, {529, 3814}, {547, 3822}, {632, 5248}, {908, 5852}, {952, 1125}, {1145, 1698}, {1317, 3616}, {1737, 5855}, {1768, 5437}, {2800, 3812}, {2801, 3848}, {2802, 3634}, {3589, 5848}, {3614, 5253}, {3742, 5083}, {3911, 5087}, {4187, 4999}, {4193, 5433}, {4996, 5047}, {5080, 5298}, {5187, 5204}, {5220, 5328}

X(6667) = midpoint of X(i) and X(j) for these {i,j}: {1,3036}, {5,6713}, {10,1387}, {11,3035}, {1125,6702}, {3911,5087}
X(6667) = complement of X(3035)

X(6668) = CENTROID OF {A,B,C,X(12)}

Barycentrics -2*a^4 + 5*a^2*b^2 - 3*b^4 + 4*a^2*b*c + 2*a*b^2*c + 5*a^2*c^2 + 2*a*b*c^2 + 6*b^2*c^2 - 3*c^4 : :
X(6668) = 3 X[2] + X[12] = 9 X[2] - X[2975] = 3 X[12] + X[2975] = X[2975] - 3 X[4999]

X(6668) lies on these lines: {2, 12}, {5, 5248}, {10, 5855}, {21, 3614}, {140, 3822}, {142, 5852}, {404, 5326}, {442, 3035}, {498, 2886}, {547, 3825}, {758, 3634}, {952, 1125}, {1001, 3090}, {1656, 3816}, {2476, 5432}, {3085, 3813}, {3295, 3829}, {3589, 5849}, {5141, 6284}

X(6668) = midpoint of X(12) and X(4999)
X(6668) = complement of X(4999)

X(6669) = CENTROID OF {A,B,C,X(13)}

Barycentrics 2*a^4 - 7*a^2*b^2 + 5*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 5*c^4 - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6669) = 3 X[2] + X[13] = 9 X[2] - X[616] = 3 X[13] + X[616] = X[616] - 3 X[618] = X[13] - 3 X[5459] = X[618] + 3 X[5459] = X[616] + 9 X[5459] = 5 X[616] - 9 X[5463] = 5 X[618] - 3 X[5463] = 5 X[2] - X[5463] = 5 X[5459] + X[5463] = 5 X[13] + 3 X[5463] = X[617] + 3 X[5469] = X[99] + 3 X[5470] = 5 X[631] - X[5473] = 5 X[1656] - X[5617]}

X(6669) lies on these lines: {2, 13}, {3, 5478}, {17, 299}, {99, 5470}, {115, 619}, {395, 1506}, {396, 533}, {397, 629}, {531, 5461}, {542, 547}, {617, 5469}, {631, 5473}, {1656, 5617}, {3642, 6300}

X(6669) = midpoint of X(i) and X(j) for these {i,j}: {2,5459}, {3,5478}, {13,618}, {115,619}, {396,623}, {624,6108}
X(6669) = reflection of X(6670) in X(67822)
X(6669) = complement of X(618)

X(6670) = CENTROID OF {A,B,C,X(14)}

Barycentrics 2*a^4 - 7*a^2*b^2 + 5*b^4 - 7*a^2*c^2 - 10*b^2*c^2 + 5*c^4 + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6670) = 3 X[2] + X[14] = 9 X[2] - X[617] = 3 X[14] + X[617] = X[617] - 3 X[619] = X[14] - 3 X[5460] = X[619] + 3 X[5460] = X[617] + 9 X[5460] = 5 X[617] - 9 X[5464] = 5 X[619] - 3 X[5464] = 5 X[2] - X[5464] = 5 X[5460] + X[5464] = 5 X[14] + 3 X[5464] = X[99] + 3 X[5469] = X[616] + 3 X[5470] = 5 X[631] - X[5474] = 5 X[1656] - X[5613]

X(6670) lies on these lines: {2, 14}, {3, 5479}, {18, 298}, {99, 5469}, {115, 618}, {395, 532}, {396, 1506}, {398, 630}, {530, 5461}, {542, 547}, {616, 5470}, {631, 5474}, {1656, 5613}, {3643, 6301}

X(6670) = midpoint of X(i) and X(j) for these {i,j}: {2, 5469), {3,5479}, {14,619}, {115,618}, {395,624}, {623,6109}
X(6670) = reflection of X(6669) in X(6722)
X(6670) = complement of X(619)

X(6671) = CENTROID OF {A,B,C,X(15)}

Barycentrics 3*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4) - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6671) = 3 X[2] + X[15] = 9 X[2] - X[621] = 3 X[15] + X[621] = X[621] - 3 X[623] = X[5318] - 3 X[5459] = 7 X[3526] + X[5611]

X(6671) lies on these lines: {2, 14}, {17, 622}, {61, 302}, {140, 143}, {187, 624}, {303, 636}, {396, 532}, {627, 3412}, {630, 6292}, {3526, 5611}, {5318, 5459}

X(6671) = midpoint of X(i) and X(j) for these {i,j}: {15,623}, {187,624}, {396,618}, {619,6109}
X(6671) = complement of X(623)
X(6671) = isogonal conjugate of X(34321)
X(6671) = cevapoint of X(396) and X(15802)
X(6671) = crosspoint of X(302) and X(8838)
X(6671) = crosssum of X(8603) and X(21461)
X(6671) = Kosnita(X(15),X(13)) point

X(6672) = CENTROID OF {A,B,C,X(16)}

Barycentrics 3*(2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4) + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6672) = 3 X[2] + X[16] = 9 X[2] - X[622] = 3 X[16] + X[622] = X[622] - 3 X[624] = X[5321] - 3 X[5460] = 7 X[3526] + X[5615]

X(6672) lies on these lines: {2, 13}, {18, 621}, {62, 303}, {140, 143}, {187, 623}, {302, 635}, {395, 533}, {628, 3411}, {629, 6292}, {3526, 5615}, {5321, 5460}

X(6672) = midpoint of X(i) and X(j) for these {i,j}: {16,624}, {187,623}, {395,619}, {618,6108}
X(6672) = complement of X(624)
X(6672) = isogonal conjugate of X(34322)
X(6672) = cevapoint of X(395) and X(15778)
X(6672) = crosspoint of X(303) and X(8836)
X(6672) = crosssum of X(8604) and X(21462)
X(6672) = Kosnita(X(16),X(14)) point

X(6673) = CENTROID OF {A,B,C,X(17)}

Barycentrics 6*a^4 - 13*a^2*b^2 + 7*b^4 - 13*a^2*c^2 - 14*b^2*c^2 + 7*c^4 - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6673) = 3 X[2] + X[17] = 9 X[2] - X[627] = 3 X[17] + X[627] = X[627] - 3 X[629]

X(6673) lies on these lines: {{2, 17}, {3589, 5965}

X(6673) = midpoint of X(17) and X(629)
X(6673) = complement of X(629)

X(6674) = CENTROID OF {A,B,C,X(18)}

Barycentrics 6*a^4 - 13*a^2*b^2 + 7*b^4 - 13*a^2*c^2 - 14*b^2*c^2 + 7*c^4 + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6674) = 3 X[2] + X[18] = 9 X[2] - X[628] = 3 X[18] + X[628] = X[628] - 3 X[630]

X(6674) lies on these lines: {2, 18}, {3589, 5965}

X(6674) = midpoint of X(18) and X(630)
X(6674) = complement of X(630)

X(6675) = CENTROID OF {A,B,C,X(21)}

Barycentrics 2*a^4 - 3*a^2*b^2 + b^4 - 4*a^2*b*c - 4*a*b^2*c - 3*a^2*c^2 - 4*a*b*c^2 - 2*b^2*c^2 + c^4 : :
X(6675) = 3 X[2] + X[21] = 9 X[2] - X[2475] = 3 X[442] - X[2475] = 3 X[21] + X[2475] = X[191] + 7 X[3624] = 7 X[3624] - X[3649] = 5 X[631] - X[3651] = 5 X[1698] + 3 X[5426] = 5 X[632] - X[5499] = 5 X[2475] - 9 X[6175] = 5 X[442] - 3 X[6175] = 5 X[2] - X[6175] = 5 X[21] + 3 X[6175]

X(6675) lies on these lines: {1, 5791}, {2, 3}, {11, 5259}, {12, 5251}, {35, 3925}, {57, 191}, {63, 6147}, {72, 5719}, {79, 4679}, {141, 5138}, {142, 3647}, {274, 6390}, {284, 1213}, {392, 5771}, {495, 958}, {496, 1001}, {498, 3820}, {499, 4423}, {620, 2795}, {758, 942}, {1212, 3002}, {1698, 1837}, {1724, 5718}, {1834, 4653}, {1901, 4877}, {2771, 5972}, {2886, 5248}, {3035, 3634}, {3487, 3927}, {3576, 5787}, {3589, 4260}, {3616, 5730}, {3712, 4647}, {3819, 5482}, {3824, 4292}, {3916, 5249}, {3940, 5703}, {4369, 5592}, {5087, 5122}, {5283, 5305}, {5436, 5705}, {5550, 5708}, {5709, 5886}, {5768, 5789}

X(6675) = midpoint of X(i) and X(j) for these {i,j}: {5,5428}, {21,442}, {191,3649}
X(6675) = complement of X(442)
X(6675) = center of bicevian ellipse of X(2) and X(21)

X(6676) = CENTROID OF {A,B,C,X(22)}

Barycentrics (a^2 - b^2 - c^2)*(2*a^4 + a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :
X(6676) = 3 X[2] + X[22] = X[378] - 5 X[631]

As a point on the Euler line, X(6676) has Shinagawa coefficients (E + 3F, -E - F).

Three constructions of X(6676) are given at Hyacinthos 24128, Hyacinthos 24100, and Hyacinthos 24098. (Antreas Hatzipolakis, August 29, 2016)

X(6676) lies on these lines: {2, 3}, {69, 3167}, {141, 206}, {154, 1352}, {184, 343}, {216, 230}, {305, 6390}, {577, 3815}, {612, 1062}, {614, 1060}, {1038, 5272}, {1040, 5268}, {1194, 5305}, {1353, 6515}, {1447, 6356}, {1578, 5420}, {1579, 5418}, {1899, 3796}, {2781, 3819}, {3580, 5012}, {3589, 5943}, {5158, 5306}

X(6676) = midpoint of X(i) and X(j) for these {i,j}: {22,427}, {184,343}
X(6676) = complement of X(427)
X(6676) = isotomic conjugate of polar conjugate of X(7745)
X(6676) = crossdifference of every pair of points on the radical axis of the Brocard circle and tangential circle
X(6676) = circumcircle-inverse of X(37972)
X(6676) = homothetic center of orthocevian triangle of X(2) and medial triangle

X(6677) = CENTROID OF {A,B,C,X(25)}

Barycentrics 2*a^6 - a^4*b^2 - 2*a^2*b^4 + b^6 - a^4*c^2 + 8*a^2*b^2*c^2 - b^4*c^2 - 2*a^2*c^4 - b^2*c^4 + c^6 : :
X(6677) = 3 X[2] + X[25] = 9 X[2] - X[1370] = 3 X[1368] - X[1370] = 3 X[25] + X[1370] = 3 X[6090] + X[6515]

As a point on the Euler line, X(6677) has Shinagawa coefficients (E - 3F, E + F).

X(6677) lies on these lines: {2, 3}, {6, 6387}, {182, 1660}, {230, 800}, {343, 5651}, {1196, 5305}, {1353, 3167}, {1627, 5913}, {2393, 3589}, {2790, 6036}, {3815, 5065}, {5943, 5972}, {6090, 6515}

X(6677) = midpoint of X(i) and X(j) for these {i,j}: {{3,1596}, {5,6644}, 25,1368}
X(6677) = complement of X(1368)
X(6677) = circumcircle-inverse of X(37973)
X(6677) = X(57)-of-submedial-triangle if ABC is acute

X(6678) = CENTROID OF {A,B,C,X(27)}

Barycentrics 2*a^6 - a^4*b^2 + 2*a^3*b^3 - 2*a^2*b^4 - 2*a*b^5 + b^6 + 2*a^3*b^2*c + 2*a^2*b^3*c - 2*a*b^4*c - 2*b^5*c - a^4*c^2 + 2*a^3*b*c^2 + 8*a^2*b^2*c^2 + 4*a*b^3*c^2 - b^4*c^2 + 2*a^3*c^3 + 2*a^2*b*c^3 + 4*a*b^2*c^3 + 4*b^3*c^3 - 2*a^2*c^4 - 2*a*b*c^4 - b^2*c^4 - 2*a*c^5 - 2*b*c^5 + c^6 : :
X(6678) = 3 X[2] + X[27] = 9 X[2] - X[3151] = 3 X[440] - X[3151] = 3 X[27] + X[3151]

As a point on the Euler line, X(6678) has Shinagawa coefficients (E - 3F + $bc$, E + F + $bc$).

X(6678) lies on these lines: {2, 3}, {57, 1723}, {1427, 3002}, {1730, 5755}, {3739, 5745}, {3925, 5285}

X(6678) = midpoint of X(27) and X(440)
X(6678) = complement of X(440)

X(6679) = CENTROID OF {A,B,C,X(31)}

Barycentrics 2*a^3 + b^3 + c^3 : :
X(6679) = 3 X[2] + X[31] = 5 X[1698] - X[4680] = 9 X[2] - X[6327] = 3 X[2887] - X[6327] = 3 X[31] + X[6327]

X(6679) lies on these lines: {1, 4438}, {2, 31}, {6, 3771}, {10, 1104}, {55, 4085}, {226, 4672}, {306, 3791}, {333, 3775}, {674, 3589}, {734, 3934}, {744, 3739}, {758, 942}, {902, 4972}, {1215, 3011}, {1698, 4680}, {1707, 4655}, {2308, 3936}, {3052, 4660}, {3550, 4429}, {3662, 4650}, {3683, 4425}, {3687, 4974}, {3712, 4970}, {3772, 3923}, {3773, 4362}, {3792, 3794}, {3821, 4640}, {3944, 4676}, {4697, 5249}

X(6675) = midpoint of X(i) and X(j) for these {i,j}: {31,2887}, {306,3791}
X(6679) = isogonal conjugate of X(38831)
X(6679) = complement of X(2887)
X(6679) = complementary conjugate of complement of X(38813)
X(6679) = crosssum of X(6) and X(3778)
X(6679) = trilinear product X(57)*X(4168)
X(6679) = trilinear quotient X(4168)/X(9)
X(6679) = barycentric product X(7)*X(4168)
X(6679) = barycentric quotient X(4168)/X(8)

X(6680) = CENTROID OF {A,B,C,X(32)}

Barycentrics 2*a^4 + b^4 + c^4 : :
X(6680) = 3 X[2] + X[32] = 9 X[2] - X[315] = 3 X[32] + X[315] = X[315] - 3 X[626] = 5 X[1698] - X[4769] = 5 X[3618] - X[5028] = X[3933] + 3 X[5306] = X[1975] + 3 X[5309]

X(6680) lies on these lines: {2, 32}, {3, 4045}, {5, 2794}, {6, 3788}, {39, 620}, {114, 3398}, {115, 384}, {140, 143}, {194, 5355}, {230, 736}, {325, 5007}, {538, 5305}, {543, 5254}, {746, 3739}, {760, 1125}, {1235, 6103}, {1698, 4769}, {1975, 5309}, {2031, 3055}, {3314, 6179}, {3491, 5972}, {3618, 5028}, {3734, 3767}, {3926, 5319}, {3933, 5306}, {3972, 5025}

X(6680) = midpoint of X(32) and X(626j
X(6680) = complement of X(626)
X(6680) = complementary conjugate of complement of X(38826)

X(6681) = CENTROID OF {A,B,C,X(36)}

Barycentrics 2*a^4 - 3*a^2*b^2 + b^4 + 2*a^2*b*c + a*b^2*c - 3*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 + c^4 : :
X(6681) = 3 X[2] + X[36] = 5 X[631] - X[2077] = X[100] + 3 X[3582] = X[484] + 7 X[3624] = 2 X[3660] + X[3678] = X[80] + 3 X[4881] = 3 X[551] - X[5048] = 9 X[2] - X[5080] = 3 X[3814] - X[5080] = 3 X[36] + X[5080] = 2 X[3634] + X[5126] = X[5057] + 3 X[5131] = 5 X[1698] - X[5176] = X[3421] + 3 X[5193] = X[3245] + 11 X[5550]

X(6681) lies on these lines: {2, 36}, {3, 3825}, {10, 1319}, {80, 4881}, {100, 3582}, {140, 517}, {214, 1737}, {468, 1878}, {474, 3841}, {484, 3624}, {499, 3434}, {519, 3035}, {549, 3816}, {551, 5048}, {631, 2077}, {758, 3911}, {908, 4973}, {1001, 5054}, {1698, 5176}, {3245, 5550}, {3421, 5193}, {3634, 4999}, {3660, 3678}, {4187, 5267}, {5057, 5131}, {5087, 5122}, {5437, 5535}

X(6681) = midpoint of X(i) and X(j) for these {i,j}: {10,1319}, {36,3814}, {214,1737}, {908,4973}, {5087,5122}, {5128,5126}
X(6681) = reflection of X(5123) in X(3634)
X(6681) = complement of X(3814)

X(6682) = CENTROID OF {A,B,C,X(38)}

Barycentrics a^2*b + 2*a*b^2 + a^2*c + b^2*c + 2*a*c^2 + b*c^2 : :
X(6682) = 3 X[2] + X[38] = 3 X[3666] + X[3706] = X[3706] - 3 X[3741] = 5 X[1698] - X[4692] = 3 X[3666] - X[4970] = 3 X[3741] + X[4970]

X(6682) lies on these lines: {1, 3769}, {2, 38}, {10, 3752}, {11, 4425}, {37, 3840}, {63, 4672}, {325, 3847}, {333, 4974}, {341, 1698}, {714, 3739}, {740, 3666}, {758, 942}, {846, 4432}, {899, 4981}, {1107, 3229}, {1150, 3791}, {1647, 6536}, {2886, 3821}, {3005, 4369}, {3218, 4697}, {3634, 6532}, {3687, 3775}, {3816, 4364}, {3848, 4698}, {3920, 4434}, {3944, 4389}, {3989, 4358}, {4038, 5625}, {4085, 4847}, {4104, 4407}, {4732, 4850}

X(6682) = midpoint of X(i) and X(j) for these {i,j}: {38,1215}, {3666,3741}, {3706,4970}
X(6682) = complement of X(1215)

X(6683) = CENTROID OF {A,B,C,X(39)}

Barycentrics 3*a^2*b^2 + 3*a^2*c^2 + 2*b^2*c^2 : :
X(6683) = 3 X[2] + X[39] = 9 X[2] - X[76] = 3 X[39] + X[76] = 5 X[39] - X[194] = 15 X[2] + X[194] = 5 X[76] + 3 X[194] = 3 X[262] + 5 X[631] = X[3095] + 7 X[3526] = X[76] - 3 X[3934] = X[194] + 5 X[3934] = 5 X[3618] - X[5052] = 5 X[631] - X[5188] = 3 X[262] + X[5188] = 5 X[1656] - X[6248]

X(6683) lies on these lines: {2, 39}, {5, 4045}, {83, 187}, {140, 143}, {182, 3202}, {262, 631}, {325, 6292}, {385, 5041}, {620, 2023}, {625, 1506}, {626, 3815}, {726, 4698}, {730, 3634}, {1078, 3329}, {1656, 6248}, {2782, 3628}, {3095, 3526}, {3618, 5052}, {3734, 5013}

X(6683) = midpoint of X(i) and X(j) for these {i,j}: {39,3934}, {620,2023}, {625,2021}
X(6683) = complement of X(3934)

X(6684) = CENTROID OF {A,B,C,X(40)}

Barycentrics 2*a^4 + a^3*b - 3*a^2*b^2 - a*b^3 + b^4 + a^3*c - 2*a^2*b*c + a*b^2*c - 3*a^2*c^2 + a*b*c^2 - 2*b^2*c^2 - a*c^3 + c^4 : :
X(6684) = 3 X[2] + X[40] = X[4] + 3 X[165] = 3 X[10] - X[355] = 3 X[3] + X[355] = X[1] - 5 X[631] = 9 X[2] - X[962] = 3 X[946] - X[962] = 3 X[40] + X[962] = 3 X[210] + X[1071] = 3 X[549] - X[1385] = 3 X[551] - X[1482] = 9 X[549] - X[1483] = 3 X[1385] - X[1483] = X[4] - 5 X[1698] = 3 X[165] + 5 X[1698] = 3 X[1699] - 7 X[3090] = X[8] + 7 X[3523] = X[944] - 9 X[3524] = 7 X[3523] - 3 X[3576] = X[8] + 3 X[3576] = 11 X[3525] - 7 X[3624] = 4 X[3530] + X[3626] = X[3579] + 2 X[3634] = X[1482] + 3 X[3654] = 3 X[3524] + X[3679] = X[944] + 3 X[3679] = 5 X[1656] - 3 X[3817] = 3 X[3] - X[4297] = 3 X[10] + X[4297] = 7 X[3526] - X[4301] = X[1482] - 9 X[5054] = X[551] - 3 X[5054] = X[3654] + 3 X[5054] = X[3416] + 3 X[5085] = 5 X[1656] + X[5493] = 3 X[3817] + X[5493] = X[20] + 3 X[5587] = 11 X[3525] - 3 X[5603] = 7 X[3624] - 3 X[5603] = X[1] + 3 X[5657] = 5 X[631] + 3 X[5657] = 3 X[549] + X[5690] = X[1483] + 3 X[5690] = 3 X[376] + X[5691] = 5 X[3876] - X[5693] = 5 X[3617] + 3 X[5731] = X[5534] + 3 X[5770] = 3 X[3740] - X[5777] = 19 X[4438] - 13 X[5790] = 5 X[355] - 9 X[5790] = 5 X[10] - 3 X[5790] = 5 X[3] + 3 X[5790] = 5 X[4297] + 9 X[5790] = X[2951] + 3 X[5817] = X[5691] - 5 X[5818] = 3 X[376] + 5 X[5818] = 5 X[3617] - X[5881] = 3 X[5731] + X[5881] = 7 X[3523] - X[5882] = 3 X[3576] - X[5882] = 7 X[3526] - 3 X[5886] = X[4301] - 3 X[5886] = 3 X[1125] - 2 X[5901] = 3 X[140] - X[5901] = 3 X[1699] + X[6361] = 7 X[3090] + X[6361]

X(6684) lies on these lines: {1, 631}, {2, 40}, {3, 10}, {4, 165}, {5, 516}, {8, 3523}, {9, 1158}, {12, 1155}, {20, 5587}, {21, 2077}, {30, 3828}, {35, 950}, {43, 581}, {46, 226}, {55, 1210}, {57, 3085}, {63, 5552}, {65, 5432}, {71, 1715}, {72, 5884}, {84, 5273}, {100, 5178}, {104, 5258}, {109, 3074}, {117, 122}, {119, 3647}, {140, 517}, {142, 5709}, {171, 580}, {182, 5847}, {201, 1735}, {210, 1071}, {212, 1771}, {376, 5691}, {390, 5704}, {406, 1753}, {468, 1902}, {474, 3428}, {495, 4298}, {499, 5119}, {518, 5771}, {519, 549}, {551, 1482}, {553, 3336}, {572, 3686}, {573, 5750}, {601, 1724}, {602, 5264}, {899, 4300}, {912, 3678}, {936, 6261}, {938, 5281}, {944, 3524}, {952, 3530}, {960, 2800}, {997, 5837}, {1056, 3361}, {1064, 3216}, {1276, 5242}, {1277, 5243}, {1329, 4640}, {1503, 3844}, {1512, 5251}, {1532, 3683}, {1571, 3767}, {1656, 3817}, {1697, 3086}, {1699, 3090}, {1702, 3069}, {1703, 3068}, {1766, 5257}, {1785, 1940}, {1837, 4304}, {1838, 2954}, {1848, 6197}, {2093, 3485}, {2550, 5705}, {2551, 6256}, {2792, 3454}, {2801, 4015}, {2951, 5817}, {3057, 5433}, {3149, 4413}, {3245, 5443}, {3333, 5435}, {3339, 3487}, {3359, 3452}, {3416, 5085}, {3436, 4652}, {3515, 5090}, {3525, 3624}, {3526, 4301}, {3617, 5731}, {3635, 5844}, {3655, 4669}, {3701, 3977}, {3740, 5777}, {3755, 5292}, {3824, 5762}, {3876, 5693}, {4295, 5128}, {4303, 4551}, {4305, 5727}, {4311, 5204}, {4312, 5714}, {4314, 5722}, {4357, 6211}, {4512, 5084}, {4847, 5687}, {4999, 5836}, {5044, 6001}, {5082, 5231}, {5131, 5270}, {5176, 5303}, {5183, 5326}, {5249, 5535}, {5259, 5537}, {5442, 5563}, {5534, 5770}, {5542, 5708}, {5715, 5759}, {5812, 5880}

X(6684) = midpoint of X(i) and X(j) for these {i,j}: {3,10}, {5,3579}, {8,5882}, {40,946}, {72,5884}, {355,4297}, {551,3654}, {1158,6260}, {1385,5690}, {3359,3452}, {3655,4669}
X(6684) = reflection of X(i) in X(j) for thesse (i,j): (5,3634), (1125,140)
X(6684) = complement of X(946)
X(6684) = centroid of {X(4),A′,B′,C′}, where A′B′C′ = excentral triangle
X(6684) = homothetic center of anti-Euler triangle and cross-triangle of ABC and Aquila triangle

X(6685) = CENTROID OF {A,B,C,X(42)}

Barycentrics 2*a^2*b + a*b^2 + 2*a^2*c + b^2*c + a*c^2 + b*c^2 : :
X(6685) = 3 X[2] + X[42]

X(6685) lies on these lines: {1, 2}, {35, 4203}, {37, 6375}, {171, 4279}, {181, 3911}, {192, 4135}, {226, 1403}, {312, 3993}, {321, 4970}, {350, 4021}, {516, 2051}, {674, 3589}, {726, 1215}, {908, 4425}, {968, 4011}, {984, 4090}, {994, 3919}, {1155, 4697}, {1575, 2092}, {1962, 4358}, {2886, 4085}, {2887, 5718}, {3683, 4759}, {3740, 3842}, {3745, 4434}, {3758, 4650}, {3791, 4991}, {3847, 4026}, {3952, 3989}, {4260, 5745}, {4640, 4672}, {4704, 4903}

X(6685) = midpoint of X(i) and X(j) for these {i,j}: {42,3741}, {321,4970}, {1215,3666}
X(6685) = complement of X(3741)

X(6686) = CENTROID OF {A,B,C,X(43)}

Barycentrics 2*a^2*b + a*b^2 + 2*a^2*c - 4*a*b*c + b^2*c + a*c^2 + b*c^2 : :
X(6686) =3 X[2] + X[43] = 3 X[3752] + X[3967]

X(6686) lies on these lines: {1, 2}, {726, 3752}, {982, 4090}, {1401, 3911}, {3210, 4135}, {3452, 3821}, {3816, 4085}, {3971, 4850}, {4358, 4970}, {4640, 4759}

X(6686) = midpoint of X(i) and X(j) for these {i,j}: {10,995}, {43,3840}, {982,4090}, {3210,4135}
X(6686) = complement of X(3840)

X(6687) = CENTROID OF {A,B,C,X(44)}

Barycentrics 4*a^2 - 3*a*b + 2*b^2 - 3*a*c - 2*b*c + 2*c^2 : :
X(6687) = 3 X[2] + X[44] = 9 X[2] - X[320] = 3 X[44] + X[320] = X[8] + 3 X[1279] = 3 X[238] + 5 X[1698] = X[2325] + 3 X[3008] = 5 X[1698] - 3 X[3823] = X[320] - 3 X[3834] = X[1266] + 3 X[4370] = 3 X[3008] - X[4395] = X[2325] - 3 X[4422] = X[4395] + 3 X[4422] = 3 X[1086] + X[4480] = 3 X[239] + X[4727] = 7 X[3622] - 3 X[4864] = 3 X[3912] + X[4969]

X(6687) lies on these lines: {2, 44}, {8, 1279}, {10, 3246}, {141, 3707}, {238, 1698}, {239, 4727}, {344, 4852}, {518, 1125}, {536, 2325}, {752, 3634}, {1086, 4480}, {1266, 4370}, {2321, 4405}, {3622, 4864}, {3759, 4889}, {3912, 4725}, {4361, 4873}

X(6687) = midpoint of X(i) and X(j) for these {i,j}: {10,3246}, {44,3834}, {238,3823}, {2325,4395}, {3008,4422}
X(6687) = complement of X(3834)

X(6688) = CENTROID OF {A,B,C,X(51)}

Barycentrics a^2*(a^2*b^2 - b^4 + a^2*c^2 + 8*b^2*c^2 - c^4) : :
X(6688) = 3 X[2] + X[51] = X[51] - 9 X[373] = X[2] + 3 X[373] = X[389] + 5 X[1656] = 9 X[2] - X[2979] = 3 X[51] + X[2979] = 7 X[51] - 3 X[3060] = 7 X[2] + X[3060] = 7 X[2979] + 9 X[3060] = X[2979] - 3 X[3819] = 9 X[373] + X[3819] = 3 X[3060] + 7 X[3819] = 5 X[2979] - 9 X[3917] = 5 X[3819] - 3 X[3917] = 5 X[2] - X[3917] = 15 X[373] + X[3917] = 5 X[51] + 3 X[3917] = 5 X[3060] + 7 X[3917] = X[185] + 11 X[5056] = X[52] + 11 X[5070] = 5 X[632] + X[5446] = 2 X[3628] + X[5462] = 13 X[5067] - X[5562] = 5 X[51] - 9 X[5640] = 5 X[373] - X[5640] = 5 X[2] + 3 X[5640] = X[3917] + 3 X[5640] = 5 X[3819] + 9 X[5640] = 7 X[3917] - 15 X[5650] = 7 X[3819] - 9 X[5650] = 7 X[2] - 3 X[5650] = 7 X[373] + X[5650] = X[3060] + 3 X[5650] = 7 X[5640] + 5 X[5650] = 7 X[51] + 9 X[5650] = 7 X[3090] + X[5890] = 5 X[1656] - X[5891] = 7 X[3090] - X[5907] = X[3060] - 7 X[5943] = 3 X[5640] - 5 X[5943] = X[51] - 3 X[5943] = 3 X[373] - X[5943] = X[3819] + 3 X[5943] = X[3917] + 5 X[5943] = 3 X[5650] + 7 X[5943] = X[2979] + 9 X[5943]

X(6688) lies on these lines: {2, 51}, {5, 2883}, {6, 5644}, {25, 5092}, {52, 5070}, {154, 182}, {185, 5056}, {375, 2810}, {389, 1656}, {394, 5097}, {632, 5446}, {1154, 3628}, {1611, 5039}, {1915, 2030}, {2390, 3812}, {2393, 3589}, {3030, 3750}, {3090, 5890}, {3292, 5643}, {5067, 5562}, {5422, 5651}

X(6688) = midpoint of X(i) and X(j) for these {i,j}: {2,5943}, {5,5892}, {51, 3819}, {375,3742}, {389,5891}, {5890,5907}
X(6688) = complement of X(3819)

X(6689) = CENTROID OF {A,B,C,X(54)}

Barycentrics 2*a^10 - 6*a^8*b^2 + 5*a^6*b^4 + a^4*b^6 - 3*a^2*b^8 + b^10 - 6*a^8*c^2 + 4*a^6*b^2*c^2 + 5*a^4*b^4*c^2 - 3*b^8*c^2 + 5*a^6*c^4 + 5*a^4*b^2*c^4 + 6*a^2*b^4*c^4 + 2*b^6*c^4 + a^4*c^6 + 2*b^4*c^6 - 3*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(6689) = 3 X[2] + X[54] = 5 X[632] + X[1493] = 9 X[2] - X[2888] = 3 X[1209] - X[2888] = 3 X[54] + X[2888] = X[195] + 7 X[3526] = 5 X[1656] - X[6288]

X(6689) lies on these lines: {2, 54}, {3, 3574}, {5, 5944}, {140, 389}, {141, 575}, {195, 394}, {973, 5462}, {1656, 6288}, {2917, 6642}, {3589, 6153}, {3628, 5972}, {4999, 5044}

X(6689) = midpoint of X(i) and X(j) for these {i,j}: {3,3574}, {54,1209}, {973,5462}
X(6689) = complement of X(1209)

X(6690) = CENTROID OF {A,B,C,X(55)}

Barycentrics 2*a^3 - 2*a^2*b - a*b^2 + b^3 - 2*a^2*c - b^2*c - a*c^2 - b*c^2 + c^3 : :
X(6690) = 3 X[2] + X[55] = 5 X[1698] - X[3419] = 5 X[631] - X[3428] = 9 X[2] - X[3434] = 3 X[2886] - X[3434] = 3 X[55] + X[3434] = X[2099] - 5 X[3616] = 7 X[3624] + X[5119] = X[5080] + 3 X[5172] = 3 X[3742] - X[5173]

X(6690) lies on these lines: {1, 4999}, {2, 11}, {5, 5248}, {12, 21}, {30, 3822}, {31, 5718}, {35, 442}, {37, 230}, {63, 5852}, {80, 5426}, {140, 517}, {141, 3771}, {165, 5880}, {226, 4640}, {321, 3712}, {377, 5217}, {405, 498}, {468, 1824}, {495, 529}, {516, 3838}, {518, 5745}, {631, 3428}, {674, 3589}, {758, 5719}, {846, 4415}, {908, 3683}, {958, 3085}, {1104, 5530}, {1155, 5249}, {1387, 3898}, {1698, 3419}, {1758, 6354}, {1788, 2099}, {2346, 6067}, {2476, 6284}, {3006, 4030}, {3011, 3666}, {3295, 3813}, {3475, 5744}, {3584, 5251}, {3601, 5794}, {3614, 5046}, {3624, 5119}, {3628, 3825}, {3742, 3911}, {3782, 4414}, {3811, 5791}, {3914, 4689}, {4023, 5278}, {4187, 5259}, {4512, 5219}, {5220, 5273}, {5226, 5698}

X(6690) = midpoint of X(i) and X(j) for these {i,j}: {55,2886}, {226,4640}, {495,993}
X(6690) = complement of X(2886)
X(6690) = center of bicevian ellipse of X(2) and X(55)

X(6691) = CENTROID OF {A,B,C,X(56)}

Barycentrics 2*a^4 - 3*a^2*b^2 + b^4 + 4*a^2*b*c + 2*a*b^2*c - 3*a^2*c^2 + 2*a*b*c^2 - 2*b^2*c^2 + c^4 : :
X(6691) = 3 X[2] + X[56] = 9 X[2] - X[3436] = 3 X[1329] - X[3436] = 3 X[56] + X[3436] = X[2098] - 5 X[3616] = X[46] + 7 X[3624]

X(6691) lies on these lines: {1, 1145}, {2, 12}, {3, 3816}, {5, 2829}, {11, 404}, {30, 3825}, {36, 4187}, {46, 3624}, {140, 517}, {142, 3647}, {468, 1828}, {474, 499}, {496, 528}, {549, 5248}, {631, 1001}, {960, 3911}, {1058, 4421}, {1104, 5121}, {1376, 3086}, {1388, 5554}, {1788, 5289}, {2098, 3616}, {2478, 5204}, {3304, 5552}, {3628, 3822}, {3871, 6174}, {4188, 6284}, {4292, 5087}, {4652, 4679}, {5302, 5316}

X(6691) = midpoint of X(56) and X(1329)
X(6691) = complement of X(1329)
X(6691) = center of bicevian ellipse of X(2) and X(56)

X(6692) = CENTROID OF {A,B,C,X(57)}

Barycentrics 2*a^3 - a^2*b - 2*a*b^2 + b^3 - a^2*c + 8*a*b*c - b^2*c - 2*a*c^2 - b*c^2 + c^3 : :
X(6692) = 3 X[2] + X[57] = 9 X[2] - X[329] = 3 X[57] + X[329] = 7 X[57] - 3 X[2094] = 7 X[2] + X[2094] = 7 X[329] + 9 X[2094] = X[2096] + 7 X[3090] = 5 X[1698] - X[3421] = X[329] - 3 X[3452] = 3 X[2094] + 7 X[3452] = X[2095] + 7 X[3526] = X[2093] + 7 X[3624] = X[5176] + 3 X[5193] = 5 X[631] - X[6282]

X(6692) lies on these lines: {2, 7}, {10, 999}, {56, 5795}, {140, 517}, {171, 5121}, {404, 950}, {474, 1210}, {516, 3816}, {631, 6282}, {938, 5438}, {946, 3359}, {1001, 6244}, {1329, 4298}, {1376, 5853}, {1476, 5176}, {1698, 3421}, {1788, 5837}, {1997, 3729}, {2093, 3624}, {2095, 3526}, {2096, 3090}, {2551, 3361}, {3035, 3742}, {3523, 5436}, {3628, 3824}, {3634, 3820}, {3748, 6174}, {3752, 3946}, {3817, 5880}, {4187, 4292}, {4413, 4847}, {4454, 6557}

X(6692) = midpoint of X(i) and X(j) for these {i,j}: {10,999}, {57,3452}, {946,3359}, {3820,3634}
X(6692) = complement of X(3452)

X(6693) = CENTROID OF {A,B,C,X(58)}

Barycentrics ). 2*a^4 + 2*a^3*b + a*b^3 + b^4 + 2*a^3*c + 2*a^2*b*c + a*b^2*c + b^3*c + a*b*c^2 + a*c^3 + b*c^3 + c^4 : :
X(6693) = 3 X[2] + X[58] = 9 X[2] - X[1330] = 3 X[58] + X[1330] = X[1043] + 3 X[3017] = 5 X[631] - X[3430] = X[1330] - 3 X[3454] = X[1046] + 7 X[3624] = 5 X[1698] + 3 X[5429]

X(6693) lies on these lines: {2, 58}, {10, 4434}, {140, 143}, {631, 3430}, {758, 942}, {1043, 3017}, {1046, 3624}, {1698, 5429}, {2842, 5972}, {3647, 4425}, {3712, 4065}

X(6693) = midpoint of X(58) and X(3454)
X(6693) = complement of X(3454)
X(6693) = center of bicevian ellipse of X(2) and X(58)

X(6694) = CENTROID OF {A,B,C,X(61)}

Barycentrics 2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6694) = 3 X[2] + X[61] = 9 X[2] - X[633] = 3 X[61] + X[633] = X[633] - 3 X[635]

X(6694) lies on these lines: {2, 18}, {17, 83}, {62, 618}, {140, 143}, {299, 3412}, {395, 629}, {396, 636}, {397, 530}, {398, 623}

X(6694) = midpoint of X(i) and X(j) for these {i,j}: {61,635}, {636,5007}
X(6694) = complement of X(635)

X(6695) = CENTROID OF {A,B,C,X(62)}

Barycentrics 2*a^4 - 3*a^2*b^2 + b^4 - 3*a^2*c^2 - 2*b^2*c^2 + c^4 + 2*Sqrt[3]*(2*a^2 + b^2 + c^2)*S : :
X(6695) = 3 X[2] + X[62] = 9 X[2] - X[634] = 3 X[62] + X[634] = X[634] - 3 X[636]

X(6695) lies on these lines: {2, 17}, {18, 83}, {61, 619}, {140, 143}, {298, 3411}, {395, 635}, {396, 630}, {397, 624}, {398, 531}

X(6695) = midpoint of X(i) and X(j) for these {i,j}: {62,636}, {635,5007}
X(6695) = complement of X(636)

X(6696) = CENTROID OF {A,B,C,X(64)}

Barycentrics 2*a^10 - 3*a^8*b^2 - 4*a^6*b^4 + 10*a^4*b^6 - 6*a^2*b^8 + b^10 - 3*a^8*c^2 + 16*a^6*b^2*c^2 - 10*a^4*b^4*c^2 - 3*b^8*c^2 - 4*a^6*c^4 - 10*a^4*b^2*c^4 + 12*a^2*b^4*c^4 + 2*b^6*c^4 + 10*a^4*c^6 + 2*b^4*c^6 - 6*a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(6696) = 3 X[2] + X[64] = 5 X[631] - X[1498] = X[20] + 3 X[1853] = 3 X[154] - 7 X[3523] = X[2892] + 3 X[5621] = 11 X[3525] - 3 X[5656] = 5 X[1656] - X[5878] = 2 X[3357] + X[5893] = 5 X[3091] - X[5895] = 3 X[5894] - X[5925] = 3 X[4] + X[5925] = 9 X[2] - X[6225] = 3 X[2883] - X[6225] = 3 X[64] + X[6225]

X(6696) lies on these lines: {2, 64}, {3, 66}, {4, 1192}, {5, 3357}, {20, 1853}, {30, 5449}, {74, 1594}, {125, 1885}, {140, 6000}, {154, 3523}, {221, 5218}, {389, 2781}, {427, 1204}, {546, 2777}, {631, 1498}, {1656, 5878}, {1788, 1854}, {1899, 3516}, {2071, 2888}, {2892, 5621}, {3088, 5480}, {3091, 5895}, {3525, 5656}, {5044, 6001}, {5433, 6285}

X(6696) = midpoint of X(i) and X(j) for these {i,j}: {3,6247}, {4,5894}, {5,3357}, {64,2883}
X(6696) = reflection of X(5893) in X(5)
X(6696) = complement of X(2883)

X(6697) = CENTROID OF {A,B,C,X(66)}

Barycentrics a^4*b^4 - b^8 + a^4*c^4 + 2*b^4*c^4 - c^8 : :
X(6697) = 3 X[2] + X[66] = X[159] + 3 X[1853] = X[159] - 5 X[3763] = 3 X[1853] + 5 X[3763] = 9 X[2] - X[5596] = 3 X[206] - X[5596] = 3 X[66] + X[5596]

X(6697) lies on these lines: {2, 66}, {6, 67}, {140, 1503}, {141, 1368}, {159, 1853}, {511, 5449}, {626, 3852}, {858, 3313}, {868, 1879}, {1352, 3548}, {3619, 5888}, {3818, 6644}, {3827, 3844}

X(6697) = midpoint of X(66) and X(206)
X(6697) = complement of X(206)
X(6697) = homothetic center of orthocevian triangle of X(2) and 3rd antipedal triangle of X(4)

X(6698) = CENTROID OF {A,B,C,X(67)}

Barycentrics -(a^6*b^2) + 2*a^4*b^4 + a^2*b^6 - 2*b^8 - a^6*c^2 - 2*a^2*b^4*c^2 + 2*a^4*c^4 - 2*a^2*b^2*c^4 + 4*b^4*c^4 + a^2*c^6 - 2*c^8 : :
X(6698) = 3 X[2] + X[67] = 3 X[599] + X[895] = X[3448] + 7 X[3619] = X[110] - 5 X[3763] = 3 X[597] - X[5095] = 3 X[141] - X[5181] = 3 X[125] + X[5181]

X(6698) lies on these lines: {2, 67}, {5, 2781}, {110, 3763}, {125, 126}, {140, 542}, {524, 5159}, {597, 5095}, {599, 895}, {2836, 5044}, {3448, 3619}

X(6698) = midpoint of X(i) and X(j) for these {i,j}: {67,6593}, {125,141}
X(6698) = complement of X(6593)

X(6699) = CENTROID OF {A,B,C,X(74)}

Barycentrics (a^2 - b^2 - c^2)*(2*a^8 - 2*a^6*b^2 - 3*a^4*b^4 + 4*a^2*b^6 - b^8 - 2*a^6*c^2 + 8*a^4*b^2*c^2 - 4*a^2*b^4*c^2 + 4*b^6*c^2 - 3*a^4*c^4 - 4*a^2*b^2*c^4 - 6*b^4*c^4 + 4*a^2*c^6 + 4*b^2*c^6 - c^8) : :
Barycentrics 2a/(cos A - 2 cos B cos C) + b/(cos B - 2 cos C cos A) + c/(cos C - 2 cos A cos B) : :
X(6699) = 3 X[2] + X[74] = 9 X[2] - X[146] = 3 X[113] - X[146] = 3 X[74] + X[146] = 3 X[125] - X[265] = 3 X[3] + X[265] = X[110] - 5 X[631] = 3 X[549] - X[1511] = 3 X[5] - X[1539] = X[1531] - 3 X[2072] = X[3448] + 7 X[3523] = X[399] - 9 X[5054] = X[67] + 3 X[5085] = 3 X[5050] - X[5095] = 5 X[3763] + 3 X[5621] = X[69] + 3 X[5622] = X[399] - 3 X[5642] = 3 X[5054] - X[5642] = 6 X[140] - X[6053] = 3 X[5972] - X[6053] = X[1539] - 6 X[6723

X(6699) lies on these lines: {2, 74}, {3, 125}, {5, 1539}, {67, 5085}, {69, 5504}, {110, 631}, {140, 5663}, {141, 542}, {399, 5054}, {690, 6036}, {914, 5440}, {1112, 5462}, {1204, 5448}, {1531, 2072}, {2771, 3035}, {2778, 3812}, {2781, 3589}, {3024, 5433}, {3028, 5432}, {3043, 5012}, {3448, 3523}, {3763, 5621}, {3818, 6644}, {5050, 5095}

X(6699) = midpoint of X(i) and X(j) for these {i,j}: {3,125}, {74,113}, {1112,5462}
X(6699) = reflection of X(i) in X(jk) for these (i,j): (5,6723), (5972,140)
X(6699) = complement of X(113)
X(6699) = isotomic conjugate of polar conjugate of X(6128)

X(6700) = CENTROID OF {A,B,C,X(78)}

Barycentrics 2*a^4 - a^3*b - 3*a^2*b^2 + a*b^3 + b^4 - a^3*c + 2*a^2*b*c + 3*a*b^2*c - 3*a^2*c^2 + 3*a*b*c^2 - 2*b^2*c^2 + a*c^3 + c^4 : :
X(6700) = 3 X[2] + X[78]

X(6700) lies on these lines: {1, 2}, {3, 3452}, {4, 5438}, {9, 631}, {20, 5328}, {72, 3911}, {140, 912}, {210, 5433}, {226, 474}, {404, 908}, {405, 5316}, {443, 5219}, {496, 5853}, {515, 1329}, {516, 3149}, {946, 1376}, {950, 4187}, {960, 2800}, {1058, 3158}, {1385, 3820}, {1420, 3421}, {1445, 5850}, {1706, 5603}, {1795, 3074}, {2478, 4304}, {2551, 3576}, {3159, 3998}, {3436, 4311}, {3487, 5437}, {3526, 5791}, {3601, 5084}, {3660, 3678}, {3740, 4999}, {4679, 5217}, {4848, 5730}, {5054, 5325}, {5257, 5783}, {5265, 5815}, {5267, 5660}, {5720, 6245}

X(6700) = midpoint of X(i) and X(j) for these {i,j}: {78,1210}, {3436,4311}, {4848,5730}
X(6700) = complement of X(1210)

X(6701) = CENTROID OF {A,B,C,X(79)}

Barycentrics -((b + c)*(-a^3 - 2*a^2*b + a*b^2 + 2*b^3 - 2*a^2*c - 5*a*b*c - 2*b^2*c + a*c^2 - 2*b*c^2 + 2*c^3)) : :
X(6701) = 3 X[2] + X[79] = X[10] - 3 X[442] = 3 X[2475] + 5 X[3616] = 3 X[21] - 7 X[3624] = 9 X[2] - X[3648] = 3 X[3647] - X[3648] = 3 X[79] + X[3648] = 3 X[442] + X[3649] = 5 X[1656] - X[3652] = 5 X[3616] - X[5441] = 3 X[2475] + X[5441] = X[1] + 3 X[6175]

X(6701) lies on these lines: {1, 6175}, {2, 79}, {5, 3833}, {10, 12}, {21, 3624}, {30, 1125}, {142, 3825}, {191, 3305}, {1387, 3636}, {1479, 2475}, {1656, 3652}, {2476, 5883}, {2771, 3812}, {2886, 3881}, {3120, 3743}, {3651, 5248}, {3884, 5499}, {5087, 5122}, {5586, 5705}

X(6701) = midpoint of X(i) and X(j) for these {i,j}: {10,3649}, {79,3647}
X(6701) = complement of X(3647)

X(6702) = CENTROID OF {A,B,C,X(80)}

Barycentrics -(a^3*b) + 2*a^2*b^2 + a*b^3 - 2*b^4 - a^3*c - 2*a*b^2*c + 2*a^2*c^2 - 2*a*b*c^2 + 4*b^2*c^2 + a*c^3 - 2*c^4 : :
X(6702) = 3 X[2] + X[80] = 3 X[10] - X[1145] = 3 X[11] + X[1145] = 3 X[551] - X[1317] = X[100] - 5 X[1698] = X[908] + 3 X[1737] = X[1320] + 3 X[3679] = X[908] - 3 X[3814] = X[1537] - 3 X[3817] = 3 X[3582] + X[5176] = X[104] + 3 X[5587] = X[3065] + 3 X[6175] = 9 X[2] - X[6224] = 3 X[214] - X[6224] = 3 X[80] + X[6224] = 5 X[1656] - X[6265]

Let A′B′C′ be the Fuhrmann triangle. Let A″ be the orthogonal projection of A on line B′C′, and define B″, C″ cyclically. Then A″B″C″ is perspective to the medial triangle at X(6702). (Randy Hutson, April 11, 2015)

X(6702) lies on these lines: {2, 80}, {3, 6246}, {5, 2800}, {10, 11}, {12, 5083}, {100, 1698}, {104, 5587}, {116, 119}, {519, 1387}, {528, 3828}, {535, 3911}, {537, 4013}, {551, 1317}, {758, 908}, {952, 1125}, {1210, 3881}, {1320, 3679}, {1329, 3678}, {1484, 3816}, {1537, 3817}, {1656, 6265}, {2771, 3812}, {2805, 3846}, {2886, 3968}, {2932, 4413}, {3035, 3634}, {3065, 6175}, {3582, 5176}, {3626, 5854}, {3820, 3956}, {3878, 4193}, {4973, 5080}, {4996, 5251}, {5046, 5445}, {5154, 5903}

X(6702) = midpoint of X(i) and X(j) for these {i,j}: {3,6246}, {10,11}, {80,214}, {1387,3036}, {1737,3814}, {4973,5080}
X(6702) = reflection of X(i) in X(j) for these (i,j): (1125,6667), (3035,3634)
X(6702) = complement of X(214)

X(6703) = CENTROID OF {A,B,C,X(81)}

Barycentrics 2*a^3 + 2*a^2*b + a*b^2 + b^3 + 2*a^2*c + 4*a*b*c + b^2*c + a*c^2 + b*c^2 + c^3 : :
X(6703) = 3 X[2] + X[81] = 9 X[2] - X[2895] = 3 X[1211] - X[2895] = 3 X[81] + X[2895] = 3 X[3745] + X[4914]

X(6703) lies on these lines: {1, 3704}, {2, 6}, {3, 5799}, {10, 4682}, {57, 1761}, {58, 4205}, {63, 4364}, {140, 970}, {171, 4026}, {226, 4670}, {551, 4906}, {594, 1999}, {758, 942}, {894, 4415}, {896, 6536}, {1010, 1834}, {1100, 3687}, {1375, 5437}, {1961, 3932}, {1962, 3712}, {2049, 5292}, {2160, 2339}, {2836, 3848}, {3187, 4399}, {3286, 4199}, {3703, 5311}, {3745, 4914}, {3986, 5325}, {4038, 4966}, {4104, 4663}, {4359, 4395}, {4418, 4854}, {4422, 5294}, {4425, 4697}, {4886, 4969}

X(6703) = midpoint of X(i) and X(j) for these {i,j}: {81,1211}, {4418,4854}, {4425,4697}
X(6703) = complement of X(1211)

X(6704) = CENTROID OF {A,B,C,X(83)}

Barycentrics 2*a^4 + 4*a^2*b^2 + b^4 + 4*a^2*c^2 + 4*b^2*c^2 + c^4 : :
X(6704) = 3 X[2] + X[83] = 9 X[2] - X[2896] = 3 X[83] + X[2896] = 5 X[1656] - X[6287] = X[2896] - 3 X[6292]}.

X(6704) lies on these lines: {2, 32}, {3, 6249}, {5, 5092}, {620, 2023}, {732, 3589}, {1656, 6287}, {3246, 3634}, {3628, 6036}

X(6704) = midpoint of X(i) and X(j) for these {i,j}: {3,6249}, {83,6292}
X(6704) = complement of X(6292)

X(6705) = CENTROID OF {A,B,C,X(84)}

Barycentrics 2*a^7 - a^6*b - 6*a^5*b^2 + 3*a^4*b^3 + 6*a^3*b^4 - 3*a^2*b^5 - 2*a*b^6 + b^7 - a^6*c + 8*a^5*b*c + a^4*b^2*c - 8*a^3*b^3*c + a^2*b^4*c - b^6*c - 6*a^5*c^2 + a^4*b*c^2 + 4*a^3*b^2*c^2 + 2*a^2*b^3*c^2 + 2*a*b^4*c^2 - 3*b^5*c^2 + 3*a^4*c^3 - 8*a^3*b*c^3 + 2*a^2*b^2*c^3 + 3*b^4*c^3 + 6*a^3*c^4 + a^2*b*c^4 + 2*a*b^2*c^4 + 3*b^3*c^4 - 3*a^2*c^5 - 3*b^2*c^5 - 2*a*c^6 - b*c^6 + c^7 : :
X(6705) = 3 X[2] + X[84] = 5 X[631] - X[1490] = 11 X[3525] - 3 X[5658] = 3 X[3928] + X[5758] = 3 X[3] + X[5787] = 9 X[2] - X[6223] = 3 X[84] + X[6223] = X[5787] - 3 X[6245] = 5 X[1656] - X[6259] = X[6223] - 3 X[6260]

X(6705) lies on these lines: {2, 84}, {3, 10}, {4, 3911}, {40, 5744}, {57, 946}, {140, 971}, {142, 3358}, {443, 6256}, {499, 1709}, {631, 1490}, {916, 5482}, {942, 1387}, {1012, 1210}, {1125, 6001}, {1656, 6259}, {2095, 4301}, {3525, 5658}, {3601, 5768}, {3928, 5758}, {4853, 6282}

X(6705) = midpoint of X(i) and X(j) for these {i,j}: {3,6245}, {84,6260}, {142,3358}, {946,1158}
X(6705) = complement of X(6260)

X(6706) = CENTROID OF {A,B,C,X(85)}

Barycentrics a^3*b - 2*a^2*b^2 + a*b^3 + a^3*c - 2*a^2*b*c - a*b^2*c + 2*b^3*c - 2*a^2*c^2 - a*b*c^2 - 4*b^2*c^2 + a*c^3 + 2*b*c^3 : :
X(6706) = 3 X[2] + X[85] = 9 X[2] - X[3177] = 3 X[1212] - X[3177] = 3 X[85] + X[3177]

X(6706) lies on these lines: {2, 85}, {10, 141}, {37, 3673}, {75, 4515}, {277, 3085}, {517, 2140}, {536, 3991}, {672, 4059}, {938, 4648}, {4000, 4646}, {5437, 5737}

X(6706) = midpoint of X(85) and X(1212)
X(6706) = complement of X(1212)

X(6707) = CENTROID OF {A,B,C,X(86)}

Barycentrics 2*a^2 + 4*a*b + b^2 + 4*a*c + 4*b*c + c^2 : :
X(6707) = 3 X[2] + X[86] = 9 X[2] - X[1654] = 3 X[1213] - X[1654] = 3 X[86] + X[1654]

X(6707) lies on these lines: {1, 4399}, {2, 6}, {9, 4798}, {10, 4478}, {37, 4472}, {142, 3647}, {551, 4852}, {620, 2796}, {740, 1125}, {1086, 6651}, {1268, 6542}, {1698, 4851}, {3616, 4361}, {3622, 4405}, {3624, 4657}, {3626, 4889}, {3664, 4708}, {3723, 4967}, {3729, 4499}, {4000, 5550}, {4021, 4739}, {4422, 4698}, {4670, 5257}, {4725, 4909}

X(6707) = midpoint of X(i) and X(j) for these {i,j}: {1,4733}, {10,5625}, {86,1213}, {1086,6651}, {8723,4967}
X(6707) = complement of X(1213)

X(6708) = CENTROID OF {A,B,C,X(92)}

Barycentrics (a - b - c)*(a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + a^4*c + a^2*b^2*c - 2*b^4*c + a^3*c^2 + a^2*b*c^2 + 2*a*b^2*c^2 + 2*b^3*c^2 - a^2*c^3 + 2*b^2*c^3 - a*c^4 - 2*b*c^4) : :
X(6708) = 3 X[2] + X[92] = 9 X[2] - X[6360] = 3 X[1214] - X[6360] = 3 X[92] + X[6360]

X(6708) lies on these lines: {2, 92}, {5, 10}, {9, 1730}, {123, 132}, {124, 3838}, {200, 3706}, {219, 5271}, {312, 3694}, {333, 1944}, {1746, 2182}, {1824, 1985}, {1829, 3142}, {1943, 6510}, {3085, 6051}, {3666, 4858}, {3739, 5745}, {3767, 3772}

X(6708) = midpoint of X(92) and X(1214)
X(6708) = complement of X(1214)

X(6709) = CENTROID OF {A,B,C,X(95)}

Barycentrics 2*a^8 - 8*a^6*b^2 + 11*a^4*b^4 - 6*a^2*b^6 + b^8 - 8*a^6*c^2 + 8*a^4*b^2*c^2 + 6*a^2*b^4*c^2 - 6*b^6*c^2 + 11*a^4*c^4 + 6*a^2*b^2*c^4 + 10*b^4*c^4 - 6*a^2*c^6 - 6*b^2*c^6 + c^8 : :
X(6709) = 3 X[2] + X[95]

X(6709) lies on these lines: {2, 95}, {141, 575}, {3525, 6389}

X(6709) = midpoint of X(95) and X(233)
X(6709) = complement of X(233)
X(6709) = isotomic conjugate of polar conjugate of X(35884)

X(6710) = CENTROID OF {A,B,C,X(101)}

Barycentrics 2*a^4 - 2*a^3*b - a*b^3 + b^4 - 2*a^3*c + 2*a^2*b*c + a*b^2*c - b^3*c + a*b*c^2 - a*c^3 - b*c^3 + c^4 : :
X(6710) = 3 X[2] + X[101] = 9 X[2] - X[150] = 3 X[116] - X[150] = 3 X[101] + X[150] = X[103] - 5 X[631] = X[152] + 7 X[3523] = X[1282] + 7 X[3624]

X(6710) lies on these lines: {2, 101}, {3, 118}, {103, 631}, {140, 2808}, {152, 3523}, {468, 5185}, {620, 2786}, {910, 5074}, {1125, 2809}, {1282, 3624}, {1362, 5433}, {2774, 5972}, {2784, 3634}, {2810, 3589}, {3022, 5432}, {3035, 3887}, {3041, 4999}, {3046, 5012}

X(6710) = midpoint of X(i) and X(j) for these {i,j}: {3,118}, {101,116}, {910,5074}
X(6710) = reflection of X(6712) in X(140)
X(6710) = complement of X(116)

X(6711) = CENTROID OF {A,B,C,X(102)}

Barycentrics 2*a^10 - 2*a^9*b - 4*a^8*b^2 + 7*a^7*b^3 - a^6*b^4 - 9*a^5*b^5 + 7*a^4*b^6 + 5*a^3*b^7 - 5*a^2*b^8 - a*b^9 + b^10 - 2*a^9*c + 6*a^8*b*c - 3*a^7*b^2*c - 11*a^6*b^3*c + 15*a^5*b^4*c + 3*a^4*b^5*c - 13*a^3*b^6*c + 3*a^2*b^7*c + 3*a*b^8*c - b^9*c - 4*a^8*c^2 - 3*a^7*b*c^2 + 20*a^6*b^2*c^2 - 6*a^5*b^3*c^2 - 19*a^4*b^4*c^2 + 9*a^3*b^5*c^2 + 6*a^2*b^6*c^2 - 3*b^8*c^2 + 7*a^7*c^3 - 11*a^6*b*c^3 - 6*a^5*b^2*c^3 + 18*a^4*b^3*c^3 - a^3*b^4*c^3 - 3*a^2*b^5*c^3 - 8*a*b^6*c^3 + 4*b^7*c^3 - a^6*c^4 + 15*a^5*b*c^4 - 19*a^4*b^2*c^4 - a^3*b^3*c^4 - 2*a^2*b^4*c^4 + 6*a*b^5*c^4 + 2*b^6*c^4 - 9*a^5*c^5 + 3*a^4*b*c^5 + 9*a^3*b^2*c^5 - 3*a^2*b^3*c^5 + 6*a*b^4*c^5 - 6*b^5*c^5 + 7*a^4*c^6 - 13*a^3*b*c^6 + 6*a^2*b^2*c^6 - 8*a*b^3*c^6 + 2*b^4*c^6 + 5*a^3*c^7 + 3*a^2*b*c^7 + 4*b^3*c^7 - 5*a^2*c^8 + 3*a*b*c^8 - 3*b^2*c^8 - a*c^9 - b*c^9 + c^10 : :
X(6711) = 3 X[2] + X[102] = 9 X[2] - X[151] = 3 X[117] - X[151] = 3 X[102] + X[151] = X[109] - 5 X[631]

X(6711) lies on these lines: {2, 102}, {3, 124}, {109, 631}, {140, 2818}, {620, 2792}, {960, 2800}, {1125, 2817}, {1361, 5432}, {1364, 5433}, {2779, 5972}, {2785, 6036}, {3042, 4999}

X(6711) = midpoint of X(i) and X(j) for these {i,j}: {3,124}, {102,117}
X(6711) = refelction of X(6718) in X(140)
X(6711) = complement of X(117)

X(6712) = CENTROID OF {A,B,C,X(103)}

Barycentrics 2*a^8-2*(b+c)*a^7-2*(b^2-b*c+c^2)*a^6-(b+c)*(b^2-6*b*c+c^2)*a^5+(3*b^4+3*c^4-5*b*c*(b^2+c^2))*a^4+4*(b^4-c^4)*(b-c)*a^3-4*(b^3-c^3)*(b-c)*(b^2+c^2)*a^2-(b^2-c^2)^3*(b-c)*a+(b^2-c^2)^2*(b-c)*(b^3-c^3) : :
X(6712) = 3 X[2] + X[103] = 9 X[2] - X[152] = 3 X[118] - X[152] = 3 X[103] + X[152] = X[101] - 5 X[631] = X[150] + 7 X[3523]

X(6712) lies on these lines: {2, 103}, {3, 116}, {101, 631}, {140, 2808}, {150, 3523}, {544, 549}, {620, 2784}, {1362, 5432}, {2772, 5972}, {2786, 6036}, {2801, 3035}, {3022, 5433}

X(6712) = midpoint of X(i) and X(j) for these {i,j}: {3,116}, {103,118}
X(6712) = reflection of X(6710) in X(140)
X(6712) = complement of X(118)

X(6713) = CENTROID OF {A,B,C,X(104)}

Barycentrics 2*a^8 - 2*a^7*b - 2*a^6*b^2 - a^5*b^3 + 3*a^4*b^4 + 4*a^3*b^5 - 4*a^2*b^6 - a*b^7 + b^8 - 2*a^7*c + 2*a^6*b*c + 5*a^5*b^2*c - 5*a^4*b^3*c - 4*a^3*b^4*c + 4*a^2*b^5*c + a*b^6*c - b^7*c - 2*a^6*c^2 + 5*a^5*b*c^2 - 4*a^2*b^4*c^2 + 3*a*b^5*c^2 - 2*b^6*c^2 - a^5*c^3 - 5*a^4*b*c^3 + 8*a^2*b^3*c^3 - 3*a*b^4*c^3 + b^5*c^3 + 3*a^4*c^4 - 4*a^3*b*c^4 - 4*a^2*b^2*c^4 - 3*a*b^3*c^4 + 2*b^4*c^4 + 4*a^3*c^5 + 4*a^2*b*c^5 + 3*a*b^2*c^5 + b^3*c^5 - 4*a^2*c^6 + a*b*c^6 - 2*b^2*c^6 - a*c^7 - b*c^7 + c^8 : :
X(6713) = 3 X[2] + X[104] = 9 X[2] - X[153] = 3 X[119] - X[153] = 3 X[104] + X[153] = X[100] - 5 X[631] = 3 X[549] + X[1484] = X[149] + 7 X[3523] = X[80] + 3 X[3576] = X[2077] + 3 X[3582] = X[1768] + 7 X[3624] = X[1320] + 3 X[5657] = X[1537] - 3 X[5886] = 3 X[5054] - X[6174]

X(6713) lies on these lines: {2, 104}, {3, 11}, {5, 2829}, {10, 140}, {35, 5533}, {36, 5841}, {40, 5442}, {80, 3576}, {100, 631}, {149, 3523}, {182, 5848}, {517, 1387}, {528, 549}, {620, 2783}, {1006, 4996}, {1125, 2800}, {1145, 3872}, {1317, 5432}, {1320, 5657}, {1537, 3306}, {1768, 3624}, {2077, 3582}, {2771, 5972}, {2787, 6036}, {4297, 6246}, {5054, 6174}, {5690, 5854}, {5883, 5901}

X(6713) = midpoint of X(i) and X(j) for these {i,j}: {3,11}, {104,119}, {4297,6246}
X(6713) = reflection of of X(i) in X(j) for these (i,j): (5,6667), (3035,140)
X(6713) = complement of X(119)

X(6714) = CENTROID OF {A,B,C,X(105)}

Barycentrics 2*a^5 - 2*a^4*b + a^3*b^2 - a^2*b^3 - a*b^4 + b^5 - 2*a^4*c - 4*a^3*b*c + 5*a^2*b^2*c - 4*a*b^3*c - b^4*c + a^3*c^2 + 5*a^2*b*c^2 + 6*a*b^2*c^2 - a^2*c^3 - 4*a*b*c^3 - a*c^4 - b*c^4 + c^5 : :
X(6714) = 3 X[2] + X[105] = 5 X[631] - X[1292] = 7 X[3624] + X[5540]

X(6714) lies on these lines: {2, 11}, {3, 5511}, {348, 1358}, {620, 2795}, {631, 1292}, {1083, 4904}, {1125, 2809}, {1212, 3039}, {2788, 6036}, {2836, 3848}, {3624, 5540}, {3756, 5272}

X(6714) = midpoint of X(i) and X(j) for these {i,j}: {3, 5511}, {105, 120}, {1083,4904}
X(6714) = complement of X(120)

X(6715) = CENTROID OF {A,B,C,X(106)}

Barycentrics 2*a^4 - 2*a^3*b - 6*a^2*b^2 - a*b^3 + b^4 - 2*a^3*c + 10*a^2*b*c + 5*a*b^2*c - b^3*c - 6*a^2*c^2 + 5*a*b*c^2 - 4*b^2*c^2 - a*c^3 - b*c^3 + c^4 : :
X(6715) = 3 X[2] + X[106] = 5 X[631] - X[1293] = X[1054] + 7 X[3624]

X(6715) lies on these lines: {2, 106}, {3, 5510}, {620, 2796}, {631, 1293}, {1054, 3624}, {1125, 1387}, {1357, 5433}, {2789, 6036}, {2810, 3589}, {2842, 5972}, {3038, 4999}, {3872, 6095}, {5432, 6018}

X(6715) = midpoint of X(i) and X(j) for these {i,j}: {3,5510}, {106,121}
X(6715) = complement of X(121)

X(6716) = CENTROID OF {A,B,C,X(107)}

Barycentrics 2*a^12 - 2*a^10*b^2 - 7*a^8*b^4 + 12*a^6*b^6 - 4*a^4*b^8 - 2*a^2*b^10 + b^12 - 2*a^10*c^2 + 16*a^8*b^2*c^2 - 12*a^6*b^4*c^2 - 16*a^4*b^6*c^2 + 14*a^2*b^8*c^2 - 7*a^8*c^4 - 12*a^6*b^2*c^4 + 40*a^4*b^4*c^4 - 12*a^2*b^6*c^4 - 9*b^8*c^4 + 12*a^6*c^6 - 16*a^4*b^2*c^6 - 12*a^2*b^4*c^6 + 16*b^6*c^6 - 4*a^4*c^8 + 14*a^2*b^2*c^8 - 9*b^4*c^8 - 2*a^2*c^10 + c^12 : :
X(6716) = 3 X[2] + X[107] = 5 X[631] - X[1294] = 7 X[3090] + X[5667]

X(6716) lies on these lines: {2, 107}, {3, 133}, {4, 3184}, {5, 1539}, {402, 5972}, {620, 2797}, {631, 1294}, {2790, 6036}, {2803, 3035}, {3090, 5667}, {3324, 5433}

X(6716) = midpoint of X(i) and X(j) for these {i,j}: {3,133}, {4,3184}, {107,122}
X(6716) = complement of X(122)

X(6717) = CENTROID OF {A,B,C,X(108)}

Barycentrics 2*a^9 - 2*a^8*b - 3*a^7*b^2 + 3*a^6*b^3 - a^5*b^4 + a^4*b^5 + 3*a^3*b^6 - 3*a^2*b^7 - a*b^8 + b^9 - 2*a^8*c + 8*a^7*b*c - 3*a^6*b^2*c - 6*a^5*b^3*c + 7*a^4*b^4*c - 8*a^3*b^5*c - a^2*b^6*c + 6*a*b^7*c - b^8*c - 3*a^7*c^2 - 3*a^6*b*c^2 + 14*a^5*b^2*c^2 - 8*a^4*b^3*c^2 - 7*a^3*b^4*c^2 + 13*a^2*b^5*c^2 - 4*a*b^6*c^2 - 2*b^7*c^2 + 3*a^6*c^3 - 6*a^5*b*c^3 - 8*a^4*b^2*c^3 + 24*a^3*b^3*c^3 - 9*a^2*b^4*c^3 - 6*a*b^5*c^3 + 2*b^6*c^3 - a^5*c^4 + 7*a^4*b*c^4 - 7*a^3*b^2*c^4 - 9*a^2*b^3*c^4 + 10*a*b^4*c^4 + a^4*c^5 - 8*a^3*b*c^5 + 13*a^2*b^2*c^5 - 6*a*b^3*c^5 + 3*a^3*c^6 - a^2*b*c^6 - 4*a*b^2*c^6 + 2*b^3*c^6 - 3*a^2*c^7 + 6*a*b*c^7 - 2*b^2*c^7 - a*c^8 - b*c^8 + c^9 : :
X(6717) = 3 X[2] + X[108] = 5 X[631] - X[1295]

X(6717) lies on these lines: {2, 108}, {5, 2829}, {620, 2798}, {631, 1295}, {676, 2804}, {1125, 2817}, {1359, 5433}, {2778, 3812}, {2791, 6036}, {2850, 5972}, {3318, 5432}

X(6717) = midpoint of X(108) and X(123)
X(6717) = complement of X(123)

X(6718) = CENTROID OF {A,B,C,X(109)}

Barycentrics 2*a^6 - 2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 - a^2*b^4 - a*b^5 + b^6 - 2*a^5*c + 6*a^4*b*c - 3*a^3*b^2*c - 3*a^2*b^3*c + 3*a*b^4*c - b^5*c - 2*a^4*c^2 - 3*a^3*b*c^2 + 8*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + 3*a^3*c^3 - 3*a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + 3*a*b*c^4 - b^2*c^4 - a*c^5 - b*c^5 + c^6 : :
X(6718) = 3 X[2] + X[109] = X[102] - 5 X[631] = X[151] + 7 X[3523]

X(6718) lies on these lines: {{2, 109}, {3, 117}, {102, 631}, {140, 2818}, {151, 3523}, {498, 1795}, {620, 2785}, {1125, 2800}, {1361, 5433}, {1364, 5432}, {2773, 5972}, {2792, 6036}, {3035, 3042}, {3040, 4999}

X(6718) = midpoint of X(i) and X(j) for these {i,j}: {3,117}, {109,124}
X(6718) = reflection of X(6711) in X(140)
X(6718) = complement of X(124)

X(6719) = CENTROID OF {A,B,C,X(111)}

Barycentrics 2*a^6 - 4*a^4*b^2 - 5*a^2*b^4 + b^6 - 4*a^4*c^2 + 20*a^2*b^2*c^2 - 3*b^4*c^2 - 5*a^2*c^4 - 3*b^2*c^4 + c^6 : :
X(6719) = 3 X[2] + X[111] = 5 X[631] - X[1296]

X(6719) lies on these lines: {2, 99}, {3, 5512}, {69, 6387}, {468, 5140}, {631, 1296}, {754, 5913}, {2793, 6036}, {2805, 3035}, {2854, 3589}, {3048, 5012}, {3325, 5433}, {5432, 6019}

X(6719) = midpoint of X(i) and X(j) for these {i,j}: {3,5512}, {111,126}
X(6719) = complement of X(126)
X(6719) = orthoptic-circle-of-Steiner-circumellipe-inverse of X(35369)

X(6720) = CENTROID OF {A,B,C,X(112)}

Barycentrics 2*a^10 - 2*a^8*b^2 - a^6*b^4 + a^4*b^6 - a^2*b^8 + b^10 - 2*a^8*c^2 + 4*a^6*b^2*c^2 - a^4*b^4*c^2 - b^8*c^2 - a^6*c^4 - a^4*b^2*c^4 + 2*a^2*b^4*c^4 + a^4*c^6 - a^2*c^8 - b^2*c^8 + c^10 : :
X(6720) = 3 X[2] + X[112] = 5 X[631] - X[1297]

X(6720) lies on these lines: {2, 112}, {3, 132}, {5, 2794}, {339, 6103}, {620, 2492}, {631, 1297}, {1485, 5961}, {2781, 3589}, {2806, 3035}, {3320, 5433}, {5432, 6020}

X(6720) = midpoint of X(i) and X(j) for these {i,j}: {3,132}, {112,127}
X(6720) = complement of X(127)

X(6721) = CENTROID OF {A,B,C,X(114)}

Barycentrics -2*a^8 + 8*a^6*b^2 - 11*a^4*b^4 + 8*a^2*b^6 - 3*b^8 + 8*a^6*c^2 - 4*a^4*b^2*c^2 - 2*a^2*b^4*c^2 + 8*b^6*c^2 - 11*a^4*c^4 - 2*a^2*b^2*c^4 - 10*b^4*c^4 + 8*a^2*c^6 + 8*b^2*c^6 - 3*c^8 : :
X(6721) = 9 X[2] - X[98] = 3 X[2] + X[114] = X[98] + 3 X[114] = 5 X[114] - X[147] = 15 X[2] + X[147] = 5 X[98] + 3 X[147] = X[115] - 5 X[1656] = X[99] + 7 X[3090] = X[2482] + 3 X[5055] = 11 X[98] - 3 X[5984] = 11 X[114] + X[5984] = 11 X[147] + 5 X[5984] = 7 X[3526] + X[6033] = X[5984] - 11 X[6036] = X[98] - 3 X[6036] = X[147] + 5 X[6036] = 7 X[147] - 15 X[6054] = 7 X[114] - 3 X[6054] = 7 X[2] + X[6054] = 7 X[6036] + 3 X[6054] = 7 X[98] + 9 X[6054] = 5 X[98] - 9 X[6055] = 5 X[6036] - 3 X[6055] = 5 X[2] - X[6055] = 5 X[114] + 3 X[6055] = X[147] + 3 X[6055] = 5 X[6054] + 7 X[6055] = 9 X[5055] - X[6321] = 3 X[2482] + X[6321]

X(6721) lies on these lines: {2, 98}, {5, 620}, {99, 3090}, {115, 1656}, {140, 2794}, {230, 5965}, {543, 547}, {576, 1007}, {2023, 3055}, {2482, 5055}, {2782, 3628}, {3526, 6033}

X(6721) = midpoint of X(i) and X(j) for these {i,j}: {5,620}, {114,6036}
X(6721) = reflection of X(6722) in X(3628)
X(6721) = complement of X(6036)

X(6722) = CENTROID OF {A,B,C,X(115)}

Barycentrics 2*a^4 - 2*a^2*b^2 + 3*b^4 - 2*a^2*c^2 - 4*b^2*c^2 + 3*c^4 : :
X(6722) = 9 X[2] - X[99] = 3 X[2] + X[115] = X[99] + 3 X[115] = 5 X[115] - X[148] = 15 X[2] + X[148] = 5 X[99] + 3 X[148] = X[99] - 3 X[620] = X[148] + 5 X[620] = 7 X[148] - 15 X[671] = 7 X[115] - 3 X[671] = 7 X[2] + X[671] = 7 X[620] + 3 X[671] = 7 X[99] + 9 X[671] = X[114] - 5 X[1656] = 5 X[99] - 9 X[2482] = 5 X[620] - 3 X[2482] = 5 X[2] - X[2482] = 5 X[115] + 3 X[2482] = X[148] + 3 X[2482] = 5 X[671] + 7 X[2482] = X[98] + 7 X[3090] = 5 X[230] - X[3793] = 5 X[625] + X[3793] = X[148] - 15 X[5461] = X[671] - 7 X[5461] = X[115] - 3 X[5461] = X[620] + 3 X[5461] = X[2482] + 5 X[5461] = X[99] + 9 X[5461] = 5 X[3618] - X[5477] = 9 X[5055] - X[6033] = 5 X[3763] + 3 X[6034] = 3 X[5055] + X[6055] = X[6033] + 3 X[6055] = 7 X[3526] + X[6321]

Let θ be a variable angle. Construct isosceles triangle BA′C on BC (outwardly if θ is positive, inwardly if θ is negative), such that ∠A′BC = ∠A′CB = θ. Define B′ and C′ cyclically. Let (Ia) be the incircle of BA′C, and define (Ib) and (Ic) cyclically. As θ varies, the locus of the radical center of (Ia), (Ib), (Ic) is the Kiepert hyperbola of the medial-of-medial triangle, with center X(6722). The same locus is obtained if nine-point circle is substituted for incircle. (Randy Hutson, July 20, 2016)

X(6722) lies on these lines: {2, 99}, {3, 23514}, {5, 2794}, {10, 11725}, {30, 15092}, {32, 32961}, {39, 33249}, {98, 3090}, {114, 1656}, {140, 7861}, {141, 14645}, {147, 7486}, {183, 626}, {187, 33228}, {230, 625}, {315, 33277}, {325, 31275}, {485, 13967}, {486, 8980}, {524, 9165}, {542, 547}, {549, 22515}, {597, 19662}, {631, 7872}, {632, 33813}, {690, 6723}, {1003, 18424}, {1007, 3767}, {1506, 3329}, {1569, 7786}, {1916, 14065}, {2023, 3934}, {2548, 32988}, {2782, 3628}, {2787, 6667}, {3054, 33184}, {3091, 34473}, {3523, 10723}, {3525, 21166}, {3526, 6321}, {3533, 13172}, {3545, 10722}, {3618, 5477}, {3619, 10754}, {3624, 13178}, {3763, 6034}, {3788, 13881}, {3815, 7817}, {3845, 26614}, {3849, 8355}, {5025, 7749}, {5054, 9880}, {5055, 6033}, {5056, 10991}, {5067, 14651}, {5070, 15561}, {5071, 9862}, {5206, 14063}, {5215, 8352}, {5326, 15452}, {5355, 7777}, {5368, 7858}, {5459, 32553}, {5460, 32552}, {5939, 6704}, {5969, 33213}, {5972, 15359}, {5976, 8363}, {6292, 7901}, {6390, 32457}, {6777, 22490}, {6778, 22489}, {6781, 14041}, {7603, 7792}, {7735, 7775}, {7737, 32984}, {7739, 34803}, {7747, 7857}, {7748, 33233}, {7752, 7755}, {7753, 7806}, {7756, 7907}, {7761, 11318}, {7765, 7769}, {7780, 14929}, {7781, 32958}, {7794, 7899}, {7795, 32955}, {7797, 9698}, {7800, 33199}, {7803, 32998}, {7810, 7934}, {7813, 7925}, {7814, 7890}, {7815, 14064}, {7818, 17008}, {7822, 33218}, {7825, 32972}, {7826, 7912}, {7827, 17005}, {7831, 14046}, {7845, 22329}, {7846, 33002}, {7847, 16923}, {7848, 13468}, {7851, 31455}, {7859, 16922}, {7863, 7940}, {7865, 34229}, {7867, 32458}, {7869, 8781}, {7889, 7942}, {7902, 31401}, {7914, 32951}, {7915, 33186}, {7918, 33015}, {7935, 33283}, {7983, 9780}, {8029, 19598}, {8176, 15484}, {8252, 13989}, {8253, 8997}, {8588, 33017}, {8722, 9754}, {9182, 23991}, {9861, 11484}, {10150, 22110}, {10175, 11710}, {10278, 12076}, {10352, 32999}, {11230, 11724}, {11313, 13873}, {11314, 13926}, {11632, 15703}, {11711, 19862}, {12117, 15709}, {13174, 19872}, {13175, 16419}, {13846, 13968}, {13847, 13908}, {13916, 13982}, {13917, 13981}, {15031, 33225}, {15059, 15357}, {15271, 33240}, {15491, 15699}, {15513, 33229}, {15515, 33000}, {16041, 21843}, {16509, 33896}, {19108, 32786}, {19109, 32785}, {35018, 35021

X(6722) = midpoint of X(i) and X(j) for these {i,j}: {2, 5461}, {5, 6036}, {10, 11725}, {114, 11623}, {115, 620}, {230, 625}, {597, 19662}, {2023, 3934}, {5972, 15359}, {6390, 32457}, {6669, 6670}, {6704, 9478}, {6721, 20398}
X(6722) = reflection of X(i) in X(j) for these {i,j}: {6721, 3628}, {20399, 6721}, {22247, 2}, {35022, 620}
X(6722) = complement of X(620)
X(6722) = X(14728)-Ceva conjugate of X(690)
X(6722) = crosssum of X(6) and X(20976)
X(6722) = orthoptic-circle-of-Steiner-inellipse-inverse of X(20099)
X(6722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 99, 31274}, {2, 115, 620}, {2, 671, 9167}, {2, 7844, 4045}, {2, 9166, 2482}, {2, 14061, 115}, {2, 14971, 5461}, {5, 7886, 6680}, {5, 34127, 6036}, {99, 31274, 620}, {115, 2482, 148}, {115, 14061, 5461}, {115, 14971, 14061}, {115, 31274, 99}, {148, 9166, 115}, {620, 5461, 115}, {1007, 3767, 7798}, {1007, 7798, 7764}, {1506, 7828, 7829}, {3628, 20398, 20399}, {3767, 7862, 7764}, {3767, 32969, 7862}, {5025, 7749, 7830}, {7746, 7887, 626}, {7752, 7755, 7838}, {7798, 7862, 1007}, {7828, 32967, 1506}, {7852, 32992, 6704}, {7857, 32966, 7747}, {7925, 14568, 7813}, {7934, 17004, 7810}, {7942, 16921, 7889}, {14046, 17006, 7831}, {22247, 35022, 620}, {32832, 33248, 7867}

X(6723) = CENTROID OF {A,B,C,X(125)}

Barycentrics    2 sin 2A sin^2(B - C) + sin 2B sin^2(A - C) + sin 2C sin^2(A - B) : :
Barycentrics    2(b^2 + c^2 - a^2)(b^2 - c^2)^2 + (c^2 + a^2 - b^2)(c^2 - a^2)^2 + (a^2 + b^2 - c^2)(a^2 - b^2)^2 : :
Barycentrics    2*a^6 - 2*a^4*b^2 - 3*a^2*b^4 + 3*b^6 - 2*a^4*c^2 + 8*a^2*b^2*c^2 - 3*b^4*c^2 - 3*a^2*c^4 - 3*b^2*c^4 + 3*c^6
X(6723) = 9 X[2] - X[110] = 3 X[2] + X[125] = X[110] + 3 X[125] = 5 X[632] - X[1511] = 5 X[5] - X[1539] = X[113] - 5 X[1656] = X[74] + 7 X[3090] = 5 X[125] - X[3448] = 15 X[2] + X[3448] = 5 X[110] + 3 X[3448] = X[265] + 7 X[3526] = X[895] + 7 X[3619] = 5 X[3618] - X[5095] = 5 X[3763] - X[5181] = 5 X[110] - 9 X[5642] = 5 X[2] - X[5642] = 5 X[125] + 3 X[5642] = X[3448] + 3 X[5642] = X[1112] - 3 X[5943] = 3 X[5642] - 5 X[5972] = X[110] - 3 X[5972] = X[3448] + 5 X[5972] = 11 X[5070] - X[6053] = X[1539] + 5 X[6699]

X(6723) lies on these lines: {2, 98}, {5, 1539}, {74, 3090}, {113, 1656}, {265, 3526}, {511, 5159}, {541, 547}, {632, 1511}, {895, 3619}, {974, 5907}, {1112, 5943}, {3066, 5094}, {3618, 5095}, {3628, 5663}, {3763, 5181}, {5070, 6053}

X(6723) = midpoint of X(i) and X(j) for these {i,j}: {5,6699}, {125,5972}, {974,5907}
X(6723) = complement of X(5972)
X(6723) = X(5159)-of-1st-Brocard-triangle
X(6723) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5984)

Homologous images: X(6724)-X(6796)

This preamble and centers X(6724)-X(6796) were contributed by Randy Hutson, March 15, 2015.

The term "homologous" means "having the same relation or relative position to". Let T₁ and T₂ be two triangles, and P a point in the plane of triangle T₁. The point P′ in the plane of triangle T₂ is a T₁-to-T₂ homologous image of P if P′ has the same relation or relative position to T₂ as P has to T₁. If T₁ and T₂ are similar triangles, this meaning is unambiguous, and we call P′ the T₁-to-T₂ similarity image of P. If, however, T₁ and T₂ are not similar, the term "homologous images" can apply to several kinds of mappings. We present three such mappings here.

Trilinear image:

Let P be a point with trilinears x : y : z wrt T₁. Then the T₁-to-T₂ trilinear image of P is the point P′ with trilinears x : y : z wrt T₂. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points. Parallel lines do not remain parallel under this mapping, so that points on the line at infinity map to finite points. Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

Barycentric image:

Let P be a point with barycentrics x : y : z wrt T₁. Then the T₁-to-T₂ barycentric image of P is the point P′ with barycentrics x : y : z wrt T₂. This is an affine collineation, preserving collinearities as well as ratios of distances between collinear points. Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Conics map to conics of the same type (ellipses, including circles, map to ellipses, parabolas to parabolas, hyperbolas to hyperbolas). Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

Functional image:

Let the sidelengths of T₁ be denoted a₁, b₁, c₁ and the sidelengths of T₂ be denoted a₂, b₂, c₂. Let P be a point with coordinates f(a₁,b₁,c₁) : g(b₁,c₁,a₁) : h(c₁,a₁,b₁) with respect to T₁, where f, g, h either have the same degree of homogeneity, or else one is the zero function and the other two have the same degree of homogeneity. (If P is a center of T₁, then f = g = h.) Then the T₁-to-T₂-functional image of P is the point P′ with coordinates f(a₂,b₂,c₂) : g(b₂,c₂,a₂) : h(c₂,a₂,b₂) with respect to T₂. Coordinates here can be trilinears or barycentrics, as long as the same system is used for T₁ and T₂. P′ serves the same 'function' wrt T₂ (e.g., centroid, circumcenter, 1st Brocard point, A-vertex of orthic triangle, etc.) as P serves wrt T₁. For example, the orthic-to-excentral functional image of X(5) = X(40), since X(5) and X(40) are the respective circumcenters of the orthic and excentral triangles. This is a non-affine collineation, preserving collinearities, but not ratios of distances between collinear points (except where these distance ratios define the point, such as midpoints, reflections, etc.). Parallel lines remain parallel under this mapping, so that the line at infinity maps to itself. Circles map to circles. Other conics map to conics, but not necessarily of the same type (e.g., the MacBeath circumconic of T₁ may be an ellipse, but maps to the MacBeath circumconic of T₂, which may be a hyperbola.) Circumconics of T₁ map to circumconics of T₂ and inconics of T₁ map to inconics of T₂.

X(6724) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(3)

Trilinears cos(B/2 - C/2) : :
Trilinears cos(B′ - C′) : : , where A′B′C′ is the excentral triangle

Let A′B′C′ be the incentral triangle. Let O_A be the circle centered at A and passing through A′. Define O_B and O_C cyclically. X(6724) is the trilinear pole of the Monge line of O_A, O_B, O_C.

X(6724) lies on these lines: {1,6732}, {174,188}, {6727,6733}

X(6724) = isogonal conjugate of X(6727)
X(6724) = SS(a → a') of X(5), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6724) = trilinear square root of X(12)
X(6724) = {X(174),X(188)}-harmonic conjugate of X(266)

X(6725) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(6)

Trilinears    cot(A/2) cos(B/2 - C/2) : :
Trilinears    tan A′ cos(B′ - C′) : : , where A′B′C′ is the excentral triangle
Trilinears    1/|I_AJ_A| : : , where I_AI_BI_C is the incentral triangle, and J_AJ_BJ_C is the excentral triangle

X(6725) lies on line {188,259}

X(6725) = SS(a → a') of X(53), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6725) = trilinear square root of X(6057)

X(6726) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(57)

Trilinears    cos(A/2) cot(A/2) : :
Trilinears    sin A′ tan A′, where A′B′C′ is the excentral triangle
Trilinears    1/|I_AJ_A| : : , where I_AI_BI_C is the mixtilinear incentral triangle, and J_AJ_BJ_C is the mixtilinear excentral triangle

X(6726) lies on line {174,188}

X(6726) = SS(a → a') of X(25), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6726) = trilinear square root of X(480)

X(6727) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(191)

Trilinears sec(B/2 - C/2) : :
Trilinears sec(B′ - C′) : : , where A′B′C′ is the excentral triangle

The trilinear polar of X(6727) passes through X(6729). (Randy Hutson, April 11, 2015)

X(6727) lies on these lines: {6724,6733}

X(6727) = isogonal conjugate of X(6724)
X(6727) = SS(a → a') of X(54), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6727) = trilinear square root of X(60)

X(6728) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(513)

Trilinears sin(B/2 - C/2) : :
Trilinears sin(B′ - C′) : : , where A′B′C′ is the excentral triangle

X(6728) lies on these lines: {6729,6730}

X(6728) = isogonal conjugate of X(6733)
X(6728) = SS(a → a') of X(523), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6728) = trilinear square root of X(11)
X(6728) = (trilinear polar of X(174))∩(trilinear polar of X(188))
X(6728) = crossdifference of every pair of points on line X(259)X(260)

X(6729) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(649)

Trilinears a sin(B/2 - C/2) : :
Trilinears sin 2A′ sin(B′ - C′) : : , where A′B′C′ is the excentral triangle

X(6729) lies on these lines: {6728,6730}

X(6729) = SS(a → a') of X(647), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6729) = trilinear square root of X(3271)
X(6729) = crossdifference of every pair of points on line X(174)X(188)

X(6730) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(650)

Trilinears cot(A/2) sin(B/2 - C/2) : :
Trilinears tan A′ sin(B′ - C′) : : , where A′B′C′ is the excentral triangle

X(6730) lies on these lines: {6728,6729}

X(6730) = SS(a → a') of X(2501), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6730) = trilinear square root of X(4081)

X(6731) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(1743)

Trilinears csc(A/2) cot(A/2) : :
Trilinears sec A′ tan A′ : : , where A′B′C′ is the excentral triangle

The trilinear polar of X(6731) passes through X(6730).

X(6731) lies on these lines: {8,178}, {174,556}, {188,259}

X(6731) = SS(a → a') of X(393), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6731) = {X(483),X(3082)}-harmonic conjugate of X(8)

X(6732) = EXCENTRAL-TO-ABC TRILINEAR IMAGE OF X(3659)

Trilinears    1 - cos(B/2 - C/2) : :
Trilinears    1 - cos(B′ - C′) : : , where A′B′C′ is the excentral triangle
Trilinears    sin²(B′/2 - C′/2) : : , where A′B′C′ is the excentral triangle

Let A′,B′,C′ be the second points of intersection of the angle bisectors of ABC with its incircle. The tangents at A′, B′, C′ form a triangle A″B″C″, introduced here as the outer tangential mid-arc triangle. A″B″C″ is the reflection in X(1) of the (inner) tangential mid-arc triangle. A″B″C″ is homothetic to the inner Hutson triangle (defined at X(363)), and the center of homothety is X(6732).

X(6732) lies on these lines: {1,6724}, {174,7670}, {266,8092}, {5934,8080}, {7588,8109}, {8076,8107}, {8082,8111}, {8090,8138}

X(6732) = SS(a → a') of X(11), where A′B′C′ is the excentral triangle (trilinear substitution)

X(6733) = INTOUCH-TO-ABC TRILINEAR IMAGE OF X(11)

Trilinears csc(B/2 - C/2) : :
Trilinears csc(B′ - C′) : : , where A′B′C′ is the excentral triangle

X(6733) lies on these lines: {6724,6727}

X(6733) = isogonal conjugate of X(6728)
X(6733) = excentral-to-ABC trilinear image of X(1768)
X(6733) = SS(a → a') of X(110), where A′B′C′ is the excentral triangle (trilinear substitution)
X(6733) = trilinear square root of X(59)
X(6733) = trilinear pole of line X(259)X(260)

X(6734) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(35)

Trilinears (b+c-a)*((b+c)*a^2+2*a*b*c-(b^2-c^2)*(b-c))/a
Barycentrics 2*(s-a)*(R+r)+r*a : :

X(6734) lies on these lines: {1,2}, {3,3419}, {4,63}, {5,72}, {11,960}, {12,518}, {20,5175}, {21,950}, {29,270}, {37,5742}, {40,3434}, {56,5792}, {57,377}, {75,225}

X(6734) = complement of X(34772)
X(6734) = {X(2),X(8)}-harmonic conjugate of X(78)
X(6734) = {X(8),X(10)}-harmonic conjugate of X(6735)

X(6735) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(36)

Trilinears (b+c-a)*((b+c)*a^2-2*a*b*c-(b^2-c^2)*(b-c))/a : :
Barycentrics 2*(s-a)*(R-r)-r*a : :
X(6735) = X[908] + 2 X[1145] = 5 X[1698] - 3 X[3582] = 2 X[3036] + X[3689] = 5X[3617] + X[3935] = X[8] + 2 X[6745]

The trilinear polar of X(6735) passes through X(2804).

X(6735) lies on these lines: {1,2}, {11,3880}, {12,3838}, {21,5795}, {40,3436}, {63,2096}, {72,5690}, {100,515}, {119,517}, {153,2950}, {224,5534}, {280,6556}, {281,3692}, {307,322}, {318,341}, {355,1012}, {377,1706}, {392,3820}, {495,3753}, {497,3895}, {516,5080}, {522,3717}, {529,1155}, {594,3965}, {765,1861}, {944,4855}, {950,3871}, {952,5440}, {962,5828}, {1018,5179}, {1056,3306}, {1146,3693}, {1260,3419}, {1319,3035}, {1329,3057}, {1376,1470}, {1697,2478}, {1738,4695}, {1837,3913}, {2093,5905}, {2321,4165}, {2551,5250}, {2802,3814}, {2975,6684}, {3036,3689}, {3158,5727}, {3434,5587}, {3452,3877}, {3583,5541}, {3813,3893}, {3816,5919}, {3826,4731}, {3868,4848}, {3876,5837}, {3885,4193}, {3911,5193}, {3951,5815}, {4004,6147}, {4095,4167}, {4188,4311}, {5048,5854}, {5082,5818}, {5288,5445}

X(6735) = midpoint of X(i) and X(j) for these {i,j}: {8,4511}, {100,5176}, {3583,5541}
X(6735) = reflection of X(i) ini X(j) for these (i,j): (11,5123), (1319,3035), (1519,119), (1737,10), (4511, 6745)
X(6735) = AC-incircle-inverse of X(6790)
X(6735) = circumconic-centered-at-X(1)-inverse of X(78)
X(6735) = X(2743)-complementary conjugate of X(513)
X(6735) = X(3262)-Ceva conjugate of X(908)
X(6735) = X(1145)-cross conjugate of X(8)
X(6735) = crossdifference of every pair of points on line X(604)X(649)
X(6735) = X(i)-beth conjugate of X(j) for these (i,j): (8,1737}, (341,6735}, (643,2077}, (1043,519}, (3699,6735}, (6735,517}
X(6735) = X(i)-gimel conjugate of X(j) for these (i,j): (513,6735), (3536,6735)
X(6735) = X(i)-isoconjugate of X(j) for these (i,j): {34,1795}, {56,104}, {57,909}, {104,56}, {269,2342}, {513,2720}, {651,2423}, {909,57}, {1398,1809}, {1412,2250}, {1415,2401}, {1795,34}, {1809,1398}, {2250,1412}, {2342,269}, {2401,1415}, {2423,651}, {2720,513}
X(6735) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2057,78), (2,8,3872), (8,10,6734), (8,4420,6737), (8,5552,1), (10,6736,8), (200,3679,8), (495,3753,5249), (3421,5657,63), (3626,6737,8)

X(6736) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(56)

Trilinears (b+c-a)^2*((b+c)*a+(b-c)^2)/a : :
Barycentrics 2*(s-a)*(2*R-r)-r*a : :

X(6736) lies on these lines: {1,2}, {11,3893}, {40,2123}, {55,5795}, {72,1145}, {100,4297}, {142,3698}, {210,5837}, {226,5836}, {322,3668}, {341,4082}, {346,6556}, {388,1706}, {404,4315}, {480,1837}, {495,3824}, {497,2136}, {515,5687}, {516,3436}, {518,4848}

X(6736) = complement of X(36846)
X(6736) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10,8582), (2,8,4853), (8,10,4847)

X(6737) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(65)

Trilinears (b+c-a)^2*(2*a^2+(b+c)*a-(b-c)^2)/a : :
Barycentrics 2*(s-a)*(2*R+r)-r*a : :

X(6737) lies on these lines: {1,2}, {72,515}, {210,5793}, {219,2321}, {950,960}, {3057,3059}

X(6737) = {X(1),X(8)}-harmonic conjugate of X(4847)
X(6737) = anticomplement of X(6738)

X(6738) = COMPLEMENT OF X(6737)

Trilinears    (2*a^4-3*(b+c)*a^3-(b+c)^2*a^2+3*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)/a : :
Trilinears    2*cos(A)+3*cos(B)+3*cos(C)-cos(B-C)+1 : :
Barycentrics    4*a*R-(2*s-3*a)*r : :

X(6738) lies on these lines: {1,2}, {4,3671}, {7,5691}, {20,3339}, {40,3488}, {46,4304}, {57,3486}, {65,516}

X(6738) = midpoint of X(65) and X(950)
X(6738) = reflection of X(4298) in X(942)
X(6738) = complement of X(6737)
X(6738) = X(8)-of-3rd-pedal-triangle-of-X(1)

X(6739) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(74)

Trilinears cos((B-C)/2)*(1-2*cos(A))*(cos(B-C)-2*cos(A))*csc(A/2) : :
Barycentrics (b+c)*(a^2-b^2+b*c-c^2)*(2*a^4-(b^2+c^2)*a^2-(b^2-c^2)^2) : :

The locus of the trilinear square of a point on the line at infinity is the inellipse centered at X(10). This inellipse passes through X(244), X(1099), X(1109), X(1111), X(4712), X(4736) and X(4738), and its Brianchon point (perspector) is X(75). The tangents to this inellipse at X(1099) and X(4736) intersect at X(6739).

X(6739) lies on these lines: {2,6740}, {8,6742}, {10,6741}, {11,214}, {30,113}, {4707,4736}

X(6739) = reflection of X(3109) in X(5972)
X(6739) = crossdifference of every pair of points on line X(2159)X(2433)
X(6739) = complement of X(6740)

X(6740) = ISOGONAL CONJUGATE OF X(1464)

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

X(6740) lies on these lines: {1,564}, {2,6739}, {8,643}, {10,21}, {30,74}

X(6740) = isogonal conjugate of X(1464)
X(6740) = anticomplement of X(6739)
X(6740) = trilinear pole of line X(9)X(1021)

X(6741) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(110)

Barycentrics (b+c-a)*(b-c) *(b^2-c^2)*(a^2-b^2-b*c-c^2) : :

X(6741) lies on these lines: {2,6742}, {8,643}, {10,6739}, {11,4092}, {125,523}

X(6741) = center of hyperbola {A,B,C,X(8),X(10)}
X(6741) = crossdifference of every pair of points on line X(163)X(2420)
X(6741) = complement of X(6742)

X(6742) = TRILINEAR POLE OF LINE X(9)X(46)

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

Line X(9)X(46) is the trilinear polar of the Feuerbach point, X(11), wrt the Feuerbach triangle.

X(6742) lies on these lines: {1,564}, {2,6741}, {8,6739}, {110,476}

X(6742) = isogonal conjugate of X(2605)
X(6742) = isotomic conjugate of X(4467)
X(6742) = anticomplement of X(6741)

X(6743) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(942)

Barycentrics 2*(s-a)*(4*R+r)-r*a : :
Trilinears b c (-a+b+c)*(2*a^3-(b+c)*a^2-2*(b+c)^2*a+(b^2-c^2)*(b-c)) : : X(6743) lies on these lines: {1,2}, {72,516}, {210,950}, {220,2321}, {960,5853}

X(6743) = anticomplement of X(6744)

X(6744) = COMPLEMENT OF X(6743)

Trilinears    (2*a^4-3*(b+c)*a^3-(b^2+c^2+10*b*c)*a^2+3*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2)/a : :
Trilinears    2*cos(A)+3*cos(B)+3*cos(C)-cos(B-C)+5 : :
Barycentrics    8*a*R-(2*s-3*a)*r : :

X(6744) lies on these lines: {1,2}, {4,5542}, {516,942}, {3812,5853}

X(6744) = complement of X(6743)
X(6744) = incircle-inverse of X(5529)
X(6744) = X(10) of 3rd pedal triangle of X(1)

X(6745) = EXCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(1155)

Trilinears    (b+c-a)*(2*a^2-a*(b+c)-(b-c)^2)/a : :
Trilinears    (cos(B)+cos(C)-2*cos(A))*csc(A/2)^2 : :
Trilinears    csc²(A/2)[2 sin²(A/2) - sin²(B/2) - sin²(C/2)] : :
Trilinears    csc²(A/2)[sin²(B/2) + sin²(C/2)] - 2 : :
Trilinears    sec² A′ (2 cos² A′ - cos² B′ - cos² C′) : : , where A′B′C′ is the excentral triangle
Trilinears    sec² A′ (cos² B′ + cos² C′) - 2 : : , where A′B′C′ is the excentral triangle
Barycentrics    (b + c - a)(2a² - b² - c² - ab - ac + 2bc) : :
Barycentrics    2*(s-a)*(2*R-r)-3*r*a : :

X(6745) is the point of intersection of the line X(1)X(8) and the trilinear polar of X(8); c.f. X(1323).

X(6745) lies on these lines: {1,2}, {9,1776}, {11,3689}, {55,3452}, {72,5884}, {100,516}, {522,656}

X(6745) = midpoint of X(i) and X(j) for these (i,j): (11,3689), (100,908)
X(6745) = reflection of X(3911) in X(3035)
X(6745) = inverse-in-circumconic-centered-at-X(1) of X(200)
X(6745) = trilinear pole of line X(6068)X(6366) (the tangent to the Mandart inellipse at the antipode of X(11), i.e. X(6068))
X(6745) = crossdifference of every pair of points on line X(56)X(649)
X(6745) = centroid of {A,B,C,X(3935)}
X(6745) = perspector of ABC and side-triangle of extouch and Yff contact triangles

X(6746) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(12)

Trilinears (sec A)[1 - cos 3A cos(B - C)] : :

Let A′B′C′ be the reflection triangle. Let A″ be the orthogonal projection of A on line B′C′, and define B″, C″ cyclically. Triangle A″B″C″ is perspective to the orthic triangle, and the perspector is X(6746).

X(6746) lies on these lines: {4,94}, {24,5944}, {25,1614}, {51,235}, {52,427}, {1593,3060}

X(6746) = X(12) of orthic triangle if ABC is acute
X(6746) = {X(4),X(143)}-harmonic conjugate of X(1112)

X(6747) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(31)

Trilinears (sec A)(sin² 2B + sin² 2C) : :

X(6747) lies on these lines: {4,54}, {53,232}, {125,2052}

X(6747) = X(31) of orthic triangle if ABC is acute

X(6748) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(37)

Trilinears (tan A)(cos A + 2 sin B sin C) : :
Trilinears 2 sec A + csc B csc C : :

X(6748) lies on these lines: {4,6}, {5,577}, {25,160}, {30,216}, {32,1595}, {39,6756}, {50,252}, {51,6752}, {125,138}, {140,233}, {141,317}, {184,6755}, {230,427}, {232,428}, {235,5063}, {264,524}, {275,1971}, {297,3589}, {340,3631}, {395,473}, {396,472}, {460,1843}, {468,3055}, {546,3284}, {566,6240}, {570,3575}, {590,1586}, {594,5081}, {615,1585}, {1100,1785}, {1321,1322}, {1593,1609}, {1596,5065}, {1598,2548}, {1841,1877}, {1885,3003}, {1907,1968}, {3053,3088}, {3054,5094}, {3627,5158}, {5064,5306}

Let A′B′C′ be the orthic triangle. Let O_A be the circle with center A passing through A′. Let R_A be the radical axis of O_A and the nine-point circle; define R_B, R_C cyclically. Let A″ = R_B∩R_C, B″ = R_C∩R_A, C″ = R_A∩R_B. The lines A′A″, B′B″, C′C′ concur in X(6748).

X(6748) = isogonal conjugate of X(31626)
X(6748) = crosssum of X(3) and X(216)
X(6748) = crosspoint of X(4) and X(275)
X(6748) = {X(4),X(6)}-harmonic conjugate of X(53)
X(6748) = inverse-in-orthosymmedial-circle of X(53)
X(6748) = X(37) of orthic triangle if ABC is acute
X(6748) = {X(1321),X(1322)}-harmonic conjugate of X(7507)

X(6749) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(45)

Trilinears    4 sin A + tan A cos(B - C) : :
                     = 5 sin A + 2 tan A cos B cos C : :
                     = 3 sin A + 2 tan A sin B sin C : :
                     = (tan A)( 2 sec A + 5 sec B sec C) : :

X(6749) lies on these lines: {4,6}, {5,3284}, {140,577}, {216,550}, {239,5094}

X(6749) = {X(4),X(6)}-harmonic conjugate of X(1990)
X(6749) = inverse-in-orthosymmedial-circle of X(1990)
X(6749) = X(45) of orthic triangle if ABC is acute

X(6750) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(58)

Trilinears sin(A + T)[cos A sin(A + T) - cos B sin(B + T) - cos C sin(C + T)] : : , T as at X(389)

X(6750) lies on these lines: {4,54}, {5,53}

X(6750) = X(58) of orthic triangle if ABC is acute
X(6750) = orthic isogonal conjugate of X(389)
X(6750) = X(4)-Ceva conjugate of X(389)
X(6750) = inverse-in-polar-circle of X(3484)

X(6751) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(69)

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

X(6751) is the QA-P19 center (AntiComplement of QA-P16 wrt the QA-Diagonal Triangle) of quadrangle ABCX(4); see QA-P19.

X(6751) lies on these lines: {51,53}, {185,1503}

X(6751) = X(69) of orthic triangle
X(6751) = orthic isotomic conjugate of X(52)
X(6751) = anticomplement of X(53) wrt orthic triangle
X(6751) = {X(185),X(1843)}-harmonic conjugate of X(6752)

X(6752) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(75)

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

X(6752) lies on these lines: {6,1987}, {185,1503}

X(6752) = orthic isotomic conjugate of X(4)
X(6752) = X(75) of orthic triangle if ABC is acute
X(6752) = {X(185),X(1843)}-harmonic conjugate of X(6751)

X(6753) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(652)

Trilinears    (tan A)[b cos(A - B) - c cos(A - C)] : :
Barycentrics    (tan A)(sec 2B - sec 2C) : :
Barycentrics    (sin A tan A)[b cos(A - B) - c cos(A - C)] : :
Barycentrics    a²(b² - c²)(a⁴ + b⁴ + c⁴ - 2a²b² - 2a²c²)/(b² + c² - a²) : :

X(6753) is the perspector of the circumconic centered at X(135), which is hyperbola {A,B,C,X(4),X(24)}. This hyperbola is the isogonal conjugate of line X(3)X(68), and intersects the circumcircle (other than at A, B, and C) at X(1299).
The trilinear polar of X(6753) passes through X(6754).

X(6753) lies on these lines: {230,231}, {512,2623}

X(6753) = X(652) of orthic triangle if ABC is acute
X(6753) = (trilinear polar of X(4))∩(trilinear polar of X(24))
X(6753) = crossdifference of every pair of points on line X(3)X(68)
X(6753) = X(2)-Ceva conjugate of X(135)
X(6753) = X(4)-Ceva conjugate of X(6754)
X(6753) = polar conjugate of isogonal conjugate of X(34952)
X(6753) = polar conjugate of anticomplement of X(39013)
X(6753) = orthic-isogonal conjugate of X(6754)
X(6753) = X(63)-isoconjugate of X(925)

X(6754) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(653)

Barycentrics (tan A)(sec 2B - sec 2C)² : :
Barycentrics [a²(b² - c²)(a⁴ + b⁴ + c⁴ - 2a²b² - 2a²c²)]²/(b² + c² - a²) : :

X(6754) lies on these lines: {52,5095} et al.

X(6754) = isogonal conjugate of isotomic conjugate of X(34338)
X(6754) = X(653) of orthic triangle if ABC is acute
X(6754) = trilinear pole, wrt orthic triangle, of line X(4)X(52)
X(6754) = X(4)-Ceva conjugate of X(6753)
X(6754) = orthic isogonal conjugate of X(6753)
X(6754) = polar conjugate of isotomic conjugate of X(39013)

X(6755) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(940)

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

Let A′B′C′ be the Euler triangle. Let A″B″C″ be the triangle bounded by the trilinear polars of A′, B′, C′. The lines AA″, BB″, CC″ concur in X(53). Let A*B*C* be the triangle bounded by the trilinear polars of A″, B″, C″. The lines AA*, BB*, CC* concur in X(6755).

X(6755) lies on these lines: {2,3}, {51,53}

X(6755) = X(940) of orthic triangle if ABC is acute

X(6756) = EXCENTRAL-TO-ABC FUNCTIONAL IMAGE OF X(942)

Trilinears 2 sec A - 2 cos A + cos(B - C) : :

As a point of the Euler line, X(6756) has Shingawa coefficients (F, -2E - 3F).

X(6756) lies on these lines: {2,3}, {52,1843}

X(6756) = X(942) of orthic triangle if ABC is acute
X(6756) = X(5) of 2nd pedal triangle of X(4)
X(6756) = X(3) of 3rd pedal triangle of X(4)
X(6756) = {X(3),X(4)}-harmonic conjugate of X(1595)
X(6756) = homothetic center of orthocevian triangle of X(2) and Euler triangle
X(6756) = {X(4),X(24)}-harmonic conjugate of X(427)

X(6757) = INCENTRAL-TO-ABC BARYCENTRIC IMAGE OF X(500)

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

A construction for X(6757) appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.

X(6757) lies on these lines: {1,564}, {5,523}, {8,79}, {92,5248}, {265,355}, {281,451}, {993,1789}, {1441,3841}, {2321,4053}, {3678,6358}

X(6757) = Feuerbach-to-ABC barycentric image of X(5948)
X(6757) = trilinear product of vertices of X(1)-altimedial triangle

X(6758) = FEUERBACH-TO-ABC BARYCENTRIC IMAGE OF X(11)

Barycentrics sin A sin²(B - C) - sin B sin²(C - A) - sin C sin²(A - B) : :

X(6758) lies on these lines: {2,1109}, {8,4736}, {100,523}, {6327,6360}

X(6758) = isotomic conjugate of X(7372)
X(6758) = anticomplement of X(1109)
X(6758) = polar conjugate of isogonal conjugate of X(23084)

X(6759) = INTOUCH-TO-ABC FUNCTIONAL IMAGE OF X(10)

Barycentrics a²[a⁸ - 3a⁶(b² + c²) + 3a⁴(b⁴ + c⁴) - a²(b² - c²)²(b² + c²) + 2b²c²(b² - c²)²] : :

Let A′B′C′ be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); X(6759) = X(4)-of-A′B′C′. (Randy Hutson, July 21, 2017)

X(6759) lies on these lines: {3,64}, {4,54}, {5,182}, {6,1598}, {20,110}, {22,5562}, {23,5889}, {24,185}, {25,389}, {30,156}, {52,161}, {155,159}

X(6759) = X(10) of tangential triangle if ABC is acute
X(6759) = inverse-in-circumcircle of X(6760)

X(6760) = INVERSE-IN-CIRCUMCIRCLE OF X(6759)

Barycentrics a²(b² + c² - a²)[a¹² - 2a¹⁰(b² + c²) - a⁸(2b⁴ - 7b²c² + 2c⁴) + 8a⁶(b² - c²)²(b² + c²) - a⁴(b² - c²)²(7b⁴ + 12b²c² + 7c⁴) + 2a²(b² - c²)²(b² + c²)³ - b²c²(b² - c²)⁴] : :

Let A′B′C′ be the reflection of the anticevian triangle of X(3) in the trilinear polar of X(3). The lines AA′, BB′, CC′ concur in X(6760).

X(6760) lies on these lines: {2,6761}, {3,64}, {30,1294}

X(6760) = complement of X(6761)
X(6760) = inverse-in-circumcircle of X(6759)
X(6760) = crosssum of X(36302) and X(36303)
X(6760) = crossdifference of every pair of points on line X(53)X(6587)

X(6761) = ANTICOMPLEMENT OF X(6760)

Barycentrics [a¹² - 3a¹⁰(b² + c²) + a⁸(3b⁴ + 7b²c² + 3c⁴) - 2a⁶(b² + c²)(b⁴ + c⁴) + 3a⁴(b² - c²)²(b⁴ + c⁴) - a²(b² - c²)²(3b⁶ + b⁴c² + b²c⁴ + 3c⁶) + (b² - c²)⁴(b⁴ + 3b²c² + c⁴)]/(b² + c² - a²) : :

X(6761) lies on these lines: {2,6760}, {4,51}, {5,3462}, {30,5667}

X(6761) = anticomplement of X(6760)
X(6761) = inverse-in-polar-circle of X(389)

X(6762) = INTOUCH-TO-EXCENTRAL SIMILARITY IMAGE OF X(8)

Barycentrics a*(a^3+(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b+c)^3) : :

X(6762) lies on these lines: {1,6}, {3,3158}, {8,57}, {10,1056}, {12,5231}, {20,5853}, {40,376}, {56,200}, {63,145}, {65,4853}, {84,517}

X(6762) = mixtilinear-incentral-to-mixtilinear-excentral similarity image of X(9)
X(6762) = {X(8),X(57)}-harmonic conjugate of X(1706)
X(6762) = X(8) of tangential triangle of excentral triangle
X(6762) = X(1498) of excentral triangle

X(6763) = INTOUCH-TO-EXCENTRAL SIMILARITY IMAGE OF X(12)

Trilinears S_A - rR : :

X(6763) lies on these lines: {1,21}, {2,3337}, {3,5904}, {4,5536}, {8,484}, {9,583}, {10,3218}, {12,57}, {35,518}, {36,72}, {40,550}, {56,3927}, {65,5258}, {75,267}, {79,2886}

X(6763) = {X(1),X(63)}-harmonic conjugate of X(191)
X(6763) = X(12) of tangential triangle of excentral triangle

X(6764) = MIXTILINEAR-INCENTRAL-TO-MIXTILINEAR-EXCENTRAL SIMILARITY IMAGE OF X(8)

Barycentrics a^4-2*(b+c)*a^3-12*b*c*a^2+2*(b+c)^3*a-(b^2-c^2)^2 : :

X(6764) lies on these lines: {1,2}, {7,3555}, {20,5853}, {518,962}

X(6764) = anticomplement of X(6765)

X(6765) = COMPLEMENT OF X(6764)

Barycentrics a*(a^3-(b+c)*a^2-(b^2+6*b*c+c^2)*a+(b+c)^3) : :

X(6765) lies on these lines: {1,2}, {3,3158}, {4,5853}, {6,4515}, {9,3295}, {40,518}, {56,3689}, {57,3555}, {1058,3452}

X(6765) = complement of X(6764)
X(6765) = perspector of pedal and antipedal triangles of X(84)
X(6765) = X(10) of 3rd antipedal triangle of X(1)

X(6766) = MIXTILINEAR-INCENTRAL-TO-MIXTILINEAR-EXCENTRAL SIMILARITY IMAGE OF X(40)

Barycentrics a*(a^6-(3*b^2+2*b*c+3*c^2)*a^4-12*(b+c)*b*c*a^3+(3*b^4+26*b^2*c^2+3*c^4)*a^2+12*(b^2-c^2)*(b-c)*b*c*a-(b^2-c^2)^2*(b-c)^2) : :

Let A′B′C′ be the Hutson extouch triangle (defined at X(5731)). Let L_A be the tangent to the A-excircle at A′, and define L_B, L_C cyclically. Let A″ = L_B∩L_C, B″ = L_C∩L_A, C″ = L_A∩L_B. Triangle A″B″C″ is homothetic to ABC at X(57), and X(6766) = X(40) of A″B″C″.

X(6766) lies on these lines: {1,3}, {9,4301}

X(6767) = CENTER OF THE APOLLONIAN CIRCLE OF THE MIXTILINEAR CIRCLES

Trilinears 4 + cos A : 4 + cos B : 4 + cos C
X(6767) = 4X(1) - X(1159) = 4R*X(1) + r*X(3)

Let A′B′C′ be a triangle in the poristic system of triangles (triangles that share the same incircle and circumcircle with ABC). Let X be the trilinear pole of line X(1)X(3) wrt A′B′C′. The locus of X as A′B′C′ varies is an ellipse, E, centered at X(3) with vertices X(1381) and X(1382). E is the poristic locus of X(651). Let F₁ and F₂ be the foci of E. Then X(6767) is the {F₁,F₂}-harmonic conjugate of X(1).

Let L be a line through X(1), and let P and Q be the circumcircle intercepts of L. Let U be the {X(1),P}-harmonic conjugate of Q. The locus of U as L varies is an ellipse centered at X(6767).

Let (Ap) denote the Apollonian circle of the mixtilinear circles. The touch point of (Ap) and the A-mixtilinear circle is the point with barycentrics a(a^2+2 a b-3 b^2+2 a c+6 b c-3 c^2) : -2 b^2 (-a+b-c) : 2 c^2 (a+b-c)}. (Peter Moses, October 4, 2015)

The vertices of the 8th mixtilinear triangle are the touchpoints of the inner Apollonian circle of the mixtilinear incircles, with A-vertex, A′, given by these trilinears:

A′ = a^2 + 2*a*b - 3*b^2 + 2*a*c + 6*b*c - 3*c^2 : -2*b*(-a + b - c) : 2*(a + b - c)*c (Dan Reznik and Peter Moses, September 10, 2021).

If you have The Geometer's Sketchpad, you can view X(6767).

X(6767) lies on these lines: {1,3}, {5,1058}, {8,5284}, {11,5055}, {12,3851}, {21,3623}, {30,390}, {145,405}, {381,495}, {382,388}, {386,1616}, {392,3870}, {474,3622}, {480,4915}, {496,1656}, {498,5070}, {499,5326}, {519,1001}, {546,5261}, {549,5281}, {550,3600}, {551,1376}, {611,5093}, {758,4068}, {938,5690}, {943,7320}, {952,954}, {956,1621}, {958,3244}, {962,6147}, {993,4428}, {1000,2346}, {1012,7967}, {1015,5024}, {1057,1807}, {1059,7100}, {1124,6418}, {1125,3913}, {1149,2177}, {1203,2334}, {1260,3872}, {1335,6417}, {1384,2242}, {1478,3058}, {1479,3843}, {1480,2293}, {1483,3560}, {1597,1870}, {1598,6198}, {1657,4294}, {2066,6199}, {2308,3915}, {3086,3526}, {3297,3312}, {3298,3311}, {3299,6501}, {3301,6500}, {3530,5265}, {3534,4293}, {3555,3927}, {3616,5687}, {3621,5047}, {3633,5259}, {3635,5248}, {3646,4882}, {3656,4342}, {3679,4423}, {3753,3895}, {3877,3957}, {3880,6600}, {3890,5730}, {3898,5289}, {3920,5020}, {4302,5434}, {4309,7354}, {4387,4692}, {5044,6765}, {5054,5218}, {5073,6284}, {5076,5229}, {5219,7743}, {5414,6395}, {5603,5719}, {5703,5901}, {5722,5790}

X(6767) = midpoint of X(1) and X(6766)
X(6767) = reflection of X(6769) in X(1385)
X(6767) = X(3)-of-8th-mixtilinear-triangle
X(6767) = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X(3)
X(6767) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,7373), (1,35,3304), (1,40,5045), (1,55,999), (1,57,5049), (1,1697,942), (1,3295,3), (1,3303,3295), (1,3746,56), (1,5119,354), (40,5045,5708), (55,56,5010), (55,999,3), (392,3870,3940), (495,497,381), (496,3085,1656), (954,3488,6913), (999,3295,55), (1621,3241,956), (1870,7071,1597), (3555,5250,3927), (3576,6244,3), (3622,3871,474), (3746,5010,55), (3748,5919,1), (3895,4666,3753)

X(6768) = MIXTILINEAR-INCENTRAL-TO-ABC TRILINEAR IMAGE OF X(1)

Trilinears Sqrt[(3 a^3-3 a^2 b-3 a b^2+3 b^3-5 a^2 c+10 a b c-5 b^2 c+a c^2+b c^2+c^3)/(a+b-c)] Sqrt[(3 a^3-5 a^2 b+a b^2+b^3-3 a^2 c+10 a b c+b^2 c-3 a c^2-5 b c^2+3 c^3)/(a-b+c)] : :

Trilinears are 1/|A′A″| : 1/|B′B″| : 1/|C′C″| , where A′B′C′ and A″B″C″ are the pedal and antipedal triangles of X(1)

X(6768) lies on these lines: {}

X(6768) = mixtilinear-excentral-to-ABC trilinear image of X(1)

X(6769) = INTANGENTS-TO-EXTANGENTS-SIMILARITY IMAGE OF X(40)

Barycentrics a*(a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3-(b+c)^4*a^2-2*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b-c)^2) : :

X(6769) lies on these lines: {1,3}, {4,200}, {19,1802}

X(6769) = X(3) of 3rd antipedal triangle of X(1)

X(6770) = OUTER-NAPOLEON-TO-INNER-NAPOLEON SIMILARITY IMAGE OF X(4)

Barycentrics Sqrt[3] (b^2+c^2-a^2) (3 a^4+b^4-2 b^2 c^2+c^4)+8 S^3 : :
X(6770) = 4 X[618] - 5 X[631] = 3 X[3545] - 4 X[5459] = 3 X[3524] - 2 X[5463] = 3 X[376] - 2 X[5473] = 3 X[4] - 4 X[5478] = 3 X[13] - 2 X[5478] = 3 X[5470] - 2 X[5479] = 7 X[3090] - 8 X[6669] = 3 X[2] - 4 X[6771] = 3 X[7967] - 2 X[7975]

X(6770) lies on these lines: {2,98}, {3,299}, {4,13}, {6,383}, {15,6778}, {30,5611}, {69,5980}, {115,5334}, {298,3564}, {376,530}, {396,1080}, {511,3180}, {617,2782}, {618,631}, {1350,5859}, {2549,5335}, {2794,6772}, {3090,6669}, {3107,7709}, {3524,5463}, {3545,5459}, {5470,5479}, {6268,6306}, {6270,6302}, {7967,7975}

X(6770) = reflection of X(i) in X(j) for these (i,j): (4, 13), (147, 5613), (616, 3), (1080, 396), (5617, 6771), (6773, 98)
X(6770) = anticomplement of X(5617)
X(6770) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6776,6773), (3,5873,634), (182,5613,2), (5617,6771,2)

X(6771) = OUTER-NAPOLEON-TO-INNER-NAPOLEON SIMILARITY IMAGE OF X(5)

Barycentrics 4S³ + 3^1/2(S_B + S_C)(S² + S_A(S_A + S_B + S_C)) : :
= 4S³ + 3^1/2a²(S² + S_A(a² + S_A)) : :

X(6771) lies on these lines: {2,98}, {3,13}, {5,6669}, {15,115}, {16,5472}, {30,5459}, {298,5965}, {396,511}

X(6771) = complement of X(5617)
X(6771) = midpoint of X(5617) and X(6770)
X(6771) = reflection of X(6774) in X(6036)
X(6771) = {X(2),X(182)}-harmonic conjugate of X(6774)

X(6772) = OUTER-NAPOLEON-TO-INNER-NAPOLEON SIMILARITY IMAGE OF X(6)

Barycentrics -2*(a^2+b^2+c^2)*S+sqrt(3)*(a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2) : :

Let A′ be the orthocenter of BCX(13), and define B′, C′ cyclically. X(6772) is X(6) of A′B′C′.

X(6772) lies on these lines: {2,99}, {3,6108}, {6,530}, {13,15}, {14,2782}, {62,6779}, {298,538}, {2794,6770}

X(6772) = X(298) of 1st Brocard triangle
X(6772) = reflection of X(6775) in X(115)
X(6772) = {X(2),X(2549)}-harmonic conjugate of X(6775)
X(6772) = intersection, other than X(15), of line X(13)X(15) and circle {{X(6), X(14), X(15)}}

X(6773) = INNER-NAPOLEON-TO-OUTER-NAPOLEON SIMILARITY IMAGE OF X(4)

Barycentrics Sqrt[3] (b^2+c^2-a^2) (3 a^4+b^4-2 b^2 c^2+c^4)-8 S^3 : :

X(6773) lies on these lines: {2,98}, {3,298}, {4,14}, {6,1080}, {16,6777}, {30,5615}, {69,5981}, {299,3564}, {511,3181}, {2794,6775}

X(6773) = anticomplement of X(5613)
X(6773) = reflection of X(5613) in X(6774)
X(6773) = reflection of X(6770) in X(98)
X(6773) = {X(2),X(6776)}-harmonic conjugate of X(6770)

X(6774) = INNER-NAPOLEON-TO-OUTER-NAPOLEON SIMILARITY IMAGE OF X(5)

Barycentrics 4S³ - 3^1/2(S_B + S_C)(S² + S_A(S_A + S_B + S_C)) : :
= 4S³ - 3^1/2a²(S² + S_A(a² + S_A)) : :

X(6774) lies on these lines: {2,98}, {3,14}, {5,6670}, {15,5471}, {16,115}, {30,5460}, {299,5965}, {395,511}

X(6774) = complement of X(5613)
X(6774) = midpoint of X(5613) and X(6773)
X(6774) = reflection of X(6771) in X(6036)
X(6774) = {X(2),X(182)}-harmonic conjugate of X(6771)

X(6775) = INNER-NAPOLEON-TO-OUTER-NAPOLEON SIMILARITY IMAGE OF X(6)

Barycentrics 2*(a^2+b^2+c^2)*S+sqrt(3)*(a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2) : :

Let A′ be the orthocenter of BCX(14), and define B′, C′ cyclically. X(6775) is X(6) of A′B′C′.

X(6775) lies on these lines: {2,99}, {3,6109}, {6,531}, {13,2782}, {14,16}, {61,6780}, {299,538}, {2794,6773}

X(6775) = X(299) of 1st Brocard triangle
X(6775) = reflection of X(6772) in X(115)
X(6775) = {X(2),X(2549)}-harmonic conjugate of X(6772)
X(6775) = intersection, other than X(16), of line X(14)X(16) and circle {{X(6), X(13),X(16)}}

X(6776) = REFLECTION OF X(4) IN X(6)

Trilinears    sin A tan ω - cos B cos C : :
Barycentrics    (b² + c² - a²)[3a⁴ + (b² - c²)²] : :
Barycentrics    S_A(a⁴ - S_BS_C) : : (Bui Quang Tuan, Hyacinthos #20428, 11/26/2011)
Barycentrics    (tan A)(tan B + tan C - cot ω) : :
X(6776) = X(4) - 2X(6)

Let L_A be the polar of X(4) wrt the circle centered at A and passing through X(6). Define L_B, L_C cyclically. (X(4) is the perspector of any circle centered at a vertex of ABC.) Let A′ = L_B∩L_C, and define B′, C′ cyclically. The orthocenter of triangle A′B′C′ is X(6776).

From Hyacinthos #20428, Bui Quang Tuan, 11/26/2011:
Let A*B*C* be the orthic triangle and L the de Longchamps point, X(20). Let A″, B″, C″ be the midpoints of LA*, LB*, LC*. Triangle A″B″C″ is perspective to the circumcevian triangle of X(3), and the perspector is X(6776).

X(6776) is the perspector of the Artzt triangle and the intriangle of X(2). (The intriangle of a point given by trilinears x : y : z is the central triangle having A-vertex 0 : y + z cos A : z + y cos A; see TCCT, p. 196. Thus, the A-vertex of the intriangle of X(2) is 0 : c + b cos A : b + c cos A.) Contributed by César Lozada, February 11, 2017.

X(6776) lies on these lines: {2,98}, {3,69}, {4,6}, {20,185}, {30,1351}

X(6776) = complement of X(5921)
X(6776) = anticomplement of X(1352)
X(6776) = reflection of X(i) in X(j) for these (i,j): (4,6), (69,3), (1352,182)
X(6776) = X(20) of 1st Brocard triangle
X(6776) = crossdifference of every pair of points on line X(520)X(2451)
X(6776) = {X(6770),X(6773)}-harmonic conjugate of X(2)
X(6776) = homothetic center of 2nd Hyacinth triangle and cross-triangle of Aries and 2nd Hyacinth triangles
X(6776) = X(9)-of-circumorthic-triangle if ABC is acute
X(6776) = X(4)-of-obverse-triangle-of-X(69)
X(6776) = Ehrmann-mid-to-Johnson similarity image of X(6)
X(6776) = second-Lemoine-circle-inverse of X(34137)

X(6777) = REFLECTION OF X(13) IN X(14)

Trilinears csc(A + π/3)(a csc(A - π/3) - b csc(B - π/3) - c csc(C - π/3)) + 2 csc(A - π/3)(b csc(B + π/3) + c csc(C + π/3)) : :
X(6777) = X(13) - 2X(14)

X(6777) lies on these lines: {6,13}, {15,5617}, {16,6773}, {18,98}, {61,147}, {99,298}

X(6777) = reflection of X(i) in X(j) for these (i,j): (13,14), (6778,115)
X(6777) = anticomplement of X(32552)
X(6777) = X(16)-Fuhrmann-to-X(15)-Fuhrmann similarity image of X(16)
X(6777) = inverse-in-X(15)-Fuhrmann-circle of X(15)
X(6777) = {X(13),X(14)}-harmonic conjugate of X(5469)

X(6778) = REFLECTION OF X(14) IN X(13)

Trilinears csc(A - π/3)(a csc(A + π/3) - b csc(B + π/3) - c csc(C + π/3)) + 2 csc(A + π/3)(b csc(B - π/3) + c csc(C - π/3)) : :
X(6778) = 2X(13) - X(14)

X(6778) lies on these lines: {6,13}, {15,6770}, {16,5613}, {17,98}, {62,147}, {99,299}

X(6778) = reflection of X(i) in X(j) for these (i,j): (14,13), (6777,115)
X(6778) = anticomplement of X(32553)
X(6778) = X(15)-Fuhrmann-to-X(16)-Fuhrmann similarity image of X(15)
X(6778) = inverse-in-X(16)-Fuhrmann-circle of X(16)
X(6778) = {X(13),X(14)}-harmonic conjugate of X(5470)

X(6779) = REFLECTION OF X(13) IN X(16)

Trilinears csc(A + π/3)(a sin(A - π/3) - b sin(B - π/3) - c sin(C - π/3)) + 2 sin(A - π/3)(b csc(B + π/3) + c csc(C + π/3)) : :
X(6779) = X(13) - 2X(16)

X(6779) lies on these lines: {2,13}, {14,5615}, {62,6772}, {99,532}, {6780,6781}

X(6779) = reflection of X(i) in X(j) for these (i,j): (13,16), (6780,6781)
X(6779) = X(14)-antipedal-to-X(13)-antipedal similarity image of X(14)
X(6779) = inverse-in-antipedal-circle-of-X(13) of X(13)

X(6780) = REFLECTION OF X(14) IN X(15)

Trilinears csc(A - π/3)(a sin(A + π/3) - b sin(B + π/3) - c sin(C + π/3)) + 2 sin(A + π/3)(b csc(B - π/3) + c csc(C - π/3)) : :
X(6780) = X(14) - 2X(15)

X(6780) lies on these lines: {2,14}, {13,5611}, {61,6775}, {99,533}, {6779,6781}

X(6780) = reflection of X(i) in X(j) for these (i,j): (14,15), (6779,6781)
X(6780) = X(13)-antipedal-to-X(14)-antipedal similarity image of X(13)
X(6780) = inverse-in-antipedal-circle-of-X(14) of X(14)

X(6781) = REFLECTION OF X(115) IN X(187)

Barycentrics (4a⁴ - b⁴ - c⁴ - 2a²b² - 2a²c² + 2b²c²) : :
X(6781) = X(115) - 2X(187) = X(13) + X(14) - 2X(15) - 2X(16)

X(6781) lies on these lines: {3,1506}, {4,5206}, {6,3534}, {15,5472}, {16,5471}, {30,115}, {99,754}, {511,1569}, {6779,6780}

X(6781) = midpoint of X(6779) and X(6780)

X(6782) = X(15)-PEDAL-TO-X(16)-PEDAL SIMILARITY IMAGE OF X(15)

Barycentrics (2*S+(-a^2+b^2+c^2)*sqrt(3))*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2) : :

Let O* be the circle with segment X(13)X(16) as diameter (and center X(6108)). Let P be the perspector of O*. Then X(6782) is the trilinear pole of the polar of P with respect to O*. See also X(3292), X(5642), and X(6783).

X(6782) lies on these lines: {6,5617}, {13,3545}, {14,530}, {15,298}, {16,6773}, {114,230}, {395,542}, {524,6109}

X(6782) = reflection of X(6783) in X(230)
X(6782) = {X(114),X(5477)}-harmonic conjugate of X(6783)

X(6783) = X(16)-PEDAL-TO-X(15)-PEDAL SIMILARITY IMAGE OF X(16)

Barycentrics (-2*S+(-a^2+b^2+c^2)*sqrt(3))*(2*a^4-(b^2+c^2)*a^2+(b^2-c^2)^2) : :

Let O* be the circle with segment X(14)X(15) as diameter (and center X(6109)). Let P be the perspector of O*. Then X(6783) is the trilinear pole of the polar of P with respect to O*. See also X(3292), X(5642), and X(6782).

X(6783) lies on these lines: {6,5613}, {13,531}, {14,3545}, {15,6770}, {16,299}, {114,230}, {396,542}, {524,6108}

X(6783) = reflection of X(6782) in X(230)
X(6783) = {X(114),X(5477)}-harmonic conjugate of X(6782)

X(6784) = CENTROID OF PEDAL TRIANGLE OF X(98)

Barycentrics a²(b² - c²)²(a⁴ - a²b² - a²c² - 2b²c²) : :

Let P and P* be antipodes on the circumcircle. Let Q be the intersection, other than P, of the circumcircle and line X(6)P. Let G' be the centroid of the (degenerate) pedal triangle of P. Then G' is the crossdifference of P* and Q. The locus of G' as P varies is an ellipse centered at X(2), with major axis line X(2)X(2574), and passing through X(125), X(3756), X(5642), X(6784), X(6786), X(6791) and X(6793). This ellipse is homothetic to the Hutson centroidal ellipse (defined at X(5943)), with insimilicenter X(373) and exsimilicenter X(51).
The reflection of P in G' lies on the orthocentroidal circle, and is P of the orthocentroidal triangle.

X(6784) lies on the above described ellipse and these lines: {2,6786}, {51,2871}, {98,6785}, {115,512}, {125,526}, {373,597}, {511,6055}

X(6784) = reflection of X(6786) in X(2)
X(6784) = midpoint of X(98) and X(6785)
X(6784) = crossdifference of every pair of points on line X(99)X(1625)

X(6785) = X(98) OF ORTHOCENTROIDAL TRIANGLE

Barycentrics a²(a⁴ + b⁴ + c⁴ - a²b² - a²c² - b²c²)[a⁸(b⁴ + b²c² + c⁴) - 3a⁶(b⁶ + c⁶) + 3a⁴(b⁸ - b⁶c² + b⁴c⁴ - b²c⁶ + c⁸) - a²(b² - c²)²(b⁶ - 2b⁴c² - 2b²c⁴ + c⁶) - b²c²(b² - c²)²(2b⁴ - b²c² + 2c⁴)] : :

X(6785) lies on the orthocentroidal circle and these lines: {2,51}, {4,512}, {381,6787}, {98,6784}

X(6785) = reflection of X(98) in X(6784)
X(6785) = 1st-Brocard-to-orthocentroidal similarity image of X(3)
X(6785) = X(6233)-of-4th-Brocard-triangle
X(6785) = Λ(X(6), X(13)) wrt orthocentroidal triangle

X(6786) = CENTROID OF PEDAL TRIANGLE OF X(99)

Barycentrics a²(2b²c² - a²b² - a²c²)(b⁴ + c⁴ - a²b² - a²c²) : :

X(6786) lies on the ellipse described at X(6784) and these lines: {2,6784}, {99,6787}, {114,325}, {125,126}, {373,3815}, {512,2482}, {526,5642}

X(6786) = reflection of X(6784) in X(2)
X(6786) = midpoint of X(99) and X(6787)
X(6786) = crossdifference of every pair of points on line X(98)X(729)
X(6786) = pole of orthic axis wrt Thomson-Gibert-Moses hyperbola

X(6787) = X(99) OF ORTHOCENTROIDAL TRIANGLE

Barycentrics a²(a⁴ + b⁴ + c⁴ - a²b² - a²c² - b²c²)(a⁴b⁴ + a⁴c⁴ - a⁴b²c² - a²b⁶ - a²c⁶ + 2b⁶c² + 2b²c⁶ - 3b⁴c⁴) : :

X(6787) lies on the orthocentroidal circle and these lines: {2,512}, {4,69}, {99,6786}, {381,6785}

X(6787) = reflection of X(99) in X(6786)
X(6787) = 1st-Brocard-to-orthocentroidal similarity image of X(6)
X(6787) = Λ(X(115), X(125)) wrt orthocentroidal triangle

X(6788) = X(106) OF ORTHOCENTROIDAL TRIANGLE

Barycentrics a^4-a^3 b+a^2 b^2+2 a b^3-b^4-a^3 c-a^2 b c-2 a b^2 c+a^2 c^2-2 a b c^2+2 b^2 c^2+2 a c^3-c^4 : :

X(6788) lies on the orthocentroidal circle, the Fuhrmann circle, the circle O(1,4), and these lines: {1,2}, {4,2457}, {106,952}, {121,3699}

X(6788) = complement of X(6790)
X(6788) = anticomplement of X(6789)
X(6788) = X(107) of Fuhrmann triangle
X(6788) = reflection of X(106) in X(3756)
X(6788) = inverse-in-polar-circle of X(7649)
X(6788) = intersection, other than X(4), of the orthocentroidal and Fuhrmann circles
X(6788) = intersection of Nagel lines of 1st and 2nd Ehrmann circumscribing triangles
X(6788) = intersection of Nagel lines of anticevian triangles of PU(4)

X(6789) = COMPLEMENT OF X(6788)

Barycentrics 2 a^4-2 a^3 b-3 a^2 b^2+a b^3-2 a^3 c+4 a^2 b c+3 a b^2 c-b^3 c-3 a^2 c^2+3 a b c^2-2 b^2 c^2+a c^3-b c^3 : :

X(6789) lies on circles O(1,3), O(2,3), and these lines: {1,2}, {3,3667}, {121,952}, {3756,6715}

X(6789) = complement of X(6788)
X(6789) = X(106) of X(1)-Brocard triangle
X(6789) = X(106) of X(2)-Brocard triangle
X(6789) = intersection, other than X(2), of Nagel line and circle O(2,3) (circle {{X(2), X(3), X(3111), X(5108)}})
X(6789) = intersection, other than X(3), of circles O(1,3) and O(2,3)
X(6789) = midpoint of segment cut off by intersection of circumcircle and Nagel line
X(6789) = inverse-in-circumcircle of X(4057)
X(6789) = inverse-in-{circumcircle, nine-point circle}-inverter of X(3006)

X(6790) = ANTICOMPLEMENT OF X(6788)

Barycentrics 3 a^4-3 a^3 b-2 a^2 b^2+3 a b^3-b^4-3 a^3 c+3 a^2 b c+a b^2 c-b^3 c-2 a^2 c^2+a b c^2+3 a c^3-b c^3-c^4 : :

X(6790) lies on circle O(1,20) and these lines: {1,2}, {20,3667}, {952,3699}

X(6790) = anticomplement of X(6788)
X(6790) = AC-incircle-inverse of X(6735)
X(6790) = intersection, other than X(1), of Nagel line and circle O(1,20) (circle {{X(1), X(20), X(3109)}})

X(6791) = CENTROID OF PEDAL TRIANGLE OF X(111)

Barycentrics (b² - c²)²(5a² - b² - c²) : :

Let L_A, L_B, L_C be the lines through A, B, C, resp. parallel to line X(2)X(6). Let M_A, M_B, M_C be the reflections of lines BC, CA, AB in L_A, L_B, L_C, resp. Let A′ = M_B∩M_C, and define B′, C′ cyclically. Triangle A′B′C′ is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A′B′C′ in line X(2)X(6). The triangle A″B″C″ is homothetic to ABC, with center of homothety X(6791). (See Hyacinthos #16741/16782, Sep 2008.)

X(6791) lies on the ellipse described at X(6784) and these lines: {2,5503), {6,5642}, {111,542}, {115,125}, {126,5969}, {373,3815}, {1499,2686}

X(6791) = midpoint of X(111) and X(6792)
X(6791) = crossdifference of every pair of points on line X(110)X(1296)

X(6792) = X(111) OF ORTHOCENTROIDAL TRIANGLE

Barycentrics a^6-(b^2+c^2)*a^4+(3*b^4-5*b^2*c^2+3*c^4)*a^2-(b^4-c^4)*(b^2-c^2) : :

Let A′ be the intersection, other than X(4), of the A-altitude and the orthosymmedial circle, and define B′ and C′ cyclically. Triangle A′B′C′ is here introduced as the 1st orthosymmedial triangle. A′B′C′ is inversely similar to ABC, with similitude center X(251). X(6792) is X(111) of A′B′C′.

Let A″ be the intersection, other than X(6), of the A-symmedian and the orthosymmedial circle, and define B″ and C″ cyclically. Triangle A″B″C″ is here introduced as the 2nd orthosymmedial triangle. A″B″C″ is the circummedial triangle of A′B′C′. A′B′C′ and A″B″C″ are perspective at X(51).

Let A′ be the trilinear product of the vertices of the A-adjunct anti-altimedial triangle, and define B′ and C′ cyclically. Triangle A′B′C′ is the anticomplementary triangle of the 1st Brocard triangle and is similar to the orthocentroidal triangle, with similitude center X(6792). (Randy Hutson, November 2, 2017)

X(6792) lies on the orthocentroidal circle, the orthosymmedial circle, the Hutson-Parry circle, and these lines: {2,6}, {4,1499}, {110,5477}, {111,542}

X(6792) = complement of X(38940)
X(6792) = orthogonal projection of X(4) on line X(2)X(6)
X(6792) = intersection, other than X(2), of line X(2)X(6) and orthocentroidal circle
X(6792) = intersection, other than X(6), of line X(2)X(6) and orthosymmedial circle
X(6792) = intersection, other than X(4), of orthocentroidal and orthosymmedial circles
X(6792) = anticomplement of X(5108)
X(6792) = reflection of X(111) in X(6791)
X(6792) = reflection of X(111) in line X(115)X(125)
X(6792) = X(110) of 4th Brocard triangle
X(6792) = X(111) of orthocentroidal triangle
X(6792) = X(111) of 1st orthosymmedial triangle
X(6792) = inverse-in-polar-circle of X(2501)
X(6792) = inverse-in-{circumcircle, nine-point circle}-inverter of X(230)
X(6792) = 1st-Brocard-to-orthocentroidal similarity image of X(5108)
X(6792) = Λ(X(6), X(110)) wrt orthocentroidal triangle
X(6792) = intersection, other than X(5640), of Hutson-Parry circles of ABC and orthocentroidal triangle
X(6792) = intersection, other than X(4), of orthocentroidal circle and X(2)-Fuhrmann circle (aka X(2)-Hagge circle)
X(6792) = circummedial-to-X(2)-Fuhrmann similarity image of X(111)
X(6792) = intersection of lines X(2)X(6) of 1st and 2nd Ehrmann circumscribing triangles
X(6792) = intersection of lines X(2)X(6) of anticevian triangles of PU(4)
X(6792) = trilinear pole, wrt 4th Brocard triangle, of Fermat axis

X(6793) = CENTROID OF PEDAL TRIANGLE OF X(112)

Barycentrics (2a⁶ - b⁶ - c⁶ - a⁴b² - a⁴c² + b⁴c² + b²c⁴)(2a⁴ - b⁴ - c⁴ - a²b² - a²c² + 2b²c²) : :

X(6793) lies on the ellipse described at X(6784) and these lines: {6,67}, {51,2871}, {112,1562}, {132,1503}

X(6793) = midpoint of X(112) and X(6794)
X(6793) = crossdifference of every pair of points on line X(74)X(1297)

X(6794) = X(112)-OF-ORTHOCENTROIDAL-TRIANGLE

Barycentrics [a¹⁰ - a⁸(b² + c²) + a⁶(2b⁴ - 3b²c² + 2c⁴) - 4a⁴(b² - c²)²(b² + c²) + a²(b² - c²)²(b⁴ + 5b²c² + c⁴) + (b² - c²)⁴(b² + c²)][a¹⁰ + b¹⁰ + c¹⁰ - a⁸(b² + c²) + a⁶b²c² - a²(b⁸ - b⁶c² - b²c⁶ + c⁸) - b⁸c² - b²c⁸] : :
Barycentrics a^10 - a^8*b^2 + 2*a^6*b^4 - 4*a^4*b^6 + a^2*b^8 + b^10 - a^8*c^2 - 3*a^6*b^2*c^2 + 4*a^4*b^4*c^2 + 3*a^2*b^6*c^2 - 3*b^8*c^2 + 2*a^6*c^4 + 4*a^4*b^2*c^4 - 8*a^2*b^4*c^4 + 2*b^6*c^4 - 4*a^4*c^6 + 3*a^2*b^2*c^6 + 2*b^4*c^6 + a^2*c^8 - 3*b^2*c^8 + c^10 : :
X(6794)= X[4] + 2 X[18338], 4 X[5] - X[18337], 2 X[6] + X[35902], X[112] + 2 X[1562], 2 X[5523] + X[13509], 4 X[5523] - X[15340], X[5523] + 2 X[15341], 2 X[13509] + X[15340], X[13509] - 4 X[15341], X[15340] + 8 X[15341]

X(6794) lies on the orthocentroidal circle and these lines: {2, 525}, {4, 6}, {5, 18337}, {20, 2420}, {39, 18304}, {54, 7765}, {74, 6103}, {112, 1562}, {115, 11005}, {247, 39024}, {477, 2549}, {868, 6792}, {1316, 15048}, {2088, 3767}, {2132, 5158}, {2409, 35260}, {2697, 4846}, {3163, 10706}, {4235, 30227}, {5024, 15000}, {5309, 5890}, {6324, 11226}, {7422, 7735}, {10413, 26917}, {16278, 36696}, {18809, 33504}, {18911, 38971}, {22146, 32423}, {32640, 39008}, {34211, 35923}

X(6794) = midpoint of X(1562) and X(6793)
X(6794) = reflection of X(112) in X(6793)
X(6794) = intersection, other than X(4), of van Aubel line and orthocentroidal circle
X(6794) = polar-circle-inversee of X(1990)
X(6794) = orthoptic-circle-of-Steiner-inellipse-inverse of X(9209)
X(6794) = psi-transform of X(1637)
X(6794) = crossdifference of every pair of points on line {520, 1495}
X(6794) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5523, 13509, 15340}, {5523, 15341, 13509}

X(6795) = ORTHOCENTROIDAL-TO-1st-BROCARD SIMILARITY IMAGE OF X(4)

Trilinears [a cos(A - ω) - b cos(B - ω) - c cos(C - ω)][bc sin²(B - C) + a² sin(A - B) sin(A - C)] + 2 cos(A - ω) [abc(sin²(A - B) - sin²(A - C)) - sin(B - C)(b³ sin(A - B) + c³ sin(A - C))] : :

X(6795) lies on the Brocard circle and these lines: {3,523}, {6,30}, {182,1316}

X(6795) = Brocard circle antipode of X(1316)
X(6795) = reflection of X(1316) in X(182)
X(6795) = X(74) of 1st Brocard triangle
X(6795) = 1st-Brocard-isogonal-conjugate of X(542)
X(6795) = intersection, other than X(6), of Brocard circle and line X(6)X(30)

X(6796) = ABC-TO-EXCENTRAL BARYCENTRIC IMAGE OF X(5)

Trilinears - a cos(B - C)/(b + c - a) + b cos(C - A)/(c + a - b) + c cos(A - B)/(a + b - c) : :
Trilinears a⁶ - a⁵(b + c) - 2a⁴(b² + c²) + a³(2b³ + b²c + bc² + 2c³) + a²(b⁴ - b³c - bc³ + c⁴) - a(b⁵ - b³c² - b²c³ + c⁵) + bc(b² - c²)² : :

X(6796) lies on these lines: {1,1389}, {3,10}, {4,35}, {5,5248}, {21,5587}, {36,944}, {40,78}, {55,946}, {56,5882}, {73,1771}, {165,191}

X(6796) = excentral isogonal conjugate of X(3460)
X(6796) = {X(165),X(1490)}-harmonic conjugate of X(1158)
X(6796) = X(5449)-of-excentral-triangle

X(6797) = MIDPOINT OF X(65) AND X(80)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c-4 a^4 b c+5 a^3 b^2 c-6 a b^4 c+4 b^5 c-a^4 c^2+5 a^3 b c^2-6 a^2 b^2 c^2+5 a b^3 c^2+b^4 c^2-2 a^3 c^3+5 a b^2 c^3-8 b^3 c^3+2 a^2 c^4-6 a b c^4+b^2 c^4+a c^5+4 b c^5-c^6) : :   (P. Moses)
Trilinears    4*cos(A)+2*(cos(A)-2)*cos(B-C)+2*(6*sin(A/2)-sin(3*A/2))*cos((B-C)/2)-3 : :   (C. Lozada)
Trilinears    (b+c)*a^5-(c^2+b^2+4*b*c)*a^4-(b+c)*(2*c^2-7*b*c+2*b^2)*a^3+2*(-b^2*c^2+(b^2-c^2)^2)*a^2+(b^2-c^2)*(b-c)*a*(c^2-5*b*c+b^2)-(b^2-c^2)^2*(b^2-4*b*c+c^2) : :    (C. Lozada)
X(6797) = (2R² + R*r - 2*r²)*X(1) - 6*R*r*X(2) + (R*r + 2r²)*X(3) = X(100) - 3X(3753) = 3X(942) - 2X(5083)

Let I be the incenter of a triangle ABC. Let N_A be the nine-point center of IBC, and define N_B and N_C cyclically. Let

A′ = reflection of N_A in AI;
B′ = reflection of N_B in BI;
C′ = reflection of N_C in CI.

Let (N_A) be the nine-point circle of A′B′C′, and define (N_B) and (N_C) cyclically. The circles (N_A), (N_B), (N_C) concur in X(6797). See Hyacinthos 23071 (January-March 2015, Antreas Hatzipolakis, Peter Moses, César Lozada)

X(6797) lies on these lines: {11,517}, {65,79}, {100,3753}, {149,5722}, {214,3812}, {942,952}, {960,6702}, {999,6264}, {1125,1387}, {1210,1484}, {1317,5045}, {1537,5806}, {3295,5541}, {6001,6246}

X(6798) = 1st ATRIA CYCLOLOGIC CENTER

Trilinears (cos(2*B)+cos(2*C))/(2*cos(A)*(cos(2*(B-C))+1)+cos(B-C)+cos(3*A)) : : (César Lozada)

Let H_A be the reflection of the orthocenter in BC, and define H_B and H_C cyclically. Let H_AB be the reflection of H_A in HB, and define H_BC and H_CA cyclically. Let H_AC be the reflection of H_A in HC, and define H_BA and H_CB cyclically. Let N_A be the nine-point center of triangle H_AH_ABH_AC, and define N_B and N_C cyclically. Then ABC and N_AN_BN_C are cyclologic triangles. The (ABC, N_AN_BN_C)-cyclologic center is X(6798), and the (N_AN_BN_C, ABC)-cyclologic center is X(6799). See Hyacinthos 23138 (March 2, 2015, Antreas Hatzipolakis, César Lozada)

X(6798) lies on the cubic K464 and these lines: {49,52}, {389,3481}

X(6799) = 2nd ATRIA CYCLOLOGIC CENTER

Trilinears 1/(sin(3*(B-C))*cos(2*A)+sin(2*(B-C))*(-cos(A)+2*cos(B-C))-sin(B-C)*cos(4*A)) : : (César Lozada)

See X(6798) and Hyacinthos 23138 (March 2, 2015, Antreas Hatzipolakis, César Lozada)

X(6799) lies on the circumcircle and this line: {1141,6240}

X(6800) = X(3)X(74)∩X(6)X(23)

Trilinears (3*a^4-2*(b^2+c^2)*a^2-b^4-c^4)*a : :
Trilinears 2*cos(2*A)*cos(B-C)+5*cos(A)-3*cos(3*A) : :
X(6800) = 2*X(3)+X(11456), X(6)+2*X(35707), X(22)+2*X(184), 2*X(22)+X(1993), 5*X(22)+4*X(34986), 4*X(184)-X(1993), 5*X(184)-2*X(34986), X(378)-4*X(18475), 5*X(1656)-2*X(34514), 5*X(1993)-8*X(34986), X(1993)+4*X(35268), 4*X(6676)-X(11442), X(7391)-4*X(23292), 2*X(7502)+X(18445), 4*X(7555)-X(37494), X(11456)+4*X(34513), 2*X(11550)-5*X(31236), X(12082)+2*X(13352), 4*X(19127)-X(41614), 2*X(34986)+5*X(35268)

Let P be a point in the plane of a triangle ABC. Let P_A be the reflection of P in BC, and define P_B and P_C cyclically. Let P_AB be the reflection of P in BP_A, and define P_BC and P_CA cyclically. Let P_AC be the reflection of P in CP_A, and define P_BA and P_CB cyclically. At Hyacinthos 23148 (March 6, 2015) Antreas Hatzipolakis asks for the locus of P such that the perpendicular bisectors of the segements P_ABP_AC, P_BCP_BA, P_CAP_CB concur and for the locus of the point Z(P) of concurrence. César Lozada responds that the first locus is the the union of {A,B,C} and the Euler line of ABC, and that the second locus is the line of the points X(3), X(74), and X(110). The appearance of (i,j) in the following list means that Z(X(i)) = X(j): (2,6800), (3,3), (4,3), (5,5944), (25,6090). The point Z(P), for P = t*|OH| and X = X(110), is given by Z(P) = t(t - 1)|OQ|/[t - (r/|OH|)]².

For another construction see Antreas P. Hatzipolakis and César Lozada, euclid 2823.

X(6800) lies on these lines: 2, 154}, {3, 74}, {4, 14389}, {6, 23}, {20, 19357}, {22, 184}, {24, 9730}, {25, 5012}, {26, 568}, {39, 44116}, {49, 13340}, {51, 39561}, {54, 7387}, {155, 7512}, {159, 1176}, {182, 373}, {183, 35356}, {185, 38444}, {206, 26206}, {323, 1350}, {352, 5210}, {376, 40112}, {378, 14915}, {394, 6636}, {428, 38136}, {468, 18911}, {567, 7530}, {569, 10594}, {575, 32237}, {576, 44109}, {599, 9143}, {1112, 15074}, {1147, 10323}, {1181, 7488}, {1204, 38438}, {1351, 11422}, {1352, 7495}, {1498, 14118}, {1501, 5359}, {1598, 13434}, {1599, 35299}, {1600, 35300}, {1656, 34514}, {1915, 13331}, {1975, 10330}, {1976, 44420}, {1994, 5102}, {2360, 37301}, {2502, 20481}, {2715, 35901}, {2854, 16165}, {2916, 18882}, {2937, 12161}, {2979, 3167}, {3053, 9463}, {3060, 5093}, {3066, 14002}, {3091, 37476}, {3098, 3292}, {3146, 11425}, {3148, 11171}, {3291, 5033}, {3398, 20897}, {3431, 7464}, {3448, 37638}, {3455, 30541}, {3515, 10574}, {3516, 12279}, {3517, 15043}, {3518, 36752}, {3522, 35602}, {3547, 14516}, {3549, 34224}, {3564, 44210}, {3567, 9714}, {3580, 6776}, {3917, 44108}, {4189, 26637}, {4550, 41450}, {4563, 7782}, {4846, 10295}, {5017, 8627}, {5059, 14528}, {5092, 5651}, {5133, 31383}, {5169, 36990}, {5408, 13617}, {5409, 13616}, {5480, 7519}, {5486, 32220}, {5544, 30734}, {5650, 7485}, {5878, 34005}, {5889, 9715}, {5890, 14070}, {5899, 39522}, {5943, 44082}, {5967, 9832}, {5968, 32729}, {6515, 10565}, {6642, 26882}, {6644, 40114}, {6660, 32447}, {6676, 11442}, {6759, 7503}, {6795, 7471}, {7391, 23292}, {7393, 43598}, {7394, 37649}, {7395, 14530}, {7426, 11179}, {7484, 8780}, {7494, 37636}, {7500, 11427}, {7502, 18445}, {7506, 13363}, {7509, 10170}, {7514, 10540}, {7516, 18350}, {7517, 32046}, {7529, 43651}, {7542, 11457}, {7547, 11750}, {7552, 14852}, {7555, 37494}, {7556, 15032}, {7558, 12134}, {7566, 13419}, {7691, 12164}, {8548, 12310}, {8550, 32269}, {8717, 10564}, {8718, 12085}, {8719, 35933}, {8836, 41041}, {8838, 41040}, {9140, 45082}, {9183, 21163}, {9465, 40825}, {9545, 37498}, {9704, 13564}, {9777, 20850}, {9818, 14157}, {9833, 13160}, {10117, 34117}, {10154, 11245}, {10282, 10984}, {10298, 10605}, {10299, 44833}, {10301, 18583}, {10541, 16042}, {10546, 11284}, {10575, 35477}, {10601, 13595}, {11004, 11477}, {11064, 16063}, {11202, 15078}, {11317, 35706}, {11410, 13445}, {11413, 13367}, {11414, 34148}, {11417, 19356}, {11418, 19355}, {11420, 19364}, {11421, 19363}, {11423, 37493}, {11472, 12112}, {11550, 31236}, {11842, 37914}, {11935, 37496}, {12045, 20190}, {12082, 13352}, {12088, 36747}, {12107, 37490}, {12173, 41482}, {12203, 41238}, {12220, 19125}, {12233, 31304}, {12254, 12293}, {12315, 15062}, {12824, 19153}, {13198, 45237}, {13321, 37956}, {13346, 33524}, {13353, 13861}, {13366, 15520}, {13383, 18912}, {13754, 44837}, {14683, 15069}, {14805, 31861}, {14927, 31099}, {15033, 18534}, {15139, 44883}, {15246, 17811}, {15305, 32063}, {15329, 15920}, {15448, 37648}, {15531, 19121}, {15647, 41670}, {15750, 43601}, {16776, 20987}, {17714, 36749}, {17810, 34545}, {17814, 37126}, {17821, 22467}, {17834, 38435}, {17845, 34007}, {18437, 44888}, {18451, 32620}, {20998, 39560}, {21525, 41330}, {21663, 38446}, {21734, 40911}, {21850, 37899}, {22234, 44107}, {23061, 33878}, {23515, 44795}, {24981, 34507}, {26257, 39141}, {26913, 37453}, {29012, 31133}, {31608, 35890}, {31670, 37900}, {32235, 32305}, {32321, 32379}, {32423, 44262}, {32534, 40647}, {33532, 37477}, {35006, 42295}, {35228, 37978}, {35243, 43574}, {36181, 44526}, {36753, 37440}, {37454, 39884}, {37514, 44802}, {37902, 44415}, {37925, 44413}, {38010, 41394}, {38225, 41275}, {39899, 41724}, {40130, 41412}, {40709, 41021}, {40710, 41020}, {42154, 44462}, {42155, 44466}

X(6800) = midpoint of X(184) and X(35268)
X(6800) = reflection of X(i) in X(j) for these (i, j): (2, 13394), (3, 34513), (22, 35268), (378, 39242), (39242, 18475), (39588, 5050), (44795, 23515)
X(6800) = anticomplement of X(45303)
X(6800) = isogonal conjugate of the isotomic conjugate of X(14907)
X(6800) = crossdifference of every pair of points on line {X(1637), X(3906)}
X(6800) = X(206)-Dao conjugate of-X(14906)
X(6800) = X(75)-isoconjugate-of-X(14906)
X(6800) = X(32)-reciprocal conjugate of-X(14906)
X(6800) = perspector of the circumconic {{A, B, C, X(11636), X(44769)}}
X(6800) = intersection, other than A, B, C, of circumconics {{A, B, C, X(23), X(30541)}} and {{A, B, C, X(74), X(1383)}}
X(6800) = X(20481)-of-circumsymmedial triangle
X(6800) = X(22111)-of-anti-McCay triangle
X(6800) = X(34513)-of-X3-ABC reflections triangle
X(6800) = barycentric product X(6)*X(14907)
X(6800) = barycentric quotient X(32)/X(14906)
X(6800) = trilinear product X(31)*X(14907)
X(6800) = trilinear quotient X(31)/X(14906)
X(6800) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i, j, k): (2, 154, 35264), (2, 35265, 35259), (3, 110, 15066), (3, 1614, 11441), (3, 6090, 7998), (3, 12174, 11440), (3, 15066, 21766), (3, 26864, 110), (22, 184, 1993), (23, 11003, 6), (25, 5012, 5422), (110, 7998, 6090), (110, 15080, 3), (154, 3796, 2), (154, 5085, 35259), (154, 23041, 35260), (154, 35259, 35265), (3796, 35259, 5085), (5012, 5640, 5050), (5012, 26881, 25), (5050, 5640, 5422), (5085, 35259, 2), (6090, 7998, 15066), (6636, 9544, 394), (6776, 7493, 3580), (7712, 11003, 23), (7712, 43697, 32124), (8546, 32217, 6), (9306, 17508, 5650), (9715, 19347, 5889), (9909, 11402, 3060), (11422, 15107, 1351), (14002, 15018, 3066), (14169, 14170, 3), (15080, 26864, 15066), (17809, 33586, 1994), (25406, 35260, 2), (35259, 35265, 35264)

X(6801) = 1st AVIOR CYCLOLOGIC CENTER

Barycentrics (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+5 a^6 b^2 c^2-4 a^4 b^4 c^2+5 a^2 b^6 c^2-3 b^8 c^2+4 a^6 c^4+4 b^6 c^4-4 a^4 c^6-5 a^2 b^2 c^6-4 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10) (a^10-3 a^8 b^2+4 a^6 b^4-4 a^4 b^6+3 a^2 b^8-b^10-3 a^8 c^2+5 a^6 b^2 c^2-2 a^4 b^4 c^2-3 a^2 b^6 c^2+3 b^8 c^2+4 a^6 c^4-2 a^4 b^2 c^4-2 b^6 c^4-4 a^4 c^6-3 a^2 b^2 c^6-2 b^4 c^6+3 a^2 c^8+3 b^2 c^8-c^10) (a^10-3 a^8 b^2+4 a^6 b^4-4 a^4 b^6+3 a^2 b^8-b^10-3 a^8 c^2+5 a^6 b^2 c^2-5 a^2 b^6 c^2+3 b^8 c^2+2 a^6 c^4-4 a^4 b^2 c^4-4 b^6 c^4+2 a^4 c^6+5 a^2 b^2 c^6+4 b^4 c^6-3 a^2 c^8-3 b^2 c^8+c^10) : : (Peter Moses, June 13, 2018)

Let A′B′C′ be the anticevian triangle of X(54). Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The triangles ABC and A″B″C″ are cyclologic. The (ABC, A″B″C″)-cyclologic center is X(6801), and the (A″B″C″, ABC)-cyclologic center is X(6802). See Hyacinthos 23162 (March 19, 2015, Randy Hutson, César Lozada)

If you have The Geometer's Sketchpad, you can view X(6801) and X(6802).

X(6801) lies on these lines: {54,136}, {1986,14111}

X(6802) = 2nd AVIOR CYCLOLOGIC CENTER

Trilinears    (2*cos(4*A)*cos(4*(B-C))+2-16*cos(A)*cos(B-C)-4*cos(A)*cos(3*(B-C))+10*cos(2*A)*cos(2*(B-C))+4*cos(2*A)*cos(4*(B-C))-8*cos(3*A)*cos(B-C)+4*cos(4*A)*cos(2*(B-C))-4*cos(5*A)*cos(B-C)+4*cos(2*A)+2*cos(6*A)-cos(8*A)+2*cos(2*(B-C))+2*cos(4*(B-C))+cos(4*A))/cos(A)/cos(B-C)/((cos(4*A)+1)*cos(B-C)+cos(2*A)*cos(3*(B-C))-cos(A)-cos(3*A)-cos(A)*cos(2*(B-C))) : :    (César Lozada)
Trilinears    [2(F₄ + 2F₂ + 1)G₄ - 4F₁G₃ + 2(2F₄ + 5F₂ + 1)G₂ - 4(F₅ + 2F₃ + 4F₁)G₁ - F₈ + 2F₆ + F₄ + 4F₂ + 2]/[F₁G₁(F₂G₃ - F₁G₂ + (F₄ + 1)G₁ - F₃ - F₁)] : : , where F_n = cos nA and G_n = cos(nB - nC)    (César Lozada)
Trilinears    (a^24-8*(b^2+c^2)*a^22+2*(23*b^2*c^2+14*b^4+14*c^4)*a^20-2*(b^2+c^2)*(28*c^4+27*b^2*c^2+28*b^4)*a^18+(69*b^8+69*c^8+140*b^6*c^2+140*b^2*c^6+167*b^4*c^4)*a^16-2*(b^2+c^2)*(3*c^4-b^2*c^2+3*b^4)*(8*c^4+11*b^2*c^2+8*b^4)*a^14+b^2*c^2*(49*b^2*c^6+50*b^4*c^4+49*b^6*c^2+34*c^8+34*b^8)*a^12+2*(b^4-c^4)*(b^2-c^2)*a^10*(24*c^8+13*b^2*c^6+31*b^4*c^4+13*b^6*c^2+24*b^8)+(b^2-c^2)^2*a^8*(-69*b^12-68*b^10*c^2-80*b^8*c^4-78*b^6*c^6-80*b^4*c^8-68*b^2*c^10-69*c^12)+2*(b^4-c^4)*(b^2-c^2)*a^6*(28*c^12-27*b^2*c^10+31*b^4*c^8-32*b^6*c^6+31*b^8*c^4-27*b^10*c^2+28*b^12)+(b^2-c^2)^4*a^4*(-28*b^12-24*b^10*c^2-17*b^8*c^4-14*b^6*c^6-17*b^4*c^8-24*b^2*c^10-28*c^12)+2*(b^4-c^4)*(b^2-c^2)^5*a^2*(4*c^8+2*b^2*c^6+3*b^4*c^4+2*b^6*c^2+4*b^8)-(b^8-c^8)*(b^4-c^4)*(b^2-c^2)^6)*a/(-b^2-c^2+a^2)/((b^2+c^2)*a^2-(b^2-c^2)^2)/((b^2+c^2)*a^14+(-5*b^4-4*b^2*c^2-5*c^4)*a^12+(b^2+c^2)*(11*c^4-7*b^2*c^2+11*b^4)*a^10+(4*b^2*c^6-15*c^8-15*b^8+4*b^6*c^2+4*b^4*c^4)*a^8+(b^4-c^4)*(b^2-c^2)*a^6*(15*c^4-2*b^2*c^2+15*b^4)+(b^2-c^2)^2*a^4*(-11*b^8+4*b^6*c^2+4*b^2*c^6-11*c^8)+5*(b^4-c^4)*(b^2-c^2)^3*a^2*(b^4-b^2*c^2+c^4)-(b^4+c^4)*(b^2-c^2)^6) : :   (César Lozada)

Let A′B′C′ be the anticevian triangle of X(54). Let A″ be the reflection of A′ in BC, and define B″ and C″ cyclically. The triangles A″B″C″ are cyclologic. The (ABC, A″B″C″)-cyclologic center is X(6801), and the (A″B″C″, ABC)-cyclologic center is X(6802). See X(6801) and Hyacinthos 23162 (March 19, 2015, Randy Hutson, César Lozada)

X(6802) lies on these lines: {}

X(6803) = (EULER LINE)∩X(69)X(389)

Barycentrics a^10-(b^2+c^2)*a^8-2*((b^2-c^2)^2-4*b^2*c^2)*a^6+2*(b^2+c^2)*(b^4-10*b^2*c^2+c^4)*a^4+(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

As a point on the Euler line, X(6803) has Shinagawa coefficients (E, F).

X(6803) lies on these lines: {2, 3}, {68, 5892}, {69, 389}, {578, 3618}, {1192, 3763}, {1217, 6530}

X(6804) = (EULER LINE)∩X(800)X(2548)

Barycentrics a^10-(b^2+c^2)*a^8-2*(b^2+c^2)^2*a^6+2*(b^2+c^2)*(b^4+6*b^2*c^2+c^4)*a^4+(b^4-10*b^2*c^2+c^4)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

As a point on the Euler line, X(6804) has Shinagawa coefficients (E, -F).

X(6804) lies on these lines: {2, 3}, {800, 2548}, {3767, 5065}

X(6805) = (EULER LINE)∩X(388)X(3084)

Barycentrics 1 + sin A cos B cos C : :

As a point on the Euler line, X(6805) has Shinagawa coefficients (E, S).

X(6805) lies on these lines: {2, 3}, {388, 3084}, {394, 1587}, {497, 3083}, {2550, 6348}, {2551, 6347}, {3068, 5409}, {3316, 6504}, {5408, 6460}
X(6805) = orthocentroidal-circle-inverse of X(6806)
X(6805) = {X(2),X(4)}-harmonic conjugate of X(6806)

X(6806) = (EULER LINE)∩X(388)X(3083)

Barycentrics 1 - sin A cos B cos C : :

As a point on the Euler line, X(6806) has Shinagawa coefficients (E, -S).

X(6806) lies on these lines: {2, 3}, {388, 3083}, {394, 1588}, {497, 3084}, {2550, 6347}, {2551, 6348}, {3069, 5408}, {3317, 6504}, {5409, 6459}
X(6806) = orthocentroidal-circle-inverse of X(6805)
X(6806) = {X(2),X(4)}-harmonic conjugate of X(6805)

X(6807) = (EULER LINE)∩X(590)X(1498)

Barycentrics 16*S^5+4*a^2*b^2*c^2*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

As a point on the Euler line, X(6807) has Shinagawa coefficients (S, E).

X(6807) lies on these lines: {2, 3}, {590, 1498}, {1181, 3068}

X(6808) = (EULER LINE)∩X(615)X(1498)

Barycentrics 16*S^5-4*a^2*b^2*c^2*(a^2+b^2-c^2)*(a^2-b^2+c^2) : :

As a point on the Euler line, X(6808) has Shinagawa coefficients (S, -E).

X(6808) lies on these lines: {2, 3}, {615, 1498}, {1181, 3069}

X(6809) = (EULER LINE)∩X(492)X(5562)

Barycentrics 32*S^5+(-a^2+b^2+c^2)*(a^2+b^2-c^2)^2*(a^2-b^2+c^2)^2 : :

As a point on the Euler line, X(6809) has Shinagawa coefficients (S, F).

X(6809) lies on these lines: {2, 3}, {492, 5562}

X(6810) = (EULER LINE)∩X(491)X(5562)

Barycentrics 32*S^5-(-a^2+b^2+c^2)*(a^2+b^2-c^2)^2*(a^2-b^2+c^2)^2 : :

As a point on the Euler line, X(6810) has Shinagawa coefficients (S, -F).

X(6810) lies on these lines: {2, 3}, {491, 5562}

X(6811) = (EULER LINE)∩X(98)X(485)

Barycentrics 2*S*(a^2+b^2+c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2)+a^8-(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2 : :

As a point on the Euler line, X(6811) has Shinagawa coefficients (S, E + F), and also (1, cot ω).

X(6811) lies on these lines: {2, 3}, {98, 485}, {114, 489}, {183, 638}, {230, 3070}, {262, 486}, {325, 637}, {491, 1352}, {492, 511}, {590, 1503}, {615, 5480}, {1587, 6423}, {1588, 6422}, {2459, 6560}, {3071, 3815}, {3316, 3424}

X(6812) = (EULER LINE)∩X(491)X(5907)

Barycentrics 32*S^5+(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-10*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

As a point on the Euler line, X(6812) has Shinagawa coefficients (S, E - F).

X(6812) lies on these lines: {2, 3}, {491, 5907}, {590, 2883}

X(6813) = (EULER LINE)∩X(98)X(486)

Barycentrics a^8-(b^2+c^2)*a^6+(3*b^4+4*b^2*c^2+3*c^4)*a^4-3*(b^4-c^4)*(b^2-c^2)*a^2-2*(b^2-c^2)^2*b^2*c^2-2*S*(a^2+b^2+c^2)*((b^2+c^2)*a^2-(b^2-c^2)^2) : :

As a point on the Euler line, X(6813) has Shinagawa coefficients (S, -E - F), and also (1, -cot ω).

X(6813) lies on these lines: {2, 3}, {98, 486}, {114, 490}, {183, 637}, {230, 3071}, {262, 485}, {325, 638}, {491, 511}, {492, 1352}, {590, 5480}, {615, 1503}, {1587, 6421}, {1588, 6424}, {2460, 6561}, {3070, 3815}, {3317, 3424}

X(6814) = (EULER LINE)∩X(492)X(5907)

Barycentrics 32*S^5-(a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^6-(b^2+c^2)*a^4-(b^4-10*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2)) : :

As a point on the Euler line, X(6814) has Shinagawa coefficients (S, -E + F).

X(6814) lies on these lines: {2, 3}, {492, 5907}, {615, 2883}

X(6815) = (EULER LINE)∩X(69)X(5889)

Barycentrics a^10-(b^2+c^2)*a^8-2*(b^4-4*b^2*c^2+c^4)*a^6+2*((b^2-c^2)^2-4*b^2*c^2)*(b^2+c^2)*a^4+(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

As a point on the Euler line, X(6815) has Shinagawa coefficients (E, 2F).

X(6815) lies on these lines: {2, 3}, {69, 5889}, {185, 1352}, {389, 6515}

X(6815) = {X(3),X(4)}-harmonic conjugate of X(1370)

X(6816) = (EULER LINE)∩X(68)X(5891)

Barycentrics a^10-(b^2+c^2)*a^8-2*(b^4+c^4)*a^6+2*(b^2+c^2)^3*a^4+((b^2-c^2)^2-4*b^2*c^2)*(b^2-c^2)^2*a^2-(b^4-c^4)*(b^2-c^2)^3 : :

As a point on the Euler line, X(6816) has Shinagawa coefficients (E, -2F).

X(6816) lies on these lines: {2, 3}, {68, 5891}, {569, 5654}, {1352, 6467}, {1478, 4320}, {1479, 4319}, {1899, 5907}, {2165, 5063}, {5562, 6515}

X(6817) = (EULER LINE)∩X(42)X(388)

Barycentrics (b+c)*a^5+b*c*a^4+4*b^2*c^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c : :

As a point on the Euler line, X(6817) has Shinagawa coefficients (E, $bc$).

X(6817) lies on these lines: {2, 3}, {42, 388}, {43, 1478}, {69, 310}, {497, 2293}, {899, 5229}, {3421, 4651}, {3434, 4645}

X(6818) = (EULER LINE)∩X(42)X(497)

Barycentrics (b+c)*a^5+b*c*a^4-4*b^2*c^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c : :

As a point on the Euler line, X(6818) has Shinagawa coefficients (E, -$bc$).

X(6818) lies on these lines: {2, 3}, {42, 497}, {43, 1479}, {388, 3720}, {899, 5225}, {4651, 5082}

X(6819) = (EULER LINE)∩X(394)X(3087)

Barycentrics (a^8-4*(b^2+c^2)*a^6+2*(3*b^4-2*b^2*c^2+3*c^4)*a^4-4*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2+(b^2-c^2)^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

As a point on the Euler line, X(6819) has Shinagawa coefficients (EF, S²).

X(6819) lies on these lines: {2, 3}, {394, 3087}, {1249, 5422}, {3066, 6525}, {5943, 6524}

X(6820) = (EULER LINE)∩X(69)X(2052)

Barycentrics (a^8-4*(b^2+c^2)*a^6+6*(b^2+c^2)^2*a^4-4*(b^2+c^2)*(b^4+c^4)*a^2+(b^2-c^2)^4)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :

As a point on the Euler line, X(6820) has Shinagawa coefficients (EF, -S²).

X(6820) lies on these lines: {2, 3}, {69, 2052}, {196, 5905}, {275, 3618}, {393, 394}, {459,3580}, {511,6524}, {1249,1993}

X(6821) = (EULER LINE)∩X(42)X(1056)

Barycentrics (b+c)*a^5+b*c*a^4+8*b^2*c^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c : :

As a point on the Euler line, X(6821) has Shinagawa coefficients (2E, $bc$).

X(6821) lies on these lines: {2, 3}, {42, 1056}, {43, 388}, {1058, 3720}, {2550, 3741}

X(6822) = (EULER LINE)∩X(42)X(1058)

Barycentrics (b+c)*a^5+b*c*a^4-8*b^2*c^2*a^2-(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c : :

As a point on the Euler line, X(6822) has Shinagawa coefficients (2E, -$bc$).

X(6822) lies on these lines: {2, 3}, {42, 1058}, {43, 497}, {1056, 3720}, {2551, 3741}

X(6823) = (EULER LINE)∩X(11)X(1038)

Barycentrics (-a^2+b^2+c^2)*((b^2+c^2)*a^6-(b^4-10*b^2*c^2+c^4)*a^4-(b^4-c^4)*(b^2-c^2)*a^2+(b^2-c^2)^4) : :

As a point on the Euler line, X(6823) has Shinagawa coefficients (E + F, -E + F).

X(6823) lies on these lines: {2, 3}, {11, 1038}, {12, 1040}, {131, 5522}, {141, 2883}, {185, 343}, {216, 5254}, {485, 1579}, {486, 1578}, {495, 1062}, {496, 1060}, {1181, 3564}, {1352, 1498}, {2968, 4385}, {3673, 6356}, {5447, 5448}

X(6823) = complement of X(1593)

X(6824) = (EULER LINE)∩X(57)X(499)

Barycentrics a^7-(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6824) has Shinagawa coefficients ($aS_A$, abc).

X(6824) lies on these lines: {2, 3}, {57, 499}, {58, 5713}, {68, 1175}, {72, 5761}, {81, 155}, {142, 3358}, {284, 5816}, {355, 3085}, {498, 3601}, {517, 5791}, {912, 3487}, {942, 3086}, {946, 5709}, {1125, 6245}, {1352, 5138}, {1385, 5787}, {1698, 6282}, {3587, 6684}, {3616, 5768}, {3822, 6256}, {3869, 5603}, {5086, 5552}, {5226, 5811}, {5273, 5758}, {5698, 5805}, {5789, 5901}, {6523, 6708}

X(6825) = (EULER LINE)∩X(10)X(5720)

Barycentrics a^7 - a^6(b + c) - a^5(3b^2 + 2bc + 3c^2) + a^4(b + c)(3b^2 - 2bc + 3c^2) + a^3(3b^4 + 2b^2c^2 + 3c^4) - a^2(b - c)^2(b + c)(3b^2 + 2bc + 3c^2) - a(b - c)^4(b + c)^2 + (b - c)^4(b + c)^3 : :

As a point on the Euler line, X(6825) has Shinagawa coefficients ($aS_A$, -abc).

X(6825) lies on these lines: {2, 3}, {10, 5720}, {12, 3428}, {40, 498}, {119, 2551}, {226, 5709}, {387, 5396}, {499, 3576}, {517, 3085}, {573, 5747}, {581, 5292}, {944, 5086}, {966, 5778}, {993, 6256}, {1000, 1482}, {1064, 5230}, {1071, 5770}, {1214, 1217}, {1385, 3086}, {1490, 5705}, {1519, 5250}, {2095, 6147}, {2193, 3087}, {3359, 3452}, {3616, 5804}, {3869, 5552}, {3890, 5603}, {3927, 5771}, {3940, 5690}, {4847, 5534}, {5044, 5887}, {5226, 5758}, {5229, 5841}, {5273, 5811}, {5706, 5718}, {5707, 5712}, {5742, 5776}, {5743, 6247}, {5745, 6260}, {5746, 5755}, {5777, 5791}, {5806, 5886}

X(6825) = complement of X(6847)
X(6825) = anticomplement of X(6862)

X(6826) = (EULER LINE)∩X(7)X(912)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6826) has Shinagawa coefficients (abc, $aS_A$).

X(6826) lies on these lines: {2, 3}, {7, 912}, {10, 5709}, {33, 1074}, {57, 1478}, {69, 1243}, {78, 5761}, {142, 515}, {226, 5720}, {278, 1060}, {355, 388}, {386, 5713}, {387, 5707}, {497, 5886}, {516, 3587}, {517, 2550}, {579, 5816}, {612, 1072}, {936, 5715}, {940, 5721}, {946, 997}, {952, 1056}, {965, 5798}, {966, 5755}, {1001, 5842}, {1038, 1838}, {1058, 5901}, {1352, 4260}, {1479, 3601}, {1482, 5082}, {1699, 6282}, {1708, 4292}, {2095, 3421}, {2551, 5791}, {3428, 3925}, {3434, 4511}, {3436, 5818}, {4295, 5887}, {5044, 5812}, {5080, 5744}, {5175, 5804}, {5396, 5712}, {5746, 5778}, {5880, 6001}, {6245, 6256}

X(6827) = (EULER LINE)∩X(1)X(5761)

Barycentrics a^7-(b+c)*a^6-(b+c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6827) has Shinagawa coefficients (abc, -$aS_A$).

X(6827) lies on these lines: {1, 5761}, {2, 3}, {11, 3428}, {34, 1076}, {40, 1479}, {63, 5770}, {165, 3583}, {182, 5800}, {329, 912}, {355, 2551}, {388, 1385}, {497, 517}, {515, 997}, {516, 3359}, {572, 5747}, {580, 5292}, {602, 5230}, {614, 1072}, {938, 5758}, {942, 5812}, {944, 3436}, {952, 3421}, {962, 5804}, {1040, 1785}, {1056, 5719}, {1058, 1482}, {1210, 1708}, {1376, 5842}, {1478, 3576}, {2077, 4302}, {2095, 5762}, {3434, 5657}, {3579, 5225}, {3586, 6282}, {4293, 5841}, {4297, 6256}, {4383, 5721}, {5080, 5731}, {5082, 5690}, {5777, 5787}

X(6828) = (EULER LINE)∩X(11)X(938)

Barycentrics (b^2-b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6828) has Shinagawa coefficients ($aS_A$abc, abc + $aS_A$).

X(6828) lies on these lines: {2, 3}, {11, 938}, {12, 3486}, {63, 5715}, {78, 5086}, {81, 5713}, {165, 3841}, {920, 1445}, {936, 3814}, {946, 3869}, {962, 2886}, {1210, 3671}, {1699, 5705}, {2287, 5816}, {3219, 5812}, {3813, 5734}, {3822, 5691}, {4018, 5887}, {5057, 5535}, {5221, 5704}, {5249, 6245}, {6253, 6690}

X(6829) = (EULER LINE)∩X(9)X(3814)

Barycentrics (b^2+b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3+c^3)*(b+c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6829) has Shinagawa coefficients (abc + $aS_A$, $aS_A$).

X(6829) lies on these lines: {2, 3}, {9, 3814}, {11, 3488}, {12, 3487}, {40, 3841}, {226, 1737}, {498, 943}, {1071, 3824}, {1213, 5798}, {1698, 5715}, {1708, 3336}, {1714, 5713}, {1751, 5397}, {2886, 5289}, {3419, 4511}, {3614, 5221}, {3822, 5587}, {3825, 5436}, {3826, 5759}, {3925, 5657}, {5776, 5949}, {5777, 5885}, {5817, 5851}

X(6830) = (EULER LINE)∩X(11)X(2099)

Barycentrics (b^2-b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3-c^3)*(b-c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6830) has Shinagawa coefficients (-abc + $aS_A$, $aS_A$).

X(6830) lies on these lines: {2, 3}, {11, 2099}, {12, 944}, {104, 1478}, {119, 6224}, {355, 4511}, {484, 1699}, {946, 1737}, {997, 3814}, {1329, 5818}, {1512, 5316}, {1519, 3817}, {1708, 5715}, {2886, 5657}, {3576, 3822}, {3585, 5450}, {5226, 5768}, {5432, 5842}, {5770, 5905}

X(6831) = (EULER LINE)∩X(11)X(65)

Barycentrics (b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3-c^3)*(b-c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6831) has Shinagawa coefficients (-abc + $aS_A$, abc + $aS_A$).

X(6831) lies on these lines: {2, 3}, {11, 65}, {12, 515}, {35, 5842}, {40, 2886}, {46, 1699}, {57, 5715}, {63, 5812}, {78, 355}, {79, 1768}, {216, 1865}, {226, 1071}, {386, 5721}, {495, 944}, {496, 938}, {573, 5742}, {579, 5798}, {581, 5718}, {908, 5777}, {936, 1329}, {940, 5713}, {942, 1425}, {965, 5816}, {1076, 1214}, {1158, 1454}, {1181, 5707}, {1445, 5805}, {1446, 1565}, {1465, 1838}, {1490, 5219}, {1503, 5135}, {1765, 1901}, {2829, 3585}, {3468, 6357}, {3487, 5768}, {3612, 5691}, {3649, 5884}, {3814, 6700}, {3817, 3825}, {3820, 5818}, {3822, 4297}, {3925, 6684}, {3927, 5789}, {4259, 5480}, {4292, 6705}, {5292, 5706}, {5432, 6253}, {5747, 5776}

X(6831) = complement of X(411)

X(6832) = (EULER LINE)∩X(72)X(5886)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6832) has Shinagawa coefficients (abc + $aS_A$, abc).

X(6832) lies on these lines: {2, 3}, {72, 5886}, {226, 499}, {354, 3086}, {496, 954}, {497, 943}, {498, 950}, {583, 5746}, {584, 5802}, {960, 5603}, {1490, 3624}, {1713, 5747}, {1724, 5713}, {1837, 3085}, {3419, 5552}, {3698, 5657}, {3876, 5761}, {5436, 5587}

X(6832) = complement of X(37112)

X(6833) = (EULER LINE)∩X(46)X(499)

Barycentrics a^7-(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(b-c)^2*(3*b^2+2*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6833) has Shinagawa coefficients (-abc + $aS_A$, abc).

X(6833) lies on these lines: {2, 3}, {46, 499}, {65, 3086}, {84, 5219}, {104, 388}, {224, 5720}, {226, 6705}, {355, 5440}, {498, 515}, {940, 1181}, {944, 2646}, {988, 1072}, {1454, 4295}, {1478, 5450}, {1512, 1698}, {1519, 5437}, {1765, 5747}, {2096, 5714}, {3075, 3215}, {3428, 4999}, {3868, 5761}, {3916, 5812}, {3940, 5789}, {4255, 5721}, {5123, 5794}, {5135, 5820}, {5217, 5842}, {5438, 5587}, {5439, 5886}, {5657, 5836}, {5703, 5768}, {5704, 5804}, {5705, 6282}, {5744, 5758}, {5748, 5811}, {5759, 5832}, {5784, 5817}

X(6834) = (EULER LINE)∩X(1)X(1512)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(b-c)^2*(3*b^2+2*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6834) has Shinagawa coefficients (-abc + $aS_A$, -abc).

X(6834) lies on these lines: {1, 1512}, {2, 3}, {36, 6256}, {40, 1519}, {119, 3436}, {498, 946}, {499, 515}, {517, 5552}, {908, 5709}, {944, 1319}, {960, 5657}, {1068, 1465}, {1145, 5730}, {1181, 4383}, {1329, 3428}, {1538, 3579}, {1728, 3911}, {1737, 6261}, {2829, 5204}, {3057, 3085}, {5087, 6361}, {5316, 6684}, {5660, 5904}, {5703, 5804}, {5704, 5768}, {5744, 5811}, {5748, 5758}

X(6835) = (EULER LINE)∩X(72)X(5805)

Barycentrics a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6835) has Shinagawa coefficients (abc, abc + $aS_A$).

X(6835) lies on these lines: {2, 3}, {72, 5805}, {78, 946}, {354, 388}, {355, 3555}, {497, 5703}, {908, 5715}, {936, 1699}, {960, 962}, {1001, 6253}, {1210, 1478}, {1445, 1728}, {1479, 3817}, {1490, 5249}, {3306, 6245}, {3419, 5806}, {3436, 5587}, {3826, 5584}, {3876, 5758}, {5082, 5730}, {5229, 5704}, {5234, 5705}, {5439, 5787}, {5777, 5905}

X(6836) = (EULER LINE)∩X(1)X(1076)

Barycentrics a^7-(b+c)*a^6-(b^2+4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6836) has Shinagawa coefficients (abc, -abc - $aS_A$).

X(6836) lies on these lines: {1, 1076}, {2, 3}, {8, 3427}, {40, 3434}, {46, 516}, {63, 6245}, {65, 497}, {72, 5787}, {78, 515}, {165, 5705}, {224, 908}, {225, 1040}, {273, 4329}, {355, 3697}, {388, 2646}, {936, 5691}, {1062, 1068}, {1071, 5812}, {1155, 5225}, {1181, 3193}, {1376, 6253}, {1454, 3474}, {1478, 3612}, {1754, 5292}, {1936, 3215}, {2551, 3740}, {2886, 5584}, {3868, 5758}, {4652, 6705}, {5135, 5800}, {5249, 5715}, {5439, 5805}

X(6837) = (EULER LINE)∩X(7)X(90)

Barycentrics a^7-(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2+4*b*c+c^2)*(b^2-c^2)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6837) has Shinagawa coefficients ($aS_A$, 2abc).

X(6837) lies on these lines: {2, 3}, {7, 90}, {63, 946}, {84, 5249}, {153, 5261}, {354, 1858}, {498, 4304}, {499, 3817}, {920, 4295}, {962, 5273}, {1071, 5886}, {1259, 3434}, {1780, 3332}, {2327, 5816}, {3085, 4313}, {3219, 5758}, {3306, 6705}, {3486, 5252}, {3616, 6261}, {3624, 5732}, {3868, 5603}, {3916, 5805}, {5552, 5587}, {5698, 5832}, {5820, 5921}

X(6838) = (EULER LINE)∩X(8)X(6261)

Barycentrics a^7-(b+c)*a^6-(3*b^2+4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6838) has Shinagawa coefficients ($aS_A$, -2abc).

X(6838) lies on these lines: {2, 3}, {8, 6261}, {40, 908}, {63, 6260}, {498, 516}, {499, 4297}, {962, 3085}, {1319, 3486}, {1329, 5584}, {1512, 5554}, {1750, 5705}, {1788, 1858}, {3057, 3485}, {3086, 5731}, {3219, 5811}, {3305, 6684}, {3428, 3436}, {3876, 5657}, {3916, 6259}, {5709, 5905}, {5744, 6223}, {5791, 5927}

X(6839) = (EULER LINE)∩X(7)X(80)

Barycentrics a^7-(b+c)*a^6-(b^2-b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6839) has Shinagawa coefficients (abc, 2$aS_A$).

X(6839) lies on these lines: {2, 3}, {7, 80}, {8, 2894}, {63, 5080}, {78, 5715}, {81, 5721}, {145, 1389}, {149, 5603}, {355, 2888}, {515, 5249}, {946, 4511}, {962, 3878}, {997, 1699}, {1071, 5885}, {1072, 3920}, {1074, 3100}, {1146, 5829}, {1479, 4313}, {1621, 5842}, {1737, 3336}, {1754, 5127}, {1812, 5799}, {1838, 4296}, {2287, 5798}, {3419, 5805}, {3583, 3817}, {3679, 5735}, {3876, 5812}, {5221, 5229}, {5800, 5921}

X(6840) = (EULER LINE)∩X(100)X(5842)

Barycentrics a^7-(b+c)*a^6-(b^2+b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6840) has Shinagawa coefficients (abc, -2$aS_A$).

X(6840) lies on these lines: {2, 3}, {100, 5842}, {104, 5841}, {149, 517}, {153, 515}, {264, 2897}, {355, 3876}, {484, 516}, {497, 2099}, {962, 1479}, {997, 5691}, {1029, 5397}, {1076, 4296}, {1329, 6253}, {1478, 5226}, {1577, 6003}, {1785, 3100}, {2800, 5180}, {2894, 2949}, {3305, 5587}, {3585, 4297}, {3868, 5812}, {3984, 5881}, {4301, 4857}, {5057, 6001}, {5768, 5905}

X(6841) = (EULER LINE)∩X(11)X(113)

Barycentrics (b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6841) has Shinagawa coefficients ($aS_A$, 2abc + $aS_A$).

X(6841) lies on these lines: {2, 3}, {11, 113}, {57, 79}, {114, 2795}, {119, 6246}, {123, 133}, {131, 5521}, {142, 3825}, {191, 1699}, {265, 1175}, {355, 3811}, {495, 3486}, {496, 3485}, {758, 946}, {920, 1836}, {1698, 3587}, {2095, 5789}, {2886, 5791}, {3579, 3925}, {3601, 5441}, {3647, 5745}, {3648, 5744}, {3650, 5057}, {3652, 5805}, {3656, 3813}, {3817, 6245}, {3818, 5138}, {4420, 5086}, {5426, 5691}, {5787, 5886}

X(6842) = (EULER LINE)∩X(10)X(119)

Barycentrics (b+c)^2*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6842) has Shinagawa coefficients ($aS_A$, -2abc + $aS_A$), and also (r + R, r - R).

X(6842) lies on these lines: {2, 3}, {10, 119}, {11, 1385}, {12, 517}, {35, 5840}, {79, 5535}, {355, 2886}, {495, 1482}, {496, 3486}, {946, 3822}, {952, 4861}, {1698, 3359}, {1737, 1858}, {1834, 5396}, {1901, 5755}, {2829, 4999}, {3579, 3614}, {3585, 5841}, {3655, 3829}, {3814, 6684}, {3869, 5690}, {5226, 5761}

X(6843) = (EULER LINE)∩X(10)X(5715)

Barycentrics a^7-(b+c)*a^6+(b+c)^2*a^5-(b+c)^3*a^4-(5*b^2-6*b*c+5*c^2)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6843) has Shinagawa coefficients ($aS_A$, -2abc + $aS_A$).

X(6843) lies on these lines: {2, 3}, {10, 5715}, {72, 5818}, {226, 5587}, {329, 5775}, {355, 3487}, {387, 5713}, {966, 5798}, {997, 3817}, {1737, 3339}, {3017, 5733}, {3419, 5603}, {3488, 5274}, {3824, 5787}, {4511, 5175}, {5226, 5720}, {5249, 5768}, {5712, 5721}, {5714, 5777}, {5746, 5816}

X(6844) = (EULER LINE)∩X(145)X(5761)

Barycentrics a^7-(b+c)*a^6+(b-c)^2*a^5-(b+c)^3*a^4-(b-c)^2*(5*b^2+6*b*c+5*c^2)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2-2*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6844) has Shinagawa coefficients (-abc + $aS_A$, 2$aS_A$).

X(6844) lies on these lines: {2, 3}, {145, 5761}, {226, 5768}, {515, 5219}, {944, 5261}, {946, 3340}, {1071, 5714}, {1210, 5715}, {1699, 1737}, {3452, 5587}, {4511, 5720}, {5044, 5818}, {5218, 5842}, {5274, 5603}

X(6845) = (EULER LINE)∩X(11)X(4295)

Barycentrics (b^2-3*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3-c^3)*(b-c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6845) has Shinagawa coefficients (-abc + $aS_A$, 2abc + $aS_A$).

X(6845) lies on these lines: {2, 3}, {11, 4295}, {12, 4305}, {104, 4317}, {355, 4420}, {946, 5902}, {1158, 1699}, {1770, 3911}, {2886, 6361}, {3811, 5881}, {3814, 5438}, {3825, 5437}, {4325, 5450}

X(6846) = (EULER LINE)∩X(9)X(946)

Barycentrics a^7-(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6846) has Shinagawa coefficients (abc + $aS_A$, 2abc).

X(6846) lies on these lines: {2, 3}, {9, 946}, {72, 5603}, {84, 142}, {226, 3086}, {355, 3488}, {390, 943}, {498, 3586}, {499, 3361}, {515, 5436}, {580, 3332}, {950, 3085}, {954, 1058}, {1125, 1490}, {1260, 5082}, {1708, 4295}, {1713, 4253}, {1750, 3624}, {1901, 5022}, {3073, 4307}, {3419, 5818}, {3487, 5045}, {3577, 5837}, {3817, 5715}, {3824, 6259}, {4002, 5657}, {4251, 5802}, {5175, 5552}, {5273, 5709}, {5342, 6350}, {5437, 6705}, {5703, 5720}, {5779, 6147}, {5791, 5806}

X(6846) = complement of X(37108)
X(6846) = anticomplement of X(6989)

X(6847) = (EULER LINE)∩X(1)X(3427)

Barycentrics a^7 - a^6(b + c) - 3a^5(b - c)^2 + a^4(b + c)(3b^2 - 2bc + 3c^2) + a^3(b - c)^2(3b^2 + 2bc + 3c^2) - a^2(b - c)^2(b + c)(3b^2 + 2bc + 3c^2) - a(b - c)^2(b + c)^4 + (b - c)^4(b + c)^3 : :

As a point on the Euler line, X(6847) has Shinagawa coefficients (-abc + $aS_A$, 2abc).

X(6847) lies on these lines: {1, 3427}, {2, 3}, {7, 3358}, {10, 6282}, {11, 1466}, {40, 5745}, {57, 946}, {63, 5758}, {81, 1181}, {84, 226}, {104, 3600}, {498, 5691}, {499, 1699}, {515, 3085}, {601, 4307}, {908, 5811}, {912, 5761}, {940, 1498}, {942, 5603}, {944, 5787}, {962, 5709}, {1071, 3487}, {1210, 5804}, {1439, 1440}, {1537, 5708}, {1765, 5746}, {2829, 5229}, {3485, 6001}, {3577, 4848}, {3582, 4338}, {3927, 5763}, {4292, 5715}, {4293, 5450}, {4340, 5713}, {5217, 6253}, {5219, 6260}, {5226, 6223}, {5657, 5791}

X(6847) = complement of X(37421)
X(6847) = anticomplement of X(6825)

X(6848) = (EULER LINE)∩X(1)X(5804)

Barycentrics a^7-(b+c)*a^6-(3*b^2+2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(b-c)^2*(3*b^2+2*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6848) has Shinagawa coefficients (-abc + $aS_A$, -2abc).

X(6848) lies on these lines: {1, 5804}, {2, 3}, {8, 1512}, {40, 3452}, {57, 6260}, {63, 5811}, {84, 3911}, {104, 5265}, {329, 5709}, {391, 5778}, {498, 1699}, {499, 5691}, {515, 1420}, {908, 5758}, {944, 5722}, {946, 1697}, {962, 1519}, {1071, 5658}, {1210, 1467}, {1498, 4383}, {1528, 1753}, {1537, 5763}, {1714, 5400}, {1750, 6245}, {1788, 6001}, {2096, 6259}, {2551, 3428}, {4266, 5747}, {4293, 6256}, {5044, 5657}, {5225, 5842}, {5435, 6223}, {5587, 5795}, {5603, 5806}, {5690, 5780}, {6282, 6700}

X(6849) = (EULER LINE)∩X(355)X(5806)

Barycentrics a^7-(b+c)*a^6-(b+c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6849) has Shinagawa coefficients (abc, 2abc + $aS_A$).

X(6849) lies on these lines: {2, 3}, {355, 5806}, {388, 5045}, {500, 4648}, {946, 3811}, {1058, 5719}, {1478, 3333}, {1479, 5219}, {2550, 5044}, {3361, 3585}, {3434, 4420}, {3818, 5800}, {3940, 5082}, {4251, 5747}, {5763, 5780}, {5777, 5805}

X(6850) = (EULER LINE)∩X(8)X(912)

Barycentrics a^7-(b+c)*a^6-(b^2-6*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6850) has Shinagawa coefficients (abc, -2abc + $aS_A$).

X(6850) lies on these lines: {2, 3}, {8, 912}, {10, 1158}, {34, 1074}, {40, 1478}, {153, 3617}, {165, 3585}, {226, 5761}, {355, 971}, {388, 517}, {497, 1385}, {498, 2077}, {511, 5800}, {601, 5230}, {944, 3434}, {952, 5082}, {958, 2829}, {1038, 1785}, {1056, 1482}, {1068, 4296}, {1070, 4320}, {1071, 3419}, {1479, 3576}, {3421, 3927}, {3436, 5657}, {3579, 5229}, {3824, 5886}, {4292, 5709}, {4294, 5840}, {4338, 5270}, {4340, 5707}, {5175, 5768}, {5720, 6260}, {5777, 6259}, {5794, 6001}

X(6850) = anticomplement of X(3560)

X(6851) = (EULER LINE)∩X(10)X(3587)

Barycentrics a^7-(b+c)*a^6-(b^2+6*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6851) has Shinagawa coefficients (abc, -2abc - $aS_A$).

X(6851) lies on these lines: {2, 3}, {10, 3587}, {33, 1076}, {57, 1479}, {278, 1062}, {388, 4305}, {497, 942}, {500, 5712}, {515, 3811}, {516, 1158}, {517, 5787}, {912, 5758}, {962, 5768}, {971, 5812}, {991, 5713}, {1040, 1838}, {1068, 3100}, {1070, 4319}, {1478, 3601}, {2550, 3579}, {3332, 5707}, {3434, 6361}, {3436, 4420}, {3583, 4333}, {4338, 4857}, {5138, 5800}, {5691, 6282}, {5715, 5732}, {5771, 5789}

X(6851) = anticomplement of X(6985)

X(6852) = (EULER LINE)∩X(80)X(498)

Barycentrics a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b+c)*(b^2-c^2)*(b^3-c^3)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6852) has Shinagawa coefficients (2$aS_A$, abc).

X(6852) lies on these lines: {2, 3}, {80, 498}, {499, 3485}, {1698, 5538}, {2077, 3841}, {3336, 3911}, {3616, 6265}, {3624, 6261}, {3869, 5886}, {3878, 5603}, {5086, 5440}, {5439, 5885}

X(6853) = (EULER LINE)∩X(10)X(6326)

Barycentrics a^7-(b+c)*a^6-(3*b^2+b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+2*b^2*c^2+3*c^4)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*(b^3+c^3)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6853) has Shinagawa coefficients (2$aS_A$, -abc).

X(6853) lies on these lines: {2, 3}, {10, 6326}, {104, 4999}, {119, 5260}, {226, 2949}, {484, 6684}, {498, 3485}, {499, 3486}, {1385, 5086}, {1698, 6261}, {2099, 3085}, {2950, 5316}, {3884, 5603}

X(6854) = (EULER LINE)∩X(354)X(1056)

Barycentrics a^7-(b+c)*a^6-(b^2-4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(4*b^2-b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6854) has Shinagawa coefficients (2abc, $aS_A$).

X(6854) lies on these lines: {2, 3}, {354, 1056}, {355, 5439}, {388, 1737}, {997, 2550}, {1072, 5268}, {1478, 3911}, {2096, 5817}, {3216, 5713}, {3428, 3826}, {3434, 5440}, {4423, 5842}, {4511, 5082}, {5249, 5720}, {5437, 5587}

X(6855) = (EULER LINE)∩X(10)X(3577)

Barycentrics a^7-(b+c)*a^6-(5*b^2-2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+2*b^2*c^2+7*c^4)*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6855) has Shinagawa coefficients (2$aS_A$, abc + $aS_A$).

X(6855) lies on these lines: {2, 3}, {10, 3577}, {78, 5818}, {355, 5703}, {938, 5886}, {946, 5705}, {1210, 3485}, {3486, 5587}, {3869, 5775}, {5226, 5777}, {5273, 5812}, {5704, 5887}, {5715, 5745}, {5758, 5791}, {5789, 6147}

X(6856) = (EULER LINE)∩X(10)X(3340)

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

As a point on the Euler line, X(6856) has Shinagawa coefficients (2$aS_A$, -abc + $aS_A$).

X(6856) lies on these lines: {2, 3}, {10, 3340}, {12, 3421}, {63, 5714}, {72, 5226}, {145, 5719}, {226, 5705}, {274, 1007}, {329, 5791}, {387, 5718}, {388, 3822}, {392, 5806}, {498, 2550}, {956, 5261}, {966, 5747}, {993, 5229}, {1125, 3486}, {1376, 6668}, {1698, 2093}, {2886, 3085}, {3256, 3841}, {3419, 5703}, {3616, 5086}, {3617, 3940}, {3634, 5128}, {3753, 3869}, {3838, 4295}, {4018, 5775}, {4293, 4999}, {4294, 6690}, {4658, 5292}, {5225, 5248}, {5439, 5704}, {5698, 6666}, {5720, 5818}, {5742, 5746}, {5804, 5886}

X(6857) = (EULER LINE)∩X(1)X(5745)

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

As a point on the Euler line, X(6857) has Shinagawa coefficients (2$aS_A$, abc - $aS_A$).

X(6857) lies on these lines: {1, 5745}, {2, 3}, {7, 3916}, {8, 5791}, {10, 3486}, {35, 2550}, {55, 5082}, {57, 1125}, {58, 5712}, {63, 3487}, {69, 261}, {72, 5273}, {142, 3624}, {274, 6337}, {284, 966}, {388, 993}, {392, 942}, {497, 5248}, {498, 2551}, {499, 5259}, {610, 5257}, {943, 1259}, {946, 4512}, {950, 5705}, {958, 3085}, {1000, 4861}, {1001, 3086}, {1056, 2975}, {1058, 1621}, {1210, 5436}, {1385, 5768}, {1466, 4423}, {1482, 5771}, {1780, 4648}, {2095, 5901}, {2886, 4294}, {3419, 4313}, {3576, 6245}, {3618, 4260}, {3813, 4428}, {3822, 5229}, {3925, 5217}, {3927, 5719}, {4252, 4340}, {4295, 4640}, {4305, 5794}, {4652, 5249}, {4653, 5292}, {4877, 5747}, {5057, 5122}, {5250, 5603}, {5260, 5552}, {5281, 5687}, {5435, 5439}, {5731, 5787}, {5742, 5802}, {6261, 6705}, {6282, 6684}

X(6857) = complement of X(5177)

X(6858) = (EULER LINE)∩X(10)X(5761)

Barycentrics a^7-(b+c)*a^6-(5*b^2+2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+7*c^4+2*b*c*(2*b^2+b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2+2*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6858) has Shinagawa coefficients (abc + 2$aS_A$, $aS_A$).

X(6858) lies on these lines: {2, 3}, {10, 5761}, {1737, 5219}, {5249, 5770}, {5719, 5790}, {5748, 5775}

X(6859) = (EULER LINE)∩X(226)X(5770)

Barycentrics a^7-(b+c)*a^6-(5*b^2-2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6859) has Shinagawa coefficients (-abc + 2$aS_A$, $aS_A$).

X(6859) lies on these lines: {2, 3}, {226, 5770}, {912, 5226}, {1737, 3340}, {4511, 5818}

X(6860) = (EULER LINE)∩X(46)X(3817)

Barycentrics a^7-(b+c)*a^6-(5*b^2-4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-3*(b^4-c^4)*(b^2-c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6860) has Shinagawa coefficients (-abc + 2$aS_A$, abc + $aS_A$).

X(6860) lies on these lines: {2, 3}, {46, 3817}, {65, 5704}, {5252, 5703}

X(6861) = (EULER LINE)∩X(498)X(1837)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6861) has Shinagawa coefficients (abc + 2$aS_A$, abc).

X(6861) lies on these lines: {2, 3}, {498, 1837}, {499, 942}, {960, 5791}, {1728, 5219}, {3085, 5790}, {5398, 5713}, {5550, 5768}, {5730, 5901}

X(6862) = (EULER LINE)∩X(65)X(499)

Barycentrics a^7 - a^6(b + c) - a^5(3b^2 - 2bc + 3c^2) + a^4(3b^3 + b^2c + bc^2 + 3c^3) + a^3(3b^4 - 2b^3c + 2b^2c^2 - 2bc^3 + 3c^4) - a^2(b - c)^2(3b^3 + 5b^2c + 5bc^2 + 3c^3) - a(b^2 - c^2)^2(b^2 + c^2) + (b - c)^4(b + c)^3 : :

As a point on the Euler line, X(6862) has Shinagawa coefficients (-abc + 2$aS_A$, abc).

X(6862) lies on these lines: {2, 3}, {65, 499}, {155, 940}, {355, 498}, {944, 2320}, {952, 3085}, {1058, 1484}, {1352, 5135}, {2278, 5816}, {3086, 5901}, {3487, 5770}, {3584, 5881}, {3612, 5587}, {3822, 5450}, {5552, 5790}, {5763, 5771}

X(6862) = complement of X(6825)

X(6863) = (EULER LINE)∩X(119)X(958)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(b^2-b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6863) has Shinagawa coefficients (-abc + 2$aS_A$, -abc).

X(6863) lies on these lines: {2, 3}, {119, 958}, {498, 517}, {499, 1385}, {1482, 3085}, {5217, 5840}, {5219, 5709}, {5231, 5534}, {5292, 5396}, {5552, 5690}, {5705, 5720}, {5707, 5718}, {5742, 5778}, {5747, 5755}

X(6864) = (EULER LINE)∩X(7)X(5777)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(4*b^2-b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6864) has Shinagawa coefficients (2abc, abc + $aS_A$).

X(6864) lies on these lines: {2, 3}, {7, 5777}, {78, 5082}, {142, 1490}, {355, 938}, {388, 1210}, {581, 4648}, {936, 946}, {1058, 5703}, {1445, 5817}, {1478, 3361}, {2551, 5705}, {3419, 5804}, {3421, 5818}, {3452, 5715}, {3487, 5720}, {3811, 6601}, {3817, 6700}, {4253, 5816}, {4423, 6253}, {5044, 5758}, {5437, 6245}, {5439, 5768}

X(6864) = complement of X(37423)

X(6865) = (EULER LINE)∩X(7)X(5812)

Barycentrics a^7-(b+c)*a^6-(b^2+6*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6865) has Shinagawa coefficients (2abc, -abc - $aS_A$).

X(6865) lies on these lines: {2, 3}, {7, 5812}, {9, 6245}, {11, 5584}, {40, 497}, {72, 5768}, {78, 944}, {142, 5715}, {165, 1479}, {278, 1076}, {329, 1071}, {388, 3576}, {515, 936}, {517, 938}, {942, 5758}, {950, 6282}, {971, 5811}, {1056, 1385}, {1158, 5698}, {1445, 5709}, {1490, 3452}, {2550, 5705}, {3086, 3428}, {3359, 6361}, {3436, 5731}, {3579, 5704}, {4297, 6700}, {4309, 5537}, {4413, 6253}, {4648, 5713}, {5044, 5787}, {5082, 5657}, {5085, 5800}, {5708, 5762}, {5732, 6260}, {5909, 5932}

X(6866) = (EULER LINE)∩X(355)X(5761)

Barycentrics a^7-(b+c)*a^6+(b-c)^2*a^5-(b+c)^3*a^4-(5*b^4-2*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2+2*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6866) has Shinagawa coefficients ($aS_A$, abc + 2$aS_A$).

X(6866) lies on these lines: {2, 3}, {355, 5761}, {3485, 5722}, {3817, 6261}, {4658, 5713}, {5261, 5719}, {5763, 5790}, {5806, 5887}

X(6867) = (EULER LINE)∩X(355)X(3485)

Barycentrics a^7-(b+c)*a^6+(b+c)^2*a^5-(b+c)^3*a^4-(5*b^4-2*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2-2*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6867) has Shinagawa coefficients ($aS_A$, -abc + 2$aS_A$).

X(6867) lies on these lines: {2, 3}, {355, 3485}, {912, 5714}, {952, 5261}, {3340, 5587}, {3419, 5761}, {3486, 5886}, {3869, 5818}, {3947, 5534}, {5086, 5603}, {5274, 5901}

X(6868) = (EULER LINE)∩X(40)X(920)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+6*b^2*c^2+c^4)*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6868) has Shinagawa coefficients ($aS_A$, abc - 2$aS_A$).

X(6868) lies on these lines: {2, 3}, {40, 920}, {165, 4324}, {387, 5398}, {388, 5841}, {390, 1482}, {393, 2193}, {515, 5837}, {517, 3486}, {527, 5882}, {573, 5822}, {912, 944}, {950, 5709}, {952, 3927}, {958, 5842}, {971, 5698}, {1385, 3485}, {3428, 6284}, {3576, 4299}, {3600, 6147}, {3916, 5770}, {3929, 5881}, {4297, 6261}, {4313, 5758}, {4317, 4654}, {4653, 5713}, {4877, 5816}, {5086, 5657}, {5755, 5802}, {5761, 5812}

X(6869) = (EULER LINE)∩X(57)X(4299)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2+2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+6*b^2*c^2+c^4)*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6869) has Shinagawa coefficients ($aS_A$, -abc - 2$aS_A$).

X(6869) lies on these lines: {2, 3}, {57, 4299}, {500, 4340}, {515, 5709}, {516, 6261}, {517, 3189}, {942, 3486}, {944, 3873}, {1175, 4846}, {1385, 5805}, {3296, 3600}, {3428, 6253}, {3485, 4294}, {3601, 4302}, {3869, 6361}, {3945, 5453}, {5770, 5787}

X(6870) = (EULER LINE)∩X(90)X(5556)

Barycentrics a^7-(b+c)*a^6+(b^2-4*b*c+c^2)*a^5-(b+c)^3*a^4-(5*b^4-2*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2+4*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6870) has Shinagawa coefficients (a$S_A$, 2abc + 2$aS_A$).

X(6870) lies on these lines: {2, 3}, {90, 5556}, {3485, 5274}, {3486, 5261}, {5715, 5905}

X(6871) = (EULER LINE)∩X(12)X(3434)

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

As a point on the Euler line, X(6871) has Shinagawa coefficients ($aS_A$, -2abc + 2$aS_A$).

X(6871) lies on these lines: {2, 3}, {12, 3434}, {144, 5832}, {145, 3485}, {193, 5820}, {210, 3617}, {1376, 3614}, {1479, 3822}, {1621, 5225}, {1837, 3838}, {1898, 3812}, {2886, 3436}, {2900, 5175}, {2975, 5229}, {3304, 3829}, {3486, 3622}, {3868, 5714}, {3870, 3947}, {4881, 5550}, {5057, 5123}, {5217, 6668}, {5554, 5587}, {5818, 5887}

X(6872) = (EULER LINE)∩X(1)X(5905)

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

As a point on the Euler line, X(6872) has Shinagawa coefficients ($aS_A$, 2abc - 2$aS_A$).

X(6872) lies on these lines: {1, 5905}, {2, 3}, {8, 90}, {10, 4302}, {35, 5552}, {40, 5554}, {55, 3436}, {63, 950}, {78, 4304}, {100, 2551}, {144, 145}, {148, 5985}, {329, 4313}, {345, 5016}, {387, 1780}, {388, 1621}, {391, 4271}, {497, 2975}, {499, 5267}, {515, 5250}, {519, 3951}, {529, 3303}, {551, 4317}, {908, 3601}, {938, 3218}, {944, 3877}, {958, 3434}, {960, 1898}, {993, 1479}, {1043, 5739}, {1125, 4299}, {1145, 4678}, {1210, 4652}, {1319, 3485}, {1329, 5217}, {1478, 5248}, {1698, 4324}, {1837, 4640}, {2550, 5260}, {2647, 4331}, {3085, 5080}, {3189, 3681}, {3421, 3871}, {3452, 4855}, {3476, 3890}, {3488, 3868}, {3616, 4293}, {3617, 5086}, {3624, 4316}, {3679, 4330}, {3683, 5794}, {3816, 5204}, {3870, 4314}, {3897, 5603}, {3916, 5722}, {3920, 4339}, {3935, 5815}, {4298, 4666}, {4305, 4511}, {4512, 5691}, {5175, 5273}, {5249, 5436}, {5441, 5692}, {5731, 6223}

X(6872) = complement of X(31295)

X(6873) = (EULER LINE)∩X(946)X(3899)

Barycentrics (2*b^2-b*c+2*c^2)*a^5-2*(b+c)*(b^2+c^2)*a^4-4*(b^4+c^4)*a^3+4*(b^3-c^3)*(b^2-c^2)*a^2+(2*b^2+b*c+2*c^2)*(b^2-c^2)^2*a-2*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6873) has Shinagawa coefficients (2$aS_A$, abc + 2$aS_A$).

X(6873) lies on these lines: {2, 3}, {946, 3899}, {3485, 5425}

X(6874) = (EULER LINE)∩X(944)X(3822)

Barycentrics (2*b^2+b*c+2*c^2)*a^5-2*(b+c)*(b^2+c^2)*a^4-4*(b^4+c^4)*a^3+4*(b^3-c^3)*(b^2-c^2)*a^2+(2*b^2-b*c+2*c^2)*(b^2-c^2)^2*a-2*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6874) has Shinagawa coefficients (2$aS_A$, -abc + 2$aS_A$).

X(6874) lies on these lines: {2, 3}, {944, 3822}, {3485, 5818}, {4004, 5887}, {5086, 5886}

X(6875) = (EULER LINE)∩X(35)X(3486)

Barycentrics a*(2*a^6-2*(b+c)*a^5-(4*b^2-b*c+4*c^2)*a^4+4*(b^3+c^3)*a^3+2*(b^2+c^2)^2*a^2-2*(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6875) has Shinagawa coefficients (2$aS_A$, abc - 2$aS_A$).

X(6875) lies on these lines: {2, 3}, {35, 3486}, {36, 3485}, {920, 3612}, {944, 993}, {970, 3567}, {1385, 3869}, {3085, 5172}, {3576, 5267}, {3647, 5693}, {5010, 6684}, {5248, 5603}, {5251, 5818}

X(6875) = {X(3),X(4)}-harmonic conjugate of X(6876)

X(6876) = (EULER LINE)∩X(35)X(3485)

Barycentrics a*(2*a^6-2*(b+c)*a^5-(4*b^2+b*c+4*c^2)*a^4+4*(b^3+c^3)*a^3+2*(b^2+c^2)^2*a^2-2*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6876) has Shinagawa coefficients (2$aS_A$, -abc - 2$aS_A$).

X(6876) lies on these lines: {2, 3}, {35, 3485}, {36, 3486}, {74, 931}, {165, 6261}, {970, 5890}, {3428, 3913}, {3579, 3869}, {3587, 4855}, {4294, 5172}

X(6876) = {X(3),X(4)}-harmonic conjugate of X(6875)

X(6877) = (EULER LINE)∩X(1708)X(5714)

Barycentrics a^7-(b+c)*a^6-(5*b^2+4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^2-6*b*c+7*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2+4*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6877) has Shinagawa coefficients (2abc + 2$aS_A$, $aS_A$).

X(6877) lies on these lines: {2, 3}, {1708, 5714}, {1737, 3487}

X(6878) = (EULER LINE)∩X(943)X(3086)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2+4*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^2-2*b*c+5*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6878) has Shinagawa coefficients (2abc + 2$aS_A$, -$aS_A$).

X(6878) lies on these lines: {2, 3}, {943, 3086}, {1708, 3338}, {1737, 3488}, {5550, 5758}

X(6879) = (EULER LINE)∩X(997)X(5818)

Barycentrics a^7-(b+c)*a^6-(5*b^2-4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b-c)^2*(7*b^2+6*b*c+7*c^2)*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2-4*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6879 has Shinagawa coefficients (-2abc + 2$aS_A$, $aS_A$).

X(6879) lies on these lines: {2, 3}, {997, 5818}, {1737, 5603}

X(6880) = (EULER LINE)∩X(165)X(1519)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-4*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(b-c)^2*(5*b^2+2*b*c+5*c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6880) has Shinagawa coefficients (-2abc + 2$aS_A$, -$aS_A$).

X(6880) lies on these lines: {2, 3}, {165, 1519}, {944, 1737}, {997, 5657}, {1512, 3576}, {3035, 3428}, {5119, 5218}

X(6881) = (EULER LINE)∩X(12)X(942)

Barycentrics (b+c)^2*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4+2*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6881) has Shinagawa coefficients (2abc + $aS_A$, $aS_A$).

X(6881) lies on these lines: {2, 3}, {12, 942}, {116, 119}, {495, 3475}, {517, 3925}, {912, 5249}, {946, 3841}, {997, 2886}, {1213, 5755}, {1329, 5791}, {1441, 2973}, {1698, 5709}, {1699, 3587}, {1714, 5707}, {2095, 3820}, {3649, 5694}, {3814, 5745}, {3824, 5777}, {3826, 5805}, {4511, 5178}, {5251, 5841}

X(6882) = (EULER LINE)∩X(11)X(517)

Barycentrics (b-c)^2*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-2*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6882) has Shinagawa coefficients (-2abc + $aS_A$, $aS_A$)and also (r - R, r + R).

X(6882) lies on these lines: {2, 3}, {11, 517}, {12, 1385}, {36, 5841}, {40, 5445}, {104, 5080}, {119, 214}, {329, 5770}, {355, 997}, {495, 3476}, {496, 1482}, {908, 912}, {938, 5761}, {946, 3754}, {952, 4511}, {1320, 1484}, {1699, 3359}, {1708, 5812}, {2077, 3583}, {3035, 5842}, {3576, 5444}, {3649, 5885}, {3654, 3829}, {3816, 5886}, {3820, 5790}, {5087, 6001}, {5179, 6506}, {5704, 5758}, {5748, 5768}

X(6883) = (EULER LINE)∩X(9)X(912)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2+b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2+4*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a-2*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6883) has Shinagawa coefficients (2abc + $aS_A$, -$aS_A$).

X(6883) lies on these lines: {2, 3}, {9, 912}, {36, 5219}, {40, 5259}, {55, 1737}, {182, 6176}, {495, 1617}, {515, 6666}, {517, 1001}, {572, 5778}, {580, 5707}, {582, 5706}, {748, 1064}, {938, 943}, {940, 5398}, {942, 1708}, {944, 5260}, {956, 3681}, {958, 997}, {993, 3452}, {999, 3475}, {1000, 2346}, {1482, 3890}, {1621, 5657}, {3074, 3157}, {3295, 5690}, {3359, 4512}, {3428, 4423}, {3576, 5251}, {3579, 5806}, {3616, 5761}, {3826, 5842}, {4383, 5396}, {5178, 5687}, {5248, 6684}, {5273, 5770}, {5284, 5603}, {5462, 5752}, {5763, 5901}

X(6884) = (EULER LINE)∩X(7)X(499)

Barycentrics a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4+2*b*c*(b^2+b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+3*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6884) has Shinagawa coefficients (abc + 2$aS_A$, 2abc).

X(6884) lies on these lines: {2, 3}, {7, 499}, {80, 3085}, {498, 4313}, {1259, 2894}, {3086, 5443}, {3622, 6265}, {3868, 5694}, {5273, 5536}

X(6885) = (EULER LINE)∩X(57)X(4317)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+c^4-2*b*c*(4*b^2-3*b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b+c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6885) has Shinagawa coefficients ($aS_A$ - abc, abc - 2$aS_A$).

X(6885) lies on these lines: {2,3}, {57,4317}, {119,5229}, {355,1788}, {390,5901}, {942,3476}, {952,3600}, {2551,5841}, {3474,5887}, {3601,4309}, {4256,5713}, {4292,5720}, {4294,5886}, {4298,5534}, {4299,5587}, {4340,5396}, {5440,5761}

X(6886) = (EULER LINE)∩X(7)X(1728)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4+2*b*c*(4*b^2+b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6886) has Shinagawa coefficients (2abc + $aS_A$, 2abc).

X(6886) lies on these lines: {2, 3}, {7, 1728}, {498, 4314}, {499, 4298}, {946, 3305}, {1837, 3748}, {3086, 5226}, {3555, 5886}, {3876, 5603}

X(6887) = (EULER LINE)∩X(499)X(3333)

Barycentrics a^7-(b+c)*a^6-(3*b^2+2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4+2*b*c*(4*b^2+b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2+6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6887) has Shinagawa coefficients (2abc + $aS_A$, abc).

X(6887) lies on these lines: {2, 3}, {499, 3333}, {582, 3332}, {946, 6666}, {1058, 2346}, {1125, 5720}, {3085, 5722}, {3086, 3475}, {3940, 5901}, {4253, 5747}, {5044, 5761}, {5178, 5552}, {5439, 5770}, {5719, 5780}

X(6887) = complement of X(37407)

X(6888) = (EULER LINE)∩X(10)X(5538)

Barycentrics a^7-(b+c)*a^6-3*(b^2-b*c+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(b^2-b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b+c)*(b^2-c^2)*(b^3-c^3)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6888) has Shinagawa coefficients (-abc + 2$aS_A$, 2abc).

X(6888) lies on these lines: {2, 3}, {10, 5538}, {12, 153}, {498, 4305}, {499, 3336}, {946, 5180}, {1158, 3306}, {1770, 3817}, {3086, 5902}, {4652, 5715}, {4855, 5587}, {5249, 6705}, {5885, 5886}

X(6889) = (EULER LINE)∩X(9)X(2252)

Barycentrics a^7-(b+c)*a^6-(3*b^2+4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6889) has Shinagawa coefficients (abc + $aS_A$, -abc).

X(6889) lies on these lines: {2,3}, {9,2252}, {46,226}, {65,3085}, {72,5552}, {165,5715}, {224,6734}, {499,950}, {581,1714}, {943,5218}, {944,5794}, {1068,1214}, {1071,5791}, {1155,5714}, {1213,5776}, {1385,3419}, {1490,1698}, {1754,5713}, {2245,5746}, {2278,5802}, {2646,3086}, {2949,3336}, {3074,3215}, {5249,5709}, {5251,6256}, {5759,5880}

X(6890) = (EULER LINE)∩X(46)X(962)

Barycentrics a^7-(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(4*b^2-b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6890) has Shinagawa coefficients (-2abc + $aS_A$, 2abc).

X(6890) lies on these lines: {2, 3}, {46, 962}, {63, 6705}, {78, 6245}, {84, 908}, {498, 4297}, {499, 516}, {515, 4855}, {946, 3306}, {2646, 3476}, {3085, 3612}, {3218, 5758}, {4999, 5584}, {5440, 5787}, {5748, 6223}

X(6891) = (EULER LINE)∩X(40)X(499)

Barycentrics a^7-(b+c)*a^6-3*(b-c)^2*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(4*b^2-b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6891) has Shinagawa coefficients (-2abc + $aS_A$, abc).

X(6891) lies on these lines: {2, 3}, {40, 499}, {72, 5770}, {104, 3436}, {498, 3576}, {517, 1788}, {942, 5761}, {944, 5176}, {946, 3359}, {1056, 1476}, {1385, 3085}, {1479, 2077}, {2095, 5763}, {3428, 5433}, {3452, 6705}, {3814, 6256}, {3911, 5709}, {5225, 5840}, {5328, 5811}, {5435, 5758}, {5720, 6245}

X(6892) = (EULER LINE)∩X(1)X(5770)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-6*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6892) has Shinagawa coefficients (2$aS_A$ - abc, 2abc - $aS_A$).

X(6892) lies on these lines: {1,5770}, {2,3}, {63,5761}, {355,4305}, {912,5703}, {952,5789}, {1125,1158}, {2320,5768}, {3601,5881}, {4257,5713}, {4295,5886}, {4301,5709}, {5708,5901}, {5734,5744}

X(6893) = (EULER LINE)∩X(119)X(3085)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6893) has Shinagawa coefficients (abc, 2abc - $aS_A$).

X(6893) lies on these lines: {2, 3}, {119, 3085}, {153, 3622}, {329, 5804}, {355, 497}, {388, 5886}, {517, 2551}, {908, 5761}, {912, 938}, {946, 5795}, {950, 5720}, {952, 1058}, {1056, 5901}, {1125, 6256}, {1210, 5770}, {1420, 1478}, {1479, 1697}, {1482, 3421}, {1512, 5250}, {3434, 5818}, {3436, 4861}, {4266, 5816}, {5082, 5790}, {5126, 5229}, {5722, 5777}, {5778, 5802}, {5806, 5812}

X(6894) = (EULER LINE)∩X(72)X(2894)

Barycentrics a^7-(b+c)*a^6-(b^2+b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6894) has Shinagawa coefficients (abc, 2abc + 2$aS_A$).

X(6894) lies on these lines: {2, 3}, {72, 2894}, {78, 1699}, {149, 946}, {153, 6583}, {938, 1478}, {1210, 3337}, {1479, 5703}, {1621, 6253}, {2949, 3219}, {3868, 5805}, {3951, 5735}, {4860, 5229}, {5080, 5536}

X(6895) = (EULER LINE)∩X(78)X(5080)

Barycentrics a^7-(b+c)*a^6-(b^2+3*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6895) has Shinagawa coefficients (abc, -2abc - 2$aS_A$).

X(6895) lies on these lines: {2, 3}, {78, 5080}, {100, 6253}, {149, 151}, {225, 3100}, {938, 1479}, {1158, 5535}, {1210, 1770}, {1446, 4872}, {1478, 4305}, {3218, 6245}, {3868, 5787}, {4333, 5131}, {5221, 5225}

X(6895) = anticomplement of X(411)

X(6896) = (EULER LINE)∩X(1056)X(1837)

Barycentrics a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+8*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6896) has Shinagawa coefficients (2abc, 2abc + $aS_A$).

X(6896) lies on these lines: {2, 3}, {1056, 1837}, {1058, 3748}, {3811, 5603}, {4420, 5082}, {5400, 5713}

X(6897) = (EULER LINE)∩X(46)X(388)

Barycentrics a^7-(b+c)*a^6-(b^2-8*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6897) has Shinagawa coefficients (2abc, -2abc + $aS_A$).

X(6897) lies on these lines: {2, 3}, {46, 388}, {65, 1056}, {355, 4002}, {497, 3612}, {944, 2550}, {1038, 1068}, {1058, 2646}, {1385, 3434}, {1478, 6684}, {1698, 6256}, {3436, 3916}, {4861, 5082}

X(6898) = (EULER LINE)∩X(497)X(5818)

Barycentrics a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-8*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6898) has Shinagawa coefficients (2abc, 2abc - $aS_A$).

X(6898) lies on these lines: {2, 3}, {497, 5818}, {1058, 1837}, {2551, 5603}, {3421, 4861}, {3436, 5886}, {3624, 6256}

X(6899) = (EULER LINE)∩X(46)X(497)

Barycentrics a^7-(b+c)*a^6-(b^2+8*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6899) has Shinagawa coefficients (2abc, -2abc - $aS_A$).

X(6899) lies on these lines: {2, 3}, {46, 497}, {65, 1058}, {355, 3921}, {388, 3612}, {944, 3811}, {1040, 1068}, {1056, 2646}, {1479, 3911}, {3421, 4420}, {3434, 3579}, {3436, 5440}

X(6900) = (EULER LINE)∩X(80)X(388)

Barycentrics a^7-(b+c)*a^6-(b^2-b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6900) has Shinagawa coefficients (2abc, abc + 2$aS_A$).

X(6900) lies on these lines: {2, 3}, {80, 388}, {355, 3873}, {497, 5443}, {1389, 6601}, {1478, 3337}, {2550, 3878}, {2894, 3940}, {3189, 5603}, {3585, 3911}, {5536, 5818}

X(6901) = (EULER LINE)∩X(10)X(5535)

Barycentrics a^7-(b+c)*a^6-(b^2-3*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b+c)*(b^2-c^2)*(b^3-c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6901) has Shinagawa coefficients (2abc, -abc + 2$aS_A$).

X(6901) lies on these lines: {2, 3}, {10, 5535}, {149, 5901}, {355, 5885}, {388, 5902}, {553, 5270}, {946, 5538}, {1074, 6198}, {1478, 1788}, {3585, 5131}, {3916, 5080}, {5260, 5841}, {5881, 6173}

X(6902) = (EULER LINE)∩X(149)X(5690)

Barycentrics a^7-(b+c)*a^6-(b^2+b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6902) has Shinagawa coefficients (2abc, abc - 2$aS_A$).

X(6902) lies on these lines: {2, 3}, {149, 5690}, {497, 5697}, {944, 6326}, {1056, 1388}, {1058, 2098}, {1199, 3193}, {1385, 5080}, {1479, 5657}, {3583, 6684}, {5253, 5841}, {5303, 6713}, {5506, 5818}

X(6903) = (EULER LINE)∩X(484)X(1479)

Barycentrics a^7-(b+c)*a^6-(b^2+3*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6903) has Shinagawa coefficients (2abc, -abc -2$aS_A$).

X(6903) lies on these lines: {2, 3}, {484, 1479}, {497, 5903}, {1058, 2099}, {1076, 1870}, {3436, 6224}

X(6904) = (EULER LINE)∩X(7)X(78)

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

As a point on the Euler line, X(6904) has Shinagawa coefficients (2R - r, 2r).

X(6904) lies on these lines: {2, 3}, {7, 78}, {8, 57}, {10, 4293}, {56, 2550}, {69, 1434}, {142, 390}, {144, 3876}, {145, 942}, {193, 4260}, {200, 4298}, {226, 5438}, {274, 3785}, {329, 936}, {347, 1038}, {354, 3189}, {386, 4340}, {388, 1376}, {391, 579}, {392, 6361}, {610, 5749}, {614, 4339}, {938, 3306}, {944, 3753}, {950, 5437}, {960, 3474}, {962, 6282}, {976, 4310}, {997, 4295}, {999, 5082}, {1056, 5687}, {1125, 4294}, {1193, 4307}, {1210, 5175}, {1329, 5229}, {1467, 3872}, {1698, 4299}, {1788, 5794}, {2096, 5777}, {2551, 4413}, {3241, 3680}, {3339, 6737}, {3361, 4847}, {3434, 5253}, {3476, 5836}, {3485, 5880}, {3486, 3812}, {3487, 5440}, {3488, 5439}, {3621, 5708}, {3624, 4302}, {3679, 4317}, {3752, 5716}, {3816, 5225}, {3925, 5204}, {4255, 5712}, {4315, 4853}, {4652, 5273}, {4855, 5249}, {5122, 5791}, {5128, 5837}, {5261, 5552}, {5277, 5286}, {5290, 6745}, {5435, 6734}, {5554, 5768}, {5587, 6705}, {5748, 6700}

X(6905) = (EULER LINE)∩X(1)X(1389)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b-c)^2*(b^2+c^2)*a^2-(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6905) has Shinagawa coefficients (r, -R - r).

X(6905) lies on these lines: {1, 1389}, {2, 3}, {11, 5172}, {35, 946}, {36, 80}, {40, 997}, {46, 6261}, {54, 1437}, {55, 5603}, {56, 944}, {63, 5720}, {73, 3075}, {78, 5709}, {81, 5396}, {95, 286}, {100, 517}, {108, 2734}, {119, 4996}, {171, 1064}, {216, 1172}, {355, 2975}, {484, 2800}, {516, 1519}, {580, 3216}, {602, 978}, {603, 1745}, {656, 3737}, {758, 5535}, {912, 3218}, {958, 5818}, {970, 1812}, {971, 5122}, {972, 2728}, {993, 5587}, {1155, 6001}, {1193, 3072}, {1292, 2726}, {1376, 3428}, {1385, 5253}, {1465, 1870}, {1470, 4293}, {1482, 3871}, {1490, 1708}, {1503, 5096}, {1610, 3417}, {1621, 5886}, {1699, 5010}, {1749, 1768}, {1783, 1951}, {1809, 5081}, {2051, 4276}, {2096, 5658}, {2287, 5755}, {2801, 4973}, {3336, 5884}, {3576, 3833}, {3916, 5777}, {4255, 5706}, {4265, 5480}, {4299, 6256}, {4313, 5804}, {5088, 6516}, {5433, 6253}, {5435, 5768}, {5450, 5691}, {5563, 5882}, {6127, 6149}

X(6905) = inverse-in-excentral-hexyl-ellipse of X(4)
X(6905) = Thomson-isogonal conjugate of X(33883)

X(6906) = (EULER LINE)∩X(1)X(104)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-3*b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b-c)^2*(b^2+c^2)*a^2-(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6906) has Shinagawa coefficients (r, R - r).

X(6906) lies on these lines: {1, 104}, {2, 3}, {10, 2077}, {12, 2829}, {35, 515}, {36, 946}, {40, 993}, {55, 944}, {56, 4295}, {84, 943}, {100, 355}, {191, 5538}, {284, 1765}, {318, 1809}, {498, 6256}, {516, 5267}, {517, 2975}, {553, 5563}, {577, 1172}, {950, 6705}, {952, 3871}, {958, 5657}, {1125, 1519}, {1193, 3073}, {1376, 5818}, {1385, 1621}, {1389, 5903}, {1437, 1614}, {1470, 3086}, {1503, 4265}, {1512, 5251}, {1537, 5901}, {1633, 3417}, {1699, 4333}, {1709, 3612}, {2078, 4311}, {2094, 5734}, {2096, 3487}, {2646, 6001}, {2654, 3075}, {2687, 2689}, {3428, 6361}, {3576, 5248}, {3577, 5128}, {3652, 5694}, {3746, 5882}, {4252, 5706}, {4304, 6245}, {4313, 5768}, {4511, 5887}, {4652, 5709}, {4855, 5720}, {4996, 5840}, {5010, 5691}, {5096, 5480}, {5122, 5806}, {5253, 5886}, {5258, 5537}, {5435, 5804}, {5440, 5777}, {5761, 5905}

X(6907) = (EULER LINE)∩X(7)X(2095)

Barycentrics (b^2+4*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3+c^3)*(b+c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6907) has Shinagawa coefficients (r + 2R, r - 2R).

X(6907) lies on these lines: {2, 3}, {7, 2095}, {9, 119}, {10, 5777}, {11, 3576}, {12, 40}, {52, 6045}, {63, 5771}, {72, 5690}, {84, 5705}, {226, 495}, {329, 5657}, {342, 6356}, {355, 1490}, {392, 1519}, {496, 950}, {515, 2886}, {516, 3822}, {573, 1901}, {581, 1834}, {944, 5175}, {946, 3838}, {952, 3419}, {958, 6256}, {993, 2829}, {1071, 6734}, {1074, 1465}, {1214, 1785}, {1329, 4640}, {1478, 3428}, {1482, 3487}, {1512, 3753}, {1537, 3877}, {1698, 1709}, {1727, 5445}, {1737, 1864}, {1750, 3925}, {1765, 5742}, {2077, 5432}, {2096, 5744}, {3584, 5537}, {3813, 5882}, {3824, 5806}, {4999, 5450}, {5219, 6282}, {5658, 5790}, {5714, 5758}, {5791, 6259}, {5794, 6261}

X(6908) = (EULER LINE)∩X(7)X(5709)

Barycentrics a^7-(b+c)*a^6-3*(b+c)^2*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6908) has Shinagawa coefficients (r + 2R, -2R).

X(6908) lies on these lines: {2, 3}, {7, 5709}, {8, 5534}, {9, 1158}, {10, 1490}, {12, 5584}, {40, 226}, {72, 5657}, {84, 5745}, {165, 498}, {318, 6350}, {329, 3359}, {347, 1068}, {387, 581}, {388, 3428}, {499, 3586}, {516, 5715}, {517, 3487}, {572, 5802}, {573, 1715}, {943, 5281}, {944, 3419}, {950, 3086}, {966, 5776}, {971, 5791}, {991, 5292}, {1211, 6247}, {1214, 3346}, {1385, 3488}, {1698, 1750}, {2096, 3916}, {3072, 4307}, {3332, 5713}, {3579, 5714}, {3584, 4338}, {3587, 5226}, {3824, 5805}, {3945, 5707}, {4300, 5230}, {5175, 5731}, {5273, 6223}, {5658, 5777}, {5704, 5809}, {5705, 5732}, {5706, 5712}, {5768, 6734}

X(6909) = (EULER LINE)∩X(1)X(1106)

Barycentrics a*(a^6-(b+c)*a^5-(2*b-c)*(b-2*c)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-4*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6909) has Shinagawa coefficients (- r + R, r - 2R).

X(6909) lies on these lines: {1, 1106}, {2, 3}, {35, 4297}, {36, 516}, {40, 2975}, {55, 3476}, {56, 962}, {63, 6282}, {78, 84}, {100, 515}, {104, 517}, {105, 2737}, {153, 2932}, {165, 993}, {497, 1470}, {519, 5537}, {603, 3562}, {758, 1768}, {934, 2739}, {938, 1466}, {944, 3871}, {946, 5253}, {956, 6244}, {971, 5440}, {997, 1709}, {1019, 6003}, {1158, 3869}, {1261, 4737}, {1262, 3100}, {1295, 1309}, {1490, 4855}, {1621, 3576}, {1754, 4257}, {1765, 2287}, {2096, 5905}, {2829, 5080}, {3304, 5734}, {4301, 5563}, {4511, 6001}, {4872, 6516}, {4973, 5536}, {4996, 5842}, {5260, 6684}, {6705, 6734}

X(6910) = (EULER LINE)∩X(8)X(2320)

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

As a point on the Euler line, X(6910) has Shinagawa coefficients (2 r + R, -r).

X(6910) lies on these lines: {2, 3}, {8, 2320}, {10, 3612}, {35, 3434}, {46, 1125}, {55, 3813}, {56, 6690}, {65, 3616}, {69, 5135}, {78, 5745}, {224, 936}, {226, 4652}, {498, 993}, {499, 5248}, {940, 3193}, {958, 5432}, {966, 2278}, {988, 3011}, {1001, 5433}, {1155, 5180}, {1454, 3485}, {1478, 5267}, {1621, 3086}, {1714, 4256}, {2182, 5296}, {2886, 5217}, {2975, 3085}, {3218, 3487}, {3305, 6700}, {3601, 6734}, {3618, 4259}, {3622, 5330}, {3624, 4512}, {3822, 4299}, {3868, 5703}, {3871, 5281}, {3876, 5273}, {3913, 4995}, {3916, 5905}, {4252, 5718}, {4293, 5303}, {4305, 5086}, {4423, 6691}, {5440, 5791}

X(6911) = (EULER LINE)∩X(36)X(5587)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-4*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+2*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6911) has Shinagawa coefficients (R - r, R + r), and also (2abc - $aSA$, $aSA$).

X(6911) lies on these lines: {2, 3}, {36, 5587}, {46, 5887}, {55, 5886}, {56, 355}, {57, 912}, {100, 5603}, {119, 1470}, {155, 3216}, {386, 5707}, {515, 6692}, {517, 997}, {569, 1437}, {579, 5778}, {750, 1064}, {936, 5709}, {940, 5396}, {944, 5253}, {952, 999}, {956, 5176}, {965, 5755}, {978, 3072}, {1060, 1465}, {1125, 6796}, {1216, 5752}, {1482, 4511}, {1699, 2077}, {1708, 5777}, {2095, 3940}, {2099, 6265}, {2975, 5818}, {3075, 3157}, {3295, 5901}, {3333, 5534}, {3336, 5693}, {3428, 4413}, {3816, 5842}, {3927, 5780}, {4383, 5398}, {5435, 5770}, {5535, 5692}, {5563, 5881}, {5902, 6326}

X(6912) = (EULER LINE)∩X(1)X(651)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-3*b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)^2*a^2-(b^4-c^4)*(b-c)*a-3*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6912) has Shinagawa coefficients (R + r, 2R - r).

X(6912) lies on these lines: {1, 651}, {2, 3}, {10, 5537}, {36, 3817}, {40, 5260}, {65, 1776}, {100, 5587}, {104, 5886}, {144, 956}, {153, 495}, {355, 3871}, {515, 1621}, {516, 5251}, {517, 3219}, {774, 2647}, {946, 2975}, {958, 962}, {993, 1699}, {1001, 5731}, {1259, 5175}, {1466, 5704}, {1768, 5883}, {1935, 2654}, {3303, 3486}, {3304, 3485}, {3305, 6282}, {3555, 5887}, {3576, 5284}, {3616, 6223}, {3746, 4314}, {3869, 3872}, {3916, 5806}, {4256, 5400}, {4297, 5259}, {4298, 5563}, {4301, 5258}, {5086, 6735}, {5248, 5691}, {5253, 5450}, {5603, 5905}

X(6913) = (EULER LINE)∩X(1)X(1864)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b+c)^2*a^2-(b^4-c^4)*(b-c)*a-4*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6913) has Shinagawa coefficients (2R + r, 2R - r).

X(6913) lies on these lines: {1, 1864}, {2, 3}, {9, 374}, {40, 3683}, {55, 3586}, {63, 2095}, {65, 1728}, {72, 1482}, {78, 5780}, {226, 999}, {329, 956}, {355, 950}, {515, 1001}, {912, 5728}, {946, 958}, {952, 954}, {962, 5260}, {993, 3817}, {1125, 6260}, {1158, 3812}, {1159, 5729}, {1260, 3419}, {1385, 1490}, {1478, 1617}, {1699, 3428}, {1724, 5706}, {1750, 3576}, {1901, 5120}, {2077, 4413}, {3073, 5711}, {3303, 5881}, {3304, 4870}, {3487, 5811}, {4254, 5816}, {5175, 5687}, {5259, 5691}, {5273, 5771}, {5284, 5731}, {5709, 5806}

X(6914) = (EULER LINE)∩X(1)X(1399)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6914) has Shinagawa coefficients (R + 2r, R - 2r).

X(6914) lies on these lines: {1, 1399}, {2, 3}, {35, 355}, {36, 1836}, {55, 952}, {56, 5901}, {100, 5790}, {104, 1621}, {119, 5432}, {155, 5453}, {156, 1437}, {497, 1484}, {517, 993}, {946, 5267}, {954, 5843}, {956, 5844}, {958, 5690}, {970, 5446}, {1030, 5816}, {1352, 4265}, {1385, 5248}, {1478, 5172}, {1482, 2975}, {1483, 3295}, {1709, 3576}, {1768, 5426}, {2077, 5251}, {2646, 5887}, {2829, 6690}, {2886, 5840}, {3065, 6326}, {3488, 5770}, {3652, 5693}, {3654, 5537}, {3816, 6713}, {4252, 5707}, {5010, 5587}, {5180, 5603}

X(6914) = circumcircle-inverse of X(37976)

X(6915) = (EULER LINE)∩X(10)X(5659)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-4*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+3*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6915) has Shinagawa coefficients (R - r, 2R + r).

X(6915) lies on these lines: {2, 3}, {10, 5659}, {35, 3817}, {56, 5704}, {58, 5400}, {80, 1210}, {100, 946}, {515, 5253}, {651, 3075}, {936, 3878}, {938, 3304}, {962, 1376}, {1014, 5740}, {1259, 5748}, {1261, 5100}, {1320, 1389}, {1470, 5229}, {1490, 3306}, {1621, 6796}, {2801, 3337}, {2975, 5587}, {3218, 5777}, {3303, 5703}, {3678, 5536}, {3746, 5443}, {3816, 6253}, {3868, 5720}, {3871, 5603}, {3876, 5709}, {3881, 5531}, {3889, 5534}, {3913, 5734}, {5176, 6734}, {5440, 5806}, {5537, 6700}

X(6916) = (EULER LINE)∩X(7)X(517)

Barycentrics a^7-(b+c)*a^6-(b^2-10*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(4*b^2-b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6916) has Shinagawa coefficients (2R, R - 2r).

X(6916) lies on these lines: {2, 3}, {7, 517}, {8, 1071}, {10, 84}, {40, 388}, {63, 2096}, {165, 1478}, {226, 6282}, {278, 1074}, {495, 6244}, {497, 3576}, {515, 2550}, {936, 6260}, {944, 3872}, {962, 3890}, {966, 1765}, {1058, 1385}, {1350, 5800}, {1422, 5930}, {2077, 5218}, {2551, 6256}, {3419, 5768}, {3428, 4293}, {3434, 5731}, {3436, 5828}, {3587, 5759}, {4340, 5706}, {5044, 5811}, {5249, 5603}, {5439, 5804}, {5658, 5720}, {5705, 6705}, {5777, 6223}, {5784, 6001}

X(6917) = (EULER LINE)∩X(46)X(3585)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6917) has Shinagawa coefficients (R, R + 2r).

X(6917) lies on these lines: {2, 3}, {46, 3585}, {65, 68}, {79, 5693}, {155, 5706}, {225, 1060}, {388, 952}, {497, 5901}, {517, 5794}, {958, 5841}, {971, 5880}, {1056, 1483}, {1062, 1074}, {1352, 4259}, {1385, 3824}, {1454, 1737}, {1479, 2646}, {1482, 3434}, {1714, 5398}, {1834, 5707}, {1836, 5887}, {1901, 5778}, {2245, 5816}, {2550, 5690}, {3436, 3927}, {3564, 5800}, {3583, 3612}, {3822, 6796}, {4654, 5270}, {5080, 5818}, {5082, 5844}, {5290, 5534}, {5784, 5805}

X(6918) = (EULER LINE)∩X(40)X(4413)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-6*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+4*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6918) has Shinagawa coefficients (2R - r, 2R + r).

X(6918) lies on these lines: {2, 3}, {40, 4413}, {56, 5587}, {57, 5777}, {72, 2095}, {78, 1482}, {355, 999}, {499, 1617}, {517, 936}, {912, 5708}, {938, 952}, {942, 5720}, {946, 1376}, {956, 5818}, {1001, 6796}, {1071, 3306}, {1445, 5779}, {1490, 5437}, {1698, 3428}, {3216, 5706}, {3295, 5886}, {3304, 5881}, {3452, 5812}, {3812, 6261}, {5044, 5709}, {5045, 5534}, {5120, 5816}, {5221, 5693}, {5603, 5687}, {5703, 5901}, {5790, 6734}, {6245, 6692}

X(6919) = (EULER LINE)∩X(1)X(5748)

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

As a point on the Euler line, X(6919) has Shinagawa coefficients (2R - r, -2r).

X(6919) lies on these lines: {1, 5748}, {2, 3}, {8, 3452}, {11, 2551}, {63, 5704}, {78, 5328}, {145, 5722}, {329, 1210}, {388, 3816}, {390, 5552}, {392, 5818}, {496, 3421}, {497, 1329}, {908, 938}, {936, 5175}, {1376, 5225}, {1706, 5250}, {2899, 3705}, {3085, 3814}, {3086, 3825}, {3485, 5087}, {3586, 6700}, {3600, 5080}, {3614, 4423}, {3616, 5219}, {3617, 3877}, {3621, 3940}, {3634, 4512}, {3820, 5082}, {4847, 4866}, {5439, 5714}, {5554, 5804}

X(6919) = complement of X(37267)
X(6919) = anticomplement of X(17567)

X(6920) = (EULER LINE)∩X(9)X(1389)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b+c)^2*a^2-(b^4-c^4)*(b-c)*a-3*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6920) has Shinagawa coefficients (2R + r, R - r).

X(6920) lies on these lines: {2, 3}, {9, 1389}, {40, 3918}, {55, 5818}, {56, 5714}, {72, 1173}, {80, 943}, {104, 1125}, {226, 5443}, {355, 1621}, {515, 5259}, {517, 5260}, {944, 1001}, {946, 5251}, {958, 5603}, {1385, 5284}, {2077, 3634}, {2654, 3074}, {2975, 5886}, {3303, 3488}, {3304, 3487}, {3614, 5172}, {3616, 5811}, {3624, 5450}, {3647, 5535}, {3652, 5885}, {3871, 5790}, {4423, 5658}, {5248, 5587}, {5273, 5804}, {5506, 5538}, {5777, 6265}

X(6920) = complement of X(37163)

X(6921) = (EULER LINE)∩X(8)X(1319)

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

As a point on the Euler line, X(6921) has Shinagawa coefficients (R - 2r, r).

X(6921) lies on these lines: {2, 3}, {8, 1319}, {36, 3436}, {55, 6691}, {56, 3035}, {63, 6700}, {78, 3911}, {100, 3086}, {145, 1145}, {499, 3434}, {944, 4881}, {966, 4268}, {1125, 5119}, {1210, 4855}, {1329, 5204}, {1376, 5433}, {1385, 5554}, {1420, 6735}, {1788, 4511}, {3057, 3616}, {3085, 5253}, {3452, 4652}, {3814, 4299}, {3816, 5217}, {3825, 4302}, {3868, 5435}, {3876, 5744}, {3913, 6174}, {4413, 4999}, {5438, 6734}, {5442, 5692}

X(6922) = (EULER LINE)∩X(11)X(40)

Barycentrics (b^2-4*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-3*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6922) has Shinagawa coefficients (r - 2R, r + 2R).

X(6922) lies on these lines: {2, 3}, {11, 40}, {12, 3576}, {57, 5812}, {78, 952}, {355, 936}, {495, 1385}, {496, 517}, {499, 3428}, {515, 1329}, {516, 3825}, {908, 1071}, {938, 1482}, {946, 3812}, {1076, 1465}, {1445, 5762}, {1706, 5705}, {2077, 6284}, {2095, 5758}, {2886, 6684}, {3035, 6796}, {3216, 5721}, {3452, 5777}, {3814, 4297}, {3927, 5770}, {4857, 5537}, {5437, 5715}, {5690, 6734}, {5703, 6049}, {5720, 5787}

X(6923) = (EULER LINE)∩X(40)X(3585)

Barycentrics a^7-(b+c)*a^6-(b^2-4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(b^2-b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6923) has Shinagawa coefficients (R, 2r - R).

X(6923) lies on these lines: {2, 3}, {40, 3585}, {55, 5840}, {119, 1376}, {355, 5836}, {388, 1482}, {497, 1387}, {511, 5820}, {517, 1478}, {912, 3419}, {952, 3434}, {971, 5832}, {1060, 1785}, {1074, 1877}, {1351, 5800}, {1385, 1479}, {1709, 3359}, {2096, 5770}, {2550, 5779}, {2829, 2886}, {3428, 5841}, {3436, 5690}, {3576, 3583}, {3838, 5886}, {4640, 5123}, {5080, 5657}, {5714, 5761}, {5794, 5887}

X(6924) = (EULER LINE)∩X(35)X(5886)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-3*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6924) has Shinagawa coefficients (R - 2r, R + 2r).

X(6924) lies on these lines: {2, 3}, {35, 5886}, {36, 355}, {55, 5901}, {56, 952}, {100, 1482}, {499, 5172}, {970, 1216}, {999, 1483}, {1155, 5887}, {1329, 5841}, {1352, 5096}, {1376, 5690}, {1385, 3812}, {1484, 3086}, {2975, 5790}, {3216, 5398}, {3336, 6326}, {3361, 5534}, {4255, 5707}, {5122, 5777}, {5124, 5816}, {5438, 5709}, {5687, 5844}, {5752, 6101}, {5842, 6691}, {5903, 6265}

X(6925) = (EULER LINE)∩X(1)X(5555)

Barycentrics a^7-(b+c)*a^6-(b^2-8*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6925) has Shinagawa coefficients (R, r - 2R).

X(6925) lies on these lines: {1, 5555}, {2, 3}, {8, 6001}, {10, 1709}, {40, 3436}, {72, 6259}, {78, 6260}, {84, 6734}, {144, 153}, {388, 962}, {497, 1319}, {515, 3434}, {516, 1478}, {517, 5905}, {908, 6282}, {971, 3419}, {1040, 1877}, {1479, 4297}, {1512, 3359}, {2096, 3218}, {2551, 4640}, {2829, 3428}, {3219, 5657}, {3586, 5732}, {3876, 5811}

X(6926) = (EULER LINE)∩X(9)X(6705)

Barycentrics a^7-(b+c)*a^6-(3*b-c)*(b-3*c)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6926) has Shinagawa coefficients (r - 2R, 2R).

X(6926) lies on these lines: {2, 3}, {9, 6705}, {40, 3086}, {57, 5758}, {78, 5768}, {84, 3452}, {165, 499}, {496, 6244}, {498, 5726}, {515, 5438}, {936, 6245}, {944, 5440}, {946, 5437}, {962, 3359}, {1210, 6282}, {1490, 6700}, {1706, 6684}, {2077, 4294}, {3085, 3576}, {5328, 6223}, {5433, 5584}, {5435, 5709}, {5439, 5603}, {5552, 5731}, {5708, 5763}

X(6927) = (EULER LINE)∩X(40)X(6700)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-2*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(b-c)^2*(5*b^2+2*b*c+5*c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2-6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6927) has Shinagawa coefficients (2r, -r - 2R).

X(6927) lies on these lines: {2, 3}, {40, 6700}, {497, 6796}, {936, 5657}, {938, 6049}, {944, 1210}, {946, 5218}, {971, 5825}, {1071, 5435}, {1490, 3911}, {1519, 6361}, {1697, 5603}, {1750, 6705}, {1788, 6261}, {2096, 6260}, {3916, 5811}, {4345, 5703}, {5053, 5822}, {5122, 6259}, {5126, 5704}, {5705, 5795}, {5744, 5777}, {5748, 5812}, {5766, 5805}, {5771, 5780}

X(6928) = (EULER LINE)∩X(40)X(3583)

Barycentrics a^7-(b+c)*a^6-(b^2+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6928) has Shinagawa coefficients (R, -R - 2r).

X(6928) lies on these lines: {2, 3}, {40, 3583}, {56, 5841}, {355, 960}, {497, 1482}, {517, 1479}, {944, 5080}, {952, 3436}, {1062, 1785}, {1076, 1877}, {1329, 5842}, {1385, 1478}, {1728, 5709}, {2551, 5790}, {3434, 5690}, {3488, 5761}, {3576, 3585}, {3814, 6796}, {5050, 5800}, {5087, 6256}, {5204, 6713}, {5292, 5398}, {5722, 5812}, {5729, 5762}

X(6929) = (EULER LINE)∩X(55)X(119)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(b^2+b*c+c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6929) has Shinagawa coefficients (R, R - 2r).

X(6929) lies on these lines: {2, 3}, {55, 119}, {355, 1479}, {388, 5901}, {497, 952}, {912, 1864}, {1058, 1483}, {1060, 1877}, {1145, 3434}, {1319, 1478}, {1376, 5840}, {1385, 6256}, {1482, 3436}, {1484, 5274}, {1837, 5887}, {2551, 5690}, {2829, 3816}, {3421, 5844}, {3583, 5119}, {3586, 5720}, {3825, 5450}, {4271, 5816}, {4857, 5881}, {5080, 5603}

X(6930) = (EULER LINE)∩X(119)X(5218)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+6*b^2*c^2+c^4)*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-6*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6930) has Shinagawa coefficients (R + r, R - 2r).

X(6930) lies on these lines: {2, 3}, {119, 5218}, {355, 4294}, {390, 952}, {517, 5698}, {912, 3488}, {944, 3890}, {1001, 2829}, {1385, 6259}, {1420, 4317}, {1478, 2078}, {1697, 4309}, {2550, 5840}, {3486, 5887}, {3600, 5901}, {4293, 5126}, {4302, 5587}, {4304, 5720}, {4313, 5811}, {4314, 5534}, {5057, 5603}, {5248, 6256}, {5722, 5770}

X(6931) = (EULER LINE)∩X(8)X(1392)

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

As a point on the Euler line, X(6931) has Shinagawa coefficients (R - 2r, -r).

X(6931) lies on these lines: {2, 3}, {8, 1392}, {11, 3913}, {56, 6667}, {145, 1387}, {498, 3825}, {499, 3436}, {3193, 4383}, {3616, 5252}, {3617, 5330}, {3618, 5820}, {3634, 5250}, {3680, 6735}, {3868, 5704}, {3871, 5274}, {3876, 5328}, {3877, 4731}, {3897, 5550}, {4299, 6681}, {4423, 6668}, {4866, 5231}, {5554, 5886}

X(6932) = (EULER LINE)∩X(11)X(5731)

Barycentrics (b^2+3*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6932) has Shinagawa coefficients (r + R, r - 2R).

X(6932) lies on these lines: {2, 3}, {11, 5731}, {12, 962}, {40, 5057}, {119, 5657}, {153, 956}, {165, 3814}, {392, 1538}, {495, 1000}, {946, 3890}, {1329, 5698}, {1388, 3486}, {1519, 3877}, {1699, 3822}, {2098, 3485}, {2975, 6256}, {3428, 5080}, {3697, 5887}, {3869, 6735}, {3872, 5086}, {3947, 4301}, {6260, 6734}

X(6933) = (EULER LINE)∩X(55)X(6668)

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

As a point on the Euler line, X(6933) has Shinagawa coefficients (R - 2r, r).

X(6933) lies on these lines: {2, 3}, {55, 6668}, {498, 3434}, {499, 3822}, {908, 5705}, {958, 3614}, {960, 3698}, {1728, 3306}, {1837, 3616}, {2320, 5550}, {2886, 5552}, {3218, 5714}, {3303, 3829}, {3436, 5258}, {3617, 5730}, {3817, 5250}, {3868, 5226}, {3876, 5748}, {5087, 5180}, {5219, 6734}

X(6934) = (EULER LINE)∩X(46)X(515)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b-c)^4*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6934) has Shinagawa coefficients (r, -R - 2r).

X(6934) lies on these lines: {2, 3}, {46, 515}, {56, 5842}, {65, 944}, {355, 3916}, {553, 4317}, {946, 3612}, {1478, 6796}, {1512, 4316}, {1699, 4324}, {1770, 6261}, {2646, 4294}, {3436, 5841}, {3928, 5881}, {4252, 5721}, {4259, 6776}, {5440, 5812}, {5657, 5794}, {5759, 5784}

X(6935) = (EULER LINE)∩X(1)X(6705)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-10*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(b-c)^2*(5*b^2+2*b*c+5*c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6935) has Shinagawa coefficients (2r, 2R - r).

X(6935) lies on these lines: {1, 6705}, {2, 3}, {57, 5603}, {104, 1056}, {142, 1519}, {226, 2096}, {388, 5450}, {515, 5218}, {517, 5744}, {942, 4323}, {944, 3601}, {946, 3474}, {1071, 5703}, {1158, 3485}, {1466, 3086}, {2077, 2550}, {3916, 5758}, {5122, 5805}, {5657, 5745}

X(6936) = (EULER LINE)∩X(9)X(5881)

Barycentrics 3*a^7-3*(b+c)*a^6-5*(b^2+c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b+c)^4*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6936) has Shinagawa coefficients (2R + r, -R - 2r).

X(6936) lies on these lines: {2, 3}, {9, 5881}, {226, 4317}, {390, 5729}, {944, 960}, {950, 1728}, {1319, 3487}, {1837, 4294}, {3057, 3488}, {3586, 4330}, {3922, 6361}, {4268, 5746}, {4271, 5802}, {4293, 5714}, {4302, 6684}, {4679, 5658}, {5731, 5811}, {5734, 5758}

X(6937) = (EULER LINE)∩X(10)X(5693)

Barycentrics (b^2+3*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3+c^3)*(b+c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6937) has Shinagawa coefficients (2R + r, -R + r).

X(6937) lies on these lines: {2, 3}, {10, 5693}, {11, 4305}, {12, 4295}, {40, 3822}, {119, 5811}, {226, 5903}, {484, 4338}, {944, 2886}, {1158, 1698}, {1770, 6684}, {1901, 5036}, {2099, 3487}, {3419, 4861}, {3826, 5817}, {3841, 5587}, {3925, 5658}, {5175, 6224}

X(6938) = (EULER LINE)∩X(1)X(5553)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-8*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b-c)^4*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6938) has Shinagawa coefficients (r, R - 2r).

X(6938) lies on these lines: {1, 5553}, {2, 3}, {35, 6256}, {55, 2829}, {104, 497}, {165, 1512}, {515, 1709}, {944, 3057}, {946, 4299}, {1319, 1836}, {1479, 5450}, {1519, 3576}, {1699, 4316}, {2096, 3488}, {3434, 5840}, {4309, 5882}, {4324, 5691}, {4640, 5657}

X(6939) = (EULER LINE)∩X(72)X(5804)

Barycentrics a^7-(b+c)*a^6-(b-c)^2*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-10*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6939) has Shinagawa coefficients (2R, 2R - r).

X(6939) lies on these lines: {2, 3}, {72, 5804}, {144, 2095}, {355, 1058}, {497, 5587}, {908, 3421}, {938, 5777}, {942, 5811}, {946, 2551}, {1056, 5226}, {2096, 3306}, {3305, 5657}, {3488, 5720}, {5082, 5818}, {5316, 6282}, {5758, 5806}, {5768, 5927}

X(6940) = (EULER LINE)∩X(10)X(104)

Barycentrics a*(a^6-(b+c)*a^5-(2*b-c)*(b-2*c)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-6*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6940) has Shinagawa coefficients (2R - r, r - R).

X(6940) lies on these lines: {2, 3}, {10, 104}, {36, 6684}, {40, 3884}, {56, 5657}, {100, 1385}, {517, 5253}, {601, 978}, {944, 1376}, {1125, 2077}, {1389, 3754}, {1466, 3487}, {1470, 3085}, {1698, 5450}, {1706, 3576}, {4413, 5818}, {5559, 5563}

X(6941) = (EULER LINE)∩X(8)X(119)

Barycentrics (b^2+b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3-c^3)*(b-c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6941) has Shinagawa coefficients (r, r - R).

X(6941) lies on these lines: {2, 3}, {8, 119}, {10, 1519}, {11, 944}, {12, 2098}, {40, 3814}, {104, 499}, {355, 4861}, {946, 1512}, {1329, 5657}, {1537, 5690}, {1538, 4002}, {2829, 5433}, {2886, 5818}, {3576, 3825}, {3583, 6796}, {5226, 5804}

X(6942) = (EULER LINE)∩X(35)X(5603)

Barycentrics a*(2*a^6-2*(b+c)*a^5-(4*b^2-3*b*c+4*c^2)*a^4+4*(b^3+c^3)*a^3+2*(b-c)^2*(b^2+c^2)*a^2-2*(b^4-c^4)*(b-c)*a+(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6942) has Shinagawa coefficients (2r, -R - 2r).

X(6942) lies on these lines: {2, 3}, {35, 5603}, {36, 944}, {104, 5204}, {946, 5010}, {993, 5818}, {1071, 5122}, {1512, 4297}, {2077, 6361}, {3086, 5172}, {3436, 4996}, {4652, 5720}, {4855, 5709}, {5082, 6585}, {5096, 6776}, {5267, 5587}, {5433, 5842}

X(6943) = (EULER LINE)∩X(11)X(962)

Barycentrics (b^2-3*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-2*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6943) has Shinagawa coefficients (R - r, -2R - r).

X(6943) lies on these lines: {2, 3}, {11, 962}, {12, 5731}, {78, 5176}, {273, 3007}, {908, 6245}, {938, 2099}, {1158, 5057}, {1210, 4301}, {1445, 5735}, {1699, 3825}, {3035, 6253}, {3218, 5812}, {3306, 5715}, {3476, 5703}, {3814, 5691}

X(6944) = (EULER LINE)∩X(119)X(388)

Barycentrics a^7-(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(4*b^2-b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6944) has Shinagawa coefficients (R - r, R).

X(6944) lies on these lines: {2, 3}, {119, 388}, {355, 3086}, {498, 1697}, {499, 1420}, {1210, 5720}, {1788, 5887}, {3085, 5886}, {3452, 5709}, {5053, 5816}, {5176, 5818}, {5328, 5758}, {5435, 5811}, {5552, 5603}, {5770, 5777}, {6260, 6692}

X(6945) = (EULER LINE)∩X(11)X(3476)

Barycentrics (b^2+b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-2*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-5*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6945) has Shinagawa coefficients (R - r, 2R - r).

X(6945) lies on these lines: {2, 3}, {11, 3476}, {119, 1320}, {153, 999}, {946, 6735}, {962, 1329}, {1476, 3086}, {1512, 3877}, {1538, 3753}, {1699, 3814}, {3816, 5731}, {3817, 4342}, {3825, 5691}, {3872, 5176}, {5253, 6256}, {5819, 6506}

X(6946) = (EULER LINE)∩X(56)X(5818)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2-3*b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2-6*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a+3*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6946) has Shinagawa coefficients (2R - r, R + r).

X(6946) lies on these lines: {2, 3}, {56, 5818}, {100, 5886}, {104, 5587}, {355, 5253}, {946, 5537}, {997, 1706}, {1376, 5603}, {1466, 5714}, {1737, 5563}, {2077, 3817}, {3306, 5720}, {3624, 6796}, {3871, 5901}, {4413, 5657}, {5883, 6326}

X(6947) = (EULER LINE)∩X(497)X(1737)

Barycentrics a^7-(b+c)*a^6-(b^2+4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6947) has Shinagawa coefficients (2R, -R - r).

X(6947) lies on these lines: {2, 3}, {497, 1737}, {515, 5316}, {944, 997}, {1056, 1319}, {1058, 3057}, {1072, 5272}, {1385, 3436}, {1479, 6684}, {3219, 5770}, {3421, 4511}, {3428, 3816}, {4413, 5842}, {4679, 6001}, {5439, 5812}

X(6948) = (EULER LINE)∩X(40)X(4299)

Barycentrics 3*a^7-3*(b+c)*a^6-5*(b-c)^2*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+c^4-2*b*c*(4*b^2-3*b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6948) has Shinagawa coefficients (R - r, 2r - R).

X(6948) lies on these lines: {2, 3}, {40, 4299}, {104, 3434}, {165, 4316}, {497, 5840}, {515, 3359}, {517, 3474}, {912, 2096}, {1376, 2829}, {1385, 4294}, {1478, 2077}, {1482, 3600}, {3419, 5770}, {3576, 4302}, {5176, 5657}

X(6949) = (EULER LINE)∩X(8)X(6265)

Barycentrics a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(b-c)^2*(3*b^2+2*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-3*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6949) has Shinagawa coefficients (2r, -R), and also (4S², -abc$a$).

X(6949) lies on these lines: {2, 3}, {8, 6265}, {80, 499}, {104, 5433}, {119, 2975}, {498, 5443}, {1000, 1389}, {1125, 1512}, {1388, 3086}, {1519, 6684}, {1768, 3467}, {2800, 5445}, {3582, 5882}, {3878, 5657}, {3890, 5886}

X(6950) = (EULER LINE)∩X(35)X(944)

Barycentrics a*(2*a^6-2*(b+c)*a^5-(4*b^2-5*b*c+4*c^2)*a^4+4*(b^3+c^3)*a^3+2*(b-c)^2*(b^2+c^2)*a^2-2*(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6950) has Shinagawa coefficients (2r, R - 2r).

X(6950) lies on these lines: {2, 3}, {35, 944}, {36, 3474}, {40, 5267}, {55, 104}, {497, 5533}, {515, 5010}, {993, 2077}, {1158, 3612}, {1385, 3890}, {1946, 2401}, {2829, 5432}, {3434, 4996}, {4265, 6776}, {4293, 5172}

X(6951) = (EULER LINE)∩X(104)X(2886)

Barycentrics a^7-(b+c)*a^6-(b^2-5*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4+2*b*c*(2*b^2-b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-c^2)*(b-c)*(b^3+c^3)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6951) has Shinagawa coefficients (2R, 2r - R).

X(6951) lies on these lines: {2, 3}, {104, 2886}, {153, 5790}, {388, 5903}, {484, 1478}, {1056, 2099}, {1074, 1870}, {1621, 5840}, {2077, 3822}, {2550, 2801}, {2829, 3925}, {3434, 6224}, {3585, 6684}, {5818, 6256}

X(6952) = (EULER LINE)∩X(12)X(104)

Barycentrics a^7-(b+c)*a^6-3*(b^2-b*c+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(b-c)^2*(3*b^2+2*b*c+3*c^2)*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*(b^3+c^3)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6952) has Shinagawa coefficients (2r, R), and also (4S², abc$a$).

X(6952) lies on these lines: {2, 3}, {12, 104}, {81, 1199}, {355, 6224}, {484, 946}, {498, 944}, {499, 1788}, {1512, 3634}, {2099, 3086}, {2800, 5443}, {2829, 3614}, {3584, 5882}, {5253, 6713}, {5303, 5841}

X(6953) = (EULER LINE)∩X(153)X(3600)

Barycentrics a^7-(b+c)*a^6-3*(b^2+c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(4*b^2-b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6953) has Shinagawa coefficients (R - r, 2R).

X(6953) lies on these lines: {2, 3}, {153, 3600}, {498, 3817}, {499, 4311}, {946, 5552}, {962, 5328}, {1728, 5435}, {1837, 3476}, {3086, 4308}, {3218, 5811}, {3306, 6260}, {3885, 5603}, {5292, 5400}

X(6954) = (EULER LINE)∩X(517)X(5218)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-2*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6954) has Shinagawa coefficients (R + 2r, -R - r).

X(6954) lies on these lines: {2, 3}, {517, 5218}, {572, 5822}, {912, 5744}, {997, 5837}, {1737, 3576}, {2095, 5719}, {3428, 5432}, {3940, 5771}, {4511, 5657}, {5267, 6256}, {5709, 5761}, {5720, 5745}

X(6955) = (EULER LINE)∩X(46)X(4317)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-12*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+c^4-6*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6955) has Shinagawa coefficients (2R - r, -R + 2r).

X(6955) lies on these lines: {2, 3}, {46, 4317}, {104, 2550}, {390, 1387}, {944, 5836}, {1155, 4293}, {1706, 5881}, {2096, 5784}, {2829, 4413}, {3612, 4309}, {4299, 6684}, {5603, 5880}, {5735, 6282}

X(6956) = (EULER LINE)∩X(936)X(5818)

Barycentrics a^7-(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b-c)^2*(7*b^2+6*b*c+7*c^2)*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b^2-2*b*c+3*c^2)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6956) has Shinagawa coefficients (2r, r + 2R).

X(6956) lies on these lines: {2, 3}, {936, 5818}, {946, 1788}, {1071, 5226}, {1210, 3340}, {2096, 6705}, {3911, 5715}, {5219, 6245}, {5229, 5450}, {5587, 6700}, {5657, 5705}, {5744, 5812}, {5748, 5777}

X(6957) = (EULER LINE)∩X(153)X(1056)

Barycentrics a^7-(b+c)*a^6-(b^2-4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-8*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6957) has Shinagawa coefficients (R, 2R - r).

X(6957) lies on these lines: {2, 3}, {153, 1056}, {497, 5252}, {938, 1864}, {946, 3436}, {962, 2551}, {1478, 3817}, {2550, 5123}, {3434, 5587}, {3868, 5804}, {5439, 6259}, {5722, 5927}

X(6958) = (EULER LINE)∩X(56)X(6713)

Barycentrics a^7-(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(3*b^2-b*c+3*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6958) has Shinagawa coefficients (2r - R, R).

X(6958) lies on these lines: {2, 3}, {56, 6713}, {182, 5820}, {355, 5123}, {498, 1385}, {499, 517}, {952, 5552}, {1387, 1482}, {3812, 5886}, {3814, 5450}, {5204, 5841}, {5780, 5789}

X(6959) = (EULER LINE)∩X(56)X(119)

Barycentrics a^7-(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(3*b^2-b*c+3*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6959) has Shinagawa coefficients (R - 2r, R).

X(6959) lies on these lines: {2, 3}, {56, 119}, {155, 4383}, {355, 499}, {498, 3057}, {952, 3086}, {1145, 1482}, {3085, 5901}, {3582, 5881}, {3825, 6796}, {4268, 5816}, {5450, 6681}

X(6960) = (EULER LINE)∩X(8)X(6326)

Barycentrics a^7-(b+c)*a^6-(3*b^2+b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(b^2-b*c+c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-3*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6960) has Shinagawa coefficients (R + 2r, -2R).

X(6960) lies on these lines: {2, 3}, {8, 6326}, {40, 5180}, {153, 2975}, {329, 2949}, {498, 962}, {499, 5731}, {2829, 5303}, {3085, 5697}, {3584, 4301}, {3678, 5660}, {4015, 5659}

X(6961) = (EULER LINE)∩X(57)X(5761)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-10*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-6*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6961) has Shinagawa coefficients (R - 2r, r - R).

X(6961) lies on these lines: {2, 3}, {57, 5761}, {78, 5770}, {104, 5552}, {499, 2077}, {1125, 3359}, {1385, 5218}, {3086, 6713}, {4861, 5657}, {5122, 5812}, {5720, 6705}

X(6962) = (EULER LINE)∩X(938)X(1319)

Barycentrics 3*a^7-3*(b+c)*a^6-7*(b^2+c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-2*(2*b^2-3*b*c+2*c^2)*b*c)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6962) has Shinagawa coefficients (R + 2r, -2R - r).

X(6962) lies on these lines: {2, 3}, {938, 1319}, {962, 5218}, {1837, 5704}, {3035, 5584}, {3057, 5703}, {3428, 5552}, {3434, 6796}, {4301, 5119}, {4652, 6260}, {5881, 6734}

X(6963) = (EULER LINE)∩X(11)X(5657)

Barycentrics (b^2-3*b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-3*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6963) has Shinagawa coefficients (r - 2R, r + R).

X(6963) lies on these lines: {2, 3}, {11, 5657}, {40, 3825}, {119, 5731}, {944, 1329}, {997, 5881}, {1072, 5121}, {1737, 5697}, {3576, 3814}, {3816, 5603}, {5328, 5768}

X(6964) = (EULER LINE)∩X(40)X(5316)

Barycentrics a^7-(b+c)*a^6-(3*b^2-2*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-10*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6964) has Shinagawa coefficients (2R - r, 2R).

X(6964) lies on these lines: {2, 3}, {40, 5316}, {57, 5811}, {78, 5804}, {84, 6692}, {119, 5261}, {938, 5720}, {946, 1706}, {3086, 5587}, {3452, 5758}, {5437, 6260}

X(6965) = (EULER LINE)∩X(80)X(497)

Barycentrics a^7-(b+c)*a^6-(b^2-b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^4+c^4-2*b*c*(2*b^2+b*c+2*c^2))*a^3+(b^2-c^2)^2*(b+c)*a^2+(b^2-5*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6965) has Shinagawa coefficients (2R, R - 2r).

X(6965) lies on these lines: {2, 3}, {80, 497}, {104, 3816}, {119, 1621}, {149, 5790}, {355, 3890}, {388, 5443}, {1479, 5818}, {2551, 3878}, {5080, 5886}

X(6966) = (EULER LINE)∩X(46)X(4301)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-12*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-6*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6966) has Shinagawa coefficients (R - 2r, r - 2R).

X(6966) lies on these lines: {2, 3}, {46, 4301}, {65, 5734}, {78, 6705}, {962, 1155}, {2077, 3434}, {3436, 5450}, {4855, 6245}, {5218, 5252}, {5881, 6735}

X(6967) = (EULER LINE)∩X(104)X(2551)

Barycentrics a^7-(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6967) has Shinagawa coefficients (r - 2Rr, R).

X(6967) lies on these lines: {2, 3}, {104, 2551}, {499, 5119}, {1319, 3085}, {1385, 5552}, {3057, 3086}, {3428, 6691}, {3812, 5603}, {3876, 5770}, {5316, 6705}

X(6968) = (EULER LINE)∩X(119)X(3434)

Barycentrics a^7-(b+c)*a^6+(b^2+4*b*c+c^2)*a^5-(b+c)^3*a^4-(b-c)^2*(5*b^2+6*b*c+5*c^2)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2-8*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6968) has Shinagawa coefficients (r, 2r - R).

X(6968) lies on these lines: {2, 3}, {119, 3434}, {1387, 5274}, {1512, 1699}, {1519, 5587}, {1537, 5790}, {5048, 5252}, {5123, 5657}, {5817, 5832}, {5818, 5836}

X(6969) = (EULER LINE)∩X(119)X(3421)

Barycentrics a^7-(b+c)*a^6-(5*b^2+2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(b-c)^2*(7*b^2+6*b*c+7*c^2)*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b-c)*(b-3*c)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6969) has Shinagawa coefficients (2r, r - 2R).

X(6969) lies on these lines: {2, 3}, {119, 3421}, {517, 5748}, {1000, 1512}, {1071, 5704}, {1519, 3452}, {2096, 3911}, {3617, 5780}, {3872, 5720}, {5225, 6796}

X(6970) = (EULER LINE)∩X(119)X(4293)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-6*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-6*b*c*(2*b^2-b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2-6*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6970) has Shinagawa coefficients (R - 2r, R + r).

X(6970) lies on these lines: {2, 3}, {119, 4293}, {355, 5126}, {499, 2078}, {912, 5435}, {1420, 1737}, {3911, 5720}, {5218, 5886}, {5709, 6700}

X(6971) = (EULER LINE)∩X(11)X(1482)

Barycentrics (b^2-b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(2*b^4+2*c^4-3*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6971) has Shinagawa coefficients (2r - R, 2r + R).

X(6971) lies on these lines: {2, 3}, {11, 1482}, {145, 1484}, {355, 3814}, {1329, 5790}, {3825, 5886}, {5087, 5887}, {5433, 5841}

X(6971) = {X(3),X(5)}-harmonic conjugate of X(6980)

X(6972) = (EULER LINE)∩X(484)X(499)

Barycentrics a^7-(b+c)*a^6-(3*b^2-5*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(3*b^2-b*c+3*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*(b^3+c^3)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6972) has Shinagawa coefficients (2r - R, 2R).

X(6972) lies on these lines: {2, 3}, {484, 499}, {498, 5731}, {908, 6705}, {3086, 5903}, {3582, 4301}, {5080, 5450}, {5552, 6224}

X(6973) = (EULER LINE)∩X(119)X(497)

Barycentrics a^7-(b+c)*a^6+(b+c)^2*a^5-(b+c)^3*a^4-(5*b^4+5*c^4-2*b*c*(4*b^2+b*c+4*c^2))*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(b^2-c^2)^2*(3*b-c)*(b-3*c)*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6973) has Shinagawa coefficients (R - r, R - 2r).

X(6973) lies on these lines: {2, 3}, {119, 497}, {952, 5274}, {1478, 5193}, {3476, 5886}, {3825, 6256}, {5176, 5603}, {5261, 5901}

X(6974) = (EULER LINE)∩X(946)X(4652)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-8*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+5*c^4-2*b*c*(2*b^2-3*b*c+2*c^2))*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b^2+4*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6974) has Shinagawa coefficients (R + 2r, 2R - r).

X(6974) lies on these lines: {2, 3}, {946, 4652}, {962, 4640}, {1125, 1709}, {1727, 3338}, {2320, 3427}, {3218, 5603}, {3616, 6001}

X(6975) = (EULER LINE)∩X(11)X(5818)

Barycentrics (b^2-b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4-3*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-5*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6975) has Shinagawa coefficients (2R - r, R - r).

X(6975) lies on these lines: {2, 3}, {11, 5818}, {119, 3616}, {944, 3816}, {1329, 5603}, {3825, 5587}, {4861, 5886}, {5328, 5804}

X(6976) = (EULER LINE)∩X(1512)X(4512)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2-4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b+c)^4*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-8*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6976) has Shinagawa coefficients (2R + r, R - 2r).

X(6976) lies on these lines: {2, 3}, {1512, 4512}, {1864, 3488}, {2829, 4423}, {3683, 5657}, {4294, 5818}, {5259, 6256}

X(6977) = (EULER LINE)∩X(46)X(5603)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2-8*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(b-c)^2*(5*b^2+2*b*c+5*c^2)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6977) has Shinagawa coefficients (2r, R - r).

X(6977) lies on these lines: {2, 3}, {46, 5603}, {104, 3085}, {498, 5450}, {944, 3612}, {1519, 3624}, {3218, 5761}

X(6978) = (EULER LINE)∩X(908)X(5770)

Barycentrics a^7-(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-3*(b^2-c^2)^2*(b-c)^2*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6978) has Shinagawa coefficients (2r - R, r + R).

X(6978) lies on these lines: {2, 3}, {908, 5770}, {912, 5748}, {1210, 5761}, {3359, 3817}, {4293, 6713}

X(6979) = (EULER LINE)∩X(56)X(153)

Barycentrics a^7-(b+c)*a^6-(3*b^2-b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(3*b^2-b*c+3*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-5*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6979) has Shinagawa coefficients (R - 2r, 2R).

X(6979) lies on these lines: {2, 3}, {56, 153}, {80, 3086}, {145, 6265}, {3085, 5443}, {3874, 5660}

X(6980) = (EULER LINE)∩X(12)X(1482)

Barycentrics (b^2+b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-(2*b^4+2*c^4-b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6980) has Shinagawa coefficients (2r + R, 2r - R).

X(6980) lies on these lines: {2, 3}, {12, 1482}, {119, 2886}, {3822, 5886}, {5180, 5657}, {5432, 5840}

X(6980) = {X(3),X(5)}-harmonic conjugate of X(6971)

X(6981) = (EULER LINE)∩X(119)X(3086)

Barycentrics a^7-(b+c)*a^6-(5*b^2-2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-2*b*c+5*c^2)*a^4+(7*b^4+7*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2*(3*b-c)*(b-3*c)*a+3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6981) has Shinagawa coefficients (R - 2r, R - r).

X(6981) lies on these lines: {2, 3}, {119, 3086}, {499, 5193}, {912, 5704}, {4861, 5818}

X(6982) = (EULER LINE)∩X(119)X(2550)

Barycentrics a^7-(b+c)*a^6+(b^2+6*b*c+c^2)*a^5-(b+c)^3*a^4-(5*b^4-2*b^2*c^2+5*c^4)*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+3*(b^2-c^2)^2*(b-c)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6982) has Shinagawa coefficients (R + r, 2r - R).

X(6982) lies on these lines: {2, 3}, {119, 2550}, {1482, 5261}, {5057, 5657}, {5218, 5840}

X(6983) = (EULER LINE)∩X(3085)X(5919)

Barycentrics a^7-(b+c)*a^6-(3*b^2-4*b*c+3*c^2)*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*b*c*(6*b^2-b*c+6*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b^2-8*b*c+c^2)*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6983) has Shinagawa coefficients (2R - r, R).

X(6983) lies on these lines: {2, 3}, {3085, 5919}, {3086, 5252}, {5552, 5886}, {5603, 5836}

X(6984) = (EULER LINE)∩X(65)X(5714)

Barycentrics a^7-(b+c)*a^6+(b^2+4*b*c+c^2)*a^5-(b+c)^3*a^4-(5*b^2-6*b*c+5*c^2)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+3*(b^4-c^4)*(b^2-c^2)*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6984) has Shinagawa coefficients (2R + r, R + 2r).

X(6984) lies on these lines: {2, 3}, {65, 5714}, {3487, 5252}, {5603, 5794}, {5817, 5880}

X(6985) = (EULER LINE)∩X(35)X(5219)

Barycentrics a*(a^6-(b+c)*a^5-2*(b^2+b*c+c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)^2*a^2-(b^4-c^4)*(b-c)*a+2*(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6985) has Shinagawa coefficients ($aS_A$, -2abc - $aS_A$).

X(6985) lies on these lines: {2, 3}, {35, 5219}, {40, 5692}, {46, 1858}, {56, 5722}, {100, 6361}, {255, 2635}, {355, 3428}, {496, 1617}, {500, 940}, {516, 6796}, {517, 3811}, {573, 5778}, {581, 4658}, {582, 1780}, {912, 1490}, {920, 1155}, {936, 3587}, {943, 5226}, {956, 5086}, {962, 5761}, {970, 6000}, {999, 3486}, {1062, 1465}, {1376, 3579}, {1385, 5806}, {1470, 4299}, {1480, 5255}, {1745, 1936}, {3295, 3485}, {3303, 3656}, {3304, 3655}, {3869, 3940}, {5396, 5706}, {5731, 5804}, {5755, 5776}, {5841, 6256}

X(6985) = complement of X(6851)
X(6985) = anticomplement of X(37356)

X(6986) = (EULER LINE)∩X(1)X(1170)

Barycentrics a*(a^6-(b+c)*a^5-(2*b^2+3*b*c+2*c^2)*a^4+2*(b^3+c^3)*a^3+(b^2+c^2)*(b^2+4*b*c+c^2)*a^2-(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*b*c) : :

As a point on the Euler line, X(6986) has Shinagawa coefficients (2abc + $aS_A$, -abc - $aS_A$).

X(6986) lies on these lines: {1, 1170}, {2, 3}, {35, 1210}, {36, 4298}, {40, 1621}, {55, 938}, {56, 3475}, {78, 947}, {81, 580}, {100, 5178}, {165, 3833}, {212, 3562}, {238, 4300}, {515, 5260}, {516, 5259}, {572, 2287}, {651, 3074}, {936, 993}, {942, 943}, {946, 5284}, {958, 5731}, {962, 1001}, {991, 1724}, {1014, 5736}, {1071, 3219}, {1257, 2224}, {1259, 5744}, {1385, 3555}, {1490, 3305}, {1768, 3647}, {3428, 3616}, {3826, 6253}, {3871, 5657}, {4015, 5531}, {4297, 5251}, {5217, 5704}, {5267, 5660}

X(6987) = (EULER LINE)∩X(1)X(5758)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2+2*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b+c)^4*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6987) has Shinagawa coefficients (abc + $aS_A$, -2$aS_A$).

X(6987) lies on these lines: {1, 5758}, {2, 3}, {9, 515}, {40, 950}, {63, 5768}, {72, 944}, {144, 912}, {165, 1737}, {226, 3576}, {329, 4511}, {347, 1870}, {387, 580}, {390, 517}, {497, 3428}, {572, 5746}, {573, 1713}, {938, 5709}, {946, 5436}, {954, 1056}, {997, 1490}, {1125, 5715}, {1260, 3421}, {1385, 3487}, {2550, 5842}, {3419, 5657}, {3587, 5809}, {4004, 6361}, {4304, 6282}, {4314, 6769}, {5081, 6350}, {5534, 5815}, {5584, 6284}, {5698, 6001}

X(6988) = (EULER LINE)∩X(40)X(3485)

Barycentrics 3*a^7-3*(b+c)*a^6-(7*b^2+2*b*c+7*c^2)*a^5+(b+c)*(7*b^2-6*b*c+7*c^2)*a^4+(5*b^4+6*b^2*c^2+5*c^4)*a^3-(b^2-c^2)*(b-c)*(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)^2*(b-c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6988) has Shinagawa coefficients (2$aS_A$, -abc - $aS_A$).

X(6988) lies on these lines: {2, 3}, {40, 3485}, {78, 5657}, {515, 5705}, {517, 4323}, {936, 6261}, {938, 1385}, {944, 6734}, {1071, 5744}, {1210, 3486}, {1490, 5745}, {2096, 4652}, {2550, 6796}, {3085, 3428}, {3452, 5924}, {3487, 5709}, {5086, 5731}, {5226, 5812}, {5273, 5777}, {5432, 5584}, {5732, 6705}

X(6989) = (EULER LINE)∩X(10)X(5534)

Barycentrics a^7-(b+c)*a^6-3*(b+c)^2*a^5+(b+c)*(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4+2*b*c*(4*b^2+b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)*(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2*(b+c)^2*a+(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6989) has Shinagawa coefficients (2abc + $aS_A$, -abc).

X(6989) lies on these lines: {2, 3}, {10, 5534}, {57, 498}, {142, 5709}, {499, 3601}, {942, 1788}, {946, 3587}, {3358, 6260}, {3579, 5805}, {3624, 6282}, {3634, 6245}, {3681, 5552}, {3824, 5812}, {4340, 5398}, {4648, 5707}, {5708, 5771}, {5770, 5791}

X(6989) = complement of X(6846)

X(6990) = (EULER LINE)∩X(11)X(3487)

Barycentrics (b^2-b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^3+c^3)*(b+c)*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6990) has Shinagawa coefficients (abc + $aS_A$, 2abc + $aS_A$).

X(6990) lies on these lines: {2, 3}, {11, 3487}, {12, 3488}, {943, 1479}, {946, 5692}, {1901, 5043}, {3419, 4420}, {3811, 5587}, {3813, 5603}, {3822, 5436}, {3925, 6361}, {4860, 5714}, {5536, 5715}, {5777, 6583}, {5817, 5852}

X(6991) = (EULER LINE)∩X(11)X(5703)

Barycentrics (b^2+b*c+c^2)*a^5-(b+c)*(b^2+c^2)*a^4-2*(b^4+c^4+2*b*c*(b^2+c^2))*a^3+2*(b^3-c^3)*(b^2-c^2)*a^2+(b^2+3*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6991) has Shinagawa coefficients (2abc + $aS_A$, abc + $aS_A$).

X(6991) lies on these lines: {2, 3}, {11, 5703}, {12, 938}, {78, 5178}, {962, 3925}, {1210, 3947}, {1260, 2894}, {1699, 3841}, {2346, 3085}, {3305, 5715}, {3337, 4355}, {3614, 4860}, {3681, 6734}, {3814, 5536}, {3824, 5927}

X(6992) = (EULER LINE)∩X(390)X(5119)

Barycentrics 3*a^7-3*(b+c)*a^6-(5*b^2+4*b*c+5*c^2)*a^5+(b+c)*(5*b^2-6*b*c+5*c^2)*a^4+(b^4+c^4+2*b*c*(4*b^2+3*b*c+4*c^2))*a^3-(b^2-c^2)*(b-c)^3*a^2+(b^2-4*b*c+c^2)*(b^2-c^2)^2*a-(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6992) has Shinagawa coefficients (2abc + $aS_A$, -2$aS_A$).

X(6992) lies on these lines: {2, 3}, {390, 5119}, {515, 3305}, {908, 3576}, {938, 1708}, {944, 3876}, {997, 5731}, {1319, 3475}, {1737, 4294}, {3219, 5768}, {3984, 5882}, {4293, 5226}

X(6993) = (EULER LINE)∩X(7)X(1737)

Barycentrics a^7-(b+c)*a^6+(b^2+4*b*c+c^2)*a^5-(b+c)^3*a^4-(5*b^4+5*c^4+2*b*c*(4*b^2-b*c+4*c^2))*a^3+(b^2-c^2)*(b-c)*(5*b^2+6*b*c+5*c^2)*a^2+(3*b^2+4*b*c+3*c^2)*(b^2-c^2)^2*a-3*(b^2-c^2)^3*(b-c) : :

As a point on the Euler line, X(6993) has Shinagawa coefficients (2abc + $aS_A$, 2$aS_A$).

X(6993) lies on these lines: {2, 3}, {7, 1737}, {3475, 5252}, {3868, 5818}, {5123, 5832}, {5249, 5587}

X(6994) = (EULER LINE)∩X(19)X(3219)

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

As a point on the Euler line, X(6994) has Shinagawa coefficients (-E, $a²$).

X(6994) lies on these lines: {2, 3}, {19, 3219}, {92, 144}, {193, 1839}, {393, 1171}

X(6995) = (EULER LINE)∩X(19)X(346)

Trilinears    2 sec A - csc A tan ω : :
Trilinears    csc A - 2 sec A cot ω : :
Barycentrics   2 tan A - tan ω : :
Barycentrics    1 - 2 tan A cot ω : :
Barycentrics    (3a² + b² + c²)/(b² + c² - a²) : :

As a point on the Euler line, X(6995) has Shinagawa coefficients (-F, $a²$).

The trilinear polar of X(6995) meets the line at infinity at X(3800). (Randy Hutson, April 11, 2015)

X(6995) lies on these lines: {2, 3}, {19, 346}, {33, 390}, {34, 3600}, {51, 6776}, {145, 1829}, {154, 5480}, {193, 1843}, {232, 1180}, {251, 393}, {273, 3598}, {499, 5345}, {612, 4294}, {614, 4293}, {1194, 3199}, {1219, 1891}, {1627, 1968}, {1824, 3995}, {1861, 5338}, {1974, 5012}, {2052, 3424}, {2207, 5359}, {2333, 4651}, {2355, 5101}, {3085, 5310}, {3086, 5322}, {3162, 5354}, {3617, 5090}, {3618, 3796}, {4299, 5272}, {4302, 5268}, {5446, 6193}, {5921, 6515}

X(6995) = isogonal conjugate of X(34817)
X(6995) = anticomplement of X(7386)
X(6995) = circumcircle-inverse of X(37977)
X(6995) = polar conjugate of X(18840)
X(6995) = pole wrt polar circle of trilinear polar of X(18840) (line X(523)X(2525))
X(6995) = orthocentroidal-circle-inverse of X(7378)
X(6995) = {X(2),X(4)}-harmonic conjugate of X(7386)

X(6996) = (EULER LINE)∩X(40)X(4384)

Barycentrics a^5+b*c*a^3-(b+c)*b*c*a^2-(b^3-c^3)*(b-c)*a-(b^2-c^2)*(b-c)*b*c : :

As a point on the Euler line, X(6996) has Shinagawa coefficients ($a²$, -(a + b + c)²).

X(6996) lies on these lines: {2, 3}, {40, 4384}, {57, 3673}, {75, 1766}, {83, 2051}, {86, 572}, {220, 5792}, {226, 4911}, {238, 516}, {239, 517}, {241, 5088}, {315, 4417}, {333, 1746}, {355, 3661}, {515, 3912}, {952, 6542}, {962, 5222}, {1019, 1577}, {1229, 5279}, {1482, 4393}, {1944, 2182}, {1999, 3555}, {3687, 5015}, {5021, 5286}, {5135, 5327}, {5195, 5723}, {5224, 5816}, {5308, 5731}

X(6997) = (EULER LINE)∩X(51)X(1352)

Barycentrics a^6+(b^2+c^2)*a^4-((b^2-c^2)^2-4*b^2*c^2)*a^2-(b^4-c^4)*(b^2-c^2) : :

As a point on the Euler line, X(6997) has Shinagawa coefficients (E, $a²$).

X(6997) lies on these lines: {2, 3}, {51, 1352}, {69, 3060}, {206, 3618}, {251, 2165}, {262, 6504}, {312, 3434}, {324, 6524}, {394, 5480}, {497, 3920}, {498, 5310}, {499, 5322}, {612, 1479}, {614, 1478}, {1180, 5421}, {1194, 2548}, {1196, 5475}, {1899, 3818}, {3583, 5268}, {3585, 5272}, {3589, 3796}, {5225, 5297}, {5422, 6776}

X(6997) = anticomplement of X(7484)
X(6997) = orthocentroidal circle inverse of X(1370)
X(6997) = {X(2),X(4)}-harmonic conjugate of X(1370)

X(6998) = (EULER LINE)∩X(10)X(98)

Barycentrics a^6+(b+c)*a^5+b*c*a^4-2*(b+c)*(b^2+c^2)*a^3-(b^2+c^2)*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*b*c : :

As a point on the Euler line, X(6998) has Shinagawa coefficients (-(a + b + c)², $a²$).

X(6998) lies on these lines: {2, 3}, {10, 98}, {86, 511}, {114, 6626}, {230, 1834}, {325, 1330}, {387, 2271}, {842, 2690}, {942, 1447}, {966, 6776}, {1125, 3430}, {1213, 1503}, {1305, 3563}, {1352, 5224}, {1654, 3564}, {2245, 5327}, {3705, 5814}, {3842, 6211}, {5275, 5706}

X(6998) = Euler line intercept, other than X(27), of circle {X(27),PU(4)}

X(6999) = (EULER LINE)∩X(40)X(3661)

Barycentrics a^5+2*(b+c)*a^4-(b^2+b*c+c^2)*a^3-(b^3+c^3)*a^2-(b-c)^2*b*c*a-(b^2-c^2)*(b^3-c^3) : :

As a point on the Euler line, X(6999) has Shinagawa coefficients (-$bc$, (a + b + c)²).

X(6999) lies on these lines: {2, 3}, {40, 3661}, {239, 515}, {241, 4872}, {516, 3685}, {517, 6542}, {573, 1654}, {661, 4560}, {944, 4393}, {1764, 2896}, {1975, 4417}, {2051, 6625}, {2356, 3100}, {3177, 5813}, {4352, 5712}, {4384, 5691}

X(7000) = (EULER LINE)∩X(98)X(1132)

Barycentrics 2*a^8-2*(b^2+c^2)*a^6+2*(3*b^4+4*b^2*c^2+3*c^4)*a^4-6*(b^4-c^4)*(b^2-c^2)*a^2-4*(b^2-c^2)^2*b^2*c^2-S*(a^2+b^2+c^2)*(3*a^4+2*(b^2+c^2)*a^2-5*(b^2-c^2)^2) : :

As a point on the Euler line, X(7000) has Shinagawa coefficients (-S, $a²$).

X(7000) lies on these lines: {2, 3}, {98, 1132}, {147, 6463}, {262, 1131}, {371, 6202}, {372, 5871}, {485, 6201}, {486, 3424}, {511, 1271}, {1160, 6215}, {1270, 1352}, {1350, 5591}, {1503, 3069}, {1588, 5304}, {3068, 5480}, {3593, 3818}

X(7000) = orthocentroidal-circle-inverse of X(7374)
X(7000) = {X(2),X(4)}-harmonic conjugate of X(7374)

This is the end of PART 4: Centers X(5001) - X(7000)

PART 1:	Introduction and Centers X(1) - X(1000)	PART 2:	Centers X(1001) - X(3000)	PART 3:	Centers X(3001) - X(5000)
PART 4:	Centers X(5001) - X(7000)	PART 5:	Centers X(7001) - X(10000)	PART 6:	Centers X(10001) - X(12000)
PART 7:	Centers X(12001) - X(14000)	PART 8:	Centers X(14001) - X(16000)	PART 9:	Centers X(16001) - X(18000)
PART 10:	Centers X(18001) - X(20000)	PART 11:	Centers X(20001) - X(22000)	PART 12:	Centers X(22001) - X(24000)
PART 13:	Centers X(24001) - X(26000)	PART 14:	Centers X(26001) - X(28000)	PART 15:	Centers X(28001) - X(30000)
PART 16:	Centers X(30001) - X(32000)	PART 17:	Centers X(32001) - X(34000)	PART 18:	Centers X(34001) - X(36000)
PART 19:	Centers X(36001) - X(38000)	PART 20:	Centers X(38001) - X(40000)	PART 21:	Centers X(40001) - X(42000)
PART 22:	Centers X(42001) - X(44000)	PART 23:	Centers X(44001) - X(46000)	PART 24:	Centers X(46001) - X(48000)
PART 25:	Centers X(48001) - X(50000)	PART 26:	Centers X(50001) - X(52000)	PART 27:	Centers X(52001) - X(54000)
PART 28:	Centers X(54001) - X(56000)	PART 29:	Centers X(56001) - X(58000)	PART 30:	Centers X(58001) - X(60000)
PART 31:	Centers X(60001) - X(62000)	PART 32:	Centers X(62001) - X(64000)	PART 33:	Centers X(64001) - X(66000)
PART 34:	Centers X(66001) - X(68000)	PART 35:	Centers X(68001) - X(70000)	PART 36:	Centers X(70001) - X(72000)