Clark Kimberling's

Encyclopedia of Triangle Centers - ETC

Parallels-Conics and related points: X(10001)-X(10014)

X(10001) =  REFLECTION OF X(664) IN X(1275)

Barycentrics    (a-b) (a-c) (a+b-c) (a-b+c) (a^4-a^3 b-2 a^2 b^2+3 a b^3-b^4-a^3 c+5 a^2 b c-3 a b^2 c-b^3 c-2 a^2 c^2-3 a b c^2+4 b^2 c^2+3 a c^3-b c^3-c^4) : :

X(10001) lies on these lines: {190,4130}, {522,664}, {927,3667}, {4887,9436}

X(10001) = reflection of X(664) in X(1275)
X(10001) = X(2)-Ceva conjugate of X(664)
X(10001) = crosssum of PU(103)
X(10001) = X(663)-isoconjugate of X(9357)
X(10001) = X(i)-complementary conjugate of X(j) for these (i,j): (31,664}, {9355,141)

X(10002) =  MIDPOINT OF X(4) AND X(1249)

X(10002) lies on the cubics K281 and K677 and these lines: {2,107}, {4,6}, {5,6523}, {253,264}, {427,6524}, {648,5921}, {1093,8801}, {1217,7401}, {1529,7710}, {1629,7408}, {1848,1857}, {2052,7378}, {3183,6696}, {6529,7694}, {6621,8888}

X(10002) = midpoint of X(4) and X(1249)
X(10002) = X(255)-isoconjugate of X(3424)
X(10002) = {X(4),X(6530)}-harmonic conjugate of X(393)
X(10002) = center of the parallels-conic Cpar(4)

X(10003) =  MIDPOINT OF X(5) AND X(216)

X(10003) lies on these lines: {2,1972}, {5,53}, {140,143}, {264,1656}, {3090,3164}, {3628,6663}

X(10003) = midpoint of X(5) and X(216)
X(10003) = center of the parallels-conic Cpar(5)

X(10004) =  MIDPOINT OF X(7) AND X(3160)

X(10004) lies on these lines: {1,7}, {2,658}, {9,7177}, {144,348}, {226,479}, {934,1001}, {1439,5232}

X(10004) = midpoint of X(7) and X(3160)
X(10004) = {X(2),X(7056)}-harmonic conjugate of X(9533)
X(10004) = center of the parallels-conic Cpar(7)

X(10005) =  MIDPOINT OF X(8) AND X(3161)

X(10005) lies on these lines: {2,1280}, {7,4899}, {8,9}, {10,4310}, {75,3617}, {145,344}, {341,1229}, {497,4126}, {518,4869}, {537,7613}, {1001,4578}, {1654,4461}, {1762,4427}, {2550,4454}, {3008,4929}, {3434,4756}, {4847,5423}

X(10005) = midpoint of X(8) and X(3161)
X(10005) = reflection of X(i) in X(j) for these (i,j): (4779, 3161), (4859, 10)
X(10005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,3717,346), (8,5686,391)
X(10005) = center of the parallels-conic Cpar(8)

X(10006) =  MIDPOINT OF X(11) AND X(650)

X(10006) lies on these lines:
{2,885}, {11,650}, {100,8641}, {513,3911}, {663,899}, {905,5121}, {3900,6745}, {4763,6139}, {4885,6667}, {6713,8760}

X(10006) = midpoint of X(11) and X(650)
X(10006) = reflection of X(4885) in X(6667)
X(10006) = center of the parallels-conic Cpar(11)

X(10007) =  MIDPOINT OF X(39) AND X(141)

X(10007) lies on these lines:
{2,694}, {6,1078}, {39,141}, {76,3763}, {83,2076}, {140,143}, {194,3619}, {262,1350}, {574,4048}, {597,5052}, {698,3934}, {730,3844}, {1506,5103}, {1691,7824}, {2021,8359}, {2782,4045}, {3098,7808}, {5026,5116}, {5031,6656}, {7815,8177}, {7914,8149}

X(10007) = midpoint of X(39) and X(141)
X(10007) = reflection of X(3589) in X(6683)
X(10017) = crosspoint of X(515) and X(522)
X(10007) = {X(2),X(8041)}-harmonic conjugate of X(4074)
X(10007) = center of the parallels-conic Cpar(39)

X(10008) =  MIDPOINT OF X(69) AND X(6337)

X(10008) lies on these lines:
{2,2987}, {3,69}, {76,2996}, {99,5921}, {141,5490}, {183,7612}, {193,1692}, {343,4176}, {1007,1351}, {3619,8361}, {5033,7793}, {6194,9742}

X(10008) = midpoint of X(69) and X(6337)
X(10008) = X(1973)-isoconjugate of X(7612)
X(10008) = center of the parallels-conic Cpar(69)

X(10009) =  MIDPOINT OF X(75) AND X(6376)

X(10009) lies on these lines:
{2,1978}, {10,75}, {274,330}, {561,4359}, {874,1001}, {1965,3980}, {3403,4384}, {3739,6374}, {3795,3993}

X(10009) = midpoint of X(75) and X(6376)
X(10009) = {X(75),X(1921)}-harmonic conjugate of X(76)
X(10009) = center of the parallels-conic Cpar(75)

X(10010) =  MIDPOINT OF X(76) AND X(6374)

X(10010) lies on these lines:
{2,4609}, {75,7034}, {76,141}, {182,880}, {308,2998}, {3934,6375}, {6379,9466}

X(10010) = midpoint of X(76) and X(6374)
X(10010) = reflection of X(6375) iin X(3934)
X(10010) = center of the parallels-conic Cpar(76)

X(10011) =  MIDPOINT OF X(114) AND X(230)

X(10011) lies on these lines:
{2,3}, {114,230}, {155,1611}, {183,9754}, {511,6721}, {1007,1351}, {1353,7735}, {1503,6036}, {3054,5033}, {3815,5028}, {5476,9771}, {5921,7612}

X(10011) = midpoint of X(114) and X(230)
X(10011) = reflection of X(8355) iin X(547)
X(10011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140,3628,8364), (1007,9752,1351), (6039,6040,3146)
X(10011) = center of the parallels-conic Cpar(114)

X(10012) =  MIDPOINT OF X(142) AND X(1212)

X(10012) lies on these lines:
{2,3119}, {142,1212}, {518,1125}, {5199,6706}

X(10012) = midpoint of X(142) and X(1212)
X(10012) = center of the parallels-conic Cpar(142)

X(10013) =  PERSPECTOR OF PARALLELS-CONIC Cpar(X(1))

X(10013) lies on these lines:
{1,3696}, {6,748}, {34,1893}, {37,2279}, {56,7225}, {58,1001}, {86,4441}, {106,6013}, {354,2215}, {584,1438}, {1500,7241}, {1918,9345}, {3616,5331}, {4492,4890}

X(10013) = isogonal conjugate of X(17018)

X(10014) =  PERSPECTOR OF PARALLELS-CONIC Cpar(X(6))

The trilinear polar of X(10014) passes through X(669). (Randy Hutson, July 20, 2016)

Let Oa be the center of the circle formed by inverting line BC in the Moses circle, and define Ob and Oc cyclically. The lines AOa, BOb, COc concur in X(10014). (Randy Hutson, July 20, 2016)

X(10014) lies on these lines:
{6,3934}, {729,7787}, {1207,3224}, {1974,5034}, {3114,7878}, {7772,9468}

X(10014) = perspector of 1st Lemoine circle
X(10014) = isogonal conjugate of X(7786)

X(10015) = TRILINEAR POLE OF THE SHERMAN LINE

Centers X(10015)-X(10017) contributed by César Eliud Lozada, June 8, 2016. See X(3259) and Forum Geometricorum.

X(10015) lies on these lines:{1,676}, {2,3904}, {40,9521}, {65,928}, {80,900}, {88,2401}, {241,514}, {297,525}, {521,7649}, {651,653}, {918,1086}, {1145,1769}, {1577,3910}, {1734,6362}, {2254,2826}, {2785,3716}, {3679,4528}, {3700,4791}, {3907,4142}, {4049,4927}, {4809,4922}, {5540,6084}, {7661,8058}

X(10015) = midpoint of X(3762) and X(4707)
X(10015) = reflection of X(i) in X(j) for these (i,j): (1,676), (3700,4791), (4927,4049)
X(10015) = isogonal conjugate of X(32641)
X(10015) = complement of X(3904)
X(10015) = crossdifference of every pair of points on line X(55)X(184)
X(10015) = trilinear pole of the line X(3259)X(3326)
X(10015) = pole wrt polar circle of trilinear polar of X(1309) (line X(6)X(281))
X(10015) = barycentric product of Steiner circumellipse intercepts of Sherman line
X(10015) = X(48)-isoconjugate (polar conjugate) of X(1309)

X(10016) = CIRCUMCIRCLE-POLE OF THE SHERMAN LINE

X(10016) lies on the tangential circle and these lines:{3,2222}, {22,901}, {24,953}, {25,3259}, {36,1455}, {1155,5370}, {1407,3025}, {1617,2717}, {5520,7354}

X(10016) = circumcircle-inverse of X(10017)
X(10016) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2222,2716,10017)

X(10017) = MIDPOINT OF THE SHERMAN CHORD IN THE CIRCUMCIRCLE

X(10017) lies on the nine-point circle and these lines:{2,1309}, {3,2222}, {4,2734}, {11,6129}, {116,7658}, {117,515}, {124,522}, {125,656}, {132,243}, {650,5514}, {1465,1532}, {3259,3326}

X(10017) = midpoint of X(4) and X(2734)
X(10017) = complement of X(1309)
X(10017) = crosspoint of X(515) and X(522)
X(10017) = crosssum of X(102) and X(109)
X(10017) = circumcircle-inverse of X(10016)
X(10017) = Stevanovic-circle-inverse of X(5514)
X(10017) = polar-circle-inverse of X(36067)
X(10017) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2222,2716,10016)
X(10017) = orthogonal projection of X(3) on the Sherman line

X(10018) = 20th HATZIPOLAKIS-MONTESDEOCA POINT

Centers X(10018)-X(10021) lie on the Euler line. For constructions and properties, see Hyacinthos messages beginning with 23422 (May 30, 2016). See also X(6102).

X(10018) lies on these lines: {2,3}, {74,2883}, {232,7749}, {394,2904}, {597,8537}, {1147,3580}, {1870,5433}, {1899,9707}, {1986,5562}, {3589,6403}, {5432,6198}, {5523,7746}, {5889,9820}, {6152,6689}, {6242,8254}, {6749,8882}

X(10018) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(2,24,1594), (2,3147,24), (2,7401,7569), (3,7505,403), (4,3518,7715), (5,186,6240), (24,1594,7576), (26,6640,858), (140,468,4), (235,549,3520), (427,632,6143), (427,7715,4), (470,471,467), (631,3542,378), (1656,3515,4), (2045,2046,7503), (3517,5094,4), (3518,6143,427), (3525,6353,3541), (3575,3628,7577)

X(10019) = 21st HATZIPOLAKIS-MONTESDEOCA POINT

Centers X(10018)-X(10021) lie on the Euler line. For constructions and properties, see Hyacinthos messages beginning with 23422 (May 30, 2016).

X(10019) lies on these lines: {2,3}, {125,5893}, {1112,5907}, {1879,1990}, {6146,7687}

X(10019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(4,5056,3516), (4,5068,5094), (403,546,3575), (1596,3857,7547), (3855,6623,7507), (6623,7507,1906)

X(10020) = 22nd HATZIPOLAKIS-MONTESDEOCA POINT

Centers X(10018)-X(10021) lie on the Euler line. For constructions and properties, see Hyacinthos messages beginning with 23422, especially 23429

X(10020) lies on these lines: {2,3}, {49,3580}, {973,8254}, {1154,9820}, {1216,5972}, {5432,8144}, {5943,6689}, {5944,6146}

X(10020) = centroid of {A,B,C,X(26)}
X(10020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(2,3518,5576), (2,7506,5), (2,7568,140), (3,7505,5), (5,549,7503), (5,3575,546), (5,3627,7547), (5,7542,140), (5,7575,3575), (24,6639,5), (140,6677,3628), (468,7499,1995), (468,7542,5), (632,7499,140), (1656,7544,5), (3147,3549,6644)

X(10021) = 23rd HATZIPOLAKIS-MONTESDEOCA POINT

Centers X(10018)-X(10021) lie on the Euler line. For constructions and properties, see Hyacinthos messages beginning with 23422, especially 23436

X(10021) lies on these lines: {2,3}, {79,5433}, {191,5886}, {355,5426}, {758,5901}, {1125,2771}, {1749,3337}, {3624,7701}, {3647,4999}, {5432,5441}, {6691,6701}

X(10021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6884,5), (5,550,6839), (5,632,6949), (1656,6965,5), (1749,5443,3649), (6852,7489,5)

X(10022) = X(551)comT(a,c)

The notation Xcom(T), where X is a triangle center and T is a triangle, is defined in the preamble to X(3663).

X(10022) lies on these lines: {2,45}, {10,4715}, {519,4665}, {524,3416}, {527,3820}, {536,551}, {597,742}, {3241,4971}, {3629,4967}, {3729,6707}, {4659,4798}, {4667,4669}, {4690,4745}, {5750,7263}

X(10022) = midpoint of X(i) and X(j) for these {i,j}: (2,4363), (3679,4795), (4667,4669)
X(10022) = reflection of X(i) in X(j) for these (i,j): (2,4472), (4364,2), (4690,4745)
X(10022) = complement of X(24441)
X(10022) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4363,4470,4472), (4363,4472,4364)

X(10023) =  POINT BECRUX 1

See Peter Moses, Hyacinthos 23783.

X(10023) lies on this line: {40,164}

X(10023) = X(13750)-of-excentral-triangle

X(10024) =  COMPLEMENT OF X(3520)

Let O_AO_BO_C be the tangential triangle of a triangle ABC. Let H_A be the orthocenter of O_ABC, and define H_B and H_C cyclically. Let L_A be the Euler line of O_AH_BH_C, and define L_B and L_C cyclically. The lines L_A, L_B, L_C concur in X(10024); see X(10023) and Hyacinthos 23764. (Antreas Hatzipolakis, July 12, 2016)

The triangle H_AH_BH_C in the construction just above is the Johnson triangle. (Randy Hutson, July 20, 2016)

X(10024) lies on these lines: {2,3}, {113,1209}, {127,3934}, {131,137}, {184,9927}, {185,5449}, {216,1879}, {265,6146}, {1060,7741}, {1062,7951}, {1216,1568}, {3574,5446}, {3580,6102}, {5448,5562}, {5476,8538}, {5893,7728}

X(10024) = midpoint of X(4) and X(7488)
X(10024) = reflection of X(i) in X(j) for these (i,j): (3,7542), (1594,5)
X(10024) = complement of X(3520)
X(10024) = orthocentroidal-circle-inverse of X(7526)
X(10024) = X(3521)-complementary conjugate of X(10)
X(10024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,4,7526), (3,5,2072), (3,1656,6640), (4,5,5576), (4,3091,7564), (4,3549,3), (4,7517,7540), (4,7552,7488), (5,235,381), (5,546,5133), (5,1596,7403), (5,7399,1656), (5,7405,5055), (113,1209,5907), (381,7517,4), (1596,7403,3843), (7530,7564,4)

H-transforms and K-transforms: X(10025)-X(10030)

X(10025) =  H-TRANSFORM: H(X(2), X(1))

The H-transform and K-transform are introduced just before X(10024).

X(10025) lies on these lines: {1,3177}, {2,7}, {85,220}, {101,5088}, {150,5179}, {152,516}, {190,3693}, {192,3870}, {200,1721}, {218,3673}, {228,7411}, {239,294}, {517,3732}, {522,3935}, {664,6603}, {673,2348}, {760,3869}, {1088,6180}, {1111,5526}, {1281,4712}, {1536,5762}, {2098,9311}, {3980,8580}, {4388,4416}, {4480,6745}, {4872,5845}, {6554,6604}, {7176,9310}

X(10025) = reflection of X(i) in X(j) for these (i,j): (150,5179), (239,666), (664),6603), (5088,101)
X(10025) = isotomic conjugate of anticomplement of X(36905)
X(10025) = X(518)-Ceva conjugate of X(239)
X(10025) = crosspoint of X(666) and X(1275)
X(10025) = anticomplement of X(9436)
X(10025) = X(i)-anticomplementary conjugate of X(j) for these (i,j): (105,3434), (294,69), (884,149), (919,693), (1024,150), (1438,7), (1462,6604), (2195,8), (5377,3888)
X(10025) = X(6)-isoconjugate of X(9442)
X(10025) = {X(672),X(9318)}-harmonic conjugate of X(1447)
X(10025) = inverse-in-Steiner-circumellipse of X(9)
X(10025) = X(2)-Hirst inverse of X(9)

X(10026) =  H-TRANSFORM: H(X(10), X(2))

The H-transform and K-transform are introduced just before X(10024).

X(10026) lies on these lines: {2,6}, {115,519}, {187,540}, {523,661}, {594,1215}, {620,6629}, {758,5164}, {1100,3846}, {1125,6537}, {2653,4109}, {2886,5949}, {3124,3726}, {4771,4892}, {6542,6543}

X(10026) = reflection of X(6629) in X(620)
X(10026) = complement of X(17731)
X(10026) = complementary conjugate of X(20339)
X(10026) = X(i)-Ceva conjugate of X(j) for these (i,j): (6542,740), (6543,1213)
X(10026) = crosspoint of X(2) and X(11599)
X(10026) = crosssum of X(6) and X(1326)
X(10026) = X(i)-complementary conjugate of X(j) for these (i,j): (213,6651), (1929,3741), (2054,10), (2702,4369), (9278,141)
X(10026) = {X(6189),X(6190)}-harmonic conjugate of X(1654)

X(10027) =  H-TRANSFORM: H(X(6), X(1))

The H-transform and K-transform are introduced just before X(10024).

X(10027) lies on these lines: {{1,2}, {194,3208}, {330,3501}, {335,517}, {668,3230}, {672,9263}, {730,3685}, {1016,5383}, {1575,4595}, {2106,7257}, {5255,6645}

X(10028) =  K-TRANSFORM: K(X(1),X(1))

The H-transform and K-transform are introduced just before X(10024).

Let V = P(8)-Ceva conjugate of U(8), and W = U(8)-Ceva conjugate of P(8). Then X(10028) is the crossdifference of every pair of points on the line VW. (Randy Hutson, July 20, 2016)

X(10028) lies on this line: {6,75}

X(10029) =  K-TRANSFORM: K(X(8), X(7))

The H-transform and K-transform are introduced just before X(10024).

X(10029) lies on these lines: {7,145}, {279,4488}, {1275,5382}, {4899,9436}

X(10029) = X(3912)-cross conjugate of X(9436)
X(10029) = X(i)-isoconjugate of X(j) for these (i,j): (294,3052), (919,4162), (1416,4936), (1438,3158), (1743,2195)

X(10030) =  K-TRANSFORM: K(X(1),X(75))

The H-transform and K-transform are introduced just before X(10024).

X(10030) lies on the cubics K356, K623, K767, and these lines: {7,8}, {19,331}, {57,6063}, {76,1423}, {274,1432}, {279,330}, {291,1738}, {308,349}, {348,2275}, {350,1281}, {516,2481}, {664,1458}, {668,4899}, {1111,1733}, {1921,3975}, {3598,4441}, {3669,4560}, {3673,4008}, {3911,4554}, {4334,9312}, {4858,7112}, {6649,9364}

X(10030) = X(7233)-Ceva conjugate of X(7196)
X(10030) = X(239)-cross conjugate of X(350)
X(10030) = cevapoint of X(239) and X(1447)
X(10030) = crosspoint of X(i) and X(j) for these {i,j}: {3500,7167}, {7233,7249}
X(10030) = X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,3212,1469), (57,6063,7196), (57,7243,6063)
X(10030) = X(i)-isoconjugate of X(j) for these {i,j}: {6,7077}, {8,1922}, {9,1911}, {31,4876}, {32,4518}, {33,2196}, {41,291}, {42,2311}, {55,292}, {295,607}, {334,9447}, {335,2175}, {644,875}, {660,3063}, {663,813}, {694,2330}, {741,1334}, {1808,2333}, {1967,2329}, {2195,3252}, {3572,3939}, {7081,9468}

X(10031) =  MIDPOINT OF X(3241) AND X(6224)

Let I be the incenter of a triangle ABC. Let A' = reflection of I in BC, and define B' and C' cyclically. Let A'' be the reflection of I in A', and define B'' and C'' cyclically. Let Ea be the Euler line of A''BC, and define Eb and Ec cyclically. The lines Ea, Eb, Ec concur in X(10031). (Seiichi Kirikami, July 5, 2016: Hyacinthos 23726)

X(10031) lies on these lines: {2,952}, {7,528}, {8,6174}, {21,5882}, {36,100}, {80,551}, {104,3655}, {149,1056}, {214,3679}, {944,5330}, {1537,3543}, {2787,9884}, {2801,3877}, {2802,3873}, {3623,6147}

X(10031) = midpoint of X(3241) and X(6224)
X(10031) = reflection of X(i) in X(j) for these (i,j): (8, 6174), (80, 551), (104, 3655), (1320, 3241), (3241, 1317), (3543, 1537), (3679, 214)
X(10031) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1317,6224,1320), (1320,6224,9963)

X(10032) =  X(8)X(30)∩X(21)X(551)

Let Ia be the A-excenter of a triangle ABC, and define Ib and Ic cyclically. LEt A' = reflection of Ia in BC, and define B' and C' cyclically. Let A'' = reflection of IA in A', and define B'' and C'' cyclically. The Euler lines of A''BC, B''CA,C''AB concur in X(10032). (Tran Quang Hung and Angel Montesdeoca, July 19, 2016: Hyacinthos 23831)

X(10032) lies on these lines: {8,30}, {21,551}, {79,3634}, {191,6175}, {527,2346}, {553,5284}, {1281,6054}, {2796,4921}, {2975,3656}, {3624,3647}

X(10032) = reflection of X(6175) in X(191)

X(10033) =  24th HATZIPOLAKIS-MONTESDEOCA POINT

Let ABC be a triangle and A'B'C' the cevian triangle of the centroid, G, and let

Oa = circumcenter of GB'C'; define Ob and Oc cyclically
Oab = circumcenter of AB'G; define Obc and Oca cyclically
Oac = circumcenter of AC'G; define Oba and Ocb cyclically
Ga = centroid of OaObOc; define Gb and Gc cyclically
Na = nine-point center of GB'C'; define Nb and Nc cyclically
Nab = nine-point center of AB'G; define Nbc and Nca cyclically
Nac = nine-point center of AC'G; define Nba and Ncb cyclically

The triangles GaGbGc, G1G2G3 are perspective, and their perspector is X(10033). Let Ea be the Euler line of OaOabOac, and define Eb and Ec cyclically; then Ea, Eb, Ec are parallel, and they concur in X(524). Let Fa be the Euler line of NaNabNac, and define Fb and Fc cyclically. Let A'' = Fb∩Fc, and define B'' and C'' cyclically. Then the triangles ABC and A''B''C'' are parallelogic, and the parallelogic center of ABC with respect to A''B''C'' is X(6094), the 11th Hatzipolakis-Montesdeoca point, and the parallelogic center of A''B''C'' with respect to ABC is X(10034). (Antreas Hatzipolakis and Angel Montesdeoca, August 1, 2016; see Hyacinthos 23907)

X(10033) lies on these lines: {2,1495}, {4,3849}, {30,7697}, {98,381}, {114,8592}, {183,3830}, {262,542}, {3545,7694}, {3839,9753}, {3845,9993}, {5066,7792}, {6054,9830}, {8370,9873}

X(10034) =  25th HATZIPOLAKIS-MONTESDEOCA POINT

Let A''B''C'' be as at X(10035). The parallelogic center of A''B''C'' with respect to ABC is X(10034). (Antreas Hatzipolakis and Angel Montesdeoca, August 1, 2016; see Hyacinthos 23907)

X(10035) =  26th HATZIPOLAKIS-MONTESDEOCA POINT

Let ABC be a triangle and A'B'C' the cevian triangle of the incenter, I, and let

Oa = circumcenter of IB'C'; define Ob and Oc cyclically
Oab = circumcenter of AB'I; define Obc and Oca cyclically
Oac = circumcenter of AC'I; define Oba and Ocb cyclically
Ga = centroid of OaObOc; define Gb and Gc cyclically
Na = nine-point center of IB'C'; define Nb and Nc cyclically
Nab = nine-point center of AB'I; define Nbc and Nca cyclically
Nac = nine-point center of AC'I; define Nba and Ncb cyclically

Let Ea be the Euler line of NaNabNac, and define Eb and Ec cyclically; then Ea, Eb, Ec concur in X(10035). Let Fa be the Euler line of OaOabOac, and define Fb and Fc cyclically; the lines Fa, Fb, Fc are parallel, and they meet in X(517). (Antreas Hatzipolakis and Angel Montesdeoca, August 1, 2016; see Hyacinthos 23914)

X(10035) lies on these lines: {11,500}, {30,1319}, {496,5495}, {511,6713}, {549,4271}, {952,5453}

X(10035) = midpoint of X(11) and X(500)
X(10035) = center of the rectangular hyperbola through X(11), X(500) and the vertices of the incentral triangle
X(10035) = QA-P36 (Complement of QA-P30 wrt the QA-Diagonal Triangle) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/64-qa-p36.html)

X(10036) =  27th HATZIPOLAKIS-MONTESDEOCA POINT

Let ABC be a triangle and A'B'C' the cevian triangle of the incenter, I. Continuing from X(10035), let Pa be the line through A' parallel to Ea, and define Pb and Pc cyclically. Then Pa,Pb,Pc concur in X(10036). (Antreas Hatzipolakis and Angel Montesdeoca, August 1, 2016; see Hyacinthos 23914)

X(10036) lies on these lines: {11,8143}, {115,119}, {952,5492}, {1317,2771}

X(10036) = reflection of X(11) in X(8143)
X(10036) = antipode of X(11) in incentral circle
X(10036) = antipode of X(500) in rectangular hyperbola passing through X(214), X(500), X(4065), and the vertices of the incentral triangle
X(10036) = QA-P30 (Reflection of QA-P2 in QA-P11) of quadrangle ABCX(1) (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

Inner- and outer- Yff triangles: X(10037)-X(10094)

X(10037) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF AND ARA

X(10037) lies on these lines:{1,25}, {3,12}, {11,7529}, {22,3085}, {24,388}, {26,495}, {47,1460}, {55,7387}, {56,6642}, {159,611}, {378,5229}, {499,5020}, {999,7506}, {1056,3518}, {1479,1598}, {1593,3585}, {1709,9910}, {1995,3086}, {2067,8276}, {3074,5329}, {3157,9937}, {3295,7517}, {3303,9673}, {3515,5270}, {3583,5198}, {5119,9911}, {5160,5899}, {5261,7488}, {5290,9590}, {6502,8277}, {7395,7951}, {7512,8164}, {9672,9818}

X(10037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55,9658,7387)

X(10038) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF AND 5TH BROCARD

X(10038) lies on these lines:{1,32}, {12,9996}, {35,1469}, {55,9821}, {388,9862}, {499,7846}, {611,3094}, {1478,9873}, {1479,9993}, {2896,3085}, {3295,9301}, {3584,7865}

X(10039) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF AND OUTER-GARCIA

X(10039) = (R-r)*X(1)+3*r*X(2)

X(10039) lies on these lines:{1,2}, {3,5252}, {4,5119}, {5,3057}, {11,9956}, {12,517}, {21,5176}, {35,515}, {36,6684}, {40,1478}, {46,388}, {47,5264}, {55,355}, {65,495}, {79,3245}, {80,943}, {140,1319}, {165,4299}, {392,1329}, {442,1145}, {484,4292}, {497,5818}, {516,3585}, {529,3916}, {611,3416}, {631,3476}, {908,3878}, {920,3421}, {944,3612}, {946,1512}, {952,2646}, {958,8069}, {1000,3090}, {1056,1788}, {1109,2292}, {1320,7504}, {1334,5179}, {1376,8071}, {1387,3628}, {1389,5559}, {1479,1697}, {1571,9597}, {1572,9596}, {1738,3987}, {1772,3670}, {1836,9654}, {1837,3295}, {2093,5290}, {2098,5886}, {2802,8068}, {3074,5255}, {3262,4357}, {3303,5722}, {3336,4298}, {3419,3913}, {3485,8164}, {3576,6961}, {3579,7354}, {3586,4309}, {3601,5881}, {3614,9955}, {3698,8728}, {3717,4710}, {3753,8256}, {3754,5249}, {3812,5570}, {3814,3884}, {3825,3898}, {3826,4002}, {3871,5086}, {3890,4193}, {3911,5445}, {3966,5827}, {4187,5123}, {4295,5261}, {4297,5010}, {4302,5691}, {4305,5281}, {4311,7280}, {4317,9588}, {4333,9778}, {4748,7961}, {4848,5902}, {4904,6706}, {5048,5901}, {5219,5761}, {5229,6361}, {5251,5795}, {5258,5745}, {5266,5724}, {5533,6702}, {5687,5794}, {5692,5837}, {5710,5725}, {5726,5758}, {5854,6668}, {6975,7741}, {6981,7962}, {7173,7743}, {7969,9646}, {7989,9614}, {8193,10037}, {9857,10038}

X(10039) = complement of X(4861)
X(10039) = midpoint of X(i),X(j) for these {i,j}: {3871,5086}
X(10039) = reflection of X(i) in X(j) for these (i,j): (6734,10)
X(10039) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10,1737), (1,1698,499), (8,3085,1), (10,6736,3679), (40,1478,1770), (40,9578,1478), (80,3746,950), (165,9613,4299), (388,5657,46), (442,1145,5836), (484,5270,4292), (495,5690,65), (944,5218,3612), (1056,1788,3338), (1697,5587,1479), (3295,5790,1837), (5445,5563,3911), (5697,7951,946), (5726,7991,9612), (7989,9819,9614), (9956,9957,11)

X(10040) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF AND INNER-GREBE

X(10040) = R*(2*SW-S)*X(1)+2*r*SW*X(6)

X(10040) lies on these lines:{1,6}, {12,6215}, {55,1161}, {495,5875}, {498,5591}, {1271,3085}, {1478,5871}, {1479,6202}, {1709,6258}, {5595,10037}, {5689,10039}, {9994,10038}

X(10041) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-YFF AND OUTER-GREBE

X(10041) = R*(2*SW+S)*X(1)+2*r*SW*X(6)

X(10041) lies on these lines:{1,6}, {12,6214}, {55,1160}, {495,5874}, {498,5590}, {1270,3085}, {1478,5870}, {1479,6201}, {1709,6257}, {5594,10037}, {5688,10039}, {9995,10038}

X(10041) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,611,10040)

X(10042) = PERSPECTOR OF THESE TRIANGLES: INNER-YFF AND HEXYL

X(10042) lies on these lines:{35,5732}, {40,4299}, {84,1478}, {1768,9613}, {3340,6264}

X(10043) = PERSPECTOR OF THESE TRIANGLES: INNER-YFF AND HUTSON INTOUCH

Trilinears    4*p^2*(p^2-3*p*q+3*q^2)-(2*q^2+1)*(2*p*q+1)+4 : : , where p=sin(A/2), q=cos((B-C)/2) p>X(10043) lies on these lines:{1,6833}, {8,5187}, {11,1482}, {355,1479}, {3434,5697}, {8275,9614}

X(10044) = PERSPECTOR OF THESE TRIANGLES: INNER-YFF AND INTOUCH

X(10044) = (R+r)*(4*R+r)*X(7)-R*(R-r)*X(46)

X(10044) lies on these lines:{1,6934}, {3,3649}, {7,46}, {79,6836}, {377,5902}, {498,1454}, {1479,5805}, {1709,6847}, {1768,6833}, {2949,3336}, {3522,3612}, {3652,6862}, {4004,5794}

X(10045) = PERSPECTOR OF THESE TRIANGLES: INNER-YFF AND 6TH MIXTILINEAR

X(10045) lies on these lines:{1478,7992}, {1699,10092}, {1770,7991}, {3062,9612}

X(10046) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF AND ARA

X(10046) lies on these lines:{1,25}, {3,11}, {12,7529}, {22,3086}, {24,497}, {26,496}, {46,9911}, {47,7083}, {55,6642}, {56,7387}, {159,613}, {378,5225}, {498,5020}, {999,7517}, {1058,3518}, {1069,9937}, {1478,1598}, {1593,3583}, {1737,8193}, {1995,3085}, {2066,8276}, {2933,8069}, {3075,7295}, {3295,7506}, {3304,9658}, {3515,4857}, {3585,5198}, {3813,9713}, {3816,9712}, {5274,7488}, {5414,8277}, {7395,7741}, {9659,9818}

X(10046) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,25,10037), (56,9673,7387)

X(10047) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF AND 5TH BROCARD

X(10047) lies on these lines:{1,32}, {11,9996}, {36,3056}, {56,9821}, {497,9862}, {498,7846}, {499,3096}, {613,3094}, {999,9301}, {1478,9993}, {1479,9873}, {1737,9857}, {2896,3086}, {3582,7865}

X(10047) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,32,10038)

X(10048) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF AND INNER-GREBE

X(10048) = R*(2*SW-S)*X(1)-2*r*SW*X(6)

X(10048) lies on these lines: {1,6}, {11,6215}, {56,1161}, {496,5875}, {499,5591}, {1271,3086}, {1478,6202}, {1479,5871}, {1737,5689}, {5595,10046}, {9994,10047}

X(10049) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-YFF AND OUTER-GREBE

X(10049) = R*(2*SW+S)*X(1)-2*r*SW*X(6)

X(10049) lies on these lines: {1,6}, {11,6214}, {56,1160}, {496,5874}, {499,5590}, {1270,3086}, {1478,6201}, {1479,5870}, {1737,5688}, {5594,10046}, {9995,10047}

X(10050) = PERSPECTOR OF THESE TRIANGLES: OUTER-YFF AND HEXYL

X(10050) lies on these lines: {4,10042}, {36,5732}, {40,920}, {84,1479}, {90,5692}, {1768,3586}, {6264,7962}

X(10051) = PERSPECTOR OF THESE TRIANGLES: OUTER-YFF AND HUTSON INTOUCH

X(10051) lies on these lines: {1,1512}, {3,5427}, {11,5790}, {390,5119}, {497,10073}, {1001,1145}, {1478,5805}, {1768,6938}, {2478,5692}, {2949,6936}, {3057,5722}, {5768,10085}, {5902,6925}, {6887,10039}

X(10052) = PERSPECTOR OF THESE TRIANGLES: OUTER-YFF AND INTOUCH

X(10052) = (4*R+r)*X(7)-R*X(90)

X(10052) lies on these lines: 1,5553}, {7,90}, {46,5552}, {65,68}, {145,4295}, {224,4292}, {226,10044}, {499,7082}, {999,3649}, {1317,1482}, {2800,10043}, {3174,4312}, {5586,9612}, {5708,7173}

X(10053) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1ST ANTI-BROCARD

X(10053) = R*X(1)+r*X(98) = (R+2*r)*X(35)-r*X(99)

X(10053) lies on these lines:{1,98}, {3,3023}, {8,5985}, {12,6033}, {35,99}, {55,2782}, {114,498}, {115,1479}, {147,3085}, {148,4294}, {388,9862}, {499,6036}, {1478,2794}, {3027,3295}, {3584,6054}, {3920,5986}, {4995,8724}, {5697,7983}, {6226,10041}, {6227,10040}, {6284,6321}, {9861,10037}, {9864,10039}

X(10053) ={X(1),X(98)}-harmonic conjugate of X(10069)

X(10054) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO ANTI-MCCAY

X(10054) = R*X(1)+r*X(671)

X(10054) lies on these lines:{1,671}, {12,8724}, {30,10053}, {99,3584}, {381,3023}, {498,2482}, {499,5461}, {542,1478}, {611,9830}, {1479,9880}, {3058,6321}, {3085,8591}, {3582,9166}, {9876,10037}, {9878,10038}, {9881,10039}, {9882,10040}, {9883,10041}

X(10054) ={X(1),X(671)}-harmonic conjugate of X(10070)

X(10055) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO ARIES

X(10055) = R*X(1)+r*X(68)

X(10055) lies on these lines:{1,68}, {4,6238}, {5,1069}, {12,155}, {388,7352}, {495,611}, {498,1147}, {499,5449}, {912,5252}, {942,5820}, {1060,1899}, {1479,9927}, {3085,6193}, {4299,7689}, {5654,7951}, {8909,9646}, {9659,9932}, {9908,10037}, {9923,10038}, {9928,10039}, {9929,10040}, {9930,10041}

X(10055) = reflection of X(i) in X(j) for these (i,j): (3157,495)

X(10055) ={X(1),X(68)}-harmonic conjugate of X(10071)

X(10056) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO ARTZT

X(10056) = R*X(1)+r*X(2)

X(10056) lies on these lines:{1,2}, {3,4317}, {4,3746}, {5,3303}, {7,484}, {11,5055}, {12,381}, {20,5270}, {30,55}, {35,376}, {36,1056}, {40,4654}, {46,553}, {56,549}, {65,3654}, {79,6361}, {80,3488}, {140,3304}, {226,5119}, {281,1784}, {344,3992}, {345,4692}, {348,7278}, {377,8715}, {390,3583}, {442,3913}, {496,547}, {497,3545}, {524,611}, {528,954}, {529,8069}, {539,10055}, {542,10053}, {543,10054}, {546,9670}, {550,9657}, {597,613}, {631,5563}, {920,3929}, {956,6690}, {984,1725}, {999,5054}, {1058,5071}, {1319,3653}, {1500,5309}, {1770,5290}, {2099,5719}, {2241,7753}, {2276,7739}, {2551,5259}, {2646,3655}, {3023,8724}, {3057,3656}, {3091,4857}, {3146,4330}, {3298,9646}, {3338,6684}, {3421,5251}, {3434,3822}, {3436,5248}, {3475,5657}, {3485,5697}, {3522,4325}, {3534,7354}, {3543,3585}, {3586,5726}, {3600,7280}, {3614,9669}, {3627,9656}, {3743,6757}, {3744,5725}, {3748,5722}, {3820,4423}, {3830,6284}, {3850,9671}, {3871,6175}, {4293,5010}, {4338,5493}, {5217,8703}, {5258,6857}, {5433,7373}, {5537,6916}, {5559,6853}, {5734,6960}, {5860,10041}, {5861,10040}, {5881,6824}, {5882,6833}, {5886,5919}, {6825,7982}, {6908,7991}, {6910,8666}, {6944,9624}, {6979,7320}, {7811,10038}, {9909,10037}

X(10056) = X(2)-of-inner-Yff-triangle
X(10056) = X(381)-of-1st Johnson-Yff triangle
X(10056) = outer-Johnson-to-ABC similarity image of X(381)
X(10056) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,10072), (1,498,499), (1,3085,498), (1,3584,2), (2,3085,3584), (2,3584,498), (4,3746,4309), (12,3058,381), (12,3295,1479), (35,388,4299), (55,495,1478), (55,1478,4302), (381,3058,1479), (381,3295,3058), (497,8164,7951), (999,5054,5298), (1056,5218,36), (3057,4870,3656), (3475,5657,5902), (4293,5281,5010), (4294,5261,3585), (4995,5434,3), (5298,5432,5054),((12941,12942,12588

X(10057) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO FUHRMANN

X(10057) = R*X(1)+r*X(80)

X(10057) lies on these lines:{1,5}, {4,10043}, {100,10039}, {214,498}, {388,10044}, {499,6702}, {1145,5794}, {1320,5086}, {1478,2800}, {1709,2829}, {1737,5176}, {1768,9613}, {2802,3434}, {3085,6224}, {3419,5854}, {5119,5840}, {6262,10041}, {6263,10040}, {9912,10037}

X(10057) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,10073), (11,355,80)

X(10058) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INNER-GARCIA

X(10058) = R*X(1)+r*X(104)

X(10058) lies on these lines:{1,104}, {3,11}, {4,8068}, {9,10050}, {10,21}, {30,5172}, {36,516}, {55,952}, {56,1387}, {78,90}, {119,498}, {149,4189}, {153,3085}, {214,3612}, {404,7741}, {405,2932}, {474,6667}, {497,5533}, {515,10057}, {758,1727}, {855,1324}, {943,3065}, {954,5851}, {956,5854}, {958,1145}, {993,2802}, {1006,3586}, {1012,1478}, {1317,3295}, {1320,2975}, {1537,10044}, {1621,5197}, {1697,6264}, {1737,2077}, {1955,3465}, {2787,10053}, {2801,7675}, {3036,5687}, {3422,7295}, {3579,6797}, {3583,6905}, {3601,6326}, {3651,4324}, {3746,7972}, {4305,6224}, {4313,9803}, {4351,8758}, {5225,6942}, {5284,5444}, {5432,7489}, {5703,9809}, {6912,7951}, {6952,8070}, {7280,9614}, {9913,10037}

X(10058) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,104,10074), (11,6713,499), (35,80,100), (149,4189,4996), (405,2932,3035), (1012,8069,1478)

X(10059) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO HUTSON EXTOUCH

X(10059) = R*X(1)+r*X(7160)

X(10059) lies on these lines:{1,5920}, {3085,9874}

X(10059) = {X(1),X(7160)}-harmonic conjugate of X(10075)

X(10060) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO MIDHEIGHT

X(10060) = R*X(1)+r*X(64)

X(10060) lies on these lines:{1,64}, {3,6285}, {12,5878}, {30,10055}, {35,1498}, {36,2192}, {55,6000}, {56,3357}, {154,5010}, {221,3746}, {498,2883}, {499,6696}, {1479,6247}, {1503,4302}, {1717,1854}, {1853,3583}, {2935,7727}, {3085,6225}, {3157,5663}, {3295,7355}, {3585,5895}, {4299,5894}, {5119,6001}, {5217,6759}, {5218,5656}, {6266,10041}, {6267,10040}, {7280,8567}, {7689,9645}, {9914,10037}

X(10060) = {X(1),X(64)}-harmonic conjugate of X(10076)

X(10061) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INNER-NAPOLEON

X(10061) = R*X(1)+r*X(14)

X(10061) lies on these lines:{1,14}, {12,5613}, {35,5474}, {56,6774}, {115,10078}, {388,6773}, {498,619}, {499,6670}, {530,10054}, {531,10056}, {542,611}, {617,3085}, {1479,5479}, {3023,5617}, {3584,5464}, {5357,9113}, {6269,10041}, {6271,10040}, {9915,10037}, {9981,10038}

X(10061) = {X(1),X(14)}-harmonic conjugate of X(10077)

X(10062) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO OUTER-NAPOLEON

X(10062) = R*X(1)+r*X(13)

X(10062) lies on these lines:{1,13}, {12,5617}, {35,5473}, {56,6771}, {115,10077}, {388,6770}, {498,618}, {499,6669}, {530,10056}, {531,10054}, {542,611}, {616,3085}, {1479,5478}, {3023,5613}, {3584,5463}, {6268,10041}, {6270,10040}, {9916,10037}, {9982,10038}

X(10062) = {X(1),X(13)}-harmonic conjugate of X(10078)

X(10063) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1ST NEUBERG

X(10063) = R*X(1)+r*X(76)

X(10063) lies on these lines:{1,76}, {11,7697}, {12,3095}, {39,498}, {55,2782}, {194,3085}, {262,7951}, {499,3934}, {511,1478}, {538,10056}, {611,732}, {1479,6248}, {1733,4692}, {3584,7757}, {4293,6194}, {4299,5188}, {5218,7709}, {5969,10054}, {6272,10041}, {6273,10040}, {7354,9821}, {9917,10037}, {9983,10038}

X(10063) = {X(1),X(76)}-harmonic conjugate of X(10079)

X(10064) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 2ND NEUBERG

X(10064) = R*X(1)+r*X(83)

X(10064) lies on these lines:{1,83}, {12,6287}, {36,9751}, {495,10053}, {498,6292}, {499,6704}, {611,732}, {754,10056}, {1479,6249}, {2896,3085}, {6274,10041}, {6275,10040}, {7354,8725}, {9918,10037}

X(10064) = {X(1),X(83)}-harmonic conjugate of X(10080)

X(10065) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO ORTHOCENTROIDAL

X(10065) = R*X(1)+r*X(74)

X(10065) lies on these lines:{1,74}, {12,7728}, {35,110}, {55,5663}, {113,498}, {125,1479}, {146,3085}, {265,6284}, {499,6699}, {541,10056}, {611,2781}, {690,10053}, {1478,2777}, {1511,5217}, {2330,9970}, {3028,3295}, {3448,4294}, {3581,5160}, {4302,10055}, {4995,5655}, {5504,6238}, {5697,7984}, {6285,9934}, {7725,10040}, {7726,10041}, {9919,10037}, {9984,10038}

X(10065) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,74,10081), (35,7727,110)

X(10066) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO REFLECTION

X(10066) = R*X(1)+r*X(54)

X(10066) lies on these lines:{1,54}, {12,6288}, {35,7356}, {55,1154}, {195,3295}, {496,8254}, {498,1209}, {499,6689}, {539,10055}, {1479,3574}, {1493,3303}, {2293,3746}, {2888,3085}, {6276,10041}, {6277,10040}, {9920,10037}, {9985,10038}

X(10066) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,54,10082), (35,7356,7691)

X(10067) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INNER-VECTEN

X(10067) = R*X(1)+r*X(486)

X(10067) lies on these lines:{1,486}, {12,6290}, {487,3085}, {495,611}, {498,642}, {499,6119}, {1479,6251}, {3289,10084}, {6280,10041}, {6281,10040}, {9921,10037}, {9986,10038}

X(10067) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,486,10083), (495,611,10068)

X(10068) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO OUTER-VECTEN

X(10068) = R*X(1)+r*X(485)

X(10068) lies on these lines:{1,485}, {12,6289}, {488,3085}, {495,611}, {498,641}, {499,6118}, {1479,6250}, {3297,10083}, {6278,10041}, {6279,10040}, {9922,10037}, {9987,10038}

X(10068) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,485,10084), (495,611,10067)

X(10069) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1ST ANTI-BROCARD

X(10069) = R*X(1)-r*X(98)

X(10069) lies on these lines:{1,98}, {3,3027}, {11,6033}, {36,99}, {56,2782}, {114,499}, {115,1478}, {147,3086}, {148,4293}, {497,9862}, {498,6036}, {999,3023}, {1210,2784}, {1479,2794}, {1737,9864}, {1795,2792}, {3582,6054}, {3616,5985}, {5298,8724}, {5434,10054}, {5986,7191}, {6055,10056}, {6226,10049}, {6227,10048}, {6321,7354}, {9861,10046}

X(10069) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,98,10053), (3,3027,10086)

X(10070) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO ANTI-MCCAY

X(10070) = R*X(1)-r*X(671)

X(10070) lies on these lines:{1,671}, {11,8724}, {30,10069}, {99,3582}, {115,10056}, {498,5461}, {499,2482}, {542,1479}, {613,9830}, {1210,2796}, {1478,9880}, {1737,9881}, {3058,10053}, {3086,8591}, {3584,9166}, {5434,6321}, {9876,10046}, {9878,10047}, {9882,10048}, {9883,10049}

X(10070) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,671,10054)

X(10071) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO ARIES

X(10071) = R*X(1)-r*X(68)

X(10071) lies on these lines:{1,68}, {4,7352}, {5,3157}, {11,155}, {496,613}, {497,6238}, {498,5449}, {499,1147}, {912,1837}, {1062,1899}, {1478,9927}, {1503,9645}, {1737,9928}, {3086,6193}, {3448,9538}, {4302,7689}, {5654,7741}, {8909,9661}, {9672,9932}, {9908,10046}, {9923,10047}, {9929,10048}, {9930,10049}

X(10071) = reflection of X(i) in X(j) for these (i,j): (1069,496)
X(10071) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,68,10055)

X(10072) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO ARTZT

X(10072) = R*X(1)-r*X(2)

X(10072) lies on these lines:{1,2}, {3,3058}, {4,4317}, {5,3304}, {7,3065}, {11,381}, {12,5055}, {20,4857}, {30,56}, {35,1058}, {36,376}, {55,549}, {65,3656}, {80,3476}, {115,10054}, {140,3303}, {278,1784}, {345,4975}, {348,7264}, {354,912}, {388,3545}, {390,5010}, {474,3813}, {484,5435}, {495,547}, {524,613}, {528,2932}, {539,10071}, {542,10069}, {543,10070}, {546,9657}, {550,9670}, {553,946}, {597,611}, {631,3746}, {920,3928}, {956,3816}, {962,3336}, {982,1725}, {1015,5309}, {1056,5071}, {1387,2099}, {1440,7271}, {1470,10058}, {1770,3361}, {1788,5697}, {1836,7743}, {1997,3992}, {2242,7753}, {2275,7739}, {2478,8666}, {2551,5288}, {2646,3653}, {2800,5603}, {3027,8724}, {3028,5655}, {3057,3654}, {3091,5270}, {3146,4325}, {3295,4995}, {3297,9661}, {3333,4654}, {3337,4295}, {3436,3825}, {3487,5443}, {3522,4330}, {3534,6284}, {3543,3583}, {3585,3600}, {3627,9671}, {3655,5722}, {3830,7354}, {3850,9656}, {3911,5119}, {4294,5265}, {4301,6890}, {5084,5258}, {5131,9778}, {5204,8703}, {5432,6767}, {5459,10062}, {5460,10061}, {5687,6691}, {5734,6972}, {5860,10049}, {5861,10048}, {5881,6944}, {5882,6834}, {6055,10053}, {6824,9624}, {6891,7982}, {6921,8715}, {6926,7991}, {7173,9654}, {7811,10047}, {9466,10063}, {9909,10046}

X(10072) = X(2)-of-outer-Yff-triangle
X(10072) = X(381)-of-2nd Johnson-Yff triangle
X(10072) = inner-Johnson-to-ABC similarity image of X(381)
X(10072) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,2,10056), (1,499,498), (1,3086,499), (1,3582,2), (2,3086,3582), (2,3582,499), (2,10056,498), (4,5563,4317), (11,999,1478), (11,5434,381), (36,497,4302), (56,496,1479), (56,1479,4299), (381,999,5434), (381,5434,1478), (499,10056,2), (1058,7288,35), (3058,5298,3), (3295,5054,4995), (3361,9614,1770), (4293,5274,3583), (4294,5265,7280), (4995,5433,5054), (12951,12952,12589)

X(10073) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO FUHRMANN

X(10073) = R*X(1)-r*X(80)

X(10073) lies on these lines:{1,5}, {4,10052}, {46,5840}, {100,1737}, {214,499}, {497,10051}, {498,6702}, {950,10058}, {1478,5083}, {1479,2800}, {1768,3586}, {1772,6788}, {1898,2771}, {3035,3419}, {3086,6224}, {3583,6001}, {3612,6713}, {6262,10049}, {6263,10048}, {9912,10046}

X(10073) = reflection of X(i) in X(j) for these (i,j): (80,1837)
X(10073) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,10057)

X(10074) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO INNER-GARCIA

X(10074) = R*X(1)-r*X(104)

X(10074) lies on these lines:{1,104}, {3,1317}, {4,5533}, {11,381}, {36,100}, {46,2802}, {56,952}, {57,6264}, {78,214}, {80,1210}, {119,499}, {149,4293}, {153,3086}, {388,8068}, {474,3036}, {498,6713}, {515,10073}, {912,1319}, {956,3035}, {1387,3304}, {1420,6326}, {1479,2829}, {1537,10052}, {1737,5193}, {2787,10069}, {2932,5854}, {2975,5692}, {3361,7993}, {4299,5840}, {4308,9803}, {5010,8275}, {5258,6700}, {5288,6736}, {7091,10042}, {9913,10046}

X(10074) = reflection of X(i) in X(j) for these (i,j): (78,214), (80,1210)
X(10074) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,104,10058), (36,7972,100)

X(10075) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO HUTSON EXTOUCH

X(10075) = R*X(1)-r*X(7160)

X(10075) lies on these lines:{1,5920}, {3086,9874}

X(10075) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7160,10059)

X(10076) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO MIDHEIGHT

X(10076) = R*X(1)-r*X(64)

X(10076) lies on these lines:{1,64}, {3,7355}, {11,5878}, {30,10071}, {35,221}, {36,1498}, {46,3149}, {55,3357}, {56,6000}, {154,7280}, {498,6696}, {499,2883}, {999,6285}, {1069,5663}, {1478,6247}, {1503,4299}, {1853,3585}, {1854,5902}, {2192,5563}, {3086,6225}, {3583,5895}, {4295,7513}, {4302,5894}, {5010,8567}, {5204,6759}, {5656,7288}, {6266,10049}, {6267,10048}, {9914,10046}

X(10076) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,64,10060)

X(10077) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO INNER-NAPOLEON

X(10077) = R*X(1)-r*X(14)

X(10077) lies on these lines:{1,14}, {11,5613}, {36,5474}, {55,6774}, {115,10062}, {497,6773}, {498,6670}, {499,619}, {530,10070}, {531,10072}, {542,613}, {617,3086}, {1478,5479}, {3027,5617}, {3582,5464}, {5353,9113}, {5460,10056}, {6269,10049}, {6271,10048}, {9915,10046}, {9981,10047}

X(10077) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,14,10061)

X(10078) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO OUTER-NAPOLEON

X(10078) = R*X(1)-r*X(13)

X(10078) lies on these lines:{1,13}, {11,5617}, {36,5473}, {55,6771}, {115,10061}, {497,6770}, {498,6669}, {499,618}, {530,10072}, {531,10070}, {542,613}, {616,3086}, {1478,5478}, {3027,5613}, {3582,5463}, {5357,9112}, {5459,10056}, {6268,10049}, {6270,10048}, {9916,10046}, {9982,10047}

X(10078) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,13,10062)

X(10079) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1ST NEUBERG

X(10079) = R*X(1)-r*X(76)

X(10079) lies on these lines:{1,76}, {11,3095}, {12,7697}, {39,499}, {56,2782}, {194,3086}, {262,7741}, {498,3934}, {511,1479}, {538,10072}, {613,732}, {726,1210}, {1478,6248}, {1733,7264}, {3582,7757}, {4294,6194}, {4302,5188}, {5969,10070}, {6272,10049}, {6273,10048}, {6284,9821}, {7288,7709}, {9466,10056}, {9917,10046}, {9983,10047}

X(10079) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,76,10063)

X(10080) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 2ND NEUBERG

X(10080) = R*X(1)-r*X(83)

X(10080) lies on these lines:{1,83}, {11,6287}, {35,9751}, {496,10069}, {498,6704}, {613,732}, {754,10072}, {1478,6249}, {2896,3086}, {6274,10049}, {6275,10048}, {6284,8725}, {9918,10046}

X(10080) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,83,10064)

X(10081) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO ORTHOCENTROIDAL

X(10081) = R*X(1)-r*X(74)

X(10081) lies on these lines:{1,74}, {3,3028}, {11,7728}, {36,110}, {56,5663}, {113,499}, {125,1478}, {146,3086}, {265,7354}, {498,6699}, {541,10072}, {613,2781}, {690,10069}, {1428,9970}, {1479,2777}, {1511,5204}, {1795,2779}, {3448,4293}, {3581,7286}, {4299,10071}, {5298,5655}, {5504,7352}, {5563,7727}, {7355,9934}, {7725,10048}, {7726,10049}, {9919,10046}, {9984,10047}

X(10081) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,74,10065)

X(10082) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO REFLECTION

X(10082) = R*X(1)-r*X(54)

X(10082) lies on these lines:{1,54}, {11,6288}, {36,6286}, {56,1154}, {195,999}, {495,8254}, {498,6689}, {499,1209}, {539,10071}, {1201,1203}, {1478,3574}, {1493,3304}, {2888,3086}, {5563,7356}, {6276,10049}, {6277,10048}, {9920,10046}, {9985,10047}

X(10082) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,54,10066), (36,6286,7691)

X(10083) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO INNER-VECTEN

X(10083) = R*X(1)-r*X(486)

X(10083) lies on these lines:{1,486}, {11,6290}, {487,3086}, {496,613}, {498,6119}, {499,642}, {1478,6251}, {3297,10068}, {6280,10049}, {6281,10048}, {9921,10046}, {9986,10047}

X(10083) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,486,10067), (496,613,10084)

X(10084) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO OUTER-VECTEN

X(10084) = R*X(1)-r*X(485)

X(10084) lies on these lines:{1,485}, {11,6289}, {488,3086}, {496,613}, {498,6118}, {499,641}, {1478,6250}, {3298,10067}, {6278,10049}, {6279,10048}, {8299,9661}, {9922,10046}

X(10084) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,485,10068), (496,613,10083)

X(10085) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO EXTOUCH

X(10085) = R*X(1)-r*X(84)

X(10085) lies on these lines:{1,84}, {3,210}, {4,3338}, {9,3207}, {36,1490}, {40,550}, {46,515}, {56,971}, {57,1837}, {63,4297}, {78,2801}, {90,104}, {165,3916}, {496,1699}, {498,6705}, {499,6260}, {944,1158}, {946,3982}, {956,9943}, {990,1468}, {1697,9845}, {1723,1951}, {1750,3361}, {1770,2096}, {2077,5534}, {2829,10073}, {2932,5531}, {2950,7972}, {3062,7091}, {3304,9856}, {3358,3601}, {3359,5881}, {3576,7330}, {3612,5450}, {3680,7993}, {3811,6909}, {3880,6762}, {3894,7982}, {4293,9799}, {4333,5842}, {4413,9947}, {4860,5806}, {5251,8726}, {5426,7701}, {5437,7989}, {5537,6765}, {5658,7288}, {5696,5732}, {5768,10051}, {5904,6282}, {6257,10049}, {6258,10048}, {6950,7162}, {8069,10075}, {9910,10046}

X(10085) = reflection of X(i) in X(j) for these (i,j): (5531,2932), (5691,1837)
X(10085) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,84,1709), (944,1158,5119), (5787,7354,5691)

X(10086) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO 1ST ANTI-BROCARD

X(10086) = R*X(1)+r*X(99)

X(10086) lies on these lines:{1,99}, {2,10070}, {3,3027}, {12,6321}, {35,98}, {55,2782}, {114,1479}, {115,498}, {147,4294}, {148,3085}, {499,620}, {542,10065}, {543,10054}, {611,5969}, {613,5026}, {618,10077}, {619,10078}, {671,3584}, {1569,2241}, {2482,10072}, {2783,10058}, {2784,4304}, {2794,4302}, {3023,3295}, {3058,8724}, {5697,7970}, {5976,10079}, {6033,6284}, {6319,10040}, {6320,10041}, {8290,10080}, {8782,10038}

X(10086) = reflection of X(i) in X(j) for these (i,j): (10053,55), (10054,10056)
X(10086) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,99,10089), (3,3027,10069)

X(10087) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO INNER-GARCIA

X(10087) = R*X(1)+r*X(100)

X(10087) lies on these lines:{1,88}, {2,5533}, {3,1317}, {10,10073}, {11,498}, {35,104}, {46,5083}, {55,952}, {80,943}, {119,1479}, {145,4996}, {149,3085}, {153,4294}, {499,3035}, {528,954}, {611,9024}, {920,6765}, {1145,3913}, {1387,3303}, {1478,5840}, {1697,6326}, {1737,5853}, {1768,9898}, {2771,10065}, {2783,10053}, {2800,5119}, {2829,4302}, {2932,8071}, {3601,6264}, {4571,4738}, {5172,5844}, {5703,9802}, {5854,8069}, {6174,10072}

X(10087) = reflection of X(i) in X(j) for these (i,j): (10058,55)
X(10087) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,100,10090), (3,1317,10074), (35,7972,104), (149,3085,8068), (214,8715,100)

X(10088) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER-YFF TO ORTHOCENTROIDAL

X(10088) = R*X(1)+r*X(110)

X(10088) lies on these lines:{1,60}, {3,3028}, {12,265}, {35,73}, {55,5663}, {56,1511}, {59,484}, {65,5504}, {113,1479}, {125,498}, {146,4294}, {221,2935}, {495,10066}, {499,5972}, {542,10053}, {611,2854}, {613,6593}, {690,10086}, {2771,10058}, {2777,4302}, {3056,9970}, {3058,5655}, {3085,3448}, {3303,5609}, {3584,9140}, {3746,7727}, {4354,9934}, {5465,10070}, {5642,10072}, {5697,7978}, {6284,7728}, {7732,10040}, {7733,10041}

X(10088) = reflection of X(i) in X(j) for these (i,j): (10065,55)
X(10088) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,110,10091), (3,3028,10081)

X(10089) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO 1ST ANTI-BROCARD

X(10089) = R*X(1)-r*X(99)

X(10089) lies on these lines:{1,99}, {2,10054}, {3,3023}, {11,6321}, {36,98}, {56,2782}, {114,1478}, {115,499}, {147,4293}, {148,3086}, {498,620}, {542,10081}, {543,10070}, {611,5026}, {613,5969}, {618,10061}, {619,10062}, {671,3582}, {999,3027}, {1569,2242}, {1795,2785}, {2482,10056}, {2783,10074}, {2784,4311}, {2794,4299}, {5434,8724}, {5976,10063}, {6033,7354}, {6319,10048}, {6320,10049}, {8290,10064}, {8782,10047}

X(10089) = reflection of X(i) in X(j) for these (i,j): (10069,56), (10070,10072)
X(10089) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,99,10086), (3,3023,10053)

X(10090) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO INNER-GARCIA

X(10090) = R*X(1)-r*X(100)

X(10090) lies on these lines:{1,88}, {2,4996}, {3,11}, {10,10057}, {21,3825}, {35,6940}, {36,80}, {46,2800}, {47,978}, {55,1387}, {56,952}, {57,6326}, {58,2617}, {119,1470}, {149,3086}, {153,4293}, {405,6667}, {411,7280}, {474,498}, {528,2932}, {613,9024}, {651,6127}, {759,3658}, {920,1768}, {956,3036}, {993,6702}, {999,1317}, {1111,6516}, {1145,1376}, {1156,3065}, {1210,10073}, {1385,6797}, {1420,6264}, {1445,2801}, {1465,4351}, {1484,5172}, {1795,3738}, {2475,8070}, {2771,10081}, {2783,10069}, {2787,10089}, {2829,3149}, {3338,5083}, {3361,5531}, {3583,6909}, {3585,6915}, {5010,7676}, {5435,9803}, {5440,5570}, {5563,7972}, {5687,5854}, {6174,10056}, {6906,7741}, {6942,7288}, {6946,7951}

X(10090) = reflection of X(i) in X(j) for these (i,j): (1479,11), (10073,1210), (10074,56)
X(10090) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,100,10087), (1,1054,1772), (3,11,10058), (36,80,104), (149,3086,5533), (474,8071,498), (1470,6911,1478)

X(10091) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TO ORTHOCENTROIDAL

X(10091) = R*X(1)-r*X(110)

X(10091) lies on these lines:{1,60}, {11,265}, {36,74}, {55,1511}, {56,5663}, {113,1478}, {125,499}, {146,4293}, {399,999}, {496,10082}, {498,5972}, {542,10069}, {611,6593}, {613,2854}, {690,10089}, {1469,9970}, {1795,2773}, {2192,2935}, {2771,10074}, {2777,4299}, {3086,3448}, {3304,5609}, {3582,9140}, {4351,9934}, {5434,5655}, {5465,10054}, {5642,10056}, {7354,7728}, {7732,10048}, {7733,10049}

X(10091) = reflection of X(i) in X(j) for these (i,j): (10081,56)
X(10091) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,110,10088), (36,7343,7727), (36,7727,74), (399,999,3028)

X(10092) = PERSPECTOR OF THESE TRIANGLES: OUTER-YFF AND 6TH MIXTILINEAR

X(10092) lies on these lines:{1479,7992}, {1699,10045}, {3062,9614}, {3586,7991}

X(10093) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: INNER-YFF AND INNER-GARCIA

X(10093) lies on these lines:{8,10087}, {35,1158}, {55,355}, {214,10094}, {390,6884}, {498,943}, {920,3811}, {2475,3085}, {4302,6256}, {5703,9782}

X(10094) = CENTER OF INVERSE SIMILITUDE OF THESE TRIANGLES: OUTER-YFF AND INNER-GARCIA

X(10094) lies on these lines:{8,10090}, {36,1158}, {56,355}, {104,499}, {214,10093}, {2475,3086}, {5265,6979}

X(10095) =  X(9)-HATZIPOLAKIS-LOZADA MIDPOINT

Let A'B'C' be the pedal triangle of a point P in the plane of a triangle ABC, and let N denote the nine-point center. Let

Na = N(AB'C'), Nb = N(BC'A'), Nc = N(CA'B')
Ab = N(AB'P), Bc = N(BC'P), Ca = N(CA'P)
Ac = N(AC'P), Ba = N(BA'P), Cb = N(CB'P)
Oa = circumcircle of NaAbAc, Ob = circumcircle of NbBcBa, Oc = circumcircle of NcCaCb

The locus of P such that Oa, Ob, Oc concur is a quintic that passes through X(3), X(5), X(523), and X(2070), given by the barycentric equation

f(a,b,c)xy²z² + f(b,c,a)yz²x² + f(c,a,b)zx²y² = 0, where

f(a,b,c) = a^2*((b^2-c^2)^2*b^4*c^4*x^5-((a^8*c^2-(3*b^2+4*c^2)*a^6*c^2+3*(b^4+b^2*c^2+2*c^4)*a^4*c^2-(b^6-b^4*c^2-3*b^2*c^4+4*c^6)*a^2*c^2+(b^6-4*b^4*c^2-3*b^2*c^4+c^6)*c^4)*y^3+(a^8*b^2-(4*b^2+3*c^2)*a^6*b^2+3*(2*b^4+b^2*c^2+c^4)*a^4*b^2-(4*b^6-3*b^4*c^2-b^2*c^4+c^6)*a^2*b^2+(b^6-3*b^4*c^2-4*b^2*c^4+c^6)*b^4)*z^3)*a^2*y*z+((2*a^6+(-5*b^2-8*c^2)*a^4+(4*b^4-b^2*c^2+10*c^4)*a^2-(b^2+4*c^2)*(b^4-4*b^2*c^2+c^4))*y+(2*a^6+(-8*b^2-5*c^2)*a^4+(10*b^4-b^2*c^2+4*c^4)*a^2-(4*b^2+c^2)*(b^4-4*b^2*c^2+c^4))*z)*a^2*b^2*c^2*y^2*z^2-(a^8+(-6*b^2-6*c^2)*a^6+(12*b^4+17*b^2*c^2+12*c^4)*a^4-(b^2+c^2)*(10*b^4-11*b^2*c^2+10*c^4)*a^2+(3*b^4-2*b^2*c^2+3*c^4)*(b^2-c^2)^2)*b^2*c^2*x^3*y*z-(a^12+(-6*b^2-6*c^2)*a^10+(15*b^4+17*b^2*c^2+15*c^4)*a^8-(b^2+c^2)*(20*b^4-7*b^2*c^2+20*c^4)*a^6+(15*b^8-6*b^6*c^2+18*b^4*c^4-6*b^2*c^6+15*c^8)*a^4-(b^2+c^2)*(6*b^8-19*b^6*c^2+23*b^4*c^4-19*b^2*c^6+6*c^8)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^4)

This curve is here named the Hatzipolakis-Lozada quintic, and if P is a point on it, then the point of concurrence of the three circles, denoted by HL(P), is the Hatzipolakis-Lozada midpoint. Examples include Z(X(3)) = X(140), Z(X(9)) = X(10095), and Z(X(2070)) = X(10096). (Based on notes by Antreas Hatzipolakis and César Lozada, July 29, 2016; see Hyacinthos 23900)

X(10095) lies on these lines: {3,5640}, {4,3521}, {5,51}, {6,156}, {30,5462}, {110,1173}, {140,5446}, {185,3845}, {373,632}, {381,3567}, {389,546}, {403,6746}, {511,3628}, {547,1216}, {548,5892}, {567,3518}, {568,3091}, {1112,1594}, {1614,7545}, {1656,3060}, {2979,5070}, {3090,6243}, {3527,6642}, {3627,9730}, {3843,5890}, {3851,5889}, {3861,6000}, {5066,5907}, {5422,7517}, {5447,6688}, {7529,9777}

X(10095) = midpoint of X(i) and X(j) for these {i,j}: {5,143}, {140,5446}, {389,546}
X(10095) = complement of X(10627)
X(10095) = X(140)-of-orthic-triangle
X(10095) = centroid of X(5) and vertices of the orthic triangle
X(10095) = pedal isogonal conjugate of X(5)

X(10096) =  X(2070)-HATZIPOLAKIS-LOZADA MIDPOINT

X(10096) was, on July 29, 2016, the most recently discovered point on the Euler line. See X(10037) and Hyacinthos 23900)

X(10096) lies on these lines: {2,3}, {1287,5966}

X(10096) = midpoint of X(i) and X(j) for these {i,j}: {5,2070}, {403,7575}
X(10096) = reflection of X(i) in X(j) for these (i,j): (546,403), (2071,3530)
X(10096) = polar-circle inverse of X(6143)

Points associated with Dao circles: X(10097)-X(10103)

X(10097) =  1st (X(3),X(111))-DAO POINT

See the preamble to X(10097). If you have The Geometer's Sketchpad, you can view X(10097).

X(10097) lies on the Jerabek hyperbola and these lines: {3,647}, {4,1499}, {6,512}, {30,2395}, {54,2623}, {66,3566}, {67,690}, {69,525}, {74,111}, {248,878}, {290,671}, {353,3288}, {523,2549}, {526,5505}, {691,2420}, {850,7841}, {895,9517}, {1177,2492}, {1384,6041}, {1648,3143}, {2502,5653}, {3053,8574}, {5968,6787}, {5996,9745}, {6391,8673}, {8617,9210}

X(10097) = reflection of X(2444) in X(9178)
X(10097) = isogonal conjugate of X(4235)
X(10097) = crosspoint of X(671) and X(691)
X(10097) = X(i)-isoconjugate of X(j) for these {i,j}: {1,4235}, {19,5468}, {92,5467}, {162,524}, {187,811}, {468,662}, {648,896}, {823,3292}, {922,6331}, {1783,6629}, {4750,5379}
X(10097) = crossdifference of every pair of points on line X(468)X(524)
X(10097) = orthocenter of X(4)X(6)X(69)

X(10098) =  2nd (X(3),X(111))-DAO POINT

See the preamble to X(10097). The sketch accessible at X(10097) shows X(10098) also.

X(10098) lies on the circumcircle and these lines: {4,2770}, {30,2373}, {74,2393}, {110,7482}, {111,186}, {376,2697}, {378,842}, {403,2374}, {468,9084}, {476,4235}, {925,7472}, {1297,7464}, {1302,7473}, {2074,9061}, {2752,4227}, {4230,9060}, {7476,9058}

X(10098) = Λ(X(4), X(9979))

X(10099) =  1st (X(3),X(105))-DAO POINT

See the preamble to X(10097). If you have The Geometer's Sketchpad, you can view X(10099).

X(10099) lies on the Jerabek hyperbola and these lines: {3,905}, {4,885}, {6,513}, {65,512}, {67,8674}, {69,521}, {71,656}, {72,525}, {73,810}, {74,105}, {290,2481}, {895,1814}, {919,2722}, {927,2714}, {1245,4822}, {5486,9001}

X(10099) = isogonal conjugate of X(4238)
X(10099) = orthocenter of X(3)x(6)X(65)
X(10099) = X(i)-isoconjugate of X(j) for these {i,j}: {1,4238}, {27,2284}, {28,1026}, {29,2283}, {99,2356}, {107,1818}, {110,1861}, {112,3912}, {162,518}, {250,4088}, {643,1876}, {648,672}, {662,5089}, {811,2223}, {883,2299}, {1025,1172}, {1897,3286}, {2254,5379}, {5236,5546}, {6331,9454}

X(10100) =  3rd (X(3),X(105))-DAO POINT

See the preamble to X(10097). The sketch accessible at X(10099) shows X(10100) also.

X(10100) lies on the Jerabek hyperbola and these lines: {3,2836}, {74,3827}, {265,518}, {895,9004}, {2771,4846}, {2778,3426}

X(10100) = isogonal conjugate of X(7469)

X(10101) =  2nd (X(3),X(105))-DAO POINT

See the preamble to X(10097). The sketch accessible at X(10099) shows X(10101) also.

X(10101) lies on the circumcircle and these lines: {4,2752}, {74,3827}, {105,186}, {110,7476}, {376,2694}, {378,2687}, {468,9061}, {476,4238}, {842,7414}, {925,7475}, {1295,7464}, {1304,4244}, {2697,3651}, {2770,4231}, {4246,9060}

X(10101) = X(2775)-cross conjugate of X(4)
X(10101) = X(656)-isoconjugate of X(7469)

X(10102) =  2nd (X(2),X(112))-DAO POINT

See the preamble to X(10097). If you have The Geometer's Sketchpad, you can view X(10102).

X(10102) lies on the circumcircle and these lines: {2,2696}, {23,1296}, {69,6082}, {99,7426}, {110,9027}, {111,9137}, {477,7417}, {523,9084}, {691,1995}, {841,7418}, {935,4232}, {2687,7458}, {2691,4239}

X(10103) =  3rd (X(2),X(112))-DAO POINT

See the preamble to X(10097). The sketch accessible at X(10102) shows X(10103) also.

X(10103) lies on the conic {{A,B,C,X(2),X(6)}} and these lines: {6,2780}, {25,6088}, {111,9137}

X(10104) =  CENTER OF THE 1st-NEUBERG-VAN LAMOEN CIRCLE

Definition (from X(8181)): Suppose that T = TaTbTc is a triangle in the plane of ABC that is perspective to ABC. Let P be the perspector. If the circumcenters of PBTc, PCTa, PATb, PCTb, PATc, PBTa lie on a circle, that circle is the T-van Lamoen circle. The radius-squared of the 1st-Neuberg-van-Lamoen circle is (SA SB SC - S^2 SW) (3 S^2 - SW^2) (S^2 + SW^2) / (64 S^6). (Peter Moses, August 9, 2016)

If you have The Geometer's Sketchpad, you can view X(10104).

X(10104) lies on these lines: {2,3398}, {3,76}, {4,2080}, {5,32}, {26,157}, {30,5171}, {69,2456}, {83,1656}, {114,7749}, {140,141}, {187,6248}, {262,6179}, {384,7697}, {385,3095}, {511,7780}, {549,7801}, {550,8722}, {575,6683}, {626,6036}, {631,7836}, {1352,1691}, {1353,5034}, {3090,7787}, {3526,7868}, {3628,7808}, {4027,7907}, {5054,7870}, {5965,7764}, {5999,9821}, {6033,9863}, {6055,7810}, {6308,9756}, {7751,9737}, {7781,9734}

X(10104) = midpoint of X(7751) and X(9737)
X(10104) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7793,2080), (98,1078,3), (182,7815,140), (6177,6178,182)

X(10105) =  28th HATZIPOLAKIS-MONTESDEOCA POINT

Let I be the incenter of a triangle ABC, and
A'B'C' = intouch triangle (the pedal triangle of I)
A''B''C'' = cevian triangle of I
Ab = orthogonal projection of A'' on IB, and define Bc and Ca cyclically
Ac = orthogonal projection of A'' on IC, and define Ba and Cb cyclically
A'b = orthogonal projection of A'' on IB', and define B'c and C'a cyclically
A'c = orthogonal projection of A'' on IC', and define B'a and C'b cyclically
(Nab) = nine-point cricle of A''AbA'b, and define (Nbc) and (Nca) cyclically
(Nac) = nine-point cricle of A''AcA'c, and define (Nba) and (Ncb) cyclically
Ra = radical axis of (Nab) and (Nac) = perpendicular bisector of segment NabNac.

The lines Ra, Rb, Rc concur in X(10105). Also, the parallels to Ra, Rb, Rc through A', B', C, respectively, concur in X(942); and the parallels to Ra, Rb, Rc through A'', B'', C'', respectively, concur in X(500). (Antreas Hatzipolakis and Angel Montesdeoca, August 8, 2016; see Hyacinthos 23972)

X(10105) lies on these lines: {73,500}, {511,9940}

X(10106) =  29th HATZIPOLAKIS-MONTESDEOCA POINT

Let I be the incenter of a triangle ABC, and
A'B'C' = intouch triangle (the pedal triangle of I)
H' = X(4)-of-A'B'C'
Ab = orthogonal projection of A on H'B', and define Bc and Ca cyclically
Ac = orthogonal projection of A on H'C', and define Ba and Cb cyclically
La = Euler line of AAbAc, and define Lb and Lc cyclically
Pa = line through A'' parallel to La, and define Pb and Pc cyclically

The lines La, Lb, Lc concur in X(10106). Let

Qa = line through A parallel to La, and define Qb and Qc cyclically
Ra = line through A' parallel to La, and define Rb and Rc cyclically

The lines La, Lb, Lc concur in X(5836), the midpoint of X(8) and X(65). The lines Qa, Qb, Qc concur in X(8), and the lines Ra, Rb, Rc concur in X(145). (Antreas Hatzipolakis and Angel Montesdeoca, August 9, 2016; see Hyacinthos 23990)

X(10106) lies on these lines:
{1,4}, {2,1420}, {3,4311}, {4,33}, {7,145}, {8,57}, {10,56}, {12,1125}, {20,1697}, {21,2078}, {30,9957}, {36,6684}, {40,4293}, {55,4297}, {65,519}, {104,6705} .

X(10106) = reflection of X(950) in X(1)
X(10106) = anticomplement of X(5795)
X(10106) = X(185)-of-intouch-triangle
X(10106) = excentral-to-intouch similarity image of X(8)

X(10107) =  30th HATZIPOLAKIS-MONTESDEOCA POINT

Let I be the incenter of a triangle ABC, and
A'B'C' = intouch triangle (the pedal triangle of I)
H' = X(4)-of-A'B'C'
Ab = orthogonal projection of A on H'B', and define Bc and Ca cyclically
Ac = orthogonal projection of A on H'C', and define Ba and Cb cyclically
La = Euler line of AAbAc, and define Lb and Lc cyclically
Ia = excenter of ABC, and define Ib and Ic cyclically
Pa = orthogonal projection of Ia on BC, and define Pb and Pc cyclically
A* = midpoint of AbAc, and define B* and C* cyclically
Qa = line through A* parallel to La, and define Qb and Qc cyclically

The lines Qa, Qb, Qc concur in X(10107). Let

Qa = line through Ia parallel to La, and define Qb and Qc cyclically

The lines Qa, Qb, Qc concur in X(2136), which is the X(145)-Ceva conjugate of X(1). (Antreas Hatzipolakis and Angel Montesdeoca, August 10, 2016; see Hyacinthos 23998)

X(10107) lies on these lines:
{1, 4004}, {2, 3922}, {7, 8}, {10, 3838}, {21, 5183}, {140, 517}, {226, 8256}, {354, 3623}, {528, 6738}, {758, 4662}, {942, 3244}, {946, 3847}, {958, 2093}, {960, 1698}, {999, 8668}, {1001, 7991}, {1155, 5303}, {1159, 3811}, {1357, 4767}, {1376, 3340}, {1836, 5554}, {2098, 3306}, {2802, 5045}, {2886, 4848}, {3057, 3622}, {3617, 3962}, {3626, 4757}, {3633, 5902}, {3649, 6735}, {3679, 4018}, {3698, 3740}, {3816, 4301}, {3826, 5837}, {3844, 5835}, {3872, 5221}, {3873, 3893}, {3876, 4731}, {3918, 5044}, {4002, 5692}, {4127, 4745}, {4711, 5904}, {5048, 5253}, {5439, 5697}, {5883, 9957}, {6928, 7686}

X(10107) = midpoint of X(i) and X(j) for these {i,j}: {65, 5836}, {960, 5903}, {3626, 4757}

X(10108) =  31st HATZIPOLAKIS-MONTESDEOCA POINT

Let I be the incenter of a triangle ABC, and
A'B'C' = cevian triangle of X(I)
Ab = orthogonal projection of A' on BB', and define Bc and Ca cyclically
Ac = orthogonal projection of A' on CC', and define Ba and Cb cyclically
Abc = orthogonal projection of Ab on CC', and define Bca and Cab cyclically
Acb = orthogonal projection of Ac on BB', and define Bac and Cba cyclically
La = Euler line of IAbcAcb, and define Lb and Lc cyclically

The lines La, Lb, Lc concur in X(10108). (Antreas Hatzipolakis and Angel Montesdeoca, August 10, 2016; see Hyacinthos 23949)

X(10108) lies on these lines:
{500,1066}, {511,5045}, {524,5044}, {3945,5752}

X(10109) =  MIDPOINT OF X(5)X(547)

Let P be a point in the plane of a triangle ABC, and
Nab = nine-point center of ABA', and define Nbc and Nca cyclically
Nac = nine-point center of ACA', and define Nba and Ncb cyclically
Ma = midpoint of NabNac, and define Mb and Mc cyclically

The locus of P such that the centroid of MaMbMc lies on the Euler line is given by the barycentric equation

(SA^2+8*SB*SC-3*S^2)*x*(y^2-z^ 2) + (SB^2+8*SC*SA-3*S^2)*y*(z^2-x^ 2) + (SC^2+8*SA*SB-3*S^2)*z*(x^2-y^ 2) = 0.

Only one point X(n), for 1 <= n <= 10108, lies on the cubic, and that is the centroid, X(2). For P = X(2), the centroid of MaMbMc is X(10109). (Antreas Hatzipolakis and César Lozada, August 10, 2016; see Hyacinthos 23955)

X(10109) lies on these lines:
{2,3}, {1327,8252}, {1328,8253}, {1587,6495}, {1588,6494}, {3582,3614}, {3584,7173}, {3655,7989}, {3656,7988}, {3828,9955}, {4669,5844}, {4677,8227}, {4745,9956}, {5355,7603}, {5663,6688}

X(10109) = midpoint of X(i),X(j) for these {i,j}: {2,5066}, {5,547}, {140,381}, {376,3853}, {546,549}, {3828,9955}
X(10109) = reflection of X(i) in X(j) for these (i,j): (3628,547), (3860,5066), (3861,381)
X(10109) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5,5066), (2,381,8703), (2,3545,3830), (2,3830,549), (2,8703,140),

X(10110) =  MIDPOINT OF X(5)X(5446)

Let P be a point in the plane of a triangle ABC, and
A'B'C' = pedal triangle of P
Ba = orthogonal projection of B' on AB, and define Cb and Ac cyclically
Ca = orthogonal projection of C' on AC, and define Ab and Bc cyclically
Ea = Euler line of ABaCa, and define Eb and Ec cyclically
La = locus of P such that Ea is perpendicular to BC, and define Lb and Lc cyclically

The point X(5) lies on all three lines, La, Lb, Lc. The point of concurrence of Ea, Eb, Ec is X(10110). (Antreas Hatzipolakis and César Lozada, August 10, 2016; see Hyacinthos 238320)

A second construction of X(10110), this time as the center of a circle, follows:

Let A'B'C' be the pedal triangle of the orthocenter, H, of a triangle ABC, and let
Ab = orthogonal projecton of A' on HB, and define Bc and Ca cyclically
Ab = orthogonal projecton of A' on HB, and define Bc and Ca cyclically
Abc = midpoint of AbC', and define Bca and Cab cyclically
Acb = midpoint of AcB', and define Bac and Cba cyclically
The six points, Abc, Bca, Cab, AcB, Bac, Cba, lie on a circle, of which the center is X(10110). (Antreas Hatzipolakis and Peter Moses Lozada, August 13, 2016; see Hyacinthos 24022)

X(10110) lies on these lines:
{3,5943}, {4,51}, {5,141}, {6,1598}, {20,5640}, {25,578}, {30,5462}, {49,7545), {52,381}, {54,1495}, {64,3531}, {68,3818}, {140,6688}, {143,546}, {155,576}, {181,3073}, {182,7387}, {373,631}, {382,9730}, {403,3574}, {428,6146}, {517,5795}, {550,5892}, {568,3843}, {569,7517}, {575,7530}, {970,3560}, {973,1112}, {1092,1995}, {1154,3850}, {1173,1199}, {1181,5198}, {1597,3357}, {1656,3819}, {1843,3089}, {1864,1871}, {1872,2262}, {2818,7686}, {2979,5056}, {3060,3091}, {3072,3271}, {3090,3917}, {3098,7393}, {3627,5946}, {3628,5447}, {3832,5889}, {3845,6102}, {3851,5891}, {3858,5876}, {3861,5663}, {5067,5650}, {5071,7999}, {5752,6913}, {6403,6622}, {6530,6750}, {6995,9833}, {7486,7998}, {7529,9306}

X(10110) = midpoint of X(i),X(j) for these {i,j}: {4,389}, {5,5446}, {52,5907}, {143,546}, {1112,7687}, {5480,9969}
X(10110) = reflection of X(i) in X(j) for these (i,j): (5447,3628), (9729,5462), (11793,5)
X(10110) = X(1125)-of-orthic-triangle if ABC is acute
X(10110) = X(10)-of-2nd-anti-Conway-triangle if ABC is acute
X(10110) = X(12512)-of-2nd-Euler-triangle if ABC is acute
X(10110) = crosssum of X(3) and X(140)
X(10110) = crosspoint of X(4) and X(1173)
X(10110) = X(1125)-of-orthic-triangle if ABC is acute
X(10110) = center of conic that is the locus of centers of conics passing through X(4) and the vertices of the orthic triangle
X(10110) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,51,389), (4,1093,8887), (4,3567,185), (4,9781,51), (6,1598,6759), (51,185,3567), (52,381,5907), (185,3567,389), (1597,9786,3357), (1598,3527,6), (3060,3091,5562), (3851,6243,5891), (5198,9777,1181)

X(10111) =  HATZIPOLAKIS-MOSES NINE-POINT-CIRCLES POINT

Let A'B'C' be the orthic triangle of a triangle ABC, and
O = circumcenter of ABC
Ab = reflection of A' in BO, and define Bc and Ca cyclically
Ac = reflection of A' in CO, and define Ba and Cb cyclically
(Na) = nine-point circle of A'AbAc, and define (Nb) and (Nc) cyclically
Na = nine-point center of A'AbAc, and define Nb and Nc cyclically
The triangle NaNbNc is here named the Hyacinth triangle.

The circles (Na), (Nb), (Nc) concur in X(10111). (Antreas Hatzipolakis and Peter Moses, August 11, 2016. See Hyacinthos 24008. For a related general theorem about isogonal conjugate pairs of points, see Hyacinthos 24006).

X(265) = ABC-to-NaNbNC orthologic center
X(6102) = NaNbNc-to-ABC orthologic center
X(1147) = perspector of NaNbNc and medial triangle
X(1511) = ABC-to-NaNbNc orthologic center
X(6102) = NaNbNc-to-ABC orthologic center
X(113) = A'B'C'-to-ABC orthologic center
X(10112) = NaNbNc-to-orthic orthologic center
X(2931) = tangential-triangle-to-NaNbNc orthologic center
X(10112) = NaNbNc-to-tangential-triangle orthologic center
X(10113) = Euler-triangle-to-NaNbNc orthologic center
X(6102) = NaNbNc-to-Euler-triangle orthologic center
X(10114) = HMT-NaNbNc orthologic center, where HMT = Hatzipolakis-Moses triangle
X(10115) = NaNbNc-to-HMT orthologic center
X(125) = orthic-triangle-to-NaNbNc paralogic center
X(10116) = NaNbNc-to-orthic-triangle paralogic center
X(10117) = tangential-triangle-to-NaNbNc paralogic center
X(10116) = NaNbNc-to-tangential-triangle paralogic center
X(10118) = intangents-triangle-to NaNbNc paralogic center
X(10116) = NaNbNc-to-intangents-triangle paralogic center
X(10119) = extangents-triangle-to-NaNbNc paralogic center
X(10115) = NaNbNc-to-extangents-triangle paralogic center

Barycentrics for the A-vertex of NaNbNc:

-2 a^4 (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^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^2-3 a^4 b^4 c^2-3 a^2 b^6 c^2+4 b^8 c^2+2 a^6 c^4-5 a^4 b^2 c^4-4 a^2 b^4 c^4-7 b^6 c^4+2 a^4 c^6+7 a^2 b^2 c^6+7 b^4 c^6-3 a^2 c^8-4 b^2 c^8+c^10 :

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^2-5 a^4 b^4 c^2+7 a^2 b^6 c^2-4 b^8 c^2+4 a^6 c^4-3 a^4 b^2 c^4-4 a^2 b^4 c^4+7 b^6 c^4-4 a^4 c^6-3 a^2 b^2 c^6-7 b^4 c^6+3 a^2 c^8+4 b^2 c^8-c^10

X(10111) lies on these lines:
{6,13}, {49,5972}, {125,1147}, {539,6699}, {1885,5663}, {1899,5504}, {3448,3541}

X(10112) =  HYACINTH-TO-ORTHIC TRIANGLES ORTHOLOGIC CENTER

X(10112) is also the tangential-to-Hyacinth triangles orthologic center. See X(10111).

X(10112) lies on these lines:
{4,542}, {5,539}, {68,578}, {182,7383}, {195,265}, {511,6146}, {567,1209}, {575,7399}, {1147,5972}, {1994,3574}, {3564,5907}, {5448,7687}, {5562,5965}, {6193,9306}, {6823,8550}

X(10112) = X(104)-of-1st-Hyacinth-triangle if ABC is acute

X(10113) =  EULER-TO-HYACINTH TRIANGLES ORTHOLOGIC CENTER

See X(10111).

Let Na be the reflection of X(5) in the A-altitude, and define Nb and Nc cyclically. NaNbNc is inversely similar to ABC, with similitude center X(195), and X(10113) = X(186)-of-NaNbNc. (See Hyacinthos #21522, 2/11/2013, Antreas Hatzipolakis)

X(10113) lies on these lines:
{4,94}, {5,1511}, {30,125}, {74,382}, {110,381}, {113,137}, {399,3843}, {542,1353}, {549,6723}, {550,6699}, {1154,7723}, {2771,6246}, {2777,3627}, {2854,3818}, {2931,7526}, {3024,3583}, {3028,3585}, {3098,6698}, {3830,9140}, {3839,5655}, {5066,5642}, {5876,9927}, {5946,7706}, {9815,9826}

X(10113) = polar-circle-inverse of X(7722)
X(10113) = X(6265)-of-orthic-triangle if ABC is acute

X(10114) =  HATZIPOLAKIS-MOSES-TO-HYACINTH TRIANGLES ORTHOLOGIC CENTER

See X(10111).

X(10114) lies on these lines:
{6,13}, {54,125}, {389,6153}, {539,1511}, {578,3448}, {2777,6241}, {2929,5898}, {5449,5972}

X(10115) =  HYACINTH-TO-HATZIPOLAKIS-MOSES-TRIANGLES ORTHOLOGIC CENTER

See X(10111).

X(10115) lies on these lines:
{51,6288}, {52,54}, {140,389}, {195,568}, {539,973}, {1209,3580}, {1493,7575}, {2888,3567}, {3574,5576}, {7691,9730}

X(10115) = midpoint of X(52) and X(54)
X(10115) = reflection of X(i) and X(j) for these (i,j): (1209, 5462), (1216, 6689), (6153, 973)
X(10115) = X(11604)-of-1st-Hyacinth-triangle if ABC is acute

X(10116) =  HYACINTH-TO-ORTHIC-TRIANGLES PARALOGIC CENTER

See X(10111).

X(10116) lies on these lines:
{2,9705}, {3,539}, {5,542}, {49,125}, {54,3448}, {68,1176}, {184,5449}, {1147,1899}, {1181,9927}, {1209,5012}, {1216,3564}, {1503,5446}, {5965,6101}, {6143,9140}, {6643,9936}

X(10116) = {X(9140,X(9706)}-harmonic conjugate of X(6143)
X(10116) = Hyacinth-to-orthic-triangles paralogic center
X(10116) = orthic-to-Hyacinth-triangles paralogic center
X(10116) = Hyacinth-to-tangential-triangles paralogic center
X(10116) = Hyacinth-to-extangents-triangles paralogic center
X(10116) = X(100)-of-1st-Hyacinth-triangle if ABC is acute

X(10117) =  TANGENTIAL-TO-HYACINTH-TRIANGLES PARALOGIC CENTER

See X(10111).

X(10117) lies on the Walsmith rectangular hyperbola and these lines: {3,113}, {6,1112}, {22,110}, {23,1503}, {24,64}, {25,125}, {26,1498}, {40,2778}, {141,2892}, {146,2883}, {159,2930}, {161,542}, {206,2916}, {221,3028}, {265,7517}, {378,7699}, {399,2917}, {924,3447}, {974,9786}, {1181,1986}, {1204,2929}, {1205,1974}, {1539,7526}, {1598,7687}, {1614,7731}, {1658,5878}, {1993,3047}, {1995,7703}, {2070,6000}, {2079,3569}, {2192,3024}, {2854,9924}, {2948,9591}, {3031,9571}, {3043,9707}, {3515,9914}, {3518,6247}, {4549,7502}, {5020,6723}, {5656,7556}, {6001,9625}, {6642,6699}, {8276,8994}, {9590,9904}

X(10117) = midpoint of X(3) and X(9919)
X(10117) = reflection of X(i) in X(j) for these (i,j): (6, 1177), (64, 74), (146, 2883), (399, 6759}, {1498, 9934), (2892, 141), (2930, 159), (2931, 26), (2935, 3), (32125, 468)
X(10117) = crosssum of de Longchamps circle intercepts of Euler line
X(10117) = circumcircle-inverse of X(122)
X(10117) = X(100)-of-tangential-triangle if ABC is acute
X(10117) = X(525)-Ceva conjugate of X(6)
X(10117) = X(122)-vertex conjugate of X(9033)
X(10117) = antipode of X(32125) in Walsmith rectangular hyperbola
X(10117) = orthocenter of X(6)X(2931)X(3569)
X(10117) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1619,9909,161), (2937,6759,2917), (10681,10682,6), (13287,13288,6)
X(10117) = X(i)-line conjugate of X(j) for these (i,j): (3,5972), (113,5972), (122,5972), (1624,5972), (2777,5972), (2935,5972), (3184,5972), (5895,5972), (5925,5972), (7728,5972), (9919,5972)

X(10118) =  INTANGENTS-TO-HYACINTH-TRIANGLES PARALOGIC CENTER

See X(10111).

X(10118) lies on these lines:
{1,2777}, {33,125}, {56,2935}, {65,74}, {73,9627}, {110,3100}, {113,1062}, {146,9538}, {399,9641}, {1040,5972}, {1177,2330}, {1469,2781}, {1503,5160}, {2931,9645}, {2948,9576}, {3028,7355}, {3031,9550}, {3043,9638}, {3047,9637}, {3295,9919}, {3448,9539}, {3465,4551}, {4354,9934}, {5663,6285}, {6723,9817}, {7356,7727}, {9577,9904}, {9642,10060}

X(10118) = reflection of X(7355) in X(3028)
X(10118) = X(100)-of-intangents triangle if ABC is acute

X(10119) =  EXTANGENTS-TO-HYACINTH-TRIANGLES PARALOGIC CENTER

See X(10111).

X(10119) lies on these lines:
{19,125}, {40,2777}, {65,2906}, {74,6197}, {110,3101}, {113,8251}, {146,9537}, {2778,7957}, {2935,5584}, {2948,9573}, {3448,9536}, {5663,6254}, {6723,9816}, {9572,9904}

X(10119) = X(100)-of extangents triangle if ABC is acute

X(10120) =  COMPLEMENT OF COMPLEMENT OF X(10121)

Let O by the circumcenter of a triangle ABC, and let
Ma = midpoint of OA, and define Mb and Mc cyclically
Aa = orthogonal projection of Ma on NA, and define Bb and Cc cyclically
Ba = orthogonal projection of Mb on NA, and define Cb and Ac cyclically
Ca = orthogonal projection of Mc on NA, and define Ab and Bc cyclically
Ea = Euler line of AaAbAc, and define Eb and Ec cyclically
Pa = line through A parallel to E1

Then Ea, Eb, Ec concur in X(10120), and Pa, Pb, Pc concur on the circumcircle in X(10121). (Antreas Hatzipolakis and Peter Moses, August 14, 2016. See Hyacinthos 24032.)

X(10120) lies on the nine-point circle of the medial trtiangle and the line {2, 10121}

X(10121) =  ANTICOMPLEMENT OF ANTICOMPLEMENT OF X(10120)

For a construction and reference, see X(10120).

X(10121) lies on the circumcircle and the line {2,10120}

X(10122) =  32nd HATZIPOLAKIS-MONTESDEOCA POINT

Let ABC be a triangle, and let
A'b = orthogonal projection of A on the external bisector of angle ABC
A'c = orthogonal projection of A on the external bisector of angle ACB
Ea = Euler line of AA'bA'c, and define Eb and Ec cyclically
Ia = A-excenter of ABC, and define Ib and Ic cyclically
A'B'C' = intouch triangle (the pedal triangle of the incenter)
A'' = orthogonal projection of IA on B'C', and define B'' and C'' cyclically
Pa = line through A'' parallel to Ea, and define Pb and Pc cyclically
A''' = orthogonal projections of A' on IbIc, and define B''' and C''' cyclically
Qa = line through A''' parallel to Ea, and define Qb and Qc cyclically

The lines Pa, Pb, Pc concur in X(10122). The lines Qa, Qb, Qc concur in X(10123). The lines Ea, Eb, Ec concur in X(442). (Antreas Hatzipolakis and Angel Montesdeoca, August 14, 2016. See Hyacinthos 24025 and href="http://www.hyacinthos.epizy.com/message.php?msg=24015" Hyacinthos 24015.)

X(10122) lies on these lines:
{1,21}, {7,79}, {20,5441}, {27,1844}, {30,553}, {46,7675}, {57,3651}, {65,4304}, {72,5325}, {142,442}, {226,6841}, {354,946}, {377,5883}, {938,2475}, {1012,5884}, {1100,3284}, {1387,2771}, {1697,8000}, {1699,9960}, {1729,2280}, {2646,5427}, {3085,5686}, {3336,7411}, {3555,5837}, {3584,4015}, {3648,9965}, {3833,4197}, {4313,5903}, {5044,6675}, {5249,6701}, {5273,5904}, {5570,6744}, {5719,10021}, {5735,7671}, {5836,8261}

X(10122) = midpoint of X(3647) and X(3874)

X(10123) =  33rd HATZIPOLAKIS-MONTESDEOCA POINT

For a construction and references, see X(10122).

X(10123) lies on these lines:
{4, 1768}, {9, 3648}, {21, 142}, {30, 553}, {35, 79}, {63, 2475}, {191, 3474}, {442, 1155}, {516, 3649}, {1708, 7701}, {1836, 5248}, {3874, 7354}, {3911, 6841}, {5122, 10021}, {5325, 6175}, {5441, 5557}

X(10123) = midpoint of X(79) and X(1770)

X(10124) =  34th HATZIPOLAKIS-MONTESDEOCA POINT

Let O be the circumcenter and N the nine-point center of a triangle ABC. Let
Ma = midpoint of OA, and define Mb and Mc cyclically
Aa = orthogonal projection of Ma on NA, and define Bb and Cc cyclically
Ba = orthogonal projection of Mb on NA, and define Cb and Ac cyclically
Ca = orthogonal projection of Mc on NA, and define Cb and Ac cyclically
M1 = midpoint of NA, and define M2 and M3 cyclically
A1 = orthogonal projection of M1 on OA, and define A2 and A3 cyclically
B1 = orthogonal projection of M2 on OA, and define B2 and B3 cyclically
C1 = orthogonal projection of M3 on OA, and define C2 and C3 cyclically
Oa = circumcenter of AaAbAc, and define Ob and Oc cyclically
O1 = circumcenter of A1A2A3, and define O2 and O3 cyclically
La = Euler line of OaO2O3, and define Lb and Lc cyclically
L1 = Euler line of O1ObOC, and define L2 and L3 cyclically

Then L1, L2, L3 concur in X(10124), which lies on the Euler line, and La, Lb, Lc concur in X(140). (Antreas Hatzipolakis and Angel Montesdeoca, August 14, 2016. See Hyacinthos 24036.)

X(10124) lies on these lines:
{2,3}, {551,5844}, {952,3828}, {1698,3653}, {3054,5309}, {3055,7753}, {3582,7294}, {3584,5326}, {3624,3654}, {5418,6470}, {5420,6471}, {5650,5946}, {7749,9300}

X(10124) = {X(2),X(3)}-harmonic conjugate of X(15699)

X(10125) =  1st HATZIPOLAKIS-MOSES-EULER POINT

Suppose that P is a point in the plane of a triangle ABC, and let
Oa = circumcenters of PBC
Oab = orthogonal projection of Oa on AC
Oac = orthogonal projection of Oa on AB
O'a = circumcenter of OaOabOac, and define O'b and O'c cyclically
If P = X(3), then the circumcenter of O'aO'bO'c is X(10125), which is on the Euler line of ABC. The circumcircles of OaOabOac, ObObcOba, OcOcaOcb, ABC concur in X(110). (Antreas Hatzipolakis and Peter Moses, August 17, 2016. See Hyacinthos 24042.)

Let La be the polar of X(4) wrt the circle centered at A and passing through X(5). Define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = Lb∩Lc, and define B' and C'cyclically. A'B'C' is homothetic to ABC, and X(10125) = X(5) of A'B'C'. (Randy Hutson, September 14, 2016)

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(10125) = X(5)-of-O_AO_BO_C. (Randy Hutson, June 7, 2019)

X(10125) lies on these lines: {2,3}, {125,5944}, {5946,8254}

X(10126) =  2nd HATZIPOLAKIS-MOSES-EULER POINT

Suppose that P is a point in the plane of a triangle ABC. In the construction at X(10125), if P = X(5), then the circumcenter of O'aO'bO'c is X(10126), which is on the Euler line of ABC. The circumcircles of OaOabOac, ObObcOba, OcOcaOcb, ABC concur in X(110). (Antreas Hatzipolakis and Peter Moses, August 17, 2016. See Hyacinthos 24042.

X(10126) lies on these lines: {2,3}, {1209,6592}, {6150,8254}

X(10127) =  3rd HATZIPOLAKIS-MOSES-EULER POINT

Let A'B'C' be the pedal triangle of the nine-point center, N, and let
Ab = orthogonal projection of A' on NB'
Ac = orthogonal projection of A' on NC'
Ha = orthocenter of A'AbAc, and cyclically for Hb and Hc
(Antreas Hatzipolakis and Peter Moses, August 17, 2016. See Hyacinthos 24047.

X(10127) lies on these lines: {2,3}, {155,9815}, {539,5462}, {542,9822}, {1503,5892}, {3564,59 46}, {5421,6128}

X(10128) =  4th HATZIPOLAKIS-MOSES-EULER POINT

Let A'B'C' be the pedal triangle of the nine-point center, N, and let
Ab = orthogonal projection of A' on NB'
Ac = orthogonal projection of A' on NC'
M1 = midpoint of AbAC, and define M2 and M3 cyclically
Ma = midpoint of A2A3, and define Mb and Mc cyclically
X(10128) = centroid of M1M2M3
X(10129) = centroid of MaMbMc
Both points lie on the Euler line. (Antreas Hatzipolakis and Peter Moses, August 18, 2016. See Hyacinthos 24074).

X(10128) lies on these lines:
{2,3}, {524,9822}, {1503,6688}, {2548,8770}, {3058,9817}, {3564,5943}, {3618,8780}, {7583,8855}, {7584,8854}, {8584,9813}

ATFF points of pairs of triangles: X(10129)-X(10136)

X(10129) =  ATFF(ABC, 3rd EULER TRIANGLE)

See the preamble to X(10129).

X(10129) lies on these lines:
{1,10031}, {2,1155}, {4,1385}, {10,908}, {21,3624}, {46,7504}, {65,5141}, {85,693}, {100,5219}, {145,3485}, {226,3873}, {946,3890}, {997,6175}, {1005,5284}, {1156,6173}, {1621,1699}, {1768,3306}, {2140,3835}, {2886,3681}, {2975,9612}, {3120,4850}, {3434,5226}, {3585,3897}, {3754,7705}, {3812,5154}, {3817,5249}, {3822,3877}, {3824,5550}, {4004,5887}, {4295,6933}, {4861,9654}, {5253,5450}, {5303,9579}, {5603,6982}, {5901,7704}, {6261,7548}, {6361,6825}, {6828,9948}, {6831,9961}

X(10129) = {X(3485),X(6871)}-harmonic conjugate of X(5086)

X(10130) =  ATFF(ABC, CIRCUMMEDIAL TRIANGLE)

Let A'B'C' be the circummedial triangle. Let A" be the trilinear pole, wrt A'B'C', of line BC, and define B" and C" cyclically. Then A", B", C" are collinear on line X(647)X(4108), and the the lines AA", BB", CC" concur in X(10130). (Randy Hutson, August 21, 2016)

X(10130) lies on these lines:
{2,32}, {23,3934}, {67,110}, {111,308}, {183,9465}, {574,9464}, {620,5987}, {625,7570}, {699,9102}, {733,9066}, {827,6325}, {1180,7754}, {3266,7824}, {3734,7492}, {5169,7761}, {5189,7830}, {5354,7780}, {6636,7816}, {6676,8793}, {7783,8024}

X(10131) =  ATFF(ABC, 6th BROCARD TRIANGLE)

See the preamble to X(10129).

X(10131) lies on these lines:
{3,4027}, {20,2456}, {32,7757}, {39,3407}, {76,8150}, {83,7748}, {98,7907}, {182,384}, {194,1691}, {385,6309}, {1078,5569}, {2558,3414}, {2559,3413}, {2782,3406}, {3398,3552}, {5038,7738}, {5182,7833}, {6337,7793}, {7808,7923}

X(10131) = cevapoint of X(574) and X(599)
X(10131) = X(i)-isoconjugate of X(j) for these {i,j}: {38,1383}, {598,1964}
X(10131) = {X(2),{X(1799)}-harmonic conjugate of X(251)
X(10131) = center of inverse similitude of ABC and 6th Brocard triangle
X(10131) = homothetic center of 6th Brocard and 6th anti-Brocard triangles

X(10132) =  ATFF(ABC, LUCAS CENTRAL TRIANGLE)

See X(10133) and the preamble to X(10129).

X(10132) lies on the cubic K171 and these lines:
{2,6222}, {3,49}, {6,3156}, {22,9732}, {25,371}, {31,2067}, {48,2066}, {51,3311}, {110,1599}, {154,1151}, {182,1584}, {216,8908}, {487,7494}, {488,8222}, {1184,6424}, {1495,6221}, {1578,6466}, {1583,9306}, {1589,6776}, {1600,5012}, {1993,9733}, {2351,8825}, {3068,5200}, {6419,9777}, {8576,8770}

X(10132) = X(i)-Ceva conjugate of X(j) for these (i,j): (371,6), (3068,6423)
X(10132) = crosspoint of X(i) and X(j) for these {i,j}: {3,6415}, {488,3068}
X(10132) = X(i)-isoconjugate of X(j) for these {i,j}: {19,5490}, {75,8948}, {92,493}
X(10132) = pole wrt circumcircle of trilinear polar of X(371)
X(10132) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,184,10133), (3,3167,5408), (154,1151,3155), (371,8854,493), (3796,5407,3), (6413,8911,6)

X(10133) =  ATFF(ABC, LUCAS(-1) CENTRAL TRIANGLE)

See X(10132) and the preamble to X(10129). (Contributed by Randy Hutson, August 21, 2016.)

X(10133) lies on the cubic K171 and these lines:
{2,6290}, {3,49}, {6,3155}, {22,9733}, {25,372}, {31,6502}, {48,5414}, {51,3312}, {110,1600}, {154,1152}, {182,1583}, {487,8223}, {488,7494}, {1184,6423}, {1495,6398}, {1579,6465}, {1584,9306}, {1586,8982}, {1590,6776}, {1599,5012}, {1993,9732}, {3564,8964}, {5200,6460}, {6420,9777}, {8577,8770}, {8909,8961}

X(10133) = X(i)-Ceva conjugate of X(j) for these (i,j): (25, 10132), (372, 6), (3069, 6424)
X(10133) = X(i)-isoconjugate of X(j) for these {i,j}: {19,5491}, {75,8946}, {92,494}
X(10133) = crosspoint of X(i) and X(j) for these {i,j}: {3,6416}, {487,3069}
X(10133) = pole wrt circumcircle of trilinear polar of X(372)
X(10133) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,184,10133), (3,3167,5409), (154,1152,3156), (372,8855,494), (3796,5406,3).

X(10134) =  ATFF(ABC, INNER SODDY TRIANGLE)

X(10134) lies on these lines: {57,482}, {176,9778}, {1373,1699}

X(10134) = crosspoint of X(7) and X(176)

X(10135) =  ATFF(ABC, OUTER SODDY TRIANGLE)

X(10135) lies on these lines: {57,481}, {175,9778}, {1374,1699}

X(10135) = crosspoint of X(7) and X(175)

X(10136) =  ATFF(INNER SODDY, OUTER SODDY TRIANGLE)

X(10136) lies on these lines: {7,1699}, {103,5542}, {165,10004}, {222,553}, {479,4312}, {516,7056}, {658,3817}, {1323,3474}, {1709,7177}

X(10136) = {X(7),X(9633)}-harmonic conjugate of X(1699)

Hex2T circles: X(10137)-X(10148)

X(10137) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS INNER, CIRCUMSYMMEDIAL)

X(10137) lies on these lines:{3,6}, {3545,9543}, {3853,9692}

X(10137) = Brocard circle-inverse-of-X(10138)
X(10137) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10138), (3,6500,6430), (1151,6429,6486), (1151,6453,6455), (1151,6468,372), (1151,6519,10145), (1152,6453,6221), (6221,6449,6419), (6221,6484,3), (6418,6473,6395), (6419,6473,6418), (6437,6449,3), (6449,6480,10143), (6453,6455,6474), (6480,6482,6429), (6481,6496,3), (6482,6486,1151), (6519,10141,3), (9690,10145,1151)

X(10138) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) INNER, CIRCUMSYMMEDIAL)

X(10138) lies on these lines:{3,6}

X(10138) = Brocard circle-inverse-of-X(10137)
X(10138) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10137), (3,6501,6429), (1151,6454,6398), (1152,6430,6487), (1152,6454,6456), (1152,6522,10146), (6398,6450,6420), (6398,6485,3), (6420,6472,6417), (6438,6450,3), (6450,6481,10144), (6454,6456,6475), (6456,6475,6395), (6480,6497,3), (6481,6483,6430), (6483,6487,1152), (6522,10142,3)

X(10139) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS INNER, LUCAS TANGENTS)

X(10139) lies on these lines:{3,6}, {8253,9693}

X(10139) = Brocard circle-inverse-of-X(10140)
X(10139) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10140), (6,6488,6409), (372,9690,1151), (1151,6437,6486), (1151,10145,6439), (3311,6409,6410), (6221,6470,6425), (6221,10147,6470), (6409,6468,6453), (6429,6433,6425), (6429,6480,10141), (6429,6482,6434), (6429,10141,6433), (6430,6486,6409), (6430,10145,10141), (6437,6453,6429), (6437,6480,6439), (6437,6486,6430), (6439,6453,6409), (6453,6519,6488), (6453,10145,1151), (6468,6480,6433), (6468,10141,6429), (6480,6484,6519), (10141,10147,6482)

X(10140) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) INNER, LUCAS(-1) TANGENTS)

X(10140) lies on these lines:{3,6}

X(10140) = Brocard circle-inverse-of-X(10139)
X(10140) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10139), (6,6489,6410), (1152,6438,6487), (1152,6493,6396), (1152,10146,6440), (3312,6410,6409), (6398,6471,6426), (6398,10148,6471), (6426,6468,372), (6429,6487,6410), (6429,10146,10142), (6430,6434,6426), (6430,6481,10142), (6430,6483,6433), (6430,10142,6434), (6438,6454,6430), (6438,6481,6440), (6438,6487,6429), (6440,6454,6410), (6454,6522,6489), (6454,10146,1152), (6481,6485,6522), (10142,10148,6483)

X(10141) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS INNER, LUCAS(-1) TANGENTS)

X(10141) lies on these lines:{3,6}, {547,9680}, {3533,9693}, {3543,9692}, {3850,9681}

X(10141) = Brocard circle-inverse-of-X(10142)
X(10141) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10142), (3,6419,6438), (3,6428,6485), (3,6429,6425), (3,6482,1151), (3,6519,6480), (3,10137,6519), (1151,6425,10147), (1151,6429,6433), (1151,6437,6484), (1151,6468,6409), (1151,6480,6429), (6221,6449,6500), (6221,6481,6437), (6221,6486,6431), (6410,6420,6426), (6420,6453,6221), (6425,6433,3), (6425,10147,6409), (6429,6480,10139), (6429,10139,6468), (6430,10145,10139), (6432,6497,6430), (6433,6480,6468), (6433,10139,6429), (6468,10147,6425), (6480,6482,3), (6481,6484,6486), (6482,10139,10147)

X(10142) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) INNER, LUCAS TANGENTS)

X(10142) lies on these lines:{3,6}

X(10142) = Brocard circle-inverse-of-X(10141)
X(10142) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10141), (3,6420,6437), (3,6427,6484), (3,6430,6426), (3,6483,1152), (3,6522,6481), (3,10138,6522), (372,6396,6496), (1152,6426,10148), (1152,6430,6434), (1152,6438,6485), (1152,6481,6430), (6398,6450,6501), (6398,6480,6438), (6398,6487,6432), (6409,6419,6425), (6419,6454,6398), (6426,6434,3), (6426,10148,6410), (6429,10146,10140), (6430,6481,10140), (6431,6496,6429), (6434,10140,6430), (6480,6485,6487), (6481,6483,3), (6483,10140,10148)

X(10143) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS INNER, LUCAS INNER)

X(10143) lies on these lines:{3,6}

X(10143) = Brocard circle-inverse-of-X(10144)
X(10143) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10144), (6449,6480,10137), (9690,9691,6501

X(10144) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) INNER, LUCAS(-1) INNER)

X(10144) lies on these lines:{3,6}

X(10144) = Brocard circle-inverse-of-X(10143)
X(10144) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10143), (6450,6481,10138)

X(10145) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS INNER, LUCAS(-1) INNER)

X(10145) lies on these lines:{3,6}, {381,9543}, {382,9542}, {1482,9617}, {3843,9692}, {5070,9693}

X(10145) = Brocard circle-inverse-of-X(10146)
X(10145) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10146), (3,6500,6473), (3,9691,6474), (371,6430,3311), (372,6486,6409), (372,6497,6446), (1151,3311,6445), (1151,6410,6484), (1151,6432,10147), (1151,6453,3311), (1151,6519,10137), (1151,9691,3), (1151,10137,9690), (1151,10139,6453), (3311,6396,6418), (3592,6409,6396), (6221,6449,3592), (6409,6453,6221), (6410,6477,6450), (6418,6468,9691), (6439,6445,9690), (6439,10139,1151), (6455,6500,3), (6474,9691,6472), (9690,9691,1151), (10139,10141,6430)

X(10146) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) INNER, LUCAS INNER)

X(10146) lies on these lines:{3,6}

X(10146) = Brocard circle-inverse-of-X(10145)
X(10146) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10145), (3,6501,6472), (371,6487,6410), (371,6496,6445), (372,6429,3312), (1152,3312,6446), (1152,6409,6485), (1152,6431,10148), (1152,6454,3312), (1152,6522,10138), (1152,10140,6454), (6398,6450,3594), (6409,6476,6449), (6410,6454,6398), (6440,10140,1152), (6445,6447,9691), (6456,6501,3), (6497,9690,3), (10140,10142,6429)

X(10147) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS TANGENTS, LUCAS(-1) INNER)

X(10147) lies on these lines:{3,6}, {546,9680}, {3146,3590}, {3544,8253}, {3628,9681}

X(10147) = Brocard circle-inverse-of-X(10148)
X(10147) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10148), (6,6439,6490), (6,6453,6425), (371,1151,6439), (371,6483,6419), (1151,1152,6480), (1151,3592,6519), (1151,6409,6468), (1151,6425,10141), (1151,6432,10145), (1151,6433,6409), (1151,6449,6), (1151,6488,3), (3592,6429,6425), (3592,6454,6), (3592,6519,6429), (3594,6434,6426), (6409,6425,6426), (6409,10141,6425), (6410,6425,3592), (6425,10141,6468), (6426,6468,6425), (6436,6480,6221), (6445,6484,1151), (6470,10139,6221), (6482,10139,10141)

X(10148) = CENTER OF CIRCUMCIRCLE OF HEX2T(LUCAS(-1) TANGENTS, LUCAS INNER)

X(10148) lies on these lines:{3,6}, {3146,3591}, {3544,8252}

X(10148) = Brocard circle-inverse-of-X(10147)
X(10148) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,6,10147), (6,6440,6491), (6,6454,6426), (372,1152,6440), (372,6482,6420), (1151,1152,6481), (1152,3594,6522), (1152,6396,6430), (1152,6426,10142), (1152,6431,10146), (1152,6434,6410), (1152,6450,6), (1152,6489,3), (3592,6433,6425), (3594,6430,6426), (3594,6453,6), (3594,6522,6430), (6396,6409,6410), (6396,6430,6409), (6396,6454,6453), (6396,6522,3594), (6409,6426,3594), (6409,6468,6486), (6410,6426,6425), (6410,10142,6426), (6435,6481,6398), (6446,6485,1152), (6471,10140,6398), (6483,10140,10142)

X(10149) = INCIRCLE-INVERSE OF X(6284)

Let P = x : y : z (barycentrics) be a point in the plane of a triangle ABC, and let
A'B'C' = cevian triangle of P
(Na) = nine-point circle of PB'C', and define (Nb) and (Nc) cyclically
R1 = radical axis of (Nb) and (Nc), and define R2 and R3 cyclically
R'1 = reflection of R1 in AA', and define R'2 and R'3 cyclically
P1 = line through A parallel to R1, and define P2 and P3 cyclically

The locus of P for which R'1, R'2, R'3 concur is the union of the Gibert quintic curve Q003 (which passes through X(i) for these i: 1,2,4,13,14,1113,1114,1156) and a curve of degree 10 in x,y,z, denoted here by Q, given by the barycentric equation

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) = y*z*((y^2+z^2)*a^4*y^3*z^3-2*( c^2*y^2+b^2*z^2)*(a^2-b^2-c^2) *x^6+(a^4-2*a^2*b^2-2*a^2*c^2+ b^4+4*b^2*c^2+c^4)*x^6*y*z+2* a^4*y^4*z^4-2*((a^2-b^2-2*c^2) *c^2*y^3+(a^2-2*b^2-c^2)*b^2* z^3)*x^5+2*((a^4-2*a^2*b^2-3* a^2*c^2+b^4+b^2*c^2+2*c^4)*y+( a^4-3*a^2*b^2-2*a^2*c^2+2*b^4+ b^2*c^2+c^4)*z)*x^5*y*z+(7*a^ 4-6*a^2*b^2-6*a^2*c^2+2*b^4-4* b^2*c^2+2*c^4)*x^2*y^3*z^3+2*( 3*a^4-6*a^2*b^2-6*a^2*c^2+3*b^ 4-4*b^2*c^2+3*c^4)*x^4*y^2*z^ 2)

The locus of P for which P1, P2, P3 concur is the union of the Lucas cubic and Q.

X(10149) = point of concurrence of R'1, R'2, R'3 for P = X(1)
X(10150) = point of concurrence of R'1, R'2, R'3 for P = X(2)
X(10151) = point of concurrence of R'1, R'2, R'3 for P = X(4)
X(10152) = point of concurrence of P1, P2, P3 for P = X(20)

The appearance of (i,j) in the following list means that if P = X(i), then the lins P1, P2, P3 concur in X(j): (2,671), (4,74), (7,1156), (8,1320).

Triangle centers X(10149)-X(10152) are contributed by Antreas Hatzipolakis and César Lozada, August 24, 2016. See Hyacinthos 24151).

X(10149) lies on these lines: {1,30}, {11,2072}, {12,403}, {23,3303}, {55,186}, {56,2071}, {468,612}, {497,3153}, {523,663}, {858,7191}, {1062,5433}, {1478,9642}, {1870,9629}, {2070,3295}, {3028,6000}, {3304,7464}, {3746,7575}, {3920,7426}, {4081,4511}, {4299,9641}, {5148,6020}, {5159,5272}, {5252,9577}, {5899,6767}

X(10149) = incircle-inverse of X(6284)
X(10149) = X(186)-of-Mandart-incircle-triangle

X(10150) = COMPLEMENT OF X(5215)

See X(10149) for a construction and reference.

X(10150) lies on these lines: {2,187}, {620,8355}, {3788,7615}, {7617,7880}, {7622,7872}, {7817,7862}, {7848,8860}

X(10150) = complement of X(5215)

X(10151) = MIDOINT OF X(4) AND X(403)

X(10151) lies on the Euler line. See X(10149) for a construction and reference.

X(10151) lies on these lines: {2,3}, {185,5893}, {974,1514}, {1990,6128}, {2452,10002}, {5186,5203}, {5318,8739}, {5321,8740}, {5480,8541}, {6746,10110}

X(10151) = midpoint of X(4) and X(403)
X(10151) = reflection of X(i) in X(j) for these (i,j): (468,403), (2071,5159)
X(10151) = circumcircle-inverse of X(3515)
X(10151) = half-altitude-circle-inverse of X(185)
X(10151) = polar-circle-inverse of X(20)
X(10151) = orthoptic-circle-of-Steiner-inellipse-inverse of X(6353)
X(10151) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(113)
X(10151) = crosspoint, wrt orthic triangle, of X(4) and X(113)
X(10151) = orthic-isogonal conjugate of X(13202)
X(10151) = X(1319)-of-orthic-triangle if ABC is acute

X(10152) = REFLECTION OF X(107) IN X(4)

See X(10149) for a construction and reference.

Let HaHbHc be the anticevian triangle of X(4). Let Pa = X(4)-of-BCOa, and define Pb and Pc cyclically. Then X(10231) = PaPbPc-to-ABC cyclologic center. X(10152) can be regarded as the 2nd Schmidt cyclologic center; see X(10229) and Quadri-Figures-Group #1936, September 9, 2016.

X(10152) lies on the Gibert curves K025, K447, Q001, Q107 and these lines: {2,3184}, {4,74}, {20,122}, {30,1294}, {146,648}, {253,317}, {1249,1562}, {2349,2816}, {2394,2848}, {2790,5186}, {2822,3668}, {3087,8749}, {3091,6716}, {3146,3346}

X(10152) = reflection of X(i) in X(j) for these (i,j): (20,122), (107,4), (1304,1552), (5667,133)
X(10152) = anticomplement of X(3184)
X(10152) = trilinear pole of the line X(1249)X(6587)
X(10152) = syngonal conjugate of X(4)
X(10152) = antigonal image of X(20)
X(10152) = antipode of X(20) in hyperbola {{A,B,C,X(4),X(20)}}

Centroidal conics and related centers: X(10153)-X(10200)

X(10153) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE ANTI-MCCAY TRIANGLE

X(10153) lies on the Kiepert Hyperbola and these lines:{4,5461}, {76,8860}, {230,5503}, {2482,5485}, {2996,8591}, {3424,6055}, {5466,9125}, {6054,7612}, {7607,8787}

X(10154) = CENTER OF THE CENTROIDAL CONIC OF THE ARA TRIANGLE

X(10154) = (8*R^2-SW)*X(2)-SW*X(3)

As a point on the Euler line, X(10154) has Shinagawa coefficients (-E-5*F, 3*F+3*E)

Let A'B'C' be the medial triangle of a triangle ABC. Let
Ba, Ca be the circumcircle intercepts of line B'C'; define Cb and Ab cyclically, and define Ac and Bc cyclically.
Let Ba' be the {Ba,Ca}-harmonic conjugate of B', and define Cb' and Ac' cyclically.
Let Ca' be the {Ba,Ca}-harmonic conjugate of C', and define Ab' and Bc' cyclically.
Then X(10154) is the centroid of {Ba',Ca',Cb',Ab',Ac',Bc'}. (Randy Hutson, September 14, 2016)

X(10154) lies on these lines:{2,3}, {69,8780}, {154,3564}, {184,1353}, {206,524}, {343,1495}, {511,10192}, {597,9969}, {612,8144}, {800,5306}, {5065,9300}, {8705,10169}, {8854,8981}

X(10154) = midpoint of X(i),X(j) for these {i,j}: {2,9909}
X(10153) = antigonal conjugate of X(32532)
X(10153) = antipode of X(32532) in the Kiepert hyperbola
X(10154) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,3,7734), (2,22,7667), (2,7667,1368), (2,7714,381), (3,6353,6677), (22,468,1368), (22,1368,550), (25,6676,5), (25,7493,6676)

X(10155) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE ARTZT TRIANGLE

X(10155) lies on the Kiepert hyperbola and these lines:{2,5093}, {4,8719}, {76,5067}, {83,3525}, {631,5395}, {671,5071}, {2996,3090}, {3055,9752}, {3815,7612}, {5485,9771}, {7607,7736}, {7608,9754}, {8796,8889}

X(10156) = CENTER OF THE CENTROIDAL CONIC OF THE ASCELLA TRIANGLE

X(10156) lies on these lines:{2,971}, {3,5436}, {140,912}, {516,3848}, {517,549}, {631,942}, {1006,5122}, {1376,1385}, {3523,5439}, {3526,5777}, {3824,6922}, {5045,6684}, {5049,5657}

X(10156) = midpoint of X(5049) and X(5657)
X(10156) = complement of X(10157)

X(10157) = CENTER OF THE CENTROIDAL CONIC OF THE ATIK TRIANGLE

X(10157) lies on these lines:{2,971}, {3,1750}, {4,5044}, {5,226}, {57,5779}, {65,7989}, {72,3091}, {210,381}, {329,5805}, {354,7988}, {355,497}, {375,2807}, {516,3740}, {518,3817}, {553,5843}, {908,8226}, {916,5943}, {962,3697}, {1385,4423}, {1465,7069}, {1538,2886}, {1709,4413}, {1829,7559}, {1858,3614}, {1864,5219}, {2801,3742}, {2808,6688}, {3057,9671}, {3305,7580}, {3419,6957}, {3452,8727}, {3475,5049}, {3666,5400}, {3681,9779}, {3748,5531}, {3832,3876}, {3868,5068}, {3916,6915}, {3925,6842}, {3983,7991}, {4301,4662}, {4679,6928}, {4847,7956}, {4866,6766}, {5045,8227}, {5056,5439}, {5084,5787}, {5122,6911}, {5226,5728}, {5440,6912}, {5720,6913}, {5722,6939}, {5791,6848}, {5811,6864}, {5812,6849}, {5887,6867}, {6244,8580}, {6260,8728}

X(10157) = midpoint of X(i),X(j) for these {i,j}: {2,5927}, {210,1699}
X(10157) = reflection of X(i) in X(j) for these (i,j): (5049,5886)
X(10157) = center of the centroidal conic of the 2nd extouch triangle
X(10157) = complement of X(10167)
X(10157) = anticomplement of X(10156)
X(10157) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,5777,942), (72,3091,5806), (1750,7308,3)
X(10157) = X(3)-of-triangle-A'B'C' as defined at X(5658)

X(10158) = CENTER OF THE CENTROIDAL CONIC OF THE AYME TRIANGLE

The Ayme triangle is defined at X(3610)

X(10158) lies on these lines:{758,3740}, {2294,8580}, {3743,9709}

X(10159) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE 1ST BROCARD TRIANGLE

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that angle(A'BC) = angle(A'CB) = ω define B' and C'cyclically. Let A" be the centroid of BA'C, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(10159). (Randy Hutson, September 14, 2016)

X(10159) lies on the Kiepert hyperbola and these lines:{2,3108}, {4,3096}, {13,636}, {14,635}, {76,3763}, {83,141}, {98,140}, {99,6292}, {262,1656}, {315,5395}, {550,6287}, {598,7770}, {599,7878}, {671,6656}, {1327,7389}, {1328,7388}, {1916,3934}, {2996,7790}, {3424,3523}, {3456,6636}, {3533,7612}, {3620,7877}, {3851,7934}, {5073,7910}, {5503,8361}, {6704,7779}, {7607,7940}, {7808,7917}, {7809,7849}, {7948,9466}

X(10159) = isogonal conjugate of X(5007)
X(10159) = isotomic conjugate of X(3589)
X(10159) = polar conjugate of X(428)
X(10159) = trilinear pole of line X(523)X(2528)

X(10160) = CENTER OF THE CENTROIDAL CONIC OF THE 2ND BROCARD TRIANGLE

X(10160) lies on these lines:{2,353}, {373,468}, {858,9128}

X(10160) = midpoint of X(i),X(j) for these {i,j}: {858,9128}

X(10161) = CENTER OF THE CENTROIDAL CONIC OF THE 3RD BROCARD TRIANGLE

X(10161) lies on these lines:{2,736}, {3229,7820}

X(10162) = CENTER OF THE CENTROIDAL CONIC OF THE 4TH BROCARD TRIANGLE

X(10162) lies on these lines:{2,187}, {5,9172}, {125,597}, {126,9771}, {7617,9745}

X(10162) = midpoint of X(2) and X(6032)
X(10162) = X(2)-of-5th-Euler-triangle

X(10163) = CENTER OF THE CENTROIDAL CONIC OF THE CIRCUMMEDIAL TRIANGLE

X(10163) lies on these lines:{2,187}, {126,549}, {141,5642}, {524,10160}, {620,7495}, {827,6325}, {6676,8891}, {7664,9466}

X(10163) = midpoint of X(60312) and X(6032)
X(10163) = complement of X(6032)
X(10163) = anticomplement-of-X(10173)
X(10163) = X(353)-of-X(2)-Brocard triangle
X(10163) = inverse of X(316) in the orthoptic circle of the Steiner Inellipse
X(10163) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6031,6032)

X(10164) = CENTER OF THE CENTROIDAL CONIC OF THE 1ST CIRCUMPERP TRIANGLE

X(10164) = 2*X(3)+X(10)

X(10164) lies on these lines:{1,3523}, {2,165}, {3,10}, {4,3634}, {8,7987}, {9,2272}, {20,1698}, {21,8582}, {35,1210}, {36,4315}, {40,631}, {43,991}, {46,3671}, {55,3911}, {57,3475}, {63,6745}, {100,4847}, {140,946}, {142,6690}, {171,4349}, {200,5744}, {210,2801}, {226,1155}, {354,4995}, {376,3828}, {474,5584}, {495,5122}, {498,3947}, {517,549}, {519,3158}, {572,3684}, {573,6685}, {750,1754}, {912,4134}, {944,3626}, {950,5217}, {952,4669}, {962,3624}, {971,3740}, {990,5268}, {1001,6244}, {1006,2077}, {1150,4061}, {1385,3244}, {1447,3663}, {1512,6950}, {1588,9582}, {1621,5537}, {1697,7288}, {1703,9540}, {1709,3305}, {1737,4304}, {1766,3986}, {1768,3219}, {1788,3601}, {2646,4848}, {2807,3819}, {2820,4763}, {2975,6736}, {3035,3452}, {3069,9616}, {3085,4298}, {3097,6194}, {3146,7989}, {3216,4300}, {3245,5444}, {3339,5703}, {3359,6954}, {3430,8258}, {3474,5219}, {3485,5128}, {3522,5691}, {3525,6361}, {3528,5818}, {3584,5131}, {3616,7991}, {3625,5690}, {3636,7982}, {3647,6260}, {3667,4448}, {3679,5731}, {3683,5316}, {3746,5442}, {3811,8726}, {3814,6907}, {3826,8727}, {3840,8299}, {3841,6831}, {3928,5850}, {3977,4082}, {4052,7612}, {4067,5884}, {4311,7280}, {4312,5226}, {4342,5119}, {4414,4656}, {4421,5853}, {4652,5552}, {4691,5881}, {4855,6737}, {5085,5847}, {5121,8616}, {5249,9352}, {5273,5732}, {5298,5919}, {5439,7957}, {5687,8273}, {5918,5927}, {6210,6686}, {7735,9574}

X(10164) = midpoint of X(i),X(j) for these {i,j}: {2,165}, {40,5603}, {376,5587}, {1699,9778}, {3097,6194}, {3576,5657}, {3679,5731}, {5918,5927}
X(10164) = reflection of X(i) in X(j) for these (i,j): (3742,10156), (3817,2), (4301,5603), (5587,3828), (5603,1125)
X(10164) = complement of X(1699)
X(10164) = anticomplement of X(10171)
X(10164) = X(2) of half-altitude triangle of excentral triangle
X(10164) = X(5943)-of-excentral-triangle
X(10164) = center of the centroidal conic of the excentral triangle
X(10164) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9778,1699), (2,9812,7988), (3,10,4297), (3,6684,10), (35,1210,4314), (40,631,1125), (40,1125,4301), (140,3579,946), (165,1699,9778), (498,4292,3947), (946,3579,5493), (1155,5432,226), (1376,5745,10), (1737,5010,4304), (1788,3601,6738), (3035,4640,3452), (3522,9780,5691), (3524,5657,3576), (5281,5435,1), (5690,5882,3625), (7987,9588,8)

X(10165) = CENTER OF THE CENTROIDAL CONIC OF THE 2ND CIRCUMPERP TRIANGLE

X(10165) = 2*(R+r)*X(3)+(4*R+r)*X(142) = X(10)-4*X(140)

X(10165) lies on these lines:{1,631}, {2,515}, {3,142}, {4,3624}, {5,4297}, {10,140}, {11,4304}, {12,4311}, {20,5550}, {30,3817}, {35,6940}, {36,226}, {40,3306}, {84,3646}, {104,5251}, {165,3524}, {354,5298}, {355,3526}, {372,8983}, {376,1699}, {392,2800}, {405,5450}, {474,6796}, {495,4315}, {496,4314}, {498,6967}, {499,950}, {517,549}, {519,3653}, {572,5257}, {581,978}, {912,5325}, {944,1698}, {956,6745}, {958,6700}, {960,5884}, {962,9624}, {993,3452}, {997,5745}, {1012,4423}, {1210,2646}, {1319,5432}, {1387,4342}, {1420,3085}, {1445,3333}, {1478,6947}, {1479,6897}, {1482,3636}, {1483,3625}, {1490,5817}, {1519,6950}, {1588,9615}, {1621,2077}, {1770,5443}, {1838,7531}, {1848,7501}, {3069,9583}, {3086,3601}, {3090,5691}, {3149,8273}, {3244,5690}, {3361,3487}, {3530,3579}, {3533,5818}, {3583,6951}, {3585,6902}, {3622,7982}, {3655,3828}, {3679,7967}, {3755,4256}, {3816,6907}, {3822,6882}, {3825,6842}, {3840,6176}, {4084,5885}, {4292,5204}, {4293,5219}, {4305,9581}, {4667,5398}, {4847,5440}, {5542,5719}, {5881,9780}

X(10165) = midpoint of X(i),X(j) for these {i,j}: {1,5657}, {2,3576}, {3,5886}, {104,5660}, {165,5603}, {376,1699}, {551,10164}, {3653,5054}, {3655,5790}, {3679,7967}, {5587,5731}
X(10165) = reflection of X(i) in X(j) for these (i,j): (946,5886), (5657,6684), (5790,3828), (5886,1125)
X(10165) = complement of X(5587)
X(10165) = anticomplement of X(10172)
X(10165) = center of the centroidal conic of the hexyl triangle
X(10165) = homothetic center of anti-Euler triangle and cross-triangle of these triangles: Aquila and anti-Aquila
X(10165) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,631,6684), (2,5731,5587), (3,1125,946), (10,1385,5882), (20,5550,8227), (140,1385,10), (355,3526,3634), (495,5126,4315), (499,3612,950), (944,3525,1698), (2646,5433,1210), (3523,3616,40), (3524,5603,165), (3530,5901,3579), (3576,5587,5731), (3579,5901,4301), (3624,7987,4), (5265,5703,3333)

X(10166) = CENTER OF THE CENTROIDAL CONIC OF THE CIRCUMSYMMEDIAL TRIANGLE

X(10166) = midpoint of X(2) and X(353)
X(10166) = X(2)-of-2nd-Brocard-triangle
X(10166) = reflection of X(i) in X(j) for these (i,j): (2,353), (111,7606), (549,9127), (597,5640), (5108,7622)

X(10167) = CENTER OF THE CENTROIDAL CONIC OF THE 1ST CONWAY TRIANGLE

X(10167) = (4*R+r)*X(3)-(R+r)*X(63)

X(10167) lies on these lines:{1,1407}, {2,971}, {3,63}, {4,5439}, {20,942}, {40,3555}, {57,5728}, {65,4297}, {84,405}, {142,8226}, {154,392}, {165,518}, {210,2801}, {222,1040}, {354,516}, {355,4002}, {376,517}, {377,5787}, {443,9799}, {474,1490}, {515,3753}, {631,5777}, {916,3917}, {940,990}, {960,7987}, {962,5045}, {982,1742}, {991,3666}, {1001,1709}, {1012,7171}, {1214,7004}, {1385,1621}, {1699,3742}, {1750,5437}, {1768,4640}, {1858,5204}, {1864,3911}, {1871,7554}, {1898,5433}, {2096,6987}, {2203,4227}, {2808,3819}, {2951,5572}, {3146,5806}, {3218,7411}, {3243,7994}, {3305,5779}, {3419,5768}, {3474,5173}, {3522,3868}, {3523,5044}, {3697,6684}, {3812,5691}, {3824,6828}, {3848,7988}, {3870,6244}, {3874,7957}, {3881,5493}, {4004,6934}, {4018,5884}, {4187,6260}, {4229,5208}, {4302,5570}, {4662,9588}, {5249,8727}, {5722,6925}, {5745,5784}, {5812,6899}, {5887,6875}, {6705,7483}

X(10167) = anticomplement of X(10157)
X(10167) = midpoint of X(i),X(j) for these {i,j}: {354,5918}, {3873,9778}
X(10167) = reflection of X(i) in X(j) for these (i,j): (210,10164), (392,3576), (1699,3742), (5927,2)
X(10167) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,5732,7580), (84,8726,405), (5768,6916,3419), (10156,10157,2)

X(10168) = CENTER OF THE CENTROIDAL CONIC OF THE 1ST EHRMANN TRIANGLE

X(10168) lies on these lines:{2,98}, {3,5476}, {6,5054}, {30,3589}, {140,524}, {373,7426}, {381,5085}, {511,549}, {543,7606}, {547,1503}, {576,631}, {599,5050}, {632,8550}, {1428,3584}, {1691,7753}, {2030,3815}, {2330,3582}, {3098,3524}, {3525,7909}, {3545,7919}, {3818,5055}, {5038,7749}, {5116,6034}, {6791,7708}, {7820,8724}, {9172,10166}

X(10168) = midpoint of X(i),X(j) for these {i,j}: {2,182}, {3,5476}, {549,597}, {5480,8703}
X(10168) = Artzt-to-McCay similarity image of X(5)

X(10169) = CENTER OF THE CENTROIDAL CONIC OF THE 2ND EHRMANN TRIANGLE

X(10169) = 5*X(6)+X(66)

X(10169) lies on these lines:{6,66}, {206,6329}, {597,2393}, {1503,3845}, {3629,6697}, {3827,3919}, {8705,10154}

X(10170) = CENTER OF THE CENTROIDAL CONIC OF THE 2ND EULER TRIANGLE

X(10170) = (6*R^2-SW)*X(5)+SW*X(141)

X(10170) lies on these lines:{2,5654}, {3,1495}, {4,5447}, {5,141}, {30,3819}, {51,5055}, {52,3090}, {68,6804}, {110,7550}, {140,5663}, {155,5050}, {185,3526}, {373,568}, {381,3917}, {389,3628}, {547,1154}, {549,6000}, {567,3292}, {632,5876}, {1147,6090}, {2979,3545}, {3060,5071}, {3091,7999}, {3098,7530}, {3567,7486}, {5067,5889}, {5079,6243}, {5085,7393}, {5448,7399}, {5946,6688}, {6759,7516}, {6800,7509}, {7514,9306}

X(10170) = complement of X(9730)
X(10170) = midpoint of X(i),X(j) for these {i,j}: {2,5891}, {381,3917}, {568,5562}
X(10170) = reflection of X(i) in X(j) for these (i,j): (568,5462), (5892,2), (5943,547), (5946,6688)
X(10170) = X(2)-of-X(5)-Brocard-triangle
X(10170) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,1216,5446), (373,568,5462), (373,5562,568), (568,1656,373), (632,5876,9729), (1656,5562,5462)

X(10171) = CENTER OF THE CENTROIDAL CONIC OF THE 3RD EULER TRIANGLE

X(10171) = 5*X(2)-X(165)

X(10171) lies on these lines:{1,5056}, {2,165}, {5,515}, {10,3090}, {11,3748}, {40,5067}, {226,4860}, {355,3636}, {499,4298}, {517,3828}, {519,5790}, {551,5587}, {950,7173}, {971,3848}, {1482,4691}, {1698,4301}, {2801,3742}, {3086,3947}, {3244,5818}, {3475,5219}, {3616,7989}, {3624,4297}, {3626,5844}, {3635,5901}, {3720,5400}, {3829,5853}, {3833,6001}, {3838,6667}, {5087,5745}, {5226,5542}, {5231,5748}, {5537,9342}, {5550,5691}, {6701,6705

X(10171) = complement of X(10164)
X(10171) = midpoint of X(i),X(j) for these {i,j}: {2,3817}, {10,5603}, {381,10165}, {551,5587}, {1699,10164}, {3742,10157}
X(10171) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,1699,10164), (2,7988,3817), (2,9779,165), (3090,8227,10), (3091,3624,4297), (3817,10164,1699), (3838,6667,6692), (5068,5550,5691), (5818,9624,3244)

X(10172) = CENTER OF THE CENTROIDAL CONIC OF THE 4TH EULER TRIANGLE

X(10172) lies on these lines:{1,5067}, {2,515}, {5,516}, {10,1482}, {40,5056}, {142,5770}, {165,3545}, {226,6877}, {517,3828}, {551,5790}, {912,3833}, {946,1698}, {952,1125}, {1737,5425}, {2077,9342}, {3614,4292}, {3617,9624}, {3624,5818}, {3626,5901}, {3814,5745}, {3822,6692}, {3824,5843}, {3911,7951}, {4745,5844}, {5550,5881}, {5817,6260}, {8169,9708}, {8227,9780}

X(10172) = complement of X(10165)
X(10172) = midpoint of X(i),X(j) for these {i,j}: {10,5886}, {381,10164}, {551,5790}, {946,5657}, {3828,10171}, {5587,10165}
X(10172) = center of the Vu pedal-centroidal circle of X(10)
X(10172) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5587,10165), (5,3634,6684), (1698,3090,946), (1698,7988,5657), (3090,5657,7988), (3624,5818,5882), (5657,7988,946), (7486,9780,8227)

X(10173) = CENTER OF THE CENTROIDAL CONIC OF THE 5TH EULER TRIANGLE

X(10173) lies on these lines:{2,187}, {547,6719}

X(10173) = complement of X(10163)
X(10173) = midpoint of X(i),X(j) for these {i,j}: {2,10162}, {6032,10163}
X(10173) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6032,10163), (10162,10163,6032)

X(10174) = CENTER OF THE CENTROIDAL CONIC OF THE EXTANGENTS TRIANGLE

X(10174) lies on these lines:{154,4421}, {1376,3197}, {3740,6001}

X(10175) = CENTER OF THE CENTROIDAL CONIC OF THE FUHRMANN TRIANGLE

X(10175) = 2*X(5)+X(10)

X(10175) lies on these lines:{1,3090}, {2,515}, {3,3634}, {4,165}, {5,10}, {8,5056}, {9,6843}, {11,5919}, {12,354}, {30,10164}, {35,6920}, {36,6946}, {40,3091}, {46,6984}, {63,6993}, {65,3614}, {84,4208}, {116,119}, {117,374}, {121,5511}, {140,4297}, {145,9624}, {226,1737}, {355,1125}, {381,516}, {405,6796}, {442,5927}, {443,6256}, {474,5450}, {495,5049}, {498,950}, {499,6983}, {519,5055}, {546,3579}, {547,551}, {631,5691}, {726,7697}, {912,5883}, {936,6855}, {942,3947}, {944,3624}, {958,6918}, {962,5068}, {993,6911}, {997,6859}, {1012,4413}, {1056,5726}, {1064,5400}, {1158,5177}, {1376,6913}, {1385,3628}, {1478,3911}, {1479,6898}, {1482,3626}, {1512,5316}, {1532,3925}, {1699,3545}, {1735,7069}, {1770,5445}, {1788,9612}, {1891,7537}, {2077,6912}, {2095,5220}, {2476,7705}, {2550,6939}, {2551,5705}, {2800,3753}, {3057,7173}, {3085,9581}, {3086,9578}, {3219,5535}, {3244,5901}, {3333,5261}, {3339,5714}, {3359,6982}, {3419,6745}, {3421,5231}, {3525,7987}, {3544,7991}, {3577,5328}, {3583,6965}, {3585,5131}, {3586,5218}, {3616,5881}, {3617,7982}, {3656,4745}, {3679,5071}, {3681,6734}, {3754,5887}, {3812,5777}, {3826,6907}, {3829,3880}, {3839,9778}, {3841,6842}, {3844,5480}, {3850,5493}, {3855,6361}, {4084,5694}, {4298,9654}, {4304,5432}, {4311,5433}, {4342,7743}, {4669,5844}, {4678,5734}, {5794,6700}, {6001,10157}, {7288,9613}, {7671,7679}

X(10175) = midpoint of X(i),X(j) for these {i,j}: {2,5587}, {4,165}, {10,3817}, {40,9812}, {1699,5657}, {3679,5603}, {5790,5886}
X(10175) = reflection of X(i) in X(j) for these (i,j): (2,10172), (165,6684), (946,3817), (3817,5), (5886,10171)
X(10175) = complement of X(3576)
X(10175) = X(2)-of-4th-Euler-triangle
X(10175) = centroid of X(3)X(4)X(10)
X(10175) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,1698,6684), (5,10,946), (5,3820,7680), (8,5056,8227), (10,3814,3452), (10,4301,5690), (119,6881,3822), (355,1125,5882), (355,1656,1125), (944,5067,3624), (1698,7989,4), (1737,7951,226), (2886,5123,10), (3090,5818,1), (3091,9780,40), (3545,5657,1699), (3679,7988,5603), (3812,5777,5884), (5055,5790,5886), (5055,5886,10171), (5071,5603,7988), (5261,5704,3333), (5587,10172,10165), (7680,7682,946)

X(10176) = CENTER OF THE CENTROIDAL CONIC OF THE INNER-GARCIA TRIANGLE

X(10176) = 2*r*X(5)-(3*R+2*r)*X(10)

X(10176) lies on these lines:{1,748}, {2,758}, {3,3647}, {5,10}, {8,3884}, {9,48}, {21,3467}, {36,3219}, {43,4868}, {63,4973}, {65,3634}, {72,354}, {78,5248}, {140,5694}, {141,2836}, {145,4547}, {165,411}, {191,404}, {210,392}, {386,3725}, {518,551}, {549,2771}, {594,3032}, {595,5293}, {631,5693}, {632,5885}, {846,4256}, {899,4424}, {908,3822}, {912,5325}, {942,3848}, {956,3715}, {984,995}, {999,5220}, {1001,3940}, {1193,3989}, {1385,5302}, {1698,3754}, {2292,3216}, {2392,3917}, {2551,6902}, {2703,2758}, {2779,5891}, {2802,3679}, {2809,3789}, {2842,5650}, {3057,3626}, {3338,3951}, {3419,4679}, {3555,3636}, {3616,3881}, {3617,4540}, {3624,3868}, {3625,4662}, {3632,3890}, {3635,4533}, {3683,5440}, {3696,4717}, {3742,4525}, {3746,4420}, {3753,3828}, {3786,4653}, {3812,4084}, {3825,6734}, {3873,4532}, {3880,4669}, {3885,4668}, {3893,4746}, {3899,3968}, {3920,5315}, {3921,4745}, {3952,4692}, {3962,5439}, {3983,4691}, {4009,4125}, {4197,6701}, {4257,7262}, {4297,5777}, {4511,5251}, {4536,5550}, {4537,5045}, {4660,9519}, {4857,5178}, {5250,8715}, {5253,6763}, {5289,9708}, {5535,6946}, {5538,6912}, {5887,6684}, {6001,10164}, {6894,9812}

X(10176) = midpoint of X(i),X(j) for these {i,j}: {1,3681}, {2,5692}, {72,354}, {210,392}, {551,4134}, {960,3740}, {3679,3877}, {4430,5904}
X(10176) = reflection of X(i) in X(j) for these (i,j): (10,3740), (354,1125), (942,3848), (3679,3956), (3681,3678), (3740,5044), (3753,3828), (3874,354), (3892,551), (3898,392), (4430,3881), (5883,2), (5902,3833)
X(10176) = anticomplement of X(3833)
X(10176) = complement of X(5902)
X(10176) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3876,3678), (1,5506,5047), (2,5902,3833), (9,997,993), (10,960,3878), (10,3452,3814), (72,1125,3874), (140,5694,5884), (846,5529,4256), (960,5044,10), (993,997,214), (1698,3869,3754), (3057,3697,3626), (3616,5904,3881), (3833,5902,5883), (3881,3988,5904), (3884,4015,8), (5690,9711,10)

X(10177) = CENTER OF THE CENTROIDAL CONIC OF THE HONSBERGER TRIANGLE

X(10177) = ((4*R+r)*SW-6*R*s^2)*X(1)+SW*(2*R-r)*X(6)

X(10177) lies on these lines:{1,6}, {2,7671}, {7,3660}, {11,142}, {55,8257}, {354,527}, {374,2809}, {516,5883}, {528,3753}, {551,2801}, {942,5698}, {971,5886}, {1125,5784}, {1479,5439}, {1699,3742}, {2550,5722}, {3059,6666}, {3624,5696}, {3816,8255}, {3873,6172}, {4666,8545}, {5805,6851}

X(10177) = midpoint of X(i),X(j) for these {i,j}: {2,7671}, {392,5728}, {3873,6172}
X(10177) = reflection of X(i) in X(j) for these (i,j): (392,1001), (6173,3742)

X(10178) = CENTER OF THE CENTROIDAL CONIC OF THE HUTSON-EXTOUCH TRIANGLE

X(10178) = 4*X(3)-X(960)

X(10178) lies on these lines:{2,5918}, {3,960}, {9,1615}, {20,3812}, {57,4326}, {63,480}, {65,3522}, {84,5302}, {165,518}, {354,8236}, {516,3742}, {517,3892}, {550,7686}, {971,3740}, {990,4682}, {1155,7411}, {1376,5732}, {1699,3848}, {1742,3752}, {2951,5437}, {3057,6049}, {3218,7964}, {3576,4428}, {3817,10156}, {3880,5731}, {4297,5836}, {4300,4719}, {4711,5657}, {5045,5493}, {5281,8581}

X(10178) = midpoint of X(i),X(j) for these {i,j}: {2,5918}, {165,10167}, {354,9778}
X(10178) = reflection of X(i) in X(j) for these (i,j): (1699,3848), (3740,10164), (3817,10156), (4711,5657)

X(10179) = CENTER OF THE CENTROIDAL CONIC OF THE HUTSON-INTOUCH TRIANGLE

X(10179) = (3*r^2-3*s^2+2*SW)*X(1)+SW*X(6)

X(10179) lies on these lines:{1,6}, {2,3880}, {65,3622}, {142,4342}, {145,4662}, {210,3241}, {354,3877}, {495,5087}, {517,549}, {519,3740}, {758,5049}, {942,3636}, {988,3445}, {997,6767}, {999,4640}, {1058,5794}, {1125,1387}, {1149,3666}, {1201,4719}, {1319,1621}, {1385,8717}, {1475,4520}, {2099,4666}, {2771,5609}, {3057,3616}, {3244,5044}, {3304,5250}, {3576,4428}, {3621,3983}, {3633,3697}, {3635,4547}, {3698,3885}, {3706,4742}, {3748,4511}, {3753,3848}, {3816,5123}, {3822,7743}, {3870,8162}, {3878,5045}, {3889,3962}, {3893,9780}, {3895,4413}, {3913,8583}, {3921,4677}, {5439,5697}, {5734,7957}, {5901,7686}, {8168,8580}

X(10179) = midpoint of X(i),X(j) for these {i,j}: {1,392}, {2,5919}, {210,3241}, {354,3877}, {551,3898}
X(10179) = reflection of X(i) in X(j) for these (i,j): (960,392), (3742,551), (3753,3848), (4711,3740)
X(10179) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3057,3616,3812), (3622,3890,65), (3636,3884,942), (3885,5550,3698)

X(10180) = CENTER OF THE CENTROIDAL CONIC OF THE INCENTRAL TRIANGLE

Let La be the line that is the locus of trilinear poles of tangents to the A-excircle, and define Lb and Lc cyclically. (Note that the Gergonne line is the locus of trilinear poles of tangents to the incircle, so that La is the A-extraversion of the Gergonne line.) Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. (Note that A' is also the insimilicenter of the circles having as diameters segments CA and AB; and cyclically for B' and C'.) X(10180) = X(2)-of-A'B'C'. The triangle A'B'C' is here named the Gergonne line extraversion triangle; A'B'C' is also the complement of the cevian triangle of X(75). (Randy Hutson, September 14, 2016)

The Gergonne line extraversion triangle is also the Gemini triangle 15. (Randy Hutson, November 30, 2018)

X(10180) lies on these lines:{1,333}, {2,740}, {10,4046}, {37,714}, {42,3842}, {43,2667}, {81,5625}, {86,846}, {171,3747}, {351,812}, {354,392}, {537,3989}, {804,4928}, {896,8025}, {982,2292}, {1125,3666}, {1213,4771}, {1376,4068}, {1621,3724}, {1961,4434}, {2352,5248}, {2650,3622}, {2664,9401}, {3178,4205}, {3624,4647}, {3636,4883}, {3720,6682}, {3739,4970}, {3740,4755}, {3741,4891}, {3896,4732}, {3936,6536}, {3986,4104}, {4028,5257}, {4062,8040}, {4155,4763}, {4418,5333}, {4425,4892}, {4703,5712}, {4728,9147}, {6703,9507}

X(10180) = midpoint of X(i),X(j) for these {i,j}: {2,1962}, {4728,9147}
X(10180) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (86,846,4697)

X(10181) = CENTER OF THE CENTROIDAL CONIC OF THE INTANGENTS TRIANGLE

X(10181) lies on these lines:{154,4428}, {551,6000}, {1001,2192}, {3616,6285}, {3622,7355}, {6001,10179}

X(10182) = CENTER OF THE CENTROIDAL CONIC OF THE KOSNITA TRIANGLE

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(10182) is the centroid of O_AO_BO_C. (Randy Hutson, June 7, 2019)

X(10182) lies on these lines:{3,113}, {140,1503}, {154,5054}, {549,6000}, {578,3147}, {631,5651}, {1147,5965}, {2393,10168}, {3357,3523}

X(10182) = midpoint of X(i),X(j) for these {i,j}: {3357,5656}

X(10183) = CENTER OF THE CENTROIDAL CONIC OF THE LEMOINE TRIANGLE

X(10183) = lies on no line X(i)X(j) for 1 < i < j < 10183

X(10184) = CENTER OF THE CENTROIDAL CONIC OF THE MACBEATH TRIANGLE

X(10184) lies on these lines:{5,51}, {1656,2052}, {3168,5056}

X(10184) = complement of X(32078)

X(10185) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE MCCAY TRIANGLE

X(10185) lies on the Kiepert hyperbola and these lines:{4,8588}, {140,671}, {598,1656}, {3054,7608}, {3533,5485}

X(10185) = isogonal conjugate of X(22330)

X(10186) = CENTER OF THE CENTROIDAL CONIC OF THE MIXTILINEAR TRIANGLE

X(10186) lies on these lines:{1,348}, {376,516}, {997,1064}, {2784,7967}, {3755,9592}

X(10186) = reflection of X(i) in X(j) for these (i,j): (9746,10165)

X(10187) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE INNER-NAPOLEON TRIANGLE

X(10187) lies on the Kiepert hyperbola and these lines: {2,3412}, {6,10188}, {13,1656}, {14,140}, {1327,2046}, {1328,2045}, {3858,5351}, {5068,5237}

X(10187) = complement of X(33405)

X(10188) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE INNER-NAPOLEON TRIANGLE

X(10188) lies on the Kiepert hyperbola and these lines: {2,3411}, {6,10187}, {13,140}, {14,1656}, {1327,2045}, {1328,2046}, {3858,5352}, {5068,5238}

X(10188) = complement of X(33404)

X(10189) = CENTER OF THE CENTROIDAL CONIC OF THE SCHROETER TRIANGLE

X(10189) lies on these lines:{2,523}, {512,6688}, {1499,5066}, {1637,9479}

X(10189) = complement of X(10190)
X(10189) = inverse-in-Hutson-Parry-circle of X(11123)
X(10189) = {X(2),X(5466)}-harmonic conjugate of X(11123)

X(10190) = CENTER OF THE CENTROIDAL CONIC OF THE STEINER TRIANGLE

X(10190) lies on the cubic K700 and these lines:{2,523}, {99,9293}, {140,8151}, {154,3566}, {351,3268}, {512,3819}, {647,1194}, {669,6636}, {1499,8703}, {4226,5502}, {8723,9306}, {9131,9148}

X(10190) = anticomplement of X(10189)
X(10190) = complement of X(8029)
X(10190) = midpoint of X(i),X(j) for these {i,j}: {351,3268}, {1649,9168}, {9131,9148}
X(10190) = reflection of X(i) in X(j) for these (i,j): (8029,10189)
X(10190) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,8029,10189)

X(10191) = CENTER OF THE CENTROIDAL CONIC OF THE SYMMEDIAL TRIANGLE

X(10191) lies on these lines:{2,732}, {6,1799}, {39,4074}, {51,597}, {826,10190}, {1194,3589}, {1613,7786}, {1915,3329}, {3618,3981}

X(10191) = centroid of the cevian traces of PU(1)

X(10192) = CENTER OF THE CENTROIDAL CONIC OF THE TANGENTIAL TRIANGLE

Let Ab be the point of intersection of the circle {{X(3),B,C}} and the line AB, and define Bc and Ca cyclically. Let Ac be the point of intersection of the circle {{X(3),B,C}} and the line AC, and define Ba and Cb cyclically. X(10192) is the centroid of {Ab,Ac,Bc,Ba,Ca,Cb}; see the preamble before X(8537). (Randy Hutson, September 14, 2016)

Let A'B'C' be the tangential triangle. Let Ba be the orthogonal projection of B' on line BC, and define Cb and Ac cyclically. Let Ca be the orthogonal projection of C' on line BC, and define Ab and Bc cyclically. Then X(10192) is the centroid of {Ba,Ca,Cb,Ab,Ac,Bc}. (Randy Hutson, September 14, 2016)

X(10192) lies on these lines:{2,154}, {3,1661}, {5,5944}, {6,6353}, {20,5893}, {25,5480}, {28,5799}, {53,436}, {64,3523}, {110,343}, {140,6247}, {141,206}, {159,3589}, {160,6638}, {161,1995}, {182,1660}, {184,468}, {221,7288}, {394,7493}, {418,1624}, {419,5254}, {427,1495}, {511,10154}, {524,3167}, {549,6000}, {597,2393}, {631,1498}, {1181,3147}, {1352,8780}, {1368,5972}, {1375,1754}, {1619,7484}, {1971,3815}, {2192,5218}, {2328,7536}, {2360,7515}, {2777,8703}, {2781,3917}, {2917,3518}, {3060,7426}, {3357,3530}, {3522,5895}, {3524,5656}, {3528,5925}, {3580,9544}, {3742,3827}, {3763,5596}, {5651,7499}, {5706,7521}, {5786,7498}, {6001,10165}, {6146,7505}, {6225,8567}, {10174,10181}

X(10192) = complement of X(1853)
X(10192) = midpoint of X(i),X(j) for these {i,j}: {2,154}, {10174,10181}
X(10192) = reflection of X(i) in X(j) for these (i,j): (549,10182)
X(10192) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,2883,5894), (140,6759,6247), (631,1498,6696), (6676,9306,141), (7505,9707,6146)

X(10193) = CENTER OF THE CENTROIDAL CONIC OF THE TRINH TRIANGLE

X(10193) = (9*R^2+SW)*X(3)+(3*R^2-SW)*X(161)

X(10193) lies on these lines:{2,2777}, {3,161}, {511,10169}, {549,6000}, {631,3357}, {632,5894}, {2781,10168}, {3523,6759}, {3526,8567}, {3530,6696}, {5070,5925}, {5448,5498}

X(10194) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE INNER-VECTEN TRIANGLE

X(10194) = 8*S*X(3)+3*(2*S-SW)*X(1328)

X(10194) lies on the Kiepert hyperbola and these lines:{2,6419}, {3,1328}, {4,5420}, {5,1327}, {6,10195}, {13,2046}, {14,2045}, {140,486}, {371,3317}, {372,1131}, {485,615}, {546,6489}, {547,3594}, {590,6500}, {598,7389}, {641,2996}, {671,7388}, {1132,3523}, {1152,3850}, {1588,3591}, {1657,6497}, {3069,3316}, {3071,6451}, {3522,6565}, {3525,9680}, {3526,6447}, {3545,6454}, {3590,6436}, {3851,6408}, {5054,9681}, {5067,6420}, {5418,6199}, {5485,7376}, {5491,6119}, {6441,8981}, {6460,6479}, {6470,7584}

X(10194) = isogonal conjugate of X(6420)
X(10194) = isotomic conjugate of anticomplement of X(32790)
X(10194) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,6426,1327)

X(10195) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE OUTER-VECTEN TRIANGLE

X(10195) = 8*S*X(3)+3*(2*S+SW)*X(1327)

X(10195) lies on the Kiepert hyperbola and these lines:{2,6420}, {3,1327}, {4,5418}, {5,1328}, {6,10194}, {13,2045}, {14,2046}, {140,485}, {371,1132}, {372,3316}, {381,9680}, {486,590}, {546,6488}, {547,3592}, {598,7388}, {615,6501}, {642,2996}, {671,7389}, {1131,3523}, {1151,3850}, {1587,3590}, {1657,6496}, {3068,3317}, {3070,6452}, {3091,9681}, {3522,6564}, {3526,6448}, {3545,6453}, {3591,6435}, {3851,6407}, {5067,6419}, {5068,9540}, {5420,6395}, {5485,7375}, {5490,6118}, {6459,6478}, {6471,7583}

X(10195) = isogonal conjugate of X(6419)
X(10195) = isotomic conjugate of X(32807)
X(10195) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,6425,1328)

X(10196) = CENTER OF THE CENTROIDAL CONIC OF THE YFF-CONTACT TRIANGLE

X(10196) = 2*X(10)+X(5592)

X(10196) lies on the cubic K700 and these lines:{2,514}, {10,5592}, {165,3667}, {190,6634}, {513,3740}, {522,3971}, {523,10180}, {649,3219}, {650,824}, {663,3961}, {812,1639}, {918,4763}, {1027,5268}, {1635,2786}, {2490,4369}, {2977,3716}, {3239,4375}, {3452,3835}, {4928,6084}

X(10196) = complement of X(6545)
X(10196) = midpoint of X(i),X(j) for these {i,j}: {2,6546}
X(10196) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6544,6546,2)

X(10197) = CENTER OF THE CENTROIDAL CONIC OF THE INNER-YFF TRIANGLE

X(10197) = R*X(1)+(R-2*r)*X(2)

X(10197) lies on these lines:{1,2}, {12,5248}, {30,7680}, {55,3822}, {381,4428}, {388,5267}, {442,8715}, {495,529}, {496,6668}, {535,5172}, {553,1454}, {1001,3814}, {1621,7951}, {2476,3746}, {3487,4084}, {3772,4868}, {3841,5687}, {3890,5443}, {3898,5886}, {3919,5657}, {4189,5270}, {4301,6825}, {4309,6871}, {4857,5141}, {5493,6908}, {5535,6173}, {5881,6852}, {5882,6862}, {6853,7982}, {7483,8666}

X(10197) = midpoint of X(i),X(j) for these {i,j}: {2,10056}
X(10197) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,551,10199), (495,6690,993), (3085,10198,10)

X(10198) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE INNER-YFF TRIANGLE

X(10198) = R*X(1)+3*(R+r)*X(2)

X(10198) lies on these lines:{1,2}, {3,6690}, {4,3822}, {5,1001}, {12,405}, {21,1478}, {35,377}, {36,6910}, {37,3767}, {40,6889}, {46,5249}, {55,442}, {56,7483}, {100,4197}, {142,5709}, {191,5905}, {197,7535}, {225,406}, {281,451}, {346,4066}, {355,6861}, {388,993}, {443,5218}, {474,5432}, {495,958}, {497,6856}, {515,6824}, {516,5715}, {518,5791}, {758,3487}, {944,6852}, {946,6825}, {999,4999}, {1056,8666}, {1104,5725}, {1376,8728}, {1385,6862}, {1479,1621}, {1656,3816}, {1788,5883}, {2077,6897}, {2078,3814}, {2475,4302}, {2478,5259}, {2550,3841}, {2551,8164}, {2886,3295}, {3072,5713}, {3090,3825}, {3189,6598}, {3193,5333}, {3434,3746}, {3436,5251}, {3475,3874}, {3485,3878}, {3525,6681}, {3526,6691}, {3560,6256}, {3576,6833}, {3579,3824}, {3583,6871}, {3585,6872}, {3654,10107}, {3739,5955}, {3754,5657}, {3772,3931}, {3813,6767}, {3817,6848}, {3826,6600}, {3884,5603}, {3925,5687}, {4187,4423}, {4189,4299}, {4190,5010}, {4193,5284}, {4208,5281}, {4228,8185}, {4293,5267}, {4294,5177}, {4297,6847}, {4512,9612}, {5070,6667}, {5257,8557}, {5436,5587}, {5450,6892}, {5691,6837}, {5698,5714}, {5731,6888}, {5886,6863}, {6361,6701}, {6585,6883}, {6796,6826}, {6834,8227}, {6886,7989}, {6887,10175}, {6890,7987}, {6891,10165}, {6933,7741}, {6953,7988}, {6964,10171}

X(10198) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1698,6734), (2,1125,10200), (2,3085,10), (2,3616,499), (2,5552,1698), (10,551,6738), (10,10197,3085), (200,1698,10), (388,6857,993), (495,6675,958), (1125,3634,9843), (1621,2476,1479), (1698,3584,5552), (3579,3824,5880), (3816,6668,1656), (3822,5248,4), (3841,8715,2550), (5259,7951,2478)

X(10199) = CENTER OF THE CENTROIDAL CONIC OF THE OUTER-YFF TRIANGLE

X(10199) = R*X(1)+(R-2*r)*X(2)

X(10199) lies on these lines:{1,2}, {30,7681}, {55,6681}, {56,535}, {214,5722}, {495,6667}, {496,528}, {912,3742}, {993,3816}, {999,3814}, {2800,5883}, {3890,5445}, {3919,5603}, {4187,8666}, {4188,4857}, {4193,5563}, {4301,6891}, {4317,5187}, {4428,5054}, {4466,6173}, {5154,5270}, {5248,5433}, {5252,6702}, {5253,7741}, {5267,7288}, {5493,6926}, {5882,6959}, {6174,8715}, {6952,9624}, {7951,10074}

X(10199) = midpoint of X(i),X(j) for these {i,j}: {2,10072}
X(10199) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,551,10197), (3086,10200,10)

X(10200) = PERSPECTOR OF THESE TRIANGLES: ABC AND 2ND-CENTROIDAL TRIANGLE OF THE OUTER-YFF TRIANGLE

X(10200) = R*X(1)+3*(R-r)*X(2)

X(10200) lies on these lines:{1,2}, {3,3816}, {4,3825}, {5,6256}, {11,474}, {35,6921}, {36,2478}, {40,6967}, {56,4187}, {104,6975}, {119,1656}, {140,1001}, {142,3358}, {214,3486}, {377,7741}, {388,3814}, {404,1479}, {405,1470}, {406,1877}, {452,5267}, {496,1376}, {515,6944}, {516,6926}, {631,2077}, {908,3338}, {946,3359}, {993,5084}, {999,1329}, {1058,8715}, {1385,6959}, {1478,4193}, {1519,5437}, {1699,6890}, {1788,3878}, {2551,8666}, {3035,3295}, {3090,3822}, {3337,5905}, {3436,5563}, {3485,5883}, {3526,6690}, {3560,6713}, {3576,6834}, {3583,4190}, {3585,5187}, {3656,10107}, {3754,5603}, {3812,5886}, {3813,9709}, {3817,6847}, {3833,6952}, {3848,6861}, {3884,5657}, {3916,4679}, {4188,4302}, {4293,6919}, {4297,6848}, {4299,5046}, {4317,5080}, {4423,7483}, {5070,6668}, {5259,6910}, {5277,9599}, {5436,6889}, {5450,6893}, {5587,6983}, {5691,6953}, {5731,6979}, {5818,6702}, {5880,9955}, {6675,8167}, {6796,6970}, {6825,10165}, {6837,7988}, {6846,10171}, {6872,7280}, {6931,7951}

X(10200) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1698,6735), (2,1125,10198), (2,3086,10), (2,3616,498), (10,10199,3086), (1698,4853,10), (3616,5554,1), (3816,6691,3), (4193,5253,1478), (5084,7288,993), (5248,6681,631)

X(10201) = 5th HATZIPOLAKIS-MOSES-EULER POINT

In the plane of a triangle ABC, let
A'B'C' = orthic triangle
A''B''C'' = tangential triangle
Oa = circumcenter of A''B'C', and define Ob and Oc cyclically.
Then X(10201) = centroid of OaObOc, on the Euler line. (Antreas Hatzipolakis and Peter Moses, August 29, 2016; see 24126).

X(10201) lies on these lines: {2,3}, {68,156}, {206,542}, {498,8144}, {1154,5654}, {1989,2165}, {5449,6759}, {5476,9969}, {6723,8717}, {9627,10056}

X(10201) = midpoint of X(5) and X(10154)
X(10201) = reflection of X(26) in X(10154)
X(10201) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,6676,7514), (5,6756,7564), (5,7568,7395), (5,7715,546), (235,7542,7526), (3542,3549,5)

X(10202) = X(1)X(3)∩X(2)X(912)

In the plane of a triangle ABC, let
A'B'C' = intouch triangle
A''B''C'' = orthic triangle of A'B'C'
Oa = circumcenter of A''B'C', and define Ob and Oc cyclically.
Then X(10202) = centroid of OAObOc, on the line X(1)X(3). For a related point and reference, see X(10201).

X(10202) lies on these lines: {1,3}, {2,912}, {5,1071}, {7,6827}, {30,10167}, {63,6883}, {72,140}, {116,119}, {226,6882}, {244,1064}, {355,3812}, {381,971}, {382,5806}, {443,5554}, {499,1858}, {515,5883}, {551,2800}, {631,3868}, {758,10165}, {916,5891}, {938,6850}, {944,6885}, {952,3753}, {1006,3218}, {1125,5884}, {1210,6842}, {1393,4303}, {1519,8727}, {1656,5777}, {1864,6980}, {1877,7510}, {1898,7741}, {2096,6930}, {3306,6911}, {3487,6891}, {3488,6948}, {3525,3876}, {3526,5044}, {3555,5690}, {3616,6892}, {3624,5693}, {3681,5552}, {3740,5791}, {3742,5886}, {3752,5396}, {3754,5882}, {3817,6245}, {3848,6861}, {3873,5657}, {3874,6684}, {4292,7491}, {5054,10156}, {5055,10157}, {5226,6978}, {5435,6954}, {5437,5720}, {5703,6961}, {5722,6923}, {5728,6907}, {5745,10176}, {5761,6926}, {5768,6826}, {5787,6256}, {5905,6947}, {6147,6922}, {6849,9799}, {6851,9812}, {6866,9960}}

X(10202) = midpoint of X(i) and X(j) for these {i,j}: {1071, 5927}, {3576, 5902}, {3873, 5657}
X(10202) = reflection of X(i) in X(j) for these (i,j): (5886,3742), (5927,5), (10175, 3833)
X(10202) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (942,9940,3), (1071,5439,5), (1125,5884,5887), (1385,5885,65), (5709,8726,3), (5768,9776,6826)

X(10203) =  X(5)-OF-ANTIPEDAL-TRIANGLE OF X(54)

X(10203) is the nine-point center of the antipedal triangle of the Kosnita point. (Tran Quang Hung and Angel Montesdeoca, August 31, 2016; see 24193).

X(10203) lies on these lines: {3, 54}, {23, 6153}, {1176, 5965}, {1614, 2888}

X(10204) =  MIDPOINT OF X(6) AND X(6)-OF-ANTIPEDAL-TRIANGLE OF X(6)

In the plane of a triangle ABC, let
K = X(6)
K' = X(6)-of-antipedial-triangle of X(6)
X(10204) = midpoint of K and K'. X(10204) lies on the Euler line of ABC. (Tran Quang Hung and Angel Montesdeoca, August 31, 2016; see 24193).

X(10204) lies on these lines: {2,3}, {6235,9871}, {8546,9830}

X(10204) = X(6)-of-1st-Ehrmann-triangle

X(10205) =  POINT BECRUX 2

X(5)-of-antipedal-triangle of X(4) = X(10205) lies on the Euler line of ABC.(Tran Quang Hung and Angel Montesdeoca, August 31, 2016; see 24193).

X(10205) lies on these lines: {2,3}, {252,1263}

X(10205) = anticomplement of X(5501)
X(10205) = X(5)-of-antipedal triangle of X(4)

X(10206) =  HUNG-MONTESDEOCA RADICAL CENTER

In the plane of a triangle ABC, let
HaHbHc = orthic triangle
Ja = incenter of AHbHc, and define Jb and Jc cyclically
Ab = JbJc∩CA, and define Bc and Ca cyclically
Ac = JcJa∩BA, and define Ba and Cb cyclically

The points Jb,Jc,Ca,Ba lie on a circle, (Oa); define (Ob) and (Oc) cyclically.

X(10206) = radical center of (Oa), (Ob), (Oc); X(10206) lies on the Euler line of OaObOc.

Let Go = X(5902) = X(2)-of-OaObOc and Ho = X(4)-of-OaObOc (used above in a combo for X(10206)). (Tran Quang Hung and Angel Montesdeoca, August 31, 2016; see 24219).

X(10206) lies on these lines: {1,201}, {942,1838}

X(10207) =  HUNG-MONTESDEOCA PERSPECTOR

In the plane of a triangle ABC, let
HaHbHc = orthic triangle
Ja = incenter of AHbHc, and define Jb and Jc cyclically
Ab = JbJc∩CA, and define Bc and Ca cyclically
Ac = JcJa⩋BA, and define Ba and Cb cyclically

The points Jb,Jc,Ca,Ba lie on a circle, (Oa); define (Ob) and (Oc) cyclically.

The triangles JaJbJc and OaObOc are perspective, and their perspector is X(10207); see X(10206). (Tran Quang Hung and Angel Montesdeoca, August 31, 2016; see 24219.

X(10207) lies on these lines: {1,71}, {942,1888}, {950,1770}, {1844,5728}

X(10208) =  1st HUNG-MONTESDEOCA-MOSES POINT

In the plane of a triangle ABC, let
FaFbFc = Feuerbach triangle
U = projection of A on line FbFc, and define V and W cyclically

The lines UFa, VFb, WFc concur in X(10208); see 23529.

X(10208) lies on these lines: {12,79}, {3614,5947}

X(10209) =  2nd HUNG-MONTESDEOCA-MOSES POINT

In the plane of a triangle ABC, let
FaFbFc = Feuerbach triangle
U = AFbFc-isogonal conjugates of Fa, and define V and W cyclically

The lines UFa, VFb, WFc concur in X(10209); see 23530. The construction was originally posted in ADGEOM #1550 by Tran Quang Hung, 9/1/2014.

X(10209) lies on these lines: {5,191}, {30,5948}, {442,5947}

X(10210) =  HUNG-LOZADA-EULER POINT

Let A'B'C' be the orthic triangle of a ABC; let Fa = X(13)-of AB'C', and define Fb and Fc cyclically. Then X(3) = X(10210)-of-FaFbFc. (Tran Quang Hung and César Lozada), 24225).

Let A'B'C' be the orthic triangle of a ABC; let F'a = X(14)-of AB'C', and define F'b and F'c cyclically. Then X(3) = X(10210)-of-FaFbFc. (Peter Moses, September 4, 2016)

X(10210) lies on these lines: {2,3}, {634,3410}, {1993,5868}

X(10210) = anticomplement of X(34008)

X(10211) =  KIRIKAMI-KOSNITA-EULER POINT

In the plane of a triangle ABC, let
P = X(54), the Kosnita point
MaMbMc = pedal triangle of X(3)
HaHbHc = pedal triangle of X(4)
The Euler lines of PMaHa, PMbHb, PMcHc concur in X(10211).

(Seiichi Kirikami, 24229).

X(10211) lies on the line {5, 49}

X(10212) =  35th HATZIPOLAKIS-MONTESDEOCA POINT

Let O be the circumcenter of a triangle ABC, and let
Na = X(5)-of-OBC, and define Nb and Nc cyclically
Aa = orthogonal projection of Na on OA, and define Ab and Ac cyclically
Ba = orthogonal projection of Nb on OA, and define OB and OC cyclically
Ca = orthogonal projection of Nc on OA, and define OB and OC cyclically
Oa = circumcenter of AaAbAc, and define Ob and Oc cyclically.

X(10212) = X(5)-of-OaObOc; X(10212) lies on the Euler line of ABC. (Antreas Hatzipolakis and Angel Montesdeoca, 24189).

X(10212) lies on these lines: {2, 3}, {10193, 32171}, {10610, 14049}, {15806, 21663}, {32165, 43394}

X(10212) = midpoint of X(3) and X(5498)

X(10213) =  36th HATZIPOLAKIS-MONTESDEOCA POINT

In the plane of a triangle ABC, let
O = circumcenter, X(3)
N = nine-point center, X(5)
Na = N-of-OBC, and define Nb and Nc cyclically
Aa = orthogonal projection of Na on OA, and define Ab and Ac cyclically
Ba = orthogonal projection of Nb on OA, and define OB and OC cyclically
Ca = orthogonal projection of Nc on OA, and define OB and OC cyclically
Ea = Euler line of AaAbAc, and define Eb and Ec cyclically
A' = Eb∩Ec, and define B' and C' cyclically

Then ABC and A'B'C' are parallelogic, and
X(10213) = (A'B'C',ABC)-parallelogic center
X(1141) = (ABC,A'B'C')-parallelogic center.

(Antreas Hatzipolakis and Angel Montesdeoca, 24183).

X(10213) lies on these lines: {140, 6368}, {539, 12100}, {6592, 17702}

X(10214) =  HUNG-LOZADA CYCLOLOGIC CENTER

In the plane of a triangle ABC, let
A'B'C' = orthic triangle
Na = nine-point center of AB'C', and define Nb and Nc cyclically
X(10214) = NaNbNc-to-A'B'C' cyclologic center
X(125) = A'B'C'-to-NaNbNc cyclologic center.

(Tran Quang Hung and César Lozada, 24287).

X(10214) lies on these lines: {4,7730}, {25,8157}, {571,2079}, {933,3518}

X(10214) = X(5620)-of-orthic-triangle if ABC is acute

X(10215) =  SCHMIDT-LOZADA CYCLOLOGIC CENTER

In the plane of a triangle ABC, let
IaIbIc = excentral triangle
Ja = incenter of IaBC
X(10215) = JaJbJc-to-ABC cyclologic center, on the circumcircle of JaJbJc
X(3659) = ABC-to-JaJbJc cyclologic center, on the circumcircle of ABC.

(Eckart Schmidt and César Lozada, 24266).

Let Aa be the Ia-excenter of IaBC, and define Bb and Cc cyclically. The lines IaAa, IbBb, IcCc concur in X(164), and the lines AAa, BBb, CCc concur in X(10215).

X(10215) lies on these lines: {1, 6724}, {10, 188}, {80, 1128}, {177, 1130}, {505, 8078}, {1488, 8241}

X(10215) = isogonal conjugate of X(10231)

X(10216) =  HATZIPOLAKIS-LOZADA-X(5) POINT

In the plane of a triangle ABC, let P be a point. Let
Ba = orthogonal projection of B on AP, and define Cb and Ac cyclically
Ca = orthogonal projection of C on AP, and define Ab and Bc cyclically
Ra = radical axis of the nine-point-circles of ABaC and ACaB, and define Rb and Rc cyclically
Sa = radical axis of the nine-point-circles of ABaB and AcaC

The locus of P for which the lines Ra, Rb, Rc concur is the Gibert quintic Q038, which passes through X(i) for i = 1, 4, 5 80, 1113, 1114, 1263, 2009, 2010.

The locus of P for which the lines Ra, Rb, Rc concur is the Gibert quintic Q066, which passes through X(i) for i = 1, 2,4,254, 1113, 1114, 1138, 2184, 3223, 3346, 3459, 8049, 9510.

For P = X(5) the point of concurrence of Ra, Rb, Rc is X(10216). For details on points of concurrence and for other choices of P, see 24453.

X(10216) lies on these lines: {4,250}, {137,143}

X(10217) =  1st HATZIPOLAKIS-LOZADA-FERMAT POINT

In the plane of a triangle ABC, let P be a point. Let
Ba = orthogonal projection of B on AP, and define Cb and Ac cyclically
Ca = orthogonal projection of C on AP, and define Ab and Bc cyclically
Ra = radical axis of the nine-point-circles of ABaC and ACaB, and define Rb and Rc cyclically

The lines Ra, Rb, Rc concur in a point, HLF(P), and HLF(X(13)) = X(10217).

In general, if P = p : q : r (trilinears), then HLF(P) = p² cos A : q² cos B : r² cos C. The appearance of (i,j) in the following list means that HLF(X(i)) = X(j):

(1,1), (2,75), (3,255), (4,158), (5,1087), (6,31), (7,1088), (8,341), (9,200), (10,1089), (11,1090), (12,1091), (13,10217), (14,10218), (15,1094), (16,1095), (19,1096), (20,1097), (21,1098), (30,1099), (31,560), (32,1917), (37,756), (40,1103), (42,872), (44,678), (46,1079), (55,1253), (56,1106), (57,269), (58,849), (63,326), (65,1254), (73,7138), (75,561), (76,1928), (81,757), (84,1256), (86,873), (88,679), (90,7042), (100,765), (101,1110), (110,1101), (174,7), (188,8), (190,7035), (192,8026), (238,8300), (259,55), (266,56), (365,6), (366,2), (483,179), (507,174), (508,85), (509,57), (513,244), (514,1111), (518,4712), (519,4738), (523,1109), (556,3596), (649,3248), (650,2310), (651,7045), (652,2638), (656,2632), (661,2643), (758,4736), (798,4117), (1049,1085), (1077,1028), (1125,6533), (1488,7002), (2089,7022), (2238,4094), (2292,6042), (3082,400), (4146,6063), (4166,220), (4179,594), (4182,346), (4367,7207), (6724,12), (6725,6057), (6726,480), (6727,60), (6728,11), (6729,3271), (6730,4081), (6731,5423), (6733,59), (7025,188), (7039,7044), (7041,7036), (7370,7023), (7371,479), (7591,7066), (9326,2226)

See 24284.

X(10217) lies on these lines: {5,8919}, {13,15}, {470,8838}, {5158,10218}

X(10217) = X(6699)-cross conjugate of X(10218)

X(10218) =  2nd HATZIPOLAKIS-LOZADA-FERMAT POINT

HLF(X(14)) = X(10218); see X(10217) and 24284.

X(10218) lies on these lines: {5,8918}, {14,16}, {471,8836}, {5158,10217}, {5619,6774}

X(10218) = X(6699)-cross conjugate of X(10217)

X(10219) =  POINT BECRUX 3

Let A'B'C' be the pedal triangle of a point P in the plane of a triangle ABC, and let
A''B''C'' = medial triangle of ABC
Ab = orthogonal projection of A'' on AC, and define Bc and Ca cyclically
Ac = orthogonal projection of A'' on AC, and define Ba and Cb cyclically
Ea = Euler line of AAbAc, and define Eb and Ec cyclically
A* = Eb∩Ec, and define B* and C* cyclically

The locus of P such that ABC and A*B*C are parallelogic is the union of two curves, K364 (a cubic through X(i) for i = 1,5,20,24,54,64,68,155,254,2917) and a curve of degree 14 in a,b,c, denoted by q4 in the reference below.

X(10219) is the A*B*C*-to-ABC parallelogic center for P = X(5). The ABC-to-A*B*C* parallelogic center is X(2).

See 24278.

X(10219) lies on these lines: {2,51}, {182,8780}, {575,3167}, {2810,3848}, {3589,8681}, {5097,5544} , {10168,10192}

X(10219) = midpoint of X(2) and X(6688)

X(10220) =  POINT BECRUX 4

Let A*B*C* be as at X(10219). Then X(10220) = ABC-to-A*B*C* parallelogic center for P = X(54). The A*B*C*-to-ABC parallelogic center is X(6).

See 24278.

X(10220) lies on the line {6709, 40136}

X(10221) =  HATZIPOLAKIS-MONTESDEOCA-EULER-PEDAL POINT

Suppose that W is a triangle center on the Euler line of a triangle ABC. Let A'B'C' be the pedal triangle of W. Then W(ABC) = W(A'B'C') if and only if W = X(10221). (Regarding the notation, recall that a triangle center is a function defined on a set of triangles, so that the notation W(T) is analogous to the notation f(x); i.e., W-of-T.) (Antreas Hatzipolakis and Angel Montesdeoca, September 13, 2016.) See 24354 and HG100916.

X(10221) lies on the cubic K019 and these lines: {2,3}

X(10222) =  CENTER OF HATZIPOLAKIS-LOZADA CIRCLE

In the plane of a triangle ABC, let I = X(1) and N = X(5); let
A'B'C' = intouch triangle
N1 = N-of-IBC, and define N2 and N3 cyclically
Na = reflection of N1 in IA, and define Nb and Nc cyclically
N'a = reflection of N1 in IA', and define N'b and N'c cyclically
R1 = perpendicular bisector of NaN'a, and define R2 and R3 cyclically

The points Na, Nb, Nc, N'a, N'b, N'c lie on a circle, here named the Hatzipolakis-Lozada circle, which has radius |r - R/2|. See 24303 (Antreas Hatzipolakis and César Lozada, September 6, 2016).

See also Antreas Hatzipolakis, César Lozada, Hyacinthos 28285.

X(10222) lies on these lines: {1, 3}, {4, 1392}, {5, 519}, {8, 3090}, {10, 3628}, {19, 23073}, {20, 3655}, {30, 4301}, {42, 19546}, {72, 1173}, {79, 14217}, {140, 551}, {145, 355}, {381, 5881}, {388, 10525}, {392, 5047}, {497, 10526}, {515, 1483}, {516, 13607}, {518, 576}, {546, 946}, {547, 4669}, {548, 5493}, {573, 3723}, {575, 1386}, {631, 3654}, {632, 1125}, {758, 11260}, {912, 15083}, {936, 11530}, {944, 3146}, {956, 3951}, {960, 18233}, {962, 3529}, {1000, 5703}, {1012, 11520}, {1056, 4323}, {1058, 4345}, {1210, 1387}, {1320, 1389}, {1457, 5399}, {1656, 3679}, {1657, 9589}, {1699, 18525}, {1766, 16884}, {1829, 10594}, {1837, 7743}, {1870, 1872}, {1953, 22356}, {2102, 15157}, {2103, 15156}, {2771, 7984}, {2800, 3881}, {2802, 19907}, {3058, 7491}, {3083, 21549}, {3084, 21546}, {3242, 11477}, {3243, 18761}, {3419, 6984}, {3434, 10597}, {3436, 10596}, {3485, 6982}, {3488, 5812}, {3518, 11363}, {3523, 3653}, {3525, 3616}, {3544, 20050}, {3555, 5887}, {3560, 12513}, {3577, 18491}, {3584, 5559}, {3585, 7972}, {3621, 5818}, {3622, 5657}, {3625, 10175}, {3632, 5079}, {3633, 5072}, {3636, 6684}, {3680, 6918}, {3751, 11482}, {3753, 17531}, {3811, 10912}, {3817, 12811}, {3827, 15581}, {3857, 19925}, {3874, 14988}, {3877, 16865}, {3880, 13374}, {3892, 5884}, {3893, 11524}, {3913, 6911}, {3915, 5398}, {3940, 4853}, {3957, 21740}, {3962, 5288}, {3991, 4919}, {4004, 5253}, {4297, 12103}, {4342, 15172}, {4511, 6946}, {4658, 15952}, {4663, 22330}, {4677, 5055}, {4691, 10172}, {4701, 10171}, {4745, 15699}, {4848, 15325}, {4864, 15310}, {4870, 6980}, {4930, 6913}, {5044, 5289}, {5054, 9588}, {5068, 20049}, {5070, 19875}, {5076, 5691}, {5198, 11396}, {5250, 19526}, {5258, 7489}, {5396, 19646}, {5440, 14923}, {5497, 10700}, {5609, 11699}, {5722, 5761}, {5727, 9669}, {5731, 17538}, {5763, 15935}, {5840, 12735}, {5853, 20330}, {5854, 10915}, {5855, 10916}, {6419, 7969}, {6420, 7968}, {6427, 18991}, {6428, 18992}, {6447, 9583}, {6519, 9616}, {6797, 12740}, {6833, 11240}, {6834, 11239}, {6842, 15888}, {6860, 12649}, {6863, 10056}, {6865, 15933}, {6914, 8666}, {6924, 8715}, {6958, 10072}, {6978, 11373}, {6988, 7320}, {7377, 17389}, {7419, 18180}, {7680, 10943}, {7681, 10942}, {7978, 15054}, {7983, 23235}, {9619, 22332}, {9708, 15829}, {10039, 15950}, {10165, 12108}, {10446, 17393}, {10573, 11376}, {10696, 19904}, {10895, 11928}, {10896, 11929}, {10944, 12047}, {11041, 14986}, {11272, 22475}, {11375, 12647}, {11496, 12559}, {11551, 11826}, {11552, 12119}, {11705, 20415}, {11706, 20416}, {11707, 21401}, {11708, 21402}, {11717, 13497}, {11724, 20399}, {11725, 20398}, {11728, 20401}, {11735, 20397}, {12053, 18527}, {12104, 22937}, {12331, 12653}, {12610, 17390}, {12773, 13253}, {12778, 15034}, {13211, 15027}, {13743, 16126}, {14563, 21625}, {16173, 19914}, {16239, 19883}, {16842, 19860}, {16862, 19861}, {17018, 19647}, {17438, 22357}, {17572, 17614}

X(10222) = midpoint of X(i) and X(j) for these {i,j}: {1, 1482}, {145, 355}, {944, 12699}, {962, 18481}, {1320, 6265}, {1657, 9589}, {3555, 5887}, {3633, 12645}, {3811, 10912}, {11496, 12559}, {12331, 12653}, {12773, 13253}, {13743, 16126}
X(10222) = reflection of X(i) in X(j) for these (i,j): (5, 13464), (8, 9956), (10, 5901)
X(10222) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 3), (1, 8148, 13624), (1, 11224, 40), (1, 11278, 3579), (1, 11531, 3576), (40, 11224, 8148), (65, 11567, 1385), (1482, 8148, 11224), (1482, 10246, 8148), (1482, 10247, 1), (3576, 11531, 12702), (7373, 10306, 10269), (7982, 16189, 1482), (7982, 16200, 16189), (8148, 10246, 40), (10680, 12000, 55)

X(10223) =  37th HATZIPOLAKIS-MONTESDEOCA POINT

Let A'B'C' be the pedal triangle of a point P in the plane of a triangle ABC. Let
A'' = reflection of A' in the Euler line, and define B'' and C'' cyclically
Na = X(5)-of-A''B''C'', and define Nb and Nc cyclically

The locus of P such that Na, Nb, Nc are collinear is the union of the cubic K187 and a circum-quintic that passes through X(74) and X(1304). If P = X(4), the line NaNbNc meets the Euler line in X(10223). (Antreas Hatzipolakis and Angel Montesdeoca, September 14, 2016.) See 24377.

X(10223) lies on these lines: {2,3}, {143,523}

X(10224) =  COMPLEMENT OF X(1658)

As a pont of the Euler line, X(10224) has Shinagawa coefficients X(10224) = (E-8*F, -3*E-8*F).

In the plane of a triangle ABC, let A'B'C' = pedal triangle of N (the nine-point center, X(5)), and let
Na = N-of-PBC), and define Nb and Nc cyclically
N1 = reflection of Na in NA', and define N2 and N3 cyclically

The locus of a point P such that the circumcenter of N1N2N3 lies on the Euler line is an excentral circumquintic that passes through X(i) for these i: 1,3,4,54,110. The appearance of (i,j) in the following list means that if P = X(i), then X(j) is the circumcenter of N1N2N3: (1,5), (3, 10224), (4,5), 54,5576), (110,403). (Antreas Hatzipolakis and César Lozada, September 10, 2016.) See 24351.

Let A' be the reflection in BC of the A-vertex of the anticevian triangle of X(5). Let Oa be the circumcenter of AB'C', and define Ob and Oc cyclically. X(10224) is the circumcenter of OaObOc. (Randy Hutson, December 10, 2016)

For another construction of X(10224), see Antreas P. Hatzipolakis and Peter Moses, Euclid 2615 .

X(10224) lies on these lines: {2,3}, {125,6102}, {569,8254}, {1154,5449}, {1568,5876}, {3574,5946}, {5448,5663}, {7741,8144}

X(10224) = reflection of X(i) in X(j) for these (i,j): (3,5498), (1658,10125), (10020,3628)
X(10224) = anticomplement of X(10125)
X(10224) = complement of X(1658)
X(10224) = X(26285)-of-orthic-triangle if ABC is acute

X(10225) =  MIDPOINT OF X(3) AND X(484)

Let P be a point in the plane of a triangle ABC, and let
Na = X(5)-of-PBC), and define Nb and Nc cyclically
Nab = reflection of Na in AC, and define Nbc and Nca cyclically
Nac = reflection of Na in AB, and define Nba and Ncb cyclically
Sa = prependicular bisector of NbaNca, and define Sb and Sc cyclically
The locus of P for which Sa, Sb, Sc concur is the union of several curves (see the reference). For P = X(1), the point of concurrence is X(10225). (Antreas Hatzipolakis and César Lozada, September 14, 2016.) See 24351.

X(10225) lies on these lines:
{1,3}, {631,5180}, {2475,9956}, {3814,4640}, {3916,5176}, {4973,5844}, {5057,6853}, {5080,6951}, {5499,6684}, {5886,9352}, {6952,9955}, {6972,7704}

X(10226) =  6th HATZIPOLAKIS-MOSES-EULER POINT

Let P be a point in the plane of a triangle ABC, and let
O = X(3)
Na = X(5)-of-OBC, and define Nb and Nc cyclically
A' = reflection of Na in OA, and define B' and C' cyclically
Then X(10026) = X(3)-of-A'B'C'; this point lies on the Euler line of ABC. (Antreas Hatzipolakis and Peter Moses, September 14, 2016.) See 24286.

X(10226) lies on these lines: {2,3}, {49,74}, {156,3357}, {5449,10193}, {7280,8144}

X(10226) = midpoint of X(156) and X(3357)
X(10226) = reflection of X(i) in X(j) for these (i,j): (5,5498), (10020,3530), (10125,10212)
X(10226) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,2071,550), (3,3516,6644), (3,3520,5), (10125,10212,549)

X(10227) =  HUNG-MONTESDEOCA-EULER POINT

Let N be the nine-point center of a triangle ABC. Let A'B'C' be the circumcevian triangle of N, and let A''B''C'' be the pedal triangle of N with respect to A'B'C'. The Euler lines of A'B''C'', B'C''A'', C'A''B'' concur in X(10227). (Tran Quang Hung and Angel Montesdeoca, September 14, 2016.) See 24387.

X(10227) lies on these lines: {5, 14367}, {110, 10203}

X(10228) =  HUNG-MONTESDEOCA CENTER OF SIMILITUDE

a^2 (a^26-8 a^24 (b^2+c^2)+28 a^22 (b^2+c^2)^2-6 a^20 (9 b^6+28 b^4 c^2+28 b^2c^4+9 c^6)+a^18 (53 b^8+277 b^6 c^2+406 b^4 c^4+277 b^2 c^6+53 c^8)+a^16 (6 b^10-273 b^8 c^2-499 b^6 c^4-499 b^4 c^6-273 b^2 c^8+6 c^10)+a^14 (-96 b^12+184 b^10 c^2+307 b^8 c^4+386 b^6 c^6+307 b^4 c^8+184 b^2 c^10-96 c^12)+2 a^12 (66 b^14-56 b^12 c^2-9 b^10 c^4-38 b^8 c^6-38 b^6 c^8-9 b^4 c^10-56 b^2 c^12+66 c^14) -a^10 (69 b^16+6 b^14 c^2+97 b^12 c^4+43 b^10 c^6+5 b^8 c^8+43 b^6 c^10+97 b^4 c^12+6 b^2 c^14+69 c^16)-a^8 (28 b^18-238 b^16 c^2+176 b^14 c^4-11 b^12c^6+63b^10 c^8+63 b^8 c^10-11 b^6 c^12+176 b^4 c^14-238 b^2 c^16+28 c^18)+a^6 (b^2-c^2)^2 (68 b^16-256 b^14 c^2+89 b^12 c^4+65 b^10 c^6+97 b^8 c^8+65 b^6 c^10+89 b^4 c^12-256 b^2 c^14+68 c^16)-a^4 (b^2-c^2)^4 (46 b^14-120 b^12 c^2+26 b^10 c^4+73 b^8 c^6+73 b^6 c^8+26 b^4 c^10-120 b^2 c^12+46 c^14) +a^2 (b^2-c^2)^6 (15 b^12-29 b^10 c^2+16 b^8 c^4+32 b^6 c^6+16 b^4 c^8-29 b^2 c^10+15 c^12)-(b^2-c^2)^8 (2 b^10-3 b^8 c^2+5 b^6 c^4+5 b^4 c^6-3 b^2 c^8+2 c^10) )

Let N be the nine-point center of a triangle ABC. Let A'B'C' be the circumcevian triangle of N, and let A''B''C'' be the pedal triangle of N with respect to A'B'C'. The triangle A''B''C'' is similar to ABC, and the center of similitude is X(10228). (Tran Quang Hung and Angel Montesdeoca, September 14, 2016.) See 24387.

X(10228) lies on the line {1157, 13621}

X(10229) =  1st SCHMIDT CYCLOLOGIC CENTER

Let OaObOc be the anticevian triangle of X(3). Let Pa = X(3)-of-BCOa, and define Pb and Pc cyclically. Then X(10229) = PaPbPc-to-ABC cyclologic center; see X(10152 for the 2nd Schmidt cyclologic center and Quadri-Figures-Group #1936

X(10229) lies on these lines: {25, 125}, {1593, 13526}, {2867, 30737}

X(10230) =  POINT BECRUX 5

Let I = X(1). Let A' be the I-excenter of BCI, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(10230). (Randy Hutson, December 10, 2016)

X(10230) lies on these lines: {79,1127}, {174,482}, {483,8351}, {3082,8092}

X(10230) = isogonal conjugate of X(10232)

X(10231) =  ISOGONAL CONJUGATE OF X(10215)

X(10231) lies on these lines: {1,164}, {43,361}, {168,1743}, {188,8092}, {1128,10023}, {6726,8090}

X(10231) = isogonal conjugate of X(10215)
X(10231) = Gibert-Burek-Moses concurrent circles image of X(164)
X(10231) = X(1324)-of-intouch triangle
X(10231) = {X(1),X(266)}-harmonic conjugate of X(1130)

X(10232) =  ISOGONAL CONJUGATE OF X(10230)

X(10232) lies on this line: {1,168}

X(10232) = isogonal conjugate of X(10230)
X(10232) = {X(1),X(259)}-harmonic conjugate of X(1129)

X(10233) =  X(1)X(188)∩X(164)X(166)

X(10233) lies on these lines: {1,188}, {164,166}, {165,505}, {167,8140}, {6726,8423}, {8089,10215}

X(10233) = {X(1),X(188)}-harmonic conjugate of X(10234)

X(10234) =  X(1)X(188)∩X(164)X(165)

X(10234) lies on these lines: {1,188}, {164,165}, {505,7991}, {6726,8090}, {7587,8078}

X(10234) = {X(1),X(188)}-harmonic conjugate of X(10233)

X(10235) =  POINT BECRUX 6

X(10235) lies on these lines: {1,167}, {482,9795}, {1129,7370}

X(10235) = {X(1),X(174)}-harmonic conjugate of X(10236)

X(10236) =  POINT BECRUX 7

X(10236) lies on these lines: {1,167}, {481,9795}

X(10236) = {X(1),X(174)}-harmonic conjugate of X(10235)

Eulerologic centers: X(10237)-X(10259)

X(10237) = (ABC, LUCAS TANGENTS)-EULEROLOGIC CENTER

X(10237) lies on these lines:{2,3}, {371,8825}, {8939,9922}

X(10238) = (LUCAS TANGENTS, ABC)-EULEROLOGIC CENTER

X(10238) lies on these lines:{3,485}, {1151,8155}

X(10239) = (ABC, LUCAS(-1) TANGENTS)-EULEROLOGIC CENTER

X(10239) lies on these lines:{2,3}, {8943,9921}

X(10240) = (LUCAS(-1) TANGENTS, ABC)-EULEROLOGIC CENTER

X(10240) lies on these lines:{3,486}, {1152,8156}

X(10241) = (ANTICOMPLEMENTARY, ATIK)-EULEROLOGIC CENTER

X(10241) lies on these lines:{4,8}, {57,3062}, {971,7956}, {1538,5249}, {1699,8581}, {6244,8580}, {7682,9948}

X(10242) = (2ND NEUBERG, ANTICOMPLEMENTARY)-EULEROLOGIC CENTER

X(10242) lies on these lines:{4,2896}, {316,6321}, {1656,6781}, {8597,8724}

X(10243) = (2ND EULER, ARA)-EULEROLOGIC CENTER

X(10243) lies on the line {2,3}

X(10243) = midpoint of X(7387) and X(7393)
X(10243) = X(7401)-of-Ara-triangle

X(10244) = (KOSNITA, ARA)-EULEROLOGIC CENTER

Shinagawa coefficients: (-2*E-7*F, 6*E+7*F)

X(10244) lies on the line {2,3}

X(10244) = X(3090)-of-Ara-triangle
X(10244) = orthic-to-Kosnita similarity image of X(3528)
X(10244) = {X(3),X(26)}-harmonic conjugate of X(10245)

X(10245) = (TRINH, ARA)-EULEROLOGIC CENTER

Shinagawa coefficients: (-2*E-9*F, 6*E+9*F)

X(10245) lies on the line {2,3}

X(10245) = {X(3),X(26)}-harmonic conjugate of X(10244)
X(10245) = X(3545)-of-Ara-triangle

X(10246) = (5TH MIXTILINEAR, 2ND CIRCUMPERP)-EULEROLOGIC CENTER

The point G' = X(10246) point is collinear with the incenter I = X(1) and the circumcenter O = X(3), and |G'O|/|G'I} = 2:1. Moreover, G' is center of the circular the locus of the centroid of certain poristic triangles; see The Triangles Web, result 195. (César Lozada, November 7, 2021).

X(10246) lies on these lines: {1,3}, {2,952}, {4,3622}, {5,944}, {8,140}, {10,3526}, {11,6980}, {12,6971}, {30,5603}, {104,1621}, {119,3816}, {145,631}, {149,6951}, {153,6965}, {182,3242}, {214,1376}, {278,7510}, {355,1125}, {381,515}, {382,946}, {388,6928}, {390,6948}, {392,912}, {405,3897}, {495,3476}, {496,3486}, {497,1387}, {516,3534}, {518,5050}, {519,3653}, {548,5734}, {549,3241}, {550,962}, {632,9780}, {758,4930}, {945,7100}, {956,3681}, {958,10176}, {993,5289}, {995,5396}, {997,3740}, {1000,5281}, {1001,2801}, {1006,4430}, {1056,5719}, {1058,6850}, {1064,1149}, {1317,5432}, {1351,1386}, {1384,1572}, {1484,6224}, {1511,7984}, {1537,6938}, {1595,7718}, {1657,4297}, {1699,3830}, {1766,3723}, {1829,3517}, {2649,9374}, {2771,5426}, {2800,3898}, {2802,4421}, {2975,3927}, {3058,5840}, {3085,6958}, {3086,6863}, {3158,10156}, {3244,6684}, {3251,3309}, {3311,7968}, {3312,7969}, {3357,7973}, {3485,7491}, {3487,4308}, {3488,6907}, {3523,3623}, {3525,3617}, {3586,7743}, {3600,6147}, {3624,5070}, {3628,5550}, {3649,4317}, {3654,10164}, {3843,5691}, {3845,9779}, {3851,8227}, {3871,6940}, {3884,5884}, {4189,5330}, {4413,6264}, {4423,6326}, {4861,5687}, {5024,9620}, {5055,5587}, {5253,6924}, {5274,6982}, {5436,5777}, {5437,7966}, {5441,9670}, {5444,7972}, {5453,9840}, {5534,8583}, {5703,6049}, {5927,6913}, {6001,10179}, {6199,9583}, {6407,9615}, {6445,9616}, {6474,9585}, {6642,8192}, {6771,7975}, {6774,7974}, {6978,8164}, {7506,9798}, {8703,9778}, {9605,9619}, {9618,10145}

X(10246) = midpoint of X(i),X(j) for these {i,j}: {1,3576}, {2,7967}, {3241,5657}, {3655,5886}, {5603,5731}, {5882,10175}
X(10246) = center of circle that is the poristic locus of X(2)
X(10246) = reflection of X(i) in X(j) for these (i,j): (3,3576), (355,10175), (381,5886), (3576,1385), (3654,10164), (3830,1699), (5054,3653), (5657,549), (5790,2), (5886,551), (9778,8703), (10175,1125)
X(10246) = anticomplement of X(38042)
X(10246) = X(355)-of-2nd-circumperp-triangle
X(10246) = X(10247)-of-5th-mixtilinear-triangle
X(10246) = excentral-to-2nd-circumperp similarity image of X(3576)
X(10246) = trisector nearest X(1) of segment X(1)X(3)
X(10246) = X(3845)-of-excentral-triangle
X(10246) = X(20) of cross-triangle of Fuhrmann and Ai (aka K798i) triangles
X(10246) = {X(1),X(3)}-harmonic conjugate of X(1482)

X(10247) = (JOHNSON, 5TH MIXTILINEAR)-EULEROLOGIC CENTER

X(10247) lies on these lines:{1,3}, {2,5844}, {4,1483}, {5,145}, {8,1656}, {10,5070}, {30,7967}, {140,3622}, {195,7979}, {355,3244}, {381,952}, {382,944}, {392,5644}, {399,7984}, {405,5330}, {495,6980}, {496,6971}, {515,3656}, {516,3655}, {518,4930}, {519,5055}, {946,3635}, {956,7489}, {962,1657}, {1056,6923}, {1058,6928}, {1317,1478}, {1320,6911}, {1351,3242}, {1389,1392}, {1484,6830}, {2800,3892}, {3090,3621}, {3243,5779}, {3488,4345}, {3526,3616}, {3534,5731}, {3560,4430}, {3617,3628}, {3632,9624}, {3633,8227}, {3653,10164}, {3654,10165}, {3681,4861}, {3871,6924}, {4323,6147}, {4669,10172}, {4678,5067}, {5054,5657}, {5073,5882}, {5079,5818}, {5289,9708}, {5434,5840}, {5727,7743}, {5762,8236}, {5881,9955}, {6417,7969}, {6418,7968}, {7517,8192}, {9301,9997}, {9616,9690}

X(10247) = midpoint of X(i),X(j) for these {i,j}: {165,7982}, {944,9812}, {1482,10246}, {3241,5603}, {3244,3817}
X(10247) = reflection of X(i) in X(j) for these (i,j): (3,10246), (165,1385), (355,3817), (381,5603), (382,9812), (3534,5731), (3654,10165), (4669,10172), (5790,5886), (10202,5049), (10246,1)
X(10247) = centroid of X(3)X(5)X(8)
X(10247) = X(10246)-of-5th-mixtilinear-triangle
X(10247) = {X(1),X(3)}-harmonic conjugate of X(37624)

X(10248) = (OUTER-GARCIA, 2ND CONWAY)-EULEROLOGIC CENTER

X(10248) lies on these lines:{4,8}, {7,5225}, {20,5550}, {40,3839}, {149,9797}, {165,5068}, {382,5731}, {516,3832}, {546,6361}, {938,3583}, {944,3830}, {946,3543}, {1125,5059}, {1483,3853}, {1698,3854}, {1699,3146}, {1770,5704}, {2476,7965}, {3091,6684}, {3241,5691}, {3522,3817}, {3529,9955}, {3579,3855}, {3617,9589}, {3627,5603}, {3648,5789}, {3843,5657}, {3845,5818}, {5226,6284}, {5229,9785}, {5261,9580}, {5274,9579}, {5714,9668}, {5806,9961}, {6895,9782}, {8236,9670}

X(10248) = X(3090)-of-2nd-Conway-triangle

X(10249) = (2ND EHRMANN, TRINH)-EULEROLOGIC CENTER

X(10249) lies on these lines:{2,154}, {3,2393}, {6,74}, {64,7527}, {159,5092}, {182,6000}, {575,3357}, {1350,2071}, {2777,5476}, {3066,10117}, {6696,8550}

X(10250) = (TRINH, 2ND EHRMANN)-EULEROLOGIC CENTER

X(10250) lies on these lines:{6,1597}, {154,5050}, {182,2393}, {511,10249}, {542,5654}, {575,6759}, {576,2781}, {1204,8537}, {1503,3845}, {5622,8541}

X(10250) = reflection of X(5476) in X(10169)

X(10251) = (EXTANGENTS, OUTER-GARCIA)-EULEROLOGIC CENTER

X(10251) lies on these lines:{1,7561}, {8,5285}, {10,1848}, {30,40}

X(10251) = midpoint of X(8) and X(7520)
X(10251) = reflection of X(i) in X(j) for these (i,j): (1,7561), (5142,10)
X(10251) = X(5142)-of-outer-Garcia-triangle

X(10252) = (INTOUCH, INNER-SODDY)-EULEROLOGIC CENTER

Let S be the inner Soddy center, X(176). Let Ia be the incenter of BCS, and define Ib and Ic cyclically. Let Oa be the circumcenter of BCS, and define Ob and Oc cyclically. Triangles IaIbIc and OaObOc are homothetic, and the center of homothety is X(10252). (Randy Hutson, December 10, 2016)

X(10252) lies on these lines:{1,3}, {176,1123}

X(10253) = (INTOUCH, OUTER-SODDY)-EULEROLOGIC CENTER

Let S' be the outer Soddy center, X(175). Let Ia' be the incenter of BCS', and define Ib' and Ic' cyclically. Let Oa' be the circumcenter of BCS', and define Ob' and Oc' cyclically. Triangles Ia'Ib'Ic' and Oa'Ob'Oc' are homothetic, and the center of homothety is X(10253). (Randy Hutson, December 10, 2016)

X(10253) lies on these lines:{1,3}, {175,1336}

X(10254) = (KOSNITA, JOHNSON)-EULEROLOGIC CENTER

Shinagawa coefficients: (E+8*F, -E+8*F)

X(10254) lies on these lines:{2,3}, {49,9927}, {184,265}, {216,9220}, {217,8571}, {1204,3521}, {1989,5158}, {3060,7699}

X(10254) = reflection of X(7577) in X(5)
X(10254) = X(7577)-of-Johnson-triangle
X(10254) = {X(3),X(5)}-harmonic conjugate of X(10255)
X(10254) = homothetic center of X(3)-Ehrmann triangle and cross-triangle of Ehrmann side- and Ehrmann vertex-triangles

X(10255) = (TRINH, JOHNSON)-EULEROLOGIC CENTER

X(10255) = (9*R^2-2*SW)*X(3)+2*(4*R^2-SW)*X(4)

Shinagawa coefficients: (E-8*F, -E-8*F)

X(10255) lies on these lines:{2,3}, {113,7729}, {125,5448}, {265,1147}, {339,7752}, {1568,5449}, {3289,8571}, {3357,7728}, {6102,7723}, {6146,9704}, {6288,9306}, {9703,9820}

X(10255) = {X(3),X(5)}-harmonic conjugate of X(10254)

X(10256) = (1ST NEUBERG, MEDIAL)-EULEROLOGIC CENTER

X(10256) lies on these lines:{3,7694}, {30,9167}, {39,140}, {262,7807}, {325,631}, {523,7663}, {524,5050}, {549,7818}, {3523,7925}, {3926,7612}, {6036,6390}, {6194,7907}, {7697,7789}, {7763,9755}

X(10257) = (TRINH, MEDIAL)-EULEROLOGIC CENTER

Shinagawa coefficients: (E-6*F, -E+2*F)

X(10257) lies on these lines:{2,3}, {185,9820}, {339,6390}, {523,7663}, {974,6699}, {1060,5432}, {1062,5433}, {1352,10249}, {3564,5622}, {5972,6000}, {6389,6795}

X(10257) = midpoint of X(i),X(j) for these {i,j}: {3,2072}, {186,858}, {403,2071}
X(10257) = reflection of X(i) in X(j) for these (i,j): (2072,5159), (10151,5)
X(10257) = complement of X(403)
X(10257) = circumcircle-inverse-of-X(7387)
X(10257) = first Droz-Farny circle-inverse-of-X(3548)
X(10257) = orthoptic circle of Steiner inellipse-inverse-of-X(1370)
X(10257) = polar circle-inverse-of-X(3542)
X(10257) = inverse-in-complement-of-polar-circle of X(5)
X(10257) = radical trace of nine-point circle and first Droz-Farny circle

X(10258) = (ABC, 2ND MORLEY)-EULEROLOGIC CENTER

X(10258) lies on these lines:{1134,1136}, {3278,5390}

X(10259) = (ABC, 3RD MORLEY)-EULEROLOGIC CENTER

X(10259) lies on these lines:{357,1134}, {3282,5390}

X(10260) =  CIRCUMCIRCLE-INVERSE OF X(80)

X(10260) lies on these lines: {1,3417}, {3,80}, {35,1807}, {36,1411}, {759,4225}, {1054,6187}, {2070,5172}, {2166,5961}, {2222,9590}

X(10261) =  1st HUNG-LOZADA-GREBE POINT

In the plane of a triangle ABC, let A'B'C' be the inner Grebe triangle. Let A'' = reflection of A' in B'C', and define B'' and C'' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is X(10261).

Choices of a triangle T = A'B'C' other than the inner Grebe triangle provide reflection triangles A''B''C'' that are also perpsective to ABC. The appearance of (T,n) in the following list means that ABC and A''B''C'' are perspective with perspector X(n): (ABC,4), (2nd circumperp,1), (Euler, 4), (excentral, 1), (2nd extouch, 943), (inner-Grebe, 10261), (outer-Grebe, 10262), (incentral, 3065), (1st Morley, 1134), (2nd Morley, 357), (3rd Morley, 1136), (inner-Napoleon, 4), (outer-Napoleon, 4), (orthic, 186), (orthocentroidal, 3), (inner-Vecten, 486), (outer-Vecten, 485). See 24193).

X(10261) lies on the Jerabek hyperbola and these lines: {54,5871}, {1173,6202}

X(10261) =isogonal conjugate of X(13616)

X(10262) =  2nd HUNG-LOZADA-GREBE POINT

In the plane of a triangle ABC, let A'B'C' be the outer Grebe triangle. Let A'' = reflection of A' in B'C', and define B'' and C'' cyclically. Then A''B''C'' is perspective to ABC, and the perspector is X(10262). See X(10261).

X(10262) lies on the Jerabek hyperbola and these lines: {54,5870}, {1173,6201}

X(10262) =isogonal conjugate of X(13617)

X(10263) =  HATZIPOLAKIS-MOSES IMAGE OF X(3)

Let P be a point in the plane of a triangle ABC, and let
Ab = orthogonal projection of A on PB, and define Bc and Ca cyclically
Ac = orthogonal projection of A on PC, and define Ca and Ab cyclically
Ma = midpoint of Ab and Ac, and define Mb and Mc cyclically
M1 = midpoint of segment AP
L1 = line MaM1, and define L2 and L3 cyclically
Let HM7 denote the circular circumseptic curve given by this barycentric equation:

a^4 b^2 c^4 x^5 y^2-b^6 c^4 x^5 y^2-3 a^2 b^2 c^6 x^5 y^2-b^4 c^6 x^5 y^2+2 b^2 c^8 x^5 y^2+a^6 c^4 x^4 y^3+2 a^4 b^2 c^4 x^4 y^3-a^2 b^4 c^4 x^4 y^3-2 b^6 c^4 x^4 y^3-3 a^4 c^6 x^4 y^3-4 a^2 b^2 c^6 x^4 y^3+3 b^4 c^6 x^4 y^3+3 a^2 c^8 x^4 y^3-c^10 x^4 y^3+2 a^6 c^4 x^3 y^4+a^4 b^2 c^4 x^3 y^4-2 a^2 b^4 c^4 x^3 y^4-b^6 c^4 x^3 y^4-3 a^4 c^6 x^3 y^4+4 a^2 b^2 c^6 x^3 y^4+3 b^4 c^6 x^3 y^4-3 b^2 c^8 x^3 y^4+c^10 x^3 y^4+a^6 c^4 x^2 y^5-a^2 b^4 c^4 x^2 y^5+a^4 c^6 x^2 y^5+3 a^2 b^2 c^6 x^2 y^5-2 a^2 c^8 x^2 y^5-a^4 b^4 c^2 x^5 y z+2 a^2 b^6 c^2 x^5 y z-b^8 c^2 x^5 y z+a^4 b^2 c^4 x^5 y z-3 b^6 c^4 x^5 y z-2 a^2 b^2 c^6 x^5 y z+3 b^4 c^6 x^5 y z+b^2 c^8 x^5 y z-2 a^4 b^4 c^2 x^4 y^2 z+4 a^2 b^6 c^2 x^4 y^2 z-2 b^8 c^2 x^4 y^2 z-7 a^2 b^4 c^4 x^4 y^2 z+3 b^6 c^4 x^4 y^2 z+a^2 b^2 c^6 x^4 y^2 z-b^2 c^8 x^4 y^2 z+a^8 c^2 x^3 y^3 z-2 a^6 b^2 c^2 x^3 y^3 z+2 a^2 b^6 c^2 x^3 y^3 z-b^8 c^2 x^3 y^3 z-3 a^6 c^4 x^3 y^3 z-3 a^4 b^2 c^4 x^3 y^3 z+3 a^2 b^4 c^4 x^3 y^3 z+3 b^6 c^4 x^3 y^3 z+3 a^4 c^6 x^3 y^3 z-3 b^4 c^6 x^3 y^3 z-a^2 c^8 x^3 y^3 z+b^2 c^8 x^3 y^3 z+2 a^8 c^2 x^2 y^4 z-4 a^6 b^2 c^2 x^2 y^4 z+2 a^4 b^4 c^2 x^2 y^4 z-3 a^6 c^4 x^2 y^4 z+7 a^4 b^2 c^4 x^2 y^4 z-a^2 b^2 c^6 x^2 y^4 z+a^2 c^8 x^2 y^4 z+a^8 c^2 x y^5 z-2 a^6 b^2 c^2 x y^5 z+a^4 b^4 c^2 x y^5 z+3 a^6 c^4 x y^5 z-a^2 b^4 c^4 x y^5 z-3 a^4 c^6 x y^5 z+2 a^2 b^2 c^6 x y^5 z-a^2 c^8 x y^5 z-a^4 b^4 c^2 x^5 z^2+3 a^2 b^6 c^2 x^5 z^2-2 b^8 c^2 x^5 z^2+b^6 c^4 x^5 z^2+b^4 c^6 x^5 z^2-a^2 b^6 c^2 x^4 y z^2+b^8 c^2 x^4 y z^2+2 a^4 b^2 c^4 x^4 y z^2+7 a^2 b^4 c^4 x^4 y z^2-4 a^2 b^2 c^6 x^4 y z^2-3 b^4 c^6 x^4 y z^2+2 b^2 c^8 x^4 y z^2-8 a^4 b^4 c^2 x^3 y^2 z^2+8 a^2 b^6 c^2 x^3 y^2 z^2+8 a^4 b^2 c^4 x^3 y^2 z^2-8 a^2 b^2 c^6 x^3 y^2 z^2-8 a^6 b^2 c^2 x^2 y^3 z^2+8 a^4 b^4 c^2 x^2 y^3 z^2-8 a^2 b^4 c^4 x^2 y^3 z^2+8 a^2 b^2 c^6 x^2 y^3 z^2-a^8 c^2 x y^4 z^2+a^6 b^2 c^2 x y^4 z^2-7 a^4 b^2 c^4 x y^4 z^2-2 a^2 b^4 c^4 x y^4 z^2+3 a^4 c^6 x y^4 z^2+4 a^2 b^2 c^6 x y^4 z^2-2 a^2 c^8 x y^4 z^2+2 a^8 c^2 y^5 z^2-3 a^6 b^2 c^2 y^5 z^2+a^4 b^4 c^2 y^5 z^2-a^6 c^4 y^5 z^2-a^4 c^6 y^5 z^2-a^6 b^4 x^4 z^3+3 a^4 b^6 x^4 z^3-3 a^2 b^8 x^4 z^3+b^10 x^4 z^3-2 a^4 b^4 c^2 x^4 z^3+4 a^2 b^6 c^2 x^4 z^3+a^2 b^4 c^4 x^4 z^3-3 b^6 c^4 x^4 z^3+2 b^4 c^6 x^4 z^3-a^8 b^2 x^3 y z^3+3 a^6 b^4 x^3 y z^3-3 a^4 b^6 x^3 y z^3+a^2 b^8 x^3 y z^3+2 a^6 b^2 c^2 x^3 y z^3+3 a^4 b^4 c^2 x^3 y z^3-b^8 c^2 x^3 y z^3-3 a^2 b^4 c^4 x^3 y z^3+3 b^6 c^4 x^3 y z^3-2 a^2 b^2 c^6 x^3 y z^3-3 b^4 c^6 x^3 y z^3+b^2 c^8 x^3 y z^3+8 a^6 b^2 c^2 x^2 y^2 z^3-8 a^2 b^6 c^2 x^2 y^2 z^3-8 a^4 b^2 c^4 x^2 y^2 z^3+8 a^2 b^4 c^4 x^2 y^2 z^3-a^8 b^2 x y^3 z^3+3 a^6 b^4 x y^3 z^3-3 a^4 b^6 x y^3 z^3+a^2 b^8 x y^3 z^3+a^8 c^2 x y^3 z^3-3 a^4 b^4 c^2 x y^3 z^3-2 a^2 b^6 c^2 x y^3 z^3-3 a^6 c^4 x y^3 z^3+3 a^4 b^2 c^4 x y^3 z^3+3 a^4 c^6 x y^3 z^3+2 a^2 b^2 c^6 x y^3 z^3-a^2 c^8 x y^3 z^3-a^10 y^4 z^3+3 a^8 b^2 y^4 z^3-3 a^6 b^4 y^4 z^3+a^4 b^6 y^4 z^3-4 a^6 b^2 c^2 y^4 z^3+2 a^4 b^4 c^2 y^4 z^3+3 a^6 c^4 y^4 z^3-a^4 b^2 c^4 y^4 z^3-2 a^4 c^6 y^4 z^3-2 a^6 b^4 x^3 z^4+3 a^4 b^6 x^3 z^4-b^10 x^3 z^4-a^4 b^4 c^2 x^3 z^4-4 a^2 b^6 c^2 x^3 z^4+3 b^8 c^2 x^3 z^4+2 a^2 b^4 c^4 x^3 z^4-3 b^6 c^4 x^3 z^4+b^4 c^6 x^3 z^4-2 a^8 b^2 x^2 y z^4+3 a^6 b^4 x^2 y z^4-a^2 b^8 x^2 y z^4+4 a^6 b^2 c^2 x^2 y z^4-7 a^4 b^4 c^2 x^2 y z^4+a^2 b^6 c^2 x^2 y z^4-2 a^4 b^2 c^4 x^2 y z^4+a^8 b^2 x y^2 z^4-3 a^4 b^6 x y^2 z^4+2 a^2 b^8 x y^2 z^4-a^6 b^2 c^2 x y^2 z^4+7 a^4 b^4 c^2 x y^2 z^4-4 a^2 b^6 c^2 x y^2 z^4+2 a^2 b^4 c^4 x y^2 z^4+a^10 y^3 z^4-3 a^6 b^4 y^3 z^4+2 a^4 b^6 y^3 z^4-3 a^8 c^2 y^3 z^4+4 a^6 b^2 c^2 y^3 z^4+a^4 b^4 c^2 y^3 z^4+3 a^6 c^4 y^3 z^4-2 a^4 b^2 c^4 y^3 z^4-a^4 c^6 y^3 z^4-a^6 b^4 x^2 z^5-a^4 b^6 x^2 z^5+2 a^2 b^8 x^2 z^5-3 a^2 b^6 c^2 x^2 z^5+a^2 b^4 c^4 x^2 z^5-a^8 b^2 x y z^5-3 a^6 b^4 x y z^5+3 a^4 b^6 x y z^5+a^2 b^8 x y z^5+2 a^6 b^2 c^2 x y z^5-2 a^2 b^6 c^2 x y z^5-a^4 b^2 c^4 x y z^5+a^2 b^4 c^4 x y z^5-2 a^8 b^2 y^2 z^5+a^6 b^4 y^2 z^5+a^4 b^6 y^2 z^5+3 a^6 b^2 c^2 y^2 z^5-a^4 b^2 c^4 y^2 z^5 = 0.

The curve HM7 passes through X(i) for these i: 1,3,5,13,14,30,1113,1114,1157,5000,5001. The curve also circumscribes ABC and the excentral triangle. The locus of P for which the lines L1, L2, L3 concur is the union of the circumcircle and HM7. For P on that locus, the point of concurrence is here named the Hatzipolakis-Moses image of P. The appearance of (i,j) in the following list means that X(j) = Hatzipolakis-Moses image of X(i): (1,946), (3,10263), (13,13), (14,14), (74,10264), (80,10265).

X(10263) lies on these lines: {2,10095}, {3,143}, {4,93}, {5,141}, {6,9683}, {20,568}, {23,49}, {24,1112}, {26,5944}, {30,52}, {51,140}, {54,2937}, {155,2930}, {156,1993}, {184,1493}, {195,1614}, {378,6746}, {382,5663}, {389,550}, {546,5562}, {548,9730}, {549,5462}, {567,7512}, {569,7525}, {576,8546}, {578,7502}, {632,5447}, {1337,3442}, {1338,3443}, {1350,7516}, {1351,7387}, {1598,6403}, {1656,2979}, {1657,5890}, {3520,3581}, {3526,5640}, {3527,7393}, {3628,3917}, {3845,5907}, {3850,5891}, {5055,7999}, {5070,7998}, {5073,6241}, {5752,6914}, {8703,9729}, {9936,9973}

X(10263) = midpoint of X(i) and X(j) for these {i,j}: {4, 6243}, {382, 5889}, {5073, 6241}
X(10263) = reflection of X(i) in X(j) for these (i,j): (3, 143), (5, 5446), (550, 389), (1216, 10110), (1511, 1112), (5562, 546), (5876, 4), (5946, 3060), (6101, 5), (6102, 52)
X(10263) = complement of X(37484)
X(10263) = anticomplement of X(10627)
X(10263) = inverse of X(5899) in circumcircle of the reflected triangle, A'B'C'; A' = reflection of A in BC, etc.
X(10263) = isogonal conjugate of X(4) wrt reflection triangle
X(10263) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,143,5946), (3,3060,143), (195,5899,1614), (1216,5446,10110), (1216,10110,5), (1993,7517,156), (2979,9781,1656), (5447,5943,632)

X(10264) =  HATZIPOLAKIS-MOSES IMAGE OF X(74)

See X(10263).

Let AeBeCe and AiBiCi be the Ae and Ai triangles (aka K798e and K798i triangles). The circumcircles of triangles AeBiCi, BeCiAi, CeAiBi concur in X(10264). (Randy Hutson, June 7, 2019)

X(10264) lies on these lines: {2,399}, {3,2888}, {5,113}, {10,2771}, {11,7727}, {30,74}, {49,5498}, {67,3564}, {110,140}, {141,542}, {146,381}, {343,8703}, {427,1986}, {495,3028}, {496,3024}, {524,9976}, {541,1539}, {546,7728}, {547,5655}, {568,7731}, {632,5609}, {1112,1595}, {1484,8674}, {1503,7575}, {1594,7722}, {1614,10125}, {2777,3627}, {2916,2931}, {3582,7343}, {3584,6126}, {3925,7724}, {5054,9143}, {5844,7984}, {5890,7703}, {6102,10115}, {7530,9919}

X(10264) = midpoint of X(i) and X(j) for these {i,j}: {3,3448}, {74,265}
X(10264) = reflection of X(i) in X(j) for these (i,j): (5,125), (110,140 ), (1511,6699 ), (1539,7687 ), (3627,10113 ), (5609,5972 ), (5655,547 ), (7728,546)
X(10264) = complement of X(399)
X(10264) = X(1138)-complementary conjugate of X(10)
X(10264) = crosspoint of X(94) and X(1494)
X(10264) = crosssum of X(i) and X(j) for these (i,j): {50,1495}, {2088,6140}
X(10264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (74,9140,265), (1511,6699,549), (1539,7687,3845)

X(10265) =  HATZIPOLAKIS-MOSES IMAGE OF X(80)

See X(10263).

X(10265) lies on these lines: {1,6952}, {2,6326}, {4,1768}, {5,2771}, {8,6264}, {10,140}, {11,65}, {36,80}, {40,149}, {100,5178}, {116,119}, {153,3306}, {191,6902}, {226,8068}, {517,1484}, {758,6882}, {912,3814}, {944,9897}, {950,10058}, {1125,6265}, {1145,4847}, {1158,9581}, {1387,9952}, {1538,6001}, {1698,5531}, {1837,5450}, {2826,4049}, {2829,6245}, {2932,3419}, {3091,9809}, {3576,6224}, {3679,7993}, {3825,5887}, {4193,5693}, {4973,5841}, {5249,9964}, {5535,6840}, {5541,5657}, {5660,10172}, {5692,6963}, {5902,6830}, {6667,9843}, {7972,10039}, {10057,10074}

X(10265) = midpoint of X(i) and X(j) for these {i,j}: {4,1768}, {8,6264}, {40,149}, {80,104}, {944,9897}, {1387,9952}, {5535,6840}, {6326,9803}, {17638,17654}
X(10265) = reflection of X(i) in X(j) for these (i,j): (100,6684), (119,6702), (214,6713), (946,11), (5660,10172), (6265,1125)
X(10265) = complement of X(6326)
X(10265) = crosssum of X(2183) and X(2361)
X(10265) = X(1138)-complementary conjugate of X(10)
X(10265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,9803,6326), (119,6702,10175), (214,6713,10165), (10057,10074,10106), (10058,10073,950)

X(10266) = (ABC, 1ST SCHIFFLER)-EULEROLOGIC CENTER

This pair of triangles are mutually eulerologic. The (1st Schiffler, ABC)-eulerologic center is X(79)

X(10266) lies on the Feuerbach hyperbola and these lines: {1,5180}, {2,3467}, {4,5885}, {11,6595}, {30,1389}, {80,2475}, {104,5606}, {191,7161}, {758,5559}, {943,3648}, {1156,7173}, {1476,3649}, {6888,7701}

X(10266) = reflection of X(6595) in X(11)
X(10266) = antigonal conjugate of X(6595)
X(10266) = Feuerbach hyperbola-antipode of X(6595)
X(10266) = orthologic center of these triangles: ABC to 1st Schiffler

X(10267) = (INNER YFF, 2ND CIRCUMPERP)-EULEROLOGIC CENTER

X(10267) lies on these lines:{1,3}, {4,1621}, {5,1001}, {8,1006}, {9,5534}, {10,6883}, {11,6863}, {12,6928}, {21,944}, {25,1871}, {30,4428}, {42,602}, {100,631}, {104,4189}, {119,2478}, {140,1376}, {155,916}, {182,9052}, {197,6642}, {219,2302}, {355,405}, {390,6908}, {411,5603}, {497,6825}, {498,6882}, {515,3560}, {549,4421}, {581,595}, {601,902}, {943,6987}, {946,6985}, {947,1069}, {952,958}, {954,5812}, {956,1259}, {962,3651}, {993,5837}, {1058,6988}, {1064,3915}, {1068,7412}, {1125,6796}, {1283,8235}, {1478,7491}, {1479,6842}, {1483,5428}, {1486,7387}, {1656,4423}, {1872,7071}, {2346,5759}, {2550,6989}, {2975,6875}, {3073,8616}, {3085,6827}, {3086,6954}, {3090,5284}, {3149,5886}, {3185,9798}, {3193,4184}, {3434,6889}, {3436,6936}, {3526,4413}, {3533,9342}, {3616,6905}, {3628,8167}, {3816,6959}, {3871,5657}, {3913,5690}, {4254,8557}, {4294,5840}, {4512,7330}, {5047,5818}, {5132,5292}, {5218,6891}, {5225,6982}, {5250,5887}, {5251,5881}, {5253,6942}, {5259,5587}, {5281,6926}, {5399,7078}, {5432,6958}, {5550,6946}, {5552,6947}, {5687,6734}, {5705,9709}, {5714,8543}, {5731,6906}, {5842,6917}, {5853,6684}, {6284,6923}, {6361,7411}, {6690,6862}

X(10267) = midpoint of X(i),X(j) for these {i,j}: {3,3295}, {388,6868}, {4294,6850}, {5837,5882}
X(10267) = reflection of X(3560) in X(5248)
X(10267) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3072,5707), (3,1482,3428), (3,10246,56), (35,3576,3), (1125,6796,6911), (1385,3579,9940), (2077,7987,3), (3303,3428,1482), (6875,7967,2975)

X(10268) = (INNER YFF, EXCENTRAL)-EULEROLOGIC CENTER

X(10268) lies on these lines:{1,3}, {4,4512}, {10,6987}, {19,7412}, {71,2261}, {84,4640}, {212,1103}, {355,5234}, {380,573}, {602,2999}, {936,6796}, {1158,5732}, {1702,5416}, {1703,5415}, {2328,9121}, {2550,5705}, {2949,3174}, {4652,5731}, {5223,5534}, {5436,7686}, {5698,6260}, {5715,10198}

X(10268) = reflection of X(5715) in X(10198)
X(10268) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40,55,6769), (40,3576,65), (3579,10267,5709), (5709,10267,1)

X(10269) = (OUTER YFF, 2ND CIRCUMPERP)-EULEROLOGIC CENTER

X(10269) lies on these lines:{1,3}, {2,104}, {4,5253}, {5,6256}, {8,6940}, {11,6923}, {12,6958}, {21,2096}, {30,7956}, {100,7967}, {106,1480}, {140,958}, {182,2810}, {329,1006}, {355,474}, {388,6891}, {404,944}, {497,5840}, {499,6842}, {515,6692}, {572,2178}, {601,1201}, {631,2975}, {859,1790}, {912,997}, {952,1376}, {956,6735}, {957,4216}, {993,3452}, {1001,6914}, {1012,1519}, {1125,3560}, {1478,6882}, {1483,3913}, {1621,6950}, {2829,3816}, {3085,6961}, {3086,6850}, {3434,6955}, {3436,6967}, {3525,5260}, {3600,6926}, {3616,6906}, {4297,6985}, {4299,7491}, {4413,5790}, {4423,7489}, {5080,6963}, {5265,6908}, {5303,6875}, {5433,6863}, {5438,5534}, {5550,6920}, {5603,6909}, {5731,6905}, {6684,8666}, {6691,6959}, {6825,7288}, {6928,7354}, {7330,8583}

X(10269) = midpoint of X(i),X(j) for these {i,j}: {1,3359}, {3,999}, {497,6948}, {4293,6827}, {4297,7682}
X(10269) = reflection of X(i) in X(j) for these (i,j):(3820,140), (6929,3816)
X(10269) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,36,1470), (1,5193,999), (3,1385,10267), (3,10246,55), (36,3576,3), (993,10165,6883), (1125,5450,3560), (6256,10200,5)

X(10270) = (OUTER YFF, EXCENTRAL)-EULEROLOGIC CENTER

X(10270) lies on these lines:{1,3}, {63,2057}, {84,1376}, {104,4853}, {223,1167}, {580,937}, {601,2999}, {603,1103}, {936,1158}, {2550,6705}, {2551,6256}, {4652,6735}, {5438,6001}, {5450,9623}, {5720,7992}, {5732,6796}, {7330,8580}

X(10270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,165,10268), (3,3359,1), (40,3576,3057), (1697,5193,1)

X(10271) =  X(1)X(1537)∩X(55)X(108)

Let A'B'C' be the intouch triangle; Ma = midpoint of AA', and define Mb and Mc cyclically; M1 = midpoint of A'X(1), and define M2 and M3 cyclically. The circumcircles of MaM2M3, MbM3M1, McM1M2 concur in X(10271); the circumcircles of M1MbMc, M2McMa, M3MaMb concur in X(1387). See Antreas Hatzipolakis and Peter Moses, 24436).

X(10271) lies on these lines: {{1,1537}, {55,108}, {123,3816}, {676,2804}, {1359,1388}, {2817,9957}, {2823,4353}

X(10271) = midpoint of X(108) and X(3318)
X(10271) = incircle-inverse of X(34789)

X(10272) =  MIDPOINT OF X(5) AND X(110)

In the plane of a triangle ABC, let NaNbNc be the pedal triangle of N = X(5) and OaObOc the pedal triangle of O = X(3). Let N1 = reflection of N in BC, and define N2 and N3 cyclically. Let O1 = reflection of O in BC, and define O2 and O3 cyclically. Let La be the line through Na parallel to O1N1, and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(10272). See Antreas Hatzipolakis and Peter Moses, 24449).

Let (Oa) be the circle centered at A and tangent to the Euler line. Define (Ob) and (Oc) cyclically. Let La be the polar of X(4) wrt (Oa), and define Lb and Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A' = Lb∩Lc, and define B' and C' cyclically. Triangle A'B'C' is the reflection of ABC in X(5972), which is the radical center of (Oa), (Ob), (Oc); and X(10272) = X(140)-of-A'B'C'. (Randy Hutson, December 10, 2016)

X(10272) lies on these lines: {2,399}, {3,146}, {4,7666}, {5,49}, {30,113}, {69,10201}, {74,549 }, {125,3628}, {140,5663}, {403,3043}, {468,1986}, {495,10091}, {496,10088}, {542,547}, {546,9820} ,{548,2777}, {550,7728}, {597, 9976}, {1125,2771}, {1154,10096}, {1656,3448}, {2931,5654}, {2948,5886}, {3564,6593}, {3582, 6126}, {3584,7343}, {3850,10113} ,{5055,9143}, {5066,7687}, { 5432,7727}, {5876,10125}, {5898, 7693}, {6140,6592}, {6153,10095} ,{6677,9826}, {7525,10117}, {7542,7723}, {7722,10018}

X(10272) = midpoint of X(i) and X(j) for these {i,j}: {5,110}, {113,1511}, {125,5609} ,{549,5655}, {550,7728}, {6053,6 699}
X(10272) = reflection of X(i) in X(j) for these (i,j): (125,3628), (140,5972), (10113, 3850)
X(10272) = crossdifference of every pair of points on the line X(2081)(X(2433)
X(10272) = complement of the complement of X(399)
X(10272) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (113,5642,1511), (5972,6053,6699)

X(10273) =  MIDPOINT OF X(3576) AND X(5903)

In the plane of a triangle ABC, let
I = X(1) = incenter, N = X(5) = nine-point center,
A' = reflection of I in BC, and define B' and C' cyclically
Na = X(5)-of-IBC, and define Nb and Nc cyclically
Then X(10273) is the centroid of N1N2N3. See Antreas Hatzipolakis and Peter Moses, 24611).

X(10273) lies on these lines: {1,3}, {5,4004}, {355,10107}, {1 864,6797}, {2800,3817}, {3698,56 94}, {3754,5887}, {3922,9956}, { 4018,5690}, {4323,6961}, {4848,6 842}, {5927,9952}

X(10273) = midpoint of X(3576) and X(5903)
X(10273) = reflection of X(i) in X(j) for these (i,j): (5887, 10175), (10175, 3754), (10247, 942)

X(10274) =  MIDPOINT OF X(195) AND X(2917)

Let A'B'C' be the pedal triangle of O = X(3). Let O' be the circumcenter of A'B'C'; let Oa be the circumcenter of OBC, and define Ob and OC cyclically. Let Ra = reflection of OOA in BC, and define Rb and Rc cyclically. Then Ra, Rb, Rc concur in X(10274). See Antreas Hatzipolakis, Peter Moses, and César Lozada, 24566) and 24574).

X(10274) lies on these lines: {3,8157}, {4,54}, {49,52}, {110, 2888}, {154,9704}, {156,9927}, { 182,6689}, {206,576}, {539, 10201}, {569,6145}, {1092,7691}, {1147,1154}, {1209,6639}, {1971, 9697}, {2904,9707}, {3518,7730}, {6288,10254}, {9813,9827}, { 10182,10203}

X(10274) = midpoint of X(195) and X(2917)
X(10274) = X(324)-Ceva conjugate of X(571)
X(10274) = {X(54),X(3574)}-harmonic conjugate of X(578)

X(10275) =  MIDPOINT OF X(140) AND X(1487)

In the plane of a triangle ABC, let N = X(5) = nine-point center, and
Na = nine-point center of NBC, and define Nb and Nc cyclically
Nab = orthogonal projection of Na on BNb, and define Nbc and Nca cyclically
Nac = orthogonal projection of Na on CNc, and define Nba and Ncb cyclically.
Let Oa = circumcenter of NaNabNac, and define Ob and Oc cyclically. The triangles ABC and OaObOC are orthologic; X(10275) = OaObOc-to-ABC-orthologic center, and X(1263) = ABC-to-OaObOc-orthologic center. See Antreas Hatzipolakis and Peter Moses, 24515).

X(10275) lies on this line: {140, 930}

X(10275) = midpoint of X(140) and X(1487)

Feuerbach quadrangle and related centers: X(10276)-X(10281)

X(10276) = CENTROID OF THE FEUERBACH QUADRANGLE

X(10276) lies on these lines:{5,399}, {11,5947}, {111,3055}

X(10276) = midpoint of X(5) and X(10277)
X(10276) = inverse-in-nine-point-circle of X(3448)

X(10277) = ANTICENTER OF THE FEUERBACH QUADRANGLE

X(10277) lies on these lines:{5,399}, {11,5948}, {119,5947}, {549,2079}

X(10277) = midpoint of X(11) and X(5948)
X(10277) = reflection of X(5) in X(10276)

X(10278) = CENTROID OF THE DIAGONAL TRIANGLE OF THE FEUERBACH QUADRANGLE

Note: (diagonal triangle of the Feuerbach quadrangle) = (tangential triangle of Kiepert hyperbola) = (Schroeter triangle) (Randy Hutson, December 10, 2016)

X(10278) lies on these lines:{2,523}, {115,9293}, {427,2501}, {512,5943}, {804,1637}, {1499,3845}, {1640,1853}, {2395,6587}, {3628,8151}, {9148,9479}

X(10278) = midpoint of X(i),X(j) for these {i,j}: {2,8029}, {1637,9134}, {5466,8371}, {9148,9979}
X(10278) = reflection of X(i) in X(j) for these (i,j): (2,10189), (10190,2)
X(10278) = isotomic conjugate of X(37880)
X(10278) = complement of X(11123)
X(10278) = X(115)-Ceva conjugate of X(523)
X(10278) = X(2)-of-Schroeter-triangle
X(10278) = inverse-in-Hutson-Parry-circle of X(8029)
X(10278) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5466,8029), (2,8371,10189), (8029,8371,2), (8029,10189,10190)

X(10279) = CIRCUMCENTER OF THE DIAGONAL TRIANGLE OF THE FEUERBACH QUADRANGLE

X(10279) lies on these lines:{2,8151}, {3,8029}, {4,1499}, {5,10278}, {140,523}, {512,5462}, {632,10190}, {1656,8371}, {3533,9168}, {3628,10189}

X(10279) = complement of X(8151)
X(10279) = X(3)-of-Schroeter-triangle

X(10280) = X(5) OF THE DIAGONAL TRIANGLE OF THE FEUERBACH QUADRANGLE

Let ABC be a triangle, P a point and A'B'C' the mid-cevian triangle of P. The locus of P such that the nine-point-circles of PNA', PNB', PNC' are coaxial is the Euler line. The locus of the 2nd intersection (other than the midpoint of PN) is the circle centered at X(10280), passing through X(i) for these i: 111, 476, 6722, 10276, 11746, 15366. See Antreas Hatzipolakis and Peter Moses, euclid 1761.

X(10280) lies on these lines:{3,8371}, {5,10278}, {140,10189}, {523,3628}, {525,5449}, {546,1499}, {1594,2501}, {1656,8029}, {3090,5466}

X(10280) = midpoint of X(5) and X(10279)
X(10280) = complement of X(32204)
X(10280) = X(5)-of-Schroeter-triangle

X(10280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,10278,10279), (1656,8029,8151)

X(10281) = QA-NINE-POINT HOMOTHETIC CENTER OF THE FEUERBACH QUADRANGLE

X(10281) lies on these lines:{5,399}, {952,5947}

X(10282) =  X(3)X(64)∩X(51)X(54)

Let O be the circumcenter of a triangle ABC, and let
Oa = circumcenter of OBC, and define Ob and OC cyclically
N1 = nine-point center of OObOC, and define N2 and N3 cyclically.
Then ABC and N1N2N3 are orthologic triangles, and X(10282) = (N1N2N3,ABC)-orthologic center, and X(74) = (ABC,N1N2N3)-othologic center. X(10282) lies on the circumcircle of N1N2N3. See Antreas Hatzipolakis and Angel Montesdeoca, 24665.

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. Triangle O_AO_BO_C is parallelogic to the orthic triangle at X(10282). (Randy Hutson, June 7, 2019)

X(10282) lies on these lines: {2, 9833}, {3, 64}, {4, 1495}, {5, 5944}, {6, 3517}, {22, 1092}, {24, 184}, {25, 578}, {26, 206}, {30, 5448}, {39, 1971}, {49, 52}, {51, 54}, {110, 5562}, {125, 10018}, {140, 1503}, {143, 5097}, {156, 1658}, {159, 182}, {161, 569}, {185, 186}, {216, 3463}, {376, 5878}, {394, 9715}, {436, 8884}, {468, 6146}, {549, 6247}, {550, 1511}, {567, 9920}, {568, 9704}, {575, 2393}, {1181, 3515}, {1216, 7502}, {1660, 6644}, {1853, 3526}, {1899, 3147}, {1970, 3199}, {1994, 9706}, {2781, 7555}, {3060, 9545}, {3270, 9638}, {3292, 7556}, {3522, 5656}, {3528, 6225}, {3530, 6696}, {3534, 5895}, {3574, 7576}, {3917, 7512}, {5010, 6285}, {5050, 9924}, {5447, 7525}, {5449, 10020}, {5480, 7715}, {5651, 7509}, {5889, 9544}, {5894, 8703}, {6001, 7508}, {6102, 7575}, {6243, 9703}, {7280, 7355}, {8681, 9937}, {8718, 9934}

X(10282) = complement of complement of X(9833)
X(10282) = anticomplement of X(32767)
X(10282) = {X(8837),X(8839)}-harmonic conjugate of X(216)
X(10282) = homothetic center of Kosnita triangle and cross-triangle of 1st and 2nd Kenmotu diagonals triangles

X(10283) =  REFLECTION OF X(5) IN X(5886)

Let I = X(1) = incenter of a triangle ABC, and let
Na = nine-point center of Ibc, and define Nb and Nc cyclically
Oia = circumcenter of INbNc, and define Oib and Oic cyclically.
Then X(10283) = centroid of OiaOibOic. See Antreas Hatzipolakis and Peter Moses, 24667.

X(10283) lies on these lines: {1,5}, {2,5844}, {3,3622}, {8, 3628}, {30,5603}, {140,1482}, { 145,1656}, {165,3653}

X(10283) = midpoint of X(i) and X(j) for these {i,j}: {1,5886}, {2,10247}, {381,7967} ,{1482,5657}, {1699,3655}, { 3241,5790}, {3576,3656}, {5603, 10246}
X(10283) = reflection of X(i) in X(j) for these (i,j): (5,5886), (5657,140), (5790, 547), (5886,5901), (8703,3576)
X(10283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,1483), (1,5901,5), (1,7951, 1317), (1,9624,355), (355,5886, 7988), (1482,3616,140), (7988, 9624,5886)

X(10284) =  X(1)X(3)∩X(5)X(2802)

Let A1B1C1 be the intouch triangle of a triangle ABC, and let
A2 = reflection of A1 in X(1), and define B2 and C2 cyclically
A3 = reflection of A in A2, and define B3 and C3 cyclically.
Then X(10284) = nine-point center of A3B3C3. See Tran Quang Hung and Angel Montesdeoca, 24438.

X(10284) lies on these lines: {1,3}, {5,2802}, {8,6965}, {140,3898}, {149,355}, {519,5694}, {1483,2800} .

X(10284) = midpoint of X(i) and X(j) for these {i,j}: {355, 3885}, {1482, 5697}
X(10284) = reflection of X(i) in X(j) for these (i,j): (1385,9957), (5690,3884), (5903,6583)
X(10284) = {X(1),X(40)}-harmonic conjugate of X(37535)

X(10285) =  EULER-LINE INTERCEPT OF X(54)X(1263)

Let N be the nine-point center of triangle ABC, and let
Ha = orthocenter of NBC, and define Hb and Hc cyclically
Oa = circumcenters of NBC, and define Ob and Oc cyclically
Nha = nine-point center of NHbHc, and define Nhb and Nhc cyclically
Noa = nine-point center of NObOc, and define Nob and Noc cyclically.
Then X(10285) = nine-point center of NhaNhbNhc. This point and X(10286) lie on the Euler line of ABC. See Tran Quang Hung and Peter Moses, 24664.

For another construction see Hyacinthos 28911

Let Na be the reflection of X(5) in BC, and define Nb and Nc cyclically. Let Oa' be the circumcenter of NaBC, and define Ob' and Oc' cyclically; then X(10285) is the circumcenter of Oa'Ob'Oc'. (Randy Hutson, December 10, 2016)

X(10285) lies on these lines: {2,3}, {54,1263}, {1154,20327}, {11671,16766}, {12026,14051}, {14140,24573}, {19552,21230}, {24385,25044}

x(10285)="reflection" of="" x(i)="" in="" x(j)="" for="" these="" (i,j):="" (5,5501),="" (10205,140)
X(10285) = anticomplement fo X(10126)

X(10286) =  MIDPOINT OF X(5) AND X(5500)

X(10286) = nine-point center of the triangle NoaNobNoc constructed at X(10285); both points lie on the Euler line of ABC. See Tran Quang Hung and Peter Moses, 24664.

X(10286) lies on this line: {2,3}

x(10286) = midpoint of x(5) and x(5500)

X(10287) =  X(3)X(2575)∩X(5)X(523)

Let H be the orthocenter of a triangle ABC. Let La be the Euler line of AHX(1113), and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(10287). See X(10288) and Seiichi Kirikami and Angel Montesdeoca, 24541 and 24545.

X(10287) lies on these lines: {3,2575}, {5,523}

X(10287) = crossdifference of every pair of points on line X(50)X(8105)

X(10288) =  X(3)X(2574)∩X(5)X(523)

Let H be the orthocenter of a triangle ABC. Let La be the Euler line of AHX(1114), and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(10288).

See X(10287) and Seiichi Kirikami and Angel Montesdeoca, 24541 and 24545.

X(10288) lies on these lines: {3,2574}, {5,523}

X(10288) = crossdifference of every pair of points on line X(50)X(8106)

X(10289) =  7th HATZIPOLAKIS-MOSES-EULER POINT

Let A'B'C' the pedal triangle of the nine-point center, N = X(5), of a triangle ABC. Let
Oa = circumcenter of NB'C', and define Ob and Oc cyclically
Ooa = circumcenter of NObOc, and define Oob and Ooc cyclically.
Then X(10289) = nine-point center of OoaOobOoc; this point lies on the Euler line of ABC. See Antreas Hatzipolakis and Peter Moses, 24670.

X(10289) lies on this line: {2,3}

X(10289) = complement of X(36837)

Points associated with mid-triangles and cross-triangles: X(10290)-X(10608)

X(10290) =  PERSPECTOR OF ABC AND MID-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10290) lies on the Kiepert hyperbola and these lines: {99,3407}

X(10290) = trilinear pole of line X(523)X(3314)

X(10291) =  PERSPECTOR OF MEDIAL TRIANGLE AND MID-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10291) lies on these lines: {2,4159}, {99,3407}, {187,736}

X(10292) =  HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND MID-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10292) lies on these lines: {2,32}, {114,7930}

X10292) = complement of X(10345)

X(10293) =  PERSPECTOR OF ABC AND MID-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

The anti-orthocentroidal triangle is here defined as the triangle of which ABC is the orthocentroidal triangle. It is also the unary cofactor triangle of the orthocentroidal triangle, and the orthocentroidal triangle is the unary cofactor triangle of the anti-orthocentroidal triangle.

The trilinear polar of X(10293) passes through X(647).

Peter Moses, November 1, 2016, gives barycentrics for the A-vertex of the anti-orthocentroidal triangle:

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

Let ABC be a triangle and HaHbHc the orthic triangle. The segment AHa has two trisectors; let A' be the trisector closer to A. Let (Ha) be the (rectangular) hyperbola through A, A', X(6), with asymptotes parallel to those of the Jerabek hyperbola. Let Ao be the center of (Ha), and define Bo and Co cyclically. The lines AAo, BBo, CCo concur in X(10293). Let Ta be the line through A tangent to (Ha), and define Tb and Tc cyclically. The lines Ta, Tb, Tc concur in X(3426). (Angel Montesdeoca, January 1, 2017)

An equation for the hyperbola (Ha), in barycentrics:
(2S^2-3SB SC) (c^2y^2-b^2z^2) + S^2(b^2-c^2)y z - b^2(2S^2-3SA SB)z x + c^2(2S^2-3SA SC)x y = 0. (Angel Montesdeoca, January 1, 2017)

X(10293) lies on the Jerabek hyperbola and these lines: {3,541}, {6,2777}, {30,895}, {69,5663}

X(10293) = isogonal conjugate of X(7464)

X(10294) =  PERSPECTOR OF ORTHIC TRIANGLE AND MID-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

X(10294) lies on these lines: {4,5609}, {6,7699}

X(10294) = X(4)-Ceva conjugate of X(10295)
X(10294) = orthic isogonal conjugate of X(10295)

X(10295) =  REFLECTION OF X(4) IN ORTHIC AXIS

X(10295) lies on these lines: {2,3}, {50,112}, {99,340}, {107,841}, {511,1986}, {523,9409}

X(10295) = midpoint of X(20) and X(23)
X(10295) = reflection of X(4) in X(468)
X(10295) = reflection of X(4) in its trilinear polar
X(10295) = X(4)-Ceva conjugate of X(10294)
X(10295) = complement of X(10296)
X(10295) = anticomplement of X(10297)
X(10295) = orthic isogonal conjugate of X(10294)
X(10295) = inverse-in-circumcircle of X(378)
X(10295) = inverse-in-nine-point-circle of X(7577)
X(10295) = inverse-in-polar-circle of X(381)
X(10295) = inverse-in-{circumcircle, nine-point circle}-inverter of X(5094)
X(10295) = inverse-in-circle-O(PU(4)) of X(2)
X(10295) = X(3245)-of-orthic-triangle if ABC is acute
X(10295) = Ehrmann-vertex-to-orthic similarity image of X(7574)
X(10295) = {X(3),X(4)}-harmonic conjugate of X(37118)

X(10296) = REFLECTION OF X(20) IN DE LONGCHAMPS LINE

Let P4' and U4' be the anticomplements of P(4) and U(4). P4' and U4' are also the circumcircle intercepts of the de Longchamps circle; then X(10296) is the orthocenter of triangle X(2)P4'U4'.

X(10296) lies on these lines: {2,3}, {110,1531}

X(10296) = reflection of X(20) in X(858)
X(10296) = reflection of X(23) in X(4)
X(10296) = anticomplement of X(10295)
X(10295) = isogonal conjugate of X(34802)
X(10295) = crossdifference of every pair of points on line X(647)X(5158)
X(10296) = inverse-in-circumcircle of X(10298)
X(10296) = inverse-in-de-Longchamps-circle of X(376)
X(10296) = anticomplementary conjugate of anticomplement of X(34802)

X(10297) = RADICAL TRACE OF NINE-POINT CIRCLE AND JOHNSON CIRCLE

The Johnson circle is the circumcircle of the Johnson triangle, and it is the reflection of the circumcircle of ABC in X(5).

Let P be a point not on the circumcircle of ABC. Let A'B'C' be the pedal triangle of P. Define the quasi-Simson line of P as the orthic axis of A'B'C'. As P approaches a point Q on the circumcircle, the quasi-Simson line of P approaches the Simson line of Q. See ADGEOM #2040, Dec 10, 2014. X(10297) is the reflection of X(3) in its quasi-Simson line (line X(523)X(4885)). (Randy Hutson, January 29, 2018)

X(10297) lies on these lines: {2,3}, {523,6334}

X(10297) = reflection of X(468) in X(5)
X(10297) = complement of X(10295)
X(10297) = complementary conjugate of complement of X(34802)
X(10297) = inverse-in-{circumcircle, nine-point circle}-inverter of X(7493)

X(10298) = HARMONIC CENTER OF CIRCUMCIRCLE AND DE LONGCHAMPS CIRCLE

X(10298) lies on this line: {2,3}

X(10298) = anticomplement of X(7577)
X(10298) = inverse-in-circumcircle of X(10296)
X(10298) = inverse-in-de-Longchamps-circle of X(7574)
X(10298) = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X(37969)

X(10299) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND MID-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

The anti-Euler triangle A'B'C' is the triangle of which ABC is the Euler triangle; also, A'B'C' = reflection of X(4) in ABC = reflection of the anticomplementary triangle in X(3).

Barycentrics for A'B'C' are given by

A' = tan A + 2 tan B + 2 tan C : - tan B : - tan C
A' = 3a⁴ - 4a²(b² + c²) + (b² - c²)² : b⁴ - (c² - a²)² : c⁴ - (a² - b²)²

The appearance of (T,i) in the following list means that the anti-Euler triangle and T are homothetic, with center of homothety X(i):

• (ABC, 4)
• (medial, 631)
• (anticomplementary, 3)
• (Euler, 4)
• (X(3) cevian, 376)
• (Johnson, 2)
• (Caelum, 7967)
• (Ara, 24)
• (outer Garcia, 5657)
• (T(1,2), 515)
• (X(2) orthocevian, 3542)
• (Mandart-incircle, 4294)
• (5th Brocard, 9862)
• (inner Yff, 388)
• (outer Yff, 497)
• (1st Johnson-Yff, 3085)
• (2nd Johnson-Yff, 3086)

X(10299) lies on these lines: {2,3}, {944,3626}, {1350,6329}, {3244,3576}, {3629,5085}, {3632,5657}

X(10299) = anticomplement of X(5079)
X(10299) = {X(2),X(3)}-harmonic conjugate of X(3528)
X(10299) = {X(2),X(20)}-harmonic conjugate of X(546)

X(10300) = HOMOTHETIC CENTER OF 3rd PEDAL TRIANGLE OF X(3) AND MID-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

X(10300) lies on these lines: {2,3}

X(10300) = complement of X(10301)

X(10301) = HOMOTHETIC CENTER OF 3rd PEDAL TRIANGLE OF X(4) AND MID-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

X(10301) lies on these lines: {2,3}

X(10301) = anticomplement of X(10300)
X(10301) = pole wrt polar circle of trilinear polar of X(10302) (line X(523)X(7840))
X(10301) = X(48)-isoconjugate (polar conjugate) of X(10302)

X(10302) =  ISOTOMIC CONJUGATE OF X(597)

X(10302) lies on the Kiepert hyperbola and these lines: {2,5355}, {4,7883}, {83,524}, {141,671}

X(10302) = isogonal conjugate of X(5008)
X(10302) = isotomic conjugate of X(597
X(10302) = polar conjugate of X(10301)
X(10302) = trilinear pole of line X(523)X(7840)

X(10303) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND MID-TRIANGLE OF EULER AND ANTI-EULER TRIANGLES

As a point on the Euler line, X(10303) has Shinagawa coefficients (5, -2).

X(10303) lies on these lines: {2,3}, {141,10541}, {182,3620}, {372,8972}, {390,499}, {487,3593}, {488,3595}, {498,3600}, {1131,10576}, {1132,9541}, {3068,3594}, {3069,3592}

X(10303) = complement of X(5068)
X(10303) = anticomplement of X(5067)

X(10304) = HOMOTHETIC CENTER OF CEVIAN TRIANGLE OF X(3) AND MID-TRIANGLE OF EULER AND ANTI-EULER TRIANGLES

X(10304) lies on these lines: {2,3}, {56,10385}, {98,8591}, {193,3098}, {538,6194}, {944,3654}, {3679,4297}

X(10304) = anticomplement of X(3545)
X(10304) = Thomson isogonal conjugate of X(5544)
X(10304) = {X(2),X(3)}-harmonic conjugate of X(15692)
X(10304) = trisector nearest X(2) of segment X(2)X(20)

X(10305) = PERSPECTOR OF ABC AND MID-TRIANGLE OF INTOUCH AND HEXYL TRIANGLES

The trilinear polar of X(10305) passes through X(650).

X(10305) lies on these lines: {1,2096}, {8,1071}, {9,631}, {20,1320}

X(10305) = isogonal conjugate of X(10306)

X(10306) = INTANGENTS-TO-EXTANGENTS SIMILARITY IMAGE OF X(3)

X(10306) lies on these lines: {4,1260}, {84,6765}, {382,5840}, {3311,5415}, {3312,5416}

X(10306) = isogonal conjugate of X(10305)

X(10307) = PERSPECTOR OF ABC AND MID-TRIANGLE OF INTOUCH AND 6th MIXTILINEAR TRIANGLES

The trilinear polar of X(10307) passes through X(650).

X(10307) lies on these lines: {1,7955}, {8,971}, {9,2272}, {516,3680}

X(10307) = isogonal conjugate of X(6244)

X(10308) = PERSPECTOR OF ABC AND MID-TRIANGLE OF HEXYL AND 6th MIXTILINEAR TRIANGLES

The trilinear polar of X(10308) passes through X(650) and the tripolar centroid of X(28).

X(10308) lies on these lines: {4,5221}, {7,496}, {8,30}, {9,2173}, {80,1770}

X(10308) = isogonal conjugate of X(3579)

X(10309) = PERSPECTOR OF ABC AND MID-TRIANGLE OF 6th MIXTILINEAR TRIANGLE AND REFLECTION TRIANGLE OF X(1)

The trilinear polar of X(10309) passes through X(650).

X(10309) lies on the Feuerbach hyperbola and these lines: {8,6001}, {9,1158}, {329,10310}

X(10309) = isogonal conjugate of X(10310)
X(10309) = antigonal conjugate of X(34256)
X(10309) = perspector of ABC and reflection of extouch triangle in X(1158)

X(10310) = REFLECTION OF X(56) IN X(3)

X(10310) lies on these lines: {1,3}, {2,7681}, {4,1329}, {5,4413}, {6,601}, {8,6909}, {10,1012}, {11,6891}, {12,6850}, {20,100}, {25,1753}, {219,1436}, {329,10309}, {377,7680}

X(10310) = reflection of X(56) in X(3)
X(10310) = isogonal conjugate of X(10309)
X(10310) = anticomplement of X(7681)
X(10310) = homothetic center of tangential triangle and reflection of intangents triangle in X(3)
X(10310) = homothetic center of anti-Mandart incircle triangle and ABC-X(3) reflections triangle

X(10311) = HOMOTHETIC CENTER OF ORTHIC TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

The trilinear polar of X(10311) passes through X(3288).

X(10311) lies on these lines: {2,95}, {4,32}, {5,10316}, {6,25}, {19,10315}, {22,216}, {23,5158}, {24,39}, {53,428}, {187,378}, {217,6759}, {340,3314}, {381,10317}, {1995,3284}

X(10311) = polar conjugate of X(327)
X(10311) = crossdifference of every pair of points on line X(525)X(684)

X(10312) = HOMOTHETIC CENTER OF CIRCUMORTHIC TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

The trilinear polar of X(10312) passes through X(3050).

X(10312) lies on these lines: {2,10316}. {3,5481}, {4,32}, {5,10317}, {6,24}, {25,251}, {39,186}, {187,3520}, {3199,5008}, {3090,10314}

X(10312) = isogonal conjugate of X(36952)
X(10312) = crosspoint of X(16813) and X(23964)
X(10312) = crosssum of X(15526) and X(17434)
X(10312) = crossdifference of every pair of points on line X(684)X(2525) (the isotomic conjugate, wrt the MacBeath triangle, of the MacBeath inconic)

X(10313) = HOMOTHETIC CENTER OF DUAL OF ORTHIC TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

X(10313) lies on these lines: {2,95}, {3,5481}, {4,10316}, {6,22}, {20,32}, {23,232}, {30,112}, {39,7488}, {111,10420}, {3101,10315}

X(10313) = crossdifference of every pair of points on line X(826)X(3574)

X(10314) = HOMOTHETIC CENTER OF SUBMEDIAL TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

X(10314) lies on these lines: {2,95}, {5,32}, {6,1196}, {25,216}, {112,3545}, {1656,10316}, {3090,10312}, {5055,10317}, {9816,10315}

X(10315) = HOMOTHETIC CENTER OF EXTANGENTS TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

X(10315) lies on these lines: {6,31}, {19,10311}, {32,40}, {48,1403}, {65,172}, {187,7688}, {230,3925}, {893,2259}, {8251,10316}, {3101,10313}, {9816,10314}

X(10316) = HOMOTHETIC CENTER OF 2nd EULER TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

X(10316) lies on these lines: {2,10312}, {3,6}, {4,10313}, {5,10311}, {20,112}, {22,8743}, {26,232}, {30,1968}, {53,7553}, {127,315}, {1062,1914}, {1656,10314}, {8251,10315}

X(10316) = complement of isotomic conjugate of X(18124)
X(10316) = X(92)-isoconjugate of X(66)
X(10316) = inverse-in-circle-{{X(371)X(372),PU(1),PU(39)}} of X(19161)
X(10316) = {X(371)X(372)}-harmonic conjugate of X(19161)

X(10317) = HOMOTHETIC CENTER OF 2nd ISOGONAL TRIANGLE OF X(4) AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

X(10317) lies on these lines: {3,6}, {5,10312}, {23,8744}, {26,8743}, {30,112}, {53,7540}, {186,10098}, {381,10311}, {5055,10314}

2nd isogonal triangles are defined at X(36).

X(10317) = X(92)-isoconjugate of X(67)
X(10317) = complement of isotomic conjugate of X(18125)
X(10317) = circle-{{X(371),X(372),PU(1),PU(39)}}-inverse of X(37473)
X(10317) = {X(371)X(372)}-harmonic conjugate of X(37473)
X(10317) = crossdifference of every pair of points on line X(427)X(523) (the radical axis of anticomplementary circle and tangential circle)
X(10317) = homothetic center of Ehrmann side-triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

X(10318) = HOMOTHETIC CENTER OF ABC AND MID-TRIANGLE OF LUCAS HOMOTHETIC AND LUCAS(-1) HOMOTHETIC TRIANGLES

X(10318) lies on these lines: {39,493}, {3199,5008}

X(10318) = {X(493),X(494)}-harmonic conjugate of X(6464)
X(10318) = barycentric product X(493)*X(494)

X(10319) = HOMOTHETIC CENTER OF EXTANGENTS TRIANGLE AND MID-TRIANGLE OF ORTHIC AND DUAL OF ORTHIC TRIANGLES

X(10319) lies on these lines: {1,3}, {2,19}, {9,440}, {20,1891}, {33,4220}, {63,69}, {77,2359}, {226,1766}, {1072,1074}

X(10319) = perspector of Gemini triangle 36 and cross-triangle of Gemini triangles 35 and 36

X(10320) = HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER YFF TRIANGLES

Let A₁B₁C₁ and A₂B₂C₂ be the inner- and outer- Yff triangles, resp. Let A' be the centroid of A₁A₂BC, and define B', C' cyclically. Triangle A'B'C' is homothetic to the medial triangle at X(10320).

X(10320) lies on these lines: {1,2}, {3,10523}, {4,8068}, {5,8609}, {11,6959}, {12,6862}, {35,6825}, {36,6891}, {55,6863}, {56,6713}

X(10321) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER YFF TRIANGLES

Let A₁B₁C₁ and A₂B₂C₂ be the inner- and outer- Yff triangles, resp. Let A' be the center of conic {A₁,B₁,C₁,B₂,C₂}, and define B', C' cyclically. Triangle A'B'C' is homothetic to ABC at X(10321).

X(10321) lies on these lines: {1,2}, {3,10629}, {4,8069}, {11,6944}, {12,6824}, {35,6908}, {36,6926}, {55,6825}, {56,6891}, {65,5761}, {104,388}, {495,6862}, {496,6959}, {497,6834}, {1478,6847}, {1479,6848}, {3091,8068}, {6852,8164}

X(10322) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ANDROMEDA AND ANTLIA TRIANGLES

Let A'B'C' be the Andromeda triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10322).
Let A'B'C' be the Antlia triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10322).

X(10322) lies on these lines: {1,1462}, {4907,5575}, {5573,5574}

X(10322) = trilinear product of vertices of Andromeda triangle
X(10322) = trilinear product of vertices of Antlia triangle
X(10322) = perspector of ABC and cross-triangle of ABC and Andromeda triangle
X(10322) = perspector of ABC and cross-triangle of ABC and Antlia triangle

X(10323) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND ARA TRIANGLE

X(10323) lies on these lines: {2,3}, {159,10519}, {944,8193}, {5218,10037}, {5594,10518}, {5595,10517}, {5603,9911}, {5656,9914}, {5657,9798}, {5658,9910}, {6684,8185}, {7288,10046}

X(10323) = anticomplement of X(7403)

X(10324) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND ATIK TRIANGLE

X(10324) lies on these lines: {1,2}, {346,3062}, {3695,9948}

X(10324) = {X(3062),X(10326)}-harmonic conjugate of X(10325)

X(10325) = PERSPECTOR OF ATIK TRIANGLE AND CROSS-TRIANGLE OF ABC AND ATIK TRIANGLE

X(10325) lies on these lines: {7,8}, {346,3062}, {391,2297}

X(10325) = {X(3062),X(10326)}-harmonic conjugate of X(10324)

X(10326) = {X(10324),X(10325)}-HARMONIC CONJUGATE OF X(3062)

X(10326) lies on these lines: {8,5542}, {346,3062}

X(10327) = PERSPECTOR OF AYME TRIANGLE AND CROSS-TRIANGLE OF ABC AND AYME TRIANGLE

The trilinear polar of X(10327) passes through X(2509).

X(10327) lies on these lines: {1,2}, {4,3701}, {55,3932}

X(10327) = perspector of Ayme triangle and cevian triangle of X(304)
X(10327) = anticomplement of X(614)
X(10327) = homothetic center of Ayme triangle and Gemini triangle 18

X(10328) = PERSPECTOR OF 1st BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd BROCARD TRIANGLES

X(10328) lies on these lines: {2,4048}, {6,6664}, {76,1501}, {99,8041}, {110,4074}, {384,3051}

X(10328) = X(251)-of-1st-Brocard-triangle
X(10328) = 1st-Brocard-isogonal conjugate of X(4045)
X(10328) = {X(2),X(10330)}-harmonic conjugate of X(10329)

X(10329) = PERSPECTOR OF 2nd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd BROCARD TRIANGLES

X(10329) lies on these lines: {2,4048}, {3,695}, {6,22}, {39,1915}, {83,10191}, {99,4074}, {110,8041}, {141,8788}, {182,3981}, {2076,3051}

X(10329) = isogonal conjugate of X(1031)
X(10329) = {X(2),X(10330)}-harmonic conjugate of X(10328)
X(10329) = X(39)-Ceva conjugate of X(6)
X(10329) = pole wrt circumcircle of trilinear polar of X(39)

X(10330) = {X(10328),X(10329)}-HARMONIC CONJUGATE OF X(2)

X(10330) lies on these lines: {2,4048}, {69,2916}, {99,110}

X(10330) = {X(99),X(110)}-harmonic conjugate of X(4576)

X(10331) = PERSPECTOR OF 1st BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th BROCARD TRIANGLES

X(10331) lies on these lines: {32,76}, {2896,10332}

X(10332) = PERSPECTOR OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th BROCARD TRIANGLES

X(10332) lies on these lines: {32,694}, {2896,10331}

X(10333) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 6th BROCARD TRIANGLES

X(10333) lies on these lines: {2,3398}, {3,10334}, {6,9983}, {32,3314}, {39,10347}, {98,7901}, {182,7876}, {315,3407}, {384,511}, {4027,6656}, {7791,10131}, {7877,10348}

X(10333) = homothetic center of 6th Brocard triangle and cross-triangle of 1st Brocard and 6th anti-Brocard triangles
X(10333) = homothetic center of 1st Brocard triangle and cross-triangle of 6th Brocard and 6th anti-Brocard triangles

X(10334) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10334) lies on these lines: {2,98}, {3,10333}, {6,9865}, {32,7779}, {39,10000}, {83,194}

X(10334) = homothetic center of 1st anti-Brocard triangle and cross-triangle of 1st Brocard and 6th anti-Brocard triangles
X(10334) = homothetic center of 1st Brocard triangle and cross-triangle of 1st anti-Brocard and 6th anti-Brocard triangles

X(10335) = PERSPECTOR OF MEDIAL TRIANGLE AND CROSS-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10335) lies on these lines: {2,698}, {3,194}, {6,8290}, {99,3407}

X(10336) = PERSPECTOR OF ANTICOMPLEMENTARY TRIANGLE AND CROSS-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10336) lies on these lines: {2,4121}, {6,147}, {20,32}, {148,3407}, {194,10583}

X(10337) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th BROCARD TRIANGLES

X(10337) lies on these lines: {2,32}, {384,3118}, {9983,10339}

X(10338) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th BROCARD TRIANGLES

X(10338) lies on these lines: {32,10339}, {69,194}, {384,3118}

X(10338) = perspector of 5th Brocard triangle and cross-triangle of 3rd and 6th Brocard triangles
X(10338) = perspector of 3rd Brocard triangle and cross-triangle of 5th and 6th Brocard triangles

X(10339) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 6th BROCARD TRIANGLES

X(10339) lies on these lines: {5,5354}, {6,2896}, {32,10338}, {384,3051}, {1369,6656}, {9983,10337}

X(10340) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 6th BROCARD TRIANGLES

X(10340) lies on these lines: {4,8878}, {32,10344}, {69,194}, {384,3051}

X(10341) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10341) lies on these lines: {32,3978}, {39,83}, {194,1186}

X(10342) = PERSPECTOR OF 5th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10342) lies on these lines: {32,3978}, {6,194}, {83,3117}

X(10343) = PERSPECTOR OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th AND 6th BROCARD TRIANGLES

X(10343) lies on these lines: {32,10338}, {141,384}, {9983,10344}

X(10344) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th AND 6th BROCARD TRIANGLES

X(10344) lies on these lines: {20,1352}, {32,10340}, {384,3118}, {9983,10343}

X(10345) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10345) lies on these lines: {2,32}, {6,9983}, {39,10000}, {98,7932}, {182,9873}, {3329,10349}, {3398,7875}, {3407,6656}, {5182,9878}, {9862,10359}

X(10345) = anticomplement of X(10292)
X(10345) = homothetic center of 5th anti-Brocard triangle and cross-triangle of ABC and 5th Brocard triangle
X(10345) = homothetic center of 5th Brocard triangle and cross-triangle of ABC and 5th anti-Brocard triangle

X(10346) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10346) lies on these lines: {2,32}, {39,10353}, {194,10000}, {2080,10357}, {3398,9862}, {3399,9821}, {3407,6655}, {7827,9878}

X(10347) = HOMOTHETIC CENTER OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10347) lies on these lines: {2,32}, {39,10333}, {98,7919}, {99,737}, {182,9862}, {3398,7859}, {3407,7761}, {5171,10357}, {7760,9983}

X(10347) = homothetic center of 5th Brocard triangle and 1st Brocard of 1st Brocard triangle

X(10348) = HOMOTHETIC CENTER OF 5th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

X(10348) lies on these lines: {2,32}, {6,10000}, {7877,10333}

X(10349) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 6th BROCARD AND 1st ANTI-BROCARD TRIANGLES

X(10349) lies on these lines: {2,3398}, {3,10350}, {6,76}, {32,325}, {98,7887}, {182,6656}, {3329,10345}, {3407,7785}

X(10349) = homothetic center of 1st anti-Brocard triangle triangle and cross-triangle of 6th Brocard and 6th anti-Brocard triangles
X(10349) = homothetic center of 6th Brocard triangle and cross-triangle of 1st and 6th anti-Brocard triangles

X(10350) = HOMOTHETIC CENTER OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 6th BROCARD AND 6th ANTI-BROCARD TRIANGLES

X(10350) lies on these lines: {2,32}, {3,10349}, {384,511}, {5182,7833}

X(10351) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 6th BROCARD AND 6th ANTI-BROCARD TRIANGLES

X(10351) lies on these lines: {3,10353}, {6,194}, {32,7799}, {83,5309}, {1692,9983}, {5182,7833}}

X(10352) = HOMOTHETIC CENTER OF 1st ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st and 6th ANTI-BROCARD TRIANGLES

X(10352) lies on these lines: {2,98}, {3,10349}, {6,5976}, {32,620}, {39,83}, {115,7803}

X(10353) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st and 6th ANTI-BROCARD TRIANGLES

X(10353) lies on these lines: {2,98}, {3,10351}, {6,8290}, {39,10346}, {83,148}, {99,7772}, {194,5149}

X(10354) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 4th ANTI-BROCARD TRIANGLE

X(10354) is the intersection of the Simson line of X(1296) (line X(126)X(524)) and the circumcircle normal at X(1296) (line X(3)X(111)).

X(10354) lies on these lines: {3,111}, {126,524}

X(10354) = complement of X(34166)
X(10354) = crossdifference of every pair of points on line X(2444)X(9125)

X(10355) = PERSPECTOR OF 4th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF ABC AND 4th ANTI-BROCARD TRIANGLE

X(10355) lies on these lines: {3,111}

X(10355) = isogonal conjugate of isotomic conjugate of X(34164)
X(10355) = X(25)-of-4th-anti-Brocard-triangle

X(10356) = HOMOTHETIC CENTER OF EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th BROCARD TRIANGLE

X(10356) lies on these lines: {2,9873}, {3,7914}, {4,3096}, {5,32}, {30,7822}, {98,7932}, {182,7859}, {576,7877}

X(10357) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th BROCARD TRIANGLE

X(10357) lies on these lines: {2,9821}, {3,147}, {4,3096}, {20,9996}, {30,7928}, {32,631}, {140,9301}, {141,7470}, {944,9857}, {2080,10346}, {5171,10347}, {5218,10038}, {7288,10047}

X(10358) = HOMOTHETIC CENTER OF EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th ANTI-BROCARD TRIANGLE

X(10358) lies on these lines: {2,5171}, {3,6683}, {4,83}, {5,32}, {6,6248}, {76,576}

X(10359) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th ANTI-BROCARD TRIANGLE

X(10359) lies on these lines: {2,3398}, {3,3329}, {4,83}, {5,7875}, {32,631}, {76,575}, {98,3090}, {140,3793}, {9862,10345}

X(10360) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

X(10360) lies on these lines: {4,65}, {7,1899}, {8,1425}, {57,6776}, {69,1439}, {2898,5929}, {5930,8815}

X(10360) = perspector of 4th extouch triangle and cross-triangle of ABC and 3rd extouch triangle
X(10360) = perspector of 3rd extouch triangle and cross-triangle of ABC and 4th extouch triangle
X(10360) = {X(5932),X(5933)}-harmonic conjugate of X(5929)

X(10361) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

Let A' be the intersection of the tangents to the Yiu conic at the points where it intersects line BC. Define B', C' cyclically. (i.e., A'B'C' is the polar triangle of the Yiu conic.) Let A" be the intersection of the tangents to the Yiu conic at the points where it intersects the A-excircle. Define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(10361).

X(10361) lies on these lines: {4,65}, {5929,10362}, {5930,6737}

X(10362) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

X(10362) lies on these lines: {4,5933}, {65,10363}, {5929,10361}, {5930,10370}

X(10363) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

X(10363) lies on these lines: {4,10369}, {65,10362}, {5929,10366}, {5930,6737}

X(10363) = perspector of 4th extouch triangle and cross-triangle of 3rd and 5th extouch triangles
X(10363) = perspector of 3rd extouch triangle and cross-triangle of 4th and 5th extouch triangles

X(10364) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

X(10364) lies on these lines: {4,69}, {7,1899}, {65,10362}, {5930,10372}

X(10364) = perspector of 4th extouch triangle and cross-triangle of intouch and 3rd extouch triangles
X(10364) = perspector of 3rd extouch triangle and cross-triangle of intouch and 4th extouch triangles
X(10364) = {X(4),X(69)}-harmonic conjugate of X(5929)

X(10365) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

X(10365) lies on these lines: {1,8808}, {4,65}, {7,10368}, {8,253}, {388,1439}, {5929,10369}

X(10365) = perspector of 5th extouch triangle and cross-triangle of ABC and 3rd extouch triangle
X(10365) = perspector of 3rd extouch triangle and cross-triangle of ABC and 5th extouch triangle

X(10366) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

X(10366) lies on these lines: {4,65}, {12,223}, {56,8808}, {225,1853}, {1439,10368}, {5252,5930}, {5929,10363}

X(10367) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

X(10367) lies on these lines: {1,8282}, {4,8}, {65,10362}, {223,9578}, {5252,5930}

X(10368) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

X(10368) lies on these lines: {1,4}, {7,10365}, {65,10362}, {1439,10366}, {1836,10373}, {5929,10372}

X(10368) = perspector of 5th extouch triangle and cross-triangle of intouch and 3rd extouch triangles
X(10368) = perspector of 3rd extouch triangle and cross-triangle of intouch and 5th extouch triangles

X(10369) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

X(10369) lies on these lines: {4,10363}, {7,8}, {5929,10365}, {5930,8815}

X(10369) = perspector of 5th extouch triangle and cross-triangle of ABC and 4th extouch triangle
X(10369) = perspector of 4th extouch triangle and cross-triangle of ABC and 5th extouch triangle

X(10370) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

X(10370) lies on these lines: {7,8}, {5929,10363}, {5930,10362}

X(10371) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

Let A"B"C" be as defined at X(10361). A"B"C" is perspective to the 5th extouch triangle at X(10371).

X(10371) lies on these lines: {1,1211}, {7,8}, {56,3687}, {72,5928}, {5930,6737}

X(10372) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

X(10372) lies on these lines: {7,8}, {56,1211}, {5929,10368}, {5930,10364}

X(10372) = perspector of 5th extouch triangle and cross-triangle of intouch and 4th extouch triangles
X(10372) = perspector of 4th extouch triangle and cross-triangle of intouch and 5th extouch triangles

X(10373) = PERSPECTOR OF EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 1st AND 3rd EXTOUCH TRIANGLES

X(10373) lies on these lines: {1,64}, {4,8}, {1836,10368}, {3057,5930}, {4295,10400}

X(10374) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 1st AND 3rd EXTOUCH TRIANGLES

X(10374) lies on these lines: {4,65}, {11,8808}, {55,223}, {56,3182}, {3057,5930}

X(10375) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 1st AND 5th EXTOUCH TRIANGLES

X(10375) lies on these lines: {8,253}, {33,64}

X(10376) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th EXTOUCH TRIANGLES

X(10376) lies on these lines: {8,479}, {34,6059}, {65,1439}, {388,7197}, {1460,4320}

X(10377) = PERSPECTOR OF EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th EXTOUCH TRIANGLE

X(10377) lies on these lines: {65,497}, {388,7197}, {4012,5930}

X(10378) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

X(10378) lies on the Jerabek hyperbola and these lines: {6,1712}, {72,10379}, {1439,10380}

X(10378) =isogonal conjugate of X(13618)

X(10379) = PERSPECTOR OF 2nd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

X(10379) lies on these lines: {4,51}, {72,10378}

X(10380) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

X(10380) lies on these lines: {4,51}, {72,5930}, {1439,10378}

X(10381) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th EXTOUCH TRIANGLES

X(10381) lies on these lines: {1,4199}, {4,69}, {9,1046}, {10,12}

X(10381) = crossdifference if every pair of points on line X(3049)X(7252)

X(10382) = HOMOTHETIC CENTER OF 2nd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

X(10382) lies on these lines: {1,4}, {2,5809}, {3,10396}, {6,7070}, {9,55}, {11,10582}, {35,1728}, {40,9786}, {42,4319}, {57,5728}, {63,1005}, {25,610}, {41,28070}, {46,10399}, {329,390}, {405,936}, {954,5927}, {5218,8580}, {5219,8226}, {8232,10578}

X(10382) = homothetic center of excentral triangle and cross-triangle of intouch and 2nd extouch triangles
X(10382) = homothetic center of intouch triangle and cross-triangle of excentral and 2nd extouch triangles
X(10382) = crossdifference of every pair of points on line X(652)X(3669)
X(10382) = {X(2),X(7675)}-harmonic conjugate of X(10383)

X(10383) = HOMOTHETIC CENTER OF ASCELLA TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

Note: the Ascella triangle is the mid-triangle of the intouch and excentral triangles.

X(10383) lies on these lines: {1,3}, {2,5809}, {9,10391}, {33,7490}, {77,479}, {78,5273}, {142,497}, {200,5218}, {223,991}, {226,5732}, {284,5324}, {390,4666},...}

X(10383) = {X(2),X(7675)}-harmonic conjugate of X(10382)

X(10384) = HOMOTHETIC CENTER OF ATIK TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

X(10384) lies on these lines: {1,971}, {7,738}, {8,9}, {55,7308}, {56,2951}, {57,497}, {518,7962}, {954,5927}, {1001,3601}, {1864,9954}, {2550,8582}, {3295,9947}, {5768,9948}, {8543,10393}

X(10385) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(2)

X(10385) lies on these lines: {1,376}, {2,11}, {4,3746}, {7,3748}, {8,3683}, {12,3839}, {35,1058}, {56,10304}, {524,10387}, {527,4326}

X(10385) = X(51) of cross-triangle of intouch and excentral triangles

X(10386) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(5)

X(10386) lies on these lines: {1,550}, {3,390}, {5,55}, {11,632}, {12,3845}, {20,6767}, {35,496}, {56,8703}, {495,3585}, {546,3085}, {548,999}, {3583,3858}, {3628,5432}

X(10386) = X(143) of cross-triangle of intouch and excentral triangles

X(10387) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(6)

X(10387) lies on these lines: {1,1350}, {6,31}, {11,3763}, {524,10385}

X(10387) = X(53) of cross-triangle of intouch and excentral triangles

X(10388) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(57)

X(10388) lies on these lines: {1,3}, {11,8580}, {329,390}, {527,4326}, {1864,9954}

X(10388) = X(25) of cross-triangle of intouch and excentral triangles

X(10389) = X(2) OF CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

X(10389) lies on these lines: {1,3}, {2,3158}, {8,5436}, {9,1174}, {21,6762}, {100,4666}, {105,612}, {3616,5438}

X(10389) = isogonal conjugate of X(10390)
X(10389) = {X(1),X(55)}-harmonic conjugate of X(57)

X(10390) = ISOGONAL CONJUGATE OF X(10389)

Let A'B'C' be the reflection of ABC in X(1). ABC and A'B'C' intersect at 6 points, which lie on an ellipse, centered at X(1). X(10390) is the perspector of this ellipse. The 6 points of intersection are the same as A'b, A'c, B'c, B'a, C'a, C'b as described in preamble before X(7955).

The trilinear polar of X(10390) passes through X(650).

X(10390) lies on these lines: {1,1418}, {4,5542}, {8,142}, {9,354}, {57,2346}

X(10390) = isogonal conjugate of X(10389)

X(10391) = HOMOTHETIC CENTER OF ASCELLA TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

Let A'B'C' be the intangents triangle. Let L_A be the line through the points of contact of the incircle and lines BC and B'C'. 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 the 1st Conway triangle at X(10391).

X(10391) lies on these lines: {1,84}, {2,1864}, {3,1708}, {4,9942}, {6,1040}, {7,354}, {9,10383}, {11,3742}, {20,65}, {55,63}, {57,5728}, {226,971}, {517,4304}, {8726,10396}

X(10392) = HOMOTHETIC CENTER OF ATIK TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10392) lies on these lines: {1,5817}, {4,3062}, {7,3832}, {8,9}, {10,9844}, {11,118}, {65,9949}, {142,2476}, {3826,8582}, {5218,8580}, {8583,10393}

X(10393) = HOMOTHETIC CENTER OF 2nd CIRCUMPERP TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10393) lies on these lines: {1,4}, {3,1708}, {8,2900}, {9,21}, {35,920}, {55,72}, {56,5728}, {57,411}, {3576,10396}, {7987,10398}, {8543,10384}, {8583,10392}

X(10393) = crossdifference of every pair of points on line X(652)X(4017)

X(10394) = HOMOTHETIC CENTER OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10394) lies on these lines: {1,651}, {2,1864}, {3,5729}, {4,7}, {6,3100}, {9,21}, {10,5696}, {11,10129}, {12,8255}, {55,1776}, {57,8544}, {72,4313}, {142,2476}, {144,145}, {404,8257}, {6173,9581}

X(10394) = anticomplement of X(5784)

X(10395) = HOMOTHETIC CENTER OF 4th EULER TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10395) lies on these lines: {1,6832}, {2,224}, {4,46}, {5,226}, {9,2478}, {10,55}, {11,72}, {12,5728}, {57,6835}, {65,8226}

X(10395) = complement of X(224)

X(10396) = HOMOTHETIC CENTER OF HEXYL TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10396) lies on these lines: {1,6}, {3,10382}, {4,57}, {11,5715}, {19,3176}, {20,1445}, {40,950}, {56,1490}, {1709,3339}, {3576,10393}, {8726,10391}

X(10396) = crossdifference of every pair of points on line X(513)X(10397)

X(10397) = CROSSDIFFERENCE OF X(4) AND X(57)

X(10397) lies on these lines: {520,647}, {650,663}

X(10397) = crossdifference of every pair of points on line X(4)X(57)
X(10397) = intersection of trilinear polars of X(3) and X(9)
X(10397) = perspector of hyperbola {{A,B,C,X(3),X(9)}}

X(10398) = HOMOTHETIC CENTER OF 6th MIXTILINEAR TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10398) lies on these lines: {1,6}, {2,5785}, {4,3062}, {7,1210}, {57,971}, {226,5817}, {7987,10393}

X(10399) = HOMOTHETIC CENTER OF REFLECTION TRIANGLE OF X(1) AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

X(10399) lies on these lines: {1,6}, {2,10122}, {4,79}, {57,6985}

X(10400) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 3rd EXTOUCH TRIANGLES

X(10400) lies on these lines: {4,7}, {5,7013}, {57,1901}, {65,10362}, {79,8809}, {198,226}, {553,8808}, {4295,10373}

X(10400) = {X(10904),X(10905)}-harmonic conjugate of X(1)

X(10401) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 4th EXTOUCH TRIANGLES

Let A_B, A_C be the points where the A-excircle touches lines CA and AB resp., and define B_C, B_A, C_A, C_B cyclically. A_B, A_C, B_C, B_A, C_A, C_B lie on the Yiu conic (defined at X(478)). Let T_A be the intersection of the tangents to the Yiu conic at B_C and C_A, and define T_B, T_C cyclically. Let T_A' be the intersection of the tangents to the Yiu conic at B_A and C_B, and define T_B', T_C' cyclically. Let S_A = T_BT_C∩T_B'T_C', S_B = T_CT_A∩T_C'T_A', S_C = T_AT_B∩T_A'T_B' (i.e. S_AS_BS_C is the side triangle of T_AT_BT_C and T_A'T_B'T_C'). S_AS_BS_C is perspective to ABC at X(65), and to the intouch triangle at X(10401).

Following are barycentric representations (Peter Moses, October 29, 2016): The Yiu conic is given by 2 S^2 x^2+(a+b-c) (a-b+c) (a^2+b^2+c^2+2 b c) y z + (cyclic) = 0, and
A_B = -a+b-c : 0 : a+b+c
A_C = -a-b+c : a+b+c : 0
T_A = a (a+b-c) (a-b+c) (b+c) : -(a+b-c) (a^3-b^3+a^2 c-a c^2-c^3) : (a+b) (a-b-c) c (a-b+c)
T_A' = a (a+b-c) (a-b+c) (b+c) : -b (a+b-c) (a+c) (-a+b+c) : -(a-b+c) (a^3+a^2 b-a b^2-b^3-c^3)
S_A = (a+b-c) (a-b+c) (a^3+b^3+2 a b c+b^2 c+b c^2+c^3) : -b (a+b-c) (a+c) (-a+b+c) (a+b+c) : (a+b) (a-b-c) c (a-b+c) (a+b+c)
V_A = a^6-a^5 b-a^4 b^2+a^2 b^4+a b^5-b^6-a^5 c+a b^4 c-a^4 c^2-2 a^2 b^2 c^2-2 a b^3 c^2+b^4 c^2-2 a b^2 c^3+a^2 c^4+a b c^4+b^2 c^4+a c^5-c^6
         : b (a+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)
         : -(a+b) (a-b-c) 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(10401) lies on these lines: {7,8}, {56,4357}, {57,1211}, {77,1464}, {86,1408}, {141,2285}, {222,226}, {1439,10361}, {5929,10400}, {10402,10403}

X(10401) = {X(10907),X(10908)}-harmonic conjugate of X(1)

X(10402) = X(4)X(65) ∩ X(7)X(1903)

Continuing from X(10401), let V_A = T_BT_B'∩T_CT_C', V_B = T_CT_C'∩T_AT_A', V_C = T_AT_A'∩T_BT_B' (i.e., V_AV_BV_C is the vertex triangle of T_AT_BT_C and T_A'T_B'T_C'). V_AV_BV_C is perspective to ABC at X(1903), and to the intouch triangle at X(10402).

X(10402) lies on these lines: {4,65}, {7,1903}, {9,1020}, {10401,10403}

X(10403) = X(9)X(478) ∩ X(65)X(1826)

Continuing from X(10401) and X(10402), triangles S_AS_BS_C and V_AV_BV_C are perspective at X(10403).

X(10403) lies on these lines: {{9,478}, {65,1826}, {10401,10402}}.

X(10404) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 5th EXTOUCH TRIANGLES

X(10404) lies on these lines: {1,30}, {2,5302}, {4,354}, {5,3338}, {7,8}, {10,553}, {11,3333}, {12,57}, {20,3475}, {55,4292}, {56,226}, {1439,10366}

X(10404) = anticomplement of X(5302)
X(10404) = {X(10910),X(10911)}-harmonic conjugate of X(1)

X(10405) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF EXTOUCH AND ANTICOMPLEMENTARY TRIANGLES

X(10405) lies on these lines: {2,3160}, {8,144}, {175,7090}, {333,5790}

X(10405) = isogonal conjugate of X(3207)
X(10405) = isotomic conjugate of X(144)
X(10405) = anticomplement of X(3160)
X(10405) = X(7)-cross conjugate of X(2)
X(10405) = X(19)-isoconjugate of X(22117)
X(10405) = trilinear pole of line X(522)X(676) (the radical axis of incircle and polar circle)

X(10406) = PERSPECTOR OF APOLLONIUS TRIANGLE AND CROSS-TRIANGLE OF FEUERBACH AND APOLLONIUS TRIANGLES

X(10406) lies on these lines: {10,3614}, {11,181}, {12,970}, {386,7354}, {1682,9569}, {3058,9554}

X(10407) = PERSPECTOR OF FEUERBACH TRIANGLE AND CROSS-TRIANGLE OF FEUERBACH AND APOLLONIUS TRIANGLES

X(10407) lies on these lines: {2,9553}, {5,181}, {10,3614}, {11,10408}, {12,1682}, {970,7951}, {3085,9554}, {3091,9555}

X(10408) = {X(12),X(181)}-HARMONIC CONJUGATE OF X(10)

Let F_AF_BF_C be the Feuerbach triangle and P_AP_BP_C the Apollonius triangle. Let A' = {F_A,P_A}-harmonic conjugate of X(10), and define B', C' cyclically. The lines AA', BB', CC' concur in X(10408)

X(10408) lies on these lines: {1,2051}, {10,12}, {11,10407}, {55,9553}, {495,970}, {1682,9569}, {3303,9554}, {5261,9534}, {9570,10037}

X(10409) = COLLINGS TRANSFORM OF X(623)

X(10409) lies on the circumcircle and these lines: {15,2380}, {511,1337}, {1634,10410}

X(10409) = perspector of circumcevian triangle of X(16) and cross-triangle of ABC and circumcevian triangle of X(15)
X(10409) = trilinear pole of line X(6)X(2981)
X(10409) = trilinear pole wrt circumtangential triangle of line X(3)X(13)
X(10409) = X(10411)-cross conjugate of X(10410)
X(10409) = perspector of ABC and the triangle formed by reflecting line X(3)X(13) in the sides of ABC
X(10409) = anticomplement of center of cevian circle of X(13)

X(10410) = COLLINGS TRANSFORM OF X(624)

X(10410) lies on the circumcircle and these lines: {16,2381}, {511,1338}, {1634,10409}

X(10410) = perspector of circumcevian triangle of X(15) and cross-triangle of ABC and circumcevian triangle of X(16)
X(10410) = trilinear pole of line X(6)X(6151)
X(10410) = trilinear pole wrt circumtangential triangle of line X(3)X(14)
X(10410) = X(10411)-cross conjugate of X(10409)
X(10410) = perspector of ABC and the triangle formed by reflecting line X(3)X(14) in the sides of ABC
X(10410) = anticomplement of center of cevian circle of X(14)

X(10411) = CROSSPOINT OF X(10409) AND X(10410)

X(10411) lies on these lines: {2,10413}, {54,69}, {99,110}

X(10411) = isotomic conjugate of X(10412)
X(10411) = anticomplement of X(10413)

X(10412) = ISOTOMIC CONJUGATE OF X(10411)

X(10412) lies on the tangent to circumcircle at X(476), and these lines: {2,8562}, {4,1510}, {5,523}

X(10412) = isotomic connjugate of X(10411)
X(10412) = anticomplement of X(8562)
X(10412) = barycentric product of Kiepert hyperbola intercepts of Hatzipolakis axis
X(10412) = trilinear pole of tangent to hyperbola {A,B,C,X(6),X(115)} at X(115) (line X(115)[X(2)-Ceva conjugate of X(137)])

X(10413) = INVERSE-IN-LESTER-CIRCLE OF X(115)

X(10413) lies on these lines: {2,10411}, {6,17}, {115,125}

X(10413) = complement of X(10411)
X(10413) = intersection of tangents to Lester circle at X(13) and X(14)
X(10413) = pole of Fermat axis wrt Lester circle
X(10413) = crossdifference of every pair of points on line X(110)X(1291)
X(10413) = {X(17),X(18)}-harmonic conjugate of X(10414)

X(10414) = INVERSE-IN-LESTER-CIRCLE OF X(140)

X(10414) lies on these lines: {6,17}, {140,523}

X(10414) = intersection of tangents to Lester circle at X(3) and X(5)
X(10414) = pole of Euler line wrt Lester circle
X(10414) = {X(17),X(18)}-harmonic conjugate of X(10413)

X(10415) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(23)

Let P be a point on the circumcircle. Let U = Λ(trilinear polar of P). Then P is also Λ(trilinear polar of U), and P and U are collinear with X(2). Let P* and U* be the points P and U which minimize |PU|. P* and U* have X(2) as midpoint, and are the circumcircle intercepts of the perpendicular to the Euler line through X(2). X(10415) is the cevapoint of P* and U*.

X(10415) lies on these lines: {4,10422}, {23,671}, {67,524}, {523,10562}

X(10415) = isogonal conjugate of X(6593)
X(10415) = isotomic conjugate of X(7664)

X(10416) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(23) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(23)

X(10416) lies on these lines: {2,8877}, {23,671}, {111,1287}, {316,691}

X(10416) = isogonal conjugate of X(10417)

X(10417) = ISOGONAL CONJUGATE OF X(10416)

The tangents to the Droussent central cubic (K042) at the vertices of ABC concur in X(10417).

X(10417) lies on the Droussent central cubic (K042) and these lines: {1205,3455}, {2393,2930}

X(10417) = isogonal conjugate of X(10416)

X(10418) = CROSSDIFFERENCE OF PU(63)

X(10418) is the centroid of the cross-triangle of ABC and the circumcevian triangle of X(25), which is degenerate (collinear), on the orthic axis.

X(10418) lies on these lines: {2,99}, {6,5642}, {23,6781}, {25,1560}, {110,5477}, {187,5913}, {230,231}, {1648,2502}

X(10418) = midpoint of X(1648) and X(2502)
X(10418) = crossdifference of every pair of points on line X(3)X(351)
X(10418) = PU(4)-harmonic conjugate of X(14273)

X(10419) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(30)

Let A'B'C' be the cevian triangle of X(74). Let A" be the circumcircle inverse of A', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10419).

X(10419) lies on these lines: {2,5627}, {30,2986}, {74,323}

X(10419) = isogonal conjugate of X(113)
X(10419) = trilinear pole of line X(526)X(686)
X(10419) = Orion transform of X(74)
X(10419) = X(3)-cross conjugate of X(74)

X(10420) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(3) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(30)

X(10420) lies on the circumcircle and these lines: {20,477}, {23,3563}, {30,1300}, {74,323}, {98,858}, {99,6563}, {107,250}, {110,924}, {111,10313}, {112,6753}, {476,2407}

X(10420) = isotomic conjugate of polar conjugate of X(32708)
X(10420) = circumcircle- antipode of X(32710)
X(10420) = crosssum of X(3) and X(526)
X(10420) = Λ(X(1637), X(1989))
X(10420) = Λ(tangent to hyperbola {{A,B,C,X(4),X(476)}} at X(4))
X(10420) = Λ(tangent to hyperbola {{A,B,C,X(4),X(476)}} at X(476))
X(10420) = de-Longchamps-circle-inverse of X(14731)
X(10420) = isogonal conjugate of barycentric square root of X(39021)
X(10420) = X(19)-isoconjugate of X(6334)

X(10421) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(186) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(186)

X(10421) lies on these lines: {4,1138}, {30,340}

X(10422) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(468)

X(10422) lies on these lines: {4,10415}, {23,895}, {316,691}, {468,10424}

X(10422) = isogonal conjugate of X(5181)

X(10423) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(25) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(468)

X(10423) lies on the circumcircle, circle {{X(4),X(6),X(25),X(111)}}, and these lines: {4,2697}, {24,842}, {74,1177}, {98,403}, {99,250}, {110,8673}, {111,8744}, {112,2485},

X(10423) = trilinear pole of line X(6)X(1112)
X(10423) = Ψ(X(6), X(1112))
X(10423) = polar-circle-inverse of X(38971)
X(10423) = circumcircle intercept, other than X(111), of circle {{X(4),X(6),X(25),X(111)}}
X(10423) = barycentric product X(112)*X(2373) (circumcircle-X(2) antipodes)

X(10424) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(468) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(468)

X(10424) lies on these lines: {23,935}, {468,10422}

X(10425) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(3) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(511)

X(10425) lies on the circumcircle and these lines: {98,325}, {99,3566}, {110,8651}, {111,323}, {112,249}, {476,5468}, {511,3563}, {512,3565}

X(10425) = anticomplement of X(36472)
X(10425) = isogonal conjugate of complementary conjugate of X(36472)
X(10425) = trilinear pole of line X(6)X(2987)
X(10425) = isotomic conjugate of polar conjugate of X(32697)

X(10426) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(1155)

X(10426) lies on these lines: {1, 528}, {2, 1156}, {3, 1633}, {7, 100}, {9, 1768}, {10, 1071}, {11, 142}, {119, 971}, {214, 516}, {442, 8287}, {518, 1145}, {527, 1155}, {900, 3126}, {952, 2550}, {954, 2932}, {1001, 10058}, {1317, 5853}, {2802, 5542}, {2829, 5732}, {3243, 5854}, {3647, 6700}, {4996, 5857}, {5805, 5840}

X(10426) = isogonal conjugate of X(10427)

X(10427) = ISOGONAL CONJUGATE OF X(10426)

X(10427) lies on these lines: {1,528}, {2,1156}, {3,1633}, {7,100}, {9,1768}, {10,1071}, {11,142}, {119,971}, {214,516}

X(10427) = isogonal conjugate of X(10426)
X(10427) = complement of X(1156)

X(10428) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(1319)

X(10428) lies on these lines: {4,6075}, {57,1168}, {104,517}, {106,1457}

X(10428) = isogonal conjugate of X(1145)

X(10429) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

X(10429) lies on these lines: {1,9799}, {8,7957}, {9,20}, {21,8273}, {63,9800}, {10432,10433}

X(10429) = isogonal conjugate of X(5584)

X(10430) = HOMOTHETIC CENTER OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

X(10430) lies on these lines: {1,9800}, {2,1750}, {4,5439}, {7,354}, {8,20}, {21,8273}, {69,10433}, {516,9965}, {962,1071}, {3868,9797}

X(10430) = anticomplement of X(1750)

X(10431) = HOMOTHETIC CENTER OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

X(10431) lies on these lines: {2,3}, {7,354}, {8,7957}, {63,516}, {149,9965}, {908,1750}, {962,3868}, {8822,10432}

X(10431) = anticomplement of X(7580)

X(10432) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 1st CONWAY TRIANGLE

X(10432) lies on these lines: {2,1713}, {7,7183}, {8,6527}, {21,69}, {63,4329}, {8822,10431}, {10429,10433}

X(10433) = PERSPECTOR OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 2nd CONWAY TRIANGLE

X(10433) lies on these lines: {7,2478}, {8,8822}, {63,391}, {69,10430}, {10429,10432}

X(10433) = perspector of 1st Conway triangle and extraversion triangle of X(7)

X(10434) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 3rd CONWAY TRIANGLE

Let A_b, A_c, B_c, B_a, C_a, C_b be as in the construction of the Conway circle; see http://mathworld.wolfram.com/ConwayCircle.html. Let A₃ be the intersection of the tangents to the Conway circle at B_a and C_a, and define B₃, C₃ cyclically. Triangle A₃B₃C₃ is here named the 3rd Conway triangle. A₃B₃C₃ is also the polar triangle of the Conway circle.

The appearance of (T,i) in the following list means that the 3rd Conway triangle and T are perspective with perspector X(i), where an asterisk * signifies that the two triangles are homothetic:

• (ABC, 10435)
• (intouch*, 1)
• (excentral*, 1764)
• (hexyl*, 1)
• (intangents, 1)
• (1st circumperp*, 10434)
• (1st Conway*, 10444)
• (2nd Conway*, 10446)
• (inverse-in-Conway-circle*, 10439)
• (6th mixtilinear*, 1)
• (X(1) reflection*, 1)
• (Hutson intouch*,1)

• (incentral, 1),
• (excentral, 1),
• (1st Conway, 10466),
• (2nd Conway, 10446),
• (3rd Conway, 10446),
• (5th Conway, 1)

• (ABC, 1),
• (incentral, 1),
• (excentral, 1),
• (1st Conway, 10444),
• (2nd Conway, 10468),
• (3rd Conway, 10444),
• (5th Conway, 1)

The cross-triangle of any pair of {3rd, 4th, 5th Conway triangles} is the remaining Conway triangle.

Peter Moses (October 29, 2016) contributes barycentrics for vertices of several triangles:

A_B = a+b : 0 : -a} and A_C = a+c : -a : 0

A-vertex 3rd Conway triangle: -a^2 (a+b+c)^2 : a^3 b+2 a^2 b^2+a b^3+2 a^3 c+2 b^3 c+a b c^2-2 a c^3-2 b c^3 : 2 a^3 b-2 a b^3+a^3 c+a b^2 c-2 b^3 c+2 a^2 c^2+a c^3+2 b c^3}.

A-vertex of 4th Conway triangle: 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 : a b (a+b+c) : a c (a+b+c)

A-vertex of 5th Conway triangle: a^2 b+a b^2+a^2 c+2 b^2 c+a c^2+2 b c^2 : -b (b+c) (a+b+c) : -c (b+c) (a+b+c)

X(10434) lies on these lines: {1,3}, {31,572}, {516,10478}, {3240,10440}, {6684,10479}, {10435,10436}

X(10434) = isogonal conjugate of X(10435)
X(10434) = homothetic center of 1st circumperp triangle and 3rd Conway triangle

X(10435) = PERSPECTOR OF CONWAY CIRCLE

Let A₃B₃C₃, A₄B₄C₄, A₅B₅C₅ be the 3rd, 4th and 5th Conway triangles, resp. Let A' be the trilinear product A₃*A₄, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10435), as do the lines A₃A', B₃B', C₃C'. Let A" be the trilinear product A₃*A₅, and define B", C" cyclically. The lines AA", BB", CC" concur in X(10435), as do the lines A₃A", B₃B", C₃C". Let A"' be the trilinear product A₃*A₄*A₅, and define B"', C"' cyclically. The lines AA"', BB"', CC"' concur in X(10435), as do the lines A₃A"', B₃B"', C₃C"'.

The trilinear polar of X(10435) passes through X(650).

X(10435) lies on the Feuerbach hyperbola and these lines: {8,10446}, {9,1764}, {21,10444}, {314,10439}, {941,10478}, {10434,10436}

X(10435) = isogonal conjugate of X(10434)
X(10435) = trilinear product of vertices of 3rd Conway triangle
X(10435) = perspector of ABC and 3rd Conway triangle

X(10436) = ISOGONAL CONJUGATE OF X(2258)

Let A₃B₃C₃ be the 3rd Conway triangle. Let A' be the trilinear pole of line B₃C₃, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10436).

X(10436) lies on these lines: {1,75}, {2,7}, {6,3739}, {8,3879}, {10,69}, {165,10442}, {3338,10468}, {10434,10435}, {10452,10479}

X(10436) = isogonal conjugate of X(2258)
X(10436) = complement of X(17257)
X(10436) = anticomplement of X(5257)
X(10436) = trilinear pole of polar of X(10435) wrt Conway circle
X(10436) = {X(2),X(7)}-harmonic conjugate of X(4357)

X(10437) = PERSPECTOR OF 3rd CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 3rd CONWAY TRIANGLE

Let A₃B₃C₃ be the 3rd Conway triangle. Let A' be the cevapoint of B₃ and C₃, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10437).

X(10437) lies on these lines: {1,7}, {3,10455}, {40,10447}, {69,10454}, {10434,10435} .

X(10437) = isogonal conjugate of X(10438)

X(10438) = ISOGONAL CONJUGATE OF X(10437)

Let A₃B₃C₃ be the 3rd Conway triangle. Let A' be the crosspoint of B₃ and C₃, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10438).

X(10438) lies on these lines: {200, 3588}, {1043, 1764}

X(10438) = isogonal conjugate of X(10437)

X(10439) = CENTROID OF 3rd CONWAY TRIANGLE

Let A' be the inverse-in-Conway-circle of A, and define B', C' cyclically. Triangle A'B'C' is here named the inverse-in-Conway-circle triangle. A'B'C' is homothetic to the 3rd Conway triangle at X(10439). A'B'C' is also the medial triangle of the 3rd Conway triangle.

The appearance of (T,i) in the following list means that the inverse-in-Conway-circle triangle and T are perspective with perspector X(i) as perspector, where an asterisk * signifies that the two triangles are homothetic:

• (ABC, 1),
• (medial, 10472),
• (intouch*, 10473),
• (excentral*, 1),
• (incentral, 1),
• (hexyl*, 10476),
• (1st circumperp*, 1764),
• (2nd circumperp*, 1),
• (inverse-in-incircle*, 1),
• (2nd extouch*, 10477),
• (1st Conway*, 5208),
• (2nd Conway*, 10453),
• (3rd Conway*, 10439),
• (4th Conway, 1),
• (5th Conway, 1),
• (3rd Euler*, 10478),
• (4th Euler*, 10479),
• (Hutson intouch*, 10480)

X(10439) lies on these lines: {1,3}, {2,10440}, {314,10435}, {3741,3817}, {5208,9962}, {10175,10479}

X(10439) = X(2)-of-3rd-Conway-triangle
X(10439) = pole of antiorthic axis wrt Conway circle
X(10439) = anticomplement of X(10440)
X(10439) = inverse-in-Conway-circle of X(1155)

X(10440) = COMPLEMENT OF X(10439)

Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC, and define B', C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle at X(43). The centroid of A'B'C' is X(10440).

X(10440) lies on these lines: {2,10439}, {5,10}, {43,165}, {3240,10434}, {3634,10441}

X(10440) = complement of X(10439)
X(10440) = inverse-in-excircles-radical-circle of X(5087)

X(10441) = NINE-POINT CENTER OF 3rd CONWAY TRIANGLE

X(10441) lies on these lines: {1,3}, {2,970}, {4,69}, {5,1211}, {10,3781}, {1012,10451}, {3634,10440}, {5295,10447}

X(10441) = anticomplement of X(970)
X(10441) = X(5)-of-3rd-Conway-triangle
X(10441) = center of inverse-in-Conway-circle-of-circumcircle
X(10441) = inverse-in-Conway-circle of X(36)
X(10441) = {X(10478),X(10479)}-harmonic conjugate of X(5)
X(10441) = X(13322)-of-excentral-triangle

X(10442) = SYMMEDIAN POINT OF 3rd CONWAY TRIANGLE

X(10442) lies on these lines: {1,7}, {2,10443}, {9,1764}, {165,10436}, {314,3062}, {4385,5223}

X(10442) = anticomplement of X(10443)
X(10442) = X(6)-of-3rd-Conway-triangle
X(10442) = pole of Gergonne line wrt Conway circle
X(10442) = inverse-in-Conway-circle of X(1323)

X(10443) = COMPLEMENT OF X(10442)

Let A'B'C' be as at X(10440). X(10443) = X(69)-of-A'B'C'.

X(10443) lies on these lines: {2,10442}, {4,9}, {6,4297}, {142,2051}

X(10443) = complement of X(10442)

X(10444) = HOMOTHETIC CENTER OF 1st AND 3rd CONWAY TRIANGLES

X(10444) lies on these lines: {2,10445}, {21,10435}, {63,321}, {84,309}

X(10444) = anticomplement of X(10445)
X(10444) = perspector 5th Conway triangle and 1st and 3rd Conway triangles
X(10444) = complement, wrt 3rd Conway triangle, of X(12549)

X(10445) = COMPLEMENT OF X(10444)

X(10445) lies on these lines: {2,10444}, {3,5750}, {4,9}, {5,5257}, {6,515}, {20,5749}

X(10445) = inverse-in-excircles-radical-circle of X(8074)
X(10445) = pole of Gergonne line wrt excircles-radical-circle
X(10445) = complement, wrt 2nd extouch triangle, of X(12689)

X(10446) = HOMOTHETIC CENTER OF 2nd AND 3rd CONWAY TRIANGLES

Let O_A be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius Sqrt(r_A² + s²), where r_A is the A-exradius). Let P_A be the perspector of O_A, and define P_B, P_C cyclically. Triangle P_AP_BP_C is perspective to the 2nd, 3rd and 4th Conway triangles at X(10446).

X(10446) lies on these lines: {1,7}, {2,573}, {3,86}, {4,69}, {5,5224}, {8,10435}, {10,10456}, {19,1944}, {27,394}, {40,10436}, {3146,10454}

X(10446) = anticomplement of X(573)
X(10446) = perspector of 4th Conway triangle and 2nd and 3rd Conway triangles

X(10447) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 4th CONWAY TRIANGLE

Let A₄B₄C₄ be the 4th Conway triangle. Let A' be the trilinear product B₄*C₄, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10447). A'B'C' is also the cross-triangle of ABC and 4th Conway triangle.

Let A₃B₃C₃, A₄B₄C₄, A₅B₅C₅ be the 3rd, 4th and 5th Conway triangles, resp. Let A" be the trilinear product A₃*A₄*A₅, and define B", C" cyclically. The lines A₄A", B₄B", C₄C" concur in X(10447).

X(10447) lies on these lines: {1,75}, {7,10449}, {8,10435}, {37,10472}, {40,10437}, {63,321}, {84,10450}, {4385,5223}, {5295,10441}, {5691,10464}

X(10447) = trilinear product of vertices of 4th Conway triangle
X(10447) = {X(1),X(10456)}-harmonic conjugate of X(10455)

X(10448) = BARYCENTRIC PRODUCT OF VERTICES OF 4th CONWAY TRIANGLE

Let A₄B₄C₄ be the 4th Conway circle. Let L_A be the trilinear polar of A₄, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10448).

X(10448) lies on these lines: {1,21}, {3,750}, {8,2177}, {55,10459}, {56,1011}

X(10448) = {X(21),X(10458)}-harmonic conjugate of X(10457)

X(10449) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

X(10449) lies on these lines: {1,2}, {3,1043}, {4,69}, {5,4417}, {7,10447}, {20,1764}, {40,3886}, {63,7283}, {321,3868}, {515,10465}, {3945,10455}

X(10449) = isotomic conjugate of X(1246)
X(10449) = anticomplement of X(386)
X(10449) = perspector of 4th Conway triangle and cross-triangle of ABC and 2nd Conway triangle
X(10449) = perspector of 2nd Conway triangle and cross-triangle of ABC and 4th Conway triangle
X(10449) = {X(4),X(69)}-harmonic conjugate of X(1330)

X(10450) = PERSPECTOR OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 4th CONWAY TRIANGLES

X(10450) lies on these lines: {84,10447}, {10451,10466}

X(10451) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 4th CONWAY TRIANGLES

X(10451) lies on these lines: {1,7}, {1012,10441}, {5208,10463}

X(10452) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

X(10452) lies on these lines: {1,69}, {7,10447}, {75,5902}, {79,314}, {3664,3741}, {4292,10462}, {10436,10479}

X(10453) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

X(10453) lies on these lines: {1,2}, {7,310}, {20,10476}, {38,192}, {56,1043}, {57,3886}, {63,3685}, {65,4673}, {69,350}, {75,354}, {321,3873}

X(10453) = isogonal conjugate of X(34445)
X(10453) = complement of X(20012)
X(10453) = anticomplement of X(43)
X(10453) = isotomic conjugate of isogonal conjugate of X(20992)
X(10453) = isotomic conjugate of polar conjugate of X(17920)
X(10453) = {X(2),X(145)}-harmonic conjugate of X(42)
X(10453) = polar conjugate of isogonal conjugate of X(22127)
X(10453) = homothetic center of 2nd Conway triangle and inverse-in-Conway-circle triangle

X(10454) = PERSPECTOR OF 5th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

X(10454) lies on these lines: {1,4}, {2,10470}, {3,10479}, {8,573}, {20,1764}, {69,10437}, {3146,10446}

X(10455) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 5th CONWAY TRIANGLE

Let A₅B₅C₅ be the 5th Conway triangle. Let A' be the trilinear product B₅*C₅, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10455). A'B'C' is also the cross-triangle of ABC and 5th Conway triangle.

Let A₃B₃C₃, A₄B₄C₄, A₅B₅C₅ be the 3rd, 4th and 5th Conway triangles, resp. Let A" be the trilinear product A₃*A₄*A₅, and define B", C" cyclically. The lines A₅A", B₅B", C₅C" concur in X(10455).

X(10455) lies on these lines: {1,75}, {2,573}, {3,10437}, {4,10464}, {6,10472}, {7,10461}, {21,10435}, {3664,3741}, {3945,10449}

X(10455) = trilinear product of vertices of 5th Conway triangle
X(10455) = {X(1),X(10456)}-harmonic conjugate of X(10447)
X(10655) = {X(3062),X(10980)}-harmonic conjugate of X(10656)

X(10456) = {X(10447),X(10455)}-HARMONIC CONJUGATE OF X(1)

X(10456) lies on these lines: {1,75}, {7,3741}, {9,1764}, {10,10446}

X(10656) = {X(3062),X(10980)}-harmonic conjugate of X(10655)

X(10457) = BARYCENTRIC PRODUCT OF VERTICES OF 5th CONWAY TRIANGLE

X(10457) lies on these lines: {1,21}, {750,1010}, {1011,4252}

X(10457) = {X(21),X(10458)}-harmonic conjugate of X(10448)

X(10458) = {X(10448),X(10457)}-HARMONIC CONJUGATE OF X(21)

X(10458) lies on these lines: {1,21}, {2,3736}

X(10459) = {X(1),X(8)}-HARMONIC CONJUGATE OF X(42)

Let A₅B₅C₅ be the 5th Conway triangle. Let L_A be the trilinear polar of A₅, and define L_B, L_C cyclically. Let A' = L_B∩L_C, and define B', C' cyclically. The lines AA', BB', CC' conur in X(10459).

X(10459) lies on these lines: {1,2}, {21,902}, {31,958}, {38,65}, {40,4414}, {55,10448}, {56,750}

X(10460) = {X(31),X(42)}-HARMONIC CONJUGATE OF X(2293)

X(10460) lies on these lines: {1,5273}, {6,31}, {57,959}

X(10460) = perspector of unary cofactor triangles of 3rd, 4th and 5th Conway triangles

X(10461) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 1st AND 5th CONWAY TRIANGLES

X(10461) lies on these lines: {1,21}, {3,10477}, {7,10455}, {20,1764}, {40,1043}, {57,1010}, {84,309}, {1012,10441}, {9799,10465}

X(10461) = perspector of 5th Conway triangle and cross-triangle of ABC and 1st Conway triangle
X(10461) = perspector of 1st Conway triangle and cross-triangle of ABC and 5th Conway triangle

X(10462) = PERSPECTOR OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th CONWAY TRIANGLES

X(10462) lies on these lines: {1,8822}, {7,10455}, {4292,10452}, {5208,9962}

X(10463) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th CONWAY TRIANGLES

X(10463) lies on these lines: {1,84}, {20,1764}, {21,572}, {5208,10451}

X(10464) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 5th CONWAY TRIANGLES

X(10464) lies on these lines: {4,10455}, {3146,10446}, {5691,10447}

X(10465) = PERSPECTOR OF 5th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 5th CONWAY TRIANGLES

X(10465) lies on these lines: {1,7}, {8,1764}, {515,10449}, {944,10441}, {3522,10434}, {9799,10461}

X(10466) = PERSPECTOR OF 1st AND 4th CONWAY TRIANGLES

X(10466) lies on these lines: {1,8822}, {2,10467}, {7,10447}

X(10466) = anticomplement of X(10467)
X(10466) = perspector of Conway circle wrt 1st Conway triangle

X(10467) = COMPLEMENT OF X(10466)

X(10467) lies on these lines: {2,10466}, {9,386}, {10,1901}, {573,1750}

X(10467) = complement of X(10466)
X(10467) = perspector of excircles radical circle wrt 2nd extouch triangle

X(10468) = PERSPECTOR OF 2nd AND 5th CONWAY TRIANGLES

X(10468) lies on these lines: {1,69}, {2,10469}, {7,10455}, {75,3670}, {314,4389}, {3338,10436}

X(10468) = anticomplement of X(10469)
X(10468) = perspector of Conway circle wrt 2nd Conway triangle

X(10469) = COMPLEMENT OF X(10468)

Let L_A be the radical axis of the A-excircle and the excircles radical circle, 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 perspective to ABC at X(2051), and to the excentral triangle at X(10469). A'B'C' is the complement of the 5th Conway triangle, and so is homothetic to the 5th Conway triangle at X(2).

X(10469) lies on these lines: {2,10468}, {6,10}

X(10469) = complement of X(3055)

X(10470) = HOMOTHETIC CENTER OF 3rd CONWAY TRIANGLE AND CROSS-TRIANGLE OF EXCENTRAL AND 2nd CIRCUMPERP TRIANGLES

X(10470) lies on these lines: {1,3}, {2,10454}, {20,10478}, {21,572}

X(10471) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND INVERSE-IN-CONWAY-CIRCLE TRIANGLE

X(10471) lies on these lines: {1,75}, {10,4476}, {333,1746}

X(10471) = trilinear product of vertices of inverse-in-Conway-circle triangle

X(10472) = PERSPECTOR OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND MEDIAL TRIANGLE

X(10472) lies on the nine-point circle and these lines: {1,3696}, {2,314}, {6,10455}, {9,1764}, {10,3781}, {37,10447}, {75,980}, {141,442}, {142,3741}

X(10472) = complement of X(941)

X(10473) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND INTOUCH TRIANGLE

Let A'B'C' be the intangents triangle. Let L_A be the line through the points of contact of the incircle and lines BC and B'C'. 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 the 3rd Conway triangle at X(10473).

X(10473) lies on these lines: {1,3}, {2,181}, {7,310}, {8,9552}

X(10473) = anticomplement of X(9564)
X(10473) = {X(10474),X(10475)}-harmonic conjugate of X(1)

X(10474) =  X(1)X(3)∩X(314)X(1441)

Let A₄B₄C₄ be the 4th Conway triangle. Let A' be the crosspoint of B₄ and C₄, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10474).

X(10474) lies on these lines: {1,3}, {181,22076}, {314,1441}, {1409,2171}, {1836,10454}, {1837,10478}, {1880,11396}

X(10474) = {X(1),X(10473)}-harmonic conjugate of X(10475)

X(10475) =  X(1)X(3)∩X(314)X(1014)

Let A₅B₅C₅ be the 5th Conway triangle. Let A' be the crosspoint of B₅ and C₅, and define B', C' cyclically. The lines AA', BB', CC' concur in X(10475).

X(10475) lies on these lines: {1,3}, {181,1193}, {314,1014}, {497,10465}, {604,5114}, {894,2975}, {961,4968}, {1042,1401}, {1125,10408}, {1201,1400}, {1397,1468}, {1469,10571}, {1696,15479}

X(10475) = {X(1),X(10473)}-harmonic conjugate of X(10474)

X(10476) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND HEXYL TRIANGLE

X(10476) lies on these lines: {1,3}, {2,9548}, {4,3741}, {20,10453}, {84,309}

X(10477) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND 2nd EXTOUCH TRIANGLE

X(10477) lies on these lines: {1,6}, {2,3786}, {3,10461}, {4,69}, {8,3963}, {10,3779}, {21,5138}

X(10477) = anticomplement of X(4260)

X(10478) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND 3rd EULER TRIANGLE

X(10478) lies on these lines: {1,4}, {2,573}, {5,1211}, {6,1746}, {20,10470}, {142,10442}, {314,4417}, {516,10434}, {941,10435}, {3741,3817}

X(10478) = {X(5),X(10441)}-harmonic conjugate of X(10479)

X(10479) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND 4th EULER TRIANGLE

X(10479) lies on these lines: {1,2}, {3,10454}, {4,1764}, {5,1211}, {75,3670}, {314,5224}, {6684,10434}, {10175,10439}, {10436,10452}

X(10479) = {X(5),X(10441)}-harmonic conjugate of X(10478)

X(10480) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND HUTSON INTOUCH TRIANGLE

X(10480) lies on these lines: {1,3}, {2,1682}, {8,314}

X(10480) = anticomplement of X(9565)

X(10481) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND INVERSE-IN-INCIRCLE TRIANGLE

Let L_A be the trilinear polar, wrt the intouch triangle, of the A-excenter; 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 perspective to the intouch triangle at X(10481).

Let A'B'C' be the inverse-in-incircle triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10481). The lines A'A", B'B", C'C" concur in X(7).

X(10481) lies on these lines: {1,7}, {57,169}, {222,553}, {555,10489}

X(10481) = midpoint of X(481) and X(482)
X(10481) = isogonal conjugate of X(10482)
X(10481) = complement of X(30625)
X(10481) = X(32)-of-intouch-triangle
X(10481) = {X(7),X(3668)}-harmonic conjugate of X(3663)
X(10481) = trilinear product of vertices of inverse-in-incircle triangle

X(10482) = ISOGONAL CONJUGATE OF X(10481)

Let A'B'C' be the 2nd mixtilinear triangle. X(10482) is the radical center of the circumcircles of A'BC, B'CA, C'AB.

The trilinear polar of X(10482) passes through X(657).

X(10482) lies on these lines: {1,1170}, {33,7322}, {55,218}

X(10482) = isogonal conjugate of X(10481)

X(10483) = HOMOTHETIC CENTER OF CEVIAN TRIANGLE OF X(3) AND CROSS-TRIANGLE OF ABC AND 2nd ISOGONAL TRIANGLE OF X(1)

X(10483) lies on these lines: {1,30}, {2,5370}, {3,3585}, {4,36}, {5,7280}, {20,35}, {55,1657}, {56,382}, {381,5204}

X(10483) = 3rd isogonal perspector of X(1) (see X(36))

X(10484) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF McCAY AND ANTI-McCAY TRIANGLES

Let A'B'C' be the McCay triangle. X(10484) is the radical center of the circumcircles of A'BC, B'CA, and C'AB.

The trilinear polar of X(10484) meets the line at infinity at X(523).

X(10484) lies on the Kiepert hyperbola and these lines: {2,8586}, {6,8587}

X(10484) = isogonal conjugate of X(10485)

X(10485) = ISOGONAL CONJUGATE OF X(10484)

X(10485) lies on these lines: {2,8587}, {3,6}

X(10485) = isogonal conjugate of X(10484)
X(10485) = inverse-in-1st-Brocard-circle of X(8586)

X(10486) = PERSPECTOR OF McCAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND McCAY TRIANGLE

Let A'B'C' be the McCay triangle. Let A" be the isogonal conjugate of A', and define B", C" cyclically. The lines AA", BB", CC" concur in X(576). The lines A'A", B'B", C'C" concur in X(10486). A"B"C" is also the cross-triangle of ABC and the McCay triangle.

X(10486) lies on these lines: {2,3}, {575,7608}, {576,7607}

X(10487) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND ANTI-McCAY TRIANGLE

Let A'B'C' be the anti-McCay triangle. Let A" be the isogonal conjugate, wrt A'B'C', of A. Define B", C" cyclically. The lines AA", BB", CC" concur in X(10487). A"B"C" is also the cross-triangle of ABC and anti-McCay triangle.

X(10487) lies on these lines: {2,99}, {8587,8860}

X(10488) = PERSPECTOR OF ANTI-McCAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND ANTI-McCAY TRIANGLE

Let A'B'C' be the anti-McCay triangle. Let A" be the isogonal conjugate, wrt A'B'C', of A. Define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(10488). A"B"C" is also the cross-triangle of ABC and anti-McCay triangle.

X(10488) lies on these lines: {2,8786}, {3,67}, {6,598}, {8587,8860}

X(10489) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND MID-ARC TRIANGLE

Let A'B'C' be the mid-arc triangle. Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10489).

Let (I_A) be the circle tangent to the incircle and lines CA and AB, such that (I_A) lies between the incircle and A. Define (I_B), (I_C) cyclically. Let A' be the insimilicenter of (I_B) and (I_C), and define B', C' cyclically. The lines AA', BB', CC' concur in X(10489). A'B'C' is equivalent to the cross-triangle of ABC and the mid-arc triangle.

X(10489) lies on these lines: {177,234}, {555,10481}

X(10489) = SS(A→A') of X(1089), where A'B'C' = excentral triangle
X(10489) = trilinear square of X(177)
X(10489) = trilinear product of vertices of mid-arc triangle

X(10490) = PERSPECTOR OF 1st TANGENTIAL MID-ARC TRIANGLE AND CROSS-TRIANGLE OF ABC AND MID-ARC TRIANGLE

X(10490) lies on these lines: {1,168}, {2,178}

X(10490) = SS(A→A') of X(72), where A'B'C' = excentral triangle

X(10491) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 2nd MID-ARC TRIANGLE

The 2nd mid-arc triangle is here defined as the triangle D"E"F" introduced in the preamble before X(8075). It is the incircle antipode of the mid-arc triangle.

D" has trilinears: (y - z)² : x² : x², where x = cos(A/2), y = cos(B/2), z = cos(C/2).
Let A' be the trilinear product E"*F", and define B', C' cyclically. The lines AA', BB', CC' concur in X(10491).

Let (I_A') be the circle tangent to the incircle and lines CA and AB, such that the incircle lies between (I_A') and A. Define (I_B'), (I_C') cyclically. Let A' be the insimilicenter of (I_B') and (I_C'), and define B', C' cyclically. The lines AA', BB', CC' concur in X(10491). A'B'C' is equivalent to the cross-triangle of ABC and the 2nd mid-arc triangle.

X(10491) lies on the incircle and these lines: {555,10481}

X(10491) = SS(A→A') of X(1111), where A'B'C' = excentral triangle
X(10491) = X(98)-of-mid-arc-triangle
X(10491) = X(99)-of-2nd-mid-arc-triangle
X(10491) = X(101)-of-intouch-triangle
X(10491) = X(103)-of-Hutson-intouch-triangle
X(10491) = trilinear square of X(10492)
X(10491) = trilinear product of vertices of 2nd mid-arc triangle

X(10492) = TRILINEAR SQUARE ROOT OF X(10491)

Let A'B'C' be the excentral-of-intouch triangle. Let A" = BB'∩CC', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10492).

X(10492) is the intersection of the line of the degenerate cross-triangle of ABC and 1st tangential mid-arc triangle (line X(6728)X(10495), the trilinear polar of X(174)), and the line of the degenerate cross-triangle of ABC and 2nd tangential mid-arc triangle (the trilinear polar of X(1488)).

X(10492) lies on these lines: {241,514}, {6728,10495}

X(10492) = SS(A→A') of X(514), where A'B'C' = excentral triangle

X(10493) = CENTER OF 1st TANGENTIAL MID-ARC CONIC

The 1st tangential mid-arc conic is here defined as the conic that passes through A, B, C and the vertices of the 1st tangential mid-arc triangle. It is the isogonal conjugate of line X(8076)X(10231) (the trilinear polar of X(174)), and also passes through X(10496) and X(10498). The perspector of the conic is X(7707).

X(10493) lies on these lines: {177,6724}

X(10493) = X(2)-Ceva conjugate of X(7707)

X(10494) = CENTER OF 2nd TANGENTIAL MID-ARC CONIC

The 2nd tangential mid-arc conic is here defined as the conic that passes through A, B, C and the vertices of the 2nd tangential mid-arc triangle. It is the isogonal conjugate of line X(1)X(168), and also passes through X(10497) and X(10498). The perspector of the conic is X(10495).

X(10494) lies on these lines: {244, 10501}, {8242, 22994}, {17761, 21623}

X(10494) = X(2)-Ceva conjugate of X(10495)

X(10495) = PERSPECTOR OF 2nd TANGENTIAL MID-ARC CONIC

X(10495) lies on these lines: {44,513}, {6728,10492}, {8076,10231}

X(10495) = crossdifference of every pair of points on line X(1)X(168)
X(10495) = X(2)-Ceva conjugate of X(10494)
X(10495) = trilinear square root of X(10501)

X(10496) = COLLINGS TRANSFORM OF X(10493)

Let I be the incenter and I_A, I_B, I_C the excenters. The Monge line of the circumcircles of BCII_A, CAII_B, ABII_C meets the line at infinity at the isogonal conjugate of X(10496).

X(10496) lies on the circumcircle and these lines: {55,10504}, {56,10505}

X(10496) = X(109)-of-1st-circumperp-triangle
X(10496) = X(102)-of-2nd-circumperp-triangle
X(10496) = X(124)-of-excentral-triangle
X(10496) = X(117)-of-hexyl-triangle
X(10496) = Collings transform of X(10493)
X(10496) = circumcircle intercept, other than A, B, C, of 1st tangential mid-arc conic
X(10496) = trilinear pole of line X(6)X(7707)
X(10496) = Ψ(X(6), X(7707))
X(10496) = Λ(X(8076), X(10231))

X(10497) = COLLINGS TRANSFORM OF X(10494)

Let A'B'C' be the excentral triangle. The Gergonne lines of triangles A'BC, B'CA, C'AB are the sidelines of a triangle perspective to ABC at X(10497). (Randy Hutson, June 27, 2018)

X(10497) lies on the circumcircle and these lines: {56, 10491}, {3659, 7028}

X(10497) = X(103)-of-1st-circumperp-triangle
X(10497) = X(101)-of-2nd-circumperp-triangle
X(10497) = X(118)-of-excentral-triangle
X(10497) = X(116)-of-hexyl-triangle
X(10497) = Collings transform of X(10494)
X(10497) = circumcircle intercept, other than A, B, C, of 2nd tangential mid-arc conic
X(10497) = Λ(X(177), X(3057))
X(10497) = trilinear pole of line X(6)X(10495)
X(10497) = Ψ(X(6), X(10495))
X(10497) = Λ(X(1), X(168))

X(10498) = ISOGONAL CONJUGATE OF X(8076)

X(10498) lies on these lines: {174,354}, {258,503}

X(10498) = isogonal conjugate of X(8076)
X(10498) = common point, other than A, B, C, of 1st and 2nd tangential mid-arc conics

X(10499) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND MID-ARC TRIANGLES

X(10499) lies on these lines: {1,7}, {177,234}

X(10499) = reflection of X(10507) in X(1)
X(10499) = X(58)-of-intouch-triangle

X(10500) = PERSPECTOR OF MID-ARC TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND MID-ARC TRIANGLES

Let A', B', C' be as at X(10501). Triangle A'B'C' is perspective to the mid-arc triangle at X(10500).

X(10500) lies on these lines: {1,3}, {11,10506}, {177,234}

X(10500) = X(21)-of-intouch-triangle
X(10500) = X(54)-of-mid-arc-triangle

X(10501) = PERSPECTOR OF MID-ARC TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd MID-ARC TRIANGLES

Of the 4 intersections of the incircle and incentral inellipse, let A' be the one farthest from A, and define B', C' cyclically. The remaining intersection is X(10501). (See Hyacinthos #21047-8, 6/1/2012, Randy Hutson/Barry Wolk/Nikos Dergiades)

X(10501) lies on the incircle and these lines: {1,8099}, {174,354}, {8422,10505},

X(10501) = reflection of X(10506) in X(1)
X(10501) = X(100)-of-intouch-triangle
X(10501) = X(74)-of-mid-arc-triangle
X(10501) = {X(174),X(354)}-harmonic conjugate of X(10502)
X(10501) = trilinear square of X(10495)

X(10502) = {X(174),X(354)}-HARMONIC CONJUGATE OF X(10501)

Let A', B', C' be as at X(10501). The lines AA', BB', CC' concur in X(10502).

Let A"B"C" be the intangents triangle. Let L_A be the line through the points of contact of the incircle and lines BC and B"C". 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 the Yff central triangle at X(10502).

X(10502) lies on these lines: {174,354}, {177,234}

X(10503) = X(177)X(234) ∩ X(354)X(2089)

Let A^*B^*C^* be as at X(10502). Triangle A^*B^*C^* is homothetic to the tangential mid-arc triangle at X(10503).

X(10503) lies on these lines: {1,8099}, {177,234}, {354,2089}

X(10503) = {X(1),X(8099)}-harmonic conjugate of X(10506)

X(10504) = PERSPECTOR OF HUTSON INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd MID-ARC TRIANGLES

X(10504) lies on the incircle and these lines: {1,7057}, {55,10496}, {177,10506}

X(10504) = reflection of X(10505) in X(1)
X(10504) = X(109)-of-intouch-triangle

X(10505) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF HUTSON INTOUCH AND MID-ARC TRIANGLES

X(10505) lies on the incircle and these lines: {1,7057}, {56,10496}, {8422,10501}

X(10505) = reflection of X(10504) in X(1)
X(10505) = X(102)-of-intouch-triangle

X(10506) = PERSPECTOR OF 2nd MID-ARC TRIANGLE AND CROSS-TRIANGLE OF HUTSON INTOUCH AND MID-ARC TRIANGLES

X(10506) lies on the incircle and these lines: {1,8099}, {11,10500}, {55,7597}, {56,3659}, {177,10504}

X(10506) = reflection of X(10501) in X(1)
X(10506) = X(104)-of-intouch-triangle
X(10506) = X(110)-of-mid-arc-triangle
X(10506) = {X(1),X(8099)}-harmonic conjugate of X(10503)

X(10507) = PERSPECTOR OF HUTSON INTOUCH TRIANGLE AND CROSS-TRIANGLE OF HUTSON INTOUCH AND 2nd MID-ARC TRIANGLES

X(10507) lies on these lines: {1,7}, {8422,10508}

X(10507) = reflection of X(10499) in X(1)
X(10507) = X(3430)-of-intouch-triangle

X(10508) = PERSPECTOR OF 2nd MID-ARC TRIANGLE AND CROSS-TRIANGLE OF HUTSON INTOUCH AND 2nd MID-ARC TRIANGLES

X(10508) lies on these lines: {1,3}, {8422,10507}

X(10508) = reflection of X(10500) in X(1)
X(10508) = X(3651)-of-intouch-triangle

X(10509) = PERSPECTOR OF HONSBERGER TRIANGLE AND CROSS-TRIANGLE OF ABC AND HONSBERGER TRIANGLE

X(10509) lies on these lines: {6,279}, {7,55}, {9,85}, {6604,7674}

X(10509) = isogonal conjugate of X(8012)

X(10510) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND X(2)-EHRMANN TRIANGLE

X(10510) lies on these lines: {2,8262}, {3,6}, {23,6593}, {67,524}, {2393,2930}

X(10510) = isogonal conjugate of X(10511)
X(10510) = isotomic conjugate of X(10512)
X(10510) = anticomplement of X(8262)
X(10510) = homothetic center of X(2)- and X(6)-Ehrmann triangles

X(10511) = ISOGONAL CONJUGATE OF X(10510)

X(10511) lies on the Kiepert hyperbola and these lines: {2,67}, {4,1383}, {23,671}

X(10511) = isogonal conjugate of X(10510)
X(10511) = vertex conjugate of X(23) and X(67)

X(10512) = ISOTOMIC CONJUGATE OF X(10510)

X(10512) lies on these lines: {67, 316}, {76, 7664}, {264, 598}, {7827, 8791}

X(10512) = isotomic conjugate of X(10510)
X(10512) = Brianchon point (perspector) of inconic centered at X(8262)

X(10513) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF INNER AND OUTER GREBE TRIANGLES

X(10513) lies on these lines: {2,6}, {20,3933}, {1160,6215}, {1161,6214}, {3640,5689}, {3641,5688}, {6201,10514}, {6202,10515}

X(10513) = anticomplement of X(5304)
X(10513) = {X(1270),X(1271)}-harmonic conjugate of X(6)
X(10513) = perspector of inner Grebe triangle and cross-triangle of ABC and outer Grebe triangle
X(10513) = perspector of outer Grebe triangle and cross-triangle of ABC and inner Grebe triangle

X(10514) = HOMOTHETIC CENTER OF EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND INNER GREBE TRIANGLE

X(10514) lies on these lines: {2,5871}, {4,640}, {5,6}, {355,5605}, {381,1161}, {637,1271}, {946,5689}, {5589,7989}, {5595,7395}, {6201,10513}

X(10514) = {X(5),X(10516)}-harmonic conjugate of X(10515)

X(10515) = HOMOTHETIC CENTER OF EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND OUTER GREBE TRIANGLE

X(10515) lies on these lines: {2,5870}, {4,639}, {5,6}, {355,5604}, {381,1160}, {638,1270}, {946,5688}, {5588,7989}, {5594,7395}, {6202,10513}

X(10515) = {X(5),X(10516)}-harmonic conjugate of X(10514)

X(10516) = {X(10514),X(10515)}-HARMONIC CONJUGATE OF X(5)

X(10516) lies on these lines: {2,154}, {3,2916}, {4,141}, {5,6}, {69,3091}, {159,7395}, {355,3242}, {381,511}, {946,3416}, {3751,7989}

X(10516) = reflection of X(10519) in X(141)

X(10517) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND INNER GREBE TRIANGLE

X(10517) lies on these lines: {2,1161}, {3,1271}, {4,640}, {6,631}, {20,6215}, {5595,10323}

X(10517) = {X(631),X(10519)}-harmonic conjugate of X(10518)

X(10518) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND OUTER GREBE TRIANGLE

X(10518) lies on these lines: {2,1160}, {3,1270}, {4,639}, {6,631}, {20,6214}, {5594,10323}

X(10518) = {X(631),X(10519)}-harmonic conjugate of X(10517)

X(10519) = {X(10517),X(10518)}-HARMONIC CONJUGATE OF X(631)

X(10519) lies on these lines: {2,51}, {3,69}, {4,141}, {6,631}, {20,1352}, {159,10323}

X(10519) = reflection of X(10516) in X(141)
X(10519) = X(69)-Gibert-Moses centroid; see the preamble just before X(21153)

X(10520) = PERSPECTOR OF INVERSE-IN-INCIRCLE TRIANGLE AND CROSS-TRIANGLE OF INVERSE-IN-INCIRCLE AND INVERSE-IN-EXCIRCLES TRIANGLES

X(10520) lies on these lines: {1,3598}, {7,1699}, {55,4021}, {57,1766}, {165,3672}, {354,1122}, {942,2808}, {999,1323}, {3673,4298}

X(10521) = PERSPECTOR OF INVERSE-IN-EXCIRCLES TRIANGLE AND CROSS-TRIANGLE OF INVERSE-IN-INCIRCLE AND INVERSE-IN-EXCIRCLES TRIANGLES

X(10521) lies on these lines: {1,3598}, {7,10}, {56,1323}, {57,169}, {511,942}, {516,3673}, {519,3212}, {3295,4021}

X(10522) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF INNER AND OUTER JOHNSON TRIANGLES

Of the two tangents to the B- and C-Johnson circles, let L_A be the one on the opposite side of BC from A, and let M_A be the other. Define L_B, M_B, L_C, M_C cyclically. Let A' = L_B∩L_C, and define B', C' cyclically. Let A" = M_B∩M_C, and define B", C" cyclically. Triangles A'B'C' and A"B"C" are here named the inner and outer Johnson triangles, resp.

The appearance of (T,i,j) in the following list means that triangle T is homothetic to the inner Johnson triangle with homothetic center X(i) and to the outer Johnson triangle with homothetic center X(j):

• (ABC, 11, 12)
• (medial, 1376, 958)
• (anticomplementary, 3434, 3436)
• (outer Garcia, -, 72)
• (Johnson, 355, 355)
• (1st Johnson-Yff, -, 12)
• (2nd Johnson-Yff, 11, -)
• (inner Yff, 10523, -)
• (outer Yff, -, 10523)
• (inner Yff tangents, 12, -)
• (outer Yff tangents, -, 11)

Peter Moses (November 1, 2016) gives barycentrics:

(A-vertex of inner Johnson triangle): -a (a (a - b - c)+2 b c) : (a-c)^2 (a-b+c) : (a-b)^2 (a+b-c)
(A-vertex outer of Johnson triangle): -a (a (a + b + c)+2 b c) : (a+b-c) (a+c)^2 : (a+b)^2 (a-b+c).

X(10522) lies on these lines: {2,8071}, {4,8}, {10,921}, {11,958}, {12,377}

X(10522) = homothetic center of inner Johnson triangle and cross-triangle of ABC and outer Johnson triangle
X(10522) = homothetic center of outer Johnson triangle and cross-triangle of ABC and inner Johnson triangle
X(10522) = anticomplement of X(8071)

X(10523) = HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND CROSS-TRIANGLE OF INNER AND OUTER JOHNSON TRIANGLES

The 1st Johnson-Yff triangle is here defined as the triangle of the intersections of the 1st Johnson-Yff circle and the inner Yff circles.
The 2nd Johnson-Yff triangle is here defined as the triangle of the intersections of the 2nd Johnson-Yff circle and the outer Yff circles.

The appearance of (T,i,j) in the following list means that triangle T is homothetic to the 1st Johnson-Yff triangle with homothetic center X(i) and to the 2nd Johnson-Yff triangle with homothetic center X(j):

• (ABC, 12, 11)
• (medial, 56, 55)
• (anticomplementary, 388, 497)
• (Euler, -, 5592)
• (anti-Euler, 3085, 3086)
• (Johnson, 1, 1)
• (outer Garcia, 65, 3057)
• (T(1,2), 9578, 9581)
• (Mandart-incircle, 4, 497)
• (inner Yff, 495, 5)
• (outer Yff, 5, 496)
• (2nd isogonal of X(1), 5270, 3583)
• (inner Johnson, -, 11)
• (outer Johnson, 12, -)