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This is PART 6: Centers X(10001) - X(12000)

PART 1: Introduction and Centers X(1) - X(1000)
PART 2: Centers X(1001) - X(3000)
PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000)
PART 5: Centers X(7001) - X(10000)
PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000)
PART 8: Centers X(14001) -


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

This preamble and centers X(10001)-X(10012) were contributed by Peter Moses, May 2, 2016.

Let A' be the line through a point P = p : q : r (barycentrics) parallel to line BC. Let AB = A'∩AB and AC = A'∩AC. Define BC and CA cyclically, and define BA and CB cyclically. The six points AB, BC, CA, AB, BC, CA lie on a conic, here called the parallels-conic of P, denoted by Cpar(P). The point P is here called the base-point of Cpar(P). The center of Cpar(P) is the point W(P) given by

W(P) = p*(p2 - pq - pr - 2qr) : q*(q2 - qr - qp - 2rp) : r*(r2 - rp - rq - 2pq)

The perspector of Cpar(P) is the point

W*(P) = p(2qr + 2pq + pr)(2qr + 2pr + pq) : q(2rp + 2qr + qp)(2rp + 2qp + qr) : r(2pq + 2rp + rq)(2pq + 2rq + rp)

Cpar(P) is an ellipse, parabola, or hyperbola according as P lies inside, on, or outside the Steiner inellipse.

Let P' be the reflection of P in W(P). Then P' is the base-point of another conic, Cpar(P'), also having center W(P). For example, if P = X(1), then P' = X(9).

Randy Hutson observes that W(P) is the midpoint of P and P' = X(2)-Ceva conjugate of P. (July 20, 2016)

If P lies on the orthic axis, then Cpar(P) is a rectangular hyperbola, and the locus of W(P) as P traces the orthic axis a circular cubic.

Cpar(X(6)) is the 1st Lemoine circle.

Examples follow:

P P' W(P) = W(P')
X(1) X(9) X(1001)
X(3) X(6) X(182)
X(4) X(1249) X(10002)
X(5) X(216) X(10003)
X(7) X(3160) X(10004)
X(8) X(3161) X(10005)
X(10) X(37) X(3842)
X(11) X(650) X(10006)
X(39) X(141) X(10007)
X(69) X(6337) X(10008)
X(75) X(6376) X(10009)
X(76) X(6374) X(10010)
X(114) X(230) X(10011)
X(142) X(1212) X(10012)
X(1275) X(10001) X(664)

underbar

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)

Barycentrics    (a^2+b^2-c^2)^2 (a^2-b^2+c^2)^2 (a^4+2 a^2 b^2-3 b^4+2 a^2 c^2-2 b^2 c^2-3 c^4) : :
X(10002) = X[253] - 5 X[3091] = (-S^4+2 SA SB SC SW)X[4]+(2 SA SB SC SW)X[6]

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)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+5 a^4 b^4-2 a^2 b^6-4 a^6 c^2+3 a^4 b^2 c^2+2 a^2 b^4 c^2-b^6 c^2+5 a^4 c^4+2 a^2 b^2 c^4+2 b^4 c^4-2 a^2 c^6-b^2 c^6) : :
X(10003) = X[264] - 5 X[1656] = 7 X[3090] + X[3164] = (3 S^4+SA SB SC SW)X[5]-(S^4-SA SB SC SW)X[53]

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)

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

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)

Barycentrics    (a-b-c) (a^2-4 a b+3 b^2-4 a c-2 b c+3 c^2) : :
X(10005) = 5 X[3617]-X[4373] = 2 X[8]+X[4779] = 5 X[2136]-9 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)

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

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)

Barycentrics    (b^2+c^2) (a^4+2 a^2 b^2+2 a^2 c^2+b^2 c^2) : :
X(10007) = 3 X[262] + X[1350] = 3 X[2] + X[3094] = X[194] + 7 X[3619] = X[76] - 5 X[3763] = 3 X[597] - X[5052] = X[6] - 5 X[7786]

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(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)

Barycentrics    (a^2-b^2-c^2) (a^4-4 a^2 b^2+3 b^4-4 a^2 c^2-2 b^2 c^2+3 c^4) : :
X(10008) = X[2996] - 5 X[3620]

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)

Barycentrics    b^2 c^2 (-2 a^2-a b-a c+b c) : :
X(10009) = X[330] - 5 X[4699]

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)

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

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)

Barycentrics    (2 a^4-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^4-4 a^2 b^2+3 b^4-4 a^2 c^2-2 b^2 c^2+3 c^4) : :
X(10011) = 3 X[2] + X[1513] = 9 X[2] - X[5999] = 3 X[1513] + X[5999] = 3 X[403] + X[7472] = 5 X[5071] - X[8352] = 3 X[3545] + X[8598]

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)

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

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))

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

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

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

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

Barycentrics    (b-c)*((b+c)*(a^2-(b-c)^2)-2*a*b*c) : :
Barycentrics    (b - c)(cos B + cos C - 1) : :

Centers X(10015)-X(10017) contributed by César 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) = complement of X(3904)
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) = X(48)-isoconjugate (polar conjugate) of X(1309)


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

Trilinears    16*p^7*(p-q)+8*(2*q^2-3)*p^6-16*(q^2-2)*q*p^5-(8*q^2-1)*p^4+(16*q^2-15)*q*p^3+(1-q^2)*(2*p*q+6*p^2-1) : : , where p = sin(A/2), q = cos((B-C)/2)

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

Trilinears    cos(A)*sin((B-C)/2)^2*(2*cos((B-C)/2)*sin(A/2)-1)*(2*cos((B-C)/2)*sin(A/2)-1-2*cos(B)*cos(C)+cos(A)) : :

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) = crosssum of X(102) and X(109)
X(10017) = circumcircle-inverse of X(10016)
X(10017) = Stevanovic-circle-inverse of X(5514)
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

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-5 a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+4 a^2 c^4+b^2 c^4-c^6) : :

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

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (2 a^6+a^4 b^2-8 a^2 b^4+5 b^6+a^4 c^2+8 a^2 b^2 c^2-5 b^4 c^2-8 a^2 c^4-5 b^2 c^4+5 c^6) : :

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

Barycentrics    2 a^10-5 a^8 b^2+2 a^6 b^4+4 a^4 b^6-4 a^2 b^8+b^10-5 a^8 c^2+4 a^6 b^2 c^2-2 a^4 b^4 c^2+6 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-2 a^4 b^2 c^4-4 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+6 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics    2a^2[b^2 cos 2B + c^2 cos 2C - a^2 cos 2A] + b^2[c^2 cos 2C + a^2 cos 2A - b^2 cos 2B] + c^2[b^2 cos 2B + c^2 cos 2C - a^2 cos 2A] : :

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

Barycentrics    2 a^7-2 a^6 b-5 a^5 b^2+5 a^4 b^3+4 a^3 b^4-4 a^2 b^5-a b^6+b^7-2 a^6 c+2 a^5 b c+a^4 b^2 c+a^3 b^3 c+2 a^2 b^4 c-3 a b^5 c-b^6 c-5 a^5 c^2+a^4 b c^2+4 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+5 a^4 c^3+a^3 b c^3+2 a^2 b^2 c^3+6 a b^3 c^3+3 b^4 c^3+4 a^3 c^4+2 a^2 b c^4+a b^2 c^4+3 b^3 c^4-4 a^2 c^5-3 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : :

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)

Barycentrics    4 a^2+2 a b+b^2+2 a c+8 b c+c^2 : :
X(10022) = 2 X[4363] + X[4364] = 5 X[4364] - 2 X[4419] = 5 X[2] - X[4419] = 5 X[4363] + X[4419] = 7 X[4363] - X[4454] = 7 X[2] + X[4454] = 7 X[4364] + 2 X[4454] = 7 X[4419] + 5 X[4454] = X[4364] - 10 X[4470] = X[2] - 5 X[4470] = X[4363] + 5 X[4470] = X[4419] - 10 X[4472] = X[4364] - 4 X[4472] = 5 X[4470] - 2 X[4472] = X[4363] + 2 X[4472] = X[4454] + 14 X[4472] = X[4665] + 2 X[4670] = X[4659] + 5 X[4798]

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) = {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

Trilinears    a (-a^2 + b^2 + c^2) + (-a + b + c) ((a - b + c) c Sin[B/2] + (a + b - c) b Sin[C/2]) : :
Barycentrics    2 a^2 (-a^2+b^2+c^2)+(-a+b+c) ((a+b-c) Sqrt[a b (-a+b+c) (a-b+c)]+(a-b+c) Sqrt[a c (-a+b+c) (a+b-c)]) : :      (Peter Moses, July 4, 2017) <.p>. Let IA, IB, IC be the excenters of a triangle ABC. Let JA be the incenter of IABC, and define JB and JC cyclically. Let LA be the Euler line of IAJBJC, and define LB and LC cyclically. The lines LA, LB, LC concur in X(10023); see X(10024). (Seiichi Kirikami, July 11, 2016)

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)

Barycentrics    2 sin 2B + tan B + 2 sin 2C + tan C : :
X(10024) = 3 X[381] + X[2937] = X[7488] - 3 X[7552] = X[4] + 3 X[7552]

Let OAOBOC be the tangential triangle of a triangle ABC. Let HA be the orthocenter of OABC, and define HB and HC cyclically. Let LA be the Euler line of OAHBHC, and define LB and LC cyclically. The lines LA, LB, LC concur in X(10024); see X(10023) and Hyacinthos 23764. (Antreas Hatzipolakis, July 12, 2016)

The triangle HAHBHC 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)



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

This preamble and centers X(10025)-X(10030) were contributed by Clark Kimberling and Peter Moses, July 13, 2016.

Suppose that R = r : s : t and U = u : v : w (barycentrics) are points not on a sideline of a triangle ABC. Let

LA be the line of the points 0 : t : r and - v : w : u
LB be the line of the points s : 0 : r and v : - w : u
LC be the line of the points s : t : 0 and v : w : - u
L'A be the line of the points 0 : r : s and - w : u : v
L'B be the line of the points t : 0 : s and w : - u : v
L'C be the line of the points t : r : 0 and w : u : - v

The lines LA, LB, LC concur in a point, P, and the lines L'A, L'B, L'C concur in a point, P'. If R and U are triangle centers, then P and P' are a pair of bicentric points and PP' is a central line f*x + g*y + h*z = 0, so that the point H(R,U) = f : g : h is a triangle center, here introduced as the H-transform of R and U, given by first barycentric

f = v*w*(u/r + v/s - w/t)(u/r - v/s + w/t) - u^2 (v/s + w/t - u/r)^2 .

Next, let

LA be the line of the points 0 : t : r and - w : u : v
LB be the line of the points s : 0 : r and w : - u : v
LC be the line of the points s : t : 0 and w : u : - v
L'A be the line of the points 0 : r : s and - v : w : u
L'B be the line of the points t : 0 : s and v : - w : u
L'C be the line of the points t : r : 0 and v : w : - u

The lines LA, LB, LC concur in a point, P, and the lines L'A, L'B, L'C concur in a point, P'. If R and U are triangle centers, then P and P' are a pair of bicentric points and PP' is a central line f*x + g*y + h*z = 0, so that the point K(R,U) = f : g : h is a triangle center, here introduced as the K-transform of R and U, given by first barycentric

f = v*w*(u/t + w/s - v/r)(u/s + v/t - w/r) - u^2 (v/r + w/s - u/t)(v/t + w/r - u/s)

Let G = centroid = X(2). Then H(G,P) = K(G,P) and H(P,G) = K(P,G) for all P. For fixed X, the locus of a point P satisfying H(P,X) = G is the circumconic with center X. In particular, for X = X(125), the locus is the Jerabek hyperbola; for X = X(115), the Kiepert hyperbola; and for X = X(11), the Feuerbach hyperbola.

If P = p : q : r, then H(P,P) = qr - p2 : rp - q2 : pq - r2, which is the Steiner-circumellipse-inverse of P.

If P is on the circumcircle, then K(P,X(6)) = X(384).

R U H(R,U)
X(1) X(1) X(239)
X(1) X(2) X(1575)
X(1) X(11) X(2)
X(2) X(1) X(100025)
X(2) X(3) X(385)
X(2) X(6) X(401)
X(2) X(9) X(239)
X(3) X(3) X(401)
X(4) X(2) X(230)
X(4) X(4) X(297)
X(4) X(11) X(2)
X(4) X(25) X(385)
X(6) X(1) X(19927)
X(6) X(2) X(3229)
X(6) X(6) X(385)
X(7) X(2) X(3008)
X(7) X(7) X(9436)
X(7) X(11) X(2)
X(8) X(8) X(3912)
X(8) X(11) X(2)
X(9) X(11) X(2)
X(10) X(2) X(10026)
X(10) X(10) X(6542)
X(13) X(2) X(395)
X(14) X(2) X(396)
X(20) X(20) X(441)
X(21) X(1) X(401)
X(21) X(3) X(239)
X(21) X(11) X(2)
X(9) X(11) X(2)
X(21) X(21) X(448)

R U K(R,U)
X(1) X(1) X(10028)
X(1) X(2) X(1575)
X(1) X(75) X(10030)
X(2) X(3) X(385)
X(2) X(6) X(401)
X(2) X(9) X(239)
X(2) X(37) X(6542)
X(2) X(39) X(7779)
X(2) X(76) X(9493)
X(2) X(114) X(193)
X(2) X(115) X(2)
X(4) X(2) X(230)
X(6) X(2) X(3229)
X(7) X(2) X(3008)
X(8) X(7) X(10029)
X(13) X(2) X(395)
X(14) X(2) X(396)
X(29) X(2) X(8558)
X(30) X(2) X(3163)
X(39) X(2) X(9496)
X(69) X(2) X(441)
X(74) X(6) X(384)
X(75) X(2) X(3912)
X(76) X(2) X(325)
X(83) X(2) X(385)
X(85) X(2) X(9436)
X(86) X(2) X(239)
X(88) X(1) X(894)
X(95) X(2) X(401)
X(98) X(2) X(6)
X(98) X(6) X(384)
X(99) X(2) X(2)

underbar

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

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

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) = 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))

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

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) = X(i)-Ceva conjugate of X(j) for these (i,j): (6542,740), (6543,1213)
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))

Barycentrics    a^3 b+a^3 c-3 a^2 b c+a b^2 c+a b c^2-b^2 c^2 : :

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))

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

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))

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

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))

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

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)

Barycentrics    -7a^4+6a^3(b+c)+a^2(5b^2-13bc+5c^2)+a(-6b^3+5b^2c+5bc^2-6c^3)+2(b^2-c^2)^2 : :
X(10031) = 4 X[1317] - X[1320] = 2 X[1317] + X[6224] = X[1320] + 2 X[6224] = X[100] + 2 X[7972] = 5 X[1320] - 2 X[9802] = 10 X[1317] - X[9802] = 5 X[3241] - X[9802] = 5 X[6224] + X[9802] = 4 X[6224] - X[9963] = 8 X[1317] + X[9963] = 2 X[1320] + X[9963] = 4 X[3241] + X[9963] = 4 X[9802] + 5 X[9963]

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)

Barycentrics    7a^3-a^2(b+c)-a(4b^2+b c+4c^2)-2(b-c)^2(b+c) : :
X(10032) = 5 X[21] - 4 X[551] = 5 X[79] - 8 X[3634] = 7 X[3624] - 10 X[3647] = X[8] + 5 X[3648] = X[8] - 10 X[3650] = X[3648] + 2 X[3650]

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

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

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

Barycentrics    10 a^12-33 a^10 (b^2+c^2)-6 a^8 (11 b^4-34 b^2 c^2+11 c^4)+a^6 (221 b^6-108 b^4 c^2-108 b^2 c^4+221 c^6)-3 a^4 (41 b^8-112 b^6 c^2+288 b^4 c^4-112 b^2 c^6+41 c^8)-6 a^2 (5 b^10-28 b^8 c^2+8 b^6 c^4+8 b^4 c^6-28 b^2 c^8+5 c^10)+13 b^12-81 b^10 c^2+153 b^8 c^4-154 b^6 c^6+153 b^4 c^8-81 b^2 c^10+13 c^12 : :

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

Barycentrics    2 a^9 (b+c)+b c (b^2-c^2)^4-2 a^8 (b^2+c^2)+a (b-c)^4 (b+c)^3 (b^2+b c+c^2)-a^7 (7 b^3+b^2 c+b c^2+7 c^3)+2 a^2 (b^2-c^2)^2 (b^4-2 b^3 c-2 b c^3+c^4)+a^6 (6 b^4-2 b^3 c+8 b^2 c^2-2 b c^3+6 c^4)-a^3 (b-c)^2 (5 b^5+7 b^4 c+6 b^3 c^2+6 b^2 c^3+7 b c^4+5 c^5)+a^5 (9 b^5-4 b^4 c+b^3 c^2+b^2 c^3-4 b c^4+9 c^5)-a^4 (6 b^6-5 b^5 c+2 b^4 c^2+6 b^3 c^3+2 b^2 c^4-5 b c^5+6 c^6) : :

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

Barycentrics    4 a^8 b c-2 a^9 (b+c)+b c (b^2-c^2)^4-12 a^6 b c (b^2+c^2)-6 a^2 b c (b^2-c^2)^2 (b^2+c^2)-a (b-c)^4 (b+c)^3 (b^2+3 b c+c^2)+a^7 (7 b^3+5 b^2 c+5 b c^2+7 c^3)+5 a^3 (b-c)^2 (b^5+3 b^4 c+2 b^3 c^2+2 b^2 c^3+3 b c^4+c^5)-a^5 (9 b^5+6 b^4 c-5 b^3 c^2-5 b^2 c^3+6 b c^4+9 c^5)-a^4 (-13 b^5 c+6 b^3 c^3-13 b c^5) : :
Barycentrics    [2 a^4 - a^2 (3 b^2 - 4 b c + 3 c^2) + (b^2 - c^2)^2] [a (b + c) (a^4 - a^2 (2 b^2 + b c + 2 c^2) + (b - c)^2 (b^2 + 3 b c + c^2)) - b c (2 a^4 - (3 b^2 + 4 b c + 3 c^2) a^2 + (b^2 - c^2)^2)] : : (factorization by Richard Hilton, September 20, 2016)

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)



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

This preamble and centers X(10037)-X(10094) were contributed by César Lozada, August 4, 2016.

The Yff circles are the two triplets of congruent circles in which each circle is tangent to two sides of a reference triangle (see Mathworld). The circles in each triplet have radius r1=r*R/(R+r) and r2=r*R/(R-r), respectively. The centers A1, A2 for the A-circles of the first and second triplets have respective trilinear coordinates:

A1= -(a^4-2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1
A2= +(a^4-2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2)/(2*a^2*b*c) : 1 : 1

and cyclically B1, C1 and B2, C2 for the B- and C- circles.

The triangles T1=A1B1C1 and T2=A2B2C2 are known as the inner- and outer- Yff triangles, respectively.

The appearance of (T, i) in the following list means that the inner-Yff triangle and T are perspective with perspector X(i), where an asterisk * signifies that the two triangles are homothetic: (ABC*, 1), (Andromeda, 1), (anticomplementary*, 3085), (Antlia, 1), (Aquila*, 1), (Ara*, 10037), (5th Brocard*, 10038), (2nd circumperp, 1), (Euler*, 1479), (excentral, 1), (outer-Garcia*, 10039), (inner-Grebe*, 10040), (outer-Grebe*, 10041), (hexyl, 10042), (Hutson intouch, 10043), (incentral, 1), (intouch, 10044), (Johnson*, 12), (medial*, 498), (midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear*, 1), (6th mixtilinear, 10045), (outer-Yff*, 1))
The appearance of (T, i, j) in the following list means that the inner-Yff triangle and T are orthologic with centers X(i) and X(j): (ABC, 1478, 4), (1st anti-Brocard, 10053, 5999), (anticomplementary, 1478, 20), (anti-McCay, 10054, 9855), (Aquila, 1478, 5691), (Ara, 1478, 3), (Aries, 10055, 9833), (Artzt, 10056, 2), (Ascella, 1, 3), (Atik, 1, 9856), (1st Brocard, 10053, 3), (5th Brocard, 1478, 9873), (6th Brocard, 10053, 20), (circumorthic, 55, 4), (1st circumperp, 1, 3), (2nd circumperp, 1, 3), (Conway, 1, 20), (2nd Conway, 1, 962), (1st Ehrmann, 611, 3), (2nd Ehrmann, 55, 576), (Euler, 1478, 4), (2nd Euler, 55, 3), (3rd Euler, 1, 5), (4th Euler, 1, 5), (excentral, 1, 40), (extangents, 55, 40), (extouch, 1709, 40), (2nd extouch, 1, 4), (3rd extouch, 5119, 4), (Fuhrmann, 10057, 3), (inner-Garcia, 10058, 40), (outer-Garcia, 1478, 40), (inner-Grebe, 1478, 5871), (outer-Grebe, 1478, 5870), (hexyl, 1, 1), (Honsberger, 1, 390), (Hutson extouch, 10059, 40), (Hutson intouch, 1, 1), (intangents, 55, 1), (intouch, 1, 1), (Johnson, 1478, 3), (Kosnita, 55, 3), (McCay, 10054, 3), (medial, 1478, 3), (midheight, 10060, 4), (5th mixtilinear, 1478, 944), (6th mixtilinear, 1, 1), (inner-Napoleon, 10061, 3), (outer-Napoleon, 10062, 3), (1st Neuberg, 10063, 3), (2nd Neuberg, 10064, 3), (orthic, 55, 4), (orthocentroidal, 10065, 4), (reflection, 10066, 4), (1st Sharygin, 1, 9840), (submedial, 55, 5), (tangential, 55, 3), (2nd tangential-midarc, 1, 8092), (Trinh, 55, 3), (inner-Vecten, 10067, 3), (outer-Vecten, 10068, 3), (outer-Yff, 1478, 1479)
The appearance of (T, i, j) in the following list means that the inner-Yff triangle and T are parallelogic with centers X(i) and X(j): (1st anti-Brocard, 10086, 385), (1st Brocard, 10086, 6), (6th Brocard, 10086, 194), (inner-Garcia, 10087, 1), (orthocentroidal, 10088, 2), (1st Parry, 1478, 9131), (2nd Parry, 1478, 9979), (2nd Sharygin, 1, 659)
The appearance of (i, j) in the following list means that X(i)-of-the-inner-Yff triangle=X(j)-of-ABC: (1, 1), (2, 10056), (3, 55), (4, 1478), (5, 495), (6, 611), (13, 10062), (14, 10061), (20, 4302), (40, 5119), (54, 10066), (56, 8069), (64, 10060), (68, 10055), (74, 10065), (76, 10063), (80, 10057), (83, 10064), (84, 1709), (98, 10053), (99, 10086), (100, 10087), (104, 10058), (110, 10088), (155, 3157), (355, 5252), (485, 10068), (486, 10067), (671, 10054), (946, 226), (1001, 954), (1482, 2099), (3072, 5255), (3811, 3870), (3813, 2886), (4297, 4304), (5536, 484), (5707, 5711), (5709, 40), (5715, 5290), (5735, 4312), (5901, 5719), (6734, 10039), (7160, 10059), (8666, 993))
The appearance of (T, i) in the following list means that the outer-Yff triangle and T are perspective with perspector X(i), where an asterisk * signifies that the two triangles are homothetic: (ABC*, 1), (Andromeda, 1), (anticomplementary*, 3086), (Antlia, 1), (Aquila*, 1), (Ara*, 10046), (5th Brocard*, 10047), (2nd circumperp, 1), (Euler*, 1478), (excentral, 1), (outer-Garcia*, 1737), (inner-Grebe*, 10048), (outer-Grebe*, 10049), (hexyl, 10050), (Hutson intouch, 10051), (incentral, 1), (intouch, 10052), (Johnson*, 11), (medial*, 499), (midarc, 1), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear*, 1), (6th mixtilinear, 10092), (inner-Yff*, 1)
The appearance of (T, i, j) in the following list means that the outer-Yff triangle and T are orthologic with centers X(i) and X(j): (ABC, 1479, 4), (1st anti-Brocard, 10069, 5999), (anticomplementary, 1479, 20), (anti-McCay, 10070, 9855), (Aquila, 1479, 5691), (Ara, 1479, 3), (Aries, 10071, 9833), (Artzt, 10072, 2), (Ascella, 1, 3), (Atik, 1, 9856), (1st Brocard, 10069, 3), (5th Brocard, 1479, 9873), (6th Brocard, 10069, 20), (circumorthic, 56, 4), (1st circumperp, 1, 3), (2nd circumperp, 1, 3), (Conway, 1, 20), (2nd Conway, 1, 962), (1st Ehrmann, 613, 3), (2nd Ehrmann, 56, 576), (Euler, 1479, 4), (2nd Euler, 56, 3), (3rd Euler, 1, 5), (4th Euler, 1, 5), (excentral, 1, 40), (extangents, 56, 40), (extouch, 10085, 40), (2nd extouch, 1, 4), (3rd extouch, 46, 4), (Fuhrmann, 10073, 3), (inner-Garcia, 10074, 40), (outer-Garcia, 1479, 40), (inner-Grebe, 1479, 5871), (outer-Grebe, 1479, 5870), (hexyl, 1, 1), (Honsberger, 1, 390), (Hutson extouch, 10075, 40), (Hutson intouch, 1, 1), (intangents, 56, 1), (intouch, 1, 1), (Johnson, 1479, 3), (Kosnita, 56, 3), (McCay, 10070, 3), (medial, 1479, 3), (midheight, 10076, 4), (5th mixtilinear, 1479, 944), (6th mixtilinear, 1, 1), (inner-Napoleon, 10077, 3), (outer-Napoleon, 10078, 3), (1st Neuberg, 10079, 3), (2nd Neuberg, 10080, 3), (orthic, 56, 4), (orthocentroidal, 10081, 4), (reflection, 10082, 4), (1st Sharygin, 1, 9840), (submedial, 56, 5), (tangential, 56, 3), (2nd tangential-midarc, 1, 8092), (Trinh, 56, 3), (inner-Vecten, 10083, 3), (outer-Vecten, 10084, 3), (inner-Yff, 1479, 1478)
The appearance of (T, i, j) in the following list means that the outer-Yff triangle and T are parallelogic with centers X(i) and X(j): (1st anti-Brocard, 10089, 385), (1st Brocard, 10089, 6), (6th Brocard, 10089, 194), (inner-Garcia, 10090, 1), (orthocentroidal, 10091, 2), (1st Parry, 1479, 9131), (2nd Parry, 1479, 9979), (2nd Sharygin, 1, 659)
The appearance of (i, j) in the following list means that X(i)-of-the-outer-Yff triangle=X(j)-of-ABC: (1, 1), (2, 10072), (3, 56), (4, 1479), (5, 496), (6, 613), (10, 1210), (13, 10078), (14, 10077), (20, 4299), (40, 46), (54, 10082), (55, 8069), (64, 10076), (68, 10071), (74, 10081), (76, 10079), (80, 10073), (83, 10080), (84, 10085), (98, 10069), (99, 10089), (100, 10090), (104, 10074), (109, 1795), (110, 10091), (119, 11), (155, 1069), (355, 1837), (485, 10084), (486, 10083), (671, 10070), (1482, 2098), (2077, 36), (2950, 1768), (3359, 57), (3811, 78), (3913, 5687), (4297, 4311), (5220, 5729), (5552, 499), (5553, 10052), (6256, 4), (6735, 1737), (7160, 10075), (7387, 9645), (7629, 656)

The inner-Yff triangle and outer-Yff triangle are each inversely similar to these triangles: 1st anti-Brocard, 1st Brocard, 6th Brocard, inner-Garcia, orthocentroidal, 1st Parry and 2nd Parry.

underbar

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

Trilinears    (a^8-2*(b^2+c^2)*a^6+2*((b^3-c^3)^2-(b+c)^2*b^2*c^2)*a^2-(b^2-c^2)^4)*a : :

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

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

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

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

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

Trilinears    (a^4-2*(b^2+b*c+c^2)*(a^2+b*c)+b^4+c^4+2*b*c*S)*a : :

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

Trilinears    (a^4-2*(b^2+b*c+c^2)*(a^2+b*c)+b^4+c^4-2*b*c*S)*a : :

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

Trilinears    4*p^5*(p-q)+4*p*(-q^2+1)*(p^3-p^2*q+q)+(2*q^2-3)*(2*p^2-1)-2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

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

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

Trilinears    p^5*(p+q)-(5*q^2-4)*p^4+(3*q^2-4)*q*p^3+(4*q^2-5)*p^2-(4*q^2-3)*q*p+1-2*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    (a^8-2*(b^2+c^2)*a^6+2*((b^3+c^3)^2-(b-c)^2*b^2*c^2)*a^2-(b^2-c^2)^4)*a : :

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

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

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

Trilinears    (a^4-2*(b^2-b*c+c^2)*(a^2-b*c)+b^4+c^4-2*b*c*S)*a : :

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

Trilinears    a^2*(a^4-2*(b^2-b*c+c^2)*(a^2-b*c)+b^4+c^4+2*b*c*S) : :

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

Trilinears    4*p^5*(p-q)+4*p*(1-q^2)*(p^3-p^2*q+q)+(2*q^2-1)*(2*p^2+1)-2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

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)

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

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

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

Barycentrics    a^8-(b+c)^2*a^6+(b^2+3*b*c+c^2)*(b^2-b*c+c^2)*a^4-(b^6+c^6+(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c)*a^2-(b^2-c^2)^2*b^2*c^2 : :

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

Barycentrics    a^8-3*(b-c)^2*a^6+(b^2-5*b*c+c^2)*(b^2-b*c+c^2)*a^4+(3*b^6+3*c^6+(6*b^4+6*c^4-(5*b^2+6*b*c+5*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(2*b^2-c^2)*(b^2-2*c^2) : :

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

Barycentrics    (a^8-2*(b^2-b*c+c^2)*a^6+2*(b^4+c^4)*a^4-2*(b^2-c^2)^2*(b^2+b*c+c^2)*a^2+(b^2-c^2)^4)*(a^2-b^2-c^2) : :

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

Trilinears    (S - (b + c)p)/a + 2p : : , where p = rR/(R + r)
Trilinears    2 - (a^4 - 2a^2(b^2 + bc + c^2) + (b^2 - c^2)^2)/(2a^2bc) : :
Barycentrics    a^4-2*(b^2+3*b*c+c^2)*a^2+(b^2-c^2)^2 : :

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)


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

Trilinears    8*p^4-16*q*p^3+2*(8*q^2-1)*p^2-2*(4*q^2-1)*q*p+1 : : , where p=sin(A/2), q=cos((B-C)/2)

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

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

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

Trilinears    2*p^5*(p-q)-(2*q^2+7)*p^4+2*(q^2+4)*q*p^3+(q^2+10)*p^2-2*(q^2+5)*q*p-2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    (4*cos(A)+6*cos(2*A)+10)*cos(B-C)-15*cos(A)-2*cos(2*A)-cos(3*A)-2 : :

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

Barycentrics    sqrt(3)*((SW-S*sqrt(3))*b*c+(sqrt(3)*SA+S)*S)*(SB+SC)-4*S^3 : :

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

Barycentrics    sqrt(3)*((SW+S*sqrt(3))*b*c-(sqrt(3)*SA-S)*S)*(SB+SC)+4*S^3 : :

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

Trilinears    ((2*b^2-b*c+2*c^2)*a^4+2*b*c*(b^2+b*c+c^2)*a^2-(b^2-c^2)^2*b*c)/a^2 : :

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

Barycentrics    a^8-(b+c)^2*a^6-(b^4+c^4+(6*b^2+5*b*c+6*c^2)*b*c)*a^4+(b^6+c^6-(2*b^2+b*c+c^2)*(b^2+b*c+2*c^2)*b*c)*a^2+(b^2-c^2)^2*b^2*c^2 : :

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

Trilinears    (4*cos(A)+4*cos(2*A)+5)*cos(B-C)-8*cos(A)-2*cos(2*A)-cos(3*A)-3 : :

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

Trilinears    (4*cos(A)+1)*cos(B-C)-2*cos(2*A)-cos(3*A)+1 : :

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

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

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

Trilinears    (a^4+b^4+c^4-2*(b^2+b*c+c^2)*(a^2+b*c))*a^2-2*S*((b^2+4*b*c+c^2)*a^2-(b^2-c^2)^2) : :

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

Barycentrics    a^8-(b-c)^2*a^6+(b^2+b*c+c^2)*(b^2+c^2-3*b*c)*a^4-(b^6+c^6-(2*b^2-3*b*c+2*c^2)*(b+c)^2*b*c)*a^2-(b^2-c^2)^2*b^2*c^2 : :

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

Barycentrics    a^8-3*(b+c)^2*a^6+(b^2+b*c+c^2)*(c^2+5*b*c+b^2)*a^4+(3*b^6+3*c^6-(6*b^4+6*c^4+(5*b^2-6*b*c+5*c^2)*b*c)*b*c)*a^2-(b^2-c^2)^2*(-c^2+2*b^2)*(-2*c^2+b^2) : :

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

Barycentrics    (a^8-2*(b^2+b*c+c^2)*a^6+2*(b^4+c^4)*a^4-2*(b^2-c^2)^2*(b^2-b*c+c^2)*a^2+(b^2-c^2)^4)*(a^2-b^2-c^2) : :

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

Trilinears    (S - (b + c)p)/a + 2p : : , where p = rR/(R - r)
Barycentrics    a^4-2*(b^2+c^2-3*b*c)*a^2+(b^2-c^2)^2 : :

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)


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

Trilinears    8*p^4-16*q*p^3+2*(8*q^2-5)*p^2-2*(4*q^2-5)*q*p-1 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    a^6-(b+c)*a^5-(2*b^2-7*b*c+2*c^2)*a^4+2*(b^2-3*b*c+c^2)*(b+c)*a^3+(b^4+c^4-6*(b-c)^2*b*c)*a^2-(b^2-c^2)*(b-c)*a*(b^2-4*b*c+c^2)-(b^2-c^2)^2*b*c : :

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

Trilinears    2*p^5*(p-q)-(2*q^2+5)*p^4+2*(q^2+2)*q*p^3+(3*q^2+2)*p^2-2*(q^2+1)*q*p+2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    (4*cos(A)-6*cos(2*A)-10)*cos(B-C)+15*cos(A)-2*cos(2*A)+cos(3*A)-2 : :

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

Barycentrics    sqrt(3)*(-(SW-S*sqrt(3))*b*c+(sqrt(3)*SA+S)*S)*(SB+SC)-4*S^3 : :

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

Barycentrics    sqrt(3)*(-(SW+S*sqrt(3))*b*c-(sqrt(3)*SA-S)*S)*(SB+SC)+4*S^3 : :

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

Trilinears    ((2*b^2+2*c^2+b*c)*a^4-2*b*c*(b^2-b*c+c^2)*a^2+(b^2-c^2)^2*b*c)/a^2 : :

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

Barycentrics    a^8-(b-c)^2*a^6-(b^4+c^4-(6*b^2-5*b*c+6*c^2)*b*c)*a^4+(b^6+c^6+(2*b^2-b*c+c^2)*(b^2-b*c+2*c^2)*b*c)*a^2+(b^2-c^2)^2*b^2*c^2 : :

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

Trilinears    (4*cos(A)-4*cos(2*A)-5)*cos(B-C)+8*cos(A)-2*cos(2*A)+cos(3*A)-3 : :

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

Trilinears    (4*cos(A)-1)*cos(B-C)-2*cos(2*A)+cos(3*A)+1 : :

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

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

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

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

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

Trilinears    4*sin(A/2)*cos((B-C)/2)-2*(cos(A)+1)*cos(B-C)+8*cos(A)-cos(2*A)-3 : :

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

Barycentrics    a^8-(3*b^2+2*b*c+3*c^2)*a^6+(3*b^4+3*c^4+(2*b^2+3*b*c+2*c^2)*b*c)*a^4-(b^6+c^6+(2*b^2-3*b*c+2*c^2)*(b+c)^2*b*c)*a^2+(b^2-c^2)^2*b^2*c^2 : :

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

Trilinears    2*(3*sin(A/2)+sin(3*A/2))*cos((B-C)/2)-cos(B-C)+cos(A)+cos(2*A)-4 : :

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

Trilinears    (4*cos(A)-1)*cos(B-C)-2*cos(2*A)-cos(3*A)-3 : :

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

Barycentrics    a^8-(3*b^2-2*b*c+3*c^2)*a^6+(3*b^4+3*c^4-(2*b^2-3*b*c+2*c^2)*b*c)*a^4-(b^6+c^6-(2*b^2+3*b*c+2*c^2)*(b-c)^2*b*c)*a^2+(b^2-c^2)^2*b^2*c^2 : :

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

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

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

(4*cos(A)+1)*cos(B-C)-2*cos(2*A)+cos(3*A)-3

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

Trilinears    p^5*(p+q)-(5*q^2-4)*p^4+(3*q^2-4)*q*p^3+(4*q^2-3)*p^2-(4*q^2-5)*q*p-3+2*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    4*sin(5*A/2)*cos((B-C)/2)-2*cos(2*A)*(cos(B-C)+1)+cos(3*A)-cos(A)-2 : :

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

Trilinears    16*p^5*(p-2*q)+8*(2*q^2+1)*p^4-8*q*p^3+2*q*p+1-2*q^2 : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    a*(SA^2+5*S^2)*(S^2+SB*SC) : :
X(10095) = X(5)+3*X(51)

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) = 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

Trilinears    (cos(2*A)-3/2)*cos(B-C)+cos(A)*cos(2*(B-C))+cos(A)-cos(3*A) : :
X(10096) = (23*R^2-6*SW)*X(3)+(13*R^2-2*SW)*X(4)

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)

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

This preamble and centers X(10097)-X(10103) were contributed by Peter Moses and Clark Kimberling, August 4, 2016, based findings of Dao Thanh Oai.

Suppose that P is a point in the plane of a triangle ABC, but not on a sideline (BC, CA, AB). Let P' be the isogonal conjugate of P. Let (O) be the circumcircle of ABC, and let C(P) be the conic through A,B,C,P,P'. Let D be the point in (O)∩C(P) other than A,B,C; let E be a point on (O), other than A,B,C,D, and let E' the point in DE∩C(P), other than D. The points P,P',E,E' lie on a circle, here named the Dao circle of P, denoted by D(P). The point E' is here named the 1st (P,E)-Dao point. (Based on "A generalization of the Sawayama-Thébault theorem", Dao Thanh Oai, July 21, 2016; ADGEOM 3353)

Write P = p : q : r (barycentrics). Then C(P) is given by

b2c2p2x(y - z) + c2a2q2y(z - x) + a2b2r2z(x - y) = 0.

The perspector of C(P) is

a2p(b2r2 - c2q2) : b2q(c2p2 - a2r2) : c2r(a2q2 - b2p2).

The circle D(P) meets (O) in another point, the 2nd (P,E)-Dao point, and D(P) meets C(P) in another point, the 3rd (P,E)-Dao point. These points are represented by F and F', respectively, in the following examples.

Example 1. C(X(3)) is the Jerabek conic. Taking E = X(111) gives E' = X(10097), F = X(5505), F' = X(10098).
Example 2. Continuing with C(X(3)), take E = X(98). Then E' = X(879), F = X(67), F' = X(935).
Example 3. Continuing with C(X(3)), take E = X(105). Then E' = X(10099), F = X(10100), F' = X(10101); i.e. the 1st, 2nd, 3rd (X(3),X(105))-Dao points are E', F, F'.
Example 4. 1st, 2nd, 3rd (X(2),X(112))-Dao points are X(25), X(10101), X(10102).

underbar

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

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

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

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

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

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

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

Barycentrics    a (b+c) (a^6-a^4 b^2-a^2 b^4+b^6-2 a^5 c-a^4 b c+3 a^3 b^2 c+3 a^2 b^3 c-a b^4 c-2 b^5 c+a^4 c^2-a^3 b c^2-2 a^2 b^2 c^2-a b^3 c^2+b^4 c^2+a^2 b c^3+a b^2 c^3-a^2 c^4-a b c^4-b^2 c^4+2 a c^5+2 b c^5-c^6) (a^6-2 a^5 b+a^4 b^2-a^2 b^4+2 a b^5-b^6-a^4 b c-a^3 b^2 c+a^2 b^3 c-a b^4 c+2 b^5 c-a^4 c^2+3 a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2-b^4 c^2+3 a^2 b c^3-a b^2 c^3-a^2 c^4-a b c^4+b^2 c^4-2 b c^5+c^6) : :

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

X(10010) lies on the Jerabek hyperbola and these lines:
{3,2836}, {74,3827}, {265,518}, {895,9004}, {2771,4846}, {2778,3426}

X(10010) = isogonal conjugate of X(7469)


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

Barycentrics    a (a-b) (a-c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^6-a^4 b^2-a^2 b^4+b^6-2 a^5 c-a^4 b c+3 a^3 b^2 c+3 a^2 b^3 c-a b^4 c-2 b^5 c+a^4 c^2-a^3 b c^2-2 a^2 b^2 c^2-a b^3 c^2+b^4 c^2+a^2 b c^3+a b^2 c^3-a^2 c^4-a b c^4-b^2 c^4+2 a c^5+2 b c^5-c^6) (a^6-2 a^5 b+a^4 b^2-a^2 b^4+2 a b^5-b^6-a^4 b c-a^3 b^2 c+a^2 b^3 c-a b^4 c+2 b^5 c-a^4 c^2+3 a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2-b^4 c^2+3 a^2 b c^3-a b^2 c^3-a^2 c^4-a b c^4+b^2 c^4-2 b c^5+c^6) : :

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

Barycentrics    = a^2 (a^8-2 a^4 b^4+b^8-6 a^6 c^2+10 a^4 b^2 c^2+10 a^2 b^4 c^2-6 b^6 c^2+a^4 c^4-20 a^2 b^2 c^4+b^4 c^4+6 a^2 c^6+6 b^2 c^6-2 c^8) (a^8-6 a^6 b^2+a^4 b^4+6 a^2 b^6-2 b^8+10 a^4 b^2 c^2-20 a^2 b^4 c^2+6 b^6 c^2-2 a^4 c^4+10 a^2 b^2 c^4+b^4 c^4-6 b^2 c^6+c^8) : :

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

Barycentrics    = a^2 (b-c) (b+c) (a^8-2 a^4 b^4+b^8-6 a^6 c^2+10 a^4 b^2 c^2+10 a^2 b^4 c^2-6 b^6 c^2+a^4 c^4-20 a^2 b^2 c^4+b^4 c^4+6 a^2 c^6+6 b^2 c^6-2 c^8) (a^8-6 a^6 b^2+a^4 b^4+6 a^2 b^6-2 b^8+10 a^4 b^2 c^2-20 a^2 b^4 c^2+6 b^6 c^2-2 a^4 c^4+10 a^2 b^2 c^4+b^4 c^4-6 b^2 c^6+c^8) : :

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

Barycentrics: a^8-2 a^6 b^2+2 a^4 b^4-a^2 b^6-2 a^6 c^2+2 a^2 b^4 c^2-b^6 c^2+2 a^4 c^4+2 a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6 : :
X(10104) = X[7781]-3 X[9734] = (3 S^2-SW^2) X[3]+(S^2+SW^2) X[76] = (S^2+SW^2) X[5]+(S^2-SW^2) X[32] = (S^2-SW^2) X[140]+SW^2 X[141]

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

Barycentrics    a (a^7 b^2-3 a^5 b^4+3 a^3 b^6-a b^8-2 a^7 b c-6 a^6 b^2 c+3 a^5 b^3 c+13 a^4 b^4 c-8 a^2 b^6 c-a b^7 c+b^8 c+a^7 c^2-6 a^6 b c^2-16 a^5 b^2 c^2+9 a^4 b^3 c^2+15 a^3 b^4 c^2-2 a^2 b^5 c^2-b^7 c^2+3 a^5 b c^3+9 a^4 b^2 c^3+12 a^3 b^3 c^3+10 a^2 b^4 c^3+a b^5 c^3-3 b^6 c^3-3 a^5 c^4+13 a^4 b c^4+15 a^3 b^2 c^4+10 a^2 b^3 c^4+2 a b^4 c^4+3 b^5 c^4-2 a^2 b^2 c^5+a b^3 c^5+3 b^4 c^5+3 a^3 c^6-8 a^2 b c^6-3 b^3 c^6-a b c^7-b^2 c^7-a c^8+b c^8) : :

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

Barycentrics    2 a^4-a^3 (b+c)+a (b-c)^2 (b+c)-(b^2-c^2)^2-a^2 (b^2-6 b c+c^2) : :
X(10106) = (2R - r)*X(1) + r*X(4)

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} et al.

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

Barycentrics    a(3a^2(b+c)-2a b c-3b^3+5b c(b+c)-3c^3) : :
X(10107) = 3(4R + r)*X(7) + (4R-3r)*X(8)

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

Barycentrics    a (a^4 (b-c)^2-a^2 (b^4+7 b^3 c+16 b^2 c^2+7 b c^3+c^4)-a (b^5+b^4 c+8 b^3 c^2+8 b^2 c^3+b c^4+c^5)+b c (b^2-c^2)^2+a^3 (b^3-7 b^2 c-7 b c^2+c^3)) : :

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)

Barycentrics    9*SB*SC+13*S^2 : :
X(10109) = 13 X(3) + 11 X(4) = X(2) + 3 X(5)

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), et al


X(10110) =  MIDPOINT OF X(5)X(5446)

Trilinears    (-2 + cos 2A) cos(B - C) + cos A : :
X(10110) = X(4) + 3 X(51)

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) = 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

Barycentrics    (a^2-b^2-c^2) (2 a^14-6 a^12 b^2+5 a^10 b^4+a^8 b^6-4 a^6 b^8+4 a^4 b^10-3 a^2 b^12+b^14-6 a^12 c^2+12 a^10 b^2 c^2-9 a^8 b^4 c^2+8 a^6 b^6 c^2-12 a^4 b^8 c^2+12 a^2 b^10 c^2-5 b^12 c^2+5 a^10 c^4-9 a^8 b^2 c^4-4 a^6 b^4 c^4+8 a^4 b^6 c^4-21 a^2 b^8 c^4+9 b^10 c^4+a^8 c^6+8 a^6 b^2 c^6+8 a^4 b^4 c^6+24 a^2 b^6 c^6-5 b^8 c^6-4 a^6 c^8-12 a^4 b^2 c^8-21 a^2 b^4 c^8-5 b^6 c^8+4 a^4 c^10+12 a^2 b^2 c^10+9 b^4 c^10-3 a^2 c^12-5 b^2 c^12+c^14) : :

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

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

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

Barycentrics    2 a^10-3 a^8 b^2-a^6 b^4+a^4 b^6+3 a^2 b^8-2 b^10-3 a^8 c^2+8 a^6 b^2 c^2-2 a^4 b^4 c^2-9 a^2 b^6 c^2+6 b^8 c^2-a^6 c^4-2 a^4 b^2 c^4+12 a^2 b^4 c^4-4 b^6 c^4+a^4 c^6-9 a^2 b^2 c^6-4 b^4 c^6+3 a^2 c^8+6 b^2 c^8-2 c^10 : :

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


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

Barycentrics    (pending)

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

Barycentrics    (pending)

X(10115) = X[195] + 3 X[568] = X[2888] - 5 X[3567] = 3 X[51] - X[6288] = X[7691] - 3 X[9730]

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

Barycentrics    (a^2-b^2-c^2) (2 a^8-4 a^6 b^2+3 a^4 b^4-2 a^2 b^6+b^8-4 a^6 c^2-4 a^4 b^2 c^2+2 a^2 b^4 c^2-4 b^6 c^2+3 a^4 c^4+2 a^2 b^2 c^4+6 b^4 c^4-2 a^2 c^6-4 b^2 c^6+c^8) : :

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

Barycentrics    a^2 (a^10-a^8 b^2-2 a^6 b^4+2 a^4 b^6+a^2 b^8-b^10-a^8 c^2+5 a^6 b^2 c^2-2 a^4 b^4 c^2-3 a^2 b^6 c^2+b^8 c^2-2 a^6 c^4-2 a^4 b^2 c^4+4 a^2 b^4 c^4+2 a^4 c^6-3 a^2 b^2 c^6+a^2 c^8+b^2 c^8-c^10) : :
X10117) = 2 X[110] - 3 X[154] = 4 X[125] - 3 X[1853] = X[2935] + 2 X[9919]

See X(10111).

X(10117) lies on 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)
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) = {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

Barycentrics    a (a-b-c) (a^10-a^8 b^2-2 a^6 b^4+2 a^4 b^6+a^2 b^8-b^10+a^4 b^5 c-2 a^2 b^7 c+b^9 c-a^8 c^2+5 a^6 b^2 c^2-2 a^4 b^4 c^2-3 a^2 b^6 c^2+b^8 c^2-2 a^4 b^3 c^3+2 a^2 b^5 c^3-2 a^6 c^4-2 a^4 b^2 c^4+4 a^2 b^4 c^4+a^4 b c^5+2 a^2 b^3 c^5-2 b^5 c^5+2 a^4 c^6-3 a^2 b^2 c^6-2 a^2 b c^7+a^2 c^8+b^2 c^8+b c^9-c^10) : :

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

Barycentrics    a (a^11+a^10 b-a^9 b^2-a^8 b^3-2 a^7 b^4-2 a^6 b^5+2 a^5 b^6+2 a^4 b^7+a^3 b^8+a^2 b^9-a b^10-b^11+a^10 c-a^8 b^2 c-2 a^6 b^4 c-a^5 b^5 c+3 a^4 b^6 c+2 a^3 b^7 c-a^2 b^8 c-a b^9 c-a^9 c^2-a^8 b c^2+5 a^7 b^2 c^2+5 a^6 b^3 c^2-2 a^5 b^4 c^2-a^4 b^5 c^2-3 a^3 b^6 c^2-5 a^2 b^7 c^2+a b^8 c^2+2 b^9 c^2-a^8 c^3+5 a^6 b^2 c^3+2 a^5 b^3 c^3-4 a^4 b^4 c^3-2 a^3 b^5 c^3-a^2 b^6 c^3+b^8 c^3-2 a^7 c^4-2 a^6 b c^4-2 a^5 b^2 c^4-4 a^4 b^3 c^4+4 a^3 b^4 c^4+6 a^2 b^5 c^4-2 a^6 c^5-a^5 b c^5-a^4 b^2 c^5-2 a^3 b^3 c^5+6 a^2 b^4 c^5+2 a b^5 c^5-2 b^6 c^5+2 a^5 c^6+3 a^4 b c^6-3 a^3 b^2 c^6-a^2 b^3 c^6-2 b^5 c^6+2 a^4 c^7+2 a^3 b c^7-5 a^2 b^2 c^7+a^3 c^8-a^2 b c^8+a b^2 c^8+b^3 c^8+a^2 c^9-a b c^9+2 b^2 c^9-a c^10-c^11) : :

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)

Barycentrics    (a^2-b^2) (a^2-c^2) (a^18-4 a^16 b^2-4 a^14 b^4+56 a^12 b^6-154 a^10 b^8+224 a^8 b^10-196 a^6 b^12+104 a^4 b^14-31 a^2 b^16+4 b^18-7 a^16 c^2+20 a^14 b^2 c^2+2 a^12 b^4 c^2-30 a^10 b^6 c^2-86 a^8 b^8 c^2+296 a^6 b^10 c^2-326 a^4 b^12 c^2+162 a^2 b^14 c^2-31 b^16 c^2+20 a^14 c^4-32 a^12 b^2 c^4-13 a^10 b^4 c^4+14 a^8 b^6 c^4-79 a^6 b^8 c^4+312 a^4 b^10 c^4-326 a^2 b^12 c^4+104 b^14 c^4-28 a^12 c^6+12 a^10 b^2 c^6+15 a^8 b^4 c^6+34 a^6 b^6 c^6-79 a^4 b^8 c^6+296 a^2 b^10 c^6-196 b^12 c^6+14 a^10 c^8+8 a^8 b^2 c^8+15 a^6 b^4 c^8+14 a^4 b^6 c^8-86 a^2 b^8 c^8+224 b^10 c^8+14 a^8 c^10+12 a^6 b^2 c^10-13 a^4 b^4 c^10-30 a^2 b^6 c^10-154 b^8 c^10-28 a^6 c^12-32 a^4 b^2 c^12+2 a^2 b^4 c^12+56 b^6 c^12+20 a^4 c^14+20 a^2 b^2 c^14-4 b^4 c^14-7 a^2 c^16-4 b^2 c^16+c^18) (a^18-7 a^16 b^2+20 a^14 b^4-28 a^12 b^6+14 a^10 b^8+14 a^8 b^10-28 a^6 b^12+20 a^4 b^14-7 a^2 b^16+b^18-4 a^16 c^2+20 a^14 b^2 c^2-32 a^12 b^4 c^2+12 a^10 b^6 c^2+8 a^8 b^8 c^2+12 a^6 b^10 c^2-32 a^4 b^12 c^2+20 a^2 b^14 c^2-4 b^16 c^2-4 a^14 c^4+2 a^12 b^2 c^4-13 a^10 b^4 c^4+15 a^8 b^6 c^4+15 a^6 b^8 c^4-13 a^4 b^10 c^4+2 a^2 b^12 c^4-4 b^14 c^4+56 a^12 c^6-30 a^10 b^2 c^6+14 a^8 b^4 c^6+34 a^6 b^6 c^6+14 a^4 b^8 c^6-30 a^2 b^10 c^6+56 b^12 c^6-154 a^10 c^8-86 a^8 b^2 c^8-79 a^6 b^4 c^8-79 a^4 b^6 c^8-86 a^2 b^8 c^8-154 b^10 c^8+224 a^8 c^10+296 a^6 b^2 c^10+312 a^4 b^4 c^10+296 a^2 b^6 c^10+224 b^8 c^10-196 a^6 c^12-326 a^4 b^2 c^12-326 a^2 b^4 c^12-196 b^6 c^12+104 a^4 c^14+162 a^2 b^2 c^14+104 b^4 c^14-31 a^2 c^16-31 b^2 c^16+4 c^18)+(-3 a^20+16 a^18 b^2-24 a^16 b^4-24 a^14 b^6+126 a^12 b^8-168 a^10 b^10+84 a^8 b^12+24 a^6 b^14-51 a^4 b^16+24 a^2 b^18-4 b^20+28 a^18 c^2-112 a^16 b^2 c^2+138 a^14 b^4 c^2-26 a^12 b^6 c^2-24 a^10 b^8 c^2-20 a^8 b^10 c^2-70 a^6 b^12 c^2+190 a^4 b^14 c^2-136 a^2 b^16 c^2+32 b^18 c^2-117 a^16 c^4+312 a^14 b^2 c^4-219 a^12 b^4 c^4+12 a^10 b^6 c^4+39 a^8 b^8 c^4-16 a^6 b^10 c^4-204 a^4 b^12 c^4+306 a^2 b^14 c^4-113 b^16 c^4+288 a^14 c^6-416 a^12 b^2 c^6+60 a^10 b^4 c^6+4 a^8 b^6 c^6-16 a^6 b^8 c^6+16 a^4 b^10 c^6-330 a^2 b^12 c^6+232 b^14 c^6-462 a^12 c^8+192 a^10 b^2 c^8+60 a^8 b^4 c^8+44 a^6 b^6 c^8+53 a^4 b^8 c^8+124 a^2 b^10 c^8-308 b^12 c^8+504 a^10 c^10+184 a^8 b^2 c^10+116 a^6 b^4 c^10+100 a^4 b^6 c^10+112 a^2 b^8 c^10+280 b^10 c^10-378 a^8 c^12-328 a^6 b^2 c^12-249 a^4 b^4 c^12-198 a^2 b^6 c^12-182 b^8 c^12+192 a^6 c^14+208 a^4 b^2 c^14+150 a^2 b^4 c^14+88 b^6 c^14-63 a^4 c^16-64 a^2 b^2 c^16-32 b^4 c^16+12 a^2 c^18+8 b^2 c^18-c^20) (3 a^20-28 a^18 b^2+117 a^16 b^4-288 a^14 b^6+462 a^12 b^8-504 a^10 b^10+378 a^8 b^12-192 a^6 b^14+63 a^4 b^16-12 a^2 b^18+b^20-16 a^18 c^2+112 a^16 b^2 c^2-312 a^14 b^4 c^2+416 a^12 b^6 c^2-192 a^10 b^8 c^2-184 a^8 b^10 c^2+328 a^6 b^12 c^2-208 a^4 b^14 c^2+64 a^2 b^16 c^2-8 b^18 c^2+24 a^16 c^4-138 a^14 b^2 c^4+219 a^12 b^4 c^4-60 a^10 b^6 c^4-60 a^8 b^8 c^4-116 a^6 b^10 c^4+249 a^4 b^12 c^4-150 a^2 b^14 c^4+32 b^16 c^4+24 a^14 c^6+26 a^12 b^2 c^6-12 a^10 b^4 c^6-4 a^8 b^6 c^6-44 a^6 b^8 c^6-100 a^4 b^10 c^6+198 a^2 b^12 c^6-88 b^14 c^6-126 a^12 c^8+24 a^10 b^2 c^8-39 a^8 b^4 c^8+16 a^6 b^6 c^8-53 a^4 b^8 c^8-112 a^2 b^10 c^8+182 b^12 c^8+168 a^10 c^10+20 a^8 b^2 c^10+16 a^6 b^4 c^10-16 a^4 b^6 c^10-124 a^2 b^8 c^10-280 b^10 c^10-84 a^8 c^12+70 a^6 b^2 c^12+204 a^4 b^4 c^12+330 a^2 b^6 c^12+308 b^8 c^12-24 a^6 c^14-190 a^4 b^2 c^14-306 a^2 b^4 c^14-232 b^6 c^14+51 a^4 c^16+136 a^2 b^2 c^16+113 b^4 c^16-24 a^2 c^18-32 b^2 c^18+4 c^20) : :
X(10120) = 3X(2) + 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)

Barycentrics    1/(a^18 b^2-9 a^16 b^4+36 a^14 b^6-84 a^12 b^8+126 a^10 b^10-126 a^8 b^12+84 a^6 b^14-36 a^4 b^16+9 a^2 b^18-b^20+a^18 c^2-10 a^16 b^2 c^2+39 a^14 b^4 c^2-71 a^12 b^6 c^2+45 a^10 b^8 c^2+49 a^8 b^10 c^2-115 a^6 b^12 c^2+91 a^4 b^14 c^2-34 a^2 b^16 c^2+5 b^18 c^2-9 a^16 c^4+39 a^14 b^2 c^4-66 a^12 b^4 c^4+36 a^10 b^6 c^4+15 a^8 b^8 c^4+8 a^6 b^10 c^4-51 a^4 b^12 c^4+33 a^2 b^14 c^4-5 b^16 c^4+36 a^14 c^6-71 a^12 b^2 c^6+36 a^10 b^4 c^6+16 a^8 b^6 c^6-4 a^6 b^8 c^6-8 a^4 b^10 c^6+15 a^2 b^12 c^6-20 b^14 c^6-84 a^12 c^8+45 a^10 b^2 c^8+15 a^8 b^4 c^8-4 a^6 b^6 c^8+8 a^4 b^8 c^8-23 a^2 b^10 c^8+70 b^12 c^8+126 a^10 c^10+49 a^8 b^2 c^10+8 a^6 b^4 c^10-8 a^4 b^6 c^10-23 a^2 b^8 c^10-98 b^10 c^10-126 a^8 c^12-115 a^6 b^2 c^12-51 a^4 b^4 c^12+15 a^2 b^6 c^12+70 b^8 c^12+84 a^6 c^14+91 a^4 b^2 c^14+33 a^2 b^4 c^14-20 b^6 c^14-36 a^4 c^16-34 a^2 b^2 c^16-5 b^4 c^16+9 a^2 c^18+5 b^2 c^18-c^20) : :
X(10121) = 3X(2) - 4X(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

Barycentrics    a (a^4 (b-c)^2-a^5 (b+c)+(b^2-c^2)^2 (b^2-b c+c^2)-a (b-c)^2 (b^3+4 b^2 c+4 b c^2+c^3)+a^3 (2 b^3+3 b^2 c+3 b c^2+2 c^3)+a^2 (-2 b^4+3 b^3 c+6 b^2 c^2+3 b c^3-2 c^4)) : :
X(10122) = (r+2R)*X(1) - r*X(21) = (r+4R)*X(7) - (r+2R)*X(79)

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="https://groups.yahoo.com/neo/groups/Hyacinthos/conversations/messages/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

Barycentrics    2 a^7-a^6 (b+c)-(b-c)^4 (b+c)^3+5 a b c (b^2-c^2)^2+a^3 (b+c)^2 (2 b^2-3 b c+2 c^2)-a^5 (4 b^2+6 b c+4 c^2)+a^4 (b^3-4 b^2 c-4 b c^2+c^3)+a^2 (b-c)^2 (b^3+6 b^2 c+6 b c^2+c^3) : :
X(10123) = (r+3R)*X(21) - (r+4R)*X(142) = (2r+R)*X(35) - (2r+5R)*X(79)

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

Barycentrics    10 a^4-17 a^2 (b^2+c^2)+7 (b^2-c^2)^2 : :
X(10124) = 5X(3) + 7X(5) = 7X(2) + X(3)

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

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

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)

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


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

Barycentrics    2 a^16-13 a^14 b^2+39 a^12 b^4-69 a^10 b^6+75 a^8 b^8-47 a^6 b^10+13 a^4 b^12+a^2 b^14-b^16-13 a^14 c^2+54 a^12 b^2 c^2-85 a^10 b^4 c^2+44 a^8 b^6 c^2+29 a^6 b^8 c^2-38 a^4 b^10 c^2+5 a^2 b^12 c^2+4 b^14 c^2+39 a^12 c^4-85 a^10 b^2 c^4+44 a^8 b^4 c^4+9 a^6 b^6 c^4+18 a^4 b^8 c^4-21 a^2 b^10 c^4-4 b^12 c^4-69 a^10 c^6+44 a^8 b^2 c^6+9 a^6 b^4 c^6+14 a^4 b^6 c^6+15 a^2 b^8 c^6-4 b^10 c^6+75 a^8 c^8+29 a^6 b^2 c^8+18 a^4 b^4 c^8+15 a^2 b^6 c^8+10 b^8 c^8-47 a^6 c^10-38 a^4 b^2 c^10-21 a^2 b^4 c^10-4 b^6 c^10+13 a^4 c^12+5 a^2 b^2 c^12-4 b^4 c^12+a^2 c^14+4 b^2 c^14-c^16 : :

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

Barycentrics    2 a^10-3 a^8 b^2-2 a^6 b^4+4 a^4 b^6-b^10-3 a^8 c^2+8 a^6 b^2 c^2-16 a^4 b^4 c^2+8 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4-16 a^4 b^2 c^4-16 a^2 b^4 c^4-2 b^6 c^4+4 a^4 c^6+8 a^2 b^2 c^6-2 b^4 c^6+3 b^2 c^8-c^10 : :

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

Barycentrics    2 a^6+a^4 b^2-2 a^2 b^4-b^6+a^4 c^2+16 a^2 b^2 c^2+b^4 c^2-2 a^2 c^4+b^2 c^4-c^6 : :

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}



leftri  ATFF points of pairs of triangles: X(10029) - X(10032)  rightri

This preamble and centers X(10029)-X(10032) were contributed by Peter Moses and Clark Kimberling, August 21, 2016, following the example at X(5643) contributed by Angel Montesdeoca.

Suppose that A'B'C' and A''B''C'' are distinct triangles in the plane of a triangle ABC. The finite fixed point of the affine transformation that carries A'B'C' onto A''B''C'' is here named accordingly and is denoted by ATFF(A'B'C', A''B''C'').

In the following list, A'B'C' = ABC, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "medial (TCCT 6.2): X(2)" means that ATFF(ABC, medial triangle) = X(2) and that the medial triangle is identified in the book Triangle Centers and Central Triangles, section 6.2).

medial (TCCT 6.2) : X(2)
anticomplementary (TCCT 6.3) : X(2)
orthic (TCCT 6.4) : X(6)
tangential (TCCT 6.5) : X(3)
incentral (TCCT 6.6) : X(37)
excentral (TCCT 6.7) : X(9)
intouch / Gergonne(TCCT 6.8) : X(1)
extouch (TCCT 6.9) : X(9)
Feuerbach (TCCT 6.10) : X(5949)
Euler (TCCT 6.11) : X(4)
ABC reflected about O (TCCT 6.12) : X(3)
O reflected in ABC (TCCT 6.13) : X(3)
mid-arc (TCCT 6.14) : X(2089)
intangents (TCCT 6.16) : X(7004)
extangents (TCCT 6.17) : X(71)
vertex triangle ABC & intangents / Pelletier (TCCT 6.18) : X(11)
circum-medial (TCCT 6.19) : X(10130)
circum-orthic (TCCT 6.20) : X(54)
first circumperp (TCCT 6.21) : X(100)
second circumperp (TCCT 6.22) : X(21)
tangential triangle of first circumperp (TCCT 6.23) : X(55)
tangential triangle of second circumperp (TCCT 6.24) : X(56)
first Morley (TCCT 6.25) : X(3604)
second Morley (TCCT 6.26) : X(3602)
third Morley (TCCT 6.27) : X(3603)
outer Napoleon (TCCT 6.31) : X(2)
inner Napoleon (TCCT 6.32) : X(2)
outer Fermat (TCCT p178) : X(2)
inner Fermat (TCCT p178) : X(2)
hexyl (TCCT 6.36) : X(1)
Yff central (TCCT 6.37) : X(7707)
half altitude (TCCT 6.38) : X(6)
BCI (TCCT 6.39) : X(1489)
inner Vecten (MathWorld) : X(2)
outer Vecten (MathWorld) : X(2)
first Neuberg (MathWorld) : X(2)
second Neuberg (MathWorld) : X(2)
Fuhrmann (TCCT 8.25) : X(7705)
first Brocard (CTC) : X(2)
second Brocard (CTC) : X(574)
third Brocard (CTC) : X(3117)
fourth Brocard / D-triangle (CTC) / Johnson (Quim Castellsaguer) : X(5094)
reflection A in BC : X(6)
MacBeath (MathWorld) : X(5)
Lucas central (MathWorld) : X(10132
) inner mixtilinear (MathWorld) : X(55)
outer mixtilinear (ETC X(7955), called 2nd) : X(220)
Kosnita (see ETC X(1658)) : X(3)
first Sharygin (see ETC X(8229)) : X(21)
second Sharygin (see ETC X(8229)) : X(100)
McCay (see ETC X(7606)) : X(2)
anti McCay (see ETC X(8587)) : X(2)
Honsberger (see ETC X(7670)) : X(2346)
Trinh (see ETC X(7688)) : X(3)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(5)
orthocentroidal (see ETC X(5476)) : X(6)
anti first-Brocard (see ETC X(5939)) : X(2)
Lemoine (MathWorld) : X(597)
Steiner (MathWorld) : X(2)
second Euler (see ETC X(3758)) : X(3)
third Euler (see ETC X(3758)) : X(10129)
fourth Euler (see ETC X(3758)) : X(7705)
fifth Euler (see ETC X(3758)) : X(5094)v symmedial (MathWorld) : X(39)
Gossard (ETC X(402)) : X(402)
Apollonius (excircles) : X(9560)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) : X(1)
Ara (see ETC X(5594) / excentral of tangential : X(25
) Aries ETC X(5596)) : X(110
) first Auriga (see ETC X(5597)) : X(5597)
second Auriga (see ETC X(5597)) : X(5598)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(1)
inner Grebe (see ETC X(1160)) : X(6)
outer Grebe (see ETC X(1161)) : X(6)
inner inscribed squares (MathWorld) : X(3068)
outer inscribed squares (MathWorld) : X(3069)
circum-symmedial : X(574)
inner Garcia (see ETC X(5587)) : X(3)
outer Garcia (see ETC X(5587)) : X(10)
second Pamfilos-Zhou (see ETC X(7954)) : X(7133)
second extouch (see ETC X(5927)) : X(9)
third extouch (see ETC X(5927)) : X(223)
fourth extouch (see ETC X(5927)) : X(1038)
fifth extouch (see ETC X(5927)) : X(1038)
sixth mixtilinear (see ETC X(7955)) : X(1)
seventh mixtilinear (see ETC X(8916)) : X(2124)
first Parry (see ETC X(9122)) : X(110)
second Parry (see ETC X(9122)) : X(111)
second EhrmannT (see ETC (8537)) : X(895)
second/outer tangential mid-arc (see ETC X(8075)) : X(6732)
first Schiffler (see ETC X(6595)) : X(11)
second Schiffler (see ETC X(6596)) : X(11)
Mandart-incircle (see ETC X(6018)) : X(55)
fifth Brocard (see ETC X(32)) : X(32)
sixth Brocard (see ETC X(384)) : X(10131)
first Conway (see ETC X(7411)) : X(21)
Ayme (see ETC X(3610) : X(10)
inverse in incircle : X(7)
inner Hutson (see ETC X(363)) : X(6732)
outer Hutson (see ETC X(363)) : X(7707)
T(-2,1) (TCCT 6.41) : X(9)
T(-1,3) (TCCT 6.42) : X(3680)
Hutson intouch (see ETC X(5731)) : X(1)
Hutson extouch (see ETC X(5731)) : X(9)
Atik (see ETC X(8580)) : X(8)
Hatzipolakis-Moses (see ETC X(6145)) : X(6)
first orthosymmedial (see ETC X(6792)) : X(251)
orthic of intouch : X(57)
intouch of orthic : X(25)
tangential of excentral : X(57)
Artzt (see ETC X(9742)) : X(2)
second Conway (see ETC X(9776)) : X(8)
submedial (see ETC X(9813)) : X(5544)
reflected 1st Brocard(CTC table 32) : X(2)
isogonal of reflected 1st Brocard(CTC table 32) : X(8623)
inner Yff (ETC X(10037)) : X(1)
outer Yff (ETC X(10037)) : X(1)

In the following list, A'B'C' is the medial triangle, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "anticomplementary (TCCT 6.3): X(2)" means that ATFF(medial triangle, anticomplementary triangle) = X(2) and that the anticomplementary triangle is identified in the book Triangle Centers and Central Triangles, section 6.3).

anticomplementary (TCCT 6.3) : X(2)
orthic (TCCT 6.4) : X(125)
tangential (TCCT 6.5) : X(6)
incentral (TCCT 6.6) : X(244)
excentral (TCCT 6.7) : X(1)
intouch / Gergonne(TCCT 6.8) : X(11)
extouch (TCCT 6.9) : X(11)
Euler (TCCT 6.11) : X(5)
ABC reflected about O (TCCT 6.12) : X(4)
O reflected in ABC (TCCT 6.13) : X(1656)
vertex triangle ABC & intangents / Pelletier (TCCT 6.18) : X(650)
circum-medial (TCCT 6.19) : X(9465)
first circumperp (TCCT 6.21) : X(9)
second circumperp (TCCT 6.22) : X(1)
tangential triangle of first circumperp (TCCT 6.23) : X(1376)
tangential triangle of second circumperp (TCCT 6.24) : X(958)
first Morley (TCCT 6.25) : X(3602)
second Morley (TCCT 6.26) : X(3603)
third Morley (TCCT 6.27) : X(3604)
outer Napoleon (TCCT 6.31) : X(2)
inner Napoleon (TCCT 6.32) : X(2)
outer Fermat (TCCT p178) : X(2)
inner Fermat (TCCT p178) : X(2)
pedal X(15) (TCCT 6.34) : X(115)
pedal X(16) (TCCT 6.35) : X(115)
hexyl (TCCT 6.36) : X(3646)
half altitude (TCCT 6.38) : X(5)
inner Vecten (MathWorld) : X(2)
outer Vecten (MathWorld) : X(2)
first Neuberg (MathWorld) : X(2)
second Neuberg (MathWorld) : X(2)
Fuhrmann (TCCT 8.25) : X(1)
first Brocard (CTC) : X(2)
fourth Brocard / D-triangle (CTC) / Johnson (Quim Castellsaguer) : X(9465)
MacBeath (MathWorld) : X(2972)
Kosnita (see ETC X(1658)) : X(1147)
second Sharygin (see ETC X(8229)) : X(244)
McCay (see ETC X(7606)) : X(2)
anti McCay (see ETC X(8587)) : X(2)
Trinh (see ETC X(7688)) : X(4550)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(3)
orthocentroidal (see ETC X(5476)) : X(7703)
anti first-Brocard (see ETC X(5939)) : X(2)
Lemoine (MathWorld) : X(8288)
Steiner (MathWorld) : X(1649)
second Euler (see ETC X(3758)) : X(1209)
third Euler (see ETC X(3758)) : X(11)
fourth Euler (see ETC X(3758)) : X(442)
symmedial (MathWorld) : X(3124)
Gossard (ETC X(402)) : X(1650)
Antila (see ETC X(5574)) : X(8580)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) : X(1698)
Ara (see ETC X(5594) / excentral of tangential : X(3)
Aries ETC X(5596)) : X(69)
first Auriga (see ETC X(5597)) : X(5599)
second Auriga (see ETC X(5597)) : X(5600)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(8)
inner Grebe (see ETC X(1160)) : X(5591)
outer Grebe (see ETC X(1161)) : X(5590)
inner inscribed squares (MathWorld) : X(485)
outer inscribed squares (MathWorld) : X(486)
Schroeter (see ETC X(8286) : X(2)
outer Garcia (see ETC X(5587)) : X(1)
second extouch (see ETC X(5927)) : X(442)
first EhrmannT (see ETC (8537)) : X(8542)
second EhrmannT (see ETC (8537)) : X(6)
Mandart-incircle (see ETC X(6018)) : X(11)
fifth Brocard (see ETC X(32)) : X(3096)
inverse in incircle : X(1)
T(-2,1) (TCCT 6.41) : X(3035)
T(-1,3) (TCCT 6.42) : X(1)
first orthosymmedial (see ETC X(6792)) : X(427)
Ascella (see ETC X(8726)) : X(142)
minimal area inscribed equilateral (see ETC X(9112)) : X(115)
maximal area inscribed equilateral (see ETC X(9113)) : X(115)
orthic of intouch : X(226)
intouch of orthic : X(427)
tangential of excentral : X(9)
Roussel : X(3604)
Artzt (see ETC X(9742)) : X(2)
submedial (see ETC X(9813)) : X(5)
reflected 1st Brocard(CTC table 32) : X(2)
Ae, (CTC K798) : X(10)
Ai, (CTC K798) : X(10)
inner Yff (ETC X(10037)) : X(498)
outer Yff (ETC X(10037)) : X(499)

In the following list, A'B'C' is the anticomplementary triangle, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "orthic (TCCT 6.4): X(4)" means that ATFF(anticomplementary triangle, orthic triangle) = X(4) and that the orthic triangle is identified in the book Triangle Centers and Central Triangles, section 6.4).

orthic (TCCT 6.4) : X(4)
tangential (TCCT 6.5) : X(110)
incentral (TCCT 6.6) : X(1)
excentral (TCCT 6.7) : X(100)
intouch / Gergonne(TCCT 6.8) : X(7)
extouch (TCCT 6.9) : X(8)
Euler (TCCT 6.11) : X(3091)
ABC reflected about O (TCCT 6.12) : X(20)
O reflected in ABC (TCCT 6.13) : X(5)
vertex triangle ABC & intangents / Pelletier (TCCT 6.18) : X(885)
circum-medial (TCCT 6.19) : X(7493)
circum-orthic (TCCT 6.20) : X(4)
tangential triangle of first circumperp (TCCT 6.23) : X(100)
tangential triangle of second circumperp (TCCT 6.24) : X(2975)
outer Napoleon (TCCT 6.31) : X(2)
inner Napoleon (TCCT 6.32) : X(2)
outer Fermat (TCCT p178) : X(2)
inner Fermat (TCCT p178) : X(2)
antipedal X(13) (TCCT 6.32) : X(99)
antipedal X(14) (TCCT 6.33) : X(99)
hexyl (TCCT 6.36) : X(21)
inner Vecten (MathWorld) : X(2)
outer Vecten (MathWorld) : X(2)
first Neuberg (MathWorld) : X(2)
second Neuberg (MathWorld) : X(2)
Fuhrmann (TCCT 8.25) : X(2475)
first Brocard (CTC) : X(2)
reflection A in BC : X(3448)
MacBeath (MathWorld) : X(264)
McCay (see ETC X(7606)) : X(2)
anti McCay (see ETC X(8587)) : X(2)
Honsberger (see ETC X(7670)) : X(7)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(4)
orthocentroidal (see ETC X(5476)) : X(7693)
anti first-Brocard (see ETC X(5939)) : X(2)
Lemoine (MathWorld) : X(598)
symmedial (MathWorld) : X(6)
Gossard (ETC X(402)) : X(4240)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) : X(10)
Ara (see ETC X(5594) / excentral of tangential : X(22)
Aries ETC X(5596)) : X(20)
first Auriga (see ETC X(5597)) : X(5601)
second Auriga (see ETC X(5597)) : X(5602)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(145)
inner Grebe (see ETC X(1160)) : X(1271)
outer Grebe (see ETC X(1161)) : X(1270)
Schroeter (see ETC X(8286) : X(5466)
inner Garcia (see ETC X(5587)) : X(8)
outer Garcia (see ETC X(5587)) : X(8)
sixth mixtilinear (see ETC X(7955)) : X(9)
anti fourth-Brocard (CTC) : X(1383)
Mandart-incircle (see ETC X(6018)) : X(497)
fifth Brocard (see ETC X(32)) : X(2896)
sixth Brocard (see ETC X(384)) : X(2896)
first Conway (see ETC X(7411)) : X(7)
Hutson intouch (see ETC X(5731)) : X(7320)
maximal area circumscribed equilateral (see ETC X(9112)) : X(99)
minimal area circumscribed equilateral (see ETC X(9113)) : X(99)
orthic of intouch : X(7)
intouch of orthic : X(4)
tangential of excentral : X(63)
Artzt (see ETC X(9742)) : X(2)
second Conway (see ETC X(9776)) : X(7)
reflected 1st Brocard(CTC table 32) : X(2)
inner Yff (ETC X(10037)) : X(3085)
outer Yff (ETC X(10037)) : X(3086)

In the following list, A'B'C' is the orthic triangle, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "tangential (TCCT 6.5): X(25)" means that ATFF(orthic triangle, tangential triangle) = X(25) and that the tangential triangle is identified in the book Triangle Centers and Central Triangles, section 6.5).

tangential (TCCT 6.5) : X(25)
excentral (TCCT 6.7) : X(19)
extouch (TCCT 6.9) : X(1146)
Euler (TCCT 6.11) : X(3574)
ABC reflected about O (TCCT 6.12) : X(64)
O reflected in ABC (TCCT 6.13) : X(3527)
intangents (TCCT 6.16) : X(33)
extangents (TCCT 6.17) : X(19)
circum-orthic (TCCT 6.20) : X(4)
half altitude (TCCT 6.38) : X(6)
reflection A in BC : X(6)
MacBeath (MathWorld) : X(2970)
Kosnita (see ETC X(1658)) : X(24)
Trinh (see ETC X(7688)) : X(378)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(4)
orthocentroidal (see ETC X(5476)) : X(6)
second Euler (see ETC X(3758)) : X(5)
Aries ETC X(5596)) : X(3)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(1854)
inner Grebe (see ETC X(1160)) : X(6)
outer Grebe (see ETC X(1161)) : X(6)
inner inscribed squares (MathWorld) : X(590)
outer inscribed squares (MathWorld) : X(615)
Schroeter (see ETC X(8286) : X(2501)
second extouch (see ETC X(5927)) : X(1901)
second EhrmannT (see ETC (8537)) : X(8541)
T(-2,1) (TCCT 6.41) : X(2182)
Hatzipolakis-Moses (see ETC X(6145)) : X(6)
first orthosymmedial (see ETC X(6792)) : X(51)
orthic of intouch : X(65)
intouch of orthic : X(4)
Artzt (see ETC X(9742)) : X(25)
submedial (see ETC X(9813)) : X(2)

In the following list, A'B'C' is the tangential triangle, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "incentral (TCCT 6.5): X(25)" means that ATFF(tangential triangle, incentral triangle) = X(31) and that the incentral triangle is identified in the book Triangle Centers and Central Triangles, section 6.6).

incentral (TCCT 6.6) : X(31)
excentral (TCCT 6.7) : X(109)
intouch / Gergonne(TCCT 6.8) : X(56)
extouch (TCCT 6.9) : X(55)
ABC reflected about O (TCCT 6.12) : X(3)
O reflected in ABC (TCCT 6.13) : X(3)
intangents (TCCT 6.16) : X(55)
extangents (TCCT 6.17) : X(55)
circum-medial (TCCT 6.19) : X(1995)
circum-orthic (TCCT 6.20) : X(24)
first circumperp (TCCT 6.21) : X(198)
second circumperp (TCCT 6.22) : X(56)
hexyl (TCCT 6.36) : X(2360)
inner Vecten (MathWorld) : X(10132)
Fuhrmann (TCCT 8.25) : X(8614)
MacBeath (MathWorld) : X(4)
Lucas tangents (MathWorld) : X(493)
Kosnita (see ETC X(1658)) : X(3)
first Sharygin (see ETC X(8229)) : X(3145)
Trinh (see ETC X(7688)) : X(3)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(155)
anti first-Brocard (see ETC X(5939)) : X(22)
Lemoine (MathWorld) : X(1383)
Steiner (MathWorld) : X(110)
second Euler (see ETC X(3758)) : X(3)
symmedial (MathWorld) : X(32)
Ara (see ETC X(5594) / excentral of tangential : X(159)
Aries ETC X(5596)) : X(159)
inner Grebe (see ETC X(1160)) : X(8903)
outer Grebe (see ETC X(1161)) : X(8904)
circum-symmedial : X(1384)
inner Garcia (see ETC X(5587)) : X(3)
outer Garcia (see ETC X(5587)) : X(3157)
fifth extouch (see ETC X(5927)) : X(56)
second EhrmannT (see ETC (8537)) : X(6)
tangential of excentral : X(610)
Artzt (see ETC X(9742)) : X(25)
submedial (see ETC X(9813)) : X(5020)
reflected 1st Brocard(CTC table 32) : X(1613)
isogonal of reflected 1st Brocard(CTC table 32) : X(6660)

In the following list, A'B'C' is the incentral, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "excentral (TCCT 6.7): X(6)" means that ATFF(incentral triangle, excentral triangle) = X(6) and that the excentral triangle is identified in the book Triangle Centers and Central Triangles, section 6.7).

excentral (TCCT 6.7) : X(6)
intouch / Gergonne(TCCT 6.8) : X(7004)
extouch (TCCT 6.9) : X(2170)
Feuerbach (TCCT 6.10) : X(115)
extangents (TCCT 6.17) : X(65)
vertex triangle ABC & intangents / Pelletier (TCCT 6.18) : X(663)
second circumperp (TCCT 6.22) : X(1201)
second Sharygin (see ETC X(8229)) : X(244)
Apus (see ETC X(5584)) : X(8573)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(3057)
Schroeter (see ETC X(8286) : X(661)
outer Garcia (see ETC X(5587)) : X(65)
Ayme (see ETC X(3610) : X(612)

In the following list, A'B'C' is the excentral, and A''B''C'' is the triangle indicated by name and reference. For example, the first item, "intouch [Gergonne] (TCCT 6.7): X(6)" means that ATFF(excentral triangle, intouch triangle) = X(6) and that the intouch [elsewhere called Gergonne triangle] is identified in the book Triangle Centers and Central Triangles, section 6.8).

intouch / Gergonne(TCCT 6.8) : X(57)
extouch (TCCT 6.9) : X(9)
Feuerbach (TCCT 6.10) : X(8818)
ABC reflected about O (TCCT 6.12) : X(40)
O reflected in ABC (TCCT 6.13) : X(3652)
(inner) tangential mid-arc (TCCT 6.15) : X(8078)
extangents (TCCT 6.17) : X(19)
first circumperp (TCCT 6.21) : X(165)
second circumperp (TCCT 6.22) : X(1)
tangential triangle of first circumperp (TCCT 6.23) : X(100)
hexyl (TCCT 6.36) : X(3)
Yff central (TCCT 6.37) : X(173)
half altitude (TCCT 6.38) : X(169)
Fuhrmann (TCCT 8.25) : X(1)
MacBeath (MathWorld) : X(92)
first Sharygin (see ETC X(8229)) : X(846)
second Sharygin (see ETC X(8229)) : X(1054)
Honsberger (see ETC X(7670)) : X(1445)
Carnot / Johnson (reflection of ABC about X(5). MathWorld) : X(3811)
Steiner (MathWorld) : X(662)
third Euler (see ETC X(3758)) : X(1699)
fourth Euler (see ETC X(3758)) : X(1698)
symmedial (MathWorld) : X(31)
Antila (see ETC X(5574)) : X(7271)
Aquilla (see ETC X(5586) / T(1,2) TCCT (6.40) : X(191)
Caelum / fifth mixitilinear (see ETC X(5603)) : X(2136)
inner Garcia (see ETC X(5587)) : X(191)
outer Garcia (see ETC X(5587)) : X(1)
second Pamfilos-Zhou (see ETC X(7954)) : X(8231)
second extouch (see ETC X(5927)) : X(9)
third extouch (see ETC X(5927)) : X(2270)
third mixtilinear (see ETC X(7955)) : X(1420)
sixth mixtilinear (see ETC X(7955)) : X(165)
second/outer tangential mid-arc (see ETC X(8075)) : X(258)
second Schiffler (see ETC X(6596)) : X(1768)
first Conway (see ETC X(7411)) : X(63)
incircle-inverse of ABC : X(1)
inner Hutson (see ETC X(363)) : X(363)
outer Hutson (see ETC X(363)) : X(168)
T(-2,1) (TCCT 6.41) : X(9)
T(-1,3) (TCCT 6.42) : X(1)
Hutson intouch (see ETC X(5731)) : X(1697)
Hutson extouch (see ETC X(5731)) : X(9)
Atik (see ETC X(8580)) : X(8580)
Ascella (see ETC X(8726)) : X(57)
tangential of excentral : X(40)
second Conway (see ETC X(9776)) : X(2)
reflected 1st Brocard (CTC table 32) : X(3550)

underbar

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

Barycentrics    a^2 b+a b^2-2 b^3+a^2 c-a b c+2 b^2 c+a c^2+2 b c^2-2 c^3 : :
X(10129) = 3 X[2320] - 5 X[3616] = 7 X[3624] + 3 X[5561]

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)

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

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)

Barycentrics    a^8+a^6 b^2-a^4 b^4+a^6 c^2-3 a^4 b^2 c^2-a^2 b^4 c^2-a^4 c^4-a^2 b^2 c^4+b^4 c^4 : :

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)

Barycentrics    a^2 (a^2+S) SA : :

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)

Barycentrics    a^2 (a^2-S) SA : :

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)

Barycentrics    2 a^5-a^4 b-4 a^3 b^2+2 a^2 b^3+2 a b^4-b^5-a^4 c-4 a^3 b c-2 a^2 b^2 c+4 a b^3 c+3 b^4 c-4 a^3 c^2-2 a^2 b c^2-12 a b^2 c^2-2 b^3 c^2+2 a^2 c^3+4 a b c^3-2 b^2 c^3+2 a c^4+3 b c^4-c^5-2 (a^3+3 a^2 b-3 a b^2-b^3+3 a^2 c+6 a b c+b^2 c-3 a c^2+b c^2-c^3) S : :

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)

Barycentrics    2 a^5-a^4 b-4 a^3 b^2+2 a^2 b^3+2 a b^4-b^5-a^4 c-4 a^3 b c-2 a^2 b^2 c+4 a b^3 c+3 b^4 c-4 a^3 c^2-2 a^2 b c^2-12 a b^2 c^2-2 b^3 c^2+2 a^2 c^3+4 a b c^3-2 b^2 c^3+2 a c^4+3 b c^4-c^5+(2 a^3+6 a^2 b-6 a b^2-2 b^3+6 a^2 c+12 a b c+2 b^2 c-6 a c^2+2 b c^2-2 c^3) S : :

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)

Barycentrics    (a+b-c) (a-b+c) (4 a^4-3 a^3 b-3 a^2 b^2-a b^3+3 b^4-3 a^3 c+6 a^2 b c+a b^2 c-4 b^3 c-3 a^2 c^2+a b c^2+2 b^2 c^2-a c^3-4 b c^3+3 c^4) : :

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)




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

This preamble and centers X(10137)-X(10148) were contributed by César Lozada, August 26, 2016.

Suppose that T' and T'' are (central) triangles in the plane of a triangle ABC, and let
A'B'C' = T'-of ABC
AaAbAc = T''-of-AB'C'
BbBcBa = T''-of-A'BC'
CcCaCb = T'' of A'B'C
Hex2T(T',T") = hexagon with vertices Ab, Ac, Bc, Ba, Ca, Cb.

For many choices of T' and T", the vertices of Hex2T(T',T") lie on a conic and for a few of them the conic is a circle. The following table shows a selection of pairs (T',T"_ for which Hex2T(T',T") is a circle, together with the center and radius.

T' T'' Center Radius
Lucas inner circumsymmedial X(10137) |13*S*R/(37*S+20*SW)|
Lucas(-1) inner circumsymmedial X(10138) |13*S*R/(37*S-20*SW)|
Lucas inner Lucas tangents X(10139) |sqrt(61)*S*R/(44*S+25*SW)|
Lucas(-1) inner Lucas(-1) tangents X(10140) |sqrt(61)*S*R/(44*S-25*SW)|
Lucas inner Lucas(-1) tangents X(10141) |sqrt(109)*S*R/(28*S+15*SW)|
Lucas(-1) inner Lucas tangents X(10142) |sqrt(109)*S*R/(28*S-15*SW)|
Lucas inner Lucas inner X(10143) |sqrt(481)*S*R/(209*S+120*SW)|
Lucas(-1) inner Lucas(-1) inner X(10144) |sqrt(481)*S*R/(209*S-120*SW)|
Lucas inner Lucas(-1) inner X(10145) |sqrt(73)*S*R/(29*S+16*SW)|
Lucas(-1) inner Lucas inner X(10146) |sqrt(73)*S*R/(29*S-16*SW)|
Lucas tangents Lucas(-1) inner X(10147) |sqrt(157)*S*R/(20*S+9*SW)|
Lucas(-1) tangents Lucas inner X(10148) |sqrt(157)*S*R/(20*S-9*SW)|
Lucas tangents circumsymmedial X(6409) |sqrt(13)*S*R/(4*S+SW)|
Lucas(-1) tangents circumsymmedial X(6410) |sqrt(13)*S*R/(4*S-SW)|
Lucas tangents Lucas inner X(6409) |sqrt(13)*S*R/(4*S+SW)|
Lucas(-1) tangents Lucas(-1) inner X(6410) |sqrt(13)*S*R/(4*S-SW)|
Lucas tangents Lucas(-1) tangents X(6449) |sqrt(13)*S*R/(5*S+2*SW)|
Lucas(-1) tangents Lucas tangents X(6450) |sqrt(13)*S*R/(5*S-2*SW)|
intouch 2nd Brocard X(1) r
intouch 4th Brocard X(1) r
Johnson 2nd Conway X(4) 2*R

underbar

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

Trilinears    37*cos(A)+20*sin(A) : :
X(10137) = 37*S*X(3)+20*SW*X(6)

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)

Trilinears    37*cos(A)-20*sin(A) : :
X(10138) = 37*S*X(3)-20*SW*X(6)

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)

Trilinears    44*cos(A)+25*sin(A) : :
X(10139) = 44*S*X(3)+25*SW*X(6)

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)

Trilinears    44*cos(A)-25*sin(A) : :
X(10140) = 44*S*X(3)-25*SW*X(6)

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)

Trilinears    28*cos(A)+15*sin(A) : :
X(10141) = 28*S*X(3)+15*SW*X(6)

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)

Trilinears    28*cos(A)-15*sin(A) : :
X(10142) = 28*S*X(3)-15*SW*X(6)

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)

Trilinears    209*cos(A)+120*sin(A) : :
X(10143) = 209*S*X(3)+120*SW*X(6)

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)

Trilinears    209*cos(A)-120*sin(A) : :
X(10144) = 209*S*X(3)-120*SW*X(6)

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)

Trilinears    29*cos(A)+16*sin(A) : :
X(10145) = 29*S*X(3)+16*SW*X(6)

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)

Trilinears    29*cos(A)-16*sin(A) : :
X(10146) = 29*S*X(3)-16*SW*X(6)

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)

Trilinears    20*cos(A)+9*sin(A) : :
X(10147) = 20*S*X(3)+9*SW*X(6)

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)

Trilinears    20*cos(A)-9*sin(A) : :
X(10148) = 20*S*X(3)-9*SW*X(6)

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)

Trilinears    2*a^6-2*(b^2+b*c+c^2)*a^4-(2* b-c)*(b-2*c)*(b+c)^2*a^2+(2*b^ 2+b*c+2*c^2)*(b^2-c^2)^2 : :
X(10149) = (R+r)*X(55)-r*X(186)

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)

Trilinears    (8*a^4-11*(b^2+c^2)*a^2+14*(b^ 4-b^2*c^2+c^4))/a : :
X(10150) = 7*X(2)-X(187)

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)

Trilinears    (SA-24*R^2+5*SW)*SB*SC*b*c : :
X(10151) = (4*R^2-SW)*X(3)+(14*R^2-3*SW)* X(4)

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

Trilinears    (S^2-2*SB*SC)/(SA*(S^2-3*SB* SC)*a) : :
Barycentrics    SB SC (-S^2 + 2 SB SC)/(S^2 - 3 SB SC) : :

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) = symgonal image 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)}}





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

This preamble and centers X(10153)-X(10200) were contributed by César Lozada, August 28, 2016.

Let ABC be the reference triangle and U, V, W three points, at least two of them distinct, and let

Ga = centroid of AVW, and define Gb and Gc cyclically
Gu = centroid of UBC, and define Gv and Gw cyclically.

Then the six centroids lie on a (possibly degenerate) conic, and the triangles GaGbGc and GuGvGw are congruent and homothetic, and their homothetic center is the center of the conic.

The conic is here named the UVW-centroidal conic and also the centroidal conic of UVW. The triangles GaGbGc and GuGvGw are the 1st and 2nd UVW-centroidal triangles and also the
1st and 2nd centroidal triangles of UVW.

The center O(UVW) of the centroidal-conic-of-UVW is a triangle center if U, V, W are either triangle centers or vertices of a central triangle.

If U, V, W have normalized barycentric coordinates

U = xu : yu : zu
V = xv : yv : zv
W = xw : yw : zw

then O(UVW) = xu + xv + xw + 1 : yu + yv + yw + 1 : zu + zv + zw + 1.

The triangles ABC and GaGbGc are perspective if and only if

(xu + xv)*(yv + yw)*(zw + zu) = (xu + xw)*(yv + yu)*(zw + zv).

The triangles ABC and GuGvGw are perspective if and only if

(xv + 1)*(yw + 1)*(zu + 1) = (xw + 1)*(yu + 1)*(zv + 1).

The appearance of (T,i,j,k) in the following list means that

X(i) = center of the T-centroidal conic
X(j) = perspector of ABC and 1st T-centroidal triangle
Xk) = perspector of ABC and 2nd T-centroidal triangle

The appearance of "np" means "not perspective".

(1st anti-Brocard, 2, 98, 8781) (4th anti-Brocard, 9172, 111, np) (anticomplementary,2,np,2) (anti-McCay, 2, 671, 10153) (Aquila, 10, 10, 10) (Ara, 10154, 22, 468)
(Aries, 154, np, np) (Artzt, 2, 7612, 10155) (Ascella, 10156, np, np) (Atik, 10157, np, np) (Ayme, 10158, np, np) (1st Brocard, 2, 83, 10159)
(2nd Brocard2 10160, np, np) (3rd Brocard, 10161, np, np) (4th Brocard, 10162, np, 2) (circummedial, 10163, np, 2) (circumorthic, 9730, 3, np) (1st circumperp, 10164, np, np)
(2nd circumperp, 10165, np, np) (circumsymmedial, 10166, np, np) (1st Conway, 10167, 84, np) (2nd Conway, 1699, 4, np) (1st Ehrmann, 10168, 3, np) (2nd Ehrmann, 10169, np, np)
(Euler, 5, 3091, 3090) (2nd Euler, 10170, np, np) (3rd Euler, 10171, np, np) (4th Euler, 10172, np, np) (5th Euler, 10173, np, 2) (excentral, 10164, np, np)
(extangents, 10174, np, np) (extouch, 3740, 9, np) (Fuhrmann, 10175, np, np) (inner-Garcia, 10176, 21, np) (outer-Garcia, 10, 8, 1698) (inner-Grebe, 1991, 1271, 3068)
(outer-Grebe, 591, 1270, 3069) (hexyl, 10165, 1, np) (Honsberger, 10177, 9, np) (Hutson-extouch, 10178, np, np) (Hutson-intouch, 10179, 1, np) (incentral, 10180, 37, np)
(intangents, 10181, np, np) (intouch, 3742, 1, np) (Johnson, 5, 4, 1656) (Kosnita, 10182, np, np) (Lemoine, 10183, 597, np) (Macbeath, 10184, 5, np)
(McCay, 2, 7608, 10185) (Medial, 2, 2, np) (midheight, 6688, np, np) (mixtilinear, 10186, np, np) (5th mixtilinear, 1, 145, 3616) (6th mixtilinear, 10164, 1, np)
(inner-Napoleon, 2, 17, 10187) (outer-Napoleon, 2, 18, 10188) (1st Neuberg, 2, 262, 7607) (2nd Neuberg, 2, 98, 7608) (orthic, 5943, 6, np) (orthocentroidal, 373, np, np)
(1st Parry, 9125, np, np) (2nd Parry, 9189, np, np) (reflection, 51, np, np) (Schroeter, 10189, np, np) (1st Sharygin, --, np, np) (2nd Sharygin, 4763, np, np)
(Steiner, 10190, 523, np) (symmedial, 10191, 39, np) (Trinh, 10193, np, np) (inner-Vecten, 2, 485, 10194) (outer-Vecten, 2, 486, 10195) (Yff-contact, 10196, 514, np)
(inner-Yff, 10197, 3085, 10198) (outer-Yff, 10199, 3086, 10200)

The centroidal conic of the Euler triangle is the circle with center X(5) and radius R/6.

underbar

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

Barycentrics    1/(4*a^4-7*(b^2+c^2)*a^2+7*b^4-4*b^2*c^2+7*c^4) : :

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

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

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(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

Barycentrics    1/(5*a^4-8*(b^2+c^2)*a^2+(b^2-3*c^2)*(3*b^2-c^2)) : :

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

Trilinears    (b+c)*a^4-2*(b^2-5*b*c+c^2)*a^3-8*b*c*(b+c)*a^2+2*(b^2-3*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2-6*b*c+c^2) : :

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

Trilinears    (b+c)*a^4-2*(b^2+b*c+c^2)*a^3+4*b*c*(b+c)*a^2+2*(b^2+3*b*c+c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2) : :

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

Trilinears    (b+c)*(a^4+a*b*c*(8*b+8*c+7*a)-b^4-c^4+(b^2+8*b*c+c^2)*b*c) : :

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

Trilinears    1/(3 sin A - cos A tan ω) : :
Trilinears    1/(cos A - 3 sin A cot ω) : :
Barycentrics    1/(2*a^2+b^2+c^2) : :

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

Barycentrics    4*a^6-5*(b^2+c^2)*a^4-2*(2*b^4+b^2*c^2+2*c^4)*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2) : :

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

Barycentrics    (3*b^4+4*b^2*c^2+3*c^4)*a^8+(b^2+c^2)*(2*b^4+3*b^2*c^2+2*c^4)*a^6+2*((b^2+c^2)^2-b^2*c^2)^2*a^4+b^2*c^2*(b^2+c^2)^2*(a^2*b^2+b^2*c^2+a^2*c^2) : :

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


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

Barycentrics    4*a^6-15*(b^2+c^2)*a^4-3*(3*b^4+2*b^2*c^2+3*c^4)*a^2+2*(b^2+c^2)*(5*b^4-11*b^2*c^2+5*c^4) : :

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

Barycentrics    4*a^6-3*(b^2+c^2)*a^4-(5*b^4+2*b^2*c^2+5*c^4)*a^2+2*((b^2-c^2)^2-b^2*c^2)*(b^2+c^2) : :

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

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

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

Let L be a line through X(2), and let Ia, Ib, Ic be the excenters. Let Ia' = orthogonal projection of Ia on L, and define Ib' and Ic' cyclically. Let P = X(2)-of-Ia'Ib'Ic'. As L varies, P traces a circle with center X(10164). (Randy Hutson, September 14, 2016)

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

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

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

Barycentrics    8*a^6-9*(b^2+c^2)*a^4-3*(2*b^4+b^2*c^2+2*c^4)*a^2+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2) : :

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

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

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

Barycentrics    4*a^4*(a^2-b^2-c^2)-(b^4+12*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^2-c^2)^2 : :

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

Barycentrics    4*a^8-5*(b^2+c^2)*a^6-5*(b^2-c^2)^2*a^4+5*(b^2-c^2)^2*a^2*(b^2+c^2)+(b^4-c^4)^2 : :

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

Trilinears    a*((b^2+c^2)*a^6-(3*b^4+4*b^2*c^2+3*c^4)*a^4+(b^2+c^2)*(3*b^4+4*b^2*c^2+3*c^4)*a^2-(b^4+6*b^2*c^2+c^4)*(b^2-c^2)^2) : :

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

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

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

Barycentrics    2*a^4+(b+c)*a^3-(7*b^2+2*b*c+7*c^2)*a^2-(b^2-c^2)*(b-c)*a+5*(b^2-c^2)^2 : :

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) = {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

Barycentrics    4*a^6-9*(b^2+c^2)*a^4-(7*b^4+4*b^2*c^2+7*c^4)*a^2+2*(b^2+c^2)*(3*b^4-7*b^2*c^2+3*c^4) : :

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

Trilinears    4*(2*q^2+1)*p^6-2*(4*q^2-5)*q*p^5-(10*q^2+7)*p^4+2*(7*q^2-8)*q*p^3-(2*q^4-4*q^2-3)*p^2+q*(-q^2+1)*(6*p-q) : : , where p=sin(A/2), q=cos((B-C)/2)

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


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

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

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) = {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

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

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

Trilinears    (b+c)*a^3-(3*b^2-2*b*c+3*c^2)*a^2+(b+c)*(3*b^2-8*b*c+3*c^2)*a-(b^2+c^2)*(b-c)^2 : :

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

Trilinears    (b+c)*a^4-2*(b^2-5*b*c+c^2)*a^3-6*b*c*(b+c)*a^2+2*(b^2-b*c+c^2)*(b-c)^2*a-(b^4-c^4)*(b-c) : :

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

Trilinears    (b+c)*(a^2-b^2-c^2)-10*a*b*c : :

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

Barycentrics    (b+c)*(3*a^2+2*(b+c)*a+b*c) : :
X(10180) = (4*R*r+r^2+5*s^2)*X(1)+(8*R*r+5*r^2+s^2)*X(333)

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'. (Randy Hutson, September 14, 2016)

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

Trilinears    (4*(2*q^2-3)*p^5-2*q*p^4-(16*q^2-19)*p^3-(2*q^2-3)*q*p^2+(q^2-1)*(7*p+q))/p : : , where p=sin(A/2), q=cos((B-C)/2)

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

Trilinears    (2*cos(2*A)+1)*cos(B-C)+cos(A)*cos(2*(B-C))-2*cos(A)-2*cos(3*A) : :

X(10182) = (15*R^2-4*SW)*X(3)+(9*R^2-2*SW)*X(113)

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

Barycentrics    (4*a^2+b^2+c^2)*(5*a^4+16*(b^2+c^2)*a^2+2*b^4+31*b^2*c^2+2*c^4) : :

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

Barycentrics    ((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^8-4*(b^2+c^2)*a^6+5*((b^2+c^2)^2-b^2*c^2)*a^4-2*(b^2-c^2)*(-c^4+b^4)*a^2-3*(b^2-c^2)^2*b^2*c^2) : :

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


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

Barycentrics    1/(4*a^4-9*(b^2+c^2)*a^2+5*b^4+5*c^4-8*b^2*c^2) : :

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


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

Barycentrics    a^5+3*(b+c)*a^4-6*(b^2+c^2)*a^3-2*b*c*(b+c)*a^2+(b^2-c^2)^2*a+(b^4-c^4)*(b-c) : :

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

Barycentrics    1/(SA-3*sqrt(3)*S) : :

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

Barycentrics    1/(SA+3*sqrt(3)*S) : :

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

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

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

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

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

Barycentrics    (b^2+c^2)*(a^2*(3*a^2+2*b^2+2*c^2)+b^2*c^2) : :

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

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

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

Barycentrics    (8*R^2-SA-3*SW)*S^2-(36*R^2-7*SW)*(SA-SW)*SA : :

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

Barycentrics    1/(SA-3*S) : :

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) = {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

Barycentrics    1/(SA+3*S) : :

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) = {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

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

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

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

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

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

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

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

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

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

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

Barycentrics    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+6 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-6 a^2 b^4 c^4+2 b^6 c^4+2 a^4 c^6+6 a^2 b^2 c^6+2 b^4 c^6-3 a^2 c^8-3 b^2 c^8+c^10 : :
X(10201) = 2 X[5] + X[26] = X[68] + 2 X[156] = X[4] + 2 X[1658] = 11 X[3525] - 8 X[5498] = 2 X[5449] + X[6759] = 5 X[1656] + X[7387] = 3 X[5055] + X[9909] = X[3] - 4 X[10020] = 5 X[631] - 8 X[10125]

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)

Barycentrics    a (a^5 b-a^4 b^2-2 a^3 b^3+2 a^2 b^4+a b^5-b^6+a^5 c+2 a^4 b c-4 a^2 b^3 c-a b^4 c+2 b^5 c-a^4 c^2+b^4 c^2-2 a^3 c^3-4 a^2 b c^3-4 b^3 c^3+2 a^2 c^4-a b c^4+b^2 c^4+a c^5+2 b c^5-c^6) : :
X(10202) = (2 R + r) X[1] + (R - r) X[3] = X[72] - 4 X[140] = X[3] + 2 X[942] = 2 X[5] + X[1071] = X[65] + 2 X[1385] = X[355] - 4 X[3812] = 5 X[631] + X[3868] = 11 X[3525] - 5 X[3876] = 7 X[3526] - 4 X[5044] = X[1482] - 4 X[5045] = 2 X[5] - 5 X[5439] = X[1071] + 5 X[5439] = X[3555] + 2 X[5690] = 7 X[3624] - X[5693] = 5 X[1656] - 2 X[5777] = X[382] - 4 X[5806] = 2 X[3754] + X[5882] = 2 X[1125] + X[5884] = X[65] - 4 X[5885] = X[1385] + 2 X[5885] = 4 X[1125] - X[5887] = 2 X[5884] + X[5887] = 5 X[5439] - X[5927] = X[3579] + 2 X[6583] = X[3874] + 2 X[6684] = 2 X[4292] + X[7491] = X[3] - 4 X[9940] = X[942] + 2 X[9940] = X[5787] + 2 X[9942] = 3 X[5054] - 4 X[10156] = 3 X[5055] - 2 X[10157]

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)

Barycentrics    a^2 (a^14-4 a^12 (b^2+c^2)-4 a^8 b^2 c^2 (b^2+c^2)-b^2 c^2 (b^2-c^2)^4 (b^2+c^2)+a^10 (5 b^4+8 b^2 c^2+5 c^4)-a^2 (b^2-c^2)^2 (b^8-5 b^6 c^2-7 b^4 c^4-5 b^2 c^6+c^8)-5 a^6 (b^8-b^6 c^2-b^4 c^4-b^2 c^6+c^8)+a^4 (4 b^10-11 b^8 c^2-5 b^6 c^4-5 b^4 c^6-11 b^2 c^8+4 c^10)) : :
X(10203) = 4R2X(3) - |OH|2X(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:
(pending)


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

Barycentrics    a^2 (a^8-b^8+12 a^4 b^2 c^2-7 b^6 c^2+24 b^4 c^4-7 b^2 c^6-c^8-4 a^6 (b^2+c^2)+a^2 (4 b^6+3 b^4 c^2+3 b^2 c^4+4 c^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

Barycentrics    (2 a^16-13 a^14 (b^2+c^2)-(b^2-c^2)^6 (b^4+c^4)+a^12 (37 b^4+50 b^2 c^2+37 c^4)+3 a^2 (b^2-c^2)^4 (b^6+b^4 c^2+b^2 c^4+c^6)-a^10 (59 b^6+71 b^4 c^2+71 b^2 c^4+59 c^6)+a^4 (b^2-c^2)^2 (3 b^8-4 b^6 c^2-7 b^4 c^4-4 b^2 c^6+3 c^8)+a^8 (55 b^8+34 b^6 c^2+32 b^4 c^4+34 b^2 c^6+55 c^8)+a^6 (-27 b^10+13 b^8 c^2+5 b^6 c^4+5 b^4 c^6+13 b^2 c^8-27 c^10) : :

In the plane of a triangle ABC, let
N = X(5)
X(10205) = X(5)-of-antipedial-triangle of X(5)-of-[triangle, pending]; 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(10206) =  HUNG-MONTESDEOCA RADICAL CENTER

Barycentrics    a (a^2 (b+c)+2 a b c-(b-c)^2 (b+c)) (a^6-2 a^5 (b+c)-a^4 (b^2+3 b c+c^2)+4 a^3 (b^3+b^2 c+b c^2+c^3)-a^2 (b+c)^2 (b^2-6 b c+c^2)-2 a (b-c)^2 (b+c)^3+(b^2-c^2)^2 (b^2-b c+c^2)) : :
X(10206) = 8(r + 2R)2X(942) - (3r2 + 8rR + 4R2 - s2)X(1838)
X(10206) = 3[(r + 2R)2 - s2]Go + 2s2Ho (see below)

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

Barycentrics    a (2 a^7 (b^2+b c+c^2) +7 a^6 b c (b+c)-2 a^5 (3 b^4+3 b^3 c-b^2 c^2+3 b c^3+3 c^4) -a^4 b c (15 b^3+17 b^2 c+17 b c^2+15 c^3)+2 a^3 (b+c)^2 (3 b^4-3 b^3 c-4 b^2 c^2-3 bc^3+3 c^4)+9 a^2 b (b-c)^2 c (b+c)^3-2 a (b^2-c^2)^2 (b^4+b^3 c-3 b^2 c^2+bc^3+c^4)-b (b-c)^4 c (b+c)^3) : :

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

Barycentrics    (2 a^7 (b+c)^3-(b-c)^4 (b+c)^6-2 a (b-c)^4 (b+c)^3 (b^2+3 b c+c^2)+a^8 (b^2+6 b c+c^2)-2 a^5 (b+c)^3 (3 b^2-b c+3 c^2)+2 a^3 (b-c)^2 (b+c)^3 (3 b^2+4 b c+3 c^2)-a^4 b^2 c^2 (11 b^2+18 b c+11 c^2)-2 a^6 (b^4+5 b^3 c+4 b^2 c^2+5 b c^3+c^4)+a^2 (b^2-c^2)^2 (2 b^4+6 b^3 c+19 b^2 c^2+6 b c^3+2 c^4) : :
X(10208) = 3 X[5947] - 2 X[5953]

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

Barycentrics    a^11 (b-c)^2-(b-c)^6 (b+c)^7-a (b-c)^4 (b+c)^6 (b^2-7 b c+c^2)+a^10 (b^3-5 b^2 c-5 b c^2+c^3)-a^9 (5 b^4+5 b^3 c+12 b^2 c^2+5 b c^3+5 c^4)+a^8 (-5 b^5+b^4 c+b c^4-5 c^5)+a^3 (b^2-c^2)^2 (5 b^6-4 b^5 c-42 b^4 c^2-59 b^3 c^3-42 b^2 c^4-4 b c^5+5 c^6)+a^2 (b-c)^2 (b+c)^3 (5 b^6+12 b^5 c-8 b^4 c^2-26 b^3 c^3-8 b^2 c^4+12 b c^5+5 c^6)+a^7 (10 b^6+22 b^5 c+25 b^4 c^2+14 b^3 c^3+25 b^2 c^4+22 b c^5+10 c^6)+a^6 (10 b^7+28 b^6 c+45 b^5 c^2+33 b^4 c^3+33 b^3 c^4+45 b^2 c^5+28 b c^6+10 c^7)+a^5 (-10 b^8-16 b^7 c+22 b^6 c^2+57 b^5 c^3+54 b^4 c^4+57 b^3 c^5+22 b^2 c^6-16 b c^7-10 c^8)-2 a^4 (5 b^9+20 b^8 c+20 b^7 c^2-14 b^6 c^3-41 b^5 c^4-41 b^4 c^5-14 b^3 c^6+20 b^2 c^7+20 b c^8+5 c^9) : :
X(10209) = 2 X[442] - 3 X[5947]

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

Barycentrics    sqrt(3)*((-R^2+SW)*S^2+(SA-SW) *SA*SW)+S*SB*SC : :
Barycentrics    sqrt(3)*(a^2*(a^4-b^4-c^4)+(b^ 2+c^2)*(a^4-(b^2-c^2)^2))+2*S* (a^2-b^2+c^2)*(a^2+b^2-c^2) : :
X(10210) = -6*R^2*X(3)+(-3*R^2+sqrt(3)*S+ 3*SW)*X(4) = [-(3/4)*E, 3*E+3*F+sqrt(3)*S]

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(10211) =  KIRIKAMI-KOSNITA-EULER POINT

Barycentrics    a^28 - 9 a^26 (b^2 + c^2) + (b^2 - c^2)^10 (b^2 + c^2)^2 (b^4 - b^2 c^2 + c^4) + 9 a^24 (4 b^4 + 7 b^2 c^2 + 4 c^4) - a^22 (83 b^6 + 187 b^4 c^2 + 187 b^2 c^4 + 83 c^6) + a^20 (116 b^8 + 298 b^6 c^2 + 387 b^4 c^4 + 298 b^2 c^6 + 116 c^8) - 2 a^2 (b^2 - c^2)^8 (4 b^10 + 5 b^8 c^2 + b^6 c^4 + b^4 c^6 + 5 b^2 c^8 + 4 c^10) + 2 a^14 (b^2 - c^2)^2 (69 b^10 + 104 b^8 c^2 + 127 b^6 c^4 + 127 b^4 c^6 + 104 b^2 c^8 + 69 c^10) - a^18 (82 b^10 + 258 b^8 c^2 + 389 b^6 c^4 + 389 b^4 c^6 + 258 b^2 c^8 + 82 c^10) + a^16 (-27 b^12 + 107 b^10 c^2 + 178 b^8 c^4 + 213 b^6 c^6 + 178 b^4 c^8 + 107 b^2 c^10 - 27 c^12) + a^4 (b^2 - c^2)^6 (26 b^12 + 36 b^10 c^2 + 9 b^8 c^4 - 3 b^6 c^6 + 9 b^4 c^8 + 36 b^2 c^10 + 26 c^12) - a^12 (b^2 - c^2)^2 (159 b^12 + 120 b^10 c^2 + 133 b^8 c^4 + 139 b^6 c^6 + 133 b^4 c^8 + 120 b^2 c^10 + 159 c^12) - a^6 (b^2 - c^2)^4 (39 b^14 + 69 b^12 c^2 + 5 b^10 c^4 - 13 b^8 c^6 - 13 b^6 c^8 + 5 b^4 c^10 + 69 b^2 c^12 + 39 c^14) + a^10 (b^2 - c^2)^2 (83 b^14 - 93 b^12 c^2 - 56 b^10 c^4 - 58 b^8 c^6 - 58 b^6 c^8 - 56 b^4 c^10 - 93 b^2 c^12 + 83 c^14) + a^8 (b^2 - c^2)^2 (6 b^16 + 115 b^14 c^2 - 47 b^12 c^4 - 16 b^10 c^6 - 20 b^8 c^8 - 16 b^6 c^10 - 47 b^4 c^12 + 115 b^2 c^14 + 6 c^16) : :

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 these lines: (pending)


X(10212) =  35th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    -6 a^10 + 13 a^8 (b^2 + c^2) - (b^2 - c^2)^4 (b^2 + c^2) - 2 a^6 (b^4 + 13 b^2 c^2 + c^4) + a^2 (b^2 - c^2)^2 (8 b^4 + 13 b^2 c^2 + 8 c^4) + a^4 (-12 b^6 + 13 b^4 c^2 + 13 b^2 c^4 - 12 c^6) : :

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} (more pending)

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


X(10213) =  36th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    (a^2-b^2-c^2) (2 a^20-11 a^18 (b^2+c^2)-(b^2-c^2)^8 (b^4+b^2 c^2+c^4)+a^16 (25 b^4+42 b^2 c^2+25 c^4)+a^2 (b^2-c^2)^6 (8 b^6+7 b^4 c^2+7 b^2 c^4+8 c^6)-a^14 (29 b^6+61 b^4 c^2+61 b^2 c^4+29 c^6)+6 a^12 (2 b^8+7 b^6 c^2+9 b^4 c^4+7 b^2 c^6+2 c^8)-a^4 (b^2-c^2)^4 (26 b^8+12 b^6 c^2+13 b^4 c^4+12 b^2 c^6+26 c^8)-a^8 (b^2-c^2)^2 (44 b^8+31 b^6 c^2+35 b^4 c^4+31 b^2 c^6+44 c^8)+a^10 (19 b^10-26 b^8 c^2-20 b^6 c^4-20 b^4 c^6-26 b^2 c^8+19 c^10)+a^6 (b^2-c^2)^2 (45 b^10-11 b^8 c^2+9 b^6 c^4+9 b^4 c^6-11 b^2 c^8+45 c^10)) : :

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: (pending)


X(10214) =  HUNG-LOZADA CYCLOLOGIC CENTER

Trilinears    (2*cos(2*A)*cos(B-C)-cos(3*A)) *((cos(2*A)+cos(4*A)+1)*cos(B- C)+(-cos(A)-cos(3*A))*cos(2*( B-C))-cos(A))*sec(A) : :

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

Trilinears    1 / (1 - 2*sin(A/2)) : :
Trilinears    [1 + 2 sin(A/2)]/(1 - 2 cos A) : :

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

Trilinears    (-1 + 2 cos 2A) cos3(B - C) : :

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} (more pending)


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

Trilinears    csc2(A + π/3) cos A : :

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

Trilinears    csc2(A - π/3) cos A : :

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

Trilinears    a ((b^2+c^2)*a^2-b^4+16*b^2*c^2- c^4) : :
X(10219) = 7X(2) + X(51)

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

Trilinears    (6*a^8-4*(b^2+c^2)*a^6-3*(3*b^ 4+4*b^2*c^2+3*c^4)*a^4+6*(b^2- c^2)^2*(b^2+c^2)*a^2+(b^4+c^4) *(b^2-c^2)^2)/a : :

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 these lines: (pending)


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

Barycentrics    a2/3b2/3c2/3SBSC - a2SA(SASBSC)1/3 : :
X(10221) = -2(SASBSC)1/3*X(3) + a2/3b2/3c2/3*X(4)

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} (others pending)


X(10222) =  CENTER OF HATZIPOLAKIS-LOZADA CIRCLE

Trilinear    2*a^3-3*(b+c)*a^2-2*(b^2-3*b* c+c^2)*a+3*(b^2-c^2)*(b-c) : :
X(10222) = 3X(1) - X(3)

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).

X(10222) lies on these lines:
{1, 3}, {4, 1392}, {5, 519}, {8, 3090}, {10, 3628}, {20, 3655}, {30, 4301}, {72, 1173}, {140, 551}, {145, 355}, {381, 5881}, {392, 5047}, {515, 1483}, {518, 576}, {546, 946}, {547, 4669}, {548, 5493}, {573, 3723}, {575, 1386}, {631, 3654}, {632, 1125}, {944, 3146}, {956, 3951}, {962, 3529}, {1000, 5703}, {1056, 4323}, {1058, 4345}, {1210, 1387}, {1320, 1389}, {1339, 6048}, {1457, 5399}, {1656, 3679}, {1657, 9589}, {1837, 7743}, {1870, 1872}, {2771, 7984}, {2800, 3881}, {3058, 7491}, {3419, 6984}, {3485, 6982}, {3488, 5812}, {3523, 3653}, {3525, 3616}, {3555, 5887}, {3584, 5559}, {3585, 7972}, {3621, 5818}, {3622, 5657}, {3632, 5079}, {3633, 5072}, {3636, 6684}, {3680, 6918}, {3872, 3984}, {3892, 5884}, {3913, 6911}, {3915, 5398}, {3940, 4853}, {3962, 5288}, {3991, 4919}, {4004, 5253}, {4511, 6946}, {4677, 5055}, {4870, 6980}, {4902, 5059}, {4930, 6913}, {5044, 5289}, {5054, 9588}, {5076, 5691}, {5258, 7489}, {5722, 5761}, {5727, 9669}, {6419, 7969}, {6420, 7968}, {6447, 9583}, {6519, 9616}, {6863, 10056}, {6914, 8666}, {6924, 8715}, {6958, 10072}, {6988, 7320}

X(10222) = midpoint of X(i) and X(j) for these {i,j}: {1, 1482}, {3, 7982}, {40, 8148}, {145, 355}, {381, 2487}, {946, 3244}, {1320, 6265}, {1657, 9589}, {3241, 3656), (3555, 5887}, {4301, 5882}
X(10222) = reflection of X(i) in X(j) for these (i,j): (8, 9956}, {10, 5901), (65, 6583), (355, 9955), (1385, 1), (1483, 3635), (3579, 1385), (4669, 547), (5493, 548), (5690, 1125), (6684, 3636)


X(10223) =  37th HATZIPOLAKIS-MONTESDEOCA POINT

Barycentrics    2 a^14 b^2-9 a^12 b^4+15 a^10 b^6-10 a^8 b^8+3 a^4 b^12-a^2 b^14+2 a^14 c^2-6 a^12 b^2 c^2+5 a^10 b^4 c^2-3 a^8 b^6 c^2+10 a^6 b^8 c^2-14 a^4 b^10 c^2+7 a^2 b^12 c^2-b^14 c^2-9 a^12 c^4+5 a^10 b^2 c^4+2 a^8 b^4 c^4-8 a^6 b^6 c^4+19 a^4 b^8 c^4-15 a^2 b^10 c^4+6 b^12 c^4+15 a^10 c^6-3 a^8 b^2 c^6-8 a^6 b^4 c^6-16 a^4 b^6 c^6+9 a^2 b^8 c^6-15 b^10 c^6-10 a^8 c^8+10 a^6 b^2 c^8+19 a^4 b^4 c^8+9 a^2 b^6 c^8+20 b^8 c^8-14 a^4 b^2 c^10-15 a^2 b^4 c^10-15 b^6 c^10+3 a^4 c^12+7 a^2 b^2 c^12+6 b^4 c^12-a^2 c^14-b^2 c^14 : :

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)

Trilinears    (cos(2*A)+1/2)*cos(B-C)-cos(A) *cos(2*(B-C)) : :
X(10224) = (9*R^2-2*SW)*X(3)+(7*R^2-2*SW) *X(4)

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)

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(10225) =  MIDPOINT OF X(3) AND X(484)

Trilinears    (2*sin(A/2)+sin(3*A/2))*cos(( B-C)/2)+(cos(A)-1)*cos(B-C)- cos(A)+cos(2*A)-1/2 : :

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

Barycentrics    a^2 (2 a^8-4 a^6 b^2+4 a^2 b^6-2 b^8-4 a^6 c^2+10 a^4 b^2 c^2-5 a^2 b^4 c^2-b^6 c^2-5 a^2 b^2 c^4+6 b^4 c^4+4 a^2 c^6-b^2 c^6-2 c^8) : :
X(10226) = 5 X[3] - X[26] = 3 X[26] - 5 X[1658] = 3 X[3] - X[1658] = 9 X[26] - 5 X[7387] = 9 X[3] - X[7387] = 3 X[1658] - X[7387] = 19 X[26] - 15 X[9909] = 19 X[1658] - 9 X[9909] = 19 X[3] - 3 X[9909] = 3 X[549] - 2 X[10125] = X[5449] - 3 X[10193] = 7 X[3523] - 3 X[10201] = 3 X[549] - 4 X[10212]

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

Barycentrics    2 a^28 -19 a^26 (b^2+c^2)+a^24 (77 b^4+142 b^2 c^2+77 c^4) -2 a^22 (83 b^6+215 b^4 c^2+215 b^2 c^4+83 c^6)+4 a^20 (44 b^8+161 b^6 c^2+221 b^4 c^4+161 b^2 c^6+44 c^8)+a^18 (11 b^10-421 b^8 c^2-744 b^6 c^4-744 b^4 c^6-421 b^2 c^8+11 c^10)+a^16 (-297 b^12-22 b^10 c^2+91 b^8 c^4+144 b^6 c^6+91 b^4 c^8-22 b^2 c^10-297 c^12)+2 a^14 (198 b^14+54 b^12 c^2+95 b^10 c^4+75 b^8 c^6+75 b^6 c^8+95 b^4 c^10+54 b^2 c^12+198 c^14)-2 a^12 (99 b^16-20 b^14 c^2+41 b^12 c^4-6 b^10 c^6+3 b^8 c^8-6 b^6 c^10+41 b^4 c^12-20 b^2 c^14+99 c^16)-a^10 (77 b^18-131 b^16 c^2+82 b^14 c^4+42 b^12 c^6+29 b^10 c^8+29 b^8 c^10+42 b^6 c^12+82 b^4 c^14-131 b^2 c^16+77 c^18)+a^8 (b^2-c^2)^2 (187 b^16-120 b^14 c^2+82 b^12 c^4+56 b^10 c^6+87 b^8 c^8+56 b^6 c^10+82 b^4 c^12-120 b^2 c^14+187 c^16)-2 a^6 (b^2-c^2)^4 (67 b^14-5 b^12 c^2+26 b^10 c^4+22 b^8 c^6+22 b^6 c^8+26 b^4 c^10-5 b^2 c^12+67 c^14)+2 a^4 (b^2-c^2)^6 (26 b^12+6 b^10 c^2+5 b^8 c^4+5 b^4 c^8+6 b^2 c^10+26 c^12)-a^2 (b^2-c^2)^8 (11 b^10+3 b^8 c^2-6 b^6 c^4-6 b^4 c^6+3 b^2 c^8+11 c^10)+(b^2-c^2)^12 (b^2+c^2)^2 ) ) : :

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: (pending)


X(10228) =  HUNG-MONTESDEOCA CENTER OF SIMILITUDE

Barycentrics   

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 these lines: (pending)


X(10229) =  1st SCHMIDT CYCLOLOGIC CENTER

Barycentrics    [a^2(-SA^2 S^2 + (S^2 - 2 SA SB)(S^2 - 2 SA SC))]/[(S^2 - 2 SB SC)^2 + SA^2 (S^2 - 4 SB SC)]

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: (pending)


X(10230) =  POINT BECRUX 5

Trilinears    [1 + 2 cos(A/2)]/(1 + 2 cos A) : :
Trilinears    cos(A/4)/cos(3A/4) : :
Barycentrics    (sin A)/(1 - 2 cos A/2) : :

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)

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

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)

Trilinears    [1 + 2 cos A ]/(1 + 2 cos A/2) : :
Trilinears    cos(3A/4)/cos(A/4) : :
Barycentrics    (sin A)(1 - 2 cos A/2) : :

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)

Barycentrics    (sin A)(1 + 2 csc A/2) : :

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)

Barycentrics    (sin A)(1 - 2 csc A/2) : :

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

Barycentrics    (sin A)(1 + 2 sec A/2) : :

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

Barycentrics    (sin A)(1 - 2 sec A/2) : :

X(10236) lies on these lines: {1,167}, {481,9795}

X(10236) = {X(1),X(174)}-harmonic conjugate of X(10235)



leftri  Eulerologic centers: X(10237) - X(10259)  rightri

This preamble and centers X(10237)-X(10259) were contributed by César Lozada, October 2, 2016.

Let T'= A'B'C' and T" = A"B"C" be triangles. If the Euler lines of A'B"C", B'C"A", C'A"B" concur, then the triangles T’ and T" are (T', T")-eulerologic and the point of concurrence is here named the (T', T")-eulerologic center. (Definitions given by Antreas Hatzipolakis in 3841).

Note that the existence of the (T',T")-eulerologic center does not imply the existence of a (T",T')-eulerologic center.

Clearly, if two triangles have the same circumcircle then they are mutually eulerologic. Examples:
(1) The following triangles are inscribed in the circumcircle of ABC, so that each pair are mutually eulerologic: ABC, circummedial, circumorthic, 1st circumperp, 2nd circumperp, circumsymmedial, 3rd mixtilinear, 4th mixtilinear. The eulerologic center of each pair is X(3).
(2) The following triangles are inscribed in the nine-point circle of ABC, so that each pair are mutually eulerologic: Euler, 2nd Euler, 3rd Euler, 4th Euler, 5th Euler, Feuerbach, medial, orthic. The eulerologic center of each pair is X(5).

The appearance of (T', T", n) in the following list means that X(n) is the (T',T")-eulerologic center. A question mark, ? , indicates an unspecified eulerologic center

(ABC, 1st anti-Brocard, 98) (ABC, anti-McCay, 671) (ABC, outer-Garcia, 2475) (ABC, hexyl, 1)
(ABC, Hutson intouch, 1) (ABC, intouch, 1) (ABC, Johnson, 4) (ABC, Lucas tangents, 10237)
(ABC, Lucas(-1) tangents, 10239) (ABC, medial, 30) (ABC, 6th mixtilinear, 1) (ABC, 1st Morley, 5390)
(ABC, 2nd Morley, 10258) (ABC, 3rd Morley, 10259) (ABC, orthic, 125) (Andromeda, hexyl, 1)
(Andromeda, Hutson intouch, 1) (Andromeda, intouch, 1) (Andromeda, 6th mixtilinear, 1) (anticomplementary, ABC, 30)
(anticomplementary, Atik, 10241) (Antlia, hexyl, 1) (Antlia, Hutson intouch, 1) (Antlia, intouch, 1)
(Antlia, 6th mixtilinear, 1) (Aquila, 2nd circumperp, 517) (Aquila, excentral, 40) (Aquila, outer-Garcia, 30)
(Aquila, hexyl, 1) (Aquila, Hutson intouch, 1) (Aquila, intouch, 1) (Aquila, 6th mixtilinear, 1)
(Ara, ABC, 3) (Ara, tangential, 10117) (Ascella, ABC, 3) (BCI, midarc, 1)
(1st Brocard, ABC, 3) (1st Brocard, 2nd Brocard, 182) (2nd Brocard, 1st Brocard, 182) (4th Brocard, orthocentroidal, 381)
(circumorthic, anticomplementary, 4) (circumorthic, Euler, 30) (circumorthic, Johnson, 4) (1st circumperp, extouch, 1158)
(2nd circumperp, Hutson intouch, 1) (2nd circumperp, intouch, 1) (2nd circumperp, 6th mixtilinear, 1) (2nd Conway, anticomplementary, 149)
(2nd Conway, 6th mixtilinear, 2) (1st Ehrmann, ABC, 3) (2nd Ehrmann, Trinh, 10249) (Euler, anticomplementary, 4)
(Euler, 2nd Euler, 5) (Euler, 3rd Euler, 5) (Euler, 4th Euler, 5) (Euler, 5th Euler, 5)
(Euler, Feuerbach, 5) (Euler, medial, 5) (Euler, orthic, 5) (2nd Euler, ABC, 3)
(2nd Euler, Ara, 10243) (2nd Euler, Euler, 5) (2nd Euler, 3rd Euler, 5) (2nd Euler, 4th Euler, 5)
(2nd Euler, 5th Euler, 5) (2nd Euler, Feuerbach, 5) (2nd Euler, medial, 5) (2nd Euler, orthic, 5)
(3rd Euler, Euler, 5) (3rd Euler, 2nd Euler, 5) (3rd Euler, 4th Euler, 5) (3rd Euler, Feuerbach, 5)
(3rd Euler, medial, 5) (3rd Euler, orthic, 5) (4th Euler, Euler, 5) (4th Euler, 2nd Euler, 5)
(4th Euler, 3rd Euler, 5) (4th Euler, 5th Euler, 5) (4th Euler, Feuerbach, 5) (4th Euler, medial, 5)
(4th Euler, orthic, 5) (5th Euler, Euler, 5) (5th Euler, 2nd Euler, 5) (5th Euler, 4th Euler, 5)
(5th Euler, Feuerbach, 5) (5th Euler, medial, 5) (5th Euler, orthic, 5) (excentral, ABC, 100)
(excentral, Aquila, 40) (excentral, 1st circumperp, 517) (excentral, hexyl, 1) (excentral, Hutson intouch, 1)
(excentral, intouch, 1) (extangents, Aquila, 40) (extangents, outer-Garcia, 10251) (extouch, Aquila, 40)
(2nd extouch, anticomplementary, 4) (2nd extouch, Johnson, 4) (3rd extouch, anticomplementary, 4) (3rd extouch, Johnson, 4)
(Feuerbach, Euler, 5) (Feuerbach, 2nd Euler, 5) (Feuerbach, 3rd Euler, 5) (Feuerbach, 4th Euler, 5)
(Feuerbach, 5th Euler, 5) (Feuerbach, medial, 5) (Feuerbach, orthic, 5) (Fuhrmann, ABC, 3)
(inner-Garcia, Aquila, 40) (outer-Garcia, ABC, 191) (outer-Garcia, 2nd Conway, 10248) (outer-Garcia, Johnson, 5587)
(outer-Garcia, 5th mixtilinear, 1) (hexyl, Aquila, 484) (hexyl, 2nd circumperp, 517) (hexyl, excentral, 40)
(Hutson extouch, Aquila, 40) (Hutson intouch, intouch, 1) (Hutson intouch, intouch, 1) (incentral, hexyl, 1)
(incentral, Hutson intouch, 1) (incentral, intouch, 1) (incentral, 6th mixtilinear, 1) (intouch, Hutson intouch, 1)
(intouch, Hutson intouch, 1) (intouch, midarc, 1) (intouch, inner-Soddy, 10252) (intouch, outer-Soddy, 10253)
(Johnson, ABC, 3) (Johnson, circumorthic, 186) (Johnson, Euler, 30) (Johnson, 2nd Euler, 113)
(Johnson, outer-Garcia, 5790) (Johnson, 5th mixtilinear, 10247) (Kosnita, ABC, 3) (Kosnita, Ara, 10244)
(Kosnita, Johnson, 10254) (Kosnita, medial, 7542) (Lucas antipodal, circumorthic, 3) (Lucas central, circumorthic, 3)
(Lucas central, Lucas tangents, 1151) (Lucas inner, Lucas tangents, ?) (Lucas inner tangential, Lucas inner, 6407) (Lucas tangents, ABC, 10238)
(Lucas tangents, Lucas inner, ?) (Lucas(-1) antipodal, circumorthic, 3) (Lucas(-1) central, circumorthic, 3) (Lucas(-1) central, Lucas(-1) tangents, 1152)
(Lucas(-1) inner, Lucas(-1) tangents, ?) (Lucas(-1) inner tangential, Lucas(-1) inner, 6408) (Lucas(-1) tangents, ABC, 10240) (Lucas(-1) tangents, Lucas(-1) inner, ?)
(McCay, ABC, 3) (McCay, Artzt, 7610) (McCay, 2nd Brocard, 182) (medial, Euler, 5)
(medial, 2nd Euler, 5) (medial, 3rd Euler, 5) (medial, 4th Euler, 5) (medial, 5th Euler, 5)
(medial, Feuerbach, 5) (medial, orthic, 5) (medial, inner-Vecten, 642) (medial, outer-Vecten, 641)
(midarc, hexyl, 1) (midarc, intouch, 1) (midarc, 6th mixtilinear, 1) (midheight, anticomplementary, 4)
(midheight, Johnson, 4) (mixtilinear, hexyl, 1) (mixtilinear, Hutson intouch, 1) (mixtilinear, intouch, 1)
(mixtilinear, 6th mixtilinear, 1) (2nd mixtilinear, hexyl, 1) (2nd mixtilinear, Hutson intouch, 1) (2nd mixtilinear, intouch, 1)
(5th mixtilinear, 2nd circumperp, 10246) (5th mixtilinear, excentral, 7987) (5th mixtilinear, outer-Garcia, 2) (5th mixtilinear, hexyl, 1)
(5th mixtilinear, Hutson intouch, 1) (5th mixtilinear, intouch, 1) (5th mixtilinear, Johnson, 5603) (5th mixtilinear, 6th mixtilinear, 1)
(6th mixtilinear, 2nd Conway, 1699) (6th mixtilinear, excentral, 517) (1st Morley, 2nd Morley, ?) (1st Morley, 3rd Morley, ?)
(1st Morley, Russel, ?) (1st Morley, Stammler, ?) (2nd Morley, 1st Morley, ?) (2nd Morley, 3rd Morley, ?)
(2nd Morley, Russel, ?) (2nd Morley, Stammler, ?) (3rd Morley, 1st Morley, ?) (3rd Morley, 2nd Morley, ?)
(3rd Morley, Russel, ?) (3rd Morley, Stammler, ?) (1st Morley-adjunct, 1st Morley, 356) (2nd Morley-adjunct, 2nd Morley, 3276)
(3rd Morley-adjunct, 3rd Morley, 3277) (inner-Napoleon, ABC, 3) (outer-Napoleon, ABC, 3) (1st Neuberg, ABC, 3)
(1st Neuberg, 1st anti-Brocard, 98) (1st Neuberg, medial, 10256) (2nd Neuberg, ABC, 3) (2nd Neuberg, anticomplementary, 10242)
(orthic, anticomplementary, 4) (orthic, Euler, 5) (orthic, 2nd Euler, 5) (orthic, 3rd Euler, 5)
(orthic, 4th Euler, 5) (orthic, 5th Euler, 5) (orthic, Feuerbach, 5) (orthic, Johnson, 4)
(orthic, medial, 5) (orthocentroidal, anticomplementary, 4) (orthocentroidal, 4th Brocard, 381) (orthocentroidal, Johnson, 4)
(orthocentroidal, inner-Napoleon, 381) (orthocentroidal, outer-Napoleon, 381) (1st Parry, 2nd Parry, 351) (1st Parry, 3rd Parry, 351)
(2nd Parry, 1st Parry, 351) (2nd Parry, 3rd Parry, 351) (3rd Parry, 1st Parry, 351) (3rd Parry, 2nd Parry, 351)
(reflection, ABC, 110) (reflection, anticomplementary, 4) (reflection, Johnson, 4) (Russel, 1st Morley, ?)
(Russel, 2nd Morley, ?) (Russel, 3rd Morley, ?) (Russel, Stammler, ?) (inner-Soddy, intouch, ?)
(outer-Soddy, intouch, ?) (Stammler, 1st Morley, ?) (Stammler, 2nd Morley, ?) (Stammler, 3rd Morley, ?)
(Stammler, Russel, ?) (tangential, ABC, 3) (tangential, Johnson, 10024) (tangential, medial, 6676)
(tangential, reflection, 195) (tangential-midarc, midarc, 1) (2nd tangential-midarc, midarc, 1) (Trinh, ABC, 3)
(Trinh, Ara, 10245) (Trinh, 2nd Ehrmann, 10250) (Trinh, Johnson, 10255) (Trinh, medial, 10257)
(inner-Vecten, ABC, 3) (outer-Vecten, ABC, 3) (Yiu, reflection, 195)

underbar

X(10237) = (ABC, LUCAS TANGENTS)-EULEROLOGIC CENTER

Trilinears    a*((SA+SW)*S^2+2*(R^2*SA+SB*SC)*S-(SB+SC)*SA*SW) : :
X(10237) = (-4*s^4+20*R^2*S+8*R^2*SW+8*R*S*s+4*SW*s^2+3*S^2-4*S*SW-2*SW^2)*X(3)+4*R^2*(2*S+SW)*X(4)

X(10237) lies on these lines:
{2,3}, {371,8825}, {8939,9922}


X(10238) = (LUCAS TANGENTS, ABC)-EULEROLOGIC CENTER

Trilinears    (SA*SW^3+S^2*(2*SA^2+(8*R^2+7*SW)*SA+2*R^2*SW+2*S^2)+S*((4*R^2+6*SA+SW)*S^2+(4*R^2+SA+4*SW)*SA*SW))*a : :

X(10238) lies on these lines:
{3,485}, {1151,8155}


X(10239) = (ABC, LUCAS(-1) TANGENTS)-EULEROLOGIC CENTER

Trilinears    a*((SA+SW)*S^2-2*(R^2*SA+SB*SC)*S-(SB+SC)*SA*SW) : :

X(10239) lies on these lines:
{2,3}, {8943,9921}


X(10240) = (LUCAS(-1) TANGENTS, ABC)-EULEROLOGIC CENTER

Trilinears    (SA*SW^3+S^2*(2*SA^2+(8*R^2+7*SW)*SA+2*R^2*SW+2*S^2)-S*((4*R^2+6*SA+SW)*S^2+(4*R^2+SA+4*SW)*SA*SW))*a : :

X(10240) lies on these lines:
{3,486}, {1152,8156}


X(10241) = (ANTICOMPLEMENTARY, ATIK)-EULEROLOGIC CENTER

Trilinears    (b+c)*a^7-3*(b^2+c^2)*a^6+(b+c)*(b^2+4*b*c+c^2)*a^5+(5*b^2-2*b*c+5*c^2)*(b-c)^2*a^4-5*(b^2-c^2)^2*(b+c)*a^3-(b^4+c^4-2*(7*b^2+19*b*c+7*c^2)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*a*(3*b^4+3*c^4+2*(b^2-21*b*c+c^2)*b*c)-(b^2-c^2)^2*(b-c)^2*(b^2+6*b*c+c^2) : :

X(10241) lies on these lines:
{4,8}, {57,3062}, {971,7956}, {1538,5249}, {1699,8581}, {6244,8580}, {7682,9948}

X(10241) = X(1368)-of-Atik-triangl
X(10241) = excentral-to-Atik similarity image of X(6244)

X(10242) = (2ND NEUBERG, ANTICOMPLEMENTARY)-EULEROLOGIC CENTER

Barycentrics    (21*S^2-SW^2)*SA^2-(23*S^2-SW^2)*SW*SA+2*(10*S^2-SW^2)*S^2 : :

X(10242) lies on these lines:
{4,2896}, {316,6321}, {1656,6781}, {8597,8724}


X(10243) = (2ND EULER, ARA)-EULEROLOGIC CENTER

Trilinears    4*(4*cos(2*A)-cos(4*A)-3)*cos(B-C)+(6*cos(A)+2*cos(3*A))*cos(2*(B-C))-9*cos(3*A)+cos(5*A)+32*cos(A) : :
X(10243) = (16*R^4-4*s^4+8*R*S*s+4*SW*s^2+S^2-2*SW^2)*X(3)+16*R^4*X(4)

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

Trilinears    2*(7*cos(2*A)-1)*cos(B-C)+11*cos(A)-7*cos(3*A) : :
X(10244) = (20*R^2-7*SW)*X(3)+8*R^2*X(4)

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

Trilinears    2*(9*cos(2*A)+1)*cos(B-C)+5*cos(A)-9*cos(3*A) : :
X(10245) = (28*R^2-9*SW)*X(3)+8*R^2*X(4)

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

Trilinears    3*a^3-2*(b+c)*a^2-(3*b^2-4*b*c+3*c^2)*a+2*(b^2-c^2)*(b-c) : :
X(10246) = 2*X(1)+X(3)

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) = 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(10247) = (JOHNSON, 5TH MIXTILINEAR)-EULEROLOGIC CENTER

Trilinears    3*a^3-4*(b+c)*a^2-(3*b^2-8*b*c+3*c^2)*a+4*(b^2-c^2)*(b-c) : :
X(10247) = 4*X(1)-X(3)

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) = X(10246)-of-5th-mixtilinear-triangle


X(10248) = (OUTER-GARCIA, 2ND CONWAY)-EULEROLOGIC CENTER

Barycentrics    7*a^4+2*(b+c)*a^3-4*a^2*b*c-2*(b^2-c^2)*(b-c)*a-7*(b^2-c^2)^2 : :
X(10248) = 8*X(4)-X(8)

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

Trilinears    8*(cos(2*A)+cos(4*A)-2)*cos(B-C)-2*(11*cos(A)+cos(3*A))*cos(2*(B-C))-5*cos(3*A)-3*cos(5*A)+32*cos(A) : :
X(10249) = (18*R^2-5*SW)*X(6)+2*(-2*SW+9*R^2)*X(74)

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

Trilinears    2*(4*cos(2*A)+cos(4*A)-5)*cos(B-C)-4*(4*cos(A)-cos(3*A))*cos(2*(B-C))-5*cos(3*A)-3*cos(5*A)+20*cos(A) : :

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

Barycentrics    2*(b+c)*a^6+(b^2+b*c+c^2)*a^5-(b^3+c^3)*a^4-b*c*(b+c)^2*a^3-(2*b^2-3*b*c+2*c^2)*(b+c)^3*a^2-(b^2-c^2)^2*a*(b^2+c^2)+(b^2-c^2)^2*(b+c)*(b^2+c^2) : :

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

Trilinears    2*(a^2+(b+c)*a+2*b*c)*S^2-(a+b+c)*(-(a^3+(b+c)*a^2-(b^2-4*b*c+c^2)*a-(b^2-c^2)*(b-c))*S-2*(a+b-c)*(a-b+c)*a*b*c) : :
Trilinears    4*(s-b)*(s-c)*(2*R+s)*s+((a+2*R)*s+b*c)*S : :
X(10252) = (2*R*(4*R+3*r+3*s)+(r+s)^2)*X(1)-2*r*(4*R+r+2*s)*X(3)

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

Trilinears    2*(a^2+(b+c)*a+2*b*c)*S^2-(a+b+c)*((a^3+(b+c)*a^2-(b^2-4*b*c+c^2)*a-(b^2-c^2)*(b-c))*S-2*(a+b-c)*(a-b+c)*a*b*c) : :
Trilinears    4*(s-b)*(s-c)*(2*R-s)*s+((a-2*R)*s+b*c)*S : :
X(10253) = (2*R*(4*R+3*r-3*s)+(r-s)^2)*X(1)-2*r*(4*R+r-2*s)*X(3)

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

Trilinears    cos(A)*(2*cos(A)*cos(B-C)-cos(2*(B-C))-3/2) : :
X(10254) = (7*R^2-2*SW)*X(3)+2*(4*R^2-SW)*X(4)

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(10255) = (TRINH, JOHNSON)-EULEROLOGIC CENTER

Trilinears    cos(A)*(2*cos(A)*cos(B-C)-cos(2*(B-C))-1/2) : :

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

Trilinears    (21*cos(2*A)+9*cos(4*A)+16)*cos(B-C)+(6*cos(A)+10*cos(3*A))*cos(2*(B-C))+(3*cos(2*A)-1)*cos(3*(B-C))+3*cos(5*A)+24*cos(A)+5*cos(3*A) : :

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

Trilinears    cos(A)*(2*cos(A)*cos(B-C)+cos(2*(B-C))-2*cos(2*A)-3) : :
X(10257) = (14*R^2-3*SW)*X(3)+(4*R^2-SW)*X(4)

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

Trilinears    cos((4*A-Pi)/3)*cos((B-C)/3)+cos(2*(B-C)/3)*cos((A-Pi)/3)-sin(B)*sin(C)-sin(A)*sin(2*A/3) : :
Trilinears: f(A+4*Pi, B+4*Pi, C+4*Pi) : :, where f(A,B,C) : : = X(5390)

X(10258) lies on these lines:
{1134,1136}, {3278,5390}


X(10259) = (ABC, 3RD MORLEY)-EULEROLOGIC CENTER

Trilinears    cos(4*A/3)*cos((B-C)/3)+cos(2*(B-C)/3)*cos(A/3)+sin(B)*sin(C)-sin(A)*sin((2*A-Pi)/3) : :
Trilinears: f(A-4*Pi, B-4*Pi, C-4*Pi) : :, where f(A,B,C) : : = X(5390)

X(10259) lies on these lines:
{357,1134}, {3282,5390}


X(10260) =  CIRCUMCIRCLE-INVERSE OF X(80)

Barycentrics    a^2 (a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^4-2 a^2 b^2+b^4+a^2 b c-b^3 c-2 a^2 c^2+b^2 c^2-b c^3+c^4) : :
Barycentrics    (Sin[2 A] - Sin[3 A]) / (2 Cos[A] - 1)
Trilinears    (Sin[3 A] - Sin[2 A]) / (Sin[A] - Sin[2 A]) : :

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

Barycentrics    1/(3*SA^2-2*SA*S+3*S^2-4*SB* SC) : :

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

Barycentrics    1/(3*SA^2+2*SA*S+3*S^2-4*SB* SC) : :

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)

Barycentrics    a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^4 b^2 c^2+3 b^6 c^2-3 a^4 c^4-4 b^4 c^4+3 a^2 c^6+3 b^2 c^6-c^8) : :
X(10263) = 3 X[51] - 2 X[140] = 3 X[52] - X[185] = X[20] - 3 X[568] = 3 X[5] - 2 X[1216] = 5 X[1656] - 3 X[2979] = X[3] - 3 X[3060] = 2 X[143] - 3 X[3060] = 3 X[3] - 5 X[3567] = 6 X[143] - 5 X[3567] = 9 X[3060] - 5 X[3567] = 4 X[3628] - 3 X[3917] = X[1216] - 3 X[5446] = 5 X[632] - 4 X[5447] = 3 X[549] - 4 X[5462] = 7 X[3526] - 9 X[5640] = X[1657] - 3 X[5890] = 4 X[3850] - 3 X[5891] = 3 X[3845] - 2 X[5907] = 5 X[632] - 6 X[5943] = 2 X[5447] - 3 X[5943] = 10 X[3567] - 9 X[5946] = 2 X[3] - 3 X[5946] = 4 X[143] - 3 X[5946] = 4 X[1216] - 3 X[6101] = 4 X[5446] - X[6101] = 2 X[185] - 3 X[6102] = X[5876] + 2 X[6243] = 11 X[5070] - 9 X[7998] = 9 X[5055] - 7 X[7999] = 3 X[8703] - 4 X[9729] = 2 X[548] - 3 X[9730] = 5 X[1656] - 7 X[9781] = 3 X[2979] - 7 X[9781] = 3 X[2] - 4 X[10095] = 3 X[6101] - 8 X[10110] = 3 X[5] - 4 X[10110] = 3 X[5446] - 2 X[10110] = 7 X[6101] - 12 X[10170] = 7 X[1216] - 9 X[10170] = 14 X[10110] - 9 X[10170] = 7 X[5] - 6 X[10170] = 7 X[5446] - 3 X[10170]

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) = 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)

Barycentrics    a^8 b^2-4 a^6 b^4+6 a^4 b^6-4 a^2 b^8+b^10+a^8 c^2+4 a^6 b^2 c^2-5 a^4 b^4 c^2+3 a^2 b^6 c^2-3 b^8 c^2-4 a^6 c^4-5 a^4 b^2 c^4+2 a^2 b^4 c^4+2 b^6 c^4+6 a^4 c^6+3 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10 : :
X(10264) = 3 X[5]-2 X[113] = X[113]-3 X[125] = X[146]-3 X[381],3 X[549]-2 X[1511],2 X[1539]-3 X[3845],5 X[632]-2 X[5609],5 X[632]-4 X[5972],3 X[549]-4 X[6699],3 X[3845]-4 X[7687],3 X[568]-X[7731] = X[265]-3 X[9140] = X[74]+3 X[9140],3 X[5054]-X[9143]

See X(10263).

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)

Barycentrics    a^6 b-2 a^5 b^2-a^4 b^3+4 a^3 b^4-a^2 b^5-2 a b^6+b^7+a^6 c+2 a^4 b^2 c-3 a^3 b^3 c-2 a^2 b^4 c+3 a b^5 c-b^6 c-2 a^5 c^2+2 a^4 b c^2-2 a^3 b^2 c^2+3 a^2 b^3 c^2+2 a b^4 c^2-3 b^5 c^2-a^4 c^3-3 a^3 b c^3+3 a^2 b^2 c^3-6 a b^3 c^3+3 b^4 c^3+4 a^3 c^4-2 a^2 b c^4+2 a b^2 c^4+3 b^3 c^4-a^2 c^5+3 a b c^5-3 b^2 c^5-2 a c^6-b c^6+c^7 : :
X(10265) = 3 X[946]-2 X[1537] =3 X[11]-X[1537] = X[1512]-3 X[1737],5 X[1698]-X[5531] = X[153]-3 X[5587] = X[5541]-3 X[5657] = 3 X[3576]-X[6224] = 3 X[3679]+X[7993] = 3 X[2]+X[9803] = 5 X[3091]-X[9809] = 2 X[214]-3 X[10165] = 4 X[6713]-3 X[10165] = 2 X[119]-3 X[10175] = 4 X[6702]-3 X[10175]

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}
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

Barycentrics    (-a+b+c)/(a^3-(b+c)*a^2-(b^2+b*c+c^2)*a+(b+c)^3) : :
X(10266) = (4*r^2-4*R*r+R^2)*X(4)+8*R*(2*r+R)*X(5885) = 2*(R-2*r)*X(80)+(R+6*r)*X(2475) = (R^2-16*R*r-4*r^2)*X(191)+(5*R^2+20*R*r+4*r^2)*X(7161)

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

Trilinears    (a^5-(b+c)*a^4-2*(b^2+b*c+c^2)*a^3+2*(b^2+c^2)*(b+c)*a^2+(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c))*a : :
Trilinears    a[(a + b + c)(b + c - a) + (b^2 + c^2 - a^2)R/r] : :
X(10267) = R*X(1)+(R+r)*X(3)

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

Trilinears    3*a^6-2*(b+c)*a^5-(7*b^2+6*b*c+7*c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3+(5*b^2-6*b*c+5*c^2)*(b+c)^2*a^2-2*(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*(b-c)^2 : :
X(10268) = r*X(1)-4*(R+r)*X(3)

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

Trilinears    (a^5-(b+c)*a^4-2*(b^2-3*b*c+c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(b^4+c^4-6*(b^2-b*c+c^2)*b*c)*a-(b^2-c^2)*(b-c)^3)*a : :
X(10269) = R*X(1)+(R-r)*X(3)

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

Trilinears    3*a^6-2*(b+c)*a^5-(7*b^2-10*b*c+7*c^2)*a^4+4*(b+c)*(b^2+c^2)*a^3+(5*b^4+5*c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*(b-c)^2 : :
X(10270) = r*X(1)+4*(R-r)*X(3)

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)

Barycentrics;    2 a^9-2 a^8 b-3 a^7 b^2+a^6 b^3+a^5 b^4+5 a^4 b^5-a^3 b^6-5 a^2 b^7+a b^8+b^9-2 a^8 c+8 a^7 b c-a^6 b^2 c-2 a^5 b^3 c-3 a^4 b^4 c-12 a^3 b^5 c+9 a^2 b^6 c+6 a b^7 c-3 b^8 c-3 a^7 c^2-a^6 b c^2+2 a^5 b^2 c^2-2 a^4 b^3 c^2+a^3 b^4 c^2+19 a^2 b^5 c^2-16 a b^6 c^2+a^6 c^3-2 a^5 b c^3-2 a^4 b^2 c^3+24 a^3 b^3 c^3-23 a^2 b^4 c^3-6 a b^5 c^3+8 b^6 c^3+a^5 c^4-3 a^4 b c^4+a^3 b^2 c^4-23 a^2 b^3 c^4+30 a b^4 c^4-6 b^5 c^4+5 a^4 c^5-12 a^3 b c^5+19 a^2 b^2 c^5-6 a b^3 c^5-6 b^4 c^5-a^3 c^6+9 a^2 b c^6-16 a b^2 c^6+8 b^3 c^6-5 a^2 c^7+6 a b c^7+a c^8-3 b c^8+c^9 : :

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(10272) =  MIDPOINT OF X(5) AND X(110)

Barycentrics;    (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6) : :
X(10272) = 3 X[3] + X[146] = 3 X[5] - X[265] = 3 X[110] + X[265] = 3 X[2] + X[399] = X[74] - 3 X[549] = 3 X[113] - X[1539] = 3 X[1511] + X[1539] = 5 X[1656] - X[3448] = 2 X[3628] + X[5609] = X[1511] - 3 X[5642] = X[113] + 3 X[5642] = X[1539] + 9 X[5642] = X[2931] + 3 X[5654] = X[74] + 3 X[5655] = X[2948] + 3 X[5886] = 3 X[5972] + X[6053] = 3 X[140] + 2 X[6053] = 3 X[140] - 2 X[6699] = 3 X[5972] - X[6699] = 3 X[5066] - 2 X[7687] = 3 X[5055] + X[9143] = 3 X[597] - X[9976]

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 }, {5,4 9 }, {30,113 }, {69,10201 }, {74,549 }, {125,3628 }, {140,5663 }, {403,3 043 }, {468,1986 }, {495,10091 }, {4 96,10088 }, {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 }, {754 2,7723 }, {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)

Barycentrics;    a (3 a^5 b-3 a^4 b^2-6 a^3 b^3+6 a^2 b^4+3 a b^5-3 b^6+3 a^5 c-6 a^4 b c+8 a^3 b^2 c-11 a b^4 c+6 b^5 c-3 a^4 c^2+8 a^3 b c^2-16 a^2 b^2 c^2+8 a b^3 c^2+3 b^4 c^2-6 a^3 c^3+8 a b^2 c^3-12 b^3 c^3+6 a^2 c^4-11 a b c^4+3 b^2 c^4+3 a c^5+6 b c^5-3 c^6) : :
X(10273) = (3 r + 2 R) X[1] - (3 r - R) X[3] = 2 X[942]-X[10247] = X[3057]-4 X[5885] = X[3576]+X[5903] = 3 X[10202]-2 X[10246] = 2 X[5]-5 X[4004] = X[355]-4 X[10107] = X[3817]-3 X[3919] = 5 X[3698]-2 X[5694] = 4 X[3754]-X[5887] = 2 X[3754]-X[10175] = X[5887]-2 X[10175] = 7 X[3922]-4 X[9956] = X[4018]+2 X[5690]

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)

Barycentrics;    a^4 (a^12-4 a^10 b^2+5 a^8 b^4-5 a^4 b^8+4 a^2 b^10-b^12-4 a^10 c^2+9 a^8 b^2 c^2-5 a^6 b^4 c^2+a^4 b^6 c^2-3 a^2 b^8 c^2+2 b^10 c^2+5 a^8 c^4-5 a^6 b^2 c^4+2 a^4 b^4 c^4-a^2 b^6 c^4-b^8 c^4+a^4 b^2 c^6-a^2 b^4 c^6-5 a^4 c^8-3 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+2 b^2 c^10-c^12) : :
Trilinears;    cos(2 A) cos(3A) - cos(4A) cos(B-C) : :
X(10274) = R^2*X(4) + (7*R^2-2*SW)*X(54)

Let A'B'C' be the pdeal 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)

Barycentrics;    14 a^16-103 a^14 b^2+335 a^12 b^4-633 a^10 b^6+765 a^8 b^8-609 a^6 b^10+313 a^4 b^12-95 a^2 b^14+13 b^16-103 a^14 c^2+454 a^12 b^2 c^2-729 a^10 b^4 c^2+342 a^8 b^6 c^2+459 a^6 b^8 c^2-786 a^4 b^10 c^2+469 a^2 b^12 c^2-106 b^14 c^2+335 a^12 c^4-729 a^10 b^2 c^4+360 a^8 b^4 c^4+15 a^6 b^6 c^4+480 a^4 b^8 c^4-837 a^2 b^10 c^4+376 b^12 c^4-633 a^10 c^6+342 a^8 b^2 c^6+15 a^6 b^4 c^6-14 a^4 b^6 c^6+463 a^2 b^8 c^6-758 b^10 c^6+765 a^8 c^8+459 a^6 b^2 c^8+480 a^4 b^4 c^8+463 a^2 b^6 c^8+950 b^8 c^8-609 a^6 c^10-786 a^4 b^2 c^10-837 a^2 b^4 c^10-758 b^6 c^10+313 a^4 c^12+469 a^2 b^2 c^12+376 b^4 c^12-95 a^2 c^14-106 b^2 c^14+13 c^16 : :

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)




leftri  Feuerbach quadrangle and related centers: X(10276) - X(10281)  rightri

This preamble and centers X(10276)-X(10281) were contributed by César Lozada, October 17, 2016.

Let FA, FB, FC be the A-, B-, C- Feuerbach points of ABC, respectively (i.e., the touchpoints of the nine-points-circle and the excircles). Let FD=X(11) be the Feuerbach point of ABC. The cyclic quadrangle QAF={FA,FB,FC,FD} is here named the Feuerbach quadrangle of ABC. The centroid of QAF is X(10276).

Properties:

(1) A maltitude ("midpoint altitude") is a perpendicular drawn to a side of a quadrilateral from the midpoint of the opposite side. In a cyclic quadrilateral the four maltitudes concur at the anticenter. The anticenter of QAF is X(10277).

(2) In a cyclic quadrangle the centroids of the component triangles are the vertices of another cyclic quadrangle. For QAF this last quadrangle has centroid coinciding with the centroid of QAF.

(3) The diagonal triangle A*B*C* of QAF has vertices with barycentric coordinates:
   A* = {FA,FB}∩{FC,FD} = -(SB-SC) : SA-SC : SA-SB
   B* = {FB,FC}∩{FA,FD} = SB-SC : -(SC-SA) : SB-SA
   C* = {FC,FA}∩{FB,FD} = SC-SB : SC-SA : -(SA-SB)

 A*, B*, C* lie all on the cubics K237, K238, K239, K672.

 A*B*C* has: area=area(ABC)/2, centroid = X(10278) , circumcenter = X(10279), orthocenter = X(5) and nine-point-center=X(10280)

In terms of Chris van Tienhoven's Encyclopedia of Quadri-Figures (EQF), some centers of QAF are:
 QA-P1 = Quadrangle centroid = X(10276)

 QA-P2 = Euler-Poncelot point = X(10277) = common point of the nine-point-circles of the component triangles

 QA-P3 = Gergonne-Steiner Point = X(5) = common point of the midray-circles
   The midray circles of the quadrangle {P1,P2,P3,P4} are the circumcircles of the triangles MijMikMil, for all combinations of (i,j,k,l) in {1,2,3,4}, where Mij = midpoint of {Pi,Pj}.

 QA-P4 = Isogonal Center = X(5) = common point of the lines {Oi,Qi}, where Oi is the circumcenter of the triangle PjPkPl and Qi is the isogonal conjugate of Pi w/r to PjPkPl.

 QA-P6 = Parabola Axes Crosspoint = X(10276) = intersection point of the axes of the two parabolas that can be constructed through the points

 QA-P7 = QA-Nine-point Homothetic Center = X(10281)
    Let QA'P be the quadrangle of the nine-point centers of QAP and let QA"P be the quadrangle of the nine-point centers of QA'P. Then QAP and QA"P are homothetic with homothetic center QA-P7.

 QA-P8 = Midray Homothetic Center = X(5)
   Let QA'P be the quadrangle of the midray centers of QAP and let QA"P be the quadrangle of the midray centers of QA'P. Then QAP and QA"P are homothetic with homothetic center QA-P8.

 QA-P10, QA-P11, QA-P12 and QA-P13 = centroid, circumcenter, orthocenter and nine-points-center, resp., of the diagonal triangle = X(10278), X(10279), X(5), X(10280), resp.

underbar

X(10276) = CENTROID OF THE FEUERBACH QUADRANGLE

Trilinears    (cos(4*A)+4)*cos(B-C)-(cos(A)-2*cos(3*A))*cos(2*(B-C))+(cos(2*A)-1)*cos(3*(B-C))+cos(3*A)+3*cos(A) : :
X(10276) = X(11)+3*X(5947)

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

Trilinears    (cos(2*A)-cos(4*A)-3/2)*cos(B-C)+2*(cos(A)-cos(3*A))*cos(2*(B-C))-(cos(2*A)-1)*cos(3*(B-C))-6*cos(A) : :
X(10277) = X(119)-3*X(5947)

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

Barycentrics    (a^4-(b^2+c^2)*a^2-(b^2-c^2)^2+b^2*c^2)*(b^2-c^2) : :
X(10278) = 5*X(2)-3*X(1649)

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) = 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

Barycentrics    (a^8-2*(b^2+c^2)*a^6+5*a^4*b^2*c^2+(b^2-2*c^2)*(2*b^2-c^2)*(b^2+c^2)*a^2-(b^2-c^2)^4)*(b^2-c^2) : :
X(10279) = X(4)-9*X(5466)

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

Barycentrics    (a^8-4*(b^2+c^2)*a^6+(4*b^4+5*b^2*c^2+4*c^4)*a^4-3*b^2*c^2*(b^2+c^2)*a^2-(b^2-c^2)^2*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :
Trilinears    ((cos(A)+2*cos(3*A))*cos(B-C)-(cos(2*A)-2)*cos(2*(B-C))+3/2*cos(2*A)+1/2*cos(4*A))*sin(B-C) : :
X(10280) = X(3)-9*X(8371)

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) = 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

Trilinears    (cos(2*A)-3*cos(4*A)-19/2)*cos(B-C)+2*(2*cos(A)-3*cos(3*A))*cos(2*(B-C))-3*(cos(2*A)-1)*cos(3*(B-C))-2*cos(3*A)-12*cos(A) : :
X(10281) = (63*R^4+4*R^2*SW-12*S^2)*X(5)-4*R^2*(9*R^2-2*SW)*X(399)

X(10281) lies on these lines:
{5,399}, {952,5947}


X(10282) =  X(3)X(64)∩X(51)X(54)

Barycentrics    a^2 (2 a^8-5 a^6 (b^2+c^2)+a^4 (3 b^4+4 b^2 c^2+3 c^4)+a^2 (b^2-c^2)^2 (b^2+c^2)-(b^2-c^2)^2 (b^4+c^4)) : :
X(10282) = 5 X(3) - X(64)

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.

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) = {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)

Barycentrics    4 a^4-4 a^3 b-5 a^2 b^2+4 a b^3+b^4-4 a^3 c+8 a^2 b c-4 a b^2 c-5 a^2 c^2-4 a b c^2-2 b^2 c^2+4 a c^3+c^4 : :
X(10283) = 2 X[1] + X[5] = 5 X[5] - 2 X[355] = 5 X[1] + X[355] = 2 X[140] + X[1482] = 4 X[1] - X[1483] = 2 X[5] + X[1483]

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)

Barycentrics    a (a^5 (b+c) -a^4 (b^2+6 b c+c^2) -a^3 (2 (b^3+c^3)-7 bc(b+c)) +2 a^2 (b^4+2 b^3 c-7 b^2 c^2+2 b c^3+c^4)+a (b-c)^2 (b^3-6 b c(b+c)+c^3)-(b-c)^4 (b+c)^2) : :
X(10284) = (3R - 2r)*X(1) + (2r - R)*X(3)

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} et al.

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(10285) =  EULER-LINE INTERCEPT OF X(54)X(1263)

Barycentrics    2 a^16-9 a^14 b^2+15 a^12 b^4-9 a^10 b^6-5 a^8 b^8+13 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-9 a^14 c^2+22 a^12 b^2 c^2-13 a^10 b^4 c^2-15 a^6 b^8 c^2+34 a^4 b^10 c^2-27 a^2 b^12 c^2+8 b^14 c^2+15 a^12 c^4-13 a^10 b^2 c^4+4 a^8 b^4 c^4-7 a^6 b^6 c^4-22 a^4 b^8 c^4+51 a^2 b^10 c^4-28 b^12 c^4-9 a^10 c^6-7 a^6 b^4 c^6-2 a^4 b^6 c^6-29 a^2 b^8 c^6+56 b^10 c^6-5 a^8 c^8-15 a^6 b^2 c^8-22 a^4 b^4 c^8-29 a^2 b^6 c^8-70 b^8 c^8+13 a^6 c^10+34 a^4 b^2 c^10+51 a^2 b^4 c^10+56 b^6 c^10-11 a^4 c^12-27 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16 : :

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.

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}

X(10285) = anticomplement fo X(10126)


X(10286) =  MIDPOINT OF X(5) AND X(5500)

Barycentrics    2 a^22-15 a^20 b^2+50 a^18 b^4-93 a^16 b^6+92 a^14 b^8-14 a^12 b^10-84 a^10 b^12+110 a^8 b^14-62 a^6 b^16+13 a^4 b^18+2 a^2 b^20-b^22-15 a^20 c^2+82 a^18 b^2 c^2-172 a^16 b^4 c^2+139 a^14 b^6 c^2+41 a^12 b^8 c^2-125 a^10 b^10 c^2+3 a^8 b^12 c^2+97 a^6 b^14 c^2-54 a^4 b^16 c^2-a^2 b^18 c^2+5 b^20 c^2+50 a^18 c^4-172 a^16 b^2 c^4+160 a^14 b^4 c^4+52 a^12 b^6 c^4-94 a^10 b^8 c^4-65 a^8 b^10 c^4+58 a^6 b^12 c^4+32 a^4 b^14 c^4-14 a^2 b^16 c^4-7 b^18 c^4-93 a^16 c^6+139 a^14 b^2 c^6+52 a^12 b^4 c^6-72 a^10 b^6 c^6-39 a^8 b^8 c^6-84 a^6 b^10 c^6+98 a^4 b^12 c^6+4 a^2 b^14 c^6-5 b^16 c^6+92 a^14 c^8+41 a^12 b^2 c^8-94 a^10 b^4 c^8-39 a^8 b^6 c^8-18 a^6 b^8 c^8-89 a^4 b^10 c^8+76 a^2 b^12 c^8+22 b^14 c^8-14 a^12 c^10-125 a^10 b^2 c^10-65 a^8 b^4 c^10-84 a^6 b^6 c^10-89 a^4 b^8 c^10-134 a^2 b^10 c^10-14 b^12 c^10-84 a^10 c^12+3 a^8 b^2 c^12+58 a^6 b^4 c^12+98 a^4 b^6 c^12+76 a^2 b^8 c^12-14 b^10 c^12+110 a^8 c^14+97 a^6 b^2 c^14+32 a^4 b^4 c^14+4 a^2 b^6 c^14+22 b^8 c^14-62 a^6 c^16-54 a^4 b^2 c^16-14 a^2 b^4 c^16-5 b^6 c^16+13 a^4 c^18-a^2 b^2 c^18-7 b^4 c^18+2 a^2 c^20+5 b^2 c^20-c^22 : :

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(10287) =  X(3)X(2575)∩X(5)X(523)

Barycentrics    a^2 (b^4 (a^2-b^2) (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) (-a^4+2 a^2 b^2-b^4-a^2 c^2-b^2 c^2+2 c^4+c^2 (-a^2-b^2+c^2) J)-c^4 (-a^2+c^2) (-a^2-a b-b^2+c^2) (-a^2+a b-b^2+c^2) (-a^4-a^2 b^2+2 b^4+2 a^2 c^2-b^2 c^2-c^4+b^2 (-a^2+b^2-c^2) J)) : : , where J = |OH|/R     (Peter Moses, October 23, 2016)

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)

Barycentrics    a^2 (b^4 (a^2-b^2) (-a^2+b^2-a c-c^2) (-a^2+b^2+a c-c^2) (-a^4+2 a^2 b^2-b^4-a^2 c^2-b^2 c^2+2 c^4-c^2 (-a^2-b^2+c^2) J)-c^4 (-a^2+c^2) (-a^2-a b-b^2+c^2) (-a^2+a b-b^2+c^2) (-a^4-a^2 b^2+2 b^4+2 a^2 c^2-b^2 c^2-c^4-b^2 (-a^2+b^2-c^2) J)) : : , where J = |OH|/R     (Peter Moses, October 23, 2016)

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

Barycentrics    2 a^16-13 a^14 b^2+43 a^12 b^4-89 a^10 b^6+115 a^8 b^8-87 a^6 b^10+33 a^4 b^12-3 a^2 b^14-b^16-13 a^14 c^2+62 a^12 b^2 c^2-113 a^10 b^4 c^2+64 a^8 b^6 c^2+61 a^6 b^8 c^2-94 a^4 b^10 c^2+33 a^2 b^12 c^2+43 a^12 c^4-113 a^10 b^2 c^4+68 a^8 b^4 c^4+17 a^6 b^6 c^4+46 a^4 b^8 c^4-81 a^2 b^10 c^4+20 b^12 c^4-89 a^10 c^6+64 a^8 b^2 c^6+17 a^6 b^4 c^6+30 a^4 b^6 c^6+51 a^2 b^8 c^6-64 b^10 c^6+115 a^8 c^8+61 a^6 b^2 c^8+46 a^4 b^4 c^8+51 a^2 b^6 c^8+90 b^8 c^8-87 a^6 c^10-94 a^4 b^2 c^10-81 a^2 b^4 c^10-64 b^6 c^10+33 a^4 c^12+33 a^2 b^2 c^12+20 b^4 c^12-3 a^2 c^14-c^16 : :

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}

leftri  Points associated with mid-triangles and cross-triangles: X(10290) - X(10608)  rightri

This preamble and centers X(10290)-X(10608) were contributed by Randy Hutson, October 22, 2016.

Let T1 = A1B1C1 and T2 = A2B2C2 be central triangles (or a pair of bicentric triangles) in the plane of a triangle ABC.

Let A' = midpoint of A1 and A2, and define B' and C' cyclically. The triangle A'B'C' is here named the mid-triangle of T1 and T2, denoted by MT(T1,T2).
Let A'' = B1C2∩C1B2, and B'' and C'' cyclically. The triangle A''B''C'' is here named the cross-triangle of T1 and T2, denoted by XT(T1,T2).

If T1 and T2 are homothetic, then both MT(T1,T2) and XT(T1,T2) are homothetic to T1 and T2.

If T1 and T2 are directly similar, then MT(T1,T2) is also directly similar to T1 and T2, with the same center of similitude.

If any pair in {T1, T2, XT(T1,T2)} are perspective, then every pair in the set are perspective.

If the vertices of T1 and T2 lie on a conic, then XT(T1,T2) is degenerate (consisting of 3 collinear points). If T1 and T2 are also perspective, XT(T1,T2) lies on the polar of the perspector wrt the conic. If T1 and T2 are the cevian triangles of P and Q, resp., then XT(T1,T2) is degenerate and collinear with P and Q.

If the vertices of T2 lie on the respective sidelines of T1 (e.g., A'' lies on B'C'), then XT(T1,T2) = T1.

For many choices of triangles T1, T2, T3,
(perspector of T1 and XT(T2,T3)) = (perspector of T2 and XT(T1,T3)) = (perspector of T3 and XT(T1,T2)).

If T1 is the cevian triangle of P and T2 is the anticevian triangle of Q, then XT(T1,T2) is perspective to ABC, and the perspector is collinear with these 3 points: P, P-Ceva conjugate of Q, Q-cross conjugate of P. Also, XT(T1,T2) is perspective to T1 at Q.

If T1 is the circumcevian triangle of P, then XT(ABC,T1) is perspective to the circumcevian triangle of P*, where P* is the circumcircle-inverse of P. The perspector lies on the circumcircle.

The cross-triangle of the cevian and anticevian triangles of P is perspective to ABC at P.

The cross-triangle of the cevian and circumcevian triangles of P is perspective to ABC at gcgP, where g = isogonal conjugate and c = complement.

The (degenerate) cross-triangle of the circumcevian triangles of P and Q is perspective to ABC at Λ(gP, gQ).

The appearance of (T1,T2,T3) in the following list means that MT(T1,T2) = T3:

(ABC, excentral, T(-2,1))
(ABC, Lucas(t) antipodal, Lucas(t) central)
(1st Conway, 2nd Conway, intouch)
(excentral, 6th mixtilinear, 1st circumperp)
(extouch, Hutson-extouch, excentral)
(inner Garcia, outer Garcia, extouch)
(inner Grebe, outer Grebe, ABC)
(inner Hutson, outer Hutson, 6th mixtilinear)
(inner Johnson, outer Johnson, Johnson)
(inner Napoleon, outer Napoleon, medial)
(inner Vecten, outer Vecten, medial)
(intangents, extangents, tangential)
(intouch, excentral, Ascella)
(intouch, extouch, medial)
(1st Neuberg, 2nd Neuberg, medial)

The appearance of (T1,T2,T3) in the following list means that XT(T1,T2) = T3:

(ABC, any cevian triangle, ABC)
(ABC, any pedal triangle, ABC)
(ABC, 2nd Johnson-Yff, Mandart-incircle)
(ABC, 1st Morley, 1st Morley adjunct)
(ABC, 1st Morley adjunct, 1st Morley)
(ABC, 2nd Morley, 2nd Morley adjunct)
(ABC, 2nd Morley adjunct, 2nd Morley)
(ABC, 3rd Morley, 3rd Morley adjunct)
(ABC, 3rd Morley adjunct, 3rd Morley)
(1st Brocard, 3rd Brocard, ABC)
(3rd Conway, 4th Conway, 5th Conway)
(3rd Conway, 5th Conway, 4th Conway)
(4th Conway, 5th Conway, 3rd Conway)
(X(7) extraversion, X(8) extraversion (2nd Conway), ABC)
(inner Napoleon, outer Napoleon, orthocentroidal)
(inner tri-equilateral, outer tri-equilateral, X(3)-Ehrmann)
(inner Vecten, outer Vecten, orthic)
(inner Yff, outer Yff, medial)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, X(3)-Ehrmann)
(Lucas central, Lucas(-1) central, symmedial)
(Lucas tangents, Lucas(-1) tangents, cevian of X(6) wrt symmedial)
(1st Morley, 1st Morley adjunct, ABC)
(2nd Morley, 2nd Morley adjunct, ABC)
(3rd Morley, 3rd Morley adjunct, ABC)

The appearance of (T1,T2,T3,i) in the following list means MT(T1,T2) is perspective to T3 with perspector X(i), and an asterisk * signifies that the two triangles are homothetic:

(Andromeda, Antlia, ABC, 1)
(1st Brocard, 1st anti-Brocard, ABC, 10290)
(1st Brocard, 1st anti-Brocard, medial, 10291)
(1st Brocard, 1st anti-Brocard, anticomplementary)
(1st Brocard, 1st anti-Brocard, 2nd Brocard, 2)
(5th Brocard, 5th anti-Brocard, ABC*, 32)
(5th Brocard, 5th anti-Brocard, medial, 10292)
(5th Brocard, 5th anti-Brocard, 3rd Brocard, 32)
(2nd circumperp, 1st Conway, ABC, 5557)
(2nd circumperp, Fuhrmann-of-orthic-of-Fuhrmann, ABC, 1)
(2nd circumperp, inverse-in-Conway-circle, ABC, 1)
(2nd circumperp, inverse-in-incircle, ABC, 1)
(2nd circumperp, 1st Sharygin, ABC)
(1st Conway, Honsberger, ABC, 7)
(1st Conway, 1st Sharygin, ABC)
(2nd Conway, Honsberger, ABC, 7)
(3rd Conway, Hutson-intouch, ABC)
(3rd Conway, 6th mixtilinear, ABC)
(3rd Conway, X(1) reflection, ABC)
(Euler, anti-Euler, ABC*, 4)
(Euler, anti-Euler, medial*, 3526)
(Euler, anti-Euler, anticomplementary*, 10303)
(Euler, anti-Euler, Johnson*, 5055)
(Euler, anti-Euler, X(3) cevian*, 10304)
(Euler, anti-Euler, 3rd pedal of X(4), 4)
(Euler, anti-Euler, 3rd antipedal of X(4), 4)
(excentral, 2nd circumperp, ABC, 1)
(excentral, 2nd extouch, ABC, 8)
(excentral, Fuhrmann-of-orthic-of-Fuhrmann, ABC, 1)
(excentral, inverse-in-Conway-circle, ABC, 1)
(excentral, inverse-in-incircle, ABC, 1)
(extouch, 2nd extouch, ABC, 80)
(2nd extouch, 3rd extouch, ABC, 4)
(4th extouch, 5th extouch, ABC)
(1st Hatzipolakis, 2nd Hatzipolakis, ABC, 10977)
(hexyl, 3rd Conway, ABC)
(hexyl, Hutson-intouch, ABC)
(hexyl, 6th mixtilinear, ABC, 10308)
(hexyl, X(1) reflection, ABC, 5553)
(Hutson-intouch, X(1) reflection, ABC, 3296)
(inner Yff, outer Yff, ABC*, 1)
(intouch, 1st Conway, ABC, 7)
(intouch, 2nd Conway, ABC, 7)
(intouch, 3rd Conway, ABC)
(intouch, hexyl, ABC, 10305)
(intouch, Honsberger, ABC, 7)
(intouch, 6th mixtilinear, ABC, 10307)
(intouch, X(1) reflection, ABC)
(1st Johnson-Yff, 2nd Johnson-Yff, ABC*, 10523)
(1st Johnson-Yff, 2nd Johnson-Yff, medial*, 8071)
(1st Johnson-Yff, 2nd Johnson-Yff, anticomplementary*, 10629)
(1st Johnson-Yff, 2nd Johnson-Yff, Johnson*, 1)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, orthic*, 10311)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, circumorthic*, 10312)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, dual of orthic*, 10313)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, submedial*, 10314)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, intangents*, 1914)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, extangents*, 10315)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, tangential*, 6)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita*, 39)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd Ehrmann*, 6)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, X(3)-Ehrmann*, 1971)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd Euler*, 10316)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd isogonal triangle of X(4)*, 10317)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, X(3) orthantiocevian*,6)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, Trinh*, 187)
(Lucas antipodal, Lucas(-1) antipodal, ABC, 3)
(Lucas central, Lucas(-1) central, ABC, 3)
(Lucas homothetic, Lucas(-1) homothetic, ABC*, 10318)
(McCay, anti-McCay, ABC)
(McCay, anti-McCay, medial)
(McCay, anti-McCay, anticomplementary)
(medial, anticomplementary, Euler*, 3851)
(medial, anticomplementary, anti-Euler*, 10299)
(medial, anticomplementary, X(3) cevian*, 3529)
(medial, anticomplementary, Johnson*, 382)
(medial, anticomplementary, outer Garcia*, 3632)
(medial, anticomplementary, 3rd pedal of X(3), 10300)
(medial, anticomplementary, 3rd pedal of X(4), 10301)
(1st mixtilinear, 2nd mixtilinear, ABC, 1)
(1st mixtilinear, 2nd mixtilinear, 5th mixtilinear/Caelum, 1)
(6th mixtilinear, X(1) reflection, ABC, 10309)
(6th mixtilinear, Hutson-intouch, ABC)
(orthic, circumorthic, ABC, 4)
(orthic, circumorthic, dual of orthic*, 3523)
(orthic, circumorthic, 2nd Euler*, 1656)
(orthic, circumorthic, 2nd isogonal triangle of X(4)*, 3851)
(orthic, circumorthic, Kosnita*, 3515)
(orthic, circumorthic, submedial*, 1656)
(orthic, circumorthic, Trinh*, 3516)
(orthic, circumorthic, tangential*, 3517)
(orthic, dual of orthic, ABC, 69)
(orthic, dual of orthic, circumorthic*, 631)
(orthic, dual of orthic, 2nd Euler*, 3)
(orthic, dual of orthic, extangents*, 10319)
(orthic, dual of orthic, intangents*, 1040)
(orthic, dual of orthic, 2nd isogonal triangle of X(4)*, 3)
(orthic, dual of orthic, Kosnita*, 3)
(orthic, dual of orthic, mid-triangle of 1st & 2nd Kenmotu diagonals triangles*, 577)
(orthic, dual of orthic, submedial*, 2)
(orthic, dual of orthic, tangential*, 3)
(orthocentroidal, anti-orthocentroidal, ABC, 10293)
(orthocentroidal, anti-orthocentroidal, orthic, 10294)
(Yff central, 1st tangential mid-arc, ABC, 177)
(inner tri-equilateral, outer tri-equilateral, ABC, 6)
(inner tri-equilateral, outer tri-equilateral, tangential*, 6)
(inner tri-equilateral, outer tri-equilateral, 2nd Ehrmann*, 6)
(inner tri-equilateral, outer tri-equilateral, Kosnita*, 574)
(inner tri-equilateral, outer tri-equilateral, Trinh*, 8588)
(inner Yff, outer Yff, ABC*, 1)
(inner Yff, outer Yff, medial*, 10320)
(inner Yff, outer Yff, anticomplementary*, 10321)
(inner Yff, outer Yff, Johnson*, 10523)

The appearance of (T1,T2,T3,i) in the following list means XT(T1,T2) is perspective to T3 with perspector X(i), and an asterisk * signifies that the two triangles are homothetic:

(ABC, Apollonius, ABC, 1682)
(ABC, Apollonius, Apollonius, 970)
(ABC, Apus, ABC, 56)
(ABC, Apus, Apus, 3)
(ABC, Aquila, ABC*, 1698)
(ABC, Aquila, medial*, 1)
(ABC, Aquila, anticomplementary*, 10)
(ABC, Aquila, anti-Euler*, 6684)
(ABC, Aquila, Aquila*, 10)
(ABC, Aquila, Euler*, 7989)
(ABC, Ara, ABC*, 3)
(ABC, Ara, medial*, 25)
(ABC, Ara, anticomplementary*, 22)
(ABC, Ara, Ara*, 22)
(ABC, Ara, Euler*, 7395)
(ABC, Ara, anti-Euler*, 10323)
(ABC, Artzt, ABC, 9755)
(ABC, Artzt, Artzt, 9756)
(ABC, Atik, ABC, 10324)
(ABC, Atik, Atik, 10325)
(ABC, Ayme, ABC, 3610)
(ABC, Ayme, Ayme, 10327)
(ABC, 1st Brocard, ABC, 32)
(ABC, 1st Brocard, 1st Brocard, 384)
(ABC, 1st anti-Brocard, ABC, 4027)
(ABC, 1st anti-Brocard, 1st anti-Brocard, 5989)
(ABC, 2nd Brocard, ABC, 69)
(ABC, 2nd Brocard, 2nd Brocard, 141)
(ABC, 3rd Brocard, ABC, 76)
(ABC, 3rd Brocard, 3rd Brocard, 384)
(ABC, 4th Brocard, ABC, 25)
(ABC, 4th Brocard, 4th Brocard, 427)
(ABC, 4th anti-Brocard, ABC, 10354)
(ABC, 4th anti-Brocard, 4th anti-Brocard, 10355)
(ABC, 5th Brocard, ABC*, 3096)
(ABC, 5th Brocard, medial*, 32)
(ABC, 5th Brocard, anticomplementary*, 2896)
(ABC, 5th Brocard, 5th Brocard*, 2896)
(ABC, 5th Brocard, 5th anti-Brocard*, 10345)
(ABC, 5th Brocard, Euler*, 10356)
(ABC, 5th Brocard, anti-Euler*, 10357)
(ABC, 5th anti-Brocard, ABC*, 83)
(ABC, 5th anti-Brocard, medial*, 32)
(ABC, 5th anti-Brocard, anticomplementary*, 7787)
(ABC, 5th anti-Brocard, 5th anti-Brocard*, 7787)
(ABC, 5th anti-Brocard, 5th Brocard*, 10345)
(ABC, 5th anti-Brocard, Euler*, 10358)
(ABC, 5th anti-Brocard, anti-Euler*, 10359)
(ABC, circummedial, ABC, 251)
(ABC, circummedial, circummedial, 1799)
(ABC, circummedial, X(23) circumcevian, 827)
(ABC, circumorthic, ABC, 54)
(ABC, circumorthic, circumorthic, 8884)
(ABC, circumorthic, X(186) circumcevian, 933)
(ABC, 2nd circumperp, ABC, 58)
(ABC, 2nd circumperp, 2nd circumperp, 21)
(ABC, 2nd circumperp, X(36) circumcevian, 110)
(ABC, circumsymmedial, ABC, 6)
(ABC, circumsymmedial, circumsymmedial, 6)
(ABC, circumsymmedial, X(187) circumcevian, 110)
(ABC, X(15) circumcevian, X(16) circumcevian, 10409)
(ABC, X(16) circumcevian, X(15) circumcevian, 10410)
(ABC, X(23) circumcevian, ABC, 10415)
(ABC, X(23) circumcevian, X(23) circumcevian, 10416)
(ABC, X(23) circumcevian, circummedial, 691)
(ABC, X(25) circumcevian, ABC, 2)
(ABC, X(25) circumcevian, X(25) circumcevian, 6353)
(ABC, X(25) circumcevian, X(468) circumcevian, 112)
(ABC, X(30) circumcevian, ABC, 10419)
(ABC, X(30) circumcevian, X(3) circumcevian, 10420)
(ABC, X(186) circumcevian, ABC, 5627)
(ABC, X(186) circumcevian, circumorthic, 1304)
(ABC, X(186) circumcevian, X(186) circumcevian, 10421)
(ABC, X(187) circumcevian, ABC, 10630)
(ABC, X(187) circumcevian, circumsymmedial, 691)
(ABC, X(468) circumcevian, ABC, 10422)
(ABC, X(468) circumcevian, X(25) circumcevian, 10423)
(ABC, X(468) circumcevian, X(468) circumcevian, 10424)
(ABC, X(511) circumcevian, ABC, 2065)
(ABC, X(511) circumcevian, X(3) circumcevian, 10425)
(ABC, X(1155) circumcevian, ABC, 10426)
(ABC, X(1155) circumcevian, 4th mixtilinear, 2742)
(ABC, X(1319) circumcevian, ABC, 10428)
(ABC, X(1319) circumcevian, 3rd mixtilinear, 2720)
(ABC, 1st Conway, ABC, 21)
(ABC, 1st Conway, 1st Conway, 8822)
(ABC, 1st Conway, 2nd Conway, 10432)
(ABC, 1st Conway, 3rd Conway)
(ABC, 1st Conway, 5th Conway, 10461)
(ABC, 2nd Conway, ABC, 8)
(ABC, 2nd Conway, 1st Conway, 10433)
(ABC, 2nd Conway, 2nd Conway, 69)
(ABC, 2nd Conway, 3rd Conway)
(ABC, 2nd Conway, 4th Conway, 10449)
(ABC, 3rd Conway, ABC, 10434)
(ABC, 3rd Conway, 3rd Conway, 10437)
(ABC, 4th Conway, ABC, 10447)
(ABC, 4th Conway, 2nd Conway, 10449)
(ABC, 4th Conway, 3rd Conway, 1764)
(ABC, 4th Conway, 4th Conway, 314)
(ABC, 5th Conway, ABC, 10455)
(ABC, 5th Conway, 1st Conway, 10461)
(ABC, 5th Conway, 3rd Conway, 1764)
(ABC, 5th Conway, 5th Conway, 314)
(ABC, 2nd Ehrmann, 2nd Ehrmann, 895)
(ABC, X(2)-Ehrmann, ABC, 10510)
(ABC, X(2)-Ehrmann, X(2)-Ehrmann, 858)
(ABC, Euler, ABC*, 5)
(ABC, Euler, medial*, 4)
(ABC, Euler, anticomplementary*, 3091)
(ABC, Euler, anti-Euler*, 3090)
(ABC, Euler, Euler*, 3091)
(ABC, anti-Euler, ABC*, 631)
(ABC, anti-Euler, medial*, 4)
(ABC, anti-Euler, anticomplementary*, 3)
(ABC, anti-Euler, Euler*, 3090)
(ABC, anti-Euler, anti-Euler*, 3)
(ABC, 2nd Euler, ABC, 52)
(ABC, 2nd Euler, 2nd Euler, 4)
(ABC, 5th Euler, ABC, 427)
(ABC, 5th Euler, 5th Euler, 8889)
(ABC, extangents, ABC, 55)
(ABC, extangents, extangents, 40)
(ABC, 2nd extouch, ABC, 72)
(ABC, 2nd extouch, extouch, 4)
(ABC, 2nd extouch, 2nd extouch, 329)
(ABC, 2nd extouch, 3rd extouch)
(ABC, 2nd extouch, 4th extouch)
(ABC, 3rd extouch, ABC, 1439)
(ABC, 3rd extouch, 2nd extouch)
(ABC, 3rd extouch, 3rd extouch, 5932)
(ABC, 3rd extouch, 4th extouch, 10360)
(ABC, 3rd extouch, 5th extouch, 10365)
(ABC, 4th extouch, ABC, 65)
(ABC, 4th extouch, 3rd extouch, 10360)
(ABC, 4th extouch, 4th extouch, 5933)
(ABC, 4th extouch, 5th extouch, 10369)
(ABC, 5th extouch, ABC, 65)
(ABC, 5th extouch, extouch, 10377)
(ABC, 5th extouch, 3rd extouch, 10365)
(ABC, 5th extouch, 4th extouch, 10369)
(ABC, 5th extouch, 5th extouch, 8)
(ABC, Feuerbach, ABC, 11)
(ABC, Feuerbach, Feuerbach, 5)
(ABC, inner Grebe, ABC*, 5591)
(ABC, inner Grebe, medial*, 6)
(ABC, inner Grebe, anticomplementary*, 1271)
(ABC, inner Grebe, inner Grebe*, 1271)
(ABC, inner Grebe, outer Grebe*, 10513)
(ABC, inner Grebe, Euler*, 10514)
(ABC, inner Grebe, anti-Euler*, 10517)
(ABC, outer Grebe, ABC*, 5590)
(ABC, outer Grebe, medial*, 6)
(ABC, outer Grebe, anticomplementary*, 1270)
(ABC, outer Grebe, outer Grebe*, 1270)
(ABC, outer Grebe, inner Grebe*, 10513)
(ABC, outer Grebe, Euler*, 10515)
(ABC, outer Grebe, anti-Euler*, 10518)
(ABC, half-altitude, ABC, 3)
(ABC, half-altitude, half-altitude, 2)
(ABC, hexyl, ABC, 40)
(ABC, hexyl, excentral, 1498)
(ABC, hexyl, hexyl, 20)
(ABC, Honsberger, ABC, 2346)
(ABC, Honsberger, Honsberger, 10509)
(ABC, Honsberger, excentral, 7)
(ABC, Hutson extouch, ABC, 55)
(ABC, Hutson extouch, Hutson extouch, 9778)
(ABC, Hutson intouch, ABC, 56)
(ABC, Hutson intouch, Hutson intouch, 4308)
(ABC, 2nd Hyacinth, ABC, 235)
(ABC, 2nd Hyacinth, 2nd Hyacinth, 4)
(ABC, 2nd Hyacinth, orthic, 155)
(ABC, inverse-in-Conway-circle, ABC, 10471)
(ABC, inverse-in-Conway-circle, inverse-in-Conway-circle, 314)
(ABC, inverse-in-excircles, ABC, 3663)
(ABC, inverse-in-excircles, inverse-in-excircles, 7)
(ABC, inverse-in-incircle, ABC, 10481)
(ABC, inverse-in-incircle, inverse-in-incircle, 7)
(ABC, 2nd isogonal of X(1), ABC*, 7951)
(ABC, 2nd isogonal of X(1), medial*, 36)
(ABC, 2nd isogonal of X(1), anticomplementary*, 1478)
(ABC, 2nd isogonal of X(1), 2nd isogonal of X(1)*, 1478)
(ABC, 2nd isogonal of X(1), anti-Euler*, 498)
(ABC, 2nd isogonal of X(1), Johnson*, 1)
(ABC, 2nd isogonal of X(1), Mandart-incircle*, 3583)
(ABC, 2nd isogonal of X(1), outer Yff*, 7741)
(ABC, 2nd isogonal of X(1), 1st Johnson-Yff*, 1)
(ABC, 2nd isogonal of X(1), 2nd Johnson-Yff*, 1)
(ABC, 2nd isogonal of X(1), X(3) cevian*, 10483)
(ABC, 2nd isogonal of X(4), ABC, 3581)
(ABC, 2nd isogonal of X(4), 2nd isogonal of X(4), 30)
(ABC, inner Johnson, ABC*, 1376)
(ABC, inner Johnson, medial*, 11)
(ABC, inner Johnson, anticomplementary*, 3434)
(ABC, inner Johnson, inner Johnson*, 3434)
(ABC, inner Johnson, outer Johnson*, 10522)
(ABC, inner Johnson, Johnson*, 10525)
(ABC, outer Johnson, ABC*, 958)
(ABC, outer Johnson, medial*, 12)
(ABC, outer Johnson, anticomplementary*, 3436)
(ABC, outer Johnson, outer Johnson*, 3436)
(ABC, outer Johnson, inner Johnson*, 10522)
(ABC, outer Johnson, Johnson*, 10526)
(ABC, 1st Johnson-Yff, ABC*, 56)
(ABC, 1st Johnson-Yff, medial*, 12)
(ABC, 1st Johnson-Yff, anticomplementary*, 388)
(ABC, 1st Johnson-Yff, 1st Johnson-Yff*, 388)
(ABC, 1st Johnson-Yff, 2nd Johnson-Yff*, 4)
(ABC, 1st Johnson-Yff, inner Yff*, 3)
(ABC, 1st Johnson-Yff, outer Yff*, 999)
(ABC, 1st Johnson-Yff, Euler*, 11)
(ABC, 1st Johnson-Yff, anti-Euler*, 4293)
(ABC, 1st Johnson-Yff, Johnson*, 1478)
(ABC, 1st Johnson-Yff, Mandart-incircle*, 1)
(ABC, 2nd Johnson-Yff, ABC*, 55)
(ABC, 2nd Johnson-Yff, medial*, 11)
(ABC, 2nd Johnson-Yff, anticomplementary*, 497)
(ABC, 2nd Johnson-Yff, 2nd Johnson-Yff*, 497)
(ABC, 2nd Johnson-Yff, 1st Johnson-Yff*, 4)
(ABC, 2nd Johnson-Yff, inner Yff*, 3295)
(ABC, 2nd Johnson-Yff, outer Yff*, 3)
(ABC, 2nd Johnson-Yff, Euler*, 12)
(ABC, 2nd Johnson-Yff, anti-Euler*, 4294)
(ABC, 2nd Johnson-Yff, Johnson*, 1479)
(ABC, 1st Kenmotu diagonals, 1st Kenmotu diagonals, 6413)
(ABC, 2nd Kenmotu diagonals, 2nd Kenmotu diagonals, 6414)
(ABC, Kosnita, ABC, 52)
(ABC, Kosnita, Kosnita, 7388)
(ABC, Lucas antipodal, ABC, 1588)
(ABC, Lucas antipodal, Lucas antipodal, 6459)
(ABC, Lucas(-1) antipodal, ABC, 1587)
(ABC, Lucas(-1) antipodal, Lucas(-1) antipodal, 6460)
(ABC, Lucas Brocard, ABC, 5058)
(ABC, Lucas Brocard, Lucas Brocard, 1151)
(ABC, Lucas Brocard, Lucas(-1) Brocard, 10541)
(ABC, Lucas(-1) Brocard, ABC, 5062)
(ABC, Lucas(-1) Brocard, Lucas(-1) Brocard, 1152)
(ABC, Lucas(-1) Brocard, Lucas Brocard, 10541)
(ABC, Lucas central, ABC, 3311)
(ABC, Lucas central, Lucas central, 371)
(ABC, Lucas central, Lucas(-1) central, 6)
(ABC, Lucas central, Lucas tangents, 3)
(ABC, Lucas central, Lucas(-1) tangents, 3)
(ABC, Lucas central, Lucas inner, 6221)
(ABC, Lucas central, Lucas(-1) inner, 6448)
(ABC, Lucas(-1) central, ABC, 3312)
(ABC, Lucas(-1) central, Lucas(-1) central, 372)
(ABC, Lucas(-1) central, Lucas central, 6)
(ABC, Lucas(-1) central, Lucas(-1) tangents, 3)
(ABC, Lucas(-1) central, Lucas tangents, 3)
(ABC, Lucas(-1) central, Lucas(-1) inner, 6398)
(ABC, Lucas(-1) central, Lucas inner, 6447)
(ABC, Lucas homothetic, ABC*, 8222)
(ABC, Lucas homothetic, Lucas homothetic*, 6462)
(ABC, Lucas homothetic, Lucas(-1) homothetic*, 6339)
(ABC, Lucas(-1) homothetic, ABC*, 8223)
(ABC, Lucas(-1) homothetic, Lucas(-1) homothetic*, 6463)
(ABC, Lucas(-1) homothetic, Lucas homothetic*, 6339)
(ABC, Lucas inner, ABC, 6425)
(ABC, Lucas inner, Lucas inner, 6429)
(ABC, Lucas inner, Lucas(-1) inner, 6)
(ABC, Lucas inner, Lucas central, 6221)
(ABC, Lucas inner, Lucas(-1) central, 6447)
(ABC, Lucas inner, Lucas tangents, 6221)
(ABC, Lucas inner, Lucas(-1) tangents, 6449)
(ABC, Lucas(-1) inner, ABC, 6426)
(ABC, Lucas(-1) inner, Lucas(-1) inner, 6430)
(ABC, Lucas(-1) inner, Lucas inner, 6)
(ABC, Lucas(-1) inner, Lucas(-1) central, 6398)
(ABC, Lucas(-1) inner, Lucas central, 6448)
(ABC, Lucas(-1) inner, Lucas(-1) tangents, 6398)
(ABC, Lucas(-1) inner, Lucas tangents, 6450)
(ABC, Lucas tangents, ABC, 3)
(ABC, Lucas tangents, Lucas tangents, 1151)
(ABC, Lucas tangents, Lucas(-1) tangents, 6)
(ABC, Lucas tangents, Lucas central, 3)
(ABC, Lucas tangents, Lucas(-1) central, 3)
(ABC, Lucas tangents, Lucas inner, 6221)
(ABC, Lucas tangents, Lucas(-1) inner, 6450)
(ABC, Lucas(-1) tangents, ABC, 3)
(ABC, Lucas(-1) tangents, Lucas(-1) tangents, 1152)
(ABC, Lucas(-1) tangents, Lucas tangents, 6)
(ABC, Lucas(-1) tangents, Lucas(-1) central, 3)
(ABC, Lucas(-1) tangents, Lucas central, 3)
(ABC, Lucas(-1) tangents, Lucas(-1) inner, 6398)
(ABC, Lucas(-1) tangents, Lucas inner, 6449)
(ABC, Mandart-incircle, ABC*, 11)
(ABC, Mandart-incircle, medial*, 55)
(ABC, Mandart-incircle, anticomplementary*, 497)
(ABC, Mandart-incircle, Mandart-incircle*, 497)
(ABC, Mandart-incircle, Euler*, 5592)
(ABC, Mandart-incircle, anti-Euler*, 3086)
(ABC, McCay, ABC, 576)
(ABC, McCay, McCay, 10486)
(ABC, anti-McCay, ABC, 10487)
(ABC, anti-McCay, anti-McCay, 10488)
(ABC, mid-arc, ABC, 10489)
(ABC, mid-arc, 1st tangential mid-arc, 10490)
(ABC, 2nd mid-arc, ABC, 10491)
(ABC, 1st mixtilinear, ABC, 40)
(ABC, 1st mixtilinear, 1st mixtilinear, 1697)
(ABC, 2nd mixtilinear, ABC, 1)
(ABC, 2nd mixtilinear, 2nd mixtilinear, 1)
(ABC, 3rd mixtilinear, ABC, 1)
(ABC, 3rd mixtilinear, 3rd mixtilinear, 1420)
(ABC, 3rd mixtilinear, X(1319) circumcevian, 109)
(ABC, 4th mixtilinear, ABC, 57)
(ABC, 4th mixtilinear, 4th mixtilinear, 165)
(ABC, 4th mixtilinear, X(1155) circumcevian), 101)
(ABC, 6th mixtilinear, ABC, 165)
(ABC, 6th mixtilinear, 6th mixtilinear, 2951)
(ABC, 7th mixtilinear, ABC, 9533)
(ABC, 7th mixtilinear, 7th mixtilinear)
(ABC, inner Napoleon, ABC, 62)
(ABC, inner Napoleon, inner Napoleon, 5)
(ABC, inner Napoleon, outer Napoleon, 628)
(ABC, outer Napoleon, ABC, 61)
(ABC, outer Napoleon, outer Napoleon, 5)
(ABC, outer Napoleon, inner Napoleon, 627)
(ABC, 1st Neuberg, ABC, 511)
(ABC, 1st Neuberg, 1st Neuberg, 5999)
(ABC, 1st Neuberg, 2nd Neuberg, 147)
(ABC, 2nd Neuberg, ABC, 182)
(ABC, 2nd Neuberg, 2nd Neuberg, 5999)
(ABC, 2nd Neuberg, 1st Neuberg, 6194)
(ABC, orthocentroidal, ABC, 3)
(ABC, orthocentroidal, orthocentroidal, 5)
(ABC, anti-orthocentroidal, ABC, 1511)
(ABC, anti-orthocentroidal, anti-orthocentroidal, 399)
(ABC, X(2) orthocevian, ABC*, 6676)
(ABC, X(2) orthocevian, medial*, 25)
(ABC, X(2) orthocevian, anticomplementary*, 7493)
(ABC, X(2) orthocevian, X(2) orthocevian*, 7493)
(ABC, 1st orthosymmedial, ABC, 10547)
(ABC, 1st orthosymmedial, 1st orthosymmedial, 10548)
(ABC, 2nd orthosymmedial, ABC, 10549)
(ABC, 2nd orthosymmedial, 2nd orthosymmedial, 10550)
(ABC, 1st Parry, ABC, 10552)
(ABC, 1st Parry, 1st Parry, 10553)
(ABC, 2nd Parry, ABC, 10555)
(ABC, 2nd Parry, 2nd Parry, 10556)
(ABC, 3rd Parry, ABC, 10558)
(ABC, 3rd Parry, 3rd Parry, 10559)
(ABC, X(2)-quadsquares, ABC*, 590)
(ABC, X(2)-quadsquares, medial*, 3068)
(ABC, X(2)-quadsquares, anticomplementary*, 8972)
(ABC, X(2)-quadsquares, X(2)-quadsquares*, 8972)
(ABC, X(1) reflection, ABC, 35)
(ABC, X(1) reflection, X(1) reflection, 1770)
(ABC, 1st Sharygin, ABC, 1580)
(ABC, 1st Sharygin, 1st Sharygin, 8424)
(ABC, 1st Sharygin, 2nd Sharygin, 8852)
(ABC, 2nd Sharygin, ABC, 8300)
(ABC, 2nd Sharygin, 2nd Sharygin, 8301)
(ABC, 2nd Sharygin, 1st Sharygin, 8852)
(ABC, T(-1,3), ABC, 10563)
(ABC, T(-1,3), T(-1,3), 3680)
(ABC, T(-1,3), excentral, 7991)
(ABC, T(-2,1), ABC, 519)
(ABC, T(-2,1), T(-2,1), 8)
(ABC, T(-2,1), excentral, 40)
(ABC, 1st tangential mid-arc, ABC, 1)
(ABC, 1st tangential mid-arc, 1st tangential mid-arc, 2089)
(ABC, Trinh, ABC, 10564)
(ABC, Trinh, Trinh, 2071)
(ABC, inner Vecten, ABC, 372)
(ABC, inner Vecten, inner Vecten, 4)
(ABC, inner Vecten, outer Vecten, 487)
(ABC, outer Vecten, ABC, 371)
(ABC, outer Vecten, outer Vecten, 4)
(ABC, outer Vecten, inner Vecten, 488)
(ABC, Yff central, ABC, 1)
(ABC, Yff central, Yff central, 174)
(ABC, inner Yff, ABC*, 498)
(ABC, inner Yff, medial*, 1)
(ABC, inner Yff, anticomplementary*, 3085)
(ABC, inner Yff, inner Yff*, 3085)
(ABC, inner Yff, outer Yff*, 2)
(ABC, outer Yff, ABC*, 499)
(ABC, outer Yff, medial*, 1)
(ABC, outer Yff, anticomplementary*, 3086)
(ABC, outer Yff, outer Yff*, 3086)
(ABC, outer Yff, inner Yff*, 2)
(ABC, inner Yff tangents, ABC*, 5552)
(ABC, inner Yff tangents, inner Yff tangents*, 10528)
(ABC, inner Yff tangents, outer Yff tangents*, 10530)
(ABC, outer Yff tangents, ABC*, 10527)
(ABC, outer Yff tangents, outer Yff tangents*, 10529)
(ABC, outer Yff tangents, inner Yff tangents*, 10530)
(Andromeda, Antlia, ABC, 10322)
(X(3) anticevian, X(3) circumcevian, X(3) anticevian, 1498)
(X(4) anticevian, circumorthic, X(4) anticevian, 24)
(anticomplementary, circummedial, anticomplementary, 22)
(anticomplementary, circummedial, circummedial, 10565)
(anticomplementary, Euler, ABC*, 3091)
(anticomplementary, Euler, medial*, 20)
(anticomplementary, Euler, anticomplementary*, 4)
(anticomplementary, Euler, Euler*, 3832)
(anticomplementary, Euler, anti-Euler*, 5)
(Atik, inverse-in-incircle, Atik, 8581)
(Atik, inverse-in-incircle, inverse-in-incircle, 10569)
(1st Brocard, 2nd Brocard, 1st Brocard, 10328)
(1st Brocard, 2nd Brocard, 2nd Brocard, 10329)
(1st Brocard, 5th Brocard, 1st Brocard, 10331)
(1st Brocard, 5th Brocard, 5th Brocard, 10332)
(1st Brocard, 6th Brocard, 1st Brocard*, 7876)
(1st Brocard, 6th Brocard, 6th Brocard*, 7791)
(1st Brocard, 6th Brocard, 1st anti-Brocard*, 6656)
(1st Brocard, 6th Brocard, 6th anti-Brocard*, 10333)
(1st Brocard, 1st anti-Brocard, ABC, 3407)
(1st Brocard, 1st anti-Brocard, medial, 10335)
(1st Brocard, 1st anti-Brocard, anticomplementary, 10336)
(1st Brocard, 1st anti-Brocard, 1st Brocard*, 2)
(1st Brocard, 1st anti-Brocard, 2nd Brocard*, 2)
(1st Brocard, 1st anti-Brocard, 6th Brocard*, 6656)
(1st Brocard, 1st anti-Brocard, 1st anti-Brocard*, 2)
(1st Brocard, 1st anti-Brocard, 6th anti-Brocard*, 10334)
(1st Brocard, 6th anti-Brocard, 6th Brocard*, 10333)
(1st Brocard, 6th anti-Brocard, 1st anti-Brocard*, 10334)
(2nd Brocard, 4th Brocard, ABC, 111)
(2nd Brocard, 4th Brocard, 4th anti-Brocard, 111)
(2nd Brocard, 1st anti-Brocard, 2nd Brocard)
(2nd Brocard, 1st anti-Brocard, 1st anti-Brocard)
(3rd Brocard, 5th Brocard, 1st Brocard, 2896)
(3rd Brocard, 5th Brocard, 3rd Brocard, 10337)
(3rd Brocard, 5th Brocard, 5th Brocard, 2896)
(3rd Brocard, 5th Brocard, 6th Brocard, 10338)
(3rd Brocard, 6th Brocard, 3rd Brocard, 10339)
(3rd Brocard, 6th Brocard, 5th Brocard, 10338)
(3rd Brocard, 6th Brocard, 6th Brocard, 10340)
(3rd Brocard, 5th anti-Brocard, 3rd Brocard, 10341)
(3rd Brocard, 5th anti-Brocard, 5th anti-Brocard, 10342)
(4th Brocard, orthocentroidal, 4th Brocard, 6)
(4th Brocard, orthocentroidal, orthocentroidal, 6)
(5th Brocard, 6th Brocard, 3rd Brocard, 10338)
(5th Brocard, 6th Brocard, 5th Brocard, 10343)
(5th Brocard, 6th Brocard, 6th Brocard, 10344)
(5th Brocard, 5th anti-Brocard, ABC*, 10345)
(5th Brocard, 5th anti-Brocard, medial*, 32)
(5th Brocard, 5th anti-Brocard, anticomplementary*, 10346)
(5th Brocard, 5th anti-Brocard, 5th Brocard*, 10347)
(5th Brocard, 5th anti-Brocard, 5th anti-Brocard*, 10348)
(6th Brocard, 1st anti-Brocard, ABC, 4)
(6th Brocard, 1st anti-Brocard, 1st Brocard*, 6656)
(6th Brocard, 1st anti-Brocard, 6th Brocard*, 6656)
(6th Brocard, 1st anti-Brocard, 1st anti-Brocard*, 5025)
(6th Brocard, 1st anti-Brocard, 6th anti-Brocard*, 10349)
(6th Brocard, 6th anti-Brocard, 1st Brocard*, 10333)
(6th Brocard, 6th anti-Brocard, 6th Brocard*, 10350)
(6th Brocard, 6th anti-Brocard, 1st anti-Brocard*, 10349)
(6th Brocard, 6th anti-Brocard, 6th anti-Brocard*, 10351)
(1st anti-Brocard, 6th anti-Brocard, 1st Brocard*, 10334)
(1st anti-Brocard, 6th anti-Brocard, 6th Brocard*, 10349)
(1st anti-Brocard, 6th anti-Brocard, 1st anti-Brocard*, 10352)
(1st anti-Brocard, 6th anti-Brocard, 6th anti-Brocard*, 10353)
(circummedial, circumorthic, ABC, 98)
(circummedial, circumsymmedial, ABC, 111)
(circumorthic, circumsymmedial, ABC, 74)
(2nd circumperp, circummedial, ABC, 105)
(2nd circumperp, circumorthic, ABC, 104)
(2nd circumperp, circumsymmedial, ABC, 106)
(1st Conway, 2nd Conway, ABC, 10429)
(1st Conway, 2nd Conway, 1st Conway*, 10430)
(1st Conway, 2nd Conway, 2nd Conway*, 10431)
(1st Conway, 4th Conway, 1st Conway, 10450)
(1st Conway, 4th Conway, 4th Conway, 10451)
(1st Conway, 5th Conway, ABC, 10461)
(1st Conway, 5th Conway, 1st Conway, 10462)
(1st Conway, 5th Conway, 4th Conway, 10463)
(1st Conway, 5th Conway, 5th Conway, 5208)
(2nd Conway, 4th Conway, ABC, 10449)
(2nd Conway, 4th Conway, 2nd Conway, 10452)
(2nd Conway, 4th Conway, 4th Conway, 10453)
(2nd Conway, 4th Conway, 5th Conway, 10454)
(2nd Conway, 5th Conway, 2nd Conway, 10464)
(Euler, anti-Euler, ABC*, 3090)
(2nd Conway, 5th Conway, 5th Conway, 10465)
(excentral, 2nd circumperp, excentral*, 3)
(excentral, 2nd circumperp, 1st circumperp*, 1)
(excentral, 2nd circumperp, 2nd circumperp*, 7987)
(excentral, 2nd circumperp, intouch*, 3601)
(excentral, 2nd circumperp, hexyl*, 3)
(excentral, 2nd circumperp, 2nd Conway*, 3522)
(excentral, 2nd circumperp, 3rd Conway*, 10470)
(excentral, 2nd circumperp, X(10) extraversion*, 20)
(excentral, 2nd circumperp, X(1) reflection*, 3612)
(excentral, 2nd circumperp, 3rd Euler*, 3624)
(excentral, 2nd circumperp, 4th Euler*, 5691)
(excentral, 2nd circumperp, Hutson intouch*, 1420)
(extouch, anticomplementary, ABC, 10405)
(extouch, anticomplementary, extouch, 2)
(extouch, anticomplementary, anticomplementary, 7)
(extouch, excentral, ABC, 84)
(extouch, excentral, extouch, 1)
(extouch, excentral, excentral, 56)
(extouch, tangential, ABC, 3435)
(extouch, tangential, extouch, 6)
(extouch, tangential, tangential, 1397)
(extouch, 2nd extouch, ABC, 4)
(extouch, 2nd extouch, extouch, 3419)
(extouch, 2nd extouch, 2nd extouch, 4)
(extouch, 2nd extouch, 3rd extouch, 4)
(extouch, 3rd extouch, ABC, 8809)
(extouch, 3rd extouch, extouch, 10373)
(extouch, 3rd extouch, 3rd extouch, 10374)
(extouch, 5th extouch, ABC, 10375)
(extouch, 5th extouch, extouch, 65)
(extouch, 5th extouch, 5th extouch, 10376)
(2nd extouch, 3rd extouch, ABC, 10378)
(2nd extouch, 3rd extouch, 2nd extouch, 10379)
(2nd extouch, 3rd extouch, 3rd extouch, 10380)
(2nd extouch, 3rd extouch, 4th extouch)
(2nd extouch, 4th extouch, 2nd extouch)
(2nd extouch, 4th extouch, 4th extouch, 10381)
(3rd extouch, 4th extouch, ABC, 10360)
(3rd extouch, 4th extouch, 3rd extouch, 10361)
(3rd extouch, 4th extouch, 4th extouch, 10362)
(3rd extouch, 4th extouch, 5th extouch, 10363)
(3rd extouch, 4th extouch, intouch, 10364)
(3rd extouch, 5th extouch, ABC, 10365)
(3rd extouch, 5th extouch, 3rd extouch, 10366)
(3rd extouch, 5th extouch, 4th extouch, 10363)
(3rd extouch, 5th extouch, 5th extouch, 10367)
(3rd extouch, 5th extouch, intouch, 10368)
(4th extouch, 5th extouch, ABC, 10369)
(4th extouch, 5th extouch, 3rd extouch, 10363)
(4th extouch, 5th extouch, 4th extouch, 10370)
(4th extouch, 5th extouch, 5th extouch, 10371)
(4th extouch, 5th extouch, intouch, 10372)
(Feuerbach, Apollonius, ABC, 2051)
(Feuerbach, Apollonius, Apollonius, 10406)
(Feuerbach, Apollonius, Feuerbach, 10407)
(inner Garcia, outer Garcia, ABC, 10570)
(inner Garcia, outer Garcia, outer Garcia, 10573)
(inner Garcia, outer Garcia, inner Garcia, 10572)
(inner Grebe, outer Grebe, ABC, 10513)
(half-altitude, orthic, ABC, 3)
(half-altitude, orthic, half-altitude, 185)
(half-altitude, orthic, orthic, 6000)
(half-altitude, reflection, ABC, 3)
(half-altitude, reflection, half-altitude, 10574)
(half-altitude, reflection, reflection, 10575)
(Hutson intouch, mid-arc, intouch, 10505)
(Hutson intouch, mid-arc, 2nd mid-arc, 10506)
(Hutson intouch, 2nd mid-arc, Hutson intouch, 10507)
(Hutson intouch, 2nd mid-arc, 2nd mid-arc, 10508)
(incentral, anticomplementary, ABC, 330)
(incentral, anticomplementary, incenrtal, 2)
(incentral, anticomplementary, anticomplementary, 75)
(incentral, 2nd circumperp, ABC, 58)
(incentral, tangential, ABC, 56)
(incentral, tangential, incentral, 6)
(incentral, tangential, tangential, 31)
(intouch, anticomplementary, ABC, 4373)
(intouch, anticomplementary, intouch, 2)
(intouch, anticomplementary, anticomplementary, 8)
(intouch, excentral, ABC, 9)
(intouch, excentral, Ascella*, 10383)
(intouch, excentral, Atik*, 10384)
(intouch, excentral, intouch*, 1)
(intouch, excentral, excentral*, 55)
(intouch, excentral, hexyl*, 1)
(intouch, excentral, 1st circumperp*, 57)
(intouch, excentral, 2nd circumperp*, 3601)
(intouch, excentral, 2nd extouch*, 10382)
(intouch, excentral, 1st Conway*, 7675)
(intouch, excentral, 2nd Conway*, 390)
(intouch, excentral, 3rd Conway*, 1)
(intouch, excentral, 3rd Euler*, 5219)
(intouch, excentral, 4th Euler*, 9581)
(intouch, excentral, X(10) extraversion*, 497)
(intouch, excentral, 6th mixtilinear*, 1)
(intouch, excentral, X(1) reflection*, 1)
(intouch, excentral, Hutson intouch*, 1)
(intouch, 2nd extouch, ABC, 9)
(intouch, 2nd extouch, Ascella*, 10391)
(intouch, 2nd extouch, Atik*, 10392)
(intouch, 2nd extouch, 1st circumperp*, 1708)
(intouch, 2nd extouch, 2nd circumperp*, 10393)
(intouch, 2nd extouch, 1st Conway*, 10394)
(intouch, 2nd extouch, 2nd Conway*, 5809)
(intouch, 2nd extouch, 3rd Euler*, 226)
(intouch, 2nd extouch, 4th Euler*, 10395)
(intouch, 2nd extouch, excentral, 10382)
(intouch, 2nd extouch, 2nd extouch*, 1864)
(intouch, 2nd extouch, 4th extouch)
(intouch, 2nd extouch, hexyl*, 10396)
(intouch, 2nd extouch, Hutson intouch*, 72)
(intouch, 2nd extouch, intouch*, 5728)
(intouch, 2nd extouch, inverse-in-incircle*, 226)
(intouch, 2nd extouch, 6th mixtilinear*, 10398)
(intouch, 2nd extouch, X(1) reflection*, 10399)
(intouch, 3rd extouch, ABC, 4)
(intouch, 3rd extouch, 2nd extouch, 4)
(intouch, 3rd extouch, 3rd extouch, 4)
(intouch, 3rd extouch, 4th extouch, 10364)
(intouch, 3rd extouch, 5th extouch, 10368)
(intouch, 3rd extouch, intouch, 10400)
(intouch, 4th extouch, ABC, 69)
(intouch, 4th extouch, 3rd extouch, 10364)
(intouch, 4th extouch, 4th extouch)
(intouch, 4th extouch, 5th extouch, 10372)
(intouch, 4th extouch, intouch, 10401)
(intouch, 5th extouch, ABC, 388)
(intouch, 5th extouch, intouch, 10404)
(intouch, 5th extouch, 3rd extouch, 10368)
(intouch, 5th extouch, 4th extouch, 10372)
(intouch, 5th extouch, 5th extouch, 388)
(intouch, mid-arc, ABC, 177)
(intouch, mid-arc, intouch, 10499)
(intouch, mid-arc, mid-arc, 10500)
(intouch, 2nd mid-arc, mid-arc, 10501)
(intouch, 2nd mid-arc, Hutson intouch, 10504)
(intouch, tangential, ABC, 3433)
(intouch, tangential, intouch, 6)
(intouch, tangential, tangential, 2175)
(inverse-in-incircle, inverse-in-excircles, excentral, 57)
(inverse-in-incircle, inverse-in-excircles, intouch, 57)
(inverse-in-incircle, inverse-in-excircles, inverse-in-incircle, 10520)
(inverse-in-incircle, inverse-in-excircles, inverse-in-excircles, 10521)
(inner Johnson, outer Johnson, ABC*, 10522
(inner Johnson, outer Johnson, medial*, 10523
(inner Johnson, outer Johnson, Euler*, 355)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, orthic*, 184)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, circumorthic*, 1614)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, dual of orthic*, 110)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, submedial*, 182)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 1st Kenmotu diagonals*, 10533)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd Kenmotu diagonals*, 10534)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, intangents*, 10535)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, extangents*, 10536)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, tangential*, 154)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita*, 10282)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd Euler*, 10539)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd isogonal of X(4)*, 10540)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, Trinh*, 6000)
(1st Kenmotu diagonals, 2nd Kenmotu diagonals, 2nd Ehrmann*, 2393)
(Lucas Brocard, Lucas(-1) Brocard, ABC, 10541)
(Lucas Brocard, Lucas(-1) Brocard, symmedial, 10542)
(Lucas Brocard, Lucas(-1) Brocard, tangential)
(Lucas Brocard, Lucas(-1) Brocard, circumsymmedial)
(Lucas Brocard, Lucas(-1) Brocard, Lucas Brocard, 574)
(Lucas Brocard, Lucas(-1) Brocard, Lucas(-1) Brocard, 574)
(Mandart-incircle, Hutson intouch, Hutson intouch, 10543)
(Mandart-incircle, Hutson intouch, Mandart-incircle, 10544)
(McCay, anti-McCay, ABC, 10484)
(McCay, anti-McCay, medial)
(McCay, anti-McCay, anticomplementary)
(McCay, anti-McCay, McCay*, 2)
(McCay, anti-McCay, anti-McCay*, 2)
(medial, anticomplementary, ABC*, 2)
(medial, anticomplementary, medial*, 2)
(medial, anticomplementary, anticomplementary*, 2)
(medial, anticomplementary, Euler*, 20)
(medial, anticomplementary, anti-Euler*, 5)
(medial, anticomplementary, X(3) cevian*, 3523)
(medial, anticomplementary, Johnson*, 3090)
(medial, anticomplementary, inner-Johnson*, 10584)
(medial, anticomplementary, outer-Johnson*, 10585)
(medial, anticomplementary, 5th mixtilinear/Caelum*, 3622)
(medial, anticomplementary, Ara*, 1995)
(medial, anticomplementary, outer Garcia*, 667)
(medial, anticomplementary, 3rd pedal of X(1)*, 10578)
(medial, anticomplementary, 3rd antipedal of X(1)*, 10582)
(medial, anticomplementary, 3rd pedal of X(3)*, 7494)
(medial, anticomplementary, 3rd antipedal of X(3)*, 7484)
(medial, anticomplementary, 3rd pedal of X(4)*, 7378)
(medial, anticomplementary, 3rd antipedal of X(4)*, 6997)
(medial, anticomplementary, T(1,2)*, 1125)
(medial, anticomplementary, X(2) orthocevian*, 6997)
(medial, anticomplementary, X(2)-quadsquares*, 7585)
(medial, anticomplementary, Mandart-incircle*, 5218)
(medial, anticomplementary, 5th Brocard*, 10583)
(medial, anticomplementary, 5th anti-Brocard*, 7793)
(medial, anticomplementary, inner Yff*, 3086)
(medial, anticomplementary, outer Yff*, 3085)
(medial, anticomplementary, 1st Johnson-Yff*, 10588)
(medial, anticomplementary, 2nd Johnson-Yff*, 10589)
(medial, anticomplementary, inner Yff tangents*, 10586)
(medial, anticomplementary, outer Yff tangents*, 10587)
(medial, circummedial, ABC, 251)
(medial, anti-Euler, ABC*, 4)
(medial, anti-Euler, medial*, 3090)
(medial, anti-Euler, anticomplementary*, 5)
(medial, anti-Euler, Euler*, 4)
(medial, anti-Euler, anti-Euler*, 4)
(medial, anti-Euler, Ara*, 10594)
(medial, anti-Euler, X(3) cevian*, 631)
(medial, anti-Euler, Johnson*, 3091)
(medial, anti-Euler, inner Johnson*, 10598)
(medial, anti-Euler, outer Johnson*, 10599)
(medial, anti-Euler, 5th mixtilinear/Caelum*, 10595)
(medial, anti-Euler, outer Garcia*, 5818)
(medial, anti-Euler, 3rd pedal of X(3)*, 7400)
(medial, anti-Euler, 3rd antipedal of X(3)*, 7395)
(medial, anti-Euler, 3rd pedal of X(4)*, 4)
(medial, anti-Euler, 3rd antipedal of X(4)*, 4)
(medial, anti-Euler, T(1,2)*, 946)
(medial, anti-Euler, Mandart-incircle*, 3085)
(medial, anti-Euler, inner Yff*, 497)
(medial, anti-Euler, outer Yff*, 388)
(medial, anti-Euler, inner Yff tangents*, 10596)
(medial, anti-Euler, outer Yff tangents*, 10597)
(medial, anti-Euler, 1st Johnson-Yff*, 10590)
(medial, anti-Euler, 2nd Johnson-Yff*, 10591)
(medial, anti-Euler, 2nd isogonal of X(1)*, 499)
(medial, 2nd Euler, ABC, 5)
(medial, 2nd Euler, medial, 1209)
(medial, 2nd Euler, 2nd Euler, 10600)
(medial, excentral, ABC, 57)
(medial, excentral, medial, 1)
(medial, excentral, excentral, 6)
(medial, submedial, medial, 141)
(medial, submedial, submedial, 2)
(medial, submedial, orthic, 2)
(medial, submedial, dual of orthic, 2)
(medial, tangential, ABC, 25)
(medial, tangential, medial, 6)
(medial, tangential, tangential, 32)
(inner Napoleon, outer Napoleon, inner Napoleon, 61)
(inner Napoleon, outer Napoleon, outer Napoleon, 62)
(1st Neuberg, 2nd Neuberg, ABC, 4)
(1st Neuberg, 2nd Neuberg, 1st Neuberg, 182)
(1st Neuberg, 2nd Neuberg, 2nd Neuberg, 511)
(orthic, anticomplementary, ABC, 2996)
(orthic, anticomplementary, orthic, 2)
(orthic, anticomplementary, anticomplementary, 69)
(orthic, circumorthic, ABC, 54)
(orthic, X(3)-Ehrmann, orthic*, 25)
(orthic, X(3)-Ehrmann, X(3)-Ehrmann*, 1495)
(orthic, Euler, ABC, 5)
(orthic, Euler, orthic, 6750)
(orthic, Euler, Euler, 3574)
(orthic, excentral, ABC, 90)
(orthic, excentral, orthic, 1)
(orthic, excentral, excentral, 3)
(orthic, submedial, orthic*, 5943)
(orthic, submedial, submedial*, 373)
(orthic, submedial, tangential*, 10602)
(orthic, submedial, 2nd Euler*, 5462)
(orthic, submedial, dual of orthic*, 5640)
(orthic, tangential, ABC, 3)
(orthic, tangential, orthic*, 6)
(orthic, tangential, tangential*, 184)
(orthic, tangential, submedial, 10601)
(orthic, tangential, circumorthic*, 7592)
(orthic, tangential, dual of orthic*, 1993)
(orthic, tangential, 2nd Euler*, 155)
(orthic, tangential, 2nd Ehrmann*, 10602)
(orthic, tangential, Trinh, 10605)
(orthocentroidal, anti-orthocentroidal, orthocentroidal*, 10545)
(orthocentroidal, anti-orthocentroidal, anti-orthocentroidal*, 10546)
(1st orthosymmedial, 2nd orthosymmedial, 1st orthosymmedial, 251)
(1st orthosymmedial, 2nd orthosymmedial, 2nd orthosymmedial, 10551)
(1st Parry, 3rd Parry, 2nd Parry, 23)
(2nd Parry, 3rd Parry, 2nd Parry, 10561)
(2nd Parry, 3rd Parry, 3rd Parry, 10562)
(reflection, ABC, ABC, 3)
(reflection, ABC, reflection, 30)
(1st Sharygin, 2nd Sharygin, ABC, 8852)
(inner Soddy, outer Soddy, ABC, 7)
(inner Soddy, outer Soddy, inner Soddy, 1373)
(inner Soddy, outer Soddy, outer Soddy, 1374)
(inner Soddy tangential, outer Soddy tangential, ABC, 7)
(inner Soddy tangential, outer Soddy tangential, inner Soddy tangential, 1373)
(inner Soddy tangential, outer Soddy tangential, outer Soddy tangential, 1374)
(inner Soddy tangential, outer Soddy tangential, inner Soddy, 1)
(inner Soddy tangential, outer Soddy tangential, outer Soddy, 1)
(inner-squares, outer-squares, ABC, 2)
(inner-squares, outer-squares, inner-squares, 10576)
(inner-squares, outer-squares, outer-squares, 10577)
(symmedial, anticomplementary, ABC, 2998)
(symmedial, anticomplementary, anticomplementary, 76)
(symmedial, anticomplementary, symmedial, 2)
(symmedial, circumsymmedial, ABC, 6)
(symmedial, excentral, ABC, 87)
(symmedial, excentral, symmedial, 1)
(symmedial, excentral, excentral, 2)
(tangential, circumsymmedial, ABC, 6)
(inner Yff tangents, outer Yff tangents, ABC*, 10530)
(inner Yff tangents, outer Yff tangents, medial*, 1)
(inner Yff tangents, outer Yff tangents, outer Garcia*, 1)

underbar

X(10290) =  PERSPECTOR OF ABC AND MID-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

Barycentrics    1/[2a6 + a2(b2 - c2)2 - b2c2(b2 + c2)] : :

X(10290) lies on the Kiepert hyperbola and these lines: {99,3407} et al

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

Barycentrics    [2a6 + a2(b2 - c2)2 - b2c2(b2 + c2)][a6 + a2(b4 + b2c2 + c4) - (b2 + c2)(b4 + c4)] : :

X(10291) lies on these lines: {2,4159}, {99,3407}, {187,736} et al


X(10292) =  HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND MID-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    b^8 + c^8 + 2a^6(b^2 + c^2) + 3a^4(b^2 + c^2)^2 + 2a^2(b^2 + c^2)^3 + 2b^2c^2(b^4 + 2b^2c^2 + c^4) : :

X(10292) lies on these lines: {2,32}, {114,7930} et al

X10292) = complement of X(10345)


X(10293) =  PERSPECTOR OF ABC AND MID-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

Barycentrics    1/[a^8 - 2a^6(b^2 + c^2) + 11a^4b^2c^2 + 2a^2(b^2 + c^2)(b^4 - 4b^2c^2 + c^4) - (b^2 - c^2)^2(b^4 + 5b^2c^2 + c^4)] : :

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} et al

X(10293) = isogonal conjugate of X(7464)


X(10294) =  PERSPECTOR OF ORTHIC TRIANGLE AND MID-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

Barycentrics    [a^6 - 4a^4(b^2 + c^2) + a^2(5b^4 - 3b^2c^2 + 5c^4) - 2(b^2 - c^2)^2(b^2 + c^2)]*[4a^6 - 7a^4(b^2 + c^2) + 2a^2(b^4 + 3b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - a^2) : :

X(10294) lies on these lines: {4,5609}, {6,7699} et al

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

Barycentrics    [4a^6 - 7a^4(b^2 + c^2) + (b^2 - c^2)^2(b^2 + c^2) + 2a^2(b^4 + 3b^2c^2 + c^4)]/(b^2 + c^2 - a^2) : :

X(10295) lies on these lines: {2,3}, {50,112}, {99,340}, {107,841}, {511,1986}, {523,9409} et al

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(10296) = REFLECTION OF X(20) IN DE LONGCHAMPS LINE

Barycentrics    3a^10 - 4a^8(b^2 + c^2) - a^6(4b^4 - 9b^2c^2 + 4c^4) + 2a^4(b^2 + c^2)(3b^4 - 5b^2c^2 + 3c^4) + a^2(b^2 - c^2)^2(b^4 - 5b^2c^2 + c^4) - 2(b^2 - c^2)^4(b^2 + c^2) : :

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} et al

X(10296) = reflection of X(20) in X(858)
X(10296) = anticomplement of X(10295)
X(10296) = inverse-in-circumcircle of X(10298)
X(10296) = inverse-in-de-Longchamps-circle of X(376)


X(10297) = RADICAL TRACE OF NINE-POINT CIRCLE AND JOHNSON CIRCLE

Barycentrics    [2a^8 + a^6(b^2 + c^2) - a^4(5b^4 - 6b^2c^2 + 5c^4) - a^2(b^2 - c^2)^2(b^2 + c^2) + 3(b^2 - c^2)^4](b^2 + c^2 - a^2) : :

The Johnson circle is the circumcircle of the Johnson triangle, and it is the reflection of the circumcircle of ABC in X(5).

X(10297) lies on these lines: {2,3}, {523,6334} et al

X(10927) = reflection of X(468) in X(5)
X(10927) = complement of X(10295)
X(10927) = inverse-in-{circumcircle, nine-point circle}-inverter of X(7493)


X(10298) = HARMONIC CENTER OF CIRCUMCIRCLE AND DE LONGCHAMPS CIRCLE

Barycentrics    tan B + tan C - tan A + cot D/2 : :, where cot D/2 = 2S/(a2 + b2 + c2 - 6R2)
Barycentrics    2a6 - 9R2 a4 + 3R2(b2 - c2)2 - 2a2(b4 + c4 - 3R2(b2 + c2)) : :

X(10298) lies on these lines: {2,3} et al

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(10299) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND MID-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    11a4 - 12a2(b2 + c2) + (b2 - c2)2 : :

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} et al

X(10299) = anticomplement of X(5079)


X(10300) = HOMOTHETIC CENTER OF 3rd PEDAL TRIANGLE OF X(3) AND MID-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    (b2 + c2 - a2)[2a4 + 5a2(b2 + c2) + 3(b2 - c2)2] : :

X(10300) lies on these lines: {2,3} et al

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

Barycentrics    (4a2 + b2 + c2)/(b2 + c2 - a2) : :

X(10301) lies on these lines: {2,3} et al

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)

Barycentrics    1/(4a2 + b2 + c2) : :

X(10302) lies on the Kiepert hyperbola and these lines: {2,5355}, {4,7883}, {83,524}, {141,671} et al

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

Trilinears    5 cos A + 3 cos B cos C : :
Trilinears    3 sec A + 5 sec B sec C : :
Barycentrics    7a^4 - 10a^2(b^2 + c^2) + 3(b^2 - c^2)^2 : :
Barycentrics    7a4 + 3b4 + 3c4 - 10a2(b2 + c2) - 6b2c2 : :

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} et al

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

Trilinears    3 cos B cos C - 2 sin B sin C - 3 cot A (sin B cos C + sin C cos B) : :

X(10304) lies on these lines: {2,3}, {56,10385}, {98,8591}, {193,3098}, {538,6194}, {944,3654}, {3679,4297} et al

X(10304) = anticomplement of X(3545)
X(10304) = Thomson isogonal conjugate of X(5544)


X(10305) = PERSPECTOR OF ABC AND MID-TRIANGLE OF INTOUCH AND HEXYL TRIANGLES

Barycentrics    1/(a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c+4 a^2 b^2 c-2 a b^3 c-3 b^4 c-2 a^3 c^2+4 a^2 b c^2-6 a b^2 c^2+4 b^3 c^2+2 a^2 c^3-2 a b c^3+4 b^2 c^3+a c^4-3 b c^4-c^5) : :

The trilinear polar of X(10305) passes through X(650).

X(10305) lies on these lines: {1,2096}, {8,1071}, {9,631}, {20,1320} et al

X(10305) = isogonal conjugate of X(10306)


X(10306) = INTANGENTS-TO-EXTANGENTS SIMILARITY IMAGE OF X(3)

Barycentrics    a^2 (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+2 a^3 b c+4 a^2 b^2 c-2 a b^3 c-3 b^4 c-2 a^3 c^2+4 a^2 b c^2-6 a b^2 c^2+4 b^3 c^2+2 a^2 c^3-2 a b c^3+4 b^2 c^3+a c^4-3 b c^4-c^5) : :

X(10306) lies on these lines: {4,1260}, {84,6765}, {382,5840}, {3311,5415}, {3312,5416} et al

X(10306) = isogonal conjugate of X(10305)


X(10307) = PERSPECTOR OF ABC AND MID-TRIANGLE OF INTOUCH AND 6th MIXTILINEAR TRIANGLES

Barycentrics    1/[a^4 - 2a^3(b + c) + 10a^2bc + 2a(b + c)(b^2 - 4bc + c^2) - (b^2 + 4bc + c^2)(b - c)^2] : :

The trilinear polar of X(10307) passes through X(650).

X(10307) lies on these lines: {1,7955}, {8,971}, {9,2272}, {516,3680} et al

X(10307) = isogonal conjugate of X(6244)


X(10308) = PERSPECTOR OF ABC AND MID-TRIANGLE OF HEXYL AND 6th MIXTILINEAR TRIANGLES

Trilinears    1/[2a^3 - b^3 - c^3 + (a^2 + bc)(b + c) - 2a(b^2 + c^2 + bc)] : :

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} et al

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)

Trilinears    1/[a cos2 A - (a + 2b + 2c)(cos A) + (b cos B + c cos C)(1 + cos A)] : :
Barycentrics    1/[a^5 - a^4(b + c) - 2a^3(b - c)^2 + 2a^2(b + c)(b^2 + c^2) + a(b^4 - 4b^3c - 2b^2c^2 - 4bc^3 + c^4) - (b - c)^2(b + c)^3] : :

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} et al

X(10309) = isogonal conjugate of X(10310)
X(10309) = perspector of ABC and reflection of extouch triangle in X(1158)


X(10310) = REFLECTION OF X(56) IN X(3)

Trilinears    a cos2 A - (a + 2b + 2c)(cos A) + (b cos B + c cos C)(1 + cos A) : :
Barycentrics    a^2[a^5 - a^4(b + c) - 2a^3(b - c)^2 + 2a^2(b + c)(b^2 + c^2) + a(b^4 - 4b^3c - 2b^2c^2 - 4bc^3 + c^4) - (b - c)^2(b + c)^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} et al

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(10311) = HOMOTHETIC CENTER OF ORTHIC TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    tan A cos(A - ω) : :

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} et al

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

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

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}, {3090,10314} et al

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

Barycentrics    a^2[a^8 - a^6(b^2 + c^2) - a^4(b^4 + b^2c^2 + c^4) + a^2(b^2 + c^2)(b^4 + c^4) + b^2c^2(b^2 - c^2)^2] : :

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} et al

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

Barycentrics    a^2[a^8 - a^6(b^2 + c^2) - a^4(b^2 - c^2)^2 + a^2(b^6 - 5b^4c^2 - 5b^2c^4 + c^6) + 4b^2c^2(b^2 - c^2)^2] : :

X(10314) lies on these lines: {2,95}, {5,32}, {6,1196}, {25,216}, {112,3545}, {1656,10316}, {3090,10312}, {5055,10317}, {9816,10315} et al


X(10315) = HOMOTHETIC CENTER OF EXTANGENTS TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

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

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} et al


X(10316) = HOMOTHETIC CENTER OF 2nd EULER TRIANGLE AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    (cos A)(sin 2A - tan ω) : :
Barycentrics    a4(b4 + c4 - a4)(b2 + c2 - a2) : :

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} et al

X(10316) = X(92)-isoconjugate of X(66)


X(10317) = HOMOTHETIC CENTER OF 2nd ISOGONAL TRIANGLE OF X(4) AND MID-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    (cos A)(2 sin 2A - 3 tan ω) : :
Barycentrics    a4(b4 + c4 - a4 - b2c2)(b2 + c2 - a2) : :

X(10317) lies on these lines: {3,6}, {5,10312}, {23,8744}, {26,8743}, {30,112}, {53,7540}, {186,10098}, {381,10311}, {5055,10314} et al

2nd isogonal triangles are defined at X(36).

X(10317) = X(92)-isoconjugate of X(67)
X(10317) = crossdifference of every pair of points on line X(427)X(523) (the radical axis of anticomplementary circle and tangential circle)


X(10318) = HOMOTHETIC CENTER OF ABC AND MID-TRIANGLE OF LUCAS HOMOTHETIC AND LUCAS(-1) HOMOTHETIC TRIANGLES

Barycentrics    (sin2 A)/(sin2 A - sin2 B sin2 C) : :

X(10318) lies on these lines: {39,493}, {3199,5008} et al

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

Trilinears    b2(c - a)(1 - cos A sec B) + c2(a - b)(1 - cos A sec C) : :
Trilinears    [a3 + a2(b + c) + a(b + c)2 + (b - c)2(b + c)](b2 + c2 - a2) : :

X(10319) lies on these lines: {1,3}, {2,19}, {9,440}, {20,1891}, {33,4220}, {63,69}, {77,2359}, {226,1766}, {1072,1074} et al


X(10320) = HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER YFF TRIANGLES

Barycentrics    a^7-a^6 b-3 a^5 b^2+3 a^4 b^3+3 a^3 b^4-3 a^2 b^5-a b^6+b^7-a^6 c+2 a^5 b c+a^4 b^2 c-4 a^3 b^3 c+a^2 b^4 c+2 a b^5 c-b^6 c-3 a^5 c^2+a^4 b c^2-2 a^3 b^2 c^2+2 a^2 b^3 c^2+a b^4 c^2-3 b^5 c^2+3 a^4 c^3-4 a^3 b c^3+2 a^2 b^2 c^3-4 a b^3 c^3+3 b^4 c^3+3 a^3 c^4+a^2 b c^4+a b^2 c^4+3 b^3 c^4-3 a^2 c^5+2 a b c^5-3 b^2 c^5-a c^6-b c^6+c^7 : :

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} et al


X(10321) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER YFF TRIANGLES

Barycentrics    a^7 - a^6(b + c) - a^5(3b^2 - 2bc + 3c^2) + a^4(b + c)(3b^2 - 2bc + 3c^2) + a^3(3b^4 - 4b^3c - 6b^2c^2 - 4bc^3 + 3c^4) - a^2(b - c)^2(b + c)(3b^2 + 2bc + 3c^2) - a(b - c)^4(b + c)^2 + (b - c)^4(b + c)^3 : :

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} et al


X(10322) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ANDROMEDA AND ANTLIA TRIANGLES

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

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} et al

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

Barycentrics    a^2[a^8 - 2a^6(b^2 + c^2) - 6a^4b^2c^2 + 2a^2(b^2 + c^2)^3 - (b^2 - c^2)^2(b^4 + c^4)] : :

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} et al

X(10323) = anticomplement of X(7403)


X(10324) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND ATIK TRIANGLE

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

X(10324) lies on these lines: {1,2}, {346,3062}, {3695,9948} et al

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

Barycentrics    a^5 + 3a^4(b + c) - 2a^3(7b^2 - 4bc + 7c^2) + 2a^2(b + c)(7b^2 - 4bc + 7c^2) - a(b^2 + 6bc + c^2)(3b^2 - 2bc + 3c^2) - (b - c)^2(b + c)^3 : :

X(10325) lies on these lines: {7,8}, {346,3062}, {391,2297} et al

X(10325) = {X(3062),X(10326)}-harmonic conjugate of X(10324)


X(10326) = {X(10324),X(10325)}-HARMONIC CONJUGATE OF X(3062)

Barycentrics    a^5 + 6a^4(b + c) - a^3(22b^2 - 4bc + 22c^2) + 20a^2(b + c)(b^2 + c^2) - a(3b^4 + 28b^3c + 2b^2c^2 + 28bc^3 + 3c^4) - 2(b - c)^2(b + c)^3 : :

X(10326) lies on these lines: {8,5542}, {346,3062} et al


X(10327) = PERSPECTOR OF AYME TRIANGLE AND CROSS-TRIANGLE OF ABC AND AYME TRIANGLE

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

The trilinear polar of X(10327) passes through X(2509).

X(10327) lies on these lines: {1,2}, {4,3701}, {55,3932} et al

X(10327) = perspector of Ayme triangle and cevian triangle of X(304)
X(10327) = anticomplement of X(614)


X(10328) = PERSPECTOR OF 1st BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd BROCARD TRIANGLES

Barycentrics    a6 + a2b2c2 + b4c2 + b2c4 : :

X(10328) lies on these lines: {2,4048}, {6,6664}, {76,1501}, {99,8041}, {110,4074}, {384,3051} et al

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

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

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} et al

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)

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

X(10330) lies on these lines: {2,4048}, {69,2916}, {99,110} et al

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

Barycentrics    a^14 b^2+3 a^12 b^4+4 a^10 b^6+3 a^8 b^8+a^6 b^10+a^14 c^2+5 a^12 b^2 c^2+9 a^10 b^4 c^2+10 a^8 b^6 c^2+8 a^6 b^8 c^2+3 a^4 b^10 c^2+a^2 b^12 c^2+3 a^12 c^4+9 a^10 b^2 c^4+14 a^8 b^4 c^4+17 a^6 b^6 c^4+12 a^4 b^8 c^4+5 a^2 b^10 c^4+b^12 c^4+4 a^10 c^6+10 a^8 b^2 c^6+17 a^6 b^4 c^6+16 a^4 b^6 c^6+9 a^2 b^8 c^6+2 b^10 c^6+3 a^8 c^8+8 a^6 b^2 c^8+12 a^4 b^4 c^8+9 a^2 b^6 c^8+3 b^8 c^8+a^6 c^10+3 a^4 b^2 c^10+5 a^2 b^4 c^10+2 b^6 c^10+a^2 b^2 c^12+b^4 c^12 : :

X(10331) lies on these lines: {32,76}, {2896,10332} et al


X(10332) = PERSPECTOR OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th BROCARD TRIANGLES

Barycentrics    a^2 (a^12 b^2+4 a^10 b^4+4 a^8 b^6+2 a^6 b^8-a^4 b^10-a^2 b^12+a^12 c^2+6 a^10 b^2 c^2+9 a^8 b^4 c^2+7 a^6 b^6 c^2+3 a^4 b^8 c^2-a^2 b^10 c^2+4 a^10 c^4+9 a^8 b^2 c^4+10 a^6 b^4 c^4+9 a^4 b^6 c^4+4 a^2 b^8 c^4+b^10 c^4+4 a^8 c^6+7 a^6 b^2 c^6+9 a^4 b^4 c^6+6 a^2 b^6 c^6+2 b^8 c^6+2 a^6 c^8+3 a^4 b^2 c^8+4 a^2 b^4 c^8+2 b^6 c^8-a^4 c^10-a^2 b^2 c^10+b^4 c^10-a^2 c^12) : :

X(10332) lies on these lines: {32,694}, {2896,10331} et al


X(10333) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st AND 6th BROCARD TRIANGLES

Barycentrics    a^8 + a^4(b^4 + c^4) + a^2(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) + b^2c^2(b^4 + b^2c^2 + c^4) : :

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} et al

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

Barycentrics    a^8 + a^6(b^2 + c^2) - a^4b^2c^2 + a^2(b^2 + c^2)(b^4 + c^4) + b^2c^2(b^4 + b^2c^2 + c^4) : :

X(10334) lies on these lines: {2,98}, {3,10333}, {6,9865}, {32,7779}, {39,10000}, {83,194} et al

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

Barycentrics    [(b2 + c2)2 - b2c2](2a4 + a2b2 + a2c2 - b2c2) : :

X(10335) lies on these lines: {2,698}, {3,194}, {6,8290}, {99,3407} et al


X(10336) = PERSPECTOR OF ANTICOMPLEMENTARY TRIANGLE AND CROSS-TRIANGLE OF 1st BROCARD AND 1st ANTI-BROCARD TRIANGLES

Barycentrics    3a^8 + b^8 + c^8 + 3a^6(b^2 + c^2) + 3a^4(2b^4 + b^2c^2 + 2c^4) + 3a^2(b^2 + c^2)(b^4 + c^4) - b^6c^2 - b^2c^6 : :

X(10336) lies on these lines: {2,4121}, {6,147}, {20,32}, {148,3407}, {194,10583} et al


X(10337) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th BROCARD TRIANGLES

Barycentrics    a^10(b^2 + c^2) + a^8(b^4 + 3b^2c^2 + c^4) + a^6(b^2 + c^2)(2b^4 + 3b^2c^2 + 2c^4) + a^4(b^2 + c^2)^4 + a^2b^2c^2(b^2 + c^2)^3 + b^4c^4(b^4 + b^2c^2 + c^4) : :

X(10337) lies on these lines: {2,32}, {384,3118}, {9983,10339} et al


X(10338) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th BROCARD TRIANGLES

Barycentrics    a^2 (a^6 b^6+2 a^4 b^8+a^2 b^10+a^8 b^2 c^2+2 a^6 b^4 c^2+5 a^4 b^6 c^2+3 a^2 b^8 c^2+2 a^6 b^2 c^4+6 a^4 b^4 c^4+5 a^2 b^6 c^4+b^8 c^4+a^6 c^6+5 a^4 b^2 c^6+5 a^2 b^4 c^6+b^6 c^6+2 a^4 c^8+3 a^2 b^2 c^8+b^4 c^8+a^2 c^10) : :

X(10338) lies on these lines: {32,10339}, {69,194}, {384,3118} et al

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

Barycentrics    a^2[a^6(b^2 + c^2) + a^4(2b^2 + c^2)(b^2 + 2c^2) + a^2(b^2 + c^2)^3 + b^4c^4] : :

X(10339) lies on these lines: {5,5354}, {6,2896}, {32,10338}, {384,3051}, {1369,6656}, {9983,10337} et al


X(10340) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 6th BROCARD TRIANGLES

Barycentrics    csc A cot A csc(A + ω) - csc B cot B csc(B + ω) - csc C cot C csc(A + ω) : :
Barycentrics    sin 2A csc(A + ω) csc(A - ω) - sin 2B csc(B + ω) csc(B - ω) - sin 2C csc(C + ω) csc(C - ω) : :
Barycentrics    a^6(b^4 + 3b^2c^2 + c^4) + a^4(b^2 + c^2)(b^4 + c^4) - a^2b^2c^2(b^4 + b^2c^2 + c^4) - b^4c^4(b^2 + c^2) : :

X(10340) lies on these lines: {4,8878}, {32,10344}, {69,194}, {384,3051} et al


X(10341) = PERSPECTOR OF 3rd BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    a^8(b^4 + b^2c^2 + c^4) + a^2b^4c^4(b^2 + c^2) + b^6c^6 : :

X(10341) lies on these lines: {32,3978}, {39,83}, {194,1186} et al


X(10342) = PERSPECTOR OF 5th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 3rd BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    a^8(b^4 + c^4) - a^6b^2c^2(b^2 + c^2) + a^2b^4c^4(b^2 + c^2) + b^6c^6 : :

X(10342) lies on these lines: {32,3978}, {6,194}, {83,3117} et al


X(10343) = PERSPECTOR OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th AND 6th BROCARD TRIANGLES

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

X(10343) lies on these lines: {32,10338}, {141,384}, {9983,10344} et al


X(10344) = PERSPECTOR OF 6th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th AND 6th BROCARD TRIANGLES

Barycentrics    a^12(b^2 + c^2) + a^10(2 b^4 + 3b^2c^2 + 2c^4) + a^8(b^6 + 5b^4c^2 + 5b^2c^4 + c^6) - a^6(b^2 + c^2)^2(b^4 - 3b^2c^2 + c^4) - a^4(b^10 - 3b^6c^4 - 3b^4c^6 + c^10) + a^2b^2c^2(b^8 + 3b^6c^2 + 5b^4c^4 + 3b^2c^6 + c^8) + b^4c^4(b^6 + 2b^4c^2 + 2b^2c^4 + c^6) : :

X(10344) lies on these lines: {20,1352}, {32,10340}, {384,3118}, {9983,10343} et al


X(10345) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    a^8 + a^6(b^2 + c^2) + a^4(2b^4 + 3b^2c^2 + 2c^4) + a^2(b^6 + 3b^4c^2 + 3b^2c^4 + c^6) + b^2c^2(b^4 + b^2c^2 + c^4) : :

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} et al

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

Barycentrics    2a^8 + a^6(b^2 + c^2) + 3a^4(b^4 + b^2c^2 + c^4) + a^2(b^2 + c^2)^3 + b^2c^2(b^4 + b^2 c^2 + c^4) : :

X(10346) lies on these lines: {2,32}, {39,10353}, {194,10000}, {2080,10357}, {3398,9862}, {3399,9821}, {3407,6655}, {7827,9878} et al


X(10347) = HOMOTHETIC CENTER OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    a^8 + 2a^4(b^4 + b^2c^2 + c^4) + a^2(b^2 + c^2)^3 + b^2c^2(b^4 + b^2c^2 + c^4) : :

X(10347) lies on these lines: {2,32}, {39,10333}, {98,7919}, {99,737}, {182,9862}, {3398,7859}, {3407,7761}, {5171,10357}, {7760,9983} et al


X(10348) = HOMOTHETIC CENTER OF 5th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 5th BROCARD AND 5th ANTI-BROCARD TRIANGLES

Barycentrics    2a^8 + 2a^6(b^2 + c^2) + a^4(3b^4 + 4b^2c^2 + 3c^4) + a^2(b^2 + c^2)^3 + b^2c^2(b^4 + b^2c^2 + c^4) : :

X(10348) lies on these lines: {2,32}, {6,10000}, {7877,10333} et al


X(10349) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 6th BROCARD AND 1st ANTI-BROCARD TRIANGLES

Barycentrics    a^8 - 2a^4b^2c^2 + a^2(b^6 + c^6) + b^2c^2(b^4 + c^4) : :

X(10349) lies on these lines: {2,3398}, {3,10350}, {6,76}, {32,325}, {98,7887}, {182,6656}, {3329,10345}, {3407,7785} et al

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

Barycentrics    a^8 - a^6(b^2 + c^2) + a^4(b^4 - b^2c^2 + c^4) + a^2(b^2 + c^2)(b^4 + c^4) + b^2c^2(b^4 + c^4) : :

X(10350) lies on these lines: {2,32}, {3,10349}, {384,511}, {5182,7833} et al


X(10351) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 6th BROCARD AND 6th ANTI-BROCARD TRIANGLES

Barycentrics    2a^8 - 4a^4b^2c^2 + a^2(b^6 + c^6) + b^2c^2(b^4 + b^2c^2 + c^4) : :

X(10351) lies on these lines: {3,10353}, {6,194}, {32,7799}, {83,5309}, {1692,9983}, {5182,7833}} et al


X(10352) = HOMOTHETIC CENTER OF 1st ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st and 6th ANTI-BROCARD TRIANGLES

Barycentrics    a^8 + a^6(b^2 + c^2) - a^4(b^4 + 3b^2c^2 + c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + b^2c^2(b^4 + c^4) : :

X(10352) lies on these lines: {2,98}, {3,10349}, {6,5976}, {32,620}, {39,83}, {115,7803} et al


X(10353) = HOMOTHETIC CENTER OF 6th ANTI-BROCARD TRIANGLE AND CROSS-TRIANGLE OF 1st and 6th ANTI-BROCARD TRIANGLES

Barycentrics    2a^8 + a^6(b^2 + c^2) - a^4(b^4 + 5b^2c^2 + c^4) + a^2(b^2 - c^2)^2(b^2 + c^2) + b^2c^2(b^4 + b^2c^2 + c^4) : :

X(10353) lies on these lines: {2,98}, {3,10351}, {6,8290}, {39,10346}, {83,148}, {99,7772}, {194,5149} et al


X(10354) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 4th ANTI-BROCARD TRIANGLE

Barycentrics    a^2[(a^2 + b^2 + c^2)^2 - 9b^2c^2](2a^2 - b^2 - c^2)/(5a^2 - b^2 - c^2) : :

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} et al

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

Barycentrics    a^2 (a^2+b^2-5 c^2) (a^2-5 b^2+c^2) (7 a^6-3 a^4 b^2-9 a^2 b^4+b^6-3 a^4 c^2+9 a^2 b^2 c^2+3 b^4 c^2-9 a^2 c^4+3 b^2 c^4+c^6) : :

X(10355) lies on these lines: {3,111} et al

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

Barycentrics    a^8 + a^4(b^4 + 6b^2c^2 + c^4) + 6a^2b^2c^2(b^2 + c^2) - 2(b^8 - b^6c^2 - b^2c^6 + c^8) : :

X(10356) lies on these lines: {2,9873}, {3,7914}, {4,3096}, {5,32}, {30,7822}, {98,7932}, {182,7859}, {576,7877} et al


X(10357) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th BROCARD TRIANGLE

Barycentrics    a^8+3 a^6 b^2-2 a^4 b^4-3 a^2 b^6+b^8+3 a^6 c^2-3 a^4 b^2 c^2-9 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4-9 a^2 b^2 c^4-3 a^2 c^6-b^2 c^6+c^8 : :

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} et al


X(10358) = HOMOTHETIC CENTER OF EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th ANTI-BROCARD TRIANGLE

Barycentrics    a^8 - 2a^6 (b^2 + c^2) - a^4 (b^4 + 6b^2c^2 + c^4) + 2a^2 (b^6 - 2b^4c^2 - 2b^2 c^4 + c^6) + 2b^2c^2(b^2 - c^2)^2 : :

X(10358) lies on these lines: {2,5171}, {3,6683}, {4,83}, {5,32}, {6,6248}, {76,576} et al


X(10359) = HOMOTHETIC CENTER OF ANTI-EULER TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th ANTI-BROCARD TRIANGLE

Barycentrics    2a^8 - a^6(b^2 + c^2) - a^4(2b^4 + 9b^2c^2 + 2c^4) + a^2(b^6 - 5b^4c^2 - 5b^2c^4 + c^6) + b^2c^2(b^2 - c^2)^2 : :

X(10359) lies on these lines: {2,3398}, {3,3329}, {4,83}, {5,7875}, {32,631}, {76,575}, {98,3090}, {140,3793}, {9862,10345} et al


X(10360) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

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

X(10360) lies on these lines: {4,65}, {7,1899}, {8,1425}, {57,6776}, {69,1439}, {2898,5929}, {5930,8815} et al

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

Barycentrics    [a^8 - a^7(b + c) - 2a^6(b^2 + bc + c^2) + a^5(b - c)^2(b + c) + 2a^4(b^4 + b^3c + bc^3 + c^4) + a^3(b - c)^2(b + c)^3 - 2a^2(b^2 - c^2)^2(b^2 + bc + c^2) - a(b - c)^4(b + c)^3 + (b - c)^2(b + c)^4(b^2 + c^2)]/(b + c - a) : :

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} et al


X(10362) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

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

X(10362) lies on these lines: {4,5933}, {65,10363}, {5929,10361}, {5930,10370} et al


X(10363) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 4th EXTOUCH TRIANGLES

Barycentrics    [a^11 + 2a^10(b + c) - 2a^9(b^2 + bc + c^2) - a^8(b + c)(7b^2 + 6bc + 7c^2) - 2a^7(b + c)^2(b^2 + 5bc + c^2) + 2a^6(b + c)(4b^4 + b^3c + 4b^2c^2 + bc^3 + 4c^4) + 2a^5(b + c)^2(4b^4 + 5b^3c - 4b^2c^2 + 5bc^3 + 4c^4) - 2a^4(b^7 + 2b^5c^2 + 5b^4c^3 + 5b^3c^4 + 2b^2c^5 + c^7) - a^3(b + c)^2(b^2 + c^2)(7b^4 + 4b^3c - 14b^2c^2 + 4bc^3 + 7c^4) - 2a^2(b - c)^2(b + c)^3(b^2 + c^2)(b^2 + bc + c^2) + 2a(b - c)^2(b + c)^4(b^4 + 2b^3c + 4b^2c^2 + 2bc^3 + c^4) + (b - c)^2(b + c)^7(b^2 + c^2)]/(b + c - a) : :

X(10363) lies on these lines: {4,10369}, {65,10362}, {5929,10366}, {5930,6737} et al

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

Barycentrics    a^8 + 2a^7(b + c) + 2a^6bc - 2a^5(b - c)^2(b + c) + 2a^4bc(b + c)^2 + 2a^3(b^5 - b^4c + 2b^3c^2 + 2b^2c^3 - bc^4 + c^5) - 2a^2bc(b^2 - c^2)^2 - 2a(b - c)^2(b^5 + 3b^4c + 6b^3c^2 + 6b^2c^3 + 3bc^4 + c^5) - (b - c)^2(b + c)^4(b^2 + c^2) : :

X(10364) lies on these lines: {4,69}, {7,1899}, {65,10362}, {5930,10372} et al

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

Barycentrics    [a^8 - 4a^6(b + c)^2 + 2a^4(b + c)^2(3b^2 - 4bc + 3c^2) - 4a^2(b^2 - c^2)^2(b^2 + c^2) + (b - c)^2(b + c)^6][a^3 - a^2(b + c) - a(b + c)^2 + (b - c)^2(b + c)]/(b + c - a) : :

X(10365) lies on these lines: {1,8808}, {4,65}, {7,10368}, {8,253}, {388,1439}, {5929,10369} et al

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

Barycentrics    [a^8 + a^7(b + c) - 2a^6(b + c)^2 - a^5(b + c)^3 + 2a^4(b^2 - c^2)^2 - a^3(b - c)^2(b + c)^3 - 2a^2(b^2 - c^2)^2(b^2 + c^2) + a(b - c)^2(b + c)^5 + (b - c)^2(b + c)^6]/(b + c - a) : :

X(10366) lies on these lines: {4,65}, {12,223}, {56,8808}, {225,1853}, {1439,10368}, {5252,5930}, {5929,10363} et al


X(10367) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

Barycentrics    a^7 + 2a^6(b + c) - a^4(b + c)(b^2 - 4bc + c^2) + a^3(b + c)^2(b^2 + c^2) - 2a^2bc(b - c)^2(b + c) - 2a(b^2 - c^2)^2(b^2 + bc + c^2) - (b - c)^2(b + c)^5 : :

X(10367) lies on these lines: {1,8282}, {4,8}, {65,10362}, {223,9578}, {5252,5930} et al


X(10368) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF 3rd AND 5th EXTOUCH TRIANGLES

Barycentrics    [a^8 + 2a^7(b + c) - 2a^5(b^3 + c^3) - 2a^4(b + c)^2(b^2 + c^2) - 2a^3(b^5 + 3b^4c + 3bc^4 + c^5) + 2a(b - c)^2(b + c)^3(b^2 + bc + c^2) + (b - c)^2(b + c)^6]/(b + c - a) : :

X(10368) lies on these lines: {1,4}, {7,10365}, {65,10362}, {1439,10366}, {1836,10373}, {5929,10372} et al

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

Barycentrics    [a^5 - a^4(b + c) - 2a^3(2b^2 + 3bc + 2c^2) - 2a^2(b + c)(2b + c)(b + 2c) - a(b + c)^2(b^2 + c^2) + (b + c)^3(b^2 + c^2)]/(b + c - a) : :

X(10369) lies on these lines: {4,10363}, {7,8}, {5929,10365}, {5930,8815} et al

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

Barycentrics    (a+b-c) (a-b+c) (a^5-3 a^3 b^2-3 a^2 b^3+b^5-4 a^3 b c-9 a^2 b^2 c+3 b^4 c-3 a^3 c^2-9 a^2 b c^2+4 b^3 c^2-3 a^2 c^3+4 b^2 c^3+3 b c^4+c^5) : :

X(10370) lies on these lines: {7,8}, {5929,10363}, {5930,10362} et al


X(10371) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

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

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} et al


X(10372) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF 4th AND 5th EXTOUCH TRIANGLES

Barycentrics    [a^5 + a^4(b + c) + 2a^3bc + a(b + c)^4 + (b + c)^3(b^2 + c^2)]/(b + c - a) : :

X(10372) lies on these lines: {7,8}, {56,1211}, {5929,10368}, {5930,10364} et al

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

Trilinears    a^8(b + c) + 2a^7(b^2 + c^2) - 2a^6(b + c)(b^2 + bc + c^2) - 6a^5(b^2 - c^2)^2 + 2a^4bc(b + c)(3b^2 - 2bc + 3c^2) + 6a^3(b^2 - c^2)^2(b^2 + c^2) + 2a^2(b - c)^4(b + c)(b^2 + bc + c^2) - 2a(b^2 - c^2)^2(b^4 + 6b^2c^2 + c^4) - (b - c)^4(b + c)^3(b^2 + c^2) : :

X(10373) lies on these lines: {1,64}, {4,8}, {1836,10368}, {3057,5930}, {4295,10400} et al


X(10374) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 1st AND 3rd EXTOUCH TRIANGLES

Trilinears    a^7(b + c) + a^6(b - c)^2 - 3a^5(b - c)^2(b + c) - 3a^4(b^2 - c^2)^2 + a^3(b - c)^2(b + c)(3b^2 + 2bc + 3c^2) + a^2(b^2 - c^2)^2(3b^2 - 2bc + 3c^2) - a(b - c)^2(b + c)^5 - (b - c)^6(b + c)^2 : :

X(10374) lies on these lines: {4,65}, {11,8808}, {55,223}, {56,3182}, {3057,5930} et al


X(10375) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 1st AND 5th EXTOUCH TRIANGLES

Trilinears    {(a^2 + b^2 + c^2 + 2bc)[a^6 + 2a^5(b + c) - a^4(b + c)^2 - 4a^3(b^3 + c^3) - a^2(b^2 - c^2)^2 + 2a(b - c)^2(b + c)(b^2 + c^2) + (b - c)^2(b + c)^4]}/[(b + c - a)(b^4 + c^4 - 3a^4 + 2a^2b^2 + 2a^2c^2 - 2b^2c^2)] : :

X(10375) lies on these lines: {8,253}, {33,64} et al


X(10376) = PERSPECTOR OF 5th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th EXTOUCH TRIANGLES

Trilinears    (b + c)(a^2 + b^2 + c^2 + 2bc)/(b + c - a)^3 : :

X(10376) lies on these lines: {8,479}, {34,6059}, {65,1439}, {388,7197}, {1460,4320} et al


X(10377) = PERSPECTOR OF EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF ABC AND 5th EXTOUCH TRIANGLE

Barycentrics    (a^2 + b^2 + c^2 + 2bc)[a^6 - 2a^5(b + c) - a^4(b^2 + 6bc + c^2) + 4a^3(b - c)^2(b + c) - a^2(b + c)^2(b^2 - 6bc + c^2) - 2a(b - c)^4(b + c) + (b - c)^2(b + c)^4]/(b + c - a) : :

X(10377) lies on these lines: {65,497}, {388,7197}, {4012,5930} et al


X(10378) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

Trilinears    (b + c)/[a^10 + 3a^8(b + c)^2 - 2a^6(b + c)^2(7b^2 - 6bc + 7c^2) + 2a^4(b + c)^2(7b^4 - 8b^3c + 10b^2c^2 - 8bc^3 + 7c^4) - a^2(b - c)^2(b + c)^2(3b^2 + c^2)(b^2 + 3c^2) - (b - c)^4(b + c)^6] : :

X(10378) lies on the Jerabek hyperbola and these lines: {6,1712}, {72,10379}, {1439,10380} et al

X(10378) =isogonal conjugate of X(13618)


X(10379) = PERSPECTOR OF 2nd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

Trilinears    (b + c)[a^12(b + c) + a^11(b + c)^2 - a^10(5b^3 - b^2c - bc^2 + 5c^3) - a^9(5b^4 + 2b^3c + 2bc^3 + 5c^4) + 2a^8(5b^5 - 4b^4c - 4bc^4 + 5c^5) + 2a^7(5b^6 - 2b^5c - 3b^4c^2 + 4b^3c^3 - 3b^2c^4 - 2bc^5 + 5c^6) - 2a^6(b - c)^2(5b^5 + 5b^4c + 8b^3c^2 + 8b^2c^3 + 5bc^4 + 5c^5) - 2a^5(b^2 - c^2)^2(5b^4 - 2b^3c + 6b^2c^2 - 2bc^3 + 5c^4) + a^4(b - c)^2(5b^7 + 3b^6c + 9b^5c^2 + 23b^4c^3 + 23b^3c^4 + 9b^2c^5 + 3bc^6 + 5c^7) + a^3(b^2 - c^2)^2(5b^6 + 2b^5c + 7b^4c^2 - 4b^3c^3 + 7b^2c^4 + 2bc^5 + 5c^6) - a^2(b - c)^4(b + c)^3(b^4 - 4b^3c + 2b^2c^2 - 4bc^3 + c^4) - a(b^2 - c^2)^4(b^4 + 2b^3c + 4b^2c^2 + 2bc^3 + c^4) - 2bc(b - c)^4(b + c)^3(b^4 + b^3c + 4b^2c^2 + bc^3 + c^4)] : :

X(10379) lies on these lines: {4,51}, {72,10378} et al


X(10380) = PERSPECTOR OF 3rd EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd EXTOUCH TRIANGLES

Trilinears    (b + c)[a^13(b + c) + 2a^12(b^2 + c^2) - 4a^11(b^3 + c^3) - 2a^10(5b^4 + b^3c + b^2c^2 + bc^3 + 5c^4) + a^9(5b^5 - 11b^4c + 2b^3c^2 + 2b^2c^3 - 11bc^4 + 5c^5) + 2a^8(10b^6 + 3b^5c - 5b^4c^2 - 4b^3c^3 - 5b^2c^4 + 3bc^5 + 10c^6) + 8a^7bc(3b^5 - b^4c - b^3c^2 - b^2c^3 - bc^4 + 3c^5) - 4a^6(b^2 - c^2)^2(5b^4 + b^3c + 5b^2c^2 + bc^3 + 5c^4) - a^5(b - c)^2(5b^7 + 31b^6c + 45b^5c^2 + 39b^4c^3 + 39b^3c^4 + 45b^2c^5 + 31bc^6 + 5c^7) + 2a^4(b^2 - c^2)^2(5b^6 - 2b^5c + 5b^4c^2 + 8b^3c^3 + 5b^2c^4 - 2bc^5 + 5c^6) + 4a^3(b - c)^2(b + c)^3(b^6 + b^5c - b^4c^2 + 4b^3c^3 - b^2c^4 + bc^5 + c^6) - 2a^2(b^2 - c^2)^4(b^4 - 3b^3c + 5b^2c^2 - 3bc^3 + c^4) - a(b - c)^4(b + c)^3(b^6 + 2b^5c + 3b^4c^2 + 12b^3c^3 + 3b^2c^4 + 2bc^5 + c^6) - 2bc(b^2 - c^2)^4(b^4 - b^3c + 4b^2c^2 - bc^3 + c^4)] : :

X(10380) lies on these lines: {4,51}, {72,5930}, {1439,10378} et al


X(10381) = PERSPECTOR OF 4th EXTOUCH TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th EXTOUCH TRIANGLES

Trilinears    (b + c)[a^4(b + c) + a^3(b + c)^2 - a^2(b - c)^2(b + c) - a(b^4 + 2b^3c + 2bc^3 + c^4) - 2bc(b^3 + c^3)] : :

X(10381) lies on these lines: {1,4199}, {4,69}, {9,1046}, {10,12} et al

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

Trilinears    [a^4 - b^4 - c^4 - 2a^3(b + c) - 4a^2bc + 2a(b - c)^2(b + c) + 2b^2c^2](b + c - a) : :

X(10382) lies on these lines: {1,4}, {2,5807}, {3,10396}, {4,33}, {6,7070}, {9,55}, {11,10582}, {35,1728}, {40,9786}, {42,4319}, {57,5728}, {63,1005}, {329,390}, {405,936}, {954,5927}, {5218,8580}, {5219,8226}, {8232,10578} et al

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(10383) = HOMOTHETIC CENTER OF ASCELLA TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

Trilinears    (b + c - a)[a^4 - 2a^3(b + c) - 8a^2bc + 2a(b - c)^2(b + c) - (b - c)^4] : :

Note: the Ascella triangle is the mid-triangle of the intouch and excentral triangles.

X(10383) lies on these lines: {1,3}, {2,5809}, {33,7490}, {77,479}, {78,5273}, {142,497}, {200,5218}, {223,991}, {226,5732}, {284,5324}, {390,4666},...}. et al


X(10384) = HOMOTHETIC CENTER OF ATIK TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

Trilinears    (b + c - a)[a^4 - 2a^2(b^2 - 6bc + c^2) + (b - c)^2(b^2 + 6bc + c^2)] : :

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} et al


X(10385) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(2)

Barycentrics    (b + c - a)(5a2 + b2 + c2 - 2bc) : :

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} et al

X(10385) = X(51) of cross-triangle of intouch and excentral triangles


X(10386) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(5)

Barycentrics    4a4 - a2(3b2 + 8bc + 3c2) - (b2 - c2)2 : :

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} et al

X(10386) = X(143) of cross-triangle of intouch and excentral triangles


X(10387) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(6)

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

X(10387) lies on these lines: {1,1350}, {6,31}, {11,3763}, {524,10385} et al

X(10387) = X(53) of cross-triangle of intouch and excentral triangles


X(10388) = EXTANGENTS-TO-INTANGENTS SIMILARITY IMAGE OF X(57)

Barycentrics    a (a-b-c) (a^4-2 a^3 b+2 a b^3-b^4-2 a^3 c+8 a^2 b c-2 a b^2 c+4 b^3 c-2 a b c^2-6 b^2 c^2+2 a c^3+4 b c^3-c^4) : :

X(10388) lies on these lines: {1,3}, {11,8580}, {329,390}, {527,4326}, {1864,9954} et al

X(10388) = X(25) of cross-triangle of intouch and excentral triangles


X(10389) = X(2) OF CROSS-TRIANGLE OF INTOUCH AND EXCENTRAL TRIANGLES

Trilinears    3a2 + b2 + c2 - 4ab - 4ac - 2bc : :

X(10389) lies on these lines: {1,3}, {2,3158}, {8,5436}, {9,1174}, {21,6762}, {100,4666}, {105,612}, {3616,5438} et al

X(10389) = isogonal conjugate of X(10390)


X(10390) = ISOGONAL CONJUGATE OF X(10389)

Trilinears    1/(3a2 + b2 + c2 - 4ab - 4ac - 2bc) : :

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} et al

X(10390) = isogonal conjugate of X(10389)


X(10391) = HOMOTHETIC CENTER OF ASCELLA TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

Trilinears    (b + c - a)[a^3(b + c) - a^2(b^2 - 4bc + c^2) - a(b - c)^2(b + c) + (b - c)^2(b^2 + c^2)] : :

Let A'B'C' be the intangents triangle. Let LA be the line through the points of contact of the incircle and lines BC and B'C'. Define LB, LC cyclically. Let A" = LB∩LC, B" = LC∩LA, C" = LA∩LB. 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} et al


X(10392) = HOMOTHETIC CENTER OF ATIK TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

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

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} et al


X(10393) = HOMOTHETIC CENTER OF 2nd CIRCUMPERP TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

Trilinears    (sec A - sec B)[sin B sin(A - C) tan(B/2)] - (sec A - sec C)[sin C sin(A - B) tan(C/2)] : :
Trilinears    (b + c - a)[a^5 - b^5 - c^5 - a^4(b + c) - 2a^3(b + c)^2 + 2a^2(b^3 - 2b^2c - 2bc^2 + c^3) + a(b^2 - c^2)^2 + bc(b^3 + c^3)] : :

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} et al

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

Trilinears    (b + c - a)[b^4 + c^4 + a^3(b + c) - a^2(b^2 - 3bc + c^2) - a(b - c)^2(b + c) - bc(b^2 + c^2)] : :

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} et al

X(10394) = anticomplement of X(5784)


X(10395) = HOMOTHETIC CENTER OF 4th EULER TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

Barycentrics    [cot C cos2(C/2) + cot A cos2(A/2) - cot B cos2(B/2)]cos B + [cot A cos2(A/2) + cot B cos2(B/2) - cot C cos2(C/2)]cos C : :
Barycentrics    (b + c - a)[a^5(b + c) - a^4(b - c)^2 - 2a^3(b^3 + c^3) + 2a^2(b^2 - c^2)^2 + a(b - c)^2(b + c)(b^2 + 4bc + c^2) - (b - c)^4(b + c)^2] : :

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} et al

X(10395) = complement of X(224)


X(10396) = HOMOTHETIC CENTER OF HEXYL TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

Trilinears    (a - b)[sec A tan(C/2) - sec C tan(A/2)] - (a - c)[sec A tan(B/2) - sec B tan(A/2)] : :
Trilinears    a^6 - 4a^3bc(b + c) - a^4(3b^2 - 2bc + 3c^2) + 3a^2(b^2 - c^2)^2 + 4abc(b - c)^2(b + c) - (b + c)^4(b - c)^2 : :

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} et al

X(10396) = crossdifference of every pair of points on line X(513)X(10397)


X(10397) = CROSSDIFFERENCE OF X(4) AND X(57)

Trilinears    sec B tan(C/2) - sec C tan(B/2) : :
Trilinears    a(b - c)(b + c - a)(b^2 + c^2 - a^2)[a^3 + a^2(b + c) - a(b + c)^2 - (b - c)^2(b + c)] : :

X(10397) lies on these lines: {520,647}, {650,663} et al

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

Trilinears    a^5 + a^4(b + c) - 6a^3(b^2 + c^2) + 2a^2(b - c)^2(b + c) + 5a(b^2 - c^2)^2 - (b - c)^2(b + c)(3b^2 + 2bc + 3c^2) : :

X(10398) lies on these lines: {1,6}, {2,5785}, {4,3062}, {7,1210}, {57,971}, {226,5817}, {7987,10393} et al


X(10399) = HOMOTHETIC CENTER OF REFLECTION TRIANGLE OF X(1) AND CROSS-TRIANGLE OF INTOUCH AND 2nd EXTOUCH TRIANGLES

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

X(10399) lies on these lines: {1,6}, {2,10122}, {4,79}, {57,6985} et al


X(10400) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 3rd EXTOUCH TRIANGLES

Barycentrics    (a+b-c) (a-b+c) (a^6+a^5 b-a^4 b^2-2 a^3 b^3-a^2 b^4+a b^5+b^6+a^5 c-2 a^4 b c-a b^4 c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^3 c^3-4 b^3 c^3-a^2 c^4-a b c^4-b^2 c^4+a c^5+2 b c^5+c^6) : :

X(10400) lies on these lines: {4,7}, {5,7013}, {57,1901}, {65,10362}, {79,8809}, {198,226}, {553,8808}, {4295,10373} et al

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

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

Let AB, AC be the points where the A-excircle touches lines CA and AB resp., and define BC, BA, CA, CB cyclically. AB, AC, BC, BA, CA, CB lie on the Yiu conic (defined at X(478)). Let TA be the intersection of the tangents to the Yiu conic at BC and CA, and define TB, TC cyclically. Let TA' be the intersection of the tangents to the Yiu conic at BA and CB, and define TB', TC' cyclically. Let SA = TBTC∩TB'TC', SB = TCTA∩TC'TA', SC = TATB∩TA'TB' (i.e. SASBSC is the side triangle of TATBTC and TA'TB'TC'). SASBSC 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
AB = -a+b-c : 0 : a+b+c
AC = -a-b+c : a+b+c : 0
TA = 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)
TA' = 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)
SA = (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)
VA = 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} et al

X(10401) = {X(10907),X(10908)}-harmonic conjugate of X(1)


X(10402) = (name pending)

Barycentrics    [a^6 - a^5(b + c) - a^4(b^2 + c^2) + a^2(b^2 - c^2)^2 + a(b - c)^2(b + c)^3 - (b^2 - c^2)^2(b^2 + c^2)]*[a^3 + a^2(b + c) - a(b + c)^2 - (b - c)^2(b + c)]/(b + c - a) : :

Continuing from X(10401), let VA = TBTB'∩TCTC', VB = TCTC'∩TATA', VC = TATA'∩TBTB' (i.e., VAVBVC is the vertex triangle of TATBTC and TA'TB'TC'). VAVBVC 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} et al


X(10403) = (name pending)

Trilinears    (b + c)[a^10 - a^8(3b^2 - 2bc + 3c^2) - 4a^7bc(b + c) + 2a^6(b^4 - 2b^3c + 4b^2c^2 - 2bc^3 + c^4) + 4a^5bc (b + c)(2b^2 - 3bc + 2c^2) + 2a^4(b - c)^2(b^4 + 4b^3c + 4b^2c^2 + 4bc^3 + c^4) - 4a^3bc(b - c)^2(b + c)(b^2 + c^2) - a^2(b - c)^2(b + c)^2(b^2 + c^2)(3b^2 + 4bc + 3c^2) + 4ab^2c^2(b - c)^2(b + c)^3 + (b - c)^2(b + c)^4(b^2 + c^2)^2]/(b + c - a) : :

Continuing from X(10401) and X(10402), triangles SASBSC and VAVBVC are perspective at X(10403).

X(10403) lies on these lines: {{9,478}, {65,1826}, {10401,10402}}. et al


X(10404) = PERSPECTOR OF INTOUCH TRIANGLE AND CROSS-TRIANGLE OF INTOUCH AND 5th EXTOUCH TRIANGLES

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

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} et al

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

Barycentrics    1/(tan B/2 + tan C/2 - tan A/2) : :
Barycentrics    1/[3a^2 - 2ab - 2ac - (b - c)^2] : :

X(10405) lies on these lines: {2,3160}, {8,144}, {175,7090}, {333,5790} et al

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) = 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

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

X(10406) lies on these lines: {10,3614}, {11,181}, {12,970}, {386,7354}, {1682,9569}, {3058,9554} et al


X(10407) = PERSPECTOR OF FEUERBACH TRIANGLE AND CROSS-TRIANGLE OF FEUERBACH AND APOLLONIUS TRIANGLES

Barycentrics    a^5(b + c)^2 + a^4(b + c)(3b^2 + 2bc + 3c^2) + a^3(b^4 + 2b^3c + 6b^2c^2 + 2bc^3 + c^4) - a^2(b + c)(3b^4 - 2b^3c - 6b^2c^2 - 2bc^3 + 3c^4) - 2a(b^2 - c^2)^2(b^2 + bc + c^2) - 2bc(b - c)^2(b + c)^3 : :

X(10407) lies on these lines: {2,9553}, {5,181}, {10,3614}, {11,10408}, {12,1682}, {970,7951}, {3085,9554}, {3091,9555} et al


X(10408) = {X(12),X(181)}-HARMONIC CONJUGATE OF X(10)

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

Let FAFBFC be the Feuerbach triangle and PAPBPC the Apollonius triangle. Let A' = {FA,PA}-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} et al


X(10409) = COLLINGS TRANSFORM OF X(623)

Barycentrics    Csc[B - C] / (Cos[B - C] + 2 Cos[A - Pi / 3]) : :

X(10409) lies on the circumcircle and these lines: {15,2380}, {511,1337}, {1634,10410} et al

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) = 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)

Barycentrics    Csc[B - C] / (Cos[B - C] + 2 Cos[A + Pi / 3]) : :

X(10410) lies on the circumcircle and these lines: {16,2381}, {511,1338}, {1634,10409} et al

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) = 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)

Barycentrics    csc2 A csc(B - C) sin 3A : :
Barycentrics    a2[(a2 - b2 - c2)2 - b2c2]/(b2 - c2) : :

X(10411) lies on these lines: {2,10413}, {54,69}, {99,110} et al

X(10411) = isotomic conjugate of X(10412)
X(10411) = anticomplement of X(10413)


X(10412) = ISOTOMIC CONJUGATE OF X(10411)

Trilinears    sin A sin(B - C) csc 3A : :
Barycentrics    b2c2(b2 - c2)/[(a2 - b2 - c2)2 - b2c2] : :

X(10412) lies on the tangent to circumcircle at X(476), and these lines: {2,8562}, {4,1510}, {5,523} et al

X(10412) = isotomic connjugate of X(10411)
X(10412) = anticomplement of X(8562)
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)

Barycentrics    csc2 B csc(C - A) sin 3B + csc2 C csc(A - B) sin 3C : :
Barycentrics    (b^2 - c^2)^2[a^6 - b^6 - c^6 - 3a^4(b^2 + c^2) + a^2(3b^4 - b^2c^2 + 3c^4) + b^2c^2(b^2 + c^2)] : :

X(10413) lies on these lines: {2,10411}, {6,17}, {115,125} et al

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)

Barycentrics    (csc B)[sin 3C csc2 C sin A sin(2B - 2C) - sin 3A csc2 A sin C sin(2A - 2B)] + (csc C)[sin 3A csc2 A sin B sin(2C - 2A) - sin 3B csc2 B sin A sin(2B - 2C)] : :
Barycentrics    2a^12 - 8a^10(b^2 + c^2) + a^8(13b^4 + 18b^2c^2 + 13c^4) - 12a^6(b^2 + c^2)(b^4 + c^4) + a^4(8b^8 - b^6c^2 + 4b^4c^4 - b^2c^6 + 8c^8) - a^2(b - c)^2(b + c)^2(b^2 + c^2)(4b^4 - 5b^2c^2 + 4c^4) + (b^2 - c^2)^6 : :

X(10414) lies on these lines: {6,17}, {140,523} et al

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)

Barycentrics    1/[(a^4 - b^4 - c^4 + b^2c^2)(2a^2 - b^2 - c^2)] : :

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} et al

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)

Barycentrics    [3a^8 - 2a^6(b^2 + c^2) - 2a^4(b^2 - c^2)^2 + 2a^2(b^6 + c^6 - 2R^2S^2) - (b^4 - c^4)^2 - 4(b^2 + c^2)R^2S^2]/(b^2 + c^2 - 2a^2) : :

X(10416) lies on these lines: {2,8877}, {23,671}, {111,1287}, {316,691} et al

X(10416) = isogonal conjugate of X(10417)


X(10417) = ISOGONAL CONJUGATE OF X(10416)

Barycentrics    a^2(b^2 + c^2 - 2a^2)/[3a^8 - 2a^6(b^2 + c^2) - 2a^4(b^2 - c^2)^2 + 2a^2(b^6 + c^6 - 2R^2S^2) - (b^4 - c^4)^2 - 4(b^2 + c^2)R^2S^2] : :

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} et al

X(10417) = isogonal conjugate of X(10416)


X(10418) = CROSSDIFFERENCE OF PU(63)

Barycentrics    2a^6 - 2a^4(b^2 + c^2) - a^2(3b^4 - 8b^2c^2 + 3c^4) + (b^2 - c^2)^2(b^2 + c^2) : :

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} et al

X(10418) = midpoint of X(1648) and X(2502)
X(10418) = crossdifference of every pair of points on line X(3)X(351)


X(10419) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(30)

Barycentrics    a^2/{(2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)[a^4(b^2 + c^2) - 2a^2(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)]} : :

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} et al

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)

Barycentrics    a^2/{(b^2 - c^2)[a^4(b^2 + c^2) - 2a^2(b^4 - b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)]} : :

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} et al

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(10421) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(186) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(186)

Barycentrics    [3a^10 - 7a^8(b^2 + c^2) + a^6(4b^4 + 11b^2c^2 + 4c^4) - 4a^4b^2c^2(b^2 + c^2) + a^2(b^2 - c^2)^2(b^4 - b^2c^2 + c^4) - (b^2 - c^2)^4(b^2 + c^2)]/[(b^2 + c^2 - a^2)(2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2)] : :

X(10421) lies on these lines: {4,1138}, {30,340} et al


X(10422) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(468)

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

X(10422) lies on these lines: {4,10415}, {23,895}, {316,691}, {468,10424} et al

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)

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

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}, et al

X(10423) = trilinear pole of line X(6)X(1112)
X(10423) = Ψ(X(6), X(1112))
X(10423) = circumcircle intercept, other than X(111), of circle {{X(4),X(6),X(25),X(111)}}


X(10424) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(468) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(468)

Barycentrics    [a^8 - 4a^6(b^2 + c^2) + 13a^4b^2c^2 + a^2(b^2 + c^2)(4b^4 - 11b^2c^2 + 4c^4) - (b^4 - c^4)^2]/{[a^4(b^2 + c^2) - 2a^2b^2c^2 - (b^2 - c^2)^2(b^2 + c^2)](b^2 + c^2 - a^2)} : :

X(10424) lies on these lines: {23,935}, {468,10422} et al


X(10425) = PERSPECTOR OF CIRCUMCEVIAN TRIANGLE OF X(3) AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(511)

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

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} et al

X(10425) = trilinear pole of line X(6)X(2987)


X(10426) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(1155)

Trilinears    a/[a^4(b + c) + 2a^3(b^2 - 4bc + c^2) - a^2(b + c)(3b^2 - 7bc + 3c^2) - 2a(b - c)^2(2b^2 + bc + 2c^2) + (b - c)^2(b + c)(4b^2 + bc + 4c^2)] : :

X(10426) lies on these lines: (pending)

X(10426) = isogonal conjugate of X(10427)


X(10427) = ISOGONAL CONJUGATE OF X(10426)

Barycentrics    a^4(b + c) + 2a^3(b^2 - 4bc + c^2) - a^2(b + c)(3b^2 - 7bc + 3c^2) - 2a(b - c)^2(2b^2 + bc + 2c^2) + (b - c)^2(b + c)(4b^2 + bc + 4c^2) : :

X(10427) lies on these lines: {1,528}, {2,1156}, {3,1633}, {7,100}, {9,1768}, {10,1071}, {11,142}, {119,971}, {214,516} et al

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)

Trilinears    a/{(b + c - 2a)[2abc - (b + c)(a^2 - (b - c)^2)]} : :

X(10428) lies on these lines: {4,6075}, {57,1168}, {104,517}, {106,1457} et al

X(10428) = isogonal conjugate of X(1145)


X(10429) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

Barycentrics    1/[a^5 - a^4(b + c) - 2a^3(b + c)^2 + 2a^2(b - c)^2(b + c) + a(b + c)^4 - (b - c)^4(b + c)] : :

X(10429) lies on these lines: {1,9799}, {8,7957}, {9,20}, {21,8273}, {63,9800}, {10432,10433} et al

X(10429) = isogonal conjugate of X(5584)


X(10430) = HOMOTHETIC CENTER OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

Barycentrics    -f(A,B,C) + f(B,C,A) + f(C,A,B) : :, where f(A,B,C) = X(1750)

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} et al

X(10430) = anticomplement of X(1750)


X(10431) = HOMOTHETIC CENTER OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd CONWAY TRIANGLES

Barycentrics    a^6 - 2a^5(b + c) + a^4(b^2 - 4bc + c^2) + 2a^3bc(b + c) - a^2(b - c)^2(b^2 + c^2) + 2a(b - c)^2(b + c) (b^2 + bc + c^2) - (b - c)^4(b + c)^2 : :

X(10431) lies on these lines: {2,3}, {7,354}, {8,7957}, {63,516}, {149,9965}, {908,1750}, {962,3868}, {8822,10432} et al

X(10431) = anticomplement of X(7580)


X(10432) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 1st CONWAY TRIANGLE

Barycentrics    a^8 + 2a^7(b + c) - 4a^6(b^2 + c^2) - 2a^5(b + c)(3b^2 + bc + 3c^2) + 2a^4(b - c)^2(3b^2 + 5bc + 3c^2) + 2a^3(b + c)(3b^4 + 2b^3c + 2b^2c^2 + 2bc^3 + 3c^4) - 4a^2(b + c)^2(b^2 + c^2)(b^2 - 3bc + c^2) - 2a(b - c)^2(b + c)^3(b^2 + bc + c^2) + (b - c)^4(b + c)^2(b^2 + c^2) : :

X(10432) lies on these lines: {2,1713}, {7,7183}, {8,6527}, {21,69}, {63,4329}, {8822,10431}, {10429,10433} et al


X(10433) = PERSPECTOR OF 1st CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 2nd CONWAY TRIANGLE

Barycentrics    a^8 - b^8 - c^8 - 2a^7(b + c) - 2a^6(b^2 + 8bc + c^2) + a^5(6b^3 - 26b^2c - 26bc^2 + 6c^3) - 16a^4b^2c^2 - a^3(6b^5 - 26b^4c - 12b^3c^2 - 12b^2c^3 - 26bc^4 + 6c^5) + 2a^2(b^6 + 8b^5c + 7b^4c^2 + 7b^2c^4 + 8bc^5 + c^6) + 2a(b^7 + b^6c + 7b^5c^2 - 9b^4c^3 - 9b^3c^4 + 7b^2c^5 + bc^6 + c^7) + 4b^6c^2 - 6b^4c^4 + 4b^2c^6 : :

X(10433) lies on these lines: {7,2478}, {8,8822}, {63,391}, {69,10430}, {10429,10432} et al

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

Trilinears    a[2a3(b + c) - a2bc - 2a(b3 + c3) - bc(b + c)2] : :

Let Ab, Ac, Bc, Ba, Ca, Cb 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 Ba and Ca, 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)

Let A4 be the intersection of the tangents to the Conway circle at Ab and Ac, and define B4, C4 cyclically. Triangle A4B4C4 is here named the 4th Conway triangle. A4B4C4 is also the polar triangle of the Conway circle wrt the 1st Conway triangle.

The appearance of (T,i) in the following list means that the 4th Conway triangle and T are perspective with perspector X(i):
(ABC, 1),
(incentral, 1),
(excentral, 1),
(1st Conway, 10466),
(2nd Conway, 10446),
(3rd Conway, 10446),
(5th Conway, 1)

Let A5 be the intersection of the tangents to the Conway circle at Bc and Cb, and define B5, C5 cyclically. Triangle A5B5C5 is here named the 5th Conway triangle. A5B5C5 is also the polar triangle of the Conway circle wrt the 2nd Conway triangle.

The appearance of (T,i) in the following list means that the 5th Conway triangle and T are perspective with perspector X(i):
(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:

AB = a+b : 0 : -a} and AC = 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} et al

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

Barycentrics    1/[2a3(b + c) - a2bc - 2a(b3 + c3) - bc(b + c)2] : :

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} et al

X(10435) = isogonal conjugate of X(10434)
X(10435) = trilinear product of vertices of 3rd Conway triangle


X(10436) = ISOGONAL CONJUGATE OF X(2258)

Barycentrics    b2c2(1 - cos A)(b + c + a cos A) : :
Barycentrics    a2 + ab + ac + 2bc : :

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} et al

X(10436) = isogonal conjugate of X(2258)
X(10436) = anticomplement of X(5257)
X(10436) = trilinear pole of polar of X(10435) wrt Conway circle


X(10437) = PERSPECTOR OF 3rd CONWAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 3rd CONWAY TRIANGLE

Barycentrics    a^7(b + c) + 3a^6(b + c)^2 + a^5(2b^3 + 3b^2c + 3bc^2 + 2c^3) - 2a^4(b^4 + 2b^3c - b^2c^2 + 2bc^3 + c^4) - a^3(3b^5 + b^4c + 4b^3c^2 + 4b^2c^3 + bc^4 + 3c^5) - a^2(b^6 + 2b^5c + 3b^4c^2 - 4b^3c^3 + 3b^2c^4 + 2bc^5 + c^6) - abc(b - c)^2(3b^3 + 5b^2c + 5bc^2 + 3c^3) - 2b^2c^2 (b^2 - c^2)^2 : :

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} et al.

X(10437) = isogonal conjugate of X(10438)


X(10438) = ISOGONAL CONJUGATE OF X(10437)

Barycentrics    a^2/[a^7(b + c) + 3a^6(b + c)^2 + a^5(2b^3 + 3b^2c + 3bc^2 + 2c^3) - 2a^4(b^4 + 2b^3c - b^2c^2 + 2bc^3 + c^4) - a^3(3b^5 + b^4c + 4b^3c^2 + 4b^2c^3 + bc^4 + 3c^5) - a^2(b^6 + 2b^5c + 3b^4c^2 - 4b^3c^3 + 3b^2c^4 + 2bc^5 + c^6) - abc(b - c)^2(3b^3 + 5b^2c + 5bc^2 + 3c^3) - 2b^2c^2 (b^2 - c^2)^2] : :

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: (pending)

X(10438) = isogonal conjugate of X(10437)


X(10439) = CENTROID OF 3rd CONWAY TRIANGLE

Trilinears    [a^2(b + c) + a(b^2 + bc + c^2) + bc(b + c)]^2 [a^3(2b^2 + bc + 2c^2) - a(2b^4 - b^3c + 2b^2c^2 - bc^3 + 2c^4) - 2bc(b - c)^2(b + c)] : :

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} et al

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)

Trilinears    [a^2(b + c) + a(b^2 + bc + c^2) + bc(b + c)]^2 [2a^3(b + c)^2 - 3a^2bc(b + c) - 2a(b^4 + b^3c - 2b^2c^2 + bc^3 + c^4) + bc(b - c)^2(b + c)] : :

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} et al

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

Trilinears    a^4(b^2 + bc + c^2) + a^3(b + c)(b^2 + c^2) - a^2(b^4 + c^4) - a(b + c)(b^4 + c^4) - bc(b^2 - c^2)^2 : :

X(10441) lies on these lines: {1,3}, {2,970}, {4,69}, {5,1211}, {10,3781}, {1012,10451}, {3634,10440}, {5295,10447} et al

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(10442) = SYMMEDIAN POINT OF 3rd CONWAY TRIANGLE

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

X(10442) lies on these lines: {1,7}, {2,10443}, {9,1764}, {165,10436}, {314,3062}, {4385,5223} et al

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)

Barycentrics    7a^4(b + c) - 8a^3bc - 2a^2(b + c)(3b^2 - 4bc + 3c^2) - (b - c)^2(b + c)^3 : :

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} et al

X(10443) = complement of X(10442)


X(10444) = HOMOTHETIC CENTER OF 1st AND 3rd CONWAY TRIANGLES

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

X(10444) lies on these lines: {2,10445}, {21,10435}, {63,321}, {84,309} et al

X(10444) = anticomplement of X(10445)
X(10444) = perspector 5th Conway triangle and 1st and 3rd Conway triangles


X(10445) = COMPLEMENT OF X(10444)

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

X(10445) lies on these lines: {2,10444}, {3,5750}, {4,9}, {5,5257}, {6,515}, {20,5749} et al

X(10445) = inverse-in-excircles-radical-circle of X(8074)
X(10445) = pole of Gergonne line wrt excircles-radical-circle


X(10446) = HOMOTHETIC CENTER OF 2nd AND 3rd CONWAY TRIANGLES

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

Let OA be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius Sqrt(rA2 + s2), where rA is the A-exradius). Let PA be the perspector of OA, and define PB, PC cyclically. Triangle PAPBPC 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} et al

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

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

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} et al

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

Trilinears    a^3 - a^2(b + c) - 2a(b^2 + bc + c^2) - 2bc(b + c) : :

Let A4B4C4 be the 4th Conway circle. Let LA be the trilinear polar of A4, and define LB, LC cyclically. Let A' = LB∩LC, 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} et al

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

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

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} et al

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

Barycentrics    2a^10(b + c) + a^9(4b^2 + 7bc + 4c^2) - a^8(b + c)(4b^2 - 13bc + 4c^2) - a^7(12b^4 - 7b^3c - 20b^2c^2 - 7bc^3 + 12c^4) + a^6bc(b + c)(b^2 + 12b c + c^2) + a^5(b - c)^2(12b^4 + 17b^3c + 16b^2c^2 + 17bc^3 + 12c^4) + a^4(b + c)(b^2 + c^2)(4b^4 - 9b^3c - 2b^2c^2 - 9bc^3 + 4c^4) - a^3(b + c)^2(4b^6 - 5b^5c + 22b^4c^2 - 34b^3c^3 + 22b^2c^4 - 5bc^5 + 4c^6) - a^2(b - c)^2(b + c)(2b^6 + 9b^5c + 24b^4c^2 + 26b^3c^3 + 24b^2c^4 + 9bc^5 + 2c^6) - 2abc(b^2 - c^2)^2(2b^4 + b^3c + 6b^2c^2 + bc^3 + 2c^4) - 2b^2c^2(b - c)^4(b + c)^3 : :

X(10450) lies on these lines: {84,10447}, {10451,10466} et al


X(10451) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 4th CONWAY TRIANGLES

Trilinears    a^8(b + c) - a^7(b + c)^2 - a^6(3b^3 - 2b^2c - 2bc^2 + 3c^3) + a^5(3b^4 + 8b^3c + 4b^2c^2 + 8bc^3 + 3c^4) + a^4(3b^5 + 2b^4c + 9b^3c^2 + 9b^2c^3 + 2bc^4 + 3c^5) - a^3(3b^6 + 2b^5c - 3b^4c^2 + 4b^3c^3 - 3b^2c^4 + 2bc^5 + 3c^6) - a^2(b^7 + 6b^6c + 8b^5c^2 - 7b^4c^3 - 7b^3c^4 + 8b^2c^5 + 6bc^6 + c^7) + a(b^2 - c^2)^2(b^4 - 4b^3c - 4b^2c^2 - 4bc^3 + c^4) + bc(b - c)^2(b + c)^3(b^2 - 4bc + c^2) : :

X(10451) lies on these lines: {1,7}, {1012,10441}, {5208,10463} et al


X(10452) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

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

X(10452) lies on these lines: {1,69}, {7,10447}, {75,5902}, {79,314}, {3664,3741}, {4292,10462}, {10436,10479} et al


X(10453) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 4th CONWAY TRIANGLES

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

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} et al

X(10453) = anticomplement of X(43)
X(10453) = {X(2),X(145)}-harmonic conjugate of X(42)
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

Barycentrics    3a^7 + 3a^6(b + c) - a^5(b^2 - 8bc + c^2) - a^4(b + c)(3b^2 + 2bc + 3c^2) - 3a^3(b + c)^2(b^2 + c^2) + a^2(b - c)^2(b + c)(b^2 + 4bc + c^2) + a(b - c)^4(b + c)^2 - (b - c)^2(b + c)^3(b^2 + c^2) : :

X(10454) lies on these lines: {1,4}, {2,10470}, {3,10479}, {8,573}, {20,1764}, {69,10437}, {3146,10446} et al


X(10455) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 5th CONWAY TRIANGLE

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

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} et al

X(10455) = trilinear product of vertices of 5th Conway triangle
X(10455) = {X(1),X(10456)}-harmonic conjugate of X(10447)


X(10456) = {X(10447),X(10455)}-HARMONIC CONJUGATE OF X(1)

Barycentrics    a^4(b + c) + a^3(2b^2 + bc + 2c^2) + a^2(b^3 + 5b^2c + 5bc^2 + c^3) + abc(5b^2 + 6bc + 5c^2) + 4b^2c^2(b + c) : :

X(10456) lies on these lines: {1,75}, {7,3741}, {9,1764}, {10,10446} et al


X(10457) = BARYCENTRIC PRODUCT OF VERTICES OF 5th CONWAY TRIANGLE

Trilinears    a[a^2(b + c) + a(b^2 + c^2) + 2bc(b + c)]/(b + c) : :

X(10457) lies on these lines: {1,21}, {750,1010}, {1011,4252} et al

X(10457) = {X(21),X(10458)}-harmonic conjugate of X(10448)


X(10458) = {X(10448),X(10457)}-HARMONIC CONJUGATE OF X(21)

Trilinears    (ab^2 + ac^2 + abc + b^2c + bc^2)/(b + c) : :

X(10458) lies on these lines: {1,21}, {2,3736} et al


X(10459) = {X(1),X(8)}-HARMONIC CONJUGATE OF X(42)

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

Let A5B5C5 be the 5th Conway triangle. Let LA be the trilinear polar of A5, and define LB, LC cyclically. Let A' = LB∩LC, 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} et al


X(10460) = {X(31),X(42)}-HARMONIC CONJUGATE OF X(2293)

Trilinears    a[a^3(b + c) - a^2(b^2 + 6bc + c^2) - a(b + c)^3 + (b^2 - c^2)^2] : :

X(10460) lies on these lines: {1,5273}, {6,31}, {57,959} et al

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

Trilinears    [a^3(b + c) + a^2(b^2 + c^2) - a(b - c)^2(b + c) - (b + c)^2(b^2 + c^2)]/(b + c) : :

X(10461) lies on these lines: {1,21}, {3,10477}, {7,10455}, {20,1764}, {40,1043}, {57,1010}, {84,309}, {1012,10441}, {9799,10465} et al

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

Barycentrics    [a^5(b + c) + a^4(4b^2 + 6bc + 4c^2) + 2a^3(b + c)^3 - 4a^2(b^2 - c^2)^2 - a(b + c)(3b^4 + 2b^2c^2 + 3c^4) - 2bc(b^2 - c^2)^2]/(b + c) : :

X(10462) lies on these lines: {1,8822}, {7,10455}, {4292,10452}, {5208,9962} et al


X(10463) = PERSPECTOR OF 4th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 1st AND 5th CONWAY TRIANGLES

Trilinears    a^8(b + c) + a^7(b^2 + c^2) - 3a^6(b + c)(b^2 - 3bc + c^2) - a^5(b - c)^2(3b^2 + 2bc + 3c^2) + a^4(b + c)(b^2 - 3bc + c^2)(3b^2 - 2bc + 3c^2) + a^3(3b^6 + b^4c^2 - 8b^3c^3 + b^2c^4 + 3c^6) - a^2(b - c)^2(b^5 + 3b^3c^2 + 3b^2c^3 + c^5) - a(b - c)^2(b + c)^2(b^2 + c^2)(b^2 + 4bc + c^2) - bc(b - c)^2(b + c)^5 : :

X(10463) lies on these lines: {1,84}, {20,1764}, {21,572}, {5208,10451} et al


X(10464) = PERSPECTOR OF 2nd CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 5th CONWAY TRIANGLES

Barycentrics    3a^7(b + c) + a^6(5b^2 + 6bc + 5c^2) + 7a^5bc(b + c) - 2a^4(b^4 - b^3c + b^2c^2 - bc^3 + c^4) - a^3(b^5 + 7b^4c + 7bc^4 + c^5) - a^2(3b^6 + 6b^5c - 3b^4c^2 - 4b^3c^3 - 3b^2c^4 + 6bc^5 - 3c^6) - a(b - c)^2(b + c)(2b^4 + 5b^3c + 10b^2c^2 + 5bc^3 + 2c^4) - 2bc(b^2 - c^2)^2(b^2 + bc + c^2) : :

X(10464) lies on these lines: {4,10455}, {3146,10446}, {5691,10447} et al


X(10465) = PERSPECTOR OF 5th CONWAY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 5th CONWAY TRIANGLES

Barycentrics    3a^6(b + c) + a^5(3b^2 + bc + 3c^2) - a^4(b - 2c)(2b - c)(b + c) - 2a^3(b^4 + 4b^2c^2 + c^4) - a^2(b + c)(b^4 + 4b^3c - 6b^2c^2 + 4bc^3 + c^4) - a(b^2 - c^2)^2(b^2 + bc + c^2) - bc(b - c)^2(b + c)^3 : :

X(10465) lies on these lines: {1,7}, {8,1764}, {515,10449}, {944,10441}, {3522,10434}, {9799,10461} et al


X(10466) = PERSPECTOR OF 1st AND 4th CONWAY TRIANGLES

Barycentrics    a^6(4b^2 + 7bc + 4c^2) + a^5(8b^3 + 22b^2c + 22bc^2 + 8c^3) + 2a^4bc(11b^2 + 21bc + 11c^2) - 4a^3(2b^5 - 7b^3c^2 - 7b^2c^3 + 2c^5) - a^2(4b^6 + 13b^5c - 18b^3c^3 + 13bc^5 + 4c^6) - 2abc(3b^5 + 3b^4c - 2b^3c^2 - 2b^2c^3 + 3bc^4 + 3c^5) - 2b^2c^2(b^2 - c^2)^2 : :

X(10466) lies on these lines: {1,8822}, {2,10467}, {7,10447} et al

X(10466) = anticomplement of X(10467)
X(10466) = perspector of Conway circle wrt 1st Conway triangle


X(10467) = COMPLEMENT OF X(10466)

Trilinears    (b + c)[6a^5(b + c) + a^4(8b^2 + 11bc + 8c^2) - 4a^3(b^3 + c^3) - 2a^2(4b^4 + 9b^3c + 14b^2c^2 + 9bc^3 + 4c^4) - 2a(b + c)(b^4 + 6b^3c + 8b^2c^2 + 6bc^3 + c^4) - bc(b + c)^2(b^2 + 6bc + c^2)] : :

X(10467) lies on these lines: {2,10466}, {9,386}, {10,1901}, {573,1750} et al

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

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

X(10468) lies on these lines: {1,69}, {2,10469}, {7,10455}, {75,3670}, {314,4389}, {3338,10436} et al

X(10468) = anticomplement of X(10469)
X(10468) = perspector of Conway circle wrt 2nd Conway triangle


X(10469) = COMPLEMENT OF X(10468)

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

Let LA be the radical axis of the A-excircle and the excircles radical circle, and define LB, LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. 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} et al

X(10469) = complement of X(3055)


X(10470) = HOMOTHETIC CENTER OF 3rd CONWAY TRIANGLE AND CROSS-TRIANGLE OF EXCENTRAL AND 2nd CIRCUMPERP TRIANGLES

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

X(10470) lies on these lines: {1,3}, {2,10454}, {20,10478}, {21,572} et al


X(10471) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND INVERSE-IN-CONWAY-CIRCLE TRIANGLE

Barycentrics    bc(ab^2 + ac^2 + abc + b^2c + bc^2)/(b + c) : :

X(10471) lies on these lines: {1,75}, {10,4476}, {333,1746} et al

X(10471) = trilinear product of vertices of inverse-in-Conway-circle triangle


X(10472) = PERSPECTOR OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND MEDIAL TRIANGLE

Trilinears    [a^2 + a(b + c) + 2bc][a(b^2 + 4bc + c^2) + (b + c)(b^2 + c^2)] : :

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} et al

X(10472) = complement of X(941)


X(10473) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND INTOUCH TRIANGLE

Trilinears    a^3(b^2 + c^2) + 4a^2bc(b + c) - a(b^4 - 4b^3c - 4bc^3 + c^4) - 2bc(b + c)(b^2 - 3bc + c^2) : :

Let A'B'C' be the intangents triangle. Let LA be the line through the points of contact of the incircle and lines BC and B'C'. Define LB, LC cyclically. Let A" = LB∩LC, B" = LC∩LA, C" = LA∩LB. 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} et al

X(10473) = anticomplement of X(9564)
X(10473) = {X(10474),X(10475)}-harmonic conjugate of X(1)


X(10474) = (name pending)

Trilinears    (b + c)(a^3 - a^2b - a^2c - 2ab^2 - 2ac^2 - 2abc - 2b^2c - 2bc^2)/(b + c - a) : :

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}, {314,1441} et al

X(10474) = {X(1),X(10473)}-harmonic conjugate of X(10475)


X(10475) = (name pending)

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

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}, {314,1014} et al

X(10475) = {X(1),X(10473)}-harmonic conjugate of X(10474)


X(10476) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND HEXYL TRIANGLE

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

X(10476) lies on these lines: {1,3}, {2,9548}, {4,3741}, {20,10453}, {84,309} et al


X(10477) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND 2nd EXTOUCH TRIANGLE

Trilinears    a^3(b^2 + bc + c^2) + a^2bc(b + c) - a(b^4 + b^3c + bc^3 + c^4) - bc(b + c)(b^2 + c^2) : :

X(10477) lies on these lines: {1,6}, {2,3786}, {3,10461}, {4,69}, {8,3963}, {10,3779}, {21,5138} et al

X(10477) = anticomplement of X(4260)


X(10478) = HOMOTHETIC CENTER OF INVERSE-IN-CONWAY-CIRCLE TRIANGLE AND 3rd EULER TRIANGLE

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

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} et al

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

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

X(10479) lies on these lines: {1,2}, {3,10454}, {4,1764}, {5,1211}, {75,3670}, {314,5224}, {6684,10434}, {10175,10439}, {10436,10452} et al

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

Trilinears    (b + c - a)[a^3(b^2 + b c + c^2) + a^2(2b^3 + b^2c + bc^2 + 2c^3) + a(b^4 + b^3c + bc^3 + c^4) + bc(b - c)^2(b + c)] : :

X(10480) lies on these lines: {1,3}, {2,1682}, {8,314} et al

X(10480) = anticomplement of X(9565)


X(10481) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND INVERSE-IN-INCIRCLE TRIANGLE

Trilinears    a[4 sec2(A/2) cos2(B/2) cos2(C/2) - 1] + b[4 cos2(C/2) - 1] + c[4 cos2(B/2) - 1] : :

Let LA be the trilinear polar, wrt the intouch triangle, of the A-excenter; define LB, LC cyclically. Let A' = LB∩LC, B' = LC∩LA, C' = LA∩LB. 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} et al

X(10481) = midpoint of X(481) and X(482)
X(10481) = isogonal conjugate of X(10482)
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)

Trilinears    1/{a[4 sec2(A/2) cos2(B/2) cos2(C/2) - 1] + b[4 cos2(C/2) - 1] + c[4 cos2(B/2) - 1]} : :

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} et al

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)

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

X(10483) lies on these lines: {1,30}, {2,5370}, {3,3585}, {4,36}, {5,7280}, {20,35}, {55,1657}, {56,382}, {381,5204} et al

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

Trilinears    1/[9 cos(A - ω) - sin(A + ω) cot ω - 2 cos A cos ω] : :
Barycentrics    1/[(2a2 - b2 - c2)2 - 9b2c2] : :

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} et al

X(10484) = isogonal conjugate of X(10485)


X(10485) = ISOGONAL CONJUGATE OF X(10484)

Trilinears    9 cos(A - ω) - sin(A + ω) cot ω - 2 cos A cos ω : :
Barycentrics    a2[(2a2 - b2 - c2)2 - 9b2c2] : :

X(10485) lies on these lines: {2,8587}, {3,6} et al

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

Trilinears    (3 sin B + cos B cot ω)[(3 sin C + cos C cot ω)2 - (3 sin A + cos A cot ω)2]*(3 sin C - cos C cot ω)[(3 sin A - cos A cot ω)2 - (3 sin B - cos B cot ω)2] - (3 sin B - cos B cot ω)[(3 sin C - cos C cot ω)2 - (3 sin A - cos A cot ω)2]*(3 sin C + cos C cot ω)[(3 sin A + cos A cot ω)2 - (3 sin B + cos B cot ω)2] : :

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} et al


X(10487) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND ANTI-McCAY TRIANGLE

Barycentrics    [(a^2 - 2b^2 - 2c^2)^2 - 9b^2c^2](5a^4 - 2a^2b^2 - 2a^2c^2 - 5b^2c^2 + 2b^4 + 2c^4) : :

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} et al


X(10488) = PERSPECTOR OF ANTI-McCAY TRIANGLE AND CROSS-TRIANGLE OF ABC AND ANTI-McCAY TRIANGLE

Barycentrics    11a^6 - 9a^4(b^2 + c^2) + 3a^2(b^4 - b^2c^2 + c^4) - 4b^6 - 4c^6 + 6b^4c^2 + 6b^2c^4 : :

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} et al


X(10489) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND MID-ARC TRIANGLE

Trilinears    [cos(B/2) + cos(C/2)]2 sec2(A/2) : :
Trilinears    [(b' + c')/a']2 : :, where a', b', c' are the sidelengths of the excentral 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 (IA) be the circle tangent to the incircle and lines CA and AB, such that (IA) lies between the incircle and A. Define (IB), (IC) cyclically. Let A' be the insimilicenter of (IB) and (IC), 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} et al

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

Trilinears    [cos(B/2) + cos(C/2)] tan(A/2) : :
Trilinears    (b' + c') cot A' : :, where A'B'C' is the excentral triangle

X(10490) lies on these lines: {1,168}, {2,178} et al

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

Trilinears    [cos(B/2) - cos(C/2)]2 sec2(A/2) : :
Trilinears    [(b' - c')/a']2 : :, where a', b', c' are the sidelengths of the excentral 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 (IA') be the circle tangent to the incircle and lines CA and AB, such that the incircle lies between (IA') and A. Define (IB'), (IC') cyclically. Let A' be the insimilicenter of (IB') and (IC'), 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} et al

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)

Trilinears    [cos(B/2) - cos(C/2)] sec(A/2) : :
Trilinears    (b' - c')/a' : :, where a', b', c' are the sidelengths of the excentral triangle

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} et al

X(10492) = SS(A→A') of X(514), where A'B'C' = excentral triangle


X(10493) = CENTER OF 1st TANGENTIAL MID-ARC CONIC

Trilinears    [cos(B/2) + cos(C/2)]{- a[cos(B/2) + cos(C/2)] + b[cos(C/2) + cos(A/2)] + c[cos(A/2) + cos(B/2)]} : :

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} et al

X(10493) = X(2)-Ceva conjugate of X(7707)


X(10494) = CENTER OF 2nd TANGENTIAL MID-ARC CONIC

Trilinears    [cos(B/2) - cos(C/2)]{- a[cos(B/2) - cos(C/2)] + b[cos(C/2) - cos(A/2)] + c[cos(A/2) - cos(B/2)]} : :

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: (pending)

X(10494) = X(2)-Ceva conjugate of X(10495)


X(10495) = PERSPECTOR OF 2nd TANGENTIAL MID-ARC CONIC

Trilinears    cos(B/2) - cos(C/2) : :

X(10495) lies on these lines: {44,513}, {6728,10492}, {8076,10231} et al

X(10495) = crossdifference of every pair of points on line X(1)X(168)
X(10495) = X(2)-Ceva conjugate of X(10494)


X(10496) = COLLINGS TRANSFORM OF X(10493)

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

Let I be the incenter and IA, IB, IC the excenters. The Monge line of the circumcircles of BCIIA, CAIIB, ABIIC 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} et al

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)

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

X(10497) lies on the circumcircle and these lines: (pending)

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)

Trilinears    1/[-(s-a)^2*a*sin(A/2)+(a-c)*(s-b)^2*sin(B/2)+(a-b)*(s-c)^2*sin(C/2)+S^2/(4*s)] : :

X(10498) lies on these lines: {174,354}, {258,503} et al

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

Trilinears    [cos2(B/2) - cos2(C/2)]{2 cos(A/2) - cos(B/2) - cos(C/2) - [cos2(B/2) + cos2(C/2)] sec(A/2) + cos(B/2) cos(C/2) [cos(B/2) + cos(C/2)] sec2(A/2)} : :

X(10499) lies on these lines: {1,7}, {177,234} et al

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

Trilinears    [cos(B/2) + cos(C/2)]{cos2(B/2) + cos2(C/2) + cos(A/2) [cos(B/2) + cos(C/2)]} : :

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} et al

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

Trilinears    a(b + c) - (b - c)^2 - 2 Sqrt(bc(c + a - b)(a + b - c)) : :

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}, et al

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(10502) = {X(174),X(354)}-HARMONIC CONJUGATE OF X(10501)

Trilinears    a(b + c) - (b - c)^2 + 2 Sqrt(bc(c + a - b)(a + b - c)) : :

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 LA be the line through the points of contact of the incircle and lines BC and B"C". Define LB, LC cyclically. Let A* = LB∩LC, B* = LC∩LA, C* = LA∩LB. Triangle A*B*C* is homothetic to the Yff central triangle at X(10502).

X(10502) lies on these lines: {174,354}, {177,234} et al


X(10503) = (name pending)

Trilinears    [cos(B/2) + cos(C/2)]{[cos(B/2) - cos(C/2)]^2 + cos(A/2) [cos(B/2) + cos(C/2)]} : :

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} et al

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

Barycentrics    2 (b-c) (a^3-a b^2+a b c-b^2 c-a c^2-b c^2) S- (b-c) (a b+b^2+a c-2 b c+c^2) (a^3-a b^2+a b c-b^2 c-a c^2-b c^2) Cos[A/2]- (a^4 b^2+a^3 b^3-a^2 b^4-a b^5+4 a^4 b c-a^3 b^2 c-5 a^2 b^3 c+a b^4 c+b^5 c+3 a^4 c^2-7 a^3 b c^2+6 a^2 b^2 c^2+6 a b^3 c^2-4 b^4 c^2-a^3 c^3+3 a^2 b c^3-10 a b^2 c^3+2 b^3 c^3-3 a^2 c^4+3 a b c^4+4 b^2 c^4+a c^5-3 b c^5) Cos[B/2]+ (3 a^4 b^2-a^3 b^3-3 a^2 b^4+a b^5+4 a^4 b c-7 a^3 b^2 c+3 a^2 b^3 c+3 a b^4 c-3 b^5 c+a^4 c^2-a^3 b c^2+6 a^2 b^2 c^2-10 a b^3 c^2+4 b^4 c^2+a^3 c^3-5 a^2 b c^3+6 a b^2 c^3+2 b^3 c^3-a^2 c^4+a b c^4-4 b^2 c^4-a c^5+b c^5) Cos[C/2]
Trilinears    a (b-c) (a b+a c+2 b c) x^5+(2 a^2 b^2-a b^3+2 a b^2 c-b^3 c+a c^3+b c^3) x^4 y+(a^3 b+a^2 b^2-2 a b^3-a b^2 c+a^2 c^2+a b c^2-b^2 c^2-a c^3-b c^3) x^3 y^2+b (a-b-c) (2 a^2+a b-b c) x^2 y^3+b (a-c) (a^2-2 a b-a c-b c) x y^4-b^2 (a-c)^2 y^5-(a b^3+b^3 c+2 a^2 c^2+2 a b c^2-a c^3-b c^3) x^4 z+(a-b-c) (b-c) (a b+a c-b c) x^3 y z+(a-b-c) c (-2 b^2+a c+b c) x^2 y^2 z+b (a^3-a^2 b-2 a^2 c+3 a b c+a c^2+2 b c^2) x y^3 z+b (a-c) (a^2-a c+3 b c) y^4 z-(a^2 b^2-a b^3+a^3 c+a b^2 c-b^3 c+a^2 c^2-a b c^2-b^2 c^2-2 a c^3) x^3 z^2-b (a^3-a^2 b+2 a^2 c+a b c-3 a c^2+2 b c^2) y^3 z^2-x y (a^2 b^2 x-a b^3 x-b^3 c x-2 a b c^2 x+b^2 c^2 x+2 b c^3 x+a^3 b y-a^2 b^2 y-a^3 c y-2 a b^2 c y+a^2 c^2 y+2 a b c^2 y) z^2-(a-b-c) c (2 a^2+a c-b c) x^2 z^3-c (a^3-2 a^2 b+a b^2-a^2 c+3 a b c+2 b^2 c) x y z^3+c (a^3+2 a^2 b-3 a b^2-a^2 c+a b c+2 b^2 c) y^2 z^3-(a-b) c (a^2-a b-2 a c-b c) x z^4-(a-b) c (a^2-a b+3 b c) y z^4+(a-b)^2 c^2 z^5, where x = cos(A/2), y = cos(B/2), z = cos(C/2)

X(10504) lies on the incircle and these lines: {1,7057}, {55,10496}, {177,10506} et al

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

Trilinears    a^3 (x + y) (x + z) [b (-1 + y^2) (x + y - z) - c (x - y + z) (-1 + z^2)] + a^2 {2 b c [-y^2 (x + y) (-1 + x y) - y^3 (x + y) z - (x + y^3) z^2 + (-1 + x^2 + x y + y^2) z^3 + (x + y) z^4] + b^2 (x - y) (x + y) (x + y - z) [-1 + x^2 + x (y + z) + y (y + z)] - c^2 (x - z) (x + z) (x - y + z) (-1 + x^2 + x (y + z) + z (y + z)]} + a {-b^3 (-1 + x^2) (x + y) (x + y - z) (y + z) + c^3 (-1 + x^2) (x + z) (x - y + z) (y + z) + b^2 c [2 x^5 + 2 x^4 y + x (y + z)^2 (y^2 + y z - z^2) + (y + z) (2 y^4 + y^3 z - 2 (-1 + y^2) z^2 - y z^3) - 2 x^2 y (1 + y (y + z)) - x^3 (2 + (y + z)^2)] + b c^2 [-2 x^5 - 2 x^4 z + x (y + z)^2 (y^2 - y z - z^2) + 2 x^2 z (1 + z (y + z)) + x^3 (2 + (y + z)^2) + (y + z) (y^3 z - y z^3 - 2 z^4 + 2 y^2 (-1 + z^2))]} - b c (x - y - z) (y + z) [b^2 (-1 + x^2) (x + y) - c^2 (-1 + x^2) (x + z) - b c (y - z) (1 - x^2 + (y + z)^2)] : :, where x = cos(A/2), y = cos(B/2), z = cos(C/2)

X(10505) lies on the incircle and these lines: {1,7057}, {56,10496}, {8422,10501} et al

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

Trilinears    a^2(b + c) - 8abc - (b - c)^2(b + c) + 2(a - b - c) Sqrt[bc(a + b - c)(a - b + c)] + 4b Sqrt[ca(a + b - c)(-a + b + c)] + 4c Sqrt[ab(a - b + c)(-a + b + c)] : :

X(10506) lies on the incircle and these lines: {1,8099}, {11,10500}, {55,7597}, {56,3659}, {177,10504} et al

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

Trilinears    a y^2 z^2 (y + z) (2x + y + z) + b z^2 [2x^4 + 4x^3 (y + z) - 2x y^2 (y + z) - y^2 (y + z)^2 + 2x^2 z (2y + z)] + c y^2 [2x^4 + 4x^3 (y + z) - 2 x z^2 (y + z) - z^2 (y + z)^2 + 2x^2 y (y + 2z)] : :, where x = cos(A/2), y = cos(B/2), z = cos(C/2)

X(10507) lies on these lines: {1,7}, {8422,10508} et al

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

Trilinears    a (y + z) [y^2 + z^2 + x (y + z)] + b [2 x^3 + 2x^2 (y + z) - (y - z) (y + z)^2 + x (-y^2 + 2y z + z^2)] + c [2 x^3 + 2x^2 (y + z) + (y - z) (y + z)^2 + x (y^2 + 2y z - z^2)] : :, where x = cos(A/2), y = cos(B/2), z = cos(C/2)

X(10508) lies on these lines: {1,3}, {8422,10507} et al

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

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

X(10509) lies on these lines: {6,279}, {7,55}, {9,85}, {6604,7674} et al

X(10509) = isogonal conjugate of X(8012)


X(10510) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND X(2)-EHRMANN TRIANGLE

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

X(10510) lies on these lines: {2,8262}, {3,6}, {23,6593}, {67,524}, {2393,2930} et al

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)

Barycentrics    1/[(a2 - 2b2 - 2c2)(a4 - b4 - c4 + b2c2)] : :

X(10511) lies on the Kiepert hyperbola and these lines: {2,67}, {4,1383}, {23,671} et al

X(10511) = isogonal conjugate of X(10510)
X(10511) = vertex conjugate of X(23) and X(67)


X(10512) = ISOTOMIC CONJUGATE OF X(10510)

Barycentrics    1/[a2(a2 - 2b2 - 2c2)(a4 - b4 - c4 + b2c2)] : :

X(10512) lies on these lines: (pending).

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

Barycentrics    3a4 - 5b4 - 5c4 + 2a2b2 + 2a2c2 - 6b2c2 : :

X(10513) lies on these lines: {2,6}, {20,3933}, {1160,6215}, {1161,6214}, {3640,5689}, {3641,5688}, {6201,10514}, {6202,10515} et al

X(10513) = anticomplement of X(5304)
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

Barycentrics    a6 + a2[b4 + c4 - 2S(b2 + c2) + 6b2c2] - 2(b2 - c2)2(b2 + c2 - S) : :

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} et al

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

Barycentrics    a6 + a2[b4 + c4 + 2S(b2 + c2) + 6b2c2] - 2(b2 - c2)2(b2 + c2 + S) : :

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} et al

X(10515) = {X(5),X(10516)}-harmonic conjugate of X(10514)


X(10516) = {X(10514),X(10515)}-HARMONIC CONJUGATE OF X(5)

Barycentrics    a6 + a2(b4 + 6b2c2 + c4) - 2(b2 - c2)2(b2 + c2) : :

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} et al

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

Barycentrics    3a^8 - 2a^6(5(b^2 + c^2) - 2S) + 4a^4(3b^4 + 2b^2c^2 + 3c^4 + 3(b^2 + c^2)S) - 2a^2(2(5b^4 + 6b^2c^2 + 5c^4)S + 3(b^2 - c^2)^2(b^2 + c^2)) + 4(b^2 - c^2)^2(b^2 + c^2)S + (b^2 - c^2)^4 : :

X(10517) lies on these lines: {2,1161}, {3,1271}, {4,640}, {6,631}, {20,6215}, {5595,10323} et al

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

Barycentrics    3a^8 - 2a^6(5(b^2 + c^2) + 2S) + 4a^4(3b^4 + 2b^2c^2 + 3c^4 - 3(b^2 + c^2)S) + 2a^2(2(5b^4 + 6b^2c^2 + 5c^4)S - 3(b^2 - c^2)^2(b^2 + c^2)) - 4(b^2 - c^2)^2(b^2 + c^2)S + (b^2 - c^2)^4 : :

X(10518) lies on these lines: {2,1160}, {3,1270}, {4,639}, {6,631}, {20,6214}, {5594,10323} et al

X(10518) = {X(631),X(10519)}-harmonic conjugate of X(10517)


X(10519) = {X(10517),X(10518)}-HARMONIC CONJUGATE OF X(631)

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

X(10519) lies on these lines: {2,51}, {3,69}, {4,141}, {6,631}, {20,1352}, {159,10323} et al

X(10519) = reflection of X(10516) in X(141)


X(10520) = PERSPECTOR OF INVERSE-IN-INCIRCLE TRIANGLE AND CROSS-TRIANGLE OF INVERSE-IN-INCIRCLE AND INVERSE-IN-EXCIRCLES TRIANGLES

Barycentrics    (b + c - a)[12a^6 + 7a^5(b + c) + 3a^4(7b^2 - 2bc + 7c^2) + 2a^3(b + c)(7b^2 - 2bc + 7c^2) + 2a^2(b + c)^2(7b^2 - 6bc + 7c^2) - a(b - c)^2(b + c)(5b^2 - 2bc + 5c^2) + (b - c)^4(b + c)^2] : :

X(10520) lies on these lines: {1,3598}, {7,1699}, {55,4021}, {57,1766}, {165,3672}, {354,1122}, {942,2808}, {999,1323}, {3673,4298} et al


X(10521) = PERSPECTOR OF INVERSE-IN-EXCIRCLES TRIANGLE AND CROSS-TRIANGLE OF INVERSE-IN-INCIRCLE AND INVERSE-IN-EXCIRCLES TRIANGLES

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

X(10521) lies on these lines: {1,3598}, {7,10}, {56,1323}, {57,169}, {511,942}, {516,3673}, {519,3212}, {3295,4021} et al


X(10522) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF INNER AND OUTER JOHNSON TRIANGLES

Barycentrics    a - b - c - a(cos A)(1 - cos A - 2 sin(A/2) cos(B/2 - C/2)) + b(cos B)(1 - cos B - 2 sin(B/2) cos(C/2 - A/2)) + c(cos C)(1 - cos C - 2 sin(C/2) cos(A/2 - B/2)) : :

Of the two tangents to the B- and C-Johnson circles, let LA be the one on the opposite side of BC from A, and let MA be the other. Define LB, MB, LC, MC cyclically. Let A' = LB∩LC, and define B', C' cyclically. Let A" = MB∩MC, 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} et al

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

Trilinears    [csc C - (csc C)(cos A)f(A,B,C) - csc A + (csc A)(cos C)f(C,A,B)]*[sec A + (sec A)(cos B)f(B,C,A) - sec B + (sec B)(cos A)f(A,B,C)] - [csc A - (csc A)(cos B)f(B,C,A) - csc B + (csc B)(cos A)f(A,B,C)]*[secC + (sec C)(cos A)f(A,B,C) - sec A + (sec A)(cos C)f(C,A,B)] : : , where f(A,B,C) = (cos A)(1 - cos A - 2 sin(A/2) cos(B/2 - C/2))
Barycentrics    a^5(b^2 + c^2) - a^4(b + c)(b^2 + c^2) - 2a^3(b^4 - b^3c - 2b^2c^2 - bc^3 + c^4) + 2a^2(b - c)^2(b + c)(b^2 + bc + c^2) + a(b - c)^4(b + c)^2 - (b - c)^4(b + c)^3 : :

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, -)

Let A1B1C1 and A2B2C2 be the inner and outer Yff triangles, resp. Let A' be the centroid of B1C1B2C2, and define B', C' cyclically. Triangle A'B'C' is homothetic to ABC at X(10321), to the medial triangle at X(1), and to the Euler triangle at X(10523).

Peter Moses (November 1, 2016) gives barycentrics:

(A-vertex of 1st Johnson-Yff triangle): -a (a (a - b - c)+2 b c) : (a-c)^2 (a-b+c) : (a-b)^2 (a+b-c)
(A-vertex 2nd Johnson-Yff triangle): -a^2 (a+b-c) (a-b+c) : (a-b-c) (a+b-c) (a+c)^2 : (a+b)^2 (a-b-c) (a-b+c)

X(10523) lies on these lines: {1,5}, {2,8071}, {3,10320}, {4,8069}, {35,6907}, {36,6922}, {55,6842}, {56,6882}, {1478,6831}, {1479,1532} et al

X(10523) = {X(11),X(12)}-harmonic conjugate of X(355)
X(10523) = homothetic center of inner Yff triangle and inner Johnson triangle
X(10523) = homothetic center of outer Yff triangle and outer Johnson triangle
X(10523) = homothetic center of ABC and mid-triangle of 1st and 2nd Johnson-Yff triangles
X(10523) = homothetic center of Johnson triangle and mid-triangle of inner and outer Yff triangles


X(10524) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND CROSS-TRIANGLE OF INNER AND OUTER JOHNSON TRIANGLES

Barycentrics   , a^7-a^6 b+a^5 b^2-a^4 b^3-5 a^3 b^4+5 a^2 b^5+3 a b^6-3 b^7-a^6 c+2 a^5 b c-3 a^4 b^2 c+4 a^3 b^3 c+a^2 b^4 c-6 a b^5 c+3 b^6 c+a^5 c^2-3 a^4 b c^2+14 a^3 b^2 c^2-10 a^2 b^3 c^2-3 a b^4 c^2+9 b^5 c^2-a^4 c^3+4 a^3 b c^3-10 a^2 b^2 c^3+12 a b^3 c^3-9 b^4 c^3-5 a^3 c^4+a^2 b c^4-3 a b^2 c^4-9 b^3 c^4+5 a^2 c^5-6 a b c^5+9 b^2 c^5+3 a c^6+3 b c^6-3 c^7 : :

X(10524) lies on these lines:
{2,8071}, {5,10530}, {11,3436}, {12,3434}, {145,355}, {958,10584}, {1376,10585}, {3871,6982}

X(10524) = {X(10522),X(10523)}-harmonic conjugate of X(2)


X(10525) = HOMOTHETIC CENTER OF JOHNSON TRIANGLE AND CROSS-TRIANGLE OF ABC AND INNER JOHNSON TRIANGLE

Barycentrics    a^7 - a^6(b + c) - a^5(b - c)^2 + a^4(b + c)(b^2 + c^2) - a^3(b^2 + c^2)^2 + a^2(b - c)^2(b + c)(b^2 + c^2) + a(b - c)^4(b + c)^2 - (b - c)^4(b + c)^3 : :

X(10525) lies on these lines: {1,6923}, {2,10598}, {3,11}, {4,8}, {5,1376}, {7,6583}, {10,6929}, {35,6863}, {40,3583}, {55,6842}, {100,6941}, {119,5687}, {382,5841}, {3585,7982} et al

X(10525) = reflection of X(10526) in X(4)
X(10525) = X(3)-of-inner-Johnson-triangle


X(10526) = HOMOTHETIC CENTER OF JOHNSON TRIANGLE AND CROSS-TRIANGLE OF ABC AND OUTER JOHNSON TRIANGLE

Barycentrics    a^7 - a^6(b + c) - a^5(b - c)^2 + a^4(b + c)(b^2 - 4bc + c^2) - a^3(b^4 - 6b^2c^2 + c^4) + a^2(b - c)^2(b + c)(b^2 + 4bc + c^2) + a(b - c)^4(b + c)^2 - (b - c)^4(b + c)^3 : :

X(10526) lies on these lines: {1,6928}, {2,10599}, {3,12}, {4,8}, {5,958}, {7,5885}, {10,6917}, {36,6958}, {40,3585}, {55,7491}, {56,6882}, {104,6943}, {119,3149}, {382,5840}, {3583,7982} et al

X(10526) = reflection of X(10525) in X(4)
X(10526) = X(3)-of-outer-Johnson-triangle


X(10527) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF ABC AND OUTER YFF TANGENTS TRIANGLE

Barycentrics    (a + b + c) - (R/r)(a - b - c) : :

Let LA be the line, other than BC, tangent to the B- and C-inner Yff circles. Define LB, LC cyclically. Let A' = LB∩LC and define B', C' cyclically. Triangle A'B'C' is here named the inner Yff tangents triangle.

Let MA be the line, other than BC, tangent to the B- and C-outer Yff circles. Define MB, MC cyclically. Let A" = MB∩MC and define B", C" cyclically. Triangle A"B"C" is here named the outer Yff tangents triangle.

The appearance of (T,i,j) in the following list means that triangle T is homothetic to the inner Yff tangents triangle with homothetic center X(i) and to the outer Yff tangents triangle with homothetic center X(j):
(ABC, 1, 1)
(medial, 5552, 10527)
(anticomplementary, 10528, 10529)
(Euler, 10531, 10532)
(Caelum, 1, 1)
(inner Johnson, 12, -)
(outer Johnson, -, 11)
(inner Yff, 1, 1)
(outer Yff, 1, 1)
(T(1,2), 1, 1)

(A-vertex of inner Yff tangents triangle): -(a-b-c) (a^3+a^2 b-a b^2-b^3+a^2 c+b^2 c-a c^2+b c^2-c^3) : 4 a b^2 c : 4 a b c^2
(A-vertex of outer Yff tangents triangle): (a-b-c) (a^3+a^2 b-a b^2-b^3+a^2 c+4 a b c+b^2 c-a c^2+b c^2-c^3) : 4 a b^2 c : 4 a b c^2

X(10527) lies on these lines: {1,2}, {3,3434}, {4,2975}, {5,956}, {7,7183}, {11,958}, {12,6933}, {20,5450}, {21,497}, {36,4190}, {40,6890}, {55,3813}, {56,377}, {100,631}, {140,5687}, {149,4189}, {318,475}, {3560,10531} et al

X(10527) = Garcia point G(-R/r)
X(10527) = complement of X(10528)
X(10527) = {X(2),X(8)}-harmonic conjugate of X(5552)
X(10527) = homothetic center of medial triangle and outer Yff tangents triangle


X(10528) = HOMOTHETIC CENTER OF INNER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF ABC AND INNER YFF TANGENTS TRIANGLE

Barycentrics    (a + b + c) + (R/r)(3a - b - c) : :

X(10528) lies on these lines: {1,2}, {4,3871}, {5,10596}, {12,3434}, {20,5841}, {21,3421}, {40,5905}, {55,3436}, {100,388}, {119,149}, {377,495}, {496,6931}, {497,5187}, {944,6890}, {956,6910}, {2975,5218} et al

X(10528) = anticomplement of X(10527)
X(10528) = {X(1),X(10530)}-harmonic conjugate of X(10529)
X(10528) = {X(2),X(145)}-harmonic conjugate of X(10529)
X(10528) = homothetic center of anticomplementary triangle and inner Yff tangents triangle


X(10529) = HOMOTHETIC CENTER OF OUTER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF ABC AND OUTER YFF TANGENTS TRIANGLE

Barycentrics    (a + b + c) - (R/r)(3a - b - c) : :

X(10529) lies on these lines: {1,2}, {5,10597}, {11,3436}, {20,104}, {21,1058}, {56,3434}, {100,7288}, {377,999}, {388,6871}, {390,4189}, {495,6933}, {496,956}, {497,2975}, {517,6890}, {631,3871}, {944,6838}, {946,5905}, {3091,10532}, {3295,6910} et al

X(10529) = anticomplement of X(5552)
X(10529) = {X(1),X(10530)}-harmonic conjugate of X(10528)
X(10529) = {X(2),X(145)}-harmonic conjugate of X(10528)
X(10529) = homothetic center of anticomplementary triangle and outer Yff tangents triangle


X(10530) = HOMOTHETIC CENTER OF ABC AND CROSS-TRIANGLE OF INNER AND OUTER YFF TANGENTS TRIANGLES

Barycentrics    a^7 - a^6(b + c) - a^5(3b^2 - 2bc + 3c^2) + a^4(b + c)(3b^2 - 2bc + 3c^2) + a^3(3b^4 - 4b^3c + 14b^2c^2 - 4bc^3 + 3c^4) - a^2(b + c)(b^2 + c^2)(3b^2 - 4bc + 3c^2) - a(b - c)^4(b + c)^2 + (b - c)^4(b + c)^3 : :

X(10530) lies on these lines: {1,2}, {5,10524}, {119,6953}, {149,6847} et al

X(10530) = {X(10528),X(10529)}-harmonic conjugate of X(1)
X(10530) = homothetic center of outer Yff tangents triangle and cross-triangle of ABC and inner Yff tangents triangle
X(10530) = homothetic center of inner Yff tangents triangle and cross-triangle of ABC and outer Yff tangents triangle


X(10531) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND INNER YFF TANGENTS

Barycentrics    (a^4 + b^4 + c^4 - 2a^3b - 2a^3c - 2a^2b^2 - 2a^2c^2 + 4a^2bc + 2ab^3 + 2ac^3 - 2ab^2c - 2abc^2 - 2b^2c^2)(R/r) + (c^2 + a^2 - b^2)(a^2 + b^2 - c^2) : :

X(10531) lies on these lines: {1,4}, {5,3434}, {7,5553}, {8,6893}, {10,6898}, {11,6833}, {12,6968}, {35,6880}, {40,6947}, {55,6834}, {56,6938}, {100,6944}, {104,5555}, {119,149}, {381,10599}, {517,2478}, {3560,10527} et al

X(10531) = {X(4),X(5603)}-harmonic conjugate of X(10532)


X(10532) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND OUTER YFF TANGENTS

Barycentrics    (a^4 + b^4 + c^4 - 2a^3b - 2a^3c - 2a^2b^2 - 2a^2c^2 + 4a^2bc + 2ab^3 + 2ac^3 - 2ab^2c - 2abc^2 - 2b^2c^2)(R/r) - (c^2 + a^2 - b^2)(a^2 + b^2 - c^2) : :

X(10532) lies on these lines: {1,4}, {5,956}, {8,6826}, {10,6854}, {12,6834}, {21,6585}, {36,6977}, {40,6897}, {55,6934}, {56,6833}, {100,6885}, {104,3600}, {119,6953}, {377,517}, {381,10598}, {3091,10529}, {5252,7686}, {6968,7681} et al

X(10532) = {X(4),X(5603)}-harmonic conjugate of X(10531)


X(10533) = HOMOTHETIC CENTER OF 1st KENMOTU DIAGONALS TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    a(tan B + tan C - tan A + 1) : :

The cross-triangle of the 1st and 2nd Kenmotu diagonals triangles is also the X(3)-Ehrmann triangle (see X(25) and X(31)), and is homothetic to the orthic triangle at X(184). (Peter Moses, November 1, 2016, gives these barycentrics:

A-vertex of 1st Kenmotu diagonal triangle: (a^2 (SA - S))/(SA + S) : b^2 : c^2
A-vertex of 2nd Kenmotu diagonal triangle: (a^2 (SA + S))/(SA - S) : b^2 : c^2

The cross-triangle of the 1st and 2nd Kenmotu diagonals triangles is also the cross-triangle of the inner and outer tri-equilateral triangles, and also the medial triangle of the tangential triangle. (Randy Hutson, December 10, 2016)

X(10533) lies on these lines: {6,25}, {371,6759}, {372,10282}, {485,9833}, {590,1503}, {615,10192}, {1151,1498}, {2066,10535}, {5415,10536} et al

X(10533) = {X(6),X(154)}-harmonic conjugate of X(10534)


X(10534) = HOMOTHETIC CENTER OF 2nd KENMOTU DIAGONALS TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    a(tan B + tan C - tan A - 1) : :

X(10534) lies on these lines: {6,25}, {371,8908}, {372,6759}, {486,9833}, {590,10192}, {615,1503}, {1152,1498}, {5414,10535}, {5416,10536} et al

X(10534) = {X(6),X(154)}-harmonic conjugate of X(10533)


X(10535) = HOMOTHETIC CENTER OF INTANGENTS TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    a[(tan A)(cos B + cos C - 1) - (tan B)(cos C + cos A - 1) - (tan C)(cos A + cos B - 1)] : :
Trilinears    a(b + c - a)[a^6 - a^4(2b^2 - 3bc + 2c^2) + a^2(b - c)^2(b^2 + c^2) - bc(b^2 - c^2)^2] : :

Note that the X(1)-Ehrmann triangle is the intangents triangle, and the X(3)-Ehrmann triangle is the cross triangle of the 1st and 2nd Kenmotu diagonals triangles, so that X(10535) is the homothetic center of X(1)- and X(3)-Ehrmann triangles.

X(10535) lies on these lines: {1,6759}, {3,6285}, {11,1428}, {33,184}, {35,10282}, {48,55}, {84,7335}, {652,663}, {1062,10539}, {2066,10533}, {5414,10534} et al

X(10535) = isogonal conjugate of isotomic conjugate of X(10538)
X(10535) = {X(154),X(10537)}-harmonic conjugate of X(10536)


X(10536) = HOMOTHETIC CENTER OF EXTANGENTS TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Trilinears    a[a^7 - a^6(b + c) - a^5(2b^2 + 3bc + 2c^2) + a^4(b + c)(2b^2 - 3bc + 2c^2) + a^3(b + c)^2(b^2 + c^2) - a^2(b - c)^2(b + c)(b^2 + c^2) + abc(b^2 - c^2)^2 + bc(b - c)^2(b + c)^3] : :

X(10536) lies on these lines: {3,6254}, {40,6759}, {48,55}, {5415,10533}, {5416,10534}, {8251,10539} et al

X(10536) = {X(154),X(10537)}-harmonic conjugate of X(10535)


X(10537) = {X(10535),X(10536)}-HARMONIC CONJUGATE OF X(154)

Trilinears    a[a^7 - a^6(b + c) - 2a^5(b^2 + c^2) + a^4(b + c)(2b^2 - 3bc + 2c^2) + a^3(b^2 + c^2)^2 - a^2(b - c)^2(b + c)(b^2 + c^2) + bc(b - c)^2(b + c)^3] : :

X(10537) lies on these lines: {1,1437}, {48,55}, {159,674} et al

X(10537) = X(283)-Ceva conjugate of X(6)


X(10538) = INVERSE-IN-AC-INCIRCLE OF X(40)

Barycentrics    (tan A)(cos B + cos C - 1) - (tan B)(cos C + cos A - 1) - (tan C)(cos A + cos B - 1) : :
Barycentrics    (b + c - a)[a^6 - a^4(2b^2 - 3bc + 2c^2) + a^2(b - c)^2(b^2 + c^2) - bc(b^2 - c^2)^2] : :

X(10538) lies on these lines: {2,1074}, {3,318}, {7,6527}, {8,20}, {21,243}, {522,663} et al

X(10538) = isotomic conjugate of isogonal conjugate of X(10535)
X(10538) = anticomplement of X(1785)
X(10538) = inverse-in-de-Longchamps-circle of X(8)
X(10538) = perspector of ABC and reflection of extouch triangle in line X(522)X(650) (i.e. the reflection of the cevian triangle of X(8) in the trilinear polar of X(8))


X(10539) = HOMOTHETIC CENTER OF 2nd EULER TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Barycentrics    a^2[a^8 - 3a^6(b^2 + c^2) + a^4(3b^4 + 2b^2c^2 + 3c^4) - a^2(b^2 + c^2)(b^4 + c^4) + 2b^2c^2(b^2 - c^2)^2] : :

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Johnson triangle at X(10539).

X(10539) lies on these lines: {2,1614}, {3,64}, {4,110}, {5,156}, {6,7529}, {22,1216}, {25,52}, {26,1495}, {30,1092}, {32,1625}, {49,381}, {54,3091}, {140,5651}, {182,1656}, {185,6644}, {186,7689}, {195,576}, {1062,10535}, {8251,10536} et al


X(10540) = HOMOTHETIC CENTER OF 2nd ISOGONAL TRIANGLE OF X(4) AND CROSS-TRIANGLE OF 1st AND 2nd KENMOTU DIAGONALS TRIANGLES

Barycentrics    a^2[a^8 - 3a^6(b^2 + c^2) + a^4(3b^4 + b^2c^2 + 3c^4) - a^2(b^6 + c^6) + 2b^2c^2(b^2 - c^2)^2] : :

X(10540) lies on these lines: {3,64}, {4,49}, {5,1614}, {23,1154}, {25,568}, {30,110}, {51,7545}, {54,546}, {548,8718} et al


X(10541) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF LUCAS BROCARD AND LUCAS(-1) BROCARD TRIANGLES

Trilinears    cos(A + ω) + 7 cos(A - ω) : :
Trilinears    4 cos A + 3 sin A tan ω : :
Trilinears    3 sin A + 4 cos A cot ω : :
Trilinears    3a + 8R cos A cot ω : :

X(10541) lies on these lines: {3,6}, {20,597}, {23,10601}, {141,10303}, {1657,5476}, {3529,5480}, et al

X(10541) = perspector of Lucas(-1) Brocard triangle and cross-triangle of ABC and Lucas Brocard triangle
X(10541) = perspector of Lucas Brocard triangle and cross-triangle of ABC and Lucas(-1) Brocard triangle
X(10541) = {X(1151),X(1152)}-harmonic conjugate of X(574)
X(10541) = radical center of Lucas(3/2 tan ω) circles


X(10542) = PERSPECTOR OF SYMMEDIAL TRIANGLE AND CROSS-TRIANGLE OF LUCAS BROCARD AND LUCAS(-1) BROCARD TRIANGLES

Barycentrics    a2(3a4 + 7b4 + 7c4 - 2a2b2 - 2a2c2 + 2b2c2) : :

Let A'B'C' and A"B"C" be the Lucas Brocard and Lucas(-1) Brocard triangles, resp. Let A* be the {A',A"}-harmonic conjugate of X(574), and define B*, C* cyclically. (Note: X(574) is the perspector of the Lucas Brocard and Lucas(-1) Brocard triangles.) The lines AA*, BB*, CC* concur in X(10542).

X(10542) lies on these lines: {3,6}, {69,7851} et al

X(10542) = {X(6421),X(6422)}-harmonic conjugate of X(574)


X(10543) = PERSPECTOR OF HUTSON INTOUCH TRIANGLE AND CROSS-TRIANGLE OF MANDART-INCIRCLE AND HUTSON INTOUCH TRIANGLES

Trilinears    2b2c2(1 + cos A)(cos2 B - cos2 C)[cos2 A - cos B cos C + cos A (1 + cos B + cos C)] + a2(cos A + cos B)(cos C + cos A)[b2(cos A - cos B)(1 + cos C)(1 + 2 cos C) + c2(cos C - cos A)(1 + cos B)(1 + 2 cos B)] : :
Barycentrics    (b + c - a)[4a3 + 2a2(b + c) - a(b - c)2 + (b - c)2(b + c)] : :

X(10543) lies on these lines: {1,30}, {3,5427}, {8,21}, {11,214}, {12,6841}, {35,5428}, {56,3488], {758,3057} et al

X(10543) = reflection of X(3649) in X(1)
X(10543) = X(21)-of-Mandart-incircle-triangle
X(10543) = X(54)-of-Hutson-intouch-triangle
X(10543) = X(7691)-of-intouch-triangle


X(10544) = PERSPECTOR OF MANDART-INCIRCLE TRIANGLE AND CROSS-TRIANGLE OF MANDART-INCIRCLE AND HUTSON INTOUCH TRIANGLES

Trilinears    a(b + c - a)[a2(b - c)2 + 2a(b3 + c3) + b4 + c4 + 2bc(b2 - bc + c2)] : :

X(10544) lies on these lines: {1,256}, {8,3794}, {11,3454}, {12,7683}, {55,58}, {56,3430}, {758,3057} et al

X(10544) = X(58)-of-Mandart-incircle-triangle
X(10544) = excentral-to-Hutson-intouch similarity image of X(1046)


X(10545) = HOMOTHETIC CENTER OF ORTHOCENTROIDAL TRIANGLE AND CROSS-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

Barycentrics    a2(a^4 - 2b4 - 2c4 + a2b2 + a2c2 + 7b2c2) : :

X(10545) lies on these lines: {2,3098}, {5,3581}, {6,110}, {23,373}, {74,381} et al

X(10545) = {X(6),X(1995)}-harmonic conjugate of X(10546)


X(10546) = HOMOTHETIC CENTER OF ANTI-ORTHOCENTROIDAL TRIANGLE AND CROSS-TRIANGLE OF ORTHOCENTROIDAL AND ANTI-ORTHOCENTROIDAL TRIANGLES

Barycentrics    a2(2a4 - b4 - c4 - a2b2 - a2c2 + 5b2c2) : :

X(10546) lies on these lines: {2,1495}, {3,5888}, {4,10564}, {6,110}, {23,3098}, {74,6644}, {381,1511} et al

X(10546) = {X(6),X(1995)}-harmonic conjugate of X(10545)


X(10547) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 1st ORTHOSYMMEDIAL TRIANGLE

Trilinears    a2(sin 2A) csc(A + ω) : :
Barycentrics    a4(b2 + c2 - a2)/(b2 + c2) : :

Let A'B'C' be the reflection of ABC in X(6). Let AB = BC∩C'A', AC = BC∩A'B', and define BC, BA, CA, CB cyclically. AB, AC, BC, BA, CA, CB lie on the 2nd Lemoine circle. Triangles AABAC, BABBC, CACBC are similar to one another and inversely similar to ABC. Let SA be the similitude center of BABBC and CACBC. Let SB be the similitude center of CACBC and AABAC. Let SC be the similitude center of AABAC and BABBC. SASBSC is perspective to ABC at X(6) and homothetic to the circumsymmedial triangle at X(6). X(10547) is the trilinear product SA*SB*SC.

X(10547) lies on these lines: {3,1176}, {4,10548}, {5,83}, {6,2353}, {25,251}, {32,206}, {39,1576} et al

X(10547) = isogonal conjugate of X(1235)
X(10547) = X(92)-isoconjugate of X(141)


X(10548) = PERSPECTOR OF 1st ORTHOSYMMEDIAL TRIANGLE AND CROSS-TRIANGLE OF ABC AND 1st ORTHOSYMMEDIAL TRIANGLE

Barycentrics    [2a6 - 3a4(b2 + c2) + (b2 - c2)2(b2 + c2)]/(b2 + c2) : :

X(10548) lies on these lines: {2,32}, {4,10547} et al


X(10549) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 2nd ORTHOSYMMEDIAL TRIANGLE

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

X(10549) lies on these lines: {6,10550} et al


X(10550) = PERSPECTOR OF 2nd ORTHOSYMMEDIAL TRIANGLE AND CROSS-TRIANGLE OF ABC AND 2nd ORTHOSYMMEDIAL TRIANGLE

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

X(10550) lies on these lines: {4,83}, {6,10549} et al


X(10551) = PERSPECTOR OF 2nd ORTHOSYMMEDIAL TRIANGLE AND CROSS-TRIANGLE OF 1st AND 2nd ORTHOSYMMEDIAL TRIANGLES

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

X(10551) lies on these lines: {51,251} et al


X(10552) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 1st PARRY TRIANGLE

Barycentrics    (2a2 - b2 - c2)(3a4 + b4 + c4 - 2a2b2 - 2a2c2 - b2c2) : :

X(10552) lies on these lines: {110,193}, {111,1992} et al

X(10552) = {X(111),X(10554)}-harmonic conjugate of X(10553)


X(10553) = PERSPECTOR OF 1st PARRY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 1st PARRY TRIANGLE

Barycentrics    7a6 - b6 - c6 - 8a4(b2 + c2) + a2(2b4 + 7b2c2 + 2c4) : :

Let A'B'C' be the 1st Parry triangle. Let A" be the cevapoint of B' and C'. Define B", C" cyclically. The lines AA", BB", CC" concur in X(10553).

X(10553) lies on these lines: {2,8587}, {69,110}, {111,1992}, {524,7665} et al

X(10553) = {X(111),X(10554)}-harmonic conjugate of X(10552)


X(10554) = {X(10552),X(10553)}-HARMONIC CONJUGATE OF X(111)

Barycentrics    13a6 - 15a4(b2 + c2) + a2(6b4 + 9b2c2 + 6c4) - 2(b6 + c6) : :

X(10554) lies on these lines: {2,5477}, {110,524}, {111,1992} et al


X(10555) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 2nd PARRY TRIANGLE

Barycentrics    (b^2 - c^2)^2(a^4 - b^4 - c^4 + b^2c^2)[a^8(b^2 + c^2) - a^4(b^6 - 2b^4c^2 - 2b^2c^4 + c^6) - a^6(b^2 + c^2)^2 + a^2(b^2 - c^2)^2(b^4 + c^4) + b^2c^2(b^2 - c^2)^2(b^2 + c^2)]/(2a^2 - b^2 - c^2) : :

X(10555) lies on these lines: {107,111}, {110,9214} et al

X(10555) = {X(110),X(10557)}-harmonic conjugate of X(10556)


X(10556) = PERSPECTOR OF 2nd PARRY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 2nd PARRY TRIANGLE

Barycentrics    [2a^8 - 3a^6(b^2 + c^2) + 6a^4b^2c^2 - a^2b^2c^2(b^2 + c^2) + (b^2 - c^2)^2(b^4 - b^2c^2 + c^4)]/(2a^2 - b^2 - c^2) : :

Let A'B'C' be the 2nd Parry triangle. Let A" be the cevapoint of B' and C'. Define B", C" cyclically. The lines AA", BB", CC" concur in X(10556).

X(10556) lies on these lines: {110,9214}, {111,925} et al

X(10556) = {X(110),X(10557)}-harmonic conjugate of X(10555)


X(10557) = {X(10555),X(10556)}-HARMONIC CONJUGATE OF X(110)

Barycentrics    [2a^8 - 3a^6(b^2 + c^2) - a^4(b^4 - 8b^2c^2 + c^4) - a^2b^2c^2(b^2 + c^2) + 2(b^2 - c^2)^2(b^4 - b^2c^2 + c^4)]/(2a^2 - b^2 - c^2) : :

X(10557) lies on these lines: {110,9214}, {111,230} et al


X(10558) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND 3rd PARRY TRIANGLE

Barycentrics    a4(a4 + 2b4 + 2c4 - 3a2b2 - 3a2c2 + b2c2)/(2a2 - b2 - c2) : :

X(10558) lies on these lines: {2,10559}, {51,111} et al

X(10558) = {X(2),X(10560)}-harmonic conjugate of X(10559)


X(10559) = PERSPECTOR OF 3rd PARRY TRIANGLE AND CROSS-TRIANGLE OF ABC AND 3rd PARRY TRIANGLE

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

Let A'B'C' be the 3rd Parry triangle. Let A" be the cevapoint of B' and C'. Define B", C" cyclically. The lines AA", BB", CC" concur in X(10559).

X(10559) lies on these lines: {2,10558}, {323,5968}

X(10559) = {X(2),X(10560)}-harmonic conjugate of X(10558)


X(10560) = {X(10558),X(10559)}-HARMONIC CONJUGATE OF X(2)

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

X(10560) lies on these lines: {2,10558} et al


X(10561) = PERSPECTOR OF 2nd PARRY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd PARRY TRIANGLES

Barycentrics    a^2(b^2 - c^2)(a^4 - b^4 - c^4 + b^2c^2)[a^6 - a^4(b^2 + c^2) - a^2(b^4 - 3b^2c^2 + c^4) + (b^2 - c^2)^2(b^2 + c^2)]/(b^2 + c^2 - 2a^2) : :

X(10561) lies on these lines: {23,2492}, {111,647} et al

X(10561) = {X(647),X(8430)}-harmonic conjugate of X(10562)


X(10562) = PERSPECTOR OF 3rd PARRY TRIANGLE AND CROSS-TRIANGLE OF 2nd AND 3rd PARRY TRIANGLES

Barycentrics    a2(a4 + 2b4 + 2c4 - 3a2b2 - 3a2c2 + b2c2)/(2a2 - b2 - c2) : :

X(10562) lies on these lines: {111,647}, {523,10415} et al

X(10562) = {X(647),X(8430)}-harmonic conjugate of X(10561)


X(10563) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND TRIANGLE T(-1,3)

Trilinears    (-a + 3b + 3c)(a - 3b + c)(a + b - 3c) : :

Let A'B'C' be triangle T(-1,3). Let A" be the trilinear product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(10563). The lines A'A", B'B", C'C" concur in X(3680). A"B"C" is also the cross-triangle of ABC and triangle T(-1,3).

X(10563) lies on these lines: {1,474}, {8,4373}, {57,6014} et al

X(10563) = trilinear product of vertices of triangle T(-1,3)


X(10564) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND TRINH TRIANGLE

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

X(10564) is the intersection of the Simson line of X(110) (line X(30)X(113)) and the trilinear polar of X(110) (line X(3)X(6)).

X(10564) lies on the Simson quartic (Q101) and these lines: {3,6}, {4,10546}, {30,113}, {74,323}, {110,841} et al

X(10564) = crossdifference of every pair of points on line X(523)X(2433)


X(10565) = PERSPECTOR OF CIRCUMMEDIAL TRIANGLE AND CROSS-TRIANGLE OF ANTICOMPLEMENTARY AND CIRCUMMEDIAL TRIANGLES

Barycentrics    SB SC(2 SA - SW) + SA (SB + SC) SW : :

X(10565) lies on these lines: {2,3}, {69,154} et al

X(10565) = anticomplement of X(8889)


X(10566) = ISOTOMIC CONJUGATE OF X(4568)

Barycentrics    (b - c)/(b2 + c2) : :

X(10566) is the intersection of lines X(23)X(385) and X(798)X(812). Line X(23)X(385) is the line of the degenerate cross-triangle of ABC and circummedial triangle. Line X(798)X(812) is the line of the trilinear products B'*C', C'*A', A'*B', where A'B'C' is the circummedial triangle.

X(10566) lies on these lines: {23,385}, {513,894}, {649,3261}, {798,812} et al


X(10567) = CENTROID OF CROSS-TRIANGLE OF PEDAL TRIANGLES OF PU(1)

Barycentrics    a2(b2 - c2)2[a6 + b6 + c6 + 2a4(b2 + c2) - a2(4b4 + 3b2c2 + 4c4) + 2b4c2 + 2b2c4] : :

The cross-triangle of the pedal triangles of PU(1) is degenerate (3 collinear points), on line X(39)X(512) (or PU(1)).

X(10567) lies on these lines: {6,647}, {39,512}, {523,9300} et al

X(10567) = crossdifference of every pair of points on line X(30)X(148)
X(10567) = PU(1)-harmonic conjugate of X(10568)


X(10568) = PU(1)-HARMONIC CONJUGATE OF X(10567)

Barycentrics    a^2[a^6(b^2 + c^2) + 2a^4(b^4 - 4b^2c^2 + c^4) - a^2(4b^6 - 5b^4c^2 - 5b^2c^4 + 4c^6) + (b^2 - c^2)^2(b^4 + 5b^2c^2 + c^4)] : :

X(10568) lies on these lines: {3,1495}, {5,2682}, 39,512} et al

X(10568) = crossdifference of every pair of points on line X(385)X(9209)


X(10569) = HOMOTHETIC CENTER OF INVERSE-IN-INCIRCLE TRIANGLE AND CROSS-TRIANGLE OF ATIK AND INVERSE-IN-INCIRCLE TRIANGLE

Trilinears    cos A (1 + cos B + cos C) + cos2 B + cos2 C - 2 cos B - 2 cos C - 7 : :

Note that the Atik triangle is bound by the polars of the incenter in the excircles, and the inverse-in-incircle triangle is bound by the polars of the excenters in the incircle.

X(10569) lies on these lines: {1,1407}, {2,9954}, {8,443}, {11,118} et al


X(10570) = PERSPECTOR OF ABC AND CROSS-TRIANGLE OF INNER- AND OUTER-GARCIA TRIANGLES

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

X(10570) lies on these lines: {1,5136}, {3,10}, {4,102}, {8,283}, {1771,1795}, {3422,10572} et al

X(10570) = isogonal conjugate of X(10571)


X(10571) = ISOGONAL CONJUGATE OF X(10570)

Trilinears    a sec A - (c + a) sec B - (a + b) sec C : :

X(10571) lies on these lines: {1,4}, {3,102}, {6,1630}, {8,4551}, {36,47}, {56,58}, {57,959}, {65,386} et al

X(10571) = isogonal conjugate X(10570)
X(10571) = Cundy-Parry Phi transform of X(102) (see http://bernard.gibert.pagesperso-orange.fr/Classes/cl037.html)


X(10572) = PERSPECTOR OF INNER-GARCIA TRIANGLE AND CROSS-TRIANGLE OF INNER- AND OUTER-GARCIA TRIANGLES

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

X(10572) lies on these lines: {1,4}, {2,3612}, {3,1737}, {5,2646}, {8,90}, {10,21}, {11,1385}, {12,6841}, {20,46}, {30,65}, {40,920}, {3422,10570} et al

X(10572) = {X(40),X(5727)}-harmonic conjugate of X(10573)


X(10573) = PERSPECTOR OF OUTER-GARCIA TRIANGLE AND CROSS-TRIANGLE OF INNER- AND OUTER-GARCIA TRIANGLES

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

X(10573) lies on these lines: {1,2}, {4,80}, {5,2099}, {7,5270}, {10,42}, {11,1482}, {12,5788}, {20,484}, {40,920}, {1771,1795} et al.

X(10573) = {X(40),X(5727)}-harmonic conjugate of X(10572)
X(10573) = orthologic center of these triangles: outer-Yff to excenters-midpoints
X(10573) = X(8)-of-outer-Yff-triangle


X(10574) = PERSPECTOR OF HALF-ALTITUDE TRIANGLE AND CROSS-TRIANGLE OF HALF-ALTITUDE AND REFLECTION TRIANGLES

Barycentrics    a^2[b^8 + c^8 - a^6(b^2 + c^2) + 3a^4(b^4 - b^2c^2 + c^4) - a^2(3b^6 - 5b^4c^2 - 5b^2c^4 + 3c^6) - b^6c^2 - b^2c^6] : :

X(10574) lies on these lines: {2,185}, {3,54}, {4,4846}, {5,6241}, {20,389}, {22,9786}, {30,3567}, {51,3146}, {52,376}, {74,7526}, {110,974}, {182,1204}, {373,5068}, {382,5946}, {1216,3524}, {3529,5446}, {3543,10110} {3832,5943} et al


X(10575) = PERSPECTOR OF REFLECTION TRIANGLE AND CROSS-TRIANGLE OF HALF-ALTITUDE AND REFLECTION TRIANGLES

Barycentrics    a^2(b^2 + c^2 - a^2)[a^14(b^2 + c^2) - a^12(5b^4 - 2b^2c^2 + 5c^4) + a^10(b^2 + c^2)(9b^4 - 16b^2c^2 + 9c^4) - a^8(5b^8 + 6b^6c^2 - 30b^4c^4 + 6b^2c^6 + 5c^8) - a^6(b^2 - c^2)^2(b^2 + c^2)(5b^4 - 22b^2c^2 + 5c^4) + a^4(b^2 - c^2)^2(9b^8 - 8b^6c^2 + 6b^4c^4 - 8b^2c^6 + 9c^8) - a^2(b - c)^2(b + c)^2(b^2 + c^2)(5b^8 - 6b^6c^2 + 18b^4c^4 - 6b^2c^6 + 5c^8) + (b^2 - c^2)^6(b^4 + 4b^2c^2 + c^4)] : :

X(10575) lies on these lines: {3,64}, {4,4846}, {20,6193}, {22,7689}, {26,1204}, {30,52}, {51,3627}, {74,7488}, {373,3850}, {376,1216}, {382,389}, {3146,5446}, {3529,5889}, {3543,3567}, {3830,10110}, {3843,5943}, {3853,5946} et al


X(10576) = PERSPECTOR OF INNER-SQUARES TRIANGLE AND CROSS-TRIANGLE OF INNER- AND OUTER-SQUARES TRIANGLES

Trilinears    3 cos A + 4 cos B cos C + sin A : :

X(10576) lies on these lines: {2,372}, {3,3366}, {4,5418}, {5,371}, {6,17}, {381,1151}, {486,3068}, {491,639}, {631,6560}, {632,6454}, {642,7389}, {1131,10303}, {1152,3526}, {3069,5067}, {3091,6453}, {3311,5055}, {3312,5070} et al

X(10576) = {X(6),X(1656)}-harmonic conjugate of X(10577)


X(10577) = PERSPECTOR OF OUTER-SQUARES TRIANGLE AND CROSS-TRIANGLE OF INNER- AND OUTER-SQUARES TRIANGLES

Trilinears    3 cos A + 4 cos B cos C - sin A : :

X(10577) lies on these lines: {2,371}, {3,3367}, {4,5420}, {5,372}, {6,17}, {381,1152}, {485,3069}, {492,640}, {631,6561}, {632,6453}, {641,7388}, {1132,9541}, {1151,3526}, {3068,5067}, {3091,6454}, {3311,5070}, {3312,5055} et al

X(10577) = {X(6),X(1656)}-harmonic conjugate of X(10576)


X(10578) = HOMOTHETIC CENTER OF 3rd PEDAL TRIANGLE OF X(1) AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    3a3 - 5a2(b + c) + a(b - c)2 + (b - c)2(b + c) : :

The trilinear polar of X(10578) meets the line at infinity at X(8713).

X(10578) lies on these lines: {1,2}, {7,55}, {9,1202}, {57,5281}, {354,5218}, {8232,10382} et al

X(10578) = isogonal conjugate of X(10579)
X(10578) = crossdifference of every pair of points on line X(649)X(10581)
X(10578) = {X(1),X(2)}-harmonic conjugate of X(10580)
X(10578) = {X(2),X(3622)}-harmonic conjugate of X(10582)


X(10579) = ISOGONAL CONJUGATE OF X(10578)

Trilinears    a/[3a3 - 5a2(b + c) + a(b - c)2 + (b - c)2(b + c)] : :

X(10579) lies on these lines: {1,3059}, {6,8012}, {56,2293} et al

X(10579) = isogonal conjugate of X(10578)
X(10579) = perspector of ABC and unary cofactor triangle of Hutson-extouch triangle
X(10579) = trilinear pole of line X(649)X(10581)


X(10580) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd CONWAY AND INVERSE-IN-INCIRCLE

Barycentrics    a3 - 3a2(b + c) + 3a(b - c)2 - (b - c)2(b + c) : :

X(10580) lies on these lines: {1,2}, {7,354}, {55,5435}, {8732,10383} et al

X(10580) = anticomplement of X(8580)
X(10580) = {X(1),X(2)}-harmonic conjugate of X(10578)
X(10580) = {X(2),X(145)}-harmonic conjugate of X(200)


X(10581) = PK-TRANSFORM OF X(7)

Trilinears    (sec4(B/2) - sec4(C/2)) sec2(A/2) : :
Trilinears    (1 + cos A)[(1 + cos B)2 - (1 + cos C)2] : :
Trilinears    (1 + cos A)(2 + cos B + cos C)(cos B - cos C) : :
Trilinears    a(b + c - a)[b2(c + a - b)2 - c2(a + b - c)2] : :

X(10581) lies on these lines: {241,514}, {657,663} et al

X(10581) = PK-transform of X(7)
X(10581) = PK-transform of X(55)
X(10581) = intersection of trilinear polars of X(7) and X(55)
X(10581) = perspector of hyperbola {{A,B,C,X(7),X(55),X(354)}}
X(10581) = crossdifference of every pair of points on line X(7)X(55)


X(10582) = HOMOTHETIC CENTER OF 3rd ANTIPEDAL TRIANGLE OF X(1) AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Trilinears    a2 + b2 + c2 - 2ab - 2ac - 6bc : :

X(10582) lies on these lines: {1,2}, {9,354}, {11,10382}, {40,5439}, {55,5437}, {56,5436}, {57,1001} et al

X(10582) = {X(1),X(2)}-harmonic conjugate of X(200)
X(10582) = {X(2),X(3622)}-harmonic conjugate of X(10578)


X(10583) = HOMOTHETIC CENTER OF 5th BROCARD TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    5a4 + 3a2(b2 + c2) + 3(b4 + b2c2 + c4) : :

X(10583) lies on these lines: {2,32}, {3,7875}, {4,7932}, {5,9862}, {6,7892}, {20,9993}, {39,8782}, {99,7829}, {140,9301}, {187,7859}, {194,10336}, {631,9821}, {3085,10047}, {3086,10038}, {3090,9996}, {3091,9873}, {3118,9998}, {3622,9997}, {3767,10000}, {9780,9857} et al

X(10583) = anticomplement of X(7944)


X(10584) = HOMOTHETIC CENTER OF INNER JOHNSON TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    a3 - a2(b + c) - a(3b2 - 8bc + 3c2) + 3(b - c)2(b + c) : :

X(10584) lies on these lines: {1,6931}, {2,11}, {3,10598}, {12,10586}, {56,3847}, {104,6973}, {355,3090}, {1125,6933}, {3091,5253}, {6871,7173} et al

X(10584) = {X(3090),X(3616)}-harmonic conjugate of X(10585)


X(10585) = HOMOTHETIC CENTER OF OUTER JOHNSON TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    a4 - 2a2(2b2 + 3bc + 2c2) - 2abc(b + c) + 3(b2 - c2)2 : :

X(10585) lies on these lines: {1,6933}, {2,12}, {3,10599}, {11,10587}, {55,6871}, {100,5177}, {119,6832}, {355,3090}, {1125,6931}, {1621,3091} et al

X(10585) = {X(3090),X(3616)}-harmonic conjugate of X(10584)


X(10586) = HOMOTHETIC CENTER OF INNER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    a4 - 2a2(b2 - 5bc + c2) + 2abc(b + c) + (b2 - c2)2 : :

X(10586) lies on these lines: {1,2}, {3,10596}, {11,6871}, {12,10584}, {56,6872}, {388,5187}, {497,4190}, {1621,7288} et al

X(10586) = {X(2),X(3622)}-harmonic conjugate of X(10587)


X(10587) = HOMOTHETIC CENTER OF OUTER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Barycentrics    a4 - 2a2(b2 + 5bc + c2) - 2abc(b + c) + (b2 - c2)2 : :

X(10587) lies on these lines: {1,2}, {3,10597}, {11,10585}, {12,5187}, {55,4190}, {388,1621}, {497,6871}, {6842,10596} et al

X(10587) = {X(2),X(3622)}-harmonic conjugate of X(10586)


X(10588) = HOMOTHETIC CENTER OF 1st JOHNSON-YFF TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Trilinears    sin B sin C + cos(B - C) + 1 : :

X(10588) lies on these lines: {1,3090}, {2,12}, {3,5229}, {4,35}, {5,497}, {8,6933}, {9,7098}, {10,3340}, {20,5432}, {36,3525}, {55,3091}, {57,3634}, {65,3740}, {100,6871}, {119,6824}, {140,4293}, {381,4294}, {1697,3817}, {1837,5703}, [3616,5252} et al

X(10588) = {X(1),X(3090)}-harmonic conjugate of X(10589)
X(10588) = {X(3),X(10592)}-harmonic conjugate of X(10590)
X(10588) = {X(5),X(3295)}-harmonic conjugate of X(10591)


X(10589) = HOMOTHETIC CENTER OF 2nd JOHNSON-YFF TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTICOMPLEMENTARY TRIANGLES

Trilinears    sin B sin C + cos(B - C) - 1 : :

X(10589) lies on these lines: {1,3090}, {2,11}, {3,5225}, {4,36}, {5,388}, {8,1392}, {10,7962}, {20,5433}, {35,3525}, {56,3091}, {57,1776}, {65,5704}, {104,6968}, {140,4294}, {381,4293}, {1697,3634}, {1837,3616}, {5253,6871} et al

X(10589) = {X(1),X(3090)}-harmonic conjugate of X(10588)
X(10589) = {X(3),X(10593)}-harmonic conjugate of X(10591)
X(10589) = {X(5),X(999)}-harmonic conjugate of X(10590)


X(10590) = HOMOTHETIC CENTER OF 1st JOHNSON-YFF TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Trilinears    cos B cos C + cos(B - C) + 1 : :

X(10590) lies on these lines: {1,3091}, {2,36}, {3,5229}, {4,12}, {5,388}, {7,1737}, {8,6871}, {10,329}, {11,1056}, {20,498}, {35,3146}, {56,3090}, {104,6879}, {119,6826}, {381,495}, {496,3851}, {499,3600} et al

X(10590) = {X(1),X(3091)}-harmonic conjugate of X(10591)
X(10590) = {X(3),X(10592)}-harmonic conjugate of X(10588)
X(10590) = {X(5),X(999)}-harmonic conjugate of X(10589)


X(10591) = HOMOTHETIC CENTER OF 2nd JOHNSON-YFF TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Trilinears    cos B cos C + cos(B - C) - 1 : :

X(10591 lies on these lines: {1,3091}, {2,35}, {3,5225}, {4,11}, {5,497}, {8,5187}, {10,6919}, {12,1058}, {20,499}, {36,3146}, {55,3090}, {100,6931}, {381,388}, {390,498}, {495,3851}, {546,999}, {962,1737}, {1056,3855}, {3585,3600}, {3616,6871} et al

X(10591 = {X(1),X(3091)}-harmonic conjugate of X(10590)
X(10591 = {X(3),X(10593)}-harmonic conjugate of X(10589)
X(10591 = {X(5),X(3295)}-harmonic conjugate of X(10588)


X(10592) = {X(10588),X(10590)}-HARMONIC CONJUGATE OF X(3)

Trilinears    2 + 3 cos(B - C) : :
Trilinears    1 - 6 cos2(B/2 - C/2) : :
Trilinears    5 - 6 sin2(B/2 - C/2) : :

X(10592) lies on these lines: {1,5}, {3,5229}, {4,5281}, {10,3838}, {30,498}, {35,3627}, {36,632}, {55,546}, {56,3628}, {140,1478}, {388,1656}, {497,3851}, {499,547}, {999,3090}, {1479,3850}, {3091,3295} et al

X(10592) = {X(1),X(5)}-harmonic conjugate of X(10593)


X(10593) = {X(10589),X(10591)}-HARMONIC CONJUGATE OF X(3)

Trilinears    2 - 3 cos(B - C) : :
Trilinears    5 - 6 cos2(B/2 - C/2) : :
Trilinears    1 - 6 sin2(B/2 - C/2) : :

X(10593) lies on these lines: {1,5}, {3,5225}, {4,5265}, {10,3829}, {30,499}, {35,632}, {36,3627}, {55,3628}, {56,546}, {140,1479}, {388,3851}, {497,1656}, {498,547}, {999,3091}, {1478,3850}, {3090,3295} et al

X(10593) = {X(1),X(5)}-harmonic conjugate of X(10592)


X(10594) = HOMOTHETIC CENTER OF ARA TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

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

X(10594) lies on these lines: {2,3}, {6,1173}, {51,6759}, {54,154}, {74,1192} et al

X(10594) = {X(4),X(24)}-harmonic conjugate of X(378)


X(10595) = HOMOTHETIC CENTER OF CAELUM TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Barycentrics    3a^4 - 4a^3b - 4a^2b^2 + 4ab^3 + b^4 - 4a^3c + 8a^2bc - 4ab^2c - 4a^2c^2 - 4abc^2 - 2b^2c^2 + 4ac^3 + c^4 : :

X(10595) lies on these lines: {1,4}, {2,1482}, {3,3622}, {5,145}, {8,3090}, {10,5067}, {20,10246}, {40,551}, {55,6942}, {56,6950}, {104,3296}, {119,5115}, {140,8148}, {149,6917}, {952,3091} et al

X(10595) = {X(10596),X(10597)}-harmonic conjugate of X(4)


X(10596) = HOMOTHETIC CENTER OF INNER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Barycentrics    a^7 - a^6(b + c) - a^5(b^2 + 4bc + c^2) + a^4(b + c)(b^2 + 6bc + c^2) - a^3(b - c)^2(b^2 - 6bc + c^2) + a^2(b - c)^2(b + c)(b^2 - 6bc + c^2) + a(b^2 - c^2)^2(b^2 - 4bc + c^2) - (b - c)^4(b + c)^3 : :

X(10596) lies on these lines: {1,4}, {3,10586}, {5,10528}, {8,6898}, {11,6879}, {12,10598}, {55,6880}, {119,3545}, {145,6893}, {149,6826}, {3090,5082}, {3296,5553}, {5084,5554}, {6842,10587} et al

X(10596) = {X(4),X(10595)}-harmonic conjugate of X(10597)


X(10597) = HOMOTHETIC CENTER OF OUTER YFF TANGENTS TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

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

X(10597) lies on these lines: {1,4}, {3,10587}, {5,10529}, {8,6854}, {11,10599}, {56,6977}, {145,6826}, {3086,6879}, {3090,3421} et al

X(10597) = {X(4),X(10595)}-harmonic conjugate of X(10596)


X(10598) = HOMOTHETIC CENTER OF INNER JOHNSON TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Barycentrics    a^7 - a^6(b + c) + a^5(b^2 + c^2) - a^4(b - c)^2(b + c) - a^3(b - c)^2(5b^2 + 2bc + 5c^2) + a^2(b - c)^2(b + c)(5b^2 + 2bc + 5c^2) + a(b^2 - c^2)^2(3b^2 - 8bc + 3c^2) - 3(b - c)^4(b + c)^3 : :

X(10598) lies on these lines: {1,6968}, {2,10525}, {3,10584}, {4,11}, {12,10596}, {5,3434}, {8,6973}, {100,6981}, {145,355}, {381,10532}, {1376,3090} et al

X(10598) = {X(3091),X(5603)}-harmonic conjugate of X(10599)


X(10599) = HOMOTHETIC CENTER OF OUTER JOHNSON TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND ANTI-EULER TRIANGLES

Barycentrics    a^7 - a^6(b + c) + a^5(b^2 + 4bc + c^2) - a^4(b + c)(b^2 + 6bc + c^2) - 5a^3(b^2 - c^2)^2 + 5a^2(b - c)^2(b + c)^3 + a(b^2 - c^2)^2(3b^2 - 4bc + 3c^2) - 3(b - c)^4(b + c)^3 : :

X(10599) lies on these lines: {2,10526}, {3,10585}, {4,12}, {5,956}, {8,6867}, {11,10597}, {56,6869}, {104,6956}, {119,6835}, {145,355}, {381,10531}, {958,3090} et al

X(10599) = {X(3091),X(5603)}-harmonic conjugate of X(10598)


X(10600) = PERSPECTOR OF 2nd EULER TRIANGLE AND CROSS-TRIANGLE OF MEDIAL AND 2nd EULER TRIANGLES

Barycentrics    SA^2 (S^2 + SB SC)[SA SB SC a^2 + S^2 (SB^2 + SC^2)] : :

The cross-triangle of medial and 2nd Euler triangles is degenerate (3 collinear points), on line X(523)X(2072) (the trilinear polar of X(5392)).

X(10600) lies on these lines: {2,8884}, {3,161}, {5,53}, {68,577}, {122,128}, {626,6389} et al

X(10600) = complement of X(8884)


X(10601) = HOMOTHETIC CENTER OF TANGENTIAL TRIANGLE AND CROSS-TRIANGLE OF ORTHIC AND SUBMEDIAL TRIANGLES

Trilinears    sin A + csc A : :
Trilinears    a(b2c2 + S2) : :
Barycentrics    sin2 A + 1 : :
Barycentrics    1 + (sin A)(cos B sin C + cos C sin B) : :
Barycentrics    4*R^2 + SB + SC : :
Barycentrics    a^2 + 4*R^2 : :

X(10601) lies on these lines: {2,6}, {3,51}, {5,1181}, {22,5085}, {23,10541}, {25,182}, {39,493}, {83,458}, {154,1995}, {155,1656}, {184,373}, {219,3305}, {222,3306}, {371,1584}, {372,1583}, {485,1592}, {486,1591}, {576,3819}, {611,614}, {612,613}, {967,5042}, {1124,3084}, {1151,1600}, {1152,1599}, {1196,5034}, {1199,5067}, {1335,3083}, {1350,3060}, {1351,3917}, {1370,5480}, {1498,3091}, {1578,1590}, {1579,1589}, {1585,3093}, {1586,3092}, {1853,5133}, {2003,5437}, {3157,5439}, {3167,5544}, {3311,5409}, {3312,5408}

X(10601) = homothetic center of orthic triangle and cross-triangle of tangential and submedial triangles
X(10601) = homothetic center of submedial triangle and cross-triangle of orthic and tangential triangles
X(10601) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,6,394), (2,5422,6), (6,1611,3051), (25,182,3796), (184,373,5020), (493,494,39), (371,1584,5407), (372,1583,5406), (1995,5012,154), (3066,3796,25), (5020,5050,184)


X(10602) = HOMOTHETIC CENTER OF 2nd EHRMANN TRIANGLE AND CROSS-TRIANGLE OF ORTHIC AND TANGENTIAL TRIANGLES

Barycentrics    a2(b2 + c2 - a2)[a4 - 3(b2 - c2)2 - 2a2(b2 + c2)] : :

X(10602) lies on these lines: {2,8263}, {3,895}, {6,25}, {69,1368} et al

X(10602) = isogonal conjugate of X(10603)
X(10602) = isotomic conjugate of X(10604)
X(10602) = anticomplement of X(8263)


X(10603) = VERTEX CONJUGATE OF X(25) AND X(69)

Barycentrics    1/{(b2 + c2 - a2)[a4 - 3(b2 - c2)2 - 2a2(b2 + c2)]} : :

X(10603) lies on these lines: {25,5203}, {69,468}, {305,6353} et al

X(10603) = isogonal conjugate of X(10602)
X(10603) = vertex conjugate of X(25) and X(69)


X(10604) = ISOTOMIC CONJUGATE OF X(10602)

Barycentrics    1/{a2(b2 + c2 - a2)[a4 - 3(b2 - c2)2 - 2a2(b2 + c2)]} : :

X(10604) lies on these lines: {305,6353} et al

X(10604) = isotomic conjugate of X(10602)
X(10604) = Brianchon point (perspector) of inconic centered at X(8263)


X(10605) = HOMOTHETIC CENTER OF TRINH TRIANGLE AND CROSS-TRIANGLE OF ORTHIC AND TANGENTIAL TRIANGLES

Barycentrics    a2(b2 + c2 - a2)[a6 + a4(b2 + c2) - 5a2(b2 - c2)2 + 3(b2 - c2)2(b2 + c2)] : :

Let A'B'C' be the cevian triangle of X(3). Let AB = BC∩C'A' and define BC, CA cyclically. Let AC = BC∩A'B' and define BA, CB cyclically. X(10605) is the radical center of the circumcircles of A'BCCB, B'CAAC, C'ABBA.

X(10605) lies on these lines: {3,49}, {4,64}, {6,74}, {20,6146}, {25,6000}, {30,1899} et al

X(10605) = reflection of X(394) in X(3)


X(10606) = REFLECTION OF X(154) IN X(3)

Barycentrics    a2[3a8 - 4a6(b2 + c2) - a4(6b4 - 20b2c2 + 6c4) + 12a2(b2 - c2)2(b2 + c2) - (b2 - c2)2(5b4 + 14b2c2 + 5c4)] : :

Continuing from X(10605), X(10606) is the radical center of the circumcircles of A'ABAC, B'BCBA, C'CACB.

X(10606) lies on these lines: {3,64}, {6,74}, {165,5692} et al

X(10606) = reflection of X(i) in X(j) for these (i,j): (154,3), (1498,154)
X(10606) = X(1498)-of-orthocentroidal-triangle


X(10607) = X(3)-CEVA CONJUGATE OF X(394)

Barycentrics    a2(a2 - b2 - c2)2(3a2 - b2 - c2) : :

X(10607) lies on these lines: {3,6391}, {6,2987}, {394,577} et al

X(10607) = reflection of X(10608) in X(3)


X(10608) = REFLECTION OF X(10607) in X(3)

Barycentrics    a2(a2 - b2 - c2)[3a8 - 6a6(b2 + c2) + a4(8b4 - 4b2c2 + 8c4) - 10a2(b2 - c2)2(b2 + c2) + (b2 - c2)2(5b^4 - 6b2c2 + 5c4)] : :

Continuing from X(10605), X(10608) is the radical center of the circumcircles of A'BACA, B'CBAB, C'ACBC.

X(10608) lies on these lines: {3,6391} et al

X(10608) = reflection of X(10607) in X(3)


X(10609) =  X(1)X(528)∩X(3)X(8)

Barycentrics    4 a^4-2 a^3 b-3 a^2 b^2+2 a b^3-b^4-2 a^3 c+2 a^2 b c-3 a^2 c^2+2 b^2 c^2+2 a c^3-c^4
X(10609) = 3 r X[3] - (r + R) X[8] = X[8] - 3 X[100] = 3 X[11] - 4 X[1125] = 3 X[214] - 2 X[1125]

X(10609) lies on these lines:
{1, 528}, {3, 8}, {10, 6174}, {11, 214}, {20, 5730}, {30, 4511}, {46, 2136}, {65, 1317}, {72, 2801}, {79, 6596}, {80, 1698}, {119, 6831}, {140, 5086}, {145, 10031}, {149, 377}, {153, 6836}, {190, 6790}, {224, 1537}, {320, 5088}, {355, 4855}, {392, 4304}, {405, 4305}, {474, 3486}, {484, 5855}, {515, 5440}, {519, 1155}, {550, 3869}, {662, 3109}, {900, 6161}, {997, 4679}, {999, 1004}, {1159, 3241}, {1320, 3296}, {1490, 2829}, {1768, 9841}, {1862, 4185}, {2182, 2325}, {2245, 4969}, {2771, 3650}, {3036, 9897}, {3419, 3576}, {3434, 10246}, {3555, 4311}, {3655, 3872}, {3916, 6737}, {4256, 5724}, {4259, 9024}, {4293, 5856}, {4302, 5289}, {4316, 4867}, {4707, 6366}, {5660, 5691}, {6955, 7967}

X(10609) = midpoint of X(i) and X(j) for these {i,j}: {100, 6224}, {149, 9963}, {1317, 6154}, {4316, 4867}, {5541, 7972}
X(10609) = reflection of X(i) in X(j) for these (i,j): (11, 214}, (80, 3035), (100, 9945), (149, 1387), (1145, 100), (1537, 6265), (9897, 3036)
X(10609) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3612,5794,7483), (6224,9945,1145)


X(10610) =  MIDPOINT OF X(3) AND X(54)

Trilinears    (3/2 + cos2A) cos(B - C) - cos 3A : :

Let P be a point if the plane of a triangle ABC, and let A'B'C' be the pedal triangle of P. Let
Oa = midpoint of AP, and define Ob and Oc cyclically
Ooa = circumcenter of PObOc, and define Oob and Ooc cyclically
Locus1 = locus of P for which the triangles ABC and OoaOobOoc are perspective
Locus2 = Neuberg cubic = locus of P for which OaObOc and OoaOobOoc are perspective
Locus3 = Napoleon-Feurbach cubic = locus of P for which A'B'C' and OoaOobOoc are perspective.
Locus1 is the cubic pK(X6,X382), 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,z,y) = (S2 - 5SBSC)(c2y2 - b2z2)x. Points on Locus1 include X(1), X(3), X(4), and X(382).

If P = X(3), the perspector of OoaOobOoc and OaObOc is X(10610). See Antreas Hatzipolakis and César Lozada, 24682.

X(10610) lies on these lines:
{2,6288}, {3,54}, {4,7712}, {5,5944}, {30,3574}, {35,20014}, {55,10082}, {56,10066}, {125,128}, {143,567}, {156,7503}, {182,9977}, {184,5876}, {186,6152}, {511,10115}, {539,549}, {569,973}, {578,7502}, {631,2888}, {1199,3581}, {1495,3850}, {1498,7526}, {1539,5893}, {2070,10095}, {2917,6644}, {3357,10274}, {3431,3519}, {3576,9905}, {3630,5092}, {5010,6286}, {5462,7575}, {5544,6642}, {5609,5907}, {5888,7666}, {5892,6153}, {7280,7356}, {7583,8995}, {7979,10246}, {10024,10113}

X(10610) = midpoint of X(i) and X(j) for these {i,j}: {3,54}, {195,7691}
X(10610) = reflection of X(i) in X(j) for these (i,j): (5,6689), (1209,140), (1493,54), (3574,8254)
X(10610) = complement of X(6288)
X(10610) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,195,7691), (54,7691,195), (567,7488,143), (569,1658,5946), (578,7502,10263)


X(10611) =  MIDPOINT OF X(13) AND X(17)

Trilinears    -sqrt(3)*((-3*cos(2*A)+5)*cos( B-C)+cos(A)*cos(2*(B-C))+2* cos(A)-cos(3*A))-9*sin(A)+sin( 3*A)-4*sin(A)*cos(2*(B-C))-5* sin(2*A)*cos(B-C) : :

Referring to the constructions at X(10610), if P = X(13), the perspector of OoaOobOoc and OaObOc is X(10611). See Antreas Hatzipolakis and César Lozada, 24682.

X(10611) lies on these lines:
{3,13}, {115,6783}, {396,5478}, {397,629}, {532,5459}, {618,6673}, {3054,6115}, {5472,5617}, {6770,7694}

X(10611) = midpoint of X(13) and X(17)
X(10611) = reflection of X(i) in X(j) for these (i,j): (618,6673), (629,6669)


X(10612) =  MIDPOINT OF X(14) AND X(18)

Trilinears    sqrt(3)*((-3*cos(2*A)+5)*cos( B-C)+cos(A)*cos(2*(B-C))+2* cos(A)-cos(3*A))-9*sin(A)+sin( 3*A)-4*sin(A)*cos(2*(B-C))-5* sin(2*A)*cos(B-C) : :

Referring to the constructions at X(10610), if P = X(14), the perspector of OoaOobOoc and OaObOc is X(10612). See Antreas Hatzipolakis and César Lozada, 24682.

X(10612) lies on these lines:
{3,14}, {115,6782}, {395,5479}, {398,630}, {533,5460}, {619,6674}, {3054,6114}, {5471,5613}, {6773,7694}

X(10612) = midpoint of X(14) and X(18)
X(10612) = reflection of X(i) in X(j) for these (i,j): (619,6674), (630,6670)


X(10613) =  MIDPOINT OF X(15) AND X(61)

Trilinears    (a^4-(b^2+c^2)*a^2-2*b^2*c^2- sqrt(3)*S*(b^2+c^2))*a : :

Referring to the constructions at X(10610), if P = X(15), the perspector of OoaOobOoc and OaObOc is X(10613). See Antreas Hatzipolakis and César Lozada, 24682.

X(10613) lies on these lines:
{3,6}, {23,2981}, {114,6109}, {396,7684}, {398,623}, {621,7834}, {628,7849}, {633,3788}, {635,6671}

X(10613) = midpoint of X(15) and X(61)
X(10613) = reflection of X(i) in X(j) for these (i,j): (623,6694), (635,6671)


X(10614) =  MIDPOINT OF X(16) AND X(62)

Trilinears    (a^4-(b^2+c^2)*a^2-2*b^2*c^2+ sqrt(3)*S*(b^2+c^2))*a : :

Referring to the constructions at X(10610), if P = X(16), the perspector of OoaOobOoc and OaObOc is X(10614). See Antreas Hatzipolakis and César Lozada, 24682.

X(10614) lies on these lines:
{3,6}, {23,6151}, {114,6108}, {395,7685}, {397,624}, {622,7834}, {627,7849}, {634,3788}, {636,6672}

X(10614) = midpoint of X(16) and X(62)
X(10614) = reflection of X(i) in X(j) for these (i,j): (624,6695), (636,6672)


X(10615) =  MIDPOINT OF X(5) AND X(6150)

Trilinears    (2*cos(2*A)+cos(2*(B-C)))*(2* cos(2*A)*cos(B-C)+cos(3*A)) : :

Referring to the constructions at X(10610), if P = X(5), the perspector of OoaOobOoc and A'B'C' is X(10615). See Antreas Hatzipolakis and César Lozada, 24682.

X(10615) lies on these lines:
{2,1157}, {5,6150}, {128,539}, {136,186}, {140,389}, {252,1209}

X(10615) = midpoint of X(5) and X(6150)


X(10616) =  POINT BECRUX 8

Barycentrics    (3*(SA-SW) + 2*sqrt(3)*S)*( 2*SA-SW – sqrt(3)*S) : :

Referring to the constructions at X(10610), if P = X(17), the perspector of OoaOobOoc and A'B'C' is X(10616). See Antreas Hatzipolakis and César Lozada, 24682.

X(10616) lies on these lines:
{16,396}, {141,1078}, {187,624}, {230,5981}, {395,533}, {511,8259}, {623,7749}, {5321,6109}


X(10617) =  POINT BECRUX 9

Barycentrics    (3*(SA-SW) - 2*sqrt(3)*S)*( 2*SA-SW + sqrt(3)*S) : :

Referring to the constructions at X(10610), if P = X(18), the perspector of OoaOobOoc and A'B'C' is X(10617). See Antreas Hatzipolakis and César Lozada, 24682.

X(10617) lies on these lines:
{15,395}, {141,1078}, {187,623}, {230,5980}, {396,532}, {511,8260}, {624,7749}, {5318,6108}


X(10618) =  POINT BECRUX 10

Barycentrics    (a (7a^5(b+c)+a^4(b^2+4b c+c^2)-a^3(14b^3+9b^2c+9b c^2+14c^3)-2a^2(b^4+5b^3c+7b^2c^2+5b c^3+c^4)+a(b-c)^2(7b^3+16b^2c+16b c^2+7c^3)+(b^2-c^2)^2(b^2+6b c+c^2)) : :
X(10618) = (2r^2+5rR+4s^2)*X(1) + r(2r+3R)*X(3)

Let I be the incenter of a triangle ABC, and let A'B'C' be the cevian triangle of I. Let
Na = nine-point center of IB'C', and define Nb and Nc cyclically
N1 = nine-point center of INbNc, and define N2 and N3 cyclically.
X(10618) = nine-point center of N1N2N3. See Tran Quang Hung and Angel Montesdeoca, July 19, 2016: Hyacinthos 24692.

X(10618) lies on these lines: {1,3} et al


X(10619) =  X(4)X(54)∩X(125)X(128)

Barycentrics    (a^2-b^2-c^2) (4 a^8-6 a^6 b^2+a^4 b^4+b^8-6 a^6 c^2-2 a^4 b^2 c^2-4 b^6 c^2+a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : :
X(10619) = X[4]-3 X[54] = 4 X[140]-3 X[1209] = 3 X[195]+X[1657] = 3 X[3]-X[3519] = 3 X[2888]-7 X[3523] = 2 X[4]-3 X[3574] = 3 X[5890]-X[6242] = 5 X[1656]-3 X[6288] = 5 X[1656]-6 X[6689] = 5 X[3522]-3 X[7691] = 2 X[3850]-3 X[8254]

Let H be the orthocenter of a triangle ABC, and
A' = reflection of H in A, and define B' and C' cyclically
Ab = orthogonal projection of A' on AB, and define Bc and Ca cyclically
Ac = orthogonal projection of A' on AB, and define Ba and Cb cyclically
La = Euler line of A'AbAc, and define Lb and Lc cyclically
The lines La, Lb, Lc concur in X(10619). See Antreas Hatzipolakis and Peter Moses, Hyacinthos 24692.

X(10619) lies on these lines:
{3,539}, {4,54}, {30,1493}, {49, 1568}, {125,128}, {185,550}, { 195,1181}, {389,6152}, {1141, 1487}, {1204,3098}, {1656,6288}, {1899,2888}, {2777,2914}, {2917, 3515}, {3153,9706}, {3517,9920}, {3850,8254}, {5094,6145}, {5339, 8742}, {5340,8741}, {5448,9704}, {5890,6242}, {6467,8550}, {7488, 10112}, {8960,8995}

X(10619) = reflection of X(i) in X(j) for these (i,j): (3574,54), (6152,389), (6252,389), (6288,6689)


X(10620) =  REFLECTION OF X(3) IN X(74)

Barycentrics    a^2 (a^8-6 a^4 b^4+8 a^2 b^6-3 b^8+9 a^4 b^2 c^2-7 a^2 b^4 c^2-2 b^6 c^2-6 a^4 c^4-7 a^2 b^2 c^4+10 b^4 c^4+8 a^2 c^6-2 b^2 c^6-3 c^8) : :
X(10620) = 3 X[3] - 2 X[110] = 3 X[74] - X[110] = 4 X[125] - 3 X[381] = 4 X[110] - 3 X[399] = 4 X[74] - X[399] = 5 X[399] - 8 X[1511] = 5 X[110] - 6 X[1511] = 5 X[3] - 4 X[1511] = 5 X[74] - 2 X[1511] = 4 X[113] - 5 X[1656] = 4 X[1539] - 5 X[3843] = 7 X[399] - 8 X[5609] = 7 X[110] - 6 X[5609] = 7 X[1511] - 5 X[5609] = 7 X[3] - 4 X[5609] = 7 X[74] - 2 X[5609] = 2 X[182] - 3 X[5621] = 3 X[5054] - 2 X[5655] = 9 X[5054] - 8 X[5972] = 3 X[5655] - 4 X[5972] = 7 X[3526] - 8 X[6699] = 15 X[1656] - 16 X[6723] = 3 X[113] - 4 X[6723] = 3 X[381] - 2 X[7728] = 3 X[5050] - 2 X[9970] = 3 X[3830] - 4 X[10113] = 3 X[9140] - 2 X[10113] = 2 X[7978] - 3 X[10247] = 5 X[631] - 4 X[10272]

X(10620) lies on K834, the Stammler circle, and these lines:
{3,74}, {4,10264}, {5,146}, {30,3448}, {40,2771}, {56,7727}, {64,265}, {113,1656}, {125,381}, {182,5621}, {185,567}, {195,2935}, {378,7722}, {517,9904}, {542,1350}, {550,2889}, {631,10272}, {999,3024}, {1112,1597}, {1351,2781}, {1539,3843}, {1593,1986}, {2070,6000}, {2930,3098}, {2931,2937}, {2948,3579}, {3028,3295}, {3031,9566}, {3043,9704}, {3047,9703}, {3526,6699}, {3830,9140}, {5050,9970}, {5054,5646}, {5584,7724}, {5898,7691}, {6102,7731}, {7517,9914}, {7687,9786}, {7978,10247}, {7984,8148}, {8703,9143}, {9301,9984}, {9642,10060}

X(10620) = reflection of X(i) in X(j) for these (i,j): (3, 74), (4, 10264), (146, 5), (382, 265), (399, 3), (2930, 3098), (2931, 7689), (2935, 3357), (2948, 3579), (3830, 9140), (5898, 7691), (7728, 125), (7731, 6102), (8148, 7984), (9143, 8703)


X(10621) =  POINT BECRUX 11

Barycentrics    (SA^2+SB*SC)/( a^2*(S^2*(17*R^2-4*SW)+2*S^2* SA+(R^2-2*SW)*SA^2)) : :

Let O be the circumcenter of a triangle ABC, and let
Ab = reflection of A' in OB, and define Bc and Ca cyclically
Ac = reflection of A' in OC, and define Ba and Cb cyclically
Na = nine-point center of AAbAc, and define Nb and Nc cyclically
Oa = circumcenter of AAbAc, and define Ob and Oc cyclically
E = Euler line of ABC
P = an arbitrary point on the E, regarded as a function with domain E
Ea = Euler line of AAbAc, and define Eb and Ec cyclically
Pa = P-of-Ea, and define Pb and Pc cyclically. (See "functional image" in the preamble to X(6724).)

As P ranges through E, the ABC-to-PaPbPc orthologic center OC(P), ranges through a circumconic, specifically, the isogonal conjugate of the line X(184)X(9292). Examples include X(10621) = OC(X(4)) and X(10622) = OC(X(5)), as well as X(264) = OC(X(2)) and X(1975) = OC(X(3)). See Antreas Hatzipolakis and César Lozada, Hyacinthos 24699.

X(10621) lies on these lines:


X(10622) =  POINT BECRUX 12

Barycentrics    (SA^2+SB*SC)/( a^2*(S^2*(33*R^2-8*SW)+4*S^2* SA+(-4*SW+R^2)*SA^2)) : :

See X(10621) and Antreas Hatzipolakis and César Lozada, Hyacinthos 24699.

X(10622) lies on these lines:


X(10623) =  (pending)

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

Let A'B'C' = pedal triangle of I = X(1) in the plane of a triangle ABC. Let
A"B"C" = orthic triangle of A'B'C'
A* = reflection of A'' in IA', and define B* and C* cyclically.

The triangles ABC and A*B*C* are orthologic; X(10623) = ABC-to-A*B*C* orthologic center, and X(10624) = A*B*C-to-ABC orthologic center. See Antreas Hatzipolakis and César Lozada, Hyacinthos 24739.

X(10623) lies on these lines: {3,3681}, {35,603}, {48,3730}, {56,378}, {1437,4184}, {1442,3295}

X(10623) = trilinear pole of X(6586)X(9404)


X(10624) =  (pending)

Trilinears    (2*a^4+(b+c)*a^3-(b^2+6*b*c+c^ 2)*a^2-(b^2-c^2)*(b-c)*a-(b^2- c^2)^2)/a : :
X(10624) = (6*R+r)*X(1) - (4*R+r)*X(7) = 3*X(1) - X(1770) = 3*X(1) - 2*X(4298)

Let A'B'C' = pedal triangle of I = X(1) in the plane of a triangle ABC. Let
A"B"C" = orthic triangle of A'B'C'
A* = reflection of A'' in IA', and define B* and C* cyclically.

The triangles ABC and A*B*C* are orthologic; X(10624) = A*B*C-to-ABC orthologic center. See X(10623) and Antreas Hatzipolakis and César Lozada, Hyacinthos 24739.

X(10624) lies on these lines:
{1,7}, {2,9614}, {4,1697}, {8,3586}, {9,5082}, {10,1479}, {11,6684}, {30,9957}, {35,404}, {40,497}, {46,5493}, {55,946}, {57,1058}, {65,3058}, {72,5853}, {80,3626}, {100,6700}, {140,7743}, {144,6764}, {149,6734}, {165,3086}, {226,3295}, {329,6765}, {355,9668}, {376,1420}, {389,517}, {452,9623}, {496,3579}, {498,3817}, {499,10164}, {515,3057}, {519,3869}, {527,3555}, {528,960}, {548,5126}, {551,3612}, {553,5045}, {908,3871}, {938,2093}, {944,7962}, {1056,9579}, {1361,2816}, {1367,3021}, {1490,10388}, {1496,1777}, {1497,1754}, {1698,10591}, {1699,3085}, {1706,5084}, {1737,4857}, {1836,3303}, {1837,9670}, {2078,3651}, {2098,5882}, {2136,3421}, {2792,10544}, {3146,9613}, {3159,4463}, {3333,3474}, {3340,3488}, {3419,5837}, {3436,3895}, {3452,5687}, {3485,10385}, {3486,7982}, {3487,10389}, {3583,10039}, {3601,5603}, {3634,7741}, {3635,5441}, {3649,3748}, {3710,5014}, {3717,5100}, {3813,4640}, {3878,6737}, {3914,3915}, {3947,10056}, {4652,10529}, {4848,5722}, {5046,6735}, {5173,10122}, {5217,10165}, {5218,8227}, {5223,9804}, {5225,5587}, {5267,10058}, {5316,9709}, {5657,9581}, {5691,9819}, {5692,6743}, {5759,10384}, {5902,6744}, {5919,7354}, {6745,8715}, {7173,10172}, {7264,10521}, {7672,10399}, {7682,10531}, {7957,9848}, {9612,9812}

X(10624) = midpoint of X(i) and X(j) for these {i,j}: {3057,6284}, {5697,10572}
X(10624) = reflection of X(i) in X(j) for these (i,j): (1770,4298), (4292,1), (6737,3878), (10106,9957)
X(10624) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,20,4311), (1,1770,4298), (1,4294,4304), (1,4299,4315), (1,4302,4297), (1,4309,4314), (1,4333,4317), (1,9589,4295), (20,9785,1), (40,497,1210), (390,962,1), (496,3579,3911), (1479,5119,10), (1697,9580,4), (1770,4298,4292), (3434,5250,10), (4297,4342,1), (4301,4314,1)


X(10625) =  REFLECTION OF X(52) IN X(3)

Trilinears    cos(2*A)*cos(B-C)+2*cos(A) : :
X(10625) = 2*X(3) - X(52) = 3*X(3) - 2*X(389) = 5*X(3) - 3*X(568)

Let O = X(3) be the circumcenter of a triangle ABC, and let
A'B'C' = pedal triangle of H = X(4); i.e. A'B'C = orthic triangle
A"B"C" = antipedal triangle of O
A* = reflection of A'' in OA, and define B* and C* cyclically.

The triangles A''B''C'' and A*B*C* are orthologic; X(10625) = A*B*C*-to-A''B''C'' orthologic center, and X(10626) = A''B''C''-to-A*B*C orthologic center. See Antreas Hatzipolakis and César Lozada, Hyacinthos 24739.

X(10625) lies on these lines:
{2,5446}, {3,6}, {4,1216}, {5,3917}, {20,6193}, {22,1147}, {26,1092}, {30,5562}, {51,140}, {54,6636}, {68,1370}, {141,7403}, {143,549}, {156,3292}, {185,550}, {323,1614}, {373,632}, {376,5889}, {382,5907}, {394,7387}, {427,1209}, {517,1770}, {548,6102}, {631,3060}, {858,5449}, {1595,1843}, {1656,3819}, {1657,5925}, {1993,10323}, {2888,5189}, {2937,10282}, {3072,7186}, {3073,3792}, {3088,6403}, {3090,7998}, {3091,7999}, {3520,6242}, {3522,5890}, {3523,3567}, {3525,5640}, {3526,5943}, {3528,10574}, {3530,5946}, {3628,5650}, {5480,7405}, {5944,7555}, {6030,9706}, {6689,7495}, {6923,10441}, {7404,10519}, {7492,9545}, {7517,9306}

X(10625) =reflection of X(i) in X(j) for these (i,j): (4,1216), (52,3), (185,550), (382,5907), (5446,5447), (5562,6101), (5891,2979), (6102,548), (6243,389), (9967,3313), (10263,140), (10575,20)
X(10625) = anticomplement of X(5446)
X(10625) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,52,9730), (3,568,9729), (3,6243,389), (4,1216,5891), (4,2979,1216), (140,10263,51), (389,6243,52), (394,7387,10539), (578,3098,3), (631,3060,5462), (632,10095,373), (3091,7999,10170), (3523,3567,5892), (3819,10110,1656), (5446,5447,2)


X(10626) =  POINT BECRUX 13

Trilinears    3*(2*cos(2*A)+1)*cos(B-C)+3*( cos(A)+cos(3*A))*cos(2*(B-C))+ (-cos(2*A)+cos(4*A)+1)*cos(3*( B-C))-1/2*cos(7*A)+1/2*cos(3* A)+2*cos(5*A) : :

See X(10625) and Antreas Hatzipolakis and César Lozada, Hyacinthos 24739.

X(10626) lies on these lines: (pending)


X(10627) =  REFLECTION OF X(143) IN X(140)

Barycentrics    a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-6 a^4 b^2 c^2+4 a^2 b^4 c^2+b^6 c^2-3 a^4 c^4+4 a^2 b^2 c^4+3 a^2 c^6+b^2 c^6-c^8) : :
X(10627) = (J^2 - 8) X[3] - (J^2 - 4) X[54], where J = |OH|/R
X(10627) = X[52] - 3 X[549] = 3 X[51] - 5 X[632] = X[3] + 3 X[2979] = 3 X[568] - 7 X[3523] = 3 X[3060] - 7 X[3526] = 2 X[3628] - 3 X[3819] = X[5] - 3 X[3917]

In the plane of a triangle ABC, let O = X(3), the circumcenter of ABC, and let
Ab - orthogonal projection of A on OB, and define Bc and Ca cyclically
Ac - orthogonal projection of A on OC, and define Ba and Cb cyclically
Na = nine-point center of OAbAc, and define Nb and Nc cyclically
Oa = circumcircle of NaBC, and define Ob and Oc cyclically.

The circles Oa, Ob, Oc concur in X(10627), which lies on the circumcircle of NaNbNc. See Antreas Hatzipolakis and Peter Moses, Hyacinthos 24721.

X(10627) lies on these lines:
{2,10095}, {3,54}, {5,3917}, { 20,5876}, {26,1350}, {30,1216}, { 49,6636}, {51,632}, {52,549}, { 140,143}, {156,394}, {185,8703}, {381,7999}, {389,3530}, {399, 8718}, {547,10110}, {550,5562}, { 568,3523}, {631,5946}, {1092, 7502}, {1112,10018}, {1147,3098} ,{1511,7488}, {1656,7998}, { 2781,7555}, {2889,3448}, {3060, 3526}, {3567,5054}, {3627,5891}, {3628,3819}, {3850,10170}, { 5070,9781}, {5944,7512}, {6030, 9705}

X(10627) = midpoint of X(i) and X(j) for these {i,j}: {3, 6101}, {20, 5876}, {550, 5562}
X(10627) = reflection of X(i) in X(j) for these (i,j): (140, 5447), (143, 140), (389, 3530), (5446, 3628), (10263, 10095)
X(10627) = complement of X(10263)
X(10627) = anticomplement of X(10095)
X(10627) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10263,10095), (3,2979,6101), (631,6243,5946), (1147,3098, 7525), (3819,5446,3628)


X(10628) =  INFINITY POINT OF X(54)X(74)

Barycentrics    a^2 (a^12 (b^2+c^2) - 2 a^10 (2 b^4+b^2 c^2+2 c^4) + a^8 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6) + a^6 (-5 b^6 c^2+4 b^4 c^4-5 b^2 c^6) - a^4 (b^2-c^2)^2 (5 b^6+2 b^4 c^2+2 b^2 c^4+5 c^6) + a^2 (b^2-c^2)^2 (4 b^8+3 b^6 c^2+3 b^2 c^6+4 c^8) - (b^2-c^2)^4 (b^2+c^2)^3) : :

Let H = X(4) be the orthocenter of a triangle ABC, and suppose that t > 0. Let
A' = the point on line AH such that |A'A|/|A'H| = t, and define B' and C' cyclically
Ab = orthogonal projection of A' on AB, and define Bc and Ca cyclically
Ac = orthogonal project of A' on AC, and define Ba and Cb cyclically
Ea = Euler line of A'AbAc, and define Eb and Ec cyclically
E'a = Euler line of AAbAc, and define E'b and E'c cyclically.

The lines Ea, Eb, Ec concur in a point whose locus as t varies is the line W =X(4)X(54). The lines E'a, E'b, E'c concur in a point whose locuse as t varies is the line W' = X(4)X(74). The lines W and W' are parallel, and X(10628) is their point of intersection on the line at infinity. See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 24703.

X(10628) lies on these lines:
{3, 8157}, {4, 7730}, {30, 511}, {52, 265}, {54, 74}, {110, 5562}, {113, 1209}, {125, 389}, {146, 2888}, {195, 2935}, {399, 2917}, {973, 1112}, {1205, 6776}, {1216, 1511}, {1498, 5898}, {1539, 6153}, {3448, 5889}, {5972, 7542}, {6276, 7726}, {6277, 7725}, {6288, 7728}, {6689, 6699}, {6723, 9826}, {7356, 7727}, {7978, 7979}, {8994, 8995}, {9904, 9905}, {9984, 9985}, {10065, 10066}, {10081, 10082}


X(10629) =  HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND MID-TRIANGLE OF 1ST AND 2ND JOHNSON-YFF TRIANGLES

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

X(10629) lies on these lines:
{1,4}, {2,8071}, {3,10321}, {11,6893}, {12,6826}, {20,8069}, {28,10037}, {35,6916}, {36,6865}, {55,6850}, {56,6827}, {100,377}, {255,5230}, {443,498}, {495,6917}, {496,6929}, {499,5084}, {611,5800}, {631,10320}, {938,5080}, {942,10526}, {988,6803}, {999,6928}, {1076,4320}, {1470,6891}, {1737,2551}, {2478,2975}, {2550,10039}, {3090,8068}, {3295,6923}, {3421,5904}, {3434,3885}, {3436,3868}, {3545,8070}, {3600,6840}, {4222,10046}, {4293,6836}, {4294,6925}, {5218,6897}, {5261,6839}, {5393,6805}, {5405,6806}, {5721,9370}, {6835,10590}, {6851,7354}, {6854,10588}, {6864,7951}, {6898,10589}, {6901,8164}, {6939,7741}, {6947,7288}, {6957,10591}, {6987,7742}, {9957,10525}

X(10629) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (388,497,944), (1478,1479,5691), (8071,10523,2)


X(10630) =  PERSPECTOR OF ABC AND CROSS-TRIANGLE OF ABC AND CIRCUMCEVIAN TRIANGLE OF X(187)

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

X(10630) lies on these lines: {23,111}, {316,524} et al

X(10630) = isogonal conjugate of X(2482)
X(10630) = cevapoint of X(6) and X(111)
X(10630) = X(6)-cross conjugate of X(111)
X(10630) = trilinear pole of line X(111)X(351) (the tangent to the circumcircle at X(111))
X(10630) = barycentric square of X(897)



leftri  Tri-equilateral triangles and related centers: X(10631) - X(10682)  rightri

This preamble and centers X(10631)-X(10682) were contributed by César Lozada, November 5, 2016.

As with the Kenmotu squares, we inscribe in a triangle ABC three congruent equilateral triangles PAbAc, PBcBa and PCaCb, with Ba, Ca on BC, Cb, Ab on CA and Ac, Bc on AB. There are two points P making possible this construction: P = Pi=X(15) and P = Po=X(16). The equilateral triangles obtained in each case are here named the A-, B-, C- inner/outer equilateral triangles, respectively.

In each case, the points Ba, Ca, Cb, Ab, Ac, Bc are obviously concyclic. Their circles Γi and Γo, here named the inner and outer tri-equilateral circles, are denoted and determined as follows:

Γi: center = X(15), radius = 2*R/|sqrt(3)+cot(ω)|
Γo: center = X(16), radius = 2*R/|sqrt(3)-cot(ω)|,

where R and ω are the circumradius and the Brocard angle of ABC, respectively.

No ETC center X(i), for 1 ≤ i ≤ 10682, lies on either circle.

The exsimilcenter of these circles is X(3) and their insimilcenter is X(6). Their radical trace is X(10631).

For P = Pi=X(15), the vertices of the inner equilateral triangles have trilinear coordinates:
   Ab = 1 : 0 : c*a/(SC+sqrt(3)*S)     Bc = a*b/(SA+sqrt(3)*S) : 1 : 0     Ca = 0 : b*c/(SB+sqrt(3)*S) : 1
   Ac = 1 : a*b/(SB+sqrt(3)*S) : 0     Ba = 0 : 1 : b*c/(SC+sqrt(3)*S)     Cb = c*a/(SA+sqrt(3)*S) : 0 : 1

and, for P=Po=X(16), the vertices of the outer equilateral triangles have trilinear coordinates:
   Ab = 1 : 0 : c*a/(SC-sqrt(3)*S)     Bc = a*b/(SA-sqrt(3)*S) : 1 : 0     Ca = 0 : b*c/(SB-sqrt(3)*S) : 1
   Ac = 1 : a*b/(SB-sqrt(3)*S) : 0     Ba = 0 : 1 : b*c/(SC-sqrt(3)*S)     Cb = c*a/(SA-sqrt(3)*S) : 0 : 1

For the inner-equilateral triangles, let's define the inner tri-equilateral triangle AiBiCi as the triangle bounded by the lines AbAc, BcBa and CaCb and, similarly, define the outer tri-equilateral triangle AoBoCo with the outer-equilateral triangles. The trilinear coordinates of Ai and Ao are:
Ai = (SA-sqrt(3)*S)*a/(SA+sqrt(3)*S) : b : c
Ao = (SA+sqrt(3)*S)*a/(SA-sqrt(3)*S) : b : c

The appearance of (T,i,j) in the following list means that triangle T is perspective to the inner and outer tri-equilateral triangles with perspectors X(i) and X(j), respectively. An asterisk * indicates that T and the tri-equilateral triangles are homothetic:

(ABC, 6, 6) (2nd Brocard, 6, 6) (circumorthic*, 10632, 10633) (circumsymmedial, 6, 6) (2nd Ehrmann*, 6, 6)
(2nd Euler*, 10634, 10635) (extangents*, 10636, 10637) (inner-Grebe, 6, 6) (outer-Grebe, 6, 6) (intangents*, 10638, 1250)
(1st Kenmotu diagonals*, 6, 6) (2nd Kenmotu diagonals*, 6, 6) (Kosnita*, 16, 15) (medial, 10639, 10640) (orthic*, 10641, 10642)
(2nd orthosymmedial, 6, 6) (submedial*, 10643, 10644) (symmedial, 6, 6) (tangential*, 6, 6) (Trinh*, 10645, 10646)

The appearance of (T,i,j) in the following list means that the endo-homothetic centers of T and the inner and outer tri-equilateral triangles are X(i) and X(j), respectively:

(circumorthic, 10647, 10648), (2nd Ehrmann*, 7, 7), (2nd Euler, 10649, 10650), (Kosnita, 10651, 10652), (orthic, 10653, 10654), (submedial, 10655, 10656)

The appearance of [ T, (i, j), (m, n) ] in the following list means that the orthologic centers of T and the inner and outer tri-equilateral triangles are (X(i), X(j)) and (X(m), X(n)), respectively:

[ABC, (3, 15), (3, 16)] [5th anti-Brocard, (3398, 15), (3398, 16)] [anticomplementary, (4, 15), (4, 16)] [anti-Euler, (20, 15), (20, 16)]
[anti-orthocentroidal, (3581, 10657), (3581, 10658)] [Aquila, (40, 15), (40, 16)] [Ara, (7387, 15), (7387, 16)] [Aries, (7387, 10659), (7387, 10660)]
[5th Brocard, (9821, 15), (9821, 16)] [circumorthic, (5889, 10661), (5889, 10662)] [1st Ehrmann, (576, 6), (576, 6)] [2nd Ehrmann, (8548, 10661), (8548, 10662)]
[2nd Euler, (5562, 10661), (5562, 10662)] [extangents, (6237, 10661), (6237, 10662)] [outer-Garcia, (355, 15), (355, 16)] [inner-Grebe, (1161, 15), (1161, 16)]
[outer-Grebe, (1160, 15), (1160, 16)] [1st Hyacinth, (10112, 10663), (10112, 10664)] [2nd Hyacinth, (3, 10659), (3, 10660)] [intangents, (6238, 10661), (6238, 10662)]
[Johnson, (4, 15), (4, 16)] [inner-Johnson, (10525, 15), (10525, 16)] [outer-Johnson, (10526, 15), (10526, 16)] [1st Johnson-Yff, (1478, 15), (1478, 16)]
[2nd Johnson-Yff, (1479, 15), (1479, 16)] [1st Kenmotu diag., (10665, 10661), (10665, 10662)] [2nd Kenmotu diag., (10666, 10661), (10666, 10662)] [Kosnita, (1147, 10661), (1147, 10662)]
[Lucas antipodal, (3, ?), (3, ?)] [Lucas central, (3, 10667), (3, 10668)] [Lucas homothetic, (10669, 15), (10669, 16)] [Lucas reflection, (10670, ?), (10670, ?)]
[Lucas(-1) antipodal, (3, ?), (3, ?)] [Lucas(-1) central, (3, 10671), (3, 10672)] [Lucas(-1) homothetic, (10673, 15), (10673, 16)] [Lucas(-1) reflection, (10674, ?), (10674, ?)]
[Macbeath, (4, 5318), (4, 5321)] [Mandart-incircle, (1, 15), (1, 16)] [medial, (5, 15), (5, 16)] [midheight, (389, 10675), (389, 10676)]
[5th mixtilinear, (1482, 15), (1482, 16)] [orthic, (52, 10661), (52, 10662)] [orthocentroidal, (568, 10657), (568, 10658)] [reflection, (6243, 10677), (6243, 10678)]
[submedial, (5462, 10661), (5462, 10662)] [tangential, (155, 10661), (155, 10662)] [Trinh, (7689, 10661), (7689, 10661)] [inner-Yff, (55, 15), (55, 16)]
[outer-Yff, (56, 15), (56, 16)] [inner-Yff tangents, (10679, 15), (10679, 16)] [outer-Yff tangents, (10680, 15), (10680, 16)]

The inner and outer tri-equilateral triangles are orthologic with centers (10661,10662).

The appearance of [ T, (i, j), (m, n) ] in the following list means that the parallelogic centers of T and the inner and outer tri-equilateral triangles are (X(i), X(j)) and (X(m), X(n)), respectively:

[1st Hyacinth, (10116, 10681), (10116, 10682)], [1st Parry, (351, 15), (351, 16)], [2nd Parry, (351, 15), (351, 16)]

underbar

X(10631) = RADICAL TRACE OF THE INNER AND OUTER TRI-EQUILATERAL CIRCLES

Trilinears    (2*a^6-4*(b^2+c^2)*a^4+(3*b^4-2*b^2*c^2+3*c^4)*a^2+b^2*c^2*(b^2+c^2))*a : :
X(10631) = (SW^2+9*S^2)*S^2*X(3)+(SW^2-7*S^2)*SW^2*X(6)

X(10631) lies on these lines:
{3,6}, {316,7746}, {625,1078}, {3849,8859}, {5031,7810}, {7874,10350}

X(10631) = Moses circle-inverse-of-X(575)
X(10631) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,2080,187), (187,5007,2021), (187,5107,8589), (1691,5039,1692), (2021,2031,5007), (3053,5162,187), (3053,9301,5162)


X(10632) = HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND INNER TRI-EQUILATERAL

Trilinears    sin(2*A+Pi/3)*sec(A) : :

X(10632) lies on these lines:
{4,15}, {6,24}, {16,186}, {25,2981}, {61,3518}, {396,7576}, {403,5321}, {473,8838}, {1870,7051}, {3542,5334}, {5318,6240}

X(10632) = {X(6),X(24)}-harmonic conjugate of X(10633)


X(10633) = HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND OUTER TRI-EQUILATERAL

Trilinears    sin(2*A-Pi/3)*sec(A) : :

X(10633) lies on these lines:
{4,16}, {6,24}, {15,186}, {25,6151}, {62,3518}, {395,7576}, {403,5318}, {472,8836}, {1250,6198}, {3542,5335}, {5321,6240}

X(10633) = {X(6),X(24)}-harmonic conjugate of X(10632)


X(10634) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND INNER TRI-EQUILATERAL

Trilinears    cos(A)*(3*cos(A)*cos(B-C)+sin(A)*(2*sqrt(3)*cos(B)*cos(C)+3*sin(A))) : :

X(10634) lies on these lines:
{2,10632}, {3,6}, {343,465}, {1060,7051}, {1209,8837}, {3547,5334}, {7488,10633}, {7502,8739}, {7514,8740}

X(10634) = Brocard circle-inverse-of-X(10635)


X(10635) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND OUTER TRI-EQUILATERAL

Trilinears    cos(A)*(3*cos(A)*cos(B-C)-sin(A)*(2*sqrt(3)*cos(B)*cos(C)-3*sin(A))) : :

X(10635) lies on these lines:
{2,10633}, {3,6}, {343,466}, {1062,1250}, {1209,8839}, {3547,5335}, {7488,10632}, {7502,8740}, {7514,8739}

X(10635) = Brocard circle-inverse-of-X(10634)


X(10636) = HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND INNER TRI-EQUILATERAL

Trilinears    a*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c)-2*sqrt(3)*(-a+b+c)*S) : :

X(10636) lies on these lines:
{6,31}, {15,40}, {65,7051}, {6197,10632}, {8251,10634}

X(10636) = {X(6), X(55)}-harmonic conjugate of X(10637)


X(10637) = HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND OUTER TRI-EQUILATERAL

Trilinears    a*(a^3-(b+c)*a^2-(b+c)^2*a+(b^2-c^2)*(b-c)+2*sqrt(3)*(-a+b+c)*S) : :

X(10637) lies on these lines:
{6,31}, {16,40}, {1251,8609}, {6197,10633}, {8251,10635}

X(10637) = {X(6), X(55)}-harmonic conjugate of X(10636)


X(10638) = HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND INNER TRI-EQUILATERAL

Trilinears    a*(-a+b+c)*(a^2-(b-c)^2+2*sqrt(3)*S) : :

X(10638) lies on these lines:
{1,15}, {6,31}, {12,5321}, {14,3584}, {16,35}, {17,4857}, {37,2154}, {61,3746}, {100,5367}, {202,5010}, {395,4995}, {396,3058}, {559,1442}, {1095,7343}, {1621,5362}, {2307,3295}, {3085,5334}, {4294,5335}, {5238,5563}, {5472,20011}, {6198,10632}, {6740,7043}

X(10638) = {X(6),X(55)}-harmonic conjugate of X(1250)


X(10639) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND INNER TRI-EQUILATERAL

Trilinears    (a^2-b^2-c^2+2*sqrt(3)*S)*(a^4-(b^2-c^2)^2-2*sqrt(3)*(-a^2+b^2+c^2)*S)*a : :

X(10639) lies on these lines:
{5,15}, {16,1147}, {62,1493}, {1209,8837}, {9306,10640}


X(10640) = PERSPECTOR OF THESE TRIANGLES: MEDIAL AND OUTER TRI-EQUILATERAL

Trilinears    (a^2-b^2-c^2-2*sqrt(3)*S)*(a^4-(b^2-c^2)^2+2*sqrt(3)*(-a^2+b^2+c^2)*S)*a : :

X(10640) lies on these lines:
{5,16}, {15,1147}, {61,1493}, {141,466}, {302,473}, {1209,8839}, {9306,10639}


X(10641) = HOMOTHETIC CENTER OF THESE TRIANGLES: ORTHIC AND INNER TRI-EQUILATERAL

Trilinears    (SA-sqrt(3)*S)*a*SB*SC : :

X(10641) lies on these lines:
{4,15}, {5,10634}, {6,25}, {13,7576}, {16,24}, {19,10636}, {26,10635}, {32,3442}, {33,10638}, {34,7051}, {53,462}, {61,10594}, {62,3518}, {112,2381}, {216,3129}, {235,5321}, {303,472}, {393,3458}, {396,428}, {463,6748}, {577,3130}, {2383,2902}, {3089,5334}, {3199,3490}, {3457,8882}, {3575,5318}, {5335,7487}

X(10641) = X(63)-isoconjugate of X(18)
X(10641) = {X(6), X(25)}-harmonic conjugate of X(10642)


X(10642) = HOMOTHETIC CENTER OF THESE TRIANGLES: ORTHIC AND OUTER TRI-EQUILATERAL

Trilinears    (SA+sqrt(3)*S)*a*SB*SC : :

X(10642) lies on these lines:
{4,16}, {5,10635}, {6,25}, {14,7576}, {15,24}, {19,10637}, {26,10634}, {32,3443}, {33,1250}, {53,463}, {61,3518}, {62,10594}, {112,2380}, {216,3130}, {235,5318}, {302,473}, {393,3457}, {395,428}, {462,6748}, {577,3129}, {2383,2903}, {3089,5335}, {3199,3489}, {3458,8882}, {3575,5321}, {5334,7487}

X(10642) = {X(6), X(25)}-harmonic conjugate of X(10641)
X(10642) = X(63)-isoconjugate of X(17)


X(10643) = HOMOTHETIC CENTER OF THESE TRIANGLES: SUBMEDIAL AND INNER TRI-EQUILATERAL

Trilinears    a*(2*(SA^2+S^2)-SW*SA+sqrt(3)*(4*R^2-SW)*S/3) : :

X(10643) lies on these lines:
{2,10641}, {5,15}, {6,1196}, {13,10127}, {16,6642}, {396,10128}, {1656,10634}, {3090,10632}, {5318,9825}, {7506,10635}, {9816,10636}, {9817,10638}

X(10643) = {X(6),X(5020)}-harmonic conjugate of X(10644)


X(10644) = HOMOTHETIC CENTER OF THESE TRIANGLES: SUBMEDIAL AND OUTER TRI-EQUILATERAL

Trilinears    a*(2*(SA^2+S^2)-SW*SA-sqrt(3)*(4*R^2-SW)*S/3) : :

X(10644) lies on these lines:
{2,10642}, {5,16}, {6,1196}, {14,10127}, {15,6642}, {395,10128}, {1250,9817}, {1656,10635}, {1995,10641}, {3090,10633}, {5321,9825}, {7506,10634}, {9816,10637}

X(10644) = {X(6),X(5020)}-harmonic conjugate of X(10643)


X(10645) = HOMOTHETIC CENTER OF THESE TRIANGLES: TRINH AND INNER TRI-EQUILATERAL

Trilinears    (a^4-2*(b^2+c^2)*a^2+(b^2-c^2)^2-6*sqrt(3)*(-a^2+b^2+c^2)*S)*a : :

X(10645) lies on these lines:
{3,6}, {4,10188}, {13,376}, {14,549}, {17,550}, {18,3523}, {35,7051}, {36,10638}, {69,5463}, {74,3166}, {99,6295}, {140,5321}, {141,5464}, {186,10642}, {203,1250}, {302,531}, {378,10641}, {396,8703}, {617,618}, {619,7831}, {1495,3131}, {1511,10640}, {2041,3366}, {2042,3367}, {2378,10409}, {3054,5474}, {3520,10632}, {3522,5335}, {3642,7835}, {4550,10639}, {6644,10644}, {7005,7280}, {7688,10636}, {9818,10643}

X(10645) = isogonal conjugate of X(12816)
X(10645) = Brocard circle-inverse-of-X(10646)
X(10645) = {X(3), X(6)}-harmonic conjugate of X(10646)


X(10646) = HOMOTHETIC CENTER OF THESE TRIANGLES: TRINH AND OUTER TRI-EQUILATERAL

Trilinears    (a^4-2*(b^2+c^2)*a^2+(b^2-c^2)^2-6*sqrt(3)*(-a^2+b^2+c^2)*S)*a : :

X(10646) lies on these lines:
{3,6}, {4,10187}, {13,549}, {14,376}, {17,3523}, {18,550}, {36,1250}, {69,5464}, {74,3165}, {99,6582}, {140,5318}, {141,5463}, {186,10641}, {202,5010}, {303,530}, {378,10642}, {395,8703}, {616,619}, {618,7831}, {1495,3132}, {1511,10639}, {2041,3392}, {2042,3391}, {3054,5473}, {3520,10633}, {3522,5334}, {3643,7835}, {4550,10640}, {6644,10643}, {7006,7051}, {7688,10637}, {9818,10644}

X(10646) = Brocard circle-inverse-of-X(10645)
X(10646) = {X(3), X(6)}-harmonic conjugate of X(10645)
X(10646) = isogonal conjugate of X(12817)


X(10647) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND INNER TRI-EQUILATERAL

Trilinears    a*(-a+b+c)+2*b*c-2*sqrt(3)*a*S/(-a+b+c) : :

X(10647) lies on these lines:
{1,16}, {7,21}, {36,10651}, {238,7051}, {559,1962}, {1081,6186}, {1929,7052}, {3576,10649}, {3639,5144}

X(10647) = {X(56),X(1001)}-harmonic conjugate of X(10648)
X(10647) = X(i)-vertex conjugate of X(j) for these (i,j): (4367,10648)
X(10647) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (56,1001,10648)


X(10648) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND OUTER TRI-EQUILATERAL

Trilinears    a*(-a+b+c)+2*b*c+2*sqrt(3)*a*S/(-a+b+c) : :

X(10648) lies on these lines:
{1,15}, {7,21}, {36,10652}, {554,6186}, {1082,1962}, {2307,4649}, {3576,10650}, {3638,5144}

X(10648) = mid-point of X(i)X(j) for these (i,j): (1,3179)
X(10648) = {X(56),X(1001)}-harmonic conjugate of X(10647)
X(10648) = X(i)-vertex conjugate of X(j) for these (i,j): (4367,10647)


X(10649) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND INNER TRI-EQUILATERAL

Trilinears    a*(a^4+(b-c)^2*(2*(a^2-b*c)-3*(b^2+c^2)))-(b+c)*(3*a^4-2*a^2*(b^2+c^2)-(b^2-c^2)^2)+2*sqrt(3)*(a*(a+b+c)*(b+c-a)+(b+c)*(a-b+c)*(a+b-c))*S

X(10649) lies on these lines:
{1,7}, {3,1653}, {1082,10391}, {3576,10647}, {5240,5784}, {7987,10655}

X(10649) = {X(1),X(5732)}-harmonic conjugate of X(10650)


X(10650) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND OUTER TRI-EQUILATERAL

Trilinears    a*(a^4+(b-c)^2*(2*(a^2-b*c)-3*(b^2+c^2)))-(b+c)*(3*a^4-2*a^2*(b^2+c^2)-(b^2-c^2)^2)-2*sqrt(3)*(a*(a+b+c)*(b+c-a)+(b+c)*(a-b+c)*(a+b-c))*S : :

X(10650) lies on these lines:
{1,7}, {3,1652}, {84,1251}, {559,10391}, {3576,10648}, {5239,5784}, {7987,10656}

X(10650) = {X(1),X(5732)}-harmonic conjugate of X(10649)


X(10651) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: KOSNITA AND INNER TRI-EQUILATERAL

Trilinears    (3*a^3-(b-c)^2*(a+2*(b+c))-2*sqrt(3)*a*S)/a : :

X(10651) lies on this line:
{1,7}, {36,10647}, {46,1276}, {55,1081}, {1082,1836}, {1478,9901}, {3339,10655}, {5240,5880

X(10651) = reflection of X(1) in X(3639)
X(10651) = {X(1),X(4312)}-harmonic conjugate of X(10652)


X(10652) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: KOSNITA AND OUTER TRI-EQUILATERAL

Trilinears    (3*a^3-(b-c)^2*(a+2*(b+c))+2*sqrt(3)*a*S)/a : :

X(10652) lies on these lines: {1,7}, {36,10648}, {46,1277}, {55,554}, {79,1251}, {559,1836}, {1478,9900}, {3179,5011}, {3339,10656}, {5239,5880}

X(10652) = reflection of X(1) in X(3638)
X(10652) = {X(1),X(4312)}-harmonic conjugate of X(10651)


X(10653) =  X(2)X(13)∩X(4)X(14)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+2 Sqrt[3] a^2 S : :

X(10653) and X(10654) are contributed by Peter Moses, December 19, 2016.

X(10653) lies on the cubic K876 and these lines:
{2, 13}, {3, 396}, {4, 14}, {5, 5340}, {6, 30}, {15, 376}, {17, 631}, {18, 3091}, {20, 61}, {69, 532}, {182, 6772}, {193, 533}, {194, 617}, {202, 497}, {203, 4293}, {298, 315}, {317, 11093}, {371, 2043}, {372, 2044}, {381, 395}, {382, 398}, {383, 9744}, {388, 7006}, {531, 1992}, {542, 6775}, {574, 5472}, {619, 6337}, {621, 3181}, {1080, 9753}, {1250, 10056}, {1478, 7127}, {1587, 3389}, {1588, 3390}, {2307, 4299}, {3107, 6770}, {3200, 9544}, {3206, 9545}, {3364, 6459}, {3365, 6460}, {3411, 3832}, {3412, 3528}, {3522, 5238}, {3523, 5351}, {3524, 10646}, {3543, 5334}, {3627, 5339}, {3830, 5321}, {3843, 5350}, {4294, 7005}, {5076, 5349}, {5464, 6783}, {5617, 9115}, {5978, 7774}, {6109, 7735}, {6771, 9736}, {7618, 9763}, {7791, 9989}, {9143, 10657}, {10304, 10645}

X(10653) = reflection of X(69) in X(3642)
X(10653) = anticomplement of X(3643)
X(10653) = crossdifference of every pair of points on line X(6137) X(8675)
X(10653) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5335,13), (13,16,2), (13,5463,6115), (17,5237,631), (395,5318,381), (2549,11179,10654)
X(10653) = circumcircle-of-inner-Napoleon-triangle inverse of X(6115)
X(10653) = circumcircle-of-outer-Napoleon-triangle inverse of X(6108)
X(10653) = reflection of X(10654) in X(6)
X(10653) = X(617)-of-1st-Brocard-triangle


X(10654) =  X(2)X(14)∩X(4)X(13)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-2 Sqrt[3] a^2 S : :

X(10654) and X(10653) are contributed by Peter Moses, December 19, 2016.

X(10654) lies on the cubic K876 and these lines:
{2, 14}, {3, 395}, {4, 13}, {5, 5339}, {6, 30}, {16, 376}, {17, 3091}, {18, 631}, {20, 62}, {69, 533}, {182, 6775}, {193, 532}, {194, 616}, {202, 4293}, {203, 497}, {299, 315}, {317, 11094}, {371, 2044}, {372, 2043}, {381, 396}, {382, 397}, {383, 9753}, {388, 7005}, {530, 1992}, {542, 6772}, {574, 5471}, {618, 6337}, {622, 3180}, {1080, 9744}, {1479, 2307}, {1587, 3364}, {1588, 3365}, {3106, 6773}, {3201, 9544}, {3205, 9545}, {3389, 6459}, {3390, 6460}, {3411, 3528}, {3412, 3832}, {3522, 5237}, {3523, 5352}, {3524, 10645}, {3543, 5335}, {3627, 5340}, {3830, 5318}, {3843, 5349}, {4294, 7006}, {4302, 7127}, {5076, 5350}, {5463, 6782}, {5613, 9117}, {5979, 7774}, {6108, 7735}, {6774, 9735}, {7051, 10072}, {7618, 9761}, {7791, 9988}, {9143, 10658}, {10056, 10638}, {10304, 10646}

X(10654) = reflection of X(69) in X(3643)
X(10654) = anticomplement of X(3642)
X(10654) = crossdifference of every pair of points on line X(6138) X(8675)
X(10654) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,5334,14), (14,15,2), (14,5464,6114), (18,5238,631), (396,5321,381), (2549,11179,10653)
X(10654) = circumcircle-of-inner-Napoleon-triangle inverse of X(6109)
X(10654) = circumcircle-of-outer-Napoleon-triangle inverse of X(6114)
X(10654) = reflection of X(10653) in X(6)
X(10654) = X(616)-of-1st-Brocard-triangle


X(10655) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: SUBMEDIAL AND INNER TRI-EQUILATERAL

Trilinears    a^4-(b-c)^2*(6*a^2+3*(b^2+c^2)+10*b*c-8*a*(b+c))-2*sqrt(3)*(a^2+2*a*(b+c)-3*(b-c)^2)*S : :

X(10655) lies on these lines: {7,1699}, {165,1653}, {3339,10651}, {7987,10649}


X(10656) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: SUBMEDIAL AND OUTER TRI-EQUILATERAL

Trilinears    a^4-(b-c)^2*(6*a^2+3*(b^2+c^2)+10*b*c-8*a*(b+c))+2*sqrt(3)*(a^2+2*a*(b+c)-3*(b-c)^2)*S : :

X(10656) lies on these lines: {7,1699}, {165,1652}, {3339,10652}, {7987,10650}


X(10657) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO ANTI-ORTHOCENTROIDAL

Trilinears    (3*(3*SA^2-9*R^2*SA+S^2)+S*sqrt(3)*(9*R^2-2*SW))*a : :

The reciprocal orthologic center of these triangles is X(3581).

X(10657) lies on the cubic K262b and these lines:
{6,13}, {15,5663}, {16,110}, {62,5609}, {74,3166}, {323,530}, {1511,10639}, {1986,10641}, {7722,10632}, {7723,10634}, {7724,10636}, {7727,10638}, {9826,10643}

X(10657) = orthologic center of these triangles: inner tri-equilateral to orthocentroidal
X(10657) = {X(6),X(399)}-harmonic conjugate of X(10658)
X(10657) = X(15)-of-anti-orthocentroidal-triangle
X(10657) = 4th-Brocard-to-circumsymmedial similarity image of X(15)


X(10658) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO ANTI-ORTHOCENTROIDAL

Trilinears    (3*(3*SA^2-9*R^2*SA+S^2)-S*sqrt(3)*(9*R^2-2*SW))*a : :

The reciprocal orthologic center of these triangles is X(3581).

X(10658) lies on the cubic K262a and these lines:
{6,13}, {15,110}, {16,5663}, {61,5609}, {74,3165}, {323,531}, {1250,7727}, {1511,10640}, {1986,10642}, {7722,10633}, {7723,10635}, {7724,10637}, {9826,10644}

X(10658) = orthologic center of these triangles: outer tri-equilateral to orthocentroidal
X(10658) = {X(6),X(399)}-harmonic conjugate of X(10657)
X(10658) = X(16)-of-anti-orthocentroidal-triangle
X(10658) = 4th-Brocard-to-circumsymmedial similarity image of X(16)


X(10659) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO ARIES

Trilinears    (sqrt(3)*S*(SA^2-2*R^2*SA-(2*R^2-SW)^2+S^2)+(4*R^2-SW)*(SA^2-SW*SA+S^2))*SA*a : :

The reciprocal orthologic center of these triangles is X(7387).

X(10659) lies on these lines:
{6,1147}, {16,9932}, {68,10634}, {155,10641}, {6193,10632}, {9820,10643}, {9931,10638}, {9938,10645}

X(10659) = orthologic center of these triangles: inner tri-equilateral to 2nd Hyacinth
X(10659) = {X(6),X(9937)}-harmonic conjugate of X(10660)


X(10660) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO ARIES

Trilinears    (-sqrt(3)*S*(SA^2-2*R^2*SA-(2*R^2-SW)^2+S^2)+(4*R^2-SW)*(SA^2-SW*SA+S^2))*SA*a : :

The reciprocal orthologic center of these triangles is X(7387).

X(10660) lies on these lines:
{6,1147}, {15,9932}, {68,10635}, {155,10642}, {1250,9931}, {6193,10633}, {9820,10644}, {9938,10646}

X(10660) = orthologic center of these triangles: outer tri-equilateral to 2nd Hyacinth
X(10660) = {X(6),X(9937)}-harmonic conjugate of X(10659)


X(10661) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO CIRCUMORTHIC

Trilinears    (S*sqrt(3)*(2*R^2-SA)+SB*SC)*SA*a : :

The reciprocal orthologic center of these triangles is X(5889).

X(10661) lies on these lines:
{5,6}, {13,539}, {16,1147}, {52,10641}, {110,10633}, {184,10635}, {5335,6193}, {5462,10643}, {5562,10634}, {5889,10632}, {6237,10636}, {6238,10638}, {7051,7352}, {7689,10645}, {10539,10642}

X(10661) = orthologic center of the inner tri-equilateral triangle to each of these triangles: 2nd Ehrmann, 2nd Euler, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, orthic, submedial, tangential, outer tr-equilateral,Trinh
X(10661) = {X(6),X(155)}-harmonic conjugate of X(10662)


X(10662) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO CIRCUMORTHIC

Trilinears    (-S*sqrt(3)*(2*R^2-SA)+SB*SC)*SA*a : :

The reciprocal orthologic center of these triangles is X(5889).

X(10662) lies on these lines:
{5,6}, {14,539}, {15,1147}, {52,10642}, {110,10632}, {184,10634}, {394,466}, {473,1993}, {1250,6238}, {5334,6193}, {5462,10644}, {5562,10635}, {5889,10633}, {6237,10637}, {7689,10646}, {10539,10641}

X(10662) = orthologic center of the outer tri-equilateral triangle to each of these triangles: 2nd Ehrmann, 2nd Euler, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, orthic, submedial, tangential, inner tri-equilateral,Trinh
X(10662) = {X(6),X(155)}-harmonic conjugate of X(10661)


X(10663) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO 1ST HYACINTH

Trilinears    SA*(sqrt(3)*S*(SA^2-3*R^2*SA+12*R^2*SW+S^2-18*R^4-2*SW^2)*a+SC*SB*(9*R^2-2*SW)*a) : :

The reciprocal orthologic center of these triangles is X(10112).

X(10663) lies on these lines:
{6,1511}, {110,10632}, {113,10641}, {125,10634}

X(10663) = {X(6),X(2931)}-harmonic conjugate of X(10664)


X(10664) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO 1ST HYACINTH

Trilinears    SA*(-sqrt(3)*S*(SA^2-3*R^2*SA+12*R^2*SW+S^2-18*R^4-2*SW^2)*a+SC*SB*(9*R^2-2*SW)*a) : :

The reciprocal orthologic center of these triangles is X(10112).

X(10664) lies on these lines:
{6,1511}, {110,10633}, {113,10642}, {125,10635}

X(10664) = {X(6),X(2931)}-harmonic conjugate of X(10663)


X(10665) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 1ST KENMOTU DIAGONALS TO INNER TRI-EQUILATERAL

Trilinears    SA*((2*R^2-SA)*S+SA^2+S^2-SW*SA)*a : :
Trilinears    cos A tan(A - π/4) : :
Trilinears    = (1 + Cos[2 A] - Sin[2 A]) / (Cos[A] + Sin[A]) : :

X(10665) lies on these lines:
{3,6413}, {5,6}, {26,10533}, {52,5412}, {156,10534}, {372,1147}, {494,3167}, {615,9820}, {912,7969}, {1069,1124}, {1321,3092}, {1335,3157}, {1586,1993}, {1587,6193}, {2066,6238}, {2067,7352}, {2351,10133}, {5413,10539}, {5415,6237}, {5448,6565}, {5449,10576}, {6200,7689}, {6449,8912}, {6564,9927}, {6810,7592}

X(10665) = orthologic center of these triangles: 1st Kenmotu diagonals to outer tri-equilateral
X(10665) = {X(6),X(155)}-harmonic conjugate of X(10666)


X(10666) = ORTHOLOGIC CENTER OF THESE TRIANGLES: 2ND KENMOTU DIAGONALS TO INNER TRI-EQUILATERAL

Trilinears    SA*(-(2*R^2-SA)*S+SA^2+S^2-SW*SA)*a : :
Trilinears    cos A tan(A + π/4) : :
Trilinears    = (1 + Cos[2 A] + Sin[2 A]) / (Cos[A] - Sin[A]) : :

X(10666) lies on these lines:
{3,6414}, {5,6}, {26,10534}, {52,5413}, {156,10533}, {371,1147}, {493,3167}, {590,9820}, {912,7968}, {1069,1335}, {1124,3157}, {1322,3093}, {1585,1993}, {1588,6193}, {2351,8825}, {5409,8961}, {5412,10539}, {5414,6238}, {5416,6237}, {5448,6564}, {5449,10577}, {6396,7689}, {6502,7352}, {6565,9927}, {6809,7592}

X(10666) = orthologic center of these triangles: 2nd Kenmotu diagonals to outer tri-equilateral
X(10666) = {X(6),X(155)}-harmonic conjugate of X(10665)


X(10667) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS CENTRAL

Trilinears    ((8-5*sqrt(3))*a^2-(4-3*sqrt(3))*(b^2+c^2)+2*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(10667) lies on these lines:
{3,6}, {590,621}, {623,8253}, {6239,10632}, {6252,10636}, {6283,10638}, {6291,10641}, {6671,8252}, {7051,7362}, {9823,10643}

X(10667) = {X(6),X(15)}-harmonic conjugate of X(10671)
X(10667) = {X(6),X(1151)}-harmonic conjugate of X(10668)
X(10667) = X(176)-of-inner-tri-equilateral-triangle if ABC is acute
X(10667) = orthic-to-inner-tri-equilateral similarity image of X(6291)


X(10668) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO LUCAS CENTRAL

Trilinears    ((8+5*sqrt(3))*a^2-(4+3*sqrt(3))*(b^2+c^2)+2*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(10668) lies on these lines:
{3,6}, {590,622}, {624,8253}, {1250,6283}, {6239,10633}, {6252,10637}, {6291,10642}, {6672,8252}, {9823,10644}

X(10668) = {X(6),X(16)}-harmonic conjugate of X(10672)
X(10668) = {X(6),X(1151)}-harmonic conjugate of X(10667)
X(10668) = X(176)-of-outer-tri-equilateral-triangle if ABC is acute
X(10668) = orthic-to-outer-tri-equilateral similarity image of X(6291)


X(10669) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS HOMOTHETIC TO INNER TRI-EQUILATERAL

Trilinears    (2*S*(SA^2-4*R^2*SA+2*S^2)+2*S^2*(SW+SA)-SW^2*SA)*a : :

X(10669) lies on these lines:
{3,493}, {4,6462}, {5,8212}, {30,9838}, {40,8188}, {355,8214}, {1160,8218}, {1161,8216}, {1482,8210}, {6339,8221}, {6461,10673}, {7387,8194}

X(10669) = orthologic center of these triangles: Lucas homothetic to outer tri-equilateral


X(10670) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS REFLECTION TO INNER TRI-EQUILATERAL

Trilinears    (S^2-SW*(2*R^2-SW)+(4*R^2+SB+SC)*S)*SA*a : :

orthologic center of these triangles: Lucas reflection to outer tri-equilateral

X(10670) lies on these lines:
{3,49}, {3564,6401}


X(10671) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO LUCAS(-1) CENTRAL

Trilinears    ((8+5*sqrt(3))*a^2-(4+3*sqrt(3))*(b^2+c^2)-2*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(10671) lies on these lines:
{3,6}, {615,621}, {623,8252}, {6400,10632}, {6404,10636}, {6405,10638}, {6406,10641}, {6671,8253}, {7051,7353}, {9824,10643}

X(10671) = {X(6),X(15)}-harmonic conjugate of X(10667)
X(10671) = {X(6),X(1152)}-harmonic conjugate of X(10672)
X(10671) = X(175)-of-inner-tri-equilateral-triangle if ABC is acute
X(10671) = orthic-to-inner-tri-equilateral similarity image of X(6406)


X(10672) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO LUCAS(-1) CENTRAL

Trilinears    ((8-5*sqrt(3))*a^2-(4-3*sqrt(3))*(b^2+c^2)-2*S)*a : :

The reciprocal orthologic center of these triangles is X(3).

X(10672) lies on these lines:
{3,6}, {615,622}, {624,8252}, {1250,6405}, {6400,10633}, {6404,10637}, {6406,10642}, {6672,8253}, {9824,10644}

X(10672) = {X(6),X(16)}-harmonic conjugate of X(10668)
X(10672) = {X(6),X(1152)}-harmonic conjugate of X(10671)
X(10672) = X(175)-of-outer-tri-equilateral-triangle if ABC is acute
X(10672) = orthic-to-outer-tri-equilateral similarity image of X(6406)


X(10673) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) HOMOTHETIC TO INNER TRI-EQUILATERAL

Trilinears    (-2*S*(SA^2-4*R^2*SA+2*S^2)+2*(SA+SW)*S^2-SW^2*SA)*a : :

The reciprocal orthologic center of these triangles is X(15).

X(10673) lies on these lines:
{3,494}, {4,6463}, {5,8213}, {30,9839}, {40,8189}, {355,8215}, {1160,8219}, {1161,8217}, {1482,8211}, {6339,8220}, {6461,10669}, {7387,8195}

X(10673) = orthologic center of these triangles: Lucas(-1) homothetic to outer tri-equilateral


X(10674) = ORTHOLOGIC CENTER OF THESE TRIANGLES: LUCAS(-1) REFLECTION TO INNER TRI-EQUILATERAL

Trilinears    (S^2-SW*(2*R^2-SW)-(4*R^2+SB+SC)*S)*SA*a : :

orthologic center of these triangles: Lucas(-1) reflection to outer tri-equilateral

X(10674) lies on these lines:
{3,49}, {3564,6402}


X(10675) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO MIDHEIGHT

Trilinears    a*(3*S^2+sqrt(3)*(4*R^2-SW)*S-6*(4*R^2-SA)*SA) : :

The reciprocal orthologic center of these triangles is X(389).

X(10675) lies on these lines:
{3,10639}, {4,6}, {15,6000}, {16,6759}, {185,10641}, {399,8175}, {3357,10645}, {5663,10663}, {6241,10632}, {6254,10636}, {6285,10638}, {7051,7355}, {9729,10643}, {10282,10646}

X(10675) = {X(6),X(1498)}-harmonic conjugate of X(10676)


X(10676) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO MIDHEIGHT

Trilinears    a*(3*S^2-sqrt(3)*(4*R^2-SW)*S-6*(4*R^2-SA)*SA) : :

The reciprocal orthologic center of these triangles is X(389).

X(10676) lies on these lines:
{3,10640}, {4,6}, {15,6759}, {16,6000}, {30,10662}, {154,3131}, {185,10642}, {399,8174}, {1250,6285}, {2777,10658}, {3357,10646}, {5663,10664}, {6241,10633}, {6254,10637}, {7051,10535}, {9729,10644}, {10282,10645}

X(10676) = {X(6),X(1498)}-harmonic conjugate of X(10675)


X(10677) = ORTHOLOGIC CENTER OF THESE TRIANGLES: REFLECTION TO INNER TRI-EQUILATERAL

Trilinears    (3*SA*(R^2-SA)-sqrt(3)*(5*R^2-2*SW)*S+3*S^2)*a : :

The reciprocal orthologic center of these triangles is X(6243).

X(10677) lies on the cubic K390 and these lines:
{6,17}, {13,539}, {15,1154}, {16,54}, {62,1493}, {1994,8836}, {2914,6117}, {3171,6151}, {3200,10633}, {5318,10657}, {6152,10641}, {6242,10632}, {6255,10636}, {6286,10638}, {7051,7356}, {7691,10645}, {9827,10643}, {10610,10646}

X(10677) = {X(6),X(195)}-harmonic conjugate of X(10678)


X(10678) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO REFLECTION

Trilinears    (3*SA*(R^2-SA)+sqrt(3)*(5*R^2-2*SW)*S+3*S^2)*a : :

The reciprocal orthologic center of these triangles is X(6243).

X(10678) lies on the cubic K390 and these lines:
{6,17}, {14,539}, {15,54}, {16,1154}, {61,1493}, {1250,6286}, {1994,8838}, {2914,6116}, {2981,3170}, {3201,10632}, {5321,10658}, {6152,10642}, {6242,10633}, {6255,10637}, {7691,10646}, {9827,10644}, {10610,10645}

X(10678) = {X(6),X(195)}-harmonic conjugate of X(10677)


X(10679) = ORTHOLOGIC CENTER OF THESE TRIANGLES: INNER-YFF TANGENTS TO INNER TRI-EQUILATERAL

Trilinears    (a^5-(b+c)*a^4-2*(b^2+c^2)*a^3+2*(b+c)*(b^2+b*c+c^2)*a^2+((b^2-c^2)^2-4*b^2*c^2)*a-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2))*a : :
X(10679) = 2*R*X(1)-(R-r)*X(3)

The reciprocal orthologic center of these triangles is X(15).

X(10679) lies on these lines:
{1,3}, {2,10596}, {4,3871}, {5,3434}, {8,3560}, {12,10525}, {100,5603}, {104,3241}, {119,381}, {145,6906}, {149,6830}, {221,5399}, {355,3913}, {382,5842}, {390,6827}, {404,10595}, {405,5554}, {474,5901}, {495,6923}, {496,6958}, {497,6882}, {631,10586}, {674,1351}, {912,3870}, {946,8715}, {952,1012}, {954,6907}, {956,5844}, {962,5761}, {1056,6948}, {1058,6891}, {1064,1480}, {1253,7086}, {1260,3419}, {1376,5886}, {1478,5840}, {1598,1824}, {1621,5657}, {1656,2886}, {2057,5780}, {2550,6881}, {2900,5777}, {3052,5398}, {3085,6842}, {3158,5720}, {3244,5450}, {3421,6930}, {3526,6690}, {3617,6920}, {3622,6940}, {3654,4428}, {3656,4421}, {3722,7986}, {3811,5887}, {4190,10597}, {4294,7491}, {4301,6796}, {5082,6824}, {5274,6978}, {5281,6954}, {6284,10526}, {6713,10072}, {6765,7330}, {6833,10530}, {6893,7080}, {6897,10587}, {6909,7967}, {6971,9669}, {6977,10529}, {6982,8164}, {7489,9708}

X(10679) = reflection of X(i) in X(j) for these (i,j): (3,55), (956,6914), (3434,5), (6923,495)
X(10679) = orthologic center of these triangles: inner-Yff tangents to outer tri-equilateral
X(10679) = outer-Yff-to-inner-Yff similarity image of X(3)
X(10679) = 2nd-Johnson-Yff-to-1st-Johnson-Yff similarity image of X(3)
X(10679) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1470,999), (1,2077,10269), (1,3359,10202), (3,1482,10680), (3,6767,10246), (3,10247,999), (40,3746,10267), (40,10267,3), (55,2099,8069), (100,5603,6911), (1385,10310,3), (1621,5657,6883), (2077,10269,3), (2099,8069,999), (3295,10306,3), (3303,10310,1385)


X(10680) = ORTHOLOGIC CENTER OF THESE TRIANGLES: OUTER-YFF TANGENTS TO INNER TRI-EQUILATERAL

Trilinears    (a^5-(b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2-3*b*c+c^2)*a^2+(b^4+c^4-2*(2*b-c)*(b-2*c)*b*c)*a-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2))*a : :
X(10680) = 2*R*X(1)-(R+r)*X(3)

The reciprocal orthologic center of these triangles is X(16).

X(10680) lies on these lines:
{1,3}, {2,10597}, {4,10529}, {5,956}, {8,6911}, {11,10526}, {21,10595}, {104,962}, {145,6905}, {381,529}, {382,2829}, {388,6842}, {405,5901}, {411,7967}, {474,5690}, {495,6863}, {496,6928}, {497,7491}, {582,3445}, {601,1480}, {602,1149}, {631,10587}, {859,3193}, {944,6985}, {945,1069}, {946,8666}, {952,3149}, {958,5886}, {1006,3622}, {1056,6825}, {1058,6868}, {1068,1398}, {1191,5398}, {1329,1656}, {1351,8679}, {1457,3157}, {1476,6948}, {1598,1828}, {2256,5755}, {2975,3560}, {3086,6882}, {3244,6796}, {3421,6944}, {3526,6691}, {3600,6850}, {3616,5761}, {3617,6946}, {3871,6942}, {4299,5840}, {4301,5450}, {5082,6885}, {5120,8609}, {5251,9624}, {5253,5657}, {5258,8227}, {5265,6961}, {5288,5587}, {5687,5844}, {5720,6762}, {5790,6734}, {6834,10530}, {6872,10596}, {6880,10528}, {6947,10586}, {6970,7080}, {6980,9654}, {7354,10525}, {8256,9709}

X(10680) = reflection of X(i) in X(j) for these (i,j): (3,56), (3436,5), (5687,6924), (6928,496)
X(10680) = orthologic center of these triangles: outer-Yff tangents to outer tri-equilateral
X(10680) = inner-Yff-to-outer-Yff similarity image of X(3)
X(10680) = 1st-Johnson-Yff-to-2nd-Johnson-Yff similarity image of X(3)
X(10680) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,1482,10679), (3,7373,10246), (3,8148,10306), (3,10247,3295), (40,5563,10269), (40,10269,3), (1385,3428,3), (2098,8069,3295), (2975,5603,3560), (3304,3428,1385), (3304,5049,7373), (10527,10532,5)


X(10681) = PARALLELOGIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL TO 1ST HYACINTH

Trilinears    (3*(5*R^2-SW)*S^2+sqrt(3)*(4*R^2-SW)*(9*R^2-2*SW)*S+3*((6*R^2-SW)*SA-2*(5*R^2-SW)*SW)*SA)*a : :

The reciprocal parallelogic center of these triangles is X(10116).

X(10681) lies on these lines:
{6,1112}, {15,2777}, {26,10664}, {74,10632}, {113,10634}, {125,10641}, {5663,10663}, {6723,10643}, {6759,10658}, {9934,10676}, {10118,10638}, {10119,10636}

X(10681) = {X(6),X(10117)}-harmonic conjugate of X(10682)


X(10682) = PARALLELOGIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL TO 1ST HYACINTH

Trilinears    (3*(5*R^2-SW)*S^2-sqrt(3)*(4*R^2-SW)*(9*R^2-2*SW)*S+3*((6*R^2-SW)*SA-2*(5*R^2-SW)*SW)*SA)*a : :

The reciprocal parallelogic center of these triangles is X(10116).

X(10682) lies on these lines:
{6,1112}, {16,2777}, {26,10663}, {74,10633}, {113,10635}, {125,10642}, {1250,10118}, {5663,10664}, {6723,10644}, {6759,10657}, {9934,10675}, {10119,10637}

X(10682) = {X(6),X(10117)}-harmonic conjugate of X(10681)


X(10683) =  POINT BECRUX 14

Barycentrics    2 a^10 + 4 a^9 (b+c) + a^8 (-9 b^2+2 b c-9 c^2) - 2 a^7 (8 b^3+7 b^2 c+7 b c^2+8 c^3) + a^6 (10 b^4-7 b^3 c-7 b c^3+10 c^4) + a^5 (24 b^5+15 b^4 c+11 b^3 c^2+11 b^2 c^3+15 b c^4+24 c^5) + 2 a^4 (2 b^6+6 b^5 c+3 b^4 c^2+6b^3 c^3+3 b^2 c^4+6 b c^5+2 c^6) - 2 a^3 (b-c)^2 (8 b^5+18 b^4 c+21 b^3 c^2+21 b^2 c^3+18 b c^4+8 c^5) - a^2 (b^2-c^2)^2 (12 b^4+11 b^3 c+6 b^2 c^2+11 b c^3+12 c^4) + a (b-c)^4 (b+c)^3 (4 b^2+3 b c+4 c^2) + (b^2-c^2)^4 (5 b^2+4 b c+5 c^2) : :
X(10683) = (3r^2+17rR+24R^2-7s^2)*X[2] - (r^2+5rR+6R^2-s^2)*X[3]

Let ABC be a triangle, and let
Fa = A-Feuerbach point, and define Fb and Fc cyclically
Oa = circumcenter of AFbFc, and define Ob and Oc cyclically.
Then X(10683) = centroid of OaObOc, and X(10683) lies on Euler line of triangle ABC. See Tran Quang Hung and Angel Montesdeoca, 24711 and 24719.

X(10683) lies on these lines: {2,3} (others pending)


X(10684) =  EULER LINE INTERCEPT OF X(194)X(525)

Barycentrics    a^10 (b^2+c^2) - a^8 (2 b^4+b^2 c^2+2 c^4) + a^6 (b^6+b^4c^2+b^2 c^4+c^6) - a^4 b^2 c^2 (b^4-b^2 c^2+c^4) + b^4 c^4 (b^2-c^2)^2 : :

In the plane of a triangle ABC, let
Ω1 = 1st Brocard point
Ω2 = 2nd Brocard point
T1 = cevian triangle of Ω1
Ω1* = T1-isogonal conjugate of Ω1
T2 = cevian triangle of Ω2
Ω2* = T2-isogonal conjugate of Ω2.

Then X(10684) = Ω1Ω1*∩Ω2Ω2*, and X(10684) lies on the Euler line. See Tran Quang Hung and Angel Montesdeoca, 24712.

X(10684) lies on these lines:
{2,3}, {32,2966}, {194,525}, {327,3734}, {1632,2882}, {1975,2421}, {5641,7775}


X(10685) =  POINT BECRUX 15

Barycentrics    a^10 (b^2+c^2) - a^8 (3 b^4+b^2 c^2+3 c^4) + 2 a^6 (b^6+b^4 c^2+b^2 c^4+c^6) + a^4 (-2 b^6 c^2+b^4 c^4-2 b^2 c^6) - a^2 b^2 c^2 (b^2-c^2)^2 (b^2+c^2)b^4 c^4 (b^2-c^2)^2 : :

In the plane of a triangle ABC, let
Ω1 = 1st Brocard point
Ω2 = 2nd Brocard point
T1 = anticevian triangle of Ω1
Ω1* = T1-isogonal conjugate of Ω1
T2 = anticevian triangle of Ω2
Ω2* = T2-isogonal conjugate of Ω2.

Then X(10685) = Ω1Ω1*∩Ω2Ω2*, and X(10684) lies on the Euler line. See Tran Quang Hung and Angel Montesdeoca, 24717.

X(10685) lies on these lines: {2,3} et al


X(10686) =  X(4)X(2575)∩X(1113)X(10287)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) ((2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^14-3 a^12 b^2+a^10 b^4+5 a^8 b^6-5 a^6 b^8-a^4 b^10+3 a^2 b^12-b^14-3 a^12 c^2+9 a^10 b^2 c^2-9 a^8 b^4 c^2+a^6 b^6 c^2+6 a^4 b^8 c^2-6 a^2 b^10 c^2+2 b^12 c^2+a^10 c^4-9 a^8 b^2 c^4+10 a^6 b^4 c^4-5 a^4 b^6 c^4+3 a^2 b^8 c^4+5 a^8 c^6+a^6 b^2 c^6-5 a^4 b^4 c^6-b^8 c^6-5 a^6 c^8+6 a^4 b^2 c^8+3 a^2 b^4 c^8-b^6 c^8-a^4 c^10-6 a^2 b^2 c^10+3 a^2 c^12+2 b^2 c^12-c^14)+a^2 (a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-2 a^12 b^2 c^2+2 a^10 b^4 c^2-8 a^8 b^6 c^2+17 a^6 b^8 c^2-14 a^4 b^10 c^2+4 a^2 b^12 c^2-5 a^12 c^4+2 a^10 b^2 c^4+10 a^8 b^4 c^4-10 a^6 b^6 c^4+a^4 b^8 c^4+8 a^2 b^10 c^4-6 b^12 c^4+9 a^10 c^6-8 a^8 b^2 c^6-10 a^6 b^4 c^6+8 a^4 b^6 c^6-7 a^2 b^8 c^6+8 b^10 c^6-5 a^8 c^8+17 a^6 b^2 c^8+a^4 b^4 c^8-7 a^2 b^6 c^8-6 b^8 c^8-5 a^6 c^10-14 a^4 b^2 c^10+8 a^2 b^4 c^10+8 b^6 c^10+9 a^4 c^12+4 a^2 b^2 c^12-6 b^4 c^12-5 a^2 c^14+c^16) J) : : , where J = |OH|/R

Let H be the orthocenter of a triangle ABC. Let La be the Brocard axis of AHX(1113), and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(10686). See Tran Quang Hung and Peter Moses, 24723.

X(10686) lies on these lines: {4,2575}, {52,520}, {1113, 10287}


X(10687) =  X(4)X(2574)∩X(1114)X(10288)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) ((2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^14-3 a^12 b^2+a^10 b^4+5 a^8 b^6-5 a^6 b^8-a^4 b^10+3 a^2 b^12-b^14-3 a^12 c^2+9 a^10 b^2 c^2-9 a^8 b^4 c^2+a^6 b^6 c^2+6 a^4 b^8 c^2-6 a^2 b^10 c^2+2 b^12 c^2+a^10 c^4-9 a^8 b^2 c^4+10 a^6 b^4 c^4-5 a^4 b^6 c^4+3 a^2 b^8 c^4+5 a^8 c^6+a^6 b^2 c^6-5 a^4 b^4 c^6-b^8 c^6-5 a^6 c^8+6 a^4 b^2 c^8+3 a^2 b^4 c^8-b^6 c^8-a^4 c^10-6 a^2 b^2 c^10+3 a^2 c^12+2 b^2 c^12-c^14)-a^2 (a^14 b^2-5 a^12 b^4+9 a^10 b^6-5 a^8 b^8-5 a^6 b^10+9 a^4 b^12-5 a^2 b^14+b^16+a^14 c^2-2 a^12 b^2 c^2+2 a^10 b^4 c^2-8 a^8 b^6 c^2+17 a^6 b^8 c^2-14 a^4 b^10 c^2+4 a^2 b^12 c^2-5 a^12 c^4+2 a^10 b^2 c^4+10 a^8 b^4 c^4-10 a^6 b^6 c^4+a^4 b^8 c^4+8 a^2 b^10 c^4-6 b^12 c^4+9 a^10 c^6-8 a^8 b^2 c^6-10 a^6 b^4 c^6+8 a^4 b^6 c^6-7 a^2 b^8 c^6+8 b^10 c^6-5 a^8 c^8+17 a^6 b^2 c^8+a^4 b^4 c^8-7 a^2 b^6 c^8-6 b^8 c^8-5 a^6 c^10-14 a^4 b^2 c^10+8 a^2 b^4 c^10+8 b^6 c^10+9 a^4 c^12+4 a^2 b^2 c^12-6 b^4 c^12-5 a^2 c^14+c^16) J) : : , where J = |OH|/R

Let H be the orthocenter of a triangle ABC. Let La be the Brocard axis of AHX(1114), and define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(10687). See Tran Quang Hung and Peter Moses, 24723.

X(10686) lies on these lines: {4,2574}, {52,520}, {1114, 10288}


X(10688) =  X(30)X(6344)∩X(265)X(6000)

Barycentrics    (b^10-3 b^8 c^2+2 b^6 c^4+2 b^4 c^6-3 b^2 c^8+c^10+(-2 b^8+6 b^6 c^2-8 b^4 c^4+6 b^2 c^6-2 c^8) a^2+(-b^6-2 b^4 c^2-2 b^2 c^4-c^6) a^4+(5 b^4+3 b^2 c^2+5 c^4) a^6+(-4 b^2-4 c^2) a^8+a^10) / (a^2(4SA^2-b^2c^2) : :

Let ABC be a triangle with orthocenter H and circumcenter O, and let
da = reflection of Euler line of triangle AOH in line OH, and define db and dc cyclically
A' = db∩dc, and define B' and C' cyclically.
The triangle A'B'C' is perspective to ABC, and the perspector is X(10688). See Tran Quang Hung and Angel Montesdeoca, 24745.

X(10688) lies on these lines: {30,6344}, {265,6000}, {476,1141}


X(10689) =  POINT BECRUX 16

Trilinears    (-2*cos(A)*cos(B-C)+1)*((2* cos(4*A)-3)*cos(B-C)+(-2*cos( A)-2*cos(3*A))*cos(2*(B-C))- cos(3*A)-cos(5*A)+5*cos(A)) : :

Let P be a point in the plane of a triangle ABC, and let
A'B'C' = pedal triangle of P
A'' = orthogonal projection of A' on the Euler line of ABC, and define B'' and C'' cyclically
La = reflection of BC in A'A'', and define Lb and Lc cyclically

Then ABC and A*B*C* are parallelogic. If P = u : v : w, then the (A*B*C*,ABC)-parallelogic center for P is the point

Q(P) = -2*(a^6-a^4*b^2-a^4*c^2-a^2*b^ 4+3*a^2*b^2*c^2-a^2*c^4+b^6-b^ 4*c^2-b^2*c^4+c^6)^2*a*b*c*u-( a^2-c^2)*c*(a^2-a*c-b^2+c^2)*( a^2+a*c-b^2+c^2)*(a^8-4*a^6*c^ 2-a^4*b^4+6*a^4*c^4-2*a^2*b^6+ 4*a^2*b^4*c^2-4*a^2*c^6+2*b^8- 2*b^6*c^2-b^4*c^4+c^8)*v-(a^2- b^2)*b*(a^2-a*b+b^2-c^2)*(a^2+ a*b+b^2-c^2)*(a^8-4*a^6*b^2+6* a^4*b^4-a^4*c^4-4*a^2*b^6+4*a^ 2*b^2*c^4-2*a^2*c^6+b^8-b^4*c^ 4-2*b^2*c^6+2*c^8)*w : :

The appearance of (i,j) in the following list means that the X(i) = Q(X(j)): (3,3258), (4,10689), (5,10690). See Antreas Hatzipolakis and César Lozada, 24747.

X(10689) lies on this line: {113,403}


X(10690) =  X(511)X(3233)∩X(1154)X(10096)

Trilinears    (-cos(2*A)+5*cos(4*A)+cos(6*A) -13/2)*cos(B-C)+(5*cos(A)-cos( 3*A)-cos(5*A))*cos(2*(B-C)) +(cos(2*A)+cos(4*A)+1/2)*cos( 3*(B-C))-2*cos(5*A)+4*cos(A)- 4*cos(3*A) : :

See X(10689).

X(10690) lies on these lines: {511,3233}, {1154,10096}


X(10691) =  COMPLEMENT OF X(428)

Barycentrics    (a^2-b^2-c^2) (2 a^4+3 a^2 b^2+b^4+3 a^2 c^2-2 b^2 c^2+c^4) : :
X(10691) = SA SB SC X[2] - S^2 SW X[3]
X(10691) = X[1885] + 5 X[3522] = 7 X[3523] - X[3575] = 4 X[140] - X[6756] = 7 X[3526] - X[7553] = 5 X[631] - X[7576] = X[428] + 3 X[7667] = X[428] - 6 X[7734] = X[7667] + 2 X[7734] = 5 X[631] - 2 X[9825] = 3 X[7734] - X[10128] = 3 X[7667] + 2 X[10128] = X[6240] - 13 X[10299].

Let ABC be a triangle and A'B'C' the pedal triangle of O = X(3). Let
A1 = orthogonal projection of A on OA', and define OB' and OC' cyclically
A2 = the orthogonal projection of A' on OA, and define OB and OC cyclically
M1 = midpoint of segment A1A2, and define M2 and M3 cyclically.
Then X(10691) = centroid of M1M2M3, and X(10691) lies on the Euler line of ABC. See Antreas Hatzipolakis and Peter Moses, 24756.

X(10691) lies on these lines:
{2,3}, {216,9300}, {305,7767}, {524,6665}, {539,5447}, {577, 5306}, {1038,5434}, {1040,3058}, {1503,3819}, {1611,2549}, {3564, 3917}, {5268,7354}, {5272,6284}

X(10691) = midpoint of X(2) and X(7667)
X(10691) = reflection of X(i) in X(j) for these (i,j): (2, 7734), (428, 10128), (6756, 10127), (7576, 9825), (10127, 140)
X(10691) = complement of X(428)
X(10691) = anticomplement X(10128)
X(10691) = X(i)-complementary conjugate of X(j) for these (i,j): (3108,226), (7953,8062)
X(10691) = centroid of 3rd pedal triangle of X(3)
X(10691) = inverse-in-complement-of-polar-circle of X(23)
X(10691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,20,7714), (2,376,9909), (2, 428,10128), (2,1370,5064), (2, 5064,5), (2,7714,5020), (3,1368, 6676), (3,6643,6823), (3,7386, 1368), (3,10300,5159), (427, 7485,140), (465,466,441), (548, 6677,22), (1368,6676,5159), ( 1368,7386,10300).(1370,7484,5) ,(5064,7484,2), (6676,10300, 1368).

X(10692) =  X(1)X(915)∩X(84)X(224)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4-a^3 c+a^2 b c+a b^2 c-b^3 c-a^2 c^2-b^2 c^2+a c^3+b c^3) (a^4-a^3 b-a^2 b^2+a b^3+a^2 b c+b^3 c-2 a^2 c^2+a b c^2-b^2 c^2-b c^3+c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6+2 a^4 b c+2 a^3 b^2 c-2 a^2 b^3 c-2 a b^4 c-3 a^4 c^2+2 a^3 b c^2-2 a^2 b^2 c^2+2 a b^3 c^2+b^4 c^2-2 a^2 b c^3+2 a b^2 c^3+3 a^2 c^4-2 a b c^4+b^2 c^4-c^6) : :

Let ABC be a triangle and IaIbIc the antipedal triangle of I (i.e., the excentral triangle). Let
A' = reflections of Ia in BC, and define B' and C' cyclically
Ma = midpoint of segment AA', and define Mb and Mc cyclically.

Then ABC and A'B'C' are cyclologic, and
X(10692) = point of concurrence of circumcircles of AB'C', BC'A', CA'B'
X(80) = point of concurrence of circumcircles of A'BC, B'CA, C'AB.

Also, ABC and MaMbMc are cyclologic; let
X(80) = point of concurrence of circumcircles of AMbMc, BMcMa, CMaMb
X(10693) = point of concurrence of circumcircles of MaBC, MbCA, McAB.

See Antreas Hatzipolakis and Peter Moses, 24768.

Let A'B'C' be the excentral triangle. Let L, M, N be lines through A', B', C', respectively, parallel to line X(4)X(46). Let L' be the reflection of L in BC, let M' be the reflection of M in CA, and let N' be the reflection of N in AB. The lines L', M', N' concur in X(10692). (Randy Hutson, December 10, 2016)

X(10692) lies on the cubic K269 and these lines:
{1,915}, {84,224}, {155,3435}, { 913,5190} et al


X(10693) =  MIDPOINT OF X(3448) AND X(3869)

Trilinears    (b + c)/[b^4 + c^4 - a^4 + abc(b + c - a) - 2b^2c^2] : :
Barycentrics    a (b+c) (a^4-2 a^2 b^2+b^4+a^2 b c+a b^2 c-a b c^2-c^4) (a^4-b^4+a^2 b c-a b^2 c-2 a^2 c^2+a b c^2+c^4) : :

See X(10692).

X(10693) lies on Jerabek hyperbola, the cubics K685 and K720, and these lines:
{3,191}, {4,2778}, {54,5494}, { 65,125}, {67,3827}, {69,2836}, { 72,7068}, {73,2632}, {74,2766}, { 110,960}, {265,517}, {518,895} et al

X(10693) = midpoint of X(3448) & X(3869)
X(10693) = reflection of X(i) in X(j) for these (i,j): (65, 125), (110, 960)
X(10693) = X(3724)-crossconjugate of X(37)
X(10693) = X(3)-crosssum of X(2948)
X(10693) = X(i)-isoconjugate of X(j) for these {i,j}: {1,1325}, {58,5080}, {162,2850}
X(10693) = X(3)-crosssum of X(2948)
X(10693) = trilinear pole of line X(647) X(2092)
X(10693) = isogonal conjugate of X(1325)
X(10693) = antigonal image of X(65)

X(10694) =  EULER INTERCEPT OF X(880)X(1502)

Barycentrics    b^4 c^4 (b^2-c^2)^2+2 b^2 c^2 (b^2-c^2)^2 (b^2+c^2) a^2+b^2 c^2 (b^4+b^2 c^2+c^4) a^4-(b^6+b^4 c^2+b^2 c^4+c^6)a^6-b^2 c^2 a^8+(b^2+c^2) a^10 : :

In the plane of a triangle ABC, let
Ω1 = 1st Brocard point
Ω2 = 2nd Brocard point
Ω1* = Orion transform of Ω1
Ω2* = Orion transform of Ω2.
X(10694) = Ω1Ω1*∩Ω2Ω2*, and X(10694) lies on the Euler line of ABC. (Orion transform is define just before X(2055).) See Tran Quang Hung and Angel Montesdeoca, 24769.

X(10694) lies on these lines: {2,3}, {880,1502}



leftri  Reflections of circumcircle-points in the incenter: X(10695) - X(10705)  rightri

This preamble and centers X(10695)-X(10705) were contributed by Clark Kimberling and Peter Moses, November 10, 2016.

Suppose that P is a point on the circumcircle of a triangle ABC, and let

P' = reflection of P in the incenter, I, of ABC.
Pc = complement of P
Pa = anticomplement of P
P'' = reflection of X(8) in Pc.

Then

P' = PI∩PcX(8)
P' = reflection of X(8) in Pc
P' = midpoint of X(145) and Pa, where X(145) = anticomplement of anticomplement of I.

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(1): (74,7978), (98,7970), (99,7983), (100,1320), (101,10695), (102,10696), (103,10697), (104,10698), (105,10699), (106,10700), (107,10701), (108,10702), (109,10703), (110,7984), (111, 10704), (112,10705), (1113, 2102), (1114,2103)

underbar

X(10695) =  REFLECTION OF X(101) IN X(1)

Barycentrics    a (a^4-2 a^3 b+a^2 b^2-2 a b^3+2 b^4-2 a^3 c+3 a^2 b c+a b^2 c-2 b^3 c+a^2 c^2+a b c^2-2 a c^3-2 b c^3+2 c^4 : :
X(10695) = 3 X[101] - 2 X[1282] = 3 X[1] - X[1282] = 2 X[118] - 3 X[5603] = 5 X[3616] - 4 X[6710] = 3 X[5657] - 4 X[6712]

X(10695) lies on these lines:
{1,41}, {8,116}, {103,517}, {118,5603}, {145,150}, {544,3241}, {1022,1280}, {1362,2099}, {1482,2808}, {2098,3022}, {2772,7978}, {2774,7984}, {2784,3244}, {2786,7983}, {2801,3243}, {2802,9451}, {2810,3242}, {3041,5289}, {3616,6710}, {4712,4752}, {4845,5048}, {5657,6712}

X(10695) = midpoint of X(145) and X(150)
X(10695) = reflection of X(i) in X(j) for these (i,j): (8, 116), (101, 1)

X(10696) =  REFLECTION OF X(102) IN X(1)

Barycentrics    a (a^9-3 a^8 b-a^7 b^2+7 a^6 b^3-3 a^5 b^4-3 a^4 b^5+5 a^3 b^6-3 a^2 b^7-2 a b^8+2 b^9-3 a^8 c+11 a^7 b c-9 a^6 b^2 c-11 a^5 b^3 c+23 a^4 b^4 c-11 a^3 b^5 c-7 a^2 b^6 c+11 a b^7 c-4 b^8 c-a^7 c^2-9 a^6 b c^2+28 a^5 b^2 c^2-20 a^4 b^3 c^2-17 a^3 b^4 c^2+31 a^2 b^5 c^2-10 a b^6 c^2-2 b^7 c^2+7 a^6 c^3-11 a^5 b c^3-20 a^4 b^2 c^3+46 a^3 b^3 c^3-21 a^2 b^4 c^3-11 a b^5 c^3+10 b^6 c^3-3 a^5 c^4+23 a^4 b c^4-17 a^3 b^2 c^4-21 a^2 b^3 c^4+24 a b^4 c^4-6 b^5 c^4-3 a^4 c^5-11 a^3 b c^5+31 a^2 b^2 c^5-11 a b^3 c^5-6 b^4 c^5+5 a^3 c^6-7 a^2 b c^6-10 a b^2 c^6+10 b^3 c^6-3 a^2 c^7+11 a b c^7-2 b^2 c^7-2 a c^8-4 b c^8+2 c^9) : :
X(10696) = 2 X[124] - 3 X[5603] = 5 X[3616] - 4 X[6711] = 3 X[5657] - 4 X[6718]

X(10696) lies on these lines:
{1,102}, {8,117}, {84,1320}, {109,517}, {124,5603}, {145,151}, {515,1897}, {1361,2098}, {1364,2099}, {1482,1854}, {1795,5903}, {2773,7978}, {2779,7984}, {2792,7983}, {2816,5882}, {3042,5289}, {3616,6711}, {5657,6718}

X(10696) = midpoint of X(145) and X(151)
X(10696) = reflection of X(i) in X(j) for these (i,j): (8, 117), (102, 1)

X(10697) =  REFLECTION OF X(103) IN X(1)

Barycentrics    a (a^7-3 a^6 b+4 a^4 b^3+a^3 b^4-3 a^2 b^5-2 a b^6+2 b^7-3 a^6 c+9 a^5 b c-6 a^4 b^2 c-2 a^3 b^3 c-3 a^2 b^4 c+9 a b^5 c-4 b^6 c-6 a^4 b c^2+2 a^3 b^2 c^2+6 a^2 b^3 c^2-2 a b^4 c^2+4 a^4 c^3-2 a^3 b c^3+6 a^2 b^2 c^3-10 a b^3 c^3+2 b^4 c^3+a^3 c^4-3 a^2 b c^4-2 a b^2 c^4+2 b^3 c^4-3 a^2 c^5+9 a b c^5-2 a c^6-4 b c^6+2 c^7) : :
X(10697) = 2 X[116] - 3 X[5603] = 3 X[5657] - 4 X[6710] = 5 X[3616] - 4 X[6712]

X(10697) lies on these lines:
{1,103}, {8,118}, {101,517}, {116,5603}, {145,152}, {516,664}, {1320,2801}, {1361,2099}, {1362,2098}, {1419,2823}, {1482,2808}, {2772,7984}, {2774,7978}, {2784,4301}, {2809,7982}, {2820,4895}, {3616,6712}, {5657,6710}

X(10697) = midpoint of X(145) and X(152)
X(10697) = reflection of X(i) in X(j) for these (i,j): (8, 118), (103, 1)

X(10698) =  REFLECTION OF X(104) IN X(1)

Barycentrics    a (a^6-3 a^5 b+6 a^3 b^3-3 a^2 b^4-3 a b^5+2 b^6-3 a^5 c+9 a^4 b c-8 a^3 b^2 c-6 a^2 b^3 c+11 a b^4 c-3 b^5 c-8 a^3 b c^2+18 a^2 b^2 c^2-8 a b^3 c^2-2 b^4 c^2+6 a^3 c^3-6 a^2 b c^3-8 a b^2 c^3+6 b^3 c^3-3 a^2 c^4+11 a b c^4-2 b^2 c^4-3 a c^5-3 b c^5+2 c^6) : :
X(10698) = 3 X[104] - 2 X[1768] = 3 X[1] - X[1768] =2 X[11] - 3 X[5603] =4 X[3035] - 3 X[5657] =4 X[3036] - 5 X[5818] =3 X[1699] - 2 X[6246] =5 X[3616] - 4 X[6713] =4 X[6702] - 5 X[8227] =5 X[5734] - X[9803] =3 X[3241] + X[9809] =3 X[1699] - X[9897] =4 X[1387] - 5 X[10595]

X(10698) lies on these lines:
{1,104}, {3,5330}, {4,145}, {8,119}, {11,2099}, {30,10031}, {40,214}, {80,946}, {100,517}, {390,6938}, {515,7972}, {519,1519}, {944,1317}, {962,5840}, {1006,3877}, {1012,10247}, {1145,5730}, {1387,6833}, {1484,6831}, {1512,4867}, {1532,5844}, {1699,6246}, {2136,2802}, {2771,7984}, {2783,7983}, {2801,3243}, {2932,10306}, {3035,5289}, {3036,5818}, {3149,8148}, {3241,9809}, {3616,6713}, {3622,6977}, {3940,6969}, {4345,5768}, {4861,5887}, {5048,6001}, {5180,5841}, {5441,5882}, {5554,6975}, {5690,6949}, {5697,10087}, {5734,6845}, {5901,6952}, {5903,10090}, {6702,8227}, {6934,10609}, {6950,10246}, {6956,9952}, {7962,7966}, {7978,8674}, {8192,9913}

X(10698) = midpoint of X(i) and X(j) for these {i,j}: {145, 153}, {962, 6224}, {6326, 7982}
X(10698) = reflection of X(i) in X(j) for these (i,j): (4, 1537), (8, 119), (40, 214), (80, 946), (100, 6265), (104, 1), (944, 1317), (1320, 1482), (9897, 6246)
X(10698) = {X(1699), X(9897)}-harmonic conjugate of X(6246)

X(10699) =  REFLECTION OF X(105) IN X(1)

Barycentrics    a (a^4-3 a^3 b+3 a^2 b^2-3 a b^3+2 b^4-3 a^3 c+3 a^2 b c+a b^2 c-3 b^3 c+3 a^2 c^2+a b c^2+2 b^2 c^2-3 a c^3-3 b c^3+2 c^4) : :
X(10699) = 3 X[105] - 2 X[5540] = 3 X[1] - X[5540] = 2 X[5511] - 3 X[5603] = 5 X[3616] - 4 X[6714]

X(10699) lies on these lines:
{1,41}, {7,528}, {8,120}, {100,3675}, {517,840}, {518,644}, {1721,9519}, {2098,3021}, {2775,7978}, {2795,7983}, {2820,4895}, {2835,4319}, {2836,5919}, {3039,5289}, {3243,4919}, {3616,6714}, {5511,5603}

X(10699) = reflection of X(i) in X(j) for these (i,j): (8,120), (105,1)

X(10700) =  REFLECTION OF X(106) IN X(1)

Barycentrics    a (a^3-3 a^2 b-2 a b^2+2 b^3-3 a^2 c+13 a b c-4 b^2 c-2 a c^2-4 b c^2+2 c^3) : :
X(10700) = 3 X[106] - 2 X[1054] = 3 X[1] - X[1054] = 2 X[5510] - 3 X[5603] = 5 X[3616] - 4 X[6715]

X(10700) lies on these lines:
{1,88}, {8,121}, {58,6095}, {517,1293}, {519,3699}, {990,9519}, {1357,2099}, {1364,2098}, {2087,4752}, {2776,7978}, {2796,7983}, {2810,3242}, {2842,7984}, {3038,5289}, {3616,6715}, {4653,5919}, {5497,10222}, {5510,5603}, {5854,6788}, {7004,7962}

X(10700) = reflection of X(i) in X(j) for these (i,j): (8,121), (106,1)

X(10701) =  REFLECTION OF X(107) IN X(1)

Barycentrics    a^13-a^12 b-a^11 b^2+a^10 b^3-8 a^9 b^4+2 a^8 b^5+18 a^7 b^6-2 a^6 b^7-11 a^5 b^8-a^4 b^9-a^3 b^10+a^2 b^11+2 a b^12-a^12 c+a^10 b^2 c+2 a^8 b^4 c-2 a^6 b^6 c-a^4 b^8 c+a^2 b^10 c-a^11 c^2+a^10 b c^2+17 a^9 b^2 c^2-5 a^8 b^3 c^2-18 a^7 b^4 c^2+2 a^6 b^5 c^2-14 a^5 b^6 c^2+6 a^4 b^7 c^2+19 a^3 b^8 c^2-3 a^2 b^9 c^2-3 a b^10 c^2-b^11 c^2+a^10 c^3-5 a^8 b^2 c^3+2 a^6 b^4 c^3+6 a^4 b^6 c^3-3 a^2 b^8 c^3-b^10 c^3-8 a^9 c^4+2 a^8 b c^4-18 a^7 b^2 c^4+2 a^6 b^3 c^4+50 a^5 b^4 c^4-10 a^4 b^5 c^4-18 a^3 b^6 c^4+2 a^2 b^7 c^4-6 a b^8 c^4+4 b^9 c^4+2 a^8 c^5+2 a^6 b^2 c^5-10 a^4 b^4 c^5+2 a^2 b^6 c^5+4 b^8 c^5+18 a^7 c^6-2 a^6 b c^6-14 a^5 b^2 c^6+6 a^4 b^3 c^6-18 a^3 b^4 c^6+2 a^2 b^5 c^6+14 a b^6 c^6-6 b^7 c^6-2 a^6 c^7+6 a^4 b^2 c^7+2 a^2 b^4 c^7-6 b^6 c^7-11 a^5 c^8-a^4 b c^8+19 a^3 b^2 c^8-3 a^2 b^3 c^8-6 a b^4 c^8+4 b^5 c^8-a^4 c^9-3 a^2 b^2 c^9+4 b^4 c^9-a^3 c^10+a^2 b c^10-3 a b^2 c^10-b^3 c^10+a^2 c^11-b^2 c^11+2 a c^12 : :
X(10701) = 2 X[133] - 3 X[5603] = 2 X[3184] - 3 X[5731] - 5 X[3616] - 4 X[6716] = X[5667] - 3 X[7967]

X(10701) lies on these lines:
{1,107}, {8,122}, {133,5603}, {515,10152}, {517,1294}, {944,2777}, {1320,2803}, {2098,7158}, {2099,3324}, {2797,7983}, {2816,5882}, {3184,5731}, {3241,9530}, {3616,6716}, {5667,7967}, {7984,9033}

X(10701) = reflection of X(i) in X(j) for these (i,j): (8,122), (107,1)

X(10702) =  REFLECTION OF X(108) IN X(1)

Barycentrics    a (a-b-c) (a^8-a^7 b-3 a^6 b^2+a^5 b^3+a^4 b^4+a^3 b^5+3 a^2 b^6-a b^7-2 b^8-a^7 c+7 a^6 b c-a^5 b^2 c-9 a^4 b^3 c+5 a^3 b^4 c-3 a^2 b^5 c-3 a b^6 c+5 b^7 c-3 a^6 c^2-a^5 b c^2+16 a^4 b^2 c^2-6 a^3 b^3 c^2-11 a^2 b^4 c^2+7 a b^5 c^2-2 b^6 c^2+a^5 c^3-9 a^4 b c^3-6 a^3 b^2 c^3+22 a^2 b^3 c^3-3 a b^4 c^3-5 b^5 c^3+a^4 c^4+5 a^3 b c^4-11 a^2 b^2 c^4-3 a b^3 c^4+8 b^4 c^4+a^3 c^5-3 a^2 b c^5+7 a b^2 c^5-5 b^3 c^5+3 a^2 c^6-3 a b c^6-2 b^2 c^6-a c^7+5 b c^7-2 c^8 : :
X(10702) = 5 X[3616] - 4 X[6717]

X(10702) lies on these lines:
{1,102}, {8,123}, {517,1295}, {944,1317}, {1320,2804}, {1359,2099}, {1419,2823}, {2778,3057}, {2798,7983}, {2850,7984}, {3616,6717}

X(10702) = reflection of X(i) in X(j) for these (i,j): (8,123), (108,1)

X(10703) =  REFLECTION OF X(109) IN X(1)

Barycentrics    a (a-b-c) (a^5-a^4 b-a^3 b^2+3 a^2 b^3-2 b^5-a^4 c+3 a^3 b c-3 a^2 b^2 c-3 a b^3 c+4 b^4 c-a^3 c^2-3 a^2 b c^2+6 a b^2 c^2-2 b^3 c^2+3 a^2 c^3-3 a b c^3-2 b^2 c^3+4 b c^4-2 c^5) : :
X(10703) = 2 X[117] - 3 X[5603] = 3 X[5657] - 4 X[6711] = 5 X[3616] - 4 X[6718]

X(10703) lies on these lines:
{1,104}, {8,124}, {11,6788}, {33,1845}, {102,517}, {117,5603}, {212,3899}, {1320,3738}, {1331,3869}, {1361,2099}, {1364,2098}, {1482,1854}, {1697,3722}, {2773,7984}, {2779,7727}, {2785,7983}, {2817,7982}, {2835,4319}, {3040,5289}, {3057,5497}, {3616,6718}, {5657,6711}

X(10703) = reflection of X(i) in X(j) for these (i,j): (8,124), (109,1)

X(10704) =  REFLECTION OF X(111) IN X(1)

Barycentrics    a (a^6-a^5 b-5 a^4 b^2+a^3 b^3-4 a^2 b^4+2 a b^5+2 b^6-a^5 c+a^3 b^2 c+2 a b^4 c-5 a^4 c^2+a^3 b c^2+25 a^2 b^2 c^2-5 a b^3 c^2-6 b^4 c^2+a^3 c^3-5 a b^2 c^3-4 a^2 c^4+2 a b c^4-6 b^2 c^4+2 a c^5+2 c^6) : :
X(10704) = 2 X[5512] - 3 X[5603] = 5 X[3616] - 4 X[6719]

X(10704) lies on these lines:
{1,111}, {8,126}, {517,1296}, {543,3241}, {1320,2805}, {2098,6019}, {2099,3325}, {2780,7978}, {2854,3242}, {3616,6719}, {5512,5603}

X(10704) = reflection of X(i) in X(j) for these (i,j): (8,126), (111,1)

X(10705) =  REFLECTION OF X(112) IN X(1)

Barycentrics    a (a^10-a^9 b-a^8 b^2+a^7 b^3+a^6 b^4+a^5 b^5-a^4 b^6-a^3 b^7-2 a^2 b^8+2 b^10-a^9 c+a^7 b^2 c+a^5 b^4 c-a^3 b^6 c-a^8 c^2+a^7 b c^2-a^6 b^2 c^2-3 a^5 b^3 c^2+a^4 b^4 c^2+a^3 b^5 c^2+3 a^2 b^6 c^2+a b^7 c^2-2 b^8 c^2+a^7 c^3-3 a^5 b^2 c^3+a^3 b^4 c^3+a b^6 c^3+a^6 c^4+a^5 b c^4+a^4 b^2 c^4+a^3 b^3 c^4-2 a^2 b^4 c^4-2 a b^5 c^4+a^5 c^5+a^3 b^2 c^5-2 a b^4 c^5-a^4 c^6-a^3 b c^6+3 a^2 b^2 c^6+a b^3 c^6-a^3 c^7+a b^2 c^7-2 a^2 c^8-2 b^2 c^8+2 c^10) : :
X(10705) = 2 X[132] - 3 X[5603] = 5 X[3616] - 4 X[6720]

X(10705) lies on these lines:
{1,112}, {8,127}, {132,5603}, {517,1297}, {944,2794}, {1320,2806}, {2098,6020}, {2099,3320}, {2781,3242}, {2799,7983}, {3616,6720}, {7984,9517}

X(10705) = reflection of X(i) in X(j) for these (i,j): (8,127), (112,1)
X(10705) = {X(13313),X(13314)}-harmonic conjugate of X(112)


leftri  Reflections of circumcircle-points in the centroid: X(10706) - X(10720)  rightri

This preamble and centers X(10706)-X(10720) were contributed by Clark Kimberling and Peter Moses, November 8, 2016.

Suppose that P is a point on the circumcrcle of a triangle ABC, and let

P' = reflection of P in the centroid, G, of ABC
Pc = complement of P
Pa = anticomplement of P.

Then

P' = midpoint of G and Pa
P' = reflection of G in Pc
P' = {Pc, Pa}-harmonic conjugate of P.

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(2): (74,10702), (98,6054), (99,671), (100,10707), (101,10708), (102,10709), (103,10710), (104,10711), (105,10712), (106,10713), (107,10714), (108,10715), (109,10716), (111,10717), (112,10718), (1113,10719), (1114,10720)

underbar

X(10706) =  REFLECTION OF X(74) IN X(2)

Barycentrics    a^10+3 a^8 b^2-13 a^6 b^4+11 a^4 b^6-2 b^10+3 a^8 c^2+11 a^6 b^2 c^2-7 a^4 b^4 c^2-13 a^2 b^6 c^2+6 b^8 c^2-13 a^6 c^4-7 a^4 b^2 c^4+26 a^2 b^4 c^4-4 b^6 c^4+11 a^4 c^6-13 a^2 b^2 c^6-4 b^4 c^6+6 b^2 c^8-2 c^10 : :
X(10706) = X[74] - 4 X[113] = 2 X[113] + X[146] = X[74] + 2 X[146] = X[399] + 2 X[1539] = 2 X[125] - 3 X[3545] = X[3448] - 3 X[3839] = X[382] + 2 X[5609] = 4 X[597] - 3 X[5622] = 3 X[3524] - 4 X[5972] = 5 X[74] - 8 X[6699] = 5 X[2] - 4 X[6699] = 5 X[113] - 2 X[6699] = 5 X[146] + 4 X[6699] = X[110] + 2 X[7728] = 3 X[5055] - X[10620].

X(10706) lies on these lines:
{2,74}, {4,542}, {30,110}, {98,5465}, {125,3545}, {265,3845}, {376,2777}, {381,5640}, {382,5609}, {399,1539}, {519,7978}, {524,1514}, {530,1524}, {531,1525}, {543,1561}, {597,5622}, {599,2781}, {690,6054}, {1499,1551}, {1511,3534}, {1513,9759}, {1555,3849}, {2883,8718}, {3163,6794}, {3448,3839}, {3524,5972}, {3543,9143}, {3582,10081}, {3584,10065}, {3656,7984}, {5055,10620}, {5066,10264}, {7865,9984}, {8703,10272}

X(10706) = midpoint of X(i) and X(j) for these {i,j}: {2, 146}, {399, 3830}, {3543, 9143}, {5655, 7728}
X(10706) = reflection of X(i) in X(j) for these (i,j): (2, 113}, {74, 2}, {98, 5465}, {110, 5655}, {265, 3845}, {376, 5642}, {3534, 1511}, {3830, 1539}, {7984, 3656}, {8703, 10272}, {9140, 381}, {10264, 5066)
X(10706) = {X(113),X(146)}-harmonic conjugate of X(74)
X(10706) = centroid of X(30)-Fuhrmann triangle


X(10707) =  REFLECTION OF X(100) IN X(2)

Barycentrics    a^3-a^2 b+2 a b^2-2 b^3-a^2 c-3 a b c+2 b^2 c+2 a c^2+2 b c^2-2 c^3 : :
X(10707) = 4 X[11] - X[100], 2 X[11] + X[149], X[100] + 2 X[149], 2 X[80] + X[1320], X[104] - 4 X[1484], 5 X[100] - 8 X[3035], 5 X[2] - 4 X[3035], 5 X[11] - 2 X[3035], 5 X[149] + 4 X[3035], X[1156] + 2 X[3254], 2 X[119] - 3 X[3545], X[153] - 3 X[3839], X[3935] - 4 X[5087], 14 X[3035] - 5 X[6154], 7 X[100] - 4 X[6154], 7 X[2] - 2 X[6154], 7 X[11] - X[6154], 7 X[149] + 2 X[6154], 3 X[6154] - 7 X[6174], 6 X[3035] - 5 X[6174], 3 X[100] - 4 X[6174], 3 X[11] - X[6174], 3 X[149] + 2 X[6174], 4 X[1387] - X[6224], 2 X[6246] + X[6264], 7 X[100] - 16 X[6667], 7 X[6174] - 12 X[6667], 7 X[3035] - 10 X[6667], 7 X[2] - 8 X[6667], 7 X[11] - 4 X[6667], X[6154] - 4 X[6667], 7 X[149] + 8 X[6667], X[5541] - 4 X[6702], 3 X[3524] - 4 X[6713], X[4511] - 4 X[7743], 2 X[1145] + X[9802], 2 X[1537] + X[9803], 4 X[214] - X[9963], 5 X[3616] - 2 X[10609]

X(10707) lies on these lines:
{1,10031}, {2,11}, {8,4767}, {21,4857}, {30,104}, {80,519}, {88,1647}, {119,3545}, {153,3839}, {214,9963}, {294,5375}, {376,5840}, {381,952}, {404,10199}, {496,5253}, {527,1156}, {535,3583}, {536,4956}, {551,6175}, {599,9024}, {671,2787}, {693,2481}, {900,903}, {1121,6366}, {1145,9802}, {1387,3488}, {1479,2975}, {1537,9803}, {1699,2801}, {1862,5064}, {1992,5848}, {2094,9812}, {2783,6054}, {2802,3679}, {2805,4688}, {2829,3543}, {3120,3315}, {3303,5141}, {3524,6713}, {3582,10090}, {3584,10087}, {3616,10609}, {3746,7504}, {3813,5046}, {3817,5660}, {3869,9614}, {3871,7741}, {3887,4728}, {3913,5154}, {3935,5087}, {4189,9670}, {4442,5211}, {4511,7743}, {4654,5083}, {5225,10529}, {5303,6284}, {5533,10072}, {5541,6702}, {5856,6172}, {6173,7671}, {6246,6264}, {8068,10056}, {8674,9140}

X(10707) = midpoint of X(2) and X(149)
X(10707) = reflection of X(i) in X(j) for these (i,j): {2, 11}, {100, 2}, {5660, 3817}, {10031, 1}
X(10707) = anticomplement X[6174]
X(10707) = crosssum of X(902) and X(1055)
X(10707) = crosspoint of X(903) and X(1121)
X(10707) = crossdifference of every pair of points on line X(665) X(1017)
X(10707) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,149,100), (11,6154,6667), (3058,3829,2)

X(10708) =  REFLECTION OF X(101) IN X(2)

Barycentrics    a^4-a^3 b+2 a b^3-2 b^4-a^3 c+a^2 b c-2 a b^2 c+2 b^3 c-2 a b c^2+2 a c^3+2 b c^3-2 c^4 : :
X(10708) = X[101] - 4 X[116], 2 X[116] + X[150], X[101] + 2 X[150], 2 X[118] - 3 X[3545], X[152] - 3 X[3839], 5 X[101] - 8 X[6710], 5 X[2] - 4 X[6710], 5 X[116] - 2 X[6710], 5 X[150] + 4 X[6710], 3 X[3524] - 4 X[6712]

X(10708) lies on these lines:
{2,101}, {11,4845}, {30,103}, {118,3545}, {152,3839}, {381,2808}, {514,1121}, {527,5179}, {551,2784}, {599,2810}, {671,903}, {1086,6788}, {2774,9140}, {2801,5587}, {2809,3679}, {3524,6712}, {3887,4728}, {5064,5185}

X(10708) = midpoint of X(2) and X(150)
X(10708) = reflection of X(i) in X(j) for these (i,j): (2,116), (101,2)
X(10708) = {X(116),X(150)}-harmonic conjugate of X(101)

X(10709) =  REFLECTION OF X(102) IN X(2)

Barycentrics    a^10-a^9 b+3 a^8 b^2+a^7 b^3-13 a^6 b^4+3 a^5 b^5+11 a^4 b^6-5 a^3 b^7+2 a b^9-2 b^10-a^9 c+3 a^8 b c-9 a^7 b^2 c+7 a^6 b^3 c+15 a^5 b^4 c-21 a^4 b^5 c+a^3 b^6 c+9 a^2 b^7 c-6 a b^8 c+2 b^9 c+3 a^8 c^2-9 a^7 b c^2+20 a^6 b^2 c^2-18 a^5 b^3 c^2-7 a^4 b^4 c^2+27 a^3 b^5 c^2-22 a^2 b^6 c^2+6 b^8 c^2+a^7 c^3+7 a^6 b c^3-18 a^5 b^2 c^3+34 a^4 b^3 c^3-23 a^3 b^4 c^3-9 a^2 b^5 c^3+16 a b^6 c^3-8 b^7 c^3-13 a^6 c^4+15 a^5 b c^4-7 a^4 b^2 c^4-23 a^3 b^3 c^4+44 a^2 b^4 c^4-12 a b^5 c^4-4 b^6 c^4+3 a^5 c^5-21 a^4 b c^5+27 a^3 b^2 c^5-9 a^2 b^3 c^5-12 a b^4 c^5+12 b^5 c^5+11 a^4 c^6+a^3 b c^6-22 a^2 b^2 c^6+16 a b^3 c^6-4 b^4 c^6-5 a^3 c^7+9 a^2 b c^7-8 b^3 c^7-6 a b c^8+6 b^2 c^8+2 a c^9+2 b c^9-2 c^10 : :
X(10709) = X[102] - 4 X[117], 2 X[117] + X[151], X[102] + 2 X[151], 2 X[124] - 3 X[3545], 5 X[102] - 8 X[6711], 5 X[2] - 4 X[6711], 5 X[117] - 2 X[6711], 5 X[151] + 4 X[6711], 3 X[3524] - 4 X[6718]

X(10709) lies on these lines:
{2,102}, {30,109}, {124,3545}, {381,2818}, {671,2792}, {2779,9140}, {2785,6054}, {2817,3679}, {3524,6718}

X(10709) = midpoint of X(2) and X(151)
X(10709) = reflection of X(i) in X(j) for these (i,j): (2,117), (102,2)
X(10709) = {X(117),X(151)}-harmonic conjugate of X(102)

X(10710) =  REFLECTION OF X(103) IN X(2)

Barycentrics    a^8-a^7 b+4 a^6 b^2-8 a^5 b^3-a^4 b^4+7 a^3 b^5-2 a^2 b^6+2 a b^7-2 b^8-a^7 c+a^6 b c+3 a^3 b^4 c-3 a^2 b^5 c-2 a b^6 c+2 b^7 c+4 a^6 c^2+10 a^4 b^2 c^2-10 a^3 b^3 c^2-2 a^2 b^4 c^2-6 a b^5 c^2+4 b^6 c^2-8 a^5 c^3-10 a^3 b^2 c^3+14 a^2 b^3 c^3+6 a b^4 c^3-2 b^5 c^3-a^4 c^4+3 a^3 b c^4-2 a^2 b^2 c^4+6 a b^3 c^4-4 b^4 c^4+7 a^3 c^5-3 a^2 b c^5-6 a b^2 c^5-2 b^3 c^5-2 a^2 c^6-2 a b c^6+4 b^2 c^6+2 a c^7+2 b c^7-2 c^8 : :

X(10710) lies on these lines:
{2,103}, {4,544}, {30,101}, {116,3545}, {150,3839}, {381,2808}, {527,1541}, {671,2784}, {1478,4845}, {1699,2801}, {2772,9140}, {2786,6054}, {3524,6710}

X(10710) = midpoint of X(2) and X(151)
X(10710) = reflection of X(i) in X(j) for these (i,j): (2,118), (103,2)

X(10711) =  REFLECTION OF X(104) IN X(2)

Barycentrics    a^7-a^6 b-3 a^3 b^4+3 a^2 b^5+2 a b^6-2 b^7-a^6 c+9 a^5 b c-8 a^4 b^2 c+7 a^2 b^4 c-9 a b^5 c+2 b^6 c-8 a^4 b c^2+14 a^3 b^2 c^2-10 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2-10 a^2 b^2 c^3+18 a b^3 c^3-6 b^4 c^3-3 a^3 c^4+7 a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+3 a^2 c^5-9 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 : :
X(10711) = X[104] - 4 X[119], 2 X[119] + X[153], X[104] + 2 X[153], 4 X[3035] - 3 X[3524], 2 X[11] - 3 X[3545], X[149] - 3 X[3839], X[5531] + 2 X[6246], 5 X[104] - 8 X[6713], 5 X[2] - 4 X[6713], 5 X[119] - 2 X[6713], 5 X[153] + 4 X[6713]

X(10711) lies on these lines:
{2,104}, {4,528}, {11,1056}, {30,100}, {80,226}, {149,3839}, {329,1145}, {376,2829}, {381,952}, {515,5660}, {519,1519}, {527,1512}, {535,6905}, {671,2783}, {938,9654}, {1320,3656}, {1387,5226}, {1484,5066}, {2771,3753}, {2787,6054}, {2800,3679}, {2801,5587}, {3035,3524}, {3543,5840}, {3582,10074}, {3584,10058}, {5437,6702}, {5531,6246}, {5657,6172}, {5748,6224}, {6265,10031}

X(10711) = midpoint of X(2) and X(152)
X(10711) = reflection of X(i) in X(j) for these (i,j): {2, 119}, {104, 2}, {376, 6174}, {1320, 3656}, {1484, 5066}, {10031, 6265}
X(10711) = {X(119),X(153)}-harmonic conjugate of X(104)

X(10712) =  REFLECTION OF X(105) IN X(2)

Barycentrics    a^5-a^4 b+3 a^3 b^2-3 a^2 b^3+2 a b^4-2 b^5-a^4 c+3 a^3 b c-5 a^2 b^2 c+3 a b^3 c+2 b^4 c+3 a^3 c^2-5 a^2 b c^2-2 a b^2 c^2-3 a^2 c^3+3 a b c^3+2 a c^4+2 b c^4-2 c^5 : :
X(10712) = X[105] - 4 X[120], 3 X[3545] - 2 X[5511], 5 X[105] - 8 X[6714], 5 X[2] - 4 X[6714], 5 X[120] - 2 X[6714]

X(10712) lies on these lines:
{2,11}, {30,1292}, {210,2836}, {668,3263}, {671,2795}, {1699,9519}, {2788,6054}, {2809,3679}, {3545,5511}

X(10712) = reflection of X(i) in X(j) for these (i,j):
X(10712) = inverse-in-orthoptic-circle of Steiner Inellipe of X(6174)

X(10713) =  REFLECTION OF X(106) IN X(2)

Barycentrics    a^4-a^3 b+2 a^2 b^2+2 a b^3-2 b^4-a^3 c+5 a^2 b c-10 a b^2 c+2 b^3 c+2 a^2 c^2-10 a b c^2+8 b^2 c^2+2 a c^3+2 b c^3-2 c^4 : :
X(10713) = X[106] - 4 X[121], 3 X[3545] - 2 X[5510], 5 X[106] - 8 X[6715], 5 X[2] - 4 X[6715], 5 X[121] - 2 X[6715]

X(10713) lies on these lines:
{2,106}, {10,190}, {30,1293}, {519,3699}, {599,2810}, {2789,6054}, {2802,3679}, {2842,9140}, {3545,5510}, {5587,9519}, {6788,9041}

X(10713) = reflection of X(i) in X(j) for these (i,j): (2,121), (106,2)

X(10714) =  REFLECTION OF X(107) IN X(2)

Barycentrics    a^12-a^10 b^2+4 a^8 b^4-14 a^6 b^6+13 a^4 b^8-a^2 b^10-2 b^12-a^10 c^2-7 a^8 b^2 c^2+14 a^6 b^4 c^2+2 a^4 b^6 c^2-13 a^2 b^8 c^2+5 b^10 c^2+4 a^8 c^4+14 a^6 b^2 c^4-30 a^4 b^4 c^4+14 a^2 b^6 c^4-2 b^8 c^4-14 a^6 c^6+2 a^4 b^2 c^6+14 a^2 b^4 c^6-2 b^6 c^6+13 a^4 c^8-13 a^2 b^2 c^8-2 b^4 c^8-a^2 c^10+5 b^2 c^10-2 c^12 : :
X(10714) = X[107] - 4 X[122], 2 X[133] - 3 X[3545], 3 X[3524] - X[5667], 5 X[107] - 8 X[6716], 5 X[2] - 4 X[6716], 5 X[122] - 2 X[6716], 2 X[1294] + X[10152], 2 X[3184] - 3 X[10304]

X(10714) lies on these lines:
{2,107}, {30,1294}, {133,3545}, {376,2777}, {671,2797}, {1494,3268}, {2790,6054}, {3184,10304}, {3524,5667}, {6175,9528}

X(10714) = reflection of X(i) in X(j) for these (i,j): (2,123), (108,2)

X(10715) =  REFLECTION OF X(108) IN X(2)

Barycentrics    a^9-a^8 b+a^7 b^2-a^6 b^3-3 a^5 b^4+3 a^4 b^5-a^3 b^6+a^2 b^7+2 a b^8-2 b^9-a^8 c-a^7 b c+a^6 b^2 c+7 a^5 b^3 c-9 a^4 b^4 c+a^3 b^5 c+7 a^2 b^6 c-7 a b^7 c+2 b^8 c+a^7 c^2+a^6 b c^2-8 a^5 b^2 c^2+6 a^4 b^3 c^2+9 a^3 b^4 c^2-11 a^2 b^5 c^2-2 a b^6 c^2+4 b^7 c^2-a^6 c^3+7 a^5 b c^3+6 a^4 b^2 c^3-18 a^3 b^3 c^3+3 a^2 b^4 c^3+7 a b^5 c^3-4 b^6 c^3-3 a^5 c^4-9 a^4 b c^4+9 a^3 b^2 c^4+3 a^2 b^3 c^4+3 a^4 c^5+a^3 b c^5-11 a^2 b^2 c^5+7 a b^3 c^5-a^3 c^6+7 a^2 b c^6-2 a b^2 c^6-4 b^3 c^6+a^2 c^7-7 a b c^7+4 b^2 c^7+2 a c^8+2 b c^8-2 c^9 : :
X(10715) = X[108] - 4 X[123], 5 X[108] - 8 X[6717], 5 X[2] - 4 X[6717], 5 X[123] - 2 X[6717]

X(10715) lies on these lines:
{2,108}, {30,1295}, {376,2829}, {671,2798}, {2791,6054}, {2817,3679}, {2850,9140}

X(10715) = reflection of X(i) in X(j) for these (i,j): (2,123), (108,2)

X(10716) =  REFLECTION OF X(109) IN X(2)

Barycentrics    a^6-a^5 b-a^4 b^2-a^3 b^3+2 a^2 b^4+2 a b^5-2 b^6-a^5 c+3 a^4 b c+a^3 b^2 c+a^2 b^3 c-6 a b^4 c+2 b^5 c-a^4 c^2+a^3 b c^2-6 a^2 b^2 c^2+4 a b^3 c^2+2 b^4 c^2-a^3 c^3+a^2 b c^3+4 a b^2 c^3-4 b^3 c^3+2 a^2 c^4-6 a b c^4+2 b^2 c^4+2 a c^5+2 b c^5-2 c^6 : :
X(10716) = X[109] - 4 X[124], 2 X[117] - 3 X[3545], X[151] - 3 X[3839], 3 X[3524] - 4 X[6711], 5 X[109] - 8 X[6718], 5 X[2] - 4 X[6718], 5 X[124] - 2 X[6718]

X(10716) lies on these lines:
{2,109}, {30,102}, {117,3545}, {151,3839}, {381,2818}, {671,1121}, {1795,3582}, {2773,9140}, {2792,6054}, {2800,3679}, {3524,6711}

X(10716) = reflection of X(i) in X(j) for these (i,j): (2,124), (109,2)

X(10717) =  REFLECTION OF X(111) IN X(2)

Barycentrics    a^6+3 a^4 b^2-2 b^6+3 a^4 c^2-15 a^2 b^2 c^2+6 b^4 c^2+6 b^2 c^4-2 c^6 : :
X(10717) = X[111] - 4 X[126], 3 X[3545] - 2 X[5512], 5 X[111] - 8 X[6719], 5 X[2] - 4 X[6719], 5 X[126] - 2 X[6719], 6 X[6719] - 5 X[9172], 3 X[111] - 4 X[9172], 3 X[126] - X[9172]

X(10717) lies on the cubic K794 and these lines:
{2,99}, {3,9829}, {30,1296}, {110,1641}, {524,9146}, {599,2854}, {670,3266}, {804,9156}, {2793,6054}, {2805,4688}, {3545,5512}, {3849,5971}, {5108,9830}, {5468,8593}, {5969,9169}, {6088,9148}, {8724,9759}, {9164,10415}

X(10717) = reflection of X(i) in X(j) for these (i,j): (2,126), (111,2)
X(10717) = anticomplement of X(9172)
X(10717) = inverse-in-orthoptic-circle of Steiner inellipse of X(2482)
X(10717) = inverse-in-orthoptic-circle of Steiner circumellipse of X(8591)

X(10718) =  REFLECTION OF X(112) IN X(2)

Barycentrics    a^10-a^8 b^2-3 a^6 b^4+3 a^4 b^6+2 a^2 b^8-2 b^10-a^8 c^2+7 a^6 b^2 c^2-3 a^4 b^4 c^2-5 a^2 b^6 c^2+2 b^8 c^2-3 a^6 c^4-3 a^4 b^2 c^4+6 a^2 b^4 c^4+3 a^4 c^6-5 a^2 b^2 c^6+2 a^2 c^8+2 b^2 c^8-2 c^10 : :
X(10718) = X[112] - 4 X[127], 2 X[132] - 3 X[3545], 5 X[112] - 8 X[6720], 5 X[2] - 4 X[6720], 5 X[127] - 2 X[6720]

X(10718) lies on these lines:
{2,112}, {4,9530}, {30,935}, {132,3545}, {315,4558}, {376,2482}, {599,2781}, {671,1494}, {868,8753}, {7512,7873}, {7552,7810}, {9140,9517}

X(10718) = reflection of X(i) in X(j) for these (i,j): (2,127), (112,2)

X(10719) =  REFLECTION OF X(1113) IN X(2)

Barycentrics    3 (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)+(-a^4-a^2 b^2+2 b^4-a^2 c^2-4 b^2 c^2+2 c^4) J : : , where J = |OH|/R
X(10719) = X[1113] - 4 X[1313], 2 X[1312] - 3 X[3545]

X(10719) lies on these lines:
{2,3}, {519,2102}, {524,2104}, {542,8116}, {2103,3656}, {2463,7286}, {2464,5160}, {2574,9140}

X(10719) = reflection of X(i) in X(j) for these (i,j): (2,1313), (1113,2), (2103,3656)
X(10719) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (23,1344,1113), (858,1313,1347)


X(10720) =  REFLECTION OF X(1114) IN X(2)

Barycentrics    (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4)-(-a^4-a^2 b^2+2 b^4-a^2 c^2-4 b^2 c^2+2 c^4) J : : , where J = |OH|/R
X(10720) = X[1114] - 4 X[1312], 2 X[1313] - 3 X[3545

X(10720) lies on these lines:
{2,3}, {519,2103}, {524,2105}, {542,8115}, {2102,3656}, {2463,5160}, {2464,7286}, {2575,9140}

X(10720) = reflection of X(i) in X(j) for these (i,j): (2,1311), (1114,2), (2102,3656)
X(10720) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (23,1345,1114), (858,1312,1346)

leftri  Reflections of circumcircle-points in the orthocenter: X(10721) - X(10737)  rightri

This preamble and centers X(10721)-X(10737) were contributed by Clark Kimberling and Peter Moses, November 9, 2016.

Suppose that P is a point on the circumcrcle of a triangle ABC, and let

H = X(4), the orthocenter of ABC
P' = reflection of P in H
Pc = complement of P
Pa = anticomplement of P
Ha = anticomplement of H (the orthocenter, X(4) Haa = anticomplement of Ha.

Then

P' = midpoint of Pa and Haa
P' = reflection of Ha in Pc.

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(4): (74,10721), (98,10722), (99,10723), (100,10724), (101,10725), (102,10726), (103,10727), (104,10728), (105,10729), (106,10730), (107,10152), (108,10731), (109,10732), (110,10733), (111,10734), (112,10735), (1113,10736), (1114,10737)

underbar

X(10721) =  REFLECTION OF X(74) IN X(4)

Barycentrics    3 a^10-3 a^8 b^2-7 a^6 b^4+9 a^4 b^6-2 b^10-3 a^8 c^2+17 a^6 b^2 c^2-9 a^4 b^4 c^2-11 a^2 b^6 c^2+6 b^8 c^2-7 a^6 c^4-9 a^4 b^2 c^4+22 a^2 b^4 c^4-4 b^6 c^4+9 a^4 c^6-11 a^2 b^2 c^6-4 b^4 c^6+6 b^2 c^8-2 c^10 : :
X(10721) = 3 X[74] - 4 X[125], 3 X[4] - 2 X[125], X[3448] - 3 X[3543], 4 X[974] - 5 X[3567], 4 X[5480] - 3 X[5622], 5 X[110] - 6 X[5655], 4 X[1112] - 3 X[5890], 3 X[376] - 4 X[5972], 5 X[3091] - 4 X[6699], 9 X[3545] - 8 X[6723], 5 X[74] - 8 X[7687], 5 X[125] - 6 X[7687], 5 X[4] - 4 X[7687], 3 X[5655] - 5 X[7728], 3 X[9140] - 4 X[10113], 3 X[3830] - 2 X[10113], 3 X[9140] - 2 X[10620], 3 X[3830] - X[10620]

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Euler line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(125) and orthocenter X(10721); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, December 10, 2016)

X(10721) lies on these lines:
{3,1539}, {4,74}, {20,113}, {24,2935}, {30,110}, {146,3146}, {265,3627}, {376,5972}, {378,7699}, {382,5663}, {399,5073}, {515,7978}, {541,3448}, {974,3567}, {1112,5890}, {1511,1657}, {1514,10295}, {1531,7464}, {1593,9919}, {1614,9934}, {1870,10118}, {1986,5895}, {2781,6403}, {3043,6759}, {3091,6699}, {3545,6723}, {3583,10081}, {3585,10065}, {3830,9140}, {3853,10264}, {5480,5622}, {6000,7722}, {6242,10628}, {8998,9541}, {10091,10483}

X(10721) = reflection of X(i) and X(j) for these (i,j): (3, 1539), (20, 113), (74, 4), (110, 7728), (265, 3627), (1657, 1511), (6241, 1986), (7464, 1531), (9140, 3830), (10264, 3853), (10295, 1514), (10620, 10113)
X(10721) = crosssum of X(3) and X(10620)
X(10721) = X(20)-of-X(30)-Fuhrmann-triangle
X(10721) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3830,10620,10113), (10113,10620,9140)


X(10722) =  REFLECTION OF X(98) IN X(4)

Barycentrics    3 a^8-3 a^6 b^2+a^4 b^4+a^2 b^6-2 b^8-3 a^6 c^2+a^4 b^2 c^2-a^2 b^4 c^2+5 b^6 c^2+a^4 c^4-a^2 b^2 c^4-6 b^4 c^4+a^2 c^6+5 b^2 c^6-2 c^8 : :
X(10722) = 3 X[98] - 4 X[115], 3 X[4] - 2 X[115], 3 X[376] - 4 X[620], X[148] - 3 X[3543], 5 X[3091] - 4 X[6036], 2 X[99] - 3 X[6054], 4 X[6033] - 3 X[6054], 3 X[3839] - 2 X[6055], 7 X[3523] - 8 X[6721], 9 X[3545] - 8 X[6722], 5 X[99] - 6 X[8724], 5 X[6054] - 4 X[8724], 5 X[6033] - 3 X[8724], 4 X[3845] - 3 X[9166], 3 X[98] - 2 X[9862], 3 X[4] - X[9862]

Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Brocard axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B', C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Brocard axis. The triangle A"B"C" is homothetic to ABC, with center of homothety X(115) and orthocenter X(10722); see Hyacinthos #16741 and #16782, Sep 2008. (Randy Hutson, December 10, 2016)

X(10722) lies on these lines:
{3,7899}, {4,32}, {20,114}, {30,99}, {146,148}, {147,3146}, {376,620}, {381,3972}, {382,2782}, {515,7970}, {516,9864}, {671,3830}, {1503,5111}, {1562,6776}, {1593,9861}, {2790,5186}, {3091,6036}, {3098,7898}, {3523,6721}, {3545,6722}, {3583,10069}, {3585,10053}, {3627,6321}, {3839,6055}, {3845,9166}, {5077,9774}, {5092,7924}, {5149,7470}, {6655,10352}, {7773,8781}, {8593,8597}, {8997,9541}, {10089,10483}

X(10722) = midpoint of X(147) and X(3146)
X(10722) = reflection of X(i) and X(j) for these (i,j): (20, 114), (98, 4), (99, 6033), (671, 3830), (6321, 3627), (9862, 115)
X(10722) = crossdifference of every pair of points on line X(684) X(6041)
X(10722) = X(20)-of-X(511)-Fuhrmann-triangle
X(10722) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,7737,9993), (4,9862,115), (99,6033,6054), (115,9862,98)


X(10723) =  REFLECTION OF X(99) IN X(4)

Barycentrics    3 a^8-5 a^6 b^2+3 a^4 b^4+a^2 b^6-2 b^8-5 a^6 c^2+5 a^4 b^2 c^2-3 a^2 b^4 c^2+7 b^6 c^2+3 a^4 c^4-3 a^2 b^2 c^4-10 b^4 c^4+a^2 c^6+7 b^2 c^6-2 c^8 : :
X(10723) = 3 X[99] - 4 X[114], 3 X[4] - 2 X[114], 2 X[98] - 3 X[671], 4 X[620] - 5 X[3091], X[147] - 3 X[3543], 2 X[2482] - 3 X[3839], 3 X[5182] - 4 X[5480], 3 X[148] - X[5984], 3 X[3146] + X[5984], 3 X[376] - 4 X[6036], 3 X[671] - 4 X[6321], 9 X[3545] - 8 X[6721], 7 X[3523] - 8 X[6722], 8 X[6036] - 9 X[9166], 2 X[376] - 3 X[9166], 3 X[9166] - 4 X[9880], 2 X[6036] - 3 X[9880], 4 X[5461] - 3 X[10304]

X(10723) lies on these lines:
{4,99}, {20,115}, {30,98}, {147,543}, {148,2794}, {376,6036}, {382,2782}, {515,7983}, {620,3091}, {2482,3839}, {2797,10152}, {3044,6759}, {3523,6722}, {3545,6721}, {3583,10089}, {3585,10086}, {3627,6033}, {3830,6054}, {5182,5480}, {5461,10304}, {5473,5479}, {5474,5478}, {8980,9541}, {10069,10483}

X(10723) = midpoint of X(148) and X(3146)
X(10723) = reflection of X(i) in X(j) for these (i,j): (20, 115), (98, 6321), (99, 4), (376, 9880), (5473, 5479), (5474, 5478), (6033, 3627), (6054, 3830)
X(10723) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (98,6321,671), (376,9880,9166)


X(10724) =  REFLECTION OF X(100) IN X(4)

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

X(10724) lies on these lines:
{4,100}, {11,20}, {30,104}, {40,6246}, {80,516}, {149,2829}, {153,528}, {165,6702}, {214,1699}, {376,6713}, {382,952}, {411,10058}, {515,1320}, {1012,4996}, {1387,5731}, {1537,6224}, {1621,6923}, {1770,10073}, {2802,5691}, {2803,10152}, {2975,10525}, {3035,3091}, {3045,6759}, {3523,6667}, {3583,6909}, {3585,10087}, {3839,6174}, {4299,5533}, {4301,7972}, {4302,6932}, {5083,9579}, {5284,6951}, {5556,5734}, {6326,9963}, {6965,9342}, {8988,9616}, {9589,9897}, {10074,10483}

X(10724) = midpoint of X(i) and X(j) for these {i,j}: {149, 3146}, {9589, 9897} X(10724) = reflection of X(i) in X(j) for these (i,j): (20, 11), (40, 6246), (100, 4), (6224, 1537), (6909, 3583), (7972, 4301), (9963, 6326)
X(10724) = {X(6224),X(9812)}-harmonic conjugate of X(1537)

X(10725) =  REFLECTION OF X(101) IN X(4)

Barycentrics    3 a^8-3 a^7 b-2 a^6 b^2+a^4 b^4+a^3 b^5+2 a b^7-2 b^8-3 a^7 c+3 a^6 b c+4 a^5 b^2 c-4 a^4 b^3 c+a^3 b^4 c-a^2 b^5 c-2 a b^6 c+2 b^7 c-2 a^6 c^2+4 a^5 b c^2+2 a^4 b^2 c^2-2 a^3 b^3 c^2-6 a b^5 c^2+4 b^6 c^2-4 a^4 b c^3-2 a^3 b^2 c^3+2 a^2 b^3 c^3+6 a b^4 c^3-2 b^5 c^3+a^4 c^4+a^3 b c^4+6 a b^3 c^4-4 b^4 c^4+a^3 c^5-a^2 b c^5-6 a b^2 c^5-2 b^3 c^5-2 a b c^6+4 b^2 c^6+2 a c^7+2 b c^7-2 c^8 : :
X(10725) = 3 X[101] - 4 X[118], 3 X[4] - 2 X[118], X[152] - 3 X[3543], 5 X[3091] - 4 X[6710], 3 X[376] - 4 X[6712]

X(10725) lies on these lines:
{4,101}, {20,116}, {30,103}, {150,3146}, {152,544}, {376,6712}, {382,2808}, {2809,5691}, {2811,10152}, {3046,6759}, {3091,6710}

X(10725) = midpoint of X(150) and X(3146)
X(10725) = reflection of X(i) and X(j) for these (i,j): (20, 116), (101, 4)
X(10725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):


X(10726) =  REFLECTION OF X(102) IN X(4)

Barycentrics    3 a^10-3 a^9 b-3 a^8 b^2+7 a^7 b^3-7 a^6 b^4-3 a^5 b^5+9 a^4 b^6-3 a^3 b^7+2 a b^9-2 b^10-3 a^9 c+9 a^8 b c-7 a^7 b^2 c-7 a^6 b^3 c+17 a^5 b^4 c-11 a^4 b^5 c-a^3 b^6 c+7 a^2 b^7 c-6 a b^8 c+2 b^9 c-3 a^8 c^2-7 a^7 b c^2+28 a^6 b^2 c^2-14 a^5 b^3 c^2-13 a^4 b^4 c^2+21 a^3 b^5 c^2-18 a^2 b^6 c^2+6 b^8 c^2+7 a^7 c^3-7 a^6 b c^3-14 a^5 b^2 c^3+30 a^4 b^3 c^3-17 a^3 b^4 c^3-7 a^2 b^5 c^3+16 a b^6 c^3-8 b^7 c^3-7 a^6 c^4+17 a^5 b c^4-13 a^4 b^2 c^4-17 a^3 b^3 c^4+36 a^2 b^4 c^4-12 a b^5 c^4-4 b^6 c^4-3 a^5 c^5-11 a^4 b c^5+21 a^3 b^2 c^5-7 a^2 b^3 c^5-12 a b^4 c^5+12 b^5 c^5+9 a^4 c^6-a^3 b c^6-18 a^2 b^2 c^6+16 a b^3 c^6-4 b^4 c^6-3 a^3 c^7+7 a^2 b c^7-8 b^3 c^7-6 a b c^8+6 b^2 c^8+2 a c^9+2 b c^9-2 c^10 : :

X(10726) lies on these lines:
{4,102}, {20,117}, {30,109}, {151,3146}, {376,6718}, {382,2818}, {515,1897}, {1795,10483}, {2349,2816}, {2817,5691}, {3091,6711}

X(10726) = midpoint of X(151) and X(3146)
X(10726) = reflection of X(i) in X(j) for these (i,j): (20, 117), (102, 4)

X(10727) =  REFLECTION OF X(103) IN X(4)

Barycentrics    3 a^8-3 a^7 b-4 a^5 b^3+a^4 b^4+5 a^3 b^5-2 a^2 b^6+2 a b^7-2 b^8-3 a^7 c+3 a^6 b c+4 a^5 b^2 c-4 a^4 b^3 c+a^3 b^4 c-a^2 b^5 c-2 a b^6 c+2 b^7 c+4 a^5 b c^2+6 a^4 b^2 c^2-6 a^3 b^3 c^2-2 a^2 b^4 c^2-6 a b^5 c^2+4 b^6 c^2-4 a^5 c^3-4 a^4 b c^3-6 a^3 b^2 c^3+10 a^2 b^3 c^3+6 a b^4 c^3-2 b^5 c^3+a^4 c^4+a^3 b c^4-2 a^2 b^2 c^4+6 a b^3 c^4-4 b^4 c^4+5 a^3 c^5-a^2 b c^5-6 a b^2 c^5-2 b^3 c^5-2 a^2 c^6-2 a b c^6+4 b^2 c^6+2 a c^7+2 b c^7-2 c^8 : :
X(10727) = 3 X[103] - 4 X[116], 3 X[4] - 2 X[116], X[150] - 3 X[3543], 3 X[376] - 4 X[6710], 5 X[3091] - 4 X[6712]

X(10727) lies on these lines:
{4,103}, {20,118}, {30,101}, {150,3543}, {152,3146}, {376,6710}, {382,2808}, {516,3732}, {2822,3668}, {3091,6712}

X(10727) = midpoint of X(152) and X(3146)
X(10727) = reflection of X(i) in X(j) for these (i,j): (20, 118), (103, 4)

X(10728) =  REFLECTION OF X(104) IN X(4)

Barycentrics    3 a^7-3 a^6 b-4 a^5 b^2+4 a^4 b^3-a^3 b^4+a^2 b^5+2 a b^6-2 b^7-3 a^6 c+11 a^5 b c-4 a^4 b^2 c-4 a^3 b^3 c+5 a^2 b^4 c-7 a b^5 c+2 b^6 c-4 a^5 c^2-4 a^4 b c^2+10 a^3 b^2 c^2-6 a^2 b^3 c^2-2 a b^4 c^2+6 b^5 c^2+4 a^4 c^3-4 a^3 b c^3-6 a^2 b^2 c^3+14 a b^3 c^3-6 b^4 c^3-a^3 c^4+5 a^2 b c^4-2 a b^2 c^4-6 b^3 c^4+a^2 c^5-7 a b c^5+6 b^2 c^5+2 a c^6+2 b c^6-2 c^7 : :
X(10728) = 3 X[4] - 2 X[11], 4 X[11] - 3 X[104], 3 X[376] - 4 X[3035], X[149] - 3 X[3543], 9 X[3545] - 8 X[6667], 5 X[3091] - 4 X[6713], 2 X[4316] - 3 X[6905], 3 X[5691] - X[9897]

X(10728) lies on these lines:
{4,11}, {20,119}, {30,100}, {80,1770}, {149,3543}, {153,3146}, {376,3035}, {382,952}, {515,7972}, {944,1537}, {1145,6361}, {1387,4308}, {1484,3853}, {1593,9913}, {1768,6246}, {2771,4018}, {2800,5691}, {2828,10152}, {3091,6713}, {3545,6667}, {3583,10074}, {3585,10058}, {3586,5083}, {3826,6951}, {4302,6256}, {4316,6905}, {4996,6985}, {5218,6938}, {6845,8068}, {6906,7951}, {10090,10483}

X(10728) = midpoint of X(153) and X(3146)
X(10728) = reflection of X(i) in X(j) for these (i,j): (20, 119), (104, 4), (944, 1537), (1484, 3853), (1768, 6246), (6361, 1145)

X(10729) =  REFLECTION OF X(105) IN X(4)

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

X(10729) lies on these lines:
{4,105}, {20,120}, {30,1292}, {153,528}, {2809,5691}, {2833,10152}, {3091,6714}

X(10729) = reflection of X(i) in X(j) for these (i,j): (20, 120), (105, 4)


X(10730) =  REFLECTION OF X(106) IN X(4)

Barycentrics    3 a^7-6 a^6 b-6 a^5 b^2+6 a^4 b^3-a^3 b^4+2 a^2 b^5+4 a b^6-2 b^7-6 a^6 c+27 a^5 b c-9 a^4 b^2 c-9 a^3 b^3 c+11 a^2 b^4 c-18 a b^5 c+4 b^6 c-6 a^5 c^2-9 a^4 b c^2+20 a^3 b^2 c^2-13 a^2 b^3 c^2-4 a b^4 c^2+8 b^5 c^2+6 a^4 c^3-9 a^3 b c^3-13 a^2 b^2 c^3+36 a b^3 c^3-10 b^4 c^3-a^3 c^4+11 a^2 b c^4-4 a b^2 c^4-10 b^3 c^4+2 a^2 c^5-18 a b c^5+8 b^2 c^5+4 a c^6+4 b c^6-2 c^7 : :
X(10730) = 3 X[106] - 4 X[5510], 3 X[4] - 2 X[5510], 5 X[3091] - 4 X[6715]

X(10730) lies on these lines:
{4,106}, {20,121}, {30,1293}, {2802,5691}, {2839,10152}, {3091,6715}

X(10730) = reflection of X(i) in X(j) for these (i,j): (20, 121), (106, 4)

X(10731) =  REFLECTION OF X(108) IN X(4)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^9-3 a^8 b-7 a^7 b^2+7 a^6 b^3+3 a^5 b^4-3 a^4 b^5+3 a^3 b^6-3 a^2 b^7-2 a b^8+2 b^9-3 a^8 c+13 a^7 b c-3 a^6 b^2 c-17 a^5 b^3 c+13 a^4 b^4 c-5 a^3 b^5 c-5 a^2 b^6 c+9 a b^7 c-2 b^8 c-7 a^7 c^2-3 a^6 b c^2+24 a^5 b^2 c^2-10 a^4 b^3 c^2-15 a^3 b^4 c^2+17 a^2 b^5 c^2-2 a b^6 c^2-4 b^7 c^2+7 a^6 c^3-17 a^5 b c^3-10 a^4 b^2 c^3+34 a^3 b^3 c^3-9 a^2 b^4 c^3-9 a b^5 c^3+4 b^6 c^3+3 a^5 c^4+13 a^4 b c^4-15 a^3 b^2 c^4-9 a^2 b^3 c^4+8 a b^4 c^4-3 a^4 c^5-5 a^3 b c^5+17 a^2 b^2 c^5-9 a b^3 c^5+3 a^3 c^6-5 a^2 b c^6-2 a b^2 c^6+4 b^3 c^6-3 a^2 c^7+9 a b c^7-4 b^2 c^7-2 a c^8-2 b c^8+2 c^9) : :
X(10731) = 5 X[3091] - 4 X[6717]

X(10731) lies on these lines:
{4,11}, {20,123}, {30,1295}, {2817,5691}, {2845,10152}, {3091,6717}

X(10731) = reflection of X(i) in X(j) for these (i,j): (20, 123), (108, 4)

X(10732) =  REFLECTION OF X(109) IN X(4)

Barycentrics    3 a^10-3 a^9 b-5 a^8 b^2+7 a^7 b^3-a^6 b^4-3 a^5 b^5+3 a^4 b^6-3 a^3 b^7+2 a^2 b^8+2 a b^9-2 b^10-3 a^9 c+9 a^8 b c-3 a^7 b^2 c-11 a^6 b^3 c+9 a^5 b^4 c-3 a^4 b^5 c+3 a^3 b^6 c+3 a^2 b^7 c-6 a b^8 c+2 b^9 c-5 a^8 c^2-3 a^7 b c^2+20 a^6 b^2 c^2-6 a^5 b^3 c^2-7 a^4 b^4 c^2+9 a^3 b^5 c^2-14 a^2 b^6 c^2+6 b^8 c^2+7 a^7 c^3-11 a^6 b c^3-6 a^5 b^2 c^3+14 a^4 b^3 c^3-9 a^3 b^4 c^3-3 a^2 b^5 c^3+16 a b^6 c^3-8 b^7 c^3-a^6 c^4+9 a^5 b c^4-7 a^4 b^2 c^4-9 a^3 b^3 c^4+24 a^2 b^4 c^4-12 a b^5 c^4-4 b^6 c^4-3 a^5 c^5-3 a^4 b c^5+9 a^3 b^2 c^5-3 a^2 b^3 c^5-12 a b^4 c^5+12 b^5 c^5+3 a^4 c^6+3 a^3 b c^6-14 a^2 b^2 c^6+16 a b^3 c^6-4 b^4 c^6-3 a^3 c^7+3 a^2 b c^7-8 b^3 c^7+2 a^2 c^8-6 a b c^8+6 b^2 c^8+2 a c^9+2 b c^9-2 c^10 : :
X(10732) = 3 X[109] - 4 X[117], 3 X[4] - 2 X[117], X[151] - 3 X[3543], 3 X[376] - 4 X[6711], 5 X[3091] - 4 X[6718]

X(10732) lies on these lines:
{4,109}, {20,124}, {30,102}, {151,3543}, {376,6711}, {382,2818}, {1795,3583}, {2800,5691}, {2846,10152}, {3091,6718}

X(10732) = reflection of X(i) in X(j) for these (i,j): (20, 124), (109, 4)


X(10733) =  REFLECTION OF X(110) IN X(4)

Barycentrics    3 a^10-5 a^8 b^2-a^6 b^4+3 a^4 b^6+2 a^2 b^8-2 b^10-5 a^8 c^2+13 a^6 b^2 c^2-5 a^4 b^4 c^2-9 a^2 b^6 c^2+6 b^8 c^2-a^6 c^4-5 a^4 b^2 c^4+14 a^2 b^4 c^4-4 b^6 c^4+3 a^4 c^6-9 a^2 b^2 c^6-4 b^4 c^6+2 a^2 c^8+6 b^2 c^8-2 c^10 : :
X(10733) =

X(10733) lies on these lines:
{2,7687}, {3,10113}, {4,110}, {20,125}, {30,74}, {52,7722}, {146,148}, {323,1531}, {376,6699}, {378,2931}, {381,1511}, {382,5663}, {399,1539}, {511,10296}, {515,7984}, {541,6515}, {895,1503}, {1533,9934}, {1986,3060}, {2771,4018}, {2777,3146}, {3047,6759}, {3091,5972}, {3523,6723}, {3583,10091}, {3585,10088}, {3627,7728}, {3839,5642}, {3845,10272}, {5073,10620}, {5076,5609}, {5640,9826}, {6053,9143}, {8994,9541}, {9033,10152}, {10081,10483}

X(10733) = midpoint of X(i) and X(j) for these {i,j}: {3146, 3448}, {5073, 10620}
X(10733) = reflection of X(i) in X(j) for these (i,j): (3, 10113), (20, 125), (74, 265), (110, 4), (323, 1531), (399, 1539), (7722, 52), (7728, 3627)
X(10733) = crosssum of X(2088) and X(9409)
X(10733) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (74,265,9140), (399,3830,1539)
X(10733) = anticomplement of anticomplement of X(7687)

X(10734) =  REFLECTION OF X(111) IN X(4)

Barycentrics    3 a^10-9 a^8 b^2-7 a^6 b^4+11 a^4 b^6+4 a^2 b^8-2 b^10-9 a^8 c^2+41 a^6 b^2 c^2-17 a^4 b^4 c^2-33 a^2 b^6 c^2+10 b^8 c^2-7 a^6 c^4-17 a^4 b^2 c^4+58 a^2 b^4 c^4-8 b^6 c^4+11 a^4 c^6-33 a^2 b^2 c^6-8 b^4 c^6+4 a^2 c^8+10 b^2 c^8-2 c^10 : :
X(10734) = 3 X[111] - 4 X[5512], 3 X[4] - 2 X[5512], 5 X[3091] - 4 X[6719], 3 X[3839] - 2 X[9172]

X(10734) lies on these lines:
{4,111}, {20,126}, {30,1296}, {147,543}, {2847,10152}, {3048,6759}, {3091,6719}, {3839,9172}

X(10734) = reflection of X(i) in X(j) for these (i,j): (20, 126), (111, 4)

X(10735) =  REFLECTION OF X(112) IN X(4)

Barycentrics    (a^2+b^2-c^2) (a^2-b^2+c^2) (3 a^10-5 a^8 b^2+3 a^6 b^4-a^4 b^6-2 a^2 b^8+2 b^10-5 a^8 c^2+5 a^6 b^2 c^2-a^4 b^4 c^2+3 a^2 b^6 c^2-2 b^8 c^2+3 a^6 c^4-a^4 b^2 c^4-2 a^2 b^4 c^4-a^4 c^6+3 a^2 b^2 c^6-2 a^2 c^8-2 b^2 c^8+2 c^10) : :
X(10735) = 3 X[112] - 4 X[132], 3 X[4] - 2 X[132], 5 X[3091] - 4 X[6720]

X(10735) lies on these lines:
{4,32}, {20,127}, {30,935}, {427,9157}, {2394,2848}, {2781,6403}, {3091,6720}

X(10735) = reflection of X(i) in X(j) for these (i,j): (20, 127), (112, 4)

X(10736) =  REFLECTION OF X(1113) IN X(4)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-(3 a^4-a^2 b^2-2 b^4-a^2 c^2+4 b^2 c^2-2 c^4) J : : , where J = |OH|/R
X(10736) = 3 X[1113] - 4 X[1312], 3 X[4] - 2 X[1312]

X(10736) lies on these lines: {2,3}, {515,2102}, {1503,2104}

X(10736) = reflection of X(i) in X(j) for these (i,j): (20, 1313), (1113, 4), (10737,382)
X(10736) = {X(1345),X(3830)}-harmonic conjugate of X(4)

X(10737) =  REFLECTION OF X(1114) IN X(4)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+(3 a^4-a^2 b^2-2 b^4-a^2 c^2+4 b^2 c^2-2 c^4) J : : , where J = |OH|/R
X(10737) = 3 X[1114] - 4 X[1313], 3 X[4] - 2 X[1313]

X(10737) lies on these lines: {2,3}, {515,2103}, {1503,2105}

X(10737) = reflection of X(i) in X(j) for these (i,j): (20, 1312), (1114, 4), (10736,382)
X(10737) = {X(1344),X(3830)}-harmonic conjugate of X(4)


leftri   Reflections of circumcircle-points in the nine-point center: X(10738) - X(10751)  rightri

This preamble and centers X(10738)-X(10751) were contributed by Clark Kimberling and Peter Moses, November 10, 2016.

Suppose that P is a point on the circumcrcle of a triangle ABC, and let

O = X(3), the circumcenter of ABC
N = X(5), the nine-point center of ABC
P' = reflection of P in N
Pc = complement of P
Pa = anticomplement of P.

Then

P' = midpoint of H and Pa
P' = reflection of O in Pc.

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(5): (74, 7728), (98,6033), (99,6321), (100,10738), (101,10739), (102,10740), (103,10741), (104,10742), (105,10743), (106,10744), (107,10745), (108,10746), (109,10747), (110,265), (111,10748), (112,10749), (1113,10750), (1114,10751)

underbar

X(10738) =  REFLECTION OF X(100) IN X(5)

Barycentrics    a^7 - a^6 b - a^5 b^2 + a^4 b^3 - a^3 b^4 + a^2 b^5 + a b^6 - b^7 - a^6 c + a^5 b c + a^4 b^2 c + a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + b^6 c - a^5 c^2 + a^4 b c^2 - 2 a^3 b^2 c^2 - a b^4 c^2 + 3 b^5 c^2 + a^4 c^3 + a^3 b c^3 + 4 a b^3 c^3 - 3 b^4 c^3 - a^3 c^4 - a^2 b c^4 - a b^2 c^4 - 3 b^3 c^4 + a^2 c^5 - 2 a b c^5 + 3 b^2 c^5 + a c^6 + b c^6 - c^7 : :
X(10738) = 3 X[4] - X[153], 3 X[149] + X[153], 2 X[119] - 3 X[381], 5 X[1656] - 4 X[3035], X[5541] - 3 X[5587], 2 X[1145] - 3 X[5790], 2 X[214] - 3 X[5886], 7 X[3851] - 2 X[6154], 3 X[5055] - 2 X[6174], 3 X[5603] - X[6224], 3 X[1699] - X[6326], 7 X[3526] - 8 X[6667], 3 X[3] - 4 X[6713], 3 X[11] - 2 X[6713], X[9803] + 3 X[9812], 4 X[1387] - 3 X[10246], 2 X[1317] - 3 X[10247]

X(10738) lies on these lines:
{3,11}, {4,145}, {5,100}, {12,10087}, {30,104}, {55,6980}, {56,5533}, {65,10073}, {79,6583}, {80,517}, {119,381}, {156,3045}, {214,5886}, {265,3657}, {355,2802}, {382,2829}, {390,6982}, {497,1387}, {516,10265}, {946,6265}, {971,3254}, {1145,3434}, {1156,5762}, {1317,1478}, {1351,5848}, {1352,9024}, {1385,4857}, {1656,3035}, {1699,6326}, {2475,5901}, {2771,7728}, {2783,6033}, {2787,6321}, {2886,7489}, {2932,6911}, {3057,10057}, {3065,5536}, {3526,6667}, {3585,7972}, {3851,6154}, {4294,6863}, {4996,6914}, {5046,5690}, {5055,6174}, {5080,5844}, {5225,6928}, {5274,6948}, {5541,5587}, {5603,6224}, {5691,6264}, {5779,5856}, {5805,9946}, {6826,9945}, {6839,9963}, {6917,10531}, {6958,10591}, {6959,10598}, {7354,10074}, {7982,9897}, {8148,10526}, {9670,10267}, {9671,10310}, {9803,9812}

X(10738) = midpoint of X(i) and X(j) for these {i,j}: {4, 149}, {5691, 6264}, {7982, 9897}
X(10738) = reflection of X(i) in X(j) for these (i,j): (3, 11), (100, 5), (104, 1484), (355, 6246), (6265, 946), (7972, 10222)
X(10738) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,6284,10058), (497,6923,10246), (1479,10525,3), (3434,6929,5790)


X(10739) =  REFLECTION OF X(101) IN X(5)

Barycentrics    a^8 - a^7 b - a^6 b^2 + a^5 b^3 - a^3 b^5 + a^2 b^6 + a b^7 - b^8 - a^7 c + a^6 b c + a^5 b^2 c - a^4 b^3 c + a^3 b^4 c - a^2 b^5 c - a b^6 c + b^7 c - a^6 c^2 + a^5 b c^2 + a^2 b^4 c^2 - 3 a b^5 c^2 + 2 b^6 c^2 + a^5 c^3 - a^4 b c^3 - 2 a^2 b^3 c^3 + 3 a b^4 c^3 - b^5 c^3 + a^3 b c^4 + a^2 b^2 c^4 + 3 a b^3 c^4 - 2 b^4 c^4 - a^3 c^5 - a^2 b c^5 - 3 a b^2 c^5 - b^3 c^5 + a^2 c^6 - a b c^6 + 2 b^2 c^6 + a c^7 + b c^7 - c^8 : :
X(10739) = 3 X[4] - X[152], 3 X[150] + X[152], 2 X[118] - 3 X[381], X[1282] - 3 X[5587], 5 X[1656] - 4 X[6710], 3 X[3] - 4 X[6712], 3 X[116] - 2 X[6712]

X(10739) lies on these lines:
{3,116}, {4,150}, {5,101}, {30,103}, {118,381}, {156,3046}, {265,2774}, {355,2809}, {946,2784}, {1282,5587}, {1352,2810}, {1362,1478}, {1479,3022}, {1656,6710}, {2772,7728}, {2786,6321}, {2801,5805}

X(10739) = midpoint of X(4) and X(150)
X(10739) = reflection of X(i) in X(j) for these (i,j): (3, 116), (101, 5)

X(10740) =  REFLECTION OF X(102) IN X(5)

Barycentrics    a^10 - a^9 b + 2 a^7 b^3 - 5 a^6 b^4 + 5 a^4 b^6 - 2 a^3 b^7 + a b^9 - b^10 - a^9 c + 3 a^8 b c - 4 a^7 b^2 c + 8 a^5 b^4 c - 8 a^4 b^5 c + 4 a^2 b^7 c - 3 a b^8 c + b^9 c - 4 a^7 b c^2 + 12 a^6 b^2 c^2 - 8 a^5 b^3 c^2 - 5 a^4 b^4 c^2 + 12 a^3 b^5 c^2 - 10 a^2 b^6 c^2 + 3 b^8 c^2 + 2 a^7 c^3 - 8 a^5 b^2 c^3 + 16 a^4 b^3 c^3 - 10 a^3 b^4 c^3 - 4 a^2 b^5 c^3 + 8 a b^6 c^3 - 4 b^7 c^3 - 5 a^6 c^4 + 8 a^5 b c^4 - 5 a^4 b^2 c^4 - 10 a^3 b^3 c^4 + 20 a^2 b^4 c^4 - 6 a b^5 c^4 - 2 b^6 c^4 - 8 a^4 b c^5 + 12 a^3 b^2 c^5 - 4 a^2 b^3 c^5 - 6 a b^4 c^5 + 6 b^5 c^5 + 5 a^4 c^6 - 10 a^2 b^2 c^6 + 8 a b^3 c^6 - 2 b^4 c^6 - 2 a^3 c^7 + 4 a^2 b c^7 - 4 b^3 c^7 - 3 a b c^8 + 3 b^2 c^8 + a c^9 + b c^9 - c^10 : :
X(10740) = 2 X[124] - 3 X[381], 5 X[1656] - 4 X[6711], 3 X[3] - 4 X[6718], 3 X[117] - 2 X[6718]

X(10740) lies on these lines:
{3,117}, {4,151}, {5,102}, {10,2816}, {30,109}, {124,381}, {265,2779}, {355,2817}, {1361,1479}, {1364,1478}, {1656,6711}, {1795,7354}, {1837,1845}, {2773,7728}, {2785,6033}, {2792,6321}

X(10740) = midpoint of X(4) and X(151)
X(10740) = reflection of X(i) in X(j) for these (i,j): (3, 117), (102, 5)

X(10741) =  REFLECTION OF X(103) IN X(5)

Barycentrics    a^8 - a^7 b + a^6 b^2 - 3 a^5 b^3 + 3 a^3 b^5 - a^2 b^6 + a b^7 - b^8 - a^7 c + a^6 b c + a^5 b^2 c - a^4 b^3 c + a^3 b^4 c - a^2 b^5 c - a b^6 c + b^7 c + a^6 c^2 + a^5 b c^2 + 4 a^4 b^2 c^2 - 4 a^3 b^3 c^2 - a^2 b^4 c^2 - 3 a b^5 c^2 + 2 b^6 c^2 - 3 a^5 c^3 - a^4 b c^3 - 4 a^3 b^2 c^3 + 6 a^2 b^3 c^3 + 3 a b^4 c^3 - b^5 c^3 + a^3 b c^4 - a^2 b^2 c^4 + 3 a b^3 c^4 - 2 b^4 c^4 + 3 a^3 c^5 - a^2 b c^5 - 3 a b^2 c^5 - b^3 c^5 - a^2 c^6 - a b c^6 + 2 b^2 c^6 + a c^7 + b c^7 - c^8 : :
X(10741) = 3 X[4] - X[150], X[150] + 3 X[152], 2 X[116] - 3 X[381], 3 X[3] - 4 X[6710], 3 X[118] - 2 X[6710], 5 X[1656] - 4 X[6712]

Let Q be the quadrilateral ABCX(101). Taking the vertices 3 at a time yields four triangles whose orthocenters are the vertices of a cyclic quadrilateral whose circumcenter is X(10741). (Randy Hutson, December 10, 2016)

X(10741) lies on these lines:
{3,118}, {4,150}, {5,103}, {30,101}, {116,381}, {265,2772}, {544,3830}, {1362,1479}, {1478,3022}, {1656,6712}, {2774,7728}, {2784,6321}, {2786,6033}

X(10741) = midpoint of X(4) and X(152)
X(10741) = reflection of X(i) in X(j) for these (i,j): (3, 118), (103, 5)


X(10742) =  REFLECTION OF X(104) IN X(5)

Barycentrics    a^7 - a^6 b - a^5 b^2 + a^4 b^3 - a^3 b^4 + a^2 b^5 + a b^6 - b^7 - a^6 c + 5 a^5 b c - 3 a^4 b^2 c - a^3 b^3 c + 3 a^2 b^4 c - 4 a b^5 c + b^6 c - a^5 c^2 - 3 a^4 b c^2 + 6 a^3 b^2 c^2 - 4 a^2 b^3 c^2 - a b^4 c^2 + 3 b^5 c^2 + a^4 c^3 - a^3 b c^3 - 4 a^2 b^2 c^3 + 8 a b^3 c^3 - 3 b^4 c^3 - a^3 c^4 + 3 a^2 b c^4 - a b^2 c^4 - 3 b^3 c^4 + a^2 c^5 - 4 a b c^5 + 3 b^2 c^5 + a c^6 + b c^6 - c^7 : :
X(10742) = 3 X[4] - X[149], X[149] + 3 X[153], 2 X[11] - 3 X[381], 3 X[3] - 4 X[3035], 3 X[119] - 2 X[3035], X[1768] - 3 X[5587], 3 X[382] + 2 X[6154], 3 X[1699] - X[6264], 9 X[5055] - 8 X[6667], 5 X[1656] - 4 X[6713]

Let Q be the quadrilateral ABCX(100). Taking the vertices 3 at a time yields four triangles whose orthocenters are the vertices of a cyclic quadrilateral whose circumcenter is X(10742). (Randy Hutson, December 10, 2016)

X(10742) lies on these lines:
{3,119}, {4,145}, {5,104}, {10,3652}, {11,381}, {12,10058}, {30,100}, {65,79}, {355,2800}, {382,5840}, {388,1387}, {515,6265}, {528,3830}, {546,1484}, {1145,3436}, {1317,1479}, {1656,6713}, {1699,6264}, {1768,5587}, {2550,5779}, {2783,6321}, {2787,6033}, {2801,5805}, {3534,6174}, {3583,5048}, {5055,6667}, {5083,5722}, {5229,5708}, {5260,5499}, {5270,9955}, {5570,10073}, {5691,6326}, {5787,9946}, {5854,8148}, {6284,10087}, {6851,9945}, {6929,10246}, {7354,10090}, {7728,8674}

X(10742) = midpoint of X(i) and X(j) for these {i,j}: {4,153}, {5691,6326}
X(10742) = reflection of X(i) in X(j) for these (i,j): (3, 119), (104, 5), (1482, 1537), (1484, 546), (3534, 6174)
X(10742) = {X(),X()}-harmonic conjugate of X()


X(10743) =  REFLECTION OF X(105) IN X(5)

Barycentrics    a^8 - 2 a^7 b + 2 a^6 b^2 - 2 a^5 b^3 + 2 a^3 b^5 - 2 a^2 b^6 + 2 a b^7 - b^8 - 2 a^7 c + 3 a^6 b c - 3 a^5 b^2 c + 3 a^4 b^3 c - a^3 b^4 c - 2 a b^6 c + 2 b^7 c + 2 a^6 c^2 - 3 a^5 b c^2 + 4 a^4 b^2 c^2 - 3 a^3 b^3 c^2 + 2 a^2 b^4 c^2 - 2 a b^5 c^2 - 2 a^5 c^3 + 3 a^4 b c^3 - 3 a^3 b^2 c^3 + 2 a b^4 c^3 - 2 b^5 c^3 - a^3 b c^4 + 2 a^2 b^2 c^4 + 2 a b^3 c^4 + 2 b^4 c^4 + 2 a^3 c^5 - 2 a b^2 c^5 - 2 b^3 c^5 - 2 a^2 c^6 - 2 a b c^6 + 2 a c^7 + 2 b c^7 - c^8 : :
X(10743) = 3 X[381] - 2 X[5511], X[5540] - 3 X[5587], 5 X[1656] - 4 X[6714]

X(10743) lies on these lines:
{3,120}, {5,105}, {30,1292}, {119,381}, {265,2836}, {355,2809}, {1358,1478}, {1479,3021}, {1656,6714}, {2775,7728}, {2788,6033}, {2795,6321}, {5540,5587}

X(10743) = reflection of X(i) in X(j) for these (i,j): (3, 120), (105, 5)

X(10744) =  REFLECTION OF X(106) IN X(5)

Barycentrics    a^7 - 2 a^6 b - a^5 b^2 + 2 a^4 b^3 - 2 a^3 b^4 + a^2 b^5 + 2 a b^6 - b^7 - 2 a^6 c + 9 a^5 b c - 7 a^4 b^2 c + 7 a^2 b^4 c - 9 a b^5 c + 2 b^6 c - a^5 c^2 - 7 a^4 b c^2 + 14 a^3 b^2 c^2 - 10 a^2 b^3 c^2 - 2 a b^4 c^2 + 4 b^5 c^2 + 2 a^4 c^3 - 10 a^2 b^2 c^3 + 18 a b^3 c^3 - 5 b^4 c^3 - 2 a^3 c^4 + 7 a^2 b c^4 - 2 a b^2 c^4 - 5 b^3 c^4 + a^2 c^5 - 9 a b c^5 + 4 b^2 c^5 + 2 a c^6 + 2 b c^6 - c^7 : :
X(10744) = 3 X[381] - 2 X[5510], X[1054] - 3 X[5587], 5 X[1656] - 4 X[6715]

X(10744) lies on these lines:
{3,121}, {5,106}, {30,1293}, {265,2842}, {355,2802}, {381,5510}, {1054,5587}, {1352,2810}, {1357,1478}, {1479,6018}, {1656,6715}, {2776,7728}, {2789,6033}, {2796,6321}

X(10744) = reflection of X(i) in X(j) for these (i,j): (3, 121), (106, 5)

X(10745) =  REFLECTION OF X(107) IN X(5)

Barycentrics    (a^2 - b^2 - c^2) (a^14 - a^12 b^2 - 2 a^10 b^4 - a^8 b^6 + 9 a^6 b^8 - 7 a^4 b^10 + b^14 - a^12 c^2 + 5 a^10 b^2 c^2 + a^8 b^4 c^2 - 18 a^6 b^6 c^2 + 13 a^4 b^8 c^2 + 5 a^2 b^10 c^2 - 5 b^12 c^2 - 2 a^10 c^4 + a^8 b^2 c^4 + 18 a^6 b^4 c^4 - 6 a^4 b^6 c^4 - 20 a^2 b^8 c^4 + 9 b^10 c^4 - a^8 c^6 - 18 a^6 b^2 c^6 - 6 a^4 b^4 c^6 + 30 a^2 b^6 c^6 - 5 b^8 c^6 + 9 a^6 c^8 + 13 a^4 b^2 c^8 - 20 a^2 b^4 c^8 - 5 b^6 c^8 - 7 a^4 c^10 + 5 a^2 b^2 c^10 + 9 b^4 c^10 - 5 b^2 c^12 + c^14)
X(10745) = 2 X[133] - 3 X[381], 3 X[3] - 2 X[3184], 3 X[122] - X[3184], 5 X[1656] - 4 X[6716]

X(10745) lies on these lines:
{2,5667}, {3,113}, {4,2972}, {5,107}, {10,2816}, {30,1294}, {74,1650}, {127,133}, {265,6334}, {1073,3830}, {1478,3324}, {1479,7158}, {1656,6716}, {2790,6033}, {2797,6321}

X(10745) = midpoint of X(1294) and X(10152)
X(10745) = reflection of X(i) in X(j) for these (i,j): (3, 122), (107, 5)
X(10745) = complement of X(5667)


X(10746) =  REFLECTION OF X(108) IN X(5)

Barycentrics    (a^2 - b^2 - c^2) (a^11 - a^10 b - a^9 b^2 + a^8 b^3 - 2 a^7 b^4 + 2 a^6 b^5 + 2 a^5 b^6 - 2 a^4 b^7 + a^3 b^8 - a^2 b^9 - a b^10 + b^11 - a^10 c + 3 a^9 b c - a^8 b^2 c + a^7 b^3 c - 7 a^5 b^5 c + 6 a^4 b^6 c - a^3 b^7 c - 3 a^2 b^8 c + 4 a b^9 c - b^10 c - a^9 c^2 - a^8 b c^2 + 2 a^7 b^2 c^2 - 2 a^6 b^3 c^2 + 2 a^5 b^4 c^2 + 2 a^4 b^5 c^2 - 6 a^3 b^6 c^2 + 6 a^2 b^7 c^2 + 3 a b^8 c^2 - 5 b^9 c^2 + a^8 c^3 + a^7 b c^3 - 2 a^6 b^2 c^3 + 6 a^5 b^3 c^3 - 6 a^4 b^4 c^3 + a^3 b^5 c^3 + 10 a^2 b^6 c^3 - 16 a b^7 c^3 + 5 b^8 c^3 - 2 a^7 c^4 + 2 a^5 b^2 c^4 - 6 a^4 b^3 c^4 + 10 a^3 b^4 c^4 - 12 a^2 b^5 c^4 - 2 a b^6 c^4 + 10 b^7 c^4 + 2 a^6 c^5 - 7 a^5 b c^5 + 2 a^4 b^2 c^5 + a^3 b^3 c^5 - 12 a^2 b^4 c^5 + 24 a b^5 c^5 - 10 b^6 c^5 + 2 a^5 c^6 + 6 a^4 b c^6 - 6 a^3 b^2 c^6 + 10 a^2 b^3 c^6 - 2 a b^4 c^6 - 10 b^5 c^6 - 2 a^4 c^7 - a^3 b c^7 + 6 a^2 b^2 c^7 - 16 a b^3 c^7 + 10 b^4 c^7 + a^3 c^8 - 3 a^2 b c^8 + 3 a b^2 c^8 + 5 b^3 c^8 - a^2 c^9 + 4 a b c^9 - 5 b^2 c^9 - a c^10 - b c^10 + c^11) : :
X(10746) = 5 X[1656] - 4 X[6717]

X(10746) lies on these lines:
{3,119}, {4,280}, {5,108}, {30,1295}, {265,2850}, {268,5514}, {355,2817}, {1359,1478}, {1479,3318}, {1656,6717}, {1809,5080}, {2778,5887}, {2791,6033}, {2798,6321}

X(10746) = reflection of X(i) in X(j) for these (i,j): (3, 123), (108, 5)
X(10746) = inverse-in-Stammler-circle of X(9913)

X(10747) =  REFLECTION OF X(109) IN X(5)

Barycentrics    a^10 - a^9 b - 2 a^8 b^2 + 2 a^7 b^3 + a^6 b^4 - a^4 b^6 - 2 a^3 b^7 + 2 a^2 b^8 + a b^9 - b^10 - a^9 c + 3 a^8 b c - 4 a^6 b^3 c + 4 a^3 b^6 c - 3 a b^8 c + b^9 c - 2 a^8 c^2 + 4 a^6 b^2 c^2 + a^4 b^4 c^2 - 6 a^2 b^6 c^2 + 3 b^8 c^2 + 2 a^7 c^3 - 4 a^6 b c^3 - 2 a^3 b^4 c^3 + 8 a b^6 c^3 - 4 b^7 c^3 + a^6 c^4 + a^4 b^2 c^4 - 2 a^3 b^3 c^4 + 8 a^2 b^4 c^4 - 6 a b^5 c^4 - 2 b^6 c^4 - 6 a b^4 c^5 + 6 b^5 c^5 - a^4 c^6 + 4 a^3 b c^6 - 6 a^2 b^2 c^6 + 8 a b^3 c^6 - 2 b^4 c^6 - 2 a^3 c^7 - 4 b^3 c^7 + 2 a^2 c^8 - 3 a b c^8 + 3 b^2 c^8 + a c^9 + b c^9 - c^10 : :
X(10747) = 3 X[4] - X[151], 2 X[117] - 3 X[381], 3 X[3] - 4 X[6711], 3 X[124] - 2 X[6711], 5 X[1656] - 4 X[6718]

Let Q be the quadrilateral ABCX(102). Taking the vertices 3 at a time yields four triangles whose orthocenters are the vertices of a cyclic quadrilateral whose circumcenter is X(10747). (Randy Hutson, December 10, 2016)

X(10747) lies on these lines:
{3,124}, {4,151}, {5,109}, {11,1795}, {30,102}, {117,381}, {265,2773}, {355,2800}, {1361,1478}, {1364,1479}, {1656,6718}, {1836,1845}, {2779,7728}, {2785,6321}, {2792,6033}

X(10747) = reflection of X(i) in X(j) for these (i,j): (3, 124), (109, 5)
X(10747) = {X(),X()}-harmonic conjugate of X()

X(10748) =  REFLECTION OF X(111) IN X(5)

Barycentrics    a^10 - 2 a^8 b^2 - 3 a^6 b^4 + 3 a^4 b^6 + 2 a^2 b^8 - b^10 - 2 a^8 c^2 + 3 a^6 b^2 c^2 + 4 a^4 b^4 c^2 - 14 a^2 b^6 c^2 + 5 b^8 c^2 - 3 a^6 c^4 + 4 a^4 b^2 c^4 + 16 a^2 b^4 c^4 - 4 b^6 c^4 + 3 a^4 c^6 - 14 a^2 b^2 c^6 - 4 b^4 c^6 + 2 a^2 c^8 + 5 b^2 c^8 - c^10 : :
X(10748) = 3 X[381] - 2 X[5512], 5 X[1656] - 4 X[6719], 3 X[5055] - 2 X[9172]

X(10748) lies on these lines:
{3,126}, {5,111}, {30,1296}, {114,381}, {156,3048}, {265,1352}, {1478,3325}, {1479,6019}, {1656,6719}, {2780,7728}, {2793,6033}, {5055,9172}

X(10748) = reflection of X(i) in X(j) for these (i,j): (3, 126), (111, 5)
X(10748) = singular focus of the Droussent cubic, K008

X(10749) =  REFLECTION OF X(112) IN X(5)

Barycentrics    (a^2 - b^2 - c^2) (a^12 - a^10 b^2 - a^8 b^4 + 2 a^6 b^6 - a^4 b^8 - a^2 b^10 + b^12 - a^10 c^2 + 3 a^8 b^2 c^2 - 2 a^6 b^4 c^2 + 3 a^4 b^6 c^2 + a^2 b^8 c^2 - 4 b^10 c^2 - a^8 c^4 - 2 a^6 b^2 c^4 - 4 a^4 b^4 c^4 + 7 b^8 c^4 + 2 a^6 c^6 + 3 a^4 b^2 c^6 - 8 b^6 c^6 - a^4 c^8 + a^2 b^2 c^8 + 7 b^4 c^8 - a^2 c^10 - 4 b^2 c^10 + c^12) : :
X(10749) = 2 X[132] - 3 X[381], 5 X[1656] - 4 X[6720]

X(10749) lies on these lines:
{3,114}, {4,339}, {5,112}, {30,935}, {132,381}, {265,879}, {1352,2781}, {1368,9775}, {1478,3320}, {1479,6020}, {1656,6720}, {2799,6321}, {3830,9530}, {6676,9157}

X(10749) = reflection of X(i) in X(j) for these (i,j): (3, 127), (112, 5)
X(10749) = inverse-in-Stammler-circle of X(9861)
X(10749) = X(112)-of-Johnson-triangle
X(10749) = {X(13296),X(13297)}-harmonic conjugate of X(1)

X(10750) =  REFLECTION OF X(1113) IN X(5)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4-(a^2+b^2-c^2) (a^2-b^2+c^2) J : : , where J = |OH|/R
X(10750) = 3 X[381] - 2 X[1312], X[2100] -3 X[5587]

Of the two Euler line intercepts of the Johnson circle, X(10750) is the closer to X(5). (Randy Hutson, December 10, 2016)

X(10750) lies on these lines: {2,3}, {265,2574}, {952,2102}, {2100,5587}, {2104,3564}, {2575,7728}

X(10750) = reflection of X(i) in X(j) for these (i,j): (3,1113), (1113,5), (10751,4)
X(10750) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,2552,3845), (5,3153,10751), (1313,10297,381)

X(10751) =  REFLECTION OF X(1114) IN X(5)

Barycentrics    2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4+(a^2+b^2-c^2) (a^2-b^2+c^2) J : : ,, where J = |OH|/R
X(10751) = 3 X[381] - 2 X[1313], X[2101] -3 X[5587]

Of the two Euler line intercepts of the Johnson circle, X(10751) is the farther from X(5). (Randy Hutson, December 10, 2016)

X(10751) lies on these lines: {2,3}, {265,2575}, {952,2103}, {2101,5587}, {2105,3564}, {2574,7728}

X(10751) = reflection of X(i) in X(j) for these (i,j): (3,1114), (1114,5), (10750,4)
X(10751) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,2553,3845), (5,3153,10750), (1312,10297,381)




leftri  Reflections of circumcircle-points in the symmedian point: X(10752) - X(10766)  rightri

This preamble and centers X(10752)-X(10766) were contributed by Clark Kimberling and Peter Moses, November 10, 2016.

Suppose that P is a point on the circumcrcle of a triangle ABC, and let

O = X(3), the circumcenter of ABC
K = X(6), the symmedian point of ABC
M = midpoint of Pc and O
P' = reflection of P in K
Pc = complement of P
Pa = anticomplement of P.

Then

P' = midpoint of X(193) and Pa
P' = reflection of X(69) in Pc
P' = 4M - 3X(10519).

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(6): (74, 10752), (98,10753), (99,10754), (100,10755), (101,10756), (102,10757), (103,10758), (104,10759), (105,10760), (106,10761), (107,10762), (108,10763), (109,10764), (110,895), (111,10765), (112,10766), (1113,2104), (1114,2105)

underbar

X(10752) =  REFLECTION OF X(74) IN X(6)

Barycentrics    a^2 (a^10-6 a^8 b^2+8 a^6 b^4+2 a^4 b^6-9 a^2 b^8+4 b^10-6 a^8 c^2+5 a^6 b^2 c^2-7 a^4 b^4 c^2+11 a^2 b^6 c^2-3 b^8 c^2+8 a^6 c^4-7 a^4 b^2 c^4-4 a^2 b^4 c^4-b^6 c^4+2 a^4 c^6+11 a^2 b^2 c^6-b^4 c^6-9 a^2 c^8-3 b^2 c^8+4 c^10) : :
X(10752) = 5 X[74] - 6 X[5621], 5 X[6] - 3 X[5621], 4 X[5621] - 5 X[5622], 4 X[6] - 3 X[5622], 2 X[74] - 3 X[5622], 5 X[3618] - 4 X[6699], 4 X[5972] - 3 X[10519], 3 X[5093] - X[10620]

X(10752) lies on these lines:
{4,5505}, {6,74}, {23,110}, {67,5480}, {69,113}, {146,148}, {206,3043}, {518,7978}, {524,1514}, {541,1992}, {895,1351}, {1177,3431}, {1350,6593}, {2104,2575}, {2105,2574}, {2777,5095}, {3098,10298}, {3331,8586}, {3564,7728}, {3618,6699}, {5093,10620}, {5972,10519}

X(10752) = midpoint of X(146) and X(193)
X(10752) = reflection of X(i) and X(j) for these (i,j): (67, 5480), (69, 113), (74, 6), (110, 9970), (895, 1351), (1350, 6593), (6776, 5095)
X(10752) = crossdifference of every pair of points on line X(1640) X(9033)
X(10752) = {X(6),X(74)}-harmonic conjugate of X(5622)

X(10753) =  REFLECTION OF X(98) IN X(6)

Barycentrics    a^10-6 a^8 b^2+8 a^6 b^4-6 a^4 b^6+3 a^2 b^8-6 a^8 c^2+5 a^6 b^2 c^2+a^4 b^4 c^2-5 a^2 b^6 c^2+b^8 c^2+8 a^6 c^4+a^4 b^2 c^4+4 a^2 b^4 c^4-b^6 c^4-6 a^4 c^6-5 a^2 b^2 c^6-b^4 c^6+3 a^2 c^8+b^2 c^8 : :
X(10753) = 2 X[3] - 3 X[5182], 5 X[3618] - 4 X[6036], 7 X[3619] - 8 X[6721], 4 X[5476] - 3 X[9166], 4 X[620] - 3 X[10519]

X(10753) lies on these lines:
{3,5182}, {4,542}, {6,98}, {30,8593}, {69,114}, {99,511}, {110,7417}, {147,193}, {287,1316}, {518,7970}, {524,1513}, {620,10519}, {1350,5026}, {1351,2782}, {1503,5111}, {1994,5986}, {2549,2794}, {2790,10602}, {3564,6033}, {3618,6036}, {3619,6721}, {5468,9775}, {5476,9166}, {5847,9864}, {6782,9750}, {6783,9749}, {7758,9890}, {8550,9607}

X(10753) = midpoint of X(147) and X(193)
X(10753) = reflection of X(i) in X(j) for these (i,j): (69, 114), (98, 6), (1350, 5026), (6776, 5477), (10754,1351)

X(10754) =  REFLECTION OF X(99) IN X(6)

Barycentrics    a^6-2 a^4 b^2+3 a^2 b^4-2 a^4 c^2-a^2 b^2 c^2-b^4 c^2+3 a^2 c^4-b^2 c^4 : :
X(10754) = 4 X[620] - 5 X[3618], 3 X[99] - 4 X[5026], 3 X[6] - 2 X[5026], 8 X[5026] - 9 X[5182], 4 X[6] - 3 X[5182], 2 X[99] - 3 X[5182], 3 X[1992] - 2 X[5477], 2 X[141] - 3 X[6034], 7 X[3619] - 8 X[6722], 3 X[5032] - X[8591], 4 X[5477] - 3 X[8593], 2 X[599] - 3 X[9166], 4 X[6036] - 3 X[10519]

X(10754) lies on these lines:
{2,5503}, {6,99}, {69,115}, {83,10290}, {98,385}, {111,5468}, {141,6034}, {146,148}, {187,9888}, {194,576}, {206,3044}, {287,2395}, {316,524}, {323,9870}, {518,7983}, {538,5107}, {543,1992}, {575,7783}, {599,7934}, {620,3618}, {690,895}, {698,5111}, {877,6531}, {1351,2782}, {2796,4780}, {3124,4563}, {3564,6321}, {3619,6722}, {3734,5028}, {4027,5039}, {5017,5152}, {5032,8591}, {5476,7777}, {5642,7665}, {6036,10519}, {6054,7774}, {6393,8781}, {7770,10542}, {8782,10352}, {9143,10552}

X(10754) = midpoint of X(148) and X(193)
X(10754) = reflection of X(i) in X(j) for these (i,j): (69, 115), (99, 6), (8593, 1992), (10753,1351)
X(10754) = X(9154)-Ceva conjugate of X(2)
X(10754) = crosspoint of X(671) and X(8781)
X(10754) = crosssum of X(i) and X(j) for these (i,j): {187,1692}, {351,2086}
X(10754) = crossdifference of every pair of points on line X(888)X(9135)
X(10754) = X(923)-anticomplementary conjugate of X(147)
X(10754) = X(9154)-anticomplementary conjugate of X(6327)
X(10754) = {X(6),X(99)}-harmonic conjugate of X(5182)

X(10755) =  REFLECTION OF X(100) IN X(6)

Barycentrics    a (a^4-a^3 b-3 a^2 b^2+3 a b^3-a^3 c+5 a^2 b c-a b^2 c-b^3 c-3 a^2 c^2-a b c^2+3 a c^3-b c^3) : :
X(10755) = 4 X[3035] - 5 X[3618], 7 X[3619] - 8 X[6667], 4 X[6713] - 3 X[10519]

X(10755) lies on these lines:
{6,100}, {11,69}, {80,5847}, {104,511}, {105,4585}, {149,193}, {206,3045}, {518,1156}, {528,1992}, {895,2991}, {952,1351}, {1026,2316}, {1027,1814}, {1083,1332}, {2802,3751}, {3035,3618}, {3619,6667}, {5378,7077}, {5840,6776}, {6713,10519}, {8540,9025}

X(10755) = midpoint of X(149) and X(193)
X(10755) = reflection of X(i) in X(j) for these (i,j): (69, 11), (100, 6)

X(10756) =  REFLECTION OF X(101) IN X(6)

Barycentrics    a^2 (a^4-a^3 b-a^2 b^2-a b^3+2 b^4-a^3 c+a^2 b c+3 a b^2 c-3 b^3 c-a^2 c^2+3 a b c^2-a c^3-3 b c^3+2 c^4) : :
X(10756) = 5 X[3618] - 4 X[6710], 4 X[6712] - 3 X[10519]

X(10756) lies on these lines:
{6,101}, {69,116}, {103,511}, {150,193}, {206,3046}, {544,1992}, {649,1797}, {895,2774}, {1024,1814}, {1351,2808}, {2809,3751}, {3618,6710}, {4845,8540}, {6712,10519}

X(10756) = midpoint of X(150) and X(193)
X(10756) = reflection of X(i) in X(j) for these (i,j): (69, 116), (101, 6)

X(10757) =  REFLECTION OF X(102) IN X(6)

Barycentrics    a^2 (a^10-a^9 b-6 a^8 b^2+6 a^7 b^3+8 a^6 b^4-12 a^5 b^5+2 a^4 b^6+10 a^3 b^7-9 a^2 b^8-3 a b^9+4 b^10-a^9 c+3 a^8 b c+4 a^7 b^2 c-16 a^6 b^3 c+6 a^5 b^4 c+18 a^4 b^5 c-20 a^3 b^6 c+11 a b^8 c-5 b^9 c-6 a^8 c^2+4 a^7 b c^2+6 a^6 b^2 c^2+6 a^5 b^3 c^2-20 a^4 b^4 c^2+22 a^2 b^6 c^2-10 a b^7 c^2-2 b^8 c^2+6 a^7 c^3-16 a^6 b c^3+6 a^5 b^2 c^3+10 a^3 b^4 c^3-8 a^2 b^5 c^3-6 a b^6 c^3+8 b^7 c^3+8 a^6 c^4+6 a^5 b c^4-20 a^4 b^2 c^4+10 a^3 b^3 c^4-10 a^2 b^4 c^4+8 a b^5 c^4-2 b^6 c^4-12 a^5 c^5+18 a^4 b c^5-8 a^2 b^3 c^5+8 a b^4 c^5-6 b^5 c^5+2 a^4 c^6-20 a^3 b c^6+22 a^2 b^2 c^6-6 a b^3 c^6-2 b^4 c^6+10 a^3 c^7-10 a b^2 c^7+8 b^3 c^7-9 a^2 c^8+11 a b c^8-2 b^2 c^8-3 a c^9-5 b c^9+4 c^10) : :
X(10757) = 5 X[3618] - 4 X[6711], 4 X[6718] - 3 X[10519]

X(10757) lies on these lines:
{6,102}, {69,117}, {109,511}, {151,193}, {895,2779}, {1351,2818}, {2817,3751}, {3618,6711}, {6718,10519}

X(10757) = midpoint of X(151) and X(193)
X(10757) = reflection of X(i) in X(j) for these (i,j): (69, 117), (102, 6)

X(10758) =  REFLECTION OF X(103) IN X(6)

Barycentrics    a^2 (a^8-a^7 b-5 a^6 b^2+5 a^5 b^3+3 a^4 b^4+a^3 b^5-3 a^2 b^6-5 a b^7+4 b^8-a^7 c+a^6 b c+5 a^5 b^2 c-5 a^4 b^3 c-7 a^3 b^4 c+7 a^2 b^5 c+3 a b^6 c-3 b^7 c-5 a^6 c^2+5 a^5 b c^2-6 a^4 b^2 c^2+6 a^3 b^3 c^2-5 a^2 b^4 c^2+5 a b^5 c^2+5 a^5 c^3-5 a^4 b c^3+6 a^3 b^2 c^3+2 a^2 b^3 c^3-3 a b^4 c^3-5 b^5 c^3+3 a^4 c^4-7 a^3 b c^4-5 a^2 b^2 c^4-3 a b^3 c^4+8 b^4 c^4+a^3 c^5+7 a^2 b c^5+5 a b^2 c^5-5 b^3 c^5-3 a^2 c^6+3 a b c^6-5 a c^7-3 b c^7+4 c^8) : :
X(10758) = 5 X[3618] - 4 X[6712], 4 X[6710] - 3 X[10519]

X(10758) lies on these lines:
{6,103}, {69,118}, {101,511}, {152,193}, {895,2772}, {1351,2808}, {3618,6712}, {6710,10519}

X(10758) = midpoint of X(152) and X(193)
X(10758) = reflection of X(i) in X(j) for these (i,j): (69, 118), (103, 6)

X(10759) =  REFLECTION OF X(104) IN X(6)

Barycentrics    a (a^8-a^7 b-5 a^6 b^2+5 a^5 b^3+7 a^4 b^4-7 a^3 b^5-3 a^2 b^6+3 a b^7-a^7 c+a^6 b c+5 a^5 b^2 c-13 a^4 b^3 c+a^3 b^4 c+11 a^2 b^5 c-5 a b^6 c+b^7 c-5 a^6 c^2+5 a^5 b c^2+2 a^4 b^2 c^2+6 a^3 b^3 c^2-5 a^2 b^4 c^2-3 a b^5 c^2+5 a^5 c^3-13 a^4 b c^3+6 a^3 b^2 c^3-6 a^2 b^3 c^3+5 a b^4 c^3-b^5 c^3+7 a^4 c^4+a^3 b c^4-5 a^2 b^2 c^4+5 a b^3 c^4-7 a^3 c^5+11 a^2 b c^5-3 a b^2 c^5-b^3 c^5-3 a^2 c^6-5 a b c^6+3 a c^7+b c^7) : :

X(10759) lies on these lines:
{4,5848}, {6,104}, {69,119}, {100,511}, {153,193}, {895,2771}, {952,1351}, {2800,3751}, {2829,6776}, {3035,10519}, {3618,6713}, {6905,9037}

X(10759) = midpoint of X(153) and X(193)
X(10759) = reflection of X(i) in X(j) for these (i,j): (69, 119), (104, 6)

X(10760) =  REFLECTION OF X(105) IN X(6)

Barycentrics    a (a^6-a^5 b-2 a^4 b^2+2 a^3 b^3-3 a^2 b^4+3 a b^5-a^5 c-5 a^4 b c+8 a^3 b^2 c-4 a^2 b^3 c-3 a b^4 c+b^5 c-2 a^4 c^2+8 a^3 b c^2+4 a^2 b^2 c^2-2 b^4 c^2+2 a^3 c^3-4 a^2 b c^3+2 b^3 c^3-3 a^2 c^4-3 a b c^4-2 b^2 c^4+3 a c^5+b c^5) : :
X(10760) = 5 X[3618] - 4 X[6714]

X(10760) lies on these lines:
{6,105}, {65,651}, {69,120}, {511,1292}, {518,644}, {528,1992}, {1814,5091}, {2809,3751}, {2834,10602}, {3618,6714}

X(10760) = reflection of X(i) in X(j) for these (i,j): (69, 120), (105, 6)

X(10761) =  REFLECTION OF X(106) IN X(6)

Barycentrics    a^2 (a^4-a^3 b-7 a^2 b^2-a b^3+4 b^4-a^3 c+5 a^2 b c+11 a b^2 c-7 b^3 c-7 a^2 c^2+11 a b c^2-4 b^2 c^2-a c^3-7 b c^3+4 c^4) : :
X(10761) = 5 X[3618]-4 X[6715]

X(10761) lies on these lines:
{6,101}, {69,121}, {511,1293}, {895,2842}, {902,1331}, {2796,4780}, {2802,3751}, {3618,6715}

X(10761) = reflection of X(i) in X(j) for these (i,j): (69, 121), (106, 6)

X(10762) =  REFLECTION OF X(107) IN X(6)

Barycentrics    (a^2-b^2-c^2) (a^6-3 a^4 b^2+a^2 b^4+b^6+2 a^4 c^2+2 a^2 b^2 c^2-2 b^4 c^2-3 a^2 c^4+b^2 c^4) (a^6+2 a^4 b^2-3 a^2 b^4-3 a^4 c^2+2 a^2 b^2 c^2+b^4 c^2+a^2 c^4-2 b^2 c^4+c^6) : :
X(10762) = 5 X[3618] - 4 X[6716]

X(10762) lies on these lines:
{6,107}, {69,122}, {287,879}, {511,1294}, {1503,10152}, {1992,9530}, {2777,5095}, {2790,10602}, {3618,6716}

X(10762) = reflection of X(i) in X(j) for these (i,j): (69, 122), (107, 6)

X(10763) =  REFLECTION OF X(108) IN X(6)

Barycentrics    a (a^2-b^2-c^2) (a^8-a^7 b-3 a^6 b^2+3 a^5 b^3-a^4 b^4+a^3 b^5+3 a^2 b^6-3 a b^7-a^7 c+7 a^6 b c-3 a^5 b^2 c-5 a^4 b^3 c+9 a^3 b^4 c-11 a^2 b^5 c+3 a b^6 c+b^7 c-3 a^6 c^2-3 a^5 b c^2+12 a^4 b^2 c^2-10 a^3 b^3 c^2-3 a^2 b^4 c^2+9 a b^5 c^2-2 b^6 c^2+3 a^5 c^3-5 a^4 b c^3-10 a^3 b^2 c^3+22 a^2 b^3 c^3-9 a b^4 c^3-b^5 c^3-a^4 c^4+9 a^3 b c^4-3 a^2 b^2 c^4-9 a b^3 c^4+4 b^4 c^4+a^3 c^5-11 a^2 b c^5+9 a b^2 c^5-b^3 c^5+3 a^2 c^6+3 a b c^6-2 b^2 c^6-3 a c^7+b c^7) : :
X(10763) = 5 X[3618] - 4 X[6717]

X(10763) lies on these lines:
{6,108}, {69,123}, {511,1295}, {895,1814}, {2817,3751}, {2829,6776}, {2834,10602}, {3618,6717}

X(10763) = reflection of X(i) in X(j) for these (i,j): (69, 123), (108, 6)

X(10764) =  REFLECTION OF X(109) IN X(6)

Barycentrics    a^2 (a^6-a^5 b-2 a^4 b^2+4 a^3 b^3-a^2 b^4-3 a b^5+2 b^6-a^5 c+3 a^4 b c-2 a^3 b^2 c-6 a^2 b^3 c+7 a b^4 c-b^5 c-2 a^4 c^2-2 a^3 b c^2+12 a^2 b^2 c^2-4 a b^3 c^2-4 b^4 c^2+4 a^3 c^3-6 a^2 b c^3-4 a b^2 c^3+6 b^3 c^3-a^2 c^4+7 a b c^4-4 b^2 c^4-3 a c^5-b c^5+2 c^6) : :
X(10764) = 5 X[3618] - 4 X[6718], 4 X[6711] - 3 X[10519]

X(10764) lies on these lines:
{6,109}, {69,124}, {102,511}, {895,2773}, {1027,1814}, {1351,2818}, {2800,3751}, {3618,6718}, {6711,10519}

X(10764) = reflection of X(i) in X(j) for these (i,j)}: (69, 124), (109, 6)

X(10765) =  REFLECTION OF X(111) IN X(6)

Barycentrics    a^2 (a^6-6 a^4 b^2-3 a^2 b^4+4 b^6-6 a^4 c^2+27 a^2 b^2 c^2-9 b^4 c^2-3 a^2 c^4-9 b^2 c^4+4 c^6) : :
X(10765) = 5 X[3618] - 4 X[6719]

X(10765) lies on these lines:
{6,110}, {69,126}, {187,4558}, {206,3048}, {352,9027}, {511,843}, {524,9146}, {543,1992}, {1383,6096}, {3618,6719}, {5166,8681}, {5912,9775}, {7708,8542}, {9023,9156}

X(10765) = reflection of X(i) in X(j) for these (i,j): (69, 126), (111, 6)

X(10766) =  REFLECTION OF X(112) IN X(6)

Barycentrics    a^2 (a^2-b^2-c^2) (a^8-a^6 b^2+a^4 b^4+a^2 b^6-2 b^8-a^6 c^2-a^4 b^2 c^2-a^2 b^4 c^2+3 b^6 c^2+a^4 c^4-a^2 b^2 c^4-2 b^4 c^4+a^2 c^6+3 b^2 c^6-2 c^8) : :
X(10766) = 5 X[3618] - 4 X[6720]

X(10766) lies on these lines:
{6,74}, {69,127}, {125,6792}, {184,353}, {287,2395}, {511,1297}, {542,1562}, {895,9517}, {1181,10542}, {2549,2794}, {3618,6720}

X(10766) = reflection of X(i) in X(j) for these (i,j): (69, 127), (112, 6)
X(10766) = crossdifference of every pair of points on line X(5095) X(9033)
X(10766) = inverse-in-2nd-Lemoine-circle of X(5622)


leftri  Reflections of circumcircle-points in the Feuerbach point: X(10767) - X(10782)  rightri

This preamble and centers X(107676)-X(10782) were contributed by Clark Kimberling and Peter Moses, November 10, 2016.

The appearance of (i,j) in the following list means that X(j) = reflection of X(i) in X(11): (74, 10767), (98,10768), (99,10769), (100,149), (101,10770), (102,10771), (103,10772), (104,4), (105,10773), (106,10774), (107,10775), (108,10776), (109,10777), (110,10778), (111,10779), (112,10780), (1113,10781), (1114,10782)

underbar

X(10767) =  REFLECTION OF X(74) IN X(11)

Barycentrics    a^13-a^12 b-a^11 b^2+a^10 b^3-5 a^9 b^4+5 a^8 b^5+10 a^7 b^6-10 a^6 b^7-5 a^5 b^8+5 a^4 b^9-a^3 b^10+a^2 b^11+a b^12-b^13-a^12 c+a^11 b c+a^10 b^2 c+3 a^9 b^3 c+a^8 b^4 c-13 a^7 b^5 c+11 a^5 b^7 c-a^4 b^8 c-a^2 b^10 c-2 a b^11 c+b^12 c-a^11 c^2+a^10 b c^2+3 a^9 b^2 c^2-7 a^8 b^3 c^2-2 a^7 b^4 c^2+12 a^6 b^5 c^2-3 a^5 b^6 c^2-3 a^4 b^7 c^2+5 a^3 b^8 c^2-7 a^2 b^9 c^2-2 a b^10 c^2+4 b^11 c^2+a^10 c^3+3 a^9 b c^3-7 a^8 b^2 c^3+11 a^7 b^3 c^3-2 a^6 b^4 c^3-7 a^5 b^5 c^3+5 a^4 b^6 c^3-13 a^3 b^7 c^3+7 a^2 b^8 c^3+6 a b^9 c^3-4 b^10 c^3-5 a^9 c^4+a^8 b c^4-2 a^7 b^2 c^4-2 a^6 b^3 c^4+8 a^5 b^4 c^4-6 a^4 b^5 c^4-4 a^3 b^6 c^4+16 a^2 b^7 c^4-a b^8 c^4-5 b^9 c^4+5 a^8 c^5-13 a^7 b c^5+12 a^6 b^2 c^5-7 a^5 b^3 c^5-6 a^4 b^4 c^5+26 a^3 b^5 c^5-16 a^2 b^6 c^5-4 a b^7 c^5+5 b^8 c^5+10 a^7 c^6-3 a^5 b^2 c^6+5 a^4 b^3 c^6-4 a^3 b^4 c^6-16 a^2 b^5 c^6+4 a b^6 c^6-10 a^6 c^7+11 a^5 b c^7-3 a^4 b^2 c^7-13 a^3 b^3 c^7+16 a^2 b^4 c^7-4 a b^5 c^7-5 a^5 c^8-a^4 b c^8+5 a^3 b^2 c^8+7 a^2 b^3 c^8-a b^4 c^8+5 b^5 c^8+5 a^4 c^9-7 a^2 b^2 c^9+6 a b^3 c^9-5 b^4 c^9-a^3 c^10-a^2 b c^10-2 a b^2 c^10-4 b^3 c^10+a^2 c^11-2 a b c^11+4 b^2 c^11+a c^12+b c^12-c^13: :

X(10767) lies on these lines:
{4,8674}, {11,74}, {100,113}, {104,2777}, {110,5840}, {146,149}, {952,7728}, {5533,10081}, {8068,10065}

X(10767) = midpoint of X(146) and X(149)
X(10767) = reflection of X(i) in X(j) for these (i,j): (74, 11), (100, 113)

X(10768) =  REFLECTION OF X(98) IN X(11)

Barycentrics    a^11-a^10 b-a^9 b^2+a^8 b^3-a^7 b^4+a^6 b^5+2 a^5 b^6-2 a^4 b^7-2 a^3 b^8+2 a^2 b^9+a b^10-b^11-a^10 c+a^9 b c+a^8 b^2 c+3 a^7 b^3 c-3 a^6 b^4 c-5 a^5 b^5 c+4 a^4 b^6 c+3 a^3 b^7 c-2 a^2 b^8 c-2 a b^9 c+b^10 c-a^9 c^2+a^8 b c^2-5 a^7 b^2 c^2+a^6 b^3 c^2+6 a^5 b^4 c^2-3 a^3 b^6 c^2-a^2 b^7 c^2-a b^8 c^2+3 b^9 c^2+a^8 c^3+3 a^7 b c^3+a^6 b^2 c^3-5 a^5 b^3 c^3-2 a^4 b^4 c^3+a^3 b^5 c^3+3 a^2 b^6 c^3+3 a b^7 c^3-3 b^8 c^3-a^7 c^4-3 a^6 b c^4+6 a^5 b^2 c^4-2 a^4 b^3 c^4+2 a^3 b^4 c^4-2 a^2 b^5 c^4-4 b^7 c^4+a^6 c^5-5 a^5 b c^5+a^3 b^3 c^5-2 a^2 b^4 c^5-2 a b^5 c^5+4 b^6 c^5+2 a^5 c^6+4 a^4 b c^6-3 a^3 b^2 c^6+3 a^2 b^3 c^6+4 b^5 c^6-2 a^4 c^7+3 a^3 b c^7-a^2 b^2 c^7+3 a b^3 c^7-4 b^4 c^7-2 a^3 c^8-2 a^2 b c^8-a b^2 c^8-3 b^3 c^8+2 a^2 c^9-2 a b c^9+3 b^2 c^9+a c^10+b c^10-c^11: :

X(10768) lies on these lines:
{4,2787}, {11,98}, {99,5840}, {100,114}, {104,2794}, {147,149}, {528,6054}, {952,6033}, {2802,9864}, {5533,10069}, {8068,10053}

X(10768) = midpoint of X(147) and X(149)
X(10768) = reflection of X(i) in X(j) for these (i,j): (98, 11), (100, 114)

X(10769) =  REFLECTION OF X(99) IN X(11)

Barycentrics    a^7-a^6 b-a^5 b^2+a^4 b^3+a b^6-b^7-a^6 c+a^5 b c+a^4 b^2 c-a^3 b^3 c-2 a b^5 c+b^6 c-a^5 c^2+a^4 b c^2+a^3 b^2 c^2-a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+a^4 c^3-a^3 b c^3-a^2 b^2 c^3+5 a b^3 c^3-3 b^4 c^3-a b^2 c^4-3 b^3 c^4-2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
X(10769) = 2 X[6174] - 3 X[9166]

X(10769) lies on these lines:
{4,2783}, {11,99}, {98,5840}, {100,115}, {148,149}, {528,671}, {952,6321}, {5533,10089}, {6174,9166}, {6246,9864}, {8068,10086}

X(10769) = midpoint of X(148) and X(149)
X(10769) = reflection of X(i) in X(j) for these (i,j): (99, 11), (100, 115), (9864, 6246)

X(10770) =  REFLECTION OF X(101) IN X(11)

Barycentrics    a^7-2 a^6 b+a^5 b^2-a^2 b^5+2 a b^6-b^7-2 a^6 c+4 a^5 b c-2 a^4 b^2 c+3 a^2 b^4 c-5 a b^5 c+2 b^6 c+a^5 c^2-2 a^4 b c^2+a^3 b^2 c^2-2 a^2 b^3 c^2+2 a b^4 c^2-2 a^2 b^2 c^3+2 a b^3 c^3-b^4 c^3+3 a^2 b c^4+2 a b^2 c^4-b^3 c^4-a^2 c^5-5 a b c^5+2 a c^6+2 b c^6-c^7 : :

X(10770) lies on these lines:
{4,2801}, {11,101}, {80,2809}, {100,116}, {103,5840}, {149,150}

X(10770) = midpoint of X(149) and X(150)
X(10770) = reflection of X(i) in X(j) for these (i,j): (100, 116), (101, 11)

X(10771) =  REFLECTION OF X(102) IN X(11)

Barycentrics    a^13-2 a^12 b+4 a^10 b^3-8 a^9 b^4+3 a^8 b^5+12 a^7 b^6-12 a^6 b^7-3 a^5 b^8+8 a^4 b^9-4 a^3 b^10+2 a b^12-b^13-2 a^12 c+6 a^11 b c-6 a^10 b^2 c+17 a^8 b^4 c-29 a^7 b^5 c+3 a^6 b^6 c+27 a^5 b^7 c-21 a^4 b^8 c+3 a^3 b^9 c+7 a^2 b^10 c-7 a b^11 c+2 b^12 c-6 a^10 b c^2+17 a^9 b^2 c^2-20 a^8 b^3 c^2-3 a^7 b^4 c^2+43 a^6 b^5 c^2-43 a^5 b^6 c^2+a^4 b^7 c^2+27 a^3 b^8 c^2-21 a^2 b^9 c^2+2 a b^10 c^2+3 b^11 c^2+4 a^10 c^3-20 a^8 b^2 c^3+40 a^7 b^3 c^3-34 a^6 b^4 c^3-15 a^5 b^5 c^3+59 a^4 b^6 c^3-46 a^3 b^7 c^3+21 a b^9 c^3-9 b^10 c^3-8 a^9 c^4+17 a^8 b c^4-3 a^7 b^2 c^4-34 a^6 b^3 c^4+68 a^5 b^4 c^4-47 a^4 b^5 c^4-23 a^3 b^6 c^4+56 a^2 b^7 c^4-26 a b^8 c^4+3 a^8 c^5-29 a^7 b c^5+43 a^6 b^2 c^5-15 a^5 b^3 c^5-47 a^4 b^4 c^5+86 a^3 b^5 c^5-42 a^2 b^6 c^5-14 a b^7 c^5+15 b^8 c^5+12 a^7 c^6+3 a^6 b c^6-43 a^5 b^2 c^6+59 a^4 b^3 c^6-23 a^3 b^4 c^6-42 a^2 b^5 c^6+44 a b^6 c^6-10 b^7 c^6-12 a^6 c^7+27 a^5 b c^7+a^4 b^2 c^7-46 a^3 b^3 c^7+56 a^2 b^4 c^7-14 a b^5 c^7-10 b^6 c^7-3 a^5 c^8-21 a^4 b c^8+27 a^3 b^2 c^8-26 a b^4 c^8+15 b^5 c^8+8 a^4 c^9+3 a^3 b c^9-21 a^2 b^2 c^9+21 a b^3 c^9-4 a^3 c^10+7 a^2 b c^10+2 a b^2 c^10-9 b^3 c^10-7 a b c^11+3 b^2 c^11+2 a c^12+2 b c^12-c^13 : :

X(10771) lies on these lines:
{4,3738}, {11,102}, {80,2817}, {100,117}, {109,5840}, {149,151}, {1845,10073}, {2816,10265}

X(10771) = midpoint of X(149) and X(151)
X(10771) = reflection of X(i) in X(j) for these (i,j): (100, 117), (102, 11)

X(10772) =  REFLECTION OF X(103) IN X(11)

Barycentrics    a^11-2 a^10 b+a^9 b^2-2 a^8 b^3+a^7 b^4+7 a^6 b^5-7 a^5 b^6-a^4 b^7+2 a^3 b^8-a^2 b^9+2 a b^10-b^11-2 a^10 c+4 a^9 b c+2 a^7 b^3 c-9 a^6 b^4 c-a^5 b^5 c+8 a^4 b^6 c+a^2 b^8 c-5 a b^9 c+2 b^10 c+a^9 c^2-5 a^7 b^2 c^2+2 a^6 b^3 c^2+a^5 b^4 c^2+10 a^4 b^5 c^2-11 a^3 b^6 c^2+2 a^2 b^7 c^2-2 a b^8 c^2+2 b^9 c^2-2 a^8 c^3+2 a^7 b c^3+2 a^6 b^2 c^3+14 a^5 b^3 c^3-17 a^4 b^4 c^3-4 a^3 b^5 c^3-2 a^2 b^6 c^3+12 a b^7 c^3-5 b^8 c^3+a^7 c^4-9 a^6 b c^4+a^5 b^2 c^4-17 a^4 b^3 c^4+26 a^3 b^4 c^4-2 b^7 c^4+7 a^6 c^5-a^5 b c^5+10 a^4 b^2 c^5-4 a^3 b^3 c^5-14 a b^5 c^5+4 b^6 c^5-7 a^5 c^6+8 a^4 b c^6-11 a^3 b^2 c^6-2 a^2 b^3 c^6+4 b^5 c^6-a^4 c^7+2 a^2 b^2 c^7+12 a b^3 c^7-2 b^4 c^7+2 a^3 c^8+a^2 b c^8-2 a b^2 c^8-5 b^3 c^8-a^2 c^9-5 a b c^9+2 b^2 c^9+2 a c^10+2 b c^10-c^11 : :

X(10772) lies on these lines: {4,3887}, {11,103}, {100,118}, {101,5840}, {149,152}

X(10772) = midpoint of X(149) and X(152)
X(10772) = reflection of X(i) in X(j) for these (i,j): (100, 118), (103, 11)

X(10773) =  REFLECTION OF X(105) IN X(11)

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

X(10773) lies on the orthocentroidal circle and these lines: {2,11}, {4,2826}, {80,2809}, {1292,5840}

X(10773) = reflection of X(i) in X(j) for these (i,j): (100, 120), (105, 11)

X(10774) =  REFLECTION OF X(106) IN X(11)

Barycentrics    a^7-2 a^6 b-a^5 b^2+2 a^4 b^3-2 a^3 b^4+a^2 b^5+2 a b^6-b^7-2 a^6 c+8 a^5 b c-4 a^4 b^2 c+2 a^3 b^3 c+5 a^2 b^4 c-9 a b^5 c+2 b^6 c-a^5 c^2-4 a^4 b c^2+a^3 b^2 c^2-6 a^2 b^3 c^2-2 a b^4 c^2+4 b^5 c^2+2 a^4 c^3+2 a^3 b c^3-6 a^2 b^2 c^3+18 a b^3 c^3-5 b^4 c^3 -2 a^3 c^4+5 a^2 b c^4-2 a b^2 c^4-5 b^3 c^4+a^2 c^5-9 a b c^5+4 b^2 c^5+2 a c^6+2 b c^6-c^7 : :
X(10774) =

X(10774) lies on Fuhrmann circle and these lines: {4,2827}, {8,80}, {11,106}, {100,121}, {900,6788}, {1293,5840}

X(10774) = reflection of X(i) in X(j) for these (i,j): (100, 121), (106, 11)

X(10775) =  REFLECTION OF X(107) IN X(11)

Barycentrics    a^15-a^14 b-a^13 b^2+a^12 b^3-2 a^11 b^4+2 a^10 b^5-a^9 b^6+a^8 b^7+9 a^7 b^8-9 a^6 b^9-7 a^5 b^10+7 a^4 b^11+a b^14-b^15-a^14 c+a^13 b c+a^12 b^2 c-a^11 b^3 c+2 a^10 b^4 c+4 a^9 b^5 c-5 a^8 b^6 c-14 a^7 b^7 c+7 a^6 b^8 c+13 a^5 b^9 c-5 a^4 b^10 c-a^3 b^11 c-2 a b^13 c+b^14 c-a^13 c^2+a^12 b c^2+5 a^11 b^2 c^2-5 a^10 b^3 c^2+a^9 b^4 c^2-7 a^8 b^5 c^2-6 a^7 b^6 c^2+22 a^6 b^7 c^2-7 a^5 b^8 c^2-5 a^4 b^9 c^2+9 a^3 b^10 c^2-9 a^2 b^11 c^2-a b^12 c^2+3 b^13 c^2+a^12 c^3-a^11 b c^3-5 a^10 b^2 c^3-7 a^9 b^3 c^3+11 a^8 b^4 c^3+14 a^7 b^5 c^3-10 a^6 b^6 c^3+2 a^5 b^7 c^3-a^4 b^8 c^3-13 a^3 b^9 c^3+7 a^2 b^10 c^3+5 a b^11 c^3-3 b^12 c^3-2 a^11 c^4+2 a^10 b c^4+a^9 b^2 c^4+11 a^8 b^3 c^4-6 a^7 b^4 c^4-10 a^6 b^5 c^4+14 a^5 b^6 c^4-22 a^4 b^7 c^4-4 a^3 b^8 c^4+20 a^2 b^9 c^4-3 a b^10 c^4-b^11 c^4+2 a^10 c^5+4 a^9 b c^5-7 a^8 b^2 c^5+14 a^7 b^3 c^5-10 a^6 b^4 c^5-30 a^5 b^5 c^5+26 a^4 b^6 c^5+14 a^3 b^7 c^5-12 a^2 b^8 c^5-2 a b^9 c^5+b^10 c^5-a^9 c^6-5 a^8 b c^6-6 a^7 b^2 c^6-10 a^6 b^3 c^6+14 a^5 b^4 c^6+26 a^4 b^5 c^6-10 a^3 b^6 c^6-6 a^2 b^7 c^6+3 a b^8 c^6-5 b^9 c^6+a^8 c^7-14 a^7 b c^7+22 a^6 b^2 c^7+2 a^5 b^3 c^7-22 a^4 b^4 c^7+14 a^3 b^5 c^7-6 a^2 b^6 c^7-2 a b^7 c^7+5 b^8 c^7+9 a^7 c^8+7 a^6 b c^8-7 a^5 b^2 c^8-a^4 b^3 c^8-4 a^3 b^4 c^8-12 a^2 b^5 c^8+3 a b^6 c^8+5 b^7 c^8-9 a^6 c^9+13 a^5 b c^9-5 a^4 b^2 c^9-13 a^3 b^3 c^9+20 a^2 b^4 c^9-2 a b^5 c^9-5 b^6 c^9-7 a^5 c^10-5 a^4 b c^10+9 a^3 b^2 c^10+7 a^2 b^3 c^10-3 a b^4 c^10+b^5 c^10+7 a^4 c^11-a^3 b c^11-9 a^2 b^2 c^11+5 a b^3 c^11-b^4 c^11-a b^2 c^12-3 b^3 c^12-2 a b c^13+3 b^2 c^13+a c^14+b c^14-c^15: :

X(10775) lies on these lines:
{4,2828}, {11,107}, {100,122}, {104,2777}, {149,2803}, {1294,5840}, {2816,10265}, {2829,10152}

X(10775) = reflection of X(i) in X(j) for these (i,j): (100, 122), (107, 11)

X(10776) =  REFLECTION OF X(108) IN X(11)

Barycentrics    a^12-2 a^11 b+2 a^9 b^3-3 a^8 b^4+4 a^7 b^5-4 a^5 b^7+3 a^4 b^8-2 a^3 b^9+2 a b^11-b^12-2 a^11 c+6 a^10 b c-4 a^9 b^2 c+2 a^8 b^3 c-2 a^7 b^4 c-10 a^6 b^5 c+16 a^5 b^6 c-8 a^4 b^7 c+8 a^2 b^9 c-8 a b^10 c+2 b^11 c-4 a^9 b c^2+3 a^8 b^2 c^2-2 a^7 b^3 c^2+15 a^6 b^4 c^2-16 a^5 b^5 c^2-a^4 b^6 c^2+18 a^3 b^7 c^2-19 a^2 b^8 c^2+4 a b^9 c^2+2 b^10 c^2+2 a^9 c^3+2 a^8 b c^3-2 a^7 b^2 c^3-10 a^6 b^3 c^3+4 a^5 b^4 c^3+24 a^4 b^5 c^3-34 a^3 b^6 c^3+6 a^2 b^7 c^3+14 a b^8 c^3-6 b^9 c^3-3 a^8 c^4-2 a^7 b c^4+15 a^6 b^2 c^4+4 a^5 b^3 c^4-36 a^4 b^4 c^4+18 a^3 b^5 c^4+19 a^2 b^6 c^4-16 a b^7 c^4+b^8 c^4+4 a^7 c^5-10 a^6 b c^5-16 a^5 b^2 c^5+24 a^4 b^3 c^5+18 a^3 b^4 c^5-28 a^2 b^5 c^5+4 a b^6 c^5+4 b^7 c^5+16 a^5 b c^6-a^4 b^2 c^6-34 a^3 b^3 c^6+19 a^2 b^4 c^6+4 a b^5 c^6-4 b^6 c^6-4 a^5 c^7-8 a^4 b c^7+18 a^3 b^2 c^7+6 a^2 b^3 c^7-16 a b^4 c^7+4 b^5 c^7+3 a^4 c^8-19 a^2 b^2 c^8+14 a b^3 c^8+b^4 c^8-2 a^3 c^9+8 a^2 b c^9+4 a b^2 c^9-6 b^3 c^9-8 a b c^10+2 b^2 c^10+2 a c^11+2 b c^11-c^12 : :

X(10776) lies on these lines: {4,11}, {80,2817}, {100,123}, {149,2804}, {1295,5840}

X(10776) = reflection of X(i) in X(j) for these (i,j): (100, 123), (108, 11)

X(10777) =  REFLECTION OF X(109) IN X(11)

Barycentrics    a^9-2 a^8 b+2 a^6 b^3-a^5 b^4+a^4 b^5-2 a^3 b^6+2 a b^8-b^9-2 a^8 c+6 a^7 b c-4 a^6 b^2 c-2 a^5 b^3 c-a^4 b^4 c+3 a^3 b^5 c+5 a^2 b^6 c-7 a b^7 c+2 b^8 c-4 a^6 b c^2+7 a^5 b^2 c^2+5 a^3 b^4 c^2-15 a^2 b^5 c^2+6 a b^6 c^2+b^7 c^2+2 a^6 c^3-2 a^5 b c^3-12 a^3 b^3 c^3+10 a^2 b^4 c^3+7 a b^5 c^3-5 b^6 c^3-a^5 c^4-a^4 b c^4+5 a^3 b^2 c^4+10 a^2 b^3 c^4-16 a b^4 c^4+3 b^5 c^4+a^4 c^5+3 a^3 b c^5-15 a^2 b^2 c^5+7 a b^3 c^5+3 b^4 c^5-2 a^3 c^6+5 a^2 b c^6+6 a b^2 c^6-5 b^3 c^6-7 a b c^7+b^2 c^7+2 a c^8+2 b c^8-c^9 : :

X(10777) = reflection of X(i) in X(j) for these (i,j): (100, 124), (109, 11)
X(10777) = inverse-in-polar-circle of X(1830)


X(10778) =  REFLECTION OF X(110) IN X(11)

Barycentrics    a^9-a^8 b-a^7 b^2+a^6 b^3-a^3 b^6+a^2 b^7+a b^8-b^9-a^8 c+a^7 b c+a^6 b^2 c-a^5 b^3 c+2 a^3 b^5 c-a^2 b^6 c-2 a b^7 c+b^8 c-a^7 c^2+a^6 b c^2+a^5 b^2 c^2-a^4 b^3 c^2+a^3 b^4 c^2-3 a^2 b^5 c^2+2 b^7 c^2+a^6 c^3-a^5 b c^3-a^4 b^2 c^3-3 a^3 b^3 c^3+3 a^2 b^4 c^3+2 a b^5 c^3-2 b^6 c^3+a^3 b^2 c^4+3 a^2 b^3 c^4-2 a b^4 c^4+2 a^3 b c^5-3 a^2 b^2 c^5+2 a b^3 c^5-a^3 c^6-a^2 b c^6-2 b^3 c^6+a^2 c^7-2 a b c^7+2 b^2 c^7+a c^8+b c^8-c^9: :

X(10778) lies on these lines:
{4,2771}, {11,110}, {67,9024}, {74,5840}, {80,1109}, {100,125}, {149,3448}, {265,952}, {528,9140}, {895,5848}, {5533,10091}, {8068,10088}

X(10778) = midpoint of X(149) and X(3448)
X(10778) = reflection of X(i) in X(j) for these (i,j): (100, 125), (110, 11)

X(10779) =  REFLECTION OF X(111) IN X(11)

Barycentrics    a^9-a^8 b-a^7 b^2+a^6 b^3-4 a^5 b^4+4 a^4 b^5-a^3 b^6+a^2 b^7+a b^8-b^9-a^8 c+a^7 b c+a^6 b^2 c+3 a^5 b^3 c-a^2 b^6 c-2 a b^7 c+b^8 c-a^7 c^2+a^6 b c^2+a^5 b^2 c^2-5 a^4 b^3 c^2+9 a^3 b^4 c^2-11 a^2 b^5 c^2-2 a b^6 c^2+4 b^7 c^2+a^6 c^3+3 a^5 b c^3-5 a^4 b^2 c^3-15 a^3 b^3 c^3+11 a^2 b^4 c^3+6 a b^5 c^3-4 b^6 c^3-4 a^5 c^4+9 a^3 b^2 c^4+11 a^2 b^3 c^4-6 a b^4 c^4+4 a^4 c^5-11 a^2 b^2 c^5+6 a b^3 c^5-a^3 c^6-a^2 b c^6-2 a b^2 c^6-4 b^3 c^6+a^2 c^7-2 a b c^7+4 b^2 c^7+a c^8+b c^8-c^9 : :

X(10779) lies on these lines: {4,2830}, {11,111}, {75,149}, {100,126}, {1296,5840}

X(10779) = reflection of X(i) in X(j) for these (i,j): (100, 126), (111, 11)

X(10780) =  REFLECTION OF X(112) IN X(11)

Barycentrics    a^13-a^12 b-a^11 b^2+a^10 b^3-a^9 b^4+a^8 b^5+2 a^7 b^6-2 a^6 b^7-a^5 b^8+a^4 b^9-a^3 b^10+a^2 b^11+a b^12-b^13-a^12 c+a^11 b c+a^10 b^2 c-a^9 b^3 c+a^8 b^4 c-3 a^7 b^5 c+3 a^5 b^7 c-a^4 b^8 c+2 a^3 b^9 c-a^2 b^10 c-2 a b^11 c+b^12 c-a^11 c^2+a^10 b c^2+3 a^9 b^2 c^2-3 a^8 b^3 c^2-2 a^7 b^4 c^2+4 a^6 b^5 c^2-a^5 b^6 c^2-a^4 b^7 c^2+a^3 b^8 c^2-3 a^2 b^9 c^2+2 b^11 c^2+a^10 c^3-a^9 b c^3-3 a^8 b^2 c^3+7 a^7 b^3 c^3-2 a^6 b^4 c^3-3 a^5 b^5 c^3+3 a^4 b^6 c^3-5 a^3 b^7 c^3+3 a^2 b^8 c^3+2 a b^9 c^3-2 b^10 c^3-a^9 c^4+a^8 b c^4-2 a^7 b^2 c^4-2 a^6 b^3 c^4+4 a^5 b^4 c^4-2 a^4 b^5 c^4+4 a^2 b^7 c^4-a b^8 c^4-b^9 c^4+a^8 c^5-3 a^7 b c^5+4 a^6 b^2 c^5-3 a^5 b^3 c^5-2 a^4 b^4 c^5+6 a^3 b^5 c^5-4 a^2 b^6 c^5+b^8 c^5+2 a^7 c^6-a^5 b^2 c^6+3 a^4 b^3 c^6-4 a^2 b^5 c^6 -2 a^6 c^7+3 a^5 b c^7-a^4 b^2 c^7-5 a^3 b^3 c^7+4 a^2 b^4 c^7-a^5 c^8-a^4 b c^8+a^3 b^2 c^8+3 a^2 b^3 c^8-a b^4 c^8+b^5 c^8+a^4 c^9+2 a^3 b c^9-3 a^2 b^2 c^9+2 a b^3 c^9-b^4 c^9-a^3 c^10-a^2 b c^10-2 b^3 c^10+a^2 c^11-2 a b c^11+2 b^2 c^11+a c^12+b c^12-c^13 : :

X(10780) lies on these lines: {4,2831}, {11,112}, {100,127}, {104,2794}, {149,2806}, {1297,5840}

X(10780) = midpoint of
X(10780) = reflection of X(i) in X(j) for these (i,j): (100, 127), (112, 11)

X(10781) =  REFLECTION OF X(1113) IN X(11)

Barycentrics    2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+6 a^5 b c-a^4 b^2 c-3 a^3 b^3 c+2 a^2 b^4 c-3 a b^5 c+b^6 c-3 a^5 c^2-a^4 b c^2+4 a^3 b^2 c^2-2 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-3 a^3 b c^3-2 a^2 b^2 c^3+6 a b^3 c^3-3 b^4 c^3+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4-3 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7-(a^7-a^6 b-a^5 b^2+a^4 b^3-a^3 b^4+a^2 b^5+a b^6-b^7-a^6 c+a^5 b c+a^4 b^2 c+a^3 b^3 c-a^2 b^4 c-2 a b^5 c+b^6 c-a^5 c^2+a^4 b c^2-2 a^3 b^2 c^2-a b^4 c^2+3 b^5 c^2+a^4 c^3+a^3 b c^3+4 a b^3 c^3-3 b^4 c^3-a^3 c^4-a^2 b c^4-a b^2 c^4-3 b^3 c^4+a^2 c^5-2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7) J : : , where J = |OH|/R

X(10781) lies on these lines: {11,1113}, {30,104}, {100,1313}, {952,2102}, {1114,5840}, {2104,5848}

X(10781) = reflection of X(i) in X(j) for these (i,j): (100, 1313), (1113, 11)

X(10782) =  REFLECTION OF X(1114) IN X(11)

Barycentrics    2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+6 a^5 b c-a^4 b^2 c-3 a^3 b^3 c+2 a^2 b^4 c-3 a b^5 c+b^6 c-3 a^5 c^2-a^4 b c^2+4 a^3 b^2 c^2-2 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-3 a^3 b c^3-2 a^2 b^2 c^3+6 a b^3 c^3-3 b^4 c^3+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4-3 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7+(a^7-a^6 b-a^5 b^2+a^4 b^3-a^3 b^4+a^2 b^5+a b^6-b^7-a^6 c+a^5 b c+a^4 b^2 c+a^3 b^3 c-a^2 b^4 c-2 a b^5 c+b^6 c-a^5 c^2+a^4 b c^2-2 a^3 b^2 c^2-a b^4 c^2+3 b^5 c^2+a^4 c^3+a^3 b c^3+4 a b^3 c^3-3 b^4 c^3-a^3 c^4-a^2 b c^4-a b^2 c^4-3 b^3 c^4+a^2 c^5-2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7) J : : , where J = |OH|/R

X(10782) lies on these lines: {11,1114}, {30,104}, {100,1312}, {952,2103}, {1113,5840}, {2105,5848}

X(10782) = reflection of X(i) in X(j) for these (i,j): (100, 1312), (1114, 11)





leftri  Miscellaneous perspectors: X(10783) - X(10976)  rightri

Centers X(10783)-X(10976) were contributed by César Lozada, November 15, 2016.

underbar

X(10783) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND INNER GREBE

Barycentrics    SB*SC*S-2*SA*(SW*SA-SW^2+S^2) : :
X(10783) = (S-2*SW)*X(4)+4*SW*X(6)

X(10783) lies on these lines:
{2,6215}, {3,1271}, {4,6}, {20,1161}, {24,5595}, {98,3316}, {182,7376}, {184,3536}, {376,5861}, {388,10040}, {497,10048}, {515,5589}, {631,642}, {944,3641}, {1352,7375}, {1899,3535}, {3090,10514}, {3312,7000}, {5605,7967}, {5657,5689}, {5921,7389}, {6227,9862}, {6270,6302}, {6271,6303}, {6811,8974}, {7374,7583}, {10513,10518}

X(10783) = reflection of X(4) in X(1587)
X(10783) = {X(4),X(6776)}-harmonic conjugate of X(10784)
X(10783) = {X(6), X(5871)}-harmonic conjugate of X(4)


X(10784) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND OUTER GREBE

Barycentrics    -SB*SC*S-2*SA*(SW*SA-SW^2+S^2) : :
X(10784) = (S+2*SW)*X(4)-4*SW*X(6)

X(10784) lies on these lines:
{2,6214}, {3,1270}, {4,6}, {20,1160}, {24,5594}, {98,3317}, {182,7375}, {184,3535}, {376,5860}, {388,10041}, {497,10049}, {515,5588}, {631,641}, {944,3640}, {1352,7376}, {1899,3536}, {3090,10515}, {3311,7374}, {3529,8982}, {5604,7967}, {5657,5688}, {5921,7388}, {6226,9862}, {6268,6306}, {6269,6307}, {7000,7584}, {10513,10517}

X(10784) = reflection of X(4) in X(1588)
X(10784) = {X(4),X(6776)}-harmonic conjugate of X(10783)
X(10784) = {X(6), X(5870)}-harmonic conjugate of X(4)


X(10785) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND INNER JOHNSON

Barycentrics    a^7-(b+c)*a^6-(3*b^2-8*b*c+3*c^2)*a^5+3*(b^2-c^2)*(b-c)*a^4+(3*b^2-2*b*c+3*c^2)*(b-c)^2*a^2*(-b-c+a)-(b^2-c^2)^2*a*(b^2+c^2)+(b^2-c^2)^3*(b-c) : :
X(10785) = (R-r)*X(4)-2*(R-2*r)*X(11)

X(10785) lies on these lines:
{1,6833}, {2,355}, {3,3434}, {4,11}, {5,10584}, {8,6891}, {10,6967}, {12,6879}, {20,10525}, {36,6934}, {55,6977}, {84,1519}, {100,6961}, {119,6931}, {145,6972}, {153,5154}, {377,10269}, {388,6830}, {496,1012}, {497,6906}, {498,5882}, {499,515}, {517,6890}, {553,946}, {631,1376}, {942,5603}, {952,5552}, {956,6922}, {958,6947}, {993,6936}, {999,6831}, {1056,6956}, {1058,6935}, {1125,6832}, {1479,5450}, {1621,6892}, {2550,6940}, {2551,6963}, {2886,6897}, {2975,6827}, {3085,6952}, {3088,5101}, {3304,7680}, {3428,6899}, {3436,6882}, {3476,10321}, {3576,6889}, {3582,5691}, {3600,6844}, {3616,5768}, {3622,6888}, {3813,10310}, {3816,6898}, {3869,5770}, {3873,5761}, {4294,6950}, {5126,6848}, {5204,5842}, {5253,6826}, {5433,6880}, {5550,6887}, {5587,6983}, {5657,6926}, {5731,6825}, {5927,6846}, {6256,6968}, {6713,6921}, {6862,10246}, {6905,7288}, {6910,10267}, {6941,10589}, {10057,10320}, {10265,10573}, {10305,10308}, {10524,10526}

X(10785) = midpoint of X(6890) and X(10529)
X(10785) = reflection of X(i) in X(j) for these (i,j): (5552,6958), (6834,499)
X(10785) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,944,10786), (496,1012,10531), (999,6831,10532), (1479,5450,6938), (3304,7680,10597), (5587,10200,6983), (6952,7967,3085)


X(10786) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND OUTER JOHNSON

Barycentrics    a^7-(b+c)*a^6-(3*b^2+4*b*c+3*c^2)*a^5+(b+c)*(3*b^2+2*b*c+3*c^2)*a^4+3*(b^2-c^2)^2*a^3-3*(b^2-c^2)^2*(b+c)*a^2-(b^2-c^2)^2*a*(b^2-4*b*c+c^2)+(b^2-c^2)^3*(b-c) : :
X(10786) = (R+r)*X(4)-2*(R+2*r)*X(12)

X(10786) lies on these lines:
{1,1512}, {2,355}, {3,3436}, {4,12}, {5,10585}, {8,6825}, {10,6889}, {20,10526}, {35,6256}, {56,6880}, {72,5657}, {100,6850}, {104,631}, {119,2478}, {145,6960}, {153,4189}, {227,1068}, {388,6905}, {495,3149}, {497,6941}, {498,515}, {499,5882}, {946,10056}, {952,6863}, {1001,6898}, {1006,2551}, {1056,6927}, {1058,6969}, {1125,6983}, {1259,6916}, {1329,6947}, {1376,6897}, {1388,3086}, {1478,6796}, {1479,6968}, {1519,1697}, {1532,3295}, {1621,6893}, {2550,6937}, {2829,5217}, {2975,6954}, {3088,5130}, {3090,3816}, {3303,7681}, {3359,5553}, {3421,6988}, {3434,6842}, {3486,10321}, {3576,6967}, {3584,5691}, {3616,6944}, {3622,6979}, {3822,6984}, {3871,6932}, {4293,6942}, {5080,6868}, {5218,6906}, {5248,6976}, {5253,6970}, {5432,6977}, {5436,5587}, {5534,6734}, {5603,5806}, {5687,6907}, {5731,6891}, {5768,6989}, {5804,10578}, {5812,6361}, {5886,6953}, {6830,10588}, {6921,10269}, {6959,10246}, {10524,10525}

X(10786) = reflection of X(i) in X(j) for these (i,j): (6833,498), (10527,6863)
X(10786) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,944,10785), (4,12,10599), (35,6256,6938), (495,3149,10532), (1478,6796,6934), (1532,3295,10531), (3303,7681,10596), (5587,10198,6832), (6953,10587,5886)


X(10787) = PERSPECTOR OF THESE TRIANGLES: 4TH ANTI-BROCARD AND ANTI-MCCAY

Barycentrics    a^14-4*(b^2+c^2)*a^12+(13*b^4-5*b^2*c^2+13*c^4)*a^10-2*(b^2+c^2)*(b^4+3*b^2*c^2+c^4)*a^8-(13*b^8+13*c^8-(13*b^4-5*b^2*c^2+13*c^4)*b^2*c^2)*a^6+(4*b^4-5*b^2*c^2+4*c^4)*(b^2+c^2)^3*a^4-(3*b^12+3*c^12-(16*b^8+16*c^8-(35*b^4-27*b^2*c^2+35*c^4)*b^2*c^2)*b^2*c^2)*a^2+b^2*c^2*(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*(b^4-4*b^2*c^2+c^4) : :

X(10787) lies on these lines:
{316,543}, {385,2793}, {804,9870}, {2418,8591}

X(10787) = reflection of X(385) in line X(2)X(99)
X(10787) = intersection, other than X(23), of circumcircles of anti-McCay and 4th anti-Brocard triangles


X(10788) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND ANTI-EULER

Barycentrics    2*a^8-5*(b^2+c^2)*a^6+(2*b^2-c^2)*(b^2-2*c^2)*a^4+(b^2-c^2)^2*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(10788) = (S^2+SW^2)*X(4)-(4*(S^2-SW^2))*X(32)

X(10788) lies on these lines:
{2,2080}, {3,3329}, {4,32}, {5,7793}, {6,7709}, {20,3398}, {83,631}, {99,576}, {114,7812}, {182,376}, {187,262}, {263,2698}, {381,8859}, {511,3972}, {1003,1351}, {1078,3090}, {1383,7417}, {1656,7885}, {2782,7766}, {3091,10104}, {3095,3552}, {3098,10348}, {3524,8722}, {3525,7808}, {5067,7815}, {5097,7757}, {5475,10631}, {6179,6248}, {6194,9301}, {6721,7752}, {10345,10357}

X(10788) = reflection of X(4) in X(9993)
X(10788) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,7787,10359), (83,5171,631), (1078,10358,3090), (7737,9753,4)


X(10789) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND AQUILA

Trilinears    2*a^3*(a+b+c)+(b^2+c^2)*a^2+b^2*c^2 : :
X(10789) = (S^2+SW^2)*X(1)-(4*(S^2-SW^2))*X(32)

X(10789) lies on these lines:
{1,32}, {6,3097}, {10,7787}, {31,43}, {40,3398}, {83,1698}, {98,1699}, {165,182}, {238,4386}, {291,5332}, {384,9902}, {515,10788}, {612,8616}, {614,1929}, {726,7766}, {730,3972}, {1078,3624}, {1125,7793}, {1687,2017}, {1688,2018}, {2080,3576}, {3203,9587}, {5038,9574}, {5171,7987}, {6684,10359}, {7989,10358}, {8227,10104}, {9857,10348}

X(10789) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (727,4279,31), (985,1914,1)


X(10790) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND ARA

Trilinears    ((b^2+c^2)*a^6+5*a^4*b^2*c^2-(b^2-c^2)^2*(b^2+c^2)*a^2-(b^2-c^2)^2*b^2*c^2)*a : :
X(10790) = (S^2+SW^2)*SW*X(3)-2*R^2*(S^2+5*SW^2)*X(83)

X(10790) lies on these lines:
{3,83}, {22,7787}, {24,10788}, {25,32}, {98,1598}, {217,263}, {384,9917}, {1078,5020}, {1995,7793}, {2080,6642}, {3398,7387}, {7395,10358}, {7484,7808}, {7529,10104}, {8185,10789}, {10323,10359}


X(10791) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND OUTER-GARCIA

Barycentrics    a^5+(b+c)*a^4+(b+c)*(b^2+c^2)*a^2+b^2*c^2*(b+c) : :
X(10791) = (S^2+SW^2)*X(1)-(S^2+5*SW^2)*X(83)

X(10791) lies on these lines:
{1,83}, {6,730}, {8,7787}, {10,32}, {42,4112}, {98,5587}, {99,3097}, {182,515}, {355,3398}, {726,3734}, {944,10359}, {946,10358}, {1078,1698}, {1125,7808}, {3099,10347}, {3634,7815}, {3679,10789}, {5039,5847}, {5171,6684}, {5657,10788}, {7760,9902}, {7793,9780}, {7878,7976}, {8193,10790}, {8722,10164}, {9941,10345}, {9956,10104}

X(10791) = reflection of X(9857) in X(10)


X(10792) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND INNER-GREBE

Trilinears    (a^4+2*(b^2+c^2)*a^2+b^2*c^2-a^2*S)*a : :

X(10792) lies on these lines:
{3,6}, {83,5591}, {98,6202}, {384,6273}, {1271,7787}, {4027,6319}, {5589,10789}, {5595,10790}, {5689,10791}, {10358,10514}, {10359,10517}, {10783,10788}


X(10793) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND OUTER-GREBE

Trilinears    (a^4+2*(b^2+c^2)*a^2+b^2*c^2+a^2*S)*a : :

X(10793) lies on these lines:
{3,6}, {83,5590}, {98,6201}, {384,6272}, {1270,7787}, {4027,6320}, {5588,10789}, {5594,10790}, {5688,10791}, {10358,10515}, {10359,10518}, {10784,10788}


X(10794) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND INNER-JOHNSON

Barycentrics    a^6*(-b-c+a)-(b^2-4*b*c+c^2)*a^5+(b^2-c^2)*(b-c)*a^4-(b^2+c^2)*(b-c)^2*a^3+(b^4-c^4)*a^2*(b-c)-b^2*c^2*(b-c)^2*a+(b^2-c^2)*(b-c)*b^2*c^2 : :

X(10794) lies on these lines:
{11,32}, {83,1376}, {355,10795}, {3398,10525}, {3434,7787}, {7793,10584}


X(10795) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND OUTER-JOHNSON

Barycentrics    a^8-2*(b^2+b*c+c^2)*a^6-2*b*c*(b+c)*a^5-2*b*c*(b+c)^2*a^4+(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*(b+c)^2*a^2+(b^2-c^2)^2*b^2*c^2 : :

X(10795) lies on these lines:
{12,32}, {72,10791}, {83,958}, {355,10794}, {3398,10526}, {3436,7787}, {7793,10585}, {10786,10788}


X(10796) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND JOHNSON

Barycentrics    a^8-2*(b^2+c^2)*a^6-4*a^4*b^2*c^2+((b^2-c^2)^2-b^2*c^2)*(b^2+c^2)*a^2+(b^2-c^2)^2*b^2*c^2 : :
X(10796) = (S^2+SW^2)*X(5)-(S^2-SW^2)*X(32)

X(10796) is the center of the circle that is the locus of crosssums of Brocard circle antipodes. This circle passes through X(6) and X(1316). Compare to the nine-point circle, which is the locus of crosssums of circumcircle antipodes. (Randy Hutson, December 10, 2016)

X(10796) lies on these lines:
{1,10797}, {2,2080}, {3,83}, {4,3398}, {5,32}, {6,2782}, {20,10359}, {30,182}, {98,381}, {114,7753}, {140,5171}, {156,3203}, {355,10794}, {384,3095}, {385,7697}, {476,1316}, {511,7804}, {517,10791}, {538,5097}, {547,8176}, {549,8722}, {576,3734}, {1003,10352}, {1078,1656}, {1691,7737}, {2549,5038}, {3090,7793}, {3564,5039}, {3628,7815}, {3849,7606}, {4027,6321}, {5007,6248}, {5055,8860}, {5103,7761}, {5587,10789}, {6214,10793}, {6215,10792}, {6287,9863}, {6658,10131}, {7514,10003}, {7603,10631}, {7770,10350}, {9301,10347}, {9821,10345}, {9993,10348}

X(10796) = midpoint of X(576) and X(3734)
X(10796) = reflection of X(9996) in X(5)
X(10796) = X(9996)-of-Johnson-triangle
X(10796) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10788,2080), (4,7787,3398), (5,32,10104), (32,10358,5), (262,3972,3), (5171,7808,140), (10797,10798,1)


X(10797) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND 1ST JOHNSON-YFF

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

X(10797) lies on these lines:
{1,10796}, {11,10358}, {12,32}, {56,83}, {65,10791}, {182,7354}, {388,7787}, {498,2080}, {1478,3398}, {3085,10788}, {3203,9652}, {4279,9552}, {4293,10359}, {5038,9597}, {5171,5432}, {5433,7808}, {7793,10588}, {9578,10789}

X(10797) = {X(1),X(10796)}-harmonic conjugate of X(10798)


X(10798) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND 2ND JOHNSON-YFF

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

X(10798) lies on these lines:
{1,10796}, {11,32}, {12,10358}, {55,83}, {182,6284}, {497,7787}, {499,2080}, {1479,3398}, {3057,10791}, {3086,10788}, {3203,9667}, {4279,9555}, {4294,10359}, {5038,9598}, {5171,5433}, {5432,7808}, {7741,10104}, {7793,10589}, {9581,10789}

X(10798) = {X(1), X(10796)}-harmonic conjugate of X(10797)


X(10799) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND MANDART-INCIRCLE

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

X(10799) lies on these lines:
{1,3398}, {4,10797}, {11,83}, {12,98}, {32,55}, {35,2080}, {41,7077}, {56,182}, {172,1691}, {215,3203}, {497,7787}, {498,10104}, {1078,5432}, {1342,3238}, {1343,3237}, {1479,10796}, {1682,4279}, {1687,2008}, {1688,2007}, {1697,10789}, {1837,10791}, {2053,2175}, {2275,5038}, {3023,4027}, {3056,7296}, {3086,10359}, {4294,10788}, {5171,5217}, {5218,7793}

X(10799) = {X(497), X(7787)}-harmonic conjugate of X(10798)


X(10800) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND 5TH MIXTILINEAR

Trilinears    a^3*(a+b+c)+2*(b^2+c^2)*a^2+2*b^2*c^2 : :
X(10800) = (S^2+SW^2)*X(1)-(S^2-SW^2)*X(32)

X(10800) lies on these lines:
{1,32}, {8,83}, {10,7808}, {82,2175}, {98,5603}, {145,7787}, {182,517}, {355,10358}, {384,7976}, {518,5039}, {519,10791}, {726,7798}, {730,3734}, {901,5091}, {952,10796}, {995,8301}, {1001,4279}, {1078,3616}, {1125,7815}, {1191,9565}, {1385,5171}, {1482,3398}, {2080,10246}, {2098,10799}, {3576,8722}, {3622,7793}, {4027,7983}, {5034,9620}, {5604,10793}, {5605,10792}, {5901,10104}, {7967,10788}, {8192,10790}, {10794,10797}, {10795,10798}

X(10800) = midpoint of X(1) and X(1572)
X(10800) = reflection of X(9997) in X(1)
X(10800) = {X(1), X(985)}-harmonic conjugate of X(2242)
X(10800) = {X(10803),X(10804)}-harmonic conjugate of X(32)


X(10801) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND INNER-YFF

Trilinears    (a^6-2*(b^2+c^2)*a^4+(b^4+c^4+2*(b^2-b*c+c^2)*b*c)*a^2+2*b^3*c^3)*a : :

X(10801) lies on these lines:
{1,32}, {5,10798}, {11,10104}, {12,10796}, {35,182}, {36,5171}, {55,3398}, {56,2080}, {83,498}, {98,1479}, {330,10089}, {384,10063}, {385,10079}, {388,10788}, {495,10797}, {499,1078}, {613,1691}, {3085,7787}, {3086,7793}, {3295,10799}, {4027,10086}, {5218,10359}, {7280,8722}, {7951,10358}, {10037,10790}, {10039,10791}, {10040,10792}, {10041,10793}, {10523,10794}

X(10801) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,32,10802), (3235,3236,2241)


X(10802) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND OUTER-YFF

Trilinears    (a^6-2*(b^2+c^2)*a^4+(b^4+c^4-2*(b^2+b*c+c^2)*b*c)*a^2-2*b^3*c^3)*a : :

X(10802) lies on these lines:
{1,32}, {3,10799}, {5,10797}, {11,10796}, {12,10104}, {35,5171}, {36,182}, {55,2080}, {56,3398}, {83,499}, {98,1478}, {192,10086}, {384,10079}, {385,10063}, {496,10798}, {497,10788}, {498,1078}, {611,1691}, {1737,10791}, {3085,7793}, {3086,7787}, {4027,10089}, {5010,8722}, {7288,10359}, {7741,10358}, {10046,10790}, {10048,10792}, {10049,10793}, {10523,10795}

X(10802) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,32,10801), (3235,3236,2242)


X(10803) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND INNER-YFF TANGENTS

Trilinears    (a^6-2*(b^2-b*c+c^2)*a^4+2*b*c*(b+c)*a^3+(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2+4*b^3*c^3)*a : :

X(10803) lies on these lines:
{1,32}, {12,10794}, {83,5552}, {98,10531}, {119,10358}, {3398,10679}, {5171,10269}, {7787,10528}, {7793,10586}, {7815,10200}

X(10803) = {X(32),X(10800)}-harmonic conjugate of X(10804)


X(10804) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH ANTI-BROCARD AND OUTER-YFF TANGENTS

Trilinears    (a^6-2*(b^2+b*c+c^2)*a^4-2*b*c*(b+c)*a^3+(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^2-4*b^3*c^3)*a : :

X(10804) lies on these lines:
{1,32}, {11,10795}, {83,10527}, {98,10532}, {3398,10680}, {5171,10267}, {7787,10529}, {7793,10587}, {7815,10198}

X(10804) = {X(32), X(10800)}-harmonic conjugate of X(10803)


X(10805) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND INNER-YFF TANGENTS

Barycentrics    a^6*(-b-c+a)-(b^2-12*b*c+c^2)*a^5+(b+c)*(b^2-10*b*c+c^2)*a^4-a^2*(b^2+10*b*c+c^2)*(b-c)^2*(-b-c+a)+(b^2-c^2)^2*a*(b^2-4*b*c+c^2)-(b^2-c^2)^3*(b-c) : :
X(10805) = 4*R*X(1)-(R-r)*X(4)

X(10805) lies on these lines:
{1,4}, {3,10528}, {5,10586}, {8,6897}, {12,6879}, {20,10679}, {56,6880}, {104,3085}, {119,3090}, {145,6850}, {153,3622}, {355,5439}, {377,952}, {443,5554}, {495,6833}, {496,6968}, {631,2975}, {956,6889}, {958,6878}, {999,6834}, {1000,5553}, {1385,3436}, {1470,6942}, {1482,6925}, {1483,6923}, {1512,3333}, {1532,7373}, {2077,3528}, {2478,10246}, {2829,3303}, {3295,6938}, {3560,10587}, {3600,6905}, {3616,6898}, {3655,10526}, {3868,6916}, {3871,6948}, {3897,5084}, {4317,6796}, {5067,10200}, {5082,6951}, {5261,6830}, {5450,10056}, {5687,6955}, {5731,6899}, {5842,9657}, {5901,6957}, {6825,10530}, {6842,10529}, {6940,7080}, {6952,8164}, {7966,9579}, {10788,10803}

X(10805) = reflection of X(4) in X(10532)
X(10805) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,10596), (4,1056,10597), (4,7967,10806), (104,3085,6977), (153,3622,6893), (388,944,4), (1385,3436,6947), (6256,10531,4)


X(10806) = HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EULER AND OUTER-YFF TANGENTS

Barycentrics    a^6*(-b-c+a)-(b^2+8*b*c+c^2)*a^5+(b+c)*(b^2+6*b*c+c^2)*a^4-a^2*(b^2-6*b*c+c^2)*(b-c)^2*(-b-c+a)+(b^2-c^2)^2*a*(b^2+c^2)-(b^2-c^2)^3*(b-c) : :
X(10806) = 4*R*X(1)-(R+r)*X(4)

X(10806) lies on these lines:
{1,4}, {3,10529}, {5,10587}, {8,6947}, {11,10786}, {20,10680}, {55,6977}, {100,631}, {104,4294}, {145,6827}, {149,6850}, {355,6898}, {377,10246}, {390,6906}, {496,6834}, {517,6899}, {952,2478}, {956,6936}, {999,6934}, {1385,3434}, {1482,6836}, {1483,6928}, {1484,6863}, {1616,5721}, {2829,9670}, {3085,6879}, {3086,6880}, {3295,6833}, {3304,5842}, {3421,6902}, {3616,6854}, {3622,6826}, {3623,6840}, {3655,10525}, {3871,6891}, {3885,6865}, {3957,5761}, {4309,5450}, {5067,10198}, {5274,6941}, {5687,6967}, {5768,9785}, {5886,6896}, {5901,6835}, {6585,6876}, {6767,6831}, {6796,10072}, {6882,10528}, {6890,10679}, {6911,10586}, {6963,7080}, {6968,9669}, {7966,9581}, {10788,10804}

X(10806) = reflection of X(4) in X(10531)
X(10806) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,4,10597), (4,1058,10596), (4,7967,10805), (497,944,4), (1385,3434,6897), (10267,10527,631)


X(10807) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND 1ST BROCARD

Barycentrics    18*(5*SW^2-3*S^2)*SA^2+12*(9*S^2-7*SW^2)*SW*SA-(9*S^2+SW^2)*(3*S^2-7*SW^2) : :

X(10807) lies on these lines:
{76,3849}, {598,5008}, {599,9855}, {3094,9830}


X(10808) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND INNER-NAPOLEON

Barycentrics    sqrt(3)*(-9*(15*S^2-SW^2)*SW*SA^2-9*(-14*S^2*SW^2+9*S^4+SW^4)*SA-(6*S^2-SW^2)*(9*S^2+SW^2)*SW)
-3*S*(-27*(SW^2+S^2)*SA^2+24*SW^3*SA+(9*S^2+SW^2)*(3*S^2-2*SW^2)) : :

X(10808) lies on these lines:
{5463,9855}, {8594,9885}, {8595,9761}


X(10809) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND OUTER-NAPOLEON

Barycentrics    sqrt(3)*(-9*(15*S^2-SW^2)*SW*SA^2-9*(-14*S^2*SW^2+9*S^4+SW^4)*SA-(6*S^2-SW^2)*(9*S^2+SW^2)*SW)
+3*S*(-27*(SW^2+S^2)*SA^2+24*SW^3*SA+(9*S^2+SW^2)*(3*S^2-2*SW^2)) : :

X(10809) lies on these lines:
{5464,9855}, {8594,9763}, {8595,9886}


X(10810) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND 1ST NEUBERG

Barycentrics    36*(S^2+SW^2)*SW^2*SA^2+12*(9*S^4-2*S^2*SW^2-3*SW^4)*SW*SA+(9*S^2+SW^2)*(9*S^4-2*S^2*SW^2+SW^4) : :

X(10810) lies on these lines:
{6,8587}, {183,8592}, {8182,8591}, {9887,9888}


X(10811) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND 2ND NEUBERG

Barycentrics    72*(7*S^2-SW^2)*SW^2*SA^2+24*(9*S^4-20*S^2*SW^2+3*SW^4)*SW*SA-(9*S^2+SW^2)*(9*S^4-26*S^2*SW^2+5*SW^4) : :

X(10811) lies on these lines:
{385,10488}, {2482,10807}, {8587,9830}, {9889,9890}


X(10812) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND INNER-VECTEN

Barycentrics    3*(9*(2*SW+3*SA)*S^2+(45*SA^2-42*SA*SW-SW^2)*SW)*S^2-S*(3*SA-SW)^2*(9*S^2+SW^2)-(9*SA^2-9*SA*SW+SW^2)*SW^3 : :

X(10812) lies on these lines:
{2482,10813}, {9891,9892}


X(10813) = PERSPECTOR OF THESE TRIANGLES: ANTI-MCCAY AND OUTER-VECTEN

Barycentrics    3*(9*(2*SW+3*SA)*S^2+(45*SA^2-42*SA*SW-SW^2)*SW)*S^2+S*(3*SA-SW)^2*(9*S^2+SW^2)-(9*SA^2-9*SA*SW+SW^2)*SW^3 : :

X(10813) lies on these lines:
{2482,10812}, {9893,9894}


X(10814) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND INNER-GREBE

Trilinears    a*((S^2+3*SA^2-6*(6*R^2-SW)*SA-4*(9*R^2-4*SW)*(9*R^2-2*SW))*S^2-2*((36*R^2-7*SW)*S^2+3*SA*(4*SW^2-(21*R^2+SA)*SW+9*R^2*SA))*S+12*(9*R^2-2*SW)*(SA-SW)*SA*SW) : :

X(10814) lies on these lines:
{74,5871}, {399,1161}, {2914,6277}


X(10815) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND OUTER-GREBE

Trilinears    a*((S^2+3*SA^2-6*(6*R^2-SW)*SA-4*(9*R^2-4*SW)*(9*R^2-2*SW))*S^2-2*((36*R^2-7*SW)*S^2-3*SA*(4*SW^2-(21*R^2+SA)*SW+9*R^2*SA))*S+12*(9*R^2-2*SW)*(SA-SW)*SA*SW) : :

X(10815) lies on these lines:
{74,5870}, {399,1160}, {2914,6276}


X(10816) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND 2ND HYACINTH

Trilinears    SA*((-11*R^2+2*SW)*(SA^2+S^2)-(5*R^2-2*SW)*(6*R^2-SW)*SA+3*(5*R^2-SW)*(60*R^4-21*R^2*SW+2*SW^2))*a : :

X(10816) lies on these lines:
{74,1885}, {185,399}


X(10817) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND LUCAS INNER

Trilinears    (3*SA*(48*R^2-SA-10*SW)+8*(9*R^2-2*SW)*S-S^2)*a : :

X(10817) lies on the Lucas inner circle and these lines:
{6,10818}, {74,6221}, {110,1151}, {146,9542}, {399,9690}, {1511,6445}, {2931,9695}, {2948,9584}, {3031,9558}, {3043,9687}, {3047,9686}, {3448,9543}, {5663,6407}, {7984,9616}, {8994,9541}, {9585,9904}, {9634,10118}, {9691,10620}, {9694,10117}


X(10818) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND LUCAS(-1) INNER

Trilinears    (3*SA*(48*R^2-SA-10*SW)-8*(9*R^2-2*SW)*S-S^2)*a : :

X(10818) lies on the Lucas(-1) inner circle and these lines:
{6,10817}, {74,6398}, {110,1152}, {1511,6446}, {5663,6408}


X(10819) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND LUCAS TANGENTS

Trilinears    (3*SA*(SA-2*SW+6*R^2)+2*(9*R^2-2*SW)*S+S^2)*a : :

X(10819) lies on the Lucas circles radical circle and these lines:
{6,1511}, {74,6200}, {110,371}, {113,6561}, {125,5418}, {146,9541}, {265,590}, {399,6221}, {485,8998}, {486,5972}, {542,8994}, {1151,5663}, {2066,10091}, {2067,10088}, {2948,9583}, {3031,9557}, {3043,9676}, {3047,9677}, {3448,9540}, {5609,6425}, {6396,10818}, {6449,10620}, {6480,10817}, {8276,8912}, {9582,9904}, {9631,10118}, {9683,10117}, {9934,10533}

X(10819) = reflection of X(485) in X(8998)
X(10819) = {X(6),X(1511)}-harmonic conjugate of X(10820)


X(10820) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND LUCAS(-1) TANGENTS

Trilinears    (3*SA*(SA-2*SW+6*R^2)-2*(9*R^2-2*SW)*S+S^2)*a : :

X(10820) lies on the Lucas(-1) circles radical circle and these lines:
{6,1511}, {74,6396}, {110,372}, {113,6560}, {125,5420}, {265,615}, {399,6398}, {485,5972}, {1152,5663}, {5414,10091}, {5609,6426}, {6200,10817}, {6450,10620}, {6481,10818}, {6502,10088}, {9934,10534}

X(10820) = {X(6),X(1511)}-harmonic conjugate of X(10819)


X(10821) = PERSPECTOR OF THESE TRIANGLES: ANTI-ORTHOCENTROIDAL AND MIDHEIGHT

Trilinears    (6*cos(2*A)+3)*cos(B-C)+(8*cos(A)-2*cos(3*A))*cos(2*(B-C))-cos(3*A)+cos(5*A)-15*cos(A) : :

X(10821) lies on these lines:
{5,399}, {6,5900}, {74,389}, {125,1199}, {2071,3581}, {5622,9969}


X(10822) = PERSPECTOR OF THESE TRIANGLES: APOLLONIUS AND EXTANGENTS

Trilinears    a*(b+c)*((b+c)*a^3+(b+c)^2*a^2-(b^2-c^2)*(b-c)*a-(b^2+c^2)^2) : :

X(10822) lies on these lines:
{1,3688}, {10,12}, {40,43}, {55,386}, {71,213}, {218,573}, {387,10480}, {511,5247}, {672,4300}, {674,1104}, {958,4259}, {960,4026}, {1284,3191}, {1362,4306}, {1402,3682}, {1453,3056}, {1468,3917}, {1724,3271}, {2093,6048}, {2175,8193}, {2292,3690}, {2294,3954}, {2550,9534}, {2841,3030}, {2911,3556}, {3032,6154}, {3611,5360}, {3869,4972}, {6007,7283}, {6769,9549}, {9567,10306}


X(10823) = PERSPECTOR OF THESE TRIANGLES: APOLLONIUS AND 2ND MIXTILINEAR

Trilinears    (a^3*(b+c)*(a^3*(b+c)-3*(b+c)^3*a-8*b*c*(b^2+c^2))+(3*b^6+3*c^6+(10*b^4+10*c^4+(13*b^2-4*b*c+13*c^2)*b*c)*b*c)*a^2+8*b*c*(b+c)*(b^4+c^4)*a-(b^2-c^2)^4)*a : :

X(10823) lies on these lines:
{1,181}, {10,7958}, {386,5584}, {573,8273}, {1695,4260}, {7957,10822}


X(10824) = PERSPECTOR OF THESE TRIANGLES: APOLLONIUS AND 4TH MIXTILINEAR

Trilinears    ((b+c)^2*a^5-(b+c)*(b^2+10*b*c+c^2)*a^4-2*(b^2+c^2)*(b^2+4*b*c+c^2)*a^3+(b+c)*a*(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*(2*a+b+c)-(b^2-c^2)^3*(b-c))*a : :

X(10824) lies on these lines:
{3,10823}, {10,7965}, {55,181}, {57,1682}, {165,970}, {210,10443}, {2092,10460}, {5273,9565}, {6244,9566}, {7964,10822}, {7994,9548}


X(10825) = PERSPECTOR OF THESE TRIANGLES: AQUILA AND 3RD CONWAY

Trilinears    4*(4*q^2+1)*p^8+24*(2*q^2-1)*q*p^7-(64*q^2-1)*p^6-3*(16*q^2-9)*q*p^5-2*(10*q^4-26*q^2+1)*p^4-2*(4*q^4-15*q^2+4)*q*p^3+3*(3*q^2-4)*q^2*p^2+(7*q^2-8)*q^3*p+2*(q^2-1)*q^4 : :
where p=sin(A/2), q=cos((B-C)/2)

X(10825) lies on these lines:
{191,1764}, {4385,5223}


X(10826) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND INNER-JOHNSON

Barycentrics    a^4-(b+c)*a^3+(b^2+c^2)*a^2+(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(10826) = (R+r)*X(1)-4*r*X(5) = 2*r*X(4)-(R-r)*X(46)

X(10826) lies on these lines:
{1,5}, {2,3612}, {4,46}, {8,5187}, {10,1479}, {35,405}, {36,3149}, {40,3583}, {43,1985}, {55,9956}, {57,3585}, {65,381}, {78,3814}, {79,3339}, {100,7705}, {158,7541}, {165,5445}, {354,9654}, {382,1155}, {388,6896}, {429,1717}, {497,5818}, {498,950}, {499,515}, {546,1836}, {938,10590}, {944,10589}, {946,10573}, {960,3679}, {997,4193}, {1125,6933}, {1210,1478}, {1329,3419}, {1420,3582}, {1656,2646}, {1697,4857}, {1723,1826}, {1736,1893}, {1752,7079}, {1905,7507}, {2098,7743}, {2099,9955}, {2362,6564}, {3057,5790}, {3085,6886}, {3086,4308}, {3090,3486}, {3245,9589}, {3333,5270}, {3336,9579}, {3485,3545}, {3488,10588}, {3576,6863}, {3601,6861}, {3632,5087}, {3634,4304}, {3706,5827}, {3832,4295}, {3843,4338}, {3911,4299}, {4187,5794}, {4293,5704}, {4294,9780}, {4297,6962}, {4302,6684}, {4312,5729}, {4511,5154}, {4860,9656}, {5123,5687}, {5225,5657}, {5231,5258}, {5251,5705}, {5441,6675}, {5560,7280}, {5563,9613}, {5731,7319}, {5902,5927}, {6256,10085}, {6702,10058}, {6734,10522}, {6829,10393}, {10072,10106}, {10789,10794}

X(10826) = Fuhrmann circle-inverse-of-X(10073)
X(10826) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5587,10827), (1,7988,5443), (1,7989,7951), (2,10572,3612), (4,1737,46), (4,1788,1770), (5,1837,1), (10,1479,5119), (11,355,1), (12,5722,1), (46,90,1727), (80,7741,1), (382,1155,4333), (496,5252,1), (497,5818,10039), (950,10175,498), (1210,1478,3338), (1698,3586,35), (1737,1770,1788), (1770,1788,46), (3679,9614,5697), (4193,5086,997), (5587,9581,1), (5727,8227,1), (5790,9669,3057)
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X(10827) = HOMOTHETIC CENTER OF THESE TRIANGLES: AQUILA AND OUTER-JOHNSON

Barycentrics    a^4-(b+c)*a^3+(b^2+4*b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-2*(b^2-c^2)^2 : :
X(10827) = (R-r)*X(1)+4*r*X(5) = 4*r*X(10)+(R-r)*X(46)

X(10827) lies on these lines:
{1,5}, {4,5119}, {8,6871}, {10,46}, {35,1012}, {36,474}, {40,3585}, {56,9956}, {57,5270}, {65,5790}, {72,3679}, {79,2093}, {165,10483}, {226,6984}, {381,3057}, {388,1737}, {443,7284}, {484,9579}, {498,515}, {499,6983}, {535,4652}, {944,6879}, {946,6968}, {950,10056}, {1000,3855}, {1125,6931}, {1155,9655}, {1319,1656}, {1479,6957}, {1697,3583}, {1699,5697}, {1709,6256}, {1727,7330}, {1770,5229}, {1836,5690}, {1883,5130}, {2098,9955}, {2476,5176}, {3085,4313}, {3090,3476}, {3486,8164}, {3576,6958}, {3579,4333}, {3584,3601}, {3586,3746}, {3617,4295}, {3634,4311}, {3751,5820}, {3811,5086}, {3911,4317}, {4007,4053}, {4293,9780}, {4297,6966}, {4299,6684}, {4312,5220}, {4338,9656}, {4861,5141}, {5223,5832}, {5231,5288}, {5253,7705}, {5258,5705}, {5261,6993}, {5290,5902}, {5445,5791}, {5812,7991}, {5919,9669}, {6702,10074}, {6735,10522}, {10789,10795}

X(10827) = reflection of X(3612) in X(498)
X(10827) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5587,10826), (1,7989,7741), (4,10039,5119), (5,5252,1), (10,1478,46), (12,355,1), (388,1737,3338), (388,5818,1737), (474,5123,1698), (495,1837,1), (1698,9613,36), (5219,5881,1), (5229,5657,1770), (5587,9578,1), (5790,9654,65), (10106,10175,499)


X(10828) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 5TH BROCARD

Trilinears    (a^8+(b^2+c^2)*a^6-a^4*b^2*c^2-(b^2-c^2)^2*(b^2+c^2)*a^2-((b^2+c^2)^2-b^2*c^2)*(b^2-c^2)^2)*a : :

X(10828) lies on these lines:
{3,3096}, {22,2896}, {24,9861}, {25,32}, {159,3094}, {315,6660}, {1598,9993}, {1995,10583}, {2076,5167}, {3098,5907}, {3099,8185}, {3456,7747}, {5020,7846}, {5594,9995}, {5595,9994}, {7387,9821}, {7395,10356}, {7484,7914}, {7517,9301}, {7811,9909}, {8192,9997}, {8193,9857}, {9798,9941}, {9876,9878}, {9908,9923}, {9915,9981}, {9916,9982}, {9917,9983}, {9919,9984}, {9920,9985}, {9921,9986}, {9922,9987}, {10037,10038}, {10046,10047}, {10323,10357}


X(10829) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND INNER-JOHNSON

Trilinears    a*(a^6*(a-b-c)-(b^2-4*b*c+c^2)*a^5+(b-c)^2*((b+c)*(a^3*(a-b-c)+(b+c)^2*a^2-(b^2+c^2)^2)+a*(b^4+c^4-2*(b^2+b*c+c^2)*b*c))) : :

X(10829) lies on these lines:
{3,10}, {11,25}, {19,5322}, {22,3434}, {56,1452}, {404,1603}, {1436,3433}, {1455,1470}, {1709,3220}, {1995,10584}, {7387,10525}, {8185,10826}, {10037,10523}, {10594,10598}, {10790,10794}

X(10829) = {X(3),X(9798)}-harmonic conjugate of X(10830)


X(10830) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND OUTER-JOHNSON

Trilinears    a*(a^8-2*(b^2+b*c+c^2)*a^6-2*b*c*(b+c)*a^5+2*(b^2+c^2)*(b^3+c^3)*(b+c)*a^2+2*(b^2-c^2)^2*(b+c)*a*b*c-(b^4-c^4)^2) : :

X(10830) lies on these lines:
{3,10}, {12,25}, {22,3436}, {24,10786}, {55,1829}, {72,3556}, {227,1104}, {1602,6986}, {1995,10585}, {5812,9911}, {7387,10526}, {8185,10827}, {10046,10523}, {10594,10599}, {10790,10795}

X(10830) = {X(3),X(9798)}-harmonic conjugate of X(10829)


X(10831) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 1ST JOHNSON-YFF

Trilinears    (a^6-(b+c)^2*a^4-(b^2-c^2)^2*a^2+(b+c)^2*(b^2+c^2)^2)*a/(a-b-c) : :

X(10831) lies on these lines:
{1,3}, {5,10046}, {11,7395}, {12,25}, {22,388}, {23,5261}, {24,3085}, {26,495}, {34,5310}, {172,1609}, {378,4294}, {496,7514}, {497,7503}, {498,6642}, {499,7393}, {1056,7512}, {1478,7387}, {1479,9818}, {1593,6284}, {1598,9673}, {1836,9911}, {1935,7295}, {1995,10588}, {2175,7066}, {2286,7251}, {2477,3167}, {3086,7509}, {3157,9908}, {3518,8164}, {3600,6636}, {4293,10323}, {4337,7163}, {5252,9798}, {5290,9591}, {5433,7484}, {7083,7299}, {7288,7485}, {7517,9654}, {7529,7951}, {8185,9578}, {8192,10829}, {8276,9646}, {9571,10408}, {9658,9909}, {9937,10055}, {10590,10594}, {10790,10797}

X(10831) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,10832), (55,9627,3295), (55,9659,3)


X(10832) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND 2ND JOHNSON-YFF

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

X(10832) lies on these lines:
{1,3}, {5,10037}, {11,25}, {12,7395}, {22,497}, {23,5274}, {24,3086}, {26,496}, {33,5322}, {215,3167}, {378,4293}, {388,7503}, {390,6636}, {495,7514}, {498,7393}, {499,6642}, {1058,7512}, {1069,9908}, {1364,1397}, {1478,9818}, {1479,7387}, {1593,7354}, {1598,9658}, {1609,1914}, {1619,10535}, {1837,9798}, {1995,10589}, {3085,7509}, {3813,9712}, {3816,9713}, {4294,10323}, {4548,7124}, {5096,7074}, {5218,7485}, {5432,7484}, {6503,8299}, {7517,9669}, {7529,7741}, {7550,8164}, {8185,9581}, {8192,10830}, {8276,9661}, {9673,9909}, {9937,10071}, {10591,10594}, {10790,10798}

X(10832) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,10831), (56,9672,3)


X(10833) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND MANDART-INCIRCLE

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

X(10833) lies on these lines:
{1,7387}, {3,11}, {4,10831}, {12,1598}, {19,25}, {22,497}, {23,390}, {24,4294}, {35,6642}, {65,9911}, {159,3056}, {161,2192}, {495,7530}, {498,7529}, {528,9713}, {999,4351}, {1317,9913}, {1593,9659}, {1697,8185}, {1837,8193}, {1936,7295}, {1995,5218}, {2098,8192}, {2361,7083}, {3027,9861}, {3028,9919}, {3057,9798}, {3058,9909}, {3085,10594}, {3086,10323}, {3167,9667}, {3295,7517}, {3303,9658}, {3583,9818}, {4186,10830}, {4309,9714}, {5020,5432}, {5225,7503}, {5274,6636}, {5899,6767}, {6238,9937}, {7082,7085}, {7355,9914}, {7393,7741}, {7485,10589}, {7509,10591}, {7516,10593}, {9670,9672}, {10790,10799}

X(10833) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (33,5310,55), (55,9629,7071), (55,9673,25)


X(10834) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND INNER-YFF TANGENTS

Trilinears    (a^8-2*(b^2-b*c+c^2)*a^6+2*b*c*(b+c)*a^5+2*(b^6+c^6-(b^4+c^4+(b^2+6*b*c+c^2)*b*c)*b*c)*a^2-2*(b^2-c^2)^2*(b+c)*a*b*c-(b^2-c^2)^4)*a : :

X(10834) lies on these lines:
{1,25}, {3,3436}, {12,10829}, {22,10528}, {24,10805}, {119,7395}, {1593,6256}, {1598,10531}, {7387,10679}, {10594,10596}, {10790,10803}

X(10834) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,8192,10835), (9798,10037,25)


X(10835) = HOMOTHETIC CENTER OF THESE TRIANGLES: ARA AND OUTER-YFF TANGENTS

Trilinears    (a^8-2*(b^2+b*c+c^2)*a^6-2*b*c*(b+c)*a^5+2*(b^6+c^6+(b^4+c^4-(b^2-6*b*c+c^2)*b*c)*b*c)*a^2+2*(b^2-c^2)^2*(b+c)*a*b*c-(b^2-c^2)^4)*a : :

X(10835) lies on these lines:
{1,25}, {3,3434}, {11,10830}, {22,10529}, {24,10806}, {1473,9911}, {1598,10532}, {1995,10587}, {7387,10680}, {10594,10597}, {10790,10804}

X(10835) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,8192,10834), (9798,10046,25)


X(10836) = PERSPECTOR OF THESE TRIANGLES: ARIES AND 1ST PARRY

Trilinears    (S^2*(18*R^2*S^2-2*R^2*SW^2-3*S^2*SW)-SW*(24*R^2*S^2-SW^3-3*S^2*SW)*SA+(36*R^2*S^2-SW^3-6*S^2*SW)*SA^2)*a : :

X(10836) lies on these lines:
{69,110}, {111,2393}, {154,5191}, {159,2502}, {5027,9934}


X(10837) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS BROCARD

Barycentrics    -3*(SW^2*SA^2-SW^3*SA+(S^2+SW^2)*S^2)*S+SW*((3*S^2+2*SW^2)*SA^2-2*(S^2+SW^2)*SW*SA+(2*S^2+SW^2)*S^2) : :

X(10837) lies on these lines:
{98,8375}, {262,6421}, {574,10838}, {1151,9756}, {5058,9755}, {8302,9772}


X(10838) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) BROCARD

Barycentrics    3*(SW^2*SA^2-SW^3*SA+(S^2+SW^2)*S^2)*S+SW*((3*S^2+2*SW^2)*SA^2-2*(S^2+SW^2)*SW*SA+(2*S^2+SW^2)*S^2) : :

X(10838) lies on these lines:
{98,8376}, {262,6422}, {574,10837}, {1152,9756}, {5062,9755}, {8303,9772}


X(10839) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS CENTRAL

Barycentrics    (S^2-2*SW^2)*SW^2*SA^2+(2*S^4+S^2*SW^2+2*SW^4)*SW*SA-2*(S^2+2*SW^2)*(S^2+SW^2)*S^2+SW*((3*S^2+SW^2)*SA^2-(S^2+SW^2)*SW*SA-2*(S^2+2*SW^2)*S^2)*S : :
X(10839) = (S+2*SW)*(2*S-SW)*X(3)+(S^2+5*SW^2)*X(83)

X(10839) lies on these lines:
{3,83}, {98,6199}, {371,6222}, {1132,10155}, {1161,6194}, {3311,9755}, {3316,7000}, {3815,9758}, {8304,9772}, {8396,10837}, {8407,10838}

X(10839) = {X(3),X(262)}-harmonic conjugate of X(10840)


X(10840) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) CENTRAL

Barycentrics    (S^2-2*SW^2)*SW^2*SA^2+(2*S^4+S^2*SW^2+2*SW^4)*SW*SA-2*(S^2+2*SW^2)*(S^2+SW^2)*S^2-SW*((3*S^2+SW^2)*SA^2-(S^2+SW^2)*SW*SA-2*(S^2+2*SW^2)*S^2)*S : :
X(10840) = (2*S+SW)*(S-2*SW)*X(3)+(S^2+5*SW^2)*X(83)

X(10840) lies on these lines:
{3,83}, {98,6395}, {372,6399}, {1131,10155}, {1160,6194}, {3312,9755}, {3317,7374}, {3815,9757}, {8305,9772}, {8400,10837}, {8416,10838}

X(10840) = {X(3),X(262)}-harmonic conjugate of X(10839)


X(10841) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS INNER

Barycentrics    -(SW^2*SA^2-(7*S^2+SW^2)*SW*SA+(S^2+8*SW^2)*S^2)*S+12*(SA-SW)*SW*S^2*SA : :

X(10841) lies on these lines:
{3,10842}, {98,6221}, {262,1151}, {6407,10839}, {6425,9755}, {6429,9756}, {6445,10840}, {8306,9772}, {8397,10837}, {8409,10838}, {9738,9751}


X(10842) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) INNER

Barycentrics    (SW^2*SA^2-(7*S^2+SW^2)*SW*SA+(S^2+8*SW^2)*S^2)*S+12*(SA-SW)*SW*S^2*SA : :

X(10842) lies on these lines:
{3,10841}, {98,6398}, {262,1152}, {6408,10840}, {6426,9755}, {6430,9756}, {6446,10839}, {8307,9772}, {8401,10837}, {8417,10838}, {9739,9751}


X(10843) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS INNER TANGENTIAL

Barycentrics    2*SW^2*SA^2+(7*S^2-2*SW^2)*SW*SA+(2*S^2-5*SW^2)*S^2-12*SW*SA*S*a^2 : :

X(10843) lies on these lines:
{3,10844}, {98,6433}, {262,6425}, {371,10840}, {1151,9756}, {6409,9755}, {6453,10839}, {6468,10841}, {6471,10842}, {8308,9772}, {8410,10838}


X(10844) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) INNER TANGENTIAL

Barycentrics    2*SW^2*SA^2+(7*S^2-2*SW^2)*SW*SA+(2*S^2-5*SW^2)*S^2+12*SW*SA*S*a^2 : :

X(10844) lies on these lines:
{3,10844}, {98,6434}, {262,6426}, {372,10839}, {1152,9756}, {6410,9755}, {6454,10840}, {6469,10842}, {6470,10841}, {8309,9772}, {8402,10837}


X(10845) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS SECONDARY CENTRAL

Barycentrics    2*(SB*SW-S^2)*(SC*SW-S^2)-SW*SA*S*a^2 : :
X(10845) = (2*S^2-S*SW-2*SW^2)*X(3)+(S^2+SW^2)*X(76)

X(10845) lies on these lines:
{2,6214}, {3,76}, {262,6417}, {371,6222}, {385,1160}, {1161,5999}, {3312,9755}, {6399,8667}, {6472,10841}, {6475,10842}, {6484,10843}, {6487,10844}, {6811,8976}, {8310,9772}, {8398,10837}, {8411,10838}

X(10845) = {X(3),X(98)}-harmonic conjugate of X(10846)


X(10846) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) SECONDARY CENTRAL

Barycentrics    2*(SB*SW-S^2)*(SC*SW-S^2)+SW*SA*S*a^2 : :
X(10846) = (2*S^2+S*SW-2*SW^2)*X(3)+(S^2+SW^2)*X(76)

X(10846) lies on these lines:
{2,6215}, {3,76}, {262,6418}, {372,6399}, {385,1161}, {1160,5999}, {3311,9755}, {6222,8667}, {6473,10842}, {6474,10841}, {6485,10844}, {6486,10843}, {8311,9772}, {8403,10837}, {8418,10838}

X(10846) = {X(3),X(98)}-harmonic conjugate of X(10845)


X(10847) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS 1ST SECONDARY TANGENTS

Barycentrics    5*SW^2*SA^2+(4*S^2-5*SW^2)*SW*SA+(5*S^2+SW^2)*S^2-3*SW*SA*S*a^2 : :

X(10847) lies on these lines:
{3,10848}, {98,6396}, {262,6419}, {1132,6813}, {1151,9756}, {3312,9755}, {3316,3424}, {6447,10839}, {6476,10841}, {6479,10842}, {6489,10844}, {6497,10846}, {8312,9772}, {8412,10838}


X(10848) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) 1ST SECONDARY TANGENTS

Barycentrics    5*SW^2*SA^2+(4*S^2-5*SW^2)*SW*SA+(5*S^2+SW^2)*S^2+3*SW*SA*S*a^2 : :

X(10848) lies on these lines:
{3,10847}, {98,6200}, {262,6420}, {1131,6811}, {1152,9756}, {3311,9755}, {3317,3424}, {6448,10840}, {6477,10842}, {6478,10841}, {6488,10843}, {6496,10845}, {8313,9772}, {8404,10837}


X(10849) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS 2ND SECONDARY TANGENTS

Barycentrics    7*SW^2*SA^2-(4*S^2+7*SW^2)*SW*SA+(7*S^2+11*SW^2)*S^2+3*SW*SA*S*a^2 : :

X(10849) lies on these lines:
{3,10850}, {98,6435}, {262,372}, {6199,10847}, {6427,9755}, {6431,9756}, {6449,10839}, {6452,10840}, {6480,10841}, {6483,10842}, {6490,10843}, {6493,10844}, {6494,10845}, {6499,10846}, {6501,10848}, {8314,9772}, {8399,10837}, {8413,10838}


X(10850) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) 2ND SECONDARY TANGENTS

Barycentrics    7*SW^2*SA^2-(4*S^2+7*SW^2)*SW*SA+(7*S^2+11*SW^2)*S^2-3*SW*SA*S*a^2 : :

X(10850) lies on these lines:
{3,10849}, {98,6436}, {262,371}, {6395,10848}, {6428,9755}, {6432,9756}, {6450,10840}, {6451,10839}, {6481,10842}, {6482,10841}, {6491,10844}, {6492,10843}, {6495,10846}, {6498,10845}, {6500,10847}, {8315,9772}, {8405,10837}, {8419,10838}


X(10851) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS TANGENTS

Barycentrics    (SW*SA+S^2)*(SB^2+SB*SC+SC^2)+3*SW*SA*S*a^2 : :
X(10851) = 2*(2*S^2-3*S*SW-SW^2)*X(3)-(S^2+SW^2)*X(194)

X(10851) lies on these lines:
{3,194}, {98,6200}, {262,371}, {372,10842}, {1151,9756}, {3311,10840}, {5860,10519}, {5870,6811}, {6221,10839}, {6274,6316}, {6412,10844}, {6425,10849}, {6449,10845}, {6453,10841}, {6459,6813}, {8316,9772}, {8414,10838}, {9744,9758}, {9754,9757}

X(10851) = {X(3),X(9755)}-harmonic conjugate of X(10852)


X(10852) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND LUCAS(-1) TANGENTS

Barycentrics    (SW*SA+S^2)*(SB^2+SB*SC+SC^2)+3*SW*SA*S*a^2 : :
X(10852) = 2*(2*S^2+3*S*SW-SW^2)*X(3)-(S^2+SW^2)*X(194)

X(10852) lies on these lines:
{3,194}, {98,6396}, {262,372}, {371,10841}, {1152,9756}, {3312,10839}, {5861,10519}, {5871,6813}, {6275,6312}, {6398,10840}, {6411,10843}, {6426,10850}, {6450,10846}, {6454,10842}, {6456,10845}, {6460,6811}, {8317,9772}, {8406,10837}, {9744,9757}, {9754,9758}

X(10852) = {X(3),X(9755)}-harmonic conjugate of X(10851)


X(10853) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND 1ST SHARYGIN

Barycentrics    3*a^9*c*b+3*(b^3+c^3)*a^8+(4*b^4+4*c^4-(3*b^2-13*b*c+3*c^2)*b*c)*a^7-3*(b^3+c^3)*a^6*(b^2+c^2)-(4*b^2-5*b*c+4*c^2)*(b^3+c^3)*(b+c)*a^5+(b^3+c^3)*a^4*(b^4-8*b^2*c^2+c^4)-b*c*(b^6+c^6+(5*b^4+5*c^4-(b^2-14*b*c+c^2)*b*c)*b*c)*a^3-(b^3+c^3)*a^2*(b^2-c^2)^2*(b^2+c^2)+(b^2-c^2)^2*a*b^2*c^2*(3*b^2-2*b*c+3*c^2)-2*(b^3+c^3)*b^2*c^2*(b^2-c^2)^2 : :

X(10853) lies on these lines:
{98,8296}, {256,262}, {1281,9772}, {1580,9755}, {8318,10837}, {8319,10838}, {8320,10839}, {8321,10840}, {8322,10841}, {8323,10842}, {8324,10843}, {8325,10844}, {8326,10845}, {8327,10846}, {8328,10847}, {8329,10848}, {8330,10849}, {8331,10850}, {8332,10851}, {8333,10852}, {8424,9756}, {9746,9840}


X(10854) = PERSPECTOR OF THESE TRIANGLES: ARTZT AND 2ND SHARYGIN

Barycentrics    3*a^8*c*b-3*(b+c)*(b^2+c^2)*a^7-(b^4+c^4-(3*b^2-b*c+3*c^2)*b*c)*a^6+(b+c)*(4*b^4+4*c^4-(6*b^2-7*b*c+6*c^2)*b*c)*a^5+b*c*(7*b^4+7*c^4-2*(3*b^2-5*b*c+3*c^2)*b*c)*a^4-(b+c)*(b^6+c^6+(2*b^2-3*b*c+2*c^2)*(3*b^2-2*b*c+3*c^2)*b*c)*a^3+(b^6+c^6+(b^4+6*b^2*c^2+c^4)*b*c)*(b-c)^2*a^2+(b^2-c^2)*(b-c)*a*b^2*c^2*(c^2-4*b*c+b^2)+2*(b^2-c^2)^2*b^2*c^2*(b^2-b*c+c^2) : :

X(10854) lies on these lines:
{8,7385}, {98,8297}, {262,291}, {1281,9772}, {8299,9746}, {8300,9755}, {8301,9756}, {8334,10837}, {8335,10838}, {8336,10839}, {8337,10840}, {8338,10841}, {8339,10842}, {8340,10843}, {8341,10844}, {8342,10845}, {8343,10846}, {8344,10847}, {8345,10848}, {8346,10849}, {8347,10850}, {8348,10851}, {8349,10852}


X(10855) = HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND ATIK

Trilinears    (b+c)*a^4-2*(b^2-b*c+c^2)*a^3+2*(b^4+c^4-(b^2-8*b*c+c^2)*b*c)*a-(b^2-c^2)^2*(b+c) : :
X(10855) = (2*R-r)*X(57)+(2*R+r)*X(210)

X(10855) lies on these lines:
{1,9858}, {2,971}, {3,4512}, {8,443}, {57,210}, {63,5044}, {142,2886}, {392,9778}, {517,3819}, {5437,5784}, {5790,10202}, {9940,9947}, {9942,9948}, {9943,9949}, {9944,9950}, {9945,9951}, {9946,9952}, {10391,10392}

X(10855) = {X(8580), X(8581)}-harmonic conjugate of X(9954)


X(10856) = HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND 3RD CONWAY

Trilinears    a^5+5*(b+c)*a^4+2*(b^2+c^2)*a^3-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^2-(3*b^2+c^2)*(b^2+3*c^2)*a+(b^2-c^2)*(b-c)^3 : :
X(10856) = (2*R*r+r^2)*X(1)-2*(r^2+s^2)*X(3)

X(10856) lies on these lines:
{1,3}, {2,10444}, {142,10442}, {573,2999}, {2345,5745}, {4431,5744}, {9776,10446}, {9841,10463}, {10167,10477}


X(10857) = HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND 6TH MIXTILINEAR

Trilinears    a^5-3*(b+c)*a^4+2*(b^2-4*b*c+c^2)*a^3+2*(b+c)^3*a^2-(3*b^2-2*b*c+3*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)^3 : :
X(10857) = (2*R+r)*X(1)+8*R*X(3)

X(10857) lies on these lines:
{1,3}, {2,1750}, {9,10167}, {77,9533}, {142,1699}, {200,5744}, {405,9841}, {443,5691}, {516,9776}, {549,5720}, {553,5759}, {610,2267}, {631,1490}, {936,3523}, {971,7308}, {991,2999}, {1001,10178}, {1212,1615}, {1742,5272}, {3062,10855}, {3306,7411}, {3345,3468}, {3586,6916}, {3599,4350}, {3624,6847}, {3679,5768}, {4297,6904}, {4326,8732}, {4423,5918}, {4666,9778}, {4882,9588}, {5226,8544}, {5234,10085}, {5316,5658}, {5338,7501}, {5435,7675}, {5437,7580}, {5531,6174}, {5715,6899}, {5731,9623}, {5745,8580}, {6865,9612}, {6883,7171}, {6935,10165}, {7965,8227}, {7988,8727}, {7989,8728}, {7992,9942}, {7993,9945}, {7996,9944}, {8089,8733}, {8245,8731}, {8583,9949}, {9549,10824}, {9851,9858}, {10391,10398}

X(10857) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,7994), (2,5732,1750), (3,57,165), (3,8726,1), (3,9940,40), (3,10202,3587), (1385,6244,10389), (1467,3601,1)


X(10858) = HOMOTHETIC CENTER OF THESE TRIANGLES: ASCELLA AND 2ND PAMFILOS-ZHOU

Barycentrics    2*(a+b+c)*((b+c)*a-(b-c)^2)*S-a^4*(2*a+b+c)+2*(b^2+c^2)*((b+c)*a+(b-c)^2)*a-(b^2-c^2)*(b-c)^3 : :
X(10858) = 2*s^2*X(3)+(8*R*s+S-2*r^2-8*R*r)*X(142)

X(10858) lies on these lines:
{2,8233}, {3,142}, {57,8231}, {942,9808}, {3601,8239}, {8234,8726}, {8237,8732}, {8247,8733}, {9776,9789}


X(10859) = PERSPECTOR OF THESE TRIANGLES: ATIK AND AYME

Trilinears    (b+c)*a^6-2*(b^2-b*c+c^2)*a^5+(b^3+c^3)*a^4-2*b*c*(b-c)^2*a^3-(b^2-c^2)*(b-c)^3*a^2+2*(b^2-c^2)^2*a*(b^2+c^2)-(b^2-c^2)*(b-c)*(b^4+c^4+(5*b^2+4*b*c+5*c^2)*b*c) : :

X(10859) lies on these lines:
{8,2893}, {10,971}, {19,3062}, {374,1903}, {610,8580}, {612,1419}, {1439,8581}, {3610,10324}, {10325,10327}


X(10860) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 1ST CIRCUMPERP

Trilinears    a^5-(b+c)*a^4-2*(b^2-6*b*c+c^2)*a^3+2*(b+c)*(b^2-4*b*c+c^2)*a^2+(b-c)^4*a-(b^2-c^2)^2*(b+c) : :
X(10860) = r*X(8)-(4*R-r)*X(20)

X(10860) lies on these lines:
{1,1407}, {3,4512}, {4,8582}, {8,20}, {9,165}, {30,3359}, {46,2955}, {55,5732}, {57,497}, {72,7992}, {78,9961}, {104,9951}, {109,7070}, {145,9845}, {170,846}, {171,1721}, {200,971}, {223,9371}, {517,7171}, {518,7994}, {528,1768}, {728,10324}, {946,9776}, {962,3333}, {990,5269}, {1001,10178}, {1260,1490}, {1621,3576}, {1697,3476}, {1699,3816}, {1706,5691}, {1707,9441}, {1708,10392}, {1754,2257}, {1761,2941}, {1766,9950}, {2956,7078}, {3158,5537}, {3306,9812}, {3338,9589}, {3579,7330}, {3692,10325}, {3873,5884}, {3880,6762}, {4319,9316}, {5128,7098}, {5281,8545}, {5658,6745}, {6001,6282}, {7308,10164}

X(10860) = reflection of X(i) in X(j) for these (i,j): (200,6244), (1750,1376), (3476,4297)
X(10860) = excentral-isogonal conjugate of X(1743)
X(10860) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,9856,8583), (55,5918,5732), (63,9778,40), (165,1709,9), (165,1750,1376), (165,2951,7580), (165,3062,8580), (1001,10178,10857), (3062,8580,5927), (7991,10085,6762)


X(10861) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND CONWAY

Trilinears    (b+c)*a^2*(a^2+b*c)-(2*b^2-b*c+2*c^2)*a^3+(2*b^4+2*c^4-(b^2-6*b*c+c^2)*b*c)*a-(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2) : :
X(10861) = (4*R-r)*X(7)+(2*R+r)*X(8)

X(10861) lies on these lines:
{1,9859}, {2,971}, {7,8}, {9,404}, {20,392}, {21,3062}, {63,5785}, {142,2476}, {144,3876}, {3306,10398}, {3681,5850}, {3869,4312}, {3889,5542}, {4292,5692}, {4313,10179}, {4861,9846}, {7675,10384}, {9948,9960}, {9949,9961}, {9950,9962}, {9951,9963}, {9952,9964}, {9954,9965}, {10391,10589}

X(10861) = {X(5927), X(10855)}-harmonic conjugate of X(2)


X(10862) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 3RD CONWAY

Trilinears    (2*b^2+3*b*c+2*c^2)*a^6-(b+c)*(2*b^2-b*c+2*c^2)*a^5-4*(b^2+b*c+c^2)*(b^2-3*b*c+c^2)*a^4+2*(b+c)*(2*b^4+2*c^4-(b^2+c^2)*b*c)*a^3+(2*b^4+2*c^4-(5*b^2+6*b*c+5*c^2)*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*a*(2*b^4+2*c^4+(b+3*c)*(3*b+c)*b*c)-2*(b^2-c^2)^2*(b+c)^2*b*c : :

X(10862) lies on these lines:
{1,971}, {8,10435}, {1469,1699}, {1764,8580}, {5927,10439}, {9947,10441}, {10444,10861}


X(10863) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 3RD EULER

Barycentrics    a^3*(b+c)*(a^2-(b+c)*a-2*b^2+8*b*c-2*c^2)+a*(b-c)^2*(2*(b^2+6*b*c+c^2)*a+(b+c)*(b^2-14*b*c+c^2))-(b+c)^2*(b-c)^4 : :

X(10863) lies on these lines:
{1,9842}, {2,10860}, {4,8583}, {5,1538}, {8,908}, {11,118}, {119,9951}, {142,3062}, {1699,2550}, {1709,6692}, {3086,7091}, {3755,5400}, {4413,4679}, {4847,7956}, {5219,10384}, {5231,5817}, {5658,10582}, {6260,8227}, {9947,9955}, {9948,10129}

X(10863) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,9856,8582), (908,9779,946)


X(10864) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND HEXYL

Trilinears    a^6-(3*b^2-14*b*c+3*c^2)*a^4-4*(b+c)*a^3*b*c+(3*b^2-2*b*c+3*c^2)*(b-c)^2*a^2+4*(b^2-c^2)*(b-c)*a*b*c-(b^2+6*b*c+c^2)*(b^2-c^2)^2 : :
X(10864) = (4*R+r)*X(8)-(4*R-r)*X(20)

X(10864) lies on these lines:
{1,971}, {3,5234}, {4,1435}, {8,20}, {9,4297}, {10,9841}, {56,1750}, {57,1837}, {355,7171}, {405,1490}, {443,5587}, {516,6762}, {517,7992}, {944,4314}, {958,5732}, {999,10241}, {1125,5658}, {1697,1709}, {1768,5128}, {3555,6001}, {3577,5884}, {3893,7991}, {4293,10392}, {4355,5805}, {4882,6244}, {5290,8727}, {5450,6986}, {6256,6835}, {6260,8227}, {6264,9951}, {6282,9954}, {7308,7987}, {8726,10855}, {9623,9943}

X(10864) = reflection of X(i) in X(j) for these (i,j): (8,9948), (40,84), (5691,5787), (6766,6762)
X(10864) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3062,9856), (3,9947,8580), (8,10860,40), (5691,10085,57)


X(10865) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND HONSBERGER

Trilinears    ((b+c)*a^4-(4*b^2+3*b*c+4*c^2)*a^3+(b+c)*(6*b^2-b*c+6*c^2)*a^2-(4*b^4+4*c^4+(5*b^2+6*b*c+5*c^2)*b*c)*a+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2))/(b+c-a) : :
X(10865) = (8*R^2+2*R*r-r^2)*X(7)+r*(4*R+r)*X(8)

X(10865) lies on these lines:
{1,9846}, {7,8}, {390,9856}, {1445,8580}, {2346,3062}, {5226,5572}, {5261,5728}, {5927,7671}, {7675,10864}, {7676,10860}, {7677,8583}, {7678,10863}, {7679,8582}, {8543,10384}, {8732,10855}


X(10866) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND HUTSON INTOUCH

Trilinears    ((b+c)*a^4-2*(b+c)*(b^2-4*b*c+c^2)*a^2+8*b*c*(b-c)^2*a+(b^2-c^2)^2*(b+c))*(b+c-a) : :
X(10866) = 8*R*X(8)-3*(4*R-r)*X(210)

X(10866) lies on these lines:
{1,971}, {8,210}, {11,3698}, {12,10863}, {55,5438}, {56,10860}, {65,4301}, {72,4342}, {950,5919}, {1420,5918}, {1616,4319}, {1697,8580}, {1864,2098}, {3244,9844}, {3304,7091}, {3601,10855}, {4313,10179}, {4314,10609}, {4323,5572}, {5044,9819}, {5274,5836}, {5691,10241}, {5881,9947}, {5920,9953}, {6762,8163}, {7962,9954}, {8236,10865}

X(10866) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,9856,8581), (960,9785,3057), (1837,3057,3893)


X(10867) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 2ND PAMFILOS-ZHOU

Trilinears    (b+c)*a^6-2*(b^2+c^2)*a^5+(b+c)*(b^2+3*b*c+c^2)*a^4-4*b*c*(b^2-b*c+c^2)*a^3-(b+c)*(b^4+c^4-2*(3*b^2-5*b*c+3*c^2-S)*b*c)*a^2+2*(b-c)^2*(b^4-4*b^2*c^2+c^4+2*S*b*c)*a-(b^2-c^2)*(b-c)*(b^4+c^4+2*S*b*c+3*(b^2+c^2)*b*c) : :

X(10867) lies on these lines:
{8,637}, {3062,7133}, {5927,8233}, {7596,9856}, {8224,10860}, {8225,8583}, {8228,10863}, {8230,8582}, {8231,8580}, {8234,10864}, {8237,10865}, {8239,10866}, {8243,8581}, {10855,10858}


X(10868) = HOMOTHETIC CENTER OF THESE TRIANGLES: ATIK AND 1ST SHARYGIN

Trilinears    (b-c)^2*a^2-2*(b+c)*(b^2-3*b*c+c^2)*a+b^4+c^4+2*(b^2-b*c+c^2)*b*c : :

X(10868) lies on these lines:
{1,9852}, {8,192}, {21,3062}, {846,8580}, {1284,8581}, {2310,4357}, {8235,10864}, {8238,10865}, {8240,10866}, {8246,10867}, {9947,9959}


X(10869) = PERSPECTOR OF THESE TRIANGLES: AYME AND OUTER-GARCIA

Barycentrics    a^7+(b+c)*a^6+(b+c)^2*a^5+(b+c)*(b^2-6*b*c+c^2)*a^4-(b^2+12*b*c+c^2)*(b+c)^2*a^3-(b^2+12*b*c+c^2)*(b+c)^3*a^2-(b^4+c^4+10*(b^2+b*c+c^2)*b*c)*(b+c)^2*a-(b^2+c^2)*(b+c)^5 : :

X(10869) lies on these lines:
{10,3974}


X(10870) = PERSPECTOR OF THESE TRIANGLES: 4TH BROCARD AND 1ST EHRMANN

Trilinears    (3*(9*R^2-2*SW)*S^2*SA^2+(-6*(9*R^2-2*SW)*(6*R^2-SW)*S^2+4*(3*R^2-SW)*SW^3)*SA+3*((9*R^2-2*SW)*S^2-2*R^2*SW^2)*S^2)*a : :

X(10870) lies on these lines:
{3,9745}, {182,5166}


X(10871) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND INNER-JOHNSON

Barycentrics    a^7-(b+c)*a^6+2*a^5*b*c+(b^2+c^2)*(b-c)^2*a^3-(b^4-c^4)*a^2*(b-c)+((b^2+c^2)^2-b^2*c^2)*a*(b-c)^2-(b^6-c^6)*(b-c) : :

X(10871) lies on these lines:
{11,32}, {355,9941}, {1376,3096}, {2896,3434}, {3099,10826}, {9821,10525}, {10038,10523}, {10583,10584}, {10828,10829}

X(10871) = {X(9941),X(9996)}-harmonic conjugate of X(10872)


X(10872) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND OUTER-JOHNSON

Barycentrics    a^8-(b^2+c^2)*a^6-2*b*c*(b+c)*a^5+(b^2+c^2)*(b+c)^2*a^4+b*c*(2*b^2-b*c+2*c^2)*(b+c)^2*a^2-((b^2+c^2)^2-b^2*c^2)*(b^2-c^2)^2 : :

X(10872) lies on these lines:
{12,32}, {72,9857}, {355,9941}, {958,3096}, {2896,3436}, {3099,10827}, {9821,10526}, {9862,10786}, {10047,10523}, {10583,10585}, {10828,10830}

X(10872) = {X(9941), X(9996)}-harmonic conjugate of X(10871)


X(10873) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND 1ST JOHNSON-YFF

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

X(10873) lies on these lines:
{1,9996}, {5,10047}, {11,10356}, {12,32}, {55,9873}, {56,3096}, {65,9857}, {388,2896}, {495,10038}, {1478,9821}, {3085,9862}, {3094,9597}, {3098,7354}, {3099,9578}, {3157,9923}, {4293,10357}, {5252,9941}, {5433,7914}, {5434,7865}, {9301,9654}, {9997,10871}, {10583,10588}, {10828,10831}

X(10873) = {X(1),X(9996)}-harmonic conjugate of X(10874)


X(10874) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND 2ND JOHNSON-YFF

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

X(10874) lies on these lines:
{1,9996}, {5,10038}, {11,32}, {55,3096}, {56,9873}, {496,10047}, {497,2896}, {1069,9923}, {1479,9821}, {1837,9941}, {3057,9857}, {3058,7865}, {3086,9862}, {3094,9598}, {3098,6284}, {3099,9581}, {4294,10357}, {5432,7914}, {9301,9669}, {9997,10872}, {10345,10799}, {10583,10589}, {10828,10832}

X(10874) = {X(1),X(9996)}-harmonic conjugate of X(10873)


X(10875) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND LUCAS HOMOTHETIC

Trilinears    (S^2*(8*(SA-SW)*R^2-2*S^2-SA^2+6*SW^2)-S*((S^2-4*SW^2)*SA+(S^2-2*SW^2)*SW)+3*SA^2*SW^2)*a : :

X(10875) lies on these lines:
{32,493}, {1271,2896}, {3096,8222}, {3099,8188}, {6461,10876}, {8194,10828}, {8210,9997}, {8212,9993}, {8214,9857}, {8218,9995}, {8220,9996}, {9821,10669}, {9838,9873}


X(10876) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND LUCAS(-1) HOMOTHETIC

Trilinears    (S^2*(8*(SA-SW)*R^2-2*S^2-SA^2+6*SW^2)+S*((S^2-4*SW^2)*SA+(S^2-2*SW^2)*SW)+3*SA^2*SW^2)*a : :

X(10876) lies on these lines:
{32,494}, {1270,2896}, {3096,8223}, {3099,8189}, {6461,10875}, {8195,10828}, {8211,9997}, {8213,9993}, {8215,9857}, {8217,9994}, {8221,9996}, {9821,10673}, {9839,9873}


X(10877) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND MANDART-INCIRCLE

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

X(10877) lies on these lines:
{1,9821}, {3,10047}, {4,10873}, {11,3096}, {12,9993}, {32,55}, {56,3098}, {497,2896}, {1479,9996}, {1697,3099}, {1837,9857}, {2053,3688}, {2076,10387}, {2098,9997}, {2275,3056}, {3023,8782}, {3027,4294}, {3028,9984}, {3057,9941}, {3058,7811}, {3086,10357}, {3295,9301}, {5218,10583}, {5432,7846}, {6022,9998}, {6284,9873}, {10345,10798}, {10828,10833}

X(10877) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,2896,10874), (3295,9301,10038)


X(10878) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND INNER-YFF TANGENTS

Trilinears    (a^6-2*(b^2+b*c+c^2)*a^4+2*b*c*(b+c)*a^3+(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a^2-4*((b^2+c^2)^2-b^2*c^2)*b*c)*a : :

X(10878) lies on these lines:
{1,32}, {12,10871}, {119,10356}, {2896,10528}, {3096,5552}, {9821,10679}, {9862,10805}, {9993,10531}, {10583,10586}, {10828,10834}

X(10878) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,9997,10879), (9941,10038,32)


X(10879) = HOMOTHETIC CENTER OF THESE TRIANGLES: 5TH BROCARD AND OUTER-YFF TANGENTS

Trilinears    (a^6-2*(b^2-b*c+c^2)*a^4-2*b*c*(b+c)*a^3+(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2+4*((b^2+c^2)^2-b^2*c^2)*b*c)*a : :

X(10879) lies on these lines:
{1,32}, {11,10872}, {2896,10529}, {3096,10527}, {9821,10680}, {9862,10806}, {9993,10532}, {10583,10587}, {10828,10835}

X(10879) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,9997,10878), (9941,10047,32)


X(10880) = HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 1ST KENMOTU DIAGONALS

Trilinears    a*(SA^2+2*S*SA-S^2)*SB*SC : :
Trilinears    Sec[A] (Cos[2 A]+Sin[2 A]) : :
X(10880) = SW*(4*R^2-SW)*X(6)+S*(5*R^2-SW)*X(24)

X(10880) lies on these lines:
{3,5410}, {4,371}, {6,24}, {25,588}, {110,10666}, {112,6400}, {186,372}, {232,5058}, {378,1151}, {403,3071}, {427,8981}, {468,7584}, {486,7505}, {590,1594}, {615,10018}, {1504,10311}, {1579,10323}, {1588,3542}, {1593,6221}, {1598,6199}, {1614,10533}, {1870,2067}, {1968,9675}, {1993,8909}, {2066,6198}, {3069,3147}, {3070,6240}, {3092,3592}, {3312,3515}, {3364,8739}, {3389,8740}, {3516,6449}, {3517,5411}, {3518,5413}, {3520,6200}, {3536,5409}, {3541,9540}, {3575,7583}, {5415,6197}, {5889,10665}, {6353,7582}, {6424,8743}, {7487,7585}, {7507,8976}, {7577,10576}, {8753,8946}

X(10880) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,24,10881), (371,5412,4)


X(10881) = HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 2ND KENMOTU DIAGONALS

Trilinears    a*(SA^2-2*S*SA-S^2)*SB*SC : :
Trilinears    Sec[A] (Cos[2 A]-Sin[2 A]) : :
X(10881) = SW*(4*R^2-SW)*X(6)-S*(5*R^2-SW)*X(24)

X(10881) lies on these lines:
{3,5411}, {4,372}, {6,24}, {25,589}, {110,10665}, {112,6239}, {186,371}, {232,5062}, {378,1152}, {403,3070}, {468,7583}, {485,7505}, {590,10018}, {615,1594}, {1505,10311}, {1578,10323}, {1587,3542}, {1593,6398}, {1598,6395}, {1614,10534}, {1870,6502}, {3068,3147}, {3071,6240}, {3093,3594}, {3311,3515}, {3365,8739}, {3390,8740}, {3516,6450}, {3517,5410}, {3518,5412}, {3520,6396}, {3535,5408}, {3575,7584}, {5414,6198}, {5416,6197}, {5889,10666}, {6353,7581}, {7487,7586}, {7577,10577}, {8753,8948}, {8911,8954}

X(10881) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,24,10880), (372,5413,4)


X(10882) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND CIRCUMPERP AND 3RD CONWAY

Trilinears    a*((b+c)*(2*a^4-a^2*(2*b^2-3*b*c+2*c^2)-b*c*(3*b^2-2*b*c+3*c^2))+(2*b^2+b*c+2*c^2)*a^3-(2*b^4+2*c^4+(b^2+6*b*c+c^2)*b*c)*a) : :
X(10882) = R*r*X(1)-(r^2+s^2)*X(3)

X(10882) lies on these lines:
{1,3}, {2,10465}, {21,10435}, {515,10479}, {572,1468}, {573,1193}, {970,5313}, {1001,10442}, {1125,10478}, {3216,9548}, {3616,10446}, {3741,4297}, {5731,10449}, {5732,10463}

X(10882) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,10434), (1,7987,10470), (1,10476,10439), (55,10475,1), (1319,10480,1), (1385,10441,1), (2646,10473,1), (3576,10476,1), (3741,4297,10454), (10444,10455,10435)


X(10883) = HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND 3RD EULER

Barycentrics    (b^2+c^2-3*b*c)*a^4-(b+c)*(2*b^2-3*b*c+2*c^2)*a^3+(b-c)*(b*c*(b-c)*a^2+(b^2-c^2)*(a*(2*b^2+b*c+2*c^2)-(b^2-c^2)*(b-c))) : :
X(10883) = 2*(R+r)*X(3)+(5*R+2*r)*X(4)

Shinagawa coefficients: (R+r, 4*R+r)

X(10883) lies on these lines:
{2,3}, {7,11}, {12,4313}, {63,1699}, {81,5733}, {119,9963}, {200,5086}, {226,10394}, {355,3935}, {497,8543}, {920,4338}, {946,3868}, {1776,1836}, {2886,5273}, {3218,5805}, {3485,10580}, {3486,10578}, {3816,10430}, {3817,5249}, {3869,4301}, {3870,5881}, {3925,9778}, {4292,7741}, {4304,7951}, {5087,5784}, {5208,10478}, {5218,7679}, {5219,7675}, {5550,7958}, {5732,7988}, {7956,9965}, {10861,10863}

X(10883) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,10431,7411), (4,5,6932), (4,3545,6982), (4,6828,2476), (4,6837,21), (4,6852,6985), (5,20,4197), (5,382,6937), (5,6932,2476) et als


X(10884) = HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND HEXYL

Trilinears    (b^2+c^2-a^2)*(a^4-2*(b+c)*a^3-4*a^2*b*c+2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : :
X(10884) = (4*R^2+4*R*r+r^2)*X(1)-R*(4*R+r)*X(7)

X(10884) lies on these lines:
{1,7}, {2,1490}, {3,63}, {4,5249}, {9,6986}, {21,84}, {33,412}, {40,3868}, {55,9943}, {56,10391}, {57,411}, {73,1040}, {142,6835}, {165,3811}, {226,6836}, {377,515}, {405,971}, {442,5787}, {518,5584}, {572,2172}, {581,5256}, {603,3561}, {631,5720}, {908,6865}, {936,3523}, {938,1467}, {942,7580}, {946,4666}, {950,6925}, {958,5784}, {960,8273}, {993,10085}, {997,5267}, {1006,7330}, {1012,1385}, {1125,6837}, {1212,5781}, {1319,9848}, {1420,7125}, {1621,9961}, {1709,5248}, {1750,3091}, {1765,3294}, {1766,3970}, {1768,9964}, {2478,6260}, {3149,3306}, {3305,5777}, {3338,10122}, {3522,6282}, {3562,7070}, {3601,6909}, {3616,10430}, {3624,6884}, {3651,5709}, {3877,7971}, {3885,7966}, {3911,6962}, {4084,7991}, {4197,5587}, {4512,7992}, {5084,5658}, {5208,10476}, {5219,6943}, {5234,5785}, {5250,6001}, {5436,6912}, {5437,6915}, {5531,9588}, {5534,5657}, {5691,6839}, {5715,6895}, {5768,6734}, {6264,9963}, {6284,7702}, {6705,6910}, {6769,9778}, {6840,9612}, {6906,7171}, {6932,9581}, {6953,9843}, {6985,10202}, {8227,10883}, {9845,9859}, {10461,10882}, {10861,10864}

X(10884) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1044,2263), (1,4303,77), (1,5732,20), (63,224,78), (63,4855,1259), (84,3576,21), (936,10857,3523), (1490,8726,2), (3149,9940,3306), (3868,7411,40), (4292,4297,20)


X(10885) = HOMOTHETIC CENTER OF THESE TRIANGLES: CONWAY AND 2ND PAMFILOS-ZHOU

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

X(10885) lies on these lines:
{2,8233}, {7,1659}, {20,7596}, {21,7595}, {63,8231}, {3868,9808}, {4197,8230}, {4313,8239}, {5249,10444}, {5732,8244}, {7411,8224}, {8228,10883}, {8234,10884}, {10861,10867}


X(10886) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND 3RD EULER

Barycentrics    (a^4*b*c+2*(b^3+c^3)*a^3-b*c*(b^2-6*b*c+c^2)*a^2-2*(b^4-c^4)*a*(b-c)-2*(b^2-c^2)^2*b*c)/a : :
X(10886) = R*r*X(1)-2*(r^2+s^2)*X(5)

X(10886) lies on these lines:
{1,5}, {2,10434}, {4,10882}, {946,10479}, {1125,10454}, {1203,5788}, {1699,1764}, {3624,10470}, {3679,9549}, {3741,3817}, {4357,10435}, {9779,10446}, {9955,10441}

X(10886) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5,10887), (3741,3817,10478), (3741,10478,10439)


X(10887) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND 4TH EULER

Barycentrics    a^5*b*c-(b+c)*(2*b^2+b*c+2*c^2)*a^4-(2*b^2-b*c+2*c^2)*(b+c)^2*a^3+(2*b-c)*(b-2*c)*(b+c)^3*a^2+2*(b+c)*(b^2-c^2)*(b^3-c^3)*a+2*(b^2-c^2)^2*(b+c)*b*c : :
X(10887) = R*r*X(1)+2*(r^2+s^2)*X(5)

X(10887) lies on these lines:
{1,5}, {2,10465}, {4,10434}, {10,10478}, {1698,1764}, {3826,10442}, {5691,10470}, {9780,10446}, {9956,10441}, {10175,10439}

X(10887) = {X(1), X(5)}-harmonic conjugate of X(10886)


X(10888) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND 2ND EXTOUCH

Barycentrics    a^6+(b+c)*a^5+2*(b+c)^2*a^4+2*(b^2-c^2)*(b-c)*a^3-3*(b^2-c^2)^2*a^2-(b^2-c^2)*(b-c)*a*(3*b^2+2*b*c+3*c^2)-4*(b^2-c^2)^2*b*c : :
X(10888) = r*(r+2*R)*X(1)+(r^2+s^2)*X(4)

X(10888) lies on these lines:
{1,4}, {2,10444}, {9,1764}, {57,2050}, {78,10451}, {165,7413}, {329,4416}, {1211,5587}, {1743,1746}, {1864,10473}, {2051,2999}, {2270,6708}, {3757,9812}, {4384,9535}, {5777,10441}, {5927,10439}

X(10888) = {X(2), X(10444)}-harmonic conjugate of X(10856)


X(10889) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND HONSBERGER

Barycentrics    (a^4+3*(b+c)*a^3+(3*b^2+3*c^2)*a^2+(b^2-c^2)*(b-c)*a+2*b*c*(b-c)^2)*(b+c-a) : :
X(10889) = 2*s^2*X(1)-(r^2+s^2)*X(7)

X(10889) lies on these lines:
{1,7}, {9,312}, {55,10436}, {69,950}, {75,1697}, {86,3601}, {144,1999}, {319,5727}, {380,1944}, {497,4357}, {518,10480}, {1445,1764}, {2269,4384}, {2346,10435}, {3057,3875}, {3486,3879}, {4360,7962}, {5224,9581}, {5572,10473}, {5728,10441}, {5809,10449}, {5933,6738}, {7671,10439}, {7676,10434}, {7677,10882}, {7678,10886}, {7679,10887}, {8232,10888}, {8732,10856}, {10862,10865}

X(10889) = {X(1), X(10442)}-harmonic conjugate of X(7)


X(10890) = PERSPECTOR OF THESE TRIANGLES: 3RD CONWAY AND 5TH MIXTILINEAR

Barycentrics    (14*b^2+27*b*c+14*c^2)*a^8+16*(b+c)*(b^2+c^2)*a^7-3*(2*b+3*c)*(3*b+2*c)*(b-c)^2*a^6-2*(b+c)*(8*b^4+8*c^4-(11*b^2-34*b*c+11*c^2)*b*c)*a^5+(10*b^6+10*c^6-(27*b^4+27*c^4+2*(25*b^2-11*b*c+25*c^2)*b*c)*b*c)*a^4-4*b*c*(b+c)*(3*b^4+3*c^4-2*(4*b^2-9*b*c+4*c^2)*b*c)*a^3-3*(b^2-c^2)^2*a^2*(2*b^4+2*c^4-(b^2-10*b*c+c^2)*b*c)-2*(b^2-c^2)^2*(b+c)*a*b*c*(5*b^2-2*b*c+5*c^2)-4*(b^2-c^2)^2*(b+c)^2*b^2*c^2 : :

X(10890) lies on these lines:
{1,2050}, {100,10882}, {145,10465}, {1764,2136}, {3243,10442}


X(10891) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND 2ND PAMFILOS-ZHOU

Barycentrics    (4*(b^2+b*c+c^2)*a^4+4*(b^3+c^3)*a^3-4*(b^2-c^2)^2*a^2-4*(b^2-c^2)*(b^3-c^3)*a-4*(b^2-c^2)^2*b*c)*S+a^8+(b+c)*a^7-(3*b^2+4*b*c+3*c^2)*a^6-3*(b^2-c^2)*(b-c)*a^5+(3*b^4+3*c^4+2*(3*b^2-5*b*c+3*c^2)*b*c)*a^4+3*(b^2-c^2)*(b-c)^3*a^3-(b-c)^4*(b^2+4*b*c+c^2)*a^2-(b^2-c^2)*(b-c)^5*a-2*(b^2-c^2)^2*(b-c)^2*b*c : :

X(10891) lies on these lines:
{1,7595}, {1764,8231}, {5249,10444}, {7133,10435}, {8224,10434}, {8225,10882}, {8228,10886}, {8230,10887}, {8233,10888}, {8237,10889}, {9789,10446}, {10856,10858}, {10862,10867}


X(10892) = HOMOTHETIC CENTER OF THESE TRIANGLES: 3RD CONWAY AND 1ST SHARYGIN

Trilinears    (2*b^2+3*b*c+2*c^2)*a^6+2*(b+c)*(b^2+c^2)*a^5-(2*b^4+2*c^4+(b^2-b*c+c^2)*b*c)*a^4-(b+c)*(b^2+b*c+c^2)*(2*b^2-b*c+2*c^2)*a^3-b*c*(2*b^2+b*c+2*c^2)*(b+c)^2*a^2+b*c*(b+c)*(b^4+c^4-3*(b^2+c^2)*b*c)*a+2*(b^2-c^2)^2*b^2*c^2 : :

X(10892) lies on these lines:
{1,256}, {21,10435}, {846,1764}, {3220,8424}, {4425,10478}, {8238,10889}, {8246,10891}, {9791,10446}, {9959,10441}


X(10893) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND INNER-JOHNSON

Barycentrics    a^6*(a-b-c)+2*a^5*b*c-a^2*(3*b^2+2*b*c+3*c^2)*(b-c)^2*(a-b-c)+2*(b^2-c^2)^2*a*(b^2-3*b*c+c^2)-2*(b^2-c^2)^3*(b-c) : :
X(10893) = (R-r)*X(4)+(R-2*r)*X(11)

X(10893) lies on these lines:
{3,3825}, {4,11}, {5,1376}, {12,6968}, {20,10584}, {55,6941}, {98,10794}, {119,3913}, {355,381}, {496,6256}, {515,9669}, {546,7956}, {958,6929}, {1001,6842}, {1012,7741}, {1329,6973}, {1479,1532}, {1519,1837}, {1537,10573}, {1598,10829}, {1709,3336}, {2886,6893}, {3035,6981}, {3058,10786}, {3091,3434}, {3149,3583}, {3428,5046}, {3814,10306}, {3816,6850}, {3925,6898}, {4193,10310}, {4294,6969}, {4413,6975}, {4423,6937}, {4999,6930}, {5101,7507}, {5225,5842}, {5433,6938}, {5584,6902}, {5715,5927}, {5840,6959}, {6001,9581}, {6284,6834}, {6667,6961}, {6691,6948}, {6796,9668}, {6833,7173}, {6957,10522}, {6984,7958}, {7548,9779}, {9993,10871}

X(10893) = midpoint of X(i) and X(j) for these {i,j}: {4,3086}, {5225,6848}
X(10893) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5,10525,1376), (381,946,10894)


X(10894) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND OUTER-JOHNSON

Barycentrics    a^6*(a-b-c)+2*a^5*b*c-4*b*c*(b+c)*a^4-(b^2-c^2)^2*(3*a^2*(a-b-c)-2*a*(b^2-b*c+c^2)+2*(b^2-c^2)*(b-c)) : :
X(10894) = (R+r)*X(4)+(R+2*r)*X(12)

X(10894) lies on these lines:
{3,3822}, {4,12}, {5,958}, {8,7548}, {10,5812}, {11,10532}, {20,10585}, {56,6830}, {72,5587}, {98,10795}, {104,9657}, {149,3832}, {355,381}, {388,6844}, {515,3947}, {1001,6928}, {1012,3585}, {1259,6839}, {1329,6826}, {1376,6917}, {1478,6831}, {1598,10830}, {1699,5697}, {2475,10310}, {2476,3428}, {2551,6843}, {2829,5229}, {2886,6867}, {3035,6885}, {3091,3436}, {3149,7951}, {3614,6834}, {3678,5790}, {3814,6918}, {3850,7956}, {3925,6984}, {4293,6956}, {4413,6901}, {4423,6902}, {4999,6859}, {5080,6828}, {5130,7507}, {5204,6952}, {5220,5805}, {5432,6934}, {5433,6879}, {5450,9655}, {5536,7989}, {5584,6937}, {5708,10265}, {5791,10175}, {6001,9612}, {6256,8727}, {6668,6954}, {6690,6868}, {6691,6978}, {6833,7354}, {6835,10522}, {6845,9656}, {6870,10524}, {6898,7958}, {6903,8273}, {9993,10872}

X(10894) = midpoint of X(i) and X(j) for these {i,j}: {4,3085}, {5229,6847}
X(10894) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,10599,12), (381,946,10893)


X(10895) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND 1ST JOHNSON-YFF

Barycentrics    a^4+(b+c)^2*a^2-2*(b^2-c^2)^2 : :
X(10895) = R*X(1)+3*r*X(381) = r*X(4)+(R+2*r)*X(12) = 4*r*X(5)+(R-r)*X(56)

X(10895) lies on these lines:
{1,381}, {2,3614}, {3,3585}, {4,12}, {5,56}, {10,1836}, {11,153}, {20,5432}, {25,9659}, {30,498}, {33,9627}, {34,7507}, {35,382}, {36,1656}, {46,9956}, {54,9652}, {57,5789}, {65,5587}, {98,10797}, {110,9653}, {115,9650}, {119,6917}, {140,4299}, {197,4214}, {198,407}, {215,578}, {226,1837}, {278,7559}, {354,5290}, {355,2099}, {377,1329}, {386,10407}, {405,3822}, {474,3814}, {495,546}, {496,3850}, {497,3832}, {517,10827}, {528,10528}, {529,10527}, {611,3818}, {612,5064}, {908,5794}, {942,10826}, {946,2098}, {950,3947}, {958,2476}, {999,3851}, {1001,5046}, {1056,3855}, {1062,9628}, {1069,5448}, {1124,6565}, {1155,1698}, {1210,4860}, {1254,7069}, {1319,8227}, {1335,6564}, {1376,2475}, {1388,5886}, {1402,10887}, {1420,7988}, {1454,7330}, {1466,6826}, {1469,10516}, {1470,6918}, {1506,9651}, {1539,10065}, {1598,9673}, {1657,5010}, {1697,5726}, {1699,3057}, {1737,5221}, {1853,7355}, {1858,5927}, {1868,1888}, {1870,7547}, {1909,7773}, {1935,5348}, {2051,9552}, {2477,10539}, {2478,4423}, {2551,3925}, {2646,5219}, {2829,6833}, {2886,3436}, {2911,8818}, {2975,5141}, {3035,4190}, {3058,3839}, {3072,7299}, {3086,3545}, {3090,4293}, {3146,5218}, {3157,9927}, {3295,3583}, {3428,6842}, {3474,3648}, {3485,7548}, {3486,5226}, {3523,5326}, {3526,7280}, {3543,4995}, {3560,5172}, {3584,3830}, {3600,5068}, {3627,4302}, {3649,5714}, {3679,4005}, {3746,9668}, {3761,7776}, {3815,9597}, {3816,5187}, {3817,10106}, {3829,10529}, {3845,10056}, {3854,5274}, {3858,8162}, {3861,4309}, {4292,10175}, {4295,5818}, {4324,5073}, {4325,5070}, {4647,5827}, {4857,6767}, {4999,6933}, {5056,7288}, {5066,10072}, {5067,7294}, {5071,5298}, {5072,5563}, {5154,5253}, {5173,9947}, {5198,10833}, {5254,9596}, {5322,7539}, {5340,7127}, {5418,9647}, {5443,10246}, {5584,6907}, {5603,10893}, {5694,5790}, {5703,10543}, {5919,9614}, {6256,6831}, {6561,9646}, {6668,6910}, {6690,6872}, {6691,6931}, {6827,8273}, {6841,10523}, {6849,10629}, {6923,10310}, {6939,7958}, {6968,7681}, {6971,10269}, {8614,8757}, {9534,10406}, {9555,10408}, {9672,9818}, {9993,10873}, {10088,10113}, {10475,10886}, {10597,10598}

X(10895) = midpoint of X(4) and X(10786)
X(10895) = reflection of X(i) in X(j) for these (i,j): (498,10592), (5217,498)
X(10895) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,10896), (2,5229,7354), (2,7354,5204), (4,12,55), (4,3085,6284), (4,8164,4294), (4,10590,12), (5,1478,56), (5,9656,9657), (11,388,3304), (12,6284,3085), (20,10588,5432), (56,1478,9657), (56,9656,1478), (377,1329,4413), (381,9654,1), (388,3091,11), (495,546,1479), (495,1479,3303), (946,5252,2098), (999,3851,7741), (1056,3855,10591), (1210,10404,4860), (1656,9655,36), (1698,9579,1155), (1699,9578,3057), (2476,5080,958), (2551,5177,3925), (3085,6284,55), (3086,3545,7173), (3090,4293,5433), (3295,3583,9670), (3295,3843,3583), (3436,6871,2886), (3585,7951,3), (3600,5068,10589), (3614,5229,5204), (3614,7354,2), (3832,5261,497), (5219,5691,2646), (5270,7741,999), (5290,9581,354), (5434,7173,3086), (5587,9612,65), (6842,10526,3428), (6872,10585,6690), (8227,9613,1319)


X(10896) = HOMOTHETIC CENTER OF THESE TRIANGLES: EULER AND 2ND JOHNSON-YFF

Barycentrics    a^4+(b-c)^2*a^2-2*(b^2-c^2)^2 : :
X(10896) = R*X(1)-3*r*X(381) = r*X(4)-(R-2*r)*X(11) = 4*r*X(5)-(R+r)*X(55)

X(10896) lies on these lines:
{1,381}, {2,5217}, {3,3583}, {4,11}, {5,55}, {10,4679}, {12,497}, {20,5433}, {25,9672}, {30,499}, {33,7507}, {35,1656}, {36,382}, {54,9667}, {65,1699}, {78,5087}, {79,5708}, {80,1482}, {98,10798}, {100,5154}, {110,9666}, {115,9665}, {140,4302}, {149,3913}, {215,10539}, {226,6744}, {350,7773}, {354,9612}, {355,2098}, {377,3816}, {388,3832}, {390,5068}, {442,4423}, {474,3825}, {495,3850}, {496,546}, {515,1388}, {517,10826}, {528,5552}, {529,10529}, {578,2477}, {613,3818}, {614,5064}, {938,3649}, {942,1898}, {946,1837}, {950,3817}, {958,5046}, {999,3585}, {1001,2476}, {1058,3855}, {1062,9629}, {1069,9927}, {1124,6564}, {1210,1836}, {1319,5691}, {1329,3434}, {1335,6565}, {1376,4193}, {1393,2310}, {1466,8727}, {1506,9664}, {1539,10081}, {1598,9658}, {1621,5141}, {1657,7280}, {1697,7989}, {1698,9580}, {1788,9812}, {1853,6285}, {1864,5715}, {1936,7299}, {2051,9555}, {2307,5339}, {2478,2886}, {2550,6919}, {2646,3586}, {3035,6931}, {3056,10516}, {3057,5587}, {3058,3085}, {3073,5348}, {3090,4294}, {3146,7288}, {3149,5172}, {3157,5448}, {3189,5748}, {3241,7319}, {3295,3851}, {3428,6928}, {3436,3813}, {3474,5704}, {3485,9779}, {3486,7548}, {3523,7294}, {3526,5010}, {3543,5298}, {3582,3830}, {3601,7988}, {3627,4299}, {3683,5705}, {3746,5072}, {3760,7776}, {3814,5687}, {3815,9598}, {3829,10527}, {3839,5229}, {3845,10072}, {3854,5261}, {3861,4317}, {3925,5084}, {4187,4413}, {4190,6691}, {4197,8167}, {4316,5073}, {4330,5070}, {4333,5122}, {4995,5071}, {4999,6872}, {5048,5881}, {5056,5218}, {5066,10056}, {5067,5326}, {5086,5289}, {5119,9956}, {5183,9589}, {5252,10863}, {5254,9599}, {5270,7373}, {5310,7539}, {5418,9660}, {5435,10248}, {5550,6175}, {5563,9655}, {5584,6827}, {5603,10894}, {5697,5790}, {5709,7082}, {5840,6958}, {5842,6834}, {5886,10572}, {5919,9578}, {6198,7547}, {6561,9661}, {6667,6921}, {6690,6933}, {6843,7958}, {6882,10310}, {6907,8273}, {6924,10058}, {6980,10267}, {7395,10833}, {7680,10531}, {9659,9818}, {9957,10827}, {9993,10874}, {10091,10113}, {10175,10624}, {10356,10877}, {10358,10799}, {10596,10599}

X(10896) = reflection of X(i) in X(j) for these (i,j): (499,10593), (5204,499)
X(10896) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,381,10895), (2,5225,6284), (2,6284,5217), (4,11,56), (4,3086,7354), (4,10591,11), (4,10598,7681), (5,1479,55), (5,9671,9670), (11,7354,3086), (12,497,3303), (20,10589,5433), (55,1479,9670), (55,9671,1479), (381,9669,1), (390,5068,10588), (496,546,1478), (496,1478,3304), (497,3091,12), (946,1837,2099), (999,3585,9657), (999,3843,3585), (1058,3855,10590), (1210,1836,5221), (1656,9668,35), (1699,9581,65), (3058,3614,3085), (3085,3545,3614), (3086,7354,56), (3090,4294,5432), (3295,3851,7951), (3434,5187,1329), (3583,7741,3), (3586,8227,2646), (3832,5274,388), (4190,10584,6691), (4857,7951,3295), (5225,7173,5217), (5587,9614,3057), (6284,7173,2), (6882,10525,10310)


X(10897) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND 1ST KENMOTU DIAGONALS

Trilinears    a*SA*(SA^2-SW*SA+S^2-S*(2*R^2-SW)) : :
X(10897) = S*(2*R^2-SW)*X(3)+SW*(4*R^2-SW)*X(6)

X(10897) lies on these lines:
{2,10880}, {3,6}, {5,5412}, {26,5413}, {68,6413}, {184,10666}, {343,5409}, {394,8909}, {486,3549}, {615,7542}, {1060,2067}, {1062,2066}, {1368,8981}, {1588,3547}, {1589,6515}, {2072,10576}, {3068,6643}, {3092,7387}, {3093,9818}, {3546,9540}, {3548,5418}, {5410,7395}, {5411,9715}, {5415,8251}, {5562,10665}, {6565,10024}, {6639,10577}, {6676,7584}, {7488,10881}, {7494,7582}, {10533,10539}

X(10897) = Brocard circle-inverse-of-X(10898)


X(10898) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND EULER AND 2ND KENMOTU DIAGONALS

Trilinears    a*SA*(SA^2-SW*SA+S^2+S*(2*R^2-SW)) : :
X(10898) = S*(2*R^2-SW)*X(3)-SW*(4*R^2-SW)*X(6)

X(10898) lies on these lines:
{2,10881}, {3,6}, {5,5413}, {26,5412}, {68,6414}, {184,10665}, {343,5408}, {485,3549}, {590,7542}, {1060,6502}, {1062,5414}, {1587,3547}, {1590,6515}, {2072,10577}, {3069,6643}, {3092,9818}, {3093,7387}, {3548,5420}, {5410,9715}, {5411,7395}, {5416,8251}, {5562,10666}, {6564,10024}, {6639,10576}, {6676,7583}, {7488,10880}, {7494,7581}, {8854,8976}, {10534,10539}

X(10898) = Brocard circle-inverse-of-X(10897)


X(10899) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND 1ST HATZIPOLAKIS

Trilinears    a^6-(3*b^2-b*c+3*c^2)*a^4+5*b*c*(b+c)*a^3+(3*b^2-5*b*c+3*c^2)*(b+c)^2*a^2-b*c*(b+c)^3*a-(b^2-c^2)^2*(b-c)^2 : :

The 1st Hatzipolakis triangle is the triangle A0B0C0 constructed at X(1118).

A0 = -a^2 (a-b-c) (a+b-c) (a-b+c) : (a+b-c) (a^2+b^2-c^2)^2 : (a-b+c) (a^2-b^2+c^2)^2 (barycentrics,    Peter Moses, November 16, 2016)

X(10899) lies on these lines:
{1,1259}, {40,10900}, {57,5666}, {484,1722}, {978,3336}


X(10900) = PERSPECTOR OF THESE TRIANGLES: EXCENTRAL AND 2ND HATZIPOLAKIS

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

The 2nd Hatzipolakis triangle is the triangle A1B1C1 constructed at X(1119).

A1 = 0 : (a+b-c)^2 (a^2+b^2-c^2) : (a-b+c)^2 (a^2-b^2+c^2) (barycentrics,    Peter Moses, November 16, 2016)

X(10900) lies on these lines:
{1,939}, {9,3772}, {40,10899}, {46,1707}, {57,1122}, {63,3008}, {278,1708}, {978,3338}, {1709,1738}, {1712,1714}, {2999,10601}, {4675,5437}, {5119,8616}


X(10901) = PERSPECTOR OF THESE TRIANGLES: EXTANGENTS AND 1ST JOHNSON-YFF

Trilinears    (p*(4*p^4-6*p^2+3)*q^2+(1-p^2)*(q+2*p^3*(2*p^2-1)))*q : : , where p=sin(A/2), q=cos((B-C)/2)

X(10901) lies on these lines:
{12,71}, {40,1478}, {55,1498}, {3157,6237}, {7724,10088}


X(10902) = HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND KOSNITA

Trilinears    a*(a^4*(a-b-c)-(2*b^2+b*c+2*c^2)*a^3+(2*b^2-b*c+2*c^2)*(b+c)*a^2+(b^3+c^3)*(b+c)*a-(b^2-c^2)*(b^3-c^3)) : :
X(10902) = R*X(1)+2*(R+r)*X(3)

Let A'B'C' be the triangle formed by the radical axes of the incircle and the Odehnal tritangent circles, as defined at X(3822). A'B'C' is homothetic to the 1st circumperp triangle at X(10902). (Randy Hutson, December 10, 2016)

X(10902) lies on these lines:
{1,3}, {2,6796}, {4,3822}, {5,5259}, {10,1006}, {15,10637}, {16,10636}, {19,24}, {21,515}, {31,581}, {39,10315}, {42,580}, {47,2003}, {54,71}, {100,5178}, {104,5267}, {109,4303}, {110,7724}, {140,3925}, {182,3779}, {186,6197}, {191,912}, {225,7412}, {283,947}, {355,5251}, {371,5416}, {372,5415}, {376,10532}, {380,8557}, {386,602}, {404,10165}, {405,5587}, {411,946}, {442,5842}, {497,6988}, {498,6827}, {499,6954}, {516,3651}, {575,8539}, {595,1064}, {601,991}, {631,2550}, {902,4300}, {944,993}, {952,5258}, {958,5881}, {960,6326}, {971,7701}, {1001,3149}, {1005,6260}, {1030,8609}, {1125,6905}, {1147,6237}, {1158,10884}, {1203,5396}, {1283,9840}, {1376,5705}, {1478,6868}, {1479,6825}, {1490,4512}, {1658,8141}, {1698,6883}, {1699,6985}, {1802,3730}, {2266,4253}, {2361,2594}, {2801,3647}, {2915,9625}, {2975,5882}, {3011,4220}, {3074,4551}, {3085,6987}, {3101,7488}, {3145,8235}, {3189,5657}, {3522,10587}, {3523,10527}, {3524,10806}, {3528,10597}, {3530,6154}, {3560,5691}, {3583,6842}, {3585,7491}, {3624,6911}, {3683,5777}, {3814,6902}, {4189,5450}, {4294,6908}, {4297,6906}, {4302,6850}, {4330,5840}, {4423,6918}, {5047,10175}, {5218,6865}, {5284,6915}, {5432,6922}, {5552,6992}, {5603,6876}, {5620,6011}, {5663,6097}, {5715,7580}, {6254,6759}, {6256,6872}, {6284,6907}, {6642,9816}, {6690,6831}, {6863,7741}, {6880,10200}, {6928,7951}, {6936,10786}, {7967,8666}, {8722,10804}, {9537,10298}, {10282,10536}

X(10902) = circumcircle-inverse-of-X(5535)
X(10902) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,165,5709), (1,10268,40), (3,35,2077), (3,40,7688), (3,55,40), (3,1385,36), (3,3295,3428), (3,10267,1), (3,10269,7280), (3,10306,5584), (71,2302,2323), (100,6986,6684), (165,6769,40), (411,1621,946), (944,6875,993), (1001,3149,8227), (1385,3579,5885), (3295,3428,7982), (3579,7957,40), (4189,5731,5450), (5010,7987,3), (5217,8273,3), (5584,10306,40)


X(10903) = PERSPECTOR OF THESE TRIANGLES: 3RD EXTOUCH AND 7TH MIXTILINEAR

Barycentrics    3*a^8+2*(b+c)*a^7-8*a^6*b*c-22*(b^2-c^2)*(b-c)*a^5+2*(3*b^2+8*b*c+3*c^2)*(b-c)^2*a^4+2*(b^2-c^2)*(b-c)*a^3*(11*b^2-6*b*c+11*c^2)-8*(b^2-c^2)^2*a^2*(b^2+c^2)-2*(b^2-c^2)*(b-c)^5*a-(b^2-c^2)^2*(b-c)^4 : :

X(10903) lies on these lines:
{4,3062}, {223,8916}, {1439,3945}, {2124,3182}, {2270,2272}


X(10904) = PERSPECTOR OF THESE TRIANGLES: 3RD EXTOUCH AND INNER-SODDY

Barycentrics    (-(2*a^5*(a+b+c)-2*(b+c)^2*a^4-4*(b^3+c^3)*a^3-2*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)*a*(b^2+c^2)+2*(b^2-c^2)^2*(b+c)^2)*S+a^8+(b+c)*a^7-3*(b+c)^2*a^6-3*(b^2-c^2)*(b-c)*a^5+(b+c)^2*(3*b^2-2*b*c+3*c^2)*a^4+(b^2-c^2)*(b-c)*a^3*(3*b^2+2*b*c+3*c^2)-(b^2-c^2)^2*(b-c)^2*a^2-(b^2-c^2)^2*(b+c)^3*a)/(a-b-c) : :

X(10904) lies on these lines:
{1,10400}, {4,176}, {482,1439}

X(10904) = {X(1),X(10400)}-harmonic conjugate of X(10905)


X(10905) = PERSPECTOR OF THESE TRIANGLES: 3RD EXTOUCH AND OUTER-SODDY

Barycentrics    ((2*a^5*(a+b+c)-2*(b+c)^2*a^4-4*(b^3+c^3)*a^3-2*(b^2-c^2)^2*a^2+2*(b^2-c^2)*(b-c)*a*(b^2+c^2)+2*(b^2-c^2)^2*(b+c)^2)*S+a^8+(b+c)*a^7-3*(b+c)^2*a^6-3*(b^2-c^2)*(b-c)*a^5+(b+c)^2*(3*b^2-2*b*c+3*c^2)*a^4+(b^2-c^2)*(b-c)*a^3*(3*b^2+2*b*c+3*c^2)-(b^2-c^2)^2*(b-c)^2*a^2-(b^2-c^2)^2*(b+c)^3*a)/(a-b-c) : :

X(10905) lies on these lines:
{1,10400}, {4,175}, {481,1439}

X(10905) = {X(1),X(10400)}-harmonic conjugate of X(10904)


X(10906) = PERSPECTOR OF THESE TRIANGLES: 4TH EXTOUCH AND 7TH MIXTILINEAR

Barycentrics    3*a^8-4*(b+c)*a^7+2*(2*b^2+9*b*c+2*c^2)*a^6-8*(b+c)*(b^2-4*b*c+c^2)*a^5-2*(b^4+c^4-(b^2-16*b*c+c^2)*b*c)*a^4+12*(b^2-c^2)^2*(b+c)*a^3-2*(2*b^4+2*c^4+3*(11*b^2+14*b*c+11*c^2)*b*c)*(b-c)^2*a^2-(b^2+c^2)*(b-c)^6 : :

X(10906) lies on these lines:
{65,9533}, {69,3062}, {857,1211}, {5929,10903}


X(10907) = PERSPECTOR OF THESE TRIANGLES: 4TH EXTOUCH AND INNER-SODDY

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

X(10907) lies on these lines:
{1,10401}, {65,482}, {69,176}, {940,1659}, {5929,10904}

X(10907) = {X(1),X(10401)}-harmonic conjugate of X(10908)


X(10908) = PERSPECTOR OF THESE TRIANGLES: 4TH EXTOUCH AND OUTER-SODDY

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

X(10908) lies on these lines:
{1,10401}, {65,481}, {69,175}, {5929,10905}

X(10908) = {X(1),X(10401)}-harmonic conjugate of X(10907)


X(10909) = PERSPECTOR OF THESE TRIANGLES: 5TH EXTOUCH AND 7TH MIXTILINEAR

Barycentrics    (3*a^8-4*(b+c)*a^7+2*(b^2+4*b*c+c^2)*a^6-8*(b+c)*(3*b^2-2*b*c+3*c^2)*a^5+4*(13*b^4+13*c^4+(7*b^2-24*b*c+7*c^2)*b*c)*a^4-4*(b^2-c^2)*(b-c)*a^3*(9*b^2+22*b*c+9*c^2)+2*(b+3*c)*(3*b+c)*(b^2+4*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b-c)^4)/(a-b-c) : :

X(10909) lies on these lines:
{65,9533}, {388,3062}, {1697,2124}, {5930,10903}


X(10910) = PERSPECTOR OF THESE TRIANGLES: 5TH EXTOUCH AND INNER-SODDY

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

X(10910) lies on these lines:
{1,30}, {56,1659}, {65,482}, {176,388}, {1371,5252}, {5930,10904}

X(10910) = {X(1),X(10404)}-harmonic conjugate of X(10911)


X(10911) = PERSPECTOR OF THESE TRIANGLES: 5TH EXTOUCH AND OUTER-SODDY

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

X(10911) lies on these lines:
{1,30}, {65,481}, {175,388}, {1372,5252}, {5930,10905}

X(10911) = {X(1),X(10404)}-harmonic conjugate of X(10910)


X(10912) = PERSPECTOR OF THESE TRIANGLES: FUHRMANN AND INNER-JOHNSON

Trilinears    (b+c-a)*(a^2-(b+c)*a-2*b^2+6*b*c-2*c^2) : :
X(10912) = (R-r)*X(8)-(R-2*r)*X(11)

X(10912) lies on these lines:
{1,474}, {3,2802}, {8,11}, {55,3885}, {56,8668}, {78,3893}, {100,1388}, {145,388}, {191,956}, {220,4051}, {355,381}, {499,1145}, {517,1158}, {518,5693}, {528,944}, {529,962}, {758,8148}, {952,6256}, {958,3057}, {960,4853}, {1001,9957}, {1159,3881}, {1385,4421}, {1389,6601}, {1709,6762}, {2170,4513}, {2646,3895}, {3086,8256}, {3189,3241}, {3244,4780}, {3333,10107}, {3560,10284}, {3617,10584}, {3625,3940}, {3632,5087}, {3633,9612}, {3646,9623}, {3754,7373}, {3829,5818}, {3878,5220}, {3884,9708}, {4342,5795}, {4662,4915}, {5176,10896}, {5855,6845}, {8715,10246}

X(10912) = midpoint of X(1) and X(3680)
X(10912) = reflection of X(i) in X(j) for these (i,j): (8,3813), (3913,1)
X(10912) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8,1320,2098), (8,2098,5289), (78,3893,8168), (3885,4861,55), (3893,5048,78), (4051,4919,220), (4853,7962,960)
X(10912) = orthologic center of these triangles: inner-Johnson to excenters-midpoints
X(10912) = X(8)-of-inner-Johnson-triangle


X(10913) = PERSPECTOR OF THESE TRIANGLES: INNER-GARCIA AND KOSNITA

Trilinears    a*(a^10*(-b-c+a)-3*(b^2-b*c+c^2)*a^9+3*(b^3+c^3)*a^8+2*(b^2-b*c+c^2)*(b-c)^2*a^7-(b^3+c^3)*a^6*(b-2*c)*(2*b-c)+(2*b^6+2*c^6-3*(2*b^2-3*b*c+2*c^2)*b^2*c^2)*a^5-(b+c)*(2*b^6+2*c^6+(3*b^4+3*c^4-(9*b^2-13*b*c+9*c^2)*b*c)*b*c)*a^4-(3*b^6+3*c^6-(12*b^4+12*c^4-(21*b^2-25*b*c+21*c^2)*b*c)*b*c)*(b+c)^2*a^3+(b^2-c^2)*(b-c)*a^2*(3*b^6+3*c^6+(3*b^4+5*b^2*c^2+3*c^4)*b*c)+(b^2-c^2)^2*a*(b^2-b*c+c^2)*(b^4+c^4-2*(b^2+c^2)*b*c)+(b^2-c^2)^3*(b-c)*(b^2*c^2-(b^2+c^2)^2)) : :

X(10913) lies on these lines:
{3,3583}, {4996,7488}, {7280,8279}


X(10914) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND INNER-JOHNSON

Trilinears    (b+c)*a^2-4*a*b*c-(b^2-4*b*c+c^2)*(b+c) : :
X(10914) = r*X(4)+(2*R-r)*X(8)

X(10914) lies on these lines:
{1,474}, {2,3885}, {4,8}, {5,6735}, {10,11}, {35,5541}, {40,956}, {65,519}, {78,1482}, {100,1385}, {145,942}, {169,4513}, {200,5730}, {210,3626}, {354,3244}, {374,2325}, {405,1697}, {443,9874}, {484,5288}, {518,3632}, {528,10572}, {529,1770}, {551,3918}, {758,3625}, {936,7962}, {944,10167}, {946,6736}, {958,5119}, {960,3679}, {997,2098}, {1018,1212}, {1125,3698}, {1155,8666}, {1201,4695}, {1259,10679}, {1320,6797}, {1483,10202}, {1709,4915}, {1737,3813}, {1953,3694}, {2093,6762}, {2099,3811}, {2262,2321}, {2646,8715}, {2886,10039}, {2932,6264}, {2975,3579}, {3241,5045}, {3245,6763}, {3254,4553}, {3295,3895}, {3306,7373}, {3339,4900}, {3340,6765}, {3501,4051}, {3612,4421}, {3617,3877}, {3621,3868}, {3623,5049}, {3624,10179}, {3633,5902}, {3634,3898}, {3635,3922}, {3678,4669}, {3730,4875}, {3876,4678}, {3881,3919}, {3890,9780}, {3899,4539}, {3931,10459}, {3940,8148}, {3953,4674}, {3962,4701}, {3983,4691}, {4005,4746}, {4188,5126}, {4193,7743}, {4511,6946}, {4533,4662}, {4640,5258}, {4677,5904}, {4855,10246}, {5084,9785}, {5123,7741}, {5250,9708}, {5552,5886}, {5554,5722}, {5587,10893}, {5603,6964}, {5657,6926}, {5686,7673}, {5690,6734}, {5727,9844}, {5728,5853}, {5795,10624}, {5818,10598}, {5828,9779}, {5881,6001}, {5882,10609}, {6967,10527}, {6975,9956}, {7966,8726}, {7967,9940}, {8069,8668}, {8193,10829}, {9857,10871}, {10791,10794}

X(10914) = midpoint of X(i) and X(j) for these {i,j}: {65,3893}, {3621,3868}
X(10914) = reflection of X(i) in X(j) for these (i,j): (1,5836), (72,8), (145,942), (1320,6797), (3057,10), (3244,3754), (3555,65), (3878,3626), (3885,9957), (3899,4711), (5697,960), (7982,7686), (10284,9956), (10624,5795)
X(10914) = anticomplement of X(9957)
X(10914) = complement of X(3885)
X(10914) = X(145)-of-X(1)-Brocard-triangle
X(10914) = X(1317)-of-inner-Garcia-triangle
X(10914) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1706,474), (1,3753,5439), (1,3987,3752), (1,5687,5440), (1,5836,3753), (2,3885,9957), (8,962,3421), (8,3434,355), (8,5082,3419), (10,3057,392), (40,956,3916), (40,4853,956), (100,4861,1385), (200,7982,5730), (960,3679,3697), (1376,10912,1), (1697,9623,405), (1706,3680,1), (3244,3754,354), (3434,10522,10525), (3617,3877,5044), (3617,5044,3921), (3626,3878,210), (3679,5697,960), (3698,5919,1125), (3813,8256,1737), (3899,4711,4539), (4662,5692,4533), (4668,5692,4662), (4691,10176,3983)


X(10915) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND INNER-YFF TANGENTS

Barycentrics    (b+c)*a^3-(b^2+6*b*c+c^2)*a^2-(b^2-4*b*c+c^2)*(b+c)*a+(b^2-c^2)^2 : :
X(10915) = (3*R-r)*X(1)-3*(R-r)*X(2)

X(10915) lies on these lines:
{1,2}, {5,3880}, {12,10914}, {65,1145}, {119,946}, {142,3918}, {355,3913}, {495,5836}, {496,5123}, {515,8715}, {516,6256}, {518,5690}, {529,3579}, {908,5697}, {942,8256}, {1146,3991}, {1329,9957}, {1470,10106}, {1479,3895}, {1512,7982}, {1519,4301}, {2077,4297}, {2136,5587}, {3158,5881}, {3208,5179}, {3262,3663}, {3434,10827}, {3436,5119}, {3452,3884}, {3560,8668}, {3585,5541}, {3678,5837}, {3680,8227}, {3813,9956}, {3817,7704}, {3871,5176}, {3874,4848}, {4187,5919}, {5248,5795}, {5250,7162}, {5252,5687}, {5657,10805}, {5761,7682}, {5818,10596}, {5828,9785}, {5886,10912}, {6147,10107}, {6684,8666}, {8193,10834}, {9857,10878}, {10791,10803}

X(10915) = midpoint of X(i) and X(j) for these {i,j}: {8,3811}, {355,3913}
X(10915) = reflection of X(i) in X(j) for these (i,j): (3813,9956), (8666,6684), (10916,10)
X(10915) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3679,5554), (1,6735,10), (8,10039,10), (8,10528,1), (10,551,8582), (10,3244,1210), (10,3625,4847), (3871,5176,10572)


X(10916) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GARCIA AND OUTER-YFF TANGENTS

Barycentrics    (b+c)*a^3-(b-c)^2*a^2-(b^2+c^2)*(b+c)*a+(b^2-c^2)^2 : :
X(10916) = (3*R+r)*X(1)-3*(R+r)*X(2) = (R-2*r)*X(11)-R*X(72)

X(10916) lies on these lines:
{1,2}, {5,518}, {11,72}, {12,3555}, {40,6899}, {46,3434}, {56,3419}, {63,90}, {80,5288}, {104,3651}, {142,3841}, {191,4857}, {210,4187}, {225,596}, {226,3874}, {329,10591}, {354,442}, {355,10680}, {377,3338}, {404,5178}, {496,960}, {515,6985}, {516,1158}, {517,3813}, {528,3579}, {631,3189}, {758,946}, {908,5904}, {942,2886}, {950,993}, {956,1837}, {958,5722}, {999,5794}, {1001,5791}, {1068,1861}, {1145,3893}, {1512,5881}, {1519,5693}, {1770,3218}, {1788,5082}, {2321,8609}, {2476,3873}, {2801,6260}, {2802,10265}, {2900,6889}, {2975,10572}, {2994,7042}, {3436,10826}, {3452,3678}, {3475,6856}, {3583,6763}, {3670,3914}, {3681,4193}, {3742,8728}, {3816,5044}, {3817,6990}, {3820,4662}, {3822,3881}, {3829,9955}, {3838,6147}, {3880,5690}, {3884,5837}, {3916,6284}, {3925,5439}, {3927,9669}, {4293,5175}, {4294,5744}, {4301,6845}, {4302,4652}, {4304,5267}, {4430,5141}, {4661,5154}, {4863,5687}, {5087,10593}, {5248,5745}, {5433,5440}, {5587,6762}, {5657,10806}, {5707,5847}, {5708,5880}, {5715,5811}, {5728,6067}, {5775,9785}, {5777,7681}, {5818,10597}, {5853,6684}, {6585,6796}, {6601,6865}, {6849,7682}, {6925,10085}, {8193,10835}, {9857,10879}, {10164,10902}, {10791,10804}

X(10916) = reflection of X(i) in X(j) for these (i,j): (8715,6684), (10915,10)
X(10916) = complement of X(3811)
X(10916) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,5705,10198), (1,6734,10), (8,1737,10), (8,3086,997), (8,10529,1), (10,3625,6736), (10,9843,3634), (997,3086,1125), (1210,4847,10), (3678,3825,3452), (5705,10198,3634), (5904,7741,908)


X(10917) = PERSPECTOR OF THESE TRIANGLES: INNER-GREBE AND 2ND HYACINTH

Trilinears    a*SA*(16*R^2*SA^2-8*R^2*SW*SA+20*R^2*S^2-8*R^2*SW^2-S^2*SW-2*(SA^2+(2*R^2-SW)*SA+S^2-SW^2+2*R^2*SW)*S) : :

Let A'B'C' be the orthic triangle of ABC and A", B", C" the orthogonal projections of A,B,C on B'C', C',A', A'B', respectively.The triangle A"B"C" is the 2nd Hyacinth triangle of ABC.

X(10917) lies on these lines:
{185,1161}, {1885,5871}, {6146,9929}


X(10918) = PERSPECTOR OF THESE TRIANGLES: OUTER-GREBE AND 2ND HYACINTH

Trilinears    a*SA*(16*R^2*SA^2-8*R^2*SW*SA+20*R^2*S^2-8*R^2*SW^2-S^2*SW+2*(SA^2+(2*R^2-SW)*SA+S^2-SW^2+2*R^2*SW)*S) : :

X(10918) lies on these lines:
{185,1160}, {1885,5870}, {6146,9930}


X(10919) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND INNER-JOHNSON

Barycentrics    a^5-(b+c)*a^4+2*a^3*b*c+(b-c)^2*(b^2+c^2)*a-(b^4-c^4)*(b-c)+(b-c)^2*(-a+b+c)*S : :
X(10919) = 2*SW*(R-r)*X(6)+(S-2*SW)*(R-2*r)*X(11)

X(10919) lies on these lines:
{6,11}, {355,3641}, {1161,10525}, {1271,3434}, {1376,5591}, {5589,10826}, {5595,10829}, {5689,10914}, {6202,10893}, {9994,10871}, {10040,10523}, {10792,10794}

X(10919) = {X(11),X(12586)}-harmonic conjugate of X(10920)


X(10920) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND INNER-JOHNSON

Barycentrics    a^5-(b+c)*a^4+2*a^3*b*c+(b-c)^2*(b^2+c^2)*a-(b^4-c^4)*(b-c)-(b-c)^2*(-a+b+c)*S : :
X(10920) = -2*SW*(R-r)*X(6)+(S+2*SW)*(R-2*r)*X(11)

X(10920) lies on these lines:
{6,11}, {355,3640}, {1160,10525}, {1270,3434}, {1376,5590}, {5588,10826}, {5594,10829}, {5688,10914}, {6201,10893}, {9995,10871}, {10041,10523}, {10793,10794}

X(10920) = {X(11),X(12586)}-harmonic conjugate of X(10919)


X(10921) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND OUTER-JOHNSON

Barycentrics    a^6-(b^2+c^2)*a^4-2*(b+c)*b*c*a^3+(b^2+c^2)*(b+c)^2*a^2-(b^2-c^2)^2*(b^2+c^2)-(a-b+c)*(a+b-c)*(b+c)^2*S : :
X(10921) = 2*SW*(R+r)*X(6)+(S-2*SW)*(2*r+R)*X(12)

X(10921) lies on these lines:
{6,12}, {72,5689}, {355,3641}, {958,5591}, {1161,10526}, {1271,3436}, {5589,10827}, {5595,10830}, {6202,10894}, {9994,10872}, {10048,10523}, {10783,10786}, {10792,10795}

X(10921) = {X(12),X(12587)}-harmonic conjugate of X(10922)


X(10922) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND OUTER-JOHNSON

Barycentrics    a^6-(b^2+c^2)*a^4-2*(b+c)*b*c*a^3+(b^2+c^2)*(b+c)^2*a^2-(b^2-c^2)^2*(b^2+c^2)+(a-b+c)*(a+b-c)*(b+c)^2*S : :
X(10922) = -2*SW*(R+r)*X(6)+(S+2*SW)*(2*r+R)*X(12)

X(10922) lies on these lines:
{6,12}, {72,5688}, {355,3640}, {958,5590}, {1160,10526}, {1270,3436}, {5588,10827}, {5594,10830}, {6201,10894}, {9995,10872}, {10049,10523}, {10784,10786}, {10793,10795}

X(10922) = {X(12),X(12587)}-harmonic conjugate of X(10921)


X(10923) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 1ST JOHNSON-YFF

Barycentrics    (a^4+(b+c)^2*(b^2+c^2-S))/(-a+b+c) : :
X(10923) = 2*r*SW*X(6)+(S-2*SW)*(2*r+R)*X(12)

X(10923) lies on these lines:
{1,6215}, {5,10048}, {6,12}, {11,10514}, {55,5871}, {56,5591}, {65,5689}, {388,1271}, {495,5875}, {1161,1478}, {3085,10783}, {3157,9929}, {3641,5252}, {4293,10517}, {5589,9578}, {5595,10831}, {5605,10919}, {6202,10895}, {9994,10873}, {10792,10797}

X(10923) = {X(1),X(6215)}-harmonic conjugate of X(10925)


X(10924) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 1ST JOHNSON-YFF

Barycentrics    (a^4+(b+c)^2*(b^2+c^2+S))/(-a+b+c) : :
X(10924) = -2*r*SW*X(6)+(S+2*SW)*(2*r+R)*X(12)

X(10924) lies on these lines:
{1,6214}, {5,10049}, {6,12}, {11,10515}, {55,5870}, {56,5590}, {65,5688}, {388,1270}, {495,5874}, {1160,1478}, {3085,10784}, {3157,9930}, {3640,5252}, {4293,10518}, {5588,9578}, {5594,10831}, {5604,10920}, {6201,10895}, {9995,10873}, {10793,10797}

X(10924) = {X(1),X(6214)}-harmonic conjugate of X(10926)


X(10925) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND 2ND JOHNSON-YFF

Barycentrics    (b+c-a)*(a^4+(b-c)^2*(b^2+c^2-S)) : :
X(10925) = -2*r*SW*X(6)+(S-2*SW)*(R-2*r)*X(11)

X(10925) lies on these lines:
{1,6215}, {5,10040}, {6,11}, {12,10514}, {55,5591}, {56,5871}, {496,5875}, {497,1271}, {1069,9929}, {1161,1479}, {1837,3641}, {3057,5689}, {3086,10783}, {4294,10517}, {5589,9581}, {5595,10832}, {5605,10921}, {6202,10896}, {9994,10874}, {10792,10798}

X(10925) = {X(1),X(6215)}-harmonic conjugate of X(10923)


X(10926) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND 2ND JOHNSON-YFF

Barycentrics    (b+c-a)*(a^4+(b-c)^2*(b^2+c^2+S)) : :
X(10926) = 2*r*SW*X(6)+(S+2*SW)*(R-2*r)*X(11)

X(10926) lies on these lines:
{1,6214}, {5,10041}, {6,11}, {12,10515}, {55,5590}, {56,5870}, {496,5874}, {497,1270}, {1069,9930}, {1160,1479}, {1837,3640}, {3057,5688}, {3086,10784}, {4294,10518}, {5588,9581}, {5594,10832}, {5604,10922}, {6201,10896}, {9995,10874}, {10793,10798}

X(10926) = {X(1),X(6214)}-harmonic conjugate of X(10924)


X(10927) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND MANDART-INCIRCLE

Trilinears    (b+c-a)*(b^2-b*c+c^2-S/2)*a : :

X(10927) lies on these lines:
{1,1161}, {3,10048}, {4,10923}, {6,31}, {11,5591}, {12,6202}, {497,1271}, {1479,6215}, {1697,5589}, {1837,5689}, {2098,5605}, {3023,6319}, {3027,6227}, {3028,7725}, {3057,3641}, {3058,5861}, {3086,10517}, {3295,10040}, {3304,6405}, {4294,10783}, {5595,10833}, {5871,6284}, {6267,7355}, {6281,9670}, {9994,10877}, {10513,10926}, {10514,10896}, {10792,10799}

X(10927) = {X(55),X(3056)}-harmonic conjugate of X(10928)
X(10927) = {X(2066), X(10387)}-harmonic conjugate of X(55)


X(10928) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND MANDART-INCIRCLE

Trilinears    (b+c-a)*(b^2-b*c+c^2+S/2)*a : :

X(10928) lies on these lines:
{1,1160}, {3,10049}, {4,10924}, {6,31}, {11,5590}, {12,6201}, {497,1270}, {1479,6214}, {1697,5588}, {1837,5688}, {2098,5604}, {3023,6320}, {3027,6226}, {3028,7726}, {3057,3640}, {3058,5860}, {3086,10518}, {3295,10041}, {3304,6283}, {4294,10784}, {5594,10833}, {5870,6284}, {6266,7355}, {6278,9670}, {9995,10877}, {10513,10925}, {10515,10896}, {10793,10799}

X(10928) = {X(55),X(3056)}-harmonic conjugate of X(10927)
X(10928) = {X(5414), X(10387)}-harmonic conjugate of X(55)


X(10929) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND INNER-YFF TANGENTS

Trilinears    (a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c+4*S*b*c)*a : :
X(10929) = R*(S-2*SW)*X(1)+SW*(R-r)*X(6)

X(10929) lies on these lines:
{1,6}, {12,10919}, {119,10514}, {1161,10679}, {1271,10528}, {5552,5591}, {5595,10834}, {5689,10915}, {6202,10531}, {9994,10878}, {10783,10805}, {10792,20721

X(10929) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12594,10930), (6,5605,10931), (1335,3242,10932)


X(10930) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND INNER-YFF TANGENTS

Trilinears    (a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c-4*S*b*c)*a : :
X(10930) = R*(S+2*SW)*X(1)-SW*(R-r)*X(6)

X(10930) lies on these lines:
{1,6}, {12,10920}, {119,10515}, {1160,10679}, {1270,10528}, {5552,5590}, {5594,10834}, {5688,10915}, {6201,10531}, {9995,10878}, {10784,10805}, {10793,10803}

X(10930) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12594,10929), (6,5604,10932), (1124,3242,10931)


X(10931) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-GREBE AND OUTER-YFF TANGENTS

Trilinears    (a^4-2*(b^2-b*c+c^2)*a^2-2*b*c*(b+c)*a+b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c-4*S*b*c)*a : :
X(10931) = R*(S-2*SW)*X(1)+SW*(R+r)*X(6)

X(10931) lies on these lines:
{1,6}, {11,10921}, {1161,10680}, {1271,10529}, {5591,10527}, {5595,10835}, {5689,10916}, {6202,10532}, {9994,10879}, {10783,10806}, {10792,10804}


X(10932) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-GREBE AND OUTER-YFF TANGENTS

Trilinears    (a^4-2*(b^2-b*c+c^2)*a^2-2*b*c*(b+c)*a+b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c+4*S*b*c)*a : :
X(10932) = R*(S+2*SW)*X(1)-SW*(R+r)*X(6)

X(10932) lies on these lines:
{1,6}, {11,10922}, {1160,10680}, {1270,10529}, {5590,10527}, {5594,10835}, {5688,10916}, {6201,10532}, {9995,10879}, {10784,10806}, {10793,10804}

X(10932) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12595,10931), (6,5604,10930), (1335,3242,10929)


X(10933) = PERSPECTOR OF THESE TRIANGLES: 2ND HATZIPOLAKIS AND SCHROETER

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

X(10933) lies on these lines:
{125,5521}


X(10934) = PERSPECTOR OF THESE TRIANGLES: 2ND HATZIPOLAKIS AND TANGENTIAL

Trilinears    (a^6-2*(b+c)*a^5+(b+c)^2*a^4+4*b*c*(b+c)*a^3-(b^4+c^4+2*(2*b^2-b*c+2*c^2)*b*c)*a^2+2*(b^4-c^4)*a*(b-c)-(b^2-c^2)^2*(b-c)^2)*a : :

X(10934) lies on these lines:
{3,1612}, {25,1841}, {48,7083}, {55,1108}, {56,1279}, {1033,1886}, {1602,1617}, {7071,8609}


X(10935) = PERSPECTOR OF THESE TRIANGLES: HUTSON INTOUCH AND INNER-YFF TANGENTS

Barycentrics    (b+c-a)*(a^9-3*(b+c)*a^8+16*a^7*c*b+8*(b^2-4*b*c+c^2)*(b+c)*a^6-2*(3*b^4+3*c^4+2*(5*b^2-18*b*c+5*c^2)*b*c)*a^5-6*(b+c)*(b^4+c^4-10*(b-c)^2*b*c)*a^4+4*(2*b^4+2*c^4+(2*b^2-17*b*c+2*c^2)*b*c)*(b-c)^2*a^3-4*(b^2-c^2)*(b-c)*b*c*a^2*(6*b^2-11*b*c+6*c^2)-3*(b^2-c^2)^2*(b-c)^4*a+(b^2-c^2)^3*(b-c)^3) : :

X(10935) lies on these lines:
{1,6890}, {11,5554}, {3057,3436}, {7962,10936}, {10043,10827}


X(10936) = PERSPECTOR OF THESE TRIANGLES: HUTSON INTOUCH AND OUTER-YFF TANGENTS

Barycentrics    (b+c-a)*(a^9-3*(b+c)*a^8+8*a^7*c*b+8*(b^3+c^3)*a^6-2*(3*b^4+3*c^4+2*(b^2+6*b*c+c^2)*b*c)*a^5-6*(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a^4+4*(2*b^4+3*b^2*c^2+2*c^4)*(b-c)^2*a^3-4*(b^2-c^2)*(b-c)*a^2*b^2*c^2-3*(b^2-c^2)^2*(b-c)^4*a+(b^2-c^2)^3*(b-c)^3) : :

X(10936) lies on these lines:
{7962,10935}, {9785,9803}


X(10937) = PERSPECTOR OF THESE TRIANGLES: 2ND HYACINTH AND ORTHOCENTROIDAL

Trilinears    a*SA*(SA^2+(10*R^2-3*SW)*SA-100*R^4+S^2-3*SW^2+35*R^2*SW) : :

X(10937) lies on these lines:
{6,3357}, {185,381}, {195,1204}, {974,5654}, {1885,5890}, {1899,3521}, {1992,6102}


X(10938) = PERSPECTOR OF THESE TRIANGLES: 2ND HYACINTH AND REFLECTION

Trilinears    a*SA*(SA^2+(-6*R^2+SW)*SA+36*R^4+S^2+SW^2-15*R^2*SW) : :

X(10938) lies on these lines:
{6,1597}, {184,399}, {185,382}, {265,974}, {567,1181}, {1204,8717}, {1885,6241}, {3581,10605}, {4549,9936}, {5663,6776}, {8916,10980}


X(10939) = PERSPECTOR OF THESE TRIANGLES: INTANGENTS AND 7TH MIXTILINEAR

Trilinears    (b+c-a)*((b+c)*a^4-4*a^3*(b-c)^2+6*(b^2-c^2)*(b-c)*a^2-4*(b^2-c^2)^2*a+(b^2-c^2)*(b-c)^3) : :

X(10939) lies on the cubic K666 and these lines:
{1,971}, {33,210}, {65,1360}, {354,9533}, {5543,5572}, {8916,10980}

X(10939) = {X(1), X(3062)}-harmonic conjugate of X(2124)


X(10940) = PERSPECTOR OF THESE TRIANGLES: INTOUCH AND INNER-YFF TANGENTS

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

X(10940) lies on these lines:
{1,4190}, {2,1158}, {3,1537}, {7,10528}, {46,5552}, {57,10530}, {119,5553}, {355,377}, {962,4881}, {1519,6890}, {4188,4295}, {9776,10586}


X(10941) = PERSPECTOR OF THESE TRIANGLES: INTOUCH AND OUTER-YFF TANGENTS

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

X(10941) lies on these lines:
{1,5905}, {7,10529}, {57,10530}, {65,3434}, {149,4295}, {329,10587}, {3338,10044}, {3889,10806}, {3957,5758}, {5506,10198}


X(10942) = HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND INNER-YFF TANGENTS

Barycentrics    a^4*(b^2+4*b*c+c^2)*(-b-c+a)-2*(b^4-4*b^2*c^2+c^4)*a^3+(b+c)*(b-c)^2*(2*(b^2+3*b*c+c^2)*a^2+(b+c)*(b^2-4*b*c+c^2)*a-(b^2-c^2)^2) : :
X(10942) = R*X(1)-(R-r)*X(5)

X(10942) lies on these lines:
{1,5}, {2,10805}, {3,3436}, {4,3871}, {8,6842}, {10,912}, {26,10830}, {30,4421}, {72,5690}, {140,958}, {145,6941}, {153,6906}, {381,10531}, {388,6911}, {442,5554}, {517,10915}, {546,10894}, {550,2077}, {944,6882}, {956,6863}, {999,6959}, {1056,6944}, {1058,6973}, {1329,1385}, {1470,6924}, {1482,1532}, {1595,5130}, {1656,10585}, {2551,6883}, {3085,3560}, {3090,10586}, {3091,10596}, {3295,6929}, {3421,6825}, {3600,6970}, {3617,6937}, {3622,6975}, {3628,10200}, {3814,5882}, {3913,10525}, {4187,10246}, {4193,7967}, {5080,7491}, {5082,6982}, {5187,10806}, {5220,5843}, {5261,6826}, {5687,6923}, {5770,9780}, {5818,6881}, {5840,8715}, {5874,10922}, {5875,10921}, {5887,10039}, {6214,10930}, {6215,10929}, {6824,8164}, {6834,10530}, {6850,7080}, {6898,10587}, {6917,9654}, {6945,10595}, {6953,10597}, {6968,10524}, {8728,10202}, {9996,10878}, {10796,10803}

X(10942) = midpoint of X(3913) and X(10525)
X(10942) = reflection of X(10943) in X(5)
X(10942) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,119,5), (4,10528,10679), (5,1483,496), (5,1484,10593), (12,355,5), (3436,10786,3), (10956,10958,1)


X(10943) = HOMOTHETIC CENTER OF THESE TRIANGLES: JOHNSON AND OUTER-YFF TANGENTS

Barycentrics    a^4*(b^2-4*b*c+c^2)*(-b-c+a)-2*(b^4+c^4-2*(b-c)^2*b*c)*a^3+2*(b^3+c^3)*a^2*(b-c)^2+(b^2-c^2)^2*a*(b^2+c^2)-(b^2-c^2)^3*(b-c) : :
X(10943) = R*X(1)-(R+r)*X(5)

X(10943) lies on these lines:
{1,5}, {2,10806}, {3,3434}, {4,10529}, {8,6882}, {26,10829}, {30,10525}, {140,1376}, {145,6830}, {149,6906}, {381,10532}, {390,6892}, {442,10246}, {497,3560}, {517,3813}, {546,7956}, {549,10902}, {912,946}, {944,6842}, {956,6928}, {962,5770}, {999,6917}, {1056,6867}, {1058,6824}, {1385,2886}, {1482,6831}, {1595,5101}, {1656,10584}, {2098,10043}, {2476,7967}, {2975,7491}, {3086,6911}, {3090,10587}, {3091,10597}, {3295,6862}, {3616,6881}, {3617,6963}, {3622,6829}, {3628,10198}, {3816,9956}, {3871,6952}, {4018,8727}, {4187,5790}, {5082,6891}, {5274,6893}, {5450,5840}, {5603,6841}, {5687,6958}, {5690,6734}, {5777,7743}, {5844,10912}, {5874,10920}, {5875,10919}, {6214,10932}, {6215,10931}, {6828,10595}, {6833,10530}, {6837,10596}, {6854,10586}, {6871,10805}, {6879,10528}, {6929,9669}, {6978,7080}, {7330,9614}, {9996,10879}, {10796,10804}

X(10943) = reflection of X(10942) in X(5)
X(10943) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,10529,10680), (5,1483,495), (5,1484,496), (11,355,5), (119,7741,5), (5881,7741,119), (10957,10959,1)


X(10944) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND 1ST JOHNSON-YFF

Barycentrics    (2*a*(-b-c+a)+(b+c)^2)/(b+c-a) : :
X(10944) = (R-2*r)*X(1)+2*r*X(5)

X(10944) lies on these lines:
{1,5}, {2,1388}, {4,2098}, {8,56}, {10,1319}, {30,5697}, {34,5101}, {36,5690}, {55,944}, {57,3632}, {65,519}, {145,388}, {169,4534}, {498,10246}, {499,5790}, {515,3057}, {517,1770}, {528,3885}, {529,3869}, {550,5559}, {594,604}, {664,3665}, {867,3938}, {946,5048}, {950,5919}, {956,5427}, {999,10573}, {1000,4294}, {1056,6901}, {1146,9310}, {1155,4311}, {1329,5176}, {1358,9312}, {1385,5432}, {1399,5255}, {1403,4030}, {1405,4969}, {1420,3679}, {1450,3214}, {1460,4046}, {1467,4915}, {1469,5846}, {1470,5687}, {1478,1482}, {1697,1709}, {1698,7294}, {1836,7982}, {2078,5258}, {2646,5882}, {2886,4861}, {2975,5172}, {3036,6691}, {3058,9957}, {3085,6952}, {3157,9933}, {3212,7198}, {3241,3485}, {3245,4325}, {3295,10043}, {3303,3486}, {3304,6946}, {3340,3633}, {3361,4677}, {3436,5289}, {3600,3621}, {3612,3655}, {3617,7288}, {3622,10584}, {3623,5261}, {3625,4315}, {3626,3911}, {3754,5083}, {3813,5086}, {3868,5855}, {3871,6224}, {3897,6690}, {3915,7299}, {3947,4870}, {4295,9657}, {4347,7286}, {4678,5265}, {4720,5323}, {5080,5330}, {5204,5657}, {5217,5731}, {5440,10915}, {5441,10386}, {5603,10893}, {5604,10920}, {5605,10919}, {5691,7962}, {5734,9656}, {5853,8581}, {6049,9780}, {6604,7223}, {6667,7705}, {8148,9655}, {8192,10829}, {8715,10609}, {9616,9649}, {9654,10247}, {9670,9785}, {9997,10871}, {10590,10595}, {10794,10797}

X(10944) = reflection of X(i) in X(j) for these (i,j): (65,10106), (6284,3057), (10572,9957), (10950,1)
X(10944) = inverse-in-Feuerbach-hyperbola of X(496)
X(10944) = X(944)-of-Mandart-incircle-triangle
X(10944) = homothetic center of inner Johnson and Caelum triangles
X(10944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,496), (1,355,11), (1,5252,12), (1,5443,10283), (1,5881,1837), (1,7741,1387), (1,7951,5901), (1,7972,1483), (1,10057,10523), (1,10827,5886), (8,404,8256), (8,3476,56), (8,4308,1788), (10,1319,5433), (11,10948,5), (12,1317,1), (65,10106,5434), (145,388,2099), (145,3434,10912), (388,2099,3649), (495,1483,1), (1385,10039,5432), (1788,3476,4308), (1788,4308,56), (3625,4315,4848), (3893,9850,65), (5886,10827,3614), (7982,9613,1836), (9957,10572,3058), (10283,10592,5443), (10956,10957,12)


X(10945) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND LUCAS HOMOTHETIC

Barycentrics    2*a^13-2*(b+c)*a^12-(b^2-6*b*c+c^2)*a^11-4*(b+c)*(b^2+c^2)*a^10+(6*b^2-11*b*c+6*c^2)*(b+c)^2*a^9+(b+c)*(11*b^4+11*c^4-(9*b^2-40*b*c+9*c^2)*b*c)*a^8-2*(11*b^6+11*c^6+(37*b^2+20*b*c+37*c^2)*b^2*c^2)*a^7+4*b*c*(b+c)*(3*b^4+3*c^4+(9*b^2+4*b*c+9*c^2)*b*c)*a^6+2*(11*b^8+11*c^8-(15*b^6+15*c^6-(16*b^4+16*c^4-5*(11*b^2-6*b*c+11*c^2)*b*c)*b*c)*b*c)*a^5-2*(b+c)*(4*b^8+4*c^8-(3*b^6+3*c^6-(6*b^4+6*c^4-(19*b^2-16*b*c+19*c^2)*b*c)*b*c)*b*c)*a^4-(9*b^10+9*c^10-(26*b^8+26*c^8-(29*b^6+29*c^6-2*(4*b^4+4*c^4-(13*b^2-14*b*c+13*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+4*(b^2-c^2)*(b-c)*a^2*(b^2+c^2)*(b^6+c^6-(b^4+c^4-2*(b^2+c^2)*b*c)*b*c)+(b^2-c^2)^2*(b-c)^2*a*(b^2+c^2)^2*(2*b^2+b*c+2*c^2)-(b^2-c^2)^3*(b-c)*(b^2-b*c+c^2)*(b^2+c^2)^2+(4*a^11-8*a^10*(b+c)+2*(-7*b^2+12*b*c-7*c^2)*a^9+6*(b+c)*(5*b^2-4*b*c+5*c^2)*a^8-4*(2*b^4+2*c^4+(11*b^2+12*b*c+11*c^2)*b*c)*a^7-4*(b+c)*(4*b^4+4*c^4-(9*b^2+10*b*c+9*c^2)*b*c)*a^6+4*(2*b^6+2*c^6-(3*b^4+3*c^4+(5*b^2+32*b*c+5*c^2)*b*c)*b*c)*a^5+4*(b+c)*(2*b^6+2*c^6-(b^4+c^4-(5*b^2+8*b*c+5*c^2)*b*c)*b*c)*a^4-4*(b^8+c^8+(b^6+c^6+(6*b^4+6*c^4+(7*b^2-6*b*c+7*c^2)*b*c)*b*c)*b*c)*a^3-4*(b^2-c^2)*(b-c)^3*b*c*a^2*(b^2+c^2)-2*(c^4+b^4)*(b-c)^2*(b^2+c^2)^2*a+2*(b^2-c^2)*(b-c)*(b^2+c^2)^2*(c^4+b^4))*S : :

X(10945) lies on these lines:
{11,493}, {355,8220}, {1376,8222}, {3434,6462}, {6461,10946}, {8188,10826}, {8194,10829}, {8210,10944}, {8212,10893}, {8214,10914}, {8216,10919}, {8218,10920}, {10525,10669}, {10871,10875}


X(10946) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND LUCAS(-1) HOMOTHETIC

Barycentrics    2*a^13-2*(b+c)*a^12-(b^2-6*b*c+c^2)*a^11-4*(b+c)*(b^2+c^2)*a^10+(6*b^2-11*b*c+6*c^2)*(b+c)^2*a^9+(b+c)*(11*b^4+11*c^4-(9*b^2-40*b*c+9*c^2)*b*c)*a^8-2*(11*b^6+11*c^6+(37*b^2+20*b*c+37*c^2)*b^2*c^2)*a^7+4*b*c*(b+c)*(3*b^4+3*c^4+(9*b^2+4*b*c+9*c^2)*b*c)*a^6+2*(11*b^8+11*c^8-(15*b^6+15*c^6-(16*b^4+16*c^4-5*(11*b^2-6*b*c+11*c^2)*b*c)*b*c)*b*c)*a^5-2*(b+c)*(4*b^8+4*c^8-(3*b^6+3*c^6-(6*b^4+6*c^4-(19*b^2-16*b*c+19*c^2)*b*c)*b*c)*b*c)*a^4-(9*b^10+9*c^10-(26*b^8+26*c^8-(29*b^6+29*c^6-2*(4*b^4+4*c^4-(13*b^2-14*b*c+13*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+4*(b^2-c^2)*(b-c)*a^2*(b^2+c^2)*(b^6+c^6-(b^4+c^4-2*(b^2+c^2)*b*c)*b*c)+(b^2-c^2)^2*(b-c)^2*a*(b^2+c^2)^2*(2*b^2+b*c+2*c^2)-(b^2-c^2)^3*(b-c)*(b^2-b*c+c^2)*(b^2+c^2)^2-(4*a^11-8*a^10*(b+c)+2*(-7*b^2+12*b*c-7*c^2)*a^9+6*(b+c)*(5*b^2-4*b*c+5*c^2)*a^8-4*(2*b^4+2*c^4+(11*b^2+12*b*c+11*c^2)*b*c)*a^7-4*(b+c)*(4*b^4+4*c^4-(9*b^2+10*b*c+9*c^2)*b*c)*a^6+4*(2*b^6+2*c^6-(3*b^4+3*c^4+(5*b^2+32*b*c+5*c^2)*b*c)*b*c)*a^5+4*(b+c)*(2*b^6+2*c^6-(b^4+c^4-(5*b^2+8*b*c+5*c^2)*b*c)*b*c)*a^4-4*(b^8+c^8+(b^6+c^6+(6*b^4+6*c^4+(7*b^2-6*b*c+7*c^2)*b*c)*b*c)*b*c)*a^3-4*(b^2-c^2)*(b-c)^3*b*c*a^2*(b^2+c^2)-2*(c^4+b^4)*(b-c)^2*(b^2+c^2)^2*a+2*(b^2-c^2)*(b-c)*(b^2+c^2)^2*(c^4+b^4))*S : :

X(10946) lies on these lines:
{11,494}, {355,8221}, {1376,8223}, {3434,6463}, {6461,10945}, {8189,10826}, {8195,10829}, {8211,10944}, {8213,10893}, {8215,10914}, {8217,10919}, {8219,10920}, {10525,10673}, {10871,10876}


X(10947) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND MANDART-INCIRCLE

Barycentrics    (a^5-(b+c)*a^4+2*a^3*b*c-(b-c)^4*a+(b^2-c^2)*(b-c)^3)*(b+c-a) : :
X(10947) = 3*r*(R-r)*X(2)+(R-2*r)^2*X(11)

X(10947) lies on these lines:
{1,6923}, {2,11}, {4,2098}, {12,6968}, {56,6948}, {355,1479}, {1058,6951}, {1388,6850}, {1478,3656}, {1697,4857}, {1709,9580}, {1837,10914}, {3085,10598}, {3295,6980}, {3303,6982}, {3583,7962}, {4294,6950}, {4847,7082}, {6284,6938}, {6914,10943}, {6930,9670}, {6973,10896}, {10522,10912}, {10794,10799}, {10829,10833}, {10871,10877}, {10919,10927}, {10920,10928}

X(10947) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,3434,11), (1479,3057,10953)


X(10948) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND OUTER-YFF

Barycentrics    a^4*(b^2-4*b*c+c^2)*(-b-c+a)-2*(b^4+c^4-3*(b-c)^2*b*c)*a^3+(b+c)*(b-c)^2*(2*(b^2-b*c+c^2)*a^2+(b^2-c^2)*(b-c)*a-(b^2-c^2)^2) : :
X(10948) = R*(R-2*r)*X(1)+2*r^2*X(5)

X(10948) lies on these lines:
{1,5}, {3,10947}, {56,10525}, {388,10598}, {404,3086}, {497,6906}, {499,1376}, {946,5570}, {1058,6952}, {1388,6842}, {1478,10893}, {1709,9614}, {1737,3813}, {2098,6882}, {3085,10584}, {3585,7956}, {3816,10039}, {5274,10629}, {5697,6922}, {10046,10829}, {10047,10871}, {10048,10919}, {10049,10920}, {10522,10529}, {10794,10802}

X(10948) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,11,10523), (1,5533,496), (1,8070,495), (11,10944,5)


X(10949) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER-JOHNSON AND OUTER-YFF TANGENTS

Barycentrics    (a^2*(b^2-6*b*c+c^2)*(-b-c+a)-(b+c)^2*(a*(b^2+c^2)-(b^2-c^2)*(b-c)))/(-b-c+a) : :
X(10949) = R*(R-3*r)*X(1)+2*r*(R+r)*X(5)

X(10949) lies on these lines:
{1,5}, {55,6977}, {56,3434}, {1376,5433}, {1388,2886}, {3913,10530}, {5048,6831}, {6284,6938}, {7354,10525}, {10532,10893}, {10584,10587}, {10597,10598}, {10794,10804}, {10829,10835}, {10871,10879}, {10914,10916}, {10919,10931}, {10920,10932}

X(10949) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10944,12), (496,10826,11)


X(10950) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND 2ND JOHNSON-YFF

Barycentrics    (b+c-a)*(2*a^3-(b-c)^2*a+(b^2-c^2)*(b-c)) : :
X(10950) = (R+2*r)*X(1)-2*r*X(5)

X(10950) lies on these lines:
{1,5}, {3,10573}, {4,1389}, {7,9657}, {8,21}, {10,2646}, {33,1904}, {35,5690}, {41,1146}, {52,517}, {56,944}, {57,9845}, {60,6740}, {65,515}, {72,519}, {100,8256}, {145,497}, {150,3665}, {210,5795}, {224,3925}, {243,5174}, {354,6738}, {388,6839}, {390,3621}, {398,7052}, {404,6224}, {484,550}, {498,5790}, {499,10246}, {528,10394}, {529,3868}, {535,4084}, {549,5445}, {594,2268}, {632,5444}, {938,3304}, {942,5434}, {1056,6900}, {1058,6965}, {1069,9933}, {1145,8715}, {1155,4297}, {1159,9655}, {1210,1319}, {1329,4511}, {1376,5554}, {1385,1737}, {1388,3086}, {1466,5768}, {1468,5348}, {1478,3649}, {1479,1482}, {1697,3632}, {1698,5326}, {1749,5441}, {1788,5204}, {1836,3340}, {1857,2334}, {1859,1891}, {2082,4534}, {2361,5247}, {2478,5289}, {2885,9458}, {2886,5086}, {3056,5846}, {3085,6852}, {3244,5048}, {3245,4324}, {3295,7489}, {3303,3488}, {3419,10393}, {3467,5559}, {3485,7548}, {3586,5812}, {3600,4860}, {3601,3679}, {3616,7504}, {3617,5218}, {3622,10585}, {3623,5274}, {3625,4314}, {3633,7962}, {3655,5298}, {3683,5837}, {3689,6736}, {3813,4861}, {3869,5855}, {3885,5854}, {3893,5853}, {3897,4999}, {4293,5221}, {4305,5217}, {4317,5708}, {4642,7004}, {4678,5281}, {4853,4863}, {4855,6174}, {5250,7082}, {5270,5425}, {5603,10894}, {5604,10922}, {5605,10921}, {5697,5844}, {5714,9656}, {5734,9671}, {5836,10391}, {5885,6797}, {7319,9779}, {8148,9668}, {8163,9797}, {8192,10830}, {9613,10404}, {9616,9662}, {9669,10247}, {9997,10872}, {10522,10912}, {10591,10595}, {10795,10798}

X(10950) = reflection of X(i) in X(j) for these (i,j): (3057,950), (6284,10572), (6737,5795), (7354,65), (10106,6738), (10944,1)
X(10950) = inverse-in-Feuerbach-hyperbola of X(5)
X(10950) = X(8)-of-Mandart-incircle-triangle
X(10950) = homothetic center of outer Johnson and Caelum triangles
X(10950) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,80,5), (1,355,12), (1,1837,11), (1,5727,1837), (1,5881,5252), (1,7741,5901), (1,10073,10948), (1,10826,5886), (8,3486,55), (10,2646,5432), (55,3486,10543), (145,497,2098), (496,1483,1), (497,3436,10953), (938,3476,3304), (950,3057,3058), (958,3913,1259), (1210,5882,1319), (1385,1737,5433), (1788,5731,5204), (3086,7967,1388), (3340,5691,1836), (4297,4848,1155), (4305,5657,5217), (5270,5425,6147), (5795,6737,210), (5886,10826,7173), (6738,10106,354), (10958,10959,11)


X(10951) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND LUCAS HOMOTHETIC

Barycentrics    2*a^14-(3*b^2+2*b*c+3*c^2)*a^12+(b+c)*(5*b^2-2*b*c+5*c^2)*a^11+(2*b^2+11*b*c+2*c^2)*(b-c)^2*a^10-(b+c)*(17*b^4+17*c^4+2*(4*b^2+15*b*c+4*c^2)*b*c)*a^9-(11*b^6+11*c^6+(29*b^4+29*c^4+(41*b^2+54*b*c+41*c^2)*b*c)*b*c)*a^8+2*(b+c)*(11*b^6+11*c^6+(6*b^4+6*c^4+(11*b^2+28*b*c+11*c^2)*b*c)*b*c)*a^7+2*(11*b^8+11*c^8+(25*b^6+25*c^6+(46*b^4+46*c^4+(53*b^2+82*b*c+53*c^2)*b*c)*b*c)*b*c)*a^6-2*(b+c)*(7*b^8+7*c^8-2*(2*b^6+2*c^6+(3*b^4+3*c^4+(14*b^2+9*b*c+14*c^2)*b*c)*b*c)*b*c)*a^5+(-17*b^10-17*c^10-(32*b^8+32*c^8+(37*b^6+37*c^6+2*(14*b^4+14*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+(b+c)*(5*b^10+5*c^10-(10*b^8+10*c^8+(11*b^6+11*c^6-2*(12*b^4+12*c^4+(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^2*a^2*(b^2+c^2)*(6*b^6+6*c^6+(7*b^4+7*c^4+6*(2*b^2+b*c+2*c^2)*b*c)*b*c)-(b^2-c^2)^4*(b+c)*a*(b^2+c^2)^2-(b^2+b*c+c^2)*(b^2+c^2)^2*(b^2-c^2)^4+(4*a^12+4*(b+c)*a^11-2*(11*b^2+4*b*c+11*c^2)*a^10-16*(b+c)*(b^2+b*c+c^2)*a^9+2*(11*b^4+11*c^4-2*(2*b^2+9*b*c+2*c^2)*b*c)*a^8+8*(b+c)*(3*b^4+3*c^4+(5*b^2-b*c+5*c^2)*b*c)*a^7+(-8*b^6-8*c^6+4*(10*b^4+10*c^4+(19*b^2+10*b*c+19*c^2)*b*c)*b*c)*a^6-16*(b+c)*(b^2+b*c+c^2)*(b^4+c^4-(b+c)^2*b*c)*a^5+4*(b^8+c^8-(6*b^6+6*c^6+(b^4+c^4-2*(5*b^2+16*b*c+5*c^2)*b*c)*b*c)*b*c)*a^4+4*(b+c)*(b^8+c^8+2*(b^6+c^6-(b^4+c^4-(7*b^2+5*b*c+7*c^2)*b*c)*b*c)*b*c)*a^3-2*(b^2+c^2)*(b^6+c^6-(2*b^4+2*c^4-(b^2+4*b*c+c^2)*b*c)*b*c)*(b+c)^2*a^2+(b^4-c^4)^2*(2*b^4+2*c^4))*S : :

X(10951) lies on these lines:
{12,493}, {72,8214}, {355,8220}, {958,8222}, {3436,6462}, {6461,10952}, {8188,10827}, {8194,10830}, {8210,10950}, {8212,10894}, {8216,10921}, {8218,10922}, {10526,10669}, {10872,10875}


X(10952) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND LUCAS(-1) HOMOTHETIC

Barycentrics    2*a^14-(3*b^2+2*b*c+3*c^2)*a^12+(b+c)*(5*b^2-2*b*c+5*c^2)*a^11+(2*b^2+11*b*c+2*c^2)*(b-c)^2*a^10-(b+c)*(17*b^4+17*c^4+2*(4*b^2+15*b*c+4*c^2)*b*c)*a^9-(11*b^6+11*c^6+(29*b^4+29*c^4+(41*b^2+54*b*c+41*c^2)*b*c)*b*c)*a^8+2*(b+c)*(11*b^6+11*c^6+(6*b^4+6*c^4+(11*b^2+28*b*c+11*c^2)*b*c)*b*c)*a^7+2*(11*b^8+11*c^8+(25*b^6+25*c^6+(46*b^4+46*c^4+(53*b^2+82*b*c+53*c^2)*b*c)*b*c)*b*c)*a^6-2*(b+c)*(7*b^8+7*c^8-2*(2*b^6+2*c^6+(3*b^4+3*c^4+(14*b^2+9*b*c+14*c^2)*b*c)*b*c)*b*c)*a^5+(-17*b^10-17*c^10-(32*b^8+32*c^8+(37*b^6+37*c^6+2*(14*b^4+14*c^4-(3*b^2-4*b*c+3*c^2)*b*c)*b*c)*b*c)*b*c)*a^4+(b+c)*(5*b^10+5*c^10-(10*b^8+10*c^8+(11*b^6+11*c^6-2*(12*b^4+12*c^4+(3*b^2+2*b*c+3*c^2)*b*c)*b*c)*b*c)*b*c)*a^3+(b^2-c^2)^2*a^2*(b^2+c^2)*(6*b^6+6*c^6+(7*b^4+7*c^4+6*(2*b^2+b*c+2*c^2)*b*c)*b*c)-(b^2-c^2)^4*(b+c)*a*(b^2+c^2)^2-(b^2+b*c+c^2)*(b^2+c^2)^2*(b^2-c^2)^4-(4*a^12+4*(b+c)*a^11-2*(11*b^2+4*b*c+11*c^2)*a^10-16*(b+c)*(b^2+b*c+c^2)*a^9+2*(11*b^4+11*c^4-2*(2*b^2+9*b*c+2*c^2)*b*c)*a^8+8*(b+c)*(3*b^4+3*c^4+(5*b^2-b*c+5*c^2)*b*c)*a^7+(-8*b^6-8*c^6+4*(10*b^4+10*c^4+(19*b^2+10*b*c+19*c^2)*b*c)*b*c)*a^6-16*(b+c)*(b^2+b*c+c^2)*(b^4+c^4-(b+c)^2*b*c)*a^5+4*(b^8+c^8-(6*b^6+6*c^6+(b^4+c^4-2*(5*b^2+16*b*c+5*c^2)*b*c)*b*c)*b*c)*a^4+4*(b+c)*(b^8+c^8+2*(b^6+c^6-(b^4+c^4-(7*b^2+5*b*c+7*c^2)*b*c)*b*c)*b*c)*a^3-2*(b^2+c^2)*(b^6+c^6-(2*b^4+2*c^4-(b^2+4*b*c+c^2)*b*c)*b*c)*(b+c)^2*a^2+(b^4-c^4)^2*(2*b^4+2*c^4))*S : :

X(10952) lies on these lines:
{12,494}, {72,8215}, {355,8221}, {958,8223}, {3436,6463}, {6461,10951}, {8189,10827}, {8195,10830}, {8211,10950}, {8213,10894}, {8217,10921}, {8219,10922}, {10526,10673}, {10872,10876}


X(10953) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND MANDART-INCIRCLE

Barycentrics    (b+c-a)*(a^6-(b^2+c^2)*a^4-2*b*c*(b+c)*a^3-(b^2-c^2)^2*a^2+(b^2-c^2)^2*(b-c)^2) : :
X(10953) = r*(R+r)*X(4)-R*(R+2*r)*X(12)

X(10953) lies on these lines:
{1,6928}, {3,10320}, {4,12}, {11,958}, {35,6923}, {56,6827}, {65,5812}, {72,1837}, {145,497}, {355,1479}, {377,5432}, {388,6840}, {498,6917}, {1259,1329}, {1317,10806}, {1470,6922}, {1478,2646}, {1697,3583}, {2361,5230}, {2475,5218}, {2551,3715}, {3086,6902}, {3486,5080}, {3585,3601}, {3614,6835}, {4186,10830}, {4222,9673}, {4293,6903}, {4857,7962}, {5119,10525}, {5172,6868}, {5204,6865}, {5217,6850}, {5229,6895}, {5433,6947}, {5810,10480}, {6836,7354}, {6839,10588}, {6863,8068}, {6872,10524}, {6882,8071}, {6893,10896}, {6898,7173}, {6965,10591}, {7412,9659}, {7491,8069}, {7952,9627}, {10795,10799}, {10872,10877}, {10921,10927}, {10922,10928}

X(10953) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (497,3436,10950), (1479,3057,10947)


X(10954) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND INNER-YFF

Barycentrics    a^4*(b^2+4*b*c+c^2)*(-b-c+a)-2*(b^4+c^4+(b-c)^2*b*c)*a^3+2*(b^2-c^2)*(b-c)*a^2*(b^2+3*b*c+c^2)+(b^2-c^2)^2*(b-c)^2*a-(b^2-c^2)^3*(b-c) : :
X(10954) = R*(R+2*r)*X(1)+2*r^2*X(5)

X(10954) lies on these lines:
{1,5}, {10,343}, {21,3085}, {55,7491}, {72,10039}, {388,6905}, {442,10573}, {497,10599}, {498,958}, {1389,3485}, {1479,10894}, {1737,5439}, {1785,1867}, {2099,6842}, {3086,7504}, {3295,10953}, {4428,10056}, {5119,5812}, {5261,6839}, {6852,8164}, {7548,10590}, {7680,10572}, {10037,10830}, {10040,10921}, {10041,10922}, {10522,10528}, {10795,10801}

X(10954) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,12,10523), (1,8070,496), (12,10950,5)


X(10955) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER-JOHNSON AND INNER-YFF TANGENTS

Barycentrics    (b+c-a)*((b^2+6*b*c+c^2)*a^4-2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*a*b*c+(b^2-c^2)^2*(b-c)^2) : :
X(10955) = R*(R+3*r)*X(1)-2*r*(R-r)*X(5)

X(10955) lies on these lines:
{1,5}, {55,3436}, {56,6880}, {72,10915}, {958,5432}, {1858,10039}, {3058,10953}, {3913,10522}, {3925,5554}, {4662,6735}, {6284,10526}, {6934,7354}, {10531,10894}, {10585,10586}, {10596,10599}, {10795,10803}, {10830,10834}, {10872,10878}, {10921,10929}, {10922,10930}

X(10955) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10958,11), (12,10950,11), (495,10827,12)


X(10956) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST JOHNSON-YFF AND INNER-YFF TANGENTS

Barycentrics    (a^2*(b^2+6*b*c+c^2)*(-b-c+a)-(b^2-4*b*c+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c))/(b+c-a) : :
X(10956) = R*(R-3*r)*X(1)+2*r*(R-r)*X(5)

X(10956) lies on these lines:
{1,5}, {10,5083}, {55,2829}, {56,3035}, {65,1145}, {100,388}, {104,3085}, {149,5261}, {214,10106}, {226,2802}, {498,6713}, {518,6735}, {529,5172}, {956,5433}, {1320,3485}, {1329,1388}, {1470,5434}, {1478,5840}, {1532,5048}, {1537,3057}, {2099,5854}, {3032,10408}, {3036,5554}, {5193,5298}, {5290,5541}, {5432,10269}, {6256,6284}, {8581,10427}, {10056,10058}, {10531,10895}, {10586,10588}, {10590,10596}, {10797,10803}, {10831,10834}, {10873,10878}, {10923,10929}, {10924,10930}

X(10956) = midpoint of X(1478) and X(10087)
X(10956) = outer-Johnson-to-ABC similarity image of X(11)
X(10956) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,119,11), (1,10942,10958), (12,1317,11), (12,10944,10957), (12,10949,5), (495,5252,12)


X(10957) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST JOHNSON-YFF AND OUTER-YFF TANGENTS

Barycentrics    ((b-c)^2*a^3-(b^2-c^2)*(b-c)*a^2-(b^2+c^2)*(b+c)^2*a+(b^2-c^2)^2*(b+c))/(b+c-a) : :
X(10957) = R*(R-r)*X(1)+2*r*(R+r)*X(5)

X(10957) lies on these lines:
{1,5}, {55,6833}, {56,377}, {65,10916}, {225,1883}, {226,3881}, {388,6871}, {442,1319}, {474,3925}, {497,6837}, {1012,6284}, {1476,6175}, {1478,10680}, {2078,7294}, {2098,7680}, {2099,3813}, {2476,3476}, {3057,6831}, {3085,6879}, {3086,6854}, {3303,6860}, {3304,6984}, {3434,10530}, {3816,6931}, {4999,5172}, {5204,6955}, {5217,6966}, {5432,6958}, {5832,6067}, {5836,6734}, {6882,10039}, {6923,7354}, {6957,10896}, {6968,7681}, {10587,10588}, {10590,10597}, {10797,10804}, {10831,10835}, {10873,10879}, {10923,10931}, {10924,10932}

X(10957) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10943,10959) , (5,5252,12), (12,10944,10956), (12,10949,1)


X(10958) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND JOHNSON-YFF AND INNER-YFF TANGENTS

Barycentrics    (b+c-a)*((b+c)^2*a^4-2*(b^2+b*c+c^2)*(b-c)^2*a^2+2*(b^2-c^2)*(b-c)*a*b*c+(b^2-c^2)^2*(b-c)^2) : :
X(10958) = R*(R+r)*X(1)-2*r*(R-r)*X(5)

X(10958) lies on these lines:
{1,5}, {41,6506}, {55,1329}, {56,6834}, {65,1532}, {388,6953}, {405,5432}, {429,1856}, {442,1864}, {497,5187}, {950,3814}, {960,6735}, {1470,3149}, {1479,10679}, {1519,7686}, {1728,6907}, {1737,1858}, {1788,6932}, {1857,5142}, {2099,7681}, {2646,4187}, {2886,5554}, {3057,10915}, {3085,6898}, {3086,10805}, {3256,7965}, {3436,10530}, {3485,6945}, {3486,4193}, {3913,10947}, {4305,6963}, {5204,6962}, {5217,6936}, {5326,6675}, {5433,6863}, {6284,6928}, {6835,10895}, {6882,10572}, {6886,10588}, {6896,10590}, {7680,10531}, {10586,10589}, {10591,10596}, {10798,10803}, {10832,10834}, {10874,10878}, {10925,10929}, {10926,10930}

X(10958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,10942,10956), (11,10950,10959), (11,10955,1)


X(10959) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND JOHNSON-YFF AND OUTER-YFF TANGENTS

Barycentrics    (b+c-a)*((b^2-6*b*c+c^2)*a^4-2*(b^2-b*c+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*a*b*c+(b^2-c^2)^2*(b-c)^2) : :
X(10959) = R*(R+3*r)*X(1)-2*r*(R+r)*X(5)

X(10959) lies on these lines:
{1,5}, {55,3813}, {56,5842}, {497,2975}, {946,1898}, {1836,10085}, {2098,5855}, {3057,10916}, {3086,6880}, {3880,6734}, {5432,5687}, {5433,10267}, {10532,10896}, {10587,10589}, {10591,10597}, {10798,10804}, {10832,10835}, {10874,10879}, {10925,10931}, {10926,10932}

X(10959) = inner-Johnson-to-ABC similarity image of X(12)
X(10959) = inner-Yff-to-outer-Yff similarity image of X(12)
X(10959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (11,10950,10958), (496,1837,11)


X(10960) = PERSPECTOR OF THESE TRIANGLES: 1ST KENMOTU DIAGONALS AND MEDIAL

Trilinears    a*(b^2+c^2-a^2-2*S)*(a^4-(b^2-c^2)^2-2*(b^2+c^2-a^2)*S) : :

X(10960) lies on these lines:
{2,6413}, {3,10533}, {5,371}, {110,6414}, {184,8963}, {216,9306}, {230,8855}, {317,491}, {372,1147}, {942,7969}, {1209,10897}, {1493,6420}, {1511,6396}, {4550,6200}, {6642,8954}, {8961,10539}

X(10960) = X(2)-Ceva conjugate of X(372)
X(10960) = {X(216),X(9306)}-harmonic conjugate of X(10962)
X(10960) = perspector of circumconic centered at X(372)
X(10960) = center of circumconic that is locus of trilinear poles of lines passing through X(372)


X(10961) = HOMOTHETIC CENTER OF THESE TRIANGLES: 1ST KENMOTU DIAGONALS AND SUBMEDIAL

Trilinears    ((2*a^4-2*(b^2-c^2)^2+8*b^2*c^2)*S+a^6-(b^2+c^2)*a^4-(b^2-c^2)^2*a^2+(b^2+c^2)*(b^2-c^2)^2)*a : :

X(10961) lies on these lines:
{2,5412}, {5,371}, {6,1196}, {182,10533}, {372,6642}, {485,7401}, {615,6677}, {1579,1598}, {1656,10897}, {1995,5413}, {2066,9817}, {3068,7392}, {3070,9825}, {3090,10880}, {3155,8963}, {5415,9816}, {5418,7404}, {5462,10665}, {6200,9818}, {6396,6644}, {7506,10898}

X(10961) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5020,10963), (9822,10314,10963)


X(10962) = PERSPECTOR OF THESE TRIANGLES: 2ND KENMOTU DIAGONALS AND MEDIAL

Trilinears    a*(b^2+c^2-a^2+2*S)*(a^4-(b^2-c^2)^2+(2*b^2+2*c^2-2*a^2)*S) : :

X(10962) lies on these lines:
{2,6414}, {3,10534}, {5,372}, {6,8956}, {110,6413}, {155,8954}, {216,9306}, {230,8854}, {317,492}, {371,1147}, {942,7968}, {1209,10898}, {1493,6419}, {1511,6200}, {1599,8911}, {4550,6396}, {5651,8963}

X(10962) = X(2)-Ceva conjugate of X(371)
X(10962) = {X(216),X(9306)}-harmonic conjugate of X(10960)
X(10962) = perspector of circumconic centered at X(371)
X(10962) = center of circumconic that is locus of trilinear poles of lines passing through X(371)


X(10963) = HOMOTHETIC CENTER OF THESE TRIANGLES: 2ND KENMOTU DIAGONALS AND SUBMEDIAL

Trilinears    ((2*a^4-2*(b^2-c^2)^2+8*b^2*c^2)*S-a^6+(b^2+c^2)*a^4+(b^2-c^2)^2*a^2+(b^2-c^2)^2*(-b^2-c^2))*a : :

X(10963) lies on these lines:
{2,5413}, {5,372}, {6,1196}, {182,10534}, {371,6642}, {486,7401}, {590,6677}, {1578,1598}, {1656,10898}, {1995,5412}, {3069,7392}, {3071,9825}, {3090,10881}, {5414,9817}, {5416,9816}, {5420,7404}, {5462,10666}, {6200,6644}, {6396,9818}, {7506,10897}

X(10963) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,5020,10961), (9822,10314,10961)


X(10964) = PERSPECTOR OF THESE TRIANGLES: MANDART-INCIRCLE AND MIXTILINEAR

Trilinears    (a^5+3*(b+c)*a^4-2*(b-c)^2*a^3-6*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(2*b^2-11*b*c+2*c^2)*b*c)*a+3*(b^2-c^2)^2*(b+c))*a : :

X(10964) lies on these lines:
{1,1864}, {55,603}, {354,2334}, {1193,3304}, {3303,8614}, {4322,7074}

X(10964) = X(963)-of-Mandart-incircle-triangle


X(10965) = HOMOTHETIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE AND INNER-YFF TANGENTS

Trilinears    (b+c-a)*(a^4-2*(b^2+b*c+c^2)*a^2+2*b*c*(b+c)*a+(b^2+6*b*c+c^2)*(b-c)^2)*a : :
X(10965) = R*(R-3*r)*X(1)+r*(R-r)*X(3)

X(10965) lies on these lines:
{1,3}, {4,10956}, {11,3913}, {12,6968}, {119,10896}, {497,5187}, {952,10043}, {1001,5554}, {1259,10912}, {1260,3893}, {1392,4996}, {1479,10942}, {1837,10915}, {3058,10953}, {3085,10596}, {4294,10805}, {4511,8668}, {5218,10586}, {6833,10949}, {10530,10959}, {10543,10935}, {10799,10803}, {10833,10834}, {10877,10878}, {10927,10929}, {10928,10930}

X(10965) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1470,3304), (55,2098,10966), (497,10528,10958), (3057,3295,55)


X(10966) = HOMOTHETIC CENTER OF THESE TRIANGLES: MANDART-INCIRCLE AND OUTER-YFF TANGENTS

Trilinears    (b+c-a)*(a^4-2*(b^2-b*c+c^2)*a^2-2*b*c*(b+c)*a+(b-c)^4)*a : :
X(10966) = R*(R-r)*X(1)+r*(R+r)*X(3)

X(10966) lies on these lines:
{1,3}, {4,10957}, {11,958}, {12,6834}, {104,4294}, {198,4271}, {212,1201}, {283,3478}, {411,3476}, {497,2975}, {950,8666}, {956,1837}, {1000,6942}, {1191,2361}, {1259,5289}, {1457,1496}, {1476,7411}, {1479,10943}, {1532,10895}, {2178,2269}, {3085,6880}, {3086,6947}, {3149,5252}, {3193,4267}, {3436,10530}, {3556,10535}, {3885,8668}, {4342,5267}, {5218,5253}, {5258,9581}, {5260,10589}, {5288,5727}, {5432,6921}, {5433,6967}, {5450,10624}, {6284,6938}, {6911,10039}, {6925,7354}, {6929,10896}, {10543,10936}, {10786,10956}, {10799,10804}, {10833,10835}, {10877,10879}, {10927,10931}, {10928,10932}

X(10966) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,3057,55), (55,2098,10965), (56,5584,5204), (497,10529,10959)


X(10967) = PERSPECTOR OF THESE TRIANGLES: 2ND MIDARC AND TANGENTIAL-MIDARC

Trilinears    ((b+c)*a-(b-c)^2)*cos(A/2)+((b-2*c)*a-(b-c)*(b+2*c))*cos(B/2)-((2*b-c)*a-(b-c)*(2*b+c))*cos(C/2)+S : :

X(10967) lies on these lines:
{1,8084}, {7,177}, {164,8075}, {174,10503}, {258,8089}, {8091,8094}, {8092,8099}, {8241,10508}, {8242,10506}

X(10967) = reflection of X(i) in X(j) for these (i,j): (8084,1), (8092,8099), (8094,8091)


X(10968) = PERSPECTOR OF THESE TRIANGLES: 2ND MIDARC AND 2ND TANGENTIAL-MIDARC

Trilinears    8*cos(A/2)*a*b*c*(b-c)*(a-b-c)*(-cos(B/2)+cos(C/2))+4*(a^2-2*(b+c)*a+(b-c)^2)*a*b*c*cos(B/2)*cos(C/2)-(a+b+c)*((b+c)*a*(a^2+3*(b-c)^2)-(b^2+c^2)*(3*a^2+(b-c)^2)) : :

X(10968) lies on these lines:
{1,168}, {166,258}, {5919,8241}, {8093,9957}

X(10968) = reflection of X(8083) in X(1)


X(10969) = PERSPECTOR OF THESE TRIANGLES: MIDHEIGHT AND 7TH MIXTILINEAR

Barycentrics    3*a^12-6*(b+c)*a^11-2*(11*b^2-18*b*c+11*c^2)*a^10+2*(b+c)*(43*b^2-62*b*c+43*c^2)*a^9-(103*b^4+103*c^4+26*(4*b^2-11*b*c+4*c^2)*b*c)*a^8+4*(b^2-c^2)*(b-c)*a^7*(9*b^2+62*b*c+9*c^2)+4*(7*b^2+34*b*c+7*c^2)*(b-c)^4*a^6-4*(b^2-c^2)*(b-c)*a^5*(13*b^4+13*c^4+6*(14*b^2-11*b*c+14*c^2)*b*c)+(69*b^6+69*c^6+(266*b^4+266*c^4+(235*b^2-116*b*c+235*c^2)*b*c)*b*c)*(b-c)^2*a^4-2*(b^2-c^2)*(b-c)^3*a^3*(31*b^4+31*c^4+2*(18*b^2+61*b*c+18*c^2)*b*c)+2*(b^2-c^2)^2*(b-c)^4*a^2*(13*b^2+6*b*c+13*c^2)-2*(b^2-c^2)*(b-c)^7*a*(b^2+6*b*c+c^2)-(b^2-c^2)^2*(b-c)^8 : :

X(10969) lies on these lines:
{226,3062}, {942,9533}


X(10970) = PERSPECTOR OF THESE TRIANGLES: 6TH MIXTILINEAR AND INNER-YFF TANGENTS

Trilinears    4*q^2+2*(10*q^2-9)*p^3*q-4*(8*q^2-7)*p^5*q-12*p*q+4*(6*q^2-5)*p^6-4-4*p^8-(16*q^4-11*q^2-16)*p^2+(12*q^4-12*q^2+7)*p^4 : : , where p=sin(A/2), q=cos((B-C)/2)

X(10970) lies on this line: {10971,10980}


X(10971) = PERSPECTOR OF THESE TRIANGLES: 6TH MIXTILINEAR AND OUTER-YFF TANGENTS

Trilinears    4*q^2+2*(22*q^2-25)*p^3*q-4*(8*q^2-9)*p^5*q-12*p*q+12*(2*q^2-1)*p^6-4-4*p^8-(16*q^4-27*q^2-16)*p^2+(6*q^2-23)*(2*q^2-1)*p^4 : :
where p=sin(A/2), q=cos((B-C)/2)

X(10971) lies on this line: {10970,10980}



X(10972) = PERSPECTOR OF THESE TRIANGLES: 7TH MIXTILINEAR AND INNER-SODDY

Barycentrics    ((a^6+4*(b+c)*a^5-(27*b^2-38*b*c+27*c^2)*a^4+48*(b^2-c^2)*(b-c)*a^3-(b-c)^2*(37*b^2+102*b*c+37*c^2)*a^2+4*(b^2-c^2)*(b-c)*a*(b+3*c)*(3*b+c)-(b-c)^6)*S+(a-b-c)*(a^6-6*(b+c)*a^5+(15*b^2-14*b*c+15*c^2)*a^4-20*(b^2-c^2)*(b-c)*a^3+5*(3*b+c)*(b+3*c)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*a*(b+3*c)*(3*b+c)+(b-c)^6)*a)/(b+c-a) : :

X(10972) lies on these lines:
{1,10973}, {176,3062}, {482,9533}, {10903,10904}, {10906,10907}, {10909,10910}, {10970,10980}


X(10973) = PERSPECTOR OF THESE TRIANGLES: 7TH MIXTILINEAR AND OUTER-SODDY

Barycentrics    (-(a^6+4*(b+c)*a^5-(27*b^2-38*b*c+27*c^2)*a^4+48*(b^2-c^2)*(b-c)*a^3-(b-c)^2*(37*b^2+102*b*c+37*c^2)*a^2+4*(b^2-c^2)*(b-c)*a*(b+3*c)*(3*b+c)-(b-c)^6)*S+(a-b-c)*(a^6-6*(b+c)*a^5+(15*b^2-14*b*c+15*c^2)*a^4-20*(b^2-c^2)*(b-c)*a^3+5*(3*b+c)*(b+3*c)*(b-c)^2*a^2-2*(b^2-c^2)*(b-c)*a*(b+3*c)*(3*b+c)+(b-c)^6)*a)/(b+c-a) : :

X(10973) lies on these lines:
{1,10972}, {175,3062}, {481,9533}, {10903,10905}, {10906,10908}, {10909,10911}


X(10974) = PERSPECTOR OF THESE TRIANGLES: MANDART-EXCIRCLES AND APOLLONIUS

Trilinears    (b+c)*a*((b+c)*a^3+(b+c)^2*a^2-(b^2-c^2)*(b-c)*a-b^4-c^4) : :

Let A'B'C' be the intangents triangle of ABC. The line B'C' touches the A-excircle at A*, and B* and C* are defined similarly. The triangle A*B*C* is named here the Mandart-excircles triangle of ABC.

A* = (b-c)^2 (a+b+c) : -b^2 (a+b-c) : -c^2 (a-b+c) (barycentrics,    Peter Moses, November 16, 2016)

The appearance of (T,n) in the following list means that triangles Mandart-excircles and T are perspective with perspector X(n): (ABC, 56), (Apollonius, 10974), (extangents, 3779), (inner-Grebe, 10975), (outer-Grebe, 10976), (Hutson extouch, 3555), (3rd mixtilinear, 56), (orthic, 513))

X(10974) lies on these lines:
{3,6}, {10,12}, {43,46}, {51,1724}, {169,2238}, {185,1754}, {213,4456}, {377,1330}, {407,1829}, {429,1905}, {517,1834}, {674,5266}, {960,4205}, {975,3781}, {1155,2392}, {1213,5044}, {1400,3682}, {1682,2646}, {1730,3216}, {1737,3142}, {1780,3145}, {1865,1871}, {1901,5777}, {2051,6831}, {2194,2915}, {2842,3030}, {3032,10609}, {3033,5213}, {3612,5429}, {3779,3811}, {3794,6910}, {3868,3936}, {3869,5051}, {5292,10441}, {5794,5814}, {5810,6917}, {6685,8258}, {6693,7483}, {6836,9535}

X(10974) = reflection of X(10381) in X(3454)
X(10974) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (181,10822,10), (970,4260,386)


X(10975) = PERSPECTOR OF THESE TRIANGLES: MANDART-EXCIRCLES AND INNER-GREBE

Trilinears    (a^6-(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*(10*b^2+7*b*c+10*c^2)*b*c)*a^2+16*(b^3+c^3)*a*b*c-(b^2-c^2)^2*(b-c)^2+8*((b+c)*a*((b+c)*a-b*c)-(b-c)*(b^3-c^3))*S)*a : :

X(10975) lies on these lines:
{65,3641}


X(10976) = PERSPECTOR OF THESE TRIANGLES: MANDART-EXCIRCLES AND OUTER-GREBE

Trilinears    (a^6-(3*b^2-2*b*c+3*c^2)*a^4+(3*b^4+3*c^4-2*(10*b^2+7*b*c+10*c^2)*b*c)*a^2+16*(b^3+c^3)*a*b*c-(b^2-c^2)^2*(b-c)^2-8*((b+c)*a*((b+c)*a-b*c)-(b-c)*(b^3-c^3))*S)*a : :

X(10976) lies on these lines:
{65,3640}




leftri  Centers related to recent advances: X(10977) - X(10989)  rightri

Centers X(10977)-X(10798) were contributed by Randy Hutson, November 15, 2016.

underbar

X(10977) = PERSPECTOR OF ABC AND MID-TRIANGLE OF 1st AND 2nd HATZIPOLAKIS TRIANGLES

Barycentrics    1/((-a+b+c) (-a^2+b^2+c^2) (3 a^2-3 b^2-2 b c-3 c^2)) : :
Trilinears    (Sec[A]-1)/(1+3 Cos[A]) : :

See X(10899) and X(10900).

X(10977) lies on these lines: {4,5556}

X(10977) = isotomic conjugate of X(10978)
X(10977) = trilinear product of vertices of mid-triangle of 1st and 2nd Hatzipolakis triangles


X(10978) = ISOTOMIC CONJUGATE OF X(10977)

Barycentrics    (-a+b+c) (-a^2+b^2+c^2) (3 a^2-3 b^2-2 b c-3 c^2) : :
Trilinears    Csc[A](3 Cot[A]+Csc[A])/(Sec[A]-1) : :

X(10978) lies on these lines: {69,72}, {319,5423}

X(10978) = isotomic conjugate of X(10977)
X(10978) = {X(1264), X(1265)}-harmonic conjugate of X(69)


X(10979) = {X(10639),X(10640)}-HARMONIC CONJUGATE OF X(9306)

Trilinears    (sin 2A)(3 cos A + 4 cos B cos C) : :

X(10979) lies on these lines: {3,6}, {4,233}, {22,10314} et al


X(10980) = {X(10655),X(10656)}-HARMONIC CONJUGATE OF X(3062)

Trilinears    a2 + 2a(b + c) - 3(b - c)2 : :

X(10980) lies on these lines: {1,3}, {2,5223}, {4,4355}, {6,5573}, {7,1699}, {9,3742}, {11,4654}, {20,6744}, {1706,8168}, {8916,10939}, {10970,10971}


X(10981) = {X(10669),X(10673)}-HARMONIC CONJUGATE OF X(6461)

Barycentrics    (pending)

X(10981) lies on these lines: {4,6339}, {6461,10669}, {8212,8223}, {8213,8222} et al

X(10981) = exsimilicenter of circumcircles of Lucas homothetic and Lucas(-1) homothetic triangles; the insimilicenter is X(6461)


X(10982) = X(3)X(51)∩X(4)X(6)

Barycentrics    a^2(SA^2 - 4*R^2*SA + 2*S^2) : :

X(10982) lies on these lines: {3,51}, {4,6}, {5,394}, {20,5422}, {24,9781}, {25,578}, {26,10610}, {52,9818}, {54,154}, {155,195}, {378,3567}, {511,7395}, {6461,10669} et al

X(10982) = {X(10669),X(10673)}-harmonic conjugate of X(10983)


X(10983) = X(3)X(6)∩X(5)X(1007)

Barycentrics    a^2[2*S^2*(SW + SA) - SW^2*SA] : :

X(10983) lies on these lines: {3,6}, {5,1007}, {20,7921}, {381,9767}, {6461,10669} et al

X(10983) = {X(10669),X(10673)}-harmonic conjugate of X(10982)


X(10984) = {X(10670),X(10674)}-HARMONIC CONJUGATE OF X(3)

Trilinears    (sin 2A)(sin A + csc A) : :
Trilinears    (cos A)(sin2 A + 1) : :
Barycentrics    a2(4R2 + a2)SA : :

X(10984) lies on these lines: {2,6759}, {3,49}, {4,83}, {20,578}, {22,389}, {24,9729}, {26,9730}, {30,569}, {110,3523}, {125,3549}, {140,5651} et al


X(10985) = HOMOTHETIC CENTER OF ORTHIC TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

Trilinears    (sin A)(3 tan A + cot ω) : :
Trilinears    (sin A)(1 + 3 tan A tan ω) : :
Barycentrics    a2(a2 - 2b2 - 2c2)/(a2 - b2 - c2) : :

X(10985) lies on these lines: {4,187}, {6,25}, {19,10988}, {22,10314}, {23,216}, {24,574}, {32,10594}, {33,10987}, {39,3518}, {53,6103}, {111,933}, {115,7576}, {3199,5008}, {7545,10317} et al


X(10986) = HOMOTHETIC CENTER OF CIRCUMORTHIC TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

Trilinears    a(tan A)[sin2 B sin(2C - 2A) (cot A - cot B)(cot A + cot B - 2 cot C) - sin2 C sin(2A - 2B) (cot C - cot A)(cot C + cot A - 2 cot B)] : :
Barycentrics    a2[2(a4 - b2c2) - (3a2 - b2 - c2)(b2 + c2) - b2c2]/(a2 - b2 - c2) : :

X(10986) lies on these lines: {4,187}, {6,24}, {25,111}, {32,3518}, {186,574}, {6197,10988}, {6198,10987}, {6644,10313} et al


X(10987) = HOMOTHETIC CENTER OF INTANGENTS TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

Trilinears    a(b - c)(2a2 - b2 - c2 - 3bc)(a2 - ab - ac + bc) : : : :

X(10987) lies on these lines: {1,187}, {6,31}, {11,3054}, {32,3746}, {33,10985}, {35,574}, {36,8588}, {56,5210}, {6198,10986} et al


X(10988) = HOMOTHETIC CENTER OF EXTANGENTS TRIANGLE AND MID-TRIANGLE OF INNER AND OUTER TRI-EQUILATERAL TRIANGLES

Trilinears    a[2a(a^6 + 3a^4bc - 4b^3c^3)(b - c) - (4a^6 + 2a^4bc - 9a^2b^2c^2 - 5b^3c^3)(b^2 - c^2) - a(a^4 + 7a^2bc + 6b^2c^2)(b^3 - c^3) + (6a^4 + 3a^2bc - b^2c^2)(b^4 - c^4) - a(2a^2 - 3bc)(b^5 - c^5) - (2a^2 + bc)(b^6 - c^6) + a(b^7 - c^7)] : :

X(10988) lies on these lines: {6,31}, {19,10985}, {40,187}, {6197,10986}, {10387,10460} et al


X(10989) = {X(10719),X(10720)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a6 + 2a4(b2 + c2) - a2(b4 + 3b2c2 + c4) - 2(b6 - b4c2 - b2c4 + c6) : :

X(10989) lies on these lines: {2,3}, {316,5971}, {323,542}, {511,9140}, {523,7840}, {524,3448}, {599,8705} et al

X(10989) = reflection of X(i) in X(j) for these (i,j): (2,858), (23,2)
X(10989) = anticomplement of X(7426)
X(10989) = reflection of X(2) in the de Longchamps line
X(10989) = McCay-to-Artzt similarity image of X(110)


X(10990) =  REFLECTION OF X(125) IN X(74)

Trilinears    (8*cos(2*A)+10)*cos(B-C)-cos( A)*cos(2*(B-C))-15*cos(A)-2* cos(3*A) : :

X(10990) lies on these lines:
{3,541}, {4,74}, {20,542}, {30,6070}, {64,67}, {110,3522}, {113,140}, {146,3523}, {185,1205}, {265,5073}, {548,5609}, {550,5562}, {690,5489}

Let P be a point in the plane of a triangle ABC, and let
A'B'C; = pedal triangle of P
Oa = circumcenter of AB'C', and define Ob and Oc cyclically
O1 = reflection of Oa in PA', and define O2 and O3 cyclically.

If P = X(98), then X(10990) = O1O2O3-to-ABC orthologic center.

If P = u : v : w (trilinears), then

O1O2O3-to-ABC orthologic center = (a*(SA^2-SW*SA-2*S^2)*u+SB*SC* (b*v+c*w))/a : :

ABC-to-O1O2O3 orthologic center = a/(SA*a^3*u-(-SW*SA+SA^2+2*S^ 2)*(b*v+c*w)) : :

The appearance of (i,j) in the following list means that if P = X(i), then X(j) = O1O2O3-to-ABC orthologic center:

(1,5882), (2,3), (3,550), (4,4), (5,140), (6,8550), (20,1657), (40,5493), (51,389), (54,10619), (186,10295), (262,39), (376,20), (381,5), (403,468), (428,6756), (546,3850), (547,3530), (549,548), (568,6102), (631,3522), (1598,7715), (1699,946), (1853,6247), (2394,5489), (2888,3519), (3060,52), (3089,3517), (3090,3523), (3091,1656), (3146,5073), (3153,7574), (3524,376), (3529,5059), (3541,3516), (3542,3515), (3543,382), (3544,3533), (3545,2), (3576,4297), (3817,1125), (3830,3627), (3832,3851), (3839,381), (3843,3858), (3845,546), (3855,5056), (5054,8703), (5055,549), (5064,1595), (5066,3628), (5067,10299), (5071,631), (5093,1353), (5102,3629), (5476,575), (5587,10), (5603,1), (5627,6070), (5640,9730), (5654,1147), (5655,5609), (5656,1498), (5657,40), (5658,1490), (5790,5690), (5817,9), (5886,1385), (5890,185), (5891,1216), (5902,5884), (5927,5777), (5943,9729), (6194,9821), (6530,1990), (7552,7488), (7565,5576), (7576,3575), (7710,8721), (7714,7487), (7775,7764), (7967,944), (7988,10165), (9166,6055), (9752,3053), (9753,32), (9754,5206), (9779,5886), (10157,5044), (10170,5447), (10175,6684), (10201,1658), (10247,1483), (10304,3534), (10516,141), (10519,1350), (10606,5894)

The appearance of (i,j) in the following list means that if P = X(i), then X(j) = ABC-to-O1O2O3 orthologic center:

(1,3417), (2,3431), (3,3), (4,4), (5,54), (6,3425), (10,947), (52,8884), (64,5879), (68,254), (69,7612), (76,3406), (113,250), (114,249), (115,2065), (119,59), (125,10419), (141,5481), (155,24), (195,3432), (265,523), (355,1), (381,6), (399,3447), (546,1173), (946,58), (1351,25), (1352,2), (1482,56), (1657,3532), (2072,5504), (2080,3455), (2095,1436), (2888,3459), (3095,32), (3448,1138), (3574,1166), (3652,35), (3656,2163), (3818,262), (3830,3426), (3843,3527), (5403,1343), (5404,1342), (5446,1179), (5480,251), (5562,96), (5611,3438), (5613,16), (5615,3439), (5617,15), (5777,943), (5779,55), (5805,57), (5887,21), (5891,95), (6033,511), (6246,1168), (6248,83), (6265,36), (6287,39), (6288,5), (6289,371), (6290,372), (6321,512), (6841,1175), (7574,67), (7680,3449), (7681,3450), (7682,3451), (7683,3453), (7686,961), (7697,182), (7728,30), (8724,187), (8905,8883), (8906,6193), (9927,847), (9967,1799), (9970,23), (10113,5627), (10297,895), (10525,5553), (20749,3433), (20750,3435)

The points X(i) for i = 10990 to 10993, lie on the circle with center X(550) and radius 3R/2. See Antreas Hatzipolakis and César Lozada, 24776 (November 7, 2016).

X(10990) = reflection of X(i) in X(j) for these (i,j): (125,74), (146,5972)


X(10991) =  REFLECTION OF X(115) IN X(98)

Barycentrics    SA*(3*S^2-2*SW^2)*(SA-2*SW)- SW^2*(S^2+2*SA^2) : :

X(10991) lies on these lines:
{3,67}, {4,32}, {5,6055}, {20,543}, {39,5477}, {99,3522}, {114,140}, {125,5191}, {147,620}, {148,5059}, {187,1503}, {446,9420}, {550,2782}, {574,6776}, {626,9863}, {631,6054}, {671,3146}, {690,5489}, {754,5999}

In the notation at X(10990), if P = X(98), then X(10991) = O1O2O3-to-ABC orthologic center.

Antreas Hatzipolakis and César Lozada, 24776 (November 7, 2016).

X(10991) = midpoint of X(i) and X(j) for these {i,j}: {98,9862}, {99,5984}
X(10991) = reflection of X(i) in X(j) for these (i,j): (115,98), (147,620)


X(10992) =  REFLECTION OF X(114) IN X(99)

Barycentrics    (12*S^2-SW^2)*SA^2-(9*S^2-SW^ 2)*SW*SA+(6*S^2-SW^2)*S^2 : :

See Antreas Hatzipolakis and César Lozada, 24776 (November 7, 2016).

In the notation at X(10990), if P = X(99), then X(10992) = O1O2O3-to-ABC orthologic center.

X(10992) lies on these lines:
{3,543}, {4,99}, {5,2482}, {20,542}, {98,3522}, {115,140}, {147,5059}, {148,3523}, {382,8724}, {550,2782}, {620,1656}, {631,671}, {1657,2794}

X(10992) = reflection of X(i) and X(j) for these (i,j): (114,99), (148,6036), (6321,620), (9880,2482)


X(10993) =  REFLECTION OF X(119) IN X(100)

Barycentrics    4*a^6*(-b-c+a)-(7*b^2-6*b*c+7* c^2)*a^5+(b+c)*(7*b^2-2*b*c+7* c^2)*a^4+2*(b^2+b*c+c^2)*(b^2- 4*b*c+c^2)*a^3-(b+c)^2*(b-c)^ 2*(2*(b+c)*a^2-a*(b^2+c^2)+(b^ 2-c^2)*(b-c)) : :

See Antreas Hatzipolakis and César Lozada, 24776 (November 7, 2016).

In the notation at X(10990), if P = X(100), then X(10993) = O1O2O3-to-ABC orthologic center.

X(10993) lies on these lines:
{3,528}, {4,100}, {5,6174}, {11,35}, {30,5537}, {40,550}, {104,3522}, {149,3523}, {153,5059}, {517,10609}

X(10993) = reflection of X(i) and X(j) for these (i,j): (119,100), (149,6713), (6265,9945), (10738,3035)


X(10994) =  POINT BECRUX 17

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

Let IaIbIc be the excentral triangle of a triangle ABC; let Sa be the point in which the Euler lines of ABC, IaBC, IaBA, IaCA concur, and define Sb and Sc cyclically. Then X(10994) = centroid of SaSbSc, which lies on the Euler line of ABC. See Seiichi Kirikami, 24793 (November 10, 2016).

X(10994) lies on this line: {2,3}

X(10994) = reflection of X(2) in X(10995)
X(10994) = centroid of (degenerate) extraversion triangle of X(21)


X(10995) =  POINT BECRUX 18

Barycentrics    4 a^12-21 a^10 b^2+45 a^8 b^4-50 a^6 b^6+30 a^4 b^8-9 a^2 b^10+b^12+10 a^10 b c+10 a^9 b^2 c-34 a^8 b^3 c-34 a^7 b^4 c+42 a^6 b^5 c+42 a^5 b^6 c-22 a^4 b^7 c-22 a^3 b^8 c+4 a^2 b^9 c+4 a b^10 c-21 a^10 c^2+10 a^9 b c^2+30 a^8 b^2 c^2+14 a^7 b^3 c^2-15 a^6 b^4 c^2-36 a^5 b^5 c^2+8 a^3 b^7 c^2+12 a^2 b^8 c^2+4 a b^9 c^2-6 b^10 c^2-34 a^8 b c^3+14 a^7 b^2 c^3+22 a^6 b^3 c^3+4 a^5 b^4 c^3-2 a^4 b^5 c^3-2 a^3 b^6 c^3+14 a^2 b^7 c^3-16 a b^8 c^3+45 a^8 c^4-34 a^7 b c^4-15 a^6 b^2 c^4+4 a^5 b^3 c^4-12 a^4 b^4 c^4+16 a^3 b^5 c^4-3 a^2 b^6 c^4-16 a b^7 c^4+15 b^8 c^4+42 a^6 b c^5-36 a^5 b^2 c^5-2 a^4 b^3 c^5+16 a^3 b^4 c^5-36 a^2 b^5 c^5+24 a b^6 c^5-50 a^6 c^6+42 a^5 b c^6-2 a^3 b^3 c^6-3 a^2 b^4 c^6+24 a b^5 c^6-20 b^6 c^6-22 a^4 b c^7+8 a^3 b^2 c^7+14 a^2 b^3 c^7-16 a b^4 c^7+30 a^4 c^8-22 a^3 b c^8+12 a^2 b^2 c^8-16 a b^3 c^8+15 b^4 c^8+4 a^2 b c^9+4 a b^2 c^9-9 a^2 c^10+4 a b c^10-6 b^2 c^10+c^12 : :

Let IaIbIc be the excentral triangle of a triangle ABC; let Sa be the point in which the Euler lines of ABC, IaBC, IaBA, IaCA concur, and define Sb and Sc cyclically. Let Ma = midpoint of ASa, and define Mb and Mc cyclically. Then X(10995) = centroid of MaMbMc, which lies on the Euler line of ABC. See Antreas Hatzipolakis and Peter Moses, 24797 (November 11, 2016).

X(10995) lies on this line: {2,3}

X(10995) = midpoint of X(2) and X(10994)


X(10996) =  X(2)X(3)∩X(64)X(141)

Barycentrics    (a^2-b^2-c^2) (a^8-2 a^4 b^4+b^8+20 a^4 b^2 c^2-4 b^6 c^2-2 a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : :
X(10996) = 5 X[631] - 4 X[7393]

See point P152, Bernard Gibert, Table 27.

X(10996) lies on the cubic K152 and these lines:
{2,3}, {64,141}, {69,185}, {216,7738}, {388,1040}, {497,1038}, {1056,1062}, {1058,1060}, {1216,4846}, {1249,1941}, {1285,10316}, {1578,1588}, {1579,1587}, {5907,6225}, {6515,10574}

X(10996) = reflection of X(4) in X(7401)
X(10996) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,20,1593), (3,4,7386), (3,3547,631), (3,6643,3538), (3,6676,3523), (3,6823,2), (3,7400,7494), (4,631,6804), (4,3537,3), (4,3538,6643), (4,6803,7392), (20,6815,4), (376,631,3520), (631,6622,2), (3146,6997,4), (3538,6643,7386)


X(10997) =  X(2)X(3)∩X(99)X(736)

Barycentrics    a^8+a^6 b^2-a^2 b^6+a^6 c^2-a^2 b^4 c^2-a^2 b^2 c^4+b^4 c^4-a^2 c^6 : :

See point Z1, Bernard Gibert, Table 27.

X(10997) lies on these lines:
{2,3}, {99,736}, {187,5152}, {316,5149}, {325,8290}, {385,698}, {511,4027}, {1691,1916}, {3094,3407}, {3095,10131}, {3314,4048}, {3329,5116}, {5017,7766}, {5104,8289}, {7761,10000}

X(10997) = midpoint of X(7924) in X(9855)
X(10997) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (384,7470,6655), (384,7824,7819), (384,7948,7770), (3552,6655,384)


X(10998) =  X(2)X(3)∩X(99)X(8295)

Barycentrics    a^12-a^10 b^2+a^8 b^4-a^2 b^10-a^10 c^2-2 a^8 b^2 c^2+2 a^6 b^4 c^2+4 a^4 b^6 c^2-3 a^2 b^8 c^2+a^8 c^4+2 a^6 b^2 c^4+11 a^4 b^4 c^4+2 a^2 b^6 c^4-b^8 c^4+4 a^4 b^2 c^6+2 a^2 b^4 c^6+2 b^6 c^6-3 a^2 b^2 c^8-b^4 c^8-a^2 c^10 : :

See point Z2, Bernard Gibert, Table 27.

X(10998) lies on these lines:
{2,3}, {99,8295}, {3095,3406}, {3398,3399}, {9983,10104}


X(10999) =  X(2)X(3)∩X(6)X(2546)

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

See point Z3, Bernard Gibert, Table 27.

X(10999) lies on these lines:
{2,3}, {6,2546}, {76,1671}, {83,1343}, {2547,5085}, {6248,8161}


X(11000) =  X(2)X(3)∩X(6)X(2547)

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

See point Z4, Bernard Gibert, Table 27.

X(11000) lies on these lines:
{2,3}, {6,2547}, {76,1670}, {83,1342}, {2546,5085}, {6248,8160}


X(11001) =  X(2)X(3)∩X(40)X(4669)

Barycentrics    13 a^4-8 a^2 b^2-5 b^4-8 a^2 c^2+10 b^2 c^2-5 c^4 : :
X(11001) = 5 X[2] - 6 X[3], 8 X[3] - 5 X[4], 4 X[2] - 3 X[4], 13 X[4] - 16 X[5], 13 X[2] - 12 X[5], 13 X[3] - 10 X[5], 4 X[5] - 13 X[20], 2 X[3] - 5 X[20], X[4] - 4 X[20], X[2] - 3 X[20], 8 X[5] - 13 X[376], 4 X[3] - 5 X[376], 2 X[2] - 3 X[376], 14 X[5] - 13 X[381], 7 X[4] - 8 X[381], 7 X[2] - 6 X[381], 7 X[3] - 5 X[381], 7 X[376] - 4 X[381], 7 X[20] - 2 X[381], 11 X[4] - 8 X[382], 11 X[381] - 7 X[382], 11 X[2] - 6 X[382], 11 X[3] - 5 X[382], 11 X[376] - 4 X[382], 11 X[20] - 2 X[382], 5 X[3] - 4 X[547], 17 X[376] - 16 X[548], 17 X[20] - 8 X[548], 11 X[4] - 16 X[549]

See point X5067', Bernard Gibert, Table 27.

X(11001) lies on these lines:
{2,3}, {40,4669}, {388,4324}, {485,6486}, {486,6487}, {497,4316}, {515,4677}, {516,7967}, {519,6361}, {530,5863}, {531,5862}, {543,9862}, {553,3488}, {962,3655}, {1056,4302}, {1058,4299}, {1131,6449}, {1132,6450}, {1285,2549}, {1327,6200}, {1328,6396}, {2482,10722}, {3058,4293}, {3068,6480}, {3069,6481}, {3070,6429}, {3071,6430}, {3316,6409}, {3317,6410}, {3486,4333}, {3582,5225}, {3584,5229}, {3654,9778}, {3656,5731}, {3849,9741}, {4294,5434}, {4297,10595}, {4304,4654}, {4745,5657}, {4995,8164}, {5012,8717}, {5041,7738}, {5102,8584}, {5485,8667}, {5642,10721}, {6055,10723}, {6174,10728}, {6337,7809}, {6431,7581}, {6432,7582}, {6437,9541}, {6781,7735}, {7736,9774}, {7739,7756}, {10056,10483}

X(11001) = midpoint of X(i) and X(j) for these {i,j}: {376, 3529}, {3543, 5059}
X(11001) = reflection of X(i) in X(j) for these (i,j): (2, 3534), (4, 376), (376, 20), (381, 550), (382, 549), (962, 3655), (3146, 381), (3543, 3), (3830, 8703), (10721, 5642), (10722, 2482), (10723, 6055), (10728, 6174)
X(11001) = anticomplement X(3830)
X(11001) = crosssum of X(6221) and X(6398)
X(11001) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k):
(2,20,3534), (2,3091,10109), (2,3534,376), (2,3543,3845), (2,3845,3545), (3,4,5067), (3,382,3850), (3,3543,3545), (3,3832,3533), (3,3845,2), (3,3853,5056), (3,5056,631), (4,376,3524), (4,3524,5071), (4,3528,3525), (20,1657,3529), (20,3146,550), (20,3529,4), (20,5059,3), (376,631,10304), (376,3524,3528), (376,3545,3), (381,550,10304), (381,5056,3545), (381,10304,631), (382,549,3839), (382,3090,4), (382,3522,3090), (547,3832,3545), (548,3091,10299), (548,5073,3091), (549,3839,3090), (550,3146,631), (550,3853,3), (550,10304,376), (631,3146,4), (2043,2044,3522), (3146,5056,3853), (3146,10304,381), (3522,3839,549), (3523,3627,3855), (3524,5071,3525), (3528,5071,3524), (3530,5076,5068), (3533,3545,547), (3534,3830,8703), (3543,3545,4), (3543,10304,5056), (3545,5067,5071), (3627,3855,4), (3830,8703,2), (5054,10109,2), (5073,10299,4), (6951,10431,4)


X(11002) =  MIDPOINT OF X(3060) AND X(5640)

Barycentrics    a^2 (2 a^2 b^2-2 b^4+2 a^2 c^2+3 b^2 c^2-2 c^4) : :
X(11002) = X[2] - 4 X[51], X[4] + 8 X[143], 3 X[2] - 4 X[373], 3 X[51] - X[373], 4 X[143] - X[568], X[4] + 2 X[568], 10 X[373] - 3 X[2979], 5 X[2] - 2 X[2979], 10 X[51] - X[2979], 2 X[51] + X[3060], X[2] + 2 X[3060], 2 X[373] + 3 X[3060]

See point Q300, Bernard Gibert, CL039: Droz-Farny cubics.

Let PaPbPc be the reflection triangle of a point P. Let Q be the isogonal conjugate of P wrt PaPbPc. When P = X(2) or X(381), Q = X(11002). (Quang Tuan Bui, Hyacinthos #20331, November 10, 2011)

Let A'B'C' be the orthocentroidal triangle. Let A" be the reflection of A' in line BC, and define B" and C" cyclically. Then X(11002) is the centroid of A"B"C"; see X(23) and X(9140). (Randy Hutson, December 10, 2016)

X(11002) lies on the cubic K300 and these lines:
{2,51}, {3,5645}, {4,94}, {6,23}, {20,3567}, {22,5050}, {25,1994}, {52,3091}, {61,3457}, {62,3458}, {69,7693}, {110,576}, {111,10560}, {182,7492}, {193,9027}, {323,1351}, {376,5946}, {389,3146}, {567,7556}, {569,1173}, {631,10263}, {888,9485}, {1154,3545}, {1199,7517}, {1216,7486}, {1350,7496}, {1352,7533}, {1353,10301}, {1495,5097}, {1992,2854}, {1993,5102}, {2393,5032}, {2888,7528}, {3047,8537}, {3090,6243}, {3095,9155}, {3124,9463}, {3292,10546}, {3410,6515}, {3431,7575}, {3518,9545}, {3523,5462}, {3527,7503}, {3533,10627}, {3543,5890}, {3580,5169}, {3832,5889}, {3854,5907}, {4226,10788}, {4232,6403}, {4430,9026}, {4661,9049}, {5052,9465}, {5056,10170}, {5059,10574}, {5067,6101}, {5068,5562}, {5085,5422}, {5104,7708}, {5651,10545}, {5654,7730}, {6776,7519}, {10303,10625}

X(11002) = midpoint of X(3060) and X(5640)
X(11002) = reflection of X(i) in X(j) for these (i,j): (2, 5640), (2979, 5650), (5640, 51), (5650, 5943), (7998, 373)
X(11002) = crossdifference of every pair of points on line X(3288) X(3906)
X(11002) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (25,1994,9544), (51,3060,2), (52,9781,3091), (373,7998,2), (1351,1995,323), (2979,5943,2), (3567,5446,20), (3580,5480,5169), (5640,7998,373), (5889,10110,3832), (6243,10095,3090), (6515,7394,3410)


X(11003) =  X(2)X(98)∩X(6)X(23)

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

See point Q302, Bernard Gibert, CL039: Droz-Farny cubics.

X(11003) lies on the cubic K302 and these lines:
{2,98}, {3,323}, {4,567}, {6,23}, {20,54}, {22,1351}, {26,1199}, {49,631}, {140,9704}, {154,3066}, {156,3090}, {193,1176}, {195,7525}, {206,7693}, {215,5218}, {373,10546}, {511,7492}, {549,9703}, {568,7556}, {569,1614}, {575,1495}, {578,3146}, {691,10560}, {1092,9706}, {1125,9587}, {1147,3523}, {1350,1993}, {1437,4188}, {1503,5169}, {1588,9677}, {1691,9463}, {1692,9465}, {1995,5050}, {2477,7288}, {2502,7708}, {2888,7558}, {3060,5097}, {3167,7485}, {3203,7793}, {3292,5092}, {3522,10984}, {3545,10540}, {3564,7495}, {3580,8550}, {3618,7605}, {3620,5157}, {3832,6759}, {4226,7709}, {5027,9485}, {5056,10539}, {5085,7496}, {5544,8780}, {5890,10298}, {6684,9622}, {7488,7592}, {7570,10516}, {7622,10554}, {7735,9604}, {8566,8588}, {8617,9225}, {8787,10166}, {9586,10164}, {9621,10165}, {9652,10588}, {9667,10589}

X(11003) = inverse-in Brocard-circle of X(9140)
X(11003) = crossdifference of every pair of points on line X(3569) X(3906)
X(11003) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,184,9544), (2,6776,3448), (6,6800,23), (23,6800,7712), (110,182,2), (110,5012,182), (182,184,110), (184,5012,2), (569,1614,3091), (575,1495,5640), (1993,3796,6636), (2502,10485,7708), (3292,5092,7998)


X(11004) =  X(2)X(6)∩X(4)X(195)

Trilinears    3 csc A - 8 sin A : :
Barycentrics    a^2 (2 a^4-4 a^2 b^2+2 b^4-4 a^2 c^2-b^2 c^2+2 c^4) : :

See point Q304, Bernard Gibert, CL039: Droz-Farny cubics.

X(11004) lies on the cubic K304 and these lines:
{2,6}, {4,195}, {23,1351}, {51,10546}, {52,9545}, {97,10979}, {110,576}, {155,3832}, {184,7712}, {353,8586}, {511,7492}, {568,1511}, {575,7998}, {858,1353}, {895,10560}, {1181,5059}, {1199,3523}, {1383,2987}, {1493,6243}, {1495,3060}, {1570,9465}, {1599,6199}, {1600,6395}, {1995,5093}, {2904,7487}, {2979,5092}, {3095,5191}, {3098,5012}, {3170,3458}, {3171,3457}, {3292,5097}, {3431,3581}, {3448,9976}, {3522,7592}, {3564,5169}, {3854,10982}, {5050,7496}, {5189,6776}, {5406,6437}, {5407,6438}, {5643,5645}, {5890,10564}, {5987,10753}, {9306,10545}

X(11004) = X(7578)-anticomplementary conjugate of X(6327)
X(11004) = X(7578)-Ceva conjugate of X(2)
X(11004) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,323,2), (6,1993,323), (323,1994,6), (1993,1994,2), (3292,5097,5640), (3431,3581,10298)


X(11005) =  REFLECTION OF X(98) IN X(125)

Barycentrics    a^14-2 a^12 b^2+2 a^10 b^4-4 a^8 b^6+6 a^6 b^8-5 a^4 b^10+3 a^2 b^12-b^14-2 a^12 c^2+2 a^10 b^2 c^2+2 a^8 b^4 c^2-4 a^6 b^6 c^2+5 a^4 b^8 c^2-6 a^2 b^10 c^2+3 b^12 c^2+2 a^10 c^4+2 a^8 b^2 c^4-3 a^6 b^4 c^4+4 a^2 b^8 c^4-3 b^10 c^4-4 a^8 c^6-4 a^6 b^2 c^6-2 a^2 b^6 c^6+b^8 c^6+6 a^6 c^8+5 a^4 b^2 c^8+4 a^2 b^4 c^8+b^6 c^8-5 a^4 c^10-6 a^2 b^2 c^10-3 b^4 c^10+3 a^2 c^12+3 b^2 c^12-c^14 : :
X(11005) = 3 X[3545] - 2 X[5465]

See point P450, Bernard Gibert, Orthopivotal cubics.

X(11005) lies on the cubic K450, the orthocentroidal circle, and these lines:
{2,98}, {4,690}, {74,2794}, {115,6794}, {265,2782}, {381,9144}, {523,1550}, {524,1551}, {526,6785}, {1637,6792}, {2777,10722}, {2783,10778}, {3545,5465}, {3580,7471}, {5477,6103}, {5663,6033}, {5877,6130}, {6321,10113}, {8674,10768}

X(11005) = midpoint of X(147) and X(3448)
X(11005) = reflection of X(i) in X(j) for these (i,j): (98,125), (110,114), (6321,10113), (9144,381)
X(11005) - inverse-in-Dao-Moses-Telv-circle of X(6792)
X(11005) = radical trace of X(15)- and X(16)-Fuhrmann circles
X(11005) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,323,2), (6,1993,323), (323,1994,6), (1993,1994,2), (3292,5097,5640), (3431,3581,10298)


X(11006) =  REFLECTION OF X(671) IN X(125)

Barycentrics    a^10-2 a^8 b^2-5 a^6 b^4+10 a^4 b^6-5 a^2 b^8+b^10-2 a^8 c^2+16 a^6 b^2 c^2-12 a^4 b^4 c^2-2 a^2 b^6 c^2+b^8 c^2-5 a^6 c^4-12 a^4 b^2 c^4+15 a^2 b^4 c^4-2 b^6 c^4+10 a^4 c^6-2 a^2 b^2 c^6-2 b^4 c^6-5 a^2 c^8+b^2 c^8+c^10 : :
X(11006) = 4 X[5465] - 3 X[9144]

See point P451, Bernard Gibert, Orthopivotal cubics.

X(11006) lies on the cubic K451 and these lines:
{2,690}, {110, 2482}, {114,10706}, {125,671}

X(11006) = midpoint of X(3448) and X(8591)
X(11006) = reflection of X(i) in X(j) for these (i,j): (110,2482), (671,125), (9144,2), (10706,114)
X(11006) = anticomplement of X(5465)


X(11007) =  COMPLEMENT OF X(1316)

Barycentrics    a^6 b^2-2 a^4 b^4+b^8+a^6 c^2+a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+a^2 b^2 c^4-b^2 c^6+c^8 : :
X(11007) = X[2453] - 5 X[3763]

Let Ω1 and Ω2 be the 1st and 2nd Brocard points of a triangle ABC, and let
Ω1* = isotomic conjugate of Ω1
Ω2* = isotomic conjugate of Ω2
L = the line Ω12*
E = Euler line of ABC

Then X(11007) = L∩E; moreover, L is perpendicular to E. See Tran Quang Hung and Peter Moses, 24815 (November 16, 2016).

X(11007) lies on these lines:
{2,3}, {69,2452}, {125,2782}, {141,523}, {691,7831}, {1352, 6795}, {2396,3933}, {2453,3763}, {2794,5972}, {3258,5650}, {5099, 7853}, {5651,9996}, {5664,8371}

X(11007) = midpoint of X(i) and X(j) for these {i,j}: {69, 2452}, {858, 5112}, {1352, 6795}
X(11007) = complement of X(1316)
X(11007) = crossdifference of every pair of points on line X(647),X(1691)
X(11007) = X(9513)-complementary conjugate of X(10)
X(11007) = inverse-in-circumcircle of X(6660)
X(11007) = inverse-in-polar-circle of X(419)
X(11007) = inverse-in-orthoptic-circle-of-Steiner-inellipe of X(5999)
X(11007) = X(125)-of-1st-Brocard-triangle
X(11007) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,868,5), (2,9832,468), (1113, 1114,6660)


X(11008) =  REFLECTION OF X(69) IN X(193)

Barycentrics    7 a^2-3 (b^2+c^2) : :
X(11008) = 9 X[2] - 10 X[6], 6 X[2] - 5 X[69], 4 X[6] - 3 X[69], 7 X[69] - 8 X[141], 7 X[6] - 6 X[141], 4 X[141] - 7 X[193], 3 X[2] - 5 X[193], 2 X[6] - 3 X[193], 19 X[6] - 18 X[597], 19 X[193] - 12 X[597], 11 X[69] - 12 X[599], 11 X[2] - 10 X[599], 11 X[6] - 9 X[599], 11 X[193] - 6 X[599]

See Antreas Hatzipolakis and Angel Montesdeoca, 24817 (November 16, 2016).

X(11008) lies on the cubic K117 and these lines:
{2,6}, {4,5965}, {187,6337}, {340,1249}, {344,3973}, {382,3564}, {487,6398}, {488,6221}, {511,3529}, {518,3644}, {546,1351}, {550,6776}, {574,7890}, {631,7905}, {1154,10938}, {1352,3855}, {1353,3530}, {1384,3926}, {3090,5097}, {3091,5102}, {3098,3528}, {3247,4416}, {3545,7926}, {3626,3751}, {3632,5847}, {3731,3879}, {3785,5024}, {5008,7855}, {5079,5093}, {5092,10299}, {5107,5207}, {5319,7882}, {7738,7893}

X(11008) = reflection of X(i) in X(j) for these (i,j): (69, 193), (193, 6144)
X(11008) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2,193,3629), (6,69,3619), (6,3619,3618), (6,3630,3620), (6,3631,2), (69,193,1992), (69,1992,3618), (1992,3619,6), (3620,3630,69), (3629,3631,6)


X(11009) =  REFLECTION OF X(35) IN X(1)

Barycentrics    a (a^3-2 a^2 b-a b^2+2 b^3-2 a^2 c+3 a b c-2 b^2 c-a c^2-2 b c^2+2 c^3) : :
X(11009) = 3 X[35] - 4 X[2646] = 3 X[1] - 2 X[2646] = (4r + R)X[1] - 2 r X[3]

In the plane of a triangle ABC, let I = incenter = X(1), and let
Na = nine-point cneter of IBC, and define Nb and Nc cyclically
A' = reflection of I in BC, and define B' and C' cyclically
La = reflection of A'Na in IA', and define Lb and Lc cyclically
Pa = line through A' parallel to La, and define Pb and Pc cyclically
IaIbIc = excentral triangle
Qa = line through Ia parallel to La, and define Qb and Qc cyclically

Then La, Lb, Lc concur in X(11009); also, Pa, Pb, Pc concur in X(11009), and Qa, Qb, Qc concur in X11010). See also X(11010)-X(11015) and Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11009) lies on these lines:
{1,3}, {8,4867}, {10,5443}, {12, 5844}, {79,1320}, {80,946}, {145, 1478}, {499,10595}, {519,5086}, { 758,4861}, {944,10483}, {952, 3585}, {958,3899}, {1125,5330}, { 1483,7354}, {1698,5289}, {1731, 1953}, {1770,5882}, {1837,3656}, {1845,6198}, {2779,7727}, {3241, 4295}, {3242,9047}, {3583,10950} ,{3621,10590}, {3623,4293}, { 3632,10827}, {3633,9612}, {3635, 4292}, {3636,5442}, {3679,5730}, {3869,5258}, {3872,5904}, {3877, 5259}, {3878,5251}, {3919,5253}, {3940,4668}, {3970,4919}, {4299, 7967}, {4301,10572}, {4677,4930} ,{4880,8666}, {5270,10944}, { 5433,10283}, {5444,6684}, {5603, 7741}, {7356,7979}

X(11009) = reflection of X(i) in X(j) for these (i,j): (35,1), (5299,4861)
X(11009) = homothetic center of Caelum triangle and 2nd isogonal triangle of X(1)
X(11009) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,65,5563), (1,484,1385), (1, 3336,1319), (1,3340,5902), (1, 5697,3746), (1,5903,36), (1, 7280,10246), (1,7982,5697), (1, 7991,3612), (56,10247,1), (65, 10222,1), (942,5048,1), (1389, 10698,946), (1482,2099,1), (5603,10573,7741)


X(11010) =  REFLECTION OF X(1) IN X(35)

Barycentrics    a (a^3+a^2 b-a b^2-b^3+a^2 c-3 a b c+b^2 c-a c^2+b c^2-c^3) : :
X(11010) = 3 X[1] - 4 X[2646] = 3 X[35] - 2 X[2646] = 3 X[3679] - 2 X[5086] = (2r - R)X[1] - 4 r X[3]

See X(11009) and Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11010) lies on these lines:
{1,3}, {4,7161}, {8,191}, {9, 5560}, {10,3583}, {11,5445}, {63, 3632}, {71,1731}, {78,3899}, {79, 495}, {80,3467}, {100,3878}, { 214,5330}, {238,3987}, {404, 3884}, {498,962}, {515,4324}, { 516,3585}, {519,6763}, {550, 5559}, {595,4642}, {758,3871}, { 944,1768}, {946,6949}, {1018, 3496}, {1203,4646}, {1320,5303}, {1334,5011}, {1478,6361}, {1479, 5657}, {1621,3754}, {1698,4193}, {1699,6941}, {1737,4857}, {1749, 5441}, {1759,3208}, {1770,5270}, {1776,4330}, {1837,3654}, {1900, 7713}, {2779,9904}, {2802,2975}, {2943,6127}, {3218,3244}, {3219, 3626}, {3555,4880}, {3633,3895}, {3679,5086}, {3730,5540}, {3751, 9047}, {3753,5259}, {3869,8715}, {3870,3901}, {3880,3916}, {3885, 8666}, {3898,5253}, {3913,5904}, {3918,5047}, {3935,4067}, {4063, 6161}, {4294,10573}, {4295, 10056}, {4299,9778}, {4325, 10106}, {4333,9613}, {4338,5290}, {4421,5730}, {4640,5258}, {4861,5267}, {5251,5836}, {5252, 10483}, {5432,5443}, {5444,5901} ,{5506,9780}, {5531,5693}, { 5561,5726}, {5687,5692}, {6192, 7150}, {6932,9589}, {6963,9588}, {7031,9620}, {9580,10826}, {9785,10072}

X(11010) = reflection of X(i) in X(j) for these (i,j): (1, 35), (3585, 10039), (4861, 5267), (5288, 3916)
X(11010) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,40,484), (1,46,3337), (1,165, 7280), (1,484,3336), (1,5131,56) ,(3,5697,1), (36,3057,1), (40, 1697,46), (40,5119,1), (46,1697, 1), (46,3337,3336), (46,5119, 1697), (55,5903,1), (65,3746,1), (191,5541,8), (484,3337,46), ( 1155,9957,5563), (1737,10624, 4857), (1759,3208,5525), (3057, 3579,36), (3245,3746,65), (3295, 5902,1), (3612,7982,1), (4424, 5255,1), (4640,10914,5258), ( 5563,9957,1), (5690,6284,80).


X(11011) =  POINT BECRUX 19

Barycentrics    a (2 a-3 b-3 c) (a+b-c) (a-b+c) : :
X(11011) = 3 X[1] - X[35], 2 X[35] - 3 X[2646], 2 X[12] - 3 X[4870], X[2646] + 2 X[11009], X[35] + 3 X[11009], 5 X[35] - 3 X[11010], 5 X[2646] - 2 X[11010], 5 X[1] - X[11010], 5 X[11009] + X[11010]

Continuing from X(11009), let DEF = intouch triangle of ABC. Let Ua = line through D parallel to La, and define Ub and Uc cyclically. Then Ua, Ub, Uc concur in X(11011). See Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11011) lies on these lines:
{1,3}, {7,1392}, {8,6933}, {12,519}, {34,1900}, {37,1405}, {80,9955}, {145,3485}, {210,5730}, {226,3244}, {388,3241}, {404,10107}, {474,3922}, {497,5734}, {518,4861}, {551,4848}, {664,4059}, {936,4731}, {944,1836}, {946,10950}, {956,3962}, {960,5260}, {997,3698}, {1100,1404}, {1104,2599}, {1122,7190}, {1125,7294}, {1126,1411}, {1317,1365}, {1361,2779}, {1389,7686}, {1400,3723}, {1457,2594}, {1463,4864}, {1469,9047}, {1479,3656}, {1532,10955}, {1737,5901}, {1788,3622}, {1837,5603}, {1858,10698}, {1870,1887}, {1875,6198}, {1953,2264}, {2089,11013}, {2611,2650}, {3476,3623}, {3632,5219}, {3633,9578}, {3636,3911}, {3655,4299}, {3671,5434}, {3683,3878}, {3689,10914}, {3811,3893}, {3870,10912}, {3897,4640}, {3983,9623}, {4018,8666}, {4295,7967}, {4301,6284}, {4511,5836}, {4955,7176}, {5270,7972}, {5443,9956}, {5543,7195}, {5727,10896}, {5844,10039}, {5855,6734}, {5881,10895}, {5882,7354}, {5886,10573}, {6831,10959}, {6879,10595}, {10543,10624}

X(11011) = midpoint of X(i) and X(j) for these {i,j}: {1, 11009}, {145, 5086}, {7982, 11012}
X(11011) = reflection of X(2646) in X(1)
X(11011) = crosspoint of X(1) and X(1389)
X(11011) = crosssum of X(1) and X(1385)
X(11011) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,46,10246), (1,57,1388), (1,65,1319), (1,1482,3057), (1,2098,5919), (1,2099,65), (1,3340,56), (1,5425,5045), (1,5903,1385), (1,7962,3303), (1,7982,55), (1,9957,3748), (1,10222,5048), (56,2099,3340), (56,3340,65), (145,3485,5252), (226,3244,10944), (551,4848,5433), (1317,3649,10106), (1385,5903,1155), (3241,4323,388), (3635,10106,1317)


X(11012) =  POINT BECRUX 20

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

Continuing from X(11009), let DEF = 1st circumperp triangle of ABC. Let Ua = line through D parallel to La, and define Ub and Uc cyclically. Then Ua, Ub, Uc concur in X(11012). See Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11012) lies on these lines:
{1,3}, {4,993}, {5,5251}, {8,6796}, {10,6905}, {20,5450}, {21,946}, {48,573}, {58,1064}, {63,5693}, {72,2949}, {102,110}, {104,3651}, {191,5887}, {255,10571}, {355,5258}, {378,1753}, {388,6988}, {404,6684}, {405,5715}, {411,515}, {498,6954}, {499,6827}, {516,5267}, {550,10943}, {572,2260}, {580,1193}, {581,1468}, {601,4257}, {602,995}, {631,10198}, {912,6763}, {944,6876}, {952,5288}, {956,5881}, {958,3149}, {962,4189}, {1001,5735}, {1006,1125}, {1158,4652}, {1203,5398}, {1290,2716}, {1350,9047}, {1478,6825}, {1479,6868}, {1593,1900}, {1698,6911}, {1699,3560}, {2551,6927}, {2800,4996}, {2814,8648}, {2915,9626}, {3086,6987}, {3218,5884}, {3220,6210}, {3430,7430}, {3436,6962}, {3522,10529}, {3524,10597}, {3528,10806}, {3583,7491}, {3585,5841}, {3624,6883}, {3634,6946}, {3715,5780}, {3814,6949}, {3817,6920}, {3822,6853}, {3825,6902}, {3841,6901}, {3916,6001}, {4220,5322}, {4221,4278}, {4293,6908}, {4299,6850}, {4324,5840}, {4999,6831}, {5080,6960}, {5124,8609}, {5231,7580}, {5248,5603}, {5253,6986}, {5259,5886}, {5260,6915}, {5303,6909}, {5428,5901}, {5433,6922}, {5657,6942}, {5691,6985}, {6256,6838}, {6361,6950}, {6863,7951}, {6865,7288}, {6907,7354}, {6923,10483}, {6928,7741}, {6940,10164}, {6947,10200}, {7411,11015}, {7483,7680}, {7489,9955}, {8075,11013}

X(11012) = midpoint of X(i) and X(j) for these {i,j}: {40, 11014}, {411, 2975}
X(11012) = reflection of X(i) in X(j) for these (i,j): (35, 3), (3585, 6842), (6831, 4999), (6906, 5267), (7982, 11011)
X(11012) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3,10902), (3,40,2077), (3,56,3576), (3,3428,40), (3,10269,7987), (3,10306,5217), (3,10680,10267), (56,354,5563), (63,6261,5693), (104,3651,4297), (165,7280,3), (631,10532,10198), (958,3149,5587), (5204,5584,3), (5253,6986,10165), (5260,6915,10175), (5603,6875,5248), (6863,10526,7951), (10267,10680,1)


X(11013) =  POINT BECRUX 21

Barycentrics    a ((a+b+c) (a^3-a^2 b-a b^2+b^3-a^2 c+a b c-b^2 c-a c^2-b c^2+c^3)+2 a b (2 a-3 b-3 c) c Sin[A/2]+2 c (a^3-a^2 b+2 a b^2-b^3+a^2 c+b^2 c-a c^2+b c^2-c^3) Sin[B/2]+2 b (a^3+a^2 b-a b^2-b^3-a^2 c+b^2 c+2 a c^2+b c^2-c^3) Sin[C/2]) : :

Continuing from X(11009), let DEF = tangential mid-arc triangle of ABC. Let Ua = line through D parallel to La, and define Ub and Uc cyclically. Then Ua, Ub, Uc concur in X(11013). See Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11013) lies on these lines:
{35,8077}, {517,8091}, {2089,11011}, {2646,8241}, {8075,11012}, {8078,11010}, {8081,11014}



X(11014) =  POINT BECRUX 22

Barycentrics    a (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6-2 a^5 c+5 a^4 b c-5 a^3 b^2 c-3 a^2 b^3 c+7 a b^4 c-2 b^5 c-a^4 c^2-5 a^3 b c^2+12 a^2 b^2 c^2-5 a b^3 c^2-b^4 c^2+4 a^3 c^3-3 a^2 b c^3-5 a b^2 c^3+4 b^3 c^3-a^2 c^4+7 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6) : :
X(11014) = 2 X[35] - 3 X[3576]

Continuing from X(11009), let DEF = hexyl triangle of ABC. Let Ua = line through D parallel to La, and define Ub and Uc cyclically. Then Ua, Ub, Uc concur in X(11014). See Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11014) lies on these lines:
{1,3}, {8,6326}, {10,6949}, {515,4861}, {912,5288}, {944,6264}, {946,5046}, {956,5693}, {1006,3884}, {2800,2975}, {3872,5086}, {3878,10698}, {3918,6946}, {4193,8227}, {5258,5887}, {5587,6941}, {5603,6902}, {5690,6265}, {6907,10944}, {6963,9624}, {8081,11013}, {10884,11015}

X(11014) = reflection of X(i) in X(j) for these (i,j): (40, 11012), (5881, 5086), (7982, 11009), (11010, 3) X(11014) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3576,7982,1697), (3872,6261,5881).


X(11015) =  POINT BECRUX 23

Barycentrics    3 a^4-a^3 b-2 a^2 b^2+a b^3-b^4-a^3 c-a^2 b c-2 a^2 c^2+2 b^2 c^2+a c^3-c^4 : :
X)11015) = 2 X[10] - 3 X[35], 6 X[2646] - 5 X[3616], 4 X[10] - 3 X[5086], 4 X[3635] - 3 X[11009], X[3632] - 3 X[11010]

Continuing from X(11009), let DEF = Conway triangle of ABC. Let Ua = line through D parallel to La, and define Ub and Uc cyclically. Then Ua, Ub, Uc concur in X(11015). See Antreas Hatzipolakis and Peter Moses, 24851 (November 21, 2016).

X(11015) lies on these lines:
{7,1392}, {8,4640}, {10,21}, {20,145}, {63,3632}, {149,1385}, {214,4857}, {377,497}, {390,10861}, {404,950}, {496,4881}, {515,3871}, {528,4861}, {550,3218}, {758,4324}, {993,5178}, {1320,5882}, {1900,4198}, {2476,3601}, {3057,6224}, {3146,5658}, {3419,4189}, {3434,3897}, {3488,4190}, {3529,5905}, {3586,4193}, {3624,4197}, {3635,4292}, {3636,5249}, {3648,3962}, {3841,5426}, {3869,4302}, {3873,4299}, {3874,4316}, {3876,6872}, {3877,4294}, {3878,4330}, {3881,4325}, {3889,4293}, {3890,4309}, {3943,5279}, {4187,9945}, {4188,5722}, {4511,6284}, {5046,5440}, {5175,6910}, {5176,8715}, {5303,10916}, {5330,10624}, {6839,9955}, {6916,10806}, {7411,11012}, {10884,11014}

X(11015) = reflection of X(5086) in X(35)
X(11015) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3434,4305,3897), (3586,4855,4193)


X(11016) =  POINT BECRUX 24

Barycentrics    a^16-3 a^14 b^2-a^12 b^4+14 a^10 b^6-20 a^8 b^8+9 a^6 b^10+3 a^4 b^12-4 a^2 b^14+b^16-3 a^14 c^2+16 a^10 b^4 c^2-10 a^8 b^6 c^2-18 a^6 b^8 c^2+17 a^4 b^10 c^2+a^2 b^12 c^2-3 b^14 c^2-a^12 c^4+16 a^10 b^2 c^4-3 a^8 b^4 c^4-18 a^6 b^6 c^4-13 a^4 b^8 c^4+21 a^2 b^10 c^4-2 b^12 c^4+14 a^10 c^6-10 a^8 b^2 c^6-18 a^6 b^4 c^6-14 a^4 b^6 c^6-18 a^2 b^8 c^6+19 b^10 c^6-20 a^8 c^8-18 a^6 b^2 c^8-13 a^4 b^4 c^8-18 a^2 b^6 c^8-30 b^8 c^8+9 a^6 c^10+17 a^4 b^2 c^10+21 a^2 b^4 c^10+19 b^6 c^10+3 a^4 c^12+a^2 b^2 c^12-2 b^4 c^12-4 a^2 c^14-3 b^2 c^14+c^16 : :

In the plane of a triangle ABC, let N = X(5) = nine-point center, O = X(3) = circumcenter, and
Na = nine-point center of NBC, and define Nb and Nc cyclically
Oa = reflection of O in BC, and define Ob and Oc cyclically
Then NaNbNc are OaObOc are perspective, and X(11016) is their perspector. See Antreas Hatzipolakis, Peter Moses, and Angel Montesdeoca, 24853 (November 21, 2016).

X(11016) lies on these lines:
{5,7691}, {140,1141}



X(11017) =  POINT BECRUX 25

Barycentrics    a^2 (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2+2 a^4 b^2 c^2+6 a^2 b^4 c^2-9 b^6 c^2-3 a^4 c^4+6 a^2 b^2 c^4+20 b^4 c^4+3 a^2 c^6-9 b^2 c^6-c^8) : :
X(11017) = (J^2 - 8) X[5] - J^2 X[113], where J = |OH|/R
X(11017) = 9 X[5] - X[185], 17 X[5] - 9 X[373], 3 X[546] + X[1216], X[143] - 5 X[3091], 5 X[3859] - X[5446], 7 X[3857] + X[5562], 7 X[3851] + X[5876], 5 X[3567] + 3 X[5876], 5 X[3858] + 3 X[5891], 3 X[5066] + X[5907], 11 X[5072] - 3 X[5946], 7 X[3832] + X[6101], 9 X[3545] - X[6102], 17 X[3854] - X[6243], 11 X[5] - 3 X[9730], 3 X[5066] - X[10095], X[9729] - 3 X[10109], 3 X[3850] - X[10110], X[3853] + 3 X[10170], 9 X[381] - X[10263], 3 X[3845] + X[10627].

In the plane of a triangle ABC, let N = X(5) = nine-point center and O = X(3) = circumcenter, and
A'B'C' = pedal triangle of N
Oa = reflection of O in BC, and define OB and Oc cyclically
Oaa = orthogonal projection of Oa on NA'
Oab = orthogonal projection of Oa on NB'
Oac = orthogonal projection of Oa on NC'
Oba = orthogonal projection of Ob on NA'
Obb = orthogonal projection of Ob on NB'
Obc = orthogonal projection of Ob on NC'
Oca = orthogonal projection of Oc on NA'
Ocb = orthogonal projection of Oc on NB'
Occ = orthogonal projection of Oc on NC'
A'' = Oba = Oca, and define B'' and C'' cyclically
Na = nine-point center of OaaB''C'', and define Nb and Nc cyclically

The the triangles A'B'C' and NaNbNc are congruent and homothetic, with homothetic center X(5663); the triangles A''B''C'' and NaNbNc are inversely homothetic, with homothetic center X(11017). See Antreas Hatzipolakis and Peter Moses, 24855 (November 21, 2016).

X(11017) lies on these lines:
{5,113}, {143,3091}, {381,10263}, {511,3856}, {546,1216}, {1154,3850}, {3545,6102}, {3567,3851}, {3832,6101}, {3845,10627}, {3853,10170}, {3854,6243}, {3857,5562}, {3858,5891}, {3859,5446}, {5066,5907}, {5072,5946}, {9729,10109}

X(11017) = midpoint of X(5907) and X(10095)
X(11017) = {X(5066),X(5907)}-harmonic conjugate of X(10095)


X(11018) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND ASCELLA

Trilinears    (b+c)*a^4-2*(b^2-b*c+c^2)*a^3-4*b*c*(b+c)*a^2+2*(b^3-c^3)*(b-c)*a-(b+c)*(b-c)^4 : :
X(11018) = (r+2*R)^2*X(1)-r^2*X(3)

The inverse-in-incircle triangle has vertices the incircle-inverses of A, B, C. The A-vertex of this triangle has trilinear coordinates:
A'=((b+c)*a-(b-c)^2)/a/(-a+b+c) : 1 : 1

The appearance of (T,n) in the following list means that T and the inverse-in-circle triangles are perspective with perspector X(n), where an asterisk * indicates homothetic triangles and symbols $- inversely similar triangles:
(ABC, 1), (Andromeda, 1), (Antlia, 1), (Aquila, 1), (Ascella *, 11018), (Atik *, 11019), (1st circumperp *, 57), (2nd circumperp *, 1), (Conway *, 11020), (2nd Conway *, 10580), (3rd Conway *, 11021), (4th Conway, 1), (5th Conway, 1), (3rd Euler *, 226), (4th Euler *, 1210), (excentral *, 1), (2nd extouch *, 5728), (3rd extouch, 11022), (Fuhrmann $-, 11023), (outer-Garcia, 11024), (hexyl *, 3333), (Honsberger *, 11025), (inner-Hutson *, 11026), (Hutson intouch *, 65), (outer-Hutson *, 11027), (incentral, 1), (incircle-circles *, 5045), (intouch *, 354), (medial, 142), (midarc, 1), (2nd midarc, 1), (midheight, 11028), (mixtilinear, 1), (2nd mixtilinear, 1), (5th mixtilinear, 1), (6th mixtilinear *, 10980), (7th mixtilinear, 11029), (2nd Pamfilos-Zhou *, 11030), (1st Sharygin *, 11031), (tangential-midarc *, 11032), (2nd tangential-midarc *, 11033), (Yff central *, 8083), (inner-Yff, 1), (outer-Yff, 1), (inner-Yff tangents, 1), (outer-Yff tangents, 1)

The center of inverse similitud of triangles inverse-in-incircle and Fuhrmann is X(1). (Explanation and centers X(11018)-X(11033) were contributed by César Lozada, November 26, 2016)

X(11018) lies on these lines:
{1,3}, {2,955}, {7,10167}, {12,9947}, {63,954}, {72,5273}, {142,2886}, {169,1190}, {226,971}, {284,910}, {388,5787}, {443,938}, {497,5805}, {518,5745}, {672,8731}, {774,6051}, {912,5719}, {946,9942}, {950,5806}, {991,1427}, {1056,5768}, {1210,8728}, {1387,9946}, {1439,3945}, {1824,7490}, {1864,5219}, {1876,4219}, {1905,5338}, {2266,2289}, {2272,2294}, {2900,5437}, {3085,5791}, {3091,9844}, {3305,5729}, {3434,9776}, {3485,9856}, {3663,9944}, {3671,9943}, {3812,6738}, {3873,5744}, {3911,10156}, {4312,5918}, {5044,6675}, {5226,5927}, {5274,7671}, {5281,7672}, {5722,6826}, {5777,6824}, {7308,10398}, {7580,7675}

X(11018) = midpoint of X(i) and X(j) for these {i,j}: {55,5173}, {226,10391}, {5572,8255}
X(11018) = incircle-inverse-of-X(5536)
X(11018) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55,354,5173), (57,354,942), (57,3601,165), (1864,5219,10157), (5045,9940,942), (5226,10394,5927), (10857,10980,57)


X(11019) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND ATIK

Trilinears    ((b+c)*a^2-2*(b-c)^2*a+(b+c)*(b-c)^2)/a : :
X(11019) = (4*R+r)*X(1)-3*r*X(2)

X(11019) lies on these lines:
{1,2}, {3,4314}, {4,1435}, {5,3947}, {7,1699}, {11,118}, {20,3361}, {36,4304}, {38,1736}, {40,1058}, {46,5493}, {55,3911}, {56,950}, {57,497}, {65,4301}, {142,2886}, {165,390}, {171,1416}, {204,461}, {210,5316}, {244,3914}, {329,5850}, {355,7373}, {388,9581}, {495,5049}, {496,942}, {515,999}, {517,4342}, {518,3452}, {553,1836}, {774,3670}, {908,3873}, {940,4349}, {954,4423}, {962,3339}, {971,7956}, {982,3663}, {1001,5745}, {1056,5587}, {1088,3673}, {1099,5620}, {1155,3058}, {1159,3656}, {1203,3562}, {1376,5853}, {1387,9952}, {1400,10443}, {1420,3486}, {1427,1536}, {1471,1754}, {1479,3338}, {1496,1724}, {1497,1771}, {1512,7967}, {1697,1788}, {1750,4321}, {1770,3337}, {1837,3304}, {1838,1895}, {2192,8808}, {2260,8804}, {2346,5659}, {2481,7196}, {2550,5437}, {2551,6762}, {3057,4848}, {3091,5290}, {3189,5438}, {3295,6684}, {3296,5714}, {3306,3434}, {3475,5219}, {3476,5727}, {3487,8227}, {3488,3576}, {3555,4187}, {3586,4293}, {3600,5691}, {3601,7288}, {3666,4356}, {3677,4353}, {3693,3950}, {3748,5432}, {3752,3755}, {3782,3999}, {3812,3813}, {3814,3892}, {3825,3881}, {3826,3848}, {3829,3838}, {3889,4193}, {3919,9951}, {3928,5698}, {3971,4712}, {4000,5573}, {4003,4854}, {4035,4966}, {4082,4358}, {4295,9614}, {4311,5563}, {4312,9812}, {4313,5265}, {4326,8732}, {4327,5807}, {4413,4863}, {4417,4684}, {4425,10868}, {4512,5744}, {4679,5729}, {4869,10325}, {4883,5718}, {5068,5558}, {5129,5234}, {5218,10389}, {5225,9579}, {5226,7988}, {5249,10861}, {5261,7989}, {5267,8071}, {5281,8236}, {5534,6944}, {5658,8166}, {6147,9955}, {6260,7681}, {6666,8167}, {6769,6926}, {7991,9785}, {9612,10591}, {10404,10896}, {10478,10862}

X(11019) = midpoint of X(i) and X(j) for these {i,j}: {57,497}, {999,5722}, {3474,9580}, {3476,5727}, {3586,4293}, {4321,5809}
X(11019) = reflection of X(i) in X(j) for these (i,j): (997,1125), (1376,6692), (3452,3816), (4315,999)
X(11019) = complement of X(200)
X(11019) = incircle-inverse-of-X(5121)
X(11019) = homothetic center of medial triangle and 3rd pedal triangle of X(1)
X(11019) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,938,6738), (1,1210,10), (1,3086,1125), (1,3624,5703), (2,8,8580), (2,3870,6745), (2,4847,10), (2,10580,1), (2,10582,1125), (4,3333,4298), (7,5274,1699), (8,3912,10324), (8,8582,10), (11,226,3817), (11,354,226), (46,10624,5493), (55,3911,10164), (56,950,4297), (57,9580,3474), (57,10384,10860), (226,354,5542), (354,8581,10569), (390,5435,165), (496,942,946), (497,3474,9580), (942,946,3671), (946,9948,9856), (1125,6744,1), (1125,10916,10), (1479,3338,4292), (1699,10980,7), (1836,4860,553), (1837,3304,10106), (2886,3742,142), (3333,10864,7091), (3337,4857,1770), (3475,10589,5219), (3811,10200,6700), (3817,5542,226), (4313,5265,7987), (5219,10589,10171), (5231,10582,2), (5563,10572,4311), (5927,10569,8581)


X(11020) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND CONWAY

Trilinears    (b+c)*a^4-(2*b^2-b*c+2*c^2)*a^3-3*b*c*(b+c)*a^2+(2*b^2+3*b*c+2*c^2)*(b-c)^2*a-(b^3+c^3)*(b-c)^2 : :
X(11020) = 2*(r+2*R)^2*X(1)-r*(3*R+2*r)*X(21)

X(11020) lies on these lines:
{1,21}, {2,955}, {7,354}, {20,942}, {55,7672}, {57,7411}, {65,4313}, {224,5253}, {226,10394}, {377,938}, {390,5173}, {518,5273}, {946,9960}, {954,3219}, {1210,4197}, {1387,9964}, {1864,5226}, {3305,10398}, {3333,10884}, {3487,6837}, {3663,9962}, {3671,9961}, {3742,5784}, {3832,9844}, {3876,5703}, {3925,8255}, {4208,5439}, {4292,6744}, {4304,5902}, {5249,10861}, {5558,10429}, {5722,6839}, {5732,10980}, {6916,10202}, {7673,10385}

X(11020) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (57,7675,7411), (63,3873,3868), (354,5572,10580), (354,10391,7), (5728,11018,2)


X(11021) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 3RD CONWAY

Trilinears    (2*b^2+3*b*c+2*c^2)*a^3+4*b*c*(b+c)*a^2-(2*b^4+2*c^4-b*c*(3*b^2+2*b*c+3*c^2))*a-2*b*c*(b+c)*(b-c)^2 : :

X(11021) lies on these lines:
{1,3}, {7,10435}, {226,10886}, {1210,10887}, {1401,4888}, {3741,5542}, {4298,10454}, {5208,10455}, {5572,10442}, {5728,10888}, {10444,11020}, {10446,10580}, {10447,10453}, {10478,10862}

X(11021) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,57,10434), (1,3361,10470), (1,10473,10439), (354,10473,1), (3304,10474,1), (5045,10441,1)


X(11022) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 3RD EXTOUCH

Trilinears    (7*sin(A/2)-8*sin(3*A/2)+sin(5*A/2))*cos((B-C)/2)+(-8*cos(A)+2*cos(2*A)-6)*cos(B-C)+(-sin(A/2)-sin(3*A/2))*cos(3*(B-C)/2)+4*cos(2*A)+4*cos(A)+4 : :
X(11022) = 2*R*r*X(4)+3*(s^2-4*R^2-2*R*r)*X(354)

X(11022) lies on these lines:
{1,198}, {4,354}, {223,5045}, {942,3182}, {10903,10939}


X(11023) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND FUHRMANN

Barycentrics    a^7+(b+c)*a^6-a^2*(3*b^2-2*b*c+3*c^2)*(a^3-(b-c)^2*a-(b+c)*(b-c)^2)-3*(b+c)*(b-c)^2*a^4-(b^2-c^2)^2*(b^2-6*b*c+c^2)*a-(b^2-c^2)^3*(b-c) : :

X(11023) lies on these lines:
{1,6904}, {3,1387}, {4,10305}, {7,1210}, {46,8732}, {57,946}, {142,3085}, {149,10940}, {158,1119}, {191,499}, {226,6964}, {355,388}, {443,5836}, {497,9940}, {938,2475}, {962,5265}, {1086,7952}, {1329,5220}, {1466,5603}, {1467,4293}, {1728,9965}, {1788,2095}, {3487,6946}, {3911,5758}, {4294,8726}, {5154,5704}, {5435,6972}, {5572,5880}, {5708,6841}, {9782,10580}, {10090,10093}, {10624,10857}

X(11023) = midpoint of X(4) and X(10305)


X(11024) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND OUTER-GARCIA

Barycentrics    a^4+2*(b+c)*a^3+4*b*c*a^2-2*(b+c)*(b^2-6*b*c+c^2)*a-(b^2-c^2)^2 : :
X(11024) = 3*(4*R+r)*X(2)-2*r*X(40) = (4*R+r)*X(7)+8*R*X(10)

X(11024) lies on these lines:
{2,40}, {7,10}, {8,443}, {46,5273}, {100,1387}, {142,1706}, {169,5749}, {329,3820}, {354,6764}, {388,3698}, {938,2550}, {997,4323}, {1125,5281}, {1376,5703}, {1698,4295}, {1722,4307}, {1788,3925}, {2551,5880}, {2802,7320}, {2886,5704}, {3421,4002}, {3485,4413}, {3487,9709}, {3600,9623}, {3671,8580}, {3730,5296}, {3824,8164}, {3931,5308}, {4646,4648}, {4731,10404}, {4882,5542}, {5045,9797}, {5082,5439}, {5222,5711}, {5249,7080}, {5274,9843}, {5554,9803}, {5687,10578}, {5731,8726}, {5811,9956}, {5818,10711}, {8165,9612}

X(11024) = midpoint of X(4866) and X(5586)
X(11024) = reflection of X(4866) in X(10)
X(11024) = anticomplement of X(3646)
X(11024) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7,10,5815), (10,5261,5828), (443,3753,8), (2550,3812,938), (5082,5439,10580)


X(11025) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND HONSBERGER

Trilinears    (b+c)*a^3-(3*b^2+b*c+3*c^2)*a^2+3*(b+c)*(b-c)^2*a-(b^2-b*c+c^2)*(b-c)^2 : :
X(11025) = X(7)-6*X(354)

X(11025) lies on these lines:
{1,1170}, {7,354}, {9,3873}, {57,7676}, {65,7673}, {144,10177}, {226,7678}, {390,942}, {480,4511}, {518,3616}, {938,9846}, {982,4343}, {1001,3868}, {1156,5083}, {1210,7679}, {1449,9502}, {3059,3742}, {3174,3306}, {3333,7675}, {3487,5045}, {3555,5686}, {3681,6666}, {3881,5223}, {3957,6600}, {4326,10980}, {5542,10394}, {5571,7670}, {8732,11018}, {10865,11019}, {10889,11021}

X(11025) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,1445,2346), (7,5572,7671), (65,8236,7673), (354,5572,7)


X(11026) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND INNER-HUTSON

Trilinears    2*((b+c)*a-(b-c)^2)*sin(A/2)+2*(a-b+c)*b*sin(B/2)+2*(a+b-c)*c*sin(C/2)-a^2-2*(b+c)*a+3*(b-c)^2 : :

X(11026) lies on these lines:
{1,289}, {57,8107}, {65,8390}, {226,8377}, {354,8113}, {938,9847}, {942,9836}, {1210,8380}, {3333,8111}, {5728,5934}, {8140,10980}, {9783,10580}


X(11027) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND OUTER-HUTSON

Trilinears    2*((b+c)*a-(b-c)^2)*sin(A/2)+2*(a-b+c)*b*sin(B/2)+2*(a+b-c)*c*sin(C/2)+a^2+2*(b+c)*a-3*(b-c)^2 : :

X(11027) lies on these lines:
{1,168}, {7,8372}, {57,8108}, {65,8392}, {226,8378}, {354,8114}, {938,9849}, {942,9837}, {1210,8381}, {3333,8112}, {5728,5935}, {8140,10980}, {9787,10580}


X(11028) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND MIDHEIGHT

Trilinears    (b+c)*a^6-2*(b^2+c^2)*a^5+(b^3+c^3)*a^4-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a^2+2*(b^4-c^4)*(b^2-c^2)*a-(b^2-c^2)*(b-c)^2*(b^3-c^3) : :
X(11028) = (4*R+r)*X(7)-r*X(152)

X(11028) lies on these lines:
{1,41}, {7,152}, {11,118}, {33,57}, {65,1360}, {116,1210}, {150,938}, {518,3041}, {672,1736}, {676,926}, {910,8554}, {942,2808}, {955,2006}, {971,1541}, {1020,2310}, {1155,5160}, {1827,3668}, {2807,5173}, {2825,5185}, {3340,10697}, {3586,10725}, {3911,6712}, {4654,10710}, {5722,10739}, {6678,11018}, {9579,10727}

X(11028) = midpoint of X(65) and X(3022)
X(11028) = incircle-inverse-of-X(105)
X(11028) = X(114)-of-intouch-triangle


X(11029) = PERSPECTOR OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 7TH MIXTILINEAR

Trilinears    a^8-4*(7*b^2+6*b*c+7*c^2)*a^6+16*(b+c)*(7*b^2-10*b*c+7*c^2)*a^5-6*(35*b^2+34*b*c+35*c^2)*(b-c)^2*a^4+32*(b^2-c^2)*(b-c)*(7*b^2+2*b*c+7*c^2)*a^3-4*(35*b^4+35*c^4+2*b*c*(32*b^2-3*b*c+32*c^2))*(b-c)^2*a^2+48*(b^2-c^2)^3*(b-c)*a-(7*b^2+2*b*c+7*c^2)*(b-c)^6 : :

X(11029) lies on these lines:
{1,1615}, {354,3062}, {8916,10939}


X(11030) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 2ND PAMFILOS-ZHOU

Trilinears    -(2*b*c*(b+c)*a^2-4*b*c*(b-c)^2*a+2*(b^2-c^2)*(b-c)*b*c)*S+(b+c)*a^6-2*(b^2+c^2)*a^5+(b^3+c^3)*a^4-4*b*c*(b^2-b*c+c^2)*a^3-(b^2-c^2)*(b-c)*(b^2-4*b*c+c^2)*a^2+2*(b^4+c^4)*(b-c)^2*a-(b^2-c^2)*(b-c)^2*(b^3-c^3) : :

X(11030) lies on these lines:
{1,372}, {7,7595}, {57,8224}, {65,8239}, {354,8243}, {942,7596}, {1210,8230}, {3333,8234}, {5728,8233}, {8237,11025}, {8244,10980}, {9789,10580}, {10858,11018}, {10867,11019}, {10885,11020}, {10891,11021}


X(11031) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 1ST SHARYGIN

Trilinears    (b-c)^2*a^3-(b+c)*(b^2+c^2)*a^2-(b^2+c^2)^2*a+(b^4-c^4)*(b-c) : :

X(11031) lies on these lines:
{1,21}, {7,256}, {20,986}, {27,240}, {57,4220}, {65,8240}, {226,8229}, {244,5249}, {284,2312}, {354,1284}, {672,8731}, {938,9852}, {942,9840}, {976,1259}, {984,5273}, {988,10884}, {1210,5051}, {1473,3145}, {1708,10383}, {1755,2294}, {1762,5324}, {3333,8235}, {3670,4292}, {3752,5784}, {3944,10883}, {4199,5728}, {4304,4424}, {4392,9965}, {4425,10868}, {5045,9959}, {7004,10391}, {8238,11025}, {8245,10980}, {8246,11030}, {8391,11026}, {9791,10580}, {10892,11021}


X(11032) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND TANGENTIAL-MIDARC

Trilinears    2*(-a+b+c)*((b+c)*a-(b-c)^2)*sin(A/2)+2*(a-b+c)^2*b*sin(B/2)+2*(a+b-c)^2*c*sin(C/2)+(-a+b+c)*(a-b+c)*(a+b-c) : :

X(11032) lies on these lines:
{1,164}, {7,177}, {57,8075}, {65,8241}, {167,173}, {174,8084}, {188,518}, {226,8085}, {354,2089}, {503,10490}, {938,9853}, {942,8091}, {946,8095}, {1071,9836}, {1210,8087}, {3057,10968}, {3333,8081}, {5045,8099}, {5083,8103}, {5728,8079}, {8089,10980}, {9793,10580}

X(11032) = X(8)-of-Yff-central-triangle
X(11032) = {X(1),X(5571)}-harmonic conjugate of X(11033)


X(11033) = HOMOTHETIC CENTER OF THESE TRIANGLES: INVERSE-IN-INCIRCLE AND 2ND TANGENTIAL-MIDARC

Trilinears    -2*b*c*sin(A/2)+(b+c)*a-(b-c)^2 : :
Trilinears    cos(B/2) cos(C/2) - cos^2(C/2) - cos^2(B/2) : :

X(11033) lies on these lines:
{1,164}, {57,8076}, {65,8242}, {173,8090}, {174,354}, {177,8084}, {226,8086}, {236,3742}, {518,7028}, {938,9854}, {942,8092}, {946,8096}, {1210,8088}, {2089,10967}, {3333,7587}, {5045,8100}, {5083,8104}, {5728,8080}, {9795,10580}

X(11033) = {X(1),X(5571)}-harmonic conjugate of X(11032)


X(11034) = PERSPECTOR OF THESE TRIANGLES: INCIRCLE-CIRCLES AND AQUILA

Barycentrics    3*a^3+12*(b+c)*a^2-13*(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The incircle-inverses of the sidelines of ABC are three congruent circles with radii=r/2 and concurrent at X(1). The triangle with vertices at the centers of these circles is here named here the incircle-circles triangle and its A-vertex has trilinear coordinates:
A' = 2*a : (a^2+4*a*b+b^2-c^2)/b : (a^2+4*a*c+c^2-b^2)/c

The appearance of (T,n) in the following list means that T and the incircle-circles triangles are perspective with perspector X(n), where an asterisk * indicates homothetic triangles:
(ABC, 3296), (Aquila, 11034), (Ascella *, 942), (Atik *, 11035), (1st circumperp *, 3295), (2nd circumperp *, 999), (Conway *, 11036), (2nd Conway *, 11037), (3rd Conway *, 1), (3rd Euler *, 496), (4th Euler *, 495), (excentral *, 3333), (2nd extouch *, 3487), (hexyl *, 1), (Honsberger *, 11038), (inner-Hutson *, 11039), (Hutson intouch *, 1), (outer-Hutson *, 11040), (intangents, 1), (intouch *, 1), (inverse-in-incircle *, 5045), (5th mixtilinear, 11041), (6th mixtilinear *, 1), (2nd Pamfilos-Zhou *, 11042), (1st Sharygin *, 11043), (tangential-midarc *, 11044), (2nd tangential-midarc *, 8351), (Yff central *, 8092), (inner-Yff, 11045), (outer-Yff, 11046), (inner-Yff tangents, 11047), (outer-Yff tangents, 11048)

(Explanation and centers X(11034)-X(11048) were contributed by César Lozada, November 26, 2016)

X(11034) lies on these lines:
{1,3528}, {2,5223}, {191,3333}, {354,4312}, {382,4355}, {942,3632}, {1699,3982}, {5083,9897}


X(11035) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND ATIK

Trilinears    (b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b^3+c^3)*a^3+2*(b^4+c^4-b*c*(b^2-16*b*c+c^2))*a^2+(b^4-c^4)*(b-c)*a-(b^2-c^2)^2*(b+c)^2 : :
X(11035) = (4*R+r)*(2*R-r)*X(8)-(6*R+r)*(4*R-r)*X(443)

X(11035) lies on these lines:
{1,971}, {3,7091}, {5,3947}, {8,443}, {20,9957}, {354,9578}, {388,5806}, {495,8582}, {496,10863}, {517,4298}, {946,10241}, {999,5044}, {3085,10156}, {3295,10860}, {3333,8580}, {3487,5049}, {3889,5177}, {4308,5766}, {5083,9952}, {5126,6986}, {5542,9953}, {5777,7373}, {6765,9858}

X(11035) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,8581,9856), (8,10569,942), (5045,9947,11019), (8583,9954,5044)


X(11036) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND CONWAY

Barycentrics    a^4-4*(b+c)*a^3-2*(b^2+4*b*c+c^2)*a^2+4*(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(11036) = (4*R+2*r)*X(1)+(4*R+r)*X(7) = 3*(4*R+r)*X(2)-4*R*X(72) = r*(3*R+2*r)*X(21)+2*(4*R^2-r^2)*X(999)

X(11036) lies on these lines:
{1,7}, {2,72}, {4,6147}, {8,4208}, {21,999}, {40,10578}, {46,5281}, {57,3523}, {63,3333}, {65,3475}, {78,9776}, {84,5558}, {144,405}, {145,377}, {200,11024}, {226,938}, {329,5129}, {346,3970}, {354,1858}, {388,6839}, {452,5905}, {495,3617}, {496,10883}, {497,3649}, {527,5436}, {553,3601}, {631,5708}, {912,6846}, {946,9799}, {950,3543}, {954,6986}, {1004,3871}, {1012,7373}, {1058,10431}, {1104,4644}, {1119,7513}, {1125,5273}, {1210,5056}, {1259,5253}, {1482,6916}, {1699,6744}, {1870,4198}, {2094,4652}, {2095,6988}, {3085,5902}, {3086,5443}, {3146,3488}, {3189,5880}, {3243,6764}, {3295,7411}, {3338,5265}, {3486,10404}, {3832,5714}, {3839,9612}, {3976,11031}, {3982,9579}, {4454,7283}, {4860,7288}, {5083,9964}, {5218,5221}, {5219,5704}, {5234,5850}, {5244,5712}, {5261,6993}, {5262,5813}, {5290,6738}, {5328,9843}, {5435,10303}, {5658,5806}, {5686,5904}, {5761,6926}, {6886,8232}, {8000,9804}, {9848,10391}, {10390,10429}, {10861,11035}

X(11036) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,20), (1,1044,2293), (1,3671,962), (1,4292,4313), (1,4295,390), (1,4298,5731), (1,4312,4314), (1,4355,4297), (1,10624,8236), (7,4313,4292), (8,5249,4208), (57,5703,3523), (226,938,3091), (942,3487,2), (962,10884,20), (1210,5226,5056), (3622,9965,21), (4292,4313,20), (5219,5704,7486), (5708,5719,631), (5714,5722,3832)


X(11037) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND 2ND CONWAY

Barycentrics    a^4+2*(b+c)*a^3+12*b*c*a^2-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(11037) = 4*R*X(1)+(4*R+r)*X(7) = (4*R+r)*X(8)-2*(6*R+r)*X(443)

X(11037) lies on these lines:
{1,7}, {2,3333}, {3,10578}, {4,5045}, {8,443}, {10,10980}, {12,5704}, {56,3475}, {142,6762}, {145,8000}, {329,405}, {354,388}, {377,3889}, {452,4666}, {495,9780}, {496,5714}, {497,10404}, {553,1697}, {1058,5049}, {1125,5234}, {1191,4644}, {1210,5261}, {1219,4869}, {1385,5758}, {1387,9809}, {1699,9851}, {1788,4860}, {2550,6764}, {2551,3742}, {3085,3338}, {3086,5226}, {3091,5290}, {3218,10587}, {3295,9778}, {3303,3474}, {3304,3485}, {3306,7080}, {3337,10056}, {3361,3523}, {3421,5439}, {3486,5434}, {3622,5905}, {3626,11034}, {3870,6904}, {3957,4190}, {3976,7385}, {4031,5128}, {4208,4847}, {5082,9797}, {5083,9803}, {5129,10582}, {5250,9965}, {5274,9612}, {5557,5697}, {5586,9819}, {5657,5708}, {5691,6744}, {5768,10532}, {5809,11025}, {5811,5886}, {6361,6767}, {8165,9843}, {9799,11020}, {10449,11021}

X(11037) = midpoint of X(1) and X(4355)
X(11037) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,962), (1,3600,5731), (1,4292,390), (1,4293,4313), (1,4294,8236), (1,4295,9785), (1,4298,20), (1,4317,4305), (1,4321,10884), (1,4334,4300), (1,4340,4344), (1,5542,11036), (4,5045,10580), (7,9785,4295), (8,9776,11024), (56,3475,5703), (354,388,938), (443,3555,8), (496,5714,9779), (942,1056,8), (942,11035,3555), (999,3487,3616), (1056,3296,942), (3085,3338,5435), (4295,9785,962), (5290,11019,3091), (5558,10580,5045)


X(11038) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND HONSBERGER

Barycentrics    a^3-5*(b+c)*a^2+3*(b-c)^2*a+(b^2-c^2)*(b-c) : :
X(11038) = 2*X(1)+X(7) = (8*R+r)*X(2)-4*R*X(210) = X(8)-4*X(142)

X(11038) lies on these lines:
{1,7}, {2,210}, {3,2346}, {8,142}, {9,1475}, {57,5281}, {144,1001}, {145,2550}, {226,5274}, {329,4666}, {377,6601}, {388,6894}, {404,6600}, {495,7679}, {496,7678}, {551,5850}, {553,9778}, {938,5261}, {942,5657}, {944,5805}, {952,1056}, {954,999}, {971,5049}, {1000,1159}, {1058,6147}, {1100,5819}, {1125,5223}, {1156,1387}, {1279,4644}, {1320,10427}, {1385,5759}, {1420,5766}, {1445,3333}, {1449,5838}, {1471,9440}, {1621,9965}, {1898,3485}, {2099,8255}, {3085,5445}, {3146,10404}, {3241,5853}, {3242,4648}, {3254,6224}, {3295,7676}, {3338,3523}, {3474,3748}, {3487,5045}, {3560,5843}, {3617,3826}, {3623,5880}, {3685,4454}, {3870,9776}, {3982,9580}, {4654,9812}, {4675,4864}, {4860,5218}, {5226,7988}, {5290,6744}, {5308,7174}, {5435,10980}, {5493,5586}, {5528,9802}, {5550,6666}, {5704,10172}, {5762,10246}, {5779,5901}, {6765,11024}, {6904,7674}, {7673,9957}, {9446,9533}, {10569,11018}, {10865,11035}

X(11038) = midpoint of X(7) and X(8236)
X(11038) = reflection of X(i) in X(j) for these (i,j): (390,8236), (5686,2), (5817,5886), (8236,1)
X(11038) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,390), (1,3664,4344), (1,3671,9785), (1,4298,4313), (1,4310,3672), (1,4321,7675), (1,4327,3100), (1,4334,2293), (1,4355,4314), (1,5542,7), (1,11037,3600), (57,10578,5281), (142,3243,8), (144,3622,1001), (226,10580,5274), (354,3475,2), (553,10389,9778), (954,999,7677), (3333,5703,5265), (3487,5728,8232), (5045,5728,11025), (5572,8581,10394)


X(11039) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND INNER-HUTSON

Trilinears    4*a*b*c*sin(A/2)+2*(a^2+4*b*a+b^2-c^2)*c*sin(B/2)+2*(a^2+4*c*a-b^2+c^2)*b*sin(C/2)+a^3-(b+c)*a^2-(b^2+14*b*c+c^2)*a+(b^2-c^2)*(b-c) : :

X(11039) lies on these lines:
{1,8111}, {363,3333}, {495,8380}, {496,8377}, {942,9805}, {999,8109}, {3295,8107}, {3487,5934}, {6732,8351}


X(11040) = HOMOTHETIC CENTER OF THESE TRIANGLES: INCIRCLE-CIRCLES AND OUTER-HUTSON

Trilinears    4*a*b*c*sin(A/2)+2*(a^2+4*b*a+b^2-c^2)*c*sin(B/2)+2*(a^2+4*c*a-b^2+c^2)*b*sin(C/2)-(a^3-(b+c)*a^2-(b^2+14*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

X(11040) lies on these lines:
{1,8111}, {168,3333}, {495,8381}, {496,8378}, {942,9806}, {999,8110}, {3295,8108}, {3296,7707}, {3487,5935}, {8138,8351}


X(11041) = PERSPECTOR OF THESE TRIANGLES: INCIRCLE-CIRCLES AND 5TH MIXTILINEAR

Barycentrics    3*a^4-6*(b+c)*a^3-2*(b-c)^2*a^2+6*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(11041) = (2*R+3*r)*X(8)-(6*R+4*r)*X(442)

The poristic locus of the Gergonne point is a circle here named the Greenhill circle, with the Greenhill point, X(1159), as center; see Hyacinthos #6535.

X(11041) is the antipode of X(7) in the Greenhill circle. (Randy Hutson, December 10, 2016)

Let P be a point on the circumcircle. The tangents at P to the incircle intersect the incircle and circumcircle at four points. Let X(P) be the intersection of its diagonals (see Dominik Burek, ADGEOM #1427, July 21, 2014.

The locus of X(P) as P moves around the circumcircle is the Greenhill circle, with segment X(7)X(11041) as diameter. Let Pa be the point P on arc BC such that X(P) = X(11041), and define Pb and Pc cyclically. Then Pa, Pb, Pc are also the points on the circumcircle, other than the antipodes of A, B, C, whose Simson lines are tangent to the incircle; also, X(11041) = X(7)-of-PaPbPc. (Randy Hutson, December 10, 2016)

X(11041) lies on these lines:
{1,631}, {4,3340}, {5,4323}, {7,952}, {8,442}, {11,2099}, {55,5427}, {57,7966}, {65,944}, {100,999}, {145,942}, {355,5714}, {376,2093}, {390,517}, {496,5734}, {514,4644}, {515,4312}, {519,1056}, {938,1482}, {962,9668}, {1002,2401}, {1058,4342}, {1210,10595}, {1317,4860}, {1389,3427}, {1478,9897}, {1483,4308}, {2136,3244}, {3256,6950}, {3296,3633}, {3474,4316}, {3475