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X(2), X(519), X(3679), X(4370), X(4908), X(4945), X(24858), X(36909), X(36910), X(36911), X(36912), X(36913), X(36914), X(36915), X(36916)

vertices of the medial triangle

other points below

Geometric properties :

K1149 is the locus of poles of pivotal cubics pK(Ω, P) passing through X(369), Z1, Z2 and also X(519). See Table 42.

The locus of pivots is K311 and the locus of isopivots is K1150.

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Other centers on K1149 :

Ω1 = (a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-5 a b c+3 b^2 c-a c^2+3 b c^2-c^3) : : , SEARCH = -2.491442945923647

Ω2 = (-a+b+c) (a^2-a b+b^2-c^2) (a^2-b^2-a c+c^2) : : , SEARCH = 3.57972978700963

Ω3 = (a-2 b-2 c) (5 a-b-c) : : , SEARCH = 1.256805208988318

Ω4 = (a-2 b-2 c) (2 a-b-c) (a^2+2 a b+b^2+2 a c-7 b c+c^2) : : , SEARCH = 5.183085428266677

Ω5 = (a-2 b-2 c) (a+b-2 c) (a-2 b+c) (7 a^3-3 a^2 b-9 a b^2+b^3-3 a^2 c+9 a b c+3 b^2 c-9 a c^2+3 b c^2+c^3) : : , SEARCH = -0.9597056500446528

Ω6 = (a-2 b-2 c) (2 a-b-c) (a+b-c) (a-b+c) (a^2-b^2+b c-c^2) : : , SEARCH = 1.471655678550038

Ω7 = (a-2 b-2 c) (2 a-b-c) (a+b-c) (a-b+c) (a^2-b^2+4 b c-c^2) : : , SEARCH = -2.574904180238664

Ω8 = (a-2 b-2 c) (a^3-a^2 b-a b^2+b^3+2 a b c-a c^2-b c^2) (a^3-a b^2-a^2 c+2 a b c-b^2 c-a c^2+c^3) (3 a^3-a^2 b-3 a b^2+b^3-a^2 c+2 a b c-b^2 c-3 a c^2-b c^2+c^3) : : , SEARCH = 4.001219684455818

Ω9 = (-2 a+b+c) (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) (a^4-2 a^2 b^2+b^4-2 a^3 c+a^2 b c+a b^2 c-2 b^3 c+2 a^2 c^2-3 a b c^2+2 b^2 c^2+2 a c^3+2 b c^3-3 c^4) (a^4-2 a^3 b+2 a^2 b^2+2 a b^3-3 b^4+a^2 b c-3 a b^2 c+2 b^3 c-2 a^2 c^2+a b c^2+2 b^2 c^2-2 b c^3+c^4) : : , SEARCH = 7.535567447365711

Ω10 = (a+b-5 c) (2 a-b-c) (a-5 b+c) : : , SEARCH = 9.228346193939886

Ω11 = (a-2 b-2 c) (2 a-b-c) (a^2-b^2+b c-c^2) (a^3-3 a^2 b-3 a b^2+b^3+a^2 c+5 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-3 a^2 c+5 a b c-b^2 c-3 a c^2+b c^2+c^3) : : , SEARCH = 5.918635110972674

Ω12 = (a^2-4 a b+b^2-c^2) (a^2-b^2-4 a c+c^2) (a^3+a^2 b-a b^2-b^3+a^2 c-26 a b c+9 b^2 c-a c^2+9 b c^2-c^3) : : , SEARCH = -0.7341664692173939

Ω13 = (a+b-5 c) (2 a-b-c) (a-5 b+c) (4 a^3-9 a^2 b-12 a b^2+b^3-9 a^2 c+30 a b c+3 b^2 c-12 a c^2+3 b c^2+c^3) : : , SEARCH = 6.775877689340537

Ω14 = (a-b-c) (a^2-4 a b+b^2-c^2) (a^2-b^2-4 a c+c^2) : : , SEARCH = 4.018628806672074

Ω15 = (2 a-b-c)^2 (a^3-9 a^2 b-3 a b^2+7 b^3+3 a^2 c+9 a b c-3 b^2 c+3 a c^2-9 b c^2+c^3) (a^3+3 a^2 b+3 a b^2+b^3-9 a^2 c+9 a b c-9 b^2 c-3 a c^2-3 b c^2+7 c^3) : : , SEARCH = 4.323406256298219

Ω16 = (2 a-b-c) (a^3-3 a^2 b-a b^2+3 b^3-a^2 c+2 a b c-b^2 c-a c^2-3 b c^2+c^3) (a^3-a^2 b-a b^2+b^3-3 a^2 c+2 a b c-3 b^2 c-a c^2-b c^2+3 c^3) (3 a^4-8 a^3 b-2 a^2 b^2+8 a b^3-b^4-8 a^3 c+24 a^2 b c-16 a b^2 c-2 a^2 c^2-16 a b c^2+2 b^2 c^2+8 a c^3-c^4) : : , SEARCH = 1.801800208427052

Ω17 = (a^2+2 a b+b^2-7 a c+2 b c+c^2) (a^2-7 a b+b^2+2 a c+2 b c+c^2) (a^4+4 a^3 b+6 a^2 b^2+4 a b^3+b^4+4 a^3 c-87 a^2 b c+57 a b^2 c-14 b^3 c+6 a^2 c^2+57 a b c^2-30 b^2 c^2+4 a c^3-14 b c^3+c^4) : : , SEARCH = 2.722011817061126

Ω18 = (a-2 b-2 c) (5 a-b-c) (a^3-12 a^2 b-9 a b^2+4 b^3+3 a^2 c+30 a b c-9 b^2 c+3 a c^2-12 b c^2+c^3) (a^3+3 a^2 b+3 a b^2+b^3-12 a^2 c+30 a b c-12 b^2 c-9 a c^2-9 b c^2+4 c^3) : : , SEARCH = -1.506985377129427

Ω19 = (a-2 b-2 c) (a^3-a^2 b-a b^2+b^3+2 a b c-a c^2-b c^2) (a^3-a b^2-a^2 c+2 a b c-b^2 c-a c^2+c^3) (3 a^4-2 a^3 b-2 a^2 b^2+2 a b^3-b^4-2 a^3 c+3 a^2 b c-a b^2 c-2 a^2 c^2-a b c^2+2 b^2 c^2+2 a c^3-c^4) : : , SEARCH = -0.4466493147356433

Ω20 = (2 a-b-c) (a^2 b-b^3+a^2 c-2 a b c+b^2 c+b c^2-c^3) (a^3-3 a^2 b-a b^2+3 b^3-a^2 c+2 a b c-b^2 c-a c^2-3 b c^2+c^3) (a^3-a^2 b-a b^2+b^3-3 a^2 c+2 a b c-3 b^2 c-a c^2-b c^2+3 c^3) : : , SEARCH = -3.779576630004337

Ω21 = (a-2 b-2 c) (2 a-b-c) (a^2-b^2+4 b c-c^2) (a^3-9 a^2 b-9 a b^2+b^3+a^2 c+26 a b c+b^2 c-a c^2-b c^2-c^3) (a^3+a^2 b-a b^2-b^3-9 a^2 c+26 a b c-b^2 c-9 a c^2+b c^2+c^3) : : , SEARCH = 7.323569365475301

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Triads of collinear points on K1149 :

X2, X519, X3679

X2, X4370, X4945

X2, X24858, Ω4

X2, Ω1, Ω11

X2, Ω2, Ω6

X2, Ω3, Ω10

X2, Ω5, Ω15

X2, Ω7, Ω14

X2, Ω8, Ω20

X2, Ω9, Ω19

X2, Ω12, Ω21

X2, Ω13, Ω18

X519, X4370, X4908

X519, X4945, Ω1

X519, X24858, Ω5

X519, Ω2, Ω19

X519, Ω3, Ω14

X519, Ω4, Ω10

X519, Ω7, Ω20

X519, Ω13, Ω21

X519, Ω17, Ω18

X3679, X4370, Ω2

X3679, X4908, Ω3

X3679, X4945, Ω4

X3679, X24858, Ω13

X3679, Ω1, Ω9

X3679, Ω5, Ω11

X3679, Ω6, Ω8

X3679, Ω10, Ω12

X3679, Ω14, Ω16

X3679, Ω15, Ω17

X4370, X24858, Ω3

X4370, Ω4, Ω15

X4370, Ω6, Ω9

X4370, Ω7, Ω10

X4370, Ω8, Ω14

X4370, Ω12, Ω18

X4370, Ω16, Ω21

X4908, X4945, Ω5

X4908, X24858, Ω17

X4908, Ω1, Ω2

X4908, Ω6, Ω7

X4908, Ω8, Ω19

X4908, Ω10, Ω13

X4908, Ω12, Ω14

X4908, Ω16, Ω20

X4945, Ω3, Ω6

X4945, Ω7, Ω19

X4945, Ω10, Ω17

X4945, Ω12, Ω20

X4945, Ω13, Ω14

X24858, Ω1, Ω7

X24858, Ω6, Ω12

X24858, Ω16, Ω19

Ω1, Ω3, Ω15

Ω1, Ω8, Ω10

Ω1, Ω16, Ω18

Ω2, Ω3, Ω20

Ω2, Ω4, Ω14

Ω2, Ω5, Ω10

Ω2, Ω17, Ω21

Ω3, Ω4, Ω18

Ω3, Ω11, Ω19

Ω4, Ω6, Ω11

Ω4, Ω7, Ω21

Ω4, Ω9, Ω20

Ω5, Ω7, Ω18

Ω5, Ω8, Ω21

Ω5, Ω9, Ω14

Ω6, Ω10, Ω16

Ω6, Ω13, Ω15

Ω11, Ω13, Ω20

Ω11, Ω14, Ω17

Ω12, Ω15, Ω19

Remarks :

• two points collinear with X(2) are obvioulsy X(4908)-isoconjugates.

• two points collinear with X(4908) are X(2)-Ceva conjugates since X(4908) is the isopivot of K1149 hence the tangential of X(2) on the cubic.

• two points collinear with Ω4 are X(4908)-crossconjugates since Ω4 is the X(2)-Ceva conjugate of X(4908).

• two points collinear with Ω6 are the poles of two pKs whose pivots are isotomic conjugates on K311 hence collinear with X(320).

• the tangentials of X(519), X(3679), X(4370), X(4908) are Ω6, Ω7, Ω11, Ω4 respectively.