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X(8), X(80), X(369), X(519), X(1000), X(1145), X(3679), X(36593), X(36919), X(36920), X(36921), X(36922), X(36923), X(36924), X(36925)

vertices of the excentral triangle

other points below

Geometric properties :

K1150 is the locus of isopivots 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 poles is K1149.

Note that K311 and K1150 are two of these pivotal cubics. See also K1147 and K1148.

The polar conic of X(8) in K1150 is also the polar conic of X(2) in K1149. This is the diagonal hyperbola (H) passing through X(2), X(8), etc, and the square roots of X(4908) which obviously lie on K1149 and K1150, More generally, (H) contains the square roots of all the points on the line (L) passing through X(2), X(37), X(75), etc. (L) is the trilinear polar of X(668), the center of (H).

(H) is the anticomplement of the circum-conic with perspector X(513) which is the isotomic transform of (L).

K311 and K1150 generate a pencil of cubics passing through A, B, C, X(8), X(80), X(369), X(519), Z1, Z2 which contains a third pK namely K1151 and a circular cubic.


Other points on K1150

Q1 = (a-2 b-2 c) (2 a-b-c) (a^2-b^2+4 b c-c^2) : : , SEARCH = 14.68219738416062

Q2 = (a-2 b-2 c) (2 a-b-c) (a+b-c) (a-b+c) : : , SEARCH = -2.802431211141678

Q3 = -(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) : : , SEARCH = 5.28966003234214

Q4 = (a-2 b-2 c) (3 a^3-a^2 b-3 a b^2+b^3-a^2 c+2 a b c-b^2 c-3 a c^2-b c^2+c^3) : : , SEARCH = 5.208367505105926

Q5 = (a-2 b-2 c) (2 a-b-c) (a^2-b^2+b c-c^2) : : , SEARCH = 5.011645081932613

Q6 = (a+b-5 c) (2 a-b-c)^2 (a-5 b+c) : : , SEARCH = 6.591675852814204

Q7 = (a+b-2 c) (a-2 b+c) (-a+2 b+2 c) (3 a^4-2 a^3 b-2 a^2 b^2+2 a b^3-b^4-2 a^3 c+3 a^2 b c-a b^2 c-2 a^2 c^2-a b c^2+2 b^2 c^2+2 a c^3-c^4) : : , SEARCH = -0.09043022279830697

Q8 = (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^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 = 0.3688354468455077

Q9 = (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^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 = 1.968895793575072

Q4* = (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) : : , SEARCH = 1.881946869711703

Q7* = -(2 a-b-c)^2 (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 = 7.460884794003621

Q8* = (a-2 b-2 c) (2 a-b-c) (a^2+2 a b+b^2+2 a c-7 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 = 8.656303643346797

Q9* = (a-2 b-2 c) (2 a-b-c) (a^2+2 a b+b^2+2 a c-7 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 = 5.70570150859634


Collinear points on K1150

X8, X80, Q5

X8, X519, X3679

X8, X1000, Q1

X8, X1145, Q3

X8, X36593, Q6

X8, Q4, Q4*

X8, Q7, Q7*

X8, Q8, Q8*

X8, Q9, Q9*

X80, X519, Q7

X80, X1145, X3679

X80, Q1, Q9

X80, Q4, Q6

X519, X1000, Q4

X519, X1145, Q2

X519, X36593, Q9

X519, Q1, Q6

X1000, X1145, X36593

X1000, Q5, Q8

X1145, Q1, Q4*

X1145, Q5, Q7*

X1145, Q6, Q9*

X3679, X36593, Q5

X3679, Q2, Q4

X3679, Q6, Q8

X3679, Q7, Q9*

X36593, Q2, Q7

X36593, Q4, Q8*

Q1, Q2, Q5

Q1, Q3, Q7

Q2, Q8, Q9

Q3, Q4, Q5

Q4, Q9, Q7*

Q7, Q8, Q4*

Note : X* is the X(4908)-isoconjugate of X hence X(8), X, X* are collinear on K1150.