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X(2), X(4), X(6), X(39), X(51), X(262), X(1640), X(3148), X(40814), X(45207), X(56395)

X(60495) → X(60501), X(60586) → X(60589)

other points below

Geometric properties :

There are two sets of cubics pK(Ω, P) which pass through the centers X(2), X(4), X(6).

• Those with pivot P on the Kiepert hyperbola and isopivot Q = X(6). Their pole Ω lies on the circum-conic with perspector X(512).

• Those with pole Ω on the line {X4, X32}, with pivot P on the line {X6, X264} such that the points X(2), Ω, P are collinear. Their isopivot Q lies on the nodal cubic K1352 with node X(6).

K677 = pK(X4, X458), K777 = pK(X98, X287), K790 = pK(X7735, X6) are examples of such cubics with isopivots X(262), X(4), X(40814) respectively. See also K1353, K1354 with isopivots X(1640), X(51) respectively.

With Q = X(2), we find pK(X6531, X6531) and with Q = X(6), we find the two cubics pK(X41200, X41194) and pK(X41201, X41195).

With P = p : q : r, this point Q is :

q r [(b-c) (b+c) (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6) p - 2(c-a) (c+a) SB (a^4-a^2 b^2-b^2 c^2+c^4) q - 2(a-b) (a+b) SC (a^4+b^4-a^2 c^2-b^2 c^2) r] : : .

***

Points on K1352

Q1 = a^2 (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) (a^4 b^2-b^6+a^4 c^2-2 a^2 b^2 c^2+b^4 c^2+b^2 c^4-c^6) : : , SEARCH = 1.78232200627713

Q2 = a^2 (a^2+b^2-2 c^2) (a^2-2 b^2+c^2) (a^4 b^2-2 a^2 b^4+b^6+a^4 c^2+2 a^2 b^2 c^2-b^4 c^2-2 a^2 c^4-b^2 c^4+c^6) : : , SEARCH = 1.43784726235391

Q3 = a^2 (a^4-a^2 b^2-a^2 c^2-2 b^2 c^2) (a^2 b^2-b^4+a^2 c^2+b^2 c^2) (a^2 b^2+a^2 c^2+b^2 c^2-c^4) : : , SEARCH = 1.82143189898734

Q4 = (a^4-a^2 b^2+b^4-c^4) (2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^4-b^4-a^2 c^2+c^4) : : , SEARCH = 1.61699282357178

Q5 = a^2 (a^2-b^2-c^2) (a^4+b^4-c^4) (a^4-b^4+c^4) : : , SEARCH = 1.65830978994201

Q6 = (2 a^2-b^2-c^2) (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) : : , SEARCH = 0.0415450789077273

Q7 = a^2 (a^4-2 a^2 b^2+b^4-2 b^2 c^2+c^4) (a^4+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : : , SEARCH = -0.574786915859974

Q8 = a^2 (-a^2 b^6+b^8+a^6 c^2+a^2 b^4 c^2-b^6 c^2-2 a^4 c^4+a^2 c^6) (a^6 b^2-2 a^4 b^4+a^2 b^6+a^2 b^2 c^4-a^2 c^6-b^2 c^6+c^8) (a^10 b^4-2 a^8 b^6+2 a^4 b^10-a^2 b^12+4 a^10 b^2 c^2-4 a^8 b^4 c^2+a^6 b^6 c^2-3 a^4 b^8 c^2+3 a^2 b^10 c^2-b^12 c^2+a^10 c^4-4 a^8 b^2 c^4+4 a^6 b^4 c^4+a^4 b^6 c^4-a^2 b^8 c^4+3 b^10 c^4-2 a^8 c^6+a^6 b^2 c^6+a^4 b^4 c^6-2 a^2 b^6 c^6-2 b^8 c^6-3 a^4 b^2 c^8-a^2 b^4 c^8-2 b^6 c^8+2 a^4 c^10+3 a^2 b^2 c^10+3 b^4 c^10-a^2 c^12-b^2 c^12) : : , SEARCH = -0.362143835957635

Q9 = (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-2 a^2 b^2 c^2-b^4 c^2+a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2-2 a^2 b^2 c^2+b^4 c^2-a^2 c^4-b^2 c^4+c^6) (a^8-a^6 b^2+3 a^4 b^4-3 a^2 b^6-a^6 c^2+4 a^4 b^2 c^2+3 a^2 b^4 c^2-2 b^6 c^2+3 a^4 c^4+3 a^2 b^2 c^4+4 b^4 c^4-3 a^2 c^6-2 b^2 c^6) : : , SEARCH = 8.93830653265617

Q10 = (a^8-a^6 b^2-a^6 c^2+a^4 b^2 c^2+b^6 c^2-2 b^4 c^4+b^2 c^6) (a^12-2 a^10 b^2+5 a^8 b^4-8 a^6 b^6+5 a^4 b^8-2 a^2 b^10+b^12-2 a^10 c^2+3 a^8 b^2 c^2-a^6 b^4 c^2-a^4 b^6 c^2+3 a^2 b^8 c^2-2 b^10 c^2+a^8 c^4-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-a^4 c^8-a^2 b^2 c^8-b^4 c^8+2 a^2 c^10+2 b^2 c^10-c^12) (a^12-2 a^10 b^2+a^8 b^4-a^4 b^8+2 a^2 b^10-b^12-2 a^10 c^2+3 a^8 b^2 c^2-a^6 b^4 c^2-a^4 b^6 c^2-a^2 b^8 c^2+2 b^10 c^2+5 a^8 c^4-a^6 b^2 c^4+2 a^4 b^4 c^4-a^2 b^6 c^4-b^8 c^4-8 a^6 c^6-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^8-2 a^2 c^10-2 b^2 c^10+c^12) : : , SEARCH = -0.217464634755797

Q11 = (b-c)^2 (b+c)^2 (a^6 b^2-a^4 b^4-a^2 b^6+b^8+3 a^6 c^2-a^4 b^2 c^2+3 a^2 b^4 c^2-b^6 c^2-6 a^4 c^4-a^2 b^2 c^4-b^4 c^4+3 a^2 c^6+b^2 c^6) (3 a^6 b^2-6 a^4 b^4+3 a^2 b^6+a^6 c^2-a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2-a^4 c^4+3 a^2 b^2 c^4-b^4 c^4-a^2 c^6-b^2 c^6+c^8) : : , SEARCH = 2.49869413873033

Q12 = a^2 (a-b)^2 (a+b)^2 (a-c)^2 (a+c)^2 (a^10-a^8 b^2-a^2 b^8+b^10-a^8 c^2+a^6 b^2 c^2-a^2 b^6 c^2-3 b^8 c^2+4 a^2 b^4 c^4+2 b^6 c^4-a^2 b^2 c^6+2 b^4 c^6-a^2 c^8-3 b^2 c^8+c^10) : : , SEARCH = -0.0518501191166856

Q13 = a^2 (a^6-a^4 b^2-a^2 b^4+b^6-a^4 c^2-b^4 c^2+2 a^2 c^4+2 b^2 c^4-2 c^6) (a^6-a^4 b^2+a^2 b^4-b^6-a^4 c^2-a^2 b^2 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6) (a^6-a^4 b^2+2 a^2 b^4-2 b^6-a^4 c^2+2 b^4 c^2-a^2 c^4-b^2 c^4+c^6) : : , SEARCH = 1.66225200922899

T1 = a^2 (a^4+3 b^4-2 a^2 c^2+c^4) (a^4+b^4-2 b^2 c^2+c^4) (a^4-2 a^2 b^2+b^4+3 c^4) : : , SEARCH = 1.58234927467567

T2 = a^4 (a^6 b^2-a^4 b^4-a^2 b^6+b^8+a^6 c^2-a^4 b^2 c^2-3 a^2 b^4 c^2-b^6 c^2-2 a^4 c^4-a^2 b^2 c^4-b^4 c^4+a^2 c^6+b^2 c^6) (a^6 b^2-2 a^4 b^4+a^2 b^6+a^6 c^2-a^4 b^2 c^2-a^2 b^4 c^2+b^6 c^2-a^4 c^4-3 a^2 b^2 c^4-b^4 c^4-a^2 c^6-b^2 c^6+c^8) : : , SEARCH = 0.805127080885323

T3 = a^2 (a^4 b^4-a^2 b^6+a^4 b^2 c^2+a^2 b^4 c^2-b^6 c^2+a^4 c^4+a^2 b^2 c^4+2 b^4 c^4-a^2 c^6-b^2 c^6) (3 a^6 b^2-3 a^4 b^4+a^2 b^6-b^8+2 a^6 c^2-3 a^4 b^2 c^2-4 a^2 b^4 c^2+b^6 c^2-4 a^4 c^4-3 a^2 b^2 c^4-3 b^4 c^4+2 a^2 c^6+3 b^2 c^6) (2 a^6 b^2-4 a^4 b^4+2 a^2 b^6+3 a^6 c^2-3 a^4 b^2 c^2-3 a^2 b^4 c^2+3 b^6 c^2-3 a^4 c^4-4 a^2 b^2 c^4-3 b^4 c^4+a^2 c^6+b^2 c^6-c^8) : : , SEARCH = : : , SEARCH = 1.59182874548856

T1, T2, T3 are the common tangentials of X(2) and X(4), X(39) and X(51), X(262) and X(40814) respectively.