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X(2), X(525), X(14846), X(34290)

A', B', C' : traces of (L), the trilinear polar of X(1989)

other points below

Q153 is the locus of the radial center X of the stelloidal nK0s. It is analogous to Q004 obtained with stelloidal pKs.

Recall that a stelloidal nK0 must have its pole Ω on K396 and its root R on K397.

Q153 is a bicircular quintic with triple point G. One tangent at G is parallel to (L) and the two other tangents are real when ABC is obtuse. In this case, these tangents meet (L) at two real points M1, M2 on Q153, on the circum-conic (C) passing through X(1576) which is the transform of (L) under the isoconjugation that swaps X(32) and X(381). Note that the midpoint of M1, M2 is X(51).

The following table shows some of these cubics with related points Ω, R and X.

Ω on K396










R on K397












note 1








X on Q153










note 1 : the cubic splits into the line at infinity and the Kiepert hyperbola.


Miscellaneous related points

P0 = (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) (a^12-5 a^10 b^2+9 a^8 b^4-8 a^6 b^6+5 a^4 b^8-3 a^2 b^10+b^12-5 a^10 c^2+12 a^8 b^2 c^2-12 a^6 b^4 c^2+6 a^4 b^6 c^2-b^10 c^2+9 a^8 c^4-12 a^6 b^2 c^4+3 a^4 b^4 c^4+b^8 c^4-8 a^6 c^6+6 a^4 b^2 c^6-2 b^6 c^6+5 a^4 c^8+b^4 c^8-3 a^2 c^10-b^2 c^10+c^12) : : , SEARCH = 0.0650713418471199

P1 = (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6) (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) : : , SEARCH = 1.20826067234074

P2 = (a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) (-2 a^2+b^2+c^2) (a^2-b^2-a c+c^2) (a^2-b^2+a c+c^2) (a^16-6 a^14 b^2+11 a^12 b^4-6 a^10 b^6-3 a^8 b^8+6 a^6 b^10-7 a^4 b^12+6 a^2 b^14-2 b^16-6 a^14 c^2+29 a^12 b^2 c^2-49 a^10 b^4 c^2+41 a^8 b^6 c^2-25 a^6 b^8 c^2+20 a^4 b^10 c^2-16 a^2 b^12 c^2+6 b^14 c^2+11 a^12 c^4-49 a^10 b^2 c^4+59 a^8 b^4 c^4-29 a^6 b^6 c^4+11 a^4 b^8 c^4+2 a^2 b^10 c^4-7 b^12 c^4-6 a^10 c^6+41 a^8 b^2 c^6-29 a^6 b^4 c^6-11 a^4 b^6 c^6+5 a^2 b^8 c^6+12 b^10 c^6-3 a^8 c^8-25 a^6 b^2 c^8+11 a^4 b^4 c^8+5 a^2 b^6 c^8-18 b^8 c^8+6 a^6 c^10+20 a^4 b^2 c^10+2 a^2 b^4 c^10+12 b^6 c^10-7 a^4 c^12-16 a^2 b^2 c^12-7 b^4 c^12+6 a^2 c^14+6 b^2 c^14-2 c^16) : : , SEARCH = 0.0701992449232681

P3 = (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) (a^2 b^2-b^4+a^2 c^2-c^4) (a^18-3 a^16 b^2+6 a^14 b^4-13 a^12 b^6+20 a^10 b^8-21 a^8 b^10+18 a^6 b^12-11 a^4 b^14+3 a^2 b^16-3 a^16 c^2+3 a^14 b^2 c^2+3 a^12 b^4 c^2-8 a^10 b^6 c^2+15 a^8 b^8 c^2-24 a^6 b^10 c^2+24 a^4 b^12 c^2-11 a^2 b^14 c^2+b^16 c^2+6 a^14 c^4+3 a^12 b^2 c^4-9 a^10 b^4 c^4+3 a^8 b^6 c^4+3 a^6 b^8 c^4-18 a^4 b^10 c^4+18 a^2 b^12 c^4-2 b^14 c^4-13 a^12 c^6-8 a^10 b^2 c^6+3 a^8 b^4 c^6+7 a^6 b^6 c^6+5 a^4 b^8 c^6-18 a^2 b^10 c^6+20 a^10 c^8+15 a^8 b^2 c^8+3 a^6 b^4 c^8+5 a^4 b^6 c^8+16 a^2 b^8 c^8+b^10 c^8-21 a^8 c^10-24 a^6 b^2 c^10-18 a^4 b^4 c^10-18 a^2 b^6 c^10+b^8 c^10+18 a^6 c^12+24 a^4 b^2 c^12+18 a^2 b^4 c^12-11 a^4 c^14-11 a^2 b^2 c^14-2 b^4 c^14+3 a^2 c^16+b^2 c^16) : : , SEARCH = 0.172228815586573

Ω1 = (a^4+b^4-a^2 c^2-b^2 c^2) (a^4-a^2 b^2-b^2 c^2+c^4) (2 a^6-2 a^4 b^2+a^2 b^4-b^6-2 a^4 c^2+b^4 c^2+a^2 c^4+b^2 c^4-c^6) : : , SEARCH = 0.615439289833560

Ω3 = a^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) (a^2 b^2-b^4+a^2 c^2-c^4) (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+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 = -0.0828390549533283

These six points above are now in ETC (2019-10-09) as X(34365), X(34366), X(34367), X(34368), X(34369), X(34370).

M1 =-3 a^8-3 a^6 b^2+8 a^4 b^4-a^2 b^6-b^8-6 a^4 b^2 c^2+2 a^2 b^4 c^2+2 b^6 c^2+4 a^4 c^4+a^2 b^2 c^4-2 a^2 c^6-2 b^2 c^6+c^8+(a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) Sqrt[(a^2-b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2+b^2+c^2)] : : , SEARCH = -0.466415967914056

M2 =-3 a^8-3 a^6 b^2+8 a^4 b^4-a^2 b^6-b^8-6 a^4 b^2 c^2+2 a^2 b^4 c^2+2 b^6 c^2+4 a^4 c^4+a^2 b^2 c^4-2 a^2 c^6-2 b^2 c^6+c^8-(a^2-a b+b^2-c^2) (a^2+a b+b^2-c^2) Sqrt[(a^2-b^2-c^2) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^2+b^2+c^2)] : : , SEARCH = 0.707667133329048

M1 x M2 = X(34416) = a^4 (a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4) : : , SEARCH = -0.0450596653477169