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X(3), X(76), X(298), X(299), X(511), X(599), X(3413), X(3414), X(32452), X(39785), X(42009), X(42060)

X(44771) → X(44777)

other points below

Geometric properties :

See K1238 for general explanations.

K1239 is an acnodal cubic with node X(76).

It has one real point of inflexion namely X(599) with tangent passing through X(115).

K1239 has three real asymptotes namely those of the Kiepert hyperbola (K) and a parallel to the Brocard axis.

The tangential of X(511) is Q = {(a^2-b c) (a^2+b c) (-b^6+a^2 b^2 c^2-b^4 c^2-b^2 c^4-c^6) : : , SEARCH = 9.09416286489454. Q lies on the tangent at X(76) to the Kiepert hyperbola passing through X(141). Q is now X(44771).

K1239 meets the sidelines of ABC at three real points A', B', C', on the trilinear polar (L) of X(850) passing through X(115), X(127), and three pairs of always imaginary points.