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X(3), X(8), X(517), X(2098), X(3307), X(3308), X(4511), X(42014)

X(44782) → X(44785)

other points below

Geometric properties :

See K1238 for general explanations.

K1240 is a cuspidal cubic with cusp X(8) and cuspidal tangent passing thtough X(80) on the Feuerbach hyperbola.

K1240 has three real asymptotes namely those of the Feuerbach hyperbola (F) and a parallel to the line {4,8}.

The tangential of X(517) is Q = (a-b-c) (2 a-b-c) (a^2-a b-2 b^2-a c+4 b c-2 c^2) : : , SEARCH = 7.55181463212915, on the line {8,210}.

K1240 has one real point of inflexion namely S = (a-b-c) (a^2+a b-2 b^2+a c+4 b c-2 c^2) (2 a^2-a b-b^2-a c+2 b c-c^2) : : , on the line {7,8}. It is the barycentric product of X(527), X(5231) and also X(6173), X(6745).

K1240 meets the sidelines of ABC at three real points A', B', C', on the trilinear polar (L) of X(4391) passing through X(1), X(123), and three pairs of always imaginary points.