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K1199

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X(2), X(6), X(15), X(16), X(30), X(41100), X(41101)

Geometric properties :

K1199 is a KHO-cubic. See K1191 for explanations and also CL075. See K1197, K1198, other Cullen cubics.

Its KHO-equation is : x^2 (2y + z) + (2y - z) (y + z)^2 = 0.

The KHO-equation of its Hessian is : x^2 (2y + 3z) - 3 (2y + z) (y + z)^2 = 0. This cubic passes through X(6), X(15), X(16), X(20), X(30), X(3070), X(3071), X(16964), X(16965).

K1199 is an acnodal cubic with an isolated point X(30) at infinity. The tangents at X(30) are imaginary and meet the Brocard axis at the imaginary foci of the Brocard inellipse. These foci also lie on the Kiepert hyperbola and their KHO-coordinates are (±i √3, 0, 1).

X(6), X(15), X(16) are three points of inflexion on K1199.

The inflexional tangent at X(6) meets the Euler line (which is the harmonic polar) at X(20).

The inflexional tangents at X(15), X(16) concur at X(5) on the Euler line. The harmonic polars are the parallels at X(62), X(61) respectively to the Euler line.

All these properties are very similar to those of K1198.