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K1200

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X(2), X(5), X(6), X(13), X(14), X(2023), X(8787), X(16966), X(16967), X(24256), X(31414), X(39641), X(39642)

X(39641), X(39642) are the imaginary foci of the Brocard inellipse, on the Brocard axis and the Kiepert hyperbola

X(42472) → X(42481), X(42534) → X(42536)

Geometric properties :

K1200 is a KHO-cubic. See K1191 for explanations and also CL075. See K458, a similar cubic.

Its KHO-equation is : x^2 z - 3 (2y - z) (y - z)^2 = 0 and its Hessian is : x^2 (6y - 5z) - 3 z (y - z)^2 = 0 which passes through X(4), X(5), X(6).

K1200 is a crunodal cubic with node X(5). The nodal tangents are the lines through X(590), X(3071) and X(615), X(3070), these latter four points on the Evans conic.

X(6) is a point of inflexion with tangent passing through H and harmonic polar the Euler line. It follows that a line passing through X(6) and meeting the Euler line at E must meet K1200 at two points which are harmonic conjugates with respect to X(6) and E. The tangents at these two points meet on the Euler line.

K1200 and its Hessian meet at X(6) – counting for six – X(6) and two remaining imaginary points of inflexion on the line X(6)X(140). Note that the polar conic of X(5) in these two cubics is the same.

The polar conic of X(4) in K1200 splits into the line X(5)X(6) and the Napoleon axis which meets K1200 at X(6), X(16966), X(16967). These latter two points are harmonic conjugates with respect to X(6) and X(1656).