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K1229

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X(4), X(6), X(13), X(14), X(15), X(16), X(550)

X(43416) → X(43439)

other points below

Geometric properties :

K1229 is another acnodal KHO-cubic analogous to K1225 and K1227, see K1191 for explanations and also CL075.

Its KHO-equation is : x^2 (y - 9 z) + 9 z^2 (3 y + z) = 0.

H is a node with imaginary tangents passing through the KHO-points (± 3√3i ,0,1).

K1229 has three real points of inflexion : K with tangent passing through (0,9,1), X(13) and X(14) with tangents passing through X(3146). See K458 for analogous properties.

The tangents at X(15), X(16) meet at (0,9,-14) and the real asymptote is the parallel at K to the Euler line.

K1229 meets the Kiepert hyperbola at H (twice), X(13), X(14) and the two KHO-points K5, K6 = (±√15,3,1) on the line X(6)X(546).

K1229 meets the Evans conic at X(13), X(14), X(15), X(16) and the two KHO-points E5, E6 = (±27√2,23,3).

Parametrization :

For any real (sometimes complex) number t or infinity (giving X550), the KHO-point P(t) = (27 t^2 + 1, 9 t (1 - t^2), t (27 t^2 + 1)) lies on K1229. This gives a lot of simple points on the curve. Note that X(6), P(t), P(-t) are collinear.

The third point of the cubic on the line passing through P(t1), P(t2) is P(t3) where t3 = (t1 + t2) / (27 t1 t2 - 1).

Hence, the tangential of P(t) is P(T) where T = 2 t / (27 t^2 - 1).

KHO-points on the cubic :

(27,20,3), (-27,20,3), (81,26,3), (-81,26,3) on the lines passing through X(550) and X(13), X(14), X(15), X(16) respectively.

(±6,5,1) and (±15,8,1) remaining points on the sidelines of the quadrilateral X(13), X(14), X(15), X(16).

(±91,54,7) tangentials of X(15), X(16).

(±5√3,3,5), (±39,10,26).