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too complicated to be written here. Click on the link to download a text file. 

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 KHOcubic analogous to K1225 and K1227, see K1191 for explanations and also CL075. Its KHOequation is : x^2 (y  9 z) + 9 z^2 (3 y + z) = 0. H is a node with imaginary tangents passing through the KHOpoints (± 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 KHOpoints 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 KHOpoints E5, E6 = (±27√2,23,3). Parametrization : For any real (sometimes complex) number t or infinity (giving X550), the KHOpoint 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). KHOpoints 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).
