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too complicated to be written here. Click on the link to download a text file. |
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X(4), X(6), X(13), X(14), X(15), X(16), X(20), X(395), X(396), X(13886), X(13939) other points below |
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Geometric properties : |
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K1227 is another acnodal KHO-cubic analogous to K1225, see K1191 for explanations and also CL075. Its KHO-equation is : x^2 (2 y - 9 z) + 9 z^2 (2 y + z) = 0. H is a node with imaginary tangents passing through the KHO-points (± 3i ,0,1). K1227 has three real points of inflexion : K with tangent passing through (0,9,2), X(13886) and X(13939) with tangents passing through X(11541). See K458 for analogous properties. The tangents at X(13), X(14) meet at X(20) and the real asymptote is the parallel at K to the Euler line. K1227 meets the Kiepert hyperbola at H (twice), X(13), X(14) and the two KHO-points K5, K6 = (±2√6,3,2) on the line X(6)X(3091). K1227 meets the Evans conic at X(13), X(14), X(15), X(16) and the two KHO-points E5, E6 = (±9√11,19,6).
For any real (sometimes complex) number t or infinity (giving X20), the KHO-point P(t) = (2 (9 t^2 + 1), 9 t (1 - t^2), 2 t (9 t^2 + 1)) lies on K1227. 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) / (9 t1 t2 - 1). Hence, the tangential of P(t) is P(T) where T = 2 t / (9 t^2 - 1). |