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K1226

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X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(41963), X(41964), X(42940), X(42941)

X(43364) → X(43387)

other points below

Geometric properties :

K1226 is a crunodal KHO-cubic related to K1207, see K1191 for explanations and also CL075.

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

H is a node with nodal tangents passing through X(371), X(372) on the Brocard axis and X(485), X(486) on the Kiepert hyperbola.

K is a point of inflexion with tangent passing through X(3091).

K1226 meets the Evans conic at X(13), X(14), X(15), X(16), X(41963), X(41964).

The KHO-points (-4,9,2) and (4,9,2) are the common tangentials of X(13), X(15) and X(14), X(16) respectively.

Parametrization :

For any real (sometimes complex) number t or infinity (giving X2), the KHO-point P(t) = (2 (3 t^2 - 1), 3 t (t^2 - 1), 2 t (3 t^2 - 1)) lies on K1226. 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) / (3 t1 t2 + 1).

Hence, the tangential of P(t) is P(T) where T = - 2 t / (3 t^2 + 1).