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X(4), X(6), X(13), X(14), X(15), X(16), X(30), X(485), X(486)

X(43324) → X(43343)

other points below

Geometric properties :

K1225 is an acnodal KHO-cubic related to K1207, see K1191 for explanations and also CL075.

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

H is a node with imaginary tangents passing through X(39641), X(39642), the imaginary foci of the Brocard inellipse which lie on the Brocard axis and the Kiepert hyperbola. These are the KHO-points (±i √3,0,1).

K1225 has three real points of inflexion : K with tangent passing through X(546), X(15) and X(16) with tangents passing through X(3146). See K458 for analogous properties.

The tangents at X(13), X(14) meet at G and the real asymptote is the parallel at K to the Euler line. The tangents at X(485), X(486) are also parallel to the Euler line.

K1225 meets the Evans conic at X(13), X(14), X(15), X(16) and the two KHO-points (±3√6,5,3), on the line X(6)X(3523).

Parametrization :

For any real (sometimes complex) number t or infinity (giving X30), the KHO-point P(t) = (3 t^2 + 1, 3 t (1 - t^2), t (3 t^2 + 1)) lies on K1225. 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).