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K1233

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X(3), X(4), X(6), X(11542), X(11543), X(42924), X(42925), X(42944), X(42945)

other points below

Geometric properties :

K1233 is a crunodal KHO-cubic, see K1191 for explanations and CL075. See also K1232, a very similar cubic.

Its KHO-equation is (1) : x^2 (2y + z) - 12 y^2 z = 0 or (2) : 3 z^3 - (2y + z) [x^2 - 3z (2y - z)] = 0.

X(3) is a node with two real nodal tangents passing through X(1587) and X(1588).

X(6) is a point of inflexion with tangent passing through X(20) and harmonic polar the Euler line. Note that the polar conic of X(20) splits into the Brocard axis and the line {5,6}. This latter line meets the cubic again at X(11542), X(11543) with KHO-coordinates (±2,1,1).

X(4) is a sextactic point with tangent passing through X(6) and sextactic conic the Kiepert hyperbola.

Parametrization :

For any real (sometimes complex) number t or infinity (giving X4), the KHO-point P(t) = (12 t^2 - 1, t (12 t^2 - 1), 2 t) lies on K1233. This gives a lot of simple points on the curve. Note that P(t), P(-t) and X(6) = P(0) are collinear for every t.