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K1203

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X(3), X(6), X(17), X(18), X(397), X(398), X(485), X(486)

other points below.

Geometric properties :

K1203 is a cuspidal KHO-cubic. See K1191 for explanations and also CL075. K1202 is another similar cubic.

Its KHO-equation is : x^2 (y + 2z) - 9y^3 = 0.

K1203 has a cusp at O with cuspidal tangent the Euler line and a point of inflexion at K with inflexional tangent passing through X(382).K1203 meets the Kiepert hyperbola at X(17), X(18), X(485), X(486) and two imaginary points on the line X(6)X(30) with KHO-coordinates (±3i,1,-1).

Q1, Q2 lie on the cubic, on the line X(6)X(631), and their KHO-coordinates are (±1,1,4).

The tangentials of :

• X(397), X(398) are X(17), X(18).

• X(17), X(18) are (±3,4,30).

• X(486), X(485) are (±√3,2,11).

• Q1, Q2 are (±1,2,35).

• (±3,5,60) are (±3,10,495).

***

More generally, if u : v: w are the KHO-coordinates of a point T (different from O and K) on K1203, then the point

T(n) = (2 (-1)^n u^3 , 2^(1 + n) u^2 v , 2^n v (-u + 3 × 2^n v) (u + 3 × 2^n v))

is its nth tangential. Obviously, T(0) = T and T(n + 1) is the tangential of T(n). Conversely, T(n - 1) is the pretangential of T(n).