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K1204

too complicated to be written here. Click on the link to download a text file.

X(2), X(6), X(13), X(14), X(17), X(18), X(41945), X(41946)

X(42594) → X(42609)

other points below

Geometric properties :

K1204 is a cuspidal KHO-cubic. See K1191 for explanations and also CL075. K1195 is a similar cubic.

Its KHO-equation is : x^2 (20y - 13z) - 9 (2y - z)^3 = 0.

K1204 has a cusp at G with cuspidal tangent the Euler line and a point of inflexion at K.

Q1, Q2 are the respective tangentials of X(17), X(18) having KHO-coordinates (±3,3,4).

Q1 lies on the lines {X2,X398}, {X5,X13}, {X6,X3090}, {X14,X140}, {X15,X632}, {X16,X546}, {X17,X547}, SEARCH = 2.77542389646872.

Q2 lies on the lines {X2,X397}, {X5,X14}, {X6,X3090}, {X13,X140}, {X15,X546}, {X16,X632}, {X18,X547}, SEARCH = 1.25034051338884.

The tangentials of Q1, Q2 are (±3,2,8), on the line {X6,X631} and on {X3,X13}, {X3,X14} respectively.

K1204 meets the Brocard axis again at (±3,0,√13) and the line HK again at (±6,√10,0).

The tangentials of X(13), X(14) are (±3,23,52).

***

More generally, if u : v: w are the KHO-coordinates of a point T (different from G and K) on K1204, then the point

T(n) = (3 ( - 1)^n 2^(1 - n) u^3 : (2v - w)[13 u^2 - 9 × 4^n (2v - w)^2] : (2v - w)[20 u^2 - 9 × 2^(1 + 2 n) (2v - w)^2])

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).

The limit points are T(+∞) = G and T(-∞) = K.