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K1202

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X(3), X(6), X(13), X(14), X(395), X(396), X(485), X(486), X(42087), X(42088)

other points below

Geometric properties :

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

Its KHO-equation is : x^2 (5y - 2z) - 9 y^3 = 0.

K1202 has a cusp at O with cuspidal tangent the Euler line and a point of inflexion at K with inflexional tangent passing through X(3526). The Hessian of K1202 splits into the Brocard axis and the Euler line counted twice. The polar conic of X(30) is the Brocard axis counted twice.

K1202 meets the Kiepert hyperbola at X(13), X(14), X(485), X(486) and two imaginary points on the line X(6)X(140) with KHO-coordinates (± 3i, 1, 3).

Other KHO-points :

• (3, -4, 22) and (3, 4, -22) : tangentials of X(13) and ( X(14). These points (3, 4, -22) and (3, -4, 22) lie on the line X(6)X(15688) and on the lines X(395)X(42087), X(396)X(42088) respectively.

• (√3, -2, 7) and (√3, 2, -7) : tangentials of X(485) and ( X(486).

• (± 3, √5, 0) on the line HK.

• (± 3, √7, -√7) on the line X(6)X(30).

• (3, 5, -50) and (3, -5, 50) on the line X(6)X(10304) and on the lines X(14)X(42087), X(13)X(42088) respectively.

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More generally, if u : v: w are the KHO-coordinates of a point T (different from O and K) on K1202, then the point

T(n) = (2 (-1)^n u^3 , 2^(1 + n) u^2 v , 2^n v (5 u^2 - 9 × 2^(2 n) v^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).