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on both curves : X(2), X(4), X(6), X(13), X(14), X(15), X(16), X(30)

on K1221a : X(397), X(42146), X(43227), X(43228), X(43231), X(43232), X(43235), X(43237)

on K1221b : X(398), X(42143), X(43226), X(43229), X(43230), X(43233), X(43234), X(43236)

other points below

Geometric properties :

K1221a and K1221b are the pivotal KHO-cubics obtained when P = X(397) and P = X(398), with respective KHO-equations :

x^2 (y - z) - 3 x y (y - 2 z) - z (2 y - z) (y + z) = 0

x^2 (y - z) + 3 x y (y - 2 z) - z (2 y - z) (y + z) = 0

See K1191 for explanations and also CL075.

These two cubics belong to the pencil generated by K1207 and the union of the Euler line, Brocard axis, Fermat line or, equivalently, the sidelines of triangle X(3)X(6)X(381).

KHO-points on these curves :

on K1221a : (-1,6,1), (5,6,5), (9,1,2), (9,2,-2), (21,4,-7), (24,-5,36), (33,4,11)

on K1221b : (1,6,1), (-5,6,5), (-9,1,2), (9,-2,2), (21,-4,7), (24,5,-36), (-33,4,11)

Q1 = (9,1,2) = 3√3 a^2 + 2S : : and Q2 = (-9,1,2) = 3√3 a^2 - 2S : : lie on the line X(2), X(6) and the parallels to the Euler line at X(397), X(398) respectively.

Note that the polar conics of the pivots X(397), X(398) in the respective cubics K1221a, K1221b coincide and pass through X(13), X(14), X(15), X(16), X(397), X(398).