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X(2), X(3), X(4), X(6), X(9), X(40), X(57), X(84), X(14550), X(14551), X(14552), X(14553)

points described below

K384 is an isogonal nK passing through G, O, H, K. Its root is X(14544), the intersection of the lines X(100)X(934) and X(107)X(110), these four points on the circumcircle. See Table 30 for other cubics of the same type. It is a member of the class CL061.

The polar conic of K meets the cubic at K (twice), O, H, X(9), X(57) hence the tangents to the cubic at O, H, X(9), X(57) pass through K. The diagonal triangle of this latter quadrilateral is X(2) X(40) X(84). It follows that K384 is a pivotal cubic with pivot K, isopivot M2 with respect to this diagonal triangle.

M2 is the tangential of K, a point lying on the lines X(2)X(1901), X(6)X(2360), X(37)X(40).

Similarly, K384 contains E1 = GX(40) /\ KX(84), E2 = GX(84) /\ KX(40), E3 = GK /\ X(40)X(84) and the polar conic of M2 meets the cubic at M2 (twice), G, K, X(40), X(84). Thus, K384 is also a pivotal cubic with pivot M2 with respect to the triangle E1 E2 E3.

These points E1, E2, E3, M2 are now X(14550), X(14551), X(14552), X(14553) in ETC (2017-09-27).