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see the general equation in CL011

X(99), X(805), X(877), X(880), X(892), X(5458), X(17929), X(17930), X(17931), X(17932), X(17933), X(17934), X(17935), X(17936), X(17937)

points at infinity of the sidelines of ABC

K052 = A2(X115) is the locus of centers of conics circumscribed to the antimedial triangle having an asymptote passing through X(99). These conics are actually hyperbolas.

K052 is the X(99)-Hirst inverse of the Steiner circum-ellipse.

K052 is a member of the class CL011 of cubics.

K052 has three real asymptotes which are the parallels at E = X(4590) to the sidelines of ABC.

The complement of K052 is K203. Its isogonal transform is K978 and its isotomic transform is K979.

See here for a family of related cubics.


A generalization by Angel Montesdeoca

Let GaGbGc be the antimedial triangle and Q a fixed point. L(Q) is a variable line passing through Q.

The locus of the center of the hyperbola circumscribed to GaGbGc and having L(Q) as an asymptote is the cubic cK(#Q, X2).

This cubic is also the locus of the intersection S of L(Q) and the circum-parabola whose axis is parallel to L(Q), S being the center of the hyperbola above.

Examples :

K015 = cK(#X2, X2) = nK(X2, X2, X2)

K052 = cK(#X99, X2) = nK(X4590, X2, X99)

K406 = cK(#X4, X2) = nK(X393, X2, X4)