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X(2), X(67), X(98), X(524)

K394 is a member of the class CL045 of tripolar centroidal cubics. It is the only circular cubic of this type. The singular focus is X(14666).

It is the locus of point M whose tripolar centroid lies on the trilinear polar of X(67).


K394 is the Psi-image of the conic passing through G, X(1319) and the vertices of the second Brocard triangle.

Psi is the involution that is the product of a symmetry about one axis of the Steiner inellipse and the inversion in the circle with diameter F1F2 (the foci of the ellipse).

More details in "Orthocorrespondence and Orthopivotal Cubics", §5.

It follows that for any point M on the conic, the bisectors of the lines G M and G Psi(M) are the axes of the ellipse.

Examples of such pairs of lines :

– Euler line and G X(98) X(110),

– lines GK and G X(99) X(111),

– lines through G and the Fermat points,

– lines through G and the isodynamic points.