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X(2), X(6), X(98), X(182), X(187), X(385), X(598)

foci of the inscribed Steiner ellipse

imaginary common points of the circumcircle and the Lemoine axis

K705 is another example of circum-cubic passing through the center and the foci of the Steiner inscribed ellipse.

It is the isogonal transform of K353.

K705 is spK(P, X2) where P is the reflection of X(385) about X(2). See CL055 and also Table 48.