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X(3), X(4), X(4240) = Z6

isogonal conjugate of X(4240), the common tangential of O and H

projections of O on the altitudes

points at infinity of ABC sidelines

We meet this cubic in the FG paper "The Lemoine cubic and its generalizations" §6.3 (see the Downloads page)

It is an isogonal nKwith root at G.

If (C) is the homothetic of the polar circle under h(H,1/sqrt(2)), KH is the locus of point M whose pedal circle is orthogonal to (C).