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X(4), X(6), X(4846), X(8743)
vertices of the Grebe triangle
infinite points of the McCay cubic
F, F' : foci of the in-conic with center X(182)
imaginary foci of the Brocard ellipse i.e. common points of the Brocard axis and the Kiepert hyperbola
K643 is the only circum-stelloid passing through the vertices of the Grebe triangle. Its asymptotes are parallel to those of the McCay cubic K003 and concur at X = X(2)X(51) /\ X(4)X(182) /\ X(5)X(6), etc, which is actually the centroid of the triangle OHK. This point is now X(14561) in ETC (2017-09-28).
K643 meets its asymptotes on the line X(26)X(32), the satellite line of the line at infinity.
K643 is also spK(X3, X182) in CL055.
K643 is a member of several pencils of circum-cubics, in particular those generated by :
More generally, if P is a point on the Euler line, Q the midpoint of OP, S the reflection of K about Q, then K643 is in the pencil generated by the union of the line K, P and (O), spK(S, Q) of CL055. See also Table 57.