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K643

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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 :

  1. the McCay cubic K003, the union of the line at infinity and the Kiepert hyperbola. See Table 51.
  2. the third Brocard cubic K019, pK(X6, X182).
  3. the Grebe cubic K102, the union of the Brocard axis KO and the circumcircle (O).
  4. K177, the union of the line K, X(5) and (O).
  5. K281, the union of the line K, X(30) and (O).
  6. K644, the union of the line K, X(4) and (O).

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.