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K804

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X(2), X(3), X(4), X(3167), X(7612), X(10002)*

Q1, Q2, Q3 vertices of the Thomson triangle

A'B'C' pedal triangle of X(2)

infinite points of K006

O1, O2 real foci of T(X3) as below

foci of the inconic with center X(381)

K078 and K463 have nine identified common points namely X(2), X(3), Q1, Q2, Q3 and the four foci of the conic T(X3) with center X(3) inscribed in the Thomson triangle. Recall that K078 is a stelloid and that K463 is a focal cubic which are respectively K003 and K187 for the Thomson triangle.

They generate a pencil of cubics which contains one circum-cubic namely K804.

K804 also belongs to the pencils generated by :

K002 and the union of the circumcircle with the Euler line, also containing K581, K615, K759. Each cubic of this pencil is spK(P, Q) with P on the Euler line and Q = midpoint(X2, P). See CL055.

K006 and the union of the Jerabek hyperbola with the line at infinity, also containing K009, K443. Each cubic of this pencil is spK(X4, Q) with Q on the Euler line.