X(3), X(4), X(64)
infinite points of the altitudes
A', B', C' : common points of the Orthocubic K006 and the circumcircle
midpoints of ABC
pedals of H with respect to A'B'C'
foci of the inconic with center O, perspector X(69)
more generally, all the points of the Orthocubic with respect to A'B'C' and in particular the in/excenters of A'B'C' whose reflections about O lie on the Stammler hyperbola
The two triangles ABC and A'B'C' share the same orthocenter, circumcenter hence the same Euler line and centroid. The orthocubic of A'B'C' is K376, the Orthocubic's sister. See other analogous cubics in Table 58.
The tangents at A', B', C' and H pass through O. The polar conic of O is the rectangular hyperbola (H) passing through X(3), X(4), X(110), X(155), X(1351), X(1352), X(2574), X(2575) and obviously A', B', C'.
The tangents at A, B, C, H pass through X(25) and the polar conic of X(25) is the rectangular circum-hyperbola through X(378).
See also another characterization and a generalization at Q098.
From Table 54, it can be seen that K376 belongs to the pencils generated by :
• K004 and the union of the Jerabek hyperbola with the line at infinity.
K376 also belongs to the pencils generated by :
• K006 and the union of the circumcircle with the Euler line,
More generally, K376 and one cubic of the Euler pencil generate a pencil which always contains a circular cubic, a stelloid, a central cubic with center O, a psK tangent at A, B, C to the symmedians.