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X(1), X(3), X(4), X(46), X(90), X(155), X(254), X(371), X(372), X(485), X(486), X(487), X(488), E(555), E(556)

vertices of orthic triangle

excenters

six intersections of a perpendicular bisector and the circle with diameter the corresponding side of ABC i.e. the centers of the six squares erected externally or internally on the sides of ABC

A', B', C' on the circumcircle and the midpoints of A'B'C'.

The Orthocubic is the isogonal pK with pivot H = X(4). See Table 27. It is a member of the class CL043 : it meets the circumcircle at A, B, C and three other points A', B', C' lying on the rectangular hyperbola through X(3), X(4), X(110), X(155), X(1351), X(1352), X(2574), X(2575) where the tangents are concurrent at the point X(25). See also Q063. The orthocubic of A'B'C' is K376.

Locus properties :

1. Let P be a point and PaPbPc its pedal triangle. A' is the reflection of P in the line PbPc and B', C' are defined similarly. The triangles ABC and A'B'C' are perspective if and only if P lies on the orthocubic (together with the line at infinity and the circumcircle).
2. Let A'B'C' be the prepedal/antipedal triangle of P and A''B''C'' the reflections of P through the sides of ABC. The locus of P such that A'B'C' and A''B''C'' are perspective is the orthocubic (together with the line at infinity and the circumcircle). (Floor van Lamoen)
3. Locus of point P such that P, GSC(P), X(155) are collinear. (X(155) is the orthocenter of the tangential triangle). GSC is defined here.
4. Locus of point P such that the reflection triangles of P and P* (isogonal conjugate of P) in the sidelines of ABC are perspective.
5. The perpendicular at P to BC meets the line AP* at A'. La is the parallel at A' to AP. Define B', C', Lb, Lc similarly. These lines La, Lb, Lc concur if and only if P lies on the union of the McCay cubic and the orthocubic. Now, if La is the parallel at A' to BC and if Lb, Lc are defined likewise, the lines concur if and only if P lies on the quartic Q023 (Floor van Lamoen and friends, Hyacinthos #5360 & sq.).

Other properties :

The centers of the three osculating circles at A, B, C to this cubic are collinear on the trilinear polar of X(847).

More generally, the locus of the pivots of all isogonal pK with the same property is the nonic Q031.