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X(3), X(110), X(136), X(523)

midpoints of ABC

K567 is a circular cubic with focus X(1511). See Pseudo-Pivotal Cubics and Poristic Triangles.

The isogonal transform of K567 is K568. These two cubics belong to a same pencil of circular cubics generated by pK(X6, X523) and the union of the line at infinity and the circum-conic with perspector X(187).

Construction :

let (L) be a variable line passing through O and (C) the circum-conic passing through X(523) and the infinite point of (L). The perpendicular (L') at O to (L) meets (C) at two points on K567.

The figure shows these points when (L') is the Euler line and the Brocard axis.

See also Table 46.