X(6), X(53), X(216), X(1249)
midpoints of ABC
K307 is the locus of poles of all pK having the same asymptotic directions as the McCay cubic i.e. having three real asymptotes perpendicular to the sidelines of the Morley triangle. For example, with the poles X(53), X(216) we obtain the McCay orthic cubic K049, K096 respectively.
The tangents at A, B, C to K307 concur at X(51), the centroid of the orthic triangle.
See "Asymptotic Directions of Pivotal Isocubics" in the downloads page.
K307 is also psK(X51, X2, X6) in Pseudo-Pivotal Cubics and Poristic Triangles.