X(2), X(6), X(5652)
X see below
3 points at infinity of Kjp = K024
It is an equilateral cubic, locus of roots R of isogonal nK0+ cubics i.e. nK0(X6, R) cubics with concurring asymptotes (at X). When R lies on K138, this point X also lies on K138 hence the mapping R -> X is an involution and a conjugation on K138.
With R = X(5652), the point X is unlisted in ETC with SEARCH = -3.31972055021895 and first barycentric :
a^2 (a^6 b^2+2 a^4 b^4+a^2 b^6+a^6 c^2-10 a^4 b^2 c^2+2 a^2 b^4 c^2-5 b^6 c^2+2 a^4 c^4+2 a^2 b^2 c^4+8 b^4 c^4+a^2 c^6-5 b^2 c^6).
This point is actually the third point of K138 on the line GK.
The three real asymptotes of K138 are parallel to the sidelines of the Morley triangle and form a triangle whose center is the centroid of the triangle GOK and whose radius is 2/3 R.
K138 contains the in/excenters of the triangle T = X(2)X(6)X(111). These are the intersections of the parallels at X(6) to the asymptotes of the Jerabek hyperbola and the parallels at X(2) to the asymptotes of the Kiepert hyperbola. These two latter parallels are the axes of the Steiner inellipse.
In particular, the incenter Io of T is X(2)X(3413) /\ X(6)X(2575) with SEARCH = 1.70769886050969.
The Jerabek hyperbolas JT and JG of the Thomson and Grebe triangles both pass through O and G, K, X(5652), the vertices of the corresponding triangle. These six latter points are their common points with K138.
Recall that X(5652) is the orthocenter of GOK.
K138 meet the sidelines of the Thomson triangle again at three points K1, K2, K3 on the Lemoine axis L(X6) and the sidelines of the Grebe triangle again at three points H1, H2, H3 on the orthic axis L(X4).
K138 meets K024 at three points on the line at infinity and six other finite points which are always imaginary, lying on an imaginay ellipse with center X(1383).