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X(2), X(6), X(5652)

X, Io see below

3 points on the circumcircle and on the Thomson cubic K002, the vertices of the Thomson triangle

3 points on the circumcircle and on the Grebe cubic K102, the vertices of the Grebe triangle

3 points at infinity of Kjp = K024

This cubic is related to Clark Kimberling's Z+(L) cubics seen in TCCT p. 241. See also Z+(IK) and Z+(O).

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 f : R -> X is an involution and a conjugation on K138.

With R = G, we obtain K082 (asymptotes concurring at K) and with R = K, we obtain K024 (asymptotes concurring at G).

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 circum-center is the (orange) centroid of the triangle GOK and whose circum-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.

These four points lie on K881, K886, K887.

Two points on a same axis of the Steiner inellipse are Psi-inverses. See K018 for instance.


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).


K138 also contains the three singular points Q1, Q2, Q3 of the involution f mentioned above.

These are the vertices of an equilateral triangle homothetic to the Morley triangle (and to the triangle bounded by the asymptotes).

Its circumcircle passes through X(111) and is tangent at this point to the circumcircle of ABC. Its circum-center Q lies on the lines X(2)X(6), X(3)X(111).