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X(2), X(99), X(523), X(1113), X(1114), X(3413), X(3414), X(6189), X(6190)

vertices of the antimedial triangle

K242 is the isotomic pK with pivot the Steiner point X(99). It has three real asymptotes : one of them is the perpendicular at G to the Euler line and the other two are the parallels to those of the Kiepert hyperbola at X, barycentric square of X(524). X = X(2482) = (b^2 + c^2 - 2a^2)^2 : : , a point on the Steiner in-ellipse.

K242 meets the circumcircle again at X(1113), X(1114) on the Euler line and the Steiner ellipse at S1, S2 on the line GK. These two latter points are X(6189), X(6190), the isotomic conjugates of the infinite points of the Kiepert hyperbola.

The tangents at A, B, C, X(99) to K242 are parallel and so are the normals at the same points. These normals are parallel to the Euler line. Let us call orthopolar of a point M with respect to the cubic the perpendicular at M to the polar line of M. The locus of M such that this orthopolar is parallel to the Euler line is the circum-conic through X(99) and X(523) whose perspector is X(1648), the tripolar centroid of X(523).

The isogonal transform of K242 is K1067 = pK(X32, X110).

K242 is a member of the class CL042.


A property by Kadir Altintas and Ercole Suppa (personal communication, 2019-03-27)

Let P be any point, DEF the cevian triangle of P, H the orthocenter of ABC.

Na, Nb, Nc are the NPC of AFE, FBD, DEC resp. La is the line through Na parallel to AH and define Lb, Lc cyclically.

The lines La, Lb, Lc concur at a point Q. K242 is the locus of point P such that Q lies on the Euler line.