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X(1), X(105), X(243), X(296), X(518), X(1155), X(1156), X(2651), X(2652)

E = intersection of OI and the antiorthic axis = X(1155)

Foci of the Mandart ellipse (see Generalized Mandart Conics in the Downloads page and also K351)

foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle)

See at the bottom for many other centers on the curve and also Table 29 : Q-Ix-(anti)cevian points.

The Pelletier point is the perspector of the Feuerbach hyperbola : the trilinear polars of all its points pass through it. It is X(650) in ETC.

The Pelletier strophoid K040 is the isogonal circum-strophoid with root at the Pelletier point. Its singular focus is X(105). It is a member of the class CL003 of cubics. Its node is I with tangents parallel to the rectangular hyperbola centered at X(116), isogonal transform of the parallel at O to the line IX(7).

It is also the Hirst transform of the Feuerbach hyperbola under the Hirst transformation with pole X(1) and conic the circum-conic with perspector X(1), with center X(9). See CL030 for a generalization.

K040 is the locus :

1. of foci of inscribed conics whose center is a point on the line IK = X(1)X(6). See also K086, K165, K351 and Z+(O) = CL025.
2. point M such that M and its isogonal conjugate M* are conjugated with respect to the Feuerbach hyperbola.

Let f : M = u:v:w -> M' = a(a^2vw - bcu^2) / ((b-c)(b+c-a)) : : . f maps any point (except I) on K040 to a point on the circumcircle. The reciprocal transformation g of f is defined by

g : M = u:v:w -> M' = [b^2c^2(b-c)^2(b+c-a)^2u^2 - a^4(c-a)(a-b)(c+a-b)(a+b-c)vw]/a : : . It maps any point on the circumcircle to a point on K040. You may want to download a text file containing more than a hundred of centers on K040.