Home page | Catalogue | Classes | Tables | Glossary | Notations | Links | Bibliography | Thanks | Downloads | Related Curves

X(3), X(4), X(5),

E(626) = X(4)X(51) /\ X(800)X(1990), the tangential of H

its reflection in X(5)

midpoints of BC, CA, AB and AH, BH, CH

reflections of A, B, C in X(5) i.e. centers of the Johnson circles

points on the circumcircle and on the Napoleon cubic.

infinite points of the McCay cubic

We meet this K60++ cubic in a paper by Musselman (see bibliography) and in a totally different context in Special isocubics ยง6.5.3.

K026 is a central equilateral cubic with center X(5), the nine point center, with inflexional tangent X(5)X(51) i.e. the Euler line of the orthic triangle.

Its three asymptotes are perpendicular to the sidelines of the Morley triangle and parallel to those of the McCay cubic.

It is the homothetic of K080 = KO++ under h(H,1/2) and the isogonal transform of K361.

K026 is also psK(X51, X2, X3) in Pseudo-Pivotal Cubics and Poristic Triangles.

Locus properties :

  1. Denote by A1, B1, C1 the reflections of P in A, B, C and by A2, B2, C2 the reflections of P in the sidelines BC, CA, AB. The three circles AB1C1, BC1A1, CA1B1 have a common point D on the circumcircle. The three circles AB2C2, BC2A2, CA2B2 have also a common point N on the circumcircle. These points D, N coincide if and only if P lies on the cubic K026 (Musselman, Some loci connected with a triangle. Monthly, p.354-361, June-July 1940). On the other hand, they are antipodes if and only if P lies on the cubic K226 we call Musselman (fourth) cubic.
  2. K026 is also the locus of centers of central pKs whose asymptotes are perpendicular to the sidelines of the Morley triangle.
  3. K026 is the locus of the centers of central focal circum-cubics passing through H. See K530 (center H) and K465 (center X5) for instance. See the related quartics Q019 and Q089.
  4. K026 is the locus of the centers of central equilateral circum-cubics passing through H. It is the case of K026 itself (center X5), K080 (center X3) and K525 (center X4).
  5. See Table 33 where K026 is the cubic Lk when k = 1/2.