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X(3), X(4), X(54), X(1342), X(1343), E(664)

vertices of the circumnormal triangle

infinite points of the Napoleon cubic K005

see below for more details and also Table 25

The McCay cubic K003 and the Lemoine cubic K009 are two circum-cubics passing through O, H and the vertices of the circumnormal triangle. Both cubics meet the sidelines of ABC at three points which are the vertices of a cevian triangle (that of O for K003 and that of G for K009). These two cubics generate a pencil of cubics which contains a third cubic with the same properties and this cubic is K361.

K361 is the isogonal transform of K026, the (first) Musselman cubic or KN++. Recall that the Lemoine cubic K009 is the isogonal transform of K028, the (third) Musselman cubic.

K361 is also psK(X54, X95, X3) in Pseudo-Pivotal Cubics and Poristic Triangles and spK(X5, X140) in CL055. See also Table 54 where K361 is mentioned in one line and three columns showing that it belongs to four pencils of cubics generated by :

• pK(X6, X140) and K187, line Q = X140,

K002 and K028, column P = aQ,

K003 and the union of the Euler line with the circumcircle, column P = [X3], already mentioned above,

K006 and K080, column P = S.

***

K361 contains :

  1. A, B, C with tangents the symmedians of ABC
  2. the vertices of the cevian triangle of X(95), the isotomic conjugate of the nine point center
  3. O with tangent the Euler line
  4. H with tangent passing through X(64), isogonal conjugate of X(20)
  5. X(54) with tangent passing through O
  6. the vertices of the circumnormal triangle with tangents also passing through O. In fact, the polar conic of O in the cubic is the rectangular hyperbola (H) which contains O, X(54), X(110), X(182), X(1147), X(1385), X(2574), X(2575), the vertices of the circumnormal triangle. Its center is X(1511), the midpoint of OX(110) and its asymptotes are parallel to those of the Jerabek hyperbola.
  7. the infinite points of the Napoleon cubic K005
  8. E(664), the isogonal conjugate of E(626) which is the cevian quotient of G and X(53) (symmedian point of the orthic triangle)
  9. the isogonal conjugates of the midpoints of AH, BH, CH (these are the third points on the cevian lines of O)
  10. the isogonal conjugates of the midpoints of GaH, GbH, GcH where GaGbGc is the antimedial triangle
  11. the isogonal conjugates of the X3-OAP points, see Table 53

***

Locus properties :

K361 is the locus of the pivots of circumnormal pKs i.e. pKs passing through the vertices of the circumnormal triangle. With X(3), we obtain the McCay cubic and with X(54), the cubic is K373. See also Table 25.

The loci of the poles and isopivots of such cubics are K378 and K405 respectively.