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

X(2), X(4), X(7), X(8), X(20), X(69), X(189), X(253), X(329), X(1032), X(1034),

E(623), E(624) : isotomic conjugates of X(1034), X(1032)

E(625) = X(69)-Ceva conjugate of X(253)

E(636) : isotomic conjugate of E(625)

vertices of antimedial triangle

points at infinity of the Thomson cubic

foci of the Steiner circum-ellipse (see below)

four CPCC points, see Table11.

seven central cyclocevian points, see Table 24.

See also Table 28 : cevian and anticevian points.

The Lucas cubic is the isotomic pK with pivot X(69), isotomic conjugate of H. It is the Thomson cubic of the antimedial triangle. Its isogonal transform is K172. It is a member of the classes CL023, CL024 of cubics. See also Table 15. It is anharmonically equivalent to the Thomson cubic. See Table 21.

K007 meets the circumcircle at A, B, C and three other points on the rectangular hyperbola through X(2), X(20), X(54), X(69), X(110), X(2574), X(2575), X(2979).

Locus properties :

  1. Locus of point P whose cevian triangle is a pedal triangle.
  2. Locus of point P whose cevian triangle is orthologic to ABC. The locus of the centers of orthology is the Darboux cubic. See a generalization at CL023.
  3. Locus of point P whose cevian triangle is orthologic to the medial triangle. The locus of the centers of orthology is the Darboux cubic and its complement. See a generalization at Table 7.
  4. Locus of point P whose cevian triangle is orthologic to the antimedial triangle. The locus of the centers of orthology is the Darboux cubic.
  5. PaPbPc is the cevian triangle of (finite) point P. For any real k, the homothety h(P,k) maps PaPbPc to QaQbQc. The triangles QaQbQc and ABC are orthologic (for all k) if and onky if P lies on the Lucas cubic. (Fred Lang, Hyacinthos #5849).
  6. Locus of point P such that the cevian triangles of P and its isotomic conjugate P' are orthologic. The locus of the centers of orthology is K401.
  7. Let P be the perspector of an inscribed conic (C) with center Q. The circum-conic (C') passing through P and Q is a rectangular hyperbola if and only if P lies on the Lucas cubic.
  8. HaHbHc is the orthic triangle and MaMbMc the cevian triangle of M. Da is the line passing through Ma and the midpoint of AHa, Db and Dc similarly. These lines are concurrent (at Q) if and only if M lies on the Lucas cubic. The locus of Q is the Thomson cubic. (Philippe Deléham, 17 nov. 2003) More generally, if HaHbHc is replaced by the cevian triangle of point P, the locus of M is pK(X2,tP) where tP is the isotomic conjugate of P, and the locus of Q is pK(ctP, X2) where ctP is the complement of tP. For example, with P = X(67), we find K008 and K043 respectively and with P = X(69), we find K170 and K168 respectively.
  9. The cevian lines of P meet the Steiner ellipse again at A', B', C'. The pedal triangle of P and A'B'C' are perspective if and only if P lies on K007. See also CL024.
  10. Locus of point P such that the trilinear polar of P is perpendicular to the line PL where L = X(20) = de Longchamps point. See CL040 for a generalization.
  11. Locus of the {X}-cevian points where X is a center on the Thomson cubic. See Table 28 : cevian and anticevian points.
  12. Let (C) be the inconic with perspector P and H(A), H(B), H(C) the Apollonius hyperbolas of A, B, C with respect to (C). The centers of these hyperbolas form a triangle perspective to ABC if and only if P lies on the Lucas cubic (Angel Montesdeoca, 2010/03/03).
  13. If A'B'C' is the cevian triangle of X then K007 is the locus of X for which BA'^2 + CB'^2 + AC'^2 = A'C^2 + B'A^2 + C'B^2. Compare with K034 and K200.
  14. Let A' B' C' be the cevian triangle of a point P. The circumcircles (Oa), (Ob), (Oc) of AB'C', BC'A', CA'B' are concurrent at M. Denote by  O* the circumcenter  of OaObOc. K007 is the locus of point P such that M, O*, O are collinear (or M is the reflection of O in O*). When P varies on the Lucas cubic, the locus of M is the Darboux cubic K004 (Angel Montesdeoca, private message 2013-08-06). See the related cubic K279.
  15. See K645.

 

Transformations preserving the Lucas cubic

The Lucas cubic is invariant under isotomic conjugation and under several other transformations. Its is indeed the locus of point P such that :

  1. X(69), P and its isotomic conjugate P' are collinear (since X(69) is the pivot),
  2. X(4), P and the X(69)-Ceva conjugate are collinear (since X(4) is the isopivot),
  3. E(624), P and the X(4)-crossconjugate of P are collinear (since E(624) is the X(69)-Ceva conjugate of X(4)),
  4. X(1032), the X(69)-Ceva conjugates of P and P' are collinear,
  5. X(20), P and the cyclocevian conjugate of P are collinear,
  6. X(2), P and the anticomplement of the isogonal of the complement of P are collinear (derives from the definition of the Thomson cubic K002).

 

The Lucas cubic and the foci of the Steiner ellipse

The Lucas cubic is the most familiar example of isocubic which contains the (four) foci of the Steiner (circum) ellipse. See also K347 and K348. Indeed, these foci are the anticomplements of those of the inscribed Steiner ellipse which lie on the Thomson cubic since they are two isogonal conjugates colinear with the pivot G.

Let then be F1, F2 the real foci and F1', F2' the imaginary foci of the Steiner ellipse. Their isotomic conjugates tF1, tF2, tF1', tF2' obviously also lie on the Lucas cubic. Furthermore :

  • the line tF1-tF2 is the parallel at L = X(20) to the focal axis F1-F2 (this parallelism is true for any two points symmetric in G),
  • the lines F1-tF2 and F2-tF1 meet at Z1, the isotomic conjugate of the infinite point E530 of the focal axis, a point on the Steiner ellipse,
  • the lines H-F1 and G-tF2 meet at Y1 = X(69)/F1 (cevian quotient) on the cubic and similarly the lines H-F2 and G-tF1 meet at Y2 = X(69)/F2,
  • Y1, Y2, X(253) = tX(20) are colinear on the cubic,
  • tY1 = L-F1 /\ X(69)-Y2 and tY2 = L-F2 /\ X(69)-Y1 also lie on the cubic,
  • the lines G-X(253) and tY1-tY2 meet at E(624) = X(69)/H = tX(1032),
  • etc, etc.

Naturally, all these results can easily be adapted to the imaginary foci.

More informations in Wilson's page.