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Given a pole W of isoconjugation which is not the Lemoine point K, there is one and only one focal (circular) nK with pole W.

Let L be the W-isoconjugate of the circumcircle i.e. the trilinear polar of the isotomic conjugate of the isogonal conjugate of W. Denote by :

  • S the trilinear pole of the line WK (on the circumcircle) and S* its W-isoconjugate (on the line at infinity),
  • P the reflection of S in L and F = P* its W-isoconjugate,
  • R the orthocorrespondent of P,
  • X* the intersection of the lines SS*, PP* and X its W-isoconjugate, antipode of F on the circle PSF.

R is the root of the focal cubic and the trilinear polar of R meets the sidelines of ABC at U, V, W on the cubic.

F is the singular focus and X the intersection of the cubic with its real asymptote, the homothetic of L under h(F,2). The circle with diameter FX passes through P and S. Thus, the lines PF, PX and SF, SX are perpendicular.

The cubic passes through the orthogonal projection E of F on the line SP and D intersection of the lines FS, PS*. D and E are isoconjugate.

The tangents at P, S, F and the real asymptote concur at X hence the polar conic of X is the hyperbola passing through P, S, F, S* and X.

The polar conic of F is the circle passing through F, centered on PS and orthogonal to the circle with diameter PS. In other words, its center is the intersection of PS with the perpendicular at F to FX.

Locus properties :

  1. This cubic is the locus of M such that the circle with diameter MM* passes through P.
  2. A circle with center Z on L passing through P and S intersects the line ZF at M, N on the cubic. The lines MM*, NN* intersect on the perpendicular at S to PS and the following triads of points are collinear on the cubic : F, M, N - F, M*, N* - P, M, N* - P, M*, N.
  3. The contacts M, N of the tangents drawn from F to a circle centered on PS and orthogonal to the circle with diameter PS are two points on the cubic. Moreover, the lines MM* and NN* intersect on DE and the following triads of points are collinear on the cubic : X, M, N - X, M*, N* - X*, M*, N - X*, M, N*.

Other properties :

  1. The most remarkable example of such cubic is K091, the only isotomic focal nK.
  2. This cubic is a nK0 if and only if W lies on K214.
  3. When W lies on the orthic axis, this cubic decomposes into a rectangular circum-hyperbola (P = H) and the line at infinity.
  4. When W lies on the circum-conic with perspector X(184), this cubic decomposes into a line through O (P* = O) and the circum-circle. This circum-conic contains X(112), X(248), X(906), X(1415), X(1576)...
  5. This cubic is an isogonal focal pK with respect to the triangle PSF, the pivot being S*. This shows that is must contains the in/excenters of triangle PSF which are the centers of anallagmaty of the curve. These points obviously lie on the rectangular hyperbola which is the polar conic of S*. Its center is the second intersection of the line XS* with the circle PSF. The asymptotes are therefore the real asymptote of the cubic and the parallel at F to PS.
  6. Any circle with center F meets the cubic at two points collinear with X.

The table gives a short selection of non-isogonal focal nKs (the red point is the singular focus).

Pole

Centers on the cubic

Cubic / remarks

X(2)

X(67), X(99), X(316), X(523)

K091

X(44)

X(8), X(100), X(900), X(1319)

nK(X44, X1997, X8)

X(50)

X(3), X(110), X(186), X(526)

nK(X50, X1993, X3)

X(67)

X(2), X(67), X(525), X(935)

 

X(111)

X(6), X(523), X(671), X(691)

 

X(187)

X(2), X(110), X(187), X(690)

nK(X187, X1992, X2)

X(385)

X(76), X(99), X(804), X(1691)

 

X(524)

X(69), X(99), X(468), X(690)

 

X(2161)

X(1), X(80), X(522), X(2222)

nK(X2161, X2006, X1)

X(2183)

X(8), X(109), X(1457), X(2804)