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X(2), X(6), X(13), X(14), X(15), X(16), X(111), X(368), X(524), E(458)=X(368)*

vertices of the second and fourth Brocard triangles

foci F1, F2 of the inscribed Steiner ellipse.

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

other points below

K018 is a circular isogonal focal nK with root X(523) and singular focus X(111). The real asymptote is parallel to GK. It is also the orthopivotal cubic O(X6) and Z+(L) with L = X(3)X(6) in TCCT p.241. See also Z+(O) = CL025 and CL034. K018 is spK(X524, X2) in CL055.

Locus properties

  1. Locus of point M such that the line angles (MX13,MX14) and (MX15,MX16) or (MX13,MX15) and (MX14,MX16) are equal. In other words, the Fermat and isodynamic points are seen from M under equal or supplementary angles.
  2. Locus of contacts of tangents drawn from X(111) to the circles passing through G and K.
  3. Locus of foci of inscribed conics whose center lies on the line GK, hence the cubic passes through the foci of the inscribed Steiner ellipse and also those of the inscribed conic with center K.
  4. Locus of point M such that the pole of the line MM* in the circum-conic through M and M* lies on the Brocard axis OK.
  5. Locus of point M such that K, M and the orthocorrespondent of M are collinear.
  6. Locus of point M such that the three circles MBC, MCA, MAB meet the sidelines of triangle ABC again at six points lying on a same conic.
  7. (equivalently) Locus of point M such that if the circle MBC meets AB, AC again at Ab, Ac and if A' = BAc /\ CAb (B', C' similarly) then ABC and A'B'C' are perspective. The perspector also lies on the curve. Since the line AA' contains the pole A" of BC in the circle MBC, another formulation is the following : the poles A", B", C" of BC, CA, AB in the circles MBC, MCA, MAB form a triangle perspective with ABC if and only if M lies on K018.
  8. Locus of point M such that if MB, MC meets the Kiepert hyperbola again at B', C' then the lines AP*, B'C', OK are concurrent.
  9. The cevian lines of point P meet the perpendicular bisectors of ABC at A', B', C'. The locus of P such that the sum of oriented lines angles (BA',BC)+(CB',CA)+(AC',AB) = 0 (mod. pi) is K018. When this sum is pi/2 (mod. pi), we obtain the Napoleon cubic. (Jean-Pierre Ehrmann)
  10. Locus of point M such that M and M* are conjugated with respect to the Kiepert hyperbola. See also Isogonal nK0 cubics and table 4.
  11. For any point P, the locus of the perspectors of the equilateral triangles centered at P which are perspective with ABC is the orthopivotal cubic O(P). When P = K, we find K018.
  12. Let P be a point and Pa, Pb, Pc its inversive images in the Apollonian circles C_a, C_b, C_c respectively. ABC and PaPbPc are perspective if and only if P lies on the circumcircle or on the Apollonius quartic Q049. The inversive image of Q049 in the circumcircle is K018.

 

Miscellaneous properties

K018b

The polar conic of X(110) decomposes into the Brocard axis and the trilinear polar of the root X(523). It follows that the six tangents at K, X(15), X(16) and the three points on the sidelines of ABC concur at X(110).

The common tangential of X(13) and X(15) is T1 and the common tangential of X(14) and X(16) is T2. These two points are collinear with the singular focus X(111).

The third points on X(13)X(15) and X(14)X(16) are N1, N2 respectively also collinear with X(111). The midpoint of N1N2 is X(230) on GK.

The lines T1N2 and T2N1 are parallel to the asymptote.

The tangents at X(13) and X(14) met at the reflection of X(110) about G.

The polar conic of X(111) is the circle through G, O, K, X(691) and naturally X(111). The tangent at X(111) to this circle meets the asymptote at X on the cubic. Recall that this asymptote is the homothetic of the axis GK under h(X111, 2).

The polar conic of the point at infinity X(524) – that of the line GK – is a rectangular hyperbola passing through X(597), having one asymptote perpendicular at X(111) to GK and the other that of the cubic.

The bisectors of the lines X(111)X(524) and X(111)X contain X(2469), X(2470) and the centers of anallagmaty E1, E2 which obviously also lie on the previous rectangular hyperbola.

The cubic is then invariant under the two inversions with pole Ei which swap X(111) and Ej.

K018 is also invariant under the Psi transformation which is the product of the reflection about one axis of the Steiner inellipse and the inversion with circle that of diameter F1F2, the foci of the ellipse. See also "Orthocorrespondence and Orthopivotal Cubics", §5 and K022.

Psi is the involution that swaps any point of the plane to the center of its polar conic in the McCay or Kjp cubics. In particular, Psi swaps A, B, C and the vertices of the second Brocard triangle.

K018d
K018cX368

 

K018 contains the equi-Brocard center X(368), its isogonal conjugate X(368)* and its Psi image.

Recall that X(368) also lies on the Wallace hyperbola (the anticomplement of the Kiepert hyperbola) and on the three equibrocardian focal cubics K083.