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X(2), X(4), X(6), X(23), X(111), X(524), X(671), X(895), X(10415)

cevian points of X(671)

K273 is a circular pK with pole X(111), the Parry point, and pivot X(671), a point on the Steiner ellipse which is the isotomic conjugate of the isogonal conjugate of X(111). K273 is tangent at A, B, C to the symmedians. The singular focus F is X(11258), the reflection of O in X(111).

K273 is the isogonal transform of K043, the Droussent medial cubic, and also the antigonal transform of K008, the Droussent cubic.

K273 is a member of the class CL043 : it meets the circumcircle at A, B, C and three other points where the tangents are concurrent. Here, these points are X(111) and the circular points at infinity hence the isotropic tangents and the tangent at X(111) meet at F.

See a related pencil in K1156.

Locus properties :

  1. Consider a point M and let M' be the crossconjugate of K and M. The line MM' passes through a fixed point Q if and only if M lies on a pK which is circular if and only if Q = X(23). In this case, the pK is K273. (from an idea by Philippe Deléham)
  2. Locus of pivots of circular pKs which pass through K and also X(111). This is the case of K273 itself and also K043, K108.
  3. Locus of P such that P, X(69) and the DF-pole of P are collinear. See CL039.
  4. See also Table 46.