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2 S^2 xyz + ∑ a^2 y z (c^2 y + b^2 z) = 0,

or equivalently

2 S^2 xyz + ∑ a^2 x (c^2 y^2 + b^2 z^2) = 0,

with S = 2*area(ABC)

centers of the Apollonian circles

points A', B', C' on (O) whose Simson lines S(A'), S(B'), S(C') pass through O

K191 is an isogonal nK with root X(6) comparable to the Kjp cubic K024. See also Table 44.

Locus properties

  1. K191 is the locus of point M whose pedal circle is orthogonal to the circumcircle. See K024 property 3.
  2. Let A'B'C' be the pedal triangle of a point P. Let Ba, Ca be the orthogonal projections of A' onto lines CA, AB, resp. Define Cb, Ab, Ac, Bc cyclically. The locus of P such that the lines CbBc, AcCa and AbBa concur (at Q) is K191 (Angel Montesdeoca, private message, 2019-03-03). The locus of Q is a rather complicated sextic passing through the vertices of the medial and orthic triangles and meeting the line at infinity at three double points on the cubic K006.