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K407

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X(1), X(23703), X(23704), X(23705), X(23706), X(23838)

K407 is a member of the class CL046. See also Table 69.

K407 is an isogonal cK with singularity I and root X(57), the orthocorrespondent of I.

The pivotal conic is the MacBeath conic of the excentral triangle. Its center is O, its real foci are I and X(40). Its principal circle is the circumcircle. Its excentricity is OI / R.

Since I lies on Q003, the trilinear polar (L2) of the root X(57) is parallel to the line (L1) which contains the three real inflexion points. (L1) is the image of (L2) under the homothety h(I, 2).

For any point u:v:w distinct of I, the point a(b+c-a)[-bcu+(c+a-b)cv+(a+b-c)bw] / (cv-bw) : : lies on K407.

Locus properties

  1. K407 is the locus of centers of anallagmaty of the isogonal strophoids in CL003.
  2. Let (C) be a circum-conic through I = X(1) and (N) its normal at I. (N) meets (C) at I and another point M which lies on K407.
  3. Let M be a variable point on the incircle and T(M) its tangent at M. The perpendiculars at the incenter I to AI, BI, CI meet T(M) at A', B', C'. The lines AA', BB', CC' concur at P. When M varies on the incircle, the locus of M is K407 (Angel Montesdeoca, 2020-03-20).