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K932

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X(3), X(4), X(110), X(523), X(7471), X(14264), X(7471)*, X(14264)*

imaginary foci of the MacBeath inellipse

in/excenters of triangle OHX(110)

A', B', C' : traces of the line X(113), X(131), the trilinear polar of the root

Geometric properties :

We meet K932 in Hyacinthos #26812 (A. P. Hatzipolakis) and here (Angel Montesdeoca, in Spanish).

K932 has the remarkable property to be an isogonal focal nK in ABC and also an isogonal focal pK in the triangle OHX(110). Compare with K019 = nK0(X6, X647) = spK(X511, X6).

K932 is also spK(X523, X5) as in CL055 where a generalization is given.

The singular focus is X(110). The orthic line (L) is the perpendicular bisector of OH. The real asymptote is the homothetic of (L) under h(X110, 2) and meets K932 again at X(7471)*, the last common point with the Jerabek hyperbola. Note that X(7471)* is the antipode of X(110) on (C), the circumcircle of OHX(110). X(7471)* is the common tangential of X(3), X(4), X(110), X(523).

The polar conic of X(523) is the rectangular hyperbola (H) passing through X(5), X(30), X(355), X(523), X(3233) and the centers of anallagmaty. These are the in/excenters J0, J1, J2, J3 of triangle OHX(110).

The inverse of K932 in the circumcircle is K933, also a focal cubic.

The points X(7471)*, X(14264)* are now X(15453), X(15454) in ETC (2017-12-05).

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Generalization

Let Q, Q* be two finite distinct isogonal conjugate points with midpoint M. Let (L) be the perpendicular bisector of QQ* with infinite point P and let F = P* on the circumcircle (O).

K(Q) = spK(P, M) is an isogonal focal nK in ABC and also an isogonal focal pK in the triangle QQ*F. It is the locus of the common points of a variable circle passing through Q, Q* with center S on (L) and the line FS.