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K410

X(1), X(194), X(3224)

excenters

vertices of A'B'C', the only triply bilogic triangle inscribed in the circumcircle (Jean-Pierre Ehrmann, Hyacinthos #14350)

projections of X(3224) on the sidelines of A'B'C'

K410 is a pK+ with three real asymptotes concurring at X, a point on the line OX(194). The isogonal conjugates of the infinite points of K410 are the vertices of A'B'C'. These asymptotes are parallel to the altitudes of A'B'C' since X(194) is the orthocenter of A'B'C'.

These points A', B', C' also lie on (H), the rectangular hyperbola through X(32), X(194), X(805), X(511), X(512), and on the cubic K411.

The conic inscribed in both triangles ABC and A'B'C' is the Brocard ellipse.

See also the McCay cubic K003 and CL017, CL018.