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K410a

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) now called the ITB triangle

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 lines {3,194}, {39,3224}. X = a^2 (a^6 b^4-a^4 b^6-2 a^6 b^2 c^2+a^4 b^4 c^2+4 a^2 b^6 c^2+a^6 c^4+a^4 b^2 c^4+7 a^2 b^4 c^4-3 b^6 c^4-a^4 c^6+4 a^2 b^2 c^6-3 b^4 c^6) : : , SEARCH = -1.090028621380516. X is now X(32524) in ETC.

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 cubics K411, K1098.

The conic inscribed in both triangles ABC and A'B'C' is the Brocard ellipse (E). See a generalization below.

See also the McCay cubic K003 and CL017, CL018.

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Generalization

A'B'C' is a poristic triangle inscribed in (O) with same Lemoine point X(6) as ABC, such that the Brocard ellipse (E) is also inscribed in A'B'C'.

Any pK(Ω, P) such that equivalently :

• P is the barycentric product of X(76) and the X(32)–Ceva conjugate of Ω, also the X(6)–Ceva conjugate of the barycentric quotient of Ω and X(6),

• Ω is the X(32)–Ceva conjugate of the barycentric product of X(6) and P, also the barycentric product of X(6) and the X(6)–Ceva conjugate of P,

has analogous properties although A'B'C' is not necessarily the ITB triangle itself anymore but a similar triangle circumscribing (E) unless P lies on K444 and Ω lies on psK(X9233, X32, X6) in which case the isopivot lies on K1104.

With P = X(4), X(32), X(194) on K444 we find K1100, K411, K410 respectively.